id	sid	tid	token	lemma	pos
ejpam-3441	1	1	european	european	PROPN
ejpam-3441	1	2	journal	journal	PROPN
ejpam-3441	1	3	of	of	ADP
ejpam-3441	1	4	pure	pure	ADJ
ejpam-3441	1	5	and	and	CCONJ
ejpam-3441	1	6	applied	apply	VERB
ejpam-3441	1	7	mathematics	mathematic	NOUN
ejpam-3441	1	8	vol	vol	NOUN
ejpam-3441	1	9	.	.	PROPN
ejpam-3441	2	1	12	12	NUM
ejpam-3441	2	2	,	,	PUNCT
ejpam-3441	2	3	no	no	INTJ
ejpam-3441	2	4	.	.	NOUN
ejpam-3441	2	5	3	3	NUM
ejpam-3441	2	6	,	,	PUNCT
ejpam-3441	2	7	2019	2019	NUM
ejpam-3441	2	8	,	,	PUNCT
ejpam-3441	2	9	906	906	NUM
ejpam-3441	2	10	-	-	SYM
ejpam-3441	2	11	943	943	NUM
ejpam-3441	2	12	issn	issn	PROPN
ejpam-3441	2	13	1307	1307	NUM
ejpam-3441	2	14	-	-	SYM
ejpam-3441	2	15	5543	5543	NUM
ejpam-3441	2	16	–	–	PUNCT
ejpam-3441	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3441	2	18	published	publish	VERB
ejpam-3441	2	19	by	by	ADP
ejpam-3441	2	20	new	new	PROPN
ejpam-3441	2	21	york	york	PROPN
ejpam-3441	2	22	business	business	PROPN
ejpam-3441	2	23	global	global	ADJ
ejpam-3441	2	24	intuitionistic	intuitionistic	ADJ
ejpam-3441	2	25	fuzzy	fuzzy	ADJ
ejpam-3441	2	26	ideals	ideal	NOUN
ejpam-3441	2	27	with	with	ADP
ejpam-3441	2	28	thresholds	threshold	NOUN
ejpam-3441	2	29	(	(	PUNCT
ejpam-3441	2	30	α	α	X
ejpam-3441	2	31	,	,	PUNCT
ejpam-3441	2	32	β	β	X
ejpam-3441	2	33	]	]	X
ejpam-3441	2	34	in	in	ADP
ejpam-3441	2	35	la	la	PROPN
ejpam-3441	2	36	-	-	PUNCT
ejpam-3441	2	37	rings	ring	NOUN
ejpam-3441	2	38	nasreen	nasreen	ADP
ejpam-3441	2	39	kausar1,∗	kausar1,∗	NOUN
ejpam-3441	2	40	,	,	PUNCT
ejpam-3441	2	41	badar	badar	PROPN
ejpam-3441	2	42	ul	ul	PROPN
ejpam-3441	2	43	islam2	islam2	PROPN
ejpam-3441	2	44	,	,	PUNCT
ejpam-3441	2	45	syed	syed	PROPN
ejpam-3441	2	46	amjad	amjad	PROPN
ejpam-3441	2	47	ahmad3	ahmad3	PROPN
ejpam-3441	2	48	,	,	PUNCT
ejpam-3441	2	49	muhammad	muhammad	PROPN
ejpam-3441	2	50	azam	azam	PROPN
ejpam-3441	2	51	waqar4	waqar4	PROPN
ejpam-3441	2	52	1	1	NUM
ejpam-3441	2	53	department	department	NOUN
ejpam-3441	2	54	of	of	ADP
ejpam-3441	2	55	mathematics	mathematic	NOUN
ejpam-3441	2	56	,	,	PUNCT
ejpam-3441	2	57	university	university	NOUN
ejpam-3441	2	58	of	of	ADP
ejpam-3441	2	59	agriculture	agriculture	PROPN
ejpam-3441	2	60	,	,	PUNCT
ejpam-3441	2	61	fsd	fsd	PROPN
ejpam-3441	2	62	,	,	PUNCT
ejpam-3441	2	63	pakistan	pakistan	PROPN
ejpam-3441	2	64	2	2	NUM
ejpam-3441	2	65	department	department	NOUN
ejpam-3441	2	66	of	of	ADP
ejpam-3441	2	67	electrical	electrical	ADJ
ejpam-3441	2	68	engineering	engineering	NOUN
ejpam-3441	2	69	,	,	PUNCT
ejpam-3441	2	70	nfc	nfc	NOUN
ejpam-3441	2	71	iefr	iefr	PROPN
ejpam-3441	2	72	fsd	fsd	PROPN
ejpam-3441	2	73	,	,	PUNCT
ejpam-3441	2	74	pakistan	pakistan	PROPN
ejpam-3441	2	75	3	3	NUM
ejpam-3441	2	76	department	department	NOUN
ejpam-3441	2	77	of	of	ADP
ejpam-3441	2	78	mechanical	mechanical	ADJ
ejpam-3441	2	79	engineering	engineering	NOUN
ejpam-3441	2	80	,	,	PUNCT
ejpam-3441	2	81	nfc	nfc	NOUN
ejpam-3441	2	82	iefr	iefr	PROPN
ejpam-3441	2	83	fsd	fsd	PROPN
ejpam-3441	2	84	,	,	PUNCT
ejpam-3441	2	85	pakistan	pakistan	PROPN
ejpam-3441	2	86	4	4	NUM
ejpam-3441	2	87	department	department	NOUN
ejpam-3441	2	88	of	of	ADP
ejpam-3441	2	89	school	school	NOUN
ejpam-3441	2	90	of	of	ADP
ejpam-3441	2	91	business	business	NOUN
ejpam-3441	2	92	manegement	manegement	NOUN
ejpam-3441	2	93	,	,	PUNCT
ejpam-3441	2	94	nfc	nfc	NOUN
ejpam-3441	2	95	iefr	iefr	PROPN
ejpam-3441	2	96	fsd	fsd	PROPN
ejpam-3441	2	97	,	,	PUNCT
ejpam-3441	2	98	pakistan	pakistan	PROPN
ejpam-3441	2	99	abstract	abstract	NOUN
ejpam-3441	2	100	.	.	PUNCT
ejpam-3441	3	1	in	in	ADP
ejpam-3441	3	2	this	this	DET
ejpam-3441	3	3	paper	paper	NOUN
ejpam-3441	3	4	,	,	PUNCT
ejpam-3441	3	5	we	we	PRON
ejpam-3441	3	6	give	give	VERB
ejpam-3441	3	7	characterizations	characterization	NOUN
ejpam-3441	3	8	of	of	ADP
ejpam-3441	3	9	regular	regular	ADJ
ejpam-3441	3	10	(	(	PUNCT
ejpam-3441	3	11	intra	intra	ADJ
ejpam-3441	3	12	-	-	ADJ
ejpam-3441	3	13	regular	regular	ADJ
ejpam-3441	3	14	,	,	PUNCT
ejpam-3441	3	15	both	both	CCONJ
ejpam-3441	3	16	regular	regular	ADJ
ejpam-3441	3	17	and	and	CCONJ
ejpam-3441	3	18	intra	intra	ADJ
ejpam-3441	3	19	-	-	ADJ
ejpam-3441	3	20	regular	regular	ADJ
ejpam-3441	3	21	)	)	PUNCT
ejpam-3441	3	22	la	la	NOUN
ejpam-3441	3	23	-	-	PUNCT
ejpam-3441	3	24	rings	ring	NOUN
ejpam-3441	3	25	by	by	ADP
ejpam-3441	3	26	the	the	DET
ejpam-3441	3	27	properties	property	NOUN
ejpam-3441	3	28	of	of	ADP
ejpam-3441	3	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	3	30	fuzzy	fuzzy	ADJ
ejpam-3441	3	31	(	(	PUNCT
ejpam-3441	3	32	left	left	ADJ
ejpam-3441	3	33	,	,	PUNCT
ejpam-3441	3	34	right	right	INTJ
ejpam-3441	3	35	,	,	PUNCT
ejpam-3441	3	36	quasi-	quasi-	INTJ
ejpam-3441	3	37	,	,	PUNCT
ejpam-3441	3	38	bi-	bi-	NUM
ejpam-3441	3	39	,	,	PUNCT
ejpam-3441	3	40	generalized	generalize	VERB
ejpam-3441	3	41	bi-	bi-	NUM
ejpam-3441	3	42	)	)	PUNCT
ejpam-3441	3	43	ideals	ideal	NOUN
ejpam-3441	3	44	with	with	ADP
ejpam-3441	3	45	thresholds	threshold	NOUN
ejpam-3441	3	46	(	(	PUNCT
ejpam-3441	3	47	α	α	X
ejpam-3441	3	48	,	,	PUNCT
ejpam-3441	3	49	β	β	X
ejpam-3441	3	50	]	]	X
ejpam-3441	3	51	.	.	PUNCT
ejpam-3441	4	1	2010	2010	NUM
ejpam-3441	4	2	mathematics	mathematic	NOUN
ejpam-3441	4	3	subject	subject	NOUN
ejpam-3441	4	4	classifications	classification	NOUN
ejpam-3441	4	5	:	:	PUNCT
ejpam-3441	4	6	03f55	03f55	NOUN
ejpam-3441	4	7	,	,	PUNCT
ejpam-3441	4	8	08a72	08a72	NUM
ejpam-3441	4	9	,	,	PUNCT
ejpam-3441	4	10	20n25	20n25	NOUN
ejpam-3441	4	11	key	key	ADJ
ejpam-3441	4	12	words	word	NOUN
ejpam-3441	4	13	and	and	CCONJ
ejpam-3441	4	14	phrases	phrase	NOUN
ejpam-3441	4	15	:	:	PUNCT
ejpam-3441	4	16	intuitionistic	intuitionistic	ADJ
ejpam-3441	4	17	fuzzy	fuzzy	ADJ
ejpam-3441	4	18	left	left	ADJ
ejpam-3441	4	19	(	(	PUNCT
ejpam-3441	4	20	right	right	ADJ
ejpam-3441	4	21	,	,	PUNCT
ejpam-3441	4	22	interior	interior	ADJ
ejpam-3441	4	23	,	,	PUNCT
ejpam-3441	4	24	quasi-	quasi-	PROPN
ejpam-3441	4	25	,	,	PUNCT
ejpam-3441	4	26	bi-	bi-	NUM
ejpam-3441	4	27	,	,	PUNCT
ejpam-3441	4	28	generalized	generalize	VERB
ejpam-3441	4	29	bi-	bi-	NUM
ejpam-3441	4	30	)	)	PUNCT
ejpam-3441	4	31	ideals	ideal	NOUN
ejpam-3441	4	32	with	with	ADP
ejpam-3441	4	33	thresholds	threshold	NOUN
ejpam-3441	4	34	(	(	PUNCT
ejpam-3441	4	35	α	α	X
ejpam-3441	4	36	,	,	PUNCT
ejpam-3441	4	37	β	β	X
ejpam-3441	4	38	]	]	X
ejpam-3441	4	39	,	,	PUNCT
ejpam-3441	4	40	regular	regular	ADJ
ejpam-3441	4	41	(	(	PUNCT
ejpam-3441	4	42	intra	intra	ADJ
ejpam-3441	4	43	-	-	ADJ
ejpam-3441	4	44	regular	regular	ADJ
ejpam-3441	4	45	)	)	PUNCT
ejpam-3441	4	46	la	la	NOUN
ejpam-3441	4	47	-	-	PUNCT
ejpam-3441	4	48	rings	ring	NOUN
ejpam-3441	4	49	.	.	PUNCT
ejpam-3441	5	1	1	1	X
ejpam-3441	5	2	.	.	X
ejpam-3441	5	3	introduction	introduction	NOUN
ejpam-3441	5	4	in	in	ADP
ejpam-3441	5	5	ternary	ternary	ADJ
ejpam-3441	5	6	operations	operation	NOUN
ejpam-3441	5	7	,	,	PUNCT
ejpam-3441	5	8	the	the	DET
ejpam-3441	5	9	commutative	commutative	ADJ
ejpam-3441	5	10	law	law	NOUN
ejpam-3441	5	11	is	be	AUX
ejpam-3441	5	12	given	give	VERB
ejpam-3441	5	13	by	by	ADP
ejpam-3441	5	14	abc	abc	PROPN
ejpam-3441	5	15	=	=	SYM
ejpam-3441	5	16	cba	cba	PROPN
ejpam-3441	5	17	.	.	PUNCT
ejpam-3441	6	1	kazim	kazim	PROPN
ejpam-3441	6	2	et	et	PROPN
ejpam-3441	6	3	al	al	PROPN
ejpam-3441	7	1	[	[	X
ejpam-3441	7	2	18	18	NUM
ejpam-3441	7	3	]	]	PUNCT
ejpam-3441	7	4	,	,	PUNCT
ejpam-3441	7	5	have	have	AUX
ejpam-3441	7	6	generalized	generalize	VERB
ejpam-3441	7	7	this	this	DET
ejpam-3441	7	8	notion	notion	NOUN
ejpam-3441	7	9	by	by	ADP
ejpam-3441	7	10	introducing	introduce	VERB
ejpam-3441	7	11	the	the	DET
ejpam-3441	7	12	paranthesis	paranthesis	NOUN
ejpam-3441	7	13	on	on	ADP
ejpam-3441	7	14	the	the	DET
ejpam-3441	7	15	left	left	ADJ
ejpam-3441	7	16	side	side	NOUN
ejpam-3441	7	17	of	of	ADP
ejpam-3441	7	18	this	this	DET
ejpam-3441	7	19	equation	equation	NOUN
ejpam-3441	7	20	to	to	PART
ejpam-3441	7	21	get	get	VERB
ejpam-3441	7	22	a	a	DET
ejpam-3441	7	23	new	new	ADJ
ejpam-3441	7	24	pseudo	pseudo	NOUN
ejpam-3441	7	25	associative	associative	NOUN
ejpam-3441	7	26	law	law	NOUN
ejpam-3441	7	27	,	,	PUNCT
ejpam-3441	7	28	that	that	ADV
ejpam-3441	7	29	is	is	ADV
ejpam-3441	7	30	(	(	PUNCT
ejpam-3441	7	31	ab)c	ab)c	PROPN
ejpam-3441	7	32	=	=	SYM
ejpam-3441	7	33	(	(	PUNCT
ejpam-3441	7	34	cb)a	cb)a	PROPN
ejpam-3441	7	35	.	.	PUNCT
ejpam-3441	8	1	this	this	DET
ejpam-3441	8	2	law	law	NOUN
ejpam-3441	8	3	(	(	PUNCT
ejpam-3441	8	4	ab)c	ab)c	PROPN
ejpam-3441	8	5	=	=	SYM
ejpam-3441	8	6	(	(	PUNCT
ejpam-3441	8	7	cb)a	cb)a	PROPN
ejpam-3441	8	8	is	be	AUX
ejpam-3441	8	9	called	call	VERB
ejpam-3441	8	10	the	the	DET
ejpam-3441	8	11	left	left	ADJ
ejpam-3441	8	12	invertive	invertive	ADJ
ejpam-3441	8	13	law	law	NOUN
ejpam-3441	8	14	.	.	PUNCT
ejpam-3441	9	1	a	a	DET
ejpam-3441	9	2	groupoid	groupoid	PROPN
ejpam-3441	9	3	s	s	X
ejpam-3441	9	4	is	be	AUX
ejpam-3441	9	5	called	call	VERB
ejpam-3441	9	6	a	a	DET
ejpam-3441	9	7	left	left	NOUN
ejpam-3441	9	8	almost	almost	ADV
ejpam-3441	9	9	semigroup	semigroup	ADJ
ejpam-3441	9	10	(	(	PUNCT
ejpam-3441	9	11	abbreviated	abbreviate	VERB
ejpam-3441	9	12	as	as	ADP
ejpam-3441	9	13	lasemigroup	lasemigroup	NOUN
ejpam-3441	9	14	)	)	PUNCT
ejpam-3441	9	15	if	if	SCONJ
ejpam-3441	9	16	it	it	PRON
ejpam-3441	9	17	satisfies	satisfy	VERB
ejpam-3441	9	18	the	the	DET
ejpam-3441	9	19	left	left	ADJ
ejpam-3441	9	20	invertive	invertive	ADJ
ejpam-3441	9	21	law	law	NOUN
ejpam-3441	9	22	.	.	PUNCT
ejpam-3441	10	1	an	an	DET
ejpam-3441	10	2	la	la	ADJ
ejpam-3441	10	3	-	-	PUNCT
ejpam-3441	10	4	semigroup	semigroup	PROPN
ejpam-3441	10	5	is	be	AUX
ejpam-3441	10	6	a	a	DET
ejpam-3441	10	7	midway	midway	NOUN
ejpam-3441	10	8	structure	structure	NOUN
ejpam-3441	10	9	between	between	ADP
ejpam-3441	10	10	a	a	DET
ejpam-3441	10	11	commutative	commutative	ADJ
ejpam-3441	10	12	semigroup	semigroup	NOUN
ejpam-3441	10	13	and	and	CCONJ
ejpam-3441	10	14	a	a	DET
ejpam-3441	10	15	groupoid	groupoid	NOUN
ejpam-3441	10	16	.	.	PUNCT
ejpam-3441	11	1	ideals	ideal	NOUN
ejpam-3441	11	2	in	in	ADP
ejpam-3441	11	3	la	la	NOUN
ejpam-3441	11	4	-	-	PUNCT
ejpam-3441	11	5	semigroups	semigroup	NOUN
ejpam-3441	11	6	have	have	AUX
ejpam-3441	11	7	been	be	AUX
ejpam-3441	11	8	investigated	investigate	VERB
ejpam-3441	11	9	by	by	ADP
ejpam-3441	11	10	protic	protic	PROPN
ejpam-3441	11	11	et	et	PROPN
ejpam-3441	11	12	al	al	PROPN
ejpam-3441	12	1	[	[	X
ejpam-3441	12	2	24	24	NUM
ejpam-3441	12	3	]	]	PUNCT
ejpam-3441	12	4	.	.	PUNCT
ejpam-3441	13	1	in	in	ADP
ejpam-3441	13	2	[	[	X
ejpam-3441	13	3	12	12	NUM
ejpam-3441	13	4	]	]	PUNCT
ejpam-3441	13	5	(	(	PUNCT
ejpam-3441	13	6	resp	resp	NOUN
ejpam-3441	13	7	.	.	PUNCT
ejpam-3441	14	1	[	[	X
ejpam-3441	14	2	8	8	NUM
ejpam-3441	14	3	]	]	NUM
ejpam-3441	14	4	)	)	PUNCT
ejpam-3441	14	5	,	,	PUNCT
ejpam-3441	14	6	a	a	DET
ejpam-3441	14	7	groupoid	groupoid	NOUN
ejpam-3441	14	8	s	s	NOUN
ejpam-3441	14	9	is	be	AUX
ejpam-3441	14	10	said	say	VERB
ejpam-3441	14	11	to	to	PART
ejpam-3441	14	12	be	be	AUX
ejpam-3441	14	13	medial	medial	ADJ
ejpam-3441	14	14	(	(	PUNCT
ejpam-3441	14	15	resp	resp	NOUN
ejpam-3441	14	16	.	.	PUNCT
ejpam-3441	15	1	paramedial	paramedial	PROPN
ejpam-3441	15	2	)	)	PUNCT
ejpam-3441	16	1	if	if	SCONJ
ejpam-3441	16	2	(	(	PUNCT
ejpam-3441	16	3	ab)(cd	ab)(cd	NOUN
ejpam-3441	16	4	)	)	PUNCT
ejpam-3441	16	5	=	=	SYM
ejpam-3441	16	6	(	(	PUNCT
ejpam-3441	16	7	ac)(bd	ac)(bd	PROPN
ejpam-3441	16	8	)	)	PUNCT
ejpam-3441	16	9	(	(	PUNCT
ejpam-3441	16	10	resp	resp	NOUN
ejpam-3441	16	11	.	.	PUNCT
ejpam-3441	17	1	(	(	PUNCT
ejpam-3441	17	2	ab)(cd	ab)(cd	PROPN
ejpam-3441	17	3	)	)	PUNCT
ejpam-3441	17	4	=	=	SYM
ejpam-3441	17	5	(	(	PUNCT
ejpam-3441	17	6	db)(ca	db)(ca	PROPN
ejpam-3441	17	7	)	)	PUNCT
ejpam-3441	17	8	)	)	PUNCT
ejpam-3441	17	9	.	.	PUNCT
ejpam-3441	18	1	in	in	ADP
ejpam-3441	18	2	[	[	X
ejpam-3441	18	3	18	18	NUM
ejpam-3441	18	4	]	]	PUNCT
ejpam-3441	18	5	,	,	PUNCT
ejpam-3441	18	6	an	an	DET
ejpam-3441	18	7	la	la	ADJ
ejpam-3441	18	8	-	-	PUNCT
ejpam-3441	18	9	semigroup	semigroup	PROPN
ejpam-3441	18	10	is	be	AUX
ejpam-3441	18	11	medial	medial	ADJ
ejpam-3441	18	12	,	,	PUNCT
ejpam-3441	18	13	but	but	CCONJ
ejpam-3441	18	14	in	in	ADP
ejpam-3441	18	15	general	general	ADJ
ejpam-3441	18	16	an	an	DET
ejpam-3441	18	17	la	la	ADJ
ejpam-3441	18	18	-	-	PUNCT
ejpam-3441	18	19	semigroup	semigroup	NOUN
ejpam-3441	18	20	needs	need	VERB
ejpam-3441	18	21	not	not	PART
ejpam-3441	18	22	to	to	PART
ejpam-3441	18	23	be	be	AUX
ejpam-3441	18	24	paramedial	paramedial	ADJ
ejpam-3441	18	25	.	.	PUNCT
ejpam-3441	19	1	every	every	DET
ejpam-3441	19	2	la	la	PROPN
ejpam-3441	19	3	-	-	PUNCT
ejpam-3441	19	4	semigroup	semigroup	NOUN
ejpam-3441	19	5	with	with	ADP
ejpam-3441	19	6	left	left	ADJ
ejpam-3441	19	7	identity	identity	NOUN
ejpam-3441	19	8	is	be	AUX
ejpam-3441	19	9	paramedial	paramedial	ADJ
ejpam-3441	19	10	by	by	ADP
ejpam-3441	19	11	protic	protic	PROPN
ejpam-3441	19	12	et	et	NOUN
ejpam-3441	19	13	al	al	PROPN
ejpam-3441	20	1	[	[	X
ejpam-3441	20	2	24	24	NUM
ejpam-3441	20	3	]	]	PUNCT
ejpam-3441	20	4	and	and	CCONJ
ejpam-3441	20	5	also	also	ADV
ejpam-3441	20	6	satisfies	satisfy	VERB
ejpam-3441	20	7	a(bc	a(bc	NOUN
ejpam-3441	20	8	)	)	PUNCT
ejpam-3441	20	9	=	=	SYM
ejpam-3441	20	10	b(ac	b(ac	PROPN
ejpam-3441	20	11	)	)	PUNCT
ejpam-3441	20	12	,	,	PUNCT
ejpam-3441	20	13	(	(	PUNCT
ejpam-3441	20	14	ab)(cd	ab)(cd	NOUN
ejpam-3441	20	15	)	)	PUNCT
ejpam-3441	20	16	=	=	SYM
ejpam-3441	20	17	(	(	PUNCT
ejpam-3441	20	18	dc)(ba	dc)(ba	PROPN
ejpam-3441	20	19	)	)	PUNCT
ejpam-3441	20	20	.	.	PUNCT
ejpam-3441	21	1	kamran	kamran	PROPN
ejpam-3441	22	1	[	[	X
ejpam-3441	22	2	14	14	NUM
ejpam-3441	22	3	]	]	PUNCT
ejpam-3441	22	4	,	,	PUNCT
ejpam-3441	22	5	extended	extend	VERB
ejpam-3441	22	6	the	the	DET
ejpam-3441	22	7	notion	notion	NOUN
ejpam-3441	22	8	of	of	ADP
ejpam-3441	22	9	la	la	NOUN
ejpam-3441	22	10	-	-	PUNCT
ejpam-3441	22	11	semigroup	semigroup	NOUN
ejpam-3441	22	12	to	to	ADP
ejpam-3441	22	13	the	the	DET
ejpam-3441	22	14	left	leave	VERB
ejpam-3441	22	15	almost	almost	ADV
ejpam-3441	22	16	group	group	NOUN
ejpam-3441	22	17	(	(	PUNCT
ejpam-3441	22	18	lagroup	lagroup	NOUN
ejpam-3441	22	19	)	)	PUNCT
ejpam-3441	22	20	.	.	PUNCT
ejpam-3441	23	1	an	an	DET
ejpam-3441	23	2	la	la	ADJ
ejpam-3441	23	3	-	-	PUNCT
ejpam-3441	23	4	semigroup	semigroup	PROPN
ejpam-3441	23	5	g	g	PROPN
ejpam-3441	23	6	is	be	AUX
ejpam-3441	23	7	called	call	VERB
ejpam-3441	23	8	a	a	DET
ejpam-3441	23	9	left	left	ADJ
ejpam-3441	23	10	almost	almost	ADV
ejpam-3441	23	11	group	group	NOUN
ejpam-3441	23	12	,	,	PUNCT
ejpam-3441	23	13	if	if	SCONJ
ejpam-3441	23	14	there	there	PRON
ejpam-3441	23	15	exists	exist	VERB
ejpam-3441	23	16	a	a	DET
ejpam-3441	23	17	left	left	ADJ
ejpam-3441	23	18	identity	identity	NOUN
ejpam-3441	23	19	∗corresponding	∗corresponde	VERB
ejpam-3441	23	20	author	author	NOUN
ejpam-3441	23	21	.	.	PUNCT
ejpam-3441	24	1	doi	doi	NOUN
ejpam-3441	24	2	:	:	PUNCT
ejpam-3441	24	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3441	https://doi.org/10.29020/nybg.ejpam.v12i3.3441	NUM
ejpam-3441	24	4	email	email	NOUN
ejpam-3441	24	5	addresses	address	NOUN
ejpam-3441	24	6	:	:	PUNCT
ejpam-3441	24	7	kausar.nasreen@gmail.com	kausar.nasreen@gmail.com	X
ejpam-3441	24	8	(	(	PUNCT
ejpam-3441	24	9	k.	k.	PROPN
ejpam-3441	24	10	nasreen	nasreen	PROPN
ejpam-3441	24	11	)	)	PUNCT
ejpam-3441	24	12	,	,	PUNCT
ejpam-3441	24	13	badar.utp@gmail.com	badar.utp@gmail.com	PROPN
ejpam-3441	24	14	(	(	PUNCT
ejpam-3441	24	15	i.	i.	PROPN
ejpam-3441	24	16	badar	badar	PROPN
ejpam-3441	24	17	)	)	PUNCT
ejpam-3441	25	1	samjadahmad67@yahoo.com	samjadahmad67@yahoo.com	X
ejpam-3441	25	2	(	(	PUNCT
ejpam-3441	25	3	s.	s.	PROPN
ejpam-3441	25	4	a.	a.	PROPN
ejpam-3441	25	5	ahmad	ahmad	PROPN
ejpam-3441	25	6	)	)	PUNCT
ejpam-3441	25	7	,	,	PUNCT
ejpam-3441	25	8	azamwaqar4@gmail.com	azamwaqar4@gmail.com	X
ejpam-3441	25	9	(	(	PUNCT
ejpam-3441	25	10	w.	w.	PROPN
ejpam-3441	25	11	azam	azam	PROPN
ejpam-3441	25	12	)	)	PUNCT
ejpam-3441	25	13	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3441	26	1	906	906	NUM
ejpam-3441	26	2	c	c	NOUN
ejpam-3441	26	3	©	©	PROPN
ejpam-3441	26	4	2019	2019	NUM
ejpam-3441	26	5	ejpam	ejpam	NOUN
ejpam-3441	26	6	all	all	DET
ejpam-3441	26	7	rights	right	NOUN
ejpam-3441	26	8	reserved	reserve	VERB
ejpam-3441	26	9	.	.	PUNCT
ejpam-3441	27	1	k.	k.	PROPN
ejpam-3441	27	2	nasreen	nasreen	PROPN
ejpam-3441	27	3	et	et	PROPN
ejpam-3441	27	4	al	al	PROPN
ejpam-3441	27	5	.	.	PUNCT
ejpam-3441	27	6	/	/	SYM
ejpam-3441	27	7	eur	eur	PROPN
ejpam-3441	27	8	.	.	PUNCT
ejpam-3441	28	1	j.	j.	PROPN
ejpam-3441	28	2	pure	pure	PROPN
ejpam-3441	28	3	appl	appl	PROPN
ejpam-3441	28	4	.	.	PROPN
ejpam-3441	28	5	math	math	PROPN
ejpam-3441	28	6	,	,	PUNCT
ejpam-3441	28	7	12	12	NUM
ejpam-3441	28	8	(	(	PUNCT
ejpam-3441	28	9	3	3	NUM
ejpam-3441	28	10	)	)	PUNCT
ejpam-3441	28	11	(	(	PUNCT
ejpam-3441	28	12	2019	2019	NUM
ejpam-3441	28	13	)	)	PUNCT
ejpam-3441	28	14	,	,	PUNCT
ejpam-3441	28	15	906	906	NUM
ejpam-3441	28	16	-	-	SYM
ejpam-3441	28	17	943	943	NUM
ejpam-3441	28	18	907	907	NUM
ejpam-3441	28	19	e	e	NOUN
ejpam-3441	28	20	∈	∈	PROPN
ejpam-3441	28	21	g	g	PROPN
ejpam-3441	28	22	such	such	ADJ
ejpam-3441	29	1	that	that	DET
ejpam-3441	29	2	ea	ea	NOUN
ejpam-3441	29	3	=	=	PUNCT
ejpam-3441	29	4	a	a	PRON
ejpam-3441	29	5	for	for	ADP
ejpam-3441	29	6	all	all	DET
ejpam-3441	29	7	a	a	DET
ejpam-3441	29	8	∈	∈	NOUN
ejpam-3441	29	9	g	g	NOUN
ejpam-3441	29	10	and	and	CCONJ
ejpam-3441	29	11	for	for	ADP
ejpam-3441	29	12	every	every	DET
ejpam-3441	29	13	a	a	DET
ejpam-3441	29	14	∈	∈	PROPN
ejpam-3441	29	15	g	g	NOUN
ejpam-3441	29	16	there	there	PRON
ejpam-3441	29	17	exists	exist	VERB
ejpam-3441	29	18	b	b	PROPN
ejpam-3441	29	19	∈	∈	PROPN
ejpam-3441	29	20	g	g	NOUN
ejpam-3441	29	21	such	such	ADJ
ejpam-3441	29	22	that	that	DET
ejpam-3441	29	23	ba	ba	PROPN
ejpam-3441	29	24	=	=	SYM
ejpam-3441	29	25	e.	e.	PROPN
ejpam-3441	29	26	shah	shah	PROPN
ejpam-3441	29	27	et	et	PROPN
ejpam-3441	29	28	al	al	PROPN
ejpam-3441	30	1	[	[	X
ejpam-3441	30	2	25	25	NUM
ejpam-3441	30	3	]	]	PUNCT
ejpam-3441	30	4	,	,	PUNCT
ejpam-3441	30	5	discussed	discuss	VERB
ejpam-3441	30	6	the	the	DET
ejpam-3441	30	7	left	left	NOUN
ejpam-3441	30	8	almost	almost	ADV
ejpam-3441	30	9	ring	ring	NOUN
ejpam-3441	30	10	(	(	PUNCT
ejpam-3441	30	11	abbreviated	abbreviate	VERB
ejpam-3441	30	12	as	as	ADP
ejpam-3441	30	13	la	la	NOUN
ejpam-3441	30	14	-	-	PUNCT
ejpam-3441	30	15	ring	ring	NOUN
ejpam-3441	30	16	)	)	PUNCT
ejpam-3441	30	17	of	of	ADP
ejpam-3441	30	18	finitely	finitely	ADV
ejpam-3441	30	19	nonzero	nonzero	PROPN
ejpam-3441	30	20	functions	function	NOUN
ejpam-3441	30	21	which	which	PRON
ejpam-3441	30	22	is	be	AUX
ejpam-3441	30	23	a	a	DET
ejpam-3441	30	24	generalization	generalization	NOUN
ejpam-3441	30	25	of	of	ADP
ejpam-3441	30	26	commutative	commutative	ADJ
ejpam-3441	30	27	semigroup	semigroup	PROPN
ejpam-3441	30	28	ring	ring	NOUN
ejpam-3441	30	29	.	.	PUNCT
ejpam-3441	31	1	by	by	ADP
ejpam-3441	31	2	a	a	DET
ejpam-3441	31	3	left	left	ADJ
ejpam-3441	31	4	almost	almost	ADV
ejpam-3441	31	5	ring	ring	NOUN
ejpam-3441	31	6	,	,	PUNCT
ejpam-3441	31	7	we	we	PRON
ejpam-3441	31	8	mean	mean	VERB
ejpam-3441	31	9	a	a	DET
ejpam-3441	31	10	non	non	ADJ
ejpam-3441	31	11	-	-	ADJ
ejpam-3441	31	12	empty	empty	ADJ
ejpam-3441	31	13	set	set	VERB
ejpam-3441	31	14	r	r	NOUN
ejpam-3441	31	15	with	with	ADP
ejpam-3441	31	16	at	at	ADV
ejpam-3441	31	17	least	least	ADV
ejpam-3441	31	18	two	two	NUM
ejpam-3441	31	19	elements	element	NOUN
ejpam-3441	31	20	such	such	ADJ
ejpam-3441	31	21	that	that	SCONJ
ejpam-3441	31	22	(	(	PUNCT
ejpam-3441	31	23	r,+	r,+	NUM
ejpam-3441	31	24	)	)	PUNCT
ejpam-3441	31	25	is	be	AUX
ejpam-3441	31	26	an	an	DET
ejpam-3441	31	27	la	la	NOUN
ejpam-3441	31	28	-	-	NOUN
ejpam-3441	31	29	group	group	NOUN
ejpam-3441	31	30	,	,	PUNCT
ejpam-3441	31	31	(	(	PUNCT
ejpam-3441	31	32	r	r	NOUN
ejpam-3441	31	33	,	,	PUNCT
ejpam-3441	31	34	·	·	PUNCT
ejpam-3441	31	35	)	)	PUNCT
ejpam-3441	31	36	is	be	AUX
ejpam-3441	31	37	an	an	DET
ejpam-3441	31	38	la	la	ADJ
ejpam-3441	31	39	-	-	PUNCT
ejpam-3441	31	40	semigroup	semigroup	NOUN
ejpam-3441	31	41	,	,	PUNCT
ejpam-3441	31	42	both	both	PRON
ejpam-3441	31	43	left	leave	VERB
ejpam-3441	31	44	and	and	CCONJ
ejpam-3441	31	45	right	right	ADJ
ejpam-3441	31	46	distributive	distributive	ADJ
ejpam-3441	31	47	laws	law	NOUN
ejpam-3441	31	48	hold	hold	VERB
ejpam-3441	31	49	.	.	PUNCT
ejpam-3441	32	1	for	for	ADP
ejpam-3441	32	2	example	example	NOUN
ejpam-3441	32	3	,	,	PUNCT
ejpam-3441	32	4	from	from	ADP
ejpam-3441	32	5	a	a	DET
ejpam-3441	32	6	commutative	commutative	ADJ
ejpam-3441	32	7	ring	ring	NOUN
ejpam-3441	32	8	(	(	PUNCT
ejpam-3441	32	9	r,+	r,+	NUM
ejpam-3441	32	10	,	,	PUNCT
ejpam-3441	32	11	·	·	PUNCT
ejpam-3441	32	12	)	)	PUNCT
ejpam-3441	32	13	,	,	PUNCT
ejpam-3441	32	14	we	we	PRON
ejpam-3441	32	15	can	can	AUX
ejpam-3441	32	16	always	always	ADV
ejpam-3441	32	17	obtain	obtain	VERB
ejpam-3441	32	18	an	an	DET
ejpam-3441	32	19	la	la	ADJ
ejpam-3441	32	20	-	-	PUNCT
ejpam-3441	32	21	ring	ring	NOUN
ejpam-3441	32	22	(	(	PUNCT
ejpam-3441	32	23	r,⊕	r,⊕	NOUN
ejpam-3441	32	24	,	,	PUNCT
ejpam-3441	32	25	·	·	PUNCT
ejpam-3441	32	26	)	)	PUNCT
ejpam-3441	32	27	by	by	ADP
ejpam-3441	32	28	defining	define	VERB
ejpam-3441	32	29	for	for	ADP
ejpam-3441	32	30	all	all	DET
ejpam-3441	32	31	a	a	DET
ejpam-3441	32	32	,	,	PUNCT
ejpam-3441	32	33	b	b	X
ejpam-3441	32	34	∈	∈	PROPN
ejpam-3441	32	35	r	r	NOUN
ejpam-3441	32	36	,	,	PUNCT
ejpam-3441	33	1	a	a	DET
ejpam-3441	33	2	⊕	⊕	PROPN
ejpam-3441	33	3	b	b	X
ejpam-3441	33	4	=	=	SYM
ejpam-3441	33	5	b	b	PROPN
ejpam-3441	33	6	−	−	PROPN
ejpam-3441	33	7	a	a	PRON
ejpam-3441	33	8	and	and	CCONJ
ejpam-3441	33	9	a	a	PRON
ejpam-3441	33	10	·	·	PUNCT
ejpam-3441	33	11	b	b	NOUN
ejpam-3441	33	12	is	be	AUX
ejpam-3441	33	13	same	same	ADJ
ejpam-3441	33	14	as	as	ADP
ejpam-3441	33	15	in	in	ADP
ejpam-3441	33	16	the	the	DET
ejpam-3441	33	17	ring	ring	NOUN
ejpam-3441	33	18	.	.	PUNCT
ejpam-3441	34	1	although	although	SCONJ
ejpam-3441	34	2	the	the	DET
ejpam-3441	34	3	structure	structure	NOUN
ejpam-3441	34	4	is	be	AUX
ejpam-3441	34	5	non	non	ADJ
ejpam-3441	34	6	-	-	ADJ
ejpam-3441	34	7	associative	associative	ADJ
ejpam-3441	34	8	and	and	CCONJ
ejpam-3441	34	9	non	non	ADJ
ejpam-3441	34	10	-	-	ADJ
ejpam-3441	34	11	commutative	commutative	ADJ
ejpam-3441	34	12	,	,	PUNCT
ejpam-3441	34	13	nevertheless	nevertheless	ADV
ejpam-3441	34	14	,	,	PUNCT
ejpam-3441	34	15	it	it	PRON
ejpam-3441	34	16	possesses	possess	VERB
ejpam-3441	34	17	many	many	ADJ
ejpam-3441	34	18	interesting	interesting	ADJ
ejpam-3441	34	19	properties	property	NOUN
ejpam-3441	34	20	which	which	PRON
ejpam-3441	34	21	we	we	PRON
ejpam-3441	34	22	usually	usually	ADV
ejpam-3441	34	23	find	find	VERB
ejpam-3441	34	24	in	in	ADP
ejpam-3441	34	25	associative	associative	ADJ
ejpam-3441	34	26	and	and	CCONJ
ejpam-3441	34	27	commutative	commutative	ADJ
ejpam-3441	34	28	algebraic	algebraic	ADJ
ejpam-3441	34	29	structures	structure	NOUN
ejpam-3441	34	30	.	.	PUNCT
ejpam-3441	35	1	a	a	DET
ejpam-3441	35	2	non	non	ADJ
ejpam-3441	35	3	-	-	ADJ
ejpam-3441	35	4	empty	empty	ADJ
ejpam-3441	35	5	subset	subset	NOUN
ejpam-3441	35	6	a	a	PRON
ejpam-3441	35	7	of	of	ADP
ejpam-3441	35	8	r	r	NOUN
ejpam-3441	35	9	is	be	AUX
ejpam-3441	35	10	called	call	VERB
ejpam-3441	35	11	an	an	DET
ejpam-3441	35	12	la	la	NOUN
ejpam-3441	35	13	-	-	PUNCT
ejpam-3441	35	14	subring	subring	NOUN
ejpam-3441	35	15	of	of	ADP
ejpam-3441	35	16	r	r	NOUN
ejpam-3441	35	17	if	if	SCONJ
ejpam-3441	35	18	a	a	DET
ejpam-3441	35	19	−	−	PROPN
ejpam-3441	35	20	b	b	NOUN
ejpam-3441	35	21	and	and	CCONJ
ejpam-3441	35	22	ab	ab	PROPN
ejpam-3441	35	23	∈	∈	PROPN
ejpam-3441	35	24	a	a	PRON
ejpam-3441	35	25	for	for	ADP
ejpam-3441	35	26	all	all	DET
ejpam-3441	35	27	a	a	DET
ejpam-3441	35	28	,	,	PUNCT
ejpam-3441	35	29	b	b	X
ejpam-3441	35	30	∈	∈	PROPN
ejpam-3441	35	31	a.	a.	NOUN
ejpam-3441	35	32	a	a	PRON
ejpam-3441	35	33	is	be	AUX
ejpam-3441	35	34	called	call	VERB
ejpam-3441	35	35	a	a	DET
ejpam-3441	35	36	left	left	ADJ
ejpam-3441	35	37	(	(	PUNCT
ejpam-3441	35	38	resp	resp	NOUN
ejpam-3441	35	39	.	.	PUNCT
ejpam-3441	36	1	right	right	ADJ
ejpam-3441	36	2	)	)	PUNCT
ejpam-3441	36	3	ideal	ideal	NOUN
ejpam-3441	36	4	of	of	ADP
ejpam-3441	36	5	r	r	NOUN
ejpam-3441	36	6	if	if	SCONJ
ejpam-3441	36	7	(	(	PUNCT
ejpam-3441	36	8	a,+	a,+	NOUN
ejpam-3441	36	9	)	)	PUNCT
ejpam-3441	36	10	is	be	AUX
ejpam-3441	36	11	an	an	DET
ejpam-3441	36	12	la	la	ADJ
ejpam-3441	36	13	-	-	NOUN
ejpam-3441	36	14	group	group	NOUN
ejpam-3441	36	15	and	and	CCONJ
ejpam-3441	36	16	ra	ra	PROPN
ejpam-3441	37	1	⊆	⊆	NUM
ejpam-3441	37	2	a	a	DET
ejpam-3441	37	3	(	(	PUNCT
ejpam-3441	37	4	resp	resp	NOUN
ejpam-3441	37	5	.	.	PUNCT
ejpam-3441	38	1	ar	ar	VERB
ejpam-3441	38	2	⊆	⊆	NUM
ejpam-3441	38	3	a	a	PRON
ejpam-3441	38	4	)	)	PUNCT
ejpam-3441	38	5	.	.	PUNCT
ejpam-3441	39	1	a	a	PRON
ejpam-3441	39	2	is	be	AUX
ejpam-3441	39	3	called	call	VERB
ejpam-3441	39	4	an	an	DET
ejpam-3441	39	5	ideal	ideal	NOUN
ejpam-3441	39	6	of	of	ADP
ejpam-3441	39	7	r	r	NOUN
ejpam-3441	39	8	if	if	SCONJ
ejpam-3441	39	9	it	it	PRON
ejpam-3441	39	10	is	be	AUX
ejpam-3441	39	11	both	both	CCONJ
ejpam-3441	39	12	a	a	DET
ejpam-3441	39	13	left	left	ADJ
ejpam-3441	39	14	ideal	ideal	NOUN
ejpam-3441	39	15	and	and	CCONJ
ejpam-3441	39	16	a	a	DET
ejpam-3441	39	17	right	right	ADJ
ejpam-3441	39	18	ideal	ideal	NOUN
ejpam-3441	39	19	of	of	ADP
ejpam-3441	39	20	r.	r.	PROPN
ejpam-3441	39	21	a	a	DET
ejpam-3441	39	22	non	non	ADJ
ejpam-3441	39	23	-	-	ADJ
ejpam-3441	39	24	empty	empty	ADJ
ejpam-3441	39	25	subset	subset	NOUN
ejpam-3441	39	26	a	a	PRON
ejpam-3441	39	27	of	of	ADP
ejpam-3441	39	28	r	r	NOUN
ejpam-3441	39	29	is	be	AUX
ejpam-3441	39	30	called	call	VERB
ejpam-3441	39	31	an	an	DET
ejpam-3441	39	32	interior	interior	ADJ
ejpam-3441	39	33	ideal	ideal	NOUN
ejpam-3441	39	34	of	of	ADP
ejpam-3441	39	35	r	r	NOUN
ejpam-3441	39	36	if	if	SCONJ
ejpam-3441	39	37	(	(	PUNCT
ejpam-3441	39	38	a,+	a,+	NOUN
ejpam-3441	39	39	)	)	PUNCT
ejpam-3441	39	40	is	be	AUX
ejpam-3441	39	41	an	an	DET
ejpam-3441	39	42	la	la	ADJ
ejpam-3441	39	43	-	-	NOUN
ejpam-3441	39	44	group	group	NOUN
ejpam-3441	39	45	and	and	CCONJ
ejpam-3441	39	46	(	(	PUNCT
ejpam-3441	39	47	ra)r	ra)r	PROPN
ejpam-3441	39	48	⊆	⊆	NUM
ejpam-3441	39	49	a.	a.	NOUN
ejpam-3441	39	50	a	a	DET
ejpam-3441	39	51	non	non	ADJ
ejpam-3441	39	52	-	-	ADJ
ejpam-3441	39	53	empty	empty	ADJ
ejpam-3441	39	54	subset	subset	NOUN
ejpam-3441	39	55	a	a	PRON
ejpam-3441	39	56	of	of	ADP
ejpam-3441	39	57	r	r	NOUN
ejpam-3441	39	58	is	be	AUX
ejpam-3441	39	59	called	call	VERB
ejpam-3441	39	60	a	a	DET
ejpam-3441	39	61	quasi	quasi	NOUN
ejpam-3441	39	62	-	-	NOUN
ejpam-3441	39	63	ideal	ideal	ADJ
ejpam-3441	39	64	of	of	ADP
ejpam-3441	39	65	r	r	NOUN
ejpam-3441	39	66	if	if	SCONJ
ejpam-3441	39	67	(	(	PUNCT
ejpam-3441	39	68	a,+	a,+	NOUN
ejpam-3441	39	69	)	)	PUNCT
ejpam-3441	39	70	is	be	AUX
ejpam-3441	39	71	an	an	DET
ejpam-3441	39	72	la	la	ADJ
ejpam-3441	39	73	-	-	NOUN
ejpam-3441	39	74	group	group	NOUN
ejpam-3441	39	75	and	and	CCONJ
ejpam-3441	39	76	ar∩ra	ar∩ra	NOUN
ejpam-3441	39	77	⊆	⊆	NUM
ejpam-3441	39	78	a.	a.	NOUN
ejpam-3441	39	79	an	an	DET
ejpam-3441	39	80	la	la	ADV
ejpam-3441	39	81	-	-	PUNCT
ejpam-3441	39	82	subring	subre	VERB
ejpam-3441	39	83	a	a	PRON
ejpam-3441	39	84	of	of	ADP
ejpam-3441	39	85	r	r	NOUN
ejpam-3441	39	86	is	be	AUX
ejpam-3441	39	87	called	call	VERB
ejpam-3441	39	88	a	a	DET
ejpam-3441	39	89	bi	bi	NOUN
ejpam-3441	39	90	-	-	NOUN
ejpam-3441	39	91	ideal	ideal	NOUN
ejpam-3441	39	92	of	of	ADP
ejpam-3441	39	93	r	r	NOUN
ejpam-3441	39	94	if	if	SCONJ
ejpam-3441	39	95	(	(	PUNCT
ejpam-3441	39	96	ar)a	ar)a	PROPN
ejpam-3441	39	97	⊆	⊆	NUM
ejpam-3441	39	98	a.	a.	NOUN
ejpam-3441	39	99	a	a	DET
ejpam-3441	39	100	non	non	ADJ
ejpam-3441	39	101	-	-	ADJ
ejpam-3441	39	102	empty	empty	ADJ
ejpam-3441	39	103	subset	subset	NOUN
ejpam-3441	39	104	a	a	PRON
ejpam-3441	39	105	of	of	ADP
ejpam-3441	39	106	r	r	NOUN
ejpam-3441	39	107	is	be	AUX
ejpam-3441	39	108	called	call	VERB
ejpam-3441	39	109	a	a	DET
ejpam-3441	39	110	generalized	generalized	ADJ
ejpam-3441	39	111	bi	bi	NOUN
ejpam-3441	39	112	-	-	NOUN
ejpam-3441	39	113	ideal	ideal	NOUN
ejpam-3441	39	114	of	of	ADP
ejpam-3441	39	115	r	r	NOUN
ejpam-3441	39	116	if	if	SCONJ
ejpam-3441	39	117	(	(	PUNCT
ejpam-3441	39	118	a,+	a,+	NOUN
ejpam-3441	39	119	)	)	PUNCT
ejpam-3441	39	120	is	be	AUX
ejpam-3441	39	121	an	an	DET
ejpam-3441	39	122	la	la	ADJ
ejpam-3441	39	123	-	-	NOUN
ejpam-3441	39	124	group	group	NOUN
ejpam-3441	39	125	and	and	CCONJ
ejpam-3441	39	126	(	(	PUNCT
ejpam-3441	39	127	ar)a	ar)a	PROPN
ejpam-3441	39	128	⊆	⊆	NUM
ejpam-3441	39	129	a.	a.	NOUN
ejpam-3441	39	130	we	we	PRON
ejpam-3441	39	131	will	will	AUX
ejpam-3441	39	132	introduce	introduce	VERB
ejpam-3441	39	133	the	the	DET
ejpam-3441	39	134	concept	concept	NOUN
ejpam-3441	39	135	intuitionistic	intuitionistic	ADJ
ejpam-3441	39	136	fuzzy	fuzzy	ADJ
ejpam-3441	39	137	left	left	NOUN
ejpam-3441	39	138	(	(	PUNCT
ejpam-3441	39	139	resp	resp	NOUN
ejpam-3441	39	140	.	.	PUNCT
ejpam-3441	40	1	right	right	ADJ
ejpam-3441	40	2	,	,	PUNCT
ejpam-3441	40	3	interior	interior	NOUN
ejpam-3441	40	4	,	,	PUNCT
ejpam-3441	40	5	quasi-	quasi-	PROPN
ejpam-3441	40	6	,	,	PUNCT
ejpam-3441	40	7	bi-	bi-	NUM
ejpam-3441	40	8	,	,	PUNCT
ejpam-3441	40	9	generalized	generalize	VERB
ejpam-3441	40	10	bi-	bi-	NUM
ejpam-3441	40	11	)	)	PUNCT
ejpam-3441	40	12	ideals	ideal	NOUN
ejpam-3441	40	13	with	with	ADP
ejpam-3441	40	14	thresholds	threshold	NOUN
ejpam-3441	40	15	(	(	PUNCT
ejpam-3441	40	16	α	α	X
ejpam-3441	40	17	,	,	PUNCT
ejpam-3441	40	18	β	β	X
ejpam-3441	40	19	]	]	PUNCT
ejpam-3441	40	20	of	of	ADP
ejpam-3441	40	21	an	an	DET
ejpam-3441	40	22	la	la	ADJ
ejpam-3441	40	23	-	-	PUNCT
ejpam-3441	40	24	ring	ring	NOUN
ejpam-3441	40	25	r.	r.	NOUN
ejpam-3441	40	26	we	we	PRON
ejpam-3441	40	27	will	will	AUX
ejpam-3441	40	28	establish	establish	VERB
ejpam-3441	40	29	a	a	DET
ejpam-3441	40	30	study	study	NOUN
ejpam-3441	40	31	by	by	ADP
ejpam-3441	40	32	describing	describe	VERB
ejpam-3441	40	33	the	the	DET
ejpam-3441	40	34	different	different	ADJ
ejpam-3441	40	35	properties	property	NOUN
ejpam-3441	40	36	in	in	ADP
ejpam-3441	40	37	terms	term	NOUN
ejpam-3441	40	38	of	of	ADP
ejpam-3441	40	39	such	such	ADJ
ejpam-3441	40	40	ideals	ideal	NOUN
ejpam-3441	40	41	,	,	PUNCT
ejpam-3441	40	42	which	which	PRON
ejpam-3441	40	43	will	will	AUX
ejpam-3441	40	44	be	be	AUX
ejpam-3441	40	45	very	very	ADV
ejpam-3441	40	46	useful	useful	ADJ
ejpam-3441	40	47	for	for	ADP
ejpam-3441	40	48	the	the	DET
ejpam-3441	40	49	characterizations	characterization	NOUN
ejpam-3441	40	50	of	of	ADP
ejpam-3441	40	51	regular	regular	ADJ
ejpam-3441	40	52	(	(	PUNCT
ejpam-3441	40	53	intra	intra	ADJ
ejpam-3441	40	54	-	-	ADJ
ejpam-3441	40	55	regular	regular	ADJ
ejpam-3441	40	56	,	,	PUNCT
ejpam-3441	40	57	both	both	CCONJ
ejpam-3441	40	58	regular	regular	ADJ
ejpam-3441	40	59	and	and	CCONJ
ejpam-3441	40	60	intra	intra	ADJ
ejpam-3441	40	61	-	-	ADJ
ejpam-3441	40	62	regular	regular	ADJ
ejpam-3441	40	63	)	)	PUNCT
ejpam-3441	40	64	la	la	NOUN
ejpam-3441	40	65	-	-	PUNCT
ejpam-3441	40	66	rings	ring	NOUN
ejpam-3441	40	67	in	in	ADP
ejpam-3441	40	68	terms	term	NOUN
ejpam-3441	40	69	of	of	ADP
ejpam-3441	40	70	intuitionistic	intuitionistic	ADJ
ejpam-3441	40	71	fuzzy	fuzzy	ADJ
ejpam-3441	40	72	left	left	NOUN
ejpam-3441	40	73	(	(	PUNCT
ejpam-3441	40	74	right	right	ADJ
ejpam-3441	40	75	,	,	PUNCT
ejpam-3441	40	76	quasi-	quasi-	INTJ
ejpam-3441	40	77	,	,	PUNCT
ejpam-3441	40	78	bi-	bi-	NUM
ejpam-3441	40	79	,	,	PUNCT
ejpam-3441	40	80	generalized	generalize	VERB
ejpam-3441	40	81	bi-	bi-	NUM
ejpam-3441	40	82	)	)	PUNCT
ejpam-3441	40	83	ideals	ideal	NOUN
ejpam-3441	40	84	with	with	ADP
ejpam-3441	40	85	thresholds	threshold	NOUN
ejpam-3441	40	86	(	(	PUNCT
ejpam-3441	40	87	α	α	X
ejpam-3441	40	88	,	,	PUNCT
ejpam-3441	40	89	β	β	X
ejpam-3441	40	90	]	]	X
ejpam-3441	40	91	.	.	PUNCT
ejpam-3441	41	1	2	2	X
ejpam-3441	41	2	.	.	X
ejpam-3441	41	3	intuitionistic	intuitionistic	ADJ
ejpam-3441	41	4	fuzzy	fuzzy	ADJ
ejpam-3441	41	5	ideals	ideal	NOUN
ejpam-3441	41	6	with	with	ADP
ejpam-3441	41	7	thresholds	threshold	NOUN
ejpam-3441	41	8	(	(	PUNCT
ejpam-3441	41	9	α	α	X
ejpam-3441	41	10	,	,	PUNCT
ejpam-3441	41	11	β	β	X
ejpam-3441	41	12	]	]	X
ejpam-3441	41	13	after	after	ADP
ejpam-3441	41	14	the	the	DET
ejpam-3441	41	15	introduction	introduction	NOUN
ejpam-3441	41	16	of	of	ADP
ejpam-3441	41	17	fuzzy	fuzzy	ADJ
ejpam-3441	41	18	set	set	VERB
ejpam-3441	41	19	by	by	ADP
ejpam-3441	41	20	zadeh	zadeh	PROPN
ejpam-3441	42	1	[	[	X
ejpam-3441	42	2	31	31	NUM
ejpam-3441	42	3	]	]	PUNCT
ejpam-3441	42	4	,	,	PUNCT
ejpam-3441	42	5	several	several	ADJ
ejpam-3441	42	6	researchers	researcher	NOUN
ejpam-3441	42	7	explored	explore	VERB
ejpam-3441	42	8	on	on	ADP
ejpam-3441	42	9	the	the	DET
ejpam-3441	42	10	generalization	generalization	NOUN
ejpam-3441	42	11	of	of	ADP
ejpam-3441	42	12	the	the	DET
ejpam-3441	42	13	notion	notion	NOUN
ejpam-3441	42	14	of	of	ADP
ejpam-3441	42	15	fuzzy	fuzzy	ADJ
ejpam-3441	42	16	set	set	NOUN
ejpam-3441	42	17	.	.	PUNCT
ejpam-3441	43	1	the	the	DET
ejpam-3441	43	2	concept	concept	NOUN
ejpam-3441	43	3	of	of	ADP
ejpam-3441	43	4	intuitionistic	intuitionistic	ADJ
ejpam-3441	43	5	fuzzy	fuzzy	ADJ
ejpam-3441	43	6	set	set	NOUN
ejpam-3441	43	7	was	be	AUX
ejpam-3441	43	8	introduced	introduce	VERB
ejpam-3441	43	9	by	by	ADP
ejpam-3441	43	10	atanassov	atanassov	NOUN
ejpam-3441	43	11	[	[	X
ejpam-3441	43	12	1	1	NUM
ejpam-3441	43	13	,	,	PUNCT
ejpam-3441	43	14	2	2	NUM
ejpam-3441	43	15	]	]	PUNCT
ejpam-3441	43	16	,	,	PUNCT
ejpam-3441	43	17	as	as	ADP
ejpam-3441	43	18	a	a	DET
ejpam-3441	43	19	generalization	generalization	NOUN
ejpam-3441	43	20	of	of	ADP
ejpam-3441	43	21	the	the	DET
ejpam-3441	43	22	notion	notion	NOUN
ejpam-3441	43	23	of	of	ADP
ejpam-3441	43	24	fuzzy	fuzzy	ADJ
ejpam-3441	43	25	set	set	NOUN
ejpam-3441	43	26	.	.	PUNCT
ejpam-3441	44	1	liu	liu	PROPN
ejpam-3441	45	1	[	[	X
ejpam-3441	45	2	20	20	NUM
ejpam-3441	45	3	]	]	PUNCT
ejpam-3441	45	4	,	,	PUNCT
ejpam-3441	45	5	introduced	introduce	VERB
ejpam-3441	45	6	the	the	DET
ejpam-3441	45	7	concept	concept	NOUN
ejpam-3441	45	8	of	of	ADP
ejpam-3441	45	9	fuzzy	fuzzy	ADJ
ejpam-3441	45	10	subrings	subring	NOUN
ejpam-3441	45	11	and	and	CCONJ
ejpam-3441	45	12	fuzzy	fuzzy	ADJ
ejpam-3441	45	13	ideals	ideal	NOUN
ejpam-3441	45	14	of	of	ADP
ejpam-3441	45	15	a	a	DET
ejpam-3441	45	16	ring	ring	NOUN
ejpam-3441	45	17	.	.	PUNCT
ejpam-3441	46	1	many	many	ADJ
ejpam-3441	46	2	authors	author	NOUN
ejpam-3441	46	3	have	have	AUX
ejpam-3441	46	4	explored	explore	VERB
ejpam-3441	46	5	the	the	DET
ejpam-3441	46	6	theory	theory	NOUN
ejpam-3441	46	7	of	of	ADP
ejpam-3441	46	8	fuzzy	fuzzy	ADJ
ejpam-3441	46	9	rings	ring	NOUN
ejpam-3441	46	10	(	(	PUNCT
ejpam-3441	46	11	for	for	ADP
ejpam-3441	46	12	example	example	NOUN
ejpam-3441	46	13	[	[	X
ejpam-3441	46	14	11	11	NUM
ejpam-3441	46	15	,	,	PUNCT
ejpam-3441	46	16	19	19	NUM
ejpam-3441	46	17	,	,	PUNCT
ejpam-3441	46	18	21	21	NUM
ejpam-3441	46	19	,	,	PUNCT
ejpam-3441	46	20	22	22	NUM
ejpam-3441	46	21	,	,	PUNCT
ejpam-3441	46	22	29	29	NUM
ejpam-3441	46	23	]	]	PUNCT
ejpam-3441	46	24	)	)	PUNCT
ejpam-3441	46	25	.	.	PUNCT
ejpam-3441	47	1	gupta	gupta	PROPN
ejpam-3441	47	2	et	et	PROPN
ejpam-3441	47	3	al	al	PROPN
ejpam-3441	48	1	[	[	X
ejpam-3441	48	2	11	11	NUM
ejpam-3441	48	3	]	]	PUNCT
ejpam-3441	48	4	,	,	PUNCT
ejpam-3441	48	5	gave	give	VERB
ejpam-3441	48	6	the	the	DET
ejpam-3441	48	7	idea	idea	NOUN
ejpam-3441	48	8	of	of	ADP
ejpam-3441	48	9	intrinsic	intrinsic	ADJ
ejpam-3441	48	10	product	product	NOUN
ejpam-3441	48	11	of	of	ADP
ejpam-3441	48	12	fuzzy	fuzzy	ADJ
ejpam-3441	48	13	subsets	subset	NOUN
ejpam-3441	48	14	of	of	ADP
ejpam-3441	48	15	a	a	DET
ejpam-3441	48	16	ring	ring	NOUN
ejpam-3441	48	17	.	.	PUNCT
ejpam-3441	49	1	kuroki	kuroki	PROPN
ejpam-3441	50	1	[	[	X
ejpam-3441	50	2	19	19	NUM
ejpam-3441	50	3	]	]	PUNCT
ejpam-3441	50	4	,	,	PUNCT
ejpam-3441	50	5	characterized	characterize	VERB
ejpam-3441	50	6	regular	regular	ADV
ejpam-3441	50	7	(	(	PUNCT
ejpam-3441	50	8	intra	intra	ADJ
ejpam-3441	50	9	-	-	ADJ
ejpam-3441	50	10	regular	regular	ADJ
ejpam-3441	50	11	,	,	PUNCT
ejpam-3441	50	12	both	both	CCONJ
ejpam-3441	50	13	regular	regular	ADJ
ejpam-3441	50	14	and	and	CCONJ
ejpam-3441	50	15	intra	intra	ADJ
ejpam-3441	50	16	-	-	ADJ
ejpam-3441	50	17	regular	regular	ADJ
ejpam-3441	50	18	)	)	PUNCT
ejpam-3441	50	19	rings	ring	NOUN
ejpam-3441	50	20	in	in	ADP
ejpam-3441	50	21	terms	term	NOUN
ejpam-3441	50	22	of	of	ADP
ejpam-3441	50	23	fuzzy	fuzzy	ADJ
ejpam-3441	50	24	left	left	NOUN
ejpam-3441	50	25	(	(	PUNCT
ejpam-3441	50	26	right	right	ADJ
ejpam-3441	50	27	,	,	PUNCT
ejpam-3441	50	28	quasi	quasi	ADJ
ejpam-3441	50	29	,	,	PUNCT
ejpam-3441	50	30	bi-	bi-	NUM
ejpam-3441	50	31	)	)	PUNCT
ejpam-3441	50	32	ideals	ideal	NOUN
ejpam-3441	50	33	.	.	PUNCT
ejpam-3441	51	1	an	an	DET
ejpam-3441	51	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	51	3	fuzzy	fuzzy	ADJ
ejpam-3441	51	4	set	set	NOUN
ejpam-3441	51	5	(	(	PUNCT
ejpam-3441	51	6	briefly	briefly	ADV
ejpam-3441	51	7	,	,	PUNCT
ejpam-3441	51	8	ifs	ifs	PROPN
ejpam-3441	51	9	)	)	PUNCT
ejpam-3441	51	10	a	a	PRON
ejpam-3441	51	11	in	in	ADP
ejpam-3441	51	12	a	a	DET
ejpam-3441	51	13	non	non	ADJ
ejpam-3441	51	14	-	-	ADJ
ejpam-3441	51	15	empty	empty	ADJ
ejpam-3441	51	16	set	set	NOUN
ejpam-3441	51	17	x	x	PUNCT
ejpam-3441	51	18	is	be	AUX
ejpam-3441	51	19	an	an	DET
ejpam-3441	51	20	object	object	NOUN
ejpam-3441	51	21	having	have	VERB
ejpam-3441	51	22	the	the	DET
ejpam-3441	51	23	form	form	NOUN
ejpam-3441	51	24	a	a	DET
ejpam-3441	51	25	=	=	X
ejpam-3441	51	26	{	{	PUNCT
ejpam-3441	51	27	(	(	PUNCT
ejpam-3441	51	28	x	x	NOUN
ejpam-3441	51	29	,	,	PUNCT
ejpam-3441	51	30	µa(x	µa(x	NOUN
ejpam-3441	51	31	)	)	PUNCT
ejpam-3441	51	32	,	,	PUNCT
ejpam-3441	51	33	γa(x	γa(x	NUM
ejpam-3441	51	34	)	)	PUNCT
ejpam-3441	51	35	)	)	PUNCT
ejpam-3441	51	36	:	:	PUNCT
ejpam-3441	52	1	x	x	X
ejpam-3441	52	2	∈	∈	NOUN
ejpam-3441	52	3	x	x	X
ejpam-3441	52	4	}	}	PUNCT
ejpam-3441	52	5	,	,	PUNCT
ejpam-3441	52	6	where	where	SCONJ
ejpam-3441	52	7	the	the	DET
ejpam-3441	52	8	functions	function	NOUN
ejpam-3441	52	9	µa	µa	VERB
ejpam-3441	52	10	:	:	PUNCT
ejpam-3441	52	11	x	x	X
ejpam-3441	52	12	→	→	SYM
ejpam-3441	52	13	[	[	X
ejpam-3441	52	14	0	0	NUM
ejpam-3441	52	15	,	,	PUNCT
ejpam-3441	52	16	1	1	NUM
ejpam-3441	52	17	]	]	PUNCT
ejpam-3441	52	18	and	and	CCONJ
ejpam-3441	52	19	γa	γa	PRON
ejpam-3441	52	20	:	:	PUNCT
ejpam-3441	52	21	x	x	X
ejpam-3441	52	22	→	→	PUNCT
ejpam-3441	52	23	[	[	X
ejpam-3441	52	24	0	0	NUM
ejpam-3441	52	25	,	,	PUNCT
ejpam-3441	52	26	1	1	NUM
ejpam-3441	52	27	]	]	PUNCT
ejpam-3441	52	28	denote	denote	VERB
ejpam-3441	52	29	the	the	DET
ejpam-3441	52	30	degree	degree	NOUN
ejpam-3441	52	31	of	of	ADP
ejpam-3441	52	32	membership	membership	NOUN
ejpam-3441	52	33	and	and	CCONJ
ejpam-3441	52	34	the	the	DET
ejpam-3441	52	35	degree	degree	NOUN
ejpam-3441	52	36	of	of	ADP
ejpam-3441	52	37	nonmembership	nonmembership	NOUN
ejpam-3441	52	38	,	,	PUNCT
ejpam-3441	52	39	respectively	respectively	ADV
ejpam-3441	52	40	and	and	CCONJ
ejpam-3441	52	41	0	0	NUM
ejpam-3441	52	42	≤	≤	NOUN
ejpam-3441	52	43	µa(x	µa(x	NOUN
ejpam-3441	52	44	)	)	PUNCT
ejpam-3441	52	45	+	+	CCONJ
ejpam-3441	52	46	γa(x	γa(x	X
ejpam-3441	52	47	)	)	PUNCT
ejpam-3441	52	48	≤	≤	NUM
ejpam-3441	52	49	1	1	NUM
ejpam-3441	52	50	for	for	ADP
ejpam-3441	52	51	all	all	DET
ejpam-3441	52	52	x	x	SYM
ejpam-3441	52	53	∈	∈	NOUN
ejpam-3441	52	54	x	x	PUNCT
ejpam-3441	53	1	[	[	X
ejpam-3441	53	2	1	1	NUM
ejpam-3441	53	3	,	,	PUNCT
ejpam-3441	53	4	2	2	NUM
ejpam-3441	53	5	]	]	PUNCT
ejpam-3441	53	6	.	.	PUNCT
ejpam-3441	54	1	an	an	DET
ejpam-3441	54	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	54	3	fuzzy	fuzzy	NOUN
ejpam-3441	54	4	set	set	VERB
ejpam-3441	54	5	a	a	PRON
ejpam-3441	54	6	=	=	X
ejpam-3441	54	7	{	{	PUNCT
ejpam-3441	54	8	(	(	PUNCT
ejpam-3441	54	9	x	x	NOUN
ejpam-3441	54	10	,	,	PUNCT
ejpam-3441	54	11	µa(x	µa(x	NOUN
ejpam-3441	54	12	)	)	PUNCT
ejpam-3441	54	13	,	,	PUNCT
ejpam-3441	54	14	γa(x	γa(x	NUM
ejpam-3441	54	15	)	)	PUNCT
ejpam-3441	54	16	)	)	PUNCT
ejpam-3441	54	17	:	:	PUNCT
ejpam-3441	55	1	x	x	X
ejpam-3441	55	2	∈	∈	NOUN
ejpam-3441	55	3	x	x	X
ejpam-3441	55	4	}	}	PUNCT
ejpam-3441	55	5	in	in	ADP
ejpam-3441	55	6	x	x	PRON
ejpam-3441	55	7	can	can	AUX
ejpam-3441	55	8	be	be	AUX
ejpam-3441	55	9	identified	identify	VERB
ejpam-3441	55	10	to	to	PART
ejpam-3441	55	11	be	be	AUX
ejpam-3441	55	12	an	an	DET
ejpam-3441	55	13	ordered	order	VERB
ejpam-3441	55	14	pair	pair	NOUN
ejpam-3441	55	15	(	(	PUNCT
ejpam-3441	55	16	µa	µa	NOUN
ejpam-3441	55	17	,	,	PUNCT
ejpam-3441	55	18	γa	γa	NOUN
ejpam-3441	55	19	)	)	PUNCT
ejpam-3441	55	20	in	in	ADP
ejpam-3441	55	21	ix	ix	ADP
ejpam-3441	55	22	×	×	NOUN
ejpam-3441	55	23	ix	ix	ADV
ejpam-3441	55	24	,	,	PUNCT
ejpam-3441	55	25	where	where	SCONJ
ejpam-3441	55	26	ix	ix	ADV
ejpam-3441	55	27	is	be	AUX
ejpam-3441	55	28	the	the	DET
ejpam-3441	55	29	set	set	NOUN
ejpam-3441	55	30	of	of	ADP
ejpam-3441	55	31	all	all	DET
ejpam-3441	55	32	functions	function	NOUN
ejpam-3441	55	33	from	from	ADP
ejpam-3441	55	34	x	x	PUNCT
ejpam-3441	55	35	to	to	ADP
ejpam-3441	55	36	[	[	X
ejpam-3441	55	37	0	0	NUM
ejpam-3441	55	38	,	,	PUNCT
ejpam-3441	55	39	1	1	NUM
ejpam-3441	55	40	]	]	PUNCT
ejpam-3441	55	41	.	.	PUNCT
ejpam-3441	56	1	for	for	ADP
ejpam-3441	56	2	the	the	DET
ejpam-3441	56	3	sake	sake	NOUN
ejpam-3441	56	4	of	of	ADP
ejpam-3441	56	5	simplicity	simplicity	NOUN
ejpam-3441	56	6	,	,	PUNCT
ejpam-3441	56	7	we	we	PRON
ejpam-3441	56	8	shall	shall	AUX
ejpam-3441	56	9	use	use	VERB
ejpam-3441	56	10	the	the	DET
ejpam-3441	56	11	symbol	symbol	NOUN
ejpam-3441	56	12	a	a	DET
ejpam-3441	56	13	=	=	X
ejpam-3441	56	14	(	(	PUNCT
ejpam-3441	56	15	µa	µa	PROPN
ejpam-3441	56	16	,	,	PUNCT
ejpam-3441	56	17	γa	γa	PROPN
ejpam-3441	56	18	)	)	PUNCT
ejpam-3441	56	19	for	for	ADP
ejpam-3441	56	20	the	the	DET
ejpam-3441	56	21	ifs	ifs	PROPN
ejpam-3441	56	22	k.	k.	PROPN
ejpam-3441	56	23	nasreen	nasreen	PROPN
ejpam-3441	56	24	et	et	PROPN
ejpam-3441	56	25	al	al	PROPN
ejpam-3441	56	26	.	.	PUNCT
ejpam-3441	56	27	/	/	SYM
ejpam-3441	56	28	eur	eur	PROPN
ejpam-3441	56	29	.	.	PUNCT
ejpam-3441	57	1	j.	j.	PROPN
ejpam-3441	57	2	pure	pure	PROPN
ejpam-3441	57	3	appl	appl	PROPN
ejpam-3441	57	4	.	.	PROPN
ejpam-3441	57	5	math	math	PROPN
ejpam-3441	57	6	,	,	PUNCT
ejpam-3441	57	7	12	12	NUM
ejpam-3441	57	8	(	(	PUNCT
ejpam-3441	57	9	3	3	NUM
ejpam-3441	57	10	)	)	PUNCT
ejpam-3441	57	11	(	(	PUNCT
ejpam-3441	57	12	2019	2019	NUM
ejpam-3441	57	13	)	)	PUNCT
ejpam-3441	57	14	,	,	PUNCT
ejpam-3441	57	15	906	906	NUM
ejpam-3441	57	16	-	-	SYM
ejpam-3441	57	17	943	943	NUM
ejpam-3441	57	18	908	908	NUM
ejpam-3441	58	1	a	a	PRON
ejpam-3441	58	2	=	=	X
ejpam-3441	58	3	{	{	PUNCT
ejpam-3441	58	4	(	(	PUNCT
ejpam-3441	58	5	x	x	NOUN
ejpam-3441	58	6	,	,	PUNCT
ejpam-3441	58	7	µa(x	µa(x	NOUN
ejpam-3441	58	8	)	)	PUNCT
ejpam-3441	58	9	,	,	PUNCT
ejpam-3441	58	10	γa(x	γa(x	NUM
ejpam-3441	58	11	)	)	PUNCT
ejpam-3441	58	12	)	)	PUNCT
ejpam-3441	58	13	:	:	PUNCT
ejpam-3441	59	1	x	x	X
ejpam-3441	59	2	∈	∈	NOUN
ejpam-3441	59	3	x	x	X
ejpam-3441	59	4	}	}	PUNCT
ejpam-3441	59	5	.	.	PUNCT
ejpam-3441	60	1	banerjee	banerjee	PROPN
ejpam-3441	60	2	et	et	PROPN
ejpam-3441	60	3	al	al	PROPN
ejpam-3441	61	1	[	[	X
ejpam-3441	61	2	3	3	NUM
ejpam-3441	61	3	]	]	PUNCT
ejpam-3441	61	4	and	and	CCONJ
ejpam-3441	61	5	hur	hur	PROPN
ejpam-3441	61	6	et	et	PROPN
ejpam-3441	61	7	al	al	PROPN
ejpam-3441	61	8	[	[	X
ejpam-3441	61	9	10	10	NUM
ejpam-3441	61	10	]	]	PUNCT
ejpam-3441	61	11	,	,	PUNCT
ejpam-3441	61	12	initiated	initiate	VERB
ejpam-3441	61	13	the	the	DET
ejpam-3441	61	14	notion	notion	NOUN
ejpam-3441	61	15	of	of	ADP
ejpam-3441	61	16	intuitionistic	intuitionistic	ADJ
ejpam-3441	61	17	fuzzy	fuzzy	ADJ
ejpam-3441	61	18	subrings	subring	NOUN
ejpam-3441	61	19	and	and	CCONJ
ejpam-3441	61	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	61	21	fuzzy	fuzzy	ADJ
ejpam-3441	61	22	ideals	ideal	NOUN
ejpam-3441	61	23	of	of	ADP
ejpam-3441	61	24	a	a	DET
ejpam-3441	61	25	ring	ring	NOUN
ejpam-3441	61	26	.	.	PUNCT
ejpam-3441	62	1	subsequently	subsequently	ADV
ejpam-3441	62	2	many	many	ADJ
ejpam-3441	62	3	authors	author	NOUN
ejpam-3441	62	4	studied	study	VERB
ejpam-3441	62	5	the	the	DET
ejpam-3441	62	6	intuitionistic	intuitionistic	ADJ
ejpam-3441	62	7	fuzzy	fuzzy	ADJ
ejpam-3441	62	8	subrings	subring	NOUN
ejpam-3441	62	9	and	and	CCONJ
ejpam-3441	62	10	intuitionistic	intuitionistic	ADJ
ejpam-3441	62	11	fuzzy	fuzzy	ADJ
ejpam-3441	62	12	ideals	ideal	NOUN
ejpam-3441	62	13	of	of	ADP
ejpam-3441	62	14	a	a	DET
ejpam-3441	62	15	ring	ring	NOUN
ejpam-3441	62	16	by	by	ADP
ejpam-3441	62	17	describing	describe	VERB
ejpam-3441	62	18	the	the	DET
ejpam-3441	62	19	different	different	ADJ
ejpam-3441	62	20	properties	property	NOUN
ejpam-3441	62	21	(	(	PUNCT
ejpam-3441	62	22	see	see	VERB
ejpam-3441	62	23	[	[	X
ejpam-3441	62	24	9	9	NUM
ejpam-3441	62	25	]	]	NUM
ejpam-3441	62	26	)	)	PUNCT
ejpam-3441	62	27	.	.	PUNCT
ejpam-3441	63	1	shah	shah	PROPN
ejpam-3441	63	2	et	et	PROPN
ejpam-3441	63	3	al	al	PROPN
ejpam-3441	64	1	[	[	X
ejpam-3441	64	2	26	26	NUM
ejpam-3441	64	3	]	]	PUNCT
ejpam-3441	64	4	,	,	PUNCT
ejpam-3441	64	5	have	have	AUX
ejpam-3441	64	6	initiated	initiate	VERB
ejpam-3441	64	7	the	the	DET
ejpam-3441	64	8	concept	concept	NOUN
ejpam-3441	64	9	of	of	ADP
ejpam-3441	64	10	intuitionistic	intuitionistic	ADJ
ejpam-3441	64	11	fuzzy	fuzzy	ADJ
ejpam-3441	64	12	normal	normal	ADJ
ejpam-3441	64	13	la	la	NOUN
ejpam-3441	64	14	-	-	PUNCT
ejpam-3441	64	15	subrings	subring	NOUN
ejpam-3441	64	16	of	of	ADP
ejpam-3441	64	17	an	an	DET
ejpam-3441	64	18	la	la	ADJ
ejpam-3441	64	19	-	-	PUNCT
ejpam-3441	64	20	ring	ring	NOUN
ejpam-3441	64	21	.	.	PUNCT
ejpam-3441	65	1	bhakat	bhakat	PROPN
ejpam-3441	65	2	et	et	PROPN
ejpam-3441	65	3	al	al	PROPN
ejpam-3441	66	1	[	[	X
ejpam-3441	66	2	4–6	4–6	X
ejpam-3441	66	3	]	]	X
ejpam-3441	66	4	,	,	PUNCT
ejpam-3441	66	5	introduced	introduce	VERB
ejpam-3441	66	6	the	the	DET
ejpam-3441	66	7	notion	notion	NOUN
ejpam-3441	66	8	of	of	ADP
ejpam-3441	66	9	(	(	PUNCT
ejpam-3441	66	10	α	α	X
ejpam-3441	66	11	,	,	PUNCT
ejpam-3441	66	12	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-3441	66	13	subgroups	subgroup	NOUN
ejpam-3441	66	14	.	.	PUNCT
ejpam-3441	67	1	it	it	PRON
ejpam-3441	67	2	is	be	AUX
ejpam-3441	67	3	a	a	DET
ejpam-3441	67	4	generalization	generalization	NOUN
ejpam-3441	67	5	of	of	ADP
ejpam-3441	67	6	rosenfeld	rosenfeld	PROPN
ejpam-3441	67	7	fuzzy	fuzzy	ADJ
ejpam-3441	67	8	subgroups	subgroup	NOUN
ejpam-3441	67	9	as	as	ADP
ejpam-3441	67	10	(	(	PUNCT
ejpam-3441	67	11	∈,∈	∈,∈	X
ejpam-3441	67	12	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-3441	67	13	subgroups	subgroup	NOUN
ejpam-3441	67	14	.	.	PUNCT
ejpam-3441	68	1	then	then	ADV
ejpam-3441	68	2	many	many	ADJ
ejpam-3441	68	3	authors	author	NOUN
ejpam-3441	68	4	studied	study	VERB
ejpam-3441	68	5	the	the	DET
ejpam-3441	68	6	algebraic	algebraic	ADJ
ejpam-3441	68	7	structures	structure	NOUN
ejpam-3441	68	8	by	by	ADP
ejpam-3441	68	9	employing	employ	VERB
ejpam-3441	68	10	the	the	DET
ejpam-3441	68	11	idea	idea	NOUN
ejpam-3441	68	12	of	of	ADP
ejpam-3441	68	13	(	(	PUNCT
ejpam-3441	68	14	α	α	NOUN
ejpam-3441	68	15	,	,	PUNCT
ejpam-3441	68	16	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-3441	68	17	subsets	subset	NOUN
ejpam-3441	68	18	(	(	PUNCT
ejpam-3441	68	19	for	for	ADP
ejpam-3441	68	20	example	example	NOUN
ejpam-3441	68	21	[	[	X
ejpam-3441	68	22	7	7	NUM
ejpam-3441	68	23	,	,	PUNCT
ejpam-3441	68	24	13	13	NUM
ejpam-3441	68	25	,	,	PUNCT
ejpam-3441	68	26	23	23	NUM
ejpam-3441	68	27	]	]	PUNCT
ejpam-3441	68	28	)	)	PUNCT
ejpam-3441	68	29	.	.	PUNCT
ejpam-3441	69	1	yuan	yuan	NOUN
ejpam-3441	69	2	et	et	NOUN
ejpam-3441	69	3	al	al	PROPN
ejpam-3441	70	1	[	[	X
ejpam-3441	70	2	30	30	NUM
ejpam-3441	70	3	]	]	PUNCT
ejpam-3441	70	4	,	,	PUNCT
ejpam-3441	70	5	initiated	initiate	VERB
ejpam-3441	70	6	the	the	DET
ejpam-3441	70	7	concept	concept	NOUN
ejpam-3441	70	8	of	of	ADP
ejpam-3441	70	9	fuzzy	fuzzy	ADJ
ejpam-3441	70	10	subgroups	subgroup	NOUN
ejpam-3441	70	11	with	with	ADP
ejpam-3441	70	12	thresholds	threshold	NOUN
ejpam-3441	70	13	.	.	PUNCT
ejpam-3441	71	1	shabir	shabir	PROPN
ejpam-3441	71	2	et	et	PROPN
ejpam-3441	71	3	al	al	PROPN
ejpam-3441	72	1	[	[	X
ejpam-3441	72	2	28	28	NUM
ejpam-3441	72	3	]	]	PUNCT
ejpam-3441	72	4	,	,	PUNCT
ejpam-3441	72	5	gave	give	VERB
ejpam-3441	72	6	the	the	DET
ejpam-3441	72	7	idea	idea	NOUN
ejpam-3441	72	8	of	of	ADP
ejpam-3441	72	9	fuzzy	fuzzy	ADJ
ejpam-3441	72	10	ideals	ideal	NOUN
ejpam-3441	72	11	with	with	ADP
ejpam-3441	72	12	thresholds	threshold	NOUN
ejpam-3441	72	13	in	in	ADP
ejpam-3441	72	14	semigroups	semigroup	NOUN
ejpam-3441	72	15	.	.	PUNCT
ejpam-3441	73	1	now	now	ADV
ejpam-3441	73	2	we	we	PRON
ejpam-3441	73	3	initiate	initiate	VERB
ejpam-3441	73	4	the	the	DET
ejpam-3441	73	5	concept	concept	NOUN
ejpam-3441	73	6	of	of	ADP
ejpam-3441	73	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	73	8	fuzzy	fuzzy	ADJ
ejpam-3441	73	9	la	la	NOUN
ejpam-3441	73	10	-	-	NOUN
ejpam-3441	73	11	subrings	subring	NOUN
ejpam-3441	73	12	with	with	ADP
ejpam-3441	73	13	thresholds	threshold	NOUN
ejpam-3441	73	14	(	(	PUNCT
ejpam-3441	73	15	α	α	X
ejpam-3441	73	16	,	,	PUNCT
ejpam-3441	73	17	β	β	NOUN
ejpam-3441	73	18	]	]	PUNCT
ejpam-3441	73	19	and	and	CCONJ
ejpam-3441	73	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	73	21	fuzzy	fuzzy	ADJ
ejpam-3441	73	22	left	left	NOUN
ejpam-3441	73	23	(	(	PUNCT
ejpam-3441	73	24	resp	resp	NOUN
ejpam-3441	73	25	.	.	PUNCT
ejpam-3441	74	1	right	right	ADJ
ejpam-3441	74	2	,	,	PUNCT
ejpam-3441	74	3	interior	interior	NOUN
ejpam-3441	74	4	,	,	PUNCT
ejpam-3441	74	5	quasi-	quasi-	PROPN
ejpam-3441	74	6	,	,	PUNCT
ejpam-3441	74	7	bi-	bi-	NUM
ejpam-3441	74	8	,	,	PUNCT
ejpam-3441	74	9	generalized	generalize	VERB
ejpam-3441	74	10	bi-	bi-	NUM
ejpam-3441	74	11	)	)	PUNCT
ejpam-3441	74	12	ideals	ideal	NOUN
ejpam-3441	74	13	with	with	ADP
ejpam-3441	74	14	thresholds	threshold	NOUN
ejpam-3441	74	15	(	(	PUNCT
ejpam-3441	74	16	α	α	X
ejpam-3441	74	17	,	,	PUNCT
ejpam-3441	74	18	β	β	X
ejpam-3441	74	19	]	]	PUNCT
ejpam-3441	74	20	of	of	ADP
ejpam-3441	74	21	an	an	DET
ejpam-3441	74	22	la	la	ADJ
ejpam-3441	74	23	-	-	PUNCT
ejpam-3441	74	24	ring	ring	NOUN
ejpam-3441	74	25	r.	r.	PROPN
ejpam-3441	74	26	an	an	DET
ejpam-3441	74	27	ifs	ifs	PROPN
ejpam-3441	74	28	a	a	X
ejpam-3441	74	29	=	=	X
ejpam-3441	74	30	(	(	PUNCT
ejpam-3441	74	31	µa	µa	PROPN
ejpam-3441	74	32	,	,	PUNCT
ejpam-3441	74	33	γa	γa	PROPN
ejpam-3441	74	34	)	)	PUNCT
ejpam-3441	74	35	of	of	ADP
ejpam-3441	74	36	an	an	DET
ejpam-3441	74	37	la	la	ADJ
ejpam-3441	74	38	-	-	PUNCT
ejpam-3441	74	39	ring	ring	NOUN
ejpam-3441	74	40	r	r	NOUN
ejpam-3441	74	41	is	be	AUX
ejpam-3441	74	42	called	call	VERB
ejpam-3441	74	43	an	an	DET
ejpam-3441	74	44	intuitionistic	intuitionistic	ADJ
ejpam-3441	74	45	fuzzy	fuzzy	ADJ
ejpam-3441	74	46	la	la	NOUN
ejpam-3441	74	47	-	-	PUNCT
ejpam-3441	74	48	subring	subre	VERB
ejpam-3441	74	49	with	with	ADP
ejpam-3441	74	50	thresholds	threshold	NOUN
ejpam-3441	74	51	(	(	PUNCT
ejpam-3441	74	52	α	α	X
ejpam-3441	74	53	,	,	PUNCT
ejpam-3441	74	54	β	β	X
ejpam-3441	74	55	]	]	PUNCT
ejpam-3441	74	56	of	of	ADP
ejpam-3441	74	57	r	r	PRON
ejpam-3441	74	58	if	if	SCONJ
ejpam-3441	74	59	(	(	PUNCT
ejpam-3441	74	60	1	1	X
ejpam-3441	74	61	)	)	PUNCT
ejpam-3441	74	62	max{µa	max{µa	X
ejpam-3441	74	63	(	(	PUNCT
ejpam-3441	74	64	x−	x−	PROPN
ejpam-3441	74	65	y	y	PROPN
ejpam-3441	74	66	)	)	PUNCT
ejpam-3441	74	67	,	,	PUNCT
ejpam-3441	74	68	α	α	X
ejpam-3441	74	69	}	}	PUNCT
ejpam-3441	74	70	≥	≥	NOUN
ejpam-3441	74	71	min{µa	min{µa	X
ejpam-3441	74	72	(	(	PUNCT
ejpam-3441	74	73	x	x	NOUN
ejpam-3441	74	74	)	)	PUNCT
ejpam-3441	74	75	,	,	PUNCT
ejpam-3441	74	76	µa(y	µa(y	NOUN
ejpam-3441	74	77	)	)	PUNCT
ejpam-3441	74	78	,	,	PUNCT
ejpam-3441	74	79	β	β	X
ejpam-3441	74	80	}	}	PUNCT
ejpam-3441	74	81	,	,	PUNCT
ejpam-3441	74	82	(	(	PUNCT
ejpam-3441	74	83	2	2	X
ejpam-3441	74	84	)	)	PUNCT
ejpam-3441	74	85	min{γa	min{γa	NOUN
ejpam-3441	74	86	(	(	PUNCT
ejpam-3441	74	87	x−	x−	PROPN
ejpam-3441	74	88	y	y	PROPN
ejpam-3441	74	89	)	)	PUNCT
ejpam-3441	74	90	,	,	PUNCT
ejpam-3441	74	91	(	(	PUNCT
ejpam-3441	74	92	1−	1−	NUM
ejpam-3441	74	93	α	α	NOUN
ejpam-3441	74	94	)	)	PUNCT
ejpam-3441	74	95	}	}	PUNCT
ejpam-3441	74	96	≤	≤	NUM
ejpam-3441	74	97	max{γa	max{γa	PUNCT
ejpam-3441	74	98	(	(	PUNCT
ejpam-3441	74	99	x	x	NOUN
ejpam-3441	74	100	)	)	PUNCT
ejpam-3441	74	101	,	,	PUNCT
ejpam-3441	74	102	γa(y	γa(y	NUM
ejpam-3441	74	103	)	)	PUNCT
ejpam-3441	74	104	,	,	PUNCT
ejpam-3441	74	105	(	(	PUNCT
ejpam-3441	74	106	1−	1−	NUM
ejpam-3441	74	107	β	β	NOUN
ejpam-3441	74	108	)	)	PUNCT
ejpam-3441	74	109	}	}	PUNCT
ejpam-3441	74	110	,	,	PUNCT
ejpam-3441	74	111	(	(	PUNCT
ejpam-3441	74	112	3	3	X
ejpam-3441	74	113	)	)	PUNCT
ejpam-3441	74	114	max{µa	max{µa	X
ejpam-3441	74	115	(	(	PUNCT
ejpam-3441	74	116	xy	xy	PROPN
ejpam-3441	74	117	)	)	PUNCT
ejpam-3441	74	118	,	,	PUNCT
ejpam-3441	74	119	α	α	X
ejpam-3441	74	120	}	}	PUNCT
ejpam-3441	74	121	≥	≥	NOUN
ejpam-3441	74	122	min{µa	min{µa	X
ejpam-3441	74	123	(	(	PUNCT
ejpam-3441	74	124	x	x	NOUN
ejpam-3441	74	125	)	)	PUNCT
ejpam-3441	74	126	,	,	PUNCT
ejpam-3441	74	127	µa	µa	ADP
ejpam-3441	74	128	(	(	PUNCT
ejpam-3441	74	129	y	y	NOUN
ejpam-3441	74	130	)	)	PUNCT
ejpam-3441	74	131	,	,	PUNCT
ejpam-3441	74	132	β	β	X
ejpam-3441	74	133	}	}	PUNCT
ejpam-3441	74	134	,	,	PUNCT
ejpam-3441	74	135	(	(	PUNCT
ejpam-3441	74	136	4	4	X
ejpam-3441	74	137	)	)	PUNCT
ejpam-3441	74	138	min{γa	min{γa	NOUN
ejpam-3441	74	139	(	(	PUNCT
ejpam-3441	74	140	xy	xy	NOUN
ejpam-3441	74	141	)	)	PUNCT
ejpam-3441	74	142	,	,	PUNCT
ejpam-3441	74	143	(	(	PUNCT
ejpam-3441	74	144	1−	1−	NUM
ejpam-3441	74	145	α	α	NOUN
ejpam-3441	74	146	)	)	PUNCT
ejpam-3441	74	147	}	}	PUNCT
ejpam-3441	74	148	≤	≤	NUM
ejpam-3441	74	149	max{γa	max{γa	PUNCT
ejpam-3441	74	150	(	(	PUNCT
ejpam-3441	74	151	x	x	X
ejpam-3441	74	152	)	)	PUNCT
ejpam-3441	74	153	,	,	PUNCT
ejpam-3441	74	154	γa	γa	PROPN
ejpam-3441	74	155	(	(	PUNCT
ejpam-3441	74	156	y	y	PROPN
ejpam-3441	74	157	)	)	PUNCT
ejpam-3441	74	158	,	,	PUNCT
ejpam-3441	74	159	(	(	PUNCT
ejpam-3441	74	160	1−	1−	NUM
ejpam-3441	74	161	β	β	NOUN
ejpam-3441	74	162	)	)	PUNCT
ejpam-3441	74	163	}	}	PUNCT
ejpam-3441	74	164	for	for	ADP
ejpam-3441	74	165	all	all	DET
ejpam-3441	74	166	x	x	NOUN
ejpam-3441	74	167	,	,	PUNCT
ejpam-3441	74	168	y	y	PROPN
ejpam-3441	74	169	∈	∈	PROPN
ejpam-3441	74	170	r	r	NOUN
ejpam-3441	74	171	and	and	CCONJ
ejpam-3441	74	172	α	α	NOUN
ejpam-3441	74	173	,	,	PUNCT
ejpam-3441	74	174	β	β	X
ejpam-3441	74	175	∈	∈	PROPN
ejpam-3441	74	176	(	(	PUNCT
ejpam-3441	74	177	0	0	NUM
ejpam-3441	74	178	,	,	PUNCT
ejpam-3441	74	179	1	1	NUM
ejpam-3441	74	180	]	]	PUNCT
ejpam-3441	74	181	such	such	ADJ
ejpam-3441	74	182	that	that	SCONJ
ejpam-3441	74	183	α	α	PRON
ejpam-3441	74	184	<	<	X
ejpam-3441	74	185	β	β	X
ejpam-3441	74	186	.	.	PUNCT
ejpam-3441	75	1	an	an	DET
ejpam-3441	75	2	ifs	ifs	PROPN
ejpam-3441	75	3	a	a	X
ejpam-3441	75	4	=	=	X
ejpam-3441	75	5	(	(	PUNCT
ejpam-3441	75	6	µa	µa	PROPN
ejpam-3441	75	7	,	,	PUNCT
ejpam-3441	75	8	γa	γa	PROPN
ejpam-3441	75	9	)	)	PUNCT
ejpam-3441	75	10	of	of	ADP
ejpam-3441	75	11	an	an	DET
ejpam-3441	75	12	la	la	ADJ
ejpam-3441	75	13	-	-	PUNCT
ejpam-3441	75	14	ring	ring	NOUN
ejpam-3441	75	15	r	r	NOUN
ejpam-3441	75	16	is	be	AUX
ejpam-3441	75	17	called	call	VERB
ejpam-3441	75	18	an	an	DET
ejpam-3441	75	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	75	20	fuzzy	fuzzy	ADJ
ejpam-3441	75	21	left	leave	VERB
ejpam-3441	75	22	ideal	ideal	NOUN
ejpam-3441	75	23	with	with	ADP
ejpam-3441	75	24	thresholds	threshold	NOUN
ejpam-3441	75	25	(	(	PUNCT
ejpam-3441	75	26	α	α	X
ejpam-3441	75	27	,	,	PUNCT
ejpam-3441	75	28	β	β	X
ejpam-3441	75	29	]	]	PUNCT
ejpam-3441	75	30	of	of	ADP
ejpam-3441	75	31	r	r	PRON
ejpam-3441	75	32	if	if	SCONJ
ejpam-3441	75	33	(	(	PUNCT
ejpam-3441	75	34	1	1	X
ejpam-3441	75	35	)	)	PUNCT
ejpam-3441	75	36	max{µa	max{µa	X
ejpam-3441	75	37	(	(	PUNCT
ejpam-3441	75	38	x−	x−	PROPN
ejpam-3441	75	39	y	y	PROPN
ejpam-3441	75	40	)	)	PUNCT
ejpam-3441	75	41	,	,	PUNCT
ejpam-3441	75	42	α	α	X
ejpam-3441	75	43	}	}	PUNCT
ejpam-3441	75	44	≥	≥	NOUN
ejpam-3441	75	45	min{µa	min{µa	X
ejpam-3441	75	46	(	(	PUNCT
ejpam-3441	75	47	x	x	NOUN
ejpam-3441	75	48	)	)	PUNCT
ejpam-3441	75	49	,	,	PUNCT
ejpam-3441	75	50	µa(y	µa(y	NOUN
ejpam-3441	75	51	)	)	PUNCT
ejpam-3441	75	52	,	,	PUNCT
ejpam-3441	75	53	β	β	X
ejpam-3441	75	54	}	}	PUNCT
ejpam-3441	75	55	,	,	PUNCT
ejpam-3441	75	56	(	(	PUNCT
ejpam-3441	75	57	2	2	X
ejpam-3441	75	58	)	)	PUNCT
ejpam-3441	75	59	min{γa	min{γa	NOUN
ejpam-3441	75	60	(	(	PUNCT
ejpam-3441	75	61	x−	x−	PROPN
ejpam-3441	75	62	y	y	PROPN
ejpam-3441	75	63	)	)	PUNCT
ejpam-3441	75	64	,	,	PUNCT
ejpam-3441	75	65	(	(	PUNCT
ejpam-3441	75	66	1−	1−	NUM
ejpam-3441	75	67	α	α	NOUN
ejpam-3441	75	68	)	)	PUNCT
ejpam-3441	75	69	}	}	PUNCT
ejpam-3441	75	70	≤	≤	NUM
ejpam-3441	75	71	max{γa	max{γa	PUNCT
ejpam-3441	75	72	(	(	PUNCT
ejpam-3441	75	73	x	x	NOUN
ejpam-3441	75	74	)	)	PUNCT
ejpam-3441	75	75	,	,	PUNCT
ejpam-3441	75	76	γa(y	γa(y	NUM
ejpam-3441	75	77	)	)	PUNCT
ejpam-3441	75	78	,	,	PUNCT
ejpam-3441	75	79	(	(	PUNCT
ejpam-3441	75	80	1−	1−	NUM
ejpam-3441	75	81	β	β	NOUN
ejpam-3441	75	82	)	)	PUNCT
ejpam-3441	75	83	}	}	PUNCT
ejpam-3441	75	84	,	,	PUNCT
ejpam-3441	75	85	(	(	PUNCT
ejpam-3441	75	86	3	3	X
ejpam-3441	75	87	)	)	PUNCT
ejpam-3441	75	88	max{µa	max{µa	X
ejpam-3441	75	89	(	(	PUNCT
ejpam-3441	75	90	xy	xy	PROPN
ejpam-3441	75	91	)	)	PUNCT
ejpam-3441	75	92	,	,	PUNCT
ejpam-3441	75	93	α	α	X
ejpam-3441	75	94	}	}	PUNCT
ejpam-3441	75	95	≥	≥	NOUN
ejpam-3441	75	96	min{µa	min{µa	X
ejpam-3441	75	97	(	(	PUNCT
ejpam-3441	75	98	y	y	NOUN
ejpam-3441	75	99	)	)	PUNCT
ejpam-3441	75	100	,	,	PUNCT
ejpam-3441	75	101	β	β	X
ejpam-3441	75	102	}	}	PUNCT
ejpam-3441	75	103	,	,	PUNCT
ejpam-3441	75	104	(	(	PUNCT
ejpam-3441	75	105	4	4	X
ejpam-3441	75	106	)	)	PUNCT
ejpam-3441	75	107	min{γa	min{γa	NOUN
ejpam-3441	75	108	(	(	PUNCT
ejpam-3441	75	109	xy	xy	NOUN
ejpam-3441	75	110	)	)	PUNCT
ejpam-3441	75	111	,	,	PUNCT
ejpam-3441	75	112	(	(	PUNCT
ejpam-3441	75	113	1	1	NUM
ejpam-3441	75	114	−	−	PROPN
ejpam-3441	75	115	α	α	X
ejpam-3441	75	116	)	)	PUNCT
ejpam-3441	75	117	}	}	PUNCT
ejpam-3441	75	118	≤	≤	NUM
ejpam-3441	75	119	max{γa	max{γa	NOUN
ejpam-3441	75	120	(	(	PUNCT
ejpam-3441	75	121	y	y	NOUN
ejpam-3441	75	122	)	)	PUNCT
ejpam-3441	75	123	,	,	PUNCT
ejpam-3441	75	124	(	(	PUNCT
ejpam-3441	75	125	1	1	NUM
ejpam-3441	75	126	−	−	NOUN
ejpam-3441	75	127	β	β	X
ejpam-3441	75	128	)	)	PUNCT
ejpam-3441	75	129	}	}	PUNCT
ejpam-3441	75	130	for	for	ADP
ejpam-3441	75	131	all	all	DET
ejpam-3441	75	132	x	x	NOUN
ejpam-3441	75	133	,	,	PUNCT
ejpam-3441	75	134	y	y	PROPN
ejpam-3441	75	135	∈	∈	PROPN
ejpam-3441	75	136	r	r	NOUN
ejpam-3441	75	137	and	and	CCONJ
ejpam-3441	75	138	α	α	NOUN
ejpam-3441	75	139	,	,	PUNCT
ejpam-3441	75	140	β	β	X
ejpam-3441	75	141	∈	∈	PROPN
ejpam-3441	75	142	(	(	PUNCT
ejpam-3441	75	143	0	0	NUM
ejpam-3441	75	144	,	,	PUNCT
ejpam-3441	75	145	1	1	NUM
ejpam-3441	75	146	]	]	PUNCT
ejpam-3441	75	147	such	such	ADJ
ejpam-3441	75	148	that	that	SCONJ
ejpam-3441	75	149	α	α	PRON
ejpam-3441	75	150	<	<	X
ejpam-3441	75	151	β	β	X
ejpam-3441	75	152	.	.	PUNCT
ejpam-3441	76	1	an	an	DET
ejpam-3441	76	2	ifs	ifs	PROPN
ejpam-3441	76	3	a	a	X
ejpam-3441	76	4	=	=	X
ejpam-3441	76	5	(	(	PUNCT
ejpam-3441	76	6	µa	µa	PROPN
ejpam-3441	76	7	,	,	PUNCT
ejpam-3441	76	8	γa	γa	PROPN
ejpam-3441	76	9	)	)	PUNCT
ejpam-3441	76	10	of	of	ADP
ejpam-3441	76	11	an	an	DET
ejpam-3441	76	12	la	la	ADJ
ejpam-3441	76	13	-	-	PUNCT
ejpam-3441	76	14	ring	ring	NOUN
ejpam-3441	76	15	r	r	NOUN
ejpam-3441	76	16	is	be	AUX
ejpam-3441	76	17	called	call	VERB
ejpam-3441	76	18	an	an	DET
ejpam-3441	76	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	76	20	fuzzy	fuzzy	ADJ
ejpam-3441	76	21	right	right	ADJ
ejpam-3441	76	22	ideal	ideal	NOUN
ejpam-3441	76	23	with	with	ADP
ejpam-3441	76	24	thresholds	threshold	NOUN
ejpam-3441	76	25	(	(	PUNCT
ejpam-3441	76	26	α	α	X
ejpam-3441	76	27	,	,	PUNCT
ejpam-3441	76	28	β	β	X
ejpam-3441	76	29	]	]	PUNCT
ejpam-3441	76	30	of	of	ADP
ejpam-3441	76	31	r	r	PRON
ejpam-3441	76	32	if	if	SCONJ
ejpam-3441	76	33	(	(	PUNCT
ejpam-3441	76	34	1	1	X
ejpam-3441	76	35	)	)	PUNCT
ejpam-3441	76	36	max{µa	max{µa	X
ejpam-3441	76	37	(	(	PUNCT
ejpam-3441	76	38	x−	x−	PROPN
ejpam-3441	76	39	y	y	PROPN
ejpam-3441	76	40	)	)	PUNCT
ejpam-3441	76	41	,	,	PUNCT
ejpam-3441	76	42	α	α	X
ejpam-3441	76	43	}	}	PUNCT
ejpam-3441	76	44	≥	≥	NOUN
ejpam-3441	76	45	min{µa	min{µa	X
ejpam-3441	76	46	(	(	PUNCT
ejpam-3441	76	47	x	x	NOUN
ejpam-3441	76	48	)	)	PUNCT
ejpam-3441	76	49	,	,	PUNCT
ejpam-3441	76	50	µa(y	µa(y	NOUN
ejpam-3441	76	51	)	)	PUNCT
ejpam-3441	76	52	,	,	PUNCT
ejpam-3441	76	53	β	β	X
ejpam-3441	76	54	}	}	PUNCT
ejpam-3441	76	55	,	,	PUNCT
ejpam-3441	76	56	(	(	PUNCT
ejpam-3441	76	57	2	2	X
ejpam-3441	76	58	)	)	PUNCT
ejpam-3441	76	59	min{γa	min{γa	NOUN
ejpam-3441	76	60	(	(	PUNCT
ejpam-3441	76	61	x−	x−	PROPN
ejpam-3441	76	62	y	y	PROPN
ejpam-3441	76	63	)	)	PUNCT
ejpam-3441	76	64	,	,	PUNCT
ejpam-3441	76	65	(	(	PUNCT
ejpam-3441	76	66	1−	1−	NUM
ejpam-3441	76	67	α	α	NOUN
ejpam-3441	76	68	)	)	PUNCT
ejpam-3441	76	69	}	}	PUNCT
ejpam-3441	76	70	≤	≤	NUM
ejpam-3441	76	71	max{γa	max{γa	PUNCT
ejpam-3441	76	72	(	(	PUNCT
ejpam-3441	76	73	x	x	NOUN
ejpam-3441	76	74	)	)	PUNCT
ejpam-3441	76	75	,	,	PUNCT
ejpam-3441	76	76	γa(y	γa(y	NUM
ejpam-3441	76	77	)	)	PUNCT
ejpam-3441	76	78	,	,	PUNCT
ejpam-3441	76	79	(	(	PUNCT
ejpam-3441	76	80	1−	1−	NUM
ejpam-3441	76	81	β	β	NOUN
ejpam-3441	76	82	)	)	PUNCT
ejpam-3441	76	83	}	}	PUNCT
ejpam-3441	76	84	,	,	PUNCT
ejpam-3441	76	85	(	(	PUNCT
ejpam-3441	76	86	3	3	X
ejpam-3441	76	87	)	)	PUNCT
ejpam-3441	76	88	max{µa	max{µa	X
ejpam-3441	76	89	(	(	PUNCT
ejpam-3441	76	90	xy	xy	PROPN
ejpam-3441	76	91	)	)	PUNCT
ejpam-3441	76	92	,	,	PUNCT
ejpam-3441	76	93	α	α	X
ejpam-3441	76	94	}	}	PUNCT
ejpam-3441	76	95	≥	≥	NOUN
ejpam-3441	76	96	min{µa	min{µa	X
ejpam-3441	76	97	(	(	PUNCT
ejpam-3441	76	98	x	x	NOUN
ejpam-3441	76	99	)	)	PUNCT
ejpam-3441	76	100	,	,	PUNCT
ejpam-3441	76	101	β	β	X
ejpam-3441	76	102	}	}	PUNCT
ejpam-3441	76	103	,	,	PUNCT
ejpam-3441	76	104	(	(	PUNCT
ejpam-3441	76	105	4	4	X
ejpam-3441	76	106	)	)	PUNCT
ejpam-3441	76	107	min{γa	min{γa	NOUN
ejpam-3441	76	108	(	(	PUNCT
ejpam-3441	76	109	xy	xy	NOUN
ejpam-3441	76	110	)	)	PUNCT
ejpam-3441	76	111	,	,	PUNCT
ejpam-3441	76	112	(	(	PUNCT
ejpam-3441	76	113	1	1	NUM
ejpam-3441	76	114	−	−	PROPN
ejpam-3441	76	115	α	α	X
ejpam-3441	76	116	)	)	PUNCT
ejpam-3441	76	117	}	}	PUNCT
ejpam-3441	76	118	≤	≤	NUM
ejpam-3441	76	119	max{γa	max{γa	PUNCT
ejpam-3441	76	120	(	(	PUNCT
ejpam-3441	76	121	x	x	NOUN
ejpam-3441	76	122	)	)	PUNCT
ejpam-3441	76	123	,	,	PUNCT
ejpam-3441	76	124	(	(	PUNCT
ejpam-3441	76	125	1	1	NUM
ejpam-3441	76	126	−	−	NOUN
ejpam-3441	76	127	β	β	X
ejpam-3441	76	128	)	)	PUNCT
ejpam-3441	76	129	}	}	PUNCT
ejpam-3441	76	130	for	for	ADP
ejpam-3441	76	131	all	all	DET
ejpam-3441	76	132	x	x	NOUN
ejpam-3441	76	133	,	,	PUNCT
ejpam-3441	76	134	y	y	PROPN
ejpam-3441	76	135	∈	∈	PROPN
ejpam-3441	76	136	r	r	NOUN
ejpam-3441	76	137	and	and	CCONJ
ejpam-3441	76	138	α	α	NOUN
ejpam-3441	76	139	,	,	PUNCT
ejpam-3441	76	140	β	β	X
ejpam-3441	76	141	∈	∈	PROPN
ejpam-3441	76	142	(	(	PUNCT
ejpam-3441	76	143	0	0	NUM
ejpam-3441	76	144	,	,	PUNCT
ejpam-3441	76	145	1	1	NUM
ejpam-3441	76	146	]	]	PUNCT
ejpam-3441	76	147	such	such	ADJ
ejpam-3441	76	148	that	that	SCONJ
ejpam-3441	76	149	α	α	PRON
ejpam-3441	76	150	<	<	X
ejpam-3441	76	151	β	β	X
ejpam-3441	76	152	.	.	PUNCT
ejpam-3441	77	1	an	an	DET
ejpam-3441	77	2	ifs	ifs	PROPN
ejpam-3441	77	3	a	a	X
ejpam-3441	77	4	=	=	X
ejpam-3441	77	5	(	(	PUNCT
ejpam-3441	77	6	µa	µa	PROPN
ejpam-3441	77	7	,	,	PUNCT
ejpam-3441	77	8	γa	γa	PROPN
ejpam-3441	77	9	)	)	PUNCT
ejpam-3441	77	10	of	of	ADP
ejpam-3441	77	11	an	an	DET
ejpam-3441	77	12	la	la	ADJ
ejpam-3441	77	13	-	-	PUNCT
ejpam-3441	77	14	ring	ring	NOUN
ejpam-3441	77	15	r	r	NOUN
ejpam-3441	77	16	is	be	AUX
ejpam-3441	77	17	called	call	VERB
ejpam-3441	77	18	an	an	DET
ejpam-3441	77	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	77	20	fuzzy	fuzzy	ADJ
ejpam-3441	77	21	ideal	ideal	NOUN
ejpam-3441	77	22	with	with	ADP
ejpam-3441	77	23	thresholds	threshold	NOUN
ejpam-3441	77	24	(	(	PUNCT
ejpam-3441	77	25	α	α	X
ejpam-3441	77	26	,	,	PUNCT
ejpam-3441	77	27	β	β	X
ejpam-3441	77	28	]	]	PUNCT
ejpam-3441	77	29	of	of	ADP
ejpam-3441	77	30	r	r	NOUN
ejpam-3441	77	31	if	if	SCONJ
ejpam-3441	77	32	it	it	PRON
ejpam-3441	77	33	is	be	AUX
ejpam-3441	77	34	both	both	CCONJ
ejpam-3441	77	35	an	an	DET
ejpam-3441	77	36	intuitionistic	intuitionistic	ADJ
ejpam-3441	77	37	fuzzy	fuzzy	ADJ
ejpam-3441	77	38	left	leave	VERB
ejpam-3441	77	39	ideal	ideal	NOUN
ejpam-3441	77	40	with	with	ADP
ejpam-3441	77	41	thresholds	threshold	NOUN
ejpam-3441	77	42	(	(	PUNCT
ejpam-3441	77	43	α	α	X
ejpam-3441	77	44	,	,	PUNCT
ejpam-3441	77	45	β	β	X
ejpam-3441	77	46	]	]	PUNCT
ejpam-3441	77	47	and	and	CCONJ
ejpam-3441	77	48	an	an	DET
ejpam-3441	77	49	intuitionistic	intuitionistic	ADJ
ejpam-3441	77	50	fuzzy	fuzzy	ADJ
ejpam-3441	77	51	right	right	ADJ
ejpam-3441	77	52	ideal	ideal	NOUN
ejpam-3441	77	53	with	with	ADP
ejpam-3441	77	54	thresholds	threshold	NOUN
ejpam-3441	77	55	(	(	PUNCT
ejpam-3441	77	56	α	α	X
ejpam-3441	77	57	,	,	PUNCT
ejpam-3441	77	58	β	β	X
ejpam-3441	77	59	]	]	PUNCT
ejpam-3441	77	60	of	of	ADP
ejpam-3441	77	61	r.	r.	PROPN
ejpam-3441	77	62	every	every	DET
ejpam-3441	77	63	intuitionistic	intuitionistic	ADJ
ejpam-3441	77	64	fuzzy	fuzzy	ADJ
ejpam-3441	77	65	left	left	NOUN
ejpam-3441	77	66	(	(	PUNCT
ejpam-3441	77	67	resp	resp	NOUN
ejpam-3441	77	68	.	.	PUNCT
ejpam-3441	78	1	right	right	ADJ
ejpam-3441	78	2	,	,	PUNCT
ejpam-3441	78	3	two	two	NUM
ejpam-3441	78	4	-	-	PUNCT
ejpam-3441	78	5	sided	sided	ADJ
ejpam-3441	78	6	)	)	PUNCT
ejpam-3441	78	7	ideal	ideal	NOUN
ejpam-3441	78	8	with	with	ADP
ejpam-3441	78	9	thresholds	threshold	NOUN
ejpam-3441	78	10	(	(	PUNCT
ejpam-3441	78	11	α	α	X
ejpam-3441	78	12	,	,	PUNCT
ejpam-3441	78	13	β	β	X
ejpam-3441	78	14	]	]	PUNCT
ejpam-3441	78	15	of	of	ADP
ejpam-3441	78	16	r	r	NOUN
ejpam-3441	78	17	is	be	AUX
ejpam-3441	78	18	an	an	DET
ejpam-3441	78	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	78	20	fuzzy	fuzzy	ADJ
ejpam-3441	78	21	la	la	NOUN
ejpam-3441	78	22	-	-	PUNCT
ejpam-3441	78	23	subring	subre	VERB
ejpam-3441	78	24	with	with	ADP
ejpam-3441	78	25	thresholds	threshold	NOUN
ejpam-3441	78	26	(	(	PUNCT
ejpam-3441	78	27	α	α	X
ejpam-3441	78	28	,	,	PUNCT
ejpam-3441	78	29	β	β	X
ejpam-3441	78	30	]	]	PUNCT
ejpam-3441	78	31	of	of	ADP
ejpam-3441	78	32	r	r	NOUN
ejpam-3441	78	33	,	,	PUNCT
ejpam-3441	78	34	but	but	CCONJ
ejpam-3441	78	35	converse	converse	NOUN
ejpam-3441	78	36	is	be	AUX
ejpam-3441	78	37	not	not	PART
ejpam-3441	78	38	true	true	ADJ
ejpam-3441	78	39	in	in	ADP
ejpam-3441	78	40	general	general	ADJ
ejpam-3441	78	41	.	.	PUNCT
ejpam-3441	79	1	k.	k.	PROPN
ejpam-3441	79	2	nasreen	nasreen	PROPN
ejpam-3441	79	3	et	et	PROPN
ejpam-3441	79	4	al	al	PROPN
ejpam-3441	79	5	.	.	PUNCT
ejpam-3441	79	6	/	/	SYM
ejpam-3441	79	7	eur	eur	PROPN
ejpam-3441	79	8	.	.	PUNCT
ejpam-3441	80	1	j.	j.	PROPN
ejpam-3441	80	2	pure	pure	PROPN
ejpam-3441	80	3	appl	appl	PROPN
ejpam-3441	80	4	.	.	PROPN
ejpam-3441	80	5	math	math	PROPN
ejpam-3441	80	6	,	,	PUNCT
ejpam-3441	80	7	12	12	NUM
ejpam-3441	80	8	(	(	PUNCT
ejpam-3441	80	9	3	3	NUM
ejpam-3441	80	10	)	)	PUNCT
ejpam-3441	80	11	(	(	PUNCT
ejpam-3441	80	12	2019	2019	NUM
ejpam-3441	80	13	)	)	PUNCT
ejpam-3441	80	14	,	,	PUNCT
ejpam-3441	80	15	906	906	NUM
ejpam-3441	80	16	-	-	SYM
ejpam-3441	80	17	943	943	NUM
ejpam-3441	80	18	909	909	NUM
ejpam-3441	80	19	example	example	NOUN
ejpam-3441	80	20	1	1	NUM
ejpam-3441	80	21	.	.	PUNCT
ejpam-3441	81	1	let	let	VERB
ejpam-3441	81	2	r	r	NOUN
ejpam-3441	81	3	=	=	SYM
ejpam-3441	81	4	{	{	PUNCT
ejpam-3441	81	5	0	0	NUM
ejpam-3441	81	6	,	,	PUNCT
ejpam-3441	81	7	1	1	NUM
ejpam-3441	81	8	,	,	PUNCT
ejpam-3441	81	9	2	2	NUM
ejpam-3441	81	10	,	,	PUNCT
ejpam-3441	81	11	3	3	NUM
ejpam-3441	81	12	,	,	PUNCT
ejpam-3441	81	13	4	4	NUM
ejpam-3441	81	14	,	,	PUNCT
ejpam-3441	81	15	5	5	NUM
ejpam-3441	81	16	,	,	PUNCT
ejpam-3441	81	17	6	6	NUM
ejpam-3441	81	18	,	,	PUNCT
ejpam-3441	81	19	7	7	NUM
ejpam-3441	81	20	}	}	PUNCT
ejpam-3441	81	21	.	.	PUNCT
ejpam-3441	82	1	define	define	VERB
ejpam-3441	82	2	+	+	CCONJ
ejpam-3441	82	3	and	and	CCONJ
ejpam-3441	82	4	·	·	PUNCT
ejpam-3441	82	5	in	in	ADP
ejpam-3441	82	6	r	r	NOUN
ejpam-3441	82	7	as	as	SCONJ
ejpam-3441	82	8	follows	follow	VERB
ejpam-3441	82	9	:	:	PUNCT
ejpam-3441	83	1	+	+	SYM
ejpam-3441	83	2	0	0	NUM
ejpam-3441	83	3	1	1	NUM
ejpam-3441	83	4	2	2	NUM
ejpam-3441	83	5	3	3	NUM
ejpam-3441	83	6	4	4	NUM
ejpam-3441	83	7	5	5	NUM
ejpam-3441	83	8	6	6	NUM
ejpam-3441	83	9	7	7	NUM
ejpam-3441	83	10	0	0	NUM
ejpam-3441	83	11	0	0	NUM
ejpam-3441	83	12	1	1	NUM
ejpam-3441	83	13	2	2	NUM
ejpam-3441	83	14	3	3	NUM
ejpam-3441	83	15	4	4	NUM
ejpam-3441	83	16	5	5	NUM
ejpam-3441	83	17	6	6	NUM
ejpam-3441	83	18	7	7	NUM
ejpam-3441	83	19	1	1	NUM
ejpam-3441	83	20	2	2	NUM
ejpam-3441	83	21	0	0	NUM
ejpam-3441	83	22	3	3	NUM
ejpam-3441	83	23	1	1	NUM
ejpam-3441	83	24	6	6	NUM
ejpam-3441	83	25	4	4	NUM
ejpam-3441	83	26	7	7	NUM
ejpam-3441	83	27	5	5	NUM
ejpam-3441	83	28	2	2	NUM
ejpam-3441	83	29	1	1	NUM
ejpam-3441	83	30	3	3	NUM
ejpam-3441	83	31	0	0	NUM
ejpam-3441	83	32	2	2	NUM
ejpam-3441	83	33	5	5	NUM
ejpam-3441	83	34	7	7	NUM
ejpam-3441	83	35	4	4	NUM
ejpam-3441	83	36	6	6	NUM
ejpam-3441	83	37	3	3	NUM
ejpam-3441	83	38	3	3	NUM
ejpam-3441	83	39	2	2	NUM
ejpam-3441	83	40	1	1	NUM
ejpam-3441	83	41	0	0	NUM
ejpam-3441	83	42	7	7	NUM
ejpam-3441	83	43	6	6	NUM
ejpam-3441	83	44	5	5	NUM
ejpam-3441	83	45	4	4	NUM
ejpam-3441	83	46	4	4	NUM
ejpam-3441	83	47	4	4	NUM
ejpam-3441	83	48	5	5	NUM
ejpam-3441	83	49	6	6	NUM
ejpam-3441	83	50	7	7	NUM
ejpam-3441	83	51	0	0	NUM
ejpam-3441	83	52	1	1	NUM
ejpam-3441	83	53	2	2	NUM
ejpam-3441	83	54	3	3	NUM
ejpam-3441	83	55	5	5	NUM
ejpam-3441	83	56	6	6	NUM
ejpam-3441	83	57	4	4	NUM
ejpam-3441	83	58	7	7	NUM
ejpam-3441	83	59	5	5	NUM
ejpam-3441	83	60	2	2	NUM
ejpam-3441	83	61	0	0	NUM
ejpam-3441	83	62	3	3	NUM
ejpam-3441	83	63	1	1	NUM
ejpam-3441	83	64	6	6	NUM
ejpam-3441	83	65	5	5	NUM
ejpam-3441	83	66	7	7	NUM
ejpam-3441	83	67	4	4	NUM
ejpam-3441	83	68	6	6	NUM
ejpam-3441	83	69	1	1	NUM
ejpam-3441	83	70	3	3	NUM
ejpam-3441	83	71	0	0	NUM
ejpam-3441	83	72	2	2	NUM
ejpam-3441	83	73	7	7	NUM
ejpam-3441	83	74	7	7	NUM
ejpam-3441	83	75	6	6	NUM
ejpam-3441	83	76	5	5	NUM
ejpam-3441	83	77	4	4	NUM
ejpam-3441	83	78	3	3	NUM
ejpam-3441	83	79	2	2	NUM
ejpam-3441	83	80	1	1	NUM
ejpam-3441	83	81	0	0	NUM
ejpam-3441	83	82	and	and	CCONJ
ejpam-3441	83	83	·	·	PUNCT
ejpam-3441	83	84	0	0	NUM
ejpam-3441	84	1	1	1	NUM
ejpam-3441	84	2	2	2	NUM
ejpam-3441	84	3	3	3	NUM
ejpam-3441	84	4	4	4	NUM
ejpam-3441	84	5	5	5	NUM
ejpam-3441	84	6	6	6	NUM
ejpam-3441	84	7	7	7	NUM
ejpam-3441	84	8	0	0	NUM
ejpam-3441	84	9	0	0	NUM
ejpam-3441	84	10	0	0	NUM
ejpam-3441	84	11	0	0	NUM
ejpam-3441	84	12	0	0	NUM
ejpam-3441	84	13	0	0	NUM
ejpam-3441	84	14	0	0	NUM
ejpam-3441	84	15	0	0	NUM
ejpam-3441	84	16	0	0	NUM
ejpam-3441	84	17	1	1	NUM
ejpam-3441	84	18	0	0	NUM
ejpam-3441	84	19	4	4	NUM
ejpam-3441	84	20	4	4	NUM
ejpam-3441	84	21	0	0	NUM
ejpam-3441	84	22	0	0	NUM
ejpam-3441	84	23	4	4	NUM
ejpam-3441	84	24	4	4	NUM
ejpam-3441	84	25	0	0	NUM
ejpam-3441	84	26	2	2	NUM
ejpam-3441	84	27	0	0	NUM
ejpam-3441	84	28	4	4	NUM
ejpam-3441	84	29	4	4	NUM
ejpam-3441	84	30	0	0	NUM
ejpam-3441	84	31	0	0	NUM
ejpam-3441	84	32	4	4	NUM
ejpam-3441	84	33	4	4	NUM
ejpam-3441	84	34	0	0	NUM
ejpam-3441	84	35	3	3	NUM
ejpam-3441	84	36	0	0	NUM
ejpam-3441	84	37	0	0	NUM
ejpam-3441	84	38	0	0	NUM
ejpam-3441	84	39	0	0	NUM
ejpam-3441	84	40	0	0	NUM
ejpam-3441	84	41	0	0	NUM
ejpam-3441	84	42	0	0	NUM
ejpam-3441	84	43	0	0	NUM
ejpam-3441	84	44	4	4	NUM
ejpam-3441	84	45	0	0	NUM
ejpam-3441	84	46	3	3	NUM
ejpam-3441	84	47	3	3	NUM
ejpam-3441	84	48	0	0	NUM
ejpam-3441	84	49	0	0	NUM
ejpam-3441	84	50	3	3	NUM
ejpam-3441	84	51	3	3	NUM
ejpam-3441	84	52	0	0	NUM
ejpam-3441	84	53	5	5	NUM
ejpam-3441	84	54	0	0	NUM
ejpam-3441	84	55	7	7	NUM
ejpam-3441	84	56	7	7	NUM
ejpam-3441	84	57	0	0	NUM
ejpam-3441	84	58	0	0	NUM
ejpam-3441	84	59	7	7	NUM
ejpam-3441	84	60	7	7	NUM
ejpam-3441	84	61	0	0	NUM
ejpam-3441	84	62	6	6	NUM
ejpam-3441	84	63	0	0	NUM
ejpam-3441	84	64	7	7	NUM
ejpam-3441	84	65	7	7	NUM
ejpam-3441	84	66	0	0	NUM
ejpam-3441	84	67	0	0	NUM
ejpam-3441	84	68	7	7	NUM
ejpam-3441	84	69	7	7	NUM
ejpam-3441	84	70	0	0	NUM
ejpam-3441	84	71	7	7	NUM
ejpam-3441	84	72	0	0	NUM
ejpam-3441	84	73	3	3	NUM
ejpam-3441	84	74	3	3	NUM
ejpam-3441	84	75	0	0	NUM
ejpam-3441	84	76	0	0	NUM
ejpam-3441	84	77	3	3	NUM
ejpam-3441	84	78	3	3	NUM
ejpam-3441	84	79	0	0	NUM
ejpam-3441	84	80	then	then	ADV
ejpam-3441	84	81	r	r	NOUN
ejpam-3441	84	82	is	be	AUX
ejpam-3441	84	83	an	an	DET
ejpam-3441	84	84	la	la	ADJ
ejpam-3441	84	85	-	-	PUNCT
ejpam-3441	84	86	ring	ring	NOUN
ejpam-3441	84	87	and	and	CCONJ
ejpam-3441	84	88	a	a	DET
ejpam-3441	84	89	=	=	X
ejpam-3441	84	90	(	(	PUNCT
ejpam-3441	84	91	µa	µa	PROPN
ejpam-3441	84	92	,	,	PUNCT
ejpam-3441	84	93	γa	γa	PROPN
ejpam-3441	84	94	)	)	PUNCT
ejpam-3441	84	95	be	be	VERB
ejpam-3441	84	96	an	an	DET
ejpam-3441	84	97	ifs	ifs	PROPN
ejpam-3441	84	98	of	of	ADP
ejpam-3441	84	99	an	an	DET
ejpam-3441	84	100	la	la	ADJ
ejpam-3441	84	101	-	-	PUNCT
ejpam-3441	84	102	ring	ring	NOUN
ejpam-3441	84	103	r.	r.	NOUN
ejpam-3441	84	104	we	we	PRON
ejpam-3441	84	105	define	define	VERB
ejpam-3441	84	106	(	(	PUNCT
ejpam-3441	84	107	α	α	NOUN
ejpam-3441	84	108	=	=	NOUN
ejpam-3441	84	109	0.1	0.1	NUM
ejpam-3441	84	110	,	,	PUNCT
ejpam-3441	84	111	β	β	X
ejpam-3441	84	112	=	=	NOUN
ejpam-3441	84	113	0.7	0.7	NUM
ejpam-3441	84	114	)	)	PUNCT
ejpam-3441	84	115	µa(0	µa(0	NOUN
ejpam-3441	84	116	)	)	PUNCT
ejpam-3441	84	117	=	=	SYM
ejpam-3441	84	118	µa(4	µa(4	NOUN
ejpam-3441	84	119	)	)	PUNCT
ejpam-3441	84	120	=	=	NOUN
ejpam-3441	84	121	0.7	0.7	NUM
ejpam-3441	84	122	,	,	PUNCT
ejpam-3441	84	123	µa(1	µa(1	NOUN
ejpam-3441	84	124	)	)	PUNCT
ejpam-3441	84	125	=	=	SYM
ejpam-3441	84	126	µa(2	µa(2	NOUN
ejpam-3441	84	127	)	)	PUNCT
ejpam-3441	84	128	=	=	SYM
ejpam-3441	84	129	µa(3	µa(3	PROPN
ejpam-3441	84	130	)	)	PUNCT
ejpam-3441	84	131	=	=	SYM
ejpam-3441	84	132	µa(5	µa(5	NOUN
ejpam-3441	84	133	)	)	PUNCT
ejpam-3441	84	134	=	=	SYM
ejpam-3441	84	135	µa(6	µa(6	NOUN
ejpam-3441	84	136	)	)	PUNCT
ejpam-3441	84	137	=	=	PUNCT
ejpam-3441	84	138	µa(7	µa(7	PROPN
ejpam-3441	84	139	)	)	PUNCT
ejpam-3441	84	140	=	=	SYM
ejpam-3441	84	141	0.1	0.1	NUM
ejpam-3441	84	142	and	and	CCONJ
ejpam-3441	84	143	γa(0	γa(0	NOUN
ejpam-3441	84	144	)	)	PUNCT
ejpam-3441	84	145	=	=	PUNCT
ejpam-3441	84	146	γa(4	γa(4	X
ejpam-3441	84	147	)	)	PUNCT
ejpam-3441	84	148	=	=	NOUN
ejpam-3441	84	149	0.1	0.1	NUM
ejpam-3441	84	150	,	,	PUNCT
ejpam-3441	84	151	γa(1	γa(1	NOUN
ejpam-3441	84	152	)	)	PUNCT
ejpam-3441	84	153	=	=	SYM
ejpam-3441	84	154	γa(2	γa(2	NUM
ejpam-3441	84	155	)	)	PUNCT
ejpam-3441	84	156	=	=	SYM
ejpam-3441	84	157	γa(3	γa(3	PROPN
ejpam-3441	84	158	)	)	PUNCT
ejpam-3441	84	159	=	=	SYM
ejpam-3441	84	160	γa(5	γa(5	NOUN
ejpam-3441	84	161	)	)	PUNCT
ejpam-3441	84	162	=	=	SYM
ejpam-3441	84	163	γa(6	γa(6	PROPN
ejpam-3441	84	164	)	)	PUNCT
ejpam-3441	84	165	=	=	PUNCT
ejpam-3441	84	166	γa(7	γa(7	X
ejpam-3441	84	167	)	)	PUNCT
ejpam-3441	84	168	=	=	PUNCT
ejpam-3441	85	1	0.7	0.7	NUM
ejpam-3441	85	2	.	.	PUNCT
ejpam-3441	85	3	since	since	SCONJ
ejpam-3441	85	4	max{µa(41	max{µa(41	PROPN
ejpam-3441	85	5	)	)	PUNCT
ejpam-3441	85	6	,	,	PUNCT
ejpam-3441	85	7	α	α	X
ejpam-3441	85	8	}	}	PUNCT
ejpam-3441	85	9	=	=	SYM
ejpam-3441	85	10	max{µa(3	max{µa(3	PROPN
ejpam-3441	85	11	)	)	PUNCT
ejpam-3441	85	12	,	,	PUNCT
ejpam-3441	85	13	α	α	X
ejpam-3441	85	14	}	}	PUNCT
ejpam-3441	85	15	=	=	SYM
ejpam-3441	85	16	max{0.1	max{0.1	PROPN
ejpam-3441	85	17	,	,	PUNCT
ejpam-3441	85	18	0.1	0.1	NUM
ejpam-3441	85	19	}	}	PUNCT
ejpam-3441	85	20	=	=	NOUN
ejpam-3441	85	21	0.1	0.1	NUM
ejpam-3441	85	22	.	.	PUNCT
ejpam-3441	86	1	min{µa(4	min{µa(4	PROPN
ejpam-3441	86	2	)	)	PUNCT
ejpam-3441	86	3	,	,	PUNCT
ejpam-3441	86	4	β	β	X
ejpam-3441	86	5	}	}	PUNCT
ejpam-3441	86	6	=	=	SYM
ejpam-3441	86	7	min{0.7	min{0.7	PROPN
ejpam-3441	86	8	,	,	PUNCT
ejpam-3441	86	9	0.7	0.7	NUM
ejpam-3441	86	10	}	}	PUNCT
ejpam-3441	86	11	=	=	SYM
ejpam-3441	86	12	0.7	0.7	NUM
ejpam-3441	86	13	.	.	PUNCT
ejpam-3441	86	14	⇒	⇒	PROPN
ejpam-3441	86	15	max{µa(41	max{µa(41	PROPN
ejpam-3441	86	16	)	)	PUNCT
ejpam-3441	86	17	,	,	PUNCT
ejpam-3441	86	18	α	α	X
ejpam-3441	86	19	}	}	PUNCT
ejpam-3441	86	20	�	�	PROPN
ejpam-3441	86	21	min{µa(4	min{µa(4	PROPN
ejpam-3441	86	22	)	)	PUNCT
ejpam-3441	86	23	,	,	PUNCT
ejpam-3441	86	24	β	β	X
ejpam-3441	86	25	}	}	PUNCT
ejpam-3441	86	26	.	.	PUNCT
ejpam-3441	87	1	and	and	CCONJ
ejpam-3441	87	2	min{γa(41	min{γa(41	PROPN
ejpam-3441	87	3	)	)	PUNCT
ejpam-3441	87	4	,	,	PUNCT
ejpam-3441	87	5	(	(	PUNCT
ejpam-3441	87	6	1−	1−	NUM
ejpam-3441	87	7	α	α	NOUN
ejpam-3441	87	8	)	)	PUNCT
ejpam-3441	87	9	}	}	PUNCT
ejpam-3441	87	10	=	=	SYM
ejpam-3441	87	11	min{γa(3	min{γa(3	PROPN
ejpam-3441	87	12	)	)	PUNCT
ejpam-3441	87	13	,	,	PUNCT
ejpam-3441	87	14	(	(	PUNCT
ejpam-3441	87	15	1−	1−	NUM
ejpam-3441	87	16	α	α	NOUN
ejpam-3441	87	17	)	)	PUNCT
ejpam-3441	87	18	}	}	PUNCT
ejpam-3441	87	19	=	=	SYM
ejpam-3441	87	20	min{0.7	min{0.7	PROPN
ejpam-3441	87	21	,	,	PUNCT
ejpam-3441	87	22	0.9	0.9	NUM
ejpam-3441	87	23	}	}	PUNCT
ejpam-3441	87	24	=	=	PUNCT
ejpam-3441	87	25	0.7	0.7	NUM
ejpam-3441	87	26	.	.	PUNCT
ejpam-3441	88	1	max{γa(4	max{γa(4	PROPN
ejpam-3441	88	2	)	)	PUNCT
ejpam-3441	88	3	,	,	PUNCT
ejpam-3441	88	4	(	(	PUNCT
ejpam-3441	88	5	1−	1−	NUM
ejpam-3441	88	6	β	β	NOUN
ejpam-3441	88	7	)	)	PUNCT
ejpam-3441	88	8	}	}	PUNCT
ejpam-3441	88	9	=	=	SYM
ejpam-3441	88	10	max{0.1	max{0.1	PROPN
ejpam-3441	88	11	,	,	PUNCT
ejpam-3441	88	12	0.3	0.3	NUM
ejpam-3441	88	13	}	}	PUNCT
ejpam-3441	88	14	=	=	SYM
ejpam-3441	88	15	0.3	0.3	NUM
ejpam-3441	88	16	.	.	PUNCT
ejpam-3441	89	1	⇒	⇒	PROPN
ejpam-3441	89	2	min{γa(41	min{γa(41	PROPN
ejpam-3441	89	3	)	)	PUNCT
ejpam-3441	89	4	,	,	PUNCT
ejpam-3441	89	5	(	(	PUNCT
ejpam-3441	89	6	1−	1−	NUM
ejpam-3441	89	7	α	α	NOUN
ejpam-3441	89	8	)	)	PUNCT
ejpam-3441	89	9	}	}	PUNCT
ejpam-3441	89	10	�	�	PROPN
ejpam-3441	89	11	max{γa(4	max{γa(4	PROPN
ejpam-3441	89	12	)	)	PUNCT
ejpam-3441	89	13	,	,	PUNCT
ejpam-3441	89	14	(	(	PUNCT
ejpam-3441	89	15	1−	1−	NUM
ejpam-3441	89	16	β	β	NOUN
ejpam-3441	89	17	)	)	PUNCT
ejpam-3441	89	18	}	}	PUNCT
ejpam-3441	89	19	.	.	PUNCT
ejpam-3441	90	1	then	then	ADV
ejpam-3441	90	2	a	a	DET
ejpam-3441	90	3	=	=	SYM
ejpam-3441	90	4	(	(	PUNCT
ejpam-3441	90	5	µa	µa	PROPN
ejpam-3441	90	6	,	,	PUNCT
ejpam-3441	90	7	γa	γa	PROPN
ejpam-3441	90	8	)	)	PUNCT
ejpam-3441	90	9	is	be	AUX
ejpam-3441	90	10	an	an	DET
ejpam-3441	90	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	90	12	fuzzy	fuzzy	ADJ
ejpam-3441	90	13	la	la	NOUN
ejpam-3441	90	14	-	-	PUNCT
ejpam-3441	90	15	subring	subre	VERB
ejpam-3441	90	16	with	with	ADP
ejpam-3441	90	17	thresholds	threshold	NOUN
ejpam-3441	90	18	(	(	PUNCT
ejpam-3441	90	19	α	α	X
ejpam-3441	90	20	,	,	PUNCT
ejpam-3441	90	21	β	β	X
ejpam-3441	90	22	]	]	PUNCT
ejpam-3441	90	23	of	of	ADP
ejpam-3441	90	24	r	r	NOUN
ejpam-3441	90	25	,	,	PUNCT
ejpam-3441	90	26	but	but	CCONJ
ejpam-3441	90	27	not	not	PART
ejpam-3441	90	28	an	an	DET
ejpam-3441	90	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	90	30	fuzzy	fuzzy	ADJ
ejpam-3441	90	31	right	right	ADJ
ejpam-3441	90	32	ideal	ideal	NOUN
ejpam-3441	90	33	with	with	ADP
ejpam-3441	90	34	thresholds	threshold	NOUN
ejpam-3441	90	35	(	(	PUNCT
ejpam-3441	90	36	α	α	X
ejpam-3441	90	37	,	,	PUNCT
ejpam-3441	90	38	β	β	X
ejpam-3441	90	39	]	]	PUNCT
ejpam-3441	90	40	of	of	ADP
ejpam-3441	90	41	r.	r.	PROPN
ejpam-3441	90	42	an	an	DET
ejpam-3441	90	43	ifs	ifs	PROPN
ejpam-3441	90	44	a	a	PRON
ejpam-3441	90	45	=	=	X
ejpam-3441	90	46	(	(	PUNCT
ejpam-3441	90	47	µa	µa	PROPN
ejpam-3441	90	48	,	,	PUNCT
ejpam-3441	90	49	γa	γa	PROPN
ejpam-3441	90	50	)	)	PUNCT
ejpam-3441	90	51	of	of	ADP
ejpam-3441	90	52	an	an	DET
ejpam-3441	90	53	la	la	ADJ
ejpam-3441	90	54	-	-	PUNCT
ejpam-3441	90	55	ring	ring	NOUN
ejpam-3441	90	56	r	r	NOUN
ejpam-3441	90	57	is	be	AUX
ejpam-3441	90	58	called	call	VERB
ejpam-3441	90	59	an	an	DET
ejpam-3441	90	60	intuitionistic	intuitionistic	ADJ
ejpam-3441	90	61	fuzzy	fuzzy	ADJ
ejpam-3441	90	62	interior	interior	ADJ
ejpam-3441	90	63	ideal	ideal	NOUN
ejpam-3441	90	64	with	with	ADP
ejpam-3441	90	65	thresholds	threshold	NOUN
ejpam-3441	90	66	(	(	PUNCT
ejpam-3441	90	67	α	α	X
ejpam-3441	90	68	,	,	PUNCT
ejpam-3441	90	69	β	β	X
ejpam-3441	90	70	]	]	PUNCT
ejpam-3441	90	71	of	of	ADP
ejpam-3441	90	72	r	r	PRON
ejpam-3441	90	73	if	if	SCONJ
ejpam-3441	90	74	(	(	PUNCT
ejpam-3441	90	75	1	1	X
ejpam-3441	90	76	)	)	PUNCT
ejpam-3441	90	77	max{µa	max{µa	X
ejpam-3441	90	78	(	(	PUNCT
ejpam-3441	90	79	x−	x−	PROPN
ejpam-3441	90	80	y	y	PROPN
ejpam-3441	90	81	)	)	PUNCT
ejpam-3441	90	82	,	,	PUNCT
ejpam-3441	90	83	α	α	X
ejpam-3441	90	84	}	}	PUNCT
ejpam-3441	90	85	≥	≥	NOUN
ejpam-3441	90	86	min{µa	min{µa	X
ejpam-3441	90	87	(	(	PUNCT
ejpam-3441	90	88	x	x	NOUN
ejpam-3441	90	89	)	)	PUNCT
ejpam-3441	90	90	,	,	PUNCT
ejpam-3441	90	91	µa(y	µa(y	NOUN
ejpam-3441	90	92	)	)	PUNCT
ejpam-3441	90	93	,	,	PUNCT
ejpam-3441	90	94	β	β	X
ejpam-3441	90	95	}	}	PUNCT
ejpam-3441	90	96	,	,	PUNCT
ejpam-3441	90	97	(	(	PUNCT
ejpam-3441	90	98	2	2	X
ejpam-3441	90	99	)	)	PUNCT
ejpam-3441	90	100	min{γa	min{γa	NOUN
ejpam-3441	90	101	(	(	PUNCT
ejpam-3441	90	102	x−	x−	PROPN
ejpam-3441	90	103	y	y	PROPN
ejpam-3441	90	104	)	)	PUNCT
ejpam-3441	90	105	,	,	PUNCT
ejpam-3441	90	106	(	(	PUNCT
ejpam-3441	90	107	1−	1−	NUM
ejpam-3441	90	108	α	α	NOUN
ejpam-3441	90	109	)	)	PUNCT
ejpam-3441	90	110	}	}	PUNCT
ejpam-3441	90	111	≤	≤	NUM
ejpam-3441	90	112	max{γa	max{γa	PUNCT
ejpam-3441	90	113	(	(	PUNCT
ejpam-3441	90	114	x	x	NOUN
ejpam-3441	90	115	)	)	PUNCT
ejpam-3441	90	116	,	,	PUNCT
ejpam-3441	90	117	γa(y	γa(y	NUM
ejpam-3441	90	118	)	)	PUNCT
ejpam-3441	90	119	,	,	PUNCT
ejpam-3441	90	120	(	(	PUNCT
ejpam-3441	90	121	1−	1−	NUM
ejpam-3441	90	122	β	β	NOUN
ejpam-3441	90	123	)	)	PUNCT
ejpam-3441	90	124	}	}	PUNCT
ejpam-3441	90	125	,	,	PUNCT
ejpam-3441	90	126	(	(	PUNCT
ejpam-3441	90	127	3	3	X
ejpam-3441	90	128	)	)	PUNCT
ejpam-3441	90	129	max{µa	max{µa	X
ejpam-3441	90	130	(	(	PUNCT
ejpam-3441	90	131	(	(	PUNCT
ejpam-3441	90	132	xy)z	xy)z	NUM
ejpam-3441	90	133	)	)	PUNCT
ejpam-3441	90	134	,	,	PUNCT
ejpam-3441	90	135	α	α	X
ejpam-3441	90	136	}	}	PUNCT
ejpam-3441	90	137	≥	≥	NOUN
ejpam-3441	90	138	min{µa	min{µa	X
ejpam-3441	90	139	(	(	PUNCT
ejpam-3441	90	140	y	y	NOUN
ejpam-3441	90	141	)	)	PUNCT
ejpam-3441	90	142	,	,	PUNCT
ejpam-3441	90	143	β	β	X
ejpam-3441	90	144	}	}	PUNCT
ejpam-3441	90	145	,	,	PUNCT
ejpam-3441	90	146	(	(	PUNCT
ejpam-3441	90	147	4	4	X
ejpam-3441	90	148	)	)	PUNCT
ejpam-3441	90	149	min{γa	min{γa	NOUN
ejpam-3441	90	150	(	(	PUNCT
ejpam-3441	90	151	(	(	PUNCT
ejpam-3441	90	152	xy)z	xy)z	NOUN
ejpam-3441	90	153	)	)	PUNCT
ejpam-3441	90	154	,	,	PUNCT
ejpam-3441	90	155	(	(	PUNCT
ejpam-3441	90	156	1−α	1−α	NUM
ejpam-3441	90	157	)	)	PUNCT
ejpam-3441	90	158	}	}	PUNCT
ejpam-3441	90	159	≤	≤	NUM
ejpam-3441	90	160	max{γa	max{γa	NOUN
ejpam-3441	90	161	(	(	PUNCT
ejpam-3441	90	162	y	y	NOUN
ejpam-3441	90	163	)	)	PUNCT
ejpam-3441	90	164	,	,	PUNCT
ejpam-3441	90	165	(	(	PUNCT
ejpam-3441	90	166	1−β	1−β	NUM
ejpam-3441	90	167	)	)	PUNCT
ejpam-3441	90	168	}	}	PUNCT
ejpam-3441	90	169	for	for	ADP
ejpam-3441	90	170	all	all	DET
ejpam-3441	90	171	x	x	NOUN
ejpam-3441	90	172	,	,	PUNCT
ejpam-3441	90	173	y	y	PROPN
ejpam-3441	90	174	,	,	PUNCT
ejpam-3441	90	175	z	z	NOUN
ejpam-3441	90	176	∈	∈	PROPN
ejpam-3441	90	177	r	r	NOUN
ejpam-3441	90	178	and	and	CCONJ
ejpam-3441	90	179	α	α	NOUN
ejpam-3441	90	180	,	,	PUNCT
ejpam-3441	90	181	β	β	X
ejpam-3441	90	182	∈	∈	PROPN
ejpam-3441	90	183	(	(	PUNCT
ejpam-3441	90	184	0	0	NUM
ejpam-3441	90	185	,	,	PUNCT
ejpam-3441	90	186	1	1	NUM
ejpam-3441	90	187	]	]	PUNCT
ejpam-3441	90	188	such	such	ADJ
ejpam-3441	90	189	that	that	SCONJ
ejpam-3441	90	190	α	α	PRON
ejpam-3441	90	191	<	<	X
ejpam-3441	90	192	β	β	X
ejpam-3441	90	193	.	.	PUNCT
ejpam-3441	91	1	an	an	DET
ejpam-3441	91	2	ifs	ifs	PROPN
ejpam-3441	91	3	a	a	X
ejpam-3441	91	4	=	=	X
ejpam-3441	91	5	(	(	PUNCT
ejpam-3441	91	6	µa	µa	PROPN
ejpam-3441	91	7	,	,	PUNCT
ejpam-3441	91	8	γa	γa	PROPN
ejpam-3441	91	9	)	)	PUNCT
ejpam-3441	91	10	of	of	ADP
ejpam-3441	91	11	an	an	DET
ejpam-3441	91	12	la	la	ADJ
ejpam-3441	91	13	-	-	PUNCT
ejpam-3441	91	14	ring	ring	NOUN
ejpam-3441	91	15	r	r	NOUN
ejpam-3441	91	16	is	be	AUX
ejpam-3441	91	17	called	call	VERB
ejpam-3441	91	18	an	an	DET
ejpam-3441	91	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	91	20	fuzzy	fuzzy	ADJ
ejpam-3441	91	21	quasi	quasi	NOUN
ejpam-3441	91	22	-	-	NOUN
ejpam-3441	91	23	ideal	ideal	ADJ
ejpam-3441	91	24	with	with	ADP
ejpam-3441	91	25	thresholds	threshold	NOUN
ejpam-3441	91	26	(	(	PUNCT
ejpam-3441	91	27	α	α	X
ejpam-3441	91	28	,	,	PUNCT
ejpam-3441	91	29	β	β	X
ejpam-3441	91	30	]	]	PUNCT
ejpam-3441	91	31	of	of	ADP
ejpam-3441	91	32	r	r	PRON
ejpam-3441	91	33	if	if	SCONJ
ejpam-3441	91	34	(	(	PUNCT
ejpam-3441	91	35	1	1	X
ejpam-3441	91	36	)	)	PUNCT
ejpam-3441	91	37	max{µa	max{µa	X
ejpam-3441	91	38	(	(	PUNCT
ejpam-3441	91	39	x−	x−	PROPN
ejpam-3441	91	40	y	y	PROPN
ejpam-3441	91	41	)	)	PUNCT
ejpam-3441	91	42	,	,	PUNCT
ejpam-3441	91	43	α	α	X
ejpam-3441	91	44	}	}	PUNCT
ejpam-3441	91	45	≥	≥	NOUN
ejpam-3441	91	46	min{µa	min{µa	X
ejpam-3441	91	47	(	(	PUNCT
ejpam-3441	91	48	x	x	NOUN
ejpam-3441	91	49	)	)	PUNCT
ejpam-3441	91	50	,	,	PUNCT
ejpam-3441	91	51	µa(y	µa(y	NOUN
ejpam-3441	91	52	)	)	PUNCT
ejpam-3441	91	53	,	,	PUNCT
ejpam-3441	91	54	β	β	X
ejpam-3441	91	55	}	}	PUNCT
ejpam-3441	91	56	,	,	PUNCT
ejpam-3441	91	57	(	(	PUNCT
ejpam-3441	91	58	2	2	X
ejpam-3441	91	59	)	)	PUNCT
ejpam-3441	91	60	min{γa	min{γa	NOUN
ejpam-3441	91	61	(	(	PUNCT
ejpam-3441	91	62	x−	x−	PROPN
ejpam-3441	91	63	y	y	PROPN
ejpam-3441	91	64	)	)	PUNCT
ejpam-3441	91	65	,	,	PUNCT
ejpam-3441	91	66	(	(	PUNCT
ejpam-3441	91	67	1−	1−	NUM
ejpam-3441	91	68	α	α	NOUN
ejpam-3441	91	69	)	)	PUNCT
ejpam-3441	91	70	}	}	PUNCT
ejpam-3441	91	71	≤	≤	NUM
ejpam-3441	91	72	max{γa	max{γa	PUNCT
ejpam-3441	91	73	(	(	PUNCT
ejpam-3441	91	74	x	x	NOUN
ejpam-3441	91	75	)	)	PUNCT
ejpam-3441	91	76	,	,	PUNCT
ejpam-3441	91	77	γa(y	γa(y	NUM
ejpam-3441	91	78	)	)	PUNCT
ejpam-3441	91	79	,	,	PUNCT
ejpam-3441	91	80	(	(	PUNCT
ejpam-3441	91	81	1−	1−	NUM
ejpam-3441	91	82	β	β	NOUN
ejpam-3441	91	83	)	)	PUNCT
ejpam-3441	91	84	}	}	PUNCT
ejpam-3441	91	85	,	,	PUNCT
ejpam-3441	91	86	(	(	PUNCT
ejpam-3441	91	87	3	3	X
ejpam-3441	91	88	)	)	PUNCT
ejpam-3441	91	89	max{µa(x	max{µa(x	NOUN
ejpam-3441	91	90	)	)	PUNCT
ejpam-3441	91	91	,	,	PUNCT
ejpam-3441	91	92	α	α	X
ejpam-3441	91	93	}	}	PUNCT
ejpam-3441	91	94	≥	≥	NOUN
ejpam-3441	91	95	min{(µa	min{(µa	X
ejpam-3441	91	96	◦	◦	NOUN
ejpam-3441	91	97	r	r	NOUN
ejpam-3441	91	98	)	)	PUNCT
ejpam-3441	91	99	(	(	PUNCT
ejpam-3441	91	100	x	x	NOUN
ejpam-3441	91	101	)	)	PUNCT
ejpam-3441	91	102	,	,	PUNCT
ejpam-3441	91	103	(	(	PUNCT
ejpam-3441	91	104	r	r	NOUN
ejpam-3441	91	105	◦	◦	NOUN
ejpam-3441	91	106	µa	µa	NOUN
ejpam-3441	91	107	)	)	PUNCT
ejpam-3441	91	108	(	(	PUNCT
ejpam-3441	91	109	x	x	NOUN
ejpam-3441	91	110	)	)	PUNCT
ejpam-3441	91	111	,	,	PUNCT
ejpam-3441	91	112	β	β	X
ejpam-3441	91	113	}	}	PUNCT
ejpam-3441	91	114	,	,	PUNCT
ejpam-3441	91	115	(	(	PUNCT
ejpam-3441	91	116	4	4	X
ejpam-3441	91	117	)	)	PUNCT
ejpam-3441	91	118	min{γa(x	min{γa(x	NOUN
ejpam-3441	91	119	)	)	PUNCT
ejpam-3441	91	120	,	,	PUNCT
ejpam-3441	91	121	(	(	PUNCT
ejpam-3441	91	122	1−α	1−α	NUM
ejpam-3441	91	123	)	)	PUNCT
ejpam-3441	91	124	}	}	PUNCT
ejpam-3441	91	125	≤	≤	NUM
ejpam-3441	91	126	max{(γa	max{(γa	ADJ
ejpam-3441	91	127	◦	◦	NOUN
ejpam-3441	91	128	r	r	NOUN
ejpam-3441	91	129	)	)	PUNCT
ejpam-3441	91	130	(	(	PUNCT
ejpam-3441	91	131	x	x	NOUN
ejpam-3441	91	132	)	)	PUNCT
ejpam-3441	91	133	,	,	PUNCT
ejpam-3441	91	134	(	(	PUNCT
ejpam-3441	91	135	r	r	NOUN
ejpam-3441	91	136	◦	◦	NOUN
ejpam-3441	91	137	γa	γa	NOUN
ejpam-3441	91	138	)	)	PUNCT
ejpam-3441	91	139	(	(	PUNCT
ejpam-3441	91	140	x	x	NOUN
ejpam-3441	91	141	)	)	PUNCT
ejpam-3441	91	142	,	,	PUNCT
ejpam-3441	91	143	(	(	PUNCT
ejpam-3441	91	144	1−β	1−β	NUM
ejpam-3441	91	145	)	)	PUNCT
ejpam-3441	91	146	}	}	PUNCT
ejpam-3441	91	147	for	for	ADP
ejpam-3441	91	148	all	all	DET
ejpam-3441	91	149	x	x	NOUN
ejpam-3441	91	150	,	,	PUNCT
ejpam-3441	91	151	y	y	PROPN
ejpam-3441	91	152	∈	∈	PROPN
ejpam-3441	91	153	r	r	NOUN
ejpam-3441	91	154	and	and	CCONJ
ejpam-3441	91	155	α	α	NOUN
ejpam-3441	91	156	,	,	PUNCT
ejpam-3441	91	157	β	β	X
ejpam-3441	91	158	∈	∈	PROPN
ejpam-3441	91	159	(	(	PUNCT
ejpam-3441	91	160	0	0	NUM
ejpam-3441	91	161	,	,	PUNCT
ejpam-3441	91	162	1	1	NUM
ejpam-3441	91	163	]	]	PUNCT
ejpam-3441	91	164	such	such	ADJ
ejpam-3441	91	165	that	that	SCONJ
ejpam-3441	91	166	α	α	PRON
ejpam-3441	91	167	<	<	X
ejpam-3441	91	168	β	β	X
ejpam-3441	91	169	.	.	PUNCT
ejpam-3441	92	1	an	an	DET
ejpam-3441	92	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	92	3	fuzzy	fuzzy	ADJ
ejpam-3441	92	4	la	la	NOUN
ejpam-3441	92	5	-	-	PUNCT
ejpam-3441	92	6	subring	subre	VERB
ejpam-3441	92	7	a	a	PRON
ejpam-3441	92	8	=	=	SYM
ejpam-3441	92	9	(	(	PUNCT
ejpam-3441	92	10	µa	µa	PROPN
ejpam-3441	92	11	,	,	PUNCT
ejpam-3441	92	12	γa	γa	PROPN
ejpam-3441	92	13	)	)	PUNCT
ejpam-3441	92	14	with	with	ADP
ejpam-3441	92	15	thresholds	threshold	NOUN
ejpam-3441	92	16	(	(	PUNCT
ejpam-3441	92	17	α	α	X
ejpam-3441	92	18	,	,	PUNCT
ejpam-3441	92	19	β	β	X
ejpam-3441	92	20	]	]	PUNCT
ejpam-3441	92	21	of	of	ADP
ejpam-3441	92	22	an	an	DET
ejpam-3441	92	23	la	la	ADJ
ejpam-3441	92	24	-	-	PUNCT
ejpam-3441	92	25	ring	ring	NOUN
ejpam-3441	92	26	r	r	NOUN
ejpam-3441	92	27	is	be	AUX
ejpam-3441	92	28	called	call	VERB
ejpam-3441	92	29	an	an	DET
ejpam-3441	92	30	intuitionistic	intuitionistic	ADJ
ejpam-3441	92	31	fuzzy	fuzzy	ADJ
ejpam-3441	92	32	bi	bi	NOUN
ejpam-3441	92	33	-	-	NOUN
ejpam-3441	92	34	ideal	ideal	ADJ
ejpam-3441	92	35	with	with	ADP
ejpam-3441	92	36	thresholds	threshold	NOUN
ejpam-3441	92	37	(	(	PUNCT
ejpam-3441	92	38	α	α	X
ejpam-3441	92	39	,	,	PUNCT
ejpam-3441	92	40	β	β	X
ejpam-3441	92	41	]	]	PUNCT
ejpam-3441	92	42	of	of	ADP
ejpam-3441	92	43	r	r	NOUN
ejpam-3441	92	44	if	if	SCONJ
ejpam-3441	92	45	k.	k.	PROPN
ejpam-3441	92	46	nasreen	nasreen	PROPN
ejpam-3441	92	47	et	et	PROPN
ejpam-3441	92	48	al	al	PROPN
ejpam-3441	92	49	.	.	PUNCT
ejpam-3441	92	50	/	/	SYM
ejpam-3441	92	51	eur	eur	PROPN
ejpam-3441	92	52	.	.	PUNCT
ejpam-3441	93	1	j.	j.	PROPN
ejpam-3441	93	2	pure	pure	PROPN
ejpam-3441	93	3	appl	appl	PROPN
ejpam-3441	93	4	.	.	PROPN
ejpam-3441	93	5	math	math	PROPN
ejpam-3441	93	6	,	,	PUNCT
ejpam-3441	93	7	12	12	NUM
ejpam-3441	93	8	(	(	PUNCT
ejpam-3441	93	9	3	3	NUM
ejpam-3441	93	10	)	)	PUNCT
ejpam-3441	93	11	(	(	PUNCT
ejpam-3441	93	12	2019	2019	NUM
ejpam-3441	93	13	)	)	PUNCT
ejpam-3441	93	14	,	,	PUNCT
ejpam-3441	93	15	906	906	NUM
ejpam-3441	93	16	-	-	SYM
ejpam-3441	93	17	943	943	NUM
ejpam-3441	93	18	910	910	NUM
ejpam-3441	93	19	(	(	PUNCT
ejpam-3441	93	20	1	1	NUM
ejpam-3441	93	21	)	)	PUNCT
ejpam-3441	93	22	max{µa	max{µa	X
ejpam-3441	93	23	(	(	PUNCT
ejpam-3441	93	24	(	(	PUNCT
ejpam-3441	93	25	xy)z	xy)z	NUM
ejpam-3441	93	26	)	)	PUNCT
ejpam-3441	93	27	,	,	PUNCT
ejpam-3441	93	28	α	α	X
ejpam-3441	93	29	}	}	PUNCT
ejpam-3441	93	30	≥	≥	NOUN
ejpam-3441	93	31	min{µa	min{µa	X
ejpam-3441	93	32	(	(	PUNCT
ejpam-3441	93	33	x	x	NOUN
ejpam-3441	93	34	)	)	PUNCT
ejpam-3441	93	35	,	,	PUNCT
ejpam-3441	93	36	µa	µa	X
ejpam-3441	93	37	(	(	PUNCT
ejpam-3441	93	38	z	z	NOUN
ejpam-3441	93	39	)	)	PUNCT
ejpam-3441	93	40	,	,	PUNCT
ejpam-3441	93	41	β	β	X
ejpam-3441	93	42	}	}	PUNCT
ejpam-3441	93	43	,	,	PUNCT
ejpam-3441	93	44	(	(	PUNCT
ejpam-3441	93	45	2	2	X
ejpam-3441	93	46	)	)	PUNCT
ejpam-3441	93	47	min{γa	min{γa	NOUN
ejpam-3441	93	48	(	(	PUNCT
ejpam-3441	93	49	(	(	PUNCT
ejpam-3441	93	50	xy)z	xy)z	NOUN
ejpam-3441	93	51	)	)	PUNCT
ejpam-3441	93	52	,	,	PUNCT
ejpam-3441	93	53	(	(	PUNCT
ejpam-3441	93	54	1	1	NUM
ejpam-3441	93	55	−	−	PROPN
ejpam-3441	93	56	α	α	X
ejpam-3441	93	57	)	)	PUNCT
ejpam-3441	93	58	}	}	PUNCT
ejpam-3441	93	59	≤	≤	NUM
ejpam-3441	93	60	max{γa	max{γa	PUNCT
ejpam-3441	93	61	(	(	PUNCT
ejpam-3441	93	62	x	x	X
ejpam-3441	93	63	)	)	PUNCT
ejpam-3441	93	64	,	,	PUNCT
ejpam-3441	93	65	γa	γa	PROPN
ejpam-3441	93	66	(	(	PUNCT
ejpam-3441	93	67	z	z	NOUN
ejpam-3441	93	68	)	)	PUNCT
ejpam-3441	93	69	,	,	PUNCT
ejpam-3441	93	70	(	(	PUNCT
ejpam-3441	93	71	1	1	NUM
ejpam-3441	93	72	−	−	NOUN
ejpam-3441	93	73	β	β	X
ejpam-3441	93	74	)	)	PUNCT
ejpam-3441	93	75	}	}	PUNCT
ejpam-3441	93	76	for	for	ADP
ejpam-3441	93	77	all	all	DET
ejpam-3441	93	78	x	x	NOUN
ejpam-3441	93	79	,	,	PUNCT
ejpam-3441	93	80	y	y	PROPN
ejpam-3441	93	81	,	,	PUNCT
ejpam-3441	93	82	z	z	NOUN
ejpam-3441	93	83	∈	∈	PROPN
ejpam-3441	93	84	r	r	NOUN
ejpam-3441	93	85	and	and	CCONJ
ejpam-3441	93	86	α	α	NOUN
ejpam-3441	93	87	,	,	PUNCT
ejpam-3441	93	88	β	β	X
ejpam-3441	93	89	∈	∈	PROPN
ejpam-3441	93	90	(	(	PUNCT
ejpam-3441	93	91	0	0	NUM
ejpam-3441	93	92	,	,	PUNCT
ejpam-3441	93	93	1	1	NUM
ejpam-3441	93	94	]	]	PUNCT
ejpam-3441	93	95	such	such	ADJ
ejpam-3441	93	96	that	that	SCONJ
ejpam-3441	93	97	α	α	PRON
ejpam-3441	93	98	<	<	X
ejpam-3441	93	99	β	β	X
ejpam-3441	93	100	.	.	PUNCT
ejpam-3441	94	1	an	an	DET
ejpam-3441	94	2	ifs	ifs	PROPN
ejpam-3441	94	3	a	a	X
ejpam-3441	94	4	=	=	X
ejpam-3441	94	5	(	(	PUNCT
ejpam-3441	94	6	µa	µa	PROPN
ejpam-3441	94	7	,	,	PUNCT
ejpam-3441	94	8	γa	γa	PROPN
ejpam-3441	94	9	)	)	PUNCT
ejpam-3441	94	10	of	of	ADP
ejpam-3441	94	11	an	an	DET
ejpam-3441	94	12	la	la	ADJ
ejpam-3441	94	13	-	-	PUNCT
ejpam-3441	94	14	ring	ring	NOUN
ejpam-3441	94	15	r	r	NOUN
ejpam-3441	94	16	is	be	AUX
ejpam-3441	94	17	called	call	VERB
ejpam-3441	94	18	an	an	DET
ejpam-3441	94	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	94	20	fuzzy	fuzzy	ADJ
ejpam-3441	94	21	generalized	generalize	VERB
ejpam-3441	94	22	bi	bi	NOUN
ejpam-3441	94	23	-	-	NOUN
ejpam-3441	94	24	ideal	ideal	NOUN
ejpam-3441	94	25	with	with	ADP
ejpam-3441	94	26	thresholds	threshold	NOUN
ejpam-3441	94	27	(	(	PUNCT
ejpam-3441	94	28	α	α	X
ejpam-3441	94	29	,	,	PUNCT
ejpam-3441	94	30	β	β	X
ejpam-3441	94	31	]	]	PUNCT
ejpam-3441	94	32	of	of	ADP
ejpam-3441	94	33	r	r	PRON
ejpam-3441	94	34	if	if	SCONJ
ejpam-3441	94	35	(	(	PUNCT
ejpam-3441	94	36	1	1	X
ejpam-3441	94	37	)	)	PUNCT
ejpam-3441	94	38	max{µa	max{µa	X
ejpam-3441	94	39	(	(	PUNCT
ejpam-3441	94	40	x−	x−	PROPN
ejpam-3441	94	41	y	y	PROPN
ejpam-3441	94	42	)	)	PUNCT
ejpam-3441	94	43	,	,	PUNCT
ejpam-3441	94	44	α	α	X
ejpam-3441	94	45	}	}	PUNCT
ejpam-3441	94	46	≥	≥	NOUN
ejpam-3441	94	47	min{µa	min{µa	X
ejpam-3441	94	48	(	(	PUNCT
ejpam-3441	94	49	x	x	NOUN
ejpam-3441	94	50	)	)	PUNCT
ejpam-3441	94	51	,	,	PUNCT
ejpam-3441	94	52	µa(y	µa(y	NOUN
ejpam-3441	94	53	)	)	PUNCT
ejpam-3441	94	54	,	,	PUNCT
ejpam-3441	94	55	β	β	X
ejpam-3441	94	56	}	}	PUNCT
ejpam-3441	94	57	,	,	PUNCT
ejpam-3441	94	58	(	(	PUNCT
ejpam-3441	94	59	2	2	X
ejpam-3441	94	60	)	)	PUNCT
ejpam-3441	94	61	min{γa	min{γa	NOUN
ejpam-3441	94	62	(	(	PUNCT
ejpam-3441	94	63	x−	x−	PROPN
ejpam-3441	94	64	y	y	PROPN
ejpam-3441	94	65	)	)	PUNCT
ejpam-3441	94	66	,	,	PUNCT
ejpam-3441	94	67	(	(	PUNCT
ejpam-3441	94	68	1−	1−	NUM
ejpam-3441	94	69	α	α	NOUN
ejpam-3441	94	70	)	)	PUNCT
ejpam-3441	94	71	}	}	PUNCT
ejpam-3441	94	72	≤	≤	NUM
ejpam-3441	94	73	max{γa	max{γa	PUNCT
ejpam-3441	94	74	(	(	PUNCT
ejpam-3441	94	75	x	x	NOUN
ejpam-3441	94	76	)	)	PUNCT
ejpam-3441	94	77	,	,	PUNCT
ejpam-3441	94	78	γa(y	γa(y	NUM
ejpam-3441	94	79	)	)	PUNCT
ejpam-3441	94	80	,	,	PUNCT
ejpam-3441	94	81	(	(	PUNCT
ejpam-3441	94	82	1−	1−	NUM
ejpam-3441	94	83	β	β	NOUN
ejpam-3441	94	84	)	)	PUNCT
ejpam-3441	94	85	}	}	PUNCT
ejpam-3441	94	86	,	,	PUNCT
ejpam-3441	94	87	(	(	PUNCT
ejpam-3441	94	88	3	3	X
ejpam-3441	94	89	)	)	PUNCT
ejpam-3441	94	90	max{µa	max{µa	X
ejpam-3441	94	91	(	(	PUNCT
ejpam-3441	94	92	(	(	PUNCT
ejpam-3441	94	93	xy)z	xy)z	NUM
ejpam-3441	94	94	)	)	PUNCT
ejpam-3441	94	95	,	,	PUNCT
ejpam-3441	94	96	α	α	X
ejpam-3441	94	97	}	}	PUNCT
ejpam-3441	94	98	≥	≥	NOUN
ejpam-3441	94	99	min{µa	min{µa	X
ejpam-3441	94	100	(	(	PUNCT
ejpam-3441	94	101	x	x	NOUN
ejpam-3441	94	102	)	)	PUNCT
ejpam-3441	94	103	,	,	PUNCT
ejpam-3441	94	104	µa	µa	X
ejpam-3441	94	105	(	(	PUNCT
ejpam-3441	94	106	z	z	NOUN
ejpam-3441	94	107	)	)	PUNCT
ejpam-3441	94	108	,	,	PUNCT
ejpam-3441	94	109	β	β	X
ejpam-3441	94	110	}	}	PUNCT
ejpam-3441	94	111	,	,	PUNCT
ejpam-3441	94	112	(	(	PUNCT
ejpam-3441	94	113	4	4	X
ejpam-3441	94	114	)	)	PUNCT
ejpam-3441	94	115	min{γa	min{γa	NOUN
ejpam-3441	94	116	(	(	PUNCT
ejpam-3441	94	117	(	(	PUNCT
ejpam-3441	94	118	xy)z	xy)z	NOUN
ejpam-3441	94	119	)	)	PUNCT
ejpam-3441	94	120	,	,	PUNCT
ejpam-3441	94	121	(	(	PUNCT
ejpam-3441	94	122	1	1	NUM
ejpam-3441	94	123	−	−	PROPN
ejpam-3441	94	124	α	α	X
ejpam-3441	94	125	)	)	PUNCT
ejpam-3441	94	126	}	}	PUNCT
ejpam-3441	94	127	≤	≤	NUM
ejpam-3441	94	128	max{γa	max{γa	PUNCT
ejpam-3441	94	129	(	(	PUNCT
ejpam-3441	94	130	x	x	X
ejpam-3441	94	131	)	)	PUNCT
ejpam-3441	94	132	,	,	PUNCT
ejpam-3441	94	133	γa	γa	PROPN
ejpam-3441	94	134	(	(	PUNCT
ejpam-3441	94	135	z	z	NOUN
ejpam-3441	94	136	)	)	PUNCT
ejpam-3441	94	137	,	,	PUNCT
ejpam-3441	94	138	(	(	PUNCT
ejpam-3441	94	139	1	1	NUM
ejpam-3441	94	140	−	−	NOUN
ejpam-3441	94	141	β	β	X
ejpam-3441	94	142	)	)	PUNCT
ejpam-3441	94	143	}	}	PUNCT
