id	sid	tid	token	lemma	pos
ejpam-3344	1	1	characterizations	characterization	NOUN
ejpam-3344	1	2	of	of	ADP
ejpam-3344	1	3	non	non	ADJ
ejpam-3344	1	4	-	-	ADJ
ejpam-3344	1	5	associative	associative	ADJ
ejpam-3344	1	6	rings	ring	NOUN
ejpam-3344	1	7	by	by	ADP
ejpam-3344	1	8	their	their	PRON
ejpam-3344	1	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	1	10	fuzzy	fuzzy	ADJ
ejpam-3344	1	11	bi	bi	ADJ
ejpam-3344	1	12	-	-	ADJ
ejpam-3344	1	13	ideals	ideal	NOUN
ejpam-3344	1	14	european	european	ADJ
ejpam-3344	1	15	journal	journal	PROPN
ejpam-3344	1	16	of	of	ADP
ejpam-3344	1	17	pure	pure	ADJ
ejpam-3344	1	18	and	and	CCONJ
ejpam-3344	1	19	applied	apply	VERB
ejpam-3344	1	20	mathematics	mathematic	NOUN
ejpam-3344	1	21	vol	vol	NOUN
ejpam-3344	1	22	.	.	PROPN
ejpam-3344	2	1	12	12	NUM
ejpam-3344	2	2	,	,	PUNCT
ejpam-3344	2	3	no	no	INTJ
ejpam-3344	2	4	.	.	NOUN
ejpam-3344	2	5	1	1	NUM
ejpam-3344	2	6	,	,	PUNCT
ejpam-3344	2	7	2019	2019	NUM
ejpam-3344	2	8	,	,	PUNCT
ejpam-3344	2	9	226	226	NUM
ejpam-3344	2	10	-	-	SYM
ejpam-3344	2	11	250	250	NUM
ejpam-3344	2	12	issn	issn	PROPN
ejpam-3344	2	13	1307	1307	NUM
ejpam-3344	2	14	-	-	SYM
ejpam-3344	2	15	5543	5543	NUM
ejpam-3344	2	16	–	–	PUNCT
ejpam-3344	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3344	2	18	published	publish	VERB
ejpam-3344	2	19	by	by	ADP
ejpam-3344	2	20	new	new	PROPN
ejpam-3344	2	21	york	york	PROPN
ejpam-3344	2	22	business	business	PROPN
ejpam-3344	2	23	global	global	ADJ
ejpam-3344	2	24	characterizations	characterization	NOUN
ejpam-3344	2	25	of	of	ADP
ejpam-3344	2	26	non	non	ADJ
ejpam-3344	2	27	-	-	ADJ
ejpam-3344	2	28	associative	associative	ADJ
ejpam-3344	2	29	rings	ring	NOUN
ejpam-3344	2	30	by	by	ADP
ejpam-3344	2	31	their	their	PRON
ejpam-3344	2	32	intuitionistic	intuitionistic	ADJ
ejpam-3344	2	33	fuzzy	fuzzy	ADJ
ejpam-3344	2	34	bi	bi	NOUN
ejpam-3344	2	35	-	-	NOUN
ejpam-3344	2	36	ideals	ideal	NOUN
ejpam-3344	2	37	nasreen	nasreen	ADP
ejpam-3344	2	38	kausar1,∗	kausar1,∗	NOUN
ejpam-3344	2	39	,	,	PUNCT
ejpam-3344	2	40	muhammad	muhammad	PROPN
ejpam-3344	2	41	azam	azam	PROPN
ejpam-3344	2	42	waqar2	waqar2	PROPN
ejpam-3344	2	43	1	1	NUM
ejpam-3344	2	44	department	department	NOUN
ejpam-3344	2	45	of	of	ADP
ejpam-3344	2	46	mathematics	mathematic	NOUN
ejpam-3344	2	47	,	,	PUNCT
ejpam-3344	2	48	quaid	quaid	PROPN
ejpam-3344	2	49	-	-	PUNCT
ejpam-3344	2	50	i	i	PROPN
ejpam-3344	2	51	-	-	PUNCT
ejpam-3344	2	52	azam	azam	PROPN
ejpam-3344	2	53	university	university	PROPN
ejpam-3344	2	54	islamabad	islamabad	PROPN
ejpam-3344	2	55	,	,	PUNCT
ejpam-3344	2	56	pakistan	pakistan	PROPN
ejpam-3344	2	57	2	2	NUM
ejpam-3344	2	58	department	department	NOUN
ejpam-3344	2	59	of	of	ADP
ejpam-3344	2	60	school	school	NOUN
ejpam-3344	2	61	of	of	ADP
ejpam-3344	2	62	business	business	NOUN
ejpam-3344	2	63	mangment	mangment	NOUN
ejpam-3344	2	64	,	,	PUNCT
ejpam-3344	2	65	nfc	nfc	NOUN
ejpam-3344	2	66	iefr	iefr	PROPN
ejpam-3344	2	67	fsd	fsd	PROPN
ejpam-3344	2	68	,	,	PUNCT
ejpam-3344	2	69	pakistan	pakistan	PROPN
ejpam-3344	2	70	abstract	abstract	NOUN
ejpam-3344	2	71	.	.	PUNCT
ejpam-3344	3	1	the	the	DET
ejpam-3344	3	2	purpose	purpose	NOUN
ejpam-3344	3	3	of	of	ADP
ejpam-3344	3	4	this	this	DET
ejpam-3344	3	5	paper	paper	NOUN
ejpam-3344	3	6	is	be	AUX
ejpam-3344	3	7	to	to	PART
ejpam-3344	3	8	initiate	initiate	VERB
ejpam-3344	3	9	and	and	CCONJ
ejpam-3344	3	10	study	study	VERB
ejpam-3344	3	11	on	on	ADP
ejpam-3344	3	12	the	the	DET
ejpam-3344	3	13	generalization	generalization	NOUN
ejpam-3344	3	14	of	of	ADP
ejpam-3344	3	15	the	the	DET
ejpam-3344	3	16	fuzzification	fuzzification	NOUN
ejpam-3344	3	17	of	of	ADP
ejpam-3344	3	18	ideals	ideal	NOUN
ejpam-3344	3	19	in	in	ADP
ejpam-3344	3	20	a	a	DET
ejpam-3344	3	21	class	class	NOUN
ejpam-3344	3	22	of	of	ADP
ejpam-3344	3	23	non	non	ADJ
ejpam-3344	3	24	-	-	ADJ
ejpam-3344	3	25	associative	associative	ADJ
ejpam-3344	3	26	and	and	CCONJ
ejpam-3344	3	27	non	non	ADJ
ejpam-3344	3	28	-	-	ADJ
ejpam-3344	3	29	commutative	commutative	ADJ
ejpam-3344	3	30	algebraic	algebraic	ADJ
ejpam-3344	3	31	structures	structure	NOUN
ejpam-3344	3	32	(	(	PUNCT
ejpam-3344	3	33	la	la	NOUN
ejpam-3344	3	34	-	-	NOUN
ejpam-3344	3	35	ring	ring	NOUN
ejpam-3344	3	36	)	)	PUNCT
ejpam-3344	3	37	.	.	PUNCT
ejpam-3344	4	1	we	we	PRON
ejpam-3344	4	2	characterize	characterize	VERB
ejpam-3344	4	3	different	different	ADJ
ejpam-3344	4	4	classes	class	NOUN
ejpam-3344	4	5	of	of	ADP
ejpam-3344	4	6	la	la	NOUN
ejpam-3344	4	7	-	-	NOUN
ejpam-3344	4	8	ring	ring	NOUN
ejpam-3344	4	9	in	in	ADP
ejpam-3344	4	10	terms	term	NOUN
ejpam-3344	4	11	of	of	ADP
ejpam-3344	4	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	4	13	fuzzy	fuzzy	ADJ
ejpam-3344	4	14	left	left	NOUN
ejpam-3344	4	15	(	(	PUNCT
ejpam-3344	4	16	resp	resp	NOUN
ejpam-3344	4	17	.	.	PUNCT
ejpam-3344	5	1	right	right	ADJ
ejpam-3344	5	2	,	,	PUNCT
ejpam-3344	5	3	bi-	bi-	NUM
ejpam-3344	5	4	,	,	PUNCT
ejpam-3344	5	5	generalized	generalize	VERB
ejpam-3344	5	6	bi-	bi-	X
ejpam-3344	5	7	,	,	PUNCT
ejpam-3344	5	8	(	(	PUNCT
ejpam-3344	5	9	1	1	NUM
ejpam-3344	5	10	,	,	PUNCT
ejpam-3344	5	11	2)-	2)-	NUM
ejpam-3344	5	12	)	)	PUNCT
ejpam-3344	5	13	ideals	ideal	NOUN
ejpam-3344	5	14	.	.	PUNCT
ejpam-3344	6	1	2010	2010	NUM
ejpam-3344	6	2	mathematics	mathematic	NOUN
ejpam-3344	6	3	subject	subject	NOUN
ejpam-3344	6	4	classifications	classification	NOUN
ejpam-3344	6	5	:	:	PUNCT
ejpam-3344	6	6	17d05	17d05	NUM
ejpam-3344	6	7	,	,	PUNCT
ejpam-3344	6	8	17d99	17d99	NUM
ejpam-3344	6	9	key	key	ADJ
ejpam-3344	6	10	words	word	NOUN
ejpam-3344	6	11	and	and	CCONJ
ejpam-3344	6	12	phrases	phrase	NOUN
ejpam-3344	6	13	:	:	PUNCT
ejpam-3344	6	14	intuitionistic	intuitionistic	ADJ
ejpam-3344	6	15	fuzzy	fuzzy	ADJ
ejpam-3344	6	16	left	left	ADJ
ejpam-3344	6	17	(	(	PUNCT
ejpam-3344	6	18	right	right	INTJ
ejpam-3344	6	19	,	,	PUNCT
ejpam-3344	6	20	bi-	bi-	NUM
ejpam-3344	6	21	,	,	PUNCT
ejpam-3344	6	22	generalized	generalize	VERB
ejpam-3344	6	23	bi-	bi-	X
ejpam-3344	6	24	,	,	PUNCT
ejpam-3344	6	25	(	(	PUNCT
ejpam-3344	6	26	1	1	NUM
ejpam-3344	6	27	,	,	PUNCT
ejpam-3344	6	28	2)-	2)-	NUM
ejpam-3344	6	29	)	)	PUNCT
ejpam-3344	6	30	ideals	ideal	NOUN
ejpam-3344	6	31	in	in	ADP
ejpam-3344	6	32	1972	1972	NUM
ejpam-3344	6	33	,	,	PUNCT
ejpam-3344	6	34	a	a	DET
ejpam-3344	6	35	generalization	generalization	NOUN
ejpam-3344	6	36	of	of	ADP
ejpam-3344	6	37	abelian	abelian	ADJ
ejpam-3344	6	38	semigroups	semigroup	NOUN
ejpam-3344	6	39	initiated	initiate	VERB
ejpam-3344	6	40	by	by	ADP
ejpam-3344	6	41	kazim	kazim	PROPN
ejpam-3344	6	42	et	et	PROPN
ejpam-3344	6	43	al	al	PROPN
ejpam-3344	7	1	[	[	X
ejpam-3344	7	2	11	11	NUM
ejpam-3344	7	3	]	]	PUNCT
ejpam-3344	7	4	.	.	PUNCT
ejpam-3344	8	1	in	in	ADP
ejpam-3344	8	2	ternary	ternary	ADJ
ejpam-3344	8	3	commutative	commutative	ADJ
ejpam-3344	8	4	(	(	PUNCT
ejpam-3344	8	5	abelian	abelian	ADJ
ejpam-3344	8	6	)	)	PUNCT
ejpam-3344	8	7	law	law	NOUN
ejpam-3344	8	8	:	:	PUNCT
ejpam-3344	8	9	abc	abc	PROPN
ejpam-3344	8	10	=	=	SYM
ejpam-3344	8	11	cba	cba	PROPN
ejpam-3344	8	12	,	,	PUNCT
ejpam-3344	8	13	they	they	PRON
ejpam-3344	8	14	introduced	introduce	VERB
ejpam-3344	8	15	braces	brace	NOUN
ejpam-3344	8	16	on	on	ADP
ejpam-3344	8	17	the	the	DET
ejpam-3344	8	18	left	left	ADJ
ejpam-3344	8	19	side	side	NOUN
ejpam-3344	8	20	of	of	ADP
ejpam-3344	8	21	this	this	DET
ejpam-3344	8	22	law	law	NOUN
ejpam-3344	8	23	and	and	CCONJ
ejpam-3344	8	24	explored	explore	VERB
ejpam-3344	8	25	a	a	DET
ejpam-3344	8	26	new	new	ADJ
ejpam-3344	8	27	pseudo	pseudo	NOUN
ejpam-3344	8	28	associative	associative	NOUN
ejpam-3344	8	29	law	law	NOUN
ejpam-3344	8	30	,	,	PUNCT
ejpam-3344	8	31	that	that	ADV
ejpam-3344	8	32	is	is	ADV
ejpam-3344	8	33	(	(	PUNCT
ejpam-3344	8	34	ab)c	ab)c	PROPN
ejpam-3344	8	35	=	=	SYM
ejpam-3344	8	36	(	(	PUNCT
ejpam-3344	8	37	cb)a	cb)a	PROPN
ejpam-3344	8	38	.	.	PUNCT
ejpam-3344	9	1	this	this	DET
ejpam-3344	9	2	law	law	NOUN
ejpam-3344	9	3	(	(	PUNCT
ejpam-3344	9	4	ab)c	ab)c	PROPN
ejpam-3344	9	5	=	=	SYM
ejpam-3344	9	6	(	(	PUNCT
ejpam-3344	9	7	cb)a	cb)a	PROPN
ejpam-3344	9	8	is	be	AUX
ejpam-3344	9	9	called	call	VERB
ejpam-3344	9	10	the	the	DET
ejpam-3344	9	11	left	left	ADJ
ejpam-3344	9	12	invertive	invertive	ADJ
ejpam-3344	9	13	law	law	NOUN
ejpam-3344	9	14	.	.	PUNCT
ejpam-3344	10	1	a	a	DET
ejpam-3344	10	2	groupoid	groupoid	PROPN
ejpam-3344	10	3	s	s	NOUN
ejpam-3344	10	4	is	be	AUX
ejpam-3344	10	5	said	say	VERB
ejpam-3344	10	6	to	to	PART
ejpam-3344	10	7	be	be	AUX
ejpam-3344	10	8	a	a	DET
ejpam-3344	10	9	left	left	NOUN
ejpam-3344	10	10	almost	almost	ADV
ejpam-3344	10	11	semigroup	semigroup	ADJ
ejpam-3344	10	12	(	(	PUNCT
ejpam-3344	10	13	abbreviated	abbreviate	VERB
ejpam-3344	10	14	as	as	ADP
ejpam-3344	10	15	la	la	NOUN
ejpam-3344	10	16	-	-	PUNCT
ejpam-3344	10	17	semigroup	semigroup	NOUN
ejpam-3344	10	18	)	)	PUNCT
ejpam-3344	10	19	,	,	PUNCT
ejpam-3344	10	20	if	if	SCONJ
ejpam-3344	10	21	it	it	PRON
ejpam-3344	10	22	satisfies	satisfy	VERB
ejpam-3344	10	23	the	the	DET
ejpam-3344	10	24	left	left	ADJ
ejpam-3344	10	25	invertive	invertive	ADJ
ejpam-3344	10	26	law	law	NOUN
ejpam-3344	10	27	:	:	PUNCT
ejpam-3344	10	28	(	(	PUNCT
ejpam-3344	10	29	ab)c	ab)c	PROPN
ejpam-3344	10	30	=	=	SYM
ejpam-3344	10	31	(	(	PUNCT
ejpam-3344	10	32	cb)a	cb)a	PROPN
ejpam-3344	10	33	.	.	PUNCT
ejpam-3344	11	1	an	an	DET
ejpam-3344	11	2	la	la	ADJ
ejpam-3344	11	3	-	-	PUNCT
ejpam-3344	11	4	semigroup	semigroup	PROPN
ejpam-3344	11	5	is	be	AUX
ejpam-3344	11	6	a	a	DET
ejpam-3344	11	7	midway	midway	NOUN
ejpam-3344	11	8	structure	structure	NOUN
ejpam-3344	11	9	between	between	ADP
ejpam-3344	11	10	an	an	DET
ejpam-3344	11	11	abelian	abelian	ADJ
ejpam-3344	11	12	semigroup	semigroup	NOUN
ejpam-3344	11	13	and	and	CCONJ
ejpam-3344	11	14	a	a	DET
ejpam-3344	11	15	groupoid	groupoid	NOUN
ejpam-3344	11	16	.	.	PUNCT
ejpam-3344	12	1	ideals	ideal	NOUN
ejpam-3344	12	2	in	in	ADP
ejpam-3344	12	3	la	la	ADJ
ejpam-3344	12	4	-	-	PUNCT
ejpam-3344	12	5	semigroup	semigroup	PROPN
ejpam-3344	12	6	have	have	AUX
ejpam-3344	12	7	been	be	AUX
ejpam-3344	12	8	investigated	investigate	VERB
ejpam-3344	12	9	by	by	ADP
ejpam-3344	12	10	[	[	X
ejpam-3344	12	11	16	16	NUM
ejpam-3344	12	12	]	]	PUNCT
ejpam-3344	12	13	.	.	PUNCT
ejpam-3344	13	1	in	in	ADP
ejpam-3344	13	2	[	[	X
ejpam-3344	13	3	9	9	NUM
ejpam-3344	13	4	]	]	PUNCT
ejpam-3344	13	5	(	(	PUNCT
ejpam-3344	13	6	resp	resp	NOUN
ejpam-3344	13	7	.	.	PUNCT
ejpam-3344	14	1	[	[	X
ejpam-3344	14	2	4	4	NUM
ejpam-3344	14	3	]	]	NUM
ejpam-3344	14	4	)	)	PUNCT
ejpam-3344	14	5	,	,	PUNCT
ejpam-3344	14	6	a	a	DET
ejpam-3344	14	7	groupoid	groupoid	NOUN
ejpam-3344	14	8	s	s	NOUN
ejpam-3344	14	9	is	be	AUX
ejpam-3344	14	10	said	say	VERB
ejpam-3344	14	11	to	to	PART
ejpam-3344	14	12	be	be	AUX
ejpam-3344	14	13	medial	medial	ADJ
ejpam-3344	14	14	(	(	PUNCT
ejpam-3344	14	15	resp	resp	NOUN
ejpam-3344	14	16	.	.	PUNCT
ejpam-3344	15	1	paramedial	paramedial	PROPN
ejpam-3344	15	2	)	)	PUNCT
ejpam-3344	16	1	if	if	SCONJ
ejpam-3344	16	2	(	(	PUNCT
ejpam-3344	16	3	ab)(cd	ab)(cd	NOUN
ejpam-3344	16	4	)	)	PUNCT
ejpam-3344	16	5	=	=	SYM
ejpam-3344	16	6	(	(	PUNCT
ejpam-3344	16	7	ac)(bd	ac)(bd	PROPN
ejpam-3344	16	8	)	)	PUNCT
ejpam-3344	16	9	(	(	PUNCT
ejpam-3344	16	10	resp	resp	NOUN
ejpam-3344	16	11	.	.	PUNCT
ejpam-3344	17	1	(	(	PUNCT
ejpam-3344	17	2	ab)(cd	ab)(cd	PROPN
ejpam-3344	17	3	)	)	PUNCT
ejpam-3344	17	4	=	=	SYM
ejpam-3344	17	5	(	(	PUNCT
ejpam-3344	17	6	db)(ca	db)(ca	PROPN
ejpam-3344	17	7	)	)	PUNCT
ejpam-3344	17	8	)	)	PUNCT
ejpam-3344	17	9	.	.	PUNCT
ejpam-3344	18	1	in	in	ADP
ejpam-3344	18	2	[	[	X
ejpam-3344	18	3	11	11	NUM
ejpam-3344	18	4	]	]	PUNCT
ejpam-3344	18	5	,	,	PUNCT
ejpam-3344	18	6	an	an	DET
ejpam-3344	18	7	la	la	ADJ
ejpam-3344	18	8	-	-	PUNCT
ejpam-3344	18	9	semigroup	semigroup	PROPN
ejpam-3344	18	10	is	be	AUX
ejpam-3344	18	11	medial	medial	ADJ
ejpam-3344	18	12	,	,	PUNCT
ejpam-3344	18	13	but	but	CCONJ
ejpam-3344	18	14	in	in	ADP
ejpam-3344	18	15	general	general	ADJ
ejpam-3344	18	16	an	an	DET
ejpam-3344	18	17	la	la	ADJ
ejpam-3344	18	18	-	-	PUNCT
ejpam-3344	18	19	semigroup	semigroup	NOUN
ejpam-3344	18	20	needs	need	VERB
ejpam-3344	18	21	not	not	PART
ejpam-3344	18	22	to	to	PART
ejpam-3344	18	23	be	be	AUX
ejpam-3344	18	24	paramedial	paramedial	ADJ
ejpam-3344	18	25	.	.	PUNCT
ejpam-3344	19	1	every	every	DET
ejpam-3344	19	2	la	la	PROPN
ejpam-3344	19	3	-	-	PUNCT
ejpam-3344	19	4	semigroup	semigroup	NOUN
ejpam-3344	19	5	with	with	ADP
ejpam-3344	19	6	left	left	ADJ
ejpam-3344	19	7	identity	identity	NOUN
ejpam-3344	19	8	is	be	AUX
ejpam-3344	19	9	paramedial	paramedial	ADJ
ejpam-3344	19	10	by	by	ADP
ejpam-3344	19	11	protic	protic	PROPN
ejpam-3344	19	12	et	et	NOUN
ejpam-3344	19	13	al	al	PROPN
ejpam-3344	20	1	[	[	X
ejpam-3344	20	2	16	16	NUM
ejpam-3344	20	3	]	]	PUNCT
ejpam-3344	20	4	and	and	CCONJ
ejpam-3344	20	5	also	also	ADV
ejpam-3344	20	6	satisfies	satisfy	VERB
ejpam-3344	20	7	a(bc	a(bc	NOUN
ejpam-3344	20	8	)	)	PUNCT
ejpam-3344	20	9	=	=	SYM
ejpam-3344	20	10	b(ac	b(ac	PROPN
ejpam-3344	20	11	)	)	PUNCT
ejpam-3344	20	12	,	,	PUNCT
ejpam-3344	20	13	(	(	PUNCT
ejpam-3344	20	14	ab)(cd	ab)(cd	NOUN
ejpam-3344	20	15	)	)	PUNCT
ejpam-3344	20	16	=	=	SYM
ejpam-3344	20	17	(	(	PUNCT
ejpam-3344	20	18	dc)(ba	dc)(ba	PROPN
ejpam-3344	20	19	)	)	PUNCT
ejpam-3344	20	20	.	.	PUNCT
ejpam-3344	21	1	kamran	kamran	PROPN
ejpam-3344	22	1	[	[	X
ejpam-3344	22	2	10	10	NUM
ejpam-3344	22	3	]	]	PUNCT
ejpam-3344	22	4	,	,	PUNCT
ejpam-3344	22	5	extended	extend	VERB
ejpam-3344	22	6	the	the	DET
ejpam-3344	22	7	notion	notion	NOUN
ejpam-3344	22	8	of	of	ADP
ejpam-3344	22	9	la	la	NOUN
ejpam-3344	22	10	-	-	PUNCT
ejpam-3344	22	11	semigroup	semigroup	NOUN
ejpam-3344	22	12	to	to	ADP
ejpam-3344	22	13	the	the	DET
ejpam-3344	22	14	left	leave	VERB
ejpam-3344	22	15	almost	almost	ADV
ejpam-3344	22	16	group	group	NOUN
ejpam-3344	22	17	(	(	PUNCT
ejpam-3344	22	18	lagroup	lagroup	NOUN
ejpam-3344	22	19	)	)	PUNCT
ejpam-3344	22	20	.	.	PUNCT
ejpam-3344	23	1	an	an	DET
ejpam-3344	23	2	la	la	ADJ
ejpam-3344	23	3	-	-	PUNCT
ejpam-3344	23	4	semigroup	semigroup	PROPN
ejpam-3344	23	5	s	s	VERB
ejpam-3344	23	6	is	be	AUX
ejpam-3344	23	7	said	say	VERB
ejpam-3344	23	8	to	to	PART
ejpam-3344	23	9	be	be	AUX
ejpam-3344	23	10	a	a	DET
ejpam-3344	23	11	left	left	ADJ
ejpam-3344	23	12	almost	almost	ADV
ejpam-3344	23	13	group	group	NOUN
ejpam-3344	23	14	,	,	PUNCT
ejpam-3344	23	15	if	if	SCONJ
ejpam-3344	23	16	there	there	PRON
ejpam-3344	23	17	exists	exist	VERB
ejpam-3344	23	18	left	leave	VERB
ejpam-3344	23	19	identity	identity	NOUN
ejpam-3344	23	20	e	e	NOUN
ejpam-3344	23	21	∈	∈	NOUN
ejpam-3344	23	22	s	s	VERB
ejpam-3344	24	1	such	such	ADJ
ejpam-3344	24	2	that	that	DET
ejpam-3344	24	3	ea	ea	NOUN
ejpam-3344	24	4	=	=	PUNCT
ejpam-3344	24	5	a	a	PRON
ejpam-3344	24	6	for	for	ADP
ejpam-3344	24	7	all	all	DET
ejpam-3344	24	8	a	a	DET
ejpam-3344	24	9	∈	∈	NOUN
ejpam-3344	24	10	s	s	NOUN
ejpam-3344	24	11	and	and	CCONJ
ejpam-3344	24	12	for	for	ADP
ejpam-3344	24	13	every	every	DET
ejpam-3344	24	14	a	a	DET
ejpam-3344	24	15	∈	∈	ADJ
ejpam-3344	24	16	s	s	NOUN
ejpam-3344	25	1	,	,	PUNCT
ejpam-3344	25	2	there	there	PRON
ejpam-3344	25	3	exists	exist	VERB
ejpam-3344	25	4	b	b	PROPN
ejpam-3344	25	5	∈	∈	PROPN
ejpam-3344	25	6	s	s	VERB
ejpam-3344	26	1	such	such	ADJ
ejpam-3344	26	2	that	that	DET
ejpam-3344	26	3	ba	ba	PROPN
ejpam-3344	26	4	=	=	SYM
ejpam-3344	26	5	e.	e.	PROPN
ejpam-3344	26	6	shah	shah	PROPN
ejpam-3344	26	7	et	et	PROPN
ejpam-3344	26	8	al	al	PROPN
ejpam-3344	27	1	[	[	X
ejpam-3344	27	2	20	20	NUM
ejpam-3344	27	3	]	]	PUNCT
ejpam-3344	27	4	,	,	PUNCT
ejpam-3344	27	5	initiated	initiate	VERB
ejpam-3344	27	6	the	the	DET
ejpam-3344	27	7	concept	concept	NOUN
ejpam-3344	27	8	of	of	ADP
ejpam-3344	27	9	left	leave	VERB
ejpam-3344	27	10	almost	almost	ADV
ejpam-3344	27	11	ring	ring	NOUN
ejpam-3344	27	12	(	(	PUNCT
ejpam-3344	27	13	abbreviated	abbreviate	VERB
ejpam-3344	27	14	as	as	ADP
ejpam-3344	27	15	la	la	NOUN
ejpam-3344	27	16	-	-	PUNCT
ejpam-3344	27	17	ring	ring	NOUN
ejpam-3344	27	18	)	)	PUNCT
ejpam-3344	27	19	of	of	ADP
ejpam-3344	27	20	finitely	finitely	ADV
ejpam-3344	27	21	nonzero	nonzero	ADJ
ejpam-3344	27	22	functions	function	NOUN
ejpam-3344	27	23	,	,	PUNCT
ejpam-3344	27	24	which	which	PRON
ejpam-3344	27	25	is	be	AUX
ejpam-3344	27	26	a	a	DET
ejpam-3344	27	27	generalization	generalization	NOUN
ejpam-3344	27	28	of	of	ADP
ejpam-3344	27	29	a	a	DET
ejpam-3344	27	30	commutative	commutative	ADJ
ejpam-3344	27	31	semigroup	semigroup	PROPN
ejpam-3344	27	32	ring	ring	NOUN
ejpam-3344	27	33	.	.	PUNCT
ejpam-3344	28	1	by	by	ADP
ejpam-3344	28	2	a	a	DET
ejpam-3344	28	3	left	left	ADJ
ejpam-3344	28	4	almost	almost	ADV
ejpam-3344	28	5	ring	ring	NOUN
ejpam-3344	28	6	,	,	PUNCT
ejpam-3344	28	7	we	we	PRON
ejpam-3344	28	8	mean	mean	VERB
ejpam-3344	28	9	a	a	DET
ejpam-3344	28	10	non	non	ADJ
ejpam-3344	28	11	-	-	ADJ
ejpam-3344	28	12	empty	empty	ADJ
ejpam-3344	28	13	set	set	VERB
ejpam-3344	28	14	r	r	NOUN
ejpam-3344	28	15	with	with	ADP
ejpam-3344	28	16	at	at	ADV
ejpam-3344	28	17	least	least	ADV
ejpam-3344	28	18	two	two	NUM
ejpam-3344	28	19	elements	element	NOUN
ejpam-3344	28	20	such	such	ADJ
ejpam-3344	28	21	that	that	SCONJ
ejpam-3344	28	22	(	(	PUNCT
ejpam-3344	28	23	r,+	r,+	NUM
ejpam-3344	28	24	)	)	PUNCT
ejpam-3344	28	25	is	be	AUX
ejpam-3344	28	26	an	an	DET
ejpam-3344	28	27	la	la	NOUN
ejpam-3344	28	28	-	-	NOUN
ejpam-3344	28	29	group	group	NOUN
ejpam-3344	28	30	,	,	PUNCT
ejpam-3344	28	31	(	(	PUNCT
ejpam-3344	28	32	r	r	NOUN
ejpam-3344	28	33	,	,	PUNCT
ejpam-3344	28	34	·	·	PUNCT
ejpam-3344	28	35	)	)	PUNCT
ejpam-3344	28	36	is	be	AUX
ejpam-3344	28	37	an	an	DET
ejpam-3344	28	38	la	la	ADJ
ejpam-3344	28	39	-	-	PUNCT
ejpam-3344	28	40	semigroup	semigroup	NOUN
ejpam-3344	28	41	,	,	PUNCT
ejpam-3344	28	42	both	both	PRON
ejpam-3344	28	43	left	leave	VERB
ejpam-3344	28	44	and	and	CCONJ
ejpam-3344	28	45	right	right	ADJ
ejpam-3344	28	46	distributive	distributive	ADJ
ejpam-3344	28	47	laws	law	NOUN
ejpam-3344	28	48	hold	hold	VERB
ejpam-3344	28	49	.	.	PUNCT
ejpam-3344	29	1	for	for	ADP
ejpam-3344	29	2	example	example	NOUN
ejpam-3344	29	3	,	,	PUNCT
ejpam-3344	29	4	from	from	ADP
ejpam-3344	29	5	a	a	DET
ejpam-3344	29	6	commutative	commutative	ADJ
ejpam-3344	29	7	ring	ring	NOUN
ejpam-3344	29	8	(	(	PUNCT
ejpam-3344	29	9	r,+	r,+	NUM
ejpam-3344	29	10	,	,	PUNCT
ejpam-3344	29	11	·	·	PUNCT
ejpam-3344	29	12	)	)	PUNCT
ejpam-3344	29	13	,	,	PUNCT
ejpam-3344	29	14	we	we	PRON
ejpam-3344	29	15	can	can	AUX
ejpam-3344	29	16	always	always	ADV
ejpam-3344	29	17	obtain	obtain	VERB
ejpam-3344	29	18	an	an	DET
ejpam-3344	29	19	la	la	ADJ
ejpam-3344	29	20	-	-	PUNCT
ejpam-3344	29	21	ring	ring	NOUN
ejpam-3344	29	22	(	(	PUNCT
ejpam-3344	29	23	r,⊕	r,⊕	NOUN
ejpam-3344	29	24	,	,	PUNCT
ejpam-3344	29	25	·	·	PUNCT
ejpam-3344	29	26	)	)	PUNCT
ejpam-3344	29	27	by	by	ADP
ejpam-3344	29	28	∗corresponding	∗corresponde	VERB
ejpam-3344	29	29	author	author	NOUN
ejpam-3344	29	30	.	.	PUNCT
ejpam-3344	30	1	doi	doi	NOUN
ejpam-3344	30	2	:	:	PUNCT
ejpam-3344	30	3	https://doi.org/10.29020/nybg.ejpam.v12i1.3344	https://doi.org/10.29020/nybg.ejpam.v12i1.3344	PROPN
ejpam-3344	30	4	email	email	NOUN
ejpam-3344	30	5	addresses	address	NOUN
ejpam-3344	30	6	:	:	PUNCT
ejpam-3344	30	7	kausar.nasreen57@gmail.com	kausar.nasreen57@gmail.com	X
ejpam-3344	30	8	(	(	PUNCT
ejpam-3344	30	9	n.	n.	PROPN
ejpam-3344	30	10	kausar	kausar	PROPN
ejpam-3344	30	11	)	)	PUNCT
ejpam-3344	30	12	,	,	PUNCT
ejpam-3344	30	13	azamwaqar4@gmail.com	azamwaqar4@gmail.com	X
ejpam-3344	30	14	(	(	PUNCT
ejpam-3344	30	15	m.	m.	NOUN
ejpam-3344	30	16	a.	a.	NOUN
ejpam-3344	30	17	waqar	waqar	PROPN
ejpam-3344	30	18	)	)	PUNCT
ejpam-3344	30	19	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3344	31	1	226	226	NUM
ejpam-3344	31	2	c	c	X
ejpam-3344	31	3	©	©	PROPN
ejpam-3344	31	4	2019	2019	NUM
ejpam-3344	31	5	ejpam	ejpam	NOUN
ejpam-3344	31	6	all	all	DET
ejpam-3344	31	7	rights	right	NOUN
ejpam-3344	31	8	reserved	reserve	VERB
ejpam-3344	31	9	.	.	PUNCT
ejpam-3344	32	1	n.	n.	PROPN
ejpam-3344	32	2	kausar	kausar	PROPN
ejpam-3344	32	3	,	,	PUNCT
ejpam-3344	32	4	m.	m.	NOUN
ejpam-3344	32	5	a.	a.	PROPN
ejpam-3344	32	6	waqar	waqar	PROPN
ejpam-3344	32	7	/	/	SYM
ejpam-3344	32	8	eur	eur	PROPN
ejpam-3344	32	9	.	.	PUNCT
ejpam-3344	33	1	j.	j.	PROPN
ejpam-3344	33	2	pure	pure	PROPN
ejpam-3344	33	3	appl	appl	PROPN
ejpam-3344	33	4	.	.	PROPN
ejpam-3344	33	5	math	math	PROPN
ejpam-3344	33	6	,	,	PUNCT
ejpam-3344	33	7	12	12	NUM
ejpam-3344	33	8	(	(	PUNCT
ejpam-3344	33	9	1	1	NUM
ejpam-3344	33	10	)	)	PUNCT
ejpam-3344	33	11	(	(	PUNCT
ejpam-3344	33	12	2019	2019	NUM
ejpam-3344	33	13	)	)	PUNCT
ejpam-3344	33	14	,	,	PUNCT
ejpam-3344	33	15	226	226	NUM
ejpam-3344	33	16	-	-	SYM
ejpam-3344	33	17	250	250	NUM
ejpam-3344	33	18	227	227	NUM
ejpam-3344	33	19	defining	define	VERB
ejpam-3344	33	20	for	for	ADP
ejpam-3344	33	21	all	all	DET
ejpam-3344	33	22	a	a	PRON
ejpam-3344	33	23	,	,	PUNCT
ejpam-3344	33	24	b	b	X
ejpam-3344	33	25	∈	∈	PROPN
ejpam-3344	33	26	r	r	NOUN
ejpam-3344	33	27	,	,	PUNCT
ejpam-3344	33	28	a	a	PRON
ejpam-3344	33	29	⊕	⊕	PROPN
ejpam-3344	33	30	b	b	X
ejpam-3344	34	1	=	=	SYM
ejpam-3344	34	2	b	b	PROPN
ejpam-3344	34	3	−	−	PROPN
ejpam-3344	34	4	a	a	PRON
ejpam-3344	34	5	and	and	CCONJ
ejpam-3344	34	6	a	a	PRON
ejpam-3344	34	7	·	·	PUNCT
ejpam-3344	34	8	b	b	NOUN
ejpam-3344	34	9	is	be	AUX
ejpam-3344	34	10	same	same	ADJ
ejpam-3344	34	11	as	as	ADP
ejpam-3344	34	12	in	in	ADP
ejpam-3344	34	13	the	the	DET
ejpam-3344	34	14	ring	ring	NOUN
ejpam-3344	34	15	.	.	PUNCT
ejpam-3344	35	1	despite	despite	SCONJ
ejpam-3344	35	2	the	the	DET
ejpam-3344	35	3	fact	fact	NOUN
ejpam-3344	35	4	that	that	SCONJ
ejpam-3344	35	5	the	the	DET
ejpam-3344	35	6	structure	structure	NOUN
ejpam-3344	35	7	is	be	AUX
ejpam-3344	35	8	non	non	ADJ
ejpam-3344	35	9	-	-	ADJ
ejpam-3344	35	10	associative	associative	ADJ
ejpam-3344	35	11	and	and	CCONJ
ejpam-3344	35	12	non	non	ADJ
ejpam-3344	35	13	-	-	ADJ
ejpam-3344	35	14	commutative	commutative	ADJ
ejpam-3344	35	15	,	,	PUNCT
ejpam-3344	35	16	however	however	ADV
ejpam-3344	35	17	it	it	PRON
ejpam-3344	35	18	possesses	possess	VERB
ejpam-3344	35	19	properties	property	NOUN
ejpam-3344	35	20	which	which	PRON
ejpam-3344	35	21	usually	usually	ADV
ejpam-3344	35	22	come	come	VERB
ejpam-3344	35	23	across	across	ADP
ejpam-3344	35	24	in	in	ADP
ejpam-3344	35	25	associative	associative	ADJ
ejpam-3344	35	26	and	and	CCONJ
ejpam-3344	35	27	commutative	commutative	ADJ
ejpam-3344	35	28	algebraic	algebraic	ADJ
ejpam-3344	35	29	structures	structure	NOUN
ejpam-3344	35	30	.	.	PUNCT
ejpam-3344	36	1	a	a	DET
ejpam-3344	36	2	non	non	ADJ
ejpam-3344	36	3	-	-	ADJ
ejpam-3344	36	4	empty	empty	ADJ
ejpam-3344	36	5	subset	subset	NOUN
ejpam-3344	36	6	a	a	PRON
ejpam-3344	36	7	of	of	ADP
ejpam-3344	36	8	an	an	DET
ejpam-3344	36	9	la	la	ADJ
ejpam-3344	36	10	-	-	PUNCT
ejpam-3344	36	11	ring	ring	NOUN
ejpam-3344	36	12	r	r	NOUN
ejpam-3344	36	13	is	be	AUX
ejpam-3344	36	14	called	call	VERB
ejpam-3344	36	15	an	an	DET
ejpam-3344	36	16	la	la	NOUN
ejpam-3344	36	17	-	-	PUNCT
ejpam-3344	36	18	subring	subring	NOUN
ejpam-3344	36	19	of	of	ADP
ejpam-3344	36	20	r	r	NOUN
ejpam-3344	36	21	if	if	SCONJ
ejpam-3344	36	22	a	a	DET
ejpam-3344	36	23	−	−	PROPN
ejpam-3344	36	24	b	b	NOUN
ejpam-3344	36	25	and	and	CCONJ
ejpam-3344	36	26	ab	ab	PROPN
ejpam-3344	36	27	∈	∈	PROPN
ejpam-3344	36	28	a	a	PRON
ejpam-3344	36	29	for	for	ADP
ejpam-3344	36	30	all	all	DET
ejpam-3344	36	31	a	a	DET
ejpam-3344	36	32	,	,	PUNCT
ejpam-3344	36	33	b	b	X
ejpam-3344	36	34	∈	∈	PROPN
ejpam-3344	36	35	a.	a.	NOUN
ejpam-3344	36	36	a	a	PRON
ejpam-3344	36	37	is	be	AUX
ejpam-3344	36	38	called	call	VERB
ejpam-3344	36	39	a	a	DET
ejpam-3344	36	40	left	left	ADJ
ejpam-3344	36	41	(	(	PUNCT
ejpam-3344	36	42	resp	resp	NOUN
ejpam-3344	36	43	.	.	PUNCT
ejpam-3344	37	1	right	right	ADJ
ejpam-3344	37	2	)	)	PUNCT
ejpam-3344	37	3	ideal	ideal	NOUN
ejpam-3344	37	4	of	of	ADP
ejpam-3344	37	5	r	r	NOUN
ejpam-3344	37	6	if	if	SCONJ
ejpam-3344	37	7	(	(	PUNCT
ejpam-3344	37	8	a,+	a,+	NOUN
ejpam-3344	37	9	)	)	PUNCT
ejpam-3344	37	10	is	be	AUX
ejpam-3344	37	11	an	an	DET
ejpam-3344	37	12	la	la	ADJ
ejpam-3344	37	13	-	-	NOUN
ejpam-3344	37	14	group	group	NOUN
ejpam-3344	37	15	and	and	CCONJ
ejpam-3344	37	16	ra	ra	PROPN
ejpam-3344	38	1	⊆	⊆	NUM
ejpam-3344	38	2	a	a	DET
ejpam-3344	38	3	(	(	PUNCT
ejpam-3344	38	4	resp	resp	NOUN
ejpam-3344	38	5	.	.	PUNCT
ejpam-3344	39	1	ar	ar	VERB
ejpam-3344	39	2	⊆	⊆	NUM
ejpam-3344	39	3	a	a	PRON
ejpam-3344	39	4	)	)	PUNCT
ejpam-3344	39	5	.	.	PUNCT
ejpam-3344	40	1	a	a	PRON
ejpam-3344	40	2	is	be	AUX
ejpam-3344	40	3	called	call	VERB
ejpam-3344	40	4	an	an	DET
ejpam-3344	40	5	ideal	ideal	NOUN
ejpam-3344	40	6	of	of	ADP
ejpam-3344	40	7	r	r	NOUN
ejpam-3344	40	8	if	if	SCONJ
ejpam-3344	40	9	it	it	PRON
ejpam-3344	40	10	is	be	AUX
ejpam-3344	40	11	both	both	CCONJ
ejpam-3344	40	12	a	a	DET
ejpam-3344	40	13	left	left	ADJ
ejpam-3344	40	14	ideal	ideal	NOUN
ejpam-3344	40	15	and	and	CCONJ
ejpam-3344	40	16	a	a	DET
ejpam-3344	40	17	right	right	ADJ
ejpam-3344	40	18	ideal	ideal	NOUN
ejpam-3344	40	19	of	of	ADP
ejpam-3344	40	20	r.	r.	PROPN
ejpam-3344	40	21	an	an	DET
ejpam-3344	40	22	la	la	ADV
ejpam-3344	40	23	-	-	PUNCT
ejpam-3344	40	24	subring	subre	VERB
ejpam-3344	40	25	a	a	PRON
ejpam-3344	40	26	of	of	ADP
ejpam-3344	40	27	r	r	NOUN
ejpam-3344	40	28	is	be	AUX
ejpam-3344	40	29	called	call	VERB
ejpam-3344	40	30	a	a	DET
ejpam-3344	40	31	bi	bi	NOUN
ejpam-3344	40	32	-	-	NOUN
ejpam-3344	40	33	ideal	ideal	NOUN
ejpam-3344	40	34	of	of	ADP
ejpam-3344	40	35	r	r	NOUN
ejpam-3344	40	36	if	if	SCONJ
ejpam-3344	40	37	(	(	PUNCT
ejpam-3344	40	38	ar)a	ar)a	PROPN
ejpam-3344	40	39	⊆	⊆	NUM
ejpam-3344	40	40	a.	a.	NOUN
ejpam-3344	40	41	a	a	DET
ejpam-3344	40	42	non	non	ADJ
ejpam-3344	40	43	-	-	ADJ
ejpam-3344	40	44	empty	empty	ADJ
ejpam-3344	40	45	subset	subset	NOUN
ejpam-3344	40	46	a	a	PRON
ejpam-3344	40	47	of	of	ADP
ejpam-3344	40	48	r	r	NOUN
ejpam-3344	40	49	is	be	AUX
ejpam-3344	40	50	called	call	VERB
ejpam-3344	40	51	a	a	DET
ejpam-3344	40	52	generalized	generalized	ADJ
ejpam-3344	40	53	bi	bi	NOUN
ejpam-3344	40	54	-	-	NOUN
ejpam-3344	40	55	ideal	ideal	NOUN
ejpam-3344	40	56	of	of	ADP
ejpam-3344	40	57	r	r	NOUN
ejpam-3344	40	58	if	if	SCONJ
ejpam-3344	40	59	(	(	PUNCT
ejpam-3344	40	60	a,+	a,+	NOUN
ejpam-3344	40	61	)	)	PUNCT
ejpam-3344	40	62	is	be	AUX
ejpam-3344	40	63	an	an	DET
ejpam-3344	40	64	la	la	ADJ
ejpam-3344	40	65	-	-	NOUN
ejpam-3344	40	66	group	group	NOUN
ejpam-3344	40	67	and	and	CCONJ
ejpam-3344	40	68	(	(	PUNCT
ejpam-3344	40	69	ar)a	ar)a	PROPN
ejpam-3344	40	70	⊆	⊆	NUM
ejpam-3344	40	71	a.	a.	NOUN
ejpam-3344	40	72	every	every	DET
ejpam-3344	40	73	bi	bi	NOUN
ejpam-3344	40	74	-	-	NOUN
ejpam-3344	40	75	ideal	ideal	NOUN
ejpam-3344	40	76	of	of	ADP
ejpam-3344	40	77	r	r	NOUN
ejpam-3344	40	78	is	be	AUX
ejpam-3344	40	79	a	a	DET
ejpam-3344	40	80	generalized	generalized	ADJ
ejpam-3344	40	81	bi	bi	NOUN
ejpam-3344	40	82	-	-	NOUN
ejpam-3344	40	83	ideal	ideal	NOUN
ejpam-3344	40	84	of	of	ADP
ejpam-3344	40	85	r.	r.	PROPN
ejpam-3344	40	86	an	an	DET
ejpam-3344	40	87	la	la	ADV
ejpam-3344	40	88	-	-	PUNCT
ejpam-3344	40	89	subring	subre	VERB
ejpam-3344	40	90	a	a	PRON
ejpam-3344	40	91	of	of	ADP
ejpam-3344	40	92	r	r	NOUN
ejpam-3344	40	93	is	be	AUX
ejpam-3344	40	94	called	call	VERB
ejpam-3344	40	95	(	(	PUNCT
ejpam-3344	40	96	1	1	NUM
ejpam-3344	40	97	,	,	PUNCT
ejpam-3344	40	98	2)-ideal	2)-ideal	NUM
ejpam-3344	40	99	of	of	ADP
ejpam-3344	40	100	r	r	PRON
ejpam-3344	40	101	if	if	SCONJ
ejpam-3344	40	102	(	(	PUNCT
ejpam-3344	40	103	ar)a2	ar)a2	NOUN
ejpam-3344	40	104	⊆	⊆	NUM
ejpam-3344	40	105	a.	a.	NOUN
ejpam-3344	40	106	we	we	PRON
ejpam-3344	40	107	will	will	AUX
ejpam-3344	40	108	initiate	initiate	VERB
ejpam-3344	40	109	the	the	DET
ejpam-3344	40	110	concept	concept	NOUN
ejpam-3344	40	111	of	of	ADP
ejpam-3344	40	112	regular	regular	ADJ
ejpam-3344	40	113	(	(	PUNCT
ejpam-3344	40	114	resp	resp	NOUN
ejpam-3344	40	115	.	.	PUNCT
ejpam-3344	41	1	left	leave	VERB
ejpam-3344	41	2	regular	regular	ADJ
ejpam-3344	41	3	,	,	PUNCT
ejpam-3344	41	4	right	right	ADV
ejpam-3344	41	5	regular	regular	ADJ
ejpam-3344	41	6	,	,	PUNCT
ejpam-3344	41	7	(	(	PUNCT
ejpam-3344	41	8	2	2	NUM
ejpam-3344	41	9	,	,	PUNCT
ejpam-3344	41	10	2)-regular	2)-regular	NUM
ejpam-3344	41	11	,	,	PUNCT
ejpam-3344	41	12	left	leave	VERB
ejpam-3344	41	13	weakly	weakly	ADV
ejpam-3344	41	14	regular	regular	ADJ
ejpam-3344	41	15	,	,	PUNCT
ejpam-3344	41	16	right	right	ADJ
ejpam-3344	41	17	weakly	weakly	ADV
ejpam-3344	41	18	regular	regular	ADJ
ejpam-3344	41	19	,	,	PUNCT
ejpam-3344	41	20	intra	intra	ADJ
ejpam-3344	41	21	-	-	ADJ
ejpam-3344	41	22	regular	regular	ADJ
ejpam-3344	41	23	)	)	PUNCT
ejpam-3344	41	24	la	la	NOUN
ejpam-3344	41	25	-	-	PUNCT
ejpam-3344	41	26	rings	ring	NOUN
ejpam-3344	41	27	.	.	PUNCT
ejpam-3344	42	1	we	we	PRON
ejpam-3344	42	2	will	will	AUX
ejpam-3344	42	3	also	also	ADV
ejpam-3344	42	4	define	define	VERB
ejpam-3344	42	5	the	the	DET
ejpam-3344	42	6	concept	concept	NOUN
ejpam-3344	42	7	of	of	ADP
ejpam-3344	42	8	intuitionistic	intuitionistic	ADJ
ejpam-3344	42	9	fuzzy	fuzzy	ADJ
ejpam-3344	42	10	left	left	NOUN
ejpam-3344	42	11	(	(	PUNCT
ejpam-3344	42	12	resp	resp	NOUN
ejpam-3344	42	13	.	.	PUNCT
ejpam-3344	43	1	right	right	ADJ
ejpam-3344	43	2	,	,	PUNCT
ejpam-3344	43	3	bi-,generalized	bi-,generalize	VERB
ejpam-3344	43	4	bi-	bi-	NUM
ejpam-3344	43	5	,	,	PUNCT
ejpam-3344	43	6	(	(	PUNCT
ejpam-3344	43	7	1	1	NUM
ejpam-3344	43	8	,	,	PUNCT
ejpam-3344	43	9	2)-	2)-	NUM
ejpam-3344	43	10	)	)	PUNCT
ejpam-3344	43	11	ideals	ideal	NOUN
ejpam-3344	43	12	.	.	PUNCT
ejpam-3344	44	1	we	we	PRON
ejpam-3344	44	2	will	will	AUX
ejpam-3344	44	3	describe	describe	VERB
ejpam-3344	44	4	a	a	DET
ejpam-3344	44	5	study	study	NOUN
ejpam-3344	44	6	of	of	ADP
ejpam-3344	44	7	regular	regular	ADJ
ejpam-3344	44	8	(	(	PUNCT
ejpam-3344	44	9	resp	resp	NOUN
ejpam-3344	44	10	.	.	PUNCT
ejpam-3344	45	1	left	leave	VERB
ejpam-3344	45	2	regular	regular	ADJ
ejpam-3344	45	3	,	,	PUNCT
ejpam-3344	45	4	right	right	ADV
ejpam-3344	45	5	regular	regular	ADJ
ejpam-3344	45	6	,	,	PUNCT
ejpam-3344	45	7	(	(	PUNCT
ejpam-3344	45	8	2	2	NUM
ejpam-3344	45	9	,	,	PUNCT
ejpam-3344	45	10	2)-regular	2)-regular	NUM
ejpam-3344	45	11	,	,	PUNCT
ejpam-3344	45	12	left	leave	VERB
ejpam-3344	45	13	weakly	weakly	ADV
ejpam-3344	45	14	regular	regular	ADJ
ejpam-3344	45	15	,	,	PUNCT
ejpam-3344	45	16	right	right	ADJ
ejpam-3344	45	17	weakly	weakly	ADV
ejpam-3344	45	18	regular	regular	ADJ
ejpam-3344	45	19	,	,	PUNCT
ejpam-3344	45	20	intra	intra	ADJ
ejpam-3344	45	21	-	-	ADJ
ejpam-3344	45	22	regular	regular	ADJ
ejpam-3344	45	23	)	)	PUNCT
ejpam-3344	45	24	la	la	NOUN
ejpam-3344	45	25	-	-	PUNCT
ejpam-3344	45	26	rings	ring	NOUN
ejpam-3344	45	27	by	by	ADP
ejpam-3344	45	28	the	the	DET
ejpam-3344	45	29	properties	property	NOUN
ejpam-3344	45	30	of	of	ADP
ejpam-3344	45	31	intuitionistic	intuitionistic	ADJ
ejpam-3344	45	32	fuzzy	fuzzy	ADJ
ejpam-3344	45	33	left	left	NOUN
ejpam-3344	45	34	(	(	PUNCT
ejpam-3344	45	35	right	right	INTJ
ejpam-3344	45	36	,	,	PUNCT
ejpam-3344	45	37	bi-	bi-	NUM
ejpam-3344	45	38	,	,	PUNCT
ejpam-3344	45	39	generalized	generalize	VERB
ejpam-3344	45	40	bi-	bi-	NUM
ejpam-3344	45	41	)	)	PUNCT
ejpam-3344	45	42	ideals	ideal	NOUN
ejpam-3344	45	43	.	.	PUNCT
ejpam-3344	46	1	in	in	ADP
ejpam-3344	46	2	this	this	DET
ejpam-3344	46	3	regard	regard	NOUN
ejpam-3344	46	4	,	,	PUNCT
ejpam-3344	46	5	we	we	PRON
ejpam-3344	46	6	will	will	AUX
ejpam-3344	46	7	prove	prove	VERB
ejpam-3344	46	8	that	that	SCONJ
ejpam-3344	46	9	in	in	ADP
ejpam-3344	46	10	regular	regular	ADJ
ejpam-3344	46	11	(	(	PUNCT
ejpam-3344	46	12	resp	resp	NOUN
ejpam-3344	46	13	.	.	PUNCT
ejpam-3344	47	1	left	leave	VERB
ejpam-3344	47	2	weakly	weakly	ADV
ejpam-3344	47	3	regular	regular	ADJ
ejpam-3344	47	4	)	)	PUNCT
ejpam-3344	47	5	la	la	NOUN
ejpam-3344	47	6	-	-	PUNCT
ejpam-3344	47	7	rings	ring	NOUN
ejpam-3344	47	8	,	,	PUNCT
ejpam-3344	47	9	the	the	DET
ejpam-3344	47	10	concept	concept	NOUN
ejpam-3344	47	11	of	of	ADP
ejpam-3344	47	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	47	13	fuzzy	fuzzy	ADJ
ejpam-3344	47	14	(	(	PUNCT
ejpam-3344	47	15	right	right	ADJ
ejpam-3344	47	16	,	,	PUNCT
ejpam-3344	47	17	two	two	NUM
ejpam-3344	47	18	-	-	PUNCT
ejpam-3344	47	19	sided	sided	ADJ
ejpam-3344	47	20	)	)	PUNCT
ejpam-3344	47	21	ideals	ideal	NOUN
ejpam-3344	47	22	coincides	coincide	NOUN
ejpam-3344	47	23	.	.	PUNCT
ejpam-3344	48	1	we	we	PRON
ejpam-3344	48	2	will	will	AUX
ejpam-3344	48	3	also	also	ADV
ejpam-3344	48	4	show	show	VERB
ejpam-3344	48	5	that	that	SCONJ
ejpam-3344	48	6	in	in	ADP
ejpam-3344	48	7	right	right	ADV
ejpam-3344	48	8	regular	regular	ADJ
ejpam-3344	48	9	(	(	PUNCT
ejpam-3344	48	10	resp	resp	NOUN
ejpam-3344	48	11	.	.	PUNCT
ejpam-3344	49	1	(	(	PUNCT
ejpam-3344	49	2	2	2	NUM
ejpam-3344	49	3	,	,	PUNCT
ejpam-3344	49	4	2)regular	2)regular	NUM
ejpam-3344	49	5	,	,	PUNCT
ejpam-3344	49	6	right	right	ADJ
ejpam-3344	49	7	weakly	weakly	ADV
ejpam-3344	49	8	regular	regular	ADJ
ejpam-3344	49	9	,	,	PUNCT
ejpam-3344	49	10	intra	intra	ADJ
ejpam-3344	49	11	-	-	ADJ
ejpam-3344	49	12	regular	regular	ADJ
ejpam-3344	49	13	)	)	PUNCT
ejpam-3344	49	14	la	la	NOUN
ejpam-3344	49	15	-	-	PUNCT
ejpam-3344	49	16	rings	ring	NOUN
ejpam-3344	49	17	,	,	PUNCT
ejpam-3344	49	18	the	the	DET
ejpam-3344	49	19	concept	concept	NOUN
ejpam-3344	49	20	of	of	ADP
ejpam-3344	49	21	intuitionistic	intuitionistic	ADJ
ejpam-3344	49	22	fuzzy	fuzzy	ADJ
ejpam-3344	49	23	(	(	PUNCT
ejpam-3344	49	24	left	left	ADJ
ejpam-3344	49	25	,	,	PUNCT
ejpam-3344	49	26	right	right	INTJ
ejpam-3344	49	27	,	,	PUNCT
ejpam-3344	49	28	two	two	NUM
ejpam-3344	49	29	-	-	PUNCT
ejpam-3344	49	30	sided	sided	ADJ
ejpam-3344	49	31	)	)	PUNCT
ejpam-3344	49	32	ideals	ideal	NOUN
ejpam-3344	49	33	coincides	coincide	NOUN
ejpam-3344	49	34	.	.	PUNCT
ejpam-3344	50	1	also	also	ADV
ejpam-3344	50	2	in	in	ADP
ejpam-3344	50	3	left	left	ADJ
ejpam-3344	50	4	regular	regular	ADJ
ejpam-3344	50	5	la	la	NOUN
ejpam-3344	50	6	-	-	PUNCT
ejpam-3344	50	7	rings	ring	NOUN
ejpam-3344	50	8	with	with	ADP
ejpam-3344	50	9	left	left	ADJ
ejpam-3344	50	10	identity	identity	NOUN
ejpam-3344	50	11	,	,	PUNCT
ejpam-3344	50	12	the	the	DET
ejpam-3344	50	13	concept	concept	NOUN
ejpam-3344	50	14	of	of	ADP
ejpam-3344	50	15	intuitionistic	intuitionistic	ADJ
ejpam-3344	50	16	fuzzy	fuzzy	ADJ
ejpam-3344	50	17	(	(	PUNCT
ejpam-3344	50	18	left	left	ADJ
ejpam-3344	50	19	,	,	PUNCT
ejpam-3344	50	20	right	right	INTJ
ejpam-3344	50	21	,	,	PUNCT
ejpam-3344	50	22	two	two	NUM
ejpam-3344	50	23	-	-	PUNCT
ejpam-3344	50	24	sided	sided	ADJ
ejpam-3344	50	25	)	)	PUNCT
ejpam-3344	50	26	ideals	ideal	NOUN
ejpam-3344	50	27	coincides	coincide	NOUN
ejpam-3344	50	28	.	.	PUNCT
ejpam-3344	51	1	we	we	PRON
ejpam-3344	51	2	will	will	AUX
ejpam-3344	51	3	also	also	ADV
ejpam-3344	51	4	characterize	characterize	VERB
ejpam-3344	51	5	left	leave	VERB
ejpam-3344	51	6	weakly	weakly	ADV
ejpam-3344	51	7	regular	regular	ADJ
ejpam-3344	51	8	la	la	NOUN
ejpam-3344	51	9	-	-	PUNCT
ejpam-3344	51	10	rings	ring	NOUN
ejpam-3344	51	11	in	in	ADP
ejpam-3344	51	12	terms	term	NOUN
ejpam-3344	51	13	of	of	ADP
ejpam-3344	51	14	intuitionistic	intuitionistic	ADJ
ejpam-3344	51	15	fuzzy	fuzzy	ADJ
ejpam-3344	51	16	right	right	NOUN
ejpam-3344	51	17	(	(	PUNCT
ejpam-3344	51	18	two	two	NUM
ejpam-3344	51	19	-	-	PUNCT
ejpam-3344	51	20	sided	sided	ADJ
ejpam-3344	51	21	,	,	PUNCT
ejpam-3344	51	22	bi-	bi-	PRON
ejpam-3344	51	23	,	,	PUNCT
ejpam-3344	51	24	generalize	generalize	VERB
ejpam-3344	51	25	bi-	bi-	NUM
ejpam-3344	51	26	)	)	PUNCT
ejpam-3344	51	27	ideals	ideal	NOUN
ejpam-3344	51	28	.	.	PUNCT
ejpam-3344	52	1	1	1	X
ejpam-3344	52	2	.	.	X
ejpam-3344	52	3	basic	basic	ADJ
ejpam-3344	52	4	definitions	definition	NOUN
ejpam-3344	52	5	and	and	CCONJ
ejpam-3344	52	6	preliminary	preliminary	ADJ
ejpam-3344	52	7	results	result	NOUN
ejpam-3344	52	8	after	after	ADP
ejpam-3344	52	9	the	the	DET
ejpam-3344	52	10	introduction	introduction	NOUN
ejpam-3344	52	11	of	of	ADP
ejpam-3344	52	12	fuzzy	fuzzy	ADJ
ejpam-3344	52	13	set	set	VERB
ejpam-3344	52	14	by	by	ADP
ejpam-3344	52	15	zadeh	zadeh	PROPN
ejpam-3344	52	16	[	[	X
ejpam-3344	52	17	22	22	NUM
ejpam-3344	52	18	]	]	PUNCT
ejpam-3344	52	19	,	,	PUNCT
ejpam-3344	52	20	several	several	ADJ
ejpam-3344	52	21	researchers	researcher	NOUN
ejpam-3344	52	22	explored	explore	VERB
ejpam-3344	52	23	on	on	ADP
ejpam-3344	52	24	the	the	DET
ejpam-3344	52	25	generalization	generalization	NOUN
ejpam-3344	52	26	of	of	ADP
ejpam-3344	52	27	the	the	DET
ejpam-3344	52	28	notion	notion	NOUN
ejpam-3344	52	29	of	of	ADP
ejpam-3344	52	30	fuzzy	fuzzy	ADJ
ejpam-3344	52	31	set	set	NOUN
ejpam-3344	52	32	.	.	PUNCT
ejpam-3344	53	1	the	the	DET
ejpam-3344	53	2	concept	concept	NOUN
ejpam-3344	53	3	of	of	ADP
ejpam-3344	53	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	53	5	fuzzy	fuzzy	ADJ
ejpam-3344	53	6	set	set	NOUN
ejpam-3344	53	7	was	be	AUX
ejpam-3344	53	8	introduced	introduce	VERB
ejpam-3344	53	9	by	by	ADP
ejpam-3344	53	10	atanassov	atanassov	NOUN
ejpam-3344	53	11	[	[	X
ejpam-3344	53	12	1	1	NUM
ejpam-3344	53	13	,	,	PUNCT
ejpam-3344	53	14	2	2	NUM
ejpam-3344	53	15	]	]	PUNCT
ejpam-3344	53	16	,	,	PUNCT
ejpam-3344	53	17	as	as	ADP
ejpam-3344	53	18	a	a	DET
ejpam-3344	53	19	generalization	generalization	NOUN
ejpam-3344	53	20	of	of	ADP
ejpam-3344	53	21	the	the	DET
ejpam-3344	53	22	notion	notion	NOUN
ejpam-3344	53	23	of	of	ADP
ejpam-3344	53	24	fuzzy	fuzzy	ADJ
ejpam-3344	53	25	set	set	NOUN
ejpam-3344	53	26	.	.	PUNCT
ejpam-3344	54	1	liu	liu	PROPN
ejpam-3344	55	1	[	[	X
ejpam-3344	55	2	13	13	NUM
ejpam-3344	55	3	]	]	PUNCT
ejpam-3344	55	4	,	,	PUNCT
ejpam-3344	55	5	introduced	introduce	VERB
ejpam-3344	55	6	the	the	DET
ejpam-3344	55	7	concept	concept	NOUN
ejpam-3344	55	8	of	of	ADP
ejpam-3344	55	9	fuzzy	fuzzy	ADJ
ejpam-3344	55	10	subrings	subring	NOUN
ejpam-3344	55	11	and	and	CCONJ
ejpam-3344	55	12	fuzzy	fuzzy	ADJ
ejpam-3344	55	13	ideals	ideal	NOUN
ejpam-3344	55	14	of	of	ADP
ejpam-3344	55	15	a	a	DET
ejpam-3344	55	16	ring	ring	NOUN
ejpam-3344	55	17	.	.	PUNCT
ejpam-3344	56	1	many	many	ADJ
ejpam-3344	56	2	authors	author	NOUN
ejpam-3344	56	3	have	have	AUX
ejpam-3344	56	4	explored	explore	VERB
ejpam-3344	56	5	the	the	DET
ejpam-3344	56	6	theory	theory	NOUN
ejpam-3344	56	7	of	of	ADP
ejpam-3344	56	8	fuzzy	fuzzy	ADJ
ejpam-3344	56	9	rings	ring	NOUN
ejpam-3344	56	10	(	(	PUNCT
ejpam-3344	56	11	for	for	ADP
ejpam-3344	56	12	example	example	NOUN
ejpam-3344	56	13	[	[	X
ejpam-3344	56	14	6	6	NUM
ejpam-3344	56	15	,	,	PUNCT
ejpam-3344	56	16	12	12	NUM
ejpam-3344	56	17	,	,	PUNCT
ejpam-3344	56	18	14	14	NUM
ejpam-3344	56	19	,	,	PUNCT
ejpam-3344	56	20	15	15	NUM
ejpam-3344	56	21	,	,	PUNCT
ejpam-3344	56	22	21	21	NUM
ejpam-3344	56	23	]	]	PUNCT
ejpam-3344	56	24	)	)	PUNCT
ejpam-3344	56	25	.	.	PUNCT
ejpam-3344	57	1	gupta	gupta	PROPN
ejpam-3344	57	2	et	et	PROPN
ejpam-3344	57	3	al	al	PROPN
ejpam-3344	58	1	[	[	X
ejpam-3344	58	2	6	6	NUM
ejpam-3344	58	3	]	]	PUNCT
ejpam-3344	58	4	,	,	PUNCT
ejpam-3344	58	5	gave	give	VERB
ejpam-3344	58	6	the	the	DET
ejpam-3344	58	7	idea	idea	NOUN
ejpam-3344	58	8	of	of	ADP
ejpam-3344	58	9	intrinsic	intrinsic	ADJ
ejpam-3344	58	10	product	product	NOUN
ejpam-3344	58	11	of	of	ADP
ejpam-3344	58	12	fuzzy	fuzzy	ADJ
ejpam-3344	58	13	subsets	subset	NOUN
ejpam-3344	58	14	of	of	ADP
ejpam-3344	58	15	a	a	DET
ejpam-3344	58	16	ring	ring	NOUN
ejpam-3344	58	17	.	.	PUNCT
ejpam-3344	59	1	kuroki	kuroki	PROPN
ejpam-3344	60	1	[	[	X
ejpam-3344	60	2	12	12	NUM
ejpam-3344	60	3	]	]	PUNCT
ejpam-3344	60	4	,	,	PUNCT
ejpam-3344	60	5	characterized	characterize	VERB
ejpam-3344	60	6	regular	regular	ADV
ejpam-3344	60	7	(	(	PUNCT
ejpam-3344	60	8	intra	intra	ADJ
ejpam-3344	60	9	-	-	ADJ
ejpam-3344	60	10	regular	regular	ADJ
ejpam-3344	60	11	,	,	PUNCT
ejpam-3344	60	12	both	both	CCONJ
ejpam-3344	60	13	regular	regular	ADJ
ejpam-3344	60	14	and	and	CCONJ
ejpam-3344	60	15	intra	intra	ADJ
ejpam-3344	60	16	-	-	ADJ
ejpam-3344	60	17	regular	regular	ADJ
ejpam-3344	60	18	)	)	PUNCT
ejpam-3344	60	19	rings	ring	NOUN
ejpam-3344	60	20	in	in	ADP
ejpam-3344	60	21	terms	term	NOUN
ejpam-3344	60	22	of	of	ADP
ejpam-3344	60	23	fuzzy	fuzzy	ADJ
ejpam-3344	60	24	left	left	NOUN
ejpam-3344	60	25	(	(	PUNCT
ejpam-3344	60	26	right	right	ADJ
ejpam-3344	60	27	,	,	PUNCT
ejpam-3344	60	28	quasi	quasi	ADJ
ejpam-3344	60	29	,	,	PUNCT
ejpam-3344	60	30	bi-	bi-	NUM
ejpam-3344	60	31	)	)	PUNCT
ejpam-3344	60	32	ideals	ideal	NOUN
ejpam-3344	60	33	.	.	PUNCT
ejpam-3344	61	1	an	an	DET
ejpam-3344	61	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	61	3	fuzzy	fuzzy	ADJ
ejpam-3344	61	4	set	set	NOUN
ejpam-3344	61	5	(	(	PUNCT
ejpam-3344	61	6	briefly	briefly	ADV
ejpam-3344	61	7	,	,	PUNCT
ejpam-3344	61	8	ifs	ifs	PROPN
ejpam-3344	61	9	)	)	PUNCT
ejpam-3344	61	10	a	a	PRON
ejpam-3344	61	11	in	in	ADP
ejpam-3344	61	12	a	a	DET
ejpam-3344	61	13	non	non	ADJ
ejpam-3344	61	14	-	-	ADJ
ejpam-3344	61	15	empty	empty	ADJ
ejpam-3344	61	16	set	set	NOUN
ejpam-3344	61	17	x	x	PUNCT
ejpam-3344	61	18	is	be	AUX
ejpam-3344	61	19	an	an	DET
ejpam-3344	61	20	object	object	NOUN
ejpam-3344	61	21	having	have	VERB
ejpam-3344	61	22	the	the	DET
ejpam-3344	61	23	form	form	NOUN
ejpam-3344	61	24	a	a	DET
ejpam-3344	61	25	=	=	X
ejpam-3344	61	26	{	{	PUNCT
ejpam-3344	61	27	(	(	PUNCT
ejpam-3344	61	28	x	x	NOUN
ejpam-3344	61	29	,	,	PUNCT
ejpam-3344	61	30	µa(x	µa(x	NOUN
ejpam-3344	61	31	)	)	PUNCT
ejpam-3344	61	32	,	,	PUNCT
ejpam-3344	61	33	γa(x	γa(x	NUM
ejpam-3344	61	34	)	)	PUNCT
ejpam-3344	61	35	)	)	PUNCT
ejpam-3344	61	36	:	:	PUNCT
ejpam-3344	62	1	x	x	X
ejpam-3344	62	2	∈	∈	NOUN
ejpam-3344	62	3	x	x	X
ejpam-3344	62	4	}	}	PUNCT
ejpam-3344	62	5	,	,	PUNCT
ejpam-3344	62	6	where	where	SCONJ
ejpam-3344	62	7	the	the	DET
ejpam-3344	62	8	functions	function	NOUN
ejpam-3344	62	9	µa	µa	VERB
ejpam-3344	62	10	:	:	PUNCT
ejpam-3344	62	11	x	x	X
ejpam-3344	62	12	→	→	SYM
ejpam-3344	62	13	[	[	X
ejpam-3344	62	14	0	0	NUM
ejpam-3344	62	15	,	,	PUNCT
ejpam-3344	62	16	1	1	NUM
ejpam-3344	62	17	]	]	PUNCT
ejpam-3344	62	18	and	and	CCONJ
ejpam-3344	62	19	γa	γa	PRON
ejpam-3344	62	20	:	:	PUNCT
ejpam-3344	62	21	x	x	X
ejpam-3344	62	22	→	→	PUNCT
ejpam-3344	62	23	[	[	X
ejpam-3344	62	24	0	0	NUM
ejpam-3344	62	25	,	,	PUNCT
ejpam-3344	62	26	1	1	NUM
ejpam-3344	62	27	]	]	PUNCT
ejpam-3344	62	28	denote	denote	VERB
ejpam-3344	62	29	the	the	DET
ejpam-3344	62	30	degree	degree	NOUN
ejpam-3344	62	31	of	of	ADP
ejpam-3344	62	32	membership	membership	NOUN
ejpam-3344	62	33	and	and	CCONJ
ejpam-3344	62	34	the	the	DET
ejpam-3344	62	35	degree	degree	NOUN
ejpam-3344	62	36	of	of	ADP
ejpam-3344	62	37	nonmembership	nonmembership	NOUN
ejpam-3344	62	38	,	,	PUNCT
ejpam-3344	62	39	respectively	respectively	ADV
ejpam-3344	62	40	and	and	CCONJ
ejpam-3344	62	41	0	0	NUM
ejpam-3344	62	42	≤	≤	NOUN
ejpam-3344	62	43	µa(x	µa(x	NOUN
ejpam-3344	62	44	)	)	PUNCT
ejpam-3344	62	45	+	+	CCONJ
ejpam-3344	62	46	γa(x	γa(x	X
ejpam-3344	62	47	)	)	PUNCT
ejpam-3344	62	48	≤	≤	NUM
ejpam-3344	62	49	1	1	NUM
ejpam-3344	62	50	for	for	ADP
ejpam-3344	62	51	all	all	DET
ejpam-3344	62	52	x	x	SYM
ejpam-3344	62	53	∈	∈	NOUN
ejpam-3344	62	54	x	x	PUNCT
ejpam-3344	63	1	[	[	X
ejpam-3344	63	2	1	1	NUM
ejpam-3344	63	3	,	,	PUNCT
ejpam-3344	63	4	2	2	NUM
ejpam-3344	63	5	]	]	PUNCT
ejpam-3344	63	6	.	.	PUNCT
ejpam-3344	64	1	an	an	DET
ejpam-3344	64	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	64	3	fuzzy	fuzzy	NOUN
ejpam-3344	64	4	set	set	VERB
ejpam-3344	64	5	a	a	PRON
ejpam-3344	64	6	=	=	X
ejpam-3344	64	7	{	{	PUNCT
ejpam-3344	64	8	(	(	PUNCT
ejpam-3344	64	9	x	x	NOUN
ejpam-3344	64	10	,	,	PUNCT
ejpam-3344	64	11	µa(x	µa(x	NOUN
ejpam-3344	64	12	)	)	PUNCT
ejpam-3344	64	13	,	,	PUNCT
ejpam-3344	64	14	γa(x	γa(x	NUM
ejpam-3344	64	15	)	)	PUNCT
ejpam-3344	64	16	)	)	PUNCT
ejpam-3344	64	17	:	:	PUNCT
ejpam-3344	65	1	x	x	X
ejpam-3344	65	2	∈	∈	NOUN
ejpam-3344	65	3	x	x	X
ejpam-3344	65	4	}	}	PUNCT
ejpam-3344	65	5	in	in	ADP
ejpam-3344	65	6	x	x	PRON
ejpam-3344	65	7	can	can	AUX
ejpam-3344	65	8	be	be	AUX
ejpam-3344	65	9	identified	identify	VERB
ejpam-3344	65	10	to	to	PART
ejpam-3344	65	11	be	be	AUX
ejpam-3344	65	12	an	an	DET
ejpam-3344	65	13	ordered	order	VERB
ejpam-3344	65	14	pair	pair	NOUN
ejpam-3344	65	15	(	(	PUNCT
ejpam-3344	65	16	µa	µa	NOUN
ejpam-3344	65	17	,	,	PUNCT
ejpam-3344	65	18	γa	γa	NOUN
ejpam-3344	65	19	)	)	PUNCT
ejpam-3344	65	20	in	in	ADP
ejpam-3344	65	21	ix	ix	ADP
ejpam-3344	65	22	×	×	NOUN
ejpam-3344	65	23	ix	ix	ADV
ejpam-3344	65	24	,	,	PUNCT
ejpam-3344	65	25	where	where	SCONJ
ejpam-3344	65	26	ix	ix	ADV
ejpam-3344	65	27	is	be	AUX
ejpam-3344	65	28	the	the	DET
ejpam-3344	65	29	set	set	NOUN
ejpam-3344	65	30	of	of	ADP
ejpam-3344	65	31	all	all	DET
ejpam-3344	65	32	functions	function	NOUN
ejpam-3344	65	33	from	from	ADP
ejpam-3344	65	34	x	x	PUNCT
ejpam-3344	65	35	to	to	ADP
ejpam-3344	65	36	[	[	X
ejpam-3344	65	37	0	0	NUM
ejpam-3344	65	38	,	,	PUNCT
ejpam-3344	65	39	1	1	NUM
ejpam-3344	65	40	]	]	PUNCT
ejpam-3344	65	41	.	.	PUNCT
ejpam-3344	66	1	for	for	ADP
ejpam-3344	66	2	the	the	DET
ejpam-3344	66	3	sake	sake	NOUN
ejpam-3344	66	4	of	of	ADP
ejpam-3344	66	5	simplicity	simplicity	NOUN
ejpam-3344	66	6	,	,	PUNCT
ejpam-3344	66	7	we	we	PRON
ejpam-3344	66	8	shall	shall	AUX
ejpam-3344	66	9	use	use	VERB
ejpam-3344	66	10	the	the	DET
ejpam-3344	66	11	symbol	symbol	NOUN
ejpam-3344	66	12	a	a	DET
ejpam-3344	66	13	=	=	X
ejpam-3344	66	14	(	(	PUNCT
ejpam-3344	66	15	µa	µa	PROPN
ejpam-3344	66	16	,	,	PUNCT
ejpam-3344	66	17	γa	γa	PROPN
ejpam-3344	66	18	)	)	PUNCT
ejpam-3344	66	19	for	for	ADP
ejpam-3344	66	20	the	the	DET
ejpam-3344	66	21	ifs	ifs	PROPN
ejpam-3344	66	22	a	a	X
ejpam-3344	66	23	=	=	X
ejpam-3344	66	24	{	{	PUNCT
ejpam-3344	66	25	(	(	PUNCT
ejpam-3344	66	26	x	x	NOUN
ejpam-3344	66	27	,	,	PUNCT
ejpam-3344	66	28	µa(x	µa(x	NOUN
ejpam-3344	66	29	)	)	PUNCT
ejpam-3344	66	30	,	,	PUNCT
ejpam-3344	66	31	γa(x	γa(x	NUM
ejpam-3344	66	32	)	)	PUNCT
ejpam-3344	66	33	)	)	PUNCT
ejpam-3344	66	34	:	:	PUNCT
ejpam-3344	67	1	x	x	X
ejpam-3344	67	2	∈	∈	NOUN
ejpam-3344	67	3	x	x	X
ejpam-3344	67	4	}	}	PUNCT
ejpam-3344	67	5	.	.	PUNCT
ejpam-3344	68	1	n.	n.	PROPN
ejpam-3344	68	2	kausar	kausar	PROPN
ejpam-3344	68	3	,	,	PUNCT
ejpam-3344	68	4	m.	m.	NOUN
ejpam-3344	68	5	a.	a.	PROPN
ejpam-3344	68	6	waqar	waqar	PROPN
ejpam-3344	68	7	/	/	SYM
ejpam-3344	68	8	eur	eur	PROPN
ejpam-3344	68	9	.	.	PUNCT
ejpam-3344	69	1	j.	j.	PROPN
ejpam-3344	69	2	pure	pure	PROPN
ejpam-3344	69	3	appl	appl	PROPN
ejpam-3344	69	4	.	.	PROPN
ejpam-3344	69	5	math	math	PROPN
ejpam-3344	69	6	,	,	PUNCT
ejpam-3344	69	7	12	12	NUM
ejpam-3344	69	8	(	(	PUNCT
ejpam-3344	69	9	1	1	NUM
ejpam-3344	69	10	)	)	PUNCT
ejpam-3344	69	11	(	(	PUNCT
ejpam-3344	69	12	2019	2019	NUM
ejpam-3344	69	13	)	)	PUNCT
ejpam-3344	69	14	,	,	PUNCT
ejpam-3344	69	15	226	226	NUM
ejpam-3344	69	16	-	-	SYM
ejpam-3344	69	17	250	250	NUM
ejpam-3344	69	18	228	228	NUM
ejpam-3344	69	19	banerjee	banerjee	NOUN
ejpam-3344	69	20	et	et	NOUN
ejpam-3344	69	21	al	al	PROPN
ejpam-3344	70	1	[	[	X
ejpam-3344	70	2	3	3	NUM
ejpam-3344	70	3	]	]	PUNCT
ejpam-3344	70	4	and	and	CCONJ
ejpam-3344	70	5	hur	hur	PROPN
ejpam-3344	70	6	et	et	PROPN
ejpam-3344	70	7	al	al	PROPN
ejpam-3344	70	8	[	[	X
ejpam-3344	70	9	7	7	NUM
ejpam-3344	70	10	]	]	PUNCT
ejpam-3344	70	11	,	,	PUNCT
ejpam-3344	70	12	initiated	initiate	VERB
ejpam-3344	70	13	the	the	DET
ejpam-3344	70	14	notion	notion	NOUN
ejpam-3344	70	15	of	of	ADP
ejpam-3344	70	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	70	17	fuzzy	fuzzy	ADJ
ejpam-3344	70	18	subrings	subring	NOUN
ejpam-3344	70	19	and	and	CCONJ
ejpam-3344	70	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	70	21	fuzzy	fuzzy	ADJ
ejpam-3344	70	22	ideals	ideal	NOUN
ejpam-3344	70	23	of	of	ADP
ejpam-3344	70	24	a	a	DET
ejpam-3344	70	25	ring	ring	NOUN
ejpam-3344	70	26	.	.	PUNCT
ejpam-3344	71	1	subsequently	subsequently	ADV
ejpam-3344	71	2	many	many	ADJ
ejpam-3344	71	3	authors	author	NOUN
ejpam-3344	71	4	studied	study	VERB
ejpam-3344	71	5	the	the	DET
ejpam-3344	71	6	intuitionistic	intuitionistic	ADJ
ejpam-3344	71	7	fuzzy	fuzzy	ADJ
ejpam-3344	71	8	subrings	subring	NOUN
ejpam-3344	71	9	and	and	CCONJ
ejpam-3344	71	10	intuitionistic	intuitionistic	ADJ
ejpam-3344	71	11	fuzzy	fuzzy	ADJ
ejpam-3344	71	12	ideals	ideal	NOUN
ejpam-3344	71	13	of	of	ADP
ejpam-3344	71	14	a	a	DET
ejpam-3344	71	15	ring	ring	NOUN
ejpam-3344	71	16	by	by	ADP
ejpam-3344	71	17	describing	describe	VERB
ejpam-3344	71	18	the	the	DET
ejpam-3344	71	19	different	different	ADJ
ejpam-3344	71	20	properties	property	NOUN
ejpam-3344	71	21	(	(	PUNCT
ejpam-3344	71	22	see	see	VERB
ejpam-3344	71	23	[	[	X
ejpam-3344	71	24	8	8	NUM
ejpam-3344	71	25	]	]	NUM
ejpam-3344	71	26	)	)	PUNCT
ejpam-3344	71	27	.	.	PUNCT
ejpam-3344	72	1	shah	shah	PROPN
ejpam-3344	72	2	et	et	PROPN
ejpam-3344	72	3	al	al	PROPN
ejpam-3344	73	1	[	[	X
ejpam-3344	73	2	18	18	NUM
ejpam-3344	73	3	]	]	PUNCT
ejpam-3344	73	4	,	,	PUNCT
ejpam-3344	73	5	have	have	AUX
ejpam-3344	73	6	initiated	initiate	VERB
ejpam-3344	73	7	the	the	DET
ejpam-3344	73	8	concept	concept	NOUN
ejpam-3344	73	9	of	of	ADP
ejpam-3344	73	10	intuitionistic	intuitionistic	ADJ
ejpam-3344	73	11	fuzzy	fuzzy	ADJ
ejpam-3344	73	12	normal	normal	ADJ
ejpam-3344	73	13	la	la	NOUN
ejpam-3344	73	14	-	-	PUNCT
ejpam-3344	73	15	subrings	subring	NOUN
ejpam-3344	73	16	of	of	ADP
ejpam-3344	73	17	an	an	DET
ejpam-3344	73	18	la	la	NOUN
ejpam-3344	73	19	-	-	PUNCT
ejpam-3344	73	20	ring	ring	NOUN
ejpam-3344	73	21	.	.	PUNCT
ejpam-3344	74	1	we	we	PRON
ejpam-3344	74	2	initiate	initiate	VERB
ejpam-3344	74	3	the	the	DET
ejpam-3344	74	4	notion	notion	NOUN
ejpam-3344	74	5	of	of	ADP
ejpam-3344	74	6	intuitionistic	intuitionistic	ADJ
ejpam-3344	74	7	fuzzy	fuzzy	ADJ
ejpam-3344	74	8	left	left	NOUN
ejpam-3344	74	9	(	(	PUNCT
ejpam-3344	74	10	resp	resp	NOUN
ejpam-3344	74	11	.	.	PUNCT
ejpam-3344	75	1	right	right	ADJ
ejpam-3344	75	2	,	,	PUNCT
ejpam-3344	75	3	bi-	bi-	NUM
ejpam-3344	75	4	,	,	PUNCT
ejpam-3344	75	5	generalized	generalized	ADJ
ejpam-3344	75	6	bi-,(1	bi-,(1	PROPN
ejpam-3344	75	7	,	,	PUNCT
ejpam-3344	75	8	2	2	NUM
ejpam-3344	75	9	)	)	PUNCT
ejpam-3344	75	10	)	)	PUNCT
ejpam-3344	75	11	ideals	ideal	NOUN
ejpam-3344	75	12	of	of	ADP
ejpam-3344	75	13	an	an	DET
ejpam-3344	75	14	la	la	ADJ
ejpam-3344	75	15	-	-	PUNCT
ejpam-3344	75	16	ring	ring	NOUN
ejpam-3344	75	17	r.	r.	NOUN
ejpam-3344	76	1	[	[	X
ejpam-3344	76	2	18	18	NUM
ejpam-3344	76	3	]	]	PUNCT
ejpam-3344	76	4	an	an	DET
ejpam-3344	76	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	76	6	fuzzy	fuzzy	ADJ
ejpam-3344	76	7	set	set	NOUN
ejpam-3344	76	8	(	(	PUNCT
ejpam-3344	76	9	ifs	ifs	PROPN
ejpam-3344	76	10	)	)	PUNCT
ejpam-3344	76	11	a	a	DET
ejpam-3344	76	12	=	=	SYM
ejpam-3344	76	13	(	(	PUNCT
ejpam-3344	76	14	µa	µa	PROPN
ejpam-3344	76	15	,	,	PUNCT
ejpam-3344	76	16	γa	γa	PROPN
ejpam-3344	76	17	)	)	PUNCT
ejpam-3344	76	18	of	of	ADP
ejpam-3344	76	19	an	an	DET
ejpam-3344	76	20	la	la	ADJ
ejpam-3344	76	21	-	-	PUNCT
ejpam-3344	76	22	ring	ring	NOUN
ejpam-3344	76	23	r	r	NOUN
ejpam-3344	76	24	is	be	AUX
ejpam-3344	76	25	called	call	VERB
ejpam-3344	76	26	an	an	DET
ejpam-3344	76	27	intuitionistic	intuitionistic	ADJ
ejpam-3344	76	28	fuzzy	fuzzy	ADJ
ejpam-3344	76	29	la	la	NOUN
ejpam-3344	76	30	-	-	PUNCT
ejpam-3344	76	31	subring	subring	NOUN
ejpam-3344	76	32	of	of	ADP
ejpam-3344	76	33	r	r	NOUN
ejpam-3344	76	34	if	if	SCONJ
ejpam-3344	76	35	(	(	PUNCT
ejpam-3344	76	36	1	1	X
ejpam-3344	76	37	)	)	PUNCT
ejpam-3344	76	38	µa	µa	NOUN
ejpam-3344	76	39	(	(	PUNCT
ejpam-3344	76	40	x−	x−	PROPN
ejpam-3344	76	41	y	y	PROPN
ejpam-3344	76	42	)	)	PUNCT
ejpam-3344	76	43	≥	≥	NOUN
ejpam-3344	76	44	min{µa	min{µa	X
ejpam-3344	76	45	(	(	PUNCT
ejpam-3344	76	46	x	x	NOUN
ejpam-3344	76	47	)	)	PUNCT
ejpam-3344	76	48	,	,	PUNCT
ejpam-3344	76	49	µa(y	µa(y	NOUN
ejpam-3344	76	50	)	)	PUNCT
ejpam-3344	76	51	}	}	PUNCT
ejpam-3344	76	52	,	,	PUNCT
ejpam-3344	76	53	(	(	PUNCT
ejpam-3344	76	54	2	2	X
ejpam-3344	76	55	)	)	PUNCT
ejpam-3344	76	56	γa	γa	NOUN
ejpam-3344	76	57	(	(	PUNCT
ejpam-3344	76	58	x−	x−	PROPN
ejpam-3344	76	59	y	y	PROPN
ejpam-3344	76	60	)	)	PUNCT
ejpam-3344	76	61	≤	≤	NUM
ejpam-3344	76	62	max{γa	max{γa	PUNCT
ejpam-3344	76	63	(	(	PUNCT
ejpam-3344	76	64	x	x	NOUN
ejpam-3344	76	65	)	)	PUNCT
ejpam-3344	76	66	,	,	PUNCT
ejpam-3344	76	67	γa(y	γa(y	NOUN
ejpam-3344	76	68	)	)	PUNCT
ejpam-3344	76	69	}	}	PUNCT
ejpam-3344	76	70	,	,	PUNCT
ejpam-3344	76	71	(	(	PUNCT
ejpam-3344	76	72	3	3	X
ejpam-3344	76	73	)	)	PUNCT
ejpam-3344	76	74	µa	µa	NOUN
ejpam-3344	76	75	(	(	PUNCT
ejpam-3344	76	76	xy	xy	NOUN
ejpam-3344	76	77	)	)	PUNCT
ejpam-3344	76	78	≥	≥	NOUN
ejpam-3344	76	79	min{µa	min{µa	X
ejpam-3344	76	80	(	(	PUNCT
ejpam-3344	76	81	x	x	NOUN
ejpam-3344	76	82	)	)	PUNCT
ejpam-3344	76	83	,	,	PUNCT
ejpam-3344	76	84	µa	µa	ADP
ejpam-3344	76	85	(	(	PUNCT
ejpam-3344	76	86	y	y	NOUN
ejpam-3344	76	87	)	)	PUNCT
ejpam-3344	76	88	}	}	PUNCT
ejpam-3344	76	89	,	,	PUNCT
ejpam-3344	76	90	(	(	PUNCT
ejpam-3344	76	91	4	4	X
ejpam-3344	76	92	)	)	PUNCT
ejpam-3344	76	93	γa	γa	NOUN
ejpam-3344	76	94	(	(	PUNCT
ejpam-3344	76	95	xy	xy	NOUN
ejpam-3344	76	96	)	)	PUNCT
ejpam-3344	76	97	≤	≤	NUM
ejpam-3344	76	98	max{γa	max{γa	PUNCT
ejpam-3344	76	99	(	(	PUNCT
ejpam-3344	76	100	x	x	X
ejpam-3344	76	101	)	)	PUNCT
ejpam-3344	76	102	,	,	PUNCT
ejpam-3344	76	103	γa	γa	PROPN
ejpam-3344	76	104	(	(	PUNCT
ejpam-3344	76	105	y	y	NOUN
ejpam-3344	76	106	)	)	PUNCT
ejpam-3344	76	107	}	}	PUNCT
ejpam-3344	76	108	for	for	ADP
ejpam-3344	76	109	all	all	DET
ejpam-3344	76	110	x	x	NOUN
ejpam-3344	76	111	,	,	PUNCT
ejpam-3344	76	112	y	y	PROPN
ejpam-3344	76	113	∈	∈	PROPN
ejpam-3344	76	114	r.	r.	PROPN
ejpam-3344	76	115	an	an	DET
ejpam-3344	76	116	ifs	ifs	PROPN
ejpam-3344	76	117	a	a	X
ejpam-3344	76	118	=	=	X
ejpam-3344	76	119	(	(	PUNCT
ejpam-3344	76	120	µa	µa	PROPN
ejpam-3344	76	121	,	,	PUNCT
ejpam-3344	76	122	γa	γa	PROPN
ejpam-3344	76	123	)	)	PUNCT
ejpam-3344	76	124	of	of	ADP
ejpam-3344	76	125	an	an	DET
ejpam-3344	76	126	la	la	ADJ
ejpam-3344	76	127	-	-	PUNCT
ejpam-3344	76	128	ring	ring	NOUN
ejpam-3344	76	129	r	r	NOUN
ejpam-3344	76	130	is	be	AUX
ejpam-3344	76	131	called	call	VERB
ejpam-3344	76	132	an	an	DET
ejpam-3344	76	133	intuitionistic	intuitionistic	ADJ
ejpam-3344	76	134	fuzzy	fuzzy	ADJ
ejpam-3344	76	135	left	leave	VERB
ejpam-3344	76	136	ideal	ideal	NOUN
ejpam-3344	76	137	of	of	ADP
ejpam-3344	76	138	r	r	NOUN
ejpam-3344	76	139	if	if	SCONJ
ejpam-3344	76	140	(	(	PUNCT
ejpam-3344	76	141	1	1	X
ejpam-3344	76	142	)	)	PUNCT
ejpam-3344	76	143	µa	µa	NOUN
ejpam-3344	76	144	(	(	PUNCT
ejpam-3344	76	145	x−	x−	PROPN
ejpam-3344	76	146	y	y	PROPN
ejpam-3344	76	147	)	)	PUNCT
ejpam-3344	76	148	≥	≥	NOUN
ejpam-3344	76	149	min{µa	min{µa	X
ejpam-3344	76	150	(	(	PUNCT
ejpam-3344	76	151	x	x	NOUN
ejpam-3344	76	152	)	)	PUNCT
ejpam-3344	76	153	,	,	PUNCT
ejpam-3344	76	154	µa(y	µa(y	NOUN
ejpam-3344	76	155	)	)	PUNCT
ejpam-3344	76	156	}	}	PUNCT
ejpam-3344	76	157	,	,	PUNCT
ejpam-3344	76	158	(	(	PUNCT
ejpam-3344	76	159	2	2	X
ejpam-3344	76	160	)	)	PUNCT
ejpam-3344	76	161	γa	γa	NOUN
ejpam-3344	76	162	(	(	PUNCT
ejpam-3344	76	163	x−	x−	PROPN
ejpam-3344	76	164	y	y	PROPN
ejpam-3344	76	165	)	)	PUNCT
ejpam-3344	76	166	≤	≤	NUM
ejpam-3344	76	167	max{γa	max{γa	PUNCT
ejpam-3344	76	168	(	(	PUNCT
ejpam-3344	76	169	x	x	NOUN
ejpam-3344	76	170	)	)	PUNCT
ejpam-3344	76	171	,	,	PUNCT
ejpam-3344	76	172	γa(y	γa(y	NOUN
