id	sid	tid	token	lemma	pos
ejpam-3427	1	1	european	european	PROPN
ejpam-3427	1	2	journal	journal	PROPN
ejpam-3427	1	3	of	of	ADP
ejpam-3427	1	4	pure	pure	ADJ
ejpam-3427	1	5	and	and	CCONJ
ejpam-3427	1	6	applied	apply	VERB
ejpam-3427	1	7	mathematics	mathematic	NOUN
ejpam-3427	1	8	vol	vol	NOUN
ejpam-3427	1	9	.	.	PROPN
ejpam-3427	2	1	12	12	NUM
ejpam-3427	2	2	,	,	PUNCT
ejpam-3427	2	3	no	no	INTJ
ejpam-3427	2	4	.	.	NOUN
ejpam-3427	2	5	2	2	NUM
ejpam-3427	2	6	,	,	PUNCT
ejpam-3427	2	7	2019	2019	NUM
ejpam-3427	2	8	,	,	PUNCT
ejpam-3427	2	9	622	622	NUM
ejpam-3427	2	10	-	-	SYM
ejpam-3427	2	11	648	648	NUM
ejpam-3427	2	12	issn	issn	PROPN
ejpam-3427	2	13	1307	1307	NUM
ejpam-3427	2	14	-	-	SYM
ejpam-3427	2	15	5543	5543	NUM
ejpam-3427	2	16	–	–	PUNCT
ejpam-3427	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3427	2	18	published	publish	VERB
ejpam-3427	2	19	by	by	ADP
ejpam-3427	2	20	new	new	PROPN
ejpam-3427	2	21	york	york	PROPN
ejpam-3427	2	22	business	business	PROPN
ejpam-3427	2	23	global	global	ADJ
ejpam-3427	2	24	direct	direct	ADJ
ejpam-3427	2	25	product	product	NOUN
ejpam-3427	2	26	of	of	ADP
ejpam-3427	2	27	finite	finite	PROPN
ejpam-3427	2	28	intuitionistic	intuitionistic	ADJ
ejpam-3427	2	29	anti	anti	ADJ
ejpam-3427	2	30	fuzzy	fuzzy	ADJ
ejpam-3427	2	31	normal	normal	ADJ
ejpam-3427	2	32	subrings	subring	NOUN
ejpam-3427	2	33	over	over	ADP
ejpam-3427	2	34	non	non	ADJ
ejpam-3427	2	35	-	-	ADJ
ejpam-3427	2	36	associative	associative	ADJ
ejpam-3427	2	37	rings	ring	NOUN
ejpam-3427	2	38	nasreen	nasreen	ADP
ejpam-3427	2	39	kausar	kausar	PROPN
ejpam-3427	2	40	department	department	PROPN
ejpam-3427	2	41	of	of	ADP
ejpam-3427	2	42	mathematics	mathematics	PROPN
ejpam-3427	2	43	,	,	PUNCT
ejpam-3427	2	44	university	university	NOUN
ejpam-3427	2	45	of	of	ADP
ejpam-3427	2	46	agriculture	agriculture	PROPN
ejpam-3427	2	47	faisalabad	faisalabad	PROPN
ejpam-3427	2	48	,	,	PUNCT
ejpam-3427	2	49	pakistan	pakistan	PROPN
ejpam-3427	2	50	abstract	abstract	NOUN
ejpam-3427	2	51	.	.	PUNCT
ejpam-3427	3	1	shal	shal	PROPN
ejpam-3427	3	2	et	et	PROPN
ejpam-3427	3	3	.	.	PUNCT
ejpam-3427	4	1	al	al	PROPN
ejpam-3427	5	1	[	[	X
ejpam-3427	5	2	17	17	NUM
ejpam-3427	5	3	]	]	PUNCT
ejpam-3427	5	4	,	,	PUNCT
ejpam-3427	5	5	have	have	AUX
ejpam-3427	5	6	introduced	introduce	VERB
ejpam-3427	5	7	the	the	DET
ejpam-3427	5	8	concept	concept	NOUN
ejpam-3427	5	9	of	of	ADP
ejpam-3427	5	10	intuitionistic	intuitionistic	ADJ
ejpam-3427	5	11	fuzzy	fuzzy	ADJ
ejpam-3427	5	12	normal	normal	ADJ
ejpam-3427	5	13	subrings	subring	NOUN
ejpam-3427	5	14	over	over	ADP
ejpam-3427	5	15	a	a	DET
ejpam-3427	5	16	non	non	ADJ
ejpam-3427	5	17	-	-	ADJ
ejpam-3427	5	18	associative	associative	ADJ
ejpam-3427	5	19	ring	ring	NOUN
ejpam-3427	5	20	.	.	PUNCT
ejpam-3427	6	1	in	in	ADP
ejpam-3427	6	2	this	this	DET
ejpam-3427	6	3	paper	paper	NOUN
ejpam-3427	6	4	,	,	PUNCT
ejpam-3427	6	5	we	we	PRON
ejpam-3427	6	6	investigate	investigate	VERB
ejpam-3427	6	7	the	the	DET
ejpam-3427	6	8	concept	concept	NOUN
ejpam-3427	6	9	of	of	ADP
ejpam-3427	6	10	intuitionistic	intuitionistic	ADJ
ejpam-3427	6	11	anti	anti	ADJ
ejpam-3427	6	12	fuzzy	fuzzy	ADJ
ejpam-3427	6	13	normal	normal	ADJ
ejpam-3427	6	14	subrings	subring	NOUN
ejpam-3427	6	15	over	over	ADP
ejpam-3427	6	16	non	non	ADJ
ejpam-3427	6	17	-	-	ADJ
ejpam-3427	6	18	associative	associative	ADJ
ejpam-3427	6	19	rings	ring	NOUN
ejpam-3427	6	20	and	and	CCONJ
ejpam-3427	6	21	give	give	VERB
ejpam-3427	6	22	some	some	DET
ejpam-3427	6	23	properties	property	NOUN
ejpam-3427	6	24	of	of	ADP
ejpam-3427	6	25	such	such	ADJ
ejpam-3427	6	26	subrings	subring	NOUN
ejpam-3427	6	27	2010	2010	NUM
ejpam-3427	6	28	mathematics	mathematic	NOUN
ejpam-3427	6	29	subject	subject	NOUN
ejpam-3427	6	30	classifications	classification	NOUN
ejpam-3427	6	31	:	:	PUNCT
ejpam-3427	6	32	03f55	03f55	NOUN
ejpam-3427	6	33	,	,	PUNCT
ejpam-3427	6	34	08a72	08a72	NUM
ejpam-3427	6	35	,	,	PUNCT
ejpam-3427	6	36	20n25	20n25	NOUN
ejpam-3427	6	37	key	key	ADJ
ejpam-3427	6	38	words	word	NOUN
ejpam-3427	6	39	and	and	CCONJ
ejpam-3427	6	40	phrases	phrase	NOUN
ejpam-3427	6	41	:	:	PUNCT
ejpam-3427	6	42	direct	direct	ADJ
ejpam-3427	6	43	product	product	NOUN
ejpam-3427	6	44	of	of	ADP
ejpam-3427	6	45	(	(	PUNCT
ejpam-3427	6	46	intuitionistic	intuitionistic	ADJ
ejpam-3427	6	47	)	)	PUNCT
ejpam-3427	6	48	fuzzy	fuzzy	ADJ
ejpam-3427	6	49	sets	set	NOUN
ejpam-3427	6	50	,	,	PUNCT
ejpam-3427	6	51	direct	direct	ADJ
ejpam-3427	6	52	product	product	NOUN
ejpam-3427	6	53	of	of	ADP
ejpam-3427	6	54	(	(	PUNCT
ejpam-3427	6	55	intuitionistic	intuitionistic	ADJ
ejpam-3427	6	56	anti	anti	ADJ
ejpam-3427	6	57	)	)	PUNCT
ejpam-3427	6	58	fuzzy	fuzzy	ADJ
ejpam-3427	6	59	la	la	NOUN
ejpam-3427	6	60	-	-	NOUN
ejpam-3427	6	61	subrings	subring	NOUN
ejpam-3427	6	62	,	,	PUNCT
ejpam-3427	6	63	direct	direct	ADJ
ejpam-3427	6	64	product	product	NOUN
ejpam-3427	6	65	of	of	ADP
ejpam-3427	6	66	(	(	PUNCT
ejpam-3427	6	67	intuitionistic	intuitionistic	ADJ
ejpam-3427	6	68	anti	anti	ADJ
ejpam-3427	6	69	)	)	PUNCT
ejpam-3427	6	70	fuzzy	fuzzy	ADJ
ejpam-3427	6	71	normal	normal	ADJ
ejpam-3427	6	72	la	la	NOUN
ejpam-3427	6	73	-	-	NOUN
ejpam-3427	6	74	subrings	subring	NOUN
ejpam-3427	6	75	.	.	PUNCT
ejpam-3427	7	1	1	1	X
ejpam-3427	7	2	.	.	X
ejpam-3427	7	3	introduction	introduction	NOUN
ejpam-3427	7	4	in	in	ADP
ejpam-3427	7	5	1972	1972	NUM
ejpam-3427	7	6	,	,	PUNCT
ejpam-3427	7	7	ageneralization	ageneralization	NOUN
ejpam-3427	7	8	of	of	ADP
ejpam-3427	7	9	commutative	commutative	ADJ
ejpam-3427	7	10	semigroups	semigroup	NOUN
ejpam-3427	7	11	has	have	AUX
ejpam-3427	7	12	been	be	AUX
ejpam-3427	7	13	established	establish	VERB
ejpam-3427	7	14	by	by	ADP
ejpam-3427	7	15	kazim	kazim	PROPN
ejpam-3427	7	16	et	et	PROPN
ejpam-3427	7	17	.	.	PUNCT
ejpam-3427	8	1	al	al	PROPN
ejpam-3427	9	1	[	[	X
ejpam-3427	9	2	9	9	NUM
ejpam-3427	9	3	]	]	PUNCT
ejpam-3427	9	4	.	.	PUNCT
ejpam-3427	10	1	in	in	ADP
ejpam-3427	10	2	ternary	ternary	ADJ
ejpam-3427	10	3	commutative	commutative	ADJ
ejpam-3427	10	4	law	law	NOUN
ejpam-3427	10	5	:	:	PUNCT
ejpam-3427	10	6	abc	abc	PROPN
ejpam-3427	10	7	=	=	SYM
ejpam-3427	10	8	cba	cba	PROPN
ejpam-3427	10	9	,	,	PUNCT
ejpam-3427	10	10	they	they	PRON
ejpam-3427	10	11	introduced	introduce	VERB
ejpam-3427	10	12	the	the	DET
ejpam-3427	10	13	braces	brace	NOUN
ejpam-3427	10	14	on	on	ADP
ejpam-3427	10	15	the	the	DET
ejpam-3427	10	16	left	left	ADJ
ejpam-3427	10	17	side	side	NOUN
ejpam-3427	10	18	of	of	ADP
ejpam-3427	10	19	this	this	DET
ejpam-3427	10	20	law	law	NOUN
ejpam-3427	10	21	and	and	CCONJ
ejpam-3427	10	22	explored	explore	VERB
ejpam-3427	10	23	a	a	DET
ejpam-3427	10	24	new	new	ADJ
ejpam-3427	10	25	pseudo	pseudo	NOUN
ejpam-3427	10	26	associative	associative	NOUN
ejpam-3427	10	27	law	law	NOUN
ejpam-3427	10	28	,	,	PUNCT
ejpam-3427	10	29	that	that	ADV
ejpam-3427	10	30	is	is	ADV
ejpam-3427	10	31	(	(	PUNCT
ejpam-3427	10	32	ab)c	ab)c	PROPN
ejpam-3427	10	33	=	=	SYM
ejpam-3427	10	34	(	(	PUNCT
ejpam-3427	10	35	cb)a	cb)a	PROPN
ejpam-3427	10	36	.	.	PUNCT
ejpam-3427	11	1	this	this	DET
ejpam-3427	11	2	law	law	NOUN
ejpam-3427	11	3	(	(	PUNCT
ejpam-3427	11	4	ab)c	ab)c	PROPN
ejpam-3427	11	5	=	=	SYM
ejpam-3427	11	6	(	(	PUNCT
ejpam-3427	11	7	cb)a	cb)a	PROPN
ejpam-3427	11	8	,	,	PUNCT
ejpam-3427	11	9	is	be	AUX
ejpam-3427	11	10	called	call	VERB
ejpam-3427	11	11	the	the	DET
ejpam-3427	11	12	left	left	ADJ
ejpam-3427	11	13	invertive	invertive	ADJ
ejpam-3427	11	14	law	law	NOUN
ejpam-3427	11	15	.	.	PUNCT
ejpam-3427	12	1	a	a	DET
ejpam-3427	12	2	groupoid	groupoid	PROPN
ejpam-3427	12	3	s	s	NOUN
ejpam-3427	12	4	is	be	AUX
ejpam-3427	12	5	said	say	VERB
ejpam-3427	12	6	to	to	PART
ejpam-3427	12	7	be	be	AUX
ejpam-3427	12	8	a	a	DET
ejpam-3427	12	9	left	left	NOUN
ejpam-3427	12	10	almost	almost	ADV
ejpam-3427	12	11	semigroup	semigroup	ADJ
ejpam-3427	12	12	(	(	PUNCT
ejpam-3427	12	13	abbreviated	abbreviate	VERB
ejpam-3427	12	14	as	as	ADP
ejpam-3427	12	15	la	la	NOUN
ejpam-3427	12	16	-	-	PUNCT
ejpam-3427	12	17	semigroup	semigroup	NOUN
ejpam-3427	12	18	)	)	PUNCT
ejpam-3427	12	19	if	if	SCONJ
ejpam-3427	12	20	it	it	PRON
ejpam-3427	12	21	satisfies	satisfy	VERB
ejpam-3427	12	22	the	the	DET
ejpam-3427	12	23	left	left	ADJ
ejpam-3427	12	24	invertive	invertive	ADJ
ejpam-3427	12	25	law	law	NOUN
ejpam-3427	12	26	:	:	PUNCT
ejpam-3427	12	27	(	(	PUNCT
ejpam-3427	12	28	ab)c	ab)c	PROPN
ejpam-3427	12	29	=	=	SYM
ejpam-3427	12	30	(	(	PUNCT
ejpam-3427	12	31	cb)a	cb)a	PROPN
ejpam-3427	12	32	.	.	PUNCT
ejpam-3427	13	1	in	in	ADP
ejpam-3427	13	2	[	[	X
ejpam-3427	13	3	7	7	NUM
ejpam-3427	13	4	]	]	X
ejpam-3427	13	5	(	(	PUNCT
ejpam-3427	13	6	resp	resp	NOUN
ejpam-3427	13	7	.	.	PUNCT
ejpam-3427	14	1	[	[	X
ejpam-3427	14	2	5	5	NUM
ejpam-3427	14	3	]	]	NUM
ejpam-3427	14	4	)	)	PUNCT
ejpam-3427	14	5	,	,	PUNCT
ejpam-3427	14	6	a	a	DET
ejpam-3427	14	7	groupoid	groupoid	NOUN
ejpam-3427	14	8	s	s	NOUN
ejpam-3427	14	9	is	be	AUX
ejpam-3427	14	10	said	say	VERB
ejpam-3427	14	11	to	to	PART
ejpam-3427	14	12	be	be	AUX
ejpam-3427	14	13	medial	medial	ADJ
ejpam-3427	14	14	(	(	PUNCT
ejpam-3427	14	15	resp	resp	NOUN
ejpam-3427	14	16	.	.	PUNCT
ejpam-3427	15	1	paramedial	paramedial	PROPN
ejpam-3427	15	2	)	)	PUNCT
ejpam-3427	16	1	if	if	SCONJ
ejpam-3427	16	2	(	(	PUNCT
ejpam-3427	16	3	ab)(cd	ab)(cd	NOUN
ejpam-3427	16	4	)	)	PUNCT
ejpam-3427	16	5	=	=	SYM
ejpam-3427	16	6	(	(	PUNCT
ejpam-3427	16	7	ac)(bd	ac)(bd	PROPN
ejpam-3427	16	8	)	)	PUNCT
ejpam-3427	16	9	(	(	PUNCT
ejpam-3427	16	10	resp	resp	NOUN
ejpam-3427	16	11	.	.	PUNCT
ejpam-3427	17	1	(	(	PUNCT
ejpam-3427	17	2	ab)(cd	ab)(cd	PROPN
ejpam-3427	17	3	)	)	PUNCT
ejpam-3427	17	4	=	=	SYM
ejpam-3427	17	5	(	(	PUNCT
ejpam-3427	17	6	db)(ca	db)(ca	PROPN
ejpam-3427	17	7	)	)	PUNCT
ejpam-3427	17	8	)	)	PUNCT
ejpam-3427	17	9	.	.	PUNCT
ejpam-3427	18	1	in	in	ADP
ejpam-3427	18	2	[	[	X
ejpam-3427	18	3	9	9	NUM
ejpam-3427	18	4	]	]	PUNCT
ejpam-3427	18	5	,	,	PUNCT
ejpam-3427	18	6	an	an	DET
ejpam-3427	18	7	la	la	ADJ
ejpam-3427	18	8	-	-	PUNCT
ejpam-3427	18	9	semigroup	semigroup	PROPN
ejpam-3427	18	10	is	be	AUX
ejpam-3427	18	11	medial	medial	ADJ
ejpam-3427	18	12	,	,	PUNCT
ejpam-3427	18	13	but	but	CCONJ
ejpam-3427	18	14	in	in	ADP
ejpam-3427	18	15	general	general	ADJ
ejpam-3427	18	16	an	an	DET
ejpam-3427	18	17	la	la	ADJ
ejpam-3427	18	18	-	-	PUNCT
ejpam-3427	18	19	semigroup	semigroup	NOUN
ejpam-3427	18	20	needs	need	VERB
ejpam-3427	18	21	not	not	PART
ejpam-3427	18	22	to	to	PART
ejpam-3427	18	23	be	be	AUX
ejpam-3427	18	24	paramedial	paramedial	ADJ
ejpam-3427	18	25	.	.	PUNCT
ejpam-3427	19	1	every	every	DET
ejpam-3427	19	2	la	la	PROPN
ejpam-3427	19	3	-	-	PUNCT
ejpam-3427	19	4	semigroup	semigroup	NOUN
ejpam-3427	19	5	with	with	ADP
ejpam-3427	19	6	left	left	ADJ
ejpam-3427	19	7	identity	identity	NOUN
ejpam-3427	19	8	is	be	AUX
ejpam-3427	19	9	paramedial	paramedial	ADJ
ejpam-3427	19	10	in	in	ADP
ejpam-3427	19	11	[	[	X
ejpam-3427	19	12	15	15	NUM
ejpam-3427	19	13	]	]	PUNCT
ejpam-3427	19	14	and	and	CCONJ
ejpam-3427	19	15	also	also	ADV
ejpam-3427	19	16	satisfies	satisfy	VERB
ejpam-3427	19	17	a(bc	a(bc	NOUN
ejpam-3427	19	18	)	)	PUNCT
ejpam-3427	19	19	=	=	SYM
ejpam-3427	19	20	b(ac	b(ac	PROPN
ejpam-3427	19	21	)	)	PUNCT
ejpam-3427	19	22	,	,	PUNCT
ejpam-3427	19	23	(	(	PUNCT
ejpam-3427	19	24	ab)(cd	ab)(cd	NOUN
ejpam-3427	19	25	)	)	PUNCT
ejpam-3427	19	26	=	=	SYM
ejpam-3427	19	27	(	(	PUNCT
ejpam-3427	19	28	dc)(ba	dc)(ba	PROPN
ejpam-3427	19	29	)	)	PUNCT
ejpam-3427	19	30	.	.	PUNCT
ejpam-3427	20	1	kamran	kamran	PROPN
ejpam-3427	21	1	[	[	X
ejpam-3427	21	2	8	8	NUM
ejpam-3427	21	3	]	]	PUNCT
ejpam-3427	21	4	,	,	PUNCT
ejpam-3427	21	5	extended	extend	VERB
ejpam-3427	21	6	the	the	DET
ejpam-3427	21	7	notion	notion	NOUN
ejpam-3427	21	8	of	of	ADP
ejpam-3427	21	9	la	la	NOUN
ejpam-3427	21	10	-	-	PUNCT
ejpam-3427	21	11	semigroup	semigroup	NOUN
ejpam-3427	21	12	to	to	ADP
ejpam-3427	21	13	the	the	DET
ejpam-3427	21	14	left	left	ADJ
ejpam-3427	21	15	almost	almost	ADV
ejpam-3427	21	16	group	group	NOUN
ejpam-3427	21	17	(	(	PUNCT
ejpam-3427	21	18	la	la	NOUN
ejpam-3427	21	19	-	-	NOUN
ejpam-3427	21	20	group	group	NOUN
ejpam-3427	21	21	)	)	PUNCT
ejpam-3427	21	22	.	.	PUNCT
ejpam-3427	22	1	an	an	DET
ejpam-3427	22	2	la	la	ADJ
ejpam-3427	22	3	-	-	PUNCT
ejpam-3427	22	4	semigroup	semigroup	PROPN
ejpam-3427	22	5	g	g	PROPN
ejpam-3427	22	6	is	be	AUX
ejpam-3427	22	7	said	say	VERB
ejpam-3427	22	8	to	to	PART
ejpam-3427	22	9	be	be	AUX
ejpam-3427	22	10	a	a	DET
ejpam-3427	22	11	left	left	ADJ
ejpam-3427	22	12	almost	almost	ADV
ejpam-3427	22	13	group	group	NOUN
ejpam-3427	22	14	,	,	PUNCT
ejpam-3427	22	15	if	if	SCONJ
ejpam-3427	22	16	there	there	PRON
ejpam-3427	22	17	exists	exist	VERB
ejpam-3427	22	18	left	leave	VERB
ejpam-3427	22	19	identity	identity	NOUN
ejpam-3427	22	20	e	e	NOUN
ejpam-3427	22	21	∈	∈	PROPN
ejpam-3427	22	22	g	g	PROPN
ejpam-3427	22	23	such	such	ADJ
ejpam-3427	23	1	that	that	DET
ejpam-3427	23	2	ea	ea	NOUN
ejpam-3427	23	3	=	=	PUNCT
ejpam-3427	23	4	a	a	PRON
ejpam-3427	23	5	for	for	ADP
ejpam-3427	23	6	all	all	DET
ejpam-3427	23	7	a	a	DET
ejpam-3427	23	8	∈	∈	NOUN
ejpam-3427	23	9	g	g	NOUN
ejpam-3427	23	10	and	and	CCONJ
ejpam-3427	23	11	for	for	ADP
ejpam-3427	23	12	every	every	DET
ejpam-3427	23	13	a	a	DET
ejpam-3427	23	14	∈	∈	PROPN
ejpam-3427	23	15	g	g	NOUN
ejpam-3427	23	16	there	there	PRON
ejpam-3427	23	17	exists	exist	VERB
ejpam-3427	23	18	b	b	PROPN
ejpam-3427	23	19	∈	∈	PROPN
ejpam-3427	23	20	g	g	NOUN
ejpam-3427	23	21	such	such	ADJ
ejpam-3427	23	22	that	that	DET
ejpam-3427	23	23	ba	ba	PROPN
ejpam-3427	23	24	=	=	PUNCT
ejpam-3427	23	25	e.	e.	PROPN
ejpam-3427	23	26	shah	shah	PROPN
ejpam-3427	23	27	et	et	PROPN
ejpam-3427	23	28	.	.	PUNCT
ejpam-3427	24	1	al	al	PROPN
ejpam-3427	25	1	[	[	X
ejpam-3427	25	2	18	18	NUM
ejpam-3427	25	3	]	]	PUNCT
ejpam-3427	25	4	,	,	PUNCT
ejpam-3427	25	5	discussed	discuss	VERB
ejpam-3427	25	6	the	the	DET
ejpam-3427	25	7	left	left	NOUN
ejpam-3427	25	8	almost	almost	ADV
ejpam-3427	25	9	ring	ring	NOUN
ejpam-3427	25	10	(	(	PUNCT
ejpam-3427	25	11	la	la	ADJ
ejpam-3427	25	12	-	-	PUNCT
ejpam-3427	25	13	ring	ring	NOUN
ejpam-3427	25	14	)	)	PUNCT
ejpam-3427	25	15	of	of	ADP
ejpam-3427	25	16	finitely	finitely	ADV
ejpam-3427	25	17	nonzero	nonzero	PROPN
ejpam-3427	25	18	functions	function	NOUN
ejpam-3427	25	19	which	which	PRON
ejpam-3427	25	20	is	be	AUX
ejpam-3427	25	21	a	a	DET
ejpam-3427	25	22	generalization	generalization	NOUN
ejpam-3427	25	23	of	of	ADP
ejpam-3427	25	24	commutative	commutative	ADJ
ejpam-3427	25	25	semigroup	semigroup	PROPN
ejpam-3427	25	26	ring	ring	NOUN
ejpam-3427	25	27	.	.	PUNCT
ejpam-3427	26	1	by	by	ADP
ejpam-3427	26	2	a	a	DET
ejpam-3427	26	3	left	left	ADJ
ejpam-3427	26	4	almost	almost	ADV
ejpam-3427	26	5	ring	ring	NOUN
ejpam-3427	26	6	,	,	PUNCT
ejpam-3427	26	7	we	we	PRON
ejpam-3427	26	8	mean	mean	VERB
ejpam-3427	26	9	a	a	DET
ejpam-3427	26	10	non	non	ADJ
ejpam-3427	26	11	-	-	ADJ
ejpam-3427	26	12	empty	empty	ADJ
ejpam-3427	26	13	set	set	VERB
ejpam-3427	26	14	r	r	NOUN
ejpam-3427	26	15	with	with	ADP
ejpam-3427	26	16	at	at	ADV
ejpam-3427	26	17	least	least	ADV
ejpam-3427	26	18	two	two	NUM
ejpam-3427	26	19	elements	element	NOUN
ejpam-3427	26	20	such	such	ADJ
ejpam-3427	26	21	that	that	SCONJ
ejpam-3427	26	22	(	(	PUNCT
ejpam-3427	26	23	r,+	r,+	NUM
ejpam-3427	26	24	)	)	PUNCT
ejpam-3427	26	25	is	be	AUX
ejpam-3427	26	26	an	an	DET
ejpam-3427	26	27	la	la	NOUN
ejpam-3427	26	28	-	-	NOUN
ejpam-3427	26	29	group	group	NOUN
ejpam-3427	26	30	,	,	PUNCT
ejpam-3427	26	31	(	(	PUNCT
ejpam-3427	26	32	r	r	NOUN
ejpam-3427	26	33	,	,	PUNCT
ejpam-3427	26	34	·	·	PUNCT
ejpam-3427	26	35	)	)	PUNCT
ejpam-3427	26	36	is	be	AUX
ejpam-3427	26	37	an	an	DET
ejpam-3427	26	38	la	la	ADJ
ejpam-3427	26	39	-	-	PUNCT
ejpam-3427	26	40	semigroup	semigroup	NOUN
ejpam-3427	26	41	,	,	PUNCT
ejpam-3427	26	42	both	both	PRON
ejpam-3427	26	43	left	leave	VERB
ejpam-3427	26	44	and	and	CCONJ
ejpam-3427	26	45	right	right	ADJ
ejpam-3427	26	46	distributive	distributive	ADJ
ejpam-3427	26	47	laws	law	NOUN
ejpam-3427	26	48	hold	hold	VERB
ejpam-3427	26	49	.	.	PUNCT
ejpam-3427	27	1	for	for	ADP
ejpam-3427	27	2	example	example	NOUN
ejpam-3427	27	3	,	,	PUNCT
ejpam-3427	27	4	from	from	ADP
ejpam-3427	27	5	a	a	DET
ejpam-3427	27	6	commutative	commutative	ADJ
ejpam-3427	27	7	ring	ring	NOUN
ejpam-3427	27	8	(	(	PUNCT
ejpam-3427	27	9	r,+	r,+	NUM
ejpam-3427	27	10	,	,	PUNCT
ejpam-3427	27	11	·	·	PUNCT
ejpam-3427	27	12	)	)	PUNCT
ejpam-3427	27	13	,	,	PUNCT
ejpam-3427	27	14	we	we	PRON
ejpam-3427	27	15	can	can	AUX
ejpam-3427	27	16	always	always	ADV
ejpam-3427	27	17	obtain	obtain	VERB
ejpam-3427	27	18	an	an	DET
ejpam-3427	27	19	la	la	ADJ
ejpam-3427	27	20	-	-	PUNCT
ejpam-3427	27	21	ring	ring	NOUN
ejpam-3427	27	22	(	(	PUNCT
ejpam-3427	27	23	r,⊕	r,⊕	NOUN
ejpam-3427	27	24	,	,	PUNCT
ejpam-3427	27	25	·	·	PUNCT
ejpam-3427	27	26	)	)	PUNCT
ejpam-3427	27	27	by	by	ADP
ejpam-3427	27	28	defining	define	VERB
ejpam-3427	27	29	for	for	ADP
ejpam-3427	27	30	all	all	DET
ejpam-3427	27	31	doi	doi	NOUN
ejpam-3427	27	32	:	:	PUNCT
ejpam-3427	27	33	https://doi.org/10.29020/nybg.ejpam.v12i2.3427	https://doi.org/10.29020/nybg.ejpam.v12i2.3427	NOUN
ejpam-3427	27	34	email	email	NOUN
ejpam-3427	27	35	address	address	NOUN
ejpam-3427	27	36	:	:	PUNCT
ejpam-3427	27	37	kausar.nasreen@gmail.com	kausar.nasreen@gmail.com	X
ejpam-3427	27	38	(	(	PUNCT
ejpam-3427	27	39	k.	k.	NOUN
ejpam-3427	27	40	nasreen	nasreen	PROPN
ejpam-3427	27	41	)	)	PUNCT
ejpam-3427	27	42	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3427	28	1	622	622	NUM
ejpam-3427	28	2	c	c	X
ejpam-3427	28	3	©	©	PROPN
ejpam-3427	28	4	2019	2019	NUM
ejpam-3427	28	5	ejpam	ejpam	NOUN
ejpam-3427	28	6	all	all	DET
ejpam-3427	28	7	rights	right	NOUN
ejpam-3427	28	8	reserved	reserve	VERB
ejpam-3427	28	9	.	.	PUNCT
ejpam-3427	29	1	k.	k.	PROPN
ejpam-3427	29	2	nasreen	nasreen	PROPN
ejpam-3427	29	3	/	/	SYM
ejpam-3427	29	4	eur	eur	PROPN
ejpam-3427	29	5	.	.	PUNCT
ejpam-3427	30	1	j.	j.	PROPN
ejpam-3427	30	2	pure	pure	PROPN
ejpam-3427	30	3	appl	appl	PROPN
ejpam-3427	30	4	.	.	PROPN
ejpam-3427	30	5	math	math	PROPN
ejpam-3427	30	6	,	,	PUNCT
ejpam-3427	30	7	12	12	NUM
ejpam-3427	30	8	(	(	PUNCT
ejpam-3427	30	9	2	2	NUM
ejpam-3427	30	10	)	)	PUNCT
ejpam-3427	30	11	(	(	PUNCT
ejpam-3427	30	12	2019	2019	NUM
ejpam-3427	30	13	)	)	PUNCT
ejpam-3427	30	14	,	,	PUNCT
ejpam-3427	30	15	622	622	NUM
ejpam-3427	30	16	-	-	SYM
ejpam-3427	30	17	648	648	NUM
ejpam-3427	30	18	623	623	NUM
ejpam-3427	30	19	a	a	PRON
ejpam-3427	30	20	,	,	PUNCT
ejpam-3427	30	21	b	b	X
ejpam-3427	30	22	∈	∈	PROPN
ejpam-3427	30	23	r	r	NOUN
ejpam-3427	30	24	,	,	PUNCT
ejpam-3427	30	25	a⊕b	a⊕b	NOUN
ejpam-3427	30	26	=	=	SYM
ejpam-3427	30	27	b−a	b−a	NOUN
ejpam-3427	30	28	and	and	CCONJ
ejpam-3427	30	29	a	a	DET
ejpam-3427	30	30	·	·	SYM
ejpam-3427	30	31	b	b	NOUN
ejpam-3427	30	32	is	be	AUX
ejpam-3427	30	33	same	same	ADJ
ejpam-3427	30	34	as	as	ADP
ejpam-3427	30	35	in	in	ADP
ejpam-3427	30	36	the	the	DET
ejpam-3427	30	37	ring	ring	NOUN
ejpam-3427	30	38	.	.	PUNCT
ejpam-3427	31	1	in	in	ADP
ejpam-3427	31	2	fact	fact	NOUN
ejpam-3427	31	3	an	an	DET
ejpam-3427	31	4	la	la	NOUN
ejpam-3427	31	5	-	-	PUNCT
ejpam-3427	31	6	ring	ring	NOUN
ejpam-3427	31	7	is	be	AUX
ejpam-3427	31	8	a	a	DET
ejpam-3427	31	9	non	non	ADJ
ejpam-3427	31	10	-	-	ADJ
ejpam-3427	31	11	associative	associative	ADJ
ejpam-3427	31	12	and	and	CCONJ
ejpam-3427	31	13	non	non	ADJ
ejpam-3427	31	14	-	-	ADJ
ejpam-3427	31	15	commutative	commutative	ADJ
ejpam-3427	31	16	ring	ring	NOUN
ejpam-3427	31	17	.	.	PUNCT
ejpam-3427	32	1	a	a	DET
ejpam-3427	32	2	non	non	ADJ
ejpam-3427	32	3	-	-	ADJ
ejpam-3427	32	4	empty	empty	ADJ
ejpam-3427	32	5	subset	subset	NOUN
ejpam-3427	32	6	a	a	PRON
ejpam-3427	32	7	of	of	ADP
ejpam-3427	32	8	an	an	DET
ejpam-3427	32	9	la	la	ADJ
ejpam-3427	32	10	-	-	PUNCT
ejpam-3427	32	11	ring	ring	NOUN
ejpam-3427	32	12	r	r	NOUN
ejpam-3427	32	13	is	be	AUX
ejpam-3427	32	14	an	an	DET
ejpam-3427	32	15	la	la	NOUN
ejpam-3427	32	16	-	-	PUNCT
ejpam-3427	32	17	subring	subring	NOUN
ejpam-3427	32	18	of	of	ADP
ejpam-3427	32	19	an	an	DET
ejpam-3427	32	20	la	la	ADJ
ejpam-3427	32	21	-	-	PUNCT
ejpam-3427	32	22	ring	ring	NOUN
ejpam-3427	32	23	r	r	NOUN
ejpam-3427	32	24	if	if	SCONJ
ejpam-3427	32	25	a	a	DET
ejpam-3427	32	26	−	−	PROPN
ejpam-3427	32	27	b	b	NOUN
ejpam-3427	32	28	and	and	CCONJ
ejpam-3427	32	29	ab	ab	PROPN
ejpam-3427	32	30	∈	∈	PROPN
ejpam-3427	32	31	a	a	PRON
ejpam-3427	32	32	for	for	ADP
ejpam-3427	32	33	all	all	DET
ejpam-3427	32	34	a	a	DET
ejpam-3427	32	35	,	,	PUNCT
ejpam-3427	32	36	b	b	X
ejpam-3427	32	37	∈	∈	PROPN
ejpam-3427	32	38	a.	a.	NOUN
ejpam-3427	32	39	a	a	PRON
ejpam-3427	32	40	is	be	AUX
ejpam-3427	32	41	a	a	DET
ejpam-3427	32	42	left	left	ADJ
ejpam-3427	32	43	(	(	PUNCT
ejpam-3427	32	44	resp	resp	NOUN
ejpam-3427	32	45	.	.	PUNCT
ejpam-3427	33	1	right	right	ADJ
ejpam-3427	33	2	)	)	PUNCT
ejpam-3427	33	3	ideal	ideal	NOUN
ejpam-3427	33	4	of	of	ADP
ejpam-3427	33	5	r	r	NOUN
ejpam-3427	33	6	if	if	SCONJ
ejpam-3427	33	7	(	(	PUNCT
ejpam-3427	33	8	a,+	a,+	NOUN
ejpam-3427	33	9	)	)	PUNCT
ejpam-3427	33	10	is	be	AUX
ejpam-3427	33	11	an	an	DET
ejpam-3427	33	12	la	la	ADJ
ejpam-3427	33	13	-	-	NOUN
ejpam-3427	33	14	group	group	NOUN
ejpam-3427	33	15	and	and	CCONJ
ejpam-3427	33	16	ra	ra	PROPN
ejpam-3427	34	1	⊆	⊆	NUM
ejpam-3427	34	2	a	a	DET
ejpam-3427	34	3	(	(	PUNCT
ejpam-3427	34	4	resp	resp	NOUN
ejpam-3427	34	5	.	.	PUNCT
ejpam-3427	35	1	ar	ar	VERB
ejpam-3427	35	2	⊆	⊆	NUM
ejpam-3427	35	3	a	a	PRON
ejpam-3427	35	4	)	)	PUNCT
ejpam-3427	35	5	.	.	PUNCT
ejpam-3427	36	1	a	a	PRON
ejpam-3427	36	2	is	be	AUX
ejpam-3427	36	3	called	call	VERB
ejpam-3427	36	4	an	an	DET
ejpam-3427	36	5	ideal	ideal	NOUN
ejpam-3427	36	6	of	of	ADP
ejpam-3427	36	7	r	r	NOUN
ejpam-3427	36	8	if	if	SCONJ
ejpam-3427	36	9	it	it	PRON
ejpam-3427	36	10	is	be	AUX
ejpam-3427	36	11	both	both	CCONJ
ejpam-3427	36	12	a	a	DET
ejpam-3427	36	13	left	left	ADJ
ejpam-3427	36	14	ideal	ideal	NOUN
ejpam-3427	36	15	and	and	CCONJ
ejpam-3427	36	16	a	a	DET
ejpam-3427	36	17	right	right	ADJ
ejpam-3427	36	18	ideal	ideal	NOUN
ejpam-3427	36	19	of	of	ADP
ejpam-3427	36	20	r.	r.	PROPN
ejpam-3427	36	21	after	after	ADP
ejpam-3427	36	22	the	the	DET
ejpam-3427	36	23	introduction	introduction	NOUN
ejpam-3427	36	24	of	of	ADP
ejpam-3427	36	25	fuzzy	fuzzy	ADJ
ejpam-3427	36	26	set	set	VERB
ejpam-3427	36	27	by	by	ADP
ejpam-3427	36	28	zadeh	zadeh	PROPN
ejpam-3427	37	1	[	[	X
ejpam-3427	37	2	22	22	NUM
ejpam-3427	37	3	]	]	PUNCT
ejpam-3427	37	4	,	,	PUNCT
ejpam-3427	37	5	several	several	ADJ
ejpam-3427	37	6	researchers	researcher	NOUN
ejpam-3427	37	7	explored	explore	VERB
ejpam-3427	37	8	on	on	ADP
ejpam-3427	37	9	the	the	DET
ejpam-3427	37	10	generalization	generalization	NOUN
ejpam-3427	37	11	of	of	ADP
ejpam-3427	37	12	the	the	DET
ejpam-3427	37	13	notion	notion	NOUN
ejpam-3427	37	14	of	of	ADP
ejpam-3427	37	15	fuzzy	fuzzy	ADJ
ejpam-3427	37	16	set	set	NOUN
ejpam-3427	37	17	.	.	PUNCT
ejpam-3427	38	1	the	the	DET
ejpam-3427	38	2	concept	concept	NOUN
ejpam-3427	38	3	of	of	ADP
ejpam-3427	38	4	intuitionistic	intuitionistic	ADJ
ejpam-3427	38	5	fuzzy	fuzzy	ADJ
ejpam-3427	38	6	set	set	NOUN
ejpam-3427	38	7	was	be	AUX
ejpam-3427	38	8	introduced	introduce	VERB
ejpam-3427	38	9	by	by	ADP
ejpam-3427	38	10	atanassov	atanassov	NOUN
ejpam-3427	38	11	[	[	X
ejpam-3427	38	12	1	1	NUM
ejpam-3427	38	13	,	,	PUNCT
ejpam-3427	38	14	2	2	NUM
ejpam-3427	38	15	]	]	PUNCT
ejpam-3427	38	16	,	,	PUNCT
ejpam-3427	38	17	as	as	ADP
ejpam-3427	38	18	a	a	DET
ejpam-3427	38	19	generalization	generalization	NOUN
ejpam-3427	38	20	of	of	ADP
ejpam-3427	38	21	the	the	DET
ejpam-3427	38	22	notion	notion	NOUN
ejpam-3427	38	23	of	of	ADP
ejpam-3427	38	24	fuzzy	fuzzy	ADJ
ejpam-3427	38	25	set	set	NOUN
ejpam-3427	38	26	.	.	PUNCT
ejpam-3427	39	1	sherwood	sherwood	PROPN
ejpam-3427	40	1	[	[	X
ejpam-3427	40	2	20	20	NUM
ejpam-3427	40	3	]	]	PUNCT
ejpam-3427	40	4	,	,	PUNCT
ejpam-3427	40	5	introduced	introduce	VERB
ejpam-3427	40	6	the	the	DET
ejpam-3427	40	7	concept	concept	NOUN
ejpam-3427	40	8	of	of	ADP
ejpam-3427	40	9	product	product	NOUN
ejpam-3427	40	10	of	of	ADP
ejpam-3427	40	11	fuzzy	fuzzy	ADJ
ejpam-3427	40	12	subgroups	subgroup	NOUN
ejpam-3427	40	13	.	.	PUNCT
ejpam-3427	41	1	after	after	ADP
ejpam-3427	41	2	this	this	PRON
ejpam-3427	41	3	,	,	PUNCT
ejpam-3427	41	4	further	further	ADJ
ejpam-3427	41	5	study	study	NOUN
ejpam-3427	41	6	on	on	ADP
ejpam-3427	41	7	this	this	DET
ejpam-3427	41	8	concept	concept	NOUN
ejpam-3427	41	9	continued	continue	VERB
ejpam-3427	41	10	by	by	ADP
ejpam-3427	41	11	osman	osman	PROPN
ejpam-3427	41	12	[	[	X
ejpam-3427	41	13	11	11	NUM
ejpam-3427	41	14	,	,	PUNCT
ejpam-3427	41	15	12	12	NUM
ejpam-3427	41	16	]	]	PUNCT
ejpam-3427	41	17	and	and	CCONJ
ejpam-3427	41	18	ray	ray	VERB
ejpam-3427	42	1	[	[	X
ejpam-3427	42	2	16	16	NUM
ejpam-3427	42	3	]	]	PUNCT
ejpam-3427	42	4	.	.	PUNCT
ejpam-3427	43	1	zaid	zaid	PROPN
ejpam-3427	43	2	[	[	X
ejpam-3427	43	3	23	23	NUM
ejpam-3427	43	4	]	]	PUNCT
ejpam-3427	43	5	,	,	PUNCT
ejpam-3427	43	6	gave	give	VERB
ejpam-3427	43	7	the	the	DET
ejpam-3427	43	8	idea	idea	NOUN
ejpam-3427	43	9	of	of	ADP
ejpam-3427	43	10	normal	normal	ADJ
ejpam-3427	43	11	fuzzy	fuzzy	ADJ
ejpam-3427	43	12	subgroups	subgroup	NOUN
ejpam-3427	43	13	.	.	PUNCT
ejpam-3427	44	1	an	an	DET
ejpam-3427	44	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	44	3	fuzzy	fuzzy	ADJ
ejpam-3427	44	4	set	set	NOUN
ejpam-3427	44	5	(	(	PUNCT
ejpam-3427	44	6	briefly	briefly	ADV
ejpam-3427	44	7	,	,	PUNCT
ejpam-3427	44	8	ifs	ifs	PROPN
ejpam-3427	44	9	)	)	PUNCT
ejpam-3427	44	10	a	a	PRON
ejpam-3427	44	11	in	in	ADP
ejpam-3427	44	12	a	a	DET
ejpam-3427	44	13	non	non	ADJ
ejpam-3427	44	14	-	-	ADJ
ejpam-3427	44	15	empty	empty	ADJ
ejpam-3427	44	16	set	set	NOUN
ejpam-3427	44	17	x	x	PUNCT
ejpam-3427	44	18	is	be	AUX
ejpam-3427	44	19	an	an	DET
ejpam-3427	44	20	object	object	NOUN
ejpam-3427	44	21	having	have	VERB
ejpam-3427	44	22	the	the	DET
ejpam-3427	44	23	form	form	NOUN
ejpam-3427	44	24	a	a	DET
ejpam-3427	44	25	=	=	X
ejpam-3427	44	26	{	{	PUNCT
ejpam-3427	44	27	(	(	PUNCT
ejpam-3427	44	28	x	x	NOUN
ejpam-3427	44	29	,	,	PUNCT
ejpam-3427	44	30	µa(x	µa(x	NOUN
ejpam-3427	44	31	)	)	PUNCT
ejpam-3427	44	32	,	,	PUNCT
ejpam-3427	44	33	γa(x	γa(x	NUM
ejpam-3427	44	34	)	)	PUNCT
ejpam-3427	44	35	)	)	PUNCT
ejpam-3427	44	36	:	:	PUNCT
ejpam-3427	45	1	x	x	X
ejpam-3427	45	2	∈	∈	NOUN
ejpam-3427	45	3	x	x	X
ejpam-3427	45	4	}	}	PUNCT
ejpam-3427	45	5	,	,	PUNCT
ejpam-3427	45	6	where	where	SCONJ
ejpam-3427	45	7	the	the	DET
ejpam-3427	45	8	functions	function	NOUN
ejpam-3427	45	9	µa	µa	VERB
ejpam-3427	45	10	:	:	PUNCT
ejpam-3427	45	11	x	x	X
ejpam-3427	45	12	→	→	SYM
ejpam-3427	45	13	[	[	X
ejpam-3427	45	14	0	0	NUM
ejpam-3427	45	15	,	,	PUNCT
ejpam-3427	45	16	1	1	NUM
ejpam-3427	45	17	]	]	PUNCT
ejpam-3427	45	18	and	and	CCONJ
ejpam-3427	45	19	γa	γa	PRON
ejpam-3427	45	20	:	:	PUNCT
ejpam-3427	45	21	x	x	X
ejpam-3427	45	22	→	→	PUNCT
ejpam-3427	45	23	[	[	X
ejpam-3427	45	24	0	0	NUM
ejpam-3427	45	25	,	,	PUNCT
ejpam-3427	45	26	1	1	NUM
ejpam-3427	45	27	]	]	PUNCT
ejpam-3427	45	28	denote	denote	VERB
ejpam-3427	45	29	the	the	DET
ejpam-3427	45	30	degree	degree	NOUN
ejpam-3427	45	31	of	of	ADP
ejpam-3427	45	32	membership	membership	NOUN
ejpam-3427	45	33	and	and	CCONJ
ejpam-3427	45	34	the	the	DET
ejpam-3427	45	35	degree	degree	NOUN
ejpam-3427	45	36	of	of	ADP
ejpam-3427	45	37	nonmembership	nonmembership	NOUN
ejpam-3427	45	38	,	,	PUNCT
ejpam-3427	45	39	respectively	respectively	ADV
ejpam-3427	45	40	and	and	CCONJ
ejpam-3427	45	41	0	0	NUM
ejpam-3427	45	42	≤	≤	NOUN
ejpam-3427	45	43	µa(x	µa(x	NOUN
ejpam-3427	45	44	)	)	PUNCT
ejpam-3427	45	45	+	+	CCONJ
ejpam-3427	45	46	γa(x	γa(x	X
ejpam-3427	45	47	)	)	PUNCT
ejpam-3427	45	48	≤	≤	NUM
ejpam-3427	45	49	1	1	NUM
ejpam-3427	45	50	for	for	ADP
ejpam-3427	45	51	all	all	DET
ejpam-3427	45	52	x	x	SYM
ejpam-3427	45	53	∈	∈	NOUN
ejpam-3427	45	54	x	x	PUNCT
ejpam-3427	46	1	[	[	X
ejpam-3427	46	2	1	1	NUM
ejpam-3427	46	3	,	,	PUNCT
ejpam-3427	46	4	2	2	NUM
ejpam-3427	46	5	]	]	PUNCT
ejpam-3427	46	6	.	.	PUNCT
ejpam-3427	47	1	an	an	DET
ejpam-3427	47	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	47	3	fuzzy	fuzzy	NOUN
ejpam-3427	47	4	set	set	VERB
ejpam-3427	47	5	a	a	PRON
ejpam-3427	47	6	=	=	X
ejpam-3427	47	7	{	{	PUNCT
ejpam-3427	47	8	(	(	PUNCT
ejpam-3427	47	9	x	x	NOUN
ejpam-3427	47	10	,	,	PUNCT
ejpam-3427	47	11	µa(x	µa(x	NOUN
ejpam-3427	47	12	)	)	PUNCT
ejpam-3427	47	13	,	,	PUNCT
ejpam-3427	47	14	γa(x	γa(x	NUM
ejpam-3427	47	15	)	)	PUNCT
ejpam-3427	47	16	)	)	PUNCT
ejpam-3427	47	17	:	:	PUNCT
ejpam-3427	48	1	x	x	X
ejpam-3427	48	2	∈	∈	NOUN
ejpam-3427	48	3	x	x	X
ejpam-3427	48	4	}	}	PUNCT
ejpam-3427	48	5	in	in	ADP
ejpam-3427	48	6	x	x	PRON
ejpam-3427	48	7	can	can	AUX
ejpam-3427	48	8	be	be	AUX
ejpam-3427	48	9	identified	identify	VERB
ejpam-3427	48	10	to	to	PART
ejpam-3427	48	11	be	be	AUX
ejpam-3427	48	12	an	an	DET
ejpam-3427	48	13	ordered	order	VERB
ejpam-3427	48	14	pair	pair	NOUN
ejpam-3427	48	15	(	(	PUNCT
ejpam-3427	48	16	µa	µa	NOUN
ejpam-3427	48	17	,	,	PUNCT
ejpam-3427	48	18	γa	γa	NOUN
ejpam-3427	48	19	)	)	PUNCT
ejpam-3427	48	20	in	in	ADP
ejpam-3427	48	21	ix	ix	ADP
ejpam-3427	48	22	×	×	NOUN
ejpam-3427	48	23	ix	ix	ADV
ejpam-3427	48	24	,	,	PUNCT
ejpam-3427	48	25	where	where	SCONJ
ejpam-3427	48	26	ix	ix	ADV
ejpam-3427	48	27	is	be	AUX
ejpam-3427	48	28	the	the	DET
ejpam-3427	48	29	set	set	NOUN
ejpam-3427	48	30	of	of	ADP
ejpam-3427	48	31	all	all	DET
ejpam-3427	48	32	functions	function	NOUN
ejpam-3427	48	33	from	from	ADP
ejpam-3427	48	34	x	x	PUNCT
ejpam-3427	48	35	to	to	ADP
ejpam-3427	48	36	[	[	X
ejpam-3427	48	37	0	0	NUM
ejpam-3427	48	38	,	,	PUNCT
ejpam-3427	48	39	1	1	NUM
ejpam-3427	48	40	]	]	PUNCT
ejpam-3427	48	41	.	.	PUNCT
ejpam-3427	49	1	for	for	ADP
ejpam-3427	49	2	the	the	DET
ejpam-3427	49	3	sake	sake	NOUN
ejpam-3427	49	4	of	of	ADP
ejpam-3427	49	5	simplicity	simplicity	NOUN
ejpam-3427	49	6	,	,	PUNCT
ejpam-3427	49	7	we	we	PRON
ejpam-3427	49	8	will	will	AUX
ejpam-3427	49	9	use	use	VERB
ejpam-3427	49	10	the	the	DET
ejpam-3427	49	11	symbol	symbol	NOUN
ejpam-3427	49	12	a	a	DET
ejpam-3427	49	13	=	=	X
ejpam-3427	49	14	(	(	PUNCT
ejpam-3427	49	15	µa	µa	PROPN
ejpam-3427	49	16	,	,	PUNCT
ejpam-3427	49	17	γa	γa	PROPN
ejpam-3427	49	18	)	)	PUNCT
ejpam-3427	49	19	for	for	ADP
ejpam-3427	49	20	the	the	DET
ejpam-3427	49	21	ifs	ifs	PROPN
ejpam-3427	49	22	a	a	X
ejpam-3427	49	23	=	=	X
ejpam-3427	49	24	{	{	PUNCT
ejpam-3427	49	25	(	(	PUNCT
ejpam-3427	49	26	x	x	NOUN
ejpam-3427	49	27	,	,	PUNCT
ejpam-3427	49	28	µa(x	µa(x	NOUN
ejpam-3427	49	29	)	)	PUNCT
ejpam-3427	49	30	,	,	PUNCT
ejpam-3427	49	31	γa(x	γa(x	NUM
ejpam-3427	49	32	)	)	PUNCT
ejpam-3427	49	33	)	)	PUNCT
ejpam-3427	49	34	:	:	PUNCT
ejpam-3427	50	1	x	x	X
ejpam-3427	50	2	∈	∈	NOUN
ejpam-3427	50	3	x	x	X
ejpam-3427	50	4	}	}	PUNCT
ejpam-3427	50	5	.	.	PUNCT
ejpam-3427	51	1	intuitionistic	intuitionistic	ADJ
ejpam-3427	51	2	fuzzy	fuzzy	ADJ
ejpam-3427	51	3	subrings	subring	NOUN
ejpam-3427	51	4	and	and	CCONJ
ejpam-3427	51	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	51	6	fuzzy	fuzzy	ADJ
ejpam-3427	51	7	ideals	ideal	NOUN
ejpam-3427	51	8	of	of	ADP
ejpam-3427	51	9	a	a	DET
ejpam-3427	51	10	ring	ring	NOUN
ejpam-3427	51	11	have	have	AUX
ejpam-3427	51	12	been	be	AUX
ejpam-3427	51	13	defined	define	VERB
ejpam-3427	51	14	in	in	ADP
ejpam-3427	51	15	[	[	X
ejpam-3427	51	16	3	3	NUM
ejpam-3427	51	17	,	,	PUNCT
ejpam-3427	51	18	6	6	NUM
ejpam-3427	51	19	]	]	PUNCT
ejpam-3427	51	20	.	.	PUNCT
ejpam-3427	52	1	palaniappan	palaniappan	PROPN
ejpam-3427	52	2	et	et	PROPN
ejpam-3427	52	3	al	al	PROPN
ejpam-3427	53	1	[	[	X
ejpam-3427	53	2	13	13	NUM
ejpam-3427	53	3	,	,	PUNCT
ejpam-3427	53	4	14	14	NUM
ejpam-3427	53	5	]	]	PUNCT
ejpam-3427	53	6	,	,	PUNCT
ejpam-3427	53	7	explored	explore	VERB
ejpam-3427	53	8	the	the	DET
ejpam-3427	53	9	notions	notion	NOUN
ejpam-3427	53	10	of	of	ADP
ejpam-3427	53	11	homomorphism	homomorphism	NOUN
ejpam-3427	53	12	,	,	PUNCT
ejpam-3427	53	13	antihomomorphism	antihomomorphism	NOUN
ejpam-3427	53	14	of	of	ADP
ejpam-3427	53	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	53	16	fuzzy	fuzzy	ADJ
ejpam-3427	53	17	normal	normal	ADJ
ejpam-3427	53	18	subrings	subring	NOUN
ejpam-3427	53	19	and	and	CCONJ
ejpam-3427	53	20	also	also	ADV
ejpam-3427	53	21	discussed	discuss	VERB
ejpam-3427	53	22	some	some	DET
ejpam-3427	53	23	properties	property	NOUN
ejpam-3427	53	24	of	of	ADP
ejpam-3427	53	25	intuitionistic	intuitionistic	ADJ
ejpam-3427	53	26	fuzzy	fuzzy	ADJ
ejpam-3427	53	27	normal	normal	ADJ
ejpam-3427	53	28	subrings	subring	NOUN
ejpam-3427	53	29	.	.	PUNCT
ejpam-3427	54	1	moreover	moreover	ADV
ejpam-3427	54	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	54	3	fuzzy	fuzzy	ADJ
ejpam-3427	54	4	ring	ring	NOUN
ejpam-3427	54	5	and	and	CCONJ
ejpam-3427	54	6	its	its	PRON
ejpam-3427	54	7	homomorphism	homomorphism	NOUN
ejpam-3427	54	8	image	image	NOUN
ejpam-3427	54	9	have	have	AUX
ejpam-3427	54	10	been	be	AUX
ejpam-3427	54	11	investigated	investigate	VERB
ejpam-3427	54	12	by	by	ADP
ejpam-3427	54	13	yan	yan	PROPN
ejpam-3427	55	1	[	[	X
ejpam-3427	55	2	21	21	NUM
ejpam-3427	55	3	]	]	PUNCT
ejpam-3427	55	4	.	.	PUNCT
ejpam-3427	56	1	shal	shal	NOUN
ejpam-3427	56	2	et	et	PROPN
ejpam-3427	56	3	al	al	PROPN
ejpam-3427	57	1	[	[	X
ejpam-3427	57	2	17	17	NUM
ejpam-3427	57	3	]	]	PUNCT
ejpam-3427	57	4	,	,	PUNCT
ejpam-3427	57	5	introduced	introduce	VERB
ejpam-3427	57	6	the	the	DET
ejpam-3427	57	7	concept	concept	NOUN
ejpam-3427	57	8	of	of	ADP
ejpam-3427	57	9	intuitionistic	intuitionistic	ADJ
ejpam-3427	57	10	fuzzy	fuzzy	ADJ
ejpam-3427	57	11	normal	normal	ADJ
ejpam-3427	57	12	subrings	subring	NOUN
ejpam-3427	57	13	over	over	ADP
ejpam-3427	57	14	a	a	DET
ejpam-3427	57	15	non	non	ADJ
ejpam-3427	57	16	-	-	ADJ
ejpam-3427	57	17	associative	associative	ADJ
ejpam-3427	57	18	ring	ring	NOUN
ejpam-3427	57	19	(	(	PUNCT
ejpam-3427	57	20	la	la	NOUN
ejpam-3427	57	21	-	-	NOUN
ejpam-3427	57	22	ring	ring	NOUN
ejpam-3427	57	23	)	)	PUNCT
ejpam-3427	57	24	.	.	PUNCT
ejpam-3427	58	1	we	we	PRON
ejpam-3427	58	2	define	define	VERB
ejpam-3427	58	3	the	the	DET
ejpam-3427	58	4	direct	direct	ADJ
ejpam-3427	58	5	product	product	NOUN
ejpam-3427	58	6	of	of	ADP
ejpam-3427	58	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	58	8	fuzzy	fuzzy	ADJ
ejpam-3427	58	9	sets	set	NOUN
ejpam-3427	58	10	a1	a1	NOUN
ejpam-3427	58	11	and	and	CCONJ
ejpam-3427	58	12	a2	a2	PROPN
ejpam-3427	58	13	of	of	ADP
ejpam-3427	58	14	la	la	PROPN
ejpam-3427	58	15	-	-	PUNCT
ejpam-3427	58	16	rings	ring	NOUN
ejpam-3427	58	17	r1	r1	NOUN
ejpam-3427	58	18	and	and	CCONJ
ejpam-3427	58	19	r2	r2	PROPN
ejpam-3427	58	20	,	,	PUNCT
ejpam-3427	58	21	respectively	respectively	ADV
ejpam-3427	58	22	and	and	CCONJ
ejpam-3427	58	23	investigate	investigate	VERB
ejpam-3427	58	24	the	the	DET
ejpam-3427	58	25	some	some	DET
ejpam-3427	58	26	basic	basic	ADJ
ejpam-3427	58	27	properties	property	NOUN
ejpam-3427	58	28	of	of	ADP
ejpam-3427	58	29	intuitionistic	intuitionistic	ADJ
ejpam-3427	58	30	anti	anti	ADJ
ejpam-3427	58	31	fuzzy	fuzzy	ADJ
ejpam-3427	58	32	normal	normal	ADJ
ejpam-3427	58	33	la	la	NOUN
ejpam-3427	58	34	-	-	PUNCT
ejpam-3427	58	35	subrings	subring	NOUN
ejpam-3427	58	36	of	of	ADP
ejpam-3427	58	37	an	an	DET
ejpam-3427	58	38	la	la	ADJ
ejpam-3427	58	39	-	-	PUNCT
ejpam-3427	58	40	ring	ring	NOUN
ejpam-3427	58	41	r1	r1	PROPN
ejpam-3427	58	42	×r2	×r2	PROPN
ejpam-3427	58	43	.	.	PUNCT
ejpam-3427	59	1	we	we	PRON
ejpam-3427	59	2	define	define	VERB
ejpam-3427	59	3	the	the	DET
ejpam-3427	59	4	direct	direct	ADJ
ejpam-3427	59	5	product	product	NOUN
ejpam-3427	59	6	of	of	ADP
ejpam-3427	59	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	59	8	fuzzy	fuzzy	ADJ
ejpam-3427	59	9	sets	set	NOUN
ejpam-3427	59	10	a1	a1	NOUN
ejpam-3427	59	11	,	,	PUNCT
ejpam-3427	59	12	a2	a2	PROPN
ejpam-3427	59	13	,	,	PUNCT
ejpam-3427	59	14	...	...	PUNCT
ejpam-3427	59	15	,	,	PUNCT
ejpam-3427	59	16	an	an	PRON
ejpam-3427	59	17	of	of	ADP
ejpam-3427	59	18	la	la	NOUN
ejpam-3427	59	19	-	-	PUNCT
ejpam-3427	59	20	rings	ring	NOUN
ejpam-3427	59	21	r1	r1	NOUN
ejpam-3427	59	22	,	,	PUNCT
ejpam-3427	59	23	r2	r2	PROPN
ejpam-3427	59	24	,	,	PUNCT
ejpam-3427	59	25	...	...	PUNCT
ejpam-3427	59	26	,	,	PUNCT
ejpam-3427	59	27	rn	rn	PROPN
ejpam-3427	59	28	,	,	PUNCT
ejpam-3427	59	29	respectively	respectively	ADV
ejpam-3427	59	30	and	and	CCONJ
ejpam-3427	59	31	examine	examine	VERB
ejpam-3427	59	32	the	the	DET
ejpam-3427	59	33	some	some	DET
ejpam-3427	59	34	fundamental	fundamental	ADJ
ejpam-3427	59	35	properties	property	NOUN
ejpam-3427	59	36	of	of	ADP
ejpam-3427	59	37	intuitionistic	intuitionistic	ADJ
ejpam-3427	59	38	anti	anti	ADJ
ejpam-3427	59	39	fuzzy	fuzzy	ADJ
ejpam-3427	59	40	normal	normal	ADJ
ejpam-3427	59	41	la	la	NOUN
ejpam-3427	59	42	-	-	PUNCT
ejpam-3427	59	43	subrings	subring	NOUN
ejpam-3427	59	44	of	of	ADP
ejpam-3427	59	45	an	an	DET
ejpam-3427	59	46	la	la	ADJ
ejpam-3427	59	47	-	-	PUNCT
ejpam-3427	59	48	ring	ring	NOUN
ejpam-3427	59	49	r1	r1	NOUN
ejpam-3427	59	50	×r2	×r2	PROPN
ejpam-3427	59	51	×	×	NOUN
ejpam-3427	59	52	...	...	PUNCT
ejpam-3427	59	53	×rn	×rn	NOUN
ejpam-3427	59	54	.	.	PUNCT
ejpam-3427	60	1	specifically	specifically	ADV
ejpam-3427	60	2	we	we	PRON
ejpam-3427	60	3	show	show	VERB
ejpam-3427	60	4	that	that	SCONJ
ejpam-3427	60	5	:	:	PUNCT
ejpam-3427	60	6	let	let	VERB
ejpam-3427	60	7	x	x	PUNCT
ejpam-3427	60	8	=	=	PUNCT
ejpam-3427	60	9	a×b	a×b	PROPN
ejpam-3427	60	10	and	and	CCONJ
ejpam-3427	60	11	y	y	PROPN
ejpam-3427	60	12	=	=	SYM
ejpam-3427	60	13	c×d	c×d	PROPN
ejpam-3427	60	14	be	be	AUX
ejpam-3427	60	15	two	two	NUM
ejpam-3427	60	16	la	la	ADJ
ejpam-3427	60	17	-	-	PUNCT
ejpam-3427	60	18	subrings	subring	NOUN
ejpam-3427	60	19	of	of	ADP
ejpam-3427	60	20	an	an	DET
ejpam-3427	60	21	la	la	ADJ
ejpam-3427	60	22	-	-	PUNCT
ejpam-3427	60	23	ring	ring	NOUN
ejpam-3427	60	24	r1×r2	r1×r2	PROPN
ejpam-3427	60	25	.	.	PUNCT
ejpam-3427	61	1	then	then	ADV
ejpam-3427	61	2	x∩y	x∩y	PROPN
ejpam-3427	61	3	is	be	AUX
ejpam-3427	61	4	an	an	DET
ejpam-3427	61	5	la	la	NOUN
ejpam-3427	61	6	-	-	PUNCT
ejpam-3427	61	7	subring	subring	NOUN
ejpam-3427	61	8	of	of	ADP
ejpam-3427	61	9	an	an	DET
ejpam-3427	61	10	la	la	ADJ
ejpam-3427	61	11	-	-	PUNCT
ejpam-3427	61	12	ring	ring	NOUN
ejpam-3427	61	13	r1	r1	NOUN
ejpam-3427	61	14	×	×	NOUN
ejpam-3427	61	15	r2	r2	NOUN
ejpam-3427	61	16	if	if	SCONJ
ejpam-3427	61	17	and	and	CCONJ
ejpam-3427	61	18	only	only	ADV
ejpam-3427	61	19	if	if	SCONJ
ejpam-3427	61	20	the	the	DET
ejpam-3427	61	21	intuitionistic	intuitionistic	ADJ
ejpam-3427	61	22	anti	anti	ADJ
ejpam-3427	61	23	characteristic	characteristic	ADJ
ejpam-3427	61	24	function	function	NOUN
ejpam-3427	61	25	χz	χz	PROPN
ejpam-3427	61	26	=	=	SYM
ejpam-3427	61	27	〈	〈	PROPN
ejpam-3427	61	28	µχz	µχz	NOUN
ejpam-3427	61	29	,	,	PUNCT
ejpam-3427	61	30	γχz	γχz	INTJ
ejpam-3427	61	31	〉	〉	NOUN
ejpam-3427	61	32	of	of	ADP
ejpam-3427	61	33	z	z	NOUN
ejpam-3427	61	34	=	=	SYM
ejpam-3427	62	1	x	x	NOUN
ejpam-3427	62	2	∩	∩	NOUN
ejpam-3427	62	3	y	y	PROPN
ejpam-3427	62	4	is	be	AUX
ejpam-3427	62	5	an	an	DET
ejpam-3427	62	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	62	7	anti	anti	ADJ
ejpam-3427	62	8	fuzzy	fuzzy	ADJ
ejpam-3427	62	9	normal	normal	ADJ
ejpam-3427	62	10	la	la	NOUN
ejpam-3427	62	11	-	-	PUNCT
ejpam-3427	62	12	subring	subring	NOUN
ejpam-3427	62	13	of	of	ADP
ejpam-3427	62	14	an	an	DET
ejpam-3427	62	15	la	la	ADJ
ejpam-3427	62	16	-	-	PUNCT
ejpam-3427	62	17	ring	ring	NOUN
ejpam-3427	62	18	r1	r1	PROPN
ejpam-3427	62	19	×r2	×r2	PROPN
ejpam-3427	62	20	.	.	PUNCT
ejpam-3427	63	1	let	let	VERB
ejpam-3427	63	2	a	a	DET
ejpam-3427	63	3	=	=	NOUN
ejpam-3427	63	4	a1×a2×	a1×a2×	NOUN
ejpam-3427	63	5	...	...	PUNCT
ejpam-3427	63	6	×an	×an	PROPN
ejpam-3427	63	7	and	and	CCONJ
ejpam-3427	63	8	b	b	X
ejpam-3427	63	9	=	=	SYM
ejpam-3427	63	10	b×b2×	b×b2×	PROPN
ejpam-3427	63	11	...	...	PUNCT
ejpam-3427	63	12	×bn	×bn	AUX
ejpam-3427	63	13	be	be	AUX
ejpam-3427	63	14	two	two	NUM
ejpam-3427	63	15	la	la	ADJ
ejpam-3427	63	16	-	-	PUNCT
ejpam-3427	63	17	subrings	subring	NOUN
ejpam-3427	63	18	of	of	ADP
ejpam-3427	63	19	an	an	DET
ejpam-3427	63	20	la	la	ADJ
ejpam-3427	63	21	-	-	PUNCT
ejpam-3427	63	22	ring	ring	NOUN
ejpam-3427	63	23	r1	r1	NOUN
ejpam-3427	63	24	×r2	×r2	PROPN
ejpam-3427	63	25	×	×	NOUN
ejpam-3427	63	26	...	...	PUNCT
ejpam-3427	63	27	×rn	×rn	NOUN
ejpam-3427	63	28	.	.	PUNCT
ejpam-3427	64	1	then	then	ADV
ejpam-3427	64	2	a	a	DET
ejpam-3427	64	3	∩b	∩b	NOUN
ejpam-3427	64	4	is	be	AUX
ejpam-3427	64	5	an	an	DET
ejpam-3427	64	6	la	la	NOUN
ejpam-3427	64	7	-	-	PUNCT
ejpam-3427	64	8	subring	subring	NOUN
ejpam-3427	64	9	of	of	ADP
ejpam-3427	64	10	an	an	DET
ejpam-3427	64	11	la	la	ADJ
ejpam-3427	64	12	-	-	PUNCT
ejpam-3427	64	13	ring	ring	NOUN
ejpam-3427	64	14	r1	r1	NOUN
ejpam-3427	64	15	×r2	×r2	PROPN
ejpam-3427	64	16	×	×	NOUN
ejpam-3427	64	17	...	...	PUNCT
ejpam-3427	64	18	×rn	×rn	NOUN
ejpam-3427	64	19	if	if	SCONJ
ejpam-3427	64	20	and	and	CCONJ
ejpam-3427	64	21	only	only	ADV
ejpam-3427	64	22	if	if	SCONJ
ejpam-3427	64	23	the	the	DET
ejpam-3427	64	24	intuitionistic	intuitionistic	ADJ
ejpam-3427	64	25	anti	anti	ADJ
ejpam-3427	64	26	characteristic	characteristic	ADJ
ejpam-3427	64	27	function	function	NOUN
ejpam-3427	64	28	χz	χz	PROPN
ejpam-3427	65	1	=	=	SYM
ejpam-3427	65	2	〈	〈	PROPN
ejpam-3427	65	3	µχz	µχz	NOUN
ejpam-3427	65	4	,	,	PUNCT
ejpam-3427	65	5	γχz	γχz	INTJ
ejpam-3427	65	6	〉	〉	NOUN
ejpam-3427	65	7	of	of	ADP
ejpam-3427	65	8	z	z	PROPN
ejpam-3427	65	9	=	=	PUNCT
ejpam-3427	65	10	a	a	DET
ejpam-3427	65	11	∩	∩	ADJ
ejpam-3427	65	12	b	b	NOUN
ejpam-3427	65	13	is	be	AUX
ejpam-3427	65	14	an	an	DET
ejpam-3427	65	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	65	16	anti	anti	ADJ
ejpam-3427	65	17	fuzzy	fuzzy	ADJ
ejpam-3427	65	18	normal	normal	ADJ
ejpam-3427	65	19	la	la	NOUN
ejpam-3427	65	20	-	-	PUNCT
ejpam-3427	65	21	subring	subring	NOUN
ejpam-3427	65	22	of	of	ADP
ejpam-3427	65	23	an	an	DET
ejpam-3427	65	24	la	la	ADJ
ejpam-3427	65	25	-	-	PUNCT
ejpam-3427	65	26	ring	ring	NOUN
ejpam-3427	65	27	r1	r1	NOUN
ejpam-3427	65	28	×r2	×r2	PROPN
ejpam-3427	65	29	×	×	NOUN
ejpam-3427	65	30	...	...	PUNCT
ejpam-3427	65	31	×rn	×rn	NOUN
ejpam-3427	65	32	.	.	PUNCT
ejpam-3427	66	1	let	let	VERB
ejpam-3427	66	2	a	a	PRON
ejpam-3427	66	3	and	and	CCONJ
ejpam-3427	66	4	b	b	NOUN
ejpam-3427	66	5	be	be	AUX
ejpam-3427	66	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	66	7	fuzzy	fuzzy	ADJ
ejpam-3427	66	8	sets	set	NOUN
ejpam-3427	66	9	of	of	ADP
ejpam-3427	66	10	la	la	NOUN
ejpam-3427	66	11	-	-	PUNCT
ejpam-3427	66	12	rings	ring	NOUN
ejpam-3427	66	13	r1	r1	NOUN
ejpam-3427	66	14	and	and	CCONJ
ejpam-3427	66	15	r2	r2	PROPN
ejpam-3427	66	16	with	with	ADP
ejpam-3427	66	17	left	left	ADJ
ejpam-3427	66	18	identities	identity	NOUN
ejpam-3427	66	19	e1	e1	PROPN
ejpam-3427	66	20	and	and	CCONJ
ejpam-3427	66	21	e2	e2	PROPN
ejpam-3427	66	22	,	,	PUNCT
ejpam-3427	66	23	respectively	respectively	ADV
ejpam-3427	66	24	and	and	CCONJ
ejpam-3427	66	25	a	a	DET
ejpam-3427	66	26	×	×	NOUN
ejpam-3427	66	27	b	b	NOUN
ejpam-3427	66	28	be	be	AUX
ejpam-3427	66	29	an	an	DET
ejpam-3427	66	30	intuitionistic	intuitionistic	ADJ
ejpam-3427	66	31	anti	anti	ADJ
ejpam-3427	66	32	fuzzy	fuzzy	ADJ
ejpam-3427	66	33	normal	normal	ADJ
ejpam-3427	66	34	la	la	NOUN
ejpam-3427	66	35	-	-	PUNCT
ejpam-3427	66	36	subring	subring	NOUN
ejpam-3427	66	37	of	of	ADP
ejpam-3427	66	38	an	an	DET
ejpam-3427	66	39	k.	k.	NOUN
ejpam-3427	66	40	nasreen	nasreen	PROPN
ejpam-3427	66	41	/	/	SYM
ejpam-3427	66	42	eur	eur	PROPN
ejpam-3427	66	43	.	.	PUNCT
ejpam-3427	67	1	j.	j.	PROPN
ejpam-3427	67	2	pure	pure	PROPN
ejpam-3427	67	3	appl	appl	PROPN
ejpam-3427	67	4	.	.	PROPN
ejpam-3427	67	5	math	math	PROPN
ejpam-3427	67	6	,	,	PUNCT
ejpam-3427	67	7	12	12	NUM
ejpam-3427	67	8	(	(	PUNCT
ejpam-3427	67	9	2	2	NUM
ejpam-3427	67	10	)	)	PUNCT
ejpam-3427	67	11	(	(	PUNCT
ejpam-3427	67	12	2019	2019	NUM
ejpam-3427	67	13	)	)	PUNCT
ejpam-3427	67	14	,	,	PUNCT
ejpam-3427	67	15	622	622	NUM
ejpam-3427	67	16	-	-	SYM
ejpam-3427	67	17	648	648	NUM
ejpam-3427	67	18	624	624	NUM
ejpam-3427	67	19	la	la	ADJ
ejpam-3427	67	20	-	-	PUNCT
ejpam-3427	67	21	ring	ring	NOUN
ejpam-3427	67	22	r1	r1	PROPN
ejpam-3427	67	23	×r2	×r2	PROPN
ejpam-3427	67	24	.	.	PUNCT
ejpam-3427	68	1	then	then	ADV
ejpam-3427	68	2	the	the	DET
ejpam-3427	68	3	following	follow	VERB
ejpam-3427	68	4	conditions	condition	NOUN
ejpam-3427	68	5	are	be	AUX
ejpam-3427	68	6	true	true	ADJ
ejpam-3427	68	7	.	.	PUNCT
ejpam-3427	69	1	1	1	X
ejpam-3427	69	2	.	.	X
ejpam-3427	70	1	if	if	SCONJ
ejpam-3427	70	2	µa	µa	PROPN
ejpam-3427	70	3	(	(	PUNCT
ejpam-3427	70	4	x	x	X
ejpam-3427	70	5	)	)	PUNCT
ejpam-3427	70	6	≥	≥	PROPN
ejpam-3427	70	7	µb	µb	PROPN
ejpam-3427	70	8	(	(	PUNCT
ejpam-3427	70	9	e2	e2	PROPN
ejpam-3427	70	10	)	)	PUNCT
ejpam-3427	70	11	and	and	CCONJ
ejpam-3427	70	12	γa	γa	PROPN
ejpam-3427	70	13	(	(	PUNCT
ejpam-3427	70	14	x	x	NOUN
ejpam-3427	70	15	)	)	PUNCT
ejpam-3427	70	16	≤	≤	NUM
ejpam-3427	70	17	γb	γb	NOUN
ejpam-3427	70	18	(	(	PUNCT
ejpam-3427	70	19	e2	e2	PROPN
ejpam-3427	70	20	)	)	PUNCT
ejpam-3427	70	21	,	,	PUNCT
ejpam-3427	70	22	for	for	ADP
ejpam-3427	70	23	all	all	DET
ejpam-3427	70	24	x	x	SYM
ejpam-3427	70	25	∈	∈	PROPN
ejpam-3427	70	26	r1	r1	NOUN
ejpam-3427	70	27	,	,	PUNCT
ejpam-3427	70	28	then	then	ADV
ejpam-3427	70	29	a	a	PRON
ejpam-3427	70	30	is	be	AUX
ejpam-3427	70	31	an	an	DET
ejpam-3427	70	32	intuitionistic	intuitionistic	ADJ
ejpam-3427	70	33	anti	anti	ADJ
ejpam-3427	70	34	fuzzy	fuzzy	ADJ
ejpam-3427	70	35	normal	normal	ADJ
ejpam-3427	70	36	la	la	NOUN
ejpam-3427	70	37	-	-	PUNCT
ejpam-3427	70	38	subring	subring	NOUN
ejpam-3427	70	39	of	of	ADP
ejpam-3427	70	40	r1	r1	PROPN
ejpam-3427	70	41	.	.	PUNCT
ejpam-3427	71	1	2	2	X
ejpam-3427	71	2	.	.	X
ejpam-3427	72	1	if	if	SCONJ
ejpam-3427	72	2	µb	µb	VERB
ejpam-3427	72	3	(	(	PUNCT
ejpam-3427	72	4	x	x	NOUN
ejpam-3427	72	5	)	)	PUNCT
ejpam-3427	72	6	≥	≥	PROPN
ejpam-3427	72	7	µa	µa	NOUN
ejpam-3427	72	8	(	(	PUNCT
ejpam-3427	72	9	e1	e1	PROPN
ejpam-3427	72	10	)	)	PUNCT
ejpam-3427	72	11	and	and	CCONJ
ejpam-3427	72	12	γb	γb	INTJ
ejpam-3427	72	13	(	(	PUNCT
ejpam-3427	72	14	x	x	NOUN
ejpam-3427	72	15	)	)	PUNCT
ejpam-3427	72	16	≤	≤	NOUN
ejpam-3427	72	17	γa	γa	PROPN
ejpam-3427	72	18	(	(	PUNCT
ejpam-3427	72	19	e1	e1	PROPN
ejpam-3427	72	20	)	)	PUNCT
ejpam-3427	72	21	,	,	PUNCT
ejpam-3427	72	22	for	for	ADP
ejpam-3427	72	23	all	all	DET
ejpam-3427	72	24	x	x	SYM
ejpam-3427	72	25	∈	∈	PROPN
ejpam-3427	72	26	r2	r2	NOUN
ejpam-3427	72	27	,	,	PUNCT
ejpam-3427	72	28	then	then	ADV
ejpam-3427	72	29	b	b	PROPN
ejpam-3427	72	30	is	be	AUX
ejpam-3427	72	31	an	an	DET
ejpam-3427	72	32	intuitionistic	intuitionistic	ADJ
ejpam-3427	72	33	anti	anti	ADJ
ejpam-3427	72	34	fuzzy	fuzzy	ADJ
ejpam-3427	72	35	normal	normal	ADJ
ejpam-3427	72	36	la	la	NOUN
ejpam-3427	72	37	-	-	PUNCT
ejpam-3427	72	38	subring	subring	NOUN
ejpam-3427	72	39	of	of	ADP
ejpam-3427	72	40	r2	r2	NOUN
ejpam-3427	72	41	.	.	PUNCT
ejpam-3427	73	1	2	2	X
ejpam-3427	73	2	.	.	X
ejpam-3427	73	3	direct	direct	ADJ
ejpam-3427	73	4	product	product	NOUN
ejpam-3427	73	5	of	of	ADP
ejpam-3427	73	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	73	7	anti	anti	ADJ
ejpam-3427	73	8	fuzzy	fuzzy	ADJ
ejpam-3427	73	9	normal	normal	ADJ
ejpam-3427	73	10	la	la	NOUN
ejpam-3427	73	11	-	-	NOUN
ejpam-3427	73	12	subrings	subring	NOUN
ejpam-3427	73	13	we	we	PRON
ejpam-3427	73	14	define	define	VERB
ejpam-3427	73	15	the	the	DET
ejpam-3427	73	16	direct	direct	ADJ
ejpam-3427	73	17	product	product	NOUN
ejpam-3427	73	18	of	of	ADP
ejpam-3427	73	19	intuitionistic	intuitionistic	ADJ
ejpam-3427	73	20	fuzzy	fuzzy	ADJ
ejpam-3427	73	21	sets	set	NOUN
ejpam-3427	73	22	a1	a1	NOUN
ejpam-3427	73	23	,	,	PUNCT
ejpam-3427	73	24	a2	a2	PROPN
ejpam-3427	73	25	of	of	ADP
ejpam-3427	73	26	la	la	PROPN
ejpam-3427	73	27	-	-	PUNCT
ejpam-3427	73	28	rings	ring	NOUN
ejpam-3427	73	29	r1	r1	NOUN
ejpam-3427	73	30	,	,	PUNCT
ejpam-3427	73	31	r2	r2	PROPN
ejpam-3427	73	32	,	,	PUNCT
ejpam-3427	73	33	respectively	respectively	ADV
ejpam-3427	73	34	and	and	CCONJ
ejpam-3427	73	35	examine	examine	VERB
ejpam-3427	73	36	the	the	DET
ejpam-3427	73	37	some	some	DET
ejpam-3427	73	38	fundamental	fundamental	ADJ
ejpam-3427	73	39	properties	property	NOUN
ejpam-3427	73	40	of	of	ADP
ejpam-3427	73	41	direct	direct	ADJ
ejpam-3427	73	42	product	product	NOUN
ejpam-3427	73	43	of	of	ADP
ejpam-3427	73	44	intuitionistic	intuitionistic	ADJ
ejpam-3427	73	45	anti	anti	ADJ
ejpam-3427	73	46	fuzzy	fuzzy	ADJ
ejpam-3427	73	47	normal	normal	ADJ
ejpam-3427	73	48	la	la	NOUN
ejpam-3427	73	49	-	-	PUNCT
ejpam-3427	73	50	subrings	subring	NOUN
ejpam-3427	73	51	of	of	ADP
ejpam-3427	73	52	an	an	DET
ejpam-3427	73	53	la	la	ADJ
ejpam-3427	73	54	-	-	PUNCT
ejpam-3427	73	55	ring	ring	NOUN
ejpam-3427	73	56	r1	r1	PROPN
ejpam-3427	73	57	×r2	×r2	PROPN
ejpam-3427	73	58	.	.	PUNCT
ejpam-3427	74	1	let	let	VERB
ejpam-3427	74	2	µ1	µ1	VERB
ejpam-3427	74	3	and	and	CCONJ
ejpam-3427	74	4	µ2	µ2	PROPN
ejpam-3427	74	5	be	be	AUX
ejpam-3427	74	6	fuzzy	fuzzy	ADJ
ejpam-3427	74	7	subsets	subset	NOUN
ejpam-3427	74	8	of	of	ADP
ejpam-3427	74	9	la	la	NOUN
ejpam-3427	74	10	-	-	PUNCT
ejpam-3427	74	11	rings	ring	NOUN
ejpam-3427	74	12	r1	r1	NOUN
ejpam-3427	74	13	and	and	CCONJ
ejpam-3427	74	14	r2	r2	PROPN
ejpam-3427	74	15	,	,	PUNCT
ejpam-3427	74	16	respectively	respectively	ADV
ejpam-3427	74	17	.	.	PUNCT
ejpam-3427	75	1	the	the	DET