ejpam-3441	94	144	for	for	ADP
ejpam-3441	94	145	all	all	DET
ejpam-3441	94	146	x	x	NOUN
ejpam-3441	94	147	,	,	PUNCT
ejpam-3441	94	148	y	y	PROPN
ejpam-3441	94	149	,	,	PUNCT
ejpam-3441	94	150	z	z	NOUN
ejpam-3441	94	151	∈	∈	PROPN
ejpam-3441	94	152	r	r	NOUN
ejpam-3441	94	153	and	and	CCONJ
ejpam-3441	94	154	α	α	NOUN
ejpam-3441	94	155	,	,	PUNCT
ejpam-3441	94	156	β	β	X
ejpam-3441	94	157	∈	∈	PROPN
ejpam-3441	94	158	(	(	PUNCT
ejpam-3441	94	159	0	0	NUM
ejpam-3441	94	160	,	,	PUNCT
ejpam-3441	94	161	1	1	NUM
ejpam-3441	94	162	]	]	PUNCT
ejpam-3441	94	163	such	such	ADJ
ejpam-3441	94	164	that	that	SCONJ
ejpam-3441	94	165	α	α	PRON
ejpam-3441	94	166	<	<	X
ejpam-3441	94	167	β	β	X
ejpam-3441	94	168	.	.	PUNCT
ejpam-3441	95	1	let	let	VERB
ejpam-3441	95	2	a	a	DET
ejpam-3441	95	3	=	=	SYM
ejpam-3441	95	4	(	(	PUNCT
ejpam-3441	95	5	µa	µa	PROPN
ejpam-3441	95	6	,	,	PUNCT
ejpam-3441	95	7	γa	γa	PROPN
ejpam-3441	95	8	)	)	PUNCT
ejpam-3441	95	9	and	and	CCONJ
ejpam-3441	95	10	b	b	X
ejpam-3441	95	11	=	=	SYM
ejpam-3441	95	12	(	(	PUNCT
ejpam-3441	95	13	µb	µb	PROPN
ejpam-3441	95	14	,	,	PUNCT
ejpam-3441	95	15	γb	γb	PROPN
ejpam-3441	95	16	)	)	PUNCT
ejpam-3441	95	17	be	be	VERB
ejpam-3441	95	18	two	two	NUM
ejpam-3441	95	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	95	20	fuzzy	fuzzy	ADJ
ejpam-3441	95	21	sets	set	NOUN
ejpam-3441	95	22	of	of	ADP
ejpam-3441	95	23	an	an	DET
ejpam-3441	95	24	la	la	ADJ
ejpam-3441	95	25	-	-	PUNCT
ejpam-3441	95	26	ring	ring	NOUN
ejpam-3441	95	27	r	r	NOUN
ejpam-3441	95	28	,	,	PUNCT
ejpam-3441	95	29	then	then	ADV
ejpam-3441	95	30	the	the	DET
ejpam-3441	95	31	product	product	NOUN
ejpam-3441	95	32	of	of	ADP
ejpam-3441	95	33	a	a	PRON
ejpam-3441	95	34	and	and	CCONJ
ejpam-3441	95	35	b	b	NOUN
ejpam-3441	95	36	is	be	AUX
ejpam-3441	95	37	denoted	denote	VERB
ejpam-3441	95	38	by	by	ADP
ejpam-3441	95	39	a	a	DET
ejpam-3441	95	40	◦	◦	NOUN
ejpam-3441	95	41	b	b	NOUN
ejpam-3441	95	42	=	=	SYM
ejpam-3441	95	43	(	(	PUNCT
ejpam-3441	95	44	µa	µa	ADP
ejpam-3441	95	45	◦	◦	NOUN
ejpam-3441	95	46	µb	µb	PROPN
ejpam-3441	95	47	,	,	PUNCT
ejpam-3441	95	48	γa	γa	PROPN
ejpam-3441	95	49	◦	◦	PROPN
ejpam-3441	95	50	γb	γb	PROPN
ejpam-3441	95	51	)	)	PUNCT
ejpam-3441	95	52	and	and	CCONJ
ejpam-3441	95	53	defined	define	VERB
ejpam-3441	95	54	by	by	ADP
ejpam-3441	95	55	:	:	PUNCT
ejpam-3441	95	56	(	(	PUNCT
ejpam-3441	95	57	µa	µa	ADP
ejpam-3441	95	58	◦	◦	NOUN
ejpam-3441	95	59	µb)(x	µb)(x	NOUN
ejpam-3441	95	60	)	)	PUNCT
ejpam-3441	95	61	=	=	PUNCT
ejpam-3441	96	1			X
ejpam-3441	96	2	∨	∨	NOUN
ejpam-3441	96	3	x=	x=	PROPN
ejpam-3441	97	1	n∑	n∑	PROPN
ejpam-3441	97	2	i=1	i=1	PROPN
ejpam-3441	97	3	aibi	aibi	PROPN
ejpam-3441	97	4	{	{	PUNCT
ejpam-3441	97	5	∧ni=1{µa(ai	∧ni=1{µa(ai	PROPN
ejpam-3441	97	6	)	)	PUNCT
ejpam-3441	97	7	∧	∧	PROPN
ejpam-3441	97	8	µb(bi	µb(bi	PROPN
ejpam-3441	97	9	)	)	PUNCT
ejpam-3441	97	10	}	}	PUNCT
ejpam-3441	97	11	}	}	PUNCT
ejpam-3441	97	12	if	if	SCONJ
ejpam-3441	97	13	x	x	X
ejpam-3441	97	14	=	=	PUNCT
ejpam-3441	97	15	n∑	n∑	PROPN
ejpam-3441	97	16	i=1	i=1	PROPN
ejpam-3441	97	17	aibi	aibi	NOUN
ejpam-3441	97	18	,	,	PUNCT
ejpam-3441	97	19	ai	ai	VERB
ejpam-3441	97	20	,	,	PUNCT
ejpam-3441	97	21	bi	bi	NOUN
ejpam-3441	97	22	∈	∈	PROPN
ejpam-3441	97	23	r	r	NOUN
ejpam-3441	97	24	0	0	PUNCT
ejpam-3441	98	1	if	if	SCONJ
ejpam-3441	98	2	x	x	PROPN
ejpam-3441	98	3	6=	6=	NUM
ejpam-3441	98	4	n∑	n∑	PROPN
ejpam-3441	98	5	i=1	i=1	PROPN
ejpam-3441	98	6	aibi	aibi	NOUN
ejpam-3441	98	7	and	and	CCONJ
ejpam-3441	98	8	(	(	PUNCT
ejpam-3441	98	9	γa	γa	NOUN
ejpam-3441	98	10	◦	◦	VERB
ejpam-3441	98	11	γb)(x	γb)(x	PROPN
ejpam-3441	98	12	)	)	PUNCT
ejpam-3441	99	1	=	=	PUNCT
ejpam-3441	99	2			PUNCT
ejpam-3441	99	3	∧	∧	NOUN
ejpam-3441	99	4	x=	x=	PUNCT
ejpam-3441	100	1	n∑	n∑	PROPN
ejpam-3441	100	2	i=1	i=1	PROPN
ejpam-3441	100	3	aibi	aibi	PROPN
ejpam-3441	100	4	{	{	PUNCT
ejpam-3441	100	5	∨ni=1{γa(ai	∨ni=1{γa(ai	PROPN
ejpam-3441	100	6	)	)	PUNCT
ejpam-3441	100	7	∨	∨	NOUN
ejpam-3441	100	8	γb(bi	γb(bi	PROPN
ejpam-3441	100	9	)	)	PUNCT
ejpam-3441	100	10	}	}	PUNCT
ejpam-3441	100	11	}	}	PUNCT
ejpam-3441	100	12	if	if	SCONJ
ejpam-3441	100	13	x	x	X
ejpam-3441	100	14	=	=	PUNCT
ejpam-3441	100	15	n∑	n∑	PROPN
ejpam-3441	100	16	i=1	i=1	PROPN
ejpam-3441	100	17	aibi	aibi	NOUN
ejpam-3441	100	18	,	,	PUNCT
ejpam-3441	100	19	ai	ai	VERB
ejpam-3441	100	20	,	,	PUNCT
ejpam-3441	100	21	bi	bi	NOUN
ejpam-3441	100	22	∈	∈	PROPN
ejpam-3441	100	23	r	r	NOUN
ejpam-3441	100	24	1	1	NUM
ejpam-3441	101	1	if	if	SCONJ
ejpam-3441	101	2	x	x	PROPN
ejpam-3441	101	3	6=	6=	PROPN
ejpam-3441	101	4	n∑	n∑	PROPN
ejpam-3441	101	5	i=1	i=1	PROPN
ejpam-3441	101	6	aibi	aibi	NOUN
ejpam-3441	101	7	let	let	VERB
ejpam-3441	101	8	a	a	PRON
ejpam-3441	101	9	=	=	SYM
ejpam-3441	101	10	(	(	PUNCT
ejpam-3441	101	11	µa	µa	PROPN
ejpam-3441	101	12	,	,	PUNCT
ejpam-3441	101	13	γa	γa	PROPN
ejpam-3441	101	14	)	)	PUNCT
ejpam-3441	101	15	be	be	VERB
ejpam-3441	101	16	an	an	DET
ejpam-3441	101	17	ifs	ifs	PROPN
ejpam-3441	101	18	of	of	ADP
ejpam-3441	101	19	an	an	DET
ejpam-3441	101	20	la	la	ADJ
ejpam-3441	101	21	-	-	PUNCT
ejpam-3441	101	22	ring	ring	NOUN
ejpam-3441	101	23	r	r	NOUN
ejpam-3441	101	24	and	and	CCONJ
ejpam-3441	101	25	α	α	NOUN
ejpam-3441	101	26	,	,	PUNCT
ejpam-3441	101	27	β	β	X
ejpam-3441	101	28	∈	∈	PROPN
ejpam-3441	101	29	(	(	PUNCT
ejpam-3441	101	30	0	0	NUM
ejpam-3441	101	31	,	,	PUNCT
ejpam-3441	101	32	1	1	NUM
ejpam-3441	101	33	]	]	PUNCT
ejpam-3441	101	34	such	such	ADJ
ejpam-3441	101	35	that	that	SCONJ
ejpam-3441	101	36	α	α	PRON
ejpam-3441	101	37	<	<	X
ejpam-3441	101	38	β	β	X
ejpam-3441	101	39	.	.	PUNCT
ejpam-3441	102	1	we	we	PRON
ejpam-3441	102	2	define	define	VERB
ejpam-3441	102	3	an	an	DET
ejpam-3441	102	4	intuitionistic	intuitionistic	ADJ
ejpam-3441	102	5	fuzzy	fuzzy	ADJ
ejpam-3441	102	6	set	set	VERB
ejpam-3441	102	7	aβα	aβα	NOUN
ejpam-3441	102	8	of	of	ADP
ejpam-3441	102	9	r	r	NOUN
ejpam-3441	102	10	as	as	ADP
ejpam-3441	102	11	follow	follow	VERB
ejpam-3441	102	12	:	:	PUNCT
ejpam-3441	102	13	(	(	PUNCT
ejpam-3441	102	14	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	102	15	)	)	PUNCT
ejpam-3441	102	16	=	=	PUNCT
ejpam-3441	102	17	(	(	PUNCT
ejpam-3441	102	18	µa(x	µa(x	NOUN
ejpam-3441	102	19	)	)	PUNCT
ejpam-3441	102	20	∧	∧	PROPN
ejpam-3441	102	21	β	β	NOUN
ejpam-3441	102	22	)	)	PUNCT
ejpam-3441	102	23	∨	∨	NUM
ejpam-3441	102	24	α	α	PROPN
ejpam-3441	102	25	and	and	CCONJ
ejpam-3441	102	26	(	(	PUNCT
ejpam-3441	102	27	γa)βα(x	γa)βα(x	PROPN
ejpam-3441	102	28	)	)	PUNCT
ejpam-3441	102	29	=	=	SYM
ejpam-3441	102	30	(	(	PUNCT
ejpam-3441	102	31	γa(x	γa(x	NUM
ejpam-3441	102	32	)	)	PUNCT
ejpam-3441	102	33	∨	∨	NUM
ejpam-3441	102	34	(	(	PUNCT
ejpam-3441	102	35	1−	1−	NUM
ejpam-3441	102	36	β	β	NOUN
ejpam-3441	102	37	)	)	PUNCT
ejpam-3441	102	38	)	)	PUNCT
ejpam-3441	103	1	∧	∧	NOUN
ejpam-3441	103	2	(	(	PUNCT
ejpam-3441	103	3	1−	1−	NUM
ejpam-3441	103	4	α	α	NOUN
ejpam-3441	103	5	)	)	PUNCT
ejpam-3441	103	6	for	for	ADP
ejpam-3441	103	7	all	all	DET
ejpam-3441	103	8	x	x	PROPN
ejpam-3441	103	9	∈	∈	PROPN
ejpam-3441	103	10	r.	r.	NOUN
ejpam-3441	103	11	let	let	VERB
ejpam-3441	103	12	a	a	DET
ejpam-3441	103	13	=	=	SYM
ejpam-3441	103	14	(	(	PUNCT
ejpam-3441	103	15	µa	µa	PROPN
ejpam-3441	103	16	,	,	PUNCT
ejpam-3441	103	17	γa	γa	PROPN
ejpam-3441	103	18	)	)	PUNCT
ejpam-3441	103	19	and	and	CCONJ
ejpam-3441	103	20	b	b	X
ejpam-3441	103	21	=	=	SYM
ejpam-3441	103	22	(	(	PUNCT
ejpam-3441	103	23	µb	µb	PROPN
ejpam-3441	103	24	,	,	PUNCT
ejpam-3441	103	25	γb	γb	PROPN
ejpam-3441	103	26	)	)	PUNCT
ejpam-3441	103	27	be	be	VERB
ejpam-3441	103	28	two	two	NUM
ejpam-3441	103	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	103	30	fuzzy	fuzzy	ADJ
ejpam-3441	103	31	sets	set	NOUN
ejpam-3441	103	32	of	of	ADP
ejpam-3441	103	33	an	an	DET
ejpam-3441	103	34	la	la	ADJ
ejpam-3441	103	35	-	-	PUNCT
ejpam-3441	103	36	ring	ring	NOUN
ejpam-3441	103	37	r.	r.	NOUN
ejpam-3441	103	38	we	we	PRON
ejpam-3441	103	39	define	define	VERB
ejpam-3441	103	40	intuitionistic	intuitionistic	ADJ
ejpam-3441	103	41	fuzzy	fuzzy	ADJ
ejpam-3441	103	42	sets	set	NOUN
ejpam-3441	103	43	a	a	DET
ejpam-3441	103	44	∧βα	∧βα	PROPN
ejpam-3441	103	45	b	b	NOUN
ejpam-3441	103	46	,	,	PUNCT
ejpam-3441	103	47	a	a	DET
ejpam-3441	103	48	∨βα	∨βα	PROPN
ejpam-3441	103	49	b	b	PROPN
ejpam-3441	103	50	,	,	PUNCT
ejpam-3441	103	51	a	a	DET
ejpam-3441	103	52	◦	◦	NOUN
ejpam-3441	103	53	βα	βα	NOUN
ejpam-3441	103	54	b	b	NOUN
ejpam-3441	103	55	and	and	CCONJ
ejpam-3441	103	56	a−βα	a−βα	PROPN
ejpam-3441	103	57	b	b	NOUN
ejpam-3441	103	58	of	of	ADP
ejpam-3441	103	59	r	r	NOUN
ejpam-3441	103	60	as	as	SCONJ
ejpam-3441	103	61	follows	follow	VERB
ejpam-3441	103	62	:	:	PUNCT
ejpam-3441	103	63	(	(	PUNCT
ejpam-3441	103	64	µa	µa	ADP
ejpam-3441	103	65	∧βα	∧βα	ADJ
ejpam-3441	103	66	µb)(x	µb)(x	NOUN
ejpam-3441	103	67	)	)	PUNCT
ejpam-3441	104	1	=	=	PRON
ejpam-3441	104	2	{	{	PUNCT
ejpam-3441	104	3	(	(	PUNCT
ejpam-3441	104	4	µa	µa	PROPN
ejpam-3441	104	5	∧	∧	PROPN
ejpam-3441	104	6	µb)(x	µb)(x	PROPN
ejpam-3441	104	7	)	)	PUNCT
ejpam-3441	104	8	∧	∧	PROPN
ejpam-3441	104	9	β	β	NOUN
ejpam-3441	104	10	}	}	PUNCT
ejpam-3441	104	11	∨	∨	NUM
ejpam-3441	104	12	α	α	PROPN
ejpam-3441	104	13	and	and	CCONJ
ejpam-3441	104	14	(	(	PUNCT
ejpam-3441	104	15	γa	γa	PROPN
ejpam-3441	104	16	∨βα	∨βα	PROPN
ejpam-3441	104	17	γb)(x	γb)(x	PROPN
ejpam-3441	104	18	)	)	PUNCT
ejpam-3441	104	19	=	=	PRON
ejpam-3441	104	20	{	{	PUNCT
ejpam-3441	104	21	(	(	PUNCT
ejpam-3441	104	22	γa	γa	PROPN
ejpam-3441	104	23	∨	∨	NUM
ejpam-3441	104	24	γb)(x	γb)(x	PROPN
ejpam-3441	104	25	)	)	PUNCT
ejpam-3441	104	26	∨	∨	NUM
ejpam-3441	104	27	(	(	PUNCT
ejpam-3441	104	28	1−	1−	NUM
ejpam-3441	104	29	β	β	NOUN
ejpam-3441	104	30	)	)	PUNCT
ejpam-3441	104	31	}	}	PUNCT
ejpam-3441	104	32	∧	∧	PROPN
ejpam-3441	104	33	(	(	PUNCT
ejpam-3441	104	34	1−	1−	NUM
ejpam-3441	104	35	α	α	NOUN
ejpam-3441	104	36	)	)	PUNCT
ejpam-3441	104	37	.	.	PUNCT
ejpam-3441	105	1	(	(	PUNCT
ejpam-3441	105	2	µa	µa	ADP
ejpam-3441	105	3	∨βα	∨βα	ADJ
ejpam-3441	105	4	µb)(x	µb)(x	NOUN
ejpam-3441	105	5	)	)	PUNCT
ejpam-3441	105	6	=	=	PRON
ejpam-3441	105	7	{	{	PUNCT
ejpam-3441	105	8	(	(	PUNCT
ejpam-3441	105	9	µa	µa	ADP
ejpam-3441	105	10	∨	∨	NUM
ejpam-3441	105	11	µb)(x	µb)(x	NOUN
ejpam-3441	105	12	)	)	PUNCT
ejpam-3441	105	13	∧	∧	PROPN
ejpam-3441	105	14	β	β	NOUN
ejpam-3441	105	15	}	}	PUNCT
ejpam-3441	105	16	∨	∨	NUM
ejpam-3441	105	17	α	α	PROPN
ejpam-3441	105	18	and	and	CCONJ
ejpam-3441	105	19	(	(	PUNCT
ejpam-3441	105	20	γa	γa	PROPN
ejpam-3441	105	21	∧βα	∧βα	ADJ
ejpam-3441	105	22	γb)(x	γb)(x	PROPN
ejpam-3441	105	23	)	)	PUNCT
ejpam-3441	105	24	=	=	PRON
ejpam-3441	105	25	{	{	PUNCT
ejpam-3441	105	26	(	(	PUNCT
ejpam-3441	105	27	γa	γa	PROPN
ejpam-3441	105	28	∧	∧	PROPN
ejpam-3441	105	29	γb)(x	γb)(x	PROPN
ejpam-3441	105	30	)	)	PUNCT
ejpam-3441	105	31	∨	∨	NUM
ejpam-3441	105	32	(	(	PUNCT
ejpam-3441	105	33	1−	1−	NUM
ejpam-3441	105	34	β	β	NOUN
ejpam-3441	105	35	)	)	PUNCT
ejpam-3441	105	36	}	}	PUNCT
ejpam-3441	105	37	∧	∧	PROPN
ejpam-3441	105	38	(	(	PUNCT
ejpam-3441	105	39	1−	1−	NUM
ejpam-3441	105	40	α	α	NOUN
ejpam-3441	105	41	)	)	PUNCT
ejpam-3441	105	42	.	.	PUNCT
ejpam-3441	106	1	(	(	PUNCT
ejpam-3441	106	2	µa	µa	ADP
ejpam-3441	106	3	◦	◦	NOUN
ejpam-3441	106	4	βα	βα	NOUN
ejpam-3441	106	5	µb)(x	µb)(x	NOUN
ejpam-3441	106	6	)	)	PUNCT
ejpam-3441	106	7	=	=	PRON
ejpam-3441	106	8	{	{	PUNCT
ejpam-3441	106	9	(	(	PUNCT
ejpam-3441	106	10	µa	µa	ADP
ejpam-3441	106	11	◦	◦	NOUN
ejpam-3441	106	12	µb)(x	µb)(x	NOUN
ejpam-3441	106	13	)	)	PUNCT
ejpam-3441	107	1	∧	∧	PROPN
ejpam-3441	107	2	β	β	NOUN
ejpam-3441	107	3	}	}	PUNCT
ejpam-3441	107	4	∨	∨	NUM
ejpam-3441	107	5	α	α	PROPN
ejpam-3441	107	6	and	and	CCONJ
ejpam-3441	107	7	(	(	PUNCT
ejpam-3441	107	8	γa	γa	PROPN
ejpam-3441	107	9	◦	◦	PROPN
ejpam-3441	107	10	βα	βα	ADJ
ejpam-3441	107	11	γb)(x	γb)(x	PROPN
ejpam-3441	107	12	)	)	PUNCT
ejpam-3441	108	1	=	=	PRON
ejpam-3441	108	2	{	{	PUNCT
ejpam-3441	108	3	(	(	PUNCT
ejpam-3441	108	4	γa	γa	AUX
ejpam-3441	108	5	◦	◦	VERB
ejpam-3441	108	6	γb)(x	γb)(x	PROPN
ejpam-3441	108	7	)	)	PUNCT
ejpam-3441	108	8	∨	∨	NUM
ejpam-3441	108	9	(	(	PUNCT
ejpam-3441	108	10	1−	1−	NUM
ejpam-3441	108	11	β	β	NOUN
ejpam-3441	108	12	)	)	PUNCT
ejpam-3441	108	13	}	}	PUNCT
ejpam-3441	108	14	∧	∧	PROPN
ejpam-3441	108	15	(	(	PUNCT
ejpam-3441	108	16	1−	1−	NUM
ejpam-3441	108	17	α	α	NOUN
ejpam-3441	108	18	)	)	PUNCT
ejpam-3441	108	19	.	.	PUNCT
ejpam-3441	109	1	(	(	PUNCT
ejpam-3441	109	2	µa	µa	NOUN
ejpam-3441	109	3	−βα	−βα	PROPN
ejpam-3441	109	4	µb)(x	µb)(x	PROPN
ejpam-3441	109	5	)	)	PUNCT
ejpam-3441	110	1	=	=	PRON
ejpam-3441	110	2	{	{	PUNCT
ejpam-3441	110	3	(	(	PUNCT
ejpam-3441	110	4	µa	µa	NOUN
ejpam-3441	110	5	−	−	PROPN
ejpam-3441	110	6	µb)(x	µb)(x	NOUN
ejpam-3441	110	7	)	)	PUNCT
ejpam-3441	111	1	∧	∧	PROPN
ejpam-3441	111	2	β	β	NOUN
ejpam-3441	111	3	}	}	PUNCT
ejpam-3441	111	4	∨	∨	NUM
ejpam-3441	111	5	α	α	PROPN
ejpam-3441	111	6	k.	k.	PROPN
ejpam-3441	111	7	nasreen	nasreen	PROPN
ejpam-3441	111	8	et	et	PROPN
ejpam-3441	111	9	al	al	PROPN
ejpam-3441	111	10	.	.	PUNCT
ejpam-3441	111	11	/	/	SYM
ejpam-3441	111	12	eur	eur	PROPN
ejpam-3441	111	13	.	.	PUNCT
ejpam-3441	112	1	j.	j.	PROPN
ejpam-3441	112	2	pure	pure	PROPN
ejpam-3441	112	3	appl	appl	PROPN
ejpam-3441	112	4	.	.	PROPN
ejpam-3441	112	5	math	math	PROPN
ejpam-3441	112	6	,	,	PUNCT
ejpam-3441	112	7	12	12	NUM
ejpam-3441	112	8	(	(	PUNCT
ejpam-3441	112	9	3	3	NUM
ejpam-3441	112	10	)	)	PUNCT
ejpam-3441	112	11	(	(	PUNCT
ejpam-3441	112	12	2019	2019	NUM
ejpam-3441	112	13	)	)	PUNCT
ejpam-3441	112	14	,	,	PUNCT
ejpam-3441	112	15	906	906	NUM
ejpam-3441	112	16	-	-	SYM
ejpam-3441	112	17	943	943	NUM
ejpam-3441	112	18	911	911	NUM
ejpam-3441	112	19	and	and	CCONJ
ejpam-3441	112	20	(	(	PUNCT
ejpam-3441	112	21	γa	γa	PROPN
ejpam-3441	112	22	−βα	−βα	PROPN
ejpam-3441	112	23	γb)(x	γb)(x	PROPN
ejpam-3441	112	24	)	)	PUNCT
ejpam-3441	113	1	=	=	PRON
ejpam-3441	113	2	{	{	PUNCT
ejpam-3441	113	3	(	(	PUNCT
ejpam-3441	113	4	γa	γa	NOUN
ejpam-3441	113	5	−	−	PROPN
ejpam-3441	113	6	γb)(x	γb)(x	PROPN
ejpam-3441	113	7	)	)	PUNCT
ejpam-3441	113	8	∨	∨	NUM
ejpam-3441	113	9	(	(	PUNCT
ejpam-3441	113	10	1−	1−	NUM
ejpam-3441	113	11	β	β	NOUN
ejpam-3441	113	12	)	)	PUNCT
ejpam-3441	113	13	}	}	PUNCT
ejpam-3441	113	14	∧	∧	PROPN
ejpam-3441	113	15	(	(	PUNCT
ejpam-3441	113	16	1−	1−	NUM
ejpam-3441	113	17	α	α	NOUN
ejpam-3441	113	18	)	)	PUNCT
ejpam-3441	113	19	,	,	PUNCT
ejpam-3441	113	20	for	for	ADP
ejpam-3441	113	21	all	all	DET
ejpam-3441	113	22	x	x	PROPN
ejpam-3441	113	23	∈	∈	PROPN
ejpam-3441	113	24	r.	r.	NOUN
ejpam-3441	113	25	now	now	ADV
ejpam-3441	113	26	we	we	PRON
ejpam-3441	113	27	are	be	AUX
ejpam-3441	113	28	giving	give	VERB
ejpam-3441	113	29	the	the	DET
ejpam-3441	113	30	central	central	ADJ
ejpam-3441	113	31	properties	property	NOUN
ejpam-3441	113	32	of	of	ADP
ejpam-3441	113	33	such	such	ADJ
ejpam-3441	113	34	ideals	ideal	NOUN
ejpam-3441	113	35	of	of	ADP
ejpam-3441	113	36	an	an	DET
ejpam-3441	113	37	la	la	ADJ
ejpam-3441	113	38	-	-	PUNCT
ejpam-3441	113	39	ring	ring	NOUN
ejpam-3441	113	40	r	r	NOUN
ejpam-3441	113	41	,	,	PUNCT
ejpam-3441	113	42	which	which	PRON
ejpam-3441	113	43	will	will	AUX
ejpam-3441	113	44	be	be	AUX
ejpam-3441	113	45	very	very	ADV
ejpam-3441	113	46	helpful	helpful	ADJ
ejpam-3441	113	47	for	for	ADP
ejpam-3441	113	48	further	further	ADJ
ejpam-3441	113	49	sections	section	NOUN
ejpam-3441	113	50	.	.	PUNCT
ejpam-3441	114	1	lemma	lemma	PROPN
ejpam-3441	114	2	1	1	X
ejpam-3441	114	3	.	.	PUNCT
ejpam-3441	115	1	let	let	VERB
ejpam-3441	115	2	a	a	PRON
ejpam-3441	115	3	and	and	CCONJ
ejpam-3441	115	4	b	b	NOUN
ejpam-3441	115	5	be	be	AUX
ejpam-3441	115	6	two	two	NUM
ejpam-3441	115	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	115	8	fuzzy	fuzzy	ADJ
ejpam-3441	115	9	sets	set	NOUN
ejpam-3441	115	10	of	of	ADP
ejpam-3441	115	11	an	an	DET
ejpam-3441	115	12	la	la	ADJ
ejpam-3441	115	13	-	-	PUNCT
ejpam-3441	115	14	ring	ring	NOUN
ejpam-3441	115	15	r.	r.	PROPN
ejpam-3441	115	16	then	then	ADV
ejpam-3441	115	17	the	the	DET
ejpam-3441	115	18	following	follow	VERB
ejpam-3441	115	19	properties	property	NOUN
ejpam-3441	115	20	holds	hold	VERB
ejpam-3441	115	21	.	.	PUNCT
ejpam-3441	116	1	(	(	PUNCT
ejpam-3441	116	2	1	1	X
ejpam-3441	116	3	)	)	PUNCT
ejpam-3441	116	4	a	a	DET
ejpam-3441	116	5	∧βα	∧βα	PROPN
ejpam-3441	116	6	b	b	NOUN
ejpam-3441	116	7	=	=	PRON
ejpam-3441	116	8	aβα	aβα	PROPN
ejpam-3441	116	9	∧bβ	∧bβ	PROPN
ejpam-3441	116	10	α	α	PRON
ejpam-3441	116	11	.	.	PUNCT
ejpam-3441	117	1	(	(	PUNCT
ejpam-3441	117	2	2	2	X
ejpam-3441	117	3	)	)	PUNCT
ejpam-3441	117	4	a	a	DET
ejpam-3441	117	5	∨βα	∨βα	PROPN
ejpam-3441	117	6	b	b	PROPN
ejpam-3441	117	7	=	=	PUNCT
ejpam-3441	117	8	aβα	aβα	NOUN
ejpam-3441	117	9	∨bβ	∨bβ	PUNCT
ejpam-3441	117	10	α	α	NOUN
ejpam-3441	117	11	.	.	PUNCT
ejpam-3441	118	1	(	(	PUNCT
ejpam-3441	118	2	3	3	X
ejpam-3441	118	3	)	)	PUNCT
ejpam-3441	118	4	a	a	DET
ejpam-3441	118	5	◦	◦	NOUN
ejpam-3441	118	6	βα	βα	X
ejpam-3441	118	7	b	b	PROPN
ejpam-3441	118	8	≥	≥	NOUN
ejpam-3441	118	9	aβα	aβα	PROPN
ejpam-3441	118	10	◦	◦	NOUN
ejpam-3441	118	11	bβ	bβ	NOUN
ejpam-3441	118	12	α	α	NOUN
ejpam-3441	118	13	.	.	PUNCT
ejpam-3441	119	1	if	if	SCONJ
ejpam-3441	119	2	every	every	DET
ejpam-3441	119	3	element	element	NOUN
ejpam-3441	119	4	x	x	PUNCT
ejpam-3441	119	5	of	of	ADP
ejpam-3441	119	6	r	r	NOUN
ejpam-3441	119	7	is	be	AUX
ejpam-3441	119	8	expressible	expressible	ADJ
ejpam-3441	119	9	as	as	ADP
ejpam-3441	119	10	x	x	PROPN
ejpam-3441	119	11	=	=	PROPN
ejpam-3441	119	12	n∑	n∑	PROPN
ejpam-3441	119	13	i=1	i=1	PROPN
ejpam-3441	119	14	aibi	aibi	NOUN
ejpam-3441	119	15	,	,	PUNCT
ejpam-3441	119	16	then	then	ADV
ejpam-3441	119	17	a	a	DET
ejpam-3441	119	18	◦	◦	NOUN
ejpam-3441	119	19	βα	βα	NOUN
ejpam-3441	119	20	b	b	NOUN
ejpam-3441	119	21	=	=	PROPN
ejpam-3441	119	22	aβα	aβα	NOUN
ejpam-3441	119	23	◦	◦	NOUN
ejpam-3441	119	24	bβ	bβ	NOUN
ejpam-3441	119	25	α	α	NOUN
ejpam-3441	119	26	.	.	PUNCT
ejpam-3441	120	1	if	if	SCONJ
ejpam-3441	120	2	χa	χa	NOUN
ejpam-3441	120	3	=	=	SYM
ejpam-3441	120	4	(	(	PUNCT
ejpam-3441	120	5	µχa	µχa	X
ejpam-3441	120	6	,	,	PUNCT
ejpam-3441	120	7	γχa	γχa	NOUN
ejpam-3441	120	8	)	)	PUNCT
ejpam-3441	120	9	is	be	AUX
ejpam-3441	120	10	an	an	DET
ejpam-3441	120	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	120	12	characteristic	characteristic	ADJ
ejpam-3441	120	13	function	function	NOUN
ejpam-3441	120	14	of	of	ADP
ejpam-3441	120	15	a	a	PRON
ejpam-3441	120	16	,	,	PUNCT
ejpam-3441	120	17	then	then	ADV
ejpam-3441	120	18	(	(	PUNCT
ejpam-3441	120	19	χa)βα	χa)βα	PROPN
ejpam-3441	120	20	is	be	AUX
ejpam-3441	120	21	defined	define	VERB
ejpam-3441	120	22	as	as	ADP
ejpam-3441	120	23	(	(	PUNCT
ejpam-3441	120	24	µχa)βα(x	µχa)βα(x	PROPN
ejpam-3441	120	25	)	)	PUNCT
ejpam-3441	120	26	=	=	PRON
ejpam-3441	120	27	{	{	PUNCT
ejpam-3441	120	28	β	β	X
ejpam-3441	120	29	if	if	SCONJ
ejpam-3441	120	30	x	x	SYM
ejpam-3441	120	31	∈	∈	PROPN
ejpam-3441	120	32	a	a	DET
ejpam-3441	120	33	α	α	NOUN
ejpam-3441	120	34	if	if	SCONJ
ejpam-3441	120	35	x	x	PROPN
ejpam-3441	120	36	/∈	/∈	NOUN
ejpam-3441	121	1	a	a	PRON
ejpam-3441	121	2	and	and	CCONJ
ejpam-3441	121	3	(	(	PUNCT
ejpam-3441	121	4	γχa)βα(x	γχa)βα(x	X
ejpam-3441	121	5	)	)	PUNCT
ejpam-3441	121	6	=	=	SYM
ejpam-3441	121	7	{	{	PUNCT
ejpam-3441	121	8	α	α	NOUN
ejpam-3441	121	9	if	if	SCONJ
ejpam-3441	121	10	x	x	SYM
ejpam-3441	121	11	∈	∈	PROPN
ejpam-3441	121	12	a	a	DET
ejpam-3441	121	13	β	β	NOUN
ejpam-3441	121	14	if	if	SCONJ
ejpam-3441	121	15	x	x	PROPN
ejpam-3441	121	16	/∈	/∈	PUNCT
ejpam-3441	121	17	a	a	DET
ejpam-3441	121	18	lemma	lemma	PROPN
ejpam-3441	121	19	2	2	NUM
ejpam-3441	121	20	.	.	PUNCT
ejpam-3441	122	1	let	let	VERB
ejpam-3441	122	2	r	r	PRON
ejpam-3441	122	3	be	be	AUX
ejpam-3441	122	4	an	an	DET
ejpam-3441	122	5	la	la	NOUN
ejpam-3441	122	6	-	-	PUNCT
ejpam-3441	122	7	ring	ring	NOUN
ejpam-3441	122	8	.	.	PUNCT
ejpam-3441	123	1	then	then	ADV
ejpam-3441	123	2	the	the	DET
ejpam-3441	123	3	following	follow	VERB
ejpam-3441	123	4	properties	property	NOUN
ejpam-3441	123	5	hold	hold	VERB
ejpam-3441	123	6	.	.	PUNCT
ejpam-3441	124	1	(	(	PUNCT
ejpam-3441	124	2	1	1	X
ejpam-3441	124	3	)	)	PUNCT
ejpam-3441	124	4	(	(	PUNCT
ejpam-3441	124	5	a	a	DET
ejpam-3441	124	6	◦	◦	NOUN
ejpam-3441	124	7	βα	βα	NOUN
ejpam-3441	124	8	b	b	NOUN
ejpam-3441	124	9	)	)	PUNCT
ejpam-3441	124	10	◦	◦	NOUN
ejpam-3441	124	11	βα	βα	X
ejpam-3441	124	12	c	c	NOUN
ejpam-3441	124	13	=	=	SYM
ejpam-3441	124	14	(	(	PUNCT
ejpam-3441	124	15	c	c	NOUN
ejpam-3441	124	16	◦	◦	PROPN
ejpam-3441	124	17	βα	βα	ADP
ejpam-3441	124	18	b	b	NOUN
ejpam-3441	124	19	)	)	PUNCT
ejpam-3441	124	20	◦	◦	NOUN
ejpam-3441	124	21	βα	βα	NOUN
ejpam-3441	124	22	a	a	PRON
ejpam-3441	124	23	,	,	PUNCT
ejpam-3441	124	24	(	(	PUNCT
ejpam-3441	124	25	2	2	NUM
ejpam-3441	124	26	)	)	PUNCT
ejpam-3441	124	27	(	(	PUNCT
ejpam-3441	124	28	a	a	DET
ejpam-3441	124	29	◦	◦	NOUN
ejpam-3441	124	30	βαb)	βαb)	NOUN
ejpam-3441	124	31	◦	◦	NOUN
ejpam-3441	124	32	βα	βα	X
ejpam-3441	124	33	(	(	PUNCT
ejpam-3441	124	34	c	c	NOUN
ejpam-3441	124	35	◦	◦	NOUN
ejpam-3441	124	36	βαd	βαd	NOUN
ejpam-3441	124	37	)	)	PUNCT
ejpam-3441	124	38	=	=	PUNCT
ejpam-3441	125	1	(	(	PUNCT
ejpam-3441	125	2	a	a	DET
ejpam-3441	125	3	◦	◦	NOUN
ejpam-3441	125	4	βαc)	βαc)	NOUN
ejpam-3441	125	5	◦	◦	NOUN
ejpam-3441	125	6	βα	βα	X
ejpam-3441	125	7	(	(	PUNCT
ejpam-3441	125	8	b	b	X
ejpam-3441	125	9	◦	◦	NOUN
ejpam-3441	125	10	βαd	βαd	NOUN
ejpam-3441	125	11	)	)	PUNCT
ejpam-3441	125	12	for	for	ADP
ejpam-3441	125	13	all	all	DET
ejpam-3441	125	14	intuitionistic	intuitionistic	ADJ
ejpam-3441	125	15	fuzzy	fuzzy	ADJ
ejpam-3441	125	16	sets	set	NOUN
ejpam-3441	125	17	a	a	DET
ejpam-3441	125	18	,	,	PUNCT
ejpam-3441	125	19	b	b	NOUN
ejpam-3441	125	20	,	,	PUNCT
ejpam-3441	125	21	c	c	PROPN
ejpam-3441	125	22	and	and	CCONJ
ejpam-3441	125	23	d	d	PROPN
ejpam-3441	125	24	of	of	ADP
ejpam-3441	125	25	r.	r.	PROPN
ejpam-3441	125	26	proof	proof	NOUN
ejpam-3441	125	27	.	.	PUNCT
ejpam-3441	126	1	let	let	VERB
ejpam-3441	126	2	a	a	PRON
ejpam-3441	126	3	=	=	SYM
ejpam-3441	126	4	(	(	PUNCT
ejpam-3441	126	5	µa	µa	PROPN
ejpam-3441	126	6	,	,	PUNCT
ejpam-3441	126	7	γa	γa	PROPN
ejpam-3441	126	8	)	)	PUNCT
ejpam-3441	126	9	,	,	PUNCT
ejpam-3441	127	1	b	b	X
ejpam-3441	127	2	=	=	PRON
ejpam-3441	127	3	(	(	PUNCT
ejpam-3441	127	4	µb	µb	PROPN
ejpam-3441	127	5	,	,	PUNCT
ejpam-3441	127	6	γb	γb	PROPN
ejpam-3441	127	7	)	)	PUNCT
ejpam-3441	127	8	and	and	CCONJ
ejpam-3441	127	9	c	c	X
ejpam-3441	127	10	=	=	SYM
ejpam-3441	127	11	(	(	PUNCT
ejpam-3441	127	12	µc	µc	INTJ
ejpam-3441	127	13	,	,	PUNCT
ejpam-3441	127	14	γc	γc	PROPN
ejpam-3441	127	15	)	)	PUNCT
ejpam-3441	127	16	be	be	AUX
ejpam-3441	127	17	intuitionistic	intuitionistic	ADJ
ejpam-3441	127	18	fuzzy	fuzzy	ADJ
ejpam-3441	127	19	sets	set	NOUN
ejpam-3441	127	20	of	of	ADP
ejpam-3441	127	21	an	an	DET
ejpam-3441	127	22	la	la	ADJ
ejpam-3441	127	23	-	-	PUNCT
ejpam-3441	127	24	ring	ring	NOUN
ejpam-3441	127	25	r.	r.	NOUN
ejpam-3441	127	26	we	we	PRON
ejpam-3441	127	27	have	have	VERB
ejpam-3441	127	28	to	to	PART
ejpam-3441	127	29	show	show	VERB
ejpam-3441	127	30	that	that	SCONJ
ejpam-3441	127	31	(	(	PUNCT
ejpam-3441	127	32	a	a	DET
ejpam-3441	127	33	◦	◦	NOUN
ejpam-3441	127	34	βα	βα	NOUN
ejpam-3441	127	35	b	b	NOUN
ejpam-3441	127	36	)	)	PUNCT
ejpam-3441	127	37	◦	◦	NOUN
ejpam-3441	127	38	βα	βα	X
ejpam-3441	127	39	c	c	NOUN
ejpam-3441	127	40	=	=	SYM
ejpam-3441	127	41	(	(	PUNCT
ejpam-3441	127	42	c	c	NOUN
ejpam-3441	127	43	◦	◦	PROPN
ejpam-3441	127	44	βα	βα	ADP
ejpam-3441	127	45	b	b	NOUN
ejpam-3441	127	46	)	)	PUNCT
ejpam-3441	127	47	◦	◦	NOUN
ejpam-3441	127	48	βα	βα	NOUN
ejpam-3441	127	49	a.	a.	NOUN
ejpam-3441	127	50	now	now	ADV
ejpam-3441	127	51	(	(	PUNCT
ejpam-3441	127	52	(	(	PUNCT
ejpam-3441	127	53	a	a	DET
ejpam-3441	127	54	◦	◦	NOUN
ejpam-3441	127	55	βα	βα	NOUN
ejpam-3441	127	56	b	b	NOUN
ejpam-3441	127	57	)	)	PUNCT
ejpam-3441	127	58	◦	◦	NOUN
ejpam-3441	127	59	βα	βα	NOUN
ejpam-3441	127	60	c)(x	c)(x	PROPN
ejpam-3441	127	61	)	)	PUNCT
ejpam-3441	128	1	=	=	PRON
ejpam-3441	128	2	{	{	PUNCT
ejpam-3441	128	3	(	(	PUNCT
ejpam-3441	128	4	(	(	PUNCT
ejpam-3441	128	5	a	a	DET
ejpam-3441	128	6	◦	◦	NOUN
ejpam-3441	128	7	b	b	NOUN
ejpam-3441	128	8	)	)	PUNCT
ejpam-3441	128	9	◦	◦	PROPN
ejpam-3441	128	10	c)(x	c)(x	PROPN
ejpam-3441	128	11	)	)	PUNCT
ejpam-3441	129	1	∧	∧	PROPN
ejpam-3441	129	2	β	β	NOUN
ejpam-3441	129	3	}	}	PUNCT
ejpam-3441	129	4	∨	∨	NUM
ejpam-3441	129	5	α	α	NOUN
ejpam-3441	129	6	=	=	SYM
ejpam-3441	129	7	{	{	PUNCT
ejpam-3441	129	8	(	(	PUNCT
ejpam-3441	129	9	(	(	PUNCT
ejpam-3441	129	10	c	c	PROPN
ejpam-3441	129	11	◦	◦	NOUN
ejpam-3441	129	12	b	b	NOUN
ejpam-3441	129	13	)	)	PUNCT
ejpam-3441	129	14	◦	◦	NOUN
ejpam-3441	129	15	a)(x	a)(x	NOUN
ejpam-3441	129	16	)	)	PUNCT
ejpam-3441	130	1	∧	∧	PROPN
ejpam-3441	130	2	β	β	NOUN
ejpam-3441	130	3	}	}	PUNCT
ejpam-3441	130	4	∨	∨	NUM
ejpam-3441	130	5	α	α	NOUN
ejpam-3441	130	6	=	=	SYM
ejpam-3441	130	7	(	(	PUNCT
ejpam-3441	130	8	(	(	PUNCT
ejpam-3441	130	9	c	c	NOUN
ejpam-3441	130	10	◦	◦	NOUN
ejpam-3441	130	11	βα	βα	ADP
ejpam-3441	130	12	b	b	NOUN
ejpam-3441	130	13	)	)	PUNCT
ejpam-3441	130	14	◦	◦	NOUN
ejpam-3441	130	15	βα	βα	NOUN
ejpam-3441	130	16	a)(x	a)(x	NOUN
ejpam-3441	130	17	)	)	PUNCT
ejpam-3441	130	18	.	.	PUNCT
ejpam-3441	131	1	in	in	ADP
ejpam-3441	131	2	same	same	ADJ
ejpam-3441	131	3	lines	line	NOUN
ejpam-3441	131	4	,	,	PUNCT
ejpam-3441	131	5	we	we	PRON
ejpam-3441	131	6	can	can	AUX
ejpam-3441	131	7	prove	prove	VERB
ejpam-3441	131	8	(	(	PUNCT
ejpam-3441	131	9	2	2	NUM
ejpam-3441	131	10	)	)	PUNCT
ejpam-3441	131	11	.	.	PUNCT
ejpam-3441	132	1	proposition	proposition	NOUN
ejpam-3441	132	2	1	1	NUM
ejpam-3441	132	3	.	.	PUNCT
ejpam-3441	133	1	let	let	VERB
ejpam-3441	133	2	r	r	PRON
ejpam-3441	133	3	be	be	AUX
ejpam-3441	133	4	an	an	DET
ejpam-3441	133	5	la	la	NOUN
ejpam-3441	133	6	-	-	NOUN
ejpam-3441	133	7	ring	ring	NOUN
ejpam-3441	133	8	with	with	ADP
ejpam-3441	133	9	left	left	ADJ
ejpam-3441	133	10	identity	identity	NOUN
ejpam-3441	133	11	e.	e.	PROPN
ejpam-3441	133	12	then	then	ADV
ejpam-3441	133	13	the	the	DET
ejpam-3441	133	14	following	follow	VERB
ejpam-3441	133	15	assertions	assertion	NOUN
ejpam-3441	133	16	hold	hold	VERB
ejpam-3441	133	17	.	.	PUNCT
ejpam-3441	134	1	(	(	PUNCT
ejpam-3441	134	2	1	1	X
ejpam-3441	134	3	)	)	PUNCT
ejpam-3441	134	4	a	a	DET
ejpam-3441	134	5	◦	◦	NOUN
ejpam-3441	134	6	βα	βα	X
ejpam-3441	134	7	(	(	PUNCT
ejpam-3441	134	8	b	b	X
ejpam-3441	134	9	◦	◦	NOUN
ejpam-3441	134	10	βα	βα	NOUN
ejpam-3441	134	11	c	c	NOUN
ejpam-3441	134	12	)	)	PUNCT
ejpam-3441	135	1	=	=	SYM
ejpam-3441	135	2	b	b	X
ejpam-3441	135	3	◦	◦	NOUN
ejpam-3441	135	4	βα	βα	X
ejpam-3441	135	5	(	(	PUNCT
ejpam-3441	135	6	a	a	DET
ejpam-3441	135	7	◦	◦	NOUN
ejpam-3441	135	8	βα	βα	NOUN
ejpam-3441	135	9	c	c	NOUN
ejpam-3441	135	10	)	)	PUNCT
ejpam-3441	135	11	,	,	PUNCT
ejpam-3441	135	12	(	(	PUNCT
ejpam-3441	135	13	2	2	X
ejpam-3441	135	14	)	)	PUNCT
ejpam-3441	135	15	(	(	PUNCT
ejpam-3441	135	16	a	a	DET
ejpam-3441	135	17	◦	◦	NOUN
ejpam-3441	135	18	βα	βα	NOUN
ejpam-3441	135	19	b	b	NOUN
ejpam-3441	135	20	)	)	PUNCT
ejpam-3441	135	21	◦	◦	NOUN
ejpam-3441	135	22	βα	βα	X
ejpam-3441	135	23	(	(	PUNCT
ejpam-3441	135	24	c	c	NOUN
ejpam-3441	135	25	◦	◦	NOUN
ejpam-3441	135	26	βα	βα	NOUN
ejpam-3441	135	27	d	d	NOUN
ejpam-3441	135	28	)	)	PUNCT
ejpam-3441	135	29	=	=	SYM
ejpam-3441	136	1	(	(	PUNCT
ejpam-3441	136	2	d	d	PART
ejpam-3441	136	3	◦	◦	NOUN
ejpam-3441	136	4	βα	βα	ADP
ejpam-3441	136	5	b	b	NOUN
ejpam-3441	136	6	)	)	PUNCT
ejpam-3441	136	7	◦	◦	NOUN
ejpam-3441	136	8	βα	βα	X
ejpam-3441	136	9	(	(	PUNCT
ejpam-3441	136	10	c	c	NOUN
ejpam-3441	136	11	◦	◦	PROPN
ejpam-3441	136	12	βα	βα	NOUN
ejpam-3441	136	13	a	a	NOUN
ejpam-3441	136	14	)	)	PUNCT
ejpam-3441	136	15	,	,	PUNCT
ejpam-3441	136	16	(	(	PUNCT
ejpam-3441	136	17	3	3	X
ejpam-3441	136	18	)	)	PUNCT
ejpam-3441	136	19	(	(	PUNCT
ejpam-3441	136	20	a	a	DET
ejpam-3441	136	21	◦	◦	NOUN
ejpam-3441	136	22	βαb)	βαb)	NOUN
ejpam-3441	136	23	◦	◦	NOUN
ejpam-3441	136	24	βα	βα	X
ejpam-3441	136	25	(	(	PUNCT
ejpam-3441	136	26	c	c	NOUN
ejpam-3441	136	27	◦	◦	NOUN
ejpam-3441	136	28	βαd	βαd	NOUN
ejpam-3441	136	29	)	)	PUNCT
ejpam-3441	136	30	=	=	PUNCT
ejpam-3441	137	1	(	(	PUNCT
ejpam-3441	137	2	d	d	PART
ejpam-3441	137	3	◦	◦	NOUN
ejpam-3441	137	4	βαc)	βαc)	SYM
ejpam-3441	137	5	◦	◦	NOUN
ejpam-3441	137	6	βα	βα	X
ejpam-3441	137	7	(	(	PUNCT
ejpam-3441	137	8	b	b	X
ejpam-3441	137	9	◦	◦	NOUN
ejpam-3441	137	10	βαa	βαa	NOUN
ejpam-3441	137	11	)	)	PUNCT
ejpam-3441	137	12	for	for	ADP
ejpam-3441	137	13	all	all	DET
ejpam-3441	137	14	intuitionistic	intuitionistic	ADJ
ejpam-3441	137	15	fuzzy	fuzzy	ADJ
ejpam-3441	137	16	sets	set	NOUN
ejpam-3441	137	17	a	a	DET
ejpam-3441	137	18	,	,	PUNCT
ejpam-3441	137	19	b	b	NOUN
ejpam-3441	137	20	,	,	PUNCT
ejpam-3441	137	21	c	c	PROPN
ejpam-3441	137	22	and	and	CCONJ
ejpam-3441	137	23	d	d	PROPN
ejpam-3441	137	24	of	of	ADP
ejpam-3441	137	25	r.	r.	PROPN
ejpam-3441	137	26	proof	proof	NOUN
ejpam-3441	137	27	.	.	PUNCT
ejpam-3441	138	1	same	same	ADJ
ejpam-3441	138	2	as	as	ADP
ejpam-3441	138	3	lemma	lemma	PROPN
ejpam-3441	138	4	2	2	NUM
ejpam-3441	138	5	.	.	PUNCT
ejpam-3441	138	6	k.	k.	PROPN
ejpam-3441	138	7	nasreen	nasreen	PROPN
ejpam-3441	138	8	et	et	PROPN
ejpam-3441	138	9	al	al	PROPN
ejpam-3441	138	10	.	.	PUNCT
ejpam-3441	138	11	/	/	SYM
ejpam-3441	138	12	eur	eur	PROPN
ejpam-3441	138	13	.	.	PUNCT
ejpam-3441	139	1	j.	j.	PROPN
ejpam-3441	139	2	pure	pure	PROPN
ejpam-3441	139	3	appl	appl	PROPN
ejpam-3441	139	4	.	.	PROPN
ejpam-3441	139	5	math	math	PROPN
ejpam-3441	139	6	,	,	PUNCT
ejpam-3441	139	7	12	12	NUM
ejpam-3441	139	8	(	(	PUNCT
ejpam-3441	139	9	3	3	NUM
ejpam-3441	139	10	)	)	PUNCT
ejpam-3441	139	11	(	(	PUNCT
ejpam-3441	139	12	2019	2019	NUM
ejpam-3441	139	13	)	)	PUNCT
ejpam-3441	139	14	,	,	PUNCT
ejpam-3441	139	15	906	906	NUM
ejpam-3441	139	16	-	-	SYM
ejpam-3441	139	17	943	943	NUM
ejpam-3441	139	18	912	912	NUM
ejpam-3441	139	19	theorem	theorem	NOUN
ejpam-3441	139	20	1	1	NUM
ejpam-3441	139	21	.	.	PUNCT
ejpam-3441	140	1	let	let	VERB
ejpam-3441	140	2	a	a	PRON
ejpam-3441	140	3	and	and	CCONJ
ejpam-3441	140	4	b	b	NOUN
ejpam-3441	140	5	be	be	AUX
ejpam-3441	140	6	two	two	NUM
ejpam-3441	140	7	non	non	ADJ
ejpam-3441	140	8	-	-	ADJ
ejpam-3441	140	9	empty	empty	ADJ
ejpam-3441	140	10	subsets	subset	NOUN
ejpam-3441	140	11	of	of	ADP
ejpam-3441	140	12	an	an	DET
ejpam-3441	140	13	la	la	ADJ
ejpam-3441	140	14	-	-	PUNCT
ejpam-3441	140	15	ring	ring	NOUN
ejpam-3441	140	16	r.	r.	PROPN
ejpam-3441	140	17	then	then	ADV
ejpam-3441	140	18	the	the	DET
ejpam-3441	140	19	following	follow	VERB
ejpam-3441	140	20	conditions	condition	NOUN
ejpam-3441	140	21	hold	hold	VERB
ejpam-3441	140	22	.	.	PUNCT
ejpam-3441	141	1	(	(	PUNCT
ejpam-3441	141	2	1	1	X
ejpam-3441	141	3	)	)	PUNCT
ejpam-3441	141	4	χa	χa	NOUN
ejpam-3441	141	5	◦	◦	NOUN
ejpam-3441	141	6	βα	βα	X
ejpam-3441	141	7	χb	χb	NOUN
ejpam-3441	141	8	=	=	PUNCT
ejpam-3441	141	9	(	(	PUNCT
ejpam-3441	141	10	χab)βα	χab)βα	NOUN
ejpam-3441	141	11	.	.	PUNCT
ejpam-3441	142	1	(	(	PUNCT
ejpam-3441	142	2	2	2	X
ejpam-3441	142	3	)	)	PUNCT
ejpam-3441	142	4	χa	χa	ADP
ejpam-3441	142	5	∨βα	∨βα	PROPN
ejpam-3441	142	6	χb	χb	PROPN
ejpam-3441	142	7	=	=	PUNCT
ejpam-3441	142	8	(	(	PUNCT
ejpam-3441	142	9	χa∪b)βα	χa∪b)βα	NOUN
ejpam-3441	142	10	.	.	PUNCT
ejpam-3441	143	1	(	(	PUNCT
ejpam-3441	143	2	3	3	X
ejpam-3441	143	3	)	)	PUNCT
ejpam-3441	143	4	χa	χa	VERB
ejpam-3441	144	1	∧βα	∧βα	NOUN
ejpam-3441	144	2	χb	χb	PROPN
ejpam-3441	144	3	=	=	PUNCT
ejpam-3441	144	4	(	(	PUNCT
ejpam-3441	144	5	χa∩b)βα	χa∩b)βα	PROPN
ejpam-3441	144	6	.	.	PUNCT
ejpam-3441	145	1	proof	proof	NOUN
ejpam-3441	145	2	.	.	PUNCT
ejpam-3441	146	1	straight	straight	ADV
ejpam-3441	146	2	forward	forward	ADV
ejpam-3441	146	3	.	.	PUNCT
ejpam-3441	147	1	theorem	theorem	NOUN
ejpam-3441	147	2	2	2	NUM
ejpam-3441	147	3	.	.	PUNCT
ejpam-3441	148	1	let	let	VERB
ejpam-3441	148	2	a	a	PRON
ejpam-3441	148	3	be	be	AUX
ejpam-3441	148	4	a	a	DET
ejpam-3441	148	5	non	non	ADJ
ejpam-3441	148	6	-	-	ADJ
ejpam-3441	148	7	empty	empty	ADJ
ejpam-3441	148	8	subset	subset	NOUN
ejpam-3441	148	9	of	of	ADP
ejpam-3441	148	10	an	an	DET
ejpam-3441	148	11	la	la	ADJ
ejpam-3441	148	12	-	-	PUNCT
ejpam-3441	148	13	ring	ring	NOUN
ejpam-3441	148	14	r.	r.	PROPN
ejpam-3441	148	15	then	then	ADV
ejpam-3441	148	16	the	the	DET
ejpam-3441	148	17	following	follow	VERB
ejpam-3441	148	18	properties	property	NOUN
ejpam-3441	148	19	hold	hold	VERB
ejpam-3441	148	20	.	.	PUNCT
ejpam-3441	149	1	(	(	PUNCT
ejpam-3441	149	2	1	1	X
ejpam-3441	149	3	)	)	PUNCT
ejpam-3441	149	4	a	a	PRON
ejpam-3441	149	5	is	be	AUX
ejpam-3441	149	6	an	an	DET
ejpam-3441	149	7	la	la	NOUN
ejpam-3441	149	8	-	-	PUNCT
ejpam-3441	149	9	subring	subring	NOUN
ejpam-3441	149	10	of	of	ADP
ejpam-3441	149	11	r	r	NOUN
ejpam-3441	149	12	if	if	SCONJ
ejpam-3441	150	1	and	and	CCONJ
ejpam-3441	150	2	only	only	ADV
ejpam-3441	150	3	if	if	SCONJ
ejpam-3441	150	4	χa	χa	PROPN
ejpam-3441	150	5	is	be	AUX
ejpam-3441	150	6	an	an	DET
ejpam-3441	150	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	150	8	fuzzy	fuzzy	ADJ
ejpam-3441	150	9	la	la	NOUN
ejpam-3441	150	10	-	-	PUNCT
ejpam-3441	150	11	subring	subre	VERB
ejpam-3441	150	12	with	with	ADP
ejpam-3441	150	13	thresholds	threshold	NOUN
ejpam-3441	150	14	(	(	PUNCT
ejpam-3441	150	15	α	α	X
ejpam-3441	150	16	,	,	PUNCT
ejpam-3441	150	17	β	β	X
ejpam-3441	150	18	]	]	PUNCT
ejpam-3441	150	19	of	of	ADP
ejpam-3441	150	20	r.	r.	PROPN
ejpam-3441	150	21	(	(	PUNCT
ejpam-3441	150	22	2	2	NUM
ejpam-3441	150	23	)	)	PUNCT
ejpam-3441	150	24	a	a	PRON
ejpam-3441	150	25	is	be	AUX
ejpam-3441	150	26	a	a	DET
ejpam-3441	150	27	left	left	ADJ
ejpam-3441	150	28	(	(	PUNCT
ejpam-3441	150	29	resp	resp	NOUN
ejpam-3441	150	30	.	.	PUNCT
ejpam-3441	151	1	right	right	ADJ
ejpam-3441	151	2	,	,	PUNCT
ejpam-3441	151	3	two	two	NUM
ejpam-3441	151	4	-	-	PUNCT
ejpam-3441	151	5	sided	sided	ADJ
ejpam-3441	151	6	)	)	PUNCT
ejpam-3441	151	7	ideal	ideal	NOUN
ejpam-3441	151	8	of	of	ADP
ejpam-3441	151	9	r	r	NOUN
ejpam-3441	151	10	if	if	SCONJ
ejpam-3441	152	1	and	and	CCONJ
ejpam-3441	152	2	only	only	ADV
ejpam-3441	152	3	if	if	SCONJ
ejpam-3441	152	4	χa	χa	PROPN
ejpam-3441	152	5	is	be	AUX
ejpam-3441	152	6	an	an	DET
ejpam-3441	152	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	152	8	fuzzy	fuzzy	ADJ
ejpam-3441	152	9	left	left	NOUN
ejpam-3441	152	10	(	(	PUNCT
ejpam-3441	152	11	resp	resp	NOUN
ejpam-3441	152	12	.	.	PUNCT
ejpam-3441	153	1	right	right	ADJ
ejpam-3441	153	2	,	,	PUNCT
ejpam-3441	153	3	two	two	NUM
ejpam-3441	153	4	-	-	PUNCT
ejpam-3441	153	5	sided	sided	ADJ
ejpam-3441	153	6	)	)	PUNCT
ejpam-3441	153	7	ideal	ideal	NOUN
ejpam-3441	153	8	with	with	ADP
ejpam-3441	153	9	thresholds	threshold	NOUN
ejpam-3441	153	10	(	(	PUNCT
ejpam-3441	153	11	α	α	X
ejpam-3441	153	12	,	,	PUNCT
ejpam-3441	153	13	β	β	X
ejpam-3441	153	14	]	]	PUNCT
ejpam-3441	153	15	of	of	ADP
ejpam-3441	153	16	r.	r.	PROPN
ejpam-3441	153	17	proof	proof	NOUN
ejpam-3441	153	18	.	.	PUNCT
ejpam-3441	154	1	(	(	PUNCT
ejpam-3441	154	2	1	1	X
ejpam-3441	154	3	)	)	PUNCT
ejpam-3441	154	4	let	let	VERB
ejpam-3441	154	5	a	a	PRON
ejpam-3441	154	6	be	be	AUX
ejpam-3441	154	7	an	an	DET
ejpam-3441	154	8	la	la	NOUN
ejpam-3441	154	9	-	-	PUNCT
ejpam-3441	154	10	subring	subring	NOUN
ejpam-3441	154	11	of	of	ADP
ejpam-3441	154	12	an	an	DET
ejpam-3441	154	13	la	la	ADJ
ejpam-3441	154	14	-	-	PUNCT
ejpam-3441	154	15	ring	ring	NOUN
ejpam-3441	154	16	r	r	NOUN
ejpam-3441	154	17	and	and	CCONJ
ejpam-3441	154	18	x	x	NOUN
ejpam-3441	154	19	,	,	PUNCT
ejpam-3441	154	20	y	y	PROPN
ejpam-3441	154	21	∈	∈	PROPN
ejpam-3441	154	22	r.	r.	PROPN
ejpam-3441	154	23	if	if	SCONJ
ejpam-3441	154	24	x	x	PROPN
ejpam-3441	154	25	,	,	PUNCT
ejpam-3441	154	26	y	y	PROPN
ejpam-3441	154	27	/∈	/∈	PUNCT
ejpam-3441	155	1	a	a	INTJ
ejpam-3441	155	2	,	,	PUNCT
ejpam-3441	155	3	then	then	ADV
ejpam-3441	155	4	by	by	ADP
ejpam-3441	155	5	definition	definition	NOUN
ejpam-3441	155	6	of	of	ADP
ejpam-3441	155	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	155	8	characteristic	characteristic	ADJ
ejpam-3441	155	9	function	function	NOUN
ejpam-3441	155	10	µχa(x	µχa(x	VERB
ejpam-3441	155	11	)	)	PUNCT
ejpam-3441	155	12	=	=	SYM
ejpam-3441	155	13	0	0	PUNCT
ejpam-3441	155	14	=	=	SYM
ejpam-3441	155	15	µχa(y	µχa(y	PROPN
ejpam-3441	155	16	)	)	PUNCT
ejpam-3441	155	17	and	and	CCONJ
ejpam-3441	155	18	γχa(x	γχa(x	PROPN
ejpam-3441	155	19	)	)	PUNCT
ejpam-3441	155	20	=	=	SYM
ejpam-3441	155	21	1	1	NUM
ejpam-3441	155	22	=	=	SYM
ejpam-3441	155	23	γχa(y	γχa(y	NOUN
ejpam-3441	155	24	)	)	PUNCT
ejpam-3441	155	25	.	.	PUNCT
ejpam-3441	156	1	thus	thus	ADV
ejpam-3441	156	2	µχa(x−	µχa(x−	ADP
ejpam-3441	156	3	y	y	PROPN
ejpam-3441	156	4	)	)	PUNCT
ejpam-3441	156	5	≥	≥	PROPN
ejpam-3441	156	6	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	7	)	)	PUNCT
ejpam-3441	156	8	,	,	PUNCT
ejpam-3441	156	9	µχa(y	µχa(y	PROPN
ejpam-3441	156	10	)	)	PUNCT
ejpam-3441	156	11	}	}	PUNCT
ejpam-3441	156	12	=	=	SYM
ejpam-3441	156	13	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	14	)	)	PUNCT
ejpam-3441	156	15	,	,	PUNCT
ejpam-3441	156	16	µχa(y	µχa(y	PROPN
ejpam-3441	156	17	)	)	PUNCT
ejpam-3441	156	18	,	,	PUNCT
ejpam-3441	156	19	β	β	X
ejpam-3441	156	20	}	}	PUNCT
ejpam-3441	156	21	⇒	⇒	NOUN
ejpam-3441	156	22	µχa(x−	µχa(x−	ADV
ejpam-3441	156	23	y	y	PROPN
ejpam-3441	156	24	)	)	PUNCT
ejpam-3441	156	25	≥	≥	PROPN
ejpam-3441	156	26	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	27	)	)	PUNCT
ejpam-3441	156	28	,	,	PUNCT
ejpam-3441	156	29	µχa(y	µχa(y	PROPN
ejpam-3441	156	30	)	)	PUNCT
ejpam-3441	156	31	,	,	PUNCT
ejpam-3441	156	32	β	β	X
ejpam-3441	156	33	}	}	PUNCT
ejpam-3441	156	34	⇒	⇒	PROPN
ejpam-3441	156	35	max{µχa(x−	max{µχa(x−	PROPN
ejpam-3441	156	36	y	y	PROPN
ejpam-3441	156	37	)	)	PUNCT
ejpam-3441	156	38	,	,	PUNCT
ejpam-3441	156	39	α	α	X
ejpam-3441	156	40	}	}	PUNCT
ejpam-3441	156	41	≥	≥	NOUN
ejpam-3441	156	42	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	43	)	)	PUNCT
ejpam-3441	156	44	,	,	PUNCT
ejpam-3441	156	45	µχa(y	µχa(y	PROPN
ejpam-3441	156	46	)	)	PUNCT
ejpam-3441	156	47	,	,	PUNCT
ejpam-3441	156	48	β	β	X
ejpam-3441	156	49	}	}	PUNCT
ejpam-3441	156	50	and	and	CCONJ
ejpam-3441	156	51	µχa(xy	µχa(xy	NUM
ejpam-3441	156	52	)	)	PUNCT
ejpam-3441	156	53	≥	≥	PROPN
ejpam-3441	156	54	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	55	)	)	PUNCT
ejpam-3441	156	56	,	,	PUNCT
ejpam-3441	156	57	µχa(y	µχa(y	PROPN
ejpam-3441	156	58	)	)	PUNCT
ejpam-3441	156	59	}	}	PUNCT
ejpam-3441	156	60	=	=	SYM
ejpam-3441	156	61	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	62	)	)	PUNCT
ejpam-3441	156	63	,	,	PUNCT
ejpam-3441	156	64	µχa(y	µχa(y	PROPN
ejpam-3441	156	65	)	)	PUNCT
ejpam-3441	156	66	,	,	PUNCT
ejpam-3441	156	67	β	β	X
ejpam-3441	156	68	}	}	PUNCT
ejpam-3441	156	69	⇒	⇒	NOUN
ejpam-3441	156	70	µχa(xy	µχa(xy	NUM
ejpam-3441	156	71	)	)	PUNCT
ejpam-3441	156	72	≥	≥	PROPN
ejpam-3441	156	73	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	74	)	)	PUNCT
ejpam-3441	156	75	,	,	PUNCT
ejpam-3441	156	76	µχa(y	µχa(y	PROPN
ejpam-3441	156	77	)	)	PUNCT
ejpam-3441	156	78	,	,	PUNCT
ejpam-3441	156	79	β	β	X
ejpam-3441	156	80	}	}	PUNCT
ejpam-3441	156	81	⇒	⇒	NOUN
ejpam-3441	156	82	max{µχa(xy	max{µχa(xy	NOUN
ejpam-3441	156	83	)	)	PUNCT
ejpam-3441	156	84	,	,	PUNCT
ejpam-3441	156	85	α	α	X
ejpam-3441	156	86	}	}	PUNCT
ejpam-3441	156	87	≥	≥	NOUN
ejpam-3441	156	88	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	156	89	)	)	PUNCT
ejpam-3441	156	90	,	,	PUNCT
ejpam-3441	156	91	µχa(y	µχa(y	PROPN
ejpam-3441	156	92	)	)	PUNCT
ejpam-3441	156	93	,	,	PUNCT
ejpam-3441	156	94	β	β	X
ejpam-3441	156	95	}	}	PUNCT
ejpam-3441	156	96	.	.	PUNCT
ejpam-3441	157	1	similarly	similarly	ADV
ejpam-3441	157	2	,	,	PUNCT
ejpam-3441	157	3	we	we	PRON
ejpam-3441	157	4	have	have	VERB
ejpam-3441	157	5	min{γχa(x−	min{γχa(x−	PROPN
ejpam-3441	157	6	y	y	NOUN
ejpam-3441	157	7	)	)	PUNCT
ejpam-3441	157	8	,	,	PUNCT
ejpam-3441	157	9	(	(	PUNCT
ejpam-3441	157	10	1−	1−	NUM
ejpam-3441	157	11	α	α	NOUN
ejpam-3441	157	12	)	)	PUNCT
ejpam-3441	157	13	}	}	PUNCT
ejpam-3441	157	14	≤	≤	NUM
ejpam-3441	158	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	158	2	)	)	PUNCT
ejpam-3441	158	3	,	,	PUNCT
ejpam-3441	158	4	γχa(y	γχa(y	NOUN
ejpam-3441	158	5	)	)	PUNCT
ejpam-3441	158	6	,	,	PUNCT
ejpam-3441	158	7	(	(	PUNCT
ejpam-3441	158	8	1−	1−	NUM
ejpam-3441	158	9	β	β	NOUN
ejpam-3441	158	10	)	)	PUNCT
ejpam-3441	158	11	}	}	PUNCT
ejpam-3441	158	12	and	and	CCONJ
ejpam-3441	158	13	min{γχa(xy	min{γχa(xy	NOUN
ejpam-3441	158	14	)	)	PUNCT
ejpam-3441	158	15	,	,	PUNCT
ejpam-3441	158	16	(	(	PUNCT
ejpam-3441	158	17	1−	1−	NUM
ejpam-3441	158	18	α	α	NOUN
ejpam-3441	158	19	)	)	PUNCT
ejpam-3441	158	20	}	}	PUNCT
ejpam-3441	158	21	≤	≤	NUM
ejpam-3441	159	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	159	2	)	)	PUNCT
ejpam-3441	159	3	,	,	PUNCT
ejpam-3441	159	4	γχa(y	γχa(y	NOUN
ejpam-3441	159	5	)	)	PUNCT
ejpam-3441	159	6	,	,	PUNCT
ejpam-3441	159	7	(	(	PUNCT
ejpam-3441	159	8	1−	1−	NUM
ejpam-3441	159	9	β	β	NOUN
ejpam-3441	159	10	)	)	PUNCT
ejpam-3441	159	11	}	}	PUNCT
ejpam-3441	159	12	.	.	PUNCT
ejpam-3441	160	1	in	in	ADP
ejpam-3441	160	2	same	same	ADJ
ejpam-3441	160	3	lines	line	NOUN
ejpam-3441	160	4	,	,	PUNCT
ejpam-3441	160	5	we	we	PRON
ejpam-3441	160	6	have	have	VERB
ejpam-3441	160	7	max{µχa(x−	max{µχa(x−	PROPN
ejpam-3441	160	8	y	y	PROPN
ejpam-3441	160	9	)	)	PUNCT
ejpam-3441	160	10	,	,	PUNCT
ejpam-3441	160	11	α	α	X
ejpam-3441	160	12	}	}	PUNCT
ejpam-3441	160	13	≥	≥	NOUN
ejpam-3441	160	14	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	160	15	)	)	PUNCT
ejpam-3441	160	16	,	,	PUNCT
ejpam-3441	160	17	µχa(y	µχa(y	PROPN
ejpam-3441	160	18	)	)	PUNCT
ejpam-3441	160	19	,	,	PUNCT
ejpam-3441	160	20	β	β	X
ejpam-3441	160	21	}	}	PUNCT
ejpam-3441	160	22	,	,	PUNCT
ejpam-3441	160	23	max{µχa(xy	max{µχa(xy	NOUN
ejpam-3441	160	24	)	)	PUNCT
ejpam-3441	160	25	,	,	PUNCT
ejpam-3441	160	26	α	α	X
ejpam-3441	160	27	}	}	PUNCT
ejpam-3441	160	28	≥	≥	NOUN
ejpam-3441	160	29	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	160	30	)	)	PUNCT
ejpam-3441	160	31	,	,	PUNCT
ejpam-3441	160	32	µχa(y	µχa(y	PROPN
ejpam-3441	160	33	)	)	PUNCT
ejpam-3441	160	34	,	,	PUNCT
ejpam-3441	160	35	β	β	X
ejpam-3441	160	36	}	}	PUNCT
ejpam-3441	160	37	,	,	PUNCT
ejpam-3441	161	1	min{γχa(x−	min{γχa(x−	PROPN
ejpam-3441	161	2	y	y	PROPN
ejpam-3441	161	3	)	)	PUNCT
ejpam-3441	161	4	,	,	PUNCT
ejpam-3441	161	5	(	(	PUNCT
ejpam-3441	161	6	1−	1−	NUM
ejpam-3441	161	7	α	α	NOUN
ejpam-3441	161	8	)	)	PUNCT
ejpam-3441	161	9	}	}	PUNCT
ejpam-3441	161	10	≤	≤	NUM
ejpam-3441	161	11	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	161	12	)	)	PUNCT
ejpam-3441	161	13	,	,	PUNCT
ejpam-3441	161	14	γχa(y	γχa(y	NOUN
ejpam-3441	161	15	)	)	PUNCT
ejpam-3441	161	16	,	,	PUNCT
ejpam-3441	161	17	(	(	PUNCT
ejpam-3441	161	18	1−	1−	NUM
ejpam-3441	161	19	β	β	NOUN
ejpam-3441	161	20	)	)	PUNCT
ejpam-3441	161	21	}	}	PUNCT
ejpam-3441	161	22	,	,	PUNCT
ejpam-3441	161	23	min{γχa(xy	min{γχa(xy	NOUN
ejpam-3441	161	24	)	)	PUNCT
ejpam-3441	161	25	,	,	PUNCT
ejpam-3441	161	26	(	(	PUNCT
ejpam-3441	161	27	1−	1−	NUM
ejpam-3441	161	28	α	α	NOUN
ejpam-3441	161	29	)	)	PUNCT
ejpam-3441	161	30	}	}	PUNCT
ejpam-3441	162	1	≤	≤	NUM
ejpam-3441	163	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	163	2	)	)	PUNCT
ejpam-3441	163	3	,	,	PUNCT
ejpam-3441	163	4	γχa(y	γχa(y	NOUN
ejpam-3441	163	5	)	)	PUNCT
ejpam-3441	163	6	,	,	PUNCT
ejpam-3441	163	7	(	(	PUNCT
ejpam-3441	163	8	1−	1−	NUM
ejpam-3441	163	9	β	β	NOUN
ejpam-3441	163	10	)	)	PUNCT
ejpam-3441	163	11	}	}	PUNCT
ejpam-3441	163	12	,	,	PUNCT
ejpam-3441	163	13	when	when	SCONJ
ejpam-3441	163	14	x	x	X
ejpam-3441	163	15	,	,	PUNCT
ejpam-3441	163	16	y	y	PROPN
ejpam-3441	163	17	∈	∈	PROPN
ejpam-3441	163	18	a.	a.	NOUN
ejpam-3441	163	19	hence	hence	ADV
ejpam-3441	163	20	the	the	DET
ejpam-3441	163	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	163	22	characteristic	characteristic	ADJ
ejpam-3441	163	23	function	function	NOUN
ejpam-3441	163	24	χa	χa	NOUN
ejpam-3441	163	25	of	of	ADP
ejpam-3441	163	26	a	a	PRON
ejpam-3441	163	27	is	be	AUX
ejpam-3441	163	28	an	an	DET
ejpam-3441	163	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	163	30	fuzzy	fuzzy	ADJ
ejpam-3441	163	31	la	la	NOUN
ejpam-3441	163	32	-	-	PUNCT
ejpam-3441	163	33	subring	subre	VERB
ejpam-3441	163	34	with	with	ADP
ejpam-3441	163	35	thresholds	threshold	NOUN
ejpam-3441	163	36	(	(	PUNCT
ejpam-3441	163	37	α	α	X
ejpam-3441	163	38	,	,	PUNCT
ejpam-3441	163	39	β	β	X
ejpam-3441	163	40	]	]	PUNCT
ejpam-3441	163	41	of	of	ADP
ejpam-3441	163	42	r.	r.	PROPN
ejpam-3441	163	43	conversely	conversely	ADV
ejpam-3441	163	44	,	,	PUNCT
ejpam-3441	163	45	suppose	suppose	VERB
ejpam-3441	163	46	that	that	SCONJ
ejpam-3441	163	47	the	the	DET
ejpam-3441	163	48	intuitionistic	intuitionistic	ADJ
ejpam-3441	163	49	characteristic	characteristic	ADJ
ejpam-3441	163	50	function	function	NOUN
ejpam-3441	163	51	χa	χa	NOUN
ejpam-3441	163	52	of	of	ADP
ejpam-3441	163	53	a	a	PRON
ejpam-3441	163	54	is	be	AUX
ejpam-3441	163	55	an	an	DET
ejpam-3441	163	56	intuitionistic	intuitionistic	ADJ
ejpam-3441	163	57	fuzzy	fuzzy	ADJ
ejpam-3441	163	58	la	la	NOUN
ejpam-3441	163	59	-	-	PUNCT
ejpam-3441	163	60	subring	subre	VERB
ejpam-3441	163	61	with	with	ADP
ejpam-3441	163	62	thresholds	threshold	NOUN
ejpam-3441	163	63	(	(	PUNCT
ejpam-3441	163	64	α	α	X
ejpam-3441	163	65	,	,	PUNCT
ejpam-3441	163	66	β	β	X
ejpam-3441	163	67	]	]	PUNCT
ejpam-3441	163	68	of	of	ADP
ejpam-3441	163	69	an	an	DET
ejpam-3441	163	70	la	la	ADJ
ejpam-3441	163	71	-	-	PUNCT
ejpam-3441	163	72	ring	ring	NOUN
ejpam-3441	163	73	r.	r.	PROPN
ejpam-3441	163	74	let	let	VERB
ejpam-3441	163	75	x	x	PRON
ejpam-3441	163	76	,	,	PUNCT
ejpam-3441	163	77	y	y	PROPN
ejpam-3441	163	78	∈	∈	PROPN
ejpam-3441	163	79	a	a	PRON
ejpam-3441	163	80	,	,	PUNCT
ejpam-3441	163	81	then	then	ADV
ejpam-3441	163	82	by	by	ADP
ejpam-3441	163	83	definition	definition	NOUN
ejpam-3441	163	84	µχa(x	µχa(x	VERB
ejpam-3441	163	85	)	)	PUNCT
ejpam-3441	163	86	=	=	SYM
ejpam-3441	163	87	1	1	NUM
ejpam-3441	163	88	=	=	SYM
ejpam-3441	163	89	µχa(y	µχa(y	PROPN
ejpam-3441	163	90	)	)	PUNCT
ejpam-3441	163	91	and	and	CCONJ
ejpam-3441	163	92	γχa(x	γχa(x	PROPN
ejpam-3441	163	93	)	)	PUNCT
ejpam-3441	163	94	=	=	SYM
ejpam-3441	163	95	0	0	PUNCT
ejpam-3441	163	96	=	=	SYM
ejpam-3441	163	97	γχa(y	γχa(y	NOUN
ejpam-3441	163	98	)	)	PUNCT
ejpam-3441	163	99	.	.	PUNCT
ejpam-3441	164	1	since	since	SCONJ
ejpam-3441	164	2	max{µχa(x−	max{µχa(x−	PROPN
ejpam-3441	164	3	y	y	PROPN
ejpam-3441	164	4	)	)	PUNCT
ejpam-3441	164	5	,	,	PUNCT
ejpam-3441	164	6	α	α	X
ejpam-3441	164	7	}	}	PUNCT
ejpam-3441	164	8	≥	≥	NOUN
ejpam-3441	164	9	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	164	10	)	)	PUNCT
ejpam-3441	164	11	,	,	PUNCT
ejpam-3441	164	12	µχa(y	µχa(y	PROPN
ejpam-3441	164	13	)	)	PUNCT
ejpam-3441	164	14	,	,	PUNCT
ejpam-3441	164	15	β	β	X
ejpam-3441	164	16	}	}	PUNCT
ejpam-3441	164	17	=	=	SYM
ejpam-3441	164	18	β	β	X
ejpam-3441	164	19	,	,	PUNCT
ejpam-3441	164	20	k.	k.	PROPN
ejpam-3441	164	21	nasreen	nasreen	PROPN
ejpam-3441	164	22	et	et	PROPN
ejpam-3441	164	23	al	al	PROPN
ejpam-3441	164	24	.	.	PUNCT
ejpam-3441	164	25	/	/	SYM
ejpam-3441	164	26	eur	eur	PROPN
ejpam-3441	164	27	.	.	PUNCT
ejpam-3441	165	1	j.	j.	PROPN
ejpam-3441	165	2	pure	pure	PROPN
ejpam-3441	165	3	appl	appl	PROPN
ejpam-3441	165	4	.	.	PROPN
ejpam-3441	165	5	math	math	PROPN
ejpam-3441	165	6	,	,	PUNCT
ejpam-3441	165	7	12	12	NUM
ejpam-3441	165	8	(	(	PUNCT
ejpam-3441	165	9	3	3	NUM
ejpam-3441	165	10	)	)	PUNCT
ejpam-3441	165	11	(	(	PUNCT
ejpam-3441	165	12	2019	2019	NUM
ejpam-3441	165	13	)	)	PUNCT
ejpam-3441	165	14	,	,	PUNCT
ejpam-3441	165	15	906	906	NUM
ejpam-3441	165	16	-	-	SYM
ejpam-3441	165	17	943	943	NUM
ejpam-3441	165	18	913	913	NUM
ejpam-3441	165	19	max{µχa(xy	max{µχa(xy	NOUN
ejpam-3441	165	20	)	)	PUNCT
ejpam-3441	165	21	,	,	PUNCT
ejpam-3441	165	22	α	α	X
ejpam-3441	165	23	}	}	PUNCT
ejpam-3441	165	24	≥	≥	NOUN
ejpam-3441	165	25	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	165	26	)	)	PUNCT
ejpam-3441	165	27	,	,	PUNCT
ejpam-3441	165	28	µχa(y	µχa(y	PROPN
ejpam-3441	165	29	)	)	PUNCT
ejpam-3441	165	30	,	,	PUNCT
ejpam-3441	165	31	β	β	X
ejpam-3441	165	32	}	}	PUNCT
ejpam-3441	165	33	=	=	SYM
ejpam-3441	165	34	β	β	X
ejpam-3441	165	35	,	,	PUNCT
ejpam-3441	165	36	min{γχa(x−	min{γχa(x−	PROPN
ejpam-3441	165	37	y	y	PROPN
ejpam-3441	165	38	)	)	PUNCT
ejpam-3441	165	39	,	,	PUNCT
ejpam-3441	165	40	(	(	PUNCT
ejpam-3441	165	41	1−	1−	NUM
ejpam-3441	165	42	α	α	NOUN
ejpam-3441	165	43	)	)	PUNCT
ejpam-3441	165	44	}	}	PUNCT
ejpam-3441	165	45	≤	≤	NUM
ejpam-3441	166	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	166	2	)	)	PUNCT
ejpam-3441	166	3	,	,	PUNCT
ejpam-3441	166	4	γχa(y	γχa(y	NOUN
ejpam-3441	166	5	)	)	PUNCT
ejpam-3441	166	6	,	,	PUNCT
ejpam-3441	166	7	(	(	PUNCT
ejpam-3441	166	8	1−	1−	NUM
ejpam-3441	166	9	β	β	NOUN
ejpam-3441	166	10	)	)	PUNCT
ejpam-3441	166	11	}	}	PUNCT
ejpam-3441	166	12	=	=	SYM
ejpam-3441	166	13	1−	1−	NUM
ejpam-3441	166	14	β	β	X
ejpam-3441	166	15	,	,	PUNCT
ejpam-3441	166	16	min{γχa(xy	min{γχa(xy	NOUN
ejpam-3441	166	17	)	)	PUNCT
ejpam-3441	166	18	,	,	PUNCT
ejpam-3441	166	19	(	(	PUNCT
ejpam-3441	166	20	1−	1−	NUM
ejpam-3441	166	21	α	α	NOUN
ejpam-3441	166	22	)	)	PUNCT
ejpam-3441	166	23	}	}	PUNCT
ejpam-3441	166	24	≤	≤	NUM
ejpam-3441	167	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	167	2	)	)	PUNCT
ejpam-3441	167	3	,	,	PUNCT
ejpam-3441	167	4	γχa(y	γχa(y	NOUN
ejpam-3441	167	5	)	)	PUNCT
ejpam-3441	167	6	,	,	PUNCT
ejpam-3441	167	7	(	(	PUNCT
ejpam-3441	167	8	1−	1−	NUM
ejpam-3441	167	9	β	β	NOUN
ejpam-3441	167	10	)	)	PUNCT
ejpam-3441	167	11	}	}	PUNCT
ejpam-3441	167	12	=	=	SYM
ejpam-3441	168	1	1−	1−	NUM
ejpam-3441	168	2	β	β	X
ejpam-3441	168	3	,	,	PUNCT
ejpam-3441	168	4	χa	χa	AUX
ejpam-3441	168	5	being	be	AUX
ejpam-3441	168	6	an	an	DET
ejpam-3441	168	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	168	8	fuzzy	fuzzy	ADJ
ejpam-3441	168	9	la	la	NOUN
ejpam-3441	168	10	-	-	PUNCT
ejpam-3441	168	11	subring	subre	VERB
ejpam-3441	168	12	with	with	ADP
ejpam-3441	168	13	thresholds	threshold	NOUN
ejpam-3441	168	14	(	(	PUNCT
ejpam-3441	168	15	α	α	X
ejpam-3441	168	16	,	,	PUNCT
ejpam-3441	168	17	β	β	X
ejpam-3441	168	18	]	]	PUNCT
ejpam-3441	168	19	of	of	ADP
ejpam-3441	168	20	r.	r.	PROPN
ejpam-3441	168	21	thus	thus	ADV
ejpam-3441	168	22	max{µχa(x−	max{µχa(x−	PROPN
ejpam-3441	168	23	y	y	PROPN
ejpam-3441	168	24	)	)	PUNCT
ejpam-3441	168	25	,	,	PUNCT
ejpam-3441	168	26	α	α	X
ejpam-3441	168	27	}	}	PUNCT
ejpam-3441	168	28	≥	≥	NUM
ejpam-3441	168	29	β	β	NOUN
ejpam-3441	168	30	and	and	CCONJ
ejpam-3441	168	31	max{µχa(xy	max{µχa(xy	NOUN
ejpam-3441	168	32	)	)	PUNCT
ejpam-3441	168	33	,	,	PUNCT
ejpam-3441	168	34	α	α	X
ejpam-3441	168	35	}	}	PUNCT
ejpam-3441	168	36	≥	≥	NUM
ejpam-3441	168	37	β	β	NOUN
ejpam-3441	168	38	.	.	PUNCT
ejpam-3441	169	1	min{γχa(x−	min{γχa(x−	PROPN
ejpam-3441	169	2	y	y	PROPN
ejpam-3441	169	3	)	)	PUNCT
ejpam-3441	169	4	,	,	PUNCT
ejpam-3441	169	5	(	(	PUNCT
ejpam-3441	169	6	1−	1−	NUM
ejpam-3441	169	7	α	α	NOUN
ejpam-3441	169	8	)	)	PUNCT
ejpam-3441	169	9	}	}	PUNCT
ejpam-3441	169	10	≤	≤	NOUN
ejpam-3441	169	11	1−	1−	NUM
ejpam-3441	169	12	β	β	X
ejpam-3441	169	13	and	and	CCONJ
ejpam-3441	169	14	min{γχa(xy	min{γχa(xy	NOUN
ejpam-3441	169	15	)	)	PUNCT
ejpam-3441	169	16	,	,	PUNCT
ejpam-3441	169	17	(	(	PUNCT
ejpam-3441	169	18	1−	1−	NUM
ejpam-3441	169	19	α	α	NOUN
ejpam-3441	169	20	)	)	PUNCT
ejpam-3441	169	21	}	}	PUNCT
ejpam-3441	169	22	≤	≤	NOUN
ejpam-3441	169	23	1−	1−	NUM
ejpam-3441	169	24	β	β	X
ejpam-3441	169	25	.	.	PUNCT
ejpam-3441	170	1	this	this	PRON
ejpam-3441	170	2	implies	imply	VERB
ejpam-3441	170	3	that	that	SCONJ
ejpam-3441	170	4	µχa(x−	µχa(x−	ADP
ejpam-3441	170	5	y	y	X
ejpam-3441	170	6	)	)	PUNCT
ejpam-3441	170	7	=	=	SYM
ejpam-3441	170	8	1	1	NUM
ejpam-3441	170	9	=	=	SYM
ejpam-3441	170	10	µχa(xy	µχa(xy	X
ejpam-3441	170	11	)	)	PUNCT
ejpam-3441	170	12	and	and	CCONJ
ejpam-3441	170	13	γχa(x−	γχa(x−	ADP
ejpam-3441	170	14	y	y	X
ejpam-3441	170	15	)	)	PUNCT
ejpam-3441	170	16	=	=	SYM
ejpam-3441	170	17	0	0	NUM
ejpam-3441	170	18	=	=	SYM
ejpam-3441	170	19	γχa(xy	γχa(xy	NOUN
ejpam-3441	170	20	)	)	PUNCT
ejpam-3441	170	21	,	,	PUNCT
ejpam-3441	170	22	i.e.	i.e.	X
ejpam-3441	170	23	,	,	PUNCT
ejpam-3441	170	24	x−	x−	PROPN
ejpam-3441	170	25	y	y	PROPN
ejpam-3441	170	26	and	and	CCONJ
ejpam-3441	170	27	xy	xy	PROPN
ejpam-3441	170	28	∈	∈	PROPN
ejpam-3441	170	29	a.	a.	NOUN
ejpam-3441	170	30	hence	hence	ADV
ejpam-3441	170	31	a	a	PRON
ejpam-3441	170	32	is	be	AUX
ejpam-3441	170	33	an	an	DET
ejpam-3441	170	34	la	la	NOUN
ejpam-3441	170	35	-	-	PUNCT
ejpam-3441	170	36	subring	subring	NOUN
ejpam-3441	170	37	of	of	ADP
ejpam-3441	170	38	r.	r.	PROPN
ejpam-3441	170	39	(	(	PUNCT
ejpam-3441	170	40	2	2	X
ejpam-3441	170	41	)	)	PUNCT
ejpam-3441	170	42	let	let	VERB
ejpam-3441	170	43	a	a	PRON
ejpam-3441	170	44	be	be	AUX
ejpam-3441	170	45	a	a	DET
ejpam-3441	170	46	left	left	ADJ
ejpam-3441	170	47	ideal	ideal	NOUN
ejpam-3441	170	48	of	of	ADP
ejpam-3441	170	49	an	an	DET
ejpam-3441	170	50	la	la	ADJ
ejpam-3441	170	51	-	-	PUNCT
ejpam-3441	170	52	ring	ring	NOUN
ejpam-3441	170	53	r	r	NOUN
ejpam-3441	170	54	and	and	CCONJ
ejpam-3441	170	55	x	x	NOUN
ejpam-3441	170	56	,	,	PUNCT
ejpam-3441	170	57	y	y	PROPN
ejpam-3441	170	58	∈	∈	PROPN
ejpam-3441	170	59	r.	r.	PROPN
ejpam-3441	170	60	if	if	SCONJ
ejpam-3441	170	61	y	y	PROPN
ejpam-3441	170	62	/∈	/∈	VERB
ejpam-3441	171	1	a	a	INTJ
ejpam-3441	171	2	,	,	PUNCT
ejpam-3441	171	3	then	then	ADV
ejpam-3441	171	4	by	by	ADP
ejpam-3441	171	5	definition	definition	NOUN
ejpam-3441	171	6	of	of	ADP
ejpam-3441	171	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	171	8	characteristic	characteristic	ADJ
ejpam-3441	171	9	function	function	NOUN
ejpam-3441	171	10	µχa(y	µχa(y	PROPN
ejpam-3441	171	11	)	)	PUNCT
ejpam-3441	171	12	=	=	SYM
ejpam-3441	171	13	0	0	NUM
ejpam-3441	171	14	and	and	CCONJ
ejpam-3441	171	15	γχa(y	γχa(y	NOUN
ejpam-3441	171	16	)	)	PUNCT
ejpam-3441	171	17	=	=	SYM
ejpam-3441	172	1	1	1	X
ejpam-3441	172	2	.	.	PUNCT
ejpam-3441	172	3	thus	thus	ADV
ejpam-3441	172	4	µχa(xy	µχa(xy	X
ejpam-3441	172	5	)	)	PUNCT
ejpam-3441	172	6	≥	≥	NOUN
ejpam-3441	172	7	µχa(y	µχa(y	PROPN
ejpam-3441	172	8	)	)	PUNCT
ejpam-3441	172	9	=	=	SYM
ejpam-3441	172	10	min{µχa(y	min{µχa(y	PROPN
ejpam-3441	172	11	)	)	PUNCT
ejpam-3441	172	12	,	,	PUNCT
ejpam-3441	172	13	β	β	X
ejpam-3441	172	14	}	}	PUNCT
ejpam-3441	172	15	⇒	⇒	NOUN
ejpam-3441	172	16	µχa(xy	µχa(xy	NUM
ejpam-3441	172	17	)	)	PUNCT
ejpam-3441	172	18	≥	≥	NOUN
ejpam-3441	172	19	min{µχa(y	min{µχa(y	PROPN
ejpam-3441	172	20	)	)	PUNCT
ejpam-3441	172	21	,	,	PUNCT
ejpam-3441	172	22	β	β	X
ejpam-3441	172	23	}	}	PUNCT
ejpam-3441	172	24	⇒	⇒	NOUN
ejpam-3441	172	25	max{µχa(xy	max{µχa(xy	NOUN
ejpam-3441	172	26	)	)	PUNCT
ejpam-3441	172	27	,	,	PUNCT
ejpam-3441	172	28	α	α	X
ejpam-3441	172	29	}	}	PUNCT
ejpam-3441	172	30	≥	≥	NOUN
ejpam-3441	172	31	min{µχa(y	min{µχa(y	PROPN
ejpam-3441	172	32	)	)	PUNCT
ejpam-3441	172	33	,	,	PUNCT
ejpam-3441	172	34	β	β	X
ejpam-3441	172	35	}	}	PUNCT
ejpam-3441	172	36	and	and	CCONJ
ejpam-3441	172	37	γχa(xy	γχa(xy	NOUN
ejpam-3441	172	38	)	)	PUNCT
ejpam-3441	172	39	≤	≤	NUM
ejpam-3441	172	40	γχa(y	γχa(y	NOUN
ejpam-3441	172	41	)	)	PUNCT
ejpam-3441	172	42	=	=	SYM
ejpam-3441	172	43	max{γχa(y	max{γχa(y	PROPN
ejpam-3441	172	44	)	)	PUNCT
ejpam-3441	172	45	,	,	PUNCT
ejpam-3441	172	46	(	(	PUNCT
ejpam-3441	172	47	1−	1−	NUM
ejpam-3441	172	48	β	β	NOUN
ejpam-3441	172	49	)	)	PUNCT
ejpam-3441	172	50	}	}	PUNCT
ejpam-3441	172	51	⇒	⇒	NOUN
ejpam-3441	172	52	γχa(xy	γχa(xy	NOUN
ejpam-3441	172	53	)	)	PUNCT
ejpam-3441	172	54	≤	≤	PROPN
ejpam-3441	172	55	max{γχa(y	max{γχa(y	PROPN
ejpam-3441	172	56	)	)	PUNCT
ejpam-3441	172	57	,	,	PUNCT
ejpam-3441	172	58	(	(	PUNCT
ejpam-3441	172	59	1−	1−	NUM
ejpam-3441	172	60	β	β	NOUN
ejpam-3441	172	61	)	)	PUNCT
ejpam-3441	172	62	}	}	PUNCT
ejpam-3441	172	63	⇒	⇒	NOUN
ejpam-3441	172	64	min{γχa(xy	min{γχa(xy	NOUN
ejpam-3441	172	65	)	)	PUNCT
ejpam-3441	172	66	,	,	PUNCT
ejpam-3441	172	67	(	(	PUNCT
ejpam-3441	172	68	1−	1−	NUM
ejpam-3441	172	69	α	α	NOUN
ejpam-3441	172	70	)	)	PUNCT
ejpam-3441	172	71	}	}	PUNCT
ejpam-3441	172	72	≤	≤	NUM
ejpam-3441	172	73	max{γχa(y	max{γχa(y	PROPN
ejpam-3441	172	74	)	)	PUNCT
ejpam-3441	172	75	,	,	PUNCT
ejpam-3441	172	76	(	(	PUNCT
ejpam-3441	172	77	1−	1−	NUM
ejpam-3441	172	78	β	β	NOUN
ejpam-3441	172	79	)	)	PUNCT
ejpam-3441	172	80	}	}	PUNCT
ejpam-3441	172	81	.	.	PUNCT
ejpam-3441	173	1	similarly	similarly	ADV
ejpam-3441	173	2	,	,	PUNCT
ejpam-3441	173	3	we	we	PRON
ejpam-3441	173	4	have	have	VERB
ejpam-3441	173	5	max{µχa(xy	max{µχa(xy	NOUN
ejpam-3441	173	6	)	)	PUNCT
ejpam-3441	173	7	,	,	PUNCT
ejpam-3441	173	8	α	α	X
ejpam-3441	173	9	}	}	PUNCT
ejpam-3441	173	10	≥	≥	NOUN
ejpam-3441	173	11	min{µχa(y	min{µχa(y	PROPN
ejpam-3441	173	12	)	)	PUNCT
ejpam-3441	173	13	,	,	PUNCT
ejpam-3441	173	14	β	β	X
ejpam-3441	173	15	}	}	PUNCT
ejpam-3441	173	16	,	,	PUNCT
ejpam-3441	173	17	min{γχa(xy	min{γχa(xy	NOUN
ejpam-3441	173	18	)	)	PUNCT
ejpam-3441	173	19	,	,	PUNCT
ejpam-3441	173	20	(	(	PUNCT
ejpam-3441	173	21	1−	1−	NUM
ejpam-3441	173	22	α	α	NOUN
ejpam-3441	173	23	)	)	PUNCT
ejpam-3441	173	24	}	}	PUNCT
ejpam-3441	173	25	≤	≤	NUM
ejpam-3441	173	26	max{γχa(y	max{γχa(y	PROPN
ejpam-3441	173	27	)	)	PUNCT
ejpam-3441	173	28	,	,	PUNCT
ejpam-3441	173	29	(	(	PUNCT
ejpam-3441	173	30	1−	1−	NUM
ejpam-3441	173	31	β	β	NOUN
ejpam-3441	173	32	)	)	PUNCT
ejpam-3441	173	33	}	}	PUNCT
ejpam-3441	173	34	,	,	PUNCT
ejpam-3441	173	35	when	when	SCONJ
ejpam-3441	173	36	y	y	PROPN
ejpam-3441	173	37	∈	∈	PROPN
ejpam-3441	173	38	a.	a.	NOUN
ejpam-3441	173	39	therefore	therefore	ADV
ejpam-3441	173	40	the	the	DET
ejpam-3441	173	41	intuitionistic	intuitionistic	ADJ
ejpam-3441	173	42	characteristic	characteristic	ADJ
ejpam-3441	173	43	function	function	NOUN
ejpam-3441	173	44	χa	χa	NOUN
ejpam-3441	173	45	of	of	ADP
ejpam-3441	173	46	a	a	PRON
ejpam-3441	173	47	is	be	AUX
ejpam-3441	173	48	an	an	DET
ejpam-3441	173	49	intuitionistic	intuitionistic	ADJ
ejpam-3441	173	50	fuzzy	fuzzy	ADJ
ejpam-3441	173	51	left	leave	VERB
ejpam-3441	173	52	ideal	ideal	NOUN
ejpam-3441	173	53	with	with	ADP
ejpam-3441	173	54	thresholds	threshold	NOUN
ejpam-3441	173	55	(	(	PUNCT
ejpam-3441	173	56	α	α	X
ejpam-3441	173	57	,	,	PUNCT
ejpam-3441	173	58	β	β	X
ejpam-3441	173	59	]	]	PUNCT
ejpam-3441	173	60	of	of	ADP
ejpam-3441	173	61	r.	r.	PROPN
ejpam-3441	173	62	conversely	conversely	ADV
ejpam-3441	173	63	,	,	PUNCT
ejpam-3441	173	64	assume	assume	VERB
ejpam-3441	173	65	that	that	SCONJ
ejpam-3441	173	66	the	the	DET
ejpam-3441	173	67	intuitionistic	intuitionistic	ADJ
ejpam-3441	173	68	characteristic	characteristic	ADJ
ejpam-3441	173	69	function	function	NOUN
ejpam-3441	173	70	χa	χa	NOUN
ejpam-3441	173	71	of	of	ADP
ejpam-3441	173	72	a	a	PRON
ejpam-3441	173	73	is	be	AUX
ejpam-3441	173	74	an	an	DET
ejpam-3441	173	75	intuitionistic	intuitionistic	ADJ
ejpam-3441	173	76	fuzzy	fuzzy	ADJ
ejpam-3441	173	77	left	leave	VERB
ejpam-3441	173	78	ideal	ideal	NOUN
ejpam-3441	173	79	with	with	ADP
ejpam-3441	173	80	thresholds	threshold	NOUN
ejpam-3441	173	81	(	(	PUNCT
ejpam-3441	173	82	α	α	X
ejpam-3441	173	83	,	,	PUNCT
ejpam-3441	173	84	β	β	X
ejpam-3441	173	85	]	]	PUNCT
ejpam-3441	173	86	of	of	ADP
ejpam-3441	173	87	an	an	DET
ejpam-3441	173	88	la	la	ADJ
ejpam-3441	173	89	-	-	PUNCT
ejpam-3441	173	90	ring	ring	NOUN
ejpam-3441	173	91	r.	r.	NOUN
ejpam-3441	173	92	let	let	VERB
ejpam-3441	173	93	y	y	PROPN
ejpam-3441	173	94	∈	∈	PROPN
ejpam-3441	173	95	a	a	PRON
ejpam-3441	173	96	and	and	CCONJ
ejpam-3441	173	97	z	z	NOUN
ejpam-3441	173	98	∈	∈	PROPN
ejpam-3441	173	99	r	r	NOUN
ejpam-3441	173	100	,	,	PUNCT
ejpam-3441	173	101	then	then	ADV
ejpam-3441	173	102	by	by	ADP
ejpam-3441	173	103	definition	definition	NOUN
ejpam-3441	173	104	µχa(y	µχa(y	PROPN
ejpam-3441	173	105	)	)	PUNCT
ejpam-3441	173	106	=	=	SYM
ejpam-3441	173	107	1	1	NUM
ejpam-3441	173	108	and	and	CCONJ
ejpam-3441	173	109	γχa(y	γχa(y	NOUN
ejpam-3441	173	110	)	)	PUNCT
ejpam-3441	174	1	=	=	SYM
ejpam-3441	175	1	0	0	X
ejpam-3441	175	2	.	.	PUNCT
ejpam-3441	175	3	since	since	SCONJ
ejpam-3441	175	4	max{µχa(zy	max{µχa(zy	NUM
ejpam-3441	175	5	)	)	PUNCT
ejpam-3441	175	6	,	,	PUNCT
ejpam-3441	175	7	α	α	X
ejpam-3441	175	8	}	}	PUNCT
ejpam-3441	175	9	≥	≥	NOUN
ejpam-3441	175	10	min{µχa(y	min{µχa(y	PROPN
ejpam-3441	175	11	)	)	PUNCT
ejpam-3441	175	12	,	,	PUNCT
ejpam-3441	175	13	β	β	X
ejpam-3441	175	14	}	}	PUNCT
ejpam-3441	175	15	=	=	SYM
ejpam-3441	175	16	β	β	NOUN
ejpam-3441	175	17	,	,	PUNCT
ejpam-3441	175	18	min{γχa(zy	min{γχa(zy	X
ejpam-3441	175	19	)	)	PUNCT
ejpam-3441	175	20	,	,	PUNCT
ejpam-3441	175	21	(	(	PUNCT
ejpam-3441	175	22	1−	1−	NUM
ejpam-3441	175	23	α	α	NOUN
ejpam-3441	175	24	)	)	PUNCT
ejpam-3441	175	25	}	}	PUNCT
ejpam-3441	175	26	≤	≤	NUM
ejpam-3441	175	27	max{γχa(y	max{γχa(y	PROPN
ejpam-3441	175	28	)	)	PUNCT
ejpam-3441	175	29	,	,	PUNCT
ejpam-3441	175	30	(	(	PUNCT
ejpam-3441	175	31	1−	1−	NUM
ejpam-3441	175	32	β	β	NOUN
ejpam-3441	175	33	)	)	PUNCT
ejpam-3441	175	34	}	}	PUNCT
ejpam-3441	175	35	=	=	SYM
ejpam-3441	175	36	1−	1−	NUM
ejpam-3441	175	37	β	β	X
ejpam-3441	175	38	,	,	PUNCT
ejpam-3441	175	39	χa	χa	ADP
ejpam-3441	175	40	being	be	AUX
ejpam-3441	175	41	an	an	DET
ejpam-3441	175	42	intuitionistic	intuitionistic	ADJ
ejpam-3441	175	43	fuzzy	fuzzy	ADJ
ejpam-3441	175	44	left	leave	VERB
ejpam-3441	175	45	ideal	ideal	NOUN
ejpam-3441	175	46	with	with	ADP
ejpam-3441	175	47	thresholds	threshold	NOUN
ejpam-3441	175	48	(	(	PUNCT
ejpam-3441	175	49	α	α	X
ejpam-3441	175	50	,	,	PUNCT
ejpam-3441	175	51	β	β	X
ejpam-3441	175	52	]	]	PUNCT
ejpam-3441	175	53	of	of	ADP
ejpam-3441	175	54	r.	r.	PROPN
ejpam-3441	175	55	thus	thus	ADV
ejpam-3441	175	56	max{µχa(zy	max{µχa(zy	NUM
ejpam-3441	175	57	)	)	PUNCT
ejpam-3441	175	58	,	,	PUNCT
ejpam-3441	175	59	α	α	X
ejpam-3441	175	60	}	}	PUNCT
ejpam-3441	175	61	≥	≥	NUM
ejpam-3441	175	62	β	β	NOUN
ejpam-3441	175	63	and	and	CCONJ
ejpam-3441	175	64	min{γχa(zy	min{γχa(zy	NUM
ejpam-3441	175	65	)	)	PUNCT
ejpam-3441	175	66	,	,	PUNCT
ejpam-3441	175	67	(	(	PUNCT
ejpam-3441	175	68	1−	1−	NUM
ejpam-3441	175	69	α	α	NOUN
ejpam-3441	175	70	)	)	PUNCT
ejpam-3441	175	71	}	}	PUNCT
ejpam-3441	175	72	≤	≤	NOUN
ejpam-3441	175	73	1−	1−	NUM
ejpam-3441	175	74	β	β	X
ejpam-3441	175	75	.	.	PUNCT
ejpam-3441	176	1	this	this	PRON
ejpam-3441	176	2	implies	imply	VERB
ejpam-3441	176	3	that	that	SCONJ
ejpam-3441	176	4	µχa(zy	µχa(zy	X
ejpam-3441	176	5	)	)	PUNCT
ejpam-3441	176	6	=	=	SYM
ejpam-3441	176	7	1	1	NUM
ejpam-3441	176	8	and	and	CCONJ
ejpam-3441	176	9	γχa(zy	γχa(zy	NOUN
ejpam-3441	176	10	)	)	PUNCT
ejpam-3441	176	11	=	=	SYM
ejpam-3441	176	12	0	0	NUM
ejpam-3441	176	13	,	,	PUNCT
ejpam-3441	176	14	i.e.	i.e.	X
ejpam-3441	176	15	,	,	PUNCT
ejpam-3441	176	16	zy	zy	PROPN
ejpam-3441	176	17	∈	∈	PROPN
ejpam-3441	176	18	a.	a.	NOUN
ejpam-3441	176	19	therefore	therefore	ADV
ejpam-3441	176	20	a	a	PRON
ejpam-3441	176	21	is	be	AUX
ejpam-3441	176	22	a	a	DET
ejpam-3441	176	23	left	left	ADJ
ejpam-3441	176	24	ideal	ideal	NOUN
ejpam-3441	176	25	of	of	ADP
ejpam-3441	176	26	r.	r.	PROPN
ejpam-3441	176	27	remark	remark	PROPN
ejpam-3441	176	28	1	1	NUM
ejpam-3441	176	29	.	.	PUNCT
ejpam-3441	177	1	(	(	PUNCT
ejpam-3441	177	2	i	i	NOUN
ejpam-3441	177	3	)	)	PUNCT
ejpam-3441	177	4	a	a	PRON
ejpam-3441	177	5	is	be	AUX
ejpam-3441	177	6	an	an	DET
ejpam-3441	177	7	additive	additive	ADJ
ejpam-3441	177	8	la	la	PROPN
ejpam-3441	177	9	-	-	NOUN
ejpam-3441	177	10	subgroup	subgroup	NOUN
ejpam-3441	177	11	of	of	ADP
ejpam-3441	177	12	r	r	NOUN
ejpam-3441	177	13	if	if	SCONJ
ejpam-3441	178	1	and	and	CCONJ
ejpam-3441	178	2	only	only	ADV
ejpam-3441	178	3	if	if	SCONJ
ejpam-3441	178	4	χa	χa	PROPN
ejpam-3441	178	5	is	be	AUX
ejpam-3441	178	6	an	an	DET
ejpam-3441	178	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	178	8	fuzzy	fuzzy	ADJ
ejpam-3441	178	9	additive	additive	ADJ
ejpam-3441	178	10	la	la	PROPN
ejpam-3441	178	11	-	-	NOUN
ejpam-3441	178	12	subgroup	subgroup	NOUN
ejpam-3441	178	13	with	with	ADP
ejpam-3441	178	14	thresholds	threshold	NOUN
ejpam-3441	178	15	(	(	PUNCT
ejpam-3441	178	16	α	α	X
ejpam-3441	178	17	,	,	PUNCT
ejpam-3441	178	18	β	β	X
ejpam-3441	178	19	]	]	PUNCT
ejpam-3441	178	20	of	of	ADP
ejpam-3441	178	21	r.	r.	PROPN
ejpam-3441	178	22	(	(	PUNCT
ejpam-3441	178	23	ii	ii	PROPN
ejpam-3441	178	24	)	)	PUNCT
ejpam-3441	178	25	a	a	PRON
ejpam-3441	178	26	is	be	AUX
ejpam-3441	178	27	an	an	DET
ejpam-3441	178	28	la	la	NOUN
ejpam-3441	178	29	-	-	PUNCT
ejpam-3441	178	30	subsemigroup	subsemigroup	NOUN
ejpam-3441	178	31	of	of	ADP
ejpam-3441	178	32	r	r	NOUN
ejpam-3441	178	33	if	if	SCONJ
ejpam-3441	179	1	and	and	CCONJ
ejpam-3441	179	2	only	only	ADV
ejpam-3441	179	3	if	if	SCONJ
ejpam-3441	179	4	χa	χa	PROPN
ejpam-3441	179	5	is	be	AUX
ejpam-3441	179	6	an	an	DET
ejpam-3441	179	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	179	8	fuzzy	fuzzy	ADJ
ejpam-3441	179	9	lasubsemigroup	lasubsemigroup	NOUN
ejpam-3441	179	10	with	with	ADP
ejpam-3441	179	11	thresholds	threshold	NOUN
ejpam-3441	179	12	(	(	PUNCT
ejpam-3441	179	13	α	α	X
ejpam-3441	179	14	,	,	PUNCT
ejpam-3441	179	15	β	β	X
ejpam-3441	179	16	]	]	PUNCT
ejpam-3441	179	17	of	of	ADP
ejpam-3441	179	18	r.	r.	PROPN
ejpam-3441	179	19	k.	k.	PROPN
ejpam-3441	179	20	nasreen	nasreen	PROPN
ejpam-3441	179	21	et	et	PROPN
ejpam-3441	179	22	al	al	PROPN
ejpam-3441	179	23	.	.	PUNCT
ejpam-3441	179	24	/	/	SYM
ejpam-3441	179	25	eur	eur	PROPN
ejpam-3441	179	26	.	.	PUNCT
ejpam-3441	180	1	j.	j.	PROPN
ejpam-3441	180	2	pure	pure	PROPN
ejpam-3441	180	3	appl	appl	PROPN
ejpam-3441	180	4	.	.	PROPN
ejpam-3441	180	5	math	math	PROPN
ejpam-3441	180	6	,	,	PUNCT
ejpam-3441	180	7	12	12	NUM
ejpam-3441	180	8	(	(	PUNCT
ejpam-3441	180	9	3	3	NUM
ejpam-3441	180	10	)	)	PUNCT
ejpam-3441	180	11	(	(	PUNCT
ejpam-3441	180	12	2019	2019	NUM
ejpam-3441	180	13	)	)	PUNCT
ejpam-3441	180	14	,	,	PUNCT
ejpam-3441	180	15	906	906	NUM
ejpam-3441	180	16	-	-	SYM
ejpam-3441	180	17	943	943	NUM
ejpam-3441	180	18	914	914	NUM
ejpam-3441	180	19	theorem	theorem	NOUN
ejpam-3441	180	20	3	3	X
ejpam-3441	180	21	.	.	PUNCT
ejpam-3441	180	22	let	let	VERB
ejpam-3441	180	23	a	a	DET
ejpam-3441	180	24	be	be	AUX
ejpam-3441	180	25	an	an	DET
ejpam-3441	180	26	ifs	ifs	PROPN
ejpam-3441	180	27	of	of	ADP
ejpam-3441	180	28	an	an	DET
ejpam-3441	180	29	la	la	ADJ
ejpam-3441	180	30	-	-	PUNCT
ejpam-3441	180	31	ring	ring	NOUN
ejpam-3441	180	32	r.	r.	PROPN
ejpam-3441	180	33	then	then	ADV
ejpam-3441	180	34	the	the	DET
ejpam-3441	180	35	following	follow	VERB
ejpam-3441	180	36	assertions	assertion	NOUN
ejpam-3441	180	37	hold	hold	VERB
ejpam-3441	180	38	.	.	PUNCT
ejpam-3441	181	1	(	(	PUNCT
ejpam-3441	181	2	1	1	X
ejpam-3441	181	3	)	)	PUNCT
ejpam-3441	181	4	a	a	PRON
ejpam-3441	181	5	is	be	AUX
ejpam-3441	181	6	an	an	DET
ejpam-3441	181	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	181	8	fuzzy	fuzzy	ADJ
ejpam-3441	181	9	la	la	NOUN
ejpam-3441	181	10	-	-	PUNCT
ejpam-3441	181	11	subring	subre	VERB
ejpam-3441	181	12	with	with	ADP
ejpam-3441	181	13	thresholds	threshold	NOUN
ejpam-3441	181	14	(	(	PUNCT
ejpam-3441	181	15	α	α	X
ejpam-3441	181	16	,	,	PUNCT
ejpam-3441	181	17	β	β	X
ejpam-3441	181	18	]	]	PUNCT
ejpam-3441	181	19	of	of	ADP
ejpam-3441	181	20	r	r	NOUN
ejpam-3441	181	21	if	if	SCONJ
ejpam-3441	182	1	and	and	CCONJ
ejpam-3441	182	2	only	only	ADV
ejpam-3441	182	3	if	if	SCONJ
ejpam-3441	182	4	a	a	DET
ejpam-3441	182	5	◦	◦	NOUN
ejpam-3441	182	6	βα	βα	NOUN
ejpam-3441	182	7	a	a	DET
ejpam-3441	182	8	⊆	⊆	NUM
ejpam-3441	182	9	aβα	aβα	NOUN
ejpam-3441	182	10	and	and	CCONJ
ejpam-3441	182	11	a−βα	a−βα	PROPN
ejpam-3441	182	12	a	a	DET
ejpam-3441	182	13	⊆	⊆	NUM
ejpam-3441	182	14	aβα	aβα	NOUN
ejpam-3441	182	15	.	.	PUNCT
ejpam-3441	183	1	(	(	PUNCT
ejpam-3441	183	2	2	2	X
ejpam-3441	183	3	)	)	PUNCT
ejpam-3441	183	4	a	a	PRON
ejpam-3441	183	5	is	be	AUX
ejpam-3441	183	6	an	an	DET
ejpam-3441	183	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	183	8	fuzzy	fuzzy	ADJ
ejpam-3441	183	9	left	left	NOUN
ejpam-3441	183	10	(	(	PUNCT
ejpam-3441	183	11	resp	resp	NOUN
ejpam-3441	183	12	.	.	PUNCT
ejpam-3441	184	1	right	right	ADJ
ejpam-3441	184	2	)	)	PUNCT
ejpam-3441	184	3	ideal	ideal	NOUN
ejpam-3441	184	4	with	with	ADP
ejpam-3441	184	5	thresholds	threshold	NOUN
ejpam-3441	184	6	(	(	PUNCT
ejpam-3441	184	7	α	α	X
ejpam-3441	184	8	,	,	PUNCT
ejpam-3441	184	9	β	β	X
ejpam-3441	184	10	]	]	PUNCT
ejpam-3441	184	11	of	of	ADP
ejpam-3441	184	12	r	r	NOUN
ejpam-3441	184	13	if	if	SCONJ
ejpam-3441	185	1	and	and	CCONJ
ejpam-3441	185	2	only	only	ADV
ejpam-3441	185	3	if	if	SCONJ
ejpam-3441	185	4	r	r	NOUN
ejpam-3441	185	5	◦	◦	NOUN
ejpam-3441	185	6	βα	βα	X
ejpam-3441	185	7	a	a	DET
ejpam-3441	185	8	⊆	⊆	NUM
ejpam-3441	185	9	aβα	aβα	NOUN
ejpam-3441	185	10	(	(	PUNCT
ejpam-3441	185	11	resp	resp	NOUN
ejpam-3441	185	12	.	.	PUNCT
ejpam-3441	186	1	a	a	DET
ejpam-3441	186	2	◦	◦	NOUN
ejpam-3441	186	3	βα	βα	NOUN
ejpam-3441	186	4	r	r	NOUN
ejpam-3441	186	5	⊆	⊆	NUM
ejpam-3441	186	6	aβα	aβα	NOUN
ejpam-3441	186	7	)	)	PUNCT
ejpam-3441	186	8	and	and	CCONJ
ejpam-3441	186	9	a−βα	a−βα	ADV
ejpam-3441	186	10	a	a	DET
ejpam-3441	186	11	⊆	⊆	NUM
ejpam-3441	186	12	aβα	aβα	NOUN
ejpam-3441	186	13	.	.	PUNCT
ejpam-3441	186	14	proof	proof	NOUN
ejpam-3441	186	15	.	.	PUNCT
ejpam-3441	187	1	(	(	PUNCT
ejpam-3441	187	2	1	1	X
ejpam-3441	187	3	)	)	PUNCT
ejpam-3441	187	4	suppose	suppose	VERB
ejpam-3441	187	5	that	that	SCONJ
ejpam-3441	187	6	a	a	DET
ejpam-3441	187	7	=	=	SYM
ejpam-3441	187	8	(	(	PUNCT
ejpam-3441	187	9	µa	µa	PROPN
ejpam-3441	187	10	,	,	PUNCT
ejpam-3441	187	11	γa	γa	PROPN
ejpam-3441	187	12	)	)	PUNCT
ejpam-3441	187	13	is	be	AUX
ejpam-3441	187	14	an	an	DET
ejpam-3441	187	15	intuitionistic	intuitionistic	ADJ
ejpam-3441	187	16	fuzzy	fuzzy	ADJ
ejpam-3441	187	17	la	la	NOUN
ejpam-3441	187	18	-	-	PUNCT
ejpam-3441	187	19	subring	subre	VERB
ejpam-3441	187	20	with	with	ADP
ejpam-3441	187	21	thresholds	threshold	NOUN
ejpam-3441	187	22	(	(	PUNCT
ejpam-3441	187	23	α	α	X
ejpam-3441	187	24	,	,	PUNCT
ejpam-3441	187	25	β	β	X
ejpam-3441	187	26	]	]	PUNCT
ejpam-3441	187	27	of	of	ADP
ejpam-3441	187	28	an	an	DET
ejpam-3441	187	29	la	la	ADJ
ejpam-3441	187	30	-	-	PUNCT
ejpam-3441	187	31	ring	ring	NOUN
ejpam-3441	187	32	r	r	NOUN
ejpam-3441	187	33	and	and	CCONJ
ejpam-3441	187	34	x	x	PROPN
ejpam-3441	187	35	∈	∈	PROPN
ejpam-3441	187	36	r.	r.	PROPN
ejpam-3441	187	37	for	for	ADP
ejpam-3441	187	38	a	a	DET
ejpam-3441	187	39	◦	◦	NOUN
ejpam-3441	187	40	βα	βα	NOUN
ejpam-3441	187	41	a	a	DET
ejpam-3441	187	42	⊆	⊆	NUM
ejpam-3441	187	43	aβα	aβα	NOUN
ejpam-3441	187	44	.	.	PUNCT
ejpam-3441	188	1	if	if	SCONJ
ejpam-3441	188	2	(	(	PUNCT
ejpam-3441	188	3	a	a	DET
ejpam-3441	188	4	◦	◦	NOUN
ejpam-3441	188	5	βα	βα	NOUN
ejpam-3441	188	6	a)(x	a)(x	NOUN
ejpam-3441	188	7	)	)	PUNCT
ejpam-3441	188	8	=	=	SYM
ejpam-3441	188	9	0	0	NUM
ejpam-3441	188	10	,	,	PUNCT
ejpam-3441	188	11	then	then	ADV
ejpam-3441	188	12	obvious	obvious	VERB
ejpam-3441	188	13	a	a	DET
ejpam-3441	188	14	◦	◦	NOUN
ejpam-3441	188	15	βα	βα	NOUN
ejpam-3441	188	16	a	a	DET
ejpam-3441	188	17	⊆	⊆	NUM
ejpam-3441	188	18	aβα	aβα	NOUN
ejpam-3441	188	19	,	,	PUNCT
ejpam-3441	188	20	otherwise	otherwise	ADV
ejpam-3441	188	21	we	we	PRON
ejpam-3441	188	22	have	have	VERB
ejpam-3441	188	23	(	(	PUNCT
ejpam-3441	188	24	µa	µa	ADP
ejpam-3441	188	25	◦	◦	NOUN
ejpam-3441	188	26	βα	βα	ADJ
ejpam-3441	188	27	µa)(x	µa)(x	NOUN
ejpam-3441	188	28	)	)	PUNCT
ejpam-3441	189	1	=	=	PRON
ejpam-3441	189	2	{	{	PUNCT
ejpam-3441	189	3	(	(	PUNCT
ejpam-3441	189	4	µa	µa	ADP
ejpam-3441	189	5	◦	◦	NOUN
ejpam-3441	189	6	µa)(x	µa)(x	NOUN
ejpam-3441	189	7	)	)	PUNCT
ejpam-3441	190	1	∧	∧	PROPN
ejpam-3441	190	2	β	β	NOUN
ejpam-3441	190	3	}	}	PUNCT
ejpam-3441	190	4	∨	∨	NUM
ejpam-3441	190	5	α	α	NOUN
ejpam-3441	190	6	=	=	X
ejpam-3441	190	7	{	{	PUNCT
ejpam-3441	190	8	(	(	PUNCT
ejpam-3441	190	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	190	10	i=1	i=1	PROPN
ejpam-3441	190	11	aibi	aibi	NOUN
ejpam-3441	190	12	{	{	PUNCT
ejpam-3441	190	13	∧ni=1	∧ni=1	X
ejpam-3441	190	14	{	{	PUNCT
ejpam-3441	190	15	µa	µa	X
ejpam-3441	190	16	(	(	PUNCT
ejpam-3441	190	17	ai	ai	NOUN
ejpam-3441	190	18	)	)	PUNCT
ejpam-3441	190	19	∧	∧	NOUN
ejpam-3441	190	20	µa	µa	NOUN
ejpam-3441	190	21	(	(	PUNCT
ejpam-3441	190	22	bi	bi	NOUN
ejpam-3441	190	23	)	)	PUNCT
ejpam-3441	190	24	}	}	PUNCT
ejpam-3441	190	25	}	}	PUNCT
ejpam-3441	190	26	)	)	PUNCT
ejpam-3441	191	1	∧	∧	PROPN
ejpam-3441	191	2	β	β	NOUN
ejpam-3441	191	3	}	}	PUNCT
ejpam-3441	191	4	∨	∨	NUM
ejpam-3441	191	5	α	α	NOUN
ejpam-3441	191	6	≤	≤	NOUN
ejpam-3441	191	7	{	{	PUNCT
ejpam-3441	191	8	(	(	PUNCT
ejpam-3441	191	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	191	10	i=1	i=1	PROPN
ejpam-3441	191	11	aibi	aibi	NOUN
ejpam-3441	191	12	{	{	PUNCT
ejpam-3441	191	13	∧ni=1µa	∧ni=1µa	NOUN
ejpam-3441	191	14	(	(	PUNCT
ejpam-3441	191	15	aibi	aibi	NOUN
ejpam-3441	191	16	)	)	PUNCT
ejpam-3441	191	17	}	}	PUNCT
ejpam-3441	191	18	)	)	PUNCT
ejpam-3441	192	1	∧	∧	PROPN
ejpam-3441	192	2	β	β	NOUN
ejpam-3441	192	3	}	}	PUNCT
ejpam-3441	192	4	∨	∨	NUM
ejpam-3441	192	5	α	α	NOUN
ejpam-3441	192	6	=	=	SYM
ejpam-3441	192	7	{	{	PUNCT
ejpam-3441	192	8	(	(	PUNCT
ejpam-3441	192	9	µa(x	µa(x	NOUN
ejpam-3441	192	10	)	)	PUNCT
ejpam-3441	192	11	∧	∧	PROPN
ejpam-3441	192	12	β	β	NOUN
ejpam-3441	192	13	)	)	PUNCT
ejpam-3441	192	14	}	}	PUNCT
ejpam-3441	192	15	∨	∨	NUM
ejpam-3441	192	16	α	α	X
ejpam-3441	192	17	=	=	SYM
ejpam-3441	192	18	(	(	PUNCT
ejpam-3441	192	19	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	192	20	)	)	PUNCT
ejpam-3441	192	21	.	.	PUNCT
ejpam-3441	193	1	⇒	⇒	PROPN
ejpam-3441	193	2	µa	µa	ADP
ejpam-3441	193	3	◦	◦	NOUN
ejpam-3441	193	4	βα	βα	NOUN
ejpam-3441	193	5	µa	µa	NOUN
ejpam-3441	193	6	⊆	⊆	NUM
ejpam-3441	193	7	(	(	PUNCT
ejpam-3441	193	8	µa)βα	µa)βα	NUM
ejpam-3441	193	9	.	.	PUNCT
ejpam-3441	194	1	similarly	similarly	ADV
ejpam-3441	194	2	,	,	PUNCT
ejpam-3441	194	3	we	we	PRON
ejpam-3441	194	4	have	have	VERB
ejpam-3441	194	5	γa	γa	PROPN
ejpam-3441	194	6	◦	◦	NOUN
ejpam-3441	194	7	βα	βα	X
ejpam-3441	194	8	γa	γa	PROPN
ejpam-3441	194	9	⊇	⊇	PROPN
ejpam-3441	194	10	(	(	PUNCT
ejpam-3441	194	11	γa)βα	γa)βα	X
ejpam-3441	194	12	.	.	PUNCT
ejpam-3441	195	1	thus	thus	ADV
ejpam-3441	195	2	a	a	DET
ejpam-3441	195	3	◦	◦	NOUN
ejpam-3441	195	4	βα	βα	VERB
ejpam-3441	195	5	a	a	DET
ejpam-3441	195	6	⊆	⊆	NUM
ejpam-3441	195	7	aβα	aβα	NOUN
ejpam-3441	195	8	.	.	PUNCT
ejpam-3441	196	1	now	now	ADV
ejpam-3441	196	2	for	for	ADP
ejpam-3441	196	3	a−βα	a−βα	PROPN
ejpam-3441	196	4	a	a	DET
ejpam-3441	196	5	⊆	⊆	NUM
ejpam-3441	196	6	aβα	aβα	NOUN
ejpam-3441	196	7	.	.	PUNCT
ejpam-3441	197	1	if	if	SCONJ
ejpam-3441	197	2	(	(	PUNCT
ejpam-3441	197	3	a−βα	a−βα	ADV
ejpam-3441	197	4	a)(x	a)(x	NOUN
ejpam-3441	197	5	)	)	PUNCT
ejpam-3441	197	6	=	=	SYM
ejpam-3441	197	7	0	0	NUM
ejpam-3441	197	8	,	,	PUNCT
ejpam-3441	197	9	then	then	ADV
ejpam-3441	197	10	obvious	obvious	ADJ
ejpam-3441	197	11	a−βα	a−βα	PROPN
ejpam-3441	197	12	a	a	DET
ejpam-3441	197	13	⊆	⊆	NUM
ejpam-3441	197	14	aβα	aβα	NOUN
ejpam-3441	197	15	,	,	PUNCT
ejpam-3441	197	16	otherwise	otherwise	ADV
ejpam-3441	197	17	we	we	PRON
ejpam-3441	197	18	have	have	VERB
ejpam-3441	197	19	(	(	PUNCT
ejpam-3441	197	20	µa	µa	NOUN
ejpam-3441	197	21	−βα	−βα	PROPN
ejpam-3441	197	22	µa)(x	µa)(x	NOUN
ejpam-3441	197	23	)	)	PUNCT
ejpam-3441	198	1	=	=	PRON
ejpam-3441	198	2	{	{	PUNCT
ejpam-3441	198	3	(	(	PUNCT
ejpam-3441	198	4	µa	µa	NOUN
ejpam-3441	198	5	−	−	PROPN
ejpam-3441	198	6	µa)(x	µa)(x	NOUN
ejpam-3441	198	7	)	)	PUNCT
ejpam-3441	199	1	∧	∧	PROPN
ejpam-3441	199	2	β	β	NOUN
ejpam-3441	199	3	}	}	PUNCT
ejpam-3441	199	4	∨	∨	NUM
ejpam-3441	199	5	α	α	NOUN
ejpam-3441	199	6	=	=	X
ejpam-3441	199	7	{	{	PUNCT
ejpam-3441	199	8	(	(	PUNCT
ejpam-3441	199	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	199	10	i=1	i=1	PROPN
ejpam-3441	199	11	ai−bi{∧	ai−bi{∧	PROPN
ejpam-3441	199	12	n	n	NOUN
ejpam-3441	199	13	i=1	i=1	PROPN
ejpam-3441	199	14	{	{	PUNCT
ejpam-3441	199	15	µa	µa	X
ejpam-3441	199	16	(	(	PUNCT
ejpam-3441	199	17	ai	ai	NOUN
ejpam-3441	199	18	)	)	PUNCT
ejpam-3441	199	19	∧	∧	NOUN
ejpam-3441	199	20	µa	µa	NOUN
ejpam-3441	199	21	(	(	PUNCT
ejpam-3441	199	22	bi	bi	NOUN
ejpam-3441	199	23	)	)	PUNCT
ejpam-3441	199	24	}	}	PUNCT
ejpam-3441	199	25	}	}	PUNCT
ejpam-3441	199	26	)	)	PUNCT
ejpam-3441	199	27	∧	∧	PROPN
ejpam-3441	199	28	β	β	NOUN
ejpam-3441	199	29	}	}	PUNCT
ejpam-3441	199	30	∨	∨	NUM
ejpam-3441	199	31	α	α	NOUN
ejpam-3441	199	32	≤	≤	NOUN
ejpam-3441	199	33	{	{	PUNCT
ejpam-3441	199	34	(	(	PUNCT
ejpam-3441	199	35	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	199	36	i=1	i=1	PROPN
ejpam-3441	199	37	ai−bi{∧	ai−bi{∧	PROPN
ejpam-3441	199	38	n	n	PRON
ejpam-3441	199	39	i=1µa	i=1µa	X
ejpam-3441	199	40	(	(	PUNCT
ejpam-3441	199	41	ai	ai	VERB
ejpam-3441	199	42	−	−	PROPN
ejpam-3441	199	43	bi	bi	NOUN
ejpam-3441	199	44	)	)	PUNCT
ejpam-3441	199	45	}	}	PUNCT
ejpam-3441	199	46	)	)	PUNCT
ejpam-3441	200	1	∧	∧	PROPN
ejpam-3441	200	2	β	β	NOUN
ejpam-3441	200	3	}	}	PUNCT
ejpam-3441	200	4	∨	∨	NUM
ejpam-3441	200	5	α	α	NOUN
ejpam-3441	200	6	=	=	SYM
ejpam-3441	200	7	{	{	PUNCT
ejpam-3441	200	8	(	(	PUNCT
ejpam-3441	200	9	µa(x	µa(x	NOUN
ejpam-3441	200	10	)	)	PUNCT
ejpam-3441	200	11	∧	∧	PROPN
ejpam-3441	200	12	β	β	NOUN
ejpam-3441	200	13	)	)	PUNCT
ejpam-3441	200	14	}	}	PUNCT
ejpam-3441	200	15	∨	∨	NUM
ejpam-3441	200	16	α	α	X
ejpam-3441	200	17	=	=	SYM
ejpam-3441	200	18	(	(	PUNCT
ejpam-3441	200	19	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	200	20	)	)	PUNCT
ejpam-3441	200	21	.	.	PUNCT
ejpam-3441	201	1	⇒	⇒	PROPN
ejpam-3441	201	2	µa	µa	ADP
ejpam-3441	201	3	−βα	−βα	X
ejpam-3441	201	4	µa	µa	ADP
ejpam-3441	201	5	⊆	⊆	NUM
ejpam-3441	201	6	(	(	PUNCT
ejpam-3441	201	7	µa)βα	µa)βα	NUM
ejpam-3441	201	8	.	.	PUNCT
ejpam-3441	202	1	similarly	similarly	ADV
ejpam-3441	202	2	,	,	PUNCT
ejpam-3441	202	3	we	we	PRON
ejpam-3441	202	4	have	have	VERB
ejpam-3441	202	5	γa	γa	PROPN
ejpam-3441	202	6	−βα	−βα	PROPN
ejpam-3441	202	7	γa	γa	PROPN
ejpam-3441	202	8	⊇	⊇	PROPN
ejpam-3441	202	9	(	(	PUNCT
ejpam-3441	202	10	γa)βα	γa)βα	X
ejpam-3441	202	11	.	.	PUNCT
ejpam-3441	203	1	thus	thus	ADV
ejpam-3441	203	2	a−βα	a−βα	ADV
ejpam-3441	203	3	a	a	DET
ejpam-3441	203	4	⊆	⊆	NUM
ejpam-3441	203	5	aβα	aβα	NOUN
ejpam-3441	203	6	.	.	PUNCT
ejpam-3441	204	1	conversely	conversely	ADV
ejpam-3441	204	2	,	,	PUNCT
ejpam-3441	204	3	assume	assume	VERB
ejpam-3441	204	4	that	that	SCONJ
ejpam-3441	204	5	a	a	DET
ejpam-3441	204	6	◦	◦	NOUN
ejpam-3441	204	7	βαa	βαa	NOUN
ejpam-3441	204	8	⊆	⊆	NUM
ejpam-3441	204	9	aβα	aβα	NOUN
ejpam-3441	204	10	and	and	CCONJ
ejpam-3441	204	11	a−βαa	a−βαa	VERB
ejpam-3441	204	12	⊆	⊆	NUM
ejpam-3441	204	13	aβα	aβα	NOUN
ejpam-3441	204	14	.	.	PUNCT
ejpam-3441	205	1	let	let	VERB
ejpam-3441	205	2	x	x	PRON
ejpam-3441	205	3	,	,	PUNCT
ejpam-3441	205	4	y	y	PROPN
ejpam-3441	205	5	∈	∈	PROPN
ejpam-3441	205	6	r	r	NOUN
ejpam-3441	206	1	such	such	DET
ejpam-3441	206	2	that	that	SCONJ
ejpam-3441	206	3	a	a	DET
ejpam-3441	206	4	=	=	X
ejpam-3441	206	5	xy	xy	PROPN
ejpam-3441	206	6	.	.	PUNCT
ejpam-3441	207	1	now	now	ADV
ejpam-3441	207	2	max{µa(xy	max{µa(xy	NOUN
ejpam-3441	207	3	)	)	PUNCT
ejpam-3441	207	4	,	,	PUNCT
ejpam-3441	207	5	α	α	X
ejpam-3441	207	6	}	}	PUNCT
ejpam-3441	207	7	=	=	SYM
ejpam-3441	207	8	max{µa(a	max{µa(a	PROPN
ejpam-3441	207	9	)	)	PUNCT
ejpam-3441	207	10	,	,	PUNCT
ejpam-3441	207	11	α	α	X
ejpam-3441	207	12	}	}	PUNCT
ejpam-3441	207	13	=	=	SYM
ejpam-3441	207	14	max{min{µa(a	max{min{µa(a	PROPN
ejpam-3441	207	15	)	)	PUNCT
ejpam-3441	207	16	,	,	PUNCT
ejpam-3441	207	17	β	β	X
ejpam-3441	207	18	}	}	PUNCT
ejpam-3441	207	19	,	,	PUNCT
ejpam-3441	207	20	α	α	NOUN
ejpam-3441	207	21	}	}	PUNCT
ejpam-3441	207	22	=	=	SYM
ejpam-3441	207	23	(	(	PUNCT
ejpam-3441	207	24	µa)βα(a	µa)βα(a	PROPN
ejpam-3441	207	25	)	)	PUNCT
ejpam-3441	207	26	≥	≥	NOUN
ejpam-3441	207	27	(	(	PUNCT
ejpam-3441	207	28	µa	µa	ADP
ejpam-3441	207	29	◦	◦	NOUN
ejpam-3441	207	30	βα	βα	NOUN
ejpam-3441	207	31	µa)(a	µa)(a	ADV
ejpam-3441	207	32	)	)	PUNCT
ejpam-3441	208	1	=	=	PRON
ejpam-3441	208	2	{	{	PUNCT
ejpam-3441	208	3	(	(	PUNCT
ejpam-3441	208	4	µa	µa	ADP
ejpam-3441	208	5	◦	◦	NOUN
ejpam-3441	208	6	µa)(a	µa)(a	NOUN
ejpam-3441	208	7	)	)	PUNCT
ejpam-3441	209	1	∧	∧	PROPN
ejpam-3441	209	2	β	β	NOUN
ejpam-3441	209	3	}	}	PUNCT
ejpam-3441	209	4	∨	∨	NUM
ejpam-3441	209	5	α	α	NOUN
ejpam-3441	209	6	=	=	X
ejpam-3441	209	7	{	{	PUNCT
ejpam-3441	209	8	(	(	PUNCT
ejpam-3441	209	9	∨a=∑n	∨a=∑n	PROPN
ejpam-3441	209	10	i=1	i=1	PROPN
ejpam-3441	209	11	aibi	aibi	NOUN
ejpam-3441	209	12	{	{	PUNCT
ejpam-3441	209	13	∧ni=1	∧ni=1	X
ejpam-3441	209	14	{	{	PUNCT
ejpam-3441	209	15	µa	µa	X
ejpam-3441	209	16	(	(	PUNCT
ejpam-3441	209	17	ai	ai	NOUN
ejpam-3441	209	18	)	)	PUNCT
ejpam-3441	209	19	∧	∧	NOUN
ejpam-3441	209	20	µa	µa	NOUN
ejpam-3441	209	21	(	(	PUNCT
ejpam-3441	209	22	bi	bi	NOUN
ejpam-3441	209	23	)	)	PUNCT
ejpam-3441	209	24	}	}	PUNCT
ejpam-3441	209	25	}	}	PUNCT
ejpam-3441	209	26	)	)	PUNCT
ejpam-3441	209	27	∧	∧	PROPN
ejpam-3441	209	28	β	β	NOUN
ejpam-3441	209	29	}	}	PUNCT
ejpam-3441	209	30	∨	∨	NUM
ejpam-3441	209	31	α	α	PROPN
ejpam-3441	209	32	≥	≥	X
ejpam-3441	209	33	{	{	PUNCT
ejpam-3441	209	34	(	(	PUNCT
ejpam-3441	209	35	µa(x	µa(x	NOUN
ejpam-3441	209	36	)	)	PUNCT
ejpam-3441	209	37	∧	∧	PROPN
ejpam-3441	209	38	µa	µa	PROPN
ejpam-3441	209	39	(	(	PUNCT
ejpam-3441	209	40	y	y	NOUN
ejpam-3441	209	41	)	)	PUNCT
ejpam-3441	209	42	)	)	PUNCT
ejpam-3441	210	1	∧	∧	PROPN
ejpam-3441	210	2	β	β	NOUN
ejpam-3441	210	3	}	}	PUNCT
ejpam-3441	210	4	∨	∨	NUM
ejpam-3441	210	5	α	α	X
ejpam-3441	210	6	=	=	SYM
ejpam-3441	210	7	(	(	PUNCT
ejpam-3441	210	8	µa(x	µa(x	NOUN
ejpam-3441	210	9	)	)	PUNCT
ejpam-3441	210	10	∧	∧	PROPN
ejpam-3441	210	11	µa	µa	PROPN
ejpam-3441	210	12	(	(	PUNCT
ejpam-3441	210	13	y	y	NOUN
ejpam-3441	210	14	)	)	PUNCT
ejpam-3441	210	15	)	)	PUNCT
ejpam-3441	211	1	∧	∧	NOUN
ejpam-3441	211	2	β	β	NOUN
ejpam-3441	211	3	=	=	SYM
ejpam-3441	211	4	min{µa(x	min{µa(x	NOUN
ejpam-3441	211	5	)	)	PUNCT
ejpam-3441	211	6	,	,	PUNCT
ejpam-3441	211	7	µa	µa	ADP
ejpam-3441	211	8	(	(	PUNCT
ejpam-3441	211	9	y	y	NOUN
ejpam-3441	211	10	)	)	PUNCT
ejpam-3441	211	11	,	,	PUNCT
ejpam-3441	211	12	β	β	NOUN
ejpam-3441	211	13	}	}	PUNCT
ejpam-3441	211	14	.	.	PUNCT
ejpam-3441	212	1	⇒	⇒	PROPN
ejpam-3441	212	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3441	212	3	)	)	PUNCT
ejpam-3441	212	4	,	,	PUNCT
ejpam-3441	212	5	α	α	X
ejpam-3441	212	6	}	}	PUNCT
ejpam-3441	212	7	≥	≥	NOUN
ejpam-3441	212	8	min{µa(x	min{µa(x	NOUN
ejpam-3441	212	9	)	)	PUNCT
ejpam-3441	212	10	,	,	PUNCT
ejpam-3441	212	11	µa	µa	ADP
ejpam-3441	212	12	(	(	PUNCT
ejpam-3441	212	13	y	y	NOUN
ejpam-3441	212	14	)	)	PUNCT
ejpam-3441	212	15	,	,	PUNCT
ejpam-3441	212	16	β	β	X
ejpam-3441	212	17	}	}	PUNCT
ejpam-3441	212	18	.	.	PUNCT
ejpam-3441	213	1	similarly	similarly	ADV
ejpam-3441	213	2	,	,	PUNCT
ejpam-3441	213	3	we	we	PRON
ejpam-3441	213	4	have	have	VERB
ejpam-3441	213	5	min{γa(xy	min{γa(xy	NUM
ejpam-3441	213	6	)	)	PUNCT
ejpam-3441	213	7	,	,	PUNCT
ejpam-3441	213	8	(	(	PUNCT
ejpam-3441	213	9	1	1	NUM
ejpam-3441	213	10	−	−	PROPN
ejpam-3441	213	11	α	α	NOUN
ejpam-3441	213	12	)	)	PUNCT
ejpam-3441	213	13	}	}	PUNCT
ejpam-3441	213	14	≤	≤	NUM
ejpam-3441	213	15	max{γa(x	max{γa(x	NOUN
ejpam-3441	213	16	)	)	PUNCT
ejpam-3441	213	17	,	,	PUNCT
ejpam-3441	213	18	γa	γa	PROPN
ejpam-3441	213	19	(	(	PUNCT
ejpam-3441	213	20	y	y	PROPN
ejpam-3441	213	21	)	)	PUNCT
ejpam-3441	213	22	,	,	PUNCT
ejpam-3441	213	23	(	(	PUNCT
ejpam-3441	213	24	1	1	NUM
ejpam-3441	213	25	−	−	NOUN
ejpam-3441	213	26	β	β	NOUN
ejpam-3441	213	27	)	)	PUNCT
ejpam-3441	213	28	}	}	PUNCT
ejpam-3441	213	29	.	.	PUNCT
ejpam-3441	214	1	now	now	ADV
ejpam-3441	214	2	we	we	PRON
ejpam-3441	214	3	set	set	VERB
ejpam-3441	214	4	a	a	DET
ejpam-3441	214	5	=	=	SYM
ejpam-3441	214	6	x−	x−	PROPN
ejpam-3441	214	7	y	y	PROPN
ejpam-3441	214	8	and	and	CCONJ
ejpam-3441	214	9	max{µa(x−	max{µa(x−	PROPN
ejpam-3441	214	10	y	y	PROPN
ejpam-3441	214	11	)	)	PUNCT
ejpam-3441	214	12	,	,	PUNCT
ejpam-3441	214	13	α	α	X
ejpam-3441	214	14	}	}	PUNCT
ejpam-3441	214	15	=	=	SYM
ejpam-3441	214	16	max{µa(a	max{µa(a	PROPN
ejpam-3441	214	17	)	)	PUNCT
ejpam-3441	214	18	,	,	PUNCT
ejpam-3441	214	19	α	α	X
ejpam-3441	214	20	}	}	PUNCT
ejpam-3441	214	21	=	=	SYM
ejpam-3441	214	22	max{min{µa(a	max{min{µa(a	PROPN
ejpam-3441	214	23	)	)	PUNCT
ejpam-3441	214	24	,	,	PUNCT
ejpam-3441	214	25	β	β	X
ejpam-3441	214	26	}	}	PUNCT
ejpam-3441	214	27	,	,	PUNCT
ejpam-3441	214	28	α	α	NOUN
ejpam-3441	214	29	}	}	PUNCT
ejpam-3441	214	30	k.	k.	PROPN
ejpam-3441	214	31	nasreen	nasreen	PROPN
ejpam-3441	214	32	et	et	PROPN
ejpam-3441	214	33	al	al	PROPN
ejpam-3441	214	34	.	.	PUNCT
ejpam-3441	214	35	/	/	SYM
ejpam-3441	214	36	eur	eur	PROPN
ejpam-3441	214	37	.	.	PUNCT
ejpam-3441	215	1	j.	j.	PROPN
ejpam-3441	215	2	pure	pure	PROPN
ejpam-3441	215	3	appl	appl	PROPN
ejpam-3441	215	4	.	.	PROPN
ejpam-3441	215	5	math	math	PROPN
ejpam-3441	215	6	,	,	PUNCT
ejpam-3441	215	7	12	12	NUM
ejpam-3441	215	8	(	(	PUNCT
ejpam-3441	215	9	3	3	NUM
ejpam-3441	215	10	)	)	PUNCT
ejpam-3441	215	11	(	(	PUNCT
ejpam-3441	215	12	2019	2019	NUM
ejpam-3441	215	13	)	)	PUNCT
ejpam-3441	215	14	,	,	PUNCT
ejpam-3441	215	15	906	906	NUM
ejpam-3441	215	16	-	-	SYM
ejpam-3441	215	17	943	943	NUM
ejpam-3441	215	18	915	915	NUM
ejpam-3441	215	19	=	=	SYM
ejpam-3441	215	20	µβα(a	µβα(a	PROPN
ejpam-3441	215	21	)	)	PUNCT
ejpam-3441	215	22	≥	≥	NOUN
ejpam-3441	215	23	(	(	PUNCT
ejpam-3441	215	24	µa	µa	NOUN
ejpam-3441	215	25	−βα	−βα	ADV
ejpam-3441	215	26	µa)(a	µa)(a	ADV
ejpam-3441	215	27	)	)	PUNCT
ejpam-3441	216	1	=	=	PRON
ejpam-3441	216	2	{	{	PUNCT
ejpam-3441	216	3	(	(	PUNCT
ejpam-3441	216	4	µa	µa	NOUN
ejpam-3441	216	5	−	−	NOUN
ejpam-3441	216	6	µa)(a	µa)(a	NOUN
ejpam-3441	216	7	)	)	PUNCT
ejpam-3441	217	1	∧	∧	PROPN
ejpam-3441	217	2	β	β	PROPN
ejpam-3441	217	3	}	}	PUNCT
ejpam-3441	217	4	∨	∨	NUM
ejpam-3441	217	5	α	α	NOUN
ejpam-3441	217	6	=	=	X
ejpam-3441	217	7	{	{	PUNCT
ejpam-3441	217	8	(	(	PUNCT
ejpam-3441	217	9	∨a=∑n	∨a=∑n	PROPN
ejpam-3441	217	10	i=1	i=1	PROPN
ejpam-3441	217	11	ai−bi{∧	ai−bi{∧	PROPN
ejpam-3441	217	12	n	n	ADJ
ejpam-3441	217	13	i=1	i=1	PROPN
ejpam-3441	217	14	{	{	PUNCT
ejpam-3441	217	15	µa	µa	X
ejpam-3441	217	16	(	(	PUNCT
ejpam-3441	217	17	ai	ai	NOUN
ejpam-3441	217	18	)	)	PUNCT
ejpam-3441	217	19	∧	∧	NOUN
ejpam-3441	217	20	µa	µa	NOUN
ejpam-3441	217	21	(	(	PUNCT
ejpam-3441	217	22	bi	bi	NOUN
ejpam-3441	217	23	)	)	PUNCT
ejpam-3441	217	24	}	}	PUNCT
ejpam-3441	217	25	}	}	PUNCT
ejpam-3441	217	26	)	)	PUNCT
ejpam-3441	217	27	∧	∧	PROPN
ejpam-3441	217	28	β	β	NOUN
ejpam-3441	217	29	}	}	PUNCT
ejpam-3441	217	30	∨	∨	NUM
ejpam-3441	217	31	α	α	PROPN
ejpam-3441	217	32	≥	≥	X
ejpam-3441	217	33	{	{	PUNCT
ejpam-3441	217	34	(	(	PUNCT
ejpam-3441	217	35	µa(x	µa(x	NOUN
ejpam-3441	217	36	)	)	PUNCT
ejpam-3441	217	37	∧	∧	PROPN
ejpam-3441	217	38	µa	µa	PROPN
ejpam-3441	217	39	(	(	PUNCT
ejpam-3441	217	40	y	y	NOUN
ejpam-3441	217	41	)	)	PUNCT
ejpam-3441	217	42	)	)	PUNCT
ejpam-3441	218	1	∧	∧	PROPN
ejpam-3441	218	2	β	β	NOUN
ejpam-3441	218	3	}	}	PUNCT
ejpam-3441	218	4	∨	∨	NUM
ejpam-3441	218	5	α	α	X
ejpam-3441	218	6	=	=	SYM
ejpam-3441	218	7	(	(	PUNCT
ejpam-3441	218	8	µa(x	µa(x	NOUN
ejpam-3441	218	9	)	)	PUNCT
ejpam-3441	218	10	∧	∧	PROPN
ejpam-3441	218	11	µa	µa	PROPN
ejpam-3441	218	12	(	(	PUNCT
ejpam-3441	218	13	y	y	NOUN
ejpam-3441	218	14	)	)	PUNCT
ejpam-3441	218	15	)	)	PUNCT
ejpam-3441	219	1	∧	∧	NOUN
ejpam-3441	219	2	β	β	NOUN
ejpam-3441	219	3	=	=	SYM
ejpam-3441	219	4	min{µa(x	min{µa(x	NOUN
ejpam-3441	219	5	)	)	PUNCT
ejpam-3441	219	6	,	,	PUNCT
ejpam-3441	219	7	µa	µa	ADP
ejpam-3441	219	8	(	(	PUNCT
ejpam-3441	219	9	y	y	NOUN
ejpam-3441	219	10	)	)	PUNCT
ejpam-3441	219	11	,	,	PUNCT
ejpam-3441	219	12	β	β	NOUN
ejpam-3441	219	13	}	}	PUNCT
ejpam-3441	219	14	.	.	PUNCT
ejpam-3441	220	1	⇒	⇒	PROPN
ejpam-3441	220	2	max{µa(x−	max{µa(x−	PROPN
ejpam-3441	220	3	y	y	PROPN
ejpam-3441	220	4	)	)	PUNCT
ejpam-3441	220	5	,	,	PUNCT
ejpam-3441	220	6	α	α	X
ejpam-3441	220	7	}	}	PUNCT
ejpam-3441	220	8	≥	≥	NOUN
ejpam-3441	220	9	min{µa(x	min{µa(x	NOUN
ejpam-3441	220	10	)	)	PUNCT
ejpam-3441	220	11	,	,	PUNCT
ejpam-3441	220	12	µa	µa	ADP
ejpam-3441	220	13	(	(	PUNCT
ejpam-3441	220	14	y	y	NOUN
ejpam-3441	220	15	)	)	PUNCT
ejpam-3441	220	16	,	,	PUNCT
ejpam-3441	220	17	β	β	X
ejpam-3441	220	18	}	}	PUNCT
ejpam-3441	220	19	.	.	PUNCT
ejpam-3441	221	1	similarly	similarly	ADV
ejpam-3441	221	2	,	,	PUNCT
ejpam-3441	221	3	we	we	PRON
ejpam-3441	221	4	have	have	VERB
ejpam-3441	221	5	min{γa(x−	min{γa(x−	PROPN
ejpam-3441	221	6	y	y	NOUN
ejpam-3441	221	7	)	)	PUNCT
ejpam-3441	221	8	,	,	PUNCT
ejpam-3441	221	9	(	(	PUNCT
ejpam-3441	221	10	1−α	1−α	NUM
ejpam-3441	221	11	)	)	PUNCT
ejpam-3441	221	12	}	}	PUNCT
ejpam-3441	221	13	≤	≤	NUM
ejpam-3441	221	14	max{γa(x	max{γa(x	NOUN
ejpam-3441	221	15	)	)	PUNCT
ejpam-3441	221	16	,	,	PUNCT
ejpam-3441	221	17	γa	γa	PROPN
ejpam-3441	221	18	(	(	PUNCT
ejpam-3441	221	19	y	y	PROPN
ejpam-3441	221	20	)	)	PUNCT
ejpam-3441	221	21	,	,	PUNCT
ejpam-3441	221	22	(	(	PUNCT
ejpam-3441	221	23	1−	1−	NUM
ejpam-3441	221	24	β	β	NOUN
ejpam-3441	221	25	)	)	PUNCT
ejpam-3441	221	26	}	}	PUNCT
ejpam-3441	221	27	.	.	PUNCT
ejpam-3441	222	1	hence	hence	ADV
ejpam-3441	222	2	a	a	PRON
ejpam-3441	222	3	is	be	AUX
ejpam-3441	222	4	an	an	DET
ejpam-3441	222	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	222	6	fuzzy	fuzzy	ADJ
ejpam-3441	222	7	la	la	NOUN
ejpam-3441	222	8	-	-	PUNCT
ejpam-3441	222	9	subring	subre	VERB
ejpam-3441	222	10	with	with	ADP
ejpam-3441	222	11	thresholds	threshold	NOUN
ejpam-3441	222	12	(	(	PUNCT
ejpam-3441	222	13	α	α	X
ejpam-3441	222	14	,	,	PUNCT
ejpam-3441	222	15	β	β	X
ejpam-3441	222	16	]	]	PUNCT
ejpam-3441	222	17	of	of	ADP
ejpam-3441	222	18	r.	r.	PROPN
ejpam-3441	222	19	(	(	PUNCT
ejpam-3441	222	20	2	2	X
ejpam-3441	222	21	)	)	PUNCT
ejpam-3441	222	22	assume	assume	VERB
ejpam-3441	222	23	that	that	SCONJ
ejpam-3441	222	24	a	a	PRON
ejpam-3441	222	25	is	be	AUX
ejpam-3441	222	26	an	an	DET
ejpam-3441	222	27	intuitionistic	intuitionistic	ADJ
ejpam-3441	222	28	fuzzy	fuzzy	ADJ
ejpam-3441	222	29	left	leave	VERB
ejpam-3441	222	30	ideal	ideal	NOUN
ejpam-3441	222	31	with	with	ADP
ejpam-3441	222	32	thresholds	threshold	NOUN
ejpam-3441	222	33	(	(	PUNCT
ejpam-3441	222	34	α	α	X
ejpam-3441	222	35	,	,	PUNCT
ejpam-3441	222	36	β	β	X
ejpam-3441	222	37	]	]	PUNCT
ejpam-3441	222	38	of	of	ADP
ejpam-3441	222	39	an	an	DET
ejpam-3441	222	40	la	la	ADJ
ejpam-3441	222	41	-	-	PUNCT
ejpam-3441	222	42	ring	ring	NOUN
ejpam-3441	222	43	r	r	NOUN
ejpam-3441	222	44	and	and	CCONJ
ejpam-3441	222	45	x	x	PROPN
ejpam-3441	222	46	∈	∈	PROPN
ejpam-3441	222	47	r.	r.	NOUN
ejpam-3441	223	1	if	if	SCONJ
ejpam-3441	223	2	(	(	PUNCT
ejpam-3441	223	3	r	r	NOUN
ejpam-3441	223	4	◦	◦	NOUN
ejpam-3441	223	5	βα	βα	NOUN
ejpam-3441	223	6	a)(x	a)(x	NOUN
ejpam-3441	223	7	)	)	PUNCT
ejpam-3441	223	8	=	=	SYM
ejpam-3441	223	9	0	0	NUM
ejpam-3441	223	10	,	,	PUNCT
ejpam-3441	223	11	then	then	ADV
ejpam-3441	223	12	obvious	obvious	ADJ
ejpam-3441	223	13	r	r	NOUN
ejpam-3441	223	14	◦	◦	NOUN
ejpam-3441	223	15	βα	βα	X
ejpam-3441	223	16	a	a	DET
ejpam-3441	223	17	⊆	⊆	NUM
ejpam-3441	223	18	aβα	aβα	NOUN
ejpam-3441	223	19	,	,	PUNCT
ejpam-3441	223	20	otherwise	otherwise	ADV
ejpam-3441	223	21	we	we	PRON
ejpam-3441	223	22	have	have	VERB
ejpam-3441	223	23	(	(	PUNCT
ejpam-3441	223	24	r	r	NOUN
ejpam-3441	223	25	◦	◦	NOUN
ejpam-3441	223	26	βα	βα	NOUN
ejpam-3441	223	27	µa)(x	µa)(x	NOUN
ejpam-3441	223	28	)	)	PUNCT
ejpam-3441	224	1	=	=	PRON
ejpam-3441	224	2	{	{	PUNCT
ejpam-3441	224	3	(	(	PUNCT
ejpam-3441	224	4	r	r	NOUN
ejpam-3441	224	5	◦	◦	NOUN
ejpam-3441	224	6	µa)(x	µa)(x	NOUN
ejpam-3441	224	7	)	)	PUNCT
ejpam-3441	224	8	∧	∧	PROPN
ejpam-3441	224	9	β	β	NOUN
ejpam-3441	224	10	}	}	PUNCT
ejpam-3441	224	11	∨	∨	NUM
ejpam-3441	224	12	α	α	NOUN
ejpam-3441	224	13	=	=	X
ejpam-3441	224	14	{	{	PUNCT
ejpam-3441	224	15	(	(	PUNCT
ejpam-3441	224	16	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	224	17	i=1	i=1	PROPN
ejpam-3441	224	18	aibi	aibi	NOUN
ejpam-3441	224	19	{	{	PUNCT
ejpam-3441	224	20	∧ni=1	∧ni=1	X
ejpam-3441	224	21	{	{	PUNCT
ejpam-3441	224	22	r	r	NOUN
ejpam-3441	224	23	(	(	PUNCT
ejpam-3441	224	24	ai	ai	NOUN
ejpam-3441	224	25	)	)	PUNCT
ejpam-3441	224	26	∧	∧	NOUN
ejpam-3441	224	27	µa	µa	NOUN
ejpam-3441	224	28	(	(	PUNCT
ejpam-3441	224	29	bi	bi	NOUN
ejpam-3441	224	30	)	)	PUNCT
ejpam-3441	224	31	}	}	PUNCT
ejpam-3441	224	32	}	}	PUNCT
ejpam-3441	224	33	)	)	PUNCT
ejpam-3441	224	34	∧	∧	PROPN
ejpam-3441	224	35	β	β	NOUN
ejpam-3441	224	36	}	}	PUNCT
ejpam-3441	224	37	∨	∨	NUM
ejpam-3441	224	38	α	α	NOUN
ejpam-3441	224	39	=	=	X
ejpam-3441	224	40	{	{	PUNCT
ejpam-3441	224	41	(	(	PUNCT
ejpam-3441	224	42	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	224	43	i=1	i=1	PROPN
ejpam-3441	224	44	aibi	aibi	NOUN
ejpam-3441	224	45	{	{	PUNCT
ejpam-3441	224	46	∧ni=1	∧ni=1	X
ejpam-3441	224	47	{	{	PUNCT
ejpam-3441	224	48	1	1	NUM
ejpam-3441	224	49	∧	∧	PROPN
ejpam-3441	224	50	µa	µa	NOUN
ejpam-3441	224	51	(	(	PUNCT
ejpam-3441	224	52	bi	bi	NOUN
ejpam-3441	224	53	)	)	PUNCT
ejpam-3441	224	54	}	}	PUNCT
ejpam-3441	224	55	}	}	PUNCT
ejpam-3441	224	56	)	)	PUNCT
ejpam-3441	224	57	∧	∧	PROPN
ejpam-3441	224	58	β	β	NOUN
ejpam-3441	224	59	}	}	PUNCT
ejpam-3441	224	60	∨	∨	NUM
ejpam-3441	224	61	α	α	NOUN
ejpam-3441	224	62	=	=	X
ejpam-3441	224	63	{	{	PUNCT
ejpam-3441	224	64	(	(	PUNCT
ejpam-3441	224	65	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	224	66	i=1	i=1	PROPN
ejpam-3441	224	67	aibi	aibi	NOUN
ejpam-3441	224	68	{	{	PUNCT
ejpam-3441	224	69	∧ni=1µa	∧ni=1µa	NOUN
ejpam-3441	224	70	(	(	PUNCT
ejpam-3441	224	71	bi	bi	NOUN
ejpam-3441	224	72	)	)	PUNCT
ejpam-3441	224	73	}	}	PUNCT
ejpam-3441	224	74	)	)	PUNCT
ejpam-3441	225	1	∧	∧	PROPN
ejpam-3441	225	2	β	β	NOUN
ejpam-3441	225	3	}	}	PUNCT
ejpam-3441	225	4	∨	∨	NUM
ejpam-3441	225	5	α	α	NOUN
ejpam-3441	225	6	≤	≤	NOUN
ejpam-3441	225	7	{	{	PUNCT
ejpam-3441	225	8	(	(	PUNCT
ejpam-3441	225	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	225	10	i=1	i=1	PROPN
ejpam-3441	225	11	aibi	aibi	NOUN
ejpam-3441	225	12	{	{	PUNCT
ejpam-3441	225	13	∧ni=1µa	∧ni=1µa	NOUN
ejpam-3441	225	14	(	(	PUNCT
ejpam-3441	225	15	aibi	aibi	NOUN
ejpam-3441	225	16	)	)	PUNCT
ejpam-3441	225	17	}	}	PUNCT
ejpam-3441	225	18	)	)	PUNCT
ejpam-3441	226	1	∧	∧	PROPN
ejpam-3441	226	2	β	β	NOUN
ejpam-3441	226	3	}	}	PUNCT
ejpam-3441	226	4	∨	∨	NUM
ejpam-3441	226	5	α	α	X
ejpam-3441	226	6	=	=	SYM
ejpam-3441	226	7	(	(	PUNCT
ejpam-3441	226	8	µa(x	µa(x	NOUN
ejpam-3441	226	9	)	)	PUNCT
ejpam-3441	226	10	∧	∧	PROPN
ejpam-3441	226	11	β	β	NOUN
ejpam-3441	226	12	)	)	PUNCT
ejpam-3441	226	13	∨	∨	NUM
ejpam-3441	226	14	α	α	NOUN
ejpam-3441	226	15	=	=	SYM
ejpam-3441	226	16	(	(	PUNCT
ejpam-3441	226	17	µ)βα(x	µ)βα(x	NUM
ejpam-3441	226	18	)	)	PUNCT
ejpam-3441	226	19	.	.	PUNCT
ejpam-3441	227	1	⇒	⇒	PROPN
ejpam-3441	227	2	r	r	PROPN
ejpam-3441	227	3	◦	◦	NOUN
ejpam-3441	227	4	βα	βα	NOUN
ejpam-3441	227	5	µa	µa	NOUN
ejpam-3441	227	6	⊆	⊆	NUM
ejpam-3441	227	7	(	(	PUNCT
ejpam-3441	227	8	µa)βα	µa)βα	NUM
ejpam-3441	227	9	.	.	PUNCT
ejpam-3441	228	1	similarly	similarly	ADV
ejpam-3441	228	2	,	,	PUNCT
ejpam-3441	228	3	we	we	PRON
ejpam-3441	228	4	have	have	VERB
ejpam-3441	228	5	r	r	NOUN
ejpam-3441	228	6	◦	◦	NOUN
ejpam-3441	228	7	βα	βα	X
ejpam-3441	228	8	γa	γa	PROPN
ejpam-3441	228	9	⊇	⊇	PROPN
ejpam-3441	228	10	(	(	PUNCT
ejpam-3441	228	11	γa)βα	γa)βα	X
ejpam-3441	228	12	.	.	PUNCT
ejpam-3441	229	1	thus	thus	ADV
ejpam-3441	229	2	r	r	X
ejpam-3441	229	3	◦	◦	NOUN
ejpam-3441	229	4	βα	βα	NOUN
ejpam-3441	229	5	a	a	DET
ejpam-3441	229	6	⊆	⊆	NUM
ejpam-3441	229	7	aβα	aβα	NOUN
ejpam-3441	229	8	.	.	PUNCT
ejpam-3441	230	1	conversely	conversely	ADV
ejpam-3441	230	2	,	,	PUNCT
ejpam-3441	230	3	suppose	suppose	VERB
ejpam-3441	230	4	that	that	SCONJ
ejpam-3441	230	5	r	r	NOUN
ejpam-3441	230	6	◦	◦	NOUN
ejpam-3441	230	7	βα	βα	VERB
ejpam-3441	230	8	a	a	DET
ejpam-3441	230	9	⊆	⊆	NUM
ejpam-3441	230	10	aβα	aβα	NOUN
ejpam-3441	230	11	.	.	PUNCT
ejpam-3441	231	1	let	let	VERB
ejpam-3441	231	2	y	y	NOUN
ejpam-3441	231	3	,	,	PUNCT
ejpam-3441	231	4	z	z	NOUN
ejpam-3441	231	5	∈	∈	NOUN
ejpam-3441	231	6	r	r	NOUN
ejpam-3441	232	1	such	such	ADJ
ejpam-3441	232	2	that	that	SCONJ
ejpam-3441	232	3	x	x	PROPN
ejpam-3441	232	4	=	=	SYM
ejpam-3441	232	5	yz	yz	PROPN
ejpam-3441	232	6	.	.	PUNCT
ejpam-3441	232	7	now	now	ADV
ejpam-3441	232	8	max{µa(yz	max{µa(yz	NOUN
ejpam-3441	232	9	)	)	PUNCT
ejpam-3441	232	10	,	,	PUNCT
ejpam-3441	232	11	α	α	NOUN
ejpam-3441	232	12	}	}	PUNCT
ejpam-3441	232	13	=	=	SYM
ejpam-3441	232	14	max{µa(x	max{µa(x	NOUN
ejpam-3441	232	15	)	)	PUNCT
ejpam-3441	232	16	,	,	PUNCT
ejpam-3441	232	17	α	α	X
ejpam-3441	232	18	}	}	PUNCT
ejpam-3441	232	19	=	=	SYM
ejpam-3441	232	20	max{min{µa(x	max{min{µa(x	NOUN
ejpam-3441	232	21	)	)	PUNCT
ejpam-3441	232	22	,	,	PUNCT
ejpam-3441	232	23	β	β	X
ejpam-3441	232	24	}	}	PUNCT
ejpam-3441	232	25	,	,	PUNCT
ejpam-3441	232	26	α	α	NOUN
ejpam-3441	232	27	}	}	PUNCT
ejpam-3441	232	28	=	=	SYM
ejpam-3441	232	29	(	(	PUNCT
ejpam-3441	232	30	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	232	31	)	)	PUNCT
ejpam-3441	232	32	≥	≥	NOUN
ejpam-3441	232	33	(	(	PUNCT
ejpam-3441	232	34	r	r	NOUN
ejpam-3441	232	35	◦	◦	NOUN
ejpam-3441	232	36	βα	βα	NOUN
ejpam-3441	232	37	µa)(x	µa)(x	NOUN
ejpam-3441	232	38	)	)	PUNCT
ejpam-3441	233	1	=	=	PRON
ejpam-3441	233	2	{	{	PUNCT
ejpam-3441	233	3	(	(	PUNCT
ejpam-3441	233	4	r	r	NOUN
ejpam-3441	233	5	◦	◦	NOUN
ejpam-3441	233	6	µa)(x	µa)(x	NOUN
ejpam-3441	233	7	)	)	PUNCT
ejpam-3441	233	8	∧	∧	PROPN
ejpam-3441	233	9	β	β	NOUN
ejpam-3441	233	10	}	}	PUNCT
ejpam-3441	233	11	∨	∨	NUM