ejpam-3344	76	173	)	)	PUNCT
ejpam-3344	76	174	}	}	PUNCT
ejpam-3344	76	175	,	,	PUNCT
ejpam-3344	76	176	(	(	PUNCT
ejpam-3344	76	177	3	3	X
ejpam-3344	76	178	)	)	PUNCT
ejpam-3344	76	179	µa	µa	NOUN
ejpam-3344	76	180	(	(	PUNCT
ejpam-3344	76	181	xy	xy	NOUN
ejpam-3344	76	182	)	)	PUNCT
ejpam-3344	76	183	≥	≥	PROPN
ejpam-3344	76	184	µa	µa	NOUN
ejpam-3344	76	185	(	(	PUNCT
ejpam-3344	76	186	y	y	NOUN
ejpam-3344	76	187	)	)	PUNCT
ejpam-3344	76	188	,	,	PUNCT
ejpam-3344	76	189	(	(	PUNCT
ejpam-3344	76	190	4	4	X
ejpam-3344	76	191	)	)	PUNCT
ejpam-3344	76	192	γa	γa	NOUN
ejpam-3344	76	193	(	(	PUNCT
ejpam-3344	76	194	xy	xy	PROPN
ejpam-3344	76	195	)	)	PUNCT
ejpam-3344	76	196	≤	≤	NOUN
ejpam-3344	76	197	γa	γa	PROPN
ejpam-3344	76	198	(	(	PUNCT
ejpam-3344	76	199	y	y	NOUN
ejpam-3344	76	200	)	)	PUNCT
ejpam-3344	76	201	for	for	ADP
ejpam-3344	76	202	all	all	DET
ejpam-3344	76	203	x	x	NOUN
ejpam-3344	76	204	,	,	PUNCT
ejpam-3344	76	205	y	y	PROPN
ejpam-3344	76	206	∈	∈	PROPN
ejpam-3344	76	207	r.	r.	PROPN
ejpam-3344	76	208	an	an	DET
ejpam-3344	76	209	ifs	ifs	PROPN
ejpam-3344	76	210	a	a	X
ejpam-3344	76	211	=	=	X
ejpam-3344	76	212	(	(	PUNCT
ejpam-3344	76	213	µa	µa	PROPN
ejpam-3344	76	214	,	,	PUNCT
ejpam-3344	76	215	γa	γa	PROPN
ejpam-3344	76	216	)	)	PUNCT
ejpam-3344	76	217	of	of	ADP
ejpam-3344	76	218	an	an	DET
ejpam-3344	76	219	la	la	ADJ
ejpam-3344	76	220	-	-	PUNCT
ejpam-3344	76	221	ring	ring	NOUN
ejpam-3344	76	222	r	r	NOUN
ejpam-3344	76	223	is	be	AUX
ejpam-3344	76	224	called	call	VERB
ejpam-3344	76	225	an	an	DET
ejpam-3344	76	226	intuitionistic	intuitionistic	ADJ
ejpam-3344	76	227	fuzzy	fuzzy	ADJ
ejpam-3344	76	228	right	right	ADJ
ejpam-3344	76	229	ideal	ideal	NOUN
ejpam-3344	76	230	of	of	ADP
ejpam-3344	76	231	r	r	NOUN
ejpam-3344	76	232	if	if	SCONJ
ejpam-3344	76	233	(	(	PUNCT
ejpam-3344	76	234	1	1	X
ejpam-3344	76	235	)	)	PUNCT
ejpam-3344	76	236	µa	µa	NOUN
ejpam-3344	76	237	(	(	PUNCT
ejpam-3344	76	238	x−	x−	PROPN
ejpam-3344	76	239	y	y	PROPN
ejpam-3344	76	240	)	)	PUNCT
ejpam-3344	76	241	≥	≥	NOUN
ejpam-3344	76	242	min{µa	min{µa	X
ejpam-3344	76	243	(	(	PUNCT
ejpam-3344	76	244	x	x	NOUN
ejpam-3344	76	245	)	)	PUNCT
ejpam-3344	76	246	,	,	PUNCT
ejpam-3344	76	247	µa(y	µa(y	NOUN
ejpam-3344	76	248	)	)	PUNCT
ejpam-3344	76	249	}	}	PUNCT
ejpam-3344	76	250	,	,	PUNCT
ejpam-3344	76	251	(	(	PUNCT
ejpam-3344	76	252	2	2	X
ejpam-3344	76	253	)	)	PUNCT
ejpam-3344	76	254	γa	γa	NOUN
ejpam-3344	76	255	(	(	PUNCT
ejpam-3344	76	256	x−	x−	PROPN
ejpam-3344	76	257	y	y	PROPN
ejpam-3344	76	258	)	)	PUNCT
ejpam-3344	76	259	≤	≤	NUM
ejpam-3344	76	260	max{γa	max{γa	PUNCT
ejpam-3344	76	261	(	(	PUNCT
ejpam-3344	76	262	x	x	NOUN
ejpam-3344	76	263	)	)	PUNCT
ejpam-3344	76	264	,	,	PUNCT
ejpam-3344	76	265	γa(y	γa(y	NOUN
ejpam-3344	76	266	)	)	PUNCT
ejpam-3344	76	267	}	}	PUNCT
ejpam-3344	76	268	,	,	PUNCT
ejpam-3344	76	269	(	(	PUNCT
ejpam-3344	76	270	3	3	X
ejpam-3344	76	271	)	)	PUNCT
ejpam-3344	76	272	µa	µa	NOUN
ejpam-3344	76	273	(	(	PUNCT
ejpam-3344	76	274	xy	xy	NOUN
ejpam-3344	76	275	)	)	PUNCT
ejpam-3344	76	276	≥	≥	PROPN
ejpam-3344	76	277	µa	µa	NOUN
ejpam-3344	76	278	(	(	PUNCT
ejpam-3344	76	279	x	x	X
ejpam-3344	76	280	)	)	PUNCT
ejpam-3344	76	281	,	,	PUNCT
ejpam-3344	76	282	(	(	PUNCT
ejpam-3344	76	283	4	4	X
ejpam-3344	76	284	)	)	PUNCT
ejpam-3344	76	285	γa	γa	NOUN
ejpam-3344	76	286	(	(	PUNCT
ejpam-3344	76	287	xy	xy	PROPN
ejpam-3344	76	288	)	)	PUNCT
ejpam-3344	76	289	≤	≤	NOUN
ejpam-3344	76	290	γa	γa	NOUN
ejpam-3344	76	291	(	(	PUNCT
ejpam-3344	76	292	x	x	X
ejpam-3344	76	293	)	)	PUNCT
ejpam-3344	76	294	for	for	ADP
ejpam-3344	76	295	all	all	DET
ejpam-3344	76	296	x	x	NOUN
ejpam-3344	76	297	,	,	PUNCT
ejpam-3344	76	298	y	y	PROPN
ejpam-3344	76	299	∈	∈	PROPN
ejpam-3344	76	300	r.	r.	PROPN
ejpam-3344	76	301	an	an	DET
ejpam-3344	76	302	ifs	ifs	PROPN
ejpam-3344	76	303	a	a	X
ejpam-3344	76	304	=	=	X
ejpam-3344	76	305	(	(	PUNCT
ejpam-3344	76	306	µa	µa	PROPN
ejpam-3344	76	307	,	,	PUNCT
ejpam-3344	76	308	γa	γa	PROPN
ejpam-3344	76	309	)	)	PUNCT
ejpam-3344	76	310	of	of	ADP
ejpam-3344	76	311	an	an	DET
ejpam-3344	76	312	la	la	ADJ
ejpam-3344	76	313	-	-	PUNCT
ejpam-3344	76	314	ring	ring	NOUN
ejpam-3344	76	315	r	r	NOUN
ejpam-3344	76	316	is	be	AUX
ejpam-3344	76	317	called	call	VERB
ejpam-3344	76	318	an	an	DET
ejpam-3344	76	319	intuitionistic	intuitionistic	ADJ
ejpam-3344	76	320	fuzzy	fuzzy	ADJ
ejpam-3344	76	321	ideal	ideal	NOUN
ejpam-3344	76	322	of	of	ADP
ejpam-3344	76	323	r	r	NOUN
ejpam-3344	76	324	if	if	SCONJ
ejpam-3344	76	325	it	it	PRON
ejpam-3344	76	326	is	be	AUX
ejpam-3344	76	327	both	both	CCONJ
ejpam-3344	76	328	an	an	DET
ejpam-3344	76	329	intuitionistic	intuitionistic	ADJ
ejpam-3344	76	330	fuzzy	fuzzy	ADJ
ejpam-3344	76	331	left	leave	VERB
ejpam-3344	76	332	ideal	ideal	NOUN
ejpam-3344	76	333	and	and	CCONJ
ejpam-3344	76	334	an	an	DET
ejpam-3344	76	335	intuitionistic	intuitionistic	ADJ
ejpam-3344	76	336	fuzzy	fuzzy	ADJ
ejpam-3344	76	337	right	right	ADJ
ejpam-3344	76	338	ideal	ideal	NOUN
ejpam-3344	76	339	of	of	ADP
ejpam-3344	76	340	r.	r.	PROPN
ejpam-3344	76	341	example	example	NOUN
ejpam-3344	77	1	1	1	NUM
ejpam-3344	77	2	.	.	PUNCT
ejpam-3344	78	1	let	let	VERB
ejpam-3344	78	2	r	r	NOUN
ejpam-3344	78	3	=	=	PUNCT
ejpam-3344	78	4	{	{	PUNCT
ejpam-3344	78	5	a	a	PRON
ejpam-3344	78	6	,	,	PUNCT
ejpam-3344	78	7	b	b	NOUN
ejpam-3344	78	8	,	,	PUNCT
ejpam-3344	78	9	c	c	NOUN
ejpam-3344	78	10	,	,	PUNCT
ejpam-3344	78	11	d	d	NOUN
ejpam-3344	78	12	}	}	PUNCT
ejpam-3344	78	13	.	.	PUNCT
ejpam-3344	79	1	define	define	VERB
ejpam-3344	79	2	+	+	CCONJ
ejpam-3344	79	3	and	and	CCONJ
ejpam-3344	79	4	·	·	PUNCT
ejpam-3344	79	5	in	in	ADP
ejpam-3344	79	6	r	r	NOUN
ejpam-3344	79	7	as	as	SCONJ
ejpam-3344	79	8	follows	follow	VERB
ejpam-3344	79	9	:	:	PUNCT
ejpam-3344	79	10	+	+	CCONJ
ejpam-3344	79	11	a	a	DET
ejpam-3344	79	12	b	b	NOUN
ejpam-3344	79	13	c	c	NOUN
ejpam-3344	79	14	d	d	NOUN
ejpam-3344	79	15	a	a	PRON
ejpam-3344	79	16	a	a	DET
ejpam-3344	79	17	b	b	NOUN
ejpam-3344	79	18	c	c	NOUN
ejpam-3344	79	19	d	d	PROPN
ejpam-3344	79	20	b	b	PROPN
ejpam-3344	79	21	d	d	NOUN
ejpam-3344	79	22	a	a	PRON
ejpam-3344	79	23	b	b	NOUN
ejpam-3344	79	24	c	c	NOUN
ejpam-3344	79	25	c	c	NOUN
ejpam-3344	79	26	c	c	NOUN
ejpam-3344	80	1	d	d	NOUN
ejpam-3344	80	2	a	a	DET
ejpam-3344	80	3	b	b	PROPN
ejpam-3344	80	4	d	d	X
ejpam-3344	80	5	b	b	PROPN
ejpam-3344	80	6	c	c	PROPN
ejpam-3344	80	7	d	d	NOUN
ejpam-3344	80	8	a	a	PROPN
ejpam-3344	80	9	and	and	CCONJ
ejpam-3344	80	10	·	·	PUNCT
ejpam-3344	80	11	a	a	DET
ejpam-3344	80	12	b	b	X
ejpam-3344	80	13	c	c	NOUN
ejpam-3344	80	14	d	d	NOUN
ejpam-3344	80	15	a	a	PRON
ejpam-3344	80	16	a	a	PRON
ejpam-3344	80	17	a	a	DET
ejpam-3344	80	18	a	a	DET
ejpam-3344	80	19	a	a	DET
ejpam-3344	80	20	b	b	NOUN
ejpam-3344	80	21	a	a	DET
ejpam-3344	80	22	b	b	NOUN
ejpam-3344	80	23	a	a	DET
ejpam-3344	80	24	b	b	NOUN
ejpam-3344	80	25	c	c	NOUN
ejpam-3344	80	26	a	a	DET
ejpam-3344	80	27	a	a	NOUN
ejpam-3344	80	28	c	c	NOUN
ejpam-3344	80	29	c	c	NOUN
ejpam-3344	81	1	d	d	NOUN
ejpam-3344	81	2	a	a	PRON
ejpam-3344	81	3	b	b	NOUN
ejpam-3344	81	4	c	c	NOUN
ejpam-3344	81	5	d	d	NOUN
ejpam-3344	81	6	then	then	ADV
ejpam-3344	81	7	r	r	NOUN
ejpam-3344	81	8	is	be	AUX
ejpam-3344	81	9	an	an	DET
ejpam-3344	81	10	la	la	ADJ
ejpam-3344	81	11	-	-	PUNCT
ejpam-3344	81	12	ring	ring	NOUN
ejpam-3344	81	13	and	and	CCONJ
ejpam-3344	81	14	a	a	DET
ejpam-3344	81	15	=	=	X
ejpam-3344	81	16	(	(	PUNCT
ejpam-3344	81	17	µa	µa	PROPN
ejpam-3344	81	18	,	,	PUNCT
ejpam-3344	81	19	γa	γa	PROPN
ejpam-3344	81	20	)	)	PUNCT
ejpam-3344	81	21	be	be	VERB
ejpam-3344	81	22	an	an	DET
ejpam-3344	81	23	ifs	ifs	PROPN
ejpam-3344	81	24	of	of	ADP
ejpam-3344	81	25	r.	r.	PROPN
ejpam-3344	81	26	we	we	PRON
ejpam-3344	81	27	define	define	VERB
ejpam-3344	81	28	µa(a	µa(a	PUNCT
ejpam-3344	81	29	)	)	PUNCT
ejpam-3344	81	30	=	=	PUNCT
ejpam-3344	82	1	µa(c	µa(c	X
ejpam-3344	82	2	)	)	PUNCT
ejpam-3344	82	3	=	=	SYM
ejpam-3344	82	4	0.7	0.7	NUM
ejpam-3344	82	5	,	,	PUNCT
ejpam-3344	82	6	µa(b	µa(b	NUM
ejpam-3344	82	7	)	)	PUNCT
ejpam-3344	82	8	=	=	SYM
ejpam-3344	82	9	µa(d	µa(d	X
ejpam-3344	82	10	)	)	PUNCT
ejpam-3344	82	11	=	=	SYM
ejpam-3344	82	12	0	0	NUM
ejpam-3344	82	13	and	and	CCONJ
ejpam-3344	82	14	γa(a	γa(a	NUM
ejpam-3344	82	15	)	)	PUNCT
ejpam-3344	83	1	=	=	PUNCT
ejpam-3344	83	2	γa(c	γa(c	X
ejpam-3344	83	3	)	)	PUNCT
ejpam-3344	83	4	=	=	SYM
ejpam-3344	83	5	0	0	NUM
ejpam-3344	83	6	,	,	PUNCT
ejpam-3344	83	7	γa(b	γa(b	NUM
ejpam-3344	83	8	)	)	PUNCT
ejpam-3344	83	9	=	=	SYM
ejpam-3344	83	10	γa(d	γa(d	X
ejpam-3344	83	11	)	)	PUNCT
ejpam-3344	83	12	=	=	PUNCT
ejpam-3344	84	1	0.7	0.7	NUM
ejpam-3344	84	2	.	.	PUNCT
ejpam-3344	85	1	then	then	ADV
ejpam-3344	85	2	a	a	DET
ejpam-3344	85	3	=	=	SYM
ejpam-3344	85	4	(	(	PUNCT
ejpam-3344	85	5	µa	µa	PROPN
ejpam-3344	85	6	,	,	PUNCT
ejpam-3344	85	7	γa	γa	PROPN
ejpam-3344	85	8	)	)	PUNCT
ejpam-3344	85	9	is	be	AUX
ejpam-3344	85	10	an	an	DET
ejpam-3344	85	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	85	12	fuzzy	fuzzy	ADJ
ejpam-3344	85	13	ideal	ideal	NOUN
ejpam-3344	85	14	of	of	ADP
ejpam-3344	85	15	r.	r.	PROPN
ejpam-3344	85	16	every	every	DET
ejpam-3344	85	17	intuitionistic	intuitionistic	ADJ
ejpam-3344	85	18	fuzzy	fuzzy	ADJ
ejpam-3344	85	19	left	left	NOUN
ejpam-3344	85	20	(	(	PUNCT
ejpam-3344	85	21	resp	resp	NOUN
ejpam-3344	85	22	.	.	PUNCT
ejpam-3344	86	1	right	right	ADJ
ejpam-3344	86	2	,	,	PUNCT
ejpam-3344	86	3	two	two	NUM
ejpam-3344	86	4	-	-	PUNCT
ejpam-3344	86	5	sided	sided	ADJ
ejpam-3344	86	6	)	)	PUNCT
ejpam-3344	86	7	ideal	ideal	NOUN
ejpam-3344	86	8	of	of	ADP
ejpam-3344	86	9	an	an	DET
ejpam-3344	86	10	la	la	ADJ
ejpam-3344	86	11	-	-	PUNCT
ejpam-3344	86	12	ring	ring	NOUN
ejpam-3344	86	13	r	r	NOUN
ejpam-3344	86	14	is	be	AUX
ejpam-3344	86	15	an	an	DET
ejpam-3344	86	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	86	17	fuzzy	fuzzy	ADJ
ejpam-3344	86	18	la	la	NOUN
ejpam-3344	86	19	-	-	PUNCT
ejpam-3344	86	20	subring	subring	NOUN
ejpam-3344	86	21	of	of	ADP
ejpam-3344	86	22	r	r	NOUN
ejpam-3344	86	23	,	,	PUNCT
ejpam-3344	86	24	but	but	CCONJ
ejpam-3344	86	25	the	the	DET
ejpam-3344	86	26	converse	converse	NOUN
ejpam-3344	86	27	is	be	AUX
ejpam-3344	86	28	not	not	PART
ejpam-3344	86	29	true	true	ADJ
ejpam-3344	86	30	.	.	PUNCT
ejpam-3344	87	1	n.	n.	PROPN
ejpam-3344	87	2	kausar	kausar	PROPN
ejpam-3344	87	3	,	,	PUNCT
ejpam-3344	87	4	m.	m.	NOUN
ejpam-3344	87	5	a.	a.	PROPN
ejpam-3344	87	6	waqar	waqar	PROPN
ejpam-3344	87	7	/	/	SYM
ejpam-3344	87	8	eur	eur	PROPN
ejpam-3344	87	9	.	.	PUNCT
ejpam-3344	88	1	j.	j.	PROPN
ejpam-3344	88	2	pure	pure	PROPN
ejpam-3344	88	3	appl	appl	PROPN
ejpam-3344	88	4	.	.	PROPN
ejpam-3344	88	5	math	math	PROPN
ejpam-3344	88	6	,	,	PUNCT
ejpam-3344	88	7	12	12	NUM
ejpam-3344	88	8	(	(	PUNCT
ejpam-3344	88	9	1	1	NUM
ejpam-3344	88	10	)	)	PUNCT
ejpam-3344	88	11	(	(	PUNCT
ejpam-3344	88	12	2019	2019	NUM
ejpam-3344	88	13	)	)	PUNCT
ejpam-3344	88	14	,	,	PUNCT
ejpam-3344	88	15	226	226	NUM
ejpam-3344	88	16	-	-	SYM
ejpam-3344	88	17	250	250	NUM
ejpam-3344	88	18	229	229	NUM
ejpam-3344	88	19	example	example	NOUN
ejpam-3344	88	20	2	2	NUM
ejpam-3344	88	21	.	.	PUNCT
ejpam-3344	88	22	r	r	NOUN
ejpam-3344	88	23	=	=	SYM
ejpam-3344	88	24	{	{	PUNCT
ejpam-3344	88	25	0	0	NUM
ejpam-3344	88	26	,	,	PUNCT
ejpam-3344	88	27	1	1	NUM
ejpam-3344	88	28	,	,	PUNCT
ejpam-3344	88	29	2	2	NUM
ejpam-3344	88	30	,	,	PUNCT
ejpam-3344	88	31	3	3	NUM
ejpam-3344	88	32	,	,	PUNCT
ejpam-3344	88	33	4	4	NUM
ejpam-3344	88	34	,	,	PUNCT
ejpam-3344	88	35	5	5	NUM
ejpam-3344	88	36	,	,	PUNCT
ejpam-3344	88	37	6	6	NUM
ejpam-3344	88	38	,	,	PUNCT
ejpam-3344	88	39	7	7	NUM
ejpam-3344	88	40	}	}	PUNCT
ejpam-3344	88	41	is	be	AUX
ejpam-3344	88	42	an	an	DET
ejpam-3344	88	43	la	la	ADJ
ejpam-3344	88	44	-	-	PUNCT
ejpam-3344	88	45	ring	ring	NOUN
ejpam-3344	88	46	.	.	PUNCT
ejpam-3344	89	1	+	+	CCONJ
ejpam-3344	89	2	0	0	NUM
ejpam-3344	89	3	1	1	NUM
ejpam-3344	89	4	2	2	NUM
ejpam-3344	89	5	3	3	NUM
ejpam-3344	89	6	4	4	NUM
ejpam-3344	89	7	5	5	NUM
ejpam-3344	89	8	6	6	NUM
ejpam-3344	89	9	7	7	NUM
ejpam-3344	89	10	0	0	NUM
ejpam-3344	89	11	0	0	NUM
ejpam-3344	89	12	1	1	NUM
ejpam-3344	89	13	2	2	NUM
ejpam-3344	89	14	3	3	NUM
ejpam-3344	89	15	4	4	NUM
ejpam-3344	89	16	5	5	NUM
ejpam-3344	89	17	6	6	NUM
ejpam-3344	89	18	7	7	NUM
ejpam-3344	89	19	1	1	NUM
ejpam-3344	89	20	2	2	NUM
ejpam-3344	89	21	0	0	NUM
ejpam-3344	89	22	3	3	NUM
ejpam-3344	89	23	1	1	NUM
ejpam-3344	89	24	6	6	NUM
ejpam-3344	89	25	4	4	NUM
ejpam-3344	89	26	7	7	NUM
ejpam-3344	89	27	5	5	NUM
ejpam-3344	89	28	2	2	NUM
ejpam-3344	89	29	1	1	NUM
ejpam-3344	89	30	3	3	NUM
ejpam-3344	89	31	0	0	NUM
ejpam-3344	89	32	2	2	NUM
ejpam-3344	89	33	5	5	NUM
ejpam-3344	89	34	7	7	NUM
ejpam-3344	89	35	4	4	NUM
ejpam-3344	89	36	6	6	NUM
ejpam-3344	89	37	3	3	NUM
ejpam-3344	89	38	3	3	NUM
ejpam-3344	89	39	2	2	NUM
ejpam-3344	89	40	1	1	NUM
ejpam-3344	89	41	0	0	NUM
ejpam-3344	89	42	7	7	NUM
ejpam-3344	89	43	6	6	NUM
ejpam-3344	89	44	5	5	NUM
ejpam-3344	89	45	4	4	NUM
ejpam-3344	89	46	4	4	NUM
ejpam-3344	89	47	4	4	NUM
ejpam-3344	89	48	5	5	NUM
ejpam-3344	89	49	6	6	NUM
ejpam-3344	89	50	7	7	NUM
ejpam-3344	89	51	0	0	NUM
ejpam-3344	89	52	1	1	NUM
ejpam-3344	89	53	2	2	NUM
ejpam-3344	89	54	3	3	NUM
ejpam-3344	89	55	5	5	NUM
ejpam-3344	89	56	6	6	NUM
ejpam-3344	89	57	4	4	NUM
ejpam-3344	89	58	7	7	NUM
ejpam-3344	89	59	5	5	NUM
ejpam-3344	89	60	2	2	NUM
ejpam-3344	89	61	0	0	NUM
ejpam-3344	89	62	3	3	NUM
ejpam-3344	89	63	1	1	NUM
ejpam-3344	89	64	6	6	NUM
ejpam-3344	89	65	5	5	NUM
ejpam-3344	89	66	7	7	NUM
ejpam-3344	89	67	4	4	NUM
ejpam-3344	89	68	6	6	NUM
ejpam-3344	89	69	1	1	NUM
ejpam-3344	89	70	3	3	NUM
ejpam-3344	89	71	0	0	NUM
ejpam-3344	89	72	2	2	NUM
ejpam-3344	89	73	7	7	NUM
ejpam-3344	89	74	7	7	NUM
ejpam-3344	89	75	6	6	NUM
ejpam-3344	89	76	5	5	NUM
ejpam-3344	89	77	4	4	NUM
ejpam-3344	89	78	3	3	NUM
ejpam-3344	89	79	2	2	NUM
ejpam-3344	89	80	1	1	NUM
ejpam-3344	89	81	0	0	NUM
ejpam-3344	89	82	and	and	CCONJ
ejpam-3344	89	83	·	·	PUNCT
ejpam-3344	89	84	0	0	NUM
ejpam-3344	90	1	1	1	NUM
ejpam-3344	90	2	2	2	NUM
ejpam-3344	90	3	3	3	NUM
ejpam-3344	90	4	4	4	NUM
ejpam-3344	90	5	5	5	NUM
ejpam-3344	90	6	6	6	NUM
ejpam-3344	90	7	7	7	NUM
ejpam-3344	90	8	0	0	NUM
ejpam-3344	90	9	0	0	NUM
ejpam-3344	90	10	0	0	NUM
ejpam-3344	90	11	0	0	NUM
ejpam-3344	90	12	0	0	NUM
ejpam-3344	90	13	0	0	NUM
ejpam-3344	90	14	0	0	NUM
ejpam-3344	90	15	0	0	NUM
ejpam-3344	90	16	0	0	NUM
ejpam-3344	90	17	1	1	NUM
ejpam-3344	90	18	0	0	NUM
ejpam-3344	90	19	4	4	NUM
ejpam-3344	90	20	4	4	NUM
ejpam-3344	90	21	0	0	NUM
ejpam-3344	90	22	0	0	NUM
ejpam-3344	90	23	4	4	NUM
ejpam-3344	90	24	4	4	NUM
ejpam-3344	90	25	0	0	NUM
ejpam-3344	90	26	2	2	NUM
ejpam-3344	90	27	0	0	NUM
ejpam-3344	90	28	4	4	NUM
ejpam-3344	90	29	4	4	NUM
ejpam-3344	90	30	0	0	NUM
ejpam-3344	90	31	0	0	NUM
ejpam-3344	90	32	4	4	NUM
ejpam-3344	90	33	4	4	NUM
ejpam-3344	90	34	0	0	NUM
ejpam-3344	90	35	3	3	NUM
ejpam-3344	90	36	0	0	NUM
ejpam-3344	90	37	0	0	NUM
ejpam-3344	90	38	0	0	NUM
ejpam-3344	90	39	0	0	NUM
ejpam-3344	90	40	0	0	NUM
ejpam-3344	90	41	0	0	NUM
ejpam-3344	90	42	0	0	NUM
ejpam-3344	90	43	0	0	NUM
ejpam-3344	90	44	4	4	NUM
ejpam-3344	90	45	0	0	NUM
ejpam-3344	90	46	3	3	NUM
ejpam-3344	90	47	3	3	NUM
ejpam-3344	90	48	0	0	NUM
ejpam-3344	90	49	0	0	NUM
ejpam-3344	90	50	3	3	NUM
ejpam-3344	90	51	3	3	NUM
ejpam-3344	90	52	0	0	NUM
ejpam-3344	90	53	5	5	NUM
ejpam-3344	90	54	0	0	NUM
ejpam-3344	90	55	7	7	NUM
ejpam-3344	90	56	7	7	NUM
ejpam-3344	90	57	0	0	NUM
ejpam-3344	90	58	0	0	NUM
ejpam-3344	90	59	7	7	NUM
ejpam-3344	90	60	7	7	NUM
ejpam-3344	90	61	0	0	NUM
ejpam-3344	90	62	6	6	NUM
ejpam-3344	90	63	0	0	NUM
ejpam-3344	90	64	7	7	NUM
ejpam-3344	90	65	7	7	NUM
ejpam-3344	90	66	0	0	NUM
ejpam-3344	90	67	0	0	NUM
ejpam-3344	90	68	7	7	NUM
ejpam-3344	90	69	7	7	NUM
ejpam-3344	90	70	0	0	NUM
ejpam-3344	90	71	7	7	NUM
ejpam-3344	90	72	0	0	NUM
ejpam-3344	90	73	3	3	NUM
ejpam-3344	90	74	3	3	NUM
ejpam-3344	90	75	0	0	NUM
ejpam-3344	90	76	0	0	NUM
ejpam-3344	90	77	3	3	NUM
ejpam-3344	90	78	3	3	NUM
ejpam-3344	90	79	0	0	NUM
ejpam-3344	90	80	let	let	VERB
ejpam-3344	90	81	a	a	PRON
ejpam-3344	90	82	=	=	SYM
ejpam-3344	90	83	(	(	PUNCT
ejpam-3344	90	84	µa	µa	PROPN
ejpam-3344	90	85	,	,	PUNCT
ejpam-3344	90	86	γa	γa	PROPN
ejpam-3344	90	87	)	)	PUNCT
ejpam-3344	90	88	be	be	VERB
ejpam-3344	90	89	an	an	DET
ejpam-3344	90	90	ifs	ifs	PROPN
ejpam-3344	90	91	of	of	ADP
ejpam-3344	90	92	an	an	DET
ejpam-3344	90	93	la	la	ADJ
ejpam-3344	90	94	-	-	PUNCT
ejpam-3344	90	95	ring	ring	NOUN
ejpam-3344	90	96	r.	r.	NOUN
ejpam-3344	90	97	we	we	PRON
ejpam-3344	90	98	define	define	VERB
ejpam-3344	90	99	µa(0	µa(0	NOUN
ejpam-3344	90	100	)	)	PUNCT
ejpam-3344	90	101	=	=	SYM
ejpam-3344	90	102	µa(4	µa(4	NOUN
ejpam-3344	90	103	)	)	PUNCT
ejpam-3344	90	104	=	=	NOUN
ejpam-3344	90	105	0.7	0.7	NUM
ejpam-3344	90	106	,	,	PUNCT
ejpam-3344	90	107	µa(1	µa(1	NOUN
ejpam-3344	90	108	)	)	PUNCT
ejpam-3344	90	109	=	=	SYM
ejpam-3344	90	110	µa(2	µa(2	NOUN
ejpam-3344	90	111	)	)	PUNCT
ejpam-3344	90	112	=	=	SYM
ejpam-3344	90	113	µa(3	µa(3	PROPN
ejpam-3344	90	114	)	)	PUNCT
ejpam-3344	90	115	=	=	SYM
ejpam-3344	90	116	µa(5	µa(5	NOUN
ejpam-3344	90	117	)	)	PUNCT
ejpam-3344	90	118	=	=	SYM
ejpam-3344	90	119	µa(6	µa(6	NOUN
ejpam-3344	90	120	)	)	PUNCT
ejpam-3344	91	1	=	=	PUNCT
ejpam-3344	91	2	µa(7	µa(7	PROPN
ejpam-3344	91	3	)	)	PUNCT
ejpam-3344	91	4	=	=	SYM
ejpam-3344	91	5	0	0	NUM
ejpam-3344	91	6	and	and	CCONJ
ejpam-3344	91	7	γa(0	γa(0	NOUN
ejpam-3344	91	8	)	)	PUNCT
ejpam-3344	91	9	=	=	PUNCT
ejpam-3344	91	10	γa(4	γa(4	X
ejpam-3344	91	11	)	)	PUNCT
ejpam-3344	91	12	=	=	SYM
ejpam-3344	91	13	0	0	NUM
ejpam-3344	91	14	,	,	PUNCT
ejpam-3344	91	15	γa(1	γa(1	NOUN
ejpam-3344	91	16	)	)	PUNCT
ejpam-3344	91	17	=	=	SYM
ejpam-3344	91	18	γa(2	γa(2	NUM
ejpam-3344	91	19	)	)	PUNCT
ejpam-3344	91	20	=	=	SYM
ejpam-3344	91	21	γa(3	γa(3	PROPN
ejpam-3344	91	22	)	)	PUNCT
ejpam-3344	91	23	=	=	SYM
ejpam-3344	91	24	γa(5	γa(5	NOUN
ejpam-3344	91	25	)	)	PUNCT
ejpam-3344	91	26	=	=	SYM
ejpam-3344	91	27	γa(6	γa(6	PROPN
ejpam-3344	91	28	)	)	PUNCT
ejpam-3344	91	29	=	=	PUNCT
ejpam-3344	92	1	γa(7	γa(7	X
ejpam-3344	92	2	)	)	PUNCT
ejpam-3344	92	3	=	=	PUNCT
ejpam-3344	93	1	0.7	0.7	NUM
ejpam-3344	93	2	.	.	PUNCT
ejpam-3344	94	1	then	then	ADV
ejpam-3344	94	2	a	a	DET
ejpam-3344	94	3	=	=	SYM
ejpam-3344	94	4	(	(	PUNCT
ejpam-3344	94	5	µa	µa	PROPN
ejpam-3344	94	6	,	,	PUNCT
ejpam-3344	94	7	γa	γa	PROPN
ejpam-3344	94	8	)	)	PUNCT
ejpam-3344	94	9	is	be	AUX
ejpam-3344	94	10	an	an	DET
ejpam-3344	94	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	94	12	fuzzy	fuzzy	ADJ
ejpam-3344	94	13	la	la	NOUN
ejpam-3344	94	14	-	-	PUNCT
ejpam-3344	94	15	subring	subring	NOUN
ejpam-3344	94	16	of	of	ADP
ejpam-3344	94	17	r	r	NOUN
ejpam-3344	94	18	,	,	PUNCT
ejpam-3344	94	19	but	but	CCONJ
ejpam-3344	94	20	not	not	PART
ejpam-3344	94	21	an	an	DET
ejpam-3344	94	22	intuitionistic	intuitionistic	ADJ
ejpam-3344	94	23	fuzzy	fuzzy	ADJ
ejpam-3344	94	24	right	right	ADJ
ejpam-3344	94	25	ideal	ideal	NOUN
ejpam-3344	94	26	of	of	ADP
ejpam-3344	94	27	r	r	NOUN
ejpam-3344	94	28	,	,	PUNCT
ejpam-3344	94	29	because	because	SCONJ
ejpam-3344	94	30	µa(41	µa(41	PROPN
ejpam-3344	94	31	)	)	PUNCT
ejpam-3344	94	32	=	=	SYM
ejpam-3344	94	33	µa(3	µa(3	PROPN
ejpam-3344	94	34	)	)	PUNCT
ejpam-3344	94	35	=	=	SYM
ejpam-3344	94	36	0	0	X
ejpam-3344	94	37	.	.	X
ejpam-3344	95	1	µa(4	µa(4	NOUN
ejpam-3344	95	2	)	)	PUNCT
ejpam-3344	95	3	=	=	NOUN
ejpam-3344	96	1	0.7	0.7	NUM
ejpam-3344	96	2	.	.	PUNCT
ejpam-3344	97	1	⇒	⇒	PROPN
ejpam-3344	97	2	µa(41	µa(41	PROPN
ejpam-3344	97	3	)	)	PUNCT
ejpam-3344	97	4	�	�	PROPN
ejpam-3344	97	5	µa(4	µa(4	PROPN
ejpam-3344	97	6	)	)	PUNCT
ejpam-3344	97	7	.	.	PUNCT
ejpam-3344	98	1	and	and	CCONJ
ejpam-3344	98	2	γa(41	γa(41	PROPN
ejpam-3344	98	3	)	)	PUNCT
ejpam-3344	98	4	=	=	SYM
ejpam-3344	98	5	γa(3	γa(3	PROPN
ejpam-3344	98	6	)	)	PUNCT
ejpam-3344	98	7	=	=	PUNCT
ejpam-3344	99	1	0.7	0.7	NUM
ejpam-3344	99	2	.	.	PUNCT
ejpam-3344	99	3	γa(4	γa(4	X
ejpam-3344	99	4	)	)	PUNCT
ejpam-3344	99	5	=	=	SYM
ejpam-3344	99	6	0	0	X
ejpam-3344	99	7	.	.	PUNCT
ejpam-3344	99	8	⇒	⇒	PROPN
ejpam-3344	99	9	γa(41	γa(41	PROPN
ejpam-3344	99	10	)	)	PUNCT
ejpam-3344	99	11	�	�	PROPN
ejpam-3344	99	12	γa(4	γa(4	PROPN
ejpam-3344	99	13	)	)	PUNCT
ejpam-3344	99	14	.	.	PUNCT
ejpam-3344	100	1	an	an	DET
ejpam-3344	100	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	100	3	fuzzy	fuzzy	ADJ
ejpam-3344	100	4	la	la	NOUN
ejpam-3344	100	5	-	-	PUNCT
ejpam-3344	100	6	subring	subre	VERB
ejpam-3344	100	7	a	a	PRON
ejpam-3344	100	8	=	=	SYM
ejpam-3344	100	9	(	(	PUNCT
ejpam-3344	100	10	µa	µa	PROPN
ejpam-3344	100	11	,	,	PUNCT
ejpam-3344	100	12	γa	γa	PROPN
ejpam-3344	100	13	)	)	PUNCT
ejpam-3344	100	14	of	of	ADP
ejpam-3344	100	15	an	an	DET
ejpam-3344	100	16	la	la	ADJ
ejpam-3344	100	17	-	-	PUNCT
ejpam-3344	100	18	ring	ring	NOUN
ejpam-3344	100	19	r	r	NOUN
ejpam-3344	100	20	is	be	AUX
ejpam-3344	100	21	called	call	VERB
ejpam-3344	100	22	an	an	DET
ejpam-3344	100	23	intuitionistic	intuitionistic	ADJ
ejpam-3344	100	24	fuzzy	fuzzy	ADJ
ejpam-3344	100	25	bi	bi	NOUN
ejpam-3344	100	26	-	-	NOUN
ejpam-3344	100	27	ideal	ideal	NOUN
ejpam-3344	100	28	of	of	ADP
ejpam-3344	100	29	r	r	NOUN
ejpam-3344	100	30	if	if	SCONJ
ejpam-3344	100	31	(	(	PUNCT
ejpam-3344	100	32	1	1	X
ejpam-3344	100	33	)	)	PUNCT
ejpam-3344	100	34	µa	µa	NOUN
ejpam-3344	100	35	(	(	PUNCT
ejpam-3344	100	36	(	(	PUNCT
ejpam-3344	100	37	xy)z	xy)z	NUM
ejpam-3344	100	38	)	)	PUNCT
ejpam-3344	100	39	≥	≥	PROPN
ejpam-3344	100	40	min	min	NOUN
ejpam-3344	100	41	{	{	PUNCT
ejpam-3344	100	42	µa(x	µa(x	NOUN
ejpam-3344	100	43	)	)	PUNCT
ejpam-3344	100	44	,	,	PUNCT
ejpam-3344	100	45	µa(z	µa(z	NUM
ejpam-3344	100	46	)	)	PUNCT
ejpam-3344	100	47	}	}	PUNCT
ejpam-3344	100	48	,	,	PUNCT
ejpam-3344	100	49	(	(	PUNCT
ejpam-3344	100	50	2	2	X
ejpam-3344	100	51	)	)	PUNCT
ejpam-3344	100	52	γa	γa	NOUN
ejpam-3344	100	53	(	(	PUNCT
ejpam-3344	100	54	(	(	PUNCT
ejpam-3344	100	55	xy)z	xy)z	NOUN
ejpam-3344	100	56	)	)	PUNCT
ejpam-3344	100	57	≤	≤	NUM
ejpam-3344	100	58	max	max	PROPN
ejpam-3344	100	59	{	{	PUNCT
ejpam-3344	100	60	γa(x	γa(x	NUM
ejpam-3344	100	61	)	)	PUNCT
ejpam-3344	100	62	,	,	PUNCT
ejpam-3344	100	63	γa(y	γa(y	NOUN
ejpam-3344	100	64	)	)	PUNCT
ejpam-3344	100	65	}	}	PUNCT
ejpam-3344	100	66	for	for	ADP
ejpam-3344	100	67	all	all	DET
ejpam-3344	100	68	x	x	NOUN
ejpam-3344	100	69	,	,	PUNCT
ejpam-3344	100	70	y	y	PROPN
ejpam-3344	100	71	,	,	PUNCT
ejpam-3344	100	72	z	z	PROPN
ejpam-3344	100	73	∈	∈	PROPN
ejpam-3344	100	74	r.	r.	PROPN
ejpam-3344	100	75	an	an	DET
ejpam-3344	100	76	ifs	ifs	PROPN
ejpam-3344	100	77	a	a	X
ejpam-3344	100	78	=	=	X
ejpam-3344	100	79	(	(	PUNCT
ejpam-3344	100	80	µa	µa	PROPN
ejpam-3344	100	81	,	,	PUNCT
ejpam-3344	100	82	γa	γa	PROPN
ejpam-3344	100	83	)	)	PUNCT
ejpam-3344	100	84	of	of	ADP
ejpam-3344	100	85	an	an	DET
ejpam-3344	100	86	la	la	ADJ
ejpam-3344	100	87	-	-	PUNCT
ejpam-3344	100	88	ring	ring	NOUN
ejpam-3344	100	89	r	r	NOUN
ejpam-3344	100	90	is	be	AUX
ejpam-3344	100	91	called	call	VERB
ejpam-3344	100	92	an	an	DET
ejpam-3344	100	93	intuitionistic	intuitionistic	ADJ
ejpam-3344	100	94	fuzzy	fuzzy	ADJ
ejpam-3344	100	95	generalized	generalize	VERB
ejpam-3344	100	96	bi	bi	NOUN
ejpam-3344	100	97	-	-	NOUN
ejpam-3344	100	98	ideal	ideal	NOUN
ejpam-3344	100	99	of	of	ADP
ejpam-3344	100	100	r	r	NOUN
ejpam-3344	100	101	if	if	SCONJ
ejpam-3344	100	102	(	(	PUNCT
ejpam-3344	100	103	1	1	X
ejpam-3344	100	104	)	)	PUNCT
ejpam-3344	100	105	µa	µa	NOUN
ejpam-3344	100	106	(	(	PUNCT
ejpam-3344	100	107	x−	x−	PROPN
ejpam-3344	100	108	y	y	PROPN
ejpam-3344	100	109	)	)	PUNCT
ejpam-3344	100	110	≥	≥	PROPN
ejpam-3344	100	111	min	min	NOUN
ejpam-3344	100	112	{	{	PUNCT
ejpam-3344	100	113	µa(x	µa(x	NOUN
ejpam-3344	100	114	)	)	PUNCT
ejpam-3344	100	115	,	,	PUNCT
ejpam-3344	100	116	µa(y	µa(y	NOUN
ejpam-3344	100	117	)	)	PUNCT
ejpam-3344	100	118	}	}	PUNCT
ejpam-3344	100	119	,	,	PUNCT
ejpam-3344	100	120	(	(	PUNCT
ejpam-3344	100	121	2	2	X
ejpam-3344	100	122	)	)	PUNCT
ejpam-3344	100	123	γa	γa	NOUN
ejpam-3344	100	124	(	(	PUNCT
ejpam-3344	100	125	x−	x−	PROPN
ejpam-3344	100	126	y	y	PROPN
ejpam-3344	100	127	)	)	PUNCT
ejpam-3344	100	128	≤	≤	NUM
ejpam-3344	100	129	max	max	PROPN
ejpam-3344	100	130	{	{	PUNCT
ejpam-3344	100	131	γa(x	γa(x	NUM
ejpam-3344	100	132	)	)	PUNCT
ejpam-3344	100	133	,	,	PUNCT
ejpam-3344	100	134	γa(y	γa(y	NOUN
ejpam-3344	100	135	)	)	PUNCT
ejpam-3344	100	136	}	}	PUNCT
ejpam-3344	100	137	,	,	PUNCT
ejpam-3344	100	138	(	(	PUNCT
ejpam-3344	100	139	3	3	X
ejpam-3344	100	140	)	)	PUNCT
ejpam-3344	100	141	µa	µa	NOUN
ejpam-3344	100	142	(	(	PUNCT
ejpam-3344	100	143	(	(	PUNCT
ejpam-3344	100	144	xy)z	xy)z	NUM
ejpam-3344	100	145	)	)	PUNCT
ejpam-3344	100	146	≥	≥	PROPN
ejpam-3344	100	147	min	min	NOUN
ejpam-3344	100	148	{	{	PUNCT
ejpam-3344	100	149	µa(x	µa(x	NOUN
ejpam-3344	100	150	)	)	PUNCT
ejpam-3344	100	151	,	,	PUNCT
ejpam-3344	100	152	µa(z	µa(z	NUM
ejpam-3344	100	153	)	)	PUNCT
ejpam-3344	100	154	}	}	PUNCT
ejpam-3344	100	155	,	,	PUNCT
ejpam-3344	100	156	(	(	PUNCT
ejpam-3344	100	157	4	4	X
ejpam-3344	100	158	)	)	PUNCT
ejpam-3344	100	159	γa	γa	NOUN
ejpam-3344	100	160	(	(	PUNCT
ejpam-3344	100	161	(	(	PUNCT
ejpam-3344	100	162	xy)z	xy)z	NOUN
ejpam-3344	100	163	)	)	PUNCT
ejpam-3344	100	164	≤	≤	NUM
ejpam-3344	100	165	max	max	PROPN
ejpam-3344	100	166	{	{	PUNCT
ejpam-3344	100	167	γa(x	γa(x	NUM
ejpam-3344	100	168	)	)	PUNCT
ejpam-3344	100	169	,	,	PUNCT
ejpam-3344	100	170	γa(z	γa(z	PROPN
ejpam-3344	100	171	)	)	PUNCT
ejpam-3344	100	172	}	}	PUNCT
ejpam-3344	100	173	for	for	ADP
ejpam-3344	100	174	all	all	DET
ejpam-3344	100	175	x	x	NOUN
ejpam-3344	100	176	,	,	PUNCT
ejpam-3344	100	177	y	y	PROPN
ejpam-3344	100	178	,	,	PUNCT
ejpam-3344	100	179	z	z	PROPN
ejpam-3344	100	180	∈	∈	PROPN
ejpam-3344	100	181	r.	r.	PROPN
ejpam-3344	100	182	an	an	DET
ejpam-3344	100	183	intuitionistic	intuitionistic	ADJ
ejpam-3344	100	184	fuzzy	fuzzy	ADJ
ejpam-3344	100	185	la	la	NOUN
ejpam-3344	100	186	-	-	PUNCT
ejpam-3344	100	187	subring	subre	VERB
ejpam-3344	100	188	a	a	PRON
ejpam-3344	100	189	=	=	SYM
ejpam-3344	100	190	(	(	PUNCT
ejpam-3344	100	191	µa	µa	PROPN
ejpam-3344	100	192	,	,	PUNCT
ejpam-3344	100	193	γa	γa	PROPN
ejpam-3344	100	194	)	)	PUNCT
ejpam-3344	100	195	of	of	ADP
ejpam-3344	100	196	an	an	DET
ejpam-3344	100	197	la	la	ADJ
ejpam-3344	100	198	-	-	PUNCT
ejpam-3344	100	199	ring	ring	NOUN
ejpam-3344	100	200	r	r	NOUN
ejpam-3344	100	201	is	be	AUX
ejpam-3344	100	202	called	call	VERB
ejpam-3344	100	203	an	an	DET
ejpam-3344	100	204	intuitionistic	intuitionistic	ADJ
ejpam-3344	100	205	fuzzy	fuzzy	ADJ
ejpam-3344	100	206	(	(	PUNCT
ejpam-3344	100	207	1	1	NUM
ejpam-3344	100	208	,	,	PUNCT
ejpam-3344	100	209	2)-ideal	2)-ideal	NUM
ejpam-3344	100	210	of	of	ADP
ejpam-3344	100	211	r	r	PRON
ejpam-3344	100	212	if	if	SCONJ
ejpam-3344	100	213	(	(	PUNCT
ejpam-3344	100	214	1	1	NUM
ejpam-3344	100	215	)	)	PUNCT
ejpam-3344	100	216	µa((xw)(yz	µa((xw)(yz	PROPN
ejpam-3344	100	217	)	)	PUNCT
ejpam-3344	100	218	)	)	PUNCT
ejpam-3344	100	219	≥	≥	PROPN
ejpam-3344	100	220	min	min	NOUN
ejpam-3344	100	221	{	{	PUNCT
ejpam-3344	100	222	µa(x	µa(x	NOUN
ejpam-3344	100	223	)	)	PUNCT
ejpam-3344	100	224	,	,	PUNCT
ejpam-3344	100	225	µa(y	µa(y	NOUN
ejpam-3344	100	226	)	)	PUNCT
ejpam-3344	100	227	,	,	PUNCT
ejpam-3344	100	228	µa(z	µa(z	NUM
ejpam-3344	100	229	)	)	PUNCT
ejpam-3344	100	230	}	}	PUNCT
ejpam-3344	100	231	,	,	PUNCT
ejpam-3344	100	232	(	(	PUNCT
ejpam-3344	100	233	2	2	X
ejpam-3344	100	234	)	)	PUNCT
ejpam-3344	100	235	γa((xw)(yz	γa((xw)(yz	NOUN
ejpam-3344	100	236	)	)	PUNCT
ejpam-3344	100	237	)	)	PUNCT
ejpam-3344	100	238	≤	≤	NUM
ejpam-3344	100	239	max	max	PROPN
ejpam-3344	100	240	{	{	PUNCT
ejpam-3344	100	241	γa(x	γa(x	NUM
ejpam-3344	100	242	)	)	PUNCT
ejpam-3344	100	243	,	,	PUNCT
ejpam-3344	100	244	γa(y	γa(y	NUM
ejpam-3344	100	245	)	)	PUNCT
ejpam-3344	100	246	,	,	PUNCT
ejpam-3344	100	247	γa(z	γa(z	PROPN
ejpam-3344	100	248	)	)	PUNCT
ejpam-3344	100	249	}	}	PUNCT
ejpam-3344	100	250	for	for	ADP
ejpam-3344	100	251	all	all	DET
ejpam-3344	100	252	x	x	NOUN
ejpam-3344	100	253	,	,	PUNCT
ejpam-3344	100	254	y	y	PROPN
ejpam-3344	100	255	,	,	PUNCT
ejpam-3344	100	256	z	z	PROPN
ejpam-3344	100	257	,	,	PUNCT
ejpam-3344	100	258	w	w	PROPN
ejpam-3344	100	259	∈	∈	PROPN
ejpam-3344	100	260	r.	r.	NOUN
ejpam-3344	100	261	we	we	PRON
ejpam-3344	100	262	note	note	VERB
ejpam-3344	100	263	that	that	SCONJ
ejpam-3344	100	264	an	an	DET
ejpam-3344	100	265	la	la	ADJ
ejpam-3344	100	266	-	-	PUNCT
ejpam-3344	100	267	ring	ring	NOUN
ejpam-3344	100	268	r	r	NOUN
ejpam-3344	100	269	can	can	AUX
ejpam-3344	100	270	be	be	AUX
ejpam-3344	100	271	considered	consider	VERB
ejpam-3344	100	272	an	an	DET
ejpam-3344	100	273	intuitionistic	intuitionistic	ADJ
ejpam-3344	100	274	fuzzy	fuzzy	ADJ
ejpam-3344	100	275	set	set	NOUN
ejpam-3344	100	276	of	of	ADP
ejpam-3344	100	277	itself	itself	PRON
ejpam-3344	100	278	and	and	CCONJ
ejpam-3344	100	279	we	we	PRON
ejpam-3344	100	280	write	write	VERB
ejpam-3344	100	281	r	r	NOUN
ejpam-3344	100	282	=	=	SYM
ejpam-3344	100	283	ir	ir	PROPN
ejpam-3344	100	284	,	,	PUNCT
ejpam-3344	100	285	i.e.	i.e.	X
ejpam-3344	100	286	,	,	PUNCT
ejpam-3344	100	287	r	r	NOUN
ejpam-3344	100	288	=	=	SYM
ejpam-3344	100	289	(	(	PUNCT
ejpam-3344	100	290	µr	µr	ADP
ejpam-3344	100	291	,	,	PUNCT
ejpam-3344	100	292	γr	γr	PROPN
ejpam-3344	100	293	)	)	PUNCT
ejpam-3344	100	294	=	=	SYM
ejpam-3344	100	295	(	(	PUNCT
ejpam-3344	100	296	1	1	NUM
ejpam-3344	100	297	,	,	PUNCT
ejpam-3344	100	298	0	0	NUM
ejpam-3344	100	299	)	)	PUNCT
ejpam-3344	100	300	for	for	SCONJ
ejpam-3344	100	301	all	all	DET
ejpam-3344	100	302	x	x	PROPN
ejpam-3344	100	303	∈	∈	PROPN
ejpam-3344	100	304	r.	r.	NOUN
ejpam-3344	100	305	let	let	VERB
ejpam-3344	100	306	a	a	PRON
ejpam-3344	100	307	and	and	CCONJ
ejpam-3344	100	308	b	b	NOUN
ejpam-3344	100	309	be	be	AUX
ejpam-3344	100	310	two	two	NUM
ejpam-3344	100	311	intuitionistic	intuitionistic	ADJ
ejpam-3344	100	312	fuzzy	fuzzy	ADJ
ejpam-3344	100	313	sets	set	NOUN
ejpam-3344	100	314	of	of	ADP
ejpam-3344	100	315	an	an	DET
ejpam-3344	100	316	la	la	ADJ
ejpam-3344	100	317	-	-	PUNCT
ejpam-3344	100	318	ring	ring	NOUN
ejpam-3344	100	319	r.	r.	PROPN
ejpam-3344	100	320	then	then	ADV
ejpam-3344	100	321	(	(	PUNCT
ejpam-3344	100	322	1	1	X
ejpam-3344	100	323	)	)	PUNCT
ejpam-3344	100	324	a	a	DET
ejpam-3344	100	325	⊆	⊆	NUM
ejpam-3344	100	326	b	b	SYM
ejpam-3344	100	327	⇔	⇔	X
ejpam-3344	100	328	µa	µa	ADP
ejpam-3344	100	329	⊆	⊆	NUM
ejpam-3344	100	330	µb	µb	NOUN
ejpam-3344	100	331	and	and	CCONJ
ejpam-3344	100	332	γa	γa	PROPN
ejpam-3344	100	333	⊇	⊇	PROPN
ejpam-3344	100	334	γb	γb	PROPN
ejpam-3344	100	335	,	,	PUNCT
ejpam-3344	100	336	(	(	PUNCT
ejpam-3344	100	337	2	2	X
ejpam-3344	100	338	)	)	PUNCT
ejpam-3344	100	339	a	a	DET
ejpam-3344	100	340	=	=	SYM
ejpam-3344	100	341	b	b	PROPN
ejpam-3344	100	342	⇔	⇔	PROPN
ejpam-3344	100	343	a	a	DET
ejpam-3344	100	344	⊆	⊆	NUM
ejpam-3344	100	345	b	b	NOUN
ejpam-3344	100	346	and	and	CCONJ
ejpam-3344	100	347	b	b	NOUN
ejpam-3344	100	348	⊆	⊆	NUM
ejpam-3344	100	349	a	a	DET
ejpam-3344	100	350	,	,	PUNCT
ejpam-3344	100	351	n.	n.	PROPN
ejpam-3344	100	352	kausar	kausar	PROPN
ejpam-3344	100	353	,	,	PUNCT
ejpam-3344	100	354	m.	m.	NOUN
ejpam-3344	100	355	a.	a.	PROPN
ejpam-3344	100	356	waqar	waqar	PROPN
ejpam-3344	100	357	/	/	SYM
ejpam-3344	100	358	eur	eur	PROPN
ejpam-3344	100	359	.	.	PUNCT
ejpam-3344	101	1	j.	j.	PROPN
ejpam-3344	101	2	pure	pure	PROPN
ejpam-3344	101	3	appl	appl	PROPN
ejpam-3344	101	4	.	.	PROPN
ejpam-3344	101	5	math	math	PROPN
ejpam-3344	101	6	,	,	PUNCT
ejpam-3344	101	7	12	12	NUM
ejpam-3344	101	8	(	(	PUNCT
ejpam-3344	101	9	1	1	NUM
ejpam-3344	101	10	)	)	PUNCT
ejpam-3344	101	11	(	(	PUNCT
ejpam-3344	101	12	2019	2019	NUM
ejpam-3344	101	13	)	)	PUNCT
ejpam-3344	101	14	,	,	PUNCT
ejpam-3344	101	15	226	226	NUM
ejpam-3344	101	16	-	-	SYM
ejpam-3344	101	17	250	250	NUM
ejpam-3344	101	18	230	230	NUM
ejpam-3344	101	19	(	(	PUNCT
ejpam-3344	101	20	3	3	NUM
ejpam-3344	101	21	)	)	PUNCT
ejpam-3344	101	22	ac	ac	NOUN
ejpam-3344	102	1	=	=	PUNCT
ejpam-3344	102	2	(	(	PUNCT
ejpam-3344	102	3	γa	γa	PROPN
ejpam-3344	102	4	,	,	PUNCT
ejpam-3344	102	5	µa	µa	ADJ
ejpam-3344	102	6	)	)	PUNCT
ejpam-3344	102	7	,	,	PUNCT
ejpam-3344	102	8	(	(	PUNCT
ejpam-3344	102	9	4	4	X
ejpam-3344	102	10	)	)	PUNCT
ejpam-3344	102	11	a	a	DET
ejpam-3344	102	12	∩b	∩b	NOUN
ejpam-3344	102	13	=	=	SYM
ejpam-3344	102	14	(	(	PUNCT
ejpam-3344	102	15	µa	µa	SCONJ
ejpam-3344	102	16	∧	∧	PROPN
ejpam-3344	102	17	µb	µb	PROPN
ejpam-3344	102	18	,	,	PUNCT
ejpam-3344	102	19	γa	γa	PROPN
ejpam-3344	102	20	∨	∨	NUM
ejpam-3344	102	21	γb	γb	PROPN
ejpam-3344	102	22	)	)	PUNCT
ejpam-3344	102	23	=	=	SYM
ejpam-3344	102	24	(	(	PUNCT
ejpam-3344	102	25	µa∧b	µa∧b	PROPN
ejpam-3344	102	26	,	,	PUNCT
ejpam-3344	102	27	γa∨b	γa∨b	NOUN
ejpam-3344	102	28	)	)	PUNCT
ejpam-3344	102	29	,	,	PUNCT
ejpam-3344	102	30	(	(	PUNCT
ejpam-3344	102	31	5	5	X
ejpam-3344	102	32	)	)	PUNCT
ejpam-3344	102	33	a	a	DET
ejpam-3344	102	34	∪b	∪b	X
ejpam-3344	102	35	=	=	SYM
ejpam-3344	102	36	(	(	PUNCT
ejpam-3344	102	37	µa	µa	PROPN
ejpam-3344	102	38	∨	∨	NUM
ejpam-3344	102	39	µb	µb	PROPN
ejpam-3344	102	40	,	,	PUNCT
ejpam-3344	102	41	γa	γa	PROPN
ejpam-3344	102	42	∧	∧	PROPN
ejpam-3344	102	43	γb	γb	PROPN
ejpam-3344	102	44	)	)	PUNCT
ejpam-3344	102	45	=	=	SYM
ejpam-3344	102	46	(	(	PUNCT
ejpam-3344	102	47	µa∨b	µa∨b	NOUN
ejpam-3344	102	48	,	,	PUNCT
ejpam-3344	102	49	γa∧b	γa∧b	PROPN
ejpam-3344	102	50	)	)	PUNCT
ejpam-3344	102	51	,	,	PUNCT
ejpam-3344	102	52	(	(	PUNCT
ejpam-3344	102	53	6	6	NUM
ejpam-3344	102	54	)	)	PUNCT
ejpam-3344	102	55	0	0	PUNCT
ejpam-3344	103	1	∼=	∼=	PROPN
ejpam-3344	103	2	(	(	PUNCT
ejpam-3344	103	3	0	0	NUM
ejpam-3344	103	4	,	,	PUNCT
ejpam-3344	103	5	1	1	NUM
ejpam-3344	103	6	)	)	PUNCT
ejpam-3344	103	7	,	,	PUNCT
ejpam-3344	103	8	1	1	NUM
ejpam-3344	103	9	∼=	∼=	PROPN
ejpam-3344	103	10	(	(	PUNCT
ejpam-3344	103	11	1	1	NUM
ejpam-3344	103	12	,	,	PUNCT
ejpam-3344	103	13	0	0	NUM
ejpam-3344	103	14	)	)	PUNCT
ejpam-3344	103	15	.	.	PUNCT
ejpam-3344	104	1	[	[	X
ejpam-3344	104	2	18	18	NUM
ejpam-3344	104	3	]	]	PUNCT
ejpam-3344	104	4	let	let	VERB
ejpam-3344	104	5	a	a	PRON
ejpam-3344	104	6	be	be	AUX
ejpam-3344	104	7	a	a	DET
ejpam-3344	104	8	non	non	ADJ
ejpam-3344	104	9	-	-	ADJ
ejpam-3344	104	10	empty	empty	ADJ
ejpam-3344	104	11	subset	subset	NOUN
ejpam-3344	104	12	of	of	ADP
ejpam-3344	104	13	an	an	DET
ejpam-3344	104	14	la	la	ADJ
ejpam-3344	104	15	-	-	PUNCT
ejpam-3344	104	16	ring	ring	NOUN
ejpam-3344	104	17	r.	r.	PROPN
ejpam-3344	104	18	then	then	ADV
ejpam-3344	104	19	the	the	DET
ejpam-3344	104	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	104	21	characteristic	characteristic	NOUN
ejpam-3344	104	22	of	of	ADP
ejpam-3344	104	23	a	a	PRON
ejpam-3344	104	24	is	be	AUX
ejpam-3344	104	25	denoted	denote	VERB
ejpam-3344	104	26	by	by	ADP
ejpam-3344	104	27	χa	χa	NOUN
ejpam-3344	105	1	=	=	SYM
ejpam-3344	105	2	〈	〈	PROPN
ejpam-3344	105	3	µχa	µχa	NOUN
ejpam-3344	105	4	,	,	PUNCT
ejpam-3344	105	5	γχa	γχa	NOUN
ejpam-3344	105	6	〉	〉	PROPN
ejpam-3344	105	7	and	and	CCONJ
ejpam-3344	105	8	defined	define	VERB
ejpam-3344	105	9	by	by	ADP
ejpam-3344	105	10	µχa	µχa	NOUN
ejpam-3344	105	11	(	(	PUNCT
ejpam-3344	105	12	x	x	NOUN
ejpam-3344	105	13	)	)	PUNCT
ejpam-3344	106	1	=	=	SYM
ejpam-3344	106	2	{	{	PUNCT
ejpam-3344	106	3	1	1	NUM
ejpam-3344	106	4	if	if	SCONJ
ejpam-3344	106	5	x	x	PROPN
ejpam-3344	106	6	∈	∈	PROPN
ejpam-3344	106	7	a	a	DET
ejpam-3344	106	8	0	0	NOUN
ejpam-3344	106	9	if	if	SCONJ
ejpam-3344	106	10	x	x	PROPN
ejpam-3344	106	11	/∈	/∈	PUNCT
ejpam-3344	106	12	a	a	PRON
ejpam-3344	106	13	and	and	CCONJ
ejpam-3344	106	14	γχa	γχa	ADJ
ejpam-3344	106	15	(	(	PUNCT
ejpam-3344	106	16	x	x	X
ejpam-3344	106	17	)	)	PUNCT
ejpam-3344	106	18	=	=	PRON
ejpam-3344	106	19	{	{	PUNCT
ejpam-3344	106	20	0	0	NUM
ejpam-3344	106	21	if	if	SCONJ
ejpam-3344	106	22	x	x	SYM
ejpam-3344	106	23	∈	∈	PROPN
ejpam-3344	106	24	a	a	DET
ejpam-3344	106	25	1	1	NUM
ejpam-3344	106	26	if	if	SCONJ
ejpam-3344	106	27	x	x	PROPN
ejpam-3344	106	28	/∈	/∈	VERB
ejpam-3344	106	29	a	a	DET
ejpam-3344	106	30	the	the	DET
ejpam-3344	106	31	product	product	NOUN
ejpam-3344	106	32	of	of	ADP
ejpam-3344	106	33	a	a	DET
ejpam-3344	106	34	=	=	X
ejpam-3344	106	35	(	(	PUNCT
ejpam-3344	106	36	µa	µa	PROPN
ejpam-3344	106	37	,	,	PUNCT
ejpam-3344	106	38	γa	γa	PROPN
ejpam-3344	106	39	)	)	PUNCT
ejpam-3344	106	40	and	and	CCONJ
ejpam-3344	106	41	b	b	X
ejpam-3344	106	42	=	=	SYM
ejpam-3344	106	43	(	(	PUNCT
ejpam-3344	106	44	µb	µb	PROPN
ejpam-3344	106	45	,	,	PUNCT
ejpam-3344	106	46	γb	γb	PROPN
ejpam-3344	106	47	)	)	PUNCT
ejpam-3344	106	48	is	be	AUX
ejpam-3344	106	49	denoted	denote	VERB
ejpam-3344	106	50	by	by	ADP
ejpam-3344	106	51	a	a	DET
ejpam-3344	106	52	◦	◦	NOUN
ejpam-3344	106	53	b	b	NOUN
ejpam-3344	106	54	=	=	SYM
ejpam-3344	106	55	(	(	PUNCT
ejpam-3344	106	56	µa	µa	ADP
ejpam-3344	106	57	◦	◦	NOUN
ejpam-3344	106	58	µb	µb	NOUN
ejpam-3344	106	59	,	,	PUNCT
ejpam-3344	106	60	γa	γa	PROPN
ejpam-3344	106	61	◦	◦	NOUN
ejpam-3344	106	62	γb	γb	NOUN
ejpam-3344	106	63	)	)	PUNCT
ejpam-3344	106	64	and	and	CCONJ
ejpam-3344	106	65	defined	define	VERB
ejpam-3344	106	66	by	by	ADP
ejpam-3344	106	67	:	:	PUNCT
ejpam-3344	106	68	(	(	PUNCT
ejpam-3344	106	69	µa	µa	ADP
ejpam-3344	106	70	◦	◦	NOUN
ejpam-3344	106	71	µb)(x	µb)(x	NOUN
ejpam-3344	106	72	)	)	PUNCT
ejpam-3344	106	73	=	=	PUNCT
ejpam-3344	106	74			X
ejpam-3344	106	75	∨	∨	NOUN
ejpam-3344	106	76	x=	x=	PROPN
ejpam-3344	107	1	n∑	n∑	PROPN
ejpam-3344	107	2	i=1	i=1	PROPN
ejpam-3344	107	3	aibi	aibi	PROPN
ejpam-3344	107	4	{	{	PUNCT
ejpam-3344	107	5	∧ni=1{µa(ai	∧ni=1{µa(ai	PROPN
ejpam-3344	107	6	)	)	PUNCT
ejpam-3344	107	7	∧	∧	PROPN
ejpam-3344	107	8	µb(bi	µb(bi	PROPN
ejpam-3344	107	9	)	)	PUNCT
ejpam-3344	107	10	}	}	PUNCT
ejpam-3344	107	11	}	}	PUNCT
ejpam-3344	107	12	if	if	SCONJ
ejpam-3344	107	13	x	x	X
ejpam-3344	107	14	=	=	PUNCT
ejpam-3344	107	15	n∑	n∑	PROPN
ejpam-3344	107	16	i=1	i=1	PROPN
ejpam-3344	107	17	aibi	aibi	NOUN
ejpam-3344	107	18	,	,	PUNCT
ejpam-3344	107	19	ai	ai	VERB
ejpam-3344	107	20	,	,	PUNCT
ejpam-3344	107	21	bi	bi	NOUN
ejpam-3344	107	22	∈	∈	PROPN
ejpam-3344	107	23	r	r	NOUN
ejpam-3344	107	24	0	0	PUNCT
ejpam-3344	108	1	if	if	SCONJ
ejpam-3344	108	2	x	x	PROPN
ejpam-3344	108	3	6=	6=	NUM
ejpam-3344	108	4	n∑	n∑	PROPN
ejpam-3344	108	5	i=1	i=1	PROPN
ejpam-3344	108	6	aibi	aibi	NOUN
ejpam-3344	108	7	and	and	CCONJ
ejpam-3344	108	8	(	(	PUNCT
ejpam-3344	108	9	γa	γa	NOUN
ejpam-3344	108	10	◦	◦	VERB
ejpam-3344	108	11	γb)(x	γb)(x	PROPN
ejpam-3344	108	12	)	)	PUNCT
ejpam-3344	109	1	=	=	PUNCT
ejpam-3344	109	2			PUNCT
ejpam-3344	109	3	∧	∧	NOUN
ejpam-3344	109	4	x=	x=	PUNCT
ejpam-3344	110	1	n∑	n∑	PROPN
ejpam-3344	110	2	i=1	i=1	PROPN
ejpam-3344	110	3	aibi	aibi	PROPN
ejpam-3344	110	4	{	{	PUNCT
ejpam-3344	110	5	∨ni=1{γa(ai	∨ni=1{γa(ai	PROPN
ejpam-3344	110	6	)	)	PUNCT
ejpam-3344	110	7	∨	∨	NOUN
ejpam-3344	110	8	γb(bi	γb(bi	PROPN
ejpam-3344	110	9	)	)	PUNCT
ejpam-3344	110	10	}	}	PUNCT
ejpam-3344	110	11	}	}	PUNCT
ejpam-3344	110	12	if	if	SCONJ
ejpam-3344	110	13	x	x	X
ejpam-3344	110	14	=	=	PUNCT
ejpam-3344	110	15	n∑	n∑	PROPN
ejpam-3344	110	16	i=1	i=1	PROPN
ejpam-3344	110	17	aibi	aibi	NOUN
ejpam-3344	110	18	,	,	PUNCT
ejpam-3344	110	19	ai	ai	VERB
ejpam-3344	110	20	,	,	PUNCT
ejpam-3344	110	21	bi	bi	NOUN
ejpam-3344	110	22	∈	∈	PROPN
ejpam-3344	110	23	r	r	NOUN
ejpam-3344	110	24	1	1	NUM
ejpam-3344	111	1	if	if	SCONJ
ejpam-3344	111	2	x	x	PROPN
ejpam-3344	111	3	6=	6=	NUM
ejpam-3344	111	4	n∑	n∑	PROPN
ejpam-3344	111	5	i=1	i=1	PROPN
ejpam-3344	111	6	aibi	aibi	NOUN
ejpam-3344	111	7	now	now	ADV
ejpam-3344	111	8	we	we	PRON
ejpam-3344	111	9	are	be	AUX
ejpam-3344	111	10	giving	give	VERB
ejpam-3344	111	11	the	the	DET
ejpam-3344	111	12	some	some	DET
ejpam-3344	111	13	fundamental	fundamental	ADJ
ejpam-3344	111	14	properties	property	NOUN
ejpam-3344	111	15	,	,	PUNCT
ejpam-3344	111	16	which	which	PRON
ejpam-3344	111	17	will	will	AUX
ejpam-3344	111	18	be	be	AUX
ejpam-3344	111	19	very	very	ADV
ejpam-3344	111	20	helpful	helpful	ADJ
ejpam-3344	111	21	for	for	ADP
ejpam-3344	111	22	next	next	ADJ
ejpam-3344	111	23	section	section	NOUN
ejpam-3344	111	24	.	.	PUNCT
ejpam-3344	112	1	theorem	theorem	NOUN
ejpam-3344	112	2	1	1	NUM
ejpam-3344	112	3	.	.	PUNCT
ejpam-3344	113	1	let	let	VERB
ejpam-3344	113	2	a	a	PRON
ejpam-3344	113	3	and	and	CCONJ
ejpam-3344	113	4	b	b	NOUN
ejpam-3344	113	5	be	be	AUX
ejpam-3344	113	6	two	two	NUM
ejpam-3344	113	7	non	non	ADJ
ejpam-3344	113	8	-	-	ADJ
ejpam-3344	113	9	empty	empty	ADJ
ejpam-3344	113	10	subsets	subset	NOUN
ejpam-3344	113	11	of	of	ADP
ejpam-3344	113	12	an	an	DET
ejpam-3344	113	13	la	la	ADJ
ejpam-3344	113	14	-	-	PUNCT
ejpam-3344	113	15	ring	ring	NOUN
ejpam-3344	113	16	r.	r.	PROPN
ejpam-3344	113	17	then	then	ADV
ejpam-3344	113	18	the	the	DET
ejpam-3344	113	19	following	follow	VERB
ejpam-3344	113	20	conditions	condition	NOUN
ejpam-3344	113	21	hold	hold	VERB
ejpam-3344	113	22	.	.	PUNCT
ejpam-3344	114	1	(	(	PUNCT
ejpam-3344	114	2	1	1	X
ejpam-3344	114	3	)	)	PUNCT
ejpam-3344	114	4	if	if	SCONJ
ejpam-3344	114	5	a	a	DET
ejpam-3344	114	6	⊆	⊆	NUM
ejpam-3344	114	7	b	b	NOUN
ejpam-3344	114	8	then	then	ADV
ejpam-3344	114	9	χa	χa	VERB
ejpam-3344	114	10	⊆	⊆	NUM
ejpam-3344	114	11	χb	χb	PROPN
ejpam-3344	114	12	.	.	PUNCT
ejpam-3344	115	1	(	(	PUNCT
ejpam-3344	115	2	2	2	X
ejpam-3344	115	3	)	)	PUNCT
ejpam-3344	115	4	χa	χa	NOUN
ejpam-3344	115	5	◦	◦	NOUN
ejpam-3344	115	6	χb	χb	PROPN
ejpam-3344	115	7	=	=	PUNCT
ejpam-3344	115	8	χab	χab	PROPN
ejpam-3344	115	9	.	.	PUNCT
ejpam-3344	116	1	(	(	PUNCT
ejpam-3344	116	2	3	3	X
ejpam-3344	116	3	)	)	PUNCT
ejpam-3344	116	4	χa	χa	NOUN
ejpam-3344	116	5	∪	∪	NOUN
ejpam-3344	116	6	χb	χb	ADP
ejpam-3344	116	7	=	=	SYM
ejpam-3344	116	8	χa∪b	χa∪b	ADJ
ejpam-3344	116	9	.	.	PUNCT
ejpam-3344	117	1	(	(	PUNCT
ejpam-3344	117	2	4	4	X
ejpam-3344	117	3	)	)	PUNCT
ejpam-3344	117	4	χa	χa	NOUN
ejpam-3344	117	5	∩	∩	NOUN
ejpam-3344	117	6	χb	χb	ADP
ejpam-3344	117	7	=	=	SYM
ejpam-3344	117	8	χa∩b	χa∩b	PROPN
ejpam-3344	117	9	.	.	PUNCT
ejpam-3344	118	1	proof	proof	NOUN
ejpam-3344	118	2	.	.	PUNCT
ejpam-3344	119	1	straight	straight	ADV
ejpam-3344	119	2	forward	forward	ADV
ejpam-3344	119	3	.	.	PUNCT
ejpam-3344	120	1	let	let	VERB
ejpam-3344	120	2	a	a	DET
ejpam-3344	120	3	=	=	SYM
ejpam-3344	120	4	(	(	PUNCT
ejpam-3344	120	5	µa	µa	PROPN
ejpam-3344	120	6	,	,	PUNCT
ejpam-3344	120	7	γa	γa	PROPN
ejpam-3344	120	8	)	)	PUNCT
ejpam-3344	120	9	and	and	CCONJ
ejpam-3344	120	10	b	b	X
ejpam-3344	120	11	=	=	SYM
ejpam-3344	120	12	(	(	PUNCT
ejpam-3344	120	13	µb	µb	PROPN
ejpam-3344	120	14	,	,	PUNCT
ejpam-3344	120	15	γb	γb	PROPN
ejpam-3344	120	16	)	)	PUNCT
ejpam-3344	120	17	be	be	VERB
ejpam-3344	120	18	two	two	NUM
ejpam-3344	120	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	120	20	fuzzy	fuzzy	ADJ
ejpam-3344	120	21	sets	set	NOUN
ejpam-3344	120	22	of	of	ADP
ejpam-3344	120	23	an	an	DET
ejpam-3344	120	24	la	la	ADJ
ejpam-3344	120	25	-	-	PUNCT
ejpam-3344	120	26	ring	ring	NOUN
ejpam-3344	120	27	r.	r.	NOUN
ejpam-3344	120	28	the	the	DET
ejpam-3344	120	29	sum	sum	NOUN
ejpam-3344	120	30	of	of	ADP
ejpam-3344	120	31	a	a	PRON
ejpam-3344	120	32	and	and	CCONJ
ejpam-3344	120	33	b	b	NOUN
ejpam-3344	120	34	is	be	AUX
ejpam-3344	120	35	denoted	denote	VERB
ejpam-3344	120	36	by	by	ADP
ejpam-3344	120	37	a+b	a+b	PROPN
ejpam-3344	120	38	=	=	PUNCT
ejpam-3344	120	39	(	(	PUNCT
ejpam-3344	120	40	µa	µa	AUX
ejpam-3344	120	41	+	+	CCONJ
ejpam-3344	120	42	µb	µb	VERB
ejpam-3344	120	43	,	,	PUNCT
ejpam-3344	120	44	γa	γa	PROPN
ejpam-3344	120	45	+	+	CCONJ
ejpam-3344	120	46	γb	γb	PROPN
ejpam-3344	120	47	)	)	PUNCT
ejpam-3344	120	48	and	and	CCONJ
ejpam-3344	120	49	defined	define	VERB
ejpam-3344	120	50	by	by	ADP
ejpam-3344	120	51	(	(	PUNCT
ejpam-3344	120	52	µa	µa	NOUN
ejpam-3344	120	53	+	+	CCONJ
ejpam-3344	120	54	µb)(x	µb)(x	NOUN
ejpam-3344	120	55	)	)	PUNCT
ejpam-3344	120	56	=	=	SYM
ejpam-3344	120	57	∨x	∨x	NOUN
ejpam-3344	120	58	=	=	SYM
ejpam-3344	120	59	y+z(µa(y	y+z(µa(y	PROPN
ejpam-3344	120	60	)	)	PUNCT
ejpam-3344	120	61	∧	∧	NOUN
ejpam-3344	120	62	µb(z	µb(z	NOUN
ejpam-3344	120	63	)	)	PUNCT
ejpam-3344	120	64	)	)	PUNCT
ejpam-3344	121	1	and	and	CCONJ
ejpam-3344	121	2	(	(	PUNCT
ejpam-3344	121	3	γa	γa	PROPN
ejpam-3344	121	4	+	+	CCONJ
ejpam-3344	121	5	γb)(x	γb)(x	PROPN
ejpam-3344	121	6	)	)	PUNCT
ejpam-3344	121	7	=	=	SYM
ejpam-3344	121	8	∧x	∧x	PROPN
ejpam-3344	121	9	=	=	SYM
ejpam-3344	121	10	y+z(γa(y	y+z(γa(y	PROPN
ejpam-3344	121	11	)	)	PUNCT
ejpam-3344	121	12	∨	∨	NUM
ejpam-3344	121	13	γb(z	γb(z	NUM
ejpam-3344	121	14	)	)	PUNCT
ejpam-3344	121	15	)	)	PUNCT
ejpam-3344	121	16	,	,	PUNCT
ejpam-3344	121	17	for	for	ADP
ejpam-3344	121	18	all	all	DET
ejpam-3344	121	19	x	x	PROPN
ejpam-3344	121	20	∈	∈	PROPN
ejpam-3344	121	21	r.	r.	PROPN
ejpam-3344	121	22	lemma	lemma	PROPN
ejpam-3344	121	23	1	1	X
ejpam-3344	121	24	.	.	PUNCT
ejpam-3344	121	25	let	let	VERB
ejpam-3344	121	26	a	a	PRON
ejpam-3344	121	27	=	=	SYM
ejpam-3344	121	28	(	(	PUNCT
ejpam-3344	121	29	µa	µa	PROPN
ejpam-3344	121	30	,	,	PUNCT
ejpam-3344	121	31	γa	γa	PROPN
ejpam-3344	121	32	)	)	PUNCT
ejpam-3344	121	33	and	and	CCONJ
ejpam-3344	121	34	b	b	X
ejpam-3344	121	35	=	=	SYM
ejpam-3344	121	36	(	(	PUNCT
ejpam-3344	121	37	µb	µb	PROPN
ejpam-3344	121	38	,	,	PUNCT
ejpam-3344	121	39	γb	γb	PROPN
ejpam-3344	121	40	)	)	PUNCT
ejpam-3344	121	41	be	be	VERB
ejpam-3344	121	42	two	two	NUM
ejpam-3344	121	43	intuitionistic	intuitionistic	ADJ
ejpam-3344	121	44	fuzzy	fuzzy	ADJ
ejpam-3344	121	45	sets	set	NOUN
ejpam-3344	121	46	of	of	ADP
ejpam-3344	121	47	an	an	DET
ejpam-3344	121	48	la	la	ADJ
ejpam-3344	121	49	-	-	PUNCT
ejpam-3344	121	50	ring	ring	NOUN
ejpam-3344	121	51	r.	r.	PROPN
ejpam-3344	121	52	then	then	ADV
ejpam-3344	121	53	a+b	a+b	PROPN
ejpam-3344	121	54	is	be	AUX
ejpam-3344	121	55	also	also	ADV
ejpam-3344	121	56	an	an	DET
ejpam-3344	121	57	intuitionistic	intuitionistic	ADJ
ejpam-3344	121	58	fuzzy	fuzzy	ADJ
ejpam-3344	121	59	set	set	NOUN
ejpam-3344	121	60	of	of	ADP
ejpam-3344	121	61	r.	r.	PROPN
ejpam-3344	121	62	proof	proof	NOUN
ejpam-3344	121	63	.	.	PUNCT
ejpam-3344	122	1	it	it	PRON
ejpam-3344	122	2	is	be	AUX
ejpam-3344	122	3	sufficient	sufficient	ADJ
ejpam-3344	122	4	to	to	PART
ejpam-3344	122	5	show	show	VERB
ejpam-3344	122	6	that	that	SCONJ
ejpam-3344	122	7	0	0	NUM
ejpam-3344	122	8	≤	≤	NUM
ejpam-3344	122	9	(	(	PUNCT
ejpam-3344	122	10	µa	µa	NOUN
ejpam-3344	122	11	+	+	CCONJ
ejpam-3344	122	12	µb)(x	µb)(x	NOUN
ejpam-3344	122	13	)	)	PUNCT
ejpam-3344	123	1	+	+	CCONJ
ejpam-3344	123	2	(	(	PUNCT
ejpam-3344	123	3	γa	γa	PROPN
ejpam-3344	123	4	+	+	CCONJ
ejpam-3344	123	5	γb)(x	γb)(x	PROPN
ejpam-3344	123	6	)	)	PUNCT
ejpam-3344	123	7	≤	≤	NUM
ejpam-3344	123	8	1	1	NUM
ejpam-3344	123	9	for	for	ADP
ejpam-3344	123	10	all	all	DET
ejpam-3344	123	11	x	x	PROPN
ejpam-3344	123	12	∈	∈	PROPN
ejpam-3344	123	13	r.	r.	NOUN
ejpam-3344	123	14	now	now	ADV
ejpam-3344	123	15	(	(	PUNCT
ejpam-3344	123	16	µa	µa	NOUN
ejpam-3344	123	17	+	+	CCONJ
ejpam-3344	123	18	µb)(x	µb)(x	NOUN
ejpam-3344	123	19	)	)	PUNCT
ejpam-3344	123	20	=	=	SYM
ejpam-3344	123	21	∨x	∨x	NOUN
ejpam-3344	123	22	=	=	SYM
ejpam-3344	123	23	y+z(µa(y	y+z(µa(y	PROPN
ejpam-3344	123	24	)	)	PUNCT
ejpam-3344	123	25	∧	∧	NOUN
ejpam-3344	123	26	µb(z	µb(z	NOUN
ejpam-3344	123	27	)	)	PUNCT
ejpam-3344	123	28	)	)	PUNCT
ejpam-3344	123	29	≤	≤	NUM
ejpam-3344	123	30	∨x	∨x	NOUN
ejpam-3344	123	31	=	=	SYM
ejpam-3344	123	32	y+z((1−	y+z((1−	X
ejpam-3344	123	33	γa(y	γa(y	NOUN
ejpam-3344	123	34	)	)	PUNCT
ejpam-3344	123	35	)	)	PUNCT
ejpam-3344	124	1	∧	∧	NOUN
ejpam-3344	124	2	(	(	PUNCT
ejpam-3344	124	3	1−	1−	NUM
ejpam-3344	124	4	γb(z	γb(z	NUM
ejpam-3344	124	5	)	)	PUNCT
ejpam-3344	124	6	)	)	PUNCT
ejpam-3344	124	7	)	)	PUNCT
ejpam-3344	125	1	n.	n.	PROPN
ejpam-3344	125	2	kausar	kausar	PROPN
ejpam-3344	125	3	,	,	PUNCT
ejpam-3344	125	4	m.	m.	NOUN
ejpam-3344	125	5	a.	a.	PROPN
ejpam-3344	125	6	waqar	waqar	PROPN
ejpam-3344	125	7	/	/	SYM
ejpam-3344	125	8	eur	eur	PROPN
ejpam-3344	125	9	.	.	PUNCT
ejpam-3344	126	1	j.	j.	PROPN
ejpam-3344	126	2	pure	pure	PROPN
ejpam-3344	126	3	appl	appl	PROPN
ejpam-3344	126	4	.	.	PROPN
ejpam-3344	126	5	math	math	PROPN
ejpam-3344	126	6	,	,	PUNCT
ejpam-3344	126	7	12	12	NUM
ejpam-3344	126	8	(	(	PUNCT
ejpam-3344	126	9	1	1	NUM
ejpam-3344	126	10	)	)	PUNCT
ejpam-3344	126	11	(	(	PUNCT
ejpam-3344	126	12	2019	2019	NUM
ejpam-3344	126	13	)	)	PUNCT
ejpam-3344	126	14	,	,	PUNCT
ejpam-3344	126	15	226	226	NUM
ejpam-3344	126	16	-	-	SYM
ejpam-3344	126	17	250	250	NUM
ejpam-3344	126	18	231	231	NUM
ejpam-3344	126	19	=	=	SYM
ejpam-3344	126	20	1−	1−	NUM
ejpam-3344	126	21	∧x	∧x	PROPN
ejpam-3344	126	22	=	=	SYM
ejpam-3344	126	23	y+z(γa(y	y+z(γa(y	PROPN
ejpam-3344	126	24	)	)	PUNCT
ejpam-3344	126	25	∨	∨	NUM
ejpam-3344	126	26	γb(z	γb(z	NUM
ejpam-3344	126	27	)	)	PUNCT
ejpam-3344	126	28	)	)	PUNCT
ejpam-3344	127	1	=	=	SYM
ejpam-3344	127	2	1−	1−	NUM
ejpam-3344	127	3	(	(	PUNCT
ejpam-3344	127	4	γa	γa	NOUN
ejpam-3344	127	5	+	+	CCONJ
ejpam-3344	127	6	γb)(x	γb)(x	PROPN
ejpam-3344	127	7	)	)	PUNCT
ejpam-3344	127	8	.	.	PUNCT
ejpam-3344	128	1	since	since	SCONJ
ejpam-3344	128	2	µa(y	µa(y	NOUN
ejpam-3344	128	3	)	)	PUNCT
ejpam-3344	128	4	≤	≤	NUM
ejpam-3344	128	5	1	1	NUM
ejpam-3344	128	6	−	−	NOUN
ejpam-3344	128	7	γa(y	γa(y	NUM
ejpam-3344	128	8	)	)	PUNCT
ejpam-3344	128	9	and	and	CCONJ
ejpam-3344	128	10	µa(z	µa(z	NUM
ejpam-3344	128	11	)	)	PUNCT
ejpam-3344	128	12	≤	≤	NUM
ejpam-3344	128	13	1	1	NUM
ejpam-3344	128	14	−	−	NOUN
ejpam-3344	128	15	γa(z	γa(z	PROPN
ejpam-3344	128	16	)	)	PUNCT
ejpam-3344	128	17	for	for	ADP
ejpam-3344	128	18	all	all	DET
ejpam-3344	128	19	y	y	PROPN
ejpam-3344	128	20	,	,	PUNCT
ejpam-3344	128	21	z	z	PROPN
ejpam-3344	128	22	∈	∈	PROPN
ejpam-3344	128	23	r.	r.	NOUN
ejpam-3344	128	24	hence	hence	ADV
ejpam-3344	128	25	a	a	PRON
ejpam-3344	128	26	+	+	X
ejpam-3344	128	27	b	b	NOUN
ejpam-3344	128	28	is	be	AUX
ejpam-3344	128	29	an	an	DET
ejpam-3344	128	30	intuitionistic	intuitionistic	ADJ
ejpam-3344	128	31	fuzzy	fuzzy	ADJ
ejpam-3344	128	32	set	set	NOUN
ejpam-3344	128	33	of	of	ADP
ejpam-3344	128	34	r.	r.	PROPN
ejpam-3344	128	35	lemma	lemma	PROPN
ejpam-3344	129	1	2	2	NUM
ejpam-3344	129	2	.	.	PUNCT
ejpam-3344	130	1	every	every	DET
ejpam-3344	130	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	130	3	fuzzy	fuzzy	ADJ
ejpam-3344	130	4	left	left	NOUN
ejpam-3344	130	5	(	(	PUNCT
ejpam-3344	130	6	resp	resp	NOUN
ejpam-3344	130	7	.	.	PUNCT
ejpam-3344	131	1	right	right	ADJ
ejpam-3344	131	2	,	,	PUNCT
ejpam-3344	131	3	two	two	NUM
ejpam-3344	131	4	-	-	PUNCT
ejpam-3344	131	5	sided	sided	ADJ
ejpam-3344	131	6	)	)	PUNCT
ejpam-3344	131	7	ideal	ideal	NOUN
ejpam-3344	131	8	of	of	ADP
ejpam-3344	131	9	an	an	DET
ejpam-3344	131	10	la	la	ADJ
ejpam-3344	131	11	-	-	PUNCT
ejpam-3344	131	12	ring	ring	NOUN
ejpam-3344	131	13	r	r	NOUN
ejpam-3344	131	14	is	be	AUX
ejpam-3344	131	15	an	an	DET
ejpam-3344	131	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	131	17	fuzzy	fuzzy	ADJ
ejpam-3344	131	18	bi	bi	NOUN
ejpam-3344	131	19	-	-	NOUN
ejpam-3344	131	20	ideal	ideal	NOUN
ejpam-3344	131	21	of	of	ADP
ejpam-3344	131	22	r.	r.	PROPN
ejpam-3344	131	23	proof	proof	NOUN
ejpam-3344	131	24	.	.	PUNCT
ejpam-3344	132	1	straight	straight	ADV
ejpam-3344	132	2	forward	forward	ADV
ejpam-3344	132	3	.	.	PUNCT
ejpam-3344	133	1	lemma	lemma	PROPN
ejpam-3344	133	2	3	3	NUM
ejpam-3344	133	3	.	.	PUNCT
ejpam-3344	134	1	every	every	DET
ejpam-3344	134	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	134	3	fuzzy	fuzzy	ADJ
ejpam-3344	134	4	bi	bi	NOUN
ejpam-3344	134	5	-	-	NOUN
ejpam-3344	134	6	ideal	ideal	NOUN
ejpam-3344	134	7	of	of	ADP
ejpam-3344	134	8	an	an	DET
ejpam-3344	134	9	la	la	ADJ
ejpam-3344	134	10	-	-	PUNCT
ejpam-3344	134	11	ring	ring	NOUN
ejpam-3344	134	12	r	r	NOUN
ejpam-3344	134	13	is	be	AUX
ejpam-3344	134	14	an	an	DET
ejpam-3344	134	15	intuitionistic	intuitionistic	ADJ
ejpam-3344	134	16	fuzzy	fuzzy	ADJ
ejpam-3344	134	17	(	(	PUNCT
ejpam-3344	134	18	1	1	NUM
ejpam-3344	134	19	,	,	PUNCT
ejpam-3344	134	20	2)-ideal	2)-ideal	NUM
ejpam-3344	134	21	of	of	ADP
ejpam-3344	134	22	r.	r.	PROPN
ejpam-3344	134	23	proof	proof	NOUN
ejpam-3344	134	24	.	.	PUNCT
ejpam-3344	135	1	let	let	VERB
ejpam-3344	135	2	a	a	DET
ejpam-3344	135	3	=	=	SYM
ejpam-3344	135	4	(	(	PUNCT
ejpam-3344	135	5	µa	µa	PROPN
ejpam-3344	135	6	,	,	PUNCT
ejpam-3344	135	7	γa	γa	PROPN
ejpam-3344	135	8	)	)	PUNCT
ejpam-3344	135	9	be	be	VERB
ejpam-3344	135	10	an	an	DET
ejpam-3344	135	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	135	12	fuzzy	fuzzy	ADJ
ejpam-3344	135	13	bi	bi	NOUN
ejpam-3344	135	14	-	-	NOUN
ejpam-3344	135	15	ideal	ideal	NOUN
ejpam-3344	135	16	of	of	ADP
ejpam-3344	135	17	r	r	NOUN
ejpam-3344	135	18	and	and	CCONJ
ejpam-3344	135	19	a	a	DET
ejpam-3344	135	20	,	,	PUNCT
ejpam-3344	135	21	x	x	NOUN
ejpam-3344	135	22	,	,	PUNCT
ejpam-3344	135	23	y	y	PROPN
ejpam-3344	135	24	,	,	PUNCT
ejpam-3344	135	25	z	z	PROPN
ejpam-3344	135	26	∈	∈	PROPN
ejpam-3344	135	27	r.	r.	NOUN
ejpam-3344	135	28	thus	thus	ADV
ejpam-3344	135	29	µa((xa)(yz	µa((xa)(yz	PROPN
ejpam-3344	135	30	)	)	PUNCT
ejpam-3344	135	31	)	)	PUNCT
ejpam-3344	135	32	≥	≥	PROPN
ejpam-3344	135	33	min{µa(x	min{µa(x	NOUN
ejpam-3344	135	34	)	)	PUNCT
ejpam-3344	135	35	,	,	PUNCT
ejpam-3344	135	36	µa(yz	µa(yz	PROPN
ejpam-3344	135	37	)	)	PUNCT
ejpam-3344	135	38	}	}	PUNCT
ejpam-3344	135	39	≥	≥	PROPN
ejpam-3344	135	40	min{µa(x	min{µa(x	NOUN
ejpam-3344	135	41	)	)	PUNCT
ejpam-3344	135	42	,	,	PUNCT
ejpam-3344	135	43	µa(y	µa(y	NOUN
ejpam-3344	135	44	)	)	PUNCT
ejpam-3344	135	45	,	,	PUNCT
ejpam-3344	135	46	µa(z	µa(z	NUM
ejpam-3344	135	47	)	)	PUNCT
ejpam-3344	135	48	}	}	PUNCT
ejpam-3344	135	49	and	and	CCONJ
ejpam-3344	135	50	γa((xa)(yz	γa((xa)(yz	PROPN
ejpam-3344	135	51	)	)	PUNCT
ejpam-3344	135	52	)	)	PUNCT
ejpam-3344	135	53	≤	≤	NUM
ejpam-3344	135	54	max{γa(x	max{γa(x	NOUN
ejpam-3344	135	55	)	)	PUNCT
ejpam-3344	135	56	,	,	PUNCT
ejpam-3344	135	57	γa(yz	γa(yz	PROPN
ejpam-3344	135	58	)	)	PUNCT
ejpam-3344	135	59	}	}	PUNCT
ejpam-3344	135	60	≤	≤	NUM
ejpam-3344	135	61	max{γa(x	max{γa(x	NOUN
ejpam-3344	135	62	)	)	PUNCT
ejpam-3344	135	63	,	,	PUNCT
ejpam-3344	135	64	γa(y	γa(y	NUM
ejpam-3344	135	65	)	)	PUNCT
ejpam-3344	135	66	,	,	PUNCT
ejpam-3344	135	67	γa(z	γa(z	PROPN
ejpam-3344	135	68	)	)	PUNCT
ejpam-3344	135	69	}	}	PUNCT
ejpam-3344	135	70	.	.	PUNCT
ejpam-3344	136	1	hence	hence	ADV
ejpam-3344	136	2	a	a	PRON
ejpam-3344	136	3	=	=	X
ejpam-3344	136	4	(	(	PUNCT
ejpam-3344	136	5	µa	µa	PROPN
ejpam-3344	136	6	,	,	PUNCT
ejpam-3344	136	7	γa	γa	PROPN
ejpam-3344	136	8	)	)	PUNCT
ejpam-3344	136	9	is	be	AUX
ejpam-3344	136	10	an	an	DET
ejpam-3344	136	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	136	12	fuzzy	fuzzy	ADJ
ejpam-3344	136	13	(	(	PUNCT
ejpam-3344	136	14	1	1	NUM
ejpam-3344	136	15	,	,	PUNCT
ejpam-3344	136	16	2)-ideal	2)-ideal	NUM
ejpam-3344	136	17	of	of	ADP
ejpam-3344	136	18	r.	r.	PROPN
ejpam-3344	136	19	remark	remark	PROPN
ejpam-3344	136	20	1	1	NUM
ejpam-3344	136	21	.	.	PUNCT
ejpam-3344	137	1	every	every	DET
ejpam-3344	137	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	137	3	fuzzy	fuzzy	ADJ
ejpam-3344	137	4	left	left	NOUN
ejpam-3344	137	5	(	(	PUNCT
ejpam-3344	137	6	resp	resp	NOUN
ejpam-3344	137	7	.	.	PUNCT
ejpam-3344	138	1	right	right	ADJ
ejpam-3344	138	2	,	,	PUNCT
ejpam-3344	138	3	two	two	NUM
ejpam-3344	138	4	-	-	PUNCT
ejpam-3344	138	5	sided	sided	ADJ
ejpam-3344	138	6	)	)	PUNCT
ejpam-3344	138	7	ideal	ideal	NOUN