ejpam-3427	75	2	direct	direct	ADJ
ejpam-3427	75	3	product	product	NOUN
ejpam-3427	75	4	of	of	ADP
ejpam-3427	75	5	fuzzy	fuzzy	ADJ
ejpam-3427	75	6	subsets	subset	NOUN
ejpam-3427	75	7	µ1	µ1	PROPN
ejpam-3427	75	8	and	and	CCONJ
ejpam-3427	75	9	µ2	µ2	PROPN
ejpam-3427	75	10	is	be	AUX
ejpam-3427	75	11	denoted	denote	VERB
ejpam-3427	75	12	by	by	ADP
ejpam-3427	75	13	µ1	µ1	PROPN
ejpam-3427	75	14	×	×	PROPN
ejpam-3427	75	15	µ2	µ2	NOUN
ejpam-3427	75	16	and	and	CCONJ
ejpam-3427	75	17	defined	define	VERB
ejpam-3427	75	18	by	by	ADP
ejpam-3427	75	19	(	(	PUNCT
ejpam-3427	75	20	µ1	µ1	PROPN
ejpam-3427	75	21	×	×	PROPN
ejpam-3427	75	22	µ2)(x1	µ2)(x1	NOUN
ejpam-3427	75	23	,	,	PUNCT
ejpam-3427	75	24	x2	x2	PROPN
ejpam-3427	75	25	)	)	PUNCT
ejpam-3427	75	26	=	=	PUNCT
ejpam-3427	76	1	min{µ1(x1	min{µ1(x1	PROPN
ejpam-3427	76	2	)	)	PUNCT
ejpam-3427	76	3	,	,	PUNCT
ejpam-3427	76	4	µ2	µ2	PROPN
ejpam-3427	76	5	(	(	PUNCT
ejpam-3427	76	6	x2	x2	PROPN
ejpam-3427	76	7	)	)	PUNCT
ejpam-3427	76	8	}	}	PUNCT
ejpam-3427	76	9	.	.	PUNCT
ejpam-3427	77	1	a	a	DET
ejpam-3427	77	2	fuzzy	fuzzy	ADJ
ejpam-3427	77	3	subset	subset	NOUN
ejpam-3427	77	4	µ1	µ1	PROPN
ejpam-3427	77	5	×	×	PROPN
ejpam-3427	77	6	µ2	µ2	NOUN
ejpam-3427	77	7	of	of	ADP
ejpam-3427	77	8	an	an	DET
ejpam-3427	77	9	la	la	ADJ
ejpam-3427	77	10	-	-	PUNCT
ejpam-3427	77	11	ring	ring	NOUN
ejpam-3427	77	12	r1	r1	PROPN
ejpam-3427	77	13	×r2	×r2	PROPN
ejpam-3427	77	14	is	be	AUX
ejpam-3427	77	15	to	to	PART
ejpam-3427	77	16	be	be	AUX
ejpam-3427	77	17	a	a	DET
ejpam-3427	77	18	fuzzy	fuzzy	ADJ
ejpam-3427	77	19	la	la	NOUN
ejpam-3427	77	20	-	-	PUNCT
ejpam-3427	77	21	subring	subring	NOUN
ejpam-3427	77	22	of	of	ADP
ejpam-3427	77	23	r1	r1	PROPN
ejpam-3427	77	24	×r2	×r2	PROPN
ejpam-3427	77	25	if	if	SCONJ
ejpam-3427	77	26	1	1	NUM
ejpam-3427	77	27	.	.	PUNCT
ejpam-3427	78	1	(	(	PUNCT
ejpam-3427	78	2	µ1	µ1	NOUN
ejpam-3427	78	3	×	×	PROPN
ejpam-3427	78	4	µ2)(x−	µ2)(x−	ADP
ejpam-3427	78	5	y	y	PROPN
ejpam-3427	78	6	)	)	PUNCT
ejpam-3427	78	7	≥	≥	NOUN
ejpam-3427	78	8	min{µ1(x	min{µ1(x	NOUN
ejpam-3427	78	9	)	)	PUNCT
ejpam-3427	78	10	,	,	PUNCT
ejpam-3427	78	11	µ2(y	µ2(y	PROPN
ejpam-3427	78	12	)	)	PUNCT
ejpam-3427	78	13	}	}	PUNCT
ejpam-3427	78	14	,	,	PUNCT
ejpam-3427	78	15	2	2	X
ejpam-3427	78	16	.	.	PUNCT
ejpam-3427	78	17	(	(	PUNCT
ejpam-3427	78	18	µ1	µ1	PROPN
ejpam-3427	78	19	×	×	PROPN
ejpam-3427	78	20	µ2)(xy	µ2)(xy	PROPN
ejpam-3427	78	21	)	)	PUNCT
ejpam-3427	78	22	≥	≥	NUM
ejpam-3427	78	23	min{µ1(x	min{µ1(x	NOUN
ejpam-3427	78	24	)	)	PUNCT
ejpam-3427	78	25	,	,	PUNCT
ejpam-3427	78	26	µ2(y	µ2(y	PROPN
ejpam-3427	78	27	)	)	PUNCT
ejpam-3427	78	28	}	}	PUNCT
ejpam-3427	78	29	for	for	ADP
ejpam-3427	78	30	all	all	PRON
ejpam-3427	78	31	x	x	X
ejpam-3427	78	32	=	=	SYM
ejpam-3427	78	33	(	(	PUNCT
ejpam-3427	78	34	x1	x1	PROPN
ejpam-3427	78	35	,	,	PUNCT
ejpam-3427	78	36	x2	x2	PROPN
ejpam-3427	78	37	)	)	PUNCT
ejpam-3427	78	38	,	,	PUNCT
ejpam-3427	79	1	y	y	PROPN
ejpam-3427	79	2	=	=	SYM
ejpam-3427	79	3	(	(	PUNCT
ejpam-3427	79	4	y1	y1	INTJ
ejpam-3427	79	5	,	,	PUNCT
ejpam-3427	79	6	y2	y2	NOUN
ejpam-3427	79	7	)	)	PUNCT
ejpam-3427	79	8	∈	∈	PROPN
ejpam-3427	79	9	r1	r1	PROPN
ejpam-3427	79	10	×r2	×r2	PROPN
ejpam-3427	79	11	.	.	PUNCT
ejpam-3427	80	1	a	a	DET
ejpam-3427	80	2	fuzzy	fuzzy	ADJ
ejpam-3427	80	3	subset	subset	NOUN
ejpam-3427	80	4	µ1	µ1	PROPN
ejpam-3427	80	5	×	×	PROPN
ejpam-3427	80	6	µ2	µ2	NOUN
ejpam-3427	80	7	of	of	ADP
ejpam-3427	80	8	an	an	DET
ejpam-3427	80	9	la	la	ADJ
ejpam-3427	80	10	-	-	PUNCT
ejpam-3427	80	11	ring	ring	NOUN
ejpam-3427	80	12	r1	r1	NOUN
ejpam-3427	80	13	×	×	NOUN
ejpam-3427	80	14	r2	r2	NOUN
ejpam-3427	80	15	is	be	AUX
ejpam-3427	80	16	to	to	PART
ejpam-3427	80	17	be	be	AUX
ejpam-3427	80	18	an	an	DET
ejpam-3427	80	19	anti	anti	ADJ
ejpam-3427	80	20	fuzzy	fuzzy	ADJ
ejpam-3427	80	21	la	la	NOUN
ejpam-3427	80	22	-	-	PUNCT
ejpam-3427	80	23	subring	subring	NOUN
ejpam-3427	80	24	of	of	ADP
ejpam-3427	80	25	r1	r1	PROPN
ejpam-3427	80	26	×r2	×r2	PROPN
ejpam-3427	80	27	if	if	SCONJ
ejpam-3427	80	28	1	1	NUM
ejpam-3427	80	29	.	.	PUNCT
ejpam-3427	81	1	(	(	PUNCT
ejpam-3427	81	2	µ1	µ1	NOUN
ejpam-3427	81	3	×	×	PROPN
ejpam-3427	81	4	µ2)(x−	µ2)(x−	ADP
ejpam-3427	81	5	y	y	NOUN
ejpam-3427	81	6	)	)	PUNCT
ejpam-3427	81	7	≤	≤	ADJ
ejpam-3427	81	8	max{µ1(x	max{µ1(x	NOUN
ejpam-3427	81	9	)	)	PUNCT
ejpam-3427	81	10	,	,	PUNCT
ejpam-3427	81	11	µ2(y	µ2(y	PROPN
ejpam-3427	81	12	)	)	PUNCT
ejpam-3427	81	13	}	}	PUNCT
ejpam-3427	81	14	2	2	NUM
ejpam-3427	81	15	.	.	PUNCT
ejpam-3427	82	1	(	(	PUNCT
ejpam-3427	82	2	µ1	µ1	PROPN
ejpam-3427	82	3	×	×	PROPN
ejpam-3427	82	4	µ2)(xy	µ2)(xy	PROPN
ejpam-3427	82	5	)	)	PUNCT
ejpam-3427	82	6	≤	≤	NUM
ejpam-3427	82	7	max{µ1(x	max{µ1(x	NOUN
ejpam-3427	82	8	)	)	PUNCT
ejpam-3427	82	9	,	,	PUNCT
ejpam-3427	82	10	µ2(y	µ2(y	PROPN
ejpam-3427	82	11	)	)	PUNCT
ejpam-3427	82	12	}	}	PUNCT
ejpam-3427	82	13	for	for	ADP
ejpam-3427	82	14	all	all	PRON
ejpam-3427	82	15	x	x	X
ejpam-3427	82	16	=	=	SYM
ejpam-3427	82	17	(	(	PUNCT
ejpam-3427	82	18	x1	x1	PROPN
ejpam-3427	82	19	,	,	PUNCT
ejpam-3427	82	20	x2	x2	PROPN
ejpam-3427	82	21	)	)	PUNCT
ejpam-3427	82	22	,	,	PUNCT
ejpam-3427	82	23	y	y	PROPN
ejpam-3427	82	24	=	=	SYM
ejpam-3427	82	25	(	(	PUNCT
ejpam-3427	82	26	y1	y1	INTJ
ejpam-3427	82	27	,	,	PUNCT
ejpam-3427	82	28	y2	y2	NOUN
ejpam-3427	82	29	)	)	PUNCT
ejpam-3427	82	30	∈	∈	PROPN
ejpam-3427	82	31	r1	r1	PROPN
ejpam-3427	82	32	×r2	×r2	PROPN
ejpam-3427	82	33	.	.	PUNCT
ejpam-3427	83	1	a	a	DET
ejpam-3427	83	2	fuzzy	fuzzy	ADJ
ejpam-3427	83	3	la	la	NOUN
ejpam-3427	83	4	-	-	PUNCT
ejpam-3427	83	5	subring	subring	NOUN
ejpam-3427	83	6	of	of	ADP
ejpam-3427	83	7	an	an	DET
ejpam-3427	83	8	la	la	ADJ
ejpam-3427	83	9	-	-	PUNCT
ejpam-3427	83	10	ring	ring	NOUN
ejpam-3427	83	11	r1	r1	NOUN
ejpam-3427	83	12	×	×	NOUN
ejpam-3427	83	13	r2	r2	NOUN
ejpam-3427	83	14	is	be	AUX
ejpam-3427	83	15	to	to	PART
ejpam-3427	83	16	be	be	AUX
ejpam-3427	83	17	a	a	DET
ejpam-3427	83	18	fuzzy	fuzzy	ADJ
ejpam-3427	83	19	normal	normal	ADJ
ejpam-3427	83	20	la	la	NOUN
ejpam-3427	83	21	-	-	PUNCT
ejpam-3427	83	22	subring	subring	NOUN
ejpam-3427	83	23	of	of	ADP
ejpam-3427	83	24	r1	r1	PROPN
ejpam-3427	83	25	×	×	PROPN
ejpam-3427	83	26	r2	r2	NOUN
ejpam-3427	83	27	if	if	SCONJ
ejpam-3427	83	28	(	(	PUNCT
ejpam-3427	83	29	µ1	µ1	PROPN
ejpam-3427	83	30	×	×	PROPN
ejpam-3427	83	31	µ2)(xy	µ2)(xy	PROPN
ejpam-3427	83	32	)	)	PUNCT
ejpam-3427	83	33	=	=	PUNCT
ejpam-3427	84	1	(	(	PUNCT
ejpam-3427	84	2	µ1	µ1	PROPN
ejpam-3427	84	3	×	×	PROPN
ejpam-3427	84	4	µ2)(yx	µ2)(yx	PROPN
ejpam-3427	84	5	)	)	PUNCT
ejpam-3427	84	6	for	for	ADP
ejpam-3427	84	7	all	all	PRON
ejpam-3427	84	8	x	x	X
ejpam-3427	85	1	=	=	SYM
ejpam-3427	86	1	(	(	PUNCT
ejpam-3427	86	2	x1	x1	PROPN
ejpam-3427	86	3	,	,	PUNCT
ejpam-3427	86	4	x2	x2	PROPN
ejpam-3427	86	5	)	)	PUNCT
ejpam-3427	86	6	,	,	PUNCT
ejpam-3427	86	7	y	y	PROPN
ejpam-3427	86	8	=	=	SYM
ejpam-3427	86	9	(	(	PUNCT
ejpam-3427	86	10	y1	y1	INTJ
ejpam-3427	86	11	,	,	PUNCT
ejpam-3427	86	12	y2	y2	NOUN
ejpam-3427	86	13	)	)	PUNCT
ejpam-3427	86	14	∈	∈	PROPN
ejpam-3427	86	15	r1	r1	NOUN
ejpam-3427	86	16	×	×	NOUN
ejpam-3427	86	17	r2	r2	NOUN
ejpam-3427	86	18	.	.	PUNCT
ejpam-3427	87	1	similarly	similarly	ADV
ejpam-3427	87	2	for	for	ADP
ejpam-3427	87	3	anti	anti	X
ejpam-3427	87	4	fuzzy	fuzzy	ADJ
ejpam-3427	87	5	normal	normal	ADJ
ejpam-3427	87	6	la	la	NOUN
ejpam-3427	87	7	-	-	PUNCT
ejpam-3427	87	8	subring	subring	NOUN
ejpam-3427	87	9	.	.	PUNCT
ejpam-3427	88	1	let	let	VERB
ejpam-3427	88	2	a	a	PRON
ejpam-3427	88	3	and	and	CCONJ
ejpam-3427	88	4	b	b	NOUN
ejpam-3427	88	5	be	be	AUX
ejpam-3427	88	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	88	7	fuzzy	fuzzy	ADJ
ejpam-3427	88	8	sets	set	NOUN
ejpam-3427	88	9	of	of	ADP
ejpam-3427	88	10	la	la	NOUN
ejpam-3427	88	11	-	-	PUNCT
ejpam-3427	88	12	rings	ring	NOUN
ejpam-3427	88	13	r1	r1	NOUN
ejpam-3427	88	14	and	and	CCONJ
ejpam-3427	88	15	r2	r2	PROPN
ejpam-3427	88	16	,	,	PUNCT
ejpam-3427	88	17	respectively	respectively	ADV
ejpam-3427	88	18	.	.	PUNCT
ejpam-3427	89	1	the	the	DET
ejpam-3427	89	2	direct	direct	ADJ
ejpam-3427	89	3	product	product	NOUN
ejpam-3427	89	4	ofa	ofa	PROPN
ejpam-3427	89	5	andb	andb	NOUN
ejpam-3427	89	6	is	be	AUX
ejpam-3427	89	7	denoted	denote	VERB
ejpam-3427	89	8	bya×b	bya×b	ADV
ejpam-3427	89	9	and	and	CCONJ
ejpam-3427	89	10	defined	define	VERB
ejpam-3427	89	11	bya×b	bya×b	PUNCT
ejpam-3427	89	12	=	=	X
ejpam-3427	89	13	{	{	PUNCT
ejpam-3427	89	14	(	(	PUNCT
ejpam-3427	89	15	(	(	PUNCT
ejpam-3427	89	16	x	x	NOUN
ejpam-3427	89	17	,	,	PUNCT
ejpam-3427	89	18	y	y	PROPN
ejpam-3427	89	19	)	)	PUNCT
ejpam-3427	89	20	,	,	PUNCT
ejpam-3427	89	21	µa×b	µa×b	PROPN
ejpam-3427	89	22	(	(	PUNCT
ejpam-3427	89	23	x	x	NOUN
ejpam-3427	89	24	,	,	PUNCT
ejpam-3427	89	25	y	y	PROPN
ejpam-3427	89	26	)	)	PUNCT
ejpam-3427	89	27	,	,	PUNCT
ejpam-3427	89	28	γa×b	γa×b	NOUN
ejpam-3427	89	29	(	(	PUNCT
ejpam-3427	89	30	x	x	NOUN
ejpam-3427	89	31	,	,	PUNCT
ejpam-3427	89	32	y	y	NOUN
ejpam-3427	89	33	)	)	PUNCT
ejpam-3427	89	34	)	)	PUNCT
ejpam-3427	90	1	|	|	ADV
ejpam-3427	90	2	for	for	ADP
ejpam-3427	90	3	all	all	DET
ejpam-3427	90	4	x	x	SYM
ejpam-3427	90	5	∈	∈	PROPN
ejpam-3427	90	6	r1	r1	NOUN
ejpam-3427	90	7	and	and	CCONJ
ejpam-3427	90	8	y	y	PROPN
ejpam-3427	90	9	∈	∈	PROPN
ejpam-3427	90	10	r2	r2	PROPN
ejpam-3427	90	11	}	}	PUNCT
ejpam-3427	90	12	,	,	PUNCT
ejpam-3427	90	13	where	where	SCONJ
ejpam-3427	90	14	µa×b(x	µa×b(x	ADJ
ejpam-3427	90	15	,	,	PUNCT
ejpam-3427	90	16	y	y	NOUN
ejpam-3427	90	17	)	)	PUNCT
ejpam-3427	90	18	=	=	SYM
ejpam-3427	90	19	max{µa(x	max{µa(x	NOUN
ejpam-3427	90	20	)	)	PUNCT
ejpam-3427	90	21	,	,	PUNCT
ejpam-3427	90	22	µb(y	µb(y	NUM
ejpam-3427	90	23	)	)	PUNCT
ejpam-3427	90	24	}	}	PUNCT
ejpam-3427	90	25	and	and	CCONJ
ejpam-3427	90	26	γa×b(x	γa×b(x	ADJ
ejpam-3427	90	27	,	,	PUNCT
ejpam-3427	90	28	y	y	NOUN
ejpam-3427	90	29	)	)	PUNCT
ejpam-3427	90	30	=	=	SYM
ejpam-3427	90	31	min{γa(x	min{γa(x	NOUN
ejpam-3427	90	32	)	)	PUNCT
ejpam-3427	90	33	,	,	PUNCT
ejpam-3427	90	34	γb(y	γb(y	ADV
ejpam-3427	90	35	)	)	PUNCT
ejpam-3427	90	36	}	}	PUNCT
ejpam-3427	90	37	.	.	PUNCT
ejpam-3427	91	1	an	an	DET
ejpam-3427	91	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	91	3	fuzzy	fuzzy	ADJ
ejpam-3427	91	4	set	set	NOUN
ejpam-3427	91	5	(	(	PUNCT
ejpam-3427	91	6	ifs	ifs	PROPN
ejpam-3427	91	7	)	)	PUNCT
ejpam-3427	91	8	a×	a×	PROPN
ejpam-3427	92	1	b	b	X
ejpam-3427	92	2	=	=	PRON
ejpam-3427	92	3	(	(	PUNCT
ejpam-3427	92	4	µa×b	µa×b	PROPN
ejpam-3427	92	5	,	,	PUNCT
ejpam-3427	92	6	γa×b	γa×b	NOUN
ejpam-3427	92	7	)	)	PUNCT
ejpam-3427	92	8	of	of	ADP
ejpam-3427	92	9	an	an	DET
ejpam-3427	92	10	la	la	ADJ
ejpam-3427	92	11	-	-	PUNCT
ejpam-3427	92	12	ring	ring	NOUN
ejpam-3427	92	13	r1	r1	NOUN
ejpam-3427	92	14	×	×	NOUN
ejpam-3427	92	15	r2	r2	NOUN
ejpam-3427	92	16	is	be	AUX
ejpam-3427	92	17	an	an	DET
ejpam-3427	92	18	intuitionistic	intuitionistic	ADJ
ejpam-3427	92	19	anti	anti	ADJ
ejpam-3427	92	20	fuzzy	fuzzy	ADJ
ejpam-3427	92	21	la	la	NOUN
ejpam-3427	92	22	-	-	PUNCT
ejpam-3427	92	23	subring	subre	VERB
ejpam-3427	92	24	(	(	PUNCT
ejpam-3427	92	25	iaflsr	iaflsr	NOUN
ejpam-3427	92	26	)	)	PUNCT
ejpam-3427	92	27	of	of	ADP
ejpam-3427	92	28	r1	r1	PROPN
ejpam-3427	92	29	×r2	×r2	PROPN
ejpam-3427	92	30	if	if	SCONJ
ejpam-3427	92	31	1	1	NUM
ejpam-3427	92	32	.	.	PUNCT
ejpam-3427	92	33	µa×b(x−	µa×b(x−	PROPN
ejpam-3427	92	34	y	y	PROPN
ejpam-3427	92	35	)	)	PUNCT
ejpam-3427	92	36	≤	≤	PROPN
ejpam-3427	92	37	max{µa×b(x	max{µa×b(x	PROPN
ejpam-3427	92	38	)	)	PUNCT
ejpam-3427	92	39	,	,	PUNCT
ejpam-3427	92	40	µa×b(y	µa×b(y	ADJ
ejpam-3427	92	41	)	)	PUNCT
ejpam-3427	92	42	}	}	PUNCT
ejpam-3427	92	43	,	,	PUNCT
ejpam-3427	92	44	2	2	X
ejpam-3427	92	45	.	.	X
ejpam-3427	92	46	µa×b(xy	µa×b(xy	PROPN
ejpam-3427	92	47	)	)	PUNCT
ejpam-3427	92	48	≤	≤	PUNCT
ejpam-3427	92	49	max{µa×b(x	max{µa×b(x	PROPN
ejpam-3427	92	50	)	)	PUNCT
ejpam-3427	92	51	,	,	PUNCT
ejpam-3427	92	52	µa×b(y	µa×b(y	ADJ
ejpam-3427	92	53	)	)	PUNCT
ejpam-3427	92	54	}	}	PUNCT
ejpam-3427	92	55	,	,	PUNCT
ejpam-3427	92	56	3	3	X
ejpam-3427	92	57	.	.	X
ejpam-3427	92	58	γa×b(x−	γa×b(x−	PROPN
ejpam-3427	92	59	y	y	NOUN
ejpam-3427	92	60	)	)	PUNCT
ejpam-3427	92	61	≥	≥	X
ejpam-3427	92	62	min{γa×b(x	min{γa×b(x	PROPN
ejpam-3427	92	63	)	)	PUNCT
ejpam-3427	92	64	,	,	PUNCT
ejpam-3427	92	65	γa×b(y	γa×b(y	NOUN
ejpam-3427	92	66	)	)	PUNCT
ejpam-3427	92	67	}	}	PUNCT
ejpam-3427	92	68	,	,	PUNCT
ejpam-3427	92	69	4	4	X
ejpam-3427	92	70	.	.	PUNCT
ejpam-3427	93	1	γa×b(xy	γa×b(xy	PROPN
ejpam-3427	93	2	)	)	PUNCT
ejpam-3427	93	3	≥	≥	X
ejpam-3427	93	4	min{γa×b(x	min{γa×b(x	PROPN
ejpam-3427	93	5	)	)	PUNCT
ejpam-3427	93	6	,	,	PUNCT
ejpam-3427	93	7	γa×b(y	γa×b(y	NOUN
ejpam-3427	93	8	)	)	PUNCT
ejpam-3427	93	9	}	}	PUNCT
ejpam-3427	93	10	,	,	PUNCT
ejpam-3427	93	11	for	for	ADP
ejpam-3427	93	12	all	all	PRON
ejpam-3427	93	13	x	x	X
ejpam-3427	93	14	=	=	SYM
ejpam-3427	93	15	(	(	PUNCT
ejpam-3427	93	16	x1	x1	PROPN
ejpam-3427	93	17	,	,	PUNCT
ejpam-3427	93	18	x2	x2	PROPN
ejpam-3427	93	19	)	)	PUNCT
ejpam-3427	93	20	,	,	PUNCT
ejpam-3427	93	21	y	y	PROPN
ejpam-3427	93	22	=	=	SYM
ejpam-3427	93	23	(	(	PUNCT
ejpam-3427	93	24	y1	y1	INTJ
ejpam-3427	93	25	,	,	PUNCT
ejpam-3427	93	26	y2	y2	NOUN
ejpam-3427	93	27	)	)	PUNCT
ejpam-3427	93	28	∈	∈	PROPN
ejpam-3427	93	29	r1	r1	PROPN
ejpam-3427	93	30	×r2	×r2	PROPN
ejpam-3427	93	31	.	.	PUNCT
ejpam-3427	94	1	an	an	DET
ejpam-3427	94	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	94	3	anti	anti	ADJ
ejpam-3427	94	4	fuzzy	fuzzy	ADJ
ejpam-3427	94	5	la	la	NOUN
ejpam-3427	94	6	-	-	PUNCT
ejpam-3427	94	7	subring	subre	VERB
ejpam-3427	94	8	a×b	a×b	PROPN
ejpam-3427	94	9	=	=	SYM
ejpam-3427	94	10	(	(	PUNCT
ejpam-3427	94	11	µa×b	µa×b	PROPN
ejpam-3427	94	12	,	,	PUNCT
ejpam-3427	94	13	γa×b	γa×b	NOUN
ejpam-3427	94	14	)	)	PUNCT
ejpam-3427	94	15	of	of	ADP
ejpam-3427	94	16	an	an	DET
ejpam-3427	94	17	la	la	ADJ
ejpam-3427	94	18	-	-	PUNCT
ejpam-3427	94	19	ring	ring	NOUN
ejpam-3427	94	20	r1×r2	r1×r2	PROPN
ejpam-3427	94	21	is	be	AUX
ejpam-3427	94	22	an	an	DET
ejpam-3427	94	23	intuitionistic	intuitionistic	ADJ
ejpam-3427	94	24	anti	anti	ADJ
ejpam-3427	94	25	fuzzy	fuzzy	ADJ
ejpam-3427	94	26	normal	normal	ADJ
ejpam-3427	94	27	la	la	NOUN
ejpam-3427	94	28	-	-	PUNCT
ejpam-3427	94	29	subring	subre	VERB
ejpam-3427	94	30	(	(	PUNCT
ejpam-3427	94	31	iafnlsr	iafnlsr	NOUN
ejpam-3427	94	32	)	)	PUNCT
ejpam-3427	94	33	of	of	ADP
ejpam-3427	94	34	r1	r1	PROPN
ejpam-3427	94	35	×	×	PROPN
ejpam-3427	94	36	r2	r2	NOUN
ejpam-3427	94	37	if	if	SCONJ
ejpam-3427	94	38	µa×b(xy	µa×b(xy	PROPN
ejpam-3427	94	39	)	)	PUNCT
ejpam-3427	94	40	=	=	PUNCT
ejpam-3427	94	41	µa×b(yx	µa×b(yx	X
ejpam-3427	94	42	)	)	PUNCT
ejpam-3427	94	43	and	and	CCONJ
ejpam-3427	94	44	γa×b(xy	γa×b(xy	PROPN
ejpam-3427	94	45	)	)	PUNCT
ejpam-3427	94	46	=	=	SYM
ejpam-3427	94	47	γa×b(yx	γa×b(yx	PROPN
ejpam-3427	94	48	)	)	PUNCT
ejpam-3427	94	49	for	for	ADP
ejpam-3427	94	50	all	all	PRON
ejpam-3427	94	51	x	x	X
ejpam-3427	94	52	=	=	SYM
ejpam-3427	94	53	(	(	PUNCT
ejpam-3427	94	54	x1	x1	PROPN
ejpam-3427	94	55	,	,	PUNCT
ejpam-3427	94	56	x2	x2	PROPN
ejpam-3427	94	57	)	)	PUNCT
ejpam-3427	94	58	,	,	PUNCT
ejpam-3427	94	59	y	y	PROPN
ejpam-3427	94	60	=	=	SYM
ejpam-3427	94	61	(	(	PUNCT
ejpam-3427	94	62	y1	y1	INTJ
ejpam-3427	94	63	,	,	PUNCT
ejpam-3427	94	64	y2	y2	NOUN
ejpam-3427	94	65	)	)	PUNCT
ejpam-3427	94	66	∈	∈	PROPN
ejpam-3427	94	67	r1	r1	PROPN
ejpam-3427	94	68	×r2	×r2	PROPN
ejpam-3427	94	69	.	.	PUNCT
ejpam-3427	95	1	let	let	VERB
ejpam-3427	95	2	a	a	DET
ejpam-3427	95	3	×	×	PROPN
ejpam-3427	95	4	b	b	NOUN
ejpam-3427	95	5	be	be	AUX
ejpam-3427	95	6	a	a	DET
ejpam-3427	95	7	non	non	ADJ
ejpam-3427	95	8	-	-	ADJ
ejpam-3427	95	9	empty	empty	ADJ
ejpam-3427	95	10	subset	subset	NOUN
ejpam-3427	95	11	of	of	ADP
ejpam-3427	95	12	an	an	DET
ejpam-3427	95	13	la	la	ADJ
ejpam-3427	95	14	-	-	PUNCT
ejpam-3427	95	15	ring	ring	NOUN
ejpam-3427	95	16	r1	r1	NOUN
ejpam-3427	95	17	×	×	NOUN
ejpam-3427	95	18	r2	r2	NOUN
ejpam-3427	95	19	.	.	PUNCT
ejpam-3427	96	1	the	the	DET
ejpam-3427	96	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	96	3	anti	anti	ADJ
ejpam-3427	96	4	characteristic	characteristic	ADJ
ejpam-3427	96	5	function	function	NOUN
ejpam-3427	96	6	of	of	ADP
ejpam-3427	96	7	a×b	a×b	PROPN
ejpam-3427	96	8	is	be	AUX
ejpam-3427	96	9	denoted	denote	VERB
ejpam-3427	96	10	by	by	ADP
ejpam-3427	96	11	χa×b	χa×b	PROPN
ejpam-3427	96	12	=	=	SYM
ejpam-3427	96	13	〈	〈	PROPN
ejpam-3427	96	14	µχa×b	µχa×b	X
ejpam-3427	96	15	,	,	PUNCT
ejpam-3427	96	16	γχa×b	γχa×b	SYM
ejpam-3427	96	17	〉	〉	NUM
ejpam-3427	96	18	and	and	CCONJ
ejpam-3427	96	19	defined	define	VERB
ejpam-3427	96	20	by	by	ADP
ejpam-3427	96	21	µχa×b	µχa×b	PROPN
ejpam-3427	96	22	(	(	PUNCT
ejpam-3427	96	23	x	x	NOUN
ejpam-3427	96	24	)	)	PUNCT
ejpam-3427	97	1	=	=	PRON
ejpam-3427	97	2	{	{	PUNCT
ejpam-3427	97	3	0	0	NUM
ejpam-3427	97	4	if	if	SCONJ
ejpam-3427	97	5	x	x	SYM
ejpam-3427	97	6	∈	∈	PROPN
ejpam-3427	97	7	a×b	a×b	PROPN
ejpam-3427	97	8	1	1	NUM
ejpam-3427	97	9	if	if	SCONJ
ejpam-3427	97	10	x	x	X
ejpam-3427	97	11	/∈	/∈	PUNCT
ejpam-3427	97	12	a×b	a×b	PROPN
ejpam-3427	97	13	and	and	CCONJ
ejpam-3427	97	14	γχa×b	γχa×b	NUM
ejpam-3427	97	15	(	(	PUNCT
ejpam-3427	97	16	x	x	X
ejpam-3427	97	17	)	)	PUNCT
ejpam-3427	97	18	=	=	SYM
ejpam-3427	97	19	{	{	PUNCT
ejpam-3427	97	20	1	1	NUM
ejpam-3427	97	21	if	if	SCONJ
ejpam-3427	97	22	x	x	SYM
ejpam-3427	97	23	∈	∈	PROPN
ejpam-3427	97	24	a×b	a×b	PROPN
ejpam-3427	97	25	0	0	PUNCT
ejpam-3427	97	26	if	if	SCONJ
ejpam-3427	97	27	x	x	PROPN
ejpam-3427	97	28	/∈	/∈	PROPN
ejpam-3427	97	29	a×b	a×b	PROPN
ejpam-3427	97	30	k.	k.	PROPN
ejpam-3427	97	31	nasreen	nasreen	PROPN
ejpam-3427	97	32	/	/	SYM
ejpam-3427	97	33	eur	eur	PROPN
ejpam-3427	97	34	.	.	PUNCT
ejpam-3427	98	1	j.	j.	PROPN
ejpam-3427	98	2	pure	pure	PROPN
ejpam-3427	98	3	appl	appl	PROPN
ejpam-3427	98	4	.	.	PROPN
ejpam-3427	98	5	math	math	PROPN
ejpam-3427	98	6	,	,	PUNCT
ejpam-3427	98	7	12	12	NUM
ejpam-3427	98	8	(	(	PUNCT
ejpam-3427	98	9	2	2	NUM
ejpam-3427	98	10	)	)	PUNCT
ejpam-3427	98	11	(	(	PUNCT
ejpam-3427	98	12	2019	2019	NUM
ejpam-3427	98	13	)	)	PUNCT
ejpam-3427	98	14	,	,	PUNCT
ejpam-3427	98	15	622	622	NUM
ejpam-3427	98	16	-	-	SYM
ejpam-3427	98	17	648	648	NUM
ejpam-3427	98	18	625	625	NUM
ejpam-3427	98	19	lemma	lemma	PROPN
ejpam-3427	98	20	1	1	NUM
ejpam-3427	98	21	.	.	PUNCT
ejpam-3427	99	1	[	[	X
ejpam-3427	99	2	17	17	NUM
ejpam-3427	99	3	,	,	PUNCT
ejpam-3427	99	4	lemma	lemma	PROPN
ejpam-3427	99	5	4.2	4.2	NUM
ejpam-3427	99	6	]	]	PUNCT
ejpam-3427	99	7	if	if	SCONJ
ejpam-3427	99	8	a	a	PRON
ejpam-3427	99	9	and	and	CCONJ
ejpam-3427	99	10	b	b	NOUN
ejpam-3427	99	11	are	be	AUX
ejpam-3427	99	12	la	la	ADJ
ejpam-3427	99	13	-	-	PUNCT
ejpam-3427	99	14	subrings	subring	NOUN
ejpam-3427	99	15	of	of	ADP
ejpam-3427	99	16	la	la	NOUN
ejpam-3427	99	17	-	-	PUNCT
ejpam-3427	99	18	rings	ring	NOUN
ejpam-3427	99	19	r1	r1	NOUN
ejpam-3427	99	20	and	and	CCONJ
ejpam-3427	99	21	r2	r2	PROPN
ejpam-3427	99	22	,	,	PUNCT
ejpam-3427	99	23	respectively	respectively	ADV
ejpam-3427	99	24	,	,	PUNCT
ejpam-3427	99	25	then	then	ADV
ejpam-3427	99	26	a	a	DET
ejpam-3427	99	27	×	×	PROPN
ejpam-3427	99	28	b	b	PROPN
ejpam-3427	99	29	is	be	AUX
ejpam-3427	99	30	an	an	DET
ejpam-3427	99	31	la	la	NOUN
ejpam-3427	99	32	-	-	PUNCT
ejpam-3427	99	33	subring	subring	NOUN
ejpam-3427	99	34	of	of	ADP
ejpam-3427	99	35	an	an	DET
ejpam-3427	99	36	la	la	ADJ
ejpam-3427	99	37	-	-	PUNCT
ejpam-3427	99	38	ring	ring	NOUN
ejpam-3427	99	39	r1	r1	NOUN
ejpam-3427	99	40	×	×	NOUN
ejpam-3427	99	41	r2	r2	NOUN
ejpam-3427	99	42	under	under	ADP
ejpam-3427	99	43	the	the	DET
ejpam-3427	99	44	same	same	ADJ
ejpam-3427	99	45	operations	operation	NOUN
ejpam-3427	99	46	defined	define	VERB
ejpam-3427	99	47	as	as	ADP
ejpam-3427	99	48	in	in	ADP
ejpam-3427	99	49	r1	r1	PROPN
ejpam-3427	99	50	×r2	×r2	PROPN
ejpam-3427	99	51	.	.	PUNCT
ejpam-3427	100	1	proposition	proposition	NOUN
ejpam-3427	100	2	1	1	NUM
ejpam-3427	100	3	.	.	PUNCT
ejpam-3427	101	1	let	let	VERB
ejpam-3427	101	2	a	a	PRON
ejpam-3427	101	3	and	and	CCONJ
ejpam-3427	101	4	b	b	NOUN
ejpam-3427	101	5	be	be	AUX
ejpam-3427	101	6	la	la	ADJ
ejpam-3427	101	7	-	-	PUNCT
ejpam-3427	101	8	subrings	subring	NOUN
ejpam-3427	101	9	of	of	ADP
ejpam-3427	101	10	la	la	NOUN
ejpam-3427	101	11	-	-	PUNCT
ejpam-3427	101	12	rings	ring	NOUN
ejpam-3427	101	13	r1	r1	NOUN
ejpam-3427	101	14	and	and	CCONJ
ejpam-3427	101	15	r2	r2	PROPN
ejpam-3427	101	16	,	,	PUNCT
ejpam-3427	101	17	respectively	respectively	ADV
ejpam-3427	101	18	.	.	PUNCT
ejpam-3427	102	1	then	then	ADV
ejpam-3427	102	2	a	a	DET
ejpam-3427	102	3	×	×	PROPN
ejpam-3427	102	4	b	b	PROPN
ejpam-3427	102	5	is	be	AUX
ejpam-3427	102	6	an	an	DET
ejpam-3427	102	7	la	la	NOUN
ejpam-3427	102	8	-	-	PUNCT
ejpam-3427	102	9	subring	subring	NOUN
ejpam-3427	102	10	of	of	ADP
ejpam-3427	102	11	an	an	DET
ejpam-3427	102	12	la	la	ADJ
ejpam-3427	102	13	-	-	PUNCT
ejpam-3427	102	14	ring	ring	NOUN
ejpam-3427	102	15	r1	r1	NOUN
ejpam-3427	102	16	×	×	NOUN
ejpam-3427	102	17	r2	r2	NOUN
ejpam-3427	102	18	if	if	SCONJ
ejpam-3427	102	19	and	and	CCONJ
ejpam-3427	102	20	only	only	ADV
ejpam-3427	102	21	if	if	SCONJ
ejpam-3427	102	22	the	the	DET
ejpam-3427	102	23	intuitionistic	intuitionistic	ADJ
ejpam-3427	102	24	anti	anti	ADJ
ejpam-3427	102	25	characteristic	characteristic	ADJ
ejpam-3427	102	26	function	function	NOUN
ejpam-3427	102	27	χc	χc	PROPN
ejpam-3427	103	1	=	=	SYM
ejpam-3427	103	2	〈	〈	PROPN
ejpam-3427	103	3	µχc	µχc	NOUN
ejpam-3427	103	4	,	,	PUNCT
ejpam-3427	103	5	γχc	γχc	PROPN
ejpam-3427	103	6	〉	〉	PROPN
ejpam-3427	103	7	of	of	ADP
ejpam-3427	103	8	c	c	PROPN
ejpam-3427	103	9	=	=	PUNCT
ejpam-3427	103	10	a	a	DET
ejpam-3427	103	11	×	×	PROPN
ejpam-3427	103	12	b	b	PROPN
ejpam-3427	103	13	is	be	AUX
ejpam-3427	103	14	an	an	DET
ejpam-3427	103	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	103	16	anti	anti	ADJ
ejpam-3427	103	17	fuzzy	fuzzy	ADJ
ejpam-3427	103	18	normal	normal	ADJ
ejpam-3427	103	19	la	la	NOUN
ejpam-3427	103	20	-	-	PUNCT
ejpam-3427	103	21	subring	subring	NOUN
ejpam-3427	103	22	of	of	ADP
ejpam-3427	103	23	an	an	DET
ejpam-3427	103	24	la	la	ADJ
ejpam-3427	103	25	-	-	PUNCT
ejpam-3427	103	26	ring	ring	NOUN
ejpam-3427	103	27	r1	r1	PROPN
ejpam-3427	103	28	×r2	×r2	PROPN
ejpam-3427	103	29	.	.	PUNCT
ejpam-3427	104	1	proof	proof	NOUN
ejpam-3427	104	2	.	.	PUNCT
ejpam-3427	105	1	let	let	VERB
ejpam-3427	105	2	c	c	NOUN
ejpam-3427	105	3	=	=	PUNCT
ejpam-3427	105	4	a×b	a×b	AUX
ejpam-3427	105	5	be	be	AUX
ejpam-3427	105	6	an	an	DET
ejpam-3427	105	7	la	la	NOUN
ejpam-3427	105	8	-	-	PUNCT
ejpam-3427	105	9	subring	subring	NOUN
ejpam-3427	105	10	of	of	ADP
ejpam-3427	105	11	an	an	DET
ejpam-3427	105	12	la	la	ADJ
ejpam-3427	105	13	-	-	PUNCT
ejpam-3427	105	14	ring	ring	NOUN
ejpam-3427	105	15	r1	r1	NOUN
ejpam-3427	105	16	×r2	×r2	PROPN
ejpam-3427	105	17	and	and	CCONJ
ejpam-3427	105	18	a	a	DET
ejpam-3427	105	19	=	=	X
ejpam-3427	105	20	(	(	PUNCT
ejpam-3427	105	21	a1	a1	PROPN
ejpam-3427	105	22	,	,	PUNCT
ejpam-3427	105	23	a2	a2	PROPN
ejpam-3427	105	24	)	)	PUNCT
ejpam-3427	105	25	,	,	PUNCT
ejpam-3427	105	26	b	b	X
ejpam-3427	105	27	=	=	SYM
ejpam-3427	105	28	(	(	PUNCT
ejpam-3427	105	29	b1	b1	NOUN
ejpam-3427	105	30	,	,	PUNCT
ejpam-3427	105	31	b2	b2	NOUN
ejpam-3427	105	32	)	)	PUNCT
ejpam-3427	105	33	∈	∈	PROPN
ejpam-3427	105	34	r1×r2	r1×r2	PROPN
ejpam-3427	105	35	.	.	PUNCT
ejpam-3427	106	1	if	if	SCONJ
ejpam-3427	106	2	a	a	DET
ejpam-3427	106	3	,	,	PUNCT
ejpam-3427	106	4	b	b	X
ejpam-3427	106	5	∈	∈	PROPN
ejpam-3427	106	6	c	c	NOUN
ejpam-3427	106	7	=	=	PUNCT
ejpam-3427	106	8	a×b	a×b	PROPN
ejpam-3427	106	9	,	,	PUNCT
ejpam-3427	106	10	then	then	ADV
ejpam-3427	106	11	by	by	ADP
ejpam-3427	106	12	definition	definition	NOUN
ejpam-3427	106	13	of	of	ADP
ejpam-3427	106	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	106	15	anti	anti	ADJ
ejpam-3427	106	16	characteristic	characteristic	ADJ
ejpam-3427	106	17	function	function	NOUN
ejpam-3427	106	18	µχc	µχc	NOUN
ejpam-3427	106	19	(	(	PUNCT
ejpam-3427	106	20	a	a	X
ejpam-3427	106	21	)	)	PUNCT
ejpam-3427	106	22	=	=	SYM
ejpam-3427	106	23	0	0	PUNCT
ejpam-3427	107	1	=	=	X
ejpam-3427	107	2	µχc	µχc	NOUN
ejpam-3427	107	3	(	(	PUNCT
ejpam-3427	107	4	b	b	NOUN
ejpam-3427	107	5	)	)	PUNCT
ejpam-3427	107	6	and	and	CCONJ
ejpam-3427	107	7	γχc	γχc	PROPN
ejpam-3427	107	8	(	(	PUNCT
ejpam-3427	107	9	a	a	NOUN
ejpam-3427	107	10	)	)	PUNCT
ejpam-3427	107	11	=	=	SYM
ejpam-3427	107	12	1	1	NUM
ejpam-3427	107	13	=	=	SYM
ejpam-3427	107	14	γχc	γχc	X
ejpam-3427	107	15	(	(	PUNCT
ejpam-3427	107	16	b	b	NOUN
ejpam-3427	107	17	)	)	PUNCT
ejpam-3427	107	18	.	.	PUNCT
ejpam-3427	108	1	since	since	SCONJ
ejpam-3427	108	2	a−	a−	PROPN
ejpam-3427	108	3	b	b	PROPN
ejpam-3427	108	4	and	and	CCONJ
ejpam-3427	108	5	ab	ab	PROPN
ejpam-3427	108	6	∈	∈	PROPN
ejpam-3427	108	7	c	c	X
ejpam-3427	108	8	,	,	PUNCT
ejpam-3427	108	9	c	c	X
ejpam-3427	108	10	being	be	AUX
ejpam-3427	108	11	an	an	DET
ejpam-3427	108	12	la	la	ADV
ejpam-3427	108	13	-	-	PUNCT
ejpam-3427	108	14	subring	subring	NOUN
ejpam-3427	108	15	of	of	ADP
ejpam-3427	108	16	r1	r1	PROPN
ejpam-3427	108	17	×r2	×r2	PROPN
ejpam-3427	108	18	.	.	PUNCT
ejpam-3427	109	1	this	this	PRON
ejpam-3427	109	2	implies	imply	VERB
ejpam-3427	109	3	that	that	DET
ejpam-3427	109	4	µχc	µχc	NOUN
ejpam-3427	109	5	(	(	PUNCT
ejpam-3427	109	6	a−	a−	PROPN
ejpam-3427	109	7	b	b	NOUN
ejpam-3427	109	8	)	)	PUNCT
ejpam-3427	109	9	=	=	SYM
ejpam-3427	109	10	0	0	PUNCT
ejpam-3427	110	1	=	=	SYM
ejpam-3427	110	2	0	0	NUM
ejpam-3427	111	1	∨	∨	NUM
ejpam-3427	111	2	0	0	NUM
ejpam-3427	112	1	=	=	SYM
ejpam-3427	112	2	µχc	µχc	NOUN
ejpam-3427	112	3	(	(	PUNCT
ejpam-3427	112	4	a	a	NOUN
ejpam-3427	112	5	)	)	PUNCT
ejpam-3427	112	6	∨	∨	NUM
ejpam-3427	112	7	µχc	µχc	NOUN
ejpam-3427	112	8	(	(	PUNCT
ejpam-3427	112	9	b	b	NOUN
ejpam-3427	112	10	)	)	PUNCT
ejpam-3427	112	11	,	,	PUNCT
ejpam-3427	112	12	µχc	µχc	NOUN
ejpam-3427	112	13	(	(	PUNCT
ejpam-3427	112	14	ab	ab	NOUN
ejpam-3427	112	15	)	)	PUNCT
ejpam-3427	112	16	=	=	SYM
ejpam-3427	112	17	0	0	PUNCT
ejpam-3427	113	1	=	=	SYM
ejpam-3427	113	2	0	0	NUM
ejpam-3427	114	1	∨	∨	NUM
ejpam-3427	114	2	0	0	NUM
ejpam-3427	115	1	=	=	SYM
ejpam-3427	115	2	µχc	µχc	NOUN
ejpam-3427	115	3	(	(	PUNCT
ejpam-3427	115	4	a	a	NOUN
ejpam-3427	115	5	)	)	PUNCT
ejpam-3427	115	6	∨	∨	NUM
ejpam-3427	115	7	µχc	µχc	NOUN
ejpam-3427	115	8	(	(	PUNCT
ejpam-3427	115	9	b	b	NOUN
ejpam-3427	115	10	)	)	PUNCT
ejpam-3427	115	11	,	,	PUNCT
ejpam-3427	115	12	γχc	γχc	X
ejpam-3427	115	13	(	(	PUNCT
ejpam-3427	115	14	a−	a−	PROPN
ejpam-3427	115	15	b	b	NOUN
ejpam-3427	115	16	)	)	PUNCT
ejpam-3427	115	17	=	=	SYM
ejpam-3427	115	18	1	1	NUM
ejpam-3427	115	19	=	=	SYM
ejpam-3427	115	20	1	1	NUM
ejpam-3427	115	21	∧	∧	PROPN
ejpam-3427	115	22	1	1	NUM
ejpam-3427	115	23	=	=	SYM
ejpam-3427	115	24	γχc	γχc	X
ejpam-3427	115	25	(	(	PUNCT
ejpam-3427	115	26	a	a	NOUN
ejpam-3427	115	27	)	)	PUNCT
ejpam-3427	115	28	∧	∧	PROPN
ejpam-3427	115	29	γχc	γχc	X
ejpam-3427	115	30	(	(	PUNCT
ejpam-3427	115	31	b	b	NOUN
ejpam-3427	115	32	)	)	PUNCT
ejpam-3427	115	33	,	,	PUNCT
ejpam-3427	115	34	γχc	γχc	PROPN
ejpam-3427	115	35	(	(	PUNCT
ejpam-3427	115	36	ab	ab	PROPN
ejpam-3427	115	37	)	)	PUNCT
ejpam-3427	115	38	=	=	SYM
ejpam-3427	115	39	1	1	NUM
ejpam-3427	115	40	=	=	SYM
ejpam-3427	115	41	1	1	NUM
ejpam-3427	115	42	∧	∧	PROPN
ejpam-3427	115	43	1	1	NUM
ejpam-3427	115	44	=	=	SYM
ejpam-3427	115	45	γχc	γχc	X
ejpam-3427	115	46	(	(	PUNCT
ejpam-3427	115	47	a	a	NOUN
ejpam-3427	115	48	)	)	PUNCT
ejpam-3427	115	49	∧	∧	PROPN
ejpam-3427	115	50	γχc	γχc	X
ejpam-3427	115	51	(	(	PUNCT
ejpam-3427	115	52	b	b	NOUN
ejpam-3427	115	53	)	)	PUNCT
ejpam-3427	115	54	.	.	PUNCT
ejpam-3427	116	1	thus	thus	ADV
ejpam-3427	116	2	µχc	µχc	VERB
ejpam-3427	116	3	(	(	PUNCT
ejpam-3427	116	4	a−	a−	PROPN
ejpam-3427	116	5	b	b	NOUN
ejpam-3427	116	6	)	)	PUNCT
ejpam-3427	116	7	≤	≤	NOUN
ejpam-3427	116	8	max{µχc	max{µχc	NOUN
ejpam-3427	116	9	(	(	PUNCT
ejpam-3427	116	10	a	a	X
ejpam-3427	116	11	)	)	PUNCT
ejpam-3427	116	12	,	,	PUNCT
ejpam-3427	116	13	µχc	µχc	NOUN
ejpam-3427	116	14	(	(	PUNCT
ejpam-3427	116	15	b	b	NOUN
ejpam-3427	116	16	)	)	PUNCT
ejpam-3427	116	17	}	}	PUNCT
ejpam-3427	116	18	,	,	PUNCT
ejpam-3427	116	19	µχc	µχc	NOUN
ejpam-3427	116	20	(	(	PUNCT
ejpam-3427	116	21	ab	ab	NOUN
ejpam-3427	116	22	)	)	PUNCT
ejpam-3427	116	23	≤	≤	NOUN
ejpam-3427	116	24	max{µχc	max{µχc	NOUN
ejpam-3427	116	25	(	(	PUNCT
ejpam-3427	116	26	a	a	X
ejpam-3427	116	27	)	)	PUNCT
ejpam-3427	116	28	,	,	PUNCT
ejpam-3427	116	29	µχc	µχc	NOUN
ejpam-3427	116	30	(	(	PUNCT
ejpam-3427	116	31	b	b	NOUN
ejpam-3427	116	32	)	)	PUNCT
ejpam-3427	116	33	}	}	PUNCT
ejpam-3427	116	34	,	,	PUNCT
ejpam-3427	116	35	γχc	γχc	X
ejpam-3427	116	36	(	(	PUNCT
ejpam-3427	116	37	a−	a−	PROPN
ejpam-3427	116	38	b	b	PROPN
ejpam-3427	116	39	)	)	PUNCT
ejpam-3427	116	40	≥	≥	NOUN
ejpam-3427	116	41	min{γχc	min{γχc	NOUN
ejpam-3427	116	42	(	(	PUNCT
ejpam-3427	116	43	a	a	NOUN
ejpam-3427	116	44	)	)	PUNCT
ejpam-3427	116	45	,	,	PUNCT
ejpam-3427	116	46	γχc	γχc	X
ejpam-3427	116	47	(	(	PUNCT
ejpam-3427	116	48	b	b	NOUN
ejpam-3427	116	49	)	)	PUNCT
ejpam-3427	116	50	}	}	PUNCT
ejpam-3427	116	51	,	,	PUNCT
ejpam-3427	116	52	γχc	γχc	PROPN
ejpam-3427	116	53	(	(	PUNCT
ejpam-3427	116	54	ab	ab	PROPN
ejpam-3427	116	55	)	)	PUNCT
ejpam-3427	116	56	≥	≥	PROPN
ejpam-3427	116	57	min{γχc	min{γχc	NOUN
ejpam-3427	116	58	(	(	PUNCT
ejpam-3427	116	59	a	a	NOUN
ejpam-3427	116	60	)	)	PUNCT
ejpam-3427	116	61	,	,	PUNCT
ejpam-3427	116	62	γχc	γχc	X
ejpam-3427	116	63	(	(	PUNCT
ejpam-3427	116	64	b	b	NOUN
ejpam-3427	116	65	)	)	PUNCT
ejpam-3427	116	66	}	}	PUNCT
ejpam-3427	116	67	.	.	PUNCT
ejpam-3427	117	1	as	as	SCONJ
ejpam-3427	117	2	ab	ab	PROPN
ejpam-3427	117	3	and	and	CCONJ
ejpam-3427	117	4	ba	ba	PROPN
ejpam-3427	117	5	∈	∈	PROPN
ejpam-3427	117	6	c	c	AUX
ejpam-3427	117	7	,	,	PUNCT
ejpam-3427	117	8	by	by	ADP
ejpam-3427	117	9	definition	definition	NOUN
ejpam-3427	117	10	we	we	PRON
ejpam-3427	117	11	have	have	VERB
ejpam-3427	117	12	µχc	µχc	NOUN
ejpam-3427	117	13	(	(	PUNCT
ejpam-3427	117	14	ab	ab	NOUN
ejpam-3427	117	15	)	)	PUNCT
ejpam-3427	117	16	=	=	SYM
ejpam-3427	117	17	0	0	PUNCT
ejpam-3427	118	1	=	=	X
ejpam-3427	118	2	µχc	µχc	NOUN
ejpam-3427	118	3	(	(	PUNCT
ejpam-3427	118	4	ba	ba	NOUN
ejpam-3427	118	5	)	)	PUNCT
ejpam-3427	118	6	and	and	CCONJ
ejpam-3427	118	7	γχc	γχc	PROPN
ejpam-3427	118	8	(	(	PUNCT
ejpam-3427	118	9	ab	ab	PROPN
ejpam-3427	118	10	)	)	PUNCT
ejpam-3427	118	11	=	=	SYM
ejpam-3427	118	12	1	1	NUM
ejpam-3427	118	13	=	=	SYM
ejpam-3427	118	14	γχc	γχc	X
ejpam-3427	118	15	(	(	PUNCT
ejpam-3427	118	16	ba	ba	PROPN
ejpam-3427	118	17	)	)	PUNCT
ejpam-3427	118	18	,	,	PUNCT
ejpam-3427	118	19	i.e.	i.e.	X
ejpam-3427	118	20	,	,	PUNCT
ejpam-3427	118	21	µχc	µχc	NOUN
ejpam-3427	118	22	(	(	PUNCT
ejpam-3427	118	23	ab	ab	NOUN
ejpam-3427	118	24	)	)	PUNCT
ejpam-3427	118	25	=	=	SYM
ejpam-3427	118	26	µχc	µχc	NOUN
ejpam-3427	118	27	(	(	PUNCT
ejpam-3427	118	28	ba	ba	NOUN
ejpam-3427	118	29	)	)	PUNCT
ejpam-3427	118	30	and	and	CCONJ
ejpam-3427	118	31	γχc	γχc	PROPN
ejpam-3427	118	32	(	(	PUNCT
ejpam-3427	118	33	ab	ab	PROPN
ejpam-3427	118	34	)	)	PUNCT
ejpam-3427	118	35	=	=	NOUN
ejpam-3427	118	36	γχc	γχc	PROPN
ejpam-3427	118	37	(	(	PUNCT
ejpam-3427	118	38	ba	ba	PROPN
ejpam-3427	118	39	)	)	PUNCT
ejpam-3427	118	40	.	.	PUNCT
ejpam-3427	119	1	similarly	similarly	ADV
ejpam-3427	119	2	,	,	PUNCT
ejpam-3427	119	3	we	we	PRON
ejpam-3427	119	4	have	have	VERB
ejpam-3427	119	5	µχc	µχc	NOUN
ejpam-3427	119	6	(	(	PUNCT
ejpam-3427	119	7	a−	a−	PROPN
ejpam-3427	119	8	b	b	NOUN
ejpam-3427	119	9	)	)	PUNCT
ejpam-3427	119	10	≤	≤	NOUN
ejpam-3427	119	11	max{µχc	max{µχc	NOUN
ejpam-3427	119	12	(	(	PUNCT
ejpam-3427	119	13	a	a	X
ejpam-3427	119	14	)	)	PUNCT
ejpam-3427	119	15	,	,	PUNCT
ejpam-3427	119	16	µχc	µχc	NOUN
ejpam-3427	119	17	(	(	PUNCT
ejpam-3427	119	18	b	b	NOUN
ejpam-3427	119	19	)	)	PUNCT
ejpam-3427	119	20	}	}	PUNCT
ejpam-3427	119	21	,	,	PUNCT
ejpam-3427	119	22	µχc	µχc	NOUN
ejpam-3427	119	23	(	(	PUNCT
ejpam-3427	119	24	ab	ab	NOUN
ejpam-3427	119	25	)	)	PUNCT
ejpam-3427	119	26	≤	≤	NOUN
ejpam-3427	119	27	max{µχc	max{µχc	NOUN
ejpam-3427	119	28	(	(	PUNCT
ejpam-3427	119	29	a	a	X
ejpam-3427	119	30	)	)	PUNCT
ejpam-3427	119	31	,	,	PUNCT
ejpam-3427	119	32	µχc	µχc	NOUN
ejpam-3427	119	33	(	(	PUNCT
ejpam-3427	119	34	b	b	NOUN
ejpam-3427	119	35	)	)	PUNCT
ejpam-3427	119	36	}	}	PUNCT
ejpam-3427	119	37	,	,	PUNCT
ejpam-3427	119	38	γχc	γχc	X
ejpam-3427	119	39	(	(	PUNCT
ejpam-3427	119	40	a−	a−	PROPN
ejpam-3427	119	41	b	b	PROPN
ejpam-3427	119	42	)	)	PUNCT
ejpam-3427	119	43	≥	≥	NOUN
ejpam-3427	119	44	min{γχc	min{γχc	NOUN
ejpam-3427	119	45	(	(	PUNCT
ejpam-3427	119	46	a	a	NOUN
ejpam-3427	119	47	)	)	PUNCT
ejpam-3427	119	48	,	,	PUNCT
ejpam-3427	119	49	γχc	γχc	X
ejpam-3427	119	50	(	(	PUNCT
ejpam-3427	119	51	b	b	NOUN
ejpam-3427	119	52	)	)	PUNCT
ejpam-3427	119	53	}	}	PUNCT
ejpam-3427	119	54	,	,	PUNCT
ejpam-3427	119	55	γχc	γχc	PROPN
ejpam-3427	119	56	(	(	PUNCT
ejpam-3427	119	57	ab	ab	PROPN
ejpam-3427	119	58	)	)	PUNCT
ejpam-3427	119	59	≥	≥	PROPN
ejpam-3427	119	60	min{γχc	min{γχc	NOUN
ejpam-3427	119	61	(	(	PUNCT
ejpam-3427	119	62	a	a	NOUN
ejpam-3427	119	63	)	)	PUNCT
ejpam-3427	119	64	,	,	PUNCT
ejpam-3427	119	65	γχc	γχc	X
ejpam-3427	119	66	(	(	PUNCT
ejpam-3427	119	67	b	b	NOUN
ejpam-3427	119	68	)	)	PUNCT
ejpam-3427	119	69	}	}	PUNCT
ejpam-3427	119	70	,	,	PUNCT
ejpam-3427	119	71	γχc	γχc	PROPN
ejpam-3427	119	72	(	(	PUNCT
ejpam-3427	119	73	ab	ab	PROPN
ejpam-3427	119	74	)	)	PUNCT
ejpam-3427	120	1	=	=	NOUN
ejpam-3427	120	2	γχc	γχc	PROPN
ejpam-3427	120	3	(	(	PUNCT
ejpam-3427	120	4	ba	ba	PROPN
ejpam-3427	120	5	)	)	PUNCT
ejpam-3427	120	6	,	,	PUNCT
ejpam-3427	120	7	γχc	γχc	PROPN
ejpam-3427	120	8	(	(	PUNCT
ejpam-3427	120	9	ab	ab	PROPN
ejpam-3427	120	10	)	)	PUNCT
ejpam-3427	120	11	=	=	NOUN
ejpam-3427	120	12	γχc	γχc	PROPN
ejpam-3427	120	13	(	(	PUNCT
ejpam-3427	120	14	ba	ba	PROPN
ejpam-3427	120	15	)	)	PUNCT
ejpam-3427	120	16	,	,	PUNCT
ejpam-3427	120	17	when	when	SCONJ
ejpam-3427	120	18	a	a	DET
ejpam-3427	120	19	,	,	PUNCT
ejpam-3427	120	20	b	b	PROPN
ejpam-3427	120	21	/∈	/∈	PROPN
ejpam-3427	120	22	c.	c.	NOUN
ejpam-3427	120	23	hence	hence	ADV
ejpam-3427	120	24	the	the	DET
ejpam-3427	120	25	intuitionistic	intuitionistic	ADJ
ejpam-3427	120	26	anti	anti	ADJ
ejpam-3427	120	27	characteristic	characteristic	ADJ
ejpam-3427	120	28	function	function	NOUN
ejpam-3427	120	29	χc	χc	PROPN
ejpam-3427	121	1	=	=	SYM
ejpam-3427	122	1	〈	〈	PROPN
ejpam-3427	122	2	µχc	µχc	NOUN
ejpam-3427	122	3	,	,	PUNCT
ejpam-3427	122	4	γχc	γχc	PROPN
ejpam-3427	122	5	〉	〉	PROPN
ejpam-3427	122	6	of	of	ADP
ejpam-3427	122	7	c	c	PROPN
ejpam-3427	122	8	is	be	AUX
ejpam-3427	122	9	an	an	DET
ejpam-3427	122	10	intuitionistic	intuitionistic	ADJ
ejpam-3427	122	11	anti	anti	ADJ
ejpam-3427	122	12	fuzzy	fuzzy	ADJ
ejpam-3427	122	13	normal	normal	ADJ
ejpam-3427	122	14	la	la	NOUN
ejpam-3427	122	15	-	-	PUNCT
ejpam-3427	122	16	subring	subring	NOUN
ejpam-3427	122	17	of	of	ADP
ejpam-3427	122	18	an	an	DET
ejpam-3427	122	19	la	la	ADJ
ejpam-3427	122	20	-	-	PUNCT
ejpam-3427	122	21	ring	ring	NOUN
ejpam-3427	122	22	r1	r1	PROPN
ejpam-3427	122	23	×r2	×r2	PROPN
ejpam-3427	122	24	.	.	PUNCT
ejpam-3427	123	1	conversely	conversely	ADV
ejpam-3427	123	2	,	,	PUNCT
ejpam-3427	123	3	suppose	suppose	VERB
ejpam-3427	123	4	that	that	SCONJ
ejpam-3427	123	5	the	the	DET
ejpam-3427	123	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	123	7	anti	anti	ADJ
ejpam-3427	123	8	characteristic	characteristic	ADJ
ejpam-3427	123	9	function	function	NOUN
ejpam-3427	123	10	χc	χc	PROPN
ejpam-3427	123	11	=	=	SYM
ejpam-3427	123	12	〈	〈	PROPN
ejpam-3427	123	13	µχc	µχc	NOUN
ejpam-3427	123	14	,	,	PUNCT
ejpam-3427	123	15	γχc	γχc	PROPN
ejpam-3427	123	16	〉	〉	PROPN
ejpam-3427	123	17	of	of	ADP
ejpam-3427	123	18	c	c	PROPN
ejpam-3427	123	19	=	=	PUNCT
ejpam-3427	123	20	a	a	DET
ejpam-3427	123	21	×	×	PROPN
ejpam-3427	123	22	b	b	PROPN
ejpam-3427	123	23	is	be	AUX
ejpam-3427	123	24	an	an	DET
ejpam-3427	123	25	intuitionistic	intuitionistic	ADJ
ejpam-3427	123	26	anti	anti	ADJ
ejpam-3427	123	27	fuzzy	fuzzy	ADJ
ejpam-3427	123	28	normal	normal	ADJ
ejpam-3427	123	29	la	la	NOUN
ejpam-3427	123	30	-	-	PUNCT
ejpam-3427	123	31	subring	subring	NOUN
ejpam-3427	123	32	of	of	ADP
ejpam-3427	123	33	an	an	DET
ejpam-3427	123	34	la	la	ADJ
ejpam-3427	123	35	-	-	PUNCT
ejpam-3427	123	36	ring	ring	NOUN
ejpam-3427	123	37	r1	r1	NOUN
ejpam-3427	123	38	×	×	NOUN
ejpam-3427	123	39	r2	r2	NOUN
ejpam-3427	123	40	.	.	PUNCT
ejpam-3427	124	1	let	let	VERB
ejpam-3427	124	2	a	a	DET
ejpam-3427	124	3	,	,	PUNCT
ejpam-3427	124	4	b	b	X
ejpam-3427	124	5	∈	∈	PROPN
ejpam-3427	124	6	c	c	NOUN
ejpam-3427	124	7	=	=	PUNCT
ejpam-3427	124	8	a×b	a×b	PROPN
ejpam-3427	124	9	,	,	PUNCT
ejpam-3427	124	10	then	then	ADV
ejpam-3427	124	11	by	by	ADP
ejpam-3427	124	12	definition	definition	NOUN
ejpam-3427	124	13	,	,	PUNCT
ejpam-3427	124	14	we	we	PRON
ejpam-3427	124	15	have	have	AUX
ejpam-3427	124	16	µχc	µχc	NOUN
ejpam-3427	124	17	(	(	PUNCT
ejpam-3427	124	18	a	a	X
ejpam-3427	124	19	)	)	PUNCT
ejpam-3427	124	20	=	=	SYM
ejpam-3427	124	21	0	0	PUNCT
ejpam-3427	125	1	=	=	X
ejpam-3427	125	2	µχc	µχc	NOUN
ejpam-3427	125	3	(	(	PUNCT
ejpam-3427	125	4	b	b	NOUN
ejpam-3427	125	5	)	)	PUNCT
ejpam-3427	125	6	and	and	CCONJ
ejpam-3427	125	7	γχc	γχc	PROPN
ejpam-3427	125	8	(	(	PUNCT
ejpam-3427	125	9	a	a	NOUN
ejpam-3427	125	10	)	)	PUNCT
ejpam-3427	125	11	=	=	SYM
ejpam-3427	125	12	1	1	NUM
ejpam-3427	125	13	=	=	SYM
ejpam-3427	125	14	γχc	γχc	X
ejpam-3427	125	15	(	(	PUNCT
ejpam-3427	125	16	b	b	NOUN
ejpam-3427	125	17	)	)	PUNCT
ejpam-3427	125	18	.	.	PUNCT
ejpam-3427	126	1	by	by	ADP
ejpam-3427	126	2	our	our	PRON
ejpam-3427	126	3	supposition	supposition	NOUN
ejpam-3427	126	4	µχc	µχc	NOUN
ejpam-3427	126	5	(	(	PUNCT
ejpam-3427	126	6	a−	a−	PROPN
ejpam-3427	126	7	b	b	NOUN
ejpam-3427	126	8	)	)	PUNCT
ejpam-3427	126	9	≤	≤	NUM
ejpam-3427	126	10	µχc	µχc	NOUN
ejpam-3427	126	11	(	(	PUNCT
ejpam-3427	126	12	a	a	X
ejpam-3427	126	13	)	)	PUNCT
ejpam-3427	126	14	∨	∨	NUM
ejpam-3427	126	15	µχc	µχc	NOUN
ejpam-3427	126	16	(	(	PUNCT
ejpam-3427	126	17	b	b	NOUN
ejpam-3427	126	18	)	)	PUNCT
ejpam-3427	126	19	=	=	SYM
ejpam-3427	126	20	0	0	NUM
ejpam-3427	126	21	∨	∨	NUM
ejpam-3427	126	22	0	0	NUM
ejpam-3427	127	1	=	=	SYM
ejpam-3427	127	2	0	0	NUM
ejpam-3427	127	3	,	,	PUNCT
ejpam-3427	127	4	µχc	µχc	NOUN
ejpam-3427	127	5	(	(	PUNCT
ejpam-3427	127	6	ab	ab	NOUN
ejpam-3427	127	7	)	)	PUNCT
ejpam-3427	127	8	≤	≤	NOUN
ejpam-3427	127	9	µχc	µχc	NOUN
ejpam-3427	127	10	(	(	PUNCT
ejpam-3427	127	11	a	a	X
ejpam-3427	127	12	)	)	PUNCT
ejpam-3427	127	13	∨	∨	NUM
ejpam-3427	127	14	µχc	µχc	NOUN
ejpam-3427	127	15	(	(	PUNCT
ejpam-3427	127	16	b	b	NOUN
ejpam-3427	127	17	)	)	PUNCT
ejpam-3427	127	18	=	=	SYM
ejpam-3427	127	19	0	0	NUM
ejpam-3427	127	20	∨	∨	NUM
ejpam-3427	127	21	0	0	NUM
ejpam-3427	128	1	=	=	SYM
ejpam-3427	128	2	0	0	PROPN
ejpam-3427	128	3	,	,	PUNCT
ejpam-3427	128	4	γχc	γχc	X
ejpam-3427	128	5	(	(	PUNCT
ejpam-3427	128	6	a−	a−	PROPN
ejpam-3427	128	7	b	b	PROPN
ejpam-3427	128	8	)	)	PUNCT
ejpam-3427	128	9	≥	≥	PROPN
ejpam-3427	128	10	γχc	γχc	NOUN
ejpam-3427	128	11	(	(	PUNCT
ejpam-3427	128	12	a	a	X
ejpam-3427	128	13	)	)	PUNCT
ejpam-3427	128	14	∧	∧	PROPN
ejpam-3427	128	15	γχc	γχc	X
ejpam-3427	128	16	(	(	PUNCT
ejpam-3427	128	17	b	b	NOUN
ejpam-3427	128	18	)	)	PUNCT
ejpam-3427	128	19	=	=	SYM
ejpam-3427	128	20	1	1	NUM
ejpam-3427	128	21	∧	∧	PROPN
ejpam-3427	128	22	1	1	NUM
ejpam-3427	128	23	=	=	SYM
ejpam-3427	128	24	1	1	NUM
ejpam-3427	128	25	,	,	PUNCT
ejpam-3427	128	26	k.	k.	PROPN
ejpam-3427	128	27	nasreen	nasreen	PROPN
ejpam-3427	128	28	/	/	SYM
ejpam-3427	128	29	eur	eur	PROPN
ejpam-3427	128	30	.	.	PUNCT
ejpam-3427	129	1	j.	j.	PROPN
ejpam-3427	129	2	pure	pure	PROPN
ejpam-3427	129	3	appl	appl	PROPN
ejpam-3427	129	4	.	.	PROPN
ejpam-3427	129	5	math	math	PROPN
ejpam-3427	129	6	,	,	PUNCT
ejpam-3427	129	7	12	12	NUM
ejpam-3427	129	8	(	(	PUNCT
ejpam-3427	129	9	2	2	NUM
ejpam-3427	129	10	)	)	PUNCT
ejpam-3427	129	11	(	(	PUNCT
ejpam-3427	129	12	2019	2019	NUM
ejpam-3427	129	13	)	)	PUNCT
ejpam-3427	129	14	,	,	PUNCT
ejpam-3427	129	15	622	622	NUM
ejpam-3427	129	16	-	-	SYM
ejpam-3427	129	17	648	648	NUM
ejpam-3427	129	18	626	626	NUM
ejpam-3427	129	19	γχc	γχc	NOUN
ejpam-3427	129	20	(	(	PUNCT
ejpam-3427	129	21	ab	ab	PROPN
ejpam-3427	129	22	)	)	PUNCT
ejpam-3427	129	23	≥	≥	PROPN
ejpam-3427	129	24	γχc	γχc	NOUN
ejpam-3427	129	25	(	(	PUNCT
ejpam-3427	129	26	a	a	X
ejpam-3427	129	27	)	)	PUNCT
ejpam-3427	129	28	∧	∧	PROPN
ejpam-3427	129	29	γχc	γχc	X
ejpam-3427	129	30	(	(	PUNCT
ejpam-3427	129	31	b	b	NOUN
ejpam-3427	129	32	)	)	PUNCT
ejpam-3427	129	33	=	=	SYM
ejpam-3427	129	34	1	1	NUM
ejpam-3427	129	35	∧	∧	PROPN
ejpam-3427	129	36	1	1	NUM
ejpam-3427	129	37	=	=	SYM
ejpam-3427	129	38	1	1	NUM
ejpam-3427	129	39	.	.	PUNCT
ejpam-3427	129	40	thus	thus	ADV
ejpam-3427	129	41	µχc	µχc	VERB
ejpam-3427	129	42	(	(	PUNCT
ejpam-3427	129	43	a−	a−	PROPN
ejpam-3427	129	44	b	b	NOUN
ejpam-3427	129	45	)	)	PUNCT
ejpam-3427	129	46	=	=	SYM
ejpam-3427	129	47	0	0	PUNCT
ejpam-3427	130	1	=	=	X
ejpam-3427	130	2	µχc	µχc	NOUN
ejpam-3427	130	3	(	(	PUNCT
ejpam-3427	130	4	ab	ab	NOUN
ejpam-3427	130	5	)	)	PUNCT
ejpam-3427	130	6	and	and	CCONJ
ejpam-3427	130	7	γχc	γχc	PROPN
ejpam-3427	130	8	(	(	PUNCT
ejpam-3427	130	9	a−	a−	PROPN
ejpam-3427	130	10	b	b	NOUN
ejpam-3427	130	11	)	)	PUNCT
ejpam-3427	130	12	=	=	SYM
ejpam-3427	130	13	1	1	NUM
ejpam-3427	130	14	=	=	SYM
ejpam-3427	130	15	γχc	γχc	X
ejpam-3427	130	16	(	(	PUNCT
ejpam-3427	130	17	ab	ab	PROPN
ejpam-3427	130	18	)	)	PUNCT
ejpam-3427	130	19	,	,	PUNCT
ejpam-3427	130	20	i.e.	i.e.	X
ejpam-3427	130	21	,	,	PUNCT
ejpam-3427	130	22	a−	a−	PROPN
ejpam-3427	130	23	b	b	PROPN
ejpam-3427	130	24	and	and	CCONJ
ejpam-3427	130	25	ab	ab	PROPN
ejpam-3427	130	26	∈	∈	PROPN
ejpam-3427	130	27	c.	c.	PROPN
ejpam-3427	131	1	hence	hence	ADV
ejpam-3427	131	2	c	c	PROPN
ejpam-3427	131	3	is	be	AUX
ejpam-3427	131	4	an	an	DET
ejpam-3427	131	5	la	la	NOUN
ejpam-3427	131	6	-	-	PUNCT
ejpam-3427	131	7	subring	subring	NOUN
ejpam-3427	131	8	of	of	ADP
ejpam-3427	131	9	an	an	DET
ejpam-3427	131	10	la	la	ADJ
ejpam-3427	131	11	-	-	PUNCT
ejpam-3427	131	12	ring	ring	NOUN
ejpam-3427	131	13	r1	r1	PROPN
ejpam-3427	131	14	×r2	×r2	PROPN
ejpam-3427	131	15	.	.	PUNCT
ejpam-3427	132	1	lemma	lemma	PROPN
ejpam-3427	132	2	2	2	NUM
ejpam-3427	132	3	.	.	PUNCT
ejpam-3427	133	1	if	if	SCONJ
ejpam-3427	133	2	x	x	X
ejpam-3427	133	3	=	=	SYM
ejpam-3427	133	4	a×	a×	PROPN
ejpam-3427	133	5	b	b	PROPN
ejpam-3427	133	6	and	and	CCONJ
ejpam-3427	133	7	y	y	PROPN
ejpam-3427	134	1	=	=	SYM
ejpam-3427	134	2	c	c	PROPN
ejpam-3427	134	3	×d	×d	NOUN
ejpam-3427	134	4	are	be	AUX
ejpam-3427	134	5	two	two	NUM
ejpam-3427	134	6	la	la	ADJ
ejpam-3427	134	7	-	-	PUNCT
ejpam-3427	134	8	subrings	subring	NOUN
ejpam-3427	134	9	of	of	ADP
ejpam-3427	134	10	an	an	DET
ejpam-3427	134	11	la	la	ADJ
ejpam-3427	134	12	-	-	PUNCT
ejpam-3427	134	13	ring	ring	NOUN
ejpam-3427	134	14	r1	r1	NOUN
ejpam-3427	134	15	×	×	NOUN
ejpam-3427	134	16	r2	r2	NOUN
ejpam-3427	134	17	,	,	PUNCT
ejpam-3427	134	18	then	then	ADV
ejpam-3427	134	19	their	their	PRON
ejpam-3427	134	20	intersection	intersection	NOUN
ejpam-3427	134	21	x	x	PUNCT
ejpam-3427	134	22	∩	∩	NOUN
ejpam-3427	134	23	y	y	PROPN
ejpam-3427	134	24	is	be	AUX
ejpam-3427	134	25	also	also	ADV
ejpam-3427	134	26	an	an	DET
ejpam-3427	134	27	la	la	ADV
ejpam-3427	134	28	-	-	PUNCT
ejpam-3427	134	29	subring	subring	NOUN
ejpam-3427	134	30	of	of	ADP
ejpam-3427	134	31	an	an	DET
ejpam-3427	134	32	la	la	ADJ
ejpam-3427	134	33	-	-	PUNCT
ejpam-3427	134	34	ring	ring	NOUN
ejpam-3427	134	35	r1	r1	PROPN
ejpam-3427	134	36	×r2	×r2	PROPN
ejpam-3427	134	37	.	.	PUNCT
ejpam-3427	135	1	proof	proof	NOUN
ejpam-3427	135	2	.	.	PUNCT
ejpam-3427	136	1	straight	straight	ADV
ejpam-3427	136	2	forward	forward	ADV
ejpam-3427	136	3	.	.	PUNCT
ejpam-3427	137	1	theorem	theorem	NOUN
ejpam-3427	137	2	1	1	NUM
ejpam-3427	137	3	.	.	PUNCT
ejpam-3427	138	1	let	let	VERB
ejpam-3427	138	2	x	x	PUNCT
ejpam-3427	138	3	=	=	PUNCT
ejpam-3427	138	4	a×b	a×b	PROPN
ejpam-3427	138	5	and	and	CCONJ
ejpam-3427	138	6	y	y	PROPN
ejpam-3427	138	7	=	=	SYM
ejpam-3427	138	8	c×d	c×d	PROPN
ejpam-3427	138	9	be	be	AUX
ejpam-3427	138	10	two	two	NUM
ejpam-3427	138	11	la	la	ADJ
ejpam-3427	138	12	-	-	PUNCT
ejpam-3427	138	13	subrings	subring	NOUN
ejpam-3427	138	14	of	of	ADP
ejpam-3427	138	15	an	an	DET
ejpam-3427	138	16	la	la	ADJ
ejpam-3427	138	17	-	-	PUNCT
ejpam-3427	138	18	ring	ring	NOUN
ejpam-3427	138	19	r1×r2	r1×r2	PROPN
ejpam-3427	138	20	.	.	PUNCT
ejpam-3427	139	1	then	then	ADV
ejpam-3427	139	2	x	x	X
ejpam-3427	139	3	∩	∩	PROPN
ejpam-3427	139	4	y	y	PROPN
ejpam-3427	139	5	is	be	AUX
ejpam-3427	139	6	an	an	DET
ejpam-3427	139	7	la	la	NOUN
ejpam-3427	139	8	-	-	PUNCT
ejpam-3427	139	9	subring	subring	NOUN
ejpam-3427	139	10	of	of	ADP
ejpam-3427	139	11	an	an	DET
ejpam-3427	139	12	la	la	ADJ
ejpam-3427	139	13	-	-	PUNCT
ejpam-3427	139	14	ring	ring	NOUN
ejpam-3427	139	15	r1	r1	NOUN
ejpam-3427	139	16	×	×	NOUN
ejpam-3427	139	17	r2	r2	NOUN
ejpam-3427	139	18	if	if	SCONJ
ejpam-3427	139	19	and	and	CCONJ
ejpam-3427	139	20	only	only	ADV
ejpam-3427	139	21	if	if	SCONJ
ejpam-3427	139	22	the	the	DET
ejpam-3427	139	23	intuitionistic	intuitionistic	ADJ
ejpam-3427	139	24	anti	anti	ADJ
ejpam-3427	139	25	characteristic	characteristic	ADJ
ejpam-3427	139	26	function	function	NOUN
ejpam-3427	139	27	χz	χz	PROPN
ejpam-3427	139	28	=	=	SYM
ejpam-3427	139	29	〈	〈	PROPN
ejpam-3427	139	30	µχz	µχz	NOUN
ejpam-3427	139	31	,	,	PUNCT
ejpam-3427	139	32	γχz	γχz	INTJ
ejpam-3427	139	33	〉	〉	NOUN
ejpam-3427	139	34	of	of	ADP
ejpam-3427	139	35	z	z	NOUN
ejpam-3427	139	36	=	=	SYM
ejpam-3427	140	1	x	x	NOUN
ejpam-3427	140	2	∩	∩	NOUN
ejpam-3427	140	3	y	y	PROPN
ejpam-3427	140	4	is	be	AUX
ejpam-3427	140	5	an	an	DET
ejpam-3427	140	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	140	7	anti	anti	ADJ
ejpam-3427	140	8	fuzzy	fuzzy	ADJ
ejpam-3427	140	9	normal	normal	ADJ
ejpam-3427	140	10	la	la	NOUN
ejpam-3427	140	11	-	-	PUNCT
ejpam-3427	140	12	subring	subring	NOUN
ejpam-3427	140	13	of	of	ADP
ejpam-3427	140	14	an	an	DET
ejpam-3427	140	15	la	la	ADJ
ejpam-3427	140	16	-	-	PUNCT
ejpam-3427	140	17	ring	ring	NOUN
ejpam-3427	140	18	r1	r1	PROPN
ejpam-3427	140	19	×r2	×r2	PROPN
ejpam-3427	140	20	.	.	PUNCT
ejpam-3427	141	1	proof	proof	NOUN
ejpam-3427	141	2	.	.	PUNCT
ejpam-3427	142	1	let	let	VERB
ejpam-3427	142	2	z	z	NOUN
ejpam-3427	142	3	=	=	SYM
ejpam-3427	142	4	x	x	NOUN
ejpam-3427	142	5	∩	∩	PROPN
ejpam-3427	142	6	y	y	PRON
ejpam-3427	142	7	be	be	AUX
ejpam-3427	142	8	an	an	DET
ejpam-3427	142	9	la	la	NOUN
ejpam-3427	142	10	-	-	PUNCT
ejpam-3427	142	11	subring	subring	NOUN
ejpam-3427	142	12	of	of	ADP
ejpam-3427	142	13	an	an	DET
ejpam-3427	142	14	la	la	ADJ
ejpam-3427	142	15	-	-	PUNCT
ejpam-3427	142	16	ring	ring	NOUN
ejpam-3427	142	17	r1	r1	NOUN
ejpam-3427	142	18	×r2	×r2	PROPN
ejpam-3427	142	19	and	and	CCONJ
ejpam-3427	142	20	a	a	DET
ejpam-3427	142	21	=	=	X
ejpam-3427	142	22	(	(	PUNCT
ejpam-3427	142	23	a1	a1	PROPN
ejpam-3427	142	24	,	,	PUNCT
ejpam-3427	142	25	a2	a2	PROPN
ejpam-3427	142	26	)	)	PUNCT
ejpam-3427	142	27	,	,	PUNCT
ejpam-3427	142	28	b	b	X
ejpam-3427	142	29	=	=	SYM
ejpam-3427	142	30	(	(	PUNCT
ejpam-3427	142	31	b1	b1	NOUN
ejpam-3427	142	32	,	,	PUNCT
ejpam-3427	142	33	b2	b2	NOUN
ejpam-3427	142	34	)	)	PUNCT
ejpam-3427	142	35	∈	∈	PROPN
ejpam-3427	142	36	r1×r2	r1×r2	PROPN
ejpam-3427	142	37	.	.	PUNCT
ejpam-3427	143	1	if	if	SCONJ
ejpam-3427	143	2	a	a	DET
ejpam-3427	143	3	,	,	PUNCT
ejpam-3427	143	4	b	b	PROPN
ejpam-3427	143	5	∈	∈	PROPN
ejpam-3427	143	6	z	z	NOUN
ejpam-3427	143	7	=	=	SYM
ejpam-3427	143	8	x∩y	x∩y	NOUN
ejpam-3427	143	9	,	,	PUNCT
ejpam-3427	143	10	then	then	ADV
ejpam-3427	143	11	by	by	ADP
ejpam-3427	143	12	definition	definition	NOUN
ejpam-3427	143	13	of	of	ADP
ejpam-3427	143	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	143	15	anti	anti	ADJ
ejpam-3427	143	16	characteristic	characteristic	ADJ
ejpam-3427	143	17	function	function	NOUN
ejpam-3427	143	18	µχz	µχz	NOUN
ejpam-3427	143	19	(	(	PUNCT
ejpam-3427	143	20	a	a	X
ejpam-3427	143	21	)	)	PUNCT
ejpam-3427	143	22	=	=	SYM
ejpam-3427	143	23	0	0	PUNCT
ejpam-3427	144	1	=	=	SYM
ejpam-3427	144	2	µχz	µχz	NOUN
ejpam-3427	144	3	(	(	PUNCT
ejpam-3427	144	4	b	b	NOUN
ejpam-3427	144	5	)	)	PUNCT
ejpam-3427	144	6	and	and	CCONJ
ejpam-3427	144	7	γχz	γχz	INTJ
ejpam-3427	144	8	(	(	PUNCT
ejpam-3427	144	9	a	a	X
ejpam-3427	144	10	)	)	PUNCT
ejpam-3427	144	11	=	=	SYM
ejpam-3427	144	12	1	1	NUM
ejpam-3427	144	13	=	=	SYM
ejpam-3427	144	14	γχz	γχz	NOUN
ejpam-3427	144	15	(	(	PUNCT
ejpam-3427	144	16	b	b	NOUN
ejpam-3427	144	17	)	)	PUNCT
ejpam-3427	144	18	.	.	PUNCT
ejpam-3427	145	1	since	since	SCONJ
ejpam-3427	145	2	a−	a−	PROPN
ejpam-3427	145	3	b	b	PROPN
ejpam-3427	145	4	and	and	CCONJ
ejpam-3427	145	5	ab	ab	PROPN
ejpam-3427	145	6	∈	∈	PROPN
ejpam-3427	145	7	z	z	PROPN
ejpam-3427	145	8	,	,	PUNCT
ejpam-3427	145	9	z	z	NOUN
ejpam-3427	145	10	being	be	AUX
ejpam-3427	145	11	an	an	DET
ejpam-3427	145	12	la	la	ADV
ejpam-3427	145	13	-	-	PUNCT
ejpam-3427	145	14	subring	subring	NOUN
ejpam-3427	145	15	of	of	ADP
ejpam-3427	145	16	an	an	DET
ejpam-3427	145	17	la	la	ADJ
ejpam-3427	145	18	-	-	PUNCT
ejpam-3427	145	19	ring	ring	NOUN
ejpam-3427	145	20	r1	r1	PROPN
ejpam-3427	145	21	×r2	×r2	PROPN
ejpam-3427	145	22	.	.	PUNCT
ejpam-3427	146	1	this	this	PRON
ejpam-3427	146	2	means	mean	VERB
ejpam-3427	146	3	that	that	SCONJ
ejpam-3427	146	4	µχz	µχz	NOUN
ejpam-3427	146	5	(	(	PUNCT