ejpam-3441	233	12	α	α	NOUN
ejpam-3441	233	13	=	=	X
ejpam-3441	233	14	{	{	PUNCT
ejpam-3441	233	15	(	(	PUNCT
ejpam-3441	233	16	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	233	17	i=1	i=1	PROPN
ejpam-3441	233	18	aibi	aibi	NOUN
ejpam-3441	233	19	{	{	PUNCT
ejpam-3441	233	20	∧ni=1	∧ni=1	X
ejpam-3441	233	21	{	{	PUNCT
ejpam-3441	233	22	r	r	NOUN
ejpam-3441	233	23	(	(	PUNCT
ejpam-3441	233	24	ai	ai	NOUN
ejpam-3441	233	25	)	)	PUNCT
ejpam-3441	233	26	∧	∧	NOUN
ejpam-3441	233	27	µa	µa	NOUN
ejpam-3441	233	28	(	(	PUNCT
ejpam-3441	233	29	bi	bi	NOUN
ejpam-3441	233	30	)	)	PUNCT
ejpam-3441	233	31	}	}	PUNCT
ejpam-3441	233	32	}	}	PUNCT
ejpam-3441	233	33	)	)	PUNCT
ejpam-3441	233	34	∧	∧	PROPN
ejpam-3441	233	35	β	β	NOUN
ejpam-3441	233	36	}	}	PUNCT
ejpam-3441	233	37	∨	∨	NUM
ejpam-3441	233	38	α	α	PROPN
ejpam-3441	233	39	≥	≥	X
ejpam-3441	233	40	(	(	PUNCT
ejpam-3441	233	41	(	(	PUNCT
ejpam-3441	233	42	r(y	r(y	ADJ
ejpam-3441	233	43	)	)	PUNCT
ejpam-3441	233	44	∧	∧	NOUN
ejpam-3441	233	45	µa	µa	NOUN
ejpam-3441	233	46	(	(	PUNCT
ejpam-3441	233	47	z	z	NOUN
ejpam-3441	233	48	)	)	PUNCT
ejpam-3441	233	49	)	)	PUNCT
ejpam-3441	234	1	∧	∧	PROPN
ejpam-3441	234	2	β	β	NOUN
ejpam-3441	234	3	)	)	PUNCT
ejpam-3441	234	4	∨	∨	NUM
ejpam-3441	234	5	α	α	X
ejpam-3441	234	6	=	=	SYM
ejpam-3441	234	7	(	(	PUNCT
ejpam-3441	234	8	1	1	NUM
ejpam-3441	234	9	∧	∧	PROPN
ejpam-3441	234	10	µa	µa	NOUN
ejpam-3441	234	11	(	(	PUNCT
ejpam-3441	234	12	z	z	NOUN
ejpam-3441	234	13	)	)	PUNCT
ejpam-3441	234	14	)	)	PUNCT
ejpam-3441	235	1	∧	∧	NOUN
ejpam-3441	235	2	β	β	X
ejpam-3441	235	3	=	=	PUNCT
ejpam-3441	235	4	min{µa	min{µa	X
ejpam-3441	235	5	(	(	PUNCT
ejpam-3441	235	6	z	z	NOUN
ejpam-3441	235	7	)	)	PUNCT
ejpam-3441	235	8	,	,	PUNCT
ejpam-3441	235	9	β	β	NOUN
ejpam-3441	235	10	}	}	PUNCT
ejpam-3441	235	11	.	.	PUNCT
ejpam-3441	236	1	⇒	⇒	PROPN
ejpam-3441	236	2	max{µa(yz	max{µa(yz	PROPN
ejpam-3441	236	3	)	)	PUNCT
ejpam-3441	236	4	,	,	PUNCT
ejpam-3441	236	5	α	α	X
ejpam-3441	236	6	}	}	PUNCT
ejpam-3441	236	7	≥	≥	NOUN
ejpam-3441	236	8	min{µa	min{µa	X
ejpam-3441	236	9	(	(	PUNCT
ejpam-3441	236	10	z	z	NOUN
ejpam-3441	236	11	)	)	PUNCT
ejpam-3441	236	12	,	,	PUNCT
ejpam-3441	236	13	β	β	NOUN
ejpam-3441	236	14	}	}	PUNCT
ejpam-3441	236	15	.	.	PUNCT
ejpam-3441	237	1	similarly	similarly	ADV
ejpam-3441	237	2	,	,	PUNCT
ejpam-3441	237	3	we	we	PRON
ejpam-3441	237	4	have	have	VERB
ejpam-3441	237	5	min{γa(yz	min{γa(yz	NOUN
ejpam-3441	237	6	)	)	PUNCT
ejpam-3441	237	7	,	,	PUNCT
ejpam-3441	237	8	(	(	PUNCT
ejpam-3441	237	9	1	1	NUM
ejpam-3441	237	10	−	−	PROPN
ejpam-3441	237	11	α	α	X
ejpam-3441	237	12	)	)	PUNCT
ejpam-3441	237	13	}	}	PUNCT
ejpam-3441	237	14	≤	≤	NUM
ejpam-3441	237	15	max{γa	max{γa	PUNCT
ejpam-3441	237	16	(	(	PUNCT
ejpam-3441	237	17	z	z	NOUN
ejpam-3441	237	18	)	)	PUNCT
ejpam-3441	237	19	,	,	PUNCT
ejpam-3441	237	20	(	(	PUNCT
ejpam-3441	237	21	1	1	NUM
ejpam-3441	237	22	−	−	NOUN
ejpam-3441	237	23	β	β	NOUN
ejpam-3441	237	24	)	)	PUNCT
ejpam-3441	237	25	}	}	PUNCT
ejpam-3441	237	26	.	.	PUNCT
ejpam-3441	238	1	therefore	therefore	ADV
ejpam-3441	238	2	a	a	PRON
ejpam-3441	238	3	is	be	AUX
ejpam-3441	238	4	an	an	DET
ejpam-3441	238	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	238	6	fuzzy	fuzzy	ADJ
ejpam-3441	238	7	left	leave	VERB
ejpam-3441	238	8	ideal	ideal	NOUN
ejpam-3441	238	9	with	with	ADP
ejpam-3441	238	10	thresholds	threshold	NOUN
ejpam-3441	238	11	(	(	PUNCT
ejpam-3441	238	12	α	α	X
ejpam-3441	238	13	,	,	PUNCT
ejpam-3441	238	14	β	β	X
ejpam-3441	238	15	]	]	PUNCT
ejpam-3441	238	16	of	of	ADP
ejpam-3441	238	17	r.	r.	PROPN
ejpam-3441	238	18	lemma	lemma	PROPN
ejpam-3441	239	1	3	3	X
ejpam-3441	239	2	.	.	PUNCT
ejpam-3441	240	1	if	if	SCONJ
ejpam-3441	240	2	a	a	PRON
ejpam-3441	240	3	and	and	CCONJ
ejpam-3441	240	4	b	b	NOUN
ejpam-3441	240	5	are	be	AUX
ejpam-3441	240	6	two	two	NUM
ejpam-3441	240	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	240	8	fuzzy	fuzzy	ADJ
ejpam-3441	240	9	la	la	NOUN
ejpam-3441	240	10	-	-	NOUN
ejpam-3441	240	11	subrings	subring	NOUN
ejpam-3441	240	12	(	(	PUNCT
ejpam-3441	240	13	resp	resp	NOUN
ejpam-3441	240	14	.	.	PUNCT
ejpam-3441	241	1	(	(	PUNCT
ejpam-3441	241	2	left	leave	VERB
ejpam-3441	241	3	,	,	PUNCT
ejpam-3441	241	4	right	right	INTJ
ejpam-3441	241	5	,	,	PUNCT
ejpam-3441	241	6	twosided	twosided	ADJ
ejpam-3441	241	7	)	)	PUNCT
ejpam-3441	241	8	ideals	ideal	NOUN
ejpam-3441	241	9	)	)	PUNCT
ejpam-3441	241	10	with	with	ADP
ejpam-3441	241	11	thresholds	threshold	NOUN
ejpam-3441	241	12	(	(	PUNCT
ejpam-3441	241	13	α	α	X
ejpam-3441	241	14	,	,	PUNCT
ejpam-3441	241	15	β	β	X
ejpam-3441	241	16	]	]	PUNCT
ejpam-3441	241	17	of	of	ADP
ejpam-3441	241	18	an	an	DET
ejpam-3441	241	19	la	la	ADJ
ejpam-3441	241	20	-	-	PUNCT
ejpam-3441	241	21	ring	ring	NOUN
ejpam-3441	241	22	r	r	NOUN
ejpam-3441	241	23	,	,	PUNCT
ejpam-3441	241	24	then	then	ADV
ejpam-3441	241	25	a∧βαb	a∧βαb	PRON
ejpam-3441	241	26	is	be	AUX
ejpam-3441	241	27	also	also	ADV
ejpam-3441	241	28	an	an	DET
ejpam-3441	241	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	241	30	fuzzy	fuzzy	ADJ
ejpam-3441	241	31	la	la	NOUN
ejpam-3441	241	32	-	-	PUNCT
ejpam-3441	241	33	subring	subre	VERB
ejpam-3441	241	34	(	(	PUNCT
ejpam-3441	241	35	resp	resp	NOUN
ejpam-3441	241	36	.	.	PUNCT
ejpam-3441	242	1	(	(	PUNCT
ejpam-3441	242	2	left	leave	VERB
ejpam-3441	242	3	,	,	PUNCT
ejpam-3441	242	4	right	right	INTJ
ejpam-3441	242	5	,	,	PUNCT
ejpam-3441	242	6	two	two	NUM
ejpam-3441	242	7	-	-	PUNCT
ejpam-3441	242	8	sided	sided	ADJ
ejpam-3441	242	9	)	)	PUNCT
ejpam-3441	242	10	ideal	ideal	NOUN
ejpam-3441	242	11	)	)	PUNCT
ejpam-3441	242	12	with	with	ADP
ejpam-3441	242	13	thresholds	threshold	NOUN
ejpam-3441	242	14	(	(	PUNCT
ejpam-3441	242	15	α	α	X
ejpam-3441	242	16	,	,	PUNCT
ejpam-3441	242	17	β	β	X
ejpam-3441	242	18	]	]	PUNCT
ejpam-3441	242	19	of	of	ADP
ejpam-3441	242	20	r.	r.	PROPN
ejpam-3441	242	21	proof	proof	NOUN
ejpam-3441	242	22	.	.	PUNCT
ejpam-3441	243	1	let	let	VERB
ejpam-3441	243	2	a	a	DET
ejpam-3441	243	3	=	=	SYM
ejpam-3441	243	4	(	(	PUNCT
ejpam-3441	243	5	µa	µa	PROPN
ejpam-3441	243	6	,	,	PUNCT
ejpam-3441	243	7	γa	γa	PROPN
ejpam-3441	243	8	)	)	PUNCT
ejpam-3441	243	9	and	and	CCONJ
ejpam-3441	243	10	b	b	X
ejpam-3441	243	11	=	=	SYM
ejpam-3441	243	12	(	(	PUNCT
ejpam-3441	243	13	µb	µb	PROPN
ejpam-3441	243	14	,	,	PUNCT
ejpam-3441	243	15	γb	γb	PROPN
ejpam-3441	243	16	)	)	PUNCT
ejpam-3441	243	17	be	be	VERB
ejpam-3441	243	18	two	two	NUM
ejpam-3441	243	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	243	20	fuzzy	fuzzy	ADJ
ejpam-3441	243	21	la	la	NOUN
ejpam-3441	243	22	-	-	NOUN
ejpam-3441	243	23	subrings	subring	NOUN
ejpam-3441	243	24	with	with	ADP
ejpam-3441	243	25	thresholds	threshold	NOUN
ejpam-3441	243	26	(	(	PUNCT
ejpam-3441	243	27	α	α	X
ejpam-3441	243	28	,	,	PUNCT
ejpam-3441	243	29	β	β	X
ejpam-3441	243	30	]	]	PUNCT
ejpam-3441	243	31	of	of	ADP
ejpam-3441	243	32	an	an	DET
ejpam-3441	243	33	la	la	ADJ
ejpam-3441	243	34	-	-	PUNCT
ejpam-3441	243	35	ring	ring	NOUN
ejpam-3441	243	36	r.	r.	NOUN
ejpam-3441	243	37	we	we	PRON
ejpam-3441	243	38	have	have	VERB
ejpam-3441	243	39	to	to	PART
ejpam-3441	243	40	show	show	VERB
ejpam-3441	243	41	that	that	SCONJ
ejpam-3441	243	42	a∧βαb	a∧βαb	PRON
ejpam-3441	243	43	is	be	AUX
ejpam-3441	243	44	also	also	ADV
ejpam-3441	243	45	an	an	DET
ejpam-3441	243	46	intuitionistic	intuitionistic	ADJ
ejpam-3441	243	47	k.	k.	PROPN
ejpam-3441	243	48	nasreen	nasreen	PROPN
ejpam-3441	243	49	et	et	PROPN
ejpam-3441	243	50	al	al	PROPN
ejpam-3441	243	51	.	.	PUNCT
ejpam-3441	243	52	/	/	SYM
ejpam-3441	243	53	eur	eur	PROPN
ejpam-3441	243	54	.	.	PUNCT
ejpam-3441	244	1	j.	j.	PROPN
ejpam-3441	244	2	pure	pure	PROPN
ejpam-3441	244	3	appl	appl	PROPN
ejpam-3441	244	4	.	.	PROPN
ejpam-3441	244	5	math	math	PROPN
ejpam-3441	244	6	,	,	PUNCT
ejpam-3441	244	7	12	12	NUM
ejpam-3441	244	8	(	(	PUNCT
ejpam-3441	244	9	3	3	NUM
ejpam-3441	244	10	)	)	PUNCT
ejpam-3441	244	11	(	(	PUNCT
ejpam-3441	244	12	2019	2019	NUM
ejpam-3441	244	13	)	)	PUNCT
ejpam-3441	244	14	,	,	PUNCT
ejpam-3441	244	15	906	906	NUM
ejpam-3441	244	16	-	-	SYM
ejpam-3441	244	17	943	943	NUM
ejpam-3441	244	18	916	916	NUM
ejpam-3441	244	19	fuzzy	fuzzy	ADJ
ejpam-3441	244	20	la	la	NOUN
ejpam-3441	244	21	-	-	PUNCT
ejpam-3441	244	22	subring	subre	VERB
ejpam-3441	244	23	with	with	ADP
ejpam-3441	244	24	thresholds	threshold	NOUN
ejpam-3441	244	25	(	(	PUNCT
ejpam-3441	244	26	α	α	X
ejpam-3441	244	27	,	,	PUNCT
ejpam-3441	244	28	β	β	X
ejpam-3441	244	29	]	]	PUNCT
ejpam-3441	244	30	of	of	ADP
ejpam-3441	244	31	r.	r.	PROPN
ejpam-3441	244	32	now	now	ADV
ejpam-3441	244	33	max{(µa	max{(µa	VERB
ejpam-3441	244	34	∧βα	∧βα	ADJ
ejpam-3441	244	35	µb)(x−	µb)(x−	ADJ
ejpam-3441	244	36	y	y	NOUN
ejpam-3441	244	37	)	)	PUNCT
ejpam-3441	244	38	,	,	PUNCT
ejpam-3441	244	39	α	α	X
ejpam-3441	244	40	}	}	PUNCT
ejpam-3441	244	41	=	=	SYM
ejpam-3441	244	42	max{{{(µa	max{{{(µa	PROPN
ejpam-3441	244	43	∧	∧	PROPN
ejpam-3441	244	44	µb)(x−	µb)(x−	ADJ
ejpam-3441	244	45	y	y	NOUN
ejpam-3441	244	46	)	)	PUNCT
ejpam-3441	244	47	∧	∧	PROPN
ejpam-3441	244	48	β	β	PROPN
ejpam-3441	244	49	}	}	PUNCT
ejpam-3441	244	50	∨	∨	NUM
ejpam-3441	244	51	α	α	NOUN
ejpam-3441	244	52	}	}	PUNCT
ejpam-3441	244	53	,	,	PUNCT
ejpam-3441	244	54	α	α	NOUN
ejpam-3441	244	55	}	}	PUNCT
ejpam-3441	244	56	=	=	PRON
ejpam-3441	244	57	{	{	PUNCT
ejpam-3441	244	58	(	(	PUNCT
ejpam-3441	244	59	µa	µa	PROPN
ejpam-3441	244	60	∧	∧	PROPN
ejpam-3441	244	61	µb)(x−	µb)(x−	ADJ
ejpam-3441	244	62	y	y	PROPN
ejpam-3441	244	63	)	)	PUNCT
ejpam-3441	244	64	∧	∧	PROPN
ejpam-3441	244	65	β	β	NOUN
ejpam-3441	244	66	}	}	PUNCT
ejpam-3441	244	67	∨	∨	NUM
ejpam-3441	244	68	α	α	X
ejpam-3441	244	69	=	=	PUNCT
ejpam-3441	244	70	{	{	PUNCT
ejpam-3441	244	71	µa(x−	µa(x−	NUM
ejpam-3441	244	72	y	y	PROPN
ejpam-3441	244	73	)	)	PUNCT
ejpam-3441	244	74	∧	∧	PROPN
ejpam-3441	244	75	µb(x−	µb(x−	X
ejpam-3441	244	76	y	y	NOUN
ejpam-3441	244	77	)	)	PUNCT
ejpam-3441	244	78	∧	∧	PROPN
ejpam-3441	244	79	β	β	PROPN
ejpam-3441	244	80	}	}	PUNCT
ejpam-3441	244	81	∨	∨	NUM
ejpam-3441	244	82	α	α	PROPN
ejpam-3441	244	83	≥	≥	X
ejpam-3441	244	84	{	{	PUNCT
ejpam-3441	244	85	µa(x	µa(x	NOUN
ejpam-3441	244	86	)	)	PUNCT
ejpam-3441	244	87	∧	∧	NOUN
ejpam-3441	244	88	µa(y	µa(y	NOUN
ejpam-3441	244	89	)	)	PUNCT
ejpam-3441	244	90	∧	∧	NOUN
ejpam-3441	244	91	µb(x	µb(x	PUNCT
ejpam-3441	244	92	)	)	PUNCT
ejpam-3441	244	93	∧	∧	PROPN
ejpam-3441	244	94	µb(y	µb(y	NUM
ejpam-3441	244	95	)	)	PUNCT
ejpam-3441	245	1	∧	∧	PROPN
ejpam-3441	245	2	β	β	NOUN
ejpam-3441	245	3	}	}	PUNCT
ejpam-3441	245	4	∨	∨	NUM
ejpam-3441	245	5	α	α	X
ejpam-3441	245	6	=	=	SYM
ejpam-3441	245	7	{	{	PUNCT
ejpam-3441	245	8	µa(x	µa(x	NOUN
ejpam-3441	245	9	)	)	PUNCT
ejpam-3441	245	10	∧	∧	NOUN
ejpam-3441	245	11	µb(x	µb(x	PUNCT
ejpam-3441	245	12	)	)	PUNCT
ejpam-3441	245	13	∧	∧	NOUN
ejpam-3441	245	14	µa(y	µa(y	NOUN
ejpam-3441	245	15	)	)	PUNCT
ejpam-3441	245	16	∧	∧	PROPN
ejpam-3441	245	17	µb(y	µb(y	NUM
ejpam-3441	245	18	)	)	PUNCT
ejpam-3441	246	1	∧	∧	PROPN
ejpam-3441	246	2	β	β	NOUN
ejpam-3441	246	3	}	}	PUNCT
ejpam-3441	246	4	∨	∨	NUM
ejpam-3441	246	5	α	α	NOUN
ejpam-3441	246	6	=	=	SYM
ejpam-3441	246	7	{	{	PUNCT
ejpam-3441	246	8	(	(	PUNCT
ejpam-3441	246	9	µa	µa	PROPN
ejpam-3441	246	10	∧	∧	PROPN
ejpam-3441	246	11	µb)(x	µb)(x	PROPN
ejpam-3441	246	12	)	)	PUNCT
ejpam-3441	246	13	∧	∧	NOUN
ejpam-3441	246	14	(	(	PUNCT
ejpam-3441	246	15	µa	µa	ADP
ejpam-3441	246	16	∧	∧	PROPN
ejpam-3441	246	17	µb)(y	µb)(y	PROPN
ejpam-3441	246	18	)	)	PUNCT
ejpam-3441	246	19	∧	∧	NOUN
ejpam-3441	246	20	β	β	X
ejpam-3441	246	21	∧	∧	PROPN
ejpam-3441	246	22	β	β	X
ejpam-3441	246	23	∧	∧	PROPN
ejpam-3441	246	24	β	β	PROPN
ejpam-3441	246	25	}	}	PUNCT
ejpam-3441	246	26	∨	∨	NUM
ejpam-3441	246	27	α	α	NOUN
ejpam-3441	246	28	=	=	SYM
ejpam-3441	246	29	{	{	PUNCT
ejpam-3441	246	30	(	(	PUNCT
ejpam-3441	246	31	(	(	PUNCT
ejpam-3441	246	32	µa	µa	ADP
ejpam-3441	246	33	∧	∧	PROPN
ejpam-3441	246	34	µb)(x	µb)(x	PROPN
ejpam-3441	246	35	)	)	PUNCT
ejpam-3441	246	36	∧	∧	PROPN
ejpam-3441	246	37	β	β	NOUN
ejpam-3441	246	38	)	)	PUNCT
ejpam-3441	246	39	∧	∧	NOUN
ejpam-3441	246	40	(	(	PUNCT
ejpam-3441	246	41	(	(	PUNCT
ejpam-3441	246	42	µa	µa	ADP
ejpam-3441	246	43	∧	∧	PROPN
ejpam-3441	246	44	µb)(y	µb)(y	PROPN
ejpam-3441	246	45	)	)	PUNCT
ejpam-3441	246	46	∧	∧	PROPN
ejpam-3441	246	47	β	β	NOUN
ejpam-3441	246	48	)	)	PUNCT
ejpam-3441	246	49	∧	∧	PROPN
ejpam-3441	246	50	β	β	NOUN
ejpam-3441	246	51	}	}	PUNCT
ejpam-3441	246	52	∨	∨	NUM
ejpam-3441	246	53	α	α	NOUN
ejpam-3441	246	54	=	=	SYM
ejpam-3441	246	55	(	(	PUNCT
ejpam-3441	246	56	{	{	PUNCT
ejpam-3441	246	57	(	(	PUNCT
ejpam-3441	246	58	µa	µa	PROPN
ejpam-3441	246	59	∧	∧	PROPN
ejpam-3441	246	60	µb)(x	µb)(x	PROPN
ejpam-3441	246	61	)	)	PUNCT
ejpam-3441	246	62	∧	∧	PROPN
ejpam-3441	246	63	β	β	NOUN
ejpam-3441	246	64	}	}	PUNCT
ejpam-3441	246	65	∨	∨	NUM
ejpam-3441	246	66	α	α	NOUN
ejpam-3441	246	67	)	)	PUNCT
ejpam-3441	246	68	∧	∧	NOUN
ejpam-3441	246	69	(	(	PUNCT
ejpam-3441	246	70	{	{	PUNCT
ejpam-3441	246	71	(	(	PUNCT
ejpam-3441	246	72	µa	µa	ADP
ejpam-3441	246	73	∧	∧	PROPN
ejpam-3441	246	74	µb)(y	µb)(y	PROPN
ejpam-3441	246	75	)	)	PUNCT
ejpam-3441	247	1	∧	∧	PROPN
ejpam-3441	247	2	β	β	PROPN
ejpam-3441	247	3	}	}	PUNCT
ejpam-3441	247	4	∨	∨	NUM
ejpam-3441	247	5	α	α	NOUN
ejpam-3441	247	6	)	)	PUNCT
ejpam-3441	247	7	∧	∧	PROPN
ejpam-3441	247	8	(	(	PUNCT
ejpam-3441	247	9	β	β	X
ejpam-3441	247	10	∨	∨	NUM
ejpam-3441	247	11	α	α	NOUN
ejpam-3441	247	12	)	)	PUNCT
ejpam-3441	247	13	=	=	SYM
ejpam-3441	247	14	(	(	PUNCT
ejpam-3441	247	15	µa	µa	ADP
ejpam-3441	247	16	∧βα	∧βα	ADJ
ejpam-3441	247	17	µb)(x	µb)(x	NOUN
ejpam-3441	247	18	)	)	PUNCT
ejpam-3441	247	19	∧	∧	NOUN
ejpam-3441	247	20	(	(	PUNCT
ejpam-3441	247	21	µa	µa	ADP
ejpam-3441	247	22	∧βα	∧βα	ADJ
ejpam-3441	247	23	µb)(y	µb)(y	PUNCT
ejpam-3441	247	24	)	)	PUNCT
ejpam-3441	248	1	∧	∧	NOUN
ejpam-3441	248	2	β	β	X
ejpam-3441	248	3	=	=	SYM
ejpam-3441	248	4	min{(µa	min{(µa	VERB
ejpam-3441	248	5	∧βα	∧βα	ADJ
ejpam-3441	248	6	µb)(x	µb)(x	NOUN
ejpam-3441	248	7	)	)	PUNCT
ejpam-3441	248	8	,	,	PUNCT
ejpam-3441	248	9	(	(	PUNCT
ejpam-3441	248	10	µa	µa	ADP
ejpam-3441	248	11	∧βα	∧βα	ADJ
ejpam-3441	248	12	µb)(y	µb)(y	PROPN
ejpam-3441	248	13	)	)	PUNCT
ejpam-3441	248	14	,	,	PUNCT
ejpam-3441	248	15	β	β	X
ejpam-3441	248	16	}	}	PUNCT
ejpam-3441	248	17	.	.	PUNCT
ejpam-3441	249	1	thus	thus	ADV
ejpam-3441	249	2	max{(µa	max{(µa	VERB
ejpam-3441	249	3	∧βα	∧βα	ADJ
ejpam-3441	249	4	µb)(x−	µb)(x−	ADJ
ejpam-3441	249	5	y	y	NOUN
ejpam-3441	249	6	)	)	PUNCT
ejpam-3441	249	7	,	,	PUNCT
ejpam-3441	249	8	α	α	PROPN
ejpam-3441	249	9	}	}	PUNCT
ejpam-3441	249	10	≥	≥	NOUN
ejpam-3441	249	11	min{(µa	min{(µa	VERB
ejpam-3441	249	12	∧βα	∧βα	ADJ
ejpam-3441	249	13	µb)(x	µb)(x	NOUN
ejpam-3441	249	14	)	)	PUNCT
ejpam-3441	249	15	,	,	PUNCT
ejpam-3441	249	16	(	(	PUNCT
ejpam-3441	249	17	µa	µa	ADP
ejpam-3441	249	18	∧βα	∧βα	ADJ
ejpam-3441	249	19	µb)(y	µb)(y	PROPN
ejpam-3441	249	20	)	)	PUNCT
ejpam-3441	249	21	,	,	PUNCT
ejpam-3441	249	22	β	β	X
ejpam-3441	249	23	}	}	PUNCT
ejpam-3441	249	24	.	.	PUNCT
ejpam-3441	250	1	similarly	similarly	ADV
ejpam-3441	250	2	,	,	PUNCT
ejpam-3441	250	3	we	we	PRON
ejpam-3441	250	4	have	have	AUX
ejpam-3441	250	5	max{(µa	max{(µa	VERB
ejpam-3441	250	6	∧βα	∧βα	ADJ
ejpam-3441	250	7	µb)(xy	µb)(xy	NUM
ejpam-3441	250	8	)	)	PUNCT
ejpam-3441	250	9	,	,	PUNCT
ejpam-3441	250	10	α	α	PROPN
ejpam-3441	250	11	}	}	PUNCT
ejpam-3441	250	12	≥	≥	NOUN
ejpam-3441	250	13	min{(µa	min{(µa	VERB
ejpam-3441	250	14	∧βα	∧βα	ADJ
ejpam-3441	250	15	µb)(x	µb)(x	NOUN
ejpam-3441	250	16	)	)	PUNCT
ejpam-3441	250	17	,	,	PUNCT
ejpam-3441	250	18	(	(	PUNCT
ejpam-3441	250	19	µa	µa	ADP
ejpam-3441	250	20	∧βα	∧βα	ADJ
ejpam-3441	250	21	µb)(y	µb)(y	PROPN
ejpam-3441	250	22	)	)	PUNCT
ejpam-3441	250	23	,	,	PUNCT
ejpam-3441	250	24	β	β	X
ejpam-3441	250	25	}	}	PUNCT
ejpam-3441	250	26	.	.	PUNCT
ejpam-3441	251	1	in	in	ADP
ejpam-3441	251	2	same	same	ADJ
ejpam-3441	251	3	lines	line	NOUN
ejpam-3441	251	4	we	we	PRON
ejpam-3441	251	5	have	have	VERB
ejpam-3441	251	6	min{(γa	min{(γa	ADJ
ejpam-3441	251	7	∨βα	∨βα	ADJ
ejpam-3441	251	8	γb)(x	γb)(x	PROPN
ejpam-3441	251	9	−	−	PROPN
ejpam-3441	251	10	y	y	NOUN
ejpam-3441	251	11	)	)	PUNCT
ejpam-3441	251	12	,	,	PUNCT
ejpam-3441	251	13	(	(	PUNCT
ejpam-3441	251	14	1	1	NUM
ejpam-3441	251	15	−	−	PROPN
ejpam-3441	251	16	α	α	X
ejpam-3441	251	17	)	)	PUNCT
ejpam-3441	251	18	}	}	PUNCT
ejpam-3441	251	19	≤	≤	NOUN
ejpam-3441	251	20	max{(γa	max{(γa	ADJ
ejpam-3441	251	21	∨βα	∨βα	ADJ
ejpam-3441	251	22	γb)(x	γb)(x	PROPN
ejpam-3441	251	23	)	)	PUNCT
ejpam-3441	251	24	,	,	PUNCT
ejpam-3441	251	25	(	(	PUNCT
ejpam-3441	251	26	γa	γa	PROPN
ejpam-3441	251	27	∨βα	∨βα	VERB
ejpam-3441	251	28	γb)(y	γb)(y	PROPN
ejpam-3441	251	29	)	)	PUNCT
ejpam-3441	251	30	,	,	PUNCT
ejpam-3441	251	31	(	(	PUNCT
ejpam-3441	251	32	1	1	NUM
ejpam-3441	251	33	−	−	NOUN
ejpam-3441	251	34	β	β	NOUN
ejpam-3441	251	35	)	)	PUNCT
ejpam-3441	251	36	}	}	PUNCT
ejpam-3441	251	37	and	and	CCONJ
ejpam-3441	251	38	min{(γa	min{(γa	ADJ
ejpam-3441	251	39	∨βα	∨βα	PROPN
ejpam-3441	251	40	γb)(xy	γb)(xy	PUNCT
ejpam-3441	251	41	)	)	PUNCT
ejpam-3441	251	42	,	,	PUNCT
ejpam-3441	251	43	(	(	PUNCT
ejpam-3441	251	44	1−	1−	NUM
ejpam-3441	251	45	α	α	NOUN
ejpam-3441	251	46	)	)	PUNCT
ejpam-3441	251	47	}	}	PUNCT
ejpam-3441	251	48	≤	≤	NOUN
ejpam-3441	252	1	max{(γa	max{(γa	ADJ
ejpam-3441	252	2	∨βα	∨βα	ADJ
ejpam-3441	252	3	γb)(x	γb)(x	PROPN
ejpam-3441	252	4	)	)	PUNCT
ejpam-3441	252	5	,	,	PUNCT
ejpam-3441	252	6	(	(	PUNCT
ejpam-3441	252	7	γa	γa	PROPN
ejpam-3441	252	8	∨βα	∨βα	VERB
ejpam-3441	252	9	γb)(y	γb)(y	PROPN
ejpam-3441	252	10	)	)	PUNCT
ejpam-3441	252	11	,	,	PUNCT
ejpam-3441	252	12	(	(	PUNCT
ejpam-3441	252	13	1−	1−	NUM
ejpam-3441	252	14	β	β	NOUN
ejpam-3441	252	15	)	)	PUNCT
ejpam-3441	252	16	}	}	PUNCT
ejpam-3441	252	17	.	.	PUNCT
ejpam-3441	253	1	hence	hence	ADV
ejpam-3441	253	2	a	a	DET
ejpam-3441	253	3	∧βα	∧βα	PROPN
ejpam-3441	253	4	b	b	NOUN
ejpam-3441	253	5	is	be	AUX
ejpam-3441	253	6	an	an	DET
ejpam-3441	253	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	253	8	fuzzy	fuzzy	ADJ
ejpam-3441	253	9	la	la	NOUN
ejpam-3441	253	10	-	-	PUNCT
ejpam-3441	253	11	subring	subre	VERB
ejpam-3441	253	12	with	with	ADP
ejpam-3441	253	13	thresholds	threshold	NOUN
ejpam-3441	253	14	(	(	PUNCT
ejpam-3441	253	15	α	α	X
ejpam-3441	253	16	,	,	PUNCT
ejpam-3441	253	17	β	β	X
ejpam-3441	253	18	]	]	PUNCT
ejpam-3441	253	19	of	of	ADP
ejpam-3441	253	20	r.	r.	PROPN
ejpam-3441	253	21	lemma	lemma	PROPN
ejpam-3441	253	22	4	4	X
ejpam-3441	253	23	.	.	PUNCT
ejpam-3441	254	1	if	if	SCONJ
ejpam-3441	254	2	a	a	PRON
ejpam-3441	254	3	and	and	CCONJ
ejpam-3441	254	4	b	b	NOUN
ejpam-3441	254	5	are	be	AUX
ejpam-3441	254	6	two	two	NUM
ejpam-3441	254	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	254	8	fuzzy	fuzzy	ADJ
ejpam-3441	254	9	la	la	NOUN
ejpam-3441	254	10	-	-	NOUN
ejpam-3441	254	11	subrings	subring	NOUN
ejpam-3441	254	12	with	with	ADP
ejpam-3441	254	13	thresholds	threshold	NOUN
ejpam-3441	254	14	(	(	PUNCT
ejpam-3441	254	15	α	α	X
ejpam-3441	254	16	,	,	PUNCT
ejpam-3441	254	17	β	β	X
ejpam-3441	254	18	]	]	PUNCT
ejpam-3441	254	19	of	of	ADP
ejpam-3441	254	20	an	an	DET
ejpam-3441	254	21	la	la	ADJ
ejpam-3441	254	22	-	-	PUNCT
ejpam-3441	254	23	ring	ring	NOUN
ejpam-3441	254	24	r	r	NOUN
ejpam-3441	254	25	,	,	PUNCT
ejpam-3441	254	26	then	then	ADV
ejpam-3441	254	27	a	a	DET
ejpam-3441	254	28	◦	◦	NOUN
ejpam-3441	254	29	βαb	βαb	NOUN
ejpam-3441	254	30	is	be	AUX
ejpam-3441	254	31	also	also	ADV
ejpam-3441	254	32	an	an	DET
ejpam-3441	254	33	intuitionistic	intuitionistic	ADJ
ejpam-3441	254	34	fuzzy	fuzzy	ADJ
ejpam-3441	254	35	la	la	NOUN
ejpam-3441	254	36	-	-	PUNCT
ejpam-3441	254	37	subring	subre	VERB
ejpam-3441	254	38	with	with	ADP
ejpam-3441	254	39	thresholds	threshold	NOUN
ejpam-3441	254	40	(	(	PUNCT
ejpam-3441	254	41	α	α	X
ejpam-3441	254	42	,	,	PUNCT
ejpam-3441	254	43	β	β	X
ejpam-3441	254	44	]	]	PUNCT
ejpam-3441	254	45	of	of	ADP
ejpam-3441	254	46	r.	r.	PROPN
ejpam-3441	254	47	proof	proof	PROPN
ejpam-3441	254	48	.	.	PUNCT
ejpam-3441	255	1	suppose	suppose	VERB
ejpam-3441	255	2	that	that	SCONJ
ejpam-3441	255	3	a	a	DET
ejpam-3441	255	4	=	=	SYM
ejpam-3441	255	5	(	(	PUNCT
ejpam-3441	255	6	µa	µa	PROPN
ejpam-3441	255	7	,	,	PUNCT
ejpam-3441	255	8	γa	γa	PROPN
ejpam-3441	255	9	)	)	PUNCT
ejpam-3441	255	10	and	and	CCONJ
ejpam-3441	255	11	b	b	X
ejpam-3441	255	12	=	=	SYM
ejpam-3441	255	13	(	(	PUNCT
ejpam-3441	255	14	µb	µb	PROPN
ejpam-3441	255	15	,	,	PUNCT
ejpam-3441	255	16	γb	γb	PROPN
ejpam-3441	255	17	)	)	PUNCT
ejpam-3441	255	18	are	be	AUX
ejpam-3441	255	19	two	two	NUM
ejpam-3441	255	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	255	21	fuzzy	fuzzy	ADJ
ejpam-3441	255	22	lasubrings	lasubring	NOUN
ejpam-3441	255	23	with	with	ADP
ejpam-3441	255	24	thresholds	threshold	NOUN
ejpam-3441	255	25	(	(	PUNCT
ejpam-3441	255	26	α	α	X
ejpam-3441	255	27	,	,	PUNCT
ejpam-3441	255	28	β	β	X
ejpam-3441	255	29	]	]	PUNCT
ejpam-3441	255	30	of	of	ADP
ejpam-3441	255	31	an	an	DET
ejpam-3441	255	32	la	la	ADJ
ejpam-3441	255	33	-	-	PUNCT
ejpam-3441	255	34	ring	ring	NOUN
ejpam-3441	255	35	r.	r.	NOUN
ejpam-3441	255	36	we	we	PRON
ejpam-3441	255	37	have	have	VERB
ejpam-3441	255	38	to	to	PART
ejpam-3441	255	39	show	show	VERB
ejpam-3441	255	40	that	that	SCONJ
ejpam-3441	255	41	a	a	DET
ejpam-3441	255	42	◦	◦	NOUN
ejpam-3441	255	43	βα	βα	NOUN
ejpam-3441	255	44	b	b	NOUN
ejpam-3441	255	45	is	be	AUX
ejpam-3441	255	46	also	also	ADV
ejpam-3441	255	47	an	an	DET
ejpam-3441	255	48	intuitionistic	intuitionistic	ADJ
ejpam-3441	255	49	fuzzy	fuzzy	ADJ
ejpam-3441	255	50	la	la	NOUN
ejpam-3441	255	51	-	-	PUNCT
ejpam-3441	255	52	subring	subre	VERB
ejpam-3441	255	53	with	with	ADP
ejpam-3441	255	54	thresholds	threshold	NOUN
ejpam-3441	255	55	(	(	PUNCT
ejpam-3441	255	56	α	α	X
ejpam-3441	255	57	,	,	PUNCT
ejpam-3441	255	58	β	β	X
ejpam-3441	255	59	]	]	PUNCT
ejpam-3441	255	60	of	of	ADP
ejpam-3441	255	61	r.	r.	PROPN
ejpam-3441	255	62	now	now	ADV
ejpam-3441	255	63	(	(	PUNCT
ejpam-3441	255	64	µa	µa	ADP
ejpam-3441	255	65	◦	◦	NOUN
ejpam-3441	255	66	βα	βα	X
ejpam-3441	255	67	µb)2	µb)2	NOUN
ejpam-3441	255	68	=	=	SYM
ejpam-3441	255	69	(	(	PUNCT
ejpam-3441	255	70	µa	µa	INTJ
ejpam-3441	255	71	◦	◦	NOUN
ejpam-3441	255	72	βα	βα	NOUN
ejpam-3441	255	73	µb	µb	NOUN
ejpam-3441	255	74	)	)	PUNCT
ejpam-3441	255	75	◦	◦	NOUN
ejpam-3441	255	76	βα	βα	X
ejpam-3441	255	77	(	(	PUNCT
ejpam-3441	255	78	µa	µa	ADP
ejpam-3441	255	79	◦	◦	NOUN
ejpam-3441	255	80	βα	βα	NOUN
ejpam-3441	255	81	µb	µb	NOUN
ejpam-3441	255	82	)	)	PUNCT
ejpam-3441	255	83	=	=	SYM
ejpam-3441	255	84	(	(	PUNCT
ejpam-3441	255	85	µa	µa	INTJ
ejpam-3441	255	86	◦	◦	NOUN
ejpam-3441	255	87	βα	βα	NOUN
ejpam-3441	255	88	µa	µa	NOUN
ejpam-3441	255	89	)	)	PUNCT
ejpam-3441	255	90	◦	◦	NOUN
ejpam-3441	255	91	βα	βα	AUX
ejpam-3441	255	92	(	(	PUNCT
ejpam-3441	255	93	µb	µb	ADP
ejpam-3441	255	94	◦	◦	NOUN
ejpam-3441	255	95	βα	βα	NOUN
ejpam-3441	255	96	µb	µb	NOUN
ejpam-3441	255	97	)	)	PUNCT
ejpam-3441	255	98	⊆	⊆	NUM
ejpam-3441	255	99	(	(	PUNCT
ejpam-3441	255	100	µa)βα	µa)βα	NUM
ejpam-3441	255	101	◦	◦	NOUN
ejpam-3441	255	102	βα	βα	X
ejpam-3441	255	103	(	(	PUNCT
ejpam-3441	255	104	µb)βα	µb)βα	X
ejpam-3441	255	105	=	=	SYM
ejpam-3441	255	106	µa	µa	NOUN
ejpam-3441	255	107	◦	◦	NOUN
ejpam-3441	255	108	βα	βα	NOUN
ejpam-3441	255	109	µb	µb	VERB
ejpam-3441	255	110	and	and	CCONJ
ejpam-3441	255	111	(	(	PUNCT
ejpam-3441	255	112	γa	γa	PROPN
ejpam-3441	255	113	◦	◦	PROPN
ejpam-3441	255	114	βα	βα	NOUN
ejpam-3441	255	115	γb)2	γb)2	PROPN
ejpam-3441	255	116	=	=	PUNCT
ejpam-3441	255	117	(	(	PUNCT
ejpam-3441	255	118	γa	γa	PROPN
ejpam-3441	255	119	◦	◦	PROPN
ejpam-3441	255	120	βα	βα	NOUN
ejpam-3441	255	121	γb	γb	PROPN
ejpam-3441	255	122	)	)	PUNCT
ejpam-3441	255	123	◦	◦	NOUN
ejpam-3441	255	124	βα	βα	X
ejpam-3441	255	125	(	(	PUNCT
ejpam-3441	255	126	γa	γa	PROPN
ejpam-3441	255	127	◦	◦	PROPN
ejpam-3441	255	128	βα	βα	NOUN
ejpam-3441	255	129	γb	γb	PROPN
ejpam-3441	255	130	)	)	PUNCT
ejpam-3441	255	131	=	=	PUNCT
ejpam-3441	256	1	(	(	PUNCT
ejpam-3441	256	2	γa	γa	PROPN
ejpam-3441	256	3	◦	◦	PROPN
ejpam-3441	256	4	βα	βα	NOUN
ejpam-3441	256	5	γa	γa	NOUN
ejpam-3441	256	6	)	)	PUNCT
ejpam-3441	256	7	◦	◦	NOUN
ejpam-3441	256	8	βα	βα	X
ejpam-3441	256	9	(	(	PUNCT
ejpam-3441	256	10	γb	γb	INTJ
ejpam-3441	256	11	◦	◦	NOUN
ejpam-3441	256	12	βα	βα	NOUN
ejpam-3441	256	13	γb	γb	PROPN
ejpam-3441	256	14	)	)	PUNCT
ejpam-3441	256	15	⊇	⊇	NOUN
ejpam-3441	256	16	(	(	PUNCT
ejpam-3441	256	17	γa)βα	γa)βα	X
ejpam-3441	256	18	◦	◦	NOUN
ejpam-3441	256	19	βα	βα	X
ejpam-3441	256	20	(	(	PUNCT
ejpam-3441	256	21	γb)βα	γb)βα	X
ejpam-3441	256	22	=	=	SYM
ejpam-3441	256	23	γa	γa	PROPN
ejpam-3441	256	24	◦	◦	PROPN
ejpam-3441	256	25	βα	βα	NOUN
ejpam-3441	256	26	γb	γb	PROPN
ejpam-3441	256	27	.	.	PUNCT
ejpam-3441	257	1	since	since	SCONJ
ejpam-3441	257	2	µb	µb	PROPN
ejpam-3441	257	3	−βα	−βα	PROPN
ejpam-3441	257	4	µb	µb	VERB
ejpam-3441	257	5	⊆	⊆	NUM
ejpam-3441	257	6	(	(	PUNCT
ejpam-3441	257	7	µb)βα	µb)βα	PRON
ejpam-3441	257	8	and	and	CCONJ
ejpam-3441	257	9	γb	γb	VERB
ejpam-3441	257	10	−βα	−βα	ADV
ejpam-3441	257	11	γb	γb	PROPN
ejpam-3441	257	12	⊇	⊇	X
ejpam-3441	257	13	(	(	PUNCT
ejpam-3441	257	14	γb)βα	γb)βα	PROPN
ejpam-3441	257	15	,	,	PUNCT
ejpam-3441	257	16	b	b	X
ejpam-3441	257	17	=	=	SYM
ejpam-3441	257	18	(	(	PUNCT
ejpam-3441	257	19	µb	µb	PROPN
ejpam-3441	257	20	,	,	PUNCT
ejpam-3441	257	21	γb	γb	NOUN
ejpam-3441	257	22	)	)	PUNCT
ejpam-3441	257	23	being	be	AUX
ejpam-3441	257	24	an	an	DET
ejpam-3441	257	25	intuitionistic	intuitionistic	ADJ
ejpam-3441	257	26	fuzzy	fuzzy	ADJ
ejpam-3441	257	27	la	la	NOUN
ejpam-3441	257	28	-	-	PUNCT
ejpam-3441	257	29	subring	subre	VERB
ejpam-3441	257	30	with	with	ADP
ejpam-3441	257	31	thresholds	threshold	NOUN
ejpam-3441	257	32	(	(	PUNCT
ejpam-3441	257	33	α	α	X
ejpam-3441	257	34	,	,	PUNCT
ejpam-3441	257	35	β	β	X
ejpam-3441	257	36	]	]	X
ejpam-3441	257	37	.	.	PUNCT
ejpam-3441	258	1	this	this	PRON
ejpam-3441	258	2	implies	imply	VERB
ejpam-3441	258	3	that	that	SCONJ
ejpam-3441	258	4	µa	µa	ADP
ejpam-3441	258	5	◦	◦	NOUN
ejpam-3441	258	6	βα(µb−βαµb	βα(µb−βαµb	X
ejpam-3441	258	7	)	)	PUNCT
ejpam-3441	258	8	⊆	⊆	NUM
ejpam-3441	258	9	µa	µa	ADP
ejpam-3441	258	10	◦	◦	NOUN
ejpam-3441	258	11	βαµb	βαµb	NOUN
ejpam-3441	258	12	and	and	CCONJ
ejpam-3441	258	13	γa	γa	NOUN
ejpam-3441	258	14	◦	◦	NOUN
ejpam-3441	258	15	βα(γb−βαγb	βα(γb−βαγb	SYM
ejpam-3441	258	16	)	)	PUNCT
ejpam-3441	258	17	⊇	⊇	PROPN
ejpam-3441	258	18	γa	γa	PROPN
ejpam-3441	258	19	◦	◦	NOUN
ejpam-3441	258	20	βαγb	βαγb	ADJ
ejpam-3441	258	21	,	,	PUNCT
ejpam-3441	258	22	i.e.	i.e.	X
ejpam-3441	258	23	,	,	PUNCT
ejpam-3441	258	24	µa	µa	ADP
ejpam-3441	258	25	◦	◦	NOUN
ejpam-3441	258	26	βαµb−βαµa	βαµb−βαµa	NOUN
ejpam-3441	258	27	◦	◦	NOUN
ejpam-3441	258	28	βαµb	βαµb	ADJ
ejpam-3441	258	29	⊆	⊆	NUM
ejpam-3441	258	30	µa	µa	ADP
ejpam-3441	258	31	◦	◦	NOUN
ejpam-3441	258	32	βαµb	βαµb	NOUN
ejpam-3441	258	33	and	and	CCONJ
ejpam-3441	258	34	γa	γa	NOUN
ejpam-3441	258	35	◦	◦	NOUN
ejpam-3441	258	36	βαγb−βαγa	βαγb−βαγa	NOUN
ejpam-3441	258	37	◦	◦	NOUN
ejpam-3441	258	38	βαγb	βαγb	ADJ
ejpam-3441	258	39	⊇	⊇	PROPN
ejpam-3441	258	40	γa	γa	PROPN
ejpam-3441	258	41	◦	◦	PROPN
ejpam-3441	258	42	βα	βα	NOUN
ejpam-3441	258	43	γb	γb	PROPN
ejpam-3441	258	44	.	.	PUNCT
ejpam-3441	259	1	therefore	therefore	ADV
ejpam-3441	259	2	a	a	DET
ejpam-3441	259	3	◦	◦	NOUN
ejpam-3441	259	4	βα	βα	NOUN
ejpam-3441	259	5	b	b	NOUN
ejpam-3441	259	6	is	be	AUX
ejpam-3441	259	7	an	an	DET
ejpam-3441	259	8	intuitionistic	intuitionistic	ADJ
ejpam-3441	259	9	fuzzy	fuzzy	ADJ
ejpam-3441	259	10	la	la	NOUN
ejpam-3441	259	11	-	-	PUNCT
ejpam-3441	259	12	subring	subre	VERB
ejpam-3441	259	13	with	with	ADP
ejpam-3441	259	14	thresholds	threshold	NOUN
ejpam-3441	259	15	(	(	PUNCT
ejpam-3441	259	16	α	α	X
ejpam-3441	259	17	,	,	PUNCT
ejpam-3441	259	18	β	β	X
ejpam-3441	259	19	]	]	PUNCT
ejpam-3441	259	20	of	of	ADP
ejpam-3441	259	21	r.	r.	PROPN
ejpam-3441	259	22	remark	remark	PROPN
ejpam-3441	259	23	2	2	NUM
ejpam-3441	259	24	.	.	PUNCT
ejpam-3441	260	1	if	if	SCONJ
ejpam-3441	260	2	a	a	PRON
ejpam-3441	260	3	is	be	AUX
ejpam-3441	260	4	an	an	DET
ejpam-3441	260	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	260	6	fuzzy	fuzzy	ADJ
ejpam-3441	260	7	la	la	NOUN
ejpam-3441	260	8	-	-	PUNCT
ejpam-3441	260	9	subring	subre	VERB
ejpam-3441	260	10	with	with	ADP
ejpam-3441	260	11	thresholds	threshold	NOUN
ejpam-3441	260	12	(	(	PUNCT
ejpam-3441	260	13	α	α	X
ejpam-3441	260	14	,	,	PUNCT
ejpam-3441	260	15	β	β	X
ejpam-3441	260	16	]	]	PUNCT
ejpam-3441	260	17	of	of	ADP
ejpam-3441	260	18	an	an	DET
ejpam-3441	260	19	la	la	ADJ
ejpam-3441	260	20	-	-	PUNCT
ejpam-3441	260	21	ring	ring	NOUN
ejpam-3441	260	22	r	r	NOUN
ejpam-3441	260	23	,	,	PUNCT
ejpam-3441	260	24	then	then	ADV
ejpam-3441	260	25	a	a	DET
ejpam-3441	260	26	◦	◦	NOUN
ejpam-3441	260	27	βα	βα	X
ejpam-3441	260	28	a	a	NOUN
ejpam-3441	260	29	is	be	AUX
ejpam-3441	260	30	also	also	ADV
ejpam-3441	260	31	an	an	DET
ejpam-3441	260	32	intuitionistic	intuitionistic	ADJ
ejpam-3441	260	33	fuzzy	fuzzy	ADJ
ejpam-3441	260	34	la	la	NOUN
ejpam-3441	260	35	-	-	PUNCT
ejpam-3441	260	36	subring	subre	VERB
ejpam-3441	260	37	with	with	ADP
ejpam-3441	260	38	thresholds	threshold	NOUN
ejpam-3441	260	39	(	(	PUNCT
ejpam-3441	260	40	α	α	X
ejpam-3441	260	41	,	,	PUNCT
ejpam-3441	260	42	β	β	X
ejpam-3441	260	43	]	]	PUNCT
ejpam-3441	260	44	of	of	ADP
ejpam-3441	260	45	r.	r.	PROPN
ejpam-3441	260	46	k.	k.	PROPN
ejpam-3441	260	47	nasreen	nasreen	PROPN
ejpam-3441	260	48	et	et	PROPN
ejpam-3441	260	49	al	al	PROPN
ejpam-3441	260	50	.	.	PUNCT
ejpam-3441	260	51	/	/	SYM
ejpam-3441	260	52	eur	eur	PROPN
ejpam-3441	260	53	.	.	PUNCT
ejpam-3441	261	1	j.	j.	PROPN
ejpam-3441	261	2	pure	pure	PROPN
ejpam-3441	261	3	appl	appl	PROPN
ejpam-3441	261	4	.	.	PROPN
ejpam-3441	261	5	math	math	PROPN
ejpam-3441	261	6	,	,	PUNCT
ejpam-3441	261	7	12	12	NUM
ejpam-3441	261	8	(	(	PUNCT
ejpam-3441	261	9	3	3	NUM
ejpam-3441	261	10	)	)	PUNCT
ejpam-3441	261	11	(	(	PUNCT
ejpam-3441	261	12	2019	2019	NUM
ejpam-3441	261	13	)	)	PUNCT
ejpam-3441	261	14	,	,	PUNCT
ejpam-3441	261	15	906	906	NUM
ejpam-3441	261	16	-	-	SYM
ejpam-3441	261	17	943	943	NUM
ejpam-3441	261	18	917	917	NUM
ejpam-3441	261	19	lemma	lemma	PROPN
ejpam-3441	261	20	5	5	NUM
ejpam-3441	261	21	.	.	PUNCT
ejpam-3441	262	1	let	let	VERB
ejpam-3441	262	2	r	r	PRON
ejpam-3441	262	3	be	be	AUX
ejpam-3441	262	4	an	an	DET
ejpam-3441	262	5	la	la	NOUN
ejpam-3441	262	6	-	-	NOUN
ejpam-3441	262	7	ring	ring	NOUN
ejpam-3441	262	8	with	with	ADP
ejpam-3441	262	9	left	left	ADJ
ejpam-3441	262	10	identity	identity	NOUN
ejpam-3441	262	11	e.	e.	PROPN
ejpam-3441	262	12	then	then	ADV
ejpam-3441	262	13	every	every	DET
ejpam-3441	262	14	intuitionistic	intuitionistic	ADJ
ejpam-3441	262	15	fuzzy	fuzzy	ADJ
ejpam-3441	262	16	right	right	ADJ
ejpam-3441	262	17	ideal	ideal	NOUN
ejpam-3441	262	18	with	with	ADP
ejpam-3441	262	19	thresholds	threshold	NOUN
ejpam-3441	262	20	(	(	PUNCT
ejpam-3441	262	21	α	α	X
ejpam-3441	262	22	,	,	PUNCT
ejpam-3441	262	23	β	β	X
ejpam-3441	262	24	]	]	PUNCT
ejpam-3441	262	25	of	of	ADP
ejpam-3441	262	26	r	r	NOUN
ejpam-3441	262	27	is	be	AUX
ejpam-3441	262	28	an	an	DET
ejpam-3441	262	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	262	30	fuzzy	fuzzy	ADJ
ejpam-3441	262	31	ideal	ideal	NOUN
ejpam-3441	262	32	with	with	ADP
ejpam-3441	262	33	thresholds	threshold	NOUN
ejpam-3441	262	34	(	(	PUNCT
ejpam-3441	262	35	α	α	X
ejpam-3441	262	36	,	,	PUNCT
ejpam-3441	262	37	β	β	X
ejpam-3441	262	38	]	]	PUNCT
ejpam-3441	262	39	of	of	ADP
ejpam-3441	262	40	r.	r.	PROPN
ejpam-3441	262	41	proof	proof	PROPN
ejpam-3441	262	42	.	.	PUNCT
ejpam-3441	263	1	suppose	suppose	VERB
ejpam-3441	263	2	that	that	SCONJ
ejpam-3441	263	3	a	a	DET
ejpam-3441	263	4	=	=	SYM
ejpam-3441	263	5	(	(	PUNCT
ejpam-3441	263	6	µa	µa	PROPN
ejpam-3441	263	7	,	,	PUNCT
ejpam-3441	263	8	γa	γa	PROPN
ejpam-3441	263	9	)	)	PUNCT
ejpam-3441	263	10	is	be	AUX
ejpam-3441	263	11	an	an	DET
ejpam-3441	263	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	263	13	fuzzy	fuzzy	ADJ
ejpam-3441	263	14	right	right	ADJ
ejpam-3441	263	15	ideal	ideal	NOUN
ejpam-3441	263	16	with	with	ADP
ejpam-3441	263	17	thresholds	threshold	NOUN
ejpam-3441	263	18	(	(	PUNCT
ejpam-3441	263	19	α	α	X
ejpam-3441	263	20	,	,	PUNCT
ejpam-3441	263	21	β	β	X
ejpam-3441	263	22	]	]	PUNCT
ejpam-3441	263	23	of	of	ADP
ejpam-3441	263	24	an	an	DET
ejpam-3441	263	25	la	la	ADJ
ejpam-3441	263	26	-	-	PUNCT
ejpam-3441	263	27	ring	ring	NOUN
ejpam-3441	263	28	r	r	NOUN
ejpam-3441	263	29	and	and	CCONJ
ejpam-3441	263	30	x	x	NOUN
ejpam-3441	263	31	,	,	PUNCT
ejpam-3441	263	32	y	y	PROPN
ejpam-3441	263	33	∈	∈	PROPN
ejpam-3441	263	34	r.	r.	NOUN
ejpam-3441	263	35	thus	thus	ADV
ejpam-3441	263	36	max{µa	max{µa	VERB
ejpam-3441	263	37	(	(	PUNCT
ejpam-3441	263	38	xy	xy	PROPN
ejpam-3441	263	39	)	)	PUNCT
ejpam-3441	263	40	,	,	PUNCT
ejpam-3441	263	41	α	α	X
ejpam-3441	263	42	}	}	PUNCT
ejpam-3441	263	43	=	=	SYM
ejpam-3441	263	44	max{µa	max{µa	X
ejpam-3441	263	45	(	(	PUNCT
ejpam-3441	263	46	(	(	PUNCT
ejpam-3441	263	47	ex	ex	NOUN
ejpam-3441	263	48	)	)	PUNCT
ejpam-3441	263	49	y	y	NOUN
ejpam-3441	263	50	)	)	PUNCT
ejpam-3441	263	51	,	,	PUNCT
ejpam-3441	263	52	α	α	X
ejpam-3441	263	53	}	}	PUNCT
ejpam-3441	263	54	=	=	SYM
ejpam-3441	263	55	max{µa	max{µa	X
ejpam-3441	263	56	(	(	PUNCT
ejpam-3441	263	57	(	(	PUNCT
ejpam-3441	263	58	yx	yx	NOUN
ejpam-3441	263	59	)	)	PUNCT
ejpam-3441	263	60	e	e	NOUN
ejpam-3441	263	61	)	)	PUNCT
ejpam-3441	263	62	,	,	PUNCT
ejpam-3441	263	63	α	α	X
ejpam-3441	263	64	}	}	PUNCT
ejpam-3441	263	65	≥	≥	NOUN
ejpam-3441	263	66	min{µa	min{µa	X
ejpam-3441	263	67	(	(	PUNCT
ejpam-3441	263	68	yx	yx	NOUN
ejpam-3441	263	69	)	)	PUNCT
ejpam-3441	263	70	,	,	PUNCT
ejpam-3441	263	71	β	β	X
ejpam-3441	263	72	}	}	PUNCT
ejpam-3441	263	73	≥	≥	X
ejpam-3441	263	74	min{µa	min{µa	X
ejpam-3441	263	75	(	(	PUNCT
ejpam-3441	263	76	y	y	NOUN
ejpam-3441	263	77	)	)	PUNCT
ejpam-3441	263	78	,	,	PUNCT
ejpam-3441	263	79	β	β	X
ejpam-3441	263	80	}	}	PUNCT
ejpam-3441	263	81	and	and	CCONJ
ejpam-3441	263	82	min{γa	min{γa	X
ejpam-3441	263	83	(	(	PUNCT
ejpam-3441	263	84	xy	xy	NOUN
ejpam-3441	263	85	)	)	PUNCT
ejpam-3441	263	86	,	,	PUNCT
ejpam-3441	263	87	(	(	PUNCT
ejpam-3441	263	88	1−	1−	NUM
ejpam-3441	263	89	α	α	NOUN
ejpam-3441	263	90	)	)	PUNCT
ejpam-3441	263	91	}	}	PUNCT
ejpam-3441	264	1	=	=	SYM
ejpam-3441	264	2	min{γa	min{γa	X
ejpam-3441	264	3	(	(	PUNCT
ejpam-3441	264	4	(	(	PUNCT
ejpam-3441	264	5	ex	ex	NOUN
ejpam-3441	264	6	)	)	PUNCT
ejpam-3441	264	7	y	y	NOUN
ejpam-3441	264	8	)	)	PUNCT
ejpam-3441	264	9	,	,	PUNCT
ejpam-3441	264	10	(	(	PUNCT
ejpam-3441	264	11	1−	1−	NUM
ejpam-3441	264	12	α	α	NOUN
ejpam-3441	264	13	)	)	PUNCT
ejpam-3441	264	14	}	}	PUNCT
ejpam-3441	264	15	=	=	SYM
ejpam-3441	264	16	min{γa	min{γa	X
ejpam-3441	264	17	(	(	PUNCT
ejpam-3441	264	18	(	(	PUNCT
ejpam-3441	264	19	yx	yx	NOUN
ejpam-3441	264	20	)	)	PUNCT
ejpam-3441	264	21	e	e	NOUN
ejpam-3441	264	22	)	)	PUNCT
ejpam-3441	264	23	,	,	PUNCT
ejpam-3441	264	24	(	(	PUNCT
ejpam-3441	264	25	1−	1−	NUM
ejpam-3441	264	26	α	α	NOUN
ejpam-3441	264	27	)	)	PUNCT
ejpam-3441	264	28	}	}	PUNCT
ejpam-3441	264	29	≤	≤	NUM
ejpam-3441	264	30	max{γa	max{γa	NOUN
ejpam-3441	264	31	(	(	PUNCT
ejpam-3441	264	32	yx	yx	NOUN
ejpam-3441	264	33	)	)	PUNCT
ejpam-3441	264	34	,	,	PUNCT
ejpam-3441	264	35	(	(	PUNCT
ejpam-3441	264	36	1−	1−	NUM
ejpam-3441	264	37	β	β	NOUN
ejpam-3441	264	38	)	)	PUNCT
ejpam-3441	264	39	}	}	PUNCT
ejpam-3441	264	40	≤	≤	NUM
ejpam-3441	264	41	max{γa	max{γa	NOUN
ejpam-3441	264	42	(	(	PUNCT
ejpam-3441	264	43	y	y	NOUN
ejpam-3441	264	44	)	)	PUNCT
ejpam-3441	264	45	,	,	PUNCT
ejpam-3441	264	46	(	(	PUNCT
ejpam-3441	264	47	1−	1−	NUM
ejpam-3441	264	48	β	β	NOUN
ejpam-3441	264	49	)	)	PUNCT
ejpam-3441	264	50	}	}	PUNCT
ejpam-3441	264	51	.	.	PUNCT
ejpam-3441	265	1	therefore	therefore	ADV
ejpam-3441	265	2	a	a	PRON
ejpam-3441	265	3	is	be	AUX
ejpam-3441	265	4	an	an	DET
ejpam-3441	265	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	265	6	fuzzy	fuzzy	ADJ
ejpam-3441	265	7	ideal	ideal	NOUN
ejpam-3441	265	8	with	with	ADP
ejpam-3441	265	9	thresholds	threshold	NOUN
ejpam-3441	265	10	(	(	PUNCT
ejpam-3441	265	11	α	α	X
ejpam-3441	265	12	,	,	PUNCT
ejpam-3441	265	13	β	β	X
ejpam-3441	265	14	]	]	PUNCT
ejpam-3441	265	15	of	of	ADP
ejpam-3441	265	16	r.	r.	PROPN
ejpam-3441	265	17	lemma	lemma	PROPN
ejpam-3441	265	18	6	6	NUM
ejpam-3441	265	19	.	.	PUNCT
ejpam-3441	266	1	if	if	SCONJ
ejpam-3441	266	2	a	a	PRON
ejpam-3441	266	3	and	and	CCONJ
ejpam-3441	266	4	b	b	NOUN
ejpam-3441	266	5	are	be	AUX
ejpam-3441	266	6	two	two	NUM
ejpam-3441	266	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	266	8	fuzzy	fuzzy	ADJ
ejpam-3441	266	9	left	left	NOUN
ejpam-3441	266	10	(	(	PUNCT
ejpam-3441	266	11	resp	resp	NOUN
ejpam-3441	266	12	.	.	PUNCT
ejpam-3441	267	1	right	right	ADJ
ejpam-3441	267	2	)	)	PUNCT
ejpam-3441	267	3	ideals	ideal	NOUN
ejpam-3441	267	4	with	with	ADP
ejpam-3441	267	5	thresholds	threshold	NOUN
ejpam-3441	267	6	(	(	PUNCT
ejpam-3441	267	7	α	α	X
ejpam-3441	267	8	,	,	PUNCT
ejpam-3441	267	9	β	β	X
ejpam-3441	267	10	]	]	PUNCT
ejpam-3441	267	11	of	of	ADP
ejpam-3441	267	12	an	an	DET
ejpam-3441	267	13	la	la	ADJ
ejpam-3441	267	14	-	-	PUNCT
ejpam-3441	267	15	ring	ring	NOUN
ejpam-3441	267	16	r	r	NOUN
ejpam-3441	267	17	with	with	ADP
ejpam-3441	267	18	left	left	ADJ
ejpam-3441	267	19	identity	identity	NOUN
ejpam-3441	267	20	e	e	NOUN
ejpam-3441	267	21	,	,	PUNCT
ejpam-3441	267	22	then	then	ADV
ejpam-3441	267	23	a	a	DET
ejpam-3441	267	24	◦	◦	NOUN
ejpam-3441	267	25	βα	βα	NOUN
ejpam-3441	267	26	b	b	NOUN
ejpam-3441	267	27	is	be	AUX
ejpam-3441	267	28	also	also	ADV
ejpam-3441	267	29	an	an	DET
ejpam-3441	267	30	intuitionistic	intuitionistic	ADJ
ejpam-3441	267	31	fuzzy	fuzzy	ADJ
ejpam-3441	267	32	left	left	NOUN
ejpam-3441	267	33	(	(	PUNCT
ejpam-3441	267	34	resp	resp	NOUN
ejpam-3441	267	35	.	.	PUNCT
ejpam-3441	268	1	right	right	ADJ
ejpam-3441	268	2	)	)	PUNCT
ejpam-3441	268	3	ideal	ideal	NOUN
ejpam-3441	268	4	with	with	ADP
ejpam-3441	268	5	thresholds	threshold	NOUN
ejpam-3441	268	6	(	(	PUNCT
ejpam-3441	268	7	α	α	X
ejpam-3441	268	8	,	,	PUNCT
ejpam-3441	268	9	β	β	X
ejpam-3441	268	10	]	]	PUNCT
ejpam-3441	268	11	of	of	ADP
ejpam-3441	268	12	r.	r.	PROPN
ejpam-3441	268	13	proof	proof	NOUN
ejpam-3441	268	14	.	.	PUNCT
ejpam-3441	269	1	let	let	VERB
ejpam-3441	269	2	a	a	DET
ejpam-3441	269	3	=	=	SYM
ejpam-3441	269	4	(	(	PUNCT
ejpam-3441	269	5	µa	µa	PROPN
ejpam-3441	269	6	,	,	PUNCT
ejpam-3441	269	7	γa	γa	PROPN
ejpam-3441	269	8	)	)	PUNCT
ejpam-3441	269	9	and	and	CCONJ
ejpam-3441	269	10	b	b	X
ejpam-3441	269	11	=	=	SYM
ejpam-3441	269	12	(	(	PUNCT
ejpam-3441	269	13	µa	µa	PROPN
ejpam-3441	269	14	,	,	PUNCT
ejpam-3441	269	15	γa	γa	PROPN
ejpam-3441	269	16	)	)	PUNCT
ejpam-3441	269	17	be	be	VERB
ejpam-3441	269	18	two	two	NUM
ejpam-3441	269	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	269	20	fuzzy	fuzzy	ADJ
ejpam-3441	269	21	left	leave	VERB
ejpam-3441	269	22	ideals	ideal	NOUN
ejpam-3441	269	23	with	with	ADP
ejpam-3441	269	24	thresholds	threshold	NOUN
ejpam-3441	269	25	(	(	PUNCT
ejpam-3441	269	26	α	α	X
ejpam-3441	269	27	,	,	PUNCT
ejpam-3441	269	28	β	β	X
ejpam-3441	269	29	]	]	PUNCT
ejpam-3441	269	30	of	of	ADP
ejpam-3441	269	31	an	an	DET
ejpam-3441	269	32	la	la	ADJ
ejpam-3441	269	33	-	-	PUNCT
ejpam-3441	269	34	ring	ring	NOUN
ejpam-3441	269	35	r.	r.	NOUN
ejpam-3441	269	36	we	we	PRON
ejpam-3441	269	37	have	have	VERB
ejpam-3441	269	38	to	to	PART
ejpam-3441	269	39	show	show	VERB
ejpam-3441	269	40	that	that	SCONJ
ejpam-3441	269	41	a	a	DET
ejpam-3441	269	42	◦	◦	NOUN
ejpam-3441	269	43	βα	βα	NOUN
ejpam-3441	269	44	b	b	NOUN
ejpam-3441	269	45	is	be	AUX
ejpam-3441	269	46	also	also	ADV
ejpam-3441	269	47	an	an	DET
ejpam-3441	269	48	intuitionistic	intuitionistic	ADJ
ejpam-3441	269	49	fuzzy	fuzzy	ADJ
ejpam-3441	269	50	left	leave	VERB
ejpam-3441	269	51	ideal	ideal	NOUN
ejpam-3441	269	52	with	with	ADP
ejpam-3441	269	53	thresholds	threshold	NOUN
ejpam-3441	269	54	(	(	PUNCT
ejpam-3441	269	55	α	α	X
ejpam-3441	269	56	,	,	PUNCT
ejpam-3441	269	57	β	β	X
ejpam-3441	269	58	]	]	PUNCT
ejpam-3441	269	59	of	of	ADP
ejpam-3441	269	60	r.	r.	PROPN
ejpam-3441	269	61	since	since	ADV
ejpam-3441	269	62	,	,	PUNCT
ejpam-3441	269	63	µa	µa	ADP
ejpam-3441	269	64	◦	◦	NOUN
ejpam-3441	269	65	βα	βα	PUNCT
ejpam-3441	269	66	µb	µb	VERB
ejpam-3441	269	67	−βα	−βα	PROPN
ejpam-3441	269	68	µa	µa	ADP
ejpam-3441	269	69	◦	◦	NOUN
ejpam-3441	269	70	βα	βα	VERB
ejpam-3441	269	71	µb	µb	VERB
ejpam-3441	269	72	⊆	⊆	NUM
ejpam-3441	269	73	µa	µa	NOUN
ejpam-3441	269	74	◦	◦	NOUN
ejpam-3441	269	75	βα	βα	NOUN
ejpam-3441	269	76	µb	µb	VERB
ejpam-3441	269	77	and	and	CCONJ
ejpam-3441	269	78	γa	γa	PROPN
ejpam-3441	269	79	◦	◦	PROPN
ejpam-3441	269	80	βα	βα	NOUN
ejpam-3441	269	81	γb	γb	PROPN
ejpam-3441	269	82	−βα	−βα	PROPN
ejpam-3441	269	83	γa	γa	PROPN
ejpam-3441	269	84	◦	◦	PROPN
ejpam-3441	269	85	βα	βα	NOUN
ejpam-3441	269	86	γb	γb	PROPN
ejpam-3441	269	87	⊇	⊇	PROPN
ejpam-3441	269	88	γa	γa	PROPN
ejpam-3441	269	89	◦	◦	PROPN
ejpam-3441	269	90	βα	βα	NOUN
ejpam-3441	269	91	γb	γb	NOUN
ejpam-3441	269	92	by	by	ADP
ejpam-3441	269	93	the	the	DET
ejpam-3441	269	94	lemma	lemma	PROPN
ejpam-3441	269	95	4	4	NUM
ejpam-3441	269	96	.	.	PUNCT
ejpam-3441	270	1	now	now	ADV
ejpam-3441	270	2	r	r	NOUN
ejpam-3441	270	3	◦	◦	NOUN
ejpam-3441	270	4	βα	βα	X
ejpam-3441	270	5	(	(	PUNCT
ejpam-3441	270	6	µa	µa	ADP
ejpam-3441	270	7	◦	◦	NOUN
ejpam-3441	270	8	βα	βα	NOUN
ejpam-3441	270	9	µb	µb	NOUN
ejpam-3441	270	10	)	)	PUNCT
ejpam-3441	270	11	=	=	SYM
ejpam-3441	271	1	(	(	PUNCT
ejpam-3441	271	2	r	r	NOUN
ejpam-3441	271	3	◦	◦	NOUN
ejpam-3441	271	4	βα	βα	NOUN
ejpam-3441	271	5	r	r	NOUN
ejpam-3441	271	6	)	)	PUNCT
ejpam-3441	271	7	◦	◦	NOUN
ejpam-3441	271	8	βα	βα	X
ejpam-3441	271	9	(	(	PUNCT
ejpam-3441	271	10	µa	µa	ADP
ejpam-3441	271	11	◦	◦	NOUN
ejpam-3441	271	12	βα	βα	NOUN
ejpam-3441	271	13	µb	µb	NOUN
ejpam-3441	271	14	)	)	PUNCT
ejpam-3441	271	15	=	=	SYM
ejpam-3441	271	16	(	(	PUNCT
ejpam-3441	271	17	r	r	NOUN
ejpam-3441	271	18	◦	◦	NOUN
ejpam-3441	271	19	βα	βα	NOUN
ejpam-3441	271	20	µa	µa	NOUN
ejpam-3441	271	21	)	)	PUNCT
ejpam-3441	271	22	◦	◦	NOUN
ejpam-3441	271	23	βα	βα	X
ejpam-3441	271	24	(	(	PUNCT
ejpam-3441	271	25	r	r	NOUN
ejpam-3441	271	26	◦	◦	NOUN
ejpam-3441	271	27	βα	βα	NOUN
ejpam-3441	271	28	µb	µb	NOUN
ejpam-3441	271	29	)	)	PUNCT
ejpam-3441	271	30	⊆	⊆	NUM
ejpam-3441	271	31	µa	µa	NOUN
ejpam-3441	271	32	◦	◦	NOUN
ejpam-3441	271	33	βα	βα	NOUN
ejpam-3441	271	34	µb	µb	VERB
ejpam-3441	271	35	and	and	CCONJ
ejpam-3441	271	36	r	r	NOUN
ejpam-3441	271	37	◦	◦	NOUN
ejpam-3441	271	38	βα	βα	X
ejpam-3441	271	39	(	(	PUNCT
ejpam-3441	271	40	γa	γa	PROPN
ejpam-3441	271	41	◦	◦	PROPN
ejpam-3441	271	42	βα	βα	NOUN
ejpam-3441	271	43	γb	γb	PROPN
ejpam-3441	271	44	)	)	PUNCT
ejpam-3441	271	45	=	=	SYM
ejpam-3441	272	1	(	(	PUNCT
ejpam-3441	272	2	r	r	NOUN
ejpam-3441	272	3	◦	◦	NOUN
ejpam-3441	272	4	βα	βα	NOUN
ejpam-3441	272	5	r	r	NOUN
ejpam-3441	272	6	)	)	PUNCT
ejpam-3441	272	7	◦	◦	NOUN
ejpam-3441	272	8	βα	βα	X
ejpam-3441	272	9	(	(	PUNCT
ejpam-3441	272	10	γa	γa	PROPN
ejpam-3441	272	11	◦	◦	PROPN
ejpam-3441	272	12	βα	βα	NOUN
ejpam-3441	272	13	γb	γb	PROPN
ejpam-3441	272	14	)	)	PUNCT
ejpam-3441	272	15	=	=	SYM
ejpam-3441	273	1	(	(	PUNCT
ejpam-3441	273	2	r	r	NOUN
ejpam-3441	273	3	◦	◦	NOUN
ejpam-3441	273	4	βα	βα	NOUN
ejpam-3441	273	5	γa	γa	NOUN
ejpam-3441	273	6	)	)	PUNCT
ejpam-3441	273	7	◦	◦	NOUN
ejpam-3441	273	8	βα	βα	X
ejpam-3441	273	9	(	(	PUNCT
ejpam-3441	273	10	r	r	NOUN
ejpam-3441	273	11	◦	◦	NOUN
ejpam-3441	273	12	βα	βα	NOUN
ejpam-3441	273	13	γb	γb	PROPN
ejpam-3441	273	14	)	)	PUNCT
ejpam-3441	273	15	⊇	⊇	PROPN
ejpam-3441	273	16	γa	γa	PROPN
ejpam-3441	273	17	◦	◦	PROPN
ejpam-3441	273	18	βα	βα	NOUN
ejpam-3441	273	19	γb	γb	PROPN
ejpam-3441	273	20	.	.	PUNCT
ejpam-3441	274	1	hence	hence	ADV
ejpam-3441	274	2	a	a	DET
ejpam-3441	274	3	◦	◦	NOUN
ejpam-3441	274	4	βαb	βαb	NOUN
ejpam-3441	274	5	is	be	AUX
ejpam-3441	274	6	an	an	DET
ejpam-3441	274	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	274	8	fuzzy	fuzzy	ADJ
ejpam-3441	274	9	left	leave	VERB
ejpam-3441	274	10	ideal	ideal	NOUN
ejpam-3441	274	11	with	with	ADP
ejpam-3441	274	12	thresholds	threshold	NOUN
ejpam-3441	274	13	(	(	PUNCT
ejpam-3441	274	14	α	α	X
ejpam-3441	274	15	,	,	PUNCT
ejpam-3441	274	16	β	β	X
ejpam-3441	274	17	]	]	PUNCT
ejpam-3441	274	18	of	of	ADP
ejpam-3441	274	19	r.	r.	PROPN
ejpam-3441	274	20	similarly	similarly	ADV
ejpam-3441	274	21	,	,	PUNCT
ejpam-3441	274	22	we	we	PRON
ejpam-3441	274	23	can	can	AUX
ejpam-3441	274	24	prove	prove	VERB
ejpam-3441	274	25	for	for	ADP
ejpam-3441	274	26	right	right	ADJ
ejpam-3441	274	27	ideals	ideal	NOUN
ejpam-3441	274	28	.	.	PUNCT
ejpam-3441	275	1	remark	remark	NOUN
ejpam-3441	275	2	3	3	NUM
ejpam-3441	275	3	.	.	PUNCT
ejpam-3441	276	1	if	if	SCONJ
ejpam-3441	276	2	a	a	PRON
ejpam-3441	276	3	is	be	AUX
ejpam-3441	276	4	an	an	DET
ejpam-3441	276	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	276	6	fuzzy	fuzzy	ADJ
ejpam-3441	276	7	left	left	NOUN
ejpam-3441	276	8	(	(	PUNCT
ejpam-3441	276	9	resp	resp	NOUN
ejpam-3441	276	10	.	.	PUNCT
ejpam-3441	277	1	right	right	ADJ
ejpam-3441	277	2	)	)	PUNCT
ejpam-3441	277	3	ideal	ideal	NOUN
ejpam-3441	277	4	with	with	ADP
ejpam-3441	277	5	thresholds	threshold	NOUN
ejpam-3441	277	6	(	(	PUNCT
ejpam-3441	277	7	α	α	X
ejpam-3441	277	8	,	,	PUNCT
ejpam-3441	277	9	β	β	X
ejpam-3441	277	10	]	]	PUNCT
ejpam-3441	277	11	of	of	ADP
ejpam-3441	277	12	an	an	DET
ejpam-3441	277	13	la	la	ADJ
ejpam-3441	277	14	-	-	PUNCT
ejpam-3441	277	15	ring	ring	NOUN
ejpam-3441	277	16	r	r	NOUN
ejpam-3441	277	17	with	with	ADP
ejpam-3441	277	18	left	left	ADJ
ejpam-3441	277	19	identity	identity	NOUN
ejpam-3441	277	20	e	e	NOUN
ejpam-3441	277	21	,	,	PUNCT
ejpam-3441	277	22	then	then	ADV
ejpam-3441	277	23	a	a	DET
ejpam-3441	277	24	◦	◦	NOUN
ejpam-3441	277	25	βαa	βαa	NOUN
ejpam-3441	277	26	is	be	AUX
ejpam-3441	277	27	an	an	DET
ejpam-3441	277	28	intuitionistic	intuitionistic	ADJ
ejpam-3441	277	29	fuzzy	fuzzy	ADJ
ejpam-3441	277	30	ideal	ideal	NOUN
ejpam-3441	277	31	with	with	ADP
ejpam-3441	277	32	thresholds	threshold	NOUN
ejpam-3441	277	33	(	(	PUNCT
ejpam-3441	277	34	α	α	X
ejpam-3441	277	35	,	,	PUNCT
ejpam-3441	277	36	β	β	X
ejpam-3441	277	37	]	]	PUNCT
ejpam-3441	277	38	of	of	ADP
ejpam-3441	277	39	r.	r.	PROPN
ejpam-3441	277	40	lemma	lemma	PROPN
ejpam-3441	277	41	7	7	X
ejpam-3441	277	42	.	.	PUNCT
ejpam-3441	278	1	if	if	SCONJ
ejpam-3441	278	2	a	a	PRON
ejpam-3441	278	3	and	and	CCONJ
ejpam-3441	278	4	b	b	NOUN
ejpam-3441	278	5	are	be	AUX
ejpam-3441	278	6	two	two	NUM
ejpam-3441	278	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	278	8	fuzzy	fuzzy	ADJ
ejpam-3441	278	9	ideals	ideal	NOUN
ejpam-3441	278	10	with	with	ADP
ejpam-3441	278	11	thresholds	threshold	NOUN
ejpam-3441	278	12	(	(	PUNCT
ejpam-3441	278	13	α	α	X
ejpam-3441	278	14	,	,	PUNCT
ejpam-3441	278	15	β	β	X
ejpam-3441	278	16	]	]	PUNCT
ejpam-3441	278	17	of	of	ADP
ejpam-3441	278	18	an	an	DET
ejpam-3441	278	19	la	la	ADJ
ejpam-3441	278	20	-	-	PUNCT
ejpam-3441	278	21	ring	ring	NOUN
ejpam-3441	278	22	r	r	NOUN
ejpam-3441	278	23	,	,	PUNCT
ejpam-3441	278	24	then	then	ADV
ejpam-3441	278	25	a	a	DET
ejpam-3441	278	26	◦	◦	NOUN
ejpam-3441	278	27	βα	βα	NOUN
ejpam-3441	278	28	b	b	NOUN
ejpam-3441	278	29	⊆	⊆	NUM
ejpam-3441	278	30	a	a	DET
ejpam-3441	278	31	∧βα	∧βα	ADJ
ejpam-3441	278	32	b.	b.	NOUN
ejpam-3441	278	33	proof	proof	NOUN
ejpam-3441	278	34	.	.	PUNCT
ejpam-3441	279	1	let	let	VERB
ejpam-3441	279	2	a	a	DET
ejpam-3441	279	3	=	=	SYM
ejpam-3441	279	4	(	(	PUNCT
ejpam-3441	279	5	µa	µa	PROPN
ejpam-3441	279	6	,	,	PUNCT
ejpam-3441	279	7	γa	γa	PROPN
ejpam-3441	279	8	)	)	PUNCT
ejpam-3441	279	9	and	and	CCONJ
ejpam-3441	279	10	b	b	X
ejpam-3441	279	11	=	=	SYM
ejpam-3441	279	12	(	(	PUNCT
ejpam-3441	279	13	µb	µb	PROPN
ejpam-3441	279	14	,	,	PUNCT
ejpam-3441	279	15	γb	γb	PROPN
ejpam-3441	279	16	)	)	PUNCT
ejpam-3441	279	17	be	be	VERB
ejpam-3441	279	18	two	two	NUM
ejpam-3441	279	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	279	20	fuzzy	fuzzy	ADJ
ejpam-3441	279	21	ideals	ideal	NOUN
ejpam-3441	279	22	with	with	ADP
ejpam-3441	279	23	thresholds	threshold	NOUN
ejpam-3441	279	24	(	(	PUNCT
ejpam-3441	279	25	α	α	X
ejpam-3441	279	26	,	,	PUNCT
ejpam-3441	279	27	β	β	X
ejpam-3441	279	28	]	]	PUNCT
ejpam-3441	279	29	of	of	ADP
ejpam-3441	279	30	an	an	DET
ejpam-3441	279	31	la	la	ADJ
ejpam-3441	279	32	-	-	PUNCT
ejpam-3441	279	33	ring	ring	NOUN
ejpam-3441	279	34	r	r	NOUN
ejpam-3441	279	35	and	and	CCONJ
ejpam-3441	279	36	x	x	PROPN
ejpam-3441	279	37	∈	∈	PROPN
ejpam-3441	279	38	r.	r.	PROPN
ejpam-3441	280	1	if	if	SCONJ
ejpam-3441	280	2	(	(	PUNCT
ejpam-3441	280	3	a	a	DET
ejpam-3441	280	4	◦	◦	NOUN
ejpam-3441	280	5	βα	βα	X
ejpam-3441	280	6	b)(x	b)(x	NOUN
ejpam-3441	280	7	)	)	PUNCT
ejpam-3441	280	8	=	=	SYM
ejpam-3441	280	9	0	0	NUM
ejpam-3441	280	10	,	,	PUNCT
ejpam-3441	280	11	then	then	ADV
ejpam-3441	280	12	obvious	obvious	VERB
ejpam-3441	280	13	a	a	DET
ejpam-3441	280	14	◦	◦	NOUN
ejpam-3441	280	15	βα	βα	NOUN
ejpam-3441	280	16	b	b	NOUN
ejpam-3441	280	17	⊆	⊆	NUM
ejpam-3441	280	18	a	a	DET
ejpam-3441	280	19	∧βα	∧βα	PROPN
ejpam-3441	280	20	b	b	NOUN
ejpam-3441	280	21	,	,	PUNCT
ejpam-3441	280	22	otherwise	otherwise	ADV
ejpam-3441	280	23	we	we	PRON
ejpam-3441	280	24	have	have	VERB
ejpam-3441	280	25	(	(	PUNCT
ejpam-3441	280	26	µa	µa	ADP
ejpam-3441	280	27	◦	◦	NOUN
ejpam-3441	280	28	βα	βα	NOUN
ejpam-3441	280	29	µb)(x	µb)(x	NOUN
ejpam-3441	280	30	)	)	PUNCT
ejpam-3441	281	1	=	=	PRON
ejpam-3441	281	2	{	{	PUNCT
ejpam-3441	281	3	(	(	PUNCT
ejpam-3441	281	4	µa	µa	ADP
ejpam-3441	281	5	◦	◦	NOUN
ejpam-3441	281	6	µb)(x	µb)(x	NOUN
ejpam-3441	281	7	)	)	PUNCT
ejpam-3441	282	1	∧	∧	PROPN
ejpam-3441	282	2	β	β	NOUN
ejpam-3441	282	3	}	}	PUNCT
ejpam-3441	282	4	∨	∨	NUM
ejpam-3441	282	5	α	α	PROPN
ejpam-3441	282	6	k.	k.	PROPN
ejpam-3441	282	7	nasreen	nasreen	PROPN
ejpam-3441	282	8	et	et	PROPN
ejpam-3441	282	9	al	al	PROPN
ejpam-3441	282	10	.	.	PUNCT
ejpam-3441	282	11	/	/	SYM
ejpam-3441	282	12	eur	eur	PROPN
ejpam-3441	282	13	.	.	PUNCT
ejpam-3441	283	1	j.	j.	PROPN
ejpam-3441	283	2	pure	pure	PROPN
ejpam-3441	283	3	appl	appl	PROPN
ejpam-3441	283	4	.	.	PROPN
ejpam-3441	283	5	math	math	PROPN
ejpam-3441	283	6	,	,	PUNCT
ejpam-3441	283	7	12	12	NUM
ejpam-3441	283	8	(	(	PUNCT
ejpam-3441	283	9	3	3	NUM
ejpam-3441	283	10	)	)	PUNCT
ejpam-3441	283	11	(	(	PUNCT
ejpam-3441	283	12	2019	2019	NUM
ejpam-3441	283	13	)	)	PUNCT
ejpam-3441	283	14	,	,	PUNCT
ejpam-3441	283	15	906	906	NUM
ejpam-3441	283	16	-	-	SYM
ejpam-3441	283	17	943	943	NUM
ejpam-3441	283	18	918	918	NUM
ejpam-3441	283	19	=	=	SYM
ejpam-3441	283	20	{	{	PUNCT
ejpam-3441	283	21	(	(	PUNCT
ejpam-3441	283	22	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	283	23	i=1	i=1	PROPN
ejpam-3441	283	24	aibi	aibi	NOUN
ejpam-3441	283	25	{	{	PUNCT
ejpam-3441	283	26	∧ni=1	∧ni=1	X
ejpam-3441	283	27	{	{	PUNCT
ejpam-3441	283	28	µa	µa	X
ejpam-3441	283	29	(	(	PUNCT
ejpam-3441	283	30	ai	ai	NOUN
ejpam-3441	283	31	)	)	PUNCT
ejpam-3441	283	32	∧	∧	NOUN
ejpam-3441	283	33	µb	µb	PROPN
ejpam-3441	283	34	(	(	PUNCT
ejpam-3441	283	35	bi	bi	NOUN
ejpam-3441	283	36	)	)	PUNCT
ejpam-3441	283	37	}	}	PUNCT
ejpam-3441	283	38	}	}	PUNCT
ejpam-3441	283	39	)	)	PUNCT
ejpam-3441	284	1	∧	∧	PROPN
ejpam-3441	284	2	β	β	NOUN
ejpam-3441	284	3	}	}	PUNCT
ejpam-3441	284	4	∨	∨	NUM
ejpam-3441	284	5	α	α	NOUN
ejpam-3441	284	6	≤	≤	NOUN
ejpam-3441	284	7	{	{	PUNCT
ejpam-3441	284	8	(	(	PUNCT
ejpam-3441	284	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	284	10	i=1	i=1	PROPN
ejpam-3441	284	11	aibi	aibi	NOUN
ejpam-3441	284	12	{	{	PUNCT
ejpam-3441	284	13	∧ni=1	∧ni=1	X
ejpam-3441	284	14	{	{	PUNCT
ejpam-3441	284	15	µa	µa	PROPN
ejpam-3441	284	16	(	(	PUNCT
ejpam-3441	284	17	aibi	aibi	NOUN
ejpam-3441	284	18	)	)	PUNCT
ejpam-3441	284	19	∧	∧	PROPN
ejpam-3441	284	20	µb	µb	PROPN
ejpam-3441	284	21	(	(	PUNCT
ejpam-3441	284	22	aibi	aibi	NOUN
ejpam-3441	284	23	)	)	PUNCT
ejpam-3441	284	24	}	}	PUNCT
ejpam-3441	284	25	}	}	PUNCT
ejpam-3441	284	26	)	)	PUNCT
ejpam-3441	285	1	∧	∧	PROPN
ejpam-3441	285	2	β	β	NOUN
ejpam-3441	285	3	}	}	PUNCT
ejpam-3441	285	4	∨	∨	NUM
ejpam-3441	285	5	α	α	NOUN
ejpam-3441	285	6	=	=	X
ejpam-3441	285	7	{	{	PUNCT
ejpam-3441	285	8	(	(	PUNCT
ejpam-3441	285	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	285	10	i=1	i=1	PROPN
ejpam-3441	285	11	aibi	aibi	NOUN
ejpam-3441	285	12	{	{	PUNCT
ejpam-3441	285	13	∧ni=1	∧ni=1	X
ejpam-3441	285	14	{	{	PUNCT
ejpam-3441	285	15	(	(	PUNCT
ejpam-3441	285	16	µa	µa	ADP
ejpam-3441	285	17	∧	∧	PROPN
ejpam-3441	285	18	µb	µb	PROPN
ejpam-3441	285	19	)	)	PUNCT
ejpam-3441	285	20	(	(	PUNCT
ejpam-3441	285	21	aibi	aibi	NOUN
ejpam-3441	285	22	)	)	PUNCT
ejpam-3441	285	23	}	}	PUNCT
ejpam-3441	285	24	}	}	PUNCT
ejpam-3441	285	25	)	)	PUNCT
ejpam-3441	285	26	∧	∧	PROPN
ejpam-3441	285	27	β	β	NOUN
ejpam-3441	285	28	}	}	PUNCT
ejpam-3441	285	29	∨	∨	NUM
ejpam-3441	285	30	α	α	NOUN
ejpam-3441	285	31	=	=	SYM
ejpam-3441	285	32	{	{	PUNCT
ejpam-3441	285	33	(	(	PUNCT
ejpam-3441	285	34	µa	µa	ADP
ejpam-3441	285	35	∧	∧	PROPN
ejpam-3441	285	36	µb	µb	PROPN
ejpam-3441	285	37	)	)	PUNCT
ejpam-3441	285	38	(	(	PUNCT
ejpam-3441	285	39	x	x	X
ejpam-3441	285	40	)	)	PUNCT
ejpam-3441	285	41	∧	∧	PROPN
ejpam-3441	285	42	β	β	NOUN
ejpam-3441	285	43	}	}	PUNCT
ejpam-3441	285	44	∨	∨	NUM
ejpam-3441	285	45	α	α	NOUN
ejpam-3441	285	46	=	=	PUNCT
ejpam-3441	285	47	(	(	PUNCT
ejpam-3441	285	48	µa	µa	ADP
ejpam-3441	285	49	∧βα	∧βα	ADJ
ejpam-3441	285	50	µb)(x	µb)(x	NOUN
ejpam-3441	285	51	)	)	PUNCT
ejpam-3441	285	52	.	.	PUNCT
ejpam-3441	286	1	⇒	⇒	PROPN
ejpam-3441	286	2	µa	µa	ADP
ejpam-3441	286	3	◦	◦	NOUN
ejpam-3441	286	4	βα	βα	VERB
ejpam-3441	286	5	µb	µb	ADP
ejpam-3441	286	6	⊆	⊆	NUM
ejpam-3441	286	7	µa	µa	NOUN
ejpam-3441	286	8	∧βα	∧βα	ADJ
ejpam-3441	286	9	µb	µb	VERB
ejpam-3441	286	10	.	.	PUNCT
ejpam-3441	287	1	similarly	similarly	ADV
ejpam-3441	287	2	,	,	PUNCT
ejpam-3441	287	3	we	we	PRON
ejpam-3441	287	4	have	have	VERB
ejpam-3441	287	5	γa	γa	PROPN
ejpam-3441	287	6	◦	◦	NOUN
ejpam-3441	287	7	βα	βα	NOUN
ejpam-3441	287	8	γb	γb	PROPN
ejpam-3441	287	9	⊇	⊇	PROPN
ejpam-3441	287	10	γa	γa	PROPN
ejpam-3441	287	11	∨βα	∨βα	PROPN
ejpam-3441	287	12	γb	γb	PROPN
ejpam-3441	287	13	.	.	PUNCT
ejpam-3441	288	1	hence	hence	ADV
ejpam-3441	288	2	a	a	DET
ejpam-3441	288	3	◦	◦	NOUN
ejpam-3441	288	4	βα	βα	NOUN
ejpam-3441	288	5	b	b	NOUN
ejpam-3441	288	6	⊆	⊆	NUM
ejpam-3441	288	7	a	a	DET
ejpam-3441	288	8	∧βα	∧βα	PROPN
ejpam-3441	288	9	b.	b.	NOUN
ejpam-3441	288	10	remark	remark	NOUN
ejpam-3441	288	11	4	4	NUM
ejpam-3441	288	12	.	.	PUNCT
ejpam-3441	289	1	if	if	SCONJ
ejpam-3441	289	2	a	a	PRON
ejpam-3441	289	3	is	be	AUX
ejpam-3441	289	4	an	an	DET
ejpam-3441	289	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	289	6	fuzzy	fuzzy	ADJ
ejpam-3441	289	7	ideal	ideal	NOUN
ejpam-3441	289	8	with	with	ADP
ejpam-3441	289	9	thresholds	threshold	NOUN
ejpam-3441	289	10	(	(	PUNCT
ejpam-3441	289	11	α	α	X
ejpam-3441	289	12	,	,	PUNCT
ejpam-3441	289	13	β	β	X
ejpam-3441	289	14	]	]	PUNCT
ejpam-3441	289	15	of	of	ADP
ejpam-3441	289	16	an	an	DET
ejpam-3441	289	17	la	la	ADJ
ejpam-3441	289	18	-	-	PUNCT
ejpam-3441	289	19	ring	ring	NOUN
ejpam-3441	289	20	r	r	NOUN
ejpam-3441	289	21	,	,	PUNCT
ejpam-3441	289	22	then	then	ADV
ejpam-3441	289	23	a	a	DET
ejpam-3441	289	24	◦	◦	NOUN
ejpam-3441	289	25	βα	βα	VERB
ejpam-3441	289	26	a	a	DET
ejpam-3441	289	27	⊆	⊆	NUM
ejpam-3441	289	28	aβα	aβα	NOUN
ejpam-3441	289	29	.	.	PUNCT
ejpam-3441	290	1	lemma	lemma	PROPN
ejpam-3441	290	2	8	8	NUM
ejpam-3441	290	3	.	.	PUNCT
ejpam-3441	291	1	let	let	VERB
ejpam-3441	291	2	r	r	PRON
ejpam-3441	291	3	be	be	AUX
ejpam-3441	291	4	an	an	DET
ejpam-3441	291	5	la	la	NOUN
ejpam-3441	291	6	-	-	PUNCT
ejpam-3441	291	7	ring	ring	NOUN
ejpam-3441	291	8	.	.	PUNCT
ejpam-3441	292	1	then	then	ADV
ejpam-3441	292	2	a	a	DET
ejpam-3441	292	3	◦	◦	NOUN
ejpam-3441	292	4	βα	βα	NOUN
ejpam-3441	292	5	b	b	NOUN
ejpam-3441	292	6	⊆	⊆	NUM
ejpam-3441	292	7	a	a	DET
ejpam-3441	292	8	∧βα	∧βα	PROPN
ejpam-3441	292	9	b	b	NOUN
ejpam-3441	292	10	for	for	ADP
ejpam-3441	292	11	every	every	DET
ejpam-3441	292	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	292	13	fuzzy	fuzzy	ADJ
ejpam-3441	292	14	right	right	ADJ
ejpam-3441	292	15	ideal	ideal	NOUN
ejpam-3441	292	16	a	a	PRON
ejpam-3441	292	17	with	with	ADP
ejpam-3441	292	18	thresholds	threshold	NOUN
ejpam-3441	292	19	(	(	PUNCT
ejpam-3441	292	20	α	α	X
ejpam-3441	292	21	,	,	PUNCT
ejpam-3441	292	22	β	β	X
ejpam-3441	292	23	]	]	PUNCT
ejpam-3441	292	24	and	and	CCONJ
ejpam-3441	292	25	every	every	DET
ejpam-3441	292	26	intuitionistic	intuitionistic	ADJ
ejpam-3441	292	27	fuzzy	fuzzy	ADJ
ejpam-3441	292	28	left	leave	VERB
ejpam-3441	292	29	ideal	ideal	PROPN
ejpam-3441	292	30	b	b	PROPN
ejpam-3441	292	31	with	with	ADP
ejpam-3441	292	32	thresholds	threshold	NOUN
ejpam-3441	292	33	(	(	PUNCT
ejpam-3441	292	34	α	α	X
ejpam-3441	292	35	,	,	PUNCT
ejpam-3441	292	36	β	β	X
ejpam-3441	292	37	]	]	PUNCT
ejpam-3441	292	38	of	of	ADP
ejpam-3441	292	39	r.	r.	PROPN
ejpam-3441	292	40	proof	proof	PROPN
ejpam-3441	292	41	.	.	PUNCT
ejpam-3441	293	1	same	same	ADJ
ejpam-3441	293	2	as	as	ADP
ejpam-3441	293	3	lemma	lemma	PROPN
ejpam-3441	293	4	7	7	NUM
ejpam-3441	293	5	.	.	PUNCT
ejpam-3441	293	6	theorem	theorem	NOUN
ejpam-3441	293	7	4	4	NUM
ejpam-3441	293	8	.	.	PUNCT
ejpam-3441	294	1	let	let	VERB
ejpam-3441	294	2	a	a	PRON
ejpam-3441	294	3	be	be	AUX
ejpam-3441	294	4	a	a	DET
ejpam-3441	294	5	non	non	ADJ
ejpam-3441	294	6	-	-	ADJ
ejpam-3441	294	7	empty	empty	ADJ
ejpam-3441	294	8	subset	subset	NOUN
ejpam-3441	294	9	of	of	ADP
ejpam-3441	294	10	an	an	DET
ejpam-3441	294	11	la	la	ADJ
ejpam-3441	294	12	-	-	PUNCT
ejpam-3441	294	13	ring	ring	NOUN
ejpam-3441	294	14	r.	r.	PROPN
ejpam-3441	294	15	then	then	ADV
ejpam-3441	294	16	the	the	DET
ejpam-3441	294	17	following	follow	VERB
ejpam-3441	294	18	conditions	condition	NOUN
ejpam-3441	294	19	are	be	AUX
ejpam-3441	294	20	true	true	ADJ
ejpam-3441	294	21	.	.	PUNCT
ejpam-3441	295	1	(	(	PUNCT
ejpam-3441	295	2	1	1	X
ejpam-3441	295	3	)	)	PUNCT
ejpam-3441	295	4	a	a	PRON
ejpam-3441	295	5	is	be	AUX
ejpam-3441	295	6	an	an	DET
ejpam-3441	295	7	interior	interior	ADJ
ejpam-3441	295	8	ideal	ideal	NOUN
ejpam-3441	295	9	of	of	ADP
ejpam-3441	295	10	r	r	NOUN
ejpam-3441	295	11	if	if	SCONJ
ejpam-3441	296	1	and	and	CCONJ
ejpam-3441	296	2	only	only	ADV
ejpam-3441	296	3	if	if	SCONJ
ejpam-3441	296	4	χa	χa	PROPN
ejpam-3441	296	5	is	be	AUX
ejpam-3441	296	6	an	an	DET
ejpam-3441	296	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	296	8	fuzzy	fuzzy	ADJ
ejpam-3441	296	9	interior	interior	ADJ
ejpam-3441	296	10	ideal	ideal	NOUN
ejpam-3441	296	11	with	with	ADP
ejpam-3441	296	12	thresholds	threshold	NOUN
ejpam-3441	296	13	(	(	PUNCT
ejpam-3441	296	14	α	α	X
ejpam-3441	296	15	,	,	PUNCT
ejpam-3441	296	16	β	β	X
ejpam-3441	296	17	]	]	PUNCT
ejpam-3441	296	18	of	of	ADP
ejpam-3441	296	19	r.	r.	PROPN
ejpam-3441	296	20	(	(	PUNCT
ejpam-3441	296	21	2	2	NUM
ejpam-3441	296	22	)	)	PUNCT
ejpam-3441	296	23	a	a	PRON
ejpam-3441	296	24	is	be	AUX
ejpam-3441	296	25	a	a	DET
ejpam-3441	296	26	quasi	quasi	NOUN
ejpam-3441	296	27	-	-	NOUN
ejpam-3441	296	28	ideal	ideal	ADJ
ejpam-3441	296	29	of	of	ADP
ejpam-3441	296	30	r	r	NOUN
ejpam-3441	296	31	if	if	SCONJ
ejpam-3441	297	1	and	and	CCONJ
ejpam-3441	297	2	only	only	ADV
ejpam-3441	297	3	if	if	SCONJ
ejpam-3441	297	4	χa	χa	PROPN
ejpam-3441	297	5	is	be	AUX
ejpam-3441	297	6	an	an	DET
ejpam-3441	297	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	297	8	fuzzy	fuzzy	ADJ
ejpam-3441	297	9	quasi	quasi	NOUN
ejpam-3441	297	10	-	-	NOUN
ejpam-3441	297	11	ideal	ideal	ADJ
ejpam-3441	297	12	with	with	ADP
ejpam-3441	297	13	thresholds	threshold	NOUN
ejpam-3441	297	14	(	(	PUNCT
ejpam-3441	297	15	α	α	X
ejpam-3441	297	16	,	,	PUNCT
ejpam-3441	297	17	β	β	X
ejpam-3441	297	18	]	]	PUNCT
ejpam-3441	297	19	of	of	ADP
ejpam-3441	297	20	r.	r.	PROPN
ejpam-3441	297	21	(	(	PUNCT
ejpam-3441	297	22	3	3	X
ejpam-3441	297	23	)	)	PUNCT
ejpam-3441	297	24	a	a	PRON
ejpam-3441	297	25	is	be	AUX
ejpam-3441	297	26	a	a	DET
ejpam-3441	297	27	bi	bi	NOUN
ejpam-3441	297	28	-	-	NOUN
ejpam-3441	297	29	ideal	ideal	NOUN
ejpam-3441	297	30	of	of	ADP
ejpam-3441	297	31	r	r	NOUN
ejpam-3441	297	32	if	if	SCONJ
ejpam-3441	298	1	and	and	CCONJ
ejpam-3441	298	2	only	only	ADV
ejpam-3441	298	3	if	if	SCONJ
ejpam-3441	298	4	χa	χa	PROPN
ejpam-3441	298	5	is	be	AUX
ejpam-3441	298	6	an	an	DET
ejpam-3441	298	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	298	8	fuzzy	fuzzy	ADJ
ejpam-3441	298	9	bi	bi	NOUN
ejpam-3441	298	10	-	-	NOUN
ejpam-3441	298	11	ideal	ideal	ADJ
ejpam-3441	298	12	with	with	ADP
ejpam-3441	298	13	thresholds	threshold	NOUN
ejpam-3441	298	14	(	(	PUNCT
ejpam-3441	298	15	α	α	X
ejpam-3441	298	16	,	,	PUNCT
ejpam-3441	298	17	β	β	X
ejpam-3441	298	18	]	]	PUNCT
ejpam-3441	298	19	of	of	ADP
ejpam-3441	298	20	r.	r.	PROPN
ejpam-3441	298	21	(	(	PUNCT
ejpam-3441	298	22	4	4	NUM
ejpam-3441	298	23	)	)	PUNCT
ejpam-3441	298	24	a	a	PRON
ejpam-3441	298	25	is	be	AUX
ejpam-3441	298	26	a	a	DET
ejpam-3441	298	27	generalized	generalized	ADJ
ejpam-3441	298	28	bi	bi	NOUN
ejpam-3441	298	29	-	-	NOUN
ejpam-3441	298	30	ideal	ideal	NOUN
ejpam-3441	298	31	of	of	ADP
ejpam-3441	298	32	r	r	NOUN
ejpam-3441	298	33	if	if	SCONJ
ejpam-3441	299	1	and	and	CCONJ
ejpam-3441	299	2	only	only	ADV
ejpam-3441	299	3	if	if	SCONJ
ejpam-3441	299	4	χa	χa	PROPN
ejpam-3441	299	5	is	be	AUX
ejpam-3441	299	6	an	an	DET
ejpam-3441	299	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	299	8	fuzzy	fuzzy	ADJ
ejpam-3441	299	9	generalized	generalize	VERB
ejpam-3441	299	10	bi	bi	NOUN
ejpam-3441	299	11	-	-	NOUN
ejpam-3441	299	12	ideal	ideal	NOUN
ejpam-3441	299	13	with	with	ADP
ejpam-3441	299	14	thresholds	threshold	NOUN
ejpam-3441	299	15	(	(	PUNCT
ejpam-3441	299	16	α	α	X
ejpam-3441	299	17	,	,	PUNCT
ejpam-3441	299	18	β	β	X
ejpam-3441	299	19	]	]	PUNCT
ejpam-3441	299	20	of	of	ADP
ejpam-3441	299	21	r.	r.	PROPN
ejpam-3441	299	22	proof	proof	NOUN
ejpam-3441	299	23	.	.	PUNCT
ejpam-3441	300	1	let	let	VERB
ejpam-3441	300	2	a	a	PRON
ejpam-3441	300	3	be	be	AUX
ejpam-3441	300	4	an	an	DET
ejpam-3441	300	5	interior	interior	ADJ
ejpam-3441	300	6	ideal	ideal	NOUN
ejpam-3441	300	7	of	of	ADP
ejpam-3441	300	8	an	an	DET
ejpam-3441	300	9	la	la	ADJ
ejpam-3441	300	10	-	-	PUNCT
ejpam-3441	300	11	ring	ring	NOUN
ejpam-3441	300	12	r	r	NOUN
ejpam-3441	300	13	,	,	PUNCT
ejpam-3441	300	14	this	this	PRON
ejpam-3441	300	15	implies	imply	VERB
ejpam-3441	300	16	that	that	SCONJ
ejpam-3441	300	17	a	a	PRON
ejpam-3441	300	18	is	be	AUX
ejpam-3441	300	19	an	an	DET
ejpam-3441	300	20	additive	additive	ADJ
ejpam-3441	300	21	la	la	PROPN
ejpam-3441	300	22	-	-	PUNCT
ejpam-3441	300	23	subgroup	subgroup	NOUN
ejpam-3441	300	24	.	.	PUNCT
ejpam-3441	301	1	then	then	ADV
ejpam-3441	301	2	χa	χa	PROPN
ejpam-3441	301	3	is	be	AUX
ejpam-3441	301	4	an	an	DET
ejpam-3441	301	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	301	6	fuzzy	fuzzy	ADJ
ejpam-3441	301	7	additive	additive	ADJ
ejpam-3441	301	8	la	la	PROPN
ejpam-3441	301	9	-	-	NOUN
ejpam-3441	301	10	subgroup	subgroup	NOUN
ejpam-3441	301	11	with	with	ADP
ejpam-3441	301	12	thresholds	threshold	NOUN
ejpam-3441	301	13	(	(	PUNCT
ejpam-3441	301	14	α	α	X
ejpam-3441	301	15	,	,	PUNCT
ejpam-3441	301	16	β	β	X
ejpam-3441	301	17	]	]	PUNCT
ejpam-3441	301	18	of	of	ADP
ejpam-3441	301	19	r	r	NOUN
ejpam-3441	301	20	by	by	ADP
ejpam-3441	301	21	the	the	DET
ejpam-3441	301	22	remark	remark	NOUN
ejpam-3441	301	23	1	1	X
ejpam-3441	301	24	.	.	PUNCT
ejpam-3441	302	1	let	let	VERB
ejpam-3441	302	2	x	x	PRON
ejpam-3441	302	3	,	,	PUNCT
ejpam-3441	302	4	y	y	PROPN
ejpam-3441	302	5	,	,	PUNCT
ejpam-3441	302	6	a	a	DET
ejpam-3441	302	7	∈	∈	PROPN
ejpam-3441	302	8	r.	r.	NOUN
ejpam-3441	302	9	if	if	SCONJ
ejpam-3441	302	10	a	a	PRON
ejpam-3441	302	11	/∈	/∈	NOUN
ejpam-3441	302	12	a	a	NOUN
ejpam-3441	302	13	,	,	PUNCT
ejpam-3441	302	14	then	then	ADV
ejpam-3441	302	15	by	by	ADP
ejpam-3441	302	16	definition	definition	NOUN
ejpam-3441	302	17	of	of	ADP
ejpam-3441	302	18	intuitionistic	intuitionistic	ADJ
ejpam-3441	302	19	characteristic	characteristic	ADJ
ejpam-3441	302	20	function	function	NOUN
ejpam-3441	302	21	µχa(a	µχa(a	PROPN
ejpam-3441	302	22	)	)	PUNCT
ejpam-3441	302	23	=	=	SYM
ejpam-3441	302	24	0	0	NUM
ejpam-3441	302	25	and	and	CCONJ
ejpam-3441	302	26	γχa(a	γχa(a	NUM
ejpam-3441	302	27	)	)	PUNCT
ejpam-3441	303	1	=	=	SYM
ejpam-3441	303	2	1	1	X
ejpam-3441	303	3	.	.	PUNCT
ejpam-3441	303	4	thus	thus	ADV
ejpam-3441	303	5	µχa((xa)y	µχa((xa)y	NOUN
ejpam-3441	303	6	)	)	PUNCT
ejpam-3441	303	7	≥	≥	NOUN
ejpam-3441	303	8	µχa(a	µχa(a	PROPN
ejpam-3441	303	9	)	)	PUNCT
ejpam-3441	303	10	=	=	SYM
ejpam-3441	303	11	min{µχa(a	min{µχa(a	PROPN
ejpam-3441	303	12	)	)	PUNCT
ejpam-3441	303	13	,	,	PUNCT
ejpam-3441	303	14	β	β	X
ejpam-3441	303	15	}	}	PUNCT
ejpam-3441	303	16	⇒	⇒	VERB
ejpam-3441	303	17	µχa((xa)y	µχa((xa)y	NOUN
ejpam-3441	303	18	)	)	PUNCT
ejpam-3441	303	19	≥	≥	NOUN
ejpam-3441	303	20	min{µχa(a	min{µχa(a	PROPN
ejpam-3441	303	21	)	)	PUNCT
ejpam-3441	303	22	,	,	PUNCT
ejpam-3441	303	23	β	β	X
ejpam-3441	303	24	}	}	PUNCT
ejpam-3441	303	25	⇒	⇒	X
ejpam-3441	303	26	max{µχa((xa)y	max{µχa((xa)y	PROPN
ejpam-3441	303	27	)	)	PUNCT
ejpam-3441	303	28	,	,	PUNCT
ejpam-3441	303	29	α	α	X
ejpam-3441	303	30	}	}	PUNCT
ejpam-3441	303	31	≥	≥	NOUN
ejpam-3441	303	32	min{µχa(a	min{µχa(a	PROPN
ejpam-3441	303	33	)	)	PUNCT
ejpam-3441	303	34	,	,	PUNCT
ejpam-3441	303	35	β	β	X
ejpam-3441	303	36	}	}	PUNCT
ejpam-3441	303	37	.	.	PUNCT
ejpam-3441	304	1	similarly	similarly	ADV
ejpam-3441	304	2	,	,	PUNCT
ejpam-3441	304	3	we	we	PRON
ejpam-3441	304	4	have	have	VERB
ejpam-3441	304	5	min{γχa((xa)y	min{γχa((xa)y	NOUN
ejpam-3441	304	6	)	)	PUNCT
ejpam-3441	304	7	,	,	PUNCT
ejpam-3441	304	8	(	(	PUNCT
ejpam-3441	304	9	1	1	NUM
ejpam-3441	304	10	−	−	PROPN
ejpam-3441	304	11	α	α	NOUN
ejpam-3441	304	12	)	)	PUNCT
ejpam-3441	304	13	}	}	PUNCT
ejpam-3441	304	14	≤	≤	NUM
ejpam-3441	304	15	max{γχa(a	max{γχa(a	PROPN
ejpam-3441	304	16	)	)	PUNCT
ejpam-3441	304	17	,	,	PUNCT
ejpam-3441	304	18	(	(	PUNCT
ejpam-3441	304	19	1	1	NUM
ejpam-3441	304	20	−	−	NOUN
ejpam-3441	304	21	β	β	NOUN
ejpam-3441	304	22	)	)	PUNCT
ejpam-3441	304	23	}	}	PUNCT
ejpam-3441	304	24	.	.	PUNCT
ejpam-3441	305	1	in	in	ADP
ejpam-3441	305	2	same	same	ADJ
ejpam-3441	305	3	lines	line	NOUN
ejpam-3441	305	4	,	,	PUNCT
ejpam-3441	305	5	we	we	PRON
ejpam-3441	305	6	have	have	VERB
ejpam-3441	305	7	max{µχa((xa)y	max{µχa((xa)y	NOUN
ejpam-3441	305	8	)	)	PUNCT
ejpam-3441	305	9	,	,	PUNCT
ejpam-3441	305	10	α	α	X
ejpam-3441	305	11	}	}	PUNCT
ejpam-3441	305	12	≥	≥	NOUN
ejpam-3441	305	13	min{µχa(a	min{µχa(a	PROPN
ejpam-3441	305	14	)	)	PUNCT
ejpam-3441	305	15	,	,	PUNCT
ejpam-3441	305	16	β	β	X
ejpam-3441	305	17	}	}	PUNCT
ejpam-3441	305	18	and	and	CCONJ
ejpam-3441	305	19	min{γχa((xa)y	min{γχa((xa)y	NOUN
ejpam-3441	305	20	)	)	PUNCT
ejpam-3441	305	21	,	,	PUNCT
ejpam-3441	305	22	(	(	PUNCT
ejpam-3441	305	23	1−	1−	NUM
ejpam-3441	305	24	α	α	NOUN
ejpam-3441	305	25	)	)	PUNCT
ejpam-3441	305	26	}	}	PUNCT
ejpam-3441	305	27	≤	≤	NUM
ejpam-3441	305	28	max{γχa(a	max{γχa(a	PROPN
ejpam-3441	305	29	)	)	PUNCT
ejpam-3441	305	30	,	,	PUNCT
ejpam-3441	305	31	(	(	PUNCT
ejpam-3441	305	32	1−	1−	NUM
ejpam-3441	305	33	β	β	NOUN
ejpam-3441	305	34	)	)	PUNCT
ejpam-3441	305	35	}	}	PUNCT
ejpam-3441	305	36	,	,	PUNCT
ejpam-3441	305	37	k.	k.	PROPN
ejpam-3441	305	38	nasreen	nasreen	PROPN
ejpam-3441	305	39	et	et	PROPN
ejpam-3441	305	40	al	al	PROPN
ejpam-3441	305	41	.	.	PUNCT
ejpam-3441	305	42	/	/	SYM
ejpam-3441	305	43	eur	eur	PROPN
ejpam-3441	305	44	.	.	PUNCT
ejpam-3441	306	1	j.	j.	PROPN
ejpam-3441	306	2	pure	pure	PROPN
ejpam-3441	306	3	appl	appl	PROPN
ejpam-3441	306	4	.	.	PROPN
ejpam-3441	306	5	math	math	PROPN
ejpam-3441	306	6	,	,	PUNCT
ejpam-3441	306	7	12	12	NUM
ejpam-3441	306	8	(	(	PUNCT
ejpam-3441	306	9	3	3	NUM
ejpam-3441	306	10	)	)	PUNCT
ejpam-3441	306	11	(	(	PUNCT
ejpam-3441	306	12	2019	2019	NUM
ejpam-3441	306	13	)	)	PUNCT
ejpam-3441	306	14	,	,	PUNCT
ejpam-3441	306	15	906	906	NUM
ejpam-3441	306	16	-	-	SYM
ejpam-3441	306	17	943	943	NUM
ejpam-3441	306	18	919	919	NUM
ejpam-3441	306	19	when	when	SCONJ
ejpam-3441	306	20	a	a	DET
ejpam-3441	306	21	∈	∈	NOUN
ejpam-3441	306	22	a.	a.	NOUN
ejpam-3441	306	23	hence	hence	ADV
ejpam-3441	306	24	the	the	DET
ejpam-3441	306	25	intuitionistic	intuitionistic	ADJ
ejpam-3441	306	26	characteristic	characteristic	ADJ
ejpam-3441	306	27	function	function	NOUN
ejpam-3441	306	28	χa	χa	NOUN
ejpam-3441	306	29	of	of	ADP
ejpam-3441	306	30	a	a	PRON
ejpam-3441	306	31	is	be	AUX
ejpam-3441	306	32	an	an	DET
ejpam-3441	306	33	i	i	PRON
ejpam-3441	306	34	ntuitionistic	ntuitionistic	ADJ
ejpam-3441	306	35	fuzzy	fuzzy	ADJ
ejpam-3441	306	36	interior	interior	ADJ
ejpam-3441	306	37	ideal	ideal	NOUN
ejpam-3441	306	38	with	with	ADP
ejpam-3441	306	39	thresholds	threshold	NOUN
ejpam-3441	306	40	(	(	PUNCT
ejpam-3441	306	41	α	α	X
ejpam-3441	306	42	,	,	PUNCT
ejpam-3441	306	43	β	β	X
ejpam-3441	306	44	]	]	PUNCT
ejpam-3441	306	45	of	of	ADP
ejpam-3441	306	46	r.	r.	PROPN
ejpam-3441	306	47	conversely	conversely	ADV
ejpam-3441	306	48	,	,	PUNCT
ejpam-3441	306	49	suppose	suppose	VERB
ejpam-3441	306	50	that	that	SCONJ
ejpam-3441	306	51	the	the	DET
ejpam-3441	306	52	intuitionistic	intuitionistic	ADJ
ejpam-3441	306	53	characteristic	characteristic	ADJ
ejpam-3441	306	54	function	function	NOUN
ejpam-3441	306	55	χa	χa	NOUN
ejpam-3441	306	56	of	of	ADP
ejpam-3441	306	57	a	a	PRON
ejpam-3441	306	58	is	be	AUX
ejpam-3441	306	59	an	an	DET
ejpam-3441	306	60	intuitionistic	intuitionistic	ADJ
ejpam-3441	306	61	fuzzy	fuzzy	ADJ
ejpam-3441	306	62	interior	interior	ADJ
ejpam-3441	306	63	ideal	ideal	NOUN
ejpam-3441	306	64	with	with	ADP
ejpam-3441	306	65	thresholds	threshold	NOUN
ejpam-3441	306	66	(	(	PUNCT
ejpam-3441	306	67	α	α	X
ejpam-3441	306	68	,	,	PUNCT
ejpam-3441	306	69	β	β	X
ejpam-3441	306	70	]	]	PUNCT
ejpam-3441	306	71	of	of	ADP
ejpam-3441	306	72	r	r	NOUN
ejpam-3441	306	73	,	,	PUNCT
ejpam-3441	306	74	this	this	PRON
ejpam-3441	306	75	means	mean	VERB
ejpam-3441	306	76	that	that	SCONJ
ejpam-3441	306	77	χa	χa	PROPN
ejpam-3441	306	78	is	be	AUX
ejpam-3441	306	79	an	an	DET
ejpam-3441	306	80	intuitionistic	intuitionistic	ADJ
ejpam-3441	306	81	fuzzy	fuzzy	ADJ
ejpam-3441	306	82	additive	additive	ADJ
ejpam-3441	306	83	la	la	PROPN
ejpam-3441	306	84	-	-	NOUN
ejpam-3441	306	85	subgroup	subgroup	NOUN
ejpam-3441	306	86	with	with	ADP
ejpam-3441	306	87	thresholds	threshold	NOUN
ejpam-3441	306	88	(	(	PUNCT
ejpam-3441	306	89	α	α	X
ejpam-3441	306	90	,	,	PUNCT
ejpam-3441	306	91	β	β	X
ejpam-3441	306	92	]	]	PUNCT
ejpam-3441	306	93	of	of	ADP
ejpam-3441	306	94	r.	r.	PROPN
ejpam-3441	306	95	then	then	ADV
ejpam-3441	306	96	a	a	PRON
ejpam-3441	306	97	is	be	AUX
ejpam-3441	306	98	an	an	DET
ejpam-3441	306	99	additive	additive	ADJ
ejpam-3441	306	100	la	la	PROPN
ejpam-3441	306	101	-	-	NOUN
ejpam-3441	306	102	subgroup	subgroup	NOUN
ejpam-3441	306	103	of	of	ADP
ejpam-3441	306	104	r	r	NOUN
ejpam-3441	306	105	by	by	ADP
ejpam-3441	306	106	the	the	DET
ejpam-3441	306	107	remark	remark	NOUN
ejpam-3441	307	1	1	1	X
ejpam-3441	307	2	.	.	PUNCT
ejpam-3441	308	1	let	let	VERB
ejpam-3441	308	2	t	t	PROPN
ejpam-3441	308	3	∈	∈	PROPN
ejpam-3441	308	4	(	(	PUNCT
ejpam-3441	308	5	ra)r	ra)r	PROPN
ejpam-3441	308	6	,	,	PUNCT
ejpam-3441	308	7	so	so	SCONJ
ejpam-3441	308	8	t	t	PROPN
ejpam-3441	308	9	=	=	SYM
ejpam-3441	308	10	(	(	PUNCT
ejpam-3441	308	11	xa)y	xa)y	PROPN
ejpam-3441	308	12	,	,	PUNCT
ejpam-3441	308	13	where	where	SCONJ
ejpam-3441	308	14	a	a	DET
ejpam-3441	308	15	∈	∈	PROPN
ejpam-3441	308	16	a	a	PRON
ejpam-3441	308	17	and	and	CCONJ
ejpam-3441	308	18	x	x	NOUN
ejpam-3441	308	19	,	,	PUNCT
ejpam-3441	308	20	y	y	PROPN
ejpam-3441	308	21	∈	∈	PROPN
ejpam-3441	308	22	r.	r.	PROPN
ejpam-3441	308	23	then	then	ADV
ejpam-3441	308	24	by	by	ADP
ejpam-3441	308	25	definition	definition	NOUN
ejpam-3441	308	26	µχa(a	µχa(a	PROPN
ejpam-3441	308	27	)	)	PUNCT
ejpam-3441	308	28	=	=	SYM
ejpam-3441	308	29	1	1	NUM
ejpam-3441	308	30	and	and	CCONJ
ejpam-3441	308	31	γχa(a	γχa(a	PRON
ejpam-3441	308	32	)	)	PUNCT
ejpam-3441	309	1	=	=	SYM
ejpam-3441	309	2	0	0	X
ejpam-3441	309	3	.	.	PUNCT
ejpam-3441	309	4	since	since	SCONJ
ejpam-3441	309	5	max{µχa((xa)y	max{µχa((xa)y	PROPN
ejpam-3441	309	6	)	)	PUNCT
ejpam-3441	309	7	,	,	PUNCT
ejpam-3441	309	8	α	α	X
ejpam-3441	309	9	}	}	PUNCT
ejpam-3441	309	10	≥	≥	NOUN
ejpam-3441	309	11	min{µχa(a	min{µχa(a	PROPN
ejpam-3441	309	12	)	)	PUNCT
ejpam-3441	309	13	,	,	PUNCT
ejpam-3441	309	14	β	β	X
ejpam-3441	309	15	}	}	PUNCT
ejpam-3441	309	16	=	=	SYM
ejpam-3441	309	17	β	β	X
ejpam-3441	309	18	and	and	CCONJ
ejpam-3441	309	19	min{γχa((xa)y	min{γχa((xa)y	NOUN
ejpam-3441	309	20	)	)	PUNCT
ejpam-3441	309	21	,	,	PUNCT
ejpam-3441	309	22	(	(	PUNCT
ejpam-3441	309	23	1−	1−	NUM
ejpam-3441	309	24	α	α	NOUN
ejpam-3441	309	25	)	)	PUNCT
ejpam-3441	309	26	}	}	PUNCT
ejpam-3441	309	27	≤	≤	NUM
ejpam-3441	309	28	max{γχa(a	max{γχa(a	PROPN
ejpam-3441	309	29	)	)	PUNCT
ejpam-3441	309	30	,	,	PUNCT
ejpam-3441	309	31	(	(	PUNCT
ejpam-3441	309	32	1−	1−	NUM
ejpam-3441	309	33	β	β	NOUN
ejpam-3441	309	34	)	)	PUNCT
ejpam-3441	309	35	}	}	PUNCT
ejpam-3441	309	36	=	=	SYM
ejpam-3441	309	37	1−	1−	NUM
ejpam-3441	309	38	β	β	X
ejpam-3441	309	39	,	,	PUNCT
ejpam-3441	309	40	χa	χa	ADP
ejpam-3441	309	41	being	be	AUX
ejpam-3441	309	42	an	an	DET
ejpam-3441	309	43	intuitionistic	intuitionistic	ADJ
ejpam-3441	309	44	fuzzy	fuzzy	ADJ
ejpam-3441	309	45	interior	interior	ADJ
ejpam-3441	309	46	ideal	ideal	NOUN
ejpam-3441	309	47	with	with	ADP
ejpam-3441	309	48	thresholds	threshold	NOUN
ejpam-3441	309	49	(	(	PUNCT
ejpam-3441	309	50	α	α	X
ejpam-3441	309	51	,	,	PUNCT
ejpam-3441	309	52	β	β	X
ejpam-3441	309	53	]	]	PUNCT
ejpam-3441	309	54	of	of	ADP
ejpam-3441	309	55	r.	r.	PROPN
ejpam-3441	309	56	this	this	PRON
ejpam-3441	309	57	implies	imply	VERB
ejpam-3441	309	58	that	that	SCONJ
ejpam-3441	309	59	µχa((xa)y	µχa((xa)y	VERB
ejpam-3441	309	60	)	)	PUNCT
ejpam-3441	309	61	≥	≥	NUM
ejpam-3441	309	62	β	β	X
ejpam-3441	309	63	and	and	CCONJ
ejpam-3441	309	64	γχa((xa)y	γχa((xa)y	NOUN
ejpam-3441	309	65	)	)	PUNCT
ejpam-3441	309	66	≤	≤	NUM
ejpam-3441	309	67	1	1	NUM
ejpam-3441	309	68	−	−	NOUN
ejpam-3441	309	69	β	β	NOUN
ejpam-3441	309	70	,	,	PUNCT
ejpam-3441	309	71	thus	thus	ADV
ejpam-3441	309	72	µχa((xa)y	µχa((xa)y	ADJ
ejpam-3441	309	73	)	)	PUNCT
ejpam-3441	309	74	=	=	SYM
ejpam-3441	309	75	1	1	NUM
ejpam-3441	309	76	and	and	CCONJ
ejpam-3441	309	77	µχa((xa)y	µχa((xa)y	NOUN
ejpam-3441	309	78	)	)	PUNCT
ejpam-3441	309	79	=	=	SYM
ejpam-3441	309	80	0	0	NUM
ejpam-3441	309	81	,	,	PUNCT
ejpam-3441	309	82	i.e.	i.e.	X
ejpam-3441	309	83	,	,	PUNCT
ejpam-3441	309	84	(	(	PUNCT
ejpam-3441	309	85	xa)y	xa)y	PROPN
ejpam-3441	309	86	∈	∈	PROPN
ejpam-3441	309	87	a.	a.	NOUN
ejpam-3441	309	88	hence	hence	ADV
ejpam-3441	309	89	a	a	PRON
ejpam-3441	309	90	is	be	AUX
ejpam-3441	309	91	an	an	DET
ejpam-3441	309	92	interior	interior	ADJ
ejpam-3441	309	93	ideal	ideal	NOUN
ejpam-3441	309	94	of	of	ADP
ejpam-3441	309	95	r.	r.	PROPN
ejpam-3441	309	96	(	(	PUNCT
ejpam-3441	309	97	2	2	X
ejpam-3441	309	98	)	)	PUNCT
ejpam-3441	309	99	let	let	VERB
ejpam-3441	309	100	a	a	PRON
ejpam-3441	309	101	be	be	AUX
ejpam-3441	309	102	a	a	DET
ejpam-3441	309	103	quasi	quasi	NOUN
ejpam-3441	309	104	-	-	NOUN
ejpam-3441	309	105	ideal	ideal	NOUN
ejpam-3441	309	106	of	of	ADP
ejpam-3441	309	107	r	r	NOUN
ejpam-3441	309	108	,	,	PUNCT
ejpam-3441	309	109	this	this	PRON
ejpam-3441	309	110	implies	imply	VERB
ejpam-3441	309	111	that	that	SCONJ
ejpam-3441	309	112	a	a	PRON
ejpam-3441	309	113	is	be	AUX
ejpam-3441	309	114	an	an	DET
ejpam-3441	309	115	additive	additive	ADJ
ejpam-3441	309	116	la	la	PROPN
ejpam-3441	309	117	-	-	PUNCT
ejpam-3441	309	118	subgroup	subgroup	NOUN
ejpam-3441	309	119	.	.	PUNCT
ejpam-3441	310	1	then	then	ADV
ejpam-3441	310	2	χa	χa	PROPN
ejpam-3441	310	3	is	be	AUX
ejpam-3441	310	4	an	an	DET
ejpam-3441	310	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	310	6	fuzzy	fuzzy	ADJ
ejpam-3441	310	7	additive	additive	ADJ
ejpam-3441	310	8	la	la	PROPN
ejpam-3441	310	9	-	-	NOUN
ejpam-3441	310	10	subgroup	subgroup	NOUN
ejpam-3441	310	11	with	with	ADP
ejpam-3441	310	12	thresholds	threshold	NOUN
ejpam-3441	310	13	(	(	PUNCT
ejpam-3441	310	14	α	α	X
ejpam-3441	310	15	,	,	PUNCT
ejpam-3441	310	16	β	β	X
ejpam-3441	310	17	]	]	PUNCT
ejpam-3441	310	18	of	of	ADP
ejpam-3441	310	19	r	r	NOUN
ejpam-3441	310	20	by	by	ADP
ejpam-3441	310	21	the	the	DET
ejpam-3441	310	22	remark	remark	NOUN
ejpam-3441	310	23	1	1	X
ejpam-3441	310	24	.	.	PUNCT
ejpam-3441	311	1	let	let	VERB
ejpam-3441	311	2	x	x	PUNCT
ejpam-3441	311	3	∈	∈	NOUN
ejpam-3441	311	4	r	r	NOUN
ejpam-3441	311	5	and	and	CCONJ
ejpam-3441	311	6	x	x	NOUN
ejpam-3441	311	7	/∈	/∈	NOUN
ejpam-3441	312	1	a	a	INTJ
ejpam-3441	312	2	,	,	PUNCT
ejpam-3441	312	3	then	then	ADV
ejpam-3441	312	4	x	x	X
ejpam-3441	312	5	/∈	/∈	PUNCT
ejpam-3441	313	1	ra	ra	PROPN
ejpam-3441	313	2	or	or	CCONJ
ejpam-3441	313	3	x	x	PROPN
ejpam-3441	313	4	/∈	/∈	PROPN
ejpam-3441	313	5	ar	ar	PROPN
ejpam-3441	314	1	.	.	PROPN
ejpam-3441	315	1	if	if	SCONJ
ejpam-3441	315	2	x	x	PROPN
ejpam-3441	315	3	/∈	/∈	PUNCT
ejpam-3441	315	4	ra	ra	PROPN
ejpam-3441	315	5	,	,	PUNCT
ejpam-3441	315	6	then	then	ADV
ejpam-3441	315	7	definition	definition	NOUN
ejpam-3441	315	8	of	of	ADP
ejpam-3441	315	9	intuitionistic	intuitionistic	ADJ
ejpam-3441	315	10	characteristic	characteristic	ADJ
ejpam-3441	315	11	function	function	NOUN
ejpam-3441	315	12	(	(	PUNCT
ejpam-3441	315	13	r	r	NOUN
ejpam-3441	315	14	◦	◦	NOUN
ejpam-3441	315	15	µχa)(x	µχa)(x	NOUN
ejpam-3441	315	16	)	)	PUNCT
ejpam-3441	316	1	=	=	SYM
ejpam-3441	316	2	0	0	PUNCT
ejpam-3441	317	1	and	and	CCONJ
ejpam-3441	317	2	(	(	PUNCT
ejpam-3441	317	3	r	r	NOUN
ejpam-3441	317	4	◦	◦	NOUN
ejpam-3441	317	5	γχa)(x	γχa)(x	NOUN
ejpam-3441	317	6	)	)	PUNCT
ejpam-3441	317	7	=	=	SYM
ejpam-3441	318	1	1	1	X
ejpam-3441	318	2	.	.	PUNCT
ejpam-3441	318	3	thus	thus	ADV
ejpam-3441	318	4	max{µχa(x	max{µχa(x	PROPN
ejpam-3441	318	5	)	)	PUNCT
ejpam-3441	318	6	,	,	PUNCT
ejpam-3441	318	7	α	α	X
ejpam-3441	318	8	}	}	PUNCT
ejpam-3441	318	9	≥	≥	NOUN
ejpam-3441	318	10	0	0	NUM
ejpam-3441	318	11	=	=	SYM
ejpam-3441	318	12	min{(µχa	min{(µχa	NOUN
ejpam-3441	318	13	◦	◦	NOUN
ejpam-3441	318	14	r	r	NOUN
ejpam-3441	318	15	)	)	PUNCT
ejpam-3441	318	16	(	(	PUNCT
ejpam-3441	318	17	x	x	NOUN
ejpam-3441	318	18	)	)	PUNCT
ejpam-3441	318	19	,	,	PUNCT
ejpam-3441	318	20	(	(	PUNCT
ejpam-3441	318	21	r	r	NOUN
ejpam-3441	318	22	◦	◦	NOUN
ejpam-3441	318	23	µχa	µχa	NOUN
ejpam-3441	318	24	)	)	PUNCT
ejpam-3441	318	25	(	(	PUNCT
ejpam-3441	318	26	x	x	X
ejpam-3441	318	27	)	)	PUNCT
ejpam-3441	318	28	,	,	PUNCT
ejpam-3441	318	29	β	β	X
ejpam-3441	318	30	}	}	PUNCT
ejpam-3441	318	31	and	and	CCONJ
ejpam-3441	318	32	min{γχa(x	min{γχa(x	PROPN
ejpam-3441	318	33	)	)	PUNCT
ejpam-3441	318	34	,	,	PUNCT
ejpam-3441	318	35	(	(	PUNCT
ejpam-3441	318	36	1−	1−	NUM
ejpam-3441	318	37	α	α	NOUN
ejpam-3441	318	38	)	)	PUNCT
ejpam-3441	318	39	}	}	PUNCT
ejpam-3441	318	40	≤	≤	NUM
ejpam-3441	318	41	1	1	NUM
ejpam-3441	318	42	=	=	SYM
ejpam-3441	318	43	max{(γχa	max{(γχa	NOUN
ejpam-3441	318	44	◦	◦	NOUN
ejpam-3441	318	45	r	r	NOUN
ejpam-3441	318	46	)	)	PUNCT
ejpam-3441	318	47	(	(	PUNCT
ejpam-3441	318	48	x	x	NOUN
ejpam-3441	318	49	)	)	PUNCT
ejpam-3441	318	50	,	,	PUNCT
ejpam-3441	318	51	(	(	PUNCT
ejpam-3441	318	52	r	r	NOUN
ejpam-3441	318	53	◦	◦	NOUN
ejpam-3441	318	54	γχa	γχa	NOUN
ejpam-3441	318	55	)	)	PUNCT
ejpam-3441	318	56	(	(	PUNCT
ejpam-3441	318	57	x	x	NOUN
ejpam-3441	318	58	)	)	PUNCT
ejpam-3441	318	59	,	,	PUNCT
ejpam-3441	318	60	(	(	PUNCT
ejpam-3441	318	61	1−	1−	NUM
ejpam-3441	318	62	β	β	NOUN
ejpam-3441	318	63	)	)	PUNCT
ejpam-3441	318	64	}	}	PUNCT
ejpam-3441	318	65	.	.	PUNCT
ejpam-3441	319	1	if	if	SCONJ
ejpam-3441	319	2	x	x	SYM
ejpam-3441	319	3	∈	∈	PROPN
ejpam-3441	319	4	a	a	DET
ejpam-3441	319	5	,	,	PUNCT
ejpam-3441	319	6	then	then	ADV
ejpam-3441	319	7	max{µχa(x	max{µχa(x	PROPN
ejpam-3441	319	8	)	)	PUNCT
ejpam-3441	319	9	,	,	PUNCT
ejpam-3441	319	10	α	α	X
ejpam-3441	319	11	}	}	PUNCT
ejpam-3441	319	12	=	=	SYM
ejpam-3441	319	13	1	1	NUM
ejpam-3441	319	14	≥	≥	NOUN
ejpam-3441	319	15	min{(µχa	min{(µχa	NOUN
ejpam-3441	319	16	◦	◦	NOUN
ejpam-3441	319	17	r	r	NOUN
ejpam-3441	319	18	)	)	PUNCT
ejpam-3441	319	19	(	(	PUNCT
ejpam-3441	319	20	x	x	X
ejpam-3441	319	21	)	)	PUNCT
ejpam-3441	319	22	,	,	PUNCT
ejpam-3441	319	23	r	r	NOUN
ejpam-3441	319	24	◦	◦	NOUN
ejpam-3441	319	25	µχa(x	µχa(x	PROPN
ejpam-3441	319	26	)	)	PUNCT
ejpam-3441	319	27	,	,	PUNCT
ejpam-3441	319	28	β	β	NOUN
ejpam-3441	319	29	}	}	PUNCT
ejpam-3441	319	30	and	and	CCONJ
ejpam-3441	319	31	min{γχa(x	min{γχa(x	PROPN
ejpam-3441	319	32	)	)	PUNCT
ejpam-3441	319	33	,	,	PUNCT
ejpam-3441	319	34	(	(	PUNCT
ejpam-3441	319	35	1−	1−	NUM
ejpam-3441	319	36	α	α	NOUN
ejpam-3441	319	37	)	)	PUNCT
ejpam-3441	319	38	}	}	PUNCT
ejpam-3441	319	39	=	=	SYM
ejpam-3441	319	40	0	0	NUM
ejpam-3441	319	41	≤	≤	NOUN
ejpam-3441	319	42	max{(γχa	max{(γχa	NOUN
ejpam-3441	319	43	◦	◦	NOUN
ejpam-3441	319	44	r	r	NOUN
ejpam-3441	319	45	)	)	PUNCT
ejpam-3441	319	46	(	(	PUNCT
ejpam-3441	319	47	x	x	X
ejpam-3441	319	48	)	)	PUNCT
ejpam-3441	319	49	,	,	PUNCT
ejpam-3441	319	50	r	r	NOUN
ejpam-3441	319	51	◦	◦	NOUN
ejpam-3441	319	52	γχa(x	γχa(x	NOUN
ejpam-3441	319	53	)	)	PUNCT
ejpam-3441	319	54	,	,	PUNCT
ejpam-3441	319	55	(	(	PUNCT
ejpam-3441	319	56	1−	1−	NUM
ejpam-3441	319	57	β	β	NOUN
ejpam-3441	319	58	)	)	PUNCT
ejpam-3441	319	59	}	}	PUNCT
ejpam-3441	319	60	.	.	PUNCT
ejpam-3441	320	1	therefore	therefore	ADV
ejpam-3441	320	2	the	the	DET
ejpam-3441	320	3	intuitionistic	intuitionistic	ADJ
ejpam-3441	320	4	characteristic	characteristic	ADJ
ejpam-3441	320	5	function	function	NOUN
ejpam-3441	320	6	χa	χa	NOUN
ejpam-3441	320	7	of	of	ADP
ejpam-3441	320	8	a	a	PRON
ejpam-3441	320	9	is	be	AUX
ejpam-3441	320	10	an	an	DET
ejpam-3441	320	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	320	12	fuzzy	fuzzy	ADJ
ejpam-3441	320	13	quasi	quasi	NOUN
ejpam-3441	320	14	-	-	NOUN
ejpam-3441	320	15	ideal	ideal	ADJ
ejpam-3441	320	16	with	with	ADP
ejpam-3441	320	17	thresholds	threshold	NOUN
ejpam-3441	320	18	(	(	PUNCT
ejpam-3441	320	19	α	α	X
ejpam-3441	320	20	,	,	PUNCT
ejpam-3441	320	21	β	β	X
ejpam-3441	320	22	]	]	PUNCT
ejpam-3441	320	23	of	of	ADP
ejpam-3441	320	24	r.	r.	PROPN
ejpam-3441	320	25	conversely	conversely	ADV
ejpam-3441	320	26	,	,	PUNCT
ejpam-3441	320	27	assume	assume	VERB
ejpam-3441	320	28	that	that	SCONJ
ejpam-3441	320	29	the	the	DET
ejpam-3441	320	30	intuitionistic	intuitionistic	ADJ
ejpam-3441	320	31	characteristic	characteristic	ADJ
ejpam-3441	320	32	function	function	NOUN
ejpam-3441	320	33	χa	χa	NOUN
ejpam-3441	320	34	of	of	ADP
ejpam-3441	320	35	a	a	PRON
ejpam-3441	320	36	is	be	AUX
ejpam-3441	320	37	an	an	DET
ejpam-3441	320	38	intuitionistic	intuitionistic	ADJ
ejpam-3441	320	39	fuzzy	fuzzy	ADJ
ejpam-3441	320	40	quasi	quasi	NOUN
ejpam-3441	320	41	-	-	NOUN
ejpam-3441	320	42	ideal	ideal	ADJ
ejpam-3441	320	43	with	with	ADP
ejpam-3441	320	44	thresholds	threshold	NOUN
ejpam-3441	320	45	(	(	PUNCT
ejpam-3441	320	46	α	α	X
ejpam-3441	320	47	,	,	PUNCT
ejpam-3441	320	48	β	β	X
ejpam-3441	320	49	]	]	PUNCT
ejpam-3441	320	50	of	of	ADP
ejpam-3441	320	51	r	r	NOUN
ejpam-3441	320	52	,	,	PUNCT
ejpam-3441	320	53	this	this	PRON
ejpam-3441	320	54	means	mean	VERB
ejpam-3441	320	55	that	that	SCONJ
ejpam-3441	320	56	χa	χa	PROPN
ejpam-3441	320	57	is	be	AUX
ejpam-3441	320	58	an	an	DET
ejpam-3441	320	59	intuitionistic	intuitionistic	ADJ
ejpam-3441	320	60	fuzzy	fuzzy	ADJ
ejpam-3441	320	61	additive	additive	ADJ
ejpam-3441	320	62	la	la	PROPN
ejpam-3441	320	63	-	-	NOUN
ejpam-3441	320	64	subgroup	subgroup	NOUN
ejpam-3441	320	65	with	with	ADP
ejpam-3441	320	66	thresholds	threshold	NOUN
ejpam-3441	320	67	(	(	PUNCT
ejpam-3441	320	68	α	α	X
ejpam-3441	320	69	,	,	PUNCT
ejpam-3441	320	70	β	β	X
ejpam-3441	320	71	]	]	PUNCT
ejpam-3441	320	72	of	of	ADP
ejpam-3441	320	73	r.	r.	PROPN
ejpam-3441	320	74	then	then	ADV
ejpam-3441	320	75	a	a	PRON
ejpam-3441	320	76	is	be	AUX
ejpam-3441	320	77	an	an	DET
ejpam-3441	320	78	additive	additive	ADJ
ejpam-3441	320	79	la	la	PROPN
ejpam-3441	320	80	-	-	NOUN
ejpam-3441	320	81	subgroup	subgroup	NOUN
ejpam-3441	320	82	of	of	ADP
ejpam-3441	320	83	r	r	NOUN
ejpam-3441	320	84	by	by	ADP
ejpam-3441	320	85	the	the	DET
ejpam-3441	320	86	remark	remark	NOUN
ejpam-3441	321	1	1	1	X
ejpam-3441	321	2	.	.	PUNCT
ejpam-3441	322	1	let	let	VERB
ejpam-3441	322	2	x	x	PRON
ejpam-3441	322	3	be	be	AUX
ejpam-3441	322	4	an	an	DET
ejpam-3441	322	5	element	element	NOUN
ejpam-3441	322	6	of	of	ADP
ejpam-3441	322	7	ar	ar	PROPN
ejpam-3441	322	8	∩	∩	PROPN
ejpam-3441	322	9	ra	ra	PROPN
ejpam-3441	322	10	,	,	PUNCT
ejpam-3441	322	11	this	this	PRON
ejpam-3441	322	12	means	mean	VERB
ejpam-3441	322	13	that	that	SCONJ
ejpam-3441	322	14	x	x	PROPN
ejpam-3441	322	15	∈	∈	PROPN
ejpam-3441	322	16	ar	ar	PROPN
ejpam-3441	322	17	and	and	CCONJ
ejpam-3441	322	18	ra	ra	PROPN
ejpam-3441	322	19	.	.	PUNCT
ejpam-3441	323	1	since	since	SCONJ
ejpam-3441	323	2	max{µχa(x	max{µχa(x	PROPN
ejpam-3441	323	3	)	)	PUNCT
ejpam-3441	323	4	,	,	PUNCT
ejpam-3441	323	5	α	α	X
ejpam-3441	323	6	}	}	PUNCT
ejpam-3441	323	7	≥	≥	NOUN
ejpam-3441	323	8	min{(µχa	min{(µχa	NOUN
ejpam-3441	323	9	◦	◦	NOUN
ejpam-3441	323	10	r)(x	r)(x	PROPN
ejpam-3441	323	11	)	)	PUNCT
ejpam-3441	323	12	,	,	PUNCT
ejpam-3441	323	13	(	(	PUNCT
ejpam-3441	323	14	r	r	NOUN
ejpam-3441	323	15	◦	◦	NOUN
ejpam-3441	323	16	µχa)(x	µχa)(x	NOUN
ejpam-3441	323	17	)	)	PUNCT
ejpam-3441	324	1	,	,	PUNCT
ejpam-3441	324	2	β	β	X
ejpam-3441	324	3	}	}	PUNCT
ejpam-3441	324	4	=	=	SYM
ejpam-3441	324	5	min{(µχa	min{(µχa	NOUN
ejpam-3441	324	6	◦	◦	NOUN
ejpam-3441	324	7	µχr)(x	µχr)(x	NOUN
ejpam-3441	324	8	)	)	PUNCT
ejpam-3441	324	9	,	,	PUNCT
ejpam-3441	324	10	(	(	PUNCT
ejpam-3441	324	11	µχr	µχr	NOUN
ejpam-3441	324	12	◦	◦	NOUN
ejpam-3441	324	13	µχa)(x	µχa)(x	NOUN
ejpam-3441	324	14	)	)	PUNCT
ejpam-3441	324	15	,	,	PUNCT
ejpam-3441	324	16	β	β	X
ejpam-3441	324	17	}	}	PUNCT
ejpam-3441	324	18	=	=	SYM
ejpam-3441	324	19	min{µχar(x	min{µχar(x	PROPN
ejpam-3441	324	20	)	)	PUNCT
ejpam-3441	324	21	,	,	PUNCT
ejpam-3441	324	22	µχra(x	µχra(x	PROPN
ejpam-3441	324	23	)	)	PUNCT
ejpam-3441	324	24	,	,	PUNCT
ejpam-3441	324	25	β	β	X
ejpam-3441	324	26	}	}	PUNCT
ejpam-3441	324	27	=	=	SYM
ejpam-3441	324	28	β	β	X
ejpam-3441	324	29	.	.	PUNCT
ejpam-3441	324	30	⇒	⇒	PROPN
ejpam-3441	324	31	max{µχa(x	max{µχa(x	PROPN
ejpam-3441	324	32	)	)	PUNCT
ejpam-3441	324	33	,	,	PUNCT
ejpam-3441	324	34	α	α	X
ejpam-3441	324	35	}	}	PUNCT
ejpam-3441	324	36	≥	≥	NUM
ejpam-3441	324	37	β	β	X
ejpam-3441	324	38	.	.	PUNCT
ejpam-3441	325	1	similarly	similarly	ADV
ejpam-3441	325	2	,	,	PUNCT
ejpam-3441	325	3	we	we	PRON
ejpam-3441	325	4	have	have	VERB
ejpam-3441	325	5	min{γχa(x	min{γχa(x	PROPN
ejpam-3441	325	6	)	)	PUNCT
ejpam-3441	325	7	,	,	PUNCT
ejpam-3441	325	8	(	(	PUNCT
ejpam-3441	325	9	1	1	NUM
ejpam-3441	325	10	−	−	PROPN
ejpam-3441	325	11	α	α	X
ejpam-3441	325	12	)	)	PUNCT
ejpam-3441	325	13	}	}	PUNCT
ejpam-3441	325	14	≤	≤	NUM
ejpam-3441	325	15	1	1	NUM
ejpam-3441	325	16	−	−	NOUN
ejpam-3441	325	17	β	β	NOUN
ejpam-3441	325	18	,	,	PUNCT
ejpam-3441	325	19	thus	thus	ADV
ejpam-3441	325	20	µχa(x	µχa(x	VERB
ejpam-3441	325	21	)	)	PUNCT
ejpam-3441	325	22	=	=	SYM
ejpam-3441	325	23	1	1	NUM
ejpam-3441	325	24	and	and	CCONJ
ejpam-3441	325	25	γχa(x	γχa(x	PROPN
ejpam-3441	325	26	)	)	PUNCT
ejpam-3441	325	27	=	=	SYM
ejpam-3441	325	28	0	0	NUM
ejpam-3441	325	29	,	,	PUNCT
ejpam-3441	325	30	i.e.	i.e.	X
ejpam-3441	325	31	,	,	PUNCT
ejpam-3441	325	32	x	x	SYM
ejpam-3441	325	33	∈	∈	NOUN
ejpam-3441	325	34	a.	a.	NOUN
ejpam-3441	325	35	therefore	therefore	ADV
ejpam-3441	325	36	a	a	PRON
ejpam-3441	325	37	is	be	AUX
ejpam-3441	325	38	a	a	DET
ejpam-3441	325	39	quasi	quasi	NOUN
ejpam-3441	325	40	-	-	NOUN
ejpam-3441	325	41	ideal	ideal	NOUN
ejpam-3441	325	42	of	of	ADP
ejpam-3441	325	43	r.	r.	PROPN
ejpam-3441	325	44	(	(	PUNCT
ejpam-3441	325	45	3	3	X
ejpam-3441	325	46	)	)	PUNCT
ejpam-3441	325	47	let	let	VERB
ejpam-3441	325	48	a	a	PRON
ejpam-3441	325	49	be	be	AUX
ejpam-3441	325	50	a	a	DET
ejpam-3441	325	51	bi	bi	NOUN
ejpam-3441	325	52	-	-	NOUN
ejpam-3441	325	53	ideal	ideal	NOUN
ejpam-3441	325	54	of	of	ADP
ejpam-3441	325	55	r	r	NOUN
ejpam-3441	325	56	,	,	PUNCT
ejpam-3441	325	57	this	this	PRON
ejpam-3441	325	58	implies	imply	VERB
ejpam-3441	325	59	that	that	SCONJ
ejpam-3441	325	60	a	a	PRON
ejpam-3441	325	61	is	be	AUX
ejpam-3441	325	62	an	an	DET
ejpam-3441	325	63	la	la	NOUN
ejpam-3441	325	64	-	-	PUNCT
ejpam-3441	325	65	subring	subring	NOUN
ejpam-3441	325	66	of	of	ADP
ejpam-3441	325	67	r.	r.	PROPN
ejpam-3441	325	68	then	then	ADV
ejpam-3441	325	69	χa	χa	PROPN
ejpam-3441	325	70	is	be	AUX
ejpam-3441	325	71	an	an	DET
ejpam-3441	325	72	intuitionistic	intuitionistic	ADJ
ejpam-3441	325	73	fuzzy	fuzzy	ADJ
ejpam-3441	325	74	la	la	NOUN
ejpam-3441	325	75	-	-	PUNCT
ejpam-3441	325	76	subring	subre	VERB
ejpam-3441	325	77	with	with	ADP
ejpam-3441	325	78	thresholds	threshold	NOUN
ejpam-3441	325	79	(	(	PUNCT
ejpam-3441	325	80	α	α	X
ejpam-3441	325	81	,	,	PUNCT
ejpam-3441	325	82	β	β	X
ejpam-3441	325	83	]	]	PUNCT
ejpam-3441	325	84	of	of	ADP
ejpam-3441	325	85	r	r	NOUN
ejpam-3441	325	86	by	by	ADP
ejpam-3441	325	87	the	the	DET
ejpam-3441	325	88	remark	remark	NOUN
ejpam-3441	325	89	1	1	X
ejpam-3441	325	90	.	.	PUNCT
ejpam-3441	326	1	let	let	VERB
ejpam-3441	326	2	k.	k.	PROPN
ejpam-3441	326	3	nasreen	nasreen	VERB
ejpam-3441	326	4	et	et	PROPN
ejpam-3441	326	5	al	al	PROPN
ejpam-3441	326	6	.	.	PUNCT
ejpam-3441	326	7	/	/	SYM
ejpam-3441	326	8	eur	eur	PROPN
ejpam-3441	326	9	.	.	PUNCT
ejpam-3441	327	1	j.	j.	PROPN
ejpam-3441	327	2	pure	pure	PROPN
ejpam-3441	327	3	appl	appl	PROPN
ejpam-3441	327	4	.	.	PROPN
ejpam-3441	327	5	math	math	PROPN
ejpam-3441	327	6	,	,	PUNCT
ejpam-3441	327	7	12	12	NUM
ejpam-3441	327	8	(	(	PUNCT
ejpam-3441	327	9	3	3	NUM
ejpam-3441	327	10	)	)	PUNCT
ejpam-3441	327	11	(	(	PUNCT
ejpam-3441	327	12	2019	2019	NUM
ejpam-3441	327	13	)	)	PUNCT
ejpam-3441	327	14	,	,	PUNCT
ejpam-3441	327	15	906	906	NUM
ejpam-3441	327	16	-	-	SYM
ejpam-3441	327	17	943	943	NUM
ejpam-3441	327	18	920	920	NUM
ejpam-3441	327	19	x	x	NOUN
ejpam-3441	327	20	,	,	PUNCT
ejpam-3441	327	21	y	y	PROPN
ejpam-3441	327	22	,	,	PUNCT
ejpam-3441	327	23	a	a	DET
ejpam-3441	327	24	∈	∈	PROPN
ejpam-3441	327	25	r.	r.	NOUN
ejpam-3441	327	26	if	if	SCONJ
ejpam-3441	327	27	x	x	PROPN
ejpam-3441	327	28	,	,	PUNCT
ejpam-3441	327	29	y	y	PROPN
ejpam-3441	327	30	/∈	/∈	PUNCT
ejpam-3441	328	1	a	a	INTJ
ejpam-3441	328	2	,	,	PUNCT
ejpam-3441	328	3	then	then	ADV
ejpam-3441	328	4	by	by	ADP
ejpam-3441	328	5	definition	definition	NOUN
ejpam-3441	328	6	of	of	ADP
ejpam-3441	328	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	328	8	characteristic	characteristic	ADJ
ejpam-3441	328	9	function	function	NOUN
ejpam-3441	328	10	µχa(x	µχa(x	VERB
ejpam-3441	328	11	)	)	PUNCT
ejpam-3441	328	12	=	=	SYM
ejpam-3441	328	13	µχa(y	µχa(y	PROPN
ejpam-3441	328	14	)	)	PUNCT
ejpam-3441	328	15	=	=	SYM
ejpam-3441	328	16	0	0	NUM
ejpam-3441	328	17	and	and	CCONJ
ejpam-3441	328	18	γχa(x	γχa(x	PROPN
ejpam-3441	328	19	)	)	PUNCT
ejpam-3441	328	20	=	=	SYM
ejpam-3441	328	21	γχa(y	γχa(y	NOUN
ejpam-3441	328	22	)	)	PUNCT
ejpam-3441	328	23	=	=	SYM
ejpam-3441	329	1	1	1	X
ejpam-3441	329	2	.	.	PUNCT
ejpam-3441	329	3	thus	thus	ADV
ejpam-3441	329	4	µχa((xa)y	µχa((xa)y	NOUN
ejpam-3441	329	5	)	)	PUNCT
ejpam-3441	329	6	≥	≥	NOUN
ejpam-3441	329	7	µχa(x	µχa(x	PROPN
ejpam-3441	329	8	)	)	PUNCT
ejpam-3441	329	9	∧	∧	PROPN
ejpam-3441	329	10	µχa(y	µχa(y	PROPN
ejpam-3441	329	11	)	)	PUNCT
ejpam-3441	329	12	=	=	SYM
ejpam-3441	329	13	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	329	14	)	)	PUNCT
ejpam-3441	329	15	,	,	PUNCT
ejpam-3441	329	16	µχa(y	µχa(y	PROPN
ejpam-3441	329	17	)	)	PUNCT
ejpam-3441	329	18	,	,	PUNCT
ejpam-3441	329	19	β	β	X
ejpam-3441	329	20	}	}	PUNCT
ejpam-3441	329	21	⇒	⇒	VERB
ejpam-3441	329	22	µχa((xa)y	µχa((xa)y	NOUN
ejpam-3441	329	23	)	)	PUNCT
ejpam-3441	329	24	≥	≥	PROPN
ejpam-3441	329	25	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	329	26	)	)	PUNCT
ejpam-3441	329	27	,	,	PUNCT
ejpam-3441	329	28	µχa(y	µχa(y	PROPN
ejpam-3441	329	29	)	)	PUNCT
ejpam-3441	329	30	,	,	PUNCT
ejpam-3441	329	31	β	β	X
ejpam-3441	329	32	}	}	PUNCT
ejpam-3441	329	33	⇒	⇒	X
ejpam-3441	329	34	max{µχa((xa)y	max{µχa((xa)y	PROPN
ejpam-3441	329	35	)	)	PUNCT
ejpam-3441	329	36	,	,	PUNCT
ejpam-3441	329	37	α	α	X
ejpam-3441	329	38	}	}	PUNCT
ejpam-3441	329	39	≥	≥	NOUN
ejpam-3441	329	40	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	329	41	)	)	PUNCT
ejpam-3441	329	42	,	,	PUNCT
ejpam-3441	329	43	µχa(y	µχa(y	PROPN
ejpam-3441	329	44	)	)	PUNCT
ejpam-3441	329	45	,	,	PUNCT
ejpam-3441	329	46	β	β	X
ejpam-3441	329	47	}	}	PUNCT
ejpam-3441	329	48	.	.	PUNCT
ejpam-3441	330	1	similarly	similarly	ADV
ejpam-3441	330	2	,	,	PUNCT
ejpam-3441	330	3	we	we	PRON
ejpam-3441	330	4	have	have	VERB
ejpam-3441	330	5	min{γχa((xa)y	min{γχa((xa)y	NOUN
ejpam-3441	330	6	)	)	PUNCT
ejpam-3441	330	7	,	,	PUNCT
ejpam-3441	330	8	(	(	PUNCT
ejpam-3441	330	9	1−α	1−α	NUM
ejpam-3441	330	10	)	)	PUNCT
ejpam-3441	330	11	}	}	PUNCT
ejpam-3441	330	12	≤	≤	NUM
ejpam-3441	331	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	331	2	)	)	PUNCT
ejpam-3441	331	3	,	,	PUNCT
ejpam-3441	331	4	µχa(y	µχa(y	PROPN
ejpam-3441	331	5	)	)	PUNCT
ejpam-3441	331	6	,	,	PUNCT
ejpam-3441	331	7	(	(	PUNCT
ejpam-3441	331	8	1−	1−	NUM
ejpam-3441	331	9	β	β	NOUN
ejpam-3441	331	10	)	)	PUNCT
ejpam-3441	331	11	}	}	PUNCT
ejpam-3441	331	12	.	.	PUNCT
ejpam-3441	332	1	in	in	ADP
ejpam-3441	332	2	same	same	ADJ
ejpam-3441	332	3	lines	line	NOUN
ejpam-3441	332	4	we	we	PRON
ejpam-3441	332	5	have	have	VERB
ejpam-3441	332	6	max{µχa((xa)y	max{µχa((xa)y	NOUN
ejpam-3441	332	7	)	)	PUNCT
ejpam-3441	332	8	,	,	PUNCT
ejpam-3441	332	9	α	α	X
ejpam-3441	332	10	}	}	PUNCT
ejpam-3441	332	11	≥	≥	NOUN
ejpam-3441	332	12	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	332	13	)	)	PUNCT
ejpam-3441	332	14	,	,	PUNCT
ejpam-3441	332	15	µχa(y	µχa(y	PROPN
ejpam-3441	332	16	)	)	PUNCT
ejpam-3441	332	17	,	,	PUNCT
ejpam-3441	332	18	β	β	NOUN
ejpam-3441	332	19	}	}	PUNCT
ejpam-3441	332	20	and	and	CCONJ
ejpam-3441	332	21	min{γχa((xa)y	min{γχa((xa)y	NOUN
ejpam-3441	332	22	)	)	PUNCT
ejpam-3441	332	23	,	,	PUNCT
ejpam-3441	332	24	(	(	PUNCT
ejpam-3441	332	25	1−	1−	NUM
ejpam-3441	332	26	α	α	NOUN
ejpam-3441	332	27	)	)	PUNCT
ejpam-3441	332	28	}	}	PUNCT
ejpam-3441	332	29	≤	≤	NUM
ejpam-3441	333	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	333	2	)	)	PUNCT
ejpam-3441	333	3	,	,	PUNCT
ejpam-3441	333	4	γχa(y	γχa(y	NOUN
ejpam-3441	333	5	)	)	PUNCT
ejpam-3441	333	6	,	,	PUNCT
ejpam-3441	333	7	(	(	PUNCT
ejpam-3441	333	8	1−	1−	NUM
ejpam-3441	333	9	β	β	NOUN
ejpam-3441	333	10	)	)	PUNCT
ejpam-3441	333	11	}	}	PUNCT
ejpam-3441	333	12	,	,	PUNCT
ejpam-3441	333	13	when	when	SCONJ
ejpam-3441	333	14	x	x	X
ejpam-3441	333	15	,	,	PUNCT
ejpam-3441	333	16	y	y	PROPN
ejpam-3441	333	17	∈	∈	PROPN
ejpam-3441	333	18	a.	a.	NOUN
ejpam-3441	333	19	ence	ence	NOUN
ejpam-3441	333	20	the	the	DET
ejpam-3441	333	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	333	22	characteristic	characteristic	ADJ
ejpam-3441	333	23	function	function	NOUN
ejpam-3441	333	24	χa	χa	NOUN
ejpam-3441	333	25	of	of	ADP
ejpam-3441	333	26	a	a	PRON
ejpam-3441	333	27	is	be	AUX
ejpam-3441	333	28	an	an	DET
ejpam-3441	333	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	333	30	fuzzy	fuzzy	ADJ
ejpam-3441	333	31	bi	bi	NOUN
ejpam-3441	333	32	-	-	NOUN
ejpam-3441	333	33	ideal	ideal	ADJ
ejpam-3441	333	34	with	with	ADP
ejpam-3441	333	35	thresholds	threshold	NOUN
ejpam-3441	333	36	(	(	PUNCT
ejpam-3441	333	37	α	α	X
ejpam-3441	333	38	,	,	PUNCT
ejpam-3441	333	39	β	β	X
ejpam-3441	333	40	]	]	PUNCT
ejpam-3441	333	41	of	of	ADP
ejpam-3441	333	42	r.	r.	PROPN
ejpam-3441	333	43	conversely	conversely	ADV
ejpam-3441	333	44	,	,	PUNCT
ejpam-3441	333	45	suppose	suppose	VERB
ejpam-3441	333	46	that	that	SCONJ
ejpam-3441	333	47	the	the	DET
ejpam-3441	333	48	intuitionistic	intuitionistic	ADJ
ejpam-3441	333	49	characteristic	characteristic	ADJ
ejpam-3441	333	50	function	function	NOUN
ejpam-3441	333	51	χa	χa	NOUN
ejpam-3441	333	52	of	of	ADP
ejpam-3441	333	53	a	a	PRON
ejpam-3441	333	54	is	be	AUX
ejpam-3441	333	55	an	an	DET
ejpam-3441	333	56	intuitionistic	intuitionistic	ADJ
ejpam-3441	333	57	fuzzy	fuzzy	ADJ
ejpam-3441	333	58	bi	bi	NOUN
ejpam-3441	333	59	-	-	NOUN
ejpam-3441	333	60	ideal	ideal	ADJ
ejpam-3441	333	61	with	with	ADP
ejpam-3441	333	62	thresholds	threshold	NOUN
ejpam-3441	333	63	(	(	PUNCT
ejpam-3441	333	64	α	α	X
ejpam-3441	333	65	,	,	PUNCT
ejpam-3441	333	66	β	β	X
ejpam-3441	333	67	]	]	PUNCT
ejpam-3441	333	68	of	of	ADP
ejpam-3441	333	69	r	r	NOUN
ejpam-3441	333	70	,	,	PUNCT
ejpam-3441	333	71	this	this	PRON
ejpam-3441	333	72	means	mean	VERB
ejpam-3441	333	73	that	that	SCONJ
ejpam-3441	333	74	χa	χa	PROPN
ejpam-3441	333	75	is	be	AUX
ejpam-3441	333	76	an	an	DET
ejpam-3441	333	77	intuitionistic	intuitionistic	ADJ
ejpam-3441	333	78	fuzzy	fuzzy	ADJ
ejpam-3441	333	79	la	la	NOUN
ejpam-3441	333	80	-	-	PUNCT
ejpam-3441	333	81	subring	subre	VERB
ejpam-3441	333	82	with	with	ADP
ejpam-3441	333	83	thresholds	threshold	NOUN
ejpam-3441	333	84	(	(	PUNCT
ejpam-3441	333	85	α	α	X
ejpam-3441	333	86	,	,	PUNCT
ejpam-3441	333	87	β	β	X
ejpam-3441	333	88	]	]	PUNCT
ejpam-3441	333	89	of	of	ADP
ejpam-3441	333	90	r.	r.	PROPN
ejpam-3441	333	91	then	then	ADV