ejpam-3344	138	8	of	of	ADP
ejpam-3344	138	9	an	an	DET
ejpam-3344	138	10	la	la	ADJ
ejpam-3344	138	11	-	-	PUNCT
ejpam-3344	138	12	ring	ring	NOUN
ejpam-3344	138	13	r	r	NOUN
ejpam-3344	138	14	is	be	AUX
ejpam-3344	138	15	an	an	DET
ejpam-3344	138	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	138	17	fuzzy	fuzzy	ADJ
ejpam-3344	138	18	(	(	PUNCT
ejpam-3344	138	19	1	1	NUM
ejpam-3344	138	20	,	,	PUNCT
ejpam-3344	138	21	2)-ideal	2)-ideal	NUM
ejpam-3344	138	22	of	of	ADP
ejpam-3344	138	23	r.	r.	PROPN
ejpam-3344	138	24	proposition	proposition	PROPN
ejpam-3344	138	25	1	1	NUM
ejpam-3344	138	26	.	.	PUNCT
ejpam-3344	139	1	let	let	VERB
ejpam-3344	139	2	r	r	PRON
ejpam-3344	139	3	be	be	AUX
ejpam-3344	139	4	an	an	DET
ejpam-3344	139	5	la	la	ADJ
ejpam-3344	139	6	-	-	NOUN
ejpam-3344	139	7	ring	ring	NOUN
ejpam-3344	139	8	having	have	VERB
ejpam-3344	139	9	the	the	DET
ejpam-3344	139	10	property	property	NOUN
ejpam-3344	139	11	a	a	DET
ejpam-3344	139	12	=	=	NOUN
ejpam-3344	139	13	a2	a2	PROPN
ejpam-3344	139	14	for	for	ADP
ejpam-3344	139	15	every	every	DET
ejpam-3344	139	16	a	a	DET
ejpam-3344	139	17	∈	∈	PROPN
ejpam-3344	139	18	r.	r.	NOUN
ejpam-3344	139	19	then	then	ADV
ejpam-3344	139	20	every	every	DET
ejpam-3344	139	21	intuitionistic	intuitionistic	ADJ
ejpam-3344	139	22	fuzzy	fuzzy	ADJ
ejpam-3344	139	23	(	(	PUNCT
ejpam-3344	139	24	1	1	NUM
ejpam-3344	139	25	,	,	PUNCT
ejpam-3344	139	26	2)-ideal	2)-ideal	NUM
ejpam-3344	139	27	of	of	ADP
ejpam-3344	139	28	r	r	NOUN
ejpam-3344	139	29	is	be	AUX
ejpam-3344	139	30	an	an	DET
ejpam-3344	139	31	intuitionistic	intuitionistic	ADJ
ejpam-3344	139	32	fuzzy	fuzzy	ADJ
ejpam-3344	139	33	bi	bi	NOUN
ejpam-3344	139	34	-	-	NOUN
ejpam-3344	139	35	ideal	ideal	NOUN
ejpam-3344	139	36	of	of	ADP
ejpam-3344	139	37	r.	r.	PROPN
ejpam-3344	139	38	proof	proof	NOUN
ejpam-3344	139	39	.	.	PUNCT
ejpam-3344	140	1	suppose	suppose	VERB
ejpam-3344	140	2	that	that	SCONJ
ejpam-3344	140	3	a	a	DET
ejpam-3344	140	4	=	=	SYM
ejpam-3344	140	5	(	(	PUNCT
ejpam-3344	140	6	µa	µa	PROPN
ejpam-3344	140	7	,	,	PUNCT
ejpam-3344	140	8	γa	γa	PROPN
ejpam-3344	140	9	)	)	PUNCT
ejpam-3344	140	10	is	be	AUX
ejpam-3344	140	11	an	an	DET
ejpam-3344	140	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	140	13	fuzzy	fuzzy	ADJ
ejpam-3344	140	14	(	(	PUNCT
ejpam-3344	140	15	1	1	NUM
ejpam-3344	140	16	,	,	PUNCT
ejpam-3344	140	17	2)-ideal	2)-ideal	NUM
ejpam-3344	140	18	of	of	ADP
ejpam-3344	140	19	r	r	NOUN
ejpam-3344	140	20	and	and	CCONJ
ejpam-3344	140	21	a	a	PRON
ejpam-3344	140	22	,	,	PUNCT
ejpam-3344	140	23	x	x	NOUN
ejpam-3344	140	24	,	,	PUNCT
ejpam-3344	140	25	y	y	PROPN
ejpam-3344	140	26	∈	∈	PROPN
ejpam-3344	140	27	r.	r.	NOUN
ejpam-3344	140	28	thus	thus	ADV
ejpam-3344	140	29	µa((xa)y	µa((xa)y	NOUN
ejpam-3344	140	30	)	)	PUNCT
ejpam-3344	140	31	=	=	SYM
ejpam-3344	140	32	µa((xa)(yy	µa((xa)(yy	ADJ
ejpam-3344	140	33	)	)	PUNCT
ejpam-3344	140	34	)	)	PUNCT
ejpam-3344	140	35	≥	≥	PROPN
ejpam-3344	140	36	min{µa(x	min{µa(x	NOUN
ejpam-3344	140	37	)	)	PUNCT
ejpam-3344	140	38	,	,	PUNCT
ejpam-3344	140	39	µa(y	µa(y	NOUN
ejpam-3344	140	40	)	)	PUNCT
ejpam-3344	140	41	,	,	PUNCT
ejpam-3344	140	42	µa(y	µa(y	NOUN
ejpam-3344	140	43	)	)	PUNCT
ejpam-3344	140	44	}	}	PUNCT
ejpam-3344	141	1	=	=	SYM
ejpam-3344	141	2	min{µa(x	min{µa(x	NOUN
ejpam-3344	141	3	)	)	PUNCT
ejpam-3344	141	4	,	,	PUNCT
ejpam-3344	141	5	µa(y	µa(y	NOUN
ejpam-3344	141	6	)	)	PUNCT
ejpam-3344	141	7	}	}	PUNCT
ejpam-3344	141	8	and	and	CCONJ
ejpam-3344	141	9	γa((xa)y	γa((xa)y	NOUN
ejpam-3344	141	10	)	)	PUNCT
ejpam-3344	141	11	=	=	SYM
ejpam-3344	141	12	γa((xa)(yy	γa((xa)(yy	NOUN
ejpam-3344	141	13	)	)	PUNCT
ejpam-3344	141	14	)	)	PUNCT
ejpam-3344	141	15	≤	≤	NUM
ejpam-3344	141	16	max{γa(x	max{γa(x	NOUN
ejpam-3344	141	17	)	)	PUNCT
ejpam-3344	141	18	,	,	PUNCT
ejpam-3344	141	19	γa(y	γa(y	NOUN
ejpam-3344	141	20	)	)	PUNCT
ejpam-3344	141	21	,	,	PUNCT
ejpam-3344	141	22	γa(y	γa(y	NOUN
ejpam-3344	141	23	)	)	PUNCT
ejpam-3344	141	24	}	}	PUNCT
ejpam-3344	141	25	=	=	SYM
ejpam-3344	141	26	max{γa(x	max{γa(x	NOUN
ejpam-3344	141	27	)	)	PUNCT
ejpam-3344	141	28	,	,	PUNCT
ejpam-3344	141	29	γa(y	γa(y	NOUN
ejpam-3344	141	30	)	)	PUNCT
ejpam-3344	141	31	}	}	PUNCT
ejpam-3344	141	32	.	.	PUNCT
ejpam-3344	142	1	therefore	therefore	ADV
ejpam-3344	142	2	a	a	DET
ejpam-3344	142	3	=	=	SYM
ejpam-3344	142	4	(	(	PUNCT
ejpam-3344	142	5	µa	µa	PROPN
ejpam-3344	142	6	,	,	PUNCT
ejpam-3344	142	7	γa	γa	PROPN
ejpam-3344	142	8	)	)	PUNCT
ejpam-3344	142	9	is	be	AUX
ejpam-3344	142	10	an	an	DET
ejpam-3344	142	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	142	12	fuzzy	fuzzy	ADJ
ejpam-3344	142	13	bi	bi	NOUN
ejpam-3344	142	14	-	-	NOUN
ejpam-3344	142	15	ideal	ideal	NOUN
ejpam-3344	142	16	of	of	ADP
ejpam-3344	142	17	r.	r.	PROPN
ejpam-3344	142	18	theorem	theorem	PROPN
ejpam-3344	142	19	2	2	X
ejpam-3344	142	20	.	.	PUNCT
ejpam-3344	143	1	if	if	SCONJ
ejpam-3344	143	2	{	{	PUNCT
ejpam-3344	143	3	ai}i∈i	ai}i∈i	ADP
ejpam-3344	143	4	is	be	AUX
ejpam-3344	143	5	a	a	DET
ejpam-3344	143	6	family	family	NOUN
ejpam-3344	143	7	of	of	ADP
ejpam-3344	143	8	intuitionistic	intuitionistic	ADJ
ejpam-3344	143	9	fuzzy	fuzzy	ADJ
ejpam-3344	143	10	(	(	PUNCT
ejpam-3344	143	11	1	1	NUM
ejpam-3344	143	12	,	,	PUNCT
ejpam-3344	143	13	2)-ideals	2)-ideals	NUM
ejpam-3344	143	14	of	of	ADP
ejpam-3344	143	15	an	an	DET
ejpam-3344	143	16	la	la	ADJ
ejpam-3344	143	17	-	-	PUNCT
ejpam-3344	143	18	ring	ring	NOUN
ejpam-3344	143	19	r	r	NOUN
ejpam-3344	143	20	,	,	PUNCT
ejpam-3344	143	21	then	then	ADV
ejpam-3344	143	22	∩ai	∩ai	NOUN
ejpam-3344	143	23	is	be	AUX
ejpam-3344	143	24	also	also	ADV
ejpam-3344	143	25	an	an	DET
ejpam-3344	143	26	intuitionistic	intuitionistic	ADJ
ejpam-3344	143	27	fuzzy	fuzzy	ADJ
ejpam-3344	143	28	(	(	PUNCT
ejpam-3344	143	29	1	1	NUM
ejpam-3344	143	30	,	,	PUNCT
ejpam-3344	143	31	2)-ideal	2)-ideal	NUM
ejpam-3344	143	32	of	of	ADP
ejpam-3344	143	33	r	r	NOUN
ejpam-3344	143	34	,	,	PUNCT
ejpam-3344	143	35	where	where	SCONJ
ejpam-3344	143	36	∩ai	∩ai	NOUN
ejpam-3344	143	37	=	=	SYM
ejpam-3344	143	38	(	(	PUNCT
ejpam-3344	143	39	∧µai	∧µai	ADV
ejpam-3344	143	40	,	,	PUNCT
ejpam-3344	143	41	∨γai	∨γai	PROPN
ejpam-3344	143	42	)	)	PUNCT
ejpam-3344	143	43	and	and	CCONJ
ejpam-3344	143	44	∧µai(x	∧µai(x	ADJ
ejpam-3344	143	45	)	)	PUNCT
ejpam-3344	143	46	=	=	PROPN
ejpam-3344	143	47	inf	inf	NOUN
ejpam-3344	143	48	{	{	PUNCT
ejpam-3344	143	49	µai(x	µai(x	PROPN
ejpam-3344	143	50	)	)	PUNCT
ejpam-3344	144	1	|	|	ADV
ejpam-3344	144	2	i	i	PRON
ejpam-3344	144	3	∈	∈	VERB
ejpam-3344	145	1	i	i	PRON
ejpam-3344	145	2	,	,	PUNCT
ejpam-3344	145	3	x	x	PUNCT
ejpam-3344	145	4	∈	∈	NOUN
ejpam-3344	145	5	r	r	NOUN
ejpam-3344	145	6	}	}	PUNCT
ejpam-3344	145	7	and	and	CCONJ
ejpam-3344	145	8	∨	∨	NUM
ejpam-3344	145	9	γai(x	γai(x	PROPN
ejpam-3344	145	10	)	)	PUNCT
ejpam-3344	145	11	=	=	SYM
ejpam-3344	145	12	sup	sup	NOUN
ejpam-3344	145	13	{	{	PUNCT
ejpam-3344	145	14	γai(x	γai(x	PROPN
ejpam-3344	145	15	)	)	PUNCT
ejpam-3344	146	1	|	|	ADV
ejpam-3344	146	2	i	i	PRON
ejpam-3344	146	3	∈	∈	VERB
ejpam-3344	147	1	i	i	PRON
ejpam-3344	147	2	,	,	PUNCT
ejpam-3344	147	3	x	x	PUNCT
ejpam-3344	147	4	∈	∈	NOUN
ejpam-3344	147	5	r	r	NOUN
ejpam-3344	147	6	}	}	PUNCT
ejpam-3344	147	7	.	.	PUNCT
ejpam-3344	148	1	proof	proof	NOUN
ejpam-3344	148	2	.	.	PUNCT
ejpam-3344	149	1	straight	straight	ADV
ejpam-3344	149	2	forward	forward	ADV
ejpam-3344	149	3	.	.	PUNCT
ejpam-3344	150	1	remark	remark	NOUN
ejpam-3344	150	2	2	2	NUM
ejpam-3344	150	3	.	.	PUNCT
ejpam-3344	151	1	intersection	intersection	NOUN
ejpam-3344	151	2	of	of	ADP
ejpam-3344	151	3	a	a	DET
ejpam-3344	151	4	family	family	NOUN
ejpam-3344	151	5	of	of	ADP
ejpam-3344	151	6	intuitionistic	intuitionistic	ADJ
ejpam-3344	151	7	fuzzy	fuzzy	ADJ
ejpam-3344	151	8	bi	bi	NOUN
ejpam-3344	151	9	-	-	NOUN
ejpam-3344	151	10	ideals	ideal	NOUN
ejpam-3344	151	11	of	of	ADP
ejpam-3344	151	12	an	an	DET
ejpam-3344	151	13	la	la	ADJ
ejpam-3344	151	14	-	-	PUNCT
ejpam-3344	151	15	ring	ring	NOUN
ejpam-3344	151	16	r	r	NOUN
ejpam-3344	151	17	,	,	PUNCT
ejpam-3344	151	18	is	be	AUX
ejpam-3344	151	19	also	also	ADV
ejpam-3344	151	20	an	an	DET
ejpam-3344	151	21	intuitionistic	intuitionistic	ADJ
ejpam-3344	151	22	fuzzy	fuzzy	ADJ
ejpam-3344	151	23	bi	bi	NOUN
ejpam-3344	151	24	-	-	NOUN
ejpam-3344	151	25	ideal	ideal	NOUN
ejpam-3344	151	26	of	of	ADP
ejpam-3344	151	27	r.	r.	PROPN
ejpam-3344	151	28	n.	n.	PROPN
ejpam-3344	151	29	kausar	kausar	PROPN
ejpam-3344	151	30	,	,	PUNCT
ejpam-3344	151	31	m.	m.	NOUN
ejpam-3344	151	32	a.	a.	PROPN
ejpam-3344	151	33	waqar	waqar	PROPN
ejpam-3344	151	34	/	/	SYM
ejpam-3344	151	35	eur	eur	PROPN
ejpam-3344	151	36	.	.	PUNCT
ejpam-3344	152	1	j.	j.	PROPN
ejpam-3344	152	2	pure	pure	PROPN
ejpam-3344	152	3	appl	appl	PROPN
ejpam-3344	152	4	.	.	PROPN
ejpam-3344	152	5	math	math	PROPN
ejpam-3344	152	6	,	,	PUNCT
ejpam-3344	152	7	12	12	NUM
ejpam-3344	152	8	(	(	PUNCT
ejpam-3344	152	9	1	1	NUM
ejpam-3344	152	10	)	)	PUNCT
ejpam-3344	152	11	(	(	PUNCT
ejpam-3344	152	12	2019	2019	NUM
ejpam-3344	152	13	)	)	PUNCT
ejpam-3344	152	14	,	,	PUNCT
ejpam-3344	152	15	226	226	NUM
ejpam-3344	152	16	-	-	SYM
ejpam-3344	152	17	250	250	NUM
ejpam-3344	152	18	232	232	NUM
ejpam-3344	152	19	lemma	lemma	PROPN
ejpam-3344	152	20	4	4	NUM
ejpam-3344	152	21	.	.	PUNCT
ejpam-3344	153	1	[	[	X
ejpam-3344	153	2	18	18	NUM
ejpam-3344	153	3	]	]	PUNCT
ejpam-3344	153	4	let	let	VERB
ejpam-3344	153	5	r	r	PRON
ejpam-3344	153	6	be	be	AUX
ejpam-3344	153	7	an	an	DET
ejpam-3344	153	8	la	la	ADJ
ejpam-3344	153	9	-	-	PUNCT
ejpam-3344	153	10	ring	ring	NOUN
ejpam-3344	153	11	and	and	CCONJ
ejpam-3344	153	12	∅	∅	NOUN
ejpam-3344	153	13	6=	6=	ADP
ejpam-3344	153	14	a	a	DET
ejpam-3344	153	15	⊆	⊆	NUM
ejpam-3344	153	16	r.	r.	NOUN
ejpam-3344	153	17	then	then	ADV
ejpam-3344	153	18	a	a	PRON
ejpam-3344	153	19	is	be	AUX
ejpam-3344	153	20	an	an	DET
ejpam-3344	153	21	la	la	NOUN
ejpam-3344	153	22	-	-	PUNCT
ejpam-3344	153	23	subring	subring	NOUN
ejpam-3344	153	24	of	of	ADP
ejpam-3344	153	25	r	r	NOUN
ejpam-3344	153	26	if	if	SCONJ
ejpam-3344	154	1	and	and	CCONJ
ejpam-3344	154	2	only	only	ADV
ejpam-3344	154	3	if	if	SCONJ
ejpam-3344	154	4	the	the	DET
ejpam-3344	154	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	154	6	characteristic	characteristic	ADJ
ejpam-3344	154	7	function	function	NOUN
ejpam-3344	154	8	χa	χa	NOUN
ejpam-3344	154	9	=	=	SYM
ejpam-3344	154	10	〈	〈	PROPN
ejpam-3344	154	11	µχa	µχa	NOUN
ejpam-3344	154	12	,	,	PUNCT
ejpam-3344	154	13	γχa	γχa	ADJ
ejpam-3344	154	14	〉	〉	NUM
ejpam-3344	154	15	of	of	ADP
ejpam-3344	154	16	a	a	PRON
ejpam-3344	154	17	is	be	AUX
ejpam-3344	154	18	an	an	DET
ejpam-3344	154	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	154	20	fuzzy	fuzzy	ADJ
ejpam-3344	154	21	la	la	NOUN
ejpam-3344	154	22	-	-	PUNCT
ejpam-3344	154	23	subring	subring	NOUN
ejpam-3344	154	24	of	of	ADP
ejpam-3344	154	25	r.	r.	PROPN
ejpam-3344	154	26	proposition	proposition	NOUN
ejpam-3344	154	27	2	2	X
ejpam-3344	154	28	.	.	PUNCT
ejpam-3344	155	1	let	let	VERB
ejpam-3344	155	2	r	r	PRON
ejpam-3344	155	3	be	be	AUX
ejpam-3344	155	4	an	an	DET
ejpam-3344	155	5	la	la	ADJ
ejpam-3344	155	6	-	-	PUNCT
ejpam-3344	155	7	ring	ring	NOUN
ejpam-3344	155	8	and	and	CCONJ
ejpam-3344	155	9	∅	∅	NOUN
ejpam-3344	155	10	6=	6=	ADP
ejpam-3344	155	11	a	a	DET
ejpam-3344	155	12	⊆	⊆	NUM
ejpam-3344	155	13	r.	r.	NOUN
ejpam-3344	155	14	then	then	ADV
ejpam-3344	155	15	a	a	PRON
ejpam-3344	155	16	is	be	AUX
ejpam-3344	155	17	a	a	DET
ejpam-3344	155	18	left	left	ADJ
ejpam-3344	155	19	(	(	PUNCT
ejpam-3344	155	20	resp	resp	NOUN
ejpam-3344	155	21	.	.	PUNCT
ejpam-3344	156	1	right	right	ADJ
ejpam-3344	156	2	)	)	PUNCT
ejpam-3344	156	3	ideal	ideal	NOUN
ejpam-3344	156	4	of	of	ADP
ejpam-3344	156	5	r	r	NOUN
ejpam-3344	156	6	if	if	SCONJ
ejpam-3344	157	1	and	and	CCONJ
ejpam-3344	157	2	only	only	ADV
ejpam-3344	157	3	if	if	SCONJ
ejpam-3344	157	4	the	the	DET
ejpam-3344	157	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	157	6	characteristic	characteristic	ADJ
ejpam-3344	157	7	function	function	NOUN
ejpam-3344	157	8	χa	χa	NOUN
ejpam-3344	157	9	=	=	SYM
ejpam-3344	157	10	〈	〈	PROPN
ejpam-3344	157	11	µχa	µχa	NOUN
ejpam-3344	157	12	,	,	PUNCT
ejpam-3344	157	13	γχa	γχa	ADJ
ejpam-3344	157	14	〉	〉	NUM
ejpam-3344	157	15	of	of	ADP
ejpam-3344	157	16	a	a	PRON
ejpam-3344	157	17	is	be	AUX
ejpam-3344	157	18	an	an	DET
ejpam-3344	157	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	157	20	fuzzy	fuzzy	ADJ
ejpam-3344	157	21	left	left	NOUN
ejpam-3344	157	22	(	(	PUNCT
ejpam-3344	157	23	resp	resp	NOUN
ejpam-3344	157	24	.	.	PUNCT
ejpam-3344	158	1	right	right	ADJ
ejpam-3344	158	2	)	)	PUNCT
ejpam-3344	158	3	ideal	ideal	NOUN
ejpam-3344	158	4	of	of	ADP
ejpam-3344	158	5	r.	r.	PROPN
ejpam-3344	158	6	proof	proof	NOUN
ejpam-3344	158	7	.	.	PUNCT
ejpam-3344	159	1	straight	straight	ADV
ejpam-3344	159	2	forward	forward	ADV
ejpam-3344	159	3	.	.	PUNCT
ejpam-3344	160	1	theorem	theorem	VERB
ejpam-3344	160	2	3	3	X
ejpam-3344	160	3	.	.	PUNCT
ejpam-3344	161	1	let	let	VERB
ejpam-3344	161	2	r	r	PRON
ejpam-3344	161	3	be	be	AUX
ejpam-3344	161	4	an	an	DET
ejpam-3344	161	5	la	la	ADJ
ejpam-3344	161	6	-	-	PUNCT
ejpam-3344	161	7	ring	ring	NOUN
ejpam-3344	161	8	and	and	CCONJ
ejpam-3344	161	9	∅	∅	NOUN
ejpam-3344	161	10	6=	6=	ADP
ejpam-3344	161	11	a	a	DET
ejpam-3344	161	12	⊆	⊆	NUM
ejpam-3344	161	13	r.	r.	NOUN
ejpam-3344	161	14	then	then	ADV
ejpam-3344	161	15	a	a	PRON
ejpam-3344	161	16	is	be	AUX
ejpam-3344	161	17	a	a	DET
ejpam-3344	161	18	(	(	PUNCT
ejpam-3344	161	19	1	1	NUM
ejpam-3344	161	20	,	,	PUNCT
ejpam-3344	161	21	2)-ideal	2)-ideal	NUM
ejpam-3344	161	22	of	of	ADP
ejpam-3344	161	23	r	r	NOUN
ejpam-3344	161	24	if	if	SCONJ
ejpam-3344	162	1	and	and	CCONJ
ejpam-3344	162	2	only	only	ADV
ejpam-3344	162	3	if	if	SCONJ
ejpam-3344	162	4	the	the	DET
ejpam-3344	162	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	162	6	characteristic	characteristic	ADJ
ejpam-3344	162	7	function	function	NOUN
ejpam-3344	162	8	χa	χa	NOUN
ejpam-3344	162	9	=	=	SYM
ejpam-3344	162	10	〈	〈	PROPN
ejpam-3344	162	11	µχa	µχa	NOUN
ejpam-3344	162	12	,	,	PUNCT
ejpam-3344	162	13	γχa	γχa	ADJ
ejpam-3344	162	14	〉	〉	NUM
ejpam-3344	162	15	of	of	ADP
ejpam-3344	162	16	a	a	PRON
ejpam-3344	162	17	is	be	AUX
ejpam-3344	162	18	an	an	DET
ejpam-3344	162	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	162	20	fuzzy	fuzzy	ADJ
ejpam-3344	162	21	(	(	PUNCT
ejpam-3344	162	22	1	1	NUM
ejpam-3344	162	23	,	,	PUNCT
ejpam-3344	162	24	2)-ideal	2)-ideal	NUM
ejpam-3344	162	25	of	of	ADP
ejpam-3344	162	26	r.	r.	PROPN
ejpam-3344	162	27	proof	proof	NOUN
ejpam-3344	162	28	.	.	PUNCT
ejpam-3344	163	1	let	let	VERB
ejpam-3344	163	2	a	a	DET
ejpam-3344	163	3	be	be	AUX
ejpam-3344	163	4	a	a	DET
ejpam-3344	163	5	(	(	PUNCT
ejpam-3344	163	6	1	1	NUM
ejpam-3344	163	7	,	,	PUNCT
ejpam-3344	163	8	2)-ideal	2)-ideal	NUM
ejpam-3344	163	9	of	of	ADP
ejpam-3344	163	10	r	r	NOUN
ejpam-3344	163	11	,	,	PUNCT
ejpam-3344	163	12	this	this	PRON
ejpam-3344	163	13	implies	imply	VERB
ejpam-3344	163	14	that	that	SCONJ
ejpam-3344	163	15	a	a	PRON
ejpam-3344	163	16	is	be	AUX
ejpam-3344	163	17	an	an	DET
ejpam-3344	163	18	la	la	NOUN
ejpam-3344	163	19	-	-	PUNCT
ejpam-3344	163	20	subring	subring	NOUN
ejpam-3344	163	21	of	of	ADP
ejpam-3344	163	22	r.	r.	PROPN
ejpam-3344	163	23	then	then	ADV
ejpam-3344	163	24	χa	χa	PROPN
ejpam-3344	163	25	is	be	AUX
ejpam-3344	163	26	an	an	DET
ejpam-3344	163	27	intuitionistic	intuitionistic	ADJ
ejpam-3344	163	28	fuzzy	fuzzy	ADJ
ejpam-3344	163	29	la	la	NOUN
ejpam-3344	163	30	-	-	PUNCT
ejpam-3344	163	31	subring	subring	NOUN
ejpam-3344	163	32	of	of	ADP
ejpam-3344	163	33	r	r	NOUN
ejpam-3344	163	34	by	by	ADP
ejpam-3344	163	35	the	the	DET
ejpam-3344	163	36	lemma	lemma	PROPN
ejpam-3344	163	37	4	4	X
ejpam-3344	163	38	.	.	PUNCT
ejpam-3344	164	1	let	let	VERB
ejpam-3344	164	2	a	a	DET
ejpam-3344	164	3	,	,	PUNCT
ejpam-3344	164	4	x	x	NOUN
ejpam-3344	164	5	,	,	PUNCT
ejpam-3344	164	6	y	y	PROPN
ejpam-3344	164	7	,	,	PUNCT
ejpam-3344	164	8	z	z	PROPN
ejpam-3344	164	9	∈	∈	PROPN
ejpam-3344	164	10	r.	r.	NOUN
ejpam-3344	164	11	if	if	SCONJ
ejpam-3344	164	12	x	x	PROPN
ejpam-3344	164	13	,	,	PUNCT
ejpam-3344	164	14	y	y	PROPN
ejpam-3344	164	15	,	,	PUNCT
ejpam-3344	164	16	z	z	PROPN
ejpam-3344	164	17	∈	∈	PROPN
ejpam-3344	164	18	a	a	PRON
ejpam-3344	164	19	,	,	PUNCT
ejpam-3344	164	20	then	then	ADV
ejpam-3344	164	21	by	by	ADP
ejpam-3344	164	22	definition	definition	NOUN
ejpam-3344	164	23	of	of	ADP
ejpam-3344	164	24	intuitionistic	intuitionistic	ADJ
ejpam-3344	164	25	characteristic	characteristic	ADJ
ejpam-3344	164	26	function	function	NOUN
ejpam-3344	164	27	µχa(x	µχa(x	VERB
ejpam-3344	164	28	)	)	PUNCT
ejpam-3344	164	29	=	=	SYM
ejpam-3344	164	30	1	1	NUM
ejpam-3344	164	31	=	=	SYM
ejpam-3344	164	32	µχa(y	µχa(y	PROPN
ejpam-3344	164	33	)	)	PUNCT
ejpam-3344	164	34	=	=	PUNCT
ejpam-3344	164	35	µχa(z	µχa(z	PROPN
ejpam-3344	164	36	)	)	PUNCT
ejpam-3344	164	37	and	and	CCONJ
ejpam-3344	164	38	γχa(x	γχa(x	PROPN
ejpam-3344	164	39	)	)	PUNCT
ejpam-3344	164	40	=	=	SYM
ejpam-3344	164	41	0	0	PUNCT
ejpam-3344	165	1	=	=	SYM
ejpam-3344	165	2	γχa(y	γχa(y	NOUN
ejpam-3344	165	3	)	)	PUNCT
ejpam-3344	165	4	=	=	PUNCT
ejpam-3344	165	5	µχa(z	µχa(z	PROPN
ejpam-3344	165	6	)	)	PUNCT
ejpam-3344	165	7	.	.	PUNCT
ejpam-3344	166	1	since	since	SCONJ
ejpam-3344	166	2	(	(	PUNCT
ejpam-3344	166	3	xa)(yz	xa)(yz	NOUN
ejpam-3344	166	4	)	)	PUNCT
ejpam-3344	166	5	∈	∈	PROPN
ejpam-3344	166	6	a	a	PRON
ejpam-3344	166	7	,	,	PUNCT
ejpam-3344	166	8	a	a	DET
ejpam-3344	166	9	being	be	AUX
ejpam-3344	166	10	a	a	DET
ejpam-3344	166	11	(	(	PUNCT
ejpam-3344	166	12	1	1	NUM
ejpam-3344	166	13	,	,	PUNCT
ejpam-3344	166	14	2)-ideal	2)-ideal	NUM
ejpam-3344	166	15	of	of	ADP
ejpam-3344	166	16	r	r	NOUN
ejpam-3344	166	17	,	,	PUNCT
ejpam-3344	166	18	so	so	ADV
ejpam-3344	166	19	µχa((xa)(yz	µχa((xa)(yz	NOUN
ejpam-3344	166	20	)	)	PUNCT
ejpam-3344	166	21	)	)	PUNCT
ejpam-3344	166	22	=	=	SYM
ejpam-3344	166	23	1	1	NUM
ejpam-3344	166	24	and	and	CCONJ
ejpam-3344	166	25	γχa((xa)(yz	γχa((xa)(yz	NOUN
ejpam-3344	166	26	)	)	PUNCT
ejpam-3344	166	27	)	)	PUNCT
ejpam-3344	167	1	=	=	PUNCT
ejpam-3344	167	2	0	0	X
ejpam-3344	167	3	.	.	PUNCT
ejpam-3344	167	4	thus	thus	ADV
ejpam-3344	167	5	µχa((xa)(yz	µχa((xa)(yz	NOUN
ejpam-3344	167	6	)	)	PUNCT
ejpam-3344	167	7	)	)	PUNCT
ejpam-3344	167	8	≥	≥	PROPN
ejpam-3344	167	9	min{µχa(x	min{µχa(x	PROPN
ejpam-3344	167	10	)	)	PUNCT
ejpam-3344	167	11	,	,	PUNCT
ejpam-3344	167	12	µχa(y	µχa(y	PROPN
ejpam-3344	167	13	)	)	PUNCT
ejpam-3344	167	14	,	,	PUNCT
ejpam-3344	167	15	µχa(z	µχa(z	NOUN
ejpam-3344	167	16	)	)	PUNCT
ejpam-3344	167	17	}	}	PUNCT
ejpam-3344	167	18	and	and	CCONJ
ejpam-3344	167	19	γχa((xa)(yz	γχa((xa)(yz	NOUN
ejpam-3344	167	20	)	)	PUNCT
ejpam-3344	167	21	)	)	PUNCT
ejpam-3344	168	1	≤	≤	PROPN
ejpam-3344	169	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3344	169	2	)	)	PUNCT
ejpam-3344	169	3	,	,	PUNCT
ejpam-3344	169	4	γχa(y	γχa(y	PROPN
ejpam-3344	169	5	)	)	PUNCT
ejpam-3344	169	6	,	,	PUNCT
ejpam-3344	169	7	γχa(z	γχa(z	PROPN
ejpam-3344	169	8	)	)	PUNCT
ejpam-3344	169	9	}	}	PUNCT
ejpam-3344	169	10	.	.	PUNCT
ejpam-3344	170	1	similarly	similarly	ADV
ejpam-3344	170	2	,	,	PUNCT
ejpam-3344	170	3	we	we	PRON
ejpam-3344	170	4	have	have	VERB
ejpam-3344	170	5	µχa((xa)(yz	µχa((xa)(yz	NOUN
ejpam-3344	170	6	)	)	PUNCT
ejpam-3344	170	7	)	)	PUNCT
ejpam-3344	170	8	≥	≥	PROPN
ejpam-3344	170	9	min{µχa(x	min{µχa(x	PROPN
ejpam-3344	170	10	)	)	PUNCT
ejpam-3344	170	11	,	,	PUNCT
ejpam-3344	170	12	µχa(y	µχa(y	PROPN
ejpam-3344	170	13	)	)	PUNCT
ejpam-3344	170	14	,	,	PUNCT
ejpam-3344	170	15	µχa(z	µχa(z	NOUN
ejpam-3344	170	16	)	)	PUNCT
ejpam-3344	170	17	}	}	PUNCT
ejpam-3344	170	18	and	and	CCONJ
ejpam-3344	170	19	γχa((xa)(yz	γχa((xa)(yz	NOUN
ejpam-3344	170	20	)	)	PUNCT
ejpam-3344	170	21	)	)	PUNCT
ejpam-3344	171	1	≤	≤	PROPN
ejpam-3344	172	1	max{γχa(x	max{γχa(x	PROPN
ejpam-3344	172	2	)	)	PUNCT
ejpam-3344	172	3	,	,	PUNCT
ejpam-3344	172	4	γχa(y	γχa(y	PROPN
ejpam-3344	172	5	)	)	PUNCT
ejpam-3344	172	6	,	,	PUNCT
ejpam-3344	172	7	γχa(z	γχa(z	PROPN
ejpam-3344	172	8	)	)	PUNCT
ejpam-3344	172	9	}	}	PUNCT
ejpam-3344	172	10	,	,	PUNCT
ejpam-3344	172	11	when	when	SCONJ
ejpam-3344	172	12	x	x	X
ejpam-3344	172	13	,	,	PUNCT
ejpam-3344	172	14	y	y	PROPN
ejpam-3344	172	15	,	,	PUNCT
ejpam-3344	172	16	z	z	NOUN
ejpam-3344	172	17	/∈	/∈	PUNCT
ejpam-3344	172	18	a.	a.	NOUN
ejpam-3344	172	19	hence	hence	ADV
ejpam-3344	172	20	the	the	DET
ejpam-3344	172	21	intuitionistic	intuitionistic	ADJ
ejpam-3344	172	22	characteristic	characteristic	ADJ
ejpam-3344	172	23	function	function	NOUN
ejpam-3344	172	24	χa	χa	NOUN
ejpam-3344	172	25	=	=	SYM
ejpam-3344	172	26	〈	〈	PROPN
ejpam-3344	172	27	µχa	µχa	NOUN
ejpam-3344	172	28	,	,	PUNCT
ejpam-3344	172	29	γχa	γχa	ADJ
ejpam-3344	172	30	〉	〉	NUM
ejpam-3344	172	31	of	of	ADP
ejpam-3344	172	32	a	a	PRON
ejpam-3344	172	33	is	be	AUX
ejpam-3344	172	34	an	an	DET
ejpam-3344	172	35	intuitionistic	intuitionistic	ADJ
ejpam-3344	172	36	fuzzy	fuzzy	ADJ
ejpam-3344	172	37	(	(	PUNCT
ejpam-3344	172	38	1	1	NUM
ejpam-3344	172	39	,	,	PUNCT
ejpam-3344	172	40	2)-ideal	2)-ideal	NUM
ejpam-3344	172	41	of	of	ADP
ejpam-3344	172	42	r.	r.	PROPN
ejpam-3344	172	43	conversely	conversely	ADV
ejpam-3344	172	44	,	,	PUNCT
ejpam-3344	172	45	suppose	suppose	VERB
ejpam-3344	172	46	that	that	SCONJ
ejpam-3344	172	47	the	the	DET
ejpam-3344	172	48	intuitionistic	intuitionistic	ADJ
ejpam-3344	172	49	characteristic	characteristic	ADJ
ejpam-3344	172	50	function	function	NOUN
ejpam-3344	172	51	χa	χa	NOUN
ejpam-3344	172	52	=	=	SYM
ejpam-3344	172	53	〈	〈	PROPN
ejpam-3344	172	54	µχa	µχa	NOUN
ejpam-3344	172	55	,	,	PUNCT
ejpam-3344	172	56	γχa	γχa	ADJ
ejpam-3344	172	57	〉	〉	NUM
ejpam-3344	172	58	of	of	ADP
ejpam-3344	172	59	a	a	PRON
ejpam-3344	172	60	is	be	AUX
ejpam-3344	172	61	an	an	DET
ejpam-3344	172	62	intuitionistic	intuitionistic	ADJ
ejpam-3344	172	63	fuzzy	fuzzy	ADJ
ejpam-3344	172	64	(	(	PUNCT
ejpam-3344	172	65	1	1	NUM
ejpam-3344	172	66	,	,	PUNCT
ejpam-3344	172	67	2)-ideal	2)-ideal	NUM
ejpam-3344	172	68	of	of	ADP
ejpam-3344	172	69	r	r	NOUN
ejpam-3344	172	70	,	,	PUNCT
ejpam-3344	172	71	this	this	PRON
ejpam-3344	172	72	means	mean	VERB
ejpam-3344	172	73	that	that	SCONJ
ejpam-3344	172	74	χa	χa	PROPN
ejpam-3344	172	75	is	be	AUX
ejpam-3344	172	76	an	an	DET
ejpam-3344	172	77	intuitionistic	intuitionistic	ADJ
ejpam-3344	172	78	fuzzy	fuzzy	ADJ
ejpam-3344	172	79	la	la	NOUN
ejpam-3344	172	80	-	-	PUNCT
ejpam-3344	172	81	subring	subring	NOUN
ejpam-3344	172	82	of	of	ADP
ejpam-3344	172	83	r.	r.	PROPN
ejpam-3344	172	84	then	then	ADV
ejpam-3344	172	85	a	a	PRON
ejpam-3344	172	86	is	be	AUX
ejpam-3344	172	87	an	an	DET
ejpam-3344	172	88	la	la	NOUN
ejpam-3344	172	89	-	-	PUNCT
ejpam-3344	172	90	subring	subring	NOUN
ejpam-3344	172	91	of	of	ADP
ejpam-3344	172	92	r	r	NOUN
ejpam-3344	172	93	by	by	ADP
ejpam-3344	172	94	the	the	DET
ejpam-3344	172	95	lemma	lemma	PROPN
ejpam-3344	172	96	4	4	X
ejpam-3344	172	97	.	.	PUNCT
ejpam-3344	173	1	let	let	VERB
ejpam-3344	173	2	t	t	PROPN
ejpam-3344	173	3	∈	∈	PROPN
ejpam-3344	173	4	(	(	PUNCT
ejpam-3344	173	5	ar)a2	ar)a2	PROPN
ejpam-3344	173	6	,	,	PUNCT
ejpam-3344	173	7	this	this	PRON
ejpam-3344	173	8	implies	imply	VERB
ejpam-3344	173	9	that	that	SCONJ
ejpam-3344	173	10	t	t	NOUN
ejpam-3344	173	11	=	=	SYM
ejpam-3344	173	12	(	(	PUNCT
ejpam-3344	173	13	xa)(yz	xa)(yz	PROPN
ejpam-3344	173	14	)	)	PUNCT
ejpam-3344	173	15	,	,	PUNCT
ejpam-3344	173	16	where	where	SCONJ
ejpam-3344	173	17	x	x	X
ejpam-3344	173	18	,	,	PUNCT
ejpam-3344	173	19	y	y	PROPN
ejpam-3344	173	20	,	,	PUNCT
ejpam-3344	173	21	z	z	PROPN
ejpam-3344	173	22	∈	∈	PROPN
ejpam-3344	173	23	a	a	PRON
ejpam-3344	173	24	and	and	CCONJ
ejpam-3344	173	25	a	a	DET
ejpam-3344	173	26	∈	∈	PROPN
ejpam-3344	173	27	r.	r.	NOUN
ejpam-3344	173	28	then	then	ADV
ejpam-3344	173	29	by	by	ADP
ejpam-3344	173	30	definition	definition	NOUN
ejpam-3344	173	31	µχa(x	µχa(x	VERB
ejpam-3344	173	32	)	)	PUNCT
ejpam-3344	173	33	=	=	SYM
ejpam-3344	173	34	1	1	NUM
ejpam-3344	173	35	=	=	SYM
ejpam-3344	173	36	µχa(y	µχa(y	PROPN
ejpam-3344	173	37	)	)	PUNCT
ejpam-3344	173	38	=	=	PUNCT
ejpam-3344	173	39	µχa(z	µχa(z	PROPN
ejpam-3344	173	40	)	)	PUNCT
ejpam-3344	173	41	and	and	CCONJ
ejpam-3344	173	42	γχa(x	γχa(x	PROPN
ejpam-3344	173	43	)	)	PUNCT
ejpam-3344	173	44	=	=	SYM
ejpam-3344	173	45	0	0	PUNCT
ejpam-3344	174	1	=	=	SYM
ejpam-3344	174	2	γχa(y	γχa(y	NOUN
ejpam-3344	174	3	)	)	PUNCT
ejpam-3344	174	4	=	=	SYM
ejpam-3344	174	5	γχa(z	γχa(z	PROPN
ejpam-3344	174	6	)	)	PUNCT
ejpam-3344	174	7	.	.	PUNCT
ejpam-3344	175	1	now	now	ADV
ejpam-3344	175	2	µχa((xa)(yz	µχa((xa)(yz	NOUN
ejpam-3344	175	3	)	)	PUNCT
ejpam-3344	175	4	)	)	PUNCT
ejpam-3344	175	5	≥	≥	PROPN
ejpam-3344	175	6	µχa(x	µχa(x	PROPN
ejpam-3344	175	7	)	)	PUNCT
ejpam-3344	175	8	∧	∧	PROPN
ejpam-3344	175	9	µχa(y	µχa(y	PROPN
ejpam-3344	175	10	)	)	PUNCT
ejpam-3344	175	11	∧	∧	NOUN
ejpam-3344	175	12	µχa(z	µχa(z	PROPN
ejpam-3344	175	13	)	)	PUNCT
ejpam-3344	175	14	=	=	SYM
ejpam-3344	175	15	1	1	NUM
ejpam-3344	175	16	and	and	CCONJ
ejpam-3344	175	17	γχa((xa)(yz	γχa((xa)(yz	NOUN
ejpam-3344	175	18	)	)	PUNCT
ejpam-3344	175	19	)	)	PUNCT
ejpam-3344	175	20	≤	≤	NUM
ejpam-3344	175	21	γχa(x	γχa(x	PROPN
ejpam-3344	175	22	)	)	PUNCT
ejpam-3344	175	23	∨	∨	NUM
ejpam-3344	175	24	γχa(y	γχa(y	NOUN
ejpam-3344	175	25	)	)	PUNCT
ejpam-3344	175	26	∨	∨	NUM
ejpam-3344	175	27	γχa(z	γχa(z	PROPN
ejpam-3344	175	28	)	)	PUNCT
ejpam-3344	175	29	=	=	SYM
ejpam-3344	175	30	0	0	NUM
ejpam-3344	175	31	,	,	PUNCT
ejpam-3344	175	32	χa	χa	ADP
ejpam-3344	175	33	being	be	AUX
ejpam-3344	175	34	an	an	DET
ejpam-3344	175	35	intuitionistic	intuitionistic	ADJ
ejpam-3344	175	36	fuzzy	fuzzy	ADJ
ejpam-3344	175	37	(	(	PUNCT
ejpam-3344	175	38	1	1	NUM
ejpam-3344	175	39	,	,	PUNCT
ejpam-3344	175	40	2)-ideal	2)-ideal	NUM
ejpam-3344	175	41	ofr	ofr	NOUN
ejpam-3344	175	42	.	.	PUNCT
ejpam-3344	176	1	thus	thus	ADV
ejpam-3344	176	2	µχa((xa)(yz	µχa((xa)(yz	NOUN
ejpam-3344	176	3	)	)	PUNCT
ejpam-3344	176	4	)	)	PUNCT
ejpam-3344	177	1	=	=	SYM
ejpam-3344	177	2	1	1	NUM
ejpam-3344	177	3	and	and	CCONJ
ejpam-3344	177	4	γχa((xa)(yz	γχa((xa)(yz	NOUN
ejpam-3344	177	5	)	)	PUNCT
ejpam-3344	177	6	)	)	PUNCT
ejpam-3344	178	1	=	=	SYM
ejpam-3344	178	2	0	0	NUM
ejpam-3344	178	3	,	,	PUNCT
ejpam-3344	178	4	i.e.	i.e.	X
ejpam-3344	178	5	,	,	PUNCT
ejpam-3344	178	6	(	(	PUNCT
ejpam-3344	178	7	xa)(yz	xa)(yz	NOUN
ejpam-3344	178	8	)	)	PUNCT
ejpam-3344	178	9	∈	∈	PROPN
ejpam-3344	178	10	a.	a.	NOUN
ejpam-3344	178	11	hence	hence	ADV
ejpam-3344	178	12	a	a	PRON
ejpam-3344	178	13	is	be	AUX
ejpam-3344	178	14	a	a	DET
ejpam-3344	178	15	(	(	PUNCT
ejpam-3344	178	16	1	1	NUM
ejpam-3344	178	17	,	,	PUNCT
ejpam-3344	178	18	2)-ideal	2)-ideal	NUM
ejpam-3344	178	19	of	of	ADP
ejpam-3344	178	20	r.	r.	PROPN
ejpam-3344	178	21	remark	remark	PROPN
ejpam-3344	178	22	3	3	NUM
ejpam-3344	178	23	.	.	PUNCT
ejpam-3344	179	1	let	let	VERB
ejpam-3344	179	2	r	r	PRON
ejpam-3344	179	3	be	be	AUX
ejpam-3344	179	4	an	an	DET
ejpam-3344	179	5	la	la	ADJ
ejpam-3344	179	6	-	-	PUNCT
ejpam-3344	179	7	ring	ring	NOUN
ejpam-3344	179	8	and	and	CCONJ
ejpam-3344	179	9	∅	∅	NOUN
ejpam-3344	179	10	6=	6=	ADP
ejpam-3344	179	11	a	a	DET
ejpam-3344	179	12	⊆	⊆	NUM
ejpam-3344	179	13	r.	r.	NOUN
ejpam-3344	179	14	then	then	ADV
ejpam-3344	179	15	a	a	PRON
ejpam-3344	179	16	is	be	AUX
ejpam-3344	179	17	a	a	DET
ejpam-3344	179	18	bi	bi	NOUN
ejpam-3344	179	19	-	-	NOUN
ejpam-3344	179	20	ideal	ideal	NOUN
ejpam-3344	179	21	of	of	ADP
ejpam-3344	179	22	r	r	NOUN
ejpam-3344	179	23	if	if	SCONJ
ejpam-3344	180	1	and	and	CCONJ
ejpam-3344	180	2	only	only	ADV
ejpam-3344	180	3	if	if	SCONJ
ejpam-3344	180	4	the	the	DET
ejpam-3344	180	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	180	6	characteristic	characteristic	ADJ
ejpam-3344	180	7	function	function	NOUN
ejpam-3344	180	8	χa	χa	NOUN
ejpam-3344	180	9	=	=	SYM
ejpam-3344	180	10	〈	〈	PROPN
ejpam-3344	180	11	µχa	µχa	NOUN
ejpam-3344	180	12	,	,	PUNCT
ejpam-3344	180	13	γχa	γχa	ADJ
ejpam-3344	180	14	〉	〉	NUM
ejpam-3344	180	15	of	of	ADP
ejpam-3344	180	16	a	a	PRON
ejpam-3344	180	17	is	be	AUX
ejpam-3344	180	18	an	an	DET
ejpam-3344	180	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	180	20	fuzzy	fuzzy	ADJ
ejpam-3344	180	21	bi	bi	NOUN
ejpam-3344	180	22	-	-	NOUN
ejpam-3344	180	23	ideal	ideal	NOUN
ejpam-3344	180	24	of	of	ADP
ejpam-3344	180	25	r.	r.	PROPN
ejpam-3344	180	26	n.	n.	PROPN
ejpam-3344	180	27	kausar	kausar	PROPN
ejpam-3344	180	28	,	,	PUNCT
ejpam-3344	180	29	m.	m.	NOUN
ejpam-3344	180	30	a.	a.	PROPN
ejpam-3344	180	31	waqar	waqar	PROPN
ejpam-3344	180	32	/	/	SYM
ejpam-3344	180	33	eur	eur	PROPN
ejpam-3344	180	34	.	.	PUNCT
ejpam-3344	181	1	j.	j.	PROPN
ejpam-3344	181	2	pure	pure	PROPN
ejpam-3344	181	3	appl	appl	PROPN
ejpam-3344	181	4	.	.	PROPN
ejpam-3344	181	5	math	math	PROPN
ejpam-3344	181	6	,	,	PUNCT
ejpam-3344	181	7	12	12	NUM
ejpam-3344	181	8	(	(	PUNCT
ejpam-3344	181	9	1	1	NUM
ejpam-3344	181	10	)	)	PUNCT
ejpam-3344	181	11	(	(	PUNCT
ejpam-3344	181	12	2019	2019	NUM
ejpam-3344	181	13	)	)	PUNCT
ejpam-3344	181	14	,	,	PUNCT
ejpam-3344	181	15	226	226	NUM
ejpam-3344	181	16	-	-	SYM
ejpam-3344	181	17	250	250	NUM
ejpam-3344	181	18	233	233	NUM
ejpam-3344	181	19	zadeh	zadeh	NOUN
ejpam-3344	182	1	[	[	X
ejpam-3344	182	2	22	22	NUM
ejpam-3344	182	3	]	]	PUNCT
ejpam-3344	182	4	,	,	PUNCT
ejpam-3344	182	5	introduced	introduce	VERB
ejpam-3344	182	6	the	the	DET
ejpam-3344	182	7	concept	concept	NOUN
ejpam-3344	182	8	of	of	ADP
ejpam-3344	182	9	level	level	NOUN
ejpam-3344	182	10	set	set	NOUN
ejpam-3344	182	11	.	.	PUNCT
ejpam-3344	183	1	das	das	PROPN
ejpam-3344	184	1	[	[	X
ejpam-3344	184	2	5	5	NUM
ejpam-3344	184	3	]	]	PUNCT
ejpam-3344	184	4	,	,	PUNCT
ejpam-3344	184	5	studied	study	VERB
ejpam-3344	184	6	the	the	DET
ejpam-3344	184	7	fuzzy	fuzzy	ADJ
ejpam-3344	184	8	groups	group	NOUN
ejpam-3344	184	9	,	,	PUNCT
ejpam-3344	184	10	level	level	NOUN
ejpam-3344	184	11	subgroups	subgroup	NOUN
ejpam-3344	184	12	and	and	CCONJ
ejpam-3344	184	13	gave	give	VERB
ejpam-3344	184	14	the	the	DET
ejpam-3344	184	15	proper	proper	ADJ
ejpam-3344	184	16	definition	definition	NOUN
ejpam-3344	184	17	of	of	ADP
ejpam-3344	184	18	a	a	DET
ejpam-3344	184	19	level	level	NOUN
ejpam-3344	184	20	set	set	NOUN
ejpam-3344	184	21	such	such	ADJ
ejpam-3344	184	22	that	that	PRON
ejpam-3344	184	23	:	:	PUNCT
ejpam-3344	184	24	let	let	VERB
ejpam-3344	184	25	µ	µ	X
ejpam-3344	184	26	be	be	AUX
ejpam-3344	184	27	a	a	DET
ejpam-3344	184	28	fuzzy	fuzzy	ADJ
ejpam-3344	184	29	subset	subset	NOUN
ejpam-3344	184	30	of	of	ADP
ejpam-3344	184	31	a	a	DET
ejpam-3344	184	32	non	non	ADJ
ejpam-3344	184	33	-	-	ADJ
ejpam-3344	184	34	empty	empty	ADJ
ejpam-3344	184	35	set	set	NOUN
ejpam-3344	184	36	s	s	PROPN
ejpam-3344	184	37	,	,	PUNCT
ejpam-3344	184	38	for	for	ADP
ejpam-3344	184	39	t	t	PROPN
ejpam-3344	184	40	∈	∈	PROPN
ejpam-3344	185	1	[	[	X
ejpam-3344	185	2	0	0	NUM
ejpam-3344	185	3	,	,	PUNCT
ejpam-3344	185	4	1	1	NUM
ejpam-3344	185	5	]	]	PUNCT
ejpam-3344	185	6	,	,	PUNCT
ejpam-3344	185	7	the	the	DET
ejpam-3344	185	8	set	set	NOUN
ejpam-3344	185	9	µt	µt	X
ejpam-3344	185	10	=	=	PUNCT
ejpam-3344	185	11	{	{	PUNCT
ejpam-3344	185	12	x	x	PUNCT
ejpam-3344	185	13	∈	∈	PROPN
ejpam-3344	185	14	s	s	X
ejpam-3344	185	15	|	|	ADV
ejpam-3344	185	16	µ(x	µ(x	NOUN
ejpam-3344	185	17	)	)	PUNCT
ejpam-3344	185	18	≥	≥	NOUN
ejpam-3344	185	19	t	t	PROPN
ejpam-3344	185	20	}	}	PUNCT
ejpam-3344	185	21	,	,	PUNCT
ejpam-3344	185	22	is	be	AUX
ejpam-3344	185	23	called	call	VERB
ejpam-3344	185	24	a	a	DET
ejpam-3344	185	25	level	level	NOUN
ejpam-3344	185	26	subset	subset	NOUN
ejpam-3344	185	27	of	of	ADP
ejpam-3344	185	28	the	the	DET
ejpam-3344	185	29	fuzzy	fuzzy	ADJ
ejpam-3344	185	30	subset	subset	NOUN
ejpam-3344	185	31	µ.	µ.	NOUN
ejpam-3344	185	32	now	now	ADV
ejpam-3344	185	33	we	we	PRON
ejpam-3344	185	34	give	give	VERB
ejpam-3344	185	35	the	the	DET
ejpam-3344	185	36	definition	definition	NOUN
ejpam-3344	185	37	of	of	ADP
ejpam-3344	185	38	strong	strong	ADJ
ejpam-3344	185	39	level	level	NOUN
ejpam-3344	185	40	set	set	NOUN
ejpam-3344	185	41	.	.	PUNCT
ejpam-3344	186	1	let	let	VERB
ejpam-3344	186	2	a	a	DET
ejpam-3344	186	3	=	=	SYM
ejpam-3344	186	4	(	(	PUNCT
ejpam-3344	186	5	µa	µa	PROPN
ejpam-3344	186	6	,	,	PUNCT
ejpam-3344	186	7	γa	γa	PROPN
ejpam-3344	186	8	)	)	PUNCT
ejpam-3344	186	9	be	be	VERB
ejpam-3344	186	10	an	an	DET
ejpam-3344	186	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	186	12	fuzzy	fuzzy	ADJ
ejpam-3344	186	13	set	set	NOUN
ejpam-3344	186	14	of	of	ADP
ejpam-3344	186	15	an	an	DET
ejpam-3344	186	16	la	la	ADJ
ejpam-3344	186	17	-	-	PUNCT
ejpam-3344	186	18	ring	ring	NOUN
ejpam-3344	186	19	r	r	NOUN
ejpam-3344	186	20	,	,	PUNCT
ejpam-3344	186	21	then	then	ADV
ejpam-3344	186	22	for	for	ADP
ejpam-3344	186	23	all	all	DET
ejpam-3344	186	24	r	r	NOUN
ejpam-3344	186	25	,	,	PUNCT
ejpam-3344	186	26	t	t	PROPN
ejpam-3344	186	27	∈	∈	PROPN
ejpam-3344	186	28	(	(	PUNCT
ejpam-3344	186	29	0	0	NUM
ejpam-3344	186	30	,	,	PUNCT
ejpam-3344	186	31	1	1	NUM
ejpam-3344	186	32	]	]	PUNCT
ejpam-3344	186	33	,	,	PUNCT
ejpam-3344	186	34	we	we	PRON
ejpam-3344	186	35	define	define	VERB
ejpam-3344	186	36	a	a	DET
ejpam-3344	186	37	set	set	NOUN
ejpam-3344	186	38	a(r	a(r	NOUN
ejpam-3344	186	39	,	,	PUNCT
ejpam-3344	186	40	t	t	PROPN
ejpam-3344	186	41	)	)	PUNCT
ejpam-3344	186	42	=	=	PRON
ejpam-3344	187	1	{	{	PUNCT
ejpam-3344	187	2	x	x	PUNCT
ejpam-3344	187	3	∈	∈	PROPN
ejpam-3344	187	4	r	r	NOUN
ejpam-3344	187	5	|	|	NOUN
ejpam-3344	187	6	µa(x	µa(x	PUNCT
ejpam-3344	187	7	)	)	PUNCT
ejpam-3344	187	8	≥	≥	NOUN
ejpam-3344	187	9	r	r	NOUN
ejpam-3344	187	10	and	and	CCONJ
ejpam-3344	187	11	γa(x	γa(x	NUM
ejpam-3344	187	12	)	)	PUNCT
ejpam-3344	187	13	≤	≤	NOUN
ejpam-3344	187	14	t	t	PROPN
ejpam-3344	187	15	}	}	PUNCT
ejpam-3344	187	16	,	,	PUNCT
ejpam-3344	187	17	which	which	PRON
ejpam-3344	187	18	is	be	AUX
ejpam-3344	187	19	called	call	VERB
ejpam-3344	187	20	the	the	DET
ejpam-3344	187	21	(	(	PUNCT
ejpam-3344	187	22	r	r	NOUN
ejpam-3344	187	23	,	,	PUNCT
ejpam-3344	187	24	t)-strong	t)-strong	PUNCT
ejpam-3344	187	25	level	level	NOUN
ejpam-3344	187	26	set	set	NOUN
ejpam-3344	187	27	of	of	ADP
ejpam-3344	187	28	a.	a.	NOUN
ejpam-3344	188	1	it	it	PRON
ejpam-3344	188	2	is	be	AUX
ejpam-3344	188	3	clear	clear	ADJ
ejpam-3344	188	4	that	that	SCONJ
ejpam-3344	188	5	a(r	a(r	PROPN
ejpam-3344	188	6	,	,	PUNCT
ejpam-3344	188	7	t	t	PROPN
ejpam-3344	188	8	)	)	PUNCT
ejpam-3344	188	9	=	=	SYM
ejpam-3344	188	10	u(µa	u(µa	NOUN
ejpam-3344	188	11	;	;	PUNCT
ejpam-3344	188	12	r	r	X
ejpam-3344	188	13	)	)	PUNCT
ejpam-3344	188	14	∩	∩	ADJ
ejpam-3344	188	15	l(γa	l(γa	PROPN
ejpam-3344	188	16	;	;	PUNCT
ejpam-3344	188	17	t	t	PROPN
ejpam-3344	188	18	)	)	PUNCT
ejpam-3344	188	19	for	for	ADP
ejpam-3344	188	20	all	all	DET
ejpam-3344	188	21	r	r	NOUN
ejpam-3344	188	22	,	,	PUNCT
ejpam-3344	188	23	t	t	PROPN
ejpam-3344	188	24	∈	∈	PROPN
ejpam-3344	188	25	(	(	PUNCT
ejpam-3344	188	26	0	0	NUM
ejpam-3344	188	27	,	,	PUNCT
ejpam-3344	188	28	1	1	NUM
ejpam-3344	188	29	]	]	PUNCT
ejpam-3344	188	30	.	.	PUNCT
ejpam-3344	189	1	lemma	lemma	PROPN
ejpam-3344	189	2	5	5	X
ejpam-3344	189	3	.	.	PUNCT
ejpam-3344	189	4	let	let	VERB
ejpam-3344	189	5	a	a	PRON
ejpam-3344	189	6	=	=	SYM
ejpam-3344	189	7	(	(	PUNCT
ejpam-3344	189	8	µa	µa	PROPN
ejpam-3344	189	9	,	,	PUNCT
ejpam-3344	189	10	γa	γa	PROPN
ejpam-3344	189	11	)	)	PUNCT
ejpam-3344	189	12	be	be	VERB
ejpam-3344	189	13	an	an	DET
ejpam-3344	189	14	ifs	ifs	PROPN
ejpam-3344	189	15	of	of	ADP
ejpam-3344	189	16	an	an	DET
ejpam-3344	189	17	la	la	ADJ
ejpam-3344	189	18	-	-	PUNCT
ejpam-3344	189	19	ring	ring	NOUN
ejpam-3344	189	20	r.	r.	PROPN
ejpam-3344	189	21	then	then	ADV
ejpam-3344	189	22	a	a	PRON
ejpam-3344	189	23	is	be	AUX
ejpam-3344	189	24	an	an	DET
ejpam-3344	189	25	intuitionistic	intuitionistic	ADJ
ejpam-3344	189	26	fuzzy	fuzzy	ADJ
ejpam-3344	189	27	la	la	NOUN
ejpam-3344	189	28	-	-	PUNCT
ejpam-3344	189	29	subring	subring	NOUN
ejpam-3344	189	30	of	of	ADP
ejpam-3344	189	31	r	r	NOUN
ejpam-3344	189	32	if	if	SCONJ
ejpam-3344	190	1	and	and	CCONJ
ejpam-3344	190	2	only	only	ADV
ejpam-3344	190	3	if	if	SCONJ
ejpam-3344	190	4	a(r	a(r	PROPN
ejpam-3344	190	5	,	,	PUNCT
ejpam-3344	190	6	t	t	PROPN
ejpam-3344	190	7	)	)	PUNCT
ejpam-3344	190	8	is	be	AUX
ejpam-3344	190	9	an	an	DET
ejpam-3344	190	10	la	la	NOUN
ejpam-3344	190	11	-	-	PUNCT
ejpam-3344	190	12	subring	subring	NOUN
ejpam-3344	190	13	of	of	ADP
ejpam-3344	190	14	r	r	NOUN
ejpam-3344	190	15	for	for	ADP
ejpam-3344	190	16	all	all	DET
ejpam-3344	190	17	r	r	NOUN
ejpam-3344	190	18	,	,	PUNCT
ejpam-3344	190	19	t	t	PROPN
ejpam-3344	190	20	∈	∈	PROPN
ejpam-3344	190	21	(	(	PUNCT
ejpam-3344	190	22	0	0	NUM
ejpam-3344	190	23	,	,	PUNCT
ejpam-3344	190	24	1	1	NUM
ejpam-3344	190	25	]	]	PUNCT
ejpam-3344	190	26	.	.	PUNCT
ejpam-3344	191	1	proof	proof	NOUN
ejpam-3344	191	2	.	.	PUNCT
ejpam-3344	192	1	straight	straight	ADV
ejpam-3344	192	2	forward	forward	ADV
ejpam-3344	192	3	.	.	PUNCT
ejpam-3344	193	1	proposition	proposition	NOUN
ejpam-3344	193	2	3	3	NUM
ejpam-3344	193	3	.	.	PUNCT
ejpam-3344	194	1	let	let	VERB
ejpam-3344	194	2	a	a	PRON
ejpam-3344	194	3	=	=	SYM
ejpam-3344	194	4	(	(	PUNCT
ejpam-3344	194	5	µa	µa	PROPN
ejpam-3344	194	6	,	,	PUNCT
ejpam-3344	194	7	γa	γa	PROPN
ejpam-3344	194	8	)	)	PUNCT
ejpam-3344	194	9	be	be	VERB
ejpam-3344	194	10	an	an	DET
ejpam-3344	194	11	ifs	ifs	PROPN
ejpam-3344	194	12	of	of	ADP
ejpam-3344	194	13	an	an	DET
ejpam-3344	194	14	la	la	ADJ
ejpam-3344	194	15	-	-	PUNCT
ejpam-3344	194	16	ring	ring	NOUN
ejpam-3344	194	17	r.	r.	PROPN
ejpam-3344	194	18	then	then	ADV
ejpam-3344	194	19	a	a	PRON
ejpam-3344	194	20	is	be	AUX
ejpam-3344	194	21	an	an	DET
ejpam-3344	194	22	intuitionistic	intuitionistic	ADJ
ejpam-3344	194	23	fuzzy	fuzzy	ADJ
ejpam-3344	194	24	left	left	NOUN
ejpam-3344	194	25	(	(	PUNCT
ejpam-3344	194	26	resp	resp	NOUN
ejpam-3344	194	27	.	.	PUNCT
ejpam-3344	195	1	right	right	ADJ
ejpam-3344	195	2	)	)	PUNCT
ejpam-3344	195	3	ideal	ideal	NOUN
ejpam-3344	195	4	of	of	ADP
ejpam-3344	195	5	r	r	NOUN
ejpam-3344	195	6	if	if	SCONJ
ejpam-3344	196	1	and	and	CCONJ
ejpam-3344	196	2	only	only	ADV
ejpam-3344	196	3	if	if	SCONJ
ejpam-3344	196	4	a(r	a(r	PROPN
ejpam-3344	196	5	,	,	PUNCT
ejpam-3344	196	6	t	t	PROPN
ejpam-3344	196	7	)	)	PUNCT
ejpam-3344	196	8	is	be	AUX
ejpam-3344	196	9	a	a	DET
ejpam-3344	196	10	left	left	ADJ
ejpam-3344	196	11	(	(	PUNCT
ejpam-3344	196	12	resp	resp	NOUN
ejpam-3344	196	13	.	.	PUNCT
ejpam-3344	197	1	right	right	ADJ
ejpam-3344	197	2	)	)	PUNCT
ejpam-3344	197	3	ideal	ideal	NOUN
ejpam-3344	197	4	of	of	ADP
ejpam-3344	197	5	r	r	NOUN
ejpam-3344	197	6	for	for	ADP
ejpam-3344	197	7	all	all	DET
ejpam-3344	197	8	r	r	NOUN
ejpam-3344	197	9	,	,	PUNCT
ejpam-3344	197	10	t	t	PROPN
ejpam-3344	197	11	∈	∈	PROPN
ejpam-3344	197	12	(	(	PUNCT
ejpam-3344	197	13	0	0	NUM
ejpam-3344	197	14	,	,	PUNCT
ejpam-3344	197	15	1	1	NUM
ejpam-3344	197	16	]	]	PUNCT
ejpam-3344	197	17	.	.	PUNCT
ejpam-3344	198	1	proof	proof	NOUN
ejpam-3344	198	2	.	.	PUNCT
ejpam-3344	199	1	straight	straight	ADV
ejpam-3344	199	2	forward	forward	ADV
ejpam-3344	199	3	.	.	PUNCT
ejpam-3344	200	1	theorem	theorem	ADJ
ejpam-3344	200	2	4	4	NUM
ejpam-3344	200	3	.	.	PUNCT
ejpam-3344	201	1	let	let	VERB
ejpam-3344	201	2	a	a	PRON
ejpam-3344	201	3	=	=	SYM
ejpam-3344	201	4	(	(	PUNCT
ejpam-3344	201	5	µa	µa	PROPN
ejpam-3344	201	6	,	,	PUNCT
ejpam-3344	201	7	γa	γa	PROPN
ejpam-3344	201	8	)	)	PUNCT
ejpam-3344	201	9	be	be	VERB
ejpam-3344	201	10	an	an	DET
ejpam-3344	201	11	ifs	ifs	PROPN
ejpam-3344	201	12	of	of	ADP
ejpam-3344	201	13	an	an	DET
ejpam-3344	201	14	la	la	ADJ
ejpam-3344	201	15	-	-	PUNCT
ejpam-3344	201	16	ring	ring	NOUN
ejpam-3344	201	17	r.	r.	PROPN
ejpam-3344	201	18	then	then	ADV
ejpam-3344	201	19	a	a	PRON
ejpam-3344	201	20	is	be	AUX
ejpam-3344	201	21	an	an	DET
ejpam-3344	201	22	intuitionistic	intuitionistic	ADJ
ejpam-3344	201	23	fuzzy	fuzzy	ADJ
ejpam-3344	201	24	(	(	PUNCT
ejpam-3344	201	25	1	1	NUM
ejpam-3344	201	26	,	,	PUNCT
ejpam-3344	201	27	2)-ideal	2)-ideal	NUM
ejpam-3344	201	28	of	of	ADP
ejpam-3344	201	29	r	r	NOUN
ejpam-3344	201	30	if	if	SCONJ
ejpam-3344	202	1	and	and	CCONJ
ejpam-3344	202	2	only	only	ADV
ejpam-3344	202	3	if	if	SCONJ
ejpam-3344	202	4	a(r	a(r	PROPN
ejpam-3344	202	5	,	,	PUNCT
ejpam-3344	202	6	t	t	PROPN
ejpam-3344	202	7	)	)	PUNCT
ejpam-3344	202	8	is	be	AUX
ejpam-3344	202	9	a	a	DET
ejpam-3344	202	10	(	(	PUNCT
ejpam-3344	202	11	1	1	NUM
ejpam-3344	202	12	,	,	PUNCT
ejpam-3344	202	13	2)-ideal	2)-ideal	NUM
ejpam-3344	202	14	of	of	ADP
ejpam-3344	202	15	r	r	NOUN
ejpam-3344	202	16	for	for	ADP
ejpam-3344	202	17	all	all	DET
ejpam-3344	202	18	r	r	NOUN
ejpam-3344	202	19	,	,	PUNCT
ejpam-3344	202	20	t	t	PROPN
ejpam-3344	202	21	∈	∈	PROPN
ejpam-3344	202	22	(	(	PUNCT
ejpam-3344	202	23	0	0	NUM
ejpam-3344	202	24	,	,	PUNCT
ejpam-3344	202	25	1	1	NUM
ejpam-3344	202	26	]	]	PUNCT
ejpam-3344	202	27	.	.	PUNCT
ejpam-3344	203	1	proof	proof	NOUN
ejpam-3344	203	2	.	.	PUNCT
ejpam-3344	204	1	let	let	VERB
ejpam-3344	204	2	a	a	DET
ejpam-3344	204	3	=	=	SYM
ejpam-3344	204	4	(	(	PUNCT
ejpam-3344	204	5	µa	µa	PROPN
ejpam-3344	204	6	,	,	PUNCT
ejpam-3344	204	7	γa	γa	PROPN
ejpam-3344	204	8	)	)	PUNCT
ejpam-3344	204	9	be	be	VERB
ejpam-3344	204	10	an	an	DET
ejpam-3344	204	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	204	12	fuzzy	fuzzy	ADJ
ejpam-3344	204	13	(	(	PUNCT
ejpam-3344	204	14	1	1	NUM
ejpam-3344	204	15	,	,	PUNCT
ejpam-3344	204	16	2)-ideal	2)-ideal	NUM
ejpam-3344	204	17	of	of	ADP
ejpam-3344	204	18	r	r	NOUN
ejpam-3344	204	19	,	,	PUNCT
ejpam-3344	204	20	this	this	PRON
ejpam-3344	204	21	implies	imply	VERB
ejpam-3344	204	22	that	that	SCONJ
ejpam-3344	204	23	a	a	PRON
ejpam-3344	204	24	is	be	AUX
ejpam-3344	204	25	an	an	DET
ejpam-3344	204	26	intuitionistic	intuitionistic	ADJ
ejpam-3344	204	27	fuzzy	fuzzy	ADJ
ejpam-3344	204	28	la	la	NOUN
ejpam-3344	204	29	-	-	PUNCT
ejpam-3344	204	30	subring	subring	NOUN
ejpam-3344	204	31	of	of	ADP
ejpam-3344	204	32	r.	r.	PROPN
ejpam-3344	204	33	then	then	ADV
ejpam-3344	204	34	a(r	a(r	PROPN
ejpam-3344	204	35	,	,	PUNCT
ejpam-3344	204	36	t	t	PROPN
ejpam-3344	204	37	)	)	PUNCT
ejpam-3344	204	38	is	be	AUX
ejpam-3344	204	39	an	an	DET
ejpam-3344	204	40	la	la	NOUN
ejpam-3344	204	41	-	-	PUNCT
ejpam-3344	204	42	subring	subring	NOUN
ejpam-3344	204	43	of	of	ADP
ejpam-3344	204	44	r	r	NOUN
ejpam-3344	204	45	by	by	ADP
ejpam-3344	204	46	the	the	DET
ejpam-3344	204	47	lemma	lemma	PROPN
ejpam-3344	204	48	5	5	NUM
ejpam-3344	204	49	.	.	PUNCT
ejpam-3344	205	1	let	let	VERB
ejpam-3344	205	2	x	x	PRON
ejpam-3344	205	3	,	,	PUNCT
ejpam-3344	205	4	y	y	PROPN
ejpam-3344	205	5	,	,	PUNCT
ejpam-3344	205	6	z	z	PROPN
ejpam-3344	205	7	∈	∈	PROPN
ejpam-3344	205	8	a(r	a(r	NOUN
ejpam-3344	205	9	,	,	PUNCT
ejpam-3344	205	10	t	t	PROPN
ejpam-3344	205	11	)	)	PUNCT
ejpam-3344	205	12	and	and	CCONJ
ejpam-3344	205	13	a	a	DET
ejpam-3344	205	14	∈	∈	PROPN
ejpam-3344	205	15	r	r	NOUN
ejpam-3344	205	16	,	,	PUNCT
ejpam-3344	205	17	so	so	ADV
ejpam-3344	205	18	µa(y	µa(y	NOUN
ejpam-3344	205	19	)	)	PUNCT
ejpam-3344	205	20	,	,	PUNCT
ejpam-3344	205	21	µa(y	µa(y	NOUN
ejpam-3344	205	22	)	)	PUNCT
ejpam-3344	205	23	,	,	PUNCT
ejpam-3344	205	24	µa(z	µa(z	NUM
ejpam-3344	205	25	)	)	PUNCT
ejpam-3344	205	26	≥	≥	NOUN
ejpam-3344	205	27	r	r	NOUN
ejpam-3344	205	28	and	and	CCONJ
ejpam-3344	205	29	γa(y	γa(y	NUM
ejpam-3344	205	30	)	)	PUNCT
ejpam-3344	205	31	,	,	PUNCT
ejpam-3344	205	32	γa(y	γa(y	NUM
ejpam-3344	205	33	)	)	PUNCT
ejpam-3344	205	34	,	,	PUNCT
ejpam-3344	205	35	γa(z	γa(z	PROPN
ejpam-3344	205	36	)	)	PUNCT
ejpam-3344	205	37	≤	≤	NUM
ejpam-3344	206	1	t.	t.	NOUN
ejpam-3344	206	2	by	by	ADP
ejpam-3344	206	3	our	our	PRON
ejpam-3344	206	4	assumption	assumption	NOUN
ejpam-3344	206	5	µa((xy)(az	µa((xy)(az	NOUN
ejpam-3344	206	6	)	)	PUNCT
ejpam-3344	206	7	)	)	PUNCT
ejpam-3344	206	8	≥	≥	NOUN
ejpam-3344	206	9	µa(y	µa(y	NOUN
ejpam-3344	206	10	)	)	PUNCT
ejpam-3344	206	11	∧	∧	NOUN
ejpam-3344	206	12	µa(y	µa(y	NOUN
ejpam-3344	206	13	)	)	PUNCT
ejpam-3344	206	14	∧	∧	NOUN
ejpam-3344	206	15	µa(z	µa(z	PUNCT
ejpam-3344	206	16	)	)	PUNCT
ejpam-3344	206	17	≥	≥	NOUN
ejpam-3344	206	18	r	r	NOUN
ejpam-3344	206	19	and	and	CCONJ
ejpam-3344	206	20	γa((xy)(az	γa((xy)(az	PROPN
ejpam-3344	206	21	)	)	PUNCT
ejpam-3344	206	22	)	)	PUNCT
ejpam-3344	206	23	≤	≤	NOUN
ejpam-3344	206	24	µa(x	µa(x	NOUN
ejpam-3344	206	25	)	)	PUNCT
ejpam-3344	206	26	∨	∨	NUM
ejpam-3344	206	27	µa(y	µa(y	NOUN
ejpam-3344	206	28	)	)	PUNCT
ejpam-3344	206	29	∨	∨	NUM
ejpam-3344	206	30	µa(z	µa(z	NUM
ejpam-3344	206	31	)	)	PUNCT
ejpam-3344	206	32	≤	≤	NOUN
ejpam-3344	207	1	t.	t.	NOUN
ejpam-3344	207	2	thus	thus	ADV
ejpam-3344	207	3	µa((xy)(az	µa((xy)(az	NOUN
ejpam-3344	207	4	)	)	PUNCT
ejpam-3344	207	5	)	)	PUNCT
ejpam-3344	207	6	≥	≥	NOUN
ejpam-3344	207	7	r	r	NOUN
ejpam-3344	207	8	and	and	CCONJ
ejpam-3344	207	9	γa((xy)(az	γa((xy)(az	PROPN
ejpam-3344	207	10	)	)	PUNCT
ejpam-3344	207	11	)	)	PUNCT
ejpam-3344	207	12	≤	≤	PROPN
ejpam-3344	207	13	t	t	PROPN
ejpam-3344	207	14	,	,	PUNCT
ejpam-3344	207	15	i.e.	i.e.	X
ejpam-3344	207	16	,	,	PUNCT
ejpam-3344	207	17	(	(	PUNCT
ejpam-3344	207	18	xy)(az	xy)(az	NOUN
ejpam-3344	207	19	)	)	PUNCT
ejpam-3344	207	20	∈	∈	PROPN
ejpam-3344	207	21	a(r	a(r	NOUN
ejpam-3344	207	22	,	,	PUNCT
ejpam-3344	207	23	t	t	PROPN
ejpam-3344	207	24	)	)	PUNCT
ejpam-3344	207	25	.	.	PUNCT
ejpam-3344	208	1	so	so	ADV
ejpam-3344	208	2	a(r	a(r	PROPN
ejpam-3344	208	3	,	,	PUNCT
ejpam-3344	208	4	t	t	PROPN
ejpam-3344	208	5	)	)	PUNCT
ejpam-3344	208	6	is	be	AUX
ejpam-3344	208	7	a	a	DET
ejpam-3344	208	8	(	(	PUNCT
ejpam-3344	208	9	1	1	NUM
ejpam-3344	208	10	,	,	PUNCT
ejpam-3344	208	11	2)-ideal	2)-ideal	NUM
ejpam-3344	208	12	of	of	ADP
ejpam-3344	208	13	r.	r.	PROPN
ejpam-3344	208	14	conversely	conversely	ADV
ejpam-3344	208	15	,	,	PUNCT
ejpam-3344	208	16	suppose	suppose	VERB
ejpam-3344	208	17	that	that	SCONJ
ejpam-3344	208	18	a(r	a(r	PROPN
ejpam-3344	208	19	,	,	PUNCT
ejpam-3344	208	20	t	t	PROPN
ejpam-3344	208	21	)	)	PUNCT
ejpam-3344	208	22	is	be	AUX
ejpam-3344	208	23	a	a	DET
ejpam-3344	208	24	(	(	PUNCT
ejpam-3344	208	25	1	1	NUM
ejpam-3344	208	26	,	,	PUNCT
ejpam-3344	208	27	2)-ideal	2)-ideal	NUM
ejpam-3344	208	28	of	of	ADP
ejpam-3344	208	29	r	r	NOUN
ejpam-3344	208	30	,	,	PUNCT
ejpam-3344	208	31	this	this	PRON
ejpam-3344	208	32	means	mean	VERB
ejpam-3344	208	33	that	that	SCONJ
ejpam-3344	208	34	a(r	a(r	NOUN
ejpam-3344	208	35	,	,	PUNCT
ejpam-3344	208	36	t	t	PROPN
ejpam-3344	208	37	)	)	PUNCT
ejpam-3344	208	38	is	be	AUX
ejpam-3344	208	39	an	an	DET
ejpam-3344	208	40	lasubring	lasubring	NOUN
ejpam-3344	208	41	of	of	ADP
ejpam-3344	208	42	r.	r.	PROPN
ejpam-3344	208	43	then	then	ADV
ejpam-3344	208	44	a	a	PRON
ejpam-3344	208	45	is	be	AUX
ejpam-3344	208	46	an	an	DET
ejpam-3344	208	47	intuitionistic	intuitionistic	ADJ
ejpam-3344	208	48	fuzzy	fuzzy	ADJ
ejpam-3344	208	49	la	la	NOUN
ejpam-3344	208	50	-	-	PUNCT
ejpam-3344	208	51	subring	subring	NOUN
ejpam-3344	208	52	of	of	ADP
ejpam-3344	208	53	r	r	NOUN
ejpam-3344	208	54	by	by	ADP
ejpam-3344	208	55	the	the	DET
ejpam-3344	208	56	lemma	lemma	PROPN
ejpam-3344	208	57	5	5	NUM
ejpam-3344	208	58	.	.	PUNCT
ejpam-3344	209	1	let	let	VERB
ejpam-3344	209	2	x	x	PRON
ejpam-3344	209	3	,	,	PUNCT
ejpam-3344	209	4	y	y	PROPN
ejpam-3344	209	5	,	,	PUNCT
ejpam-3344	209	6	z	z	PROPN
ejpam-3344	209	7	,	,	PUNCT
ejpam-3344	209	8	a	a	DET
ejpam-3344	209	9	∈	∈	PROPN
ejpam-3344	209	10	r.	r.	NOUN
ejpam-3344	209	11	we	we	PRON
ejpam-3344	209	12	have	have	VERB
ejpam-3344	209	13	to	to	PART
ejpam-3344	209	14	show	show	VERB
ejpam-3344	209	15	that	that	SCONJ
ejpam-3344	209	16	µa((xy)(az	µa((xy)(az	NOUN
ejpam-3344	209	17	)	)	PUNCT
ejpam-3344	209	18	)	)	PUNCT
ejpam-3344	209	19	≥	≥	NOUN
ejpam-3344	209	20	µa(x	µa(x	NOUN
ejpam-3344	209	21	)	)	PUNCT
ejpam-3344	209	22	∧	∧	NOUN
ejpam-3344	209	23	µa(y	µa(y	NOUN
ejpam-3344	209	24	)	)	PUNCT
ejpam-3344	209	25	∧	∧	NOUN
ejpam-3344	209	26	µa(y	µa(y	NOUN
ejpam-3344	209	27	)	)	PUNCT
ejpam-3344	209	28	and	and	CCONJ
ejpam-3344	209	29	γa((xy)(az	γa((xy)(az	VERB
ejpam-3344	209	30	)	)	PUNCT
ejpam-3344	209	31	)	)	PUNCT
ejpam-3344	209	32	≤	≤	NOUN
ejpam-3344	209	33	γa(x	γa(x	NUM
ejpam-3344	209	34	)	)	PUNCT
ejpam-3344	209	35	∨	∨	NUM
ejpam-3344	209	36	γa(y	γa(y	NUM
ejpam-3344	209	37	)	)	PUNCT
ejpam-3344	209	38	∨	∨	NUM
ejpam-3344	209	39	γa(y	γa(y	NUM
ejpam-3344	209	40	)	)	PUNCT
ejpam-3344	209	41	.	.	PUNCT
ejpam-3344	210	1	we	we	PRON
ejpam-3344	210	2	assume	assume	VERB
ejpam-3344	210	3	a	a	DET
ejpam-3344	210	4	contradiction	contradiction	NOUN
ejpam-3344	210	5	µa((xy)(az	µa((xy)(az	NOUN
ejpam-3344	210	6	)	)	PUNCT
ejpam-3344	210	7	)	)	PUNCT
ejpam-3344	210	8	≤	≤	NOUN
ejpam-3344	210	9	µa(x	µa(x	NOUN
ejpam-3344	210	10	)	)	PUNCT
ejpam-3344	210	11	∨	∨	NUM
ejpam-3344	210	12	µa(y	µa(y	NOUN
ejpam-3344	210	13	)	)	PUNCT
ejpam-3344	210	14	∨	∨	NUM
ejpam-3344	210	15	µa(y	µa(y	NOUN
ejpam-3344	210	16	)	)	PUNCT
ejpam-3344	210	17	and	and	CCONJ
ejpam-3344	210	18	γa((xy)(az	γa((xy)(az	PROPN
ejpam-3344	210	19	)	)	PUNCT
ejpam-3344	210	20	)	)	PUNCT
ejpam-3344	210	21	≥	≥	NOUN
ejpam-3344	210	22	γa(x	γa(x	NUM
ejpam-3344	210	23	)	)	PUNCT
ejpam-3344	210	24	∧	∧	NOUN
ejpam-3344	210	25	γa(y	γa(y	NOUN
ejpam-3344	210	26	)	)	PUNCT
ejpam-3344	210	27	∧	∧	NOUN
ejpam-3344	210	28	γa(y	γa(y	NOUN
ejpam-3344	210	29	)	)	PUNCT
ejpam-3344	210	30	.	.	PUNCT
ejpam-3344	211	1	n.	n.	PROPN
ejpam-3344	211	2	kausar	kausar	PROPN
ejpam-3344	211	3	,	,	PUNCT
ejpam-3344	211	4	m.	m.	NOUN
ejpam-3344	211	5	a.	a.	PROPN
ejpam-3344	211	6	waqar	waqar	PROPN
ejpam-3344	211	7	/	/	SYM
ejpam-3344	211	8	eur	eur	PROPN
ejpam-3344	211	9	.	.	PUNCT
ejpam-3344	212	1	j.	j.	PROPN
ejpam-3344	212	2	pure	pure	PROPN
ejpam-3344	212	3	appl	appl	PROPN
ejpam-3344	212	4	.	.	PROPN
ejpam-3344	212	5	math	math	PROPN
ejpam-3344	212	6	,	,	PUNCT
ejpam-3344	212	7	12	12	NUM
ejpam-3344	212	8	(	(	PUNCT
ejpam-3344	212	9	1	1	NUM
ejpam-3344	212	10	)	)	PUNCT
ejpam-3344	212	11	(	(	PUNCT
ejpam-3344	212	12	2019	2019	NUM
ejpam-3344	212	13	)	)	PUNCT
ejpam-3344	212	14	,	,	PUNCT
ejpam-3344	212	15	226	226	NUM
ejpam-3344	212	16	-	-	SYM
ejpam-3344	212	17	250	250	NUM
ejpam-3344	212	18	234	234	NUM
ejpam-3344	212	19	let	let	NOUN
ejpam-3344	212	20	µa(x	µa(x	PUNCT
ejpam-3344	212	21	)	)	PUNCT
ejpam-3344	213	1	=	=	SYM
ejpam-3344	213	2	r	r	NOUN
ejpam-3344	213	3	=	=	PUNCT
ejpam-3344	213	4	µa(y	µa(y	NOUN
ejpam-3344	213	5	)	)	PUNCT
ejpam-3344	213	6	=	=	NOUN
ejpam-3344	213	7	µa(z	µa(z	PUNCT
ejpam-3344	213	8	)	)	PUNCT
ejpam-3344	213	9	and	and	CCONJ
ejpam-3344	213	10	γa(x	γa(x	NUM
ejpam-3344	213	11	)	)	PUNCT
ejpam-3344	213	12	=	=	SYM
ejpam-3344	213	13	t	t	NOUN
ejpam-3344	213	14	=	=	PUNCT
ejpam-3344	213	15	γa(y	γa(y	X
ejpam-3344	213	16	)	)	PUNCT
ejpam-3344	213	17	=	=	SYM
ejpam-3344	213	18	γa(z	γa(z	PROPN
ejpam-3344	213	19	)	)	PUNCT
ejpam-3344	213	20	,	,	PUNCT
ejpam-3344	213	21	this	this	PRON
ejpam-3344	213	22	implies	imply	VERB
ejpam-3344	213	23	that	that	SCONJ
ejpam-3344	213	24	µa(x	µa(x	ADP
ejpam-3344	213	25	)	)	PUNCT
ejpam-3344	213	26	,	,	PUNCT
ejpam-3344	213	27	µa(y	µa(y	NOUN
ejpam-3344	213	28	)	)	PUNCT
ejpam-3344	213	29	,	,	PUNCT
ejpam-3344	213	30	µa(z	µa(z	NUM
ejpam-3344	213	31	)	)	PUNCT
ejpam-3344	213	32	≥	≥	NOUN
ejpam-3344	213	33	r	r	NOUN
ejpam-3344	213	34	and	and	CCONJ
ejpam-3344	213	35	γa(x	γa(x	NUM
ejpam-3344	213	36	)	)	PUNCT
ejpam-3344	213	37	,	,	PUNCT
ejpam-3344	213	38	γa(y	γa(y	NUM
ejpam-3344	213	39	)	)	PUNCT
ejpam-3344	213	40	,	,	PUNCT
ejpam-3344	213	41	γa(z	γa(z	PROPN
ejpam-3344	213	42	)	)	PUNCT
ejpam-3344	213	43	≤	≤	NOUN
ejpam-3344	213	44	t	t	PROPN
ejpam-3344	213	45	,	,	PUNCT
ejpam-3344	213	46	i.e.	i.e.	X
ejpam-3344	213	47	,	,	PUNCT
ejpam-3344	213	48	x	x	X
ejpam-3344	213	49	,	,	PUNCT
ejpam-3344	213	50	y	y	PROPN
ejpam-3344	213	51	,	,	PUNCT
ejpam-3344	213	52	z	z	PROPN
ejpam-3344	213	53	∈	∈	PROPN
ejpam-3344	213	54	a(r	a(r	NOUN
ejpam-3344	213	55	,	,	PUNCT
ejpam-3344	213	56	t	t	PROPN
ejpam-3344	213	57	)	)	PUNCT
ejpam-3344	213	58	.	.	PUNCT
ejpam-3344	214	1	but	but	CCONJ
ejpam-3344	214	2	µa((xy)(az	µa((xy)(az	NOUN
ejpam-3344	214	3	)	)	PUNCT
ejpam-3344	214	4	)	)	PUNCT
ejpam-3344	214	5	≤	≤	NUM
ejpam-3344	214	6	r	r	NOUN
ejpam-3344	214	7	and	and	CCONJ
ejpam-3344	214	8	γa((xy)(az	γa((xy)(az	PROPN
ejpam-3344	214	9	)	)	PUNCT
ejpam-3344	214	10	)	)	PUNCT
ejpam-3344	214	11	≥	≥	PROPN
ejpam-3344	214	12	t	t	PROPN
ejpam-3344	214	13	,	,	PUNCT
ejpam-3344	214	14	i.e.	i.e.	X
ejpam-3344	214	15	,	,	PUNCT
ejpam-3344	214	16	(	(	PUNCT
ejpam-3344	214	17	xy)(az	xy)(az	NOUN
ejpam-3344	214	18	)	)	PUNCT
ejpam-3344	214	19	/∈	/∈	PUNCT
ejpam-3344	215	1	a(r	a(r	PROPN
ejpam-3344	215	2	,	,	PUNCT
ejpam-3344	215	3	t	t	PROPN
ejpam-3344	215	4	)	)	PUNCT
ejpam-3344	215	5	,	,	PUNCT
ejpam-3344	215	6	which	which	PRON
ejpam-3344	215	7	is	be	AUX
ejpam-3344	215	8	a	a	DET
ejpam-3344	215	9	contradiction	contradiction	NOUN
ejpam-3344	215	10	.	.	PUNCT
ejpam-3344	216	1	so	so	ADV
ejpam-3344	216	2	µa((xy)(az	µa((xy)(az	NOUN
ejpam-3344	216	3	)	)	PUNCT
ejpam-3344	216	4	)	)	PUNCT
ejpam-3344	216	5	≥	≥	NOUN
ejpam-3344	216	6	µa(x	µa(x	NOUN
ejpam-3344	216	7	)	)	PUNCT
ejpam-3344	216	8	∧	∧	NOUN
ejpam-3344	216	9	µa(y	µa(y	NOUN
ejpam-3344	216	10	)	)	PUNCT
ejpam-3344	216	11	∧	∧	NOUN
ejpam-3344	216	12	µa(y	µa(y	NOUN
ejpam-3344	216	13	)	)	PUNCT
ejpam-3344	216	14	and	and	CCONJ
ejpam-3344	216	15	γa((xy)(az	γa((xy)(az	VERB
ejpam-3344	216	16	)	)	PUNCT
ejpam-3344	216	17	)	)	PUNCT
ejpam-3344	216	18	≤	≤	NOUN
ejpam-3344	216	19	γa(x	γa(x	NUM
ejpam-3344	216	20	)	)	PUNCT
ejpam-3344	216	21	∨	∨	NUM
ejpam-3344	216	22	γa(y	γa(y	NUM
ejpam-3344	216	23	)	)	PUNCT
ejpam-3344	216	24	∨	∨	NUM
ejpam-3344	216	25	γa(y	γa(y	NUM
ejpam-3344	216	26	)	)	PUNCT
ejpam-3344	216	27	.	.	PUNCT
ejpam-3344	217	1	remark	remark	PROPN
ejpam-3344	217	2	4	4	NUM
ejpam-3344	217	3	.	.	PUNCT
ejpam-3344	218	1	let	let	VERB
ejpam-3344	218	2	a	a	PRON
ejpam-3344	218	3	=	=	SYM
ejpam-3344	218	4	(	(	PUNCT
ejpam-3344	218	5	µa	µa	PROPN
ejpam-3344	218	6	,	,	PUNCT
ejpam-3344	218	7	γa	γa	PROPN
ejpam-3344	218	8	)	)	PUNCT
ejpam-3344	218	9	be	be	VERB
ejpam-3344	218	10	an	an	DET
ejpam-3344	218	11	ifs	ifs	PROPN
ejpam-3344	218	12	of	of	ADP
ejpam-3344	218	13	an	an	DET
ejpam-3344	218	14	la	la	ADJ
ejpam-3344	218	15	-	-	PUNCT
ejpam-3344	218	16	ring	ring	NOUN
ejpam-3344	218	17	r.	r.	PROPN
ejpam-3344	218	18	then	then	ADV
ejpam-3344	218	19	a	a	PRON
ejpam-3344	218	20	is	be	AUX
ejpam-3344	218	21	an	an	DET
ejpam-3344	218	22	intuitionistic	intuitionistic	ADJ
ejpam-3344	218	23	fuzzy	fuzzy	ADJ
ejpam-3344	218	24	bi	bi	NOUN
ejpam-3344	218	25	-	-	NOUN
ejpam-3344	218	26	ideal	ideal	NOUN
ejpam-3344	218	27	of	of	ADP
ejpam-3344	218	28	r	r	NOUN
ejpam-3344	218	29	if	if	SCONJ
ejpam-3344	219	1	and	and	CCONJ
ejpam-3344	219	2	only	only	ADV
ejpam-3344	219	3	if	if	SCONJ
ejpam-3344	219	4	a(r	a(r	PROPN
ejpam-3344	219	5	,	,	PUNCT
ejpam-3344	219	6	t	t	PROPN
ejpam-3344	219	7	)	)	PUNCT
ejpam-3344	219	8	is	be	AUX
ejpam-3344	219	9	a	a	DET
ejpam-3344	219	10	bi	bi	NOUN
ejpam-3344	219	11	-	-	NOUN
ejpam-3344	219	12	ideal	ideal	NOUN
ejpam-3344	219	13	of	of	ADP
ejpam-3344	219	14	r	r	NOUN
ejpam-3344	219	15	for	for	ADP
ejpam-3344	219	16	all	all	DET
ejpam-3344	219	17	r	r	NOUN
ejpam-3344	219	18	,	,	PUNCT
ejpam-3344	219	19	t	t	PROPN
ejpam-3344	219	20	∈	∈	PROPN
ejpam-3344	219	21	(	(	PUNCT
ejpam-3344	219	22	0	0	NUM
ejpam-3344	219	23	,	,	PUNCT
ejpam-3344	219	24	1	1	NUM