ejpam-3427	146	6	a−	a−	PROPN
ejpam-3427	146	7	b	b	NOUN
ejpam-3427	146	8	)	)	PUNCT
ejpam-3427	146	9	=	=	SYM
ejpam-3427	146	10	0	0	PUNCT
ejpam-3427	147	1	=	=	SYM
ejpam-3427	147	2	0	0	NUM
ejpam-3427	148	1	∨	∨	NUM
ejpam-3427	148	2	0	0	NUM
ejpam-3427	149	1	=	=	SYM
ejpam-3427	149	2	µχz	µχz	NOUN
ejpam-3427	149	3	(	(	PUNCT
ejpam-3427	149	4	a	a	X
ejpam-3427	149	5	)	)	PUNCT
ejpam-3427	149	6	∨	∨	NUM
ejpam-3427	149	7	µχz	µχz	NOUN
ejpam-3427	149	8	(	(	PUNCT
ejpam-3427	149	9	b	b	NOUN
ejpam-3427	149	10	)	)	PUNCT
ejpam-3427	149	11	,	,	PUNCT
ejpam-3427	149	12	µχz	µχz	NOUN
ejpam-3427	149	13	(	(	PUNCT
ejpam-3427	149	14	ab	ab	NOUN
ejpam-3427	149	15	)	)	PUNCT
ejpam-3427	149	16	=	=	SYM
ejpam-3427	149	17	0	0	PUNCT
ejpam-3427	150	1	=	=	SYM
ejpam-3427	150	2	0	0	NUM
ejpam-3427	151	1	∨	∨	NUM
ejpam-3427	151	2	0	0	NUM
ejpam-3427	151	3	=	=	SYM
ejpam-3427	151	4	µχz	µχz	NOUN
ejpam-3427	151	5	(	(	PUNCT
ejpam-3427	151	6	a	a	X
ejpam-3427	151	7	)	)	PUNCT
ejpam-3427	151	8	∨	∨	NUM
ejpam-3427	151	9	µχz	µχz	NOUN
ejpam-3427	151	10	(	(	PUNCT
ejpam-3427	151	11	b	b	NOUN
ejpam-3427	151	12	)	)	PUNCT
ejpam-3427	151	13	,	,	PUNCT
ejpam-3427	151	14	γχz	γχz	INTJ
ejpam-3427	151	15	(	(	PUNCT
ejpam-3427	151	16	a−	a−	PROPN
ejpam-3427	151	17	b	b	NOUN
ejpam-3427	151	18	)	)	PUNCT
ejpam-3427	151	19	=	=	SYM
ejpam-3427	151	20	1	1	NUM
ejpam-3427	151	21	=	=	SYM
ejpam-3427	151	22	1	1	NUM
ejpam-3427	151	23	∧	∧	PROPN
ejpam-3427	151	24	1	1	NUM
ejpam-3427	151	25	=	=	SYM
ejpam-3427	151	26	γχz	γχz	NOUN
ejpam-3427	151	27	(	(	PUNCT
ejpam-3427	151	28	a	a	X
ejpam-3427	151	29	)	)	PUNCT
ejpam-3427	151	30	∧	∧	NOUN
ejpam-3427	151	31	γχz	γχz	NOUN
ejpam-3427	151	32	(	(	PUNCT
ejpam-3427	151	33	b	b	NOUN
ejpam-3427	151	34	)	)	PUNCT
ejpam-3427	151	35	,	,	PUNCT
ejpam-3427	151	36	γχz	γχz	INTJ
ejpam-3427	151	37	(	(	PUNCT
ejpam-3427	151	38	ab	ab	NOUN
ejpam-3427	151	39	)	)	PUNCT
ejpam-3427	151	40	=	=	SYM
ejpam-3427	151	41	1	1	NUM
ejpam-3427	151	42	=	=	SYM
ejpam-3427	151	43	1	1	NUM
ejpam-3427	151	44	∧	∧	PROPN
ejpam-3427	151	45	1	1	NUM
ejpam-3427	151	46	=	=	SYM
ejpam-3427	151	47	γχz	γχz	NOUN
ejpam-3427	151	48	(	(	PUNCT
ejpam-3427	151	49	a	a	X
ejpam-3427	151	50	)	)	PUNCT
ejpam-3427	151	51	∧	∧	NOUN
ejpam-3427	151	52	γχz	γχz	NOUN
ejpam-3427	151	53	(	(	PUNCT
ejpam-3427	151	54	b	b	NOUN
ejpam-3427	151	55	)	)	PUNCT
ejpam-3427	151	56	.	.	PUNCT
ejpam-3427	152	1	thus	thus	ADV
ejpam-3427	152	2	µχz	µχz	NOUN
ejpam-3427	152	3	(	(	PUNCT
ejpam-3427	152	4	a−	a−	PROPN
ejpam-3427	152	5	b	b	NOUN
ejpam-3427	152	6	)	)	PUNCT
ejpam-3427	152	7	≤	≤	NUM
ejpam-3427	152	8	max{µχz	max{µχz	NOUN
ejpam-3427	152	9	(	(	PUNCT
ejpam-3427	152	10	a	a	X
ejpam-3427	152	11	)	)	PUNCT
ejpam-3427	152	12	,	,	PUNCT
ejpam-3427	152	13	µχz	µχz	NOUN
ejpam-3427	152	14	(	(	PUNCT
ejpam-3427	152	15	b	b	NOUN
ejpam-3427	152	16	)	)	PUNCT
ejpam-3427	152	17	}	}	PUNCT
ejpam-3427	152	18	,	,	PUNCT
ejpam-3427	152	19	µχz	µχz	NOUN
ejpam-3427	152	20	(	(	PUNCT
ejpam-3427	152	21	ab	ab	NOUN
ejpam-3427	152	22	)	)	PUNCT
ejpam-3427	152	23	≤	≤	NOUN
ejpam-3427	152	24	max{µχz	max{µχz	NOUN
ejpam-3427	152	25	(	(	PUNCT
ejpam-3427	152	26	a	a	X
ejpam-3427	152	27	)	)	PUNCT
ejpam-3427	152	28	,	,	PUNCT
ejpam-3427	152	29	µχz	µχz	NOUN
ejpam-3427	152	30	(	(	PUNCT
ejpam-3427	152	31	b	b	NOUN
ejpam-3427	152	32	)	)	PUNCT
ejpam-3427	152	33	}	}	PUNCT
ejpam-3427	152	34	,	,	PUNCT
ejpam-3427	152	35	γχz	γχz	INTJ
ejpam-3427	152	36	(	(	PUNCT
ejpam-3427	152	37	a−	a−	PROPN
ejpam-3427	152	38	b	b	PROPN
ejpam-3427	152	39	)	)	PUNCT
ejpam-3427	152	40	≥	≥	NOUN
ejpam-3427	152	41	min{γχz	min{γχz	NOUN
ejpam-3427	152	42	(	(	PUNCT
ejpam-3427	152	43	a	a	NOUN
ejpam-3427	152	44	)	)	PUNCT
ejpam-3427	152	45	,	,	PUNCT
ejpam-3427	152	46	γχz	γχz	INTJ
ejpam-3427	152	47	(	(	PUNCT
ejpam-3427	152	48	b	b	NOUN
ejpam-3427	152	49	)	)	PUNCT
ejpam-3427	152	50	}	}	PUNCT
ejpam-3427	152	51	,	,	PUNCT
ejpam-3427	152	52	γχz	γχz	INTJ
ejpam-3427	152	53	(	(	PUNCT
ejpam-3427	152	54	a−	a−	PROPN
ejpam-3427	152	55	b	b	PROPN
ejpam-3427	152	56	)	)	PUNCT
ejpam-3427	152	57	≥	≥	NOUN
ejpam-3427	152	58	min{γχz	min{γχz	NOUN
ejpam-3427	152	59	(	(	PUNCT
ejpam-3427	152	60	a	a	NOUN
ejpam-3427	152	61	)	)	PUNCT
ejpam-3427	152	62	,	,	PUNCT
ejpam-3427	152	63	γχz	γχz	INTJ
ejpam-3427	152	64	(	(	PUNCT
ejpam-3427	152	65	b	b	NOUN
ejpam-3427	152	66	)	)	PUNCT
ejpam-3427	152	67	}	}	PUNCT
ejpam-3427	152	68	.	.	PUNCT
ejpam-3427	153	1	as	as	SCONJ
ejpam-3427	153	2	ab	ab	PROPN
ejpam-3427	153	3	and	and	CCONJ
ejpam-3427	153	4	ba	ba	PROPN
ejpam-3427	153	5	∈	∈	PROPN
ejpam-3427	153	6	z	z	PROPN
ejpam-3427	153	7	,	,	PUNCT
ejpam-3427	153	8	by	by	ADP
ejpam-3427	153	9	definition	definition	NOUN
ejpam-3427	153	10	we	we	PRON
ejpam-3427	153	11	get	get	VERB
ejpam-3427	153	12	µχz	µχz	NOUN
ejpam-3427	153	13	(	(	PUNCT
ejpam-3427	153	14	ab	ab	NOUN
ejpam-3427	153	15	)	)	PUNCT
ejpam-3427	153	16	=	=	SYM
ejpam-3427	153	17	0	0	NUM
ejpam-3427	154	1	=	=	SYM
ejpam-3427	154	2	µχz	µχz	NOUN
ejpam-3427	154	3	(	(	PUNCT
ejpam-3427	154	4	ba	ba	NOUN
ejpam-3427	154	5	)	)	PUNCT
ejpam-3427	154	6	and	and	CCONJ
ejpam-3427	154	7	γχz	γχz	INTJ
ejpam-3427	154	8	(	(	PUNCT
ejpam-3427	154	9	ab	ab	NOUN
ejpam-3427	154	10	)	)	PUNCT
ejpam-3427	154	11	=	=	SYM
ejpam-3427	155	1	1	1	NUM
ejpam-3427	155	2	=	=	SYM
ejpam-3427	155	3	γχz	γχz	NOUN
ejpam-3427	155	4	(	(	PUNCT
ejpam-3427	155	5	ba	ba	NOUN
ejpam-3427	155	6	)	)	PUNCT
ejpam-3427	155	7	,	,	PUNCT
ejpam-3427	155	8	i.e.	i.e.	X
ejpam-3427	155	9	,	,	PUNCT
ejpam-3427	155	10	µχz	µχz	NOUN
ejpam-3427	155	11	(	(	PUNCT
ejpam-3427	155	12	ab	ab	NOUN
ejpam-3427	155	13	)	)	PUNCT
ejpam-3427	155	14	=	=	PRON
ejpam-3427	155	15	µχz	µχz	NOUN
ejpam-3427	155	16	(	(	PUNCT
ejpam-3427	155	17	ba	ba	NOUN
ejpam-3427	155	18	)	)	PUNCT
ejpam-3427	155	19	and	and	CCONJ
ejpam-3427	155	20	γχz	γχz	INTJ
ejpam-3427	155	21	(	(	PUNCT
ejpam-3427	155	22	ab	ab	NOUN
ejpam-3427	155	23	)	)	PUNCT
ejpam-3427	155	24	=	=	SYM
ejpam-3427	156	1	γχz	γχz	NOUN
ejpam-3427	156	2	(	(	PUNCT
ejpam-3427	156	3	ba	ba	NOUN
ejpam-3427	156	4	)	)	PUNCT
ejpam-3427	156	5	.	.	PUNCT
ejpam-3427	157	1	similarly	similarly	ADV
ejpam-3427	157	2	,	,	PUNCT
ejpam-3427	157	3	we	we	PRON
ejpam-3427	157	4	have	have	VERB
ejpam-3427	157	5	µχz	µχz	NOUN
ejpam-3427	157	6	(	(	PUNCT
ejpam-3427	157	7	a−	a−	PROPN
ejpam-3427	157	8	b	b	NOUN
ejpam-3427	157	9	)	)	PUNCT
ejpam-3427	157	10	≤	≤	NUM
ejpam-3427	157	11	max{µχz	max{µχz	NOUN
ejpam-3427	157	12	(	(	PUNCT
ejpam-3427	157	13	a	a	X
ejpam-3427	157	14	)	)	PUNCT
ejpam-3427	157	15	,	,	PUNCT
ejpam-3427	157	16	µχz	µχz	NOUN
ejpam-3427	157	17	(	(	PUNCT
ejpam-3427	157	18	b	b	NOUN
ejpam-3427	157	19	)	)	PUNCT
ejpam-3427	157	20	}	}	PUNCT
ejpam-3427	157	21	,	,	PUNCT
ejpam-3427	157	22	µχz	µχz	NOUN
ejpam-3427	157	23	(	(	PUNCT
ejpam-3427	157	24	ab	ab	NOUN
ejpam-3427	157	25	)	)	PUNCT
ejpam-3427	157	26	≤	≤	NOUN
ejpam-3427	157	27	max{µχz	max{µχz	NOUN
ejpam-3427	157	28	(	(	PUNCT
ejpam-3427	157	29	a	a	X
ejpam-3427	157	30	)	)	PUNCT
ejpam-3427	157	31	,	,	PUNCT
ejpam-3427	157	32	µχz	µχz	NOUN
ejpam-3427	157	33	(	(	PUNCT
ejpam-3427	157	34	b	b	NOUN
ejpam-3427	157	35	)	)	PUNCT
ejpam-3427	157	36	}	}	PUNCT
ejpam-3427	157	37	,	,	PUNCT
ejpam-3427	157	38	γχz	γχz	INTJ
ejpam-3427	157	39	(	(	PUNCT
ejpam-3427	157	40	a−	a−	PROPN
ejpam-3427	157	41	b	b	PROPN
ejpam-3427	157	42	)	)	PUNCT
ejpam-3427	157	43	≥	≥	NOUN
ejpam-3427	157	44	min{γχz	min{γχz	NOUN
ejpam-3427	157	45	(	(	PUNCT
ejpam-3427	157	46	a	a	NOUN
ejpam-3427	157	47	)	)	PUNCT
ejpam-3427	157	48	,	,	PUNCT
ejpam-3427	157	49	γχz	γχz	INTJ
ejpam-3427	157	50	(	(	PUNCT
ejpam-3427	157	51	b	b	NOUN
ejpam-3427	157	52	)	)	PUNCT
ejpam-3427	157	53	}	}	PUNCT
ejpam-3427	157	54	,	,	PUNCT
ejpam-3427	157	55	γχz	γχz	INTJ
ejpam-3427	157	56	(	(	PUNCT
ejpam-3427	157	57	ab	ab	PROPN
ejpam-3427	157	58	)	)	PUNCT
ejpam-3427	157	59	≥	≥	NOUN
ejpam-3427	157	60	min{γχz	min{γχz	NOUN
ejpam-3427	157	61	(	(	PUNCT
ejpam-3427	157	62	a	a	NOUN
ejpam-3427	157	63	)	)	PUNCT
ejpam-3427	157	64	,	,	PUNCT
ejpam-3427	157	65	γχz	γχz	INTJ
ejpam-3427	157	66	(	(	PUNCT
ejpam-3427	157	67	b	b	NOUN
ejpam-3427	157	68	)	)	PUNCT
ejpam-3427	157	69	}	}	PUNCT
ejpam-3427	157	70	,	,	PUNCT
ejpam-3427	157	71	γχz	γχz	INTJ
ejpam-3427	157	72	(	(	PUNCT
ejpam-3427	157	73	ab	ab	NOUN
ejpam-3427	157	74	)	)	PUNCT
ejpam-3427	157	75	=	=	SYM
ejpam-3427	158	1	γχz	γχz	NOUN
ejpam-3427	158	2	(	(	PUNCT
ejpam-3427	158	3	ba	ba	PROPN
ejpam-3427	158	4	)	)	PUNCT
ejpam-3427	158	5	,	,	PUNCT
ejpam-3427	158	6	γχz	γχz	INTJ
ejpam-3427	158	7	(	(	PUNCT
ejpam-3427	158	8	ab	ab	NOUN
ejpam-3427	158	9	)	)	PUNCT
ejpam-3427	158	10	=	=	SYM
ejpam-3427	159	1	γχz	γχz	NOUN
ejpam-3427	159	2	(	(	PUNCT
ejpam-3427	159	3	ba	ba	NOUN
ejpam-3427	159	4	)	)	PUNCT
ejpam-3427	159	5	,	,	PUNCT
ejpam-3427	159	6	when	when	SCONJ
ejpam-3427	159	7	a	a	DET
ejpam-3427	159	8	,	,	PUNCT
ejpam-3427	159	9	b	b	PROPN
ejpam-3427	159	10	/∈	/∈	PUNCT
ejpam-3427	159	11	z.	z.	PROPN
ejpam-3427	160	1	hence	hence	ADV
ejpam-3427	160	2	the	the	DET
ejpam-3427	160	3	intuitionistic	intuitionistic	ADJ
ejpam-3427	160	4	anti	anti	ADJ
ejpam-3427	160	5	characteristic	characteristic	ADJ
ejpam-3427	160	6	function	function	NOUN
ejpam-3427	160	7	χz	χz	PROPN
ejpam-3427	160	8	=	=	SYM
ejpam-3427	160	9	〈	〈	PROPN
ejpam-3427	160	10	µχz	µχz	NOUN
ejpam-3427	160	11	,	,	PUNCT
ejpam-3427	160	12	γχz	γχz	INTJ
ejpam-3427	160	13	〉	〉	PROPN
ejpam-3427	160	14	of	of	ADP
ejpam-3427	160	15	z	z	PROPN
ejpam-3427	160	16	is	be	AUX
ejpam-3427	160	17	an	an	DET
ejpam-3427	160	18	intuitionistic	intuitionistic	ADJ
ejpam-3427	160	19	anti	anti	ADJ
ejpam-3427	160	20	fuzzy	fuzzy	ADJ
ejpam-3427	160	21	normal	normal	ADJ
ejpam-3427	160	22	la	la	NOUN
ejpam-3427	160	23	-	-	PUNCT
ejpam-3427	160	24	subring	subring	NOUN
ejpam-3427	160	25	of	of	ADP
ejpam-3427	160	26	an	an	DET
ejpam-3427	160	27	la	la	ADJ
ejpam-3427	160	28	-	-	PUNCT
ejpam-3427	160	29	ring	ring	NOUN
ejpam-3427	160	30	r1	r1	PROPN
ejpam-3427	160	31	×r2	×r2	PROPN
ejpam-3427	160	32	.	.	PUNCT
ejpam-3427	161	1	conversely	conversely	ADV
ejpam-3427	161	2	,	,	PUNCT
ejpam-3427	161	3	assume	assume	VERB
ejpam-3427	161	4	that	that	SCONJ
ejpam-3427	161	5	the	the	DET
ejpam-3427	161	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	161	7	anti	anti	ADJ
ejpam-3427	161	8	characteristic	characteristic	ADJ
ejpam-3427	161	9	function	function	NOUN
ejpam-3427	161	10	χz	χz	PROPN
ejpam-3427	161	11	=	=	SYM
ejpam-3427	161	12	〈	〈	PROPN
ejpam-3427	161	13	µχz	µχz	NOUN
ejpam-3427	161	14	,	,	PUNCT
ejpam-3427	161	15	γχz	γχz	INTJ
ejpam-3427	161	16	〉	〉	NOUN
ejpam-3427	161	17	of	of	ADP
ejpam-3427	161	18	z	z	NOUN
ejpam-3427	161	19	=	=	SYM
ejpam-3427	162	1	x	x	NOUN
ejpam-3427	162	2	∩	∩	NOUN
ejpam-3427	162	3	y	y	PROPN
ejpam-3427	162	4	is	be	AUX
ejpam-3427	162	5	an	an	DET
ejpam-3427	162	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	162	7	anti	anti	ADJ
ejpam-3427	162	8	fuzzy	fuzzy	ADJ
ejpam-3427	162	9	normal	normal	ADJ
ejpam-3427	162	10	la	la	NOUN
ejpam-3427	162	11	-	-	PUNCT
ejpam-3427	162	12	subring	subring	NOUN
ejpam-3427	162	13	of	of	ADP
ejpam-3427	162	14	an	an	DET
ejpam-3427	162	15	la	la	ADJ
ejpam-3427	162	16	-	-	PUNCT
ejpam-3427	162	17	ring	ring	NOUN
ejpam-3427	162	18	r1	r1	NOUN
ejpam-3427	162	19	×	×	NOUN
ejpam-3427	162	20	r2	r2	NOUN
ejpam-3427	162	21	.	.	PUNCT
ejpam-3427	163	1	k.	k.	PROPN
ejpam-3427	163	2	nasreen	nasreen	PROPN
ejpam-3427	163	3	/	/	SYM
ejpam-3427	163	4	eur	eur	PROPN
ejpam-3427	163	5	.	.	PUNCT
ejpam-3427	164	1	j.	j.	PROPN
ejpam-3427	164	2	pure	pure	PROPN
ejpam-3427	164	3	appl	appl	PROPN
ejpam-3427	164	4	.	.	PROPN
ejpam-3427	164	5	math	math	PROPN
ejpam-3427	164	6	,	,	PUNCT
ejpam-3427	164	7	12	12	NUM
ejpam-3427	164	8	(	(	PUNCT
ejpam-3427	164	9	2	2	NUM
ejpam-3427	164	10	)	)	PUNCT
ejpam-3427	164	11	(	(	PUNCT
ejpam-3427	164	12	2019	2019	NUM
ejpam-3427	164	13	)	)	PUNCT
ejpam-3427	164	14	,	,	PUNCT
ejpam-3427	164	15	622	622	NUM
ejpam-3427	164	16	-	-	SYM
ejpam-3427	164	17	648	648	NUM
ejpam-3427	164	18	627	627	NUM
ejpam-3427	164	19	let	let	VERB
ejpam-3427	164	20	a	a	DET
ejpam-3427	164	21	,	,	PUNCT
ejpam-3427	164	22	b	b	X
ejpam-3427	164	23	∈	∈	PROPN
ejpam-3427	164	24	z	z	NOUN
ejpam-3427	165	1	=	=	SYM
ejpam-3427	165	2	x	x	PROPN
ejpam-3427	165	3	∩	∩	PROPN
ejpam-3427	165	4	y	y	PROPN
ejpam-3427	165	5	,	,	PUNCT
ejpam-3427	165	6	then	then	ADV
ejpam-3427	165	7	by	by	ADP
ejpam-3427	165	8	definition	definition	NOUN
ejpam-3427	165	9	,	,	PUNCT
ejpam-3427	165	10	we	we	PRON
ejpam-3427	165	11	have	have	VERB
ejpam-3427	165	12	µχz	µχz	NOUN
ejpam-3427	165	13	(	(	PUNCT
ejpam-3427	165	14	a	a	X
ejpam-3427	165	15	)	)	PUNCT
ejpam-3427	165	16	=	=	SYM
ejpam-3427	165	17	0	0	PUNCT
ejpam-3427	166	1	=	=	SYM
ejpam-3427	166	2	µχz	µχz	NOUN
ejpam-3427	166	3	(	(	PUNCT
ejpam-3427	166	4	b	b	NOUN
ejpam-3427	166	5	)	)	PUNCT
ejpam-3427	166	6	and	and	CCONJ
ejpam-3427	166	7	γχz	γχz	INTJ
ejpam-3427	166	8	(	(	PUNCT
ejpam-3427	166	9	a	a	X
ejpam-3427	166	10	)	)	PUNCT
ejpam-3427	166	11	=	=	SYM
ejpam-3427	166	12	1	1	NUM
ejpam-3427	166	13	=	=	SYM
ejpam-3427	166	14	γχz	γχz	NOUN
ejpam-3427	166	15	(	(	PUNCT
ejpam-3427	166	16	b	b	NOUN
ejpam-3427	166	17	)	)	PUNCT
ejpam-3427	166	18	.	.	PUNCT
ejpam-3427	167	1	by	by	ADP
ejpam-3427	167	2	our	our	PRON
ejpam-3427	167	3	assumption	assumption	NOUN
ejpam-3427	167	4	µχz	µχz	NOUN
ejpam-3427	167	5	(	(	PUNCT
ejpam-3427	167	6	a−	a−	PROPN
ejpam-3427	167	7	b	b	NOUN
ejpam-3427	167	8	)	)	PUNCT
ejpam-3427	167	9	≤	≤	NOUN
ejpam-3427	167	10	µχz	µχz	NOUN
ejpam-3427	167	11	(	(	PUNCT
ejpam-3427	167	12	a	a	X
ejpam-3427	167	13	)	)	PUNCT
ejpam-3427	167	14	∨	∨	NUM
ejpam-3427	167	15	µχz	µχz	NOUN
ejpam-3427	167	16	(	(	PUNCT
ejpam-3427	167	17	b	b	NOUN
ejpam-3427	167	18	)	)	PUNCT
ejpam-3427	167	19	=	=	SYM
ejpam-3427	167	20	0	0	NUM
ejpam-3427	168	1	∨	∨	NUM
ejpam-3427	168	2	0	0	NUM
ejpam-3427	168	3	=	=	SYM
ejpam-3427	168	4	0	0	NUM
ejpam-3427	168	5	,	,	PUNCT
ejpam-3427	168	6	µχz	µχz	NOUN
ejpam-3427	168	7	(	(	PUNCT
ejpam-3427	168	8	ab	ab	NOUN
ejpam-3427	168	9	)	)	PUNCT
ejpam-3427	168	10	≤	≤	NOUN
ejpam-3427	168	11	µχz	µχz	NOUN
ejpam-3427	168	12	(	(	PUNCT
ejpam-3427	168	13	a	a	X
ejpam-3427	168	14	)	)	PUNCT
ejpam-3427	168	15	∨	∨	NUM
ejpam-3427	168	16	µχz	µχz	NOUN
ejpam-3427	168	17	(	(	PUNCT
ejpam-3427	168	18	b	b	NOUN
ejpam-3427	168	19	)	)	PUNCT
ejpam-3427	168	20	=	=	SYM
ejpam-3427	168	21	0	0	NUM
ejpam-3427	168	22	∨	∨	NUM
ejpam-3427	168	23	0	0	NUM
ejpam-3427	168	24	=	=	SYM
ejpam-3427	168	25	0	0	NUM
ejpam-3427	168	26	,	,	PUNCT
ejpam-3427	168	27	γχz	γχz	INTJ
ejpam-3427	168	28	(	(	PUNCT
ejpam-3427	168	29	a−	a−	PROPN
ejpam-3427	168	30	b	b	PROPN
ejpam-3427	168	31	)	)	PUNCT
ejpam-3427	168	32	≥	≥	NOUN
ejpam-3427	168	33	γχz	γχz	INTJ
ejpam-3427	168	34	(	(	PUNCT
ejpam-3427	168	35	a	a	X
ejpam-3427	168	36	)	)	PUNCT
ejpam-3427	168	37	∧	∧	NOUN
ejpam-3427	168	38	γχz	γχz	NOUN
ejpam-3427	168	39	(	(	PUNCT
ejpam-3427	168	40	b	b	NOUN
ejpam-3427	168	41	)	)	PUNCT
ejpam-3427	168	42	=	=	SYM
ejpam-3427	168	43	1	1	NUM
ejpam-3427	168	44	∧	∧	PROPN
ejpam-3427	168	45	1	1	NUM
ejpam-3427	168	46	=	=	SYM
ejpam-3427	168	47	1	1	NUM
ejpam-3427	168	48	,	,	PUNCT
ejpam-3427	168	49	γχz	γχz	INTJ
ejpam-3427	168	50	(	(	PUNCT
ejpam-3427	168	51	ab	ab	PROPN
ejpam-3427	168	52	)	)	PUNCT
ejpam-3427	168	53	≥	≥	NOUN
ejpam-3427	168	54	γχz	γχz	INTJ
ejpam-3427	168	55	(	(	PUNCT
ejpam-3427	168	56	a	a	X
ejpam-3427	168	57	)	)	PUNCT
ejpam-3427	168	58	∧	∧	NOUN
ejpam-3427	168	59	γχz	γχz	NOUN
ejpam-3427	168	60	(	(	PUNCT
ejpam-3427	168	61	b	b	NOUN
ejpam-3427	168	62	)	)	PUNCT
ejpam-3427	168	63	=	=	SYM
ejpam-3427	168	64	1	1	NUM
ejpam-3427	168	65	∧	∧	PROPN
ejpam-3427	168	66	1	1	NUM
ejpam-3427	168	67	=	=	SYM
ejpam-3427	168	68	1	1	NUM
ejpam-3427	168	69	.	.	PUNCT
ejpam-3427	168	70	thus	thus	ADV
ejpam-3427	168	71	µχz	µχz	NOUN
ejpam-3427	168	72	(	(	PUNCT
ejpam-3427	168	73	a−	a−	PROPN
ejpam-3427	168	74	b	b	NOUN
ejpam-3427	168	75	)	)	PUNCT
ejpam-3427	168	76	=	=	SYM
ejpam-3427	168	77	0	0	PUNCT
ejpam-3427	169	1	=	=	SYM
ejpam-3427	169	2	µχz	µχz	NOUN
ejpam-3427	169	3	(	(	PUNCT
ejpam-3427	169	4	ab	ab	NOUN
ejpam-3427	169	5	)	)	PUNCT
ejpam-3427	169	6	and	and	CCONJ
ejpam-3427	169	7	γχz	γχz	INTJ
ejpam-3427	169	8	(	(	PUNCT
ejpam-3427	169	9	a−	a−	PROPN
ejpam-3427	169	10	b	b	NOUN
ejpam-3427	169	11	)	)	PUNCT
ejpam-3427	169	12	=	=	SYM
ejpam-3427	169	13	1	1	NUM
ejpam-3427	169	14	=	=	SYM
ejpam-3427	169	15	γχz	γχz	NOUN
ejpam-3427	169	16	(	(	PUNCT
ejpam-3427	169	17	ab	ab	NOUN
ejpam-3427	169	18	)	)	PUNCT
ejpam-3427	169	19	,	,	PUNCT
ejpam-3427	169	20	i.e.	i.e.	X
ejpam-3427	169	21	,	,	PUNCT
ejpam-3427	169	22	a−	a−	PROPN
ejpam-3427	169	23	b	b	PROPN
ejpam-3427	169	24	and	and	CCONJ
ejpam-3427	169	25	ab	ab	PROPN
ejpam-3427	169	26	∈	∈	PROPN
ejpam-3427	169	27	z.	z.	PROPN
ejpam-3427	170	1	hence	hence	ADV
ejpam-3427	170	2	z	z	PROPN
ejpam-3427	170	3	is	be	AUX
ejpam-3427	170	4	an	an	DET
ejpam-3427	170	5	la	la	NOUN
ejpam-3427	170	6	-	-	PUNCT
ejpam-3427	170	7	subring	subring	NOUN
ejpam-3427	170	8	of	of	ADP
ejpam-3427	170	9	an	an	DET
ejpam-3427	170	10	la	la	ADJ
ejpam-3427	170	11	-	-	PUNCT
ejpam-3427	170	12	ring	ring	NOUN
ejpam-3427	170	13	r1	r1	PROPN
ejpam-3427	170	14	×r2	×r2	PROPN
ejpam-3427	170	15	.	.	PUNCT
ejpam-3427	171	1	corollary	corollary	ADJ
ejpam-3427	171	2	1	1	NUM
ejpam-3427	171	3	.	.	PUNCT
ejpam-3427	172	1	let	let	VERB
ejpam-3427	172	2	{	{	PUNCT
ejpam-3427	172	3	ci}i∈i	ci}i∈i	NOUN
ejpam-3427	172	4	=	=	SYM
ejpam-3427	172	5	{	{	PUNCT
ejpam-3427	172	6	ai	ai	AUX
ejpam-3427	172	7	×bi}i∈i	×bi}i∈i	NOUN
ejpam-3427	172	8	be	be	AUX
ejpam-3427	172	9	a	a	DET
ejpam-3427	172	10	family	family	NOUN
ejpam-3427	172	11	of	of	ADP
ejpam-3427	172	12	la	la	ADJ
ejpam-3427	172	13	-	-	PUNCT
ejpam-3427	172	14	subrings	subring	NOUN
ejpam-3427	172	15	of	of	ADP
ejpam-3427	172	16	an	an	DET
ejpam-3427	172	17	la	la	ADJ
ejpam-3427	172	18	-	-	PUNCT
ejpam-3427	172	19	ring	ring	NOUN
ejpam-3427	172	20	r1	r1	NOUN
ejpam-3427	172	21	×	×	NOUN
ejpam-3427	172	22	r2	r2	NOUN
ejpam-3427	172	23	.	.	PUNCT
ejpam-3427	173	1	then	then	ADV
ejpam-3427	173	2	c	c	X
ejpam-3427	173	3	=	=	SYM
ejpam-3427	173	4	∩ci	∩ci	NOUN
ejpam-3427	173	5	is	be	AUX
ejpam-3427	173	6	an	an	DET
ejpam-3427	173	7	la	la	NOUN
ejpam-3427	173	8	-	-	PUNCT
ejpam-3427	173	9	subring	subring	NOUN
ejpam-3427	173	10	of	of	ADP
ejpam-3427	173	11	an	an	DET
ejpam-3427	173	12	la	la	ADJ
ejpam-3427	173	13	-	-	PUNCT
ejpam-3427	173	14	ring	ring	NOUN
ejpam-3427	173	15	r1	r1	NOUN
ejpam-3427	173	16	×	×	NOUN
ejpam-3427	173	17	r2	r2	NOUN
ejpam-3427	173	18	if	if	SCONJ
ejpam-3427	173	19	and	and	CCONJ
ejpam-3427	173	20	only	only	ADV
ejpam-3427	173	21	if	if	SCONJ
ejpam-3427	173	22	the	the	DET
ejpam-3427	173	23	intuitionistic	intuitionistic	ADJ
ejpam-3427	173	24	anti	anti	ADJ
ejpam-3427	173	25	characteristic	characteristic	ADJ
ejpam-3427	173	26	function	function	NOUN
ejpam-3427	173	27	χc	χc	PROPN
ejpam-3427	174	1	=	=	SYM
ejpam-3427	174	2	〈	〈	PROPN
ejpam-3427	174	3	µχc	µχc	NOUN
ejpam-3427	174	4	,	,	PUNCT
ejpam-3427	174	5	γχc	γχc	PROPN
ejpam-3427	174	6	〉	〉	NOUN
ejpam-3427	174	7	of	of	ADP
ejpam-3427	174	8	c	c	NOUN
ejpam-3427	174	9	=	=	SYM
ejpam-3427	174	10	∩ci	∩ci	NOUN
ejpam-3427	174	11	is	be	AUX
ejpam-3427	174	12	an	an	DET
ejpam-3427	174	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	174	14	anti	anti	ADJ
ejpam-3427	174	15	fuzzy	fuzzy	ADJ
ejpam-3427	174	16	normal	normal	ADJ
ejpam-3427	174	17	la	la	NOUN
ejpam-3427	174	18	-	-	PUNCT
ejpam-3427	174	19	subring	subring	NOUN
ejpam-3427	174	20	of	of	ADP
ejpam-3427	174	21	an	an	DET
ejpam-3427	174	22	la	la	ADJ
ejpam-3427	174	23	-	-	PUNCT
ejpam-3427	174	24	ring	ring	NOUN
ejpam-3427	174	25	r1	r1	PROPN
ejpam-3427	174	26	×r2	×r2	PROPN
ejpam-3427	174	27	.	.	PUNCT
ejpam-3427	175	1	lemma	lemma	PROPN
ejpam-3427	175	2	3	3	X
ejpam-3427	175	3	.	.	PUNCT
ejpam-3427	176	1	if	if	SCONJ
ejpam-3427	176	2	a	a	PRON
ejpam-3427	176	3	and	and	CCONJ
ejpam-3427	176	4	b	b	NOUN
ejpam-3427	176	5	are	be	AUX
ejpam-3427	176	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	176	7	anti	anti	ADJ
ejpam-3427	176	8	fuzzy	fuzzy	ADJ
ejpam-3427	176	9	normal	normal	ADJ
ejpam-3427	176	10	la	la	ADJ
ejpam-3427	176	11	-	-	PUNCT
ejpam-3427	176	12	subrings	subring	NOUN
ejpam-3427	176	13	of	of	ADP
ejpam-3427	176	14	la	la	NOUN
ejpam-3427	176	15	-	-	PUNCT
ejpam-3427	176	16	rings	ring	NOUN
ejpam-3427	176	17	r1	r1	NOUN
ejpam-3427	176	18	and	and	CCONJ
ejpam-3427	176	19	r2	r2	PROPN
ejpam-3427	176	20	,	,	PUNCT
ejpam-3427	176	21	respectively	respectively	ADV
ejpam-3427	176	22	,	,	PUNCT
ejpam-3427	176	23	then	then	ADV
ejpam-3427	176	24	a×b	a×b	PROPN
ejpam-3427	176	25	is	be	AUX
ejpam-3427	176	26	also	also	ADV
ejpam-3427	176	27	an	an	DET
ejpam-3427	176	28	intuitionistic	intuitionistic	ADJ
ejpam-3427	176	29	anti	anti	ADJ
ejpam-3427	176	30	fuzzy	fuzzy	ADJ
ejpam-3427	176	31	normal	normal	ADJ
ejpam-3427	176	32	la	la	NOUN
ejpam-3427	176	33	-	-	PUNCT
ejpam-3427	176	34	subring	subring	NOUN
ejpam-3427	176	35	of	of	ADP
ejpam-3427	176	36	an	an	DET
ejpam-3427	176	37	la	la	ADJ
ejpam-3427	176	38	-	-	PUNCT
ejpam-3427	176	39	ring	ring	NOUN
ejpam-3427	176	40	r1	r1	PROPN
ejpam-3427	176	41	×r2	×r2	PROPN
ejpam-3427	176	42	.	.	PUNCT
ejpam-3427	177	1	proof	proof	NOUN
ejpam-3427	177	2	.	.	PUNCT
ejpam-3427	178	1	let	let	VERB
ejpam-3427	178	2	a	a	PRON
ejpam-3427	178	3	=	=	PUNCT
ejpam-3427	178	4	{	{	PUNCT
ejpam-3427	178	5	(	(	PUNCT
ejpam-3427	178	6	x	x	NOUN
ejpam-3427	178	7	,	,	PUNCT
ejpam-3427	178	8	µa(x	µa(x	NOUN
ejpam-3427	178	9	)	)	PUNCT
ejpam-3427	178	10	,	,	PUNCT
ejpam-3427	178	11	γa(x	γa(x	NUM
ejpam-3427	178	12	)	)	PUNCT
ejpam-3427	178	13	)	)	PUNCT
ejpam-3427	179	1	|	|	ADV
ejpam-3427	179	2	x	x	SYM
ejpam-3427	179	3	∈	∈	PROPN
ejpam-3427	179	4	r1	r1	PROPN
ejpam-3427	179	5	}	}	PUNCT
ejpam-3427	179	6	and	and	CCONJ
ejpam-3427	179	7	b	b	X
ejpam-3427	179	8	=	=	SYM
ejpam-3427	179	9	{	{	PUNCT
ejpam-3427	179	10	(	(	PUNCT
ejpam-3427	179	11	y	y	NOUN
ejpam-3427	179	12	,	,	PUNCT
ejpam-3427	179	13	µb(y	µb(y	NUM
ejpam-3427	179	14	)	)	PUNCT
ejpam-3427	179	15	,	,	PUNCT
ejpam-3427	179	16	γb(y	γb(y	NUM
ejpam-3427	179	17	)	)	PUNCT
ejpam-3427	179	18	)	)	PUNCT
ejpam-3427	180	1	|	|	ADV
ejpam-3427	180	2	y	y	PROPN
ejpam-3427	180	3	∈	∈	PROPN
ejpam-3427	180	4	r2	r2	PROPN
ejpam-3427	180	5	}	}	PUNCT
ejpam-3427	180	6	be	be	AUX
ejpam-3427	180	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	180	8	anti	anti	ADJ
ejpam-3427	180	9	fuzzy	fuzzy	ADJ
ejpam-3427	180	10	normal	normal	ADJ
ejpam-3427	180	11	la	la	ADJ
ejpam-3427	180	12	-	-	PUNCT
ejpam-3427	180	13	subrings	subring	NOUN
ejpam-3427	180	14	of	of	ADP
ejpam-3427	180	15	la	la	NOUN
ejpam-3427	180	16	-	-	PUNCT
ejpam-3427	180	17	rings	ring	NOUN
ejpam-3427	180	18	r1	r1	NOUN
ejpam-3427	180	19	and	and	CCONJ
ejpam-3427	180	20	r2	r2	PROPN
ejpam-3427	180	21	,	,	PUNCT
ejpam-3427	180	22	respectively	respectively	ADV
ejpam-3427	180	23	.	.	PUNCT
ejpam-3427	181	1	now	now	ADV
ejpam-3427	181	2	a×b	a×b	PROPN
ejpam-3427	181	3	=	=	SYM
ejpam-3427	181	4	{	{	PUNCT
ejpam-3427	181	5	(	(	PUNCT
ejpam-3427	181	6	(	(	PUNCT
ejpam-3427	181	7	x	x	NOUN
ejpam-3427	181	8	,	,	PUNCT
ejpam-3427	181	9	y	y	PROPN
ejpam-3427	181	10	)	)	PUNCT
ejpam-3427	181	11	,	,	PUNCT
ejpam-3427	181	12	µa×b(x	µa×b(x	ADJ
ejpam-3427	181	13	,	,	PUNCT
ejpam-3427	181	14	y	y	NOUN
ejpam-3427	181	15	)	)	PUNCT
ejpam-3427	181	16	,	,	PUNCT
ejpam-3427	181	17	γa×b(x	γa×b(x	ADJ
ejpam-3427	181	18	,	,	PUNCT
ejpam-3427	181	19	y	y	NOUN
ejpam-3427	181	20	)	)	PUNCT
ejpam-3427	181	21	)	)	PUNCT
ejpam-3427	181	22	|	|	ADV
ejpam-3427	181	23	for	for	ADP
ejpam-3427	181	24	all	all	DET
ejpam-3427	181	25	x	x	SYM
ejpam-3427	181	26	∈	∈	PROPN
ejpam-3427	181	27	r1	r1	NOUN
ejpam-3427	181	28	and	and	CCONJ
ejpam-3427	181	29	y	y	PROPN
ejpam-3427	181	30	∈	∈	PROPN
ejpam-3427	181	31	r2	r2	PROPN
ejpam-3427	181	32	}	}	PUNCT
ejpam-3427	181	33	,	,	PUNCT
ejpam-3427	181	34	where	where	SCONJ
ejpam-3427	181	35	µa×b(x	µa×b(x	ADJ
ejpam-3427	181	36	,	,	PUNCT
ejpam-3427	181	37	y	y	NOUN
ejpam-3427	181	38	)	)	PUNCT
ejpam-3427	181	39	=	=	SYM
ejpam-3427	181	40	max{µa(x	max{µa(x	NOUN
ejpam-3427	181	41	)	)	PUNCT
ejpam-3427	181	42	,	,	PUNCT
ejpam-3427	181	43	µb(y	µb(y	NUM
ejpam-3427	181	44	)	)	PUNCT
ejpam-3427	181	45	and	and	CCONJ
ejpam-3427	181	46	γa×b(x	γa×b(x	ADJ
ejpam-3427	181	47	,	,	PUNCT
ejpam-3427	181	48	y	y	NOUN
ejpam-3427	181	49	)	)	PUNCT
ejpam-3427	181	50	=	=	SYM
ejpam-3427	181	51	min{γa(x	min{γa(x	NOUN
ejpam-3427	181	52	)	)	PUNCT
ejpam-3427	181	53	,	,	PUNCT
ejpam-3427	181	54	γb(y	γb(y	ADV
ejpam-3427	181	55	)	)	PUNCT
ejpam-3427	181	56	}	}	PUNCT
ejpam-3427	181	57	.	.	PUNCT
ejpam-3427	182	1	we	we	PRON
ejpam-3427	182	2	have	have	VERB
ejpam-3427	182	3	to	to	PART
ejpam-3427	182	4	show	show	VERB
ejpam-3427	182	5	that	that	SCONJ
ejpam-3427	182	6	a	a	DET
ejpam-3427	182	7	×	×	PROPN
ejpam-3427	182	8	b	b	PROPN
ejpam-3427	182	9	is	be	AUX
ejpam-3427	182	10	an	an	DET
ejpam-3427	182	11	intuitionistic	intuitionistic	ADJ
ejpam-3427	182	12	anti	anti	ADJ
ejpam-3427	182	13	fuzzy	fuzzy	ADJ
ejpam-3427	182	14	normal	normal	ADJ
ejpam-3427	182	15	la	la	NOUN
ejpam-3427	182	16	-	-	PUNCT
ejpam-3427	182	17	subring	subring	NOUN
ejpam-3427	182	18	of	of	ADP
ejpam-3427	182	19	an	an	DET
ejpam-3427	182	20	la	la	ADJ
ejpam-3427	182	21	-	-	PUNCT
ejpam-3427	182	22	ring	ring	NOUN
ejpam-3427	182	23	r1	r1	PROPN
ejpam-3427	182	24	×r2	×r2	PROPN
ejpam-3427	182	25	.	.	PUNCT
ejpam-3427	183	1	now	now	ADV
ejpam-3427	183	2	µa×b((a	µa×b((a	VERB
ejpam-3427	183	3	,	,	PUNCT
ejpam-3427	183	4	b)−	b)−	PROPN
ejpam-3427	183	5	(	(	PUNCT
ejpam-3427	183	6	c	c	X
ejpam-3427	183	7	,	,	PUNCT
ejpam-3427	183	8	d	d	NOUN
ejpam-3427	183	9	)	)	PUNCT
ejpam-3427	183	10	)	)	PUNCT
ejpam-3427	184	1	=	=	PUNCT
ejpam-3427	185	1	µa×b(a−	µa×b(a−	ADP
ejpam-3427	185	2	c	c	X
ejpam-3427	185	3	,	,	PUNCT
ejpam-3427	185	4	b−	b−	PROPN
ejpam-3427	185	5	d	d	PROPN
ejpam-3427	185	6	)	)	PUNCT
ejpam-3427	185	7	=	=	PUNCT
ejpam-3427	186	1	max{µa(a−	max{µa(a−	PROPN
ejpam-3427	186	2	c	c	PROPN
ejpam-3427	186	3	)	)	PUNCT
ejpam-3427	186	4	,	,	PUNCT
ejpam-3427	186	5	µb(b−	µb(b−	ADP
ejpam-3427	186	6	d	d	NOUN
ejpam-3427	186	7	)	)	PUNCT
ejpam-3427	186	8	}	}	PUNCT
ejpam-3427	186	9	=	=	SYM
ejpam-3427	186	10	µa(a−	µa(a−	PUNCT
ejpam-3427	186	11	c	c	X
ejpam-3427	186	12	)	)	PUNCT
ejpam-3427	186	13	∨	∨	NOUN
ejpam-3427	186	14	µb(b−	µb(b−	ADP
ejpam-3427	186	15	d	d	NOUN
ejpam-3427	186	16	)	)	PUNCT
ejpam-3427	186	17	≤	≤	NOUN
ejpam-3427	186	18	{	{	PUNCT
ejpam-3427	186	19	µa(a	µa(a	NUM
ejpam-3427	186	20	)	)	PUNCT
ejpam-3427	186	21	∨	∨	NOUN
ejpam-3427	186	22	µa(c	µa(c	NUM
ejpam-3427	186	23	)	)	PUNCT
ejpam-3427	186	24	}	}	PUNCT
ejpam-3427	186	25	∨	∨	X
ejpam-3427	186	26	{	{	PUNCT
ejpam-3427	186	27	µb(b	µb(b	NOUN
ejpam-3427	186	28	)	)	PUNCT
ejpam-3427	186	29	∨	∨	NOUN
ejpam-3427	186	30	µb(d	µb(d	PUNCT
ejpam-3427	186	31	)	)	PUNCT
ejpam-3427	186	32	}	}	PUNCT
ejpam-3427	186	33	=	=	SYM
ejpam-3427	186	34	µa(a	µa(a	X
ejpam-3427	186	35	)	)	PUNCT
ejpam-3427	186	36	∨	∨	X
ejpam-3427	186	37	{	{	PUNCT
ejpam-3427	186	38	µa(c	µa(c	NOUN
ejpam-3427	186	39	)	)	PUNCT
ejpam-3427	186	40	∨	∨	NOUN
ejpam-3427	186	41	µb(b	µb(b	NUM
ejpam-3427	186	42	)	)	PUNCT
ejpam-3427	186	43	}	}	PUNCT
ejpam-3427	186	44	∨	∨	NUM
ejpam-3427	186	45	µb(d	µb(d	PUNCT
ejpam-3427	186	46	)	)	PUNCT
ejpam-3427	186	47	=	=	SYM
ejpam-3427	186	48	µa(a	µa(a	X
ejpam-3427	186	49	)	)	PUNCT
ejpam-3427	186	50	∨	∨	X
ejpam-3427	186	51	{	{	PUNCT
ejpam-3427	186	52	µb(b	µb(b	NOUN
ejpam-3427	186	53	)	)	PUNCT
ejpam-3427	186	54	∨	∨	NOUN
ejpam-3427	186	55	µa(c	µa(c	NUM
ejpam-3427	186	56	)	)	PUNCT
ejpam-3427	186	57	}	}	PUNCT
ejpam-3427	186	58	∨	∨	NUM
ejpam-3427	186	59	µb(d	µb(d	PUNCT
ejpam-3427	186	60	)	)	PUNCT
ejpam-3427	186	61	=	=	SYM
ejpam-3427	186	62	{	{	PUNCT
ejpam-3427	186	63	µa(a	µa(a	NOUN
ejpam-3427	186	64	)	)	PUNCT
ejpam-3427	186	65	∨	∨	NUM
ejpam-3427	186	66	µb(b	µb(b	NUM
ejpam-3427	186	67	)	)	PUNCT
ejpam-3427	186	68	}	}	PUNCT
ejpam-3427	186	69	∨	∨	X
ejpam-3427	186	70	{	{	PUNCT
ejpam-3427	186	71	µa(c	µa(c	NOUN
ejpam-3427	186	72	)	)	PUNCT
ejpam-3427	186	73	∨	∨	NOUN
ejpam-3427	186	74	µb(d	µb(d	PUNCT
ejpam-3427	186	75	)	)	PUNCT
ejpam-3427	186	76	}	}	PUNCT
ejpam-3427	186	77	=	=	NOUN
ejpam-3427	186	78	µa×b(a	µa×b(a	NOUN
ejpam-3427	186	79	,	,	PUNCT
ejpam-3427	186	80	b	b	NOUN
ejpam-3427	186	81	)	)	PUNCT
ejpam-3427	186	82	∨	∨	NOUN
ejpam-3427	186	83	µa×b(c	µa×b(c	NOUN
ejpam-3427	186	84	,	,	PUNCT
ejpam-3427	186	85	d	d	NOUN
ejpam-3427	186	86	)	)	PUNCT
ejpam-3427	186	87	and	and	CCONJ
ejpam-3427	186	88	µa×b((a	µa×b((a	NOUN
ejpam-3427	186	89	,	,	PUNCT
ejpam-3427	186	90	b	b	NOUN
ejpam-3427	186	91	)	)	PUNCT
ejpam-3427	186	92	◦	◦	NOUN
ejpam-3427	186	93	(	(	PUNCT
ejpam-3427	186	94	c	c	X
ejpam-3427	186	95	,	,	PUNCT
ejpam-3427	186	96	d	d	NOUN
ejpam-3427	186	97	)	)	PUNCT
ejpam-3427	186	98	)	)	PUNCT
ejpam-3427	187	1	=	=	NOUN
ejpam-3427	187	2	µa×b(a	µa×b(a	NOUN
ejpam-3427	187	3	◦	◦	NOUN
ejpam-3427	187	4	c	c	NOUN
ejpam-3427	187	5	,	,	PUNCT
ejpam-3427	187	6	b	b	X
ejpam-3427	187	7	◦	◦	NOUN
ejpam-3427	187	8	d	d	NOUN
ejpam-3427	187	9	)	)	PUNCT
ejpam-3427	187	10	=	=	SYM
ejpam-3427	188	1	max{µa(a	max{µa(a	NOUN
ejpam-3427	188	2	◦	◦	NOUN
ejpam-3427	188	3	c	c	NOUN
ejpam-3427	188	4	)	)	PUNCT
ejpam-3427	188	5	,	,	PUNCT
ejpam-3427	188	6	µb(b	µb(b	PUNCT
ejpam-3427	188	7	◦	◦	NOUN
ejpam-3427	188	8	d	d	NOUN
ejpam-3427	188	9	)	)	PUNCT
ejpam-3427	188	10	}	}	PUNCT
ejpam-3427	188	11	=	=	PRON
ejpam-3427	188	12	µa(a	µa(a	PUNCT
ejpam-3427	188	13	◦	◦	NOUN
ejpam-3427	188	14	c	c	NUM
ejpam-3427	188	15	)	)	PUNCT
ejpam-3427	188	16	∨	∨	NOUN
ejpam-3427	188	17	µb(b	µb(b	PUNCT
ejpam-3427	188	18	◦	◦	NOUN
ejpam-3427	188	19	d	d	NOUN
ejpam-3427	188	20	)	)	PUNCT
ejpam-3427	188	21	k.	k.	NOUN
ejpam-3427	189	1	nasreen	nasreen	PROPN
ejpam-3427	189	2	/	/	SYM
ejpam-3427	189	3	eur	eur	PROPN
ejpam-3427	189	4	.	.	PUNCT
ejpam-3427	190	1	j.	j.	PROPN
ejpam-3427	190	2	pure	pure	PROPN
ejpam-3427	190	3	appl	appl	PROPN
ejpam-3427	190	4	.	.	PROPN
ejpam-3427	190	5	math	math	PROPN
ejpam-3427	190	6	,	,	PUNCT
ejpam-3427	190	7	12	12	NUM
ejpam-3427	190	8	(	(	PUNCT
ejpam-3427	190	9	2	2	NUM
ejpam-3427	190	10	)	)	PUNCT
ejpam-3427	190	11	(	(	PUNCT
ejpam-3427	190	12	2019	2019	NUM
ejpam-3427	190	13	)	)	PUNCT
ejpam-3427	190	14	,	,	PUNCT
ejpam-3427	190	15	622	622	NUM
ejpam-3427	190	16	-	-	SYM
ejpam-3427	190	17	648	648	NUM
ejpam-3427	190	18	628	628	NUM
ejpam-3427	190	19	≤	≤	NOUN
ejpam-3427	190	20	{	{	PUNCT
ejpam-3427	190	21	µa(a	µa(a	NUM
ejpam-3427	190	22	)	)	PUNCT
ejpam-3427	190	23	∨	∨	NOUN
ejpam-3427	190	24	µa(c	µa(c	NUM
ejpam-3427	190	25	)	)	PUNCT
ejpam-3427	190	26	}	}	PUNCT
ejpam-3427	190	27	∨	∨	X
ejpam-3427	190	28	{	{	PUNCT
ejpam-3427	190	29	µb(b	µb(b	NOUN
ejpam-3427	190	30	)	)	PUNCT
ejpam-3427	190	31	∨	∨	NOUN
ejpam-3427	190	32	µb(d	µb(d	PUNCT
ejpam-3427	190	33	)	)	PUNCT
ejpam-3427	190	34	}	}	PUNCT
ejpam-3427	190	35	=	=	SYM
ejpam-3427	190	36	µa(a	µa(a	X
ejpam-3427	190	37	)	)	PUNCT
ejpam-3427	190	38	∨	∨	X
ejpam-3427	190	39	{	{	PUNCT
ejpam-3427	190	40	µa(c	µa(c	NOUN
ejpam-3427	190	41	)	)	PUNCT
ejpam-3427	190	42	∨	∨	NOUN
ejpam-3427	190	43	µb(b	µb(b	NUM
ejpam-3427	190	44	)	)	PUNCT
ejpam-3427	190	45	}	}	PUNCT
ejpam-3427	190	46	∨	∨	NUM
ejpam-3427	190	47	µb(d	µb(d	PUNCT
ejpam-3427	190	48	)	)	PUNCT
ejpam-3427	190	49	=	=	SYM
ejpam-3427	190	50	µa(a	µa(a	X
ejpam-3427	190	51	)	)	PUNCT
ejpam-3427	190	52	∨	∨	X
ejpam-3427	190	53	{	{	PUNCT
ejpam-3427	190	54	µb(b	µb(b	NOUN
ejpam-3427	190	55	)	)	PUNCT
ejpam-3427	190	56	∨	∨	NOUN
ejpam-3427	190	57	µa(c	µa(c	NUM
ejpam-3427	190	58	)	)	PUNCT
ejpam-3427	190	59	}	}	PUNCT
ejpam-3427	190	60	∨	∨	NUM
ejpam-3427	190	61	µb(d	µb(d	PUNCT
ejpam-3427	190	62	)	)	PUNCT
ejpam-3427	190	63	=	=	SYM
ejpam-3427	190	64	{	{	PUNCT
ejpam-3427	190	65	µa(a	µa(a	NOUN
ejpam-3427	190	66	)	)	PUNCT
ejpam-3427	190	67	∨	∨	NUM
ejpam-3427	190	68	µb(b	µb(b	NUM
ejpam-3427	190	69	)	)	PUNCT
ejpam-3427	190	70	}	}	PUNCT
ejpam-3427	190	71	∨	∨	X
ejpam-3427	190	72	{	{	PUNCT
ejpam-3427	190	73	µa(c	µa(c	NOUN
ejpam-3427	190	74	)	)	PUNCT
ejpam-3427	190	75	∨	∨	NOUN
ejpam-3427	190	76	µb(d	µb(d	PUNCT
ejpam-3427	190	77	)	)	PUNCT
ejpam-3427	190	78	}	}	PUNCT
ejpam-3427	190	79	=	=	NOUN
ejpam-3427	190	80	µa×b(a	µa×b(a	NOUN
ejpam-3427	190	81	,	,	PUNCT
ejpam-3427	190	82	b	b	NOUN
ejpam-3427	190	83	)	)	PUNCT
ejpam-3427	190	84	∨	∨	NOUN
ejpam-3427	190	85	µa×b(c	µa×b(c	NOUN
ejpam-3427	190	86	,	,	PUNCT
ejpam-3427	190	87	d	d	NOUN
ejpam-3427	190	88	)	)	PUNCT
ejpam-3427	190	89	.	.	PUNCT
ejpam-3427	191	1	thus	thus	ADV
ejpam-3427	191	2	µa×b((a	µa×b((a	VERB
ejpam-3427	191	3	,	,	PUNCT
ejpam-3427	191	4	b)−	b)−	PROPN
ejpam-3427	191	5	(	(	PUNCT
ejpam-3427	191	6	c	c	X
ejpam-3427	191	7	,	,	PUNCT
ejpam-3427	191	8	d	d	NOUN
ejpam-3427	191	9	)	)	PUNCT
ejpam-3427	191	10	)	)	PUNCT
ejpam-3427	191	11	≤	≤	NUM
ejpam-3427	191	12	µa×b(a	µa×b(a	NOUN
ejpam-3427	191	13	,	,	PUNCT
ejpam-3427	191	14	b	b	NOUN
ejpam-3427	191	15	)	)	PUNCT
ejpam-3427	191	16	∨	∨	NOUN
ejpam-3427	191	17	µa×b(c	µa×b(c	NOUN
ejpam-3427	191	18	,	,	PUNCT
ejpam-3427	191	19	d	d	NOUN
ejpam-3427	191	20	)	)	PUNCT
ejpam-3427	191	21	and	and	CCONJ
ejpam-3427	191	22	µa×b((a	µa×b((a	NOUN
ejpam-3427	191	23	,	,	PUNCT
ejpam-3427	191	24	b	b	NOUN
ejpam-3427	191	25	)	)	PUNCT
ejpam-3427	191	26	◦	◦	NOUN
ejpam-3427	191	27	(	(	PUNCT
ejpam-3427	191	28	c	c	X
ejpam-3427	191	29	,	,	PUNCT
ejpam-3427	191	30	d	d	NOUN
ejpam-3427	191	31	)	)	PUNCT
ejpam-3427	191	32	)	)	PUNCT
ejpam-3427	191	33	≤	≤	NUM
ejpam-3427	191	34	µa×b(a	µa×b(a	NOUN
ejpam-3427	191	35	,	,	PUNCT
ejpam-3427	191	36	b	b	NOUN
ejpam-3427	191	37	)	)	PUNCT
ejpam-3427	191	38	∨	∨	NOUN
ejpam-3427	191	39	µa×b(c	µa×b(c	NOUN
ejpam-3427	191	40	,	,	PUNCT
ejpam-3427	191	41	d	d	NOUN
ejpam-3427	191	42	)	)	PUNCT
ejpam-3427	191	43	.	.	PUNCT
ejpam-3427	192	1	similarly	similarly	ADV
ejpam-3427	192	2	,	,	PUNCT
ejpam-3427	192	3	we	we	PRON
ejpam-3427	192	4	have	have	VERB
ejpam-3427	192	5	γa×b((a	γa×b((a	NOUN
ejpam-3427	192	6	,	,	PUNCT
ejpam-3427	192	7	b)−	b)−	PROPN
ejpam-3427	192	8	(	(	PUNCT
ejpam-3427	192	9	c	c	X
ejpam-3427	192	10	,	,	PUNCT
ejpam-3427	192	11	d	d	NOUN
ejpam-3427	192	12	)	)	PUNCT
ejpam-3427	192	13	)	)	PUNCT
ejpam-3427	192	14	≥	≥	NOUN
ejpam-3427	192	15	γa×b(a	γa×b(a	ADJ
ejpam-3427	192	16	,	,	PUNCT
ejpam-3427	192	17	b	b	NOUN
ejpam-3427	192	18	)	)	PUNCT
ejpam-3427	192	19	∧	∧	NOUN
ejpam-3427	192	20	γa×b(c	γa×b(c	NOUN
ejpam-3427	192	21	,	,	PUNCT
ejpam-3427	192	22	d	d	NOUN
ejpam-3427	192	23	)	)	PUNCT
ejpam-3427	192	24	and	and	CCONJ
ejpam-3427	192	25	γa×b((a	γa×b((a	NOUN
ejpam-3427	192	26	,	,	PUNCT
ejpam-3427	192	27	b	b	NOUN
ejpam-3427	192	28	)	)	PUNCT
ejpam-3427	192	29	◦	◦	NOUN
ejpam-3427	192	30	(	(	PUNCT
ejpam-3427	192	31	c	c	X
ejpam-3427	192	32	,	,	PUNCT
ejpam-3427	192	33	d	d	NOUN
ejpam-3427	192	34	)	)	PUNCT
ejpam-3427	192	35	)	)	PUNCT
ejpam-3427	192	36	≥	≥	NOUN
ejpam-3427	192	37	γa×b(a	γa×b(a	ADJ
ejpam-3427	192	38	,	,	PUNCT
ejpam-3427	192	39	b	b	NOUN
ejpam-3427	192	40	)	)	PUNCT
ejpam-3427	192	41	∧	∧	NOUN
ejpam-3427	192	42	γa×b(c	γa×b(c	NOUN
ejpam-3427	192	43	,	,	PUNCT
ejpam-3427	192	44	d	d	NOUN
ejpam-3427	192	45	)	)	PUNCT
ejpam-3427	192	46	.	.	PUNCT
ejpam-3427	193	1	therefore	therefore	ADV
ejpam-3427	193	2	a×b	a×b	PROPN
ejpam-3427	193	3	is	be	AUX
ejpam-3427	193	4	an	an	DET
ejpam-3427	193	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	193	6	anti	anti	ADJ
ejpam-3427	193	7	fuzzy	fuzzy	ADJ
ejpam-3427	193	8	la	la	NOUN
ejpam-3427	193	9	-	-	PUNCT
ejpam-3427	193	10	subring	subring	NOUN
ejpam-3427	193	11	of	of	ADP
ejpam-3427	193	12	an	an	DET
ejpam-3427	193	13	la	la	ADJ
ejpam-3427	193	14	-	-	PUNCT
ejpam-3427	193	15	ring	ring	NOUN
ejpam-3427	193	16	r1×r2	r1×r2	PROPN
ejpam-3427	193	17	.	.	PUNCT
ejpam-3427	194	1	now	now	ADV
ejpam-3427	194	2	µa×b((a	µa×b((a	VERB
ejpam-3427	194	3	,	,	PUNCT
ejpam-3427	194	4	b	b	NOUN
ejpam-3427	194	5	)	)	PUNCT
ejpam-3427	194	6	◦	◦	NOUN
ejpam-3427	194	7	(	(	PUNCT
ejpam-3427	194	8	c	c	X
ejpam-3427	194	9	,	,	PUNCT
ejpam-3427	194	10	d	d	NOUN
ejpam-3427	194	11	)	)	PUNCT
ejpam-3427	194	12	)	)	PUNCT
ejpam-3427	195	1	=	=	SYM
ejpam-3427	196	1	µa×b(ac	µa×b(ac	PROPN
ejpam-3427	196	2	,	,	PUNCT
ejpam-3427	196	3	bd	bd	PROPN
ejpam-3427	196	4	)	)	PUNCT
ejpam-3427	196	5	=	=	SYM
ejpam-3427	196	6	max{µa(ac	max{µa(ac	NOUN
ejpam-3427	196	7	)	)	PUNCT
ejpam-3427	196	8	,	,	PUNCT
ejpam-3427	196	9	µb(bd	µb(bd	NOUN
ejpam-3427	196	10	)	)	PUNCT
ejpam-3427	196	11	}	}	PUNCT
ejpam-3427	196	12	=	=	SYM
ejpam-3427	196	13	max{µa(ca	max{µa(ca	NOUN
ejpam-3427	196	14	)	)	PUNCT
ejpam-3427	196	15	,	,	PUNCT
ejpam-3427	196	16	µb(db	µb(db	ADJ
ejpam-3427	196	17	)	)	PUNCT
ejpam-3427	196	18	}	}	PUNCT
ejpam-3427	196	19	=	=	SYM
ejpam-3427	196	20	µa×b(ca	µa×b(ca	PROPN
ejpam-3427	196	21	,	,	PUNCT
ejpam-3427	196	22	db	db	PROPN
ejpam-3427	196	23	)	)	PUNCT
ejpam-3427	196	24	=	=	SYM
ejpam-3427	196	25	µa×b((c	µa×b((c	X
ejpam-3427	196	26	,	,	PUNCT
ejpam-3427	196	27	d	d	NOUN
ejpam-3427	196	28	)	)	PUNCT
ejpam-3427	196	29	◦	◦	NOUN
ejpam-3427	196	30	(	(	PUNCT
ejpam-3427	196	31	a	a	DET
ejpam-3427	196	32	,	,	PUNCT
ejpam-3427	196	33	b	b	NOUN
ejpam-3427	196	34	)	)	PUNCT
ejpam-3427	196	35	)	)	PUNCT
ejpam-3427	196	36	.	.	PUNCT
ejpam-3427	197	1	similarly	similarly	ADV
ejpam-3427	197	2	,	,	PUNCT
ejpam-3427	197	3	γa×b((a	γa×b((a	NOUN
ejpam-3427	197	4	,	,	PUNCT
ejpam-3427	197	5	b	b	NOUN
ejpam-3427	197	6	)	)	PUNCT
ejpam-3427	197	7	◦	◦	NOUN
ejpam-3427	197	8	(	(	PUNCT
ejpam-3427	197	9	c	c	X
ejpam-3427	197	10	,	,	PUNCT
ejpam-3427	197	11	d	d	NOUN
ejpam-3427	197	12	)	)	PUNCT
ejpam-3427	197	13	)	)	PUNCT
ejpam-3427	198	1	=	=	SYM
ejpam-3427	198	2	γa×b((c	γa×b((c	NOUN
ejpam-3427	198	3	,	,	PUNCT
ejpam-3427	198	4	d	d	NOUN
ejpam-3427	198	5	)	)	PUNCT
ejpam-3427	198	6	◦	◦	NOUN
ejpam-3427	198	7	(	(	PUNCT
ejpam-3427	198	8	a	a	DET
ejpam-3427	198	9	,	,	PUNCT
ejpam-3427	198	10	b	b	NOUN
ejpam-3427	198	11	)	)	PUNCT
ejpam-3427	198	12	)	)	PUNCT
ejpam-3427	198	13	.	.	PUNCT
ejpam-3427	199	1	hence	hence	ADV
ejpam-3427	199	2	a×	a×	PROPN
ejpam-3427	199	3	b	b	NOUN
ejpam-3427	199	4	is	be	AUX
ejpam-3427	199	5	an	an	DET
ejpam-3427	199	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	199	7	anti	anti	ADJ
ejpam-3427	199	8	fuzzy	fuzzy	ADJ
ejpam-3427	199	9	normal	normal	ADJ
ejpam-3427	199	10	la	la	NOUN
ejpam-3427	199	11	-	-	PUNCT
ejpam-3427	199	12	subring	subring	NOUN
ejpam-3427	199	13	of	of	ADP
ejpam-3427	199	14	an	an	DET
ejpam-3427	199	15	la	la	ADJ
ejpam-3427	199	16	-	-	PUNCT
ejpam-3427	199	17	ring	ring	NOUN
ejpam-3427	199	18	r1	r1	PROPN
ejpam-3427	199	19	×r2	×r2	PROPN
ejpam-3427	199	20	.	.	PUNCT
ejpam-3427	200	1	proposition	proposition	NOUN
ejpam-3427	200	2	2	2	NUM
ejpam-3427	200	3	.	.	PUNCT
ejpam-3427	201	1	if	if	SCONJ
ejpam-3427	201	2	x	x	PROPN
ejpam-3427	201	3	=	=	PUNCT
ejpam-3427	201	4	a	a	DET
ejpam-3427	201	5	×	×	PROPN
ejpam-3427	201	6	b	b	NOUN
ejpam-3427	201	7	and	and	CCONJ
ejpam-3427	201	8	y	y	PROPN
ejpam-3427	202	1	=	=	SYM
ejpam-3427	202	2	c	c	PROPN
ejpam-3427	202	3	×d	×d	NOUN
ejpam-3427	202	4	are	be	AUX
ejpam-3427	202	5	two	two	NUM
ejpam-3427	202	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	202	7	anti	anti	ADJ
ejpam-3427	202	8	fuzzy	fuzzy	ADJ
ejpam-3427	202	9	normal	normal	ADJ
ejpam-3427	202	10	la	la	NOUN
ejpam-3427	202	11	-	-	PUNCT
ejpam-3427	202	12	subrings	subring	NOUN
ejpam-3427	202	13	of	of	ADP
ejpam-3427	202	14	an	an	DET
ejpam-3427	202	15	la	la	ADJ
ejpam-3427	202	16	-	-	PUNCT
ejpam-3427	202	17	ring	ring	NOUN
ejpam-3427	202	18	r1×r2	r1×r2	PROPN
ejpam-3427	202	19	,	,	PUNCT
ejpam-3427	202	20	then	then	ADV
ejpam-3427	202	21	their	their	PRON
ejpam-3427	202	22	intersection	intersection	NOUN
ejpam-3427	202	23	x	x	X
ejpam-3427	202	24	∩y	∩y	NOUN
ejpam-3427	202	25	is	be	AUX
ejpam-3427	202	26	also	also	ADV
ejpam-3427	202	27	an	an	DET
ejpam-3427	202	28	intuitionistic	intuitionistic	ADJ
ejpam-3427	202	29	anti	anti	ADJ
ejpam-3427	202	30	fuzzy	fuzzy	ADJ
ejpam-3427	202	31	normal	normal	ADJ
ejpam-3427	202	32	la	la	NOUN
ejpam-3427	202	33	-	-	PUNCT
ejpam-3427	202	34	subring	subring	NOUN
ejpam-3427	202	35	of	of	ADP
ejpam-3427	202	36	an	an	DET
ejpam-3427	202	37	la	la	ADJ
ejpam-3427	202	38	-	-	PUNCT
ejpam-3427	202	39	ring	ring	NOUN
ejpam-3427	202	40	r1	r1	PROPN
ejpam-3427	202	41	×r2	×r2	PROPN
ejpam-3427	202	42	.	.	PUNCT
ejpam-3427	203	1	proof	proof	NOUN
ejpam-3427	203	2	.	.	PUNCT
ejpam-3427	204	1	let	let	VERB
ejpam-3427	204	2	x	x	X
ejpam-3427	204	3	=	=	PUNCT
ejpam-3427	204	4	a×b	a×b	PROPN
ejpam-3427	204	5	=	=	PRON
ejpam-3427	204	6	{	{	PUNCT
ejpam-3427	204	7	(	(	PUNCT
ejpam-3427	204	8	(	(	PUNCT
ejpam-3427	204	9	x1	x1	PROPN
ejpam-3427	204	10	,	,	PUNCT
ejpam-3427	204	11	x2	x2	PROPN
ejpam-3427	204	12	)	)	PUNCT
ejpam-3427	204	13	,	,	PUNCT
ejpam-3427	204	14	µa×b(x1	µa×b(x1	X
ejpam-3427	204	15	,	,	PUNCT
ejpam-3427	204	16	x2	x2	PROPN
ejpam-3427	204	17	)	)	PUNCT
ejpam-3427	204	18	,	,	PUNCT
ejpam-3427	204	19	γa×b(x1	γa×b(x1	NUM
ejpam-3427	204	20	,	,	PUNCT
ejpam-3427	204	21	x2	x2	PROPN
ejpam-3427	204	22	)	)	PUNCT
ejpam-3427	204	23	)	)	PUNCT
ejpam-3427	204	24	|	|	ADV
ejpam-3427	204	25	for	for	ADP
ejpam-3427	204	26	all	all	PRON
ejpam-3427	204	27	(	(	PUNCT
ejpam-3427	204	28	x1	x1	PROPN
ejpam-3427	204	29	,	,	PUNCT
ejpam-3427	204	30	x2	x2	ADJ
ejpam-3427	204	31	)	)	PUNCT
ejpam-3427	204	32	∈	∈	PROPN
ejpam-3427	204	33	r1×	r1×	NOUN
ejpam-3427	204	34	r2	r2	PROPN
ejpam-3427	204	35	}	}	PUNCT
ejpam-3427	204	36	and	and	CCONJ
ejpam-3427	204	37	y	y	PROPN
ejpam-3427	204	38	=	=	SYM
ejpam-3427	204	39	c	c	NOUN
ejpam-3427	204	40	×d	×d	NOUN
ejpam-3427	204	41	=	=	SYM
ejpam-3427	204	42	{	{	PUNCT
ejpam-3427	204	43	(	(	PUNCT
ejpam-3427	204	44	(	(	PUNCT
ejpam-3427	204	45	y1	y1	INTJ
ejpam-3427	204	46	,	,	PUNCT
ejpam-3427	204	47	y2	y2	PROPN
ejpam-3427	204	48	)	)	PUNCT
ejpam-3427	204	49	,	,	PUNCT
ejpam-3427	204	50	µc×d(y1	µc×d(y1	X
ejpam-3427	204	51	,	,	PUNCT
ejpam-3427	204	52	y2	y2	PROPN
ejpam-3427	204	53	)	)	PUNCT
ejpam-3427	204	54	,	,	PUNCT
ejpam-3427	204	55	γc×d(y1	γc×d(y1	NUM
ejpam-3427	204	56	,	,	PUNCT
ejpam-3427	204	57	y2	y2	PROPN
ejpam-3427	204	58	)	)	PUNCT
ejpam-3427	204	59	)	)	PUNCT
ejpam-3427	205	1	|	|	ADV
ejpam-3427	205	2	for	for	ADP
ejpam-3427	205	3	all	all	DET
ejpam-3427	205	4	(	(	PUNCT
ejpam-3427	205	5	y1	y1	NOUN
ejpam-3427	205	6	,	,	PUNCT
ejpam-3427	205	7	y2	y2	NOUN
ejpam-3427	205	8	)	)	PUNCT
ejpam-3427	205	9	∈	∈	PROPN
ejpam-3427	205	10	r1	r1	PROPN
ejpam-3427	205	11	×r2	×r2	PROPN
ejpam-3427	205	12	}	}	PUNCT
ejpam-3427	205	13	be	be	VERB
ejpam-3427	205	14	two	two	NUM
ejpam-3427	205	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	205	16	anti	anti	ADJ
ejpam-3427	205	17	fuzzy	fuzzy	ADJ
ejpam-3427	205	18	normal	normal	ADJ
ejpam-3427	205	19	la	la	NOUN
ejpam-3427	205	20	-	-	PUNCT
ejpam-3427	205	21	subrings	subring	NOUN
ejpam-3427	205	22	of	of	ADP
ejpam-3427	205	23	an	an	DET
ejpam-3427	205	24	la	la	ADJ
ejpam-3427	205	25	-	-	PUNCT
ejpam-3427	205	26	ring	ring	NOUN
ejpam-3427	205	27	r1×r2	r1×r2	PROPN
ejpam-3427	205	28	.	.	PUNCT
ejpam-3427	206	1	let	let	VERB
ejpam-3427	206	2	z	z	NOUN
ejpam-3427	206	3	=	=	PUNCT
ejpam-3427	206	4	x∩y	x∩y	PROPN
ejpam-3427	206	5	and	and	CCONJ
ejpam-3427	206	6	z	z	NOUN
ejpam-3427	206	7	=	=	PRON
ejpam-3427	206	8	{	{	PUNCT
ejpam-3427	206	9	(	(	PUNCT
ejpam-3427	206	10	(	(	PUNCT
ejpam-3427	206	11	z1	z1	PROPN
ejpam-3427	206	12	,	,	PUNCT
ejpam-3427	206	13	z2	z2	PROPN
ejpam-3427	206	14	)	)	PUNCT
ejpam-3427	206	15	,	,	PUNCT
ejpam-3427	206	16	µz(z1	µz(z1	ADJ
ejpam-3427	206	17	,	,	PUNCT
ejpam-3427	206	18	z2	z2	NOUN
ejpam-3427	206	19	)	)	PUNCT
ejpam-3427	206	20	,	,	PUNCT
ejpam-3427	206	21	γz(z1	γz(z1	NOUN
ejpam-3427	206	22	,	,	PUNCT
ejpam-3427	206	23	z2	z2	NOUN
ejpam-3427	206	24	)	)	PUNCT
ejpam-3427	206	25	)	)	PUNCT
ejpam-3427	207	1	|	|	ADV
ejpam-3427	207	2	(	(	PUNCT
ejpam-3427	207	3	z1	z1	PROPN
ejpam-3427	207	4	,	,	PUNCT
ejpam-3427	207	5	z2	z2	NUM
ejpam-3427	207	6	)	)	PUNCT
ejpam-3427	207	7	∈	∈	PROPN
ejpam-3427	207	8	r1	r1	PROPN
ejpam-3427	207	9	×r2	×r2	PROPN
ejpam-3427	207	10	}	}	PUNCT
ejpam-3427	207	11	,	,	PUNCT
ejpam-3427	207	12	where	where	SCONJ
ejpam-3427	207	13	µz(z1	µz(z1	ADJ
ejpam-3427	207	14	,	,	PUNCT
ejpam-3427	207	15	z2	z2	NUM
ejpam-3427	207	16	)	)	PUNCT
ejpam-3427	207	17	=	=	PUNCT
ejpam-3427	208	1	µx∩y	µx∩y	NOUN
ejpam-3427	208	2	(	(	PUNCT
ejpam-3427	208	3	z1	z1	NOUN
ejpam-3427	208	4	,	,	PUNCT
ejpam-3427	208	5	z2	z2	NUM
ejpam-3427	208	6	)	)	PUNCT
ejpam-3427	208	7	=	=	SYM
ejpam-3427	208	8	max{µx(z1	max{µx(z1	PROPN
ejpam-3427	208	9	,	,	PUNCT
ejpam-3427	208	10	z2	z2	PROPN
ejpam-3427	208	11	)	)	PUNCT
ejpam-3427	208	12	,	,	PUNCT
ejpam-3427	208	13	µy	µy	X
ejpam-3427	208	14	(	(	PUNCT
ejpam-3427	208	15	z1	z1	PROPN
ejpam-3427	208	16	,	,	PUNCT
ejpam-3427	208	17	z2	z2	NOUN
ejpam-3427	208	18	)	)	PUNCT
ejpam-3427	208	19	}	}	PUNCT
ejpam-3427	208	20	and	and	CCONJ
ejpam-3427	208	21	γz(z1	γz(z1	NOUN
ejpam-3427	208	22	,	,	PUNCT
ejpam-3427	208	23	z2	z2	NUM
ejpam-3427	208	24	)	)	PUNCT
ejpam-3427	208	25	=	=	PRON
ejpam-3427	208	26	γx∩y	γx∩y	X
ejpam-3427	208	27	(	(	PUNCT
ejpam-3427	208	28	z1	z1	NOUN
ejpam-3427	208	29	,	,	PUNCT
ejpam-3427	208	30	z2	z2	NUM
ejpam-3427	208	31	)	)	PUNCT
ejpam-3427	208	32	=	=	SYM
ejpam-3427	208	33	min{γx(z1	min{γx(z1	NOUN
ejpam-3427	208	34	,	,	PUNCT
ejpam-3427	208	35	z2	z2	PROPN
ejpam-3427	208	36	)	)	PUNCT
ejpam-3427	208	37	,	,	PUNCT
ejpam-3427	208	38	γy	γy	PROPN
ejpam-3427	208	39	(	(	PUNCT
ejpam-3427	208	40	z1	z1	PROPN
ejpam-3427	208	41	,	,	PUNCT
ejpam-3427	208	42	z2	z2	PROPN
ejpam-3427	208	43	)	)	PUNCT
ejpam-3427	208	44	}	}	PUNCT
ejpam-3427	208	45	.	.	PUNCT
ejpam-3427	209	1	now	now	ADV
ejpam-3427	209	2	µz((z1	µz((z1	NOUN
ejpam-3427	209	3	,	,	PUNCT
ejpam-3427	209	4	z2)−	z2)−	X
ejpam-3427	209	5	(	(	PUNCT
ejpam-3427	209	6	z3	z3	PROPN
ejpam-3427	209	7	,	,	PUNCT
ejpam-3427	209	8	z4	z4	PROPN
ejpam-3427	209	9	)	)	PUNCT
ejpam-3427	209	10	)	)	PUNCT
ejpam-3427	210	1	=	=	PUNCT
ejpam-3427	210	2	µx∩y	µx∩y	NOUN
ejpam-3427	210	3	(	(	PUNCT
ejpam-3427	210	4	(	(	PUNCT
ejpam-3427	210	5	z1	z1	PROPN
ejpam-3427	210	6	,	,	PUNCT
ejpam-3427	210	7	z2)−	z2)−	PROPN
ejpam-3427	210	8	(	(	PUNCT
ejpam-3427	210	9	z3	z3	PROPN
ejpam-3427	210	10	,	,	PUNCT
ejpam-3427	210	11	z4	z4	PROPN
ejpam-3427	210	12	)	)	PUNCT
ejpam-3427	210	13	)	)	PUNCT
ejpam-3427	211	1	=	=	SYM
ejpam-3427	211	2	max{µx((z1	max{µx((z1	PROPN
ejpam-3427	211	3	,	,	PUNCT
ejpam-3427	211	4	z2)−	z2)−	X
ejpam-3427	211	5	(	(	PUNCT
ejpam-3427	211	6	z3	z3	PROPN
ejpam-3427	211	7	,	,	PUNCT
ejpam-3427	211	8	z4	z4	PROPN
ejpam-3427	211	9	)	)	PUNCT
ejpam-3427	211	10	)	)	PUNCT
ejpam-3427	211	11	,	,	PUNCT
ejpam-3427	211	12	µy	µy	X
ejpam-3427	211	13	(	(	PUNCT
ejpam-3427	211	14	(	(	PUNCT
ejpam-3427	211	15	z1	z1	PROPN
ejpam-3427	211	16	,	,	PUNCT
ejpam-3427	211	17	z2)−	z2)−	PROPN
ejpam-3427	211	18	(	(	PUNCT
ejpam-3427	211	19	z3	z3	PROPN
ejpam-3427	211	20	,	,	PUNCT
ejpam-3427	211	21	z4	z4	PROPN
ejpam-3427	211	22	)	)	PUNCT
ejpam-3427	211	23	)	)	PUNCT
ejpam-3427	211	24	}	}	PUNCT
ejpam-3427	211	25	≤	≤	NOUN
ejpam-3427	211	26	{	{	PUNCT
ejpam-3427	211	27	µx(z1	µx(z1	NOUN
ejpam-3427	211	28	,	,	PUNCT
ejpam-3427	211	29	z2	z2	NUM
ejpam-3427	211	30	)	)	PUNCT
ejpam-3427	211	31	∨	∨	NUM
ejpam-3427	211	32	µx(z3	µx(z3	NOUN
ejpam-3427	211	33	,	,	PUNCT
ejpam-3427	211	34	z4	z4	PROPN
ejpam-3427	211	35	)	)	PUNCT
ejpam-3427	211	36	}	}	PUNCT
ejpam-3427	211	37	∨	∨	X
ejpam-3427	211	38	{	{	PUNCT
ejpam-3427	211	39	µy	µy	X
ejpam-3427	211	40	(	(	PUNCT
ejpam-3427	211	41	z1	z1	PROPN
ejpam-3427	211	42	,	,	PUNCT
ejpam-3427	211	43	z2	z2	PROPN
ejpam-3427	211	44	)	)	PUNCT
ejpam-3427	211	45	∨	∨	NUM
ejpam-3427	211	46	µy	µy	X
ejpam-3427	211	47	(	(	PUNCT
ejpam-3427	211	48	z3	z3	PROPN
ejpam-3427	211	49	,	,	PUNCT
ejpam-3427	211	50	z4	z4	PROPN
ejpam-3427	211	51	)	)	PUNCT
ejpam-3427	211	52	}	}	PUNCT
ejpam-3427	211	53	k.	k.	ADV
ejpam-3427	212	1	nasreen	nasreen	PROPN
ejpam-3427	212	2	/	/	SYM
ejpam-3427	212	3	eur	eur	PROPN
ejpam-3427	212	4	.	.	PUNCT
ejpam-3427	213	1	j.	j.	PROPN
ejpam-3427	213	2	pure	pure	PROPN
ejpam-3427	213	3	appl	appl	PROPN
ejpam-3427	213	4	.	.	PROPN
ejpam-3427	213	5	math	math	PROPN
ejpam-3427	213	6	,	,	PUNCT
ejpam-3427	213	7	12	12	NUM
ejpam-3427	213	8	(	(	PUNCT
ejpam-3427	213	9	2	2	NUM
ejpam-3427	213	10	)	)	PUNCT
ejpam-3427	213	11	(	(	PUNCT
ejpam-3427	213	12	2019	2019	NUM
ejpam-3427	213	13	)	)	PUNCT
ejpam-3427	213	14	,	,	PUNCT
ejpam-3427	213	15	622	622	NUM
ejpam-3427	213	16	-	-	SYM
ejpam-3427	213	17	648	648	NUM
ejpam-3427	213	18	629	629	NUM
ejpam-3427	213	19	=	=	SYM
ejpam-3427	213	20	{	{	PUNCT
ejpam-3427	213	21	µx(z1	µx(z1	NOUN
ejpam-3427	213	22	,	,	PUNCT
ejpam-3427	213	23	z2	z2	NUM
ejpam-3427	213	24	)	)	PUNCT
ejpam-3427	213	25	∨	∨	NOUN
ejpam-3427	213	26	{	{	PUNCT
ejpam-3427	213	27	µx(z3	µx(z3	NOUN
ejpam-3427	213	28	,	,	PUNCT
ejpam-3427	213	29	z4	z4	PROPN
ejpam-3427	213	30	)	)	PUNCT
ejpam-3427	213	31	∨	∨	NUM
ejpam-3427	213	32	µy	µy	X
ejpam-3427	213	33	(	(	PUNCT
ejpam-3427	213	34	z1	z1	PROPN
ejpam-3427	213	35	,	,	PUNCT
ejpam-3427	213	36	z2	z2	NOUN
ejpam-3427	213	37	)	)	PUNCT
ejpam-3427	213	38	}	}	PUNCT
ejpam-3427	213	39	∨	∨	NUM
ejpam-3427	213	40	µy	µy	X
ejpam-3427	213	41	(	(	PUNCT
ejpam-3427	213	42	z3	z3	PROPN
ejpam-3427	213	43	,	,	PUNCT
ejpam-3427	213	44	z4	z4	PROPN
ejpam-3427	213	45	)	)	PUNCT
ejpam-3427	213	46	}	}	PUNCT
ejpam-3427	213	47	=	=	SYM
ejpam-3427	213	48	{	{	PUNCT
ejpam-3427	213	49	µx(z1	µx(z1	NOUN
ejpam-3427	213	50	,	,	PUNCT
ejpam-3427	213	51	z2	z2	NUM
ejpam-3427	213	52	)	)	PUNCT
ejpam-3427	213	53	∨	∨	NOUN
ejpam-3427	213	54	{	{	PUNCT
ejpam-3427	213	55	µy	µy	X
ejpam-3427	213	56	(	(	PUNCT
ejpam-3427	213	57	z1	z1	PROPN
ejpam-3427	213	58	,	,	PUNCT
ejpam-3427	213	59	z2	z2	PROPN
ejpam-3427	213	60	)	)	PUNCT
ejpam-3427	213	61	∨	∨	NUM
ejpam-3427	213	62	µx(z3	µx(z3	NOUN
ejpam-3427	213	63	,	,	PUNCT
ejpam-3427	213	64	z4	z4	PROPN