ejpam-3441	333	92	a	a	PRON
ejpam-3441	333	93	is	be	AUX
ejpam-3441	333	94	an	an	DET
ejpam-3441	333	95	la	la	NOUN
ejpam-3441	333	96	-	-	PUNCT
ejpam-3441	333	97	subring	subring	NOUN
ejpam-3441	333	98	of	of	ADP
ejpam-3441	333	99	r	r	NOUN
ejpam-3441	333	100	by	by	ADP
ejpam-3441	333	101	the	the	DET
ejpam-3441	333	102	remark	remark	NOUN
ejpam-3441	334	1	1	1	X
ejpam-3441	334	2	.	.	PUNCT
ejpam-3441	335	1	let	let	VERB
ejpam-3441	335	2	t	t	PROPN
ejpam-3441	335	3	∈	∈	PROPN
ejpam-3441	335	4	(	(	PUNCT
ejpam-3441	335	5	ar)a	ar)a	PROPN
ejpam-3441	335	6	,	,	PUNCT
ejpam-3441	335	7	so	so	ADV
ejpam-3441	335	8	t	t	PROPN
ejpam-3441	335	9	=	=	SYM
ejpam-3441	335	10	(	(	PUNCT
ejpam-3441	335	11	xa)y	xa)y	PROPN
ejpam-3441	335	12	,	,	PUNCT
ejpam-3441	335	13	where	where	SCONJ
ejpam-3441	335	14	x	x	X
ejpam-3441	335	15	,	,	PUNCT
ejpam-3441	335	16	y	y	PROPN
ejpam-3441	335	17	∈	∈	PROPN
ejpam-3441	335	18	a	a	PRON
ejpam-3441	335	19	and	and	CCONJ
ejpam-3441	335	20	a	a	DET
ejpam-3441	335	21	∈	∈	PROPN
ejpam-3441	335	22	r.	r.	NOUN
ejpam-3441	335	23	then	then	ADV
ejpam-3441	335	24	the	the	DET
ejpam-3441	335	25	definition	definition	NOUN
ejpam-3441	335	26	µχa(x	µχa(x	VERB
ejpam-3441	335	27	)	)	PUNCT
ejpam-3441	336	1	=	=	SYM
ejpam-3441	336	2	µχa(y	µχa(y	PROPN
ejpam-3441	336	3	)	)	PUNCT
ejpam-3441	336	4	=	=	NOUN
ejpam-3441	336	5	1	1	NUM
ejpam-3441	336	6	and	and	CCONJ
ejpam-3441	336	7	γχa(x	γχa(x	PROPN
ejpam-3441	336	8	)	)	PUNCT
ejpam-3441	336	9	=	=	SYM
ejpam-3441	336	10	γχa(y	γχa(y	NOUN
ejpam-3441	336	11	)	)	PUNCT
ejpam-3441	336	12	=	=	SYM
ejpam-3441	337	1	0	0	X
ejpam-3441	337	2	.	.	PUNCT
ejpam-3441	337	3	as	as	ADP
ejpam-3441	337	4	max{µχa((xa)y	max{µχa((xa)y	ADJ
ejpam-3441	337	5	)	)	PUNCT
ejpam-3441	337	6	,	,	PUNCT
ejpam-3441	337	7	α	α	X
ejpam-3441	337	8	}	}	PUNCT
ejpam-3441	337	9	≥	≥	NOUN
ejpam-3441	337	10	min{µχa(x	min{µχa(x	PROPN
ejpam-3441	337	11	)	)	PUNCT
ejpam-3441	337	12	,	,	PUNCT
ejpam-3441	337	13	µχa(y	µχa(y	PROPN
ejpam-3441	337	14	)	)	PUNCT
ejpam-3441	337	15	,	,	PUNCT
ejpam-3441	337	16	β	β	X
ejpam-3441	337	17	}	}	PUNCT
ejpam-3441	337	18	=	=	SYM
ejpam-3441	337	19	β	β	X
ejpam-3441	337	20	and	and	CCONJ
ejpam-3441	337	21	min{γχa((xa)y	min{γχa((xa)y	NOUN
ejpam-3441	337	22	)	)	PUNCT
ejpam-3441	337	23	,	,	PUNCT
ejpam-3441	337	24	(	(	PUNCT
ejpam-3441	337	25	1−	1−	NUM
ejpam-3441	337	26	α	α	NOUN
ejpam-3441	337	27	)	)	PUNCT
ejpam-3441	337	28	}	}	PUNCT
ejpam-3441	337	29	≤	≤	NUM
ejpam-3441	337	30	max{γχa(x	max{γχa(x	PROPN
ejpam-3441	337	31	)	)	PUNCT
ejpam-3441	337	32	,	,	PUNCT
ejpam-3441	337	33	γχa(y	γχa(y	NOUN
ejpam-3441	337	34	)	)	PUNCT
ejpam-3441	337	35	,	,	PUNCT
ejpam-3441	337	36	(	(	PUNCT
ejpam-3441	337	37	1−	1−	NUM
ejpam-3441	337	38	β	β	NOUN
ejpam-3441	337	39	)	)	PUNCT
ejpam-3441	337	40	}	}	PUNCT
ejpam-3441	337	41	=	=	SYM
ejpam-3441	337	42	1−	1−	NUM
ejpam-3441	337	43	β	β	X
ejpam-3441	337	44	,	,	PUNCT
ejpam-3441	337	45	χa	χa	ADP
ejpam-3441	337	46	being	be	AUX
ejpam-3441	337	47	an	an	DET
ejpam-3441	337	48	intuitionistic	intuitionistic	ADJ
ejpam-3441	337	49	fuzzy	fuzzy	ADJ
ejpam-3441	337	50	bi	bi	NOUN
ejpam-3441	337	51	-	-	NOUN
ejpam-3441	337	52	ideal	ideal	ADJ
ejpam-3441	337	53	with	with	ADP
ejpam-3441	337	54	thresholds	threshold	NOUN
ejpam-3441	337	55	(	(	PUNCT
ejpam-3441	337	56	α	α	X
ejpam-3441	337	57	,	,	PUNCT
ejpam-3441	337	58	β	β	X
ejpam-3441	337	59	]	]	PUNCT
ejpam-3441	337	60	of	of	ADP
ejpam-3441	337	61	r.	r.	PROPN
ejpam-3441	337	62	this	this	PRON
ejpam-3441	337	63	implies	imply	VERB
ejpam-3441	337	64	that	that	SCONJ
ejpam-3441	337	65	µχa((xa)y	µχa((xa)y	VERB
ejpam-3441	337	66	)	)	PUNCT
ejpam-3441	337	67	≥	≥	NUM
ejpam-3441	337	68	β	β	X
ejpam-3441	337	69	and	and	CCONJ
ejpam-3441	337	70	γχa((xa)y	γχa((xa)y	NOUN
ejpam-3441	337	71	)	)	PUNCT
ejpam-3441	337	72	≤	≤	NUM
ejpam-3441	337	73	1	1	NUM
ejpam-3441	337	74	−	−	NOUN
ejpam-3441	337	75	β	β	NOUN
ejpam-3441	337	76	,	,	PUNCT
ejpam-3441	337	77	thus	thus	ADV
ejpam-3441	337	78	µχa((xa)y	µχa((xa)y	ADJ
ejpam-3441	337	79	)	)	PUNCT
ejpam-3441	337	80	=	=	SYM
ejpam-3441	337	81	1	1	NUM
ejpam-3441	337	82	and	and	CCONJ
ejpam-3441	337	83	µχa((xa)y	µχa((xa)y	NOUN
ejpam-3441	337	84	)	)	PUNCT
ejpam-3441	337	85	=	=	SYM
ejpam-3441	337	86	0	0	NUM
ejpam-3441	337	87	,	,	PUNCT
ejpam-3441	337	88	i.e.	i.e.	X
ejpam-3441	337	89	,	,	PUNCT
ejpam-3441	337	90	(	(	PUNCT
ejpam-3441	337	91	xa)y	xa)y	PROPN
ejpam-3441	337	92	∈	∈	PROPN
ejpam-3441	337	93	a.	a.	NOUN
ejpam-3441	337	94	hence	hence	ADV
ejpam-3441	337	95	a	a	PRON
ejpam-3441	337	96	is	be	AUX
ejpam-3441	337	97	bi	bi	NOUN
ejpam-3441	337	98	-	-	NOUN
ejpam-3441	337	99	ideal	ideal	NOUN
ejpam-3441	337	100	of	of	ADP
ejpam-3441	337	101	r.	r.	PROPN
ejpam-3441	337	102	similarly	similarly	ADV
ejpam-3441	337	103	,	,	PUNCT
ejpam-3441	337	104	we	we	PRON
ejpam-3441	337	105	can	can	AUX
ejpam-3441	337	106	prove	prove	VERB
ejpam-3441	337	107	(	(	PUNCT
ejpam-3441	337	108	4	4	NUM
ejpam-3441	337	109	)	)	PUNCT
ejpam-3441	337	110	.	.	PUNCT
ejpam-3441	338	1	theorem	theorem	ADJ
ejpam-3441	338	2	5	5	NUM
ejpam-3441	338	3	.	.	PUNCT
ejpam-3441	339	1	let	let	VERB
ejpam-3441	339	2	a	a	PRON
ejpam-3441	339	3	=	=	SYM
ejpam-3441	339	4	(	(	PUNCT
ejpam-3441	339	5	µa	µa	PROPN
ejpam-3441	339	6	,	,	PUNCT
ejpam-3441	339	7	γa	γa	PROPN
ejpam-3441	339	8	)	)	PUNCT
ejpam-3441	339	9	be	be	VERB
ejpam-3441	339	10	an	an	DET
ejpam-3441	339	11	ifs	ifs	PROPN
ejpam-3441	339	12	of	of	ADP
ejpam-3441	339	13	an	an	DET
ejpam-3441	339	14	la	la	ADJ
ejpam-3441	339	15	-	-	PUNCT
ejpam-3441	339	16	ring	ring	NOUN
ejpam-3441	339	17	r.	r.	PROPN
ejpam-3441	339	18	then	then	ADV
ejpam-3441	339	19	a	a	PRON
ejpam-3441	339	20	is	be	AUX
ejpam-3441	339	21	an	an	DET
ejpam-3441	339	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	339	23	fuzzy	fuzzy	ADJ
ejpam-3441	339	24	interior	interior	ADJ
ejpam-3441	339	25	ideal	ideal	NOUN
ejpam-3441	339	26	with	with	ADP
ejpam-3441	339	27	thresholds	threshold	NOUN
ejpam-3441	339	28	(	(	PUNCT
ejpam-3441	339	29	α	α	X
ejpam-3441	339	30	,	,	PUNCT
ejpam-3441	339	31	β	β	X
ejpam-3441	339	32	]	]	PUNCT
ejpam-3441	339	33	of	of	ADP
ejpam-3441	339	34	r	r	NOUN
ejpam-3441	339	35	if	if	SCONJ
ejpam-3441	340	1	and	and	CCONJ
ejpam-3441	340	2	only	only	ADV
ejpam-3441	340	3	if	if	SCONJ
ejpam-3441	340	4	(	(	PUNCT
ejpam-3441	340	5	r	r	NOUN
ejpam-3441	340	6	◦	◦	NOUN
ejpam-3441	340	7	βα	βα	X
ejpam-3441	340	8	a	a	PRON
ejpam-3441	340	9	)	)	PUNCT
ejpam-3441	340	10	◦	◦	NOUN
ejpam-3441	340	11	βα	βα	NOUN
ejpam-3441	340	12	r	r	NOUN
ejpam-3441	340	13	⊆	⊆	NUM
ejpam-3441	340	14	aβα	aβα	NOUN
ejpam-3441	340	15	and	and	CCONJ
ejpam-3441	340	16	a−βα	a−βα	PROPN
ejpam-3441	340	17	a	a	DET
ejpam-3441	340	18	⊆	⊆	NUM
ejpam-3441	340	19	aβα	aβα	NOUN
ejpam-3441	340	20	.	.	PUNCT
ejpam-3441	340	21	proof	proof	NOUN
ejpam-3441	340	22	.	.	PUNCT
ejpam-3441	341	1	suppose	suppose	VERB
ejpam-3441	341	2	that	that	SCONJ
ejpam-3441	341	3	a	a	DET
ejpam-3441	341	4	=	=	SYM
ejpam-3441	341	5	(	(	PUNCT
ejpam-3441	341	6	µa	µa	PROPN
ejpam-3441	341	7	,	,	PUNCT
ejpam-3441	341	8	γa	γa	PROPN
ejpam-3441	341	9	)	)	PUNCT
ejpam-3441	341	10	is	be	AUX
ejpam-3441	341	11	an	an	DET
ejpam-3441	341	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	341	13	fuzzy	fuzzy	ADJ
ejpam-3441	341	14	interior	interior	ADJ
ejpam-3441	341	15	ideal	ideal	NOUN
ejpam-3441	341	16	with	with	ADP
ejpam-3441	341	17	thresholds	threshold	NOUN
ejpam-3441	341	18	(	(	PUNCT
ejpam-3441	341	19	α	α	X
ejpam-3441	341	20	,	,	PUNCT
ejpam-3441	341	21	β	β	X
ejpam-3441	341	22	]	]	PUNCT
ejpam-3441	341	23	of	of	ADP
ejpam-3441	341	24	an	an	DET
ejpam-3441	341	25	la	la	ADJ
ejpam-3441	341	26	-	-	PUNCT
ejpam-3441	341	27	ring	ring	NOUN
ejpam-3441	341	28	r	r	NOUN
ejpam-3441	341	29	and	and	CCONJ
ejpam-3441	341	30	x	x	PROPN
ejpam-3441	341	31	∈	∈	PROPN
ejpam-3441	341	32	r.	r.	NOUN
ejpam-3441	342	1	if	if	SCONJ
ejpam-3441	342	2	(	(	PUNCT
ejpam-3441	342	3	(	(	PUNCT
ejpam-3441	342	4	r	r	NOUN
ejpam-3441	342	5	◦	◦	NOUN
ejpam-3441	342	6	βαa)	βαa)	SYM
ejpam-3441	342	7	◦	◦	NOUN
ejpam-3441	342	8	βαr)(x	βαr)(x	NOUN
ejpam-3441	342	9	)	)	PUNCT
ejpam-3441	342	10	=	=	SYM
ejpam-3441	342	11	0	0	NUM
ejpam-3441	342	12	,	,	PUNCT
ejpam-3441	342	13	then	then	ADV
ejpam-3441	342	14	obvious	obvious	ADJ
ejpam-3441	342	15	(	(	PUNCT
ejpam-3441	342	16	r	r	NOUN
ejpam-3441	342	17	◦	◦	NOUN
ejpam-3441	342	18	βαa)	βαa)	SYM
ejpam-3441	342	19	◦	◦	NOUN
ejpam-3441	342	20	βαr	βαr	NOUN
ejpam-3441	342	21	⊆	⊆	NUM
ejpam-3441	342	22	aβα	aβα	NOUN
ejpam-3441	342	23	.	.	PUNCT
ejpam-3441	343	1	otherwise	otherwise	ADV
ejpam-3441	343	2	there	there	PRON
ejpam-3441	343	3	exist	exist	VERB
ejpam-3441	343	4	ai	ai	NOUN
ejpam-3441	343	5	,	,	PUNCT
ejpam-3441	343	6	bi	bi	PROPN
ejpam-3441	343	7	,	,	PUNCT
ejpam-3441	343	8	ci	ci	PROPN
ejpam-3441	343	9	,	,	PUNCT
ejpam-3441	343	10	di	di	NOUN
ejpam-3441	343	11	∈	∈	PROPN
ejpam-3441	343	12	r	r	NOUN
ejpam-3441	343	13	such	such	ADJ
ejpam-3441	343	14	that	that	SCONJ
ejpam-3441	343	15	x	x	NOUN
ejpam-3441	343	16	=	=	PUNCT
ejpam-3441	343	17	∑n	∑n	PROPN
ejpam-3441	343	18	i=1	i=1	PROPN
ejpam-3441	343	19	aibi	aibi	NOUN
ejpam-3441	343	20	and	and	CCONJ
ejpam-3441	343	21	ai	ai	VERB
ejpam-3441	343	22	=	=	ADJ
ejpam-3441	344	1	∑n	∑n	PROPN
ejpam-3441	344	2	i=1	i=1	PROPN
ejpam-3441	344	3	cidi	cidi	NOUN
ejpam-3441	344	4	.	.	PUNCT
ejpam-3441	345	1	since	since	SCONJ
ejpam-3441	345	2	a	a	PRON
ejpam-3441	345	3	is	be	AUX
ejpam-3441	345	4	an	an	DET
ejpam-3441	345	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	345	6	fuzzy	fuzzy	ADJ
ejpam-3441	345	7	interior	interior	ADJ
ejpam-3441	345	8	ideal	ideal	NOUN
ejpam-3441	345	9	with	with	ADP
ejpam-3441	345	10	thresholds	threshold	NOUN
ejpam-3441	345	11	(	(	PUNCT
ejpam-3441	345	12	α	α	X
ejpam-3441	345	13	,	,	PUNCT
ejpam-3441	345	14	β	β	X
ejpam-3441	345	15	]	]	PUNCT
ejpam-3441	345	16	of	of	ADP
ejpam-3441	345	17	r	r	NOUN
ejpam-3441	345	18	,	,	PUNCT
ejpam-3441	345	19	this	this	PRON
ejpam-3441	345	20	implies	imply	VERB
ejpam-3441	345	21	that	that	SCONJ
ejpam-3441	345	22	max{µa((cidi)bi	max{µa((cidi)bi	PROPN
ejpam-3441	345	23	)	)	PUNCT
ejpam-3441	345	24	,	,	PUNCT
ejpam-3441	345	25	α	α	X
ejpam-3441	345	26	}	}	PUNCT
ejpam-3441	345	27	≥	≥	NOUN
ejpam-3441	345	28	min{µa(di	min{µa(di	NUM
ejpam-3441	345	29	)	)	PUNCT
ejpam-3441	345	30	,	,	PUNCT
ejpam-3441	345	31	β	β	X
ejpam-3441	345	32	}	}	PUNCT
ejpam-3441	345	33	andmin{γa((cidi)bi	andmin{γa((cidi)bi	ADJ
ejpam-3441	345	34	)	)	PUNCT
ejpam-3441	345	35	,	,	PUNCT
ejpam-3441	345	36	(	(	PUNCT
ejpam-3441	345	37	1−α	1−α	NUM
ejpam-3441	345	38	)	)	PUNCT
ejpam-3441	345	39	}	}	PUNCT
ejpam-3441	345	40	≤	≤	NOUN
ejpam-3441	345	41	max{γa(di	max{γa(di	NOUN
ejpam-3441	345	42	)	)	PUNCT
ejpam-3441	345	43	,	,	PUNCT
ejpam-3441	345	44	(	(	PUNCT
ejpam-3441	345	45	1−	1−	NUM
ejpam-3441	345	46	β	β	NOUN
ejpam-3441	345	47	)	)	PUNCT
ejpam-3441	345	48	}	}	PUNCT
ejpam-3441	345	49	.	.	PUNCT
ejpam-3441	346	1	now	now	ADV
ejpam-3441	346	2	(	(	PUNCT
ejpam-3441	346	3	(	(	PUNCT
ejpam-3441	346	4	r	r	NOUN
ejpam-3441	346	5	◦	◦	NOUN
ejpam-3441	346	6	βα	βα	NOUN
ejpam-3441	346	7	µa	µa	NOUN
ejpam-3441	346	8	)	)	PUNCT
ejpam-3441	346	9	◦	◦	NOUN
ejpam-3441	346	10	βα	βα	NOUN
ejpam-3441	346	11	r)(x	r)(x	PROPN
ejpam-3441	346	12	)	)	PUNCT
ejpam-3441	346	13	=	=	PRON
ejpam-3441	346	14	{	{	PUNCT
ejpam-3441	346	15	(	(	PUNCT
ejpam-3441	346	16	(	(	PUNCT
ejpam-3441	346	17	r	r	NOUN
ejpam-3441	346	18	◦	◦	NOUN
ejpam-3441	346	19	µa	µa	NOUN
ejpam-3441	346	20	)	)	PUNCT
ejpam-3441	346	21	◦	◦	NOUN
ejpam-3441	346	22	r)(x	r)(x	NOUN
ejpam-3441	346	23	)	)	PUNCT
ejpam-3441	346	24	∧	∧	PROPN
ejpam-3441	346	25	β	β	NOUN
ejpam-3441	346	26	}	}	PUNCT
ejpam-3441	346	27	∨	∨	NUM
ejpam-3441	346	28	α	α	NOUN
ejpam-3441	346	29	=	=	X
ejpam-3441	346	30	{	{	PUNCT
ejpam-3441	346	31	(	(	PUNCT
ejpam-3441	346	32	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	346	33	i=1	i=1	PROPN
ejpam-3441	346	34	aibi	aibi	NOUN
ejpam-3441	346	35	{	{	PUNCT
ejpam-3441	346	36	∧ni=1	∧ni=1	X
ejpam-3441	346	37	{	{	PUNCT
ejpam-3441	346	38	(	(	PUNCT
ejpam-3441	346	39	r	r	NOUN
ejpam-3441	346	40	◦	◦	NOUN
ejpam-3441	346	41	µa	µa	NOUN
ejpam-3441	346	42	)	)	PUNCT
ejpam-3441	346	43	(	(	PUNCT
ejpam-3441	346	44	ai	ai	NOUN
ejpam-3441	346	45	)	)	PUNCT
ejpam-3441	346	46	∧r	∧r	PROPN
ejpam-3441	346	47	(	(	PUNCT
ejpam-3441	346	48	bi	bi	NOUN
ejpam-3441	346	49	)	)	PUNCT
ejpam-3441	346	50	}	}	PUNCT
ejpam-3441	346	51	}	}	PUNCT
ejpam-3441	346	52	)	)	PUNCT
ejpam-3441	346	53	∧	∧	PROPN
ejpam-3441	346	54	β	β	NOUN
ejpam-3441	346	55	}	}	PUNCT
ejpam-3441	346	56	∨	∨	NUM
ejpam-3441	346	57	α	α	NOUN
ejpam-3441	346	58	=	=	X
ejpam-3441	346	59	{	{	PUNCT
ejpam-3441	346	60	(	(	PUNCT
ejpam-3441	346	61	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	346	62	i=1	i=1	PROPN
ejpam-3441	346	63	aibi	aibi	NOUN
ejpam-3441	346	64	{	{	PUNCT
ejpam-3441	346	65	∧ni=1	∧ni=1	X
ejpam-3441	346	66	{	{	PUNCT
ejpam-3441	346	67	(	(	PUNCT
ejpam-3441	346	68	r	r	NOUN
ejpam-3441	346	69	◦	◦	NOUN
ejpam-3441	346	70	µa	µa	NOUN
ejpam-3441	346	71	)	)	PUNCT
ejpam-3441	346	72	(	(	PUNCT
ejpam-3441	346	73	ai	ai	NOUN
ejpam-3441	346	74	)	)	PUNCT
ejpam-3441	346	75	∧	∧	NOUN
ejpam-3441	346	76	1	1	NUM
ejpam-3441	346	77	}	}	PUNCT
ejpam-3441	346	78	}	}	PUNCT
ejpam-3441	346	79	)	)	PUNCT
ejpam-3441	346	80	∧	∧	PROPN
ejpam-3441	346	81	β	β	NOUN
ejpam-3441	346	82	}	}	PUNCT
ejpam-3441	346	83	∨	∨	NUM
ejpam-3441	346	84	α	α	PROPN
ejpam-3441	346	85	k.	k.	PROPN
ejpam-3441	346	86	nasreen	nasreen	PROPN
ejpam-3441	346	87	et	et	PROPN
ejpam-3441	346	88	al	al	PROPN
ejpam-3441	346	89	.	.	PUNCT
ejpam-3441	346	90	/	/	SYM
ejpam-3441	346	91	eur	eur	PROPN
ejpam-3441	346	92	.	.	PUNCT
ejpam-3441	347	1	j.	j.	PROPN
ejpam-3441	347	2	pure	pure	PROPN
ejpam-3441	347	3	appl	appl	PROPN
ejpam-3441	347	4	.	.	PROPN
ejpam-3441	347	5	math	math	PROPN
ejpam-3441	347	6	,	,	PUNCT
ejpam-3441	347	7	12	12	NUM
ejpam-3441	347	8	(	(	PUNCT
ejpam-3441	347	9	3	3	NUM
ejpam-3441	347	10	)	)	PUNCT
ejpam-3441	347	11	(	(	PUNCT
ejpam-3441	347	12	2019	2019	NUM
ejpam-3441	347	13	)	)	PUNCT
ejpam-3441	347	14	,	,	PUNCT
ejpam-3441	347	15	906	906	NUM
ejpam-3441	347	16	-	-	SYM
ejpam-3441	347	17	943	943	NUM
ejpam-3441	347	18	921	921	NUM
ejpam-3441	347	19	=	=	SYM
ejpam-3441	347	20	{	{	PUNCT
ejpam-3441	347	21	(	(	PUNCT
ejpam-3441	347	22	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	347	23	i=1	i=1	PROPN
ejpam-3441	347	24	aibi	aibi	NOUN
ejpam-3441	347	25	{	{	PUNCT
ejpam-3441	347	26	∧ni=1(r	∧ni=1(r	PROPN
ejpam-3441	347	27	◦	◦	NOUN
ejpam-3441	347	28	µa	µa	NOUN
ejpam-3441	347	29	)	)	PUNCT
ejpam-3441	347	30	(	(	PUNCT
ejpam-3441	347	31	ai	ai	NOUN
ejpam-3441	347	32	)	)	PUNCT
ejpam-3441	347	33	}	}	PUNCT
ejpam-3441	347	34	)	)	PUNCT
ejpam-3441	348	1	∧	∧	PROPN
ejpam-3441	348	2	β	β	NOUN
ejpam-3441	348	3	}	}	PUNCT
ejpam-3441	348	4	∨	∨	NUM
ejpam-3441	348	5	α	α	NOUN
ejpam-3441	348	6	=	=	X
ejpam-3441	348	7	{	{	PUNCT
ejpam-3441	348	8	(	(	PUNCT
ejpam-3441	348	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	348	10	i=1	i=1	PROPN
ejpam-3441	348	11	aibi	aibi	NOUN
ejpam-3441	348	12	{	{	PUNCT
ejpam-3441	348	13	∧ni=1	∧ni=1	X
ejpam-3441	348	14	(	(	PUNCT
ejpam-3441	348	15	∨ai=∑n	∨ai=∑n	PROPN
ejpam-3441	348	16	i=1	i=1	PROPN
ejpam-3441	348	17	cidi	cidi	NOUN
ejpam-3441	348	18	{	{	PUNCT
ejpam-3441	348	19	∧ni=1	∧ni=1	X
ejpam-3441	348	20	{	{	PUNCT
ejpam-3441	348	21	r	r	NOUN
ejpam-3441	348	22	(	(	PUNCT
ejpam-3441	348	23	ci	ci	NOUN
ejpam-3441	348	24	)	)	PUNCT
ejpam-3441	348	25	∧	∧	NOUN
ejpam-3441	348	26	µa	µa	PROPN
ejpam-3441	348	27	(	(	PUNCT
ejpam-3441	348	28	di	di	NOUN
ejpam-3441	348	29	)	)	PUNCT
ejpam-3441	348	30	}	}	PUNCT
ejpam-3441	348	31	}	}	PUNCT
ejpam-3441	348	32	)	)	PUNCT
ejpam-3441	348	33	}	}	PUNCT
ejpam-3441	348	34	)	)	PUNCT
ejpam-3441	349	1	∧	∧	PROPN
ejpam-3441	349	2	β	β	NOUN
ejpam-3441	349	3	}	}	PUNCT
ejpam-3441	349	4	∨	∨	NUM
ejpam-3441	349	5	α	α	NOUN
ejpam-3441	349	6	=	=	X
ejpam-3441	349	7	{	{	PUNCT
ejpam-3441	349	8	(	(	PUNCT
ejpam-3441	349	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	349	10	i=1	i=1	PROPN
ejpam-3441	349	11	aibi	aibi	NOUN
ejpam-3441	349	12	{	{	PUNCT
ejpam-3441	349	13	∧ni=1	∧ni=1	X
ejpam-3441	349	14	(	(	PUNCT
ejpam-3441	349	15	∨ai=∑n	∨ai=∑n	PROPN
ejpam-3441	349	16	i=1	i=1	PROPN
ejpam-3441	349	17	cidi	cidi	NOUN
ejpam-3441	349	18	{	{	PUNCT
ejpam-3441	349	19	∧ni=1	∧ni=1	X
ejpam-3441	349	20	{	{	PUNCT
ejpam-3441	349	21	1	1	NUM
ejpam-3441	349	22	∧	∧	PROPN
ejpam-3441	349	23	µa	µa	PROPN
ejpam-3441	349	24	(	(	PUNCT
ejpam-3441	349	25	di	di	NOUN
ejpam-3441	349	26	)	)	PUNCT
ejpam-3441	349	27	}	}	PUNCT
ejpam-3441	349	28	}	}	PUNCT
ejpam-3441	349	29	)	)	PUNCT
ejpam-3441	349	30	}	}	PUNCT
ejpam-3441	349	31	)	)	PUNCT
ejpam-3441	350	1	∧	∧	PROPN
ejpam-3441	350	2	β	β	NOUN
ejpam-3441	350	3	}	}	PUNCT
ejpam-3441	350	4	∨	∨	NUM
ejpam-3441	350	5	α	α	NOUN
ejpam-3441	350	6	=	=	X
ejpam-3441	350	7	{	{	PUNCT
ejpam-3441	350	8	(	(	PUNCT
ejpam-3441	350	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	350	10	i=1	i=1	PROPN
ejpam-3441	350	11	aibi	aibi	NOUN
ejpam-3441	350	12	{	{	PUNCT
ejpam-3441	350	13	∧ni=1	∧ni=1	X
ejpam-3441	350	14	(	(	PUNCT
ejpam-3441	350	15	∨ai=∑n	∨ai=∑n	PROPN
ejpam-3441	350	16	i=1	i=1	PRON
ejpam-3441	350	17	cidi	cidi	NOUN
ejpam-3441	350	18	{	{	PUNCT
ejpam-3441	350	19	∧ni=1µa	∧ni=1µa	NOUN
ejpam-3441	350	20	(	(	PUNCT
ejpam-3441	350	21	di	di	NOUN
ejpam-3441	350	22	)	)	PUNCT
ejpam-3441	350	23	}	}	PUNCT
ejpam-3441	350	24	)	)	PUNCT
ejpam-3441	350	25	}	}	PUNCT
ejpam-3441	350	26	)	)	PUNCT
ejpam-3441	351	1	∧	∧	PROPN
ejpam-3441	351	2	β	β	NOUN
ejpam-3441	351	3	}	}	PUNCT
ejpam-3441	351	4	∨	∨	NUM
ejpam-3441	351	5	α	α	NOUN
ejpam-3441	351	6	=	=	X
ejpam-3441	351	7	{	{	PUNCT
ejpam-3441	351	8	(	(	PUNCT
ejpam-3441	351	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	351	10	i=1(cidi)bi	i=1(cidi)bi	NOUN
ejpam-3441	351	11	{	{	PUNCT
ejpam-3441	351	12	∧ni=1µa	∧ni=1µa	NOUN
ejpam-3441	351	13	(	(	PUNCT
ejpam-3441	351	14	di	di	NOUN
ejpam-3441	351	15	)	)	PUNCT
ejpam-3441	351	16	}	}	PUNCT
ejpam-3441	351	17	)	)	PUNCT
ejpam-3441	352	1	∧	∧	PROPN
ejpam-3441	352	2	β	β	NOUN
ejpam-3441	352	3	}	}	PUNCT
ejpam-3441	352	4	∨	∨	NUM
ejpam-3441	352	5	α	α	NOUN
ejpam-3441	352	6	≤	≤	NOUN
ejpam-3441	352	7	{	{	PUNCT
ejpam-3441	352	8	(	(	PUNCT
ejpam-3441	352	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	352	10	i=1(cidi)bi	i=1(cidi)bi	NOUN
ejpam-3441	352	11	{	{	PUNCT
ejpam-3441	352	12	∧ni=1µa	∧ni=1µa	NOUN
ejpam-3441	352	13	(	(	PUNCT
ejpam-3441	352	14	(	(	PUNCT
ejpam-3441	352	15	cidi	cidi	NOUN
ejpam-3441	352	16	)	)	PUNCT
ejpam-3441	352	17	bi	bi	NOUN
ejpam-3441	352	18	)	)	PUNCT
ejpam-3441	352	19	}	}	PUNCT
ejpam-3441	352	20	)	)	PUNCT
ejpam-3441	353	1	∧	∧	PROPN
ejpam-3441	353	2	β	β	NOUN
ejpam-3441	353	3	}	}	PUNCT
ejpam-3441	353	4	∨	∨	NUM
ejpam-3441	353	5	α	α	X
ejpam-3441	353	6	=	=	SYM
ejpam-3441	353	7	{	{	PUNCT
ejpam-3441	353	8	µa(x	µa(x	NOUN
ejpam-3441	353	9	)	)	PUNCT
ejpam-3441	353	10	∧	∧	PROPN
ejpam-3441	353	11	β	β	NOUN
ejpam-3441	353	12	}	}	PUNCT
ejpam-3441	353	13	∨	∨	NUM
ejpam-3441	353	14	α	α	NOUN
ejpam-3441	353	15	=	=	SYM
ejpam-3441	353	16	(	(	PUNCT
ejpam-3441	353	17	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	353	18	)	)	PUNCT
ejpam-3441	353	19	.	.	PUNCT
ejpam-3441	354	1	⇒	⇒	NOUN
ejpam-3441	354	2	(	(	PUNCT
ejpam-3441	354	3	r	r	NOUN
ejpam-3441	354	4	◦	◦	NOUN
ejpam-3441	354	5	βα	βα	NOUN
ejpam-3441	354	6	µa	µa	NOUN
ejpam-3441	354	7	)	)	PUNCT
ejpam-3441	354	8	◦	◦	NOUN
ejpam-3441	354	9	βα	βα	NOUN
ejpam-3441	354	10	r	r	NOUN
ejpam-3441	354	11	⊆	⊆	NUM
ejpam-3441	354	12	(	(	PUNCT
ejpam-3441	354	13	µa)βα	µa)βα	NUM
ejpam-3441	354	14	.	.	PUNCT
ejpam-3441	355	1	similarly	similarly	ADV
ejpam-3441	355	2	,	,	PUNCT
ejpam-3441	355	3	we	we	PRON
ejpam-3441	355	4	have	have	AUX
ejpam-3441	355	5	(	(	PUNCT
ejpam-3441	355	6	r	r	NOUN
ejpam-3441	355	7	◦	◦	NOUN
ejpam-3441	355	8	βα	βα	NOUN
ejpam-3441	355	9	γa	γa	NOUN
ejpam-3441	355	10	)	)	PUNCT
ejpam-3441	355	11	◦	◦	NOUN
ejpam-3441	355	12	βα	βα	NOUN
ejpam-3441	355	13	r	r	NOUN
ejpam-3441	355	14	⊇	⊇	NOUN
ejpam-3441	355	15	(	(	PUNCT
ejpam-3441	355	16	γa)βα	γa)βα	X
ejpam-3441	355	17	.	.	PUNCT
ejpam-3441	356	1	hence	hence	ADV
ejpam-3441	356	2	(	(	PUNCT
ejpam-3441	356	3	r	r	NOUN
ejpam-3441	356	4	◦	◦	NOUN
ejpam-3441	356	5	βα	βα	X
ejpam-3441	356	6	a	a	PRON
ejpam-3441	356	7	)	)	PUNCT
ejpam-3441	356	8	◦	◦	NOUN
ejpam-3441	356	9	βα	βα	NOUN
ejpam-3441	356	10	r	r	NOUN
ejpam-3441	356	11	⊆	⊆	NUM
ejpam-3441	356	12	aβα	aβα	NOUN
ejpam-3441	356	13	.	.	PUNCT
ejpam-3441	357	1	conversely	conversely	ADV
ejpam-3441	357	2	,	,	PUNCT
ejpam-3441	357	3	assume	assume	VERB
ejpam-3441	357	4	that	that	SCONJ
ejpam-3441	357	5	(	(	PUNCT
ejpam-3441	357	6	r	r	NOUN
ejpam-3441	357	7	◦	◦	NOUN
ejpam-3441	357	8	βαa	βαa	NOUN
ejpam-3441	357	9	)	)	PUNCT
ejpam-3441	357	10	◦	◦	NOUN
ejpam-3441	357	11	βαr	βαr	NOUN
ejpam-3441	357	12	⊆	⊆	NUM
ejpam-3441	357	13	aβα	aβα	NOUN
ejpam-3441	357	14	and	and	CCONJ
ejpam-3441	357	15	x	x	NOUN
ejpam-3441	357	16	,	,	PUNCT
ejpam-3441	357	17	y	y	PROPN
ejpam-3441	357	18	,	,	PUNCT
ejpam-3441	357	19	z	z	NOUN
ejpam-3441	357	20	∈	∈	NOUN
ejpam-3441	357	21	r	r	NOUN
ejpam-3441	357	22	such	such	ADJ
ejpam-3441	357	23	that	that	SCONJ
ejpam-3441	357	24	a	a	PRON
ejpam-3441	357	25	=	=	X
ejpam-3441	357	26	(	(	PUNCT
ejpam-3441	357	27	xy)z	xy)z	PROPN
ejpam-3441	357	28	.	.	PUNCT
ejpam-3441	357	29	now	now	ADV
ejpam-3441	357	30	max{µa((xy)z	max{µa((xy)z	PROPN
ejpam-3441	357	31	)	)	PUNCT
ejpam-3441	357	32	,	,	PUNCT
ejpam-3441	357	33	α	α	X
ejpam-3441	357	34	}	}	PUNCT
ejpam-3441	357	35	=	=	SYM
ejpam-3441	357	36	max{min{µa((xy)z	max{min{µa((xy)z	PROPN
ejpam-3441	357	37	)	)	PUNCT
ejpam-3441	357	38	,	,	PUNCT
ejpam-3441	357	39	β	β	X
ejpam-3441	357	40	}	}	PUNCT
ejpam-3441	357	41	,	,	PUNCT
ejpam-3441	357	42	α	α	NOUN
ejpam-3441	357	43	}	}	PUNCT
ejpam-3441	357	44	=	=	SYM
ejpam-3441	357	45	max{min{µa(a	max{min{µa(a	PROPN
ejpam-3441	357	46	)	)	PUNCT
ejpam-3441	357	47	,	,	PUNCT
ejpam-3441	357	48	β	β	X
ejpam-3441	357	49	}	}	PUNCT
ejpam-3441	357	50	,	,	PUNCT
ejpam-3441	357	51	α	α	NOUN
ejpam-3441	357	52	}	}	PUNCT
ejpam-3441	357	53	=	=	SYM
ejpam-3441	357	54	(	(	PUNCT
ejpam-3441	357	55	µa)βα(a	µa)βα(a	PROPN
ejpam-3441	357	56	)	)	PUNCT
ejpam-3441	357	57	≥	≥	NOUN
ejpam-3441	357	58	(	(	PUNCT
ejpam-3441	357	59	(	(	PUNCT
ejpam-3441	357	60	r	r	NOUN
ejpam-3441	357	61	◦	◦	NOUN
ejpam-3441	357	62	βα	βα	NOUN
ejpam-3441	357	63	µa	µa	NOUN
ejpam-3441	357	64	)	)	PUNCT
ejpam-3441	357	65	◦	◦	NOUN
ejpam-3441	357	66	βα	βα	NOUN
ejpam-3441	357	67	r)(a	r)(a	NUM
ejpam-3441	357	68	)	)	PUNCT
ejpam-3441	358	1	=	=	PRON
ejpam-3441	358	2	{	{	PUNCT
ejpam-3441	358	3	(	(	PUNCT
ejpam-3441	358	4	(	(	PUNCT
ejpam-3441	358	5	r	r	NOUN
ejpam-3441	358	6	◦	◦	NOUN
ejpam-3441	358	7	µa	µa	NOUN
ejpam-3441	358	8	)	)	PUNCT
ejpam-3441	358	9	◦	◦	NOUN
ejpam-3441	358	10	r)(a	r)(a	NUM
ejpam-3441	358	11	)	)	PUNCT
ejpam-3441	359	1	∧	∧	PROPN
ejpam-3441	359	2	β	β	NOUN
ejpam-3441	359	3	}	}	PUNCT
ejpam-3441	359	4	∨	∨	NUM
ejpam-3441	359	5	α	α	NOUN
ejpam-3441	359	6	=	=	X
ejpam-3441	359	7	{	{	PUNCT
ejpam-3441	359	8	(	(	PUNCT
ejpam-3441	359	9	∨a=∑n	∨a=∑n	PROPN
ejpam-3441	359	10	i=1	i=1	PROPN
ejpam-3441	359	11	aibi	aibi	NOUN
ejpam-3441	359	12	{	{	PUNCT
ejpam-3441	359	13	∧ni=1	∧ni=1	X
ejpam-3441	359	14	{	{	PUNCT
ejpam-3441	359	15	(	(	PUNCT
ejpam-3441	359	16	r	r	NOUN
ejpam-3441	359	17	◦	◦	NOUN
ejpam-3441	359	18	µa	µa	NOUN
ejpam-3441	359	19	)	)	PUNCT
ejpam-3441	359	20	(	(	PUNCT
ejpam-3441	359	21	ai	ai	NOUN
ejpam-3441	359	22	)	)	PUNCT
ejpam-3441	359	23	∧r	∧r	PROPN
ejpam-3441	359	24	(	(	PUNCT
ejpam-3441	359	25	bi	bi	NOUN
ejpam-3441	359	26	)	)	PUNCT
ejpam-3441	359	27	}	}	PUNCT
ejpam-3441	359	28	}	}	PUNCT
ejpam-3441	359	29	)	)	PUNCT
ejpam-3441	360	1	∧	∧	PROPN
ejpam-3441	360	2	β	β	NOUN
ejpam-3441	360	3	}	}	PUNCT
ejpam-3441	360	4	∨	∨	NUM
ejpam-3441	360	5	α	α	PROPN
ejpam-3441	360	6	≥	≥	X
ejpam-3441	360	7	{	{	PUNCT
ejpam-3441	360	8	(	(	PUNCT
ejpam-3441	360	9	(	(	PUNCT
ejpam-3441	360	10	r	r	NOUN
ejpam-3441	360	11	◦	◦	NOUN
ejpam-3441	360	12	µa	µa	NOUN
ejpam-3441	360	13	)	)	PUNCT
ejpam-3441	360	14	(	(	PUNCT
ejpam-3441	360	15	xy	xy	NOUN
ejpam-3441	360	16	)	)	PUNCT
ejpam-3441	360	17	∧r	∧r	PROPN
ejpam-3441	360	18	(	(	PUNCT
ejpam-3441	360	19	z	z	NOUN
ejpam-3441	360	20	)	)	PUNCT
ejpam-3441	360	21	)	)	PUNCT
ejpam-3441	361	1	∧	∧	PROPN
ejpam-3441	361	2	β	β	NOUN
ejpam-3441	361	3	}	}	PUNCT
ejpam-3441	361	4	∨	∨	NUM
ejpam-3441	361	5	α	α	NOUN
ejpam-3441	361	6	=	=	SYM
ejpam-3441	361	7	{	{	PUNCT
ejpam-3441	361	8	(	(	PUNCT
ejpam-3441	361	9	(	(	PUNCT
ejpam-3441	361	10	r	r	NOUN
ejpam-3441	361	11	◦	◦	NOUN
ejpam-3441	361	12	µa	µa	NOUN
ejpam-3441	361	13	)	)	PUNCT
ejpam-3441	361	14	(	(	PUNCT
ejpam-3441	361	15	xy	xy	NOUN
ejpam-3441	361	16	)	)	PUNCT
ejpam-3441	361	17	∧	∧	NOUN
ejpam-3441	361	18	1	1	NUM
ejpam-3441	361	19	)	)	PUNCT
ejpam-3441	361	20	∧	∧	PROPN
ejpam-3441	361	21	β	β	NOUN
ejpam-3441	361	22	}	}	PUNCT
ejpam-3441	361	23	∨	∨	NUM
ejpam-3441	361	24	α	α	NOUN
ejpam-3441	361	25	=	=	SYM
ejpam-3441	361	26	{	{	PUNCT
ejpam-3441	361	27	(	(	PUNCT
ejpam-3441	361	28	r	r	NOUN
ejpam-3441	361	29	◦	◦	NOUN
ejpam-3441	361	30	µa	µa	NOUN
ejpam-3441	361	31	)	)	PUNCT
ejpam-3441	361	32	(	(	PUNCT
ejpam-3441	361	33	xy	xy	NOUN
ejpam-3441	361	34	)	)	PUNCT
ejpam-3441	361	35	∧	∧	PROPN
ejpam-3441	361	36	β	β	NOUN
ejpam-3441	361	37	}	}	PUNCT
ejpam-3441	361	38	∨	∨	NUM
ejpam-3441	361	39	α	α	NOUN
ejpam-3441	361	40	=	=	X
ejpam-3441	361	41	{	{	PUNCT
ejpam-3441	361	42	(	(	PUNCT
ejpam-3441	361	43	∨xy=∑n	∨xy=∑n	NUM
ejpam-3441	361	44	i=1	i=1	PROPN
ejpam-3441	361	45	cidi	cidi	PROPN
ejpam-3441	361	46	{	{	PUNCT
ejpam-3441	361	47	∧ni=1	∧ni=1	X
ejpam-3441	361	48	{	{	PUNCT
ejpam-3441	361	49	r	r	NOUN
ejpam-3441	361	50	(	(	PUNCT
ejpam-3441	361	51	ai	ai	NOUN
ejpam-3441	361	52	)	)	PUNCT
ejpam-3441	361	53	∧	∧	NOUN
ejpam-3441	361	54	µa	µa	NOUN
ejpam-3441	361	55	(	(	PUNCT
ejpam-3441	361	56	bi	bi	NOUN
ejpam-3441	361	57	)	)	PUNCT
ejpam-3441	361	58	}	}	PUNCT
ejpam-3441	361	59	}	}	PUNCT
ejpam-3441	361	60	)	)	PUNCT
ejpam-3441	361	61	∧	∧	PROPN
ejpam-3441	361	62	β	β	NOUN
ejpam-3441	361	63	}	}	PUNCT
ejpam-3441	361	64	∨	∨	NUM
ejpam-3441	361	65	α	α	PROPN
ejpam-3441	361	66	≥	≥	X
ejpam-3441	361	67	{	{	PUNCT
ejpam-3441	361	68	(	(	PUNCT
ejpam-3441	361	69	r	r	NOUN
ejpam-3441	361	70	(	(	PUNCT
ejpam-3441	361	71	x	x	NOUN
ejpam-3441	361	72	)	)	PUNCT
ejpam-3441	361	73	∧	∧	NOUN
ejpam-3441	361	74	µa	µa	PROPN
ejpam-3441	361	75	(	(	PUNCT
ejpam-3441	361	76	y	y	NOUN
ejpam-3441	361	77	)	)	PUNCT
ejpam-3441	361	78	)	)	PUNCT
ejpam-3441	362	1	∧	∧	PROPN
ejpam-3441	362	2	β	β	NOUN
ejpam-3441	362	3	}	}	PUNCT
ejpam-3441	362	4	∨	∨	NUM
ejpam-3441	362	5	α	α	NOUN
ejpam-3441	362	6	=	=	SYM
ejpam-3441	362	7	{	{	PUNCT
ejpam-3441	362	8	(	(	PUNCT
ejpam-3441	362	9	1	1	NUM
ejpam-3441	362	10	∧	∧	PROPN
ejpam-3441	362	11	µa	µa	NOUN
ejpam-3441	362	12	(	(	PUNCT
ejpam-3441	362	13	y	y	NOUN
ejpam-3441	362	14	)	)	PUNCT
ejpam-3441	362	15	)	)	PUNCT
ejpam-3441	363	1	∧	∧	PROPN
ejpam-3441	363	2	β	β	NOUN
ejpam-3441	363	3	}	}	PUNCT
ejpam-3441	363	4	∨	∨	NUM
ejpam-3441	363	5	α	α	NOUN
ejpam-3441	363	6	=	=	NOUN
ejpam-3441	363	7	µa(y	µa(y	NOUN
ejpam-3441	363	8	)	)	PUNCT
ejpam-3441	363	9	∧	∧	PROPN
ejpam-3441	363	10	β	β	X
ejpam-3441	363	11	=	=	SYM
ejpam-3441	363	12	min{µa(y	min{µa(y	PROPN
ejpam-3441	363	13	)	)	PUNCT
ejpam-3441	363	14	,	,	PUNCT
ejpam-3441	363	15	β	β	NOUN
ejpam-3441	363	16	}	}	PUNCT
ejpam-3441	363	17	.	.	PUNCT
ejpam-3441	364	1	⇒	⇒	PROPN
ejpam-3441	364	2	max{µa((xy)z	max{µa((xy)z	PROPN
ejpam-3441	364	3	)	)	PUNCT
ejpam-3441	364	4	,	,	PUNCT
ejpam-3441	364	5	α	α	X
ejpam-3441	364	6	}	}	PUNCT
ejpam-3441	364	7	≥	≥	NOUN
ejpam-3441	364	8	min{µa(y	min{µa(y	NOUN
ejpam-3441	364	9	)	)	PUNCT
ejpam-3441	364	10	,	,	PUNCT
ejpam-3441	364	11	β	β	X
ejpam-3441	364	12	}	}	PUNCT
ejpam-3441	364	13	.	.	PUNCT
ejpam-3441	365	1	similarly	similarly	ADV
ejpam-3441	365	2	,	,	PUNCT
ejpam-3441	365	3	we	we	PRON
ejpam-3441	365	4	have	have	VERB
ejpam-3441	365	5	min{γa((xy)z	min{γa((xy)z	NOUN
ejpam-3441	365	6	)	)	PUNCT
ejpam-3441	365	7	,	,	PUNCT
ejpam-3441	365	8	(	(	PUNCT
ejpam-3441	365	9	1−	1−	NUM
ejpam-3441	365	10	α	α	NOUN
ejpam-3441	365	11	)	)	PUNCT
ejpam-3441	365	12	}	}	PUNCT
ejpam-3441	365	13	≤	≤	NUM
ejpam-3441	365	14	max{γa(y	max{γa(y	PROPN
ejpam-3441	365	15	)	)	PUNCT
ejpam-3441	365	16	,	,	PUNCT
ejpam-3441	365	17	(	(	PUNCT
ejpam-3441	365	18	1−	1−	NUM
ejpam-3441	365	19	β	β	NOUN
ejpam-3441	365	20	)	)	PUNCT
ejpam-3441	365	21	}	}	PUNCT
ejpam-3441	365	22	.	.	PUNCT
ejpam-3441	366	1	therefore	therefore	ADV
ejpam-3441	366	2	a	a	PRON
ejpam-3441	366	3	is	be	AUX
ejpam-3441	366	4	an	an	DET
ejpam-3441	366	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	366	6	fuzzy	fuzzy	ADJ
ejpam-3441	366	7	interior	interior	ADJ
ejpam-3441	366	8	ideal	ideal	NOUN
ejpam-3441	366	9	with	with	ADP
ejpam-3441	366	10	thresholds	threshold	NOUN
ejpam-3441	366	11	(	(	PUNCT
ejpam-3441	366	12	α	α	X
ejpam-3441	366	13	,	,	PUNCT
ejpam-3441	366	14	β	β	X
ejpam-3441	366	15	]	]	PUNCT
ejpam-3441	366	16	of	of	ADP
ejpam-3441	366	17	r.	r.	PROPN
ejpam-3441	366	18	theorem	theorem	VERB
ejpam-3441	366	19	6	6	NUM
ejpam-3441	366	20	.	.	PUNCT
ejpam-3441	367	1	let	let	VERB
ejpam-3441	367	2	a	a	PRON
ejpam-3441	367	3	=	=	SYM
ejpam-3441	367	4	(	(	PUNCT
ejpam-3441	367	5	µa	µa	PROPN
ejpam-3441	367	6	,	,	PUNCT
ejpam-3441	367	7	γa	γa	PROPN
ejpam-3441	367	8	)	)	PUNCT
ejpam-3441	367	9	be	be	VERB
ejpam-3441	367	10	an	an	DET
ejpam-3441	367	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	367	12	fuzzy	fuzzy	ADJ
ejpam-3441	367	13	la	la	NOUN
ejpam-3441	367	14	-	-	PUNCT
ejpam-3441	367	15	subring	subre	VERB
ejpam-3441	367	16	with	with	ADP
ejpam-3441	367	17	thresholds	threshold	NOUN
ejpam-3441	367	18	(	(	PUNCT
ejpam-3441	367	19	α	α	X
ejpam-3441	367	20	,	,	PUNCT
ejpam-3441	367	21	β	β	X
ejpam-3441	367	22	]	]	PUNCT
ejpam-3441	367	23	of	of	ADP
ejpam-3441	367	24	an	an	DET
ejpam-3441	367	25	la	la	ADJ
ejpam-3441	367	26	-	-	PUNCT
ejpam-3441	367	27	ring	ring	NOUN
ejpam-3441	367	28	r.	r.	PROPN
ejpam-3441	367	29	then	then	ADV
ejpam-3441	367	30	a	a	PRON
ejpam-3441	367	31	is	be	AUX
ejpam-3441	367	32	an	an	DET
ejpam-3441	367	33	intuitionistic	intuitionistic	ADJ
ejpam-3441	367	34	fuzzy	fuzzy	ADJ
ejpam-3441	367	35	bi	bi	NOUN
ejpam-3441	367	36	-	-	NOUN
ejpam-3441	367	37	ideal	ideal	ADJ
ejpam-3441	367	38	with	with	ADP
ejpam-3441	367	39	thresholds	threshold	NOUN
ejpam-3441	367	40	(	(	PUNCT
ejpam-3441	367	41	α	α	X
ejpam-3441	367	42	,	,	PUNCT
ejpam-3441	367	43	β	β	X
ejpam-3441	367	44	]	]	PUNCT
ejpam-3441	367	45	of	of	ADP
ejpam-3441	367	46	r	r	NOUN
ejpam-3441	367	47	if	if	SCONJ
ejpam-3441	368	1	and	and	CCONJ
ejpam-3441	368	2	only	only	ADV
ejpam-3441	368	3	if	if	SCONJ
ejpam-3441	368	4	(	(	PUNCT
ejpam-3441	368	5	a	a	DET
ejpam-3441	368	6	◦	◦	NOUN
ejpam-3441	368	7	βα	βα	NOUN
ejpam-3441	368	8	r	r	NOUN
ejpam-3441	368	9	)	)	PUNCT
ejpam-3441	368	10	◦	◦	NOUN
ejpam-3441	368	11	βα	βα	NOUN
ejpam-3441	368	12	a	a	DET
ejpam-3441	368	13	⊆	⊆	NUM
ejpam-3441	368	14	aβα	aβα	NOUN
ejpam-3441	368	15	.	.	PUNCT
ejpam-3441	369	1	proof	proof	NOUN
ejpam-3441	369	2	.	.	PUNCT
ejpam-3441	370	1	same	same	ADJ
ejpam-3441	370	2	as	as	SCONJ
ejpam-3441	370	3	theorem	theorem	ADJ
ejpam-3441	370	4	5	5	NUM
ejpam-3441	370	5	.	.	PUNCT
ejpam-3441	370	6	theorem	theorem	VERB
ejpam-3441	370	7	7	7	NUM
ejpam-3441	370	8	.	.	PUNCT
ejpam-3441	371	1	let	let	VERB
ejpam-3441	371	2	a	a	PRON
ejpam-3441	371	3	=	=	SYM
ejpam-3441	371	4	(	(	PUNCT
ejpam-3441	371	5	µa	µa	PROPN
ejpam-3441	371	6	,	,	PUNCT
ejpam-3441	371	7	γa	γa	PROPN
ejpam-3441	371	8	)	)	PUNCT
ejpam-3441	371	9	be	be	VERB
ejpam-3441	371	10	an	an	DET
ejpam-3441	371	11	ifs	ifs	PROPN
ejpam-3441	371	12	of	of	ADP
ejpam-3441	371	13	an	an	DET
ejpam-3441	371	14	la	la	ADJ
ejpam-3441	371	15	-	-	PUNCT
ejpam-3441	371	16	ring	ring	NOUN
ejpam-3441	371	17	r.	r.	PROPN
ejpam-3441	371	18	then	then	ADV
ejpam-3441	371	19	a	a	PRON
ejpam-3441	371	20	is	be	AUX
ejpam-3441	371	21	an	an	DET
ejpam-3441	371	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	371	23	fuzzy	fuzzy	ADJ
ejpam-3441	371	24	generalized	generalize	VERB
ejpam-3441	371	25	bi	bi	NOUN
ejpam-3441	371	26	-	-	NOUN
ejpam-3441	371	27	ideal	ideal	NOUN
ejpam-3441	371	28	with	with	ADP
ejpam-3441	371	29	thresholds	threshold	NOUN
ejpam-3441	371	30	(	(	PUNCT
ejpam-3441	371	31	α	α	X
ejpam-3441	371	32	,	,	PUNCT
ejpam-3441	371	33	β	β	X
ejpam-3441	371	34	]	]	PUNCT
ejpam-3441	371	35	of	of	ADP
ejpam-3441	371	36	r	r	NOUN
ejpam-3441	371	37	if	if	SCONJ
ejpam-3441	372	1	and	and	CCONJ
ejpam-3441	372	2	only	only	ADV
ejpam-3441	372	3	if	if	SCONJ
ejpam-3441	372	4	(	(	PUNCT
ejpam-3441	372	5	a	a	DET
ejpam-3441	372	6	◦	◦	NOUN
ejpam-3441	372	7	βαr	βαr	NOUN
ejpam-3441	372	8	)	)	PUNCT
ejpam-3441	372	9	◦	◦	NOUN
ejpam-3441	372	10	βαa	βαa	NOUN
ejpam-3441	372	11	⊆	⊆	NUM
ejpam-3441	372	12	aβα	aβα	NOUN
ejpam-3441	372	13	and	and	CCONJ
ejpam-3441	372	14	a−βα	a−βα	PROPN
ejpam-3441	372	15	a	a	DET
ejpam-3441	372	16	⊆	⊆	NUM
ejpam-3441	372	17	aβα	aβα	NOUN
ejpam-3441	372	18	.	.	PUNCT
ejpam-3441	372	19	proof	proof	NOUN
ejpam-3441	372	20	.	.	PUNCT
ejpam-3441	373	1	same	same	ADJ
ejpam-3441	373	2	as	as	ADP
ejpam-3441	373	3	theorem	theorem	ADJ
ejpam-3441	373	4	5	5	NUM
ejpam-3441	373	5	.	.	PUNCT
ejpam-3441	373	6	k.	k.	PROPN
ejpam-3441	373	7	nasreen	nasreen	PROPN
ejpam-3441	373	8	et	et	PROPN
ejpam-3441	373	9	al	al	PROPN
ejpam-3441	373	10	.	.	PUNCT
ejpam-3441	373	11	/	/	SYM
ejpam-3441	373	12	eur	eur	PROPN
ejpam-3441	373	13	.	.	PUNCT
ejpam-3441	374	1	j.	j.	PROPN
ejpam-3441	374	2	pure	pure	PROPN
ejpam-3441	374	3	appl	appl	PROPN
ejpam-3441	374	4	.	.	PROPN
ejpam-3441	374	5	math	math	PROPN
ejpam-3441	374	6	,	,	PUNCT
ejpam-3441	374	7	12	12	NUM
ejpam-3441	374	8	(	(	PUNCT
ejpam-3441	374	9	3	3	NUM
ejpam-3441	374	10	)	)	PUNCT
ejpam-3441	374	11	(	(	PUNCT
ejpam-3441	374	12	2019	2019	NUM
ejpam-3441	374	13	)	)	PUNCT
ejpam-3441	374	14	,	,	PUNCT
ejpam-3441	374	15	906	906	NUM
ejpam-3441	374	16	-	-	SYM
ejpam-3441	374	17	943	943	NUM
ejpam-3441	374	18	922	922	NUM
ejpam-3441	374	19	lemma	lemma	PROPN
ejpam-3441	374	20	9	9	NUM
ejpam-3441	374	21	.	.	PUNCT
ejpam-3441	375	1	if	if	SCONJ
ejpam-3441	375	2	a	a	PRON
ejpam-3441	375	3	and	and	CCONJ
ejpam-3441	375	4	b	b	NOUN
ejpam-3441	375	5	are	be	AUX
ejpam-3441	375	6	two	two	NUM
ejpam-3441	375	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	375	8	fuzzy	fuzzy	ADJ
ejpam-3441	375	9	bi(resp	bi(resp	PROPN
ejpam-3441	375	10	.	.	PUNCT
ejpam-3441	376	1	generalized	generalize	VERB
ejpam-3441	376	2	bi-	bi-	PROPN
ejpam-3441	376	3	,	,	PUNCT
ejpam-3441	376	4	quasi-	quasi-	ADJ
ejpam-3441	376	5	,	,	PUNCT
ejpam-3441	376	6	interior	interior	ADJ
ejpam-3441	376	7	)	)	PUNCT
ejpam-3441	376	8	ideals	ideal	NOUN
ejpam-3441	376	9	with	with	ADP
ejpam-3441	376	10	thresholds	threshold	NOUN
ejpam-3441	376	11	(	(	PUNCT
ejpam-3441	376	12	α	α	X
ejpam-3441	376	13	,	,	PUNCT
ejpam-3441	376	14	β	β	X
ejpam-3441	376	15	]	]	PUNCT
ejpam-3441	376	16	of	of	ADP
ejpam-3441	376	17	an	an	DET
ejpam-3441	376	18	la	la	ADJ
ejpam-3441	376	19	-	-	PUNCT
ejpam-3441	376	20	ring	ring	NOUN
ejpam-3441	376	21	r	r	NOUN
ejpam-3441	376	22	,	,	PUNCT
ejpam-3441	376	23	then	then	ADV
ejpam-3441	376	24	a∧βαb	a∧βαb	PRON
ejpam-3441	376	25	is	be	AUX
ejpam-3441	376	26	also	also	ADV
ejpam-3441	376	27	an	an	DET
ejpam-3441	376	28	intuitionistic	intuitionistic	ADJ
ejpam-3441	376	29	fuzzy	fuzzy	ADJ
ejpam-3441	376	30	bi(resp	bi(resp	PROPN
ejpam-3441	376	31	.	.	PUNCT
ejpam-3441	377	1	generalized	generalize	VERB
ejpam-3441	377	2	bi-	bi-	PROPN
ejpam-3441	377	3	,	,	PUNCT
ejpam-3441	377	4	quasi-	quasi-	ADJ
ejpam-3441	377	5	,	,	PUNCT
ejpam-3441	377	6	interior	interior	ADJ
ejpam-3441	377	7	)	)	PUNCT
ejpam-3441	377	8	ideal	ideal	NOUN
ejpam-3441	377	9	with	with	ADP
ejpam-3441	377	10	thresholds	threshold	NOUN
ejpam-3441	377	11	(	(	PUNCT
ejpam-3441	377	12	α	α	X
ejpam-3441	377	13	,	,	PUNCT
ejpam-3441	377	14	β	β	X
ejpam-3441	377	15	]	]	PUNCT
ejpam-3441	377	16	of	of	ADP
ejpam-3441	377	17	r.	r.	PROPN
ejpam-3441	377	18	proof	proof	NOUN
ejpam-3441	377	19	.	.	PUNCT
ejpam-3441	378	1	let	let	VERB
ejpam-3441	378	2	a	a	DET
ejpam-3441	378	3	=	=	SYM
ejpam-3441	378	4	(	(	PUNCT
ejpam-3441	378	5	µa	µa	PROPN
ejpam-3441	378	6	,	,	PUNCT
ejpam-3441	378	7	γa	γa	PROPN
ejpam-3441	378	8	)	)	PUNCT
ejpam-3441	378	9	and	and	CCONJ
ejpam-3441	378	10	b	b	X
ejpam-3441	378	11	=	=	SYM
ejpam-3441	378	12	(	(	PUNCT
ejpam-3441	378	13	µb	µb	PROPN
ejpam-3441	378	14	,	,	PUNCT
ejpam-3441	378	15	γb	γb	PROPN
ejpam-3441	378	16	)	)	PUNCT
ejpam-3441	378	17	be	be	VERB
ejpam-3441	378	18	two	two	NUM
ejpam-3441	378	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	378	20	fuzzy	fuzzy	ADJ
ejpam-3441	378	21	bi	bi	NOUN
ejpam-3441	378	22	-	-	NOUN
ejpam-3441	378	23	ideals	ideal	NOUN
ejpam-3441	378	24	with	with	ADP
ejpam-3441	378	25	thresholds	threshold	NOUN
ejpam-3441	378	26	(	(	PUNCT
ejpam-3441	378	27	α	α	X
ejpam-3441	378	28	,	,	PUNCT
ejpam-3441	378	29	β	β	X
ejpam-3441	378	30	]	]	PUNCT
ejpam-3441	378	31	of	of	ADP
ejpam-3441	378	32	an	an	DET
ejpam-3441	378	33	la-ringr.we	la-ringr.we	NOUN
ejpam-3441	378	34	have	have	VERB
ejpam-3441	378	35	to	to	PART
ejpam-3441	378	36	show	show	VERB
ejpam-3441	378	37	thata∧βαb	thata∧βαb	PROPN
ejpam-3441	378	38	is	be	AUX
ejpam-3441	378	39	also	also	ADV
ejpam-3441	378	40	an	an	DET
ejpam-3441	378	41	intuitionistic	intuitionistic	ADJ
ejpam-3441	378	42	fuzzy	fuzzy	ADJ
ejpam-3441	378	43	bi	bi	NOUN
ejpam-3441	378	44	-	-	NOUN
ejpam-3441	378	45	ideal	ideal	ADJ
ejpam-3441	378	46	with	with	ADP
ejpam-3441	378	47	thresholds	threshold	NOUN
ejpam-3441	378	48	(	(	PUNCT
ejpam-3441	378	49	α	α	X
ejpam-3441	378	50	,	,	PUNCT
ejpam-3441	378	51	β	β	X
ejpam-3441	378	52	]	]	PUNCT
ejpam-3441	378	53	of	of	ADP
ejpam-3441	378	54	r.	r.	PROPN
ejpam-3441	378	55	since	since	SCONJ
ejpam-3441	378	56	a	a	PRON
ejpam-3441	378	57	and	and	CCONJ
ejpam-3441	378	58	b	b	NOUN
ejpam-3441	378	59	are	be	AUX
ejpam-3441	378	60	intuitionistic	intuitionistic	ADJ
ejpam-3441	378	61	fuzzy	fuzzy	ADJ
ejpam-3441	378	62	la	la	NOUN
ejpam-3441	378	63	-	-	NOUN
ejpam-3441	378	64	subrings	subring	NOUN
ejpam-3441	378	65	with	with	ADP
ejpam-3441	378	66	thresholds	threshold	NOUN
ejpam-3441	378	67	(	(	PUNCT
ejpam-3441	378	68	α	α	X
ejpam-3441	378	69	,	,	PUNCT
ejpam-3441	378	70	β	β	X
ejpam-3441	378	71	]	]	PUNCT
ejpam-3441	378	72	of	of	ADP
ejpam-3441	378	73	r	r	NOUN
ejpam-3441	378	74	,	,	PUNCT
ejpam-3441	378	75	then	then	ADV
ejpam-3441	378	76	a	a	DET
ejpam-3441	378	77	∧βα	∧βα	PROPN
ejpam-3441	378	78	b	b	NOUN
ejpam-3441	378	79	is	be	AUX
ejpam-3441	378	80	also	also	ADV
ejpam-3441	378	81	an	an	DET
ejpam-3441	378	82	intuitionistic	intuitionistic	ADJ
ejpam-3441	378	83	fuzzy	fuzzy	ADJ
ejpam-3441	378	84	la	la	NOUN
ejpam-3441	378	85	-	-	PUNCT
ejpam-3441	378	86	subring	subre	VERB
ejpam-3441	378	87	with	with	ADP
ejpam-3441	378	88	thresholds	threshold	NOUN
ejpam-3441	378	89	(	(	PUNCT
ejpam-3441	378	90	α	α	X
ejpam-3441	378	91	,	,	PUNCT
ejpam-3441	378	92	β	β	X
ejpam-3441	378	93	]	]	X
ejpam-3441	378	94	ofr	ofr	NOUN
ejpam-3441	378	95	by	by	ADP
ejpam-3441	378	96	the	the	DET
ejpam-3441	378	97	lemma	lemma	PROPN
ejpam-3441	378	98	3	3	X
ejpam-3441	378	99	.	.	X
ejpam-3441	379	1	we	we	PRON
ejpam-3441	379	2	have	have	VERB
ejpam-3441	379	3	to	to	PART
ejpam-3441	379	4	show	show	VERB
ejpam-3441	379	5	thatmax{(µa∧βαµb)((xa)y	thatmax{(µa∧βαµb)((xa)y	NUM
ejpam-3441	379	6	)	)	PUNCT
ejpam-3441	379	7	,	,	PUNCT
ejpam-3441	379	8	α	α	PROPN
ejpam-3441	379	9	}	}	PUNCT
ejpam-3441	379	10	≥	≥	NUM
ejpam-3441	379	11	min{(µa∧βαµb)(x	min{(µa∧βαµb)(x	NOUN
ejpam-3441	379	12	)	)	PUNCT
ejpam-3441	379	13	,	,	PUNCT
ejpam-3441	379	14	(	(	PUNCT
ejpam-3441	379	15	µa∧βαµb)(y	µa∧βαµb)(y	NOUN
ejpam-3441	379	16	)	)	PUNCT
ejpam-3441	379	17	,	,	PUNCT
ejpam-3441	379	18	β	β	X
ejpam-3441	379	19	}	}	PUNCT
ejpam-3441	379	20	and	and	CCONJ
ejpam-3441	379	21	min{(γa∨βαγb)((xa)y	min{(γa∨βαγb)((xa)y	PROPN
ejpam-3441	379	22	)	)	PUNCT
ejpam-3441	379	23	,	,	PUNCT
ejpam-3441	379	24	(	(	PUNCT
ejpam-3441	379	25	1−α	1−α	NUM
ejpam-3441	379	26	)	)	PUNCT
ejpam-3441	379	27	}	}	PUNCT
ejpam-3441	379	28	≤	≤	NUM
ejpam-3441	379	29	max{(γa∨βα	max{(γa∨βα	PROPN
ejpam-3441	379	30	γb)(x	γb)(x	PROPN
ejpam-3441	379	31	)	)	PUNCT
ejpam-3441	379	32	,	,	PUNCT
ejpam-3441	379	33	(	(	PUNCT
ejpam-3441	379	34	γa	γa	PROPN
ejpam-3441	379	35	∨βα	∨βα	VERB
ejpam-3441	379	36	γb)(y	γb)(y	PROPN
ejpam-3441	379	37	)	)	PUNCT
ejpam-3441	379	38	,	,	PUNCT
ejpam-3441	379	39	(	(	PUNCT
ejpam-3441	379	40	1−	1−	NUM
ejpam-3441	379	41	β	β	NOUN
ejpam-3441	379	42	)	)	PUNCT
ejpam-3441	379	43	}	}	PUNCT
ejpam-3441	379	44	.	.	PUNCT
ejpam-3441	380	1	now	now	ADV
ejpam-3441	380	2	max{(µa	max{(µa	VERB
ejpam-3441	380	3	∧βα	∧βα	ADJ
ejpam-3441	380	4	µb)((xa)y	µb)((xa)y	NOUN
ejpam-3441	380	5	)	)	PUNCT
ejpam-3441	380	6	,	,	PUNCT
ejpam-3441	380	7	α	α	NOUN
ejpam-3441	380	8	}	}	PUNCT
ejpam-3441	380	9	=	=	SYM
ejpam-3441	380	10	max{{{(µa	max{{{(µa	PROPN
ejpam-3441	380	11	∧	∧	PROPN
ejpam-3441	380	12	µb)((xa)y	µb)((xa)y	NOUN
ejpam-3441	380	13	)	)	PUNCT
ejpam-3441	380	14	∧	∧	PROPN
ejpam-3441	380	15	β	β	NOUN
ejpam-3441	380	16	}	}	PUNCT
ejpam-3441	380	17	∨	∨	NUM
ejpam-3441	380	18	α	α	NOUN
ejpam-3441	380	19	}	}	PUNCT
ejpam-3441	380	20	,	,	PUNCT
ejpam-3441	380	21	α	α	NOUN
ejpam-3441	380	22	}	}	PUNCT
ejpam-3441	380	23	=	=	PRON
ejpam-3441	380	24	{	{	PUNCT
ejpam-3441	380	25	(	(	PUNCT
ejpam-3441	380	26	µa	µa	ADP
ejpam-3441	380	27	∧	∧	PROPN
ejpam-3441	380	28	µb)((xa)y	µb)((xa)y	NOUN
ejpam-3441	380	29	)	)	PUNCT
ejpam-3441	380	30	∧	∧	PROPN
ejpam-3441	380	31	β	β	NOUN
ejpam-3441	380	32	}	}	PUNCT
ejpam-3441	380	33	∨	∨	NUM
ejpam-3441	380	34	α	α	NOUN
ejpam-3441	380	35	=	=	PUNCT
ejpam-3441	380	36	{	{	PUNCT
ejpam-3441	380	37	µa((xa)y	µa((xa)y	NOUN
ejpam-3441	380	38	)	)	PUNCT
ejpam-3441	380	39	∧	∧	PROPN
ejpam-3441	380	40	µb((xa)y	µb((xa)y	NOUN
ejpam-3441	380	41	)	)	PUNCT
ejpam-3441	380	42	∧	∧	PROPN
ejpam-3441	380	43	β	β	NOUN
ejpam-3441	380	44	}	}	PUNCT
ejpam-3441	380	45	∨	∨	NUM
ejpam-3441	380	46	α	α	PROPN
ejpam-3441	380	47	≥	≥	X
ejpam-3441	380	48	{	{	PUNCT
ejpam-3441	380	49	µa(x	µa(x	NOUN
ejpam-3441	380	50	)	)	PUNCT
ejpam-3441	380	51	∧	∧	NOUN
ejpam-3441	380	52	µa(y	µa(y	NOUN
ejpam-3441	380	53	)	)	PUNCT
ejpam-3441	380	54	∧	∧	NOUN
ejpam-3441	380	55	µb(x	µb(x	PUNCT
ejpam-3441	380	56	)	)	PUNCT
ejpam-3441	380	57	∧	∧	PROPN
ejpam-3441	380	58	µb(y	µb(y	NUM
ejpam-3441	380	59	)	)	PUNCT
ejpam-3441	381	1	∧	∧	PROPN
ejpam-3441	381	2	β	β	NOUN
ejpam-3441	381	3	}	}	PUNCT
ejpam-3441	381	4	∨	∨	NUM
ejpam-3441	381	5	α	α	X
ejpam-3441	381	6	=	=	SYM
ejpam-3441	381	7	{	{	PUNCT
ejpam-3441	381	8	µa(x	µa(x	NOUN
ejpam-3441	381	9	)	)	PUNCT
ejpam-3441	381	10	∧	∧	NOUN
ejpam-3441	381	11	µb(x	µb(x	PUNCT
ejpam-3441	381	12	)	)	PUNCT
ejpam-3441	381	13	∧	∧	NOUN
ejpam-3441	381	14	µa(y	µa(y	NOUN
ejpam-3441	381	15	)	)	PUNCT
ejpam-3441	381	16	∧	∧	PROPN
ejpam-3441	381	17	µb(y	µb(y	NUM
ejpam-3441	381	18	)	)	PUNCT
ejpam-3441	382	1	∧	∧	PROPN
ejpam-3441	382	2	β	β	NOUN
ejpam-3441	382	3	}	}	PUNCT
ejpam-3441	382	4	∨	∨	NUM
ejpam-3441	382	5	α	α	NOUN
ejpam-3441	382	6	=	=	SYM
ejpam-3441	382	7	{	{	PUNCT
ejpam-3441	382	8	(	(	PUNCT
ejpam-3441	382	9	µa	µa	PROPN
ejpam-3441	382	10	∧	∧	PROPN
ejpam-3441	382	11	µb)(x	µb)(x	PROPN
ejpam-3441	382	12	)	)	PUNCT
ejpam-3441	382	13	∧	∧	NOUN
ejpam-3441	382	14	(	(	PUNCT
ejpam-3441	382	15	µa	µa	ADP
ejpam-3441	382	16	∧	∧	PROPN
ejpam-3441	382	17	µb)(y	µb)(y	PROPN
ejpam-3441	382	18	)	)	PUNCT
ejpam-3441	382	19	∧	∧	NOUN
ejpam-3441	382	20	β	β	X
ejpam-3441	382	21	∧	∧	PROPN
ejpam-3441	382	22	β	β	X
ejpam-3441	382	23	∧	∧	PROPN
ejpam-3441	382	24	β	β	PROPN
ejpam-3441	382	25	}	}	PUNCT
ejpam-3441	382	26	∨	∨	NUM
ejpam-3441	382	27	α	α	NOUN
ejpam-3441	382	28	=	=	SYM
ejpam-3441	382	29	{	{	PUNCT
ejpam-3441	382	30	(	(	PUNCT
ejpam-3441	382	31	(	(	PUNCT
ejpam-3441	382	32	µa	µa	ADP
ejpam-3441	382	33	∧	∧	PROPN
ejpam-3441	382	34	µb)(x	µb)(x	PROPN
ejpam-3441	382	35	)	)	PUNCT
ejpam-3441	382	36	∧	∧	PROPN
ejpam-3441	382	37	β	β	NOUN
ejpam-3441	382	38	)	)	PUNCT
ejpam-3441	382	39	∧	∧	NOUN
ejpam-3441	382	40	(	(	PUNCT
ejpam-3441	382	41	(	(	PUNCT
ejpam-3441	382	42	µa	µa	ADP
ejpam-3441	382	43	∧	∧	PROPN
ejpam-3441	382	44	µb)(y	µb)(y	PROPN
ejpam-3441	382	45	)	)	PUNCT
ejpam-3441	382	46	∧	∧	PROPN
ejpam-3441	382	47	β	β	NOUN
ejpam-3441	382	48	)	)	PUNCT
ejpam-3441	382	49	∧	∧	PROPN
ejpam-3441	382	50	β	β	NOUN
ejpam-3441	382	51	}	}	PUNCT
ejpam-3441	382	52	∨	∨	NUM
ejpam-3441	382	53	α	α	NOUN
ejpam-3441	382	54	=	=	SYM
ejpam-3441	382	55	(	(	PUNCT
ejpam-3441	382	56	{	{	PUNCT
ejpam-3441	382	57	(	(	PUNCT
ejpam-3441	382	58	µa	µa	PROPN
ejpam-3441	382	59	∧	∧	PROPN
ejpam-3441	382	60	µb)(x	µb)(x	PROPN
ejpam-3441	382	61	)	)	PUNCT
ejpam-3441	382	62	∧	∧	PROPN
ejpam-3441	382	63	β	β	NOUN
ejpam-3441	382	64	}	}	PUNCT
ejpam-3441	382	65	∨	∨	NUM
ejpam-3441	382	66	α	α	NOUN
ejpam-3441	382	67	)	)	PUNCT
ejpam-3441	382	68	∧	∧	NOUN
ejpam-3441	382	69	(	(	PUNCT
ejpam-3441	382	70	{	{	PUNCT
ejpam-3441	382	71	(	(	PUNCT
ejpam-3441	382	72	µa	µa	ADP
ejpam-3441	382	73	∧	∧	PROPN
ejpam-3441	382	74	µb)(y	µb)(y	PROPN
ejpam-3441	382	75	)	)	PUNCT
ejpam-3441	383	1	∧	∧	PROPN
ejpam-3441	383	2	β	β	PROPN
ejpam-3441	383	3	}	}	PUNCT
ejpam-3441	383	4	∨	∨	NUM
ejpam-3441	383	5	α	α	NOUN
ejpam-3441	383	6	)	)	PUNCT
ejpam-3441	383	7	∧	∧	PROPN
ejpam-3441	383	8	(	(	PUNCT
ejpam-3441	383	9	β	β	X
ejpam-3441	383	10	∨	∨	NUM
ejpam-3441	383	11	α	α	NOUN
ejpam-3441	383	12	)	)	PUNCT
ejpam-3441	383	13	=	=	SYM
ejpam-3441	383	14	(	(	PUNCT
ejpam-3441	383	15	µa	µa	ADP
ejpam-3441	383	16	∧βα	∧βα	ADJ
ejpam-3441	383	17	µb)(x	µb)(x	NOUN
ejpam-3441	383	18	)	)	PUNCT
ejpam-3441	383	19	∧	∧	NOUN
ejpam-3441	383	20	(	(	PUNCT
ejpam-3441	383	21	µa	µa	ADP
ejpam-3441	383	22	∧βα	∧βα	ADJ
ejpam-3441	383	23	µb)(y	µb)(y	PUNCT
ejpam-3441	383	24	)	)	PUNCT
ejpam-3441	384	1	∧	∧	NOUN
ejpam-3441	384	2	β	β	X
ejpam-3441	384	3	=	=	SYM
ejpam-3441	384	4	min{(µa	min{(µa	VERB
ejpam-3441	384	5	∧βα	∧βα	ADJ
ejpam-3441	384	6	µb)(x	µb)(x	NOUN
ejpam-3441	384	7	)	)	PUNCT
ejpam-3441	384	8	,	,	PUNCT
ejpam-3441	384	9	(	(	PUNCT
ejpam-3441	384	10	µa	µa	ADP
ejpam-3441	384	11	∧βα	∧βα	ADJ
ejpam-3441	384	12	µb)(y	µb)(y	PROPN
ejpam-3441	384	13	)	)	PUNCT
ejpam-3441	384	14	,	,	PUNCT
ejpam-3441	384	15	β	β	X
ejpam-3441	384	16	}	}	PUNCT
ejpam-3441	384	17	.	.	PUNCT
ejpam-3441	385	1	thus	thus	ADV
ejpam-3441	385	2	max{(µa	max{(µa	VERB
ejpam-3441	385	3	∧βα	∧βα	ADJ
ejpam-3441	385	4	µb)((xa)y	µb)((xa)y	NOUN
ejpam-3441	385	5	)	)	PUNCT
ejpam-3441	385	6	,	,	PUNCT
ejpam-3441	385	7	α	α	PROPN
ejpam-3441	385	8	}	}	PUNCT
ejpam-3441	385	9	≥	≥	NOUN
ejpam-3441	385	10	min{(µa	min{(µa	VERB
ejpam-3441	385	11	∧βα	∧βα	ADJ
ejpam-3441	385	12	µb)(x	µb)(x	NOUN
ejpam-3441	385	13	)	)	PUNCT
ejpam-3441	385	14	,	,	PUNCT
ejpam-3441	385	15	(	(	PUNCT
ejpam-3441	385	16	µa	µa	ADP
ejpam-3441	385	17	∧βα	∧βα	ADJ
ejpam-3441	385	18	µb)(y	µb)(y	PROPN
ejpam-3441	385	19	)	)	PUNCT
ejpam-3441	385	20	,	,	PUNCT
ejpam-3441	385	21	β	β	X
ejpam-3441	385	22	}	}	PUNCT
ejpam-3441	385	23	.	.	PUNCT
ejpam-3441	386	1	similarly	similarly	ADV
ejpam-3441	386	2	,	,	PUNCT
ejpam-3441	386	3	we	we	PRON
ejpam-3441	386	4	have	have	VERB
ejpam-3441	386	5	min{(γa	min{(γa	VERB
ejpam-3441	386	6	∨βα	∨βα	NOUN
ejpam-3441	386	7	γb)((xa)y	γb)((xa)y	NOUN
ejpam-3441	386	8	)	)	PUNCT
ejpam-3441	386	9	,	,	PUNCT
ejpam-3441	386	10	(	(	PUNCT
ejpam-3441	386	11	1−	1−	NUM
ejpam-3441	386	12	α	α	NOUN
ejpam-3441	386	13	)	)	PUNCT
ejpam-3441	386	14	}	}	PUNCT
ejpam-3441	386	15	≤	≤	NOUN
ejpam-3441	386	16	max{(γa	max{(γa	ADJ
ejpam-3441	386	17	∨βα	∨βα	ADJ
ejpam-3441	386	18	γb)(x	γb)(x	PROPN
ejpam-3441	386	19	)	)	PUNCT
ejpam-3441	386	20	,	,	PUNCT
ejpam-3441	386	21	(	(	PUNCT
ejpam-3441	386	22	γa	γa	PROPN
ejpam-3441	386	23	∨βα	∨βα	VERB
ejpam-3441	386	24	γb)(y	γb)(y	PROPN
ejpam-3441	386	25	)	)	PUNCT
ejpam-3441	386	26	,	,	PUNCT
ejpam-3441	386	27	(	(	PUNCT
ejpam-3441	386	28	1−	1−	NUM
ejpam-3441	386	29	β	β	NOUN
ejpam-3441	386	30	)	)	PUNCT
ejpam-3441	386	31	}	}	PUNCT
ejpam-3441	386	32	.	.	PUNCT
ejpam-3441	387	1	hence	hence	ADV
ejpam-3441	387	2	a	a	DET
ejpam-3441	387	3	∧βα	∧βα	PROPN
ejpam-3441	387	4	b	b	NOUN
ejpam-3441	387	5	is	be	AUX
ejpam-3441	387	6	an	an	DET
ejpam-3441	387	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	387	8	fuzzy	fuzzy	ADJ
ejpam-3441	387	9	bi	bi	NOUN
ejpam-3441	387	10	-	-	NOUN
ejpam-3441	387	11	ideal	ideal	ADJ
ejpam-3441	387	12	with	with	ADP
ejpam-3441	387	13	thresholds	threshold	NOUN
ejpam-3441	387	14	(	(	PUNCT
ejpam-3441	387	15	α	α	X
ejpam-3441	387	16	,	,	PUNCT
ejpam-3441	387	17	β	β	X
ejpam-3441	387	18	]	]	PUNCT
ejpam-3441	387	19	of	of	ADP
ejpam-3441	387	20	r.	r.	PROPN
ejpam-3441	387	21	lemma	lemma	PROPN
ejpam-3441	387	22	10	10	NUM
ejpam-3441	387	23	.	.	PUNCT
ejpam-3441	388	1	if	if	SCONJ
ejpam-3441	388	2	a	a	PRON
ejpam-3441	388	3	and	and	CCONJ
ejpam-3441	388	4	b	b	NOUN
ejpam-3441	388	5	are	be	AUX
ejpam-3441	388	6	two	two	NUM
ejpam-3441	388	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	388	8	fuzzy	fuzzy	ADJ
ejpam-3441	388	9	bi(resp	bi(resp	PROPN
ejpam-3441	388	10	.	.	PUNCT
ejpam-3441	389	1	generalized	generalize	VERB
ejpam-3441	389	2	bi-	bi-	NUM
ejpam-3441	389	3	,	,	PUNCT
ejpam-3441	389	4	interior	interior	ADJ
ejpam-3441	389	5	)	)	PUNCT
ejpam-3441	389	6	ideals	ideal	NOUN
ejpam-3441	389	7	with	with	ADP
ejpam-3441	389	8	thresholds	threshold	NOUN
ejpam-3441	389	9	(	(	PUNCT
ejpam-3441	389	10	α	α	X
ejpam-3441	389	11	,	,	PUNCT
ejpam-3441	389	12	β	β	X
ejpam-3441	389	13	]	]	PUNCT
ejpam-3441	389	14	of	of	ADP
ejpam-3441	389	15	an	an	DET
ejpam-3441	389	16	la	la	ADJ
ejpam-3441	389	17	-	-	PUNCT
ejpam-3441	389	18	ring	ring	NOUN
ejpam-3441	389	19	r	r	NOUN
ejpam-3441	389	20	with	with	ADP
ejpam-3441	389	21	left	left	ADJ
ejpam-3441	389	22	identity	identity	NOUN
ejpam-3441	389	23	e	e	NOUN
ejpam-3441	389	24	,	,	PUNCT
ejpam-3441	389	25	then	then	ADV
ejpam-3441	389	26	a	a	DET
ejpam-3441	389	27	◦	◦	NOUN
ejpam-3441	389	28	βα	βα	NOUN
ejpam-3441	389	29	b	b	NOUN
ejpam-3441	389	30	is	be	AUX
ejpam-3441	389	31	also	also	ADV
ejpam-3441	389	32	an	an	DET
ejpam-3441	389	33	intuitionistic	intuitionistic	ADJ
ejpam-3441	389	34	fuzzy	fuzzy	ADJ
ejpam-3441	389	35	bi(resp	bi(resp	PROPN
ejpam-3441	389	36	.	.	PUNCT
ejpam-3441	390	1	generalized	generalize	VERB
ejpam-3441	390	2	bi-	bi-	NUM
ejpam-3441	390	3	,	,	PUNCT
ejpam-3441	390	4	interior	interior	ADJ
ejpam-3441	390	5	)	)	PUNCT
ejpam-3441	390	6	ideal	ideal	NOUN
ejpam-3441	390	7	with	with	ADP
ejpam-3441	390	8	thresholds	threshold	NOUN
ejpam-3441	390	9	(	(	PUNCT
ejpam-3441	390	10	α	α	X
ejpam-3441	390	11	,	,	PUNCT
ejpam-3441	390	12	β	β	X
ejpam-3441	390	13	]	]	PUNCT
ejpam-3441	390	14	of	of	ADP
ejpam-3441	390	15	r.	r.	PROPN
ejpam-3441	390	16	proof	proof	NOUN
ejpam-3441	390	17	.	.	PUNCT
ejpam-3441	391	1	let	let	VERB
ejpam-3441	391	2	a	a	DET
ejpam-3441	391	3	=	=	SYM
ejpam-3441	391	4	(	(	PUNCT
ejpam-3441	391	5	µa	µa	PROPN
ejpam-3441	391	6	,	,	PUNCT
ejpam-3441	391	7	γa	γa	PROPN
ejpam-3441	391	8	)	)	PUNCT
ejpam-3441	391	9	and	and	CCONJ
ejpam-3441	391	10	b	b	X
ejpam-3441	391	11	=	=	SYM
ejpam-3441	391	12	(	(	PUNCT
ejpam-3441	391	13	µb	µb	PROPN
ejpam-3441	391	14	,	,	PUNCT
ejpam-3441	391	15	γb	γb	PROPN
ejpam-3441	391	16	)	)	PUNCT
ejpam-3441	391	17	be	be	VERB
ejpam-3441	391	18	two	two	NUM
ejpam-3441	391	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	391	20	fuzzy	fuzzy	ADJ
ejpam-3441	391	21	bi	bi	NOUN
ejpam-3441	391	22	-	-	NOUN
ejpam-3441	391	23	ideals	ideal	NOUN
ejpam-3441	391	24	with	with	ADP
ejpam-3441	391	25	thresholds	threshold	NOUN
ejpam-3441	391	26	(	(	PUNCT
ejpam-3441	391	27	α	α	X
ejpam-3441	391	28	,	,	PUNCT
ejpam-3441	391	29	β	β	X
ejpam-3441	391	30	]	]	PUNCT
ejpam-3441	391	31	of	of	ADP
ejpam-3441	391	32	an	an	DET
ejpam-3441	391	33	la	la	ADJ
ejpam-3441	391	34	-	-	PUNCT
ejpam-3441	391	35	ring	ring	NOUN
ejpam-3441	391	36	r.	r.	NOUN
ejpam-3441	391	37	we	we	PRON
ejpam-3441	391	38	have	have	VERB
ejpam-3441	391	39	to	to	PART
ejpam-3441	391	40	show	show	VERB
ejpam-3441	391	41	that	that	SCONJ
ejpam-3441	391	42	a	a	DET
ejpam-3441	391	43	◦	◦	NOUN
ejpam-3441	391	44	βα	βα	NOUN
ejpam-3441	391	45	b	b	NOUN
ejpam-3441	391	46	is	be	AUX
ejpam-3441	391	47	also	also	ADV
ejpam-3441	391	48	an	an	DET
ejpam-3441	391	49	intuitionistic	intuitionistic	ADJ
ejpam-3441	391	50	fuzzy	fuzzy	ADJ
ejpam-3441	391	51	bi	bi	NOUN
ejpam-3441	391	52	-	-	NOUN
ejpam-3441	391	53	ideal	ideal	ADJ
ejpam-3441	391	54	with	with	ADP
ejpam-3441	391	55	thresholds	threshold	NOUN
ejpam-3441	391	56	(	(	PUNCT
ejpam-3441	391	57	α	α	X
ejpam-3441	391	58	,	,	PUNCT
ejpam-3441	391	59	β	β	X
ejpam-3441	391	60	]	]	PUNCT
ejpam-3441	391	61	of	of	ADP
ejpam-3441	391	62	r.	r.	PROPN
ejpam-3441	391	63	since	since	SCONJ
ejpam-3441	391	64	a	a	PRON
ejpam-3441	391	65	and	and	CCONJ
ejpam-3441	391	66	b	b	NOUN
ejpam-3441	391	67	are	be	AUX
ejpam-3441	391	68	intuitionistic	intuitionistic	ADJ
ejpam-3441	391	69	fuzzy	fuzzy	ADJ
ejpam-3441	391	70	lasubrings	lasubring	NOUN
ejpam-3441	391	71	with	with	ADP
ejpam-3441	391	72	thresholds	threshold	NOUN
ejpam-3441	391	73	(	(	PUNCT
ejpam-3441	391	74	α	α	X
ejpam-3441	391	75	,	,	PUNCT
ejpam-3441	391	76	β	β	X
ejpam-3441	391	77	]	]	PUNCT
ejpam-3441	391	78	of	of	ADP
ejpam-3441	391	79	r	r	NOUN
ejpam-3441	391	80	,	,	PUNCT
ejpam-3441	391	81	then	then	ADV
ejpam-3441	391	82	a	a	DET
ejpam-3441	391	83	◦	◦	NOUN
ejpam-3441	392	1	βαb	βαb	NOUN
ejpam-3441	392	2	is	be	AUX
ejpam-3441	392	3	also	also	ADV
ejpam-3441	392	4	an	an	DET
ejpam-3441	392	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	392	6	fuzzy	fuzzy	ADJ
ejpam-3441	392	7	la	la	NOUN
ejpam-3441	392	8	-	-	PUNCT
ejpam-3441	392	9	subring	subre	VERB
ejpam-3441	392	10	with	with	ADP
ejpam-3441	392	11	thresholds	threshold	NOUN
ejpam-3441	392	12	(	(	PUNCT
ejpam-3441	392	13	α	α	X
ejpam-3441	392	14	,	,	PUNCT
ejpam-3441	392	15	β	β	X
ejpam-3441	392	16	]	]	PUNCT
ejpam-3441	392	17	of	of	ADP
ejpam-3441	392	18	r	r	NOUN
ejpam-3441	392	19	by	by	ADP
ejpam-3441	392	20	the	the	DET
ejpam-3441	392	21	lemma	lemma	PROPN
ejpam-3441	392	22	4	4	NUM
ejpam-3441	392	23	.	.	PUNCT
ejpam-3441	393	1	now	now	ADV
ejpam-3441	393	2	(	(	PUNCT
ejpam-3441	393	3	(	(	PUNCT
ejpam-3441	393	4	µa	µa	ADP
ejpam-3441	393	5	◦	◦	NOUN
ejpam-3441	393	6	βα	βα	NOUN
ejpam-3441	393	7	µb	µb	NOUN
ejpam-3441	393	8	)	)	PUNCT
ejpam-3441	393	9	◦	◦	NOUN
ejpam-3441	393	10	βα	βα	NOUN
ejpam-3441	393	11	r	r	NOUN
ejpam-3441	393	12	)	)	PUNCT
ejpam-3441	393	13	◦	◦	NOUN
ejpam-3441	393	14	βα	βα	X
ejpam-3441	393	15	(	(	PUNCT
ejpam-3441	393	16	µa	µa	ADP
ejpam-3441	393	17	◦	◦	NOUN
ejpam-3441	393	18	βα	βα	NOUN
ejpam-3441	393	19	µb	µb	NOUN
ejpam-3441	393	20	)	)	PUNCT
ejpam-3441	393	21	=	=	SYM
ejpam-3441	393	22	(	(	PUNCT
ejpam-3441	393	23	(	(	PUNCT
ejpam-3441	393	24	µa	µa	ADP
ejpam-3441	393	25	◦	◦	NOUN
ejpam-3441	393	26	βα	βα	NOUN
ejpam-3441	393	27	µb	µb	NOUN
ejpam-3441	393	28	)	)	PUNCT
ejpam-3441	393	29	◦	◦	NOUN
ejpam-3441	393	30	βα	βα	X
ejpam-3441	393	31	(	(	PUNCT
ejpam-3441	393	32	r	r	NOUN
ejpam-3441	393	33	◦	◦	NOUN
ejpam-3441	393	34	βα	βα	NOUN
ejpam-3441	393	35	r	r	NOUN
ejpam-3441	393	36	)	)	PUNCT
ejpam-3441	393	37	)	)	PUNCT
ejpam-3441	393	38	◦	◦	NOUN
ejpam-3441	393	39	βα	βα	X
ejpam-3441	393	40	(	(	PUNCT
ejpam-3441	393	41	µa	µa	ADP
ejpam-3441	393	42	◦	◦	NOUN
ejpam-3441	393	43	βα	βα	NOUN
ejpam-3441	393	44	µb	µb	NOUN
ejpam-3441	393	45	)	)	PUNCT
ejpam-3441	393	46	=	=	SYM
ejpam-3441	393	47	(	(	PUNCT
ejpam-3441	393	48	(	(	PUNCT
ejpam-3441	393	49	µa	µa	ADP
ejpam-3441	393	50	◦	◦	NOUN
ejpam-3441	393	51	βα	βα	NOUN
ejpam-3441	393	52	r	r	NOUN
ejpam-3441	393	53	)	)	PUNCT
ejpam-3441	393	54	◦	◦	NOUN
ejpam-3441	393	55	βα	βα	X
ejpam-3441	393	56	(	(	PUNCT
ejpam-3441	393	57	µb	µb	ADP
ejpam-3441	393	58	◦	◦	NOUN
ejpam-3441	393	59	βα	βα	NOUN
ejpam-3441	393	60	r	r	NOUN
ejpam-3441	393	61	)	)	PUNCT
ejpam-3441	393	62	)	)	PUNCT
ejpam-3441	393	63	◦	◦	NOUN
ejpam-3441	393	64	βα	βα	X
ejpam-3441	393	65	(	(	PUNCT
ejpam-3441	393	66	µa	µa	ADP
ejpam-3441	393	67	◦	◦	NOUN
ejpam-3441	393	68	βα	βα	NOUN
ejpam-3441	393	69	µb	µb	NOUN
ejpam-3441	393	70	)	)	PUNCT
ejpam-3441	393	71	=	=	SYM
ejpam-3441	393	72	(	(	PUNCT
ejpam-3441	393	73	(	(	PUNCT
ejpam-3441	393	74	µa	µa	ADP
ejpam-3441	393	75	◦	◦	NOUN
ejpam-3441	393	76	βα	βα	NOUN
ejpam-3441	393	77	r	r	NOUN
ejpam-3441	393	78	)	)	PUNCT
ejpam-3441	393	79	◦	◦	NOUN
ejpam-3441	393	80	βα	βα	NOUN
ejpam-3441	393	81	µa	µa	NOUN
ejpam-3441	393	82	)	)	PUNCT
ejpam-3441	393	83	◦	◦	NOUN
ejpam-3441	393	84	βα	βα	X
ejpam-3441	393	85	(	(	PUNCT
ejpam-3441	393	86	(	(	PUNCT
ejpam-3441	393	87	µb	µb	NOUN
ejpam-3441	393	88	◦	◦	NOUN
ejpam-3441	393	89	βα	βα	NOUN
ejpam-3441	393	90	r	r	NOUN
ejpam-3441	393	91	)	)	PUNCT
ejpam-3441	393	92	◦	◦	NOUN
ejpam-3441	393	93	βα	βα	NOUN
ejpam-3441	393	94	µb	µb	PROPN
ejpam-3441	393	95	)	)	PUNCT
ejpam-3441	393	96	k.	k.	PROPN
ejpam-3441	393	97	nasreen	nasreen	PROPN
ejpam-3441	393	98	et	et	PROPN
ejpam-3441	393	99	al	al	PROPN
ejpam-3441	393	100	.	.	PUNCT
ejpam-3441	393	101	/	/	SYM
ejpam-3441	393	102	eur	eur	PROPN
ejpam-3441	393	103	.	.	PUNCT
ejpam-3441	394	1	j.	j.	PROPN
ejpam-3441	394	2	pure	pure	PROPN
ejpam-3441	394	3	appl	appl	PROPN
ejpam-3441	394	4	.	.	PROPN
ejpam-3441	394	5	math	math	PROPN
ejpam-3441	394	6	,	,	PUNCT
ejpam-3441	394	7	12	12	NUM
ejpam-3441	394	8	(	(	PUNCT
ejpam-3441	394	9	3	3	NUM
ejpam-3441	394	10	)	)	PUNCT
ejpam-3441	394	11	(	(	PUNCT
ejpam-3441	394	12	2019	2019	NUM
ejpam-3441	394	13	)	)	PUNCT
ejpam-3441	394	14	,	,	PUNCT
ejpam-3441	394	15	906	906	NUM
ejpam-3441	394	16	-	-	SYM
ejpam-3441	394	17	943	943	NUM
ejpam-3441	394	18	923	923	NUM
ejpam-3441	394	19	⊆	⊆	NUM
ejpam-3441	394	20	(	(	PUNCT
ejpam-3441	394	21	µa)βα	µa)βα	NUM
ejpam-3441	394	22	◦	◦	NOUN
ejpam-3441	394	23	βα	βα	X
ejpam-3441	394	24	(	(	PUNCT
ejpam-3441	394	25	µb)βα	µb)βα	X
ejpam-3441	394	26	=	=	SYM
ejpam-3441	394	27	µa	µa	PROPN
ejpam-3441	394	28	◦	◦	NOUN
ejpam-3441	394	29	βα	βα	NOUN
ejpam-3441	394	30	µb	µb	PROPN
ejpam-3441	394	31	.	.	PUNCT
ejpam-3441	395	1	similarly	similarly	ADV
ejpam-3441	395	2	,	,	PUNCT
ejpam-3441	395	3	we	we	PRON
ejpam-3441	395	4	have	have	VERB
ejpam-3441	395	5	(	(	PUNCT
ejpam-3441	395	6	(	(	PUNCT
ejpam-3441	395	7	γa	γa	PROPN
ejpam-3441	395	8	◦	◦	PROPN
ejpam-3441	395	9	βα	βα	NOUN
ejpam-3441	395	10	γb	γb	PROPN
ejpam-3441	395	11	)	)	PUNCT
ejpam-3441	395	12	◦	◦	NOUN
ejpam-3441	395	13	βα	βα	NOUN
ejpam-3441	395	14	r	r	NOUN
ejpam-3441	395	15	)	)	PUNCT
ejpam-3441	395	16	◦	◦	NOUN
ejpam-3441	395	17	βα	βα	X
ejpam-3441	395	18	(	(	PUNCT
ejpam-3441	395	19	γa	γa	PROPN
ejpam-3441	395	20	◦	◦	PROPN
ejpam-3441	395	21	βα	βα	NOUN
ejpam-3441	395	22	γb	γb	PROPN
ejpam-3441	395	23	)	)	PUNCT
ejpam-3441	395	24	⊇	⊇	PROPN
ejpam-3441	395	25	γa	γa	PROPN
ejpam-3441	395	26	◦	◦	PROPN
ejpam-3441	395	27	βα	βα	NOUN
ejpam-3441	395	28	γb	γb	PROPN
ejpam-3441	395	29	.	.	PUNCT
ejpam-3441	396	1	therefore	therefore	ADV
ejpam-3441	396	2	a	a	DET
ejpam-3441	396	3	◦	◦	NOUN
ejpam-3441	396	4	βα	βα	NOUN
ejpam-3441	396	5	b	b	NOUN
ejpam-3441	396	6	is	be	AUX
ejpam-3441	396	7	an	an	DET
ejpam-3441	396	8	intuitionistic	intuitionistic	ADJ
ejpam-3441	396	9	fuzzy	fuzzy	ADJ
ejpam-3441	396	10	bi	bi	NOUN
ejpam-3441	396	11	-	-	NOUN
ejpam-3441	396	12	ideal	ideal	ADJ
ejpam-3441	396	13	with	with	ADP
ejpam-3441	396	14	thresholds	threshold	NOUN
ejpam-3441	396	15	(	(	PUNCT
ejpam-3441	396	16	α	α	X
ejpam-3441	396	17	,	,	PUNCT
ejpam-3441	396	18	β	β	X
ejpam-3441	396	19	]	]	PUNCT
ejpam-3441	396	20	of	of	ADP
ejpam-3441	396	21	r.	r.	PROPN
ejpam-3441	396	22	lemma	lemma	PROPN
ejpam-3441	396	23	11	11	NUM
ejpam-3441	396	24	.	.	PUNCT
ejpam-3441	397	1	every	every	DET
ejpam-3441	397	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	397	3	fuzzy	fuzzy	ADJ
ejpam-3441	397	4	ideal	ideal	NOUN
ejpam-3441	397	5	with	with	ADP
ejpam-3441	397	6	thresholds	threshold	NOUN
ejpam-3441	397	7	(	(	PUNCT
ejpam-3441	397	8	α	α	X
ejpam-3441	397	9	,	,	PUNCT
ejpam-3441	397	10	β	β	X
ejpam-3441	397	11	]	]	PUNCT
ejpam-3441	397	12	of	of	ADP
ejpam-3441	397	13	an	an	DET
ejpam-3441	397	14	la	la	ADJ
ejpam-3441	397	15	-	-	PUNCT
ejpam-3441	397	16	ring	ring	NOUN
ejpam-3441	397	17	r	r	NOUN
ejpam-3441	397	18	is	be	AUX
ejpam-3441	397	19	an	an	DET
ejpam-3441	397	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	397	21	fuzzy	fuzzy	ADJ
ejpam-3441	397	22	interior	interior	ADJ
ejpam-3441	397	23	ideal	ideal	NOUN
ejpam-3441	397	24	with	with	ADP
ejpam-3441	397	25	thresholds	threshold	NOUN
ejpam-3441	397	26	(	(	PUNCT
ejpam-3441	397	27	α	α	X
ejpam-3441	397	28	,	,	PUNCT
ejpam-3441	397	29	β	β	X
ejpam-3441	397	30	]	]	PUNCT
ejpam-3441	397	31	of	of	ADP
ejpam-3441	397	32	r.	r.	PROPN
ejpam-3441	397	33	the	the	DET
ejpam-3441	397	34	converse	converse	NOUN
ejpam-3441	397	35	is	be	AUX
ejpam-3441	397	36	not	not	PART
ejpam-3441	397	37	true	true	ADJ
ejpam-3441	397	38	in	in	ADP
ejpam-3441	397	39	general	general	ADJ
ejpam-3441	397	40	.	.	PUNCT
ejpam-3441	398	1	proof	proof	NOUN
ejpam-3441	398	2	.	.	PUNCT
ejpam-3441	399	1	let	let	VERB
ejpam-3441	399	2	a	a	DET
ejpam-3441	399	3	=	=	SYM
ejpam-3441	399	4	(	(	PUNCT
ejpam-3441	399	5	µa	µa	PROPN
ejpam-3441	399	6	,	,	PUNCT
ejpam-3441	399	7	γa	γa	PROPN
ejpam-3441	399	8	)	)	PUNCT
ejpam-3441	399	9	be	be	VERB
ejpam-3441	399	10	an	an	DET
ejpam-3441	399	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	399	12	fuzzy	fuzzy	ADJ
ejpam-3441	399	13	ideal	ideal	NOUN
ejpam-3441	399	14	with	with	ADP
ejpam-3441	399	15	thresholds	threshold	NOUN
ejpam-3441	399	16	(	(	PUNCT
ejpam-3441	399	17	α	α	X
ejpam-3441	399	18	,	,	PUNCT
ejpam-3441	399	19	β	β	X
ejpam-3441	399	20	]	]	PUNCT
ejpam-3441	399	21	of	of	ADP
ejpam-3441	399	22	an	an	DET
ejpam-3441	399	23	la	la	ADJ
ejpam-3441	399	24	-	-	PUNCT
ejpam-3441	399	25	ring	ring	NOUN
ejpam-3441	399	26	r	r	NOUN
ejpam-3441	399	27	and	and	CCONJ
ejpam-3441	399	28	x	x	NOUN
ejpam-3441	399	29	,	,	PUNCT
ejpam-3441	399	30	y	y	PROPN
ejpam-3441	399	31	,	,	PUNCT
ejpam-3441	399	32	z	z	PROPN
ejpam-3441	399	33	∈	∈	PROPN
ejpam-3441	399	34	r.	r.	NOUN
ejpam-3441	399	35	thus	thus	ADV
ejpam-3441	399	36	max{µa	max{µa	VERB
ejpam-3441	399	37	(	(	PUNCT
ejpam-3441	399	38	(	(	PUNCT
ejpam-3441	399	39	xy)z	xy)z	NUM
ejpam-3441	399	40	)	)	PUNCT
ejpam-3441	399	41	,	,	PUNCT
ejpam-3441	399	42	α	α	X
ejpam-3441	399	43	}	}	PUNCT
ejpam-3441	399	44	≥	≥	NOUN
ejpam-3441	399	45	min{µa	min{µa	X
ejpam-3441	399	46	(	(	PUNCT
ejpam-3441	399	47	xy	xy	PROPN
ejpam-3441	399	48	)	)	PUNCT
ejpam-3441	399	49	,	,	PUNCT
ejpam-3441	399	50	β	β	X
ejpam-3441	399	51	}	}	PUNCT
ejpam-3441	399	52	≥	≥	X
ejpam-3441	399	53	min{µa	min{µa	X
ejpam-3441	399	54	(	(	PUNCT
ejpam-3441	399	55	y	y	NOUN
ejpam-3441	399	56	)	)	PUNCT
ejpam-3441	399	57	,	,	PUNCT
ejpam-3441	399	58	β	β	X
ejpam-3441	399	59	}	}	PUNCT
ejpam-3441	399	60	and	and	CCONJ
ejpam-3441	399	61	min{γa	min{γa	X
ejpam-3441	399	62	(	(	PUNCT
ejpam-3441	399	63	(	(	PUNCT
ejpam-3441	399	64	xy)z	xy)z	NOUN
ejpam-3441	399	65	)	)	PUNCT
ejpam-3441	399	66	,	,	PUNCT
ejpam-3441	399	67	(	(	PUNCT
ejpam-3441	399	68	1−	1−	NUM
ejpam-3441	399	69	α	α	NOUN
ejpam-3441	399	70	)	)	PUNCT
ejpam-3441	399	71	}	}	PUNCT
ejpam-3441	399	72	≤	≤	NUM
ejpam-3441	399	73	max{γa	max{γa	NOUN
ejpam-3441	399	74	(	(	PUNCT
ejpam-3441	399	75	xy	xy	NOUN
ejpam-3441	399	76	)	)	PUNCT
ejpam-3441	399	77	,	,	PUNCT
ejpam-3441	399	78	(	(	PUNCT
ejpam-3441	399	79	1−	1−	NUM
ejpam-3441	399	80	β	β	NOUN
ejpam-3441	399	81	)	)	PUNCT
ejpam-3441	399	82	}	}	PUNCT
ejpam-3441	399	83	≤	≤	NUM
ejpam-3441	399	84	max{γa	max{γa	NOUN
ejpam-3441	399	85	(	(	PUNCT
ejpam-3441	399	86	y	y	NOUN
ejpam-3441	399	87	)	)	PUNCT
ejpam-3441	399	88	,	,	PUNCT
ejpam-3441	399	89	(	(	PUNCT
ejpam-3441	399	90	1−	1−	NUM
ejpam-3441	399	91	β	β	NOUN
ejpam-3441	399	92	)	)	PUNCT
ejpam-3441	399	93	}	}	PUNCT
ejpam-3441	399	94	.	.	PUNCT
ejpam-3441	400	1	hence	hence	ADV
ejpam-3441	400	2	a	a	PRON
ejpam-3441	400	3	is	be	AUX
ejpam-3441	400	4	an	an	DET
ejpam-3441	400	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	400	6	fuzzy	fuzzy	ADJ
ejpam-3441	400	7	interior	interior	ADJ
ejpam-3441	400	8	ideal	ideal	NOUN
ejpam-3441	400	9	with	with	ADP
ejpam-3441	400	10	thresholds	threshold	NOUN
ejpam-3441	400	11	(	(	PUNCT
ejpam-3441	400	12	α	α	X
ejpam-3441	400	13	,	,	PUNCT
ejpam-3441	400	14	β	β	X
ejpam-3441	400	15	]	]	PUNCT
ejpam-3441	400	16	of	of	ADP
ejpam-3441	400	17	r.	r.	PROPN
ejpam-3441	400	18	proposition	proposition	PROPN
ejpam-3441	400	19	2	2	X
ejpam-3441	400	20	.	.	PUNCT
ejpam-3441	401	1	let	let	VERB
ejpam-3441	401	2	a	a	PRON
ejpam-3441	401	3	=	=	SYM
ejpam-3441	401	4	(	(	PUNCT
ejpam-3441	401	5	µa	µa	PROPN
ejpam-3441	401	6	,	,	PUNCT
ejpam-3441	401	7	γa	γa	PROPN
ejpam-3441	401	8	)	)	PUNCT
ejpam-3441	401	9	be	be	VERB
ejpam-3441	401	10	an	an	DET
ejpam-3441	401	11	ifs	ifs	PROPN
ejpam-3441	401	12	of	of	ADP
ejpam-3441	401	13	an	an	DET
ejpam-3441	401	14	la	la	ADJ
ejpam-3441	401	15	-	-	PUNCT
ejpam-3441	401	16	ring	ring	NOUN
ejpam-3441	401	17	r	r	NOUN
ejpam-3441	401	18	with	with	ADP
ejpam-3441	401	19	left	left	ADJ
ejpam-3441	401	20	identity	identity	NOUN
ejpam-3441	402	1	e.	e.	PROPN
ejpam-3441	402	2	then	then	ADV
ejpam-3441	402	3	a	a	PRON
ejpam-3441	402	4	is	be	AUX
ejpam-3441	402	5	an	an	DET
ejpam-3441	402	6	intuitionistic	intuitionistic	ADJ
ejpam-3441	402	7	fuzzy	fuzzy	ADJ
ejpam-3441	402	8	ideal	ideal	NOUN
ejpam-3441	402	9	with	with	ADP
ejpam-3441	402	10	thresholds	threshold	NOUN
ejpam-3441	402	11	(	(	PUNCT
ejpam-3441	402	12	α	α	X
ejpam-3441	402	13	,	,	PUNCT
ejpam-3441	402	14	β	β	X
ejpam-3441	402	15	]	]	PUNCT
ejpam-3441	402	16	of	of	ADP
ejpam-3441	402	17	r	r	NOUN
ejpam-3441	402	18	if	if	SCONJ
ejpam-3441	403	1	and	and	CCONJ
ejpam-3441	403	2	only	only	ADV
ejpam-3441	403	3	if	if	SCONJ
ejpam-3441	403	4	a	a	PRON
ejpam-3441	403	5	is	be	AUX
ejpam-3441	403	6	an	an	DET
ejpam-3441	403	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	403	8	fuzzy	fuzzy	ADJ
ejpam-3441	403	9	interior	interior	ADJ
ejpam-3441	403	10	ideal	ideal	NOUN
ejpam-3441	403	11	with	with	ADP
ejpam-3441	403	12	thresholds	threshold	NOUN
ejpam-3441	403	13	(	(	PUNCT
ejpam-3441	403	14	α	α	X
ejpam-3441	403	15	,	,	PUNCT
ejpam-3441	403	16	β	β	X
ejpam-3441	403	17	]	]	PUNCT
ejpam-3441	403	18	of	of	ADP
ejpam-3441	403	19	r.	r.	PROPN
ejpam-3441	403	20	proof	proof	PROPN
ejpam-3441	403	21	.	.	PUNCT
ejpam-3441	404	1	suppose	suppose	VERB
ejpam-3441	404	2	that	that	SCONJ
ejpam-3441	404	3	a	a	DET
ejpam-3441	404	4	=	=	SYM
ejpam-3441	404	5	(	(	PUNCT
ejpam-3441	404	6	µa	µa	PROPN
ejpam-3441	404	7	,	,	PUNCT
ejpam-3441	404	8	γa	γa	PROPN
ejpam-3441	404	9	)	)	PUNCT
ejpam-3441	404	10	is	be	AUX
ejpam-3441	404	11	an	an	DET
ejpam-3441	404	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	404	13	fuzzy	fuzzy	ADJ
ejpam-3441	404	14	interior	interior	ADJ
ejpam-3441	404	15	ideal	ideal	NOUN
ejpam-3441	404	16	with	with	ADP
ejpam-3441	404	17	thresholds	threshold	NOUN
ejpam-3441	404	18	(	(	PUNCT
ejpam-3441	404	19	α	α	X
ejpam-3441	404	20	,	,	PUNCT
ejpam-3441	404	21	β	β	X
ejpam-3441	404	22	]	]	PUNCT
ejpam-3441	404	23	of	of	ADP
ejpam-3441	404	24	an	an	DET
ejpam-3441	404	25	la	la	ADJ
ejpam-3441	404	26	-	-	PUNCT
ejpam-3441	404	27	ring	ring	NOUN
ejpam-3441	404	28	r	r	NOUN
ejpam-3441	404	29	and	and	CCONJ
ejpam-3441	404	30	x	x	NOUN
ejpam-3441	404	31	,	,	PUNCT
ejpam-3441	404	32	y	y	PROPN
ejpam-3441	404	33	∈	∈	PROPN
ejpam-3441	404	34	r.	r.	PROPN
ejpam-3441	404	35	thus	thus	ADV
ejpam-3441	404	36	max{µa(xy	max{µa(xy	PROPN
ejpam-3441	404	37	)	)	PUNCT
ejpam-3441	404	38	,	,	PUNCT
ejpam-3441	404	39	α	α	X
ejpam-3441	404	40	}	}	PUNCT
ejpam-3441	404	41	=	=	SYM
ejpam-3441	404	42	max{µa((ex)y	max{µa((ex)y	PROPN
ejpam-3441	404	43	)	)	PUNCT
ejpam-3441	404	44	,	,	PUNCT
ejpam-3441	404	45	α	α	X
ejpam-3441	404	46	}	}	PUNCT
ejpam-3441	404	47	≥	≥	NOUN
ejpam-3441	404	48	min{µa(x	min{µa(x	NOUN
ejpam-3441	404	49	)	)	PUNCT
ejpam-3441	404	50	,	,	PUNCT
ejpam-3441	404	51	β	β	X
ejpam-3441	404	52	}	}	PUNCT
ejpam-3441	404	53	and	and	CCONJ
ejpam-3441	404	54	min{γa(xy	min{γa(xy	NUM
ejpam-3441	404	55	)	)	PUNCT
ejpam-3441	404	56	,	,	PUNCT
ejpam-3441	404	57	(	(	PUNCT
ejpam-3441	404	58	1−	1−	NUM
ejpam-3441	404	59	α	α	NOUN
ejpam-3441	404	60	)	)	PUNCT
ejpam-3441	404	61	}	}	PUNCT
ejpam-3441	404	62	=	=	SYM
ejpam-3441	404	63	min{γa((ex)y	min{γa((ex)y	PROPN
ejpam-3441	404	64	)	)	PUNCT
ejpam-3441	404	65	,	,	PUNCT
ejpam-3441	404	66	(	(	PUNCT
ejpam-3441	404	67	1−	1−	NUM
ejpam-3441	404	68	α	α	NOUN
ejpam-3441	404	69	)	)	PUNCT
ejpam-3441	404	70	}	}	PUNCT
ejpam-3441	404	71	≤	≤	NUM
ejpam-3441	404	72	max{γa(x	max{γa(x	NOUN
ejpam-3441	404	73	)	)	PUNCT
ejpam-3441	404	74	,	,	PUNCT
ejpam-3441	404	75	(	(	PUNCT
ejpam-3441	404	76	1−	1−	NUM
ejpam-3441	404	77	β	β	NOUN
ejpam-3441	404	78	)	)	PUNCT
ejpam-3441	404	79	}	}	PUNCT
ejpam-3441	404	80	.	.	PUNCT
ejpam-3441	405	1	so	so	ADV
ejpam-3441	405	2	a	a	PRON
ejpam-3441	405	3	is	be	AUX
ejpam-3441	405	4	an	an	DET
ejpam-3441	405	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	405	6	fuzzy	fuzzy	ADJ
ejpam-3441	405	7	right	right	ADJ
ejpam-3441	405	8	ideal	ideal	NOUN
ejpam-3441	405	9	with	with	ADP
ejpam-3441	405	10	thresholds	threshold	NOUN
ejpam-3441	405	11	(	(	PUNCT
ejpam-3441	405	12	α	α	X
ejpam-3441	405	13	,	,	PUNCT
ejpam-3441	405	14	β	β	X
ejpam-3441	405	15	]	]	PUNCT
ejpam-3441	405	16	of	of	ADP
ejpam-3441	405	17	r.	r.	PROPN
ejpam-3441	405	18	therefore	therefore	ADV
ejpam-3441	405	19	a	a	PRON
ejpam-3441	405	20	is	be	AUX
ejpam-3441	405	21	an	an	DET
ejpam-3441	405	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	405	23	fuzzy	fuzzy	ADJ
ejpam-3441	405	24	ideal	ideal	NOUN
ejpam-3441	405	25	with	with	ADP
ejpam-3441	405	26	thresholds	threshold	NOUN
ejpam-3441	405	27	(	(	PUNCT
ejpam-3441	405	28	α	α	X
ejpam-3441	405	29	,	,	PUNCT
ejpam-3441	405	30	β	β	X
ejpam-3441	405	31	]	]	PUNCT
ejpam-3441	405	32	of	of	ADP
ejpam-3441	405	33	r	r	NOUN
ejpam-3441	405	34	by	by	ADP
ejpam-3441	405	35	the	the	DET
ejpam-3441	405	36	lemma	lemma	PROPN
ejpam-3441	405	37	5	5	NUM
ejpam-3441	405	38	.	.	PUNCT
ejpam-3441	406	1	converse	converse	NOUN
ejpam-3441	406	2	is	be	AUX
ejpam-3441	406	3	true	true	ADJ
ejpam-3441	406	4	by	by	ADP
ejpam-3441	406	5	the	the	DET
ejpam-3441	406	6	lemma	lemma	PROPN
ejpam-3441	406	7	11	11	NUM
ejpam-3441	406	8	.	.	PUNCT
ejpam-3441	407	1	lemma	lemma	PROPN
ejpam-3441	407	2	12	12	NUM
ejpam-3441	407	3	.	.	PUNCT
ejpam-3441	408	1	every	every	DET
ejpam-3441	408	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	408	3	fuzzy	fuzzy	ADJ
ejpam-3441	408	4	left	left	NOUN
ejpam-3441	408	5	(	(	PUNCT
ejpam-3441	408	6	resp	resp	NOUN
ejpam-3441	408	7	.	.	PUNCT
ejpam-3441	409	1	right	right	ADJ
ejpam-3441	409	2	,	,	PUNCT
ejpam-3441	409	3	two	two	NUM
ejpam-3441	409	4	-	-	PUNCT
ejpam-3441	409	5	sided	sided	ADJ
ejpam-3441	409	6	)	)	PUNCT
ejpam-3441	409	7	ideal	ideal	NOUN
ejpam-3441	409	8	with	with	ADP
ejpam-3441	409	9	thresholds	threshold	NOUN
ejpam-3441	409	10	(	(	PUNCT
ejpam-3441	409	11	α	α	X
ejpam-3441	409	12	,	,	PUNCT
ejpam-3441	409	13	β	β	X
ejpam-3441	409	14	]	]	PUNCT
ejpam-3441	409	15	of	of	ADP
ejpam-3441	409	16	an	an	DET
ejpam-3441	409	17	la	la	ADJ
ejpam-3441	409	18	-	-	PUNCT
ejpam-3441	409	19	ring	ring	NOUN
ejpam-3441	409	20	r	r	NOUN
ejpam-3441	409	21	is	be	AUX
ejpam-3441	409	22	an	an	DET
ejpam-3441	409	23	intuitionistic	intuitionistic	ADJ
ejpam-3441	409	24	fuzzy	fuzzy	ADJ
ejpam-3441	409	25	bi	bi	NOUN
ejpam-3441	409	26	-	-	NOUN
ejpam-3441	409	27	ideal	ideal	ADJ
ejpam-3441	409	28	with	with	ADP
ejpam-3441	409	29	thresholds	threshold	NOUN
ejpam-3441	409	30	(	(	PUNCT
ejpam-3441	409	31	α	α	X
ejpam-3441	409	32	,	,	PUNCT
ejpam-3441	409	33	β	β	X
ejpam-3441	409	34	]	]	PUNCT
ejpam-3441	409	35	of	of	ADP
ejpam-3441	409	36	r.	r.	PROPN
ejpam-3441	409	37	the	the	DET
ejpam-3441	409	38	converse	converse	NOUN
ejpam-3441	409	39	is	be	AUX
ejpam-3441	409	40	not	not	PART
ejpam-3441	409	41	true	true	ADJ
ejpam-3441	409	42	in	in	ADP
ejpam-3441	409	43	general	general	ADJ
ejpam-3441	409	44	.	.	PUNCT
ejpam-3441	410	1	proof	proof	NOUN
ejpam-3441	410	2	.	.	PUNCT
ejpam-3441	411	1	assume	assume	VERB
ejpam-3441	411	2	that	that	SCONJ
ejpam-3441	411	3	a	a	DET
ejpam-3441	411	4	=	=	SYM
ejpam-3441	411	5	(	(	PUNCT
ejpam-3441	411	6	µa	µa	PROPN
ejpam-3441	411	7	,	,	PUNCT
ejpam-3441	411	8	γa	γa	PROPN
ejpam-3441	411	9	)	)	PUNCT
ejpam-3441	411	10	is	be	AUX
ejpam-3441	411	11	an	an	DET
ejpam-3441	411	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	411	13	fuzzy	fuzzy	ADJ
ejpam-3441	411	14	right	right	ADJ
ejpam-3441	411	15	ideal	ideal	NOUN
ejpam-3441	411	16	with	with	ADP
ejpam-3441	411	17	thresholds	threshold	NOUN
ejpam-3441	411	18	(	(	PUNCT
ejpam-3441	411	19	α	α	X
ejpam-3441	411	20	,	,	PUNCT
ejpam-3441	411	21	β	β	X
ejpam-3441	411	22	]	]	PUNCT
ejpam-3441	411	23	of	of	ADP
ejpam-3441	411	24	an	an	DET
ejpam-3441	411	25	la	la	ADJ
ejpam-3441	411	26	-	-	PUNCT
ejpam-3441	411	27	ring	ring	NOUN
ejpam-3441	411	28	r	r	NOUN
ejpam-3441	411	29	and	and	CCONJ
ejpam-3441	411	30	x	x	NOUN
ejpam-3441	411	31	,	,	PUNCT
ejpam-3441	411	32	y	y	PROPN
ejpam-3441	411	33	,	,	PUNCT
ejpam-3441	411	34	z	z	PROPN
ejpam-3441	411	35	∈	∈	PROPN
ejpam-3441	411	36	r.	r.	NOUN
ejpam-3441	411	37	thus	thus	ADV
ejpam-3441	411	38	max{µa	max{µa	VERB
ejpam-3441	411	39	(	(	PUNCT
ejpam-3441	411	40	(	(	PUNCT
ejpam-3441	411	41	xy)z	xy)z	NUM
ejpam-3441	411	42	)	)	PUNCT
ejpam-3441	411	43	,	,	PUNCT
ejpam-3441	411	44	α	α	X
ejpam-3441	411	45	}	}	PUNCT
ejpam-3441	411	46	≥	≥	NOUN
ejpam-3441	411	47	min{µa	min{µa	X
ejpam-3441	411	48	(	(	PUNCT
ejpam-3441	411	49	xy	xy	PROPN
ejpam-3441	411	50	)	)	PUNCT
ejpam-3441	411	51	,	,	PUNCT
ejpam-3441	411	52	β	β	X
ejpam-3441	411	53	}	}	PUNCT
ejpam-3441	411	54	≥	≥	X
ejpam-3441	411	55	min{µa	min{µa	X
ejpam-3441	411	56	(	(	PUNCT
ejpam-3441	411	57	x	x	NOUN
ejpam-3441	411	58	)	)	PUNCT
ejpam-3441	411	59	,	,	PUNCT
ejpam-3441	411	60	β	β	X
ejpam-3441	411	61	}	}	PUNCT
ejpam-3441	411	62	and	and	CCONJ
ejpam-3441	411	63	max{µa((xy)z	max{µa((xy)z	PROPN
ejpam-3441	411	64	)	)	PUNCT
ejpam-3441	411	65	,	,	PUNCT
ejpam-3441	411	66	α	α	X
ejpam-3441	411	67	}	}	PUNCT
ejpam-3441	411	68	=	=	SYM
ejpam-3441	411	69	max{µa((zy)x	max{µa((zy)x	NOUN
ejpam-3441	411	70	)	)	PUNCT
ejpam-3441	411	71	,	,	PUNCT
ejpam-3441	411	72	α	α	X
ejpam-3441	411	73	}	}	PUNCT
ejpam-3441	411	74	≥	≥	NOUN
ejpam-3441	411	75	min{µa(zy	min{µa(zy	NOUN
ejpam-3441	411	76	)	)	PUNCT
ejpam-3441	411	77	,	,	PUNCT
ejpam-3441	411	78	β	β	X
ejpam-3441	411	79	}	}	PUNCT
ejpam-3441	411	80	≥	≥	NUM
ejpam-3441	411	81	min{µa(z	min{µa(z	NOUN
ejpam-3441	411	82	)	)	PUNCT
ejpam-3441	411	83	,	,	PUNCT
ejpam-3441	411	84	β	β	X
ejpam-3441	411	85	}	}	PUNCT
ejpam-3441	411	86	.	.	PUNCT
ejpam-3441	412	1	this	this	PRON
ejpam-3441	412	2	implies	imply	VERB
ejpam-3441	412	3	that	that	DET
ejpam-3441	412	4	max{µa((xy)z	max{µa((xy)z	PROPN
ejpam-3441	412	5	)	)	PUNCT
ejpam-3441	412	6	,	,	PUNCT
ejpam-3441	412	7	α	α	X
ejpam-3441	412	8	}	}	PUNCT
ejpam-3441	412	9	≥	≥	NOUN
ejpam-3441	412	10	min{µa(x	min{µa(x	NOUN
ejpam-3441	412	11	)	)	PUNCT
ejpam-3441	412	12	,	,	PUNCT
ejpam-3441	412	13	µa(z	µa(z	NUM
ejpam-3441	412	14	)	)	PUNCT
ejpam-3441	412	15	,	,	PUNCT
ejpam-3441	412	16	β	β	X
ejpam-3441	412	17	}	}	PUNCT
ejpam-3441	412	18	.	.	PUNCT
ejpam-3441	413	1	similarly	similarly	ADV
ejpam-3441	413	2	,	,	PUNCT
ejpam-3441	413	3	we	we	PRON
ejpam-3441	413	4	have	have	VERB
ejpam-3441	413	5	min{γa((xy)z	min{γa((xy)z	NOUN
ejpam-3441	413	6	)	)	PUNCT
ejpam-3441	413	7	,	,	PUNCT
ejpam-3441	413	8	(	(	PUNCT
ejpam-3441	413	9	1−	1−	NUM
ejpam-3441	413	10	α	α	NOUN
ejpam-3441	413	11	)	)	PUNCT
ejpam-3441	413	12	}	}	PUNCT
ejpam-3441	413	13	≤	≤	NUM
ejpam-3441	413	14	max{γa(x	max{γa(x	NOUN
ejpam-3441	413	15	)	)	PUNCT
ejpam-3441	413	16	,	,	PUNCT
ejpam-3441	413	17	γa(z	γa(z	PROPN
ejpam-3441	413	18	)	)	PUNCT
ejpam-3441	413	19	,	,	PUNCT
ejpam-3441	413	20	(	(	PUNCT
ejpam-3441	413	21	1−	1−	NUM
ejpam-3441	413	22	β	β	NOUN
ejpam-3441	413	23	)	)	PUNCT
ejpam-3441	413	24	}	}	PUNCT
ejpam-3441	413	25	.	.	PUNCT
ejpam-3441	414	1	so	so	ADV
ejpam-3441	414	2	a	a	PRON
ejpam-3441	414	3	is	be	AUX
ejpam-3441	414	4	an	an	DET
ejpam-3441	414	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	414	6	fuzzy	fuzzy	ADJ
ejpam-3441	414	7	bi	bi	NOUN
ejpam-3441	414	8	-	-	NOUN
ejpam-3441	414	9	ideal	ideal	ADJ
ejpam-3441	414	10	with	with	ADP
ejpam-3441	414	11	thresholds	threshold	NOUN
ejpam-3441	414	12	(	(	PUNCT
ejpam-3441	414	13	α	α	X
ejpam-3441	414	14	,	,	PUNCT
ejpam-3441	414	15	β	β	X
ejpam-3441	414	16	]	]	PUNCT
ejpam-3441	414	17	of	of	ADP
ejpam-3441	414	18	r.	r.	PROPN
ejpam-3441	414	19	k.	k.	PROPN
ejpam-3441	414	20	nasreen	nasreen	PROPN
ejpam-3441	414	21	et	et	PROPN
ejpam-3441	414	22	al	al	PROPN
ejpam-3441	414	23	.	.	PUNCT
ejpam-3441	414	24	/	/	SYM
ejpam-3441	414	25	eur	eur	PROPN
ejpam-3441	414	26	.	.	PUNCT
ejpam-3441	415	1	j.	j.	PROPN
ejpam-3441	415	2	pure	pure	PROPN
ejpam-3441	415	3	appl	appl	PROPN
ejpam-3441	415	4	.	.	PROPN
ejpam-3441	415	5	math	math	PROPN
ejpam-3441	415	6	,	,	PUNCT
ejpam-3441	415	7	12	12	NUM
ejpam-3441	415	8	(	(	PUNCT
ejpam-3441	415	9	3	3	NUM
ejpam-3441	415	10	)	)	PUNCT
ejpam-3441	415	11	(	(	PUNCT
ejpam-3441	415	12	2019	2019	NUM
ejpam-3441	415	13	)	)	PUNCT
ejpam-3441	415	14	,	,	PUNCT
ejpam-3441	415	15	906	906	NUM
ejpam-3441	415	16	-	-	SYM
ejpam-3441	415	17	943	943	NUM
ejpam-3441	415	18	924	924	NUM
ejpam-3441	415	19	lemma	lemma	PROPN
ejpam-3441	415	20	13	13	NUM
ejpam-3441	415	21	.	.	PUNCT
ejpam-3441	416	1	every	every	DET
ejpam-3441	416	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	416	3	fuzzy	fuzzy	ADJ
ejpam-3441	416	4	bi	bi	NOUN
ejpam-3441	416	5	-	-	NOUN
ejpam-3441	416	6	ideal	ideal	ADJ
ejpam-3441	416	7	with	with	ADP
ejpam-3441	416	8	thresholds	threshold	NOUN
ejpam-3441	416	9	(	(	PUNCT
ejpam-3441	416	10	α	α	X
ejpam-3441	416	11	,	,	PUNCT
ejpam-3441	416	12	β	β	X
ejpam-3441	416	13	]	]	PUNCT
ejpam-3441	416	14	of	of	ADP
ejpam-3441	416	15	an	an	DET
ejpam-3441	416	16	la	la	ADJ
ejpam-3441	416	17	-	-	PUNCT
ejpam-3441	416	18	ring	ring	NOUN
ejpam-3441	416	19	r	r	NOUN
ejpam-3441	416	20	is	be	AUX
ejpam-3441	416	21	an	an	DET
ejpam-3441	416	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	416	23	fuzzy	fuzzy	ADJ
ejpam-3441	416	24	generalized	generalize	VERB
ejpam-3441	416	25	bi	bi	NOUN
ejpam-3441	416	26	-	-	NOUN
ejpam-3441	416	27	ideal	ideal	NOUN
ejpam-3441	416	28	with	with	ADP
ejpam-3441	416	29	thresholds	threshold	NOUN
ejpam-3441	416	30	(	(	PUNCT
ejpam-3441	416	31	α	α	X
ejpam-3441	416	32	,	,	PUNCT
ejpam-3441	416	33	β	β	X
ejpam-3441	416	34	]	]	PUNCT
ejpam-3441	416	35	of	of	ADP
ejpam-3441	416	36	r.	r.	PROPN
ejpam-3441	416	37	the	the	DET
ejpam-3441	416	38	converse	converse	NOUN
ejpam-3441	416	39	is	be	AUX
ejpam-3441	416	40	not	not	PART
ejpam-3441	416	41	true	true	ADJ
ejpam-3441	416	42	in	in	ADP
ejpam-3441	416	43	general	general	ADJ
ejpam-3441	416	44	.	.	PUNCT
ejpam-3441	417	1	proof	proof	NOUN
ejpam-3441	417	2	.	.	PUNCT
ejpam-3441	418	1	obvious	obvious	ADJ
ejpam-3441	418	2	.	.	PUNCT
ejpam-3441	419	1	lemma	lemma	PROPN
ejpam-3441	419	2	14	14	NUM
ejpam-3441	419	3	.	.	PUNCT
ejpam-3441	420	1	every	every	DET
ejpam-3441	420	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	420	3	fuzzy	fuzzy	ADJ
ejpam-3441	420	4	left	left	NOUN
ejpam-3441	420	5	(	(	PUNCT
ejpam-3441	420	6	resp	resp	NOUN
ejpam-3441	420	7	.	.	PUNCT
ejpam-3441	421	1	right	right	ADJ
ejpam-3441	421	2	,	,	PUNCT
ejpam-3441	421	3	two	two	NUM
ejpam-3441	421	4	-	-	PUNCT
ejpam-3441	421	5	sided	sided	ADJ
ejpam-3441	421	6	)	)	PUNCT
ejpam-3441	421	7	ideal	ideal	NOUN
ejpam-3441	421	8	with	with	ADP
ejpam-3441	421	9	thresholds	threshold	NOUN
ejpam-3441	421	10	(	(	PUNCT
ejpam-3441	421	11	α	α	X
ejpam-3441	421	12	,	,	PUNCT
ejpam-3441	421	13	β	β	X
ejpam-3441	421	14	]	]	PUNCT
ejpam-3441	421	15	of	of	ADP
ejpam-3441	421	16	an	an	DET
ejpam-3441	421	17	la	la	ADJ
ejpam-3441	421	18	-	-	PUNCT
ejpam-3441	421	19	ring	ring	NOUN
ejpam-3441	421	20	r	r	NOUN
ejpam-3441	421	21	is	be	AUX
ejpam-3441	421	22	an	an	DET
ejpam-3441	421	23	intuitionistic	intuitionistic	ADJ
ejpam-3441	421	24	fuzzy	fuzzy	ADJ
ejpam-3441	421	25	quasi	quasi	NOUN
ejpam-3441	421	26	-	-	NOUN
ejpam-3441	421	27	ideal	ideal	ADJ
ejpam-3441	421	28	with	with	ADP
ejpam-3441	421	29	thresholds	threshold	NOUN
ejpam-3441	421	30	(	(	PUNCT
ejpam-3441	421	31	α	α	X
ejpam-3441	421	32	,	,	PUNCT
ejpam-3441	421	33	β	β	X
ejpam-3441	421	34	]	]	PUNCT
ejpam-3441	421	35	of	of	ADP
ejpam-3441	421	36	r.	r.	PROPN
ejpam-3441	421	37	the	the	DET
ejpam-3441	421	38	converse	converse	NOUN
ejpam-3441	421	39	is	be	AUX
ejpam-3441	421	40	not	not	PART
ejpam-3441	421	41	true	true	ADJ
ejpam-3441	421	42	in	in	ADP
ejpam-3441	421	43	general	general	ADJ
ejpam-3441	421	44	.	.	PUNCT
ejpam-3441	422	1	proof	proof	NOUN
ejpam-3441	422	2	.	.	PUNCT
ejpam-3441	423	1	let	let	VERB
ejpam-3441	423	2	a	a	DET
ejpam-3441	423	3	=	=	SYM
ejpam-3441	423	4	(	(	PUNCT
ejpam-3441	423	5	µa	µa	PROPN
ejpam-3441	423	6	,	,	PUNCT
ejpam-3441	423	7	γa	γa	PROPN
ejpam-3441	423	8	)	)	PUNCT
ejpam-3441	423	9	be	be	VERB
ejpam-3441	423	10	an	an	DET
ejpam-3441	423	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	423	12	fuzzy	fuzzy	ADJ
ejpam-3441	423	13	left	leave	VERB
ejpam-3441	423	14	ideal	ideal	NOUN
ejpam-3441	423	15	with	with	ADP
ejpam-3441	423	16	thresholds	threshold	NOUN
ejpam-3441	423	17	(	(	PUNCT
ejpam-3441	423	18	α	α	X
ejpam-3441	423	19	,	,	PUNCT
ejpam-3441	423	20	β	β	X
ejpam-3441	423	21	]	]	PUNCT
ejpam-3441	423	22	of	of	ADP
ejpam-3441	423	23	an	an	DET
ejpam-3441	423	24	la	la	ADJ
ejpam-3441	423	25	-	-	PUNCT
ejpam-3441	423	26	ring	ring	NOUN
ejpam-3441	423	27	r.	r.	PROPN
ejpam-3441	423	28	thus	thus	ADV
ejpam-3441	423	29	max{µa(x	max{µa(x	NOUN
ejpam-3441	423	30	)	)	PUNCT
ejpam-3441	423	31	,	,	PUNCT
ejpam-3441	423	32	α	α	X
ejpam-3441	423	33	}	}	PUNCT
ejpam-3441	423	34	≥	≥	NOUN
ejpam-3441	423	35	min{(r	min{(r	NOUN
ejpam-3441	423	36	◦	◦	NOUN
ejpam-3441	423	37	µa)(x	µa)(x	NOUN
ejpam-3441	423	38	)	)	PUNCT
ejpam-3441	423	39	,	,	PUNCT
ejpam-3441	423	40	β	β	X
ejpam-3441	423	41	}	}	PUNCT
ejpam-3441	423	42	≥	≥	NOUN
ejpam-3441	423	43	min{(µa	min{(µa	X
ejpam-3441	423	44	◦	◦	NOUN
ejpam-3441	423	45	r)(x	r)(x	NOUN
ejpam-3441	423	46	)	)	PUNCT
ejpam-3441	423	47	,	,	PUNCT
ejpam-3441	423	48	(	(	PUNCT
ejpam-3441	423	49	r	r	NOUN
ejpam-3441	423	50	◦	◦	NOUN
ejpam-3441	423	51	µa)(x	µa)(x	NOUN
ejpam-3441	423	52	)	)	PUNCT
ejpam-3441	423	53	,	,	PUNCT
ejpam-3441	423	54	β	β	X
ejpam-3441	423	55	}	}	PUNCT
ejpam-3441	423	56	and	and	CCONJ
ejpam-3441	423	57	min{γa(x	min{γa(x	NOUN
ejpam-3441	423	58	)	)	PUNCT
ejpam-3441	423	59	,	,	PUNCT
ejpam-3441	423	60	(	(	PUNCT
ejpam-3441	423	61	1−	1−	NUM
ejpam-3441	423	62	α	α	NOUN
ejpam-3441	423	63	)	)	PUNCT
ejpam-3441	423	64	}	}	PUNCT
ejpam-3441	423	65	≤	≤	NUM
ejpam-3441	423	66	max{(r	max{(r	NOUN
ejpam-3441	423	67	◦	◦	PROPN
ejpam-3441	423	68	γa)(x	γa)(x	PROPN
ejpam-3441	423	69	)	)	PUNCT
ejpam-3441	423	70	,	,	PUNCT
ejpam-3441	423	71	(	(	PUNCT
ejpam-3441	423	72	1−	1−	NUM
ejpam-3441	423	73	β	β	NOUN
ejpam-3441	423	74	)	)	PUNCT
ejpam-3441	423	75	}	}	PUNCT
ejpam-3441	423	76	≤	≤	NUM
ejpam-3441	423	77	max{(γa	max{(γa	ADJ
ejpam-3441	423	78	◦	◦	NOUN
ejpam-3441	423	79	r)(x	r)(x	PROPN
ejpam-3441	423	80	)	)	PUNCT
ejpam-3441	423	81	,	,	PUNCT
ejpam-3441	423	82	(	(	PUNCT
ejpam-3441	423	83	r	r	NOUN
ejpam-3441	423	84	◦	◦	NOUN
ejpam-3441	423	85	γa)(x	γa)(x	PROPN
ejpam-3441	423	86	)	)	PUNCT
ejpam-3441	423	87	,	,	PUNCT
ejpam-3441	423	88	(	(	PUNCT
ejpam-3441	423	89	1−	1−	NUM
ejpam-3441	423	90	β	β	NOUN
ejpam-3441	423	91	)	)	PUNCT
ejpam-3441	423	92	}	}	PUNCT
ejpam-3441	423	93	.	.	PUNCT
ejpam-3441	424	1	hence	hence	ADV
ejpam-3441	424	2	a	a	PRON
ejpam-3441	424	3	is	be	AUX
ejpam-3441	424	4	an	an	DET
ejpam-3441	424	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	424	6	fuzzy	fuzzy	ADJ
ejpam-3441	424	7	quasi	quasi	NOUN
ejpam-3441	424	8	-	-	NOUN
ejpam-3441	424	9	ideal	ideal	ADJ
ejpam-3441	424	10	with	with	ADP
ejpam-3441	424	11	thresholds	threshold	NOUN
ejpam-3441	424	12	(	(	PUNCT
ejpam-3441	424	13	α	α	X
ejpam-3441	424	14	,	,	PUNCT
ejpam-3441	424	15	β	β	X
ejpam-3441	424	16	]	]	PUNCT
ejpam-3441	424	17	of	of	ADP
ejpam-3441	424	18	r.	r.	PROPN
ejpam-3441	424	19	proposition	proposition	PROPN
ejpam-3441	424	20	3	3	X
ejpam-3441	424	21	.	.	PUNCT
ejpam-3441	425	1	every	every	DET
ejpam-3441	425	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	425	3	fuzzy	fuzzy	ADJ
ejpam-3441	425	4	quasi	quasi	NOUN
ejpam-3441	425	5	-	-	NOUN
ejpam-3441	425	6	ideal	ideal	ADJ
ejpam-3441	425	7	with	with	ADP
ejpam-3441	425	8	thresholds	threshold	NOUN
ejpam-3441	425	9	(	(	PUNCT
ejpam-3441	425	10	α	α	X
ejpam-3441	425	11	,	,	PUNCT
ejpam-3441	425	12	β	β	X
ejpam-3441	425	13	]	]	PUNCT
ejpam-3441	425	14	of	of	ADP
ejpam-3441	425	15	an	an	DET
ejpam-3441	425	16	la	la	ADJ
ejpam-3441	425	17	-	-	PUNCT
ejpam-3441	425	18	ring	ring	NOUN
ejpam-3441	425	19	r	r	NOUN
ejpam-3441	425	20	is	be	AUX
ejpam-3441	425	21	an	an	DET
ejpam-3441	425	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	425	23	fuzzy	fuzzy	ADJ
ejpam-3441	425	24	la	la	NOUN
ejpam-3441	425	25	-	-	PUNCT
ejpam-3441	425	26	subring	subre	VERB
ejpam-3441	425	27	with	with	ADP
ejpam-3441	425	28	thresholds	threshold	NOUN
ejpam-3441	425	29	(	(	PUNCT
ejpam-3441	425	30	α	α	X
ejpam-3441	425	31	,	,	PUNCT
ejpam-3441	425	32	β	β	X
ejpam-3441	425	33	]	]	PUNCT
ejpam-3441	425	34	of	of	ADP
ejpam-3441	425	35	r.	r.	PROPN
ejpam-3441	425	36	proof	proof	PROPN
ejpam-3441	425	37	.	.	PUNCT
ejpam-3441	426	1	suppose	suppose	VERB
ejpam-3441	426	2	that	that	SCONJ
ejpam-3441	426	3	a	a	DET
ejpam-3441	426	4	=	=	SYM
ejpam-3441	426	5	(	(	PUNCT
ejpam-3441	426	6	µa	µa	PROPN
ejpam-3441	426	7	,	,	PUNCT
ejpam-3441	426	8	γa	γa	PROPN
ejpam-3441	426	9	)	)	PUNCT
ejpam-3441	426	10	is	be	AUX
ejpam-3441	426	11	an	an	DET
ejpam-3441	426	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	426	13	fuzzy	fuzzy	ADJ
ejpam-3441	426	14	quasi	quasi	NOUN
ejpam-3441	426	15	-	-	NOUN
ejpam-3441	426	16	ideal	ideal	ADJ
ejpam-3441	426	17	with	with	ADP
ejpam-3441	426	18	thresholds	threshold	NOUN
ejpam-3441	426	19	(	(	PUNCT
ejpam-3441	426	20	α	α	X
ejpam-3441	426	21	,	,	PUNCT
ejpam-3441	426	22	β	β	X
ejpam-3441	426	23	]	]	PUNCT
ejpam-3441	426	24	of	of	ADP
ejpam-3441	426	25	an	an	DET
ejpam-3441	426	26	la	la	ADJ
ejpam-3441	426	27	-	-	PUNCT
ejpam-3441	426	28	ring	ring	NOUN
ejpam-3441	426	29	r.	r.	PROPN
ejpam-3441	426	30	since	since	SCONJ
ejpam-3441	426	31	µa	µa	NOUN
ejpam-3441	426	32	◦	◦	NOUN
ejpam-3441	426	33	βαµa	βαµa	ADJ
ejpam-3441	426	34	⊆	⊆	NUM
ejpam-3441	426	35	µa	µa	NOUN
ejpam-3441	426	36	◦	◦	NOUN
ejpam-3441	426	37	βαr	βαr	NOUN
ejpam-3441	426	38	and	and	CCONJ
ejpam-3441	426	39	µa	µa	ADP
ejpam-3441	426	40	◦	◦	NOUN
ejpam-3441	426	41	βαµa	βαµa	ADJ
ejpam-3441	426	42	⊆	⊆	NUM
ejpam-3441	426	43	r	r	NOUN
ejpam-3441	426	44	◦	◦	NOUN
ejpam-3441	426	45	βαµa	βαµa	NOUN
ejpam-3441	426	46	,	,	PUNCT
ejpam-3441	426	47	this	this	PRON
ejpam-3441	426	48	implies	imply	VERB
ejpam-3441	426	49	that	that	SCONJ
ejpam-3441	426	50	µa	µa	ADP
ejpam-3441	426	51	◦	◦	NOUN
ejpam-3441	426	52	βα	βα	NOUN
ejpam-3441	426	53	µa	µa	NOUN
ejpam-3441	426	54	⊆	⊆	NUM
ejpam-3441	426	55	µa	µa	NOUN
ejpam-3441	426	56	◦	◦	NOUN
ejpam-3441	426	57	βαr∧r	βαr∧r	NOUN
ejpam-3441	426	58	◦	◦	NOUN
ejpam-3441	426	59	βα	βα	NOUN
ejpam-3441	426	60	µa	µa	ADJ
ejpam-3441	426	61	⊆	⊆	NUM
ejpam-3441	426	62	(	(	PUNCT
ejpam-3441	426	63	µa)βα	µa)βα	NUM
ejpam-3441	426	64	.	.	PUNCT
ejpam-3441	427	1	similarly	similarly	ADV
ejpam-3441	427	2	we	we	PRON
ejpam-3441	427	3	have	have	VERB
ejpam-3441	427	4	,	,	PUNCT
ejpam-3441	427	5	γa	γa	PROPN
ejpam-3441	427	6	◦	◦	PROPN
ejpam-3441	427	7	βα	βα	X
ejpam-3441	427	8	γa	γa	PROPN
ejpam-3441	427	9	⊇	⊇	PROPN
ejpam-3441	427	10	γa	γa	PROPN
ejpam-3441	427	11	◦	◦	PROPN
ejpam-3441	427	12	βαr∨r	βαr∨r	PROPN
ejpam-3441	427	13	◦	◦	NOUN
ejpam-3441	427	14	βα	βα	X
ejpam-3441	427	15	γa	γa	PROPN
ejpam-3441	427	16	⊇	⊇	PROPN
ejpam-3441	427	17	(	(	PUNCT
ejpam-3441	427	18	γa)βα	γa)βα	X
ejpam-3441	427	19	.	.	PUNCT
ejpam-3441	428	1	therefore	therefore	ADV
ejpam-3441	428	2	a	a	PRON
ejpam-3441	428	3	is	be	AUX
ejpam-3441	428	4	an	an	DET
ejpam-3441	428	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	428	6	fuzzy	fuzzy	ADJ
ejpam-3441	428	7	la	la	NOUN
ejpam-3441	428	8	-	-	PUNCT
ejpam-3441	428	9	subring	subre	VERB
ejpam-3441	428	10	with	with	ADP
ejpam-3441	428	11	thresholds	threshold	NOUN
ejpam-3441	428	12	(	(	PUNCT
ejpam-3441	428	13	α	α	X
ejpam-3441	428	14	,	,	PUNCT
ejpam-3441	428	15	β	β	X
ejpam-3441	428	16	]	]	PUNCT
ejpam-3441	428	17	of	of	ADP
ejpam-3441	428	18	r.	r.	PROPN
ejpam-3441	428	19	proposition	proposition	PROPN
ejpam-3441	428	20	4	4	NUM
ejpam-3441	428	21	.	.	PUNCT
ejpam-3441	429	1	let	let	VERB
ejpam-3441	429	2	a	a	PRON
ejpam-3441	429	3	=	=	SYM
ejpam-3441	429	4	(	(	PUNCT
ejpam-3441	429	5	µa	µa	PROPN
ejpam-3441	429	6	,	,	PUNCT
ejpam-3441	429	7	γa	γa	PROPN
ejpam-3441	429	8	)	)	PUNCT
ejpam-3441	429	9	be	be	VERB
ejpam-3441	429	10	an	an	DET
ejpam-3441	429	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	429	12	fuzzy	fuzzy	ADJ
ejpam-3441	429	13	right	right	ADJ
ejpam-3441	429	14	ideal	ideal	NOUN
ejpam-3441	429	15	with	with	ADP
ejpam-3441	429	16	thresholds	threshold	NOUN
ejpam-3441	429	17	(	(	PUNCT
ejpam-3441	429	18	α	α	X
ejpam-3441	429	19	,	,	PUNCT
ejpam-3441	429	20	β	β	X
ejpam-3441	429	21	]	]	PUNCT
ejpam-3441	429	22	and	and	CCONJ
ejpam-3441	429	23	b	b	X
ejpam-3441	429	24	=	=	SYM
ejpam-3441	429	25	(	(	PUNCT
ejpam-3441	429	26	µb	µb	PROPN
ejpam-3441	429	27	,	,	PUNCT
ejpam-3441	429	28	γb	γb	PROPN
ejpam-3441	429	29	)	)	PUNCT
ejpam-3441	429	30	be	be	VERB
ejpam-3441	429	31	an	an	DET
ejpam-3441	429	32	intuitionistic	intuitionistic	ADJ
ejpam-3441	429	33	fuzzy	fuzzy	ADJ
ejpam-3441	429	34	left	leave	VERB
ejpam-3441	429	35	ideal	ideal	NOUN
ejpam-3441	429	36	with	with	ADP
ejpam-3441	429	37	thresholds	threshold	NOUN
ejpam-3441	429	38	(	(	PUNCT
ejpam-3441	429	39	α	α	X
ejpam-3441	429	40	,	,	PUNCT
ejpam-3441	429	41	β	β	X
ejpam-3441	429	42	]	]	PUNCT
ejpam-3441	429	43	of	of	ADP
ejpam-3441	429	44	an	an	DET
ejpam-3441	429	45	la	la	ADJ
ejpam-3441	429	46	-	-	PUNCT
ejpam-3441	429	47	ring	ring	NOUN
ejpam-3441	429	48	r	r	NOUN
ejpam-3441	429	49	,	,	PUNCT
ejpam-3441	429	50	respectively	respectively	ADV
ejpam-3441	429	51	.	.	PUNCT
ejpam-3441	430	1	then	then	ADV
ejpam-3441	430	2	a∧βαb	a∧βαb	PRON
ejpam-3441	430	3	is	be	AUX
ejpam-3441	430	4	an	an	DET
ejpam-3441	430	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	430	6	fuzzy	fuzzy	ADJ
ejpam-3441	430	7	quasi	quasi	NOUN
ejpam-3441	430	8	-	-	NOUN
ejpam-3441	430	9	ideal	ideal	ADJ
ejpam-3441	430	10	with	with	ADP
ejpam-3441	430	11	thresholds	threshold	NOUN
ejpam-3441	430	12	(	(	PUNCT
ejpam-3441	430	13	α	α	X
ejpam-3441	430	14	,	,	PUNCT
ejpam-3441	430	15	β	β	X
ejpam-3441	430	16	]	]	PUNCT
ejpam-3441	430	17	of	of	ADP
ejpam-3441	430	18	r.	r.	PROPN
ejpam-3441	430	19	proof	proof	NOUN
ejpam-3441	430	20	.	.	PUNCT
ejpam-3441	431	1	we	we	PRON
ejpam-3441	431	2	have	have	VERB
ejpam-3441	431	3	to	to	PART
ejpam-3441	431	4	show	show	VERB
ejpam-3441	431	5	that	that	SCONJ
ejpam-3441	431	6	a∧βαb	a∧βαb	PRON
ejpam-3441	431	7	is	be	AUX
ejpam-3441	431	8	an	an	DET
ejpam-3441	431	9	intuitionistic	intuitionistic	ADJ
ejpam-3441	431	10	fuzzy	fuzzy	ADJ
ejpam-3441	431	11	quasi	quasi	NOUN
ejpam-3441	431	12	-	-	NOUN
ejpam-3441	431	13	ideal	ideal	ADJ
ejpam-3441	431	14	with	with	ADP
ejpam-3441	431	15	thresholds	threshold	NOUN
ejpam-3441	431	16	(	(	PUNCT
ejpam-3441	431	17	α	α	X
ejpam-3441	431	18	,	,	PUNCT
ejpam-3441	431	19	β	β	X
ejpam-3441	431	20	]	]	PUNCT
ejpam-3441	431	21	of	of	ADP
ejpam-3441	431	22	an	an	DET
ejpam-3441	431	23	la	la	ADJ
ejpam-3441	431	24	-	-	PUNCT
ejpam-3441	431	25	ring	ring	NOUN
ejpam-3441	431	26	r.	r.	NOUN
ejpam-3441	431	27	since	since	SCONJ
ejpam-3441	431	28	max{(µa	max{(µa	VERB
ejpam-3441	431	29	∧βα	∧βα	ADJ
ejpam-3441	431	30	µb)(x−	µb)(x−	ADJ
ejpam-3441	431	31	y	y	NOUN
ejpam-3441	431	32	)	)	PUNCT
ejpam-3441	431	33	,	,	PUNCT
ejpam-3441	431	34	α	α	PROPN
ejpam-3441	431	35	}	}	PUNCT
ejpam-3441	431	36	≥	≥	NOUN
ejpam-3441	431	37	min{(µa	min{(µa	VERB
ejpam-3441	431	38	∧βα	∧βα	ADJ
ejpam-3441	431	39	µb)(x	µb)(x	NOUN
ejpam-3441	431	40	)	)	PUNCT
ejpam-3441	431	41	,	,	PUNCT
ejpam-3441	431	42	(	(	PUNCT
ejpam-3441	431	43	µa	µa	ADP
ejpam-3441	431	44	∧βα	∧βα	ADJ
ejpam-3441	431	45	µb)(y	µb)(y	PROPN
ejpam-3441	431	46	)	)	PUNCT
ejpam-3441	431	47	,	,	PUNCT
ejpam-3441	431	48	β	β	X
ejpam-3441	431	49	}	}	PUNCT
ejpam-3441	431	50	and	and	CCONJ
ejpam-3441	431	51	min{(γa	min{(γa	ADJ
ejpam-3441	431	52	∨βα	∨βα	ADJ
ejpam-3441	431	53	γb)(x−	γb)(x−	ADJ
ejpam-3441	431	54	y	y	PROPN
ejpam-3441	431	55	)	)	PUNCT
ejpam-3441	431	56	,	,	PUNCT
ejpam-3441	431	57	(	(	PUNCT
ejpam-3441	431	58	1−	1−	NUM
ejpam-3441	431	59	α	α	NOUN
ejpam-3441	431	60	)	)	PUNCT
ejpam-3441	431	61	}	}	PUNCT
ejpam-3441	431	62	≤	≤	NOUN
ejpam-3441	432	1	max{(γa	max{(γa	ADJ
ejpam-3441	432	2	∨βα	∨βα	ADJ
ejpam-3441	432	3	γb)(x	γb)(x	PROPN
ejpam-3441	432	4	)	)	PUNCT
ejpam-3441	432	5	,	,	PUNCT
ejpam-3441	432	6	(	(	PUNCT
ejpam-3441	432	7	γa	γa	PROPN
ejpam-3441	432	8	∨βα	∨βα	VERB
ejpam-3441	432	9	γb)(y	γb)(y	PROPN
ejpam-3441	432	10	)	)	PUNCT
ejpam-3441	432	11	,	,	PUNCT
ejpam-3441	432	12	(	(	PUNCT
ejpam-3441	432	13	1−	1−	NUM
ejpam-3441	432	14	β	β	NOUN
ejpam-3441	432	15	)	)	PUNCT
ejpam-3441	432	16	}	}	PUNCT
ejpam-3441	432	17	,	,	PUNCT
ejpam-3441	432	18	by	by	ADP
ejpam-3441	432	19	the	the	DET
ejpam-3441	432	20	lemma	lemma	PROPN
ejpam-3441	432	21	3	3	NUM
ejpam-3441	432	22	and	and	CCONJ
ejpam-3441	432	23	(	(	PUNCT
ejpam-3441	432	24	(	(	PUNCT
ejpam-3441	432	25	µa	µa	ADP
ejpam-3441	432	26	∧βα	∧βα	ADJ
ejpam-3441	432	27	µb	µb	NOUN
ejpam-3441	432	28	)	)	PUNCT
ejpam-3441	432	29	◦	◦	NOUN
ejpam-3441	432	30	βα	βα	NOUN
ejpam-3441	432	31	r	r	NOUN
ejpam-3441	432	32	)	)	PUNCT
ejpam-3441	432	33	∧	∧	NOUN
ejpam-3441	432	34	(	(	PUNCT
ejpam-3441	432	35	r	r	NOUN
ejpam-3441	432	36	◦	◦	NOUN
ejpam-3441	432	37	βα	βα	X
ejpam-3441	432	38	(	(	PUNCT
ejpam-3441	432	39	µa	µa	ADP
ejpam-3441	432	40	∧βα	∧βα	ADJ
ejpam-3441	432	41	µb	µb	NOUN
ejpam-3441	432	42	)	)	PUNCT
ejpam-3441	432	43	)	)	PUNCT
ejpam-3441	432	44	k.	k.	PROPN
ejpam-3441	432	45	nasreen	nasreen	PROPN
ejpam-3441	433	1	et	et	PROPN
ejpam-3441	433	2	al	al	PROPN
ejpam-3441	433	3	.	.	PUNCT
ejpam-3441	433	4	/	/	SYM
ejpam-3441	433	5	eur	eur	PROPN
ejpam-3441	433	6	.	.	PUNCT
ejpam-3441	434	1	j.	j.	PROPN
ejpam-3441	434	2	pure	pure	PROPN
ejpam-3441	434	3	appl	appl	PROPN
ejpam-3441	434	4	.	.	PROPN
ejpam-3441	434	5	math	math	PROPN
ejpam-3441	434	6	,	,	PUNCT
ejpam-3441	434	7	12	12	NUM
ejpam-3441	434	8	(	(	PUNCT
ejpam-3441	434	9	3	3	NUM
ejpam-3441	434	10	)	)	PUNCT
ejpam-3441	434	11	(	(	PUNCT
ejpam-3441	434	12	2019	2019	NUM
ejpam-3441	434	13	)	)	PUNCT
ejpam-3441	434	14	,	,	PUNCT
ejpam-3441	434	15	906	906	NUM
ejpam-3441	434	16	-	-	SYM
ejpam-3441	434	17	943	943	NUM
ejpam-3441	434	18	925	925	NUM
ejpam-3441	434	19	⊆	⊆	NUM
ejpam-3441	434	20	(	(	PUNCT
ejpam-3441	434	21	µa	µa	ADP
ejpam-3441	434	22	◦	◦	NOUN
ejpam-3441	434	23	βα	βα	NOUN
ejpam-3441	434	24	r	r	NOUN
ejpam-3441	434	25	)	)	PUNCT
ejpam-3441	434	26	∧	∧	NOUN
ejpam-3441	434	27	(	(	PUNCT
ejpam-3441	434	28	r	r	NOUN
ejpam-3441	434	29	◦	◦	NOUN
ejpam-3441	434	30	βα	βα	NOUN
ejpam-3441	434	31	µb	µb	NOUN
ejpam-3441	434	32	)	)	PUNCT
ejpam-3441	434	33	⊆	⊆	NUM
ejpam-3441	434	34	(	(	PUNCT
ejpam-3441	434	35	µa)βα	µa)βα	NUM
ejpam-3441	434	36	∧	∧	PROPN
ejpam-3441	434	37	(	(	PUNCT
ejpam-3441	434	38	µb)βα	µb)βα	X
ejpam-3441	434	39	=	=	SYM
ejpam-3441	434	40	µa	µa	NOUN
ejpam-3441	434	41	∧βα	∧βα	PROPN
ejpam-3441	434	42	µb	µb	VERB
ejpam-3441	434	43	and	and	CCONJ
ejpam-3441	434	44	(	(	PUNCT
ejpam-3441	434	45	(	(	PUNCT
ejpam-3441	434	46	γa	γa	PROPN
ejpam-3441	434	47	∨βα	∨βα	PROPN
ejpam-3441	434	48	γb	γb	PROPN
ejpam-3441	434	49	)	)	PUNCT
ejpam-3441	434	50	◦	◦	NOUN
ejpam-3441	434	51	βα	βα	NOUN
ejpam-3441	434	52	r	r	NOUN
ejpam-3441	434	53	)	)	PUNCT
ejpam-3441	434	54	∨	∨	NOUN
ejpam-3441	434	55	(	(	PUNCT
ejpam-3441	434	56	r	r	NOUN
ejpam-3441	434	57	◦	◦	NOUN
ejpam-3441	434	58	βα	βα	X
ejpam-3441	434	59	(	(	PUNCT
ejpam-3441	434	60	γa	γa	PROPN
ejpam-3441	434	61	∨βα	∨βα	PROPN
ejpam-3441	434	62	γb	γb	PROPN
ejpam-3441	434	63	)	)	PUNCT
ejpam-3441	434	64	)	)	PUNCT
ejpam-3441	434	65	⊇	⊇	NOUN
ejpam-3441	434	66	(	(	PUNCT
ejpam-3441	434	67	γa	γa	PROPN
ejpam-3441	434	68	◦	◦	PROPN
ejpam-3441	434	69	βα	βα	X
ejpam-3441	434	70	r	r	NOUN
ejpam-3441	434	71	)	)	PUNCT
ejpam-3441	434	72	∨	∨	NOUN
ejpam-3441	434	73	(	(	PUNCT
ejpam-3441	434	74	r	r	NOUN
ejpam-3441	434	75	◦	◦	NOUN
ejpam-3441	434	76	βα	βα	NOUN
ejpam-3441	434	77	γb	γb	NOUN
ejpam-3441	434	78	)	)	PUNCT
ejpam-3441	434	79	⊇	⊇	NOUN
ejpam-3441	434	80	(	(	PUNCT
ejpam-3441	434	81	γa)βα	γa)βα	X
ejpam-3441	434	82	∨	∨	X
ejpam-3441	434	83	(	(	PUNCT
ejpam-3441	434	84	γb)βα	γb)βα	X
ejpam-3441	434	85	=	=	SYM
ejpam-3441	434	86	γa	γa	VERB
ejpam-3441	434	87	∧βα	∧βα	ADJ
ejpam-3441	434	88	γb	γb	VERB
ejpam-3441	434	89	.	.	PUNCT
ejpam-3441	435	1	thus	thus	ADV
ejpam-3441	435	2	a	a	DET
ejpam-3441	435	3	∧βα	∧βα	PROPN
ejpam-3441	435	4	b	b	NOUN
ejpam-3441	435	5	is	be	AUX
ejpam-3441	435	6	an	an	DET
ejpam-3441	435	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	435	8	fuzzy	fuzzy	ADJ
ejpam-3441	435	9	quasi	quasi	NOUN
ejpam-3441	435	10	-	-	NOUN
ejpam-3441	435	11	ideal	ideal	ADJ
ejpam-3441	435	12	with	with	ADP
ejpam-3441	435	13	thresholds	threshold	NOUN
ejpam-3441	435	14	(	(	PUNCT
ejpam-3441	435	15	α	α	X
ejpam-3441	435	16	,	,	PUNCT
ejpam-3441	435	17	β	β	X
ejpam-3441	435	18	]	]	PUNCT
ejpam-3441	435	19	of	of	ADP
ejpam-3441	435	20	r.	r.	PROPN
ejpam-3441	435	21	lemma	lemma	PROPN
ejpam-3441	435	22	15	15	NUM
ejpam-3441	435	23	.	.	PUNCT
ejpam-3441	436	1	let	let	VERB
ejpam-3441	436	2	r	r	PRON
ejpam-3441	436	3	be	be	AUX
ejpam-3441	436	4	an	an	DET
ejpam-3441	436	5	la	la	NOUN
ejpam-3441	436	6	-	-	NOUN
ejpam-3441	436	7	ring	ring	NOUN
ejpam-3441	436	8	with	with	ADP
ejpam-3441	436	9	left	left	ADJ
ejpam-3441	436	10	identity	identity	NOUN
ejpam-3441	436	11	e	e	NOUN
ejpam-3441	436	12	,	,	PUNCT
ejpam-3441	436	13	such	such	ADJ
ejpam-3441	436	14	that	that	SCONJ
ejpam-3441	436	15	(	(	PUNCT