ejpam-3344	219	25	]	]	PUNCT
ejpam-3344	219	26	.	.	PUNCT
ejpam-3344	220	1	2	2	X
ejpam-3344	220	2	.	.	X
ejpam-3344	220	3	characterizations	characterization	NOUN
ejpam-3344	220	4	of	of	ADP
ejpam-3344	220	5	la	la	NOUN
ejpam-3344	220	6	-	-	PUNCT
ejpam-3344	220	7	rings	ring	NOUN
ejpam-3344	220	8	in	in	ADP
ejpam-3344	220	9	this	this	DET
ejpam-3344	220	10	section	section	NOUN
ejpam-3344	220	11	,	,	PUNCT
ejpam-3344	220	12	we	we	PRON
ejpam-3344	220	13	characterize	characterize	VERB
ejpam-3344	220	14	different	different	ADJ
ejpam-3344	220	15	classes	class	NOUN
ejpam-3344	220	16	of	of	ADP
ejpam-3344	220	17	la	la	NOUN
ejpam-3344	220	18	-	-	NOUN
ejpam-3344	220	19	ring	ring	NOUN
ejpam-3344	220	20	in	in	ADP
ejpam-3344	220	21	terms	term	NOUN
ejpam-3344	220	22	of	of	ADP
ejpam-3344	220	23	intuitionistic	intuitionistic	ADJ
ejpam-3344	220	24	fuzzy	fuzzy	ADJ
ejpam-3344	220	25	left	left	NOUN
ejpam-3344	220	26	(	(	PUNCT
ejpam-3344	220	27	right	right	INTJ
ejpam-3344	220	28	,	,	PUNCT
ejpam-3344	220	29	bi-	bi-	NUM
ejpam-3344	220	30	,	,	PUNCT
ejpam-3344	220	31	generalized	generalize	VERB
ejpam-3344	220	32	bi-	bi-	NUM
ejpam-3344	220	33	)	)	PUNCT
ejpam-3344	220	34	ideals	ideal	NOUN
ejpam-3344	220	35	.	.	PUNCT
ejpam-3344	221	1	an	an	DET
ejpam-3344	221	2	la	la	ADJ
ejpam-3344	221	3	-	-	PUNCT
ejpam-3344	221	4	ring	ring	NOUN
ejpam-3344	221	5	r	r	NOUN
ejpam-3344	221	6	is	be	AUX
ejpam-3344	221	7	called	call	VERB
ejpam-3344	221	8	regular	regular	ADJ
ejpam-3344	221	9	,	,	PUNCT
ejpam-3344	221	10	if	if	SCONJ
ejpam-3344	221	11	for	for	ADP
ejpam-3344	221	12	every	every	DET
ejpam-3344	221	13	element	element	NOUN
ejpam-3344	221	14	x	x	SYM
ejpam-3344	221	15	∈	∈	PROPN
ejpam-3344	221	16	r	r	NOUN
ejpam-3344	221	17	,	,	PUNCT
ejpam-3344	221	18	there	there	PRON
ejpam-3344	221	19	exists	exist	VERB
ejpam-3344	221	20	an	an	DET
ejpam-3344	221	21	element	element	NOUN
ejpam-3344	221	22	a	a	DET
ejpam-3344	221	23	∈	∈	NOUN
ejpam-3344	221	24	r	r	NOUN
ejpam-3344	221	25	such	such	ADJ
ejpam-3344	221	26	that	that	PRON
ejpam-3344	221	27	x	x	SYM
ejpam-3344	221	28	=	=	SYM
ejpam-3344	221	29	(	(	PUNCT
ejpam-3344	221	30	xa)x	xa)x	PROPN
ejpam-3344	221	31	.	.	PUNCT
ejpam-3344	222	1	an	an	DET
ejpam-3344	222	2	la	la	ADJ
ejpam-3344	222	3	-	-	PUNCT
ejpam-3344	222	4	ring	ring	NOUN
ejpam-3344	222	5	r	r	NOUN
ejpam-3344	222	6	is	be	AUX
ejpam-3344	222	7	called	call	VERB
ejpam-3344	222	8	intra	intra	ADJ
ejpam-3344	222	9	-	-	ADJ
ejpam-3344	222	10	regular	regular	ADJ
ejpam-3344	222	11	,	,	PUNCT
ejpam-3344	222	12	if	if	SCONJ
ejpam-3344	222	13	for	for	ADP
ejpam-3344	222	14	every	every	DET
ejpam-3344	222	15	element	element	NOUN
ejpam-3344	222	16	x	x	SYM
ejpam-3344	222	17	∈	∈	PROPN
ejpam-3344	222	18	r	r	NOUN
ejpam-3344	222	19	,	,	PUNCT
ejpam-3344	222	20	there	there	PRON
ejpam-3344	222	21	exist	exist	VERB
ejpam-3344	222	22	elements	element	NOUN
ejpam-3344	222	23	ai	ai	VERB
ejpam-3344	222	24	,	,	PUNCT
ejpam-3344	222	25	bi	bi	NOUN
ejpam-3344	222	26	∈	∈	PROPN
ejpam-3344	222	27	r	r	NOUN
ejpam-3344	222	28	such	such	ADJ
ejpam-3344	222	29	that	that	SCONJ
ejpam-3344	222	30	x	x	NOUN
ejpam-3344	223	1	=	=	PUNCT
ejpam-3344	223	2	∑n	∑n	PROPN
ejpam-3344	223	3	i=1(aix	i=1(aix	PROPN
ejpam-3344	223	4	2)bi	2)bi	NUM
ejpam-3344	223	5	.	.	PUNCT
ejpam-3344	224	1	an	an	DET
ejpam-3344	224	2	la	la	ADJ
ejpam-3344	224	3	-	-	PUNCT
ejpam-3344	224	4	ring	ring	NOUN
ejpam-3344	224	5	r	r	NOUN
ejpam-3344	224	6	is	be	AUX
ejpam-3344	224	7	called	call	VERB
ejpam-3344	224	8	left	left	ADJ
ejpam-3344	224	9	(	(	PUNCT
ejpam-3344	224	10	resp	resp	NOUN
ejpam-3344	224	11	.	.	PUNCT
ejpam-3344	225	1	right	right	ADJ
ejpam-3344	225	2	)	)	PUNCT
ejpam-3344	226	1	regular	regular	ADJ
ejpam-3344	226	2	,	,	PUNCT
ejpam-3344	226	3	if	if	SCONJ
ejpam-3344	226	4	for	for	ADP
ejpam-3344	226	5	every	every	DET
ejpam-3344	226	6	element	element	NOUN
ejpam-3344	226	7	x	x	SYM
ejpam-3344	226	8	∈	∈	PROPN
ejpam-3344	226	9	r	r	NOUN
ejpam-3344	226	10	,	,	PUNCT
ejpam-3344	226	11	there	there	PRON
ejpam-3344	226	12	exists	exist	VERB
ejpam-3344	226	13	an	an	DET
ejpam-3344	226	14	element	element	NOUN
ejpam-3344	226	15	a	a	DET
ejpam-3344	226	16	∈	∈	NOUN
ejpam-3344	226	17	r	r	NOUN
ejpam-3344	227	1	such	such	ADJ
ejpam-3344	227	2	that	that	SCONJ
ejpam-3344	227	3	x	x	X
ejpam-3344	227	4	=	=	SYM
ejpam-3344	227	5	ax2	ax2	NOUN
ejpam-3344	227	6	(	(	PUNCT
ejpam-3344	227	7	resp	resp	NOUN
ejpam-3344	227	8	.	.	PUNCT
ejpam-3344	227	9	x2a	x2a	PROPN
ejpam-3344	227	10	)	)	PUNCT
ejpam-3344	227	11	.	.	PUNCT
ejpam-3344	228	1	an	an	DET
ejpam-3344	228	2	la	la	ADJ
ejpam-3344	228	3	-	-	PUNCT
ejpam-3344	228	4	ring	ring	NOUN
ejpam-3344	228	5	r	r	NOUN
ejpam-3344	228	6	is	be	AUX
ejpam-3344	228	7	called	call	VERB
ejpam-3344	228	8	completely	completely	ADV
ejpam-3344	228	9	regular	regular	ADJ
ejpam-3344	228	10	,	,	PUNCT
ejpam-3344	228	11	if	if	SCONJ
ejpam-3344	228	12	it	it	PRON
ejpam-3344	228	13	is	be	AUX
ejpam-3344	228	14	regular	regular	ADJ
ejpam-3344	228	15	,	,	PUNCT
ejpam-3344	228	16	left	leave	VERB
ejpam-3344	228	17	regular	regular	ADV
ejpam-3344	228	18	and	and	CCONJ
ejpam-3344	228	19	right	right	ADV
ejpam-3344	228	20	regular	regular	ADV
ejpam-3344	228	21	.	.	PUNCT
ejpam-3344	229	1	an	an	DET
ejpam-3344	229	2	la	la	ADJ
ejpam-3344	229	3	-	-	PUNCT
ejpam-3344	229	4	ring	ring	NOUN
ejpam-3344	229	5	r	r	NOUN
ejpam-3344	229	6	is	be	AUX
ejpam-3344	229	7	called	call	VERB
ejpam-3344	229	8	(	(	PUNCT
ejpam-3344	229	9	2	2	NUM
ejpam-3344	229	10	,	,	PUNCT
ejpam-3344	229	11	2)-regular	2)-regular	NUM
ejpam-3344	229	12	,	,	PUNCT
ejpam-3344	229	13	if	if	SCONJ
ejpam-3344	229	14	for	for	ADP
ejpam-3344	229	15	every	every	DET
ejpam-3344	229	16	element	element	NOUN
ejpam-3344	229	17	x	x	SYM
ejpam-3344	229	18	∈	∈	PROPN
ejpam-3344	229	19	r	r	NOUN
ejpam-3344	229	20	,	,	PUNCT
ejpam-3344	229	21	there	there	PRON
ejpam-3344	229	22	exists	exist	VERB
ejpam-3344	229	23	an	an	DET
ejpam-3344	229	24	element	element	NOUN
ejpam-3344	229	25	a	a	DET
ejpam-3344	229	26	∈	∈	NOUN
ejpam-3344	229	27	r	r	NOUN
ejpam-3344	229	28	such	such	ADJ
ejpam-3344	229	29	that	that	PRON
ejpam-3344	229	30	x	x	X
ejpam-3344	229	31	=	=	PRON
ejpam-3344	229	32	(	(	PUNCT
ejpam-3344	229	33	x2a)x2	x2a)x2	PROPN
ejpam-3344	229	34	.	.	PUNCT
ejpam-3344	230	1	an	an	DET
ejpam-3344	230	2	la	la	ADJ
ejpam-3344	230	3	-	-	PUNCT
ejpam-3344	230	4	ring	ring	NOUN
ejpam-3344	230	5	r	r	NOUN
ejpam-3344	230	6	is	be	AUX
ejpam-3344	230	7	called	call	VERB
ejpam-3344	230	8	locally	locally	ADV
ejpam-3344	230	9	associative	associative	ADJ
ejpam-3344	230	10	la	la	ADJ
ejpam-3344	230	11	-	-	PUNCT
ejpam-3344	230	12	ring	ring	NOUN
ejpam-3344	230	13	if	if	SCONJ
ejpam-3344	230	14	(	(	PUNCT
ejpam-3344	230	15	a.a).a	a.a).a	PROPN
ejpam-3344	230	16	=	=	SYM
ejpam-3344	230	17	a.(a.a	a.(a.a	PROPN
ejpam-3344	230	18	)	)	PUNCT
ejpam-3344	230	19	for	for	ADP
ejpam-3344	230	20	all	all	DET
ejpam-3344	230	21	a	a	DET
ejpam-3344	230	22	∈	∈	PROPN
ejpam-3344	230	23	r.	r.	NOUN
ejpam-3344	230	24	a	a	DET
ejpam-3344	230	25	ring	ring	NOUN
ejpam-3344	230	26	r	r	NOUN
ejpam-3344	230	27	is	be	AUX
ejpam-3344	230	28	called	call	VERB
ejpam-3344	230	29	left	left	ADJ
ejpam-3344	230	30	(	(	PUNCT
ejpam-3344	230	31	resp	resp	NOUN
ejpam-3344	230	32	.	.	PUNCT
ejpam-3344	231	1	right	right	ADJ
ejpam-3344	231	2	)	)	PUNCT
ejpam-3344	231	3	weakly	weakly	ADV
ejpam-3344	231	4	regular	regular	ADV
ejpam-3344	231	5	if	if	SCONJ
ejpam-3344	231	6	i2	i2	PROPN
ejpam-3344	231	7	=	=	PROPN
ejpam-3344	231	8	i	i	PROPN
ejpam-3344	231	9	,	,	PUNCT
ejpam-3344	231	10	for	for	ADP
ejpam-3344	231	11	every	every	DET
ejpam-3344	231	12	left	left	NOUN
ejpam-3344	231	13	(	(	PUNCT
ejpam-3344	231	14	resp	resp	NOUN
ejpam-3344	231	15	.	.	PUNCT
ejpam-3344	232	1	right	right	ADJ
ejpam-3344	232	2	)	)	PUNCT
ejpam-3344	232	3	ideal	ideal	NOUN
ejpam-3344	232	4	i	i	PRON
ejpam-3344	232	5	of	of	ADP
ejpam-3344	232	6	r	r	NOUN
ejpam-3344	232	7	,	,	PUNCT
ejpam-3344	232	8	equivalently	equivalently	ADV
ejpam-3344	232	9	x	x	SYM
ejpam-3344	232	10	∈	∈	PROPN
ejpam-3344	232	11	rxrx(x	rxrx(x	AUX
ejpam-3344	232	12	∈	∈	PROPN
ejpam-3344	232	13	xrxr	xrxr	PROPN
ejpam-3344	232	14	)	)	PUNCT
ejpam-3344	232	15	for	for	ADP
ejpam-3344	232	16	every	every	DET
ejpam-3344	232	17	x	x	PROPN
ejpam-3344	232	18	∈	∈	PROPN
ejpam-3344	232	19	r.	r.	NOUN
ejpam-3344	232	20	an	an	DET
ejpam-3344	232	21	la	la	ADJ
ejpam-3344	232	22	-	-	PUNCT
ejpam-3344	232	23	ring	ring	NOUN
ejpam-3344	232	24	r	r	NOUN
ejpam-3344	232	25	is	be	AUX
ejpam-3344	232	26	called	call	VERB
ejpam-3344	232	27	weakly	weakly	ADV
ejpam-3344	232	28	regular	regular	ADV
ejpam-3344	232	29	if	if	SCONJ
ejpam-3344	232	30	it	it	PRON
ejpam-3344	232	31	is	be	AUX
ejpam-3344	232	32	both	both	PRON
ejpam-3344	232	33	left	leave	VERB
ejpam-3344	232	34	weakly	weakly	ADV
ejpam-3344	232	35	regular	regular	ADJ
ejpam-3344	232	36	and	and	CCONJ
ejpam-3344	232	37	right	right	ADV
ejpam-3344	232	38	weakly	weakly	ADV
ejpam-3344	232	39	regular	regular	ADV
ejpam-3344	232	40	[	[	X
ejpam-3344	232	41	17	17	NUM
ejpam-3344	232	42	]	]	PUNCT
ejpam-3344	232	43	.	.	PUNCT
ejpam-3344	233	1	now	now	ADV
ejpam-3344	233	2	we	we	PRON
ejpam-3344	233	3	define	define	VERB
ejpam-3344	233	4	this	this	DET
ejpam-3344	233	5	notion	notion	NOUN
ejpam-3344	233	6	in	in	ADP
ejpam-3344	233	7	a	a	DET
ejpam-3344	233	8	class	class	NOUN
ejpam-3344	233	9	of	of	ADP
ejpam-3344	233	10	non	non	ADJ
ejpam-3344	233	11	-	-	ADJ
ejpam-3344	233	12	associative	associative	ADJ
ejpam-3344	233	13	and	and	CCONJ
ejpam-3344	233	14	non	non	ADJ
ejpam-3344	233	15	-	-	ADJ
ejpam-3344	233	16	commutative	commutative	ADJ
ejpam-3344	233	17	rings	ring	NOUN
ejpam-3344	233	18	(	(	PUNCT
ejpam-3344	233	19	la	la	NOUN
ejpam-3344	233	20	-	-	NOUN
ejpam-3344	233	21	ring	ring	NOUN
ejpam-3344	233	22	)	)	PUNCT
ejpam-3344	233	23	.	.	PUNCT
ejpam-3344	234	1	an	an	DET
ejpam-3344	234	2	la	la	ADJ
ejpam-3344	234	3	-	-	PUNCT
ejpam-3344	234	4	ring	ring	NOUN
ejpam-3344	234	5	r	r	NOUN
ejpam-3344	234	6	is	be	AUX
ejpam-3344	234	7	called	call	VERB
ejpam-3344	234	8	left	left	ADJ
ejpam-3344	234	9	(	(	PUNCT
ejpam-3344	234	10	resp	resp	NOUN
ejpam-3344	234	11	.	.	PUNCT
ejpam-3344	235	1	right	right	ADJ
ejpam-3344	235	2	)	)	PUNCT
ejpam-3344	235	3	weakly	weakly	ADV
ejpam-3344	235	4	regular	regular	ADV
ejpam-3344	235	5	,	,	PUNCT
ejpam-3344	235	6	if	if	SCONJ
ejpam-3344	235	7	for	for	ADP
ejpam-3344	235	8	every	every	DET
ejpam-3344	235	9	element	element	NOUN
ejpam-3344	235	10	x	x	SYM
ejpam-3344	235	11	∈	∈	PROPN
ejpam-3344	235	12	r	r	NOUN
ejpam-3344	235	13	,	,	PUNCT
ejpam-3344	235	14	there	there	PRON
ejpam-3344	235	15	exist	exist	VERB
ejpam-3344	235	16	elements	element	NOUN
ejpam-3344	235	17	a	a	PRON
ejpam-3344	235	18	,	,	PUNCT
ejpam-3344	235	19	b	b	X
ejpam-3344	235	20	∈	∈	NOUN
ejpam-3344	235	21	r	r	NOUN
ejpam-3344	235	22	such	such	ADJ
ejpam-3344	235	23	that	that	PRON
ejpam-3344	235	24	x	x	X
ejpam-3344	235	25	=	=	SYM
ejpam-3344	235	26	(	(	PUNCT
ejpam-3344	235	27	ax)(bx	ax)(bx	NOUN
ejpam-3344	235	28	)	)	PUNCT
ejpam-3344	235	29	(	(	PUNCT
ejpam-3344	235	30	resp	resp	NOUN
ejpam-3344	235	31	.	.	PUNCT
ejpam-3344	236	1	x	x	X
ejpam-3344	236	2	=	=	PUNCT
ejpam-3344	236	3	(	(	PUNCT
ejpam-3344	236	4	xa)(xb	xa)(xb	PROPN
ejpam-3344	236	5	)	)	PUNCT
ejpam-3344	236	6	)	)	PUNCT
ejpam-3344	236	7	.	.	PUNCT
ejpam-3344	237	1	an	an	DET
ejpam-3344	237	2	la	la	ADJ
ejpam-3344	237	3	-	-	PUNCT
ejpam-3344	237	4	ring	ring	NOUN
ejpam-3344	237	5	r	r	NOUN
ejpam-3344	237	6	is	be	AUX
ejpam-3344	237	7	called	call	VERB
ejpam-3344	237	8	weakly	weakly	ADV
ejpam-3344	237	9	regular	regular	ADV
ejpam-3344	237	10	if	if	SCONJ
ejpam-3344	237	11	it	it	PRON
ejpam-3344	237	12	is	be	AUX
ejpam-3344	237	13	both	both	PRON
ejpam-3344	237	14	left	leave	VERB
ejpam-3344	237	15	weakly	weakly	ADV
ejpam-3344	237	16	regular	regular	ADJ
ejpam-3344	237	17	and	and	CCONJ
ejpam-3344	237	18	right	right	ADJ
ejpam-3344	237	19	weakly	weakly	ADV
ejpam-3344	237	20	regular	regular	ADV
ejpam-3344	237	21	.	.	PUNCT
ejpam-3344	238	1	lemma	lemma	PROPN
ejpam-3344	238	2	6	6	NUM
ejpam-3344	238	3	.	.	PUNCT
ejpam-3344	239	1	every	every	DET
ejpam-3344	239	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	239	3	fuzzy	fuzzy	ADJ
ejpam-3344	239	4	right	right	ADJ
ejpam-3344	239	5	ideal	ideal	NOUN
ejpam-3344	239	6	of	of	ADP
ejpam-3344	239	7	an	an	DET
ejpam-3344	239	8	la	la	ADJ
ejpam-3344	239	9	-	-	PUNCT
ejpam-3344	239	10	ring	ring	NOUN
ejpam-3344	239	11	r	r	NOUN
ejpam-3344	239	12	with	with	ADP
ejpam-3344	239	13	left	left	ADJ
ejpam-3344	239	14	identity	identity	NOUN
ejpam-3344	239	15	e	e	NOUN
ejpam-3344	239	16	,	,	PUNCT
ejpam-3344	239	17	is	be	AUX
ejpam-3344	239	18	an	an	DET
ejpam-3344	239	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	239	20	fuzzy	fuzzy	ADJ
ejpam-3344	239	21	ideal	ideal	NOUN
ejpam-3344	239	22	of	of	ADP
ejpam-3344	239	23	r.	r.	PROPN
ejpam-3344	239	24	proof	proof	NOUN
ejpam-3344	239	25	.	.	PUNCT
ejpam-3344	240	1	let	let	VERB
ejpam-3344	240	2	a	a	DET
ejpam-3344	240	3	=	=	SYM
ejpam-3344	240	4	(	(	PUNCT
ejpam-3344	240	5	µa	µa	PROPN
ejpam-3344	240	6	,	,	PUNCT
ejpam-3344	240	7	γa	γa	PROPN
ejpam-3344	240	8	)	)	PUNCT
ejpam-3344	240	9	be	be	VERB
ejpam-3344	240	10	an	an	DET
ejpam-3344	240	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	240	12	fuzzy	fuzzy	ADJ
ejpam-3344	240	13	right	right	ADJ
ejpam-3344	240	14	ideal	ideal	NOUN
ejpam-3344	240	15	of	of	ADP
ejpam-3344	240	16	r	r	NOUN
ejpam-3344	240	17	and	and	CCONJ
ejpam-3344	240	18	x	x	NOUN
ejpam-3344	240	19	,	,	PUNCT
ejpam-3344	240	20	y	y	PROPN
ejpam-3344	240	21	∈	∈	PROPN
ejpam-3344	240	22	r.	r.	NOUN
ejpam-3344	240	23	thus	thus	ADV
ejpam-3344	240	24	µa	µa	X
ejpam-3344	240	25	(	(	PUNCT
ejpam-3344	240	26	xy	xy	NOUN
ejpam-3344	240	27	)	)	PUNCT
ejpam-3344	240	28	=	=	SYM
ejpam-3344	241	1	µa	µa	NOUN
ejpam-3344	241	2	(	(	PUNCT
ejpam-3344	241	3	(	(	PUNCT
ejpam-3344	241	4	ex	ex	NOUN
ejpam-3344	241	5	)	)	PUNCT
ejpam-3344	241	6	y	y	NOUN
ejpam-3344	241	7	)	)	PUNCT
ejpam-3344	241	8	=	=	SYM
ejpam-3344	241	9	µa	µa	NOUN
ejpam-3344	241	10	(	(	PUNCT
ejpam-3344	241	11	(	(	PUNCT
ejpam-3344	241	12	yx	yx	NOUN
ejpam-3344	241	13	)	)	PUNCT
ejpam-3344	241	14	e	e	NOUN
ejpam-3344	241	15	)	)	PUNCT
ejpam-3344	241	16	≥	≥	NOUN
ejpam-3344	241	17	µa	µa	NOUN
ejpam-3344	241	18	(	(	PUNCT
ejpam-3344	241	19	yx	yx	NOUN
ejpam-3344	241	20	)	)	PUNCT
ejpam-3344	241	21	≥	≥	NOUN
ejpam-3344	241	22	µa	µa	NOUN
ejpam-3344	241	23	(	(	PUNCT
ejpam-3344	241	24	y	y	NOUN
ejpam-3344	241	25	)	)	PUNCT
ejpam-3344	241	26	and	and	CCONJ
ejpam-3344	241	27	γa	γa	PROPN
ejpam-3344	241	28	(	(	PUNCT
ejpam-3344	241	29	xy	xy	PROPN
ejpam-3344	241	30	)	)	PUNCT
ejpam-3344	242	1	=	=	PRON
ejpam-3344	242	2	γa	γa	PROPN
ejpam-3344	242	3	(	(	PUNCT
ejpam-3344	242	4	(	(	PUNCT
ejpam-3344	242	5	ex	ex	NOUN
ejpam-3344	242	6	)	)	PUNCT
ejpam-3344	242	7	y	y	NOUN
ejpam-3344	242	8	)	)	PUNCT
ejpam-3344	243	1	=	=	PRON
ejpam-3344	243	2	γa	γa	PROPN
ejpam-3344	243	3	(	(	PUNCT
ejpam-3344	243	4	(	(	PUNCT
ejpam-3344	243	5	yx	yx	NOUN
ejpam-3344	243	6	)	)	PUNCT
ejpam-3344	243	7	e	e	NOUN
ejpam-3344	243	8	)	)	PUNCT
ejpam-3344	243	9	≤	≤	NOUN
ejpam-3344	243	10	γa	γa	NOUN
ejpam-3344	243	11	(	(	PUNCT
ejpam-3344	243	12	yx	yx	NOUN
ejpam-3344	243	13	)	)	PUNCT
ejpam-3344	243	14	≤	≤	NOUN
ejpam-3344	243	15	γa	γa	NOUN
ejpam-3344	243	16	(	(	PUNCT
ejpam-3344	243	17	y	y	PROPN
ejpam-3344	243	18	)	)	PUNCT
ejpam-3344	243	19	.	.	PUNCT
ejpam-3344	244	1	hence	hence	ADV
ejpam-3344	244	2	a	a	PRON
ejpam-3344	244	3	is	be	AUX
ejpam-3344	244	4	an	an	DET
ejpam-3344	244	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	244	6	fuzzy	fuzzy	ADJ
ejpam-3344	244	7	ideal	ideal	NOUN
ejpam-3344	244	8	of	of	ADP
ejpam-3344	244	9	r.	r.	PROPN
ejpam-3344	244	10	lemma	lemma	PROPN
ejpam-3344	244	11	7	7	NUM
ejpam-3344	244	12	.	.	PUNCT
ejpam-3344	245	1	every	every	DET
ejpam-3344	245	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	245	3	fuzzy	fuzzy	ADJ
ejpam-3344	245	4	right	right	ADJ
ejpam-3344	245	5	ideal	ideal	NOUN
ejpam-3344	245	6	of	of	ADP
ejpam-3344	245	7	a	a	DET
ejpam-3344	245	8	regular	regular	ADJ
ejpam-3344	245	9	la	la	ADJ
ejpam-3344	245	10	-	-	PUNCT
ejpam-3344	245	11	ring	ring	NOUN
ejpam-3344	245	12	r	r	NOUN
ejpam-3344	245	13	,	,	PUNCT
ejpam-3344	245	14	is	be	AUX
ejpam-3344	245	15	an	an	DET
ejpam-3344	245	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	245	17	fuzzy	fuzzy	ADJ
ejpam-3344	245	18	ideal	ideal	NOUN
ejpam-3344	245	19	of	of	ADP
ejpam-3344	245	20	r.	r.	PROPN
ejpam-3344	245	21	n.	n.	PROPN
ejpam-3344	245	22	kausar	kausar	PROPN
ejpam-3344	245	23	,	,	PUNCT
ejpam-3344	245	24	m.	m.	NOUN
ejpam-3344	245	25	a.	a.	PROPN
ejpam-3344	245	26	waqar	waqar	PROPN
ejpam-3344	245	27	/	/	SYM
ejpam-3344	245	28	eur	eur	PROPN
ejpam-3344	245	29	.	.	PUNCT
ejpam-3344	246	1	j.	j.	PROPN
ejpam-3344	246	2	pure	pure	PROPN
ejpam-3344	246	3	appl	appl	PROPN
ejpam-3344	246	4	.	.	PROPN
ejpam-3344	246	5	math	math	PROPN
ejpam-3344	246	6	,	,	PUNCT
ejpam-3344	246	7	12	12	NUM
ejpam-3344	246	8	(	(	PUNCT
ejpam-3344	246	9	1	1	NUM
ejpam-3344	246	10	)	)	PUNCT
ejpam-3344	246	11	(	(	PUNCT
ejpam-3344	246	12	2019	2019	NUM
ejpam-3344	246	13	)	)	PUNCT
ejpam-3344	246	14	,	,	PUNCT
ejpam-3344	246	15	226	226	NUM
ejpam-3344	246	16	-	-	SYM
ejpam-3344	246	17	250	250	NUM
ejpam-3344	246	18	235	235	NUM
ejpam-3344	246	19	proof	proof	NOUN
ejpam-3344	246	20	.	.	PUNCT
ejpam-3344	246	21	suppose	suppose	VERB
ejpam-3344	246	22	that	that	SCONJ
ejpam-3344	246	23	a	a	DET
ejpam-3344	246	24	=	=	SYM
ejpam-3344	246	25	(	(	PUNCT
ejpam-3344	246	26	µa	µa	PROPN
ejpam-3344	246	27	,	,	PUNCT
ejpam-3344	246	28	γa	γa	PROPN
ejpam-3344	246	29	)	)	PUNCT
ejpam-3344	246	30	is	be	AUX
ejpam-3344	246	31	an	an	DET
ejpam-3344	246	32	intuitionistic	intuitionistic	ADJ
ejpam-3344	246	33	fuzzy	fuzzy	ADJ
ejpam-3344	246	34	right	right	ADJ
ejpam-3344	246	35	ideal	ideal	NOUN
ejpam-3344	246	36	of	of	ADP
ejpam-3344	246	37	r.	r.	PROPN
ejpam-3344	246	38	let	let	VERB
ejpam-3344	246	39	x	x	PRON
ejpam-3344	246	40	,	,	PUNCT
ejpam-3344	246	41	y	y	PROPN
ejpam-3344	246	42	∈	∈	PROPN
ejpam-3344	246	43	r	r	NOUN
ejpam-3344	246	44	,	,	PUNCT
ejpam-3344	246	45	this	this	PRON
ejpam-3344	246	46	implies	imply	VERB
ejpam-3344	246	47	that	that	SCONJ
ejpam-3344	246	48	there	there	PRON
ejpam-3344	246	49	exists	exist	VERB
ejpam-3344	246	50	a	a	DET
ejpam-3344	246	51	∈	∈	PROPN
ejpam-3344	246	52	r	r	NOUN
ejpam-3344	246	53	,	,	PUNCT
ejpam-3344	246	54	such	such	ADJ
ejpam-3344	246	55	that	that	SCONJ
ejpam-3344	246	56	x	x	SYM
ejpam-3344	246	57	=	=	SYM
ejpam-3344	246	58	(	(	PUNCT
ejpam-3344	246	59	xa)x	xa)x	PROPN
ejpam-3344	246	60	.	.	PUNCT
ejpam-3344	246	61	thus	thus	ADV
ejpam-3344	246	62	µa(xy	µa(xy	NUM
ejpam-3344	246	63	)	)	PUNCT
ejpam-3344	246	64	=	=	SYM
ejpam-3344	246	65	µa(((xa)x)y	µa(((xa)x)y	ADJ
ejpam-3344	246	66	)	)	PUNCT
ejpam-3344	246	67	=	=	SYM
ejpam-3344	246	68	µa((yx)(xa	µa((yx)(xa	NOUN
ejpam-3344	246	69	)	)	PUNCT
ejpam-3344	246	70	)	)	PUNCT
ejpam-3344	246	71	≥	≥	NUM
ejpam-3344	246	72	µa(yx	µa(yx	NOUN
ejpam-3344	246	73	)	)	PUNCT
ejpam-3344	246	74	≥	≥	NOUN
ejpam-3344	246	75	µa(y	µa(y	NOUN
ejpam-3344	246	76	)	)	PUNCT
ejpam-3344	246	77	and	and	CCONJ
ejpam-3344	246	78	γa(xy	γa(xy	PROPN
ejpam-3344	246	79	)	)	PUNCT
ejpam-3344	246	80	=	=	SYM
ejpam-3344	247	1	γa(((xa)x)y	γa(((xa)x)y	PROPN
ejpam-3344	247	2	)	)	PUNCT
ejpam-3344	247	3	=	=	SYM
ejpam-3344	247	4	γa((yx)(xa	γa((yx)(xa	NOUN
ejpam-3344	247	5	)	)	PUNCT
ejpam-3344	247	6	)	)	PUNCT
ejpam-3344	247	7	≤	≤	NUM
ejpam-3344	248	1	γa(yx	γa(yx	PROPN
ejpam-3344	248	2	)	)	PUNCT
ejpam-3344	248	3	≤	≤	NOUN
ejpam-3344	248	4	γa(y	γa(y	NUM
ejpam-3344	248	5	)	)	PUNCT
ejpam-3344	248	6	.	.	PUNCT
ejpam-3344	249	1	therefore	therefore	ADV
ejpam-3344	249	2	a	a	PRON
ejpam-3344	249	3	is	be	AUX
ejpam-3344	249	4	an	an	DET
ejpam-3344	249	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	249	6	fuzzy	fuzzy	ADJ
ejpam-3344	249	7	ideal	ideal	NOUN
ejpam-3344	249	8	of	of	ADP
ejpam-3344	249	9	r.	r.	PROPN
ejpam-3344	249	10	proposition	proposition	PROPN
ejpam-3344	249	11	4	4	NUM
ejpam-3344	249	12	.	.	PUNCT
ejpam-3344	250	1	let	let	VERB
ejpam-3344	250	2	r	r	PRON
ejpam-3344	250	3	be	be	AUX
ejpam-3344	250	4	a	a	DET
ejpam-3344	250	5	regular	regular	ADJ
ejpam-3344	250	6	la	la	ADJ
ejpam-3344	250	7	-	-	PUNCT
ejpam-3344	250	8	ring	ring	NOUN
ejpam-3344	250	9	having	have	VERB
ejpam-3344	250	10	the	the	DET
ejpam-3344	250	11	property	property	NOUN
ejpam-3344	250	12	a	a	DET
ejpam-3344	250	13	=	=	NOUN
ejpam-3344	250	14	a2	a2	PROPN
ejpam-3344	250	15	for	for	ADP
ejpam-3344	250	16	every	every	DET
ejpam-3344	250	17	a	a	DET
ejpam-3344	250	18	∈	∈	PROPN
ejpam-3344	250	19	r	r	NOUN
ejpam-3344	250	20	,	,	PUNCT
ejpam-3344	250	21	with	with	ADP
ejpam-3344	250	22	left	left	ADJ
ejpam-3344	250	23	identity	identity	NOUN
ejpam-3344	250	24	e.	e.	PROPN
ejpam-3344	250	25	then	then	ADV
ejpam-3344	250	26	every	every	DET
ejpam-3344	250	27	intuitionistic	intuitionistic	ADJ
ejpam-3344	250	28	fuzzy	fuzzy	ADJ
ejpam-3344	250	29	generalized	generalize	VERB
ejpam-3344	250	30	bi	bi	NOUN
ejpam-3344	250	31	-	-	NOUN
ejpam-3344	250	32	ideal	ideal	NOUN
ejpam-3344	250	33	of	of	ADP
ejpam-3344	250	34	r	r	NOUN
ejpam-3344	250	35	is	be	AUX
ejpam-3344	250	36	an	an	DET
ejpam-3344	250	37	intuitionistic	intuitionistic	ADJ
ejpam-3344	250	38	fuzzy	fuzzy	ADJ
ejpam-3344	250	39	bi	bi	NOUN
ejpam-3344	250	40	-	-	NOUN
ejpam-3344	250	41	ideal	ideal	NOUN
ejpam-3344	250	42	of	of	ADP
ejpam-3344	250	43	r.	r.	PROPN
ejpam-3344	250	44	proof	proof	NOUN
ejpam-3344	250	45	.	.	PUNCT
ejpam-3344	251	1	let	let	VERB
ejpam-3344	251	2	a	a	DET
ejpam-3344	251	3	=	=	SYM
ejpam-3344	251	4	(	(	PUNCT
ejpam-3344	251	5	µa	µa	PROPN
ejpam-3344	251	6	,	,	PUNCT
ejpam-3344	251	7	γa	γa	PROPN
ejpam-3344	251	8	)	)	PUNCT
ejpam-3344	251	9	be	be	VERB
ejpam-3344	251	10	an	an	DET
ejpam-3344	251	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	251	12	fuzzy	fuzzy	ADJ
ejpam-3344	251	13	generalized	generalize	VERB
ejpam-3344	251	14	bi	bi	NOUN
ejpam-3344	251	15	-	-	NOUN
ejpam-3344	251	16	ideal	ideal	NOUN
ejpam-3344	251	17	of	of	ADP
ejpam-3344	251	18	r	r	NOUN
ejpam-3344	251	19	and	and	CCONJ
ejpam-3344	251	20	x	x	NOUN
ejpam-3344	251	21	,	,	PUNCT
ejpam-3344	251	22	y	y	PROPN
ejpam-3344	251	23	∈	∈	PROPN
ejpam-3344	251	24	r	r	NOUN
ejpam-3344	251	25	,	,	PUNCT
ejpam-3344	251	26	this	this	PRON
ejpam-3344	251	27	implies	imply	VERB
ejpam-3344	251	28	that	that	SCONJ
ejpam-3344	251	29	there	there	PRON
ejpam-3344	251	30	exists	exist	VERB
ejpam-3344	251	31	a	a	DET
ejpam-3344	251	32	∈	∈	NOUN
ejpam-3344	251	33	r	r	NOUN
ejpam-3344	252	1	such	such	ADJ
ejpam-3344	252	2	that	that	PRON
ejpam-3344	252	3	x	x	SYM
ejpam-3344	252	4	=	=	SYM
ejpam-3344	252	5	(	(	PUNCT
ejpam-3344	252	6	xa)x	xa)x	PROPN
ejpam-3344	252	7	.	.	PUNCT
ejpam-3344	253	1	we	we	PRON
ejpam-3344	253	2	have	have	VERB
ejpam-3344	253	3	to	to	PART
ejpam-3344	253	4	show	show	VERB
ejpam-3344	253	5	that	that	SCONJ
ejpam-3344	253	6	a	a	PRON
ejpam-3344	253	7	is	be	AUX
ejpam-3344	253	8	an	an	DET
ejpam-3344	253	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	253	10	fuzzy	fuzzy	ADJ
ejpam-3344	253	11	la	la	NOUN
ejpam-3344	253	12	-	-	PUNCT
ejpam-3344	253	13	subring	subring	NOUN
ejpam-3344	253	14	of	of	ADP
ejpam-3344	253	15	r.	r.	PROPN
ejpam-3344	253	16	thus	thus	ADV
ejpam-3344	253	17	µa(xy	µa(xy	NUM
ejpam-3344	253	18	)	)	PUNCT
ejpam-3344	253	19	=	=	SYM
ejpam-3344	253	20	µa(((xa)x)y	µa(((xa)x)y	ADJ
ejpam-3344	253	21	)	)	PUNCT
ejpam-3344	253	22	=	=	SYM
ejpam-3344	253	23	µa(((xa)x2)y	µa(((xa)x2)y	X
ejpam-3344	253	24	)	)	PUNCT
ejpam-3344	253	25	=	=	SYM
ejpam-3344	253	26	µa(((xa)(xx))y	µa(((xa)(xx))y	NOUN
ejpam-3344	253	27	)	)	PUNCT
ejpam-3344	253	28	=	=	SYM
ejpam-3344	253	29	µa((x((xa)x))y	µa((x((xa)x))y	X
ejpam-3344	253	30	)	)	PUNCT
ejpam-3344	253	31	≥	≥	NOUN
ejpam-3344	253	32	min{µa(x	min{µa(x	NOUN
ejpam-3344	253	33	)	)	PUNCT
ejpam-3344	253	34	,	,	PUNCT
ejpam-3344	253	35	µa(y	µa(y	NOUN
ejpam-3344	253	36	)	)	PUNCT
ejpam-3344	253	37	}	}	PUNCT
ejpam-3344	253	38	and	and	CCONJ
ejpam-3344	253	39	γa(xy	γa(xy	PROPN
ejpam-3344	253	40	)	)	PUNCT
ejpam-3344	253	41	=	=	SYM
ejpam-3344	253	42	γa(((xa)x)y	γa(((xa)x)y	PROPN
ejpam-3344	253	43	)	)	PUNCT
ejpam-3344	253	44	=	=	SYM
ejpam-3344	253	45	γa(((xa)x2)y	γa(((xa)x2)y	PROPN
ejpam-3344	253	46	)	)	PUNCT
ejpam-3344	253	47	=	=	SYM
ejpam-3344	254	1	γa(((xa)(xx))y	γa(((xa)(xx))y	X
ejpam-3344	254	2	)	)	PUNCT
ejpam-3344	254	3	=	=	SYM
ejpam-3344	254	4	γa((x((xa)x))y	γa((x((xa)x))y	X
ejpam-3344	254	5	)	)	PUNCT
ejpam-3344	254	6	≤	≤	NUM
ejpam-3344	254	7	max{γa(x	max{γa(x	NOUN
ejpam-3344	254	8	)	)	PUNCT
ejpam-3344	254	9	,	,	PUNCT
ejpam-3344	254	10	γa(y	γa(y	NOUN
ejpam-3344	254	11	)	)	PUNCT
ejpam-3344	254	12	}	}	PUNCT
ejpam-3344	254	13	.	.	PUNCT
ejpam-3344	255	1	hence	hence	ADV
ejpam-3344	255	2	a	a	PRON
ejpam-3344	255	3	is	be	AUX
ejpam-3344	255	4	an	an	DET
ejpam-3344	255	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	255	6	fuzzy	fuzzy	ADJ
ejpam-3344	255	7	la	la	NOUN
ejpam-3344	255	8	-	-	PUNCT
ejpam-3344	255	9	subring	subring	NOUN
ejpam-3344	255	10	of	of	ADP
ejpam-3344	255	11	r.	r.	PROPN
ejpam-3344	255	12	lemma	lemma	PROPN
ejpam-3344	255	13	8	8	NUM
ejpam-3344	255	14	.	.	PUNCT
ejpam-3344	256	1	let	let	VERB
ejpam-3344	256	2	r	r	PRON
ejpam-3344	256	3	be	be	AUX
ejpam-3344	256	4	an	an	DET
ejpam-3344	256	5	la	la	NOUN
ejpam-3344	256	6	-	-	NOUN
ejpam-3344	256	7	ring	ring	NOUN
ejpam-3344	256	8	with	with	ADP
ejpam-3344	256	9	left	left	ADJ
ejpam-3344	256	10	identity	identity	NOUN
ejpam-3344	256	11	e.	e.	PROPN
ejpam-3344	256	12	then	then	ADV
ejpam-3344	256	13	ra	ra	PROPN
ejpam-3344	256	14	is	be	AUX
ejpam-3344	256	15	the	the	DET
ejpam-3344	256	16	smallest	small	ADJ
ejpam-3344	256	17	left	leave	VERB
ejpam-3344	256	18	ideal	ideal	NOUN
ejpam-3344	256	19	of	of	ADP
ejpam-3344	256	20	r	r	NOUN
ejpam-3344	256	21	containing	contain	VERB
ejpam-3344	256	22	a.	a.	NOUN
ejpam-3344	256	23	proof	proof	NOUN
ejpam-3344	256	24	.	.	PUNCT
ejpam-3344	257	1	let	let	VERB
ejpam-3344	257	2	x	x	PRON
ejpam-3344	257	3	,	,	PUNCT
ejpam-3344	257	4	y	y	PROPN
ejpam-3344	257	5	∈	∈	PROPN
ejpam-3344	257	6	ra	ra	PROPN
ejpam-3344	257	7	and	and	CCONJ
ejpam-3344	257	8	r	r	PROPN
ejpam-3344	257	9	∈	∈	PROPN
ejpam-3344	257	10	r.	r.	NOUN
ejpam-3344	257	11	this	this	PRON
ejpam-3344	257	12	implies	imply	VERB
ejpam-3344	257	13	that	that	SCONJ
ejpam-3344	258	1	x	x	X
ejpam-3344	258	2	=	=	PUNCT
ejpam-3344	258	3	r1a	r1a	NOUN
ejpam-3344	258	4	and	and	CCONJ
ejpam-3344	258	5	y	y	PROPN
ejpam-3344	258	6	=	=	SYM
ejpam-3344	258	7	r2a	r2a	PROPN
ejpam-3344	258	8	,	,	PUNCT
ejpam-3344	258	9	where	where	SCONJ
ejpam-3344	258	10	r1	r1	PROPN
ejpam-3344	258	11	,	,	PUNCT
ejpam-3344	258	12	r2	r2	PROPN
ejpam-3344	258	13	∈	∈	PROPN
ejpam-3344	258	14	r.	r.	PROPN
ejpam-3344	258	15	now	now	ADV
ejpam-3344	258	16	x−	x−	PROPN
ejpam-3344	259	1	y	y	PROPN
ejpam-3344	259	2	=	=	PUNCT
ejpam-3344	260	1	r1a−	r1a−	ADJ
ejpam-3344	260	2	r2a	r2a	NOUN
ejpam-3344	260	3	=	=	PUNCT
ejpam-3344	260	4	(	(	PUNCT
ejpam-3344	260	5	r1	r1	PROPN
ejpam-3344	260	6	−	−	PROPN
ejpam-3344	260	7	r2)a	r2)a	NOUN
ejpam-3344	260	8	∈	∈	PROPN
ejpam-3344	260	9	ra	ra	PROPN
ejpam-3344	260	10	and	and	CCONJ
ejpam-3344	260	11	rx	rx	VERB
ejpam-3344	260	12	=	=	NOUN
ejpam-3344	260	13	r(r1a	r(r1a	NOUN
ejpam-3344	260	14	)	)	PUNCT
ejpam-3344	260	15	=	=	SYM
ejpam-3344	260	16	(	(	PUNCT
ejpam-3344	260	17	er)(r1a	er)(r1a	NOUN
ejpam-3344	260	18	)	)	PUNCT
ejpam-3344	260	19	=	=	SYM
ejpam-3344	260	20	(	(	PUNCT
ejpam-3344	260	21	(	(	PUNCT
ejpam-3344	260	22	r1a)r)e	r1a)r)e	NOUN
ejpam-3344	260	23	=	=	SYM
ejpam-3344	260	24	(	(	PUNCT
ejpam-3344	260	25	(	(	PUNCT
ejpam-3344	260	26	r1a)(er))e	r1a)(er))e	NOUN
ejpam-3344	260	27	=	=	SYM
ejpam-3344	260	28	(	(	PUNCT
ejpam-3344	260	29	(	(	PUNCT
ejpam-3344	260	30	r1e)(ar))e	r1e)(ar))e	NOUN
ejpam-3344	260	31	=	=	SYM
ejpam-3344	260	32	(	(	PUNCT
ejpam-3344	260	33	e(ar))(r1e	e(ar))(r1e	PROPN
ejpam-3344	260	34	)	)	PUNCT
ejpam-3344	260	35	=	=	SYM
ejpam-3344	260	36	(	(	PUNCT
ejpam-3344	260	37	ar)(r1e	ar)(r1e	X
ejpam-3344	260	38	)	)	PUNCT
ejpam-3344	260	39	=	=	SYM
ejpam-3344	260	40	(	(	PUNCT
ejpam-3344	260	41	(	(	PUNCT
ejpam-3344	260	42	r1e)r)a	r1e)r)a	PROPN
ejpam-3344	260	43	∈	∈	PROPN
ejpam-3344	260	44	ra	ra	PROPN
ejpam-3344	260	45	.	.	PUNCT
ejpam-3344	261	1	since	since	SCONJ
ejpam-3344	261	2	a	a	DET
ejpam-3344	261	3	=	=	SYM
ejpam-3344	261	4	ea	ea	PROPN
ejpam-3344	261	5	∈	∈	PROPN
ejpam-3344	261	6	ra	ra	PROPN
ejpam-3344	261	7	.	.	PUNCT
ejpam-3344	262	1	thus	thus	ADV
ejpam-3344	262	2	ra	ra	PROPN
ejpam-3344	262	3	is	be	AUX
ejpam-3344	262	4	a	a	DET
ejpam-3344	262	5	left	left	ADJ
ejpam-3344	262	6	ideal	ideal	NOUN
ejpam-3344	262	7	of	of	ADP
ejpam-3344	262	8	r	r	NOUN
ejpam-3344	262	9	containing	contain	VERB
ejpam-3344	262	10	a.	a.	NOUN
ejpam-3344	262	11	let	let	VERB
ejpam-3344	262	12	i	i	PRON
ejpam-3344	262	13	be	be	AUX
ejpam-3344	262	14	another	another	DET
ejpam-3344	262	15	left	left	ADJ
ejpam-3344	262	16	ideal	ideal	NOUN
ejpam-3344	262	17	of	of	ADP
ejpam-3344	262	18	r	r	NOUN
ejpam-3344	262	19	containing	contain	VERB
ejpam-3344	262	20	a.	a.	NOUN
ejpam-3344	263	1	so	so	ADV
ejpam-3344	263	2	ra	ra	PROPN
ejpam-3344	263	3	∈	∈	PROPN
ejpam-3344	264	1	i	i	PRON
ejpam-3344	264	2	,	,	PUNCT
ejpam-3344	264	3	where	where	SCONJ
ejpam-3344	264	4	ra	ra	PROPN
ejpam-3344	264	5	∈	∈	PROPN
ejpam-3344	264	6	ra	ra	PROPN
ejpam-3344	264	7	,	,	PUNCT
ejpam-3344	264	8	i.e.	i.e.	X
ejpam-3344	264	9	,	,	PUNCT
ejpam-3344	264	10	ra	ra	PROPN
ejpam-3344	264	11	⊆	⊆	NUM
ejpam-3344	264	12	i.	i.	NOUN
ejpam-3344	264	13	hence	hence	ADV
ejpam-3344	264	14	ra	ra	PROPN
ejpam-3344	264	15	is	be	AUX
ejpam-3344	264	16	the	the	DET
ejpam-3344	264	17	smallest	small	ADJ
ejpam-3344	264	18	left	leave	VERB
ejpam-3344	264	19	ideal	ideal	NOUN
ejpam-3344	264	20	of	of	ADP
ejpam-3344	264	21	r	r	NOUN
ejpam-3344	264	22	containing	contain	VERB
ejpam-3344	264	23	a.	a.	NOUN
ejpam-3344	264	24	lemma	lemma	PROPN
ejpam-3344	264	25	9	9	X
ejpam-3344	264	26	.	.	PUNCT
ejpam-3344	264	27	let	let	VERB
ejpam-3344	264	28	r	r	PRON
ejpam-3344	264	29	be	be	AUX
ejpam-3344	264	30	an	an	DET
ejpam-3344	264	31	la	la	NOUN
ejpam-3344	264	32	-	-	NOUN
ejpam-3344	264	33	ring	ring	NOUN
ejpam-3344	264	34	with	with	ADP
ejpam-3344	264	35	left	left	ADJ
ejpam-3344	264	36	identity	identity	NOUN
ejpam-3344	264	37	e.	e.	PROPN
ejpam-3344	264	38	then	then	ADV
ejpam-3344	264	39	ar	ar	PROPN
ejpam-3344	264	40	is	be	AUX
ejpam-3344	264	41	a	a	DET
ejpam-3344	264	42	left	left	ADJ
ejpam-3344	264	43	ideal	ideal	NOUN
ejpam-3344	264	44	of	of	ADP
ejpam-3344	264	45	r.	r.	PROPN
ejpam-3344	264	46	proof	proof	NOUN
ejpam-3344	264	47	.	.	PUNCT
ejpam-3344	265	1	straight	straight	ADV
ejpam-3344	265	2	forward	forward	ADV
ejpam-3344	265	3	.	.	PUNCT
ejpam-3344	266	1	proposition	proposition	NOUN
ejpam-3344	266	2	5	5	NUM
ejpam-3344	266	3	.	.	PUNCT
ejpam-3344	267	1	let	let	VERB
ejpam-3344	267	2	r	r	PRON
ejpam-3344	267	3	be	be	AUX
ejpam-3344	267	4	an	an	DET
ejpam-3344	267	5	la	la	NOUN
ejpam-3344	267	6	-	-	NOUN
ejpam-3344	267	7	ring	ring	NOUN
ejpam-3344	267	8	with	with	ADP
ejpam-3344	267	9	left	left	ADJ
ejpam-3344	267	10	identity	identity	NOUN
ejpam-3344	267	11	e.	e.	PROPN
ejpam-3344	267	12	then	then	ADV
ejpam-3344	267	13	ar	ar	PROPN
ejpam-3344	267	14	∪	∪	PROPN
ejpam-3344	267	15	ra	ra	PROPN
ejpam-3344	267	16	is	be	AUX
ejpam-3344	267	17	the	the	DET
ejpam-3344	267	18	smallest	small	ADJ
ejpam-3344	267	19	right	right	ADJ
ejpam-3344	267	20	ideal	ideal	NOUN
ejpam-3344	267	21	of	of	ADP
ejpam-3344	267	22	r	r	NOUN
ejpam-3344	267	23	containing	contain	VERB
ejpam-3344	267	24	a.	a.	NOUN
ejpam-3344	267	25	n.	n.	PROPN
ejpam-3344	267	26	kausar	kausar	PROPN
ejpam-3344	267	27	,	,	PUNCT
ejpam-3344	267	28	m.	m.	NOUN
ejpam-3344	267	29	a.	a.	PROPN
ejpam-3344	267	30	waqar	waqar	PROPN
ejpam-3344	267	31	/	/	SYM
ejpam-3344	267	32	eur	eur	PROPN
ejpam-3344	267	33	.	.	PUNCT
ejpam-3344	268	1	j.	j.	PROPN
ejpam-3344	268	2	pure	pure	PROPN
ejpam-3344	268	3	appl	appl	PROPN
ejpam-3344	268	4	.	.	PROPN
ejpam-3344	268	5	math	math	PROPN
ejpam-3344	268	6	,	,	PUNCT
ejpam-3344	268	7	12	12	NUM
ejpam-3344	268	8	(	(	PUNCT
ejpam-3344	268	9	1	1	NUM
ejpam-3344	268	10	)	)	PUNCT
ejpam-3344	268	11	(	(	PUNCT
ejpam-3344	268	12	2019	2019	NUM
ejpam-3344	268	13	)	)	PUNCT
ejpam-3344	268	14	,	,	PUNCT
ejpam-3344	268	15	226	226	NUM
ejpam-3344	268	16	-	-	SYM
ejpam-3344	268	17	250	250	NUM
ejpam-3344	268	18	236	236	NUM
ejpam-3344	268	19	proof	proof	NOUN
ejpam-3344	268	20	.	.	PUNCT
ejpam-3344	269	1	let	let	VERB
ejpam-3344	269	2	x	x	PRON
ejpam-3344	269	3	,	,	PUNCT
ejpam-3344	269	4	y	y	PROPN
ejpam-3344	269	5	∈	∈	PROPN
ejpam-3344	269	6	ar	ar	PROPN
ejpam-3344	269	7	∪	∪	PROPN
ejpam-3344	269	8	ra	ra	PROPN
ejpam-3344	269	9	,	,	PUNCT
ejpam-3344	269	10	this	this	PRON
ejpam-3344	269	11	means	mean	VERB
ejpam-3344	269	12	that	that	SCONJ
ejpam-3344	269	13	x	x	X
ejpam-3344	269	14	,	,	PUNCT
ejpam-3344	269	15	y	y	PROPN
ejpam-3344	269	16	∈	∈	PROPN
ejpam-3344	269	17	ar	ar	PROPN
ejpam-3344	269	18	or	or	CCONJ
ejpam-3344	269	19	ra	ra	PROPN
ejpam-3344	269	20	.	.	PUNCT
ejpam-3344	270	1	since	since	SCONJ
ejpam-3344	270	2	ar	ar	PROPN
ejpam-3344	270	3	and	and	CCONJ
ejpam-3344	270	4	ra	ra	PROPN
ejpam-3344	270	5	both	both	PRON
ejpam-3344	270	6	are	be	AUX
ejpam-3344	270	7	left	leave	VERB
ejpam-3344	270	8	ideals	ideal	NOUN
ejpam-3344	270	9	of	of	ADP
ejpam-3344	270	10	r	r	NOUN
ejpam-3344	270	11	,	,	PUNCT
ejpam-3344	270	12	so	so	ADV
ejpam-3344	270	13	x	x	SYM
ejpam-3344	270	14	−	−	PROPN
ejpam-3344	270	15	y	y	PROPN
ejpam-3344	270	16	∈	∈	PROPN
ejpam-3344	270	17	ar	ar	PROPN
ejpam-3344	270	18	and	and	CCONJ
ejpam-3344	270	19	ra	ra	PROPN
ejpam-3344	270	20	,	,	PUNCT
ejpam-3344	270	21	i.e.	i.e.	X
ejpam-3344	270	22	,	,	PUNCT
ejpam-3344	270	23	x	x	PUNCT
ejpam-3344	270	24	−	−	PROPN
ejpam-3344	270	25	y	y	PROPN
ejpam-3344	270	26	∈	∈	PROPN
ejpam-3344	270	27	ar	ar	PROPN
ejpam-3344	270	28	∪	∪	PROPN
ejpam-3344	270	29	ra	ra	PROPN
ejpam-3344	270	30	.	.	PUNCT
ejpam-3344	271	1	we	we	PRON
ejpam-3344	271	2	have	have	VERB
ejpam-3344	271	3	to	to	PART
ejpam-3344	271	4	show	show	VERB
ejpam-3344	271	5	that	that	SCONJ
ejpam-3344	271	6	(	(	PUNCT
ejpam-3344	271	7	ar	ar	PROPN
ejpam-3344	271	8	∪ra)r	∪ra)r	NOUN
ejpam-3344	271	9	⊆	⊆	NUM
ejpam-3344	271	10	(	(	PUNCT
ejpam-3344	271	11	ar	ar	NOUN
ejpam-3344	271	12	∪ra	∪ra	ADV
ejpam-3344	271	13	)	)	PUNCT
ejpam-3344	271	14	.	.	PUNCT
ejpam-3344	272	1	now	now	ADV
ejpam-3344	272	2	(	(	PUNCT
ejpam-3344	272	3	ar	ar	PROPN
ejpam-3344	272	4	∪ra)r	∪ra)r	NOUN
ejpam-3344	272	5	=	=	PUNCT
ejpam-3344	272	6	(	(	PUNCT
ejpam-3344	272	7	ar)r	ar)r	PROPN
ejpam-3344	272	8	∪	∪	ADV
ejpam-3344	272	9	(	(	PUNCT
ejpam-3344	272	10	ra)r	ra)r	PROPN
ejpam-3344	272	11	=	=	SYM
ejpam-3344	272	12	(	(	PUNCT
ejpam-3344	272	13	rr)a	rr)a	PROPN
ejpam-3344	272	14	∪	∪	X
ejpam-3344	272	15	(	(	PUNCT
ejpam-3344	272	16	ra)(er	ra)(er	X
ejpam-3344	272	17	)	)	PUNCT
ejpam-3344	272	18	⊆	⊆	NUM
ejpam-3344	272	19	ra	ra	PROPN
ejpam-3344	272	20	∪	∪	X
ejpam-3344	272	21	(	(	PUNCT
ejpam-3344	272	22	re)(ar	re)(ar	NOUN
ejpam-3344	272	23	)	)	PUNCT
ejpam-3344	272	24	=	=	SYM
ejpam-3344	272	25	ra	ra	PROPN
ejpam-3344	272	26	∪r(ar	∪r(ar	PROPN
ejpam-3344	272	27	)	)	PUNCT
ejpam-3344	273	1	=	=	SYM
ejpam-3344	273	2	ra	ra	PROPN
ejpam-3344	273	3	∪	∪	ADP
ejpam-3344	273	4	a(rr	a(rr	PROPN
ejpam-3344	273	5	)	)	PUNCT
ejpam-3344	273	6	⊆	⊆	NUM
ejpam-3344	273	7	ra	ra	PROPN
ejpam-3344	273	8	∪	∪	X
ejpam-3344	273	9	ar	ar	PROPN
ejpam-3344	273	10	=	=	PROPN
ejpam-3344	273	11	ar	ar	PROPN
ejpam-3344	273	12	∪ra	∪ra	ADV
ejpam-3344	273	13	.	.	PUNCT
ejpam-3344	274	1	⇒	⇒	PROPN
ejpam-3344	274	2	(	(	PUNCT
ejpam-3344	274	3	ar	ar	PROPN
ejpam-3344	274	4	∪ra)r	∪ra)r	PROPN
ejpam-3344	274	5	⊆	⊆	NUM
ejpam-3344	274	6	ar	ar	NOUN
ejpam-3344	274	7	∪ra	∪ra	ADV
ejpam-3344	274	8	.	.	PUNCT
ejpam-3344	275	1	as	as	SCONJ
ejpam-3344	275	2	a	a	DET
ejpam-3344	275	3	∈	∈	PROPN
ejpam-3344	275	4	ra	ra	PROPN
ejpam-3344	275	5	,	,	PUNCT
ejpam-3344	275	6	i.e.	i.e.	X
ejpam-3344	275	7	,	,	PUNCT
ejpam-3344	275	8	a	a	DET
ejpam-3344	275	9	∈	∈	PROPN
ejpam-3344	275	10	ar	ar	PROPN
ejpam-3344	275	11	∪	∪	PROPN
ejpam-3344	275	12	ra	ra	PROPN
ejpam-3344	275	13	.	.	PUNCT
ejpam-3344	275	14	let	let	VERB
ejpam-3344	275	15	i	i	PRON
ejpam-3344	275	16	be	be	AUX
ejpam-3344	275	17	another	another	DET
ejpam-3344	275	18	right	right	ADJ
ejpam-3344	275	19	ideal	ideal	NOUN
ejpam-3344	275	20	of	of	ADP
ejpam-3344	275	21	r	r	NOUN
ejpam-3344	275	22	containing	contain	VERB
ejpam-3344	275	23	a.	a.	NOUN
ejpam-3344	275	24	since	since	SCONJ
ejpam-3344	275	25	ar	ar	PROPN
ejpam-3344	275	26	∈	∈	PROPN
ejpam-3344	275	27	ir	ir	PROPN
ejpam-3344	275	28	⊆	⊆	NUM
ejpam-3344	275	29	i	i	PROPN
ejpam-3344	275	30	and	and	CCONJ
ejpam-3344	275	31	ra	ra	PROPN
ejpam-3344	276	1	=	=	PUNCT
ejpam-3344	277	1	(	(	PUNCT
ejpam-3344	277	2	rr)a	rr)a	PROPN
ejpam-3344	277	3	=	=	SYM
ejpam-3344	277	4	(	(	PUNCT
ejpam-3344	277	5	ar)r	ar)r	PROPN
ejpam-3344	277	6	∈	∈	PROPN
ejpam-3344	277	7	(	(	PUNCT
ejpam-3344	277	8	ir)r	ir)r	PROPN
ejpam-3344	277	9	⊆	⊆	NUM
ejpam-3344	277	10	ir	ir	PROPN
ejpam-3344	277	11	⊆	⊆	NUM
ejpam-3344	277	12	i	i	PRON
ejpam-3344	277	13	,	,	PUNCT
ejpam-3344	277	14	i.e.	i.e.	X
ejpam-3344	277	15	,	,	PUNCT
ejpam-3344	277	16	ar	ar	PROPN
ejpam-3344	277	17	∪ra	∪ra	PROPN
ejpam-3344	277	18	⊆	⊆	NUM
ejpam-3344	277	19	i.	i.	NOUN
ejpam-3344	277	20	therefore	therefore	ADV
ejpam-3344	277	21	ar	ar	PROPN
ejpam-3344	277	22	∪ra	∪ra	PROPN
ejpam-3344	277	23	is	be	AUX
ejpam-3344	277	24	the	the	DET
ejpam-3344	277	25	smallest	small	ADJ
ejpam-3344	277	26	right	right	ADJ
ejpam-3344	277	27	ideal	ideal	NOUN
ejpam-3344	277	28	of	of	ADP
ejpam-3344	277	29	r	r	NOUN
ejpam-3344	277	30	containing	contain	VERB
ejpam-3344	277	31	a.	a.	NOUN
ejpam-3344	277	32	lemma	lemma	PROPN
ejpam-3344	277	33	10	10	NUM
ejpam-3344	277	34	.	.	PUNCT
ejpam-3344	278	1	let	let	VERB
ejpam-3344	278	2	r	r	PRON
ejpam-3344	278	3	be	be	AUX
ejpam-3344	278	4	an	an	DET
ejpam-3344	278	5	la	la	NOUN
ejpam-3344	278	6	-	-	PUNCT
ejpam-3344	278	7	ring	ring	NOUN
ejpam-3344	278	8	.	.	PUNCT
ejpam-3344	279	1	then	then	ADV
ejpam-3344	279	2	a	a	DET
ejpam-3344	279	3	◦	◦	NOUN
ejpam-3344	279	4	b	b	NOUN
ejpam-3344	279	5	⊆	⊆	NUM
ejpam-3344	279	6	a∩b	a∩b	NOUN
ejpam-3344	279	7	for	for	ADP
ejpam-3344	279	8	every	every	DET
ejpam-3344	279	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	279	10	fuzzy	fuzzy	ADJ
ejpam-3344	279	11	right	right	ADJ
ejpam-3344	279	12	ideal	ideal	NOUN
ejpam-3344	279	13	a	a	PRON
ejpam-3344	279	14	and	and	CCONJ
ejpam-3344	279	15	every	every	DET
ejpam-3344	279	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	279	17	fuzzy	fuzzy	ADJ
ejpam-3344	279	18	left	leave	VERB
ejpam-3344	279	19	ideal	ideal	PROPN
ejpam-3344	279	20	b	b	PROPN
ejpam-3344	279	21	of	of	ADP
ejpam-3344	279	22	r.	r.	PROPN
ejpam-3344	279	23	proof	proof	NOUN
ejpam-3344	279	24	.	.	PUNCT
ejpam-3344	280	1	let	let	VERB
ejpam-3344	280	2	a	a	DET
ejpam-3344	280	3	=	=	SYM
ejpam-3344	280	4	(	(	PUNCT
ejpam-3344	280	5	µa	µa	PROPN
ejpam-3344	280	6	,	,	PUNCT
ejpam-3344	280	7	γa	γa	PROPN
ejpam-3344	280	8	)	)	PUNCT
ejpam-3344	280	9	be	be	VERB
ejpam-3344	280	10	an	an	DET
ejpam-3344	280	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	280	12	fuzzy	fuzzy	ADJ
ejpam-3344	280	13	right	right	NOUN
ejpam-3344	280	14	and	and	CCONJ
ejpam-3344	280	15	b	b	X
ejpam-3344	280	16	=	=	PUNCT
ejpam-3344	280	17	(	(	PUNCT
ejpam-3344	280	18	µb	µb	PROPN
ejpam-3344	280	19	,	,	PUNCT
ejpam-3344	280	20	γb	γb	PROPN
ejpam-3344	280	21	)	)	PUNCT
ejpam-3344	280	22	be	be	VERB
ejpam-3344	280	23	an	an	DET
ejpam-3344	280	24	intuitionistic	intuitionistic	ADJ
ejpam-3344	280	25	fuzzy	fuzzy	ADJ
ejpam-3344	280	26	left	leave	VERB
ejpam-3344	280	27	ideal	ideal	NOUN
ejpam-3344	280	28	of	of	ADP
ejpam-3344	280	29	r	r	NOUN
ejpam-3344	280	30	and	and	CCONJ
ejpam-3344	280	31	x	x	PROPN
ejpam-3344	280	32	∈	∈	PROPN
ejpam-3344	280	33	r.	r.	NOUN
ejpam-3344	280	34	if	if	SCONJ
ejpam-3344	280	35	x	x	PRON
ejpam-3344	280	36	can	can	AUX
ejpam-3344	280	37	not	not	PART
ejpam-3344	280	38	be	be	AUX
ejpam-3344	280	39	expressible	expressible	ADJ
ejpam-3344	280	40	as	as	SCONJ
ejpam-3344	280	41	x	x	X
ejpam-3344	280	42	=	=	SYM
ejpam-3344	280	43	∑n	∑n	PROPN
ejpam-3344	280	44	i=1	i=1	PROPN
ejpam-3344	280	45	aibi	aibi	NOUN
ejpam-3344	280	46	,	,	PUNCT
ejpam-3344	280	47	where	where	SCONJ
ejpam-3344	280	48	ai	ai	VERB
ejpam-3344	280	49	,	,	PUNCT
ejpam-3344	280	50	bi	bi	NOUN
ejpam-3344	280	51	∈	∈	PROPN
ejpam-3344	280	52	r	r	NOUN
ejpam-3344	280	53	and	and	CCONJ
ejpam-3344	280	54	n	n	NOUN
ejpam-3344	280	55	is	be	AUX
ejpam-3344	280	56	any	any	DET
ejpam-3344	280	57	positive	positive	ADJ
ejpam-3344	280	58	integer	integer	NOUN
ejpam-3344	280	59	,	,	PUNCT
ejpam-3344	280	60	then	then	ADV
ejpam-3344	280	61	obvious	obvious	ADJ
ejpam-3344	280	62	a	a	DET
ejpam-3344	280	63	◦	◦	NOUN
ejpam-3344	280	64	b	b	NOUN
ejpam-3344	280	65	⊆	⊆	NUM
ejpam-3344	280	66	a	a	DET
ejpam-3344	280	67	∩b	∩b	NOUN
ejpam-3344	280	68	,	,	PUNCT
ejpam-3344	280	69	otherwise	otherwise	ADV
ejpam-3344	280	70	we	we	PRON
ejpam-3344	280	71	have	have	AUX
ejpam-3344	280	72	(	(	PUNCT
ejpam-3344	280	73	µa	µa	ADP
ejpam-3344	280	74	◦	◦	NOUN
ejpam-3344	280	75	µb	µb	NOUN
ejpam-3344	280	76	)	)	PUNCT
ejpam-3344	280	77	(	(	PUNCT
ejpam-3344	280	78	x	x	X
ejpam-3344	280	79	)	)	PUNCT
ejpam-3344	280	80	=	=	PUNCT
ejpam-3344	281	1	∨x=∑n	∨x=∑n	NOUN
ejpam-3344	281	2	i=1	i=1	PROPN
ejpam-3344	281	3	aibi	aibi	NOUN
ejpam-3344	281	4	{	{	PUNCT
ejpam-3344	281	5	∧ni=1	∧ni=1	X
ejpam-3344	281	6	{	{	PUNCT
ejpam-3344	281	7	µa	µa	X
ejpam-3344	281	8	(	(	PUNCT
ejpam-3344	281	9	ai	ai	NOUN
ejpam-3344	281	10	)	)	PUNCT
ejpam-3344	281	11	∧	∧	NOUN
ejpam-3344	281	12	µb	µb	PROPN
ejpam-3344	281	13	(	(	PUNCT
ejpam-3344	281	14	bi	bi	NOUN
ejpam-3344	281	15	)	)	PUNCT
ejpam-3344	281	16	}	}	PUNCT
ejpam-3344	281	17	}	}	PUNCT
ejpam-3344	281	18	≤	≤	NUM
ejpam-3344	281	19	∨x=∑n	∨x=∑n	NUM
ejpam-3344	281	20	i=1	i=1	PROPN
ejpam-3344	281	21	aibi	aibi	NOUN
ejpam-3344	281	22	{	{	PUNCT
ejpam-3344	281	23	∧ni=1	∧ni=1	X
ejpam-3344	281	24	{	{	PUNCT
ejpam-3344	281	25	µa	µa	PROPN
ejpam-3344	281	26	(	(	PUNCT
ejpam-3344	281	27	aibi	aibi	NOUN
ejpam-3344	281	28	)	)	PUNCT
ejpam-3344	281	29	∧	∧	PROPN
ejpam-3344	281	30	µb	µb	PROPN
ejpam-3344	281	31	(	(	PUNCT
ejpam-3344	281	32	aibi	aibi	NOUN
ejpam-3344	281	33	)	)	PUNCT
ejpam-3344	281	34	}	}	PUNCT
ejpam-3344	281	35	}	}	PUNCT
ejpam-3344	281	36	=	=	PUNCT
ejpam-3344	281	37	∨x=∑n	∨x=∑n	NUM
ejpam-3344	281	38	i=1	i=1	PROPN
ejpam-3344	281	39	aibi	aibi	NOUN
ejpam-3344	281	40	{	{	PUNCT
ejpam-3344	281	41	∧ni=1(µa	∧ni=1(µa	PUNCT
ejpam-3344	281	42	∧	∧	NOUN
ejpam-3344	281	43	µb	µb	NOUN
ejpam-3344	281	44	)	)	PUNCT
ejpam-3344	281	45	(	(	PUNCT
ejpam-3344	281	46	aibi	aibi	NOUN
ejpam-3344	281	47	)	)	PUNCT
ejpam-3344	281	48	}	}	PUNCT
ejpam-3344	281	49	=	=	SYM
ejpam-3344	281	50	(	(	PUNCT
ejpam-3344	281	51	µa	µa	ADP
ejpam-3344	281	52	∧	∧	PROPN
ejpam-3344	281	53	µb	µb	PROPN
ejpam-3344	281	54	)	)	PUNCT
ejpam-3344	281	55	(	(	PUNCT
ejpam-3344	281	56	x	x	X
ejpam-3344	281	57	)	)	PUNCT
ejpam-3344	281	58	=	=	SYM
ejpam-3344	281	59	(	(	PUNCT
ejpam-3344	281	60	µa	µa	ADP
ejpam-3344	281	61	∩	∩	NOUN
ejpam-3344	281	62	µb	µb	VERB
ejpam-3344	281	63	)	)	PUNCT
ejpam-3344	281	64	(	(	PUNCT
ejpam-3344	281	65	x	x	X
ejpam-3344	281	66	)	)	PUNCT
ejpam-3344	281	67	⇒	⇒	NOUN
ejpam-3344	281	68	µa	µa	ADP
ejpam-3344	281	69	◦	◦	VERB
ejpam-3344	281	70	µb	µb	ADP
ejpam-3344	281	71	⊆	⊆	NUM
ejpam-3344	281	72	µa	µa	NOUN
ejpam-3344	281	73	∩	∩	NOUN
ejpam-3344	281	74	µb	µb	VERB
ejpam-3344	281	75	similarly	similarly	ADV
ejpam-3344	281	76	,	,	PUNCT
ejpam-3344	281	77	we	we	PRON
ejpam-3344	281	78	have	have	VERB
ejpam-3344	281	79	γa	γa	PRON
ejpam-3344	281	80	◦	◦	VERB
ejpam-3344	281	81	γb	γb	PROPN
ejpam-3344	281	82	⊇	⊇	PROPN
ejpam-3344	281	83	γa	γa	PROPN
ejpam-3344	281	84	∪	∪	PROPN
ejpam-3344	281	85	γb	γb	PROPN
ejpam-3344	281	86	.	.	PUNCT
ejpam-3344	282	1	hence	hence	ADV
ejpam-3344	282	2	a	a	DET
ejpam-3344	282	3	◦	◦	NOUN
ejpam-3344	282	4	b	b	NOUN
ejpam-3344	282	5	⊆	⊆	NUM
ejpam-3344	282	6	a	a	DET
ejpam-3344	282	7	∩	∩	ADJ
ejpam-3344	282	8	b	b	NOUN
ejpam-3344	282	9	for	for	ADP
ejpam-3344	282	10	every	every	DET
ejpam-3344	282	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	282	12	fuzzy	fuzzy	ADJ
ejpam-3344	282	13	right	right	ADJ
ejpam-3344	282	14	ideal	ideal	NOUN
ejpam-3344	282	15	a	a	PRON
ejpam-3344	282	16	and	and	CCONJ
ejpam-3344	282	17	every	every	DET
ejpam-3344	282	18	intuitionistic	intuitionistic	ADJ
ejpam-3344	282	19	fuzzy	fuzzy	ADJ
ejpam-3344	282	20	left	leave	VERB
ejpam-3344	282	21	ideal	ideal	PROPN
ejpam-3344	282	22	b	b	PROPN
ejpam-3344	282	23	of	of	ADP
ejpam-3344	282	24	r.	r.	PROPN
ejpam-3344	282	25	theorem	theorem	PROPN
ejpam-3344	282	26	5	5	X
ejpam-3344	282	27	.	.	PUNCT
ejpam-3344	283	1	let	let	VERB
ejpam-3344	283	2	r	r	PRON
ejpam-3344	283	3	be	be	AUX
ejpam-3344	283	4	an	an	DET
ejpam-3344	283	5	la	la	NOUN
ejpam-3344	283	6	-	-	NOUN
ejpam-3344	283	7	ring	ring	NOUN
ejpam-3344	283	8	with	with	ADP
ejpam-3344	283	9	left	left	ADJ
ejpam-3344	283	10	identity	identity	NOUN
ejpam-3344	283	11	e	e	NOUN
ejpam-3344	283	12	,	,	PUNCT
ejpam-3344	283	13	such	such	ADJ
ejpam-3344	283	14	that	that	SCONJ
ejpam-3344	283	15	(	(	PUNCT
ejpam-3344	283	16	xe)r	xe)r	PROPN
ejpam-3344	283	17	=	=	SYM
ejpam-3344	283	18	xr	xr	PROPN
ejpam-3344	283	19	for	for	ADP
ejpam-3344	283	20	all	all	DET
ejpam-3344	283	21	x	x	PROPN
ejpam-3344	283	22	∈	∈	PROPN
ejpam-3344	283	23	r.	r.	NOUN
ejpam-3344	283	24	then	then	ADV
ejpam-3344	283	25	the	the	DET
ejpam-3344	283	26	following	follow	VERB
ejpam-3344	283	27	conditions	condition	NOUN
ejpam-3344	283	28	are	be	AUX
ejpam-3344	283	29	equivalent	equivalent	ADJ
ejpam-3344	283	30	.	.	PUNCT
ejpam-3344	284	1	(	(	PUNCT
ejpam-3344	284	2	1	1	X
ejpam-3344	284	3	)	)	PUNCT
ejpam-3344	284	4	r	r	NOUN
ejpam-3344	284	5	is	be	AUX
ejpam-3344	284	6	a	a	DET
ejpam-3344	284	7	regular	regular	NOUN
ejpam-3344	284	8	.	.	PUNCT
ejpam-3344	285	1	(	(	PUNCT
ejpam-3344	285	2	2	2	X
ejpam-3344	285	3	)	)	PUNCT
ejpam-3344	285	4	a	a	DET
ejpam-3344	285	5	∩	∩	ADJ
ejpam-3344	285	6	b	b	X
ejpam-3344	285	7	=	=	SYM
ejpam-3344	285	8	a	a	DET
ejpam-3344	285	9	◦	◦	NOUN
ejpam-3344	285	10	b	b	NOUN
ejpam-3344	285	11	for	for	ADP
ejpam-3344	285	12	every	every	DET
ejpam-3344	285	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	285	14	fuzzy	fuzzy	ADJ
ejpam-3344	285	15	right	right	ADJ
ejpam-3344	285	16	ideal	ideal	NOUN
ejpam-3344	285	17	a	a	PRON
ejpam-3344	285	18	and	and	CCONJ
ejpam-3344	285	19	every	every	DET
ejpam-3344	285	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	285	21	fuzzy	fuzzy	ADJ
ejpam-3344	285	22	left	leave	VERB
ejpam-3344	285	23	ideal	ideal	PROPN
ejpam-3344	285	24	b	b	PROPN
ejpam-3344	285	25	of	of	ADP
ejpam-3344	285	26	r.	r.	PROPN
ejpam-3344	285	27	proof	proof	NOUN
ejpam-3344	285	28	.	.	PUNCT
ejpam-3344	286	1	suppose	suppose	VERB
ejpam-3344	286	2	that	that	SCONJ
ejpam-3344	286	3	(	(	PUNCT
ejpam-3344	286	4	1	1	X
ejpam-3344	286	5	)	)	PUNCT
ejpam-3344	286	6	holds	hold	VERB
ejpam-3344	286	7	.	.	PUNCT
ejpam-3344	287	1	since	since	SCONJ
ejpam-3344	287	2	a	a	DET
ejpam-3344	287	3	◦	◦	NOUN
ejpam-3344	287	4	b	b	NOUN
ejpam-3344	287	5	⊆	⊆	NUM
ejpam-3344	287	6	a∩b	a∩b	PROPN
ejpam-3344	287	7	,	,	PUNCT
ejpam-3344	287	8	for	for	ADP
ejpam-3344	287	9	every	every	DET
ejpam-3344	287	10	intuitionistic	intuitionistic	ADJ
ejpam-3344	287	11	fuzzy	fuzzy	ADJ
ejpam-3344	287	12	right	right	ADJ
ejpam-3344	287	13	ideal	ideal	NOUN
ejpam-3344	287	14	a	a	PRON
ejpam-3344	287	15	and	and	CCONJ
ejpam-3344	287	16	every	every	DET
ejpam-3344	287	17	intuitionistic	intuitionistic	ADJ
ejpam-3344	287	18	fuzzy	fuzzy	ADJ
ejpam-3344	287	19	left	leave	VERB
ejpam-3344	287	20	ideal	ideal	PROPN
ejpam-3344	287	21	b	b	PROPN
ejpam-3344	287	22	of	of	ADP
ejpam-3344	287	23	r	r	NOUN
ejpam-3344	287	24	by	by	ADP
ejpam-3344	287	25	the	the	DET
ejpam-3344	287	26	lemma	lemma	PROPN
ejpam-3344	287	27	10	10	NUM
ejpam-3344	287	28	.	.	PUNCT
ejpam-3344	288	1	let	let	VERB
ejpam-3344	288	2	x	x	PUNCT
ejpam-3344	288	3	∈	∈	PROPN
ejpam-3344	288	4	r	r	NOUN
ejpam-3344	288	5	,	,	PUNCT
ejpam-3344	288	6	this	this	PRON
ejpam-3344	288	7	implies	imply	VERB
ejpam-3344	288	8	that	that	SCONJ
ejpam-3344	288	9	there	there	PRON
ejpam-3344	288	10	exists	exist	VERB
ejpam-3344	288	11	an	an	DET
ejpam-3344	288	12	element	element	NOUN
ejpam-3344	288	13	a	a	DET
ejpam-3344	288	14	∈	∈	NOUN
ejpam-3344	288	15	r	r	NOUN
ejpam-3344	289	1	such	such	ADJ
ejpam-3344	289	2	that	that	PRON
ejpam-3344	289	3	x	x	SYM
ejpam-3344	289	4	=	=	SYM
ejpam-3344	289	5	(	(	PUNCT
ejpam-3344	289	6	xa)x	xa)x	PROPN
ejpam-3344	289	7	.	.	PUNCT
ejpam-3344	290	1	thus	thus	ADV
ejpam-3344	290	2	(	(	PUNCT
ejpam-3344	290	3	µa	µa	ADP
ejpam-3344	290	4	◦	◦	NOUN
ejpam-3344	290	5	µb)(x	µb)(x	NOUN
ejpam-3344	290	6	)	)	PUNCT
ejpam-3344	290	7	=	=	PUNCT
ejpam-3344	291	1	∨x=∑n	∨x=∑n	NOUN
ejpam-3344	291	2	i=1	i=1	PROPN
ejpam-3344	291	3	aibi	aibi	NOUN
ejpam-3344	291	4	{	{	PUNCT
ejpam-3344	291	5	∧ni=1	∧ni=1	X
ejpam-3344	291	6	{	{	PUNCT
ejpam-3344	291	7	µa	µa	X
ejpam-3344	291	8	(	(	PUNCT
ejpam-3344	291	9	ai	ai	NOUN
ejpam-3344	291	10	)	)	PUNCT
ejpam-3344	291	11	∧	∧	NOUN
ejpam-3344	291	12	µb	µb	PROPN
ejpam-3344	291	13	(	(	PUNCT
ejpam-3344	291	14	bi	bi	NOUN
ejpam-3344	291	15	)	)	PUNCT
ejpam-3344	291	16	}	}	PUNCT
ejpam-3344	291	17	}	}	PUNCT
ejpam-3344	291	18	≥	≥	NOUN
ejpam-3344	291	19	min{µa(xa	min{µa(xa	NOUN
ejpam-3344	291	20	)	)	PUNCT
ejpam-3344	291	21	,	,	PUNCT
ejpam-3344	291	22	µb(x	µb(x	PUNCT
ejpam-3344	291	23	)	)	PUNCT
ejpam-3344	291	24	}	}	PUNCT
ejpam-3344	291	25	≥	≥	PROPN
ejpam-3344	291	26	min{µa(x	min{µa(x	NOUN
ejpam-3344	291	27	)	)	PUNCT
ejpam-3344	291	28	,	,	PUNCT
ejpam-3344	291	29	µb(x	µb(x	PUNCT
ejpam-3344	291	30	)	)	PUNCT
ejpam-3344	291	31	}	}	PUNCT
ejpam-3344	291	32	=	=	SYM
ejpam-3344	291	33	(	(	PUNCT
ejpam-3344	291	34	µa	µa	ADP
ejpam-3344	291	35	∧	∧	PROPN
ejpam-3344	291	36	µb)(x	µb)(x	PROPN
ejpam-3344	291	37	)	)	PUNCT
ejpam-3344	291	38	=	=	PRON
ejpam-3344	291	39	(	(	PUNCT
ejpam-3344	291	40	µa	µa	ADP
ejpam-3344	291	41	∩	∩	ADJ
ejpam-3344	291	42	µb)(x	µb)(x	PROPN
ejpam-3344	291	43	)	)	PUNCT
ejpam-3344	291	44	.	.	PUNCT
ejpam-3344	292	1	n.	n.	PROPN
ejpam-3344	292	2	kausar	kausar	PROPN
ejpam-3344	292	3	,	,	PUNCT
ejpam-3344	292	4	m.	m.	NOUN
ejpam-3344	292	5	a.	a.	PROPN
ejpam-3344	292	6	waqar	waqar	PROPN
ejpam-3344	292	7	/	/	SYM
ejpam-3344	292	8	eur	eur	PROPN
ejpam-3344	292	9	.	.	PUNCT
ejpam-3344	293	1	j.	j.	PROPN
ejpam-3344	293	2	pure	pure	PROPN
ejpam-3344	293	3	appl	appl	PROPN
ejpam-3344	293	4	.	.	PROPN
ejpam-3344	293	5	math	math	PROPN
ejpam-3344	293	6	,	,	PUNCT
ejpam-3344	293	7	12	12	NUM
ejpam-3344	293	8	(	(	PUNCT
ejpam-3344	293	9	1	1	NUM
ejpam-3344	293	10	)	)	PUNCT
ejpam-3344	293	11	(	(	PUNCT
ejpam-3344	293	12	2019	2019	NUM
ejpam-3344	293	13	)	)	PUNCT
ejpam-3344	293	14	,	,	PUNCT
ejpam-3344	293	15	226	226	NUM
ejpam-3344	293	16	-	-	SYM
ejpam-3344	293	17	250	250	NUM
ejpam-3344	293	18	237	237	NUM
ejpam-3344	293	19	⇒	⇒	NOUN
ejpam-3344	293	20	µa	µa	ADP
ejpam-3344	293	21	∩	∩	NOUN
ejpam-3344	293	22	µb	µb	VERB
ejpam-3344	293	23	⊆	⊆	NUM
ejpam-3344	293	24	µa	µa	ADP
ejpam-3344	293	25	◦	◦	NOUN
ejpam-3344	293	26	µb	µb	NOUN
ejpam-3344	293	27	.	.	PUNCT
ejpam-3344	294	1	similarly	similarly	ADV
ejpam-3344	294	2	,	,	PUNCT
ejpam-3344	294	3	we	we	PRON
ejpam-3344	294	4	have	have	VERB
ejpam-3344	294	5	γa	γa	PROPN
ejpam-3344	294	6	∪	∪	X
ejpam-3344	294	7	γb	γb	PROPN
ejpam-3344	294	8	⊇	⊇	PROPN
ejpam-3344	294	9	γa	γa	PROPN
ejpam-3344	294	10	◦	◦	PROPN
ejpam-3344	294	11	γb	γb	PROPN
ejpam-3344	294	12	.	.	PUNCT
ejpam-3344	295	1	hence	hence	ADV
ejpam-3344	295	2	a	a	DET
ejpam-3344	295	3	∩	∩	ADJ
ejpam-3344	295	4	b	b	NOUN
ejpam-3344	295	5	=	=	SYM
ejpam-3344	295	6	a	a	DET
ejpam-3344	295	7	◦	◦	NOUN
ejpam-3344	295	8	b	b	NUM
ejpam-3344	295	9	,	,	PUNCT
ejpam-3344	295	10	i.e.	i.e.	X
ejpam-3344	295	11	,	,	PUNCT
ejpam-3344	295	12	(	(	PUNCT
ejpam-3344	295	13	1)⇒	1)⇒	NUM
ejpam-3344	295	14	(	(	PUNCT
ejpam-3344	295	15	2	2	NUM
ejpam-3344	295	16	)	)	PUNCT
ejpam-3344	295	17	.	.	PUNCT
ejpam-3344	296	1	assume	assume	VERB
ejpam-3344	296	2	that	that	SCONJ
ejpam-3344	296	3	(	(	PUNCT
ejpam-3344	296	4	2	2	X
ejpam-3344	296	5	)	)	PUNCT
ejpam-3344	296	6	is	be	AUX
ejpam-3344	296	7	true	true	ADJ
ejpam-3344	296	8	and	and	CCONJ
ejpam-3344	296	9	a	a	DET
ejpam-3344	296	10	∈	∈	PROPN
ejpam-3344	296	11	r.	r.	NOUN
ejpam-3344	296	12	then	then	ADV
ejpam-3344	296	13	ra	ra	PROPN
ejpam-3344	296	14	is	be	AUX
ejpam-3344	296	15	a	a	DET
ejpam-3344	296	16	left	left	ADJ
ejpam-3344	296	17	ideal	ideal	NOUN
ejpam-3344	296	18	of	of	ADP
ejpam-3344	296	19	r	r	NOUN
ejpam-3344	296	20	containing	contain	VERB
ejpam-3344	296	21	a	a	PRON
ejpam-3344	296	22	by	by	ADP
ejpam-3344	296	23	the	the	DET
ejpam-3344	296	24	lemma	lemma	PROPN
ejpam-3344	296	25	8	8	NUM
ejpam-3344	296	26	and	and	CCONJ
ejpam-3344	296	27	ar∪ra	ar∪ra	PROPN
ejpam-3344	296	28	is	be	AUX
ejpam-3344	296	29	a	a	DET
ejpam-3344	296	30	right	right	ADJ
ejpam-3344	296	31	ideal	ideal	NOUN
ejpam-3344	296	32	of	of	ADP
ejpam-3344	296	33	r	r	NOUN
ejpam-3344	296	34	containing	contain	VERB
ejpam-3344	296	35	a	a	PRON
ejpam-3344	296	36	by	by	ADP
ejpam-3344	296	37	the	the	DET
ejpam-3344	296	38	proposition	proposition	NOUN
ejpam-3344	296	39	5	5	NUM
ejpam-3344	296	40	.	.	PUNCT
ejpam-3344	297	1	so	so	ADV
ejpam-3344	297	2	χra	χra	PROPN
ejpam-3344	297	3	is	be	AUX
ejpam-3344	297	4	an	an	DET
ejpam-3344	297	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	297	6	fuzzy	fuzzy	ADJ
ejpam-3344	297	7	left	leave	VERB
ejpam-3344	297	8	ideal	ideal	NOUN
ejpam-3344	297	9	and	and	CCONJ
ejpam-3344	297	10	χar∪ra	χar∪ra	ADV
ejpam-3344	297	11	is	be	AUX
ejpam-3344	297	12	an	an	DET
ejpam-3344	297	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	297	14	fuzzy	fuzzy	ADJ
ejpam-3344	297	15	right	right	ADJ
ejpam-3344	297	16	ideal	ideal	NOUN
ejpam-3344	297	17	of	of	ADP
ejpam-3344	297	18	r	r	NOUN
ejpam-3344	297	19	,	,	PUNCT
ejpam-3344	297	20	by	by	ADP
ejpam-3344	297	21	the	the	DET
ejpam-3344	297	22	lemma	lemma	PROPN
ejpam-3344	297	23	2	2	NUM
ejpam-3344	297	24	.	.	PUNCT
ejpam-3344	298	1	by	by	ADP
ejpam-3344	298	2	our	our	PRON
ejpam-3344	298	3	assumption	assumption	NOUN
ejpam-3344	298	4	χar∪ra	χar∪ra	PROPN
ejpam-3344	298	5	∩	∩	PROPN
ejpam-3344	298	6	χra	χra	PROPN
ejpam-3344	298	7	=	=	SYM
ejpam-3344	298	8	χar∪ra	χar∪ra	PUNCT
ejpam-3344	298	9	◦	◦	NOUN
ejpam-3344	298	10	χra	χra	PROPN
ejpam-3344	298	11	,	,	PUNCT
ejpam-3344	298	12	i.e.	i.e.	X
ejpam-3344	298	13	,	,	PUNCT
ejpam-3344	298	14	χ(ar∪ra)∩ra	χ(ar∪ra)∩ra	PROPN
ejpam-3344	298	15	=	=	SYM
ejpam-3344	298	16	χ(ar∪ra)ra	χ(ar∪ra)ra	PROPN
ejpam-3344	298	17	.	.	PUNCT
ejpam-3344	299	1	thus	thus	ADV
ejpam-3344	299	2	(	(	PUNCT
ejpam-3344	299	3	ar	ar	PROPN
ejpam-3344	299	4	∪	∪	PROPN
ejpam-3344	299	5	ra	ra	PROPN
ejpam-3344	299	6	)	)	PUNCT
ejpam-3344	299	7	∩	∩	PROPN
ejpam-3344	299	8	ra	ra	PROPN
ejpam-3344	300	1	=	=	SYM
ejpam-3344	300	2	(	(	PUNCT
ejpam-3344	300	3	ar	ar	PROPN
ejpam-3344	300	4	∪	∪	X
ejpam-3344	300	5	ra)ra	ra)ra	PROPN
ejpam-3344	300	6	.	.	PUNCT
ejpam-3344	301	1	since	since	SCONJ
ejpam-3344	301	2	a	a	DET
ejpam-3344	301	3	∈	∈	PROPN
ejpam-3344	301	4	(	(	PUNCT
ejpam-3344	301	5	ar	ar	NOUN
ejpam-3344	301	6	∪	∪	PROPN
ejpam-3344	301	7	ra	ra	PROPN
ejpam-3344	301	8	)	)	PUNCT
ejpam-3344	301	9	∩	∩	PROPN
ejpam-3344	301	10	ra	ra	PROPN