ejpam-3427	213	65	)	)	PUNCT
ejpam-3427	213	66	}	}	PUNCT
ejpam-3427	213	67	∨	∨	NUM
ejpam-3427	213	68	µy	µy	X
ejpam-3427	213	69	(	(	PUNCT
ejpam-3427	213	70	z3	z3	PROPN
ejpam-3427	213	71	,	,	PUNCT
ejpam-3427	213	72	z4	z4	PROPN
ejpam-3427	213	73	)	)	PUNCT
ejpam-3427	213	74	}	}	PUNCT
ejpam-3427	213	75	=	=	SYM
ejpam-3427	213	76	{	{	PUNCT
ejpam-3427	213	77	µx(z1	µx(z1	NOUN
ejpam-3427	213	78	,	,	PUNCT
ejpam-3427	213	79	z2	z2	PROPN
ejpam-3427	213	80	)	)	PUNCT
ejpam-3427	213	81	∨	∨	NUM
ejpam-3427	213	82	µy	µy	X
ejpam-3427	213	83	(	(	PUNCT
ejpam-3427	213	84	z1	z1	PROPN
ejpam-3427	213	85	,	,	PUNCT
ejpam-3427	213	86	z2	z2	NOUN
ejpam-3427	213	87	)	)	PUNCT
ejpam-3427	213	88	}	}	PUNCT
ejpam-3427	213	89	∨	∨	X
ejpam-3427	213	90	{	{	PUNCT
ejpam-3427	213	91	µx(z3	µx(z3	NOUN
ejpam-3427	213	92	,	,	PUNCT
ejpam-3427	213	93	z4	z4	PROPN
ejpam-3427	213	94	)	)	PUNCT
ejpam-3427	213	95	∨	∨	NUM
ejpam-3427	213	96	µy	µy	X
ejpam-3427	213	97	(	(	PUNCT
ejpam-3427	213	98	z3	z3	PROPN
ejpam-3427	213	99	,	,	PUNCT
ejpam-3427	213	100	z4	z4	PROPN
ejpam-3427	213	101	)	)	PUNCT
ejpam-3427	213	102	}	}	PUNCT
ejpam-3427	213	103	=	=	SYM
ejpam-3427	213	104	max{µx∩y	max{µx∩y	NOUN
ejpam-3427	213	105	(	(	PUNCT
ejpam-3427	213	106	z1	z1	PROPN
ejpam-3427	213	107	,	,	PUNCT
ejpam-3427	213	108	z2	z2	PROPN
ejpam-3427	213	109	)	)	PUNCT
ejpam-3427	213	110	,	,	PUNCT
ejpam-3427	213	111	µx∩y	µx∩y	NOUN
ejpam-3427	213	112	(	(	PUNCT
ejpam-3427	213	113	z3	z3	PROPN
ejpam-3427	213	114	,	,	PUNCT
ejpam-3427	213	115	z4	z4	PROPN
ejpam-3427	213	116	)	)	PUNCT
ejpam-3427	213	117	}	}	PUNCT
ejpam-3427	213	118	=	=	SYM
ejpam-3427	213	119	max{µz(z1	max{µz(z1	PROPN
ejpam-3427	213	120	,	,	PUNCT
ejpam-3427	213	121	z2	z2	PROPN
ejpam-3427	213	122	)	)	PUNCT
ejpam-3427	213	123	,	,	PUNCT
ejpam-3427	213	124	µz(z3	µz(z3	NOUN
ejpam-3427	213	125	,	,	PUNCT
ejpam-3427	213	126	z4	z4	PROPN
ejpam-3427	213	127	)	)	PUNCT
ejpam-3427	213	128	}	}	PUNCT
ejpam-3427	213	129	and	and	CCONJ
ejpam-3427	213	130	µz((z1	µz((z1	NOUN
ejpam-3427	213	131	,	,	PUNCT
ejpam-3427	213	132	z2	z2	NUM
ejpam-3427	213	133	)	)	PUNCT
ejpam-3427	213	134	◦	◦	NOUN
ejpam-3427	213	135	(	(	PUNCT
ejpam-3427	213	136	z3	z3	PROPN
ejpam-3427	213	137	,	,	PUNCT
ejpam-3427	213	138	z4	z4	PROPN
ejpam-3427	213	139	)	)	PUNCT
ejpam-3427	213	140	)	)	PUNCT
ejpam-3427	214	1	=	=	PUNCT
ejpam-3427	214	2	µx∩y	µx∩y	NOUN
ejpam-3427	214	3	(	(	PUNCT
ejpam-3427	214	4	(	(	PUNCT
ejpam-3427	214	5	z1	z1	ADJ
ejpam-3427	214	6	,	,	PUNCT
ejpam-3427	214	7	z2	z2	PROPN
ejpam-3427	214	8	)	)	PUNCT
ejpam-3427	214	9	◦	◦	NOUN
ejpam-3427	214	10	(	(	PUNCT
ejpam-3427	214	11	z3	z3	PROPN
ejpam-3427	214	12	,	,	PUNCT
ejpam-3427	214	13	z4	z4	PROPN
ejpam-3427	214	14	)	)	PUNCT
ejpam-3427	214	15	)	)	PUNCT
ejpam-3427	215	1	=	=	SYM
ejpam-3427	215	2	max{µx((z1	max{µx((z1	PROPN
ejpam-3427	215	3	,	,	PUNCT
ejpam-3427	215	4	z2	z2	NOUN
ejpam-3427	215	5	)	)	PUNCT
ejpam-3427	215	6	◦	◦	NOUN
ejpam-3427	215	7	(	(	PUNCT
ejpam-3427	215	8	z3	z3	PROPN
ejpam-3427	215	9	,	,	PUNCT
ejpam-3427	215	10	z4	z4	PROPN
ejpam-3427	215	11	)	)	PUNCT
ejpam-3427	215	12	)	)	PUNCT
ejpam-3427	215	13	,	,	PUNCT
ejpam-3427	215	14	µy	µy	X
ejpam-3427	215	15	(	(	PUNCT
ejpam-3427	215	16	(	(	PUNCT
ejpam-3427	215	17	z1	z1	PROPN
ejpam-3427	215	18	,	,	PUNCT
ejpam-3427	215	19	z2	z2	PROPN
ejpam-3427	215	20	)	)	PUNCT
ejpam-3427	215	21	◦	◦	NOUN
ejpam-3427	215	22	(	(	PUNCT
ejpam-3427	215	23	z3	z3	PROPN
ejpam-3427	215	24	,	,	PUNCT
ejpam-3427	215	25	z4	z4	PROPN
ejpam-3427	215	26	)	)	PUNCT
ejpam-3427	215	27	)	)	PUNCT
ejpam-3427	215	28	}	}	PUNCT
ejpam-3427	215	29	≤	≤	NOUN
ejpam-3427	215	30	{	{	PUNCT
ejpam-3427	215	31	µx(z1	µx(z1	NOUN
ejpam-3427	215	32	,	,	PUNCT
ejpam-3427	215	33	z2	z2	NUM
ejpam-3427	215	34	)	)	PUNCT
ejpam-3427	215	35	∨	∨	NUM
ejpam-3427	215	36	µx(z3	µx(z3	NOUN
ejpam-3427	215	37	,	,	PUNCT
ejpam-3427	215	38	z4	z4	PROPN
ejpam-3427	215	39	)	)	PUNCT
ejpam-3427	215	40	}	}	PUNCT
ejpam-3427	215	41	∨	∨	X
ejpam-3427	215	42	{	{	PUNCT
ejpam-3427	215	43	µy	µy	X
ejpam-3427	215	44	(	(	PUNCT
ejpam-3427	215	45	z1	z1	PROPN
ejpam-3427	215	46	,	,	PUNCT
ejpam-3427	215	47	z2	z2	PROPN
ejpam-3427	215	48	)	)	PUNCT
ejpam-3427	215	49	∨	∨	NUM
ejpam-3427	215	50	µy	µy	X
ejpam-3427	215	51	(	(	PUNCT
ejpam-3427	215	52	z3	z3	PROPN
ejpam-3427	215	53	,	,	PUNCT
ejpam-3427	215	54	z4	z4	PROPN
ejpam-3427	215	55	)	)	PUNCT
ejpam-3427	215	56	}	}	PUNCT
ejpam-3427	215	57	=	=	SYM
ejpam-3427	215	58	{	{	PUNCT
ejpam-3427	215	59	µx(z1	µx(z1	NOUN
ejpam-3427	215	60	,	,	PUNCT
ejpam-3427	215	61	z2	z2	NUM
ejpam-3427	215	62	)	)	PUNCT
ejpam-3427	215	63	∨	∨	NOUN
ejpam-3427	215	64	{	{	PUNCT
ejpam-3427	215	65	µx(z3	µx(z3	NOUN
ejpam-3427	215	66	,	,	PUNCT
ejpam-3427	215	67	z4	z4	PROPN
ejpam-3427	215	68	)	)	PUNCT
ejpam-3427	215	69	∨	∨	NUM
ejpam-3427	215	70	µy	µy	X
ejpam-3427	215	71	(	(	PUNCT
ejpam-3427	215	72	z1	z1	PROPN
ejpam-3427	215	73	,	,	PUNCT
ejpam-3427	215	74	z2	z2	NOUN
ejpam-3427	215	75	)	)	PUNCT
ejpam-3427	215	76	}	}	PUNCT
ejpam-3427	215	77	∨	∨	NUM
ejpam-3427	215	78	µy	µy	X
ejpam-3427	215	79	(	(	PUNCT
ejpam-3427	215	80	z3	z3	PROPN
ejpam-3427	215	81	,	,	PUNCT
ejpam-3427	215	82	z4	z4	PROPN
ejpam-3427	215	83	)	)	PUNCT
ejpam-3427	215	84	}	}	PUNCT
ejpam-3427	215	85	=	=	SYM
ejpam-3427	215	86	{	{	PUNCT
ejpam-3427	215	87	µx(z1	µx(z1	NOUN
ejpam-3427	215	88	,	,	PUNCT
ejpam-3427	215	89	z2	z2	NUM
ejpam-3427	215	90	)	)	PUNCT
ejpam-3427	215	91	∨	∨	NOUN
ejpam-3427	215	92	{	{	PUNCT
ejpam-3427	215	93	µy	µy	X
ejpam-3427	215	94	(	(	PUNCT
ejpam-3427	215	95	z1	z1	PROPN
ejpam-3427	215	96	,	,	PUNCT
ejpam-3427	215	97	z2	z2	PROPN
ejpam-3427	215	98	)	)	PUNCT
ejpam-3427	215	99	∨	∨	NUM
ejpam-3427	215	100	µx(z3	µx(z3	NOUN
ejpam-3427	215	101	,	,	PUNCT
ejpam-3427	215	102	z4	z4	PROPN
ejpam-3427	215	103	)	)	PUNCT
ejpam-3427	215	104	}	}	PUNCT
ejpam-3427	215	105	∨	∨	NUM
ejpam-3427	215	106	µy	µy	X
ejpam-3427	215	107	(	(	PUNCT
ejpam-3427	215	108	z3	z3	PROPN
ejpam-3427	215	109	,	,	PUNCT
ejpam-3427	215	110	z4	z4	PROPN
ejpam-3427	215	111	)	)	PUNCT
ejpam-3427	215	112	}	}	PUNCT
ejpam-3427	215	113	=	=	SYM
ejpam-3427	215	114	{	{	PUNCT
ejpam-3427	215	115	µx(z1	µx(z1	NOUN
ejpam-3427	215	116	,	,	PUNCT
ejpam-3427	215	117	z2	z2	PROPN
ejpam-3427	215	118	)	)	PUNCT
ejpam-3427	215	119	∨	∨	NUM
ejpam-3427	215	120	µy	µy	X
ejpam-3427	215	121	(	(	PUNCT
ejpam-3427	215	122	z1	z1	PROPN
ejpam-3427	215	123	,	,	PUNCT
ejpam-3427	215	124	z2	z2	NOUN
ejpam-3427	215	125	)	)	PUNCT
ejpam-3427	215	126	}	}	PUNCT
ejpam-3427	215	127	∨	∨	X
ejpam-3427	215	128	{	{	PUNCT
ejpam-3427	215	129	µx(z3	µx(z3	NOUN
ejpam-3427	215	130	,	,	PUNCT
ejpam-3427	215	131	z4	z4	PROPN
ejpam-3427	215	132	)	)	PUNCT
ejpam-3427	215	133	∨	∨	NUM
ejpam-3427	215	134	µy	µy	X
ejpam-3427	215	135	(	(	PUNCT
ejpam-3427	215	136	z3	z3	PROPN
ejpam-3427	215	137	,	,	PUNCT
ejpam-3427	215	138	z4	z4	PROPN
ejpam-3427	215	139	)	)	PUNCT
ejpam-3427	215	140	}	}	PUNCT
ejpam-3427	215	141	=	=	SYM
ejpam-3427	215	142	max{µx∩y	max{µx∩y	NOUN
ejpam-3427	215	143	(	(	PUNCT
ejpam-3427	215	144	z1	z1	PROPN
ejpam-3427	215	145	,	,	PUNCT
ejpam-3427	215	146	z2	z2	PROPN
ejpam-3427	215	147	)	)	PUNCT
ejpam-3427	215	148	,	,	PUNCT
ejpam-3427	215	149	µx∩y	µx∩y	NOUN
ejpam-3427	215	150	(	(	PUNCT
ejpam-3427	215	151	z3	z3	PROPN
ejpam-3427	215	152	,	,	PUNCT
ejpam-3427	215	153	z4	z4	PROPN
ejpam-3427	215	154	)	)	PUNCT
ejpam-3427	215	155	}	}	PUNCT
ejpam-3427	215	156	=	=	SYM
ejpam-3427	215	157	max{µz(z1	max{µz(z1	PROPN
ejpam-3427	215	158	,	,	PUNCT
ejpam-3427	215	159	z2	z2	PROPN
ejpam-3427	215	160	)	)	PUNCT
ejpam-3427	215	161	,	,	PUNCT
ejpam-3427	215	162	µz(z3	µz(z3	NOUN
ejpam-3427	215	163	,	,	PUNCT
ejpam-3427	215	164	z4	z4	PROPN
ejpam-3427	215	165	)	)	PUNCT
ejpam-3427	215	166	}	}	PUNCT
ejpam-3427	215	167	.	.	PUNCT
ejpam-3427	216	1	thus	thus	ADV
ejpam-3427	216	2	µz((z1	µz((z1	NOUN
ejpam-3427	216	3	,	,	PUNCT
ejpam-3427	216	4	z2)−	z2)−	PROPN
ejpam-3427	216	5	(	(	PUNCT
ejpam-3427	216	6	z3	z3	PROPN
ejpam-3427	216	7	,	,	PUNCT
ejpam-3427	216	8	z4	z4	PROPN
ejpam-3427	216	9	)	)	PUNCT
ejpam-3427	216	10	)	)	PUNCT
ejpam-3427	216	11	≤	≤	NUM
ejpam-3427	216	12	max{µz(z1	max{µz(z1	PROPN
ejpam-3427	216	13	,	,	PUNCT
ejpam-3427	216	14	z2	z2	PROPN
ejpam-3427	216	15	)	)	PUNCT
ejpam-3427	216	16	,	,	PUNCT
ejpam-3427	216	17	µz(z3	µz(z3	NOUN
ejpam-3427	216	18	,	,	PUNCT
ejpam-3427	216	19	z4	z4	PROPN
ejpam-3427	216	20	)	)	PUNCT
ejpam-3427	216	21	}	}	PUNCT
ejpam-3427	216	22	and	and	CCONJ
ejpam-3427	216	23	µz((z1	µz((z1	NOUN
ejpam-3427	216	24	,	,	PUNCT
ejpam-3427	216	25	z2	z2	NUM
ejpam-3427	216	26	)	)	PUNCT
ejpam-3427	216	27	◦	◦	NOUN
ejpam-3427	216	28	(	(	PUNCT
ejpam-3427	216	29	z3	z3	PROPN
ejpam-3427	216	30	,	,	PUNCT
ejpam-3427	216	31	z4	z4	PROPN
ejpam-3427	216	32	)	)	PUNCT
ejpam-3427	216	33	)	)	PUNCT
ejpam-3427	216	34	≤	≤	NUM
ejpam-3427	216	35	max{µz(z1	max{µz(z1	PROPN
ejpam-3427	216	36	,	,	PUNCT
ejpam-3427	216	37	z2	z2	PROPN
ejpam-3427	216	38	)	)	PUNCT
ejpam-3427	216	39	,	,	PUNCT
ejpam-3427	216	40	µz(z3	µz(z3	NOUN
ejpam-3427	216	41	,	,	PUNCT
ejpam-3427	216	42	z4	z4	PROPN
ejpam-3427	216	43	)	)	PUNCT
ejpam-3427	216	44	}	}	PUNCT
ejpam-3427	216	45	.	.	PUNCT
ejpam-3427	217	1	similarly	similarly	ADV
ejpam-3427	217	2	,	,	PUNCT
ejpam-3427	217	3	we	we	PRON
ejpam-3427	217	4	have	have	VERB
ejpam-3427	217	5	γz((z1	γz((z1	NOUN
ejpam-3427	217	6	,	,	PUNCT
ejpam-3427	217	7	z2)−	z2)−	X
ejpam-3427	217	8	(	(	PUNCT
ejpam-3427	217	9	z3	z3	PROPN
ejpam-3427	217	10	,	,	PUNCT
ejpam-3427	217	11	z4	z4	PROPN
ejpam-3427	217	12	)	)	PUNCT
ejpam-3427	217	13	)	)	PUNCT
ejpam-3427	217	14	≥	≥	NOUN
ejpam-3427	217	15	min{γz(z1	min{γz(z1	PROPN
ejpam-3427	217	16	,	,	PUNCT
ejpam-3427	217	17	z2	z2	PROPN
ejpam-3427	217	18	)	)	PUNCT
ejpam-3427	217	19	,	,	PUNCT
ejpam-3427	217	20	γz(z3	γz(z3	PROPN
ejpam-3427	217	21	,	,	PUNCT
ejpam-3427	217	22	z4	z4	PROPN
ejpam-3427	217	23	)	)	PUNCT
ejpam-3427	217	24	}	}	PUNCT
ejpam-3427	217	25	and	and	CCONJ
ejpam-3427	217	26	γz((z1	γz((z1	NOUN
ejpam-3427	217	27	,	,	PUNCT
ejpam-3427	217	28	z2	z2	ADJ
ejpam-3427	217	29	)	)	PUNCT
ejpam-3427	217	30	◦	◦	NOUN
ejpam-3427	217	31	(	(	PUNCT
ejpam-3427	217	32	z3	z3	PROPN
ejpam-3427	217	33	,	,	PUNCT
ejpam-3427	217	34	z4	z4	PROPN
ejpam-3427	217	35	)	)	PUNCT
ejpam-3427	217	36	)	)	PUNCT
ejpam-3427	217	37	≥	≥	NOUN
ejpam-3427	217	38	min{γz(z1	min{γz(z1	PROPN
ejpam-3427	217	39	,	,	PUNCT
ejpam-3427	217	40	z2	z2	PROPN
ejpam-3427	217	41	)	)	PUNCT
ejpam-3427	217	42	,	,	PUNCT
ejpam-3427	217	43	γz(z3	γz(z3	PROPN
ejpam-3427	217	44	,	,	PUNCT
ejpam-3427	217	45	z4	z4	PROPN
ejpam-3427	217	46	)	)	PUNCT
ejpam-3427	217	47	}	}	PUNCT
ejpam-3427	217	48	.	.	PUNCT
ejpam-3427	218	1	therefore	therefore	ADV
ejpam-3427	218	2	z	z	X
ejpam-3427	218	3	=	=	SYM
ejpam-3427	218	4	(	(	PUNCT
ejpam-3427	218	5	µz	µz	PROPN
ejpam-3427	218	6	,	,	PUNCT
ejpam-3427	218	7	γz	γz	X
ejpam-3427	218	8	)	)	PUNCT
ejpam-3427	218	9	is	be	AUX
ejpam-3427	218	10	an	an	DET
ejpam-3427	218	11	intuitionistic	intuitionistic	ADJ
ejpam-3427	218	12	anti	anti	ADJ
ejpam-3427	218	13	fuzzy	fuzzy	ADJ
ejpam-3427	218	14	la	la	NOUN
ejpam-3427	218	15	-	-	PUNCT
ejpam-3427	218	16	subring	subring	NOUN
ejpam-3427	218	17	of	of	ADP
ejpam-3427	218	18	an	an	DET
ejpam-3427	218	19	la	la	ADJ
ejpam-3427	218	20	-	-	PUNCT
ejpam-3427	218	21	ring	ring	NOUN
ejpam-3427	218	22	r1	r1	PROPN
ejpam-3427	218	23	×r2	×r2	PROPN
ejpam-3427	218	24	.	.	PUNCT
ejpam-3427	219	1	now	now	ADV
ejpam-3427	219	2	µz((z1	µz((z1	NOUN
ejpam-3427	219	3	,	,	PUNCT
ejpam-3427	219	4	z2	z2	NUM
ejpam-3427	219	5	)	)	PUNCT
ejpam-3427	219	6	◦	◦	NOUN
ejpam-3427	219	7	(	(	PUNCT
ejpam-3427	219	8	z3	z3	PROPN
ejpam-3427	219	9	,	,	PUNCT
ejpam-3427	219	10	z4	z4	PROPN
ejpam-3427	219	11	)	)	PUNCT
ejpam-3427	219	12	)	)	PUNCT
ejpam-3427	220	1	=	=	PUNCT
ejpam-3427	220	2	µx∩y	µx∩y	NOUN
ejpam-3427	220	3	(	(	PUNCT
ejpam-3427	220	4	z1z3	z1z3	NOUN
ejpam-3427	220	5	,	,	PUNCT
ejpam-3427	220	6	z2z4	z2z4	NUM
ejpam-3427	220	7	)	)	PUNCT
ejpam-3427	220	8	=	=	SYM
ejpam-3427	220	9	max{µx(z1z3	max{µx(z1z3	X
ejpam-3427	220	10	,	,	PUNCT
ejpam-3427	220	11	z2z4	z2z4	NUM
ejpam-3427	220	12	)	)	PUNCT
ejpam-3427	220	13	,	,	PUNCT
ejpam-3427	220	14	µy	µy	X
ejpam-3427	220	15	(	(	PUNCT
ejpam-3427	220	16	z1z3	z1z3	NOUN
ejpam-3427	220	17	,	,	PUNCT
ejpam-3427	220	18	z2z4	z2z4	NOUN
ejpam-3427	220	19	)	)	PUNCT
ejpam-3427	220	20	}	}	PUNCT
ejpam-3427	220	21	=	=	SYM
ejpam-3427	220	22	max{µx(z3z1	max{µx(z3z1	NOUN
ejpam-3427	220	23	,	,	PUNCT
ejpam-3427	220	24	z4z2	z4z2	NOUN
ejpam-3427	220	25	)	)	PUNCT
ejpam-3427	220	26	,	,	PUNCT
ejpam-3427	220	27	µy	µy	X
ejpam-3427	220	28	(	(	PUNCT
ejpam-3427	220	29	z3z1	z3z1	INTJ
ejpam-3427	220	30	,	,	PUNCT
ejpam-3427	220	31	z4z2	z4z2	NOUN
ejpam-3427	220	32	)	)	PUNCT
ejpam-3427	220	33	}	}	PUNCT
ejpam-3427	220	34	=	=	PUNCT
ejpam-3427	220	35	µx∩y	µx∩y	NOUN
ejpam-3427	220	36	(	(	PUNCT
ejpam-3427	220	37	z3z1	z3z1	INTJ
ejpam-3427	220	38	,	,	PUNCT
ejpam-3427	220	39	z4z2	z4z2	X
ejpam-3427	220	40	)	)	PUNCT
ejpam-3427	220	41	=	=	SYM
ejpam-3427	220	42	µz((z3	µz((z3	PROPN
ejpam-3427	220	43	,	,	PUNCT
ejpam-3427	220	44	z4	z4	PROPN
ejpam-3427	220	45	)	)	PUNCT
ejpam-3427	220	46	◦	◦	NOUN
ejpam-3427	220	47	(	(	PUNCT
ejpam-3427	220	48	z1	z1	PROPN
ejpam-3427	220	49	,	,	PUNCT
ejpam-3427	220	50	z2	z2	PROPN
ejpam-3427	220	51	)	)	PUNCT
ejpam-3427	220	52	)	)	PUNCT
ejpam-3427	220	53	.	.	PUNCT
ejpam-3427	221	1	similarly	similarly	ADV
ejpam-3427	221	2	,	,	PUNCT
ejpam-3427	221	3	γz((z1	γz((z1	NOUN
ejpam-3427	221	4	,	,	PUNCT
ejpam-3427	221	5	z2	z2	ADJ
ejpam-3427	221	6	)	)	PUNCT
ejpam-3427	221	7	◦	◦	NOUN
ejpam-3427	221	8	(	(	PUNCT
ejpam-3427	221	9	z3	z3	PROPN
ejpam-3427	221	10	,	,	PUNCT
ejpam-3427	221	11	z4	z4	PROPN
ejpam-3427	221	12	)	)	PUNCT
ejpam-3427	221	13	)	)	PUNCT
ejpam-3427	222	1	=	=	SYM
ejpam-3427	222	2	γz((z3	γz((z3	PROPN
ejpam-3427	222	3	,	,	PUNCT
ejpam-3427	222	4	z4	z4	PROPN
ejpam-3427	222	5	)	)	PUNCT
ejpam-3427	222	6	◦	◦	NOUN
ejpam-3427	222	7	(	(	PUNCT
ejpam-3427	222	8	z1	z1	PROPN
ejpam-3427	222	9	,	,	PUNCT
ejpam-3427	222	10	z2	z2	PROPN
ejpam-3427	222	11	)	)	PUNCT
ejpam-3427	222	12	)	)	PUNCT
ejpam-3427	222	13	.	.	PUNCT
ejpam-3427	223	1	hence	hence	ADV
ejpam-3427	223	2	z	z	NOUN
ejpam-3427	223	3	=	=	PUNCT
ejpam-3427	223	4	x	x	NOUN
ejpam-3427	223	5	∩	∩	NOUN
ejpam-3427	223	6	y	y	PROPN
ejpam-3427	223	7	is	be	AUX
ejpam-3427	223	8	an	an	DET
ejpam-3427	223	9	intuitionistic	intuitionistic	ADJ
ejpam-3427	223	10	anti	anti	ADJ
ejpam-3427	223	11	fuzzy	fuzzy	ADJ
ejpam-3427	223	12	normal	normal	ADJ
ejpam-3427	223	13	la	la	NOUN
ejpam-3427	223	14	-	-	PUNCT
ejpam-3427	223	15	subring	subring	NOUN
ejpam-3427	223	16	of	of	ADP
ejpam-3427	223	17	an	an	DET
ejpam-3427	223	18	la	la	ADJ
ejpam-3427	223	19	-	-	PUNCT
ejpam-3427	223	20	ring	ring	NOUN
ejpam-3427	223	21	r1	r1	PROPN
ejpam-3427	223	22	×r2	×r2	PROPN
ejpam-3427	223	23	.	.	PUNCT
ejpam-3427	224	1	corollary	corollary	ADJ
ejpam-3427	224	2	2	2	NUM
ejpam-3427	224	3	.	.	PUNCT
ejpam-3427	225	1	if	if	SCONJ
ejpam-3427	225	2	{	{	PUNCT
ejpam-3427	225	3	ci}i∈i	ci}i∈i	NOUN
ejpam-3427	225	4	=	=	PUNCT
ejpam-3427	225	5	{	{	PUNCT
ejpam-3427	225	6	ai	ai	VERB
ejpam-3427	225	7	×	×	NOUN
ejpam-3427	225	8	bi}i∈i	bi}i∈i	VERB
ejpam-3427	225	9	is	be	AUX
ejpam-3427	225	10	a	a	DET
ejpam-3427	225	11	family	family	NOUN
ejpam-3427	225	12	of	of	ADP
ejpam-3427	225	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	225	14	anti	anti	ADJ
ejpam-3427	225	15	fuzzy	fuzzy	ADJ
ejpam-3427	225	16	normal	normal	ADJ
ejpam-3427	225	17	la	la	NOUN
ejpam-3427	225	18	-	-	PUNCT
ejpam-3427	225	19	subrings	subring	NOUN
ejpam-3427	225	20	of	of	ADP
ejpam-3427	225	21	an	an	DET
ejpam-3427	225	22	la	la	ADJ
ejpam-3427	225	23	-	-	PUNCT
ejpam-3427	225	24	ring	ring	NOUN
ejpam-3427	225	25	r1	r1	NOUN
ejpam-3427	225	26	×	×	NOUN
ejpam-3427	225	27	r2	r2	NOUN
ejpam-3427	225	28	,	,	PUNCT
ejpam-3427	225	29	then	then	ADV
ejpam-3427	225	30	c	c	NOUN
ejpam-3427	225	31	=	=	SYM
ejpam-3427	225	32	∩ci	∩ci	NOUN
ejpam-3427	225	33	is	be	AUX
ejpam-3427	225	34	also	also	ADV
ejpam-3427	225	35	an	an	DET
ejpam-3427	225	36	intuitionistic	intuitionistic	ADJ
ejpam-3427	225	37	anti	anti	ADJ
ejpam-3427	225	38	fuzzy	fuzzy	ADJ
ejpam-3427	225	39	normal	normal	ADJ
ejpam-3427	225	40	la	la	NOUN
ejpam-3427	225	41	-	-	PUNCT
ejpam-3427	225	42	subring	subring	NOUN
ejpam-3427	225	43	of	of	ADP
ejpam-3427	225	44	an	an	DET
ejpam-3427	225	45	la	la	ADJ
ejpam-3427	225	46	-	-	PUNCT
ejpam-3427	225	47	ring	ring	NOUN
ejpam-3427	225	48	r1	r1	PROPN
ejpam-3427	225	49	×r2	×r2	PROPN
ejpam-3427	225	50	.	.	PUNCT
ejpam-3427	226	1	k.	k.	PROPN
ejpam-3427	226	2	nasreen	nasreen	PROPN
ejpam-3427	226	3	/	/	SYM
ejpam-3427	226	4	eur	eur	PROPN
ejpam-3427	226	5	.	.	PUNCT
ejpam-3427	227	1	j.	j.	PROPN
ejpam-3427	227	2	pure	pure	PROPN
ejpam-3427	227	3	appl	appl	PROPN
ejpam-3427	227	4	.	.	PROPN
ejpam-3427	227	5	math	math	PROPN
ejpam-3427	227	6	,	,	PUNCT
ejpam-3427	227	7	12	12	NUM
ejpam-3427	227	8	(	(	PUNCT
ejpam-3427	227	9	2	2	NUM
ejpam-3427	227	10	)	)	PUNCT
ejpam-3427	227	11	(	(	PUNCT
ejpam-3427	227	12	2019	2019	NUM
ejpam-3427	227	13	)	)	PUNCT
ejpam-3427	227	14	,	,	PUNCT
ejpam-3427	227	15	622	622	NUM
ejpam-3427	227	16	-	-	SYM
ejpam-3427	227	17	648	648	NUM
ejpam-3427	227	18	630	630	NUM
ejpam-3427	227	19	theorem	theorem	NOUN
ejpam-3427	227	20	2	2	NUM
ejpam-3427	227	21	.	.	PUNCT
ejpam-3427	228	1	if	if	SCONJ
ejpam-3427	228	2	x	x	PROPN
ejpam-3427	228	3	=	=	PUNCT
ejpam-3427	228	4	a×b	a×b	PROPN
ejpam-3427	228	5	and	and	CCONJ
ejpam-3427	228	6	y	y	PROPN
ejpam-3427	228	7	=	=	PROPN
ejpam-3427	228	8	c×d	c×d	PROPN
ejpam-3427	228	9	are	be	AUX
ejpam-3427	228	10	intuitionistic	intuitionistic	ADJ
ejpam-3427	228	11	anti	anti	ADJ
ejpam-3427	228	12	fuzzy	fuzzy	ADJ
ejpam-3427	228	13	normal	normal	ADJ
ejpam-3427	228	14	la	la	ADJ
ejpam-3427	228	15	-	-	PUNCT
ejpam-3427	228	16	subrings	subring	NOUN
ejpam-3427	228	17	of	of	ADP
ejpam-3427	228	18	la	la	NOUN
ejpam-3427	228	19	-	-	PUNCT
ejpam-3427	228	20	rings	ring	NOUN
ejpam-3427	228	21	r′	r′	PROPN
ejpam-3427	228	22	=	=	PROPN
ejpam-3427	228	23	r1	r1	PROPN
ejpam-3427	228	24	×	×	NOUN
ejpam-3427	228	25	r2	r2	NOUN
ejpam-3427	228	26	and	and	CCONJ
ejpam-3427	228	27	r′′	r′′	VERB
ejpam-3427	228	28	=	=	SYM
ejpam-3427	228	29	r3	r3	PROPN
ejpam-3427	228	30	×	×	PROPN
ejpam-3427	228	31	r4	r4	NOUN
ejpam-3427	228	32	,	,	PUNCT
ejpam-3427	228	33	respectively	respectively	ADV
ejpam-3427	228	34	,	,	PUNCT
ejpam-3427	228	35	then	then	ADV
ejpam-3427	228	36	z	z	NOUN
ejpam-3427	229	1	=	=	PUNCT
ejpam-3427	229	2	x	x	SYM
ejpam-3427	229	3	×	×	NOUN
ejpam-3427	229	4	y	y	PROPN
ejpam-3427	229	5	is	be	AUX
ejpam-3427	229	6	also	also	ADV
ejpam-3427	229	7	an	an	DET
ejpam-3427	229	8	intuitionistic	intuitionistic	ADJ
ejpam-3427	229	9	anti	anti	ADJ
ejpam-3427	229	10	fuzzy	fuzzy	ADJ
ejpam-3427	229	11	normal	normal	ADJ
ejpam-3427	229	12	la	la	NOUN
ejpam-3427	229	13	-	-	PUNCT
ejpam-3427	229	14	subring	subring	NOUN
ejpam-3427	229	15	of	of	ADP
ejpam-3427	229	16	an	an	DET
ejpam-3427	229	17	la	la	ADJ
ejpam-3427	229	18	-	-	PUNCT
ejpam-3427	229	19	ring	ring	NOUN
ejpam-3427	229	20	r′×r′′	r′×r′′	VERB
ejpam-3427	229	21	=	=	SYM
ejpam-3427	229	22	(	(	PUNCT
ejpam-3427	229	23	r1×r2)×(r3×r4	r1×r2)×(r3×r4	NOUN
ejpam-3427	229	24	)	)	PUNCT
ejpam-3427	229	25	.	.	PUNCT
ejpam-3427	230	1	proof	proof	NOUN
ejpam-3427	230	2	.	.	PUNCT
ejpam-3427	231	1	let	let	VERB
ejpam-3427	231	2	x	x	PUNCT
ejpam-3427	231	3	=	=	PUNCT
ejpam-3427	231	4	a	a	DET
ejpam-3427	231	5	×	×	NOUN
ejpam-3427	231	6	b	b	X
ejpam-3427	231	7	=	=	PRON
ejpam-3427	231	8	{	{	PUNCT
ejpam-3427	231	9	(	(	PUNCT
ejpam-3427	231	10	(	(	PUNCT
ejpam-3427	231	11	x1	x1	PROPN
ejpam-3427	231	12	,	,	PUNCT
ejpam-3427	231	13	x2	x2	PROPN
ejpam-3427	231	14	)	)	PUNCT
ejpam-3427	231	15	,	,	PUNCT
ejpam-3427	231	16	µa×b(x1	µa×b(x1	X
ejpam-3427	231	17	,	,	PUNCT
ejpam-3427	231	18	x2	x2	PROPN
ejpam-3427	231	19	)	)	PUNCT
ejpam-3427	231	20	,	,	PUNCT
ejpam-3427	231	21	γa×b(x1	γa×b(x1	NUM
ejpam-3427	231	22	,	,	PUNCT
ejpam-3427	231	23	x2	x2	PROPN
ejpam-3427	231	24	)	)	PUNCT
ejpam-3427	231	25	)	)	PUNCT
ejpam-3427	231	26	|	|	ADV
ejpam-3427	231	27	for	for	ADP
ejpam-3427	231	28	all	all	DET
ejpam-3427	231	29	(	(	PUNCT
ejpam-3427	231	30	x1	x1	PROPN
ejpam-3427	231	31	,	,	PUNCT
ejpam-3427	231	32	x2	x2	ADJ
ejpam-3427	231	33	)	)	PUNCT
ejpam-3427	231	34	∈	∈	PROPN
ejpam-3427	231	35	r1	r1	NOUN
ejpam-3427	231	36	×	×	NOUN
ejpam-3427	231	37	r2	r2	PROPN
ejpam-3427	231	38	}	}	PUNCT
ejpam-3427	231	39	and	and	CCONJ
ejpam-3427	231	40	y	y	PROPN
ejpam-3427	231	41	=	=	SYM
ejpam-3427	232	1	c	c	PROPN
ejpam-3427	232	2	×	×	NOUN
ejpam-3427	232	3	d	d	NOUN
ejpam-3427	232	4	=	=	PRON
ejpam-3427	232	5	{	{	PUNCT
ejpam-3427	232	6	(	(	PUNCT
ejpam-3427	232	7	(	(	PUNCT
ejpam-3427	232	8	y1	y1	INTJ
ejpam-3427	232	9	,	,	PUNCT
ejpam-3427	232	10	y2	y2	PROPN
ejpam-3427	232	11	)	)	PUNCT
ejpam-3427	232	12	,	,	PUNCT
ejpam-3427	232	13	µc×d(y1	µc×d(y1	X
ejpam-3427	232	14	,	,	PUNCT
ejpam-3427	232	15	y2	y2	PROPN
ejpam-3427	232	16	)	)	PUNCT
ejpam-3427	232	17	,	,	PUNCT
ejpam-3427	232	18	γc×d(y1	γc×d(y1	NUM
ejpam-3427	232	19	,	,	PUNCT
ejpam-3427	232	20	y2	y2	PROPN
ejpam-3427	232	21	)	)	PUNCT
ejpam-3427	232	22	)	)	PUNCT
ejpam-3427	233	1	|	|	ADV
ejpam-3427	233	2	for	for	ADP
ejpam-3427	233	3	all	all	DET
ejpam-3427	233	4	(	(	PUNCT
ejpam-3427	233	5	y1	y1	NOUN
ejpam-3427	233	6	,	,	PUNCT
ejpam-3427	233	7	y2	y2	NOUN
ejpam-3427	233	8	)	)	PUNCT
ejpam-3427	233	9	∈	∈	PROPN
ejpam-3427	233	10	r3	r3	PROPN
ejpam-3427	233	11	×	×	PROPN
ejpam-3427	233	12	r4	r4	PROPN
ejpam-3427	233	13	}	}	PUNCT
ejpam-3427	233	14	be	be	AUX
ejpam-3427	233	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	233	16	anti	anti	ADJ
ejpam-3427	233	17	fuzzy	fuzzy	ADJ
ejpam-3427	233	18	normal	normal	ADJ
ejpam-3427	233	19	la	la	ADJ
ejpam-3427	233	20	-	-	PUNCT
ejpam-3427	233	21	subrings	subring	NOUN
ejpam-3427	233	22	of	of	ADP
ejpam-3427	233	23	la	la	NOUN
ejpam-3427	233	24	-	-	PUNCT
ejpam-3427	233	25	rings	ring	NOUN
ejpam-3427	233	26	r′	r′	PROPN
ejpam-3427	233	27	=	=	PROPN
ejpam-3427	233	28	r1	r1	PROPN
ejpam-3427	233	29	×	×	NOUN
ejpam-3427	233	30	r2	r2	NOUN
ejpam-3427	233	31	and	and	CCONJ
ejpam-3427	233	32	r′′	r′′	VERB
ejpam-3427	233	33	=	=	SYM
ejpam-3427	233	34	r3	r3	PROPN
ejpam-3427	233	35	×	×	PROPN
ejpam-3427	233	36	r4	r4	NOUN
ejpam-3427	233	37	,	,	PUNCT
ejpam-3427	233	38	respectively	respectively	ADV
ejpam-3427	233	39	.	.	PUNCT
ejpam-3427	234	1	let	let	VERB
ejpam-3427	234	2	z	z	NOUN
ejpam-3427	235	1	=	=	PUNCT
ejpam-3427	236	1	x	x	SYM
ejpam-3427	236	2	×	×	PROPN
ejpam-3427	236	3	y	y	PROPN
ejpam-3427	236	4	and	and	CCONJ
ejpam-3427	236	5	z	z	NOUN
ejpam-3427	236	6	=	=	PRON
ejpam-3427	236	7	{	{	PUNCT
ejpam-3427	236	8	(	(	PUNCT
ejpam-3427	236	9	(	(	PUNCT
ejpam-3427	236	10	z′	z′	NOUN
ejpam-3427	236	11	,	,	PUNCT
ejpam-3427	236	12	z′′	z′′	NOUN
ejpam-3427	236	13	)	)	PUNCT
ejpam-3427	236	14	,	,	PUNCT
ejpam-3427	236	15	µz(z′	µz(z′	NOUN
ejpam-3427	236	16	,	,	PUNCT
ejpam-3427	236	17	z′′	z′′	NOUN
ejpam-3427	236	18	)	)	PUNCT
ejpam-3427	236	19	,	,	PUNCT
ejpam-3427	236	20	γz(z′	γz(z′	PROPN
ejpam-3427	236	21	,	,	PUNCT
ejpam-3427	236	22	z′′	z′′	NOUN
ejpam-3427	236	23	)	)	PUNCT
ejpam-3427	236	24	)	)	PUNCT
ejpam-3427	237	1	|	|	ADV
ejpam-3427	237	2	(	(	PUNCT
ejpam-3427	237	3	z′	z′	NOUN
ejpam-3427	237	4	,	,	PUNCT
ejpam-3427	237	5	z′′	z′′	NOUN
ejpam-3427	237	6	)	)	PUNCT
ejpam-3427	237	7	=	=	SYM
ejpam-3427	237	8	(	(	PUNCT
ejpam-3427	237	9	(	(	PUNCT
ejpam-3427	237	10	z1	z1	PROPN
ejpam-3427	237	11	,	,	PUNCT
ejpam-3427	237	12	z2	z2	PROPN
ejpam-3427	237	13	)	)	PUNCT
ejpam-3427	237	14	,	,	PUNCT
ejpam-3427	237	15	(	(	PUNCT
ejpam-3427	237	16	z3	z3	PROPN
ejpam-3427	237	17	,	,	PUNCT
ejpam-3427	237	18	z4	z4	PROPN
ejpam-3427	237	19	)	)	PUNCT
ejpam-3427	237	20	)	)	PUNCT
ejpam-3427	238	1	∈	∈	PROPN
ejpam-3427	238	2	r′	r′	PRON
ejpam-3427	238	3	×r′′	×r′′	PROPN
ejpam-3427	238	4	}	}	PUNCT
ejpam-3427	238	5	,	,	PUNCT
ejpam-3427	238	6	where	where	SCONJ
ejpam-3427	238	7	µz(z′	µz(z′	NOUN
ejpam-3427	238	8	,	,	PUNCT
ejpam-3427	238	9	z′′	z′′	NOUN
ejpam-3427	238	10	)	)	PUNCT
ejpam-3427	238	11	=	=	SYM
ejpam-3427	238	12	µx×y	µx×y	PROPN
ejpam-3427	238	13	(	(	PUNCT
ejpam-3427	238	14	(	(	PUNCT
ejpam-3427	238	15	z1	z1	PROPN
ejpam-3427	238	16	,	,	PUNCT
ejpam-3427	238	17	z2	z2	PROPN
ejpam-3427	238	18	)	)	PUNCT
ejpam-3427	238	19	,	,	PUNCT
ejpam-3427	238	20	(	(	PUNCT
ejpam-3427	238	21	z3	z3	PROPN
ejpam-3427	238	22	,	,	PUNCT
ejpam-3427	238	23	z4	z4	PROPN
ejpam-3427	238	24	)	)	PUNCT
ejpam-3427	238	25	)	)	PUNCT
ejpam-3427	239	1	=	=	SYM
ejpam-3427	239	2	max{µx(z1	max{µx(z1	PROPN
ejpam-3427	239	3	,	,	PUNCT
ejpam-3427	239	4	z2	z2	PROPN
ejpam-3427	239	5	)	)	PUNCT
ejpam-3427	239	6	,	,	PUNCT
ejpam-3427	239	7	µy	µy	X
ejpam-3427	239	8	(	(	PUNCT
ejpam-3427	239	9	z3	z3	PROPN
ejpam-3427	239	10	,	,	PUNCT
ejpam-3427	239	11	z4	z4	PROPN
ejpam-3427	239	12	)	)	PUNCT
ejpam-3427	239	13	}	}	PUNCT
ejpam-3427	239	14	,	,	PUNCT
ejpam-3427	239	15	and	and	CCONJ
ejpam-3427	239	16	γz(z′	γz(z′	PROPN
ejpam-3427	239	17	,	,	PUNCT
ejpam-3427	239	18	z′′	z′′	NOUN
ejpam-3427	239	19	)	)	PUNCT
ejpam-3427	239	20	=	=	SYM
ejpam-3427	239	21	γx×y	γx×y	NOUN
ejpam-3427	239	22	(	(	PUNCT
ejpam-3427	239	23	(	(	PUNCT
ejpam-3427	239	24	z1	z1	PROPN
ejpam-3427	239	25	,	,	PUNCT
ejpam-3427	239	26	z2	z2	PROPN
ejpam-3427	239	27	)	)	PUNCT
ejpam-3427	239	28	,	,	PUNCT
ejpam-3427	239	29	(	(	PUNCT
ejpam-3427	239	30	z3	z3	PROPN
ejpam-3427	239	31	,	,	PUNCT
ejpam-3427	239	32	z4	z4	PROPN
ejpam-3427	239	33	)	)	PUNCT
ejpam-3427	239	34	)	)	PUNCT
ejpam-3427	240	1	=	=	SYM
ejpam-3427	240	2	min{γx(z1	min{γx(z1	NOUN
ejpam-3427	240	3	,	,	PUNCT
ejpam-3427	240	4	z2	z2	PROPN
ejpam-3427	240	5	)	)	PUNCT
ejpam-3427	240	6	,	,	PUNCT
ejpam-3427	240	7	γy	γy	PROPN
ejpam-3427	240	8	(	(	PUNCT
ejpam-3427	240	9	z3	z3	PROPN
ejpam-3427	240	10	,	,	PUNCT
ejpam-3427	240	11	z4	z4	PROPN
ejpam-3427	240	12	)	)	PUNCT
ejpam-3427	240	13	}	}	PUNCT
ejpam-3427	240	14	.	.	PUNCT
ejpam-3427	241	1	now	now	ADV
ejpam-3427	241	2	µz(((z1	µz(((z1	PROPN
ejpam-3427	241	3	,	,	PUNCT
ejpam-3427	241	4	z2	z2	NOUN
ejpam-3427	241	5	)	)	PUNCT
ejpam-3427	241	6	,	,	PUNCT
ejpam-3427	241	7	(	(	PUNCT
ejpam-3427	241	8	z3	z3	PROPN
ejpam-3427	241	9	,	,	PUNCT
ejpam-3427	241	10	z4))−	z4))−	PROPN
ejpam-3427	241	11	(	(	PUNCT
ejpam-3427	241	12	(	(	PUNCT
ejpam-3427	241	13	z5	z5	X
ejpam-3427	241	14	,	,	PUNCT
ejpam-3427	241	15	z6	z6	PROPN
ejpam-3427	241	16	)	)	PUNCT
ejpam-3427	241	17	,	,	PUNCT
ejpam-3427	241	18	(	(	PUNCT
ejpam-3427	241	19	z7	z7	PROPN
ejpam-3427	241	20	,	,	PUNCT
ejpam-3427	241	21	z8	z8	NOUN
ejpam-3427	241	22	)	)	PUNCT
ejpam-3427	241	23	)	)	PUNCT
ejpam-3427	241	24	)	)	PUNCT
ejpam-3427	242	1	=	=	PRON
ejpam-3427	242	2	µx×y	µx×y	PROPN
ejpam-3427	242	3	(	(	PUNCT
ejpam-3427	242	4	(	(	PUNCT
ejpam-3427	242	5	(	(	PUNCT
ejpam-3427	242	6	z1	z1	PROPN
ejpam-3427	242	7	,	,	PUNCT
ejpam-3427	242	8	z2	z2	PROPN
ejpam-3427	242	9	)	)	PUNCT
ejpam-3427	242	10	,	,	PUNCT
ejpam-3427	242	11	(	(	PUNCT
ejpam-3427	242	12	z3	z3	PROPN
ejpam-3427	242	13	,	,	PUNCT
ejpam-3427	242	14	z4))−	z4))−	PROPN
ejpam-3427	242	15	(	(	PUNCT
ejpam-3427	242	16	(	(	PUNCT
ejpam-3427	242	17	z5	z5	X
ejpam-3427	242	18	,	,	PUNCT
ejpam-3427	242	19	z6	z6	PROPN
ejpam-3427	242	20	)	)	PUNCT
ejpam-3427	242	21	,	,	PUNCT
ejpam-3427	242	22	(	(	PUNCT
ejpam-3427	242	23	z7	z7	PROPN
ejpam-3427	242	24	,	,	PUNCT
ejpam-3427	242	25	z8	z8	NOUN
ejpam-3427	242	26	)	)	PUNCT
ejpam-3427	242	27	)	)	PUNCT
ejpam-3427	242	28	)	)	PUNCT
ejpam-3427	243	1	=	=	PRON
ejpam-3427	243	2	µx×y	µx×y	PROPN
ejpam-3427	243	3	(	(	PUNCT
ejpam-3427	243	4	(	(	PUNCT
ejpam-3427	243	5	(	(	PUNCT
ejpam-3427	243	6	z1	z1	PROPN
ejpam-3427	243	7	,	,	PUNCT
ejpam-3427	243	8	z2)−	z2)−	PROPN
ejpam-3427	243	9	(	(	PUNCT
ejpam-3427	243	10	z5	z5	PROPN
ejpam-3427	243	11	,	,	PUNCT
ejpam-3427	243	12	z6	z6	PROPN
ejpam-3427	243	13	)	)	PUNCT
ejpam-3427	243	14	)	)	PUNCT
ejpam-3427	243	15	,	,	PUNCT
ejpam-3427	243	16	(	(	PUNCT
ejpam-3427	243	17	(	(	PUNCT
ejpam-3427	243	18	z3	z3	PROPN
ejpam-3427	243	19	,	,	PUNCT
ejpam-3427	243	20	z4)−	z4)−	PROPN
ejpam-3427	243	21	(	(	PUNCT
ejpam-3427	243	22	z7	z7	PROPN
ejpam-3427	243	23	,	,	PUNCT
ejpam-3427	243	24	z8	z8	NOUN
ejpam-3427	243	25	)	)	PUNCT
ejpam-3427	243	26	)	)	PUNCT
ejpam-3427	243	27	)	)	PUNCT
ejpam-3427	244	1	=	=	SYM
ejpam-3427	244	2	max{µx((z1	max{µx((z1	PROPN
ejpam-3427	244	3	,	,	PUNCT
ejpam-3427	244	4	z2)−	z2)−	X
ejpam-3427	244	5	(	(	PUNCT
ejpam-3427	244	6	z5	z5	PROPN
ejpam-3427	244	7	,	,	PUNCT
ejpam-3427	244	8	z6	z6	PROPN
ejpam-3427	244	9	)	)	PUNCT
ejpam-3427	244	10	)	)	PUNCT
ejpam-3427	244	11	,	,	PUNCT
ejpam-3427	244	12	µy	µy	X
ejpam-3427	244	13	(	(	PUNCT
ejpam-3427	244	14	(	(	PUNCT
ejpam-3427	244	15	z3	z3	PROPN
ejpam-3427	244	16	,	,	PUNCT
ejpam-3427	244	17	z4)−	z4)−	PROPN
ejpam-3427	244	18	(	(	PUNCT
ejpam-3427	244	19	z7	z7	PROPN
ejpam-3427	244	20	,	,	PUNCT
ejpam-3427	244	21	z8	z8	NOUN
ejpam-3427	244	22	)	)	PUNCT
ejpam-3427	244	23	)	)	PUNCT
ejpam-3427	244	24	}	}	PUNCT
ejpam-3427	244	25	≤	≤	NUM
ejpam-3427	244	26	max{(µx(z1	max{(µx(z1	NOUN
ejpam-3427	244	27	,	,	PUNCT
ejpam-3427	244	28	z2	z2	PROPN
ejpam-3427	244	29	)	)	PUNCT
ejpam-3427	244	30	∨	∨	NUM
ejpam-3427	244	31	µx(z5	µx(z5	NOUN
ejpam-3427	244	32	,	,	PUNCT
ejpam-3427	244	33	z6	z6	PROPN
ejpam-3427	244	34	)	)	PUNCT
ejpam-3427	244	35	)	)	PUNCT
ejpam-3427	244	36	,	,	PUNCT
ejpam-3427	244	37	(	(	PUNCT
ejpam-3427	244	38	µy	µy	X
ejpam-3427	244	39	(	(	PUNCT
ejpam-3427	244	40	z3	z3	PROPN
ejpam-3427	244	41	,	,	PUNCT
ejpam-3427	244	42	z4	z4	PROPN
ejpam-3427	244	43	)	)	PUNCT
ejpam-3427	244	44	∨	∨	PROPN
ejpam-3427	244	45	µy	µy	X
ejpam-3427	244	46	(	(	PUNCT
ejpam-3427	244	47	z7	z7	PROPN
ejpam-3427	244	48	,	,	PUNCT
ejpam-3427	244	49	z8	z8	NOUN
ejpam-3427	244	50	)	)	PUNCT
ejpam-3427	244	51	)	)	PUNCT
ejpam-3427	244	52	}	}	PUNCT
ejpam-3427	244	53	=	=	SYM
ejpam-3427	244	54	(	(	PUNCT
ejpam-3427	244	55	(	(	PUNCT
ejpam-3427	244	56	µx(z1	µx(z1	NOUN
ejpam-3427	244	57	,	,	PUNCT
ejpam-3427	244	58	z2	z2	PROPN
ejpam-3427	244	59	)	)	PUNCT
ejpam-3427	244	60	∨	∨	PROPN
ejpam-3427	244	61	µy	µy	X
ejpam-3427	244	62	(	(	PUNCT
ejpam-3427	244	63	z5	z5	X
ejpam-3427	244	64	,	,	PUNCT
ejpam-3427	244	65	z6	z6	PROPN
ejpam-3427	244	66	)	)	PUNCT
ejpam-3427	244	67	)	)	PUNCT
ejpam-3427	244	68	∨	∨	NUM
ejpam-3427	244	69	(	(	PUNCT
ejpam-3427	244	70	µx(z3	µx(z3	ADJ
ejpam-3427	244	71	,	,	PUNCT
ejpam-3427	244	72	z4	z4	PROPN
ejpam-3427	244	73	)	)	PUNCT
ejpam-3427	244	74	∨	∨	PROPN
ejpam-3427	244	75	µy	µy	X
ejpam-3427	244	76	(	(	PUNCT
ejpam-3427	244	77	z7	z7	PROPN
ejpam-3427	244	78	,	,	PUNCT
ejpam-3427	244	79	z8	z8	NOUN
ejpam-3427	244	80	)	)	PUNCT
ejpam-3427	244	81	)	)	PUNCT
ejpam-3427	244	82	)	)	PUNCT
ejpam-3427	245	1	=	=	PUNCT
ejpam-3427	245	2	(	(	PUNCT
ejpam-3427	245	3	(	(	PUNCT
ejpam-3427	245	4	µx(z1	µx(z1	NOUN
ejpam-3427	245	5	,	,	PUNCT
ejpam-3427	245	6	z2	z2	PROPN
ejpam-3427	245	7	)	)	PUNCT
ejpam-3427	245	8	∨	∨	NUM
ejpam-3427	245	9	µy	µy	X
ejpam-3427	245	10	(	(	PUNCT
ejpam-3427	245	11	z3	z3	PROPN
ejpam-3427	245	12	,	,	PUNCT
ejpam-3427	245	13	z4	z4	PROPN
ejpam-3427	245	14	)	)	PUNCT
ejpam-3427	245	15	)	)	PUNCT
ejpam-3427	246	1	∨	∨	NUM
ejpam-3427	246	2	(	(	PUNCT
ejpam-3427	246	3	µx(z5	µx(z5	X
ejpam-3427	246	4	,	,	PUNCT
ejpam-3427	246	5	z6	z6	PROPN
ejpam-3427	246	6	)	)	PUNCT
ejpam-3427	246	7	∨	∨	PROPN
ejpam-3427	246	8	µy	µy	X
ejpam-3427	246	9	(	(	PUNCT
ejpam-3427	246	10	z7	z7	PROPN
ejpam-3427	246	11	,	,	PUNCT
ejpam-3427	246	12	z8	z8	NOUN
ejpam-3427	246	13	)	)	PUNCT
ejpam-3427	246	14	)	)	PUNCT
ejpam-3427	246	15	)	)	PUNCT
ejpam-3427	247	1	=	=	SYM
ejpam-3427	247	2	max{(µx(z1	max{(µx(z1	PROPN
ejpam-3427	247	3	,	,	PUNCT
ejpam-3427	247	4	z2	z2	PROPN
ejpam-3427	247	5	)	)	PUNCT
ejpam-3427	247	6	∨	∨	NUM
ejpam-3427	247	7	µy	µy	X
ejpam-3427	247	8	(	(	PUNCT
ejpam-3427	247	9	z3	z3	PROPN
ejpam-3427	247	10	,	,	PUNCT
ejpam-3427	247	11	z4	z4	PROPN
ejpam-3427	247	12	)	)	PUNCT
ejpam-3427	247	13	)	)	PUNCT
ejpam-3427	247	14	,	,	PUNCT
ejpam-3427	247	15	(	(	PUNCT
ejpam-3427	247	16	µx(z5	µx(z5	X
ejpam-3427	247	17	,	,	PUNCT
ejpam-3427	247	18	z6	z6	PROPN
ejpam-3427	247	19	)	)	PUNCT
ejpam-3427	247	20	∨	∨	PROPN
ejpam-3427	247	21	µy	µy	X
ejpam-3427	247	22	(	(	PUNCT
ejpam-3427	247	23	z7	z7	PROPN
ejpam-3427	247	24	,	,	PUNCT
ejpam-3427	247	25	z8	z8	NOUN
ejpam-3427	247	26	)	)	PUNCT
ejpam-3427	247	27	)	)	PUNCT
ejpam-3427	247	28	}	}	PUNCT
ejpam-3427	248	1	=	=	PUNCT
ejpam-3427	248	2	max{µx×y	max{µx×y	X
ejpam-3427	248	3	(	(	PUNCT
ejpam-3427	248	4	(	(	PUNCT
ejpam-3427	248	5	z1	z1	PROPN
ejpam-3427	248	6	,	,	PUNCT
ejpam-3427	248	7	z2	z2	PROPN
ejpam-3427	248	8	)	)	PUNCT
ejpam-3427	248	9	,	,	PUNCT
ejpam-3427	248	10	(	(	PUNCT
ejpam-3427	248	11	z3	z3	PROPN
ejpam-3427	248	12	,	,	PUNCT
ejpam-3427	248	13	z4	z4	PROPN
ejpam-3427	248	14	)	)	PUNCT
ejpam-3427	248	15	)	)	PUNCT
ejpam-3427	248	16	,	,	PUNCT
ejpam-3427	248	17	µx×y	µx×y	PROPN
ejpam-3427	248	18	(	(	PUNCT
ejpam-3427	248	19	(	(	PUNCT
ejpam-3427	248	20	z5	z5	X
ejpam-3427	248	21	,	,	PUNCT
ejpam-3427	248	22	z6	z6	PROPN
ejpam-3427	248	23	)	)	PUNCT
ejpam-3427	248	24	,	,	PUNCT
ejpam-3427	248	25	(	(	PUNCT
ejpam-3427	248	26	z7	z7	PROPN
ejpam-3427	248	27	,	,	PUNCT
ejpam-3427	248	28	z8	z8	NOUN
ejpam-3427	248	29	)	)	PUNCT
ejpam-3427	248	30	)	)	PUNCT
ejpam-3427	248	31	}	}	PUNCT
ejpam-3427	249	1	=	=	PUNCT
ejpam-3427	249	2	max{µz((z1	max{µz((z1	ADJ
ejpam-3427	249	3	,	,	PUNCT
ejpam-3427	249	4	z2	z2	NOUN
ejpam-3427	249	5	)	)	PUNCT
ejpam-3427	249	6	,	,	PUNCT
ejpam-3427	249	7	(	(	PUNCT
ejpam-3427	249	8	z3	z3	PROPN
ejpam-3427	249	9	,	,	PUNCT
ejpam-3427	249	10	z4	z4	PROPN
ejpam-3427	249	11	)	)	PUNCT
ejpam-3427	249	12	)	)	PUNCT
ejpam-3427	249	13	,	,	PUNCT
ejpam-3427	250	1	µz((z5	µz((z5	ADP
ejpam-3427	250	2	,	,	PUNCT
ejpam-3427	250	3	z6	z6	PROPN
ejpam-3427	250	4	)	)	PUNCT
ejpam-3427	250	5	,	,	PUNCT
ejpam-3427	250	6	(	(	PUNCT
ejpam-3427	250	7	z7	z7	PROPN
ejpam-3427	250	8	,	,	PUNCT
ejpam-3427	250	9	z8	z8	NOUN
ejpam-3427	250	10	)	)	PUNCT
ejpam-3427	250	11	)	)	PUNCT
ejpam-3427	250	12	}	}	PUNCT
ejpam-3427	250	13	.	.	PUNCT
ejpam-3427	251	1	and	and	CCONJ
ejpam-3427	251	2	µz(((z1	µz(((z1	PROPN
ejpam-3427	251	3	,	,	PUNCT
ejpam-3427	251	4	z2	z2	NOUN
ejpam-3427	251	5	)	)	PUNCT
ejpam-3427	251	6	,	,	PUNCT
ejpam-3427	251	7	(	(	PUNCT
ejpam-3427	251	8	z3	z3	PROPN
ejpam-3427	251	9	,	,	PUNCT
ejpam-3427	251	10	z4	z4	PROPN
ejpam-3427	251	11	)	)	PUNCT
ejpam-3427	251	12	)	)	PUNCT
ejpam-3427	252	1	◦	◦	NOUN
ejpam-3427	252	2	(	(	PUNCT
ejpam-3427	252	3	(	(	PUNCT
ejpam-3427	252	4	z5	z5	X
ejpam-3427	252	5	,	,	PUNCT
ejpam-3427	252	6	z6	z6	PROPN
ejpam-3427	252	7	)	)	PUNCT
ejpam-3427	252	8	,	,	PUNCT
ejpam-3427	252	9	(	(	PUNCT
ejpam-3427	252	10	z7	z7	PROPN
ejpam-3427	252	11	,	,	PUNCT
ejpam-3427	252	12	z8	z8	NOUN
ejpam-3427	252	13	)	)	PUNCT
ejpam-3427	252	14	)	)	PUNCT
ejpam-3427	252	15	)	)	PUNCT
ejpam-3427	253	1	=	=	PRON
ejpam-3427	253	2	µx×y	µx×y	PROPN
ejpam-3427	253	3	(	(	PUNCT
ejpam-3427	253	4	(	(	PUNCT
ejpam-3427	253	5	(	(	PUNCT
ejpam-3427	253	6	z1	z1	PROPN
ejpam-3427	253	7	,	,	PUNCT
ejpam-3427	253	8	z2	z2	PROPN
ejpam-3427	253	9	)	)	PUNCT
ejpam-3427	253	10	,	,	PUNCT
ejpam-3427	253	11	(	(	PUNCT
ejpam-3427	253	12	z3	z3	PROPN
ejpam-3427	253	13	,	,	PUNCT
ejpam-3427	253	14	z4	z4	PROPN
ejpam-3427	253	15	)	)	PUNCT
ejpam-3427	253	16	)	)	PUNCT
ejpam-3427	253	17	◦	◦	NOUN
ejpam-3427	253	18	(	(	PUNCT
ejpam-3427	253	19	(	(	PUNCT
ejpam-3427	253	20	z5	z5	X
ejpam-3427	253	21	,	,	PUNCT
ejpam-3427	253	22	z6	z6	PROPN
ejpam-3427	253	23	)	)	PUNCT
ejpam-3427	253	24	,	,	PUNCT
ejpam-3427	253	25	(	(	PUNCT
ejpam-3427	253	26	z7	z7	PROPN
ejpam-3427	253	27	,	,	PUNCT
ejpam-3427	253	28	z8	z8	NOUN
ejpam-3427	253	29	)	)	PUNCT
ejpam-3427	253	30	)	)	PUNCT
ejpam-3427	253	31	)	)	PUNCT
ejpam-3427	254	1	=	=	PRON
ejpam-3427	254	2	µx×y	µx×y	PROPN
ejpam-3427	254	3	(	(	PUNCT
ejpam-3427	254	4	(	(	PUNCT
ejpam-3427	254	5	(	(	PUNCT
ejpam-3427	254	6	z1	z1	ADJ
ejpam-3427	254	7	,	,	PUNCT
ejpam-3427	254	8	z2	z2	PROPN
ejpam-3427	254	9	)	)	PUNCT
ejpam-3427	254	10	◦	◦	NOUN
ejpam-3427	254	11	(	(	PUNCT
ejpam-3427	254	12	z5	z5	X
ejpam-3427	254	13	,	,	PUNCT
ejpam-3427	254	14	z6	z6	PROPN
ejpam-3427	254	15	)	)	PUNCT
ejpam-3427	254	16	)	)	PUNCT
ejpam-3427	254	17	,	,	PUNCT
ejpam-3427	254	18	(	(	PUNCT
ejpam-3427	254	19	(	(	PUNCT
ejpam-3427	254	20	z3	z3	PROPN
ejpam-3427	254	21	,	,	PUNCT
ejpam-3427	254	22	z4	z4	PROPN
ejpam-3427	254	23	)	)	PUNCT
ejpam-3427	254	24	◦	◦	NOUN
ejpam-3427	254	25	(	(	PUNCT
ejpam-3427	254	26	z7	z7	PROPN
ejpam-3427	254	27	,	,	PUNCT
ejpam-3427	254	28	z8	z8	NOUN
ejpam-3427	254	29	)	)	PUNCT
ejpam-3427	254	30	)	)	PUNCT
ejpam-3427	254	31	)	)	PUNCT
ejpam-3427	255	1	=	=	SYM
ejpam-3427	255	2	max{µx((z1	max{µx((z1	PROPN
ejpam-3427	255	3	,	,	PUNCT
ejpam-3427	255	4	z2	z2	NOUN
ejpam-3427	255	5	)	)	PUNCT
ejpam-3427	255	6	◦	◦	NOUN
ejpam-3427	255	7	(	(	PUNCT
ejpam-3427	255	8	z5	z5	X
ejpam-3427	255	9	,	,	PUNCT
ejpam-3427	255	10	z6	z6	PROPN
ejpam-3427	255	11	)	)	PUNCT
ejpam-3427	255	12	)	)	PUNCT
ejpam-3427	255	13	,	,	PUNCT
ejpam-3427	255	14	µy	µy	X
ejpam-3427	255	15	(	(	PUNCT
ejpam-3427	255	16	(	(	PUNCT
ejpam-3427	255	17	z3	z3	PROPN
ejpam-3427	255	18	,	,	PUNCT
ejpam-3427	255	19	z4	z4	PROPN
ejpam-3427	255	20	)	)	PUNCT
ejpam-3427	255	21	◦	◦	NOUN
ejpam-3427	255	22	(	(	PUNCT
ejpam-3427	255	23	z7	z7	PROPN
ejpam-3427	255	24	,	,	PUNCT
ejpam-3427	255	25	z8	z8	NOUN
ejpam-3427	255	26	)	)	PUNCT
ejpam-3427	255	27	)	)	PUNCT
ejpam-3427	255	28	}	}	PUNCT
ejpam-3427	255	29	≤	≤	NUM
ejpam-3427	255	30	max{(µx(z1	max{(µx(z1	NOUN
ejpam-3427	255	31	,	,	PUNCT
ejpam-3427	255	32	z2	z2	PROPN
ejpam-3427	255	33	)	)	PUNCT
ejpam-3427	255	34	∨	∨	NUM
ejpam-3427	255	35	µx(z5	µx(z5	NOUN
ejpam-3427	255	36	,	,	PUNCT
ejpam-3427	255	37	z6	z6	PROPN
ejpam-3427	255	38	)	)	PUNCT
ejpam-3427	255	39	)	)	PUNCT
ejpam-3427	255	40	,	,	PUNCT
ejpam-3427	255	41	(	(	PUNCT
ejpam-3427	255	42	µy	µy	X
ejpam-3427	255	43	(	(	PUNCT
ejpam-3427	255	44	z3	z3	PROPN
ejpam-3427	255	45	,	,	PUNCT
ejpam-3427	255	46	z4	z4	PROPN
ejpam-3427	255	47	)	)	PUNCT
ejpam-3427	255	48	∨	∨	PROPN
ejpam-3427	255	49	µy	µy	X
ejpam-3427	255	50	(	(	PUNCT
ejpam-3427	255	51	z7	z7	PROPN
ejpam-3427	255	52	,	,	PUNCT
ejpam-3427	255	53	z8	z8	NOUN
ejpam-3427	255	54	)	)	PUNCT
ejpam-3427	255	55	)	)	PUNCT
ejpam-3427	255	56	}	}	PUNCT
ejpam-3427	256	1	=	=	SYM
ejpam-3427	256	2	(	(	PUNCT
ejpam-3427	256	3	(	(	PUNCT
ejpam-3427	256	4	µx(z1	µx(z1	NOUN
ejpam-3427	256	5	,	,	PUNCT
ejpam-3427	256	6	z2	z2	PROPN
ejpam-3427	256	7	)	)	PUNCT
ejpam-3427	256	8	∨	∨	PROPN
ejpam-3427	256	9	µy	µy	X
ejpam-3427	256	10	(	(	PUNCT
ejpam-3427	256	11	z5	z5	X
ejpam-3427	256	12	,	,	PUNCT
ejpam-3427	256	13	z6	z6	PROPN
ejpam-3427	256	14	)	)	PUNCT
ejpam-3427	256	15	)	)	PUNCT
ejpam-3427	257	1	∨	∨	NUM
ejpam-3427	257	2	(	(	PUNCT
ejpam-3427	257	3	µx(z3	µx(z3	ADJ
ejpam-3427	257	4	,	,	PUNCT
ejpam-3427	257	5	z4	z4	PROPN
ejpam-3427	257	6	)	)	PUNCT
ejpam-3427	257	7	∨	∨	PROPN
ejpam-3427	257	8	µy	µy	X
ejpam-3427	257	9	(	(	PUNCT
ejpam-3427	257	10	z7	z7	PROPN
ejpam-3427	257	11	,	,	PUNCT
ejpam-3427	257	12	z8	z8	NOUN
ejpam-3427	257	13	)	)	PUNCT
ejpam-3427	257	14	)	)	PUNCT
ejpam-3427	257	15	)	)	PUNCT
ejpam-3427	258	1	=	=	PUNCT
ejpam-3427	258	2	(	(	PUNCT
ejpam-3427	258	3	(	(	PUNCT
ejpam-3427	258	4	µx(z1	µx(z1	NOUN
ejpam-3427	258	5	,	,	PUNCT
ejpam-3427	258	6	z2	z2	PROPN
ejpam-3427	258	7	)	)	PUNCT
ejpam-3427	258	8	∨	∨	NUM
ejpam-3427	258	9	µy	µy	X
ejpam-3427	258	10	(	(	PUNCT
ejpam-3427	258	11	z3	z3	PROPN
ejpam-3427	258	12	,	,	PUNCT
ejpam-3427	258	13	z4	z4	PROPN
ejpam-3427	258	14	)	)	PUNCT
ejpam-3427	258	15	)	)	PUNCT
ejpam-3427	259	1	∨	∨	NUM
ejpam-3427	259	2	(	(	PUNCT
ejpam-3427	259	3	µx(z5	µx(z5	X
ejpam-3427	259	4	,	,	PUNCT
ejpam-3427	259	5	z6	z6	PROPN
ejpam-3427	259	6	)	)	PUNCT
ejpam-3427	259	7	∨	∨	PROPN
ejpam-3427	259	8	µy	µy	X
ejpam-3427	259	9	(	(	PUNCT
ejpam-3427	259	10	z7	z7	PROPN
ejpam-3427	259	11	,	,	PUNCT
ejpam-3427	259	12	z8	z8	NOUN
ejpam-3427	259	13	)	)	PUNCT
ejpam-3427	259	14	)	)	PUNCT
ejpam-3427	259	15	)	)	PUNCT
ejpam-3427	260	1	=	=	SYM
ejpam-3427	260	2	max{(µx(z1	max{(µx(z1	PROPN
ejpam-3427	260	3	,	,	PUNCT
ejpam-3427	260	4	z2	z2	PROPN
ejpam-3427	260	5	)	)	PUNCT
ejpam-3427	260	6	∨	∨	NUM
ejpam-3427	260	7	µy	µy	X
ejpam-3427	260	8	(	(	PUNCT
ejpam-3427	260	9	z3	z3	PROPN
ejpam-3427	260	10	,	,	PUNCT
ejpam-3427	260	11	z4	z4	PROPN
ejpam-3427	260	12	)	)	PUNCT
ejpam-3427	260	13	)	)	PUNCT
ejpam-3427	260	14	,	,	PUNCT
ejpam-3427	260	15	(	(	PUNCT
ejpam-3427	260	16	µx(z5	µx(z5	X
ejpam-3427	260	17	,	,	PUNCT
ejpam-3427	260	18	z6	z6	PROPN
ejpam-3427	260	19	)	)	PUNCT
ejpam-3427	260	20	∨	∨	PROPN
ejpam-3427	260	21	µy	µy	X
ejpam-3427	260	22	(	(	PUNCT
ejpam-3427	260	23	z7	z7	PROPN
ejpam-3427	260	24	,	,	PUNCT
ejpam-3427	260	25	z8	z8	NOUN
ejpam-3427	260	26	)	)	PUNCT
ejpam-3427	260	27	)	)	PUNCT
ejpam-3427	260	28	}	}	PUNCT
ejpam-3427	261	1	=	=	PUNCT
ejpam-3427	261	2	max{µx×y	max{µx×y	X
ejpam-3427	261	3	(	(	PUNCT
ejpam-3427	261	4	(	(	PUNCT
ejpam-3427	261	5	z1	z1	PROPN
ejpam-3427	261	6	,	,	PUNCT
ejpam-3427	261	7	z2	z2	PROPN
ejpam-3427	261	8	)	)	PUNCT
ejpam-3427	261	9	,	,	PUNCT
ejpam-3427	261	10	(	(	PUNCT
ejpam-3427	261	11	z3	z3	PROPN
ejpam-3427	261	12	,	,	PUNCT
ejpam-3427	261	13	z4	z4	PROPN
ejpam-3427	261	14	)	)	PUNCT
ejpam-3427	261	15	)	)	PUNCT
ejpam-3427	261	16	,	,	PUNCT
ejpam-3427	261	17	µx×y	µx×y	PROPN
ejpam-3427	261	18	(	(	PUNCT
ejpam-3427	261	19	(	(	PUNCT
ejpam-3427	261	20	z5	z5	X
ejpam-3427	261	21	,	,	PUNCT
ejpam-3427	261	22	z6	z6	PROPN
ejpam-3427	261	23	)	)	PUNCT
ejpam-3427	261	24	,	,	PUNCT
ejpam-3427	261	25	(	(	PUNCT
ejpam-3427	261	26	z7	z7	PROPN
ejpam-3427	261	27	,	,	PUNCT
ejpam-3427	261	28	z8	z8	NOUN
ejpam-3427	261	29	)	)	PUNCT
ejpam-3427	261	30	)	)	PUNCT
ejpam-3427	261	31	}	}	PUNCT
ejpam-3427	262	1	=	=	PUNCT
ejpam-3427	262	2	max{µz((z1	max{µz((z1	ADJ
ejpam-3427	262	3	,	,	PUNCT
ejpam-3427	262	4	z2	z2	NOUN
ejpam-3427	262	5	)	)	PUNCT
ejpam-3427	262	6	,	,	PUNCT
ejpam-3427	262	7	(	(	PUNCT
ejpam-3427	262	8	z3	z3	PROPN
ejpam-3427	262	9	,	,	PUNCT
ejpam-3427	262	10	z4	z4	PROPN
ejpam-3427	262	11	)	)	PUNCT
ejpam-3427	262	12	)	)	PUNCT
ejpam-3427	262	13	,	,	PUNCT
ejpam-3427	263	1	µz((z5	µz((z5	ADP
ejpam-3427	263	2	,	,	PUNCT
ejpam-3427	263	3	z6	z6	PROPN
ejpam-3427	263	4	)	)	PUNCT
ejpam-3427	263	5	,	,	PUNCT
ejpam-3427	263	6	(	(	PUNCT
ejpam-3427	263	7	z7	z7	PROPN
ejpam-3427	263	8	,	,	PUNCT
ejpam-3427	263	9	z8	z8	NOUN
ejpam-3427	263	10	)	)	PUNCT
ejpam-3427	263	11	)	)	PUNCT
ejpam-3427	263	12	}	}	PUNCT
ejpam-3427	263	13	.	.	PUNCT
ejpam-3427	264	1	similarly	similarly	ADV
ejpam-3427	264	2	γz(((z1	γz(((z1	PROPN
ejpam-3427	264	3	,	,	PUNCT
ejpam-3427	264	4	z2	z2	PROPN
ejpam-3427	264	5	)	)	PUNCT
ejpam-3427	264	6	,	,	PUNCT
ejpam-3427	264	7	(	(	PUNCT
ejpam-3427	264	8	z3	z3	PROPN
ejpam-3427	264	9	,	,	PUNCT
ejpam-3427	264	10	z4))−	z4))−	PROPN
ejpam-3427	264	11	(	(	PUNCT
ejpam-3427	264	12	(	(	PUNCT
ejpam-3427	264	13	z5	z5	X
ejpam-3427	264	14	,	,	PUNCT
ejpam-3427	264	15	z6	z6	PROPN
ejpam-3427	264	16	)	)	PUNCT
ejpam-3427	264	17	,	,	PUNCT
ejpam-3427	264	18	(	(	PUNCT
ejpam-3427	264	19	z7	z7	PROPN
ejpam-3427	264	20	,	,	PUNCT
ejpam-3427	264	21	z8	z8	NOUN
ejpam-3427	264	22	)	)	PUNCT
ejpam-3427	264	23	)	)	PUNCT
ejpam-3427	264	24	)	)	PUNCT
ejpam-3427	265	1	k.	k.	PROPN
ejpam-3427	266	1	nasreen	nasreen	PROPN
ejpam-3427	266	2	/	/	SYM
ejpam-3427	266	3	eur	eur	PROPN
ejpam-3427	266	4	.	.	PUNCT
ejpam-3427	267	1	j.	j.	PROPN
ejpam-3427	267	2	pure	pure	PROPN
ejpam-3427	267	3	appl	appl	PROPN
ejpam-3427	267	4	.	.	PROPN
ejpam-3427	267	5	math	math	PROPN
ejpam-3427	267	6	,	,	PUNCT
ejpam-3427	267	7	12	12	NUM
ejpam-3427	267	8	(	(	PUNCT
ejpam-3427	267	9	2	2	NUM
ejpam-3427	267	10	)	)	PUNCT
ejpam-3427	267	11	(	(	PUNCT
ejpam-3427	267	12	2019	2019	NUM
ejpam-3427	267	13	)	)	PUNCT
ejpam-3427	267	14	,	,	PUNCT
ejpam-3427	267	15	622	622	NUM
ejpam-3427	267	16	-	-	SYM
ejpam-3427	267	17	648	648	NUM
ejpam-3427	267	18	631	631	NUM
ejpam-3427	267	19	≥	≥	NOUN
ejpam-3427	267	20	min{γz((z1	min{γz((z1	PROPN
ejpam-3427	267	21	,	,	PUNCT
ejpam-3427	267	22	z2	z2	PROPN
ejpam-3427	267	23	)	)	PUNCT
ejpam-3427	267	24	,	,	PUNCT
ejpam-3427	267	25	(	(	PUNCT
ejpam-3427	267	26	z3	z3	PROPN
ejpam-3427	267	27	,	,	PUNCT
ejpam-3427	267	28	z4	z4	PROPN
ejpam-3427	267	29	)	)	PUNCT
ejpam-3427	267	30	)	)	PUNCT
ejpam-3427	267	31	,	,	PUNCT
ejpam-3427	267	32	γz((z5	γz((z5	ADJ
ejpam-3427	267	33	,	,	PUNCT
ejpam-3427	267	34	z6	z6	PROPN
ejpam-3427	267	35	)	)	PUNCT
ejpam-3427	267	36	,	,	PUNCT
ejpam-3427	267	37	(	(	PUNCT
ejpam-3427	267	38	z7	z7	PROPN
ejpam-3427	267	39	,	,	PUNCT
ejpam-3427	267	40	z8	z8	NOUN
ejpam-3427	267	41	)	)	PUNCT
ejpam-3427	267	42	)	)	PUNCT
ejpam-3427	267	43	}	}	PUNCT
ejpam-3427	267	44	and	and	CCONJ
ejpam-3427	267	45	γz(((z1	γz(((z1	PROPN
ejpam-3427	267	46	,	,	PUNCT
ejpam-3427	267	47	z2	z2	PROPN
ejpam-3427	267	48	)	)	PUNCT
ejpam-3427	267	49	,	,	PUNCT
ejpam-3427	267	50	(	(	PUNCT
ejpam-3427	267	51	z3	z3	PROPN
ejpam-3427	267	52	,	,	PUNCT
ejpam-3427	267	53	z4	z4	PROPN
ejpam-3427	267	54	)	)	PUNCT
ejpam-3427	267	55	)	)	PUNCT
ejpam-3427	268	1	◦	◦	NOUN
ejpam-3427	268	2	(	(	PUNCT
ejpam-3427	268	3	(	(	PUNCT
ejpam-3427	268	4	z5	z5	X
ejpam-3427	268	5	,	,	PUNCT
ejpam-3427	268	6	z6	z6	PROPN
ejpam-3427	268	7	)	)	PUNCT
ejpam-3427	268	8	,	,	PUNCT
ejpam-3427	268	9	(	(	PUNCT
ejpam-3427	268	10	z7	z7	PROPN
ejpam-3427	268	11	,	,	PUNCT
ejpam-3427	268	12	z8	z8	NOUN
ejpam-3427	268	13	)	)	PUNCT
ejpam-3427	268	14	)	)	PUNCT
ejpam-3427	268	15	)	)	PUNCT
ejpam-3427	268	16	≥	≥	PROPN
ejpam-3427	268	17	min{γz((z1	min{γz((z1	NOUN
ejpam-3427	268	18	,	,	PUNCT
ejpam-3427	268	19	z2	z2	PROPN
ejpam-3427	268	20	)	)	PUNCT
ejpam-3427	268	21	,	,	PUNCT
ejpam-3427	268	22	(	(	PUNCT
ejpam-3427	268	23	z3	z3	PROPN
ejpam-3427	268	24	,	,	PUNCT
ejpam-3427	268	25	z4	z4	PROPN
ejpam-3427	268	26	)	)	PUNCT
ejpam-3427	268	27	)	)	PUNCT
ejpam-3427	268	28	,	,	PUNCT
ejpam-3427	268	29	γz((z5	γz((z5	ADJ
ejpam-3427	268	30	,	,	PUNCT
ejpam-3427	268	31	z6	z6	PROPN
ejpam-3427	268	32	)	)	PUNCT
ejpam-3427	268	33	,	,	PUNCT
ejpam-3427	268	34	(	(	PUNCT
ejpam-3427	268	35	z7	z7	PROPN
ejpam-3427	268	36	,	,	PUNCT
ejpam-3427	268	37	z8	z8	NOUN
ejpam-3427	268	38	)	)	PUNCT
ejpam-3427	268	39	)	)	PUNCT
ejpam-3427	268	40	}	}	PUNCT
ejpam-3427	268	41	thus	thus	ADV
ejpam-3427	268	42	z	z	X
ejpam-3427	268	43	=	=	SYM
ejpam-3427	268	44	(	(	PUNCT
ejpam-3427	268	45	µz	µz	PROPN
ejpam-3427	268	46	,	,	PUNCT
ejpam-3427	268	47	γz	γz	X
ejpam-3427	268	48	)	)	PUNCT
ejpam-3427	268	49	is	be	AUX
ejpam-3427	268	50	an	an	DET
ejpam-3427	268	51	intuitionistic	intuitionistic	ADJ
ejpam-3427	268	52	anti	anti	ADJ
ejpam-3427	268	53	fuzzy	fuzzy	ADJ
ejpam-3427	268	54	la	la	NOUN
ejpam-3427	268	55	-	-	PUNCT
ejpam-3427	268	56	subring	subring	NOUN
ejpam-3427	268	57	of	of	ADP
ejpam-3427	268	58	an	an	DET
ejpam-3427	268	59	la	la	ADJ
ejpam-3427	268	60	-	-	PUNCT
ejpam-3427	268	61	ring	ring	NOUN
ejpam-3427	268	62	r′	r′	NOUN
ejpam-3427	268	63	×r′′.	×r′′.	NOUN
ejpam-3427	268	64	now	now	ADV
ejpam-3427	268	65	µz(((z1	µz(((z1	PROPN
ejpam-3427	268	66	,	,	PUNCT
ejpam-3427	268	67	z2	z2	NOUN
ejpam-3427	268	68	)	)	PUNCT
ejpam-3427	268	69	,	,	PUNCT
ejpam-3427	268	70	(	(	PUNCT
ejpam-3427	268	71	z3	z3	PROPN
ejpam-3427	268	72	,	,	PUNCT
ejpam-3427	268	73	z4	z4	PROPN
ejpam-3427	268	74	)	)	PUNCT
ejpam-3427	268	75	)	)	PUNCT
ejpam-3427	269	1	◦	◦	NOUN
ejpam-3427	269	2	(	(	PUNCT
ejpam-3427	269	3	(	(	PUNCT
ejpam-3427	269	4	z5	z5	X
ejpam-3427	269	5	,	,	PUNCT
ejpam-3427	269	6	z6	z6	PROPN
ejpam-3427	269	7	)	)	PUNCT
ejpam-3427	269	8	,	,	PUNCT
ejpam-3427	269	9	(	(	PUNCT
ejpam-3427	269	10	z7	z7	PROPN
ejpam-3427	269	11	,	,	PUNCT
ejpam-3427	269	12	z8	z8	NOUN
ejpam-3427	269	13	)	)	PUNCT
ejpam-3427	269	14	)	)	PUNCT
ejpam-3427	269	15	)	)	PUNCT
ejpam-3427	270	1	=	=	PRON
ejpam-3427	270	2	µx×y	µx×y	PROPN
ejpam-3427	270	3	(	(	PUNCT
ejpam-3427	270	4	(	(	PUNCT
ejpam-3427	270	5	(	(	PUNCT
ejpam-3427	270	6	z1	z1	ADJ
ejpam-3427	270	7	,	,	PUNCT
ejpam-3427	270	8	z2	z2	PROPN
ejpam-3427	270	9	)	)	PUNCT
ejpam-3427	270	10	◦	◦	NOUN
ejpam-3427	270	11	(	(	PUNCT
ejpam-3427	270	12	z5	z5	X
ejpam-3427	270	13	,	,	PUNCT
ejpam-3427	270	14	z6	z6	PROPN
ejpam-3427	270	15	)	)	PUNCT
ejpam-3427	270	16	)	)	PUNCT
ejpam-3427	270	17	,	,	PUNCT
ejpam-3427	270	18	(	(	PUNCT
ejpam-3427	270	19	(	(	PUNCT
ejpam-3427	270	20	z3	z3	PROPN
ejpam-3427	270	21	,	,	PUNCT
ejpam-3427	270	22	z4	z4	PROPN
ejpam-3427	270	23	)	)	PUNCT
ejpam-3427	270	24	◦	◦	NOUN
ejpam-3427	270	25	(	(	PUNCT
ejpam-3427	270	26	z7	z7	PROPN
ejpam-3427	270	27	,	,	PUNCT
ejpam-3427	270	28	z8	z8	NOUN
ejpam-3427	270	29	)	)	PUNCT
ejpam-3427	270	30	)	)	PUNCT
ejpam-3427	270	31	)	)	PUNCT
ejpam-3427	271	1	=	=	SYM
ejpam-3427	271	2	max{µx((z1	max{µx((z1	PROPN
ejpam-3427	271	3	,	,	PUNCT
ejpam-3427	271	4	z2	z2	NOUN
ejpam-3427	271	5	)	)	PUNCT
ejpam-3427	271	6	◦	◦	NOUN
ejpam-3427	271	7	(	(	PUNCT
ejpam-3427	271	8	z5	z5	X
ejpam-3427	271	9	,	,	PUNCT
ejpam-3427	271	10	z6	z6	PROPN
ejpam-3427	271	11	)	)	PUNCT
ejpam-3427	271	12	)	)	PUNCT
ejpam-3427	271	13	,	,	PUNCT
ejpam-3427	271	14	µy	µy	X
ejpam-3427	271	15	(	(	PUNCT
ejpam-3427	271	16	(	(	PUNCT
ejpam-3427	271	17	z3	z3	PROPN
ejpam-3427	271	18	,	,	PUNCT
ejpam-3427	271	19	z4	z4	PROPN
ejpam-3427	271	20	)	)	PUNCT
ejpam-3427	271	21	◦	◦	NOUN
ejpam-3427	271	22	(	(	PUNCT
ejpam-3427	271	23	z7	z7	PROPN
ejpam-3427	271	24	,	,	PUNCT
ejpam-3427	271	25	z8	z8	NOUN
ejpam-3427	271	26	)	)	PUNCT
ejpam-3427	271	27	)	)	PUNCT
ejpam-3427	271	28	}	}	PUNCT
ejpam-3427	272	1	=	=	SYM
ejpam-3427	272	2	max{µx((z5	max{µx((z5	PROPN
ejpam-3427	272	3	,	,	PUNCT
ejpam-3427	272	4	z6	z6	PROPN
ejpam-3427	272	5	)	)	PUNCT
ejpam-3427	272	6	◦	◦	NOUN
ejpam-3427	272	7	(	(	PUNCT
ejpam-3427	272	8	z1	z1	PROPN
ejpam-3427	272	9	,	,	PUNCT
ejpam-3427	272	10	z2	z2	PROPN