ejpam-3441	436	16	xe)r	xe)r	PROPN
ejpam-3441	436	17	=	=	SYM
ejpam-3441	436	18	xr	xr	PROPN
ejpam-3441	436	19	for	for	ADP
ejpam-3441	436	20	all	all	DET
ejpam-3441	436	21	x	x	PROPN
ejpam-3441	436	22	∈	∈	PROPN
ejpam-3441	436	23	r.	r.	NOUN
ejpam-3441	436	24	then	then	ADV
ejpam-3441	436	25	every	every	DET
ejpam-3441	436	26	intuitionistic	intuitionistic	ADJ
ejpam-3441	436	27	fuzzy	fuzzy	ADJ
ejpam-3441	436	28	quasi	quasi	NOUN
ejpam-3441	436	29	-	-	NOUN
ejpam-3441	436	30	ideal	ideal	ADJ
ejpam-3441	436	31	with	with	ADP
ejpam-3441	436	32	thresholds	threshold	NOUN
ejpam-3441	436	33	(	(	PUNCT
ejpam-3441	436	34	α	α	X
ejpam-3441	436	35	,	,	PUNCT
ejpam-3441	436	36	β	β	X
ejpam-3441	436	37	]	]	PUNCT
ejpam-3441	436	38	of	of	ADP
ejpam-3441	436	39	r	r	NOUN
ejpam-3441	436	40	is	be	AUX
ejpam-3441	436	41	an	an	DET
ejpam-3441	436	42	intuitionistic	intuitionistic	ADJ
ejpam-3441	436	43	fuzzy	fuzzy	ADJ
ejpam-3441	436	44	bi	bi	NOUN
ejpam-3441	436	45	-	-	NOUN
ejpam-3441	436	46	ideal	ideal	ADJ
ejpam-3441	436	47	with	with	ADP
ejpam-3441	436	48	thresholds	threshold	NOUN
ejpam-3441	436	49	(	(	PUNCT
ejpam-3441	436	50	α	α	X
ejpam-3441	436	51	,	,	PUNCT
ejpam-3441	436	52	β	β	X
ejpam-3441	436	53	]	]	PUNCT
ejpam-3441	436	54	of	of	ADP
ejpam-3441	436	55	r.	r.	PROPN
ejpam-3441	436	56	proof	proof	PROPN
ejpam-3441	436	57	.	.	PUNCT
ejpam-3441	437	1	assume	assume	VERB
ejpam-3441	437	2	that	that	SCONJ
ejpam-3441	437	3	a	a	DET
ejpam-3441	437	4	=	=	SYM
ejpam-3441	437	5	(	(	PUNCT
ejpam-3441	437	6	µa	µa	PROPN
ejpam-3441	437	7	,	,	PUNCT
ejpam-3441	437	8	γa	γa	PROPN
ejpam-3441	437	9	)	)	PUNCT
ejpam-3441	437	10	is	be	AUX
ejpam-3441	437	11	an	an	DET
ejpam-3441	437	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	437	13	fuzzy	fuzzy	ADJ
ejpam-3441	437	14	quasi	quasi	NOUN
ejpam-3441	437	15	-	-	NOUN
ejpam-3441	437	16	ideal	ideal	ADJ
ejpam-3441	437	17	with	with	ADP
ejpam-3441	437	18	thresholds	threshold	NOUN
ejpam-3441	437	19	(	(	PUNCT
ejpam-3441	437	20	α	α	X
ejpam-3441	437	21	,	,	PUNCT
ejpam-3441	437	22	β	β	X
ejpam-3441	437	23	]	]	PUNCT
ejpam-3441	437	24	of	of	ADP
ejpam-3441	437	25	an	an	DET
ejpam-3441	437	26	la	la	ADJ
ejpam-3441	437	27	-	-	PUNCT
ejpam-3441	437	28	ring	ring	NOUN
ejpam-3441	437	29	r.	r.	NOUN
ejpam-3441	437	30	this	this	PRON
ejpam-3441	437	31	implies	imply	VERB
ejpam-3441	437	32	that	that	SCONJ
ejpam-3441	437	33	a	a	PRON
ejpam-3441	437	34	is	be	AUX
ejpam-3441	437	35	an	an	DET
ejpam-3441	437	36	intuitionistic	intuitionistic	ADJ
ejpam-3441	437	37	fuzzy	fuzzy	ADJ
ejpam-3441	437	38	la	la	NOUN
ejpam-3441	437	39	-	-	PUNCT
ejpam-3441	437	40	subring	subre	VERB
ejpam-3441	437	41	with	with	ADP
ejpam-3441	437	42	thresholds	threshold	NOUN
ejpam-3441	437	43	(	(	PUNCT
ejpam-3441	437	44	α	α	X
ejpam-3441	437	45	,	,	PUNCT
ejpam-3441	437	46	β	β	X
ejpam-3441	437	47	]	]	PUNCT
ejpam-3441	437	48	of	of	ADP
ejpam-3441	437	49	r.	r.	PROPN
ejpam-3441	437	50	we	we	PRON
ejpam-3441	437	51	have	have	VERB
ejpam-3441	437	52	to	to	PART
ejpam-3441	437	53	show	show	VERB
ejpam-3441	437	54	that	that	SCONJ
ejpam-3441	437	55	(	(	PUNCT
ejpam-3441	437	56	µa	µa	ADP
ejpam-3441	437	57	◦	◦	NOUN
ejpam-3441	437	58	βαr)	βαr)	NOUN
ejpam-3441	437	59	◦	◦	NOUN
ejpam-3441	437	60	βαµa	βαµa	NOUN
ejpam-3441	438	1	⊆	⊆	X
ejpam-3441	438	2	(	(	PUNCT
ejpam-3441	438	3	µa)βα	µa)βα	NUM
ejpam-3441	438	4	and	and	CCONJ
ejpam-3441	438	5	(	(	PUNCT
ejpam-3441	438	6	γa	γa	NOUN
ejpam-3441	438	7	◦	◦	NOUN
ejpam-3441	438	8	βαr)	βαr)	NOUN
ejpam-3441	438	9	◦	◦	NOUN
ejpam-3441	438	10	βαγa	βαγa	ADJ
ejpam-3441	438	11	⊇	⊇	NOUN
ejpam-3441	438	12	(	(	PUNCT
ejpam-3441	438	13	γa)βα	γa)βα	X
ejpam-3441	438	14	.	.	PUNCT
ejpam-3441	439	1	now	now	ADV
ejpam-3441	439	2	(	(	PUNCT
ejpam-3441	439	3	µa	µa	ADP
ejpam-3441	439	4	◦	◦	NOUN
ejpam-3441	439	5	βα	βα	NOUN
ejpam-3441	439	6	r	r	NOUN
ejpam-3441	439	7	)	)	PUNCT
ejpam-3441	439	8	◦	◦	NOUN
ejpam-3441	439	9	βα	βα	NOUN
ejpam-3441	439	10	µa	µa	NOUN
ejpam-3441	439	11	⊆	⊆	NUM
ejpam-3441	439	12	(	(	PUNCT
ejpam-3441	439	13	r	r	NOUN
ejpam-3441	439	14	◦	◦	NOUN
ejpam-3441	439	15	βα	βα	NOUN
ejpam-3441	439	16	r	r	NOUN
ejpam-3441	439	17	)	)	PUNCT
ejpam-3441	439	18	◦	◦	NOUN
ejpam-3441	439	19	βα	βα	NOUN
ejpam-3441	439	20	µa	µa	NOUN
ejpam-3441	439	21	⊆	⊆	NUM
ejpam-3441	439	22	r	r	NOUN
ejpam-3441	439	23	◦	◦	NOUN
ejpam-3441	439	24	βα	βα	NOUN
ejpam-3441	439	25	µa	µa	NOUN
ejpam-3441	439	26	and	and	CCONJ
ejpam-3441	439	27	(	(	PUNCT
ejpam-3441	439	28	µa	µa	ADP
ejpam-3441	439	29	◦	◦	NOUN
ejpam-3441	439	30	βα	βα	NOUN
ejpam-3441	439	31	r	r	NOUN
ejpam-3441	439	32	)	)	PUNCT
ejpam-3441	439	33	◦	◦	NOUN
ejpam-3441	439	34	βα	βα	NOUN
ejpam-3441	439	35	µa	µa	NOUN
ejpam-3441	439	36	⊆	⊆	NUM
ejpam-3441	439	37	(	(	PUNCT
ejpam-3441	439	38	µa	µa	ADP
ejpam-3441	439	39	◦	◦	NOUN
ejpam-3441	439	40	βα	βα	NOUN
ejpam-3441	439	41	r	r	NOUN
ejpam-3441	439	42	)	)	PUNCT
ejpam-3441	439	43	◦	◦	NOUN
ejpam-3441	439	44	βα	βα	NOUN
ejpam-3441	439	45	r	r	NOUN
ejpam-3441	439	46	=	=	PUNCT
ejpam-3441	439	47	(	(	PUNCT
ejpam-3441	439	48	µa	µa	INTJ
ejpam-3441	439	49	◦	◦	NOUN
ejpam-3441	439	50	βα	βα	NOUN
ejpam-3441	439	51	r	r	NOUN
ejpam-3441	439	52	)	)	PUNCT
ejpam-3441	439	53	◦	◦	NOUN
ejpam-3441	439	54	βα	βα	X
ejpam-3441	439	55	(	(	PUNCT
ejpam-3441	439	56	e	e	X
ejpam-3441	439	57	◦	◦	NOUN
ejpam-3441	439	58	βα	βα	NOUN
ejpam-3441	439	59	r	r	NOUN
ejpam-3441	439	60	)	)	PUNCT
ejpam-3441	439	61	=	=	SYM
ejpam-3441	439	62	(	(	PUNCT
ejpam-3441	439	63	µa	µa	INTJ
ejpam-3441	439	64	◦	◦	NOUN
ejpam-3441	439	65	βα	βα	X
ejpam-3441	439	66	e	e	NOUN
ejpam-3441	439	67	)	)	PUNCT
ejpam-3441	439	68	◦	◦	NOUN
ejpam-3441	439	69	βα	βα	X
ejpam-3441	439	70	(	(	PUNCT
ejpam-3441	439	71	r	r	NOUN
ejpam-3441	439	72	◦	◦	NOUN
ejpam-3441	439	73	βα	βα	NOUN
ejpam-3441	439	74	r	r	NOUN
ejpam-3441	439	75	)	)	PUNCT
ejpam-3441	439	76	⊆	⊆	NUM
ejpam-3441	439	77	(	(	PUNCT
ejpam-3441	439	78	µa	µa	ADP
ejpam-3441	439	79	◦	◦	NOUN
ejpam-3441	439	80	βα	βα	X
ejpam-3441	439	81	e	e	NOUN
ejpam-3441	439	82	)	)	PUNCT
ejpam-3441	439	83	◦	◦	NOUN
ejpam-3441	439	84	βα	βα	NOUN
ejpam-3441	439	85	rβα	rβα	NOUN
ejpam-3441	440	1	=	=	SYM
ejpam-3441	440	2	(	(	PUNCT
ejpam-3441	440	3	µa)βα	µa)βα	NUM
ejpam-3441	440	4	◦	◦	NOUN
ejpam-3441	440	5	βα	βα	NOUN
ejpam-3441	440	6	rβα	rβα	NOUN
ejpam-3441	440	7	=	=	SYM
ejpam-3441	440	8	µa	µa	PROPN
ejpam-3441	440	9	◦	◦	PROPN
ejpam-3441	440	10	βα	βα	X
ejpam-3441	440	11	r.	r.	PROPN
ejpam-3441	440	12	⇒	⇒	PROPN
ejpam-3441	440	13	(	(	PUNCT
ejpam-3441	440	14	µa	µa	ADP
ejpam-3441	440	15	◦	◦	NOUN
ejpam-3441	440	16	βα	βα	NOUN
ejpam-3441	440	17	r	r	NOUN
ejpam-3441	440	18	)	)	PUNCT
ejpam-3441	440	19	◦	◦	NOUN
ejpam-3441	440	20	βα	βα	NOUN
ejpam-3441	440	21	µa	µa	NOUN
ejpam-3441	440	22	⊆	⊆	NUM
ejpam-3441	440	23	µa	µa	NOUN
ejpam-3441	440	24	◦	◦	NOUN
ejpam-3441	440	25	βα	βα	NOUN
ejpam-3441	440	26	r	r	NOUN
ejpam-3441	440	27	∧r	∧r	NUM
ejpam-3441	440	28	◦	◦	NOUN
ejpam-3441	440	29	βα	βα	NOUN
ejpam-3441	440	30	µa	µa	ADP
ejpam-3441	440	31	⊆	⊆	NUM
ejpam-3441	440	32	(	(	PUNCT
ejpam-3441	440	33	µa)βα	µa)βα	NUM
ejpam-3441	440	34	.	.	PUNCT
ejpam-3441	441	1	similarly	similarly	ADV
ejpam-3441	441	2	,	,	PUNCT
ejpam-3441	441	3	we	we	PRON
ejpam-3441	441	4	have	have	VERB
ejpam-3441	441	5	(	(	PUNCT
ejpam-3441	441	6	γa	γa	NOUN
ejpam-3441	441	7	◦	◦	NOUN
ejpam-3441	441	8	βαr)	βαr)	PROPN
ejpam-3441	441	9	◦	◦	NOUN
ejpam-3441	441	10	βα	βα	X
ejpam-3441	441	11	γa	γa	PROPN
ejpam-3441	441	12	⊇	⊇	PROPN
ejpam-3441	441	13	(	(	PUNCT
ejpam-3441	441	14	γa)βα	γa)βα	X
ejpam-3441	441	15	.	.	PUNCT
ejpam-3441	442	1	so	so	ADV
ejpam-3441	442	2	a	a	PRON
ejpam-3441	442	3	is	be	AUX
ejpam-3441	442	4	an	an	DET
ejpam-3441	442	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	442	6	fuzzy	fuzzy	ADJ
ejpam-3441	442	7	bi	bi	NOUN
ejpam-3441	442	8	-	-	NOUN
ejpam-3441	442	9	ideal	ideal	ADJ
ejpam-3441	442	10	with	with	ADP
ejpam-3441	442	11	thresholds	threshold	NOUN
ejpam-3441	442	12	(	(	PUNCT
ejpam-3441	442	13	α	α	X
ejpam-3441	442	14	,	,	PUNCT
ejpam-3441	442	15	β	β	X
ejpam-3441	442	16	]	]	PUNCT
ejpam-3441	442	17	of	of	ADP
ejpam-3441	442	18	r.	r.	PROPN
ejpam-3441	442	19	proposition	proposition	NOUN
ejpam-3441	442	20	5	5	NUM
ejpam-3441	442	21	.	.	PUNCT
ejpam-3441	443	1	if	if	SCONJ
ejpam-3441	443	2	a	a	PRON
ejpam-3441	443	3	and	and	CCONJ
ejpam-3441	443	4	b	b	NOUN
ejpam-3441	443	5	are	be	AUX
ejpam-3441	443	6	two	two	NUM
ejpam-3441	443	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	443	8	fuzzy	fuzzy	ADJ
ejpam-3441	443	9	quasi	quasi	NOUN
ejpam-3441	443	10	-	-	NOUN
ejpam-3441	443	11	ideals	ideal	NOUN
ejpam-3441	443	12	with	with	ADP
ejpam-3441	443	13	thresholds	threshold	NOUN
ejpam-3441	443	14	(	(	PUNCT
ejpam-3441	443	15	α	α	X
ejpam-3441	443	16	,	,	PUNCT
ejpam-3441	443	17	β	β	X
ejpam-3441	443	18	]	]	PUNCT
ejpam-3441	443	19	of	of	ADP
ejpam-3441	443	20	an	an	DET
ejpam-3441	443	21	la	la	ADJ
ejpam-3441	443	22	-	-	PUNCT
ejpam-3441	443	23	ring	ring	NOUN
ejpam-3441	443	24	r	r	NOUN
ejpam-3441	443	25	with	with	ADP
ejpam-3441	443	26	left	left	ADJ
ejpam-3441	443	27	identity	identity	NOUN
ejpam-3441	443	28	e	e	NOUN
ejpam-3441	443	29	,	,	PUNCT
ejpam-3441	443	30	such	such	ADJ
ejpam-3441	443	31	that	that	SCONJ
ejpam-3441	443	32	(	(	PUNCT
ejpam-3441	443	33	xe)r	xe)r	PROPN
ejpam-3441	443	34	=	=	SYM
ejpam-3441	443	35	xr	xr	PROPN
ejpam-3441	443	36	for	for	ADP
ejpam-3441	443	37	all	all	DET
ejpam-3441	443	38	x	x	SYM
ejpam-3441	443	39	∈	∈	PROPN
ejpam-3441	443	40	r	r	NOUN
ejpam-3441	443	41	,	,	PUNCT
ejpam-3441	443	42	then	then	ADV
ejpam-3441	443	43	a	a	DET
ejpam-3441	443	44	◦	◦	NOUN
ejpam-3441	443	45	βα	βα	NOUN
ejpam-3441	443	46	b	b	NOUN
ejpam-3441	443	47	is	be	AUX
ejpam-3441	443	48	an	an	DET
ejpam-3441	443	49	intuitionistic	intuitionistic	ADJ
ejpam-3441	443	50	fuzzy	fuzzy	ADJ
ejpam-3441	443	51	bi	bi	NOUN
ejpam-3441	443	52	-	-	NOUN
ejpam-3441	443	53	ideal	ideal	ADJ
ejpam-3441	443	54	with	with	ADP
ejpam-3441	443	55	thresholds	threshold	NOUN
ejpam-3441	443	56	(	(	PUNCT
ejpam-3441	443	57	α	α	X
ejpam-3441	443	58	,	,	PUNCT
ejpam-3441	443	59	β	β	X
ejpam-3441	443	60	]	]	PUNCT
ejpam-3441	443	61	of	of	ADP
ejpam-3441	443	62	r.	r.	PROPN
ejpam-3441	443	63	proof	proof	NOUN
ejpam-3441	443	64	.	.	PUNCT
ejpam-3441	444	1	let	let	VERB
ejpam-3441	444	2	a	a	PRON
ejpam-3441	444	3	and	and	CCONJ
ejpam-3441	444	4	b	b	NOUN
ejpam-3441	444	5	be	be	AUX
ejpam-3441	444	6	two	two	NUM
ejpam-3441	444	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	444	8	fuzzy	fuzzy	ADJ
ejpam-3441	444	9	quasi	quasi	NOUN
ejpam-3441	444	10	-	-	NOUN
ejpam-3441	444	11	ideals	ideal	NOUN
ejpam-3441	444	12	with	with	ADP
ejpam-3441	444	13	thresholds	threshold	NOUN
ejpam-3441	444	14	(	(	PUNCT
ejpam-3441	444	15	α	α	X
ejpam-3441	444	16	,	,	PUNCT
ejpam-3441	444	17	β	β	X
ejpam-3441	444	18	]	]	PUNCT
ejpam-3441	444	19	of	of	ADP
ejpam-3441	444	20	an	an	DET
ejpam-3441	444	21	la	la	ADJ
ejpam-3441	444	22	-	-	PUNCT
ejpam-3441	444	23	ring	ring	NOUN
ejpam-3441	444	24	r	r	NOUN
ejpam-3441	444	25	,	,	PUNCT
ejpam-3441	444	26	this	this	PRON
ejpam-3441	444	27	implies	imply	VERB
ejpam-3441	444	28	that	that	SCONJ
ejpam-3441	444	29	a	a	PRON
ejpam-3441	444	30	and	and	CCONJ
ejpam-3441	444	31	b	b	NOUN
ejpam-3441	444	32	be	be	AUX
ejpam-3441	444	33	two	two	NUM
ejpam-3441	444	34	intuitionistic	intuitionistic	ADJ
ejpam-3441	444	35	fuzzy	fuzzy	ADJ
ejpam-3441	444	36	bi	bi	NOUN
ejpam-3441	444	37	-	-	NOUN
ejpam-3441	444	38	ideals	ideal	NOUN
ejpam-3441	444	39	with	with	ADP
ejpam-3441	444	40	thresholds	threshold	NOUN
ejpam-3441	444	41	(	(	PUNCT
ejpam-3441	444	42	α	α	X
ejpam-3441	444	43	,	,	PUNCT
ejpam-3441	444	44	β	β	X
ejpam-3441	444	45	]	]	PUNCT
ejpam-3441	444	46	of	of	ADP
ejpam-3441	444	47	r	r	NOUN
ejpam-3441	444	48	,	,	PUNCT
ejpam-3441	444	49	by	by	ADP
ejpam-3441	444	50	the	the	DET
ejpam-3441	444	51	lemma	lemma	PROPN
ejpam-3441	444	52	15	15	NUM
ejpam-3441	444	53	.	.	PUNCT
ejpam-3441	445	1	then	then	ADV
ejpam-3441	445	2	a	a	DET
ejpam-3441	445	3	◦	◦	NOUN
ejpam-3441	445	4	βα	βα	NOUN
ejpam-3441	445	5	b	b	NOUN
ejpam-3441	445	6	is	be	AUX
ejpam-3441	445	7	also	also	ADV
ejpam-3441	445	8	an	an	DET
ejpam-3441	445	9	intuitionistic	intuitionistic	ADJ
ejpam-3441	445	10	fuzzy	fuzzy	ADJ
ejpam-3441	445	11	bi	bi	NOUN
ejpam-3441	445	12	-	-	NOUN
ejpam-3441	445	13	ideal	ideal	ADJ
ejpam-3441	445	14	with	with	ADP
ejpam-3441	445	15	thresholds	threshold	NOUN
ejpam-3441	445	16	(	(	PUNCT
ejpam-3441	445	17	α	α	X
ejpam-3441	445	18	,	,	PUNCT
ejpam-3441	445	19	β	β	X
ejpam-3441	445	20	]	]	PUNCT
ejpam-3441	445	21	of	of	ADP
ejpam-3441	445	22	r	r	NOUN
ejpam-3441	445	23	by	by	ADP
ejpam-3441	445	24	the	the	DET
ejpam-3441	445	25	lemma	lemma	PROPN
ejpam-3441	445	26	10	10	NUM
ejpam-3441	445	27	.	.	PUNCT
ejpam-3441	446	1	3	3	X
ejpam-3441	446	2	.	.	X
ejpam-3441	446	3	regular	regular	ADJ
ejpam-3441	446	4	la	la	NOUN
ejpam-3441	446	5	-	-	PUNCT
ejpam-3441	446	6	rings	ring	NOUN
ejpam-3441	446	7	in	in	ADP
ejpam-3441	446	8	this	this	DET
ejpam-3441	446	9	section	section	NOUN
ejpam-3441	446	10	,	,	PUNCT
ejpam-3441	446	11	we	we	PRON
ejpam-3441	446	12	characterize	characterize	VERB
ejpam-3441	446	13	regular	regular	ADJ
ejpam-3441	446	14	la	la	NOUN
ejpam-3441	446	15	-	-	PUNCT
ejpam-3441	446	16	rings	ring	NOUN
ejpam-3441	446	17	by	by	ADP
ejpam-3441	446	18	the	the	DET
ejpam-3441	446	19	properties	property	NOUN
ejpam-3441	446	20	of	of	ADP
ejpam-3441	446	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	446	22	fuzzy	fuzzy	ADJ
ejpam-3441	446	23	left	left	NOUN
ejpam-3441	446	24	(	(	PUNCT
ejpam-3441	446	25	right	right	ADJ
ejpam-3441	446	26	,	,	PUNCT
ejpam-3441	446	27	quasi-	quasi-	INTJ
ejpam-3441	446	28	,	,	PUNCT
ejpam-3441	446	29	bi-	bi-	NUM
ejpam-3441	446	30	,	,	PUNCT
ejpam-3441	446	31	generalized	generalize	VERB
ejpam-3441	446	32	bi-	bi-	NUM
ejpam-3441	446	33	)	)	PUNCT
ejpam-3441	446	34	ideals	ideal	NOUN
ejpam-3441	446	35	with	with	ADP
ejpam-3441	446	36	thresholds	threshold	NOUN
ejpam-3441	446	37	(	(	PUNCT
ejpam-3441	446	38	α	α	X
ejpam-3441	446	39	,	,	PUNCT
ejpam-3441	446	40	β	β	X
ejpam-3441	446	41	]	]	X
ejpam-3441	446	42	.	.	PUNCT
ejpam-3441	447	1	an	an	DET
ejpam-3441	447	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	447	3	fuzzy	fuzzy	ADJ
ejpam-3441	447	4	ideal	ideal	NOUN
ejpam-3441	447	5	a	a	PRON
ejpam-3441	447	6	=	=	X
ejpam-3441	447	7	(	(	PUNCT
ejpam-3441	447	8	µa	µa	PROPN
ejpam-3441	447	9	,	,	PUNCT
ejpam-3441	447	10	γa	γa	PROPN
ejpam-3441	447	11	)	)	PUNCT
ejpam-3441	447	12	with	with	ADP
ejpam-3441	447	13	thresholds	threshold	NOUN
ejpam-3441	447	14	(	(	PUNCT
ejpam-3441	447	15	α	α	X
ejpam-3441	447	16	,	,	PUNCT
ejpam-3441	447	17	β	β	X
ejpam-3441	447	18	]	]	PUNCT
ejpam-3441	447	19	of	of	ADP
ejpam-3441	447	20	an	an	DET
ejpam-3441	447	21	la	la	ADJ
ejpam-3441	447	22	-	-	PUNCT
ejpam-3441	447	23	ring	ring	NOUN
ejpam-3441	447	24	r	r	NOUN
ejpam-3441	447	25	is	be	AUX
ejpam-3441	447	26	an	an	DET
ejpam-3441	447	27	intuitionistic	intuitionistic	ADJ
ejpam-3441	447	28	fuzzy	fuzzy	ADJ
ejpam-3441	447	29	idempotent	idempotent	NOUN
ejpam-3441	447	30	with	with	ADP
ejpam-3441	447	31	thresholds	threshold	NOUN
ejpam-3441	447	32	(	(	PUNCT
ejpam-3441	447	33	α	α	X
ejpam-3441	447	34	,	,	PUNCT
ejpam-3441	447	35	β	β	X
ejpam-3441	447	36	]	]	PUNCT
ejpam-3441	447	37	of	of	ADP
ejpam-3441	447	38	r	r	PRON
ejpam-3441	447	39	if	if	SCONJ
ejpam-3441	447	40	a	a	DET
ejpam-3441	447	41	◦	◦	NOUN
ejpam-3441	447	42	βα	βα	VERB
ejpam-3441	447	43	a	a	DET
ejpam-3441	447	44	=	=	SYM
ejpam-3441	447	45	aβα	aβα	PROPN
ejpam-3441	447	46	.	.	PUNCT
ejpam-3441	448	1	lemma	lemma	PROPN
ejpam-3441	448	2	16	16	NUM
ejpam-3441	448	3	.	.	PUNCT
ejpam-3441	449	1	every	every	DET
ejpam-3441	449	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	449	3	fuzzy	fuzzy	ADJ
ejpam-3441	449	4	right	right	ADJ
ejpam-3441	449	5	ideal	ideal	NOUN
ejpam-3441	449	6	with	with	ADP
ejpam-3441	449	7	thresholds	threshold	NOUN
ejpam-3441	449	8	(	(	PUNCT
ejpam-3441	449	9	α	α	X
ejpam-3441	449	10	,	,	PUNCT
ejpam-3441	449	11	β	β	X
ejpam-3441	449	12	]	]	PUNCT
ejpam-3441	449	13	of	of	ADP
ejpam-3441	449	14	a	a	DET
ejpam-3441	449	15	regular	regular	ADJ
ejpam-3441	449	16	laring	laring	NOUN
ejpam-3441	449	17	r	r	NOUN
ejpam-3441	449	18	is	be	AUX
ejpam-3441	449	19	an	an	DET
ejpam-3441	449	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	449	21	fuzzy	fuzzy	ADJ
ejpam-3441	449	22	ideal	ideal	NOUN
ejpam-3441	449	23	with	with	ADP
ejpam-3441	449	24	thresholds	threshold	NOUN
ejpam-3441	449	25	(	(	PUNCT
ejpam-3441	449	26	α	α	X
ejpam-3441	449	27	,	,	PUNCT
ejpam-3441	449	28	β	β	X
ejpam-3441	449	29	]	]	PUNCT
ejpam-3441	449	30	of	of	ADP
ejpam-3441	449	31	r.	r.	PROPN
ejpam-3441	449	32	k.	k.	PROPN
ejpam-3441	449	33	nasreen	nasreen	PROPN
ejpam-3441	449	34	et	et	PROPN
ejpam-3441	449	35	al	al	PROPN
ejpam-3441	449	36	.	.	PUNCT
ejpam-3441	449	37	/	/	SYM
ejpam-3441	449	38	eur	eur	PROPN
ejpam-3441	449	39	.	.	PUNCT
ejpam-3441	450	1	j.	j.	PROPN
ejpam-3441	450	2	pure	pure	PROPN
ejpam-3441	450	3	appl	appl	PROPN
ejpam-3441	450	4	.	.	PROPN
ejpam-3441	450	5	math	math	PROPN
ejpam-3441	450	6	,	,	PUNCT
ejpam-3441	450	7	12	12	NUM
ejpam-3441	450	8	(	(	PUNCT
ejpam-3441	450	9	3	3	NUM
ejpam-3441	450	10	)	)	PUNCT
ejpam-3441	450	11	(	(	PUNCT
ejpam-3441	450	12	2019	2019	NUM
ejpam-3441	450	13	)	)	PUNCT
ejpam-3441	450	14	,	,	PUNCT
ejpam-3441	450	15	906	906	NUM
ejpam-3441	450	16	-	-	SYM
ejpam-3441	450	17	943	943	NUM
ejpam-3441	450	18	926	926	NUM
ejpam-3441	450	19	proof	proof	NOUN
ejpam-3441	450	20	.	.	PUNCT
ejpam-3441	450	21	suppose	suppose	VERB
ejpam-3441	450	22	that	that	SCONJ
ejpam-3441	450	23	a	a	DET
ejpam-3441	450	24	=	=	SYM
ejpam-3441	450	25	(	(	PUNCT
ejpam-3441	450	26	µa	µa	PROPN
ejpam-3441	450	27	,	,	PUNCT
ejpam-3441	450	28	γa	γa	PROPN
ejpam-3441	450	29	)	)	PUNCT
ejpam-3441	450	30	is	be	AUX
ejpam-3441	450	31	an	an	DET
ejpam-3441	450	32	intuitionistic	intuitionistic	ADJ
ejpam-3441	450	33	fuzzy	fuzzy	ADJ
ejpam-3441	450	34	right	right	ADJ
ejpam-3441	450	35	ideal	ideal	NOUN
ejpam-3441	450	36	with	with	ADP
ejpam-3441	450	37	thresholds	threshold	NOUN
ejpam-3441	450	38	(	(	PUNCT
ejpam-3441	450	39	α	α	X
ejpam-3441	450	40	,	,	PUNCT
ejpam-3441	450	41	β	β	X
ejpam-3441	450	42	]	]	PUNCT
ejpam-3441	450	43	of	of	ADP
ejpam-3441	450	44	r.	r.	PROPN
ejpam-3441	450	45	let	let	VERB
ejpam-3441	450	46	x	x	PRON
ejpam-3441	450	47	,	,	PUNCT
ejpam-3441	450	48	y	y	PROPN
ejpam-3441	450	49	∈	∈	PROPN
ejpam-3441	450	50	r	r	NOUN
ejpam-3441	450	51	,	,	PUNCT
ejpam-3441	450	52	this	this	PRON
ejpam-3441	450	53	implies	imply	VERB
ejpam-3441	450	54	that	that	SCONJ
ejpam-3441	450	55	there	there	PRON
ejpam-3441	450	56	exists	exist	VERB
ejpam-3441	450	57	a	a	DET
ejpam-3441	450	58	∈	∈	PROPN
ejpam-3441	450	59	r	r	NOUN
ejpam-3441	450	60	,	,	PUNCT
ejpam-3441	450	61	such	such	ADJ
ejpam-3441	450	62	that	that	SCONJ
ejpam-3441	450	63	x	x	SYM
ejpam-3441	450	64	=	=	SYM
ejpam-3441	450	65	(	(	PUNCT
ejpam-3441	450	66	xa)x	xa)x	PROPN
ejpam-3441	450	67	.	.	PUNCT
ejpam-3441	451	1	thus	thus	ADV
ejpam-3441	451	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3441	451	3	)	)	PUNCT
ejpam-3441	451	4	,	,	PUNCT
ejpam-3441	451	5	α	α	X
ejpam-3441	451	6	}	}	PUNCT
ejpam-3441	451	7	=	=	SYM
ejpam-3441	451	8	max{µa(((xa)x)y	max{µa(((xa)x)y	PROPN
ejpam-3441	451	9	)	)	PUNCT
ejpam-3441	451	10	,	,	PUNCT
ejpam-3441	451	11	α	α	X
ejpam-3441	451	12	}	}	PUNCT
ejpam-3441	451	13	=	=	SYM
ejpam-3441	451	14	max{µa((yx)(xa	max{µa((yx)(xa	NOUN
ejpam-3441	451	15	)	)	PUNCT
ejpam-3441	451	16	)	)	PUNCT
ejpam-3441	451	17	,	,	PUNCT
ejpam-3441	451	18	α	α	X
ejpam-3441	451	19	}	}	PUNCT
ejpam-3441	451	20	≥	≥	NOUN
ejpam-3441	451	21	min{µa(yx	min{µa(yx	NOUN
ejpam-3441	451	22	)	)	PUNCT
ejpam-3441	451	23	,	,	PUNCT
ejpam-3441	451	24	β	β	X
ejpam-3441	451	25	}	}	PUNCT
ejpam-3441	451	26	≥	≥	NOUN
ejpam-3441	451	27	min{µa(y	min{µa(y	NOUN
ejpam-3441	451	28	)	)	PUNCT
ejpam-3441	451	29	,	,	PUNCT
ejpam-3441	451	30	β	β	X
ejpam-3441	451	31	}	}	PUNCT
ejpam-3441	451	32	and	and	CCONJ
ejpam-3441	451	33	min{γa(xy	min{γa(xy	NUM
ejpam-3441	451	34	)	)	PUNCT
ejpam-3441	451	35	,	,	PUNCT
ejpam-3441	451	36	(	(	PUNCT
ejpam-3441	451	37	1−	1−	NUM
ejpam-3441	451	38	α	α	NOUN
ejpam-3441	451	39	)	)	PUNCT
ejpam-3441	451	40	}	}	PUNCT
ejpam-3441	451	41	=	=	SYM
ejpam-3441	451	42	min{γa(((xa)x)y	min{γa(((xa)x)y	PROPN
ejpam-3441	451	43	)	)	PUNCT
ejpam-3441	451	44	,	,	PUNCT
ejpam-3441	451	45	(	(	PUNCT
ejpam-3441	451	46	1−	1−	NUM
ejpam-3441	451	47	α	α	NOUN
ejpam-3441	451	48	)	)	PUNCT
ejpam-3441	451	49	}	}	PUNCT
ejpam-3441	451	50	=	=	SYM
ejpam-3441	451	51	min{γa((yx)(xa	min{γa((yx)(xa	NOUN
ejpam-3441	451	52	)	)	PUNCT
ejpam-3441	451	53	)	)	PUNCT
ejpam-3441	451	54	,	,	PUNCT
ejpam-3441	451	55	(	(	PUNCT
ejpam-3441	451	56	1−	1−	NUM
ejpam-3441	451	57	α	α	NOUN
ejpam-3441	451	58	)	)	PUNCT
ejpam-3441	451	59	}	}	PUNCT
ejpam-3441	451	60	≤	≤	NOUN
ejpam-3441	451	61	max{γa(yx	max{γa(yx	NOUN
ejpam-3441	451	62	)	)	PUNCT
ejpam-3441	451	63	,	,	PUNCT
ejpam-3441	451	64	(	(	PUNCT
ejpam-3441	451	65	1−	1−	NUM
ejpam-3441	451	66	β	β	NOUN
ejpam-3441	451	67	)	)	PUNCT
ejpam-3441	451	68	}	}	PUNCT
ejpam-3441	451	69	≤	≤	NUM
ejpam-3441	451	70	max{γa(y	max{γa(y	PROPN
ejpam-3441	451	71	)	)	PUNCT
ejpam-3441	451	72	,	,	PUNCT
ejpam-3441	451	73	(	(	PUNCT
ejpam-3441	451	74	1−	1−	NUM
ejpam-3441	451	75	β	β	NOUN
ejpam-3441	451	76	)	)	PUNCT
ejpam-3441	451	77	}	}	PUNCT
ejpam-3441	451	78	.	.	PUNCT
ejpam-3441	452	1	hence	hence	ADV
ejpam-3441	452	2	a	a	PRON
ejpam-3441	452	3	is	be	AUX
ejpam-3441	452	4	an	an	DET
ejpam-3441	452	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	452	6	fuzzy	fuzzy	ADJ
ejpam-3441	452	7	ideal	ideal	NOUN
ejpam-3441	452	8	with	with	ADP
ejpam-3441	452	9	thresholds	threshold	NOUN
ejpam-3441	452	10	(	(	PUNCT
ejpam-3441	452	11	α	α	X
ejpam-3441	452	12	,	,	PUNCT
ejpam-3441	452	13	β	β	X
ejpam-3441	452	14	]	]	PUNCT
ejpam-3441	452	15	of	of	ADP
ejpam-3441	452	16	r.	r.	PROPN
ejpam-3441	452	17	lemma	lemma	PROPN
ejpam-3441	452	18	17	17	NUM
ejpam-3441	452	19	.	.	PUNCT
ejpam-3441	453	1	every	every	DET
ejpam-3441	453	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	453	3	fuzzy	fuzzy	ADJ
ejpam-3441	453	4	ideal	ideal	NOUN
ejpam-3441	453	5	with	with	ADP
ejpam-3441	453	6	thresholds	threshold	NOUN
ejpam-3441	453	7	(	(	PUNCT
ejpam-3441	453	8	α	α	X
ejpam-3441	453	9	,	,	PUNCT
ejpam-3441	453	10	β	β	X
ejpam-3441	453	11	]	]	PUNCT
ejpam-3441	453	12	of	of	ADP
ejpam-3441	453	13	a	a	DET
ejpam-3441	453	14	regular	regular	ADJ
ejpam-3441	453	15	la	la	ADJ
ejpam-3441	453	16	-	-	PUNCT
ejpam-3441	453	17	ring	ring	NOUN
ejpam-3441	453	18	r	r	NOUN
ejpam-3441	453	19	is	be	AUX
ejpam-3441	453	20	an	an	DET
ejpam-3441	453	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	453	22	fuzzy	fuzzy	ADJ
ejpam-3441	453	23	idempotent	idempotent	NOUN
ejpam-3441	453	24	with	with	ADP
ejpam-3441	453	25	thresholds	threshold	NOUN
ejpam-3441	453	26	(	(	PUNCT
ejpam-3441	453	27	α	α	X
ejpam-3441	453	28	,	,	PUNCT
ejpam-3441	453	29	β	β	X
ejpam-3441	453	30	]	]	PUNCT
ejpam-3441	453	31	.	.	PUNCT
ejpam-3441	454	1	proof	proof	NOUN
ejpam-3441	454	2	.	.	PUNCT
ejpam-3441	455	1	assume	assume	VERB
ejpam-3441	455	2	that	that	SCONJ
ejpam-3441	455	3	a	a	DET
ejpam-3441	455	4	=	=	SYM
ejpam-3441	455	5	(	(	PUNCT
ejpam-3441	455	6	µa	µa	PROPN
ejpam-3441	455	7	,	,	PUNCT
ejpam-3441	455	8	γa	γa	PROPN
ejpam-3441	455	9	)	)	PUNCT
ejpam-3441	455	10	is	be	AUX
ejpam-3441	455	11	an	an	DET
ejpam-3441	455	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	455	13	fuzzy	fuzzy	ADJ
ejpam-3441	455	14	ideal	ideal	NOUN
ejpam-3441	455	15	with	with	ADP
ejpam-3441	455	16	thresholds	threshold	NOUN
ejpam-3441	455	17	(	(	PUNCT
ejpam-3441	455	18	α	α	X
ejpam-3441	455	19	,	,	PUNCT
ejpam-3441	455	20	β	β	X
ejpam-3441	455	21	]	]	PUNCT
ejpam-3441	455	22	of	of	ADP
ejpam-3441	455	23	r	r	NOUN
ejpam-3441	455	24	and	and	CCONJ
ejpam-3441	455	25	a	a	DET
ejpam-3441	455	26	◦	◦	NOUN
ejpam-3441	455	27	βα	βα	NOUN
ejpam-3441	455	28	a	a	DET
ejpam-3441	455	29	⊆	⊆	NUM
ejpam-3441	455	30	aβα	aβα	NOUN
ejpam-3441	455	31	.	.	PUNCT
ejpam-3441	456	1	we	we	PRON
ejpam-3441	456	2	have	have	VERB
ejpam-3441	456	3	to	to	PART
ejpam-3441	456	4	show	show	VERB
ejpam-3441	456	5	that	that	SCONJ
ejpam-3441	456	6	aβα	aβα	NOUN
ejpam-3441	456	7	⊆	⊆	NUM
ejpam-3441	456	8	a	a	DET
ejpam-3441	456	9	◦	◦	NOUN
ejpam-3441	456	10	βα	βα	NOUN
ejpam-3441	456	11	a.	a.	NOUN
ejpam-3441	456	12	let	let	VERB
ejpam-3441	456	13	x	x	X
ejpam-3441	456	14	∈	∈	PROPN
ejpam-3441	456	15	r	r	NOUN
ejpam-3441	456	16	,	,	PUNCT
ejpam-3441	456	17	this	this	PRON
ejpam-3441	456	18	means	mean	VERB
ejpam-3441	456	19	that	that	SCONJ
ejpam-3441	456	20	there	there	PRON
ejpam-3441	456	21	exists	exist	VERB
ejpam-3441	456	22	a	a	DET
ejpam-3441	456	23	∈	∈	NOUN
ejpam-3441	456	24	r	r	NOUN
ejpam-3441	456	25	such	such	ADJ
ejpam-3441	456	26	that	that	PRON
ejpam-3441	456	27	x	x	SYM
ejpam-3441	456	28	=	=	SYM
ejpam-3441	456	29	(	(	PUNCT
ejpam-3441	456	30	xa)x	xa)x	PROPN
ejpam-3441	456	31	.	.	PUNCT
ejpam-3441	457	1	thus	thus	ADV
ejpam-3441	457	2	(	(	PUNCT
ejpam-3441	457	3	µa	µa	ADP
ejpam-3441	457	4	◦	◦	NOUN
ejpam-3441	457	5	βα	βα	ADJ
ejpam-3441	457	6	µa)(x	µa)(x	NOUN
ejpam-3441	457	7	)	)	PUNCT
ejpam-3441	458	1	=	=	PRON
ejpam-3441	458	2	{	{	PUNCT
ejpam-3441	458	3	(	(	PUNCT
ejpam-3441	458	4	µa	µa	ADP
ejpam-3441	458	5	◦	◦	NOUN
ejpam-3441	458	6	µa)(x	µa)(x	NOUN
ejpam-3441	458	7	)	)	PUNCT
ejpam-3441	459	1	∧	∧	PROPN
ejpam-3441	459	2	β	β	NOUN
ejpam-3441	459	3	}	}	PUNCT
ejpam-3441	459	4	∨	∨	NUM
ejpam-3441	459	5	α	α	NOUN
ejpam-3441	459	6	=	=	X
ejpam-3441	459	7	{	{	PUNCT
ejpam-3441	459	8	(	(	PUNCT
ejpam-3441	459	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	459	10	i=1	i=1	PROPN
ejpam-3441	459	11	aibi	aibi	NOUN
ejpam-3441	459	12	{	{	PUNCT
ejpam-3441	459	13	∧ni=1	∧ni=1	X
ejpam-3441	459	14	{	{	PUNCT
ejpam-3441	459	15	µa	µa	X
ejpam-3441	459	16	(	(	PUNCT
ejpam-3441	459	17	ai	ai	NOUN
ejpam-3441	459	18	)	)	PUNCT
ejpam-3441	459	19	∧	∧	NOUN
ejpam-3441	459	20	µa	µa	NOUN
ejpam-3441	459	21	(	(	PUNCT
ejpam-3441	459	22	bi	bi	NOUN
ejpam-3441	459	23	)	)	PUNCT
ejpam-3441	459	24	}	}	PUNCT
ejpam-3441	459	25	}	}	PUNCT
ejpam-3441	459	26	)	)	PUNCT
ejpam-3441	460	1	∧	∧	PROPN
ejpam-3441	460	2	β	β	NOUN
ejpam-3441	460	3	}	}	PUNCT
ejpam-3441	460	4	∨	∨	NUM
ejpam-3441	460	5	α	α	PROPN
ejpam-3441	460	6	≥	≥	X
ejpam-3441	460	7	{	{	PUNCT
ejpam-3441	460	8	{	{	PUNCT
ejpam-3441	460	9	µa	µa	PROPN
ejpam-3441	460	10	(	(	PUNCT
ejpam-3441	460	11	xa	xa	NOUN
ejpam-3441	460	12	)	)	PUNCT
ejpam-3441	460	13	∧	∧	PROPN
ejpam-3441	460	14	µa	µa	INTJ
ejpam-3441	460	15	(	(	PUNCT
ejpam-3441	460	16	x	x	NOUN
ejpam-3441	460	17	)	)	PUNCT
ejpam-3441	460	18	}	}	PUNCT
ejpam-3441	460	19	∧	∧	PROPN
ejpam-3441	460	20	β	β	NOUN
ejpam-3441	460	21	}	}	PUNCT
ejpam-3441	460	22	∨	∨	NUM
ejpam-3441	460	23	α	α	NOUN
ejpam-3441	460	24	=	=	SYM
ejpam-3441	460	25	(	(	PUNCT
ejpam-3441	460	26	µa	µa	X
ejpam-3441	460	27	(	(	PUNCT
ejpam-3441	460	28	xa	xa	PROPN
ejpam-3441	460	29	)	)	PUNCT
ejpam-3441	460	30	∨	∨	NUM
ejpam-3441	460	31	α	α	NOUN
ejpam-3441	460	32	)	)	PUNCT
ejpam-3441	460	33	∧	∧	PROPN
ejpam-3441	460	34	(	(	PUNCT
ejpam-3441	460	35	µa	µa	X
ejpam-3441	460	36	(	(	PUNCT
ejpam-3441	460	37	x	x	NOUN
ejpam-3441	460	38	)	)	PUNCT
ejpam-3441	460	39	∨	∨	NUM
ejpam-3441	460	40	α	α	NOUN
ejpam-3441	460	41	)	)	PUNCT
ejpam-3441	460	42	∧	∧	PROPN
ejpam-3441	460	43	(	(	PUNCT
ejpam-3441	460	44	β	β	X
ejpam-3441	460	45	∨	∨	NUM
ejpam-3441	460	46	α	α	NOUN
ejpam-3441	460	47	)	)	PUNCT
ejpam-3441	460	48	≥	≥	NOUN
ejpam-3441	460	49	(	(	PUNCT
ejpam-3441	460	50	µa	µa	PROPN
ejpam-3441	460	51	(	(	PUNCT
ejpam-3441	460	52	x	x	NOUN
ejpam-3441	460	53	)	)	PUNCT
ejpam-3441	460	54	∧	∧	PROPN
ejpam-3441	460	55	β	β	NOUN
ejpam-3441	460	56	)	)	PUNCT
ejpam-3441	460	57	∧	∧	PROPN
ejpam-3441	460	58	µa	µa	INTJ
ejpam-3441	460	59	(	(	PUNCT
ejpam-3441	460	60	x	x	X
ejpam-3441	460	61	)	)	PUNCT
ejpam-3441	460	62	∧	∧	NOUN
ejpam-3441	460	63	β	β	X
ejpam-3441	460	64	=	=	SYM
ejpam-3441	460	65	µa	µa	X
ejpam-3441	460	66	(	(	PUNCT
ejpam-3441	460	67	x	x	X
ejpam-3441	460	68	)	)	PUNCT
ejpam-3441	460	69	∧	∧	NOUN
ejpam-3441	460	70	β	β	X
ejpam-3441	460	71	=	=	SYM
ejpam-3441	460	72	(	(	PUNCT
ejpam-3441	460	73	µa	µa	INTJ
ejpam-3441	460	74	(	(	PUNCT
ejpam-3441	460	75	x	x	NOUN
ejpam-3441	460	76	)	)	PUNCT
ejpam-3441	460	77	∧	∧	PROPN
ejpam-3441	460	78	β	β	NOUN
ejpam-3441	460	79	)	)	PUNCT
ejpam-3441	460	80	∨	∨	NUM
ejpam-3441	460	81	α	α	NOUN
ejpam-3441	460	82	=	=	SYM
ejpam-3441	460	83	(	(	PUNCT
ejpam-3441	460	84	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	460	85	)	)	PUNCT
ejpam-3441	460	86	.	.	PUNCT
ejpam-3441	461	1	⇒	⇒	NOUN
ejpam-3441	461	2	(	(	PUNCT
ejpam-3441	461	3	µa)βα	µa)βα	NUM
ejpam-3441	461	4	⊆	⊆	NUM
ejpam-3441	461	5	µa	µa	NOUN
ejpam-3441	461	6	◦	◦	NOUN
ejpam-3441	461	7	βα	βα	NOUN
ejpam-3441	461	8	µa	µa	PROPN
ejpam-3441	461	9	.	.	PROPN
ejpam-3441	462	1	similarly	similarly	ADV
ejpam-3441	462	2	,	,	PUNCT
ejpam-3441	462	3	we	we	PRON
ejpam-3441	462	4	have	have	VERB
ejpam-3441	462	5	(	(	PUNCT
ejpam-3441	462	6	γa)βα	γa)βα	X
ejpam-3441	462	7	⊇	⊇	PROPN
ejpam-3441	462	8	γa	γa	PROPN
ejpam-3441	462	9	◦	◦	PROPN
ejpam-3441	462	10	βα	βα	PROPN
ejpam-3441	462	11	γa	γa	PROPN
ejpam-3441	462	12	.	.	PUNCT
ejpam-3441	463	1	therefore	therefore	ADV
ejpam-3441	463	2	aβα	aβα	PROPN
ejpam-3441	463	3	=	=	PUNCT
ejpam-3441	463	4	a	a	DET
ejpam-3441	463	5	◦	◦	NOUN
ejpam-3441	463	6	βα	βα	NOUN
ejpam-3441	463	7	a.	a.	NOUN
ejpam-3441	463	8	remark	remark	NOUN
ejpam-3441	463	9	5	5	NUM
ejpam-3441	463	10	.	.	PUNCT
ejpam-3441	464	1	every	every	DET
ejpam-3441	464	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	464	3	fuzzy	fuzzy	ADJ
ejpam-3441	464	4	right	right	ADJ
ejpam-3441	464	5	ideal	ideal	NOUN
ejpam-3441	464	6	with	with	ADP
ejpam-3441	464	7	thresholds	threshold	NOUN
ejpam-3441	464	8	(	(	PUNCT
ejpam-3441	464	9	α	α	X
ejpam-3441	464	10	,	,	PUNCT
ejpam-3441	464	11	β	β	X
ejpam-3441	464	12	]	]	PUNCT
ejpam-3441	464	13	of	of	ADP
ejpam-3441	464	14	a	a	DET
ejpam-3441	464	15	regular	regular	ADJ
ejpam-3441	464	16	la	la	ADJ
ejpam-3441	464	17	-	-	PUNCT
ejpam-3441	464	18	ring	ring	NOUN
ejpam-3441	464	19	r	r	NOUN
ejpam-3441	464	20	is	be	AUX
ejpam-3441	464	21	an	an	DET
ejpam-3441	464	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	464	23	fuzzy	fuzzy	ADJ
ejpam-3441	464	24	idempotent	idempotent	NOUN
ejpam-3441	464	25	with	with	ADP
ejpam-3441	464	26	thresholds	threshold	NOUN
ejpam-3441	464	27	(	(	PUNCT
ejpam-3441	464	28	α	α	X
ejpam-3441	464	29	,	,	PUNCT
ejpam-3441	464	30	β	β	X
ejpam-3441	464	31	]	]	PUNCT
ejpam-3441	464	32	.	.	PUNCT
ejpam-3441	465	1	proposition	proposition	NOUN
ejpam-3441	465	2	6	6	NUM
ejpam-3441	465	3	.	.	PUNCT
ejpam-3441	466	1	let	let	VERB
ejpam-3441	466	2	a	a	PRON
ejpam-3441	466	3	=	=	SYM
ejpam-3441	466	4	(	(	PUNCT
ejpam-3441	466	5	µa	µa	PROPN
ejpam-3441	466	6	,	,	PUNCT
ejpam-3441	466	7	γa	γa	PROPN
ejpam-3441	466	8	)	)	PUNCT
ejpam-3441	466	9	be	be	VERB
ejpam-3441	466	10	an	an	DET
ejpam-3441	466	11	ifs	ifs	PROPN
ejpam-3441	466	12	of	of	ADP
ejpam-3441	466	13	a	a	DET
ejpam-3441	466	14	regular	regular	ADJ
ejpam-3441	466	15	la	la	ADJ
ejpam-3441	466	16	-	-	PUNCT
ejpam-3441	466	17	ring	ring	NOUN
ejpam-3441	466	18	r.	r.	PROPN
ejpam-3441	466	19	then	then	ADV
ejpam-3441	466	20	a	a	PRON
ejpam-3441	466	21	is	be	AUX
ejpam-3441	466	22	an	an	DET
ejpam-3441	466	23	intuitionistic	intuitionistic	ADJ
ejpam-3441	466	24	fuzzy	fuzzy	ADJ
ejpam-3441	466	25	ideal	ideal	NOUN
ejpam-3441	466	26	with	with	ADP
ejpam-3441	466	27	thresholds	threshold	NOUN
ejpam-3441	466	28	(	(	PUNCT
ejpam-3441	466	29	α	α	X
ejpam-3441	466	30	,	,	PUNCT
ejpam-3441	466	31	β	β	X
ejpam-3441	466	32	]	]	PUNCT
ejpam-3441	466	33	of	of	ADP
ejpam-3441	466	34	r	r	NOUN
ejpam-3441	466	35	if	if	SCONJ
ejpam-3441	467	1	and	and	CCONJ
ejpam-3441	467	2	only	only	ADV
ejpam-3441	467	3	if	if	SCONJ
ejpam-3441	467	4	a	a	PRON
ejpam-3441	467	5	is	be	AUX
ejpam-3441	467	6	an	an	DET
ejpam-3441	467	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	467	8	fuzzy	fuzzy	ADJ
ejpam-3441	467	9	interior	interior	ADJ
ejpam-3441	467	10	ideal	ideal	NOUN
ejpam-3441	467	11	with	with	ADP
ejpam-3441	467	12	thresholds	threshold	NOUN
ejpam-3441	467	13	(	(	PUNCT
ejpam-3441	467	14	α	α	X
ejpam-3441	467	15	,	,	PUNCT
ejpam-3441	467	16	β	β	X
ejpam-3441	467	17	]	]	PUNCT
ejpam-3441	467	18	of	of	ADP
ejpam-3441	467	19	r.	r.	PROPN
ejpam-3441	467	20	proof	proof	NOUN
ejpam-3441	467	21	.	.	PUNCT
ejpam-3441	468	1	consider	consider	VERB
ejpam-3441	468	2	that	that	PRON
ejpam-3441	468	3	a	a	DET
ejpam-3441	468	4	=	=	SYM
ejpam-3441	468	5	(	(	PUNCT
ejpam-3441	468	6	µa	µa	PROPN
ejpam-3441	468	7	,	,	PUNCT
ejpam-3441	468	8	γa	γa	PROPN
ejpam-3441	468	9	)	)	PUNCT
ejpam-3441	468	10	is	be	AUX
ejpam-3441	468	11	an	an	DET
ejpam-3441	468	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	468	13	fuzzy	fuzzy	ADJ
ejpam-3441	468	14	interior	interior	ADJ
ejpam-3441	468	15	ideal	ideal	NOUN
ejpam-3441	468	16	with	with	ADP
ejpam-3441	468	17	thresholds	threshold	NOUN
ejpam-3441	468	18	(	(	PUNCT
ejpam-3441	468	19	α	α	X
ejpam-3441	468	20	,	,	PUNCT
ejpam-3441	468	21	β	β	X
ejpam-3441	468	22	]	]	PUNCT
ejpam-3441	468	23	of	of	ADP
ejpam-3441	468	24	r.	r.	PROPN
ejpam-3441	468	25	let	let	VERB
ejpam-3441	468	26	x	x	PRON
ejpam-3441	468	27	,	,	PUNCT
ejpam-3441	468	28	y	y	PROPN
ejpam-3441	468	29	∈	∈	PROPN
ejpam-3441	468	30	r	r	NOUN
ejpam-3441	468	31	,	,	PUNCT
ejpam-3441	468	32	then	then	ADV
ejpam-3441	468	33	there	there	PRON
ejpam-3441	468	34	exists	exist	VERB
ejpam-3441	468	35	an	an	DET
ejpam-3441	468	36	element	element	NOUN
ejpam-3441	468	37	a	a	DET
ejpam-3441	468	38	∈	∈	PROPN
ejpam-3441	468	39	r	r	NOUN
ejpam-3441	468	40	,	,	PUNCT
ejpam-3441	468	41	such	such	ADJ
ejpam-3441	468	42	that	that	SCONJ
ejpam-3441	468	43	x	x	SYM
ejpam-3441	468	44	=	=	SYM
ejpam-3441	468	45	(	(	PUNCT
ejpam-3441	468	46	xa)x	xa)x	PROPN
ejpam-3441	468	47	.	.	PUNCT
ejpam-3441	469	1	thus	thus	ADV
ejpam-3441	469	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3441	469	3	)	)	PUNCT
ejpam-3441	469	4	,	,	PUNCT
ejpam-3441	469	5	α	α	X
ejpam-3441	469	6	}	}	PUNCT
ejpam-3441	469	7	=	=	SYM
ejpam-3441	469	8	max{µa(((xa)x)y	max{µa(((xa)x)y	PROPN
ejpam-3441	469	9	)	)	PUNCT
ejpam-3441	469	10	,	,	PUNCT
ejpam-3441	469	11	α	α	X
ejpam-3441	469	12	}	}	PUNCT
ejpam-3441	469	13	=	=	SYM
ejpam-3441	469	14	max{µa((yx)(xa	max{µa((yx)(xa	NOUN
ejpam-3441	469	15	)	)	PUNCT
ejpam-3441	469	16	)	)	PUNCT
ejpam-3441	469	17	,	,	PUNCT
ejpam-3441	469	18	α	α	X
ejpam-3441	469	19	}	}	PUNCT
ejpam-3441	469	20	≥	≥	NOUN
ejpam-3441	469	21	min{µa(x	min{µa(x	NOUN
ejpam-3441	469	22	)	)	PUNCT
ejpam-3441	469	23	,	,	PUNCT
ejpam-3441	469	24	β	β	X
ejpam-3441	469	25	}	}	PUNCT
ejpam-3441	469	26	and	and	CCONJ
ejpam-3441	469	27	min{γa(xy	min{γa(xy	NUM
ejpam-3441	469	28	)	)	PUNCT
ejpam-3441	469	29	,	,	PUNCT
ejpam-3441	469	30	(	(	PUNCT
ejpam-3441	469	31	1−	1−	NUM
ejpam-3441	469	32	α	α	NOUN
ejpam-3441	469	33	)	)	PUNCT
ejpam-3441	469	34	}	}	PUNCT
ejpam-3441	469	35	=	=	SYM
ejpam-3441	469	36	min{γa(((xa)x)y	min{γa(((xa)x)y	PROPN
ejpam-3441	469	37	)	)	PUNCT
ejpam-3441	469	38	,	,	PUNCT
ejpam-3441	469	39	(	(	PUNCT
ejpam-3441	469	40	1−	1−	NUM
ejpam-3441	469	41	α	α	NOUN
ejpam-3441	469	42	)	)	PUNCT
ejpam-3441	469	43	}	}	PUNCT
ejpam-3441	469	44	k.	k.	PROPN
ejpam-3441	470	1	nasreen	nasreen	PROPN
ejpam-3441	470	2	et	et	PROPN
ejpam-3441	470	3	al	al	PROPN
ejpam-3441	470	4	.	.	PUNCT
ejpam-3441	470	5	/	/	SYM
ejpam-3441	470	6	eur	eur	PROPN
ejpam-3441	470	7	.	.	PUNCT
ejpam-3441	471	1	j.	j.	PROPN
ejpam-3441	471	2	pure	pure	PROPN
ejpam-3441	471	3	appl	appl	PROPN
ejpam-3441	471	4	.	.	PROPN
ejpam-3441	471	5	math	math	PROPN
ejpam-3441	471	6	,	,	PUNCT
ejpam-3441	471	7	12	12	NUM
ejpam-3441	471	8	(	(	PUNCT
ejpam-3441	471	9	3	3	NUM
ejpam-3441	471	10	)	)	PUNCT
ejpam-3441	471	11	(	(	PUNCT
ejpam-3441	471	12	2019	2019	NUM
ejpam-3441	471	13	)	)	PUNCT
ejpam-3441	471	14	,	,	PUNCT
ejpam-3441	471	15	906	906	NUM
ejpam-3441	471	16	-	-	SYM
ejpam-3441	471	17	943	943	NUM
ejpam-3441	471	18	927	927	NUM
ejpam-3441	471	19	=	=	NOUN
ejpam-3441	471	20	min{γa((yx)(xa	min{γa((yx)(xa	NOUN
ejpam-3441	471	21	)	)	PUNCT
ejpam-3441	471	22	)	)	PUNCT
ejpam-3441	471	23	,	,	PUNCT
ejpam-3441	471	24	(	(	PUNCT
ejpam-3441	471	25	1−	1−	NUM
ejpam-3441	471	26	α	α	NOUN
ejpam-3441	471	27	)	)	PUNCT
ejpam-3441	471	28	}	}	PUNCT
ejpam-3441	471	29	≤	≤	NUM
ejpam-3441	471	30	max{γa(x	max{γa(x	NOUN
ejpam-3441	471	31	)	)	PUNCT
ejpam-3441	471	32	,	,	PUNCT
ejpam-3441	471	33	(	(	PUNCT
ejpam-3441	471	34	1−	1−	NUM
ejpam-3441	471	35	β	β	NOUN
ejpam-3441	471	36	)	)	PUNCT
ejpam-3441	471	37	}	}	PUNCT
ejpam-3441	471	38	.	.	PUNCT
ejpam-3441	472	1	consequently	consequently	ADV
ejpam-3441	472	2	a	a	PRON
ejpam-3441	472	3	is	be	AUX
ejpam-3441	472	4	an	an	DET
ejpam-3441	472	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	472	6	fuzzy	fuzzy	ADJ
ejpam-3441	472	7	right	right	ADJ
ejpam-3441	472	8	ideal	ideal	NOUN
ejpam-3441	472	9	with	with	ADP
ejpam-3441	472	10	thresholds	threshold	NOUN
ejpam-3441	472	11	(	(	PUNCT
ejpam-3441	472	12	α	α	X
ejpam-3441	472	13	,	,	PUNCT
ejpam-3441	472	14	β	β	X
ejpam-3441	472	15	]	]	PUNCT
ejpam-3441	472	16	of	of	ADP
ejpam-3441	472	17	r.	r.	PROPN
ejpam-3441	472	18	so	so	ADV
ejpam-3441	472	19	a	a	PRON
ejpam-3441	472	20	is	be	AUX
ejpam-3441	472	21	an	an	DET
ejpam-3441	472	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	472	23	fuzzy	fuzzy	ADJ
ejpam-3441	472	24	ideal	ideal	NOUN
ejpam-3441	472	25	with	with	ADP
ejpam-3441	472	26	thresholds	threshold	NOUN
ejpam-3441	472	27	(	(	PUNCT
ejpam-3441	472	28	α	α	X
ejpam-3441	472	29	,	,	PUNCT
ejpam-3441	472	30	β	β	X
ejpam-3441	472	31	]	]	PUNCT
ejpam-3441	472	32	of	of	ADP
ejpam-3441	472	33	r	r	NOUN
ejpam-3441	472	34	by	by	ADP
ejpam-3441	472	35	the	the	DET
ejpam-3441	472	36	lemma	lemma	PROPN
ejpam-3441	472	37	16	16	NUM
ejpam-3441	472	38	.	.	PUNCT
ejpam-3441	473	1	converse	converse	NOUN
ejpam-3441	473	2	is	be	AUX
ejpam-3441	473	3	true	true	ADJ
ejpam-3441	473	4	by	by	ADP
ejpam-3441	473	5	the	the	DET
ejpam-3441	473	6	lemma	lemma	PROPN
ejpam-3441	473	7	11	11	NUM
ejpam-3441	473	8	.	.	PUNCT
ejpam-3441	474	1	remark	remark	NOUN
ejpam-3441	474	2	6	6	NUM
ejpam-3441	474	3	.	.	PUNCT
ejpam-3441	475	1	the	the	DET
ejpam-3441	475	2	concept	concept	NOUN
ejpam-3441	475	3	of	of	ADP
ejpam-3441	475	4	intuitionistic	intuitionistic	ADJ
ejpam-3441	475	5	fuzzy	fuzzy	ADJ
ejpam-3441	475	6	(	(	PUNCT
ejpam-3441	475	7	interior	interior	ADJ
ejpam-3441	475	8	,	,	PUNCT
ejpam-3441	475	9	two	two	NUM
ejpam-3441	475	10	-	-	PUNCT
ejpam-3441	475	11	sided	sided	ADJ
ejpam-3441	475	12	)	)	PUNCT
ejpam-3441	475	13	ideals	ideal	NOUN
ejpam-3441	475	14	with	with	ADP
ejpam-3441	475	15	thresholds	threshold	NOUN
ejpam-3441	475	16	(	(	PUNCT
ejpam-3441	475	17	α	α	X
ejpam-3441	475	18	,	,	PUNCT
ejpam-3441	475	19	β	β	X
ejpam-3441	475	20	]	]	X
ejpam-3441	475	21	coincides	coincide	VERB
ejpam-3441	475	22	in	in	ADP
ejpam-3441	475	23	regular	regular	ADJ
ejpam-3441	475	24	la	la	ADJ
ejpam-3441	475	25	-	-	PUNCT
ejpam-3441	475	26	rings	ring	NOUN
ejpam-3441	475	27	.	.	PUNCT
ejpam-3441	476	1	proposition	proposition	NOUN
ejpam-3441	476	2	7	7	NUM
ejpam-3441	476	3	.	.	PUNCT
ejpam-3441	477	1	let	let	VERB
ejpam-3441	477	2	r	r	PRON
ejpam-3441	477	3	be	be	AUX
ejpam-3441	477	4	a	a	DET
ejpam-3441	477	5	regular	regular	ADJ
ejpam-3441	477	6	la	la	ADJ
ejpam-3441	477	7	-	-	PUNCT
ejpam-3441	477	8	ring	ring	NOUN
ejpam-3441	477	9	.	.	PUNCT
ejpam-3441	478	1	then	then	ADV
ejpam-3441	478	2	(	(	PUNCT
ejpam-3441	478	3	a	a	DET
ejpam-3441	478	4	◦	◦	NOUN
ejpam-3441	478	5	βα	βα	NOUN
ejpam-3441	478	6	r	r	NOUN
ejpam-3441	478	7	)	)	PUNCT
ejpam-3441	478	8	∧	∧	NOUN
ejpam-3441	478	9	(	(	PUNCT
ejpam-3441	478	10	r	r	NOUN
ejpam-3441	478	11	◦	◦	NOUN
ejpam-3441	478	12	βα	βα	X
ejpam-3441	478	13	a	a	NOUN
ejpam-3441	478	14	)	)	PUNCT
ejpam-3441	478	15	=	=	VERB
ejpam-3441	478	16	aβα	aβα	NOUN
ejpam-3441	478	17	for	for	ADP
ejpam-3441	478	18	every	every	DET
ejpam-3441	478	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	478	20	fuzzy	fuzzy	ADJ
ejpam-3441	478	21	right	right	ADJ
ejpam-3441	478	22	ideal	ideal	NOUN
ejpam-3441	478	23	a	a	PRON
ejpam-3441	478	24	with	with	ADP
ejpam-3441	478	25	thresholds	threshold	NOUN
ejpam-3441	478	26	(	(	PUNCT
ejpam-3441	478	27	α	α	X
ejpam-3441	478	28	,	,	PUNCT
ejpam-3441	478	29	β	β	X
ejpam-3441	478	30	]	]	PUNCT
ejpam-3441	478	31	of	of	ADP
ejpam-3441	478	32	r.	r.	PROPN
ejpam-3441	478	33	proof	proof	PROPN
ejpam-3441	478	34	.	.	PUNCT
ejpam-3441	479	1	suppose	suppose	VERB
ejpam-3441	479	2	that	that	SCONJ
ejpam-3441	479	3	a	a	DET
ejpam-3441	479	4	=	=	SYM
ejpam-3441	479	5	(	(	PUNCT
ejpam-3441	479	6	µa	µa	PROPN
ejpam-3441	479	7	,	,	PUNCT
ejpam-3441	479	8	γa	γa	PROPN
ejpam-3441	479	9	)	)	PUNCT
ejpam-3441	479	10	is	be	AUX
ejpam-3441	479	11	an	an	DET
ejpam-3441	479	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	479	13	fuzzy	fuzzy	ADJ
ejpam-3441	479	14	right	right	ADJ
ejpam-3441	479	15	ideal	ideal	NOUN
ejpam-3441	479	16	with	with	ADP
ejpam-3441	479	17	thresholds	threshold	NOUN
ejpam-3441	479	18	(	(	PUNCT
ejpam-3441	479	19	α	α	X
ejpam-3441	479	20	,	,	PUNCT
ejpam-3441	479	21	β	β	X
ejpam-3441	479	22	]	]	PUNCT
ejpam-3441	479	23	of	of	ADP
ejpam-3441	479	24	r.	r.	PROPN
ejpam-3441	479	25	this	this	PRON
ejpam-3441	479	26	implies	imply	VERB
ejpam-3441	479	27	that	that	SCONJ
ejpam-3441	479	28	(	(	PUNCT
ejpam-3441	479	29	a	a	DET
ejpam-3441	479	30	◦	◦	NOUN
ejpam-3441	479	31	βαr)∧	βαr)∧	X
ejpam-3441	479	32	(	(	PUNCT
ejpam-3441	479	33	r	r	NOUN
ejpam-3441	479	34	◦	◦	NOUN
ejpam-3441	479	35	βαa	βαa	NOUN
ejpam-3441	479	36	)	)	PUNCT
ejpam-3441	479	37	⊆	⊆	NUM
ejpam-3441	479	38	aβα	aβα	PROPN
ejpam-3441	479	39	,	,	PUNCT
ejpam-3441	479	40	because	because	SCONJ
ejpam-3441	479	41	every	every	DET
ejpam-3441	479	42	intuitionistic	intuitionistic	ADJ
ejpam-3441	479	43	fuzzy	fuzzy	ADJ
ejpam-3441	479	44	right	right	ADJ
ejpam-3441	479	45	ideal	ideal	NOUN
ejpam-3441	479	46	with	with	ADP
ejpam-3441	479	47	thresholds	threshold	NOUN
ejpam-3441	479	48	(	(	PUNCT
ejpam-3441	479	49	α	α	X
ejpam-3441	479	50	,	,	PUNCT
ejpam-3441	479	51	β	β	X
ejpam-3441	479	52	]	]	PUNCT
ejpam-3441	479	53	of	of	ADP
ejpam-3441	479	54	r	r	NOUN
ejpam-3441	479	55	is	be	AUX
ejpam-3441	479	56	an	an	DET
ejpam-3441	479	57	intuitionistic	intuitionistic	ADJ
ejpam-3441	479	58	fuzzy	fuzzy	ADJ
ejpam-3441	479	59	quasi	quasi	NOUN
ejpam-3441	479	60	-	-	NOUN
ejpam-3441	479	61	ideal	ideal	ADJ
ejpam-3441	479	62	with	with	ADP
ejpam-3441	479	63	thresholds	threshold	NOUN
ejpam-3441	479	64	(	(	PUNCT
ejpam-3441	479	65	α	α	X
ejpam-3441	479	66	,	,	PUNCT
ejpam-3441	479	67	β	β	X
ejpam-3441	479	68	]	]	PUNCT
ejpam-3441	479	69	of	of	ADP
ejpam-3441	479	70	r	r	NOUN
ejpam-3441	479	71	by	by	ADP
ejpam-3441	479	72	the	the	DET
ejpam-3441	479	73	lemma	lemma	PROPN
ejpam-3441	479	74	14	14	NUM
ejpam-3441	479	75	.	.	PUNCT
ejpam-3441	480	1	let	let	VERB
ejpam-3441	480	2	x	x	PUNCT
ejpam-3441	480	3	∈	∈	PROPN
ejpam-3441	480	4	r	r	NOUN
ejpam-3441	480	5	,	,	PUNCT
ejpam-3441	480	6	this	this	PRON
ejpam-3441	480	7	implies	imply	VERB
ejpam-3441	480	8	that	that	SCONJ
ejpam-3441	480	9	there	there	PRON
ejpam-3441	480	10	exists	exist	VERB
ejpam-3441	480	11	a	a	DET
ejpam-3441	480	12	∈	∈	PROPN
ejpam-3441	480	13	r	r	NOUN
ejpam-3441	480	14	,	,	PUNCT
ejpam-3441	481	1	such	such	ADJ
ejpam-3441	481	2	that	that	SCONJ
ejpam-3441	481	3	x	x	SYM
ejpam-3441	481	4	=	=	SYM
ejpam-3441	481	5	(	(	PUNCT
ejpam-3441	481	6	xa)x	xa)x	PROPN
ejpam-3441	481	7	.	.	PUNCT
ejpam-3441	482	1	thus	thus	ADV
ejpam-3441	482	2	(	(	PUNCT
ejpam-3441	482	3	µa	µa	ADP
ejpam-3441	482	4	◦	◦	NOUN
ejpam-3441	482	5	βα	βα	NOUN
ejpam-3441	482	6	r)(x	r)(x	PROPN
ejpam-3441	482	7	)	)	PUNCT
ejpam-3441	483	1	=	=	PRON
ejpam-3441	483	2	{	{	PUNCT
ejpam-3441	483	3	(	(	PUNCT
ejpam-3441	483	4	µa	µa	AUX
ejpam-3441	483	5	◦	◦	NOUN
ejpam-3441	483	6	r)(x	r)(x	NOUN
ejpam-3441	483	7	)	)	PUNCT
ejpam-3441	484	1	∧	∧	PROPN
ejpam-3441	484	2	β	β	NOUN
ejpam-3441	484	3	}	}	PUNCT
ejpam-3441	484	4	∨	∨	NUM
ejpam-3441	484	5	α	α	NOUN
ejpam-3441	484	6	=	=	X
ejpam-3441	484	7	{	{	PUNCT
ejpam-3441	484	8	(	(	PUNCT
ejpam-3441	484	9	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	484	10	i=1	i=1	PROPN
ejpam-3441	484	11	aibi	aibi	NOUN
ejpam-3441	484	12	{	{	PUNCT
ejpam-3441	484	13	∧ni=1	∧ni=1	X
ejpam-3441	484	14	{	{	PUNCT
ejpam-3441	484	15	µa	µa	X
ejpam-3441	484	16	(	(	PUNCT
ejpam-3441	484	17	ai	ai	NOUN
ejpam-3441	484	18	)	)	PUNCT
ejpam-3441	484	19	∧r	∧r	PROPN
ejpam-3441	484	20	(	(	PUNCT
ejpam-3441	484	21	bi	bi	NOUN
ejpam-3441	484	22	)	)	PUNCT
ejpam-3441	484	23	}	}	PUNCT
ejpam-3441	484	24	}	}	PUNCT
ejpam-3441	484	25	)	)	PUNCT
ejpam-3441	485	1	∧	∧	PROPN
ejpam-3441	485	2	β	β	NOUN
ejpam-3441	485	3	}	}	PUNCT
ejpam-3441	485	4	∨	∨	NUM
ejpam-3441	485	5	α	α	PROPN
ejpam-3441	485	6	≥	≥	X
ejpam-3441	485	7	{	{	PUNCT
ejpam-3441	485	8	{	{	PUNCT
ejpam-3441	485	9	µa	µa	PROPN
ejpam-3441	485	10	(	(	PUNCT
ejpam-3441	485	11	xa	xa	NOUN
ejpam-3441	485	12	)	)	PUNCT
ejpam-3441	485	13	∧r	∧r	PROPN
ejpam-3441	485	14	(	(	PUNCT
ejpam-3441	485	15	x	x	NOUN
ejpam-3441	485	16	)	)	PUNCT
ejpam-3441	485	17	}	}	PUNCT
ejpam-3441	485	18	∧	∧	PROPN
ejpam-3441	485	19	β	β	NOUN
ejpam-3441	485	20	}	}	PUNCT
ejpam-3441	485	21	∨	∨	NUM
ejpam-3441	485	22	α	α	X
ejpam-3441	485	23	=	=	SYM
ejpam-3441	485	24	{	{	PUNCT
ejpam-3441	485	25	µa	µa	X
ejpam-3441	485	26	(	(	PUNCT
ejpam-3441	485	27	xa	xa	NOUN
ejpam-3441	485	28	)	)	PUNCT
ejpam-3441	485	29	∧	∧	PROPN
ejpam-3441	485	30	β	β	PROPN
ejpam-3441	485	31	}	}	PUNCT
ejpam-3441	485	32	∨	∨	NUM
ejpam-3441	485	33	α	α	NOUN
ejpam-3441	485	34	=	=	SYM
ejpam-3441	485	35	(	(	PUNCT
ejpam-3441	485	36	µa	µa	X
ejpam-3441	485	37	(	(	PUNCT
ejpam-3441	485	38	xa	xa	PROPN
ejpam-3441	485	39	)	)	PUNCT
ejpam-3441	485	40	∨	∨	NUM
ejpam-3441	485	41	α	α	NOUN
ejpam-3441	485	42	)	)	PUNCT
ejpam-3441	485	43	∧	∧	PROPN
ejpam-3441	485	44	(	(	PUNCT
ejpam-3441	485	45	β	β	X
ejpam-3441	485	46	∨	∨	NUM
ejpam-3441	485	47	α	α	NOUN
ejpam-3441	485	48	)	)	PUNCT
ejpam-3441	485	49	≥	≥	NOUN
ejpam-3441	485	50	(	(	PUNCT
ejpam-3441	485	51	µa	µa	PROPN
ejpam-3441	485	52	(	(	PUNCT
ejpam-3441	485	53	x	x	NOUN
ejpam-3441	485	54	)	)	PUNCT
ejpam-3441	485	55	∧	∧	PROPN
ejpam-3441	485	56	β	β	NOUN
ejpam-3441	485	57	)	)	PUNCT
ejpam-3441	485	58	∧	∧	NOUN
ejpam-3441	485	59	β	β	X
ejpam-3441	485	60	=	=	SYM
ejpam-3441	485	61	µa	µa	X
ejpam-3441	485	62	(	(	PUNCT
ejpam-3441	485	63	x	x	X
ejpam-3441	485	64	)	)	PUNCT
ejpam-3441	485	65	∧	∧	NOUN
ejpam-3441	485	66	β	β	X
ejpam-3441	485	67	=	=	SYM
ejpam-3441	485	68	(	(	PUNCT
ejpam-3441	485	69	µa	µa	INTJ
ejpam-3441	485	70	(	(	PUNCT
ejpam-3441	485	71	x	x	NOUN
ejpam-3441	485	72	)	)	PUNCT
ejpam-3441	485	73	∧	∧	PROPN
ejpam-3441	485	74	β	β	NOUN
ejpam-3441	485	75	)	)	PUNCT
ejpam-3441	485	76	∨	∨	NUM
ejpam-3441	485	77	α	α	NOUN
ejpam-3441	485	78	=	=	SYM
ejpam-3441	485	79	(	(	PUNCT
ejpam-3441	485	80	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	485	81	)	)	PUNCT
ejpam-3441	485	82	.	.	PUNCT
ejpam-3441	486	1	⇒	⇒	NOUN
ejpam-3441	486	2	(	(	PUNCT
ejpam-3441	486	3	µa)βα	µa)βα	NUM
ejpam-3441	486	4	⊆	⊆	NUM
ejpam-3441	486	5	µa	µa	NOUN
ejpam-3441	486	6	◦	◦	NOUN
ejpam-3441	486	7	βα	βα	NOUN
ejpam-3441	486	8	r.	r.	PROPN
ejpam-3441	486	9	similarly	similarly	ADV
ejpam-3441	486	10	,	,	PUNCT
ejpam-3441	486	11	we	we	PRON
ejpam-3441	486	12	have	have	VERB
ejpam-3441	486	13	(	(	PUNCT
ejpam-3441	486	14	µa)βα	µa)βα	NUM
ejpam-3441	486	15	⊆	⊆	NUM
ejpam-3441	486	16	r	r	NOUN
ejpam-3441	486	17	◦	◦	NOUN
ejpam-3441	486	18	βα	βα	NOUN
ejpam-3441	486	19	µa	µa	NOUN
ejpam-3441	486	20	,	,	PUNCT
ejpam-3441	486	21	i.e.	i.e.	X
ejpam-3441	486	22	,	,	PUNCT
ejpam-3441	486	23	(	(	PUNCT
ejpam-3441	486	24	µa)βα	µa)βα	NUM
ejpam-3441	486	25	⊆	⊆	NUM
ejpam-3441	486	26	(	(	PUNCT
ejpam-3441	486	27	µa	µa	ADP
ejpam-3441	486	28	◦	◦	NOUN
ejpam-3441	486	29	βαr)∧	βαr)∧	X
ejpam-3441	486	30	(	(	PUNCT
ejpam-3441	486	31	r	r	NOUN
ejpam-3441	486	32	◦	◦	NOUN
ejpam-3441	486	33	βα	βα	NOUN
ejpam-3441	486	34	µa	µa	NOUN
ejpam-3441	486	35	)	)	PUNCT
ejpam-3441	486	36	.	.	PUNCT
ejpam-3441	487	1	in	in	ADP
ejpam-3441	487	2	same	same	ADJ
ejpam-3441	487	3	lines	line	NOUN
ejpam-3441	487	4	,	,	PUNCT
ejpam-3441	487	5	we	we	PRON
ejpam-3441	487	6	have	have	VERB
ejpam-3441	487	7	(	(	PUNCT
ejpam-3441	487	8	γa)βα	γa)βα	X
ejpam-3441	487	9	⊇	⊇	X
ejpam-3441	487	10	(	(	PUNCT
ejpam-3441	487	11	γa	γa	PROPN
ejpam-3441	487	12	◦	◦	PROPN
ejpam-3441	487	13	βα	βα	X
ejpam-3441	487	14	r	r	NOUN
ejpam-3441	487	15	)	)	PUNCT
ejpam-3441	487	16	∨	∨	NOUN
ejpam-3441	487	17	(	(	PUNCT
ejpam-3441	487	18	r	r	NOUN
ejpam-3441	487	19	◦	◦	NOUN
ejpam-3441	487	20	βα	βα	NOUN
ejpam-3441	487	21	γa	γa	NOUN
ejpam-3441	487	22	)	)	PUNCT
ejpam-3441	487	23	.	.	PUNCT
ejpam-3441	488	1	hence	hence	ADV
ejpam-3441	488	2	(	(	PUNCT
ejpam-3441	488	3	a	a	DET
ejpam-3441	488	4	◦	◦	NOUN
ejpam-3441	488	5	βα	βα	NOUN
ejpam-3441	488	6	r	r	NOUN
ejpam-3441	488	7	)	)	PUNCT
ejpam-3441	488	8	∧	∧	NOUN
ejpam-3441	488	9	(	(	PUNCT
ejpam-3441	488	10	r	r	NOUN
ejpam-3441	488	11	◦	◦	NOUN
ejpam-3441	488	12	βα	βα	X
ejpam-3441	488	13	a	a	NOUN
ejpam-3441	488	14	)	)	PUNCT
ejpam-3441	488	15	=	=	SYM
ejpam-3441	488	16	aβα	aβα	PROPN
ejpam-3441	488	17	.	.	PUNCT
ejpam-3441	489	1	lemma	lemma	PROPN
ejpam-3441	489	2	18	18	NUM
ejpam-3441	489	3	.	.	PUNCT
ejpam-3441	490	1	let	let	VERB
ejpam-3441	490	2	r	r	PRON
ejpam-3441	490	3	be	be	AUX
ejpam-3441	490	4	a	a	DET
ejpam-3441	490	5	regular	regular	ADJ
ejpam-3441	490	6	la	la	ADJ
ejpam-3441	490	7	-	-	PUNCT
ejpam-3441	490	8	ring	ring	NOUN
ejpam-3441	490	9	.	.	PUNCT
ejpam-3441	491	1	then	then	ADV
ejpam-3441	491	2	a	a	DET
ejpam-3441	491	3	◦	◦	NOUN
ejpam-3441	491	4	βα	βα	NOUN
ejpam-3441	491	5	b	b	NOUN
ejpam-3441	491	6	=	=	PUNCT
ejpam-3441	491	7	a	a	DET
ejpam-3441	491	8	∧βα	∧βα	PROPN
ejpam-3441	491	9	b	b	NOUN
ejpam-3441	491	10	for	for	ADP
ejpam-3441	491	11	every	every	DET
ejpam-3441	491	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	491	13	fuzzy	fuzzy	ADJ
ejpam-3441	491	14	right	right	ADJ
ejpam-3441	491	15	ideal	ideal	NOUN
ejpam-3441	491	16	a	a	PRON
ejpam-3441	491	17	=	=	X
ejpam-3441	491	18	(	(	PUNCT
ejpam-3441	491	19	µa	µa	PROPN
ejpam-3441	491	20	,	,	PUNCT
ejpam-3441	491	21	γa	γa	PROPN
ejpam-3441	491	22	)	)	PUNCT
ejpam-3441	491	23	with	with	ADP
ejpam-3441	491	24	thresholds	threshold	NOUN
ejpam-3441	491	25	(	(	PUNCT
ejpam-3441	491	26	α	α	X
ejpam-3441	491	27	,	,	PUNCT
ejpam-3441	491	28	β	β	X
ejpam-3441	491	29	]	]	PUNCT
ejpam-3441	491	30	and	and	CCONJ
ejpam-3441	491	31	every	every	DET
ejpam-3441	491	32	intuitionistic	intuitionistic	ADJ
ejpam-3441	491	33	fuzzy	fuzzy	ADJ
ejpam-3441	491	34	left	leave	VERB
ejpam-3441	491	35	ideal	ideal	PROPN
ejpam-3441	491	36	b	b	PROPN
ejpam-3441	492	1	=	=	PUNCT
ejpam-3441	492	2	(	(	PUNCT
ejpam-3441	492	3	µb	µb	PROPN
ejpam-3441	492	4	,	,	PUNCT
ejpam-3441	492	5	γb	γb	PROPN
ejpam-3441	492	6	)	)	PUNCT
ejpam-3441	492	7	with	with	ADP
ejpam-3441	492	8	thresholds	threshold	NOUN
ejpam-3441	492	9	(	(	PUNCT
ejpam-3441	492	10	α	α	X
ejpam-3441	492	11	,	,	PUNCT
ejpam-3441	492	12	β	β	X
ejpam-3441	492	13	]	]	PUNCT
ejpam-3441	492	14	of	of	ADP
ejpam-3441	492	15	r.	r.	PROPN
ejpam-3441	492	16	proof	proof	NOUN
ejpam-3441	492	17	.	.	PUNCT
ejpam-3441	493	1	since	since	SCONJ
ejpam-3441	493	2	a	a	DET
ejpam-3441	493	3	◦	◦	NOUN
ejpam-3441	493	4	βα	βα	NOUN
ejpam-3441	493	5	b	b	NOUN
ejpam-3441	493	6	⊆	⊆	NUM
ejpam-3441	493	7	a	a	DET
ejpam-3441	493	8	∧βα	∧βα	PROPN
ejpam-3441	493	9	b	b	NOUN
ejpam-3441	493	10	,	,	PUNCT
ejpam-3441	493	11	for	for	ADP
ejpam-3441	493	12	every	every	DET
ejpam-3441	493	13	intuitionistic	intuitionistic	ADJ
ejpam-3441	493	14	fuzzy	fuzzy	ADJ
ejpam-3441	493	15	right	right	ADJ
ejpam-3441	493	16	ideal	ideal	NOUN
ejpam-3441	493	17	a	a	PRON
ejpam-3441	493	18	=	=	X
ejpam-3441	493	19	(	(	PUNCT
ejpam-3441	493	20	µa	µa	PROPN
ejpam-3441	493	21	,	,	PUNCT
ejpam-3441	493	22	γa	γa	PROPN
ejpam-3441	493	23	)	)	PUNCT
ejpam-3441	493	24	with	with	ADP
ejpam-3441	493	25	thresholds	threshold	NOUN
ejpam-3441	493	26	(	(	PUNCT
ejpam-3441	493	27	α	α	X
ejpam-3441	493	28	,	,	PUNCT
ejpam-3441	493	29	β	β	X
ejpam-3441	493	30	]	]	PUNCT
ejpam-3441	493	31	and	and	CCONJ
ejpam-3441	493	32	every	every	DET
ejpam-3441	493	33	intuitionistic	intuitionistic	ADJ
ejpam-3441	493	34	fuzzy	fuzzy	ADJ
ejpam-3441	493	35	left	leave	VERB
ejpam-3441	493	36	ideal	ideal	PROPN
ejpam-3441	493	37	b	b	PROPN
ejpam-3441	493	38	=	=	PUNCT
ejpam-3441	493	39	(	(	PUNCT
ejpam-3441	493	40	µb	µb	PROPN
ejpam-3441	493	41	,	,	PUNCT
ejpam-3441	493	42	γb	γb	PROPN
ejpam-3441	493	43	)	)	PUNCT
ejpam-3441	493	44	with	with	ADP
ejpam-3441	493	45	thresholds	threshold	NOUN
ejpam-3441	493	46	(	(	PUNCT
ejpam-3441	493	47	α	α	X
ejpam-3441	493	48	,	,	PUNCT
ejpam-3441	493	49	β	β	X
ejpam-3441	493	50	]	]	PUNCT
ejpam-3441	493	51	of	of	ADP
ejpam-3441	493	52	r	r	NOUN
ejpam-3441	493	53	by	by	ADP
ejpam-3441	493	54	the	the	DET
ejpam-3441	493	55	lemma	lemma	PROPN
ejpam-3441	493	56	8	8	NUM
ejpam-3441	493	57	.	.	PUNCT
ejpam-3441	493	58	let	let	VERB
ejpam-3441	493	59	x	x	PUNCT
ejpam-3441	493	60	∈	∈	PROPN
ejpam-3441	493	61	r	r	NOUN
ejpam-3441	493	62	,	,	PUNCT
ejpam-3441	493	63	this	this	PRON
ejpam-3441	493	64	means	mean	VERB
ejpam-3441	493	65	that	that	SCONJ
ejpam-3441	493	66	there	there	PRON
ejpam-3441	493	67	exists	exist	VERB
ejpam-3441	493	68	a	a	DET
ejpam-3441	493	69	∈	∈	NOUN
ejpam-3441	493	70	r	r	NOUN
ejpam-3441	493	71	such	such	ADJ
ejpam-3441	493	72	that	that	PRON
ejpam-3441	493	73	x	x	SYM
ejpam-3441	493	74	=	=	SYM
ejpam-3441	493	75	(	(	PUNCT
ejpam-3441	493	76	xa)x	xa)x	PROPN
ejpam-3441	493	77	.	.	PUNCT
ejpam-3441	494	1	thus	thus	ADV
ejpam-3441	494	2	(	(	PUNCT
ejpam-3441	494	3	µa	µa	ADP
ejpam-3441	494	4	◦	◦	NOUN
ejpam-3441	494	5	βα	βα	NOUN
ejpam-3441	494	6	µb)(x	µb)(x	NOUN
ejpam-3441	494	7	)	)	PUNCT
ejpam-3441	494	8	=	=	PRON
ejpam-3441	494	9	{	{	PUNCT
ejpam-3441	494	10	(	(	PUNCT
ejpam-3441	494	11	µa	µa	ADP
ejpam-3441	494	12	◦	◦	NOUN
ejpam-3441	494	13	µb)(x	µb)(x	NOUN
ejpam-3441	494	14	)	)	PUNCT
ejpam-3441	494	15	∧	∧	PROPN
ejpam-3441	494	16	β	β	NOUN
ejpam-3441	494	17	}	}	PUNCT
ejpam-3441	494	18	∨	∨	NUM
ejpam-3441	494	19	α	α	NOUN
ejpam-3441	494	20	=	=	X
ejpam-3441	494	21	{	{	PUNCT
ejpam-3441	494	22	(	(	PUNCT
ejpam-3441	494	23	∨x=∑n	∨x=∑n	NOUN
ejpam-3441	494	24	i=1	i=1	PROPN
ejpam-3441	494	25	aibi	aibi	NOUN
ejpam-3441	494	26	{	{	PUNCT
ejpam-3441	494	27	∧ni=1	∧ni=1	X
ejpam-3441	494	28	{	{	PUNCT
ejpam-3441	494	29	µa	µa	X
ejpam-3441	494	30	(	(	PUNCT
ejpam-3441	494	31	ai	ai	NOUN
ejpam-3441	494	32	)	)	PUNCT
ejpam-3441	494	33	∧	∧	NOUN
ejpam-3441	494	34	µb	µb	PROPN
ejpam-3441	494	35	(	(	PUNCT
ejpam-3441	494	36	bi	bi	NOUN
ejpam-3441	494	37	)	)	PUNCT
ejpam-3441	494	38	}	}	PUNCT
ejpam-3441	494	39	}	}	PUNCT
ejpam-3441	494	40	)	)	PUNCT
ejpam-3441	494	41	∧	∧	PROPN
ejpam-3441	494	42	β	β	NOUN
ejpam-3441	494	43	}	}	PUNCT
ejpam-3441	494	44	∨	∨	NUM
ejpam-3441	494	45	α	α	PROPN
ejpam-3441	494	46	≥	≥	X
ejpam-3441	494	47	{	{	PUNCT
ejpam-3441	494	48	{	{	PUNCT
ejpam-3441	494	49	µa	µa	PROPN
ejpam-3441	494	50	(	(	PUNCT
ejpam-3441	494	51	xa	xa	NOUN
ejpam-3441	494	52	)	)	PUNCT
ejpam-3441	494	53	∧	∧	PROPN
ejpam-3441	494	54	µb	µb	VERB
ejpam-3441	494	55	(	(	PUNCT
ejpam-3441	494	56	x	x	NOUN
ejpam-3441	494	57	)	)	PUNCT
ejpam-3441	494	58	}	}	PUNCT
ejpam-3441	494	59	∧	∧	PROPN
ejpam-3441	494	60	β	β	NOUN
ejpam-3441	494	61	}	}	PUNCT
ejpam-3441	494	62	∨	∨	NUM
ejpam-3441	494	63	α	α	NOUN
ejpam-3441	494	64	=	=	SYM
ejpam-3441	494	65	(	(	PUNCT
ejpam-3441	494	66	µa	µa	X
ejpam-3441	494	67	(	(	PUNCT
ejpam-3441	494	68	xa	xa	PROPN
ejpam-3441	494	69	)	)	PUNCT
ejpam-3441	494	70	∨	∨	NUM
ejpam-3441	494	71	α	α	NOUN
ejpam-3441	494	72	)	)	PUNCT
ejpam-3441	494	73	∧	∧	NOUN
ejpam-3441	494	74	(	(	PUNCT
ejpam-3441	494	75	µb	µb	PROPN
ejpam-3441	494	76	(	(	PUNCT
ejpam-3441	494	77	x	x	NOUN
ejpam-3441	494	78	)	)	PUNCT
ejpam-3441	494	79	∨	∨	NUM
ejpam-3441	494	80	α	α	NOUN
ejpam-3441	494	81	)	)	PUNCT
ejpam-3441	494	82	∧	∧	PROPN
ejpam-3441	494	83	(	(	PUNCT
ejpam-3441	494	84	β	β	X
ejpam-3441	494	85	∨	∨	NUM
ejpam-3441	494	86	α	α	NOUN
ejpam-3441	494	87	)	)	PUNCT
ejpam-3441	494	88	k.	k.	PROPN
ejpam-3441	495	1	nasreen	nasreen	PROPN
ejpam-3441	495	2	et	et	PROPN
ejpam-3441	495	3	al	al	PROPN
ejpam-3441	495	4	.	.	PUNCT
ejpam-3441	495	5	/	/	SYM
ejpam-3441	495	6	eur	eur	PROPN
ejpam-3441	495	7	.	.	PUNCT
ejpam-3441	496	1	j.	j.	PROPN
ejpam-3441	496	2	pure	pure	PROPN
ejpam-3441	496	3	appl	appl	PROPN
ejpam-3441	496	4	.	.	PROPN
ejpam-3441	496	5	math	math	PROPN
ejpam-3441	496	6	,	,	PUNCT
ejpam-3441	496	7	12	12	NUM
ejpam-3441	496	8	(	(	PUNCT
ejpam-3441	496	9	3	3	NUM
ejpam-3441	496	10	)	)	PUNCT
ejpam-3441	496	11	(	(	PUNCT
ejpam-3441	496	12	2019	2019	NUM
ejpam-3441	496	13	)	)	PUNCT
ejpam-3441	496	14	,	,	PUNCT
ejpam-3441	496	15	906	906	NUM
ejpam-3441	496	16	-	-	SYM
ejpam-3441	496	17	943	943	NUM
ejpam-3441	496	18	928	928	NUM
ejpam-3441	496	19	≥	≥	NOUN
ejpam-3441	496	20	(	(	PUNCT
ejpam-3441	496	21	µa	µa	X
ejpam-3441	496	22	(	(	PUNCT
ejpam-3441	496	23	x	x	NOUN
ejpam-3441	496	24	)	)	PUNCT
ejpam-3441	496	25	∧	∧	PROPN
ejpam-3441	496	26	β	β	NOUN
ejpam-3441	496	27	)	)	PUNCT
ejpam-3441	496	28	∧	∧	NOUN
ejpam-3441	496	29	µb	µb	ADP
ejpam-3441	496	30	(	(	PUNCT
ejpam-3441	496	31	x	x	NOUN
ejpam-3441	496	32	)	)	PUNCT
ejpam-3441	496	33	∧	∧	NOUN
ejpam-3441	496	34	β	β	X
ejpam-3441	496	35	=	=	SYM
ejpam-3441	496	36	µa	µa	X
ejpam-3441	496	37	(	(	PUNCT
ejpam-3441	496	38	x	x	X
ejpam-3441	496	39	)	)	PUNCT
ejpam-3441	496	40	∧	∧	NOUN
ejpam-3441	496	41	µb	µb	ADP
ejpam-3441	496	42	(	(	PUNCT
ejpam-3441	496	43	x	x	NOUN
ejpam-3441	496	44	)	)	PUNCT
ejpam-3441	496	45	∧	∧	NOUN
ejpam-3441	496	46	β	β	X
ejpam-3441	496	47	=	=	SYM
ejpam-3441	496	48	(	(	PUNCT
ejpam-3441	496	49	µa	µa	SCONJ
ejpam-3441	496	50	∧	∧	PROPN
ejpam-3441	496	51	µb	µb	PROPN
ejpam-3441	496	52	)	)	PUNCT
ejpam-3441	496	53	(	(	PUNCT
ejpam-3441	496	54	x	x	X
ejpam-3441	496	55	)	)	PUNCT
ejpam-3441	496	56	∧	∧	NOUN
ejpam-3441	496	57	β	β	X
ejpam-3441	496	58	=	=	SYM
ejpam-3441	496	59	{	{	PUNCT
ejpam-3441	496	60	(	(	PUNCT
ejpam-3441	496	61	µa	µa	ADP
ejpam-3441	496	62	∧	∧	PROPN
ejpam-3441	496	63	µb	µb	PROPN
ejpam-3441	496	64	)	)	PUNCT
ejpam-3441	496	65	(	(	PUNCT
ejpam-3441	496	66	x	x	X
ejpam-3441	496	67	)	)	PUNCT
ejpam-3441	496	68	∧	∧	PROPN
ejpam-3441	496	69	β	β	NOUN
ejpam-3441	496	70	}	}	PUNCT
ejpam-3441	496	71	∨	∨	NUM
ejpam-3441	496	72	α	α	NOUN
ejpam-3441	496	73	=	=	PUNCT
ejpam-3441	496	74	(	(	PUNCT
ejpam-3441	496	75	µa	µa	ADP
ejpam-3441	496	76	∧βα	∧βα	ADJ
ejpam-3441	496	77	µb)(x	µb)(x	NOUN
ejpam-3441	496	78	)	)	PUNCT
ejpam-3441	496	79	.	.	PUNCT
ejpam-3441	497	1	⇒	⇒	NOUN
ejpam-3441	497	2	µa	µa	ADP
ejpam-3441	497	3	∧βα	∧βα	PROPN
ejpam-3441	497	4	µb	µb	VERB
ejpam-3441	497	5	⊆	⊆	NUM
ejpam-3441	497	6	µa	µa	NOUN
ejpam-3441	497	7	◦	◦	NOUN
ejpam-3441	497	8	βα	βα	NOUN
ejpam-3441	497	9	µb	µb	PROPN
ejpam-3441	497	10	.	.	PUNCT
ejpam-3441	498	1	similarly	similarly	ADV
ejpam-3441	498	2	,	,	PUNCT
ejpam-3441	498	3	we	we	PRON
ejpam-3441	498	4	have	have	VERB
ejpam-3441	498	5	γa∨βαγb	γa∨βαγb	PUNCT
ejpam-3441	499	1	⊇	⊇	PROPN
ejpam-3441	499	2	γa	γa	PROPN
ejpam-3441	499	3	◦	◦	NOUN
ejpam-3441	499	4	βαγb	βαγb	ADJ
ejpam-3441	499	5	,	,	PUNCT
ejpam-3441	499	6	i.e.	i.e.	X
ejpam-3441	499	7	,	,	PUNCT
ejpam-3441	499	8	a∧βαb	a∧βαb	CCONJ
ejpam-3441	499	9	⊆	⊆	NUM
ejpam-3441	499	10	a	a	DET
ejpam-3441	499	11	◦	◦	NOUN
ejpam-3441	499	12	βαb	βαb	NOUN
ejpam-3441	499	13	.	.	PUNCT
ejpam-3441	500	1	therefore	therefore	ADV
ejpam-3441	500	2	a	a	DET
ejpam-3441	500	3	◦	◦	NOUN
ejpam-3441	500	4	βαb	βαb	NOUN
ejpam-3441	500	5	=	=	NOUN
ejpam-3441	500	6	a∧βαb	a∧βαb	NUM
ejpam-3441	500	7	.	.	PUNCT
ejpam-3441	501	1	lemma	lemma	PROPN
ejpam-3441	501	2	19	19	NUM
ejpam-3441	501	3	.	.	PUNCT
ejpam-3441	502	1	let	let	VERB
ejpam-3441	502	2	r	r	PRON
ejpam-3441	502	3	be	be	AUX
ejpam-3441	502	4	an	an	DET
ejpam-3441	502	5	la	la	NOUN
ejpam-3441	502	6	-	-	NOUN
ejpam-3441	502	7	ring	ring	NOUN
ejpam-3441	502	8	with	with	ADP
ejpam-3441	502	9	left	left	ADJ
ejpam-3441	502	10	identity	identity	NOUN
ejpam-3441	502	11	e.	e.	PROPN
ejpam-3441	502	12	then	then	ADV
ejpam-3441	502	13	ra	ra	PROPN
ejpam-3441	502	14	is	be	AUX
ejpam-3441	502	15	the	the	DET
ejpam-3441	502	16	smallest	small	ADJ
ejpam-3441	502	17	left	leave	VERB
ejpam-3441	502	18	ideal	ideal	NOUN
ejpam-3441	502	19	of	of	ADP
ejpam-3441	502	20	r	r	NOUN
ejpam-3441	502	21	containing	contain	VERB
ejpam-3441	502	22	a.	a.	NOUN
ejpam-3441	502	23	proof	proof	NOUN
ejpam-3441	502	24	.	.	PUNCT
ejpam-3441	503	1	let	let	VERB
ejpam-3441	503	2	x	x	PRON
ejpam-3441	503	3	,	,	PUNCT
ejpam-3441	503	4	y	y	PROPN
ejpam-3441	503	5	∈	∈	PROPN
ejpam-3441	503	6	ra	ra	PROPN
ejpam-3441	503	7	and	and	CCONJ
ejpam-3441	503	8	r	r	PROPN
ejpam-3441	503	9	∈	∈	PROPN
ejpam-3441	503	10	r.	r.	NOUN
ejpam-3441	503	11	this	this	PRON
ejpam-3441	503	12	implies	imply	VERB
ejpam-3441	503	13	that	that	SCONJ
ejpam-3441	504	1	x	x	X
ejpam-3441	504	2	=	=	PUNCT
ejpam-3441	504	3	r1a	r1a	NOUN
ejpam-3441	504	4	and	and	CCONJ
ejpam-3441	504	5	y	y	PROPN
ejpam-3441	504	6	=	=	SYM
ejpam-3441	504	7	r2a	r2a	PROPN
ejpam-3441	504	8	,	,	PUNCT
ejpam-3441	504	9	where	where	SCONJ
ejpam-3441	504	10	r1	r1	PROPN
ejpam-3441	504	11	,	,	PUNCT
ejpam-3441	504	12	r2	r2	PROPN
ejpam-3441	504	13	∈	∈	PROPN
ejpam-3441	504	14	r.	r.	PROPN
ejpam-3441	504	15	now	now	ADV
ejpam-3441	504	16	x−	x−	PROPN
ejpam-3441	505	1	y	y	PROPN
ejpam-3441	505	2	=	=	PUNCT
ejpam-3441	506	1	r1a−	r1a−	ADJ
ejpam-3441	506	2	r2a	r2a	NOUN
ejpam-3441	506	3	=	=	PUNCT
ejpam-3441	506	4	(	(	PUNCT
ejpam-3441	506	5	r1	r1	PROPN
ejpam-3441	506	6	−	−	PROPN
ejpam-3441	506	7	r2)a	r2)a	NOUN
ejpam-3441	506	8	∈	∈	PROPN
ejpam-3441	506	9	ra	ra	PROPN
ejpam-3441	506	10	and	and	CCONJ
ejpam-3441	506	11	rx	rx	VERB
ejpam-3441	506	12	=	=	NOUN
ejpam-3441	506	13	r(r1a	r(r1a	NOUN
ejpam-3441	506	14	)	)	PUNCT
ejpam-3441	506	15	=	=	SYM
ejpam-3441	506	16	(	(	PUNCT
ejpam-3441	506	17	er)(r1a	er)(r1a	NOUN
ejpam-3441	506	18	)	)	PUNCT
ejpam-3441	506	19	=	=	SYM
ejpam-3441	506	20	(	(	PUNCT
ejpam-3441	506	21	(	(	PUNCT
ejpam-3441	506	22	r1a)r)e	r1a)r)e	NOUN
ejpam-3441	506	23	=	=	SYM
ejpam-3441	506	24	(	(	PUNCT
ejpam-3441	506	25	(	(	PUNCT
ejpam-3441	506	26	r1a)(er))e	r1a)(er))e	NOUN
ejpam-3441	506	27	=	=	SYM
ejpam-3441	506	28	(	(	PUNCT
ejpam-3441	506	29	(	(	PUNCT
ejpam-3441	506	30	r1e)(ar))e	r1e)(ar))e	NOUN
ejpam-3441	506	31	=	=	SYM
ejpam-3441	506	32	(	(	PUNCT
ejpam-3441	506	33	e(ar))(r1e	e(ar))(r1e	PROPN
ejpam-3441	506	34	)	)	PUNCT
ejpam-3441	506	35	=	=	SYM
ejpam-3441	506	36	(	(	PUNCT
ejpam-3441	506	37	ar)(r1e	ar)(r1e	X
ejpam-3441	506	38	)	)	PUNCT
ejpam-3441	506	39	=	=	SYM
ejpam-3441	506	40	(	(	PUNCT
ejpam-3441	506	41	(	(	PUNCT
ejpam-3441	506	42	r1e)r)a	r1e)r)a	PROPN
ejpam-3441	506	43	∈	∈	PROPN
ejpam-3441	506	44	ra	ra	PROPN
ejpam-3441	506	45	.	.	PUNCT
ejpam-3441	507	1	since	since	SCONJ
ejpam-3441	507	2	a	a	DET
ejpam-3441	507	3	=	=	SYM
ejpam-3441	507	4	ea	ea	PROPN
ejpam-3441	507	5	∈	∈	PROPN
ejpam-3441	507	6	ra	ra	PROPN
ejpam-3441	507	7	.	.	PUNCT
ejpam-3441	508	1	thus	thus	ADV
ejpam-3441	508	2	ra	ra	PROPN
ejpam-3441	508	3	is	be	AUX
ejpam-3441	508	4	a	a	DET
ejpam-3441	508	5	left	left	ADJ
ejpam-3441	508	6	ideal	ideal	NOUN
ejpam-3441	508	7	of	of	ADP
ejpam-3441	508	8	r	r	NOUN
ejpam-3441	508	9	containing	contain	VERB
ejpam-3441	508	10	a.	a.	NOUN
ejpam-3441	508	11	let	let	VERB
ejpam-3441	508	12	i	i	PRON
ejpam-3441	508	13	be	be	AUX
ejpam-3441	508	14	another	another	DET
ejpam-3441	508	15	left	left	ADJ
ejpam-3441	508	16	ideal	ideal	NOUN
ejpam-3441	508	17	of	of	ADP
ejpam-3441	508	18	r	r	NOUN
ejpam-3441	508	19	containing	contain	VERB
ejpam-3441	508	20	a.	a.	NOUN
ejpam-3441	508	21	since	since	SCONJ
ejpam-3441	508	22	ra	ra	PROPN
ejpam-3441	508	23	∈	∈	PROPN
ejpam-3441	509	1	i	i	PRON
ejpam-3441	509	2	,	,	PUNCT
ejpam-3441	509	3	where	where	SCONJ
ejpam-3441	509	4	ra	ra	PROPN
ejpam-3441	509	5	∈	∈	PROPN
ejpam-3441	509	6	ra	ra	PROPN
ejpam-3441	509	7	,	,	PUNCT
ejpam-3441	509	8	i.e.	i.e.	X
ejpam-3441	509	9	,	,	PUNCT
ejpam-3441	509	10	ra	ra	PROPN
ejpam-3441	509	11	⊆	⊆	NUM
ejpam-3441	509	12	i.	i.	NOUN
ejpam-3441	509	13	hence	hence	ADV
ejpam-3441	509	14	ra	ra	PROPN
ejpam-3441	509	15	is	be	AUX
ejpam-3441	509	16	the	the	DET
ejpam-3441	509	17	smallest	small	ADJ
ejpam-3441	509	18	left	leave	VERB
ejpam-3441	509	19	ideal	ideal	NOUN
ejpam-3441	509	20	of	of	ADP
ejpam-3441	509	21	r	r	NOUN
ejpam-3441	509	22	containing	contain	VERB
ejpam-3441	509	23	a.	a.	NOUN
ejpam-3441	509	24	lemma	lemma	PROPN
ejpam-3441	509	25	20	20	NUM
ejpam-3441	509	26	.	.	PUNCT
ejpam-3441	510	1	let	let	VERB
ejpam-3441	510	2	r	r	PRON
ejpam-3441	510	3	be	be	AUX
ejpam-3441	510	4	an	an	DET
ejpam-3441	510	5	la	la	NOUN
ejpam-3441	510	6	-	-	NOUN
ejpam-3441	510	7	ring	ring	NOUN
ejpam-3441	510	8	with	with	ADP
ejpam-3441	510	9	left	left	ADJ
ejpam-3441	510	10	identity	identity	NOUN
ejpam-3441	510	11	e.	e.	PROPN
ejpam-3441	510	12	then	then	ADV
ejpam-3441	510	13	ar	ar	PROPN
ejpam-3441	510	14	is	be	AUX
ejpam-3441	510	15	a	a	DET
ejpam-3441	510	16	left	left	ADJ
ejpam-3441	510	17	ideal	ideal	NOUN
ejpam-3441	510	18	of	of	ADP
ejpam-3441	510	19	r.	r.	PROPN
ejpam-3441	510	20	proof	proof	NOUN
ejpam-3441	510	21	.	.	PUNCT
ejpam-3441	511	1	straight	straight	ADV
ejpam-3441	511	2	forward	forward	ADV
ejpam-3441	511	3	.	.	PUNCT
ejpam-3441	512	1	proposition	proposition	NOUN
ejpam-3441	512	2	8	8	NUM
ejpam-3441	512	3	.	.	PUNCT
ejpam-3441	513	1	let	let	VERB
ejpam-3441	513	2	r	r	PRON
ejpam-3441	513	3	be	be	AUX
ejpam-3441	513	4	an	an	DET
ejpam-3441	513	5	la	la	NOUN
ejpam-3441	513	6	-	-	NOUN
ejpam-3441	513	7	ring	ring	NOUN
ejpam-3441	513	8	with	with	ADP
ejpam-3441	513	9	left	left	ADJ
ejpam-3441	513	10	identity	identity	NOUN
ejpam-3441	513	11	e.	e.	PROPN
ejpam-3441	513	12	then	then	ADV
ejpam-3441	513	13	ar	ar	PROPN
ejpam-3441	513	14	∪	∪	PROPN
ejpam-3441	513	15	ra	ra	PROPN
ejpam-3441	513	16	is	be	AUX
ejpam-3441	513	17	the	the	DET
ejpam-3441	513	18	smallest	small	ADJ
ejpam-3441	513	19	right	right	ADJ
ejpam-3441	513	20	ideal	ideal	NOUN
ejpam-3441	513	21	of	of	ADP
ejpam-3441	513	22	r	r	NOUN
ejpam-3441	513	23	containing	contain	VERB
ejpam-3441	513	24	a.	a.	NOUN
ejpam-3441	513	25	proof	proof	NOUN
ejpam-3441	513	26	.	.	PUNCT
ejpam-3441	514	1	let	let	VERB
ejpam-3441	514	2	x	x	PRON
ejpam-3441	514	3	,	,	PUNCT
ejpam-3441	514	4	y	y	PROPN
ejpam-3441	514	5	∈	∈	PROPN
ejpam-3441	514	6	ar	ar	PROPN
ejpam-3441	514	7	∪	∪	PROPN
ejpam-3441	514	8	ra	ra	PROPN
ejpam-3441	514	9	,	,	PUNCT
ejpam-3441	514	10	this	this	PRON
ejpam-3441	514	11	means	mean	VERB
ejpam-3441	514	12	that	that	SCONJ
ejpam-3441	514	13	x	x	X
ejpam-3441	514	14	,	,	PUNCT
ejpam-3441	514	15	y	y	PROPN
ejpam-3441	514	16	∈	∈	PROPN
ejpam-3441	514	17	ar	ar	PROPN
ejpam-3441	514	18	or	or	CCONJ
ejpam-3441	514	19	ra	ra	PROPN
ejpam-3441	514	20	.	.	PUNCT
ejpam-3441	515	1	since	since	SCONJ
ejpam-3441	515	2	ar	ar	PROPN
ejpam-3441	515	3	and	and	CCONJ
ejpam-3441	515	4	ra	ra	PROPN
ejpam-3441	515	5	both	both	PRON
ejpam-3441	515	6	are	be	AUX
ejpam-3441	515	7	left	leave	VERB
ejpam-3441	515	8	ideals	ideal	NOUN
ejpam-3441	515	9	of	of	ADP
ejpam-3441	515	10	r	r	NOUN
ejpam-3441	515	11	,	,	PUNCT
ejpam-3441	515	12	so	so	ADV
ejpam-3441	515	13	x	x	SYM
ejpam-3441	515	14	−	−	PROPN
ejpam-3441	515	15	y	y	PROPN
ejpam-3441	515	16	∈	∈	PROPN
ejpam-3441	515	17	ar	ar	PROPN
ejpam-3441	515	18	and	and	CCONJ
ejpam-3441	515	19	ra	ra	PROPN
ejpam-3441	515	20	,	,	PUNCT
ejpam-3441	515	21	i.e.	i.e.	X
ejpam-3441	515	22	,	,	PUNCT
ejpam-3441	515	23	x	x	PUNCT
ejpam-3441	515	24	−	−	PROPN
ejpam-3441	515	25	y	y	PROPN
ejpam-3441	515	26	∈	∈	PROPN
ejpam-3441	515	27	ar	ar	PROPN
ejpam-3441	515	28	∪	∪	PROPN
ejpam-3441	515	29	ra	ra	PROPN
ejpam-3441	515	30	.	.	PUNCT
ejpam-3441	516	1	we	we	PRON
ejpam-3441	516	2	have	have	VERB
ejpam-3441	516	3	to	to	PART
ejpam-3441	516	4	show	show	VERB
ejpam-3441	516	5	that	that	SCONJ
ejpam-3441	516	6	(	(	PUNCT
ejpam-3441	516	7	ar	ar	PROPN
ejpam-3441	516	8	∪ra)r	∪ra)r	NOUN
ejpam-3441	516	9	⊆	⊆	NUM
ejpam-3441	516	10	(	(	PUNCT
ejpam-3441	516	11	ar	ar	NOUN
ejpam-3441	516	12	∪ra	∪ra	ADV
ejpam-3441	516	13	)	)	PUNCT
ejpam-3441	516	14	.	.	PUNCT
ejpam-3441	517	1	now	now	ADV
ejpam-3441	517	2	(	(	PUNCT
ejpam-3441	517	3	ar	ar	PROPN
ejpam-3441	517	4	∪ra)r	∪ra)r	NOUN
ejpam-3441	517	5	=	=	PUNCT
ejpam-3441	517	6	(	(	PUNCT
ejpam-3441	517	7	ar)r	ar)r	PROPN
ejpam-3441	517	8	∪	∪	ADV
ejpam-3441	517	9	(	(	PUNCT
ejpam-3441	517	10	ra)r	ra)r	PROPN
ejpam-3441	517	11	=	=	SYM
ejpam-3441	517	12	(	(	PUNCT
ejpam-3441	517	13	rr)a	rr)a	PROPN
ejpam-3441	517	14	∪	∪	X
ejpam-3441	517	15	(	(	PUNCT
ejpam-3441	517	16	ra)(er	ra)(er	X
ejpam-3441	517	17	)	)	PUNCT
ejpam-3441	517	18	⊆	⊆	NUM
ejpam-3441	517	19	ra	ra	PROPN
ejpam-3441	517	20	∪	∪	X
ejpam-3441	517	21	(	(	PUNCT
ejpam-3441	517	22	re)(ar	re)(ar	NOUN
ejpam-3441	517	23	)	)	PUNCT
ejpam-3441	517	24	=	=	SYM
ejpam-3441	517	25	ra	ra	PROPN
ejpam-3441	517	26	∪r(ar	∪r(ar	PROPN
ejpam-3441	517	27	)	)	PUNCT
ejpam-3441	518	1	=	=	SYM
ejpam-3441	518	2	ra	ra	PROPN
ejpam-3441	518	3	∪	∪	ADP
ejpam-3441	518	4	a(rr	a(rr	PROPN
ejpam-3441	518	5	)	)	PUNCT
ejpam-3441	518	6	⊆	⊆	NUM
ejpam-3441	518	7	ra	ra	PROPN
ejpam-3441	518	8	∪	∪	X
ejpam-3441	518	9	ar	ar	PROPN
ejpam-3441	518	10	=	=	PROPN
ejpam-3441	518	11	ar	ar	PROPN
ejpam-3441	518	12	∪ra	∪ra	ADV
ejpam-3441	518	13	.	.	PUNCT
ejpam-3441	519	1	⇒	⇒	PROPN
ejpam-3441	519	2	(	(	PUNCT
ejpam-3441	519	3	ar	ar	PROPN
ejpam-3441	519	4	∪ra)r	∪ra)r	PROPN
ejpam-3441	519	5	⊆	⊆	NUM
ejpam-3441	519	6	ar	ar	NOUN
ejpam-3441	519	7	∪ra	∪ra	ADV
ejpam-3441	519	8	.	.	PUNCT
ejpam-3441	520	1	since	since	SCONJ
ejpam-3441	520	2	a	a	DET
ejpam-3441	520	3	∈	∈	PROPN
ejpam-3441	520	4	ra	ra	PROPN
ejpam-3441	520	5	,	,	PUNCT
ejpam-3441	520	6	i.e.	i.e.	X
ejpam-3441	520	7	,	,	PUNCT
ejpam-3441	520	8	a	a	DET
ejpam-3441	520	9	∈	∈	PROPN
ejpam-3441	520	10	ar	ar	PROPN
ejpam-3441	520	11	∪	∪	PROPN
ejpam-3441	520	12	ra	ra	PROPN
ejpam-3441	520	13	.	.	PUNCT
ejpam-3441	520	14	let	let	VERB
ejpam-3441	520	15	i	i	PRON
ejpam-3441	520	16	be	be	AUX
ejpam-3441	520	17	another	another	DET
ejpam-3441	520	18	right	right	ADJ
ejpam-3441	520	19	ideal	ideal	NOUN
ejpam-3441	520	20	of	of	ADP
ejpam-3441	520	21	r	r	NOUN
ejpam-3441	520	22	containing	contain	VERB
ejpam-3441	520	23	a.	a.	NOUN
ejpam-3441	520	24	since	since	SCONJ
ejpam-3441	520	25	ar	ar	PROPN
ejpam-3441	520	26	∈	∈	PROPN
ejpam-3441	520	27	ir	ir	PROPN
ejpam-3441	520	28	⊆	⊆	NUM
ejpam-3441	520	29	i	i	PROPN
ejpam-3441	520	30	and	and	CCONJ
ejpam-3441	520	31	ra	ra	PROPN
ejpam-3441	521	1	=	=	PUNCT
ejpam-3441	522	1	(	(	PUNCT
ejpam-3441	522	2	rr)a	rr)a	PROPN
ejpam-3441	522	3	=	=	SYM
ejpam-3441	522	4	(	(	PUNCT
ejpam-3441	522	5	ar)r	ar)r	PROPN
ejpam-3441	522	6	∈	∈	PROPN
ejpam-3441	522	7	(	(	PUNCT
ejpam-3441	522	8	ir)r	ir)r	PROPN
ejpam-3441	522	9	⊆	⊆	NUM
ejpam-3441	522	10	ir	ir	PROPN
ejpam-3441	522	11	⊆	⊆	NUM
ejpam-3441	522	12	i	i	PRON
ejpam-3441	522	13	,	,	PUNCT
ejpam-3441	522	14	i.e.	i.e.	X
ejpam-3441	522	15	,	,	PUNCT
ejpam-3441	522	16	ar	ar	PROPN
ejpam-3441	522	17	∪ra	∪ra	PROPN
ejpam-3441	522	18	⊆	⊆	NUM
ejpam-3441	522	19	i.	i.	NOUN
ejpam-3441	522	20	therefore	therefore	ADV
ejpam-3441	522	21	ar	ar	PROPN
ejpam-3441	522	22	∪ra	∪ra	PROPN
ejpam-3441	522	23	is	be	AUX
ejpam-3441	522	24	the	the	DET
ejpam-3441	522	25	smallest	small	ADJ
ejpam-3441	522	26	right	right	ADJ
ejpam-3441	522	27	ideal	ideal	NOUN
ejpam-3441	522	28	of	of	ADP
ejpam-3441	522	29	r	r	NOUN
ejpam-3441	522	30	containing	contain	VERB
ejpam-3441	522	31	a.	a.	PROPN
ejpam-3441	522	32	k.	k.	PROPN
ejpam-3441	522	33	nasreen	nasreen	PROPN
ejpam-3441	522	34	et	et	PROPN
ejpam-3441	523	1	al	al	PROPN
ejpam-3441	523	2	.	.	PUNCT
ejpam-3441	523	3	/	/	SYM
ejpam-3441	523	4	eur	eur	PROPN
ejpam-3441	523	5	.	.	PUNCT
ejpam-3441	524	1	j.	j.	PROPN
ejpam-3441	524	2	pure	pure	PROPN
ejpam-3441	524	3	appl	appl	PROPN
ejpam-3441	524	4	.	.	PROPN
ejpam-3441	524	5	math	math	PROPN
ejpam-3441	524	6	,	,	PUNCT
ejpam-3441	524	7	12	12	NUM
ejpam-3441	524	8	(	(	PUNCT
ejpam-3441	524	9	3	3	NUM
ejpam-3441	524	10	)	)	PUNCT
ejpam-3441	524	11	(	(	PUNCT
ejpam-3441	524	12	2019	2019	NUM
ejpam-3441	524	13	)	)	PUNCT
ejpam-3441	524	14	,	,	PUNCT
ejpam-3441	524	15	906	906	NUM
ejpam-3441	524	16	-	-	SYM
ejpam-3441	524	17	943	943	NUM
ejpam-3441	524	18	929	929	NUM
ejpam-3441	524	19	theorem	theorem	NOUN
ejpam-3441	524	20	8	8	NUM
ejpam-3441	524	21	.	.	PUNCT
ejpam-3441	525	1	let	let	VERB
ejpam-3441	525	2	r	r	PRON
ejpam-3441	525	3	be	be	AUX
ejpam-3441	525	4	an	an	DET
ejpam-3441	525	5	la	la	NOUN
ejpam-3441	525	6	-	-	NOUN
ejpam-3441	525	7	ring	ring	NOUN
ejpam-3441	525	8	with	with	ADP
ejpam-3441	525	9	left	left	ADJ
ejpam-3441	525	10	identity	identity	NOUN
ejpam-3441	525	11	e	e	NOUN
ejpam-3441	525	12	,	,	PUNCT
ejpam-3441	525	13	such	such	ADJ
ejpam-3441	525	14	that	that	SCONJ
ejpam-3441	525	15	(	(	PUNCT
ejpam-3441	525	16	xe)r	xe)r	PROPN
ejpam-3441	525	17	=	=	SYM
ejpam-3441	525	18	xr	xr	PROPN
ejpam-3441	525	19	for	for	ADP
ejpam-3441	525	20	all	all	DET
ejpam-3441	525	21	x	x	PROPN
ejpam-3441	525	22	∈	∈	PROPN
ejpam-3441	525	23	r.	r.	NOUN
ejpam-3441	525	24	then	then	ADV
ejpam-3441	525	25	the	the	DET
ejpam-3441	525	26	following	follow	VERB
ejpam-3441	525	27	conditions	condition	NOUN
ejpam-3441	525	28	are	be	AUX
ejpam-3441	525	29	equivalent	equivalent	ADJ
ejpam-3441	525	30	.	.	PUNCT
ejpam-3441	526	1	(	(	PUNCT
ejpam-3441	526	2	1	1	X
ejpam-3441	526	3	)	)	PUNCT
ejpam-3441	526	4	r	r	NOUN
ejpam-3441	526	5	is	be	AUX
ejpam-3441	526	6	a	a	DET
ejpam-3441	526	7	regular	regular	NOUN
ejpam-3441	526	8	.	.	PUNCT
ejpam-3441	527	1	(	(	PUNCT
ejpam-3441	527	2	2	2	X
ejpam-3441	527	3	)	)	PUNCT
ejpam-3441	527	4	a∧βαb	a∧βαb	NOUN
ejpam-3441	527	5	=	=	PUNCT
ejpam-3441	527	6	a	a	DET
ejpam-3441	527	7	◦	◦	NOUN
ejpam-3441	527	8	βαb	βαb	NOUN
ejpam-3441	527	9	for	for	ADP
ejpam-3441	527	10	every	every	DET
ejpam-3441	527	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	527	12	fuzzy	fuzzy	ADJ
ejpam-3441	527	13	right	right	ADJ
ejpam-3441	527	14	ideal	ideal	NOUN
ejpam-3441	527	15	a	a	PRON
ejpam-3441	527	16	with	with	ADP
ejpam-3441	527	17	thresholds	threshold	NOUN
ejpam-3441	527	18	(	(	PUNCT
ejpam-3441	527	19	α	α	X
ejpam-3441	527	20	,	,	PUNCT
ejpam-3441	527	21	β	β	X
ejpam-3441	527	22	]	]	PUNCT
ejpam-3441	527	23	and	and	CCONJ
ejpam-3441	527	24	every	every	DET
ejpam-3441	527	25	intuitionistic	intuitionistic	ADJ
ejpam-3441	527	26	fuzzy	fuzzy	ADJ
ejpam-3441	527	27	left	leave	VERB
ejpam-3441	527	28	ideal	ideal	PROPN
ejpam-3441	527	29	b	b	PROPN
ejpam-3441	527	30	with	with	ADP
ejpam-3441	527	31	thresholds	threshold	NOUN
ejpam-3441	527	32	(	(	PUNCT
ejpam-3441	527	33	α	α	X
ejpam-3441	527	34	,	,	PUNCT
ejpam-3441	527	35	β	β	X
ejpam-3441	527	36	]	]	PUNCT
ejpam-3441	527	37	of	of	ADP
ejpam-3441	527	38	r.	r.	PROPN
ejpam-3441	527	39	(	(	PUNCT
ejpam-3441	527	40	3	3	NUM
ejpam-3441	527	41	)	)	PUNCT
ejpam-3441	527	42	cβα	cβα	NOUN
ejpam-3441	527	43	=	=	SYM
ejpam-3441	527	44	(	(	PUNCT
ejpam-3441	527	45	c	c	NOUN
ejpam-3441	527	46	◦	◦	NOUN
ejpam-3441	527	47	βα	βα	NOUN
ejpam-3441	527	48	r	r	NOUN
ejpam-3441	527	49	)	)	PUNCT
ejpam-3441	527	50	◦	◦	NOUN
ejpam-3441	527	51	βα	βα	NOUN
ejpam-3441	527	52	c	c	NOUN
ejpam-3441	527	53	for	for	ADP
ejpam-3441	527	54	every	every	DET
ejpam-3441	527	55	intuitionistic	intuitionistic	ADJ
ejpam-3441	527	56	fuzzy	fuzzy	ADJ
ejpam-3441	527	57	quasi	quasi	ADJ
ejpam-3441	527	58	-	-	ADJ
ejpam-3441	527	59	ideal	ideal	ADJ
ejpam-3441	527	60	c	c	NOUN
ejpam-3441	527	61	with	with	ADP
ejpam-3441	527	62	thresholds	threshold	NOUN
ejpam-3441	527	63	(	(	PUNCT
ejpam-3441	527	64	α	α	X
ejpam-3441	527	65	,	,	PUNCT
ejpam-3441	527	66	β	β	X
ejpam-3441	527	67	]	]	PUNCT
ejpam-3441	527	68	of	of	ADP
ejpam-3441	527	69	r.	r.	PROPN
ejpam-3441	527	70	proof	proof	NOUN
ejpam-3441	527	71	.	.	PUNCT
ejpam-3441	528	1	consider	consider	VERB
ejpam-3441	528	2	that	that	PRON
ejpam-3441	528	3	(	(	PUNCT
ejpam-3441	528	4	1	1	X
ejpam-3441	528	5	)	)	PUNCT
ejpam-3441	528	6	holds	hold	VERB
ejpam-3441	528	7	and	and	CCONJ
ejpam-3441	528	8	c	c	NOUN
ejpam-3441	529	1	=	=	PUNCT
ejpam-3441	529	2	(	(	PUNCT
ejpam-3441	529	3	µc	µc	INTJ
ejpam-3441	529	4	,	,	PUNCT
ejpam-3441	529	5	γc	γc	PROPN
ejpam-3441	529	6	)	)	PUNCT
ejpam-3441	529	7	be	be	AUX
ejpam-3441	529	8	an	an	DET
ejpam-3441	529	9	intuitionistic	intuitionistic	ADJ
ejpam-3441	529	10	fuzzy	fuzzy	ADJ
ejpam-3441	529	11	quasiideal	quasiideal	NOUN
ejpam-3441	529	12	with	with	ADP
ejpam-3441	529	13	thresholds	threshold	NOUN
ejpam-3441	529	14	(	(	PUNCT
ejpam-3441	529	15	α	α	X
ejpam-3441	529	16	,	,	PUNCT
ejpam-3441	529	17	β	β	X
ejpam-3441	529	18	]	]	PUNCT
ejpam-3441	529	19	of	of	ADP
ejpam-3441	529	20	r.	r.	PROPN
ejpam-3441	529	21	this	this	PRON
ejpam-3441	529	22	implies	imply	VERB
ejpam-3441	529	23	that	that	SCONJ
ejpam-3441	529	24	(	(	PUNCT
ejpam-3441	529	25	c	c	NOUN
ejpam-3441	529	26	◦	◦	NOUN
ejpam-3441	529	27	βα	βα	NOUN
ejpam-3441	529	28	r	r	NOUN
ejpam-3441	529	29	)	)	PUNCT
ejpam-3441	529	30	◦	◦	NOUN
ejpam-3441	529	31	βα	βα	X
ejpam-3441	529	32	c	c	NOUN
ejpam-3441	529	33	⊆	⊆	NUM
ejpam-3441	529	34	cβα	cβα	NOUN
ejpam-3441	529	35	,	,	PUNCT
ejpam-3441	529	36	because	because	SCONJ
ejpam-3441	529	37	every	every	DET
ejpam-3441	529	38	intuitionistic	intuitionistic	ADJ
ejpam-3441	529	39	fuzzy	fuzzy	ADJ
ejpam-3441	529	40	quasi	quasi	NOUN
ejpam-3441	529	41	-	-	NOUN
ejpam-3441	529	42	ideal	ideal	ADJ
ejpam-3441	529	43	with	with	ADP
ejpam-3441	529	44	thresholds	threshold	NOUN
ejpam-3441	529	45	(	(	PUNCT
ejpam-3441	529	46	α	α	X
ejpam-3441	529	47	,	,	PUNCT
ejpam-3441	529	48	β	β	X
ejpam-3441	529	49	]	]	PUNCT
ejpam-3441	529	50	of	of	ADP
ejpam-3441	529	51	r	r	NOUN
ejpam-3441	529	52	is	be	AUX
ejpam-3441	529	53	an	an	DET
ejpam-3441	529	54	intuitionistic	intuitionistic	ADJ
ejpam-3441	529	55	fuzzy	fuzzy	ADJ
ejpam-3441	529	56	bi	bi	NOUN
ejpam-3441	529	57	-	-	NOUN
ejpam-3441	529	58	ideal	ideal	ADJ
ejpam-3441	529	59	with	with	ADP
ejpam-3441	529	60	thresholds	threshold	NOUN
ejpam-3441	529	61	(	(	PUNCT
ejpam-3441	529	62	α	α	X
ejpam-3441	529	63	,	,	PUNCT
ejpam-3441	529	64	β	β	X
ejpam-3441	529	65	]	]	PUNCT
ejpam-3441	529	66	of	of	ADP
ejpam-3441	529	67	r	r	NOUN
ejpam-3441	529	68	by	by	ADP
ejpam-3441	529	69	the	the	DET
ejpam-3441	529	70	lemma	lemma	PROPN
ejpam-3441	529	71	15	15	NUM
ejpam-3441	529	72	.	.	PUNCT
ejpam-3441	530	1	let	let	VERB
ejpam-3441	530	2	x	x	PUNCT
ejpam-3441	530	3	∈	∈	PROPN
ejpam-3441	530	4	r	r	NOUN
ejpam-3441	530	5	,	,	PUNCT
ejpam-3441	530	6	then	then	ADV
ejpam-3441	530	7	there	there	PRON
ejpam-3441	530	8	exists	exist	VERB
ejpam-3441	530	9	an	an	DET
ejpam-3441	530	10	element	element	NOUN
ejpam-3441	530	11	a	a	DET
ejpam-3441	530	12	∈	∈	NOUN
ejpam-3441	530	13	r	r	NOUN
ejpam-3441	530	14	such	such	ADJ
ejpam-3441	530	15	that	that	PRON
ejpam-3441	530	16	x	x	SYM
ejpam-3441	530	17	=	=	SYM
ejpam-3441	530	18	(	(	PUNCT
ejpam-3441	530	19	xa)x	xa)x	PROPN
ejpam-3441	530	20	.	.	PUNCT
ejpam-3441	531	1	thus	thus	ADV
ejpam-3441	531	2	(	(	PUNCT
ejpam-3441	531	3	(	(	PUNCT
ejpam-3441	531	4	µc	µc	INTJ
ejpam-3441	531	5	◦	◦	NOUN
ejpam-3441	531	6	βα	βα	NOUN
ejpam-3441	531	7	r	r	NOUN
ejpam-3441	531	8	)	)	PUNCT
ejpam-3441	531	9	◦	◦	NOUN
ejpam-3441	531	10	βα	βα	NOUN
ejpam-3441	531	11	µc)(x	µc)(x	NOUN
ejpam-3441	531	12	)	)	PUNCT
ejpam-3441	532	1	=	=	PRON
ejpam-3441	532	2	{	{	PUNCT
ejpam-3441	532	3	(	(	PUNCT
ejpam-3441	532	4	(	(	PUNCT
ejpam-3441	532	5	µc	µc	INTJ
ejpam-3441	532	6	◦	◦	NOUN
ejpam-3441	532	7	r	r	NOUN
ejpam-3441	532	8	)	)	PUNCT
ejpam-3441	532	9	◦	◦	NOUN
ejpam-3441	532	10	µc)(x	µc)(x	NOUN
ejpam-3441	532	11	)	)	PUNCT
ejpam-3441	533	1	∧	∧	PROPN
ejpam-3441	533	2	β	β	NOUN
ejpam-3441	533	3	}	}	PUNCT
ejpam-3441	533	4	∨	∨	NUM
ejpam-3441	533	5	α	α	NOUN
ejpam-3441	533	6	=	=	X
ejpam-3441	533	7	{	{	PUNCT
ejpam-3441	533	8	(	(	PUNCT
ejpam-3441	533	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	533	10	i=1	i=1	PROPN
ejpam-3441	533	11	piqi	piqi	NOUN
ejpam-3441	533	12	{	{	PUNCT
ejpam-3441	533	13	∧ni=1	∧ni=1	X
ejpam-3441	533	14	{	{	PUNCT
ejpam-3441	533	15	(	(	PUNCT
ejpam-3441	533	16	µc	µc	INTJ
ejpam-3441	533	17	◦	◦	NOUN
ejpam-3441	533	18	r	r	NOUN
ejpam-3441	533	19	)	)	PUNCT
ejpam-3441	533	20	(	(	PUNCT
ejpam-3441	533	21	pi	pi	NOUN
ejpam-3441	533	22	)	)	PUNCT
ejpam-3441	533	23	∧	∧	NOUN
ejpam-3441	533	24	µc	µc	INTJ
ejpam-3441	533	25	(	(	PUNCT
ejpam-3441	533	26	qi	qi	NOUN
ejpam-3441	533	27	)	)	PUNCT
ejpam-3441	533	28	}	}	PUNCT
ejpam-3441	533	29	}	}	PUNCT
ejpam-3441	533	30	)	)	PUNCT
ejpam-3441	534	1	∧	∧	PROPN
ejpam-3441	534	2	β	β	NOUN
ejpam-3441	534	3	}	}	PUNCT
ejpam-3441	534	4	∨	∨	NUM
ejpam-3441	534	5	α	α	PROPN
ejpam-3441	534	6	≥	≥	X
ejpam-3441	534	7	{	{	PUNCT
ejpam-3441	534	8	{	{	PUNCT
ejpam-3441	534	9	(	(	PUNCT
ejpam-3441	534	10	µc	µc	INTJ
ejpam-3441	534	11	◦	◦	NOUN
ejpam-3441	534	12	r	r	NOUN
ejpam-3441	534	13	)	)	PUNCT
ejpam-3441	534	14	(	(	PUNCT
ejpam-3441	534	15	xa	xa	NOUN
ejpam-3441	534	16	)	)	PUNCT
ejpam-3441	534	17	∧	∧	PROPN
ejpam-3441	534	18	µc	µc	INTJ
ejpam-3441	534	19	(	(	PUNCT
ejpam-3441	534	20	x	x	NOUN
ejpam-3441	534	21	)	)	PUNCT
ejpam-3441	534	22	}	}	PUNCT
ejpam-3441	534	23	∧	∧	PROPN
ejpam-3441	534	24	β	β	NOUN
ejpam-3441	534	25	}	}	PUNCT
ejpam-3441	534	26	∨	∨	NUM
ejpam-3441	534	27	α	α	NOUN
ejpam-3441	534	28	=	=	SYM
ejpam-3441	534	29	(	(	PUNCT
ejpam-3441	534	30	(	(	PUNCT
ejpam-3441	534	31	µc	µc	INTJ
ejpam-3441	534	32	◦	◦	NOUN
ejpam-3441	534	33	r	r	NOUN
ejpam-3441	534	34	)	)	PUNCT
ejpam-3441	534	35	(	(	PUNCT
ejpam-3441	534	36	xa	xa	PROPN
ejpam-3441	534	37	)	)	PUNCT
ejpam-3441	534	38	∨	∨	NUM
ejpam-3441	534	39	α	α	NOUN
ejpam-3441	534	40	)	)	PUNCT
ejpam-3441	534	41	∧	∧	NOUN
ejpam-3441	534	42	(	(	PUNCT
ejpam-3441	534	43	µc	µc	INTJ
ejpam-3441	534	44	(	(	PUNCT
ejpam-3441	534	45	x	x	NOUN
ejpam-3441	534	46	)	)	PUNCT
ejpam-3441	534	47	∨	∨	NUM
ejpam-3441	534	48	α	α	NOUN
ejpam-3441	534	49	)	)	PUNCT
ejpam-3441	534	50	∧	∧	PROPN
ejpam-3441	534	51	(	(	PUNCT
ejpam-3441	534	52	β	β	X
ejpam-3441	534	53	∨	∨	NUM
ejpam-3441	534	54	α	α	NOUN
ejpam-3441	534	55	)	)	PUNCT
ejpam-3441	534	56	=	=	SYM
ejpam-3441	534	57	(	(	PUNCT
ejpam-3441	534	58	(	(	PUNCT
ejpam-3441	534	59	µc	µc	INTJ
ejpam-3441	534	60	◦	◦	NOUN
ejpam-3441	534	61	r	r	NOUN
ejpam-3441	534	62	)	)	PUNCT
ejpam-3441	534	63	(	(	PUNCT
ejpam-3441	534	64	xa	xa	PROPN
ejpam-3441	534	65	)	)	PUNCT
ejpam-3441	534	66	∨	∨	NUM
ejpam-3441	534	67	α	α	NOUN
ejpam-3441	534	68	)	)	PUNCT
ejpam-3441	534	69	∧	∧	PROPN
ejpam-3441	534	70	µc(x	µc(x	NOUN
ejpam-3441	534	71	)	)	PUNCT
ejpam-3441	534	72	∧	∧	NOUN
ejpam-3441	534	73	β	β	X
ejpam-3441	534	74	=	=	SYM
ejpam-3441	534	75	(	(	PUNCT
ejpam-3441	534	76	(	(	PUNCT
ejpam-3441	534	77	∨xa=∑n	∨xa=∑n	PROPN
ejpam-3441	534	78	i=1mini	i=1mini	PROPN
ejpam-3441	534	79	{	{	PUNCT
ejpam-3441	534	80	∧ni=1	∧ni=1	X
ejpam-3441	534	81	{	{	PUNCT
ejpam-3441	534	82	µc	µc	PROPN
ejpam-3441	534	83	(	(	PUNCT
ejpam-3441	534	84	mi	mi	NOUN
ejpam-3441	534	85	)	)	PUNCT
ejpam-3441	534	86	∧r	∧r	PROPN
ejpam-3441	534	87	(	(	PUNCT
ejpam-3441	534	88	ni	ni	NOUN
ejpam-3441	534	89	)	)	PUNCT
ejpam-3441	534	90	}	}	PUNCT
ejpam-3441	534	91	}	}	PUNCT
ejpam-3441	534	92	)	)	PUNCT
ejpam-3441	534	93	∨	∨	NUM
ejpam-3441	534	94	α	α	NOUN
ejpam-3441	534	95	)	)	PUNCT
ejpam-3441	534	96	∧	∧	PROPN
ejpam-3441	534	97	µc(x	µc(x	NOUN
ejpam-3441	534	98	)	)	PUNCT
ejpam-3441	534	99	∧	∧	PROPN
ejpam-3441	534	100	β	β	X
ejpam-3441	534	101	≥	≥	X
ejpam-3441	534	102	(	(	PUNCT
ejpam-3441	534	103	{	{	PUNCT
ejpam-3441	534	104	µc(x	µc(x	NOUN
ejpam-3441	534	105	)	)	PUNCT
ejpam-3441	534	106	∧r(a	∧r(a	NUM
ejpam-3441	534	107	)	)	PUNCT
ejpam-3441	534	108	}	}	PUNCT
ejpam-3441	534	109	∨	∨	NUM
ejpam-3441	534	110	α	α	NOUN
ejpam-3441	534	111	)	)	PUNCT
ejpam-3441	534	112	∧	∧	PROPN
ejpam-3441	534	113	µc(x	µc(x	NOUN
ejpam-3441	534	114	)	)	PUNCT
ejpam-3441	534	115	∧	∧	NOUN
ejpam-3441	534	116	β	β	X
ejpam-3441	534	117	=	=	SYM
ejpam-3441	534	118	(	(	PUNCT
ejpam-3441	534	119	{	{	PUNCT
ejpam-3441	534	120	µc(x	µc(x	NOUN
ejpam-3441	534	121	)	)	PUNCT
ejpam-3441	534	122	∧	∧	NOUN
ejpam-3441	534	123	1	1	NUM
ejpam-3441	534	124	}	}	PUNCT
ejpam-3441	534	125	∨	∨	NUM
ejpam-3441	534	126	α	α	NOUN
ejpam-3441	534	127	)	)	PUNCT
ejpam-3441	534	128	∧	∧	PROPN
ejpam-3441	534	129	µc(x	µc(x	NOUN
ejpam-3441	534	130	)	)	PUNCT
ejpam-3441	534	131	∧	∧	NOUN
ejpam-3441	534	132	β	β	X
ejpam-3441	534	133	=	=	SYM
ejpam-3441	534	134	(	(	PUNCT
ejpam-3441	534	135	µc(x	µc(x	NOUN
ejpam-3441	534	136	)	)	PUNCT
ejpam-3441	534	137	∨	∨	NUM
ejpam-3441	534	138	α	α	NOUN
ejpam-3441	534	139	)	)	PUNCT
ejpam-3441	534	140	∧	∧	PROPN
ejpam-3441	534	141	µc(x	µc(x	NOUN
ejpam-3441	534	142	)	)	PUNCT
ejpam-3441	534	143	∧	∧	PROPN
ejpam-3441	534	144	β	β	X
ejpam-3441	534	145	=	=	SYM
ejpam-3441	534	146	µc(x	µc(x	NOUN
ejpam-3441	534	147	)	)	PUNCT
ejpam-3441	534	148	∧	∧	NOUN
ejpam-3441	534	149	β	β	X
ejpam-3441	534	150	=	=	SYM
ejpam-3441	534	151	(	(	PUNCT
ejpam-3441	534	152	µc(x	µc(x	NOUN
ejpam-3441	534	153	)	)	PUNCT
ejpam-3441	534	154	∧	∧	PROPN
ejpam-3441	534	155	β	β	NOUN
ejpam-3441	534	156	)	)	PUNCT
ejpam-3441	534	157	∨	∨	NUM
ejpam-3441	534	158	α	α	NOUN
ejpam-3441	534	159	=	=	SYM
ejpam-3441	534	160	(	(	PUNCT
ejpam-3441	534	161	µc)βα(x	µc)βα(x	PROPN
ejpam-3441	534	162	)	)	PUNCT
ejpam-3441	534	163	.	.	PUNCT
ejpam-3441	535	1	⇒	⇒	NOUN
ejpam-3441	535	2	(	(	PUNCT
ejpam-3441	535	3	µc)βα	µc)βα	X
ejpam-3441	535	4	⊆	⊆	NUM
ejpam-3441	535	5	(	(	PUNCT
ejpam-3441	535	6	µc	µc	INTJ
ejpam-3441	535	7	◦	◦	NOUN
ejpam-3441	535	8	βα	βα	NOUN
ejpam-3441	535	9	r	r	NOUN
ejpam-3441	535	10	)	)	PUNCT
ejpam-3441	535	11	◦	◦	NOUN
ejpam-3441	535	12	βα	βα	NOUN
ejpam-3441	535	13	µc	µc	INTJ
ejpam-3441	535	14	.	.	PUNCT
ejpam-3441	536	1	similarly	similarly	ADV
ejpam-3441	536	2	,	,	PUNCT
ejpam-3441	536	3	we	we	PRON
ejpam-3441	536	4	have	have	VERB
ejpam-3441	536	5	(	(	PUNCT
ejpam-3441	536	6	γc)βα	γc)βα	PROPN
ejpam-3441	536	7	⊇	⊇	X
ejpam-3441	536	8	(	(	PUNCT
ejpam-3441	536	9	γc	γc	SYM
ejpam-3441	536	10	◦	◦	NOUN
ejpam-3441	536	11	βα	βα	NOUN
ejpam-3441	536	12	r	r	NOUN
ejpam-3441	536	13	)	)	PUNCT
ejpam-3441	536	14	◦	◦	NOUN
ejpam-3441	536	15	βα	βα	NOUN
ejpam-3441	536	16	γc	γc	PROPN
ejpam-3441	536	17	.	.	PUNCT
ejpam-3441	537	1	so	so	ADV
ejpam-3441	537	2	cβα	cβα	VERB
ejpam-3441	537	3	=	=	SYM
ejpam-3441	537	4	(	(	PUNCT
ejpam-3441	537	5	c	c	NOUN
ejpam-3441	537	6	◦	◦	NOUN
ejpam-3441	537	7	βα	βα	NOUN
ejpam-3441	537	8	r	r	NOUN
ejpam-3441	537	9	)	)	PUNCT
ejpam-3441	537	10	◦	◦	NOUN
ejpam-3441	537	11	βα	βα	X
ejpam-3441	537	12	c	c	NOUN
ejpam-3441	537	13	,	,	PUNCT
ejpam-3441	537	14	i.e.	i.e.	X
ejpam-3441	537	15	,	,	PUNCT
ejpam-3441	537	16	(	(	PUNCT
ejpam-3441	537	17	1	1	X
ejpam-3441	537	18	)	)	PUNCT
ejpam-3441	537	19	implies	imply	VERB
ejpam-3441	537	20	(	(	PUNCT
ejpam-3441	537	21	3	3	NUM
ejpam-3441	537	22	)	)	PUNCT
ejpam-3441	537	23	.	.	PUNCT
ejpam-3441	538	1	suppose	suppose	VERB
ejpam-3441	538	2	that	that	SCONJ
ejpam-3441	538	3	(	(	PUNCT
ejpam-3441	538	4	3	3	X
ejpam-3441	538	5	)	)	PUNCT
ejpam-3441	538	6	holds	hold	VERB
ejpam-3441	538	7	.	.	PUNCT
ejpam-3441	539	1	let	let	VERB
ejpam-3441	539	2	a	a	PRON
ejpam-3441	539	3	be	be	AUX
ejpam-3441	539	4	an	an	DET
ejpam-3441	539	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	539	6	fuzzy	fuzzy	ADJ
ejpam-3441	539	7	right	right	ADJ
ejpam-3441	539	8	ideal	ideal	NOUN
ejpam-3441	539	9	with	with	ADP
ejpam-3441	539	10	thresholds	threshold	NOUN
ejpam-3441	539	11	(	(	PUNCT
ejpam-3441	539	12	α	α	X
ejpam-3441	539	13	,	,	PUNCT
ejpam-3441	539	14	β	β	X
ejpam-3441	539	15	]	]	PUNCT
ejpam-3441	539	16	and	and	CCONJ
ejpam-3441	539	17	b	b	X
ejpam-3441	539	18	be	be	AUX
ejpam-3441	539	19	an	an	DET
ejpam-3441	539	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	539	21	fuzzy	fuzzy	ADJ
ejpam-3441	539	22	left	leave	VERB
ejpam-3441	539	23	ideal	ideal	NOUN
ejpam-3441	539	24	with	with	ADP
ejpam-3441	539	25	thresholds	threshold	NOUN
ejpam-3441	539	26	(	(	PUNCT
ejpam-3441	539	27	α	α	X
ejpam-3441	539	28	,	,	PUNCT
ejpam-3441	539	29	β	β	X
ejpam-3441	539	30	]	]	PUNCT
ejpam-3441	539	31	of	of	ADP
ejpam-3441	539	32	r.	r.	PROPN
ejpam-3441	539	33	this	this	PRON
ejpam-3441	539	34	implies	imply	VERB
ejpam-3441	539	35	that	that	SCONJ
ejpam-3441	539	36	a	a	PRON
ejpam-3441	539	37	and	and	CCONJ
ejpam-3441	539	38	b	b	NOUN
ejpam-3441	539	39	be	be	AUX
ejpam-3441	539	40	intuitionistic	intuitionistic	ADJ
ejpam-3441	539	41	fuzzy	fuzzy	ADJ
ejpam-3441	539	42	quasi	quasi	NOUN
ejpam-3441	539	43	-	-	NOUN
ejpam-3441	539	44	ideals	ideal	NOUN
ejpam-3441	539	45	with	with	ADP
ejpam-3441	539	46	thresholds	threshold	NOUN
ejpam-3441	539	47	(	(	PUNCT
ejpam-3441	539	48	α	α	X
ejpam-3441	539	49	,	,	PUNCT
ejpam-3441	539	50	β	β	X
ejpam-3441	539	51	]	]	PUNCT
ejpam-3441	539	52	of	of	ADP
ejpam-3441	539	53	r	r	NOUN
ejpam-3441	539	54	by	by	ADP
ejpam-3441	539	55	the	the	DET
ejpam-3441	539	56	lemma	lemma	PROPN
ejpam-3441	539	57	14	14	NUM
ejpam-3441	539	58	,	,	PUNCT
ejpam-3441	539	59	so	so	SCONJ
ejpam-3441	539	60	a	a	DET
ejpam-3441	539	61	∧βα	∧βα	PROPN
ejpam-3441	539	62	b	b	NOUN
ejpam-3441	539	63	be	be	AUX
ejpam-3441	539	64	also	also	ADV
ejpam-3441	539	65	an	an	DET
ejpam-3441	539	66	intuitionistic	intuitionistic	ADJ
ejpam-3441	539	67	fuzzy	fuzzy	ADJ
ejpam-3441	539	68	quasi	quasi	NOUN
ejpam-3441	539	69	-	-	NOUN
ejpam-3441	539	70	ideal	ideal	ADJ
ejpam-3441	539	71	with	with	ADP
ejpam-3441	539	72	thresholds	threshold	NOUN
ejpam-3441	539	73	(	(	PUNCT
ejpam-3441	539	74	α	α	X
ejpam-3441	539	75	,	,	PUNCT
ejpam-3441	539	76	β	β	X
ejpam-3441	539	77	]	]	PUNCT
ejpam-3441	539	78	of	of	ADP
ejpam-3441	539	79	r.	r.	PROPN
ejpam-3441	539	80	then	then	ADV
ejpam-3441	539	81	by	by	ADP
ejpam-3441	539	82	our	our	PRON
ejpam-3441	539	83	supposition	supposition	NOUN
ejpam-3441	539	84	,	,	PUNCT
ejpam-3441	539	85	a	a	DET
ejpam-3441	539	86	∧βα	∧βα	PROPN
ejpam-3441	539	87	b	b	NOUN
ejpam-3441	539	88	=	=	SYM
ejpam-3441	539	89	(	(	PUNCT
ejpam-3441	539	90	(	(	PUNCT
ejpam-3441	539	91	a	a	DET
ejpam-3441	539	92	∧βα	∧βα	ADJ
ejpam-3441	539	93	b	b	NOUN
ejpam-3441	539	94	)	)	PUNCT
ejpam-3441	539	95	◦	◦	NOUN
ejpam-3441	539	96	βα	βα	NOUN
ejpam-3441	539	97	r	r	NOUN
ejpam-3441	539	98	)	)	PUNCT
ejpam-3441	539	99	◦	◦	NOUN
ejpam-3441	539	100	βα	βα	X
ejpam-3441	539	101	(	(	PUNCT
ejpam-3441	539	102	a	a	DET
ejpam-3441	539	103	∧βα	∧βα	PROPN
ejpam-3441	539	104	b	b	NOUN
ejpam-3441	539	105	)	)	PUNCT
ejpam-3441	539	106	⊆	⊆	NUM
ejpam-3441	539	107	(	(	PUNCT
ejpam-3441	539	108	a	a	DET
ejpam-3441	539	109	◦	◦	NOUN
ejpam-3441	539	110	βα	βα	NOUN
ejpam-3441	539	111	r	r	NOUN
ejpam-3441	539	112	)	)	PUNCT
ejpam-3441	539	113	◦	◦	NOUN
ejpam-3441	539	114	βα	βα	NOUN
ejpam-3441	539	115	b	b	NOUN
ejpam-3441	539	116	⊆	⊆	NUM
ejpam-3441	539	117	a	a	DET
ejpam-3441	539	118	◦	◦	NOUN
ejpam-3441	539	119	βα	βα	NOUN
ejpam-3441	539	120	b	b	NOUN
ejpam-3441	539	121	,	,	PUNCT
ejpam-3441	539	122	i.e.	i.e.	X
ejpam-3441	539	123	,	,	PUNCT
ejpam-3441	539	124	a∧βαb	a∧βαb	NOUN
ejpam-3441	539	125	⊆	⊆	NUM
ejpam-3441	539	126	a	a	DET
ejpam-3441	539	127	◦	◦	NOUN
ejpam-3441	539	128	βαb	βαb	NOUN
ejpam-3441	539	129	.	.	PUNCT
ejpam-3441	540	1	since	since	SCONJ
ejpam-3441	540	2	a	a	DET
ejpam-3441	540	3	◦	◦	NOUN
ejpam-3441	540	4	βαb	βαb	NOUN
ejpam-3441	540	5	⊆	⊆	NUM
ejpam-3441	540	6	a∧βαb	a∧βαb	NUM
ejpam-3441	540	7	,	,	PUNCT
ejpam-3441	540	8	so	so	ADV
ejpam-3441	540	9	a	a	DET
ejpam-3441	540	10	◦	◦	NOUN
ejpam-3441	540	11	βαb	βαb	NOUN
ejpam-3441	540	12	=	=	SYM
ejpam-3441	540	13	a∧βαb	a∧βαb	NOUN
ejpam-3441	540	14	,	,	PUNCT
ejpam-3441	540	15	i.e.	i.e.	X
ejpam-3441	540	16	,	,	PUNCT
ejpam-3441	540	17	(	(	PUNCT
ejpam-3441	540	18	3)⇒	3)⇒	NUM
ejpam-3441	540	19	(	(	PUNCT
ejpam-3441	540	20	2	2	NUM
ejpam-3441	540	21	)	)	PUNCT
ejpam-3441	540	22	.	.	PUNCT
ejpam-3441	540	23	assume	assume	VERB
ejpam-3441	540	24	that	that	SCONJ
ejpam-3441	540	25	(	(	PUNCT
ejpam-3441	540	26	2	2	X
ejpam-3441	540	27	)	)	PUNCT
ejpam-3441	540	28	is	be	AUX
ejpam-3441	540	29	true	true	ADJ
ejpam-3441	540	30	and	and	CCONJ
ejpam-3441	540	31	a	a	DET
ejpam-3441	540	32	∈	∈	PROPN
ejpam-3441	540	33	r.	r.	NOUN
ejpam-3441	540	34	then	then	ADV
ejpam-3441	540	35	ra	ra	PROPN
ejpam-3441	540	36	is	be	AUX
ejpam-3441	540	37	a	a	DET
ejpam-3441	540	38	left	left	ADJ
ejpam-3441	540	39	ideal	ideal	NOUN
ejpam-3441	540	40	of	of	ADP
ejpam-3441	540	41	r	r	NOUN
ejpam-3441	540	42	containing	contain	VERB
ejpam-3441	540	43	a	a	PRON
ejpam-3441	540	44	by	by	ADP
ejpam-3441	540	45	the	the	DET
ejpam-3441	540	46	lemma	lemma	PROPN
ejpam-3441	540	47	19	19	NUM
ejpam-3441	540	48	and	and	CCONJ
ejpam-3441	540	49	ar∪ra	ar∪ra	PROPN
ejpam-3441	540	50	is	be	AUX
ejpam-3441	540	51	a	a	DET
ejpam-3441	540	52	right	right	ADJ
ejpam-3441	540	53	ideal	ideal	NOUN
ejpam-3441	540	54	of	of	ADP
ejpam-3441	540	55	r	r	NOUN
ejpam-3441	540	56	containing	contain	VERB
ejpam-3441	540	57	a	a	PRON
ejpam-3441	540	58	by	by	ADP
ejpam-3441	540	59	the	the	DET
ejpam-3441	540	60	proposition	proposition	NOUN
ejpam-3441	540	61	8	8	NUM
ejpam-3441	540	62	.	.	PUNCT
ejpam-3441	541	1	this	this	PRON
ejpam-3441	541	2	means	mean	VERB
ejpam-3441	541	3	that	that	SCONJ
ejpam-3441	541	4	χra	χra	PROPN
ejpam-3441	541	5	is	be	AUX
ejpam-3441	541	6	an	an	DET
ejpam-3441	541	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	541	8	fuzzy	fuzzy	ADJ
ejpam-3441	541	9	left	leave	VERB
ejpam-3441	541	10	ideal	ideal	NOUN
ejpam-3441	541	11	with	with	ADP
ejpam-3441	541	12	thresholds	threshold	NOUN
ejpam-3441	541	13	(	(	PUNCT
ejpam-3441	541	14	α	α	X
ejpam-3441	541	15	,	,	PUNCT
ejpam-3441	541	16	β	β	X
ejpam-3441	541	17	]	]	PUNCT
ejpam-3441	541	18	and	and	CCONJ
ejpam-3441	541	19	χar∪ra	χar∪ra	PROPN
ejpam-3441	541	20	is	be	AUX
ejpam-3441	541	21	an	an	DET
ejpam-3441	541	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	541	23	fuzzy	fuzzy	ADJ
ejpam-3441	541	24	right	right	ADJ
ejpam-3441	541	25	ideal	ideal	NOUN
ejpam-3441	541	26	with	with	ADP
ejpam-3441	541	27	thresholds	threshold	NOUN
ejpam-3441	541	28	(	(	PUNCT
ejpam-3441	541	29	α	α	X
ejpam-3441	541	30	,	,	PUNCT
ejpam-3441	541	31	β	β	X
ejpam-3441	541	32	]	]	PUNCT
ejpam-3441	541	33	of	of	ADP
ejpam-3441	541	34	r	r	NOUN
ejpam-3441	541	35	,	,	PUNCT
ejpam-3441	541	36	by	by	ADP
ejpam-3441	541	37	the	the	DET
ejpam-3441	541	38	theorem	theorem	NOUN
ejpam-3441	541	39	2	2	NUM
ejpam-3441	541	40	.	.	PUNCT
ejpam-3441	541	41	then	then	ADV
ejpam-3441	541	42	by	by	ADP
ejpam-3441	541	43	our	our	PRON
ejpam-3441	541	44	assumption	assumption	NOUN
ejpam-3441	541	45	χar∪ra	χar∪ra	ADJ
ejpam-3441	541	46	∧βα	∧βα	ADJ
ejpam-3441	541	47	χra	χra	PROPN
ejpam-3441	541	48	=	=	SYM
ejpam-3441	541	49	χar∪ra	χar∪ra	NUM
ejpam-3441	541	50	◦	◦	PROPN
ejpam-3441	541	51	βα	βα	PUNCT
ejpam-3441	541	52	χra	χra	PROPN
ejpam-3441	541	53	,	,	PUNCT
ejpam-3441	541	54	i.e.	i.e.	X
ejpam-3441	541	55	,	,	PUNCT
ejpam-3441	541	56	(	(	PUNCT
ejpam-3441	541	57	χ(ar∪ra)∩ra	χ(ar∪ra)∩ra	PROPN
ejpam-3441	541	58	)	)	PUNCT
ejpam-3441	541	59	β	β	PROPN
ejpam-3441	541	60	α	α	X
ejpam-3441	541	61	=	=	SYM
ejpam-3441	541	62	(	(	PUNCT
ejpam-3441	541	63	χ(ar∪ra)ra	χ(ar∪ra)ra	PROPN
ejpam-3441	541	64	)	)	PUNCT
ejpam-3441	541	65	β	β	PROPN
ejpam-3441	541	66	α	α	NOUN
ejpam-3441	541	67	by	by	ADP
ejpam-3441	541	68	the	the	DET
ejpam-3441	541	69	theorem	theorem	NOUN
ejpam-3441	541	70	1	1	NUM
ejpam-3441	541	71	.	.	PUNCT
ejpam-3441	542	1	thus	thus	ADV
ejpam-3441	542	2	(	(	PUNCT
ejpam-3441	542	3	ar∪ra)∩ra	ar∪ra)∩ra	PROPN
ejpam-3441	542	4	=	=	SYM
ejpam-3441	542	5	(	(	PUNCT
ejpam-3441	542	6	ar∪ra)ra	ar∪ra)ra	PROPN
ejpam-3441	542	7	.	.	PUNCT
ejpam-3441	543	1	since	since	SCONJ
ejpam-3441	543	2	a	a	DET
ejpam-3441	543	3	∈	∈	PROPN
ejpam-3441	543	4	(	(	PUNCT
ejpam-3441	543	5	ar∪ra)∩ra	ar∪ra)∩ra	PROPN
ejpam-3441	543	6	,	,	PUNCT
ejpam-3441	543	7	i.e.	i.e.	X
ejpam-3441	543	8	,	,	PUNCT
ejpam-3441	543	9	a	a	DET
ejpam-3441	543	10	∈	∈	PROPN
ejpam-3441	543	11	(	(	PUNCT
ejpam-3441	543	12	ar∪ra)ra	ar∪ra)ra	PROPN
ejpam-3441	543	13	,	,	PUNCT
ejpam-3441	543	14	so	so	SCONJ
ejpam-3441	543	15	a	a	DET
ejpam-3441	543	16	∈	∈	PROPN
ejpam-3441	543	17	k.	k.	NOUN
ejpam-3441	543	18	nasreen	nasreen	PROPN
ejpam-3441	543	19	et	et	PROPN
ejpam-3441	543	20	al	al	PROPN
ejpam-3441	543	21	.	.	PUNCT
ejpam-3441	543	22	/	/	SYM
ejpam-3441	543	23	eur	eur	PROPN
ejpam-3441	543	24	.	.	PUNCT
ejpam-3441	544	1	j.	j.	PROPN
ejpam-3441	544	2	pure	pure	PROPN
ejpam-3441	544	3	appl	appl	PROPN
ejpam-3441	544	4	.	.	PROPN
ejpam-3441	544	5	math	math	PROPN
ejpam-3441	544	6	,	,	PUNCT
ejpam-3441	544	7	12	12	NUM
ejpam-3441	544	8	(	(	PUNCT
ejpam-3441	544	9	3	3	NUM
ejpam-3441	544	10	)	)	PUNCT
ejpam-3441	544	11	(	(	PUNCT
ejpam-3441	544	12	2019	2019	NUM
ejpam-3441	544	13	)	)	PUNCT
ejpam-3441	544	14	,	,	PUNCT
ejpam-3441	544	15	906	906	NUM
ejpam-3441	544	16	-	-	SYM
ejpam-3441	544	17	943	943	NUM
ejpam-3441	544	18	930	930	NUM
ejpam-3441	544	19	(	(	PUNCT
ejpam-3441	544	20	ar)(ra	ar)(ra	NOUN
ejpam-3441	544	21	)	)	PUNCT
ejpam-3441	544	22	∪	∪	NOUN
ejpam-3441	544	23	(	(	PUNCT
ejpam-3441	544	24	ra)(ra	ra)(ra	X
ejpam-3441	544	25	)	)	PUNCT
ejpam-3441	544	26	.	.	PUNCT
ejpam-3441	545	1	this	this	PRON
ejpam-3441	545	2	implies	imply	VERB
ejpam-3441	545	3	that	that	SCONJ
ejpam-3441	545	4	a	a	DET
ejpam-3441	545	5	∈	∈	PROPN
ejpam-3441	545	6	(	(	PUNCT
ejpam-3441	545	7	ar)(ra	ar)(ra	PROPN
ejpam-3441	545	8	)	)	PUNCT
ejpam-3441	545	9	or	or	CCONJ
ejpam-3441	545	10	a	a	DET
ejpam-3441	545	11	∈	∈	PROPN
ejpam-3441	545	12	(	(	PUNCT
ejpam-3441	545	13	ra)(ra	ra)(ra	NOUN
ejpam-3441	545	14	)	)	PUNCT
ejpam-3441	545	15	.	.	PUNCT
ejpam-3441	546	1	if	if	SCONJ
ejpam-3441	546	2	a	a	DET
ejpam-3441	546	3	∈	∈	PROPN
ejpam-3441	546	4	(	(	PUNCT
ejpam-3441	546	5	ar)(ra	ar)(ra	PROPN
ejpam-3441	546	6	)	)	PUNCT
ejpam-3441	546	7	,	,	PUNCT
ejpam-3441	546	8	then	then	ADV
ejpam-3441	546	9	a	a	PRON
ejpam-3441	546	10	=	=	PUNCT
ejpam-3441	546	11	(	(	PUNCT
ejpam-3441	546	12	ax)(ya	ax)(ya	NOUN
ejpam-3441	546	13	)	)	PUNCT
ejpam-3441	546	14	=	=	SYM
ejpam-3441	546	15	(	(	PUNCT
ejpam-3441	546	16	(	(	PUNCT
ejpam-3441	546	17	ya)x)a	ya)x)a	NOUN
ejpam-3441	546	18	=	=	SYM
ejpam-3441	546	19	(	(	PUNCT
ejpam-3441	546	20	(	(	PUNCT
ejpam-3441	546	21	(	(	PUNCT