ejpam-3344	301	11	,	,	PUNCT
ejpam-3344	301	12	i.e.	i.e.	X
ejpam-3344	301	13	,	,	PUNCT
ejpam-3344	301	14	a	a	DET
ejpam-3344	301	15	∈	∈	PROPN
ejpam-3344	301	16	(	(	PUNCT
ejpam-3344	301	17	ar	ar	NOUN
ejpam-3344	301	18	∪	∪	X
ejpam-3344	301	19	ra)ra	ra)ra	PROPN
ejpam-3344	301	20	,	,	PUNCT
ejpam-3344	301	21	so	so	SCONJ
ejpam-3344	301	22	a	a	DET
ejpam-3344	301	23	∈	∈	PROPN
ejpam-3344	301	24	(	(	PUNCT
ejpam-3344	301	25	ar)(ra	ar)(ra	PROPN
ejpam-3344	301	26	)	)	PUNCT
ejpam-3344	301	27	∪	∪	NOUN
ejpam-3344	301	28	(	(	PUNCT
ejpam-3344	301	29	ra)(ra	ra)(ra	X
ejpam-3344	301	30	)	)	PUNCT
ejpam-3344	301	31	.	.	PUNCT
ejpam-3344	302	1	now	now	ADV
ejpam-3344	302	2	(	(	PUNCT
ejpam-3344	302	3	ra)(ra	ra)(ra	X
ejpam-3344	302	4	)	)	PUNCT
ejpam-3344	302	5	=	=	SYM
ejpam-3344	302	6	(	(	PUNCT
ejpam-3344	302	7	(	(	PUNCT
ejpam-3344	302	8	re)a)(ra	re)a)(ra	PROPN
ejpam-3344	302	9	)	)	PUNCT
ejpam-3344	302	10	=	=	SYM
ejpam-3344	302	11	(	(	PUNCT
ejpam-3344	302	12	(	(	PUNCT
ejpam-3344	302	13	ae)r)(ra	ae)r)(ra	PROPN
ejpam-3344	302	14	)	)	PUNCT
ejpam-3344	302	15	=	=	PUNCT
ejpam-3344	302	16	(	(	PUNCT
ejpam-3344	302	17	ar)(ra	ar)(ra	PROPN
ejpam-3344	302	18	)	)	PUNCT
ejpam-3344	302	19	.	.	PUNCT
ejpam-3344	303	1	this	this	PRON
ejpam-3344	303	2	implies	imply	VERB
ejpam-3344	303	3	that	that	SCONJ
ejpam-3344	303	4	(	(	PUNCT
ejpam-3344	303	5	ar)(ra	ar)(ra	NOUN
ejpam-3344	303	6	)	)	PUNCT
ejpam-3344	303	7	∪	∪	NOUN
ejpam-3344	303	8	(	(	PUNCT
ejpam-3344	303	9	ra)(ra	ra)(ra	X
ejpam-3344	303	10	)	)	PUNCT
ejpam-3344	303	11	=	=	SYM
ejpam-3344	303	12	(	(	PUNCT
ejpam-3344	303	13	ar)(ra	ar)(ra	NOUN
ejpam-3344	303	14	)	)	PUNCT
ejpam-3344	303	15	∪	∪	ADV
ejpam-3344	303	16	(	(	PUNCT
ejpam-3344	303	17	ar)(ra	ar)(ra	NOUN
ejpam-3344	303	18	)	)	PUNCT
ejpam-3344	303	19	=	=	SYM
ejpam-3344	303	20	(	(	PUNCT
ejpam-3344	303	21	ar)(ra	ar)(ra	PROPN
ejpam-3344	303	22	)	)	PUNCT
ejpam-3344	303	23	.	.	PUNCT
ejpam-3344	304	1	thus	thus	ADV
ejpam-3344	304	2	a	a	DET
ejpam-3344	304	3	∈	∈	PROPN
ejpam-3344	304	4	(	(	PUNCT
ejpam-3344	304	5	ar)(ra	ar)(ra	PROPN
ejpam-3344	304	6	)	)	PUNCT
ejpam-3344	304	7	.	.	PUNCT
ejpam-3344	305	1	then	then	ADV
ejpam-3344	305	2	a	a	PRON
ejpam-3344	305	3	=	=	X
ejpam-3344	305	4	(	(	PUNCT
ejpam-3344	305	5	ax)(ya	ax)(ya	NOUN
ejpam-3344	305	6	)	)	PUNCT
ejpam-3344	305	7	=	=	SYM
ejpam-3344	305	8	(	(	PUNCT
ejpam-3344	305	9	(	(	PUNCT
ejpam-3344	305	10	ya)x)a	ya)x)a	NOUN
ejpam-3344	305	11	=	=	SYM
ejpam-3344	305	12	(	(	PUNCT
ejpam-3344	305	13	(	(	PUNCT
ejpam-3344	305	14	(	(	PUNCT
ejpam-3344	305	15	ey)a)x)a	ey)a)x)a	PROPN
ejpam-3344	305	16	=	=	SYM
ejpam-3344	305	17	(	(	PUNCT
ejpam-3344	305	18	(	(	PUNCT
ejpam-3344	305	19	(	(	PUNCT
ejpam-3344	305	20	ay)e)x)a	ay)e)x)a	PROPN
ejpam-3344	305	21	=	=	SYM
ejpam-3344	305	22	(	(	PUNCT
ejpam-3344	305	23	(	(	PUNCT
ejpam-3344	305	24	xe)(ay))a	xe)(ay))a	PROPN
ejpam-3344	305	25	=	=	SYM
ejpam-3344	305	26	(	(	PUNCT
ejpam-3344	305	27	a((xe)y))a	a((xe)y))a	PROPN
ejpam-3344	305	28	∈	∈	PROPN
ejpam-3344	305	29	(	(	PUNCT
ejpam-3344	305	30	ar)a	ar)a	PROPN
ejpam-3344	305	31	,	,	PUNCT
ejpam-3344	305	32	for	for	ADP
ejpam-3344	305	33	any	any	DET
ejpam-3344	305	34	x	x	NOUN
ejpam-3344	305	35	,	,	PUNCT
ejpam-3344	305	36	y	y	PROPN
ejpam-3344	305	37	∈	∈	PROPN
ejpam-3344	305	38	r.	r.	PROPN
ejpam-3344	305	39	this	this	PRON
ejpam-3344	305	40	means	mean	VERB
ejpam-3344	305	41	that	that	SCONJ
ejpam-3344	305	42	a	a	DET
ejpam-3344	305	43	∈	∈	PROPN
ejpam-3344	305	44	(	(	PUNCT
ejpam-3344	305	45	ar)a	ar)a	PROPN
ejpam-3344	305	46	,	,	PUNCT
ejpam-3344	305	47	i.e.	i.e.	X
ejpam-3344	305	48	,	,	PUNCT
ejpam-3344	305	49	a	a	PRON
ejpam-3344	305	50	is	be	AUX
ejpam-3344	305	51	regular	regular	ADJ
ejpam-3344	305	52	.	.	PUNCT
ejpam-3344	306	1	hence	hence	ADV
ejpam-3344	306	2	r	r	NOUN
ejpam-3344	306	3	is	be	AUX
ejpam-3344	306	4	a	a	DET
ejpam-3344	306	5	regular	regular	ADJ
ejpam-3344	306	6	,	,	PUNCT
ejpam-3344	306	7	i.e.	i.e.	X
ejpam-3344	306	8	,	,	PUNCT
ejpam-3344	306	9	(	(	PUNCT
ejpam-3344	306	10	2)⇒	2)⇒	NUM
ejpam-3344	306	11	(	(	PUNCT
ejpam-3344	306	12	1	1	NUM
ejpam-3344	306	13	)	)	PUNCT
ejpam-3344	306	14	.	.	PUNCT
ejpam-3344	307	1	theorem	theorem	ADJ
ejpam-3344	307	2	6	6	NUM
ejpam-3344	307	3	.	.	PUNCT
ejpam-3344	308	1	let	let	VERB
ejpam-3344	308	2	r	r	PRON
ejpam-3344	308	3	be	be	AUX
ejpam-3344	308	4	a	a	DET
ejpam-3344	308	5	regular	regular	ADJ
ejpam-3344	308	6	locally	locally	ADV
ejpam-3344	308	7	associative	associative	ADJ
ejpam-3344	308	8	la	la	ADJ
ejpam-3344	308	9	-	-	NOUN
ejpam-3344	308	10	ring	ring	NOUN
ejpam-3344	308	11	having	have	VERB
ejpam-3344	308	12	the	the	DET
ejpam-3344	308	13	property	property	NOUN
ejpam-3344	308	14	a	a	DET
ejpam-3344	308	15	=	=	NOUN
ejpam-3344	308	16	a2	a2	PROPN
ejpam-3344	308	17	for	for	ADP
ejpam-3344	308	18	every	every	DET
ejpam-3344	308	19	a	a	DET
ejpam-3344	308	20	∈	∈	PROPN
ejpam-3344	308	21	r.	r.	NOUN
ejpam-3344	308	22	then	then	ADV
ejpam-3344	308	23	for	for	ADP
ejpam-3344	308	24	every	every	DET
ejpam-3344	308	25	intuitionistic	intuitionistic	ADJ
ejpam-3344	308	26	fuzzy	fuzzy	ADJ
ejpam-3344	308	27	bi	bi	NOUN
ejpam-3344	308	28	-	-	NOUN
ejpam-3344	308	29	ideal	ideal	ADJ
ejpam-3344	308	30	a	a	PRON
ejpam-3344	308	31	=	=	X
ejpam-3344	308	32	(	(	PUNCT
ejpam-3344	308	33	µa	µa	PROPN
ejpam-3344	308	34	,	,	PUNCT
ejpam-3344	308	35	γa	γa	PROPN
ejpam-3344	308	36	)	)	PUNCT
ejpam-3344	308	37	of	of	ADP
ejpam-3344	308	38	r	r	NOUN
ejpam-3344	308	39	,	,	PUNCT
ejpam-3344	308	40	a(an	a(an	PROPN
ejpam-3344	308	41	)	)	PUNCT
ejpam-3344	308	42	=	=	PUNCT
ejpam-3344	308	43	a(a2n	a(a2n	VERB
ejpam-3344	308	44	)	)	PUNCT
ejpam-3344	308	45	for	for	ADP
ejpam-3344	308	46	all	all	DET
ejpam-3344	308	47	a	a	DET
ejpam-3344	308	48	∈	∈	NOUN
ejpam-3344	308	49	r	r	NOUN
ejpam-3344	308	50	,	,	PUNCT
ejpam-3344	308	51	where	where	SCONJ
ejpam-3344	308	52	n	n	PRON
ejpam-3344	308	53	is	be	AUX
ejpam-3344	308	54	any	any	DET
ejpam-3344	308	55	positive	positive	ADJ
ejpam-3344	308	56	integer	integer	NOUN
ejpam-3344	308	57	.	.	PUNCT
ejpam-3344	309	1	proof	proof	NOUN
ejpam-3344	309	2	.	.	PUNCT
ejpam-3344	310	1	for	for	ADP
ejpam-3344	310	2	n	n	NOUN
ejpam-3344	310	3	=	=	SYM
ejpam-3344	310	4	1	1	X
ejpam-3344	310	5	.	.	PUNCT
ejpam-3344	310	6	let	let	VERB
ejpam-3344	310	7	a	a	DET
ejpam-3344	310	8	∈	∈	ADJ
ejpam-3344	310	9	r	r	NOUN
ejpam-3344	310	10	,	,	PUNCT
ejpam-3344	310	11	this	this	PRON
ejpam-3344	310	12	implies	imply	VERB
ejpam-3344	310	13	that	that	SCONJ
ejpam-3344	310	14	there	there	PRON
ejpam-3344	310	15	exists	exist	VERB
ejpam-3344	310	16	an	an	DET
ejpam-3344	310	17	element	element	NOUN
ejpam-3344	310	18	x	x	SYM
ejpam-3344	310	19	∈	∈	NOUN
ejpam-3344	310	20	r	r	NOUN
ejpam-3344	311	1	such	such	DET
ejpam-3344	311	2	that	that	SCONJ
ejpam-3344	311	3	a	a	DET
ejpam-3344	311	4	=	=	X
ejpam-3344	311	5	(	(	PUNCT
ejpam-3344	311	6	ax)a	ax)a	PROPN
ejpam-3344	311	7	.	.	PUNCT
ejpam-3344	311	8	now	now	ADV
ejpam-3344	311	9	a	a	DET
ejpam-3344	311	10	=	=	X
ejpam-3344	311	11	(	(	PUNCT
ejpam-3344	311	12	ax)a	ax)a	PROPN
ejpam-3344	311	13	=	=	SYM
ejpam-3344	311	14	(	(	PUNCT
ejpam-3344	311	15	a2x)a2	a2x)a2	ADP
ejpam-3344	311	16	,	,	PUNCT
ejpam-3344	311	17	because	because	SCONJ
ejpam-3344	311	18	a	a	DET
ejpam-3344	311	19	=	=	NOUN
ejpam-3344	311	20	a2	a2	PROPN
ejpam-3344	311	21	.	.	PUNCT
ejpam-3344	312	1	thus	thus	ADV
ejpam-3344	312	2	µa	µa	X
ejpam-3344	312	3	(	(	PUNCT
ejpam-3344	312	4	a	a	X
ejpam-3344	312	5	)	)	PUNCT
ejpam-3344	312	6	=	=	SYM
ejpam-3344	312	7	µa	µa	NOUN
ejpam-3344	312	8	(	(	PUNCT
ejpam-3344	312	9	(	(	PUNCT
ejpam-3344	312	10	a2x)a2	a2x)a2	NOUN
ejpam-3344	312	11	)	)	PUNCT
ejpam-3344	312	12	≥	≥	NOUN
ejpam-3344	312	13	min{µa	min{µa	VERB
ejpam-3344	312	14	(	(	PUNCT
ejpam-3344	312	15	a2	a2	PROPN
ejpam-3344	312	16	)	)	PUNCT
ejpam-3344	312	17	,	,	PUNCT
ejpam-3344	312	18	µa	µa	ADP
ejpam-3344	312	19	(	(	PUNCT
ejpam-3344	312	20	a2	a2	PROPN
ejpam-3344	312	21	)	)	PUNCT
ejpam-3344	312	22	}	}	PUNCT
ejpam-3344	312	23	=	=	SYM
ejpam-3344	312	24	µa	µa	ADP
ejpam-3344	312	25	(	(	PUNCT
ejpam-3344	312	26	a2	a2	PROPN
ejpam-3344	312	27	)	)	PUNCT
ejpam-3344	312	28	=	=	SYM
ejpam-3344	312	29	µa	µa	INTJ
ejpam-3344	312	30	(	(	PUNCT
ejpam-3344	312	31	aa	aa	NOUN
ejpam-3344	312	32	)	)	PUNCT
ejpam-3344	312	33	≥	≥	NOUN
ejpam-3344	312	34	min{µa	min{µa	X
ejpam-3344	312	35	(	(	PUNCT
ejpam-3344	312	36	a	a	NOUN
ejpam-3344	312	37	)	)	PUNCT
ejpam-3344	312	38	,	,	PUNCT
ejpam-3344	312	39	µa	µa	X
ejpam-3344	312	40	(	(	PUNCT
ejpam-3344	312	41	a	a	NOUN
ejpam-3344	312	42	)	)	PUNCT
ejpam-3344	312	43	}	}	PUNCT
ejpam-3344	312	44	=	=	SYM
ejpam-3344	312	45	µa(a	µa(a	NUM
ejpam-3344	312	46	)	)	PUNCT
ejpam-3344	312	47	.	.	PUNCT
ejpam-3344	313	1	similarly	similarly	ADV
ejpam-3344	313	2	,	,	PUNCT
ejpam-3344	313	3	γa(a	γa(a	PUNCT
ejpam-3344	313	4	)	)	PUNCT
ejpam-3344	314	1	=	=	SYM
ejpam-3344	314	2	γa	γa	PROPN
ejpam-3344	314	3	(	(	PUNCT
ejpam-3344	314	4	a2	a2	PROPN
ejpam-3344	314	5	)	)	PUNCT
ejpam-3344	314	6	,	,	PUNCT
ejpam-3344	314	7	therefore	therefore	ADV
ejpam-3344	314	8	a(a	a(a	PROPN
ejpam-3344	314	9	)	)	PUNCT
ejpam-3344	315	1	=	=	SYM
ejpam-3344	315	2	a	a	PRON
ejpam-3344	315	3	(	(	PUNCT
ejpam-3344	315	4	a2	a2	PROPN
ejpam-3344	315	5	)	)	PUNCT
ejpam-3344	315	6	.	.	PUNCT
ejpam-3344	316	1	now	now	ADV
ejpam-3344	316	2	a2	a2	PROPN
ejpam-3344	316	3	=	=	SYM
ejpam-3344	316	4	aa	aa	PROPN
ejpam-3344	316	5	=	=	PUNCT
ejpam-3344	316	6	(	(	PUNCT
ejpam-3344	316	7	(	(	PUNCT
ejpam-3344	316	8	a2x)a2)((a2x)a2	a2x)a2)((a2x)a2	PROPN
ejpam-3344	316	9	)	)	PUNCT
ejpam-3344	316	10	=	=	SYM
ejpam-3344	316	11	(	(	PUNCT
ejpam-3344	316	12	a4x2)a4	a4x2)a4	NUM
ejpam-3344	316	13	,	,	PUNCT
ejpam-3344	316	14	then	then	ADV
ejpam-3344	316	15	the	the	DET
ejpam-3344	316	16	result	result	NOUN
ejpam-3344	316	17	is	be	AUX
ejpam-3344	316	18	true	true	ADJ
ejpam-3344	316	19	for	for	ADP
ejpam-3344	316	20	n	n	NOUN
ejpam-3344	316	21	=	=	SYM
ejpam-3344	316	22	2	2	X
ejpam-3344	316	23	.	.	PUNCT
ejpam-3344	316	24	suppose	suppose	VERB
ejpam-3344	316	25	that	that	SCONJ
ejpam-3344	316	26	the	the	DET
ejpam-3344	316	27	result	result	NOUN
ejpam-3344	316	28	is	be	AUX
ejpam-3344	316	29	true	true	ADJ
ejpam-3344	316	30	for	for	ADP
ejpam-3344	316	31	n	n	PROPN
ejpam-3344	316	32	=	=	SYM
ejpam-3344	316	33	k	k	NOUN
ejpam-3344	316	34	,	,	PUNCT
ejpam-3344	316	35	i.e.	i.e.	X
ejpam-3344	316	36	,	,	PUNCT
ejpam-3344	316	37	a(ak	a(ak	PROPN
ejpam-3344	316	38	)	)	PUNCT
ejpam-3344	316	39	=	=	SYM
ejpam-3344	316	40	a	a	PRON
ejpam-3344	316	41	(	(	PUNCT
ejpam-3344	316	42	a2k	a2k	PROPN
ejpam-3344	316	43	)	)	PUNCT
ejpam-3344	316	44	.	.	PUNCT
ejpam-3344	317	1	now	now	ADV
ejpam-3344	317	2	ak+1	ak+1	VERB
ejpam-3344	317	3	=	=	PUNCT
ejpam-3344	317	4	aka	aka	ADV
ejpam-3344	317	5	=	=	SYM
ejpam-3344	317	6	(	(	PUNCT
ejpam-3344	317	7	(	(	PUNCT
ejpam-3344	317	8	a2kxk)a2k)((a2x)a2	a2kxk)a2k)((a2x)a2	NOUN
ejpam-3344	317	9	)	)	PUNCT
ejpam-3344	317	10	=	=	SYM
ejpam-3344	317	11	(	(	PUNCT
ejpam-3344	317	12	a2(k+1)xk+1)a2(k+1	a2(k+1)xk+1)a2(k+1	NOUN
ejpam-3344	317	13	)	)	PUNCT
ejpam-3344	317	14	.	.	PUNCT
ejpam-3344	318	1	thus	thus	ADV
ejpam-3344	318	2	µa	µa	X
ejpam-3344	318	3	(	(	PUNCT
ejpam-3344	318	4	ak+1	ak+1	X
ejpam-3344	318	5	)	)	PUNCT
ejpam-3344	318	6	=	=	SYM
ejpam-3344	318	7	µa	µa	NOUN
ejpam-3344	318	8	(	(	PUNCT
ejpam-3344	318	9	(	(	PUNCT
ejpam-3344	318	10	a2(k+1)xk+1)a2(k+1	a2(k+1)xk+1)a2(k+1	NOUN
ejpam-3344	318	11	)	)	PUNCT
ejpam-3344	318	12	)	)	PUNCT
ejpam-3344	318	13	≥	≥	NOUN
ejpam-3344	318	14	min{µa(a2(k+1	min{µa(a2(k+1	NOUN
ejpam-3344	318	15	)	)	PUNCT
ejpam-3344	318	16	)	)	PUNCT
ejpam-3344	318	17	,	,	PUNCT
ejpam-3344	318	18	µa	µa	ADP
ejpam-3344	318	19	(	(	PUNCT
ejpam-3344	318	20	a2(k+1	a2(k+1	X
ejpam-3344	318	21	)	)	PUNCT
ejpam-3344	318	22	)	)	PUNCT
ejpam-3344	318	23	}	}	PUNCT
ejpam-3344	319	1	=	=	SYM
ejpam-3344	319	2	µa	µa	NOUN
ejpam-3344	319	3	(	(	PUNCT
ejpam-3344	319	4	a2(k+1	a2(k+1	NOUN
ejpam-3344	319	5	)	)	PUNCT
ejpam-3344	319	6	)	)	PUNCT
ejpam-3344	320	1	=	=	SYM
ejpam-3344	320	2	µa	µa	NOUN
ejpam-3344	320	3	(	(	PUNCT
ejpam-3344	320	4	ak+1ak+1	ak+1ak+1	NOUN
ejpam-3344	320	5	)	)	PUNCT
ejpam-3344	320	6	)	)	PUNCT
ejpam-3344	320	7	≥	≥	NOUN
ejpam-3344	320	8	min{µa	min{µa	VERB
ejpam-3344	320	9	(	(	PUNCT
ejpam-3344	320	10	ak+1	ak+1	NOUN
ejpam-3344	320	11	)	)	PUNCT
ejpam-3344	320	12	)	)	PUNCT
ejpam-3344	320	13	,	,	PUNCT
ejpam-3344	320	14	µa	µa	ADP
ejpam-3344	320	15	(	(	PUNCT
ejpam-3344	320	16	ak+1	ak+1	NOUN
ejpam-3344	320	17	)	)	PUNCT
ejpam-3344	320	18	)	)	PUNCT
ejpam-3344	320	19	}	}	PUNCT
ejpam-3344	320	20	=	=	SYM
ejpam-3344	320	21	µa(ak+1	µa(ak+1	NOUN
ejpam-3344	320	22	)	)	PUNCT
ejpam-3344	320	23	)	)	PUNCT
ejpam-3344	320	24	.	.	PUNCT
ejpam-3344	321	1	similarly	similarly	ADV
ejpam-3344	321	2	,	,	PUNCT
ejpam-3344	321	3	γa(ak+1	γa(ak+1	ADJ
ejpam-3344	321	4	)	)	PUNCT
ejpam-3344	321	5	=	=	SYM
ejpam-3344	321	6	γa	γa	NOUN
ejpam-3344	321	7	(	(	PUNCT
ejpam-3344	321	8	a2(k+1	a2(k+1	X
ejpam-3344	321	9	)	)	PUNCT
ejpam-3344	321	10	)	)	PUNCT
ejpam-3344	321	11	,	,	PUNCT
ejpam-3344	321	12	therefore	therefore	ADV
ejpam-3344	321	13	a(ak+1	a(ak+1	VERB
ejpam-3344	321	14	)	)	PUNCT
ejpam-3344	321	15	=	=	SYM
ejpam-3344	321	16	a(a2(k+1	a(a2(k+1	PROPN
ejpam-3344	321	17	)	)	PUNCT
ejpam-3344	321	18	)	)	PUNCT
ejpam-3344	321	19	.	.	PUNCT
ejpam-3344	322	1	hence	hence	ADV
ejpam-3344	322	2	by	by	ADP
ejpam-3344	322	3	induction	induction	NOUN
ejpam-3344	322	4	method	method	NOUN
ejpam-3344	322	5	,	,	PUNCT
ejpam-3344	322	6	the	the	DET
ejpam-3344	322	7	result	result	NOUN
ejpam-3344	322	8	is	be	AUX
ejpam-3344	322	9	true	true	ADJ
ejpam-3344	322	10	for	for	ADP
ejpam-3344	322	11	all	all	DET
ejpam-3344	322	12	positive	positive	ADJ
ejpam-3344	322	13	integers	integer	NOUN
ejpam-3344	322	14	.	.	PUNCT
ejpam-3344	323	1	lemma	lemma	PROPN
ejpam-3344	323	2	11	11	NUM
ejpam-3344	323	3	.	.	PUNCT
ejpam-3344	324	1	every	every	DET
ejpam-3344	324	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	324	3	fuzzy	fuzzy	ADJ
ejpam-3344	324	4	left	left	ADJ
ejpam-3344	324	5	(	(	PUNCT
ejpam-3344	324	6	right	right	ADJ
ejpam-3344	324	7	)	)	PUNCT
ejpam-3344	324	8	ideal	ideal	NOUN
ejpam-3344	324	9	of	of	ADP
ejpam-3344	324	10	(	(	PUNCT
ejpam-3344	324	11	2	2	NUM
ejpam-3344	324	12	,	,	PUNCT
ejpam-3344	324	13	2)-regular	2)-regular	NUM
ejpam-3344	324	14	la	la	ADJ
ejpam-3344	324	15	-	-	PUNCT
ejpam-3344	324	16	ring	ring	NOUN
ejpam-3344	324	17	r	r	NOUN
ejpam-3344	324	18	,	,	PUNCT
ejpam-3344	324	19	is	be	AUX
ejpam-3344	324	20	an	an	DET
ejpam-3344	324	21	intuitionistic	intuitionistic	ADJ
ejpam-3344	324	22	fuzzy	fuzzy	ADJ
ejpam-3344	324	23	ideal	ideal	NOUN
ejpam-3344	324	24	of	of	ADP
ejpam-3344	324	25	r.	r.	PROPN
ejpam-3344	324	26	n.	n.	PROPN
ejpam-3344	324	27	kausar	kausar	PROPN
ejpam-3344	324	28	,	,	PUNCT
ejpam-3344	324	29	m.	m.	NOUN
ejpam-3344	324	30	a.	a.	PROPN
ejpam-3344	324	31	waqar	waqar	PROPN
ejpam-3344	324	32	/	/	SYM
ejpam-3344	324	33	eur	eur	PROPN
ejpam-3344	324	34	.	.	PUNCT
ejpam-3344	325	1	j.	j.	PROPN
ejpam-3344	325	2	pure	pure	PROPN
ejpam-3344	325	3	appl	appl	PROPN
ejpam-3344	325	4	.	.	PROPN
ejpam-3344	325	5	math	math	PROPN
ejpam-3344	325	6	,	,	PUNCT
ejpam-3344	325	7	12	12	NUM
ejpam-3344	325	8	(	(	PUNCT
ejpam-3344	325	9	1	1	NUM
ejpam-3344	325	10	)	)	PUNCT
ejpam-3344	325	11	(	(	PUNCT
ejpam-3344	325	12	2019	2019	NUM
ejpam-3344	325	13	)	)	PUNCT
ejpam-3344	325	14	,	,	PUNCT
ejpam-3344	325	15	226	226	NUM
ejpam-3344	325	16	-	-	SYM
ejpam-3344	325	17	250	250	NUM
ejpam-3344	325	18	238	238	NUM
ejpam-3344	325	19	proof	proof	NOUN
ejpam-3344	325	20	.	.	PUNCT
ejpam-3344	325	21	suppose	suppose	VERB
ejpam-3344	325	22	that	that	SCONJ
ejpam-3344	325	23	a	a	DET
ejpam-3344	325	24	=	=	SYM
ejpam-3344	325	25	(	(	PUNCT
ejpam-3344	325	26	µa	µa	PROPN
ejpam-3344	325	27	,	,	PUNCT
ejpam-3344	325	28	γa	γa	PROPN
ejpam-3344	325	29	)	)	PUNCT
ejpam-3344	325	30	is	be	AUX
ejpam-3344	325	31	an	an	DET
ejpam-3344	325	32	intuitionistic	intuitionistic	ADJ
ejpam-3344	325	33	fuzzy	fuzzy	ADJ
ejpam-3344	325	34	right	right	ADJ
ejpam-3344	325	35	ideal	ideal	NOUN
ejpam-3344	325	36	of	of	ADP
ejpam-3344	325	37	r	r	NOUN
ejpam-3344	325	38	and	and	CCONJ
ejpam-3344	325	39	x	x	NOUN
ejpam-3344	325	40	,	,	PUNCT
ejpam-3344	325	41	y	y	PROPN
ejpam-3344	325	42	∈	∈	PROPN
ejpam-3344	325	43	r	r	NOUN
ejpam-3344	325	44	,	,	PUNCT
ejpam-3344	325	45	this	this	PRON
ejpam-3344	325	46	means	mean	VERB
ejpam-3344	325	47	that	that	SCONJ
ejpam-3344	325	48	there	there	PRON
ejpam-3344	325	49	exists	exist	VERB
ejpam-3344	325	50	a	a	DET
ejpam-3344	325	51	∈	∈	NOUN
ejpam-3344	325	52	r	r	NOUN
ejpam-3344	325	53	such	such	ADJ
ejpam-3344	325	54	that	that	PRON
ejpam-3344	325	55	x	x	X
ejpam-3344	325	56	=	=	PRON
ejpam-3344	325	57	(	(	PUNCT
ejpam-3344	325	58	x2a)x2	x2a)x2	PROPN
ejpam-3344	325	59	.	.	PUNCT
ejpam-3344	326	1	thus	thus	ADV
ejpam-3344	326	2	µa(xy	µa(xy	NUM
ejpam-3344	326	3	)	)	PUNCT
ejpam-3344	326	4	=	=	SYM
ejpam-3344	326	5	µa(((x2a)x2)y	µa(((x2a)x2)y	PROPN
ejpam-3344	326	6	)	)	PUNCT
ejpam-3344	326	7	=	=	SYM
ejpam-3344	326	8	µa((yx2)(x2a	µa((yx2)(x2a	PROPN
ejpam-3344	326	9	)	)	PUNCT
ejpam-3344	326	10	)	)	PUNCT
ejpam-3344	326	11	≥	≥	X
ejpam-3344	326	12	µa(yx2	µa(yx2	NUM
ejpam-3344	326	13	)	)	PUNCT
ejpam-3344	326	14	≥	≥	NOUN
ejpam-3344	326	15	µa(y	µa(y	NOUN
ejpam-3344	326	16	)	)	PUNCT
ejpam-3344	326	17	and	and	CCONJ
ejpam-3344	326	18	γa(xy	γa(xy	PROPN
ejpam-3344	326	19	)	)	PUNCT
ejpam-3344	327	1	=	=	SYM
ejpam-3344	327	2	γa(((x2a)x2)y	γa(((x2a)x2)y	PROPN
ejpam-3344	327	3	)	)	PUNCT
ejpam-3344	327	4	=	=	SYM
ejpam-3344	327	5	γa((yx2)(x2a	γa((yx2)(x2a	PROPN
ejpam-3344	327	6	)	)	PUNCT
ejpam-3344	327	7	)	)	PUNCT
ejpam-3344	327	8	≤	≤	NUM
ejpam-3344	327	9	γa(yx2	γa(yx2	NOUN
ejpam-3344	327	10	)	)	PUNCT
ejpam-3344	327	11	≤	≤	NOUN
ejpam-3344	327	12	γa(y	γa(y	NUM
ejpam-3344	327	13	)	)	PUNCT
ejpam-3344	327	14	.	.	PUNCT
ejpam-3344	328	1	therefore	therefore	ADV
ejpam-3344	328	2	a	a	PRON
ejpam-3344	328	3	is	be	AUX
ejpam-3344	328	4	an	an	DET
ejpam-3344	328	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	328	6	fuzzy	fuzzy	ADJ
ejpam-3344	328	7	ideal	ideal	NOUN
ejpam-3344	328	8	of	of	ADP
ejpam-3344	328	9	r.	r.	PROPN
ejpam-3344	328	10	similarly	similarly	ADV
ejpam-3344	328	11	,	,	PUNCT
ejpam-3344	328	12	for	for	ADP
ejpam-3344	328	13	left	left	ADJ
ejpam-3344	328	14	ideal	ideal	NOUN
ejpam-3344	328	15	.	.	PUNCT
ejpam-3344	329	1	remark	remark	NOUN
ejpam-3344	329	2	5	5	NUM
ejpam-3344	329	3	.	.	PUNCT
ejpam-3344	330	1	the	the	DET
ejpam-3344	330	2	concept	concept	NOUN
ejpam-3344	330	3	of	of	ADP
ejpam-3344	330	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	330	5	fuzzy	fuzzy	ADJ
ejpam-3344	330	6	(	(	PUNCT
ejpam-3344	330	7	left	left	ADJ
ejpam-3344	330	8	,	,	PUNCT
ejpam-3344	330	9	right	right	INTJ
ejpam-3344	330	10	,	,	PUNCT
ejpam-3344	330	11	two	two	NUM
ejpam-3344	330	12	-	-	PUNCT
ejpam-3344	330	13	sided	sided	ADJ
ejpam-3344	330	14	)	)	PUNCT
ejpam-3344	330	15	ideals	ideal	NOUN
ejpam-3344	330	16	coincides	coincide	VERB
ejpam-3344	330	17	in	in	ADP
ejpam-3344	330	18	(	(	PUNCT
ejpam-3344	330	19	2	2	NUM
ejpam-3344	330	20	,	,	PUNCT
ejpam-3344	330	21	2)-regular	2)-regular	NUM
ejpam-3344	330	22	la	la	NOUN
ejpam-3344	330	23	-	-	PUNCT
ejpam-3344	330	24	rings	ring	NOUN
ejpam-3344	330	25	.	.	PUNCT
ejpam-3344	331	1	proposition	proposition	NOUN
ejpam-3344	331	2	6	6	NUM
ejpam-3344	331	3	.	.	PUNCT
ejpam-3344	332	1	every	every	DET
ejpam-3344	332	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	332	3	fuzzy	fuzzy	ADJ
ejpam-3344	332	4	generalized	generalize	VERB
ejpam-3344	332	5	bi	bi	NOUN
ejpam-3344	332	6	-	-	NOUN
ejpam-3344	332	7	ideal	ideal	NOUN
ejpam-3344	332	8	of	of	ADP
ejpam-3344	332	9	(	(	PUNCT
ejpam-3344	332	10	2	2	NUM
ejpam-3344	332	11	,	,	PUNCT
ejpam-3344	332	12	2)-regular	2)-regular	NUM
ejpam-3344	332	13	la	la	ADJ
ejpam-3344	332	14	-	-	PUNCT
ejpam-3344	332	15	ring	ring	NOUN
ejpam-3344	332	16	r	r	NOUN
ejpam-3344	332	17	with	with	ADP
ejpam-3344	332	18	left	left	ADJ
ejpam-3344	332	19	identity	identity	NOUN
ejpam-3344	332	20	e	e	NOUN
ejpam-3344	332	21	,	,	PUNCT
ejpam-3344	332	22	is	be	AUX
ejpam-3344	332	23	an	an	DET
ejpam-3344	332	24	intuitionistic	intuitionistic	ADJ
ejpam-3344	332	25	fuzzy	fuzzy	ADJ
ejpam-3344	332	26	bi	bi	NOUN
ejpam-3344	332	27	-	-	NOUN
ejpam-3344	332	28	ideal	ideal	NOUN
ejpam-3344	332	29	of	of	ADP
ejpam-3344	332	30	r.	r.	PROPN
ejpam-3344	332	31	proof	proof	PROPN
ejpam-3344	332	32	.	.	PUNCT
ejpam-3344	333	1	assume	assume	VERB
ejpam-3344	333	2	that	that	SCONJ
ejpam-3344	333	3	a	a	DET
ejpam-3344	333	4	=	=	SYM
ejpam-3344	333	5	(	(	PUNCT
ejpam-3344	333	6	µa	µa	PROPN
ejpam-3344	333	7	,	,	PUNCT
ejpam-3344	333	8	γa	γa	PROPN
ejpam-3344	333	9	)	)	PUNCT
ejpam-3344	333	10	is	be	AUX
ejpam-3344	333	11	an	an	DET
ejpam-3344	333	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	333	13	fuzzy	fuzzy	ADJ
ejpam-3344	333	14	generalized	generalize	VERB
ejpam-3344	333	15	bi	bi	NOUN
ejpam-3344	333	16	-	-	NOUN
ejpam-3344	333	17	ideal	ideal	NOUN
ejpam-3344	333	18	of	of	ADP
ejpam-3344	333	19	r	r	NOUN
ejpam-3344	333	20	and	and	CCONJ
ejpam-3344	333	21	x	x	NOUN
ejpam-3344	333	22	,	,	PUNCT
ejpam-3344	333	23	y	y	PROPN
ejpam-3344	333	24	∈	∈	PROPN
ejpam-3344	333	25	r	r	NOUN
ejpam-3344	333	26	,	,	PUNCT
ejpam-3344	333	27	then	then	ADV
ejpam-3344	333	28	there	there	PRON
ejpam-3344	333	29	exists	exist	VERB
ejpam-3344	333	30	an	an	DET
ejpam-3344	333	31	element	element	NOUN
ejpam-3344	333	32	a	a	DET
ejpam-3344	333	33	∈	∈	NOUN
ejpam-3344	333	34	r	r	NOUN
ejpam-3344	333	35	such	such	ADJ
ejpam-3344	333	36	that	that	PRON
ejpam-3344	333	37	x	x	X
ejpam-3344	333	38	=	=	PRON
ejpam-3344	333	39	(	(	PUNCT
ejpam-3344	333	40	x2a)x2	x2a)x2	PROPN
ejpam-3344	333	41	.	.	X
ejpam-3344	334	1	we	we	PRON
ejpam-3344	334	2	have	have	VERB
ejpam-3344	334	3	to	to	PART
ejpam-3344	334	4	show	show	VERB
ejpam-3344	334	5	that	that	SCONJ
ejpam-3344	334	6	a	a	PRON
ejpam-3344	334	7	is	be	AUX
ejpam-3344	334	8	an	an	DET
ejpam-3344	334	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	334	10	fuzzy	fuzzy	ADJ
ejpam-3344	334	11	la	la	NOUN
ejpam-3344	334	12	-	-	PUNCT
ejpam-3344	334	13	subring	subring	NOUN
ejpam-3344	334	14	of	of	ADP
ejpam-3344	334	15	r.	r.	PROPN
ejpam-3344	334	16	thus	thus	ADV
ejpam-3344	334	17	µa(xy	µa(xy	NOUN
ejpam-3344	334	18	)	)	PUNCT
ejpam-3344	334	19	=	=	SYM
ejpam-3344	334	20	µa(((x2a)x2)y	µa(((x2a)x2)y	PROPN
ejpam-3344	334	21	)	)	PUNCT
ejpam-3344	334	22	=	=	SYM
ejpam-3344	334	23	µa(((x2a)(xx))y	µa(((x2a)(xx))y	ADJ
ejpam-3344	334	24	)	)	PUNCT
ejpam-3344	334	25	=	=	SYM
ejpam-3344	334	26	µa((x((x2a)x))y	µa((x((x2a)x))y	X
ejpam-3344	334	27	)	)	PUNCT
ejpam-3344	334	28	≥	≥	PROPN
ejpam-3344	334	29	min{µa(x	min{µa(x	NOUN
ejpam-3344	334	30	)	)	PUNCT
ejpam-3344	334	31	,	,	PUNCT
ejpam-3344	334	32	µa(y	µa(y	NOUN
ejpam-3344	334	33	)	)	PUNCT
ejpam-3344	334	34	}	}	PUNCT
ejpam-3344	334	35	and	and	CCONJ
ejpam-3344	334	36	γa(xy	γa(xy	PROPN
ejpam-3344	334	37	)	)	PUNCT
ejpam-3344	335	1	=	=	SYM
ejpam-3344	335	2	γa(((x2a)x2)y	γa(((x2a)x2)y	PROPN
ejpam-3344	335	3	)	)	PUNCT
ejpam-3344	335	4	=	=	SYM
ejpam-3344	335	5	γa(((x2a)(xx))y	γa(((x2a)(xx))y	NOUN
ejpam-3344	335	6	)	)	PUNCT
ejpam-3344	335	7	=	=	SYM
ejpam-3344	335	8	γa((x((x2a)x))y	γa((x((x2a)x))y	X
ejpam-3344	335	9	)	)	PUNCT
ejpam-3344	335	10	≤	≤	NUM
ejpam-3344	335	11	max{γa(x	max{γa(x	NOUN
ejpam-3344	335	12	)	)	PUNCT
ejpam-3344	335	13	,	,	PUNCT
ejpam-3344	335	14	γa(y	γa(y	NOUN
ejpam-3344	335	15	)	)	PUNCT
ejpam-3344	335	16	}	}	PUNCT
ejpam-3344	335	17	.	.	PUNCT
ejpam-3344	336	1	so	so	ADV
ejpam-3344	336	2	a	a	PRON
ejpam-3344	336	3	is	be	AUX
ejpam-3344	336	4	an	an	DET
ejpam-3344	336	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	336	6	fuzzy	fuzzy	ADJ
ejpam-3344	336	7	la	la	NOUN
ejpam-3344	336	8	-	-	PUNCT
ejpam-3344	336	9	subring	subring	NOUN
ejpam-3344	336	10	of	of	ADP
ejpam-3344	336	11	r.	r.	PROPN
ejpam-3344	336	12	theorem	theorem	VERB
ejpam-3344	336	13	7	7	NUM
ejpam-3344	336	14	.	.	PUNCT
ejpam-3344	337	1	let	let	VERB
ejpam-3344	337	2	r	r	PRON
ejpam-3344	337	3	be	be	AUX
ejpam-3344	337	4	a	a	DET
ejpam-3344	337	5	(	(	PUNCT
ejpam-3344	337	6	2	2	NUM
ejpam-3344	337	7	,	,	PUNCT
ejpam-3344	337	8	2)-regular	2)-regular	NUM
ejpam-3344	337	9	locally	locally	ADV
ejpam-3344	337	10	associative	associative	ADJ
ejpam-3344	337	11	la	la	ADJ
ejpam-3344	337	12	-	-	PUNCT
ejpam-3344	337	13	ring	ring	NOUN
ejpam-3344	337	14	.	.	PUNCT
ejpam-3344	338	1	then	then	ADV
ejpam-3344	338	2	for	for	ADP
ejpam-3344	338	3	every	every	DET
ejpam-3344	338	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	338	5	fuzzy	fuzzy	ADJ
ejpam-3344	338	6	bi	bi	NOUN
ejpam-3344	338	7	-	-	NOUN
ejpam-3344	338	8	ideal	ideal	ADJ
ejpam-3344	338	9	a	a	PRON
ejpam-3344	338	10	=	=	X
ejpam-3344	338	11	(	(	PUNCT
ejpam-3344	338	12	µa	µa	PROPN
ejpam-3344	338	13	,	,	PUNCT
ejpam-3344	338	14	γa	γa	PROPN
ejpam-3344	338	15	)	)	PUNCT
ejpam-3344	338	16	of	of	ADP
ejpam-3344	338	17	r	r	NOUN
ejpam-3344	338	18	,	,	PUNCT
ejpam-3344	338	19	a(an	a(an	PROPN
ejpam-3344	338	20	)	)	PUNCT
ejpam-3344	338	21	=	=	PUNCT
ejpam-3344	338	22	a(a2n	a(a2n	VERB
ejpam-3344	338	23	)	)	PUNCT
ejpam-3344	338	24	for	for	ADP
ejpam-3344	338	25	all	all	DET
ejpam-3344	338	26	a	a	DET
ejpam-3344	338	27	∈	∈	NOUN
ejpam-3344	338	28	r	r	NOUN
ejpam-3344	338	29	,	,	PUNCT
ejpam-3344	338	30	where	where	SCONJ
ejpam-3344	338	31	n	n	PRON
ejpam-3344	338	32	is	be	AUX
ejpam-3344	338	33	any	any	DET
ejpam-3344	338	34	positive	positive	ADJ
ejpam-3344	338	35	integer	integer	NOUN
ejpam-3344	338	36	.	.	PUNCT
ejpam-3344	339	1	proof	proof	NOUN
ejpam-3344	339	2	.	.	PUNCT
ejpam-3344	340	1	same	same	ADJ
ejpam-3344	340	2	as	as	SCONJ
ejpam-3344	340	3	theorem	theorem	ADJ
ejpam-3344	340	4	6	6	NUM
ejpam-3344	340	5	.	.	PUNCT
ejpam-3344	341	1	lemma	lemma	PROPN
ejpam-3344	341	2	12	12	NUM
ejpam-3344	341	3	.	.	PUNCT
ejpam-3344	342	1	let	let	VERB
ejpam-3344	342	2	r	r	PRON
ejpam-3344	342	3	be	be	AUX
ejpam-3344	342	4	a	a	DET
ejpam-3344	342	5	right	right	ADJ
ejpam-3344	342	6	regular	regular	ADJ
ejpam-3344	342	7	la	la	NOUN
ejpam-3344	342	8	-	-	PUNCT
ejpam-3344	342	9	ring	ring	NOUN
ejpam-3344	342	10	.	.	PUNCT
ejpam-3344	343	1	then	then	ADV
ejpam-3344	343	2	every	every	DET
ejpam-3344	343	3	intuitionistic	intuitionistic	ADJ
ejpam-3344	343	4	fuzzy	fuzzy	ADJ
ejpam-3344	343	5	left	left	ADJ
ejpam-3344	343	6	(	(	PUNCT
ejpam-3344	343	7	right	right	ADJ
ejpam-3344	343	8	)	)	PUNCT
ejpam-3344	343	9	ideal	ideal	NOUN
ejpam-3344	343	10	of	of	ADP
ejpam-3344	343	11	r	r	NOUN
ejpam-3344	343	12	is	be	AUX
ejpam-3344	343	13	an	an	DET
ejpam-3344	343	14	intuitionistic	intuitionistic	ADJ
ejpam-3344	343	15	fuzzy	fuzzy	ADJ
ejpam-3344	343	16	ideal	ideal	NOUN
ejpam-3344	343	17	of	of	ADP
ejpam-3344	343	18	r.	r.	PROPN
ejpam-3344	343	19	proof	proof	NOUN
ejpam-3344	343	20	.	.	PUNCT
ejpam-3344	344	1	let	let	VERB
ejpam-3344	344	2	a	a	DET
ejpam-3344	344	3	=	=	SYM
ejpam-3344	344	4	(	(	PUNCT
ejpam-3344	344	5	µa	µa	PROPN
ejpam-3344	344	6	,	,	PUNCT
ejpam-3344	344	7	γa	γa	PROPN
ejpam-3344	344	8	)	)	PUNCT
ejpam-3344	344	9	be	be	VERB
ejpam-3344	344	10	an	an	DET
ejpam-3344	344	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	344	12	fuzzy	fuzzy	ADJ
ejpam-3344	344	13	right	right	ADJ
ejpam-3344	344	14	ideal	ideal	NOUN
ejpam-3344	344	15	of	of	ADP
ejpam-3344	344	16	r	r	NOUN
ejpam-3344	344	17	and	and	CCONJ
ejpam-3344	344	18	x	x	NOUN
ejpam-3344	344	19	,	,	PUNCT
ejpam-3344	344	20	y	y	PROPN
ejpam-3344	344	21	∈	∈	PROPN
ejpam-3344	344	22	r	r	NOUN
ejpam-3344	344	23	,	,	PUNCT
ejpam-3344	344	24	this	this	PRON
ejpam-3344	344	25	implies	imply	VERB
ejpam-3344	344	26	that	that	SCONJ
ejpam-3344	344	27	there	there	PRON
ejpam-3344	344	28	exists	exist	VERB
ejpam-3344	344	29	a	a	DET
ejpam-3344	344	30	∈	∈	NOUN
ejpam-3344	344	31	r	r	NOUN
ejpam-3344	344	32	such	such	ADJ
ejpam-3344	344	33	that	that	PRON
ejpam-3344	344	34	x	x	X
ejpam-3344	344	35	=	=	SYM
ejpam-3344	344	36	x2a	x2a	PROPN
ejpam-3344	344	37	.	.	PUNCT
ejpam-3344	345	1	thus	thus	ADV
ejpam-3344	345	2	µa(xy	µa(xy	NOUN
ejpam-3344	345	3	)	)	PUNCT
ejpam-3344	345	4	=	=	SYM
ejpam-3344	345	5	µa((x2a)y	µa((x2a)y	NOUN
ejpam-3344	345	6	)	)	PUNCT
ejpam-3344	345	7	=	=	SYM
ejpam-3344	345	8	µa(((xx)a)y	µa(((xx)a)y	ADJ
ejpam-3344	345	9	)	)	PUNCT
ejpam-3344	345	10	=	=	SYM
ejpam-3344	345	11	µa(((ax)x)y	µa(((ax)x)y	ADJ
ejpam-3344	345	12	)	)	PUNCT
ejpam-3344	346	1	=	=	SYM
ejpam-3344	346	2	µa((yx)(ax	µa((yx)(ax	ADJ
ejpam-3344	346	3	)	)	PUNCT
ejpam-3344	346	4	)	)	PUNCT
ejpam-3344	347	1	≥	≥	NUM
ejpam-3344	347	2	µa(yx	µa(yx	NOUN
ejpam-3344	347	3	)	)	PUNCT
ejpam-3344	347	4	≥	≥	NOUN
ejpam-3344	347	5	µa(y	µa(y	NOUN
ejpam-3344	347	6	)	)	PUNCT
ejpam-3344	347	7	and	and	CCONJ
ejpam-3344	347	8	γa(xy	γa(xy	PROPN
ejpam-3344	347	9	)	)	PUNCT
ejpam-3344	348	1	=	=	PUNCT
ejpam-3344	348	2	γa((x2a)y	γa((x2a)y	PROPN
ejpam-3344	348	3	)	)	PUNCT
ejpam-3344	348	4	=	=	SYM
ejpam-3344	348	5	γa(((xx)a)y	γa(((xx)a)y	ADJ
ejpam-3344	348	6	)	)	PUNCT
ejpam-3344	348	7	=	=	SYM
ejpam-3344	348	8	γa(((ax)x)y	γa(((ax)x)y	ADJ
ejpam-3344	348	9	)	)	PUNCT
ejpam-3344	348	10	=	=	SYM
ejpam-3344	348	11	γa((yx)(ax	γa((yx)(ax	PROPN
ejpam-3344	348	12	)	)	PUNCT
ejpam-3344	348	13	)	)	PUNCT
ejpam-3344	348	14	≤	≤	NUM
ejpam-3344	348	15	γa(yx	γa(yx	PROPN
ejpam-3344	348	16	)	)	PUNCT
ejpam-3344	348	17	≤	≤	NOUN
ejpam-3344	348	18	γa(y	γa(y	NUM
ejpam-3344	348	19	)	)	PUNCT
ejpam-3344	348	20	.	.	PUNCT
ejpam-3344	349	1	hence	hence	ADV
ejpam-3344	349	2	a	a	PRON
ejpam-3344	349	3	is	be	AUX
ejpam-3344	349	4	an	an	DET
ejpam-3344	349	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	349	6	fuzzy	fuzzy	ADJ
ejpam-3344	349	7	ideal	ideal	NOUN
ejpam-3344	349	8	of	of	ADP
ejpam-3344	349	9	r.	r.	PROPN
ejpam-3344	349	10	similarly	similarly	ADV
ejpam-3344	349	11	,	,	PUNCT
ejpam-3344	349	12	for	for	ADP
ejpam-3344	349	13	left	left	ADJ
ejpam-3344	349	14	ideal	ideal	NOUN
ejpam-3344	349	15	.	.	PUNCT
ejpam-3344	350	1	n.	n.	PROPN
ejpam-3344	350	2	kausar	kausar	PROPN
ejpam-3344	350	3	,	,	PUNCT
ejpam-3344	350	4	m.	m.	NOUN
ejpam-3344	350	5	a.	a.	PROPN
ejpam-3344	350	6	waqar	waqar	PROPN
ejpam-3344	350	7	/	/	SYM
ejpam-3344	350	8	eur	eur	PROPN
ejpam-3344	350	9	.	.	PUNCT
ejpam-3344	351	1	j.	j.	PROPN
ejpam-3344	351	2	pure	pure	PROPN
ejpam-3344	351	3	appl	appl	PROPN
ejpam-3344	351	4	.	.	PROPN
ejpam-3344	351	5	math	math	PROPN
ejpam-3344	351	6	,	,	PUNCT
ejpam-3344	351	7	12	12	NUM
ejpam-3344	351	8	(	(	PUNCT
ejpam-3344	351	9	1	1	NUM
ejpam-3344	351	10	)	)	PUNCT
ejpam-3344	351	11	(	(	PUNCT
ejpam-3344	351	12	2019	2019	NUM
ejpam-3344	351	13	)	)	PUNCT
ejpam-3344	351	14	,	,	PUNCT
ejpam-3344	351	15	226	226	NUM
ejpam-3344	351	16	-	-	SYM
ejpam-3344	351	17	250	250	NUM
ejpam-3344	351	18	239	239	NUM
ejpam-3344	351	19	remark	remark	NOUN
ejpam-3344	351	20	6	6	NUM
ejpam-3344	351	21	.	.	PUNCT
ejpam-3344	352	1	the	the	DET
ejpam-3344	352	2	concept	concept	NOUN
ejpam-3344	352	3	of	of	ADP
ejpam-3344	352	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	352	5	fuzzy	fuzzy	ADJ
ejpam-3344	352	6	(	(	PUNCT
ejpam-3344	352	7	left	left	ADJ
ejpam-3344	352	8	,	,	PUNCT
ejpam-3344	352	9	right	right	INTJ
ejpam-3344	352	10	,	,	PUNCT
ejpam-3344	352	11	two	two	NUM
ejpam-3344	352	12	-	-	PUNCT
ejpam-3344	352	13	sided	sided	ADJ
ejpam-3344	352	14	)	)	PUNCT
ejpam-3344	352	15	ideals	ideal	NOUN
ejpam-3344	352	16	coincides	coincide	VERB
ejpam-3344	352	17	in	in	ADP
ejpam-3344	352	18	right	right	ADJ
ejpam-3344	352	19	regular	regular	ADJ
ejpam-3344	352	20	la	la	PROPN
ejpam-3344	352	21	-	-	PUNCT
ejpam-3344	352	22	rings	ring	NOUN
ejpam-3344	352	23	.	.	PUNCT
ejpam-3344	353	1	proposition	proposition	NOUN
ejpam-3344	353	2	7	7	NUM
ejpam-3344	353	3	.	.	PUNCT
ejpam-3344	354	1	let	let	VERB
ejpam-3344	354	2	r	r	PRON
ejpam-3344	354	3	be	be	AUX
ejpam-3344	354	4	a	a	DET
ejpam-3344	354	5	right	right	ADJ
ejpam-3344	354	6	regular	regular	ADJ
ejpam-3344	354	7	la	la	NOUN
ejpam-3344	354	8	-	-	NOUN
ejpam-3344	354	9	ring	ring	NOUN
ejpam-3344	354	10	with	with	ADP
ejpam-3344	354	11	left	left	ADJ
ejpam-3344	354	12	identity	identity	NOUN
ejpam-3344	354	13	e.	e.	PROPN
ejpam-3344	354	14	then	then	ADV
ejpam-3344	354	15	every	every	DET
ejpam-3344	354	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	354	17	fuzzy	fuzzy	ADJ
ejpam-3344	354	18	generalized	generalize	VERB
ejpam-3344	354	19	bi	bi	NOUN
ejpam-3344	354	20	-	-	NOUN
ejpam-3344	354	21	ideal	ideal	NOUN
ejpam-3344	354	22	of	of	ADP
ejpam-3344	354	23	r	r	NOUN
ejpam-3344	354	24	is	be	AUX
ejpam-3344	354	25	an	an	DET
ejpam-3344	354	26	intuitionistic	intuitionistic	ADJ
ejpam-3344	354	27	fuzzy	fuzzy	ADJ
ejpam-3344	354	28	bi	bi	NOUN
ejpam-3344	354	29	-	-	NOUN
ejpam-3344	354	30	ideal	ideal	NOUN
ejpam-3344	354	31	of	of	ADP
ejpam-3344	354	32	r.	r.	PROPN
ejpam-3344	354	33	proof	proof	NOUN
ejpam-3344	354	34	.	.	PUNCT
ejpam-3344	355	1	suppose	suppose	VERB
ejpam-3344	355	2	that	that	SCONJ
ejpam-3344	355	3	a	a	DET
ejpam-3344	355	4	=	=	SYM
ejpam-3344	355	5	(	(	PUNCT
ejpam-3344	355	6	µa	µa	PROPN
ejpam-3344	355	7	,	,	PUNCT
ejpam-3344	355	8	γa	γa	PROPN
ejpam-3344	355	9	)	)	PUNCT
ejpam-3344	355	10	is	be	AUX
ejpam-3344	355	11	an	an	DET
ejpam-3344	355	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	355	13	fuzzy	fuzzy	ADJ
ejpam-3344	355	14	generalized	generalize	VERB
ejpam-3344	355	15	bi	bi	NOUN
ejpam-3344	355	16	-	-	NOUN
ejpam-3344	355	17	ideal	ideal	NOUN
ejpam-3344	355	18	of	of	ADP
ejpam-3344	355	19	r	r	NOUN
ejpam-3344	355	20	and	and	CCONJ
ejpam-3344	355	21	x	x	NOUN
ejpam-3344	355	22	,	,	PUNCT
ejpam-3344	355	23	y	y	PROPN
ejpam-3344	355	24	∈	∈	PROPN
ejpam-3344	355	25	r	r	NOUN
ejpam-3344	355	26	,	,	PUNCT
ejpam-3344	355	27	this	this	PRON
ejpam-3344	355	28	means	mean	VERB
ejpam-3344	355	29	that	that	SCONJ
ejpam-3344	355	30	there	there	PRON
ejpam-3344	355	31	exists	exist	VERB
ejpam-3344	355	32	a	a	DET
ejpam-3344	355	33	∈	∈	NOUN
ejpam-3344	355	34	r	r	NOUN
ejpam-3344	355	35	such	such	ADJ
ejpam-3344	355	36	that	that	PRON
ejpam-3344	355	37	x	x	X
ejpam-3344	355	38	=	=	SYM
ejpam-3344	355	39	x2a	x2a	PROPN
ejpam-3344	355	40	.	.	PUNCT
ejpam-3344	356	1	we	we	PRON
ejpam-3344	356	2	have	have	VERB
ejpam-3344	356	3	to	to	PART
ejpam-3344	356	4	show	show	VERB
ejpam-3344	356	5	that	that	SCONJ
ejpam-3344	356	6	a	a	PRON
ejpam-3344	356	7	is	be	AUX
ejpam-3344	356	8	an	an	DET
ejpam-3344	356	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	356	10	fuzzy	fuzzy	ADJ
ejpam-3344	356	11	la	la	NOUN
ejpam-3344	356	12	-	-	PUNCT
ejpam-3344	356	13	subring	subring	NOUN
ejpam-3344	356	14	of	of	ADP
ejpam-3344	356	15	r.	r.	PROPN
ejpam-3344	356	16	thus	thus	ADV
ejpam-3344	356	17	µa(xy	µa(xy	NOUN
ejpam-3344	356	18	)	)	PUNCT
ejpam-3344	356	19	=	=	SYM
ejpam-3344	356	20	µa((x2a)y	µa((x2a)y	PROPN
ejpam-3344	356	21	)	)	PUNCT
ejpam-3344	356	22	=	=	SYM
ejpam-3344	357	1	µa(((xx)(ea))y	µa(((xx)(ea))y	PROPN
ejpam-3344	357	2	)	)	PUNCT
ejpam-3344	357	3	=	=	SYM
ejpam-3344	357	4	µa(((ae)(xx))y	µa(((ae)(xx))y	PROPN
ejpam-3344	357	5	)	)	PUNCT
ejpam-3344	357	6	=	=	PUNCT
ejpam-3344	357	7	µa((x((ae)x))y	µa((x((ae)x))y	NUM
ejpam-3344	357	8	)	)	PUNCT
ejpam-3344	357	9	≥	≥	PROPN
ejpam-3344	357	10	min{µa(x	min{µa(x	NOUN
ejpam-3344	357	11	)	)	PUNCT
ejpam-3344	357	12	,	,	PUNCT
ejpam-3344	357	13	µa(y	µa(y	NOUN
ejpam-3344	357	14	)	)	PUNCT
ejpam-3344	357	15	}	}	PUNCT
ejpam-3344	357	16	and	and	CCONJ
ejpam-3344	357	17	γa(xy	γa(xy	PROPN
ejpam-3344	357	18	)	)	PUNCT
ejpam-3344	357	19	=	=	SYM
ejpam-3344	358	1	γa((x2a)y	γa((x2a)y	PROPN
ejpam-3344	358	2	)	)	PUNCT
ejpam-3344	358	3	=	=	SYM
ejpam-3344	358	4	γa(((xx)(ea))y	γa(((xx)(ea))y	PROPN
ejpam-3344	358	5	)	)	PUNCT
ejpam-3344	358	6	=	=	PUNCT
ejpam-3344	358	7	γa(((ae)(xx))y	γa(((ae)(xx))y	VERB
ejpam-3344	358	8	)	)	PUNCT
ejpam-3344	358	9	=	=	PUNCT
ejpam-3344	358	10	γa((x((ae)x))y	γa((x((ae)x))y	NUM
ejpam-3344	358	11	)	)	PUNCT
ejpam-3344	358	12	≤	≤	NUM
ejpam-3344	358	13	max{γa(x	max{γa(x	NOUN
ejpam-3344	358	14	)	)	PUNCT
ejpam-3344	358	15	,	,	PUNCT
ejpam-3344	358	16	γa(y	γa(y	NOUN
ejpam-3344	358	17	)	)	PUNCT
ejpam-3344	358	18	}	}	PUNCT
ejpam-3344	358	19	.	.	PUNCT
ejpam-3344	359	1	therefore	therefore	ADV
ejpam-3344	359	2	a	a	PRON
ejpam-3344	359	3	is	be	AUX
ejpam-3344	359	4	an	an	DET
ejpam-3344	359	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	359	6	fuzzy	fuzzy	ADJ
ejpam-3344	359	7	la	la	NOUN
ejpam-3344	359	8	-	-	PUNCT
ejpam-3344	359	9	subring	subring	NOUN
ejpam-3344	359	10	of	of	ADP
ejpam-3344	359	11	r.	r.	PROPN
ejpam-3344	359	12	lemma	lemma	PROPN
ejpam-3344	359	13	13	13	NUM
ejpam-3344	359	14	.	.	PUNCT
ejpam-3344	360	1	let	let	VERB
ejpam-3344	360	2	r	r	PRON
ejpam-3344	360	3	be	be	AUX
ejpam-3344	360	4	a	a	DET
ejpam-3344	360	5	left	left	ADJ
ejpam-3344	360	6	regular	regular	ADJ
ejpam-3344	360	7	la	la	ADJ
ejpam-3344	360	8	-	-	NOUN
ejpam-3344	360	9	ring	ring	NOUN
ejpam-3344	360	10	with	with	ADP
ejpam-3344	360	11	left	left	ADJ
ejpam-3344	360	12	identity	identity	NOUN
ejpam-3344	360	13	e.	e.	PROPN
ejpam-3344	360	14	then	then	ADV
ejpam-3344	360	15	every	every	DET
ejpam-3344	360	16	intuitionistic	intuitionistic	ADJ
ejpam-3344	360	17	fuzzy	fuzzy	ADJ
ejpam-3344	360	18	left	left	ADJ
ejpam-3344	360	19	(	(	PUNCT
ejpam-3344	360	20	right	right	ADJ
ejpam-3344	360	21	)	)	PUNCT
ejpam-3344	360	22	ideal	ideal	NOUN
ejpam-3344	360	23	of	of	ADP
ejpam-3344	360	24	r	r	NOUN
ejpam-3344	360	25	is	be	AUX
ejpam-3344	360	26	an	an	DET
ejpam-3344	360	27	intuitionistic	intuitionistic	ADJ
ejpam-3344	360	28	fuzzy	fuzzy	ADJ
ejpam-3344	360	29	ideal	ideal	NOUN
ejpam-3344	360	30	of	of	ADP
ejpam-3344	360	31	r.	r.	PROPN
ejpam-3344	360	32	proof	proof	PROPN
ejpam-3344	360	33	.	.	PUNCT
ejpam-3344	361	1	assume	assume	VERB
ejpam-3344	361	2	that	that	SCONJ
ejpam-3344	361	3	a	a	DET
ejpam-3344	361	4	=	=	SYM
ejpam-3344	361	5	(	(	PUNCT
ejpam-3344	361	6	µa	µa	PROPN
ejpam-3344	361	7	,	,	PUNCT
ejpam-3344	361	8	γa	γa	PROPN
ejpam-3344	361	9	)	)	PUNCT
ejpam-3344	361	10	is	be	AUX
ejpam-3344	361	11	an	an	DET
ejpam-3344	361	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	361	13	fuzzy	fuzzy	ADJ
ejpam-3344	361	14	right	right	ADJ
ejpam-3344	361	15	ideal	ideal	NOUN
ejpam-3344	361	16	of	of	ADP
ejpam-3344	361	17	r	r	NOUN
ejpam-3344	361	18	and	and	CCONJ
ejpam-3344	361	19	x	x	NOUN
ejpam-3344	361	20	,	,	PUNCT
ejpam-3344	361	21	y	y	PROPN
ejpam-3344	361	22	∈	∈	PROPN
ejpam-3344	361	23	r	r	NOUN
ejpam-3344	361	24	,	,	PUNCT
ejpam-3344	361	25	then	then	ADV
ejpam-3344	361	26	there	there	PRON
ejpam-3344	361	27	exists	exist	VERB
ejpam-3344	361	28	an	an	DET
ejpam-3344	361	29	element	element	NOUN
ejpam-3344	361	30	a	a	DET
ejpam-3344	361	31	∈	∈	NOUN
ejpam-3344	361	32	r	r	NOUN
ejpam-3344	361	33	such	such	ADJ
ejpam-3344	361	34	that	that	SCONJ
ejpam-3344	361	35	x	x	X
ejpam-3344	361	36	=	=	SYM
ejpam-3344	361	37	ax2	ax2	NOUN
ejpam-3344	361	38	.	.	PUNCT
ejpam-3344	362	1	thus	thus	ADV
ejpam-3344	362	2	µa(xy	µa(xy	NUM
ejpam-3344	362	3	)	)	PUNCT
ejpam-3344	362	4	=	=	SYM
ejpam-3344	363	1	µa((ax2)y	µa((ax2)y	X
ejpam-3344	363	2	)	)	PUNCT
ejpam-3344	363	3	=	=	SYM
ejpam-3344	363	4	µa((a(xx))y	µa((a(xx))y	NOUN
ejpam-3344	363	5	)	)	PUNCT
ejpam-3344	363	6	=	=	SYM
ejpam-3344	363	7	µa((x(ax))y	µa((x(ax))y	NOUN
ejpam-3344	363	8	)	)	PUNCT
ejpam-3344	363	9	=	=	SYM
ejpam-3344	363	10	µa((y(ax))x	µa((y(ax))x	NOUN
ejpam-3344	363	11	)	)	PUNCT
ejpam-3344	363	12	≥	≥	PROPN
ejpam-3344	363	13	µa(y(ax	µa(y(ax	NOUN
ejpam-3344	363	14	)	)	PUNCT
ejpam-3344	363	15	)	)	PUNCT
ejpam-3344	363	16	≥	≥	NOUN
ejpam-3344	363	17	µa(y	µa(y	NOUN
ejpam-3344	363	18	)	)	PUNCT
ejpam-3344	363	19	and	and	CCONJ
ejpam-3344	363	20	γa(xy	γa(xy	PROPN
ejpam-3344	363	21	)	)	PUNCT
ejpam-3344	364	1	=	=	SYM
ejpam-3344	364	2	γa((ax2)y	γa((ax2)y	PROPN
ejpam-3344	364	3	)	)	PUNCT
ejpam-3344	364	4	=	=	SYM
ejpam-3344	364	5	γa((a(xx))y	γa((a(xx))y	ADJ
ejpam-3344	364	6	)	)	PUNCT
ejpam-3344	364	7	=	=	SYM
ejpam-3344	364	8	γa((x(ax))y	γa((x(ax))y	ADJ
ejpam-3344	364	9	)	)	PUNCT
ejpam-3344	364	10	=	=	SYM
ejpam-3344	364	11	γa((y(ax))x	γa((y(ax))x	ADJ
ejpam-3344	364	12	)	)	PUNCT
ejpam-3344	364	13	≤	≤	PROPN
ejpam-3344	364	14	γa(y(ax	γa(y(ax	PROPN
ejpam-3344	364	15	)	)	PUNCT
ejpam-3344	364	16	)	)	PUNCT
ejpam-3344	364	17	≤	≤	NUM
ejpam-3344	364	18	γa(y	γa(y	NUM
ejpam-3344	364	19	)	)	PUNCT
ejpam-3344	364	20	.	.	PUNCT
ejpam-3344	365	1	so	so	ADV
ejpam-3344	365	2	a	a	PRON
ejpam-3344	365	3	is	be	AUX
ejpam-3344	365	4	an	an	DET
ejpam-3344	365	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	365	6	fuzzy	fuzzy	ADJ
ejpam-3344	365	7	ideal	ideal	NOUN
ejpam-3344	365	8	of	of	ADP
ejpam-3344	365	9	r.	r.	PROPN
ejpam-3344	365	10	similarly	similarly	ADV
ejpam-3344	365	11	,	,	PUNCT
ejpam-3344	365	12	for	for	ADP
ejpam-3344	365	13	left	left	ADJ
ejpam-3344	365	14	ideal	ideal	NOUN
ejpam-3344	365	15	.	.	PUNCT
ejpam-3344	366	1	remark	remark	PROPN
ejpam-3344	366	2	7	7	NUM
ejpam-3344	366	3	.	.	PUNCT
ejpam-3344	367	1	the	the	DET
ejpam-3344	367	2	concept	concept	NOUN
ejpam-3344	367	3	of	of	ADP
ejpam-3344	367	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	367	5	fuzzy	fuzzy	ADJ
ejpam-3344	367	6	(	(	PUNCT
ejpam-3344	367	7	left	left	ADJ
ejpam-3344	367	8	,	,	PUNCT
ejpam-3344	367	9	right	right	INTJ
ejpam-3344	367	10	,	,	PUNCT
ejpam-3344	367	11	two	two	NUM
ejpam-3344	367	12	-	-	PUNCT
ejpam-3344	367	13	sided	sided	ADJ
ejpam-3344	367	14	)	)	PUNCT
ejpam-3344	367	15	ideals	ideal	NOUN
ejpam-3344	367	16	coincides	coincide	VERB
ejpam-3344	367	17	in	in	ADP
ejpam-3344	367	18	left	left	ADJ
ejpam-3344	367	19	regular	regular	ADJ
ejpam-3344	367	20	la	la	NOUN
ejpam-3344	367	21	-	-	PUNCT
ejpam-3344	367	22	rings	ring	NOUN
ejpam-3344	367	23	with	with	ADP
ejpam-3344	367	24	left	left	ADJ
ejpam-3344	367	25	identity	identity	NOUN
ejpam-3344	367	26	.	.	PUNCT
ejpam-3344	368	1	proposition	proposition	NOUN
ejpam-3344	368	2	8	8	NUM
ejpam-3344	368	3	.	.	PUNCT
ejpam-3344	369	1	every	every	DET
ejpam-3344	369	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	369	3	fuzzy	fuzzy	ADJ
ejpam-3344	369	4	generalized	generalize	VERB
ejpam-3344	369	5	bi	bi	NOUN
ejpam-3344	369	6	-	-	NOUN
ejpam-3344	369	7	ideal	ideal	NOUN
ejpam-3344	369	8	of	of	ADP
ejpam-3344	369	9	a	a	DET
ejpam-3344	369	10	left	left	ADJ
ejpam-3344	369	11	regular	regular	ADJ
ejpam-3344	369	12	la	la	ADJ
ejpam-3344	369	13	-	-	PUNCT
ejpam-3344	369	14	ring	ring	NOUN
ejpam-3344	369	15	r	r	NOUN
ejpam-3344	369	16	with	with	ADP
ejpam-3344	369	17	left	left	ADJ
ejpam-3344	369	18	identity	identity	NOUN
ejpam-3344	369	19	e	e	NOUN
ejpam-3344	369	20	,	,	PUNCT
ejpam-3344	369	21	is	be	AUX
ejpam-3344	369	22	an	an	DET
ejpam-3344	369	23	intuitionistic	intuitionistic	ADJ
ejpam-3344	369	24	fuzzy	fuzzy	ADJ
ejpam-3344	369	25	bi	bi	NOUN
ejpam-3344	369	26	-	-	NOUN
ejpam-3344	369	27	ideal	ideal	NOUN
ejpam-3344	369	28	of	of	ADP
ejpam-3344	369	29	r.	r.	PROPN
ejpam-3344	369	30	proof	proof	NOUN
ejpam-3344	369	31	.	.	PUNCT
ejpam-3344	370	1	let	let	VERB
ejpam-3344	370	2	a	a	PRON
ejpam-3344	370	3	be	be	AUX
ejpam-3344	370	4	an	an	DET
ejpam-3344	370	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	370	6	fuzzy	fuzzy	ADJ
ejpam-3344	370	7	generalized	generalize	VERB
ejpam-3344	370	8	bi	bi	NOUN
ejpam-3344	370	9	-	-	NOUN
ejpam-3344	370	10	ideal	ideal	NOUN
ejpam-3344	370	11	of	of	ADP
ejpam-3344	370	12	r	r	NOUN
ejpam-3344	370	13	and	and	CCONJ
ejpam-3344	370	14	x	x	NOUN
ejpam-3344	370	15	,	,	PUNCT
ejpam-3344	371	1	y	y	PROPN
ejpam-3344	371	2	∈	∈	PROPN
ejpam-3344	371	3	r	r	NOUN
ejpam-3344	371	4	,	,	PUNCT
ejpam-3344	371	5	this	this	PRON
ejpam-3344	371	6	implies	imply	VERB
ejpam-3344	371	7	that	that	SCONJ
ejpam-3344	371	8	there	there	PRON
ejpam-3344	371	9	exists	exist	VERB
ejpam-3344	371	10	a	a	DET
ejpam-3344	371	11	∈	∈	NOUN
ejpam-3344	371	12	r	r	NOUN
ejpam-3344	371	13	such	such	ADJ
ejpam-3344	371	14	that	that	SCONJ
ejpam-3344	371	15	x	x	X
ejpam-3344	371	16	=	=	SYM
ejpam-3344	371	17	ax2	ax2	NOUN
ejpam-3344	371	18	.	.	PUNCT
ejpam-3344	372	1	we	we	PRON
ejpam-3344	372	2	have	have	VERB
ejpam-3344	372	3	to	to	PART
ejpam-3344	372	4	show	show	VERB
ejpam-3344	372	5	that	that	SCONJ
ejpam-3344	372	6	a	a	PRON
ejpam-3344	372	7	is	be	AUX
ejpam-3344	372	8	an	an	DET
ejpam-3344	372	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	372	10	fuzzy	fuzzy	ADJ
ejpam-3344	372	11	la	la	NOUN
ejpam-3344	372	12	-	-	PUNCT
ejpam-3344	372	13	subring	subring	NOUN
ejpam-3344	372	14	of	of	ADP
ejpam-3344	372	15	r.	r.	PROPN
ejpam-3344	372	16	thus	thus	ADV
ejpam-3344	372	17	µa(xy	µa(xy	PROPN
ejpam-3344	372	18	)	)	PUNCT
ejpam-3344	372	19	=	=	SYM
ejpam-3344	372	20	µa((ax2)y	µa((ax2)y	X
ejpam-3344	372	21	)	)	PUNCT
ejpam-3344	372	22	=	=	SYM
ejpam-3344	372	23	µa((a(xx))y	µa((a(xx))y	NOUN
ejpam-3344	372	24	)	)	PUNCT
ejpam-3344	373	1	=	=	SYM
ejpam-3344	373	2	µa((x(ax))y	µa((x(ax))y	PROPN
ejpam-3344	373	3	)	)	PUNCT
ejpam-3344	373	4	≥	≥	PROPN
ejpam-3344	373	5	min{µa(x	min{µa(x	NOUN
ejpam-3344	373	6	)	)	PUNCT
ejpam-3344	373	7	,	,	PUNCT
ejpam-3344	373	8	µa(y	µa(y	NOUN
ejpam-3344	373	9	)	)	PUNCT
ejpam-3344	373	10	}	}	PUNCT
ejpam-3344	373	11	and	and	CCONJ
ejpam-3344	373	12	γa(xy	γa(xy	PROPN
ejpam-3344	373	13	)	)	PUNCT
ejpam-3344	374	1	=	=	SYM
ejpam-3344	374	2	γa((ax2)y	γa((ax2)y	PROPN
ejpam-3344	374	3	)	)	PUNCT
ejpam-3344	374	4	=	=	SYM
ejpam-3344	374	5	γa((a(xx))y	γa((a(xx))y	ADJ
ejpam-3344	374	6	)	)	PUNCT
ejpam-3344	374	7	=	=	SYM
ejpam-3344	374	8	γa((x(ax))y	γa((x(ax))y	ADJ
ejpam-3344	374	9	)	)	PUNCT
ejpam-3344	374	10	≤	≤	NUM
ejpam-3344	374	11	max{γa(x	max{γa(x	NOUN
ejpam-3344	374	12	)	)	PUNCT
ejpam-3344	374	13	,	,	PUNCT
ejpam-3344	374	14	γa(y	γa(y	NOUN
ejpam-3344	374	15	)	)	PUNCT
ejpam-3344	374	16	}	}	PUNCT
ejpam-3344	374	17	.	.	PUNCT
ejpam-3344	375	1	hence	hence	ADV
ejpam-3344	375	2	a	a	PRON
ejpam-3344	375	3	is	be	AUX
ejpam-3344	375	4	an	an	DET
ejpam-3344	375	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	375	6	fuzzy	fuzzy	ADJ
ejpam-3344	375	7	la	la	NOUN
ejpam-3344	375	8	-	-	PUNCT
ejpam-3344	375	9	subring	subring	NOUN
ejpam-3344	375	10	of	of	ADP
ejpam-3344	375	11	r.	r.	PROPN
ejpam-3344	375	12	n.	n.	PROPN
ejpam-3344	375	13	kausar	kausar	PROPN
ejpam-3344	375	14	,	,	PUNCT
ejpam-3344	375	15	m.	m.	NOUN
ejpam-3344	375	16	a.	a.	PROPN
ejpam-3344	375	17	waqar	waqar	PROPN
ejpam-3344	375	18	/	/	SYM
ejpam-3344	375	19	eur	eur	PROPN
ejpam-3344	375	20	.	.	PUNCT
ejpam-3344	376	1	j.	j.	PROPN
ejpam-3344	376	2	pure	pure	PROPN
ejpam-3344	376	3	appl	appl	PROPN
ejpam-3344	376	4	.	.	PROPN
ejpam-3344	376	5	math	math	PROPN
ejpam-3344	376	6	,	,	PUNCT
ejpam-3344	376	7	12	12	NUM
ejpam-3344	376	8	(	(	PUNCT
ejpam-3344	376	9	1	1	NUM
ejpam-3344	376	10	)	)	PUNCT
ejpam-3344	376	11	(	(	PUNCT
ejpam-3344	376	12	2019	2019	NUM
ejpam-3344	376	13	)	)	PUNCT
ejpam-3344	376	14	,	,	PUNCT
ejpam-3344	376	15	226	226	NUM
ejpam-3344	376	16	-	-	SYM
ejpam-3344	376	17	250	250	NUM
ejpam-3344	376	18	240	240	NUM
ejpam-3344	376	19	theorem	theorem	NOUN
ejpam-3344	376	20	8	8	NUM
ejpam-3344	376	21	.	.	PUNCT
ejpam-3344	377	1	let	let	VERB
ejpam-3344	377	2	r	r	PRON
ejpam-3344	377	3	be	be	AUX
ejpam-3344	377	4	a	a	DET
ejpam-3344	377	5	regular	regular	ADJ
ejpam-3344	377	6	and	and	CCONJ
ejpam-3344	377	7	right	right	ADV
ejpam-3344	377	8	regular	regular	ADJ
ejpam-3344	377	9	locally	locally	ADV
ejpam-3344	377	10	associative	associative	ADJ
ejpam-3344	377	11	la	la	ADJ
ejpam-3344	377	12	-	-	PUNCT
ejpam-3344	377	13	ring	ring	NOUN
ejpam-3344	377	14	.	.	PUNCT
ejpam-3344	378	1	then	then	ADV
ejpam-3344	378	2	for	for	ADP
ejpam-3344	378	3	every	every	DET
ejpam-3344	378	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	378	5	fuzzy	fuzzy	ADJ
ejpam-3344	378	6	right	right	ADJ
ejpam-3344	378	7	ideal	ideal	NOUN
ejpam-3344	378	8	a	a	PRON
ejpam-3344	378	9	=	=	X
ejpam-3344	378	10	(	(	PUNCT
ejpam-3344	378	11	µa	µa	PROPN
ejpam-3344	378	12	,	,	PUNCT
ejpam-3344	378	13	γa	γa	PROPN
ejpam-3344	378	14	)	)	PUNCT
ejpam-3344	378	15	of	of	ADP
ejpam-3344	378	16	r	r	NOUN
ejpam-3344	378	17	,	,	PUNCT
ejpam-3344	378	18	a(an	a(an	PROPN
ejpam-3344	378	19	)	)	PUNCT
ejpam-3344	378	20	=	=	SYM
ejpam-3344	378	21	a(a3n	a(a3n	PROPN
ejpam-3344	378	22	)	)	PUNCT
ejpam-3344	378	23	for	for	ADP
ejpam-3344	378	24	all	all	DET
ejpam-3344	378	25	a	a	DET
ejpam-3344	378	26	∈	∈	NOUN
ejpam-3344	378	27	r	r	NOUN
ejpam-3344	378	28	,	,	PUNCT
ejpam-3344	378	29	where	where	SCONJ
ejpam-3344	378	30	n	n	PRON
ejpam-3344	378	31	is	be	AUX
ejpam-3344	378	32	any	any	DET
ejpam-3344	378	33	positive	positive	ADJ
ejpam-3344	378	34	integer	integer	NOUN
ejpam-3344	378	35	.	.	PUNCT
ejpam-3344	379	1	proof	proof	NOUN
ejpam-3344	379	2	.	.	PUNCT
ejpam-3344	380	1	for	for	ADP
ejpam-3344	380	2	n	n	NOUN
ejpam-3344	380	3	=	=	SYM
ejpam-3344	380	4	1	1	X
ejpam-3344	380	5	.	.	PUNCT
ejpam-3344	380	6	let	let	VERB
ejpam-3344	380	7	a	a	DET
ejpam-3344	380	8	∈	∈	ADJ
ejpam-3344	380	9	r	r	NOUN
ejpam-3344	380	10	,	,	PUNCT
ejpam-3344	380	11	this	this	PRON
ejpam-3344	380	12	means	mean	VERB
ejpam-3344	380	13	that	that	SCONJ
ejpam-3344	380	14	there	there	PRON
ejpam-3344	380	15	exists	exist	VERB
ejpam-3344	380	16	an	an	DET
ejpam-3344	380	17	element	element	NOUN
ejpam-3344	380	18	x	x	SYM
ejpam-3344	380	19	∈	∈	NOUN
ejpam-3344	380	20	r	r	NOUN
ejpam-3344	380	21	such	such	DET
ejpam-3344	380	22	that	that	SCONJ
ejpam-3344	380	23	a	a	DET
ejpam-3344	380	24	=	=	X
ejpam-3344	380	25	(	(	PUNCT
ejpam-3344	380	26	ax)a	ax)a	PROPN
ejpam-3344	380	27	and	and	CCONJ
ejpam-3344	380	28	a	a	DET
ejpam-3344	380	29	=	=	PUNCT
ejpam-3344	380	30	a2x	a2x	PROPN
ejpam-3344	380	31	.	.	PUNCT
ejpam-3344	381	1	now	now	ADV
ejpam-3344	381	2	a	a	DET
ejpam-3344	381	3	=	=	X
ejpam-3344	381	4	(	(	PUNCT
ejpam-3344	381	5	ax)a	ax)a	PROPN
ejpam-3344	381	6	=	=	SYM
ejpam-3344	381	7	(	(	PUNCT
ejpam-3344	381	8	ax)(a2x	ax)(a2x	NOUN
ejpam-3344	381	9	)	)	PUNCT
ejpam-3344	381	10	=	=	PUNCT
ejpam-3344	381	11	a3x2	a3x2	X
ejpam-3344	381	12	.	.	PUNCT
ejpam-3344	381	13	thus	thus	ADV
ejpam-3344	381	14	µa(a	µa(a	PUNCT
ejpam-3344	381	15	)	)	PUNCT
ejpam-3344	381	16	=	=	SYM
ejpam-3344	381	17	µa(a3x2	µa(a3x2	PROPN
ejpam-3344	381	18	)	)	PUNCT
ejpam-3344	381	19	≥	≥	NOUN
ejpam-3344	381	20	µa(a3	µa(a3	NOUN
ejpam-3344	381	21	)	)	PUNCT
ejpam-3344	381	22	=	=	PUNCT
ejpam-3344	381	23	µa(aa2	µa(aa2	NUM
ejpam-3344	381	24	)	)	PUNCT
ejpam-3344	381	25	≥	≥	NOUN
ejpam-3344	381	26	min{µa	min{µa	X
ejpam-3344	381	27	(	(	PUNCT
ejpam-3344	381	28	a	a	NOUN
ejpam-3344	381	29	)	)	PUNCT
ejpam-3344	381	30	,	,	PUNCT
ejpam-3344	381	31	µa	µa	X
ejpam-3344	381	32	(	(	PUNCT
ejpam-3344	381	33	a2	a2	PROPN
ejpam-3344	381	34	)	)	PUNCT
ejpam-3344	381	35	}	}	PUNCT
ejpam-3344	381	36	≥	≥	VERB
ejpam-3344	381	37	min{µa	min{µa	X
ejpam-3344	381	38	(	(	PUNCT
ejpam-3344	381	39	a	a	NOUN
ejpam-3344	381	40	)	)	PUNCT
ejpam-3344	381	41	,	,	PUNCT
ejpam-3344	381	42	µa	µa	X
ejpam-3344	381	43	(	(	PUNCT
ejpam-3344	381	44	a	a	NOUN
ejpam-3344	381	45	)	)	PUNCT
ejpam-3344	381	46	,	,	PUNCT
ejpam-3344	381	47	µa	µa	X
ejpam-3344	381	48	(	(	PUNCT
ejpam-3344	381	49	a	a	NOUN
ejpam-3344	381	50	)	)	PUNCT
ejpam-3344	381	51	}	}	PUNCT
ejpam-3344	381	52	=	=	SYM
ejpam-3344	381	53	µa	µa	NOUN
ejpam-3344	381	54	(	(	PUNCT
ejpam-3344	381	55	a	a	NOUN
ejpam-3344	381	56	)	)	PUNCT
ejpam-3344	381	57	.	.	PUNCT
ejpam-3344	382	1	similarly	similarly	ADV
ejpam-3344	382	2	,	,	PUNCT
ejpam-3344	382	3	γa	γa	PROPN
ejpam-3344	382	4	(	(	PUNCT
ejpam-3344	382	5	a	a	X
ejpam-3344	382	6	)	)	PUNCT
ejpam-3344	382	7	=	=	SYM
ejpam-3344	382	8	γa	γa	PROPN
ejpam-3344	382	9	(	(	PUNCT
ejpam-3344	382	10	a3	a3	PROPN
ejpam-3344	382	11	)	)	PUNCT
ejpam-3344	382	12	,	,	PUNCT
ejpam-3344	382	13	so	so	ADV
ejpam-3344	382	14	a	a	DET
ejpam-3344	382	15	(	(	PUNCT
ejpam-3344	382	16	a	a	NOUN
ejpam-3344	382	17	)	)	PUNCT
ejpam-3344	382	18	=	=	SYM
ejpam-3344	382	19	a	a	PRON
ejpam-3344	382	20	(	(	PUNCT
ejpam-3344	382	21	a3	a3	NOUN
ejpam-3344	382	22	)	)	PUNCT
ejpam-3344	382	23	.	.	PUNCT
ejpam-3344	383	1	here	here	ADV
ejpam-3344	383	2	a2	a2	PROPN
ejpam-3344	383	3	=	=	SYM
ejpam-3344	383	4	aa	aa	PROPN
ejpam-3344	383	5	=	=	PUNCT
ejpam-3344	383	6	(	(	PUNCT
ejpam-3344	383	7	a3x2)(a3x2	a3x2)(a3x2	NOUN
ejpam-3344	383	8	)	)	PUNCT
ejpam-3344	383	9	=	=	SYM
ejpam-3344	384	1	a6x4	a6x4	PROPN
ejpam-3344	384	2	,	,	PUNCT
ejpam-3344	384	3	then	then	ADV
ejpam-3344	384	4	the	the	DET
ejpam-3344	384	5	result	result	NOUN
ejpam-3344	384	6	is	be	AUX
ejpam-3344	384	7	true	true	ADJ
ejpam-3344	384	8	for	for	ADP
ejpam-3344	384	9	n	n	NOUN
ejpam-3344	384	10	=	=	SYM
ejpam-3344	384	11	2	2	X
ejpam-3344	384	12	.	.	X
ejpam-3344	384	13	assume	assume	VERB
ejpam-3344	384	14	that	that	SCONJ
ejpam-3344	384	15	the	the	DET
ejpam-3344	384	16	result	result	NOUN
ejpam-3344	384	17	is	be	AUX
ejpam-3344	384	18	true	true	ADJ
ejpam-3344	384	19	for	for	ADP
ejpam-3344	384	20	n	n	PROPN
ejpam-3344	384	21	=	=	SYM
ejpam-3344	384	22	k	k	NOUN
ejpam-3344	384	23	,	,	PUNCT
ejpam-3344	384	24	i.e.	i.e.	X
ejpam-3344	384	25	,	,	PUNCT
ejpam-3344	384	26	a(ak	a(ak	PROPN
ejpam-3344	384	27	)	)	PUNCT
ejpam-3344	384	28	=	=	SYM
ejpam-3344	384	29	a(a3k	a(a3k	ADJ
ejpam-3344	384	30	)	)	PUNCT
ejpam-3344	384	31	.	.	PUNCT
ejpam-3344	385	1	now	now	ADV
ejpam-3344	385	2	ak+1	ak+1	VERB
ejpam-3344	385	3	=	=	PUNCT
ejpam-3344	385	4	aka	aka	ADV
ejpam-3344	385	5	=	=	X
ejpam-3344	385	6	(	(	PUNCT
ejpam-3344	385	7	a3kx2k)(a3x2	a3kx2k)(a3x2	NOUN
ejpam-3344	385	8	)	)	PUNCT
ejpam-3344	385	9	=	=	PUNCT
ejpam-3344	385	10	a3(k+1)x2(k+1	a3(k+1)x2(k+1	PROPN
ejpam-3344	385	11	)	)	PUNCT
ejpam-3344	385	12	.	.	PUNCT
ejpam-3344	386	1	thus	thus	ADV
ejpam-3344	386	2	µa(ak+1	µa(ak+1	VERB
ejpam-3344	386	3	)	)	PUNCT
ejpam-3344	386	4	=	=	SYM
ejpam-3344	386	5	µa(a3(k+1)x2(k+1	µa(a3(k+1)x2(k+1	NUM
ejpam-3344	386	6	)	)	PUNCT
ejpam-3344	386	7	)	)	PUNCT
ejpam-3344	386	8	≥	≥	NOUN
ejpam-3344	386	9	µa(a3(k+1	µa(a3(k+1	NOUN
ejpam-3344	386	10	)	)	PUNCT
ejpam-3344	386	11	)	)	PUNCT
ejpam-3344	387	1	=	=	SYM
ejpam-3344	387	2	µa(a3k+3	µa(a3k+3	NUM
ejpam-3344	387	3	)	)	PUNCT
ejpam-3344	387	4	=	=	PUNCT
ejpam-3344	387	5	µa(ak+1a2k+2	µa(ak+1a2k+2	NOUN
ejpam-3344	387	6	)	)	PUNCT
ejpam-3344	387	7	≥	≥	NOUN
ejpam-3344	387	8	min{µa	min{µa	X
ejpam-3344	387	9	(	(	PUNCT
ejpam-3344	387	10	ak+1	ak+1	NUM
ejpam-3344	387	11	)	)	PUNCT
ejpam-3344	387	12	,	,	PUNCT
ejpam-3344	387	13	µa	µa	ADP
ejpam-3344	387	14	(	(	PUNCT
ejpam-3344	387	15	a2k+2	a2k+2	NOUN
ejpam-3344	387	16	)	)	PUNCT
ejpam-3344	387	17	}	}	PUNCT
ejpam-3344	387	18	≥	≥	X
ejpam-3344	387	19	min{µa	min{µa	X
ejpam-3344	387	20	(	(	PUNCT
ejpam-3344	387	21	ak+1	ak+1	NUM
ejpam-3344	387	22	)	)	PUNCT
ejpam-3344	387	23	,	,	PUNCT
ejpam-3344	387	24	µa	µa	ADP
ejpam-3344	387	25	(	(	PUNCT
ejpam-3344	387	26	ak+1	ak+1	X
ejpam-3344	387	27	)	)	PUNCT
ejpam-3344	387	28	,	,	PUNCT
ejpam-3344	387	29	µa	µa	ADP
ejpam-3344	387	30	(	(	PUNCT
ejpam-3344	387	31	ak+1	ak+1	X
ejpam-3344	387	32	)	)	PUNCT
ejpam-3344	387	33	}	}	PUNCT
ejpam-3344	387	34	=	=	SYM
ejpam-3344	387	35	µa	µa	NOUN
ejpam-3344	387	36	(	(	PUNCT
ejpam-3344	387	37	ak+1	ak+1	X
ejpam-3344	387	38	)	)	PUNCT
ejpam-3344	387	39	.	.	PUNCT
ejpam-3344	388	1	similarly	similarly	ADV
ejpam-3344	388	2	,	,	PUNCT
ejpam-3344	388	3	γa	γa	PROPN
ejpam-3344	388	4	(	(	PUNCT
ejpam-3344	388	5	ak+1	ak+1	X
ejpam-3344	388	6	)	)	PUNCT
ejpam-3344	388	7	=	=	SYM
ejpam-3344	388	8	γa	γa	PROPN
ejpam-3344	388	9	(	(	PUNCT
ejpam-3344	388	10	a3(k+1	a3(k+1	PROPN
ejpam-3344	388	11	)	)	PUNCT
ejpam-3344	388	12	)	)	PUNCT
ejpam-3344	388	13	,	,	PUNCT
ejpam-3344	388	14	so	so	ADV
ejpam-3344	388	15	a(ak+1	a(ak+1	ADJ
ejpam-3344	388	16	)	)	PUNCT
ejpam-3344	388	17	=	=	SYM
ejpam-3344	388	18	a(a3(k+1	a(a3(k+1	NOUN
ejpam-3344	388	19	)	)	PUNCT