ejpam-3427	272	11	)	)	PUNCT
ejpam-3427	272	12	)	)	PUNCT
ejpam-3427	272	13	,	,	PUNCT
ejpam-3427	272	14	µy	µy	X
ejpam-3427	272	15	(	(	PUNCT
ejpam-3427	272	16	(	(	PUNCT
ejpam-3427	272	17	z7	z7	PROPN
ejpam-3427	272	18	,	,	PUNCT
ejpam-3427	272	19	z8	z8	NOUN
ejpam-3427	272	20	)	)	PUNCT
ejpam-3427	272	21	◦	◦	NOUN
ejpam-3427	272	22	(	(	PUNCT
ejpam-3427	272	23	z3	z3	PROPN
ejpam-3427	272	24	,	,	PUNCT
ejpam-3427	272	25	z4	z4	PROPN
ejpam-3427	272	26	)	)	PUNCT
ejpam-3427	272	27	)	)	PUNCT
ejpam-3427	272	28	}	}	PUNCT
ejpam-3427	273	1	=	=	SYM
ejpam-3427	273	2	µx×y	µx×y	PROPN
ejpam-3427	273	3	(	(	PUNCT
ejpam-3427	273	4	(	(	PUNCT
ejpam-3427	273	5	(	(	PUNCT
ejpam-3427	273	6	z5	z5	X
ejpam-3427	273	7	,	,	PUNCT
ejpam-3427	273	8	z6	z6	PROPN
ejpam-3427	273	9	)	)	PUNCT
ejpam-3427	273	10	◦	◦	NOUN
ejpam-3427	273	11	(	(	PUNCT
ejpam-3427	273	12	z1	z1	PROPN
ejpam-3427	273	13	,	,	PUNCT
ejpam-3427	273	14	z2	z2	PROPN
ejpam-3427	273	15	)	)	PUNCT
ejpam-3427	273	16	)	)	PUNCT
ejpam-3427	273	17	,	,	PUNCT
ejpam-3427	273	18	(	(	PUNCT
ejpam-3427	273	19	(	(	PUNCT
ejpam-3427	273	20	z7	z7	PROPN
ejpam-3427	273	21	,	,	PUNCT
ejpam-3427	273	22	z8	z8	NOUN
ejpam-3427	273	23	)	)	PUNCT
ejpam-3427	273	24	◦	◦	NOUN
ejpam-3427	273	25	(	(	PUNCT
ejpam-3427	273	26	z3	z3	PROPN
ejpam-3427	273	27	,	,	PUNCT
ejpam-3427	273	28	z4	z4	PROPN
ejpam-3427	273	29	)	)	PUNCT
ejpam-3427	273	30	)	)	PUNCT
ejpam-3427	273	31	)	)	PUNCT
ejpam-3427	274	1	=	=	PUNCT
ejpam-3427	274	2	µz(((z5	µz(((z5	PROPN
ejpam-3427	274	3	,	,	PUNCT
ejpam-3427	274	4	z6	z6	PROPN
ejpam-3427	274	5	)	)	PUNCT
ejpam-3427	274	6	,	,	PUNCT
ejpam-3427	274	7	(	(	PUNCT
ejpam-3427	274	8	z7	z7	PROPN
ejpam-3427	274	9	,	,	PUNCT
ejpam-3427	274	10	z8	z8	NOUN
ejpam-3427	274	11	)	)	PUNCT
ejpam-3427	274	12	)	)	PUNCT
ejpam-3427	274	13	◦	◦	NOUN
ejpam-3427	274	14	(	(	PUNCT
ejpam-3427	274	15	(	(	PUNCT
ejpam-3427	274	16	z1	z1	PROPN
ejpam-3427	274	17	,	,	PUNCT
ejpam-3427	274	18	z2	z2	PROPN
ejpam-3427	274	19	)	)	PUNCT
ejpam-3427	274	20	,	,	PUNCT
ejpam-3427	274	21	(	(	PUNCT
ejpam-3427	274	22	z3	z3	PROPN
ejpam-3427	274	23	,	,	PUNCT
ejpam-3427	274	24	z4	z4	PROPN
ejpam-3427	274	25	)	)	PUNCT
ejpam-3427	274	26	)	)	PUNCT
ejpam-3427	274	27	)	)	PUNCT
ejpam-3427	274	28	.	.	PUNCT
ejpam-3427	275	1	similarly	similarly	ADV
ejpam-3427	275	2	γz(((z1	γz(((z1	PROPN
ejpam-3427	275	3	,	,	PUNCT
ejpam-3427	275	4	z2	z2	PROPN
ejpam-3427	275	5	)	)	PUNCT
ejpam-3427	275	6	,	,	PUNCT
ejpam-3427	275	7	(	(	PUNCT
ejpam-3427	275	8	z3	z3	PROPN
ejpam-3427	275	9	,	,	PUNCT
ejpam-3427	275	10	z4	z4	PROPN
ejpam-3427	275	11	)	)	PUNCT
ejpam-3427	275	12	)	)	PUNCT
ejpam-3427	276	1	◦	◦	NOUN
ejpam-3427	276	2	(	(	PUNCT
ejpam-3427	276	3	(	(	PUNCT
ejpam-3427	276	4	z5	z5	X
ejpam-3427	276	5	,	,	PUNCT
ejpam-3427	276	6	z6	z6	PROPN
ejpam-3427	276	7	)	)	PUNCT
ejpam-3427	276	8	,	,	PUNCT
ejpam-3427	276	9	(	(	PUNCT
ejpam-3427	276	10	z7	z7	PROPN
ejpam-3427	276	11	,	,	PUNCT
ejpam-3427	276	12	z8	z8	NOUN
ejpam-3427	276	13	)	)	PUNCT
ejpam-3427	276	14	)	)	PUNCT
ejpam-3427	276	15	)	)	PUNCT
ejpam-3427	277	1	=	=	PUNCT
ejpam-3427	277	2	γz(((z5	γz(((z5	PROPN
ejpam-3427	277	3	,	,	PUNCT
ejpam-3427	277	4	z6	z6	PROPN
ejpam-3427	277	5	)	)	PUNCT
ejpam-3427	277	6	,	,	PUNCT
ejpam-3427	277	7	(	(	PUNCT
ejpam-3427	277	8	z7	z7	PROPN
ejpam-3427	277	9	,	,	PUNCT
ejpam-3427	277	10	z8	z8	NOUN
ejpam-3427	277	11	)	)	PUNCT
ejpam-3427	277	12	)	)	PUNCT
ejpam-3427	278	1	◦	◦	NOUN
ejpam-3427	278	2	(	(	PUNCT
ejpam-3427	278	3	(	(	PUNCT
ejpam-3427	278	4	z1	z1	PROPN
ejpam-3427	278	5	,	,	PUNCT
ejpam-3427	278	6	z2	z2	PROPN
ejpam-3427	278	7	)	)	PUNCT
ejpam-3427	278	8	,	,	PUNCT
ejpam-3427	278	9	(	(	PUNCT
ejpam-3427	278	10	z3	z3	PROPN
ejpam-3427	278	11	,	,	PUNCT
ejpam-3427	278	12	z4	z4	PROPN
ejpam-3427	278	13	)	)	PUNCT
ejpam-3427	278	14	)	)	PUNCT
ejpam-3427	278	15	)	)	PUNCT
ejpam-3427	278	16	.	.	PUNCT
ejpam-3427	279	1	hence	hence	ADV
ejpam-3427	279	2	z	z	NOUN
ejpam-3427	280	1	=	=	PUNCT
ejpam-3427	280	2	x	x	SYM
ejpam-3427	280	3	×	×	NOUN
ejpam-3427	280	4	y	y	PROPN
ejpam-3427	280	5	is	be	AUX
ejpam-3427	280	6	an	an	DET
ejpam-3427	280	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	280	8	anti	anti	ADJ
ejpam-3427	280	9	fuzzy	fuzzy	ADJ
ejpam-3427	280	10	normal	normal	ADJ
ejpam-3427	280	11	la	la	NOUN
ejpam-3427	280	12	-	-	PUNCT
ejpam-3427	280	13	subring	subring	NOUN
ejpam-3427	280	14	of	of	ADP
ejpam-3427	280	15	an	an	DET
ejpam-3427	280	16	la	la	ADJ
ejpam-3427	280	17	-	-	PUNCT
ejpam-3427	280	18	ring	ring	NOUN
ejpam-3427	280	19	r′	r′	NOUN
ejpam-3427	280	20	×r′′.	×r′′.	NOUN
ejpam-3427	280	21	proposition	proposition	NOUN
ejpam-3427	280	22	3	3	NUM
ejpam-3427	280	23	.	.	PUNCT
ejpam-3427	281	1	if	if	SCONJ
ejpam-3427	281	2	an	an	DET
ejpam-3427	281	3	ifs	ifs	PROPN
ejpam-3427	281	4	a×b	a×b	PROPN
ejpam-3427	281	5	is	be	AUX
ejpam-3427	281	6	an	an	DET
ejpam-3427	281	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	281	8	anti	anti	ADJ
ejpam-3427	281	9	fuzzy	fuzzy	ADJ
ejpam-3427	281	10	normal	normal	ADJ
ejpam-3427	281	11	la	la	NOUN
ejpam-3427	281	12	-	-	PUNCT
ejpam-3427	281	13	subring	subring	NOUN
ejpam-3427	281	14	of	of	ADP
ejpam-3427	281	15	an	an	DET
ejpam-3427	281	16	la	la	ADJ
ejpam-3427	281	17	-	-	PUNCT
ejpam-3427	281	18	ring	ring	NOUN
ejpam-3427	281	19	r1	r1	PROPN
ejpam-3427	281	20	×r2	×r2	PROPN
ejpam-3427	281	21	,	,	PUNCT
ejpam-3427	281	22	then	then	ADV
ejpam-3427	281	23	�	�	PROPN
ejpam-3427	281	24	a×b	a×b	PROPN
ejpam-3427	281	25	=	=	PROPN
ejpam-3427	281	26	(	(	PUNCT
ejpam-3427	281	27	µa×b	µa×b	PROPN
ejpam-3427	281	28	,	,	PUNCT
ejpam-3427	281	29	µa×b	µa×b	PROPN
ejpam-3427	281	30	)	)	PUNCT
ejpam-3427	281	31	(	(	PUNCT
ejpam-3427	281	32	resp	resp	NOUN
ejpam-3427	281	33	.	.	PUNCT
ejpam-3427	282	1	♦	♦	PROPN
ejpam-3427	282	2	a×b	a×b	PROPN
ejpam-3427	282	3	=	=	PROPN
ejpam-3427	282	4	(	(	PUNCT
ejpam-3427	282	5	γa×b	γa×b	NOUN
ejpam-3427	282	6	,	,	PUNCT
ejpam-3427	282	7	γa×b	γa×b	NOUN
ejpam-3427	282	8	)	)	PUNCT
ejpam-3427	282	9	)	)	PUNCT
ejpam-3427	282	10	is	be	AUX
ejpam-3427	282	11	also	also	ADV
ejpam-3427	282	12	an	an	DET
ejpam-3427	282	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	282	14	anti	anti	ADJ
ejpam-3427	282	15	fuzzy	fuzzy	ADJ
ejpam-3427	282	16	normal	normal	ADJ
ejpam-3427	282	17	la	la	NOUN
ejpam-3427	282	18	-	-	PUNCT
ejpam-3427	282	19	subring	subring	NOUN
ejpam-3427	282	20	of	of	ADP
ejpam-3427	282	21	an	an	DET
ejpam-3427	282	22	la	la	ADJ
ejpam-3427	282	23	-	-	PUNCT
ejpam-3427	282	24	ring	ring	NOUN
ejpam-3427	282	25	r1	r1	PROPN
ejpam-3427	282	26	×r2	×r2	PROPN
ejpam-3427	282	27	.	.	PUNCT
ejpam-3427	283	1	proof	proof	NOUN
ejpam-3427	283	2	.	.	PUNCT
ejpam-3427	284	1	let	let	VERB
ejpam-3427	284	2	a	a	DET
ejpam-3427	284	3	×	×	PROPN
ejpam-3427	284	4	b	b	NOUN
ejpam-3427	284	5	be	be	AUX
ejpam-3427	284	6	an	an	DET
ejpam-3427	284	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	284	8	anti	anti	ADJ
ejpam-3427	284	9	fuzzy	fuzzy	ADJ
ejpam-3427	284	10	normal	normal	ADJ
ejpam-3427	284	11	la	la	NOUN
ejpam-3427	284	12	-	-	PUNCT
ejpam-3427	284	13	subring	subring	NOUN
ejpam-3427	284	14	of	of	ADP
ejpam-3427	284	15	an	an	DET
ejpam-3427	284	16	la	la	ADJ
ejpam-3427	284	17	-	-	PUNCT
ejpam-3427	284	18	ring	ring	NOUN
ejpam-3427	284	19	r1	r1	NOUN
ejpam-3427	284	20	×	×	NOUN
ejpam-3427	284	21	r2	r2	NOUN
ejpam-3427	284	22	.	.	PUNCT
ejpam-3427	285	1	we	we	PRON
ejpam-3427	285	2	have	have	VERB
ejpam-3427	285	3	to	to	PART
ejpam-3427	285	4	show	show	VERB
ejpam-3427	285	5	that	that	SCONJ
ejpam-3427	285	6	�	�	NOUN
ejpam-3427	285	7	a	a	DET
ejpam-3427	285	8	×	×	PROPN
ejpam-3427	285	9	b	b	NOUN
ejpam-3427	285	10	=	=	PUNCT
ejpam-3427	285	11	(	(	PUNCT
ejpam-3427	285	12	µa×b	µa×b	PROPN
ejpam-3427	285	13	,	,	PUNCT
ejpam-3427	285	14	µa×b	µa×b	PROPN
ejpam-3427	285	15	)	)	PUNCT
ejpam-3427	285	16	is	be	AUX
ejpam-3427	285	17	also	also	ADV
ejpam-3427	285	18	an	an	DET
ejpam-3427	285	19	intuitionistic	intuitionistic	ADJ
ejpam-3427	285	20	anti	anti	ADJ
ejpam-3427	285	21	fuzzy	fuzzy	ADJ
ejpam-3427	285	22	normal	normal	ADJ
ejpam-3427	285	23	la	la	NOUN
ejpam-3427	285	24	-	-	PUNCT
ejpam-3427	285	25	subring	subring	NOUN
ejpam-3427	285	26	of	of	ADP
ejpam-3427	285	27	an	an	DET
ejpam-3427	285	28	la	la	ADJ
ejpam-3427	285	29	-	-	PUNCT
ejpam-3427	285	30	ring	ring	NOUN
ejpam-3427	285	31	r1	r1	PROPN
ejpam-3427	285	32	×r2	×r2	PROPN
ejpam-3427	285	33	.	.	PUNCT
ejpam-3427	286	1	now	now	ADV
ejpam-3427	286	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	286	3	,	,	PUNCT
ejpam-3427	286	4	x2)−	x2)−	NOUN
ejpam-3427	286	5	(	(	PUNCT
ejpam-3427	286	6	y1	y1	NOUN
ejpam-3427	286	7	,	,	PUNCT
ejpam-3427	286	8	y2	y2	PROPN
ejpam-3427	286	9	)	)	PUNCT
ejpam-3427	286	10	)	)	PUNCT
ejpam-3427	287	1	=	=	SYM
ejpam-3427	288	1	1−	1−	NUM
ejpam-3427	288	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	288	3	,	,	PUNCT
ejpam-3427	288	4	x2)−	x2)−	NOUN
ejpam-3427	288	5	(	(	PUNCT
ejpam-3427	288	6	y1	y1	NOUN
ejpam-3427	288	7	,	,	PUNCT
ejpam-3427	288	8	y2	y2	PROPN
ejpam-3427	288	9	)	)	PUNCT
ejpam-3427	288	10	)	)	PUNCT
ejpam-3427	288	11	≥	≥	PROPN
ejpam-3427	289	1	1−max	1−max	NUM
ejpam-3427	289	2	{	{	PUNCT
ejpam-3427	289	3	µa×b(x1	µa×b(x1	NOUN
ejpam-3427	289	4	,	,	PUNCT
ejpam-3427	289	5	x2	x2	PROPN
ejpam-3427	289	6	)	)	PUNCT
ejpam-3427	289	7	,	,	PUNCT
ejpam-3427	289	8	µa×b(y1	µa×b(y1	NUM
ejpam-3427	289	9	,	,	PUNCT
ejpam-3427	289	10	y2	y2	PROPN
ejpam-3427	289	11	)	)	PUNCT
ejpam-3427	289	12	}	}	PUNCT
ejpam-3427	289	13	=	=	SYM
ejpam-3427	289	14	min	min	X
ejpam-3427	289	15	{	{	PUNCT
ejpam-3427	289	16	1−	1−	NUM
ejpam-3427	289	17	µa×b(x1	µa×b(x1	X
ejpam-3427	289	18	,	,	PUNCT
ejpam-3427	289	19	x2	x2	PROPN
ejpam-3427	289	20	)	)	PUNCT
ejpam-3427	289	21	,	,	PUNCT
ejpam-3427	289	22	1−	1−	NUM
ejpam-3427	289	23	µa×b(y1	µa×b(y1	NUM
ejpam-3427	289	24	,	,	PUNCT
ejpam-3427	289	25	y2	y2	PROPN
ejpam-3427	289	26	)	)	PUNCT
ejpam-3427	289	27	}	}	PUNCT
ejpam-3427	289	28	=	=	SYM
ejpam-3427	290	1	min{µa×b(x1	min{µa×b(x1	PROPN
ejpam-3427	290	2	,	,	PUNCT
ejpam-3427	290	3	x2	x2	PROPN
ejpam-3427	290	4	)	)	PUNCT
ejpam-3427	290	5	,	,	PUNCT
ejpam-3427	290	6	µa×b(y1	µa×b(y1	NUM
ejpam-3427	290	7	,	,	PUNCT
ejpam-3427	290	8	y2	y2	PROPN
ejpam-3427	290	9	)	)	PUNCT
ejpam-3427	290	10	}	}	PUNCT
ejpam-3427	290	11	.	.	PUNCT
ejpam-3427	291	1	and	and	CCONJ
ejpam-3427	291	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	291	3	,	,	PUNCT
ejpam-3427	291	4	x2	x2	ADJ
ejpam-3427	291	5	)	)	PUNCT
ejpam-3427	291	6	◦	◦	NOUN
ejpam-3427	291	7	(	(	PUNCT
ejpam-3427	291	8	y1	y1	INTJ
ejpam-3427	291	9	,	,	PUNCT
ejpam-3427	291	10	y2	y2	PROPN
ejpam-3427	291	11	)	)	PUNCT
ejpam-3427	291	12	)	)	PUNCT
ejpam-3427	292	1	=	=	SYM
ejpam-3427	292	2	1−	1−	NUM
ejpam-3427	292	3	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	292	4	,	,	PUNCT
ejpam-3427	292	5	x2	x2	PROPN
ejpam-3427	292	6	)	)	PUNCT
ejpam-3427	292	7	◦	◦	NOUN
ejpam-3427	292	8	(	(	PUNCT
ejpam-3427	292	9	y1	y1	INTJ
ejpam-3427	292	10	,	,	PUNCT
ejpam-3427	292	11	y2	y2	PROPN
ejpam-3427	292	12	)	)	PUNCT
ejpam-3427	292	13	)	)	PUNCT
ejpam-3427	292	14	≥	≥	PROPN
ejpam-3427	293	1	1−max	1−max	NUM
ejpam-3427	293	2	{	{	PUNCT
ejpam-3427	293	3	µa×b(x1	µa×b(x1	NOUN
ejpam-3427	293	4	,	,	PUNCT
ejpam-3427	293	5	x2	x2	PROPN
ejpam-3427	293	6	)	)	PUNCT
ejpam-3427	293	7	,	,	PUNCT
ejpam-3427	293	8	µa×b(y1	µa×b(y1	NUM
ejpam-3427	293	9	,	,	PUNCT
ejpam-3427	293	10	y2	y2	PROPN
ejpam-3427	293	11	)	)	PUNCT
ejpam-3427	293	12	}	}	PUNCT
ejpam-3427	293	13	=	=	SYM
ejpam-3427	293	14	min	min	X
ejpam-3427	293	15	{	{	PUNCT
ejpam-3427	293	16	1−	1−	NUM
ejpam-3427	293	17	µa×b(x1	µa×b(x1	X
ejpam-3427	293	18	,	,	PUNCT
ejpam-3427	293	19	x2	x2	PROPN
ejpam-3427	293	20	)	)	PUNCT
ejpam-3427	293	21	,	,	PUNCT
ejpam-3427	293	22	1−	1−	NUM
ejpam-3427	293	23	µa×b(y1	µa×b(y1	NUM
ejpam-3427	293	24	,	,	PUNCT
ejpam-3427	293	25	y2	y2	PROPN
ejpam-3427	293	26	)	)	PUNCT
ejpam-3427	293	27	}	}	PUNCT
ejpam-3427	293	28	=	=	SYM
ejpam-3427	294	1	min{µa×b(x1	min{µa×b(x1	PROPN
ejpam-3427	294	2	,	,	PUNCT
ejpam-3427	294	3	x2	x2	PROPN
ejpam-3427	294	4	)	)	PUNCT
ejpam-3427	294	5	,	,	PUNCT
ejpam-3427	294	6	µa×b(y1	µa×b(y1	NUM
ejpam-3427	294	7	,	,	PUNCT
ejpam-3427	294	8	y2	y2	PROPN
ejpam-3427	294	9	)	)	PUNCT
ejpam-3427	294	10	}	}	PUNCT
ejpam-3427	294	11	.	.	PUNCT
ejpam-3427	295	1	thus	thus	ADV
ejpam-3427	295	2	�	�	PROPN
ejpam-3427	295	3	a×b	a×b	PROPN
ejpam-3427	295	4	=	=	PRON
ejpam-3427	295	5	(	(	PUNCT
ejpam-3427	295	6	µa×b	µa×b	PROPN
ejpam-3427	295	7	,	,	PUNCT
ejpam-3427	295	8	µa×b	µa×b	PROPN
ejpam-3427	295	9	)	)	PUNCT
ejpam-3427	295	10	is	be	AUX
ejpam-3427	295	11	an	an	DET
ejpam-3427	295	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	295	13	anti	anti	ADJ
ejpam-3427	295	14	fuzzy	fuzzy	ADJ
ejpam-3427	295	15	la	la	NOUN
ejpam-3427	295	16	-	-	PUNCT
ejpam-3427	295	17	subring	subring	NOUN
ejpam-3427	295	18	of	of	ADP
ejpam-3427	295	19	an	an	DET
ejpam-3427	295	20	la	la	ADJ
ejpam-3427	295	21	-	-	PUNCT
ejpam-3427	295	22	ring	ring	NOUN
ejpam-3427	295	23	r1	r1	PROPN
ejpam-3427	295	24	×r2	×r2	PROPN
ejpam-3427	295	25	.	.	PUNCT
ejpam-3427	296	1	now	now	ADV
ejpam-3427	296	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	296	3	,	,	PUNCT
ejpam-3427	296	4	x2	x2	ADJ
ejpam-3427	296	5	)	)	PUNCT
ejpam-3427	296	6	◦	◦	NOUN
ejpam-3427	296	7	(	(	PUNCT
ejpam-3427	296	8	y1	y1	INTJ
ejpam-3427	296	9	,	,	PUNCT
ejpam-3427	296	10	y2	y2	PROPN
ejpam-3427	296	11	)	)	PUNCT
ejpam-3427	296	12	)	)	PUNCT
ejpam-3427	297	1	=	=	SYM
ejpam-3427	297	2	1−	1−	NUM
ejpam-3427	297	3	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	297	4	,	,	PUNCT
ejpam-3427	297	5	x2	x2	PROPN
ejpam-3427	297	6	)	)	PUNCT
ejpam-3427	297	7	◦	◦	NOUN
ejpam-3427	297	8	(	(	PUNCT
ejpam-3427	297	9	y1	y1	INTJ
ejpam-3427	297	10	,	,	PUNCT
ejpam-3427	297	11	y2	y2	PROPN
ejpam-3427	297	12	)	)	PUNCT
ejpam-3427	297	13	)	)	PUNCT
ejpam-3427	298	1	k.	k.	PROPN
ejpam-3427	299	1	nasreen	nasreen	PROPN
ejpam-3427	299	2	/	/	SYM
ejpam-3427	299	3	eur	eur	PROPN
ejpam-3427	299	4	.	.	PUNCT
ejpam-3427	300	1	j.	j.	PROPN
ejpam-3427	300	2	pure	pure	PROPN
ejpam-3427	300	3	appl	appl	PROPN
ejpam-3427	300	4	.	.	PROPN
ejpam-3427	300	5	math	math	PROPN
ejpam-3427	300	6	,	,	PUNCT
ejpam-3427	300	7	12	12	NUM
ejpam-3427	300	8	(	(	PUNCT
ejpam-3427	300	9	2	2	NUM
ejpam-3427	300	10	)	)	PUNCT
ejpam-3427	300	11	(	(	PUNCT
ejpam-3427	300	12	2019	2019	NUM
ejpam-3427	300	13	)	)	PUNCT
ejpam-3427	300	14	,	,	PUNCT
ejpam-3427	300	15	622	622	NUM
ejpam-3427	300	16	-	-	SYM
ejpam-3427	300	17	648	648	NUM
ejpam-3427	300	18	632	632	NUM
ejpam-3427	300	19	=	=	SYM
ejpam-3427	300	20	1−	1−	NUM
ejpam-3427	300	21	µa×b((y1	µa×b((y1	PROPN
ejpam-3427	300	22	,	,	PUNCT
ejpam-3427	300	23	y2	y2	NOUN
ejpam-3427	300	24	)	)	PUNCT
ejpam-3427	300	25	◦	◦	NOUN
ejpam-3427	300	26	(	(	PUNCT
ejpam-3427	300	27	x1	x1	PROPN
ejpam-3427	300	28	,	,	PUNCT
ejpam-3427	300	29	x2	x2	PROPN
ejpam-3427	300	30	)	)	PUNCT
ejpam-3427	300	31	)	)	PUNCT
ejpam-3427	301	1	=	=	PUNCT
ejpam-3427	301	2	µa×b((y1	µa×b((y1	PROPN
ejpam-3427	301	3	,	,	PUNCT
ejpam-3427	301	4	y2	y2	NOUN
ejpam-3427	301	5	)	)	PUNCT
ejpam-3427	301	6	◦	◦	NOUN
ejpam-3427	301	7	(	(	PUNCT
ejpam-3427	301	8	x1	x1	PROPN
ejpam-3427	301	9	,	,	PUNCT
ejpam-3427	301	10	x2	x2	PROPN
ejpam-3427	301	11	)	)	PUNCT
ejpam-3427	301	12	)	)	PUNCT
ejpam-3427	301	13	.	.	PUNCT
ejpam-3427	302	1	hence	hence	ADV
ejpam-3427	302	2	�	�	VERB
ejpam-3427	302	3	a	a	DET
ejpam-3427	302	4	×	×	PROPN
ejpam-3427	302	5	b	b	NOUN
ejpam-3427	302	6	=	=	PUNCT
ejpam-3427	302	7	(	(	PUNCT
ejpam-3427	302	8	µa×b	µa×b	PROPN
ejpam-3427	302	9	,	,	PUNCT
ejpam-3427	302	10	µa×b	µa×b	PROPN
ejpam-3427	302	11	)	)	PUNCT
ejpam-3427	302	12	is	be	AUX
ejpam-3427	302	13	an	an	DET
ejpam-3427	302	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	302	15	anti	anti	ADJ
ejpam-3427	302	16	fuzzy	fuzzy	ADJ
ejpam-3427	302	17	normal	normal	ADJ
ejpam-3427	302	18	la	la	NOUN
ejpam-3427	302	19	-	-	PUNCT
ejpam-3427	302	20	subring	subring	NOUN
ejpam-3427	302	21	of	of	ADP
ejpam-3427	302	22	an	an	DET
ejpam-3427	302	23	la	la	ADJ
ejpam-3427	302	24	-	-	PUNCT
ejpam-3427	302	25	ring	ring	NOUN
ejpam-3427	302	26	r1	r1	PROPN
ejpam-3427	302	27	×r2	×r2	PROPN
ejpam-3427	302	28	.	.	PUNCT
ejpam-3427	302	29	corollary	corollary	ADJ
ejpam-3427	302	30	3	3	NUM
ejpam-3427	302	31	.	.	PUNCT
ejpam-3427	302	32	an	an	DET
ejpam-3427	302	33	ifs	ifs	PROPN
ejpam-3427	302	34	a×b	a×b	PROPN
ejpam-3427	302	35	is	be	AUX
ejpam-3427	302	36	an	an	DET
ejpam-3427	302	37	intuitionistic	intuitionistic	ADJ
ejpam-3427	302	38	anti	anti	ADJ
ejpam-3427	302	39	fuzzy	fuzzy	ADJ
ejpam-3427	302	40	normal	normal	ADJ
ejpam-3427	302	41	la	la	NOUN
ejpam-3427	302	42	-	-	PUNCT
ejpam-3427	302	43	subring	subring	NOUN
ejpam-3427	302	44	of	of	ADP
ejpam-3427	302	45	an	an	DET
ejpam-3427	302	46	laring	laring	NOUN
ejpam-3427	302	47	r1	r1	NOUN
ejpam-3427	302	48	×r2	×r2	PROPN
ejpam-3427	303	1	if	if	SCONJ
ejpam-3427	303	2	and	and	CCONJ
ejpam-3427	303	3	only	only	ADV
ejpam-3427	303	4	if	if	SCONJ
ejpam-3427	303	5	�	�	PROPN
ejpam-3427	303	6	a×b	a×b	PROPN
ejpam-3427	303	7	=	=	PRON
ejpam-3427	303	8	(	(	PUNCT
ejpam-3427	303	9	µa×b	µa×b	PROPN
ejpam-3427	303	10	,	,	PUNCT
ejpam-3427	303	11	µa×b	µa×b	PROPN
ejpam-3427	303	12	)	)	PUNCT
ejpam-3427	303	13	(	(	PUNCT
ejpam-3427	303	14	resp	resp	NOUN
ejpam-3427	303	15	.	.	PUNCT
ejpam-3427	304	1	♦	♦	PROPN
ejpam-3427	304	2	a×b	a×b	PROPN
ejpam-3427	304	3	=	=	PROPN
ejpam-3427	304	4	(	(	PUNCT
ejpam-3427	304	5	γa×b	γa×b	NOUN
ejpam-3427	304	6	,	,	PUNCT
ejpam-3427	304	7	γa×b	γa×b	NOUN
ejpam-3427	304	8	)	)	PUNCT
ejpam-3427	304	9	)	)	PUNCT
ejpam-3427	304	10	is	be	AUX
ejpam-3427	304	11	an	an	DET
ejpam-3427	304	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	304	13	anti	anti	ADJ
ejpam-3427	304	14	fuzzy	fuzzy	ADJ
ejpam-3427	304	15	normal	normal	ADJ
ejpam-3427	304	16	la	la	NOUN
ejpam-3427	304	17	-	-	PUNCT
ejpam-3427	304	18	subring	subring	NOUN
ejpam-3427	304	19	of	of	ADP
ejpam-3427	304	20	an	an	DET
ejpam-3427	304	21	la	la	ADJ
ejpam-3427	304	22	-	-	PUNCT
ejpam-3427	304	23	ring	ring	NOUN
ejpam-3427	304	24	r1	r1	PROPN
ejpam-3427	304	25	×r2	×r2	PROPN
ejpam-3427	304	26	.	.	PUNCT
ejpam-3427	305	1	theorem	theorem	NOUN
ejpam-3427	305	2	3	3	NUM
ejpam-3427	305	3	.	.	PUNCT
ejpam-3427	306	1	an	an	DET
ejpam-3427	306	2	ifs	ifs	PROPN
ejpam-3427	306	3	a	a	DET
ejpam-3427	306	4	×	×	PROPN
ejpam-3427	306	5	b	b	X
ejpam-3427	306	6	=	=	PUNCT
ejpam-3427	306	7	(	(	PUNCT
ejpam-3427	306	8	µa×b	µa×b	PROPN
ejpam-3427	306	9	,	,	PUNCT
ejpam-3427	306	10	γa×b	γa×b	NOUN
ejpam-3427	306	11	)	)	PUNCT
ejpam-3427	306	12	is	be	AUX
ejpam-3427	306	13	an	an	DET
ejpam-3427	306	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	306	15	anti	anti	ADJ
ejpam-3427	306	16	fuzzy	fuzzy	ADJ
ejpam-3427	306	17	normal	normal	ADJ
ejpam-3427	306	18	lasubring	lasubring	NOUN
ejpam-3427	306	19	of	of	ADP
ejpam-3427	306	20	an	an	DET
ejpam-3427	306	21	la	la	ADJ
ejpam-3427	306	22	-	-	PUNCT
ejpam-3427	306	23	ring	ring	NOUN
ejpam-3427	306	24	r1	r1	NOUN
ejpam-3427	306	25	×	×	NOUN
ejpam-3427	306	26	r2	r2	NOUN
ejpam-3427	306	27	if	if	SCONJ
ejpam-3427	306	28	and	and	CCONJ
ejpam-3427	306	29	only	only	ADV
ejpam-3427	306	30	if	if	SCONJ
ejpam-3427	306	31	the	the	DET
ejpam-3427	306	32	fuzzy	fuzzy	ADJ
ejpam-3427	306	33	subsets	subset	NOUN
ejpam-3427	306	34	µa×b	µa×b	PROPN
ejpam-3427	306	35	and	and	CCONJ
ejpam-3427	306	36	γa×b	γa×b	NOUN
ejpam-3427	306	37	are	be	AUX
ejpam-3427	306	38	anti	anti	X
ejpam-3427	306	39	fuzzy	fuzzy	ADJ
ejpam-3427	306	40	normal	normal	ADJ
ejpam-3427	306	41	la	la	ADJ
ejpam-3427	306	42	-	-	PUNCT
ejpam-3427	306	43	subrings	subring	NOUN
ejpam-3427	306	44	of	of	ADP
ejpam-3427	306	45	an	an	DET
ejpam-3427	306	46	la	la	ADJ
ejpam-3427	306	47	-	-	PUNCT
ejpam-3427	306	48	ring	ring	NOUN
ejpam-3427	306	49	r1	r1	PROPN
ejpam-3427	306	50	×r2	×r2	PROPN
ejpam-3427	306	51	.	.	PUNCT
ejpam-3427	307	1	proof	proof	NOUN
ejpam-3427	307	2	.	.	PUNCT
ejpam-3427	308	1	let	let	VERB
ejpam-3427	308	2	a	a	DET
ejpam-3427	308	3	×	×	PROPN
ejpam-3427	308	4	b	b	NOUN
ejpam-3427	308	5	=	=	PUNCT
ejpam-3427	308	6	(	(	PUNCT
ejpam-3427	308	7	µa×b	µa×b	PROPN
ejpam-3427	308	8	,	,	PUNCT
ejpam-3427	308	9	γa×b	γa×b	NOUN
ejpam-3427	308	10	)	)	PUNCT
ejpam-3427	308	11	be	be	VERB
ejpam-3427	308	12	an	an	DET
ejpam-3427	308	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	308	14	anti	anti	ADJ
ejpam-3427	308	15	fuzzy	fuzzy	ADJ
ejpam-3427	308	16	normal	normal	ADJ
ejpam-3427	308	17	la	la	NOUN
ejpam-3427	308	18	-	-	PUNCT
ejpam-3427	308	19	subring	subring	NOUN
ejpam-3427	308	20	of	of	ADP
ejpam-3427	308	21	an	an	DET
ejpam-3427	308	22	la	la	ADJ
ejpam-3427	308	23	-	-	PUNCT
ejpam-3427	308	24	ring	ring	NOUN
ejpam-3427	308	25	r1	r1	PROPN
ejpam-3427	308	26	×r2	×r2	PROPN
ejpam-3427	308	27	.	.	PUNCT
ejpam-3427	309	1	this	this	PRON
ejpam-3427	309	2	implies	imply	VERB
ejpam-3427	309	3	that	that	SCONJ
ejpam-3427	309	4	µa×b	µa×b	PROPN
ejpam-3427	309	5	is	be	AUX
ejpam-3427	309	6	an	an	DET
ejpam-3427	309	7	anti	anti	ADJ
ejpam-3427	309	8	fuzzy	fuzzy	ADJ
ejpam-3427	309	9	normal	normal	ADJ
ejpam-3427	309	10	la	la	NOUN
ejpam-3427	309	11	-	-	PUNCT
ejpam-3427	309	12	subring	subring	NOUN
ejpam-3427	309	13	of	of	ADP
ejpam-3427	309	14	an	an	DET
ejpam-3427	309	15	la	la	ADJ
ejpam-3427	309	16	-	-	PUNCT
ejpam-3427	309	17	ring	ring	NOUN
ejpam-3427	309	18	r1	r1	PROPN
ejpam-3427	309	19	×r2	×r2	PROPN
ejpam-3427	309	20	.	.	PUNCT
ejpam-3427	310	1	we	we	PRON
ejpam-3427	310	2	have	have	VERB
ejpam-3427	310	3	to	to	PART
ejpam-3427	310	4	show	show	VERB
ejpam-3427	310	5	that	that	SCONJ
ejpam-3427	310	6	γa×b	γa×b	NOUN
ejpam-3427	310	7	is	be	AUX
ejpam-3427	310	8	also	also	ADV
ejpam-3427	310	9	an	an	DET
ejpam-3427	310	10	anti	anti	ADJ
ejpam-3427	310	11	fuzzy	fuzzy	ADJ
ejpam-3427	310	12	normal	normal	ADJ
ejpam-3427	310	13	la	la	NOUN
ejpam-3427	310	14	-	-	PUNCT
ejpam-3427	310	15	subring	subring	NOUN
ejpam-3427	310	16	of	of	ADP
ejpam-3427	310	17	an	an	DET
ejpam-3427	310	18	la	la	ADJ
ejpam-3427	310	19	-	-	PUNCT
ejpam-3427	310	20	ring	ring	NOUN
ejpam-3427	310	21	r1	r1	PROPN
ejpam-3427	310	22	×r2	×r2	PROPN
ejpam-3427	310	23	.	.	PUNCT
ejpam-3427	311	1	now	now	ADV
ejpam-3427	311	2	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	311	3	,	,	PUNCT
ejpam-3427	311	4	x2)−	x2)−	X
ejpam-3427	311	5	(	(	PUNCT
ejpam-3427	311	6	y1	y1	NOUN
ejpam-3427	311	7	,	,	PUNCT
ejpam-3427	311	8	y2	y2	PROPN
ejpam-3427	311	9	)	)	PUNCT
ejpam-3427	311	10	)	)	PUNCT
ejpam-3427	312	1	=	=	SYM
ejpam-3427	313	1	1−	1−	NUM
ejpam-3427	313	2	γa×b((x1	γa×b((x1	PROPN
ejpam-3427	313	3	,	,	PUNCT
ejpam-3427	313	4	x2)−	x2)−	X
ejpam-3427	313	5	(	(	PUNCT
ejpam-3427	313	6	y1	y1	NOUN
ejpam-3427	313	7	,	,	PUNCT
ejpam-3427	313	8	y2	y2	PROPN
ejpam-3427	313	9	)	)	PUNCT
ejpam-3427	313	10	)	)	PUNCT
ejpam-3427	313	11	≤	≤	NUM
ejpam-3427	314	1	1−min{γa×b(x1	1−min{γa×b(x1	NUM
ejpam-3427	314	2	,	,	PUNCT
ejpam-3427	314	3	x2	x2	PROPN
ejpam-3427	314	4	)	)	PUNCT
ejpam-3427	314	5	,	,	PUNCT
ejpam-3427	314	6	γa×b(y1	γa×b(y1	NUM
ejpam-3427	314	7	,	,	PUNCT
ejpam-3427	314	8	y2	y2	PROPN
ejpam-3427	314	9	)	)	PUNCT
ejpam-3427	314	10	}	}	PUNCT
ejpam-3427	314	11	=	=	SYM
ejpam-3427	314	12	max{1−	max{1−	PROPN
ejpam-3427	314	13	γa×b(x1	γa×b(x1	PROPN
ejpam-3427	314	14	,	,	PUNCT
ejpam-3427	314	15	x2	x2	PROPN
ejpam-3427	314	16	)	)	PUNCT
ejpam-3427	314	17	,	,	PUNCT
ejpam-3427	314	18	1−	1−	NUM
ejpam-3427	314	19	γa×b(y1	γa×b(y1	NUM
ejpam-3427	314	20	,	,	PUNCT
ejpam-3427	314	21	y2	y2	PROPN
ejpam-3427	314	22	)	)	PUNCT
ejpam-3427	314	23	}	}	PUNCT
ejpam-3427	315	1	=	=	SYM
ejpam-3427	315	2	max{γa×b(x1	max{γa×b(x1	PROPN
ejpam-3427	315	3	,	,	PUNCT
ejpam-3427	315	4	x2	x2	PROPN
ejpam-3427	315	5	)	)	PUNCT
ejpam-3427	315	6	,	,	PUNCT
ejpam-3427	315	7	γa×b(y1	γa×b(y1	NUM
ejpam-3427	315	8	,	,	PUNCT
ejpam-3427	315	9	y2	y2	PROPN
ejpam-3427	315	10	)	)	PUNCT
ejpam-3427	315	11	}	}	PUNCT
ejpam-3427	315	12	.	.	PUNCT
ejpam-3427	316	1	and	and	CCONJ
ejpam-3427	316	2	γa×b((x1	γa×b((x1	PROPN
ejpam-3427	316	3	,	,	PUNCT
ejpam-3427	316	4	x2	x2	ADJ
ejpam-3427	316	5	)	)	PUNCT
ejpam-3427	316	6	◦	◦	NOUN
ejpam-3427	316	7	(	(	PUNCT
ejpam-3427	316	8	y1	y1	INTJ
ejpam-3427	316	9	,	,	PUNCT
ejpam-3427	316	10	y2	y2	PROPN
ejpam-3427	316	11	)	)	PUNCT
ejpam-3427	316	12	)	)	PUNCT
ejpam-3427	317	1	=	=	SYM
ejpam-3427	318	1	1−	1−	NUM
ejpam-3427	318	2	γa×b((x1	γa×b((x1	PROPN
ejpam-3427	318	3	,	,	PUNCT
ejpam-3427	318	4	x2	x2	PROPN
ejpam-3427	318	5	)	)	PUNCT
ejpam-3427	318	6	◦	◦	NOUN
ejpam-3427	318	7	(	(	PUNCT
ejpam-3427	318	8	y1	y1	INTJ
ejpam-3427	318	9	,	,	PUNCT
ejpam-3427	318	10	y2	y2	PROPN
ejpam-3427	318	11	)	)	PUNCT
ejpam-3427	318	12	)	)	PUNCT
ejpam-3427	319	1	≤	≤	NUM
ejpam-3427	320	1	1−min{γa×b(x1	1−min{γa×b(x1	NUM
ejpam-3427	320	2	,	,	PUNCT
ejpam-3427	320	3	x2	x2	PROPN
ejpam-3427	320	4	)	)	PUNCT
ejpam-3427	320	5	,	,	PUNCT
ejpam-3427	320	6	γa×b(y1	γa×b(y1	NUM
ejpam-3427	320	7	,	,	PUNCT
ejpam-3427	320	8	y2	y2	PROPN
ejpam-3427	320	9	)	)	PUNCT
ejpam-3427	320	10	}	}	PUNCT
ejpam-3427	320	11	=	=	SYM
ejpam-3427	320	12	max{1−	max{1−	PROPN
ejpam-3427	320	13	γa×b(x1	γa×b(x1	PROPN
ejpam-3427	320	14	,	,	PUNCT
ejpam-3427	320	15	x2	x2	PROPN
ejpam-3427	320	16	)	)	PUNCT
ejpam-3427	320	17	,	,	PUNCT
ejpam-3427	320	18	1−	1−	NUM
ejpam-3427	320	19	γa×b(y1	γa×b(y1	NUM
ejpam-3427	320	20	,	,	PUNCT
ejpam-3427	320	21	y2	y2	PROPN
ejpam-3427	320	22	)	)	PUNCT
ejpam-3427	320	23	}	}	PUNCT
ejpam-3427	321	1	=	=	SYM
ejpam-3427	321	2	max{γa×b(x1	max{γa×b(x1	PROPN
ejpam-3427	321	3	,	,	PUNCT
ejpam-3427	321	4	x2	x2	PROPN
ejpam-3427	321	5	)	)	PUNCT
ejpam-3427	321	6	,	,	PUNCT
ejpam-3427	321	7	γa×b(y1	γa×b(y1	NUM
ejpam-3427	321	8	,	,	PUNCT
ejpam-3427	321	9	y2	y2	PROPN
ejpam-3427	321	10	)	)	PUNCT
ejpam-3427	321	11	}	}	PUNCT
ejpam-3427	321	12	.	.	PUNCT
ejpam-3427	322	1	thus	thus	ADV
ejpam-3427	322	2	γa×b	γa×b	NOUN
ejpam-3427	322	3	is	be	AUX
ejpam-3427	322	4	an	an	DET
ejpam-3427	322	5	anti	anti	ADJ
ejpam-3427	322	6	fuzzy	fuzzy	ADJ
ejpam-3427	322	7	la	la	NOUN
ejpam-3427	322	8	-	-	PUNCT
ejpam-3427	322	9	subring	subring	NOUN
ejpam-3427	322	10	of	of	ADP
ejpam-3427	322	11	an	an	DET
ejpam-3427	322	12	la	la	ADJ
ejpam-3427	322	13	-	-	PUNCT
ejpam-3427	322	14	ring	ring	NOUN
ejpam-3427	322	15	r1	r1	PROPN
ejpam-3427	322	16	×r2	×r2	PROPN
ejpam-3427	322	17	.	.	PUNCT
ejpam-3427	323	1	now	now	ADV
ejpam-3427	323	2	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	323	3	,	,	PUNCT
ejpam-3427	323	4	x2	x2	ADJ
ejpam-3427	323	5	)	)	PUNCT
ejpam-3427	323	6	◦	◦	NOUN
ejpam-3427	323	7	(	(	PUNCT
ejpam-3427	323	8	y1	y1	INTJ
ejpam-3427	323	9	,	,	PUNCT
ejpam-3427	323	10	y2	y2	PROPN
ejpam-3427	323	11	)	)	PUNCT
ejpam-3427	323	12	)	)	PUNCT
ejpam-3427	324	1	=	=	SYM
ejpam-3427	325	1	1−	1−	NUM
ejpam-3427	325	2	γa×b((x1	γa×b((x1	PROPN
ejpam-3427	325	3	,	,	PUNCT
ejpam-3427	325	4	x2	x2	PROPN
ejpam-3427	325	5	)	)	PUNCT
ejpam-3427	325	6	◦	◦	NOUN
ejpam-3427	325	7	(	(	PUNCT
ejpam-3427	325	8	y1	y1	INTJ
ejpam-3427	325	9	,	,	PUNCT
ejpam-3427	325	10	y2	y2	PROPN
ejpam-3427	325	11	)	)	PUNCT
ejpam-3427	325	12	)	)	PUNCT
ejpam-3427	326	1	=	=	SYM
ejpam-3427	326	2	1−	1−	NUM
ejpam-3427	326	3	γa×b((y1	γa×b((y1	NOUN
ejpam-3427	326	4	,	,	PUNCT
ejpam-3427	326	5	y2	y2	NUM
ejpam-3427	326	6	)	)	PUNCT
ejpam-3427	326	7	◦	◦	NOUN
ejpam-3427	326	8	(	(	PUNCT
ejpam-3427	326	9	x1	x1	PROPN
ejpam-3427	326	10	,	,	PUNCT
ejpam-3427	326	11	x2	x2	PROPN
ejpam-3427	326	12	)	)	PUNCT
ejpam-3427	326	13	)	)	PUNCT
ejpam-3427	327	1	=	=	SYM
ejpam-3427	327	2	γa×b((y1	γa×b((y1	PROPN
ejpam-3427	327	3	,	,	PUNCT
ejpam-3427	327	4	y2	y2	NUM
ejpam-3427	327	5	)	)	PUNCT
ejpam-3427	327	6	◦	◦	NOUN
ejpam-3427	327	7	(	(	PUNCT
ejpam-3427	327	8	x1	x1	PROPN
ejpam-3427	327	9	,	,	PUNCT
ejpam-3427	327	10	x2	x2	PROPN
ejpam-3427	327	11	)	)	PUNCT
ejpam-3427	327	12	)	)	PUNCT
ejpam-3427	327	13	.	.	PUNCT
ejpam-3427	328	1	hence	hence	ADV
ejpam-3427	328	2	γa×b	γa×b	PROPN
ejpam-3427	328	3	is	be	AUX
ejpam-3427	328	4	an	an	DET
ejpam-3427	328	5	anti	anti	ADJ
ejpam-3427	328	6	fuzzy	fuzzy	ADJ
ejpam-3427	328	7	normal	normal	ADJ
ejpam-3427	328	8	la	la	NOUN
ejpam-3427	328	9	-	-	PUNCT
ejpam-3427	328	10	subring	subring	NOUN
ejpam-3427	328	11	of	of	ADP
ejpam-3427	328	12	an	an	DET
ejpam-3427	328	13	la	la	ADJ
ejpam-3427	328	14	-	-	PUNCT
ejpam-3427	328	15	ring	ring	NOUN
ejpam-3427	328	16	r1	r1	PROPN
ejpam-3427	328	17	×r2	×r2	PROPN
ejpam-3427	328	18	.	.	PUNCT
ejpam-3427	329	1	conversely	conversely	ADV
ejpam-3427	329	2	,	,	PUNCT
ejpam-3427	329	3	suppose	suppose	VERB
ejpam-3427	329	4	that	that	SCONJ
ejpam-3427	329	5	µa×b	µa×b	PROPN
ejpam-3427	329	6	and	and	CCONJ
ejpam-3427	329	7	γa×b	γa×b	NOUN
ejpam-3427	329	8	are	be	AUX
ejpam-3427	329	9	anti	anti	X
ejpam-3427	329	10	fuzzy	fuzzy	ADJ
ejpam-3427	329	11	normal	normal	ADJ
ejpam-3427	329	12	la	la	ADJ
ejpam-3427	329	13	-	-	PUNCT
ejpam-3427	329	14	subrings	subring	NOUN
ejpam-3427	329	15	of	of	ADP
ejpam-3427	329	16	an	an	DET
ejpam-3427	329	17	la	la	ADJ
ejpam-3427	329	18	-	-	PUNCT
ejpam-3427	329	19	ring	ring	NOUN
ejpam-3427	329	20	r1	r1	NOUN
ejpam-3427	329	21	×	×	NOUN
ejpam-3427	329	22	r2	r2	NOUN
ejpam-3427	329	23	.	.	PUNCT
ejpam-3427	330	1	we	we	PRON
ejpam-3427	330	2	have	have	VERB
ejpam-3427	330	3	to	to	PART
ejpam-3427	330	4	show	show	VERB
ejpam-3427	330	5	that	that	SCONJ
ejpam-3427	330	6	a	a	DET
ejpam-3427	330	7	×	×	NOUN
ejpam-3427	330	8	b	b	X
ejpam-3427	330	9	=	=	PUNCT
ejpam-3427	330	10	(	(	PUNCT
ejpam-3427	330	11	µa×b	µa×b	PROPN
ejpam-3427	330	12	,	,	PUNCT
ejpam-3427	330	13	γa×b	γa×b	NOUN
ejpam-3427	330	14	)	)	PUNCT
ejpam-3427	330	15	is	be	AUX
ejpam-3427	330	16	an	an	DET
ejpam-3427	330	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	330	18	anti	anti	ADJ
ejpam-3427	330	19	fuzzy	fuzzy	ADJ
ejpam-3427	330	20	normal	normal	ADJ
ejpam-3427	330	21	la	la	NOUN
ejpam-3427	330	22	-	-	PUNCT
ejpam-3427	330	23	subring	subring	NOUN
ejpam-3427	330	24	of	of	ADP
ejpam-3427	330	25	an	an	DET
ejpam-3427	330	26	la	la	ADJ
ejpam-3427	330	27	-	-	PUNCT
ejpam-3427	330	28	ring	ring	NOUN
ejpam-3427	330	29	r1	r1	PROPN
ejpam-3427	330	30	×r2	×r2	PROPN
ejpam-3427	330	31	.	.	PUNCT
ejpam-3427	331	1	now	now	ADV
ejpam-3427	331	2	1−	1−	NUM
ejpam-3427	332	1	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	332	2	,	,	PUNCT
ejpam-3427	332	3	x2)−	x2)−	X
ejpam-3427	332	4	(	(	PUNCT
ejpam-3427	332	5	y1	y1	NOUN
ejpam-3427	332	6	,	,	PUNCT
ejpam-3427	332	7	y2	y2	PROPN
ejpam-3427	332	8	)	)	PUNCT
ejpam-3427	332	9	)	)	PUNCT
ejpam-3427	333	1	=	=	PUNCT
ejpam-3427	333	2	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	333	3	,	,	PUNCT
ejpam-3427	333	4	x2)−	x2)−	X
ejpam-3427	333	5	(	(	PUNCT
ejpam-3427	333	6	y1	y1	NOUN
ejpam-3427	333	7	,	,	PUNCT
ejpam-3427	333	8	y2	y2	PROPN
ejpam-3427	333	9	)	)	PUNCT
ejpam-3427	333	10	)	)	PUNCT
ejpam-3427	333	11	≤	≤	NUM
ejpam-3427	334	1	max{γa×b(x1	max{γa×b(x1	PROPN
ejpam-3427	334	2	,	,	PUNCT
ejpam-3427	334	3	x2	x2	PROPN
ejpam-3427	334	4	)	)	PUNCT
ejpam-3427	334	5	,	,	PUNCT
ejpam-3427	334	6	γa×b(y1	γa×b(y1	NUM
ejpam-3427	334	7	,	,	PUNCT
ejpam-3427	334	8	y2	y2	PROPN
ejpam-3427	334	9	)	)	PUNCT
ejpam-3427	334	10	}	}	PUNCT
ejpam-3427	334	11	=	=	SYM
ejpam-3427	334	12	max{1−	max{1−	PROPN
ejpam-3427	334	13	γa×b(x1	γa×b(x1	PROPN
ejpam-3427	334	14	,	,	PUNCT
ejpam-3427	334	15	x2	x2	PROPN
ejpam-3427	334	16	)	)	PUNCT
ejpam-3427	334	17	,	,	PUNCT
ejpam-3427	334	18	1−	1−	NUM
ejpam-3427	334	19	γa×b(y1	γa×b(y1	NUM
ejpam-3427	334	20	,	,	PUNCT
ejpam-3427	334	21	y2	y2	PROPN
ejpam-3427	334	22	)	)	PUNCT
ejpam-3427	334	23	}	}	PUNCT
ejpam-3427	334	24	k.	k.	PROPN
ejpam-3427	335	1	nasreen	nasreen	PROPN
ejpam-3427	335	2	/	/	SYM
ejpam-3427	335	3	eur	eur	PROPN
ejpam-3427	335	4	.	.	PUNCT
ejpam-3427	336	1	j.	j.	PROPN
ejpam-3427	336	2	pure	pure	PROPN
ejpam-3427	336	3	appl	appl	PROPN
ejpam-3427	336	4	.	.	PROPN
ejpam-3427	336	5	math	math	PROPN
ejpam-3427	336	6	,	,	PUNCT
ejpam-3427	336	7	12	12	NUM
ejpam-3427	336	8	(	(	PUNCT
ejpam-3427	336	9	2	2	NUM
ejpam-3427	336	10	)	)	PUNCT
ejpam-3427	336	11	(	(	PUNCT
ejpam-3427	336	12	2019	2019	NUM
ejpam-3427	336	13	)	)	PUNCT
ejpam-3427	336	14	,	,	PUNCT
ejpam-3427	336	15	622	622	NUM
ejpam-3427	336	16	-	-	SYM
ejpam-3427	336	17	648	648	NUM
ejpam-3427	336	18	633	633	NUM
ejpam-3427	336	19	=	=	SYM
ejpam-3427	336	20	1−min{γa×b(x1	1−min{γa×b(x1	NUM
ejpam-3427	336	21	,	,	PUNCT
ejpam-3427	336	22	x2	x2	PROPN
ejpam-3427	336	23	)	)	PUNCT
ejpam-3427	336	24	,	,	PUNCT
ejpam-3427	336	25	γa×b(y1	γa×b(y1	NUM
ejpam-3427	336	26	,	,	PUNCT
ejpam-3427	336	27	y2	y2	PROPN
ejpam-3427	336	28	)	)	PUNCT
ejpam-3427	336	29	}	}	PUNCT
ejpam-3427	336	30	and	and	CCONJ
ejpam-3427	336	31	1−	1−	NUM
ejpam-3427	336	32	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	336	33	,	,	PUNCT
ejpam-3427	336	34	x2	x2	PROPN
ejpam-3427	336	35	)	)	PUNCT
ejpam-3427	336	36	◦	◦	NOUN
ejpam-3427	336	37	(	(	PUNCT
ejpam-3427	336	38	y1	y1	INTJ
ejpam-3427	336	39	,	,	PUNCT
ejpam-3427	336	40	y2	y2	PROPN
ejpam-3427	336	41	)	)	PUNCT
ejpam-3427	336	42	)	)	PUNCT
ejpam-3427	337	1	=	=	PUNCT
ejpam-3427	338	1	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	338	2	,	,	PUNCT
ejpam-3427	338	3	x2	x2	PROPN
ejpam-3427	338	4	)	)	PUNCT
ejpam-3427	338	5	◦	◦	NOUN
ejpam-3427	338	6	(	(	PUNCT
ejpam-3427	338	7	y1	y1	INTJ
ejpam-3427	338	8	,	,	PUNCT
ejpam-3427	338	9	y2	y2	PROPN
ejpam-3427	338	10	)	)	PUNCT
ejpam-3427	338	11	)	)	PUNCT
ejpam-3427	338	12	≤	≤	NUM
ejpam-3427	339	1	max{γa×b(x1	max{γa×b(x1	PROPN
ejpam-3427	339	2	,	,	PUNCT
ejpam-3427	339	3	x2	x2	PROPN
ejpam-3427	339	4	)	)	PUNCT
ejpam-3427	339	5	,	,	PUNCT
ejpam-3427	339	6	γa×b(y1	γa×b(y1	NUM
ejpam-3427	339	7	,	,	PUNCT
ejpam-3427	339	8	y2	y2	PROPN
ejpam-3427	339	9	)	)	PUNCT
ejpam-3427	339	10	}	}	PUNCT
ejpam-3427	340	1	=	=	SYM
ejpam-3427	340	2	max{1−	max{1−	NOUN
ejpam-3427	340	3	γa(x1	γa(x1	NOUN
ejpam-3427	340	4	,	,	PUNCT
ejpam-3427	340	5	x2	x2	PROPN
ejpam-3427	340	6	)	)	PUNCT
ejpam-3427	340	7	,	,	PUNCT
ejpam-3427	340	8	1−	1−	NUM
ejpam-3427	340	9	γa(y1	γa(y1	NOUN
ejpam-3427	340	10	,	,	PUNCT
ejpam-3427	340	11	y2	y2	PROPN
ejpam-3427	340	12	)	)	PUNCT
ejpam-3427	340	13	}	}	PUNCT
ejpam-3427	340	14	=	=	SYM
ejpam-3427	340	15	1−min{γa(x1	1−min{γa(x1	NUM
ejpam-3427	340	16	,	,	PUNCT
ejpam-3427	340	17	x2	x2	PROPN
ejpam-3427	340	18	)	)	PUNCT
ejpam-3427	340	19	,	,	PUNCT
ejpam-3427	340	20	γa(y1	γa(y1	NOUN
ejpam-3427	340	21	,	,	PUNCT
ejpam-3427	340	22	y2	y2	PROPN
ejpam-3427	340	23	)	)	PUNCT
ejpam-3427	340	24	}	}	PUNCT
ejpam-3427	340	25	.	.	PUNCT
ejpam-3427	341	1	thus	thus	ADV
ejpam-3427	341	2	a	a	DET
ejpam-3427	341	3	×	×	PROPN
ejpam-3427	341	4	b	b	X
ejpam-3427	341	5	=	=	PUNCT
ejpam-3427	341	6	(	(	PUNCT
ejpam-3427	341	7	µa×b	µa×b	PROPN
ejpam-3427	341	8	,	,	PUNCT
ejpam-3427	341	9	γa×b	γa×b	NOUN
ejpam-3427	341	10	)	)	PUNCT
ejpam-3427	341	11	is	be	AUX
ejpam-3427	341	12	an	an	DET
ejpam-3427	341	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	341	14	anti	anti	ADJ
ejpam-3427	341	15	fuzzy	fuzzy	ADJ
ejpam-3427	341	16	la	la	NOUN
ejpam-3427	341	17	-	-	PUNCT
ejpam-3427	341	18	subring	subring	NOUN
ejpam-3427	341	19	of	of	ADP
ejpam-3427	341	20	an	an	DET
ejpam-3427	341	21	la	la	ADJ
ejpam-3427	341	22	-	-	PUNCT
ejpam-3427	341	23	ring	ring	NOUN
ejpam-3427	341	24	r1	r1	PROPN
ejpam-3427	341	25	×r2	×r2	PROPN
ejpam-3427	341	26	.	.	PUNCT
ejpam-3427	342	1	now	now	ADV
ejpam-3427	342	2	1−	1−	NUM
ejpam-3427	343	1	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	343	2	,	,	PUNCT
ejpam-3427	343	3	x2	x2	PROPN
ejpam-3427	343	4	)	)	PUNCT
ejpam-3427	343	5	◦	◦	NOUN
ejpam-3427	343	6	(	(	PUNCT
ejpam-3427	343	7	y1	y1	INTJ
ejpam-3427	343	8	,	,	PUNCT
ejpam-3427	343	9	y2	y2	PROPN
ejpam-3427	343	10	)	)	PUNCT
ejpam-3427	343	11	)	)	PUNCT
ejpam-3427	344	1	=	=	PUNCT
ejpam-3427	345	1	γa×b((x1	γa×b((x1	ADJ
ejpam-3427	345	2	,	,	PUNCT
ejpam-3427	345	3	x2	x2	PROPN
ejpam-3427	345	4	)	)	PUNCT
ejpam-3427	345	5	◦	◦	NOUN
ejpam-3427	345	6	(	(	PUNCT
ejpam-3427	345	7	y1	y1	INTJ
ejpam-3427	345	8	,	,	PUNCT
ejpam-3427	345	9	y2	y2	PROPN
ejpam-3427	345	10	)	)	PUNCT
ejpam-3427	345	11	)	)	PUNCT
ejpam-3427	346	1	=	=	PUNCT
ejpam-3427	346	2	γa×b((y1	γa×b((y1	PROPN
ejpam-3427	346	3	,	,	PUNCT
ejpam-3427	346	4	y2	y2	NUM
ejpam-3427	346	5	)	)	PUNCT
ejpam-3427	346	6	◦	◦	NOUN
ejpam-3427	346	7	(	(	PUNCT
ejpam-3427	346	8	x1	x1	PROPN
ejpam-3427	346	9	,	,	PUNCT
ejpam-3427	346	10	x2	x2	PROPN
ejpam-3427	346	11	)	)	PUNCT
ejpam-3427	346	12	)	)	PUNCT
ejpam-3427	347	1	=	=	SYM
ejpam-3427	348	1	1−	1−	NUM
ejpam-3427	348	2	γa((y1	γa((y1	NOUN
ejpam-3427	348	3	,	,	PUNCT
ejpam-3427	348	4	y2	y2	NOUN
ejpam-3427	348	5	)	)	PUNCT
ejpam-3427	348	6	◦	◦	NOUN
ejpam-3427	348	7	(	(	PUNCT
ejpam-3427	348	8	x1	x1	PROPN
ejpam-3427	348	9	,	,	PUNCT
ejpam-3427	348	10	x2	x2	PROPN
ejpam-3427	348	11	)	)	PUNCT
ejpam-3427	348	12	)	)	PUNCT
ejpam-3427	348	13	.	.	PUNCT
ejpam-3427	349	1	hence	hence	ADV
ejpam-3427	349	2	a×	a×	PROPN
ejpam-3427	349	3	b	b	X
ejpam-3427	349	4	=	=	PUNCT
ejpam-3427	349	5	(	(	PUNCT
ejpam-3427	349	6	µa×b	µa×b	PROPN
ejpam-3427	349	7	,	,	PUNCT
ejpam-3427	349	8	γa×b	γa×b	NOUN
ejpam-3427	349	9	)	)	PUNCT
ejpam-3427	349	10	is	be	AUX
ejpam-3427	349	11	an	an	DET
ejpam-3427	349	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	349	13	anti	anti	ADJ
ejpam-3427	349	14	fuzzy	fuzzy	ADJ
ejpam-3427	349	15	normal	normal	ADJ
ejpam-3427	349	16	la	la	NOUN
ejpam-3427	349	17	-	-	PUNCT
ejpam-3427	349	18	subring	subring	NOUN
ejpam-3427	349	19	of	of	ADP
ejpam-3427	349	20	an	an	DET
ejpam-3427	349	21	la	la	ADJ
ejpam-3427	349	22	-	-	PUNCT
ejpam-3427	349	23	ring	ring	NOUN
ejpam-3427	349	24	r1	r1	PROPN
ejpam-3427	349	25	×r2	×r2	PROPN
ejpam-3427	349	26	.	.	PUNCT
ejpam-3427	350	1	theorem	theorem	VERB
ejpam-3427	350	2	4	4	NUM
ejpam-3427	350	3	.	.	PUNCT
ejpam-3427	351	1	an	an	DET
ejpam-3427	351	2	ifs	ifs	PROPN
ejpam-3427	351	3	a	a	DET
ejpam-3427	351	4	×	×	PROPN
ejpam-3427	351	5	b	b	X
ejpam-3427	351	6	=	=	PUNCT
ejpam-3427	351	7	(	(	PUNCT
ejpam-3427	351	8	µa×b	µa×b	PROPN
ejpam-3427	351	9	,	,	PUNCT
ejpam-3427	351	10	γa×b	γa×b	NOUN
ejpam-3427	351	11	)	)	PUNCT
ejpam-3427	351	12	is	be	AUX
ejpam-3427	351	13	an	an	DET
ejpam-3427	351	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	351	15	anti	anti	ADJ
ejpam-3427	351	16	fuzzy	fuzzy	ADJ
ejpam-3427	351	17	normal	normal	ADJ
ejpam-3427	351	18	lasubring	lasubring	NOUN
ejpam-3427	351	19	of	of	ADP
ejpam-3427	351	20	an	an	DET
ejpam-3427	351	21	la	la	ADJ
ejpam-3427	351	22	-	-	PUNCT
ejpam-3427	351	23	ring	ring	NOUN
ejpam-3427	351	24	r1	r1	NOUN
ejpam-3427	351	25	×r2	×r2	PROPN
ejpam-3427	351	26	if	if	SCONJ
ejpam-3427	351	27	and	and	CCONJ
ejpam-3427	351	28	only	only	ADV
ejpam-3427	351	29	if	if	SCONJ
ejpam-3427	351	30	the	the	DET
ejpam-3427	351	31	fuzzy	fuzzy	ADJ
ejpam-3427	351	32	subsets	subset	NOUN
ejpam-3427	351	33	µa×b	µa×b	PROPN
ejpam-3427	351	34	and	and	CCONJ
ejpam-3427	351	35	γa×b	γa×b	NOUN
ejpam-3427	351	36	are	be	AUX
ejpam-3427	351	37	fuzzy	fuzzy	ADJ
ejpam-3427	351	38	normal	normal	ADJ
ejpam-3427	351	39	la	la	ADJ
ejpam-3427	351	40	-	-	PUNCT
ejpam-3427	351	41	subrings	subring	NOUN
ejpam-3427	351	42	of	of	ADP
ejpam-3427	351	43	an	an	DET
ejpam-3427	351	44	la	la	ADJ
ejpam-3427	351	45	-	-	PUNCT
ejpam-3427	351	46	ring	ring	NOUN
ejpam-3427	351	47	r1	r1	PROPN
ejpam-3427	351	48	×r2	×r2	PROPN
ejpam-3427	351	49	.	.	PUNCT
ejpam-3427	352	1	proof	proof	NOUN
ejpam-3427	352	2	.	.	PUNCT
ejpam-3427	353	1	let	let	VERB
ejpam-3427	353	2	a	a	DET
ejpam-3427	353	3	×	×	PROPN
ejpam-3427	353	4	b	b	NOUN
ejpam-3427	353	5	=	=	PUNCT
ejpam-3427	353	6	(	(	PUNCT
ejpam-3427	353	7	µa×b	µa×b	PROPN
ejpam-3427	353	8	,	,	PUNCT
ejpam-3427	353	9	γa×b	γa×b	NOUN
ejpam-3427	353	10	)	)	PUNCT
ejpam-3427	353	11	be	be	VERB
ejpam-3427	353	12	an	an	DET
ejpam-3427	353	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	353	14	anti	anti	ADJ
ejpam-3427	353	15	fuzzy	fuzzy	ADJ
ejpam-3427	353	16	normal	normal	ADJ
ejpam-3427	353	17	la	la	NOUN
ejpam-3427	353	18	-	-	PUNCT
ejpam-3427	353	19	subring	subring	NOUN
ejpam-3427	353	20	of	of	ADP
ejpam-3427	353	21	an	an	DET
ejpam-3427	353	22	la	la	ADJ
ejpam-3427	353	23	-	-	PUNCT
ejpam-3427	353	24	ring	ring	NOUN
ejpam-3427	353	25	r1×r2	r1×r2	PROPN
ejpam-3427	353	26	.	.	PUNCT
ejpam-3427	354	1	this	this	PRON
ejpam-3427	354	2	means	mean	VERB
ejpam-3427	354	3	that	that	SCONJ
ejpam-3427	354	4	γa×b	γa×b	NOUN
ejpam-3427	354	5	is	be	AUX
ejpam-3427	354	6	a	a	DET
ejpam-3427	354	7	fuzzy	fuzzy	ADJ
ejpam-3427	354	8	normal	normal	ADJ
ejpam-3427	354	9	la	la	NOUN
ejpam-3427	354	10	-	-	PUNCT
ejpam-3427	354	11	subring	subring	NOUN
ejpam-3427	354	12	of	of	ADP
ejpam-3427	354	13	an	an	DET
ejpam-3427	354	14	la	la	ADJ
ejpam-3427	354	15	-	-	PUNCT
ejpam-3427	354	16	ring	ring	NOUN
ejpam-3427	354	17	r1	r1	NOUN
ejpam-3427	354	18	×	×	NOUN
ejpam-3427	354	19	r2	r2	NOUN
ejpam-3427	354	20	.	.	PUNCT
ejpam-3427	355	1	we	we	PRON
ejpam-3427	355	2	have	have	VERB
ejpam-3427	355	3	to	to	PART
ejpam-3427	355	4	show	show	VERB
ejpam-3427	355	5	that	that	SCONJ
ejpam-3427	355	6	µa×b	µa×b	PROPN
ejpam-3427	355	7	is	be	AUX
ejpam-3427	355	8	also	also	ADV
ejpam-3427	355	9	a	a	DET
ejpam-3427	355	10	fuzzy	fuzzy	ADJ
ejpam-3427	355	11	normal	normal	ADJ
ejpam-3427	355	12	la	la	NOUN
ejpam-3427	355	13	-	-	PUNCT
ejpam-3427	355	14	subring	subring	NOUN
ejpam-3427	355	15	of	of	ADP
ejpam-3427	355	16	an	an	DET
ejpam-3427	355	17	la	la	ADJ
ejpam-3427	355	18	-	-	PUNCT
ejpam-3427	355	19	ring	ring	NOUN
ejpam-3427	355	20	r1	r1	PROPN
ejpam-3427	355	21	×r2	×r2	PROPN
ejpam-3427	355	22	.	.	PUNCT
ejpam-3427	356	1	now	now	ADV
ejpam-3427	356	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	356	3	,	,	PUNCT
ejpam-3427	356	4	x2)−	x2)−	NOUN
ejpam-3427	356	5	(	(	PUNCT
ejpam-3427	356	6	y1	y1	NOUN
ejpam-3427	356	7	,	,	PUNCT
ejpam-3427	356	8	y2	y2	PROPN
ejpam-3427	356	9	)	)	PUNCT
ejpam-3427	356	10	)	)	PUNCT
ejpam-3427	357	1	=	=	SYM
ejpam-3427	358	1	1−	1−	NUM
ejpam-3427	358	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	358	3	,	,	PUNCT
ejpam-3427	358	4	x2)−	x2)−	NOUN
ejpam-3427	358	5	(	(	PUNCT
ejpam-3427	358	6	y1	y1	NOUN
ejpam-3427	358	7	,	,	PUNCT
ejpam-3427	358	8	y2	y2	PROPN
ejpam-3427	358	9	)	)	PUNCT
ejpam-3427	358	10	)	)	PUNCT
ejpam-3427	358	11	≥	≥	NOUN
ejpam-3427	359	1	1−max{µa×b(x1	1−max{µa×b(x1	NUM
ejpam-3427	359	2	,	,	PUNCT
ejpam-3427	359	3	x2	x2	PROPN
ejpam-3427	359	4	)	)	PUNCT
ejpam-3427	359	5	,	,	PUNCT
ejpam-3427	359	6	µa×b(y1	µa×b(y1	NUM
ejpam-3427	359	7	,	,	PUNCT
ejpam-3427	359	8	y2	y2	PROPN
ejpam-3427	359	9	)	)	PUNCT
ejpam-3427	359	10	}	}	PUNCT
ejpam-3427	359	11	=	=	PUNCT
ejpam-3427	360	1	min{1−	min{1−	VERB
ejpam-3427	360	2	µa×b(x1	µa×b(x1	NOUN
ejpam-3427	360	3	,	,	PUNCT
ejpam-3427	360	4	x2	x2	PROPN
ejpam-3427	360	5	)	)	PUNCT
ejpam-3427	360	6	,	,	PUNCT
ejpam-3427	360	7	1−	1−	NUM
ejpam-3427	360	8	µa×b(y1	µa×b(y1	NUM
ejpam-3427	360	9	,	,	PUNCT
ejpam-3427	360	10	y2	y2	PROPN
ejpam-3427	360	11	)	)	PUNCT
ejpam-3427	360	12	}	}	PUNCT
ejpam-3427	361	1	=	=	SYM
ejpam-3427	361	2	min{µa×b(x1	min{µa×b(x1	PROPN
ejpam-3427	361	3	,	,	PUNCT
ejpam-3427	361	4	x2	x2	PROPN
ejpam-3427	361	5	)	)	PUNCT
ejpam-3427	361	6	,	,	PUNCT
ejpam-3427	361	7	µa×b(y1	µa×b(y1	NUM
ejpam-3427	361	8	,	,	PUNCT
ejpam-3427	361	9	y2	y2	PROPN
ejpam-3427	361	10	)	)	PUNCT
ejpam-3427	361	11	}	}	PUNCT
ejpam-3427	361	12	.	.	PUNCT
ejpam-3427	362	1	and	and	CCONJ
ejpam-3427	362	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	362	3	,	,	PUNCT
ejpam-3427	362	4	x2	x2	ADJ
ejpam-3427	362	5	)	)	PUNCT
ejpam-3427	362	6	◦	◦	NOUN
ejpam-3427	362	7	(	(	PUNCT
ejpam-3427	362	8	y1	y1	INTJ
ejpam-3427	362	9	,	,	PUNCT
ejpam-3427	362	10	y2	y2	PROPN
ejpam-3427	362	11	)	)	PUNCT
ejpam-3427	362	12	)	)	PUNCT
ejpam-3427	363	1	=	=	SYM
ejpam-3427	363	2	1−	1−	NUM
ejpam-3427	363	3	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	363	4	,	,	PUNCT
ejpam-3427	363	5	x2	x2	PROPN
ejpam-3427	363	6	)	)	PUNCT
ejpam-3427	363	7	◦	◦	NOUN
ejpam-3427	363	8	(	(	PUNCT
ejpam-3427	363	9	y1	y1	INTJ
ejpam-3427	363	10	,	,	PUNCT
ejpam-3427	363	11	y2	y2	PROPN
ejpam-3427	363	12	)	)	PUNCT
ejpam-3427	363	13	)	)	PUNCT
ejpam-3427	363	14	≥	≥	NOUN
ejpam-3427	363	15	1−max{µa(x1	1−max{µa(x1	NUM
ejpam-3427	363	16	,	,	PUNCT
ejpam-3427	363	17	x2	x2	PROPN
ejpam-3427	363	18	)	)	PUNCT
ejpam-3427	363	19	,	,	PUNCT
ejpam-3427	363	20	µa(y1	µa(y1	NOUN
ejpam-3427	363	21	,	,	PUNCT
ejpam-3427	363	22	y2	y2	NOUN
ejpam-3427	363	23	)	)	PUNCT
ejpam-3427	363	24	}	}	PUNCT
ejpam-3427	364	1	=	=	PUNCT
ejpam-3427	364	2	min{1−	min{1−	ADJ
ejpam-3427	364	3	µa(x1	µa(x1	NOUN
ejpam-3427	364	4	,	,	PUNCT
ejpam-3427	364	5	x2	x2	PROPN
ejpam-3427	364	6	)	)	PUNCT
ejpam-3427	364	7	,	,	PUNCT
ejpam-3427	364	8	1−	1−	NUM
ejpam-3427	364	9	µa(y1	µa(y1	NOUN
ejpam-3427	364	10	,	,	PUNCT
ejpam-3427	364	11	y2	y2	NOUN
ejpam-3427	364	12	)	)	PUNCT
ejpam-3427	364	13	}	}	PUNCT
ejpam-3427	364	14	=	=	SYM
ejpam-3427	365	1	min{µa×b(x1	min{µa×b(x1	PROPN
ejpam-3427	365	2	,	,	PUNCT
ejpam-3427	365	3	x2	x2	PROPN
ejpam-3427	365	4	)	)	PUNCT
ejpam-3427	365	5	,	,	PUNCT
ejpam-3427	365	6	µa×b(y1	µa×b(y1	NUM
ejpam-3427	365	7	,	,	PUNCT
ejpam-3427	365	8	y2	y2	PROPN
ejpam-3427	365	9	)	)	PUNCT
ejpam-3427	365	10	}	}	PUNCT
ejpam-3427	365	11	.	.	PUNCT
ejpam-3427	366	1	thus	thus	ADV
ejpam-3427	366	2	µa×b	µa×b	PROPN
ejpam-3427	366	3	is	be	AUX
ejpam-3427	366	4	a	a	DET
ejpam-3427	366	5	fuzzy	fuzzy	ADJ
ejpam-3427	366	6	la	la	NOUN
ejpam-3427	366	7	-	-	PUNCT
ejpam-3427	366	8	subring	subring	NOUN
ejpam-3427	366	9	of	of	ADP
ejpam-3427	366	10	an	an	DET
ejpam-3427	366	11	la	la	ADJ
ejpam-3427	366	12	-	-	PUNCT
ejpam-3427	366	13	ring	ring	NOUN
ejpam-3427	366	14	r1	r1	PROPN
ejpam-3427	366	15	×r2	×r2	PROPN
ejpam-3427	366	16	.	.	PUNCT
ejpam-3427	367	1	µa×b((x1	µa×b((x1	ADJ
ejpam-3427	367	2	,	,	PUNCT
ejpam-3427	367	3	x2	x2	ADJ
ejpam-3427	367	4	)	)	PUNCT
ejpam-3427	367	5	◦	◦	NOUN
ejpam-3427	367	6	(	(	PUNCT
ejpam-3427	367	7	y1	y1	INTJ
ejpam-3427	367	8	,	,	PUNCT
ejpam-3427	367	9	y2	y2	PROPN
ejpam-3427	367	10	)	)	PUNCT
ejpam-3427	367	11	)	)	PUNCT
ejpam-3427	368	1	=	=	SYM
ejpam-3427	368	2	1−	1−	NUM
ejpam-3427	368	3	µa((x1	µa((x1	PROPN
ejpam-3427	368	4	,	,	PUNCT
ejpam-3427	368	5	x2	x2	ADJ
ejpam-3427	368	6	)	)	PUNCT
ejpam-3427	368	7	◦	◦	NOUN
ejpam-3427	368	8	(	(	PUNCT
ejpam-3427	368	9	y1	y1	INTJ
ejpam-3427	368	10	,	,	PUNCT
ejpam-3427	368	11	y2	y2	PROPN
ejpam-3427	368	12	)	)	PUNCT
ejpam-3427	368	13	)	)	PUNCT
ejpam-3427	369	1	=	=	SYM
ejpam-3427	369	2	1−	1−	NUM
ejpam-3427	369	3	µa((y1	µa((y1	NOUN
ejpam-3427	369	4	,	,	PUNCT
ejpam-3427	369	5	y2	y2	NOUN
ejpam-3427	369	6	)	)	PUNCT
ejpam-3427	369	7	◦	◦	NOUN
ejpam-3427	369	8	(	(	PUNCT
ejpam-3427	369	9	x1	x1	PROPN
ejpam-3427	369	10	,	,	PUNCT
ejpam-3427	369	11	x2	x2	PROPN
ejpam-3427	369	12	)	)	PUNCT
ejpam-3427	369	13	)	)	PUNCT
ejpam-3427	370	1	=	=	PUNCT
ejpam-3427	370	2	µa×b((y1	µa×b((y1	PROPN
ejpam-3427	370	3	,	,	PUNCT
ejpam-3427	370	4	y2	y2	NOUN
ejpam-3427	370	5	)	)	PUNCT
ejpam-3427	370	6	◦	◦	NOUN
ejpam-3427	370	7	(	(	PUNCT
ejpam-3427	370	8	x1	x1	PROPN
ejpam-3427	370	9	,	,	PUNCT
ejpam-3427	370	10	x2	x2	PROPN
ejpam-3427	370	11	)	)	PUNCT
ejpam-3427	370	12	)	)	PUNCT
ejpam-3427	370	13	.	.	PUNCT
ejpam-3427	371	1	hence	hence	ADV
ejpam-3427	371	2	µa×b	µa×b	PROPN
ejpam-3427	371	3	is	be	AUX
ejpam-3427	371	4	a	a	DET
ejpam-3427	371	5	fuzzy	fuzzy	ADJ
ejpam-3427	371	6	normal	normal	ADJ
ejpam-3427	371	7	la	la	NOUN
ejpam-3427	371	8	-	-	PUNCT
ejpam-3427	371	9	subring	subring	NOUN
ejpam-3427	371	10	of	of	ADP
ejpam-3427	371	11	an	an	DET
ejpam-3427	371	12	la	la	ADJ
ejpam-3427	371	13	-	-	PUNCT
ejpam-3427	371	14	ring	ring	NOUN
ejpam-3427	371	15	r1	r1	PROPN
ejpam-3427	371	16	×r2	×r2	PROPN
ejpam-3427	371	17	.	.	PUNCT
ejpam-3427	372	1	k.	k.	PROPN
ejpam-3427	372	2	nasreen	nasreen	PROPN
ejpam-3427	372	3	/	/	SYM
ejpam-3427	372	4	eur	eur	PROPN
ejpam-3427	372	5	.	.	PUNCT
ejpam-3427	373	1	j.	j.	PROPN
ejpam-3427	373	2	pure	pure	PROPN
ejpam-3427	373	3	appl	appl	PROPN
ejpam-3427	373	4	.	.	PROPN
ejpam-3427	373	5	math	math	PROPN
ejpam-3427	373	6	,	,	PUNCT
ejpam-3427	373	7	12	12	NUM
ejpam-3427	373	8	(	(	PUNCT
ejpam-3427	373	9	2	2	NUM
ejpam-3427	373	10	)	)	PUNCT
ejpam-3427	373	11	(	(	PUNCT
ejpam-3427	373	12	2019	2019	NUM
ejpam-3427	373	13	)	)	PUNCT
ejpam-3427	373	14	,	,	PUNCT
ejpam-3427	373	15	622	622	NUM
ejpam-3427	373	16	-	-	SYM
ejpam-3427	373	17	648	648	NUM
ejpam-3427	373	18	634	634	NUM
ejpam-3427	373	19	conversely	conversely	ADV
ejpam-3427	373	20	,	,	PUNCT
ejpam-3427	373	21	assume	assume	VERB
ejpam-3427	373	22	that	that	SCONJ
ejpam-3427	373	23	µa×band	µa×band	ADV
ejpam-3427	373	24	γa×b	γa×b	NOUN
ejpam-3427	373	25	are	be	AUX
ejpam-3427	373	26	fuzzy	fuzzy	ADJ
ejpam-3427	373	27	normal	normal	ADJ
ejpam-3427	373	28	la	la	ADJ
ejpam-3427	373	29	-	-	PUNCT
ejpam-3427	373	30	subrings	subring	NOUN
ejpam-3427	373	31	of	of	ADP
ejpam-3427	373	32	an	an	DET
ejpam-3427	373	33	la	la	ADJ
ejpam-3427	373	34	-	-	PUNCT
ejpam-3427	373	35	ring	ring	NOUN
ejpam-3427	373	36	r1	r1	NOUN
ejpam-3427	373	37	×	×	NOUN
ejpam-3427	373	38	r2	r2	NOUN
ejpam-3427	373	39	.	.	PUNCT
ejpam-3427	374	1	we	we	PRON
ejpam-3427	374	2	have	have	VERB
ejpam-3427	374	3	to	to	PART
ejpam-3427	374	4	show	show	VERB
ejpam-3427	374	5	that	that	SCONJ
ejpam-3427	374	6	a	a	DET
ejpam-3427	374	7	×	×	NOUN
ejpam-3427	374	8	b	b	X
ejpam-3427	374	9	=	=	PUNCT
ejpam-3427	374	10	(	(	PUNCT
ejpam-3427	374	11	µa×b	µa×b	PROPN
ejpam-3427	374	12	,	,	PUNCT
ejpam-3427	374	13	γa×b	γa×b	NOUN
ejpam-3427	374	14	)	)	PUNCT
ejpam-3427	374	15	is	be	AUX
ejpam-3427	374	16	an	an	DET
ejpam-3427	374	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	374	18	anti	anti	ADJ
ejpam-3427	374	19	fuzzy	fuzzy	ADJ
ejpam-3427	374	20	normal	normal	ADJ
ejpam-3427	374	21	la	la	NOUN
ejpam-3427	374	22	-	-	PUNCT
ejpam-3427	374	23	subring	subring	NOUN
ejpam-3427	374	24	of	of	ADP
ejpam-3427	374	25	an	an	DET