ejpam-3441	546	22	ey)a)x)a	ey)a)x)a	PROPN
ejpam-3441	546	23	=	=	SYM
ejpam-3441	546	24	(	(	PUNCT
ejpam-3441	546	25	(	(	PUNCT
ejpam-3441	546	26	(	(	PUNCT
ejpam-3441	546	27	ay)e)x)a	ay)e)x)a	PROPN
ejpam-3441	546	28	=	=	SYM
ejpam-3441	546	29	(	(	PUNCT
ejpam-3441	546	30	(	(	PUNCT
ejpam-3441	546	31	xe)(ay))a	xe)(ay))a	PROPN
ejpam-3441	546	32	=	=	SYM
ejpam-3441	546	33	(	(	PUNCT
ejpam-3441	546	34	a((xe)y))a	a((xe)y))a	PROPN
ejpam-3441	546	35	for	for	ADP
ejpam-3441	546	36	any	any	DET
ejpam-3441	546	37	x	x	NOUN
ejpam-3441	546	38	,	,	PUNCT
ejpam-3441	546	39	y	y	PROPN
ejpam-3441	546	40	∈	∈	PROPN
ejpam-3441	546	41	r.	r.	PROPN
ejpam-3441	546	42	if	if	SCONJ
ejpam-3441	546	43	a	a	DET
ejpam-3441	546	44	∈	∈	PROPN
ejpam-3441	546	45	(	(	PUNCT
ejpam-3441	546	46	ra)(ra	ra)(ra	NOUN
ejpam-3441	546	47	)	)	PUNCT
ejpam-3441	546	48	,	,	PUNCT
ejpam-3441	546	49	then	then	ADV
ejpam-3441	546	50	(	(	PUNCT
ejpam-3441	546	51	ra)(ra	ra)(ra	X
ejpam-3441	546	52	)	)	PUNCT
ejpam-3441	546	53	=	=	SYM
ejpam-3441	546	54	(	(	PUNCT
ejpam-3441	546	55	(	(	PUNCT
ejpam-3441	546	56	re)a)(ra	re)a)(ra	PROPN
ejpam-3441	546	57	)	)	PUNCT
ejpam-3441	546	58	=	=	SYM
ejpam-3441	546	59	(	(	PUNCT
ejpam-3441	546	60	(	(	PUNCT
ejpam-3441	546	61	ae)r)(ra	ae)r)(ra	PROPN
ejpam-3441	546	62	)	)	PUNCT
ejpam-3441	546	63	=	=	PUNCT
ejpam-3441	546	64	(	(	PUNCT
ejpam-3441	546	65	ar)(ra	ar)(ra	PROPN
ejpam-3441	546	66	)	)	PUNCT
ejpam-3441	546	67	,	,	PUNCT
ejpam-3441	546	68	i.e.	i.e.	X
ejpam-3441	546	69	,	,	PUNCT
ejpam-3441	546	70	a	a	DET
ejpam-3441	546	71	∈	∈	PROPN
ejpam-3441	546	72	(	(	PUNCT
ejpam-3441	546	73	ar)(ra	ar)(ra	PROPN
ejpam-3441	546	74	)	)	PUNCT
ejpam-3441	546	75	.	.	PUNCT
ejpam-3441	547	1	therefore	therefore	ADV
ejpam-3441	547	2	a	a	PRON
ejpam-3441	547	3	is	be	AUX
ejpam-3441	547	4	a	a	DET
ejpam-3441	547	5	regular	regular	ADJ
ejpam-3441	547	6	,	,	PUNCT
ejpam-3441	547	7	i.e.	i.e.	X
ejpam-3441	547	8	,	,	PUNCT
ejpam-3441	547	9	r	r	NOUN
ejpam-3441	547	10	is	be	AUX
ejpam-3441	547	11	a	a	DET
ejpam-3441	547	12	regular	regular	NOUN
ejpam-3441	547	13	.	.	PUNCT
ejpam-3441	548	1	so	so	ADV
ejpam-3441	548	2	(	(	PUNCT
ejpam-3441	548	3	2)⇒	2)⇒	NUM
ejpam-3441	548	4	(	(	PUNCT
ejpam-3441	548	5	1	1	NUM
ejpam-3441	548	6	)	)	PUNCT
ejpam-3441	548	7	.	.	PUNCT
ejpam-3441	549	1	theorem	theorem	ADJ
ejpam-3441	549	2	9	9	NUM
ejpam-3441	549	3	.	.	PUNCT
ejpam-3441	550	1	let	let	VERB
ejpam-3441	550	2	r	r	PRON
ejpam-3441	550	3	be	be	AUX
ejpam-3441	550	4	an	an	DET
ejpam-3441	550	5	la	la	NOUN
ejpam-3441	550	6	-	-	NOUN
ejpam-3441	550	7	ring	ring	NOUN
ejpam-3441	550	8	with	with	ADP
ejpam-3441	550	9	left	left	ADJ
ejpam-3441	550	10	identity	identity	NOUN
ejpam-3441	550	11	e	e	NOUN
ejpam-3441	550	12	,	,	PUNCT
ejpam-3441	550	13	such	such	ADJ
ejpam-3441	550	14	that	that	SCONJ
ejpam-3441	550	15	(	(	PUNCT
ejpam-3441	550	16	xe)r	xe)r	PROPN
ejpam-3441	550	17	=	=	SYM
ejpam-3441	550	18	xr	xr	PROPN
ejpam-3441	550	19	for	for	ADP
ejpam-3441	550	20	all	all	DET
ejpam-3441	550	21	x	x	PROPN
ejpam-3441	550	22	∈	∈	PROPN
ejpam-3441	550	23	r.	r.	NOUN
ejpam-3441	550	24	then	then	ADV
ejpam-3441	550	25	the	the	DET
ejpam-3441	550	26	following	follow	VERB
ejpam-3441	550	27	conditions	condition	NOUN
ejpam-3441	550	28	are	be	AUX
ejpam-3441	550	29	equivalent	equivalent	ADJ
ejpam-3441	550	30	.	.	PUNCT
ejpam-3441	551	1	(	(	PUNCT
ejpam-3441	551	2	1	1	X
ejpam-3441	551	3	)	)	PUNCT
ejpam-3441	551	4	r	r	NOUN
ejpam-3441	551	5	is	be	AUX
ejpam-3441	551	6	a	a	DET
ejpam-3441	551	7	regular	regular	NOUN
ejpam-3441	551	8	.	.	PUNCT
ejpam-3441	552	1	(	(	PUNCT
ejpam-3441	552	2	2	2	X
ejpam-3441	552	3	)	)	PUNCT
ejpam-3441	552	4	aβα	aβα	NOUN
ejpam-3441	552	5	=	=	PUNCT
ejpam-3441	552	6	(	(	PUNCT
ejpam-3441	552	7	a	a	DET
ejpam-3441	552	8	◦	◦	NOUN
ejpam-3441	552	9	βα	βα	NOUN
ejpam-3441	552	10	r	r	NOUN
ejpam-3441	552	11	)	)	PUNCT
ejpam-3441	552	12	◦	◦	NOUN
ejpam-3441	552	13	βα	βα	NOUN
ejpam-3441	552	14	a	a	PRON
ejpam-3441	552	15	for	for	ADP
ejpam-3441	552	16	every	every	DET
ejpam-3441	552	17	intuitionistic	intuitionistic	ADJ
ejpam-3441	552	18	fuzzy	fuzzy	ADJ
ejpam-3441	552	19	quasi	quasi	NOUN
ejpam-3441	552	20	-	-	NOUN
ejpam-3441	552	21	ideal	ideal	ADJ
ejpam-3441	552	22	a	a	PRON
ejpam-3441	552	23	with	with	ADP
ejpam-3441	552	24	thresholds	threshold	NOUN
ejpam-3441	552	25	(	(	PUNCT
ejpam-3441	552	26	α	α	X
ejpam-3441	552	27	,	,	PUNCT
ejpam-3441	552	28	β	β	X
ejpam-3441	552	29	]	]	PUNCT
ejpam-3441	552	30	of	of	ADP
ejpam-3441	552	31	r.	r.	PROPN
ejpam-3441	552	32	(	(	PUNCT
ejpam-3441	552	33	3	3	NUM
ejpam-3441	552	34	)	)	PUNCT
ejpam-3441	552	35	bβ	bβ	NOUN
ejpam-3441	552	36	α	α	NOUN
ejpam-3441	552	37	=	=	PUNCT
ejpam-3441	552	38	(	(	PUNCT
ejpam-3441	552	39	b	b	PROPN
ejpam-3441	552	40	◦	◦	NOUN
ejpam-3441	552	41	βαr)	βαr)	NOUN
ejpam-3441	552	42	◦	◦	NOUN
ejpam-3441	552	43	βαb	βαb	NOUN
ejpam-3441	552	44	for	for	ADP
ejpam-3441	552	45	every	every	DET
ejpam-3441	552	46	intuitionistic	intuitionistic	ADJ
ejpam-3441	552	47	fuzzy	fuzzy	ADJ
ejpam-3441	552	48	bi	bi	ADJ
ejpam-3441	552	49	-	-	ADJ
ejpam-3441	552	50	ideal	ideal	ADJ
ejpam-3441	552	51	b	b	PROPN
ejpam-3441	552	52	with	with	ADP
ejpam-3441	552	53	thresholds	threshold	NOUN
ejpam-3441	552	54	(	(	PUNCT
ejpam-3441	552	55	α	α	X
ejpam-3441	552	56	,	,	PUNCT
ejpam-3441	552	57	β	β	X
ejpam-3441	552	58	]	]	PUNCT
ejpam-3441	552	59	of	of	ADP
ejpam-3441	552	60	r.	r.	PROPN
ejpam-3441	552	61	(	(	PUNCT
ejpam-3441	552	62	4	4	NUM
ejpam-3441	552	63	)	)	PUNCT
ejpam-3441	552	64	cβα	cβα	NOUN
ejpam-3441	552	65	=	=	SYM
ejpam-3441	552	66	(	(	PUNCT
ejpam-3441	552	67	c	c	NOUN
ejpam-3441	552	68	◦	◦	NOUN
ejpam-3441	552	69	βα	βα	NOUN
ejpam-3441	552	70	r	r	NOUN
ejpam-3441	552	71	)	)	PUNCT
ejpam-3441	552	72	◦	◦	NOUN
ejpam-3441	552	73	βα	βα	NOUN
ejpam-3441	552	74	c	c	NOUN
ejpam-3441	552	75	for	for	ADP
ejpam-3441	552	76	every	every	DET
ejpam-3441	552	77	intuitionistic	intuitionistic	ADJ
ejpam-3441	552	78	fuzzy	fuzzy	ADJ
ejpam-3441	552	79	generalized	generalize	VERB
ejpam-3441	552	80	bi	bi	ADJ
ejpam-3441	552	81	-	-	ADJ
ejpam-3441	552	82	ideal	ideal	ADJ
ejpam-3441	552	83	c	c	NOUN
ejpam-3441	552	84	with	with	ADP
ejpam-3441	552	85	thresholds	threshold	NOUN
ejpam-3441	552	86	(	(	PUNCT
ejpam-3441	552	87	α	α	X
ejpam-3441	552	88	,	,	PUNCT
ejpam-3441	552	89	β	β	X
ejpam-3441	552	90	]	]	PUNCT
ejpam-3441	552	91	of	of	ADP
ejpam-3441	552	92	r.	r.	PROPN
ejpam-3441	552	93	proof	proof	NOUN
ejpam-3441	552	94	.	.	PUNCT
ejpam-3441	553	1	(	(	PUNCT
ejpam-3441	553	2	1	1	X
ejpam-3441	553	3	)	)	PUNCT
ejpam-3441	553	4	⇒	⇒	NOUN
ejpam-3441	553	5	(	(	PUNCT
ejpam-3441	553	6	4	4	NUM
ejpam-3441	553	7	)	)	PUNCT
ejpam-3441	553	8	,	,	PUNCT
ejpam-3441	553	9	is	be	AUX
ejpam-3441	553	10	obvious	obvious	ADJ
ejpam-3441	553	11	.	.	PUNCT
ejpam-3441	554	1	(	(	PUNCT
ejpam-3441	554	2	4	4	X
ejpam-3441	554	3	)	)	PUNCT
ejpam-3441	554	4	⇒	⇒	NOUN
ejpam-3441	554	5	(	(	PUNCT
ejpam-3441	554	6	3	3	NUM
ejpam-3441	554	7	)	)	PUNCT
ejpam-3441	554	8	,	,	PUNCT
ejpam-3441	554	9	since	since	SCONJ
ejpam-3441	554	10	every	every	DET
ejpam-3441	554	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	554	12	fuzzy	fuzzy	ADJ
ejpam-3441	554	13	bi	bi	NOUN
ejpam-3441	554	14	-	-	NOUN
ejpam-3441	554	15	ideal	ideal	ADJ
ejpam-3441	554	16	with	with	ADP
ejpam-3441	554	17	thresholds	threshold	NOUN
ejpam-3441	554	18	(	(	PUNCT
ejpam-3441	554	19	α	α	X
ejpam-3441	554	20	,	,	PUNCT
ejpam-3441	554	21	β	β	X
ejpam-3441	554	22	]	]	PUNCT
ejpam-3441	554	23	of	of	ADP
ejpam-3441	554	24	r	r	NOUN
ejpam-3441	554	25	is	be	AUX
ejpam-3441	554	26	an	an	DET
ejpam-3441	554	27	intuitionistic	intuitionistic	ADJ
ejpam-3441	554	28	fuzzy	fuzzy	ADJ
ejpam-3441	554	29	generalized	generalize	VERB
ejpam-3441	554	30	bi	bi	NOUN
ejpam-3441	554	31	-	-	NOUN
ejpam-3441	554	32	ideal	ideal	NOUN
ejpam-3441	554	33	with	with	ADP
ejpam-3441	554	34	thresholds	threshold	NOUN
ejpam-3441	554	35	(	(	PUNCT
ejpam-3441	554	36	α	α	X
ejpam-3441	554	37	,	,	PUNCT
ejpam-3441	554	38	β	β	X
ejpam-3441	554	39	]	]	PUNCT
ejpam-3441	554	40	of	of	ADP
ejpam-3441	554	41	r	r	NOUN
ejpam-3441	554	42	by	by	ADP
ejpam-3441	554	43	the	the	DET
ejpam-3441	554	44	lemma	lemma	PROPN
ejpam-3441	554	45	13	13	NUM
ejpam-3441	554	46	.	.	PUNCT
ejpam-3441	555	1	(	(	PUNCT
ejpam-3441	555	2	3)⇒	3)⇒	NUM
ejpam-3441	555	3	(	(	PUNCT
ejpam-3441	555	4	2	2	NUM
ejpam-3441	555	5	)	)	PUNCT
ejpam-3441	555	6	,	,	PUNCT
ejpam-3441	555	7	since	since	SCONJ
ejpam-3441	555	8	every	every	DET
ejpam-3441	555	9	intuitionistic	intuitionistic	ADJ
ejpam-3441	555	10	fuzzy	fuzzy	ADJ
ejpam-3441	555	11	quasi	quasi	NOUN
ejpam-3441	555	12	-	-	NOUN
ejpam-3441	555	13	ideal	ideal	ADJ
ejpam-3441	555	14	with	with	ADP
ejpam-3441	555	15	thresholds	threshold	NOUN
ejpam-3441	555	16	(	(	PUNCT
ejpam-3441	555	17	α	α	X
ejpam-3441	555	18	,	,	PUNCT
ejpam-3441	555	19	β	β	X
ejpam-3441	555	20	]	]	PUNCT
ejpam-3441	555	21	of	of	ADP
ejpam-3441	555	22	r	r	NOUN
ejpam-3441	555	23	is	be	AUX
ejpam-3441	555	24	an	an	DET
ejpam-3441	555	25	intuitionistic	intuitionistic	ADJ
ejpam-3441	555	26	fuzzy	fuzzy	ADJ
ejpam-3441	555	27	bi	bi	NOUN
ejpam-3441	555	28	-	-	NOUN
ejpam-3441	555	29	ideal	ideal	ADJ
ejpam-3441	555	30	with	with	ADP
ejpam-3441	555	31	thresholds	threshold	NOUN
ejpam-3441	555	32	(	(	PUNCT
ejpam-3441	555	33	α	α	X
ejpam-3441	555	34	,	,	PUNCT
ejpam-3441	555	35	β	β	X
ejpam-3441	555	36	]	]	PUNCT
ejpam-3441	555	37	of	of	ADP
ejpam-3441	555	38	r	r	NOUN
ejpam-3441	555	39	by	by	ADP
ejpam-3441	555	40	the	the	DET
ejpam-3441	555	41	lemma	lemma	PROPN
ejpam-3441	555	42	15	15	NUM
ejpam-3441	555	43	.	.	PUNCT
ejpam-3441	556	1	(	(	PUNCT
ejpam-3441	556	2	2)⇒	2)⇒	NUM
ejpam-3441	556	3	(	(	PUNCT
ejpam-3441	556	4	1	1	NUM
ejpam-3441	556	5	)	)	PUNCT
ejpam-3441	556	6	,	,	PUNCT
ejpam-3441	556	7	by	by	ADP
ejpam-3441	556	8	the	the	DET
ejpam-3441	556	9	theorem	theorem	ADJ
ejpam-3441	556	10	8	8	NUM
ejpam-3441	556	11	.	.	PUNCT
ejpam-3441	556	12	theorem	theorem	NOUN
ejpam-3441	556	13	10	10	NUM
ejpam-3441	556	14	.	.	PUNCT
ejpam-3441	557	1	let	let	VERB
ejpam-3441	557	2	r	r	PRON
ejpam-3441	557	3	be	be	AUX
ejpam-3441	557	4	an	an	DET
ejpam-3441	557	5	la	la	NOUN
ejpam-3441	557	6	-	-	NOUN
ejpam-3441	557	7	ring	ring	NOUN
ejpam-3441	557	8	with	with	ADP
ejpam-3441	557	9	left	left	ADJ
ejpam-3441	557	10	identity	identity	NOUN
ejpam-3441	557	11	e	e	NOUN
ejpam-3441	557	12	,	,	PUNCT
ejpam-3441	557	13	such	such	ADJ
ejpam-3441	557	14	that	that	SCONJ
ejpam-3441	557	15	(	(	PUNCT
ejpam-3441	557	16	xe)r	xe)r	PROPN
ejpam-3441	557	17	=	=	SYM
ejpam-3441	557	18	xr	xr	PROPN
ejpam-3441	557	19	for	for	ADP
ejpam-3441	557	20	all	all	DET
ejpam-3441	557	21	x	x	PROPN
ejpam-3441	557	22	∈	∈	PROPN
ejpam-3441	557	23	r.	r.	NOUN
ejpam-3441	557	24	then	then	ADV
ejpam-3441	557	25	the	the	DET
ejpam-3441	557	26	following	follow	VERB
ejpam-3441	557	27	conditions	condition	NOUN
ejpam-3441	557	28	are	be	AUX
ejpam-3441	557	29	equivalent	equivalent	ADJ
ejpam-3441	557	30	.	.	PUNCT
ejpam-3441	558	1	(	(	PUNCT
ejpam-3441	558	2	1	1	X
ejpam-3441	558	3	)	)	PUNCT
ejpam-3441	558	4	r	r	NOUN
ejpam-3441	558	5	is	be	AUX
ejpam-3441	558	6	a	a	DET
ejpam-3441	558	7	regular	regular	NOUN
ejpam-3441	558	8	.	.	PUNCT
ejpam-3441	559	1	(	(	PUNCT
ejpam-3441	559	2	2	2	X
ejpam-3441	559	3	)	)	PUNCT
ejpam-3441	559	4	a	a	DET
ejpam-3441	559	5	∧βα	∧βα	NOUN
ejpam-3441	559	6	i	i	NOUN
ejpam-3441	559	7	=	=	PUNCT
ejpam-3441	559	8	(	(	PUNCT
ejpam-3441	559	9	a	a	DET
ejpam-3441	559	10	◦	◦	NOUN
ejpam-3441	559	11	βα	βα	X
ejpam-3441	559	12	i	i	NOUN
ejpam-3441	559	13	)	)	PUNCT
ejpam-3441	559	14	◦	◦	NOUN
ejpam-3441	559	15	βα	βα	NOUN
ejpam-3441	559	16	a	a	PRON
ejpam-3441	559	17	for	for	ADP
ejpam-3441	559	18	every	every	DET
ejpam-3441	559	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	559	20	fuzzy	fuzzy	ADJ
ejpam-3441	559	21	quasi	quasi	NOUN
ejpam-3441	559	22	-	-	NOUN
ejpam-3441	559	23	ideal	ideal	ADJ
ejpam-3441	559	24	a	a	PRON
ejpam-3441	559	25	with	with	ADP
ejpam-3441	559	26	thresholds	threshold	NOUN
ejpam-3441	559	27	(	(	PUNCT
ejpam-3441	559	28	α	α	X
ejpam-3441	559	29	,	,	PUNCT
ejpam-3441	559	30	β	β	X
ejpam-3441	559	31	]	]	PUNCT
ejpam-3441	559	32	and	and	CCONJ
ejpam-3441	559	33	every	every	DET
ejpam-3441	559	34	intuitionistic	intuitionistic	ADJ
ejpam-3441	559	35	fuzzy	fuzzy	ADJ
ejpam-3441	559	36	ideal	ideal	NOUN
ejpam-3441	559	37	i	i	PRON
ejpam-3441	559	38	with	with	ADP
ejpam-3441	559	39	thresholds	threshold	NOUN
ejpam-3441	559	40	(	(	PUNCT
ejpam-3441	559	41	α	α	X
ejpam-3441	559	42	,	,	PUNCT
ejpam-3441	559	43	β	β	X
ejpam-3441	559	44	]	]	PUNCT
ejpam-3441	559	45	of	of	ADP
ejpam-3441	559	46	r.	r.	PROPN
ejpam-3441	559	47	(	(	PUNCT
ejpam-3441	559	48	3	3	NUM
ejpam-3441	559	49	)	)	PUNCT
ejpam-3441	559	50	b	b	NOUN
ejpam-3441	560	1	∧βα	∧βα	NOUN
ejpam-3441	560	2	i	i	NOUN
ejpam-3441	560	3	=	=	PUNCT
ejpam-3441	560	4	(	(	PUNCT
ejpam-3441	560	5	b	b	X
ejpam-3441	560	6	◦	◦	NOUN
ejpam-3441	560	7	βα	βα	NOUN
ejpam-3441	560	8	i	i	NOUN
ejpam-3441	560	9	)	)	PUNCT
ejpam-3441	560	10	◦	◦	PROPN
ejpam-3441	560	11	βα	βα	NOUN
ejpam-3441	560	12	b	b	NOUN
ejpam-3441	560	13	for	for	ADP
ejpam-3441	560	14	every	every	DET
ejpam-3441	560	15	intuitionistic	intuitionistic	ADJ
ejpam-3441	560	16	fuzzy	fuzzy	ADJ
ejpam-3441	560	17	bi	bi	ADJ
ejpam-3441	560	18	-	-	ADJ
ejpam-3441	560	19	ideal	ideal	ADJ
ejpam-3441	560	20	b	b	PROPN
ejpam-3441	560	21	with	with	ADP
ejpam-3441	560	22	thresholds	threshold	NOUN
ejpam-3441	560	23	(	(	PUNCT
ejpam-3441	560	24	α	α	X
ejpam-3441	560	25	,	,	PUNCT
ejpam-3441	560	26	β	β	X
ejpam-3441	560	27	]	]	PUNCT
ejpam-3441	560	28	and	and	CCONJ
ejpam-3441	560	29	every	every	DET
ejpam-3441	560	30	intuitionistic	intuitionistic	ADJ
ejpam-3441	560	31	fuzzy	fuzzy	ADJ
ejpam-3441	560	32	ideal	ideal	NOUN
ejpam-3441	560	33	i	i	PRON
ejpam-3441	560	34	with	with	ADP
ejpam-3441	560	35	thresholds	threshold	NOUN
ejpam-3441	560	36	(	(	PUNCT
ejpam-3441	560	37	α	α	X
ejpam-3441	560	38	,	,	PUNCT
ejpam-3441	560	39	β	β	X
ejpam-3441	560	40	]	]	PUNCT
ejpam-3441	560	41	of	of	ADP
ejpam-3441	560	42	r.	r.	PROPN
ejpam-3441	560	43	(	(	PUNCT
ejpam-3441	560	44	4	4	NUM
ejpam-3441	560	45	)	)	PUNCT
ejpam-3441	560	46	c	c	NOUN
ejpam-3441	561	1	∧βα	∧βα	ADJ
ejpam-3441	561	2	i	i	NOUN
ejpam-3441	561	3	=	=	PUNCT
ejpam-3441	561	4	(	(	PUNCT
ejpam-3441	561	5	c	c	NOUN
ejpam-3441	561	6	◦	◦	NOUN
ejpam-3441	561	7	βα	βα	X
ejpam-3441	561	8	i	i	NOUN
ejpam-3441	561	9	)	)	PUNCT
ejpam-3441	561	10	◦	◦	NOUN
ejpam-3441	561	11	βα	βα	X
ejpam-3441	561	12	c	c	NOUN
ejpam-3441	561	13	for	for	ADP
ejpam-3441	561	14	every	every	DET
ejpam-3441	561	15	intuitionistic	intuitionistic	ADJ
ejpam-3441	561	16	fuzzy	fuzzy	ADJ
ejpam-3441	561	17	generalized	generalize	VERB
ejpam-3441	561	18	bi	bi	ADJ
ejpam-3441	561	19	-	-	ADJ
ejpam-3441	561	20	ideal	ideal	ADJ
ejpam-3441	561	21	c	c	NOUN
ejpam-3441	561	22	with	with	ADP
ejpam-3441	561	23	thresholds	threshold	NOUN
ejpam-3441	561	24	(	(	PUNCT
ejpam-3441	561	25	α	α	X
ejpam-3441	561	26	,	,	PUNCT
ejpam-3441	561	27	β	β	X
ejpam-3441	561	28	]	]	PUNCT
ejpam-3441	561	29	and	and	CCONJ
ejpam-3441	561	30	every	every	DET
ejpam-3441	561	31	intuitionistic	intuitionistic	ADJ
ejpam-3441	561	32	fuzzy	fuzzy	ADJ
ejpam-3441	561	33	ideal	ideal	NOUN
ejpam-3441	561	34	i	i	PRON
ejpam-3441	561	35	with	with	ADP
ejpam-3441	561	36	thresholds	threshold	NOUN
ejpam-3441	561	37	(	(	PUNCT
ejpam-3441	561	38	α	α	X
ejpam-3441	561	39	,	,	PUNCT
ejpam-3441	561	40	β	β	X
ejpam-3441	561	41	]	]	PUNCT
ejpam-3441	561	42	of	of	ADP
ejpam-3441	561	43	r.	r.	PROPN
ejpam-3441	561	44	proof	proof	PROPN
ejpam-3441	561	45	.	.	PUNCT
ejpam-3441	561	46	suppose	suppose	VERB
ejpam-3441	561	47	that	that	SCONJ
ejpam-3441	561	48	(	(	PUNCT
ejpam-3441	561	49	1	1	X
ejpam-3441	561	50	)	)	PUNCT
ejpam-3441	561	51	holds	hold	VERB
ejpam-3441	561	52	.	.	PUNCT
ejpam-3441	562	1	let	let	VERB
ejpam-3441	562	2	c	c	NOUN
ejpam-3441	562	3	=	=	PUNCT
ejpam-3441	562	4	(	(	PUNCT
ejpam-3441	562	5	µc	µc	INTJ
ejpam-3441	562	6	,	,	PUNCT
ejpam-3441	562	7	γc	γc	PROPN
ejpam-3441	562	8	)	)	PUNCT
ejpam-3441	562	9	be	be	AUX
ejpam-3441	562	10	an	an	DET
ejpam-3441	562	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	562	12	fuzzy	fuzzy	ADJ
ejpam-3441	562	13	generalized	generalize	VERB
ejpam-3441	562	14	bi	bi	NOUN
ejpam-3441	562	15	-	-	NOUN
ejpam-3441	562	16	ideal	ideal	NOUN
ejpam-3441	562	17	with	with	ADP
ejpam-3441	562	18	thresholds	threshold	NOUN
ejpam-3441	562	19	(	(	PUNCT
ejpam-3441	562	20	α	α	X
ejpam-3441	562	21	,	,	PUNCT
ejpam-3441	562	22	β	β	X
ejpam-3441	562	23	]	]	PUNCT
ejpam-3441	563	1	and	and	CCONJ
ejpam-3441	563	2	i	i	PRON
ejpam-3441	563	3	=	=	PUNCT
ejpam-3441	563	4	(	(	PUNCT
ejpam-3441	563	5	µi	µi	INTJ
ejpam-3441	563	6	,	,	PUNCT
ejpam-3441	563	7	γi	γi	INTJ
ejpam-3441	563	8	)	)	PUNCT
ejpam-3441	563	9	be	be	VERB
ejpam-3441	563	10	an	an	DET
ejpam-3441	563	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	563	12	fuzzy	fuzzy	ADJ
ejpam-3441	563	13	ideal	ideal	NOUN
ejpam-3441	563	14	with	with	ADP
ejpam-3441	563	15	thresholds	threshold	NOUN
ejpam-3441	563	16	(	(	PUNCT
ejpam-3441	563	17	α	α	X
ejpam-3441	563	18	,	,	PUNCT
ejpam-3441	563	19	β	β	X
ejpam-3441	563	20	]	]	PUNCT
ejpam-3441	563	21	of	of	ADP
ejpam-3441	563	22	r.	r.	PROPN
ejpam-3441	563	23	now	now	ADV
ejpam-3441	563	24	(	(	PUNCT
ejpam-3441	563	25	c	c	NOUN
ejpam-3441	563	26	◦	◦	NOUN
ejpam-3441	563	27	βα	βα	X
ejpam-3441	563	28	i	i	NOUN
ejpam-3441	563	29	)	)	PUNCT
ejpam-3441	563	30	◦	◦	NOUN
ejpam-3441	563	31	βα	βα	X
ejpam-3441	564	1	c	c	NOUN
ejpam-3441	564	2	⊆	⊆	NUM
ejpam-3441	564	3	(	(	PUNCT
ejpam-3441	564	4	r	r	NOUN
ejpam-3441	564	5	◦	◦	NOUN
ejpam-3441	564	6	βα	βα	X
ejpam-3441	564	7	i	i	NOUN
ejpam-3441	564	8	)	)	PUNCT
ejpam-3441	564	9	◦	◦	AUX
ejpam-3441	564	10	βα	βα	NOUN
ejpam-3441	564	11	r	r	NOUN
ejpam-3441	564	12	⊆	⊆	NUM
ejpam-3441	564	13	i	i	SYM
ejpam-3441	564	14	◦	◦	NOUN
ejpam-3441	564	15	βα	βα	NOUN
ejpam-3441	564	16	r	r	NOUN
ejpam-3441	564	17	⊆	⊆	NUM
ejpam-3441	564	18	iβα	iβα	NOUN
ejpam-3441	564	19	and	and	CCONJ
ejpam-3441	564	20	(	(	PUNCT
ejpam-3441	564	21	c	c	PROPN
ejpam-3441	564	22	◦	◦	NOUN
ejpam-3441	564	23	βα	βα	X
ejpam-3441	564	24	i	i	NOUN
ejpam-3441	564	25	)	)	PUNCT
ejpam-3441	564	26	◦	◦	NOUN
ejpam-3441	564	27	βα	βα	X
ejpam-3441	564	28	c	c	NOUN
ejpam-3441	564	29	⊆	⊆	NUM
ejpam-3441	564	30	(	(	PUNCT
ejpam-3441	564	31	c	c	NOUN
ejpam-3441	564	32	◦	◦	NOUN
ejpam-3441	564	33	βα	βα	NOUN
ejpam-3441	564	34	r	r	NOUN
ejpam-3441	564	35	)	)	PUNCT
ejpam-3441	564	36	◦	◦	NOUN
ejpam-3441	564	37	βα	βα	X
ejpam-3441	564	38	c	c	NOUN
ejpam-3441	564	39	⊆	⊆	NUM
ejpam-3441	564	40	cβα	cβα	NOUN
ejpam-3441	564	41	,	,	PUNCT
ejpam-3441	564	42	i.e.	i.e.	X
ejpam-3441	564	43	,	,	PUNCT
ejpam-3441	564	44	(	(	PUNCT
ejpam-3441	564	45	c	c	NOUN
ejpam-3441	564	46	◦	◦	NOUN
ejpam-3441	564	47	βα	βα	X
ejpam-3441	564	48	i	i	NOUN
ejpam-3441	564	49	)	)	PUNCT
ejpam-3441	564	50	◦	◦	NOUN
ejpam-3441	564	51	βα	βα	X
ejpam-3441	564	52	c	c	NOUN
ejpam-3441	564	53	⊆	⊆	NUM
ejpam-3441	564	54	cβα	cβα	NOUN
ejpam-3441	564	55	∧	∧	NOUN
ejpam-3441	564	56	iβα	iβα	NOUN
ejpam-3441	564	57	=	=	SYM
ejpam-3441	564	58	c	c	NOUN
ejpam-3441	564	59	∧βα	∧βα	ADJ
ejpam-3441	564	60	i.	i.	NOUN
ejpam-3441	564	61	let	let	VERB
ejpam-3441	564	62	x	x	X
ejpam-3441	564	63	∈	∈	PROPN
ejpam-3441	564	64	r	r	NOUN
ejpam-3441	564	65	,	,	PUNCT
ejpam-3441	564	66	this	this	PRON
ejpam-3441	564	67	implies	imply	VERB
ejpam-3441	564	68	that	that	SCONJ
ejpam-3441	564	69	there	there	PRON
ejpam-3441	564	70	exists	exist	VERB
ejpam-3441	564	71	a	a	DET
ejpam-3441	564	72	∈	∈	NOUN
ejpam-3441	564	73	r	r	NOUN
ejpam-3441	565	1	such	such	ADJ
ejpam-3441	565	2	that	that	PRON
ejpam-3441	565	3	x	x	SYM
ejpam-3441	565	4	=	=	SYM
ejpam-3441	565	5	(	(	PUNCT
ejpam-3441	565	6	xa)x	xa)x	PROPN
ejpam-3441	565	7	.	.	PUNCT
ejpam-3441	566	1	now	now	ADV
ejpam-3441	566	2	xa	xa	PROPN
ejpam-3441	566	3	=	=	PRON
ejpam-3441	566	4	(	(	PUNCT
ejpam-3441	566	5	(	(	PUNCT
ejpam-3441	566	6	xa)x)a	xa)x)a	PROPN
ejpam-3441	566	7	=	=	SYM
ejpam-3441	566	8	(	(	PUNCT
ejpam-3441	566	9	ax)(xa	ax)(xa	PROPN
ejpam-3441	566	10	)	)	PUNCT
ejpam-3441	566	11	=	=	PUNCT
ejpam-3441	566	12	x((ax)a	x((ax)a	PROPN
ejpam-3441	566	13	)	)	PUNCT
ejpam-3441	566	14	.	.	PUNCT
ejpam-3441	567	1	thus	thus	ADV
ejpam-3441	567	2	(	(	PUNCT
ejpam-3441	567	3	(	(	PUNCT
ejpam-3441	567	4	µc	µc	INTJ
ejpam-3441	567	5	◦	◦	NOUN
ejpam-3441	567	6	βα	βα	NOUN
ejpam-3441	567	7	µi	µi	NOUN
ejpam-3441	567	8	)	)	PUNCT
ejpam-3441	567	9	◦	◦	NOUN
ejpam-3441	567	10	βα	βα	NOUN
ejpam-3441	567	11	µc)(x	µc)(x	NOUN
ejpam-3441	567	12	)	)	PUNCT
ejpam-3441	568	1	=	=	PRON
ejpam-3441	568	2	{	{	PUNCT
ejpam-3441	568	3	(	(	PUNCT
ejpam-3441	568	4	(	(	PUNCT
ejpam-3441	568	5	µc	µc	INTJ
ejpam-3441	568	6	◦	◦	VERB
ejpam-3441	568	7	µi	µi	PART
ejpam-3441	568	8	)	)	PUNCT
ejpam-3441	568	9	◦	◦	NOUN
ejpam-3441	568	10	µc)(x	µc)(x	NOUN
ejpam-3441	568	11	)	)	PUNCT
ejpam-3441	569	1	∧	∧	PROPN
ejpam-3441	569	2	β	β	NOUN
ejpam-3441	569	3	}	}	PUNCT
ejpam-3441	569	4	∨	∨	NUM
ejpam-3441	569	5	α	α	NOUN
ejpam-3441	569	6	=	=	X
ejpam-3441	569	7	{	{	PUNCT
ejpam-3441	569	8	(	(	PUNCT
ejpam-3441	569	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	569	10	i=1	i=1	PROPN
ejpam-3441	569	11	piqi	piqi	NOUN
ejpam-3441	569	12	{	{	PUNCT
ejpam-3441	569	13	∧ni=1	∧ni=1	X
ejpam-3441	569	14	{	{	PUNCT
ejpam-3441	569	15	(	(	PUNCT
ejpam-3441	569	16	µc	µc	INTJ
ejpam-3441	569	17	◦	◦	NOUN
ejpam-3441	569	18	µi	µi	PROPN
ejpam-3441	569	19	)	)	PUNCT
ejpam-3441	569	20	(	(	PUNCT
ejpam-3441	569	21	pi	pi	NOUN
ejpam-3441	569	22	)	)	PUNCT
ejpam-3441	569	23	∧	∧	NOUN
ejpam-3441	569	24	µc	µc	INTJ
ejpam-3441	569	25	(	(	PUNCT
ejpam-3441	569	26	qi	qi	NOUN
ejpam-3441	569	27	)	)	PUNCT
ejpam-3441	569	28	}	}	PUNCT
ejpam-3441	569	29	}	}	PUNCT
ejpam-3441	569	30	)	)	PUNCT
ejpam-3441	570	1	∧	∧	PROPN
ejpam-3441	570	2	β	β	NOUN
ejpam-3441	570	3	}	}	PUNCT
ejpam-3441	570	4	∨	∨	NUM
ejpam-3441	570	5	α	α	PROPN
ejpam-3441	570	6	≥	≥	X
ejpam-3441	570	7	{	{	PUNCT
ejpam-3441	570	8	{	{	PUNCT
ejpam-3441	570	9	(	(	PUNCT
ejpam-3441	570	10	µc	µc	INTJ
ejpam-3441	570	11	◦	◦	NOUN
ejpam-3441	570	12	µi	µi	PROPN
ejpam-3441	570	13	)	)	PUNCT
ejpam-3441	570	14	(	(	PUNCT
ejpam-3441	570	15	xa	xa	X
ejpam-3441	570	16	)	)	PUNCT
ejpam-3441	570	17	∧	∧	PROPN
ejpam-3441	570	18	µc	µc	INTJ
ejpam-3441	570	19	(	(	PUNCT
ejpam-3441	570	20	x	x	NOUN
ejpam-3441	570	21	)	)	PUNCT
ejpam-3441	570	22	}	}	PUNCT
ejpam-3441	570	23	∧	∧	PROPN
ejpam-3441	570	24	β	β	NOUN
ejpam-3441	570	25	}	}	PUNCT
ejpam-3441	570	26	∨	∨	NUM
ejpam-3441	570	27	α	α	NOUN
ejpam-3441	570	28	=	=	SYM
ejpam-3441	570	29	(	(	PUNCT
ejpam-3441	570	30	(	(	PUNCT
ejpam-3441	570	31	µc	µc	INTJ
ejpam-3441	570	32	◦	◦	NOUN
ejpam-3441	570	33	µi	µi	PROPN
ejpam-3441	570	34	)	)	PUNCT
ejpam-3441	570	35	(	(	PUNCT
ejpam-3441	570	36	xa	xa	PROPN
ejpam-3441	570	37	)	)	PUNCT
ejpam-3441	570	38	∨	∨	NUM
ejpam-3441	570	39	α	α	NOUN
ejpam-3441	570	40	)	)	PUNCT
ejpam-3441	570	41	∧	∧	NOUN
ejpam-3441	570	42	(	(	PUNCT
ejpam-3441	570	43	µc	µc	INTJ
ejpam-3441	570	44	(	(	PUNCT
ejpam-3441	570	45	x	x	NOUN
ejpam-3441	570	46	)	)	PUNCT
ejpam-3441	570	47	∨	∨	NUM
ejpam-3441	570	48	α	α	NOUN
ejpam-3441	570	49	)	)	PUNCT
ejpam-3441	570	50	∧	∧	PROPN
ejpam-3441	570	51	(	(	PUNCT
ejpam-3441	570	52	β	β	X
ejpam-3441	570	53	∨	∨	NUM
ejpam-3441	570	54	α	α	NOUN
ejpam-3441	570	55	)	)	PUNCT
ejpam-3441	570	56	k.	k.	PROPN
ejpam-3441	571	1	nasreen	nasreen	PROPN
ejpam-3441	571	2	et	et	PROPN
ejpam-3441	571	3	al	al	PROPN
ejpam-3441	571	4	.	.	PUNCT
ejpam-3441	571	5	/	/	SYM
ejpam-3441	571	6	eur	eur	PROPN
ejpam-3441	571	7	.	.	PUNCT
ejpam-3441	572	1	j.	j.	PROPN
ejpam-3441	572	2	pure	pure	PROPN
ejpam-3441	572	3	appl	appl	PROPN
ejpam-3441	572	4	.	.	PROPN
ejpam-3441	572	5	math	math	PROPN
ejpam-3441	572	6	,	,	PUNCT
ejpam-3441	572	7	12	12	NUM
ejpam-3441	572	8	(	(	PUNCT
ejpam-3441	572	9	3	3	NUM
ejpam-3441	572	10	)	)	PUNCT
ejpam-3441	572	11	(	(	PUNCT
ejpam-3441	572	12	2019	2019	NUM
ejpam-3441	572	13	)	)	PUNCT
ejpam-3441	572	14	,	,	PUNCT
ejpam-3441	572	15	906	906	NUM
ejpam-3441	572	16	-	-	SYM
ejpam-3441	572	17	943	943	NUM
ejpam-3441	572	18	931	931	NUM
ejpam-3441	572	19	=	=	SYM
ejpam-3441	572	20	(	(	PUNCT
ejpam-3441	572	21	(	(	PUNCT
ejpam-3441	572	22	µc	µc	AUX
ejpam-3441	572	23	◦	◦	NOUN
ejpam-3441	572	24	µi	µi	PROPN
ejpam-3441	572	25	)	)	PUNCT
ejpam-3441	572	26	(	(	PUNCT
ejpam-3441	572	27	xa	xa	PROPN
ejpam-3441	572	28	)	)	PUNCT
ejpam-3441	572	29	∨	∨	NUM
ejpam-3441	572	30	α	α	NOUN
ejpam-3441	572	31	)	)	PUNCT
ejpam-3441	572	32	∧	∧	PROPN
ejpam-3441	572	33	µc(x	µc(x	NOUN
ejpam-3441	572	34	)	)	PUNCT
ejpam-3441	572	35	∧	∧	NOUN
ejpam-3441	572	36	β	β	X
ejpam-3441	572	37	=	=	SYM
ejpam-3441	572	38	(	(	PUNCT
ejpam-3441	572	39	(	(	PUNCT
ejpam-3441	572	40	∨xa=∑n	∨xa=∑n	PROPN
ejpam-3441	572	41	i=1mini	i=1mini	PROPN
ejpam-3441	572	42	{	{	PUNCT
ejpam-3441	572	43	∧ni=1	∧ni=1	X
ejpam-3441	572	44	{	{	PUNCT
ejpam-3441	572	45	µc	µc	PROPN
ejpam-3441	572	46	(	(	PUNCT
ejpam-3441	572	47	mi	mi	NOUN
ejpam-3441	572	48	)	)	PUNCT
ejpam-3441	572	49	∧	∧	PROPN
ejpam-3441	572	50	µi	µi	PROPN
ejpam-3441	572	51	(	(	PUNCT
ejpam-3441	572	52	ni	ni	PROPN
ejpam-3441	572	53	)	)	PUNCT
ejpam-3441	572	54	}	}	PUNCT
ejpam-3441	572	55	}	}	PUNCT
ejpam-3441	572	56	)	)	PUNCT
ejpam-3441	572	57	∨	∨	NUM
ejpam-3441	572	58	α	α	NOUN
ejpam-3441	572	59	)	)	PUNCT
ejpam-3441	572	60	∧	∧	PROPN
ejpam-3441	572	61	µc(x	µc(x	NOUN
ejpam-3441	572	62	)	)	PUNCT
ejpam-3441	572	63	∧	∧	PROPN
ejpam-3441	572	64	β	β	X
ejpam-3441	572	65	≥	≥	X
ejpam-3441	572	66	(	(	PUNCT
ejpam-3441	572	67	{	{	PUNCT
ejpam-3441	572	68	µc(x	µc(x	NOUN
ejpam-3441	572	69	)	)	PUNCT
ejpam-3441	572	70	∧	∧	NOUN
ejpam-3441	572	71	µi((ax)a	µi((ax)a	NOUN
ejpam-3441	572	72	)	)	PUNCT
ejpam-3441	572	73	}	}	PUNCT
ejpam-3441	572	74	∨	∨	NUM
ejpam-3441	572	75	α	α	NOUN
ejpam-3441	572	76	)	)	PUNCT
ejpam-3441	572	77	∧	∧	PROPN
ejpam-3441	572	78	µc(x	µc(x	NOUN
ejpam-3441	572	79	)	)	PUNCT
ejpam-3441	572	80	∧	∧	NOUN
ejpam-3441	572	81	β	β	X
ejpam-3441	572	82	=	=	SYM
ejpam-3441	572	83	(	(	PUNCT
ejpam-3441	572	84	µc(x	µc(x	NOUN
ejpam-3441	572	85	)	)	PUNCT
ejpam-3441	572	86	∨	∨	NUM
ejpam-3441	572	87	α	α	NOUN
ejpam-3441	572	88	)	)	PUNCT
ejpam-3441	572	89	∧	∧	PROPN
ejpam-3441	572	90	(	(	PUNCT
ejpam-3441	572	91	µi((ax)a	µi((ax)a	NOUN
ejpam-3441	572	92	)	)	PUNCT
ejpam-3441	572	93	∨	∨	NUM
ejpam-3441	572	94	α	α	NOUN
ejpam-3441	572	95	)	)	PUNCT
ejpam-3441	572	96	∧	∧	PROPN
ejpam-3441	572	97	µc(x	µc(x	NOUN
ejpam-3441	572	98	)	)	PUNCT
ejpam-3441	572	99	∧	∧	PROPN
ejpam-3441	572	100	β	β	X
ejpam-3441	572	101	≥	≥	NOUN
ejpam-3441	572	102	µc(x	µc(x	NOUN
ejpam-3441	572	103	)	)	PUNCT
ejpam-3441	572	104	∧	∧	PROPN
ejpam-3441	572	105	(	(	PUNCT
ejpam-3441	572	106	µi(x	µi(x	NOUN
ejpam-3441	572	107	)	)	PUNCT
ejpam-3441	573	1	∧	∧	NOUN
ejpam-3441	573	2	β	β	NOUN
ejpam-3441	573	3	)	)	PUNCT
ejpam-3441	573	4	∧	∧	PROPN
ejpam-3441	573	5	µc(x	µc(x	NOUN
ejpam-3441	573	6	)	)	PUNCT
ejpam-3441	573	7	∧	∧	PROPN
ejpam-3441	573	8	β	β	X
ejpam-3441	573	9	=	=	SYM
ejpam-3441	573	10	µc(x	µc(x	NOUN
ejpam-3441	573	11	)	)	PUNCT
ejpam-3441	573	12	∧	∧	NOUN
ejpam-3441	573	13	µi(x	µi(x	NUM
ejpam-3441	573	14	)	)	PUNCT
ejpam-3441	573	15	∧	∧	NOUN
ejpam-3441	573	16	β	β	X
ejpam-3441	573	17	=	=	SYM
ejpam-3441	573	18	(	(	PUNCT
ejpam-3441	573	19	µc	µc	INTJ
ejpam-3441	573	20	∧	∧	PROPN
ejpam-3441	573	21	µi)(x	µi)(x	PROPN
ejpam-3441	573	22	)	)	PUNCT
ejpam-3441	573	23	∧	∧	NOUN
ejpam-3441	573	24	β	β	X
ejpam-3441	573	25	=	=	SYM
ejpam-3441	573	26	{	{	PUNCT
ejpam-3441	573	27	(	(	PUNCT
ejpam-3441	573	28	µc	µc	INTJ
ejpam-3441	573	29	∧	∧	PROPN
ejpam-3441	573	30	µi)(x	µi)(x	PROPN
ejpam-3441	573	31	)	)	PUNCT
ejpam-3441	573	32	∧	∧	PROPN
ejpam-3441	573	33	β	β	PROPN
ejpam-3441	573	34	}	}	PUNCT
ejpam-3441	573	35	∨	∨	NUM
ejpam-3441	573	36	α	α	NOUN
ejpam-3441	573	37	=	=	PUNCT
ejpam-3441	573	38	(	(	PUNCT
ejpam-3441	573	39	µc	µc	INTJ
ejpam-3441	573	40	∧βα	∧βα	ADJ
ejpam-3441	573	41	µi)(x	µi)(x	NOUN
ejpam-3441	573	42	)	)	PUNCT
ejpam-3441	573	43	.	.	PUNCT
ejpam-3441	574	1	⇒	⇒	NOUN
ejpam-3441	574	2	µc	µc	VERB
ejpam-3441	574	3	∧βα	∧βα	PROPN
ejpam-3441	574	4	µi	µi	PROPN
ejpam-3441	574	5	⊆	⊆	NUM
ejpam-3441	574	6	(	(	PUNCT
ejpam-3441	574	7	µc	µc	INTJ
ejpam-3441	574	8	◦	◦	NOUN
ejpam-3441	574	9	βα	βα	NOUN
ejpam-3441	574	10	µi	µi	NOUN
ejpam-3441	574	11	)	)	PUNCT
ejpam-3441	574	12	◦	◦	NOUN
ejpam-3441	574	13	βα	βα	NOUN
ejpam-3441	574	14	µc	µc	INTJ
ejpam-3441	574	15	.	.	PUNCT
ejpam-3441	575	1	similarly	similarly	ADV
ejpam-3441	575	2	,	,	PUNCT
ejpam-3441	575	3	we	we	PRON
ejpam-3441	575	4	have	have	VERB
ejpam-3441	575	5	γc∨βαγi	γc∨βαγi	PUNCT
ejpam-3441	576	1	⊇	⊇	X
ejpam-3441	576	2	(	(	PUNCT
ejpam-3441	576	3	γc	γc	NOUN
ejpam-3441	576	4	◦	◦	NOUN
ejpam-3441	576	5	βαγi)	βαγi)	NOUN
ejpam-3441	576	6	◦	◦	NOUN
ejpam-3441	576	7	βαγc	βαγc	NOUN
ejpam-3441	576	8	.	.	PUNCT
ejpam-3441	577	1	therefore	therefore	ADV
ejpam-3441	577	2	c∧βαi	c∧βαi	PUNCT
ejpam-3441	577	3	=	=	PUNCT
ejpam-3441	577	4	(	(	PUNCT
ejpam-3441	577	5	c	c	X
ejpam-3441	577	6	◦	◦	NOUN
ejpam-3441	577	7	βαi)	βαi)	NOUN
ejpam-3441	577	8	◦	◦	NOUN
ejpam-3441	577	9	βαc	βαc	ADJ
ejpam-3441	577	10	,	,	PUNCT
ejpam-3441	577	11	i.e.	i.e.	X
ejpam-3441	577	12	,	,	PUNCT
ejpam-3441	577	13	(	(	PUNCT
ejpam-3441	577	14	1)⇒	1)⇒	NUM
ejpam-3441	577	15	(	(	PUNCT
ejpam-3441	577	16	4	4	NUM
ejpam-3441	577	17	)	)	PUNCT
ejpam-3441	577	18	.	.	PUNCT
ejpam-3441	578	1	since	since	SCONJ
ejpam-3441	578	2	(	(	PUNCT
ejpam-3441	578	3	4)⇒	4)⇒	X
ejpam-3441	578	4	(	(	PUNCT
ejpam-3441	578	5	3	3	NUM
ejpam-3441	578	6	)	)	PUNCT
ejpam-3441	578	7	and	and	CCONJ
ejpam-3441	578	8	(	(	PUNCT
ejpam-3441	578	9	3)⇒	3)⇒	NUM
ejpam-3441	578	10	(	(	PUNCT
ejpam-3441	578	11	2	2	NUM
ejpam-3441	578	12	)	)	PUNCT
ejpam-3441	578	13	.	.	PUNCT
ejpam-3441	579	1	assume	assume	VERB
ejpam-3441	579	2	that	that	SCONJ
ejpam-3441	579	3	(	(	PUNCT
ejpam-3441	579	4	2	2	X
ejpam-3441	579	5	)	)	PUNCT
ejpam-3441	579	6	holds	hold	VERB
ejpam-3441	579	7	.	.	PUNCT
ejpam-3441	580	1	then	then	ADV
ejpam-3441	580	2	a∧βαr	a∧βαr	PRON
ejpam-3441	580	3	=	=	SYM
ejpam-3441	580	4	(	(	PUNCT
ejpam-3441	580	5	a	a	DET
ejpam-3441	580	6	◦	◦	NOUN
ejpam-3441	580	7	βαr	βαr	NOUN
ejpam-3441	580	8	)	)	PUNCT
ejpam-3441	580	9	◦	◦	NOUN
ejpam-3441	580	10	βαa	βαa	PROPN
ejpam-3441	580	11	,	,	PUNCT
ejpam-3441	580	12	where	where	SCONJ
ejpam-3441	580	13	r	r	NOUN
ejpam-3441	580	14	itself	itself	PRON
ejpam-3441	580	15	is	be	AUX
ejpam-3441	580	16	an	an	DET
ejpam-3441	580	17	intuitionistic	intuitionistic	ADJ
ejpam-3441	580	18	fuzzy	fuzzy	ADJ
ejpam-3441	580	19	two	two	NUM
ejpam-3441	580	20	-	-	PUNCT
ejpam-3441	580	21	sided	sided	ADJ
ejpam-3441	580	22	ideal	ideal	NOUN
ejpam-3441	580	23	with	with	ADP
ejpam-3441	580	24	thresholds	threshold	NOUN
ejpam-3441	580	25	(	(	PUNCT
ejpam-3441	580	26	α	α	X
ejpam-3441	580	27	,	,	PUNCT
ejpam-3441	580	28	β	β	X
ejpam-3441	580	29	]	]	PUNCT
ejpam-3441	580	30	of	of	ADP
ejpam-3441	580	31	r	r	NOUN
ejpam-3441	580	32	,	,	PUNCT
ejpam-3441	580	33	i.e.	i.e.	X
ejpam-3441	580	34	,	,	PUNCT
ejpam-3441	580	35	aβα	aβα	NOUN
ejpam-3441	580	36	=	=	PUNCT
ejpam-3441	580	37	(	(	PUNCT
ejpam-3441	580	38	a	a	DET
ejpam-3441	580	39	◦	◦	NOUN
ejpam-3441	580	40	βα	βα	NOUN
ejpam-3441	580	41	r	r	NOUN
ejpam-3441	580	42	)	)	PUNCT
ejpam-3441	580	43	◦	◦	NOUN
ejpam-3441	580	44	βα	βα	NOUN
ejpam-3441	580	45	a.	a.	NOUN
ejpam-3441	580	46	hence	hence	ADV
ejpam-3441	580	47	r	r	NOUN
ejpam-3441	580	48	is	be	AUX
ejpam-3441	580	49	a	a	DET
ejpam-3441	580	50	regular	regular	NOUN
ejpam-3441	580	51	by	by	ADP
ejpam-3441	580	52	the	the	DET
ejpam-3441	580	53	theorem	theorem	NOUN
ejpam-3441	580	54	8	8	NUM
ejpam-3441	580	55	,	,	PUNCT
ejpam-3441	580	56	i.e.	i.e.	X
ejpam-3441	580	57	,	,	PUNCT
ejpam-3441	580	58	(	(	PUNCT
ejpam-3441	580	59	2)⇒	2)⇒	NUM
ejpam-3441	580	60	(	(	PUNCT
ejpam-3441	580	61	1	1	NUM
ejpam-3441	580	62	)	)	PUNCT
ejpam-3441	580	63	.	.	PUNCT
ejpam-3441	581	1	theorem	theorem	VERB
ejpam-3441	581	2	11	11	NUM
ejpam-3441	581	3	.	.	PUNCT
ejpam-3441	582	1	let	let	VERB
ejpam-3441	582	2	r	r	PRON
ejpam-3441	582	3	be	be	AUX
ejpam-3441	582	4	an	an	DET
ejpam-3441	582	5	la	la	NOUN
ejpam-3441	582	6	-	-	NOUN
ejpam-3441	582	7	ring	ring	NOUN
ejpam-3441	582	8	with	with	ADP
ejpam-3441	582	9	left	left	ADJ
ejpam-3441	582	10	identity	identity	NOUN
ejpam-3441	582	11	e	e	NOUN
ejpam-3441	582	12	,	,	PUNCT
ejpam-3441	582	13	such	such	ADJ
ejpam-3441	582	14	that	that	SCONJ
ejpam-3441	582	15	(	(	PUNCT
ejpam-3441	582	16	xe)r	xe)r	PROPN
ejpam-3441	582	17	=	=	SYM
ejpam-3441	582	18	xr	xr	PROPN
ejpam-3441	582	19	for	for	ADP
ejpam-3441	582	20	all	all	DET
ejpam-3441	582	21	x	x	PROPN
ejpam-3441	582	22	∈	∈	PROPN
ejpam-3441	582	23	r.	r.	NOUN
ejpam-3441	582	24	then	then	ADV
ejpam-3441	582	25	the	the	DET
ejpam-3441	582	26	following	follow	VERB
ejpam-3441	582	27	conditions	condition	NOUN
ejpam-3441	582	28	are	be	AUX
ejpam-3441	582	29	equivalent	equivalent	ADJ
ejpam-3441	582	30	.	.	PUNCT
ejpam-3441	583	1	(	(	PUNCT
ejpam-3441	583	2	1	1	X
ejpam-3441	583	3	)	)	PUNCT
ejpam-3441	583	4	r	r	NOUN
ejpam-3441	583	5	is	be	AUX
ejpam-3441	583	6	a	a	DET
ejpam-3441	583	7	regular	regular	NOUN
ejpam-3441	583	8	.	.	PUNCT
ejpam-3441	584	1	(	(	PUNCT
ejpam-3441	584	2	2	2	X
ejpam-3441	584	3	)	)	PUNCT
ejpam-3441	584	4	a	a	DET
ejpam-3441	584	5	∧βα	∧βα	PROPN
ejpam-3441	584	6	d	d	ADP
ejpam-3441	584	7	⊆	⊆	PROPN
ejpam-3441	584	8	d	d	PROPN
ejpam-3441	584	9	◦	◦	NOUN
ejpam-3441	584	10	βα	βα	X
ejpam-3441	584	11	a	a	PRON
ejpam-3441	584	12	for	for	ADP
ejpam-3441	584	13	every	every	DET
ejpam-3441	584	14	intuitionistic	intuitionistic	ADJ
ejpam-3441	584	15	fuzzy	fuzzy	ADJ
ejpam-3441	584	16	quasi	quasi	NOUN
ejpam-3441	584	17	-	-	NOUN
ejpam-3441	584	18	ideal	ideal	ADJ
ejpam-3441	584	19	a	a	PRON
ejpam-3441	584	20	with	with	ADP
ejpam-3441	584	21	thresholds	threshold	NOUN
ejpam-3441	584	22	(	(	PUNCT
ejpam-3441	584	23	α	α	X
ejpam-3441	584	24	,	,	PUNCT
ejpam-3441	584	25	β	β	X
ejpam-3441	584	26	]	]	PUNCT
ejpam-3441	584	27	and	and	CCONJ
ejpam-3441	584	28	every	every	DET
ejpam-3441	584	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	584	30	fuzzy	fuzzy	ADJ
ejpam-3441	584	31	right	right	ADJ
ejpam-3441	584	32	ideal	ideal	NOUN
ejpam-3441	585	1	d	d	NOUN
ejpam-3441	585	2	with	with	ADP
ejpam-3441	585	3	thresholds	threshold	NOUN
ejpam-3441	585	4	(	(	PUNCT
ejpam-3441	585	5	α	α	X
ejpam-3441	585	6	,	,	PUNCT
ejpam-3441	585	7	β	β	X
ejpam-3441	585	8	]	]	PUNCT
ejpam-3441	585	9	of	of	ADP
ejpam-3441	585	10	r.	r.	PROPN
ejpam-3441	585	11	(	(	PUNCT
ejpam-3441	585	12	3	3	NUM
ejpam-3441	585	13	)	)	PUNCT
ejpam-3441	585	14	b∧βαd	b∧βαd	NOUN
ejpam-3441	585	15	⊆	⊆	NUM
ejpam-3441	585	16	d	d	PROPN
ejpam-3441	585	17	◦	◦	NOUN
ejpam-3441	585	18	βαb	βαb	NOUN
ejpam-3441	585	19	for	for	ADP
ejpam-3441	585	20	every	every	DET
ejpam-3441	585	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	585	22	fuzzy	fuzzy	ADJ
ejpam-3441	585	23	bi	bi	ADJ
ejpam-3441	585	24	-	-	ADJ
ejpam-3441	585	25	ideal	ideal	ADJ
ejpam-3441	585	26	b	b	PROPN
ejpam-3441	585	27	with	with	ADP
ejpam-3441	585	28	thresholds	threshold	NOUN
ejpam-3441	585	29	(	(	PUNCT
ejpam-3441	585	30	α	α	X
ejpam-3441	585	31	,	,	PUNCT
ejpam-3441	585	32	β	β	X
ejpam-3441	585	33	]	]	PUNCT
ejpam-3441	585	34	and	and	CCONJ
ejpam-3441	585	35	every	every	DET
ejpam-3441	585	36	intuitionistic	intuitionistic	ADJ
ejpam-3441	585	37	fuzzy	fuzzy	ADJ
ejpam-3441	585	38	right	right	ADJ
ejpam-3441	585	39	ideal	ideal	NOUN
ejpam-3441	585	40	d	d	NOUN
ejpam-3441	585	41	with	with	ADP
ejpam-3441	585	42	thresholds	threshold	NOUN
ejpam-3441	585	43	(	(	PUNCT
ejpam-3441	585	44	α	α	X
ejpam-3441	585	45	,	,	PUNCT
ejpam-3441	585	46	β	β	X
ejpam-3441	585	47	]	]	PUNCT
ejpam-3441	585	48	of	of	ADP
ejpam-3441	585	49	r.	r.	PROPN
ejpam-3441	585	50	(	(	PUNCT
ejpam-3441	585	51	4	4	NUM
ejpam-3441	585	52	)	)	PUNCT
ejpam-3441	585	53	c∧βαd	c∧βαd	NUM
ejpam-3441	586	1	⊆	⊆	NUM
ejpam-3441	586	2	d	d	PUNCT
ejpam-3441	586	3	◦	◦	NOUN
ejpam-3441	586	4	βαc	βαc	NOUN
ejpam-3441	586	5	for	for	ADP
ejpam-3441	586	6	every	every	DET
ejpam-3441	586	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	586	8	fuzzy	fuzzy	ADJ
ejpam-3441	586	9	generalized	generalize	VERB
ejpam-3441	586	10	bi	bi	ADJ
ejpam-3441	586	11	-	-	ADJ
ejpam-3441	586	12	ideal	ideal	ADJ
ejpam-3441	586	13	c	c	NOUN
ejpam-3441	586	14	with	with	ADP
ejpam-3441	586	15	thresholds	threshold	NOUN
ejpam-3441	586	16	(	(	PUNCT
ejpam-3441	586	17	α	α	X
ejpam-3441	586	18	,	,	PUNCT
ejpam-3441	586	19	β	β	X
ejpam-3441	586	20	]	]	PUNCT
ejpam-3441	586	21	and	and	CCONJ
ejpam-3441	586	22	every	every	DET
ejpam-3441	586	23	intuitionistic	intuitionistic	ADJ
ejpam-3441	586	24	fuzzy	fuzzy	ADJ
ejpam-3441	586	25	right	right	ADJ
ejpam-3441	586	26	ideal	ideal	NOUN
ejpam-3441	586	27	d	d	NOUN
ejpam-3441	586	28	with	with	ADP
ejpam-3441	586	29	thresholds	threshold	NOUN
ejpam-3441	586	30	(	(	PUNCT
ejpam-3441	586	31	α	α	X
ejpam-3441	586	32	,	,	PUNCT
ejpam-3441	586	33	β	β	X
ejpam-3441	586	34	]	]	PUNCT
ejpam-3441	586	35	of	of	ADP
ejpam-3441	586	36	r.	r.	PROPN
ejpam-3441	586	37	proof	proof	NOUN
ejpam-3441	586	38	.	.	PUNCT
ejpam-3441	587	1	(	(	PUNCT
ejpam-3441	587	2	1)⇒	1)⇒	NUM
ejpam-3441	587	3	(	(	PUNCT
ejpam-3441	587	4	4	4	NUM
ejpam-3441	587	5	)	)	PUNCT
ejpam-3441	587	6	,	,	PUNCT
ejpam-3441	587	7	is	be	AUX
ejpam-3441	587	8	obvious	obvious	ADJ
ejpam-3441	587	9	.	.	PUNCT
ejpam-3441	588	1	it	it	PRON
ejpam-3441	588	2	is	be	AUX
ejpam-3441	588	3	clear	clear	ADJ
ejpam-3441	588	4	that	that	SCONJ
ejpam-3441	588	5	(	(	PUNCT
ejpam-3441	588	6	4)⇒	4)⇒	X
ejpam-3441	588	7	(	(	PUNCT
ejpam-3441	588	8	3	3	NUM
ejpam-3441	588	9	)	)	PUNCT
ejpam-3441	588	10	and	and	CCONJ
ejpam-3441	588	11	(	(	PUNCT
ejpam-3441	588	12	3)⇒	3)⇒	NUM
ejpam-3441	588	13	(	(	PUNCT
ejpam-3441	588	14	2	2	NUM
ejpam-3441	588	15	)	)	PUNCT
ejpam-3441	588	16	.	.	PUNCT
ejpam-3441	589	1	assume	assume	VERB
ejpam-3441	589	2	that	that	SCONJ
ejpam-3441	589	3	(	(	PUNCT
ejpam-3441	589	4	2	2	X
ejpam-3441	589	5	)	)	PUNCT
ejpam-3441	589	6	holds	hold	VERB
ejpam-3441	589	7	,	,	PUNCT
ejpam-3441	589	8	this	this	PRON
ejpam-3441	589	9	means	mean	VERB
ejpam-3441	589	10	that	that	SCONJ
ejpam-3441	589	11	d	d	X
ejpam-3441	589	12	∧βα	∧βα	ADJ
ejpam-3441	589	13	a	a	PRON
ejpam-3441	589	14	=	=	PUNCT
ejpam-3441	589	15	a∧βαd	a∧βαd	NOUN
ejpam-3441	589	16	⊆	⊆	NUM
ejpam-3441	589	17	d	d	PROPN
ejpam-3441	589	18	◦	◦	NOUN
ejpam-3441	589	19	βα	βα	X
ejpam-3441	589	20	a	a	PRON
ejpam-3441	589	21	,	,	PUNCT
ejpam-3441	589	22	where	where	SCONJ
ejpam-3441	589	23	a	a	PRON
ejpam-3441	589	24	is	be	AUX
ejpam-3441	589	25	an	an	DET
ejpam-3441	589	26	intuitionistic	intuitionistic	ADJ
ejpam-3441	589	27	fuzzy	fuzzy	ADJ
ejpam-3441	589	28	left	leave	VERB
ejpam-3441	589	29	ideal	ideal	NOUN
ejpam-3441	589	30	with	with	ADP
ejpam-3441	589	31	thresholds	threshold	NOUN
ejpam-3441	589	32	(	(	PUNCT
ejpam-3441	589	33	α	α	X
ejpam-3441	589	34	,	,	PUNCT
ejpam-3441	589	35	β	β	X
ejpam-3441	589	36	]	]	PUNCT
ejpam-3441	589	37	of	of	ADP
ejpam-3441	589	38	r.	r.	PROPN
ejpam-3441	589	39	since	since	SCONJ
ejpam-3441	589	40	d	d	PROPN
ejpam-3441	589	41	◦	◦	VERB
ejpam-3441	589	42	βα	βα	X
ejpam-3441	589	43	a	a	DET
ejpam-3441	589	44	⊆	⊆	NUM
ejpam-3441	589	45	d	d	X
ejpam-3441	589	46	∧βα	∧βα	PROPN
ejpam-3441	589	47	a	a	PRON
ejpam-3441	589	48	,	,	PUNCT
ejpam-3441	590	1	so	so	ADV
ejpam-3441	590	2	d	d	ADP
ejpam-3441	590	3	∧βα	∧βα	ADJ
ejpam-3441	590	4	a	a	DET
ejpam-3441	590	5	=	=	SYM
ejpam-3441	590	6	d	d	NOUN
ejpam-3441	590	7	◦	◦	NOUN
ejpam-3441	590	8	βα	βα	NOUN
ejpam-3441	590	9	a.	a.	NOUN
ejpam-3441	590	10	therefore	therefore	ADV
ejpam-3441	590	11	r	r	NOUN
ejpam-3441	590	12	is	be	AUX
ejpam-3441	590	13	a	a	DET
ejpam-3441	590	14	regular	regular	NOUN
ejpam-3441	590	15	by	by	ADP
ejpam-3441	590	16	the	the	DET
ejpam-3441	590	17	theorem	theorem	NOUN
ejpam-3441	590	18	8	8	NUM
ejpam-3441	590	19	,	,	PUNCT
ejpam-3441	590	20	i.e.	i.e.	X
ejpam-3441	590	21	,	,	PUNCT
ejpam-3441	590	22	(	(	PUNCT
ejpam-3441	590	23	2)⇒	2)⇒	NUM
ejpam-3441	590	24	(	(	PUNCT
ejpam-3441	590	25	1	1	NUM
ejpam-3441	590	26	)	)	PUNCT
ejpam-3441	590	27	.	.	PUNCT
ejpam-3441	591	1	theorem	theorem	NOUN
ejpam-3441	591	2	12	12	NUM
ejpam-3441	591	3	.	.	PUNCT
ejpam-3441	592	1	let	let	VERB
ejpam-3441	592	2	r	r	PRON
ejpam-3441	592	3	be	be	AUX
ejpam-3441	592	4	an	an	DET
ejpam-3441	592	5	la	la	NOUN
ejpam-3441	592	6	-	-	NOUN
ejpam-3441	592	7	ring	ring	NOUN
ejpam-3441	592	8	with	with	ADP
ejpam-3441	592	9	left	left	ADJ
ejpam-3441	592	10	identity	identity	NOUN
ejpam-3441	592	11	e	e	NOUN
ejpam-3441	592	12	,	,	PUNCT
ejpam-3441	592	13	such	such	ADJ
ejpam-3441	592	14	that	that	SCONJ
ejpam-3441	592	15	(	(	PUNCT
ejpam-3441	592	16	xe)r	xe)r	PROPN
ejpam-3441	592	17	=	=	SYM
ejpam-3441	592	18	xr	xr	PROPN
ejpam-3441	592	19	for	for	ADP
ejpam-3441	592	20	all	all	DET
ejpam-3441	592	21	x	x	PROPN
ejpam-3441	592	22	∈	∈	PROPN
ejpam-3441	592	23	r.	r.	NOUN
ejpam-3441	592	24	then	then	ADV
ejpam-3441	592	25	the	the	DET
ejpam-3441	592	26	following	follow	VERB
ejpam-3441	592	27	conditions	condition	NOUN
ejpam-3441	592	28	are	be	AUX
ejpam-3441	592	29	equivalent	equivalent	ADJ
ejpam-3441	592	30	.	.	PUNCT
ejpam-3441	593	1	(	(	PUNCT
ejpam-3441	593	2	1	1	X
ejpam-3441	593	3	)	)	PUNCT
ejpam-3441	593	4	r	r	NOUN
ejpam-3441	593	5	is	be	AUX
ejpam-3441	593	6	a	a	DET
ejpam-3441	593	7	regular	regular	NOUN
ejpam-3441	593	8	.	.	PUNCT
ejpam-3441	594	1	(	(	PUNCT
ejpam-3441	594	2	2	2	X
ejpam-3441	594	3	)	)	PUNCT
ejpam-3441	594	4	a	a	DET
ejpam-3441	594	5	∧βα	∧βα	ADJ
ejpam-3441	594	6	d	d	NOUN
ejpam-3441	594	7	∧βα	∧βα	ADJ
ejpam-3441	594	8	l	l	NOUN
ejpam-3441	594	9	⊆	⊆	NUM
ejpam-3441	594	10	(	(	PUNCT
ejpam-3441	594	11	a	a	DET
ejpam-3441	594	12	◦	◦	NOUN
ejpam-3441	594	13	βα	βα	NOUN
ejpam-3441	594	14	d	d	NOUN
ejpam-3441	594	15	)	)	PUNCT
ejpam-3441	594	16	◦	◦	NOUN
ejpam-3441	594	17	βα	βα	NOUN
ejpam-3441	594	18	l	l	NOUN
ejpam-3441	594	19	for	for	ADP
ejpam-3441	594	20	every	every	DET
ejpam-3441	594	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	594	22	fuzzy	fuzzy	ADJ
ejpam-3441	594	23	quasi	quasi	NOUN
ejpam-3441	594	24	-	-	NOUN
ejpam-3441	594	25	ideal	ideal	ADJ
ejpam-3441	594	26	a	a	PRON
ejpam-3441	594	27	with	with	ADP
ejpam-3441	594	28	thresholds	threshold	NOUN
ejpam-3441	594	29	(	(	PUNCT
ejpam-3441	594	30	α	α	X
ejpam-3441	594	31	,	,	PUNCT
ejpam-3441	594	32	β	β	X
ejpam-3441	594	33	]	]	X
ejpam-3441	594	34	,	,	PUNCT
ejpam-3441	594	35	every	every	DET
ejpam-3441	594	36	intuitionistic	intuitionistic	ADJ
ejpam-3441	594	37	fuzzy	fuzzy	ADJ
ejpam-3441	594	38	right	right	ADJ
ejpam-3441	594	39	ideal	ideal	NOUN
ejpam-3441	594	40	d	d	NOUN
ejpam-3441	594	41	with	with	ADP
ejpam-3441	594	42	thresholds	threshold	NOUN
ejpam-3441	594	43	(	(	PUNCT
ejpam-3441	594	44	α	α	X
ejpam-3441	594	45	,	,	PUNCT
ejpam-3441	594	46	β	β	X
ejpam-3441	594	47	]	]	PUNCT
ejpam-3441	594	48	and	and	CCONJ
ejpam-3441	594	49	every	every	DET
ejpam-3441	594	50	intuitionistic	intuitionistic	ADJ
ejpam-3441	594	51	fuzzy	fuzzy	ADJ
ejpam-3441	594	52	left	leave	VERB
ejpam-3441	594	53	ideal	ideal	NOUN
ejpam-3441	594	54	l	l	PROPN
ejpam-3441	594	55	with	with	ADP
ejpam-3441	594	56	thresholds	threshold	NOUN
ejpam-3441	594	57	(	(	PUNCT
ejpam-3441	594	58	α	α	X
ejpam-3441	594	59	,	,	PUNCT
ejpam-3441	594	60	β	β	X
ejpam-3441	594	61	]	]	PUNCT
ejpam-3441	594	62	of	of	ADP
ejpam-3441	594	63	r.	r.	PROPN
ejpam-3441	594	64	(	(	PUNCT
ejpam-3441	594	65	3	3	NUM
ejpam-3441	594	66	)	)	PUNCT
ejpam-3441	594	67	b∧βαd∧βαl	b∧βαd∧βαl	NOUN
ejpam-3441	595	1	⊆	⊆	NUM
ejpam-3441	595	2	(	(	PUNCT
ejpam-3441	595	3	b	b	NOUN
ejpam-3441	595	4	◦	◦	NOUN
ejpam-3441	595	5	βαd)	βαd)	NOUN
ejpam-3441	595	6	◦	◦	NOUN
ejpam-3441	595	7	βαl	βαl	NOUN
ejpam-3441	595	8	for	for	ADP
ejpam-3441	595	9	every	every	DET
ejpam-3441	595	10	intuitionistic	intuitionistic	ADJ
ejpam-3441	595	11	fuzzy	fuzzy	ADJ
ejpam-3441	595	12	bi	bi	ADJ
ejpam-3441	595	13	-	-	ADJ
ejpam-3441	595	14	ideal	ideal	ADJ
ejpam-3441	595	15	b	b	PROPN
ejpam-3441	595	16	with	with	ADP
ejpam-3441	595	17	thresholds	threshold	NOUN
ejpam-3441	595	18	(	(	PUNCT
ejpam-3441	595	19	α	α	X
ejpam-3441	595	20	,	,	PUNCT
ejpam-3441	595	21	β	β	X
ejpam-3441	595	22	]	]	X
ejpam-3441	595	23	,	,	PUNCT
ejpam-3441	595	24	every	every	DET
ejpam-3441	595	25	intuitionistic	intuitionistic	ADJ
ejpam-3441	595	26	fuzzy	fuzzy	ADJ
ejpam-3441	595	27	right	right	ADJ
ejpam-3441	595	28	ideal	ideal	NOUN
ejpam-3441	595	29	d	d	NOUN
ejpam-3441	595	30	with	with	ADP
ejpam-3441	595	31	thresholds	threshold	NOUN
ejpam-3441	595	32	(	(	PUNCT
ejpam-3441	595	33	α	α	X
ejpam-3441	595	34	,	,	PUNCT
ejpam-3441	595	35	β	β	X
ejpam-3441	595	36	]	]	PUNCT
ejpam-3441	595	37	and	and	CCONJ
ejpam-3441	595	38	every	every	DET
ejpam-3441	595	39	intuitionistic	intuitionistic	ADJ
ejpam-3441	595	40	fuzzy	fuzzy	ADJ
ejpam-3441	595	41	left	leave	VERB
ejpam-3441	595	42	ideal	ideal	NOUN
ejpam-3441	595	43	l	l	PROPN
ejpam-3441	595	44	with	with	ADP
ejpam-3441	595	45	thresholds	threshold	NOUN
ejpam-3441	595	46	(	(	PUNCT
ejpam-3441	595	47	α	α	X
ejpam-3441	595	48	,	,	PUNCT
ejpam-3441	595	49	β	β	X
ejpam-3441	595	50	]	]	PUNCT
ejpam-3441	595	51	of	of	ADP
ejpam-3441	595	52	r.	r.	PROPN
ejpam-3441	595	53	(	(	PUNCT
ejpam-3441	595	54	4	4	NUM
ejpam-3441	595	55	)	)	PUNCT
ejpam-3441	595	56	c	c	NOUN
ejpam-3441	596	1	∧βα	∧βα	NOUN
ejpam-3441	596	2	d	d	NOUN
ejpam-3441	596	3	∧βα	∧βα	ADJ
ejpam-3441	596	4	l	l	NOUN
ejpam-3441	596	5	⊆	⊆	NUM
ejpam-3441	596	6	(	(	PUNCT
ejpam-3441	596	7	c	c	NOUN
ejpam-3441	596	8	◦	◦	NOUN
ejpam-3441	596	9	βα	βα	X
ejpam-3441	596	10	d	d	NOUN
ejpam-3441	596	11	)	)	PUNCT
ejpam-3441	596	12	◦	◦	NOUN
ejpam-3441	596	13	βα	βα	NOUN
ejpam-3441	596	14	l	l	NOUN
ejpam-3441	596	15	for	for	ADP
ejpam-3441	596	16	every	every	DET
ejpam-3441	596	17	intuitionistic	intuitionistic	ADJ
ejpam-3441	596	18	fuzzy	fuzzy	ADJ
ejpam-3441	596	19	generalized	generalize	VERB
ejpam-3441	596	20	bi	bi	ADJ
ejpam-3441	596	21	-	-	ADJ
ejpam-3441	596	22	ideal	ideal	ADJ
ejpam-3441	596	23	c	c	NOUN
ejpam-3441	596	24	with	with	ADP
ejpam-3441	596	25	thresholds	threshold	NOUN
ejpam-3441	596	26	(	(	PUNCT
ejpam-3441	596	27	α	α	X
ejpam-3441	596	28	,	,	PUNCT
ejpam-3441	596	29	β	β	X
ejpam-3441	596	30	]	]	X
ejpam-3441	596	31	,	,	PUNCT
ejpam-3441	596	32	every	every	DET
ejpam-3441	596	33	intuitionistic	intuitionistic	ADJ
ejpam-3441	596	34	fuzzy	fuzzy	ADJ
ejpam-3441	596	35	right	right	ADJ
ejpam-3441	596	36	ideal	ideal	NOUN
ejpam-3441	596	37	d	d	NOUN
ejpam-3441	596	38	with	with	ADP
ejpam-3441	596	39	thresholds	threshold	NOUN
ejpam-3441	596	40	(	(	PUNCT
ejpam-3441	596	41	α	α	X
ejpam-3441	596	42	,	,	PUNCT
ejpam-3441	596	43	β	β	X
ejpam-3441	596	44	]	]	PUNCT
ejpam-3441	596	45	and	and	CCONJ
ejpam-3441	596	46	every	every	DET
ejpam-3441	596	47	intuitionistic	intuitionistic	ADJ
ejpam-3441	596	48	fuzzy	fuzzy	ADJ
ejpam-3441	596	49	left	leave	VERB
ejpam-3441	596	50	ideal	ideal	NOUN
ejpam-3441	596	51	l	l	PROPN
ejpam-3441	596	52	with	with	ADP
ejpam-3441	596	53	thresholds	threshold	NOUN
ejpam-3441	596	54	(	(	PUNCT
ejpam-3441	596	55	α	α	X
ejpam-3441	596	56	,	,	PUNCT
ejpam-3441	596	57	β	β	X
ejpam-3441	596	58	]	]	PUNCT
ejpam-3441	596	59	of	of	ADP
ejpam-3441	596	60	r.	r.	PROPN
ejpam-3441	596	61	k.	k.	PROPN
ejpam-3441	596	62	nasreen	nasreen	PROPN
ejpam-3441	596	63	et	et	PROPN
ejpam-3441	596	64	al	al	PROPN
ejpam-3441	596	65	.	.	PUNCT
ejpam-3441	596	66	/	/	SYM
ejpam-3441	596	67	eur	eur	PROPN
ejpam-3441	596	68	.	.	PUNCT
ejpam-3441	597	1	j.	j.	PROPN
ejpam-3441	597	2	pure	pure	PROPN
ejpam-3441	597	3	appl	appl	PROPN
ejpam-3441	597	4	.	.	PROPN
ejpam-3441	597	5	math	math	PROPN
ejpam-3441	597	6	,	,	PUNCT
ejpam-3441	597	7	12	12	NUM
ejpam-3441	597	8	(	(	PUNCT
ejpam-3441	597	9	3	3	NUM
ejpam-3441	597	10	)	)	PUNCT
ejpam-3441	597	11	(	(	PUNCT
ejpam-3441	597	12	2019	2019	NUM
ejpam-3441	597	13	)	)	PUNCT
ejpam-3441	597	14	,	,	PUNCT
ejpam-3441	597	15	906	906	NUM
ejpam-3441	597	16	-	-	SYM
ejpam-3441	597	17	943	943	NUM
ejpam-3441	597	18	932	932	NUM
ejpam-3441	597	19	proof	proof	NOUN
ejpam-3441	597	20	.	.	PUNCT
ejpam-3441	598	1	consider	consider	VERB
ejpam-3441	598	2	that	that	PRON
ejpam-3441	598	3	(	(	PUNCT
ejpam-3441	598	4	1	1	X
ejpam-3441	598	5	)	)	PUNCT
ejpam-3441	598	6	holds	hold	VERB
ejpam-3441	598	7	.	.	PUNCT
ejpam-3441	599	1	let	let	AUX
ejpam-3441	599	2	c	c	NOUN
ejpam-3441	599	3	=	=	PUNCT
ejpam-3441	599	4	(	(	PUNCT
ejpam-3441	599	5	µc	µc	INTJ
ejpam-3441	599	6	,	,	PUNCT
ejpam-3441	599	7	γc	γc	PROPN
ejpam-3441	599	8	)	)	PUNCT
ejpam-3441	599	9	be	be	AUX
ejpam-3441	599	10	an	an	DET
ejpam-3441	599	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	599	12	fuzzy	fuzzy	ADJ
ejpam-3441	599	13	generalized	generalize	VERB
ejpam-3441	599	14	bi	bi	NOUN
ejpam-3441	599	15	-	-	NOUN
ejpam-3441	599	16	ideal	ideal	NOUN
ejpam-3441	599	17	with	with	ADP
ejpam-3441	599	18	thresholds	threshold	NOUN
ejpam-3441	599	19	(	(	PUNCT
ejpam-3441	599	20	α	α	X
ejpam-3441	599	21	,	,	PUNCT
ejpam-3441	599	22	β	β	X
ejpam-3441	599	23	]	]	X
ejpam-3441	599	24	,	,	PUNCT
ejpam-3441	599	25	l	l	NOUN
ejpam-3441	599	26	=	=	SYM
ejpam-3441	599	27	(	(	PUNCT
ejpam-3441	599	28	µl	µl	NOUN
ejpam-3441	599	29	,	,	PUNCT
ejpam-3441	599	30	γl	γl	NUM
ejpam-3441	599	31	)	)	PUNCT
ejpam-3441	599	32	be	be	VERB
ejpam-3441	599	33	an	an	DET
ejpam-3441	599	34	intuitionistic	intuitionistic	ADJ
ejpam-3441	599	35	fuzzy	fuzzy	ADJ
ejpam-3441	599	36	left	leave	VERB
ejpam-3441	599	37	ideal	ideal	NOUN
ejpam-3441	599	38	with	with	ADP
ejpam-3441	599	39	thresholds	threshold	NOUN
ejpam-3441	599	40	(	(	PUNCT
ejpam-3441	599	41	α	α	X
ejpam-3441	599	42	,	,	PUNCT
ejpam-3441	599	43	β	β	X
ejpam-3441	599	44	]	]	PUNCT
ejpam-3441	599	45	and	and	CCONJ
ejpam-3441	599	46	d	d	NOUN
ejpam-3441	599	47	=	=	PUNCT
ejpam-3441	599	48	(	(	PUNCT
ejpam-3441	599	49	µd	µd	ADP
ejpam-3441	599	50	,	,	PUNCT
ejpam-3441	599	51	γd	γd	ADV
ejpam-3441	599	52	)	)	PUNCT
ejpam-3441	599	53	be	be	AUX
ejpam-3441	599	54	an	an	DET
ejpam-3441	599	55	intuitionistic	intuitionistic	ADJ
ejpam-3441	599	56	fuzzy	fuzzy	ADJ
ejpam-3441	599	57	right	right	ADJ
ejpam-3441	599	58	ideal	ideal	NOUN
ejpam-3441	599	59	with	with	ADP
ejpam-3441	599	60	thresholds	threshold	NOUN
ejpam-3441	599	61	(	(	PUNCT
ejpam-3441	599	62	α	α	X
ejpam-3441	599	63	,	,	PUNCT
ejpam-3441	599	64	β	β	X
ejpam-3441	599	65	]	]	PUNCT
ejpam-3441	599	66	of	of	ADP
ejpam-3441	599	67	r.	r.	PROPN
ejpam-3441	599	68	let	let	VERB
ejpam-3441	599	69	x	x	X
ejpam-3441	599	70	∈	∈	PROPN
ejpam-3441	599	71	r	r	NOUN
ejpam-3441	599	72	,	,	PUNCT
ejpam-3441	599	73	then	then	ADV
ejpam-3441	599	74	there	there	PRON
ejpam-3441	599	75	exists	exist	VERB
ejpam-3441	599	76	an	an	DET
ejpam-3441	599	77	element	element	NOUN
ejpam-3441	599	78	a	a	DET
ejpam-3441	599	79	∈	∈	NOUN
ejpam-3441	599	80	r	r	NOUN
ejpam-3441	599	81	such	such	ADJ
ejpam-3441	599	82	that	that	PRON
ejpam-3441	599	83	x	x	SYM
ejpam-3441	599	84	=	=	SYM
ejpam-3441	599	85	(	(	PUNCT
ejpam-3441	599	86	xa)x	xa)x	PROPN
ejpam-3441	599	87	.	.	PUNCT
ejpam-3441	600	1	now	now	ADV
ejpam-3441	600	2	x	x	X
ejpam-3441	600	3	=	=	SYM
ejpam-3441	600	4	(	(	PUNCT
ejpam-3441	600	5	xa)x	xa)x	PROPN
ejpam-3441	600	6	.	.	PUNCT
ejpam-3441	601	1	xa	xa	PROPN
ejpam-3441	601	2	=	=	PRON
ejpam-3441	601	3	(	(	PUNCT
ejpam-3441	601	4	(	(	PUNCT
ejpam-3441	601	5	xa)x)a	xa)x)a	PROPN
ejpam-3441	601	6	=	=	SYM
ejpam-3441	601	7	(	(	PUNCT
ejpam-3441	601	8	ax)(xa	ax)(xa	PROPN
ejpam-3441	601	9	)	)	PUNCT
ejpam-3441	601	10	=	=	PUNCT
ejpam-3441	601	11	x((ax)a	x((ax)a	PROPN
ejpam-3441	601	12	)	)	PUNCT
ejpam-3441	601	13	.	.	PUNCT
ejpam-3441	602	1	(	(	PUNCT
ejpam-3441	602	2	ax)a	ax)a	PROPN
ejpam-3441	602	3	=	=	SYM
ejpam-3441	602	4	(	(	PUNCT
ejpam-3441	602	5	a((xa)x))a	a((xa)x))a	PROPN
ejpam-3441	602	6	=	=	SYM
ejpam-3441	602	7	(	(	PUNCT
ejpam-3441	602	8	(	(	PUNCT
ejpam-3441	602	9	xa)(ax))a	xa)(ax))a	X
ejpam-3441	602	10	=	=	SYM
ejpam-3441	602	11	(	(	PUNCT
ejpam-3441	602	12	a(ax))(xa	a(ax))(xa	PROPN
ejpam-3441	602	13	)	)	PUNCT
ejpam-3441	602	14	=	=	SYM
ejpam-3441	602	15	x((a(ax))a	x((a(ax))a	PROPN
ejpam-3441	602	16	)	)	PUNCT
ejpam-3441	602	17	=	=	SYM
ejpam-3441	602	18	x(((ea)(ax))a	x(((ea)(ax))a	PROPN
ejpam-3441	602	19	)	)	PUNCT
ejpam-3441	602	20	=	=	SYM
ejpam-3441	602	21	x(((xa)(ae))a	x(((xa)(ae))a	PROPN
ejpam-3441	602	22	)	)	PUNCT
ejpam-3441	602	23	=	=	PUNCT
ejpam-3441	603	1	x((((ae)a)x)a	x((((ae)a)x)a	PROPN
ejpam-3441	603	2	)	)	PUNCT
ejpam-3441	603	3	=	=	SYM
ejpam-3441	603	4	x((nx)a	x((nx)a	NUM
ejpam-3441	603	5	)	)	PUNCT
ejpam-3441	603	6	=	=	PUNCT
ejpam-3441	603	7	x((nx)(ea	x((nx)(ea	PROPN
ejpam-3441	603	8	)	)	PUNCT
ejpam-3441	603	9	)	)	PUNCT
ejpam-3441	604	1	=	=	PUNCT
ejpam-3441	604	2	x((ae)(xn	x((ae)(xn	X
ejpam-3441	604	3	)	)	PUNCT
ejpam-3441	604	4	)	)	PUNCT
ejpam-3441	605	1	=	=	PUNCT
ejpam-3441	605	2	x(x((ae)n	x(x((ae)n	PROPN
ejpam-3441	605	3	)	)	PUNCT
ejpam-3441	605	4	)	)	PUNCT
ejpam-3441	606	1	=	=	SYM
ejpam-3441	606	2	x(xm	x(xm	PROPN
ejpam-3441	606	3	)	)	PUNCT
ejpam-3441	606	4	.	.	PUNCT
ejpam-3441	607	1	⇒	⇒	PROPN
ejpam-3441	607	2	xa	xa	PROPN
ejpam-3441	608	1	=	=	PUNCT
ejpam-3441	608	2	x((ax)a	x((ax)a	PROPN
ejpam-3441	608	3	)	)	PUNCT
ejpam-3441	609	1	=	=	SYM
ejpam-3441	609	2	x(x(xm	x(x(xm	X
ejpam-3441	609	3	)	)	PUNCT
ejpam-3441	609	4	)	)	PUNCT
ejpam-3441	610	1	=	=	SYM
ejpam-3441	610	2	(	(	PUNCT
ejpam-3441	610	3	ex)(x(xm	ex)(x(xm	PROPN
ejpam-3441	610	4	)	)	PUNCT
ejpam-3441	610	5	)	)	PUNCT
ejpam-3441	611	1	=	=	PUNCT
ejpam-3441	611	2	(	(	PUNCT
ejpam-3441	611	3	(	(	PUNCT
ejpam-3441	611	4	xm)x)(xe	xm)x)(xe	PROPN
ejpam-3441	611	5	)	)	PUNCT
ejpam-3441	611	6	.	.	PUNCT
ejpam-3441	612	1	thus	thus	ADV
ejpam-3441	612	2	(	(	PUNCT
ejpam-3441	612	3	(	(	PUNCT
ejpam-3441	612	4	µc	µc	INTJ
ejpam-3441	612	5	◦	◦	NOUN
ejpam-3441	612	6	βα	βα	NOUN
ejpam-3441	612	7	µd	µd	NOUN
ejpam-3441	612	8	)	)	PUNCT
ejpam-3441	612	9	◦	◦	NOUN
ejpam-3441	612	10	βα	βα	NOUN
ejpam-3441	612	11	µl)(x	µl)(x	PROPN
ejpam-3441	612	12	)	)	PUNCT
ejpam-3441	613	1	=	=	PRON
ejpam-3441	613	2	{	{	PUNCT
ejpam-3441	613	3	(	(	PUNCT
ejpam-3441	613	4	(	(	PUNCT
ejpam-3441	613	5	µc	µc	INTJ
ejpam-3441	613	6	◦	◦	VERB
ejpam-3441	613	7	µd	µd	PRON
ejpam-3441	613	8	)	)	PUNCT
ejpam-3441	613	9	◦	◦	NOUN
ejpam-3441	613	10	µl)(x	µl)(x	PROPN
ejpam-3441	613	11	)	)	PUNCT
ejpam-3441	614	1	∧	∧	PROPN
ejpam-3441	614	2	β	β	NOUN
ejpam-3441	614	3	}	}	PUNCT
ejpam-3441	614	4	∨	∨	NUM
ejpam-3441	614	5	α	α	NOUN
ejpam-3441	614	6	=	=	X
ejpam-3441	614	7	{	{	PUNCT
ejpam-3441	614	8	(	(	PUNCT
ejpam-3441	614	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	614	10	i=1	i=1	PROPN
ejpam-3441	614	11	piqi	piqi	NOUN
ejpam-3441	614	12	{	{	PUNCT
ejpam-3441	614	13	∧ni=1	∧ni=1	X
ejpam-3441	614	14	{	{	PUNCT
ejpam-3441	614	15	(	(	PUNCT
ejpam-3441	614	16	µc	µc	INTJ
ejpam-3441	614	17	◦	◦	NOUN
ejpam-3441	614	18	µd	µd	ADP
ejpam-3441	614	19	)	)	PUNCT
ejpam-3441	614	20	(	(	PUNCT
ejpam-3441	614	21	pi	pi	NOUN
ejpam-3441	614	22	)	)	PUNCT
ejpam-3441	614	23	∧	∧	NOUN
ejpam-3441	614	24	µl	µl	ADP
ejpam-3441	614	25	(	(	PUNCT
ejpam-3441	614	26	qi	qi	NOUN
ejpam-3441	614	27	)	)	PUNCT
ejpam-3441	614	28	}	}	PUNCT
ejpam-3441	614	29	}	}	PUNCT
ejpam-3441	614	30	)	)	PUNCT
ejpam-3441	614	31	∧	∧	PROPN
ejpam-3441	614	32	β	β	NOUN
ejpam-3441	614	33	}	}	PUNCT
ejpam-3441	614	34	∨	∨	NUM
ejpam-3441	614	35	α	α	PROPN
ejpam-3441	614	36	≥	≥	X
ejpam-3441	614	37	{	{	PUNCT
ejpam-3441	614	38	{	{	PUNCT
ejpam-3441	614	39	(	(	PUNCT
ejpam-3441	614	40	µc	µc	INTJ
ejpam-3441	614	41	◦	◦	NOUN
ejpam-3441	614	42	µd	µd	ADP
ejpam-3441	614	43	)	)	PUNCT
ejpam-3441	614	44	(	(	PUNCT
ejpam-3441	614	45	xa	xa	NOUN
ejpam-3441	614	46	)	)	PUNCT
ejpam-3441	614	47	∧	∧	NOUN
ejpam-3441	614	48	µl	µl	ADP
ejpam-3441	614	49	(	(	PUNCT
ejpam-3441	614	50	x	x	NOUN
ejpam-3441	614	51	)	)	PUNCT
ejpam-3441	614	52	}	}	PUNCT
ejpam-3441	614	53	∧	∧	PROPN
ejpam-3441	614	54	β	β	NOUN
ejpam-3441	614	55	}	}	PUNCT
ejpam-3441	614	56	∨	∨	NUM
ejpam-3441	614	57	α	α	NOUN
ejpam-3441	614	58	=	=	SYM
ejpam-3441	614	59	(	(	PUNCT
ejpam-3441	614	60	(	(	PUNCT
ejpam-3441	614	61	µc	µc	INTJ
ejpam-3441	614	62	◦	◦	NOUN
ejpam-3441	614	63	µd	µd	ADP
ejpam-3441	614	64	)	)	PUNCT
ejpam-3441	614	65	(	(	PUNCT
ejpam-3441	614	66	xa	xa	PROPN
ejpam-3441	614	67	)	)	PUNCT
ejpam-3441	614	68	∨	∨	NUM
ejpam-3441	614	69	α	α	NOUN
ejpam-3441	614	70	)	)	PUNCT
ejpam-3441	614	71	∧	∧	NOUN
ejpam-3441	614	72	(	(	PUNCT
ejpam-3441	614	73	µl	µl	PART
ejpam-3441	614	74	(	(	PUNCT
ejpam-3441	614	75	x	x	NOUN
ejpam-3441	614	76	)	)	PUNCT
ejpam-3441	614	77	∨	∨	NUM
ejpam-3441	614	78	α	α	NOUN
ejpam-3441	614	79	)	)	PUNCT
ejpam-3441	614	80	∧	∧	PROPN
ejpam-3441	614	81	(	(	PUNCT
ejpam-3441	614	82	β	β	X
ejpam-3441	614	83	∨	∨	NUM
ejpam-3441	614	84	α	α	NOUN
ejpam-3441	614	85	)	)	PUNCT
ejpam-3441	614	86	=	=	SYM
ejpam-3441	614	87	(	(	PUNCT
ejpam-3441	614	88	(	(	PUNCT
ejpam-3441	614	89	µc	µc	INTJ
ejpam-3441	614	90	◦	◦	NOUN
ejpam-3441	614	91	µd	µd	ADP
ejpam-3441	614	92	)	)	PUNCT
ejpam-3441	614	93	(	(	PUNCT
ejpam-3441	614	94	xa	xa	PROPN
ejpam-3441	614	95	)	)	PUNCT
ejpam-3441	614	96	∨	∨	NUM
ejpam-3441	614	97	α	α	NOUN
ejpam-3441	614	98	)	)	PUNCT
ejpam-3441	614	99	∧	∧	NOUN
ejpam-3441	614	100	µl(x	µl(x	NOUN
ejpam-3441	614	101	)	)	PUNCT
ejpam-3441	614	102	∧	∧	PROPN
ejpam-3441	614	103	β	β	X
ejpam-3441	614	104	=	=	SYM
ejpam-3441	614	105	(	(	PUNCT
ejpam-3441	614	106	(	(	PUNCT
ejpam-3441	614	107	∨xa=∑n	∨xa=∑n	PROPN
ejpam-3441	614	108	i=1mini	i=1mini	PROPN
ejpam-3441	614	109	{	{	PUNCT
ejpam-3441	614	110	∧ni=1	∧ni=1	X
ejpam-3441	614	111	{	{	PUNCT
ejpam-3441	614	112	µc	µc	PROPN
ejpam-3441	614	113	(	(	PUNCT
ejpam-3441	614	114	mi	mi	NOUN
ejpam-3441	614	115	)	)	PUNCT
ejpam-3441	614	116	∧	∧	NOUN
ejpam-3441	614	117	µd	µd	ADP
ejpam-3441	614	118	(	(	PUNCT
ejpam-3441	614	119	ni	ni	NOUN
ejpam-3441	614	120	)	)	PUNCT
ejpam-3441	614	121	}	}	PUNCT
ejpam-3441	614	122	}	}	PUNCT
ejpam-3441	614	123	)	)	PUNCT
ejpam-3441	614	124	∨	∨	NUM
ejpam-3441	614	125	α	α	NOUN
ejpam-3441	614	126	)	)	PUNCT
ejpam-3441	614	127	∧	∧	NOUN
ejpam-3441	614	128	µl(x	µl(x	NOUN
ejpam-3441	614	129	)	)	PUNCT
ejpam-3441	614	130	∧	∧	PROPN
ejpam-3441	614	131	β	β	X
ejpam-3441	614	132	≥	≥	X
ejpam-3441	614	133	(	(	PUNCT
ejpam-3441	614	134	{	{	PUNCT
ejpam-3441	614	135	µc((xm)x	µc((xm)x	ADJ
ejpam-3441	614	136	)	)	PUNCT
ejpam-3441	614	137	∧	∧	PROPN
ejpam-3441	614	138	µd(xe	µd(xe	NOUN
ejpam-3441	614	139	)	)	PUNCT
ejpam-3441	614	140	}	}	PUNCT
ejpam-3441	614	141	∨	∨	NUM
ejpam-3441	614	142	α	α	NOUN
ejpam-3441	614	143	)	)	PUNCT
ejpam-3441	614	144	∧	∧	NOUN
ejpam-3441	614	145	µl(x	µl(x	NOUN
ejpam-3441	614	146	)	)	PUNCT
ejpam-3441	614	147	∧	∧	PROPN
ejpam-3441	614	148	β	β	X
ejpam-3441	614	149	=	=	SYM
ejpam-3441	614	150	(	(	PUNCT
ejpam-3441	614	151	µc((xm)x	µc((xm)x	PROPN
ejpam-3441	614	152	)	)	PUNCT
ejpam-3441	614	153	∨	∨	NUM
ejpam-3441	614	154	α	α	NOUN
ejpam-3441	614	155	)	)	PUNCT
ejpam-3441	614	156	∧	∧	PROPN
ejpam-3441	614	157	(	(	PUNCT
ejpam-3441	614	158	µd(xe	µd(xe	PROPN
ejpam-3441	614	159	)	)	PUNCT
ejpam-3441	614	160	∨	∨	NUM
ejpam-3441	614	161	α	α	NOUN
ejpam-3441	614	162	)	)	PUNCT
ejpam-3441	614	163	∧	∧	NOUN
ejpam-3441	614	164	µl(x	µl(x	NOUN
ejpam-3441	614	165	)	)	PUNCT
ejpam-3441	614	166	∧	∧	PROPN
ejpam-3441	614	167	β	β	X
ejpam-3441	614	168	≥	≥	X
ejpam-3441	614	169	(	(	PUNCT
ejpam-3441	614	170	µc(x	µc(x	NOUN
ejpam-3441	614	171	)	)	PUNCT
ejpam-3441	614	172	∧	∧	PROPN
ejpam-3441	614	173	µc(x	µc(x	NOUN
ejpam-3441	614	174	)	)	PUNCT
ejpam-3441	614	175	∧	∧	PROPN
ejpam-3441	614	176	β	β	NOUN
ejpam-3441	614	177	)	)	PUNCT
ejpam-3441	614	178	∧	∧	NOUN
ejpam-3441	614	179	(	(	PUNCT
ejpam-3441	614	180	µd(x	µd(x	ADJ
ejpam-3441	614	181	)	)	PUNCT
ejpam-3441	614	182	∧	∧	PROPN
ejpam-3441	614	183	β	β	NOUN
ejpam-3441	614	184	)	)	PUNCT
ejpam-3441	614	185	∧	∧	NOUN
ejpam-3441	614	186	µl(x	µl(x	NOUN
ejpam-3441	614	187	)	)	PUNCT
ejpam-3441	614	188	∧	∧	PROPN
ejpam-3441	614	189	β	β	X
ejpam-3441	614	190	=	=	SYM
ejpam-3441	614	191	µc(x	µc(x	NOUN
ejpam-3441	614	192	)	)	PUNCT
ejpam-3441	614	193	∧	∧	NOUN
ejpam-3441	614	194	µd(x	µd(x	NOUN
ejpam-3441	614	195	)	)	PUNCT
ejpam-3441	614	196	∧	∧	NOUN
ejpam-3441	614	197	µl(x	µl(x	NOUN
ejpam-3441	614	198	)	)	PUNCT
ejpam-3441	614	199	∧	∧	PROPN
ejpam-3441	614	200	β	β	X
ejpam-3441	614	201	=	=	SYM
ejpam-3441	614	202	(	(	PUNCT
ejpam-3441	614	203	µc(x	µc(x	NOUN
ejpam-3441	614	204	)	)	PUNCT
ejpam-3441	614	205	∧	∧	NOUN
ejpam-3441	614	206	µd(x	µd(x	NOUN
ejpam-3441	614	207	)	)	PUNCT
ejpam-3441	614	208	∧	∧	NOUN
ejpam-3441	614	209	µl(x	µl(x	NOUN
ejpam-3441	614	210	)	)	PUNCT
ejpam-3441	614	211	∧	∧	PROPN
ejpam-3441	614	212	β	β	NOUN
ejpam-3441	614	213	)	)	PUNCT
ejpam-3441	614	214	∨	∨	NUM
ejpam-3441	614	215	α	α	NOUN
ejpam-3441	614	216	=	=	PUNCT
ejpam-3441	614	217	(	(	PUNCT
ejpam-3441	614	218	µc	µc	INTJ
ejpam-3441	614	219	∧βα	∧βα	ADJ
ejpam-3441	614	220	µd	µd	DET
ejpam-3441	614	221	∧βα	∧βα	PROPN
ejpam-3441	614	222	µl)(x	µl)(x	NUM
ejpam-3441	614	223	)	)	PUNCT
ejpam-3441	614	224	.	.	PUNCT
ejpam-3441	615	1	⇒	⇒	NOUN
ejpam-3441	615	2	µc	µc	VERB
ejpam-3441	615	3	∧βα	∧βα	ADJ
ejpam-3441	615	4	µd	µd	DET
ejpam-3441	615	5	∧βα	∧βα	NOUN
ejpam-3441	615	6	µl	µl	ADP
ejpam-3441	615	7	⊆	⊆	NUM
ejpam-3441	615	8	(	(	PUNCT
ejpam-3441	615	9	µc	µc	INTJ
ejpam-3441	615	10	◦	◦	NOUN
ejpam-3441	615	11	βα	βα	NOUN
ejpam-3441	615	12	µd	µd	NOUN
ejpam-3441	615	13	)	)	PUNCT
ejpam-3441	615	14	◦	◦	NOUN
ejpam-3441	615	15	βα	βα	NOUN
ejpam-3441	615	16	µl	µl	NOUN
ejpam-3441	615	17	.	.	PUNCT
ejpam-3441	616	1	similarly	similarly	ADV
ejpam-3441	616	2	,	,	PUNCT
ejpam-3441	616	3	we	we	PRON
ejpam-3441	616	4	have	have	VERB
ejpam-3441	616	5	γc∨βαγd∨βαγl	γc∨βαγd∨βαγl	ADP
ejpam-3441	616	6	⊇	⊇	X
ejpam-3441	616	7	(	(	PUNCT
ejpam-3441	616	8	γc	γc	NOUN
ejpam-3441	616	9	◦	◦	NOUN
ejpam-3441	616	10	βαγd)	βαγd)	NOUN
ejpam-3441	616	11	◦	◦	NOUN
ejpam-3441	616	12	βαγl	βαγl	ADJ
ejpam-3441	616	13	.	.	PUNCT
ejpam-3441	617	1	hence	hence	ADV
ejpam-3441	617	2	c∧βαd∧βαl	c∧βαd∧βαl	NOUN
ejpam-3441	617	3	⊆	⊆	NUM
ejpam-3441	617	4	(	(	PUNCT
ejpam-3441	617	5	c	c	NOUN
ejpam-3441	617	6	◦	◦	NOUN
ejpam-3441	617	7	βαd)	βαd)	NOUN
ejpam-3441	617	8	◦	◦	NOUN
ejpam-3441	617	9	βαl	βαl	NOUN
ejpam-3441	617	10	,	,	PUNCT
ejpam-3441	617	11	i.e.	i.e.	X
ejpam-3441	617	12	,	,	PUNCT
ejpam-3441	617	13	(	(	PUNCT
ejpam-3441	617	14	1	1	X
ejpam-3441	617	15	)	)	PUNCT
ejpam-3441	617	16	⇒	⇒	NOUN
ejpam-3441	617	17	(	(	PUNCT
ejpam-3441	617	18	4	4	NUM
ejpam-3441	617	19	)	)	PUNCT
ejpam-3441	617	20	.	.	PUNCT
ejpam-3441	618	1	it	it	PRON
ejpam-3441	618	2	is	be	AUX
ejpam-3441	618	3	clear	clear	ADJ
ejpam-3441	618	4	that	that	SCONJ
ejpam-3441	618	5	(	(	PUNCT
ejpam-3441	618	6	4	4	X
ejpam-3441	618	7	)	)	PUNCT
ejpam-3441	618	8	⇒	⇒	NOUN
ejpam-3441	618	9	(	(	PUNCT
ejpam-3441	618	10	3	3	NUM
ejpam-3441	618	11	)	)	PUNCT
ejpam-3441	618	12	and	and	CCONJ
ejpam-3441	618	13	(	(	PUNCT
ejpam-3441	618	14	3	3	X
ejpam-3441	618	15	)	)	PUNCT
ejpam-3441	618	16	⇒	⇒	NOUN
ejpam-3441	618	17	(	(	PUNCT
ejpam-3441	618	18	2	2	NUM
ejpam-3441	618	19	)	)	PUNCT
ejpam-3441	618	20	.	.	PUNCT
ejpam-3441	619	1	assume	assume	VERB
ejpam-3441	619	2	that	that	SCONJ
ejpam-3441	619	3	(	(	PUNCT
ejpam-3441	619	4	2	2	X
ejpam-3441	619	5	)	)	PUNCT
ejpam-3441	619	6	holds	hold	VERB
ejpam-3441	619	7	.	.	PUNCT
ejpam-3441	620	1	then	then	ADV
ejpam-3441	620	2	a∧βαr∧βα	a∧βαr∧βα	INTJ
ejpam-3441	620	3	l	l	NOUN
ejpam-3441	620	4	⊆	⊆	NUM
ejpam-3441	620	5	(	(	PUNCT
ejpam-3441	620	6	a	a	DET
ejpam-3441	620	7	◦	◦	NOUN
ejpam-3441	620	8	βαr	βαr	NOUN
ejpam-3441	620	9	)	)	PUNCT
ejpam-3441	620	10	◦	◦	NOUN
ejpam-3441	620	11	βα	βα	NOUN
ejpam-3441	620	12	l	l	NOUN
ejpam-3441	620	13	,	,	PUNCT
ejpam-3441	620	14	where	where	SCONJ
ejpam-3441	620	15	a	a	PRON
ejpam-3441	620	16	is	be	AUX
ejpam-3441	620	17	an	an	DET
ejpam-3441	620	18	intuitionistic	intuitionistic	ADJ
ejpam-3441	620	19	fuzzy	fuzzy	ADJ
ejpam-3441	620	20	right	right	ADJ
ejpam-3441	620	21	ideal	ideal	NOUN
ejpam-3441	620	22	with	with	ADP
ejpam-3441	620	23	thresholds	threshold	NOUN
ejpam-3441	620	24	(	(	PUNCT
ejpam-3441	620	25	α	α	X
ejpam-3441	620	26	,	,	PUNCT
ejpam-3441	620	27	β	β	X
ejpam-3441	620	28	]	]	PUNCT
ejpam-3441	620	29	of	of	ADP
ejpam-3441	620	30	r	r	NOUN
ejpam-3441	620	31	,	,	PUNCT
ejpam-3441	620	32	i.e.	i.e.	X
ejpam-3441	620	33	,	,	PUNCT
ejpam-3441	620	34	a	a	DET
ejpam-3441	620	35	∧βα	∧βα	ADJ
ejpam-3441	620	36	l	l	NOUN
ejpam-3441	620	37	⊆	⊆	NUM
ejpam-3441	620	38	a	a	DET
ejpam-3441	620	39	◦	◦	NOUN
ejpam-3441	620	40	βα	βα	X
ejpam-3441	620	41	l.	l.	NOUN
ejpam-3441	620	42	since	since	SCONJ
ejpam-3441	620	43	a	a	DET
ejpam-3441	620	44	◦	◦	NOUN
ejpam-3441	620	45	βα	βα	NOUN
ejpam-3441	620	46	l	l	NOUN
ejpam-3441	620	47	⊆	⊆	NUM
ejpam-3441	620	48	a	a	DET
ejpam-3441	620	49	∧βα	∧βα	ADJ
ejpam-3441	620	50	l	l	NOUN
ejpam-3441	620	51	,	,	PUNCT
ejpam-3441	620	52	thus	thus	ADV
ejpam-3441	620	53	a	a	DET
ejpam-3441	620	54	◦	◦	NOUN
ejpam-3441	620	55	βα	βα	NOUN
ejpam-3441	620	56	l	l	NOUN
ejpam-3441	620	57	=	=	PUNCT
ejpam-3441	620	58	a	a	DET
ejpam-3441	620	59	∧βα	∧βα	ADJ
ejpam-3441	620	60	l.	l.	NOUN
ejpam-3441	621	1	so	so	ADV
ejpam-3441	621	2	r	r	NOUN
ejpam-3441	621	3	is	be	AUX
ejpam-3441	621	4	regular	regular	ADJ
ejpam-3441	621	5	by	by	ADP
ejpam-3441	621	6	the	the	DET
ejpam-3441	621	7	theorem	theorem	NOUN
ejpam-3441	621	8	8	8	NUM
ejpam-3441	621	9	,	,	PUNCT
ejpam-3441	621	10	i.e.	i.e.	X
ejpam-3441	621	11	,	,	PUNCT
ejpam-3441	621	12	(	(	PUNCT
ejpam-3441	621	13	2)⇒	2)⇒	NUM
ejpam-3441	621	14	(	(	PUNCT
ejpam-3441	621	15	1	1	NUM
ejpam-3441	621	16	)	)	PUNCT
ejpam-3441	621	17	.	.	PUNCT
ejpam-3441	622	1	4	4	X
ejpam-3441	622	2	.	.	X
ejpam-3441	622	3	intra	intra	ADJ
ejpam-3441	622	4	-	-	ADJ
ejpam-3441	622	5	regular	regular	ADJ
ejpam-3441	622	6	la	la	NOUN
ejpam-3441	622	7	-	-	PUNCT
ejpam-3441	622	8	rings	ring	NOUN
ejpam-3441	622	9	in	in	ADP
ejpam-3441	622	10	this	this	DET
ejpam-3441	622	11	section	section	NOUN
ejpam-3441	622	12	,	,	PUNCT
ejpam-3441	622	13	we	we	PRON
ejpam-3441	622	14	characterize	characterize	VERB
ejpam-3441	622	15	intra	intra	ADJ
ejpam-3441	622	16	-	-	ADJ
ejpam-3441	622	17	regular	regular	ADJ
ejpam-3441	622	18	la	la	NOUN
ejpam-3441	622	19	-	-	PUNCT
ejpam-3441	622	20	rings	ring	NOUN
ejpam-3441	622	21	in	in	ADP
ejpam-3441	622	22	terms	term	NOUN
ejpam-3441	622	23	of	of	ADP
ejpam-3441	622	24	intuitionistic	intuitionistic	ADJ
ejpam-3441	622	25	fuzzy	fuzzy	ADJ
ejpam-3441	622	26	left	left	NOUN
ejpam-3441	622	27	(	(	PUNCT
ejpam-3441	622	28	right	right	ADJ
ejpam-3441	622	29	,	,	PUNCT
ejpam-3441	622	30	quasi-	quasi-	INTJ
ejpam-3441	622	31	,	,	PUNCT
ejpam-3441	622	32	bi-	bi-	NUM
ejpam-3441	622	33	,	,	PUNCT
ejpam-3441	622	34	generalized	generalize	VERB
ejpam-3441	622	35	bi-	bi-	NUM
ejpam-3441	622	36	)	)	PUNCT
ejpam-3441	622	37	ideals	ideal	NOUN
ejpam-3441	622	38	with	with	ADP
ejpam-3441	622	39	thresholds	threshold	NOUN
ejpam-3441	622	40	(	(	PUNCT
ejpam-3441	622	41	α	α	X
ejpam-3441	622	42	,	,	PUNCT
ejpam-3441	622	43	β	β	X
ejpam-3441	622	44	]	]	PUNCT
ejpam-3441	622	45	.	.	PUNCT
ejpam-3441	623	1	k.	k.	PROPN
ejpam-3441	623	2	nasreen	nasreen	PROPN
ejpam-3441	623	3	et	et	PROPN
ejpam-3441	623	4	al	al	PROPN
ejpam-3441	623	5	.	.	PUNCT
ejpam-3441	623	6	/	/	SYM
ejpam-3441	623	7	eur	eur	PROPN
ejpam-3441	623	8	.	.	PUNCT
ejpam-3441	624	1	j.	j.	PROPN
ejpam-3441	624	2	pure	pure	PROPN
ejpam-3441	624	3	appl	appl	PROPN
ejpam-3441	624	4	.	.	PROPN
ejpam-3441	624	5	math	math	PROPN
ejpam-3441	624	6	,	,	PUNCT
ejpam-3441	624	7	12	12	NUM
ejpam-3441	624	8	(	(	PUNCT
ejpam-3441	624	9	3	3	NUM
ejpam-3441	624	10	)	)	PUNCT
ejpam-3441	624	11	(	(	PUNCT
ejpam-3441	624	12	2019	2019	NUM
ejpam-3441	624	13	)	)	PUNCT
ejpam-3441	624	14	,	,	PUNCT
ejpam-3441	624	15	906	906	NUM
ejpam-3441	624	16	-	-	SYM
ejpam-3441	624	17	943	943	NUM
ejpam-3441	624	18	933	933	NUM
ejpam-3441	624	19	lemma	lemma	PROPN
ejpam-3441	624	20	21	21	NUM
ejpam-3441	624	21	.	.	PUNCT
ejpam-3441	625	1	every	every	DET
ejpam-3441	625	2	intuitionistic	intuitionistic	ADJ
ejpam-3441	625	3	fuzzy	fuzzy	ADJ
ejpam-3441	625	4	left	left	ADJ
ejpam-3441	625	5	(	(	PUNCT
ejpam-3441	625	6	right	right	ADJ
ejpam-3441	625	7	)	)	PUNCT
ejpam-3441	625	8	ideal	ideal	NOUN
ejpam-3441	625	9	with	with	ADP
ejpam-3441	625	10	thresholds	threshold	NOUN
ejpam-3441	625	11	(	(	PUNCT
ejpam-3441	625	12	α	α	X
ejpam-3441	625	13	,	,	PUNCT
ejpam-3441	625	14	β	β	X
ejpam-3441	625	15	]	]	PUNCT
ejpam-3441	625	16	of	of	ADP
ejpam-3441	625	17	an	an	DET
ejpam-3441	625	18	intraregular	intraregular	ADJ
ejpam-3441	625	19	la	la	ADJ
ejpam-3441	625	20	-	-	PUNCT
ejpam-3441	625	21	ring	ring	NOUN
ejpam-3441	625	22	r	r	NOUN
ejpam-3441	625	23	is	be	AUX
ejpam-3441	625	24	an	an	DET
ejpam-3441	625	25	intuitionistic	intuitionistic	ADJ
ejpam-3441	625	26	fuzzy	fuzzy	ADJ
ejpam-3441	625	27	ideal	ideal	NOUN
ejpam-3441	625	28	with	with	ADP
ejpam-3441	625	29	thresholds	threshold	NOUN
ejpam-3441	625	30	(	(	PUNCT
ejpam-3441	625	31	α	α	X
ejpam-3441	625	32	,	,	PUNCT
ejpam-3441	625	33	β	β	X
ejpam-3441	625	34	]	]	PUNCT
ejpam-3441	625	35	of	of	ADP
ejpam-3441	625	36	r.	r.	PROPN
ejpam-3441	625	37	proof	proof	PROPN
ejpam-3441	625	38	.	.	PUNCT
ejpam-3441	626	1	suppose	suppose	VERB
ejpam-3441	626	2	that	that	SCONJ
ejpam-3441	626	3	a	a	DET
ejpam-3441	626	4	=	=	SYM
ejpam-3441	626	5	(	(	PUNCT
ejpam-3441	626	6	µa	µa	PROPN
ejpam-3441	626	7	,	,	PUNCT
ejpam-3441	626	8	γa	γa	PROPN
ejpam-3441	626	9	)	)	PUNCT
ejpam-3441	626	10	is	be	AUX
ejpam-3441	626	11	an	an	DET
ejpam-3441	626	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	626	13	fuzzy	fuzzy	ADJ
ejpam-3441	626	14	left	leave	VERB
ejpam-3441	626	15	ideal	ideal	NOUN
ejpam-3441	626	16	with	with	ADP
ejpam-3441	626	17	thresholds	threshold	NOUN
ejpam-3441	626	18	(	(	PUNCT
ejpam-3441	626	19	α	α	X
ejpam-3441	626	20	,	,	PUNCT
ejpam-3441	626	21	β	β	X
ejpam-3441	626	22	]	]	PUNCT
ejpam-3441	626	23	of	of	ADP
ejpam-3441	626	24	r.	r.	PROPN
ejpam-3441	626	25	let	let	VERB
ejpam-3441	626	26	x	x	PRON
ejpam-3441	626	27	,	,	PUNCT
ejpam-3441	626	28	y	y	PROPN
ejpam-3441	626	29	∈	∈	PROPN
ejpam-3441	626	30	r	r	NOUN
ejpam-3441	626	31	,	,	PUNCT
ejpam-3441	626	32	this	this	PRON
ejpam-3441	626	33	implies	imply	VERB
ejpam-3441	626	34	that	that	SCONJ
ejpam-3441	626	35	there	there	PRON
ejpam-3441	626	36	exist	exist	VERB
ejpam-3441	626	37	ai	ai	NOUN
ejpam-3441	626	38	,	,	PUNCT
ejpam-3441	626	39	bi	bi	NOUN
ejpam-3441	626	40	∈	∈	PROPN
ejpam-3441	626	41	r	r	PROPN
ejpam-3441	626	42	,	,	PUNCT
ejpam-3441	626	43	such	such	ADJ
ejpam-3441	626	44	that	that	SCONJ
ejpam-3441	626	45	x	x	NOUN
ejpam-3441	626	46	=	=	PUNCT
ejpam-3441	626	47	∑n	∑n	PROPN
ejpam-3441	626	48	i=1(aix	i=1(aix	ADJ
ejpam-3441	626	49	2)bi	2)bi	NUM
ejpam-3441	626	50	.	.	PUNCT
ejpam-3441	627	1	thus	thus	ADV
ejpam-3441	627	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3441	627	3	)	)	PUNCT
ejpam-3441	627	4	,	,	PUNCT
ejpam-3441	627	5	α	α	X
ejpam-3441	627	6	}	}	PUNCT
ejpam-3441	627	7	=	=	SYM
ejpam-3441	627	8	max{µa(((aix	max{µa(((aix	NOUN
ejpam-3441	627	9	2)bi)y	2)bi)y	NOUN
ejpam-3441	627	10	)	)	PUNCT
ejpam-3441	627	11	,	,	PUNCT
ejpam-3441	627	12	α	α	X
ejpam-3441	627	13	}	}	PUNCT
ejpam-3441	627	14	=	=	PUNCT
ejpam-3441	627	15	max{µa((ybi)(aix	max{µa((ybi)(aix	NOUN
ejpam-3441	627	16	2	2	NUM
ejpam-3441	627	17	)	)	PUNCT
ejpam-3441	627	18	)	)	PUNCT
ejpam-3441	627	19	,	,	PUNCT
ejpam-3441	627	20	α	α	X
ejpam-3441	627	21	}	}	PUNCT
ejpam-3441	627	22	≥	≥	NOUN
ejpam-3441	627	23	min{µa(ai(xx	min{µa(ai(xx	PROPN
ejpam-3441	627	24	)	)	PUNCT
ejpam-3441	627	25	)	)	PUNCT
ejpam-3441	627	26	,	,	PUNCT
ejpam-3441	627	27	β	β	X
ejpam-3441	627	28	}	}	PUNCT
ejpam-3441	627	29	≥	≥	NOUN
ejpam-3441	627	30	min{µa(xx	min{µa(xx	NOUN
ejpam-3441	627	31	)	)	PUNCT
ejpam-3441	627	32	,	,	PUNCT
ejpam-3441	627	33	β	β	X
ejpam-3441	627	34	}	}	PUNCT
ejpam-3441	627	35	≥	≥	PROPN
ejpam-3441	627	36	min{µa(x	min{µa(x	NOUN
ejpam-3441	627	37	)	)	PUNCT
ejpam-3441	627	38	,	,	PUNCT
ejpam-3441	627	39	β	β	X
ejpam-3441	627	40	}	}	PUNCT
ejpam-3441	627	41	and	and	CCONJ
ejpam-3441	627	42	min{γa(xy	min{γa(xy	NUM
ejpam-3441	627	43	)	)	PUNCT
ejpam-3441	627	44	,	,	PUNCT
ejpam-3441	627	45	(	(	PUNCT
ejpam-3441	627	46	1−	1−	NUM
ejpam-3441	627	47	α	α	NOUN
ejpam-3441	627	48	)	)	PUNCT
ejpam-3441	627	49	}	}	PUNCT
ejpam-3441	628	1	=	=	SYM
ejpam-3441	628	2	min{γa(((aix	min{γa(((aix	PROPN
ejpam-3441	628	3	2)bi)y	2)bi)y	NUM
ejpam-3441	628	4	)	)	PUNCT
ejpam-3441	628	5	,	,	PUNCT
ejpam-3441	628	6	(	(	PUNCT
ejpam-3441	628	7	1−	1−	NUM
ejpam-3441	628	8	α	α	NOUN
ejpam-3441	628	9	)	)	PUNCT
ejpam-3441	628	10	}	}	PUNCT
ejpam-3441	628	11	=	=	PUNCT
ejpam-3441	628	12	min{γa((ybi)(aix	min{γa((ybi)(aix	NOUN
ejpam-3441	628	13	2	2	NUM
ejpam-3441	628	14	)	)	PUNCT
ejpam-3441	628	15	)	)	PUNCT
ejpam-3441	628	16	,	,	PUNCT
ejpam-3441	628	17	(	(	PUNCT
ejpam-3441	628	18	1−	1−	NUM
ejpam-3441	628	19	α	α	NOUN
ejpam-3441	628	20	)	)	PUNCT
ejpam-3441	628	21	}	}	PUNCT
ejpam-3441	628	22	≤	≤	NUM
ejpam-3441	628	23	max{γa(ai(xx	max{γa(ai(xx	NOUN
ejpam-3441	628	24	)	)	PUNCT
ejpam-3441	628	25	)	)	PUNCT
ejpam-3441	628	26	,	,	PUNCT
ejpam-3441	628	27	(	(	PUNCT
ejpam-3441	628	28	1−	1−	NUM
ejpam-3441	628	29	β	β	NOUN
ejpam-3441	628	30	)	)	PUNCT
ejpam-3441	628	31	}	}	PUNCT
ejpam-3441	628	32	≤	≤	NUM
ejpam-3441	628	33	max{γa(xx	max{γa(xx	PROPN
ejpam-3441	628	34	)	)	PUNCT
ejpam-3441	628	35	,	,	PUNCT
ejpam-3441	628	36	(	(	PUNCT
ejpam-3441	628	37	1−	1−	NUM
ejpam-3441	628	38	β	β	NOUN
ejpam-3441	628	39	)	)	PUNCT
ejpam-3441	628	40	}	}	PUNCT
ejpam-3441	628	41	≤	≤	NUM
ejpam-3441	628	42	max{γa(x	max{γa(x	NOUN
ejpam-3441	628	43	)	)	PUNCT
ejpam-3441	628	44	,	,	PUNCT
ejpam-3441	628	45	(	(	PUNCT
ejpam-3441	628	46	1−	1−	NUM
ejpam-3441	628	47	β	β	NOUN
ejpam-3441	628	48	)	)	PUNCT
ejpam-3441	628	49	}	}	PUNCT
ejpam-3441	628	50	.	.	PUNCT
ejpam-3441	629	1	hence	hence	ADV
ejpam-3441	629	2	a	a	PRON
ejpam-3441	629	3	is	be	AUX
ejpam-3441	629	4	an	an	DET
ejpam-3441	629	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	629	6	fuzzy	fuzzy	ADJ
ejpam-3441	629	7	ideal	ideal	NOUN
ejpam-3441	629	8	with	with	ADP
ejpam-3441	629	9	thresholds	threshold	NOUN
ejpam-3441	629	10	(	(	PUNCT
ejpam-3441	629	11	α	α	X
ejpam-3441	629	12	,	,	PUNCT
ejpam-3441	629	13	β	β	X
ejpam-3441	629	14	]	]	PUNCT
ejpam-3441	629	15	of	of	ADP
ejpam-3441	629	16	r.	r.	PROPN
ejpam-3441	629	17	lemma	lemma	PROPN
ejpam-3441	629	18	22	22	NUM
ejpam-3441	629	19	.	.	PUNCT
ejpam-3441	630	1	let	let	VERB
ejpam-3441	630	2	r	r	PRON
ejpam-3441	630	3	be	be	AUX
ejpam-3441	630	4	an	an	DET
ejpam-3441	630	5	intra	intra	ADJ
ejpam-3441	630	6	-	-	ADJ
ejpam-3441	630	7	regular	regular	ADJ
ejpam-3441	630	8	la	la	NOUN
ejpam-3441	630	9	-	-	NOUN
ejpam-3441	630	10	ring	ring	NOUN
ejpam-3441	630	11	with	with	ADP
ejpam-3441	630	12	left	left	ADJ
ejpam-3441	630	13	identity	identity	NOUN
ejpam-3441	630	14	e.	e.	PROPN
ejpam-3441	630	15	then	then	ADV
ejpam-3441	630	16	every	every	DET
ejpam-3441	630	17	intuitionistic	intuitionistic	ADJ
ejpam-3441	630	18	fuzzy	fuzzy	ADJ
ejpam-3441	630	19	ideal	ideal	NOUN
ejpam-3441	630	20	with	with	ADP
ejpam-3441	630	21	thresholds	threshold	NOUN
ejpam-3441	630	22	(	(	PUNCT
ejpam-3441	630	23	α	α	X
ejpam-3441	630	24	,	,	PUNCT
ejpam-3441	630	25	β	β	X
ejpam-3441	630	26	]	]	PUNCT
ejpam-3441	630	27	of	of	ADP
ejpam-3441	630	28	r	r	NOUN
ejpam-3441	630	29	is	be	AUX
ejpam-3441	630	30	an	an	DET
ejpam-3441	630	31	intuitionistic	intuitionistic	ADJ
ejpam-3441	630	32	fuzzy	fuzzy	ADJ
ejpam-3441	630	33	idempotent	idempotent	NOUN
ejpam-3441	630	34	with	with	ADP
ejpam-3441	630	35	thresholds	threshold	NOUN
ejpam-3441	630	36	(	(	PUNCT
ejpam-3441	630	37	α	α	X
ejpam-3441	630	38	,	,	PUNCT
ejpam-3441	630	39	β	β	X
ejpam-3441	630	40	]	]	PUNCT
ejpam-3441	630	41	.	.	PUNCT
ejpam-3441	631	1	proof	proof	NOUN
ejpam-3441	631	2	.	.	PUNCT
ejpam-3441	632	1	assume	assume	VERB
ejpam-3441	632	2	that	that	SCONJ
ejpam-3441	632	3	a	a	DET
ejpam-3441	632	4	=	=	SYM
ejpam-3441	632	5	(	(	PUNCT
ejpam-3441	632	6	µa	µa	PROPN
ejpam-3441	632	7	,	,	PUNCT
ejpam-3441	632	8	γa	γa	PROPN
ejpam-3441	632	9	)	)	PUNCT
ejpam-3441	632	10	is	be	AUX
ejpam-3441	632	11	an	an	DET
ejpam-3441	632	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	632	13	fuzzy	fuzzy	ADJ
ejpam-3441	632	14	ideal	ideal	NOUN
ejpam-3441	632	15	with	with	ADP
ejpam-3441	632	16	thresholds	threshold	NOUN
ejpam-3441	632	17	(	(	PUNCT
ejpam-3441	632	18	α	α	X
ejpam-3441	632	19	,	,	PUNCT
ejpam-3441	632	20	β	β	X
ejpam-3441	632	21	]	]	PUNCT
ejpam-3441	632	22	of	of	ADP
ejpam-3441	632	23	r	r	NOUN
ejpam-3441	632	24	and	and	CCONJ
ejpam-3441	632	25	a	a	DET
ejpam-3441	632	26	◦	◦	NOUN
ejpam-3441	632	27	βα	βα	NOUN
ejpam-3441	632	28	a	a	DET
ejpam-3441	632	29	⊆	⊆	NUM
ejpam-3441	632	30	aβα	aβα	NOUN
ejpam-3441	632	31	.	.	PUNCT
ejpam-3441	633	1	let	let	VERB
ejpam-3441	633	2	x	x	PUNCT
ejpam-3441	633	3	∈	∈	PROPN
ejpam-3441	633	4	r	r	NOUN
ejpam-3441	633	5	,	,	PUNCT
ejpam-3441	633	6	this	this	PRON
ejpam-3441	633	7	means	mean	VERB
ejpam-3441	633	8	that	that	SCONJ
ejpam-3441	633	9	there	there	PRON
ejpam-3441	633	10	exist	exist	VERB
ejpam-3441	633	11	ai	ai	NOUN
ejpam-3441	633	12	,	,	PUNCT
ejpam-3441	633	13	bi	bi	NOUN
ejpam-3441	633	14	∈	∈	PROPN
ejpam-3441	633	15	r	r	PROPN
ejpam-3441	633	16	,	,	PUNCT
ejpam-3441	633	17	such	such	ADJ
ejpam-3441	633	18	that	that	SCONJ
ejpam-3441	633	19	x	x	NOUN
ejpam-3441	634	1	=	=	PUNCT
ejpam-3441	634	2	∑n	∑n	PROPN
ejpam-3441	634	3	i=1(aix	i=1(aix	PROPN
ejpam-3441	634	4	2)bi	2)bi	NUM
ejpam-3441	634	5	.	.	PUNCT
ejpam-3441	635	1	now	now	ADV
ejpam-3441	635	2	x	x	X
ejpam-3441	635	3	=	=	SYM
ejpam-3441	635	4	(	(	PUNCT
ejpam-3441	635	5	aix	aix	NOUN
ejpam-3441	635	6	2)bi	2)bi	NUM
ejpam-3441	635	7	=	=	SYM
ejpam-3441	635	8	(	(	PUNCT
ejpam-3441	635	9	ai(xx))bi	ai(xx))bi	NOUN
ejpam-3441	635	10	=	=	PUNCT
ejpam-3441	635	11	(	(	PUNCT
ejpam-3441	635	12	x(aix))bi	x(aix))bi	PROPN
ejpam-3441	635	13	=	=	SYM
ejpam-3441	635	14	(	(	PUNCT
ejpam-3441	635	15	x(aix))(ebi	x(aix))(ebi	PROPN
ejpam-3441	635	16	)	)	PUNCT
ejpam-3441	635	17	=	=	SYM
ejpam-3441	635	18	(	(	PUNCT
ejpam-3441	635	19	xe)((aix)bi	xe)((aix)bi	PROPN
ejpam-3441	635	20	)	)	PUNCT
ejpam-3441	636	1	=	=	PRON
ejpam-3441	636	2	(	(	PUNCT
ejpam-3441	636	3	aix)((xe)bi	aix)((xe)bi	NOUN
ejpam-3441	636	4	)	)	PUNCT
ejpam-3441	636	5	.	.	PUNCT
ejpam-3441	637	1	thus	thus	ADV
ejpam-3441	637	2	(	(	PUNCT
ejpam-3441	637	3	µa	µa	ADP
ejpam-3441	637	4	◦	◦	NOUN
ejpam-3441	637	5	βα	βα	ADJ
ejpam-3441	637	6	µa)(x	µa)(x	NOUN
ejpam-3441	637	7	)	)	PUNCT
ejpam-3441	638	1	=	=	PRON
ejpam-3441	638	2	{	{	PUNCT
ejpam-3441	638	3	(	(	PUNCT
ejpam-3441	638	4	µa	µa	ADP
ejpam-3441	638	5	◦	◦	NOUN
ejpam-3441	638	6	µa)(x	µa)(x	NOUN
ejpam-3441	638	7	)	)	PUNCT
ejpam-3441	639	1	∧	∧	PROPN
ejpam-3441	639	2	β	β	NOUN
ejpam-3441	639	3	}	}	PUNCT
ejpam-3441	639	4	∨	∨	NUM
ejpam-3441	639	5	α	α	NOUN
ejpam-3441	639	6	=	=	X
ejpam-3441	639	7	{	{	PUNCT
ejpam-3441	639	8	(	(	PUNCT
ejpam-3441	639	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	639	10	i=1	i=1	PROPN
ejpam-3441	639	11	piqi	piqi	NOUN
ejpam-3441	639	12	{	{	PUNCT
ejpam-3441	639	13	∧ni=1	∧ni=1	X
ejpam-3441	639	14	{	{	PUNCT
ejpam-3441	639	15	µa	µa	X
ejpam-3441	639	16	(	(	PUNCT
ejpam-3441	639	17	pi	pi	NOUN
ejpam-3441	639	18	)	)	PUNCT
ejpam-3441	639	19	∧	∧	PROPN
ejpam-3441	639	20	µa	µa	PROPN
ejpam-3441	639	21	(	(	PUNCT
ejpam-3441	639	22	qi	qi	NOUN
ejpam-3441	639	23	)	)	PUNCT
ejpam-3441	639	24	}	}	PUNCT
ejpam-3441	639	25	}	}	PUNCT
ejpam-3441	639	26	)	)	PUNCT
ejpam-3441	640	1	∧	∧	PROPN
ejpam-3441	640	2	β	β	NOUN
ejpam-3441	640	3	}	}	PUNCT
ejpam-3441	640	4	∨	∨	NUM
ejpam-3441	640	5	α	α	PROPN
ejpam-3441	640	6	≥	≥	X
ejpam-3441	640	7	{	{	PUNCT
ejpam-3441	640	8	{	{	PUNCT
ejpam-3441	640	9	µa	µa	PROPN
ejpam-3441	640	10	(	(	PUNCT
ejpam-3441	640	11	aix	aix	PROPN
ejpam-3441	640	12	)	)	PUNCT
ejpam-3441	640	13	∧	∧	PROPN
ejpam-3441	640	14	µa	µa	NOUN
ejpam-3441	640	15	(	(	PUNCT
ejpam-3441	640	16	(	(	PUNCT
ejpam-3441	640	17	xe)bi	xe)bi	NOUN
ejpam-3441	640	18	)	)	PUNCT
ejpam-3441	640	19	}	}	PUNCT
ejpam-3441	640	20	∧	∧	PROPN
ejpam-3441	640	21	β	β	NOUN
ejpam-3441	640	22	}	}	PUNCT
ejpam-3441	640	23	∨	∨	NUM
ejpam-3441	640	24	α	α	NOUN
ejpam-3441	640	25	=	=	SYM
ejpam-3441	640	26	(	(	PUNCT
ejpam-3441	640	27	µa	µa	INTJ
ejpam-3441	640	28	(	(	PUNCT
ejpam-3441	640	29	aix	aix	PROPN
ejpam-3441	640	30	)	)	PUNCT
ejpam-3441	640	31	∨	∨	NUM
ejpam-3441	640	32	α	α	NOUN
ejpam-3441	640	33	)	)	PUNCT
ejpam-3441	640	34	∧	∧	PROPN
ejpam-3441	640	35	(	(	PUNCT
ejpam-3441	640	36	µa	µa	X
ejpam-3441	640	37	(	(	PUNCT
ejpam-3441	640	38	(	(	PUNCT
ejpam-3441	640	39	xe)bi	xe)bi	PROPN
ejpam-3441	640	40	)	)	PUNCT
ejpam-3441	640	41	∨	∨	NUM
ejpam-3441	640	42	α	α	NOUN
ejpam-3441	640	43	)	)	PUNCT
ejpam-3441	640	44	∧	∧	PROPN
ejpam-3441	640	45	(	(	PUNCT
ejpam-3441	640	46	β	β	X
ejpam-3441	640	47	∨	∨	NUM
ejpam-3441	640	48	α	α	NOUN
ejpam-3441	640	49	)	)	PUNCT
ejpam-3441	640	50	≥	≥	NOUN
ejpam-3441	640	51	(	(	PUNCT
ejpam-3441	640	52	µa	µa	PROPN
ejpam-3441	640	53	(	(	PUNCT
ejpam-3441	640	54	x	x	NOUN
ejpam-3441	640	55	)	)	PUNCT
ejpam-3441	640	56	∧	∧	PROPN
ejpam-3441	640	57	β	β	NOUN
ejpam-3441	640	58	)	)	PUNCT
ejpam-3441	640	59	∧	∧	PROPN
ejpam-3441	640	60	(	(	PUNCT
ejpam-3441	640	61	µa	µa	X
ejpam-3441	640	62	(	(	PUNCT
ejpam-3441	640	63	x	x	NOUN
ejpam-3441	640	64	)	)	PUNCT
ejpam-3441	640	65	∧	∧	PROPN
ejpam-3441	640	66	β	β	NOUN
ejpam-3441	640	67	)	)	PUNCT
ejpam-3441	640	68	∧	∧	NOUN
ejpam-3441	640	69	β	β	X
ejpam-3441	640	70	=	=	SYM
ejpam-3441	640	71	µa	µa	X
ejpam-3441	640	72	(	(	PUNCT
ejpam-3441	640	73	x	x	X
ejpam-3441	640	74	)	)	PUNCT
ejpam-3441	640	75	∧	∧	NOUN
ejpam-3441	640	76	β	β	X
ejpam-3441	640	77	=	=	SYM
ejpam-3441	640	78	(	(	PUNCT
ejpam-3441	640	79	µa	µa	INTJ
ejpam-3441	640	80	(	(	PUNCT
ejpam-3441	640	81	x	x	NOUN
ejpam-3441	640	82	)	)	PUNCT
ejpam-3441	640	83	∧	∧	PROPN
ejpam-3441	640	84	β	β	NOUN
ejpam-3441	640	85	)	)	PUNCT
ejpam-3441	640	86	∨	∨	NUM
ejpam-3441	640	87	α	α	NOUN
ejpam-3441	640	88	=	=	SYM
ejpam-3441	640	89	(	(	PUNCT
ejpam-3441	640	90	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	640	91	)	)	PUNCT
ejpam-3441	640	92	.	.	PUNCT
ejpam-3441	641	1	⇒	⇒	NOUN
ejpam-3441	641	2	(	(	PUNCT
ejpam-3441	641	3	µa)βα	µa)βα	NUM
ejpam-3441	641	4	⊆	⊆	NUM
ejpam-3441	641	5	µa	µa	NOUN
ejpam-3441	641	6	◦	◦	NOUN
ejpam-3441	641	7	βα	βα	NOUN
ejpam-3441	641	8	µa	µa	PROPN
ejpam-3441	641	9	.	.	PROPN
ejpam-3441	642	1	similarly	similarly	ADV
ejpam-3441	642	2	,	,	PUNCT
ejpam-3441	642	3	we	we	PRON
ejpam-3441	642	4	have	have	VERB
ejpam-3441	642	5	(	(	PUNCT
ejpam-3441	642	6	γa)βα	γa)βα	X
ejpam-3441	642	7	⊇	⊇	PROPN
ejpam-3441	642	8	γa	γa	PROPN
ejpam-3441	642	9	◦	◦	PROPN
ejpam-3441	642	10	βα	βα	PROPN
ejpam-3441	642	11	γa	γa	PROPN
ejpam-3441	642	12	.	.	PUNCT
ejpam-3441	643	1	therefore	therefore	ADV
ejpam-3441	643	2	aβα	aβα	PROPN
ejpam-3441	643	3	=	=	PUNCT
ejpam-3441	643	4	a	a	DET
ejpam-3441	643	5	◦	◦	NOUN
ejpam-3441	643	6	βα	βα	NOUN
ejpam-3441	643	7	a.	a.	PROPN
ejpam-3441	643	8	k.	k.	PROPN
ejpam-3441	643	9	nasreen	nasreen	PROPN
ejpam-3441	643	10	et	et	PROPN
ejpam-3441	643	11	al	al	PROPN
ejpam-3441	643	12	.	.	PUNCT
ejpam-3441	643	13	/	/	SYM
ejpam-3441	643	14	eur	eur	PROPN
ejpam-3441	643	15	.	.	PUNCT
ejpam-3441	644	1	j.	j.	PROPN
ejpam-3441	644	2	pure	pure	PROPN
ejpam-3441	644	3	appl	appl	PROPN
ejpam-3441	644	4	.	.	PROPN
ejpam-3441	644	5	math	math	PROPN
ejpam-3441	644	6	,	,	PUNCT
ejpam-3441	644	7	12	12	NUM
ejpam-3441	644	8	(	(	PUNCT
ejpam-3441	644	9	3	3	NUM
ejpam-3441	644	10	)	)	PUNCT
ejpam-3441	644	11	(	(	PUNCT
ejpam-3441	644	12	2019	2019	NUM
ejpam-3441	644	13	)	)	PUNCT
ejpam-3441	644	14	,	,	PUNCT
ejpam-3441	644	15	906	906	NUM
ejpam-3441	644	16	-	-	SYM
ejpam-3441	644	17	943	943	NUM
ejpam-3441	644	18	934	934	NUM
ejpam-3441	644	19	proposition	proposition	NOUN
ejpam-3441	644	20	9	9	NUM
ejpam-3441	644	21	.	.	PUNCT
ejpam-3441	645	1	let	let	VERB
ejpam-3441	645	2	a	a	DET
ejpam-3441	645	3	be	be	AUX
ejpam-3441	645	4	an	an	DET
ejpam-3441	645	5	ifs	ifs	PROPN
ejpam-3441	645	6	of	of	ADP
ejpam-3441	645	7	an	an	DET
ejpam-3441	645	8	intra	intra	ADJ
ejpam-3441	645	9	-	-	ADJ
ejpam-3441	645	10	regular	regular	ADJ
ejpam-3441	645	11	la	la	ADJ
ejpam-3441	645	12	-	-	PUNCT
ejpam-3441	645	13	ring	ring	NOUN
ejpam-3441	645	14	r	r	NOUN
ejpam-3441	645	15	with	with	ADP
ejpam-3441	645	16	left	left	ADJ
ejpam-3441	645	17	identity	identity	NOUN
ejpam-3441	645	18	e.	e.	PROPN
ejpam-3441	645	19	then	then	ADV
ejpam-3441	645	20	a	a	PRON
ejpam-3441	645	21	is	be	AUX
ejpam-3441	645	22	an	an	DET
ejpam-3441	645	23	intuitionistic	intuitionistic	ADJ
ejpam-3441	645	24	fuzzy	fuzzy	ADJ
ejpam-3441	645	25	ideal	ideal	NOUN
ejpam-3441	645	26	with	with	ADP
ejpam-3441	645	27	thresholds	threshold	NOUN
ejpam-3441	645	28	(	(	PUNCT
ejpam-3441	645	29	α	α	X
ejpam-3441	645	30	,	,	PUNCT
ejpam-3441	645	31	β	β	X
ejpam-3441	645	32	]	]	PUNCT
ejpam-3441	645	33	of	of	ADP
ejpam-3441	645	34	r	r	NOUN
ejpam-3441	645	35	if	if	SCONJ
ejpam-3441	646	1	and	and	CCONJ
ejpam-3441	646	2	only	only	ADV
ejpam-3441	646	3	if	if	SCONJ
ejpam-3441	646	4	a	a	PRON
ejpam-3441	646	5	is	be	AUX
ejpam-3441	646	6	an	an	DET
ejpam-3441	646	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	646	8	fuzzy	fuzzy	ADJ
ejpam-3441	646	9	interior	interior	ADJ
ejpam-3441	646	10	ideal	ideal	NOUN
ejpam-3441	646	11	with	with	ADP
ejpam-3441	646	12	thresholds	threshold	NOUN
ejpam-3441	646	13	(	(	PUNCT
ejpam-3441	646	14	α	α	X
ejpam-3441	646	15	,	,	PUNCT
ejpam-3441	646	16	β	β	X
ejpam-3441	646	17	]	]	PUNCT
ejpam-3441	646	18	of	of	ADP
ejpam-3441	646	19	r.	r.	PROPN
ejpam-3441	646	20	proof	proof	NOUN
ejpam-3441	646	21	.	.	PUNCT
ejpam-3441	647	1	consider	consider	VERB
ejpam-3441	647	2	that	that	PRON
ejpam-3441	647	3	a	a	DET
ejpam-3441	647	4	=	=	SYM
ejpam-3441	647	5	(	(	PUNCT
ejpam-3441	647	6	µa	µa	PROPN
ejpam-3441	647	7	,	,	PUNCT
ejpam-3441	647	8	γa	γa	PROPN
ejpam-3441	647	9	)	)	PUNCT
ejpam-3441	647	10	is	be	AUX
ejpam-3441	647	11	an	an	DET
ejpam-3441	647	12	intuitionistic	intuitionistic	ADJ
ejpam-3441	647	13	fuzzy	fuzzy	ADJ
ejpam-3441	647	14	interior	interior	ADJ
ejpam-3441	647	15	ideal	ideal	NOUN
ejpam-3441	647	16	with	with	ADP
ejpam-3441	647	17	thresholds	threshold	NOUN
ejpam-3441	647	18	(	(	PUNCT
ejpam-3441	647	19	α	α	X
ejpam-3441	647	20	,	,	PUNCT
ejpam-3441	647	21	β	β	X
ejpam-3441	647	22	]	]	PUNCT
ejpam-3441	647	23	of	of	ADP
ejpam-3441	647	24	r.	r.	PROPN
ejpam-3441	647	25	let	let	VERB
ejpam-3441	647	26	x	x	PRON
ejpam-3441	647	27	,	,	PUNCT
ejpam-3441	647	28	y	y	PROPN
ejpam-3441	647	29	∈	∈	PROPN
ejpam-3441	647	30	r	r	NOUN
ejpam-3441	647	31	,	,	PUNCT
ejpam-3441	647	32	then	then	ADV
ejpam-3441	647	33	there	there	PRON
ejpam-3441	647	34	exist	exist	VERB
ejpam-3441	647	35	elements	element	NOUN
ejpam-3441	647	36	ai	ai	VERB
ejpam-3441	647	37	,	,	PUNCT
ejpam-3441	647	38	bi	bi	NOUN
ejpam-3441	647	39	∈	∈	PROPN
ejpam-3441	647	40	r	r	PROPN
ejpam-3441	647	41	,	,	PUNCT
ejpam-3441	647	42	such	such	ADJ
ejpam-3441	647	43	that	that	SCONJ
ejpam-3441	647	44	x	x	NOUN
ejpam-3441	647	45	=	=	PUNCT
ejpam-3441	647	46	∑n	∑n	PROPN
ejpam-3441	647	47	i=1(aix	i=1(aix	ADJ
ejpam-3441	647	48	2)bi	2)bi	NUM
ejpam-3441	647	49	.	.	PUNCT
ejpam-3441	648	1	thus	thus	ADV
ejpam-3441	648	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3441	648	3	)	)	PUNCT
ejpam-3441	648	4	,	,	PUNCT
ejpam-3441	648	5	α	α	X
ejpam-3441	648	6	}	}	PUNCT
ejpam-3441	648	7	=	=	SYM
ejpam-3441	648	8	max{µa(((aix	max{µa(((aix	NOUN
ejpam-3441	648	9	2)bi)y	2)bi)y	NOUN
ejpam-3441	648	10	)	)	PUNCT
ejpam-3441	648	11	,	,	PUNCT
ejpam-3441	648	12	α	α	X
ejpam-3441	648	13	}	}	PUNCT
ejpam-3441	648	14	=	=	PUNCT
ejpam-3441	648	15	max{µa((ybi)(aix	max{µa((ybi)(aix	NOUN
ejpam-3441	648	16	2	2	NUM
ejpam-3441	648	17	)	)	PUNCT
ejpam-3441	648	18	)	)	PUNCT
ejpam-3441	648	19	,	,	PUNCT
ejpam-3441	648	20	α	α	X
ejpam-3441	648	21	}	}	PUNCT
ejpam-3441	648	22	=	=	SYM
ejpam-3441	648	23	max{µa((ybi)(ai(xx	max{µa((ybi)(ai(xx	NOUN
ejpam-3441	648	24	)	)	PUNCT
ejpam-3441	648	25	)	)	PUNCT
ejpam-3441	648	26	)	)	PUNCT
ejpam-3441	648	27	,	,	PUNCT
ejpam-3441	648	28	α	α	X
ejpam-3441	648	29	}	}	PUNCT
ejpam-3441	648	30	=	=	SYM
ejpam-3441	648	31	max{µa((ybi)(x(aix	max{µa((ybi)(x(aix	NOUN
ejpam-3441	648	32	)	)	PUNCT
ejpam-3441	648	33	)	)	PUNCT
ejpam-3441	648	34	)	)	PUNCT
ejpam-3441	648	35	,	,	PUNCT
ejpam-3441	648	36	α	α	X
ejpam-3441	648	37	}	}	PUNCT
ejpam-3441	648	38	=	=	SYM
ejpam-3441	648	39	max{µa((yx)(bi(aix	max{µa((yx)(bi(aix	NOUN
ejpam-3441	648	40	)	)	PUNCT
ejpam-3441	648	41	)	)	PUNCT
ejpam-3441	648	42	)	)	PUNCT
ejpam-3441	648	43	,	,	PUNCT
ejpam-3441	648	44	α	α	X
ejpam-3441	648	45	}	}	PUNCT
ejpam-3441	648	46	≥	≥	NOUN
ejpam-3441	648	47	min{µa(x	min{µa(x	NOUN
ejpam-3441	648	48	)	)	PUNCT
ejpam-3441	648	49	,	,	PUNCT
ejpam-3441	648	50	β	β	NOUN
ejpam-3441	648	51	}	}	PUNCT
ejpam-3441	648	52	.	.	PUNCT
ejpam-3441	649	1	⇒	⇒	PROPN
ejpam-3441	649	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3441	649	3	)	)	PUNCT
ejpam-3441	649	4	,	,	PUNCT
ejpam-3441	649	5	α	α	X
ejpam-3441	649	6	}	}	PUNCT
ejpam-3441	649	7	≥	≥	NOUN
ejpam-3441	649	8	min{µa(x	min{µa(x	NOUN
ejpam-3441	649	9	)	)	PUNCT
ejpam-3441	649	10	,	,	PUNCT
ejpam-3441	649	11	β	β	X
ejpam-3441	649	12	}	}	PUNCT
ejpam-3441	649	13	.	.	PUNCT
ejpam-3441	650	1	similarly	similarly	ADV
ejpam-3441	650	2	,	,	PUNCT
ejpam-3441	650	3	we	we	PRON
ejpam-3441	650	4	have	have	VERB
ejpam-3441	650	5	min{γa(xy	min{γa(xy	NUM
ejpam-3441	650	6	)	)	PUNCT
ejpam-3441	650	7	,	,	PUNCT
ejpam-3441	650	8	(	(	PUNCT
ejpam-3441	650	9	1−α	1−α	NUM
ejpam-3441	650	10	)	)	PUNCT
ejpam-3441	650	11	}	}	PUNCT
ejpam-3441	650	12	≤	≤	NUM
ejpam-3441	650	13	max{γa(x	max{γa(x	NOUN
ejpam-3441	650	14	)	)	PUNCT
ejpam-3441	650	15	,	,	PUNCT
ejpam-3441	650	16	(	(	PUNCT
ejpam-3441	650	17	1−β	1−β	NUM
ejpam-3441	650	18	)	)	PUNCT
ejpam-3441	650	19	}	}	PUNCT
ejpam-3441	650	20	,	,	PUNCT
ejpam-3441	650	21	i.e.	i.e.	X
ejpam-3441	650	22	,	,	PUNCT
ejpam-3441	650	23	a	a	PRON
ejpam-3441	650	24	is	be	AUX
ejpam-3441	650	25	an	an	DET
ejpam-3441	650	26	intuitionistic	intuitionistic	ADJ
ejpam-3441	650	27	fuzzy	fuzzy	ADJ
ejpam-3441	650	28	right	right	ADJ
ejpam-3441	650	29	ideal	ideal	NOUN
ejpam-3441	650	30	with	with	ADP
ejpam-3441	650	31	thresholds	threshold	NOUN
ejpam-3441	650	32	(	(	PUNCT
ejpam-3441	650	33	α	α	X
ejpam-3441	650	34	,	,	PUNCT
ejpam-3441	650	35	β	β	X
ejpam-3441	650	36	]	]	PUNCT
ejpam-3441	650	37	of	of	ADP
ejpam-3441	650	38	r.	r.	PROPN
ejpam-3441	650	39	therefore	therefore	ADV
ejpam-3441	650	40	a	a	PRON
ejpam-3441	650	41	is	be	AUX
ejpam-3441	650	42	an	an	DET
ejpam-3441	650	43	intuitionistic	intuitionistic	ADJ
ejpam-3441	650	44	fuzzy	fuzzy	ADJ
ejpam-3441	650	45	ideal	ideal	NOUN
ejpam-3441	650	46	with	with	ADP
ejpam-3441	650	47	thresholds	threshold	NOUN
ejpam-3441	650	48	(	(	PUNCT
ejpam-3441	650	49	α	α	X
ejpam-3441	650	50	,	,	PUNCT
ejpam-3441	650	51	β	β	X
ejpam-3441	650	52	]	]	PUNCT
ejpam-3441	650	53	of	of	ADP
ejpam-3441	650	54	r	r	NOUN
ejpam-3441	650	55	by	by	ADP
ejpam-3441	650	56	the	the	DET
ejpam-3441	650	57	lemma	lemma	PROPN
ejpam-3441	650	58	21	21	NUM
ejpam-3441	650	59	.	.	PUNCT
ejpam-3441	651	1	converse	converse	NOUN
ejpam-3441	651	2	is	be	AUX
ejpam-3441	651	3	true	true	ADJ
ejpam-3441	651	4	by	by	ADP
ejpam-3441	651	5	the	the	DET
ejpam-3441	651	6	lemma	lemma	PROPN
ejpam-3441	651	7	11	11	NUM
ejpam-3441	651	8	.	.	PUNCT
ejpam-3441	652	1	remark	remark	PROPN
ejpam-3441	652	2	7	7	NUM
ejpam-3441	652	3	.	.	PUNCT
ejpam-3441	653	1	the	the	DET
ejpam-3441	653	2	concept	concept	NOUN
ejpam-3441	653	3	of	of	ADP
ejpam-3441	653	4	intuitionistic	intuitionistic	ADJ
ejpam-3441	653	5	fuzzy	fuzzy	ADJ
ejpam-3441	653	6	(	(	PUNCT
ejpam-3441	653	7	interior	interior	ADJ
ejpam-3441	653	8	,	,	PUNCT
ejpam-3441	653	9	two	two	NUM
ejpam-3441	653	10	-	-	PUNCT
ejpam-3441	653	11	sided	sided	ADJ
ejpam-3441	653	12	)	)	PUNCT
ejpam-3441	653	13	ideals	ideal	NOUN
ejpam-3441	653	14	with	with	ADP
ejpam-3441	653	15	thresholds	threshold	NOUN
ejpam-3441	653	16	(	(	PUNCT
ejpam-3441	653	17	α	α	X
ejpam-3441	653	18	,	,	PUNCT
ejpam-3441	653	19	β	β	X
ejpam-3441	653	20	]	]	X
ejpam-3441	653	21	coincides	coincide	VERB
ejpam-3441	653	22	in	in	ADP
ejpam-3441	653	23	intra	intra	ADJ
ejpam-3441	653	24	-	-	ADJ
ejpam-3441	653	25	regular	regular	ADJ
ejpam-3441	653	26	la	la	NOUN
ejpam-3441	653	27	-	-	PUNCT
ejpam-3441	653	28	rings	ring	NOUN
ejpam-3441	653	29	with	with	ADP
ejpam-3441	653	30	left	left	ADJ
ejpam-3441	653	31	identity	identity	NOUN
ejpam-3441	653	32	.	.	PUNCT
ejpam-3441	654	1	lemma	lemma	PROPN
ejpam-3441	654	2	23	23	NUM
ejpam-3441	654	3	.	.	PUNCT
ejpam-3441	655	1	let	let	VERB
ejpam-3441	655	2	r	r	PRON
ejpam-3441	655	3	be	be	AUX
ejpam-3441	655	4	an	an	DET
ejpam-3441	655	5	intra	intra	ADJ
ejpam-3441	655	6	-	-	ADJ
ejpam-3441	655	7	regular	regular	ADJ
ejpam-3441	655	8	la	la	NOUN
ejpam-3441	655	9	-	-	NOUN
ejpam-3441	655	10	ring	ring	NOUN
ejpam-3441	655	11	with	with	ADP
ejpam-3441	655	12	left	left	ADJ
ejpam-3441	655	13	identity	identity	NOUN
ejpam-3441	655	14	e.	e.	PROPN
ejpam-3441	656	1	then	then	ADV
ejpam-3441	656	2	b	b	X
ejpam-3441	657	1	∧βα	∧βα	DET
ejpam-3441	657	2	a	a	DET
ejpam-3441	657	3	⊆	⊆	NUM
ejpam-3441	657	4	a	a	DET
ejpam-3441	657	5	◦	◦	NOUN
ejpam-3441	657	6	βα	βα	NOUN
ejpam-3441	657	7	b	b	NOUN
ejpam-3441	657	8	for	for	ADP
ejpam-3441	657	9	every	every	DET
ejpam-3441	657	10	intuitionistic	intuitionistic	ADJ
ejpam-3441	657	11	fuzzy	fuzzy	ADJ
ejpam-3441	657	12	left	leave	VERB
ejpam-3441	657	13	ideal	ideal	NOUN
ejpam-3441	657	14	a	a	PRON
ejpam-3441	657	15	=	=	X
ejpam-3441	657	16	(	(	PUNCT
ejpam-3441	657	17	µa	µa	PROPN
ejpam-3441	657	18	,	,	PUNCT
ejpam-3441	657	19	γa	γa	PROPN
ejpam-3441	657	20	)	)	PUNCT
ejpam-3441	657	21	with	with	ADP
ejpam-3441	657	22	thresholds	threshold	NOUN
ejpam-3441	657	23	(	(	PUNCT
ejpam-3441	657	24	α	α	X
ejpam-3441	657	25	,	,	PUNCT
ejpam-3441	657	26	β	β	X
ejpam-3441	657	27	]	]	PUNCT
ejpam-3441	657	28	and	and	CCONJ
ejpam-3441	657	29	every	every	DET
ejpam-3441	657	30	intuitionistic	intuitionistic	ADJ
ejpam-3441	657	31	fuzzy	fuzzy	ADJ
ejpam-3441	657	32	right	right	ADJ
ejpam-3441	657	33	ideal	ideal	NOUN
ejpam-3441	658	1	b	b	PROPN
ejpam-3441	659	1	=	=	PUNCT
ejpam-3441	659	2	(	(	PUNCT
ejpam-3441	659	3	µb	µb	PROPN
ejpam-3441	659	4	,	,	PUNCT
ejpam-3441	659	5	γb	γb	PROPN
ejpam-3441	659	6	)	)	PUNCT
ejpam-3441	659	7	with	with	ADP
ejpam-3441	659	8	thresholds	threshold	NOUN
ejpam-3441	659	9	(	(	PUNCT
ejpam-3441	659	10	α	α	X
ejpam-3441	659	11	,	,	PUNCT
ejpam-3441	659	12	β	β	X
ejpam-3441	659	13	]	]	PUNCT
ejpam-3441	659	14	of	of	ADP
ejpam-3441	659	15	r.	r.	PROPN
ejpam-3441	659	16	proof	proof	PROPN
ejpam-3441	659	17	.	.	PUNCT
ejpam-3441	659	18	suppose	suppose	VERB
ejpam-3441	659	19	that	that	SCONJ
ejpam-3441	659	20	a	a	DET
ejpam-3441	659	21	=	=	SYM
ejpam-3441	659	22	(	(	PUNCT
ejpam-3441	659	23	µa	µa	PROPN
ejpam-3441	659	24	,	,	PUNCT
ejpam-3441	659	25	γa	γa	PROPN
ejpam-3441	659	26	)	)	PUNCT
ejpam-3441	659	27	is	be	AUX
ejpam-3441	659	28	an	an	DET
ejpam-3441	659	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	659	30	fuzzy	fuzzy	ADJ
ejpam-3441	659	31	left	leave	VERB
ejpam-3441	659	32	ideal	ideal	NOUN
ejpam-3441	659	33	with	with	ADP
ejpam-3441	659	34	thresholds	threshold	NOUN
ejpam-3441	659	35	(	(	PUNCT
ejpam-3441	659	36	α	α	X
ejpam-3441	659	37	,	,	PUNCT
ejpam-3441	659	38	β	β	X
ejpam-3441	659	39	]	]	PUNCT
ejpam-3441	659	40	and	and	CCONJ
ejpam-3441	659	41	b	b	X
ejpam-3441	659	42	=	=	SYM
ejpam-3441	659	43	(	(	PUNCT
ejpam-3441	659	44	µb	µb	PROPN
ejpam-3441	659	45	,	,	PUNCT
ejpam-3441	659	46	γb	γb	PROPN
ejpam-3441	659	47	)	)	PUNCT
ejpam-3441	659	48	is	be	AUX
ejpam-3441	659	49	an	an	DET
ejpam-3441	659	50	intuitionistic	intuitionistic	ADJ
ejpam-3441	659	51	fuzzy	fuzzy	ADJ
ejpam-3441	659	52	right	right	ADJ
ejpam-3441	659	53	ideal	ideal	NOUN
ejpam-3441	659	54	with	with	ADP
ejpam-3441	659	55	thresholds	threshold	NOUN
ejpam-3441	659	56	(	(	PUNCT
ejpam-3441	659	57	α	α	X
ejpam-3441	659	58	,	,	PUNCT
ejpam-3441	659	59	β	β	X
ejpam-3441	659	60	]	]	PUNCT
ejpam-3441	659	61	of	of	ADP
ejpam-3441	659	62	r.	r.	PROPN
ejpam-3441	659	63	let	let	VERB
ejpam-3441	659	64	x	x	X
ejpam-3441	659	65	∈	∈	PROPN
ejpam-3441	659	66	r	r	NOUN
ejpam-3441	659	67	,	,	PUNCT
ejpam-3441	659	68	this	this	PRON
ejpam-3441	659	69	implies	imply	VERB
ejpam-3441	659	70	that	that	SCONJ
ejpam-3441	659	71	there	there	PRON
ejpam-3441	659	72	exist	exist	VERB
ejpam-3441	659	73	ai	ai	NOUN
ejpam-3441	659	74	,	,	PUNCT
ejpam-3441	659	75	bi	bi	NOUN
ejpam-3441	659	76	∈	∈	PROPN
ejpam-3441	659	77	r	r	NOUN
ejpam-3441	659	78	such	such	ADJ
ejpam-3441	659	79	that	that	SCONJ
ejpam-3441	659	80	x	x	NOUN
ejpam-3441	660	1	=	=	PUNCT
ejpam-3441	660	2	∑n	∑n	PROPN
ejpam-3441	660	3	i=1(aix	i=1(aix	PROPN
ejpam-3441	660	4	2)bi	2)bi	NUM
ejpam-3441	660	5	.	.	PUNCT
ejpam-3441	661	1	now	now	ADV
ejpam-3441	661	2	x	x	X
ejpam-3441	661	3	=	=	SYM
ejpam-3441	661	4	(	(	PUNCT
ejpam-3441	661	5	aix	aix	NOUN
ejpam-3441	661	6	2)bi	2)bi	NUM
ejpam-3441	661	7	=	=	SYM
ejpam-3441	661	8	(	(	PUNCT
ejpam-3441	661	9	ai(xx))bi	ai(xx))bi	NOUN
ejpam-3441	661	10	=	=	PUNCT
ejpam-3441	661	11	(	(	PUNCT
ejpam-3441	661	12	x(aix))bi	x(aix))bi	PROPN
ejpam-3441	661	13	=	=	SYM
ejpam-3441	661	14	(	(	PUNCT
ejpam-3441	661	15	x(aix))(ebi	x(aix))(ebi	PROPN
ejpam-3441	661	16	)	)	PUNCT
ejpam-3441	661	17	=	=	SYM
ejpam-3441	661	18	(	(	PUNCT
ejpam-3441	661	19	xe)((aix)bi	xe)((aix)bi	PROPN
ejpam-3441	661	20	)	)	PUNCT
ejpam-3441	662	1	=	=	PRON
ejpam-3441	662	2	(	(	PUNCT
ejpam-3441	662	3	aix)((xe)bi	aix)((xe)bi	NOUN
ejpam-3441	662	4	)	)	PUNCT
ejpam-3441	662	5	.	.	PUNCT
ejpam-3441	663	1	thus	thus	ADV
ejpam-3441	663	2	(	(	PUNCT
ejpam-3441	663	3	µa	µa	ADP
ejpam-3441	663	4	◦	◦	NOUN
ejpam-3441	663	5	βα	βα	NOUN
ejpam-3441	663	6	µb)(x	µb)(x	NOUN
ejpam-3441	663	7	)	)	PUNCT
ejpam-3441	663	8	=	=	PRON
ejpam-3441	663	9	{	{	PUNCT
ejpam-3441	663	10	(	(	PUNCT
ejpam-3441	663	11	µa	µa	ADP
ejpam-3441	663	12	◦	◦	NOUN
ejpam-3441	663	13	µb)(x	µb)(x	NOUN
ejpam-3441	663	14	)	)	PUNCT
ejpam-3441	663	15	∧	∧	PROPN
ejpam-3441	663	16	β	β	NOUN
ejpam-3441	663	17	}	}	PUNCT
ejpam-3441	663	18	∨	∨	NUM
ejpam-3441	663	19	α	α	NOUN
ejpam-3441	663	20	=	=	X
ejpam-3441	663	21	{	{	PUNCT
ejpam-3441	663	22	(	(	PUNCT
ejpam-3441	663	23	∨x=∑n	∨x=∑n	NUM
ejpam-3441	663	24	i=1	i=1	PROPN
ejpam-3441	663	25	piqi	piqi	NOUN
ejpam-3441	663	26	{	{	PUNCT