ejpam-3344	388	20	)	)	PUNCT
ejpam-3344	388	21	.	.	PUNCT
ejpam-3344	389	1	hence	hence	ADV
ejpam-3344	389	2	by	by	ADP
ejpam-3344	389	3	induction	induction	NOUN
ejpam-3344	389	4	method	method	NOUN
ejpam-3344	389	5	,	,	PUNCT
ejpam-3344	389	6	the	the	DET
ejpam-3344	389	7	result	result	NOUN
ejpam-3344	389	8	is	be	AUX
ejpam-3344	389	9	true	true	ADJ
ejpam-3344	389	10	for	for	ADP
ejpam-3344	389	11	all	all	DET
ejpam-3344	389	12	positive	positive	ADJ
ejpam-3344	389	13	integers	integer	NOUN
ejpam-3344	389	14	.	.	PUNCT
ejpam-3344	390	1	theorem	theorem	NOUN
ejpam-3344	390	2	9	9	NUM
ejpam-3344	390	3	.	.	PUNCT
ejpam-3344	391	1	let	let	VERB
ejpam-3344	391	2	r	r	PRON
ejpam-3344	391	3	be	be	AUX
ejpam-3344	391	4	a	a	DET
ejpam-3344	391	5	right	right	ADJ
ejpam-3344	391	6	regular	regular	ADJ
ejpam-3344	391	7	locally	locally	ADV
ejpam-3344	391	8	associative	associative	ADJ
ejpam-3344	391	9	la	la	ADJ
ejpam-3344	391	10	-	-	PUNCT
ejpam-3344	391	11	ring	ring	NOUN
ejpam-3344	391	12	.	.	PUNCT
ejpam-3344	392	1	then	then	ADV
ejpam-3344	392	2	for	for	ADP
ejpam-3344	392	3	every	every	DET
ejpam-3344	392	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	392	5	fuzzy	fuzzy	ADJ
ejpam-3344	392	6	right	right	ADJ
ejpam-3344	392	7	ideal	ideal	NOUN
ejpam-3344	392	8	a	a	PRON
ejpam-3344	392	9	=	=	X
ejpam-3344	392	10	(	(	PUNCT
ejpam-3344	392	11	µa	µa	PROPN
ejpam-3344	392	12	,	,	PUNCT
ejpam-3344	392	13	γa	γa	PROPN
ejpam-3344	392	14	)	)	PUNCT
ejpam-3344	392	15	of	of	ADP
ejpam-3344	392	16	r	r	NOUN
ejpam-3344	392	17	,	,	PUNCT
ejpam-3344	392	18	a(an	a(an	PROPN
ejpam-3344	392	19	)	)	PUNCT
ejpam-3344	392	20	=	=	PUNCT
ejpam-3344	392	21	a(a2n	a(a2n	VERB
ejpam-3344	392	22	)	)	PUNCT
ejpam-3344	392	23	for	for	ADP
ejpam-3344	392	24	all	all	DET
ejpam-3344	392	25	a	a	DET
ejpam-3344	392	26	∈	∈	NOUN
ejpam-3344	392	27	r	r	NOUN
ejpam-3344	392	28	,	,	PUNCT
ejpam-3344	392	29	where	where	SCONJ
ejpam-3344	392	30	n	n	PRON
ejpam-3344	392	31	is	be	AUX
ejpam-3344	392	32	any	any	DET
ejpam-3344	392	33	positive	positive	ADJ
ejpam-3344	392	34	integer	integer	NOUN
ejpam-3344	392	35	.	.	PUNCT
ejpam-3344	393	1	proof	proof	NOUN
ejpam-3344	393	2	.	.	PUNCT
ejpam-3344	394	1	for	for	ADP
ejpam-3344	394	2	n	n	NOUN
ejpam-3344	394	3	=	=	SYM
ejpam-3344	394	4	1	1	X
ejpam-3344	394	5	.	.	PUNCT
ejpam-3344	394	6	let	let	VERB
ejpam-3344	394	7	a	a	DET
ejpam-3344	394	8	∈	∈	ADJ
ejpam-3344	394	9	r	r	NOUN
ejpam-3344	394	10	,	,	PUNCT
ejpam-3344	394	11	then	then	ADV
ejpam-3344	394	12	there	there	PRON
ejpam-3344	394	13	exists	exist	VERB
ejpam-3344	394	14	an	an	DET
ejpam-3344	394	15	element	element	NOUN
ejpam-3344	394	16	x	x	SYM
ejpam-3344	394	17	∈	∈	NOUN
ejpam-3344	394	18	r	r	NOUN
ejpam-3344	394	19	such	such	DET
ejpam-3344	394	20	that	that	SCONJ
ejpam-3344	394	21	a	a	DET
ejpam-3344	394	22	=	=	SYM
ejpam-3344	394	23	a2x	a2x	PROPN
ejpam-3344	394	24	.	.	PUNCT
ejpam-3344	395	1	thus	thus	ADV
ejpam-3344	395	2	µa(a	µa(a	PUNCT
ejpam-3344	395	3	)	)	PUNCT
ejpam-3344	395	4	=	=	SYM
ejpam-3344	395	5	µa(a2x	µa(a2x	PROPN
ejpam-3344	395	6	)	)	PUNCT
ejpam-3344	395	7	≥	≥	NOUN
ejpam-3344	395	8	µa(a2	µa(a2	NOUN
ejpam-3344	395	9	)	)	PUNCT
ejpam-3344	395	10	=	=	SYM
ejpam-3344	395	11	µa(aa	µa(aa	PROPN
ejpam-3344	395	12	)	)	PUNCT
ejpam-3344	395	13	≥	≥	NOUN
ejpam-3344	395	14	min{µa	min{µa	X
ejpam-3344	395	15	(	(	PUNCT
ejpam-3344	395	16	a	a	NOUN
ejpam-3344	395	17	)	)	PUNCT
ejpam-3344	395	18	,	,	PUNCT
ejpam-3344	395	19	µa	µa	X
ejpam-3344	395	20	(	(	PUNCT
ejpam-3344	395	21	a	a	NOUN
ejpam-3344	395	22	)	)	PUNCT
ejpam-3344	395	23	}	}	PUNCT
ejpam-3344	396	1	=	=	SYM
ejpam-3344	396	2	µa	µa	NOUN
ejpam-3344	396	3	(	(	PUNCT
ejpam-3344	396	4	a	a	NOUN
ejpam-3344	396	5	)	)	PUNCT
ejpam-3344	396	6	.	.	PUNCT
ejpam-3344	397	1	similarly	similarly	ADV
ejpam-3344	397	2	,	,	PUNCT
ejpam-3344	397	3	γa(a	γa(a	ADJ
ejpam-3344	397	4	)	)	PUNCT
ejpam-3344	397	5	=	=	SYM
ejpam-3344	397	6	γa(a2	γa(a2	NOUN
ejpam-3344	397	7	)	)	PUNCT
ejpam-3344	397	8	,	,	PUNCT
ejpam-3344	397	9	therefore	therefore	ADV
ejpam-3344	397	10	a	a	DET
ejpam-3344	397	11	(	(	PUNCT
ejpam-3344	397	12	a	a	NOUN
ejpam-3344	397	13	)	)	PUNCT
ejpam-3344	397	14	=	=	SYM
ejpam-3344	398	1	a	a	PRON
ejpam-3344	398	2	(	(	PUNCT
ejpam-3344	398	3	a2	a2	PROPN
ejpam-3344	398	4	)	)	PUNCT
ejpam-3344	398	5	.	.	PUNCT
ejpam-3344	399	1	now	now	ADV
ejpam-3344	399	2	a2	a2	PROPN
ejpam-3344	399	3	=	=	SYM
ejpam-3344	399	4	aa	aa	PROPN
ejpam-3344	399	5	=	=	SYM
ejpam-3344	399	6	(	(	PUNCT
ejpam-3344	399	7	a2x)(a2x	a2x)(a2x	PROPN
ejpam-3344	399	8	)	)	PUNCT
ejpam-3344	399	9	=	=	SYM
ejpam-3344	400	1	a4x2	a4x2	PROPN
ejpam-3344	400	2	,	,	PUNCT
ejpam-3344	400	3	then	then	ADV
ejpam-3344	400	4	the	the	DET
ejpam-3344	400	5	result	result	NOUN
ejpam-3344	400	6	is	be	AUX
ejpam-3344	400	7	true	true	ADJ
ejpam-3344	400	8	for	for	ADP
ejpam-3344	400	9	n	n	NOUN
ejpam-3344	400	10	=	=	SYM
ejpam-3344	400	11	2	2	X
ejpam-3344	400	12	.	.	PUNCT
ejpam-3344	400	13	suppose	suppose	VERB
ejpam-3344	400	14	that	that	SCONJ
ejpam-3344	400	15	the	the	DET
ejpam-3344	400	16	result	result	NOUN
ejpam-3344	400	17	is	be	AUX
ejpam-3344	400	18	true	true	ADJ
ejpam-3344	400	19	for	for	ADP
ejpam-3344	400	20	n	n	PROPN
ejpam-3344	400	21	=	=	SYM
ejpam-3344	400	22	k	k	NOUN
ejpam-3344	400	23	,	,	PUNCT
ejpam-3344	400	24	i.e.	i.e.	X
ejpam-3344	400	25	,	,	PUNCT
ejpam-3344	400	26	a(ak	a(ak	PROPN
ejpam-3344	400	27	)	)	PUNCT
ejpam-3344	400	28	=	=	PUNCT
ejpam-3344	400	29	a(a2k	a(a2k	NOUN
ejpam-3344	400	30	)	)	PUNCT
ejpam-3344	400	31	.	.	PUNCT
ejpam-3344	401	1	now	now	ADV
ejpam-3344	401	2	ak+1	ak+1	VERB
ejpam-3344	401	3	=	=	PUNCT
ejpam-3344	401	4	aka	aka	ADV
ejpam-3344	401	5	=	=	SYM
ejpam-3344	401	6	(	(	PUNCT
ejpam-3344	401	7	a2kxk)(a2x	a2kxk)(a2x	PROPN
ejpam-3344	401	8	)	)	PUNCT
ejpam-3344	401	9	=	=	SYM
ejpam-3344	401	10	a2(k+1)x(k+1	a2(k+1)x(k+1	PROPN
ejpam-3344	401	11	)	)	PUNCT
ejpam-3344	401	12	.	.	PUNCT
ejpam-3344	402	1	thus	thus	ADV
ejpam-3344	402	2	µa(ak+1	µa(ak+1	VERB
ejpam-3344	402	3	)	)	PUNCT
ejpam-3344	402	4	=	=	SYM
ejpam-3344	402	5	µa(a2(k+1)x(k+1	µa(a2(k+1)x(k+1	NOUN
ejpam-3344	402	6	)	)	PUNCT
ejpam-3344	402	7	)	)	PUNCT
ejpam-3344	402	8	≥	≥	NOUN
ejpam-3344	402	9	µa(a2(k+1	µa(a2(k+1	NOUN
ejpam-3344	402	10	)	)	PUNCT
ejpam-3344	402	11	)	)	PUNCT
ejpam-3344	402	12	=	=	SYM
ejpam-3344	402	13	µa(a2k+2	µa(a2k+2	NOUN
ejpam-3344	402	14	)	)	PUNCT
ejpam-3344	402	15	=	=	PUNCT
ejpam-3344	402	16	µa(ak+1ak+1	µa(ak+1ak+1	X
ejpam-3344	402	17	)	)	PUNCT
ejpam-3344	402	18	≥	≥	NOUN
ejpam-3344	402	19	min{µa	min{µa	X
ejpam-3344	402	20	(	(	PUNCT
ejpam-3344	402	21	ak+1	ak+1	NUM
ejpam-3344	402	22	)	)	PUNCT
ejpam-3344	402	23	,	,	PUNCT
ejpam-3344	402	24	µa	µa	ADP
ejpam-3344	402	25	(	(	PUNCT
ejpam-3344	402	26	ak+1	ak+1	X
ejpam-3344	402	27	)	)	PUNCT
ejpam-3344	402	28	}	}	PUNCT
ejpam-3344	402	29	=	=	SYM
ejpam-3344	402	30	µa	µa	NOUN
ejpam-3344	402	31	(	(	PUNCT
ejpam-3344	402	32	ak+1	ak+1	X
ejpam-3344	402	33	)	)	PUNCT
ejpam-3344	402	34	.	.	PUNCT
ejpam-3344	403	1	similarly	similarly	ADV
ejpam-3344	403	2	,	,	PUNCT
ejpam-3344	403	3	γa(ak+1	γa(ak+1	ADJ
ejpam-3344	403	4	)	)	PUNCT
ejpam-3344	403	5	=	=	SYM
ejpam-3344	403	6	γa(a2(k+1	γa(a2(k+1	ADJ
ejpam-3344	403	7	)	)	PUNCT
ejpam-3344	403	8	)	)	PUNCT
ejpam-3344	403	9	,	,	PUNCT
ejpam-3344	403	10	therefore	therefore	ADV
ejpam-3344	403	11	a(ak+1	a(ak+1	VERB
ejpam-3344	403	12	)	)	PUNCT
ejpam-3344	403	13	=	=	SYM
ejpam-3344	403	14	a(a2(k+1	a(a2(k+1	PROPN
ejpam-3344	403	15	)	)	PUNCT
ejpam-3344	403	16	)	)	PUNCT
ejpam-3344	403	17	.	.	PUNCT
ejpam-3344	404	1	hence	hence	ADV
ejpam-3344	404	2	by	by	ADP
ejpam-3344	404	3	induction	induction	NOUN
ejpam-3344	404	4	method	method	NOUN
ejpam-3344	404	5	,	,	PUNCT
ejpam-3344	404	6	the	the	DET
ejpam-3344	404	7	result	result	NOUN
ejpam-3344	404	8	is	be	AUX
ejpam-3344	404	9	true	true	ADJ
ejpam-3344	404	10	for	for	ADP
ejpam-3344	404	11	all	all	DET
ejpam-3344	404	12	positive	positive	ADJ
ejpam-3344	404	13	integers	integer	NOUN
ejpam-3344	404	14	.	.	PUNCT
ejpam-3344	405	1	n.	n.	PROPN
ejpam-3344	405	2	kausar	kausar	PROPN
ejpam-3344	405	3	,	,	PUNCT
ejpam-3344	405	4	m.	m.	NOUN
ejpam-3344	405	5	a.	a.	PROPN
ejpam-3344	405	6	waqar	waqar	PROPN
ejpam-3344	405	7	/	/	SYM
ejpam-3344	405	8	eur	eur	PROPN
ejpam-3344	405	9	.	.	PUNCT
ejpam-3344	406	1	j.	j.	PROPN
ejpam-3344	406	2	pure	pure	PROPN
ejpam-3344	406	3	appl	appl	PROPN
ejpam-3344	406	4	.	.	PROPN
ejpam-3344	406	5	math	math	PROPN
ejpam-3344	406	6	,	,	PUNCT
ejpam-3344	406	7	12	12	NUM
ejpam-3344	406	8	(	(	PUNCT
ejpam-3344	406	9	1	1	NUM
ejpam-3344	406	10	)	)	PUNCT
ejpam-3344	406	11	(	(	PUNCT
ejpam-3344	406	12	2019	2019	NUM
ejpam-3344	406	13	)	)	PUNCT
ejpam-3344	406	14	,	,	PUNCT
ejpam-3344	406	15	226	226	NUM
ejpam-3344	406	16	-	-	SYM
ejpam-3344	406	17	250	250	NUM
ejpam-3344	406	18	241	241	NUM
ejpam-3344	406	19	lemma	lemma	PROPN
ejpam-3344	406	20	14	14	NUM
ejpam-3344	406	21	.	.	PUNCT
ejpam-3344	407	1	let	let	VERB
ejpam-3344	407	2	r	r	PRON
ejpam-3344	407	3	be	be	AUX
ejpam-3344	407	4	a	a	DET
ejpam-3344	407	5	right	right	ADJ
ejpam-3344	407	6	regular	regular	ADJ
ejpam-3344	407	7	locally	locally	ADV
ejpam-3344	407	8	associative	associative	ADJ
ejpam-3344	407	9	la	la	NOUN
ejpam-3344	407	10	-	-	NOUN
ejpam-3344	407	11	ring	ring	NOUN
ejpam-3344	407	12	with	with	ADP
ejpam-3344	407	13	left	left	ADJ
ejpam-3344	407	14	identity	identity	NOUN
ejpam-3344	407	15	e.	e.	PROPN
ejpam-3344	407	16	then	then	ADV
ejpam-3344	407	17	for	for	ADP
ejpam-3344	407	18	every	every	DET
ejpam-3344	407	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	407	20	fuzzy	fuzzy	ADJ
ejpam-3344	407	21	right	right	ADJ
ejpam-3344	407	22	ideal	ideal	NOUN
ejpam-3344	407	23	a	a	PRON
ejpam-3344	407	24	=	=	X
ejpam-3344	407	25	(	(	PUNCT
ejpam-3344	407	26	µa	µa	PROPN
ejpam-3344	407	27	,	,	PUNCT
ejpam-3344	407	28	γa	γa	PROPN
ejpam-3344	407	29	)	)	PUNCT
ejpam-3344	407	30	of	of	ADP
ejpam-3344	407	31	r	r	NOUN
ejpam-3344	407	32	,	,	PUNCT
ejpam-3344	407	33	a(ab	a(ab	NOUN
ejpam-3344	407	34	)	)	PUNCT
ejpam-3344	407	35	=	=	SYM
ejpam-3344	407	36	a(ba	a(ba	NOUN
ejpam-3344	407	37	)	)	PUNCT
ejpam-3344	407	38	for	for	ADP
ejpam-3344	407	39	all	all	DET
ejpam-3344	407	40	a	a	DET
ejpam-3344	407	41	,	,	PUNCT
ejpam-3344	407	42	b	b	PROPN
ejpam-3344	407	43	∈	∈	PROPN
ejpam-3344	407	44	r.	r.	NOUN
ejpam-3344	407	45	proof	proof	NOUN
ejpam-3344	407	46	.	.	PUNCT
ejpam-3344	408	1	let	let	VERB
ejpam-3344	408	2	a	a	DET
ejpam-3344	408	3	,	,	PUNCT
ejpam-3344	408	4	b	b	PROPN
ejpam-3344	408	5	∈	∈	PROPN
ejpam-3344	408	6	r.	r.	NOUN
ejpam-3344	408	7	by	by	ADP
ejpam-3344	408	8	using	use	VERB
ejpam-3344	408	9	theorem	theorem	ADJ
ejpam-3344	408	10	9	9	NUM
ejpam-3344	408	11	(	(	PUNCT
ejpam-3344	408	12	for	for	ADP
ejpam-3344	408	13	n	n	NOUN
ejpam-3344	408	14	=	=	SYM
ejpam-3344	408	15	1	1	NUM
ejpam-3344	408	16	)	)	PUNCT
ejpam-3344	408	17	.	.	PUNCT
ejpam-3344	409	1	now	now	ADV
ejpam-3344	409	2	µa(ab	µa(ab	PROPN
ejpam-3344	409	3	)	)	PUNCT
ejpam-3344	409	4	=	=	SYM
ejpam-3344	409	5	µa((ab)2	µa((ab)2	ADJ
ejpam-3344	409	6	)	)	PUNCT
ejpam-3344	409	7	=	=	SYM
ejpam-3344	409	8	µa((ab)(ab	µa((ab)(ab	PROPN
ejpam-3344	409	9	)	)	PUNCT
ejpam-3344	409	10	)	)	PUNCT
ejpam-3344	410	1	=	=	SYM
ejpam-3344	410	2	µa((ba)(ba	µa((ba)(ba	NOUN
ejpam-3344	410	3	)	)	PUNCT
ejpam-3344	410	4	)	)	PUNCT
ejpam-3344	411	1	=	=	PUNCT
ejpam-3344	411	2	µa((ba)2	µa((ba)2	PROPN
ejpam-3344	411	3	)	)	PUNCT
ejpam-3344	411	4	=	=	SYM
ejpam-3344	411	5	µa(ba	µa(ba	NOUN
ejpam-3344	411	6	)	)	PUNCT
ejpam-3344	411	7	and	and	CCONJ
ejpam-3344	411	8	γa(ab	γa(ab	PROPN
ejpam-3344	411	9	)	)	PUNCT
ejpam-3344	411	10	=	=	SYM
ejpam-3344	411	11	γa((ab)2	γa((ab)2	ADJ
ejpam-3344	411	12	)	)	PUNCT
ejpam-3344	411	13	=	=	SYM
ejpam-3344	411	14	γa((ab)(ab	γa((ab)(ab	PROPN
ejpam-3344	411	15	)	)	PUNCT
ejpam-3344	411	16	)	)	PUNCT
ejpam-3344	412	1	=	=	SYM
ejpam-3344	412	2	γa((ba)(ba	γa((ba)(ba	NOUN
ejpam-3344	412	3	)	)	PUNCT
ejpam-3344	412	4	)	)	PUNCT
ejpam-3344	413	1	=	=	SYM
ejpam-3344	413	2	γa((ba)2	γa((ba)2	PROPN
ejpam-3344	413	3	)	)	PUNCT
ejpam-3344	413	4	=	=	SYM
ejpam-3344	413	5	γa(ba	γa(ba	NOUN
ejpam-3344	413	6	)	)	PUNCT
ejpam-3344	413	7	.	.	PUNCT
ejpam-3344	414	1	thus	thus	ADV
ejpam-3344	414	2	a(ab	a(ab	NOUN
ejpam-3344	414	3	)	)	PUNCT
ejpam-3344	414	4	=	=	SYM
ejpam-3344	414	5	a(ba	a(ba	NOUN
ejpam-3344	414	6	)	)	PUNCT
ejpam-3344	414	7	.	.	PUNCT
ejpam-3344	415	1	remark	remark	PROPN
ejpam-3344	415	2	8	8	NUM
ejpam-3344	415	3	.	.	PUNCT
ejpam-3344	416	1	it	it	PRON
ejpam-3344	416	2	is	be	AUX
ejpam-3344	416	3	easy	easy	ADJ
ejpam-3344	416	4	to	to	PART
ejpam-3344	416	5	see	see	VERB
ejpam-3344	416	6	that	that	SCONJ
ejpam-3344	416	7	,	,	PUNCT
ejpam-3344	416	8	if	if	SCONJ
ejpam-3344	416	9	r	r	NOUN
ejpam-3344	416	10	is	be	AUX
ejpam-3344	416	11	a	a	DET
ejpam-3344	416	12	left	left	ADJ
ejpam-3344	416	13	regular	regular	NOUN
ejpam-3344	416	14	locally	locally	ADV
ejpam-3344	416	15	associative	associative	ADJ
ejpam-3344	416	16	la	la	NOUN
ejpam-3344	416	17	-	-	NOUN
ejpam-3344	416	18	ring	ring	NOUN
ejpam-3344	416	19	with	with	ADP
ejpam-3344	416	20	left	left	ADJ
ejpam-3344	416	21	identity	identity	NOUN
ejpam-3344	416	22	e.	e.	PROPN
ejpam-3344	416	23	then	then	ADV
ejpam-3344	416	24	for	for	SCONJ
ejpam-3344	416	25	every	every	DET
ejpam-3344	416	26	intuitionistic	intuitionistic	ADJ
ejpam-3344	416	27	fuzzy	fuzzy	ADJ
ejpam-3344	416	28	left	leave	VERB
ejpam-3344	416	29	ideal	ideal	NOUN
ejpam-3344	416	30	a	a	PRON
ejpam-3344	416	31	=	=	X
ejpam-3344	416	32	(	(	PUNCT
ejpam-3344	416	33	µa	µa	PROPN
ejpam-3344	416	34	,	,	PUNCT
ejpam-3344	416	35	γa	γa	PROPN
ejpam-3344	416	36	)	)	PUNCT
ejpam-3344	416	37	of	of	ADP
ejpam-3344	416	38	r	r	NOUN
ejpam-3344	416	39	,	,	PUNCT
ejpam-3344	416	40	a(an	a(an	PROPN
ejpam-3344	416	41	)	)	PUNCT
ejpam-3344	416	42	=	=	PUNCT
ejpam-3344	416	43	a(a2n	a(a2n	VERB
ejpam-3344	416	44	)	)	PUNCT
ejpam-3344	416	45	for	for	ADP
ejpam-3344	416	46	all	all	DET
ejpam-3344	416	47	a	a	DET
ejpam-3344	416	48	∈	∈	NOUN
ejpam-3344	416	49	r	r	NOUN
ejpam-3344	416	50	,	,	PUNCT
ejpam-3344	416	51	where	where	SCONJ
ejpam-3344	416	52	n	n	PRON
ejpam-3344	416	53	is	be	AUX
ejpam-3344	416	54	any	any	DET
ejpam-3344	416	55	positive	positive	ADJ
ejpam-3344	416	56	integer	integer	NOUN
ejpam-3344	416	57	.	.	PUNCT
ejpam-3344	417	1	and	and	CCONJ
ejpam-3344	417	2	also	also	ADV
ejpam-3344	417	3	for	for	ADP
ejpam-3344	417	4	every	every	DET
ejpam-3344	417	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	417	6	fuzzy	fuzzy	ADJ
ejpam-3344	417	7	left	leave	VERB
ejpam-3344	417	8	ideal	ideal	NOUN
ejpam-3344	417	9	a	a	PRON
ejpam-3344	417	10	=	=	X
ejpam-3344	417	11	(	(	PUNCT
ejpam-3344	417	12	µa	µa	PROPN
ejpam-3344	417	13	,	,	PUNCT
ejpam-3344	417	14	γa	γa	PROPN
ejpam-3344	417	15	)	)	PUNCT
ejpam-3344	417	16	of	of	ADP
ejpam-3344	417	17	r	r	NOUN
ejpam-3344	417	18	,	,	PUNCT
ejpam-3344	417	19	a(ab	a(ab	NOUN
ejpam-3344	417	20	)	)	PUNCT
ejpam-3344	417	21	=	=	SYM
ejpam-3344	417	22	a(ba	a(ba	NOUN
ejpam-3344	417	23	)	)	PUNCT
ejpam-3344	417	24	for	for	ADP
ejpam-3344	417	25	all	all	DET
ejpam-3344	417	26	a	a	DET
ejpam-3344	417	27	,	,	PUNCT
ejpam-3344	417	28	b	b	X
ejpam-3344	417	29	∈	∈	PROPN
ejpam-3344	417	30	r.	r.	PROPN
ejpam-3344	417	31	lemma	lemma	PROPN
ejpam-3344	417	32	15	15	NUM
ejpam-3344	417	33	.	.	PUNCT
ejpam-3344	418	1	let	let	VERB
ejpam-3344	418	2	r	r	PRON
ejpam-3344	418	3	be	be	AUX
ejpam-3344	418	4	a	a	DET
ejpam-3344	418	5	right	right	ADJ
ejpam-3344	418	6	weakly	weakly	ADJ
ejpam-3344	418	7	regular	regular	ADJ
ejpam-3344	418	8	la	la	ADJ
ejpam-3344	418	9	-	-	PUNCT
ejpam-3344	418	10	ring	ring	NOUN
ejpam-3344	418	11	.	.	PUNCT
ejpam-3344	419	1	then	then	ADV
ejpam-3344	419	2	every	every	DET
ejpam-3344	419	3	intuitionistic	intuitionistic	ADJ
ejpam-3344	419	4	fuzzy	fuzzy	ADJ
ejpam-3344	419	5	left	left	ADJ
ejpam-3344	419	6	(	(	PUNCT
ejpam-3344	419	7	right	right	ADJ
ejpam-3344	419	8	)	)	PUNCT
ejpam-3344	419	9	ideal	ideal	NOUN
ejpam-3344	419	10	of	of	ADP
ejpam-3344	419	11	r	r	NOUN
ejpam-3344	419	12	is	be	AUX
ejpam-3344	419	13	an	an	DET
ejpam-3344	419	14	intuitionistic	intuitionistic	ADJ
ejpam-3344	419	15	fuzzy	fuzzy	ADJ
ejpam-3344	419	16	ideal	ideal	NOUN
ejpam-3344	419	17	of	of	ADP
ejpam-3344	419	18	r.	r.	PROPN
ejpam-3344	419	19	proof	proof	NOUN
ejpam-3344	419	20	.	.	PUNCT
ejpam-3344	420	1	suppose	suppose	VERB
ejpam-3344	420	2	that	that	SCONJ
ejpam-3344	420	3	a	a	DET
ejpam-3344	420	4	=	=	SYM
ejpam-3344	420	5	(	(	PUNCT
ejpam-3344	420	6	µa	µa	PROPN
ejpam-3344	420	7	,	,	PUNCT
ejpam-3344	420	8	γa	γa	PROPN
ejpam-3344	420	9	)	)	PUNCT
ejpam-3344	420	10	is	be	AUX
ejpam-3344	420	11	an	an	DET
ejpam-3344	420	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	420	13	fuzzy	fuzzy	ADJ
ejpam-3344	420	14	right	right	ADJ
ejpam-3344	420	15	ideal	ideal	NOUN
ejpam-3344	420	16	of	of	ADP
ejpam-3344	420	17	r	r	NOUN
ejpam-3344	420	18	and	and	CCONJ
ejpam-3344	420	19	x	x	NOUN
ejpam-3344	420	20	,	,	PUNCT
ejpam-3344	420	21	y	y	PROPN
ejpam-3344	420	22	∈	∈	PROPN
ejpam-3344	420	23	r	r	NOUN
ejpam-3344	420	24	,	,	PUNCT
ejpam-3344	420	25	this	this	PRON
ejpam-3344	420	26	means	mean	VERB
ejpam-3344	420	27	that	that	SCONJ
ejpam-3344	420	28	there	there	PRON
ejpam-3344	420	29	exist	exist	VERB
ejpam-3344	420	30	a	a	DET
ejpam-3344	420	31	,	,	PUNCT
ejpam-3344	420	32	b	b	X
ejpam-3344	420	33	∈	∈	NOUN
ejpam-3344	420	34	r	r	NOUN
ejpam-3344	420	35	such	such	ADJ
ejpam-3344	420	36	that	that	PRON
ejpam-3344	420	37	x	x	X
ejpam-3344	420	38	=	=	SYM
ejpam-3344	420	39	(	(	PUNCT
ejpam-3344	420	40	xa)(xb	xa)(xb	PROPN
ejpam-3344	420	41	)	)	PUNCT
ejpam-3344	420	42	.	.	PUNCT
ejpam-3344	421	1	thus	thus	ADV
ejpam-3344	421	2	µa(xy	µa(xy	NUM
ejpam-3344	421	3	)	)	PUNCT
ejpam-3344	421	4	=	=	SYM
ejpam-3344	421	5	µa(((xa)(xb))y	µa(((xa)(xb))y	X
ejpam-3344	421	6	)	)	PUNCT
ejpam-3344	421	7	=	=	PUNCT
ejpam-3344	421	8	µa((((xb)a)x)y	µa((((xb)a)x)y	X
ejpam-3344	421	9	)	)	PUNCT
ejpam-3344	421	10	=	=	SYM
ejpam-3344	421	11	µa((((ab)x)x)y	µa((((ab)x)x)y	X
ejpam-3344	421	12	)	)	PUNCT
ejpam-3344	421	13	=	=	SYM
ejpam-3344	421	14	µa((yx)((ab)x	µa((yx)((ab)x	NUM
ejpam-3344	421	15	)	)	PUNCT
ejpam-3344	421	16	)	)	PUNCT
ejpam-3344	422	1	=	=	SYM
ejpam-3344	422	2	µa((yx)(nx	µa((yx)(nx	NOUN
ejpam-3344	422	3	)	)	PUNCT
ejpam-3344	422	4	)	)	PUNCT
ejpam-3344	422	5	say	say	VERB
ejpam-3344	422	6	ab	ab	PROPN
ejpam-3344	422	7	=	=	PUNCT
ejpam-3344	422	8	n	n	PROPN
ejpam-3344	422	9	≥	≥	NOUN
ejpam-3344	422	10	µa(yx	µa(yx	NOUN
ejpam-3344	422	11	)	)	PUNCT
ejpam-3344	422	12	≥	≥	NOUN
ejpam-3344	422	13	µa(y	µa(y	NOUN
ejpam-3344	422	14	)	)	PUNCT
ejpam-3344	422	15	and	and	CCONJ
ejpam-3344	422	16	γa(xy	γa(xy	PROPN
ejpam-3344	422	17	)	)	PUNCT
ejpam-3344	423	1	=	=	SYM
ejpam-3344	423	2	γa(((xa)(xb))y	γa(((xa)(xb))y	PROPN
ejpam-3344	423	3	)	)	PUNCT
ejpam-3344	423	4	=	=	PUNCT
ejpam-3344	423	5	γa((((xb)a)x)y	γa((((xb)a)x)y	X
ejpam-3344	423	6	)	)	PUNCT
ejpam-3344	423	7	=	=	SYM
ejpam-3344	423	8	γa((((ab)x)x)y	γa((((ab)x)x)y	X
ejpam-3344	423	9	)	)	PUNCT
ejpam-3344	423	10	=	=	NOUN
ejpam-3344	423	11	γa((yx)((ab)x	γa((yx)((ab)x	NUM
ejpam-3344	423	12	)	)	PUNCT
ejpam-3344	423	13	)	)	PUNCT
ejpam-3344	423	14	=	=	SYM
ejpam-3344	424	1	γa((yx)(nx	γa((yx)(nx	NOUN
ejpam-3344	424	2	)	)	PUNCT
ejpam-3344	424	3	)	)	PUNCT
ejpam-3344	424	4	≤	≤	NUM
ejpam-3344	424	5	γa(yx	γa(yx	PROPN
ejpam-3344	424	6	)	)	PUNCT
ejpam-3344	424	7	≤	≤	NUM
ejpam-3344	424	8	γ(y	γ(y	PROPN
ejpam-3344	424	9	)	)	PUNCT
ejpam-3344	424	10	.	.	PUNCT
ejpam-3344	425	1	therefore	therefore	ADV
ejpam-3344	425	2	a	a	PRON
ejpam-3344	425	3	is	be	AUX
ejpam-3344	425	4	an	an	DET
ejpam-3344	425	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	425	6	fuzzy	fuzzy	ADJ
ejpam-3344	425	7	ideal	ideal	NOUN
ejpam-3344	425	8	of	of	ADP
ejpam-3344	425	9	r.	r.	PROPN
ejpam-3344	425	10	similarly	similarly	ADV
ejpam-3344	425	11	,	,	PUNCT
ejpam-3344	425	12	for	for	ADP
ejpam-3344	425	13	left	left	ADJ
ejpam-3344	425	14	ideal	ideal	NOUN
ejpam-3344	425	15	.	.	PUNCT
ejpam-3344	426	1	remark	remark	NOUN
ejpam-3344	426	2	9	9	NUM
ejpam-3344	426	3	.	.	PUNCT
ejpam-3344	427	1	the	the	DET
ejpam-3344	427	2	concept	concept	NOUN
ejpam-3344	427	3	of	of	ADP
ejpam-3344	427	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	427	5	fuzzy	fuzzy	ADJ
ejpam-3344	427	6	(	(	PUNCT
ejpam-3344	427	7	left	left	ADJ
ejpam-3344	427	8	,	,	PUNCT
ejpam-3344	427	9	right	right	INTJ
ejpam-3344	427	10	,	,	PUNCT
ejpam-3344	427	11	two	two	NUM
ejpam-3344	427	12	-	-	PUNCT
ejpam-3344	427	13	sided	sided	ADJ
ejpam-3344	427	14	)	)	PUNCT
ejpam-3344	427	15	ideals	ideal	NOUN
ejpam-3344	427	16	coincides	coincide	VERB
ejpam-3344	427	17	in	in	ADP
ejpam-3344	427	18	right	right	ADJ
ejpam-3344	427	19	weakly	weakly	ADJ
ejpam-3344	427	20	regular	regular	ADJ
ejpam-3344	427	21	la	la	ADJ
ejpam-3344	427	22	-	-	PUNCT
ejpam-3344	427	23	rings	ring	NOUN
ejpam-3344	427	24	.	.	PUNCT
ejpam-3344	428	1	proposition	proposition	NOUN
ejpam-3344	428	2	9	9	NUM
ejpam-3344	428	3	.	.	PUNCT
ejpam-3344	429	1	every	every	DET
ejpam-3344	429	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	429	3	fuzzy	fuzzy	ADJ
ejpam-3344	429	4	generalized	generalize	VERB
ejpam-3344	429	5	bi	bi	NOUN
ejpam-3344	429	6	-	-	NOUN
ejpam-3344	429	7	ideal	ideal	NOUN
ejpam-3344	429	8	of	of	ADP
ejpam-3344	429	9	a	a	DET
ejpam-3344	429	10	right	right	ADJ
ejpam-3344	429	11	weakly	weakly	ADJ
ejpam-3344	429	12	regular	regular	ADJ
ejpam-3344	429	13	la	la	ADJ
ejpam-3344	429	14	-	-	PUNCT
ejpam-3344	429	15	ring	ring	NOUN
ejpam-3344	429	16	r	r	NOUN
ejpam-3344	429	17	with	with	ADP
ejpam-3344	429	18	left	left	ADJ
ejpam-3344	429	19	identity	identity	NOUN
ejpam-3344	429	20	e	e	NOUN
ejpam-3344	429	21	,	,	PUNCT
ejpam-3344	429	22	is	be	AUX
ejpam-3344	429	23	an	an	DET
ejpam-3344	429	24	intuitionistic	intuitionistic	ADJ
ejpam-3344	429	25	fuzzy	fuzzy	ADJ
ejpam-3344	429	26	bi	bi	NOUN
ejpam-3344	429	27	-	-	NOUN
ejpam-3344	429	28	ideal	ideal	NOUN
ejpam-3344	429	29	of	of	ADP
ejpam-3344	429	30	r.	r.	PROPN
ejpam-3344	429	31	proof	proof	PROPN
ejpam-3344	429	32	.	.	PUNCT
ejpam-3344	430	1	assume	assume	VERB
ejpam-3344	430	2	that	that	SCONJ
ejpam-3344	430	3	a	a	DET
ejpam-3344	430	4	=	=	SYM
ejpam-3344	430	5	(	(	PUNCT
ejpam-3344	430	6	µa	µa	PROPN
ejpam-3344	430	7	,	,	PUNCT
ejpam-3344	430	8	γa	γa	PROPN
ejpam-3344	430	9	)	)	PUNCT
ejpam-3344	430	10	is	be	AUX
ejpam-3344	430	11	an	an	DET
ejpam-3344	430	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	430	13	fuzzy	fuzzy	ADJ
ejpam-3344	430	14	generalized	generalize	VERB
ejpam-3344	430	15	bi	bi	NOUN
ejpam-3344	430	16	-	-	NOUN
ejpam-3344	430	17	ideal	ideal	NOUN
ejpam-3344	430	18	of	of	ADP
ejpam-3344	430	19	r	r	NOUN
ejpam-3344	430	20	and	and	CCONJ
ejpam-3344	430	21	x	x	NOUN
ejpam-3344	430	22	,	,	PUNCT
ejpam-3344	430	23	y	y	PROPN
ejpam-3344	430	24	∈	∈	PROPN
ejpam-3344	430	25	r	r	NOUN
ejpam-3344	430	26	,	,	PUNCT
ejpam-3344	430	27	then	then	ADV
ejpam-3344	430	28	there	there	PRON
ejpam-3344	430	29	exist	exist	VERB
ejpam-3344	430	30	elements	element	NOUN
ejpam-3344	430	31	a	a	PRON
ejpam-3344	430	32	,	,	PUNCT
ejpam-3344	430	33	b	b	X
ejpam-3344	430	34	∈	∈	NOUN
ejpam-3344	430	35	r	r	NOUN
ejpam-3344	431	1	such	such	ADJ
ejpam-3344	431	2	that	that	PRON
ejpam-3344	431	3	x	x	X
ejpam-3344	431	4	=	=	SYM
ejpam-3344	431	5	(	(	PUNCT
ejpam-3344	431	6	xa)(xb	xa)(xb	PROPN
ejpam-3344	431	7	)	)	PUNCT
ejpam-3344	431	8	.	.	PUNCT
ejpam-3344	432	1	we	we	PRON
ejpam-3344	432	2	have	have	VERB
ejpam-3344	432	3	to	to	PART
ejpam-3344	432	4	show	show	VERB
ejpam-3344	432	5	that	that	SCONJ
ejpam-3344	432	6	a	a	PRON
ejpam-3344	432	7	is	be	AUX
ejpam-3344	432	8	an	an	DET
ejpam-3344	432	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	432	10	fuzzy	fuzzy	ADJ
ejpam-3344	432	11	la	la	NOUN
ejpam-3344	432	12	-	-	PUNCT
ejpam-3344	432	13	subring	subring	NOUN
ejpam-3344	432	14	of	of	ADP
ejpam-3344	432	15	r.	r.	PROPN
ejpam-3344	432	16	thus	thus	ADV
ejpam-3344	432	17	µa(xy	µa(xy	PROPN
ejpam-3344	432	18	)	)	PUNCT
ejpam-3344	432	19	=	=	SYM
ejpam-3344	432	20	µa(((xa)(xb))y	µa(((xa)(xb))y	X
ejpam-3344	432	21	)	)	PUNCT
ejpam-3344	432	22	=	=	SYM
ejpam-3344	432	23	µa((x((xa)b))y	µa((x((xa)b))y	NUM
ejpam-3344	432	24	)	)	PUNCT
ejpam-3344	432	25	≥	≥	NOUN
ejpam-3344	432	26	min{µa(x	min{µa(x	NOUN
ejpam-3344	432	27	)	)	PUNCT
ejpam-3344	432	28	,	,	PUNCT
ejpam-3344	432	29	µa(y	µa(y	NOUN
ejpam-3344	432	30	)	)	PUNCT
ejpam-3344	432	31	}	}	PUNCT
ejpam-3344	432	32	and	and	CCONJ
ejpam-3344	432	33	γa(xy	γa(xy	PROPN
ejpam-3344	432	34	)	)	PUNCT
ejpam-3344	433	1	=	=	SYM
ejpam-3344	433	2	γa(((xa)(xb))y	γa(((xa)(xb))y	PROPN
ejpam-3344	433	3	)	)	PUNCT
ejpam-3344	433	4	=	=	PUNCT
ejpam-3344	433	5	γa((x((xa)b))y	γa((x((xa)b))y	NOUN
ejpam-3344	433	6	)	)	PUNCT
ejpam-3344	433	7	≤	≤	NUM
ejpam-3344	433	8	max{γa(x	max{γa(x	NOUN
ejpam-3344	433	9	)	)	PUNCT
ejpam-3344	433	10	,	,	PUNCT
ejpam-3344	433	11	γa(y	γa(y	NOUN
ejpam-3344	433	12	)	)	PUNCT
ejpam-3344	433	13	}	}	PUNCT
ejpam-3344	433	14	.	.	PUNCT
ejpam-3344	434	1	so	so	ADV
ejpam-3344	434	2	a	a	PRON
ejpam-3344	434	3	is	be	AUX
ejpam-3344	434	4	an	an	DET
ejpam-3344	434	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	434	6	fuzzy	fuzzy	ADJ
ejpam-3344	434	7	la	la	NOUN
ejpam-3344	434	8	-	-	PUNCT
ejpam-3344	434	9	subring	subring	NOUN
ejpam-3344	434	10	of	of	ADP
ejpam-3344	434	11	r.	r.	PROPN
ejpam-3344	434	12	n.	n.	PROPN
ejpam-3344	434	13	kausar	kausar	PROPN
ejpam-3344	434	14	,	,	PUNCT
ejpam-3344	434	15	m.	m.	NOUN
ejpam-3344	434	16	a.	a.	PROPN
ejpam-3344	434	17	waqar	waqar	PROPN
ejpam-3344	434	18	/	/	SYM
ejpam-3344	434	19	eur	eur	PROPN
ejpam-3344	434	20	.	.	PUNCT
ejpam-3344	435	1	j.	j.	PROPN
ejpam-3344	435	2	pure	pure	PROPN
ejpam-3344	435	3	appl	appl	PROPN
ejpam-3344	435	4	.	.	PROPN
ejpam-3344	435	5	math	math	PROPN
ejpam-3344	435	6	,	,	PUNCT
ejpam-3344	435	7	12	12	NUM
ejpam-3344	435	8	(	(	PUNCT
ejpam-3344	435	9	1	1	NUM
ejpam-3344	435	10	)	)	PUNCT
ejpam-3344	435	11	(	(	PUNCT
ejpam-3344	435	12	2019	2019	NUM
ejpam-3344	435	13	)	)	PUNCT
ejpam-3344	435	14	,	,	PUNCT
ejpam-3344	435	15	226	226	NUM
ejpam-3344	435	16	-	-	SYM
ejpam-3344	435	17	250	250	NUM
ejpam-3344	435	18	242	242	NUM
ejpam-3344	435	19	lemma	lemma	PROPN
ejpam-3344	435	20	16	16	NUM
ejpam-3344	435	21	.	.	PUNCT
ejpam-3344	436	1	let	let	VERB
ejpam-3344	436	2	r	r	PRON
ejpam-3344	436	3	be	be	AUX
ejpam-3344	436	4	a	a	DET
ejpam-3344	436	5	left	left	ADJ
ejpam-3344	436	6	weakly	weakly	ADJ
ejpam-3344	436	7	regular	regular	ADJ
ejpam-3344	436	8	la	la	ADJ
ejpam-3344	436	9	-	-	PUNCT
ejpam-3344	436	10	ring	ring	NOUN
ejpam-3344	436	11	.	.	PUNCT
ejpam-3344	437	1	then	then	ADV
ejpam-3344	437	2	every	every	DET
ejpam-3344	437	3	intuitionistic	intuitionistic	ADJ
ejpam-3344	437	4	fuzzy	fuzzy	ADJ
ejpam-3344	437	5	right	right	ADJ
ejpam-3344	437	6	ideal	ideal	NOUN
ejpam-3344	437	7	of	of	ADP
ejpam-3344	437	8	r	r	NOUN
ejpam-3344	437	9	is	be	AUX
ejpam-3344	437	10	an	an	DET
ejpam-3344	437	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	437	12	fuzzy	fuzzy	ADJ
ejpam-3344	437	13	ideal	ideal	NOUN
ejpam-3344	437	14	of	of	ADP
ejpam-3344	437	15	r.	r.	PROPN
ejpam-3344	437	16	proof	proof	NOUN
ejpam-3344	437	17	.	.	PUNCT
ejpam-3344	438	1	let	let	VERB
ejpam-3344	438	2	a	a	DET
ejpam-3344	438	3	=	=	SYM
ejpam-3344	438	4	(	(	PUNCT
ejpam-3344	438	5	µa	µa	PROPN
ejpam-3344	438	6	,	,	PUNCT
ejpam-3344	438	7	γa	γa	PROPN
ejpam-3344	438	8	)	)	PUNCT
ejpam-3344	438	9	be	be	VERB
ejpam-3344	438	10	an	an	DET
ejpam-3344	438	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	438	12	fuzzy	fuzzy	ADJ
ejpam-3344	438	13	right	right	ADJ
ejpam-3344	438	14	ideal	ideal	NOUN
ejpam-3344	438	15	of	of	ADP
ejpam-3344	438	16	r	r	NOUN
ejpam-3344	438	17	and	and	CCONJ
ejpam-3344	438	18	x	x	NOUN
ejpam-3344	438	19	,	,	PUNCT
ejpam-3344	438	20	y	y	PROPN
ejpam-3344	438	21	∈	∈	PROPN
ejpam-3344	438	22	r	r	NOUN
ejpam-3344	438	23	,	,	PUNCT
ejpam-3344	438	24	this	this	PRON
ejpam-3344	438	25	implies	imply	VERB
ejpam-3344	438	26	that	that	SCONJ
ejpam-3344	438	27	there	there	PRON
ejpam-3344	438	28	exist	exist	VERB
ejpam-3344	438	29	a	a	DET
ejpam-3344	438	30	,	,	PUNCT
ejpam-3344	438	31	b	b	X
ejpam-3344	438	32	∈	∈	NOUN
ejpam-3344	438	33	r	r	NOUN
ejpam-3344	439	1	such	such	ADJ
ejpam-3344	439	2	that	that	PRON
ejpam-3344	439	3	x	x	X
ejpam-3344	439	4	=	=	SYM
ejpam-3344	439	5	(	(	PUNCT
ejpam-3344	439	6	ax)(bx	ax)(bx	NOUN
ejpam-3344	439	7	)	)	PUNCT
ejpam-3344	439	8	.	.	PUNCT
ejpam-3344	440	1	thus	thus	ADV
ejpam-3344	440	2	µa(xy	µa(xy	NUM
ejpam-3344	440	3	)	)	PUNCT
ejpam-3344	440	4	=	=	SYM
ejpam-3344	440	5	µa(((ax)(bx))y	µa(((ax)(bx))y	PROPN
ejpam-3344	440	6	)	)	PUNCT
ejpam-3344	440	7	=	=	SYM
ejpam-3344	440	8	µa(y(bx))(ax	µa(y(bx))(ax	PROPN
ejpam-3344	440	9	)	)	PUNCT
ejpam-3344	440	10	≥	≥	NOUN
ejpam-3344	440	11	µa(y(bx	µa(y(bx	PROPN
ejpam-3344	440	12	)	)	PUNCT
ejpam-3344	440	13	)	)	PUNCT
ejpam-3344	440	14	≥	≥	NOUN
ejpam-3344	440	15	µa(y	µa(y	NOUN
ejpam-3344	440	16	)	)	PUNCT
ejpam-3344	440	17	and	and	CCONJ
ejpam-3344	440	18	γa(xy	γa(xy	PROPN
ejpam-3344	440	19	)	)	PUNCT
ejpam-3344	440	20	=	=	SYM
ejpam-3344	440	21	γa(((ax)(bx))y	γa(((ax)(bx))y	PROPN
ejpam-3344	440	22	)	)	PUNCT
ejpam-3344	440	23	=	=	SYM
ejpam-3344	440	24	γa(y(bx))(ax	γa(y(bx))(ax	PROPN
ejpam-3344	440	25	)	)	PUNCT
ejpam-3344	440	26	≤	≤	NUM
ejpam-3344	441	1	γa(y(bx	γa(y(bx	PROPN
ejpam-3344	441	2	)	)	PUNCT
ejpam-3344	441	3	)	)	PUNCT
ejpam-3344	442	1	≤	≤	NUM
ejpam-3344	442	2	γa(y	γa(y	NUM
ejpam-3344	442	3	)	)	PUNCT
ejpam-3344	442	4	.	.	PUNCT
ejpam-3344	443	1	hence	hence	ADV
ejpam-3344	443	2	a	a	PRON
ejpam-3344	443	3	is	be	AUX
ejpam-3344	443	4	an	an	DET
ejpam-3344	443	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	443	6	fuzzy	fuzzy	ADJ
ejpam-3344	443	7	ideal	ideal	NOUN
ejpam-3344	443	8	of	of	ADP
ejpam-3344	443	9	r.	r.	PROPN
ejpam-3344	443	10	remark	remark	PROPN
ejpam-3344	443	11	10	10	NUM
ejpam-3344	443	12	.	.	PUNCT
ejpam-3344	444	1	the	the	DET
ejpam-3344	444	2	concept	concept	NOUN
ejpam-3344	444	3	of	of	ADP
ejpam-3344	444	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	444	5	fuzzy	fuzzy	ADJ
ejpam-3344	444	6	(	(	PUNCT
ejpam-3344	444	7	right	right	ADJ
ejpam-3344	444	8	,	,	PUNCT
ejpam-3344	444	9	two	two	NUM
ejpam-3344	444	10	-	-	PUNCT
ejpam-3344	444	11	sided	sided	ADJ
ejpam-3344	444	12	)	)	PUNCT
ejpam-3344	444	13	ideals	ideal	NOUN
ejpam-3344	444	14	coincides	coincide	VERB
ejpam-3344	444	15	in	in	ADP
ejpam-3344	444	16	left	left	ADJ
ejpam-3344	444	17	weakly	weakly	ADV
ejpam-3344	444	18	regular	regular	ADJ
ejpam-3344	444	19	la	la	NOUN
ejpam-3344	444	20	-	-	PUNCT
ejpam-3344	444	21	rings	ring	NOUN
ejpam-3344	444	22	.	.	PUNCT
ejpam-3344	445	1	lemma	lemma	PROPN
ejpam-3344	445	2	17	17	NUM
ejpam-3344	445	3	.	.	PUNCT
ejpam-3344	446	1	let	let	VERB
ejpam-3344	446	2	r	r	PRON
ejpam-3344	446	3	be	be	AUX
ejpam-3344	446	4	a	a	DET
ejpam-3344	446	5	left	left	ADJ
ejpam-3344	446	6	weakly	weakly	ADJ
ejpam-3344	446	7	regular	regular	ADJ
ejpam-3344	446	8	la	la	NOUN
ejpam-3344	446	9	-	-	NOUN
ejpam-3344	446	10	ring	ring	NOUN
ejpam-3344	446	11	with	with	ADP
ejpam-3344	446	12	left	left	ADJ
ejpam-3344	446	13	identity	identity	NOUN
ejpam-3344	446	14	e.	e.	PROPN
ejpam-3344	446	15	then	then	ADV
ejpam-3344	446	16	every	every	DET
ejpam-3344	446	17	intuitionistic	intuitionistic	ADJ
ejpam-3344	446	18	fuzzy	fuzzy	ADJ
ejpam-3344	446	19	left	leave	VERB
ejpam-3344	446	20	ideal	ideal	NOUN
ejpam-3344	446	21	of	of	ADP
ejpam-3344	446	22	r	r	NOUN
ejpam-3344	446	23	is	be	AUX
ejpam-3344	446	24	an	an	DET
ejpam-3344	446	25	intuitionistic	intuitionistic	ADJ
ejpam-3344	446	26	fuzzy	fuzzy	ADJ
ejpam-3344	446	27	ideal	ideal	NOUN
ejpam-3344	446	28	of	of	ADP
ejpam-3344	446	29	r.	r.	PROPN
ejpam-3344	446	30	proof	proof	NOUN
ejpam-3344	446	31	.	.	PUNCT
ejpam-3344	447	1	suppose	suppose	VERB
ejpam-3344	447	2	that	that	SCONJ
ejpam-3344	447	3	a	a	DET
ejpam-3344	447	4	=	=	SYM
ejpam-3344	447	5	(	(	PUNCT
ejpam-3344	447	6	µa	µa	PROPN
ejpam-3344	447	7	,	,	PUNCT
ejpam-3344	447	8	γa	γa	PROPN
ejpam-3344	447	9	)	)	PUNCT
ejpam-3344	447	10	is	be	AUX
ejpam-3344	447	11	an	an	DET
ejpam-3344	447	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	447	13	fuzzy	fuzzy	ADJ
ejpam-3344	447	14	left	leave	VERB
ejpam-3344	447	15	ideal	ideal	NOUN
ejpam-3344	447	16	of	of	ADP
ejpam-3344	447	17	r	r	NOUN
ejpam-3344	447	18	and	and	CCONJ
ejpam-3344	447	19	x	x	NOUN
ejpam-3344	447	20	,	,	PUNCT
ejpam-3344	447	21	y	y	PROPN
ejpam-3344	447	22	∈	∈	PROPN
ejpam-3344	447	23	r	r	NOUN
ejpam-3344	447	24	,	,	PUNCT
ejpam-3344	447	25	this	this	PRON
ejpam-3344	447	26	means	mean	VERB
ejpam-3344	447	27	that	that	SCONJ
ejpam-3344	447	28	there	there	PRON
ejpam-3344	447	29	exist	exist	VERB
ejpam-3344	447	30	a	a	DET
ejpam-3344	447	31	,	,	PUNCT
ejpam-3344	447	32	b	b	X
ejpam-3344	447	33	∈	∈	NOUN
ejpam-3344	447	34	r	r	NOUN
ejpam-3344	447	35	such	such	ADJ
ejpam-3344	447	36	that	that	PRON
ejpam-3344	447	37	x	x	X
ejpam-3344	447	38	=	=	SYM
ejpam-3344	447	39	(	(	PUNCT
ejpam-3344	447	40	ax)(bx	ax)(bx	NOUN
ejpam-3344	447	41	)	)	PUNCT
ejpam-3344	447	42	.	.	PUNCT
ejpam-3344	448	1	thus	thus	ADV
ejpam-3344	448	2	µa(xy	µa(xy	NUM
ejpam-3344	448	3	)	)	PUNCT
ejpam-3344	448	4	=	=	SYM
ejpam-3344	448	5	µa(((ax)(bx))y	µa(((ax)(bx))y	PROPN
ejpam-3344	448	6	)	)	PUNCT
ejpam-3344	448	7	=	=	PUNCT
ejpam-3344	448	8	µa(((ab)(xx))y	µa(((ab)(xx))y	PROPN
ejpam-3344	448	9	)	)	PUNCT
ejpam-3344	448	10	=	=	PUNCT
ejpam-3344	448	11	µa((x((ab)x))y	µa((x((ab)x))y	X
ejpam-3344	448	12	)	)	PUNCT
ejpam-3344	448	13	=	=	SYM
ejpam-3344	448	14	µa((y((ab)x))x	µa((y((ab)x))x	X
ejpam-3344	448	15	)	)	PUNCT
ejpam-3344	448	16	≥	≥	NOUN
ejpam-3344	448	17	µa(x	µa(x	NOUN
ejpam-3344	448	18	)	)	PUNCT
ejpam-3344	448	19	and	and	CCONJ
ejpam-3344	448	20	γa(xy	γa(xy	PROPN
ejpam-3344	448	21	)	)	PUNCT
ejpam-3344	448	22	=	=	SYM
ejpam-3344	448	23	γa(((ax)(bx))y	γa(((ax)(bx))y	PROPN
ejpam-3344	448	24	)	)	PUNCT
ejpam-3344	448	25	=	=	SYM
ejpam-3344	448	26	γa(((ab)(xx))y	γa(((ab)(xx))y	NOUN
ejpam-3344	448	27	)	)	PUNCT
ejpam-3344	448	28	=	=	PUNCT
ejpam-3344	448	29	γa((x((ab)x))y	γa((x((ab)x))y	NUM
ejpam-3344	448	30	)	)	PUNCT
ejpam-3344	448	31	=	=	SYM
ejpam-3344	448	32	γa((y((ab)x))x	γa((y((ab)x))x	X
ejpam-3344	448	33	)	)	PUNCT
ejpam-3344	448	34	≤	≤	NOUN
ejpam-3344	448	35	γa(x	γa(x	NUM
ejpam-3344	448	36	)	)	PUNCT
ejpam-3344	448	37	.	.	PUNCT
ejpam-3344	449	1	therefore	therefore	ADV
ejpam-3344	449	2	a	a	PRON
ejpam-3344	449	3	is	be	AUX
ejpam-3344	449	4	an	an	DET
ejpam-3344	449	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	449	6	fuzzy	fuzzy	ADJ
ejpam-3344	449	7	ideal	ideal	NOUN
ejpam-3344	449	8	of	of	ADP
ejpam-3344	449	9	r.	r.	PROPN
ejpam-3344	449	10	remark	remark	PROPN
ejpam-3344	449	11	11	11	NUM
ejpam-3344	449	12	.	.	PUNCT
ejpam-3344	450	1	the	the	DET
ejpam-3344	450	2	concept	concept	NOUN
ejpam-3344	450	3	of	of	ADP
ejpam-3344	450	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	450	5	fuzzy	fuzzy	ADJ
ejpam-3344	450	6	(	(	PUNCT
ejpam-3344	450	7	left	left	ADJ
ejpam-3344	450	8	,	,	PUNCT
ejpam-3344	450	9	two	two	NUM
ejpam-3344	450	10	-	-	PUNCT
ejpam-3344	450	11	sided	sided	ADJ
ejpam-3344	450	12	)	)	PUNCT
ejpam-3344	450	13	ideals	ideal	NOUN
ejpam-3344	450	14	coincides	coincide	VERB
ejpam-3344	450	15	in	in	ADP
ejpam-3344	450	16	left	left	ADJ
ejpam-3344	450	17	weakly	weakly	ADV
ejpam-3344	450	18	regular	regular	ADJ
ejpam-3344	450	19	la	la	NOUN
ejpam-3344	450	20	-	-	PUNCT
ejpam-3344	450	21	rings	ring	NOUN
ejpam-3344	450	22	with	with	ADP
ejpam-3344	450	23	left	left	ADJ
ejpam-3344	450	24	identity	identity	NOUN
ejpam-3344	450	25	e.	e.	PROPN
ejpam-3344	450	26	proposition	proposition	PROPN
ejpam-3344	450	27	10	10	NUM
ejpam-3344	450	28	.	.	PUNCT
ejpam-3344	451	1	every	every	DET
ejpam-3344	451	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	451	3	fuzzy	fuzzy	ADJ
ejpam-3344	451	4	generalized	generalize	VERB
ejpam-3344	451	5	bi	bi	NOUN
ejpam-3344	451	6	-	-	NOUN
ejpam-3344	451	7	ideal	ideal	NOUN
ejpam-3344	451	8	of	of	ADP
ejpam-3344	451	9	a	a	DET
ejpam-3344	451	10	left	left	ADJ
ejpam-3344	451	11	weakly	weakly	ADJ
ejpam-3344	451	12	regular	regular	ADJ
ejpam-3344	451	13	la	la	ADJ
ejpam-3344	451	14	-	-	PUNCT
ejpam-3344	451	15	ring	ring	NOUN
ejpam-3344	451	16	r	r	NOUN
ejpam-3344	451	17	with	with	ADP
ejpam-3344	451	18	left	left	ADJ
ejpam-3344	451	19	identity	identity	NOUN
ejpam-3344	451	20	e	e	NOUN
ejpam-3344	451	21	,	,	PUNCT
ejpam-3344	451	22	is	be	AUX
ejpam-3344	451	23	an	an	DET
ejpam-3344	451	24	intuitionistic	intuitionistic	ADJ
ejpam-3344	451	25	fuzzy	fuzzy	ADJ
ejpam-3344	451	26	bi	bi	NOUN
ejpam-3344	451	27	-	-	NOUN
ejpam-3344	451	28	ideal	ideal	NOUN
ejpam-3344	451	29	of	of	ADP
ejpam-3344	451	30	r.	r.	PROPN
ejpam-3344	451	31	proof	proof	PROPN
ejpam-3344	451	32	.	.	PUNCT
ejpam-3344	452	1	assume	assume	VERB
ejpam-3344	452	2	that	that	SCONJ
ejpam-3344	452	3	a	a	DET
ejpam-3344	452	4	=	=	SYM
ejpam-3344	452	5	(	(	PUNCT
ejpam-3344	452	6	µa	µa	PROPN
ejpam-3344	452	7	,	,	PUNCT
ejpam-3344	452	8	γa	γa	PROPN
ejpam-3344	452	9	)	)	PUNCT
ejpam-3344	452	10	is	be	AUX
ejpam-3344	452	11	an	an	DET
ejpam-3344	452	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	452	13	fuzzy	fuzzy	ADJ
ejpam-3344	452	14	generalized	generalize	VERB
ejpam-3344	452	15	bi	bi	NOUN
ejpam-3344	452	16	-	-	NOUN
ejpam-3344	452	17	ideal	ideal	NOUN
ejpam-3344	452	18	of	of	ADP
ejpam-3344	452	19	r	r	NOUN
ejpam-3344	452	20	and	and	CCONJ
ejpam-3344	452	21	x	x	NOUN
ejpam-3344	452	22	,	,	PUNCT
ejpam-3344	452	23	y	y	PROPN
ejpam-3344	452	24	∈	∈	PROPN
ejpam-3344	452	25	r	r	NOUN
ejpam-3344	452	26	,	,	PUNCT
ejpam-3344	452	27	then	then	ADV
ejpam-3344	452	28	there	there	PRON
ejpam-3344	452	29	exist	exist	VERB
ejpam-3344	452	30	elements	element	NOUN
ejpam-3344	452	31	a	a	PRON
ejpam-3344	452	32	,	,	PUNCT
ejpam-3344	452	33	b	b	X
ejpam-3344	452	34	∈	∈	NOUN
ejpam-3344	452	35	r	r	NOUN
ejpam-3344	453	1	such	such	ADJ
ejpam-3344	453	2	that	that	PRON
ejpam-3344	453	3	x	x	X
ejpam-3344	453	4	=	=	SYM
ejpam-3344	453	5	(	(	PUNCT
ejpam-3344	453	6	ax)(bx	ax)(bx	NOUN
ejpam-3344	453	7	)	)	PUNCT
ejpam-3344	453	8	.	.	PUNCT
ejpam-3344	454	1	we	we	PRON
ejpam-3344	454	2	have	have	VERB
ejpam-3344	454	3	to	to	PART
ejpam-3344	454	4	show	show	VERB
ejpam-3344	454	5	that	that	SCONJ
ejpam-3344	454	6	a	a	PRON
ejpam-3344	454	7	is	be	AUX
ejpam-3344	454	8	an	an	DET
ejpam-3344	454	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	454	10	fuzzy	fuzzy	ADJ
ejpam-3344	454	11	la	la	NOUN
ejpam-3344	454	12	-	-	PUNCT
ejpam-3344	454	13	subring	subring	NOUN
ejpam-3344	454	14	of	of	ADP
ejpam-3344	454	15	r.	r.	PROPN
ejpam-3344	454	16	thus	thus	ADV
ejpam-3344	454	17	µa(xy	µa(xy	PROPN
ejpam-3344	454	18	)	)	PUNCT
ejpam-3344	454	19	=	=	SYM
ejpam-3344	454	20	µa(((ax)(bx))y	µa(((ax)(bx))y	PROPN
ejpam-3344	454	21	)	)	PUNCT
ejpam-3344	454	22	=	=	PUNCT
ejpam-3344	454	23	µa(((ab)(xx))y	µa(((ab)(xx))y	PROPN
ejpam-3344	454	24	)	)	PUNCT
ejpam-3344	454	25	=	=	PUNCT
ejpam-3344	454	26	µa((x((ab)x))y	µa((x((ab)x))y	X
ejpam-3344	454	27	)	)	PUNCT
ejpam-3344	454	28	≥	≥	PROPN
ejpam-3344	454	29	min{µa(x	min{µa(x	NOUN
ejpam-3344	454	30	)	)	PUNCT
ejpam-3344	454	31	,	,	PUNCT
ejpam-3344	454	32	µa(y	µa(y	NOUN
ejpam-3344	454	33	)	)	PUNCT
ejpam-3344	454	34	}	}	PUNCT
ejpam-3344	454	35	and	and	CCONJ
ejpam-3344	454	36	γa(xy	γa(xy	PROPN
ejpam-3344	454	37	)	)	PUNCT
ejpam-3344	454	38	=	=	SYM
ejpam-3344	454	39	γa(((ax)(bx))y	γa(((ax)(bx))y	PROPN
ejpam-3344	454	40	)	)	PUNCT
ejpam-3344	454	41	=	=	SYM
ejpam-3344	454	42	γa(((ab)(xx))y	γa(((ab)(xx))y	NOUN
ejpam-3344	454	43	)	)	PUNCT
ejpam-3344	454	44	=	=	PUNCT
ejpam-3344	454	45	γa((x((ab)x))y	γa((x((ab)x))y	CCONJ
ejpam-3344	454	46	)	)	PUNCT
ejpam-3344	454	47	≤	≤	NUM
ejpam-3344	454	48	max{γa(x	max{γa(x	NOUN
ejpam-3344	454	49	)	)	PUNCT
ejpam-3344	454	50	,	,	PUNCT
ejpam-3344	454	51	γa(y	γa(y	NOUN
ejpam-3344	454	52	)	)	PUNCT
ejpam-3344	454	53	}	}	PUNCT
ejpam-3344	454	54	.	.	PUNCT
ejpam-3344	455	1	so	so	ADV
ejpam-3344	455	2	a	a	PRON
ejpam-3344	455	3	is	be	AUX
ejpam-3344	455	4	an	an	DET
ejpam-3344	455	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	455	6	fuzzy	fuzzy	ADJ
ejpam-3344	455	7	la	la	NOUN
ejpam-3344	455	8	-	-	PUNCT
ejpam-3344	455	9	subring	subring	NOUN
ejpam-3344	455	10	of	of	ADP
ejpam-3344	455	11	r.	r.	PROPN
ejpam-3344	455	12	n.	n.	PROPN
ejpam-3344	455	13	kausar	kausar	PROPN
ejpam-3344	455	14	,	,	PUNCT
ejpam-3344	455	15	m.	m.	NOUN
ejpam-3344	455	16	a.	a.	PROPN
ejpam-3344	455	17	waqar	waqar	PROPN
ejpam-3344	455	18	/	/	SYM
ejpam-3344	455	19	eur	eur	PROPN
ejpam-3344	455	20	.	.	PUNCT
ejpam-3344	456	1	j.	j.	PROPN
ejpam-3344	456	2	pure	pure	PROPN
ejpam-3344	456	3	appl	appl	PROPN
ejpam-3344	456	4	.	.	PROPN
ejpam-3344	456	5	math	math	PROPN
ejpam-3344	456	6	,	,	PUNCT
ejpam-3344	456	7	12	12	NUM
ejpam-3344	456	8	(	(	PUNCT
ejpam-3344	456	9	1	1	NUM
ejpam-3344	456	10	)	)	PUNCT
ejpam-3344	456	11	(	(	PUNCT
ejpam-3344	456	12	2019	2019	NUM
ejpam-3344	456	13	)	)	PUNCT
ejpam-3344	456	14	,	,	PUNCT
ejpam-3344	456	15	226	226	NUM
ejpam-3344	456	16	-	-	SYM
ejpam-3344	456	17	250	250	NUM
ejpam-3344	456	18	243	243	NUM
ejpam-3344	456	19	remark	remark	NOUN
ejpam-3344	456	20	12	12	NUM
ejpam-3344	456	21	.	.	PUNCT
ejpam-3344	457	1	it	it	PRON
ejpam-3344	457	2	is	be	AUX
ejpam-3344	457	3	easy	easy	ADJ
ejpam-3344	457	4	to	to	PART
ejpam-3344	457	5	see	see	VERB
ejpam-3344	457	6	that	that	SCONJ
ejpam-3344	457	7	,	,	PUNCT
ejpam-3344	457	8	if	if	SCONJ
ejpam-3344	457	9	r	r	NOUN
ejpam-3344	457	10	is	be	AUX
ejpam-3344	457	11	a	a	DET
ejpam-3344	457	12	left	left	ADJ
ejpam-3344	457	13	(	(	PUNCT
ejpam-3344	457	14	right	right	ADJ
ejpam-3344	457	15	)	)	PUNCT
ejpam-3344	457	16	weakly	weakly	ADV
ejpam-3344	457	17	regular	regular	ADJ
ejpam-3344	457	18	locally	locally	ADV
ejpam-3344	457	19	associative	associative	ADJ
ejpam-3344	457	20	la	la	ADJ
ejpam-3344	457	21	-	-	PUNCT
ejpam-3344	457	22	ring	ring	NOUN
ejpam-3344	457	23	.	.	PUNCT
ejpam-3344	458	1	then	then	ADV
ejpam-3344	458	2	for	for	ADP
ejpam-3344	458	3	every	every	DET
ejpam-3344	458	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	458	5	fuzzy	fuzzy	ADJ
ejpam-3344	458	6	left	left	ADJ
ejpam-3344	458	7	(	(	PUNCT
ejpam-3344	458	8	right	right	ADJ
ejpam-3344	458	9	)	)	PUNCT
ejpam-3344	458	10	ideal	ideal	NOUN
ejpam-3344	458	11	a	a	PRON
ejpam-3344	458	12	=	=	X
ejpam-3344	458	13	(	(	PUNCT
ejpam-3344	458	14	µa	µa	PROPN
ejpam-3344	458	15	,	,	PUNCT
ejpam-3344	458	16	γa	γa	PROPN
ejpam-3344	458	17	)	)	PUNCT
ejpam-3344	458	18	of	of	ADP
ejpam-3344	458	19	r	r	NOUN
ejpam-3344	458	20	,	,	PUNCT
ejpam-3344	458	21	a(an	a(an	PROPN
ejpam-3344	458	22	)	)	PUNCT
ejpam-3344	458	23	=	=	PUNCT
ejpam-3344	458	24	a(a2n	a(a2n	VERB
ejpam-3344	458	25	)	)	PUNCT
ejpam-3344	458	26	for	for	ADP
ejpam-3344	458	27	all	all	DET
ejpam-3344	458	28	a	a	DET
ejpam-3344	458	29	∈	∈	NOUN
ejpam-3344	458	30	r	r	NOUN
ejpam-3344	458	31	,	,	PUNCT
ejpam-3344	458	32	where	where	SCONJ
ejpam-3344	458	33	n	n	PRON
ejpam-3344	458	34	is	be	AUX
ejpam-3344	458	35	any	any	DET
ejpam-3344	458	36	positive	positive	ADJ
ejpam-3344	458	37	integer	integer	NOUN
ejpam-3344	458	38	.	.	PUNCT
ejpam-3344	459	1	theorem	theorem	NOUN
ejpam-3344	459	2	10	10	NUM
ejpam-3344	459	3	.	.	PUNCT
ejpam-3344	460	1	let	let	VERB
ejpam-3344	460	2	r	r	PRON
ejpam-3344	460	3	be	be	AUX
ejpam-3344	460	4	an	an	DET
ejpam-3344	460	5	la	la	NOUN
ejpam-3344	460	6	-	-	NOUN
ejpam-3344	460	7	ring	ring	NOUN
ejpam-3344	460	8	with	with	ADP
ejpam-3344	460	9	left	left	ADJ
ejpam-3344	460	10	identity	identity	NOUN
ejpam-3344	460	11	e	e	NOUN
ejpam-3344	460	12	,	,	PUNCT
ejpam-3344	460	13	such	such	ADJ
ejpam-3344	460	14	that	that	SCONJ
ejpam-3344	460	15	(	(	PUNCT
ejpam-3344	460	16	xe)r	xe)r	PROPN
ejpam-3344	460	17	=	=	SYM
ejpam-3344	460	18	xr	xr	PROPN
ejpam-3344	460	19	for	for	ADP
ejpam-3344	460	20	all	all	DET
ejpam-3344	460	21	x	x	PROPN
ejpam-3344	460	22	∈	∈	PROPN
ejpam-3344	460	23	r.	r.	NOUN
ejpam-3344	460	24	then	then	ADV
ejpam-3344	460	25	the	the	DET
ejpam-3344	460	26	following	follow	VERB
ejpam-3344	460	27	conditions	condition	NOUN
ejpam-3344	460	28	are	be	AUX
ejpam-3344	460	29	equivalent	equivalent	ADJ
ejpam-3344	460	30	.	.	PUNCT
ejpam-3344	461	1	(	(	PUNCT
ejpam-3344	461	2	1	1	X
ejpam-3344	461	3	)	)	PUNCT
ejpam-3344	461	4	r	r	NOUN
ejpam-3344	461	5	is	be	AUX
ejpam-3344	461	6	a	a	DET
ejpam-3344	461	7	left	left	ADJ
ejpam-3344	461	8	weakly	weakly	ADV
ejpam-3344	461	9	regular	regular	ADJ
ejpam-3344	461	10	.	.	PUNCT
ejpam-3344	462	1	(	(	PUNCT
ejpam-3344	462	2	2	2	X
ejpam-3344	462	3	)	)	PUNCT
ejpam-3344	462	4	a	a	DET
ejpam-3344	462	5	∩	∩	ADJ
ejpam-3344	462	6	b	b	X
ejpam-3344	462	7	=	=	SYM
ejpam-3344	462	8	a	a	DET
ejpam-3344	462	9	◦	◦	NOUN
ejpam-3344	462	10	b	b	NOUN
ejpam-3344	462	11	for	for	ADP
ejpam-3344	462	12	every	every	DET
ejpam-3344	462	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	462	14	fuzzy	fuzzy	ADJ
ejpam-3344	462	15	right	right	ADJ
ejpam-3344	462	16	ideal	ideal	NOUN
ejpam-3344	462	17	a	a	PRON
ejpam-3344	462	18	and	and	CCONJ
ejpam-3344	462	19	every	every	DET
ejpam-3344	462	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	462	21	fuzzy	fuzzy	ADJ
ejpam-3344	462	22	left	leave	VERB
ejpam-3344	462	23	ideal	ideal	PROPN
ejpam-3344	462	24	b	b	PROPN
ejpam-3344	462	25	of	of	ADP
ejpam-3344	462	26	r.	r.	PROPN
ejpam-3344	462	27	proof	proof	NOUN
ejpam-3344	462	28	.	.	PUNCT
ejpam-3344	463	1	suppose	suppose	VERB
ejpam-3344	463	2	that	that	SCONJ
ejpam-3344	463	3	(	(	PUNCT
ejpam-3344	463	4	1	1	X
ejpam-3344	463	5	)	)	PUNCT
ejpam-3344	463	6	holds	hold	VERB
ejpam-3344	463	7	.	.	PUNCT
ejpam-3344	464	1	since	since	SCONJ
ejpam-3344	464	2	a	a	DET
ejpam-3344	464	3	◦	◦	NOUN
ejpam-3344	464	4	b	b	NUM
ejpam-3344	464	5	⊆	⊆	NUM
ejpam-3344	464	6	a	a	DET
ejpam-3344	464	7	∩	∩	ADJ
ejpam-3344	464	8	b	b	NOUN
ejpam-3344	464	9	for	for	ADP
ejpam-3344	464	10	every	every	DET
ejpam-3344	464	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	464	12	fuzzy	fuzzy	ADJ
ejpam-3344	464	13	right	right	ADJ
ejpam-3344	464	14	ideal	ideal	NOUN
ejpam-3344	464	15	a	a	PRON
ejpam-3344	464	16	=	=	X
ejpam-3344	464	17	(	(	PUNCT
ejpam-3344	464	18	µa	µa	PROPN
ejpam-3344	464	19	,	,	PUNCT
ejpam-3344	464	20	γa	γa	PROPN
ejpam-3344	464	21	)	)	PUNCT
ejpam-3344	464	22	and	and	CCONJ
ejpam-3344	464	23	every	every	DET
ejpam-3344	464	24	intuitionistic	intuitionistic	ADJ
ejpam-3344	464	25	fuzzy	fuzzy	ADJ
ejpam-3344	464	26	left	leave	VERB
ejpam-3344	464	27	ideal	ideal	PROPN
ejpam-3344	464	28	b	b	PROPN
ejpam-3344	465	1	=	=	PUNCT
ejpam-3344	465	2	(	(	PUNCT
ejpam-3344	465	3	µb	µb	PROPN
ejpam-3344	465	4	,	,	PUNCT
ejpam-3344	465	5	γb	γb	PROPN
ejpam-3344	465	6	)	)	PUNCT
ejpam-3344	465	7	of	of	ADP
ejpam-3344	465	8	r	r	NOUN
ejpam-3344	465	9	by	by	ADP
ejpam-3344	465	10	the	the	DET
ejpam-3344	465	11	lemma	lemma	PROPN
ejpam-3344	465	12	10	10	NUM
ejpam-3344	465	13	.	.	PUNCT
ejpam-3344	466	1	let	let	VERB
ejpam-3344	466	2	x	x	PUNCT
ejpam-3344	466	3	∈	∈	PROPN
ejpam-3344	466	4	r	r	NOUN
ejpam-3344	466	5	,	,	PUNCT
ejpam-3344	466	6	this	this	PRON
ejpam-3344	466	7	implies	imply	VERB
ejpam-3344	466	8	that	that	SCONJ
ejpam-3344	466	9	there	there	PRON
ejpam-3344	466	10	exist	exist	VERB
ejpam-3344	466	11	a	a	DET
ejpam-3344	466	12	,	,	PUNCT
ejpam-3344	466	13	b	b	X
ejpam-3344	466	14	∈	∈	NOUN
ejpam-3344	466	15	r	r	NOUN
ejpam-3344	467	1	such	such	ADJ
ejpam-3344	467	2	that	that	PRON
ejpam-3344	467	3	x	x	X
ejpam-3344	467	4	=	=	SYM
ejpam-3344	467	5	(	(	PUNCT
ejpam-3344	467	6	ax)(bx	ax)(bx	PROPN
ejpam-3344	467	7	)	)	PUNCT
ejpam-3344	467	8	=	=	SYM
ejpam-3344	467	9	(	(	PUNCT
ejpam-3344	467	10	ab)(xx	ab)(xx	PROPN
ejpam-3344	467	11	)	)	PUNCT
ejpam-3344	467	12	=	=	SYM
ejpam-3344	467	13	x((ab)x	x((ab)x	PROPN
ejpam-3344	467	14	)	)	PUNCT
ejpam-3344	467	15	.	.	PUNCT
ejpam-3344	468	1	now	now	ADV
ejpam-3344	468	2	(	(	PUNCT
ejpam-3344	468	3	µa	µa	ADP
ejpam-3344	468	4	◦	◦	NOUN
ejpam-3344	468	5	µb)(x	µb)(x	NOUN
ejpam-3344	468	6	)	)	PUNCT
ejpam-3344	468	7	=	=	PUNCT
ejpam-3344	469	1	∨x=∑n	∨x=∑n	NOUN
ejpam-3344	469	2	i=1	i=1	PROPN
ejpam-3344	469	3	aibi	aibi	NOUN
ejpam-3344	469	4	{	{	PUNCT
ejpam-3344	469	5	∧ni=1	∧ni=1	X
ejpam-3344	469	6	{	{	PUNCT
ejpam-3344	469	7	µa	µa	X
ejpam-3344	469	8	(	(	PUNCT
ejpam-3344	469	9	ai	ai	NOUN
ejpam-3344	469	10	)	)	PUNCT
ejpam-3344	469	11	∧	∧	NOUN
ejpam-3344	469	12	µb	µb	PROPN
ejpam-3344	469	13	(	(	PUNCT
ejpam-3344	469	14	bi	bi	NOUN
ejpam-3344	469	15	)	)	PUNCT
ejpam-3344	469	16	}	}	PUNCT
ejpam-3344	469	17	}	}	PUNCT
ejpam-3344	469	18	≥	≥	NOUN
ejpam-3344	469	19	µa(x	µa(x	NOUN
ejpam-3344	469	20	)	)	PUNCT
ejpam-3344	469	21	∧	∧	PROPN
ejpam-3344	469	22	µb((ab)x	µb((ab)x	NOUN
ejpam-3344	469	23	)	)	PUNCT
ejpam-3344	469	24	≥	≥	NOUN
ejpam-3344	469	25	µa(x	µa(x	NOUN
ejpam-3344	469	26	)	)	PUNCT
ejpam-3344	469	27	∧	∧	NOUN
ejpam-3344	469	28	µb(x	µb(x	PUNCT
ejpam-3344	469	29	)	)	PUNCT
ejpam-3344	469	30	=	=	SYM
ejpam-3344	469	31	(	(	PUNCT
ejpam-3344	469	32	µa	µa	ADP
ejpam-3344	469	33	∩	∩	ADJ
ejpam-3344	469	34	µb)(x	µb)(x	NOUN
ejpam-3344	469	35	)	)	PUNCT
ejpam-3344	469	36	and	and	CCONJ
ejpam-3344	469	37	(	(	PUNCT
ejpam-3344	469	38	γa	γa	AUX
ejpam-3344	469	39	◦	◦	VERB
ejpam-3344	469	40	γb)(x	γb)(x	PROPN
ejpam-3344	469	41	)	)	PUNCT
ejpam-3344	470	1	=	=	NOUN
ejpam-3344	471	1	∧x=∑n	∧x=∑n	NOUN
ejpam-3344	471	2	i=1	i=1	PROPN
ejpam-3344	471	3	aibi	aibi	PROPN
ejpam-3344	471	4	{	{	PUNCT
ejpam-3344	471	5	∨ni=1	∨ni=1	PRON
ejpam-3344	471	6	{	{	PUNCT
ejpam-3344	471	7	γa	γa	PROPN
ejpam-3344	471	8	(	(	PUNCT
ejpam-3344	471	9	ai	ai	PROPN
ejpam-3344	471	10	)	)	PUNCT
ejpam-3344	471	11	∨	∨	NUM
ejpam-3344	471	12	γb	γb	X
ejpam-3344	471	13	(	(	PUNCT
ejpam-3344	471	14	bi	bi	NOUN
ejpam-3344	471	15	)	)	PUNCT
ejpam-3344	471	16	}	}	PUNCT
ejpam-3344	471	17	}	}	PUNCT
ejpam-3344	471	18	≤	≤	NOUN
ejpam-3344	471	19	γa(x	γa(x	NUM
ejpam-3344	471	20	)	)	PUNCT
ejpam-3344	471	21	∨	∨	NUM
ejpam-3344	471	22	γb((ab)x	γb((ab)x	NOUN
ejpam-3344	471	23	)	)	PUNCT
ejpam-3344	471	24	≤	≤	NOUN
ejpam-3344	471	25	γa(x	γa(x	NUM
ejpam-3344	471	26	)	)	PUNCT
ejpam-3344	471	27	∨	∨	NOUN
ejpam-3344	471	28	γb(x	γb(x	NUM
ejpam-3344	471	29	)	)	PUNCT
ejpam-3344	472	1	=	=	PUNCT
ejpam-3344	472	2	(	(	PUNCT
ejpam-3344	472	3	γa	γa	NOUN
ejpam-3344	472	4	∪	∪	VERB
ejpam-3344	472	5	γb)(x	γb)(x	PROPN
ejpam-3344	472	6	)	)	PUNCT
ejpam-3344	472	7	.	.	PUNCT
ejpam-3344	473	1	thus	thus	ADV
ejpam-3344	473	2	µa∩µb	µa∩µb	PROPN
ejpam-3344	473	3	⊆	⊆	NUM
ejpam-3344	473	4	µa	µa	NOUN
ejpam-3344	473	5	◦	◦	NOUN
ejpam-3344	473	6	µb	µb	NOUN
ejpam-3344	473	7	and	and	CCONJ
ejpam-3344	473	8	γa∪γb	γa∪γb	NUM
ejpam-3344	473	9	⊇	⊇	PROPN
ejpam-3344	473	10	γa	γa	PROPN
ejpam-3344	473	11	◦	◦	NOUN
ejpam-3344	473	12	γb	γb	PROPN
ejpam-3344	473	13	,	,	PUNCT
ejpam-3344	473	14	i.e.	i.e.	X
ejpam-3344	473	15	,	,	PUNCT
ejpam-3344	473	16	a∩b	a∩b	PROPN
ejpam-3344	473	17	⊆	⊆	NUM
ejpam-3344	473	18	a	a	DET
ejpam-3344	473	19	◦	◦	NOUN
ejpam-3344	473	20	b.	b.	NOUN
ejpam-3344	473	21	hence	hence	ADV
ejpam-3344	473	22	a∩b	a∩b	PROPN
ejpam-3344	474	1	=	=	PUNCT
ejpam-3344	474	2	a	a	DET
ejpam-3344	474	3	◦	◦	NOUN
ejpam-3344	474	4	b	b	NOUN
ejpam-3344	474	5	,	,	PUNCT
ejpam-3344	474	6	i.e.	i.e.	X
ejpam-3344	474	7	,	,	PUNCT
ejpam-3344	474	8	(	(	PUNCT
ejpam-3344	474	9	1)⇒	1)⇒	NUM
ejpam-3344	474	10	(	(	PUNCT
ejpam-3344	474	11	2	2	NUM
ejpam-3344	474	12	)	)	PUNCT
ejpam-3344	474	13	.	.	PUNCT
ejpam-3344	475	1	assume	assume	VERB
ejpam-3344	475	2	that	that	SCONJ
ejpam-3344	475	3	(	(	PUNCT
ejpam-3344	475	4	2	2	X
ejpam-3344	475	5	)	)	PUNCT
ejpam-3344	475	6	is	be	AUX
ejpam-3344	475	7	true	true	ADJ
ejpam-3344	475	8	and	and	CCONJ
ejpam-3344	475	9	a	a	DET
ejpam-3344	475	10	∈	∈	PROPN
ejpam-3344	475	11	r.	r.	NOUN
ejpam-3344	475	12	then	then	ADV
ejpam-3344	475	13	ra	ra	PROPN
ejpam-3344	475	14	is	be	AUX
ejpam-3344	475	15	a	a	DET
ejpam-3344	475	16	left	left	ADJ
ejpam-3344	475	17	ideal	ideal	NOUN
ejpam-3344	475	18	of	of	ADP
ejpam-3344	475	19	r	r	NOUN
ejpam-3344	475	20	containing	contain	VERB
ejpam-3344	475	21	a	a	PRON
ejpam-3344	475	22	by	by	ADP
ejpam-3344	475	23	the	the	DET
ejpam-3344	475	24	lemma	lemma	PROPN
ejpam-3344	475	25	8	8	NUM
ejpam-3344	475	26	and	and	CCONJ
ejpam-3344	475	27	ar	ar	PROPN
ejpam-3344	475	28	∪	∪	PROPN
ejpam-3344	475	29	ra	ra	PROPN
ejpam-3344	475	30	is	be	AUX
ejpam-3344	475	31	a	a	DET
ejpam-3344	475	32	right	right	ADJ
ejpam-3344	475	33	ideal	ideal	NOUN
ejpam-3344	475	34	of	of	ADP
ejpam-3344	475	35	r	r	NOUN
ejpam-3344	475	36	containing	contain	VERB
ejpam-3344	475	37	a	a	PRON
ejpam-3344	475	38	by	by	ADP
ejpam-3344	475	39	the	the	DET
ejpam-3344	475	40	proposition	proposition	NOUN
ejpam-3344	475	41	5	5	NUM
ejpam-3344	475	42	.	.	PUNCT
ejpam-3344	476	1	so	so	ADV
ejpam-3344	476	2	χra	χra	PROPN
ejpam-3344	476	3	is	be	AUX
ejpam-3344	476	4	an	an	DET
ejpam-3344	476	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	476	6	fuzzy	fuzzy	ADJ
ejpam-3344	476	7	left	leave	VERB
ejpam-3344	476	8	ideal	ideal	NOUN
ejpam-3344	476	9	and	and	CCONJ
ejpam-3344	476	10	χar∪ra	χar∪ra	ADV
ejpam-3344	476	11	is	be	AUX
ejpam-3344	476	12	an	an	DET
ejpam-3344	476	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	476	14	fuzzy	fuzzy	ADJ
ejpam-3344	476	15	right	right	ADJ
ejpam-3344	476	16	ideal	ideal	NOUN
ejpam-3344	476	17	of	of	ADP
ejpam-3344	476	18	r	r	NOUN
ejpam-3344	476	19	,	,	PUNCT
ejpam-3344	476	20	by	by	ADP
ejpam-3344	476	21	the	the	DET
ejpam-3344	476	22	proposition	proposition	NOUN
ejpam-3344	476	23	2	2	NUM
ejpam-3344	476	24	.	.	PUNCT
ejpam-3344	476	25	then	then	ADV
ejpam-3344	476	26	by	by	ADP
ejpam-3344	476	27	our	our	PRON
ejpam-3344	476	28	assumption	assumption	NOUN
ejpam-3344	476	29	χar∪ra	χar∪ra	PROPN
ejpam-3344	476	30	∩	∩	PROPN
ejpam-3344	476	31	χra	χra	PROPN
ejpam-3344	476	32	=	=	SYM
ejpam-3344	476	33	χar∪ra	χar∪ra	PUNCT
ejpam-3344	476	34	◦	◦	NOUN
ejpam-3344	476	35	χra	χra	PROPN
ejpam-3344	476	36	,	,	PUNCT
ejpam-3344	476	37	i.e.	i.e.	X
ejpam-3344	476	38	,	,	PUNCT
ejpam-3344	476	39	χ(ar∪ra)∩ra	χ(ar∪ra)∩ra	PROPN
ejpam-3344	476	40	=	=	SYM
ejpam-3344	476	41	χ(ar∪ra)ra	χ(ar∪ra)ra	PROPN
ejpam-3344	476	42	by	by	ADP
ejpam-3344	476	43	the	the	DET
ejpam-3344	476	44	theorem	theorem	NOUN
ejpam-3344	476	45	1	1	NUM
ejpam-3344	476	46	.	.	PUNCT
ejpam-3344	477	1	thus	thus	ADV
ejpam-3344	477	2	(	(	PUNCT
ejpam-3344	477	3	ar∪ra)∩ra	ar∪ra)∩ra	PROPN
ejpam-3344	477	4	=	=	SYM
ejpam-3344	477	5	(	(	PUNCT
ejpam-3344	477	6	ar∪ra)ra	ar∪ra)ra	PROPN
ejpam-3344	477	7	.	.	PUNCT
ejpam-3344	478	1	since	since	SCONJ
ejpam-3344	478	2	a	a	DET
ejpam-3344	478	3	∈	∈	PROPN
ejpam-3344	478	4	(	(	PUNCT
ejpam-3344	478	5	ar	ar	NOUN
ejpam-3344	478	6	∪	∪	PROPN
ejpam-3344	478	7	ra	ra	PROPN
ejpam-3344	478	8	)	)	PUNCT
ejpam-3344	478	9	∩	∩	PROPN
ejpam-3344	478	10	ra	ra	PROPN
ejpam-3344	478	11	,	,	PUNCT
ejpam-3344	478	12	i.e.	i.e.	X
ejpam-3344	478	13	,	,	PUNCT
ejpam-3344	478	14	a	a	DET
ejpam-3344	478	15	∈	∈	PROPN
ejpam-3344	478	16	(	(	PUNCT
ejpam-3344	478	17	ar	ar	NOUN
ejpam-3344	478	18	∪	∪	X
ejpam-3344	478	19	ra)ra	ra)ra	PROPN
ejpam-3344	478	20	,	,	PUNCT
ejpam-3344	478	21	so	so	SCONJ
ejpam-3344	478	22	a	a	DET
ejpam-3344	478	23	∈	∈	PROPN
ejpam-3344	478	24	(	(	PUNCT
ejpam-3344	478	25	ar)(ra	ar)(ra	PROPN
ejpam-3344	478	26	)	)	PUNCT
ejpam-3344	478	27	∪	∪	NOUN
ejpam-3344	478	28	(	(	PUNCT
ejpam-3344	478	29	ra)(ra	ra)(ra	X
ejpam-3344	478	30	)	)	PUNCT
ejpam-3344	478	31	.	.	PUNCT
ejpam-3344	479	1	this	this	PRON
ejpam-3344	479	2	implies	imply	VERB
ejpam-3344	479	3	that	that	SCONJ
ejpam-3344	479	4	a	a	DET
ejpam-3344	479	5	∈	∈	PROPN
ejpam-3344	479	6	(	(	PUNCT
ejpam-3344	479	7	ar)(ra	ar)(ra	PROPN
ejpam-3344	479	8	)	)	PUNCT