ejpam-3427	374	26	la	la	ADJ
ejpam-3427	374	27	-	-	PUNCT
ejpam-3427	374	28	ring	ring	NOUN
ejpam-3427	374	29	r1	r1	PROPN
ejpam-3427	374	30	×r2	×r2	PROPN
ejpam-3427	374	31	.	.	PUNCT
ejpam-3427	375	1	now	now	ADV
ejpam-3427	375	2	1−	1−	NUM
ejpam-3427	376	1	µa×b((x1	µa×b((x1	ADJ
ejpam-3427	376	2	,	,	PUNCT
ejpam-3427	376	3	x2)−	x2)−	NOUN
ejpam-3427	376	4	(	(	PUNCT
ejpam-3427	376	5	y1	y1	NOUN
ejpam-3427	376	6	,	,	PUNCT
ejpam-3427	376	7	y2	y2	PROPN
ejpam-3427	376	8	)	)	PUNCT
ejpam-3427	376	9	)	)	PUNCT
ejpam-3427	377	1	=	=	PUNCT
ejpam-3427	378	1	µa×b((x1	µa×b((x1	ADJ
ejpam-3427	378	2	,	,	PUNCT
ejpam-3427	378	3	x2)−	x2)−	NOUN
ejpam-3427	378	4	(	(	PUNCT
ejpam-3427	378	5	y1	y1	NOUN
ejpam-3427	378	6	,	,	PUNCT
ejpam-3427	378	7	y2	y2	PROPN
ejpam-3427	378	8	)	)	PUNCT
ejpam-3427	378	9	)	)	PUNCT
ejpam-3427	378	10	≥	≥	PRON
ejpam-3427	379	1	min{µa×b(x1	min{µa×b(x1	PROPN
ejpam-3427	379	2	,	,	PUNCT
ejpam-3427	379	3	x2	x2	PROPN
ejpam-3427	379	4	)	)	PUNCT
ejpam-3427	379	5	,	,	PUNCT
ejpam-3427	379	6	µa×b(y1	µa×b(y1	NUM
ejpam-3427	379	7	,	,	PUNCT
ejpam-3427	379	8	y2	y2	PROPN
ejpam-3427	379	9	)	)	PUNCT
ejpam-3427	379	10	}	}	PUNCT
ejpam-3427	379	11	=	=	PUNCT
ejpam-3427	379	12	min{1−	min{1−	VERB
ejpam-3427	379	13	µa×b(x1	µa×b(x1	NOUN
ejpam-3427	379	14	,	,	PUNCT
ejpam-3427	379	15	x2	x2	PROPN
ejpam-3427	379	16	)	)	PUNCT
ejpam-3427	379	17	,	,	PUNCT
ejpam-3427	379	18	1−	1−	NUM
ejpam-3427	379	19	µa×b(y1	µa×b(y1	NUM
ejpam-3427	379	20	,	,	PUNCT
ejpam-3427	379	21	y2	y2	PROPN
ejpam-3427	379	22	)	)	PUNCT
ejpam-3427	379	23	}	}	PUNCT
ejpam-3427	380	1	=	=	SYM
ejpam-3427	380	2	1−max{µa×b(x1	1−max{µa×b(x1	NUM
ejpam-3427	380	3	,	,	PUNCT
ejpam-3427	380	4	x2	x2	PROPN
ejpam-3427	380	5	)	)	PUNCT
ejpam-3427	380	6	,	,	PUNCT
ejpam-3427	380	7	µa×b(y1	µa×b(y1	NUM
ejpam-3427	380	8	,	,	PUNCT
ejpam-3427	380	9	y2	y2	PROPN
ejpam-3427	380	10	)	)	PUNCT
ejpam-3427	380	11	}	}	PUNCT
ejpam-3427	380	12	and	and	CCONJ
ejpam-3427	380	13	1−	1−	NUM
ejpam-3427	380	14	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	380	15	,	,	PUNCT
ejpam-3427	380	16	x2	x2	PROPN
ejpam-3427	380	17	)	)	PUNCT
ejpam-3427	380	18	◦	◦	NOUN
ejpam-3427	380	19	(	(	PUNCT
ejpam-3427	380	20	y1	y1	INTJ
ejpam-3427	380	21	,	,	PUNCT
ejpam-3427	380	22	y2	y2	PROPN
ejpam-3427	380	23	)	)	PUNCT
ejpam-3427	380	24	)	)	PUNCT
ejpam-3427	381	1	=	=	PUNCT
ejpam-3427	381	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	381	3	,	,	PUNCT
ejpam-3427	381	4	x2	x2	ADJ
ejpam-3427	381	5	)	)	PUNCT
ejpam-3427	381	6	◦	◦	NOUN
ejpam-3427	381	7	(	(	PUNCT
ejpam-3427	381	8	y1	y1	INTJ
ejpam-3427	381	9	,	,	PUNCT
ejpam-3427	381	10	y2	y2	PROPN
ejpam-3427	381	11	)	)	PUNCT
ejpam-3427	381	12	)	)	PUNCT
ejpam-3427	381	13	≥	≥	PRON
ejpam-3427	382	1	min{µa×b(x1	min{µa×b(x1	PROPN
ejpam-3427	382	2	,	,	PUNCT
ejpam-3427	382	3	x2	x2	PROPN
ejpam-3427	382	4	)	)	PUNCT
ejpam-3427	382	5	,	,	PUNCT
ejpam-3427	382	6	µa×b(y1	µa×b(y1	NUM
ejpam-3427	382	7	,	,	PUNCT
ejpam-3427	382	8	y2	y2	PROPN
ejpam-3427	382	9	)	)	PUNCT
ejpam-3427	382	10	}	}	PUNCT
ejpam-3427	382	11	=	=	PUNCT
ejpam-3427	382	12	min{1−	min{1−	VERB
ejpam-3427	382	13	µa×b(x1	µa×b(x1	NOUN
ejpam-3427	382	14	,	,	PUNCT
ejpam-3427	382	15	x2	x2	PROPN
ejpam-3427	382	16	)	)	PUNCT
ejpam-3427	382	17	,	,	PUNCT
ejpam-3427	382	18	1−	1−	NUM
ejpam-3427	382	19	µa×b(y1	µa×b(y1	NUM
ejpam-3427	382	20	,	,	PUNCT
ejpam-3427	382	21	y2	y2	PROPN
ejpam-3427	382	22	)	)	PUNCT
ejpam-3427	382	23	}	}	PUNCT
ejpam-3427	383	1	=	=	SYM
ejpam-3427	383	2	1−max{µa×b(x1	1−max{µa×b(x1	NUM
ejpam-3427	383	3	,	,	PUNCT
ejpam-3427	383	4	x2	x2	PROPN
ejpam-3427	383	5	)	)	PUNCT
ejpam-3427	383	6	,	,	PUNCT
ejpam-3427	383	7	µa×b(y1	µa×b(y1	NUM
ejpam-3427	383	8	,	,	PUNCT
ejpam-3427	383	9	y2	y2	PROPN
ejpam-3427	383	10	)	)	PUNCT
ejpam-3427	383	11	}	}	PUNCT
ejpam-3427	383	12	.	.	PUNCT
ejpam-3427	384	1	thus	thus	ADV
ejpam-3427	384	2	a	a	DET
ejpam-3427	384	3	×	×	PROPN
ejpam-3427	384	4	b	b	X
ejpam-3427	384	5	=	=	PUNCT
ejpam-3427	384	6	(	(	PUNCT
ejpam-3427	384	7	µa×b	µa×b	PROPN
ejpam-3427	384	8	,	,	PUNCT
ejpam-3427	384	9	γa×b	γa×b	NOUN
ejpam-3427	384	10	)	)	PUNCT
ejpam-3427	384	11	is	be	AUX
ejpam-3427	384	12	an	an	DET
ejpam-3427	384	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	384	14	anti	anti	ADJ
ejpam-3427	384	15	fuzzy	fuzzy	ADJ
ejpam-3427	384	16	la	la	NOUN
ejpam-3427	384	17	-	-	PUNCT
ejpam-3427	384	18	subring	subring	NOUN
ejpam-3427	384	19	of	of	ADP
ejpam-3427	384	20	an	an	DET
ejpam-3427	384	21	la	la	ADJ
ejpam-3427	384	22	-	-	PUNCT
ejpam-3427	384	23	ring	ring	NOUN
ejpam-3427	384	24	r1	r1	PROPN
ejpam-3427	384	25	×r2	×r2	PROPN
ejpam-3427	384	26	.	.	PUNCT
ejpam-3427	385	1	now	now	ADV
ejpam-3427	385	2	1−	1−	NUM
ejpam-3427	386	1	µa×b((x1	µa×b((x1	ADJ
ejpam-3427	386	2	,	,	PUNCT
ejpam-3427	386	3	x2	x2	PROPN
ejpam-3427	386	4	)	)	PUNCT
ejpam-3427	386	5	◦	◦	NOUN
ejpam-3427	386	6	(	(	PUNCT
ejpam-3427	386	7	y1	y1	INTJ
ejpam-3427	386	8	,	,	PUNCT
ejpam-3427	386	9	y2	y2	PROPN
ejpam-3427	386	10	)	)	PUNCT
ejpam-3427	386	11	)	)	PUNCT
ejpam-3427	387	1	=	=	PUNCT
ejpam-3427	387	2	µa×b((x1	µa×b((x1	PROPN
ejpam-3427	387	3	,	,	PUNCT
ejpam-3427	387	4	x2	x2	ADJ
ejpam-3427	387	5	)	)	PUNCT
ejpam-3427	387	6	◦	◦	NOUN
ejpam-3427	387	7	(	(	PUNCT
ejpam-3427	387	8	y1	y1	INTJ
ejpam-3427	387	9	,	,	PUNCT
ejpam-3427	387	10	y2	y2	PROPN
ejpam-3427	387	11	)	)	PUNCT
ejpam-3427	387	12	)	)	PUNCT
ejpam-3427	388	1	=	=	PUNCT
ejpam-3427	388	2	µa×b((y1	µa×b((y1	PROPN
ejpam-3427	388	3	,	,	PUNCT
ejpam-3427	388	4	y2	y2	NOUN
ejpam-3427	388	5	)	)	PUNCT
ejpam-3427	388	6	◦	◦	NOUN
ejpam-3427	388	7	(	(	PUNCT
ejpam-3427	388	8	x1	x1	PROPN
ejpam-3427	388	9	,	,	PUNCT
ejpam-3427	388	10	x2	x2	PROPN
ejpam-3427	388	11	)	)	PUNCT
ejpam-3427	388	12	)	)	PUNCT
ejpam-3427	389	1	=	=	SYM
ejpam-3427	389	2	1−	1−	NUM
ejpam-3427	389	3	µa×b((y1	µa×b((y1	PROPN
ejpam-3427	389	4	,	,	PUNCT
ejpam-3427	389	5	y2	y2	NOUN
ejpam-3427	389	6	)	)	PUNCT
ejpam-3427	389	7	◦	◦	NOUN
ejpam-3427	389	8	(	(	PUNCT
ejpam-3427	389	9	x1	x1	PROPN
ejpam-3427	389	10	,	,	PUNCT
ejpam-3427	389	11	x2	x2	PROPN
ejpam-3427	389	12	)	)	PUNCT
ejpam-3427	389	13	)	)	PUNCT
ejpam-3427	389	14	.	.	PUNCT
ejpam-3427	390	1	hence	hence	ADV
ejpam-3427	390	2	a×	a×	PROPN
ejpam-3427	390	3	b	b	X
ejpam-3427	390	4	=	=	PUNCT
ejpam-3427	390	5	(	(	PUNCT
ejpam-3427	390	6	µa×b	µa×b	PROPN
ejpam-3427	390	7	,	,	PUNCT
ejpam-3427	390	8	γa×b	γa×b	NOUN
ejpam-3427	390	9	)	)	PUNCT
ejpam-3427	390	10	is	be	AUX
ejpam-3427	390	11	an	an	DET
ejpam-3427	390	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	390	13	anti	anti	ADJ
ejpam-3427	390	14	fuzzy	fuzzy	ADJ
ejpam-3427	390	15	normal	normal	ADJ
ejpam-3427	390	16	la	la	NOUN
ejpam-3427	390	17	-	-	PUNCT
ejpam-3427	390	18	subring	subring	NOUN
ejpam-3427	390	19	of	of	ADP
ejpam-3427	390	20	an	an	DET
ejpam-3427	390	21	la	la	ADJ
ejpam-3427	390	22	-	-	PUNCT
ejpam-3427	390	23	ring	ring	NOUN
ejpam-3427	390	24	r1	r1	PROPN
ejpam-3427	390	25	×r2	×r2	PROPN
ejpam-3427	390	26	.	.	PUNCT
ejpam-3427	391	1	lemma	lemma	PROPN
ejpam-3427	391	2	4	4	X
ejpam-3427	391	3	.	.	PUNCT
ejpam-3427	392	1	let	let	VERB
ejpam-3427	392	2	a	a	PRON
ejpam-3427	392	3	and	and	CCONJ
ejpam-3427	392	4	b	b	NOUN
ejpam-3427	392	5	be	be	AUX
ejpam-3427	392	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	392	7	fuzzy	fuzzy	ADJ
ejpam-3427	392	8	sets	set	NOUN
ejpam-3427	392	9	of	of	ADP
ejpam-3427	392	10	la	la	NOUN
ejpam-3427	392	11	-	-	PUNCT
ejpam-3427	392	12	rings	ring	NOUN
ejpam-3427	392	13	r1	r1	NOUN
ejpam-3427	392	14	and	and	CCONJ
ejpam-3427	392	15	r2	r2	PROPN
ejpam-3427	392	16	with	with	ADP
ejpam-3427	392	17	left	left	ADJ
ejpam-3427	392	18	identities	identity	NOUN
ejpam-3427	392	19	e1	e1	PROPN
ejpam-3427	392	20	and	and	CCONJ
ejpam-3427	392	21	e2	e2	PROPN
ejpam-3427	392	22	,	,	PUNCT
ejpam-3427	392	23	respectively	respectively	ADV
ejpam-3427	392	24	.	.	PUNCT
ejpam-3427	393	1	if	if	SCONJ
ejpam-3427	393	2	a×b	a×b	PROPN
ejpam-3427	393	3	is	be	AUX
ejpam-3427	393	4	an	an	DET
ejpam-3427	393	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	393	6	anti	anti	ADJ
ejpam-3427	393	7	fuzzy	fuzzy	ADJ
ejpam-3427	393	8	la	la	NOUN
ejpam-3427	393	9	-	-	PUNCT
ejpam-3427	393	10	subring	subring	NOUN
ejpam-3427	393	11	of	of	ADP
ejpam-3427	393	12	an	an	DET
ejpam-3427	393	13	la	la	ADJ
ejpam-3427	393	14	-	-	PUNCT
ejpam-3427	393	15	ring	ring	NOUN
ejpam-3427	393	16	r1	r1	PROPN
ejpam-3427	393	17	×r2	×r2	PROPN
ejpam-3427	393	18	,	,	PUNCT
ejpam-3427	393	19	then	then	ADV
ejpam-3427	393	20	at	at	ADP
ejpam-3427	393	21	least	least	ADJ
ejpam-3427	393	22	one	one	NUM
ejpam-3427	393	23	of	of	ADP
ejpam-3427	393	24	the	the	DET
ejpam-3427	393	25	following	follow	VERB
ejpam-3427	393	26	two	two	NUM
ejpam-3427	393	27	statements	statement	NOUN
ejpam-3427	393	28	must	must	AUX
ejpam-3427	393	29	hold	hold	VERB
ejpam-3427	393	30	.	.	PUNCT
ejpam-3427	394	1	1	1	X
ejpam-3427	394	2	.	.	X
ejpam-3427	394	3	µa	µa	NOUN
ejpam-3427	394	4	(	(	PUNCT
ejpam-3427	394	5	x	x	X
ejpam-3427	394	6	)	)	PUNCT
ejpam-3427	394	7	≥	≥	PROPN
ejpam-3427	394	8	µb	µb	PROPN
ejpam-3427	394	9	(	(	PUNCT
ejpam-3427	394	10	e2	e2	PROPN
ejpam-3427	394	11	)	)	PUNCT
ejpam-3427	394	12	and	and	CCONJ
ejpam-3427	394	13	γa	γa	PROPN
ejpam-3427	394	14	(	(	PUNCT
ejpam-3427	394	15	x	x	NOUN
ejpam-3427	394	16	)	)	PUNCT
ejpam-3427	394	17	≤	≤	NUM
ejpam-3427	394	18	γb	γb	NOUN
ejpam-3427	394	19	(	(	PUNCT
ejpam-3427	394	20	e2	e2	PROPN
ejpam-3427	394	21	)	)	PUNCT
ejpam-3427	394	22	,	,	PUNCT
ejpam-3427	394	23	for	for	ADP
ejpam-3427	394	24	all	all	DET
ejpam-3427	394	25	x	x	SYM
ejpam-3427	394	26	∈	∈	PROPN
ejpam-3427	394	27	r1	r1	NOUN
ejpam-3427	394	28	.	.	PUNCT
ejpam-3427	395	1	2	2	X
ejpam-3427	395	2	.	.	X
ejpam-3427	395	3	µb	µb	PROPN
ejpam-3427	395	4	(	(	PUNCT
ejpam-3427	395	5	x	x	NOUN
ejpam-3427	395	6	)	)	PUNCT
ejpam-3427	395	7	≥	≥	PROPN
ejpam-3427	395	8	µa	µa	NOUN
ejpam-3427	395	9	(	(	PUNCT
ejpam-3427	395	10	e1	e1	PROPN
ejpam-3427	395	11	)	)	PUNCT
ejpam-3427	395	12	and	and	CCONJ
ejpam-3427	395	13	γb	γb	INTJ
ejpam-3427	395	14	(	(	PUNCT
ejpam-3427	395	15	x	x	NOUN
ejpam-3427	395	16	)	)	PUNCT
ejpam-3427	395	17	≤	≤	NOUN
ejpam-3427	395	18	γa	γa	PROPN
ejpam-3427	395	19	(	(	PUNCT
ejpam-3427	395	20	e1	e1	PROPN
ejpam-3427	395	21	)	)	PUNCT
ejpam-3427	395	22	,	,	PUNCT
ejpam-3427	395	23	for	for	ADP
ejpam-3427	395	24	all	all	DET
ejpam-3427	395	25	x	x	SYM
ejpam-3427	395	26	∈	∈	PROPN
ejpam-3427	395	27	r2	r2	NOUN
ejpam-3427	395	28	.	.	PUNCT
ejpam-3427	396	1	proof	proof	NOUN
ejpam-3427	396	2	.	.	PUNCT
ejpam-3427	397	1	let	let	VERB
ejpam-3427	397	2	a	a	DET
ejpam-3427	397	3	×	×	PROPN
ejpam-3427	397	4	b	b	NOUN
ejpam-3427	397	5	be	be	AUX
ejpam-3427	397	6	an	an	DET
ejpam-3427	397	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	397	8	anti	anti	ADJ
ejpam-3427	397	9	fuzzy	fuzzy	ADJ
ejpam-3427	397	10	la	la	NOUN
ejpam-3427	397	11	-	-	PUNCT
ejpam-3427	397	12	subring	subring	NOUN
ejpam-3427	397	13	of	of	ADP
ejpam-3427	397	14	an	an	DET
ejpam-3427	397	15	la	la	ADJ
ejpam-3427	397	16	-	-	PUNCT
ejpam-3427	397	17	ring	ring	NOUN
ejpam-3427	397	18	r1	r1	NOUN
ejpam-3427	397	19	×	×	NOUN
ejpam-3427	397	20	r2	r2	NOUN
ejpam-3427	397	21	.	.	PUNCT
ejpam-3427	398	1	by	by	ADP
ejpam-3427	398	2	contraposition	contraposition	NOUN
ejpam-3427	398	3	,	,	PUNCT
ejpam-3427	398	4	suppose	suppose	VERB
ejpam-3427	398	5	that	that	SCONJ
ejpam-3427	398	6	none	none	NOUN
ejpam-3427	398	7	of	of	ADP
ejpam-3427	398	8	the	the	DET
ejpam-3427	398	9	statements	statement	NOUN
ejpam-3427	398	10	(	(	PUNCT
ejpam-3427	398	11	i	i	NOUN
ejpam-3427	398	12	)	)	PUNCT
ejpam-3427	398	13	and	and	CCONJ
ejpam-3427	398	14	(	(	PUNCT
ejpam-3427	398	15	ii	ii	NOUN
ejpam-3427	398	16	)	)	PUNCT
ejpam-3427	398	17	holds	hold	VERB
ejpam-3427	398	18	.	.	PUNCT
ejpam-3427	399	1	then	then	ADV
ejpam-3427	399	2	we	we	PRON
ejpam-3427	399	3	can	can	AUX
ejpam-3427	399	4	find	find	VERB
ejpam-3427	399	5	a	a	PRON
ejpam-3427	399	6	and	and	CCONJ
ejpam-3427	399	7	b	b	NOUN
ejpam-3427	399	8	in	in	ADP
ejpam-3427	399	9	r1	r1	NOUN
ejpam-3427	399	10	and	and	CCONJ
ejpam-3427	399	11	r2	r2	PROPN
ejpam-3427	399	12	,	,	PUNCT
ejpam-3427	399	13	respectively	respectively	ADV
ejpam-3427	399	14	such	such	ADJ
ejpam-3427	399	15	that	that	SCONJ
ejpam-3427	399	16	µa	µa	NOUN
ejpam-3427	399	17	(	(	PUNCT
ejpam-3427	399	18	a	a	X
ejpam-3427	399	19	)	)	PUNCT
ejpam-3427	399	20	≤	≤	PROPN
ejpam-3427	399	21	µb	µb	PROPN
ejpam-3427	399	22	(	(	PUNCT
ejpam-3427	399	23	e2	e2	PROPN
ejpam-3427	399	24	)	)	PUNCT
ejpam-3427	399	25	and	and	CCONJ
ejpam-3427	399	26	γa	γa	PROPN
ejpam-3427	399	27	(	(	PUNCT
ejpam-3427	399	28	a	a	PRON
ejpam-3427	399	29	)	)	PUNCT
ejpam-3427	399	30	≥	≥	NOUN
ejpam-3427	399	31	γb	γb	PROPN
ejpam-3427	399	32	(	(	PUNCT
ejpam-3427	399	33	e2	e2	PROPN
ejpam-3427	399	34	)	)	PUNCT
ejpam-3427	399	35	,	,	PUNCT
ejpam-3427	399	36	µb	µb	VERB
ejpam-3427	399	37	(	(	PUNCT
ejpam-3427	399	38	b	b	NOUN
ejpam-3427	399	39	)	)	PUNCT
ejpam-3427	399	40	≤	≤	NOUN
ejpam-3427	399	41	µa	µa	NOUN
ejpam-3427	399	42	(	(	PUNCT
ejpam-3427	399	43	e1	e1	PROPN
ejpam-3427	399	44	)	)	PUNCT
ejpam-3427	399	45	and	and	CCONJ
ejpam-3427	399	46	γb	γb	INTJ
ejpam-3427	399	47	(	(	PUNCT
ejpam-3427	399	48	b	b	NOUN
ejpam-3427	399	49	)	)	PUNCT
ejpam-3427	399	50	≥	≥	NOUN
ejpam-3427	399	51	γa	γa	PROPN
ejpam-3427	399	52	(	(	PUNCT
ejpam-3427	399	53	e1	e1	PROPN
ejpam-3427	399	54	)	)	PUNCT
ejpam-3427	399	55	.	.	PUNCT
ejpam-3427	400	1	thus	thus	ADV
ejpam-3427	400	2	µa×b(a	µa×b(a	NOUN
ejpam-3427	400	3	,	,	PUNCT
ejpam-3427	400	4	b	b	NOUN
ejpam-3427	400	5	)	)	PUNCT
ejpam-3427	400	6	=	=	SYM
ejpam-3427	400	7	max{µa(a	max{µa(a	PROPN
ejpam-3427	400	8	)	)	PUNCT
ejpam-3427	400	9	,	,	PUNCT
ejpam-3427	400	10	µb(b	µb(b	NUM
ejpam-3427	400	11	)	)	PUNCT
ejpam-3427	400	12	}	}	PUNCT
ejpam-3427	400	13	≤	≤	NUM
ejpam-3427	400	14	max{µa(e1	max{µa(e1	NOUN
ejpam-3427	400	15	)	)	PUNCT
ejpam-3427	400	16	,	,	PUNCT
ejpam-3427	400	17	µb(e2	µb(e2	NOUN
ejpam-3427	400	18	)	)	PUNCT
ejpam-3427	400	19	}	}	PUNCT
ejpam-3427	400	20	=	=	SYM
ejpam-3427	400	21	µa×b(e1	µa×b(e1	X
ejpam-3427	400	22	,	,	PUNCT
ejpam-3427	400	23	e2	e2	PROPN
ejpam-3427	400	24	)	)	PUNCT
ejpam-3427	400	25	and	and	CCONJ
ejpam-3427	400	26	γa×b(a	γa×b(a	ADJ
ejpam-3427	400	27	,	,	PUNCT
ejpam-3427	400	28	b	b	NOUN
ejpam-3427	400	29	)	)	PUNCT
ejpam-3427	400	30	=	=	SYM
ejpam-3427	400	31	min{γa(a	min{γa(a	PROPN
ejpam-3427	400	32	)	)	PUNCT
ejpam-3427	400	33	,	,	PUNCT
ejpam-3427	400	34	γb(b	γb(b	NOUN
ejpam-3427	400	35	)	)	PUNCT
ejpam-3427	400	36	}	}	PUNCT
ejpam-3427	400	37	k.	k.	ADV
ejpam-3427	401	1	nasreen	nasreen	PROPN
ejpam-3427	401	2	/	/	SYM
ejpam-3427	401	3	eur	eur	PROPN
ejpam-3427	401	4	.	.	PUNCT
ejpam-3427	402	1	j.	j.	PROPN
ejpam-3427	402	2	pure	pure	PROPN
ejpam-3427	402	3	appl	appl	PROPN
ejpam-3427	402	4	.	.	PROPN
ejpam-3427	402	5	math	math	PROPN
ejpam-3427	402	6	,	,	PUNCT
ejpam-3427	402	7	12	12	NUM
ejpam-3427	402	8	(	(	PUNCT
ejpam-3427	402	9	2	2	NUM
ejpam-3427	402	10	)	)	PUNCT
ejpam-3427	402	11	(	(	PUNCT
ejpam-3427	402	12	2019	2019	NUM
ejpam-3427	402	13	)	)	PUNCT
ejpam-3427	402	14	,	,	PUNCT
ejpam-3427	402	15	622	622	NUM
ejpam-3427	402	16	-	-	SYM
ejpam-3427	402	17	648	648	NUM
ejpam-3427	402	18	635	635	NUM
ejpam-3427	402	19	≥	≥	NOUN
ejpam-3427	402	20	min(γa(e1	min(γa(e1	PROPN
ejpam-3427	402	21	)	)	PUNCT
ejpam-3427	402	22	,	,	PUNCT
ejpam-3427	402	23	γb(e2	γb(e2	PROPN
ejpam-3427	402	24	)	)	PUNCT
ejpam-3427	402	25	)	)	PUNCT
ejpam-3427	403	1	=	=	SYM
ejpam-3427	403	2	γa×b(e1	γa×b(e1	NUM
ejpam-3427	403	3	,	,	PUNCT
ejpam-3427	403	4	e2	e2	PROPN
ejpam-3427	403	5	)	)	PUNCT
ejpam-3427	403	6	.	.	PUNCT
ejpam-3427	404	1	this	this	PRON
ejpam-3427	404	2	implies	imply	VERB
ejpam-3427	404	3	that	that	SCONJ
ejpam-3427	404	4	a	a	DET
ejpam-3427	404	5	×	×	PROPN
ejpam-3427	404	6	b	b	PROPN
ejpam-3427	404	7	is	be	AUX
ejpam-3427	404	8	not	not	PART
ejpam-3427	404	9	an	an	DET
ejpam-3427	404	10	intuitionistic	intuitionistic	ADJ
ejpam-3427	404	11	anti	anti	ADJ
ejpam-3427	404	12	fuzzy	fuzzy	ADJ
ejpam-3427	404	13	la	la	NOUN
ejpam-3427	404	14	-	-	PUNCT
ejpam-3427	404	15	subring	subring	NOUN
ejpam-3427	404	16	of	of	ADP
ejpam-3427	404	17	an	an	DET
ejpam-3427	404	18	laring	laring	NOUN
ejpam-3427	404	19	r1	r1	NOUN
ejpam-3427	404	20	×	×	NOUN
ejpam-3427	404	21	r2	r2	NOUN
ejpam-3427	404	22	.	.	PUNCT
ejpam-3427	405	1	hence	hence	ADV
ejpam-3427	405	2	either	either	CCONJ
ejpam-3427	405	3	µa	µa	NOUN
ejpam-3427	405	4	(	(	PUNCT
ejpam-3427	405	5	x	x	X
ejpam-3427	405	6	)	)	PUNCT
ejpam-3427	405	7	≥	≥	PROPN
ejpam-3427	405	8	µb	µb	PROPN
ejpam-3427	405	9	(	(	PUNCT
ejpam-3427	405	10	e2	e2	PROPN
ejpam-3427	405	11	)	)	PUNCT
ejpam-3427	405	12	and	and	CCONJ
ejpam-3427	405	13	γa	γa	PROPN
ejpam-3427	405	14	(	(	PUNCT
ejpam-3427	405	15	x	x	NOUN
ejpam-3427	405	16	)	)	PUNCT
ejpam-3427	405	17	≤	≤	NUM
ejpam-3427	405	18	γb	γb	NOUN
ejpam-3427	405	19	(	(	PUNCT
ejpam-3427	405	20	e2	e2	PROPN
ejpam-3427	405	21	)	)	PUNCT
ejpam-3427	405	22	,	,	PUNCT
ejpam-3427	405	23	for	for	ADP
ejpam-3427	405	24	all	all	DET
ejpam-3427	405	25	x	x	SYM
ejpam-3427	405	26	∈	∈	PROPN
ejpam-3427	405	27	r1	r1	NOUN
ejpam-3427	405	28	or	or	CCONJ
ejpam-3427	405	29	µb	µb	VERB
ejpam-3427	405	30	(	(	PUNCT
ejpam-3427	405	31	x	x	NOUN
ejpam-3427	405	32	)	)	PUNCT
ejpam-3427	405	33	≥	≥	PROPN
ejpam-3427	405	34	µa	µa	NOUN
ejpam-3427	405	35	(	(	PUNCT
ejpam-3427	405	36	e1	e1	PROPN
ejpam-3427	405	37	)	)	PUNCT
ejpam-3427	405	38	and	and	CCONJ
ejpam-3427	405	39	γb	γb	INTJ
ejpam-3427	405	40	(	(	PUNCT
ejpam-3427	405	41	x	x	NOUN
ejpam-3427	405	42	)	)	PUNCT
ejpam-3427	405	43	≤	≤	NOUN
ejpam-3427	405	44	γa	γa	PROPN
ejpam-3427	405	45	(	(	PUNCT
ejpam-3427	405	46	e1	e1	PROPN
ejpam-3427	405	47	)	)	PUNCT
ejpam-3427	405	48	,	,	PUNCT
ejpam-3427	405	49	for	for	ADP
ejpam-3427	405	50	all	all	DET
ejpam-3427	405	51	x	x	SYM
ejpam-3427	405	52	∈	∈	PROPN
ejpam-3427	405	53	r2	r2	NOUN
ejpam-3427	405	54	.	.	PUNCT
ejpam-3427	406	1	theorem	theorem	NOUN
ejpam-3427	406	2	5	5	NUM
ejpam-3427	406	3	.	.	PUNCT
ejpam-3427	407	1	let	let	VERB
ejpam-3427	407	2	a	a	PRON
ejpam-3427	407	3	and	and	CCONJ
ejpam-3427	407	4	b	b	NOUN
ejpam-3427	407	5	be	be	AUX
ejpam-3427	407	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	407	7	fuzzy	fuzzy	ADJ
ejpam-3427	407	8	sets	set	NOUN
ejpam-3427	407	9	of	of	ADP
ejpam-3427	407	10	la	la	NOUN
ejpam-3427	407	11	-	-	PUNCT
ejpam-3427	407	12	rings	ring	NOUN
ejpam-3427	407	13	r1	r1	NOUN
ejpam-3427	407	14	and	and	CCONJ
ejpam-3427	407	15	r2	r2	PROPN
ejpam-3427	407	16	with	with	ADP
ejpam-3427	407	17	left	left	ADJ
ejpam-3427	407	18	identities	identity	NOUN
ejpam-3427	407	19	e1	e1	PROPN
ejpam-3427	407	20	and	and	CCONJ
ejpam-3427	407	21	e2	e2	PROPN
ejpam-3427	407	22	,	,	PUNCT
ejpam-3427	407	23	respectively	respectively	ADV
ejpam-3427	407	24	and	and	CCONJ
ejpam-3427	407	25	a	a	DET
ejpam-3427	407	26	×	×	PROPN
ejpam-3427	407	27	b	b	NOUN
ejpam-3427	407	28	is	be	AUX
ejpam-3427	407	29	an	an	DET
ejpam-3427	407	30	intuitionistic	intuitionistic	ADJ
ejpam-3427	407	31	anti	anti	ADJ
ejpam-3427	407	32	fuzzy	fuzzy	ADJ
ejpam-3427	407	33	normal	normal	ADJ
ejpam-3427	407	34	lasubring	lasubring	NOUN
ejpam-3427	407	35	of	of	ADP
ejpam-3427	407	36	an	an	DET
ejpam-3427	407	37	la	la	ADJ
ejpam-3427	407	38	-	-	PUNCT
ejpam-3427	407	39	ring	ring	NOUN
ejpam-3427	407	40	r1	r1	PROPN
ejpam-3427	407	41	×r2	×r2	PROPN
ejpam-3427	407	42	.	.	PUNCT
ejpam-3427	408	1	then	then	ADV
ejpam-3427	408	2	the	the	DET
ejpam-3427	408	3	following	follow	VERB
ejpam-3427	408	4	conditions	condition	NOUN
ejpam-3427	408	5	are	be	AUX
ejpam-3427	408	6	true	true	ADJ
ejpam-3427	408	7	.	.	PUNCT
ejpam-3427	409	1	1	1	X
ejpam-3427	409	2	.	.	X
ejpam-3427	410	1	if	if	SCONJ
ejpam-3427	410	2	µa	µa	PROPN
ejpam-3427	410	3	(	(	PUNCT
ejpam-3427	410	4	x	x	X
ejpam-3427	410	5	)	)	PUNCT
ejpam-3427	410	6	≥	≥	PROPN
ejpam-3427	410	7	µb	µb	PROPN
ejpam-3427	410	8	(	(	PUNCT
ejpam-3427	410	9	e2	e2	PROPN
ejpam-3427	410	10	)	)	PUNCT
ejpam-3427	410	11	and	and	CCONJ
ejpam-3427	410	12	γa	γa	PROPN
ejpam-3427	410	13	(	(	PUNCT
ejpam-3427	410	14	x	x	NOUN
ejpam-3427	410	15	)	)	PUNCT
ejpam-3427	410	16	≤	≤	NUM
ejpam-3427	410	17	γb	γb	NOUN
ejpam-3427	410	18	(	(	PUNCT
ejpam-3427	410	19	e2	e2	PROPN
ejpam-3427	410	20	)	)	PUNCT
ejpam-3427	410	21	,	,	PUNCT
ejpam-3427	410	22	for	for	ADP
ejpam-3427	410	23	all	all	DET
ejpam-3427	410	24	x	x	SYM
ejpam-3427	410	25	∈	∈	PROPN
ejpam-3427	410	26	r1	r1	NOUN
ejpam-3427	410	27	,	,	PUNCT
ejpam-3427	410	28	then	then	ADV
ejpam-3427	410	29	a	a	PRON
ejpam-3427	410	30	is	be	AUX
ejpam-3427	410	31	an	an	DET
ejpam-3427	410	32	intuitionistic	intuitionistic	ADJ
ejpam-3427	410	33	anti	anti	ADJ
ejpam-3427	410	34	fuzzy	fuzzy	ADJ
ejpam-3427	410	35	normal	normal	ADJ
ejpam-3427	410	36	la	la	NOUN
ejpam-3427	410	37	-	-	PUNCT
ejpam-3427	410	38	subring	subring	NOUN
ejpam-3427	410	39	of	of	ADP
ejpam-3427	410	40	r1	r1	PROPN
ejpam-3427	410	41	.	.	PUNCT
ejpam-3427	411	1	2	2	X
ejpam-3427	411	2	.	.	X
ejpam-3427	412	1	if	if	SCONJ
ejpam-3427	412	2	µb	µb	VERB
ejpam-3427	412	3	(	(	PUNCT
ejpam-3427	412	4	x	x	NOUN
ejpam-3427	412	5	)	)	PUNCT
ejpam-3427	412	6	≥	≥	PROPN
ejpam-3427	412	7	µa	µa	NOUN
ejpam-3427	412	8	(	(	PUNCT
ejpam-3427	412	9	e1	e1	PROPN
ejpam-3427	412	10	)	)	PUNCT
ejpam-3427	412	11	and	and	CCONJ
ejpam-3427	412	12	γb	γb	INTJ
ejpam-3427	412	13	(	(	PUNCT
ejpam-3427	412	14	x	x	NOUN
ejpam-3427	412	15	)	)	PUNCT
ejpam-3427	412	16	≤	≤	NOUN
ejpam-3427	412	17	γa	γa	PROPN
ejpam-3427	412	18	(	(	PUNCT
ejpam-3427	412	19	e1	e1	PROPN
ejpam-3427	412	20	)	)	PUNCT
ejpam-3427	412	21	,	,	PUNCT
ejpam-3427	412	22	for	for	ADP
ejpam-3427	412	23	all	all	DET
ejpam-3427	412	24	x	x	SYM
ejpam-3427	412	25	∈	∈	PROPN
ejpam-3427	412	26	r2	r2	NOUN
ejpam-3427	412	27	,	,	PUNCT
ejpam-3427	412	28	then	then	ADV
ejpam-3427	412	29	b	b	PROPN
ejpam-3427	412	30	is	be	AUX
ejpam-3427	412	31	an	an	DET
ejpam-3427	412	32	intuitionistic	intuitionistic	ADJ
ejpam-3427	412	33	anti	anti	ADJ
ejpam-3427	412	34	fuzzy	fuzzy	ADJ
ejpam-3427	412	35	normal	normal	ADJ
ejpam-3427	412	36	la	la	NOUN
ejpam-3427	412	37	-	-	PUNCT
ejpam-3427	412	38	subring	subring	NOUN
ejpam-3427	412	39	of	of	ADP
ejpam-3427	412	40	r2	r2	NOUN
ejpam-3427	412	41	.	.	PUNCT
ejpam-3427	413	1	proof	proof	NOUN
ejpam-3427	413	2	.	.	PUNCT
ejpam-3427	414	1	1	1	X
ejpam-3427	414	2	.	.	X
ejpam-3427	414	3	let	let	VERB
ejpam-3427	414	4	µa	µa	INTJ
ejpam-3427	414	5	(	(	PUNCT
ejpam-3427	414	6	x	x	X
ejpam-3427	414	7	)	)	PUNCT
ejpam-3427	414	8	≥	≥	PROPN
ejpam-3427	414	9	µb	µb	PROPN
ejpam-3427	414	10	(	(	PUNCT
ejpam-3427	414	11	e2	e2	PROPN
ejpam-3427	414	12	)	)	PUNCT
ejpam-3427	414	13	and	and	CCONJ
ejpam-3427	414	14	γa	γa	PROPN
ejpam-3427	414	15	(	(	PUNCT
ejpam-3427	414	16	x	x	NOUN
ejpam-3427	414	17	)	)	PUNCT
ejpam-3427	414	18	≤	≤	NUM
ejpam-3427	414	19	γb	γb	NOUN
ejpam-3427	414	20	(	(	PUNCT
ejpam-3427	414	21	e2	e2	PROPN
ejpam-3427	414	22	)	)	PUNCT
ejpam-3427	414	23	for	for	ADP
ejpam-3427	414	24	all	all	DET
ejpam-3427	414	25	x	x	SYM
ejpam-3427	414	26	∈	∈	PROPN
ejpam-3427	414	27	r1	r1	NOUN
ejpam-3427	414	28	,	,	PUNCT
ejpam-3427	414	29	and	and	CCONJ
ejpam-3427	414	30	y	y	PROPN
ejpam-3427	414	31	∈	∈	PROPN
ejpam-3427	414	32	r1	r1	PROPN
ejpam-3427	414	33	.	.	PUNCT
ejpam-3427	415	1	we	we	PRON
ejpam-3427	415	2	have	have	VERB
ejpam-3427	415	3	to	to	PART
ejpam-3427	415	4	show	show	VERB
ejpam-3427	415	5	that	that	SCONJ
ejpam-3427	415	6	a	a	PRON
ejpam-3427	415	7	is	be	AUX
ejpam-3427	415	8	an	an	DET
ejpam-3427	415	9	intuitionistic	intuitionistic	ADJ
ejpam-3427	415	10	anti	anti	ADJ
ejpam-3427	415	11	fuzzy	fuzzy	ADJ
ejpam-3427	415	12	normal	normal	ADJ
ejpam-3427	415	13	la	la	NOUN
ejpam-3427	415	14	-	-	PUNCT
ejpam-3427	415	15	subring	subring	NOUN
ejpam-3427	415	16	of	of	ADP
ejpam-3427	415	17	an	an	DET
ejpam-3427	415	18	la	la	ADJ
ejpam-3427	415	19	-	-	PUNCT
ejpam-3427	415	20	ring	ring	NOUN
ejpam-3427	415	21	r1	r1	NOUN
ejpam-3427	415	22	.	.	PUNCT
ejpam-3427	416	1	now	now	ADV
ejpam-3427	416	2	µa(x−	µa(x−	NUM
ejpam-3427	416	3	y	y	NOUN
ejpam-3427	416	4	)	)	PUNCT
ejpam-3427	416	5	=	=	SYM
ejpam-3427	416	6	µa(x+	µa(x+	PROPN
ejpam-3427	416	7	(	(	PUNCT
ejpam-3427	416	8	−y	−y	NOUN
ejpam-3427	416	9	)	)	PUNCT
ejpam-3427	416	10	)	)	PUNCT
ejpam-3427	417	1	=	=	SYM
ejpam-3427	417	2	max{µa(x+	max{µa(x+	PROPN
ejpam-3427	417	3	(	(	PUNCT
ejpam-3427	417	4	−y	−y	NOUN
ejpam-3427	417	5	)	)	PUNCT
ejpam-3427	417	6	)	)	PUNCT
ejpam-3427	417	7	,	,	PUNCT
ejpam-3427	417	8	µb(e2	µb(e2	NOUN
ejpam-3427	417	9	+	+	X
ejpam-3427	417	10	(	(	PUNCT
ejpam-3427	417	11	−e2	−e2	PROPN
ejpam-3427	417	12	)	)	PUNCT
ejpam-3427	417	13	)	)	PUNCT
ejpam-3427	417	14	}	}	PUNCT
ejpam-3427	417	15	=	=	SYM
ejpam-3427	417	16	µa×b(x+	µa×b(x+	X
ejpam-3427	417	17	(	(	PUNCT
ejpam-3427	417	18	−y	−y	NOUN
ejpam-3427	417	19	)	)	PUNCT
ejpam-3427	417	20	,	,	PUNCT
ejpam-3427	417	21	e2	e2	PROPN
ejpam-3427	417	22	+	+	CCONJ
ejpam-3427	417	23	(	(	PUNCT
ejpam-3427	417	24	−e2	−e2	PROPN
ejpam-3427	417	25	)	)	PUNCT
ejpam-3427	417	26	)	)	PUNCT
ejpam-3427	418	1	=	=	PUNCT
ejpam-3427	418	2	µa×b((x	µa×b((x	VERB
ejpam-3427	418	3	,	,	PUNCT
ejpam-3427	418	4	e2	e2	PROPN
ejpam-3427	418	5	)	)	PUNCT
ejpam-3427	418	6	+	+	CCONJ
ejpam-3427	418	7	(	(	PUNCT
ejpam-3427	418	8	−y,−e2	−y,−e2	NOUN
ejpam-3427	418	9	)	)	PUNCT
ejpam-3427	418	10	)	)	PUNCT
ejpam-3427	419	1	=	=	SYM
ejpam-3427	419	2	µa×b((x	µa×b((x	VERB
ejpam-3427	419	3	,	,	PUNCT
ejpam-3427	419	4	e2)−	e2)−	PROPN
ejpam-3427	419	5	(	(	PUNCT
ejpam-3427	419	6	y	y	PROPN
ejpam-3427	419	7	,	,	PUNCT
ejpam-3427	419	8	e2	e2	PROPN
ejpam-3427	419	9	)	)	PUNCT
ejpam-3427	419	10	)	)	PUNCT
ejpam-3427	419	11	≤	≤	NUM
ejpam-3427	419	12	µa×b(x	µa×b(x	PROPN
ejpam-3427	419	13	,	,	PUNCT
ejpam-3427	419	14	e2	e2	PROPN
ejpam-3427	419	15	)	)	PUNCT
ejpam-3427	419	16	∨	∨	NOUN
ejpam-3427	419	17	µa×b(y	µa×b(y	ADJ
ejpam-3427	419	18	,	,	PUNCT
ejpam-3427	419	19	e2	e2	PROPN
ejpam-3427	419	20	)	)	PUNCT
ejpam-3427	419	21	=	=	SYM
ejpam-3427	419	22	max{max{µa(x	max{max{µa(x	NOUN
ejpam-3427	419	23	)	)	PUNCT
ejpam-3427	419	24	,	,	PUNCT
ejpam-3427	419	25	µb(e2)},max{µa(y	µb(e2)},max{µa(y	NOUN
ejpam-3427	419	26	)	)	PUNCT
ejpam-3427	419	27	,	,	PUNCT
ejpam-3427	419	28	µb(e2	µb(e2	NOUN
ejpam-3427	419	29	)	)	PUNCT
ejpam-3427	419	30	}	}	PUNCT
ejpam-3427	419	31	}	}	PUNCT
ejpam-3427	419	32	=	=	SYM
ejpam-3427	419	33	µa(x	µa(x	NOUN
ejpam-3427	419	34	)	)	PUNCT
ejpam-3427	419	35	∨	∨	NUM
ejpam-3427	419	36	µa(y	µa(y	NOUN
ejpam-3427	419	37	)	)	PUNCT
ejpam-3427	419	38	and	and	CCONJ
ejpam-3427	419	39	µa(xy	µa(xy	NOUN
ejpam-3427	419	40	)	)	PUNCT
ejpam-3427	419	41	=	=	SYM
ejpam-3427	419	42	max{µa(xy	max{µa(xy	PROPN
ejpam-3427	419	43	)	)	PUNCT
ejpam-3427	419	44	,	,	PUNCT
ejpam-3427	419	45	µb(e2e2	µb(e2e2	PROPN
ejpam-3427	419	46	)	)	PUNCT
ejpam-3427	419	47	}	}	PUNCT
ejpam-3427	420	1	=	=	SYM
ejpam-3427	420	2	µa×b(xy	µa×b(xy	PROPN
ejpam-3427	420	3	,	,	PUNCT
ejpam-3427	420	4	e2e2	e2e2	NOUN
ejpam-3427	420	5	)	)	PUNCT
ejpam-3427	420	6	=	=	PUNCT
ejpam-3427	420	7	µa×b((x	µa×b((x	PROPN
ejpam-3427	420	8	,	,	PUNCT
ejpam-3427	420	9	e2	e2	PROPN
ejpam-3427	420	10	)	)	PUNCT
ejpam-3427	420	11	◦	◦	NOUN
ejpam-3427	420	12	(	(	PUNCT
ejpam-3427	420	13	y	y	PROPN
ejpam-3427	420	14	,	,	PUNCT
ejpam-3427	420	15	e2	e2	PROPN
ejpam-3427	420	16	)	)	PUNCT
ejpam-3427	420	17	)	)	PUNCT
ejpam-3427	420	18	≤	≤	NUM
ejpam-3427	420	19	µa×b(x	µa×b(x	PROPN
ejpam-3427	420	20	,	,	PUNCT
ejpam-3427	420	21	e2	e2	PROPN
ejpam-3427	420	22	)	)	PUNCT
ejpam-3427	420	23	∨	∨	NOUN
ejpam-3427	420	24	µa×b(y	µa×b(y	ADJ
ejpam-3427	420	25	,	,	PUNCT
ejpam-3427	420	26	e2	e2	PROPN
ejpam-3427	420	27	)	)	PUNCT
ejpam-3427	420	28	=	=	SYM
ejpam-3427	420	29	max{max{µa(x	max{max{µa(x	NOUN
ejpam-3427	420	30	)	)	PUNCT
ejpam-3427	420	31	,	,	PUNCT
ejpam-3427	420	32	µb(e2)},max{µa(y	µb(e2)},max{µa(y	NOUN
ejpam-3427	420	33	)	)	PUNCT
ejpam-3427	420	34	,	,	PUNCT
ejpam-3427	420	35	µb(e2	µb(e2	NOUN
ejpam-3427	420	36	)	)	PUNCT
ejpam-3427	420	37	}	}	PUNCT
ejpam-3427	420	38	}	}	PUNCT
ejpam-3427	420	39	=	=	SYM
ejpam-3427	420	40	µa(x	µa(x	NOUN
ejpam-3427	420	41	)	)	PUNCT
ejpam-3427	420	42	∨	∨	NUM
ejpam-3427	420	43	µa(y	µa(y	NOUN
ejpam-3427	420	44	)	)	PUNCT
ejpam-3427	420	45	.	.	PUNCT
ejpam-3427	421	1	similarly	similarly	ADV
ejpam-3427	421	2	,	,	PUNCT
ejpam-3427	421	3	we	we	PRON
ejpam-3427	421	4	have	have	VERB
ejpam-3427	421	5	γa(x−	γa(x−	PROPN
ejpam-3427	421	6	y	y	PROPN
ejpam-3427	421	7	)	)	PUNCT
ejpam-3427	421	8	≥	≥	PROPN
ejpam-3427	421	9	min{γa(x	min{γa(x	NOUN
ejpam-3427	421	10	)	)	PUNCT
ejpam-3427	421	11	,	,	PUNCT
ejpam-3427	421	12	γa(y	γa(y	NOUN
ejpam-3427	421	13	)	)	PUNCT
ejpam-3427	421	14	}	}	PUNCT
ejpam-3427	421	15	and	and	CCONJ
ejpam-3427	421	16	γa(xy	γa(xy	PROPN
ejpam-3427	421	17	)	)	PUNCT
ejpam-3427	421	18	≥	≥	PROPN
ejpam-3427	421	19	min{γa(x	min{γa(x	NOUN
ejpam-3427	421	20	)	)	PUNCT
ejpam-3427	421	21	,	,	PUNCT
ejpam-3427	421	22	γa(y	γa(y	NOUN
ejpam-3427	421	23	)	)	PUNCT
ejpam-3427	421	24	}	}	PUNCT
ejpam-3427	421	25	.	.	PUNCT
ejpam-3427	422	1	thus	thus	ADV
ejpam-3427	422	2	a	a	PRON
ejpam-3427	422	3	is	be	AUX
ejpam-3427	422	4	an	an	DET
ejpam-3427	422	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	422	6	anti	anti	ADJ
ejpam-3427	422	7	fuzzy	fuzzy	ADJ
ejpam-3427	422	8	la	la	NOUN
ejpam-3427	422	9	-	-	PUNCT
ejpam-3427	422	10	subring	subring	NOUN
ejpam-3427	422	11	of	of	ADP
ejpam-3427	422	12	an	an	DET
ejpam-3427	422	13	la	la	ADJ
ejpam-3427	422	14	-	-	PUNCT
ejpam-3427	422	15	ring	ring	NOUN
ejpam-3427	422	16	r1	r1	NOUN
ejpam-3427	422	17	.	.	PUNCT
ejpam-3427	423	1	now	now	ADV
ejpam-3427	423	2	µa(xy	µa(xy	NUM
ejpam-3427	423	3	)	)	PUNCT
ejpam-3427	424	1	=	=	SYM
ejpam-3427	424	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3427	424	3	)	)	PUNCT
ejpam-3427	424	4	,	,	PUNCT
ejpam-3427	424	5	µb(e2e2	µb(e2e2	PROPN
ejpam-3427	424	6	)	)	PUNCT
ejpam-3427	424	7	}	}	PUNCT
ejpam-3427	425	1	k.	k.	PROPN
ejpam-3427	425	2	nasreen	nasreen	PROPN
ejpam-3427	425	3	/	/	SYM
ejpam-3427	425	4	eur	eur	PROPN
ejpam-3427	425	5	.	.	PUNCT
ejpam-3427	426	1	j.	j.	PROPN
ejpam-3427	426	2	pure	pure	PROPN
ejpam-3427	426	3	appl	appl	PROPN
ejpam-3427	426	4	.	.	PROPN
ejpam-3427	426	5	math	math	PROPN
ejpam-3427	426	6	,	,	PUNCT
ejpam-3427	426	7	12	12	NUM
ejpam-3427	426	8	(	(	PUNCT
ejpam-3427	426	9	2	2	NUM
ejpam-3427	426	10	)	)	PUNCT
ejpam-3427	426	11	(	(	PUNCT
ejpam-3427	426	12	2019	2019	NUM
ejpam-3427	426	13	)	)	PUNCT
ejpam-3427	426	14	,	,	PUNCT
ejpam-3427	426	15	622	622	NUM
ejpam-3427	426	16	-	-	SYM
ejpam-3427	426	17	648	648	NUM
ejpam-3427	426	18	636	636	NUM
ejpam-3427	426	19	=	=	NOUN
ejpam-3427	426	20	µa×b	µa×b	PROPN
ejpam-3427	426	21	(	(	PUNCT
ejpam-3427	426	22	xy	xy	PROPN
ejpam-3427	426	23	,	,	PUNCT
ejpam-3427	426	24	e2e2	e2e2	NOUN
ejpam-3427	426	25	)	)	PUNCT
ejpam-3427	426	26	=	=	SYM
ejpam-3427	426	27	µa×b	µa×b	PROPN
ejpam-3427	426	28	(	(	PUNCT
ejpam-3427	426	29	(	(	PUNCT
ejpam-3427	426	30	x	x	NOUN
ejpam-3427	426	31	,	,	PUNCT
ejpam-3427	426	32	e2	e2	PROPN
ejpam-3427	426	33	)	)	PUNCT
ejpam-3427	426	34	◦	◦	NOUN
ejpam-3427	426	35	(	(	PUNCT
ejpam-3427	426	36	y	y	PROPN
ejpam-3427	426	37	,	,	PUNCT
ejpam-3427	426	38	e2	e2	PROPN
ejpam-3427	426	39	)	)	PUNCT
ejpam-3427	426	40	)	)	PUNCT
ejpam-3427	427	1	=	=	SYM
ejpam-3427	427	2	µa×b	µa×b	PROPN
ejpam-3427	427	3	(	(	PUNCT
ejpam-3427	427	4	(	(	PUNCT
ejpam-3427	427	5	y	y	PROPN
ejpam-3427	427	6	,	,	PUNCT
ejpam-3427	427	7	e2	e2	PROPN
ejpam-3427	427	8	)	)	PUNCT
ejpam-3427	427	9	◦	◦	NOUN
ejpam-3427	427	10	(	(	PUNCT
ejpam-3427	427	11	x	x	NOUN
ejpam-3427	427	12	,	,	PUNCT
ejpam-3427	427	13	e2	e2	PROPN
ejpam-3427	427	14	)	)	PUNCT
ejpam-3427	427	15	)	)	PUNCT
ejpam-3427	428	1	=	=	SYM
ejpam-3427	428	2	µa×b(yx	µa×b(yx	X
ejpam-3427	428	3	,	,	PUNCT
ejpam-3427	428	4	e2e2	e2e2	NOUN
ejpam-3427	428	5	)	)	PUNCT
ejpam-3427	428	6	=	=	SYM
ejpam-3427	428	7	max{µa(yx	max{µa(yx	X
ejpam-3427	428	8	)	)	PUNCT
ejpam-3427	428	9	,	,	PUNCT
ejpam-3427	428	10	µb(e2e2	µb(e2e2	PROPN
ejpam-3427	428	11	)	)	PUNCT
ejpam-3427	428	12	}	}	PUNCT
ejpam-3427	428	13	=	=	SYM
ejpam-3427	428	14	µa(yx	µa(yx	NOUN
ejpam-3427	428	15	)	)	PUNCT
ejpam-3427	428	16	.	.	PUNCT
ejpam-3427	429	1	similarly	similarly	ADV
ejpam-3427	429	2	,	,	PUNCT
ejpam-3427	429	3	γb(xy	γb(xy	NOUN
ejpam-3427	429	4	)	)	PUNCT
ejpam-3427	429	5	=	=	SYM
ejpam-3427	429	6	γb(yx	γb(yx	PROPN
ejpam-3427	429	7	)	)	PUNCT
ejpam-3427	429	8	.	.	PUNCT
ejpam-3427	430	1	hence	hence	ADV
ejpam-3427	430	2	a	a	PRON
ejpam-3427	430	3	is	be	AUX
ejpam-3427	430	4	an	an	DET
ejpam-3427	430	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	430	6	anti	anti	ADJ
ejpam-3427	430	7	fuzzy	fuzzy	ADJ
ejpam-3427	430	8	normal	normal	ADJ
ejpam-3427	430	9	la	la	NOUN
ejpam-3427	430	10	-	-	PUNCT
ejpam-3427	430	11	subring	subring	NOUN
ejpam-3427	430	12	of	of	ADP
ejpam-3427	430	13	an	an	DET
ejpam-3427	430	14	la	la	ADJ
ejpam-3427	430	15	-	-	PUNCT
ejpam-3427	430	16	ring	ring	NOUN
ejpam-3427	430	17	r1	r1	NOUN
ejpam-3427	430	18	.	.	PUNCT
ejpam-3427	431	1	2	2	NUM
ejpam-3427	431	2	.	.	X
ejpam-3427	431	3	is	be	AUX
ejpam-3427	431	4	same	same	ADJ
ejpam-3427	431	5	as	as	ADP
ejpam-3427	431	6	1	1	NUM
ejpam-3427	431	7	.	.	NOUN
ejpam-3427	432	1	3	3	X
ejpam-3427	432	2	.	.	X
ejpam-3427	432	3	direct	direct	ADJ
ejpam-3427	432	4	product	product	NOUN
ejpam-3427	432	5	of	of	ADP
ejpam-3427	432	6	finite	finite	PROPN
ejpam-3427	432	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	432	8	anti	anti	ADJ
ejpam-3427	432	9	fuzzy	fuzzy	ADJ
ejpam-3427	432	10	normal	normal	ADJ
ejpam-3427	432	11	la	la	NOUN
ejpam-3427	432	12	-	-	NOUN
ejpam-3427	432	13	subrings	subring	NOUN
ejpam-3427	432	14	we	we	PRON
ejpam-3427	432	15	define	define	VERB
ejpam-3427	432	16	the	the	DET
ejpam-3427	432	17	direct	direct	ADJ
ejpam-3427	432	18	product	product	NOUN
ejpam-3427	432	19	of	of	ADP
ejpam-3427	432	20	intuitionistic	intuitionistic	ADJ
ejpam-3427	432	21	fuzzy	fuzzy	ADJ
ejpam-3427	432	22	sets	set	NOUN
ejpam-3427	432	23	a1	a1	NOUN
ejpam-3427	432	24	,	,	PUNCT
ejpam-3427	432	25	a2	a2	PROPN
ejpam-3427	432	26	,	,	PUNCT
ejpam-3427	432	27	...	...	PUNCT
ejpam-3427	432	28	,	,	PUNCT
ejpam-3427	432	29	an	an	PRON
ejpam-3427	432	30	of	of	ADP
ejpam-3427	432	31	la	la	NOUN
ejpam-3427	432	32	-	-	PUNCT
ejpam-3427	432	33	rings	ring	NOUN
ejpam-3427	432	34	r1	r1	NOUN
ejpam-3427	432	35	,	,	PUNCT
ejpam-3427	432	36	r2	r2	PROPN
ejpam-3427	432	37	,	,	PUNCT
ejpam-3427	432	38	...	...	PUNCT
ejpam-3427	432	39	,	,	PUNCT
ejpam-3427	432	40	rn	rn	PROPN
ejpam-3427	432	41	,	,	PUNCT
ejpam-3427	432	42	respectively	respectively	ADV
ejpam-3427	432	43	and	and	CCONJ
ejpam-3427	432	44	examine	examine	VERB
ejpam-3427	432	45	the	the	DET
ejpam-3427	432	46	some	some	DET
ejpam-3427	432	47	fundamental	fundamental	ADJ
ejpam-3427	432	48	properties	property	NOUN
ejpam-3427	432	49	of	of	ADP
ejpam-3427	432	50	direct	direct	ADJ
ejpam-3427	432	51	product	product	NOUN
ejpam-3427	432	52	of	of	ADP
ejpam-3427	432	53	intuitionistic	intuitionistic	ADJ
ejpam-3427	432	54	anti	anti	ADJ
ejpam-3427	432	55	fuzzy	fuzzy	ADJ
ejpam-3427	432	56	normal	normal	ADJ
ejpam-3427	432	57	la	la	NOUN
ejpam-3427	432	58	-	-	PUNCT
ejpam-3427	432	59	subrings	subring	NOUN
ejpam-3427	432	60	of	of	ADP
ejpam-3427	432	61	an	an	DET
ejpam-3427	432	62	la	la	ADJ
ejpam-3427	432	63	-	-	PUNCT
ejpam-3427	432	64	ring	ring	NOUN
ejpam-3427	432	65	r1	r1	NOUN
ejpam-3427	432	66	×r2	×r2	PROPN
ejpam-3427	432	67	×	×	NOUN
ejpam-3427	432	68	...	...	PUNCT
ejpam-3427	432	69	×rn	×rn	NOUN
ejpam-3427	432	70	.	.	PUNCT
ejpam-3427	433	1	let	let	VERB
ejpam-3427	433	2	µ1	µ1	PROPN
ejpam-3427	433	3	,	,	PUNCT
ejpam-3427	433	4	µ2	µ2	PROPN
ejpam-3427	433	5	,	,	PUNCT
ejpam-3427	433	6	...	...	PUNCT
ejpam-3427	433	7	,	,	PUNCT
ejpam-3427	433	8	µn	µn	PROPN
ejpam-3427	433	9	be	be	VERB
ejpam-3427	433	10	fuzzy	fuzzy	ADJ
ejpam-3427	433	11	subsets	subset	NOUN
ejpam-3427	433	12	of	of	ADP
ejpam-3427	433	13	la	la	NOUN
ejpam-3427	433	14	-	-	PUNCT
ejpam-3427	433	15	rings	ring	NOUN
ejpam-3427	433	16	r1	r1	NOUN
ejpam-3427	433	17	,	,	PUNCT
ejpam-3427	433	18	r2	r2	PROPN
ejpam-3427	433	19	,	,	PUNCT
ejpam-3427	433	20	...	...	PUNCT
ejpam-3427	433	21	,	,	PUNCT
ejpam-3427	433	22	rn	rn	PROPN
ejpam-3427	433	23	,	,	PUNCT
ejpam-3427	433	24	respectively	respectively	ADV
ejpam-3427	433	25	.	.	PUNCT
ejpam-3427	434	1	the	the	DET
ejpam-3427	434	2	direct	direct	ADJ
ejpam-3427	434	3	product	product	NOUN
ejpam-3427	434	4	of	of	ADP
ejpam-3427	434	5	fuzzy	fuzzy	ADJ
ejpam-3427	434	6	subsets	subset	NOUN
ejpam-3427	434	7	µ1	µ1	PROPN
ejpam-3427	434	8	,	,	PUNCT
ejpam-3427	434	9	µ2	µ2	PROPN
ejpam-3427	434	10	,	,	PUNCT
ejpam-3427	434	11	...	...	PUNCT
ejpam-3427	434	12	,	,	PUNCT
ejpam-3427	434	13	µn	µn	PROPN
ejpam-3427	434	14	is	be	AUX
ejpam-3427	434	15	denoted	denote	VERB
ejpam-3427	434	16	by	by	ADP
ejpam-3427	434	17	µ1	µ1	PROPN
ejpam-3427	434	18	×	×	PROPN
ejpam-3427	434	19	µ2	µ2	PROPN
ejpam-3427	434	20	×	×	NOUN
ejpam-3427	434	21	...	...	PUNCT
ejpam-3427	434	22	×	×	NOUN
ejpam-3427	434	23	µn	µn	NOUN
ejpam-3427	434	24	and	and	CCONJ
ejpam-3427	434	25	defined	define	VERB
ejpam-3427	434	26	by	by	ADP
ejpam-3427	434	27	(	(	PUNCT
ejpam-3427	434	28	µ1	µ1	PROPN
ejpam-3427	434	29	×	×	NOUN
ejpam-3427	434	30	µ2	µ2	PROPN
ejpam-3427	434	31	×	×	NOUN
ejpam-3427	434	32	...	...	PUNCT
ejpam-3427	434	33	×	×	NOUN
ejpam-3427	434	34	µn)(x1	µn)(x1	NOUN
ejpam-3427	434	35	,	,	PUNCT
ejpam-3427	434	36	x2	x2	PROPN
ejpam-3427	434	37	,	,	PUNCT
ejpam-3427	434	38	...	...	PUNCT
ejpam-3427	434	39	,	,	PUNCT
ejpam-3427	434	40	xn	xn	PROPN
ejpam-3427	434	41	)	)	PUNCT
ejpam-3427	434	42	=	=	PUNCT
ejpam-3427	435	1	min{µ1(x1	min{µ1(x1	PROPN
ejpam-3427	435	2	)	)	PUNCT
ejpam-3427	435	3	,	,	PUNCT
ejpam-3427	435	4	µ2	µ2	PROPN
ejpam-3427	435	5	(	(	PUNCT
ejpam-3427	435	6	x2	x2	PROPN
ejpam-3427	435	7	)	)	PUNCT
ejpam-3427	435	8	,	,	PUNCT
ejpam-3427	435	9	...	...	PUNCT
ejpam-3427	435	10	,	,	PUNCT
ejpam-3427	435	11	µn(xn	µn(xn	NUM
ejpam-3427	435	12	)	)	PUNCT
ejpam-3427	435	13	}	}	PUNCT
ejpam-3427	435	14	.	.	PUNCT
ejpam-3427	436	1	a	a	DET
ejpam-3427	436	2	fuzzy	fuzzy	ADJ
ejpam-3427	436	3	subset	subset	NOUN
ejpam-3427	436	4	µ1	µ1	PROPN
ejpam-3427	436	5	×	×	NOUN
ejpam-3427	436	6	µ2	µ2	PROPN
ejpam-3427	436	7	×	×	NOUN
ejpam-3427	436	8	...	...	PUNCT
ejpam-3427	436	9	×	×	NOUN
ejpam-3427	436	10	µn	µn	NOUN
ejpam-3427	436	11	of	of	ADP
ejpam-3427	436	12	an	an	DET
ejpam-3427	436	13	la	la	ADJ
ejpam-3427	436	14	-	-	PUNCT
ejpam-3427	436	15	ring	ring	NOUN
ejpam-3427	436	16	r1	r1	NOUN
ejpam-3427	436	17	×	×	NOUN
ejpam-3427	436	18	r2	r2	PROPN
ejpam-3427	436	19	×	×	NOUN
ejpam-3427	436	20	...	...	PUNCT
ejpam-3427	437	1	×	×	PROPN
ejpam-3427	437	2	rn	rn	PROPN
ejpam-3427	437	3	is	be	AUX
ejpam-3427	437	4	to	to	PART
ejpam-3427	437	5	be	be	AUX
ejpam-3427	437	6	a	a	DET
ejpam-3427	437	7	fuzzy	fuzzy	ADJ
ejpam-3427	437	8	la	la	NOUN
ejpam-3427	437	9	-	-	PUNCT
ejpam-3427	437	10	subring	subring	NOUN
ejpam-3427	437	11	of	of	ADP
ejpam-3427	437	12	r1	r1	PROPN
ejpam-3427	437	13	×r2	×r2	PROPN
ejpam-3427	437	14	×	×	NOUN
ejpam-3427	437	15	...	...	PUNCT
ejpam-3427	437	16	×rn	×rn	NOUN
ejpam-3427	437	17	if	if	SCONJ
ejpam-3427	437	18	1	1	NUM
ejpam-3427	437	19	.	.	PUNCT
ejpam-3427	438	1	(	(	PUNCT
ejpam-3427	438	2	µ1	µ1	NOUN
ejpam-3427	438	3	×	×	NOUN
ejpam-3427	438	4	µ2	µ2	PROPN
ejpam-3427	438	5	×	×	PROPN
ejpam-3427	438	6	...	...	PUNCT
ejpam-3427	438	7	×	×	PROPN
ejpam-3427	438	8	µn)(x−	µn)(x−	ADJ
ejpam-3427	438	9	y	y	NOUN
ejpam-3427	438	10	)	)	PUNCT
ejpam-3427	438	11	≥	≥	NOUN
ejpam-3427	438	12	min{(µ1	min{(µ1	VERB
ejpam-3427	438	13	×	×	PROPN
ejpam-3427	438	14	µ2	µ2	PROPN
ejpam-3427	438	15	×	×	NOUN
ejpam-3427	438	16	...	...	PUNCT
ejpam-3427	438	17	×	×	PROPN
ejpam-3427	438	18	µn)(x	µn)(x	PROPN
ejpam-3427	438	19	)	)	PUNCT
ejpam-3427	438	20	,	,	PUNCT
ejpam-3427	438	21	(	(	PUNCT
ejpam-3427	438	22	µ1	µ1	NOUN
ejpam-3427	438	23	×	×	NOUN
ejpam-3427	438	24	µ2	µ2	PROPN
ejpam-3427	438	25	×	×	NOUN
ejpam-3427	438	26	...	...	PUNCT
ejpam-3427	438	27	×	×	NOUN
ejpam-3427	438	28	µn)(y	µn)(y	PROPN
ejpam-3427	438	29	)	)	PUNCT
ejpam-3427	438	30	}	}	PUNCT
ejpam-3427	438	31	,	,	PUNCT
ejpam-3427	438	32	2	2	X
ejpam-3427	438	33	.	.	PUNCT
ejpam-3427	439	1	(	(	PUNCT
ejpam-3427	439	2	µ1	µ1	NOUN
ejpam-3427	439	3	×	×	NOUN
ejpam-3427	439	4	µ2	µ2	PROPN
ejpam-3427	439	5	×	×	NOUN
ejpam-3427	439	6	...	...	PUNCT
ejpam-3427	439	7	×	×	NOUN
ejpam-3427	439	8	µn)(xy	µn)(xy	PUNCT
ejpam-3427	439	9	)	)	PUNCT
ejpam-3427	439	10	≥	≥	NOUN
ejpam-3427	439	11	min{(µ1	min{(µ1	VERB
ejpam-3427	439	12	×	×	PROPN
ejpam-3427	439	13	µ2	µ2	PROPN
ejpam-3427	439	14	×	×	NOUN
ejpam-3427	439	15	...	...	PUNCT
ejpam-3427	439	16	×	×	PROPN
ejpam-3427	439	17	µn)(x	µn)(x	PROPN
ejpam-3427	439	18	)	)	PUNCT
ejpam-3427	439	19	,	,	PUNCT
ejpam-3427	439	20	(	(	PUNCT
ejpam-3427	439	21	µ1	µ1	NOUN
ejpam-3427	439	22	×	×	NOUN
ejpam-3427	439	23	µ2	µ2	PROPN
ejpam-3427	439	24	×	×	NOUN
ejpam-3427	439	25	...	...	PUNCT
ejpam-3427	439	26	×	×	NOUN
ejpam-3427	439	27	µn)(y	µn)(y	PROPN
ejpam-3427	439	28	)	)	PUNCT
ejpam-3427	439	29	}	}	PUNCT
ejpam-3427	439	30	for	for	ADP
ejpam-3427	439	31	all	all	PRON
ejpam-3427	439	32	x	x	X
ejpam-3427	439	33	=	=	SYM
ejpam-3427	439	34	(	(	PUNCT
ejpam-3427	439	35	x1	x1	PROPN
ejpam-3427	439	36	,	,	PUNCT
ejpam-3427	439	37	x2	x2	PROPN
ejpam-3427	439	38	,	,	PUNCT
ejpam-3427	439	39	...	...	PUNCT
ejpam-3427	439	40	,	,	PUNCT
ejpam-3427	439	41	xn	xn	PROPN
ejpam-3427	439	42	)	)	PUNCT
ejpam-3427	439	43	,	,	PUNCT
ejpam-3427	439	44	y	y	PROPN
ejpam-3427	439	45	=	=	SYM
ejpam-3427	439	46	(	(	PUNCT
ejpam-3427	439	47	y1	y1	PROPN
ejpam-3427	439	48	,	,	PUNCT
ejpam-3427	439	49	y2	y2	PROPN
ejpam-3427	439	50	,	,	PUNCT
ejpam-3427	439	51	...	...	PUNCT
ejpam-3427	439	52	,	,	PUNCT
ejpam-3427	439	53	yn	yn	X
ejpam-3427	439	54	)	)	PUNCT
ejpam-3427	439	55	∈	∈	PROPN
ejpam-3427	439	56	r1	r1	PROPN
ejpam-3427	439	57	×r2	×r2	PROPN
ejpam-3427	439	58	×	×	NOUN
ejpam-3427	439	59	...	...	PUNCT
ejpam-3427	439	60	×rn	×rn	NOUN
ejpam-3427	439	61	.	.	PUNCT
ejpam-3427	440	1	a	a	DET
ejpam-3427	440	2	fuzzy	fuzzy	ADJ
ejpam-3427	440	3	subset	subset	VERB
ejpam-3427	440	4	µ1×µ2×	µ1×µ2×	X
ejpam-3427	440	5	...	...	PUNCT
ejpam-3427	440	6	×µn	×µn	NOUN
ejpam-3427	440	7	of	of	ADP
ejpam-3427	440	8	an	an	DET
ejpam-3427	440	9	la	la	ADJ
ejpam-3427	440	10	-	-	PUNCT
ejpam-3427	440	11	ring	ring	NOUN
ejpam-3427	440	12	r1×r2×	r1×r2×	NOUN
ejpam-3427	440	13	...	...	PUNCT
ejpam-3427	440	14	×rn	×rn	PROPN
ejpam-3427	440	15	is	be	AUX
ejpam-3427	440	16	to	to	PART
ejpam-3427	440	17	be	be	AUX
ejpam-3427	440	18	an	an	DET
ejpam-3427	440	19	anti	anti	ADJ
ejpam-3427	440	20	fuzzy	fuzzy	ADJ
ejpam-3427	440	21	la	la	NOUN
ejpam-3427	440	22	-	-	PUNCT
ejpam-3427	440	23	subring	subring	NOUN
ejpam-3427	440	24	of	of	ADP
ejpam-3427	440	25	r1	r1	PROPN
ejpam-3427	440	26	×r2	×r2	PROPN
ejpam-3427	440	27	×	×	NOUN
ejpam-3427	440	28	...	...	PUNCT
ejpam-3427	440	29	×rn	×rn	NOUN
ejpam-3427	440	30	if	if	SCONJ
ejpam-3427	440	31	1	1	NUM
ejpam-3427	440	32	.	.	PUNCT
ejpam-3427	441	1	(	(	PUNCT
ejpam-3427	441	2	µ1	µ1	NOUN
ejpam-3427	441	3	×	×	NOUN
ejpam-3427	441	4	µ2	µ2	PROPN
ejpam-3427	441	5	×	×	PROPN
ejpam-3427	441	6	...	...	PUNCT
ejpam-3427	441	7	×	×	PROPN
ejpam-3427	441	8	µn)(x−	µn)(x−	ADJ
ejpam-3427	441	9	y	y	NOUN
ejpam-3427	441	10	)	)	PUNCT
ejpam-3427	441	11	≤	≤	NUM
ejpam-3427	441	12	max{(µ1	max{(µ1	NOUN
ejpam-3427	441	13	×	×	NOUN
ejpam-3427	441	14	µ2	µ2	PROPN
ejpam-3427	441	15	×	×	PROPN
ejpam-3427	441	16	...	...	PUNCT
ejpam-3427	441	17	×	×	PROPN
ejpam-3427	441	18	µn)(x	µn)(x	PROPN
ejpam-3427	441	19	)	)	PUNCT
ejpam-3427	441	20	,	,	PUNCT
ejpam-3427	441	21	(	(	PUNCT
ejpam-3427	441	22	µ1	µ1	NOUN
ejpam-3427	441	23	×	×	NOUN
ejpam-3427	441	24	µ2	µ2	PROPN
ejpam-3427	441	25	×	×	NOUN
ejpam-3427	441	26	...	...	PUNCT
ejpam-3427	441	27	×	×	NOUN
ejpam-3427	441	28	µn)(y	µn)(y	PROPN
ejpam-3427	441	29	)	)	PUNCT
ejpam-3427	441	30	}	}	PUNCT
ejpam-3427	441	31	,	,	PUNCT
ejpam-3427	441	32	2	2	X
ejpam-3427	441	33	.	.	X
ejpam-3427	441	34	µ1	µ1	NOUN
ejpam-3427	441	35	×	×	NOUN
ejpam-3427	441	36	µ2	µ2	PROPN
ejpam-3427	441	37	×	×	NOUN
ejpam-3427	441	38	...	...	PUNCT
ejpam-3427	441	39	×	×	PROPN
ejpam-3427	441	40	µn(xy	µn(xy	PROPN
ejpam-3427	441	41	)	)	PUNCT
ejpam-3427	441	42	≤	≤	NUM
ejpam-3427	441	43	max{(µ1	max{(µ1	NOUN
ejpam-3427	441	44	×	×	NOUN
ejpam-3427	441	45	µ2	µ2	PROPN
ejpam-3427	441	46	×	×	PROPN
ejpam-3427	441	47	...	...	PUNCT
ejpam-3427	441	48	×	×	PROPN
ejpam-3427	441	49	µn)(x	µn)(x	PROPN
ejpam-3427	441	50	)	)	PUNCT
ejpam-3427	441	51	,	,	PUNCT
ejpam-3427	441	52	(	(	PUNCT
ejpam-3427	441	53	µ1	µ1	NOUN
ejpam-3427	441	54	×	×	NOUN
ejpam-3427	441	55	µ2	µ2	PROPN
ejpam-3427	441	56	×	×	NOUN
ejpam-3427	441	57	...	...	PUNCT
ejpam-3427	441	58	×	×	NOUN
ejpam-3427	441	59	µn)(y	µn)(y	PROPN
ejpam-3427	441	60	)	)	PUNCT
ejpam-3427	441	61	}	}	PUNCT
ejpam-3427	441	62	for	for	ADP
ejpam-3427	441	63	all	all	PRON
ejpam-3427	441	64	x	x	X
ejpam-3427	441	65	=	=	SYM
ejpam-3427	441	66	(	(	PUNCT
ejpam-3427	441	67	x1	x1	PROPN
ejpam-3427	441	68	,	,	PUNCT
ejpam-3427	441	69	x2	x2	PROPN
ejpam-3427	441	70	,	,	PUNCT
ejpam-3427	441	71	...	...	PUNCT
ejpam-3427	441	72	,	,	PUNCT
ejpam-3427	441	73	xn	xn	PROPN
ejpam-3427	441	74	)	)	PUNCT
ejpam-3427	441	75	,	,	PUNCT
ejpam-3427	441	76	y	y	PROPN
ejpam-3427	441	77	=	=	SYM
ejpam-3427	441	78	(	(	PUNCT
ejpam-3427	441	79	y1	y1	PROPN
ejpam-3427	441	80	,	,	PUNCT
ejpam-3427	441	81	y2	y2	PROPN
ejpam-3427	441	82	,	,	PUNCT
ejpam-3427	441	83	...	...	PUNCT
ejpam-3427	441	84	,	,	PUNCT
ejpam-3427	441	85	yn	yn	X
ejpam-3427	441	86	)	)	PUNCT
ejpam-3427	441	87	∈	∈	PROPN
ejpam-3427	441	88	r1	r1	PROPN
ejpam-3427	441	89	×r2	×r2	PROPN
ejpam-3427	441	90	×	×	NOUN
ejpam-3427	441	91	...	...	PUNCT
ejpam-3427	441	92	×rn	×rn	NOUN
ejpam-3427	441	93	.	.	PUNCT
ejpam-3427	442	1	a	a	DET
ejpam-3427	442	2	fuzzy	fuzzy	ADJ
ejpam-3427	442	3	la	la	NOUN
ejpam-3427	442	4	-	-	PUNCT
ejpam-3427	442	5	subring	subring	NOUN
ejpam-3427	442	6	of	of	ADP
ejpam-3427	442	7	an	an	DET
ejpam-3427	442	8	la	la	ADJ
ejpam-3427	442	9	-	-	PUNCT
ejpam-3427	442	10	ring	ring	NOUN
ejpam-3427	442	11	r1	r1	NOUN
ejpam-3427	442	12	×	×	NOUN
ejpam-3427	442	13	r2	r2	PROPN
ejpam-3427	442	14	×	×	NOUN
ejpam-3427	442	15	...	...	PUNCT
ejpam-3427	442	16	×	×	PROPN
ejpam-3427	442	17	rn	rn	PROPN
ejpam-3427	442	18	is	be	AUX
ejpam-3427	442	19	said	say	VERB
ejpam-3427	442	20	to	to	PART
ejpam-3427	442	21	be	be	AUX
ejpam-3427	442	22	a	a	DET
ejpam-3427	442	23	fuzzy	fuzzy	ADJ
ejpam-3427	442	24	normal	normal	ADJ
ejpam-3427	442	25	la	la	NOUN
ejpam-3427	442	26	-	-	PUNCT
ejpam-3427	442	27	subring	subring	NOUN
ejpam-3427	442	28	of	of	ADP
ejpam-3427	442	29	r1	r1	PROPN
ejpam-3427	442	30	×	×	PROPN
ejpam-3427	442	31	r2	r2	PROPN
ejpam-3427	442	32	×	×	NOUN
ejpam-3427	442	33	...	...	PUNCT
ejpam-3427	443	1	×	×	PROPN
ejpam-3427	443	2	rn	rn	NOUN
ejpam-3427	443	3	if	if	SCONJ
ejpam-3427	443	4	(	(	PUNCT
ejpam-3427	443	5	µ1	µ1	NOUN
ejpam-3427	443	6	×	×	NOUN
ejpam-3427	443	7	µ2	µ2	PROPN
ejpam-3427	443	8	×	×	NOUN
ejpam-3427	443	9	...	...	PUNCT
ejpam-3427	443	10	×	×	NOUN
ejpam-3427	443	11	µn)(xy	µn)(xy	PUNCT
ejpam-3427	443	12	)	)	PUNCT
ejpam-3427	443	13	=	=	SYM
ejpam-3427	443	14	(	(	PUNCT
ejpam-3427	443	15	µ1	µ1	NOUN
ejpam-3427	443	16	×	×	NOUN
ejpam-3427	443	17	µ2	µ2	PROPN
ejpam-3427	443	18	×	×	NOUN
ejpam-3427	443	19	...	...	PUNCT
ejpam-3427	443	20	×	×	PROPN
ejpam-3427	443	21	µn)(yx	µn)(yx	PROPN
ejpam-3427	443	22	)	)	PUNCT
ejpam-3427	443	23	for	for	ADP
ejpam-3427	443	24	all	all	PRON
ejpam-3427	443	25	x	x	X
ejpam-3427	443	26	=	=	SYM
ejpam-3427	443	27	(	(	PUNCT
ejpam-3427	443	28	x1	x1	PROPN
ejpam-3427	443	29	,	,	PUNCT
ejpam-3427	443	30	x2	x2	PROPN
ejpam-3427	443	31	,	,	PUNCT
ejpam-3427	443	32	...	...	PUNCT
ejpam-3427	443	33	,	,	PUNCT
ejpam-3427	443	34	xn	xn	PROPN
ejpam-3427	443	35	)	)	PUNCT
ejpam-3427	443	36	,	,	PUNCT
ejpam-3427	443	37	y	y	PROPN
ejpam-3427	443	38	=	=	SYM
ejpam-3427	443	39	(	(	PUNCT
ejpam-3427	443	40	y1	y1	PROPN
ejpam-3427	443	41	,	,	PUNCT
ejpam-3427	443	42	y2	y2	PROPN
ejpam-3427	443	43	,	,	PUNCT
ejpam-3427	443	44	...	...	PUNCT
ejpam-3427	443	45	,	,	PUNCT
ejpam-3427	443	46	yn	yn	X
ejpam-3427	443	47	)	)	PUNCT
ejpam-3427	443	48	∈	∈	PROPN
ejpam-3427	443	49	r1	r1	PROPN
ejpam-3427	443	50	×r2	×r2	PROPN
ejpam-3427	443	51	×	×	NOUN
ejpam-3427	443	52	...	...	PUNCT
ejpam-3427	443	53	×rn	×rn	NOUN
ejpam-3427	443	54	.	.	PUNCT
ejpam-3427	444	1	similarly	similarly	ADV
ejpam-3427	444	2	for	for	ADP
ejpam-3427	444	3	anti	anti	X
ejpam-3427	444	4	fuzzy	fuzzy	ADJ
ejpam-3427	444	5	normal	normal	ADJ
ejpam-3427	444	6	la	la	NOUN
ejpam-3427	444	7	-	-	PUNCT
ejpam-3427	444	8	subring	subring	NOUN
ejpam-3427	444	9	.	.	PUNCT
ejpam-3427	445	1	let	let	VERB
ejpam-3427	445	2	a1	a1	NOUN
ejpam-3427	445	3	,	,	PUNCT
ejpam-3427	445	4	a2	a2	PROPN
ejpam-3427	445	5	,	,	PUNCT
ejpam-3427	445	6	...	...	PUNCT
ejpam-3427	445	7	,	,	PUNCT
ejpam-3427	445	8	an	an	DET
ejpam-3427	445	9	be	be	AUX
ejpam-3427	445	10	intuitionistic	intuitionistic	ADJ
ejpam-3427	445	11	fuzzy	fuzzy	ADJ
ejpam-3427	445	12	sets	set	NOUN
ejpam-3427	445	13	of	of	ADP
ejpam-3427	445	14	la	la	NOUN
ejpam-3427	445	15	-	-	PUNCT
ejpam-3427	445	16	rings	ring	NOUN
ejpam-3427	445	17	r1	r1	NOUN
ejpam-3427	445	18	,	,	PUNCT
ejpam-3427	445	19	r2	r2	PROPN
ejpam-3427	445	20	,	,	PUNCT
ejpam-3427	445	21	...	...	PUNCT
ejpam-3427	445	22	,	,	PUNCT
ejpam-3427	445	23	rn	rn	PROPN
ejpam-3427	445	24	,	,	PUNCT
ejpam-3427	445	25	respectively	respectively	ADV
ejpam-3427	445	26	.	.	PUNCT
ejpam-3427	446	1	the	the	DET
ejpam-3427	446	2	direct	direct	ADJ
ejpam-3427	446	3	product	product	NOUN
ejpam-3427	446	4	of	of	ADP
ejpam-3427	446	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	446	6	fuzzy	fuzzy	ADJ
ejpam-3427	446	7	sets	set	NOUN
ejpam-3427	446	8	a1	a1	NOUN