ejpam-3441	663	27	∧ni=1	∧ni=1	X
ejpam-3441	663	28	{	{	PUNCT
ejpam-3441	663	29	µa	µa	X
ejpam-3441	663	30	(	(	PUNCT
ejpam-3441	663	31	pi	pi	NOUN
ejpam-3441	663	32	)	)	PUNCT
ejpam-3441	663	33	∧	∧	PROPN
ejpam-3441	663	34	µb	µb	PROPN
ejpam-3441	663	35	(	(	PUNCT
ejpam-3441	663	36	qi	qi	NOUN
ejpam-3441	663	37	)	)	PUNCT
ejpam-3441	663	38	}	}	PUNCT
ejpam-3441	663	39	}	}	PUNCT
ejpam-3441	663	40	)	)	PUNCT
ejpam-3441	664	1	∧	∧	PROPN
ejpam-3441	664	2	β	β	NOUN
ejpam-3441	664	3	}	}	PUNCT
ejpam-3441	664	4	∨	∨	NUM
ejpam-3441	664	5	α	α	PROPN
ejpam-3441	664	6	≥	≥	NUM
ejpam-3441	664	7	{	{	PUNCT
ejpam-3441	664	8	{	{	PUNCT
ejpam-3441	664	9	µa(aix	µa(aix	ADJ
ejpam-3441	664	10	)	)	PUNCT
ejpam-3441	664	11	∧	∧	PROPN
ejpam-3441	664	12	µb	µb	NOUN
ejpam-3441	664	13	(	(	PUNCT
ejpam-3441	664	14	(	(	PUNCT
ejpam-3441	664	15	xe)bi	xe)bi	NOUN
ejpam-3441	664	16	)	)	PUNCT
ejpam-3441	664	17	}	}	PUNCT
ejpam-3441	664	18	∧	∧	PROPN
ejpam-3441	664	19	β	β	NOUN
ejpam-3441	664	20	}	}	PUNCT
ejpam-3441	664	21	∨	∨	NUM
ejpam-3441	664	22	α	α	NOUN
ejpam-3441	664	23	=	=	SYM
ejpam-3441	664	24	(	(	PUNCT
ejpam-3441	664	25	µa	µa	INTJ
ejpam-3441	664	26	(	(	PUNCT
ejpam-3441	664	27	aix	aix	PROPN
ejpam-3441	664	28	)	)	PUNCT
ejpam-3441	664	29	∨	∨	NUM
ejpam-3441	664	30	α	α	NOUN
ejpam-3441	664	31	)	)	PUNCT
ejpam-3441	664	32	∧	∧	NOUN
ejpam-3441	664	33	(	(	PUNCT
ejpam-3441	664	34	µb	µb	PROPN
ejpam-3441	664	35	(	(	PUNCT
ejpam-3441	664	36	(	(	PUNCT
ejpam-3441	664	37	xe)bi	xe)bi	PROPN
ejpam-3441	664	38	)	)	PUNCT
ejpam-3441	664	39	∨	∨	NUM
ejpam-3441	664	40	α	α	NOUN
ejpam-3441	664	41	)	)	PUNCT
ejpam-3441	664	42	∧	∧	PROPN
ejpam-3441	664	43	(	(	PUNCT
ejpam-3441	664	44	β	β	X
ejpam-3441	664	45	∨	∨	NUM
ejpam-3441	664	46	α	α	NOUN
ejpam-3441	664	47	)	)	PUNCT
ejpam-3441	664	48	≥	≥	NOUN
ejpam-3441	664	49	(	(	PUNCT
ejpam-3441	664	50	µa	µa	PROPN
ejpam-3441	664	51	(	(	PUNCT
ejpam-3441	664	52	x	x	NOUN
ejpam-3441	664	53	)	)	PUNCT
ejpam-3441	664	54	∧	∧	PROPN
ejpam-3441	664	55	β	β	NOUN
ejpam-3441	664	56	)	)	PUNCT
ejpam-3441	664	57	∧	∧	PROPN
ejpam-3441	664	58	(	(	PUNCT
ejpam-3441	664	59	µb	µb	PROPN
ejpam-3441	664	60	(	(	PUNCT
ejpam-3441	664	61	x	x	NOUN
ejpam-3441	664	62	)	)	PUNCT
ejpam-3441	664	63	∧	∧	PROPN
ejpam-3441	664	64	β	β	NOUN
ejpam-3441	664	65	)	)	PUNCT
ejpam-3441	664	66	∧	∧	NOUN
ejpam-3441	664	67	β	β	X
ejpam-3441	664	68	=	=	SYM
ejpam-3441	664	69	µa	µa	X
ejpam-3441	664	70	(	(	PUNCT
ejpam-3441	664	71	x	x	X
ejpam-3441	664	72	)	)	PUNCT
ejpam-3441	664	73	∧	∧	NOUN
ejpam-3441	664	74	µb	µb	ADP
ejpam-3441	664	75	(	(	PUNCT
ejpam-3441	664	76	x	x	NOUN
ejpam-3441	664	77	)	)	PUNCT
ejpam-3441	664	78	∧	∧	NOUN
ejpam-3441	664	79	β	β	X
ejpam-3441	664	80	=	=	PUNCT
ejpam-3441	664	81	µb	µb	PROPN
ejpam-3441	664	82	(	(	PUNCT
ejpam-3441	664	83	x	x	NOUN
ejpam-3441	664	84	)	)	PUNCT
ejpam-3441	664	85	∧	∧	NOUN
ejpam-3441	664	86	µa	µa	INTJ
ejpam-3441	664	87	(	(	PUNCT
ejpam-3441	664	88	x	x	X
ejpam-3441	664	89	)	)	PUNCT
ejpam-3441	664	90	∧	∧	NOUN
ejpam-3441	664	91	β	β	X
ejpam-3441	664	92	=	=	SYM
ejpam-3441	664	93	{	{	PUNCT
ejpam-3441	664	94	(	(	PUNCT
ejpam-3441	664	95	µb	µb	ADP
ejpam-3441	664	96	∧	∧	PROPN
ejpam-3441	664	97	µa	µa	NOUN
ejpam-3441	664	98	)	)	PUNCT
ejpam-3441	664	99	(	(	PUNCT
ejpam-3441	664	100	x	x	X
ejpam-3441	664	101	)	)	PUNCT
ejpam-3441	664	102	∧	∧	NOUN
ejpam-3441	664	103	β	β	X
ejpam-3441	664	104	=	=	SYM
ejpam-3441	664	105	{	{	PUNCT
ejpam-3441	664	106	(	(	PUNCT
ejpam-3441	664	107	µb	µb	ADP
ejpam-3441	664	108	∧	∧	PROPN
ejpam-3441	664	109	µa	µa	NOUN
ejpam-3441	664	110	)	)	PUNCT
ejpam-3441	664	111	(	(	PUNCT
ejpam-3441	664	112	x	x	X
ejpam-3441	664	113	)	)	PUNCT
ejpam-3441	664	114	∧	∧	PROPN
ejpam-3441	664	115	β	β	NOUN
ejpam-3441	664	116	}	}	PUNCT
ejpam-3441	664	117	∨	∨	NUM
ejpam-3441	664	118	α	α	NOUN
ejpam-3441	664	119	=	=	PUNCT
ejpam-3441	664	120	(	(	PUNCT
ejpam-3441	664	121	µb	µb	VERB
ejpam-3441	664	122	∧βα	∧βα	PROPN
ejpam-3441	664	123	µa)(x	µa)(x	NOUN
ejpam-3441	664	124	)	)	PUNCT
ejpam-3441	664	125	.	.	PUNCT
ejpam-3441	665	1	⇒	⇒	NOUN
ejpam-3441	665	2	µb	µb	VERB
ejpam-3441	665	3	∧βα	∧βα	PROPN
ejpam-3441	665	4	µa	µa	ADP
ejpam-3441	665	5	⊆	⊆	NUM
ejpam-3441	665	6	µa	µa	NOUN
ejpam-3441	665	7	◦	◦	NOUN
ejpam-3441	665	8	βα	βα	NOUN
ejpam-3441	665	9	µb	µb	PROPN
ejpam-3441	665	10	.	.	PUNCT
ejpam-3441	666	1	similarly	similarly	ADV
ejpam-3441	666	2	,	,	PUNCT
ejpam-3441	666	3	we	we	PRON
ejpam-3441	666	4	have	have	AUX
ejpam-3441	666	5	γb	γb	VERB
ejpam-3441	666	6	∨βα	∨βα	PROPN
ejpam-3441	666	7	γa	γa	PROPN
ejpam-3441	666	8	⊇	⊇	PROPN
ejpam-3441	666	9	γa	γa	PROPN
ejpam-3441	666	10	◦	◦	PROPN
ejpam-3441	666	11	βα	βα	NOUN
ejpam-3441	666	12	γb	γb	PROPN
ejpam-3441	666	13	.	.	PUNCT
ejpam-3441	667	1	hence	hence	ADV
ejpam-3441	667	2	b	b	X
ejpam-3441	667	3	∧βα	∧βα	DET
ejpam-3441	667	4	a	a	DET
ejpam-3441	667	5	⊆	⊆	NUM
ejpam-3441	667	6	a	a	DET
ejpam-3441	667	7	◦	◦	NOUN
ejpam-3441	667	8	βα	βα	PROPN
ejpam-3441	668	1	b.	b.	PROPN
ejpam-3441	668	2	k.	k.	PROPN
ejpam-3441	668	3	nasreen	nasreen	PROPN
ejpam-3441	668	4	et	et	PROPN
ejpam-3441	668	5	al	al	PROPN
ejpam-3441	668	6	.	.	PUNCT
ejpam-3441	668	7	/	/	SYM
ejpam-3441	668	8	eur	eur	PROPN
ejpam-3441	668	9	.	.	PUNCT
ejpam-3441	669	1	j.	j.	PROPN
ejpam-3441	669	2	pure	pure	PROPN
ejpam-3441	669	3	appl	appl	PROPN
ejpam-3441	669	4	.	.	PROPN
ejpam-3441	669	5	math	math	PROPN
ejpam-3441	669	6	,	,	PUNCT
ejpam-3441	669	7	12	12	NUM
ejpam-3441	669	8	(	(	PUNCT
ejpam-3441	669	9	3	3	NUM
ejpam-3441	669	10	)	)	PUNCT
ejpam-3441	669	11	(	(	PUNCT
ejpam-3441	669	12	2019	2019	NUM
ejpam-3441	669	13	)	)	PUNCT
ejpam-3441	669	14	,	,	PUNCT
ejpam-3441	669	15	906	906	NUM
ejpam-3441	669	16	-	-	SYM
ejpam-3441	669	17	943	943	NUM
ejpam-3441	669	18	935	935	NUM
ejpam-3441	669	19	theorem	theorem	NOUN
ejpam-3441	669	20	13	13	NUM
ejpam-3441	669	21	.	.	PUNCT
ejpam-3441	670	1	let	let	VERB
ejpam-3441	670	2	r	r	PRON
ejpam-3441	670	3	be	be	AUX
ejpam-3441	670	4	an	an	DET
ejpam-3441	670	5	la	la	NOUN
ejpam-3441	670	6	-	-	NOUN
ejpam-3441	670	7	ring	ring	NOUN
ejpam-3441	670	8	with	with	ADP
ejpam-3441	670	9	left	left	ADJ
ejpam-3441	670	10	identity	identity	NOUN
ejpam-3441	670	11	e	e	NOUN
ejpam-3441	670	12	,	,	PUNCT
ejpam-3441	670	13	such	such	ADJ
ejpam-3441	670	14	that	that	SCONJ
ejpam-3441	670	15	(	(	PUNCT
ejpam-3441	670	16	xe)r	xe)r	PROPN
ejpam-3441	670	17	=	=	SYM
ejpam-3441	670	18	xr	xr	PROPN
ejpam-3441	670	19	for	for	ADP
ejpam-3441	670	20	all	all	DET
ejpam-3441	670	21	x	x	PROPN
ejpam-3441	670	22	∈	∈	PROPN
ejpam-3441	670	23	r.	r.	NOUN
ejpam-3441	670	24	then	then	ADV
ejpam-3441	670	25	the	the	DET
ejpam-3441	670	26	following	follow	VERB
ejpam-3441	670	27	conditions	condition	NOUN
ejpam-3441	670	28	are	be	AUX
ejpam-3441	670	29	equivalent	equivalent	ADJ
ejpam-3441	670	30	.	.	PUNCT
ejpam-3441	671	1	(	(	PUNCT
ejpam-3441	671	2	1	1	X
ejpam-3441	671	3	)	)	PUNCT
ejpam-3441	671	4	r	r	NOUN
ejpam-3441	671	5	is	be	AUX
ejpam-3441	671	6	an	an	DET
ejpam-3441	671	7	intra	intra	ADJ
ejpam-3441	671	8	-	-	ADJ
ejpam-3441	671	9	regular	regular	ADJ
ejpam-3441	671	10	.	.	PUNCT
ejpam-3441	672	1	(	(	PUNCT
ejpam-3441	672	2	2	2	X
ejpam-3441	672	3	)	)	PUNCT
ejpam-3441	672	4	b∧βαa	b∧βαa	NOUN
ejpam-3441	672	5	⊆	⊆	NUM
ejpam-3441	672	6	a	a	DET
ejpam-3441	672	7	◦	◦	NOUN
ejpam-3441	672	8	βαb	βαb	NOUN
ejpam-3441	672	9	for	for	ADP
ejpam-3441	672	10	every	every	DET
ejpam-3441	672	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	672	12	fuzzy	fuzzy	ADJ
ejpam-3441	672	13	left	leave	VERB
ejpam-3441	672	14	ideal	ideal	NOUN
ejpam-3441	672	15	a	a	PRON
ejpam-3441	672	16	with	with	ADP
ejpam-3441	672	17	thresholds	threshold	NOUN
ejpam-3441	672	18	(	(	PUNCT
ejpam-3441	672	19	α	α	X
ejpam-3441	672	20	,	,	PUNCT
ejpam-3441	672	21	β	β	X
ejpam-3441	672	22	]	]	PUNCT
ejpam-3441	672	23	and	and	CCONJ
ejpam-3441	672	24	every	every	DET
ejpam-3441	672	25	intuitionistic	intuitionistic	ADJ
ejpam-3441	672	26	fuzzy	fuzzy	ADJ
ejpam-3441	672	27	right	right	ADJ
ejpam-3441	672	28	ideal	ideal	PROPN
ejpam-3441	672	29	b	b	PROPN
ejpam-3441	672	30	with	with	ADP
ejpam-3441	672	31	thresholds	threshold	NOUN
ejpam-3441	672	32	(	(	PUNCT
ejpam-3441	672	33	α	α	X
ejpam-3441	672	34	,	,	PUNCT
ejpam-3441	672	35	β	β	X
ejpam-3441	672	36	]	]	PUNCT
ejpam-3441	672	37	of	of	ADP
ejpam-3441	672	38	r.	r.	PROPN
ejpam-3441	672	39	proof	proof	NOUN
ejpam-3441	672	40	.	.	PUNCT
ejpam-3441	673	1	(	(	PUNCT
ejpam-3441	673	2	1	1	X
ejpam-3441	673	3	)	)	PUNCT
ejpam-3441	673	4	⇒	⇒	NOUN
ejpam-3441	673	5	(	(	PUNCT
ejpam-3441	673	6	2	2	NUM
ejpam-3441	673	7	)	)	PUNCT
ejpam-3441	673	8	,	,	PUNCT
ejpam-3441	673	9	is	be	AUX
ejpam-3441	673	10	true	true	ADJ
ejpam-3441	673	11	by	by	ADP
ejpam-3441	673	12	the	the	DET
ejpam-3441	673	13	lemma	lemma	PROPN
ejpam-3441	673	14	23	23	NUM
ejpam-3441	673	15	.	.	PUNCT
ejpam-3441	674	1	assume	assume	VERB
ejpam-3441	674	2	that	that	SCONJ
ejpam-3441	674	3	(	(	PUNCT
ejpam-3441	674	4	2	2	X
ejpam-3441	674	5	)	)	PUNCT
ejpam-3441	674	6	holds	hold	NOUN
ejpam-3441	674	7	and	and	CCONJ
ejpam-3441	674	8	a	a	DET
ejpam-3441	674	9	∈	∈	PROPN
ejpam-3441	674	10	r.	r.	NOUN
ejpam-3441	674	11	then	then	ADV
ejpam-3441	674	12	ra	ra	PROPN
ejpam-3441	674	13	is	be	AUX
ejpam-3441	674	14	a	a	DET
ejpam-3441	674	15	left	left	ADJ
ejpam-3441	674	16	ideal	ideal	NOUN
ejpam-3441	674	17	of	of	ADP
ejpam-3441	674	18	r	r	NOUN
ejpam-3441	674	19	containing	contain	VERB
ejpam-3441	674	20	a	a	PRON
ejpam-3441	674	21	by	by	ADP
ejpam-3441	674	22	the	the	DET
ejpam-3441	674	23	lemma	lemma	PROPN
ejpam-3441	674	24	19	19	NUM
ejpam-3441	674	25	and	and	CCONJ
ejpam-3441	674	26	ar	ar	PROPN
ejpam-3441	674	27	∪	∪	PROPN
ejpam-3441	674	28	ra	ra	PROPN
ejpam-3441	674	29	is	be	AUX
ejpam-3441	674	30	a	a	DET
ejpam-3441	674	31	right	right	ADJ
ejpam-3441	674	32	ideal	ideal	NOUN
ejpam-3441	674	33	of	of	ADP
ejpam-3441	674	34	r	r	NOUN
ejpam-3441	674	35	containing	contain	VERB
ejpam-3441	674	36	a	a	PRON
ejpam-3441	674	37	by	by	ADP
ejpam-3441	674	38	the	the	DET
ejpam-3441	674	39	proposition	proposition	NOUN
ejpam-3441	674	40	8	8	NUM
ejpam-3441	674	41	.	.	PUNCT
ejpam-3441	675	1	this	this	PRON
ejpam-3441	675	2	means	mean	VERB
ejpam-3441	675	3	that	that	SCONJ
ejpam-3441	675	4	χra	χra	PROPN
ejpam-3441	675	5	is	be	AUX
ejpam-3441	675	6	an	an	DET
ejpam-3441	675	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	675	8	fuzzy	fuzzy	ADJ
ejpam-3441	675	9	left	leave	VERB
ejpam-3441	675	10	ideal	ideal	NOUN
ejpam-3441	675	11	with	with	ADP
ejpam-3441	675	12	thresholds	threshold	NOUN
ejpam-3441	675	13	(	(	PUNCT
ejpam-3441	675	14	α	α	X
ejpam-3441	675	15	,	,	PUNCT
ejpam-3441	675	16	β	β	X
ejpam-3441	675	17	]	]	PUNCT
ejpam-3441	675	18	and	and	CCONJ
ejpam-3441	675	19	χar∪ra	χar∪ra	PROPN
ejpam-3441	675	20	is	be	AUX
ejpam-3441	675	21	an	an	DET
ejpam-3441	675	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	675	23	fuzzy	fuzzy	ADJ
ejpam-3441	675	24	right	right	ADJ
ejpam-3441	675	25	ideal	ideal	NOUN
ejpam-3441	675	26	with	with	ADP
ejpam-3441	675	27	thresholds	threshold	NOUN
ejpam-3441	675	28	(	(	PUNCT
ejpam-3441	675	29	α	α	X
ejpam-3441	675	30	,	,	PUNCT
ejpam-3441	675	31	β	β	X
ejpam-3441	675	32	]	]	PUNCT
ejpam-3441	675	33	of	of	ADP
ejpam-3441	675	34	r	r	NOUN
ejpam-3441	675	35	,	,	PUNCT
ejpam-3441	675	36	by	by	ADP
ejpam-3441	675	37	the	the	DET
ejpam-3441	675	38	theorem	theorem	NOUN
ejpam-3441	675	39	2	2	NUM
ejpam-3441	675	40	.	.	PUNCT
ejpam-3441	675	41	by	by	ADP
ejpam-3441	675	42	our	our	PRON
ejpam-3441	675	43	assumption	assumption	NOUN
ejpam-3441	675	44	χar∪ra	χar∪ra	ADJ
ejpam-3441	675	45	∧βα	∧βα	ADJ
ejpam-3441	675	46	χra	χra	NUM
ejpam-3441	675	47	⊆	⊆	NUM
ejpam-3441	675	48	χra	χra	NUM
ejpam-3441	675	49	◦	◦	NOUN
ejpam-3441	675	50	βα	βα	NOUN
ejpam-3441	675	51	χar∪ra	χar∪ra	ADV
ejpam-3441	675	52	,	,	PUNCT
ejpam-3441	675	53	i.e.	i.e.	X
ejpam-3441	675	54	,	,	PUNCT
ejpam-3441	675	55	(	(	PUNCT
ejpam-3441	675	56	χ(ar∪ra)∩ra	χ(ar∪ra)∩ra	PROPN
ejpam-3441	675	57	)	)	PUNCT
ejpam-3441	675	58	β	β	PROPN
ejpam-3441	675	59	α	α	NOUN
ejpam-3441	675	60	⊆	⊆	NUM
ejpam-3441	675	61	(	(	PUNCT
ejpam-3441	675	62	χ(ra)(ar∪ra	χ(ra)(ar∪ra	ADV
ejpam-3441	675	63	)	)	PUNCT
ejpam-3441	675	64	)	)	PUNCT
ejpam-3441	676	1	β	β	PROPN
ejpam-3441	676	2	α	α	NOUN
ejpam-3441	676	3	by	by	ADP
ejpam-3441	676	4	the	the	DET
ejpam-3441	676	5	theorem	theorem	NOUN
ejpam-3441	676	6	1	1	NUM
ejpam-3441	676	7	.	.	PUNCT
ejpam-3441	677	1	thus	thus	ADV
ejpam-3441	677	2	(	(	PUNCT
ejpam-3441	677	3	ar∪ra)∩ra	ar∪ra)∩ra	PROPN
ejpam-3441	677	4	⊆	⊆	NUM
ejpam-3441	677	5	ra(ar∪ra	ra(ar∪ra	PROPN
ejpam-3441	677	6	)	)	PUNCT
ejpam-3441	677	7	.	.	PUNCT
ejpam-3441	678	1	since	since	SCONJ
ejpam-3441	678	2	a	a	DET
ejpam-3441	678	3	∈	∈	PROPN
ejpam-3441	678	4	(	(	PUNCT
ejpam-3441	678	5	ar	ar	NOUN
ejpam-3441	678	6	∪	∪	PROPN
ejpam-3441	678	7	ra	ra	PROPN
ejpam-3441	678	8	)	)	PUNCT
ejpam-3441	678	9	∩	∩	PROPN
ejpam-3441	678	10	ra	ra	PROPN
ejpam-3441	678	11	,	,	PUNCT
ejpam-3441	678	12	i.e.	i.e.	X
ejpam-3441	678	13	,	,	PUNCT
ejpam-3441	678	14	a	a	DET
ejpam-3441	678	15	∈	∈	PROPN
ejpam-3441	678	16	ra(ar	ra(ar	NOUN
ejpam-3441	678	17	∪	∪	PROPN
ejpam-3441	678	18	ra	ra	PROPN
ejpam-3441	678	19	)	)	PUNCT
ejpam-3441	678	20	=	=	SYM
ejpam-3441	678	21	(	(	PUNCT
ejpam-3441	678	22	ra)(ar	ra)(ar	NOUN
ejpam-3441	678	23	)	)	PUNCT
ejpam-3441	678	24	∪	∪	X
ejpam-3441	678	25	(	(	PUNCT
ejpam-3441	678	26	ra)(ra	ra)(ra	X
ejpam-3441	678	27	)	)	PUNCT
ejpam-3441	678	28	.	.	PUNCT
ejpam-3441	679	1	this	this	PRON
ejpam-3441	679	2	implies	imply	VERB
ejpam-3441	679	3	that	that	SCONJ
ejpam-3441	679	4	a	a	DET
ejpam-3441	679	5	∈	∈	PROPN
ejpam-3441	679	6	(	(	PUNCT
ejpam-3441	679	7	ra)(ar	ra)(ar	NOUN
ejpam-3441	679	8	)	)	PUNCT
ejpam-3441	679	9	or	or	CCONJ
ejpam-3441	679	10	a	a	DET
ejpam-3441	679	11	∈	∈	PROPN
ejpam-3441	679	12	(	(	PUNCT
ejpam-3441	679	13	ra)(ra	ra)(ra	NOUN
ejpam-3441	679	14	)	)	PUNCT
ejpam-3441	679	15	.	.	PUNCT
ejpam-3441	680	1	if	if	SCONJ
ejpam-3441	680	2	a	a	DET
ejpam-3441	680	3	∈	∈	PROPN
ejpam-3441	680	4	(	(	PUNCT
ejpam-3441	680	5	ra)(ar	ra)(ar	NOUN
ejpam-3441	680	6	)	)	PUNCT
ejpam-3441	680	7	,	,	PUNCT
ejpam-3441	680	8	then	then	ADV
ejpam-3441	680	9	(	(	PUNCT
ejpam-3441	680	10	ra)(ar	ra)(ar	NOUN
ejpam-3441	680	11	)	)	PUNCT
ejpam-3441	680	12	=	=	SYM
ejpam-3441	680	13	(	(	PUNCT
ejpam-3441	680	14	ra)((ea)(rr	ra)((ea)(rr	NOUN
ejpam-3441	680	15	)	)	PUNCT
ejpam-3441	680	16	)	)	PUNCT
ejpam-3441	681	1	=	=	SYM
ejpam-3441	681	2	(	(	PUNCT
ejpam-3441	681	3	ra)((rr)(ae	ra)((rr)(ae	PROPN
ejpam-3441	681	4	)	)	PUNCT
ejpam-3441	681	5	)	)	PUNCT
ejpam-3441	682	1	=	=	PRON
ejpam-3441	682	2	(	(	PUNCT
ejpam-3441	682	3	ra)(((ae)r)r	ra)(((ae)r)r	NOUN
ejpam-3441	682	4	)	)	PUNCT
ejpam-3441	682	5	=	=	SYM
ejpam-3441	682	6	(	(	PUNCT
ejpam-3441	682	7	ra)((ar)r	ra)((ar)r	PROPN
ejpam-3441	682	8	)	)	PUNCT
ejpam-3441	682	9	=	=	SYM
ejpam-3441	682	10	(	(	PUNCT
ejpam-3441	682	11	ra)((rr)a	ra)((rr)a	PROPN
ejpam-3441	682	12	)	)	PUNCT
ejpam-3441	682	13	=	=	SYM
ejpam-3441	682	14	(	(	PUNCT
ejpam-3441	682	15	ra)(ra	ra)(ra	X
ejpam-3441	682	16	)	)	PUNCT
ejpam-3441	682	17	=	=	SYM
ejpam-3441	682	18	(	(	PUNCT
ejpam-3441	682	19	(	(	PUNCT
ejpam-3441	682	20	ra)a)r	ra)a)r	NOUN
ejpam-3441	682	21	=	=	SYM
ejpam-3441	682	22	(	(	PUNCT
ejpam-3441	682	23	(	(	PUNCT
ejpam-3441	682	24	ra)(ea))r	ra)(ea))r	PROPN
ejpam-3441	682	25	=	=	SYM
ejpam-3441	682	26	(	(	PUNCT
ejpam-3441	682	27	(	(	PUNCT
ejpam-3441	682	28	re)(aa))r	re)(aa))r	X
ejpam-3441	682	29	=	=	SYM
ejpam-3441	682	30	(	(	PUNCT
ejpam-3441	682	31	ra2)r	ra2)r	NOUN
ejpam-3441	682	32	.	.	PUNCT
ejpam-3441	683	1	thus	thus	ADV
ejpam-3441	683	2	a	a	DET
ejpam-3441	683	3	∈	∈	NOUN
ejpam-3441	683	4	(	(	PUNCT
ejpam-3441	683	5	ra2)r	ra2)r	NOUN
ejpam-3441	683	6	.	.	PUNCT
ejpam-3441	684	1	if	if	SCONJ
ejpam-3441	684	2	a	a	DET
ejpam-3441	684	3	∈	∈	PROPN
ejpam-3441	684	4	(	(	PUNCT
ejpam-3441	684	5	ra)(ra	ra)(ra	NOUN
ejpam-3441	684	6	)	)	PUNCT
ejpam-3441	684	7	,	,	PUNCT
ejpam-3441	684	8	then	then	ADV
ejpam-3441	684	9	obvious	obvious	VERB
ejpam-3441	684	10	a	a	DET
ejpam-3441	684	11	∈	∈	NOUN
ejpam-3441	684	12	(	(	PUNCT
ejpam-3441	684	13	ra2)r	ra2)r	PROPN
ejpam-3441	684	14	.	.	PUNCT
ejpam-3441	685	1	so	so	ADV
ejpam-3441	685	2	a	a	PRON
ejpam-3441	685	3	is	be	AUX
ejpam-3441	685	4	an	an	DET
ejpam-3441	685	5	intra	intra	ADJ
ejpam-3441	685	6	regular	regular	NOUN
ejpam-3441	685	7	.	.	PUNCT
ejpam-3441	686	1	therefore	therefore	ADV
ejpam-3441	686	2	r	r	NOUN
ejpam-3441	686	3	is	be	AUX
ejpam-3441	686	4	an	an	DET
ejpam-3441	686	5	intra	intra	ADJ
ejpam-3441	686	6	-	-	ADJ
ejpam-3441	686	7	regular	regular	ADJ
ejpam-3441	686	8	,	,	PUNCT
ejpam-3441	686	9	i.e.	i.e.	X
ejpam-3441	686	10	,	,	PUNCT
ejpam-3441	686	11	(	(	PUNCT
ejpam-3441	686	12	2)⇒	2)⇒	NUM
ejpam-3441	686	13	(	(	PUNCT
ejpam-3441	686	14	1	1	NUM
ejpam-3441	686	15	)	)	PUNCT
ejpam-3441	686	16	.	.	PUNCT
ejpam-3441	687	1	theorem	theorem	VERB
ejpam-3441	687	2	14	14	NUM
ejpam-3441	687	3	.	.	PUNCT
ejpam-3441	688	1	let	let	VERB
ejpam-3441	688	2	r	r	PRON
ejpam-3441	688	3	be	be	AUX
ejpam-3441	688	4	an	an	DET
ejpam-3441	688	5	la	la	NOUN
ejpam-3441	688	6	-	-	NOUN
ejpam-3441	688	7	ring	ring	NOUN
ejpam-3441	688	8	with	with	ADP
ejpam-3441	688	9	left	left	ADJ
ejpam-3441	688	10	identity	identity	NOUN
ejpam-3441	688	11	e	e	NOUN
ejpam-3441	688	12	,	,	PUNCT
ejpam-3441	688	13	such	such	ADJ
ejpam-3441	688	14	that	that	SCONJ
ejpam-3441	688	15	(	(	PUNCT
ejpam-3441	688	16	xe)r	xe)r	PROPN
ejpam-3441	688	17	=	=	SYM
ejpam-3441	688	18	xr	xr	PROPN
ejpam-3441	688	19	for	for	ADP
ejpam-3441	688	20	all	all	DET
ejpam-3441	688	21	x	x	PROPN
ejpam-3441	688	22	∈	∈	PROPN
ejpam-3441	688	23	r.	r.	NOUN
ejpam-3441	688	24	then	then	ADV
ejpam-3441	688	25	the	the	DET
ejpam-3441	688	26	following	follow	VERB
ejpam-3441	688	27	conditions	condition	NOUN
ejpam-3441	688	28	are	be	AUX
ejpam-3441	688	29	equivalent	equivalent	ADJ
ejpam-3441	688	30	.	.	PUNCT
ejpam-3441	689	1	(	(	PUNCT
ejpam-3441	689	2	1	1	X
ejpam-3441	689	3	)	)	PUNCT
ejpam-3441	689	4	r	r	NOUN
ejpam-3441	689	5	is	be	AUX
ejpam-3441	689	6	an	an	DET
ejpam-3441	689	7	intra	intra	ADJ
ejpam-3441	689	8	-	-	ADJ
ejpam-3441	689	9	regular	regular	ADJ
ejpam-3441	689	10	.	.	PUNCT
ejpam-3441	690	1	(	(	PUNCT
ejpam-3441	690	2	2	2	X
ejpam-3441	690	3	)	)	PUNCT
ejpam-3441	690	4	a	a	DET
ejpam-3441	690	5	∧βα	∧βα	NOUN
ejpam-3441	690	6	i	i	NOUN
ejpam-3441	690	7	=	=	PUNCT
ejpam-3441	690	8	(	(	PUNCT
ejpam-3441	690	9	a	a	DET
ejpam-3441	690	10	◦	◦	NOUN
ejpam-3441	690	11	βα	βα	X
ejpam-3441	690	12	i	i	NOUN
ejpam-3441	690	13	)	)	PUNCT
ejpam-3441	690	14	◦	◦	NOUN
ejpam-3441	690	15	βα	βα	NOUN
ejpam-3441	690	16	a	a	PRON
ejpam-3441	690	17	for	for	ADP
ejpam-3441	690	18	every	every	DET
ejpam-3441	690	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	690	20	fuzzy	fuzzy	ADJ
ejpam-3441	690	21	quasi	quasi	NOUN
ejpam-3441	690	22	-	-	NOUN
ejpam-3441	690	23	ideal	ideal	ADJ
ejpam-3441	690	24	a	a	PRON
ejpam-3441	690	25	with	with	ADP
ejpam-3441	690	26	thresholds	threshold	NOUN
ejpam-3441	690	27	(	(	PUNCT
ejpam-3441	690	28	α	α	X
ejpam-3441	690	29	,	,	PUNCT
ejpam-3441	690	30	β	β	X
ejpam-3441	690	31	]	]	PUNCT
ejpam-3441	690	32	and	and	CCONJ
ejpam-3441	690	33	every	every	DET
ejpam-3441	690	34	intuitionistic	intuitionistic	ADJ
ejpam-3441	690	35	fuzzy	fuzzy	ADJ
ejpam-3441	690	36	ideal	ideal	NOUN
ejpam-3441	690	37	i	i	PRON
ejpam-3441	690	38	with	with	ADP
ejpam-3441	690	39	thresholds	threshold	NOUN
ejpam-3441	690	40	(	(	PUNCT
ejpam-3441	690	41	α	α	X
ejpam-3441	690	42	,	,	PUNCT
ejpam-3441	690	43	β	β	X
ejpam-3441	690	44	]	]	PUNCT
ejpam-3441	690	45	of	of	ADP
ejpam-3441	690	46	r.	r.	PROPN
ejpam-3441	690	47	(	(	PUNCT
ejpam-3441	690	48	3	3	NUM
ejpam-3441	690	49	)	)	PUNCT
ejpam-3441	690	50	b	b	NOUN
ejpam-3441	691	1	∧βα	∧βα	NOUN
ejpam-3441	691	2	i	i	NOUN
ejpam-3441	691	3	=	=	PUNCT
ejpam-3441	691	4	(	(	PUNCT
ejpam-3441	691	5	b	b	X
ejpam-3441	691	6	◦	◦	NOUN
ejpam-3441	691	7	βα	βα	NOUN
ejpam-3441	691	8	i	i	NOUN
ejpam-3441	691	9	)	)	PUNCT
ejpam-3441	691	10	◦	◦	PROPN
ejpam-3441	691	11	βα	βα	NOUN
ejpam-3441	691	12	b	b	NOUN
ejpam-3441	691	13	for	for	ADP
ejpam-3441	691	14	every	every	DET
ejpam-3441	691	15	intuitionistic	intuitionistic	ADJ
ejpam-3441	691	16	fuzzy	fuzzy	ADJ
ejpam-3441	691	17	bi	bi	ADJ
ejpam-3441	691	18	-	-	ADJ
ejpam-3441	691	19	ideal	ideal	ADJ
ejpam-3441	691	20	b	b	PROPN
ejpam-3441	691	21	with	with	ADP
ejpam-3441	691	22	thresholds	threshold	NOUN
ejpam-3441	691	23	(	(	PUNCT
ejpam-3441	691	24	α	α	X
ejpam-3441	691	25	,	,	PUNCT
ejpam-3441	691	26	β	β	X
ejpam-3441	691	27	]	]	PUNCT
ejpam-3441	691	28	and	and	CCONJ
ejpam-3441	691	29	every	every	DET
ejpam-3441	691	30	intuitionistic	intuitionistic	ADJ
ejpam-3441	691	31	fuzzy	fuzzy	ADJ
ejpam-3441	691	32	ideal	ideal	NOUN
ejpam-3441	691	33	i	i	PRON
ejpam-3441	691	34	with	with	ADP
ejpam-3441	691	35	thresholds	threshold	NOUN
ejpam-3441	691	36	(	(	PUNCT
ejpam-3441	691	37	α	α	X
ejpam-3441	691	38	,	,	PUNCT
ejpam-3441	691	39	β	β	X
ejpam-3441	691	40	]	]	PUNCT
ejpam-3441	691	41	of	of	ADP
ejpam-3441	691	42	r.	r.	PROPN
ejpam-3441	691	43	(	(	PUNCT
ejpam-3441	691	44	4	4	NUM
ejpam-3441	691	45	)	)	PUNCT
ejpam-3441	691	46	c	c	NOUN
ejpam-3441	692	1	∧βα	∧βα	ADJ
ejpam-3441	692	2	i	i	NOUN
ejpam-3441	692	3	=	=	PUNCT
ejpam-3441	692	4	(	(	PUNCT
ejpam-3441	692	5	c	c	NOUN
ejpam-3441	692	6	◦	◦	NOUN
ejpam-3441	692	7	βα	βα	X
ejpam-3441	692	8	i	i	NOUN
ejpam-3441	692	9	)	)	PUNCT
ejpam-3441	692	10	◦	◦	NOUN
ejpam-3441	692	11	βα	βα	X
ejpam-3441	692	12	c	c	NOUN
ejpam-3441	692	13	for	for	ADP
ejpam-3441	692	14	every	every	DET
ejpam-3441	692	15	intuitionistic	intuitionistic	ADJ
ejpam-3441	692	16	fuzzy	fuzzy	ADJ
ejpam-3441	692	17	generalized	generalize	VERB
ejpam-3441	692	18	bi	bi	ADJ
ejpam-3441	692	19	-	-	ADJ
ejpam-3441	692	20	ideal	ideal	ADJ
ejpam-3441	692	21	c	c	NOUN
ejpam-3441	692	22	with	with	ADP
ejpam-3441	692	23	thresholds	threshold	NOUN
ejpam-3441	692	24	(	(	PUNCT
ejpam-3441	692	25	α	α	X
ejpam-3441	692	26	,	,	PUNCT
ejpam-3441	692	27	β	β	X
ejpam-3441	692	28	]	]	PUNCT
ejpam-3441	692	29	and	and	CCONJ
ejpam-3441	692	30	every	every	DET
ejpam-3441	692	31	intuitionistic	intuitionistic	ADJ
ejpam-3441	692	32	fuzzy	fuzzy	ADJ
ejpam-3441	692	33	ideal	ideal	NOUN
ejpam-3441	692	34	i	i	PRON
ejpam-3441	692	35	with	with	ADP
ejpam-3441	692	36	thresholds	threshold	NOUN
ejpam-3441	692	37	(	(	PUNCT
ejpam-3441	692	38	α	α	X
ejpam-3441	692	39	,	,	PUNCT
ejpam-3441	692	40	β	β	X
ejpam-3441	692	41	]	]	PUNCT
ejpam-3441	692	42	of	of	ADP
ejpam-3441	692	43	r.	r.	PROPN
ejpam-3441	692	44	proof	proof	NOUN
ejpam-3441	692	45	.	.	PUNCT
ejpam-3441	693	1	consider	consider	VERB
ejpam-3441	693	2	that	that	PRON
ejpam-3441	693	3	(	(	PUNCT
ejpam-3441	693	4	1	1	X
ejpam-3441	693	5	)	)	PUNCT
ejpam-3441	693	6	holds	hold	VERB
ejpam-3441	693	7	.	.	PUNCT
ejpam-3441	694	1	let	let	VERB
ejpam-3441	694	2	c	c	NOUN
ejpam-3441	694	3	=	=	PUNCT
ejpam-3441	694	4	(	(	PUNCT
ejpam-3441	694	5	µc	µc	INTJ
ejpam-3441	694	6	,	,	PUNCT
ejpam-3441	694	7	γc	γc	PROPN
ejpam-3441	694	8	)	)	PUNCT
ejpam-3441	694	9	be	be	AUX
ejpam-3441	694	10	an	an	DET
ejpam-3441	694	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	694	12	fuzzy	fuzzy	ADJ
ejpam-3441	694	13	generalized	generalize	VERB
ejpam-3441	694	14	bi	bi	NOUN
ejpam-3441	694	15	-	-	NOUN
ejpam-3441	694	16	ideal	ideal	NOUN
ejpam-3441	694	17	with	with	ADP
ejpam-3441	694	18	thresholds	threshold	NOUN
ejpam-3441	694	19	(	(	PUNCT
ejpam-3441	694	20	α	α	X
ejpam-3441	694	21	,	,	PUNCT
ejpam-3441	694	22	β	β	X
ejpam-3441	694	23	]	]	PUNCT
ejpam-3441	695	1	and	and	CCONJ
ejpam-3441	695	2	i	i	PRON
ejpam-3441	695	3	=	=	PUNCT
ejpam-3441	695	4	(	(	PUNCT
ejpam-3441	695	5	µi	µi	INTJ
ejpam-3441	695	6	,	,	PUNCT
ejpam-3441	695	7	γi	γi	INTJ
ejpam-3441	695	8	)	)	PUNCT
ejpam-3441	695	9	be	be	VERB
ejpam-3441	695	10	an	an	DET
ejpam-3441	695	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	695	12	fuzzy	fuzzy	ADJ
ejpam-3441	695	13	ideal	ideal	NOUN
ejpam-3441	695	14	with	with	ADP
ejpam-3441	695	15	thresholds	threshold	NOUN
ejpam-3441	695	16	(	(	PUNCT
ejpam-3441	695	17	α	α	X
ejpam-3441	695	18	,	,	PUNCT
ejpam-3441	695	19	β	β	X
ejpam-3441	695	20	]	]	PUNCT
ejpam-3441	695	21	of	of	ADP
ejpam-3441	695	22	r.	r.	PROPN
ejpam-3441	695	23	now	now	ADV
ejpam-3441	695	24	(	(	PUNCT
ejpam-3441	695	25	c	c	NOUN
ejpam-3441	695	26	◦	◦	NOUN
ejpam-3441	695	27	βα	βα	X
ejpam-3441	695	28	i	i	NOUN
ejpam-3441	695	29	)	)	PUNCT
ejpam-3441	695	30	◦	◦	NOUN
ejpam-3441	695	31	βα	βα	X
ejpam-3441	696	1	c	c	NOUN
ejpam-3441	696	2	⊆	⊆	NUM
ejpam-3441	696	3	(	(	PUNCT
ejpam-3441	696	4	r	r	NOUN
ejpam-3441	696	5	◦	◦	NOUN
ejpam-3441	696	6	βα	βα	X
ejpam-3441	696	7	i	i	NOUN
ejpam-3441	696	8	)	)	PUNCT
ejpam-3441	696	9	◦	◦	AUX
ejpam-3441	696	10	βα	βα	NOUN
ejpam-3441	696	11	r	r	NOUN
ejpam-3441	696	12	⊆	⊆	NUM
ejpam-3441	696	13	i	i	SYM
ejpam-3441	696	14	◦	◦	NOUN
ejpam-3441	696	15	βα	βα	NOUN
ejpam-3441	696	16	r	r	NOUN
ejpam-3441	696	17	⊆	⊆	NUM
ejpam-3441	696	18	iβα	iβα	NOUN
ejpam-3441	696	19	and	and	CCONJ
ejpam-3441	696	20	(	(	PUNCT
ejpam-3441	696	21	c	c	PROPN
ejpam-3441	696	22	◦	◦	NOUN
ejpam-3441	696	23	βα	βα	X
ejpam-3441	696	24	i	i	NOUN
ejpam-3441	696	25	)	)	PUNCT
ejpam-3441	696	26	◦	◦	NOUN
ejpam-3441	696	27	βα	βα	X
ejpam-3441	696	28	c	c	NOUN
ejpam-3441	696	29	⊆	⊆	NUM
ejpam-3441	696	30	(	(	PUNCT
ejpam-3441	696	31	c	c	NOUN
ejpam-3441	696	32	◦	◦	NOUN
ejpam-3441	696	33	βα	βα	NOUN
ejpam-3441	696	34	r	r	NOUN
ejpam-3441	696	35	)	)	PUNCT
ejpam-3441	696	36	◦	◦	NOUN
ejpam-3441	696	37	βα	βα	X
ejpam-3441	696	38	c	c	NOUN
ejpam-3441	696	39	⊆	⊆	NUM
ejpam-3441	696	40	cβα	cβα	NOUN
ejpam-3441	696	41	,	,	PUNCT
ejpam-3441	696	42	thus	thus	ADV
ejpam-3441	696	43	(	(	PUNCT
ejpam-3441	696	44	c	c	NOUN
ejpam-3441	696	45	◦	◦	NOUN
ejpam-3441	696	46	βα	βα	X
ejpam-3441	696	47	i	i	NOUN
ejpam-3441	696	48	)	)	PUNCT
ejpam-3441	696	49	◦	◦	NOUN
ejpam-3441	696	50	βα	βα	X
ejpam-3441	696	51	c	c	NOUN
ejpam-3441	696	52	⊆	⊆	NUM
ejpam-3441	696	53	cβα	cβα	NOUN
ejpam-3441	696	54	∧	∧	NOUN
ejpam-3441	696	55	iβα	iβα	NOUN
ejpam-3441	696	56	=	=	SYM
ejpam-3441	696	57	c	c	NOUN
ejpam-3441	696	58	∧βα	∧βα	ADJ
ejpam-3441	696	59	i.	i.	NOUN
ejpam-3441	696	60	let	let	VERB
ejpam-3441	696	61	x	x	X
ejpam-3441	696	62	∈	∈	PROPN
ejpam-3441	696	63	r	r	NOUN
ejpam-3441	696	64	,	,	PUNCT
ejpam-3441	696	65	then	then	ADV
ejpam-3441	696	66	there	there	PRON
ejpam-3441	696	67	exist	exist	VERB
ejpam-3441	696	68	elements	element	NOUN
ejpam-3441	696	69	ai	ai	VERB
ejpam-3441	696	70	,	,	PUNCT
ejpam-3441	696	71	bi	bi	NOUN
ejpam-3441	696	72	∈	∈	PROPN
ejpam-3441	696	73	r	r	NOUN
ejpam-3441	696	74	such	such	ADJ
ejpam-3441	696	75	that	that	SCONJ
ejpam-3441	696	76	x	x	NOUN
ejpam-3441	697	1	=	=	PUNCT
ejpam-3441	697	2	∑n	∑n	PROPN
ejpam-3441	697	3	i=1(aix	i=1(aix	PROPN
ejpam-3441	697	4	2)bi	2)bi	NUM
ejpam-3441	697	5	.	.	PUNCT
ejpam-3441	698	1	now	now	ADV
ejpam-3441	698	2	x	x	X
ejpam-3441	698	3	=	=	SYM
ejpam-3441	698	4	(	(	PUNCT
ejpam-3441	698	5	aix	aix	NOUN
ejpam-3441	698	6	2)bi	2)bi	NUM
ejpam-3441	698	7	=	=	SYM
ejpam-3441	698	8	(	(	PUNCT
ejpam-3441	698	9	ai(xx))bi	ai(xx))bi	NOUN
ejpam-3441	698	10	=	=	PUNCT
ejpam-3441	698	11	(	(	PUNCT
ejpam-3441	698	12	x(aix))b	x(aix))b	PROPN
ejpam-3441	698	13	=	=	SYM
ejpam-3441	698	14	(	(	PUNCT
ejpam-3441	698	15	bi(aix))x	bi(aix))x	PROPN
ejpam-3441	698	16	.	.	PUNCT
ejpam-3441	699	1	bi(aix	bi(aix	NOUN
ejpam-3441	699	2	)	)	PUNCT
ejpam-3441	700	1	=	=	PROPN
ejpam-3441	701	1	bi(ai((aix	bi(ai((aix	PROPN
ejpam-3441	701	2	2)bi	2)bi	PROPN
ejpam-3441	701	3	)	)	PUNCT
ejpam-3441	701	4	)	)	PUNCT
ejpam-3441	702	1	=	=	SYM
ejpam-3441	702	2	bi((aix	bi((aix	ADJ
ejpam-3441	702	3	2)(aibi	2)(aibi	NOUN
ejpam-3441	702	4	)	)	PUNCT
ejpam-3441	702	5	)	)	PUNCT
ejpam-3441	703	1	=	=	SYM
ejpam-3441	703	2	bi((aix	bi((aix	PROPN
ejpam-3441	703	3	2)ci	2)ci	NUM
ejpam-3441	703	4	)	)	PUNCT
ejpam-3441	703	5	=	=	PUNCT
ejpam-3441	704	1	(	(	PUNCT
ejpam-3441	704	2	aix	aix	PROPN
ejpam-3441	704	3	2)(bici	2)(bici	NUM
ejpam-3441	704	4	)	)	PUNCT
ejpam-3441	705	1	=	=	PRON
ejpam-3441	706	1	(	(	PUNCT
ejpam-3441	706	2	aix	aix	NOUN
ejpam-3441	706	3	2)di	2)di	NUM
ejpam-3441	706	4	=	=	SYM
ejpam-3441	706	5	(	(	PUNCT
ejpam-3441	706	6	aix	aix	PROPN
ejpam-3441	706	7	2)(edi	2)(edi	PROPN
ejpam-3441	706	8	)	)	PUNCT
ejpam-3441	707	1	=	=	PUNCT
ejpam-3441	707	2	(	(	PUNCT
ejpam-3441	707	3	die)(x	die)(x	NOUN
ejpam-3441	707	4	2ai	2ai	ADJ
ejpam-3441	707	5	)	)	PUNCT
ejpam-3441	707	6	=	=	SYM
ejpam-3441	707	7	mi(x	mi(x	NUM
ejpam-3441	707	8	2ai	2ai	ADJ
ejpam-3441	707	9	)	)	PUNCT
ejpam-3441	707	10	=	=	SYM
ejpam-3441	707	11	x2(miai	x2(miai	PROPN
ejpam-3441	707	12	)	)	PUNCT
ejpam-3441	707	13	=	=	PUNCT
ejpam-3441	707	14	(	(	PUNCT
ejpam-3441	707	15	xx)li	xx)li	PUNCT
ejpam-3441	707	16	=	=	SYM
ejpam-3441	707	17	(	(	PUNCT
ejpam-3441	707	18	lix)x	lix)x	PROPN
ejpam-3441	707	19	=	=	X
ejpam-3441	707	20	(	(	PUNCT
ejpam-3441	707	21	lix)(ex	lix)(ex	PROPN
ejpam-3441	707	22	)	)	PUNCT
ejpam-3441	707	23	k.	k.	PROPN
ejpam-3441	707	24	nasreen	nasreen	PROPN
ejpam-3441	707	25	et	et	PROPN
ejpam-3441	707	26	al	al	PROPN
ejpam-3441	707	27	.	.	PUNCT
ejpam-3441	707	28	/	/	SYM
ejpam-3441	707	29	eur	eur	PROPN
ejpam-3441	707	30	.	.	PUNCT
ejpam-3441	708	1	j.	j.	PROPN
ejpam-3441	708	2	pure	pure	PROPN
ejpam-3441	708	3	appl	appl	PROPN
ejpam-3441	708	4	.	.	PROPN
ejpam-3441	708	5	math	math	PROPN
ejpam-3441	708	6	,	,	PUNCT
ejpam-3441	708	7	12	12	NUM
ejpam-3441	708	8	(	(	PUNCT
ejpam-3441	708	9	3	3	NUM
ejpam-3441	708	10	)	)	PUNCT
ejpam-3441	708	11	(	(	PUNCT
ejpam-3441	708	12	2019	2019	NUM
ejpam-3441	708	13	)	)	PUNCT
ejpam-3441	708	14	,	,	PUNCT
ejpam-3441	708	15	906	906	NUM
ejpam-3441	708	16	-	-	SYM
ejpam-3441	708	17	943	943	NUM
ejpam-3441	708	18	936	936	NUM
ejpam-3441	708	19	=	=	SYM
ejpam-3441	708	20	(	(	PUNCT
ejpam-3441	708	21	xe)(xli	xe)(xli	X
ejpam-3441	708	22	)	)	PUNCT
ejpam-3441	708	23	=	=	SYM
ejpam-3441	708	24	x((xe)li	x((xe)li	NUM
ejpam-3441	708	25	)	)	PUNCT
ejpam-3441	708	26	.	.	PUNCT
ejpam-3441	709	1	thus	thus	ADV
ejpam-3441	709	2	(	(	PUNCT
ejpam-3441	709	3	(	(	PUNCT
ejpam-3441	709	4	µc	µc	INTJ
ejpam-3441	709	5	◦	◦	NOUN
ejpam-3441	709	6	βα	βα	NOUN
ejpam-3441	709	7	µi	µi	NOUN
ejpam-3441	709	8	)	)	PUNCT
ejpam-3441	709	9	◦	◦	NOUN
ejpam-3441	709	10	βα	βα	NOUN
ejpam-3441	709	11	µc)(x	µc)(x	NOUN
ejpam-3441	709	12	)	)	PUNCT
ejpam-3441	710	1	=	=	PRON
ejpam-3441	710	2	{	{	PUNCT
ejpam-3441	710	3	(	(	PUNCT
ejpam-3441	710	4	(	(	PUNCT
ejpam-3441	710	5	µc	µc	INTJ
ejpam-3441	710	6	◦	◦	VERB
ejpam-3441	710	7	µi	µi	PART
ejpam-3441	710	8	)	)	PUNCT
ejpam-3441	710	9	◦	◦	NOUN
ejpam-3441	710	10	µc)(x	µc)(x	NOUN
ejpam-3441	710	11	)	)	PUNCT
ejpam-3441	711	1	∧	∧	PROPN
ejpam-3441	711	2	β	β	NOUN
ejpam-3441	711	3	}	}	PUNCT
ejpam-3441	711	4	∨	∨	NUM
ejpam-3441	711	5	α	α	NOUN
ejpam-3441	711	6	=	=	X
ejpam-3441	711	7	{	{	PUNCT
ejpam-3441	711	8	(	(	PUNCT
ejpam-3441	711	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	711	10	i=1	i=1	PROPN
ejpam-3441	711	11	piqi	piqi	NOUN
ejpam-3441	711	12	{	{	PUNCT
ejpam-3441	711	13	∧ni=1	∧ni=1	X
ejpam-3441	711	14	{	{	PUNCT
ejpam-3441	711	15	(	(	PUNCT
ejpam-3441	711	16	µc	µc	INTJ
ejpam-3441	711	17	◦	◦	NOUN
ejpam-3441	711	18	µi	µi	PROPN
ejpam-3441	711	19	)	)	PUNCT
ejpam-3441	711	20	(	(	PUNCT
ejpam-3441	711	21	pi	pi	NOUN
ejpam-3441	711	22	)	)	PUNCT
ejpam-3441	711	23	∧	∧	NOUN
ejpam-3441	711	24	µc	µc	INTJ
ejpam-3441	711	25	(	(	PUNCT
ejpam-3441	711	26	qi	qi	NOUN
ejpam-3441	711	27	)	)	PUNCT
ejpam-3441	711	28	}	}	PUNCT
ejpam-3441	711	29	}	}	PUNCT
ejpam-3441	711	30	)	)	PUNCT
ejpam-3441	712	1	∧	∧	PROPN
ejpam-3441	712	2	β	β	NOUN
ejpam-3441	712	3	}	}	PUNCT
ejpam-3441	712	4	∨	∨	NUM
ejpam-3441	712	5	α	α	PROPN
ejpam-3441	712	6	≥	≥	X
ejpam-3441	712	7	{	{	PUNCT
ejpam-3441	712	8	{	{	PUNCT
ejpam-3441	712	9	(	(	PUNCT
ejpam-3441	712	10	µc	µc	INTJ
ejpam-3441	712	11	◦	◦	NOUN
ejpam-3441	712	12	µi	µi	PROPN
ejpam-3441	712	13	)	)	PUNCT
ejpam-3441	712	14	(	(	PUNCT
ejpam-3441	712	15	bi(aix	bi(aix	NOUN
ejpam-3441	712	16	)	)	PUNCT
ejpam-3441	712	17	)	)	PUNCT
ejpam-3441	713	1	∧	∧	NOUN
ejpam-3441	713	2	µc	µc	INTJ
ejpam-3441	713	3	(	(	PUNCT
ejpam-3441	713	4	x	x	NOUN
ejpam-3441	713	5	)	)	PUNCT
ejpam-3441	713	6	}	}	PUNCT
ejpam-3441	713	7	∧	∧	PROPN
ejpam-3441	713	8	β	β	NOUN
ejpam-3441	713	9	}	}	PUNCT
ejpam-3441	713	10	∨	∨	NUM
ejpam-3441	713	11	α	α	NOUN
ejpam-3441	713	12	=	=	SYM
ejpam-3441	713	13	(	(	PUNCT
ejpam-3441	713	14	(	(	PUNCT
ejpam-3441	713	15	µc	µc	INTJ
ejpam-3441	713	16	◦	◦	NOUN
ejpam-3441	713	17	µi	µi	PROPN
ejpam-3441	713	18	)	)	PUNCT
ejpam-3441	713	19	(	(	PUNCT
ejpam-3441	713	20	bi(aix	bi(aix	NOUN
ejpam-3441	713	21	)	)	PUNCT
ejpam-3441	713	22	)	)	PUNCT
ejpam-3441	714	1	∨	∨	NUM
ejpam-3441	714	2	α	α	NOUN
ejpam-3441	714	3	)	)	PUNCT
ejpam-3441	714	4	∧	∧	NOUN
ejpam-3441	714	5	(	(	PUNCT
ejpam-3441	714	6	µc	µc	INTJ
ejpam-3441	714	7	(	(	PUNCT
ejpam-3441	714	8	x	x	NOUN
ejpam-3441	714	9	)	)	PUNCT
ejpam-3441	714	10	∨	∨	NUM
ejpam-3441	714	11	α	α	NOUN
ejpam-3441	714	12	)	)	PUNCT
ejpam-3441	714	13	∧	∧	PROPN
ejpam-3441	714	14	(	(	PUNCT
ejpam-3441	714	15	β	β	X
ejpam-3441	714	16	∨	∨	NUM
ejpam-3441	714	17	α	α	NOUN
ejpam-3441	714	18	)	)	PUNCT
ejpam-3441	714	19	=	=	SYM
ejpam-3441	714	20	(	(	PUNCT
ejpam-3441	714	21	(	(	PUNCT
ejpam-3441	714	22	µc	µc	INTJ
ejpam-3441	714	23	◦	◦	NOUN
ejpam-3441	714	24	µi	µi	PROPN
ejpam-3441	714	25	)	)	PUNCT
ejpam-3441	714	26	(	(	PUNCT
ejpam-3441	714	27	bi(aix	bi(aix	NOUN
ejpam-3441	714	28	)	)	PUNCT
ejpam-3441	714	29	)	)	PUNCT
ejpam-3441	715	1	∨	∨	NUM
ejpam-3441	715	2	α	α	NOUN
ejpam-3441	715	3	)	)	PUNCT
ejpam-3441	715	4	∧	∧	PROPN
ejpam-3441	715	5	µc(x	µc(x	NOUN
ejpam-3441	715	6	)	)	PUNCT
ejpam-3441	715	7	∧	∧	NOUN
ejpam-3441	715	8	β	β	X
ejpam-3441	715	9	=	=	SYM
ejpam-3441	715	10	(	(	PUNCT
ejpam-3441	715	11	(	(	PUNCT
ejpam-3441	715	12	∨bi(aix)=∑n	∨bi(aix)=∑n	NOUN
ejpam-3441	715	13	i=1mini	i=1mini	PRON
ejpam-3441	715	14	{	{	PUNCT
ejpam-3441	715	15	∧ni=1	∧ni=1	X
ejpam-3441	715	16	{	{	PUNCT
ejpam-3441	715	17	µc	µc	PROPN
ejpam-3441	715	18	(	(	PUNCT
ejpam-3441	715	19	mi	mi	NOUN
ejpam-3441	715	20	)	)	PUNCT
ejpam-3441	715	21	∧	∧	PROPN
ejpam-3441	715	22	µi	µi	PROPN
ejpam-3441	715	23	(	(	PUNCT
ejpam-3441	715	24	ni	ni	PROPN
ejpam-3441	715	25	)	)	PUNCT
ejpam-3441	715	26	}	}	PUNCT
ejpam-3441	715	27	}	}	PUNCT
ejpam-3441	715	28	)	)	PUNCT
ejpam-3441	715	29	∨	∨	NUM
ejpam-3441	715	30	α	α	NOUN
ejpam-3441	715	31	)	)	PUNCT
ejpam-3441	715	32	∧	∧	PROPN
ejpam-3441	715	33	µc(x	µc(x	NOUN
ejpam-3441	715	34	)	)	PUNCT
ejpam-3441	715	35	∧	∧	PROPN
ejpam-3441	715	36	β	β	X
ejpam-3441	715	37	≥	≥	X
ejpam-3441	715	38	(	(	PUNCT
ejpam-3441	715	39	{	{	PUNCT
ejpam-3441	715	40	µc(x	µc(x	NOUN
ejpam-3441	715	41	)	)	PUNCT
ejpam-3441	715	42	∧	∧	PROPN
ejpam-3441	715	43	µi((xe)li	µi((xe)li	PROPN
ejpam-3441	715	44	)	)	PUNCT
ejpam-3441	715	45	}	}	PUNCT
ejpam-3441	715	46	∨	∨	NUM
ejpam-3441	715	47	α	α	NOUN
ejpam-3441	715	48	)	)	PUNCT
ejpam-3441	715	49	∧	∧	PROPN
ejpam-3441	715	50	µc(x	µc(x	NOUN
ejpam-3441	715	51	)	)	PUNCT
ejpam-3441	715	52	∧	∧	NOUN
ejpam-3441	715	53	β	β	X
ejpam-3441	715	54	=	=	SYM
ejpam-3441	715	55	(	(	PUNCT
ejpam-3441	715	56	µc(x	µc(x	NOUN
ejpam-3441	715	57	)	)	PUNCT
ejpam-3441	715	58	∨	∨	NUM
ejpam-3441	715	59	α	α	NOUN
ejpam-3441	715	60	)	)	PUNCT
ejpam-3441	715	61	∧	∧	PROPN
ejpam-3441	715	62	(	(	PUNCT
ejpam-3441	715	63	µi((xe)li	µi((xe)li	PROPN
ejpam-3441	715	64	)	)	PUNCT
ejpam-3441	715	65	∨	∨	NUM
ejpam-3441	715	66	α	α	NOUN
ejpam-3441	715	67	)	)	PUNCT
ejpam-3441	715	68	∧	∧	PROPN
ejpam-3441	715	69	µc(x	µc(x	NOUN
ejpam-3441	715	70	)	)	PUNCT
ejpam-3441	715	71	∧	∧	PROPN
ejpam-3441	715	72	β	β	X
ejpam-3441	715	73	≥	≥	NOUN
ejpam-3441	715	74	µc(x	µc(x	NOUN
ejpam-3441	715	75	)	)	PUNCT
ejpam-3441	715	76	∧	∧	PROPN
ejpam-3441	715	77	(	(	PUNCT
ejpam-3441	715	78	µi(x	µi(x	NOUN
ejpam-3441	715	79	)	)	PUNCT
ejpam-3441	716	1	∧	∧	NOUN
ejpam-3441	716	2	β	β	NOUN
ejpam-3441	716	3	)	)	PUNCT
ejpam-3441	716	4	∧	∧	PROPN
ejpam-3441	716	5	µc(x	µc(x	NOUN
ejpam-3441	716	6	)	)	PUNCT
ejpam-3441	716	7	∧	∧	PROPN
ejpam-3441	716	8	β	β	X
ejpam-3441	716	9	=	=	SYM
ejpam-3441	716	10	µc(x	µc(x	NOUN
ejpam-3441	716	11	)	)	PUNCT
ejpam-3441	716	12	∧	∧	NOUN
ejpam-3441	716	13	µi(x	µi(x	NUM
ejpam-3441	716	14	)	)	PUNCT
ejpam-3441	716	15	∧	∧	NOUN
ejpam-3441	716	16	β	β	X
ejpam-3441	716	17	=	=	SYM
ejpam-3441	716	18	(	(	PUNCT
ejpam-3441	716	19	µc	µc	INTJ
ejpam-3441	716	20	∧	∧	PROPN
ejpam-3441	716	21	µi)(x	µi)(x	PROPN
ejpam-3441	716	22	)	)	PUNCT
ejpam-3441	716	23	∧	∧	NOUN
ejpam-3441	716	24	β	β	X
ejpam-3441	716	25	=	=	SYM
ejpam-3441	716	26	{	{	PUNCT
ejpam-3441	716	27	(	(	PUNCT
ejpam-3441	716	28	µc	µc	INTJ
ejpam-3441	716	29	∧	∧	PROPN
ejpam-3441	716	30	µi)(x	µi)(x	PROPN
ejpam-3441	716	31	)	)	PUNCT
ejpam-3441	716	32	∧	∧	PROPN
ejpam-3441	716	33	β	β	PROPN
ejpam-3441	716	34	}	}	PUNCT
ejpam-3441	716	35	∨	∨	NUM
ejpam-3441	716	36	α	α	NOUN
ejpam-3441	716	37	=	=	PUNCT
ejpam-3441	716	38	(	(	PUNCT
ejpam-3441	716	39	µc	µc	INTJ
ejpam-3441	716	40	∧βα	∧βα	ADJ
ejpam-3441	716	41	µi)(x	µi)(x	NOUN
ejpam-3441	716	42	)	)	PUNCT
ejpam-3441	716	43	.	.	PUNCT
ejpam-3441	717	1	⇒	⇒	NOUN
ejpam-3441	717	2	µc	µc	VERB
ejpam-3441	717	3	∧βα	∧βα	PROPN
ejpam-3441	717	4	µi	µi	PROPN
ejpam-3441	717	5	⊆	⊆	NUM
ejpam-3441	717	6	(	(	PUNCT
ejpam-3441	717	7	µc	µc	INTJ
ejpam-3441	717	8	◦	◦	NOUN
ejpam-3441	717	9	βα	βα	NOUN
ejpam-3441	717	10	µi	µi	NOUN
ejpam-3441	717	11	)	)	PUNCT
ejpam-3441	717	12	◦	◦	NOUN
ejpam-3441	717	13	βα	βα	NOUN
ejpam-3441	717	14	µc	µc	INTJ
ejpam-3441	717	15	.	.	PUNCT
ejpam-3441	718	1	similarly	similarly	ADV
ejpam-3441	718	2	,	,	PUNCT
ejpam-3441	718	3	we	we	PRON
ejpam-3441	718	4	have	have	VERB
ejpam-3441	718	5	γc	γc	NUM
ejpam-3441	718	6	∨βα	∨βα	PROPN
ejpam-3441	718	7	γi	γi	ADP
ejpam-3441	718	8	⊇	⊇	X
ejpam-3441	718	9	(	(	PUNCT
ejpam-3441	718	10	γc	γc	PROPN
ejpam-3441	718	11	◦	◦	NOUN
ejpam-3441	718	12	βα	βα	NOUN
ejpam-3441	718	13	γi	γi	NOUN
ejpam-3441	718	14	)	)	PUNCT
ejpam-3441	718	15	◦	◦	NOUN
ejpam-3441	718	16	βα	βα	NOUN
ejpam-3441	718	17	γc	γc	X
ejpam-3441	718	18	.	.	PUNCT
ejpam-3441	719	1	hence	hence	ADV
ejpam-3441	719	2	c	c	VERB
ejpam-3441	720	1	∧βα	∧βα	ADJ
ejpam-3441	720	2	i	i	NOUN
ejpam-3441	720	3	=	=	PUNCT
ejpam-3441	720	4	(	(	PUNCT
ejpam-3441	720	5	c	c	NOUN
ejpam-3441	720	6	◦	◦	NOUN
ejpam-3441	720	7	βα	βα	X
ejpam-3441	720	8	i	i	NOUN
ejpam-3441	720	9	)	)	PUNCT
ejpam-3441	720	10	◦	◦	NOUN
ejpam-3441	720	11	βα	βα	X
ejpam-3441	720	12	c	c	NOUN
ejpam-3441	720	13	,	,	PUNCT
ejpam-3441	720	14	i.e.	i.e.	X
ejpam-3441	720	15	,	,	PUNCT
ejpam-3441	720	16	(	(	PUNCT
ejpam-3441	720	17	1	1	X
ejpam-3441	720	18	)	)	PUNCT
ejpam-3441	720	19	implies	imply	VERB
ejpam-3441	720	20	(	(	PUNCT
ejpam-3441	720	21	4	4	NUM
ejpam-3441	720	22	)	)	PUNCT
ejpam-3441	720	23	.	.	PUNCT
ejpam-3441	721	1	it	it	PRON
ejpam-3441	721	2	is	be	AUX
ejpam-3441	721	3	clear	clear	ADJ
ejpam-3441	721	4	that	that	SCONJ
ejpam-3441	721	5	(	(	PUNCT
ejpam-3441	721	6	4	4	X
ejpam-3441	721	7	)	)	PUNCT
ejpam-3441	721	8	⇒	⇒	NOUN
ejpam-3441	721	9	(	(	PUNCT
ejpam-3441	721	10	3	3	NUM
ejpam-3441	721	11	)	)	PUNCT
ejpam-3441	721	12	and	and	CCONJ
ejpam-3441	721	13	(	(	PUNCT
ejpam-3441	721	14	3	3	X
ejpam-3441	721	15	)	)	PUNCT
ejpam-3441	721	16	⇒	⇒	NOUN
ejpam-3441	721	17	(	(	PUNCT
ejpam-3441	721	18	2	2	NUM
ejpam-3441	721	19	)	)	PUNCT
ejpam-3441	721	20	.	.	PUNCT
ejpam-3441	722	1	suppose	suppose	VERB
ejpam-3441	722	2	that	that	SCONJ
ejpam-3441	722	3	(	(	PUNCT
ejpam-3441	722	4	2	2	X
ejpam-3441	722	5	)	)	PUNCT
ejpam-3441	722	6	holds	hold	VERB
ejpam-3441	722	7	.	.	PUNCT
ejpam-3441	723	1	let	let	VERB
ejpam-3441	723	2	a	a	PRON
ejpam-3441	723	3	be	be	AUX
ejpam-3441	723	4	an	an	DET
ejpam-3441	723	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	723	6	fuzzy	fuzzy	ADJ
ejpam-3441	723	7	right	right	ADJ
ejpam-3441	723	8	ideal	ideal	NOUN
ejpam-3441	723	9	with	with	ADP
ejpam-3441	723	10	thresholds	threshold	NOUN
ejpam-3441	723	11	(	(	PUNCT
ejpam-3441	723	12	α	α	X
ejpam-3441	723	13	,	,	PUNCT
ejpam-3441	723	14	β	β	X
ejpam-3441	723	15	]	]	PUNCT
ejpam-3441	723	16	and	and	CCONJ
ejpam-3441	723	17	i	i	PRON
ejpam-3441	723	18	be	be	VERB
ejpam-3441	723	19	an	an	DET
ejpam-3441	723	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	723	21	fuzzy	fuzzy	ADJ
ejpam-3441	723	22	two	two	NUM
ejpam-3441	723	23	-	-	PUNCT
ejpam-3441	723	24	sided	sided	ADJ
ejpam-3441	723	25	ideal	ideal	NOUN
ejpam-3441	723	26	with	with	ADP
ejpam-3441	723	27	thresholds	threshold	NOUN
ejpam-3441	723	28	(	(	PUNCT
ejpam-3441	723	29	α	α	X
ejpam-3441	723	30	,	,	PUNCT
ejpam-3441	723	31	β	β	X
ejpam-3441	723	32	]	]	PUNCT
ejpam-3441	723	33	of	of	ADP
ejpam-3441	723	34	r.	r.	PROPN
ejpam-3441	723	35	since	since	SCONJ
ejpam-3441	723	36	every	every	DET
ejpam-3441	723	37	intuitionistic	intuitionistic	ADJ
ejpam-3441	723	38	fuzzy	fuzzy	ADJ
ejpam-3441	723	39	right	right	ADJ
ejpam-3441	723	40	ideal	ideal	NOUN
ejpam-3441	723	41	with	with	ADP
ejpam-3441	723	42	thresholds	threshold	NOUN
ejpam-3441	723	43	(	(	PUNCT
ejpam-3441	723	44	α	α	X
ejpam-3441	723	45	,	,	PUNCT
ejpam-3441	723	46	β	β	X
ejpam-3441	723	47	]	]	PUNCT
ejpam-3441	723	48	of	of	ADP
ejpam-3441	723	49	r	r	NOUN
ejpam-3441	723	50	is	be	AUX
ejpam-3441	723	51	an	an	DET
ejpam-3441	723	52	intuitionistic	intuitionistic	ADJ
ejpam-3441	723	53	fuzzy	fuzzy	ADJ
ejpam-3441	723	54	quasi	quasi	NOUN
ejpam-3441	723	55	-	-	NOUN
ejpam-3441	723	56	ideal	ideal	ADJ
ejpam-3441	723	57	with	with	ADP
ejpam-3441	723	58	thresholds	threshold	NOUN
ejpam-3441	723	59	(	(	PUNCT
ejpam-3441	723	60	α	α	X
ejpam-3441	723	61	,	,	PUNCT
ejpam-3441	723	62	β	β	X
ejpam-3441	723	63	]	]	PUNCT
ejpam-3441	723	64	of	of	ADP
ejpam-3441	723	65	r	r	NOUN
ejpam-3441	723	66	by	by	ADP
ejpam-3441	723	67	the	the	DET
ejpam-3441	723	68	lemma	lemma	PROPN
ejpam-3441	723	69	14	14	NUM
ejpam-3441	723	70	,	,	PUNCT
ejpam-3441	723	71	this	this	PRON
ejpam-3441	723	72	implies	imply	VERB
ejpam-3441	723	73	that	that	SCONJ
ejpam-3441	723	74	a	a	PRON
ejpam-3441	723	75	is	be	AUX
ejpam-3441	723	76	an	an	DET
ejpam-3441	723	77	intuitionistic	intuitionistic	ADJ
ejpam-3441	723	78	fuzzy	fuzzy	ADJ
ejpam-3441	723	79	quasi	quasi	NOUN
ejpam-3441	723	80	-	-	NOUN
ejpam-3441	723	81	ideal	ideal	ADJ
ejpam-3441	723	82	with	with	ADP
ejpam-3441	723	83	thresholds	threshold	NOUN
ejpam-3441	723	84	(	(	PUNCT
ejpam-3441	723	85	α	α	X
ejpam-3441	723	86	,	,	PUNCT
ejpam-3441	723	87	β	β	X
ejpam-3441	723	88	]	]	PUNCT
ejpam-3441	723	89	of	of	ADP
ejpam-3441	723	90	r.	r.	PROPN
ejpam-3441	723	91	by	by	ADP
ejpam-3441	723	92	our	our	PRON
ejpam-3441	723	93	supposition	supposition	NOUN
ejpam-3441	723	94	a	a	DET
ejpam-3441	723	95	∧βα	∧βα	NOUN
ejpam-3441	723	96	i	i	NOUN
ejpam-3441	723	97	=	=	PUNCT
ejpam-3441	723	98	(	(	PUNCT
ejpam-3441	723	99	a	a	DET
ejpam-3441	723	100	◦	◦	NOUN
ejpam-3441	723	101	βα	βα	X
ejpam-3441	723	102	i	i	NOUN
ejpam-3441	723	103	)	)	PUNCT
ejpam-3441	723	104	◦	◦	NOUN
ejpam-3441	723	105	βα	βα	NOUN
ejpam-3441	723	106	a	a	DET
ejpam-3441	723	107	⊆	⊆	NUM
ejpam-3441	723	108	(	(	PUNCT
ejpam-3441	723	109	r	r	NOUN
ejpam-3441	723	110	◦	◦	NOUN
ejpam-3441	723	111	βα	βα	X
ejpam-3441	723	112	i	i	NOUN
ejpam-3441	723	113	)	)	PUNCT
ejpam-3441	723	114	◦	◦	NOUN
ejpam-3441	723	115	βα	βα	NOUN
ejpam-3441	723	116	a	a	DET
ejpam-3441	723	117	⊆	⊆	NUM
ejpam-3441	723	118	i	i	SYM
ejpam-3441	723	119	◦	◦	NOUN
ejpam-3441	723	120	βα	βα	X
ejpam-3441	723	121	a	a	DET
ejpam-3441	723	122	,	,	PUNCT
ejpam-3441	723	123	i.e.	i.e.	X
ejpam-3441	723	124	,	,	PUNCT
ejpam-3441	723	125	a	a	DET
ejpam-3441	723	126	∧βα	∧βα	ADJ
ejpam-3441	723	127	i	i	NOUN
ejpam-3441	723	128	⊆	⊆	NUM
ejpam-3441	723	129	i	i	PROPN
ejpam-3441	723	130	◦	◦	PROPN
ejpam-3441	723	131	βα	βα	NOUN
ejpam-3441	723	132	a.	a.	NOUN
ejpam-3441	724	1	so	so	ADV
ejpam-3441	724	2	r	r	NOUN
ejpam-3441	724	3	is	be	AUX
ejpam-3441	724	4	an	an	DET
ejpam-3441	724	5	intra	intra	ADJ
ejpam-3441	724	6	-	-	ADJ
ejpam-3441	724	7	regular	regular	ADJ
ejpam-3441	724	8	by	by	ADP
ejpam-3441	724	9	the	the	DET
ejpam-3441	724	10	theorem	theorem	ADJ
ejpam-3441	724	11	13	13	NUM
ejpam-3441	724	12	,	,	PUNCT
ejpam-3441	724	13	i.e.	i.e.	X
ejpam-3441	724	14	,	,	PUNCT
ejpam-3441	724	15	(	(	PUNCT
ejpam-3441	724	16	2)⇒	2)⇒	NUM
ejpam-3441	724	17	(	(	PUNCT
ejpam-3441	724	18	1	1	NUM
ejpam-3441	724	19	)	)	PUNCT
ejpam-3441	724	20	.	.	PUNCT
ejpam-3441	725	1	theorem	theorem	ADJ
ejpam-3441	725	2	15	15	NUM
ejpam-3441	725	3	.	.	PUNCT
ejpam-3441	726	1	let	let	VERB
ejpam-3441	726	2	r	r	PRON
ejpam-3441	726	3	be	be	AUX
ejpam-3441	726	4	an	an	DET
ejpam-3441	726	5	la	la	NOUN
ejpam-3441	726	6	-	-	NOUN
ejpam-3441	726	7	ring	ring	NOUN
ejpam-3441	726	8	with	with	ADP
ejpam-3441	726	9	left	left	ADJ
ejpam-3441	726	10	identity	identity	NOUN
ejpam-3441	726	11	e	e	NOUN
ejpam-3441	726	12	,	,	PUNCT
ejpam-3441	726	13	such	such	ADJ
ejpam-3441	726	14	that	that	SCONJ
ejpam-3441	726	15	(	(	PUNCT
ejpam-3441	726	16	xe)r	xe)r	PROPN
ejpam-3441	726	17	=	=	SYM
ejpam-3441	726	18	xr	xr	PROPN
ejpam-3441	726	19	for	for	ADP
ejpam-3441	726	20	all	all	DET
ejpam-3441	726	21	x	x	PROPN
ejpam-3441	726	22	∈	∈	PROPN
ejpam-3441	726	23	r.	r.	NOUN
ejpam-3441	726	24	then	then	ADV
ejpam-3441	726	25	the	the	DET
ejpam-3441	726	26	following	follow	VERB
ejpam-3441	726	27	conditions	condition	NOUN
ejpam-3441	726	28	are	be	AUX
ejpam-3441	726	29	equivalent	equivalent	ADJ
ejpam-3441	726	30	.	.	PUNCT
ejpam-3441	727	1	(	(	PUNCT
ejpam-3441	727	2	1	1	X
ejpam-3441	727	3	)	)	PUNCT
ejpam-3441	727	4	r	r	NOUN
ejpam-3441	727	5	is	be	AUX
ejpam-3441	727	6	an	an	DET
ejpam-3441	727	7	intra	intra	ADJ
ejpam-3441	727	8	-	-	ADJ
ejpam-3441	727	9	regular	regular	ADJ
ejpam-3441	727	10	.	.	PUNCT
ejpam-3441	728	1	(	(	PUNCT
ejpam-3441	728	2	2	2	X
ejpam-3441	728	3	)	)	PUNCT
ejpam-3441	728	4	a∧βαl	a∧βαl	NUM
ejpam-3441	729	1	⊆	⊆	NUM
ejpam-3441	729	2	l	l	NOUN
ejpam-3441	729	3	◦	◦	NOUN
ejpam-3441	729	4	βαa	βαa	NOUN
ejpam-3441	729	5	for	for	ADP
ejpam-3441	729	6	every	every	DET
ejpam-3441	729	7	intuitionistic	intuitionistic	ADJ
ejpam-3441	729	8	fuzzy	fuzzy	ADJ
ejpam-3441	729	9	quasi	quasi	NOUN
ejpam-3441	729	10	-	-	NOUN
ejpam-3441	729	11	ideal	ideal	ADJ
ejpam-3441	729	12	a	a	DET
ejpam-3441	729	13	with	with	ADP
ejpam-3441	729	14	thresholds	threshold	NOUN
ejpam-3441	729	15	(	(	PUNCT
ejpam-3441	729	16	α	α	X
ejpam-3441	729	17	,	,	PUNCT
ejpam-3441	729	18	β	β	X
ejpam-3441	729	19	]	]	PUNCT
ejpam-3441	729	20	and	and	CCONJ
ejpam-3441	729	21	every	every	DET
ejpam-3441	729	22	intuitionistic	intuitionistic	ADJ
ejpam-3441	729	23	fuzzy	fuzzy	ADJ
ejpam-3441	729	24	left	leave	VERB
ejpam-3441	729	25	ideal	ideal	NOUN
ejpam-3441	729	26	l	l	PROPN
ejpam-3441	729	27	with	with	ADP
ejpam-3441	729	28	thresholds	threshold	NOUN
ejpam-3441	729	29	(	(	PUNCT
ejpam-3441	729	30	α	α	X
ejpam-3441	729	31	,	,	PUNCT
ejpam-3441	729	32	β	β	X
ejpam-3441	729	33	]	]	PUNCT
ejpam-3441	729	34	of	of	ADP
ejpam-3441	729	35	r.	r.	PROPN
ejpam-3441	729	36	(	(	PUNCT
ejpam-3441	729	37	3	3	NUM
ejpam-3441	729	38	)	)	PUNCT
ejpam-3441	729	39	b	b	NOUN
ejpam-3441	730	1	∧βα	∧βα	ADJ
ejpam-3441	730	2	l	l	NOUN
ejpam-3441	730	3	⊆	⊆	NUM
ejpam-3441	730	4	l	l	NOUN
ejpam-3441	730	5	◦	◦	NOUN
ejpam-3441	730	6	βαb	βαb	NOUN
ejpam-3441	730	7	for	for	ADP
ejpam-3441	730	8	every	every	DET
ejpam-3441	730	9	intuitionistic	intuitionistic	ADJ
ejpam-3441	730	10	fuzzy	fuzzy	ADJ
ejpam-3441	730	11	bi	bi	ADJ
ejpam-3441	730	12	-	-	ADJ
ejpam-3441	730	13	ideal	ideal	ADJ
ejpam-3441	730	14	b	b	PROPN
ejpam-3441	730	15	with	with	ADP
ejpam-3441	730	16	thresholds	threshold	NOUN
ejpam-3441	730	17	(	(	PUNCT
ejpam-3441	730	18	α	α	X
ejpam-3441	730	19	,	,	PUNCT
ejpam-3441	730	20	β	β	X
ejpam-3441	730	21	]	]	PUNCT
ejpam-3441	730	22	and	and	CCONJ
ejpam-3441	730	23	every	every	DET
ejpam-3441	730	24	intuitionistic	intuitionistic	ADJ
ejpam-3441	730	25	fuzzy	fuzzy	ADJ
ejpam-3441	730	26	left	leave	VERB
ejpam-3441	730	27	ideal	ideal	NOUN
ejpam-3441	730	28	l	l	PROPN
ejpam-3441	730	29	with	with	ADP
ejpam-3441	730	30	thresholds	threshold	NOUN
ejpam-3441	730	31	(	(	PUNCT
ejpam-3441	730	32	α	α	X
ejpam-3441	730	33	,	,	PUNCT
ejpam-3441	730	34	β	β	X
ejpam-3441	730	35	]	]	PUNCT
ejpam-3441	730	36	of	of	ADP
ejpam-3441	730	37	r.	r.	PROPN
ejpam-3441	730	38	(	(	PUNCT
ejpam-3441	730	39	4	4	NUM
ejpam-3441	730	40	)	)	PUNCT
ejpam-3441	730	41	c∧βαl	c∧βαl	VERB
ejpam-3441	730	42	⊆	⊆	NUM
ejpam-3441	730	43	l	l	NOUN
ejpam-3441	730	44	◦	◦	NOUN
ejpam-3441	730	45	βαc	βαc	NOUN
ejpam-3441	730	46	for	for	ADP
ejpam-3441	730	47	every	every	DET
ejpam-3441	730	48	intuitionistic	intuitionistic	ADJ
ejpam-3441	730	49	fuzzy	fuzzy	ADJ
ejpam-3441	730	50	generalized	generalize	VERB
ejpam-3441	730	51	bi	bi	ADJ
ejpam-3441	730	52	-	-	ADJ
ejpam-3441	730	53	ideal	ideal	ADJ
ejpam-3441	730	54	c	c	NOUN
ejpam-3441	730	55	with	with	ADP
ejpam-3441	730	56	thresholds	threshold	NOUN
ejpam-3441	730	57	(	(	PUNCT
ejpam-3441	730	58	α	α	X
ejpam-3441	730	59	,	,	PUNCT
ejpam-3441	730	60	β	β	X
ejpam-3441	730	61	]	]	PUNCT
ejpam-3441	730	62	and	and	CCONJ
ejpam-3441	730	63	every	every	DET
ejpam-3441	730	64	intuitionistic	intuitionistic	ADJ
ejpam-3441	730	65	fuzzy	fuzzy	ADJ
ejpam-3441	730	66	left	leave	VERB
ejpam-3441	730	67	ideal	ideal	NOUN
ejpam-3441	730	68	l	l	PROPN
ejpam-3441	730	69	with	with	ADP
ejpam-3441	730	70	thresholds	threshold	NOUN
ejpam-3441	730	71	(	(	PUNCT
ejpam-3441	730	72	α	α	X
ejpam-3441	730	73	,	,	PUNCT
ejpam-3441	730	74	β	β	X
ejpam-3441	730	75	]	]	PUNCT
ejpam-3441	730	76	of	of	ADP
ejpam-3441	730	77	r.	r.	PROPN
ejpam-3441	730	78	proof	proof	PROPN
ejpam-3441	730	79	.	.	PUNCT
ejpam-3441	731	1	suppose	suppose	VERB
ejpam-3441	731	2	that	that	SCONJ
ejpam-3441	731	3	(	(	PUNCT
ejpam-3441	731	4	1	1	X
ejpam-3441	731	5	)	)	PUNCT
ejpam-3441	731	6	holds	hold	VERB
ejpam-3441	731	7	.	.	PUNCT
ejpam-3441	732	1	let	let	VERB
ejpam-3441	732	2	c	c	NOUN
ejpam-3441	732	3	=	=	PUNCT
ejpam-3441	732	4	(	(	PUNCT
ejpam-3441	732	5	µc	µc	INTJ
ejpam-3441	732	6	,	,	PUNCT
ejpam-3441	732	7	γc	γc	PROPN
ejpam-3441	732	8	)	)	PUNCT
ejpam-3441	732	9	be	be	AUX
ejpam-3441	732	10	an	an	DET
ejpam-3441	732	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	732	12	fuzzy	fuzzy	ADJ
ejpam-3441	732	13	generalized	generalize	VERB
ejpam-3441	732	14	bi	bi	NOUN
ejpam-3441	732	15	-	-	NOUN
ejpam-3441	732	16	ideal	ideal	NOUN
ejpam-3441	732	17	with	with	ADP
ejpam-3441	732	18	thresholds	threshold	NOUN
ejpam-3441	732	19	(	(	PUNCT
ejpam-3441	732	20	α	α	X
ejpam-3441	732	21	,	,	PUNCT
ejpam-3441	732	22	β	β	NOUN
ejpam-3441	732	23	]	]	PUNCT
ejpam-3441	732	24	and	and	CCONJ
ejpam-3441	732	25	l	l	NOUN
ejpam-3441	732	26	=	=	SYM
ejpam-3441	732	27	(	(	PUNCT
ejpam-3441	732	28	µl	µl	NOUN
ejpam-3441	732	29	,	,	PUNCT
ejpam-3441	732	30	γl	γl	NUM
ejpam-3441	732	31	)	)	PUNCT
ejpam-3441	732	32	be	be	VERB
ejpam-3441	732	33	an	an	DET
ejpam-3441	732	34	intuitionistic	intuitionistic	ADJ
ejpam-3441	732	35	fuzzy	fuzzy	ADJ
ejpam-3441	732	36	left	leave	VERB
ejpam-3441	732	37	ideal	ideal	NOUN
ejpam-3441	732	38	with	with	ADP
ejpam-3441	732	39	k.	k.	PROPN
ejpam-3441	732	40	nasreen	nasreen	PROPN
ejpam-3441	732	41	et	et	PROPN
ejpam-3441	732	42	al	al	PROPN
ejpam-3441	732	43	.	.	PUNCT
ejpam-3441	732	44	/	/	SYM
ejpam-3441	732	45	eur	eur	PROPN
ejpam-3441	732	46	.	.	PUNCT
ejpam-3441	733	1	j.	j.	PROPN
ejpam-3441	733	2	pure	pure	PROPN
ejpam-3441	733	3	appl	appl	PROPN
ejpam-3441	733	4	.	.	PROPN
ejpam-3441	733	5	math	math	PROPN
ejpam-3441	733	6	,	,	PUNCT
ejpam-3441	733	7	12	12	NUM
ejpam-3441	733	8	(	(	PUNCT
ejpam-3441	733	9	3	3	NUM
ejpam-3441	733	10	)	)	PUNCT
ejpam-3441	733	11	(	(	PUNCT
ejpam-3441	733	12	2019	2019	NUM
ejpam-3441	733	13	)	)	PUNCT
ejpam-3441	733	14	,	,	PUNCT
ejpam-3441	733	15	906	906	NUM
ejpam-3441	733	16	-	-	SYM
ejpam-3441	733	17	943	943	NUM
ejpam-3441	733	18	937	937	NUM
ejpam-3441	733	19	thresholds	threshold	NOUN
ejpam-3441	733	20	(	(	PUNCT
ejpam-3441	733	21	α	α	X
ejpam-3441	733	22	,	,	PUNCT
ejpam-3441	733	23	β	β	X
ejpam-3441	733	24	]	]	PUNCT
ejpam-3441	733	25	of	of	ADP
ejpam-3441	733	26	r.	r.	PROPN
ejpam-3441	733	27	let	let	VERB
ejpam-3441	733	28	x	x	X
ejpam-3441	733	29	∈	∈	PROPN
ejpam-3441	733	30	r	r	NOUN
ejpam-3441	733	31	,	,	PUNCT
ejpam-3441	733	32	this	this	PRON
ejpam-3441	733	33	implies	imply	VERB
ejpam-3441	733	34	that	that	SCONJ
ejpam-3441	733	35	there	there	PRON
ejpam-3441	733	36	exist	exist	VERB
ejpam-3441	733	37	ai	ai	NOUN
ejpam-3441	733	38	,	,	PUNCT
ejpam-3441	733	39	bi	bi	NOUN
ejpam-3441	733	40	∈	∈	PROPN
ejpam-3441	733	41	r	r	NOUN
ejpam-3441	733	42	such	such	ADJ
ejpam-3441	733	43	that	that	SCONJ
ejpam-3441	733	44	x	x	NOUN
ejpam-3441	734	1	=	=	PUNCT
ejpam-3441	734	2	∑n	∑n	PROPN
ejpam-3441	734	3	i=1(aix	i=1(aix	PROPN
ejpam-3441	734	4	2)bi	2)bi	NUM
ejpam-3441	734	5	.	.	PUNCT
ejpam-3441	735	1	now	now	ADV
ejpam-3441	735	2	x	x	X
ejpam-3441	735	3	=	=	SYM
ejpam-3441	735	4	(	(	PUNCT
ejpam-3441	735	5	ai(xx))bi	ai(xx))bi	NOUN
ejpam-3441	735	6	=	=	PUNCT
ejpam-3441	735	7	(	(	PUNCT
ejpam-3441	735	8	x(aix))bi	x(aix))bi	PROPN
ejpam-3441	735	9	=	=	SYM
ejpam-3441	735	10	(	(	PUNCT
ejpam-3441	735	11	bi(aix))x	bi(aix))x	PROPN
ejpam-3441	735	12	.	.	PUNCT
ejpam-3441	736	1	thus	thus	ADV
ejpam-3441	736	2	(	(	PUNCT
ejpam-3441	736	3	µl	µl	ADP
ejpam-3441	736	4	◦	◦	NOUN
ejpam-3441	736	5	βα	βα	NOUN
ejpam-3441	736	6	µc)(x	µc)(x	NOUN
ejpam-3441	736	7	)	)	PUNCT
ejpam-3441	737	1	=	=	PRON
ejpam-3441	737	2	{	{	PUNCT
ejpam-3441	737	3	(	(	PUNCT
ejpam-3441	737	4	µl	µl	PART
ejpam-3441	737	5	◦	◦	VERB
ejpam-3441	737	6	µc)(x	µc)(x	NOUN
ejpam-3441	737	7	)	)	PUNCT
ejpam-3441	738	1	∧	∧	PROPN
ejpam-3441	738	2	β	β	NOUN
ejpam-3441	738	3	}	}	PUNCT
ejpam-3441	738	4	∨	∨	NUM
ejpam-3441	738	5	α	α	NOUN
ejpam-3441	738	6	=	=	X
ejpam-3441	738	7	{	{	PUNCT
ejpam-3441	738	8	(	(	PUNCT
ejpam-3441	738	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	738	10	i=1	i=1	PROPN
ejpam-3441	738	11	piqi	piqi	NOUN
ejpam-3441	738	12	{	{	PUNCT
ejpam-3441	738	13	∧ni=1	∧ni=1	X
ejpam-3441	738	14	{	{	PUNCT
ejpam-3441	738	15	µl	µl	PART
ejpam-3441	738	16	(	(	PUNCT
ejpam-3441	738	17	pi	pi	NOUN
ejpam-3441	738	18	)	)	PUNCT
ejpam-3441	738	19	∧	∧	NOUN
ejpam-3441	738	20	µc	µc	INTJ
ejpam-3441	738	21	(	(	PUNCT
ejpam-3441	738	22	qi	qi	NOUN
ejpam-3441	738	23	)	)	PUNCT
ejpam-3441	738	24	}	}	PUNCT
ejpam-3441	738	25	}	}	PUNCT
ejpam-3441	738	26	)	)	PUNCT
ejpam-3441	738	27	∧	∧	PROPN
ejpam-3441	738	28	β	β	NOUN
ejpam-3441	738	29	}	}	PUNCT
ejpam-3441	738	30	∨	∨	NUM
ejpam-3441	738	31	α	α	PROPN
ejpam-3441	738	32	≥	≥	X
ejpam-3441	738	33	{	{	PUNCT
ejpam-3441	738	34	{	{	PUNCT
ejpam-3441	738	35	µl	µl	PROPN
ejpam-3441	738	36	(	(	PUNCT
ejpam-3441	738	37	bi(aix	bi(aix	NOUN
ejpam-3441	738	38	)	)	PUNCT
ejpam-3441	738	39	)	)	PUNCT
ejpam-3441	739	1	∧	∧	NOUN
ejpam-3441	739	2	µc	µc	INTJ
ejpam-3441	739	3	(	(	PUNCT
ejpam-3441	739	4	x	x	NOUN
ejpam-3441	739	5	)	)	PUNCT
ejpam-3441	739	6	}	}	PUNCT
ejpam-3441	739	7	∧	∧	PROPN
ejpam-3441	739	8	β	β	NOUN
ejpam-3441	739	9	}	}	PUNCT
ejpam-3441	739	10	∨	∨	NUM
ejpam-3441	739	11	α	α	NOUN
ejpam-3441	739	12	=	=	PUNCT
ejpam-3441	739	13	(	(	PUNCT
ejpam-3441	739	14	µl	µl	PART
ejpam-3441	739	15	(	(	PUNCT
ejpam-3441	739	16	bi(aix	bi(aix	NOUN
ejpam-3441	739	17	)	)	PUNCT
ejpam-3441	739	18	)	)	PUNCT
ejpam-3441	740	1	∨	∨	NUM
ejpam-3441	740	2	α	α	NOUN
ejpam-3441	740	3	)	)	PUNCT
ejpam-3441	740	4	∧	∧	NOUN
ejpam-3441	740	5	(	(	PUNCT
ejpam-3441	740	6	µc	µc	INTJ
ejpam-3441	740	7	(	(	PUNCT
ejpam-3441	740	8	x	x	NOUN
ejpam-3441	740	9	)	)	PUNCT
ejpam-3441	740	10	∨	∨	NUM
ejpam-3441	740	11	α	α	NOUN
ejpam-3441	740	12	)	)	PUNCT
ejpam-3441	740	13	∧	∧	PROPN
ejpam-3441	740	14	(	(	PUNCT
ejpam-3441	740	15	β	β	X
ejpam-3441	740	16	∨	∨	NUM
ejpam-3441	740	17	α	α	NOUN
ejpam-3441	740	18	)	)	PUNCT
ejpam-3441	740	19	≥	≥	NOUN
ejpam-3441	740	20	(	(	PUNCT
ejpam-3441	740	21	µl	µl	PART
ejpam-3441	740	22	(	(	PUNCT
ejpam-3441	740	23	x	x	NOUN
ejpam-3441	740	24	)	)	PUNCT
ejpam-3441	740	25	∧	∧	PROPN
ejpam-3441	740	26	β	β	NOUN
ejpam-3441	740	27	)	)	PUNCT
ejpam-3441	740	28	∧	∧	NOUN
ejpam-3441	740	29	µc	µc	INTJ
ejpam-3441	740	30	(	(	PUNCT
ejpam-3441	740	31	x	x	NOUN
ejpam-3441	740	32	)	)	PUNCT
ejpam-3441	740	33	∧	∧	NOUN
ejpam-3441	740	34	β	β	NOUN
ejpam-3441	740	35	=	=	PRON
ejpam-3441	740	36	µl	µl	PART
ejpam-3441	740	37	(	(	PUNCT
ejpam-3441	740	38	x	x	X
ejpam-3441	740	39	)	)	PUNCT
ejpam-3441	740	40	∧	∧	NOUN
ejpam-3441	740	41	µc	µc	INTJ
ejpam-3441	740	42	(	(	PUNCT
ejpam-3441	740	43	x	x	NOUN
ejpam-3441	740	44	)	)	PUNCT
ejpam-3441	740	45	∧	∧	NOUN
ejpam-3441	740	46	β	β	NOUN
ejpam-3441	740	47	=	=	PRON
ejpam-3441	740	48	µc	µc	PROPN
ejpam-3441	740	49	(	(	PUNCT
ejpam-3441	740	50	x	x	NOUN
ejpam-3441	740	51	)	)	PUNCT
ejpam-3441	740	52	∧	∧	NOUN
ejpam-3441	740	53	µl	µl	ADP
ejpam-3441	740	54	(	(	PUNCT
ejpam-3441	740	55	x	x	X
ejpam-3441	740	56	)	)	PUNCT
ejpam-3441	740	57	∧	∧	NOUN
ejpam-3441	740	58	β	β	X
ejpam-3441	740	59	=	=	SYM
ejpam-3441	740	60	(	(	PUNCT
ejpam-3441	740	61	µc	µc	INTJ
ejpam-3441	740	62	∧	∧	NOUN
ejpam-3441	740	63	µl	µl	NOUN
ejpam-3441	740	64	)	)	PUNCT
ejpam-3441	740	65	(	(	PUNCT
ejpam-3441	740	66	x	x	X
ejpam-3441	740	67	)	)	PUNCT
ejpam-3441	740	68	∧	∧	NOUN
ejpam-3441	740	69	β	β	X
ejpam-3441	740	70	=	=	SYM
ejpam-3441	740	71	{	{	PUNCT
ejpam-3441	740	72	(	(	PUNCT
ejpam-3441	740	73	µc	µc	INTJ
ejpam-3441	740	74	∧	∧	NOUN
ejpam-3441	740	75	µl	µl	NOUN
ejpam-3441	740	76	)	)	PUNCT
ejpam-3441	740	77	(	(	PUNCT
ejpam-3441	740	78	x	x	X
ejpam-3441	740	79	)	)	PUNCT
ejpam-3441	740	80	∧	∧	PROPN
ejpam-3441	740	81	β	β	NOUN
ejpam-3441	740	82	}	}	PUNCT
ejpam-3441	740	83	∨	∨	NUM
ejpam-3441	740	84	α	α	NOUN
ejpam-3441	740	85	=	=	PUNCT
ejpam-3441	740	86	(	(	PUNCT
ejpam-3441	740	87	µc	µc	INTJ
ejpam-3441	740	88	∧βα	∧βα	PROPN
ejpam-3441	740	89	µl)(x	µl)(x	PROPN
ejpam-3441	740	90	)	)	PUNCT
ejpam-3441	740	91	.	.	PUNCT
ejpam-3441	741	1	⇒	⇒	NOUN
ejpam-3441	741	2	µc	µc	VERB
ejpam-3441	741	3	∧βα	∧βα	ADV
ejpam-3441	741	4	µl	µl	ADP
ejpam-3441	741	5	⊆	⊆	NUM
ejpam-3441	741	6	µl	µl	ADP
ejpam-3441	741	7	◦	◦	NOUN
ejpam-3441	741	8	βα	βα	NOUN
ejpam-3441	741	9	µc	µc	ADV
ejpam-3441	741	10	.	.	PUNCT
ejpam-3441	742	1	similarly	similarly	ADV
ejpam-3441	742	2	,	,	PUNCT
ejpam-3441	742	3	we	we	PRON
ejpam-3441	742	4	have	have	VERB
ejpam-3441	742	5	γc	γc	NUM
ejpam-3441	742	6	∨βα	∨βα	ADJ
ejpam-3441	742	7	γl	γl	NUM
ejpam-3441	742	8	⊇	⊇	ADJ
ejpam-3441	742	9	γl	γl	PROPN
ejpam-3441	742	10	◦	◦	PROPN
ejpam-3441	742	11	βα	βα	X
ejpam-3441	742	12	γc	γc	X
ejpam-3441	742	13	.	.	PUNCT
ejpam-3441	743	1	hence	hence	ADV
ejpam-3441	743	2	c	c	NOUN
ejpam-3441	743	3	∧βαl	∧βαl	NUM
ejpam-3441	743	4	⊆	⊆	NUM
ejpam-3441	743	5	l	l	NOUN
ejpam-3441	743	6	◦	◦	NOUN
ejpam-3441	743	7	βαc	βαc	ADJ
ejpam-3441	743	8	,	,	PUNCT
ejpam-3441	743	9	i.e.	i.e.	X
ejpam-3441	743	10	,	,	PUNCT
ejpam-3441	743	11	(	(	PUNCT
ejpam-3441	743	12	1	1	X
ejpam-3441	743	13	)	)	PUNCT
ejpam-3441	743	14	implies	imply	VERB
ejpam-3441	743	15	(	(	PUNCT
ejpam-3441	743	16	4	4	NUM
ejpam-3441	743	17	)	)	PUNCT
ejpam-3441	743	18	.	.	PUNCT
ejpam-3441	744	1	it	it	PRON
ejpam-3441	744	2	is	be	AUX
ejpam-3441	744	3	clear	clear	ADJ
ejpam-3441	744	4	that	that	SCONJ
ejpam-3441	744	5	(	(	PUNCT
ejpam-3441	744	6	4	4	X
ejpam-3441	744	7	)	)	PUNCT
ejpam-3441	744	8	⇒	⇒	NOUN
ejpam-3441	744	9	(	(	PUNCT
ejpam-3441	744	10	3	3	NUM
ejpam-3441	744	11	)	)	PUNCT
ejpam-3441	744	12	and	and	CCONJ
ejpam-3441	744	13	(	(	PUNCT
ejpam-3441	744	14	3	3	X
ejpam-3441	744	15	)	)	PUNCT
ejpam-3441	744	16	⇒	⇒	NOUN
ejpam-3441	744	17	(	(	PUNCT
ejpam-3441	744	18	2	2	NUM
ejpam-3441	744	19	)	)	PUNCT
ejpam-3441	744	20	.	.	PUNCT
ejpam-3441	745	1	assume	assume	VERB
ejpam-3441	745	2	that	that	SCONJ
ejpam-3441	745	3	(	(	PUNCT
ejpam-3441	745	4	2	2	X
ejpam-3441	745	5	)	)	PUNCT
ejpam-3441	745	6	holds	hold	VERB
ejpam-3441	745	7	.	.	PUNCT
ejpam-3441	746	1	let	let	VERB
ejpam-3441	746	2	a	a	PRON
ejpam-3441	746	3	be	be	AUX
ejpam-3441	746	4	an	an	DET
ejpam-3441	746	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	746	6	fuzzy	fuzzy	ADJ
ejpam-3441	746	7	right	right	ADJ
ejpam-3441	746	8	ideal	ideal	NOUN
ejpam-3441	746	9	with	with	ADP
ejpam-3441	746	10	thresholds	threshold	NOUN
ejpam-3441	746	11	(	(	PUNCT
ejpam-3441	746	12	α	α	X
ejpam-3441	746	13	,	,	PUNCT
ejpam-3441	746	14	β	β	X
ejpam-3441	746	15	]	]	PUNCT
ejpam-3441	746	16	and	and	CCONJ
ejpam-3441	746	17	l	l	NOUN
ejpam-3441	746	18	be	be	AUX
ejpam-3441	746	19	an	an	DET
ejpam-3441	746	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	746	21	fuzzy	fuzzy	ADJ
ejpam-3441	746	22	left	leave	VERB
ejpam-3441	746	23	ideal	ideal	NOUN
ejpam-3441	746	24	with	with	ADP
ejpam-3441	746	25	thresholds	threshold	NOUN
ejpam-3441	746	26	(	(	PUNCT
ejpam-3441	746	27	α	α	X
ejpam-3441	746	28	,	,	PUNCT
ejpam-3441	746	29	β	β	X
ejpam-3441	746	30	]	]	PUNCT
ejpam-3441	746	31	of	of	ADP
ejpam-3441	746	32	r.	r.	PROPN
ejpam-3441	746	33	since	since	SCONJ
ejpam-3441	746	34	every	every	DET
ejpam-3441	746	35	intuitionistic	intuitionistic	ADJ
ejpam-3441	746	36	fuzzy	fuzzy	ADJ
ejpam-3441	746	37	right	right	ADJ
ejpam-3441	746	38	ideal	ideal	NOUN
ejpam-3441	746	39	with	with	ADP
ejpam-3441	746	40	thresholds	threshold	NOUN
ejpam-3441	746	41	(	(	PUNCT
ejpam-3441	746	42	α	α	X
ejpam-3441	746	43	,	,	PUNCT
ejpam-3441	746	44	β	β	X
ejpam-3441	746	45	]	]	PUNCT
ejpam-3441	746	46	of	of	ADP
ejpam-3441	746	47	r	r	NOUN
ejpam-3441	746	48	is	be	AUX
ejpam-3441	746	49	an	an	DET
ejpam-3441	746	50	intuitionistic	intuitionistic	ADJ
ejpam-3441	746	51	fuzzy	fuzzy	ADJ
ejpam-3441	746	52	quasi	quasi	NOUN
ejpam-3441	746	53	-	-	NOUN
ejpam-3441	746	54	ideal	ideal	ADJ
ejpam-3441	746	55	with	with	ADP
ejpam-3441	746	56	thresholds	threshold	NOUN
ejpam-3441	746	57	(	(	PUNCT
ejpam-3441	746	58	α	α	X
ejpam-3441	746	59	,	,	PUNCT
ejpam-3441	746	60	β	β	X
ejpam-3441	746	61	]	]	PUNCT
ejpam-3441	746	62	of	of	ADP
ejpam-3441	746	63	r	r	NOUN
ejpam-3441	746	64	,	,	PUNCT
ejpam-3441	746	65	this	this	PRON
ejpam-3441	746	66	means	mean	VERB
ejpam-3441	746	67	that	that	SCONJ
ejpam-3441	746	68	a	a	PRON
ejpam-3441	746	69	is	be	AUX
ejpam-3441	746	70	an	an	DET
ejpam-3441	746	71	intuitionistic	intuitionistic	ADJ
ejpam-3441	746	72	fuzzy	fuzzy	ADJ
ejpam-3441	746	73	quasi	quasi	NOUN
ejpam-3441	746	74	-	-	NOUN
ejpam-3441	746	75	ideal	ideal	ADJ
ejpam-3441	746	76	with	with	ADP
ejpam-3441	746	77	thresholds	threshold	NOUN
ejpam-3441	746	78	(	(	PUNCT
ejpam-3441	746	79	α	α	X
ejpam-3441	746	80	,	,	PUNCT
ejpam-3441	746	81	β	β	X
ejpam-3441	746	82	]	]	PUNCT
ejpam-3441	746	83	of	of	ADP
ejpam-3441	746	84	r.	r.	PROPN
ejpam-3441	746	85	by	by	ADP
ejpam-3441	746	86	our	our	PRON
ejpam-3441	746	87	assumption	assumption	NOUN
ejpam-3441	746	88	,	,	PUNCT
ejpam-3441	746	89	a	a	DET
ejpam-3441	746	90	∧βα	∧βα	ADJ
ejpam-3441	746	91	l	l	NOUN
ejpam-3441	746	92	⊆	⊆	NUM
ejpam-3441	746	93	l	l	NOUN
ejpam-3441	746	94	◦	◦	NOUN
ejpam-3441	746	95	βα	βα	NOUN
ejpam-3441	746	96	a.	a.	NOUN
ejpam-3441	746	97	hence	hence	ADV
ejpam-3441	746	98	r	r	NOUN
ejpam-3441	746	99	is	be	AUX
ejpam-3441	746	100	an	an	DET
ejpam-3441	746	101	intra	intra	ADJ
ejpam-3441	746	102	-	-	ADJ
ejpam-3441	746	103	regular	regular	ADJ
ejpam-3441	746	104	by	by	ADP
ejpam-3441	746	105	the	the	DET
ejpam-3441	746	106	theorem	theorem	ADJ
ejpam-3441	746	107	13	13	NUM
ejpam-3441	746	108	,	,	PUNCT
ejpam-3441	746	109	i.e.	i.e.	X
ejpam-3441	746	110	,	,	PUNCT
ejpam-3441	746	111	(	(	PUNCT
ejpam-3441	746	112	2)⇒	2)⇒	NUM
ejpam-3441	746	113	(	(	PUNCT
ejpam-3441	746	114	1	1	NUM
ejpam-3441	746	115	)	)	PUNCT
ejpam-3441	746	116	.	.	PUNCT
ejpam-3441	747	1	theorem	theorem	VERB
ejpam-3441	747	2	16	16	NUM
ejpam-3441	747	3	.	.	PUNCT
ejpam-3441	748	1	let	let	VERB
ejpam-3441	748	2	r	r	PRON
ejpam-3441	748	3	be	be	AUX
ejpam-3441	748	4	an	an	DET
ejpam-3441	748	5	la	la	NOUN
ejpam-3441	748	6	-	-	NOUN
ejpam-3441	748	7	ring	ring	NOUN
ejpam-3441	748	8	with	with	ADP
ejpam-3441	748	9	left	left	ADJ
ejpam-3441	748	10	identity	identity	NOUN
ejpam-3441	748	11	e	e	NOUN
ejpam-3441	748	12	,	,	PUNCT
ejpam-3441	748	13	such	such	ADJ
ejpam-3441	748	14	that	that	SCONJ
ejpam-3441	748	15	(	(	PUNCT
ejpam-3441	748	16	xe)r	xe)r	PROPN
ejpam-3441	748	17	=	=	SYM
ejpam-3441	748	18	xr	xr	PROPN
ejpam-3441	748	19	for	for	ADP
ejpam-3441	748	20	all	all	DET
ejpam-3441	748	21	x	x	PROPN
ejpam-3441	748	22	∈	∈	PROPN
ejpam-3441	748	23	r.	r.	NOUN
ejpam-3441	748	24	then	then	ADV
ejpam-3441	748	25	the	the	DET
ejpam-3441	748	26	following	follow	VERB
ejpam-3441	748	27	conditions	condition	NOUN
ejpam-3441	748	28	are	be	AUX
ejpam-3441	748	29	equivalent	equivalent	ADJ
ejpam-3441	748	30	.	.	PUNCT
ejpam-3441	749	1	(	(	PUNCT
ejpam-3441	749	2	1	1	X
ejpam-3441	749	3	)	)	PUNCT
ejpam-3441	749	4	r	r	NOUN
ejpam-3441	749	5	is	be	AUX
ejpam-3441	749	6	an	an	DET
ejpam-3441	749	7	intra	intra	ADJ
ejpam-3441	749	8	-	-	ADJ
ejpam-3441	749	9	regular	regular	ADJ
ejpam-3441	749	10	.	.	PUNCT
ejpam-3441	750	1	(	(	PUNCT
ejpam-3441	750	2	2	2	X
ejpam-3441	750	3	)	)	PUNCT
ejpam-3441	750	4	a	a	DET
ejpam-3441	750	5	∧βα	∧βα	ADJ
ejpam-3441	750	6	l	l	NOUN
ejpam-3441	750	7	∧βα	∧βα	ADJ
ejpam-3441	750	8	d	d	NOUN
ejpam-3441	750	9	⊆	⊆	NUM
ejpam-3441	750	10	(	(	PUNCT
ejpam-3441	750	11	l	l	NOUN
ejpam-3441	750	12	◦	◦	NOUN
ejpam-3441	750	13	βα	βα	X
ejpam-3441	750	14	a	a	X
ejpam-3441	750	15	)	)	PUNCT
ejpam-3441	750	16	◦	◦	NOUN
ejpam-3441	750	17	βα	βα	NOUN
ejpam-3441	750	18	d	d	NOUN
ejpam-3441	750	19	for	for	ADP
ejpam-3441	750	20	every	every	DET
ejpam-3441	750	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	750	22	fuzzy	fuzzy	ADJ
ejpam-3441	750	23	quasi	quasi	NOUN
ejpam-3441	750	24	-	-	NOUN
ejpam-3441	750	25	ideal	ideal	ADJ
ejpam-3441	750	26	a	a	PRON
ejpam-3441	750	27	with	with	ADP
ejpam-3441	750	28	thresholds	threshold	NOUN
ejpam-3441	750	29	(	(	PUNCT
ejpam-3441	750	30	α	α	X
ejpam-3441	750	31	,	,	PUNCT
ejpam-3441	750	32	β	β	X
ejpam-3441	750	33	]	]	X
ejpam-3441	750	34	,	,	PUNCT
ejpam-3441	750	35	every	every	DET
ejpam-3441	750	36	intuitionistic	intuitionistic	ADJ
ejpam-3441	750	37	fuzzy	fuzzy	ADJ
ejpam-3441	750	38	left	leave	VERB
ejpam-3441	750	39	ideal	ideal	NOUN
ejpam-3441	750	40	l	l	PROPN
ejpam-3441	750	41	with	with	ADP
ejpam-3441	750	42	thresholds	threshold	NOUN
ejpam-3441	750	43	(	(	PUNCT
ejpam-3441	750	44	α	α	X
ejpam-3441	750	45	,	,	PUNCT
ejpam-3441	750	46	β	β	X
ejpam-3441	750	47	]	]	PUNCT
ejpam-3441	750	48	and	and	CCONJ
ejpam-3441	750	49	every	every	DET
ejpam-3441	750	50	intuitionistic	intuitionistic	ADJ
ejpam-3441	750	51	fuzzy	fuzzy	ADJ
ejpam-3441	750	52	right	right	ADJ
ejpam-3441	750	53	ideal	ideal	NOUN
ejpam-3441	751	1	d	d	NOUN
ejpam-3441	751	2	with	with	ADP
ejpam-3441	751	3	thresholds	threshold	NOUN
ejpam-3441	751	4	(	(	PUNCT
ejpam-3441	751	5	α	α	X
ejpam-3441	751	6	,	,	PUNCT
ejpam-3441	751	7	β	β	X
ejpam-3441	751	8	]	]	PUNCT
ejpam-3441	751	9	of	of	ADP
ejpam-3441	751	10	r.	r.	PROPN
ejpam-3441	751	11	(	(	PUNCT
ejpam-3441	751	12	3	3	X
ejpam-3441	751	13	)	)	PUNCT
ejpam-3441	751	14	b∧βαl∧βαd	b∧βαl∧βαd	ADV
ejpam-3441	751	15	⊆	⊆	NUM
ejpam-3441	751	16	(	(	PUNCT
ejpam-3441	751	17	l	l	NOUN
ejpam-3441	751	18	◦	◦	NOUN
ejpam-3441	751	19	βαb)	βαb)	NOUN
ejpam-3441	751	20	◦	◦	NOUN
ejpam-3441	751	21	βαd	βαd	NOUN
ejpam-3441	751	22	for	for	ADP
ejpam-3441	751	23	every	every	DET
ejpam-3441	751	24	intuitionistic	intuitionistic	ADJ
ejpam-3441	751	25	fuzzy	fuzzy	ADJ
ejpam-3441	751	26	bi	bi	ADJ
ejpam-3441	751	27	-	-	ADJ
ejpam-3441	751	28	ideal	ideal	ADJ
ejpam-3441	751	29	b	b	PROPN
ejpam-3441	751	30	with	with	ADP
ejpam-3441	751	31	thresholds	threshold	NOUN
ejpam-3441	751	32	(	(	PUNCT
ejpam-3441	751	33	α	α	X
ejpam-3441	751	34	,	,	PUNCT
ejpam-3441	751	35	β	β	X
ejpam-3441	751	36	]	]	X
ejpam-3441	751	37	,	,	PUNCT
ejpam-3441	751	38	every	every	DET
ejpam-3441	751	39	intuitionistic	intuitionistic	ADJ
ejpam-3441	751	40	fuzzy	fuzzy	ADJ
ejpam-3441	751	41	left	leave	VERB
ejpam-3441	751	42	ideal	ideal	NOUN
ejpam-3441	751	43	l	l	PROPN
ejpam-3441	751	44	with	with	ADP
ejpam-3441	751	45	thresholds	threshold	NOUN
ejpam-3441	751	46	(	(	PUNCT
ejpam-3441	751	47	α	α	X
ejpam-3441	751	48	,	,	PUNCT
ejpam-3441	751	49	β	β	X
ejpam-3441	751	50	]	]	PUNCT
ejpam-3441	751	51	and	and	CCONJ
ejpam-3441	751	52	every	every	DET
ejpam-3441	751	53	intuitionistic	intuitionistic	ADJ
ejpam-3441	751	54	fuzzy	fuzzy	ADJ
ejpam-3441	751	55	right	right	ADJ
ejpam-3441	751	56	ideal	ideal	NOUN
ejpam-3441	751	57	d	d	NOUN
ejpam-3441	751	58	with	with	ADP
ejpam-3441	751	59	thresholds	threshold	NOUN
ejpam-3441	751	60	(	(	PUNCT
ejpam-3441	751	61	α	α	X
ejpam-3441	751	62	,	,	PUNCT
ejpam-3441	751	63	β	β	X
ejpam-3441	751	64	]	]	PUNCT
ejpam-3441	751	65	of	of	ADP
ejpam-3441	751	66	r.	r.	PROPN
ejpam-3441	751	67	(	(	PUNCT
ejpam-3441	751	68	4	4	NUM
ejpam-3441	751	69	)	)	PUNCT
ejpam-3441	751	70	c	c	NOUN
ejpam-3441	752	1	∧βα	∧βα	ADJ
ejpam-3441	752	2	l	l	NOUN
ejpam-3441	752	3	∧βα	∧βα	ADJ
ejpam-3441	752	4	d	d	NOUN
ejpam-3441	752	5	⊆	⊆	NUM
ejpam-3441	752	6	(	(	PUNCT
ejpam-3441	752	7	l	l	NOUN
ejpam-3441	752	8	◦	◦	NOUN
ejpam-3441	752	9	βα	βα	NOUN
ejpam-3441	752	10	c	c	NOUN
ejpam-3441	752	11	)	)	PUNCT
ejpam-3441	752	12	◦	◦	NOUN
ejpam-3441	752	13	βα	βα	NOUN
ejpam-3441	752	14	d	d	NOUN
ejpam-3441	752	15	for	for	ADP
ejpam-3441	752	16	every	every	DET
ejpam-3441	752	17	intuitionistic	intuitionistic	ADJ
ejpam-3441	752	18	fuzzy	fuzzy	ADJ
ejpam-3441	752	19	generalized	generalize	VERB
ejpam-3441	752	20	bi	bi	ADJ
ejpam-3441	752	21	-	-	ADJ
ejpam-3441	752	22	ideal	ideal	ADJ
ejpam-3441	752	23	c	c	NOUN
ejpam-3441	752	24	with	with	ADP
ejpam-3441	752	25	thresholds	threshold	NOUN
ejpam-3441	752	26	(	(	PUNCT
ejpam-3441	752	27	α	α	X
ejpam-3441	752	28	,	,	PUNCT
ejpam-3441	752	29	β	β	X
ejpam-3441	752	30	]	]	X
ejpam-3441	752	31	,	,	PUNCT
ejpam-3441	752	32	every	every	DET
ejpam-3441	752	33	intuitionistic	intuitionistic	ADJ
ejpam-3441	752	34	fuzzy	fuzzy	ADJ
ejpam-3441	752	35	left	leave	VERB
ejpam-3441	752	36	ideal	ideal	NOUN
ejpam-3441	752	37	l	l	PROPN
ejpam-3441	752	38	with	with	ADP
ejpam-3441	752	39	thresholds	threshold	NOUN
ejpam-3441	752	40	(	(	PUNCT
ejpam-3441	752	41	α	α	X
ejpam-3441	752	42	,	,	PUNCT
ejpam-3441	752	43	β	β	X
ejpam-3441	752	44	]	]	PUNCT
ejpam-3441	752	45	and	and	CCONJ
ejpam-3441	752	46	every	every	DET
ejpam-3441	752	47	intuitionistic	intuitionistic	ADJ
ejpam-3441	752	48	fuzzy	fuzzy	ADJ
ejpam-3441	752	49	right	right	ADJ
ejpam-3441	752	50	ideal	ideal	NOUN
ejpam-3441	752	51	d	d	NOUN
ejpam-3441	752	52	with	with	ADP
ejpam-3441	752	53	thresholds	threshold	NOUN
ejpam-3441	752	54	(	(	PUNCT
ejpam-3441	752	55	α	α	X
ejpam-3441	752	56	,	,	PUNCT
ejpam-3441	752	57	β	β	X
ejpam-3441	752	58	]	]	PUNCT
ejpam-3441	752	59	of	of	ADP
ejpam-3441	752	60	r.	r.	PROPN
ejpam-3441	752	61	proof	proof	PROPN
ejpam-3441	752	62	.	.	PUNCT
ejpam-3441	753	1	assume	assume	VERB
ejpam-3441	753	2	that	that	SCONJ
ejpam-3441	753	3	(	(	PUNCT
ejpam-3441	753	4	1	1	X
ejpam-3441	753	5	)	)	PUNCT
ejpam-3441	753	6	holds	hold	VERB
ejpam-3441	753	7	.	.	PUNCT
ejpam-3441	754	1	let	let	AUX
ejpam-3441	754	2	c	c	NOUN
ejpam-3441	754	3	=	=	PUNCT
ejpam-3441	754	4	(	(	PUNCT
ejpam-3441	754	5	µc	µc	INTJ
ejpam-3441	754	6	,	,	PUNCT
ejpam-3441	754	7	γc	γc	PROPN
ejpam-3441	754	8	)	)	PUNCT
ejpam-3441	754	9	be	be	AUX
ejpam-3441	754	10	an	an	DET
ejpam-3441	754	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	754	12	fuzzy	fuzzy	ADJ
ejpam-3441	754	13	generalized	generalize	VERB
ejpam-3441	754	14	bi	bi	NOUN
ejpam-3441	754	15	-	-	NOUN
ejpam-3441	754	16	ideal	ideal	NOUN
ejpam-3441	754	17	with	with	ADP
ejpam-3441	754	18	thresholds	threshold	NOUN
ejpam-3441	754	19	(	(	PUNCT
ejpam-3441	754	20	α	α	X
ejpam-3441	754	21	,	,	PUNCT
ejpam-3441	754	22	β	β	X
ejpam-3441	754	23	]	]	X
ejpam-3441	754	24	,	,	PUNCT
ejpam-3441	754	25	l	l	NOUN
ejpam-3441	754	26	=	=	SYM
ejpam-3441	754	27	(	(	PUNCT
ejpam-3441	754	28	µl	µl	NOUN
ejpam-3441	754	29	,	,	PUNCT
ejpam-3441	754	30	γl	γl	NUM
ejpam-3441	754	31	)	)	PUNCT
ejpam-3441	754	32	be	be	VERB
ejpam-3441	754	33	an	an	DET
ejpam-3441	754	34	intuitionistic	intuitionistic	ADJ
ejpam-3441	754	35	fuzzy	fuzzy	ADJ
ejpam-3441	754	36	left	leave	VERB
ejpam-3441	754	37	ideal	ideal	NOUN
ejpam-3441	754	38	with	with	ADP
ejpam-3441	754	39	thresholds	threshold	NOUN
ejpam-3441	754	40	(	(	PUNCT
ejpam-3441	754	41	α	α	X
ejpam-3441	754	42	,	,	PUNCT
ejpam-3441	754	43	β	β	X
ejpam-3441	754	44	]	]	PUNCT
ejpam-3441	754	45	and	and	CCONJ
ejpam-3441	754	46	d	d	NOUN
ejpam-3441	754	47	=	=	PUNCT
ejpam-3441	754	48	(	(	PUNCT
ejpam-3441	754	49	µd	µd	ADP
ejpam-3441	754	50	,	,	PUNCT
ejpam-3441	754	51	γd	γd	ADV
ejpam-3441	754	52	)	)	PUNCT
ejpam-3441	754	53	be	be	AUX
ejpam-3441	754	54	an	an	DET
ejpam-3441	754	55	intuitionistic	intuitionistic	ADJ
ejpam-3441	754	56	fuzzy	fuzzy	ADJ
ejpam-3441	754	57	right	right	ADJ
ejpam-3441	754	58	ideal	ideal	NOUN
ejpam-3441	754	59	with	with	ADP
ejpam-3441	754	60	thresholds	threshold	NOUN
ejpam-3441	754	61	(	(	PUNCT
ejpam-3441	754	62	α	α	X
ejpam-3441	754	63	,	,	PUNCT
ejpam-3441	754	64	β	β	X
ejpam-3441	754	65	]	]	PUNCT
ejpam-3441	754	66	of	of	ADP
ejpam-3441	754	67	r.	r.	PROPN
ejpam-3441	754	68	let	let	VERB
ejpam-3441	754	69	x	x	X
ejpam-3441	754	70	∈	∈	PROPN
ejpam-3441	754	71	r	r	NOUN
ejpam-3441	754	72	,	,	PUNCT
ejpam-3441	754	73	this	this	PRON
ejpam-3441	754	74	means	mean	VERB
ejpam-3441	754	75	that	that	SCONJ
ejpam-3441	754	76	there	there	PRON
ejpam-3441	754	77	exist	exist	VERB
ejpam-3441	754	78	ai	ai	NOUN
ejpam-3441	754	79	,	,	PUNCT
ejpam-3441	754	80	bi	bi	NOUN
ejpam-3441	754	81	∈	∈	PROPN
ejpam-3441	754	82	r	r	NOUN
ejpam-3441	754	83	such	such	ADJ
ejpam-3441	754	84	that	that	SCONJ
ejpam-3441	754	85	x	x	NOUN
ejpam-3441	755	1	=	=	PUNCT
ejpam-3441	755	2	∑n	∑n	PROPN
ejpam-3441	755	3	i=1(aix	i=1(aix	PROPN
ejpam-3441	755	4	2)bi	2)bi	NUM
ejpam-3441	755	5	.	.	PUNCT
ejpam-3441	756	1	now	now	ADV
ejpam-3441	756	2	x	x	X
ejpam-3441	756	3	=	=	SYM
ejpam-3441	756	4	(	(	PUNCT
ejpam-3441	756	5	ai(xx))bi	ai(xx))bi	NOUN
ejpam-3441	756	6	=	=	PUNCT
ejpam-3441	756	7	(	(	PUNCT
ejpam-3441	756	8	x(aix))bi	x(aix))bi	PROPN
ejpam-3441	756	9	=	=	PRON
ejpam-3441	756	10	(	(	PUNCT
ejpam-3441	756	11	bi(aix))x	bi(aix))x	PROPN
ejpam-3441	756	12	and	and	CCONJ
ejpam-3441	756	13	bi(aix	bi(aix	NOUN
ejpam-3441	756	14	)	)	PUNCT
ejpam-3441	756	15	=	=	PROPN
ejpam-3441	757	1	bi(ai((aix	bi(ai((aix	PROPN
ejpam-3441	757	2	2)bi	2)bi	PROPN
ejpam-3441	757	3	)	)	PUNCT
ejpam-3441	757	4	)	)	PUNCT
ejpam-3441	758	1	=	=	SYM
ejpam-3441	758	2	bi((aix	bi((aix	ADJ
ejpam-3441	758	3	2)(aibi	2)(aibi	NUM
ejpam-3441	758	4	)	)	PUNCT
ejpam-3441	758	5	)	)	PUNCT
ejpam-3441	759	1	k.	k.	PROPN
ejpam-3441	760	1	nasreen	nasreen	PROPN
ejpam-3441	760	2	et	et	PROPN
ejpam-3441	760	3	al	al	PROPN
ejpam-3441	760	4	.	.	PUNCT
ejpam-3441	760	5	/	/	SYM
ejpam-3441	760	6	eur	eur	PROPN
ejpam-3441	760	7	.	.	PUNCT
ejpam-3441	761	1	j.	j.	PROPN
ejpam-3441	761	2	pure	pure	PROPN
ejpam-3441	761	3	appl	appl	PROPN
ejpam-3441	761	4	.	.	PROPN
ejpam-3441	761	5	math	math	PROPN
ejpam-3441	761	6	,	,	PUNCT
ejpam-3441	761	7	12	12	NUM
ejpam-3441	761	8	(	(	PUNCT
ejpam-3441	761	9	3	3	NUM
ejpam-3441	761	10	)	)	PUNCT
ejpam-3441	761	11	(	(	PUNCT
ejpam-3441	761	12	2019	2019	NUM
ejpam-3441	761	13	)	)	PUNCT
ejpam-3441	761	14	,	,	PUNCT
ejpam-3441	761	15	906	906	NUM
ejpam-3441	761	16	-	-	SYM
ejpam-3441	761	17	943	943	NUM
ejpam-3441	761	18	938	938	NUM
ejpam-3441	761	19	=	=	NOUN
ejpam-3441	761	20	bi((aix	bi((aix	PROPN
ejpam-3441	761	21	2)ci	2)ci	NUM
ejpam-3441	761	22	)	)	PUNCT
ejpam-3441	761	23	=	=	PUNCT
ejpam-3441	762	1	(	(	PUNCT
ejpam-3441	762	2	aix	aix	PROPN
ejpam-3441	762	3	2)(bici	2)(bici	NUM
ejpam-3441	762	4	)	)	PUNCT
ejpam-3441	763	1	=	=	PRON
ejpam-3441	764	1	(	(	PUNCT
ejpam-3441	764	2	aix	aix	NOUN
ejpam-3441	764	3	2)di	2)di	NUM
ejpam-3441	764	4	=	=	PUNCT
ejpam-3441	764	5	(	(	PUNCT
ejpam-3441	764	6	ai(xx))di	ai(xx))di	PROPN
ejpam-3441	764	7	=	=	PUNCT
ejpam-3441	764	8	(	(	PUNCT
ejpam-3441	764	9	x(aix))di	x(aix))di	NOUN
ejpam-3441	764	10	=	=	SYM
ejpam-3441	764	11	(	(	PUNCT
ejpam-3441	764	12	di(aix))x	di(aix))x	PROPN
ejpam-3441	764	13	.	.	PUNCT
ejpam-3441	765	1	thus	thus	ADV
ejpam-3441	765	2	(	(	PUNCT
ejpam-3441	765	3	(	(	PUNCT
ejpam-3441	765	4	µl	µl	ADP
ejpam-3441	765	5	◦	◦	NOUN
ejpam-3441	765	6	βα	βα	VERB
ejpam-3441	765	7	µc	µc	NOUN
ejpam-3441	765	8	)	)	PUNCT
ejpam-3441	765	9	◦	◦	NOUN
ejpam-3441	765	10	βα	βα	NOUN
ejpam-3441	765	11	µd)(x	µd)(x	PROPN
ejpam-3441	765	12	)	)	PUNCT
ejpam-3441	766	1	=	=	PRON
ejpam-3441	766	2	{	{	PUNCT
ejpam-3441	766	3	(	(	PUNCT
ejpam-3441	766	4	(	(	PUNCT
ejpam-3441	766	5	µl	µl	ADP
ejpam-3441	766	6	◦	◦	NOUN
ejpam-3441	766	7	µc	µc	NOUN
ejpam-3441	766	8	)	)	PUNCT
ejpam-3441	766	9	◦	◦	NOUN
ejpam-3441	766	10	µd)(x	µd)(x	PROPN
ejpam-3441	766	11	)	)	PUNCT
ejpam-3441	767	1	∧	∧	PROPN
ejpam-3441	767	2	β	β	NOUN
ejpam-3441	767	3	}	}	PUNCT
ejpam-3441	767	4	∨	∨	NUM
ejpam-3441	767	5	α	α	NOUN
ejpam-3441	767	6	=	=	X
ejpam-3441	767	7	{	{	PUNCT
ejpam-3441	767	8	(	(	PUNCT
ejpam-3441	767	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	767	10	i=1	i=1	PROPN
ejpam-3441	767	11	piqi	piqi	NOUN
ejpam-3441	767	12	{	{	PUNCT
ejpam-3441	767	13	∧ni=1	∧ni=1	X
ejpam-3441	767	14	{	{	PUNCT
ejpam-3441	767	15	(	(	PUNCT
ejpam-3441	767	16	µl	µl	ADP
ejpam-3441	767	17	◦	◦	NOUN
ejpam-3441	767	18	µc	µc	PUNCT
ejpam-3441	767	19	)	)	PUNCT
ejpam-3441	767	20	(	(	PUNCT
ejpam-3441	767	21	pi	pi	NOUN
ejpam-3441	767	22	)	)	PUNCT
ejpam-3441	767	23	∧	∧	NOUN
ejpam-3441	767	24	µd	µd	ADP
ejpam-3441	767	25	(	(	PUNCT
ejpam-3441	767	26	qi	qi	NOUN
ejpam-3441	767	27	)	)	PUNCT
ejpam-3441	767	28	}	}	PUNCT
ejpam-3441	767	29	}	}	PUNCT
ejpam-3441	767	30	)	)	PUNCT
ejpam-3441	767	31	∧	∧	PROPN
ejpam-3441	767	32	β	β	NOUN
ejpam-3441	767	33	}	}	PUNCT
ejpam-3441	767	34	∨	∨	NUM
ejpam-3441	767	35	α	α	PROPN
ejpam-3441	767	36	≥	≥	X
ejpam-3441	767	37	{	{	PUNCT
ejpam-3441	767	38	{	{	PUNCT
ejpam-3441	767	39	(	(	PUNCT
ejpam-3441	767	40	µl	µl	PART
ejpam-3441	767	41	◦	◦	NOUN
ejpam-3441	767	42	µc	µc	PUNCT
ejpam-3441	767	43	)	)	PUNCT
ejpam-3441	767	44	(	(	PUNCT
ejpam-3441	767	45	bi(aix	bi(aix	NOUN
ejpam-3441	767	46	)	)	PUNCT
ejpam-3441	767	47	)	)	PUNCT
ejpam-3441	768	1	∧	∧	NOUN
ejpam-3441	768	2	µd	µd	ADP
ejpam-3441	768	3	(	(	PUNCT
ejpam-3441	768	4	x	x	NOUN
ejpam-3441	768	5	)	)	PUNCT
ejpam-3441	768	6	}	}	PUNCT
ejpam-3441	768	7	∧	∧	PROPN
ejpam-3441	768	8	β	β	NOUN
ejpam-3441	768	9	}	}	PUNCT
ejpam-3441	768	10	∨	∨	NUM
ejpam-3441	768	11	α	α	NOUN
ejpam-3441	768	12	=	=	SYM
ejpam-3441	768	13	(	(	PUNCT
ejpam-3441	768	14	(	(	PUNCT
ejpam-3441	768	15	µl	µl	ADP
ejpam-3441	768	16	◦	◦	NOUN
ejpam-3441	768	17	µc	µc	PUNCT
ejpam-3441	768	18	)	)	PUNCT
ejpam-3441	768	19	(	(	PUNCT
ejpam-3441	768	20	bi(aix	bi(aix	NOUN
ejpam-3441	768	21	)	)	PUNCT
ejpam-3441	768	22	)	)	PUNCT
ejpam-3441	769	1	∨	∨	NUM
ejpam-3441	769	2	α	α	NOUN
ejpam-3441	769	3	)	)	PUNCT
ejpam-3441	769	4	∧	∧	NOUN
ejpam-3441	769	5	(	(	PUNCT
ejpam-3441	769	6	µd	µd	X
ejpam-3441	769	7	(	(	PUNCT
ejpam-3441	769	8	x	x	NOUN
ejpam-3441	769	9	)	)	PUNCT
ejpam-3441	769	10	∨	∨	NUM
ejpam-3441	769	11	α	α	NOUN
ejpam-3441	769	12	)	)	PUNCT
ejpam-3441	769	13	∧	∧	PROPN
ejpam-3441	769	14	(	(	PUNCT
ejpam-3441	769	15	β	β	X
ejpam-3441	769	16	∨	∨	NUM
ejpam-3441	769	17	α	α	NOUN
ejpam-3441	769	18	)	)	PUNCT
ejpam-3441	769	19	≥	≥	NOUN
ejpam-3441	769	20	(	(	PUNCT
ejpam-3441	769	21	(	(	PUNCT
ejpam-3441	769	22	µl	µl	ADP
ejpam-3441	769	23	◦	◦	NOUN
ejpam-3441	769	24	µc	µc	PUNCT
ejpam-3441	769	25	)	)	PUNCT
ejpam-3441	769	26	(	(	PUNCT
ejpam-3441	769	27	bi(aix	bi(aix	NOUN
ejpam-3441	769	28	)	)	PUNCT
ejpam-3441	769	29	)	)	PUNCT
ejpam-3441	770	1	∨	∨	NUM
ejpam-3441	770	2	α	α	NOUN
ejpam-3441	770	3	)	)	PUNCT
ejpam-3441	770	4	∧	∧	NOUN
ejpam-3441	770	5	µd(x	µd(x	NOUN
ejpam-3441	770	6	)	)	PUNCT
ejpam-3441	770	7	∧	∧	NOUN
ejpam-3441	770	8	β	β	X
ejpam-3441	770	9	=	=	SYM
ejpam-3441	770	10	(	(	PUNCT
ejpam-3441	770	11	(	(	PUNCT
ejpam-3441	770	12	∨bi(aix)=∑n	∨bi(aix)=∑n	NOUN
ejpam-3441	770	13	i=1mini	i=1mini	PRON
ejpam-3441	770	14	{	{	PUNCT
ejpam-3441	770	15	∧ni=1	∧ni=1	X
ejpam-3441	770	16	{	{	PUNCT
ejpam-3441	770	17	µl	µl	PROPN
ejpam-3441	770	18	(	(	PUNCT
ejpam-3441	770	19	mi	mi	NOUN
ejpam-3441	770	20	)	)	PUNCT
ejpam-3441	770	21	∧	∧	NOUN
ejpam-3441	770	22	µc	µc	PROPN
ejpam-3441	770	23	(	(	PUNCT
ejpam-3441	770	24	ni	ni	NOUN
ejpam-3441	770	25	)	)	PUNCT
ejpam-3441	770	26	}	}	PUNCT
ejpam-3441	770	27	}	}	PUNCT
ejpam-3441	770	28	)	)	PUNCT
ejpam-3441	770	29	∨	∨	NUM
ejpam-3441	770	30	α	α	NOUN
ejpam-3441	770	31	)	)	PUNCT
ejpam-3441	770	32	∧	∧	NOUN
ejpam-3441	770	33	µd(x	µd(x	NOUN
ejpam-3441	770	34	)	)	PUNCT
ejpam-3441	770	35	∧	∧	PROPN
ejpam-3441	770	36	β	β	X
ejpam-3441	770	37	≥	≥	X
ejpam-3441	770	38	(	(	PUNCT
ejpam-3441	770	39	{	{	PUNCT
ejpam-3441	770	40	µl(di(aix	µl(di(aix	ADJ
ejpam-3441	770	41	)	)	PUNCT
ejpam-3441	770	42	)	)	PUNCT
ejpam-3441	771	1	∧	∧	PROPN
ejpam-3441	771	2	µc(x	µc(x	NOUN
ejpam-3441	771	3	)	)	PUNCT
ejpam-3441	771	4	}	}	PUNCT
ejpam-3441	771	5	∨	∨	NUM
ejpam-3441	771	6	α	α	NOUN
ejpam-3441	771	7	)	)	PUNCT
ejpam-3441	771	8	∧	∧	NOUN
ejpam-3441	771	9	µd(x	µd(x	NOUN
ejpam-3441	771	10	)	)	PUNCT
ejpam-3441	772	1	∧	∧	NOUN
ejpam-3441	772	2	β	β	X
ejpam-3441	772	3	=	=	SYM
ejpam-3441	772	4	(	(	PUNCT
ejpam-3441	772	5	µl(di(aix	µl(di(aix	NOUN
ejpam-3441	772	6	)	)	PUNCT
ejpam-3441	772	7	)	)	PUNCT
ejpam-3441	773	1	∨	∨	NUM
ejpam-3441	773	2	α	α	NOUN
ejpam-3441	773	3	)	)	PUNCT
ejpam-3441	773	4	∧	∧	PROPN
ejpam-3441	773	5	(	(	PUNCT
ejpam-3441	773	6	µc(x	µc(x	NOUN
ejpam-3441	773	7	)	)	PUNCT
ejpam-3441	773	8	∨	∨	NUM
ejpam-3441	773	9	α	α	NOUN
ejpam-3441	773	10	)	)	PUNCT
ejpam-3441	773	11	∧	∧	NOUN
ejpam-3441	773	12	µd(x	µd(x	NOUN
ejpam-3441	773	13	)	)	PUNCT
ejpam-3441	773	14	∧	∧	PROPN
ejpam-3441	773	15	β	β	X
ejpam-3441	773	16	≥	≥	X
ejpam-3441	773	17	(	(	PUNCT
ejpam-3441	773	18	µl(x	µl(x	X
ejpam-3441	773	19	)	)	PUNCT
ejpam-3441	773	20	∧	∧	PROPN
ejpam-3441	773	21	β	β	NOUN
ejpam-3441	773	22	)	)	PUNCT
ejpam-3441	773	23	∧	∧	PROPN
ejpam-3441	773	24	µc(x	µc(x	NOUN
ejpam-3441	773	25	)	)	PUNCT
ejpam-3441	773	26	∧	∧	NOUN
ejpam-3441	773	27	µd(x	µd(x	NOUN
ejpam-3441	773	28	)	)	PUNCT
ejpam-3441	773	29	∧	∧	PROPN
ejpam-3441	773	30	β	β	X
ejpam-3441	773	31	=	=	SYM
ejpam-3441	773	32	µl(x	µl(x	NOUN
ejpam-3441	773	33	)	)	PUNCT
ejpam-3441	773	34	∧	∧	PROPN
ejpam-3441	773	35	µc(x	µc(x	NOUN
ejpam-3441	773	36	)	)	PUNCT
ejpam-3441	773	37	∧	∧	NOUN
ejpam-3441	773	38	µd(x	µd(x	NOUN
ejpam-3441	773	39	)	)	PUNCT
ejpam-3441	773	40	∧	∧	NOUN
ejpam-3441	773	41	β	β	X
ejpam-3441	773	42	=	=	SYM
ejpam-3441	773	43	(	(	PUNCT
ejpam-3441	773	44	µl	µl	ADP
ejpam-3441	773	45	∧	∧	PROPN
ejpam-3441	773	46	µc	µc	ADP
ejpam-3441	773	47	∧	∧	PROPN
ejpam-3441	773	48	µd)(x	µd)(x	PROPN
ejpam-3441	773	49	)	)	PUNCT
ejpam-3441	774	1	∧	∧	NOUN
ejpam-3441	774	2	β	β	X
ejpam-3441	774	3	=	=	SYM
ejpam-3441	774	4	{	{	PUNCT
ejpam-3441	774	5	(	(	PUNCT
ejpam-3441	774	6	µc	µc	INTJ
ejpam-3441	774	7	∧	∧	PROPN
ejpam-3441	774	8	µl	µl	ADP
ejpam-3441	774	9	∧	∧	PROPN
ejpam-3441	774	10	µd)(x	µd)(x	PROPN
ejpam-3441	774	11	)	)	PUNCT
ejpam-3441	775	1	∧	∧	PROPN
ejpam-3441	775	2	β	β	NOUN
ejpam-3441	775	3	}	}	PUNCT
ejpam-3441	775	4	∨	∨	NUM
ejpam-3441	775	5	α	α	NOUN
ejpam-3441	775	6	=	=	PUNCT
ejpam-3441	775	7	(	(	PUNCT
ejpam-3441	775	8	µc	µc	INTJ
ejpam-3441	775	9	∧βα	∧βα	ADJ
ejpam-3441	775	10	µl	µl	ADP
ejpam-3441	775	11	∧βα	∧βα	PROPN
ejpam-3441	775	12	µd)(x	µd)(x	PROPN
ejpam-3441	775	13	)	)	PUNCT
ejpam-3441	775	14	.	.	PUNCT
ejpam-3441	776	1	⇒	⇒	NOUN
ejpam-3441	776	2	µc	µc	VERB
ejpam-3441	776	3	∧βα	∧βα	NOUN
ejpam-3441	776	4	µl	µl	ADP
ejpam-3441	776	5	∧βα	∧βα	ADJ
ejpam-3441	776	6	µd	µd	ADP
ejpam-3441	776	7	⊆	⊆	NUM
ejpam-3441	776	8	(	(	PUNCT
ejpam-3441	776	9	µl	µl	ADP
ejpam-3441	776	10	◦	◦	NOUN
ejpam-3441	776	11	βα	βα	VERB
ejpam-3441	776	12	µc	µc	NOUN
ejpam-3441	776	13	)	)	PUNCT
ejpam-3441	776	14	◦	◦	NOUN
ejpam-3441	776	15	βα	βα	NOUN
ejpam-3441	776	16	µd	µd	NOUN
ejpam-3441	776	17	.	.	PUNCT
ejpam-3441	777	1	similarly	similarly	ADV
ejpam-3441	777	2	,	,	PUNCT
ejpam-3441	777	3	we	we	PRON
ejpam-3441	777	4	have	have	VERB
ejpam-3441	777	5	µc∨βαµl∨βαµd	µc∨βαµl∨βαµd	NOUN
ejpam-3441	777	6	⊇	⊇	X
ejpam-3441	777	7	(	(	PUNCT
ejpam-3441	777	8	µl	µl	ADP
ejpam-3441	777	9	◦	◦	NOUN
ejpam-3441	777	10	βαµc)	βαµc)	NOUN
ejpam-3441	777	11	◦	◦	NOUN
ejpam-3441	777	12	βαµd	βαµd	NOUN
ejpam-3441	777	13	.	.	PUNCT
ejpam-3441	778	1	hence	hence	ADV
ejpam-3441	778	2	c∧βαl∧βαd	c∧βαl∧βαd	NOUN
ejpam-3441	778	3	⊆	⊆	NUM
ejpam-3441	778	4	(	(	PUNCT
ejpam-3441	778	5	l	l	NOUN
ejpam-3441	778	6	◦	◦	NOUN
ejpam-3441	778	7	βαc)	βαc)	NOUN
ejpam-3441	778	8	◦	◦	NOUN
ejpam-3441	778	9	βαd	βαd	NOUN
ejpam-3441	778	10	,	,	PUNCT
ejpam-3441	778	11	i.e.	i.e.	X
ejpam-3441	778	12	,	,	PUNCT
ejpam-3441	778	13	(	(	PUNCT
ejpam-3441	778	14	1	1	X
ejpam-3441	778	15	)	)	PUNCT
ejpam-3441	778	16	implies	imply	VERB
ejpam-3441	778	17	(	(	PUNCT
ejpam-3441	778	18	4	4	NUM
ejpam-3441	778	19	)	)	PUNCT
ejpam-3441	778	20	.	.	PUNCT
ejpam-3441	779	1	since	since	SCONJ
ejpam-3441	779	2	(	(	PUNCT
ejpam-3441	779	3	4	4	X
ejpam-3441	779	4	)	)	PUNCT
ejpam-3441	779	5	⇒	⇒	NOUN
ejpam-3441	779	6	(	(	PUNCT
ejpam-3441	779	7	3	3	NUM
ejpam-3441	779	8	)	)	PUNCT
ejpam-3441	779	9	and	and	CCONJ
ejpam-3441	779	10	(	(	PUNCT
ejpam-3441	779	11	3	3	X
ejpam-3441	779	12	)	)	PUNCT
ejpam-3441	779	13	⇒	⇒	NOUN
ejpam-3441	779	14	(	(	PUNCT
ejpam-3441	779	15	2	2	NUM
ejpam-3441	779	16	)	)	PUNCT
ejpam-3441	779	17	.	.	PUNCT
ejpam-3441	779	18	suppose	suppose	VERB
ejpam-3441	779	19	that	that	SCONJ
ejpam-3441	779	20	(	(	PUNCT
ejpam-3441	779	21	2	2	X
ejpam-3441	779	22	)	)	PUNCT
ejpam-3441	779	23	holds	hold	NOUN
ejpam-3441	779	24	.	.	PUNCT
ejpam-3441	780	1	then	then	ADV
ejpam-3441	780	2	a∧βα	a∧βα	VERB
ejpam-3441	780	3	r∧βαd	r∧βαd	NUM
ejpam-3441	780	4	⊆	⊆	NUM
ejpam-3441	780	5	(	(	PUNCT
ejpam-3441	780	6	r	r	NOUN
ejpam-3441	780	7	◦	◦	NOUN
ejpam-3441	780	8	βα	βα	X
ejpam-3441	780	9	a	a	PRON
ejpam-3441	780	10	)	)	PUNCT
ejpam-3441	780	11	◦	◦	NOUN
ejpam-3441	780	12	βαd	βαd	NOUN
ejpam-3441	780	13	,	,	PUNCT
ejpam-3441	780	14	where	where	SCONJ
ejpam-3441	780	15	a	a	PRON
ejpam-3441	780	16	is	be	AUX
ejpam-3441	780	17	an	an	DET
ejpam-3441	780	18	intuitionistic	intuitionistic	ADJ
ejpam-3441	780	19	fuzzy	fuzzy	ADJ
ejpam-3441	780	20	left	leave	VERB
ejpam-3441	780	21	ideal	ideal	NOUN
ejpam-3441	780	22	with	with	ADP
ejpam-3441	780	23	thresholds	threshold	NOUN
ejpam-3441	780	24	(	(	PUNCT
ejpam-3441	780	25	α	α	X
ejpam-3441	780	26	,	,	PUNCT
ejpam-3441	780	27	β	β	X
ejpam-3441	780	28	]	]	PUNCT
ejpam-3441	780	29	of	of	ADP
ejpam-3441	780	30	r	r	NOUN
ejpam-3441	780	31	,	,	PUNCT
ejpam-3441	780	32	i.e.	i.e.	X
ejpam-3441	780	33	,	,	PUNCT
ejpam-3441	780	34	a	a	DET
ejpam-3441	780	35	∧βα	∧βα	ADJ
ejpam-3441	780	36	d	d	NOUN
ejpam-3441	780	37	⊆	⊆	NUM
ejpam-3441	780	38	a	a	DET
ejpam-3441	780	39	◦	◦	NOUN
ejpam-3441	780	40	βα	βα	X
ejpam-3441	780	41	d.	d.	PROPN
ejpam-3441	780	42	therefore	therefore	ADV
ejpam-3441	780	43	r	r	NOUN
ejpam-3441	780	44	is	be	AUX
ejpam-3441	780	45	an	an	DET
ejpam-3441	780	46	intra	intra	ADJ
ejpam-3441	780	47	-	-	ADJ
ejpam-3441	780	48	regular	regular	ADJ
ejpam-3441	780	49	,	,	PUNCT
ejpam-3441	780	50	i.e.	i.e.	X
ejpam-3441	780	51	,	,	PUNCT
ejpam-3441	780	52	(	(	PUNCT
ejpam-3441	780	53	2)⇒	2)⇒	NUM
ejpam-3441	780	54	(	(	PUNCT
ejpam-3441	780	55	1	1	NUM
ejpam-3441	780	56	)	)	PUNCT
ejpam-3441	780	57	.	.	PUNCT
ejpam-3441	781	1	5	5	X
ejpam-3441	781	2	.	.	X
ejpam-3441	781	3	regular	regular	ADJ
ejpam-3441	781	4	and	and	CCONJ
ejpam-3441	781	5	intra	intra	ADJ
ejpam-3441	781	6	-	-	ADJ
ejpam-3441	781	7	regular	regular	ADJ
ejpam-3441	781	8	la	la	NOUN
ejpam-3441	781	9	-	-	PUNCT
ejpam-3441	781	10	rings	ring	NOUN
ejpam-3441	781	11	in	in	ADP
ejpam-3441	781	12	this	this	DET
ejpam-3441	781	13	section	section	NOUN
ejpam-3441	781	14	,	,	PUNCT
ejpam-3441	781	15	we	we	PRON
ejpam-3441	781	16	characterize	characterize	VERB
ejpam-3441	781	17	both	both	CCONJ
ejpam-3441	781	18	regular	regular	ADJ
ejpam-3441	781	19	and	and	CCONJ
ejpam-3441	781	20	intra	intra	ADJ
ejpam-3441	781	21	-	-	ADJ
ejpam-3441	781	22	regular	regular	ADJ
ejpam-3441	781	23	la	la	NOUN
ejpam-3441	781	24	-	-	PUNCT
ejpam-3441	781	25	rings	ring	NOUN
ejpam-3441	781	26	in	in	ADP
ejpam-3441	781	27	terms	term	NOUN
ejpam-3441	781	28	of	of	ADP
ejpam-3441	781	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	781	30	fuzzy	fuzzy	ADJ
ejpam-3441	781	31	left	left	NOUN
ejpam-3441	781	32	(	(	PUNCT
ejpam-3441	781	33	right	right	ADJ
ejpam-3441	781	34	,	,	PUNCT
ejpam-3441	781	35	quasi-	quasi-	INTJ
ejpam-3441	781	36	,	,	PUNCT
ejpam-3441	781	37	bi-	bi-	NUM
ejpam-3441	781	38	,	,	PUNCT
ejpam-3441	781	39	generalized	generalize	VERB
ejpam-3441	781	40	bi-	bi-	NUM
ejpam-3441	781	41	)	)	PUNCT
ejpam-3441	781	42	ideals	ideal	NOUN
ejpam-3441	781	43	with	with	ADP
ejpam-3441	781	44	thresholds	threshold	NOUN
ejpam-3441	781	45	(	(	PUNCT
ejpam-3441	781	46	α	α	X
ejpam-3441	781	47	,	,	PUNCT
ejpam-3441	781	48	β	β	X
ejpam-3441	781	49	]	]	PUNCT
ejpam-3441	781	50	.	.	PUNCT
ejpam-3441	782	1	theorem	theorem	NOUN
ejpam-3441	782	2	17	17	NUM
ejpam-3441	782	3	.	.	PUNCT
ejpam-3441	783	1	let	let	VERB
ejpam-3441	783	2	r	r	PRON
ejpam-3441	783	3	be	be	AUX
ejpam-3441	783	4	an	an	DET
ejpam-3441	783	5	la	la	NOUN
ejpam-3441	783	6	-	-	NOUN
ejpam-3441	783	7	ring	ring	NOUN
ejpam-3441	783	8	with	with	ADP
ejpam-3441	783	9	left	left	ADJ
ejpam-3441	783	10	identity	identity	NOUN
ejpam-3441	783	11	e	e	NOUN
ejpam-3441	783	12	,	,	PUNCT
ejpam-3441	783	13	such	such	ADJ
ejpam-3441	783	14	that	that	SCONJ
ejpam-3441	783	15	(	(	PUNCT
ejpam-3441	783	16	xe)r	xe)r	PROPN
ejpam-3441	783	17	=	=	SYM
ejpam-3441	783	18	xr	xr	PROPN
ejpam-3441	783	19	for	for	ADP
ejpam-3441	783	20	all	all	DET
ejpam-3441	783	21	x	x	PROPN
ejpam-3441	783	22	∈	∈	PROPN
ejpam-3441	783	23	r.	r.	NOUN
ejpam-3441	783	24	then	then	ADV
ejpam-3441	783	25	the	the	DET
ejpam-3441	783	26	following	follow	VERB
ejpam-3441	783	27	conditions	condition	NOUN
ejpam-3441	783	28	are	be	AUX
ejpam-3441	783	29	equivalent	equivalent	ADJ
ejpam-3441	783	30	.	.	PUNCT
ejpam-3441	784	1	(	(	PUNCT
ejpam-3441	784	2	1	1	X
ejpam-3441	784	3	)	)	PUNCT
ejpam-3441	784	4	r	r	NOUN
ejpam-3441	784	5	is	be	AUX
ejpam-3441	784	6	both	both	CCONJ
ejpam-3441	784	7	a	a	DET
ejpam-3441	784	8	regular	regular	ADJ
ejpam-3441	784	9	and	and	CCONJ
ejpam-3441	784	10	an	an	DET
ejpam-3441	784	11	intra	intra	ADJ
ejpam-3441	784	12	-	-	ADJ
ejpam-3441	784	13	regular	regular	ADJ
ejpam-3441	784	14	.	.	PUNCT
ejpam-3441	785	1	(	(	PUNCT
ejpam-3441	785	2	2	2	X
ejpam-3441	785	3	)	)	PUNCT
ejpam-3441	785	4	every	every	DET
ejpam-3441	785	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	785	6	fuzzy	fuzzy	ADJ
ejpam-3441	785	7	quasi	quasi	NOUN
ejpam-3441	785	8	-	-	NOUN
ejpam-3441	785	9	ideal	ideal	ADJ
ejpam-3441	785	10	with	with	ADP
ejpam-3441	785	11	thresholds	threshold	NOUN
ejpam-3441	785	12	(	(	PUNCT
ejpam-3441	785	13	α	α	X
ejpam-3441	785	14	,	,	PUNCT
ejpam-3441	785	15	β	β	X
ejpam-3441	785	16	]	]	PUNCT
ejpam-3441	785	17	of	of	ADP
ejpam-3441	785	18	r	r	NOUN
ejpam-3441	785	19	is	be	AUX
ejpam-3441	785	20	an	an	DET
ejpam-3441	785	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	785	22	fuzzy	fuzzy	ADJ
ejpam-3441	785	23	idempotent	idempotent	NOUN
ejpam-3441	785	24	with	with	ADP
ejpam-3441	785	25	thresholds	threshold	NOUN
ejpam-3441	785	26	(	(	PUNCT
ejpam-3441	785	27	α	α	X
ejpam-3441	785	28	,	,	PUNCT
ejpam-3441	785	29	β	β	X
ejpam-3441	785	30	]	]	PUNCT
ejpam-3441	785	31	.	.	PUNCT
ejpam-3441	786	1	proof	proof	NOUN
ejpam-3441	786	2	.	.	PUNCT
ejpam-3441	787	1	suppose	suppose	VERB
ejpam-3441	787	2	that	that	SCONJ
ejpam-3441	787	3	r	r	NOUN
ejpam-3441	787	4	is	be	AUX
ejpam-3441	787	5	both	both	CCONJ
ejpam-3441	787	6	a	a	DET
ejpam-3441	787	7	regular	regular	ADJ
ejpam-3441	787	8	and	and	CCONJ
ejpam-3441	787	9	an	an	DET
ejpam-3441	787	10	intra	intra	ADJ
ejpam-3441	787	11	-	-	ADJ
ejpam-3441	787	12	regular	regular	ADJ
ejpam-3441	787	13	.	.	PUNCT
ejpam-3441	788	1	let	let	VERB
ejpam-3441	788	2	a	a	DET
ejpam-3441	788	3	=	=	SYM
ejpam-3441	788	4	(	(	PUNCT
ejpam-3441	788	5	µa	µa	PROPN
ejpam-3441	788	6	,	,	PUNCT
ejpam-3441	788	7	γa	γa	PROPN
ejpam-3441	788	8	)	)	PUNCT
ejpam-3441	788	9	be	be	VERB
ejpam-3441	788	10	an	an	DET
ejpam-3441	788	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	788	12	fuzzy	fuzzy	ADJ
ejpam-3441	788	13	quasi	quasi	NOUN
ejpam-3441	788	14	-	-	NOUN
ejpam-3441	788	15	ideal	ideal	ADJ
ejpam-3441	788	16	with	with	ADP
ejpam-3441	788	17	thresholds	threshold	NOUN
ejpam-3441	788	18	(	(	PUNCT
ejpam-3441	788	19	α	α	X
ejpam-3441	788	20	,	,	PUNCT
ejpam-3441	788	21	β	β	X
ejpam-3441	788	22	]	]	PUNCT
ejpam-3441	788	23	of	of	ADP
ejpam-3441	788	24	r.then	r.then	ADV
ejpam-3441	788	25	a	a	DET
ejpam-3441	788	26	be	be	AUX
ejpam-3441	788	27	an	an	DET
ejpam-3441	788	28	intuitionistic	intuitionistic	ADJ
ejpam-3441	788	29	fuzzy	fuzzy	ADJ
ejpam-3441	788	30	bi	bi	NOUN
ejpam-3441	788	31	-	-	NOUN
ejpam-3441	788	32	ideal	ideal	ADJ
ejpam-3441	788	33	with	with	ADP
ejpam-3441	788	34	thresholds	threshold	NOUN
ejpam-3441	788	35	(	(	PUNCT
ejpam-3441	788	36	α	α	X
ejpam-3441	788	37	,	,	PUNCT
ejpam-3441	788	38	β	β	X
ejpam-3441	788	39	]	]	PUNCT
ejpam-3441	788	40	of	of	ADP
ejpam-3441	788	41	r	r	NOUN
ejpam-3441	788	42	and	and	CCONJ
ejpam-3441	788	43	a	a	DET
ejpam-3441	788	44	◦	◦	NOUN
ejpam-3441	788	45	βα	βα	NOUN
ejpam-3441	788	46	a	a	DET
ejpam-3441	788	47	⊆	⊆	NUM
ejpam-3441	788	48	aβα	aβα	NOUN
ejpam-3441	788	49	.	.	PUNCT
ejpam-3441	789	1	let	let	VERB
ejpam-3441	789	2	x	x	PUNCT
ejpam-3441	789	3	∈	∈	PROPN
ejpam-3441	789	4	r	r	NOUN
ejpam-3441	789	5	,	,	PUNCT
ejpam-3441	789	6	this	this	PRON
ejpam-3441	789	7	implies	imply	VERB
ejpam-3441	789	8	k.	k.	PROPN
ejpam-3441	789	9	nasreen	nasreen	PROPN
ejpam-3441	789	10	et	et	PROPN
ejpam-3441	789	11	al	al	PROPN
ejpam-3441	789	12	.	.	PUNCT
ejpam-3441	789	13	/	/	SYM
ejpam-3441	789	14	eur	eur	PROPN
ejpam-3441	789	15	.	.	PUNCT
ejpam-3441	790	1	j.	j.	PROPN
ejpam-3441	790	2	pure	pure	PROPN
ejpam-3441	790	3	appl	appl	PROPN
ejpam-3441	790	4	.	.	PROPN
ejpam-3441	790	5	math	math	PROPN
ejpam-3441	790	6	,	,	PUNCT
ejpam-3441	790	7	12	12	NUM
ejpam-3441	790	8	(	(	PUNCT
ejpam-3441	790	9	3	3	NUM
ejpam-3441	790	10	)	)	PUNCT
ejpam-3441	790	11	(	(	PUNCT
ejpam-3441	790	12	2019	2019	NUM
ejpam-3441	790	13	)	)	PUNCT
ejpam-3441	790	14	,	,	PUNCT
ejpam-3441	790	15	906	906	NUM
ejpam-3441	790	16	-	-	SYM
ejpam-3441	790	17	943	943	NUM
ejpam-3441	790	18	939	939	NUM
ejpam-3441	790	19	that	that	SCONJ
ejpam-3441	790	20	there	there	PRON
ejpam-3441	790	21	exists	exist	VERB
ejpam-3441	790	22	a	a	DET
ejpam-3441	790	23	∈	∈	NOUN
ejpam-3441	790	24	r	r	NOUN
ejpam-3441	791	1	such	such	ADJ
ejpam-3441	791	2	that	that	PRON
ejpam-3441	791	3	x	x	SYM
ejpam-3441	791	4	=	=	SYM
ejpam-3441	791	5	(	(	PUNCT
ejpam-3441	791	6	xa)x	xa)x	PROPN
ejpam-3441	791	7	,	,	PUNCT
ejpam-3441	791	8	and	and	CCONJ
ejpam-3441	791	9	also	also	ADV
ejpam-3441	791	10	there	there	PRON
ejpam-3441	791	11	exist	exist	VERB
ejpam-3441	791	12	ai	ai	NOUN
ejpam-3441	791	13	,	,	PUNCT
ejpam-3441	791	14	bi	bi	NOUN
ejpam-3441	791	15	∈	∈	PROPN
ejpam-3441	791	16	r	r	NOUN
ejpam-3441	791	17	such	such	ADJ
ejpam-3441	791	18	that	that	SCONJ
ejpam-3441	791	19	x	x	NOUN
ejpam-3441	791	20	=	=	PUNCT
ejpam-3441	791	21	∑n	∑n	PROPN
ejpam-3441	791	22	i=1(aix	i=1(aix	PROPN
ejpam-3441	791	23	2)bi	2)bi	NUM
ejpam-3441	791	24	.	.	PUNCT
ejpam-3441	792	1	now	now	ADV
ejpam-3441	792	2	x	x	X
ejpam-3441	792	3	=	=	SYM
ejpam-3441	792	4	(	(	PUNCT
ejpam-3441	792	5	xa)x	xa)x	PROPN
ejpam-3441	792	6	xa	xa	PROPN
ejpam-3441	792	7	=	=	PRON
ejpam-3441	792	8	(	(	PUNCT
ejpam-3441	792	9	(	(	PUNCT
ejpam-3441	792	10	aix	aix	NOUN
ejpam-3441	792	11	2)bi)a	2)bi)a	NOUN
ejpam-3441	792	12	=	=	SYM
ejpam-3441	792	13	(	(	PUNCT
ejpam-3441	792	14	abi)(aix	abi)(aix	NOUN
ejpam-3441	792	15	2	2	NUM
ejpam-3441	792	16	)	)	PUNCT
ejpam-3441	792	17	=	=	SYM
ejpam-3441	792	18	ci(ai(xx	ci(ai(xx	NOUN
ejpam-3441	792	19	)	)	PUNCT
ejpam-3441	792	20	)	)	PUNCT
ejpam-3441	793	1	=	=	SYM
ejpam-3441	793	2	ci(x(aix	ci(x(aix	NOUN
ejpam-3441	793	3	)	)	PUNCT
ejpam-3441	793	4	)	)	PUNCT
ejpam-3441	794	1	=	=	PUNCT
ejpam-3441	794	2	x(ci(aix	x(ci(aix	PROPN
ejpam-3441	794	3	)	)	PUNCT
ejpam-3441	794	4	)	)	PUNCT
ejpam-3441	795	1	=	=	PUNCT
ejpam-3441	795	2	x((eci)(aix	x((eci)(aix	PROPN
ejpam-3441	795	3	)	)	PUNCT
ejpam-3441	795	4	)	)	PUNCT
ejpam-3441	796	1	=	=	SYM
ejpam-3441	796	2	x((xai)(cie	x((xai)(cie	PROPN
ejpam-3441	796	3	)	)	PUNCT
ejpam-3441	796	4	)	)	PUNCT
ejpam-3441	797	1	=	=	PUNCT
ejpam-3441	797	2	x((xai)di	x((xai)di	X
ejpam-3441	797	3	)	)	PUNCT
ejpam-3441	797	4	=	=	SYM
ejpam-3441	797	5	x((diai)x	x((diai)x	NUM
ejpam-3441	797	6	)	)	PUNCT
ejpam-3441	797	7	=	=	SYM
ejpam-3441	798	1	x(lix	x(lix	X
ejpam-3441	798	2	)	)	PUNCT
ejpam-3441	798	3	=	=	SYM
ejpam-3441	798	4	li(xx	li(xx	PROPN
ejpam-3441	798	5	)	)	PUNCT
ejpam-3441	799	1	=	=	PRON
ejpam-3441	799	2	(	(	PUNCT
ejpam-3441	799	3	eli)(xx	eli)(xx	NOUN
ejpam-3441	799	4	)	)	PUNCT
ejpam-3441	799	5	=	=	SYM
ejpam-3441	799	6	(	(	PUNCT
ejpam-3441	799	7	xx)(lie	xx)(lie	NUM
ejpam-3441	799	8	)	)	PUNCT
ejpam-3441	799	9	=	=	PUNCT
ejpam-3441	800	1	(	(	PUNCT
ejpam-3441	800	2	xx)mi	xx)mi	PUNCT
ejpam-3441	800	3	=	=	SYM
ejpam-3441	800	4	(	(	PUNCT
ejpam-3441	800	5	mix)x	mix)x	ADJ
ejpam-3441	800	6	.	.	PUNCT
ejpam-3441	800	7	mix	mix	NOUN
ejpam-3441	800	8	=	=	PUNCT
ejpam-3441	800	9	mi((aix	mi((aix	NOUN
ejpam-3441	800	10	2)bi	2)bi	PROPN
ejpam-3441	800	11	)	)	PUNCT
ejpam-3441	800	12	=	=	PUNCT
ejpam-3441	800	13	(	(	PUNCT
ejpam-3441	800	14	aix	aix	NOUN
ejpam-3441	800	15	2)(mibi	2)(mibi	NUM
ejpam-3441	800	16	)	)	PUNCT
ejpam-3441	801	1	=	=	PUNCT
ejpam-3441	801	2	(	(	PUNCT
ejpam-3441	801	3	ai(xx))ni	ai(xx))ni	NOUN
ejpam-3441	801	4	=	=	SYM
ejpam-3441	801	5	(	(	PUNCT
ejpam-3441	801	6	x(aix))ni	x(aix))ni	PROPN
ejpam-3441	801	7	=	=	SYM
ejpam-3441	801	8	(	(	PUNCT
ejpam-3441	801	9	x(aix))(eni	x(aix))(eni	NOUN
ejpam-3441	801	10	)	)	PUNCT
ejpam-3441	801	11	=	=	SYM
ejpam-3441	801	12	(	(	PUNCT
ejpam-3441	801	13	xe)((aix)ni	xe)((aix)ni	PROPN
ejpam-3441	801	14	)	)	PUNCT
ejpam-3441	801	15	=	=	SYM
ejpam-3441	801	16	(	(	PUNCT
ejpam-3441	801	17	xe)((aix)(eni	xe)((aix)(eni	NOUN
ejpam-3441	801	18	)	)	PUNCT
ejpam-3441	801	19	)	)	PUNCT
ejpam-3441	802	1	=	=	SYM
ejpam-3441	802	2	(	(	PUNCT
ejpam-3441	802	3	xe)((aie)(xni	xe)((aie)(xni	ADJ
ejpam-3441	802	4	)	)	PUNCT