ejpam-3344	479	9	or	or	CCONJ
ejpam-3344	479	10	a	a	DET
ejpam-3344	479	11	∈	∈	PROPN
ejpam-3344	479	12	(	(	PUNCT
ejpam-3344	479	13	ra)(ra	ra)(ra	NOUN
ejpam-3344	479	14	)	)	PUNCT
ejpam-3344	479	15	.	.	PUNCT
ejpam-3344	480	1	if	if	SCONJ
ejpam-3344	480	2	a	a	DET
ejpam-3344	480	3	∈	∈	PROPN
ejpam-3344	480	4	(	(	PUNCT
ejpam-3344	480	5	ra)(ra	ra)(ra	NOUN
ejpam-3344	480	6	)	)	PUNCT
ejpam-3344	480	7	,	,	PUNCT
ejpam-3344	480	8	then	then	ADV
ejpam-3344	480	9	r	r	NOUN
ejpam-3344	480	10	is	be	AUX
ejpam-3344	480	11	a	a	DET
ejpam-3344	480	12	left	left	ADJ
ejpam-3344	480	13	weakly	weakly	ADV
ejpam-3344	480	14	regular	regular	ADV
ejpam-3344	480	15	.	.	PUNCT
ejpam-3344	481	1	if	if	SCONJ
ejpam-3344	481	2	a	a	DET
ejpam-3344	481	3	∈	∈	PROPN
ejpam-3344	481	4	(	(	PUNCT
ejpam-3344	481	5	ar)(ra	ar)(ra	PROPN
ejpam-3344	481	6	)	)	PUNCT
ejpam-3344	481	7	,	,	PUNCT
ejpam-3344	481	8	then	then	ADV
ejpam-3344	481	9	(	(	PUNCT
ejpam-3344	481	10	ar)(ra	ar)(ra	PROPN
ejpam-3344	481	11	)	)	PUNCT
ejpam-3344	481	12	=	=	SYM
ejpam-3344	481	13	(	(	PUNCT
ejpam-3344	481	14	(	(	PUNCT
ejpam-3344	481	15	ea)(rr))(ra	ea)(rr))(ra	NOUN
ejpam-3344	481	16	)	)	PUNCT
ejpam-3344	481	17	=	=	PUNCT
ejpam-3344	481	18	(	(	PUNCT
ejpam-3344	481	19	(	(	PUNCT
ejpam-3344	481	20	rr)(ae))(ra	rr)(ae))(ra	PROPN
ejpam-3344	481	21	)	)	PUNCT
ejpam-3344	481	22	=	=	PUNCT
ejpam-3344	481	23	(	(	PUNCT
ejpam-3344	481	24	(	(	PUNCT
ejpam-3344	481	25	(	(	PUNCT
ejpam-3344	481	26	ae)r)r)(ra	ae)r)r)(ra	NOUN
ejpam-3344	481	27	)	)	PUNCT
ejpam-3344	481	28	=	=	SYM
ejpam-3344	481	29	(	(	PUNCT
ejpam-3344	481	30	(	(	PUNCT
ejpam-3344	481	31	ar)r)(ra	ar)r)(ra	PROPN
ejpam-3344	481	32	)	)	PUNCT
ejpam-3344	481	33	=	=	SYM
ejpam-3344	481	34	(	(	PUNCT
ejpam-3344	481	35	(	(	PUNCT
ejpam-3344	481	36	rr)a)(ra	rr)a)(ra	PROPN
ejpam-3344	481	37	)	)	PUNCT
ejpam-3344	481	38	=	=	PUNCT
ejpam-3344	481	39	(	(	PUNCT
ejpam-3344	481	40	ra)(ra	ra)(ra	NOUN
ejpam-3344	481	41	)	)	PUNCT
ejpam-3344	481	42	.	.	PUNCT
ejpam-3344	482	1	hence	hence	ADV
ejpam-3344	482	2	r	r	NOUN
ejpam-3344	482	3	is	be	AUX
ejpam-3344	482	4	a	a	DET
ejpam-3344	482	5	left	left	ADJ
ejpam-3344	482	6	weakly	weakly	ADV
ejpam-3344	482	7	regular	regular	ADJ
ejpam-3344	482	8	,	,	PUNCT
ejpam-3344	482	9	i.e.	i.e.	X
ejpam-3344	482	10	,	,	PUNCT
ejpam-3344	482	11	(	(	PUNCT
ejpam-3344	482	12	2)⇒	2)⇒	NUM
ejpam-3344	482	13	(	(	PUNCT
ejpam-3344	482	14	1	1	NUM
ejpam-3344	482	15	)	)	PUNCT
ejpam-3344	482	16	.	.	PUNCT
ejpam-3344	483	1	theorem	theorem	VERB
ejpam-3344	483	2	11	11	NUM
ejpam-3344	483	3	.	.	PUNCT
ejpam-3344	484	1	let	let	VERB
ejpam-3344	484	2	r	r	PRON
ejpam-3344	484	3	be	be	AUX
ejpam-3344	484	4	an	an	DET
ejpam-3344	484	5	la	la	NOUN
ejpam-3344	484	6	-	-	NOUN
ejpam-3344	484	7	ring	ring	NOUN
ejpam-3344	484	8	with	with	ADP
ejpam-3344	484	9	left	left	ADJ
ejpam-3344	484	10	identity	identity	NOUN
ejpam-3344	484	11	e	e	NOUN
ejpam-3344	484	12	,	,	PUNCT
ejpam-3344	484	13	such	such	ADJ
ejpam-3344	484	14	that	that	SCONJ
ejpam-3344	484	15	(	(	PUNCT
ejpam-3344	484	16	xe)r	xe)r	PROPN
ejpam-3344	484	17	=	=	SYM
ejpam-3344	484	18	xr	xr	PROPN
ejpam-3344	484	19	for	for	ADP
ejpam-3344	484	20	all	all	DET
ejpam-3344	484	21	x	x	PROPN
ejpam-3344	484	22	∈	∈	PROPN
ejpam-3344	484	23	r.	r.	NOUN
ejpam-3344	484	24	then	then	ADV
ejpam-3344	484	25	the	the	DET
ejpam-3344	484	26	following	follow	VERB
ejpam-3344	484	27	conditions	condition	NOUN
ejpam-3344	484	28	are	be	AUX
ejpam-3344	484	29	equivalent	equivalent	ADJ
ejpam-3344	484	30	.	.	PUNCT
ejpam-3344	485	1	(	(	PUNCT
ejpam-3344	485	2	1	1	X
ejpam-3344	485	3	)	)	PUNCT
ejpam-3344	485	4	r	r	NOUN
ejpam-3344	485	5	is	be	AUX
ejpam-3344	485	6	a	a	DET
ejpam-3344	485	7	left	left	ADJ
ejpam-3344	485	8	weakly	weakly	ADV
ejpam-3344	485	9	regular	regular	ADJ
ejpam-3344	485	10	.	.	PUNCT
ejpam-3344	486	1	(	(	PUNCT
ejpam-3344	486	2	2	2	X
ejpam-3344	486	3	)	)	PUNCT
ejpam-3344	486	4	a∩	a∩	PROPN
ejpam-3344	486	5	i	i	PRON
ejpam-3344	486	6	⊆	⊆	PROPN
ejpam-3344	486	7	a	a	DET
ejpam-3344	486	8	◦	◦	NOUN
ejpam-3344	486	9	i	i	PRON
ejpam-3344	486	10	for	for	ADP
ejpam-3344	486	11	every	every	DET
ejpam-3344	486	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	486	13	fuzzy	fuzzy	ADJ
ejpam-3344	486	14	bi	bi	NOUN
ejpam-3344	486	15	-	-	NOUN
ejpam-3344	486	16	ideal	ideal	ADJ
ejpam-3344	486	17	a	a	PRON
ejpam-3344	486	18	and	and	CCONJ
ejpam-3344	486	19	every	every	DET
ejpam-3344	486	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	486	21	fuzzy	fuzzy	ADJ
ejpam-3344	486	22	ideal	ideal	NOUN
ejpam-3344	486	23	i	i	PRON
ejpam-3344	486	24	of	of	ADP
ejpam-3344	486	25	r.	r.	PROPN
ejpam-3344	486	26	(	(	PUNCT
ejpam-3344	486	27	3	3	NUM
ejpam-3344	486	28	)	)	PUNCT
ejpam-3344	486	29	b	b	NOUN
ejpam-3344	486	30	∩	∩	NOUN
ejpam-3344	486	31	i	i	ADP
ejpam-3344	486	32	⊆	⊆	PROPN
ejpam-3344	486	33	b	b	X
ejpam-3344	486	34	◦	◦	NOUN
ejpam-3344	486	35	i	i	PRON
ejpam-3344	486	36	for	for	ADP
ejpam-3344	486	37	every	every	DET
ejpam-3344	486	38	intuitionistic	intuitionistic	ADJ
ejpam-3344	486	39	fuzzy	fuzzy	ADJ
ejpam-3344	486	40	generalized	generalize	VERB
ejpam-3344	486	41	bi	bi	ADJ
ejpam-3344	486	42	-	-	ADJ
ejpam-3344	486	43	ideal	ideal	ADJ
ejpam-3344	486	44	b	b	NOUN
ejpam-3344	486	45	and	and	CCONJ
ejpam-3344	486	46	every	every	DET
ejpam-3344	486	47	intuitionistic	intuitionistic	ADJ
ejpam-3344	486	48	fuzzy	fuzzy	ADJ
ejpam-3344	486	49	ideal	ideal	NOUN
ejpam-3344	486	50	i	i	PRON
ejpam-3344	486	51	of	of	ADP
ejpam-3344	486	52	r.	r.	PROPN
ejpam-3344	486	53	n.	n.	PROPN
ejpam-3344	486	54	kausar	kausar	PROPN
ejpam-3344	486	55	,	,	PUNCT
ejpam-3344	486	56	m.	m.	NOUN
ejpam-3344	486	57	a.	a.	PROPN
ejpam-3344	486	58	waqar	waqar	PROPN
ejpam-3344	486	59	/	/	SYM
ejpam-3344	486	60	eur	eur	PROPN
ejpam-3344	486	61	.	.	PUNCT
ejpam-3344	487	1	j.	j.	PROPN
ejpam-3344	487	2	pure	pure	PROPN
ejpam-3344	487	3	appl	appl	PROPN
ejpam-3344	487	4	.	.	PROPN
ejpam-3344	487	5	math	math	PROPN
ejpam-3344	487	6	,	,	PUNCT
ejpam-3344	487	7	12	12	NUM
ejpam-3344	487	8	(	(	PUNCT
ejpam-3344	487	9	1	1	NUM
ejpam-3344	487	10	)	)	PUNCT
ejpam-3344	487	11	(	(	PUNCT
ejpam-3344	487	12	2019	2019	NUM
ejpam-3344	487	13	)	)	PUNCT
ejpam-3344	487	14	,	,	PUNCT
ejpam-3344	487	15	226	226	NUM
ejpam-3344	487	16	-	-	SYM
ejpam-3344	487	17	250	250	NUM
ejpam-3344	487	18	244	244	NUM
ejpam-3344	487	19	proof	proof	NOUN
ejpam-3344	487	20	.	.	PUNCT
ejpam-3344	488	1	assume	assume	VERB
ejpam-3344	488	2	that	that	SCONJ
ejpam-3344	488	3	(	(	PUNCT
ejpam-3344	488	4	1	1	X
ejpam-3344	488	5	)	)	PUNCT
ejpam-3344	488	6	holds	hold	VERB
ejpam-3344	488	7	.	.	PUNCT
ejpam-3344	489	1	let	let	VERB
ejpam-3344	489	2	b	b	NOUN
ejpam-3344	489	3	=	=	X
ejpam-3344	489	4	(	(	PUNCT
ejpam-3344	489	5	µb	µb	PROPN
ejpam-3344	489	6	,	,	PUNCT
ejpam-3344	489	7	γb	γb	PROPN
ejpam-3344	489	8	)	)	PUNCT
ejpam-3344	489	9	be	be	VERB
ejpam-3344	489	10	an	an	DET
ejpam-3344	489	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	489	12	fuzzy	fuzzy	ADJ
ejpam-3344	489	13	generalized	generalize	VERB
ejpam-3344	489	14	bi	bi	NOUN
ejpam-3344	489	15	-	-	NOUN
ejpam-3344	489	16	ideal	ideal	NOUN
ejpam-3344	490	1	and	and	CCONJ
ejpam-3344	490	2	i	i	PRON
ejpam-3344	490	3	=	=	PUNCT
ejpam-3344	490	4	(	(	PUNCT
ejpam-3344	490	5	µi	µi	INTJ
ejpam-3344	490	6	,	,	PUNCT
ejpam-3344	490	7	γi	γi	INTJ
ejpam-3344	490	8	)	)	PUNCT
ejpam-3344	490	9	be	be	VERB
ejpam-3344	490	10	an	an	DET
ejpam-3344	490	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	490	12	fuzzy	fuzzy	ADJ
ejpam-3344	490	13	ideal	ideal	NOUN
ejpam-3344	490	14	of	of	ADP
ejpam-3344	490	15	r.	r.	PROPN
ejpam-3344	490	16	let	let	VERB
ejpam-3344	490	17	x	x	X
ejpam-3344	490	18	∈	∈	PROPN
ejpam-3344	490	19	r	r	NOUN
ejpam-3344	490	20	,	,	PUNCT
ejpam-3344	490	21	this	this	PRON
ejpam-3344	490	22	means	mean	VERB
ejpam-3344	490	23	that	that	SCONJ
ejpam-3344	490	24	there	there	PRON
ejpam-3344	490	25	exist	exist	VERB
ejpam-3344	490	26	a	a	DET
ejpam-3344	490	27	,	,	PUNCT
ejpam-3344	490	28	b	b	X
ejpam-3344	490	29	∈	∈	NOUN
ejpam-3344	490	30	r	r	NOUN
ejpam-3344	490	31	such	such	ADJ
ejpam-3344	490	32	that	that	PRON
ejpam-3344	490	33	x	x	X
ejpam-3344	490	34	=	=	SYM
ejpam-3344	490	35	(	(	PUNCT
ejpam-3344	490	36	ax)(bx	ax)(bx	PROPN
ejpam-3344	490	37	)	)	PUNCT
ejpam-3344	490	38	=	=	SYM
ejpam-3344	490	39	(	(	PUNCT
ejpam-3344	490	40	ab)(xx	ab)(xx	PROPN
ejpam-3344	490	41	)	)	PUNCT
ejpam-3344	490	42	=	=	SYM
ejpam-3344	490	43	x((ab)x	x((ab)x	PROPN
ejpam-3344	490	44	)	)	PUNCT
ejpam-3344	490	45	.	.	PUNCT
ejpam-3344	491	1	now	now	ADV
ejpam-3344	491	2	(	(	PUNCT
ejpam-3344	491	3	µb	µb	VERB
ejpam-3344	491	4	◦	◦	NOUN
ejpam-3344	491	5	µi)(x	µi)(x	NOUN
ejpam-3344	491	6	)	)	PUNCT
ejpam-3344	492	1	=	=	PUNCT
ejpam-3344	492	2	∨x=∑n	∨x=∑n	NOUN
ejpam-3344	492	3	i=1	i=1	PROPN
ejpam-3344	492	4	aibi	aibi	NOUN
ejpam-3344	492	5	{	{	PUNCT
ejpam-3344	492	6	∧ni=1	∧ni=1	X
ejpam-3344	492	7	{	{	PUNCT
ejpam-3344	492	8	µb	µb	PROPN
ejpam-3344	492	9	(	(	PUNCT
ejpam-3344	492	10	ai	ai	NOUN
ejpam-3344	492	11	)	)	PUNCT
ejpam-3344	492	12	∧	∧	PROPN
ejpam-3344	492	13	µi	µi	PROPN
ejpam-3344	492	14	(	(	PUNCT
ejpam-3344	492	15	bi	bi	NOUN
ejpam-3344	492	16	)	)	PUNCT
ejpam-3344	492	17	}	}	PUNCT
ejpam-3344	492	18	}	}	PUNCT
ejpam-3344	492	19	≥	≥	NUM
ejpam-3344	492	20	µb(x	µb(x	NUM
ejpam-3344	492	21	)	)	PUNCT
ejpam-3344	492	22	∧	∧	NOUN
ejpam-3344	492	23	µi((ab)x	µi((ab)x	NOUN
ejpam-3344	492	24	)	)	PUNCT
ejpam-3344	492	25	≥	≥	NOUN
ejpam-3344	492	26	µb(x	µb(x	PUNCT
ejpam-3344	492	27	)	)	PUNCT
ejpam-3344	492	28	∧	∧	NOUN
ejpam-3344	492	29	µi(x	µi(x	NUM
ejpam-3344	492	30	)	)	PUNCT
ejpam-3344	493	1	=	=	SYM
ejpam-3344	493	2	(	(	PUNCT
ejpam-3344	493	3	µb	µb	ADP
ejpam-3344	493	4	∩	∩	ADJ
ejpam-3344	493	5	µi)(x	µi)(x	NOUN
ejpam-3344	493	6	)	)	PUNCT
ejpam-3344	493	7	.	.	PUNCT
ejpam-3344	494	1	⇒	⇒	PROPN
ejpam-3344	494	2	µb	µb	ADP
ejpam-3344	494	3	∩	∩	NOUN
ejpam-3344	494	4	µi	µi	ADP
ejpam-3344	494	5	⊆	⊆	NUM
ejpam-3344	494	6	µb	µb	ADP
ejpam-3344	494	7	◦	◦	NOUN
ejpam-3344	494	8	µi	µi	PROPN
ejpam-3344	494	9	.	.	PUNCT
ejpam-3344	495	1	similarly	similarly	ADV
ejpam-3344	495	2	,	,	PUNCT
ejpam-3344	495	3	γa	γa	PROPN
ejpam-3344	495	4	∪	∪	VERB
ejpam-3344	495	5	γb	γb	PROPN
ejpam-3344	495	6	⊇	⊇	PROPN
ejpam-3344	495	7	γa	γa	PROPN
ejpam-3344	495	8	◦	◦	PROPN
ejpam-3344	495	9	γb	γb	PROPN
ejpam-3344	495	10	.	.	PUNCT
ejpam-3344	496	1	hence	hence	ADV
ejpam-3344	496	2	a	a	DET
ejpam-3344	496	3	∩	∩	ADJ
ejpam-3344	496	4	b	b	NOUN
ejpam-3344	496	5	⊆	⊆	SYM
ejpam-3344	496	6	a	a	DET
ejpam-3344	496	7	◦	◦	NOUN
ejpam-3344	496	8	b	b	NUM
ejpam-3344	496	9	,	,	PUNCT
ejpam-3344	496	10	i.e.	i.e.	X
ejpam-3344	496	11	,	,	PUNCT
ejpam-3344	496	12	(	(	PUNCT
ejpam-3344	496	13	1	1	X
ejpam-3344	496	14	)	)	PUNCT
ejpam-3344	496	15	⇒	⇒	NOUN
ejpam-3344	496	16	(	(	PUNCT
ejpam-3344	496	17	3	3	NUM
ejpam-3344	496	18	)	)	PUNCT
ejpam-3344	496	19	.	.	PUNCT
ejpam-3344	497	1	it	it	PRON
ejpam-3344	497	2	is	be	AUX
ejpam-3344	497	3	clear	clear	ADJ
ejpam-3344	497	4	that	that	SCONJ
ejpam-3344	497	5	(	(	PUNCT
ejpam-3344	497	6	3)⇒	3)⇒	NUM
ejpam-3344	497	7	(	(	PUNCT
ejpam-3344	497	8	2	2	NUM
ejpam-3344	497	9	)	)	PUNCT
ejpam-3344	497	10	.	.	PUNCT
ejpam-3344	498	1	suppose	suppose	VERB
ejpam-3344	498	2	that	that	SCONJ
ejpam-3344	498	3	(	(	PUNCT
ejpam-3344	498	4	2	2	X
ejpam-3344	498	5	)	)	PUNCT
ejpam-3344	498	6	holds	hold	VERB
ejpam-3344	498	7	.	.	PUNCT
ejpam-3344	499	1	then	then	ADV
ejpam-3344	499	2	a∩	a∩	PROPN
ejpam-3344	499	3	i	i	PRON
ejpam-3344	499	4	⊆	⊆	PROPN
ejpam-3344	499	5	a	a	DET
ejpam-3344	499	6	◦	◦	NOUN
ejpam-3344	500	1	i	i	PRON
ejpam-3344	500	2	,	,	PUNCT
ejpam-3344	500	3	where	where	SCONJ
ejpam-3344	500	4	a	a	PRON
ejpam-3344	500	5	is	be	AUX
ejpam-3344	500	6	an	an	DET
ejpam-3344	500	7	intuitionistic	intuitionistic	ADJ
ejpam-3344	500	8	fuzzy	fuzzy	ADJ
ejpam-3344	500	9	right	right	ADJ
ejpam-3344	500	10	ideal	ideal	NOUN
ejpam-3344	500	11	of	of	ADP
ejpam-3344	500	12	r.	r.	PROPN
ejpam-3344	500	13	since	since	SCONJ
ejpam-3344	500	14	a	a	DET
ejpam-3344	500	15	◦	◦	NOUN
ejpam-3344	500	16	i	i	PRON
ejpam-3344	500	17	⊆	⊆	NUM
ejpam-3344	500	18	a∩	a∩	PROPN
ejpam-3344	501	1	i	i	PRON
ejpam-3344	501	2	,	,	PUNCT
ejpam-3344	501	3	so	so	ADV
ejpam-3344	501	4	a	a	DET
ejpam-3344	501	5	◦	◦	NOUN
ejpam-3344	502	1	i	i	NOUN
ejpam-3344	502	2	=	=	SYM
ejpam-3344	502	3	a∩	a∩	PROPN
ejpam-3344	502	4	i.	i.	NOUN
ejpam-3344	502	5	therefore	therefore	ADV
ejpam-3344	502	6	r	r	NOUN
ejpam-3344	502	7	is	be	AUX
ejpam-3344	502	8	a	a	DET
ejpam-3344	502	9	left	left	ADJ
ejpam-3344	502	10	weakly	weakly	ADV
ejpam-3344	502	11	regular	regular	ADV
ejpam-3344	502	12	by	by	ADP
ejpam-3344	502	13	the	the	DET
ejpam-3344	502	14	theorem	theorem	NOUN
ejpam-3344	502	15	10	10	NUM
ejpam-3344	502	16	,	,	PUNCT
ejpam-3344	502	17	i.e.	i.e.	X
ejpam-3344	502	18	,	,	PUNCT
ejpam-3344	502	19	(	(	PUNCT
ejpam-3344	502	20	2)⇒	2)⇒	NUM
ejpam-3344	502	21	(	(	PUNCT
ejpam-3344	502	22	1	1	NUM
ejpam-3344	502	23	)	)	PUNCT
ejpam-3344	502	24	.	.	PUNCT
ejpam-3344	503	1	theorem	theorem	NOUN
ejpam-3344	503	2	12	12	NUM
ejpam-3344	503	3	.	.	PUNCT
ejpam-3344	504	1	let	let	VERB
ejpam-3344	504	2	r	r	PRON
ejpam-3344	504	3	be	be	AUX
ejpam-3344	504	4	an	an	DET
ejpam-3344	504	5	la	la	NOUN
ejpam-3344	504	6	-	-	NOUN
ejpam-3344	504	7	ring	ring	NOUN
ejpam-3344	504	8	with	with	ADP
ejpam-3344	504	9	left	left	ADJ
ejpam-3344	504	10	identity	identity	NOUN
ejpam-3344	504	11	e	e	NOUN
ejpam-3344	504	12	,	,	PUNCT
ejpam-3344	504	13	such	such	ADJ
ejpam-3344	504	14	that	that	SCONJ
ejpam-3344	504	15	(	(	PUNCT
ejpam-3344	504	16	xe)r	xe)r	PROPN
ejpam-3344	504	17	=	=	SYM
ejpam-3344	504	18	xr	xr	PROPN
ejpam-3344	504	19	for	for	ADP
ejpam-3344	504	20	all	all	DET
ejpam-3344	504	21	x	x	PROPN
ejpam-3344	504	22	∈	∈	PROPN
ejpam-3344	504	23	r.	r.	NOUN
ejpam-3344	504	24	then	then	ADV
ejpam-3344	504	25	the	the	DET
ejpam-3344	504	26	following	follow	VERB
ejpam-3344	504	27	conditions	condition	NOUN
ejpam-3344	504	28	are	be	AUX
ejpam-3344	504	29	equivalent	equivalent	ADJ
ejpam-3344	504	30	.	.	PUNCT
ejpam-3344	505	1	(	(	PUNCT
ejpam-3344	505	2	1	1	X
ejpam-3344	505	3	)	)	PUNCT
ejpam-3344	505	4	r	r	NOUN
ejpam-3344	505	5	is	be	AUX
ejpam-3344	505	6	a	a	DET
ejpam-3344	505	7	left	left	ADJ
ejpam-3344	505	8	weakly	weakly	ADV
ejpam-3344	505	9	regular	regular	ADJ
ejpam-3344	505	10	.	.	PUNCT
ejpam-3344	506	1	(	(	PUNCT
ejpam-3344	506	2	2	2	X
ejpam-3344	506	3	)	)	PUNCT
ejpam-3344	506	4	a∩	a∩	PROPN
ejpam-3344	506	5	i	i	PRON
ejpam-3344	506	6	∩c	∩c	VERB
ejpam-3344	507	1	⊆	⊆	X
ejpam-3344	507	2	(	(	PUNCT
ejpam-3344	507	3	a	a	DET
ejpam-3344	507	4	◦	◦	NOUN
ejpam-3344	507	5	i	i	NOUN
ejpam-3344	507	6	)	)	PUNCT
ejpam-3344	508	1	◦	◦	NOUN
ejpam-3344	508	2	c	c	NOUN
ejpam-3344	508	3	for	for	ADP
ejpam-3344	508	4	every	every	DET
ejpam-3344	508	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	508	6	fuzzy	fuzzy	ADJ
ejpam-3344	508	7	bi	bi	NOUN
ejpam-3344	508	8	-	-	NOUN
ejpam-3344	508	9	ideal	ideal	NOUN
ejpam-3344	508	10	a	a	PRON
ejpam-3344	508	11	,	,	PUNCT
ejpam-3344	508	12	every	every	DET
ejpam-3344	508	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	508	14	fuzzy	fuzzy	ADJ
ejpam-3344	508	15	ideal	ideal	NOUN
ejpam-3344	509	1	i	i	PRON
ejpam-3344	509	2	and	and	CCONJ
ejpam-3344	509	3	every	every	DET
ejpam-3344	509	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	509	5	fuzzy	fuzzy	ADJ
ejpam-3344	509	6	right	right	ADJ
ejpam-3344	509	7	ideal	ideal	NOUN
ejpam-3344	509	8	c	c	PROPN
ejpam-3344	509	9	of	of	ADP
ejpam-3344	509	10	r.	r.	PROPN
ejpam-3344	509	11	(	(	PUNCT
ejpam-3344	509	12	3	3	NUM
ejpam-3344	509	13	)	)	PUNCT
ejpam-3344	509	14	b	b	NOUN
ejpam-3344	509	15	∩	∩	NOUN
ejpam-3344	509	16	i	i	PRON
ejpam-3344	509	17	∩c	∩c	VERB
ejpam-3344	509	18	⊆	⊆	NUM
ejpam-3344	509	19	(	(	PUNCT
ejpam-3344	509	20	b	b	X
ejpam-3344	509	21	◦	◦	NOUN
ejpam-3344	509	22	i	i	NOUN
ejpam-3344	509	23	)	)	PUNCT
ejpam-3344	509	24	◦	◦	NOUN
ejpam-3344	509	25	c	c	NOUN
ejpam-3344	509	26	for	for	ADP
ejpam-3344	509	27	every	every	DET
ejpam-3344	509	28	intuitionistic	intuitionistic	ADJ
ejpam-3344	509	29	fuzzy	fuzzy	ADJ
ejpam-3344	509	30	generalized	generalize	VERB
ejpam-3344	509	31	bi	bi	ADJ
ejpam-3344	509	32	-	-	ADJ
ejpam-3344	509	33	ideal	ideal	ADJ
ejpam-3344	509	34	b	b	NOUN
ejpam-3344	509	35	,	,	PUNCT
ejpam-3344	509	36	every	every	DET
ejpam-3344	509	37	intuitionistic	intuitionistic	ADJ
ejpam-3344	509	38	fuzzy	fuzzy	ADJ
ejpam-3344	509	39	ideal	ideal	NOUN
ejpam-3344	510	1	i	i	PRON
ejpam-3344	510	2	and	and	CCONJ
ejpam-3344	510	3	every	every	DET
ejpam-3344	510	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	510	5	fuzzy	fuzzy	ADJ
ejpam-3344	510	6	right	right	ADJ
ejpam-3344	510	7	ideal	ideal	NOUN
ejpam-3344	510	8	c	c	PROPN
ejpam-3344	510	9	of	of	ADP
ejpam-3344	510	10	r.	r.	PROPN
ejpam-3344	510	11	proof	proof	PROPN
ejpam-3344	510	12	.	.	PUNCT
ejpam-3344	511	1	suppose	suppose	VERB
ejpam-3344	511	2	that	that	SCONJ
ejpam-3344	511	3	(	(	PUNCT
ejpam-3344	511	4	1	1	X
ejpam-3344	511	5	)	)	PUNCT
ejpam-3344	511	6	holds	hold	VERB
ejpam-3344	511	7	.	.	PUNCT
ejpam-3344	512	1	let	let	VERB
ejpam-3344	512	2	b	b	NOUN
ejpam-3344	512	3	=	=	X
ejpam-3344	512	4	(	(	PUNCT
ejpam-3344	512	5	µb	µb	PROPN
ejpam-3344	512	6	,	,	PUNCT
ejpam-3344	512	7	γb	γb	PROPN
ejpam-3344	512	8	)	)	PUNCT
ejpam-3344	512	9	be	be	VERB
ejpam-3344	512	10	an	an	DET
ejpam-3344	512	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	512	12	fuzzy	fuzzy	ADJ
ejpam-3344	512	13	generalized	generalize	VERB
ejpam-3344	512	14	bi	bi	NOUN
ejpam-3344	512	15	-	-	NOUN
ejpam-3344	512	16	ideal	ideal	ADJ
ejpam-3344	512	17	,	,	PUNCT
ejpam-3344	512	18	i	i	PRON
ejpam-3344	512	19	=	=	PUNCT
ejpam-3344	512	20	(	(	PUNCT
ejpam-3344	512	21	µi	µi	INTJ
ejpam-3344	512	22	,	,	PUNCT
ejpam-3344	512	23	γi	γi	INTJ
ejpam-3344	512	24	)	)	PUNCT
ejpam-3344	512	25	be	be	VERB
ejpam-3344	512	26	an	an	DET
ejpam-3344	512	27	intuitionistic	intuitionistic	ADJ
ejpam-3344	512	28	fuzzy	fuzzy	ADJ
ejpam-3344	512	29	ideal	ideal	NOUN
ejpam-3344	512	30	and	and	CCONJ
ejpam-3344	512	31	c	c	NOUN
ejpam-3344	512	32	=	=	SYM
ejpam-3344	512	33	(	(	PUNCT
ejpam-3344	512	34	µc	µc	INTJ
ejpam-3344	512	35	,	,	PUNCT
ejpam-3344	512	36	γc	γc	PROPN
ejpam-3344	512	37	)	)	PUNCT
ejpam-3344	512	38	be	be	AUX
ejpam-3344	512	39	an	an	DET
ejpam-3344	512	40	intuitionistic	intuitionistic	ADJ
ejpam-3344	512	41	fuzzy	fuzzy	ADJ
ejpam-3344	512	42	right	right	ADJ
ejpam-3344	512	43	ideal	ideal	NOUN
ejpam-3344	512	44	of	of	ADP
ejpam-3344	512	45	r.	r.	PROPN
ejpam-3344	512	46	let	let	VERB
ejpam-3344	512	47	x	x	X
ejpam-3344	512	48	∈	∈	PROPN
ejpam-3344	512	49	r	r	NOUN
ejpam-3344	512	50	,	,	PUNCT
ejpam-3344	512	51	then	then	ADV
ejpam-3344	512	52	there	there	PRON
ejpam-3344	512	53	exist	exist	VERB
ejpam-3344	512	54	elements	element	NOUN
ejpam-3344	512	55	a	a	PRON
ejpam-3344	512	56	,	,	PUNCT
ejpam-3344	512	57	b	b	X
ejpam-3344	512	58	∈	∈	NOUN
ejpam-3344	512	59	r	r	NOUN
ejpam-3344	513	1	such	such	ADJ
ejpam-3344	513	2	that	that	PRON
ejpam-3344	513	3	x	x	X
ejpam-3344	513	4	=	=	SYM
ejpam-3344	513	5	(	(	PUNCT
ejpam-3344	513	6	ax)(bx	ax)(bx	NOUN
ejpam-3344	513	7	)	)	PUNCT
ejpam-3344	513	8	.	.	PUNCT
ejpam-3344	514	1	here	here	ADV
ejpam-3344	514	2	x	x	X
ejpam-3344	514	3	=	=	SYM
ejpam-3344	514	4	(	(	PUNCT
ejpam-3344	514	5	ax)(bx	ax)(bx	PROPN
ejpam-3344	514	6	)	)	PUNCT
ejpam-3344	514	7	=	=	SYM
ejpam-3344	514	8	(	(	PUNCT
ejpam-3344	514	9	xb)(xa	xb)(xa	PROPN
ejpam-3344	514	10	)	)	PUNCT
ejpam-3344	514	11	xb	xb	X
ejpam-3344	515	1	=	=	PUNCT
ejpam-3344	515	2	(	(	PUNCT
ejpam-3344	515	3	(	(	PUNCT
ejpam-3344	515	4	ax)(bx))b	ax)(bx))b	NOUN
ejpam-3344	515	5	=	=	X
ejpam-3344	515	6	(	(	PUNCT
ejpam-3344	515	7	(	(	PUNCT
ejpam-3344	515	8	xx)(ba))b	xx)(ba))b	NOUN
ejpam-3344	515	9	=	=	SYM
ejpam-3344	515	10	(	(	PUNCT
ejpam-3344	515	11	b(ba))(xx	b(ba))(xx	PROPN
ejpam-3344	515	12	)	)	PUNCT
ejpam-3344	515	13	=	=	PUNCT
ejpam-3344	515	14	c(xx	c(xx	PROPN
ejpam-3344	515	15	)	)	PUNCT
ejpam-3344	515	16	=	=	PUNCT
ejpam-3344	515	17	x(cx	x(cx	PROPN
ejpam-3344	515	18	)	)	PUNCT
ejpam-3344	515	19	say	say	VERB
ejpam-3344	515	20	c	c	NOUN
ejpam-3344	515	21	=	=	SYM
ejpam-3344	515	22	b(ba	b(ba	X
ejpam-3344	515	23	)	)	PUNCT
ejpam-3344	515	24	now	now	ADV
ejpam-3344	515	25	(	(	PUNCT
ejpam-3344	515	26	(	(	PUNCT
ejpam-3344	515	27	µb	µb	VERB
ejpam-3344	515	28	◦	◦	NOUN
ejpam-3344	515	29	µi	µi	PROPN
ejpam-3344	515	30	)	)	PUNCT
ejpam-3344	515	31	◦	◦	NOUN
ejpam-3344	515	32	µc)(x	µc)(x	NOUN
ejpam-3344	515	33	)	)	PUNCT
ejpam-3344	515	34	=	=	PUNCT
ejpam-3344	516	1	∨x=∑n	∨x=∑n	NOUN
ejpam-3344	516	2	i=1	i=1	PROPN
ejpam-3344	516	3	aibi	aibi	NOUN
ejpam-3344	516	4	{	{	PUNCT
ejpam-3344	516	5	∧ni=1	∧ni=1	X
ejpam-3344	516	6	{	{	PUNCT
ejpam-3344	516	7	(	(	PUNCT
ejpam-3344	516	8	µb	µb	VERB
ejpam-3344	516	9	◦	◦	NOUN
ejpam-3344	516	10	µi	µi	PROPN
ejpam-3344	516	11	)	)	PUNCT
ejpam-3344	516	12	(	(	PUNCT
ejpam-3344	516	13	ai	ai	NOUN
ejpam-3344	516	14	)	)	PUNCT
ejpam-3344	516	15	∧	∧	NOUN
ejpam-3344	516	16	µc	µc	INTJ
ejpam-3344	516	17	(	(	PUNCT
ejpam-3344	516	18	bi	bi	NOUN
ejpam-3344	516	19	)	)	PUNCT
ejpam-3344	516	20	}	}	PUNCT
ejpam-3344	516	21	}	}	PUNCT
ejpam-3344	516	22	≥	≥	X
ejpam-3344	516	23	(	(	PUNCT
ejpam-3344	516	24	µb	µb	VERB
ejpam-3344	516	25	◦	◦	NOUN
ejpam-3344	516	26	µi)(xb	µi)(xb	PUNCT
ejpam-3344	516	27	)	)	PUNCT
ejpam-3344	516	28	∧	∧	NOUN
ejpam-3344	516	29	µc(xa	µc(xa	PROPN
ejpam-3344	516	30	)	)	PUNCT
ejpam-3344	516	31	≥	≥	NOUN
ejpam-3344	516	32	(	(	PUNCT
ejpam-3344	516	33	µb	µb	VERB
ejpam-3344	516	34	◦	◦	NOUN
ejpam-3344	516	35	µi)(xb	µi)(xb	PUNCT
ejpam-3344	516	36	)	)	PUNCT
ejpam-3344	516	37	∧	∧	NOUN
ejpam-3344	516	38	µc(x	µc(x	NOUN
ejpam-3344	516	39	)	)	PUNCT
ejpam-3344	516	40	=	=	PUNCT
ejpam-3344	517	1	∨xb=∑n	∨xb=∑n	NOUN
ejpam-3344	517	2	i=1	i=1	PROPN
ejpam-3344	517	3	piqi	piqi	NOUN
ejpam-3344	517	4	{	{	PUNCT
ejpam-3344	517	5	∧ni=1	∧ni=1	X
ejpam-3344	517	6	{	{	PUNCT
ejpam-3344	517	7	µb	µb	PROPN
ejpam-3344	517	8	(	(	PUNCT
ejpam-3344	517	9	pi	pi	NOUN
ejpam-3344	517	10	)	)	PUNCT
ejpam-3344	517	11	∧	∧	PROPN
ejpam-3344	517	12	µi	µi	PROPN
ejpam-3344	517	13	(	(	PUNCT
ejpam-3344	517	14	qi	qi	NOUN
ejpam-3344	517	15	)	)	PUNCT
ejpam-3344	517	16	}	}	PUNCT
ejpam-3344	517	17	}	}	PUNCT
ejpam-3344	517	18	∧	∧	PROPN
ejpam-3344	517	19	µc(x	µc(x	NOUN
ejpam-3344	517	20	)	)	PUNCT
ejpam-3344	517	21	≥	≥	NOUN
ejpam-3344	517	22	µb(x	µb(x	PUNCT
ejpam-3344	517	23	)	)	PUNCT
ejpam-3344	517	24	∧	∧	PROPN
ejpam-3344	517	25	µi(cx	µi(cx	PROPN
ejpam-3344	517	26	)	)	PUNCT
ejpam-3344	517	27	∧	∧	PROPN
ejpam-3344	517	28	µc(x	µc(x	NOUN
ejpam-3344	517	29	)	)	PUNCT
ejpam-3344	517	30	≥	≥	NOUN
ejpam-3344	517	31	µb(x	µb(x	PUNCT
ejpam-3344	517	32	)	)	PUNCT
ejpam-3344	517	33	∧	∧	NOUN
ejpam-3344	517	34	µi(x	µi(x	NUM
ejpam-3344	517	35	)	)	PUNCT
ejpam-3344	517	36	∧	∧	NOUN
ejpam-3344	517	37	µc(x	µc(x	NOUN
ejpam-3344	517	38	)	)	PUNCT
ejpam-3344	517	39	=	=	PUNCT
ejpam-3344	517	40	(	(	PUNCT
ejpam-3344	517	41	µb	µb	ADP
ejpam-3344	517	42	∩	∩	NOUN
ejpam-3344	517	43	µi	µi	ADP
ejpam-3344	517	44	∩	∩	NOUN
ejpam-3344	517	45	µc)(x	µc)(x	NOUN
ejpam-3344	517	46	)	)	PUNCT
ejpam-3344	517	47	.	.	PUNCT
ejpam-3344	518	1	⇒	⇒	PROPN
ejpam-3344	518	2	µb	µb	ADP
ejpam-3344	518	3	∩	∩	NOUN
ejpam-3344	518	4	µi	µi	ADP
ejpam-3344	518	5	∩	∩	NOUN
ejpam-3344	518	6	µc	µc	ADP
ejpam-3344	518	7	⊆	⊆	NUM
ejpam-3344	518	8	(	(	PUNCT
ejpam-3344	518	9	µb	µb	VERB
ejpam-3344	518	10	◦	◦	NOUN
ejpam-3344	518	11	µi	µi	PROPN
ejpam-3344	518	12	)	)	PUNCT
ejpam-3344	518	13	◦	◦	NOUN
ejpam-3344	518	14	µc	µc	INTJ
ejpam-3344	518	15	.	.	PUNCT
ejpam-3344	519	1	similarly	similarly	ADV
ejpam-3344	519	2	,	,	PUNCT
ejpam-3344	519	3	γb	γb	NOUN
ejpam-3344	519	4	∪	∪	NOUN
ejpam-3344	519	5	γi	γi	ADP
ejpam-3344	519	6	∪	∪	PROPN
ejpam-3344	519	7	γc	γc	PROPN
ejpam-3344	519	8	⊇	⊇	X
ejpam-3344	519	9	(	(	PUNCT
ejpam-3344	519	10	γb	γb	INTJ
ejpam-3344	519	11	◦	◦	NOUN
ejpam-3344	519	12	γi	γi	NOUN
ejpam-3344	519	13	)	)	PUNCT
ejpam-3344	519	14	◦	◦	NOUN
ejpam-3344	519	15	γc	γc	PROPN
ejpam-3344	519	16	,	,	PUNCT
ejpam-3344	519	17	i.e.	i.e.	X
ejpam-3344	519	18	,	,	PUNCT
ejpam-3344	519	19	b	b	NOUN
ejpam-3344	519	20	∩	∩	NOUN
ejpam-3344	519	21	i	i	PRON
ejpam-3344	519	22	∩c	∩c	VERB
ejpam-3344	519	23	⊆	⊆	NUM
ejpam-3344	519	24	(	(	PUNCT
ejpam-3344	519	25	b	b	X
ejpam-3344	519	26	◦	◦	VERB
ejpam-3344	519	27	i	i	NOUN
ejpam-3344	519	28	)	)	PUNCT
ejpam-3344	519	29	◦	◦	NOUN
ejpam-3344	519	30	c.	c.	NOUN
ejpam-3344	519	31	hence	hence	ADV
ejpam-3344	519	32	(	(	PUNCT
ejpam-3344	519	33	1)⇒	1)⇒	NUM
ejpam-3344	519	34	(	(	PUNCT
ejpam-3344	519	35	3	3	NUM
ejpam-3344	519	36	)	)	PUNCT
ejpam-3344	519	37	.	.	PUNCT
ejpam-3344	520	1	it	it	PRON
ejpam-3344	520	2	is	be	AUX
ejpam-3344	520	3	clear	clear	ADJ
ejpam-3344	520	4	that	that	SCONJ
ejpam-3344	520	5	(	(	PUNCT
ejpam-3344	520	6	3)⇒	3)⇒	NUM
ejpam-3344	520	7	(	(	PUNCT
ejpam-3344	520	8	2	2	NUM
ejpam-3344	520	9	)	)	PUNCT
ejpam-3344	520	10	,	,	PUNCT
ejpam-3344	520	11	every	every	DET
ejpam-3344	520	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	520	13	fuzzy	fuzzy	ADJ
ejpam-3344	520	14	bi	bi	NOUN
ejpam-3344	520	15	-	-	NOUN
ejpam-3344	520	16	ideal	ideal	NOUN
ejpam-3344	520	17	of	of	ADP
ejpam-3344	520	18	r	r	NOUN
ejpam-3344	520	19	is	be	AUX
ejpam-3344	520	20	an	an	DET
ejpam-3344	520	21	intuitionistic	intuitionistic	ADJ
ejpam-3344	520	22	fuzzy	fuzzy	ADJ
ejpam-3344	520	23	n.	n.	NOUN
ejpam-3344	520	24	kausar	kausar	PROPN
ejpam-3344	520	25	,	,	PUNCT
ejpam-3344	520	26	m.	m.	NOUN
ejpam-3344	520	27	a.	a.	PROPN
ejpam-3344	520	28	waqar	waqar	PROPN
ejpam-3344	520	29	/	/	SYM
ejpam-3344	520	30	eur	eur	PROPN
ejpam-3344	520	31	.	.	PUNCT
ejpam-3344	521	1	j.	j.	PROPN
ejpam-3344	521	2	pure	pure	PROPN
ejpam-3344	521	3	appl	appl	PROPN
ejpam-3344	521	4	.	.	PROPN
ejpam-3344	521	5	math	math	PROPN
ejpam-3344	521	6	,	,	PUNCT
ejpam-3344	521	7	12	12	NUM
ejpam-3344	521	8	(	(	PUNCT
ejpam-3344	521	9	1	1	NUM
ejpam-3344	521	10	)	)	PUNCT
ejpam-3344	521	11	(	(	PUNCT
ejpam-3344	521	12	2019	2019	NUM
ejpam-3344	521	13	)	)	PUNCT
ejpam-3344	521	14	,	,	PUNCT
ejpam-3344	521	15	226	226	NUM
ejpam-3344	521	16	-	-	SYM
ejpam-3344	521	17	250	250	NUM
ejpam-3344	521	18	245	245	NUM
ejpam-3344	521	19	generalized	generalized	ADJ
ejpam-3344	521	20	bi	bi	NOUN
ejpam-3344	521	21	-	-	NOUN
ejpam-3344	521	22	ideal	ideal	NOUN
ejpam-3344	521	23	of	of	ADP
ejpam-3344	521	24	r.	r.	PROPN
ejpam-3344	521	25	assume	assume	VERB
ejpam-3344	521	26	that	that	SCONJ
ejpam-3344	521	27	(	(	PUNCT
ejpam-3344	521	28	2	2	X
ejpam-3344	521	29	)	)	PUNCT
ejpam-3344	521	30	is	be	AUX
ejpam-3344	521	31	true	true	ADJ
ejpam-3344	521	32	.	.	PUNCT
ejpam-3344	522	1	then	then	ADV
ejpam-3344	522	2	a∩	a∩	PROPN
ejpam-3344	522	3	i	i	PRON
ejpam-3344	522	4	∩r	∩r	VERB
ejpam-3344	522	5	⊆	⊆	NUM
ejpam-3344	522	6	(	(	PUNCT
ejpam-3344	522	7	a	a	DET
ejpam-3344	522	8	◦	◦	NOUN
ejpam-3344	522	9	i	i	NOUN
ejpam-3344	522	10	)	)	PUNCT
ejpam-3344	522	11	◦	◦	NOUN
ejpam-3344	522	12	r	r	NOUN
ejpam-3344	522	13	,	,	PUNCT
ejpam-3344	522	14	where	where	SCONJ
ejpam-3344	522	15	a	a	PRON
ejpam-3344	522	16	is	be	AUX
ejpam-3344	522	17	an	an	DET
ejpam-3344	522	18	intuitionistic	intuitionistic	ADJ
ejpam-3344	522	19	right	right	ADJ
ejpam-3344	522	20	ideal	ideal	NOUN
ejpam-3344	522	21	of	of	ADP
ejpam-3344	522	22	r	r	NOUN
ejpam-3344	522	23	,	,	PUNCT
ejpam-3344	522	24	i.e.	i.e.	X
ejpam-3344	522	25	,	,	PUNCT
ejpam-3344	522	26	a∩	a∩	PROPN
ejpam-3344	522	27	i	i	PRON
ejpam-3344	522	28	⊆	⊆	PROPN
ejpam-3344	522	29	a	a	DET
ejpam-3344	522	30	◦	◦	NOUN
ejpam-3344	522	31	i.	i.	NOUN
ejpam-3344	522	32	since	since	SCONJ
ejpam-3344	522	33	a	a	DET
ejpam-3344	522	34	◦	◦	NOUN
ejpam-3344	522	35	i	i	PRON
ejpam-3344	522	36	⊆	⊆	NUM
ejpam-3344	522	37	a∩	a∩	PROPN
ejpam-3344	523	1	i	i	PRON
ejpam-3344	523	2	,	,	PUNCT
ejpam-3344	523	3	so	so	ADV
ejpam-3344	523	4	a	a	DET
ejpam-3344	523	5	◦	◦	NOUN
ejpam-3344	523	6	i	i	NOUN
ejpam-3344	523	7	=	=	SYM
ejpam-3344	523	8	a∩	a∩	PROPN
ejpam-3344	523	9	i.	i.	NOUN
ejpam-3344	523	10	therefore	therefore	ADV
ejpam-3344	523	11	r	r	NOUN
ejpam-3344	523	12	is	be	AUX
ejpam-3344	523	13	a	a	DET
ejpam-3344	523	14	left	left	ADJ
ejpam-3344	523	15	weakly	weakly	ADV
ejpam-3344	523	16	regular	regular	ADV
ejpam-3344	523	17	by	by	ADP
ejpam-3344	523	18	the	the	DET
ejpam-3344	523	19	theorem	theorem	NOUN
ejpam-3344	523	20	10	10	NUM
ejpam-3344	523	21	,	,	PUNCT
ejpam-3344	523	22	i.e.	i.e.	X
ejpam-3344	523	23	,	,	PUNCT
ejpam-3344	523	24	(	(	PUNCT
ejpam-3344	523	25	2)⇒	2)⇒	NUM
ejpam-3344	523	26	(	(	PUNCT
ejpam-3344	523	27	1	1	NUM
ejpam-3344	523	28	)	)	PUNCT
ejpam-3344	523	29	.	.	PUNCT
ejpam-3344	524	1	lemma	lemma	PROPN
ejpam-3344	524	2	18	18	NUM
ejpam-3344	524	3	.	.	PUNCT
ejpam-3344	525	1	every	every	DET
ejpam-3344	525	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	525	3	fuzzy	fuzzy	ADJ
ejpam-3344	525	4	left	left	ADJ
ejpam-3344	525	5	(	(	PUNCT
ejpam-3344	525	6	right	right	ADJ
ejpam-3344	525	7	)	)	PUNCT
ejpam-3344	525	8	ideal	ideal	NOUN
ejpam-3344	525	9	of	of	ADP
ejpam-3344	525	10	an	an	DET
ejpam-3344	525	11	intra	intra	ADJ
ejpam-3344	525	12	-	-	ADJ
ejpam-3344	525	13	regular	regular	ADJ
ejpam-3344	525	14	la	la	ADJ
ejpam-3344	525	15	-	-	PUNCT
ejpam-3344	525	16	ring	ring	NOUN
ejpam-3344	525	17	r	r	NOUN
ejpam-3344	525	18	,	,	PUNCT
ejpam-3344	525	19	is	be	AUX
ejpam-3344	525	20	an	an	DET
ejpam-3344	525	21	intuitionistic	intuitionistic	ADJ
ejpam-3344	525	22	fuzzy	fuzzy	ADJ
ejpam-3344	525	23	ideal	ideal	NOUN
ejpam-3344	525	24	of	of	ADP
ejpam-3344	525	25	r.	r.	PROPN
ejpam-3344	525	26	proof	proof	NOUN
ejpam-3344	525	27	.	.	PUNCT
ejpam-3344	526	1	suppose	suppose	VERB
ejpam-3344	526	2	that	that	SCONJ
ejpam-3344	526	3	a	a	DET
ejpam-3344	526	4	=	=	SYM
ejpam-3344	526	5	(	(	PUNCT
ejpam-3344	526	6	µa	µa	PROPN
ejpam-3344	526	7	,	,	PUNCT
ejpam-3344	526	8	γa	γa	PROPN
ejpam-3344	526	9	)	)	PUNCT
ejpam-3344	526	10	is	be	AUX
ejpam-3344	526	11	an	an	DET
ejpam-3344	526	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	526	13	fuzzy	fuzzy	ADJ
ejpam-3344	526	14	right	right	ADJ
ejpam-3344	526	15	ideal	ideal	NOUN
ejpam-3344	526	16	of	of	ADP
ejpam-3344	526	17	r.	r.	PROPN
ejpam-3344	526	18	let	let	VERB
ejpam-3344	526	19	x	x	PRON
ejpam-3344	526	20	,	,	PUNCT
ejpam-3344	526	21	y	y	PROPN
ejpam-3344	526	22	∈	∈	PROPN
ejpam-3344	526	23	r	r	NOUN
ejpam-3344	526	24	,	,	PUNCT
ejpam-3344	526	25	this	this	PRON
ejpam-3344	526	26	implies	imply	VERB
ejpam-3344	526	27	that	that	SCONJ
ejpam-3344	526	28	there	there	PRON
ejpam-3344	526	29	exist	exist	VERB
ejpam-3344	526	30	ai	ai	NOUN
ejpam-3344	526	31	,	,	PUNCT
ejpam-3344	526	32	bi	bi	NOUN
ejpam-3344	526	33	∈	∈	PROPN
ejpam-3344	526	34	r	r	PROPN
ejpam-3344	526	35	,	,	PUNCT
ejpam-3344	526	36	such	such	ADJ
ejpam-3344	526	37	that	that	SCONJ
ejpam-3344	526	38	x	x	NOUN
ejpam-3344	527	1	=	=	PUNCT
ejpam-3344	527	2	∑n	∑n	PROPN
ejpam-3344	527	3	i=1(aix	i=1(aix	PROPN
ejpam-3344	527	4	2)bi	2)bi	NUM
ejpam-3344	527	5	.	.	PUNCT
ejpam-3344	528	1	thus	thus	ADV
ejpam-3344	528	2	µa(xy	µa(xy	NUM
ejpam-3344	528	3	)	)	PUNCT
ejpam-3344	528	4	=	=	PUNCT
ejpam-3344	529	1	µa(((aix	µa(((aix	PUNCT
ejpam-3344	529	2	2)bi)y	2)bi)y	X
ejpam-3344	529	3	)	)	PUNCT
ejpam-3344	529	4	=	=	SYM
ejpam-3344	529	5	µa((ybi)(aix	µa((ybi)(aix	PROPN
ejpam-3344	529	6	2	2	NUM
ejpam-3344	529	7	)	)	PUNCT
ejpam-3344	529	8	)	)	PUNCT
ejpam-3344	529	9	≥	≥	PROPN
ejpam-3344	529	10	µa(ybi	µa(ybi	PROPN
ejpam-3344	529	11	)	)	PUNCT
ejpam-3344	529	12	≥	≥	NOUN
ejpam-3344	529	13	µa(y	µa(y	NOUN
ejpam-3344	529	14	)	)	PUNCT
ejpam-3344	529	15	and	and	CCONJ
ejpam-3344	529	16	γa(xy	γa(xy	PROPN
ejpam-3344	529	17	)	)	PUNCT
ejpam-3344	529	18	=	=	PUNCT
ejpam-3344	530	1	γa(((aix	γa(((aix	NUM
ejpam-3344	530	2	2)bi)y	2)bi)y	X
ejpam-3344	530	3	)	)	PUNCT
ejpam-3344	530	4	=	=	PUNCT
ejpam-3344	531	1	γa((ybi)(aix	γa((ybi)(aix	ADJ
ejpam-3344	531	2	2	2	NUM
ejpam-3344	531	3	)	)	PUNCT
ejpam-3344	531	4	)	)	PUNCT
ejpam-3344	531	5	≤	≤	NUM
ejpam-3344	532	1	γa(ybi	γa(ybi	X
ejpam-3344	532	2	)	)	PUNCT
ejpam-3344	532	3	≤	≤	NOUN
ejpam-3344	532	4	γa(y	γa(y	NUM
ejpam-3344	532	5	)	)	PUNCT
ejpam-3344	532	6	.	.	PUNCT
ejpam-3344	533	1	hence	hence	ADV
ejpam-3344	533	2	a	a	PRON
ejpam-3344	533	3	is	be	AUX
ejpam-3344	533	4	an	an	DET
ejpam-3344	533	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	533	6	fuzzy	fuzzy	ADJ
ejpam-3344	533	7	ideal	ideal	NOUN
ejpam-3344	533	8	of	of	ADP
ejpam-3344	533	9	r.	r.	PROPN
ejpam-3344	533	10	similarly	similarly	ADV
ejpam-3344	533	11	,	,	PUNCT
ejpam-3344	533	12	for	for	ADP
ejpam-3344	533	13	left	left	ADJ
ejpam-3344	533	14	ideal	ideal	NOUN
ejpam-3344	533	15	.	.	PUNCT
ejpam-3344	534	1	remark	remark	PROPN
ejpam-3344	534	2	13	13	NUM
ejpam-3344	534	3	.	.	PUNCT
ejpam-3344	535	1	the	the	DET
ejpam-3344	535	2	concept	concept	NOUN
ejpam-3344	535	3	of	of	ADP
ejpam-3344	535	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	535	5	fuzzy	fuzzy	ADJ
ejpam-3344	535	6	(	(	PUNCT
ejpam-3344	535	7	left	left	ADJ
ejpam-3344	535	8	,	,	PUNCT
ejpam-3344	535	9	right	right	INTJ
ejpam-3344	535	10	,	,	PUNCT
ejpam-3344	535	11	two	two	NUM
ejpam-3344	535	12	-	-	PUNCT
ejpam-3344	535	13	sided	sided	ADJ
ejpam-3344	535	14	)	)	PUNCT
ejpam-3344	535	15	ideals	ideal	NOUN
ejpam-3344	535	16	coincides	coincide	VERB
ejpam-3344	535	17	in	in	ADP
ejpam-3344	535	18	inta	inta	NOUN
ejpam-3344	535	19	-	-	PUNCT
ejpam-3344	535	20	regular	regular	ADJ
ejpam-3344	535	21	la	la	NOUN
ejpam-3344	535	22	-	-	PUNCT
ejpam-3344	535	23	rings	ring	NOUN
ejpam-3344	535	24	.	.	PUNCT
ejpam-3344	536	1	proposition	proposition	NOUN
ejpam-3344	536	2	11	11	NUM
ejpam-3344	536	3	.	.	PUNCT
ejpam-3344	537	1	every	every	DET
ejpam-3344	537	2	intuitionistic	intuitionistic	ADJ
ejpam-3344	537	3	fuzzy	fuzzy	ADJ
ejpam-3344	537	4	generalized	generalize	VERB
ejpam-3344	537	5	bi	bi	NOUN
ejpam-3344	537	6	-	-	NOUN
ejpam-3344	537	7	ideal	ideal	NOUN
ejpam-3344	537	8	of	of	ADP
ejpam-3344	537	9	an	an	DET
ejpam-3344	537	10	intra	intra	ADJ
ejpam-3344	537	11	-	-	ADJ
ejpam-3344	537	12	regular	regular	ADJ
ejpam-3344	537	13	la	la	ADJ
ejpam-3344	537	14	-	-	PUNCT
ejpam-3344	537	15	ring	ring	NOUN
ejpam-3344	537	16	r	r	NOUN
ejpam-3344	537	17	with	with	ADP
ejpam-3344	537	18	left	left	ADJ
ejpam-3344	537	19	identity	identity	NOUN
ejpam-3344	537	20	e	e	NOUN
ejpam-3344	537	21	,	,	PUNCT
ejpam-3344	537	22	is	be	AUX
ejpam-3344	537	23	an	an	DET
ejpam-3344	537	24	intuitionistic	intuitionistic	ADJ
ejpam-3344	537	25	fuzzy	fuzzy	ADJ
ejpam-3344	537	26	bi	bi	NOUN
ejpam-3344	537	27	-	-	NOUN
ejpam-3344	537	28	ideal	ideal	NOUN
ejpam-3344	537	29	of	of	ADP
ejpam-3344	537	30	r.	r.	PROPN
ejpam-3344	537	31	proof	proof	NOUN
ejpam-3344	537	32	.	.	PUNCT
ejpam-3344	538	1	let	let	VERB
ejpam-3344	538	2	a	a	DET
ejpam-3344	538	3	=	=	SYM
ejpam-3344	538	4	(	(	PUNCT
ejpam-3344	538	5	µa	µa	PROPN
ejpam-3344	538	6	,	,	PUNCT
ejpam-3344	538	7	γa	γa	PROPN
ejpam-3344	538	8	)	)	PUNCT
ejpam-3344	538	9	be	be	VERB
ejpam-3344	538	10	an	an	DET
ejpam-3344	538	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	538	12	fuzzy	fuzzy	ADJ
ejpam-3344	538	13	generalized	generalize	VERB
ejpam-3344	538	14	bi	bi	NOUN
ejpam-3344	538	15	-	-	NOUN
ejpam-3344	538	16	ideal	ideal	NOUN
ejpam-3344	538	17	of	of	ADP
ejpam-3344	538	18	r	r	NOUN
ejpam-3344	538	19	and	and	CCONJ
ejpam-3344	538	20	x	x	NOUN
ejpam-3344	538	21	,	,	PUNCT
ejpam-3344	538	22	y	y	PROPN
ejpam-3344	538	23	∈	∈	PROPN
ejpam-3344	538	24	r	r	NOUN
ejpam-3344	538	25	,	,	PUNCT
ejpam-3344	538	26	this	this	PRON
ejpam-3344	538	27	implies	imply	VERB
ejpam-3344	538	28	that	that	SCONJ
ejpam-3344	538	29	there	there	PRON
ejpam-3344	538	30	exist	exist	VERB
ejpam-3344	538	31	ai	ai	NOUN
ejpam-3344	538	32	,	,	PUNCT
ejpam-3344	538	33	bi	bi	NOUN
ejpam-3344	538	34	∈	∈	PROPN
ejpam-3344	538	35	r	r	NOUN
ejpam-3344	538	36	such	such	ADJ
ejpam-3344	538	37	that	that	SCONJ
ejpam-3344	538	38	x	x	NOUN
ejpam-3344	539	1	=	=	PUNCT
ejpam-3344	539	2	∑n	∑n	PROPN
ejpam-3344	539	3	i=1(aix	i=1(aix	PROPN
ejpam-3344	539	4	2)bi	2)bi	NUM
ejpam-3344	539	5	.	.	PUNCT
ejpam-3344	540	1	we	we	PRON
ejpam-3344	540	2	have	have	VERB
ejpam-3344	540	3	to	to	PART
ejpam-3344	540	4	show	show	VERB
ejpam-3344	540	5	that	that	SCONJ
ejpam-3344	540	6	a	a	PRON
ejpam-3344	540	7	is	be	AUX
ejpam-3344	540	8	an	an	DET
ejpam-3344	540	9	intuitionistic	intuitionistic	ADJ
ejpam-3344	540	10	fuzzy	fuzzy	ADJ
ejpam-3344	540	11	la	la	NOUN
ejpam-3344	540	12	-	-	PUNCT
ejpam-3344	540	13	subring	subring	NOUN
ejpam-3344	540	14	of	of	ADP
ejpam-3344	540	15	r.	r.	PROPN
ejpam-3344	540	16	now	now	ADV
ejpam-3344	540	17	x	x	X
ejpam-3344	540	18	=	=	SYM
ejpam-3344	540	19	(	(	PUNCT
ejpam-3344	540	20	aix	aix	NOUN
ejpam-3344	540	21	2)bi	2)bi	NUM
ejpam-3344	540	22	=	=	SYM
ejpam-3344	540	23	(	(	PUNCT
ejpam-3344	540	24	aix	aix	NOUN
ejpam-3344	540	25	2)(ebi	2)(ebi	NOUN
ejpam-3344	540	26	)	)	PUNCT
ejpam-3344	541	1	=	=	PUNCT
ejpam-3344	541	2	(	(	PUNCT
ejpam-3344	541	3	aie)(x	aie)(x	PROPN
ejpam-3344	541	4	2bi	2bi	NOUN
ejpam-3344	541	5	)	)	PUNCT
ejpam-3344	542	1	=	=	PRON
ejpam-3344	542	2	(	(	PUNCT
ejpam-3344	542	3	aie)((xx)bi	aie)((xx)bi	NOUN
ejpam-3344	542	4	)	)	PUNCT
ejpam-3344	542	5	=	=	SYM
ejpam-3344	542	6	(	(	PUNCT
ejpam-3344	542	7	aie)((bix)x	aie)((bix)x	ADV
ejpam-3344	542	8	)	)	PUNCT
ejpam-3344	542	9	=	=	SYM
ejpam-3344	542	10	(	(	PUNCT
ejpam-3344	542	11	x(bix))(eai	x(bix))(eai	NOUN
ejpam-3344	542	12	)	)	PUNCT
ejpam-3344	542	13	=	=	SYM
ejpam-3344	543	1	(	(	PUNCT
ejpam-3344	543	2	x(bix))ai	x(bix))ai	X
ejpam-3344	543	3	=	=	SYM
ejpam-3344	543	4	(	(	PUNCT
ejpam-3344	543	5	ai(bix))x	ai(bix))x	PROPN
ejpam-3344	543	6	=	=	PUNCT
ejpam-3344	543	7	(	(	PUNCT
ejpam-3344	543	8	ai(bix))(ex	ai(bix))(ex	VERB
ejpam-3344	543	9	)	)	PUNCT
ejpam-3344	543	10	=	=	SYM
ejpam-3344	543	11	(	(	PUNCT
ejpam-3344	543	12	xe)((bix)ai	xe)((bix)ai	PROPN
ejpam-3344	543	13	)	)	PUNCT
ejpam-3344	543	14	=	=	SYM
ejpam-3344	543	15	(	(	PUNCT
ejpam-3344	543	16	bix)((xe)ai	bix)((xe)ai	NOUN
ejpam-3344	543	17	)	)	PUNCT
ejpam-3344	543	18	=	=	SYM
ejpam-3344	543	19	(	(	PUNCT
ejpam-3344	543	20	bix)((aie)x	bix)((aie)x	PROPN
ejpam-3344	543	21	)	)	PUNCT
ejpam-3344	543	22	=	=	SYM
ejpam-3344	543	23	(	(	PUNCT
ejpam-3344	543	24	x(aie))(xbi	x(aie))(xbi	NOUN
ejpam-3344	543	25	)	)	PUNCT
ejpam-3344	543	26	=	=	SYM
ejpam-3344	543	27	x((x(aie))bi	x((x(aie))bi	X
ejpam-3344	543	28	)	)	PUNCT
ejpam-3344	544	1	=	=	SYM
ejpam-3344	544	2	xn	xn	PROPN
ejpam-3344	544	3	,	,	PUNCT
ejpam-3344	544	4	say	say	VERB
ejpam-3344	544	5	n	n	ADV
ejpam-3344	544	6	=	=	SYM
ejpam-3344	544	7	(	(	PUNCT
ejpam-3344	544	8	x(aie))bi	x(aie))bi	PROPN
ejpam-3344	544	9	thus	thus	ADV
ejpam-3344	544	10	µa(xy	µa(xy	NUM
ejpam-3344	544	11	)	)	PUNCT
ejpam-3344	544	12	=	=	SYM
ejpam-3344	545	1	µa((xn)y	µa((xn)y	ADJ
ejpam-3344	545	2	)	)	PUNCT
ejpam-3344	545	3	≥	≥	NOUN
ejpam-3344	545	4	min{µa(x	min{µa(x	NOUN
ejpam-3344	545	5	)	)	PUNCT
ejpam-3344	545	6	,	,	PUNCT
ejpam-3344	545	7	µa(y	µa(y	NOUN
ejpam-3344	545	8	)	)	PUNCT
ejpam-3344	545	9	}	}	PUNCT
ejpam-3344	545	10	and	and	CCONJ
ejpam-3344	545	11	γa(xy	γa(xy	PROPN
ejpam-3344	545	12	)	)	PUNCT
ejpam-3344	546	1	=	=	PUNCT
ejpam-3344	546	2	γa((xn)y	γa((xn)y	ADJ
ejpam-3344	546	3	)	)	PUNCT
ejpam-3344	546	4	≤	≤	NUM
ejpam-3344	546	5	max{γa(x	max{γa(x	NOUN
ejpam-3344	546	6	)	)	PUNCT
ejpam-3344	546	7	,	,	PUNCT
ejpam-3344	546	8	γa(y	γa(y	NOUN
ejpam-3344	546	9	)	)	PUNCT
ejpam-3344	546	10	}	}	PUNCT
ejpam-3344	546	11	.	.	PUNCT
ejpam-3344	547	1	hence	hence	ADV
ejpam-3344	547	2	a	a	PRON
ejpam-3344	547	3	is	be	AUX
ejpam-3344	547	4	an	an	DET
ejpam-3344	547	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	547	6	fuzzy	fuzzy	ADJ
ejpam-3344	547	7	la	la	NOUN
ejpam-3344	547	8	-	-	PUNCT
ejpam-3344	547	9	subring	subring	NOUN
ejpam-3344	547	10	of	of	ADP
ejpam-3344	547	11	r.	r.	PROPN
ejpam-3344	547	12	theorem	theorem	VERB
ejpam-3344	547	13	13	13	NUM
ejpam-3344	547	14	.	.	PUNCT
ejpam-3344	548	1	let	let	VERB
ejpam-3344	548	2	r	r	PRON
ejpam-3344	548	3	be	be	AUX
ejpam-3344	548	4	an	an	DET
ejpam-3344	548	5	la	la	NOUN
ejpam-3344	548	6	-	-	NOUN
ejpam-3344	548	7	ring	ring	NOUN
ejpam-3344	548	8	with	with	ADP
ejpam-3344	548	9	left	left	ADJ
ejpam-3344	548	10	identity	identity	NOUN
ejpam-3344	548	11	e	e	NOUN
ejpam-3344	548	12	,	,	PUNCT
ejpam-3344	548	13	such	such	ADJ
ejpam-3344	548	14	that	that	SCONJ
ejpam-3344	548	15	(	(	PUNCT
ejpam-3344	548	16	xe)r	xe)r	PROPN
ejpam-3344	548	17	=	=	SYM
ejpam-3344	548	18	xr	xr	PROPN
ejpam-3344	548	19	for	for	ADP
ejpam-3344	548	20	all	all	DET
ejpam-3344	548	21	x	x	PROPN
ejpam-3344	548	22	∈	∈	PROPN
ejpam-3344	548	23	r.	r.	NOUN
ejpam-3344	548	24	then	then	ADV
ejpam-3344	548	25	the	the	DET
ejpam-3344	548	26	following	follow	VERB
ejpam-3344	548	27	conditions	condition	NOUN
ejpam-3344	548	28	are	be	AUX
ejpam-3344	548	29	equivalent	equivalent	ADJ
ejpam-3344	548	30	.	.	PUNCT
ejpam-3344	549	1	(	(	PUNCT
ejpam-3344	549	2	1	1	X
ejpam-3344	549	3	)	)	PUNCT
ejpam-3344	549	4	r	r	NOUN
ejpam-3344	549	5	is	be	AUX
ejpam-3344	549	6	an	an	DET
ejpam-3344	549	7	intra	intra	ADJ
ejpam-3344	549	8	-	-	ADJ
ejpam-3344	549	9	regular	regular	ADJ
ejpam-3344	549	10	.	.	PUNCT
ejpam-3344	550	1	(	(	PUNCT
ejpam-3344	550	2	2	2	X
ejpam-3344	550	3	)	)	PUNCT
ejpam-3344	550	4	a	a	DET
ejpam-3344	550	5	∩	∩	ADJ
ejpam-3344	550	6	b	b	ADP
ejpam-3344	550	7	⊆	⊆	SYM
ejpam-3344	550	8	a	a	DET
ejpam-3344	550	9	◦	◦	NOUN
ejpam-3344	550	10	b	b	NOUN
ejpam-3344	550	11	for	for	ADP
ejpam-3344	550	12	every	every	DET
ejpam-3344	550	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	550	14	fuzzy	fuzzy	ADJ
ejpam-3344	550	15	right	right	ADJ
ejpam-3344	550	16	ideal	ideal	PROPN
ejpam-3344	550	17	b	b	PROPN
ejpam-3344	550	18	and	and	CCONJ
ejpam-3344	550	19	every	every	DET
ejpam-3344	550	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	550	21	fuzzy	fuzzy	ADJ
ejpam-3344	550	22	left	leave	VERB
ejpam-3344	550	23	ideal	ideal	NOUN
ejpam-3344	550	24	a	a	PRON
ejpam-3344	550	25	of	of	ADP
ejpam-3344	550	26	r.	r.	PROPN
ejpam-3344	550	27	n.	n.	PROPN
ejpam-3344	550	28	kausar	kausar	PROPN
ejpam-3344	550	29	,	,	PUNCT
ejpam-3344	550	30	m.	m.	NOUN
ejpam-3344	550	31	a.	a.	PROPN
ejpam-3344	550	32	waqar	waqar	PROPN
ejpam-3344	550	33	/	/	SYM
ejpam-3344	550	34	eur	eur	PROPN
ejpam-3344	550	35	.	.	PUNCT
ejpam-3344	551	1	j.	j.	PROPN
ejpam-3344	551	2	pure	pure	PROPN
ejpam-3344	551	3	appl	appl	PROPN
ejpam-3344	551	4	.	.	PROPN
ejpam-3344	551	5	math	math	PROPN
ejpam-3344	551	6	,	,	PUNCT
ejpam-3344	551	7	12	12	NUM
ejpam-3344	551	8	(	(	PUNCT
ejpam-3344	551	9	1	1	NUM
ejpam-3344	551	10	)	)	PUNCT
ejpam-3344	551	11	(	(	PUNCT
ejpam-3344	551	12	2019	2019	NUM
ejpam-3344	551	13	)	)	PUNCT
ejpam-3344	551	14	,	,	PUNCT
ejpam-3344	551	15	226	226	NUM
ejpam-3344	551	16	-	-	SYM
ejpam-3344	551	17	250	250	NUM
ejpam-3344	551	18	246	246	NUM
ejpam-3344	551	19	proof	proof	NOUN
ejpam-3344	551	20	.	.	PUNCT
ejpam-3344	552	1	assume	assume	VERB
ejpam-3344	552	2	that	that	SCONJ
ejpam-3344	552	3	(	(	PUNCT
ejpam-3344	552	4	1	1	X
ejpam-3344	552	5	)	)	PUNCT
ejpam-3344	552	6	holds	hold	VERB
ejpam-3344	552	7	.	.	PUNCT
ejpam-3344	553	1	let	let	VERB
ejpam-3344	553	2	x	x	PUNCT
ejpam-3344	553	3	∈	∈	PROPN
ejpam-3344	553	4	r	r	NOUN
ejpam-3344	553	5	,	,	PUNCT
ejpam-3344	553	6	then	then	ADV
ejpam-3344	553	7	there	there	PRON
ejpam-3344	553	8	exist	exist	VERB
ejpam-3344	553	9	elements	element	NOUN
ejpam-3344	553	10	ai	ai	VERB
ejpam-3344	553	11	,	,	PUNCT
ejpam-3344	553	12	bi	bi	NOUN
ejpam-3344	553	13	∈	∈	PROPN
ejpam-3344	553	14	r	r	NOUN
ejpam-3344	553	15	such	such	ADJ
ejpam-3344	553	16	that	that	SCONJ
ejpam-3344	553	17	x	x	NOUN
ejpam-3344	554	1	=	=	PUNCT
ejpam-3344	554	2	∑n	∑n	PROPN
ejpam-3344	554	3	i=1(aix	i=1(aix	PROPN
ejpam-3344	554	4	2)bi	2)bi	NUM
ejpam-3344	554	5	.	.	PUNCT
ejpam-3344	555	1	now	now	ADV
ejpam-3344	555	2	x	x	X
ejpam-3344	555	3	=	=	SYM
ejpam-3344	555	4	(	(	PUNCT
ejpam-3344	555	5	aix	aix	NOUN
ejpam-3344	555	6	2)bi	2)bi	NUM
ejpam-3344	555	7	=	=	SYM
ejpam-3344	555	8	(	(	PUNCT
ejpam-3344	555	9	ai(xx))bi	ai(xx))bi	NOUN
ejpam-3344	555	10	=	=	PUNCT
ejpam-3344	555	11	(	(	PUNCT
ejpam-3344	555	12	x(aix))(ebi	x(aix))(ebi	PROPN
ejpam-3344	555	13	)	)	PUNCT
ejpam-3344	555	14	=	=	SYM
ejpam-3344	555	15	(	(	PUNCT
ejpam-3344	555	16	xe)((aix)bi	xe)((aix)bi	PROPN
ejpam-3344	555	17	)	)	PUNCT
ejpam-3344	556	1	=	=	PRON
ejpam-3344	556	2	(	(	PUNCT
ejpam-3344	556	3	aix)((xe)bi	aix)((xe)bi	NOUN
ejpam-3344	556	4	)	)	PUNCT
ejpam-3344	556	5	.	.	PUNCT
ejpam-3344	557	1	thus	thus	ADV
ejpam-3344	557	2	(	(	PUNCT
ejpam-3344	557	3	µa	µa	ADP
ejpam-3344	557	4	◦	◦	NOUN
ejpam-3344	557	5	µb)(x	µb)(x	NOUN
ejpam-3344	557	6	)	)	PUNCT
ejpam-3344	557	7	=	=	PUNCT
ejpam-3344	558	1	∨x=∑n	∨x=∑n	NOUN
ejpam-3344	558	2	i=1	i=1	PROPN
ejpam-3344	558	3	aibi	aibi	NOUN
ejpam-3344	558	4	{	{	PUNCT
ejpam-3344	558	5	∧ni=1µa	∧ni=1µa	NOUN
ejpam-3344	558	6	(	(	PUNCT
ejpam-3344	558	7	ai	ai	NOUN
ejpam-3344	558	8	)	)	PUNCT
ejpam-3344	558	9	∧	∧	NOUN
ejpam-3344	558	10	µb	µb	PROPN
ejpam-3344	558	11	(	(	PUNCT
ejpam-3344	558	12	bi	bi	NOUN
ejpam-3344	558	13	)	)	PUNCT
ejpam-3344	558	14	}	}	PUNCT
ejpam-3344	558	15	≥	≥	NOUN
ejpam-3344	558	16	min{µa(aix	min{µa(aix	NUM
ejpam-3344	558	17	)	)	PUNCT
ejpam-3344	558	18	,	,	PUNCT
ejpam-3344	558	19	µb((xe)bi	µb((xe)bi	PROPN
ejpam-3344	558	20	)	)	PUNCT
ejpam-3344	558	21	}	}	PUNCT
ejpam-3344	558	22	≥	≥	PROPN
ejpam-3344	558	23	min{µa(x	min{µa(x	NOUN
ejpam-3344	558	24	)	)	PUNCT
ejpam-3344	558	25	,	,	PUNCT
ejpam-3344	558	26	µb(x	µb(x	PUNCT
ejpam-3344	558	27	)	)	PUNCT
ejpam-3344	558	28	}	}	PUNCT
ejpam-3344	558	29	=	=	SYM
ejpam-3344	558	30	(	(	PUNCT
ejpam-3344	558	31	µa	µa	ADP
ejpam-3344	558	32	∧	∧	PROPN
ejpam-3344	558	33	µb)(x	µb)(x	PROPN
ejpam-3344	558	34	)	)	PUNCT
ejpam-3344	558	35	=	=	PRON
ejpam-3344	559	1	(	(	PUNCT
ejpam-3344	559	2	µa	µa	ADP
ejpam-3344	559	3	∩	∩	ADJ
ejpam-3344	559	4	µb)(x	µb)(x	PROPN
ejpam-3344	559	5	)	)	PUNCT
ejpam-3344	559	6	.	.	PUNCT
ejpam-3344	560	1	⇒	⇒	PROPN
ejpam-3344	560	2	µa	µa	ADP
ejpam-3344	560	3	∩	∩	PROPN
ejpam-3344	560	4	µb	µb	VERB
ejpam-3344	560	5	⊆	⊆	NUM
ejpam-3344	560	6	µa	µa	ADP
ejpam-3344	560	7	◦	◦	NOUN
ejpam-3344	560	8	µb	µb	NOUN
ejpam-3344	560	9	.	.	PUNCT
ejpam-3344	561	1	similarly	similarly	ADV
ejpam-3344	561	2	,	,	PUNCT
ejpam-3344	561	3	we	we	PRON
ejpam-3344	561	4	have	have	VERB
ejpam-3344	561	5	γa	γa	PROPN
ejpam-3344	561	6	∪	∪	VERB
ejpam-3344	561	7	γb	γb	PROPN
ejpam-3344	561	8	⊇	⊇	PROPN
ejpam-3344	561	9	µa	µa	ADP
ejpam-3344	561	10	◦	◦	NOUN
ejpam-3344	561	11	µb	µb	VERB
ejpam-3344	561	12	.	.	PUNCT
ejpam-3344	562	1	hence	hence	ADV
ejpam-3344	562	2	a	a	DET
ejpam-3344	562	3	∩b	∩b	NOUN
ejpam-3344	562	4	⊆	⊆	NUM
ejpam-3344	562	5	a	a	DET
ejpam-3344	562	6	◦	◦	NOUN
ejpam-3344	562	7	b	b	NOUN
ejpam-3344	562	8	,	,	PUNCT
ejpam-3344	562	9	i.e.	i.e.	X
ejpam-3344	562	10	,	,	PUNCT
ejpam-3344	562	11	(	(	PUNCT
ejpam-3344	562	12	1)⇒	1)⇒	NUM
ejpam-3344	562	13	(	(	PUNCT
ejpam-3344	562	14	2	2	NUM
ejpam-3344	562	15	)	)	PUNCT
ejpam-3344	562	16	.	.	PUNCT
ejpam-3344	563	1	suppose	suppose	VERB
ejpam-3344	563	2	that	that	SCONJ
ejpam-3344	563	3	(	(	PUNCT
ejpam-3344	563	4	2	2	X
ejpam-3344	563	5	)	)	PUNCT
ejpam-3344	563	6	is	be	AUX
ejpam-3344	563	7	true	true	ADJ
ejpam-3344	563	8	and	and	CCONJ
ejpam-3344	563	9	a	a	DET
ejpam-3344	563	10	∈	∈	PROPN
ejpam-3344	563	11	r	r	NOUN
ejpam-3344	563	12	,	,	PUNCT
ejpam-3344	563	13	then	then	ADV
ejpam-3344	563	14	ra	ra	PROPN
ejpam-3344	563	15	is	be	AUX
ejpam-3344	563	16	a	a	DET
ejpam-3344	563	17	left	left	ADJ
ejpam-3344	563	18	ideal	ideal	NOUN
ejpam-3344	563	19	of	of	ADP
ejpam-3344	563	20	r	r	NOUN
ejpam-3344	563	21	containing	contain	VERB
ejpam-3344	563	22	a	a	PRON
ejpam-3344	563	23	by	by	ADP
ejpam-3344	563	24	the	the	DET
ejpam-3344	563	25	lemma	lemma	PROPN
ejpam-3344	563	26	8	8	NUM
ejpam-3344	563	27	and	and	CCONJ
ejpam-3344	563	28	ar	ar	PROPN
ejpam-3344	563	29	∪	∪	PROPN
ejpam-3344	563	30	ra	ra	PROPN
ejpam-3344	563	31	is	be	AUX
ejpam-3344	563	32	a	a	DET
ejpam-3344	563	33	right	right	ADJ
ejpam-3344	563	34	ideal	ideal	NOUN
ejpam-3344	563	35	of	of	ADP
ejpam-3344	563	36	r	r	NOUN
ejpam-3344	563	37	containing	contain	VERB
ejpam-3344	563	38	a	a	PRON
ejpam-3344	563	39	by	by	ADP
ejpam-3344	563	40	the	the	DET
ejpam-3344	563	41	proposition	proposition	NOUN
ejpam-3344	563	42	5	5	NUM
ejpam-3344	563	43	.	.	PUNCT
ejpam-3344	564	1	this	this	PRON
ejpam-3344	564	2	means	mean	VERB
ejpam-3344	564	3	that	that	SCONJ
ejpam-3344	564	4	χra	χra	PROPN
ejpam-3344	564	5	is	be	AUX
ejpam-3344	564	6	an	an	DET
ejpam-3344	564	7	intuitionistic	intuitionistic	ADJ
ejpam-3344	564	8	fuzzy	fuzzy	ADJ
ejpam-3344	564	9	left	leave	VERB
ejpam-3344	564	10	ideal	ideal	NOUN
ejpam-3344	564	11	and	and	CCONJ
ejpam-3344	564	12	χar∪ra	χar∪ra	ADV
ejpam-3344	564	13	is	be	AUX
ejpam-3344	564	14	an	an	DET
ejpam-3344	564	15	intuitionistic	intuitionistic	ADJ
ejpam-3344	564	16	fuzzy	fuzzy	ADJ
ejpam-3344	564	17	right	right	ADJ
ejpam-3344	564	18	ideal	ideal	NOUN
ejpam-3344	564	19	of	of	ADP
ejpam-3344	564	20	r	r	NOUN
ejpam-3344	564	21	,	,	PUNCT
ejpam-3344	564	22	by	by	ADP
ejpam-3344	564	23	the	the	DET
ejpam-3344	564	24	lemma	lemma	PROPN
ejpam-3344	564	25	2	2	NUM
ejpam-3344	564	26	.	.	PUNCT
ejpam-3344	564	27	by	by	ADP
ejpam-3344	564	28	our	our	PRON
ejpam-3344	564	29	supposition	supposition	NOUN
ejpam-3344	564	30	χar∪ra	χar∪ra	PROPN
ejpam-3344	564	31	∩	∩	PROPN
ejpam-3344	564	32	χra	χra	PROPN
ejpam-3344	564	33	⊆	⊆	NUM
ejpam-3344	564	34	χra	χra	NOUN
ejpam-3344	564	35	◦	◦	NOUN
ejpam-3344	564	36	χar∪ra	χar∪ra	ADV
ejpam-3344	564	37	,	,	PUNCT
ejpam-3344	564	38	i.e.	i.e.	X
ejpam-3344	564	39	,	,	PUNCT
ejpam-3344	564	40	χ(ar∪ra)∩ra	χ(ar∪ra)∩ra	PROPN
ejpam-3344	564	41	⊆	⊆	NUM
ejpam-3344	564	42	χ(ra)(ar∪ra	χ(ra)(ar∪ra	ADV
ejpam-3344	564	43	)	)	PUNCT
ejpam-3344	564	44	.	.	PUNCT
ejpam-3344	565	1	thus	thus	ADV
ejpam-3344	565	2	(	(	PUNCT
ejpam-3344	565	3	ar∪ra)∩ra	ar∪ra)∩ra	PROPN
ejpam-3344	565	4	⊆	⊆	NUM
ejpam-3344	565	5	ra(ar∪ra	ra(ar∪ra	PROPN
ejpam-3344	565	6	)	)	PUNCT
ejpam-3344	565	7	.	.	PUNCT
ejpam-3344	566	1	since	since	SCONJ
ejpam-3344	566	2	a	a	DET
ejpam-3344	566	3	∈	∈	PROPN
ejpam-3344	566	4	(	(	PUNCT
ejpam-3344	566	5	ar∪ra)∩ra	ar∪ra)∩ra	PROPN
ejpam-3344	566	6	,	,	PUNCT
ejpam-3344	566	7	i.e.	i.e.	X
ejpam-3344	566	8	,	,	PUNCT
ejpam-3344	566	9	a	a	DET
ejpam-3344	566	10	∈	∈	PROPN
ejpam-3344	566	11	ra(ar	ra(ar	NOUN
ejpam-3344	566	12	∪ra	∪ra	ADV
ejpam-3344	566	13	)	)	PUNCT
ejpam-3344	566	14	=	=	SYM
ejpam-3344	566	15	(	(	PUNCT
ejpam-3344	566	16	ra)(ar	ra)(ar	NOUN
ejpam-3344	566	17	)	)	PUNCT
ejpam-3344	566	18	∪	∪	X
ejpam-3344	566	19	(	(	PUNCT
ejpam-3344	566	20	ra)(ra	ra)(ra	X
ejpam-3344	566	21	)	)	PUNCT
ejpam-3344	566	22	.	.	PUNCT
ejpam-3344	567	1	now	now	ADV
ejpam-3344	567	2	(	(	PUNCT
ejpam-3344	567	3	ra)(ar	ra)(ar	NOUN
ejpam-3344	567	4	)	)	PUNCT
ejpam-3344	567	5	=	=	SYM
ejpam-3344	567	6	(	(	PUNCT
ejpam-3344	567	7	ra)((ea)(rr	ra)((ea)(rr	NOUN
ejpam-3344	567	8	)	)	PUNCT
ejpam-3344	567	9	)	)	PUNCT
ejpam-3344	568	1	=	=	SYM
ejpam-3344	568	2	(	(	PUNCT
ejpam-3344	568	3	ra)((rr)(ae	ra)((rr)(ae	PROPN
ejpam-3344	568	4	)	)	PUNCT
ejpam-3344	568	5	)	)	PUNCT
ejpam-3344	569	1	=	=	PRON
ejpam-3344	569	2	(	(	PUNCT
ejpam-3344	569	3	ra)(((ae)r)r	ra)(((ae)r)r	NOUN
ejpam-3344	569	4	)	)	PUNCT
ejpam-3344	569	5	=	=	SYM
ejpam-3344	569	6	(	(	PUNCT
ejpam-3344	569	7	ra)((ar)r	ra)((ar)r	PROPN
ejpam-3344	569	8	)	)	PUNCT
ejpam-3344	569	9	=	=	SYM
ejpam-3344	569	10	(	(	PUNCT
ejpam-3344	569	11	ra)((rr)a	ra)((rr)a	PROPN
ejpam-3344	569	12	)	)	PUNCT
ejpam-3344	569	13	=	=	SYM
ejpam-3344	569	14	(	(	PUNCT
ejpam-3344	569	15	ra)(ra	ra)(ra	NOUN
ejpam-3344	569	16	)	)	PUNCT
ejpam-3344	569	17	.	.	PUNCT
ejpam-3344	570	1	this	this	PRON
ejpam-3344	570	2	implies	imply	VERB
ejpam-3344	570	3	that	that	SCONJ
ejpam-3344	570	4	(	(	PUNCT
ejpam-3344	570	5	ra)(ar	ra)(ar	NOUN
ejpam-3344	570	6	)	)	PUNCT
ejpam-3344	570	7	∪	∪	X
ejpam-3344	570	8	(	(	PUNCT
ejpam-3344	570	9	ra)(ra	ra)(ra	X
ejpam-3344	570	10	)	)	PUNCT
ejpam-3344	570	11	=	=	SYM
ejpam-3344	570	12	(	(	PUNCT
ejpam-3344	570	13	ra)(ra	ra)(ra	NOUN
ejpam-3344	570	14	)	)	PUNCT
ejpam-3344	570	15	∪	∪	NOUN
ejpam-3344	570	16	(	(	PUNCT
ejpam-3344	570	17	ra)(ra	ra)(ra	X
ejpam-3344	570	18	)	)	PUNCT
ejpam-3344	570	19	=	=	SYM
ejpam-3344	570	20	(	(	PUNCT
ejpam-3344	570	21	ra)(ra	ra)(ra	X
ejpam-3344	570	22	)	)	PUNCT
ejpam-3344	570	23	=	=	SYM
ejpam-3344	570	24	(	(	PUNCT
ejpam-3344	570	25	(	(	PUNCT
ejpam-3344	570	26	ra)a)r	ra)a)r	NOUN
ejpam-3344	570	27	=	=	SYM
ejpam-3344	570	28	(	(	PUNCT
ejpam-3344	570	29	(	(	PUNCT
ejpam-3344	570	30	ra)(ea))r	ra)(ea))r	PROPN
ejpam-3344	570	31	=	=	SYM
ejpam-3344	570	32	(	(	PUNCT
ejpam-3344	570	33	(	(	PUNCT
ejpam-3344	570	34	re)(aa))r	re)(aa))r	X
ejpam-3344	570	35	=	=	SYM
ejpam-3344	570	36	(	(	PUNCT
ejpam-3344	570	37	ra2)r	ra2)r	NOUN
ejpam-3344	570	38	.	.	PUNCT
ejpam-3344	571	1	thus	thus	ADV
ejpam-3344	571	2	a	a	DET
ejpam-3344	571	3	∈	∈	NOUN
ejpam-3344	571	4	(	(	PUNCT
ejpam-3344	571	5	ra2)r	ra2)r	PROPN
ejpam-3344	571	6	,	,	PUNCT
ejpam-3344	571	7	i.e.	i.e.	X
ejpam-3344	571	8	,	,	PUNCT
ejpam-3344	571	9	a	a	PRON
ejpam-3344	571	10	is	be	AUX
ejpam-3344	571	11	an	an	DET
ejpam-3344	571	12	intra	intra	ADJ
ejpam-3344	571	13	regular	regular	NOUN
ejpam-3344	571	14	.	.	PUNCT
ejpam-3344	572	1	therefore	therefore	ADV
ejpam-3344	572	2	r	r	NOUN
ejpam-3344	572	3	is	be	AUX
ejpam-3344	572	4	an	an	DET
ejpam-3344	572	5	intra	intra	ADJ
ejpam-3344	572	6	-	-	ADJ
ejpam-3344	572	7	regular	regular	ADJ
ejpam-3344	572	8	,	,	PUNCT
ejpam-3344	572	9	i.e.	i.e.	X
ejpam-3344	572	10	,	,	PUNCT
ejpam-3344	572	11	(	(	PUNCT
ejpam-3344	572	12	2)⇒	2)⇒	NUM
ejpam-3344	572	13	(	(	PUNCT
ejpam-3344	572	14	1	1	NUM
ejpam-3344	572	15	)	)	PUNCT
ejpam-3344	572	16	.	.	PUNCT
ejpam-3344	573	1	theorem	theorem	VERB
ejpam-3344	573	2	14	14	NUM
ejpam-3344	573	3	.	.	PUNCT
ejpam-3344	574	1	let	let	VERB
ejpam-3344	574	2	r	r	PRON
ejpam-3344	574	3	be	be	AUX
ejpam-3344	574	4	an	an	DET
ejpam-3344	574	5	intra	intra	ADJ
ejpam-3344	574	6	-	-	ADJ
ejpam-3344	574	7	regular	regular	ADJ
ejpam-3344	574	8	locally	locally	ADV
ejpam-3344	574	9	associative	associative	ADJ
ejpam-3344	574	10	la	la	ADJ
ejpam-3344	574	11	-	-	PUNCT
ejpam-3344	574	12	ring	ring	NOUN
ejpam-3344	574	13	.	.	PUNCT
ejpam-3344	575	1	then	then	ADV
ejpam-3344	575	2	for	for	ADP
ejpam-3344	575	3	every	every	DET
ejpam-3344	575	4	intuitionistic	intuitionistic	ADJ
ejpam-3344	575	5	fuzzy	fuzzy	ADJ
ejpam-3344	575	6	right	right	ADJ
ejpam-3344	575	7	ideal	ideal	NOUN
ejpam-3344	575	8	a	a	PRON
ejpam-3344	575	9	=	=	X
ejpam-3344	575	10	(	(	PUNCT
ejpam-3344	575	11	µa	µa	PROPN
ejpam-3344	575	12	,	,	PUNCT
ejpam-3344	575	13	γa	γa	PROPN
ejpam-3344	575	14	)	)	PUNCT
ejpam-3344	575	15	of	of	ADP
ejpam-3344	575	16	r	r	NOUN
ejpam-3344	575	17	,	,	PUNCT
ejpam-3344	575	18	a(an	a(an	PROPN
ejpam-3344	575	19	)	)	PUNCT
ejpam-3344	575	20	=	=	PUNCT
ejpam-3344	575	21	a(a2n	a(a2n	VERB
ejpam-3344	575	22	)	)	PUNCT
ejpam-3344	575	23	for	for	ADP
ejpam-3344	575	24	all	all	DET
ejpam-3344	575	25	a	a	DET
ejpam-3344	575	26	∈	∈	NOUN
ejpam-3344	575	27	r	r	NOUN
ejpam-3344	575	28	,	,	PUNCT
ejpam-3344	575	29	where	where	SCONJ
ejpam-3344	575	30	n	n	PRON
ejpam-3344	575	31	is	be	AUX
ejpam-3344	575	32	a	a	DET
ejpam-3344	575	33	positive	positive	ADJ
ejpam-3344	575	34	integer	integer	NOUN
ejpam-3344	575	35	.	.	PUNCT
ejpam-3344	576	1	proof	proof	NOUN
ejpam-3344	576	2	.	.	PUNCT
ejpam-3344	577	1	for	for	ADP
ejpam-3344	577	2	n	n	NOUN
ejpam-3344	577	3	=	=	SYM
ejpam-3344	577	4	1	1	X
ejpam-3344	577	5	.	.	PUNCT
ejpam-3344	577	6	let	let	VERB
ejpam-3344	577	7	a	a	DET
ejpam-3344	577	8	∈	∈	ADJ
ejpam-3344	577	9	r	r	NOUN
ejpam-3344	577	10	,	,	PUNCT