ejpam-3427	446	9	,	,	PUNCT
ejpam-3427	446	10	a2	a2	PROPN
ejpam-3427	446	11	...	...	PUNCT
ejpam-3427	446	12	,	,	PUNCT
ejpam-3427	446	13	an	an	PRON
ejpam-3427	446	14	is	be	AUX
ejpam-3427	446	15	denoted	denote	VERB
ejpam-3427	446	16	by	by	ADP
ejpam-3427	446	17	a1×a2×	a1×a2×	NOUN
ejpam-3427	446	18	...	...	PUNCT
ejpam-3427	446	19	×an	×an	NOUN
ejpam-3427	446	20	and	and	CCONJ
ejpam-3427	446	21	defined	define	VERB
ejpam-3427	446	22	by	by	ADP
ejpam-3427	446	23	a1	a1	PROPN
ejpam-3427	446	24	×	×	PROPN
ejpam-3427	446	25	a2	a2	PROPN
ejpam-3427	446	26	×	×	NOUN
ejpam-3427	446	27	...	...	PUNCT
ejpam-3427	446	28	×	×	NOUN
ejpam-3427	447	1	an	an	PRON
ejpam-3427	447	2	=	=	X
ejpam-3427	447	3	{	{	PUNCT
ejpam-3427	447	4	(	(	PUNCT
ejpam-3427	447	5	x	x	NOUN
ejpam-3427	447	6	,	,	PUNCT
ejpam-3427	447	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	447	8	...	...	PUNCT
ejpam-3427	447	9	×an(x	×an(x	NUM
ejpam-3427	447	10	)	)	PUNCT
ejpam-3427	447	11	,	,	PUNCT
ejpam-3427	447	12	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	447	13	...	...	PUNCT
ejpam-3427	447	14	×an(x	×an(x	NOUN
ejpam-3427	447	15	)	)	PUNCT
ejpam-3427	447	16	)	)	PUNCT
ejpam-3427	448	1	|	|	ADV
ejpam-3427	448	2	for	for	ADP
ejpam-3427	448	3	all	all	PRON
ejpam-3427	448	4	x	x	X
ejpam-3427	448	5	=	=	SYM
ejpam-3427	448	6	(	(	PUNCT
ejpam-3427	448	7	x1	x1	PROPN
ejpam-3427	448	8	,	,	PUNCT
ejpam-3427	448	9	x2	x2	PROPN
ejpam-3427	448	10	,	,	PUNCT
ejpam-3427	448	11	...	...	PUNCT
ejpam-3427	448	12	,	,	PUNCT
ejpam-3427	448	13	xn	xn	X
ejpam-3427	448	14	)	)	PUNCT
ejpam-3427	448	15	∈	∈	PROPN
ejpam-3427	448	16	r1	r1	PROPN
ejpam-3427	448	17	×r2	×r2	PROPN
ejpam-3427	448	18	×	×	NOUN
ejpam-3427	448	19	...	...	PUNCT
ejpam-3427	448	20	×rn	×rn	NOUN
ejpam-3427	448	21	}	}	PUNCT
ejpam-3427	448	22	,	,	PUNCT
ejpam-3427	448	23	where	where	SCONJ
ejpam-3427	448	24	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	448	25	...	...	PUNCT
ejpam-3427	448	26	×an(x1	×an(x1	NOUN
ejpam-3427	448	27	,	,	PUNCT
ejpam-3427	448	28	x2	x2	PROPN
ejpam-3427	448	29	,	,	PUNCT
ejpam-3427	448	30	...	...	PUNCT
ejpam-3427	448	31	,	,	PUNCT
ejpam-3427	448	32	xn	xn	X
ejpam-3427	448	33	)	)	PUNCT
ejpam-3427	448	34	=	=	SYM
ejpam-3427	448	35	max{µa1(x1	max{µa1(x1	PROPN
ejpam-3427	448	36	)	)	PUNCT
ejpam-3427	448	37	,	,	PUNCT
ejpam-3427	448	38	µa2(x2	µa2(x2	NOUN
ejpam-3427	448	39	)	)	PUNCT
ejpam-3427	448	40	,	,	PUNCT
ejpam-3427	448	41	...	...	PUNCT
ejpam-3427	448	42	,	,	PUNCT
ejpam-3427	448	43	µan(xn	µan(xn	NOUN
ejpam-3427	448	44	)	)	PUNCT
ejpam-3427	448	45	}	}	PUNCT
ejpam-3427	448	46	and	and	CCONJ
ejpam-3427	448	47	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	448	48	...	...	PUNCT
ejpam-3427	448	49	×an(x1	×an(x1	NOUN
ejpam-3427	448	50	,	,	PUNCT
ejpam-3427	448	51	x2	x2	PROPN
ejpam-3427	448	52	,	,	PUNCT
ejpam-3427	448	53	...	...	PUNCT
ejpam-3427	448	54	,	,	PUNCT
ejpam-3427	448	55	xn	xn	X
ejpam-3427	448	56	)	)	PUNCT
ejpam-3427	448	57	=	=	SYM
ejpam-3427	448	58	min{γa1(x1	min{γa1(x1	PROPN
ejpam-3427	448	59	)	)	PUNCT
ejpam-3427	448	60	,	,	PUNCT
ejpam-3427	448	61	γa2(x2	γa2(x2	PROPN
ejpam-3427	448	62	)	)	PUNCT
ejpam-3427	448	63	,	,	PUNCT
ejpam-3427	448	64	...	...	PUNCT
ejpam-3427	448	65	,	,	PUNCT
ejpam-3427	448	66	γan(xn	γan(xn	X
ejpam-3427	448	67	)	)	PUNCT
ejpam-3427	448	68	}	}	PUNCT
ejpam-3427	448	69	.	.	PUNCT
ejpam-3427	449	1	k.	k.	PROPN
ejpam-3427	449	2	nasreen	nasreen	PROPN
ejpam-3427	449	3	/	/	SYM
ejpam-3427	449	4	eur	eur	PROPN
ejpam-3427	449	5	.	.	PUNCT
ejpam-3427	450	1	j.	j.	PROPN
ejpam-3427	450	2	pure	pure	PROPN
ejpam-3427	450	3	appl	appl	PROPN
ejpam-3427	450	4	.	.	PROPN
ejpam-3427	450	5	math	math	PROPN
ejpam-3427	450	6	,	,	PUNCT
ejpam-3427	450	7	12	12	NUM
ejpam-3427	450	8	(	(	PUNCT
ejpam-3427	450	9	2	2	NUM
ejpam-3427	450	10	)	)	PUNCT
ejpam-3427	450	11	(	(	PUNCT
ejpam-3427	450	12	2019	2019	NUM
ejpam-3427	450	13	)	)	PUNCT
ejpam-3427	450	14	,	,	PUNCT
ejpam-3427	450	15	622	622	NUM
ejpam-3427	450	16	-	-	SYM
ejpam-3427	450	17	648	648	NUM
ejpam-3427	450	18	637	637	NUM
ejpam-3427	450	19	an	an	DET
ejpam-3427	450	20	intuitionistic	intuitionistic	ADJ
ejpam-3427	450	21	fuzzy	fuzzy	ADJ
ejpam-3427	450	22	set	set	NOUN
ejpam-3427	450	23	(	(	PUNCT
ejpam-3427	450	24	ifs	ifs	PROPN
ejpam-3427	450	25	)	)	PUNCT
ejpam-3427	450	26	a1×a2×	a1×a2×	PROPN
ejpam-3427	450	27	...	...	PUNCT
ejpam-3427	450	28	×an	×an	PROPN
ejpam-3427	450	29	=	=	SYM
ejpam-3427	450	30	(	(	PUNCT
ejpam-3427	450	31	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	450	32	...	...	PUNCT
ejpam-3427	450	33	×an	×an	PROPN
ejpam-3427	450	34	,	,	PUNCT
ejpam-3427	450	35	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	450	36	...	...	PUNCT
ejpam-3427	450	37	×an	×an	PROPN
ejpam-3427	450	38	)	)	PUNCT
ejpam-3427	450	39	of	of	ADP
ejpam-3427	450	40	an	an	DET
ejpam-3427	450	41	la	la	ADJ
ejpam-3427	450	42	-	-	PUNCT
ejpam-3427	450	43	ring	ring	NOUN
ejpam-3427	450	44	r1×r2×	r1×r2×	NOUN
ejpam-3427	450	45	...	...	PUNCT
ejpam-3427	450	46	×rn	×rn	PROPN
ejpam-3427	450	47	is	be	AUX
ejpam-3427	450	48	to	to	PART
ejpam-3427	450	49	be	be	AUX
ejpam-3427	450	50	an	an	DET
ejpam-3427	450	51	intuitionistic	intuitionistic	ADJ
ejpam-3427	450	52	anti	anti	ADJ
ejpam-3427	450	53	fuzzy	fuzzy	ADJ
ejpam-3427	450	54	la	la	NOUN
ejpam-3427	450	55	-	-	PUNCT
ejpam-3427	450	56	subring	subre	VERB
ejpam-3427	450	57	(	(	PUNCT
ejpam-3427	450	58	iaflsr	iaflsr	NOUN
ejpam-3427	450	59	)	)	PUNCT
ejpam-3427	450	60	of	of	ADP
ejpam-3427	450	61	r1	r1	PROPN
ejpam-3427	450	62	×r2	×r2	PROPN
ejpam-3427	450	63	×	×	NOUN
ejpam-3427	450	64	...	...	PUNCT
ejpam-3427	450	65	×rn	×rn	NOUN
ejpam-3427	450	66	if	if	SCONJ
ejpam-3427	450	67	1	1	NUM
ejpam-3427	450	68	.	.	X
ejpam-3427	450	69	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	450	70	...	...	PUNCT
ejpam-3427	450	71	×an(x−	×an(x−	PROPN
ejpam-3427	450	72	y	y	NOUN
ejpam-3427	450	73	)	)	PUNCT
ejpam-3427	450	74	≤	≤	NOUN
ejpam-3427	450	75	max{µa1×a2×	max{µa1×a2×	NOUN
ejpam-3427	450	76	...	...	PUNCT
ejpam-3427	450	77	×an(x	×an(x	NUM
ejpam-3427	450	78	)	)	PUNCT
ejpam-3427	450	79	,	,	PUNCT
ejpam-3427	450	80	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	450	81	...	...	PUNCT
ejpam-3427	450	82	×an(y	×an(y	NUM
ejpam-3427	450	83	)	)	PUNCT
ejpam-3427	450	84	}	}	PUNCT
ejpam-3427	450	85	,	,	PUNCT
ejpam-3427	450	86	2	2	X
ejpam-3427	450	87	.	.	X
ejpam-3427	450	88	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	450	89	...	...	PUNCT
ejpam-3427	450	90	×an(xy	×an(xy	NOUN
ejpam-3427	450	91	)	)	PUNCT
ejpam-3427	450	92	≤	≤	NOUN
ejpam-3427	450	93	max{µa1×a2×	max{µa1×a2×	NOUN
ejpam-3427	450	94	...	...	PUNCT
ejpam-3427	450	95	×an(x	×an(x	NUM
ejpam-3427	450	96	)	)	PUNCT
ejpam-3427	450	97	,	,	PUNCT
ejpam-3427	450	98	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	450	99	...	...	PUNCT
ejpam-3427	450	100	×an(y	×an(y	NUM
ejpam-3427	450	101	)	)	PUNCT
ejpam-3427	450	102	}	}	PUNCT
ejpam-3427	450	103	,	,	PUNCT
ejpam-3427	451	1	3	3	X
ejpam-3427	451	2	.	.	PUNCT
ejpam-3427	451	3	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	451	4	...	...	PUNCT
ejpam-3427	452	1	×an(x−	×an(x−	PROPN
ejpam-3427	452	2	y	y	PROPN
ejpam-3427	452	3	)	)	PUNCT
ejpam-3427	452	4	≥	≥	NOUN
ejpam-3427	452	5	min{γa1×a2×	min{γa1×a2×	NOUN
ejpam-3427	452	6	...	...	PUNCT
ejpam-3427	452	7	×an(x	×an(x	NUM
ejpam-3427	452	8	)	)	PUNCT
ejpam-3427	452	9	,	,	PUNCT
ejpam-3427	452	10	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	452	11	...	...	PUNCT
ejpam-3427	452	12	×an(y	×an(y	X
ejpam-3427	452	13	)	)	PUNCT
ejpam-3427	452	14	}	}	PUNCT
ejpam-3427	452	15	,	,	PUNCT
ejpam-3427	452	16	4	4	X
ejpam-3427	452	17	.	.	PUNCT
ejpam-3427	452	18	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	452	19	...	...	PUNCT
ejpam-3427	452	20	×an(xy	×an(xy	X
ejpam-3427	452	21	)	)	PUNCT
ejpam-3427	452	22	≥	≥	NOUN
ejpam-3427	452	23	min{γa1×a2×	min{γa1×a2×	PROPN
ejpam-3427	452	24	...	...	PUNCT
ejpam-3427	452	25	×an(x	×an(x	NUM
ejpam-3427	452	26	)	)	PUNCT
ejpam-3427	452	27	,	,	PUNCT
ejpam-3427	452	28	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	452	29	...	...	PUNCT
ejpam-3427	452	30	×an(y	×an(y	X
ejpam-3427	452	31	)	)	PUNCT
ejpam-3427	452	32	}	}	PUNCT
ejpam-3427	452	33	,	,	PUNCT
ejpam-3427	452	34	for	for	ADP
ejpam-3427	452	35	all	all	PRON
ejpam-3427	452	36	x	x	X
ejpam-3427	452	37	=	=	SYM
ejpam-3427	452	38	(	(	PUNCT
ejpam-3427	452	39	x1	x1	PROPN
ejpam-3427	452	40	,	,	PUNCT
ejpam-3427	452	41	x2	x2	PROPN
ejpam-3427	452	42	,	,	PUNCT
ejpam-3427	452	43	...	...	PUNCT
ejpam-3427	452	44	,	,	PUNCT
ejpam-3427	452	45	xn	xn	PROPN
ejpam-3427	452	46	)	)	PUNCT
ejpam-3427	452	47	,	,	PUNCT
ejpam-3427	452	48	y	y	PROPN
ejpam-3427	452	49	=	=	SYM
ejpam-3427	452	50	(	(	PUNCT
ejpam-3427	452	51	y1	y1	PROPN
ejpam-3427	452	52	,	,	PUNCT
ejpam-3427	452	53	y2	y2	PROPN
ejpam-3427	452	54	,	,	PUNCT
ejpam-3427	452	55	...	...	PUNCT
ejpam-3427	452	56	,	,	PUNCT
ejpam-3427	452	57	yn	yn	X
ejpam-3427	452	58	)	)	PUNCT
ejpam-3427	452	59	∈	∈	PROPN
ejpam-3427	452	60	r1	r1	PROPN
ejpam-3427	452	61	×r2	×r2	PROPN
ejpam-3427	452	62	×	×	NOUN
ejpam-3427	452	63	...	...	PUNCT
ejpam-3427	452	64	×rn	×rn	NOUN
ejpam-3427	452	65	.	.	PUNCT
ejpam-3427	453	1	an	an	DET
ejpam-3427	453	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	453	3	anti	anti	ADJ
ejpam-3427	453	4	fuzzy	fuzzy	ADJ
ejpam-3427	453	5	la	la	NOUN
ejpam-3427	453	6	-	-	PUNCT
ejpam-3427	453	7	subringa1×a2×	subringa1×a2×	NOUN
ejpam-3427	453	8	...	...	PUNCT
ejpam-3427	453	9	×an	×an	NOUN
ejpam-3427	453	10	=	=	SYM
ejpam-3427	453	11	(	(	PUNCT
ejpam-3427	453	12	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	453	13	...	...	PUNCT
ejpam-3427	453	14	×an	×an	PROPN
ejpam-3427	453	15	,	,	PUNCT
ejpam-3427	453	16	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	453	17	...	...	PUNCT
ejpam-3427	453	18	×an	×an	PROPN
ejpam-3427	453	19	)	)	PUNCT
ejpam-3427	453	20	of	of	ADP
ejpam-3427	453	21	an	an	DET
ejpam-3427	453	22	la	la	ADJ
ejpam-3427	453	23	-	-	PUNCT
ejpam-3427	453	24	ring	ring	NOUN
ejpam-3427	453	25	r1×r2×	r1×r2×	NOUN
ejpam-3427	453	26	...	...	PUNCT
ejpam-3427	453	27	×rn	×rn	PROPN
ejpam-3427	453	28	is	be	AUX
ejpam-3427	453	29	said	say	VERB
ejpam-3427	453	30	to	to	PART
ejpam-3427	453	31	be	be	AUX
ejpam-3427	453	32	an	an	DET
ejpam-3427	453	33	intuitionistic	intuitionistic	ADJ
ejpam-3427	453	34	anti	anti	ADJ
ejpam-3427	453	35	fuzzy	fuzzy	ADJ
ejpam-3427	453	36	normal	normal	ADJ
ejpam-3427	453	37	la	la	NOUN
ejpam-3427	453	38	-	-	PUNCT
ejpam-3427	453	39	subring	subre	VERB
ejpam-3427	453	40	(	(	PUNCT
ejpam-3427	453	41	iafnlsr	iafnlsr	NOUN
ejpam-3427	453	42	)	)	PUNCT
ejpam-3427	453	43	of	of	ADP
ejpam-3427	453	44	r1	r1	PROPN
ejpam-3427	453	45	×r2	×r2	PROPN
ejpam-3427	453	46	×	×	NOUN
ejpam-3427	453	47	...	...	PUNCT
ejpam-3427	453	48	×rn	×rn	NOUN
ejpam-3427	453	49	if	if	SCONJ
ejpam-3427	453	50	1	1	NUM
ejpam-3427	453	51	.	.	X
ejpam-3427	453	52	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	453	53	...	...	PUNCT
ejpam-3427	453	54	×an(xy	×an(xy	NOUN
ejpam-3427	453	55	)	)	PUNCT
ejpam-3427	453	56	=	=	SYM
ejpam-3427	453	57	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	453	58	...	...	PUNCT
ejpam-3427	453	59	×an(yx	×an(yx	NOUN
ejpam-3427	453	60	)	)	PUNCT
ejpam-3427	454	1	2	2	NUM
ejpam-3427	454	2	.	.	PUNCT
ejpam-3427	454	3	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	454	4	...	...	PUNCT
ejpam-3427	454	5	×an(xy	×an(xy	X
ejpam-3427	454	6	)	)	PUNCT
ejpam-3427	454	7	=	=	VERB
ejpam-3427	455	1	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	455	2	...	...	PUNCT
ejpam-3427	455	3	×an(yx	×an(yx	NOUN
ejpam-3427	455	4	)	)	PUNCT
ejpam-3427	456	1	for	for	ADP
ejpam-3427	456	2	all	all	PRON
ejpam-3427	456	3	x	x	SYM
ejpam-3427	456	4	=	=	SYM
ejpam-3427	456	5	(	(	PUNCT
ejpam-3427	456	6	x1	x1	PROPN
ejpam-3427	456	7	,	,	PUNCT
ejpam-3427	456	8	x2	x2	PROPN
ejpam-3427	456	9	,	,	PUNCT
ejpam-3427	456	10	...	...	PUNCT
ejpam-3427	456	11	,	,	PUNCT
ejpam-3427	456	12	xn	xn	PROPN
ejpam-3427	456	13	)	)	PUNCT
ejpam-3427	456	14	,	,	PUNCT
ejpam-3427	456	15	y	y	PROPN
ejpam-3427	456	16	=	=	SYM
ejpam-3427	456	17	(	(	PUNCT
ejpam-3427	456	18	y1	y1	PROPN
ejpam-3427	456	19	,	,	PUNCT
ejpam-3427	456	20	y2	y2	PROPN
ejpam-3427	456	21	,	,	PUNCT
ejpam-3427	456	22	...	...	PUNCT
ejpam-3427	456	23	,	,	PUNCT
ejpam-3427	456	24	yn	yn	X
ejpam-3427	456	25	)	)	PUNCT
ejpam-3427	456	26	∈	∈	PROPN
ejpam-3427	456	27	r1	r1	PROPN
ejpam-3427	456	28	×r2	×r2	PROPN
ejpam-3427	456	29	×	×	NOUN
ejpam-3427	456	30	...	...	PUNCT
ejpam-3427	456	31	×rn	×rn	NOUN
ejpam-3427	456	32	.	.	PUNCT
ejpam-3427	457	1	let	let	VERB
ejpam-3427	457	2	a1	a1	VERB
ejpam-3427	457	3	×a2	×a2	PROPN
ejpam-3427	457	4	×	×	NOUN
ejpam-3427	457	5	...	...	PUNCT
ejpam-3427	457	6	×an	×an	PROPN
ejpam-3427	457	7	be	be	AUX
ejpam-3427	457	8	a	a	DET
ejpam-3427	457	9	non	non	ADJ
ejpam-3427	457	10	-	-	ADJ
ejpam-3427	457	11	empty	empty	ADJ
ejpam-3427	457	12	subset	subset	NOUN
ejpam-3427	457	13	of	of	ADP
ejpam-3427	457	14	an	an	DET
ejpam-3427	457	15	la	la	ADJ
ejpam-3427	457	16	-	-	PUNCT
ejpam-3427	457	17	ring	ring	NOUN
ejpam-3427	457	18	r	r	NOUN
ejpam-3427	457	19	=	=	SYM
ejpam-3427	457	20	r1	r1	PROPN
ejpam-3427	457	21	×r2	×r2	PROPN
ejpam-3427	457	22	×	×	NOUN
ejpam-3427	457	23	...	...	PUNCT
ejpam-3427	457	24	×rn	×rn	NOUN
ejpam-3427	457	25	.	.	PUNCT
ejpam-3427	458	1	the	the	DET
ejpam-3427	458	2	intuitionistic	intuitionistic	ADJ
ejpam-3427	458	3	anti	anti	ADJ
ejpam-3427	458	4	characteristic	characteristic	ADJ
ejpam-3427	458	5	function	function	NOUN
ejpam-3427	458	6	of	of	ADP
ejpam-3427	458	7	a	a	DET
ejpam-3427	458	8	=	=	NOUN
ejpam-3427	458	9	a1	a1	NOUN
ejpam-3427	458	10	×	×	PROPN
ejpam-3427	458	11	a2	a2	PROPN
ejpam-3427	458	12	×	×	NOUN
ejpam-3427	458	13	...	...	PUNCT
ejpam-3427	458	14	×	×	NOUN
ejpam-3427	458	15	an	an	PRON
ejpam-3427	458	16	is	be	AUX
ejpam-3427	458	17	denoted	denote	VERB
ejpam-3427	458	18	by	by	ADP
ejpam-3427	458	19	χa1×a2×	χa1×a2×	PROPN
ejpam-3427	458	20	...	...	PUNCT
ejpam-3427	458	21	×an	×an	PROPN
ejpam-3427	458	22	=	=	PUNCT
ejpam-3427	458	23	〈	〈	PROPN
ejpam-3427	458	24	µχa1×a2×	µχa1×a2×	NUM
ejpam-3427	458	25	...	...	PUNCT
ejpam-3427	458	26	×an	×an	NOUN
ejpam-3427	458	27	,	,	PUNCT
ejpam-3427	458	28	γχa1×a2×	γχa1×a2×	PROPN
ejpam-3427	458	29	..	..	PUNCT
ejpam-3427	458	30	×an	×an	PROPN
ejpam-3427	458	31	〉	〉	PROPN
ejpam-3427	458	32	and	and	CCONJ
ejpam-3427	458	33	defined	define	VERB
ejpam-3427	458	34	by	by	ADP
ejpam-3427	458	35	µχa	µχa	NOUN
ejpam-3427	458	36	(	(	PUNCT
ejpam-3427	458	37	x	x	NOUN
ejpam-3427	458	38	)	)	PUNCT
ejpam-3427	458	39	=	=	PRON
ejpam-3427	458	40	{	{	PUNCT
ejpam-3427	458	41	0	0	NUM
ejpam-3427	458	42	if	if	SCONJ
ejpam-3427	458	43	x	x	SYM
ejpam-3427	458	44	∈	∈	PROPN
ejpam-3427	458	45	a	a	DET
ejpam-3427	458	46	1	1	NUM
ejpam-3427	458	47	if	if	SCONJ
ejpam-3427	458	48	x	x	PROPN
ejpam-3427	458	49	/∈	/∈	NOUN
ejpam-3427	458	50	a	a	PRON
ejpam-3427	458	51	and	and	CCONJ
ejpam-3427	458	52	γχa	γχa	ADJ
ejpam-3427	458	53	(	(	PUNCT
ejpam-3427	458	54	x	x	X
ejpam-3427	458	55	)	)	PUNCT
ejpam-3427	458	56	=	=	SYM
ejpam-3427	458	57	{	{	PUNCT
ejpam-3427	458	58	1	1	NUM
ejpam-3427	458	59	if	if	SCONJ
ejpam-3427	458	60	x	x	PROPN
ejpam-3427	458	61	∈	∈	PROPN
ejpam-3427	458	62	a	a	DET
ejpam-3427	458	63	0	0	NOUN
ejpam-3427	459	1	if	if	SCONJ
ejpam-3427	459	2	x	x	PROPN
ejpam-3427	459	3	/∈	/∈	PUNCT
ejpam-3427	460	1	a	a	PRON
ejpam-3427	460	2	[	[	X
ejpam-3427	460	3	17	17	NUM
ejpam-3427	460	4	]	]	PUNCT
ejpam-3427	460	5	if	if	SCONJ
ejpam-3427	460	6	r1	r1	PROPN
ejpam-3427	460	7	,	,	PUNCT
ejpam-3427	460	8	r2	r2	PROPN
ejpam-3427	460	9	are	be	AUX
ejpam-3427	460	10	la	la	NOUN
ejpam-3427	460	11	-	-	PUNCT
ejpam-3427	460	12	rings	ring	NOUN
ejpam-3427	460	13	,	,	PUNCT
ejpam-3427	460	14	then	then	ADV
ejpam-3427	460	15	direct	direct	ADJ
ejpam-3427	460	16	product	product	NOUN
ejpam-3427	460	17	r1	r1	VERB
ejpam-3427	460	18	×	×	NOUN
ejpam-3427	460	19	r2	r2	NOUN
ejpam-3427	460	20	of	of	ADP
ejpam-3427	460	21	r1	r1	PROPN
ejpam-3427	460	22	and	and	CCONJ
ejpam-3427	460	23	r2	r2	PROPN
ejpam-3427	460	24	is	be	AUX
ejpam-3427	460	25	an	an	DET
ejpam-3427	460	26	la	la	NOUN
ejpam-3427	460	27	-	-	NOUN
ejpam-3427	460	28	ring	ring	NOUN
ejpam-3427	460	29	with	with	ADP
ejpam-3427	460	30	pointwise	pointwise	ADJ
ejpam-3427	460	31	addition	addition	NOUN
ejpam-3427	460	32	‘	'	PUNCT
ejpam-3427	460	33	+	+	NOUN
ejpam-3427	460	34	’	'	PUNCT
ejpam-3427	460	35	and	and	CCONJ
ejpam-3427	460	36	multiplication	multiplication	NOUN
ejpam-3427	460	37	‘	'	PUNCT
ejpam-3427	460	38	◦	◦	NOUN
ejpam-3427	460	39	’	'	PUNCT
ejpam-3427	460	40	defined	define	VERB
ejpam-3427	460	41	as	as	ADP
ejpam-3427	460	42	(	(	PUNCT
ejpam-3427	460	43	a	a	DET
ejpam-3427	460	44	,	,	PUNCT
ejpam-3427	460	45	b	b	NOUN
ejpam-3427	460	46	)	)	PUNCT
ejpam-3427	460	47	+	+	CCONJ
ejpam-3427	460	48	(	(	PUNCT
ejpam-3427	460	49	c	c	X
ejpam-3427	460	50	,	,	PUNCT
ejpam-3427	460	51	d	d	NOUN
ejpam-3427	460	52	)	)	PUNCT
ejpam-3427	460	53	=	=	SYM
ejpam-3427	460	54	(	(	PUNCT
ejpam-3427	460	55	a+	a+	PUNCT
ejpam-3427	460	56	c	c	NOUN
ejpam-3427	460	57	,	,	PUNCT
ejpam-3427	460	58	b+	b+	X
ejpam-3427	460	59	d	d	X
ejpam-3427	460	60	)	)	PUNCT
ejpam-3427	460	61	and	and	CCONJ
ejpam-3427	460	62	(	(	PUNCT
ejpam-3427	460	63	a	a	PRON
ejpam-3427	460	64	,	,	PUNCT
ejpam-3427	460	65	b	b	NOUN
ejpam-3427	460	66	)	)	PUNCT
ejpam-3427	460	67	◦	◦	NOUN
ejpam-3427	460	68	(	(	PUNCT
ejpam-3427	460	69	c	c	X
ejpam-3427	460	70	,	,	PUNCT
ejpam-3427	460	71	d	d	NOUN
ejpam-3427	460	72	)	)	PUNCT
ejpam-3427	460	73	=	=	SYM
ejpam-3427	460	74	(	(	PUNCT
ejpam-3427	460	75	ac	ac	PROPN
ejpam-3427	460	76	,	,	PUNCT
ejpam-3427	460	77	bd	bd	PROPN
ejpam-3427	460	78	)	)	PUNCT
ejpam-3427	460	79	,	,	PUNCT
ejpam-3427	460	80	respectively	respectively	ADV
ejpam-3427	460	81	for	for	ADP
ejpam-3427	460	82	every	every	DET
ejpam-3427	460	83	(	(	PUNCT
ejpam-3427	460	84	a	a	PRON
ejpam-3427	460	85	,	,	PUNCT
ejpam-3427	460	86	b	b	NOUN
ejpam-3427	460	87	)	)	PUNCT
ejpam-3427	460	88	,	,	PUNCT
ejpam-3427	460	89	(	(	PUNCT
ejpam-3427	460	90	c	c	X
ejpam-3427	460	91	,	,	PUNCT
ejpam-3427	460	92	d	d	NOUN
ejpam-3427	460	93	)	)	PUNCT
ejpam-3427	460	94	∈	∈	PROPN
ejpam-3427	460	95	r1	r1	NOUN
ejpam-3427	460	96	×	×	NOUN
ejpam-3427	460	97	r2	r2	NOUN
ejpam-3427	460	98	.	.	PUNCT
ejpam-3427	461	1	likewise	likewise	ADV
ejpam-3427	461	2	the	the	DET
ejpam-3427	461	3	direct	direct	ADJ
ejpam-3427	461	4	product	product	NOUN
ejpam-3427	461	5	r	r	NOUN
ejpam-3427	461	6	=	=	SYM
ejpam-3427	461	7	×i∈ωri	×i∈ωri	PROPN
ejpam-3427	461	8	of	of	ADP
ejpam-3427	461	9	a	a	DET
ejpam-3427	461	10	family	family	NOUN
ejpam-3427	461	11	of	of	ADP
ejpam-3427	461	12	la	la	PROPN
ejpam-3427	461	13	-	-	PUNCT
ejpam-3427	461	14	rings	ring	NOUN
ejpam-3427	461	15	{	{	PUNCT
ejpam-3427	461	16	ri	ri	NOUN
ejpam-3427	461	17	:	:	PUNCT
ejpam-3427	461	18	i	i	PROPN
ejpam-3427	461	19	∈	∈	PROPN
ejpam-3427	461	20	ω	ω	PROPN
ejpam-3427	461	21	}	}	PUNCT
ejpam-3427	461	22	has	have	VERB
ejpam-3427	461	23	the	the	DET
ejpam-3427	461	24	structure	structure	NOUN
ejpam-3427	461	25	of	of	ADP
ejpam-3427	461	26	an	an	DET
ejpam-3427	461	27	la	la	NOUN
ejpam-3427	461	28	-	-	NOUN
ejpam-3427	461	29	ring	ring	NOUN
ejpam-3427	461	30	with	with	ADP
ejpam-3427	461	31	the	the	DET
ejpam-3427	461	32	operations	operation	NOUN
ejpam-3427	461	33	of	of	ADP
ejpam-3427	461	34	addition	addition	NOUN
ejpam-3427	461	35	and	and	CCONJ
ejpam-3427	461	36	multiplication	multiplication	NOUN
ejpam-3427	461	37	defined	define	VERB
ejpam-3427	461	38	as	as	ADP
ejpam-3427	461	39	a+	a+	PRON
ejpam-3427	461	40	b	b	PROPN
ejpam-3427	461	41	=	=	SYM
ejpam-3427	461	42	(	(	PUNCT
ejpam-3427	461	43	a1	a1	PROPN
ejpam-3427	461	44	,	,	PUNCT
ejpam-3427	461	45	a2	a2	PROPN
ejpam-3427	461	46	,	,	PUNCT
ejpam-3427	461	47	a3	a3	NOUN
ejpam-3427	461	48	,	,	PUNCT
ejpam-3427	461	49	...	...	PUNCT
ejpam-3427	461	50	)	)	PUNCT
ejpam-3427	462	1	+	+	CCONJ
ejpam-3427	462	2	(	(	PUNCT
ejpam-3427	462	3	b1	b1	NOUN
ejpam-3427	462	4	,	,	PUNCT
ejpam-3427	462	5	b2	b2	NOUN
ejpam-3427	462	6	,	,	PUNCT
ejpam-3427	462	7	b3	b3	PROPN
ejpam-3427	462	8	,	,	PUNCT
ejpam-3427	462	9	...	...	PUNCT
ejpam-3427	462	10	)	)	PUNCT
ejpam-3427	463	1	=	=	PUNCT
ejpam-3427	463	2	(	(	PUNCT
ejpam-3427	463	3	a1	a1	NOUN
ejpam-3427	463	4	+	+	CCONJ
ejpam-3427	463	5	b1	b1	NOUN
ejpam-3427	463	6	,	,	PUNCT
ejpam-3427	463	7	a2	a2	PROPN
ejpam-3427	463	8	+	+	CCONJ
ejpam-3427	463	9	b2	b2	NOUN
ejpam-3427	463	10	,	,	PUNCT
ejpam-3427	463	11	a3	a3	NOUN
ejpam-3427	463	12	+	+	CCONJ
ejpam-3427	463	13	b3	b3	NOUN
ejpam-3427	463	14	,	,	PUNCT
ejpam-3427	463	15	...	...	PUNCT
ejpam-3427	463	16	)	)	PUNCT
ejpam-3427	463	17	and	and	CCONJ
ejpam-3427	463	18	a	a	DET
ejpam-3427	463	19	◦	◦	NOUN
ejpam-3427	463	20	b	b	NOUN
ejpam-3427	463	21	=	=	SYM
ejpam-3427	463	22	(	(	PUNCT
ejpam-3427	463	23	a1	a1	PROPN
ejpam-3427	463	24	,	,	PUNCT
ejpam-3427	463	25	a2	a2	PROPN
ejpam-3427	463	26	,	,	PUNCT
ejpam-3427	463	27	a3	a3	NOUN
ejpam-3427	463	28	,	,	PUNCT
ejpam-3427	463	29	...	...	PUNCT
ejpam-3427	463	30	)	)	PUNCT
ejpam-3427	464	1	◦	◦	NOUN
ejpam-3427	464	2	(	(	PUNCT
ejpam-3427	464	3	b1	b1	NOUN
ejpam-3427	464	4	,	,	PUNCT
ejpam-3427	464	5	b2	b2	NOUN
ejpam-3427	464	6	,	,	PUNCT
ejpam-3427	464	7	b3	b3	PROPN
ejpam-3427	464	8	,	,	PUNCT
ejpam-3427	464	9	...	...	PUNCT
ejpam-3427	464	10	)	)	PUNCT
ejpam-3427	465	1	=	=	PRON
ejpam-3427	465	2	(	(	PUNCT
ejpam-3427	465	3	a1b1	a1b1	ADJ
ejpam-3427	465	4	,	,	PUNCT
ejpam-3427	465	5	a2b2	a2b2	PROPN
ejpam-3427	465	6	,	,	PUNCT
ejpam-3427	465	7	a3b3	a3b3	NOUN
ejpam-3427	465	8	,	,	PUNCT
ejpam-3427	465	9	...	...	PUNCT
ejpam-3427	465	10	)	)	PUNCT
ejpam-3427	465	11	for	for	ADP
ejpam-3427	465	12	all	all	DET
ejpam-3427	465	13	a	a	DET
ejpam-3427	465	14	=	=	PUNCT
ejpam-3427	465	15	(	(	PUNCT
ejpam-3427	465	16	a1	a1	PROPN
ejpam-3427	465	17	,	,	PUNCT
ejpam-3427	465	18	a2	a2	PROPN
ejpam-3427	465	19	,	,	PUNCT
ejpam-3427	465	20	...	...	PUNCT
ejpam-3427	465	21	,	,	PUNCT
ejpam-3427	465	22	an	an	PRON
ejpam-3427	465	23	)	)	PUNCT
ejpam-3427	465	24	,	,	PUNCT
ejpam-3427	465	25	b	b	X
ejpam-3427	465	26	=	=	SYM
ejpam-3427	465	27	(	(	PUNCT
ejpam-3427	465	28	b1	b1	PROPN
ejpam-3427	465	29	,	,	PUNCT
ejpam-3427	465	30	b2	b2	NOUN
ejpam-3427	465	31	,	,	PUNCT
ejpam-3427	465	32	...	...	PUNCT
ejpam-3427	465	33	,	,	PUNCT
ejpam-3427	465	34	bn	bn	X
ejpam-3427	465	35	)	)	PUNCT
ejpam-3427	465	36	∈	∈	PROPN
ejpam-3427	465	37	r.	r.	PROPN
ejpam-3427	465	38	lemma	lemma	PROPN
ejpam-3427	465	39	5	5	X
ejpam-3427	465	40	.	.	PUNCT
ejpam-3427	466	1	if	if	SCONJ
ejpam-3427	466	2	a1	a1	PROPN
ejpam-3427	466	3	,	,	PUNCT
ejpam-3427	466	4	a2	a2	PROPN
ejpam-3427	466	5	,	,	PUNCT
ejpam-3427	466	6	...	...	PUNCT
ejpam-3427	466	7	,	,	PUNCT
ejpam-3427	466	8	an	an	PRON
ejpam-3427	466	9	are	be	AUX
ejpam-3427	466	10	la	la	ADJ
ejpam-3427	466	11	-	-	PUNCT
ejpam-3427	466	12	subrings	subring	NOUN
ejpam-3427	466	13	of	of	ADP
ejpam-3427	466	14	la	la	NOUN
ejpam-3427	466	15	-	-	PUNCT
ejpam-3427	466	16	rings	ring	NOUN
ejpam-3427	466	17	r1	r1	NOUN
ejpam-3427	466	18	,	,	PUNCT
ejpam-3427	466	19	r2	r2	PROPN
ejpam-3427	466	20	,	,	PUNCT
ejpam-3427	466	21	...	...	PUNCT
ejpam-3427	466	22	,	,	PUNCT
ejpam-3427	466	23	rn	rn	PROPN
ejpam-3427	466	24	,	,	PUNCT
ejpam-3427	466	25	respectively	respectively	ADV
ejpam-3427	466	26	,	,	PUNCT
ejpam-3427	466	27	then	then	ADV
ejpam-3427	466	28	a1	a1	NOUN
ejpam-3427	466	29	×	×	PROPN
ejpam-3427	466	30	a2	a2	PROPN
ejpam-3427	466	31	×	×	NOUN
ejpam-3427	466	32	...	...	PUNCT
ejpam-3427	466	33	×	×	NOUN
ejpam-3427	467	1	an	an	PRON
ejpam-3427	467	2	is	be	AUX
ejpam-3427	467	3	an	an	DET
ejpam-3427	467	4	la	la	NOUN
ejpam-3427	467	5	-	-	PUNCT
ejpam-3427	467	6	subring	subring	NOUN
ejpam-3427	467	7	of	of	ADP
ejpam-3427	467	8	an	an	DET
ejpam-3427	467	9	la	la	ADJ
ejpam-3427	467	10	-	-	PUNCT
ejpam-3427	467	11	ring	ring	NOUN
ejpam-3427	467	12	r1	r1	NOUN
ejpam-3427	467	13	×	×	NOUN
ejpam-3427	467	14	r2	r2	PROPN
ejpam-3427	467	15	×	×	NOUN
ejpam-3427	467	16	...	...	PUNCT
ejpam-3427	467	17	×	×	PROPN
ejpam-3427	467	18	rn	rn	PROPN
ejpam-3427	467	19	under	under	ADP
ejpam-3427	467	20	the	the	DET
ejpam-3427	467	21	same	same	ADJ
ejpam-3427	467	22	operations	operation	NOUN
ejpam-3427	467	23	defined	define	VERB
ejpam-3427	467	24	as	as	ADP
ejpam-3427	467	25	in	in	ADP
ejpam-3427	467	26	[	[	X
ejpam-3427	467	27	17	17	NUM
ejpam-3427	467	28	]	]	PUNCT
ejpam-3427	467	29	.	.	PUNCT
ejpam-3427	468	1	proof	proof	NOUN
ejpam-3427	468	2	.	.	PUNCT
ejpam-3427	469	1	straight	straight	ADV
ejpam-3427	469	2	forward	forward	ADV
ejpam-3427	469	3	.	.	PUNCT
ejpam-3427	470	1	lemma	lemma	PROPN
ejpam-3427	470	2	6	6	NUM
ejpam-3427	470	3	.	.	PUNCT
ejpam-3427	471	1	let	let	VERB
ejpam-3427	471	2	a1	a1	NOUN
ejpam-3427	471	3	,	,	PUNCT
ejpam-3427	471	4	a2	a2	PROPN
ejpam-3427	471	5	,	,	PUNCT
ejpam-3427	471	6	...	...	PUNCT
ejpam-3427	471	7	,	,	PUNCT
ejpam-3427	471	8	an	an	DET
ejpam-3427	471	9	be	be	AUX
ejpam-3427	471	10	la	la	NOUN
ejpam-3427	471	11	-	-	PUNCT
ejpam-3427	471	12	subrings	subring	NOUN
ejpam-3427	471	13	of	of	ADP
ejpam-3427	471	14	la	la	NOUN
ejpam-3427	471	15	-	-	PUNCT
ejpam-3427	471	16	rings	ring	NOUN
ejpam-3427	471	17	r1	r1	NOUN
ejpam-3427	471	18	,	,	PUNCT
ejpam-3427	471	19	r2	r2	PROPN
ejpam-3427	471	20	,	,	PUNCT
ejpam-3427	471	21	...	...	PUNCT
ejpam-3427	471	22	,	,	PUNCT
ejpam-3427	471	23	rn	rn	PROPN
ejpam-3427	471	24	,	,	PUNCT
ejpam-3427	471	25	respectively	respectively	ADV
ejpam-3427	471	26	.	.	PUNCT
ejpam-3427	472	1	then	then	ADV
ejpam-3427	472	2	a1	a1	VERB
ejpam-3427	472	3	×	×	PROPN
ejpam-3427	472	4	a2	a2	PROPN
ejpam-3427	472	5	×	×	NOUN
ejpam-3427	472	6	...	...	PUNCT
ejpam-3427	472	7	×	×	NOUN
ejpam-3427	472	8	an	an	PRON
ejpam-3427	472	9	is	be	AUX
ejpam-3427	472	10	an	an	DET
ejpam-3427	472	11	la	la	NOUN
ejpam-3427	472	12	-	-	PUNCT
ejpam-3427	472	13	subring	subring	NOUN
ejpam-3427	472	14	of	of	ADP
ejpam-3427	472	15	an	an	DET
ejpam-3427	472	16	la	la	ADJ
ejpam-3427	472	17	-	-	PUNCT
ejpam-3427	472	18	ring	ring	NOUN
ejpam-3427	472	19	r1	r1	NOUN
ejpam-3427	472	20	×	×	NOUN
ejpam-3427	472	21	r2	r2	PROPN
ejpam-3427	472	22	×	×	NOUN
ejpam-3427	472	23	...	...	PUNCT
ejpam-3427	473	1	×	×	PROPN
ejpam-3427	473	2	rn	rn	NOUN
ejpam-3427	473	3	if	if	SCONJ
ejpam-3427	474	1	and	and	CCONJ
ejpam-3427	474	2	only	only	ADV
ejpam-3427	474	3	if	if	SCONJ
ejpam-3427	474	4	the	the	DET
ejpam-3427	474	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	474	6	anti	anti	ADJ
ejpam-3427	474	7	characteristic	characteristic	ADJ
ejpam-3427	474	8	function	function	NOUN
ejpam-3427	474	9	χa	χa	NOUN
ejpam-3427	474	10	=	=	SYM
ejpam-3427	474	11	〈	〈	PROPN
ejpam-3427	474	12	µχa	µχa	NOUN
ejpam-3427	474	13	,	,	PUNCT
ejpam-3427	474	14	γχa	γχa	NOUN
ejpam-3427	474	15	〉	〉	NUM
ejpam-3427	474	16	of	of	ADP
ejpam-3427	474	17	a	a	DET
ejpam-3427	474	18	=	=	NOUN
ejpam-3427	474	19	a1×a2×	a1×a2×	NOUN
ejpam-3427	474	20	...	...	PUNCT
ejpam-3427	474	21	×an	×an	PROPN
ejpam-3427	474	22	is	be	AUX
ejpam-3427	474	23	an	an	DET
ejpam-3427	474	24	intuitionistic	intuitionistic	ADJ
ejpam-3427	474	25	anti	anti	ADJ
ejpam-3427	474	26	fuzzy	fuzzy	ADJ
ejpam-3427	474	27	normal	normal	ADJ
ejpam-3427	474	28	la	la	NOUN
ejpam-3427	474	29	-	-	PUNCT
ejpam-3427	474	30	subring	subring	NOUN
ejpam-3427	474	31	of	of	ADP
ejpam-3427	474	32	an	an	DET
ejpam-3427	474	33	la	la	ADJ
ejpam-3427	474	34	-	-	PUNCT
ejpam-3427	474	35	ring	ring	NOUN
ejpam-3427	474	36	r1	r1	NOUN
ejpam-3427	474	37	×r2	×r2	PROPN
ejpam-3427	474	38	×	×	NOUN
ejpam-3427	474	39	...	...	PUNCT
ejpam-3427	474	40	×rn	×rn	PROPN
ejpam-3427	474	41	.	.	PUNCT
ejpam-3427	475	1	k.	k.	PROPN
ejpam-3427	475	2	nasreen	nasreen	PROPN
ejpam-3427	475	3	/	/	SYM
ejpam-3427	475	4	eur	eur	PROPN
ejpam-3427	475	5	.	.	PUNCT
ejpam-3427	476	1	j.	j.	PROPN
ejpam-3427	476	2	pure	pure	PROPN
ejpam-3427	476	3	appl	appl	PROPN
ejpam-3427	476	4	.	.	PROPN
ejpam-3427	476	5	math	math	PROPN
ejpam-3427	476	6	,	,	PUNCT
ejpam-3427	476	7	12	12	NUM
ejpam-3427	476	8	(	(	PUNCT
ejpam-3427	476	9	2	2	NUM
ejpam-3427	476	10	)	)	PUNCT
ejpam-3427	476	11	(	(	PUNCT
ejpam-3427	476	12	2019	2019	NUM
ejpam-3427	476	13	)	)	PUNCT
ejpam-3427	476	14	,	,	PUNCT
ejpam-3427	476	15	622	622	NUM
ejpam-3427	476	16	-	-	SYM
ejpam-3427	476	17	648	648	NUM
ejpam-3427	476	18	638	638	NUM
ejpam-3427	476	19	proof	proof	NOUN
ejpam-3427	476	20	.	.	PUNCT
ejpam-3427	477	1	let	let	VERB
ejpam-3427	477	2	a	a	DET
ejpam-3427	477	3	=	=	PUNCT
ejpam-3427	477	4	a1×a2×	a1×a2×	NOUN
ejpam-3427	477	5	...	...	PUNCT
ejpam-3427	477	6	×an	×an	NOUN
ejpam-3427	477	7	be	be	VERB
ejpam-3427	477	8	an	an	DET
ejpam-3427	477	9	la	la	NOUN
ejpam-3427	477	10	-	-	PUNCT
ejpam-3427	477	11	subring	subring	NOUN
ejpam-3427	477	12	of	of	ADP
ejpam-3427	477	13	an	an	DET
ejpam-3427	477	14	la	la	ADJ
ejpam-3427	477	15	-	-	PUNCT
ejpam-3427	477	16	ring	ring	NOUN
ejpam-3427	477	17	r1×r2×	r1×r2×	NOUN
ejpam-3427	477	18	...	...	PUNCT
ejpam-3427	477	19	×rn	×rn	NOUN
ejpam-3427	477	20	and	and	CCONJ
ejpam-3427	477	21	a	a	DET
ejpam-3427	477	22	=	=	X
ejpam-3427	477	23	(	(	PUNCT
ejpam-3427	477	24	a1	a1	PROPN
ejpam-3427	477	25	,	,	PUNCT
ejpam-3427	477	26	a2	a2	PROPN
ejpam-3427	477	27	,	,	PUNCT
ejpam-3427	477	28	...	...	PUNCT
ejpam-3427	477	29	,	,	PUNCT
ejpam-3427	477	30	an	an	PRON
ejpam-3427	477	31	)	)	PUNCT
ejpam-3427	477	32	,	,	PUNCT
ejpam-3427	477	33	b	b	X
ejpam-3427	477	34	=	=	SYM
ejpam-3427	477	35	(	(	PUNCT
ejpam-3427	477	36	b1	b1	PROPN
ejpam-3427	477	37	,	,	PUNCT
ejpam-3427	477	38	b2	b2	NOUN
ejpam-3427	477	39	,	,	PUNCT
ejpam-3427	477	40	...	...	PUNCT
ejpam-3427	477	41	,	,	PUNCT
ejpam-3427	477	42	bn	bn	X
ejpam-3427	477	43	)	)	PUNCT
ejpam-3427	477	44	∈	∈	PROPN
ejpam-3427	477	45	r1×r2×	r1×r2×	NOUN
ejpam-3427	477	46	...	...	PUNCT
ejpam-3427	477	47	×rn	×rn	VERB
ejpam-3427	477	48	.	.	PUNCT
ejpam-3427	478	1	if	if	SCONJ
ejpam-3427	478	2	a	a	PRON
ejpam-3427	478	3	,	,	PUNCT
ejpam-3427	478	4	b	b	X
ejpam-3427	478	5	∈	∈	PROPN
ejpam-3427	478	6	a	a	DET
ejpam-3427	478	7	=	=	NOUN
ejpam-3427	478	8	a1×a2×	a1×a2×	NOUN
ejpam-3427	478	9	...	...	PUNCT
ejpam-3427	478	10	×an	×an	PROPN
ejpam-3427	478	11	,	,	PUNCT
ejpam-3427	478	12	then	then	ADV
ejpam-3427	478	13	by	by	ADP
ejpam-3427	478	14	definition	definition	NOUN
ejpam-3427	478	15	of	of	ADP
ejpam-3427	478	16	intuitionistic	intuitionistic	ADJ
ejpam-3427	478	17	anti	anti	ADJ
ejpam-3427	478	18	characteristic	characteristic	ADJ
ejpam-3427	478	19	function	function	NOUN
ejpam-3427	478	20	µχa(a	µχa(a	PROPN
ejpam-3427	478	21	)	)	PUNCT
ejpam-3427	478	22	=	=	SYM
ejpam-3427	478	23	0	0	PUNCT
ejpam-3427	478	24	=	=	SYM
ejpam-3427	478	25	µχa(b	µχa(b	PROPN
ejpam-3427	478	26	)	)	PUNCT
ejpam-3427	478	27	and	and	CCONJ
ejpam-3427	478	28	γχa(a	γχa(a	NUM
ejpam-3427	478	29	)	)	PUNCT
ejpam-3427	478	30	=	=	SYM
ejpam-3427	478	31	1	1	X
ejpam-3427	478	32	=	=	SYM
ejpam-3427	478	33	γχa(b	γχa(b	PROPN
ejpam-3427	478	34	)	)	PUNCT
ejpam-3427	478	35	.	.	PUNCT
ejpam-3427	479	1	since	since	SCONJ
ejpam-3427	479	2	a−	a−	PROPN
ejpam-3427	479	3	b	b	PROPN
ejpam-3427	479	4	and	and	CCONJ
ejpam-3427	479	5	ab	ab	PROPN
ejpam-3427	479	6	∈	∈	PROPN
ejpam-3427	479	7	a	a	PRON
ejpam-3427	479	8	,	,	PUNCT
ejpam-3427	479	9	a	a	DET
ejpam-3427	479	10	being	be	AUX
ejpam-3427	479	11	an	an	DET
ejpam-3427	479	12	la	la	ADV
ejpam-3427	479	13	-	-	PUNCT
ejpam-3427	479	14	subring	subring	NOUN
ejpam-3427	479	15	.	.	PUNCT
ejpam-3427	480	1	this	this	PRON
ejpam-3427	480	2	implies	imply	VERB
ejpam-3427	480	3	that	that	SCONJ
ejpam-3427	480	4	µχa(a−	µχa(a−	PROPN
ejpam-3427	480	5	b	b	X
ejpam-3427	480	6	)	)	PUNCT
ejpam-3427	480	7	=	=	SYM
ejpam-3427	480	8	0	0	PUNCT
ejpam-3427	481	1	=	=	SYM
ejpam-3427	481	2	0	0	NUM
ejpam-3427	482	1	∨	∨	NUM
ejpam-3427	482	2	0	0	NUM
ejpam-3427	482	3	=	=	SYM
ejpam-3427	482	4	µχa(a	µχa(a	PROPN
ejpam-3427	482	5	)	)	PUNCT
ejpam-3427	482	6	∨	∨	NUM
ejpam-3427	482	7	µχa(b	µχa(b	PROPN
ejpam-3427	482	8	)	)	PUNCT
ejpam-3427	482	9	,	,	PUNCT
ejpam-3427	482	10	µχa(ab	µχa(ab	NOUN
ejpam-3427	482	11	)	)	PUNCT
ejpam-3427	482	12	=	=	SYM
ejpam-3427	482	13	0	0	PUNCT
ejpam-3427	483	1	=	=	SYM
ejpam-3427	483	2	0	0	NUM
ejpam-3427	484	1	∨	∨	NUM
ejpam-3427	484	2	0	0	NUM
ejpam-3427	484	3	=	=	SYM
ejpam-3427	484	4	µχa(a	µχa(a	PROPN
ejpam-3427	484	5	)	)	PUNCT
ejpam-3427	484	6	∨	∨	NUM
ejpam-3427	484	7	µχa(b	µχa(b	PROPN
ejpam-3427	484	8	)	)	PUNCT
ejpam-3427	484	9	,	,	PUNCT
ejpam-3427	484	10	γχa(a−	γχa(a−	SYM
ejpam-3427	484	11	b	b	X
ejpam-3427	484	12	)	)	PUNCT
ejpam-3427	484	13	=	=	SYM
ejpam-3427	484	14	1	1	NUM
ejpam-3427	484	15	=	=	SYM
ejpam-3427	484	16	1	1	NUM
ejpam-3427	484	17	∧	∧	PROPN
ejpam-3427	484	18	1	1	NUM
ejpam-3427	484	19	=	=	SYM
ejpam-3427	484	20	γχa(a	γχa(a	PROPN
ejpam-3427	484	21	)	)	PUNCT
ejpam-3427	484	22	∧	∧	PROPN
ejpam-3427	484	23	γχa(b	γχa(b	PROPN
ejpam-3427	484	24	)	)	PUNCT
ejpam-3427	484	25	,	,	PUNCT
ejpam-3427	484	26	γχa(ab	γχa(ab	NOUN
ejpam-3427	484	27	)	)	PUNCT
ejpam-3427	484	28	=	=	SYM
ejpam-3427	484	29	1	1	NUM
ejpam-3427	484	30	=	=	SYM
ejpam-3427	484	31	1	1	NUM
ejpam-3427	484	32	∧	∧	PROPN
ejpam-3427	484	33	1	1	NUM
ejpam-3427	484	34	=	=	SYM
ejpam-3427	484	35	γχa(a	γχa(a	PROPN
ejpam-3427	484	36	)	)	PUNCT
ejpam-3427	484	37	∧	∧	PROPN
ejpam-3427	484	38	γχa(b	γχa(b	PROPN
ejpam-3427	484	39	)	)	PUNCT
ejpam-3427	484	40	.	.	PUNCT
ejpam-3427	485	1	thus	thus	ADV
ejpam-3427	485	2	µχa(a−	µχa(a−	PROPN
ejpam-3427	485	3	b	b	X
ejpam-3427	485	4	)	)	PUNCT
ejpam-3427	485	5	≤	≤	NOUN
ejpam-3427	485	6	max{µχa(a	max{µχa(a	PROPN
ejpam-3427	485	7	)	)	PUNCT
ejpam-3427	485	8	,	,	PUNCT
ejpam-3427	485	9	µχa(b	µχa(b	PROPN
ejpam-3427	485	10	)	)	PUNCT
ejpam-3427	485	11	}	}	PUNCT
ejpam-3427	485	12	,	,	PUNCT
ejpam-3427	485	13	µχa(ab	µχa(ab	NOUN
ejpam-3427	485	14	)	)	PUNCT
ejpam-3427	485	15	≤	≤	NOUN
ejpam-3427	485	16	max{µχa(a	max{µχa(a	PROPN
ejpam-3427	485	17	)	)	PUNCT
ejpam-3427	485	18	,	,	PUNCT
ejpam-3427	485	19	µχa(b	µχa(b	PROPN
ejpam-3427	485	20	)	)	PUNCT
ejpam-3427	485	21	}	}	PUNCT
ejpam-3427	485	22	,	,	PUNCT
ejpam-3427	485	23	γχa(a−	γχa(a−	SYM
ejpam-3427	485	24	b	b	X
ejpam-3427	485	25	)	)	PUNCT
ejpam-3427	485	26	≥	≥	NOUN
ejpam-3427	485	27	min{γχa(a	min{γχa(a	PROPN
ejpam-3427	485	28	)	)	PUNCT
ejpam-3427	485	29	,	,	PUNCT
ejpam-3427	485	30	γχa(b	γχa(b	PROPN
ejpam-3427	485	31	)	)	PUNCT
ejpam-3427	485	32	}	}	PUNCT
ejpam-3427	485	33	,	,	PUNCT
ejpam-3427	485	34	γχa(ab	γχa(ab	NOUN
ejpam-3427	485	35	)	)	PUNCT
ejpam-3427	485	36	≥	≥	NOUN
ejpam-3427	485	37	min{γχa(a	min{γχa(a	PROPN
ejpam-3427	485	38	)	)	PUNCT
ejpam-3427	485	39	,	,	PUNCT
ejpam-3427	485	40	γχa(b	γχa(b	PROPN
ejpam-3427	485	41	)	)	PUNCT
ejpam-3427	485	42	}	}	PUNCT
ejpam-3427	485	43	.	.	PUNCT
ejpam-3427	486	1	as	as	SCONJ
ejpam-3427	486	2	ab	ab	PROPN
ejpam-3427	486	3	and	and	CCONJ
ejpam-3427	486	4	ba	ba	PROPN
ejpam-3427	486	5	∈	∈	PROPN
ejpam-3427	486	6	a	a	DET
ejpam-3427	486	7	,	,	PUNCT
ejpam-3427	486	8	so	so	SCONJ
ejpam-3427	486	9	µχa(ab	µχa(ab	NOUN
ejpam-3427	486	10	)	)	PUNCT
ejpam-3427	486	11	=	=	SYM
ejpam-3427	486	12	0	0	NUM
ejpam-3427	486	13	=	=	SYM
ejpam-3427	486	14	µχa(ba	µχa(ba	NOUN
ejpam-3427	486	15	)	)	PUNCT
ejpam-3427	486	16	and	and	CCONJ
ejpam-3427	486	17	γχa(ab	γχa(ab	NOUN
ejpam-3427	486	18	)	)	PUNCT
ejpam-3427	486	19	=	=	SYM
ejpam-3427	486	20	1	1	NUM
ejpam-3427	486	21	=	=	NOUN
ejpam-3427	486	22	γχa(ba	γχa(ba	NOUN
ejpam-3427	486	23	)	)	PUNCT
ejpam-3427	486	24	,	,	PUNCT
ejpam-3427	486	25	i.e.	i.e.	X
ejpam-3427	486	26	,	,	PUNCT
ejpam-3427	486	27	µχa(ab	µχa(ab	NOUN
ejpam-3427	486	28	)	)	PUNCT
ejpam-3427	486	29	=	=	SYM
ejpam-3427	486	30	µχa(ba	µχa(ba	NOUN
ejpam-3427	486	31	)	)	PUNCT
ejpam-3427	486	32	and	and	CCONJ
ejpam-3427	486	33	γχa(ab	γχa(ab	NOUN
ejpam-3427	486	34	)	)	PUNCT
ejpam-3427	486	35	=	=	SYM
ejpam-3427	486	36	γχa(ba	γχa(ba	NOUN
ejpam-3427	486	37	)	)	PUNCT
ejpam-3427	486	38	.	.	PUNCT
ejpam-3427	487	1	similarly	similarly	ADV
ejpam-3427	487	2	,	,	PUNCT
ejpam-3427	487	3	we	we	PRON
ejpam-3427	487	4	have	have	VERB
ejpam-3427	487	5	µχa(a−	µχa(a−	PRON
ejpam-3427	487	6	b	b	PROPN
ejpam-3427	487	7	)	)	PUNCT
ejpam-3427	487	8	≤	≤	NOUN
ejpam-3427	487	9	max{µχa(a	max{µχa(a	PROPN
ejpam-3427	487	10	)	)	PUNCT
ejpam-3427	487	11	,	,	PUNCT
ejpam-3427	487	12	µχa(b	µχa(b	PROPN
ejpam-3427	487	13	)	)	PUNCT
ejpam-3427	487	14	}	}	PUNCT
ejpam-3427	487	15	,	,	PUNCT
ejpam-3427	487	16	µχa(ab	µχa(ab	NOUN
ejpam-3427	487	17	)	)	PUNCT
ejpam-3427	487	18	≤	≤	NOUN
ejpam-3427	487	19	max{µχa(a	max{µχa(a	PROPN
ejpam-3427	487	20	)	)	PUNCT
ejpam-3427	487	21	,	,	PUNCT
ejpam-3427	487	22	µχa(b	µχa(b	PROPN
ejpam-3427	487	23	)	)	PUNCT
ejpam-3427	487	24	}	}	PUNCT
ejpam-3427	487	25	,	,	PUNCT
ejpam-3427	487	26	γχa(a−	γχa(a−	SYM
ejpam-3427	487	27	b	b	X
ejpam-3427	487	28	)	)	PUNCT
ejpam-3427	487	29	≥	≥	NOUN
ejpam-3427	487	30	min{γχa(a	min{γχa(a	PROPN
ejpam-3427	487	31	)	)	PUNCT
ejpam-3427	487	32	,	,	PUNCT
ejpam-3427	487	33	γχa(b	γχa(b	PROPN
ejpam-3427	487	34	)	)	PUNCT
ejpam-3427	487	35	}	}	PUNCT
ejpam-3427	487	36	,	,	PUNCT
ejpam-3427	487	37	γχa(ab	γχa(ab	NOUN
ejpam-3427	487	38	)	)	PUNCT
ejpam-3427	487	39	≥	≥	NOUN
ejpam-3427	487	40	min{γχa(a	min{γχa(a	PROPN
ejpam-3427	487	41	)	)	PUNCT
ejpam-3427	487	42	,	,	PUNCT
ejpam-3427	487	43	γχa(b	γχa(b	PROPN
ejpam-3427	487	44	)	)	PUNCT
ejpam-3427	487	45	}	}	PUNCT
ejpam-3427	487	46	,	,	PUNCT
ejpam-3427	487	47	µχa(ab	µχa(ab	NOUN
ejpam-3427	487	48	)	)	PUNCT
ejpam-3427	487	49	=	=	SYM
ejpam-3427	487	50	µχa(ba	µχa(ba	NOUN
ejpam-3427	487	51	)	)	PUNCT
ejpam-3427	487	52	,	,	PUNCT
ejpam-3427	487	53	γχa(ab	γχa(ab	NOUN
ejpam-3427	487	54	)	)	PUNCT
ejpam-3427	487	55	=	=	SYM
ejpam-3427	487	56	γχa(ba	γχa(ba	NOUN
ejpam-3427	487	57	)	)	PUNCT
ejpam-3427	487	58	,	,	PUNCT
ejpam-3427	487	59	when	when	SCONJ
ejpam-3427	487	60	a	a	DET
ejpam-3427	487	61	,	,	PUNCT
ejpam-3427	487	62	b	b	NOUN
ejpam-3427	487	63	/∈	/∈	PUNCT
ejpam-3427	487	64	a.	a.	NOUN
ejpam-3427	487	65	hence	hence	ADV
ejpam-3427	487	66	the	the	DET
ejpam-3427	487	67	intuitionistic	intuitionistic	ADJ
ejpam-3427	487	68	anti	anti	ADJ
ejpam-3427	487	69	characteristic	characteristic	ADJ
ejpam-3427	487	70	function	function	NOUN
ejpam-3427	487	71	χa	χa	NOUN
ejpam-3427	487	72	=	=	SYM
ejpam-3427	487	73	〈	〈	PROPN
ejpam-3427	487	74	µχa	µχa	NOUN
ejpam-3427	487	75	,	,	PUNCT
ejpam-3427	487	76	γχa	γχa	NOUN
ejpam-3427	487	77	〉	〉	NUM
ejpam-3427	487	78	of	of	ADP
ejpam-3427	487	79	a	a	DET
ejpam-3427	487	80	=	=	NOUN
ejpam-3427	487	81	a1	a1	NOUN
ejpam-3427	487	82	×	×	PROPN
ejpam-3427	487	83	a2	a2	PROPN
ejpam-3427	487	84	×	×	NOUN
ejpam-3427	487	85	...	...	PUNCT
ejpam-3427	488	1	×	×	NOUN
ejpam-3427	488	2	an	an	PRON
ejpam-3427	488	3	is	be	AUX
ejpam-3427	488	4	an	an	DET
ejpam-3427	488	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	488	6	anti	anti	ADJ
ejpam-3427	488	7	fuzzy	fuzzy	ADJ
ejpam-3427	488	8	normal	normal	ADJ
ejpam-3427	488	9	la	la	NOUN
ejpam-3427	488	10	-	-	PUNCT
ejpam-3427	488	11	subring	subring	NOUN
ejpam-3427	488	12	of	of	ADP
ejpam-3427	488	13	an	an	DET
ejpam-3427	488	14	la	la	ADJ
ejpam-3427	488	15	-	-	PUNCT
ejpam-3427	488	16	ring	ring	NOUN
ejpam-3427	488	17	r1	r1	NOUN
ejpam-3427	488	18	×r2	×r2	PROPN
ejpam-3427	488	19	×	×	NOUN
ejpam-3427	488	20	...	...	PUNCT
ejpam-3427	488	21	×rn	×rn	NOUN
ejpam-3427	488	22	.	.	PUNCT
ejpam-3427	489	1	conversely	conversely	ADV
ejpam-3427	489	2	,	,	PUNCT
ejpam-3427	489	3	suppose	suppose	VERB
ejpam-3427	489	4	that	that	SCONJ
ejpam-3427	489	5	the	the	DET
ejpam-3427	489	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	489	7	anti	anti	ADJ
ejpam-3427	489	8	characteristic	characteristic	ADJ
ejpam-3427	489	9	function	function	NOUN
ejpam-3427	489	10	χa	χa	NOUN
ejpam-3427	489	11	=	=	SYM
ejpam-3427	489	12	〈	〈	PROPN
ejpam-3427	489	13	µχa	µχa	NOUN
ejpam-3427	489	14	,	,	PUNCT
ejpam-3427	489	15	γχa	γχa	NOUN
ejpam-3427	489	16	〉	〉	NUM
ejpam-3427	489	17	of	of	ADP
ejpam-3427	489	18	a	a	DET
ejpam-3427	489	19	=	=	NOUN
ejpam-3427	489	20	a1	a1	NOUN
ejpam-3427	489	21	×a2	×a2	PROPN
ejpam-3427	489	22	×	×	NOUN
ejpam-3427	489	23	...	...	PUNCT
ejpam-3427	489	24	×an	×an	PROPN
ejpam-3427	489	25	is	be	AUX
ejpam-3427	489	26	an	an	DET
ejpam-3427	489	27	intuitionistic	intuitionistic	ADJ
ejpam-3427	489	28	anti	anti	ADJ
ejpam-3427	489	29	fuzzy	fuzzy	ADJ
ejpam-3427	489	30	normal	normal	ADJ
ejpam-3427	489	31	la	la	NOUN
ejpam-3427	489	32	-	-	PUNCT
ejpam-3427	489	33	subring	subring	NOUN
ejpam-3427	489	34	of	of	ADP
ejpam-3427	489	35	an	an	DET
ejpam-3427	489	36	la	la	ADJ
ejpam-3427	489	37	-	-	PUNCT
ejpam-3427	489	38	ring	ring	NOUN
ejpam-3427	489	39	r1	r1	NOUN
ejpam-3427	489	40	×r2	×r2	PROPN
ejpam-3427	489	41	×	×	NOUN
ejpam-3427	489	42	...	...	PUNCT
ejpam-3427	489	43	×rn	×rn	NOUN
ejpam-3427	489	44	.	.	PUNCT
ejpam-3427	490	1	we	we	PRON
ejpam-3427	490	2	have	have	VERB
ejpam-3427	490	3	to	to	PART
ejpam-3427	490	4	show	show	VERB
ejpam-3427	490	5	that	that	SCONJ
ejpam-3427	490	6	a	a	DET
ejpam-3427	490	7	=	=	NOUN
ejpam-3427	490	8	a1	a1	NOUN
ejpam-3427	490	9	×a2	×a2	PROPN
ejpam-3427	490	10	×	×	NOUN
ejpam-3427	490	11	...	...	PUNCT
ejpam-3427	490	12	×an	×an	PROPN
ejpam-3427	490	13	is	be	AUX
ejpam-3427	490	14	an	an	DET
ejpam-3427	490	15	la	la	NOUN
ejpam-3427	490	16	-	-	PUNCT
ejpam-3427	490	17	subring	subring	NOUN
ejpam-3427	490	18	of	of	ADP
ejpam-3427	490	19	an	an	DET
ejpam-3427	490	20	la	la	ADJ
ejpam-3427	490	21	-	-	PUNCT
ejpam-3427	490	22	ring	ring	NOUN
ejpam-3427	490	23	r1	r1	NOUN
ejpam-3427	490	24	×	×	NOUN
ejpam-3427	490	25	r2	r2	PROPN
ejpam-3427	490	26	×	×	NOUN
ejpam-3427	490	27	...	...	PUNCT
ejpam-3427	490	28	×	×	PROPN
ejpam-3427	490	29	rn	rn	PROPN
ejpam-3427	490	30	.	.	PROPN
ejpam-3427	491	1	let	let	VERB
ejpam-3427	491	2	a	a	DET
ejpam-3427	491	3	,	,	PUNCT
ejpam-3427	491	4	b	b	PROPN
ejpam-3427	491	5	∈	∈	PROPN
ejpam-3427	491	6	a	a	PRON
ejpam-3427	491	7	,	,	PUNCT
ejpam-3427	491	8	where	where	SCONJ
ejpam-3427	491	9	a	a	DET
ejpam-3427	491	10	=	=	X
ejpam-3427	491	11	(	(	PUNCT
ejpam-3427	491	12	a1	a1	PROPN
ejpam-3427	491	13	,	,	PUNCT
ejpam-3427	491	14	a	a	PRON
ejpam-3427	491	15	,	,	PUNCT
ejpam-3427	491	16	...	...	PUNCT
ejpam-3427	491	17	,	,	PUNCT
ejpam-3427	491	18	an	an	X
ejpam-3427	491	19	)	)	PUNCT
ejpam-3427	491	20	and	and	CCONJ
ejpam-3427	491	21	b	b	X
ejpam-3427	491	22	=	=	SYM
ejpam-3427	491	23	(	(	PUNCT
ejpam-3427	491	24	b1	b1	PROPN
ejpam-3427	491	25	,	,	PUNCT
ejpam-3427	491	26	b2	b2	NOUN
ejpam-3427	491	27	,	,	PUNCT
ejpam-3427	491	28	...	...	PUNCT
ejpam-3427	491	29	,	,	PUNCT
ejpam-3427	491	30	bn	bn	ADJ
ejpam-3427	491	31	)	)	PUNCT
ejpam-3427	491	32	,	,	PUNCT
ejpam-3427	491	33	by	by	ADP
ejpam-3427	491	34	definition	definition	NOUN
ejpam-3427	491	35	,	,	PUNCT
ejpam-3427	491	36	we	we	PRON
ejpam-3427	491	37	have	have	VERB
ejpam-3427	491	38	µχa(a	µχa(a	PROPN
ejpam-3427	491	39	)	)	PUNCT
ejpam-3427	491	40	=	=	SYM
ejpam-3427	491	41	0	0	PUNCT
ejpam-3427	492	1	=	=	SYM
ejpam-3427	492	2	µχa(b	µχa(b	PROPN
ejpam-3427	492	3	)	)	PUNCT
ejpam-3427	492	4	and	and	CCONJ
ejpam-3427	492	5	γχa(a	γχa(a	NUM
ejpam-3427	492	6	)	)	PUNCT
ejpam-3427	492	7	=	=	SYM
ejpam-3427	492	8	1	1	X
ejpam-3427	492	9	=	=	SYM
ejpam-3427	492	10	γχa(b	γχa(b	PROPN
ejpam-3427	492	11	)	)	PUNCT
ejpam-3427	492	12	.	.	PUNCT
ejpam-3427	493	1	by	by	ADP
ejpam-3427	493	2	our	our	PRON
ejpam-3427	493	3	supposition	supposition	NOUN
ejpam-3427	493	4	µχa(a−	µχa(a−	PROPN
ejpam-3427	493	5	b	b	PROPN
ejpam-3427	493	6	)	)	PUNCT
ejpam-3427	493	7	≤	≤	NOUN
ejpam-3427	493	8	µχa(a	µχa(a	PROPN
ejpam-3427	493	9	)	)	PUNCT
ejpam-3427	493	10	∨	∨	NUM
ejpam-3427	493	11	µχa(b	µχa(b	PROPN
ejpam-3427	493	12	)	)	PUNCT
ejpam-3427	493	13	=	=	SYM
ejpam-3427	493	14	0	0	NUM
ejpam-3427	493	15	∨	∨	NUM
ejpam-3427	493	16	0	0	NUM
ejpam-3427	494	1	=	=	SYM
ejpam-3427	494	2	0	0	NUM
ejpam-3427	494	3	,	,	PUNCT
ejpam-3427	494	4	µχa(ab	µχa(ab	NOUN
ejpam-3427	494	5	)	)	PUNCT
ejpam-3427	494	6	≤	≤	NUM
ejpam-3427	494	7	µχa(a	µχa(a	PROPN
ejpam-3427	494	8	)	)	PUNCT
ejpam-3427	494	9	∨	∨	NUM
ejpam-3427	494	10	µχa(b	µχa(b	PROPN
ejpam-3427	494	11	)	)	PUNCT
ejpam-3427	494	12	=	=	SYM
ejpam-3427	494	13	0	0	NUM
ejpam-3427	494	14	∨	∨	NUM
ejpam-3427	494	15	0	0	NUM
ejpam-3427	495	1	=	=	SYM
ejpam-3427	495	2	0	0	PROPN
ejpam-3427	495	3	,	,	PUNCT
ejpam-3427	495	4	γχa(a−	γχa(a−	NOUN
ejpam-3427	495	5	b	b	X
ejpam-3427	495	6	)	)	PUNCT
ejpam-3427	495	7	≥	≥	NOUN
ejpam-3427	495	8	γχa(a	γχa(a	PRON
ejpam-3427	495	9	)	)	PUNCT
ejpam-3427	495	10	∧	∧	PROPN
ejpam-3427	495	11	γχa(b	γχa(b	PROPN
ejpam-3427	495	12	)	)	PUNCT
ejpam-3427	495	13	=	=	SYM
ejpam-3427	495	14	1	1	NUM
ejpam-3427	495	15	∧	∧	PROPN
ejpam-3427	495	16	1	1	NUM
ejpam-3427	495	17	=	=	SYM
ejpam-3427	495	18	1	1	NUM
ejpam-3427	495	19	,	,	PUNCT
ejpam-3427	495	20	γχa(ab	γχa(ab	NOUN
ejpam-3427	495	21	)	)	PUNCT
ejpam-3427	495	22	≥	≥	X
ejpam-3427	495	23	γχa(a	γχa(a	PRON
ejpam-3427	495	24	)	)	PUNCT
ejpam-3427	495	25	∧	∧	PROPN
ejpam-3427	495	26	γχa(b	γχa(b	PROPN
ejpam-3427	495	27	)	)	PUNCT
ejpam-3427	495	28	=	=	SYM
ejpam-3427	495	29	1	1	NUM
ejpam-3427	495	30	∧	∧	PROPN
ejpam-3427	495	31	1	1	NUM
ejpam-3427	495	32	=	=	SYM
ejpam-3427	495	33	1	1	NUM
ejpam-3427	495	34	.	.	PUNCT
ejpam-3427	496	1	thus	thus	ADV
ejpam-3427	496	2	µχa(a−	µχa(a−	PROPN
ejpam-3427	496	3	b	b	X
ejpam-3427	496	4	)	)	PUNCT
ejpam-3427	496	5	=	=	SYM
ejpam-3427	496	6	0	0	NUM
ejpam-3427	496	7	=	=	SYM
ejpam-3427	496	8	µχa(ab	µχa(ab	NOUN
ejpam-3427	496	9	)	)	PUNCT
ejpam-3427	496	10	and	and	CCONJ
ejpam-3427	496	11	γχa(a−	γχa(a−	ADV
ejpam-3427	496	12	b	b	X
ejpam-3427	496	13	)	)	PUNCT
ejpam-3427	496	14	=	=	SYM
ejpam-3427	496	15	1	1	NUM
ejpam-3427	496	16	=	=	SYM
ejpam-3427	496	17	γχa(ab	γχa(ab	NOUN
ejpam-3427	496	18	)	)	PUNCT
ejpam-3427	496	19	,	,	PUNCT
ejpam-3427	496	20	i.e.	i.e.	X
ejpam-3427	496	21	,	,	PUNCT
ejpam-3427	496	22	a−	a−	PROPN
ejpam-3427	496	23	b	b	PROPN
ejpam-3427	496	24	and	and	CCONJ
ejpam-3427	496	25	ab	ab	PROPN
ejpam-3427	496	26	∈	∈	PROPN
ejpam-3427	496	27	a.	a.	NOUN
ejpam-3427	496	28	hence	hence	ADV
ejpam-3427	496	29	a	a	DET
ejpam-3427	496	30	=	=	NOUN
ejpam-3427	496	31	a1	a1	NOUN
ejpam-3427	496	32	×a2	×a2	PROPN
ejpam-3427	496	33	×	×	NOUN
ejpam-3427	496	34	...	...	PUNCT
ejpam-3427	496	35	×an	×an	PROPN
ejpam-3427	496	36	is	be	AUX
ejpam-3427	496	37	an	an	DET
ejpam-3427	496	38	la	la	NOUN
ejpam-3427	496	39	-	-	PUNCT
ejpam-3427	496	40	subring	subring	NOUN
ejpam-3427	496	41	of	of	ADP
ejpam-3427	496	42	an	an	DET
ejpam-3427	496	43	la	la	ADJ
ejpam-3427	496	44	-	-	PUNCT
ejpam-3427	496	45	ring	ring	NOUN
ejpam-3427	496	46	r1	r1	NOUN
ejpam-3427	496	47	×r2	×r2	PROPN
ejpam-3427	496	48	×	×	NOUN
ejpam-3427	496	49	...	...	PUNCT
ejpam-3427	496	50	×rn	×rn	PROPN
ejpam-3427	496	51	.	.	PUNCT
ejpam-3427	497	1	lemma	lemma	PROPN
ejpam-3427	497	2	7	7	NUM
ejpam-3427	497	3	.	.	PUNCT
ejpam-3427	498	1	if	if	SCONJ
ejpam-3427	498	2	a	a	DET
ejpam-3427	498	3	=	=	NOUN
ejpam-3427	498	4	a1	a1	NOUN
ejpam-3427	498	5	×	×	PROPN
ejpam-3427	498	6	a2	a2	PROPN
ejpam-3427	498	7	×	×	NOUN
ejpam-3427	498	8	...	...	PUNCT
ejpam-3427	498	9	×	×	PROPN
ejpam-3427	498	10	an	an	PRON
ejpam-3427	498	11	and	and	CCONJ
ejpam-3427	498	12	b	b	X
ejpam-3427	498	13	=	=	SYM
ejpam-3427	498	14	b1	b1	NOUN
ejpam-3427	498	15	×	×	PROPN
ejpam-3427	498	16	b2	b2	NOUN
ejpam-3427	498	17	×	×	NOUN
ejpam-3427	498	18	...	...	PUNCT
ejpam-3427	499	1	×	×	NOUN
ejpam-3427	499	2	bn	bn	INTJ
ejpam-3427	499	3	are	be	AUX
ejpam-3427	499	4	two	two	NUM
ejpam-3427	499	5	la	la	ADJ
ejpam-3427	499	6	-	-	PUNCT
ejpam-3427	499	7	subrings	subring	NOUN
ejpam-3427	499	8	of	of	ADP
ejpam-3427	499	9	an	an	DET
ejpam-3427	499	10	la	la	ADJ
ejpam-3427	499	11	-	-	PUNCT
ejpam-3427	499	12	ring	ring	NOUN
ejpam-3427	499	13	r1	r1	NOUN
ejpam-3427	499	14	×r2	×r2	PROPN
ejpam-3427	499	15	×	×	NOUN
ejpam-3427	499	16	...	...	PUNCT
ejpam-3427	499	17	×rn	×rn	NOUN
ejpam-3427	499	18	,	,	PUNCT
ejpam-3427	499	19	then	then	ADV
ejpam-3427	499	20	their	their	PRON
ejpam-3427	499	21	intersection	intersection	NOUN
ejpam-3427	499	22	a	a	DET
ejpam-3427	499	23	∩b	∩b	NOUN
ejpam-3427	499	24	is	be	AUX
ejpam-3427	499	25	also	also	ADV
ejpam-3427	499	26	an	an	DET
ejpam-3427	499	27	la	la	ADV
ejpam-3427	499	28	-	-	PUNCT
ejpam-3427	499	29	subring	subring	NOUN
ejpam-3427	499	30	of	of	ADP
ejpam-3427	499	31	an	an	DET
ejpam-3427	499	32	la	la	ADJ
ejpam-3427	499	33	-	-	PUNCT
ejpam-3427	499	34	ring	ring	NOUN
ejpam-3427	499	35	r1	r1	NOUN
ejpam-3427	499	36	×r2	×r2	PROPN
ejpam-3427	499	37	×	×	NOUN
ejpam-3427	499	38	...	...	PUNCT
ejpam-3427	499	39	×rn	×rn	PROPN
ejpam-3427	499	40	.	.	PUNCT
ejpam-3427	500	1	k.	k.	PROPN
ejpam-3427	500	2	nasreen	nasreen	PROPN
ejpam-3427	500	3	/	/	SYM
ejpam-3427	500	4	eur	eur	PROPN
ejpam-3427	500	5	.	.	PUNCT
ejpam-3427	501	1	j.	j.	PROPN
ejpam-3427	501	2	pure	pure	PROPN
ejpam-3427	501	3	appl	appl	PROPN
ejpam-3427	501	4	.	.	PROPN
ejpam-3427	501	5	math	math	PROPN
ejpam-3427	501	6	,	,	PUNCT
ejpam-3427	501	7	12	12	NUM
ejpam-3427	501	8	(	(	PUNCT
ejpam-3427	501	9	2	2	NUM
ejpam-3427	501	10	)	)	PUNCT
ejpam-3427	501	11	(	(	PUNCT
ejpam-3427	501	12	2019	2019	NUM
ejpam-3427	501	13	)	)	PUNCT
ejpam-3427	501	14	,	,	PUNCT
ejpam-3427	501	15	622	622	NUM
ejpam-3427	501	16	-	-	SYM
ejpam-3427	501	17	648	648	NUM
ejpam-3427	501	18	639	639	NUM
ejpam-3427	501	19	proof	proof	NOUN
ejpam-3427	501	20	.	.	PUNCT
ejpam-3427	502	1	straight	straight	ADV
ejpam-3427	502	2	forward	forward	ADV
ejpam-3427	502	3	.	.	PUNCT
ejpam-3427	503	1	theorem	theorem	VERB
ejpam-3427	503	2	6	6	NUM
ejpam-3427	503	3	.	.	PUNCT
ejpam-3427	504	1	let	let	VERB
ejpam-3427	504	2	a	a	DET
ejpam-3427	504	3	=	=	PUNCT
ejpam-3427	504	4	a1×a2×	a1×a2×	PROPN
ejpam-3427	504	5	...	...	PUNCT
ejpam-3427	504	6	×an	×an	PROPN
ejpam-3427	504	7	and	and	CCONJ
ejpam-3427	504	8	b	b	PROPN
ejpam-3427	504	9	=	=	SYM
ejpam-3427	504	10	b×b2×	b×b2×	PROPN
ejpam-3427	504	11	...	...	PUNCT