ejpam-3441	802	5	)	)	PUNCT
ejpam-3441	803	1	=	=	PUNCT
ejpam-3441	803	2	(	(	PUNCT
ejpam-3441	803	3	xe)(x((aie)ni	xe)(x((aie)ni	NUM
ejpam-3441	803	4	)	)	PUNCT
ejpam-3441	803	5	)	)	PUNCT
ejpam-3441	804	1	=	=	SYM
ejpam-3441	804	2	(	(	PUNCT
ejpam-3441	804	3	xe)(xui	xe)(xui	NUM
ejpam-3441	804	4	)	)	PUNCT
ejpam-3441	804	5	=	=	SYM
ejpam-3441	804	6	x((xe)ui	x((xe)ui	PROPN
ejpam-3441	804	7	)	)	PUNCT
ejpam-3441	804	8	=	=	PUNCT
ejpam-3441	804	9	xwi	xwi	X
ejpam-3441	804	10	.	.	PUNCT
ejpam-3441	805	1	⇒	⇒	PROPN
ejpam-3441	805	2	xa	xa	PROPN
ejpam-3441	806	1	=	=	PRON
ejpam-3441	806	2	(	(	PUNCT
ejpam-3441	806	3	mix)x	mix)x	PROPN
ejpam-3441	806	4	=	=	SYM
ejpam-3441	806	5	(	(	PUNCT
ejpam-3441	806	6	xwi)x	xwi)x	PROPN
ejpam-3441	806	7	.	.	PUNCT
ejpam-3441	807	1	thus	thus	ADV
ejpam-3441	807	2	(	(	PUNCT
ejpam-3441	807	3	µa	µa	ADP
ejpam-3441	807	4	◦	◦	NOUN
ejpam-3441	807	5	βα	βα	ADJ
ejpam-3441	807	6	µa)(x	µa)(x	NOUN
ejpam-3441	807	7	)	)	PUNCT
ejpam-3441	808	1	=	=	PRON
ejpam-3441	808	2	{	{	PUNCT
ejpam-3441	808	3	(	(	PUNCT
ejpam-3441	808	4	µa	µa	ADP
ejpam-3441	808	5	◦	◦	NOUN
ejpam-3441	808	6	µa)(x	µa)(x	NOUN
ejpam-3441	808	7	)	)	PUNCT
ejpam-3441	809	1	∧	∧	PROPN
ejpam-3441	809	2	β	β	NOUN
ejpam-3441	809	3	}	}	PUNCT
ejpam-3441	809	4	∨	∨	NUM
ejpam-3441	809	5	α	α	NOUN
ejpam-3441	809	6	=	=	X
ejpam-3441	809	7	{	{	PUNCT
ejpam-3441	809	8	(	(	PUNCT
ejpam-3441	809	9	∨x=∑n	∨x=∑n	NUM
ejpam-3441	809	10	i=1	i=1	PROPN
ejpam-3441	809	11	piqi	piqi	NOUN
ejpam-3441	809	12	{	{	PUNCT
ejpam-3441	809	13	∧ni=1	∧ni=1	X
ejpam-3441	809	14	{	{	PUNCT
ejpam-3441	809	15	µa	µa	X
ejpam-3441	809	16	(	(	PUNCT
ejpam-3441	809	17	pi	pi	NOUN
ejpam-3441	809	18	)	)	PUNCT
ejpam-3441	809	19	∧	∧	PROPN
ejpam-3441	809	20	µa	µa	PROPN
ejpam-3441	809	21	(	(	PUNCT
ejpam-3441	809	22	qi	qi	NOUN
ejpam-3441	809	23	)	)	PUNCT
ejpam-3441	809	24	}	}	PUNCT
ejpam-3441	809	25	}	}	PUNCT
ejpam-3441	809	26	)	)	PUNCT
ejpam-3441	810	1	∧	∧	PROPN
ejpam-3441	810	2	β	β	NOUN
ejpam-3441	810	3	}	}	PUNCT
ejpam-3441	810	4	∨	∨	NUM
ejpam-3441	810	5	α	α	PROPN
ejpam-3441	810	6	≥	≥	X
ejpam-3441	810	7	{	{	PUNCT
ejpam-3441	810	8	{	{	PUNCT
ejpam-3441	810	9	µa	µa	X
ejpam-3441	810	10	(	(	PUNCT
ejpam-3441	810	11	(	(	PUNCT
ejpam-3441	810	12	xwi)x	xwi)x	NOUN
ejpam-3441	810	13	)	)	PUNCT
ejpam-3441	810	14	∧	∧	NOUN
ejpam-3441	810	15	µa	µa	INTJ
ejpam-3441	810	16	(	(	PUNCT
ejpam-3441	810	17	x	x	NOUN
ejpam-3441	810	18	)	)	PUNCT
ejpam-3441	810	19	}	}	PUNCT
ejpam-3441	810	20	∧	∧	PROPN
ejpam-3441	810	21	β	β	NOUN
ejpam-3441	810	22	}	}	PUNCT
ejpam-3441	810	23	∨	∨	NUM
ejpam-3441	810	24	α	α	NOUN
ejpam-3441	810	25	=	=	SYM
ejpam-3441	810	26	(	(	PUNCT
ejpam-3441	810	27	µa	µa	INTJ
ejpam-3441	810	28	(	(	PUNCT
ejpam-3441	810	29	(	(	PUNCT
ejpam-3441	810	30	xwi)x	xwi)x	PROPN
ejpam-3441	810	31	)	)	PUNCT
ejpam-3441	810	32	∨	∨	NUM
ejpam-3441	810	33	α	α	NOUN
ejpam-3441	810	34	)	)	PUNCT
ejpam-3441	810	35	∧	∧	PROPN
ejpam-3441	810	36	(	(	PUNCT
ejpam-3441	810	37	µa	µa	X
ejpam-3441	810	38	(	(	PUNCT
ejpam-3441	810	39	x	x	NOUN
ejpam-3441	810	40	)	)	PUNCT
ejpam-3441	810	41	∨	∨	NUM
ejpam-3441	810	42	α	α	NOUN
ejpam-3441	810	43	)	)	PUNCT
ejpam-3441	810	44	∧	∧	PROPN
ejpam-3441	810	45	(	(	PUNCT
ejpam-3441	810	46	β	β	X
ejpam-3441	810	47	∨	∨	NUM
ejpam-3441	810	48	α	α	NOUN
ejpam-3441	810	49	)	)	PUNCT
ejpam-3441	810	50	≥	≥	NOUN
ejpam-3441	810	51	(	(	PUNCT
ejpam-3441	810	52	µa	µa	PROPN
ejpam-3441	810	53	(	(	PUNCT
ejpam-3441	810	54	x	x	NOUN
ejpam-3441	810	55	)	)	PUNCT
ejpam-3441	810	56	∧	∧	NOUN
ejpam-3441	810	57	µa	µa	INTJ
ejpam-3441	810	58	(	(	PUNCT
ejpam-3441	810	59	x	x	X
ejpam-3441	810	60	)	)	PUNCT
ejpam-3441	810	61	∧	∧	PROPN
ejpam-3441	810	62	β	β	NOUN
ejpam-3441	810	63	)	)	PUNCT
ejpam-3441	810	64	∧	∧	PROPN
ejpam-3441	810	65	µa	µa	INTJ
ejpam-3441	810	66	(	(	PUNCT
ejpam-3441	810	67	x	x	X
ejpam-3441	810	68	)	)	PUNCT
ejpam-3441	810	69	∧	∧	NOUN
ejpam-3441	810	70	β	β	X
ejpam-3441	810	71	=	=	SYM
ejpam-3441	810	72	µa	µa	X
ejpam-3441	810	73	(	(	PUNCT
ejpam-3441	810	74	x	x	X
ejpam-3441	810	75	)	)	PUNCT
ejpam-3441	810	76	∧	∧	NOUN
ejpam-3441	810	77	β	β	X
ejpam-3441	810	78	=	=	SYM
ejpam-3441	810	79	(	(	PUNCT
ejpam-3441	810	80	µa	µa	INTJ
ejpam-3441	810	81	(	(	PUNCT
ejpam-3441	810	82	x	x	NOUN
ejpam-3441	810	83	)	)	PUNCT
ejpam-3441	810	84	∧	∧	PROPN
ejpam-3441	810	85	β	β	NOUN
ejpam-3441	810	86	)	)	PUNCT
ejpam-3441	810	87	∨	∨	NUM
ejpam-3441	810	88	α	α	NOUN
ejpam-3441	810	89	=	=	SYM
ejpam-3441	810	90	(	(	PUNCT
ejpam-3441	810	91	µa)βα(x	µa)βα(x	PROPN
ejpam-3441	810	92	)	)	PUNCT
ejpam-3441	810	93	.	.	PUNCT
ejpam-3441	811	1	⇒	⇒	NOUN
ejpam-3441	811	2	(	(	PUNCT
ejpam-3441	811	3	µa)βα	µa)βα	NUM
ejpam-3441	811	4	⊆	⊆	NUM
ejpam-3441	811	5	µa	µa	NOUN
ejpam-3441	811	6	◦	◦	NOUN
ejpam-3441	811	7	βα	βα	NOUN
ejpam-3441	811	8	µa	µa	PROPN
ejpam-3441	811	9	.	.	PROPN
ejpam-3441	812	1	similarly	similarly	ADV
ejpam-3441	812	2	,	,	PUNCT
ejpam-3441	812	3	we	we	PRON
ejpam-3441	812	4	have	have	VERB
ejpam-3441	812	5	(	(	PUNCT
ejpam-3441	812	6	γa)βα	γa)βα	X
ejpam-3441	812	7	⊇	⊇	PROPN
ejpam-3441	812	8	γa	γa	PROPN
ejpam-3441	812	9	◦	◦	PROPN
ejpam-3441	812	10	βα	βα	PROPN
ejpam-3441	812	11	γa	γa	PROPN
ejpam-3441	812	12	.	.	PUNCT
ejpam-3441	813	1	hence	hence	ADV
ejpam-3441	813	2	aβα	aβα	NOUN
ejpam-3441	813	3	=	=	PUNCT
ejpam-3441	813	4	a	a	DET
ejpam-3441	813	5	◦	◦	NOUN
ejpam-3441	813	6	βα	βα	NOUN
ejpam-3441	813	7	a.	a.	NOUN
ejpam-3441	813	8	conversely	conversely	ADV
ejpam-3441	813	9	,	,	PUNCT
ejpam-3441	813	10	assume	assume	VERB
ejpam-3441	813	11	that	that	SCONJ
ejpam-3441	813	12	every	every	DET
ejpam-3441	813	13	intuitionistic	intuitionistic	ADJ
ejpam-3441	813	14	fuzzy	fuzzy	ADJ
ejpam-3441	813	15	quasi	quasi	NOUN
ejpam-3441	813	16	-	-	NOUN
ejpam-3441	813	17	ideal	ideal	ADJ
ejpam-3441	813	18	with	with	ADP
ejpam-3441	813	19	thresholds	threshold	NOUN
ejpam-3441	813	20	(	(	PUNCT
ejpam-3441	813	21	α	α	X
ejpam-3441	813	22	,	,	PUNCT
ejpam-3441	813	23	β	β	X
ejpam-3441	813	24	]	]	PUNCT
ejpam-3441	813	25	of	of	ADP
ejpam-3441	813	26	r	r	NOUN
ejpam-3441	813	27	is	be	AUX
ejpam-3441	813	28	an	an	DET
ejpam-3441	813	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	813	30	fuzzy	fuzzy	ADJ
ejpam-3441	813	31	idempotent	idempotent	NOUN
ejpam-3441	813	32	with	with	ADP
ejpam-3441	813	33	thresholds	threshold	NOUN
ejpam-3441	813	34	(	(	PUNCT
ejpam-3441	813	35	α	α	X
ejpam-3441	813	36	,	,	PUNCT
ejpam-3441	813	37	β	β	X
ejpam-3441	813	38	]	]	PUNCT
ejpam-3441	813	39	.	.	PUNCT
ejpam-3441	814	1	let	let	VERB
ejpam-3441	814	2	a	a	DET
ejpam-3441	814	3	∈	∈	ADJ
ejpam-3441	814	4	r	r	NOUN
ejpam-3441	814	5	,	,	PUNCT
ejpam-3441	814	6	then	then	ADV
ejpam-3441	814	7	ra	ra	PROPN
ejpam-3441	814	8	is	be	AUX
ejpam-3441	814	9	a	a	DET
ejpam-3441	814	10	left	left	ADJ
ejpam-3441	814	11	ideal	ideal	NOUN
ejpam-3441	814	12	of	of	ADP
ejpam-3441	814	13	r	r	NOUN
ejpam-3441	814	14	containing	contain	VERB
ejpam-3441	814	15	a	a	PRON
ejpam-3441	814	16	by	by	ADP
ejpam-3441	814	17	the	the	DET
ejpam-3441	814	18	lemma	lemma	PROPN
ejpam-3441	814	19	19	19	NUM
ejpam-3441	814	20	.	.	PUNCT
ejpam-3441	815	1	this	this	PRON
ejpam-3441	815	2	implies	imply	VERB
ejpam-3441	815	3	that	that	SCONJ
ejpam-3441	815	4	ra	ra	PROPN
ejpam-3441	815	5	is	be	AUX
ejpam-3441	815	6	a	a	DET
ejpam-3441	815	7	quasi	quasi	NOUN
ejpam-3441	815	8	-	-	NOUN
ejpam-3441	815	9	ideal	ideal	NOUN
ejpam-3441	815	10	of	of	ADP
ejpam-3441	815	11	r	r	NOUN
ejpam-3441	815	12	,	,	PUNCT
ejpam-3441	815	13	so	so	SCONJ
ejpam-3441	815	14	χra	χra	PROPN
ejpam-3441	815	15	is	be	AUX
ejpam-3441	815	16	an	an	DET
ejpam-3441	815	17	intuitionistic	intuitionistic	ADJ
ejpam-3441	815	18	fuzzy	fuzzy	ADJ
ejpam-3441	815	19	quasi	quasi	NOUN
ejpam-3441	815	20	-	-	NOUN
ejpam-3441	815	21	ideal	ideal	ADJ
ejpam-3441	815	22	with	with	ADP
ejpam-3441	815	23	thresholds	threshold	NOUN
ejpam-3441	815	24	(	(	PUNCT
ejpam-3441	815	25	α	α	X
ejpam-3441	815	26	,	,	PUNCT
ejpam-3441	815	27	β	β	X
ejpam-3441	815	28	]	]	PUNCT
ejpam-3441	815	29	of	of	ADP
ejpam-3441	815	30	r	r	NOUN
ejpam-3441	815	31	by	by	ADP
ejpam-3441	815	32	the	the	DET
ejpam-3441	815	33	theorem	theorem	NOUN
ejpam-3441	815	34	4	4	NUM
ejpam-3441	815	35	.	.	PUNCT
ejpam-3441	815	36	by	by	ADP
ejpam-3441	815	37	our	our	PRON
ejpam-3441	815	38	assumption	assumption	NOUN
ejpam-3441	815	39	(	(	PUNCT
ejpam-3441	815	40	χra	χra	PROPN
ejpam-3441	815	41	)	)	PUNCT
ejpam-3441	815	42	β	β	X
ejpam-3441	815	43	α	α	X
ejpam-3441	815	44	=	=	SYM
ejpam-3441	815	45	χra	χra	PROPN
ejpam-3441	815	46	◦	◦	NOUN
ejpam-3441	815	47	βα	βα	NOUN
ejpam-3441	815	48	χra	χra	NOUN
ejpam-3441	815	49	=	=	SYM
ejpam-3441	815	50	(	(	PUNCT
ejpam-3441	815	51	χ(ra)(ra	χ(ra)(ra	PROPN
ejpam-3441	815	52	)	)	PUNCT
ejpam-3441	815	53	)	)	PUNCT
ejpam-3441	815	54	β	β	PROPN
ejpam-3441	815	55	α	α	PROPN
ejpam-3441	815	56	,	,	PUNCT
ejpam-3441	815	57	i.e.	i.e.	X
ejpam-3441	815	58	,	,	PUNCT
ejpam-3441	815	59	ra	ra	PROPN
ejpam-3441	815	60	=	=	SYM
ejpam-3441	815	61	(	(	PUNCT
ejpam-3441	815	62	ra)(ra	ra)(ra	NOUN
ejpam-3441	815	63	)	)	PUNCT
ejpam-3441	815	64	.	.	PUNCT
ejpam-3441	816	1	since	since	SCONJ
ejpam-3441	816	2	a	a	DET
ejpam-3441	816	3	∈	∈	PROPN
ejpam-3441	816	4	ra	ra	PROPN
ejpam-3441	816	5	,	,	PUNCT
ejpam-3441	816	6	i.e.	i.e.	X
ejpam-3441	816	7	,	,	PUNCT
ejpam-3441	816	8	a	a	DET
ejpam-3441	816	9	∈	∈	PROPN
ejpam-3441	816	10	(	(	PUNCT
ejpam-3441	816	11	ra)(ra	ra)(ra	NOUN
ejpam-3441	816	12	)	)	PUNCT
ejpam-3441	816	13	.	.	PUNCT
ejpam-3441	817	1	thus	thus	ADV
ejpam-3441	817	2	a	a	PRON
ejpam-3441	817	3	is	be	AUX
ejpam-3441	817	4	both	both	CCONJ
ejpam-3441	817	5	a	a	DET
ejpam-3441	817	6	regular	regular	ADJ
ejpam-3441	817	7	and	and	CCONJ
ejpam-3441	817	8	an	an	DET
ejpam-3441	817	9	intra	intra	ADJ
ejpam-3441	817	10	-	-	ADJ
ejpam-3441	817	11	regular	regular	ADJ
ejpam-3441	817	12	by	by	ADP
ejpam-3441	817	13	the	the	DET
ejpam-3441	817	14	theorems	theorem	NOUN
ejpam-3441	817	15	8	8	NUM
ejpam-3441	817	16	and	and	CCONJ
ejpam-3441	817	17	13	13	NUM
ejpam-3441	817	18	,	,	PUNCT
ejpam-3441	817	19	respectively	respectively	ADV
ejpam-3441	817	20	.	.	PUNCT
ejpam-3441	818	1	hence	hence	ADV
ejpam-3441	818	2	r	r	NOUN
ejpam-3441	818	3	is	be	AUX
ejpam-3441	818	4	both	both	CCONJ
ejpam-3441	818	5	a	a	DET
ejpam-3441	818	6	regular	regular	ADJ
ejpam-3441	818	7	and	and	CCONJ
ejpam-3441	818	8	an	an	DET
ejpam-3441	818	9	intra	intra	ADJ
ejpam-3441	818	10	-	-	ADJ
ejpam-3441	818	11	regular	regular	ADJ
ejpam-3441	818	12	,	,	PUNCT
ejpam-3441	818	13	i.e.	i.e.	X
ejpam-3441	818	14	,	,	PUNCT
ejpam-3441	818	15	(	(	PUNCT
ejpam-3441	818	16	2)⇒	2)⇒	NUM
ejpam-3441	818	17	(	(	PUNCT
ejpam-3441	818	18	1	1	NUM
ejpam-3441	818	19	)	)	PUNCT
ejpam-3441	818	20	.	.	PUNCT
ejpam-3441	819	1	theorem	theorem	NOUN
ejpam-3441	819	2	18	18	NUM
ejpam-3441	819	3	.	.	PUNCT
ejpam-3441	820	1	let	let	VERB
ejpam-3441	820	2	r	r	PRON
ejpam-3441	820	3	be	be	AUX
ejpam-3441	820	4	an	an	DET
ejpam-3441	820	5	la	la	NOUN
ejpam-3441	820	6	-	-	NOUN
ejpam-3441	820	7	ring	ring	NOUN
ejpam-3441	820	8	with	with	ADP
ejpam-3441	820	9	left	left	ADJ
ejpam-3441	820	10	identity	identity	NOUN
ejpam-3441	820	11	e	e	NOUN
ejpam-3441	820	12	,	,	PUNCT
ejpam-3441	820	13	such	such	ADJ
ejpam-3441	820	14	that	that	SCONJ
ejpam-3441	820	15	(	(	PUNCT
ejpam-3441	820	16	xe)r	xe)r	PROPN
ejpam-3441	820	17	=	=	SYM
ejpam-3441	820	18	xr	xr	PROPN
ejpam-3441	820	19	for	for	ADP
ejpam-3441	820	20	all	all	DET
ejpam-3441	820	21	x	x	PROPN
ejpam-3441	820	22	∈	∈	PROPN
ejpam-3441	820	23	r.	r.	NOUN
ejpam-3441	820	24	then	then	ADV
ejpam-3441	820	25	the	the	DET
ejpam-3441	820	26	following	follow	VERB
ejpam-3441	820	27	conditions	condition	NOUN
ejpam-3441	820	28	are	be	AUX
ejpam-3441	820	29	equivalent	equivalent	ADJ
ejpam-3441	820	30	.	.	PUNCT
ejpam-3441	821	1	(	(	PUNCT
ejpam-3441	821	2	1	1	X
ejpam-3441	821	3	)	)	PUNCT
ejpam-3441	821	4	r	r	NOUN
ejpam-3441	821	5	is	be	AUX
ejpam-3441	821	6	both	both	CCONJ
ejpam-3441	821	7	a	a	DET
ejpam-3441	821	8	regular	regular	ADJ
ejpam-3441	821	9	and	and	CCONJ
ejpam-3441	821	10	an	an	DET
ejpam-3441	821	11	intra	intra	ADJ
ejpam-3441	821	12	-	-	ADJ
ejpam-3441	821	13	regular	regular	ADJ
ejpam-3441	821	14	.	.	PUNCT
ejpam-3441	822	1	(	(	PUNCT
ejpam-3441	822	2	2	2	X
ejpam-3441	822	3	)	)	PUNCT
ejpam-3441	822	4	a	a	DET
ejpam-3441	822	5	∧βα	∧βα	NOUN
ejpam-3441	822	6	b	b	NOUN
ejpam-3441	822	7	⊆	⊆	NUM
ejpam-3441	822	8	a	a	DET
ejpam-3441	822	9	◦	◦	NOUN
ejpam-3441	822	10	βα	βα	NOUN
ejpam-3441	822	11	b	b	NOUN
ejpam-3441	822	12	for	for	ADP
ejpam-3441	822	13	all	all	DET
ejpam-3441	822	14	intuitionistic	intuitionistic	ADJ
ejpam-3441	822	15	fuzzy	fuzzy	ADJ
ejpam-3441	822	16	quasi	quasi	NOUN
ejpam-3441	822	17	-	-	NOUN
ejpam-3441	822	18	ideals	ideal	NOUN
ejpam-3441	822	19	a	a	PRON
ejpam-3441	822	20	and	and	CCONJ
ejpam-3441	822	21	b	b	NOUN
ejpam-3441	822	22	with	with	ADP
ejpam-3441	822	23	thresholds	threshold	NOUN
ejpam-3441	822	24	(	(	PUNCT
ejpam-3441	822	25	α	α	X
ejpam-3441	822	26	,	,	PUNCT
ejpam-3441	822	27	β	β	X
ejpam-3441	822	28	]	]	PUNCT
ejpam-3441	822	29	of	of	ADP
ejpam-3441	822	30	r.	r.	PROPN
ejpam-3441	822	31	k.	k.	PROPN
ejpam-3441	822	32	nasreen	nasreen	PROPN
ejpam-3441	822	33	et	et	PROPN
ejpam-3441	822	34	al	al	PROPN
ejpam-3441	822	35	.	.	PUNCT
ejpam-3441	822	36	/	/	SYM
ejpam-3441	822	37	eur	eur	PROPN
ejpam-3441	822	38	.	.	PUNCT
ejpam-3441	823	1	j.	j.	PROPN
ejpam-3441	823	2	pure	pure	PROPN
ejpam-3441	823	3	appl	appl	PROPN
ejpam-3441	823	4	.	.	PROPN
ejpam-3441	823	5	math	math	PROPN
ejpam-3441	823	6	,	,	PUNCT
ejpam-3441	823	7	12	12	NUM
ejpam-3441	823	8	(	(	PUNCT
ejpam-3441	823	9	3	3	NUM
ejpam-3441	823	10	)	)	PUNCT
ejpam-3441	823	11	(	(	PUNCT
ejpam-3441	823	12	2019	2019	NUM
ejpam-3441	823	13	)	)	PUNCT
ejpam-3441	823	14	,	,	PUNCT
ejpam-3441	823	15	906	906	NUM
ejpam-3441	823	16	-	-	SYM
ejpam-3441	823	17	943	943	NUM
ejpam-3441	823	18	940	940	NUM
ejpam-3441	823	19	(	(	PUNCT
ejpam-3441	823	20	3	3	NUM
ejpam-3441	823	21	)	)	PUNCT
ejpam-3441	823	22	a∧βαb	a∧βαb	NOUN
ejpam-3441	823	23	⊆	⊆	NUM
ejpam-3441	823	24	a	a	DET
ejpam-3441	823	25	◦	◦	NOUN
ejpam-3441	823	26	βαb	βαb	NOUN
ejpam-3441	823	27	for	for	ADP
ejpam-3441	823	28	every	every	DET
ejpam-3441	823	29	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	30	fuzzy	fuzzy	ADJ
ejpam-3441	823	31	quasi	quasi	NOUN
ejpam-3441	823	32	-	-	NOUN
ejpam-3441	823	33	ideal	ideal	ADJ
ejpam-3441	823	34	a	a	PRON
ejpam-3441	823	35	with	with	ADP
ejpam-3441	823	36	thresholds	threshold	NOUN
ejpam-3441	823	37	(	(	PUNCT
ejpam-3441	823	38	α	α	X
ejpam-3441	823	39	,	,	PUNCT
ejpam-3441	823	40	β	β	X
ejpam-3441	823	41	]	]	PUNCT
ejpam-3441	823	42	and	and	CCONJ
ejpam-3441	823	43	every	every	DET
ejpam-3441	823	44	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	45	fuzzy	fuzzy	ADJ
ejpam-3441	823	46	bi	bi	ADJ
ejpam-3441	823	47	-	-	ADJ
ejpam-3441	823	48	ideal	ideal	ADJ
ejpam-3441	823	49	b	b	PROPN
ejpam-3441	823	50	with	with	ADP
ejpam-3441	823	51	thresholds	threshold	NOUN
ejpam-3441	823	52	(	(	PUNCT
ejpam-3441	823	53	α	α	X
ejpam-3441	823	54	,	,	PUNCT
ejpam-3441	823	55	β	β	X
ejpam-3441	823	56	]	]	PUNCT
ejpam-3441	823	57	of	of	ADP
ejpam-3441	823	58	r.	r.	PROPN
ejpam-3441	823	59	(	(	PUNCT
ejpam-3441	823	60	4	4	NUM
ejpam-3441	823	61	)	)	PUNCT
ejpam-3441	823	62	a∧βαb	a∧βαb	NOUN
ejpam-3441	823	63	⊆	⊆	NUM
ejpam-3441	823	64	a	a	DET
ejpam-3441	823	65	◦	◦	NOUN
ejpam-3441	823	66	βαb	βαb	NOUN
ejpam-3441	823	67	for	for	ADP
ejpam-3441	823	68	every	every	DET
ejpam-3441	823	69	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	70	fuzzy	fuzzy	ADJ
ejpam-3441	823	71	bi	bi	NOUN
ejpam-3441	823	72	-	-	NOUN
ejpam-3441	823	73	ideal	ideal	NOUN
ejpam-3441	823	74	a	a	PRON
ejpam-3441	823	75	with	with	ADP
ejpam-3441	823	76	thresholds	threshold	NOUN
ejpam-3441	823	77	(	(	PUNCT
ejpam-3441	823	78	α	α	X
ejpam-3441	823	79	,	,	PUNCT
ejpam-3441	823	80	β	β	X
ejpam-3441	823	81	]	]	PUNCT
ejpam-3441	823	82	and	and	CCONJ
ejpam-3441	823	83	every	every	DET
ejpam-3441	823	84	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	85	fuzzy	fuzzy	ADJ
ejpam-3441	823	86	quasi	quasi	ADJ
ejpam-3441	823	87	-	-	ADJ
ejpam-3441	823	88	ideal	ideal	ADJ
ejpam-3441	823	89	b	b	PROPN
ejpam-3441	823	90	with	with	ADP
ejpam-3441	823	91	thresholds	threshold	NOUN
ejpam-3441	823	92	(	(	PUNCT
ejpam-3441	823	93	α	α	X
ejpam-3441	823	94	,	,	PUNCT
ejpam-3441	823	95	β	β	X
ejpam-3441	823	96	]	]	PUNCT
ejpam-3441	823	97	of	of	ADP
ejpam-3441	823	98	r.	r.	PROPN
ejpam-3441	823	99	(	(	PUNCT
ejpam-3441	823	100	5	5	NUM
ejpam-3441	823	101	)	)	PUNCT
ejpam-3441	823	102	a	a	DET
ejpam-3441	823	103	∧βα	∧βα	NOUN
ejpam-3441	823	104	b	b	NOUN
ejpam-3441	823	105	⊆	⊆	NUM
ejpam-3441	823	106	a	a	DET
ejpam-3441	823	107	◦	◦	NOUN
ejpam-3441	823	108	βα	βα	NOUN
ejpam-3441	823	109	b	b	NOUN
ejpam-3441	823	110	for	for	ADP
ejpam-3441	823	111	all	all	DET
ejpam-3441	823	112	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	113	fuzzy	fuzzy	ADJ
ejpam-3441	823	114	bi	bi	NOUN
ejpam-3441	823	115	-	-	NOUN
ejpam-3441	823	116	ideals	ideal	NOUN
ejpam-3441	823	117	a	a	PRON
ejpam-3441	823	118	and	and	CCONJ
ejpam-3441	823	119	b	b	NOUN
ejpam-3441	823	120	with	with	ADP
ejpam-3441	823	121	thresholds	threshold	NOUN
ejpam-3441	823	122	(	(	PUNCT
ejpam-3441	823	123	α	α	X
ejpam-3441	823	124	,	,	PUNCT
ejpam-3441	823	125	β	β	X
ejpam-3441	823	126	]	]	PUNCT
ejpam-3441	823	127	of	of	ADP
ejpam-3441	823	128	r.	r.	PROPN
ejpam-3441	823	129	(	(	PUNCT
ejpam-3441	823	130	6	6	NUM
ejpam-3441	823	131	)	)	PUNCT
ejpam-3441	823	132	a∧βαb	a∧βαb	NOUN
ejpam-3441	823	133	⊆	⊆	NUM
ejpam-3441	823	134	a	a	DET
ejpam-3441	823	135	◦	◦	NOUN
ejpam-3441	823	136	βαb	βαb	NOUN
ejpam-3441	823	137	for	for	ADP
ejpam-3441	823	138	every	every	DET
ejpam-3441	823	139	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	140	fuzzy	fuzzy	ADJ
ejpam-3441	823	141	bi	bi	NOUN
ejpam-3441	823	142	-	-	NOUN
ejpam-3441	823	143	ideal	ideal	NOUN
ejpam-3441	823	144	a	a	PRON
ejpam-3441	823	145	with	with	ADP
ejpam-3441	823	146	thresholds	threshold	NOUN
ejpam-3441	823	147	(	(	PUNCT
ejpam-3441	823	148	α	α	X
ejpam-3441	823	149	,	,	PUNCT
ejpam-3441	823	150	β	β	X
ejpam-3441	823	151	]	]	PUNCT
ejpam-3441	823	152	and	and	CCONJ
ejpam-3441	823	153	every	every	DET
ejpam-3441	823	154	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	155	fuzzy	fuzzy	ADJ
ejpam-3441	823	156	generalized	generalize	VERB
ejpam-3441	823	157	bi	bi	ADJ
ejpam-3441	823	158	-	-	ADJ
ejpam-3441	823	159	ideal	ideal	ADJ
ejpam-3441	823	160	b	b	PROPN
ejpam-3441	823	161	with	with	ADP
ejpam-3441	823	162	thresholds	threshold	NOUN
ejpam-3441	823	163	(	(	PUNCT
ejpam-3441	823	164	α	α	X
ejpam-3441	823	165	,	,	PUNCT
ejpam-3441	823	166	β	β	X
ejpam-3441	823	167	]	]	PUNCT
ejpam-3441	823	168	of	of	ADP
ejpam-3441	823	169	r.	r.	PROPN
ejpam-3441	823	170	(	(	PUNCT
ejpam-3441	823	171	7	7	NUM
ejpam-3441	823	172	)	)	PUNCT
ejpam-3441	823	173	a∧βαb	a∧βαb	NOUN
ejpam-3441	823	174	⊆	⊆	NUM
ejpam-3441	823	175	a	a	DET
ejpam-3441	823	176	◦	◦	NOUN
ejpam-3441	823	177	βαb	βαb	NOUN
ejpam-3441	823	178	for	for	ADP
ejpam-3441	823	179	every	every	DET
ejpam-3441	823	180	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	181	fuzzy	fuzzy	ADJ
ejpam-3441	823	182	generalized	generalize	VERB
ejpam-3441	823	183	bi	bi	NOUN
ejpam-3441	823	184	-	-	NOUN
ejpam-3441	823	185	ideal	ideal	NOUN
ejpam-3441	823	186	a	a	PRON
ejpam-3441	823	187	with	with	ADP
ejpam-3441	823	188	thresholds	threshold	NOUN
ejpam-3441	823	189	(	(	PUNCT
ejpam-3441	823	190	α	α	X
ejpam-3441	823	191	,	,	PUNCT
ejpam-3441	823	192	β	β	X
ejpam-3441	823	193	]	]	PUNCT
ejpam-3441	823	194	and	and	CCONJ
ejpam-3441	823	195	every	every	DET
ejpam-3441	823	196	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	197	fuzzy	fuzzy	ADJ
ejpam-3441	823	198	quasi	quasi	ADJ
ejpam-3441	823	199	-	-	ADJ
ejpam-3441	823	200	ideal	ideal	ADJ
ejpam-3441	823	201	b	b	PROPN
ejpam-3441	823	202	with	with	ADP
ejpam-3441	823	203	thresholds	threshold	NOUN
ejpam-3441	823	204	(	(	PUNCT
ejpam-3441	823	205	α	α	X
ejpam-3441	823	206	,	,	PUNCT
ejpam-3441	823	207	β	β	X
ejpam-3441	823	208	]	]	PUNCT
ejpam-3441	823	209	of	of	ADP
ejpam-3441	823	210	r.	r.	PROPN
ejpam-3441	823	211	(	(	PUNCT
ejpam-3441	823	212	8)	8)	NUM
ejpam-3441	823	213	a∧βαb	a∧βαb	NUM
ejpam-3441	823	214	⊆	⊆	NUM
ejpam-3441	823	215	a	a	DET
ejpam-3441	823	216	◦	◦	NOUN
ejpam-3441	823	217	βαb	βαb	NOUN
ejpam-3441	823	218	for	for	ADP
ejpam-3441	823	219	every	every	DET
ejpam-3441	823	220	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	221	fuzzy	fuzzy	ADJ
ejpam-3441	823	222	generalized	generalize	VERB
ejpam-3441	823	223	bi	bi	NOUN
ejpam-3441	823	224	-	-	NOUN
ejpam-3441	823	225	ideal	ideal	NOUN
ejpam-3441	823	226	a	a	PRON
ejpam-3441	823	227	with	with	ADP
ejpam-3441	823	228	thresholds	threshold	NOUN
ejpam-3441	823	229	(	(	PUNCT
ejpam-3441	823	230	α	α	X
ejpam-3441	823	231	,	,	PUNCT
ejpam-3441	823	232	β	β	X
ejpam-3441	823	233	]	]	PUNCT
ejpam-3441	823	234	and	and	CCONJ
ejpam-3441	823	235	every	every	DET
ejpam-3441	823	236	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	237	fuzzy	fuzzy	ADJ
ejpam-3441	823	238	bi	bi	ADJ
ejpam-3441	823	239	-	-	ADJ
ejpam-3441	823	240	ideal	ideal	ADJ
ejpam-3441	823	241	b	b	PROPN
ejpam-3441	823	242	with	with	ADP
ejpam-3441	823	243	thresholds	threshold	NOUN
ejpam-3441	823	244	(	(	PUNCT
ejpam-3441	823	245	α	α	X
ejpam-3441	823	246	,	,	PUNCT
ejpam-3441	823	247	β	β	X
ejpam-3441	823	248	]	]	PUNCT
ejpam-3441	823	249	of	of	ADP
ejpam-3441	823	250	r.	r.	PROPN
ejpam-3441	823	251	(	(	PUNCT
ejpam-3441	823	252	9	9	NUM
ejpam-3441	823	253	)	)	PUNCT
ejpam-3441	823	254	a	a	DET
ejpam-3441	823	255	∧βα	∧βα	NOUN
ejpam-3441	823	256	b	b	NOUN
ejpam-3441	823	257	⊆	⊆	NUM
ejpam-3441	823	258	a	a	DET
ejpam-3441	823	259	◦	◦	NOUN
ejpam-3441	823	260	βα	βα	NOUN
ejpam-3441	823	261	b	b	NOUN
ejpam-3441	823	262	for	for	ADP
ejpam-3441	823	263	all	all	DET
ejpam-3441	823	264	intuitionistic	intuitionistic	ADJ
ejpam-3441	823	265	fuzzy	fuzzy	ADJ
ejpam-3441	823	266	generalized	generalize	VERB
ejpam-3441	823	267	bi	bi	NOUN
ejpam-3441	823	268	-	-	NOUN
ejpam-3441	823	269	ideals	ideal	NOUN
ejpam-3441	823	270	a	a	PRON
ejpam-3441	823	271	and	and	CCONJ
ejpam-3441	823	272	b	b	NOUN
ejpam-3441	823	273	with	with	ADP
ejpam-3441	823	274	thresholds	threshold	NOUN
ejpam-3441	823	275	(	(	PUNCT
ejpam-3441	823	276	α	α	X
ejpam-3441	823	277	,	,	PUNCT
ejpam-3441	823	278	β	β	X
ejpam-3441	823	279	]	]	PUNCT
ejpam-3441	823	280	of	of	ADP
ejpam-3441	823	281	r.	r.	PROPN
ejpam-3441	823	282	proof	proof	PROPN
ejpam-3441	823	283	.	.	PUNCT
ejpam-3441	824	1	assume	assume	VERB
ejpam-3441	824	2	that	that	SCONJ
ejpam-3441	824	3	(	(	PUNCT
ejpam-3441	824	4	1	1	X
ejpam-3441	824	5	)	)	PUNCT
ejpam-3441	824	6	holds	hold	VERB
ejpam-3441	824	7	.	.	PUNCT
ejpam-3441	825	1	let	let	VERB
ejpam-3441	825	2	a	a	PRON
ejpam-3441	825	3	=	=	SYM
ejpam-3441	825	4	(	(	PUNCT
ejpam-3441	825	5	µa	µa	PROPN
ejpam-3441	825	6	,	,	PUNCT
ejpam-3441	825	7	γa	γa	PROPN
ejpam-3441	825	8	)	)	PUNCT
ejpam-3441	825	9	and	and	CCONJ
ejpam-3441	825	10	b	b	X
ejpam-3441	825	11	=	=	SYM
ejpam-3441	825	12	(	(	PUNCT
ejpam-3441	825	13	µb	µb	PROPN
ejpam-3441	825	14	,	,	PUNCT
ejpam-3441	825	15	γb	γb	PROPN
ejpam-3441	825	16	)	)	PUNCT
ejpam-3441	825	17	be	be	VERB
ejpam-3441	825	18	two	two	NUM
ejpam-3441	825	19	intuitionistic	intuitionistic	ADJ
ejpam-3441	825	20	fuzzy	fuzzy	ADJ
ejpam-3441	825	21	generalized	generalize	VERB
ejpam-3441	825	22	bi	bi	NOUN
ejpam-3441	825	23	-	-	NOUN
ejpam-3441	825	24	ideals	ideal	NOUN
ejpam-3441	825	25	with	with	ADP
ejpam-3441	825	26	thresholds	threshold	NOUN
ejpam-3441	825	27	(	(	PUNCT
ejpam-3441	825	28	α	α	X
ejpam-3441	825	29	,	,	PUNCT
ejpam-3441	825	30	β	β	X
ejpam-3441	825	31	]	]	PUNCT
ejpam-3441	825	32	of	of	ADP
ejpam-3441	825	33	r.	r.	PROPN
ejpam-3441	825	34	let	let	VERB
ejpam-3441	825	35	x	x	X
ejpam-3441	825	36	∈	∈	PROPN
ejpam-3441	825	37	r	r	NOUN
ejpam-3441	825	38	,	,	PUNCT
ejpam-3441	825	39	then	then	ADV
ejpam-3441	825	40	means	mean	VERB
ejpam-3441	825	41	that	that	SCONJ
ejpam-3441	825	42	there	there	PRON
ejpam-3441	825	43	exists	exist	VERB
ejpam-3441	825	44	an	an	DET
ejpam-3441	825	45	element	element	NOUN
ejpam-3441	825	46	a	a	DET
ejpam-3441	825	47	∈	∈	NOUN
ejpam-3441	825	48	r	r	NOUN
ejpam-3441	825	49	such	such	ADJ
ejpam-3441	825	50	that	that	PRON
ejpam-3441	825	51	x	x	SYM
ejpam-3441	825	52	=	=	SYM
ejpam-3441	825	53	(	(	PUNCT
ejpam-3441	825	54	xa)x	xa)x	PROPN
ejpam-3441	825	55	,	,	PUNCT
ejpam-3441	825	56	and	and	CCONJ
ejpam-3441	825	57	also	also	ADV
ejpam-3441	825	58	there	there	PRON
ejpam-3441	825	59	exist	exist	VERB
ejpam-3441	825	60	elements	element	NOUN
ejpam-3441	825	61	ai	ai	VERB
ejpam-3441	825	62	,	,	PUNCT
ejpam-3441	825	63	bi	bi	NOUN
ejpam-3441	825	64	∈	∈	PROPN
ejpam-3441	825	65	r	r	NOUN
ejpam-3441	826	1	such	such	ADJ
ejpam-3441	826	2	that	that	SCONJ
ejpam-3441	826	3	x	x	NOUN
ejpam-3441	827	1	=	=	PUNCT
ejpam-3441	827	2	∑n	∑n	PROPN
ejpam-3441	827	3	i=1(aix	i=1(aix	PROPN
ejpam-3441	827	4	2)bi	2)bi	NUM
ejpam-3441	827	5	.	.	PUNCT
ejpam-3441	828	1	since	since	SCONJ
ejpam-3441	828	2	x	x	X
ejpam-3441	828	3	=	=	SYM
ejpam-3441	828	4	(	(	PUNCT
ejpam-3441	828	5	xa)x	xa)x	PROPN
ejpam-3441	828	6	=	=	SYM
ejpam-3441	828	7	(	(	PUNCT
ejpam-3441	828	8	(	(	PUNCT
ejpam-3441	828	9	xwi)x)x	xwi)x)x	NUM
ejpam-3441	828	10	by	by	ADP
ejpam-3441	828	11	the	the	DET
ejpam-3441	828	12	theorem	theorem	NOUN
ejpam-3441	828	13	17	17	NUM
ejpam-3441	828	14	.	.	PUNCT
ejpam-3441	829	1	thus	thus	ADV
ejpam-3441	829	2	(	(	PUNCT
ejpam-3441	829	3	µa	µa	ADP
ejpam-3441	829	4	◦	◦	NOUN
ejpam-3441	829	5	βα	βα	NOUN
ejpam-3441	829	6	µb)(x	µb)(x	NOUN
ejpam-3441	829	7	)	)	PUNCT
ejpam-3441	829	8	=	=	PRON
ejpam-3441	829	9	{	{	PUNCT
ejpam-3441	829	10	(	(	PUNCT
ejpam-3441	829	11	µa	µa	ADP
ejpam-3441	829	12	◦	◦	NOUN
ejpam-3441	829	13	µb)(x	µb)(x	NOUN
ejpam-3441	829	14	)	)	PUNCT
ejpam-3441	829	15	∧	∧	PROPN
ejpam-3441	829	16	β	β	NOUN
ejpam-3441	829	17	}	}	PUNCT
ejpam-3441	829	18	∨	∨	NUM
ejpam-3441	829	19	α	α	NOUN
ejpam-3441	829	20	=	=	X
ejpam-3441	829	21	{	{	PUNCT
ejpam-3441	829	22	(	(	PUNCT
ejpam-3441	829	23	∨x=∑n	∨x=∑n	NUM
ejpam-3441	829	24	i=1	i=1	PROPN
ejpam-3441	829	25	piqi	piqi	NOUN
ejpam-3441	829	26	{	{	PUNCT
ejpam-3441	829	27	∧ni=1	∧ni=1	X
ejpam-3441	829	28	{	{	PUNCT
ejpam-3441	829	29	µa	µa	X
ejpam-3441	829	30	(	(	PUNCT
ejpam-3441	829	31	pi	pi	NOUN
ejpam-3441	829	32	)	)	PUNCT
ejpam-3441	829	33	∧	∧	PROPN
ejpam-3441	829	34	µb	µb	PROPN
ejpam-3441	829	35	(	(	PUNCT
ejpam-3441	829	36	qi	qi	NOUN
ejpam-3441	829	37	)	)	PUNCT
ejpam-3441	829	38	}	}	PUNCT
ejpam-3441	829	39	}	}	PUNCT
ejpam-3441	829	40	)	)	PUNCT
ejpam-3441	830	1	∧	∧	PROPN
ejpam-3441	830	2	β	β	NOUN
ejpam-3441	830	3	}	}	PUNCT
ejpam-3441	830	4	∨	∨	NUM
ejpam-3441	830	5	α	α	PROPN
ejpam-3441	830	6	≥	≥	X
ejpam-3441	830	7	{	{	PUNCT
ejpam-3441	830	8	{	{	PUNCT
ejpam-3441	830	9	µa	µa	X
ejpam-3441	830	10	(	(	PUNCT
ejpam-3441	830	11	(	(	PUNCT
ejpam-3441	830	12	xwi)x	xwi)x	NOUN
ejpam-3441	830	13	)	)	PUNCT
ejpam-3441	830	14	∧	∧	NOUN
ejpam-3441	830	15	µb	µb	PROPN
ejpam-3441	830	16	(	(	PUNCT
ejpam-3441	830	17	x	x	NOUN
ejpam-3441	830	18	)	)	PUNCT
ejpam-3441	830	19	}	}	PUNCT
ejpam-3441	830	20	∧	∧	PROPN
ejpam-3441	830	21	β	β	NOUN
ejpam-3441	830	22	}	}	PUNCT
ejpam-3441	830	23	∨	∨	NUM
ejpam-3441	830	24	α	α	NOUN
ejpam-3441	830	25	=	=	SYM
ejpam-3441	830	26	(	(	PUNCT
ejpam-3441	830	27	µa	µa	INTJ
ejpam-3441	830	28	(	(	PUNCT
ejpam-3441	830	29	(	(	PUNCT
ejpam-3441	830	30	xwi)x	xwi)x	PROPN
ejpam-3441	830	31	)	)	PUNCT
ejpam-3441	830	32	∨	∨	NUM
ejpam-3441	830	33	α	α	NOUN
ejpam-3441	830	34	)	)	PUNCT
ejpam-3441	830	35	∧	∧	NOUN
ejpam-3441	830	36	(	(	PUNCT
ejpam-3441	830	37	µb	µb	PROPN
ejpam-3441	830	38	(	(	PUNCT
ejpam-3441	830	39	x	x	NOUN
ejpam-3441	830	40	)	)	PUNCT
ejpam-3441	830	41	∨	∨	NUM
ejpam-3441	830	42	α	α	NOUN
ejpam-3441	830	43	)	)	PUNCT
ejpam-3441	830	44	∧	∧	PROPN
ejpam-3441	830	45	(	(	PUNCT
ejpam-3441	830	46	β	β	X
ejpam-3441	830	47	∨	∨	NUM
ejpam-3441	830	48	α	α	NOUN
ejpam-3441	830	49	)	)	PUNCT
ejpam-3441	830	50	≥	≥	NOUN
ejpam-3441	830	51	(	(	PUNCT
ejpam-3441	830	52	µa	µa	PROPN
ejpam-3441	830	53	(	(	PUNCT
ejpam-3441	830	54	x	x	NOUN
ejpam-3441	830	55	)	)	PUNCT
ejpam-3441	830	56	∧	∧	NOUN
ejpam-3441	830	57	µa	µa	INTJ
ejpam-3441	830	58	(	(	PUNCT
ejpam-3441	830	59	x	x	X
ejpam-3441	830	60	)	)	PUNCT
ejpam-3441	830	61	∧	∧	PROPN
ejpam-3441	830	62	β	β	NOUN
ejpam-3441	830	63	)	)	PUNCT
ejpam-3441	830	64	∧	∧	NOUN
ejpam-3441	830	65	µb	µb	ADP
ejpam-3441	830	66	(	(	PUNCT
ejpam-3441	830	67	x	x	NOUN
ejpam-3441	830	68	)	)	PUNCT
ejpam-3441	830	69	∧	∧	NOUN
ejpam-3441	830	70	β	β	X
ejpam-3441	830	71	=	=	SYM
ejpam-3441	830	72	µa	µa	X
ejpam-3441	830	73	(	(	PUNCT
ejpam-3441	830	74	x	x	X
ejpam-3441	830	75	)	)	PUNCT
ejpam-3441	830	76	∧	∧	NOUN
ejpam-3441	830	77	µb	µb	ADP
ejpam-3441	830	78	(	(	PUNCT
ejpam-3441	830	79	x	x	NOUN
ejpam-3441	830	80	)	)	PUNCT
ejpam-3441	830	81	∧	∧	NOUN
ejpam-3441	830	82	β	β	X
ejpam-3441	830	83	=	=	SYM
ejpam-3441	830	84	(	(	PUNCT
ejpam-3441	830	85	µa	µa	ADP
ejpam-3441	830	86	∧	∧	PROPN
ejpam-3441	830	87	µb	µb	PROPN
ejpam-3441	830	88	)	)	PUNCT
ejpam-3441	830	89	(	(	PUNCT
ejpam-3441	830	90	x	x	X
ejpam-3441	830	91	)	)	PUNCT
ejpam-3441	830	92	∧	∧	NOUN
ejpam-3441	830	93	β	β	X
ejpam-3441	830	94	=	=	SYM
ejpam-3441	830	95	{	{	PUNCT
ejpam-3441	830	96	(	(	PUNCT
ejpam-3441	830	97	µa	µa	ADP
ejpam-3441	830	98	∧	∧	PROPN
ejpam-3441	830	99	µb	µb	PROPN
ejpam-3441	830	100	)	)	PUNCT
ejpam-3441	830	101	(	(	PUNCT
ejpam-3441	830	102	x	x	X
ejpam-3441	830	103	)	)	PUNCT
ejpam-3441	830	104	∧	∧	PROPN
ejpam-3441	830	105	β	β	NOUN
ejpam-3441	830	106	}	}	PUNCT
ejpam-3441	830	107	∨	∨	NUM
ejpam-3441	830	108	α	α	NOUN
ejpam-3441	830	109	=	=	PUNCT
ejpam-3441	830	110	(	(	PUNCT
ejpam-3441	830	111	µa	µa	ADP
ejpam-3441	830	112	∧βα	∧βα	ADJ
ejpam-3441	830	113	µb)(x	µb)(x	NOUN
ejpam-3441	830	114	)	)	PUNCT
ejpam-3441	830	115	.	.	PUNCT
ejpam-3441	831	1	⇒	⇒	NOUN
ejpam-3441	831	2	µa	µa	ADP
ejpam-3441	831	3	∧βα	∧βα	PROPN
ejpam-3441	831	4	µb	µb	VERB
ejpam-3441	831	5	⊆	⊆	NUM
ejpam-3441	831	6	µa	µa	NOUN
ejpam-3441	831	7	◦	◦	NOUN
ejpam-3441	831	8	βα	βα	NOUN
ejpam-3441	831	9	µb	µb	PROPN
ejpam-3441	831	10	.	.	PUNCT
ejpam-3441	832	1	similarly	similarly	ADV
ejpam-3441	832	2	,	,	PUNCT
ejpam-3441	832	3	we	we	PRON
ejpam-3441	832	4	have	have	VERB
ejpam-3441	832	5	γa	γa	NOUN
ejpam-3441	832	6	∨βα	∨βα	NOUN
ejpam-3441	832	7	γb	γb	PROPN
ejpam-3441	832	8	⊇	⊇	PROPN
ejpam-3441	832	9	γa	γa	PROPN
ejpam-3441	832	10	◦	◦	PROPN
ejpam-3441	832	11	βα	βα	NOUN
ejpam-3441	832	12	γb	γb	PROPN
ejpam-3441	832	13	.	.	PUNCT
ejpam-3441	833	1	therefore	therefore	ADV
ejpam-3441	833	2	a	a	DET
ejpam-3441	833	3	∧βα	∧βα	PROPN
ejpam-3441	833	4	b	b	NOUN
ejpam-3441	833	5	⊆	⊆	NUM
ejpam-3441	833	6	a	a	DET
ejpam-3441	833	7	◦	◦	NOUN
ejpam-3441	833	8	βα	βα	NOUN
ejpam-3441	833	9	b	b	NOUN
ejpam-3441	833	10	,	,	PUNCT
ejpam-3441	833	11	i.e.	i.e.	X
ejpam-3441	833	12	,	,	PUNCT
ejpam-3441	833	13	(	(	PUNCT
ejpam-3441	833	14	1	1	X
ejpam-3441	833	15	)	)	PUNCT
ejpam-3441	833	16	implies	imply	VERB
ejpam-3441	833	17	(	(	PUNCT
ejpam-3441	833	18	9	9	NUM
ejpam-3441	833	19	)	)	PUNCT
ejpam-3441	833	20	.	.	PUNCT
ejpam-3441	834	1	it	it	PRON
ejpam-3441	834	2	is	be	AUX
ejpam-3441	834	3	clear	clear	ADJ
ejpam-3441	834	4	that	that	SCONJ
ejpam-3441	834	5	(	(	PUNCT
ejpam-3441	834	6	9	9	X
ejpam-3441	834	7	)	)	PUNCT
ejpam-3441	834	8	⇒	⇒	NOUN
ejpam-3441	834	9	(	(	PUNCT
ejpam-3441	834	10	8)	8)	NUM
ejpam-3441	834	11	⇒	⇒	NOUN
ejpam-3441	834	12	(	(	PUNCT
ejpam-3441	834	13	7	7	NUM
ejpam-3441	834	14	)	)	PUNCT
ejpam-3441	834	15	⇒	⇒	NOUN
ejpam-3441	834	16	(	(	PUNCT
ejpam-3441	834	17	4	4	NUM
ejpam-3441	834	18	)	)	PUNCT
ejpam-3441	834	19	⇒	⇒	NOUN
ejpam-3441	834	20	(	(	PUNCT
ejpam-3441	834	21	2	2	NUM
ejpam-3441	834	22	)	)	PUNCT
ejpam-3441	834	23	and	and	CCONJ
ejpam-3441	834	24	(	(	PUNCT
ejpam-3441	834	25	9	9	X
ejpam-3441	834	26	)	)	PUNCT
ejpam-3441	834	27	⇒	⇒	NOUN
ejpam-3441	834	28	(	(	PUNCT
ejpam-3441	834	29	6	6	NUM
ejpam-3441	834	30	)	)	PUNCT
ejpam-3441	834	31	⇒	⇒	NOUN
ejpam-3441	834	32	(	(	PUNCT
ejpam-3441	834	33	5	5	NUM
ejpam-3441	834	34	)	)	PUNCT
ejpam-3441	834	35	⇒	⇒	NOUN
ejpam-3441	834	36	(	(	PUNCT
ejpam-3441	834	37	3	3	NUM
ejpam-3441	834	38	)	)	PUNCT
ejpam-3441	834	39	.	.	PUNCT
ejpam-3441	835	1	suppose	suppose	VERB
ejpam-3441	835	2	that	that	SCONJ
ejpam-3441	835	3	(	(	PUNCT
ejpam-3441	835	4	2	2	X
ejpam-3441	835	5	)	)	PUNCT
ejpam-3441	835	6	holds	hold	VERB
ejpam-3441	835	7	.	.	PUNCT
ejpam-3441	836	1	let	let	VERB
ejpam-3441	836	2	a	a	PRON
ejpam-3441	836	3	be	be	AUX
ejpam-3441	836	4	an	an	DET
ejpam-3441	836	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	836	6	fuzzy	fuzzy	ADJ
ejpam-3441	836	7	right	right	ADJ
ejpam-3441	836	8	ideal	ideal	NOUN
ejpam-3441	836	9	with	with	ADP
ejpam-3441	836	10	thresholds	threshold	NOUN
ejpam-3441	836	11	(	(	PUNCT
ejpam-3441	836	12	α	α	X
ejpam-3441	836	13	,	,	PUNCT
ejpam-3441	836	14	β	β	X
ejpam-3441	836	15	]	]	PUNCT
ejpam-3441	836	16	and	and	CCONJ
ejpam-3441	836	17	b	b	X
ejpam-3441	836	18	be	be	AUX
ejpam-3441	836	19	an	an	DET
ejpam-3441	836	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	836	21	fuzzy	fuzzy	ADJ
ejpam-3441	836	22	left	leave	VERB
ejpam-3441	836	23	ideal	ideal	NOUN
ejpam-3441	836	24	with	with	ADP
ejpam-3441	836	25	thresholds	threshold	NOUN
ejpam-3441	836	26	(	(	PUNCT
ejpam-3441	836	27	α	α	X
ejpam-3441	836	28	,	,	PUNCT
ejpam-3441	836	29	β	β	X
ejpam-3441	836	30	]	]	PUNCT
ejpam-3441	836	31	of	of	ADP
ejpam-3441	836	32	r.	r.	PROPN
ejpam-3441	836	33	since	since	SCONJ
ejpam-3441	836	34	every	every	DET
ejpam-3441	836	35	intuitionistic	intuitionistic	ADJ
ejpam-3441	836	36	fuzzy	fuzzy	ADJ
ejpam-3441	836	37	right	right	ADJ
ejpam-3441	836	38	ideal	ideal	NOUN
ejpam-3441	836	39	with	with	ADP
ejpam-3441	836	40	thresholds	threshold	NOUN
ejpam-3441	836	41	(	(	PUNCT
ejpam-3441	836	42	α	α	X
ejpam-3441	836	43	,	,	PUNCT
ejpam-3441	836	44	β	β	NOUN
ejpam-3441	836	45	]	]	PUNCT
ejpam-3441	836	46	and	and	CCONJ
ejpam-3441	836	47	intuitionistic	intuitionistic	ADJ
ejpam-3441	836	48	fuzzy	fuzzy	ADJ
ejpam-3441	836	49	left	leave	VERB
ejpam-3441	836	50	ideal	ideal	NOUN
ejpam-3441	836	51	with	with	ADP
ejpam-3441	836	52	thresholds	threshold	NOUN
ejpam-3441	836	53	(	(	PUNCT
ejpam-3441	836	54	α	α	X
ejpam-3441	836	55	,	,	PUNCT
ejpam-3441	836	56	β	β	X
ejpam-3441	836	57	]	]	PUNCT
ejpam-3441	836	58	of	of	ADP
ejpam-3441	836	59	r	r	NOUN
ejpam-3441	836	60	is	be	AUX
ejpam-3441	836	61	an	an	DET
ejpam-3441	836	62	intuitionistic	intuitionistic	ADJ
ejpam-3441	836	63	fuzzy	fuzzy	ADJ
ejpam-3441	836	64	quasi	quasi	NOUN
ejpam-3441	836	65	-	-	NOUN
ejpam-3441	836	66	ideal	ideal	ADJ
ejpam-3441	836	67	with	with	ADP
ejpam-3441	836	68	thresholds	threshold	NOUN
ejpam-3441	836	69	(	(	PUNCT
ejpam-3441	836	70	α	α	X
ejpam-3441	836	71	,	,	PUNCT
ejpam-3441	836	72	β	β	X
ejpam-3441	836	73	]	]	PUNCT
ejpam-3441	836	74	of	of	ADP
ejpam-3441	836	75	r	r	NOUN
ejpam-3441	836	76	by	by	ADP
ejpam-3441	836	77	the	the	DET
ejpam-3441	836	78	lemma	lemma	PROPN
ejpam-3441	836	79	14	14	NUM
ejpam-3441	836	80	.	.	PUNCT
ejpam-3441	837	1	by	by	ADP
ejpam-3441	837	2	our	our	PRON
ejpam-3441	837	3	supposition	supposition	NOUN
ejpam-3441	837	4	,	,	PUNCT
ejpam-3441	837	5	a	a	DET
ejpam-3441	837	6	∧βα	∧βα	ADJ
ejpam-3441	837	7	b	b	NOUN
ejpam-3441	837	8	⊆	⊆	NUM
ejpam-3441	837	9	a	a	DET
ejpam-3441	837	10	◦	◦	NOUN
ejpam-3441	837	11	βα	βα	X
ejpam-3441	837	12	b.	b.	PROPN
ejpam-3441	837	13	since	since	SCONJ
ejpam-3441	837	14	a	a	DET
ejpam-3441	837	15	◦	◦	NOUN
ejpam-3441	837	16	βα	βα	NOUN
ejpam-3441	837	17	b	b	NOUN
ejpam-3441	837	18	⊆	⊆	NUM
ejpam-3441	837	19	a	a	DET
ejpam-3441	837	20	∧βα	∧βα	PROPN
ejpam-3441	837	21	b	b	NOUN
ejpam-3441	837	22	,	,	PUNCT
ejpam-3441	837	23	so	so	SCONJ
ejpam-3441	837	24	a	a	DET
ejpam-3441	837	25	∧βα	∧βα	PROPN
ejpam-3441	837	26	b	b	NOUN
ejpam-3441	837	27	=	=	PUNCT
ejpam-3441	837	28	a	a	DET
ejpam-3441	837	29	◦	◦	NOUN
ejpam-3441	837	30	βα	βα	X
ejpam-3441	837	31	b	b	NOUN
ejpam-3441	837	32	,	,	PUNCT
ejpam-3441	837	33	i.e.	i.e.	X
ejpam-3441	837	34	,	,	PUNCT
ejpam-3441	837	35	r	r	NOUN
ejpam-3441	837	36	is	be	AUX
ejpam-3441	837	37	a	a	DET
ejpam-3441	837	38	regular	regular	NOUN
ejpam-3441	837	39	.	.	PUNCT
ejpam-3441	838	1	again	again	ADV
ejpam-3441	838	2	by	by	ADP
ejpam-3441	838	3	our	our	PRON
ejpam-3441	838	4	supposition	supposition	NOUN
ejpam-3441	838	5	,	,	PUNCT
ejpam-3441	838	6	a	a	DET
ejpam-3441	838	7	∧βα	∧βα	PROPN
ejpam-3441	838	8	b	b	NOUN
ejpam-3441	838	9	=	=	SYM
ejpam-3441	838	10	b	b	X
ejpam-3441	838	11	∧βα	∧βα	DET
ejpam-3441	838	12	a	a	DET
ejpam-3441	838	13	⊆	⊆	NUM
ejpam-3441	838	14	b	b	NOUN
ejpam-3441	838	15	◦	◦	NOUN
ejpam-3441	838	16	βα	βα	X
ejpam-3441	838	17	a	a	PRON
ejpam-3441	838	18	,	,	PUNCT
ejpam-3441	838	19	i.e.	i.e.	X
ejpam-3441	838	20	,	,	PUNCT
ejpam-3441	838	21	r	r	NOUN
ejpam-3441	838	22	is	be	AUX
ejpam-3441	838	23	an	an	DET
ejpam-3441	838	24	intra	intra	ADJ
ejpam-3441	838	25	-	-	ADJ
ejpam-3441	838	26	regular	regular	ADJ
ejpam-3441	838	27	.	.	PUNCT
ejpam-3441	839	1	therefore	therefore	ADV
ejpam-3441	839	2	r	r	NOUN
ejpam-3441	839	3	is	be	AUX
ejpam-3441	839	4	both	both	CCONJ
ejpam-3441	839	5	a	a	DET
ejpam-3441	839	6	regular	regular	ADJ
ejpam-3441	839	7	and	and	CCONJ
ejpam-3441	839	8	an	an	DET
ejpam-3441	839	9	intra	intra	ADJ
ejpam-3441	839	10	-	-	ADJ
ejpam-3441	839	11	regular	regular	ADJ
ejpam-3441	839	12	,	,	PUNCT
ejpam-3441	839	13	i.e.	i.e.	X
ejpam-3441	839	14	,	,	PUNCT
ejpam-3441	839	15	(	(	PUNCT
ejpam-3441	839	16	2	2	X
ejpam-3441	839	17	)	)	PUNCT
ejpam-3441	839	18	⇒	⇒	NOUN
ejpam-3441	839	19	(	(	PUNCT
ejpam-3441	839	20	1	1	NUM
ejpam-3441	839	21	)	)	PUNCT
ejpam-3441	839	22	.	.	PUNCT
ejpam-3441	840	1	in	in	ADP
ejpam-3441	840	2	similar	similar	ADJ
ejpam-3441	840	3	way	way	NOUN
ejpam-3441	840	4	,	,	PUNCT
ejpam-3441	840	5	we	we	PRON
ejpam-3441	840	6	can	can	AUX
ejpam-3441	840	7	prove	prove	VERB
ejpam-3441	840	8	that	that	SCONJ
ejpam-3441	840	9	(	(	PUNCT
ejpam-3441	840	10	3)⇒	3)⇒	NUM
ejpam-3441	840	11	(	(	PUNCT
ejpam-3441	840	12	1	1	NUM
ejpam-3441	840	13	)	)	PUNCT
ejpam-3441	840	14	.	.	PUNCT
ejpam-3441	841	1	k.	k.	PROPN
ejpam-3441	841	2	nasreen	nasreen	PROPN
ejpam-3441	841	3	et	et	PROPN
ejpam-3441	841	4	al	al	PROPN
ejpam-3441	841	5	.	.	PUNCT
ejpam-3441	841	6	/	/	SYM
ejpam-3441	841	7	eur	eur	PROPN
ejpam-3441	841	8	.	.	PUNCT
ejpam-3441	842	1	j.	j.	PROPN
ejpam-3441	842	2	pure	pure	PROPN
ejpam-3441	842	3	appl	appl	PROPN
ejpam-3441	842	4	.	.	PROPN
ejpam-3441	842	5	math	math	PROPN
ejpam-3441	842	6	,	,	PUNCT
ejpam-3441	842	7	12	12	NUM
ejpam-3441	842	8	(	(	PUNCT
ejpam-3441	842	9	3	3	NUM
ejpam-3441	842	10	)	)	PUNCT
ejpam-3441	842	11	(	(	PUNCT
ejpam-3441	842	12	2019	2019	NUM
ejpam-3441	842	13	)	)	PUNCT
ejpam-3441	842	14	,	,	PUNCT
ejpam-3441	842	15	906	906	NUM
ejpam-3441	842	16	-	-	SYM
ejpam-3441	842	17	943	943	NUM
ejpam-3441	842	18	941	941	NUM
ejpam-3441	842	19	theorem	theorem	NOUN
ejpam-3441	842	20	19	19	NUM
ejpam-3441	842	21	.	.	PUNCT
ejpam-3441	843	1	let	let	VERB
ejpam-3441	843	2	r	r	PRON
ejpam-3441	843	3	be	be	AUX
ejpam-3441	843	4	an	an	DET
ejpam-3441	843	5	la	la	NOUN
ejpam-3441	843	6	-	-	NOUN
ejpam-3441	843	7	ring	ring	NOUN
ejpam-3441	843	8	with	with	ADP
ejpam-3441	843	9	left	left	ADJ
ejpam-3441	843	10	identity	identity	NOUN
ejpam-3441	843	11	e	e	NOUN
ejpam-3441	843	12	,	,	PUNCT
ejpam-3441	843	13	such	such	ADJ
ejpam-3441	843	14	that	that	SCONJ
ejpam-3441	843	15	(	(	PUNCT
ejpam-3441	843	16	xe)r	xe)r	PROPN
ejpam-3441	843	17	=	=	SYM
ejpam-3441	843	18	xr	xr	PROPN
ejpam-3441	843	19	for	for	ADP
ejpam-3441	843	20	all	all	DET
ejpam-3441	843	21	x	x	PROPN
ejpam-3441	843	22	∈	∈	PROPN
ejpam-3441	843	23	r.	r.	NOUN
ejpam-3441	843	24	then	then	ADV
ejpam-3441	843	25	the	the	DET
ejpam-3441	843	26	following	follow	VERB
ejpam-3441	843	27	conditions	condition	NOUN
ejpam-3441	843	28	are	be	AUX
ejpam-3441	843	29	equivalent	equivalent	ADJ
ejpam-3441	843	30	.	.	PUNCT
ejpam-3441	844	1	(	(	PUNCT
ejpam-3441	844	2	1	1	X
ejpam-3441	844	3	)	)	PUNCT
ejpam-3441	844	4	r	r	NOUN
ejpam-3441	844	5	is	be	AUX
ejpam-3441	844	6	both	both	CCONJ
ejpam-3441	844	7	a	a	DET
ejpam-3441	844	8	regular	regular	ADJ
ejpam-3441	844	9	and	and	CCONJ
ejpam-3441	844	10	an	an	DET
ejpam-3441	844	11	intra	intra	ADJ
ejpam-3441	844	12	-	-	ADJ
ejpam-3441	844	13	regular	regular	ADJ
ejpam-3441	844	14	.	.	PUNCT
ejpam-3441	845	1	(	(	PUNCT
ejpam-3441	845	2	2	2	X
ejpam-3441	845	3	)	)	PUNCT
ejpam-3441	845	4	a	a	DET
ejpam-3441	845	5	∧βα	∧βα	NOUN
ejpam-3441	845	6	b	b	NOUN
ejpam-3441	845	7	⊆	⊆	NUM
ejpam-3441	845	8	(	(	PUNCT
ejpam-3441	845	9	a	a	DET
ejpam-3441	845	10	◦	◦	NOUN
ejpam-3441	845	11	βα	βα	NOUN
ejpam-3441	845	12	b	b	NOUN
ejpam-3441	845	13	)	)	PUNCT
ejpam-3441	845	14	∧	∧	PROPN
ejpam-3441	845	15	(	(	PUNCT
ejpam-3441	845	16	b	b	X
ejpam-3441	845	17	◦	◦	NOUN
ejpam-3441	845	18	βα	βα	X
ejpam-3441	845	19	a	a	NOUN
ejpam-3441	845	20	)	)	PUNCT
ejpam-3441	845	21	for	for	ADP
ejpam-3441	845	22	every	every	DET
ejpam-3441	845	23	intuitionistic	intuitionistic	ADJ
ejpam-3441	845	24	fuzzy	fuzzy	ADJ
ejpam-3441	845	25	right	right	ADJ
ejpam-3441	845	26	ideal	ideal	NOUN
ejpam-3441	846	1	a	a	PRON
ejpam-3441	846	2	with	with	ADP
ejpam-3441	846	3	thresholds	threshold	NOUN
ejpam-3441	846	4	(	(	PUNCT
ejpam-3441	846	5	α	α	X
ejpam-3441	846	6	,	,	PUNCT
ejpam-3441	846	7	β	β	X
ejpam-3441	846	8	]	]	PUNCT
ejpam-3441	846	9	and	and	CCONJ
ejpam-3441	846	10	every	every	DET
ejpam-3441	846	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	12	fuzzy	fuzzy	ADJ
ejpam-3441	846	13	left	leave	VERB
ejpam-3441	846	14	ideal	ideal	PROPN
ejpam-3441	846	15	b	b	PROPN
ejpam-3441	846	16	with	with	ADP
ejpam-3441	846	17	thresholds	threshold	NOUN
ejpam-3441	846	18	(	(	PUNCT
ejpam-3441	846	19	α	α	X
ejpam-3441	846	20	,	,	PUNCT
ejpam-3441	846	21	β	β	X
ejpam-3441	846	22	]	]	PUNCT
ejpam-3441	846	23	of	of	ADP
ejpam-3441	846	24	r.	r.	PROPN
ejpam-3441	846	25	(	(	PUNCT
ejpam-3441	846	26	3	3	NUM
ejpam-3441	846	27	)	)	PUNCT
ejpam-3441	846	28	a	a	DET
ejpam-3441	846	29	∧βα	∧βα	NOUN
ejpam-3441	846	30	b	b	NOUN
ejpam-3441	846	31	⊆	⊆	NUM
ejpam-3441	846	32	(	(	PUNCT
ejpam-3441	846	33	a	a	DET
ejpam-3441	846	34	◦	◦	NOUN
ejpam-3441	846	35	βα	βα	NOUN
ejpam-3441	846	36	b	b	NOUN
ejpam-3441	846	37	)	)	PUNCT
ejpam-3441	846	38	∧	∧	PROPN
ejpam-3441	846	39	(	(	PUNCT
ejpam-3441	846	40	b	b	X
ejpam-3441	846	41	◦	◦	NOUN
ejpam-3441	846	42	βα	βα	X
ejpam-3441	846	43	a	a	NOUN
ejpam-3441	846	44	)	)	PUNCT
ejpam-3441	846	45	for	for	ADP
ejpam-3441	846	46	every	every	DET
ejpam-3441	846	47	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	48	fuzzy	fuzzy	ADJ
ejpam-3441	846	49	right	right	ADJ
ejpam-3441	846	50	ideal	ideal	NOUN
ejpam-3441	846	51	a	a	PRON
ejpam-3441	846	52	with	with	ADP
ejpam-3441	846	53	thresholds	threshold	NOUN
ejpam-3441	846	54	(	(	PUNCT
ejpam-3441	846	55	α	α	X
ejpam-3441	846	56	,	,	PUNCT
ejpam-3441	846	57	β	β	X
ejpam-3441	846	58	]	]	PUNCT
ejpam-3441	846	59	and	and	CCONJ
ejpam-3441	846	60	every	every	DET
ejpam-3441	846	61	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	62	fuzzy	fuzzy	ADJ
ejpam-3441	846	63	quasi	quasi	ADJ
ejpam-3441	846	64	-	-	ADJ
ejpam-3441	846	65	ideal	ideal	ADJ
ejpam-3441	846	66	b	b	PROPN
ejpam-3441	846	67	with	with	ADP
ejpam-3441	846	68	thresholds	threshold	NOUN
ejpam-3441	846	69	(	(	PUNCT
ejpam-3441	846	70	α	α	X
ejpam-3441	846	71	,	,	PUNCT
ejpam-3441	846	72	β	β	X
ejpam-3441	846	73	]	]	PUNCT
ejpam-3441	846	74	of	of	ADP
ejpam-3441	846	75	r.	r.	PROPN
ejpam-3441	846	76	(	(	PUNCT
ejpam-3441	846	77	4	4	NUM
ejpam-3441	846	78	)	)	PUNCT
ejpam-3441	846	79	a	a	DET
ejpam-3441	846	80	∧βα	∧βα	NOUN
ejpam-3441	846	81	b	b	NOUN
ejpam-3441	846	82	⊆	⊆	NUM
ejpam-3441	846	83	(	(	PUNCT
ejpam-3441	846	84	a	a	DET
ejpam-3441	846	85	◦	◦	NOUN
ejpam-3441	846	86	βα	βα	NOUN
ejpam-3441	846	87	b	b	NOUN
ejpam-3441	846	88	)	)	PUNCT
ejpam-3441	846	89	∧	∧	PROPN
ejpam-3441	846	90	(	(	PUNCT
ejpam-3441	846	91	b	b	X
ejpam-3441	846	92	◦	◦	NOUN
ejpam-3441	846	93	βα	βα	X
ejpam-3441	846	94	a	a	NOUN
ejpam-3441	846	95	)	)	PUNCT
ejpam-3441	846	96	for	for	ADP
ejpam-3441	846	97	every	every	DET
ejpam-3441	846	98	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	99	fuzzy	fuzzy	ADJ
ejpam-3441	846	100	right	right	ADJ
ejpam-3441	846	101	ideal	ideal	NOUN
ejpam-3441	846	102	a	a	PRON
ejpam-3441	846	103	with	with	ADP
ejpam-3441	846	104	thresholds	threshold	NOUN
ejpam-3441	846	105	(	(	PUNCT
ejpam-3441	846	106	α	α	X
ejpam-3441	846	107	,	,	PUNCT
ejpam-3441	846	108	β	β	X
ejpam-3441	846	109	]	]	PUNCT
ejpam-3441	846	110	and	and	CCONJ
ejpam-3441	846	111	every	every	DET
ejpam-3441	846	112	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	113	fuzzy	fuzzy	ADJ
ejpam-3441	846	114	bi	bi	ADJ
ejpam-3441	846	115	-	-	ADJ
ejpam-3441	846	116	ideal	ideal	ADJ
ejpam-3441	846	117	b	b	PROPN
ejpam-3441	846	118	with	with	ADP
ejpam-3441	846	119	thresholds	threshold	NOUN
ejpam-3441	846	120	(	(	PUNCT
ejpam-3441	846	121	α	α	X
ejpam-3441	846	122	,	,	PUNCT
ejpam-3441	846	123	β	β	X
ejpam-3441	846	124	]	]	PUNCT
ejpam-3441	846	125	of	of	ADP
ejpam-3441	846	126	r.	r.	PROPN
ejpam-3441	846	127	(	(	PUNCT
ejpam-3441	846	128	5	5	NUM
ejpam-3441	846	129	)	)	PUNCT
ejpam-3441	846	130	a	a	DET
ejpam-3441	846	131	∧βα	∧βα	NOUN
ejpam-3441	846	132	b	b	NOUN
ejpam-3441	846	133	⊆	⊆	NUM
ejpam-3441	846	134	(	(	PUNCT
ejpam-3441	846	135	a	a	DET
ejpam-3441	846	136	◦	◦	NOUN
ejpam-3441	846	137	βα	βα	NOUN
ejpam-3441	846	138	b	b	NOUN
ejpam-3441	846	139	)	)	PUNCT
ejpam-3441	846	140	∧	∧	PROPN
ejpam-3441	846	141	(	(	PUNCT
ejpam-3441	846	142	b	b	X
ejpam-3441	846	143	◦	◦	NOUN
ejpam-3441	846	144	βα	βα	X
ejpam-3441	846	145	a	a	NOUN
ejpam-3441	846	146	)	)	PUNCT
ejpam-3441	846	147	for	for	ADP
ejpam-3441	846	148	every	every	DET
ejpam-3441	846	149	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	150	fuzzy	fuzzy	ADJ
ejpam-3441	846	151	right	right	ADJ
ejpam-3441	846	152	ideal	ideal	NOUN
ejpam-3441	846	153	a	a	PRON
ejpam-3441	846	154	with	with	ADP
ejpam-3441	846	155	thresholds	threshold	NOUN
ejpam-3441	846	156	(	(	PUNCT
ejpam-3441	846	157	α	α	X
ejpam-3441	846	158	,	,	PUNCT
ejpam-3441	846	159	β	β	X
ejpam-3441	846	160	]	]	PUNCT
ejpam-3441	846	161	and	and	CCONJ
ejpam-3441	846	162	every	every	DET
ejpam-3441	846	163	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	164	fuzzy	fuzzy	ADJ
ejpam-3441	846	165	generalized	generalize	VERB
ejpam-3441	846	166	bi	bi	ADJ
ejpam-3441	846	167	-	-	ADJ
ejpam-3441	846	168	ideal	ideal	ADJ
ejpam-3441	846	169	b	b	PROPN
ejpam-3441	846	170	with	with	ADP
ejpam-3441	846	171	thresholds	threshold	NOUN
ejpam-3441	846	172	(	(	PUNCT
ejpam-3441	846	173	α	α	X
ejpam-3441	846	174	,	,	PUNCT
ejpam-3441	846	175	β	β	X
ejpam-3441	846	176	]	]	PUNCT
ejpam-3441	846	177	of	of	ADP
ejpam-3441	846	178	r.	r.	PROPN
ejpam-3441	846	179	(	(	PUNCT
ejpam-3441	846	180	6	6	NUM
ejpam-3441	846	181	)	)	PUNCT
ejpam-3441	846	182	a∧βαb	a∧βαb	NUM
ejpam-3441	846	183	⊆	⊆	NUM
ejpam-3441	846	184	(	(	PUNCT
ejpam-3441	846	185	a	a	DET
ejpam-3441	846	186	◦	◦	NOUN
ejpam-3441	846	187	βαb)∧(b	βαb)∧(b	NOUN
ejpam-3441	846	188	◦	◦	NOUN
ejpam-3441	846	189	βαa	βαa	NOUN
ejpam-3441	846	190	)	)	PUNCT
ejpam-3441	846	191	for	for	ADP
ejpam-3441	846	192	every	every	DET
ejpam-3441	846	193	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	194	fuzzy	fuzzy	ADJ
ejpam-3441	846	195	left	leave	VERB
ejpam-3441	846	196	ideal	ideal	NOUN
ejpam-3441	846	197	a	a	PRON
ejpam-3441	846	198	with	with	ADP
ejpam-3441	846	199	thresholds	threshold	NOUN
ejpam-3441	846	200	(	(	PUNCT
ejpam-3441	846	201	α	α	X
ejpam-3441	846	202	,	,	PUNCT
ejpam-3441	846	203	β	β	X
ejpam-3441	846	204	]	]	PUNCT
ejpam-3441	846	205	and	and	CCONJ
ejpam-3441	846	206	every	every	DET
ejpam-3441	846	207	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	208	fuzzy	fuzzy	ADJ
ejpam-3441	846	209	quasi	quasi	ADJ
ejpam-3441	846	210	-	-	ADJ
ejpam-3441	846	211	ideal	ideal	ADJ
ejpam-3441	846	212	b	b	PROPN
ejpam-3441	846	213	with	with	ADP
ejpam-3441	846	214	thresholds	threshold	NOUN
ejpam-3441	846	215	(	(	PUNCT
ejpam-3441	846	216	α	α	X
ejpam-3441	846	217	,	,	PUNCT
ejpam-3441	846	218	β	β	X
ejpam-3441	846	219	]	]	PUNCT
ejpam-3441	846	220	of	of	ADP
ejpam-3441	846	221	r.	r.	PROPN
ejpam-3441	846	222	(	(	PUNCT
ejpam-3441	846	223	7	7	NUM
ejpam-3441	846	224	)	)	PUNCT
ejpam-3441	846	225	a∧βαb	a∧βαb	NUM
ejpam-3441	846	226	⊆	⊆	NUM
ejpam-3441	846	227	(	(	PUNCT
ejpam-3441	846	228	a	a	DET
ejpam-3441	846	229	◦	◦	NOUN
ejpam-3441	846	230	βαb)∧(b	βαb)∧(b	NOUN
ejpam-3441	846	231	◦	◦	NOUN
ejpam-3441	846	232	βαa	βαa	NOUN
ejpam-3441	846	233	)	)	PUNCT
ejpam-3441	846	234	for	for	ADP
ejpam-3441	846	235	every	every	DET
ejpam-3441	846	236	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	237	fuzzy	fuzzy	ADJ
ejpam-3441	846	238	left	leave	VERB
ejpam-3441	846	239	ideal	ideal	NOUN
ejpam-3441	846	240	a	a	PRON
ejpam-3441	846	241	with	with	ADP
ejpam-3441	846	242	thresholds	threshold	NOUN
ejpam-3441	846	243	(	(	PUNCT
ejpam-3441	846	244	α	α	X
ejpam-3441	846	245	,	,	PUNCT
ejpam-3441	846	246	β	β	X
ejpam-3441	846	247	]	]	PUNCT
ejpam-3441	846	248	and	and	CCONJ
ejpam-3441	846	249	every	every	DET
ejpam-3441	846	250	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	251	fuzzy	fuzzy	ADJ
ejpam-3441	846	252	bi	bi	ADJ
ejpam-3441	846	253	-	-	ADJ
ejpam-3441	846	254	ideal	ideal	ADJ
ejpam-3441	846	255	b	b	PROPN
ejpam-3441	846	256	with	with	ADP
ejpam-3441	846	257	thresholds	threshold	NOUN
ejpam-3441	846	258	(	(	PUNCT
ejpam-3441	846	259	α	α	X
ejpam-3441	846	260	,	,	PUNCT
ejpam-3441	846	261	β	β	X
ejpam-3441	846	262	]	]	PUNCT
ejpam-3441	846	263	of	of	ADP
ejpam-3441	846	264	r.	r.	PROPN
ejpam-3441	846	265	(	(	PUNCT
ejpam-3441	846	266	8)	8)	NUM
ejpam-3441	846	267	a∧βαb	a∧βαb	NUM
ejpam-3441	846	268	⊆	⊆	NUM
ejpam-3441	846	269	(	(	PUNCT
ejpam-3441	846	270	a	a	DET
ejpam-3441	846	271	◦	◦	NOUN
ejpam-3441	846	272	βαb)∧(b	βαb)∧(b	NOUN
ejpam-3441	846	273	◦	◦	NOUN
ejpam-3441	846	274	βαa	βαa	NOUN
ejpam-3441	846	275	)	)	PUNCT
ejpam-3441	846	276	for	for	ADP
ejpam-3441	846	277	every	every	DET
ejpam-3441	846	278	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	279	fuzzy	fuzzy	ADJ
ejpam-3441	846	280	left	leave	VERB
ejpam-3441	846	281	ideal	ideal	NOUN
ejpam-3441	846	282	a	a	PRON
ejpam-3441	846	283	with	with	ADP
ejpam-3441	846	284	thresholds	threshold	NOUN
ejpam-3441	846	285	(	(	PUNCT
ejpam-3441	846	286	α	α	X
ejpam-3441	846	287	,	,	PUNCT
ejpam-3441	846	288	β	β	X
ejpam-3441	846	289	]	]	PUNCT
ejpam-3441	846	290	and	and	CCONJ
ejpam-3441	846	291	every	every	DET
ejpam-3441	846	292	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	293	fuzzy	fuzzy	ADJ
ejpam-3441	846	294	generalized	generalize	VERB
ejpam-3441	846	295	bi	bi	ADJ
ejpam-3441	846	296	-	-	ADJ
ejpam-3441	846	297	ideal	ideal	ADJ
ejpam-3441	846	298	b	b	PROPN
ejpam-3441	846	299	with	with	ADP
ejpam-3441	846	300	thresholds	threshold	NOUN
ejpam-3441	846	301	(	(	PUNCT
ejpam-3441	846	302	α	α	X
ejpam-3441	846	303	,	,	PUNCT
ejpam-3441	846	304	β	β	X
ejpam-3441	846	305	]	]	PUNCT
ejpam-3441	846	306	of	of	ADP
ejpam-3441	846	307	r.	r.	PROPN
ejpam-3441	846	308	(	(	PUNCT
ejpam-3441	846	309	9	9	NUM
ejpam-3441	846	310	)	)	PUNCT
ejpam-3441	846	311	a∧βαb	a∧βαb	NUM
ejpam-3441	846	312	⊆	⊆	NUM
ejpam-3441	846	313	(	(	PUNCT
ejpam-3441	846	314	a	a	DET
ejpam-3441	846	315	◦	◦	NOUN
ejpam-3441	846	316	βαb)∧	βαb)∧	PUNCT
ejpam-3441	846	317	(	(	PUNCT
ejpam-3441	846	318	b	b	PROPN
ejpam-3441	846	319	◦	◦	NOUN
ejpam-3441	846	320	βαa	βαa	NOUN
ejpam-3441	846	321	)	)	PUNCT
ejpam-3441	846	322	for	for	ADP
ejpam-3441	846	323	all	all	DET
ejpam-3441	846	324	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	325	fuzzy	fuzzy	ADJ
ejpam-3441	846	326	quasi	quasi	NOUN
ejpam-3441	846	327	-	-	NOUN
ejpam-3441	846	328	ideals	ideal	NOUN
ejpam-3441	846	329	a	a	PRON
ejpam-3441	846	330	and	and	CCONJ
ejpam-3441	846	331	b	b	NOUN
ejpam-3441	846	332	with	with	ADP
ejpam-3441	846	333	thresholds	threshold	NOUN
ejpam-3441	846	334	(	(	PUNCT
ejpam-3441	846	335	α	α	X
ejpam-3441	846	336	,	,	PUNCT
ejpam-3441	846	337	β	β	X
ejpam-3441	846	338	]	]	PUNCT
ejpam-3441	846	339	of	of	ADP
ejpam-3441	846	340	r.	r.	PROPN
ejpam-3441	846	341	(	(	PUNCT
ejpam-3441	846	342	10	10	NUM
ejpam-3441	846	343	)	)	PUNCT
ejpam-3441	846	344	a	a	DET
ejpam-3441	846	345	∧βα	∧βα	NOUN
ejpam-3441	846	346	b	b	NOUN
ejpam-3441	846	347	⊆	⊆	NUM
ejpam-3441	846	348	(	(	PUNCT
ejpam-3441	846	349	a	a	DET
ejpam-3441	846	350	◦	◦	NOUN
ejpam-3441	846	351	βα	βα	NOUN
ejpam-3441	846	352	b	b	NOUN
ejpam-3441	846	353	)	)	PUNCT
ejpam-3441	846	354	∧	∧	PROPN
ejpam-3441	846	355	(	(	PUNCT
ejpam-3441	846	356	b	b	X
ejpam-3441	846	357	◦	◦	NOUN
ejpam-3441	846	358	βα	βα	X
ejpam-3441	846	359	a	a	NOUN
ejpam-3441	846	360	)	)	PUNCT
ejpam-3441	846	361	for	for	ADP
ejpam-3441	846	362	every	every	DET
ejpam-3441	846	363	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	364	fuzzy	fuzzy	ADJ
ejpam-3441	846	365	quasi	quasi	NOUN
ejpam-3441	846	366	-	-	NOUN
ejpam-3441	846	367	ideal	ideal	ADJ
ejpam-3441	846	368	a	a	PRON
ejpam-3441	846	369	with	with	ADP
ejpam-3441	846	370	thresholds	threshold	NOUN
ejpam-3441	846	371	(	(	PUNCT
ejpam-3441	846	372	α	α	X
ejpam-3441	846	373	,	,	PUNCT
ejpam-3441	846	374	β	β	X
ejpam-3441	846	375	]	]	PUNCT
ejpam-3441	846	376	and	and	CCONJ
ejpam-3441	846	377	every	every	DET
ejpam-3441	846	378	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	379	fuzzy	fuzzy	ADJ
ejpam-3441	846	380	bi	bi	ADJ
ejpam-3441	846	381	-	-	ADJ
ejpam-3441	846	382	ideal	ideal	ADJ
ejpam-3441	846	383	b	b	PROPN
ejpam-3441	846	384	with	with	ADP
ejpam-3441	846	385	thresholds	threshold	NOUN
ejpam-3441	846	386	(	(	PUNCT
ejpam-3441	846	387	α	α	X
ejpam-3441	846	388	,	,	PUNCT
ejpam-3441	846	389	β	β	X
ejpam-3441	846	390	]	]	PUNCT
ejpam-3441	846	391	of	of	ADP
ejpam-3441	846	392	r.	r.	PROPN
ejpam-3441	846	393	(	(	PUNCT
ejpam-3441	846	394	11	11	NUM
ejpam-3441	846	395	)	)	PUNCT
ejpam-3441	846	396	a	a	DET
ejpam-3441	846	397	∧βα	∧βα	PROPN
ejpam-3441	846	398	b	b	NOUN
ejpam-3441	846	399	⊆	⊆	NUM
ejpam-3441	846	400	(	(	PUNCT
ejpam-3441	846	401	a	a	DET
ejpam-3441	846	402	◦	◦	NOUN
ejpam-3441	846	403	βα	βα	NOUN
ejpam-3441	846	404	b	b	NOUN
ejpam-3441	846	405	)	)	PUNCT
ejpam-3441	846	406	∧	∧	PROPN
ejpam-3441	846	407	(	(	PUNCT
ejpam-3441	846	408	b	b	X
ejpam-3441	846	409	◦	◦	NOUN
ejpam-3441	846	410	βα	βα	X
ejpam-3441	846	411	a	a	NOUN
ejpam-3441	846	412	)	)	PUNCT
ejpam-3441	846	413	for	for	ADP
ejpam-3441	846	414	every	every	DET
ejpam-3441	846	415	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	416	fuzzy	fuzzy	ADJ
ejpam-3441	846	417	quasi	quasi	NOUN
ejpam-3441	846	418	-	-	NOUN
ejpam-3441	846	419	ideal	ideal	ADJ
ejpam-3441	846	420	a	a	PRON
ejpam-3441	846	421	with	with	ADP
ejpam-3441	846	422	thresholds	threshold	NOUN
ejpam-3441	846	423	(	(	PUNCT
ejpam-3441	846	424	α	α	X
ejpam-3441	846	425	,	,	PUNCT
ejpam-3441	846	426	β	β	X
ejpam-3441	846	427	]	]	PUNCT
ejpam-3441	846	428	and	and	CCONJ
ejpam-3441	846	429	every	every	DET
ejpam-3441	846	430	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	431	fuzzy	fuzzy	ADJ
ejpam-3441	846	432	generalized	generalize	VERB
ejpam-3441	846	433	bi	bi	ADJ
ejpam-3441	846	434	-	-	ADJ
ejpam-3441	846	435	ideal	ideal	ADJ
ejpam-3441	846	436	b	b	PROPN
ejpam-3441	846	437	with	with	ADP
ejpam-3441	846	438	thresholds	threshold	NOUN
ejpam-3441	846	439	(	(	PUNCT
ejpam-3441	846	440	α	α	X
ejpam-3441	846	441	,	,	PUNCT
ejpam-3441	846	442	β	β	X
ejpam-3441	846	443	]	]	PUNCT
ejpam-3441	846	444	of	of	ADP
ejpam-3441	846	445	r.	r.	PROPN
ejpam-3441	846	446	(	(	PUNCT
ejpam-3441	846	447	12	12	NUM
ejpam-3441	846	448	)	)	PUNCT
ejpam-3441	846	449	a∧βαb	a∧βαb	NUM
ejpam-3441	846	450	⊆	⊆	NUM
ejpam-3441	846	451	(	(	PUNCT
ejpam-3441	846	452	a	a	DET
ejpam-3441	846	453	◦	◦	NOUN
ejpam-3441	846	454	βαb)∧(b	βαb)∧(b	NOUN
ejpam-3441	846	455	◦	◦	NOUN
ejpam-3441	846	456	βαa	βαa	PROPN
ejpam-3441	846	457	)	)	PUNCT
ejpam-3441	846	458	for	for	ADP
ejpam-3441	846	459	all	all	DET
ejpam-3441	846	460	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	461	fuzzy	fuzzy	ADJ
ejpam-3441	846	462	bi	bi	NOUN
ejpam-3441	846	463	-	-	NOUN
ejpam-3441	846	464	ideals	ideal	NOUN
ejpam-3441	846	465	a	a	PRON
ejpam-3441	846	466	with	with	ADP
ejpam-3441	846	467	thresholds	threshold	NOUN
ejpam-3441	846	468	(	(	PUNCT
ejpam-3441	846	469	α	α	X
ejpam-3441	846	470	,	,	PUNCT
ejpam-3441	846	471	β	β	X
ejpam-3441	846	472	]	]	PUNCT
ejpam-3441	846	473	and	and	CCONJ
ejpam-3441	846	474	b	b	X
ejpam-3441	846	475	with	with	ADP
ejpam-3441	846	476	thresholds	threshold	NOUN
ejpam-3441	846	477	(	(	PUNCT
ejpam-3441	846	478	α	α	X
ejpam-3441	846	479	,	,	PUNCT
ejpam-3441	846	480	β	β	X
ejpam-3441	846	481	]	]	PUNCT
ejpam-3441	846	482	of	of	ADP
ejpam-3441	846	483	r.	r.	PROPN
ejpam-3441	846	484	(	(	PUNCT
ejpam-3441	846	485	13	13	NUM
ejpam-3441	846	486	)	)	PUNCT
ejpam-3441	846	487	a∧βαb	a∧βαb	NUM
ejpam-3441	846	488	⊆	⊆	NUM
ejpam-3441	846	489	(	(	PUNCT
ejpam-3441	846	490	a	a	DET
ejpam-3441	846	491	◦	◦	NOUN
ejpam-3441	846	492	βαb)∧(b	βαb)∧(b	NOUN
ejpam-3441	846	493	◦	◦	NOUN
ejpam-3441	846	494	βαa	βαa	NOUN
ejpam-3441	846	495	)	)	PUNCT
ejpam-3441	846	496	for	for	ADP
ejpam-3441	846	497	every	every	DET
ejpam-3441	846	498	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	499	fuzzy	fuzzy	ADJ
ejpam-3441	846	500	bi	bi	NOUN
ejpam-3441	846	501	-	-	NOUN
ejpam-3441	846	502	ideal	ideal	NOUN
ejpam-3441	846	503	a	a	PRON
ejpam-3441	846	504	with	with	ADP
ejpam-3441	846	505	thresholds	threshold	NOUN
ejpam-3441	846	506	(	(	PUNCT
ejpam-3441	846	507	α	α	X
ejpam-3441	846	508	,	,	PUNCT
ejpam-3441	846	509	β	β	X
ejpam-3441	846	510	]	]	PUNCT
ejpam-3441	846	511	and	and	CCONJ
ejpam-3441	846	512	every	every	DET
ejpam-3441	846	513	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	514	fuzzy	fuzzy	ADJ
ejpam-3441	846	515	generalized	generalize	VERB
ejpam-3441	846	516	bi	bi	ADJ
ejpam-3441	846	517	-	-	ADJ
ejpam-3441	846	518	ideal	ideal	ADJ
ejpam-3441	846	519	b	b	PROPN
ejpam-3441	846	520	with	with	ADP
ejpam-3441	846	521	thresholds	threshold	NOUN
ejpam-3441	846	522	(	(	PUNCT
ejpam-3441	846	523	α	α	X
ejpam-3441	846	524	,	,	PUNCT
ejpam-3441	846	525	β	β	X
ejpam-3441	846	526	]	]	PUNCT
ejpam-3441	846	527	of	of	ADP
ejpam-3441	846	528	r.	r.	PROPN
ejpam-3441	846	529	(	(	PUNCT
ejpam-3441	846	530	14	14	NUM
ejpam-3441	846	531	)	)	PUNCT
ejpam-3441	846	532	a	a	DET
ejpam-3441	846	533	∧βα	∧βα	NOUN
ejpam-3441	846	534	b	b	NOUN
ejpam-3441	846	535	⊆	⊆	NUM
ejpam-3441	846	536	(	(	PUNCT
ejpam-3441	846	537	a	a	DET
ejpam-3441	846	538	◦	◦	NOUN
ejpam-3441	846	539	βα	βα	NOUN
ejpam-3441	846	540	b	b	NOUN
ejpam-3441	846	541	)	)	PUNCT
ejpam-3441	846	542	∧	∧	PROPN
ejpam-3441	846	543	(	(	PUNCT
ejpam-3441	846	544	b	b	X
ejpam-3441	846	545	◦	◦	NOUN
ejpam-3441	846	546	βα	βα	X
ejpam-3441	846	547	a	a	NOUN
ejpam-3441	846	548	)	)	PUNCT
ejpam-3441	846	549	for	for	ADP
ejpam-3441	846	550	all	all	DET
ejpam-3441	846	551	intuitionistic	intuitionistic	ADJ
ejpam-3441	846	552	fuzzy	fuzzy	ADJ
ejpam-3441	846	553	generalized	generalize	VERB
ejpam-3441	846	554	bi	bi	NOUN
ejpam-3441	846	555	-	-	NOUN
ejpam-3441	846	556	ideals	ideal	NOUN
ejpam-3441	846	557	a	a	PRON
ejpam-3441	846	558	and	and	CCONJ
ejpam-3441	846	559	b	b	NOUN
ejpam-3441	846	560	with	with	ADP
ejpam-3441	846	561	thresholds	threshold	NOUN
ejpam-3441	846	562	(	(	PUNCT
ejpam-3441	846	563	α	α	X
ejpam-3441	846	564	,	,	PUNCT
ejpam-3441	846	565	β	β	X
ejpam-3441	846	566	]	]	PUNCT
ejpam-3441	846	567	of	of	ADP
ejpam-3441	846	568	r.	r.	PROPN
ejpam-3441	846	569	proof	proof	NOUN
ejpam-3441	846	570	.	.	PUNCT
ejpam-3441	847	1	consider	consider	VERB
ejpam-3441	847	2	that	that	PRON
ejpam-3441	847	3	(	(	PUNCT
ejpam-3441	847	4	1	1	X
ejpam-3441	847	5	)	)	PUNCT
ejpam-3441	847	6	holds	hold	VERB
ejpam-3441	847	7	.	.	PUNCT
ejpam-3441	848	1	since	since	SCONJ
ejpam-3441	848	2	a	a	DET
ejpam-3441	848	3	∧βα	∧βα	PROPN
ejpam-3441	848	4	b	b	NOUN
ejpam-3441	848	5	⊆	⊆	NUM
ejpam-3441	848	6	a	a	DET
ejpam-3441	848	7	◦	◦	NOUN
ejpam-3441	848	8	βα	βα	NOUN
ejpam-3441	848	9	b	b	NOUN
ejpam-3441	848	10	and	and	CCONJ
ejpam-3441	848	11	a	a	DET
ejpam-3441	848	12	∧βα	∧βα	PROPN
ejpam-3441	848	13	b	b	NOUN
ejpam-3441	848	14	⊆	⊆	NUM
ejpam-3441	848	15	b	b	NOUN
ejpam-3441	848	16	◦	◦	NOUN
ejpam-3441	848	17	βα	βα	X
ejpam-3441	848	18	a	a	PRON
ejpam-3441	848	19	for	for	ADP
ejpam-3441	848	20	all	all	DET
ejpam-3441	848	21	intuitionistic	intuitionistic	ADJ
ejpam-3441	848	22	fuzzy	fuzzy	ADJ
ejpam-3441	848	23	generalized	generalize	VERB
ejpam-3441	848	24	bi	bi	NOUN
ejpam-3441	848	25	-	-	NOUN
ejpam-3441	848	26	ideals	ideal	NOUN
ejpam-3441	848	27	a	a	PRON
ejpam-3441	848	28	and	and	CCONJ
ejpam-3441	848	29	b	b	NOUN
ejpam-3441	848	30	with	with	ADP
ejpam-3441	848	31	thresholds	threshold	NOUN
ejpam-3441	848	32	(	(	PUNCT
ejpam-3441	848	33	α	α	X
ejpam-3441	848	34	,	,	PUNCT
ejpam-3441	848	35	β	β	X
ejpam-3441	848	36	]	]	PUNCT
ejpam-3441	848	37	of	of	ADP
ejpam-3441	848	38	r	r	NOUN
ejpam-3441	848	39	by	by	ADP
ejpam-3441	848	40	the	the	DET
ejpam-3441	848	41	theorem	theorem	ADJ
ejpam-3441	848	42	18	18	NUM
ejpam-3441	848	43	.	.	PUNCT
ejpam-3441	848	44	hence	hence	ADV
ejpam-3441	848	45	a∧βαb	a∧βαb	NUM
ejpam-3441	848	46	⊆	⊆	NUM
ejpam-3441	848	47	(	(	PUNCT
ejpam-3441	848	48	a	a	DET
ejpam-3441	848	49	◦	◦	NOUN
ejpam-3441	848	50	βαb)∧	βαb)∧	PUNCT
ejpam-3441	848	51	(	(	PUNCT
ejpam-3441	848	52	b	b	NOUN
ejpam-3441	848	53	◦	◦	NOUN
ejpam-3441	848	54	βαa	βαa	NOUN
ejpam-3441	848	55	)	)	PUNCT
ejpam-3441	848	56	,	,	PUNCT
ejpam-3441	848	57	i.e.	i.e.	X
ejpam-3441	848	58	,	,	PUNCT
ejpam-3441	848	59	(	(	PUNCT
ejpam-3441	848	60	1)⇒	1)⇒	NUM
ejpam-3441	848	61	(	(	PUNCT
ejpam-3441	848	62	14	14	NUM
ejpam-3441	848	63	)	)	PUNCT
ejpam-3441	848	64	.	.	PUNCT
ejpam-3441	849	1	it	it	PRON
ejpam-3441	849	2	is	be	AUX
ejpam-3441	849	3	clear	clear	ADJ
ejpam-3441	849	4	that	that	SCONJ
ejpam-3441	849	5	(	(	PUNCT
ejpam-3441	849	6	14)⇒	14)⇒	NUM
ejpam-3441	849	7	(	(	PUNCT
ejpam-3441	849	8	13)⇒	13)⇒	NUM
ejpam-3441	849	9	(	(	PUNCT
ejpam-3441	849	10	12)⇒	12)⇒	PROPN
ejpam-3441	849	11	(	(	PUNCT
ejpam-3441	849	12	9)⇒	9)⇒	NUM
ejpam-3441	849	13	(	(	PUNCT
ejpam-3441	849	14	6)⇒	6)⇒	NUM
ejpam-3441	849	15	(	(	PUNCT
ejpam-3441	849	16	2	2	NUM
ejpam-3441	849	17	)	)	PUNCT
ejpam-3441	849	18	,	,	PUNCT
ejpam-3441	849	19	(	(	PUNCT
ejpam-3441	849	20	14)⇒	14)⇒	NUM
ejpam-3441	849	21	(	(	PUNCT
ejpam-3441	849	22	11)⇒	11)⇒	NUM
ejpam-3441	849	23	(	(	PUNCT
ejpam-3441	849	24	10)⇒	10)⇒	NUM
ejpam-3441	849	25	(	(	PUNCT
ejpam-3441	849	26	9	9	NUM
ejpam-3441	849	27	)	)	PUNCT
ejpam-3441	849	28	,	,	PUNCT
ejpam-3441	849	29	(	(	PUNCT
ejpam-3441	849	30	14)⇒	14)⇒	NUM
ejpam-3441	849	31	(	(	PUNCT
ejpam-3441	849	32	8)⇒	8)⇒	NUM
ejpam-3441	849	33	(	(	PUNCT
ejpam-3441	849	34	7)⇒	7)⇒	NUM
ejpam-3441	849	35	(	(	PUNCT
ejpam-3441	849	36	6	6	NUM
ejpam-3441	849	37	)	)	PUNCT
ejpam-3441	849	38	and	and	CCONJ
ejpam-3441	849	39	(	(	PUNCT
ejpam-3441	849	40	14	14	NUM
ejpam-3441	849	41	)	)	PUNCT
ejpam-3441	849	42	⇒	⇒	NOUN
ejpam-3441	849	43	(	(	PUNCT
ejpam-3441	849	44	5	5	NUM
ejpam-3441	849	45	)	)	PUNCT
ejpam-3441	849	46	⇒	⇒	NOUN
ejpam-3441	849	47	(	(	PUNCT
ejpam-3441	849	48	4	4	NUM
ejpam-3441	849	49	)	)	PUNCT
ejpam-3441	849	50	⇒	⇒	NOUN
ejpam-3441	849	51	(	(	PUNCT
ejpam-3441	849	52	3	3	NUM
ejpam-3441	849	53	)	)	PUNCT
ejpam-3441	849	54	⇒	⇒	NOUN
ejpam-3441	849	55	(	(	PUNCT
ejpam-3441	849	56	2	2	NUM
ejpam-3441	849	57	)	)	PUNCT
ejpam-3441	849	58	.	.	PUNCT
ejpam-3441	850	1	suppose	suppose	VERB
ejpam-3441	850	2	that	that	SCONJ
ejpam-3441	850	3	(	(	PUNCT
ejpam-3441	850	4	2	2	X
ejpam-3441	850	5	)	)	PUNCT
ejpam-3441	850	6	holds	hold	VERB
ejpam-3441	850	7	.	.	PUNCT
ejpam-3441	851	1	let	let	VERB
ejpam-3441	851	2	a	a	PRON
ejpam-3441	851	3	be	be	AUX
ejpam-3441	851	4	an	an	DET
ejpam-3441	851	5	intuitionistic	intuitionistic	ADJ
ejpam-3441	851	6	fuzzy	fuzzy	ADJ
ejpam-3441	851	7	right	right	ADJ
ejpam-3441	851	8	ideal	ideal	NOUN
ejpam-3441	851	9	with	with	ADP
ejpam-3441	851	10	thresholds	threshold	NOUN
ejpam-3441	851	11	(	(	PUNCT
ejpam-3441	851	12	α	α	X
ejpam-3441	851	13	,	,	PUNCT
ejpam-3441	851	14	β	β	X
ejpam-3441	851	15	]	]	PUNCT
ejpam-3441	851	16	and	and	CCONJ
ejpam-3441	851	17	b	b	X
ejpam-3441	851	18	be	be	AUX
ejpam-3441	851	19	an	an	DET
ejpam-3441	851	20	intuitionistic	intuitionistic	ADJ
ejpam-3441	851	21	fuzzy	fuzzy	ADJ
ejpam-3441	851	22	left	leave	VERB
ejpam-3441	851	23	ideal	ideal	NOUN
ejpam-3441	851	24	with	with	ADP
ejpam-3441	851	25	thresholds	threshold	NOUN
ejpam-3441	851	26	(	(	PUNCT
ejpam-3441	851	27	α	α	X
ejpam-3441	851	28	,	,	PUNCT
ejpam-3441	851	29	β	β	X
ejpam-3441	851	30	]	]	PUNCT
ejpam-3441	851	31	of	of	ADP
ejpam-3441	851	32	r.	r.	PROPN
ejpam-3441	851	33	by	by	ADP
ejpam-3441	851	34	our	our	PRON
ejpam-3441	851	35	supposition	supposition	NOUN
ejpam-3441	851	36	a	a	DET
ejpam-3441	851	37	∧βα	∧βα	NOUN
ejpam-3441	851	38	b	b	NOUN
ejpam-3441	851	39	⊆	⊆	NUM
ejpam-3441	851	40	(	(	PUNCT
ejpam-3441	851	41	a	a	DET
ejpam-3441	851	42	◦	◦	NOUN
ejpam-3441	851	43	βα	βα	NOUN
ejpam-3441	851	44	b	b	NOUN
ejpam-3441	851	45	)	)	PUNCT
ejpam-3441	851	46	∧	∧	PROPN
ejpam-3441	851	47	(	(	PUNCT
ejpam-3441	851	48	b	b	X
ejpam-3441	851	49	◦	◦	NOUN
ejpam-3441	851	50	βα	βα	X
ejpam-3441	851	51	a	a	NOUN
ejpam-3441	851	52	)	)	PUNCT
ejpam-3441	851	53	⊆	⊆	NUM
ejpam-3441	851	54	b	b	X
ejpam-3441	851	55	◦	◦	NOUN
ejpam-3441	851	56	βα	βα	X
ejpam-3441	851	57	a	a	PRON
ejpam-3441	851	58	,	,	PUNCT
ejpam-3441	851	59	i.e.	i.e.	X
ejpam-3441	851	60	,	,	PUNCT
ejpam-3441	851	61	r	r	NOUN
ejpam-3441	851	62	is	be	AUX
ejpam-3441	851	63	an	an	DET
ejpam-3441	851	64	intra	intra	ADJ
ejpam-3441	851	65	-	-	ADJ
ejpam-3441	851	66	regular	regular	ADJ
ejpam-3441	851	67	.	.	PUNCT
ejpam-3441	852	1	again	again	ADV
ejpam-3441	852	2	a	a	DET
ejpam-3441	852	3	∧βα	∧βα	PROPN
ejpam-3441	852	4	b	b	NOUN
ejpam-3441	852	5	⊆	⊆	NUM
ejpam-3441	852	6	(	(	PUNCT
ejpam-3441	852	7	a	a	DET
ejpam-3441	852	8	◦	◦	NOUN
ejpam-3441	852	9	βα	βα	NOUN
ejpam-3441	852	10	b	b	NOUN
ejpam-3441	852	11	)	)	PUNCT
ejpam-3441	852	12	∧	∧	PROPN
ejpam-3441	852	13	(	(	PUNCT
ejpam-3441	852	14	b	b	X
ejpam-3441	852	15	◦	◦	NOUN
ejpam-3441	852	16	βα	βα	X
ejpam-3441	852	17	a	a	NOUN
ejpam-3441	852	18	)	)	PUNCT
ejpam-3441	852	19	⊆	⊆	PROPN
ejpam-3441	852	20	a	a	DET
ejpam-3441	852	21	◦	◦	NOUN
ejpam-3441	852	22	βα	βα	X
ejpam-3441	852	23	b.	b.	PROPN
ejpam-3441	852	24	since	since	SCONJ
ejpam-3441	852	25	a	a	DET
ejpam-3441	852	26	◦	◦	NOUN
ejpam-3441	852	27	βα	βα	NOUN
ejpam-3441	852	28	b	b	NOUN
ejpam-3441	852	29	⊆	⊆	NUM
ejpam-3441	852	30	a	a	DET
ejpam-3441	852	31	∧βα	∧βα	PROPN
ejpam-3441	852	32	b	b	NOUN
ejpam-3441	852	33	,	,	PUNCT
ejpam-3441	852	34	so	so	SCONJ
ejpam-3441	852	35	a	a	DET
ejpam-3441	852	36	∧βα	∧βα	PROPN
ejpam-3441	852	37	b	b	NOUN
ejpam-3441	852	38	=	=	PUNCT
ejpam-3441	852	39	a	a	DET
ejpam-3441	852	40	◦	◦	NOUN
ejpam-3441	852	41	βα	βα	X
ejpam-3441	852	42	b	b	NOUN
ejpam-3441	852	43	,	,	PUNCT
ejpam-3441	852	44	i.e.	i.e.	X
ejpam-3441	852	45	,	,	PUNCT
ejpam-3441	852	46	r	r	NOUN
ejpam-3441	852	47	is	be	AUX
ejpam-3441	852	48	a	a	DET
ejpam-3441	852	49	regular	regular	NOUN
ejpam-3441	852	50	.	.	PUNCT
ejpam-3441	853	1	hence	hence	ADV
ejpam-3441	853	2	r	r	NOUN
ejpam-3441	853	3	is	be	AUX
ejpam-3441	853	4	both	both	CCONJ
ejpam-3441	853	5	a	a	DET
ejpam-3441	853	6	regular	regular	ADJ
ejpam-3441	853	7	and	and	CCONJ
ejpam-3441	853	8	an	an	DET
ejpam-3441	853	9	inta	inta	NOUN
ejpam-3441	853	10	-	-	PUNCT
ejpam-3441	853	11	regular	regular	ADJ
ejpam-3441	853	12	,	,	PUNCT
ejpam-3441	853	13	i.e.	i.e.	X
ejpam-3441	853	14	,	,	PUNCT
ejpam-3441	853	15	(	(	PUNCT
ejpam-3441	853	16	2)⇒	2)⇒	NUM
ejpam-3441	853	17	(	(	PUNCT
ejpam-3441	853	18	1	1	NUM
ejpam-3441	853	19	)	)	PUNCT
ejpam-3441	853	20	.	.	PUNCT
ejpam-3441	854	1	references	reference	NOUN
ejpam-3441	854	2	942	942	NUM
ejpam-3441	854	3	references	reference	NOUN
ejpam-3441	854	4	[	[	X
ejpam-3441	854	5	1	1	NUM
ejpam-3441	854	6	]	]	PUNCT
ejpam-3441	854	7	k.	k.	PROPN
ejpam-3441	854	8	t.	t.	PROPN
ejpam-3441	854	9	atanassov	atanassov	PROPN
ejpam-3441	854	10	,	,	PUNCT
ejpam-3441	854	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	854	12	fuzzy	fuzzy	ADJ
ejpam-3441	854	13	sets	set	NOUN
ejpam-3441	854	14	,	,	PUNCT
ejpam-3441	854	15	fuzzy	fuzzy	ADJ
ejpam-3441	854	16	sets	set	NOUN
ejpam-3441	854	17	and	and	CCONJ
ejpam-3441	854	18	systems	system	NOUN
ejpam-3441	854	19	,	,	PUNCT
ejpam-3441	854	20	20(1986	20(1986	NUM
ejpam-3441	854	21	)	)	PUNCT
ejpam-3441	854	22	87	87	NUM
ejpam-3441	854	23	-	-	SYM
ejpam-3441	854	24	96	96	NUM
ejpam-3441	854	25	.	.	PUNCT
ejpam-3441	855	1	[	[	X
ejpam-3441	855	2	2	2	NUM
ejpam-3441	855	3	]	]	PUNCT
ejpam-3441	855	4	k.	k.	PROPN
ejpam-3441	855	5	t.	t.	PROPN
ejpam-3441	855	6	atanassov	atanassov	PROPN
ejpam-3441	855	7	,	,	PUNCT
ejpam-3441	855	8	new	new	ADJ
ejpam-3441	855	9	operations	operation	NOUN
ejpam-3441	855	10	defined	define	VERB
ejpam-3441	855	11	over	over	ADP
ejpam-3441	855	12	the	the	DET
ejpam-3441	855	13	intuitionistic	intuitionistic	ADJ
ejpam-3441	855	14	fuzzy	fuzzy	ADJ
ejpam-3441	855	15	sets	set	NOUN
ejpam-3441	855	16	,	,	PUNCT
ejpam-3441	855	17	fuzzy	fuzzy	ADJ
ejpam-3441	855	18	sets	set	NOUN
ejpam-3441	855	19	and	and	CCONJ
ejpam-3441	855	20	systems	system	NOUN
ejpam-3441	855	21	,	,	PUNCT
ejpam-3441	855	22	61(1994	61(1994	NUM
ejpam-3441	855	23	)	)	PUNCT
ejpam-3441	855	24	137	137	NUM
ejpam-3441	855	25	-	-	SYM
ejpam-3441	855	26	142	142	NUM
ejpam-3441	855	27	.	.	PUNCT
ejpam-3441	856	1	[	[	X
ejpam-3441	856	2	3	3	X
ejpam-3441	856	3	]	]	X
ejpam-3441	856	4	b.	b.	PROPN
ejpam-3441	856	5	banerjee	banerjee	PROPN
ejpam-3441	856	6	and	and	CCONJ
ejpam-3441	856	7	d.	d.	PROPN
ejpam-3441	856	8	k.	k.	PROPN
ejpam-3441	856	9	basnet	basnet	PROPN
ejpam-3441	856	10	,	,	PUNCT
ejpam-3441	856	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	856	12	fuzzy	fuzzy	ADJ
ejpam-3441	856	13	subrings	subring	NOUN
ejpam-3441	856	14	and	and	CCONJ
ejpam-3441	856	15	ideals	ideal	NOUN
ejpam-3441	856	16	,	,	PUNCT
ejpam-3441	856	17	j.	j.	PROPN
ejpam-3441	856	18	fuzzy	fuzzy	PROPN
ejpam-3441	856	19	math	math	PROPN
ejpam-3441	856	20	.	.	PUNCT
ejpam-3441	856	21	,	,	PUNCT
ejpam-3441	857	1	11(2003	11(2003	NUM
ejpam-3441	857	2	)	)	PUNCT
ejpam-3441	857	3	139	139	NUM
ejpam-3441	857	4	-	-	SYM
ejpam-3441	857	5	155	155	NUM
ejpam-3441	857	6	.	.	PUNCT
ejpam-3441	858	1	[	[	X
ejpam-3441	858	2	4	4	X
ejpam-3441	858	3	]	]	PUNCT
ejpam-3441	858	4	s.	s.	PROPN
ejpam-3441	858	5	k.	k.	PROPN
ejpam-3441	858	6	bhakat	bhakat	PROPN
ejpam-3441	858	7	and	and	CCONJ
ejpam-3441	858	8	p.	p.	NOUN
ejpam-3441	858	9	das	das	PROPN
ejpam-3441	858	10	,	,	PUNCT
ejpam-3441	858	11	on	on	ADP
ejpam-3441	858	12	the	the	DET
ejpam-3441	858	13	definition	definition	NOUN
ejpam-3441	858	14	of	of	ADP
ejpam-3441	858	15	a	a	DET
ejpam-3441	858	16	fuzzy	fuzzy	ADJ
ejpam-3441	858	17	subgroup	subgroup	NOUN
ejpam-3441	858	18	,	,	PUNCT
ejpam-3441	858	19	fuzzy	fuzzy	ADJ
ejpam-3441	858	20	sets	set	NOUN
ejpam-3441	858	21	and	and	CCONJ
ejpam-3441	858	22	systems	system	NOUN
ejpam-3441	858	23	,	,	PUNCT
ejpam-3441	858	24	51(1992	51(1992	NUM
ejpam-3441	858	25	)	)	PUNCT
ejpam-3441	858	26	235	235	NUM
ejpam-3441	858	27	-	-	SYM
ejpam-3441	858	28	241	241	NUM
ejpam-3441	858	29	.	.	PUNCT
ejpam-3441	859	1	[	[	X
ejpam-3441	859	2	5	5	X
ejpam-3441	859	3	]	]	PUNCT
ejpam-3441	859	4	s.	s.	PROPN
ejpam-3441	859	5	k.	k.	PROPN
ejpam-3441	859	6	bhakat	bhakat	PROPN
ejpam-3441	859	7	and	and	CCONJ
ejpam-3441	859	8	p.	p.	PROPN
ejpam-3441	859	9	das	das	PROPN
ejpam-3441	859	10	,	,	PUNCT
ejpam-3441	859	11	(	(	PUNCT
ejpam-3441	859	12	∈,∈	∈,∈	X
ejpam-3441	859	13	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-3441	859	14	subgroups	subgroup	NOUN
ejpam-3441	859	15	,	,	PUNCT
ejpam-3441	859	16	fuzzy	fuzzy	ADJ
ejpam-3441	859	17	sets	set	NOUN
ejpam-3441	859	18	and	and	CCONJ
ejpam-3441	859	19	systems	system	NOUN
ejpam-3441	859	20	,	,	PUNCT
ejpam-3441	859	21	80(1996	80(1996	NUM
ejpam-3441	859	22	)	)	PUNCT
ejpam-3441	859	23	359	359	NUM
ejpam-3441	859	24	-	-	SYM
ejpam-3441	859	25	368	368	NUM
ejpam-3441	859	26	.	.	PUNCT
ejpam-3441	860	1	[	[	X
ejpam-3441	860	2	6	6	NUM
ejpam-3441	860	3	]	]	PUNCT
ejpam-3441	860	4	s.	s.	PROPN
ejpam-3441	860	5	k.	k.	PROPN
ejpam-3441	860	6	bhakat	bhakat	PROPN
ejpam-3441	860	7	and	and	CCONJ
ejpam-3441	860	8	p.	p.	PROPN
ejpam-3441	860	9	das	das	PROPN
ejpam-3441	860	10	,	,	PUNCT
ejpam-3441	860	11	fuzzy	fuzzy	ADJ
ejpam-3441	860	12	subrings	subring	NOUN
ejpam-3441	860	13	and	and	CCONJ
ejpam-3441	860	14	ideals	ideal	NOUN
ejpam-3441	860	15	redefined	redefine	VERB
ejpam-3441	860	16	,	,	PUNCT
ejpam-3441	860	17	fuzzy	fuzzy	ADJ
ejpam-3441	860	18	sets	set	NOUN
ejpam-3441	860	19	and	and	CCONJ
ejpam-3441	860	20	systems	system	NOUN
ejpam-3441	860	21	,	,	PUNCT
ejpam-3441	860	22	81(1996	81(1996	NUM
ejpam-3441	860	23	)	)	PUNCT
ejpam-3441	860	24	383	383	NUM
ejpam-3441	860	25	-	-	SYM
ejpam-3441	860	26	393	393	NUM
ejpam-3441	860	27	.	.	PUNCT
ejpam-3441	861	1	[	[	X
ejpam-3441	861	2	7	7	X
ejpam-3441	861	3	]	]	X
ejpam-3441	861	4	b.	b.	PROPN
ejpam-3441	861	5	davvaz	davvaz	PROPN
ejpam-3441	861	6	,	,	PUNCT
ejpam-3441	861	7	(	(	PUNCT
ejpam-3441	861	8	∈,∈	∈,∈	X
ejpam-3441	861	9	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-3441	861	10	subnear	subnear	NOUN
ejpam-3441	861	11	-	-	PUNCT
ejpam-3441	861	12	rings	ring	NOUN
ejpam-3441	861	13	and	and	CCONJ
ejpam-3441	861	14	ideals	ideal	NOUN
ejpam-3441	861	15	,	,	PUNCT
ejpam-3441	861	16	soft	soft	ADJ
ejpam-3441	861	17	comput	comput	NOUN
ejpam-3441	861	18	.	.	PUNCT
ejpam-3441	861	19	,	,	PUNCT
ejpam-3441	861	20	10(2006	10(2006	NUM
ejpam-3441	861	21	)	)	PUNCT
ejpam-3441	861	22	206	206	NUM
ejpam-3441	861	23	-	-	SYM
ejpam-3441	861	24	211	211	NUM
ejpam-3441	861	25	.	.	PUNCT
ejpam-3441	862	1	[	[	X
ejpam-3441	862	2	8	8	X
ejpam-3441	862	3	]	]	X
ejpam-3441	862	4	j.	j.	PROPN
ejpam-3441	862	5	r.	r.	PROPN
ejpam-3441	862	6	cho	cho	PROPN
ejpam-3441	862	7	,	,	PUNCT
ejpam-3441	862	8	j.	j.	PROPN
ejpam-3441	862	9	jezek	jezek	PROPN
ejpam-3441	862	10	and	and	CCONJ
ejpam-3441	862	11	t.	t.	PROPN
ejpam-3441	862	12	kepka	kepka	NOUN
ejpam-3441	862	13	,	,	PUNCT
ejpam-3441	862	14	paramedial	paramedial	ADJ
ejpam-3441	862	15	groupoids	groupoid	NOUN
ejpam-3441	862	16	,	,	PUNCT
ejpam-3441	862	17	czechoslovak	czechoslovak	ADJ
ejpam-3441	862	18	math	math	NOUN
ejpam-3441	862	19	.	.	PUNCT
ejpam-3441	863	1	j.	j.	PROPN
ejpam-3441	863	2	,	,	PUNCT
ejpam-3441	863	3	49(1999	49(1999	PROPN
ejpam-3441	863	4	)	)	PUNCT
ejpam-3441	863	5	277	277	NUM
ejpam-3441	863	6	-	-	SYM
ejpam-3441	863	7	290	290	NUM
ejpam-3441	863	8	.	.	PUNCT
ejpam-3441	864	1	[	[	X
ejpam-3441	864	2	9	9	NUM
ejpam-3441	864	3	]	]	PUNCT
ejpam-3441	864	4	k.	k.	PROPN
ejpam-3441	864	5	hur	hur	PROPN
ejpam-3441	864	6	,	,	PUNCT
ejpam-3441	864	7	s.	s.	PROPN
ejpam-3441	864	8	y.	y.	PROPN
ejpam-3441	864	9	jang	jang	PROPN
ejpam-3441	864	10	and	and	CCONJ
ejpam-3441	864	11	h.	h.	PROPN
ejpam-3441	864	12	w.	w.	PROPN
ejpam-3441	864	13	kang	kang	PROPN
ejpam-3441	864	14	,	,	PUNCT
ejpam-3441	864	15	intuitionistic	intuitionistic	ADJ
ejpam-3441	864	16	fuzzy	fuzzy	ADJ
ejpam-3441	864	17	ideals	ideal	NOUN
ejpam-3441	864	18	of	of	ADP
ejpam-3441	864	19	a	a	DET
ejpam-3441	864	20	ring	ring	NOUN
ejpam-3441	864	21	,	,	PUNCT
ejpam-3441	864	22	j.	j.	PROPN
ejpam-3441	864	23	korea	korea	PROPN
ejpam-3441	864	24	soc	soc	PROPN
ejpam-3441	864	25	.	.	PUNCT
ejpam-3441	865	1	math	math	PROPN
ejpam-3441	865	2	.	.	PUNCT
ejpam-3441	866	1	educ	educ	PROPN
ejpam-3441	866	2	.	.	PUNCT
ejpam-3441	867	1	ser	ser	PROPN
ejpam-3441	867	2	.	.	PUNCT
ejpam-3441	868	1	b	b	X
ejpam-3441	868	2	:	:	PUNCT
ejpam-3441	868	3	pure	pure	ADJ
ejpam-3441	868	4	appl	appl	PROPN
ejpam-3441	868	5	.	.	PUNCT
ejpam-3441	868	6	math	math	PROPN
ejpam-3441	868	7	.	.	PUNCT
ejpam-3441	868	8	,	,	PUNCT
ejpam-3441	868	9	12(2005	12(2005	NUM
ejpam-3441	868	10	)	)	PUNCT
ejpam-3441	868	11	193	193	NUM
ejpam-3441	868	12	-	-	SYM
ejpam-3441	868	13	209	209	NUM
ejpam-3441	868	14	.	.	PUNCT
ejpam-3441	869	1	[	[	X
ejpam-3441	869	2	10	10	NUM
ejpam-3441	869	3	]	]	X
ejpam-3441	869	4	k.	k.	PROPN
ejpam-3441	869	5	hur	hur	PROPN
ejpam-3441	869	6	,	,	PUNCT
ejpam-3441	869	7	h.	h.	PROPN
ejpam-3441	869	8	w.	w.	PROPN
ejpam-3441	869	9	kang	kang	PROPN
ejpam-3441	869	10	and	and	CCONJ
ejpam-3441	869	11	h.	h.	PROPN
ejpam-3441	869	12	k.	k.	PROPN
ejpam-3441	869	13	song	song	PROPN
ejpam-3441	869	14	,	,	PUNCT
ejpam-3441	869	15	intuitionistic	intuitionistic	ADJ
ejpam-3441	869	16	fuzzy	fuzzy	ADJ
ejpam-3441	869	17	subgroups	subgroup	NOUN
ejpam-3441	869	18	and	and	CCONJ
ejpam-3441	869	19	subrings	subring	NOUN
ejpam-3441	869	20	,	,	PUNCT
ejpam-3441	869	21	honam	honam	PROPN
ejpam-3441	869	22	math	math	PROPN
ejpam-3441	869	23	.	.	PUNCT
ejpam-3441	870	1	j.	j.	PROPN
ejpam-3441	870	2	,	,	PUNCT
ejpam-3441	870	3	25(2003	25(2003	NUM
ejpam-3441	870	4	)	)	PUNCT
ejpam-3441	870	5	19	19	NUM
ejpam-3441	870	6	-	-	SYM
ejpam-3441	870	7	41	41	NUM
ejpam-3441	870	8	.	.	PUNCT
ejpam-3441	871	1	[	[	X
ejpam-3441	871	2	11	11	NUM
ejpam-3441	871	3	]	]	PUNCT
ejpam-3441	871	4	k.	k.	PROPN
ejpam-3441	871	5	c.	c.	PROPN
ejpam-3441	871	6	gupta	gupta	PROPN
ejpam-3441	871	7	and	and	CCONJ
ejpam-3441	871	8	m.	m.	PROPN
ejpam-3441	871	9	k.	k.	PROPN
ejpam-3441	871	10	kantroo	kantroo	PROPN
ejpam-3441	871	11	,	,	PUNCT
ejpam-3441	871	12	the	the	DET
ejpam-3441	871	13	intrinsic	intrinsic	ADJ
ejpam-3441	871	14	product	product	NOUN
ejpam-3441	871	15	of	of	ADP
ejpam-3441	871	16	fuzzy	fuzzy	ADJ
ejpam-3441	871	17	subsets	subset	NOUN
ejpam-3441	871	18	of	of	ADP
ejpam-3441	871	19	a	a	DET
ejpam-3441	871	20	ring	ring	NOUN
ejpam-3441	871	21	,	,	PUNCT
ejpam-3441	871	22	fuzzy	fuzzy	ADJ
ejpam-3441	871	23	sets	set	NOUN
ejpam-3441	871	24	and	and	CCONJ
ejpam-3441	871	25	systems	system	NOUN
ejpam-3441	871	26	,	,	PUNCT
ejpam-3441	871	27	57(1993	57(1993	NUM
ejpam-3441	871	28	)	)	PUNCT
ejpam-3441	871	29	103	103	NUM
ejpam-3441	871	30	-	-	SYM
ejpam-3441	871	31	110	110	NUM
ejpam-3441	871	32	.	.	PUNCT
ejpam-3441	872	1	[	[	X
ejpam-3441	872	2	12	12	NUM
ejpam-3441	872	3	]	]	X
ejpam-3441	872	4	j.	j.	PROPN
ejpam-3441	872	5	jezek	jezek	PROPN
ejpam-3441	872	6	and	and	CCONJ
ejpam-3441	872	7	t.	t.	PROPN
ejpam-3441	872	8	kepka	kepka	NOUN
ejpam-3441	872	9	,	,	PUNCT
ejpam-3441	872	10	medial	medial	ADJ
ejpam-3441	872	11	groupoids	groupoid	NOUN
ejpam-3441	872	12	,	,	PUNCT
ejpam-3441	872	13	rozpravy	rozpravy	PROPN
ejpam-3441	872	14	csav	csav	PROPN
ejpam-3441	872	15	rada	rada	PROPN
ejpam-3441	872	16	mat	mat	PROPN
ejpam-3441	872	17	.	.	PUNCT
ejpam-3441	873	1	a	a	DET
ejpam-3441	873	2	prir	prir	NOUN
ejpam-3441	873	3	.	.	PUNCT
ejpam-3441	874	1	ved	ve	VERB
ejpam-3441	874	2	93/2	93/2	NUM
ejpam-3441	874	3	,	,	PUNCT
ejpam-3441	874	4	1983	1983	NUM
ejpam-3441	874	5	,	,	PUNCT
ejpam-3441	874	6	93	93	NUM
ejpam-3441	874	7	pp	pp	NOUN
ejpam-3441	874	8	.	.	PUNCT
ejpam-3441	875	1	[	[	X
ejpam-3441	875	2	13	13	NUM
ejpam-3441	875	3	]	]	X
ejpam-3441	875	4	y.	y.	PROPN
ejpam-3441	875	5	b.	b.	PROPN
ejpam-3441	875	6	jun	jun	PROPN
ejpam-3441	875	7	and	and	CCONJ
ejpam-3441	875	8	s.	s.	PROPN
ejpam-3441	875	9	z.	z.	PROPN
ejpam-3441	875	10	song	song	PROPN
ejpam-3441	875	11	,	,	PUNCT
ejpam-3441	875	12	generalized	generalize	VERB
ejpam-3441	875	13	fuzzy	fuzzy	ADJ
ejpam-3441	875	14	interior	interior	ADJ
ejpam-3441	875	15	ideals	ideal	NOUN
ejpam-3441	875	16	in	in	ADP
ejpam-3441	875	17	semigroups	semigroup	NOUN
ejpam-3441	875	18	,	,	PUNCT
ejpam-3441	875	19	inf	inf	PROPN
ejpam-3441	875	20	.	.	PUNCT
ejpam-3441	875	21	sci	sci	PROPN
ejpam-3441	875	22	.	.	PROPN
ejpam-3441	875	23	,	,	PUNCT
ejpam-3441	875	24	176(2006	176(2006	NUM
ejpam-3441	875	25	)	)	PUNCT
ejpam-3441	875	26	3079	3079	NUM
ejpam-3441	875	27	-	-	SYM
ejpam-3441	875	28	3093	3093	NUM
ejpam-3441	875	29	.	.	PUNCT
ejpam-3441	876	1	[	[	X
ejpam-3441	876	2	14	14	NUM
ejpam-3441	876	3	]	]	PUNCT
ejpam-3441	876	4	m.	m.	NOUN
ejpam-3441	876	5	s.	s.	PROPN
ejpam-3441	876	6	kamran	kamran	PROPN
ejpam-3441	876	7	,	,	PUNCT
ejpam-3441	876	8	conditions	condition	NOUN
ejpam-3441	876	9	for	for	ADP
ejpam-3441	876	10	la	la	NOUN
ejpam-3441	876	11	-	-	PUNCT
ejpam-3441	876	12	semigroups	semigroup	NOUN
ejpam-3441	876	13	to	to	PART
ejpam-3441	876	14	resemble	resemble	VERB
ejpam-3441	876	15	associative	associative	ADJ
ejpam-3441	876	16	structures	structure	NOUN
ejpam-3441	876	17	,	,	PUNCT
ejpam-3441	876	18	ph.d	ph.d	PROPN
ejpam-3441	876	19	.	.	PUNCT
ejpam-3441	877	1	thesis	thesis	NOUN
ejpam-3441	877	2	,	,	PUNCT
ejpam-3441	877	3	quaid	quaid	PROPN
ejpam-3441	877	4	-	-	PUNCT
ejpam-3441	877	5	i	i	PROPN
ejpam-3441	877	6	-	-	PUNCT
ejpam-3441	877	7	azam	azam	PROPN
ejpam-3441	877	8	university	university	PROPN
ejpam-3441	877	9	,	,	PUNCT
ejpam-3441	877	10	islamabad	islamabad	PROPN
ejpam-3441	877	11	,	,	PUNCT
ejpam-3441	877	12	1993	1993	NUM
ejpam-3441	877	13	.	.	PUNCT
ejpam-3441	878	1	[	[	X
ejpam-3441	878	2	15	15	NUM
ejpam-3441	878	3	]	]	X
ejpam-3441	878	4	n.	n.	NOUN
ejpam-3441	878	5	kausar	kausar	PROPN
ejpam-3441	878	6	,	,	PUNCT
ejpam-3441	878	7	a.	a.	PROPN
ejpam-3441	878	8	waqar	waqar	PROPN
ejpam-3441	878	9	,	,	PUNCT
ejpam-3441	878	10	characterizations	characterization	NOUN
ejpam-3441	878	11	of	of	ADP
ejpam-3441	878	12	non	non	ADJ
ejpam-3441	878	13	-	-	ADJ
ejpam-3441	878	14	associative	associative	ADJ
ejpam-3441	878	15	rings	ring	NOUN
ejpam-3441	878	16	by	by	ADP
ejpam-3441	878	17	their	their	PRON
ejpam-3441	878	18	intuitionistic	intuitionistic	ADJ
ejpam-3441	878	19	fuzzy	fuzzy	ADJ
ejpam-3441	878	20	bi	bi	NOUN
ejpam-3441	878	21	-	-	NOUN
ejpam-3441	878	22	ideals	ideal	NOUN
ejpam-3441	878	23	,	,	PUNCT
ejpam-3441	878	24	european	european	ADJ
ejpam-3441	878	25	journal	journal	PROPN
ejpam-3441	878	26	of	of	ADP
ejpam-3441	878	27	pure	pure	ADJ
ejpam-3441	878	28	and	and	CCONJ
ejpam-3441	878	29	applied	applied	ADJ
ejpam-3441	878	30	mathematics	mathematic	NOUN
ejpam-3441	878	31	,	,	PUNCT
ejpam-3441	878	32	12(2019	12(2019	NUM
ejpam-3441	878	33	)	)	PUNCT
ejpam-3441	878	34	226250	226250	NUM
ejpam-3441	878	35	.	.	PUNCT
ejpam-3441	879	1	[	[	X
ejpam-3441	879	2	16	16	NUM
ejpam-3441	879	3	]	]	X
ejpam-3441	879	4	n.	n.	PROPN
ejpam-3441	879	5	kausar	kausar	PROPN
ejpam-3441	879	6	,	,	PUNCT
ejpam-3441	879	7	characterizations	characterization	NOUN
ejpam-3441	879	8	of	of	ADP
ejpam-3441	879	9	non	non	ADJ
ejpam-3441	879	10	-	-	ADJ
ejpam-3441	879	11	associative	associative	ADJ
ejpam-3441	879	12	ordered	order	VERB
ejpam-3441	879	13	semigroups	semigroup	NOUN
ejpam-3441	879	14	by	by	ADP
ejpam-3441	879	15	the	the	DET
ejpam-3441	879	16	properties	property	NOUN
ejpam-3441	879	17	of	of	ADP
ejpam-3441	879	18	their	their	PRON
ejpam-3441	879	19	fuzzy	fuzzy	ADJ
ejpam-3441	879	20	ideals	ideal	NOUN
ejpam-3441	879	21	with	with	ADP
ejpam-3441	879	22	thresholds	threshold	NOUN
ejpam-3441	879	23	,	,	PUNCT
ejpam-3441	879	24	prikladnaya	prikladnaya	NOUN
ejpam-3441	879	25	diskretnaya	diskretnaya	PROPN
ejpam-3441	879	26	matematika	matematika	PROPN
ejpam-3441	879	27	,	,	PUNCT
ejpam-3441	879	28	43(2019	43(2019	NUM
ejpam-3441	879	29	)	)	PUNCT
ejpam-3441	879	30	37	37	NUM
ejpam-3441	879	31	-	-	SYM
ejpam-3441	879	32	59	59	NUM
ejpam-3441	879	33	.	.	PUNCT
ejpam-3441	880	1	references	reference	NOUN
ejpam-3441	880	2	943	943	NUM
ejpam-3441	881	1	[	[	X
ejpam-3441	881	2	17	17	NUM
ejpam-3441	881	3	]	]	X
ejpam-3441	881	4	n.	n.	NOUN
ejpam-3441	881	5	kausar	kausar	PROPN
ejpam-3441	881	6	,	,	PUNCT
ejpam-3441	881	7	direct	direct	ADJ
ejpam-3441	881	8	product	product	NOUN
ejpam-3441	881	9	of	of	ADP
ejpam-3441	881	10	finite	finite	PROPN
ejpam-3441	881	11	intuitionistic	intuitionistic	ADJ
ejpam-3441	881	12	fuzzy	fuzzy	ADJ
ejpam-3441	881	13	normal	normal	ADJ
ejpam-3441	881	14	subrings	subring	NOUN
ejpam-3441	881	15	over	over	ADP
ejpam-3441	881	16	nonassociative	nonassociative	ADJ
ejpam-3441	881	17	rings	ring	NOUN
ejpam-3441	881	18	,	,	PUNCT
ejpam-3441	881	19	european	european	PROPN
ejpam-3441	881	20	journal	journal	PROPN
ejpam-3441	881	21	of	of	ADP
ejpam-3441	881	22	pure	pure	ADJ
ejpam-3441	881	23	and	and	CCONJ
ejpam-3441	881	24	applied	applied	ADJ
ejpam-3441	881	25	mathematics	mathematic	NOUN
ejpam-3441	881	26	,	,	PUNCT
ejpam-3441	881	27	12(2019	12(2019	NUM
ejpam-3441	881	28	)	)	PUNCT
ejpam-3441	881	29	622648	622648	NUM
ejpam-3441	881	30	.	.	PUNCT
ejpam-3441	882	1	[	[	X
ejpam-3441	882	2	18	18	NUM
ejpam-3441	882	3	]	]	PUNCT
ejpam-3441	882	4	m.	m.	NOUN
ejpam-3441	882	5	a.	a.	PROPN
ejpam-3441	882	6	kazim	kazim	PROPN
ejpam-3441	882	7	and	and	CCONJ
ejpam-3441	882	8	m.	m.	PROPN
ejpam-3441	882	9	naseerudin	naseerudin	PROPN
ejpam-3441	882	10	,	,	PUNCT
ejpam-3441	882	11	on	on	ADP
ejpam-3441	882	12	almost	almost	ADV
ejpam-3441	882	13	semigroups	semigroup	NOUN
ejpam-3441	882	14	,	,	PUNCT
ejpam-3441	882	15	alig	alig	PROPN
ejpam-3441	882	16	.	.	PUNCT
ejpam-3441	883	1	bull	bull	PROPN
ejpam-3441	883	2	.	.	PUNCT
ejpam-3441	884	1	math	math	NOUN
ejpam-3441	884	2	.	.	PUNCT
ejpam-3441	884	3	,	,	PUNCT
ejpam-3441	884	4	2(1972	2(1972	X
ejpam-3441	884	5	)	)	PUNCT
ejpam-3441	884	6	1	1	NUM
ejpam-3441	884	7	-	-	SYM
ejpam-3441	884	8	7	7	NUM
ejpam-3441	884	9	.	.	PUNCT
ejpam-3441	885	1	[	[	X
ejpam-3441	885	2	19	19	NUM
ejpam-3441	885	3	]	]	X
ejpam-3441	885	4	n.	n.	PROPN
ejpam-3441	885	5	kuroki	kuroki	PROPN
ejpam-3441	885	6	,	,	PUNCT
ejpam-3441	885	7	regular	regular	ADJ
ejpam-3441	885	8	fuzzy	fuzzy	ADJ
ejpam-3441	885	9	duo	duo	NOUN
ejpam-3441	885	10	rings	ring	NOUN
ejpam-3441	885	11	,	,	PUNCT
ejpam-3441	885	12	inform	inform	NOUN
ejpam-3441	885	13	.	.	PUNCT
ejpam-3441	886	1	sci	sci	PROPN
ejpam-3441	886	2	.	.	PROPN
ejpam-3441	886	3	,	,	PUNCT
ejpam-3441	886	4	94(1996	94(1996	X
ejpam-3441	886	5	)	)	PUNCT
ejpam-3441	886	6	119	119	NUM
ejpam-3441	886	7	-	-	SYM
ejpam-3441	886	8	139	139	NUM
ejpam-3441	886	9	.	.	PUNCT
ejpam-3441	887	1	[	[	X
ejpam-3441	887	2	20	20	NUM
ejpam-3441	887	3	]	]	PUNCT
ejpam-3441	887	4	w.	w.	PROPN
ejpam-3441	887	5	j.	j.	PROPN
ejpam-3441	887	6	liu	liu	PROPN
ejpam-3441	887	7	,	,	PUNCT
ejpam-3441	887	8	fuzzy	fuzzy	ADJ
ejpam-3441	887	9	invariant	invariant	ADJ
ejpam-3441	887	10	subgroups	subgroup	NOUN
ejpam-3441	887	11	and	and	CCONJ
ejpam-3441	887	12	ideals	ideal	NOUN
ejpam-3441	887	13	,	,	PUNCT
ejpam-3441	887	14	fuzzy	fuzzy	ADJ
ejpam-3441	887	15	sets	set	NOUN
ejpam-3441	887	16	and	and	CCONJ
ejpam-3441	887	17	systems	system	NOUN
ejpam-3441	887	18	,	,	PUNCT
ejpam-3441	887	19	8(1982	8(1982	NUM
ejpam-3441	887	20	)	)	PUNCT
ejpam-3441	887	21	133	133	NUM
ejpam-3441	887	22	-	-	SYM
ejpam-3441	887	23	139	139	NUM
ejpam-3441	887	24	.	.	PUNCT
ejpam-3441	888	1	[	[	X
ejpam-3441	888	2	21	21	NUM
ejpam-3441	888	3	]	]	PUNCT
ejpam-3441	888	4	t.	t.	PROPN
ejpam-3441	888	5	k.	k.	PROPN
ejpam-3441	888	6	mukherjee	mukherjee	PROPN
ejpam-3441	888	7	and	and	CCONJ
ejpam-3441	888	8	m.	m.	PROPN
ejpam-3441	888	9	k.	k.	PROPN
ejpam-3441	888	10	sen	sen	PROPN
ejpam-3441	888	11	,	,	PUNCT
ejpam-3441	888	12	on	on	ADP
ejpam-3441	888	13	fuzzy	fuzzy	ADJ
ejpam-3441	888	14	ideals	ideal	NOUN
ejpam-3441	888	15	of	of	ADP
ejpam-3441	888	16	a	a	DET
ejpam-3441	888	17	ring	ring	NOUN
ejpam-3441	888	18	1	1	NUM
ejpam-3441	888	19	,	,	PUNCT
ejpam-3441	888	20	fuzzy	fuzzy	ADJ
ejpam-3441	888	21	sets	set	NOUN
ejpam-3441	888	22	and	and	CCONJ
ejpam-3441	888	23	systems	system	NOUN
ejpam-3441	888	24	,	,	PUNCT
ejpam-3441	888	25	21(1987	21(1987	NUM
ejpam-3441	888	26	)	)	PUNCT
ejpam-3441	888	27	99	99	NUM
ejpam-3441	888	28	-	-	SYM
ejpam-3441	888	29	104	104	NUM
ejpam-3441	888	30	.	.	PUNCT
ejpam-3441	889	1	[	[	X
ejpam-3441	889	2	22	22	NUM
ejpam-3441	889	3	]	]	PUNCT
ejpam-3441	889	4	t.	t.	PROPN
ejpam-3441	889	5	k.	k.	PROPN
ejpam-3441	889	6	mukherjee	mukherjee	PROPN
ejpam-3441	889	7	and	and	CCONJ
ejpam-3441	889	8	m.	m.	PROPN
ejpam-3441	889	9	k.	k.	PROPN
ejpam-3441	889	10	sen	sen	PROPN
ejpam-3441	889	11	,	,	PUNCT
ejpam-3441	889	12	prime	prime	ADJ
ejpam-3441	889	13	fuzzy	fuzzy	ADJ
ejpam-3441	889	14	ideals	ideal	NOUN
ejpam-3441	889	15	in	in	ADP
ejpam-3441	889	16	rings	ring	NOUN
ejpam-3441	889	17	,	,	PUNCT
ejpam-3441	889	18	fuzzy	fuzzy	ADJ
ejpam-3441	889	19	sets	set	NOUN
ejpam-3441	889	20	and	and	CCONJ
ejpam-3441	889	21	systems	system	NOUN
ejpam-3441	889	22	,	,	PUNCT
ejpam-3441	889	23	32(1989	32(1989	NUM
ejpam-3441	889	24	)	)	PUNCT
ejpam-3441	889	25	337	337	NUM
ejpam-3441	889	26	-	-	SYM
ejpam-3441	889	27	341	341	NUM
ejpam-3441	889	28	.	.	PUNCT
ejpam-3441	890	1	[	[	X
ejpam-3441	890	2	23	23	NUM
ejpam-3441	890	3	]	]	PUNCT
ejpam-3441	890	4	a.	a.	PROPN
ejpam-3441	890	5	l.	l.	PROPN
ejpam-3441	890	6	narayanan	narayanan	PROPN
ejpam-3441	890	7	and	and	CCONJ
ejpam-3441	890	8	t.	t.	PROPN
ejpam-3441	890	9	manikantan	manikantan	PROPN
ejpam-3441	890	10	,	,	PUNCT
ejpam-3441	890	11	(	(	PUNCT
ejpam-3441	890	12	∈,∈	∈,∈	X
ejpam-3441	890	13	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-3441	890	14	subnear	subnear	NOUN
ejpam-3441	890	15	-	-	PUNCT
ejpam-3441	890	16	rings	ring	NOUN
ejpam-3441	890	17	and	and	CCONJ
ejpam-3441	890	18	(	(	PUNCT
ejpam-3441	890	19	∈,∈	∈,∈	X
ejpam-3441	890	20	∨q)fuzzy	∨q)fuzzy	ADJ
ejpam-3441	890	21	ideals	ideal	NOUN
ejpam-3441	890	22	of	of	ADP
ejpam-3441	890	23	near	near	ADJ
ejpam-3441	890	24	-	-	PUNCT
ejpam-3441	890	25	rings	ring	NOUN
ejpam-3441	890	26	,	,	PUNCT
ejpam-3441	890	27	fuzzy	fuzzy	ADJ
ejpam-3441	890	28	j.	j.	PROPN
ejpam-3441	890	29	appl	appl	PROPN
ejpam-3441	890	30	.	.	PROPN
ejpam-3441	890	31	math	math	PROPN
ejpam-3441	890	32	.	.	PUNCT
ejpam-3441	891	1	comput	comput	NOUN
ejpam-3441	891	2	.	.	PUNCT
ejpam-3441	891	3	,	,	PUNCT
ejpam-3441	891	4	18(2005	18(2005	X
ejpam-3441	891	5	)	)	PUNCT
ejpam-3441	891	6	419	419	NUM
ejpam-3441	891	7	-	-	SYM
ejpam-3441	891	8	430	430	NUM
ejpam-3441	891	9	.	.	PUNCT
ejpam-3441	892	1	[	[	X
ejpam-3441	892	2	24	24	NUM
ejpam-3441	892	3	]	]	PUNCT
ejpam-3441	892	4	p.	p.	NOUN
ejpam-3441	892	5	v.	v.	ADP
ejpam-3441	892	6	protic	protic	PROPN
ejpam-3441	892	7	and	and	CCONJ
ejpam-3441	892	8	n.	n.	PROPN
ejpam-3441	892	9	stevanovic	stevanovic	PROPN
ejpam-3441	892	10	,	,	PUNCT
ejpam-3441	892	11	ag	ag	NOUN
ejpam-3441	892	12	-	-	PUNCT
ejpam-3441	892	13	test	test	NOUN
ejpam-3441	892	14	and	and	CCONJ
ejpam-3441	892	15	some	some	DET
ejpam-3441	892	16	general	general	ADJ
ejpam-3441	892	17	properties	property	NOUN
ejpam-3441	892	18	of	of	ADP
ejpam-3441	892	19	abelgrassmann	abelgrassmann	PROPN
ejpam-3441	892	20	’s	’s	PART
ejpam-3441	892	21	groupoids	groupoid	NOUN
ejpam-3441	892	22	,	,	PUNCT
ejpam-3441	892	23	pure	pure	ADJ
ejpam-3441	892	24	math	math	NOUN
ejpam-3441	892	25	.	.	PUNCT
ejpam-3441	893	1	appl	appl	PROPN
ejpam-3441	893	2	.	.	PROPN
ejpam-3441	893	3	,	,	PUNCT
ejpam-3441	893	4	6(1995	6(1995	NUM
ejpam-3441	893	5	)	)	PUNCT
ejpam-3441	893	6	371	371	NUM
ejpam-3441	893	7	-	-	SYM
ejpam-3441	893	8	383	383	NUM
ejpam-3441	893	9	.	.	PUNCT
ejpam-3441	894	1	[	[	X
ejpam-3441	894	2	25	25	NUM
ejpam-3441	894	3	]	]	PUNCT
ejpam-3441	894	4	t.	t.	NOUN
ejpam-3441	894	5	shah	shah	PROPN
ejpam-3441	894	6	and	and	CCONJ
ejpam-3441	894	7	i.	i.	PROPN
ejpam-3441	894	8	rehman	rehman	PROPN
ejpam-3441	894	9	,	,	PUNCT
ejpam-3441	894	10	on	on	ADP
ejpam-3441	894	11	la	la	NOUN
ejpam-3441	894	12	-	-	PUNCT
ejpam-3441	894	13	rings	ring	NOUN
ejpam-3441	894	14	of	of	ADP
ejpam-3441	894	15	finitely	finitely	ADJ
ejpam-3441	894	16	non	non	ADJ
ejpam-3441	894	17	-	-	ADJ
ejpam-3441	894	18	zero	zero	NUM
ejpam-3441	894	19	functions	function	NOUN
ejpam-3441	894	20	,	,	PUNCT
ejpam-3441	894	21	int	int	NOUN
ejpam-3441	894	22	.	.	PUNCT
ejpam-3441	895	1	j.	j.	PROPN
ejpam-3441	895	2	contempt	contempt	PROPN
ejpam-3441	895	3	.	.	PUNCT
ejpam-3441	896	1	math	math	NOUN
ejpam-3441	896	2	.	.	PUNCT
ejpam-3441	897	1	sci	sci	PROPN
ejpam-3441	897	2	.	.	PROPN
ejpam-3441	897	3	,	,	PUNCT
ejpam-3441	897	4	5(2010	5(2010	NUM
ejpam-3441	897	5	)	)	PUNCT
ejpam-3441	897	6	209	209	NUM
ejpam-3441	897	7	-	-	SYM
ejpam-3441	897	8	222	222	NUM
ejpam-3441	897	9	.	.	PUNCT
ejpam-3441	898	1	[	[	X
ejpam-3441	898	2	26	26	NUM
ejpam-3441	898	3	]	]	PUNCT
ejpam-3441	898	4	t.	t.	NOUN
ejpam-3441	898	5	shah	shah	NOUN
ejpam-3441	898	6	,	,	PUNCT
ejpam-3441	898	7	n.	n.	PROPN
ejpam-3441	898	8	kausar	kausar	PROPN
ejpam-3441	898	9	and	and	CCONJ
ejpam-3441	898	10	i.	i.	PROPN
ejpam-3441	898	11	rehman	rehman	PROPN
ejpam-3441	898	12	,	,	PUNCT
ejpam-3441	898	13	intuitionistic	intuitionistic	ADJ
ejpam-3441	898	14	fuzzy	fuzzy	ADJ
ejpam-3441	898	15	normal	normal	ADJ
ejpam-3441	898	16	subrings	subring	NOUN
ejpam-3441	898	17	over	over	ADP
ejpam-3441	898	18	a	a	DET
ejpam-3441	898	19	nonassociative	nonassociative	ADJ
ejpam-3441	898	20	ring	ring	NOUN
ejpam-3441	898	21	,	,	PUNCT
ejpam-3441	898	22	an	an	PROPN
ejpam-3441	898	23	.	.	PUNCT
ejpam-3441	898	24	st	st	PROPN
ejpam-3441	898	25	.	.	PROPN
ejpam-3441	898	26	univ	univ	PROPN
ejpam-3441	898	27	.	.	PUNCT
ejpam-3441	899	1	ovidius	ovidius	PROPN
ejpam-3441	899	2	constanta	constanta	PROPN
ejpam-3441	899	3	,	,	PUNCT
ejpam-3441	899	4	1(2012	1(2012	NUM
ejpam-3441	899	5	)	)	PUNCT
ejpam-3441	899	6	369	369	NUM
ejpam-3441	899	7	-	-	SYM
ejpam-3441	899	8	386	386	NUM
ejpam-3441	899	9	.	.	PUNCT
ejpam-3441	900	1	[	[	X
ejpam-3441	900	2	27	27	NUM
ejpam-3441	900	3	]	]	PUNCT
ejpam-3441	900	4	t.	t.	NOUN
ejpam-3441	900	5	shah	shah	PROPN
ejpam-3441	900	6	,	,	PUNCT
ejpam-3441	900	7	n.	n.	PROPN
ejpam-3441	900	8	kausar	kausar	PROPN
ejpam-3441	900	9	,	,	PUNCT
ejpam-3441	900	10	characterizations	characterization	NOUN
ejpam-3441	900	11	of	of	ADP
ejpam-3441	900	12	non	non	ADJ
ejpam-3441	900	13	-	-	ADJ
ejpam-3441	900	14	associative	associative	ADJ
ejpam-3441	900	15	ordered	order	VERB
ejpam-3441	900	16	semigroups	semigroup	NOUN
ejpam-3441	900	17	by	by	ADP
ejpam-3441	900	18	their	their	PRON
ejpam-3441	900	19	fuzzy	fuzzy	ADJ
ejpam-3441	900	20	bi	bi	NOUN
ejpam-3441	900	21	-	-	NOUN
ejpam-3441	900	22	ideals	ideal	NOUN
ejpam-3441	900	23	,	,	PUNCT
ejpam-3441	900	24	theoretical	theoretical	ADJ
ejpam-3441	900	25	computer	computer	NOUN
ejpam-3441	900	26	science	science	NOUN
ejpam-3441	900	27	,	,	PUNCT
ejpam-3441	900	28	529(2014	529(2014	NUM
ejpam-3441	900	29	)	)	PUNCT
ejpam-3441	900	30	,	,	PUNCT
ejpam-3441	900	31	96	96	NUM
ejpam-3441	900	32	-	-	SYM
ejpam-3441	900	33	110	110	NUM
ejpam-3441	900	34	.	.	PUNCT
ejpam-3441	901	1	[	[	X
ejpam-3441	901	2	28	28	NUM
ejpam-3441	901	3	]	]	X
ejpam-3441	901	4	m.	m.	NOUN
ejpam-3441	901	5	shabir	shabir	PROPN
ejpam-3441	901	6	,	,	PUNCT
ejpam-3441	901	7	y.	y.	NOUN
ejpam-3441	901	8	nawaz	nawaz	NOUN
ejpam-3441	901	9	and	and	CCONJ
ejpam-3441	901	10	m.	m.	PROPN
ejpam-3441	901	11	aslam	aslam	PROPN
ejpam-3441	901	12	,	,	PUNCT
ejpam-3441	901	13	semigroups	semigroup	NOUN
ejpam-3441	901	14	characterized	characterize	VERB
ejpam-3441	901	15	by	by	ADP
ejpam-3441	901	16	the	the	DET
ejpam-3441	901	17	properties	property	NOUN
ejpam-3441	901	18	of	of	ADP
ejpam-3441	901	19	their	their	PRON
ejpam-3441	901	20	fuzzy	fuzzy	ADJ
ejpam-3441	901	21	ideals	ideal	NOUN
ejpam-3441	901	22	with	with	ADP
ejpam-3441	901	23	thresholds	threshold	NOUN
ejpam-3441	901	24	,	,	PUNCT
ejpam-3441	901	25	world	world	NOUN
ejpam-3441	901	26	appl	appl	PROPN
ejpam-3441	901	27	.	.	PUNCT
ejpam-3441	902	1	sci	sci	PROPN
ejpam-3441	902	2	.	.	PUNCT
ejpam-3441	903	1	j.	j.	PROPN
ejpam-3441	903	2	,	,	PUNCT
ejpam-3441	903	3	14(2011	14(2011	NUM
ejpam-3441	903	4	)	)	PUNCT
ejpam-3441	903	5	1851	1851	NUM
ejpam-3441	903	6	-	-	SYM
ejpam-3441	903	7	1865	1865	NUM
ejpam-3441	903	8	.	.	PUNCT
ejpam-3441	904	1	[	[	X
ejpam-3441	904	2	29	29	NUM
ejpam-3441	904	3	]	]	X
ejpam-3441	904	4	u.	u.	PROPN
ejpam-3441	904	5	m.	m.	PROPN
ejpam-3441	904	6	swamy	swamy	PROPN
ejpam-3441	904	7	and	and	CCONJ
ejpam-3441	904	8	k.	k.	PROPN
ejpam-3441	904	9	l.	l.	PROPN
ejpam-3441	904	10	n.	n.	PROPN
ejpam-3441	904	11	swamy	swamy	PROPN
ejpam-3441	904	12	,	,	PUNCT
ejpam-3441	904	13	fuzzy	fuzzy	ADJ
ejpam-3441	904	14	prime	prime	ADJ
ejpam-3441	904	15	ideals	ideal	NOUN
ejpam-3441	904	16	of	of	ADP
ejpam-3441	904	17	rings	ring	NOUN
ejpam-3441	904	18	,	,	PUNCT
ejpam-3441	904	19	j.	j.	PROPN
ejpam-3441	904	20	math	math	PROPN
ejpam-3441	904	21	.	.	PUNCT
ejpam-3441	905	1	anal	anal	PROPN
ejpam-3441	905	2	.	.	PUNCT
ejpam-3441	906	1	appl	appl	PROPN
ejpam-3441	906	2	.	.	PROPN
ejpam-3441	906	3	,	,	PUNCT
ejpam-3441	906	4	134(1988	134(1988	NUM
ejpam-3441	906	5	)	)	PUNCT
ejpam-3441	906	6	94	94	NUM
ejpam-3441	906	7	-	-	SYM
ejpam-3441	906	8	103	103	NUM
ejpam-3441	906	9	.	.	PUNCT
ejpam-3441	907	1	[	[	X
ejpam-3441	907	2	30	30	NUM
ejpam-3441	907	3	]	]	PUNCT
ejpam-3441	907	4	x.	x.	NOUN
ejpam-3441	907	5	yuan	yuan	PROPN
ejpam-3441	907	6	,	,	PUNCT
ejpam-3441	907	7	c.	c.	PROPN
ejpam-3441	907	8	zhang	zhang	PROPN
ejpam-3441	907	9	and	and	CCONJ
ejpam-3441	907	10	y.	y.	PROPN
ejpam-3441	907	11	ren	ren	PROPN
ejpam-3441	907	12	,	,	PUNCT
ejpam-3441	907	13	generalized	generalize	VERB
ejpam-3441	907	14	fuzzy	fuzzy	ADJ
ejpam-3441	907	15	groups	group	NOUN
ejpam-3441	907	16	and	and	CCONJ
ejpam-3441	907	17	many	many	ADV
ejpam-3441	907	18	-	-	PUNCT
ejpam-3441	907	19	valued	value	VERB
ejpam-3441	907	20	implications	implication	NOUN
ejpam-3441	907	21	,	,	PUNCT
ejpam-3441	907	22	fuzzy	fuzzy	ADJ
ejpam-3441	907	23	sets	set	NOUN
ejpam-3441	907	24	and	and	CCONJ
ejpam-3441	907	25	systems	system	NOUN
ejpam-3441	907	26	,	,	PUNCT
ejpam-3441	907	27	138(2003	138(2003	NUM
ejpam-3441	907	28	)	)	PUNCT
ejpam-3441	907	29	205	205	NUM
ejpam-3441	907	30	-	-	SYM
ejpam-3441	907	31	211	211	NUM
ejpam-3441	907	32	.	.	PUNCT
ejpam-3441	908	1	[	[	X
ejpam-3441	908	2	31	31	NUM
ejpam-3441	908	3	]	]	PUNCT
ejpam-3441	908	4	l.	l.	PROPN
ejpam-3441	908	5	a.	a.	PROPN
ejpam-3441	908	6	zadeh	zadeh	PROPN
ejpam-3441	908	7	,	,	PUNCT
ejpam-3441	908	8	fuzzy	fuzzy	ADJ
ejpam-3441	908	9	sets	set	NOUN
ejpam-3441	908	10	,	,	PUNCT
ejpam-3441	908	11	information	information	NOUN
ejpam-3441	908	12	and	and	CCONJ
ejpam-3441	908	13	control	control	NOUN
ejpam-3441	908	14	,	,	PUNCT
ejpam-3441	908	15	8(1965	8(1965	NUM
ejpam-3441	908	16	)	)	PUNCT
ejpam-3441	908	17	338	338	NUM
ejpam-3441	908	18	-	-	SYM
ejpam-3441	908	19	363	363	NUM
ejpam-3441	908	20	.	.	PUNCT