ejpam-3344	577	11	this	this	PRON
ejpam-3344	577	12	implies	imply	VERB
ejpam-3344	577	13	that	that	SCONJ
ejpam-3344	577	14	there	there	PRON
ejpam-3344	577	15	exist	exist	VERB
ejpam-3344	577	16	elements	element	NOUN
ejpam-3344	577	17	xi	xi	NUM
ejpam-3344	577	18	,	,	PUNCT
ejpam-3344	577	19	yi	yi	PROPN
ejpam-3344	577	20	∈	∈	PROPN
ejpam-3344	577	21	r	r	NOUN
ejpam-3344	577	22	such	such	ADJ
ejpam-3344	577	23	that	that	SCONJ
ejpam-3344	577	24	a	a	DET
ejpam-3344	577	25	=	=	SYM
ejpam-3344	577	26	∑n	∑n	PROPN
ejpam-3344	577	27	i=1(xia	i=1(xia	NOUN
ejpam-3344	577	28	2)yi	2)yi	NUM
ejpam-3344	577	29	.	.	PUNCT
ejpam-3344	578	1	thus	thus	ADV
ejpam-3344	578	2	µa	µa	X
ejpam-3344	578	3	(	(	PUNCT
ejpam-3344	578	4	a	a	X
ejpam-3344	578	5	)	)	PUNCT
ejpam-3344	578	6	=	=	SYM
ejpam-3344	578	7	µa((xia	µa((xia	PROPN
ejpam-3344	578	8	2)yi	2)yi	NUM
ejpam-3344	578	9	)	)	PUNCT
ejpam-3344	578	10	≥	≥	NOUN
ejpam-3344	578	11	µa(xia	µa(xia	PROPN
ejpam-3344	578	12	2	2	NUM
ejpam-3344	578	13	)	)	PUNCT
ejpam-3344	578	14	≥	≥	NOUN
ejpam-3344	578	15	µa(a2	µa(a2	NOUN
ejpam-3344	578	16	)	)	PUNCT
ejpam-3344	578	17	n.	n.	PROPN
ejpam-3344	578	18	kausar	kausar	PROPN
ejpam-3344	578	19	,	,	PUNCT
ejpam-3344	578	20	m.	m.	NOUN
ejpam-3344	578	21	a.	a.	PROPN
ejpam-3344	578	22	waqar	waqar	PROPN
ejpam-3344	578	23	/	/	SYM
ejpam-3344	578	24	eur	eur	PROPN
ejpam-3344	578	25	.	.	PUNCT
ejpam-3344	579	1	j.	j.	PROPN
ejpam-3344	579	2	pure	pure	PROPN
ejpam-3344	579	3	appl	appl	PROPN
ejpam-3344	579	4	.	.	PROPN
ejpam-3344	579	5	math	math	PROPN
ejpam-3344	579	6	,	,	PUNCT
ejpam-3344	579	7	12	12	NUM
ejpam-3344	579	8	(	(	PUNCT
ejpam-3344	579	9	1	1	NUM
ejpam-3344	579	10	)	)	PUNCT
ejpam-3344	579	11	(	(	PUNCT
ejpam-3344	579	12	2019	2019	NUM
ejpam-3344	579	13	)	)	PUNCT
ejpam-3344	579	14	,	,	PUNCT
ejpam-3344	579	15	226	226	NUM
ejpam-3344	579	16	-	-	SYM
ejpam-3344	579	17	250	250	NUM
ejpam-3344	579	18	247	247	NUM
ejpam-3344	579	19	=	=	SYM
ejpam-3344	579	20	µa(aa	µa(aa	PROPN
ejpam-3344	579	21	)	)	PUNCT
ejpam-3344	579	22	≥	≥	NOUN
ejpam-3344	579	23	min{µa	min{µa	X
ejpam-3344	579	24	(	(	PUNCT
ejpam-3344	579	25	a	a	NOUN
ejpam-3344	579	26	)	)	PUNCT
ejpam-3344	579	27	,	,	PUNCT
ejpam-3344	579	28	µa	µa	X
ejpam-3344	579	29	(	(	PUNCT
ejpam-3344	579	30	a	a	NOUN
ejpam-3344	579	31	)	)	PUNCT
ejpam-3344	579	32	}	}	PUNCT
ejpam-3344	579	33	=	=	SYM
ejpam-3344	579	34	µa	µa	NOUN
ejpam-3344	579	35	(	(	PUNCT
ejpam-3344	579	36	a	a	NOUN
ejpam-3344	579	37	)	)	PUNCT
ejpam-3344	579	38	.	.	PUNCT
ejpam-3344	580	1	similarly	similarly	ADV
ejpam-3344	580	2	,	,	PUNCT
ejpam-3344	580	3	γa	γa	PROPN
ejpam-3344	580	4	(	(	PUNCT
ejpam-3344	580	5	a	a	X
ejpam-3344	580	6	)	)	PUNCT
ejpam-3344	580	7	=	=	SYM
ejpam-3344	580	8	γa(a2	γa(a2	NOUN
ejpam-3344	580	9	)	)	PUNCT
ejpam-3344	580	10	,	,	PUNCT
ejpam-3344	580	11	so	so	ADV
ejpam-3344	580	12	a(a	a(a	PROPN
ejpam-3344	580	13	)	)	PUNCT
ejpam-3344	580	14	=	=	SYM
ejpam-3344	580	15	a(a2	a(a2	NOUN
ejpam-3344	580	16	)	)	PUNCT
ejpam-3344	580	17	.	.	PUNCT
ejpam-3344	581	1	result	result	NOUN
ejpam-3344	581	2	is	be	AUX
ejpam-3344	581	3	also	also	ADV
ejpam-3344	581	4	true	true	ADJ
ejpam-3344	581	5	for	for	ADP
ejpam-3344	581	6	n	n	NOUN
ejpam-3344	581	7	=	=	SYM
ejpam-3344	581	8	2	2	NUM
ejpam-3344	581	9	,	,	PUNCT
ejpam-3344	581	10	as	as	ADP
ejpam-3344	581	11	a2	a2	PROPN
ejpam-3344	581	12	=	=	SYM
ejpam-3344	581	13	aa	aa	NOUN
ejpam-3344	581	14	=	=	PUNCT
ejpam-3344	581	15	(	(	PUNCT
ejpam-3344	581	16	(	(	PUNCT
ejpam-3344	581	17	xia	xia	PROPN
ejpam-3344	581	18	2)yi)((xia	2)yi)((xia	NUM
ejpam-3344	581	19	2)yi	2)yi	NUM
ejpam-3344	581	20	)	)	PUNCT
ejpam-3344	581	21	=	=	PUNCT
ejpam-3344	582	1	(	(	PUNCT
ejpam-3344	582	2	x2i	x2i	PROPN
ejpam-3344	582	3	a	a	DET
ejpam-3344	582	4	4)y2i	4)y2i	NOUN
ejpam-3344	582	5	.	.	PUNCT
ejpam-3344	583	1	assume	assume	VERB
ejpam-3344	583	2	that	that	SCONJ
ejpam-3344	583	3	the	the	DET
ejpam-3344	583	4	result	result	NOUN
ejpam-3344	583	5	is	be	AUX
ejpam-3344	583	6	true	true	ADJ
ejpam-3344	583	7	for	for	ADP
ejpam-3344	583	8	n	n	PROPN
ejpam-3344	583	9	=	=	SYM
ejpam-3344	583	10	k	k	NOUN
ejpam-3344	583	11	,	,	PUNCT
ejpam-3344	583	12	i.e.	i.e.	X
ejpam-3344	583	13	,	,	PUNCT
ejpam-3344	583	14	a(ak	a(ak	PROPN
ejpam-3344	583	15	)	)	PUNCT
ejpam-3344	583	16	=	=	PUNCT
ejpam-3344	583	17	a(a2k	a(a2k	NOUN
ejpam-3344	583	18	)	)	PUNCT
ejpam-3344	583	19	.	.	PUNCT
ejpam-3344	584	1	now	now	ADV
ejpam-3344	584	2	ak+1	ak+1	VERB
ejpam-3344	584	3	=	=	PUNCT
ejpam-3344	584	4	aka	aka	ADV
ejpam-3344	584	5	=	=	SYM
ejpam-3344	584	6	(	(	PUNCT
ejpam-3344	584	7	(	(	PUNCT
ejpam-3344	584	8	xki	xki	VERB
ejpam-3344	584	9	a	a	DET
ejpam-3344	584	10	2k)yki	2k)yki	NOUN
ejpam-3344	584	11	)	)	PUNCT
ejpam-3344	584	12	(	(	PUNCT
ejpam-3344	584	13	(	(	PUNCT
ejpam-3344	584	14	xia	xia	PROPN
ejpam-3344	584	15	2)yi	2)yi	NUM
ejpam-3344	584	16	)	)	PUNCT
ejpam-3344	584	17	=	=	NOUN
ejpam-3344	585	1	(	(	PUNCT
ejpam-3344	585	2	xk+1	xk+1	NUM
ejpam-3344	585	3	i	i	PRON
ejpam-3344	585	4	a2(k+1))yk+1	a2(k+1))yk+1	VERB
ejpam-3344	585	5	i	i	PRON
ejpam-3344	585	6	.	.	PUNCT
ejpam-3344	585	7	thus	thus	ADV
ejpam-3344	585	8	µa(ak+1	µa(ak+1	VERB
ejpam-3344	585	9	)	)	PUNCT
ejpam-3344	585	10	=	=	PUNCT
ejpam-3344	585	11	µa((xk+1	µa((xk+1	VERB
ejpam-3344	585	12	i	i	PRON
ejpam-3344	585	13	a2(k+1))yk+1	a2(k+1))yk+1	PUNCT
ejpam-3344	585	14	i	i	PROPN
ejpam-3344	585	15	)	)	PUNCT
ejpam-3344	585	16	≥	≥	PROPN
ejpam-3344	585	17	µa(xk+1	µa(xk+1	PROPN
ejpam-3344	585	18	i	i	PRON
ejpam-3344	585	19	a2(k+1	a2(k+1	ADJ
ejpam-3344	585	20	)	)	PUNCT
ejpam-3344	585	21	)	)	PUNCT
ejpam-3344	585	22	≥	≥	NOUN
ejpam-3344	585	23	µa(a2(k+1	µa(a2(k+1	NOUN
ejpam-3344	585	24	)	)	PUNCT
ejpam-3344	585	25	)	)	PUNCT
ejpam-3344	586	1	=	=	SYM
ejpam-3344	586	2	µa(ak+1ak+1	µa(ak+1ak+1	X
ejpam-3344	586	3	)	)	PUNCT
ejpam-3344	586	4	≥	≥	NOUN
ejpam-3344	586	5	min{µa	min{µa	X
ejpam-3344	586	6	(	(	PUNCT
ejpam-3344	586	7	a(k+1	a(k+1	NOUN
ejpam-3344	586	8	)	)	PUNCT
ejpam-3344	586	9	)	)	PUNCT
ejpam-3344	586	10	,	,	PUNCT
ejpam-3344	586	11	µa	µa	ADP
ejpam-3344	586	12	(	(	PUNCT
ejpam-3344	586	13	a(k+1	a(k+1	NOUN
ejpam-3344	586	14	)	)	PUNCT
ejpam-3344	586	15	)	)	PUNCT
ejpam-3344	586	16	}	}	PUNCT
ejpam-3344	587	1	=	=	SYM
ejpam-3344	587	2	µa	µa	NOUN
ejpam-3344	587	3	(	(	PUNCT
ejpam-3344	587	4	a(k+1	a(k+1	NOUN
ejpam-3344	587	5	)	)	PUNCT
ejpam-3344	587	6	)	)	PUNCT
ejpam-3344	587	7	.	.	PUNCT
ejpam-3344	588	1	similarly	similarly	ADV
ejpam-3344	588	2	,	,	PUNCT
ejpam-3344	588	3	γa	γa	PROPN
ejpam-3344	588	4	(	(	PUNCT
ejpam-3344	588	5	a	a	X
ejpam-3344	588	6	)	)	PUNCT
ejpam-3344	588	7	=	=	SYM
ejpam-3344	588	8	γa(a2(k+1	γa(a2(k+1	ADJ
ejpam-3344	588	9	)	)	PUNCT
ejpam-3344	588	10	)	)	PUNCT
ejpam-3344	588	11	,	,	PUNCT
ejpam-3344	588	12	so	so	ADV
ejpam-3344	588	13	a(ak+1	a(ak+1	ADJ
ejpam-3344	588	14	)	)	PUNCT
ejpam-3344	588	15	=	=	SYM
ejpam-3344	588	16	a(a2(k+1	a(a2(k+1	PROPN
ejpam-3344	588	17	)	)	PUNCT
ejpam-3344	588	18	)	)	PUNCT
ejpam-3344	588	19	.	.	PUNCT
ejpam-3344	589	1	hence	hence	ADV
ejpam-3344	589	2	by	by	ADP
ejpam-3344	589	3	induction	induction	NOUN
ejpam-3344	589	4	method	method	NOUN
ejpam-3344	589	5	,	,	PUNCT
ejpam-3344	589	6	the	the	DET
ejpam-3344	589	7	result	result	NOUN
ejpam-3344	589	8	is	be	AUX
ejpam-3344	589	9	true	true	ADJ
ejpam-3344	589	10	for	for	ADP
ejpam-3344	589	11	all	all	DET
ejpam-3344	589	12	positive	positive	ADJ
ejpam-3344	589	13	integers	integer	NOUN
ejpam-3344	589	14	.	.	PUNCT
ejpam-3344	590	1	proposition	proposition	NOUN
ejpam-3344	590	2	12	12	NUM
ejpam-3344	590	3	.	.	PUNCT
ejpam-3344	591	1	let	let	VERB
ejpam-3344	591	2	r	r	PRON
ejpam-3344	591	3	be	be	AUX
ejpam-3344	591	4	an	an	DET
ejpam-3344	591	5	intra	intra	ADJ
ejpam-3344	591	6	-	-	ADJ
ejpam-3344	591	7	regular	regular	ADJ
ejpam-3344	591	8	locally	locally	ADV
ejpam-3344	591	9	associative	associative	ADJ
ejpam-3344	591	10	la	la	NOUN
ejpam-3344	591	11	-	-	NOUN
ejpam-3344	591	12	ring	ring	NOUN
ejpam-3344	591	13	with	with	ADP
ejpam-3344	591	14	left	left	ADJ
ejpam-3344	591	15	identity	identity	NOUN
ejpam-3344	591	16	e.	e.	PROPN
ejpam-3344	591	17	then	then	ADV
ejpam-3344	591	18	for	for	ADP
ejpam-3344	591	19	every	every	DET
ejpam-3344	591	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	591	21	fuzzy	fuzzy	ADJ
ejpam-3344	591	22	right	right	ADJ
ejpam-3344	591	23	ideal	ideal	NOUN
ejpam-3344	591	24	a	a	PRON
ejpam-3344	591	25	=	=	X
ejpam-3344	591	26	(	(	PUNCT
ejpam-3344	591	27	µa	µa	PROPN
ejpam-3344	591	28	,	,	PUNCT
ejpam-3344	591	29	γa	γa	PROPN
ejpam-3344	591	30	)	)	PUNCT
ejpam-3344	591	31	of	of	ADP
ejpam-3344	591	32	r	r	NOUN
ejpam-3344	591	33	,	,	PUNCT
ejpam-3344	591	34	a(ab	a(ab	NOUN
ejpam-3344	591	35	)	)	PUNCT
ejpam-3344	591	36	=	=	SYM
ejpam-3344	591	37	a(ba	a(ba	NOUN
ejpam-3344	591	38	)	)	PUNCT
ejpam-3344	591	39	for	for	ADP
ejpam-3344	591	40	all	all	DET
ejpam-3344	591	41	a	a	DET
ejpam-3344	591	42	,	,	PUNCT
ejpam-3344	591	43	b	b	PROPN
ejpam-3344	591	44	∈	∈	PROPN
ejpam-3344	591	45	r.	r.	NOUN
ejpam-3344	591	46	proof	proof	NOUN
ejpam-3344	591	47	.	.	PUNCT
ejpam-3344	592	1	same	same	ADJ
ejpam-3344	592	2	as	as	ADP
ejpam-3344	592	3	lemma	lemma	PROPN
ejpam-3344	592	4	14	14	NUM
ejpam-3344	592	5	.	.	PUNCT
ejpam-3344	593	1	theorem	theorem	VERB
ejpam-3344	593	2	15	15	NUM
ejpam-3344	593	3	.	.	PUNCT
ejpam-3344	594	1	if	if	SCONJ
ejpam-3344	594	2	an	an	DET
ejpam-3344	594	3	ifs	ifs	PROPN
ejpam-3344	594	4	a	a	X
ejpam-3344	594	5	=	=	X
ejpam-3344	594	6	(	(	PUNCT
ejpam-3344	594	7	µa	µa	PROPN
ejpam-3344	594	8	,	,	PUNCT
ejpam-3344	594	9	γa	γa	PROPN
ejpam-3344	594	10	)	)	PUNCT
ejpam-3344	594	11	of	of	ADP
ejpam-3344	594	12	an	an	DET
ejpam-3344	594	13	la	la	ADJ
ejpam-3344	594	14	-	-	PUNCT
ejpam-3344	594	15	ring	ring	NOUN
ejpam-3344	594	16	r	r	NOUN
ejpam-3344	594	17	is	be	AUX
ejpam-3344	594	18	an	an	DET
ejpam-3344	594	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	594	20	fuzzy	fuzzy	ADJ
ejpam-3344	594	21	(	(	PUNCT
ejpam-3344	594	22	1	1	NUM
ejpam-3344	594	23	,	,	PUNCT
ejpam-3344	594	24	2)ideal	2)ideal	NUM
ejpam-3344	594	25	of	of	ADP
ejpam-3344	594	26	r	r	NOUN
ejpam-3344	594	27	,	,	PUNCT
ejpam-3344	594	28	then	then	ADV
ejpam-3344	594	29	so	so	ADV
ejpam-3344	594	30	is	be	AUX
ejpam-3344	594	31	�	�	X
ejpam-3344	594	32	a	a	NOUN
ejpam-3344	594	33	=	=	PUNCT
ejpam-3344	594	34	(	(	PUNCT
ejpam-3344	594	35	µa	µa	PROPN
ejpam-3344	594	36	,	,	PUNCT
ejpam-3344	594	37	µa	µa	NOUN
ejpam-3344	594	38	)	)	PUNCT
ejpam-3344	594	39	(	(	PUNCT
ejpam-3344	594	40	resp	resp	NOUN
ejpam-3344	594	41	.	.	PUNCT
ejpam-3344	595	1	♦	♦	PROPN
ejpam-3344	595	2	a	a	PROPN
ejpam-3344	595	3	=	=	X
ejpam-3344	595	4	(	(	PUNCT
ejpam-3344	595	5	γa	γa	PROPN
ejpam-3344	595	6	,	,	PUNCT
ejpam-3344	595	7	γa	γa	NOUN
ejpam-3344	595	8	)	)	PUNCT
ejpam-3344	595	9	)	)	PUNCT
ejpam-3344	595	10	.	.	PUNCT
ejpam-3344	596	1	proof	proof	NOUN
ejpam-3344	596	2	.	.	PUNCT
ejpam-3344	597	1	let	let	VERB
ejpam-3344	597	2	a	a	DET
ejpam-3344	597	3	=	=	SYM
ejpam-3344	597	4	(	(	PUNCT
ejpam-3344	597	5	µa	µa	PROPN
ejpam-3344	597	6	,	,	PUNCT
ejpam-3344	597	7	γa	γa	PROPN
ejpam-3344	597	8	)	)	PUNCT
ejpam-3344	597	9	be	be	VERB
ejpam-3344	597	10	an	an	DET
ejpam-3344	597	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	597	12	fuzzy	fuzzy	ADJ
ejpam-3344	597	13	(	(	PUNCT
ejpam-3344	597	14	1	1	NUM
ejpam-3344	597	15	,	,	PUNCT
ejpam-3344	597	16	2)-ideal	2)-ideal	NUM
ejpam-3344	597	17	of	of	ADP
ejpam-3344	597	18	r.	r.	PROPN
ejpam-3344	597	19	we	we	PRON
ejpam-3344	597	20	have	have	VERB
ejpam-3344	597	21	to	to	PART
ejpam-3344	597	22	show	show	VERB
ejpam-3344	597	23	that	that	SCONJ
ejpam-3344	597	24	�	�	NOUN
ejpam-3344	597	25	a	a	NOUN
ejpam-3344	597	26	=	=	SYM
ejpam-3344	597	27	(	(	PUNCT
ejpam-3344	597	28	µa	µa	PROPN
ejpam-3344	597	29	,	,	PUNCT
ejpam-3344	597	30	µa	µa	NOUN
ejpam-3344	597	31	)	)	PUNCT
ejpam-3344	597	32	is	be	AUX
ejpam-3344	597	33	also	also	ADV
ejpam-3344	597	34	an	an	DET
ejpam-3344	597	35	intuitionistic	intuitionistic	ADJ
ejpam-3344	597	36	fuzzy	fuzzy	ADJ
ejpam-3344	597	37	(	(	PUNCT
ejpam-3344	597	38	1	1	NUM
ejpam-3344	597	39	,	,	PUNCT
ejpam-3344	597	40	2)-ideal	2)-ideal	NUM
ejpam-3344	597	41	of	of	ADP
ejpam-3344	597	42	r.	r.	PROPN
ejpam-3344	597	43	now	now	ADV
ejpam-3344	597	44	µa(x−	µa(x−	NUM
ejpam-3344	597	45	y	y	NOUN
ejpam-3344	597	46	)	)	PUNCT
ejpam-3344	597	47	=	=	SYM
ejpam-3344	598	1	1−	1−	NUM
ejpam-3344	598	2	µa(x−	µa(x−	NUM
ejpam-3344	598	3	y	y	PROPN
ejpam-3344	598	4	)	)	PUNCT
ejpam-3344	598	5	≤	≤	NUM
ejpam-3344	598	6	1−min	1−min	PROPN
ejpam-3344	598	7	{	{	PUNCT
ejpam-3344	598	8	µa(x	µa(x	NOUN
ejpam-3344	598	9	)	)	PUNCT
ejpam-3344	598	10	,	,	PUNCT
ejpam-3344	598	11	µa(y	µa(y	NOUN
ejpam-3344	598	12	)	)	PUNCT
ejpam-3344	598	13	}	}	PUNCT
ejpam-3344	599	1	=	=	SYM
ejpam-3344	599	2	max	max	X
ejpam-3344	599	3	{	{	PUNCT
ejpam-3344	599	4	1−	1−	NUM
ejpam-3344	599	5	µa(x	µa(x	NOUN
ejpam-3344	599	6	)	)	PUNCT
ejpam-3344	599	7	,	,	PUNCT
ejpam-3344	599	8	1−	1−	NUM
ejpam-3344	599	9	µa(y	µa(y	NOUN
ejpam-3344	599	10	)	)	PUNCT
ejpam-3344	599	11	}	}	PUNCT
ejpam-3344	599	12	=	=	SYM
ejpam-3344	599	13	max{µa(x	max{µa(x	NOUN
ejpam-3344	599	14	)	)	PUNCT
ejpam-3344	599	15	,	,	PUNCT
ejpam-3344	599	16	µa(y	µa(y	NOUN
ejpam-3344	599	17	)	)	PUNCT
ejpam-3344	599	18	}	}	PUNCT
ejpam-3344	599	19	.	.	PUNCT
ejpam-3344	600	1	µa(xy	µa(xy	NOUN
ejpam-3344	600	2	)	)	PUNCT
ejpam-3344	600	3	=	=	SYM
ejpam-3344	600	4	1−	1−	NUM
ejpam-3344	600	5	µa(xy	µa(xy	NOUN
ejpam-3344	600	6	)	)	PUNCT
ejpam-3344	600	7	≤	≤	NUM
ejpam-3344	600	8	1−min	1−min	PROPN
ejpam-3344	600	9	{	{	PUNCT
ejpam-3344	600	10	µa(x	µa(x	NOUN
ejpam-3344	600	11	)	)	PUNCT
ejpam-3344	600	12	,	,	PUNCT
ejpam-3344	600	13	µa(y	µa(y	NOUN
ejpam-3344	600	14	)	)	PUNCT
ejpam-3344	600	15	}	}	PUNCT
ejpam-3344	600	16	=	=	SYM
ejpam-3344	600	17	max	max	X
ejpam-3344	600	18	{	{	PUNCT
ejpam-3344	600	19	1−	1−	NUM
ejpam-3344	600	20	µa(x	µa(x	NOUN
ejpam-3344	600	21	)	)	PUNCT
ejpam-3344	600	22	,	,	PUNCT
ejpam-3344	600	23	1−	1−	NUM
ejpam-3344	600	24	µa(y	µa(y	NOUN
ejpam-3344	600	25	)	)	PUNCT
ejpam-3344	600	26	}	}	PUNCT
ejpam-3344	600	27	=	=	SYM
ejpam-3344	600	28	max{µa(x	max{µa(x	NOUN
ejpam-3344	600	29	)	)	PUNCT
ejpam-3344	600	30	,	,	PUNCT
ejpam-3344	600	31	µa(y	µa(y	NOUN
ejpam-3344	600	32	)	)	PUNCT
ejpam-3344	600	33	}	}	PUNCT
ejpam-3344	600	34	.	.	PUNCT
ejpam-3344	601	1	µa((xa)(yz	µa((xa)(yz	X
ejpam-3344	601	2	)	)	PUNCT
ejpam-3344	601	3	)	)	PUNCT
ejpam-3344	602	1	=	=	SYM
ejpam-3344	602	2	1−	1−	NUM
ejpam-3344	602	3	µa((xa)(yz	µa((xa)(yz	NOUN
ejpam-3344	602	4	)	)	PUNCT
ejpam-3344	602	5	)	)	PUNCT
ejpam-3344	603	1	≤	≤	NUM
ejpam-3344	603	2	1−min	1−min	NUM
ejpam-3344	603	3	{	{	PUNCT
ejpam-3344	603	4	µa(x	µa(x	NOUN
ejpam-3344	603	5	)	)	PUNCT
ejpam-3344	603	6	,	,	PUNCT
ejpam-3344	603	7	µa(y	µa(y	NOUN
ejpam-3344	603	8	)	)	PUNCT
ejpam-3344	603	9	,	,	PUNCT
ejpam-3344	603	10	µa(z	µa(z	NUM
ejpam-3344	603	11	)	)	PUNCT
ejpam-3344	603	12	}	}	PUNCT
ejpam-3344	603	13	=	=	SYM
ejpam-3344	603	14	max	max	X
ejpam-3344	603	15	{	{	PUNCT
ejpam-3344	603	16	1−	1−	NUM
ejpam-3344	603	17	µa(x	µa(x	NOUN
ejpam-3344	603	18	)	)	PUNCT
ejpam-3344	603	19	,	,	PUNCT
ejpam-3344	603	20	1−	1−	NUM
ejpam-3344	603	21	µa(y	µa(y	NOUN
ejpam-3344	603	22	)	)	PUNCT
ejpam-3344	603	23	,	,	PUNCT
ejpam-3344	603	24	1−	1−	NUM
ejpam-3344	603	25	µa(z	µa(z	NUM
ejpam-3344	603	26	)	)	PUNCT
ejpam-3344	603	27	}	}	PUNCT
ejpam-3344	603	28	=	=	SYM
ejpam-3344	603	29	max{µa(x	max{µa(x	NOUN
ejpam-3344	603	30	)	)	PUNCT
ejpam-3344	603	31	,	,	PUNCT
ejpam-3344	603	32	µa(y	µa(y	NOUN
ejpam-3344	603	33	)	)	PUNCT
ejpam-3344	603	34	,	,	PUNCT
ejpam-3344	603	35	µa(z	µa(z	NUM
ejpam-3344	603	36	)	)	PUNCT
ejpam-3344	603	37	}	}	PUNCT
ejpam-3344	603	38	.	.	PUNCT
ejpam-3344	604	1	hence	hence	ADV
ejpam-3344	604	2	�	�	PROPN
ejpam-3344	604	3	a	a	PRON
ejpam-3344	604	4	is	be	AUX
ejpam-3344	604	5	an	an	DET
ejpam-3344	604	6	intuitionistic	intuitionistic	ADJ
ejpam-3344	604	7	fuzzy	fuzzy	ADJ
ejpam-3344	604	8	(	(	PUNCT
ejpam-3344	604	9	1	1	NUM
ejpam-3344	604	10	,	,	PUNCT
ejpam-3344	604	11	2)-ideal	2)-ideal	NUM
ejpam-3344	604	12	of	of	ADP
ejpam-3344	604	13	r.	r.	PROPN
ejpam-3344	604	14	similarly	similarly	ADV
ejpam-3344	604	15	,	,	PUNCT
ejpam-3344	604	16	for	for	ADP
ejpam-3344	604	17	♦	♦	PROPN
ejpam-3344	604	18	a	a	X
ejpam-3344	604	19	=	=	X
ejpam-3344	604	20	(	(	PUNCT
ejpam-3344	604	21	γa	γa	PROPN
ejpam-3344	604	22	,	,	PUNCT
ejpam-3344	604	23	γa	γa	PROPN
ejpam-3344	604	24	)	)	PUNCT
ejpam-3344	604	25	.	.	PUNCT
ejpam-3344	605	1	remark	remark	PROPN
ejpam-3344	605	2	14	14	NUM
ejpam-3344	605	3	.	.	PUNCT
ejpam-3344	606	1	1	1	NUM
ejpam-3344	606	2	.	.	X
ejpam-3344	606	3	an	an	DET
ejpam-3344	606	4	ifs	ifs	PROPN
ejpam-3344	606	5	a	a	X
ejpam-3344	606	6	=	=	X
ejpam-3344	606	7	(	(	PUNCT
ejpam-3344	606	8	µa	µa	PROPN
ejpam-3344	606	9	,	,	PUNCT
ejpam-3344	606	10	γa	γa	PROPN
ejpam-3344	606	11	)	)	PUNCT
ejpam-3344	606	12	of	of	ADP
ejpam-3344	606	13	an	an	DET
ejpam-3344	606	14	la	la	ADJ
ejpam-3344	606	15	-	-	PUNCT
ejpam-3344	606	16	ring	ring	NOUN
ejpam-3344	606	17	r	r	NOUN
ejpam-3344	606	18	is	be	AUX
ejpam-3344	606	19	an	an	DET
ejpam-3344	606	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	606	21	fuzzy	fuzzy	ADJ
ejpam-3344	606	22	(	(	PUNCT
ejpam-3344	606	23	1	1	NUM
ejpam-3344	606	24	,	,	PUNCT
ejpam-3344	606	25	2)ideal	2)ideal	NUM
ejpam-3344	606	26	of	of	ADP
ejpam-3344	606	27	r	r	NOUN
ejpam-3344	606	28	if	if	SCONJ
ejpam-3344	607	1	and	and	CCONJ
ejpam-3344	607	2	only	only	ADV
ejpam-3344	607	3	if	if	SCONJ
ejpam-3344	607	4	�	�	NOUN
ejpam-3344	607	5	a	a	NOUN
ejpam-3344	607	6	=	=	X
ejpam-3344	607	7	(	(	PUNCT
ejpam-3344	607	8	µa	µa	PROPN
ejpam-3344	607	9	,	,	PUNCT
ejpam-3344	607	10	µa	µa	NOUN
ejpam-3344	607	11	)	)	PUNCT
ejpam-3344	607	12	(	(	PUNCT
ejpam-3344	607	13	resp	resp	NOUN
ejpam-3344	607	14	.	.	PUNCT
ejpam-3344	608	1	♦	♦	PROPN
ejpam-3344	608	2	a	a	PROPN
ejpam-3344	608	3	=	=	X
ejpam-3344	608	4	(	(	PUNCT
ejpam-3344	608	5	γa	γa	PROPN
ejpam-3344	608	6	,	,	PUNCT
ejpam-3344	608	7	γa	γa	PROPN
ejpam-3344	608	8	)	)	PUNCT
ejpam-3344	608	9	)	)	PUNCT
ejpam-3344	608	10	is	be	AUX
ejpam-3344	608	11	an	an	DET
ejpam-3344	608	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	608	13	fuzzy	fuzzy	ADJ
ejpam-3344	608	14	(	(	PUNCT
ejpam-3344	608	15	1	1	NUM
ejpam-3344	608	16	,	,	PUNCT
ejpam-3344	608	17	2)-ideal	2)-ideal	NUM
ejpam-3344	608	18	of	of	ADP
ejpam-3344	608	19	r.	r.	PROPN
ejpam-3344	608	20	2	2	NUM
ejpam-3344	608	21	.	.	PUNCT
ejpam-3344	609	1	if	if	SCONJ
ejpam-3344	609	2	an	an	DET
ejpam-3344	609	3	ifs	ifs	PROPN
ejpam-3344	609	4	a	a	X
ejpam-3344	609	5	=	=	X
ejpam-3344	609	6	(	(	PUNCT
ejpam-3344	609	7	µa	µa	PROPN
ejpam-3344	609	8	,	,	PUNCT
ejpam-3344	609	9	γa	γa	PROPN
ejpam-3344	609	10	)	)	PUNCT
ejpam-3344	609	11	of	of	ADP
ejpam-3344	609	12	an	an	DET
ejpam-3344	609	13	la	la	ADJ
ejpam-3344	609	14	-	-	PUNCT
ejpam-3344	609	15	ring	ring	NOUN
ejpam-3344	609	16	r	r	NOUN
ejpam-3344	609	17	is	be	AUX
ejpam-3344	609	18	an	an	DET
ejpam-3344	609	19	intuitionistic	intuitionistic	ADJ
ejpam-3344	609	20	fuzzy	fuzzy	ADJ
ejpam-3344	609	21	bi	bi	NOUN
ejpam-3344	609	22	-	-	NOUN
ejpam-3344	609	23	ideal	ideal	NOUN
ejpam-3344	609	24	of	of	ADP
ejpam-3344	609	25	r	r	NOUN
ejpam-3344	609	26	,	,	PUNCT
ejpam-3344	609	27	then	then	ADV
ejpam-3344	609	28	so	so	ADV
ejpam-3344	609	29	is	be	AUX
ejpam-3344	609	30	�	�	X
ejpam-3344	609	31	a	a	NOUN
ejpam-3344	609	32	=	=	PUNCT
ejpam-3344	609	33	(	(	PUNCT
ejpam-3344	609	34	µa	µa	PROPN
ejpam-3344	609	35	,	,	PUNCT
ejpam-3344	609	36	µa	µa	NOUN
ejpam-3344	609	37	)	)	PUNCT
ejpam-3344	609	38	(	(	PUNCT
ejpam-3344	609	39	resp	resp	NOUN
ejpam-3344	609	40	.	.	PUNCT
ejpam-3344	610	1	♦	♦	PROPN
ejpam-3344	610	2	a	a	PROPN
ejpam-3344	610	3	=	=	X
ejpam-3344	610	4	(	(	PUNCT
ejpam-3344	610	5	γa	γa	PROPN
ejpam-3344	610	6	,	,	PUNCT
ejpam-3344	610	7	γa	γa	PROPN
ejpam-3344	610	8	)	)	PUNCT
ejpam-3344	610	9	)	)	PUNCT
ejpam-3344	610	10	.	.	PUNCT
ejpam-3344	611	1	3	3	X
ejpam-3344	611	2	.	.	X
ejpam-3344	611	3	an	an	DET
ejpam-3344	611	4	ifs	ifs	PROPN
ejpam-3344	611	5	a	a	X
ejpam-3344	611	6	=	=	X
ejpam-3344	611	7	(	(	PUNCT
ejpam-3344	611	8	µa	µa	PROPN
ejpam-3344	611	9	,	,	PUNCT
ejpam-3344	611	10	γa	γa	PROPN
ejpam-3344	611	11	)	)	PUNCT
ejpam-3344	611	12	of	of	ADP
ejpam-3344	611	13	an	an	DET
ejpam-3344	611	14	la	la	ADJ
ejpam-3344	611	15	-	-	PUNCT
ejpam-3344	611	16	ring	ring	NOUN
ejpam-3344	611	17	r	r	NOUN
ejpam-3344	611	18	is	be	AUX
ejpam-3344	611	19	an	an	DET
ejpam-3344	611	20	intuitionistic	intuitionistic	ADJ
ejpam-3344	611	21	fuzzy	fuzzy	ADJ
ejpam-3344	611	22	bi	bi	NOUN
ejpam-3344	611	23	-	-	NOUN
ejpam-3344	611	24	ideal	ideal	NOUN
ejpam-3344	611	25	of	of	ADP
ejpam-3344	611	26	r	r	NOUN
ejpam-3344	611	27	if	if	SCONJ
ejpam-3344	611	28	and	and	CCONJ
ejpam-3344	611	29	only	only	ADV
ejpam-3344	611	30	if	if	SCONJ
ejpam-3344	611	31	�	�	NOUN
ejpam-3344	611	32	a	a	NOUN
ejpam-3344	611	33	=	=	X
ejpam-3344	611	34	(	(	PUNCT
ejpam-3344	611	35	µa	µa	PROPN
ejpam-3344	611	36	,	,	PUNCT
ejpam-3344	611	37	µa	µa	NOUN
ejpam-3344	611	38	)	)	PUNCT
ejpam-3344	611	39	(	(	PUNCT
ejpam-3344	611	40	resp	resp	NOUN
ejpam-3344	611	41	.	.	PUNCT
ejpam-3344	612	1	♦	♦	PROPN
ejpam-3344	612	2	a	a	PROPN
ejpam-3344	612	3	=	=	X
ejpam-3344	612	4	(	(	PUNCT
ejpam-3344	612	5	γa	γa	PROPN
ejpam-3344	612	6	,	,	PUNCT
ejpam-3344	612	7	γa	γa	PROPN
ejpam-3344	612	8	)	)	PUNCT
ejpam-3344	612	9	)	)	PUNCT
ejpam-3344	612	10	is	be	AUX
ejpam-3344	612	11	an	an	DET
ejpam-3344	612	12	intuitionistic	intuitionistic	ADJ
ejpam-3344	612	13	fuzzy	fuzzy	ADJ
ejpam-3344	612	14	bi	bi	NOUN
ejpam-3344	612	15	-	-	NOUN
ejpam-3344	612	16	ideal	ideal	NOUN
ejpam-3344	612	17	of	of	ADP
ejpam-3344	612	18	r.	r.	PROPN
ejpam-3344	612	19	pseudo	pseudo	NOUN
ejpam-3344	612	20	-	-	NOUN
ejpam-3344	612	21	integrality	integrality	NOUN
ejpam-3344	612	22	relative	relative	NOUN
ejpam-3344	612	23	to	to	ADP
ejpam-3344	612	24	a	a	DET
ejpam-3344	612	25	rational	rational	ADJ
ejpam-3344	612	26	cyclic	cyclic	ADJ
ejpam-3344	612	27	monoid	monoid	NOUN
ejpam-3344	612	28	248	248	NUM
ejpam-3344	612	29	theorem	theorem	VERB
ejpam-3344	612	30	16	16	NUM
ejpam-3344	612	31	.	.	PUNCT
ejpam-3344	613	1	an	an	DET
ejpam-3344	613	2	ifs	ifs	PROPN
ejpam-3344	613	3	a	a	X
ejpam-3344	613	4	=	=	X
ejpam-3344	613	5	(	(	PUNCT
ejpam-3344	613	6	µa	µa	PROPN
ejpam-3344	613	7	,	,	PUNCT
ejpam-3344	613	8	γa	γa	PROPN
ejpam-3344	613	9	)	)	PUNCT
ejpam-3344	613	10	of	of	ADP
ejpam-3344	613	11	an	an	DET
ejpam-3344	613	12	la	la	ADJ
ejpam-3344	613	13	-	-	PUNCT
ejpam-3344	613	14	ring	ring	NOUN
ejpam-3344	613	15	r	r	NOUN
ejpam-3344	613	16	is	be	AUX
ejpam-3344	613	17	an	an	DET
ejpam-3344	613	18	intuitionistic	intuitionistic	ADJ
ejpam-3344	613	19	fuzzy	fuzzy	ADJ
ejpam-3344	613	20	(	(	PUNCT
ejpam-3344	613	21	1	1	NUM
ejpam-3344	613	22	,	,	PUNCT
ejpam-3344	613	23	2)-ideal	2)-ideal	NUM
ejpam-3344	613	24	of	of	ADP
ejpam-3344	613	25	r	r	NOUN
ejpam-3344	613	26	if	if	SCONJ
ejpam-3344	614	1	and	and	CCONJ
ejpam-3344	614	2	only	only	ADV
ejpam-3344	614	3	if	if	SCONJ
ejpam-3344	614	4	the	the	DET
ejpam-3344	614	5	fuzzy	fuzzy	ADJ
ejpam-3344	614	6	subsets	subset	NOUN
ejpam-3344	614	7	µa	µa	PROPN
ejpam-3344	614	8	and	and	CCONJ
ejpam-3344	614	9	γa	γa	PROPN
ejpam-3344	614	10	are	be	AUX
ejpam-3344	614	11	fuzzy	fuzzy	ADJ
ejpam-3344	614	12	(	(	PUNCT
ejpam-3344	614	13	1	1	NUM
ejpam-3344	614	14	,	,	PUNCT
ejpam-3344	614	15	2)-ideals	2)-ideals	NUM
ejpam-3344	614	16	of	of	ADP
ejpam-3344	614	17	r.	r.	PROPN
ejpam-3344	614	18	proof	proof	NOUN
ejpam-3344	614	19	.	.	PUNCT
ejpam-3344	615	1	let	let	VERB
ejpam-3344	615	2	a	a	DET
ejpam-3344	615	3	=	=	SYM
ejpam-3344	615	4	(	(	PUNCT
ejpam-3344	615	5	µa	µa	PROPN
ejpam-3344	615	6	,	,	PUNCT
ejpam-3344	615	7	γa	γa	PROPN
ejpam-3344	615	8	)	)	PUNCT
ejpam-3344	615	9	be	be	VERB
ejpam-3344	615	10	an	an	DET
ejpam-3344	615	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	615	12	fuzzy	fuzzy	ADJ
ejpam-3344	615	13	(	(	PUNCT
ejpam-3344	615	14	1	1	NUM
ejpam-3344	615	15	,	,	PUNCT
ejpam-3344	615	16	2)-ideal	2)-ideal	NUM
ejpam-3344	615	17	of	of	ADP
ejpam-3344	615	18	r	r	NOUN
ejpam-3344	615	19	,	,	PUNCT
ejpam-3344	615	20	this	this	PRON
ejpam-3344	615	21	implies	imply	VERB
ejpam-3344	615	22	that	that	SCONJ
ejpam-3344	615	23	µa	µa	NOUN
ejpam-3344	615	24	is	be	AUX
ejpam-3344	615	25	a	a	DET
ejpam-3344	615	26	fuzzy	fuzzy	ADJ
ejpam-3344	615	27	(	(	PUNCT
ejpam-3344	615	28	1	1	NUM
ejpam-3344	615	29	,	,	PUNCT
ejpam-3344	615	30	2)-ideal	2)-ideal	NUM
ejpam-3344	615	31	of	of	ADP
ejpam-3344	615	32	r.	r.	PROPN
ejpam-3344	615	33	we	we	PRON
ejpam-3344	615	34	have	have	VERB
ejpam-3344	615	35	to	to	PART
ejpam-3344	615	36	show	show	VERB
ejpam-3344	615	37	that	that	SCONJ
ejpam-3344	615	38	γa	γa	PROPN
ejpam-3344	615	39	is	be	AUX
ejpam-3344	615	40	also	also	ADV
ejpam-3344	615	41	a	a	DET
ejpam-3344	615	42	fuzzy	fuzzy	ADJ
ejpam-3344	615	43	(	(	PUNCT
ejpam-3344	615	44	1	1	NUM
ejpam-3344	615	45	,	,	PUNCT
ejpam-3344	615	46	2)-ideal	2)-ideal	NUM
ejpam-3344	615	47	of	of	ADP
ejpam-3344	615	48	r.	r.	PROPN
ejpam-3344	615	49	now	now	ADV
ejpam-3344	615	50	γa(x−	γa(x−	PROPN
ejpam-3344	615	51	y	y	NOUN
ejpam-3344	615	52	)	)	PUNCT
ejpam-3344	615	53	=	=	SYM
ejpam-3344	615	54	1−	1−	NUM
ejpam-3344	615	55	γa(x−	γa(x−	SYM
ejpam-3344	615	56	y	y	PROPN
ejpam-3344	615	57	)	)	PUNCT
ejpam-3344	615	58	≥	≥	NOUN
ejpam-3344	615	59	1−max{γa(x	1−max{γa(x	PROPN
ejpam-3344	615	60	)	)	PUNCT
ejpam-3344	615	61	,	,	PUNCT
ejpam-3344	615	62	γa(y	γa(y	NOUN
ejpam-3344	615	63	)	)	PUNCT
ejpam-3344	615	64	}	}	PUNCT
ejpam-3344	615	65	=	=	PUNCT
ejpam-3344	615	66	min{1−	min{1−	VERB
ejpam-3344	615	67	γa(x	γa(x	NUM
ejpam-3344	615	68	)	)	PUNCT
ejpam-3344	615	69	,	,	PUNCT
ejpam-3344	615	70	1−	1−	NUM
ejpam-3344	615	71	γa(y	γa(y	NUM
ejpam-3344	615	72	)	)	PUNCT
ejpam-3344	615	73	}	}	PUNCT
ejpam-3344	615	74	=	=	SYM
ejpam-3344	615	75	min{γa(x	min{γa(x	NOUN
ejpam-3344	615	76	)	)	PUNCT
ejpam-3344	615	77	,	,	PUNCT
ejpam-3344	615	78	γa(y	γa(y	NOUN
ejpam-3344	615	79	)	)	PUNCT
ejpam-3344	615	80	}	}	PUNCT
ejpam-3344	615	81	.	.	PUNCT
ejpam-3344	616	1	γa(xy	γa(xy	PROPN
ejpam-3344	616	2	)	)	PUNCT
ejpam-3344	617	1	=	=	SYM
ejpam-3344	617	2	1−	1−	NUM
ejpam-3344	617	3	γa(xy	γa(xy	PROPN
ejpam-3344	617	4	)	)	PUNCT
ejpam-3344	617	5	≥	≥	NOUN
ejpam-3344	617	6	1−max{γa(x	1−max{γa(x	PROPN
ejpam-3344	617	7	)	)	PUNCT
ejpam-3344	617	8	,	,	PUNCT
ejpam-3344	617	9	γa(y	γa(y	NOUN
ejpam-3344	617	10	)	)	PUNCT
ejpam-3344	617	11	}	}	PUNCT
ejpam-3344	617	12	=	=	PUNCT
ejpam-3344	617	13	min{1−	min{1−	VERB
ejpam-3344	617	14	γa(x	γa(x	NUM
ejpam-3344	617	15	)	)	PUNCT
ejpam-3344	617	16	,	,	PUNCT
ejpam-3344	617	17	1−	1−	NUM
ejpam-3344	617	18	γa(y	γa(y	NUM
ejpam-3344	617	19	)	)	PUNCT
ejpam-3344	617	20	}	}	PUNCT
ejpam-3344	617	21	=	=	SYM
ejpam-3344	617	22	min{γa(x	min{γa(x	NOUN
ejpam-3344	617	23	)	)	PUNCT
ejpam-3344	617	24	,	,	PUNCT
ejpam-3344	617	25	γa(y	γa(y	NOUN
ejpam-3344	617	26	)	)	PUNCT
ejpam-3344	617	27	}	}	PUNCT
ejpam-3344	617	28	.	.	PUNCT
ejpam-3344	618	1	γa((xa)(yz	γa((xa)(yz	NOUN
ejpam-3344	618	2	)	)	PUNCT
ejpam-3344	618	3	)	)	PUNCT
ejpam-3344	619	1	=	=	SYM
ejpam-3344	619	2	1−	1−	NUM
ejpam-3344	619	3	γa((xa)(yz	γa((xa)(yz	PROPN
ejpam-3344	619	4	)	)	PUNCT
ejpam-3344	619	5	)	)	PUNCT
ejpam-3344	619	6	≥	≥	NOUN
ejpam-3344	619	7	1−max{γa(x	1−max{γa(x	PROPN
ejpam-3344	619	8	)	)	PUNCT
ejpam-3344	619	9	,	,	PUNCT
ejpam-3344	619	10	γa(y	γa(y	NUM
ejpam-3344	619	11	)	)	PUNCT
ejpam-3344	619	12	,	,	PUNCT
ejpam-3344	619	13	γa(z	γa(z	PROPN
ejpam-3344	619	14	)	)	PUNCT
ejpam-3344	619	15	}	}	PUNCT
ejpam-3344	619	16	=	=	PUNCT
ejpam-3344	619	17	min{1−	min{1−	VERB
ejpam-3344	619	18	γa(x	γa(x	NUM
ejpam-3344	619	19	)	)	PUNCT
ejpam-3344	619	20	,	,	PUNCT
ejpam-3344	619	21	1−	1−	NUM
ejpam-3344	619	22	γa(y	γa(y	NUM
ejpam-3344	619	23	)	)	PUNCT
ejpam-3344	619	24	,	,	PUNCT
ejpam-3344	619	25	1−	1−	NUM
ejpam-3344	619	26	γa(z	γa(z	NOUN
ejpam-3344	619	27	)	)	PUNCT
ejpam-3344	619	28	}	}	PUNCT
ejpam-3344	619	29	=	=	SYM
ejpam-3344	619	30	min{γa(x	min{γa(x	NOUN
ejpam-3344	619	31	)	)	PUNCT
ejpam-3344	619	32	,	,	PUNCT
ejpam-3344	619	33	γa(y	γa(y	NUM
ejpam-3344	619	34	)	)	PUNCT
ejpam-3344	619	35	,	,	PUNCT
ejpam-3344	619	36	γa(z	γa(z	PROPN
ejpam-3344	619	37	)	)	PUNCT
ejpam-3344	619	38	}	}	PUNCT
ejpam-3344	619	39	.	.	PUNCT
ejpam-3344	620	1	therefore	therefore	ADV
ejpam-3344	620	2	γa	γa	PROPN
ejpam-3344	620	3	is	be	AUX
ejpam-3344	620	4	a	a	DET
ejpam-3344	620	5	fuzzy	fuzzy	ADJ
ejpam-3344	620	6	(	(	PUNCT
ejpam-3344	620	7	1	1	NUM
ejpam-3344	620	8	,	,	PUNCT
ejpam-3344	620	9	2)-ideal	2)-ideal	NUM
ejpam-3344	620	10	of	of	ADP
ejpam-3344	620	11	r.	r.	PROPN
ejpam-3344	620	12	conversely	conversely	ADV
ejpam-3344	620	13	,	,	PUNCT
ejpam-3344	620	14	suppose	suppose	VERB
ejpam-3344	620	15	that	that	SCONJ
ejpam-3344	620	16	µa	µa	NOUN
ejpam-3344	620	17	and	and	CCONJ
ejpam-3344	620	18	γa	γa	PROPN
ejpam-3344	620	19	are	be	AUX
ejpam-3344	620	20	fuzzy	fuzzy	ADJ
ejpam-3344	620	21	(	(	PUNCT
ejpam-3344	620	22	1	1	NUM
ejpam-3344	620	23	,	,	PUNCT
ejpam-3344	620	24	2)-ideals	2)-ideals	NUM
ejpam-3344	620	25	of	of	ADP
ejpam-3344	620	26	r.	r.	PROPN
ejpam-3344	620	27	we	we	PRON
ejpam-3344	620	28	have	have	VERB
ejpam-3344	620	29	to	to	PART
ejpam-3344	620	30	show	show	VERB
ejpam-3344	620	31	that	that	SCONJ
ejpam-3344	620	32	a	a	DET
ejpam-3344	620	33	=	=	SYM
ejpam-3344	620	34	(	(	PUNCT
ejpam-3344	620	35	µa	µa	PROPN
ejpam-3344	620	36	,	,	PUNCT
ejpam-3344	620	37	γa	γa	PROPN
ejpam-3344	620	38	)	)	PUNCT
ejpam-3344	620	39	is	be	AUX
ejpam-3344	620	40	an	an	DET
ejpam-3344	620	41	intuitionistic	intuitionistic	ADJ
ejpam-3344	620	42	fuzzy	fuzzy	ADJ
ejpam-3344	620	43	(	(	PUNCT
ejpam-3344	620	44	1	1	NUM
ejpam-3344	620	45	,	,	PUNCT
ejpam-3344	620	46	2)-ideal	2)-ideal	NUM
ejpam-3344	620	47	of	of	ADP
ejpam-3344	620	48	r.	r.	PROPN
ejpam-3344	620	49	now	now	ADV
ejpam-3344	620	50	1−	1−	NUM
ejpam-3344	620	51	γa(x−	γa(x−	SYM
ejpam-3344	620	52	y	y	PROPN
ejpam-3344	620	53	)	)	PUNCT
ejpam-3344	620	54	=	=	SYM
ejpam-3344	620	55	γa(x−	γa(x−	SYM
ejpam-3344	620	56	y	y	PROPN
ejpam-3344	620	57	)	)	PUNCT
ejpam-3344	620	58	≥	≥	PROPN
ejpam-3344	620	59	min{γa(x	min{γa(x	NOUN
ejpam-3344	620	60	)	)	PUNCT
ejpam-3344	620	61	,	,	PUNCT
ejpam-3344	620	62	γa(y	γa(y	NOUN
ejpam-3344	620	63	)	)	PUNCT
ejpam-3344	620	64	}	}	PUNCT
ejpam-3344	620	65	=	=	PUNCT
ejpam-3344	620	66	min{1−	min{1−	VERB
ejpam-3344	620	67	γa(x	γa(x	NUM
ejpam-3344	620	68	)	)	PUNCT
ejpam-3344	620	69	,	,	PUNCT
ejpam-3344	620	70	1−	1−	NUM
ejpam-3344	620	71	γa(y	γa(y	NUM
ejpam-3344	620	72	)	)	PUNCT
ejpam-3344	620	73	}	}	PUNCT
ejpam-3344	620	74	=	=	SYM
ejpam-3344	620	75	1−max{γa(x	1−max{γa(x	PROPN
ejpam-3344	620	76	)	)	PUNCT
ejpam-3344	620	77	,	,	PUNCT
ejpam-3344	620	78	γa(y	γa(y	NOUN
ejpam-3344	620	79	)	)	PUNCT
ejpam-3344	620	80	}	}	PUNCT
ejpam-3344	620	81	.	.	PUNCT
ejpam-3344	621	1	1−	1−	NUM
ejpam-3344	621	2	γa(xy	γa(xy	PROPN
ejpam-3344	621	3	)	)	PUNCT
ejpam-3344	621	4	=	=	SYM
ejpam-3344	621	5	γa(xy	γa(xy	PROPN
ejpam-3344	621	6	)	)	PUNCT
ejpam-3344	621	7	≥	≥	PROPN
ejpam-3344	621	8	min{γa(x	min{γa(x	NOUN
ejpam-3344	621	9	)	)	PUNCT
ejpam-3344	621	10	,	,	PUNCT
ejpam-3344	621	11	γa(y	γa(y	NOUN
ejpam-3344	621	12	)	)	PUNCT
ejpam-3344	621	13	}	}	PUNCT
ejpam-3344	621	14	=	=	PUNCT
ejpam-3344	621	15	min{1−	min{1−	VERB
ejpam-3344	621	16	γa(x	γa(x	NUM
ejpam-3344	621	17	)	)	PUNCT
ejpam-3344	621	18	,	,	PUNCT
ejpam-3344	621	19	1−	1−	NUM
ejpam-3344	621	20	γa(y	γa(y	NUM
ejpam-3344	621	21	)	)	PUNCT
ejpam-3344	621	22	}	}	PUNCT
ejpam-3344	621	23	=	=	SYM
ejpam-3344	621	24	1−max{γa(x	1−max{γa(x	PROPN
ejpam-3344	621	25	)	)	PUNCT
ejpam-3344	621	26	,	,	PUNCT
ejpam-3344	621	27	γa(y	γa(y	NOUN
ejpam-3344	621	28	)	)	PUNCT
ejpam-3344	621	29	}	}	PUNCT
ejpam-3344	621	30	.	.	PUNCT
ejpam-3344	622	1	1−	1−	NUM
ejpam-3344	622	2	γa((xa)(yz	γa((xa)(yz	NOUN
ejpam-3344	622	3	)	)	PUNCT
ejpam-3344	622	4	)	)	PUNCT
ejpam-3344	623	1	=	=	SYM
ejpam-3344	623	2	γa((xa)(yz	γa((xa)(yz	NOUN
ejpam-3344	623	3	)	)	PUNCT
ejpam-3344	623	4	)	)	PUNCT
ejpam-3344	623	5	≥	≥	PROPN
ejpam-3344	623	6	min{γa(x	min{γa(x	NOUN
ejpam-3344	623	7	)	)	PUNCT
ejpam-3344	623	8	,	,	PUNCT
ejpam-3344	623	9	γa(y	γa(y	NUM
ejpam-3344	623	10	)	)	PUNCT
ejpam-3344	623	11	,	,	PUNCT
ejpam-3344	623	12	γa(z	γa(z	PROPN
ejpam-3344	623	13	)	)	PUNCT
ejpam-3344	623	14	}	}	PUNCT
ejpam-3344	623	15	=	=	PUNCT
ejpam-3344	623	16	min{1−	min{1−	VERB
ejpam-3344	623	17	γa(x	γa(x	NUM
ejpam-3344	623	18	)	)	PUNCT
ejpam-3344	623	19	,	,	PUNCT
ejpam-3344	623	20	1−	1−	NUM
ejpam-3344	623	21	γa(y	γa(y	NUM
ejpam-3344	623	22	)	)	PUNCT
ejpam-3344	623	23	,	,	PUNCT
ejpam-3344	623	24	1−	1−	NUM
ejpam-3344	623	25	γa(z	γa(z	NOUN
ejpam-3344	623	26	)	)	PUNCT
ejpam-3344	623	27	}	}	PUNCT
ejpam-3344	623	28	=	=	SYM
ejpam-3344	623	29	1−max{γa(x	1−max{γa(x	PROPN
ejpam-3344	623	30	)	)	PUNCT
ejpam-3344	623	31	,	,	PUNCT
ejpam-3344	623	32	γa(y	γa(y	NUM
ejpam-3344	623	33	)	)	PUNCT
ejpam-3344	623	34	,	,	PUNCT
ejpam-3344	623	35	γa(z	γa(z	PROPN
ejpam-3344	623	36	)	)	PUNCT
ejpam-3344	623	37	}	}	PUNCT
ejpam-3344	623	38	.	.	PUNCT
ejpam-3344	624	1	therefore	therefore	ADV
ejpam-3344	624	2	a	a	PRON
ejpam-3344	624	3	is	be	AUX
ejpam-3344	624	4	an	an	DET
ejpam-3344	624	5	intuitionistic	intuitionistic	ADJ
ejpam-3344	624	6	fuzzy	fuzzy	ADJ
ejpam-3344	624	7	(	(	PUNCT
ejpam-3344	624	8	1	1	NUM
ejpam-3344	624	9	,	,	PUNCT
ejpam-3344	624	10	2)-ideal	2)-ideal	NUM
ejpam-3344	624	11	of	of	ADP
ejpam-3344	624	12	r.	r.	PROPN
ejpam-3344	624	13	remark	remark	PROPN
ejpam-3344	624	14	15	15	NUM
ejpam-3344	624	15	.	.	PUNCT
ejpam-3344	625	1	an	an	DET
ejpam-3344	625	2	ifs	ifs	PROPN
ejpam-3344	625	3	a	a	X
ejpam-3344	625	4	=	=	X
ejpam-3344	625	5	(	(	PUNCT
ejpam-3344	625	6	µa	µa	PROPN
ejpam-3344	625	7	,	,	PUNCT
ejpam-3344	625	8	γa	γa	PROPN
ejpam-3344	625	9	)	)	PUNCT
ejpam-3344	625	10	of	of	ADP
ejpam-3344	625	11	an	an	DET
ejpam-3344	625	12	la	la	ADJ
ejpam-3344	625	13	-	-	PUNCT
ejpam-3344	625	14	ring	ring	NOUN
ejpam-3344	625	15	r	r	NOUN
ejpam-3344	625	16	is	be	AUX
ejpam-3344	625	17	an	an	DET
ejpam-3344	625	18	intuitionistic	intuitionistic	ADJ
ejpam-3344	625	19	fuzzy	fuzzy	ADJ
ejpam-3344	625	20	bi	bi	NOUN
ejpam-3344	625	21	-	-	NOUN
ejpam-3344	625	22	ideal	ideal	NOUN
ejpam-3344	625	23	of	of	ADP
ejpam-3344	625	24	r	r	NOUN
ejpam-3344	625	25	if	if	SCONJ
ejpam-3344	625	26	and	and	CCONJ
ejpam-3344	625	27	only	only	ADV
ejpam-3344	625	28	if	if	SCONJ
ejpam-3344	625	29	the	the	DET
ejpam-3344	625	30	fuzzy	fuzzy	ADJ
ejpam-3344	625	31	subsets	subset	NOUN
ejpam-3344	625	32	µa	µa	PROPN
ejpam-3344	625	33	and	and	CCONJ
ejpam-3344	625	34	γa	γa	PRON
ejpam-3344	625	35	are	be	AUX
ejpam-3344	625	36	fuzzy	fuzzy	ADJ
ejpam-3344	625	37	bi	bi	NOUN
ejpam-3344	625	38	-	-	NOUN
ejpam-3344	625	39	ideals	ideal	NOUN
ejpam-3344	625	40	of	of	ADP
ejpam-3344	625	41	r.	r.	PROPN
ejpam-3344	625	42	references	reference	NOUN
ejpam-3344	625	43	[	[	X
ejpam-3344	625	44	1	1	NUM
ejpam-3344	625	45	]	]	PUNCT
ejpam-3344	625	46	k.	k.	PROPN
ejpam-3344	625	47	t.	t.	PROPN
ejpam-3344	625	48	atanassov	atanassov	PROPN
ejpam-3344	625	49	,	,	PUNCT
ejpam-3344	625	50	intuitionistic	intuitionistic	ADJ
ejpam-3344	625	51	fuzzy	fuzzy	ADJ
ejpam-3344	625	52	sets	set	NOUN
ejpam-3344	625	53	,	,	PUNCT
ejpam-3344	625	54	fuzzy	fuzzy	ADJ
ejpam-3344	625	55	sets	set	NOUN
ejpam-3344	625	56	and	and	CCONJ
ejpam-3344	625	57	systems	system	NOUN
ejpam-3344	625	58	,	,	PUNCT
ejpam-3344	625	59	20(1986	20(1986	NUM
ejpam-3344	625	60	)	)	PUNCT
ejpam-3344	625	61	87	87	NUM
ejpam-3344	625	62	-	-	SYM
ejpam-3344	625	63	96	96	NUM
ejpam-3344	625	64	.	.	PUNCT
ejpam-3344	626	1	[	[	X
ejpam-3344	626	2	2	2	NUM
ejpam-3344	626	3	]	]	PUNCT
ejpam-3344	626	4	k.	k.	PROPN
ejpam-3344	626	5	t.	t.	PROPN
ejpam-3344	626	6	atanassov	atanassov	PROPN
ejpam-3344	626	7	,	,	PUNCT
ejpam-3344	626	8	new	new	ADJ
ejpam-3344	626	9	operations	operation	NOUN
ejpam-3344	626	10	defined	define	VERB
ejpam-3344	626	11	over	over	ADP
ejpam-3344	626	12	the	the	DET
ejpam-3344	626	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	626	14	fuzzy	fuzzy	ADJ
ejpam-3344	626	15	sets	set	NOUN
ejpam-3344	626	16	,	,	PUNCT
ejpam-3344	626	17	fuzzy	fuzzy	ADJ
ejpam-3344	626	18	sets	set	NOUN
ejpam-3344	626	19	and	and	CCONJ
ejpam-3344	626	20	systems	system	NOUN
ejpam-3344	626	21	,	,	PUNCT
ejpam-3344	626	22	61(1994	61(1994	NUM
ejpam-3344	626	23	)	)	PUNCT
ejpam-3344	626	24	137	137	NUM
ejpam-3344	626	25	-	-	SYM
ejpam-3344	626	26	142	142	NUM
ejpam-3344	626	27	.	.	PUNCT
ejpam-3344	627	1	references	reference	NOUN
ejpam-3344	627	2	249	249	NUM
ejpam-3344	628	1	[	[	X
ejpam-3344	628	2	3	3	NUM
ejpam-3344	628	3	]	]	X
ejpam-3344	628	4	b.	b.	PROPN
ejpam-3344	628	5	banerjee	banerjee	PROPN
ejpam-3344	628	6	and	and	CCONJ
ejpam-3344	628	7	d.	d.	PROPN
ejpam-3344	628	8	k.	k.	PROPN
ejpam-3344	628	9	basnet	basnet	PROPN
ejpam-3344	628	10	,	,	PUNCT
ejpam-3344	628	11	intuitionistic	intuitionistic	ADJ
ejpam-3344	628	12	fuzzy	fuzzy	ADJ
ejpam-3344	628	13	subrings	subring	NOUN
ejpam-3344	628	14	and	and	CCONJ
ejpam-3344	628	15	ideals	ideal	NOUN
ejpam-3344	628	16	,	,	PUNCT
ejpam-3344	628	17	j.	j.	PROPN
ejpam-3344	628	18	fuzzy	fuzzy	PROPN
ejpam-3344	628	19	math	math	PROPN
ejpam-3344	628	20	.	.	PUNCT
ejpam-3344	628	21	,	,	PUNCT
ejpam-3344	628	22	11(2003	11(2003	NUM
ejpam-3344	628	23	)	)	PUNCT
ejpam-3344	628	24	139	139	NUM
ejpam-3344	628	25	-	-	SYM
ejpam-3344	628	26	155	155	NUM
ejpam-3344	628	27	.	.	PUNCT
ejpam-3344	629	1	[	[	X
ejpam-3344	629	2	4	4	X
ejpam-3344	629	3	]	]	PUNCT
ejpam-3344	629	4	r.	r.	PROPN
ejpam-3344	629	5	j.	j.	PROPN
ejpam-3344	629	6	cho	cho	PROPN
ejpam-3344	629	7	,	,	PUNCT
ejpam-3344	629	8	j.	j.	PROPN
ejpam-3344	629	9	jezek	jezek	PROPN
ejpam-3344	629	10	and	and	CCONJ
ejpam-3344	629	11	t.	t.	PROPN
ejpam-3344	629	12	kepka	kepka	NOUN
ejpam-3344	629	13	,	,	PUNCT
ejpam-3344	629	14	paramedial	paramedial	ADJ
ejpam-3344	629	15	groupoids	groupoid	NOUN
ejpam-3344	629	16	,	,	PUNCT
ejpam-3344	629	17	czechoslovak	czechoslovak	ADJ
ejpam-3344	629	18	math	math	NOUN
ejpam-3344	629	19	.	.	PUNCT
ejpam-3344	630	1	j.	j.	PROPN
ejpam-3344	630	2	,	,	PUNCT
ejpam-3344	630	3	49(1999	49(1999	PROPN
ejpam-3344	630	4	)	)	PUNCT
ejpam-3344	630	5	277	277	NUM
ejpam-3344	630	6	-	-	SYM
ejpam-3344	630	7	290	290	NUM
ejpam-3344	630	8	.	.	PUNCT
ejpam-3344	631	1	[	[	X
ejpam-3344	631	2	5	5	X
ejpam-3344	631	3	]	]	PUNCT
ejpam-3344	631	4	p.	p.	NOUN
ejpam-3344	631	5	s.	s.	PROPN
ejpam-3344	631	6	das	das	PROPN
ejpam-3344	631	7	,	,	PUNCT
ejpam-3344	631	8	fuzzy	fuzzy	ADJ
ejpam-3344	631	9	groups	group	NOUN
ejpam-3344	631	10	and	and	CCONJ
ejpam-3344	631	11	level	level	NOUN
ejpam-3344	631	12	subgroups	subgroup	NOUN
ejpam-3344	631	13	,	,	PUNCT
ejpam-3344	631	14	j.	j.	PROPN
ejpam-3344	631	15	math	math	PROPN
ejpam-3344	631	16	.	.	PUNCT
ejpam-3344	632	1	anal	anal	PROPN
ejpam-3344	632	2	.	.	PUNCT
ejpam-3344	633	1	appli	appli	PROPN
ejpam-3344	633	2	.	.	PROPN
ejpam-3344	633	3	,	,	PUNCT
ejpam-3344	633	4	84(1981	84(1981	NUM
ejpam-3344	633	5	)	)	PUNCT
ejpam-3344	633	6	264	264	NUM
ejpam-3344	633	7	-	-	SYM
ejpam-3344	633	8	269	269	NUM
ejpam-3344	633	9	.	.	PUNCT
ejpam-3344	634	1	[	[	X
ejpam-3344	634	2	6	6	NUM
ejpam-3344	634	3	]	]	PUNCT
ejpam-3344	634	4	k.	k.	PROPN
ejpam-3344	634	5	c.	c.	PROPN
ejpam-3344	634	6	gupta	gupta	PROPN
ejpam-3344	634	7	and	and	CCONJ
ejpam-3344	634	8	m.	m.	PROPN
ejpam-3344	634	9	k.	k.	PROPN
ejpam-3344	634	10	kantroo	kantroo	PROPN
ejpam-3344	634	11	,	,	PUNCT
ejpam-3344	634	12	the	the	DET
ejpam-3344	634	13	intrinsic	intrinsic	ADJ
ejpam-3344	634	14	product	product	NOUN
ejpam-3344	634	15	of	of	ADP
ejpam-3344	634	16	fuzzy	fuzzy	ADJ
ejpam-3344	634	17	subsets	subset	NOUN
ejpam-3344	634	18	of	of	ADP
ejpam-3344	634	19	a	a	DET
ejpam-3344	634	20	ring	ring	NOUN
ejpam-3344	634	21	,	,	PUNCT
ejpam-3344	634	22	fuzzy	fuzzy	ADJ
ejpam-3344	634	23	sets	set	NOUN
ejpam-3344	634	24	and	and	CCONJ
ejpam-3344	634	25	systems	system	NOUN
ejpam-3344	634	26	,	,	PUNCT
ejpam-3344	634	27	57(1993	57(1993	NUM
ejpam-3344	634	28	)	)	PUNCT
ejpam-3344	634	29	103	103	NUM
ejpam-3344	634	30	-	-	SYM
ejpam-3344	634	31	110	110	NUM
ejpam-3344	634	32	.	.	PUNCT
ejpam-3344	635	1	[	[	X
ejpam-3344	635	2	7	7	X
ejpam-3344	635	3	]	]	X
ejpam-3344	635	4	k.	k.	PROPN
ejpam-3344	635	5	hur	hur	PROPN
ejpam-3344	635	6	,	,	PUNCT
ejpam-3344	635	7	h.	h.	PROPN
ejpam-3344	635	8	w.	w.	PROPN
ejpam-3344	635	9	kang	kang	PROPN
ejpam-3344	635	10	and	and	CCONJ
ejpam-3344	635	11	h.	h.	PROPN
ejpam-3344	635	12	k.	k.	PROPN
ejpam-3344	635	13	song	song	PROPN
ejpam-3344	635	14	,	,	PUNCT
ejpam-3344	635	15	intuitionistic	intuitionistic	ADJ
ejpam-3344	635	16	fuzzy	fuzzy	ADJ
ejpam-3344	635	17	subgroups	subgroup	NOUN
ejpam-3344	635	18	and	and	CCONJ
ejpam-3344	635	19	subrings	subring	NOUN
ejpam-3344	635	20	,	,	PUNCT
ejpam-3344	635	21	honam	honam	PROPN
ejpam-3344	635	22	math	math	PROPN
ejpam-3344	635	23	.	.	PUNCT
ejpam-3344	636	1	j.	j.	PROPN
ejpam-3344	636	2	,	,	PUNCT
ejpam-3344	636	3	25(2003	25(2003	NUM
ejpam-3344	636	4	)	)	PUNCT
ejpam-3344	636	5	19	19	NUM
ejpam-3344	636	6	-	-	SYM
ejpam-3344	636	7	41	41	NUM
ejpam-3344	636	8	.	.	PUNCT
ejpam-3344	637	1	[	[	X
ejpam-3344	637	2	8	8	NUM
ejpam-3344	637	3	]	]	PUNCT
ejpam-3344	637	4	k.	k.	PROPN
ejpam-3344	637	5	hur	hur	PROPN
ejpam-3344	637	6	,	,	PUNCT
ejpam-3344	637	7	s.	s.	PROPN
ejpam-3344	637	8	y.	y.	PROPN
ejpam-3344	637	9	jang	jang	PROPN
ejpam-3344	637	10	and	and	CCONJ
ejpam-3344	637	11	h.	h.	PROPN
ejpam-3344	637	12	w.	w.	PROPN
ejpam-3344	637	13	kang	kang	PROPN
ejpam-3344	637	14	,	,	PUNCT
ejpam-3344	637	15	intuitionistic	intuitionistic	ADJ
ejpam-3344	637	16	fuzzy	fuzzy	ADJ
ejpam-3344	637	17	ideals	ideal	NOUN
ejpam-3344	637	18	of	of	ADP
ejpam-3344	637	19	a	a	DET
ejpam-3344	637	20	ring	ring	NOUN
ejpam-3344	637	21	,	,	PUNCT
ejpam-3344	637	22	j.	j.	PROPN
ejpam-3344	637	23	korea	korea	PROPN
ejpam-3344	637	24	soc	soc	PROPN
ejpam-3344	637	25	.	.	PUNCT
ejpam-3344	638	1	math	math	PROPN
ejpam-3344	638	2	.	.	PUNCT
ejpam-3344	639	1	educ	educ	PROPN
ejpam-3344	639	2	.	.	PUNCT
ejpam-3344	640	1	ser	ser	PROPN
ejpam-3344	640	2	.	.	PUNCT
ejpam-3344	641	1	b	b	X
ejpam-3344	641	2	:	:	PUNCT
ejpam-3344	641	3	pure	pure	ADJ
ejpam-3344	641	4	appl	appl	PROPN
ejpam-3344	641	5	.	.	PUNCT
ejpam-3344	641	6	math	math	PROPN
ejpam-3344	641	7	.	.	PUNCT
ejpam-3344	642	1	,	,	PUNCT
ejpam-3344	642	2	12	12	NUM
ejpam-3344	642	3	(	(	PUNCT
ejpam-3344	642	4	2005	2005	NUM
ejpam-3344	642	5	)	)	PUNCT
ejpam-3344	642	6	193	193	NUM
ejpam-3344	642	7	-	-	SYM
ejpam-3344	642	8	209	209	NUM
ejpam-3344	642	9	.	.	PUNCT
ejpam-3344	643	1	[	[	X
ejpam-3344	643	2	9	9	NUM
ejpam-3344	643	3	]	]	X
ejpam-3344	643	4	j.	j.	PROPN
ejpam-3344	643	5	jezek	jezek	PROPN
ejpam-3344	643	6	and	and	CCONJ
ejpam-3344	643	7	t.	t.	PROPN
ejpam-3344	643	8	kepka	kepka	NOUN
ejpam-3344	643	9	,	,	PUNCT
ejpam-3344	643	10	medial	medial	ADJ
ejpam-3344	643	11	groupoids	groupoid	NOUN
ejpam-3344	643	12	,	,	PUNCT
ejpam-3344	643	13	rozpravy	rozpravy	PROPN
ejpam-3344	643	14	csav	csav	PROPN
ejpam-3344	643	15	rada	rada	PROPN
ejpam-3344	643	16	mat	mat	PROPN
ejpam-3344	643	17	.	.	PUNCT
ejpam-3344	644	1	a	a	DET
ejpam-3344	644	2	prir	prir	NOUN
ejpam-3344	644	3	.	.	PUNCT
ejpam-3344	645	1	ved	ve	VERB
ejpam-3344	645	2	93/2	93/2	NUM
ejpam-3344	645	3	,	,	PUNCT
ejpam-3344	645	4	1983	1983	NUM
ejpam-3344	645	5	,	,	PUNCT
ejpam-3344	645	6	93	93	NUM
ejpam-3344	645	7	pp	pp	NOUN
ejpam-3344	645	8	.	.	PUNCT
ejpam-3344	646	1	[	[	X
ejpam-3344	646	2	10	10	NUM
ejpam-3344	646	3	]	]	PUNCT
ejpam-3344	646	4	m.	m.	NOUN
ejpam-3344	646	5	s.	s.	PROPN
ejpam-3344	646	6	kamran	kamran	PROPN
ejpam-3344	646	7	,	,	PUNCT
ejpam-3344	646	8	conditions	condition	NOUN
ejpam-3344	646	9	for	for	ADP
ejpam-3344	646	10	la	la	NOUN
ejpam-3344	646	11	-	-	PUNCT
ejpam-3344	646	12	semigroups	semigroup	NOUN
ejpam-3344	646	13	to	to	PART
ejpam-3344	646	14	resemble	resemble	VERB
ejpam-3344	646	15	associative	associative	ADJ
ejpam-3344	646	16	structures	structure	NOUN
ejpam-3344	646	17	,	,	PUNCT
ejpam-3344	646	18	ph.d	ph.d	PROPN
ejpam-3344	646	19	.	.	PUNCT
ejpam-3344	647	1	thesis	thesis	NOUN
ejpam-3344	647	2	,	,	PUNCT
ejpam-3344	647	3	quaid	quaid	PROPN
ejpam-3344	647	4	-	-	PUNCT
ejpam-3344	647	5	i	i	PROPN
ejpam-3344	647	6	-	-	PUNCT
ejpam-3344	647	7	azam	azam	PROPN
ejpam-3344	647	8	university	university	PROPN
ejpam-3344	647	9	,	,	PUNCT
ejpam-3344	647	10	islamabad	islamabad	PROPN
ejpam-3344	647	11	,	,	PUNCT
ejpam-3344	647	12	1993	1993	NUM
ejpam-3344	647	13	.	.	PUNCT
ejpam-3344	648	1	[	[	X
ejpam-3344	648	2	11	11	NUM
ejpam-3344	648	3	]	]	PUNCT
ejpam-3344	648	4	m.	m.	NOUN
ejpam-3344	648	5	a.	a.	PROPN
ejpam-3344	648	6	kazim	kazim	PROPN
ejpam-3344	648	7	and	and	CCONJ
ejpam-3344	648	8	m.	m.	PROPN
ejpam-3344	648	9	naseerdin	naseerdin	PROPN
ejpam-3344	648	10	,	,	PUNCT
ejpam-3344	648	11	on	on	ADP
ejpam-3344	648	12	almost	almost	ADV
ejpam-3344	648	13	semigroups	semigroup	NOUN
ejpam-3344	648	14	,	,	PUNCT
ejpam-3344	648	15	alig	alig	PROPN
ejpam-3344	648	16	.	.	PUNCT
ejpam-3344	649	1	bull	bull	PROPN
ejpam-3344	649	2	.	.	PUNCT
ejpam-3344	650	1	math	math	NOUN
ejpam-3344	650	2	.	.	PUNCT
ejpam-3344	650	3	,	,	PUNCT
ejpam-3344	650	4	2(1972	2(1972	X
ejpam-3344	650	5	)	)	PUNCT
ejpam-3344	650	6	1	1	NUM
ejpam-3344	650	7	-	-	SYM
ejpam-3344	650	8	7	7	NUM
ejpam-3344	650	9	.	.	PUNCT
ejpam-3344	651	1	[	[	X
ejpam-3344	651	2	12	12	NUM
ejpam-3344	651	3	]	]	X
ejpam-3344	651	4	n.	n.	PROPN
ejpam-3344	651	5	kuroki	kuroki	PROPN
ejpam-3344	651	6	,	,	PUNCT
ejpam-3344	651	7	regular	regular	ADJ
ejpam-3344	651	8	fuzzy	fuzzy	ADJ
ejpam-3344	651	9	duo	duo	NOUN
ejpam-3344	651	10	rings	ring	NOUN
ejpam-3344	651	11	,	,	PUNCT
ejpam-3344	651	12	inform	inform	NOUN
ejpam-3344	651	13	.	.	PUNCT
ejpam-3344	652	1	sci	sci	PROPN
ejpam-3344	652	2	.	.	PROPN
ejpam-3344	652	3	,	,	PUNCT
ejpam-3344	652	4	94(1996	94(1996	X
ejpam-3344	652	5	)	)	PUNCT
ejpam-3344	652	6	119	119	NUM
ejpam-3344	652	7	-	-	SYM
ejpam-3344	652	8	139	139	NUM
ejpam-3344	652	9	.	.	PUNCT
ejpam-3344	653	1	[	[	X
ejpam-3344	653	2	13	13	NUM
ejpam-3344	653	3	]	]	PUNCT
ejpam-3344	653	4	w.	w.	PROPN
ejpam-3344	653	5	j.	j.	PROPN
ejpam-3344	653	6	liu	liu	PROPN
ejpam-3344	653	7	,	,	PUNCT
ejpam-3344	653	8	fuzzy	fuzzy	ADJ
ejpam-3344	653	9	invariant	invariant	ADJ
ejpam-3344	653	10	subgroups	subgroup	NOUN
ejpam-3344	653	11	and	and	CCONJ
ejpam-3344	653	12	ideals	ideal	NOUN
ejpam-3344	653	13	,	,	PUNCT
ejpam-3344	653	14	fuzzy	fuzzy	ADJ
ejpam-3344	653	15	sets	set	NOUN
ejpam-3344	653	16	and	and	CCONJ
ejpam-3344	653	17	systems	system	NOUN
ejpam-3344	653	18	,	,	PUNCT
ejpam-3344	653	19	8(1982	8(1982	NUM
ejpam-3344	653	20	)	)	PUNCT
ejpam-3344	653	21	133	133	NUM
ejpam-3344	653	22	-	-	SYM
ejpam-3344	653	23	139	139	NUM
ejpam-3344	653	24	.	.	PUNCT
ejpam-3344	654	1	[	[	X
ejpam-3344	654	2	14	14	NUM
ejpam-3344	654	3	]	]	PUNCT
ejpam-3344	654	4	t.	t.	PROPN
ejpam-3344	654	5	k.	k.	PROPN
ejpam-3344	654	6	mukherjee	mukherjee	PROPN
ejpam-3344	654	7	and	and	CCONJ
ejpam-3344	654	8	m.	m.	PROPN
ejpam-3344	654	9	k.	k.	PROPN
ejpam-3344	654	10	sen	sen	PROPN
ejpam-3344	654	11	,	,	PUNCT
ejpam-3344	654	12	on	on	ADP
ejpam-3344	654	13	fuzzy	fuzzy	ADJ
ejpam-3344	654	14	ideals	ideal	NOUN
ejpam-3344	654	15	of	of	ADP
ejpam-3344	654	16	a	a	DET
ejpam-3344	654	17	ring	ring	NOUN
ejpam-3344	654	18	1	1	NUM
ejpam-3344	654	19	,	,	PUNCT
ejpam-3344	654	20	fuzzy	fuzzy	ADJ
ejpam-3344	654	21	sets	set	NOUN
ejpam-3344	654	22	and	and	CCONJ
ejpam-3344	654	23	systems	system	NOUN
ejpam-3344	654	24	,	,	PUNCT
ejpam-3344	654	25	21(1987	21(1987	NUM
ejpam-3344	654	26	)	)	PUNCT
ejpam-3344	654	27	99	99	NUM
ejpam-3344	654	28	-	-	SYM
ejpam-3344	654	29	104	104	NUM
ejpam-3344	654	30	.	.	PUNCT
ejpam-3344	655	1	[	[	X
ejpam-3344	655	2	15	15	NUM
ejpam-3344	655	3	]	]	PUNCT
ejpam-3344	655	4	t.	t.	PROPN
ejpam-3344	655	5	k.	k.	PROPN
ejpam-3344	655	6	mukherjee	mukherjee	PROPN
ejpam-3344	655	7	and	and	CCONJ
ejpam-3344	655	8	m.	m.	PROPN
ejpam-3344	655	9	k.	k.	PROPN
ejpam-3344	655	10	sen	sen	PROPN
ejpam-3344	655	11	,	,	PUNCT
ejpam-3344	655	12	prime	prime	ADJ
ejpam-3344	655	13	fuzzy	fuzzy	ADJ
ejpam-3344	655	14	ideals	ideal	NOUN
ejpam-3344	655	15	in	in	ADP
ejpam-3344	655	16	rings	ring	NOUN
ejpam-3344	655	17	,	,	PUNCT
ejpam-3344	655	18	fuzzy	fuzzy	ADJ
ejpam-3344	655	19	sets	set	NOUN
ejpam-3344	655	20	and	and	CCONJ
ejpam-3344	655	21	systems	system	NOUN
ejpam-3344	655	22	,	,	PUNCT
ejpam-3344	655	23	32(1989	32(1989	NUM
ejpam-3344	655	24	)	)	PUNCT
ejpam-3344	655	25	337	337	NUM
ejpam-3344	655	26	-	-	SYM
ejpam-3344	655	27	341	341	NUM
ejpam-3344	655	28	.	.	PUNCT
ejpam-3344	656	1	[	[	X
ejpam-3344	656	2	16	16	X
ejpam-3344	656	3	]	]	PUNCT
ejpam-3344	656	4	p.	p.	NOUN
ejpam-3344	656	5	v.	v.	ADP
ejpam-3344	656	6	protic	protic	PROPN
ejpam-3344	656	7	and	and	CCONJ
ejpam-3344	656	8	n.	n.	PROPN
ejpam-3344	656	9	stevanovic	stevanovic	PROPN
ejpam-3344	656	10	,	,	PUNCT
ejpam-3344	656	11	ag	ag	NOUN
ejpam-3344	656	12	-	-	PUNCT
ejpam-3344	656	13	test	test	NOUN
ejpam-3344	656	14	and	and	CCONJ
ejpam-3344	656	15	some	some	DET
ejpam-3344	656	16	general	general	ADJ
ejpam-3344	656	17	properties	property	NOUN
ejpam-3344	656	18	of	of	ADP
ejpam-3344	656	19	abelgrassmann	abelgrassmann	PROPN
ejpam-3344	656	20	’s	’s	PART
ejpam-3344	656	21	groupoids	groupoid	NOUN
ejpam-3344	656	22	,	,	PUNCT
ejpam-3344	656	23	pure	pure	ADJ
ejpam-3344	656	24	math	math	NOUN
ejpam-3344	656	25	.	.	PUNCT
ejpam-3344	657	1	appli	appli	PROPN
ejpam-3344	657	2	.	.	PROPN
ejpam-3344	657	3	,	,	PUNCT
ejpam-3344	658	1	6(1995	6(1995	NUM
ejpam-3344	658	2	)	)	PUNCT
ejpam-3344	658	3	371	371	NUM
ejpam-3344	658	4	-	-	SYM
ejpam-3344	658	5	383	383	NUM
ejpam-3344	658	6	.	.	PUNCT
ejpam-3344	659	1	[	[	X
ejpam-3344	659	2	17	17	NUM
ejpam-3344	659	3	]	]	X
ejpam-3344	659	4	v.	v.	PROPN
ejpam-3344	659	5	s.	s.	PROPN
ejpam-3344	659	6	ramamurthi	ramamurthi	PROPN
ejpam-3344	659	7	,	,	PUNCT
ejpam-3344	659	8	weakly	weakly	ADJ
ejpam-3344	659	9	regular	regular	ADJ
ejpam-3344	659	10	rings	ring	NOUN
ejpam-3344	659	11	,	,	PUNCT
ejpam-3344	659	12	canad	canad	PROPN
ejpam-3344	659	13	.	.	PUNCT
ejpam-3344	660	1	math	math	NOUN
ejpam-3344	660	2	.	.	PUNCT
ejpam-3344	661	1	bull	bull	PROPN
ejpam-3344	661	2	.	.	PUNCT
ejpam-3344	662	1	,	,	PUNCT
ejpam-3344	662	2	16	16	NUM
ejpam-3344	662	3	(	(	PUNCT
ejpam-3344	662	4	1973	1973	NUM
ejpam-3344	662	5	)	)	PUNCT
ejpam-3344	662	6	317	317	NUM
ejpam-3344	662	7	-	-	SYM
ejpam-3344	662	8	321	321	NUM
ejpam-3344	662	9	.	.	PUNCT
ejpam-3344	663	1	[	[	X
ejpam-3344	663	2	18	18	NUM
ejpam-3344	663	3	]	]	PUNCT
ejpam-3344	663	4	t.	t.	NOUN
ejpam-3344	663	5	shah	shah	NOUN
ejpam-3344	663	6	,	,	PUNCT
ejpam-3344	663	7	n.	n.	PROPN
ejpam-3344	663	8	kausar	kausar	PROPN
ejpam-3344	663	9	and	and	CCONJ
ejpam-3344	663	10	i.	i.	PROPN
ejpam-3344	663	11	rehman	rehman	PROPN
ejpam-3344	663	12	,	,	PUNCT
ejpam-3344	663	13	intuitionistic	intuitionistic	ADJ
ejpam-3344	663	14	fuzzy	fuzzy	ADJ
ejpam-3344	663	15	normal	normal	ADJ
ejpam-3344	663	16	subrings	subring	NOUN
ejpam-3344	663	17	over	over	ADP
ejpam-3344	663	18	a	a	DET
ejpam-3344	663	19	nonassociative	nonassociative	ADJ
ejpam-3344	663	20	ring	ring	NOUN
ejpam-3344	663	21	,	,	PUNCT
ejpam-3344	663	22	an	an	PROPN
ejpam-3344	663	23	.	.	PUNCT
ejpam-3344	663	24	st	st	PROPN
ejpam-3344	663	25	.	.	PROPN
ejpam-3344	663	26	univ	univ	PROPN
ejpam-3344	663	27	.	.	PUNCT
ejpam-3344	664	1	ovidius	ovidius	PROPN
ejpam-3344	664	2	constanta	constanta	PROPN
ejpam-3344	664	3	,	,	PUNCT
ejpam-3344	664	4	1(2012	1(2012	NUM
ejpam-3344	664	5	)	)	PUNCT
ejpam-3344	664	6	369	369	NUM
ejpam-3344	664	7	-	-	SYM
ejpam-3344	664	8	386	386	NUM
ejpam-3344	664	9	.	.	PUNCT
ejpam-3344	665	1	[	[	X
ejpam-3344	665	2	19	19	NUM
ejpam-3344	665	3	]	]	X
ejpam-3344	665	4	shah	shah	NOUN
ejpam-3344	665	5	,	,	PUNCT
ejpam-3344	665	6	kausar	kausar	NOUN
ejpam-3344	665	7	,	,	PUNCT
ejpam-3344	665	8	“	"	PUNCT
ejpam-3344	665	9	characterizations	characterization	NOUN
ejpam-3344	665	10	of	of	ADP
ejpam-3344	665	11	non	non	ADJ
ejpam-3344	665	12	-	-	ADJ
ejpam-3344	665	13	associative	associative	ADJ
ejpam-3344	665	14	ordered	order	VERB
ejpam-3344	665	15	semigroups	semigroup	NOUN
ejpam-3344	665	16	by	by	ADP
ejpam-3344	665	17	their	their	PRON
ejpam-3344	665	18	fuzzy	fuzzy	ADJ
ejpam-3344	665	19	bi	bi	NOUN
ejpam-3344	665	20	-	-	NOUN
ejpam-3344	665	21	ideals	ideal	NOUN
ejpam-3344	665	22	”	"	PUNCT
ejpam-3344	665	23	,	,	PUNCT
ejpam-3344	665	24	theoretical	theoretical	ADJ
ejpam-3344	665	25	computer	computer	NOUN
ejpam-3344	665	26	science	science	NOUN
ejpam-3344	665	27	,	,	PUNCT
ejpam-3344	665	28	vol	vol	NOUN
ejpam-3344	665	29	.	.	PUNCT
ejpam-3344	666	1	529	529	NUM
ejpam-3344	666	2	(	(	PUNCT
ejpam-3344	666	3	2014	2014	NUM
ejpam-3344	666	4	)	)	PUNCT
ejpam-3344	666	5	96	96	NUM
ejpam-3344	666	6	-	-	SYM
ejpam-3344	666	7	110	110	NUM
ejpam-3344	666	8	.	.	PUNCT
ejpam-3344	667	1	references	reference	NOUN
ejpam-3344	667	2	250	250	NUM
ejpam-3344	667	3	[	[	X
ejpam-3344	667	4	20	20	NUM
ejpam-3344	667	5	]	]	PUNCT
ejpam-3344	667	6	t.	t.	NOUN
ejpam-3344	667	7	shah	shah	PROPN
ejpam-3344	667	8	and	and	CCONJ
ejpam-3344	667	9	i.	i.	PROPN
ejpam-3344	667	10	rehman	rehman	PROPN
ejpam-3344	667	11	,	,	PUNCT
ejpam-3344	667	12	on	on	ADP
ejpam-3344	667	13	la	la	NOUN
ejpam-3344	667	14	-	-	PUNCT
ejpam-3344	667	15	rings	ring	NOUN
ejpam-3344	667	16	of	of	ADP
ejpam-3344	667	17	finitely	finitely	ADJ
ejpam-3344	667	18	non	non	ADJ
ejpam-3344	667	19	-	-	ADJ
ejpam-3344	667	20	zero	zero	NUM
ejpam-3344	667	21	functions	function	NOUN
ejpam-3344	667	22	,	,	PUNCT
ejpam-3344	667	23	int	int	NOUN
ejpam-3344	667	24	.	.	PUNCT
ejpam-3344	668	1	j.	j.	PROPN
ejpam-3344	668	2	contempt	contempt	PROPN
ejpam-3344	668	3	.	.	PUNCT
ejpam-3344	669	1	math	math	NOUN
ejpam-3344	669	2	.	.	PUNCT
ejpam-3344	670	1	sci	sci	PROPN
ejpam-3344	670	2	.	.	PROPN
ejpam-3344	670	3	,	,	PUNCT
ejpam-3344	670	4	5(2010	5(2010	NUM
ejpam-3344	670	5	)	)	PUNCT
ejpam-3344	670	6	209	209	NUM
ejpam-3344	670	7	-	-	SYM
ejpam-3344	670	8	222	222	NUM
ejpam-3344	670	9	.	.	PUNCT
ejpam-3344	671	1	[	[	X
ejpam-3344	671	2	21	21	NUM
ejpam-3344	671	3	]	]	X
ejpam-3344	671	4	u.	u.	PROPN
ejpam-3344	671	5	m.	m.	PROPN
ejpam-3344	671	6	swamy	swamy	PROPN
ejpam-3344	671	7	and	and	CCONJ
ejpam-3344	671	8	k.	k.	PROPN
ejpam-3344	671	9	l.	l.	PROPN
ejpam-3344	671	10	n.	n.	PROPN
ejpam-3344	671	11	swamy	swamy	PROPN
ejpam-3344	671	12	,	,	PUNCT
ejpam-3344	671	13	fuzzy	fuzzy	ADJ
ejpam-3344	671	14	prime	prime	ADJ
ejpam-3344	671	15	ideals	ideal	NOUN
ejpam-3344	671	16	of	of	ADP
ejpam-3344	671	17	rings	ring	NOUN
ejpam-3344	671	18	,	,	PUNCT
ejpam-3344	671	19	j.	j.	PROPN
ejpam-3344	671	20	math	math	PROPN
ejpam-3344	671	21	.	.	PUNCT
ejpam-3344	672	1	anal	anal	PROPN
ejpam-3344	672	2	.	.	PUNCT
ejpam-3344	673	1	appl	appl	PROPN
ejpam-3344	673	2	.	.	PROPN
ejpam-3344	673	3	,	,	PUNCT
ejpam-3344	673	4	134(1988	134(1988	NUM
ejpam-3344	673	5	)	)	PUNCT
ejpam-3344	673	6	94	94	NUM
ejpam-3344	673	7	-	-	SYM
ejpam-3344	673	8	103	103	NUM
ejpam-3344	673	9	.	.	PUNCT
ejpam-3344	674	1	[	[	X
ejpam-3344	674	2	22	22	NUM
ejpam-3344	674	3	]	]	X
ejpam-3344	674	4	l.	l.	PROPN
ejpam-3344	674	5	a.	a.	PROPN
ejpam-3344	674	6	zadeh	zadeh	PROPN
ejpam-3344	674	7	,	,	PUNCT
ejpam-3344	674	8	fuzzy	fuzzy	ADJ
ejpam-3344	674	9	sets	set	NOUN
ejpam-3344	674	10	,	,	PUNCT
ejpam-3344	674	11	information	information	NOUN
ejpam-3344	674	12	and	and	CCONJ
ejpam-3344	674	13	control	control	NOUN
ejpam-3344	674	14	,	,	PUNCT
ejpam-3344	674	15	8(1965	8(1965	NUM
ejpam-3344	674	16	)	)	PUNCT
ejpam-3344	674	17	338	338	NUM
ejpam-3344	674	18	-	-	SYM
ejpam-3344	674	19	363	363	NUM
ejpam-3344	674	20	.	.	PUNCT