ejpam-3427	504	12	×bn	×bn	AUX
ejpam-3427	504	13	be	be	AUX
ejpam-3427	504	14	two	two	NUM
ejpam-3427	504	15	la	la	ADJ
ejpam-3427	504	16	-	-	PUNCT
ejpam-3427	504	17	subrings	subring	NOUN
ejpam-3427	504	18	of	of	ADP
ejpam-3427	504	19	an	an	DET
ejpam-3427	504	20	la	la	ADJ
ejpam-3427	504	21	-	-	PUNCT
ejpam-3427	504	22	ring	ring	NOUN
ejpam-3427	504	23	r1×r2×	r1×r2×	NOUN
ejpam-3427	504	24	...	...	PUNCT
ejpam-3427	504	25	×rn	×rn	PROPN
ejpam-3427	504	26	.	.	PUNCT
ejpam-3427	505	1	then	then	ADV
ejpam-3427	505	2	a∩b	a∩b	PROPN
ejpam-3427	505	3	is	be	AUX
ejpam-3427	505	4	an	an	DET
ejpam-3427	505	5	la	la	NOUN
ejpam-3427	505	6	-	-	PUNCT
ejpam-3427	505	7	subring	subring	NOUN
ejpam-3427	505	8	of	of	ADP
ejpam-3427	505	9	an	an	DET
ejpam-3427	505	10	la	la	ADJ
ejpam-3427	505	11	-	-	PUNCT
ejpam-3427	505	12	ring	ring	NOUN
ejpam-3427	505	13	r1×r2×	r1×r2×	NOUN
ejpam-3427	505	14	...	...	PUNCT
ejpam-3427	505	15	×rn	×rn	VERB
ejpam-3427	505	16	if	if	SCONJ
ejpam-3427	505	17	and	and	CCONJ
ejpam-3427	505	18	only	only	ADV
ejpam-3427	505	19	if	if	SCONJ
ejpam-3427	505	20	the	the	DET
ejpam-3427	505	21	intuitionistic	intuitionistic	ADJ
ejpam-3427	505	22	anti	anti	ADJ
ejpam-3427	505	23	characteristic	characteristic	ADJ
ejpam-3427	505	24	function	function	NOUN
ejpam-3427	505	25	χz	χz	PROPN
ejpam-3427	505	26	=	=	SYM
ejpam-3427	505	27	〈	〈	PROPN
ejpam-3427	505	28	µχz	µχz	NOUN
ejpam-3427	505	29	,	,	PUNCT
ejpam-3427	505	30	γχz	γχz	INTJ
ejpam-3427	505	31	〉	〉	NOUN
ejpam-3427	505	32	of	of	ADP
ejpam-3427	505	33	z	z	PROPN
ejpam-3427	505	34	=	=	PRON
ejpam-3427	506	1	a∩b	a∩b	PROPN
ejpam-3427	506	2	is	be	AUX
ejpam-3427	506	3	an	an	DET
ejpam-3427	506	4	intuitionistic	intuitionistic	ADJ
ejpam-3427	506	5	anti	anti	ADJ
ejpam-3427	506	6	fuzzy	fuzzy	ADJ
ejpam-3427	506	7	normal	normal	ADJ
ejpam-3427	506	8	la	la	NOUN
ejpam-3427	506	9	-	-	PUNCT
ejpam-3427	506	10	subring	subring	NOUN
ejpam-3427	506	11	of	of	ADP
ejpam-3427	506	12	an	an	DET
ejpam-3427	506	13	la	la	ADJ
ejpam-3427	506	14	-	-	PUNCT
ejpam-3427	506	15	ring	ring	NOUN
ejpam-3427	506	16	r1	r1	NOUN
ejpam-3427	506	17	×r2	×r2	PROPN
ejpam-3427	506	18	×	×	NOUN
ejpam-3427	506	19	...	...	PUNCT
ejpam-3427	506	20	×rn	×rn	NOUN
ejpam-3427	506	21	.	.	PUNCT
ejpam-3427	507	1	proof	proof	NOUN
ejpam-3427	507	2	.	.	PUNCT
ejpam-3427	508	1	let	let	VERB
ejpam-3427	508	2	z	z	NOUN
ejpam-3427	508	3	=	=	PUNCT
ejpam-3427	508	4	a	a	DET
ejpam-3427	508	5	∩	∩	ADJ
ejpam-3427	508	6	b	b	NOUN
ejpam-3427	508	7	be	be	AUX
ejpam-3427	508	8	an	an	DET
ejpam-3427	508	9	la	la	NOUN
ejpam-3427	508	10	-	-	PUNCT
ejpam-3427	508	11	subring	subring	NOUN
ejpam-3427	508	12	of	of	ADP
ejpam-3427	508	13	an	an	DET
ejpam-3427	508	14	la	la	ADJ
ejpam-3427	508	15	-	-	PUNCT
ejpam-3427	508	16	ring	ring	NOUN
ejpam-3427	508	17	r1	r1	NOUN
ejpam-3427	508	18	×	×	NOUN
ejpam-3427	508	19	r2	r2	PROPN
ejpam-3427	508	20	×	×	NOUN
ejpam-3427	508	21	...	...	PUNCT
ejpam-3427	508	22	×	×	PROPN
ejpam-3427	508	23	rn	rn	PROPN
ejpam-3427	508	24	and	and	CCONJ
ejpam-3427	508	25	a	a	DET
ejpam-3427	508	26	=	=	X
ejpam-3427	508	27	(	(	PUNCT
ejpam-3427	508	28	a1	a1	PROPN
ejpam-3427	508	29	,	,	PUNCT
ejpam-3427	508	30	a2	a2	PROPN
ejpam-3427	508	31	,	,	PUNCT
ejpam-3427	508	32	...	...	PUNCT
ejpam-3427	508	33	,	,	PUNCT
ejpam-3427	508	34	an	an	PRON
ejpam-3427	508	35	)	)	PUNCT
ejpam-3427	508	36	,	,	PUNCT
ejpam-3427	508	37	b	b	X
ejpam-3427	508	38	=	=	SYM
ejpam-3427	508	39	(	(	PUNCT
ejpam-3427	508	40	b1	b1	PROPN
ejpam-3427	508	41	,	,	PUNCT
ejpam-3427	508	42	b1	b1	NOUN
ejpam-3427	508	43	,	,	PUNCT
ejpam-3427	508	44	...	...	PUNCT
ejpam-3427	508	45	,	,	PUNCT
ejpam-3427	508	46	bn	bn	X
ejpam-3427	508	47	)	)	PUNCT
ejpam-3427	508	48	∈	∈	PROPN
ejpam-3427	508	49	r1×r2×	r1×r2×	NOUN
ejpam-3427	508	50	...	...	PUNCT
ejpam-3427	508	51	×rn	×rn	VERB
ejpam-3427	508	52	.	.	PUNCT
ejpam-3427	509	1	if	if	SCONJ
ejpam-3427	509	2	a	a	PRON
ejpam-3427	509	3	,	,	PUNCT
ejpam-3427	509	4	b	b	PROPN
ejpam-3427	509	5	∈	∈	PROPN
ejpam-3427	509	6	z	z	NOUN
ejpam-3427	509	7	=	=	SYM
ejpam-3427	509	8	a∩b	a∩b	PROPN
ejpam-3427	509	9	,	,	PUNCT
ejpam-3427	509	10	then	then	ADV
ejpam-3427	509	11	by	by	ADP
ejpam-3427	509	12	definition	definition	NOUN
ejpam-3427	509	13	of	of	ADP
ejpam-3427	509	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	509	15	anti	anti	ADJ
ejpam-3427	509	16	characteristic	characteristic	ADJ
ejpam-3427	509	17	function	function	NOUN
ejpam-3427	509	18	µχz	µχz	NOUN
ejpam-3427	509	19	(	(	PUNCT
ejpam-3427	509	20	a	a	X
ejpam-3427	509	21	)	)	PUNCT
ejpam-3427	509	22	=	=	SYM
ejpam-3427	509	23	0	0	PUNCT
ejpam-3427	510	1	=	=	SYM
ejpam-3427	510	2	µχz	µχz	NOUN
ejpam-3427	510	3	(	(	PUNCT
ejpam-3427	510	4	b	b	NOUN
ejpam-3427	510	5	)	)	PUNCT
ejpam-3427	510	6	and	and	CCONJ
ejpam-3427	510	7	γχz	γχz	INTJ
ejpam-3427	510	8	(	(	PUNCT
ejpam-3427	510	9	a	a	X
ejpam-3427	510	10	)	)	PUNCT
ejpam-3427	510	11	=	=	SYM
ejpam-3427	510	12	1	1	NUM
ejpam-3427	510	13	=	=	SYM
ejpam-3427	510	14	γχz	γχz	NOUN
ejpam-3427	510	15	(	(	PUNCT
ejpam-3427	510	16	b	b	NOUN
ejpam-3427	510	17	)	)	PUNCT
ejpam-3427	510	18	.	.	PUNCT
ejpam-3427	511	1	since	since	SCONJ
ejpam-3427	511	2	a−	a−	PROPN
ejpam-3427	511	3	b	b	PROPN
ejpam-3427	511	4	and	and	CCONJ
ejpam-3427	511	5	ab	ab	PROPN
ejpam-3427	511	6	∈	∈	PROPN
ejpam-3427	511	7	z	z	PROPN
ejpam-3427	511	8	,	,	PUNCT
ejpam-3427	511	9	z	z	NOUN
ejpam-3427	511	10	being	be	AUX
ejpam-3427	511	11	an	an	DET
ejpam-3427	511	12	la	la	ADV
ejpam-3427	511	13	-	-	PUNCT
ejpam-3427	511	14	subring	subring	NOUN
ejpam-3427	511	15	.	.	PUNCT
ejpam-3427	512	1	this	this	PRON
ejpam-3427	512	2	means	mean	VERB
ejpam-3427	512	3	that	that	SCONJ
ejpam-3427	512	4	µχz	µχz	NOUN
ejpam-3427	512	5	(	(	PUNCT
ejpam-3427	512	6	a−	a−	PROPN
ejpam-3427	512	7	b	b	NOUN
ejpam-3427	512	8	)	)	PUNCT
ejpam-3427	512	9	=	=	SYM
ejpam-3427	512	10	0	0	PUNCT
ejpam-3427	513	1	=	=	SYM
ejpam-3427	513	2	0	0	NUM
ejpam-3427	514	1	∨	∨	NUM
ejpam-3427	514	2	0	0	NUM
ejpam-3427	515	1	=	=	SYM
ejpam-3427	515	2	µχz	µχz	NOUN
ejpam-3427	515	3	(	(	PUNCT
ejpam-3427	515	4	a	a	X
ejpam-3427	515	5	)	)	PUNCT
ejpam-3427	515	6	∨	∨	NUM
ejpam-3427	515	7	µχz	µχz	NOUN
ejpam-3427	515	8	(	(	PUNCT
ejpam-3427	515	9	b	b	NOUN
ejpam-3427	515	10	)	)	PUNCT
ejpam-3427	515	11	,	,	PUNCT
ejpam-3427	515	12	µχz	µχz	NOUN
ejpam-3427	515	13	(	(	PUNCT
ejpam-3427	515	14	ab	ab	NOUN
ejpam-3427	515	15	)	)	PUNCT
ejpam-3427	515	16	=	=	SYM
ejpam-3427	515	17	0	0	PUNCT
ejpam-3427	516	1	=	=	SYM
ejpam-3427	516	2	0	0	NUM
ejpam-3427	517	1	∨	∨	NUM
ejpam-3427	517	2	0	0	NUM
ejpam-3427	517	3	=	=	SYM
ejpam-3427	517	4	µχz	µχz	NOUN
ejpam-3427	517	5	(	(	PUNCT
ejpam-3427	517	6	a	a	X
ejpam-3427	517	7	)	)	PUNCT
ejpam-3427	517	8	∨	∨	NUM
ejpam-3427	517	9	µχz	µχz	NOUN
ejpam-3427	517	10	(	(	PUNCT
ejpam-3427	517	11	b	b	NOUN
ejpam-3427	517	12	)	)	PUNCT
ejpam-3427	517	13	,	,	PUNCT
ejpam-3427	517	14	γχz	γχz	INTJ
ejpam-3427	517	15	(	(	PUNCT
ejpam-3427	517	16	a−	a−	PROPN
ejpam-3427	517	17	b	b	NOUN
ejpam-3427	517	18	)	)	PUNCT
ejpam-3427	517	19	=	=	SYM
ejpam-3427	517	20	1	1	NUM
ejpam-3427	517	21	=	=	SYM
ejpam-3427	517	22	1	1	NUM
ejpam-3427	517	23	∧	∧	PROPN
ejpam-3427	517	24	1	1	NUM
ejpam-3427	517	25	=	=	SYM
ejpam-3427	517	26	γχz	γχz	NOUN
ejpam-3427	517	27	(	(	PUNCT
ejpam-3427	517	28	a	a	X
ejpam-3427	517	29	)	)	PUNCT
ejpam-3427	517	30	∧	∧	NOUN
ejpam-3427	517	31	γχz	γχz	NOUN
ejpam-3427	517	32	(	(	PUNCT
ejpam-3427	517	33	b	b	NOUN
ejpam-3427	517	34	)	)	PUNCT
ejpam-3427	517	35	,	,	PUNCT
ejpam-3427	517	36	γχz	γχz	INTJ
ejpam-3427	517	37	(	(	PUNCT
ejpam-3427	517	38	ab	ab	NOUN
ejpam-3427	517	39	)	)	PUNCT
ejpam-3427	517	40	=	=	SYM
ejpam-3427	517	41	1	1	NUM
ejpam-3427	517	42	=	=	SYM
ejpam-3427	517	43	1	1	NUM
ejpam-3427	517	44	∧	∧	PROPN
ejpam-3427	517	45	1	1	NUM
ejpam-3427	517	46	=	=	SYM
ejpam-3427	517	47	γχz	γχz	NOUN
ejpam-3427	517	48	(	(	PUNCT
ejpam-3427	517	49	a	a	X
ejpam-3427	517	50	)	)	PUNCT
ejpam-3427	517	51	∧	∧	NOUN
ejpam-3427	517	52	γχz	γχz	NOUN
ejpam-3427	517	53	(	(	PUNCT
ejpam-3427	517	54	b	b	NOUN
ejpam-3427	517	55	)	)	PUNCT
ejpam-3427	517	56	.	.	PUNCT
ejpam-3427	518	1	thus	thus	ADV
ejpam-3427	518	2	µχz	µχz	NOUN
ejpam-3427	518	3	(	(	PUNCT
ejpam-3427	518	4	a−	a−	PROPN
ejpam-3427	518	5	b	b	NOUN
ejpam-3427	518	6	)	)	PUNCT
ejpam-3427	518	7	≤	≤	NUM
ejpam-3427	518	8	max{µχz	max{µχz	NOUN
ejpam-3427	518	9	(	(	PUNCT
ejpam-3427	518	10	a	a	X
ejpam-3427	518	11	)	)	PUNCT
ejpam-3427	518	12	,	,	PUNCT
ejpam-3427	518	13	µχz	µχz	NOUN
ejpam-3427	518	14	(	(	PUNCT
ejpam-3427	518	15	b	b	NOUN
ejpam-3427	518	16	)	)	PUNCT
ejpam-3427	518	17	}	}	PUNCT
ejpam-3427	518	18	,	,	PUNCT
ejpam-3427	518	19	µχz	µχz	NOUN
ejpam-3427	518	20	(	(	PUNCT
ejpam-3427	518	21	ab	ab	NOUN
ejpam-3427	518	22	)	)	PUNCT
ejpam-3427	518	23	≤	≤	NOUN
ejpam-3427	518	24	max{µχz	max{µχz	NOUN
ejpam-3427	518	25	(	(	PUNCT
ejpam-3427	518	26	a	a	X
ejpam-3427	518	27	)	)	PUNCT
ejpam-3427	518	28	,	,	PUNCT
ejpam-3427	518	29	µχz	µχz	NOUN
ejpam-3427	518	30	(	(	PUNCT
ejpam-3427	518	31	b	b	NOUN
ejpam-3427	518	32	)	)	PUNCT
ejpam-3427	518	33	}	}	PUNCT
ejpam-3427	518	34	,	,	PUNCT
ejpam-3427	518	35	γχz	γχz	INTJ
ejpam-3427	518	36	(	(	PUNCT
ejpam-3427	518	37	a−	a−	PROPN
ejpam-3427	518	38	b	b	PROPN
ejpam-3427	518	39	)	)	PUNCT
ejpam-3427	518	40	≥	≥	NOUN
ejpam-3427	518	41	min{γχz	min{γχz	NOUN
ejpam-3427	518	42	(	(	PUNCT
ejpam-3427	518	43	a	a	NOUN
ejpam-3427	518	44	)	)	PUNCT
ejpam-3427	518	45	,	,	PUNCT
ejpam-3427	518	46	γχz	γχz	INTJ
ejpam-3427	518	47	(	(	PUNCT
ejpam-3427	518	48	b	b	NOUN
ejpam-3427	518	49	)	)	PUNCT
ejpam-3427	518	50	}	}	PUNCT
ejpam-3427	518	51	,	,	PUNCT
ejpam-3427	518	52	γχz	γχz	INTJ
ejpam-3427	518	53	(	(	PUNCT
ejpam-3427	518	54	ab	ab	PROPN
ejpam-3427	518	55	)	)	PUNCT
ejpam-3427	518	56	≥	≥	NOUN
ejpam-3427	518	57	min{γχz	min{γχz	NOUN
ejpam-3427	518	58	(	(	PUNCT
ejpam-3427	518	59	a	a	NOUN
ejpam-3427	518	60	)	)	PUNCT
ejpam-3427	518	61	,	,	PUNCT
ejpam-3427	518	62	γχz	γχz	INTJ
ejpam-3427	518	63	(	(	PUNCT
ejpam-3427	518	64	b	b	NOUN
ejpam-3427	518	65	)	)	PUNCT
ejpam-3427	518	66	}	}	PUNCT
ejpam-3427	518	67	.	.	PUNCT
ejpam-3427	519	1	as	as	ADP
ejpam-3427	519	2	ab	ab	PROPN
ejpam-3427	519	3	and	and	CCONJ
ejpam-3427	519	4	ba	ba	PROPN
ejpam-3427	519	5	∈	∈	PROPN
ejpam-3427	519	6	z	z	PROPN
ejpam-3427	519	7	,	,	PUNCT
ejpam-3427	519	8	this	this	PRON
ejpam-3427	519	9	implies	imply	VERB
ejpam-3427	519	10	that	that	SCONJ
ejpam-3427	519	11	µχz	µχz	NOUN
ejpam-3427	519	12	(	(	PUNCT
ejpam-3427	519	13	ab	ab	NOUN
ejpam-3427	519	14	)	)	PUNCT
ejpam-3427	519	15	=	=	SYM
ejpam-3427	519	16	0	0	NUM
ejpam-3427	519	17	=	=	SYM
ejpam-3427	519	18	µχz	µχz	NOUN
ejpam-3427	519	19	(	(	PUNCT
ejpam-3427	519	20	ba	ba	NOUN
ejpam-3427	519	21	)	)	PUNCT
ejpam-3427	519	22	and	and	CCONJ
ejpam-3427	519	23	γχz	γχz	INTJ
ejpam-3427	519	24	(	(	PUNCT
ejpam-3427	519	25	ab	ab	NOUN
ejpam-3427	519	26	)	)	PUNCT
ejpam-3427	519	27	=	=	SYM
ejpam-3427	519	28	1	1	NUM
ejpam-3427	519	29	=	=	SYM
ejpam-3427	519	30	γχz	γχz	NOUN
ejpam-3427	519	31	(	(	PUNCT
ejpam-3427	519	32	ba	ba	NOUN
ejpam-3427	519	33	)	)	PUNCT
ejpam-3427	519	34	,	,	PUNCT
ejpam-3427	519	35	i.e.	i.e.	X
ejpam-3427	519	36	,	,	PUNCT
ejpam-3427	519	37	µχz	µχz	NOUN
ejpam-3427	519	38	(	(	PUNCT
ejpam-3427	519	39	ab	ab	NOUN
ejpam-3427	519	40	)	)	PUNCT
ejpam-3427	520	1	=	=	PRON
ejpam-3427	520	2	µχz	µχz	NOUN
ejpam-3427	520	3	(	(	PUNCT
ejpam-3427	520	4	ba	ba	NOUN
ejpam-3427	520	5	)	)	PUNCT
ejpam-3427	520	6	and	and	CCONJ
ejpam-3427	520	7	γχz	γχz	INTJ
ejpam-3427	520	8	(	(	PUNCT
ejpam-3427	520	9	ab	ab	NOUN
ejpam-3427	520	10	)	)	PUNCT
ejpam-3427	520	11	=	=	SYM
ejpam-3427	521	1	γχz	γχz	NOUN
ejpam-3427	521	2	(	(	PUNCT
ejpam-3427	521	3	ba	ba	NOUN
ejpam-3427	521	4	)	)	PUNCT
ejpam-3427	521	5	.	.	PUNCT
ejpam-3427	522	1	similarly	similarly	ADV
ejpam-3427	522	2	,	,	PUNCT
ejpam-3427	522	3	we	we	PRON
ejpam-3427	522	4	have	have	VERB
ejpam-3427	522	5	µχz	µχz	NOUN
ejpam-3427	522	6	(	(	PUNCT
ejpam-3427	522	7	a−	a−	PROPN
ejpam-3427	522	8	b	b	NOUN
ejpam-3427	522	9	)	)	PUNCT
ejpam-3427	522	10	≤	≤	NUM
ejpam-3427	522	11	max{µχz	max{µχz	NOUN
ejpam-3427	522	12	(	(	PUNCT
ejpam-3427	522	13	a	a	X
ejpam-3427	522	14	)	)	PUNCT
ejpam-3427	522	15	,	,	PUNCT
ejpam-3427	522	16	µχz	µχz	NOUN
ejpam-3427	522	17	(	(	PUNCT
ejpam-3427	522	18	b	b	NOUN
ejpam-3427	522	19	)	)	PUNCT
ejpam-3427	522	20	}	}	PUNCT
ejpam-3427	522	21	,	,	PUNCT
ejpam-3427	522	22	µχz	µχz	NOUN
ejpam-3427	522	23	(	(	PUNCT
ejpam-3427	522	24	ab	ab	NOUN
ejpam-3427	522	25	)	)	PUNCT
ejpam-3427	522	26	≤	≤	NOUN
ejpam-3427	522	27	max{µχz	max{µχz	NOUN
ejpam-3427	522	28	(	(	PUNCT
ejpam-3427	522	29	a	a	X
ejpam-3427	522	30	)	)	PUNCT
ejpam-3427	522	31	,	,	PUNCT
ejpam-3427	522	32	µχz	µχz	NOUN
ejpam-3427	522	33	(	(	PUNCT
ejpam-3427	522	34	b	b	NOUN
ejpam-3427	522	35	)	)	PUNCT
ejpam-3427	522	36	}	}	PUNCT
ejpam-3427	522	37	,	,	PUNCT
ejpam-3427	522	38	γχz	γχz	INTJ
ejpam-3427	522	39	(	(	PUNCT
ejpam-3427	522	40	a−	a−	PROPN
ejpam-3427	522	41	b	b	PROPN
ejpam-3427	522	42	)	)	PUNCT
ejpam-3427	522	43	≥	≥	NOUN
ejpam-3427	522	44	min{γχz	min{γχz	NOUN
ejpam-3427	522	45	(	(	PUNCT
ejpam-3427	522	46	a	a	NOUN
ejpam-3427	522	47	)	)	PUNCT
ejpam-3427	522	48	,	,	PUNCT
ejpam-3427	522	49	γχz	γχz	INTJ
ejpam-3427	522	50	(	(	PUNCT
ejpam-3427	522	51	b	b	NOUN
ejpam-3427	522	52	)	)	PUNCT
ejpam-3427	522	53	}	}	PUNCT
ejpam-3427	522	54	,	,	PUNCT
ejpam-3427	522	55	γχz	γχz	INTJ
ejpam-3427	522	56	(	(	PUNCT
ejpam-3427	522	57	ab	ab	PROPN
ejpam-3427	522	58	)	)	PUNCT
ejpam-3427	522	59	≥	≥	NOUN
ejpam-3427	522	60	min{γχz	min{γχz	NOUN
ejpam-3427	522	61	(	(	PUNCT
ejpam-3427	522	62	a	a	NOUN
ejpam-3427	522	63	)	)	PUNCT
ejpam-3427	522	64	,	,	PUNCT
ejpam-3427	522	65	γχz	γχz	INTJ
ejpam-3427	522	66	(	(	PUNCT
ejpam-3427	522	67	b	b	NOUN
ejpam-3427	522	68	)	)	PUNCT
ejpam-3427	522	69	}	}	PUNCT
ejpam-3427	522	70	,	,	PUNCT
ejpam-3427	522	71	γχz	γχz	INTJ
ejpam-3427	522	72	(	(	PUNCT
ejpam-3427	522	73	ab	ab	NOUN
ejpam-3427	522	74	)	)	PUNCT
ejpam-3427	522	75	=	=	SYM
ejpam-3427	523	1	γχz	γχz	NOUN
ejpam-3427	523	2	(	(	PUNCT
ejpam-3427	523	3	ba	ba	PROPN
ejpam-3427	523	4	)	)	PUNCT
ejpam-3427	523	5	,	,	PUNCT
ejpam-3427	523	6	γχz	γχz	INTJ
ejpam-3427	523	7	(	(	PUNCT
ejpam-3427	523	8	ab	ab	NOUN
ejpam-3427	523	9	)	)	PUNCT
ejpam-3427	523	10	=	=	SYM
ejpam-3427	524	1	γχz	γχz	NOUN
ejpam-3427	524	2	(	(	PUNCT
ejpam-3427	524	3	ba	ba	NOUN
ejpam-3427	524	4	)	)	PUNCT
ejpam-3427	524	5	,	,	PUNCT
ejpam-3427	524	6	when	when	SCONJ
ejpam-3427	524	7	a	a	DET
ejpam-3427	524	8	,	,	PUNCT
ejpam-3427	524	9	b	b	PROPN
ejpam-3427	524	10	/∈	/∈	PUNCT
ejpam-3427	524	11	z.	z.	PROPN
ejpam-3427	525	1	hence	hence	ADV
ejpam-3427	525	2	the	the	DET
ejpam-3427	525	3	intuitionistic	intuitionistic	ADJ
ejpam-3427	525	4	anti	anti	ADJ
ejpam-3427	525	5	characteristic	characteristic	ADJ
ejpam-3427	525	6	function	function	NOUN
ejpam-3427	525	7	χz	χz	PROPN
ejpam-3427	525	8	=	=	SYM
ejpam-3427	525	9	〈	〈	PROPN
ejpam-3427	525	10	µχz	µχz	NOUN
ejpam-3427	525	11	,	,	PUNCT
ejpam-3427	525	12	γχz	γχz	INTJ
ejpam-3427	525	13	〉	〉	PROPN
ejpam-3427	525	14	of	of	ADP
ejpam-3427	525	15	z	z	PROPN
ejpam-3427	525	16	is	be	AUX
ejpam-3427	525	17	an	an	DET
ejpam-3427	525	18	intuitionistic	intuitionistic	ADJ
ejpam-3427	525	19	anti	anti	ADJ
ejpam-3427	525	20	fuzzy	fuzzy	ADJ
ejpam-3427	525	21	normal	normal	ADJ
ejpam-3427	525	22	la	la	NOUN
ejpam-3427	525	23	-	-	PUNCT
ejpam-3427	525	24	subring	subring	NOUN
ejpam-3427	525	25	of	of	ADP
ejpam-3427	525	26	an	an	DET
ejpam-3427	525	27	la	la	ADJ
ejpam-3427	525	28	-	-	PUNCT
ejpam-3427	525	29	ring	ring	NOUN
ejpam-3427	525	30	r1	r1	NOUN
ejpam-3427	525	31	×r2	×r2	PROPN
ejpam-3427	525	32	×	×	NOUN
ejpam-3427	525	33	...	...	PUNCT
ejpam-3427	525	34	×rn	×rn	NOUN
ejpam-3427	525	35	.	.	PUNCT
ejpam-3427	526	1	conversely	conversely	ADV
ejpam-3427	526	2	,	,	PUNCT
ejpam-3427	526	3	assume	assume	VERB
ejpam-3427	526	4	that	that	SCONJ
ejpam-3427	526	5	the	the	DET
ejpam-3427	526	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	526	7	anti	anti	ADJ
ejpam-3427	526	8	characteristic	characteristic	ADJ
ejpam-3427	526	9	function	function	NOUN
ejpam-3427	526	10	χz	χz	PROPN
ejpam-3427	526	11	=	=	SYM
ejpam-3427	526	12	〈	〈	PROPN
ejpam-3427	526	13	µχz	µχz	NOUN
ejpam-3427	526	14	,	,	PUNCT
ejpam-3427	526	15	γχz	γχz	INTJ
ejpam-3427	526	16	〉	〉	NOUN
ejpam-3427	526	17	of	of	ADP
ejpam-3427	526	18	z	z	PROPN
ejpam-3427	526	19	=	=	PUNCT
ejpam-3427	526	20	a	a	DET
ejpam-3427	526	21	∩	∩	ADJ
ejpam-3427	526	22	b	b	NOUN
ejpam-3427	526	23	is	be	AUX
ejpam-3427	526	24	an	an	DET
ejpam-3427	526	25	intuitionistic	intuitionistic	ADJ
ejpam-3427	526	26	anti	anti	ADJ
ejpam-3427	526	27	fuzzy	fuzzy	ADJ
ejpam-3427	526	28	normal	normal	ADJ
ejpam-3427	526	29	la	la	NOUN
ejpam-3427	526	30	-	-	PUNCT
ejpam-3427	526	31	subring	subring	NOUN
ejpam-3427	526	32	of	of	ADP
ejpam-3427	526	33	an	an	DET
ejpam-3427	526	34	la	la	ADJ
ejpam-3427	526	35	-	-	PUNCT
ejpam-3427	526	36	ring	ring	NOUN
ejpam-3427	526	37	r1	r1	NOUN
ejpam-3427	526	38	×	×	NOUN
ejpam-3427	526	39	r2	r2	PROPN
ejpam-3427	526	40	×	×	NOUN
ejpam-3427	526	41	...	...	PUNCT
ejpam-3427	526	42	×	×	PROPN
ejpam-3427	526	43	rn	rn	PROPN
ejpam-3427	526	44	.	.	PROPN
ejpam-3427	527	1	let	let	VERB
ejpam-3427	527	2	a	a	DET
ejpam-3427	527	3	,	,	PUNCT
ejpam-3427	527	4	b	b	X
ejpam-3427	527	5	∈	∈	PROPN
ejpam-3427	527	6	z	z	NOUN
ejpam-3427	527	7	=	=	PUNCT
ejpam-3427	527	8	a	a	DET
ejpam-3427	527	9	∩	∩	ADJ
ejpam-3427	527	10	b	b	NOUN
ejpam-3427	527	11	,	,	PUNCT
ejpam-3427	527	12	by	by	ADP
ejpam-3427	527	13	definition	definition	NOUN
ejpam-3427	527	14	,	,	PUNCT
ejpam-3427	527	15	we	we	PRON
ejpam-3427	527	16	have	have	VERB
ejpam-3427	527	17	µχz	µχz	NOUN
ejpam-3427	527	18	(	(	PUNCT
ejpam-3427	527	19	a	a	X
ejpam-3427	527	20	)	)	PUNCT
ejpam-3427	528	1	=	=	SYM
ejpam-3427	528	2	0	0	PUNCT
ejpam-3427	529	1	=	=	SYM
ejpam-3427	529	2	µχz	µχz	NOUN
ejpam-3427	529	3	(	(	PUNCT
ejpam-3427	529	4	b	b	NOUN
ejpam-3427	529	5	)	)	PUNCT
ejpam-3427	529	6	and	and	CCONJ
ejpam-3427	529	7	γχz	γχz	INTJ
ejpam-3427	529	8	(	(	PUNCT
ejpam-3427	529	9	a	a	X
ejpam-3427	529	10	)	)	PUNCT
ejpam-3427	529	11	=	=	SYM
ejpam-3427	529	12	1	1	NUM
ejpam-3427	529	13	=	=	SYM
ejpam-3427	529	14	γχz	γχz	NOUN
ejpam-3427	529	15	(	(	PUNCT
ejpam-3427	529	16	b	b	NOUN
ejpam-3427	529	17	)	)	PUNCT
ejpam-3427	529	18	.	.	PUNCT
ejpam-3427	530	1	by	by	ADP
ejpam-3427	530	2	our	our	PRON
ejpam-3427	530	3	assumption	assumption	NOUN
ejpam-3427	530	4	µχz	µχz	NOUN
ejpam-3427	530	5	(	(	PUNCT
ejpam-3427	530	6	a−	a−	PROPN
ejpam-3427	530	7	b	b	NOUN
ejpam-3427	530	8	)	)	PUNCT
ejpam-3427	530	9	≤	≤	NOUN
ejpam-3427	530	10	µχz	µχz	NOUN
ejpam-3427	530	11	(	(	PUNCT
ejpam-3427	530	12	a	a	X
ejpam-3427	530	13	)	)	PUNCT
ejpam-3427	530	14	∨	∨	NUM
ejpam-3427	530	15	µχz	µχz	NOUN
ejpam-3427	530	16	(	(	PUNCT
ejpam-3427	530	17	b	b	NOUN
ejpam-3427	530	18	)	)	PUNCT
ejpam-3427	530	19	=	=	SYM
ejpam-3427	530	20	0	0	NUM
ejpam-3427	531	1	∨	∨	NUM
ejpam-3427	531	2	0	0	NUM
ejpam-3427	531	3	=	=	SYM
ejpam-3427	531	4	0	0	NUM
ejpam-3427	531	5	,	,	PUNCT
ejpam-3427	531	6	µχz	µχz	NOUN
ejpam-3427	531	7	(	(	PUNCT
ejpam-3427	531	8	ab	ab	NOUN
ejpam-3427	531	9	)	)	PUNCT
ejpam-3427	531	10	≤	≤	NOUN
ejpam-3427	531	11	µχz	µχz	NOUN
ejpam-3427	531	12	(	(	PUNCT
ejpam-3427	531	13	a	a	X
ejpam-3427	531	14	)	)	PUNCT
ejpam-3427	531	15	∨	∨	NUM
ejpam-3427	531	16	µχz	µχz	NOUN
ejpam-3427	531	17	(	(	PUNCT
ejpam-3427	531	18	b	b	NOUN
ejpam-3427	531	19	)	)	PUNCT
ejpam-3427	531	20	=	=	SYM
ejpam-3427	531	21	0	0	NUM
ejpam-3427	531	22	∨	∨	NUM
ejpam-3427	531	23	0	0	NUM
ejpam-3427	531	24	=	=	SYM
ejpam-3427	531	25	0	0	NUM
ejpam-3427	531	26	,	,	PUNCT
ejpam-3427	531	27	γχz	γχz	INTJ
ejpam-3427	531	28	(	(	PUNCT
ejpam-3427	531	29	a−	a−	PROPN
ejpam-3427	531	30	b	b	PROPN
ejpam-3427	531	31	)	)	PUNCT
ejpam-3427	531	32	≥	≥	NOUN
ejpam-3427	531	33	γχz	γχz	INTJ
ejpam-3427	531	34	(	(	PUNCT
ejpam-3427	531	35	a	a	X
ejpam-3427	531	36	)	)	PUNCT
ejpam-3427	531	37	∧	∧	NOUN
ejpam-3427	531	38	γχz	γχz	NOUN
ejpam-3427	531	39	(	(	PUNCT
ejpam-3427	531	40	b	b	NOUN
ejpam-3427	531	41	)	)	PUNCT
ejpam-3427	531	42	=	=	SYM
ejpam-3427	531	43	1	1	NUM
ejpam-3427	531	44	∧	∧	PROPN
ejpam-3427	531	45	1	1	NUM
ejpam-3427	531	46	=	=	SYM
ejpam-3427	531	47	1	1	NUM
ejpam-3427	531	48	,	,	PUNCT
ejpam-3427	531	49	γχz	γχz	INTJ
ejpam-3427	531	50	(	(	PUNCT
ejpam-3427	531	51	ab	ab	PROPN
ejpam-3427	531	52	)	)	PUNCT
ejpam-3427	531	53	≥	≥	NOUN
ejpam-3427	531	54	γχz	γχz	INTJ
ejpam-3427	531	55	(	(	PUNCT
ejpam-3427	531	56	a	a	X
ejpam-3427	531	57	)	)	PUNCT
ejpam-3427	531	58	∧	∧	NOUN
ejpam-3427	531	59	γχz	γχz	NOUN
ejpam-3427	531	60	(	(	PUNCT
ejpam-3427	531	61	b	b	NOUN
ejpam-3427	531	62	)	)	PUNCT
ejpam-3427	531	63	=	=	SYM
ejpam-3427	531	64	1	1	NUM
ejpam-3427	531	65	∧	∧	PROPN
ejpam-3427	531	66	1	1	NUM
ejpam-3427	531	67	=	=	SYM
ejpam-3427	531	68	1	1	NUM
ejpam-3427	531	69	.	.	PUNCT
ejpam-3427	531	70	thus	thus	ADV
ejpam-3427	531	71	µχz	µχz	NOUN
ejpam-3427	531	72	(	(	PUNCT
ejpam-3427	531	73	a−	a−	PROPN
ejpam-3427	531	74	b	b	NOUN
ejpam-3427	531	75	)	)	PUNCT
ejpam-3427	531	76	=	=	SYM
ejpam-3427	531	77	0	0	PUNCT
ejpam-3427	532	1	=	=	SYM
ejpam-3427	532	2	µχz	µχz	NOUN
ejpam-3427	532	3	(	(	PUNCT
ejpam-3427	532	4	ab	ab	NOUN
ejpam-3427	532	5	)	)	PUNCT
ejpam-3427	532	6	and	and	CCONJ
ejpam-3427	532	7	γχz	γχz	INTJ
ejpam-3427	532	8	(	(	PUNCT
ejpam-3427	532	9	a−	a−	PROPN
ejpam-3427	532	10	b	b	NOUN
ejpam-3427	532	11	)	)	PUNCT
ejpam-3427	532	12	=	=	SYM
ejpam-3427	532	13	1	1	NUM
ejpam-3427	532	14	=	=	SYM
ejpam-3427	532	15	γχz	γχz	NOUN
ejpam-3427	532	16	(	(	PUNCT
ejpam-3427	532	17	ab	ab	NOUN
ejpam-3427	532	18	)	)	PUNCT
ejpam-3427	532	19	,	,	PUNCT
ejpam-3427	532	20	i.e.	i.e.	X
ejpam-3427	532	21	,	,	PUNCT
ejpam-3427	532	22	a−	a−	PROPN
ejpam-3427	532	23	b	b	PROPN
ejpam-3427	532	24	and	and	CCONJ
ejpam-3427	532	25	ab	ab	PROPN
ejpam-3427	532	26	∈	∈	PROPN
ejpam-3427	532	27	z.	z.	PROPN
ejpam-3427	533	1	hence	hence	ADV
ejpam-3427	533	2	z	z	PROPN
ejpam-3427	533	3	is	be	AUX
ejpam-3427	533	4	an	an	DET
ejpam-3427	533	5	la	la	NOUN
ejpam-3427	533	6	-	-	PUNCT
ejpam-3427	533	7	subring	subring	NOUN
ejpam-3427	533	8	of	of	ADP
ejpam-3427	533	9	an	an	DET
ejpam-3427	533	10	la	la	ADJ
ejpam-3427	533	11	-	-	PUNCT
ejpam-3427	533	12	ring	ring	NOUN
ejpam-3427	533	13	r1	r1	NOUN
ejpam-3427	533	14	×r2	×r2	PROPN
ejpam-3427	533	15	×	×	NOUN
ejpam-3427	533	16	...	...	PUNCT
ejpam-3427	533	17	×rn	×rn	PROPN
ejpam-3427	533	18	.	.	PUNCT
ejpam-3427	534	1	k.	k.	PROPN
ejpam-3427	534	2	nasreen	nasreen	PROPN
ejpam-3427	534	3	/	/	SYM
ejpam-3427	534	4	eur	eur	PROPN
ejpam-3427	534	5	.	.	PUNCT
ejpam-3427	535	1	j.	j.	PROPN
ejpam-3427	535	2	pure	pure	PROPN
ejpam-3427	535	3	appl	appl	PROPN
ejpam-3427	535	4	.	.	PROPN
ejpam-3427	535	5	math	math	PROPN
ejpam-3427	535	6	,	,	PUNCT
ejpam-3427	535	7	12	12	NUM
ejpam-3427	535	8	(	(	PUNCT
ejpam-3427	535	9	2	2	NUM
ejpam-3427	535	10	)	)	PUNCT
ejpam-3427	535	11	(	(	PUNCT
ejpam-3427	535	12	2019	2019	NUM
ejpam-3427	535	13	)	)	PUNCT
ejpam-3427	535	14	,	,	PUNCT
ejpam-3427	535	15	622	622	NUM
ejpam-3427	535	16	-	-	SYM
ejpam-3427	535	17	648	648	NUM
ejpam-3427	535	18	640	640	NUM
ejpam-3427	535	19	corollary	corollary	ADJ
ejpam-3427	535	20	4	4	NUM
ejpam-3427	535	21	.	.	PUNCT
ejpam-3427	536	1	let	let	VERB
ejpam-3427	536	2	{	{	PUNCT
ejpam-3427	536	3	bi}i∈i	bi}i∈i	NOUN
ejpam-3427	536	4	=	=	X
ejpam-3427	536	5	{	{	PUNCT
ejpam-3427	536	6	ai1	ai1	VERB
ejpam-3427	536	7	×ai2	×ai2	NOUN
ejpam-3427	536	8	×	×	PROPN
ejpam-3427	536	9	...	...	PUNCT
ejpam-3427	537	1	×ain}i∈i	×ain}i∈i	NOUN
ejpam-3427	537	2	be	be	VERB
ejpam-3427	537	3	a	a	DET
ejpam-3427	537	4	family	family	NOUN
ejpam-3427	537	5	of	of	ADP
ejpam-3427	537	6	la	la	ADJ
ejpam-3427	537	7	-	-	PUNCT
ejpam-3427	537	8	subrings	subring	NOUN
ejpam-3427	537	9	of	of	ADP
ejpam-3427	537	10	an	an	DET
ejpam-3427	537	11	la	la	ADJ
ejpam-3427	537	12	-	-	PUNCT
ejpam-3427	537	13	ring	ring	NOUN
ejpam-3427	537	14	r1	r1	NOUN
ejpam-3427	537	15	×	×	NOUN
ejpam-3427	537	16	r2	r2	PROPN
ejpam-3427	537	17	×	×	NOUN
ejpam-3427	537	18	...	...	PUNCT
ejpam-3427	537	19	×	×	PROPN
ejpam-3427	537	20	rn	rn	PROPN
ejpam-3427	537	21	.	.	PROPN
ejpam-3427	538	1	then	then	ADV
ejpam-3427	538	2	b	b	X
ejpam-3427	538	3	=	=	NOUN
ejpam-3427	538	4	∩bi	∩bi	NOUN
ejpam-3427	538	5	is	be	AUX
ejpam-3427	538	6	an	an	DET
ejpam-3427	538	7	la	la	NOUN
ejpam-3427	538	8	-	-	PUNCT
ejpam-3427	538	9	subring	subring	NOUN
ejpam-3427	538	10	of	of	ADP
ejpam-3427	538	11	an	an	DET
ejpam-3427	538	12	la	la	ADJ
ejpam-3427	538	13	-	-	PUNCT
ejpam-3427	538	14	ring	ring	NOUN
ejpam-3427	538	15	r1	r1	NOUN
ejpam-3427	538	16	×	×	NOUN
ejpam-3427	538	17	r2	r2	PROPN
ejpam-3427	538	18	×	×	NOUN
ejpam-3427	538	19	...	...	PUNCT
ejpam-3427	539	1	×	×	PROPN
ejpam-3427	539	2	rn	rn	NOUN
ejpam-3427	539	3	if	if	SCONJ
ejpam-3427	540	1	and	and	CCONJ
ejpam-3427	540	2	only	only	ADV
ejpam-3427	540	3	if	if	SCONJ
ejpam-3427	540	4	the	the	DET
ejpam-3427	540	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	540	6	anti	anti	ADJ
ejpam-3427	540	7	characteristic	characteristic	ADJ
ejpam-3427	540	8	function	function	NOUN
ejpam-3427	540	9	χb	χb	PROPN
ejpam-3427	540	10	=	=	PUNCT
ejpam-3427	541	1	〈	〈	PROPN
ejpam-3427	541	2	µχb	µχb	ADJ
ejpam-3427	541	3	,	,	PUNCT
ejpam-3427	541	4	γχb	γχb	PROPN
ejpam-3427	541	5	〉	〉	NUM
ejpam-3427	541	6	of	of	ADP
ejpam-3427	541	7	b	b	PROPN
ejpam-3427	541	8	=	=	NOUN
ejpam-3427	541	9	∩bi	∩bi	NOUN
ejpam-3427	541	10	is	be	AUX
ejpam-3427	541	11	an	an	DET
ejpam-3427	541	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	541	13	anti	anti	ADJ
ejpam-3427	541	14	fuzzy	fuzzy	ADJ
ejpam-3427	541	15	normal	normal	ADJ
ejpam-3427	541	16	la	la	NOUN
ejpam-3427	541	17	-	-	PUNCT
ejpam-3427	541	18	subring	subring	NOUN
ejpam-3427	541	19	of	of	ADP
ejpam-3427	541	20	an	an	DET
ejpam-3427	541	21	la	la	ADJ
ejpam-3427	541	22	-	-	PUNCT
ejpam-3427	541	23	ring	ring	NOUN
ejpam-3427	541	24	r1×r2×	r1×r2×	NOUN
ejpam-3427	541	25	...	...	PUNCT
ejpam-3427	541	26	×rn	×rn	PROPN
ejpam-3427	541	27	.	.	PUNCT
ejpam-3427	542	1	theorem	theorem	VERB
ejpam-3427	542	2	7	7	NUM
ejpam-3427	542	3	.	.	PUNCT
ejpam-3427	543	1	if	if	SCONJ
ejpam-3427	543	2	a	a	DET
ejpam-3427	543	3	=	=	PUNCT
ejpam-3427	543	4	a1×a2×	a1×a2×	NOUN
ejpam-3427	543	5	...	...	PUNCT
ejpam-3427	543	6	×an	×an	PROPN
ejpam-3427	543	7	and	and	CCONJ
ejpam-3427	543	8	b	b	X
ejpam-3427	543	9	=	=	SYM
ejpam-3427	543	10	b1×b2×	b1×b2×	PROPN
ejpam-3427	543	11	...	...	PUNCT
ejpam-3427	543	12	×bn	×bn	VERB
ejpam-3427	543	13	are	be	AUX
ejpam-3427	543	14	two	two	NUM
ejpam-3427	543	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	543	16	anti	anti	ADJ
ejpam-3427	543	17	fuzzy	fuzzy	ADJ
ejpam-3427	543	18	normal	normal	ADJ
ejpam-3427	543	19	la	la	NOUN
ejpam-3427	543	20	-	-	PUNCT
ejpam-3427	543	21	subrings	subring	NOUN
ejpam-3427	543	22	of	of	ADP
ejpam-3427	543	23	an	an	DET
ejpam-3427	543	24	la	la	ADJ
ejpam-3427	543	25	-	-	PUNCT
ejpam-3427	543	26	ring	ring	NOUN
ejpam-3427	543	27	r1×r2×	r1×r2×	NOUN
ejpam-3427	543	28	...	...	PUNCT
ejpam-3427	543	29	×rn	×rn	PROPN
ejpam-3427	543	30	,	,	PUNCT
ejpam-3427	543	31	then	then	ADV
ejpam-3427	543	32	their	their	PRON
ejpam-3427	543	33	intersection	intersection	NOUN
ejpam-3427	543	34	a∩b	a∩b	PROPN
ejpam-3427	543	35	is	be	AUX
ejpam-3427	543	36	also	also	ADV
ejpam-3427	543	37	an	an	DET
ejpam-3427	543	38	intuitionistic	intuitionistic	ADJ
ejpam-3427	543	39	anti	anti	ADJ
ejpam-3427	543	40	fuzzy	fuzzy	ADJ
ejpam-3427	543	41	normal	normal	ADJ
ejpam-3427	543	42	la	la	NOUN
ejpam-3427	543	43	-	-	PUNCT
ejpam-3427	543	44	subring	subring	NOUN
ejpam-3427	543	45	of	of	ADP
ejpam-3427	543	46	an	an	DET
ejpam-3427	543	47	la	la	ADJ
ejpam-3427	543	48	-	-	PUNCT
ejpam-3427	543	49	ring	ring	NOUN
ejpam-3427	543	50	r1	r1	NOUN
ejpam-3427	543	51	×r2	×r2	PROPN
ejpam-3427	543	52	×	×	NOUN
ejpam-3427	543	53	...	...	PUNCT
ejpam-3427	543	54	×rn	×rn	NOUN
ejpam-3427	543	55	.	.	PUNCT
ejpam-3427	544	1	proof	proof	NOUN
ejpam-3427	544	2	.	.	PUNCT
ejpam-3427	545	1	let	let	VERB
ejpam-3427	545	2	a	a	DET
ejpam-3427	545	3	=	=	PUNCT
ejpam-3427	545	4	a1	a1	NOUN
ejpam-3427	545	5	×	×	PROPN
ejpam-3427	545	6	a2	a2	PROPN
ejpam-3427	545	7	×	×	NOUN
ejpam-3427	545	8	...	...	PUNCT
ejpam-3427	545	9	×	×	NOUN
ejpam-3427	545	10	an	an	DET
ejpam-3427	545	11	=	=	X
ejpam-3427	545	12	{	{	PUNCT
ejpam-3427	545	13	(	(	PUNCT
ejpam-3427	545	14	(	(	PUNCT
ejpam-3427	545	15	a	a	NOUN
ejpam-3427	545	16	)	)	PUNCT
ejpam-3427	545	17	,	,	PUNCT
ejpam-3427	545	18	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	545	19	...	...	PUNCT
ejpam-3427	545	20	×an(a	×an(a	NOUN
ejpam-3427	545	21	)	)	PUNCT
ejpam-3427	545	22	,	,	PUNCT
ejpam-3427	545	23	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	545	24	...	...	PUNCT
ejpam-3427	545	25	×a(a	×a(a	PROPN
ejpam-3427	545	26	)	)	PUNCT
ejpam-3427	545	27	)	)	PUNCT
ejpam-3427	546	1	|	|	ADV
ejpam-3427	546	2	for	for	ADP
ejpam-3427	546	3	all	all	DET
ejpam-3427	546	4	a	a	DET
ejpam-3427	546	5	=	=	PUNCT
ejpam-3427	546	6	(	(	PUNCT
ejpam-3427	546	7	a1	a1	PROPN
ejpam-3427	546	8	,	,	PUNCT
ejpam-3427	546	9	a2	a2	PROPN
ejpam-3427	546	10	,	,	PUNCT
ejpam-3427	546	11	...	...	PUNCT
ejpam-3427	546	12	,	,	PUNCT
ejpam-3427	546	13	an	an	DET
ejpam-3427	546	14	)	)	PUNCT
ejpam-3427	546	15	∈	∈	PROPN
ejpam-3427	546	16	r1	r1	NOUN
ejpam-3427	546	17	×	×	NOUN
ejpam-3427	546	18	r2	r2	PROPN
ejpam-3427	546	19	×	×	NOUN
ejpam-3427	546	20	...	...	PUNCT
ejpam-3427	546	21	×	×	PROPN
ejpam-3427	546	22	rn	rn	PROPN
ejpam-3427	546	23	}	}	PUNCT
ejpam-3427	546	24	and	and	CCONJ
ejpam-3427	546	25	b	b	X
ejpam-3427	546	26	=	=	SYM
ejpam-3427	546	27	b1	b1	NOUN
ejpam-3427	546	28	×	×	PROPN
ejpam-3427	546	29	b2	b2	NOUN
ejpam-3427	546	30	×	×	NOUN
ejpam-3427	546	31	...	...	PUNCT
ejpam-3427	546	32	×	×	NOUN
ejpam-3427	546	33	bn	bn	NOUN
ejpam-3427	546	34	=	=	SYM
ejpam-3427	546	35	{	{	PUNCT
ejpam-3427	546	36	(	(	PUNCT
ejpam-3427	546	37	(	(	PUNCT
ejpam-3427	546	38	b	b	NOUN
ejpam-3427	546	39	)	)	PUNCT
ejpam-3427	546	40	,	,	PUNCT
ejpam-3427	546	41	µb1×b2×	µb1×b2×	PROPN
ejpam-3427	546	42	...	...	PUNCT
ejpam-3427	546	43	×bn(b	×bn(b	NOUN
ejpam-3427	546	44	)	)	PUNCT
ejpam-3427	546	45	,	,	PUNCT
ejpam-3427	546	46	γb1×b2×	γb1×b2×	ADJ
ejpam-3427	546	47	...	...	PUNCT
ejpam-3427	546	48	×bn(b	×bn(b	NOUN
ejpam-3427	546	49	)	)	PUNCT
ejpam-3427	546	50	)	)	PUNCT
ejpam-3427	547	1	|	|	ADV
ejpam-3427	547	2	for	for	ADP
ejpam-3427	547	3	all	all	PRON
ejpam-3427	547	4	b	b	NOUN
ejpam-3427	547	5	=	=	SYM
ejpam-3427	547	6	(	(	PUNCT
ejpam-3427	547	7	b1	b1	PROPN
ejpam-3427	547	8	,	,	PUNCT
ejpam-3427	547	9	b2	b2	NOUN
ejpam-3427	547	10	,	,	PUNCT
ejpam-3427	547	11	...	...	PUNCT
ejpam-3427	547	12	,	,	PUNCT
ejpam-3427	547	13	bn	bn	X
ejpam-3427	547	14	)	)	PUNCT
ejpam-3427	547	15	∈	∈	PROPN
ejpam-3427	547	16	r1	r1	PROPN
ejpam-3427	547	17	×r2	×r2	PROPN
ejpam-3427	547	18	×	×	NOUN
ejpam-3427	547	19	...	...	PUNCT
ejpam-3427	547	20	×rn	×rn	NOUN
ejpam-3427	547	21	}	}	PUNCT
ejpam-3427	547	22	be	be	AUX
ejpam-3427	547	23	two	two	NUM
ejpam-3427	547	24	intuitionistic	intuitionistic	ADJ
ejpam-3427	547	25	anti	anti	ADJ
ejpam-3427	547	26	fuzzy	fuzzy	ADJ
ejpam-3427	547	27	normal	normal	ADJ
ejpam-3427	547	28	la	la	NOUN
ejpam-3427	547	29	-	-	PUNCT
ejpam-3427	547	30	subrings	subring	NOUN
ejpam-3427	547	31	of	of	ADP
ejpam-3427	547	32	an	an	DET
ejpam-3427	547	33	la	la	ADJ
ejpam-3427	547	34	-	-	PUNCT
ejpam-3427	547	35	ring	ring	NOUN
ejpam-3427	547	36	r1	r1	NOUN
ejpam-3427	547	37	×r2	×r2	PROPN
ejpam-3427	547	38	×	×	NOUN
ejpam-3427	547	39	...	...	PUNCT
ejpam-3427	547	40	×rn	×rn	NOUN
ejpam-3427	547	41	.	.	PUNCT
ejpam-3427	548	1	let	let	VERB
ejpam-3427	548	2	z	z	NOUN
ejpam-3427	548	3	=	=	PUNCT
ejpam-3427	548	4	a	a	DET
ejpam-3427	548	5	∩b	∩b	NOUN
ejpam-3427	548	6	and	and	CCONJ
ejpam-3427	548	7	z	z	NOUN
ejpam-3427	548	8	=	=	PRON
ejpam-3427	548	9	{	{	PUNCT
ejpam-3427	548	10	(	(	PUNCT
ejpam-3427	548	11	(	(	PUNCT
ejpam-3427	548	12	z	z	NOUN
ejpam-3427	548	13	)	)	PUNCT
ejpam-3427	548	14	,	,	PUNCT
ejpam-3427	548	15	µz(z	µz(z	NOUN
ejpam-3427	548	16	)	)	PUNCT
ejpam-3427	548	17	,	,	PUNCT
ejpam-3427	548	18	γz(z	γz(z	NUM
ejpam-3427	548	19	)	)	PUNCT
ejpam-3427	548	20	)	)	PUNCT
ejpam-3427	549	1	|	|	ADV
ejpam-3427	549	2	for	for	ADP
ejpam-3427	549	3	all	all	DET
ejpam-3427	549	4	z	z	NOUN
ejpam-3427	549	5	=	=	SYM
ejpam-3427	549	6	(	(	PUNCT
ejpam-3427	549	7	z1	z1	PROPN
ejpam-3427	549	8	,	,	PUNCT
ejpam-3427	549	9	z2	z2	PROPN
ejpam-3427	549	10	,	,	PUNCT
ejpam-3427	549	11	...	...	PUNCT
ejpam-3427	549	12	,	,	PUNCT
ejpam-3427	549	13	zn	zn	X
ejpam-3427	549	14	)	)	PUNCT
ejpam-3427	549	15	∈	∈	PROPN
ejpam-3427	549	16	r1	r1	PROPN
ejpam-3427	549	17	×r2	×r2	PROPN
ejpam-3427	549	18	×	×	NOUN
ejpam-3427	549	19	...	...	PUNCT
ejpam-3427	549	20	×rn	×rn	NOUN
ejpam-3427	549	21	}	}	PUNCT
ejpam-3427	549	22	,	,	PUNCT
ejpam-3427	549	23	where	where	SCONJ
ejpam-3427	549	24	µz(z1	µz(z1	ADJ
ejpam-3427	549	25	,	,	PUNCT
ejpam-3427	549	26	z2	z2	PROPN
ejpam-3427	549	27	,	,	PUNCT
ejpam-3427	549	28	...	...	PUNCT
ejpam-3427	549	29	,	,	PUNCT
ejpam-3427	549	30	zn	zn	X
ejpam-3427	549	31	)	)	PUNCT
ejpam-3427	549	32	=	=	SYM
ejpam-3427	549	33	µa∩b(z1	µa∩b(z1	NOUN
ejpam-3427	549	34	,	,	PUNCT
ejpam-3427	549	35	z2	z2	PROPN
ejpam-3427	549	36	,	,	PUNCT
ejpam-3427	549	37	...	...	PUNCT
ejpam-3427	549	38	,	,	PUNCT
ejpam-3427	549	39	zn	zn	X
ejpam-3427	549	40	)	)	PUNCT
ejpam-3427	549	41	=	=	SYM
ejpam-3427	549	42	max{µa(z1	max{µa(z1	NOUN
ejpam-3427	549	43	,	,	PUNCT
ejpam-3427	549	44	z2	z2	PROPN
ejpam-3427	549	45	,	,	PUNCT
ejpam-3427	549	46	...	...	PUNCT
ejpam-3427	549	47	,	,	PUNCT
ejpam-3427	549	48	zn	zn	PROPN
ejpam-3427	549	49	)	)	PUNCT
ejpam-3427	549	50	,	,	PUNCT
ejpam-3427	549	51	µb(z1	µb(z1	NOUN
ejpam-3427	549	52	,	,	PUNCT
ejpam-3427	549	53	z2	z2	PROPN
ejpam-3427	549	54	,	,	PUNCT
ejpam-3427	549	55	...	...	PUNCT
ejpam-3427	549	56	,	,	PUNCT
ejpam-3427	549	57	zn	zn	X
ejpam-3427	549	58	)	)	PUNCT
ejpam-3427	549	59	}	}	PUNCT
ejpam-3427	549	60	and	and	CCONJ
ejpam-3427	549	61	γz(z1	γz(z1	NOUN
ejpam-3427	549	62	,	,	PUNCT
ejpam-3427	549	63	z2	z2	PROPN
ejpam-3427	549	64	,	,	PUNCT
ejpam-3427	549	65	...	...	PUNCT
ejpam-3427	549	66	,	,	PUNCT
ejpam-3427	549	67	zn	zn	X
ejpam-3427	549	68	)	)	PUNCT
ejpam-3427	549	69	=	=	SYM
ejpam-3427	549	70	γa∩b(z1	γa∩b(z1	ADJ
ejpam-3427	549	71	,	,	PUNCT
ejpam-3427	549	72	z2	z2	PROPN
ejpam-3427	549	73	,	,	PUNCT
ejpam-3427	549	74	...	...	PUNCT
ejpam-3427	549	75	,	,	PUNCT
ejpam-3427	549	76	zn	zn	X
ejpam-3427	549	77	)	)	PUNCT
ejpam-3427	549	78	=	=	SYM
ejpam-3427	549	79	min{γa(z1	min{γa(z1	PROPN
ejpam-3427	549	80	,	,	PUNCT
ejpam-3427	549	81	z2	z2	PROPN
ejpam-3427	549	82	,	,	PUNCT
ejpam-3427	549	83	...	...	PUNCT
ejpam-3427	549	84	,	,	PUNCT
ejpam-3427	549	85	zn	zn	PROPN
ejpam-3427	549	86	)	)	PUNCT
ejpam-3427	549	87	,	,	PUNCT
ejpam-3427	549	88	γb(z1	γb(z1	NOUN
ejpam-3427	549	89	,	,	PUNCT
ejpam-3427	549	90	z2	z2	PROPN
ejpam-3427	549	91	,	,	PUNCT
ejpam-3427	549	92	...	...	PUNCT
ejpam-3427	549	93	,	,	PUNCT
ejpam-3427	549	94	zn	zn	NOUN
ejpam-3427	549	95	)	)	PUNCT
ejpam-3427	549	96	}	}	PUNCT
ejpam-3427	549	97	.	.	PUNCT
ejpam-3427	550	1	let	let	VERB
ejpam-3427	550	2	z	z	NOUN
ejpam-3427	550	3	=	=	SYM
ejpam-3427	550	4	(	(	PUNCT
ejpam-3427	550	5	z1	z1	PROPN
ejpam-3427	550	6	,	,	PUNCT
ejpam-3427	550	7	zn	zn	PROPN
ejpam-3427	550	8	,	,	PUNCT
ejpam-3427	550	9	...	...	PUNCT
ejpam-3427	550	10	,	,	PUNCT
ejpam-3427	550	11	zn	zn	PROPN
ejpam-3427	550	12	)	)	PUNCT
ejpam-3427	550	13	,	,	PUNCT
ejpam-3427	550	14	w	w	NOUN
ejpam-3427	550	15	=	=	SYM
ejpam-3427	550	16	(	(	PUNCT
ejpam-3427	550	17	w1	w1	NOUN
ejpam-3427	550	18	,	,	PUNCT
ejpam-3427	550	19	w2	w2	NOUN
ejpam-3427	550	20	,	,	PUNCT
ejpam-3427	550	21	...	...	PUNCT
ejpam-3427	550	22	,	,	PUNCT
ejpam-3427	550	23	wn	wn	PROPN
ejpam-3427	550	24	)	)	PUNCT
ejpam-3427	550	25	∈	∈	PROPN
ejpam-3427	550	26	r1	r1	PROPN
ejpam-3427	550	27	×r2	×r2	PROPN
ejpam-3427	550	28	×	×	NOUN
ejpam-3427	550	29	...	...	PUNCT
ejpam-3427	550	30	×rn	×rn	NOUN
ejpam-3427	550	31	.	.	PUNCT
ejpam-3427	551	1	now	now	ADV
ejpam-3427	551	2	µz(z	µz(z	PUNCT
ejpam-3427	551	3	−	−	PROPN
ejpam-3427	551	4	w	w	NOUN
ejpam-3427	551	5	)	)	PUNCT
ejpam-3427	551	6	=	=	SYM
ejpam-3427	551	7	µz(z	µz(z	NUM
ejpam-3427	551	8	−	−	PROPN
ejpam-3427	551	9	w	w	NOUN
ejpam-3427	551	10	)	)	PUNCT
ejpam-3427	551	11	=	=	SYM
ejpam-3427	551	12	max{µa(z	max{µa(z	NOUN
ejpam-3427	551	13	−	−	PROPN
ejpam-3427	551	14	w	w	NOUN
ejpam-3427	551	15	)	)	PUNCT
ejpam-3427	551	16	,	,	PUNCT
ejpam-3427	551	17	µb(z	µb(z	PUNCT
ejpam-3427	551	18	−	−	PROPN
ejpam-3427	551	19	w	w	NOUN
ejpam-3427	551	20	)	)	PUNCT
ejpam-3427	551	21	}	}	PUNCT
ejpam-3427	551	22	≤	≤	NOUN
ejpam-3427	551	23	{	{	PUNCT
ejpam-3427	551	24	µa(z	µa(z	NUM
ejpam-3427	551	25	)	)	PUNCT
ejpam-3427	551	26	∨	∨	NOUN
ejpam-3427	551	27	µa(w	µa(w	PUNCT
ejpam-3427	551	28	)	)	PUNCT
ejpam-3427	551	29	}	}	PUNCT
ejpam-3427	551	30	∨	∨	X
ejpam-3427	551	31	{	{	PUNCT
ejpam-3427	551	32	µb(z	µb(z	NOUN
ejpam-3427	551	33	)	)	PUNCT
ejpam-3427	551	34	∨	∨	NUM
ejpam-3427	551	35	µb(w	µb(w	NOUN
ejpam-3427	551	36	)	)	PUNCT
ejpam-3427	551	37	}	}	PUNCT
ejpam-3427	551	38	=	=	SYM
ejpam-3427	551	39	µa(z	µa(z	PRON
ejpam-3427	551	40	)	)	PUNCT
ejpam-3427	551	41	∨	∨	NUM
ejpam-3427	551	42	{	{	PUNCT
ejpam-3427	551	43	µa(w	µa(w	NUM
ejpam-3427	551	44	)	)	PUNCT
ejpam-3427	551	45	∨	∨	NUM
ejpam-3427	551	46	µb(z	µb(z	NUM
ejpam-3427	551	47	)	)	PUNCT
ejpam-3427	551	48	}	}	PUNCT
ejpam-3427	551	49	∨	∨	NUM
ejpam-3427	551	50	µb(w	µb(w	NOUN
ejpam-3427	551	51	)	)	PUNCT
ejpam-3427	551	52	=	=	SYM
ejpam-3427	551	53	µa(z	µa(z	PRON
ejpam-3427	551	54	)	)	PUNCT
ejpam-3427	551	55	∨	∨	NUM
ejpam-3427	551	56	{	{	PUNCT
ejpam-3427	551	57	µb(z	µb(z	NOUN
ejpam-3427	551	58	)	)	PUNCT
ejpam-3427	551	59	∨	∨	NOUN
ejpam-3427	551	60	µa(w	µa(w	PUNCT
ejpam-3427	551	61	)	)	PUNCT
ejpam-3427	551	62	}	}	PUNCT
ejpam-3427	551	63	∨	∨	NUM
ejpam-3427	551	64	µb(w	µb(w	NOUN
ejpam-3427	551	65	)	)	PUNCT
ejpam-3427	551	66	=	=	SYM
ejpam-3427	551	67	{	{	PUNCT
ejpam-3427	551	68	µa(z	µa(z	PROPN
ejpam-3427	551	69	)	)	PUNCT
ejpam-3427	551	70	∨	∨	NUM
ejpam-3427	551	71	µb(z	µb(z	NUM
ejpam-3427	551	72	)	)	PUNCT
ejpam-3427	551	73	}	}	PUNCT
ejpam-3427	551	74	∨	∨	X
ejpam-3427	551	75	{	{	PUNCT
ejpam-3427	551	76	µa(w	µa(w	NOUN
ejpam-3427	551	77	)	)	PUNCT
ejpam-3427	551	78	∨	∨	NUM
ejpam-3427	551	79	µb(w	µb(w	NOUN
ejpam-3427	551	80	)	)	PUNCT
ejpam-3427	551	81	}	}	PUNCT
ejpam-3427	552	1	=	=	SYM
ejpam-3427	552	2	max{µa∩b(z	max{µa∩b(z	NOUN
ejpam-3427	552	3	)	)	PUNCT
ejpam-3427	552	4	,	,	PUNCT
ejpam-3427	552	5	µa∩b(w	µa∩b(w	NOUN
ejpam-3427	552	6	)	)	PUNCT
ejpam-3427	552	7	}	}	PUNCT
ejpam-3427	552	8	=	=	SYM
ejpam-3427	552	9	max{µz(z	max{µz(z	NOUN
ejpam-3427	552	10	)	)	PUNCT
ejpam-3427	552	11	,	,	PUNCT
ejpam-3427	552	12	µz(w	µz(w	NUM
ejpam-3427	552	13	)	)	PUNCT
ejpam-3427	552	14	}	}	PUNCT
ejpam-3427	552	15	and	and	CCONJ
ejpam-3427	552	16	µz(z	µz(z	VERB
ejpam-3427	552	17	◦	◦	NOUN
ejpam-3427	552	18	w	w	NOUN
ejpam-3427	552	19	)	)	PUNCT
ejpam-3427	552	20	=	=	PRON
ejpam-3427	552	21	µz(z	µz(z	PUNCT
ejpam-3427	552	22	◦	◦	NOUN
ejpam-3427	552	23	w	w	NOUN
ejpam-3427	552	24	)	)	PUNCT
ejpam-3427	552	25	=	=	SYM
ejpam-3427	553	1	max{µa(z	max{µa(z	NOUN
ejpam-3427	553	2	◦	◦	NOUN
ejpam-3427	553	3	w	w	PROPN
ejpam-3427	553	4	)	)	PUNCT
ejpam-3427	553	5	,	,	PUNCT
ejpam-3427	553	6	µb(z	µb(z	PUNCT
ejpam-3427	553	7	◦	◦	NOUN
ejpam-3427	553	8	w	w	NOUN
ejpam-3427	553	9	)	)	PUNCT
ejpam-3427	553	10	}	}	PUNCT
ejpam-3427	553	11	≤	≤	NOUN
ejpam-3427	553	12	{	{	PUNCT
ejpam-3427	553	13	µa(z	µa(z	NUM
ejpam-3427	553	14	)	)	PUNCT
ejpam-3427	553	15	∨	∨	NOUN
ejpam-3427	553	16	µa(w	µa(w	PUNCT
ejpam-3427	553	17	)	)	PUNCT
ejpam-3427	553	18	}	}	PUNCT
ejpam-3427	553	19	∨	∨	X
ejpam-3427	553	20	{	{	PUNCT
ejpam-3427	553	21	µb(z	µb(z	NOUN
ejpam-3427	553	22	)	)	PUNCT
ejpam-3427	553	23	∨	∨	NUM
ejpam-3427	553	24	µb(w	µb(w	NOUN
ejpam-3427	553	25	)	)	PUNCT
ejpam-3427	553	26	}	}	PUNCT
ejpam-3427	553	27	=	=	SYM
ejpam-3427	553	28	µa(z	µa(z	PRON
ejpam-3427	553	29	)	)	PUNCT
ejpam-3427	553	30	∨	∨	NUM
ejpam-3427	553	31	{	{	PUNCT
ejpam-3427	553	32	µa(w	µa(w	NUM
ejpam-3427	553	33	)	)	PUNCT
ejpam-3427	553	34	∨	∨	NUM
ejpam-3427	553	35	µb(z	µb(z	NUM
ejpam-3427	553	36	)	)	PUNCT
ejpam-3427	553	37	}	}	PUNCT
ejpam-3427	553	38	∨	∨	NUM
ejpam-3427	553	39	µb(w	µb(w	NOUN
ejpam-3427	553	40	)	)	PUNCT
ejpam-3427	553	41	=	=	SYM
ejpam-3427	553	42	µa(z	µa(z	PRON
ejpam-3427	553	43	)	)	PUNCT
ejpam-3427	553	44	∨	∨	NUM
ejpam-3427	553	45	{	{	PUNCT
ejpam-3427	553	46	µb(z	µb(z	NOUN
ejpam-3427	553	47	)	)	PUNCT
ejpam-3427	553	48	∨	∨	NOUN
ejpam-3427	553	49	µa(w	µa(w	PUNCT
ejpam-3427	553	50	)	)	PUNCT
ejpam-3427	553	51	}	}	PUNCT
ejpam-3427	553	52	∨	∨	NUM
ejpam-3427	553	53	µb(w	µb(w	NOUN
ejpam-3427	553	54	)	)	PUNCT
ejpam-3427	553	55	=	=	SYM
ejpam-3427	553	56	{	{	PUNCT
ejpam-3427	553	57	µa(z	µa(z	PROPN
ejpam-3427	553	58	)	)	PUNCT
ejpam-3427	553	59	∨	∨	NUM
ejpam-3427	553	60	µb(z	µb(z	NUM
ejpam-3427	553	61	)	)	PUNCT
ejpam-3427	553	62	}	}	PUNCT
ejpam-3427	553	63	∨	∨	X
ejpam-3427	553	64	{	{	PUNCT
ejpam-3427	553	65	µa(w	µa(w	NOUN
ejpam-3427	553	66	)	)	PUNCT
ejpam-3427	553	67	∨	∨	NUM
ejpam-3427	553	68	µb(w	µb(w	NOUN
ejpam-3427	553	69	)	)	PUNCT
ejpam-3427	553	70	}	}	PUNCT
ejpam-3427	553	71	=	=	SYM
ejpam-3427	553	72	max{µa∩b(z	max{µa∩b(z	NOUN
ejpam-3427	553	73	)	)	PUNCT
ejpam-3427	553	74	,	,	PUNCT
ejpam-3427	553	75	µa∩b(w	µa∩b(w	NOUN
ejpam-3427	553	76	)	)	PUNCT
ejpam-3427	553	77	}	}	PUNCT
ejpam-3427	553	78	=	=	SYM
ejpam-3427	553	79	max{µz(z	max{µz(z	NOUN
ejpam-3427	553	80	)	)	PUNCT
ejpam-3427	553	81	,	,	PUNCT
ejpam-3427	553	82	µz(w	µz(w	NUM
ejpam-3427	553	83	)	)	PUNCT
ejpam-3427	553	84	}	}	PUNCT
ejpam-3427	553	85	.	.	PUNCT
ejpam-3427	554	1	thus	thus	ADV
ejpam-3427	554	2	µz((z1	µz((z1	NOUN
ejpam-3427	554	3	,	,	PUNCT
ejpam-3427	554	4	z2	z2	PROPN
ejpam-3427	554	5	,	,	PUNCT
ejpam-3427	554	6	...	...	PUNCT
ejpam-3427	554	7	,	,	PUNCT
ejpam-3427	554	8	zn)−	zn)−	NUM
ejpam-3427	554	9	(	(	PUNCT
ejpam-3427	554	10	w1	w1	NOUN
ejpam-3427	554	11	,	,	PUNCT
ejpam-3427	554	12	w2	w2	NOUN
ejpam-3427	554	13	,	,	PUNCT
ejpam-3427	554	14	...	...	PUNCT
ejpam-3427	554	15	,	,	PUNCT
ejpam-3427	554	16	wn	wn	PROPN
ejpam-3427	554	17	)	)	PUNCT
ejpam-3427	554	18	)	)	PUNCT
ejpam-3427	554	19	k.	k.	PROPN
ejpam-3427	555	1	nasreen	nasreen	PROPN
ejpam-3427	555	2	/	/	SYM
ejpam-3427	555	3	eur	eur	PROPN
ejpam-3427	555	4	.	.	PUNCT
ejpam-3427	556	1	j.	j.	PROPN
ejpam-3427	556	2	pure	pure	PROPN
ejpam-3427	556	3	appl	appl	PROPN
ejpam-3427	556	4	.	.	PROPN
ejpam-3427	556	5	math	math	PROPN
ejpam-3427	556	6	,	,	PUNCT
ejpam-3427	556	7	12	12	NUM
ejpam-3427	556	8	(	(	PUNCT
ejpam-3427	556	9	2	2	NUM
ejpam-3427	556	10	)	)	PUNCT
ejpam-3427	556	11	(	(	PUNCT
ejpam-3427	556	12	2019	2019	NUM
ejpam-3427	556	13	)	)	PUNCT
ejpam-3427	556	14	,	,	PUNCT
ejpam-3427	556	15	622	622	NUM
ejpam-3427	556	16	-	-	SYM
ejpam-3427	556	17	648	648	NUM
ejpam-3427	556	18	641	641	NUM
ejpam-3427	556	19	≤	≤	NOUN
ejpam-3427	556	20	max{µz(z1	max{µz(z1	PROPN
ejpam-3427	556	21	,	,	PUNCT
ejpam-3427	556	22	z2	z2	PROPN
ejpam-3427	556	23	,	,	PUNCT
ejpam-3427	556	24	...	...	PUNCT
ejpam-3427	556	25	,	,	PUNCT
ejpam-3427	556	26	zn	zn	X
ejpam-3427	556	27	)	)	PUNCT
ejpam-3427	556	28	,	,	PUNCT
ejpam-3427	556	29	µz(w1	µz(w1	NOUN
ejpam-3427	556	30	,	,	PUNCT
ejpam-3427	556	31	w2	w2	NOUN
ejpam-3427	556	32	,	,	PUNCT
ejpam-3427	556	33	...	...	PUNCT
ejpam-3427	556	34	,	,	PUNCT
ejpam-3427	556	35	wn	wn	PROPN
ejpam-3427	556	36	)	)	PUNCT
ejpam-3427	556	37	}	}	PUNCT
ejpam-3427	556	38	and	and	CCONJ
ejpam-3427	556	39	µz((z1	µz((z1	NOUN
ejpam-3427	556	40	,	,	PUNCT
ejpam-3427	556	41	z2	z2	PROPN
ejpam-3427	556	42	,	,	PUNCT
ejpam-3427	556	43	...	...	PUNCT
ejpam-3427	556	44	,	,	PUNCT
ejpam-3427	556	45	zn	zn	X
ejpam-3427	556	46	)	)	PUNCT
ejpam-3427	556	47	◦	◦	NOUN
ejpam-3427	556	48	(	(	PUNCT
ejpam-3427	556	49	w1	w1	NOUN
ejpam-3427	556	50	,	,	PUNCT
ejpam-3427	556	51	w2	w2	NOUN
ejpam-3427	556	52	,	,	PUNCT
ejpam-3427	556	53	...	...	PUNCT
ejpam-3427	556	54	,	,	PUNCT
ejpam-3427	556	55	wn	wn	PROPN
ejpam-3427	556	56	)	)	PUNCT
ejpam-3427	556	57	)	)	PUNCT
ejpam-3427	557	1	≤	≤	NUM
ejpam-3427	557	2	max{µz(z1	max{µz(z1	PROPN
ejpam-3427	557	3	,	,	PUNCT
ejpam-3427	557	4	z2	z2	PROPN
ejpam-3427	557	5	,	,	PUNCT
ejpam-3427	557	6	...	...	PUNCT
ejpam-3427	557	7	,	,	PUNCT
ejpam-3427	557	8	zn	zn	X
ejpam-3427	557	9	)	)	PUNCT
ejpam-3427	557	10	,	,	PUNCT
ejpam-3427	557	11	µz(w1	µz(w1	NOUN
ejpam-3427	557	12	,	,	PUNCT
ejpam-3427	557	13	w2	w2	NOUN
ejpam-3427	557	14	,	,	PUNCT
ejpam-3427	557	15	...	...	PUNCT
ejpam-3427	557	16	,	,	PUNCT
ejpam-3427	557	17	wn	wn	PROPN
ejpam-3427	557	18	)	)	PUNCT
ejpam-3427	557	19	}	}	PUNCT
ejpam-3427	557	20	.	.	PUNCT
ejpam-3427	558	1	similarly	similarly	ADV
ejpam-3427	558	2	,	,	PUNCT
ejpam-3427	558	3	we	we	PRON
ejpam-3427	558	4	have	have	VERB
ejpam-3427	558	5	γz((z1	γz((z1	NOUN
ejpam-3427	558	6	,	,	PUNCT
ejpam-3427	558	7	z2	z2	PROPN
ejpam-3427	558	8	,	,	PUNCT
ejpam-3427	558	9	...	...	PUNCT
ejpam-3427	558	10	,	,	PUNCT
ejpam-3427	558	11	zn)−	zn)−	NUM
ejpam-3427	558	12	(	(	PUNCT
ejpam-3427	558	13	w1	w1	NOUN
ejpam-3427	558	14	,	,	PUNCT
ejpam-3427	558	15	w2	w2	NOUN
ejpam-3427	558	16	,	,	PUNCT
ejpam-3427	558	17	...	...	PUNCT
ejpam-3427	558	18	,	,	PUNCT
ejpam-3427	558	19	wn	wn	PROPN
ejpam-3427	558	20	)	)	PUNCT
ejpam-3427	558	21	)	)	PUNCT
ejpam-3427	558	22	≥	≥	NOUN
ejpam-3427	558	23	min{γz(z1	min{γz(z1	PROPN
ejpam-3427	558	24	,	,	PUNCT
ejpam-3427	558	25	z2	z2	PROPN
ejpam-3427	558	26	,	,	PUNCT
ejpam-3427	558	27	...	...	PUNCT
ejpam-3427	558	28	,	,	PUNCT
ejpam-3427	558	29	zn	zn	PROPN
ejpam-3427	558	30	)	)	PUNCT
ejpam-3427	558	31	,	,	PUNCT
ejpam-3427	558	32	γz(w1	γz(w1	X
ejpam-3427	558	33	,	,	PUNCT
ejpam-3427	558	34	w2	w2	NOUN
ejpam-3427	558	35	,	,	PUNCT
ejpam-3427	558	36	...	...	PUNCT
ejpam-3427	558	37	,	,	PUNCT
ejpam-3427	558	38	wn	wn	PROPN
ejpam-3427	558	39	)	)	PUNCT
ejpam-3427	558	40	}	}	PUNCT
ejpam-3427	558	41	and	and	CCONJ
ejpam-3427	558	42	γz((z1	γz((z1	NOUN
ejpam-3427	558	43	,	,	PUNCT
ejpam-3427	558	44	z2	z2	PROPN
ejpam-3427	558	45	,	,	PUNCT
ejpam-3427	558	46	...	...	PUNCT
ejpam-3427	558	47	,	,	PUNCT
ejpam-3427	558	48	zn	zn	X
ejpam-3427	558	49	)	)	PUNCT
ejpam-3427	558	50	◦	◦	NOUN
ejpam-3427	558	51	(	(	PUNCT
ejpam-3427	558	52	w1	w1	NOUN
ejpam-3427	558	53	,	,	PUNCT
ejpam-3427	558	54	w2	w2	NOUN
ejpam-3427	558	55	,	,	PUNCT
ejpam-3427	558	56	...	...	PUNCT
ejpam-3427	558	57	,	,	PUNCT
ejpam-3427	558	58	wn	wn	PROPN
ejpam-3427	558	59	)	)	PUNCT
ejpam-3427	558	60	)	)	PUNCT
ejpam-3427	558	61	≥	≥	NOUN
ejpam-3427	558	62	min{γz(z1	min{γz(z1	PROPN
ejpam-3427	558	63	,	,	PUNCT
ejpam-3427	558	64	z2	z2	PROPN
ejpam-3427	558	65	,	,	PUNCT
ejpam-3427	558	66	...	...	PUNCT
ejpam-3427	558	67	,	,	PUNCT
ejpam-3427	558	68	zn	zn	PROPN
ejpam-3427	558	69	)	)	PUNCT
ejpam-3427	558	70	,	,	PUNCT
ejpam-3427	558	71	γz(w1	γz(w1	X
ejpam-3427	558	72	,	,	PUNCT
ejpam-3427	558	73	w2	w2	NOUN
ejpam-3427	558	74	,	,	PUNCT
ejpam-3427	558	75	...	...	PUNCT
ejpam-3427	558	76	,	,	PUNCT
ejpam-3427	558	77	wn	wn	PROPN
ejpam-3427	558	78	)	)	PUNCT
ejpam-3427	558	79	}	}	PUNCT
ejpam-3427	559	1	thus	thus	ADV
ejpam-3427	559	2	z	z	X
ejpam-3427	559	3	=	=	SYM
ejpam-3427	559	4	(	(	PUNCT
ejpam-3427	559	5	µz	µz	PROPN
ejpam-3427	559	6	,	,	PUNCT
ejpam-3427	559	7	γz	γz	X
ejpam-3427	559	8	)	)	PUNCT
ejpam-3427	559	9	is	be	AUX
ejpam-3427	559	10	an	an	DET
ejpam-3427	559	11	intuitionistic	intuitionistic	ADJ
ejpam-3427	559	12	anti	anti	ADJ
ejpam-3427	559	13	fuzzy	fuzzy	ADJ
ejpam-3427	559	14	la	la	NOUN
ejpam-3427	559	15	-	-	PUNCT
ejpam-3427	559	16	subring	subring	NOUN
ejpam-3427	559	17	of	of	ADP
ejpam-3427	559	18	an	an	DET
ejpam-3427	559	19	la	la	ADJ
ejpam-3427	559	20	-	-	PUNCT
ejpam-3427	559	21	ring	ring	NOUN
ejpam-3427	559	22	r1	r1	NOUN
ejpam-3427	559	23	×r2	×r2	PROPN
ejpam-3427	559	24	×	×	NOUN
ejpam-3427	559	25	...	...	PUNCT
ejpam-3427	559	26	×rn	×rn	NOUN
ejpam-3427	559	27	.	.	PUNCT
ejpam-3427	560	1	now	now	ADV
ejpam-3427	560	2	µz((z1	µz((z1	NOUN
ejpam-3427	560	3	,	,	PUNCT
ejpam-3427	560	4	z2	z2	PROPN
ejpam-3427	560	5	,	,	PUNCT
ejpam-3427	560	6	...	...	PUNCT
ejpam-3427	560	7	,	,	PUNCT
ejpam-3427	560	8	zn	zn	X
ejpam-3427	560	9	)	)	PUNCT
ejpam-3427	560	10	◦	◦	NOUN
ejpam-3427	560	11	(	(	PUNCT
ejpam-3427	560	12	w1	w1	NOUN
ejpam-3427	560	13	,	,	PUNCT
ejpam-3427	560	14	w2	w2	NOUN
ejpam-3427	560	15	,	,	PUNCT
ejpam-3427	560	16	...	...	PUNCT
ejpam-3427	560	17	,	,	PUNCT
ejpam-3427	560	18	wn	wn	PROPN
ejpam-3427	560	19	)	)	PUNCT
ejpam-3427	560	20	)	)	PUNCT
ejpam-3427	561	1	=	=	PUNCT
ejpam-3427	561	2	µa∩b(z1w1	µa∩b(z1w1	PROPN
ejpam-3427	561	3	,	,	PUNCT
ejpam-3427	561	4	z2w2	z2w2	X
ejpam-3427	561	5	,	,	PUNCT
ejpam-3427	561	6	...	...	PUNCT
ejpam-3427	561	7	,	,	PUNCT
ejpam-3427	561	8	znwn	znwn	ADJ
ejpam-3427	561	9	)	)	PUNCT
ejpam-3427	561	10	=	=	SYM
ejpam-3427	561	11	max{µa(z1w1	max{µa(z1w1	PROPN
ejpam-3427	561	12	,	,	PUNCT
ejpam-3427	561	13	z2w2	z2w2	NOUN
ejpam-3427	561	14	,	,	PUNCT
ejpam-3427	561	15	...	...	PUNCT
ejpam-3427	561	16	,	,	PUNCT
ejpam-3427	561	17	znwn	znwn	ADJ
ejpam-3427	561	18	)	)	PUNCT
ejpam-3427	561	19	,	,	PUNCT
ejpam-3427	561	20	µb(z1w1	µb(z1w1	PROPN
ejpam-3427	561	21	,	,	PUNCT
ejpam-3427	561	22	z2w2	z2w2	NOUN
ejpam-3427	561	23	,	,	PUNCT
ejpam-3427	561	24	...	...	PUNCT
ejpam-3427	561	25	,	,	PUNCT
ejpam-3427	561	26	znwn	znwn	ADJ
ejpam-3427	561	27	)	)	PUNCT
ejpam-3427	561	28	}	}	PUNCT
ejpam-3427	561	29	=	=	SYM
ejpam-3427	561	30	max{µa(w1z1	max{µa(w1z1	PROPN
ejpam-3427	561	31	,	,	PUNCT
ejpam-3427	561	32	w2z2	w2z2	PRON
ejpam-3427	561	33	,	,	PUNCT
ejpam-3427	561	34	...	...	PUNCT
ejpam-3427	561	35	,	,	PUNCT
ejpam-3427	561	36	wnzn	wnzn	NOUN
ejpam-3427	561	37	)	)	PUNCT
ejpam-3427	561	38	,	,	PUNCT
ejpam-3427	561	39	µb(w1z1	µb(w1z1	PROPN
ejpam-3427	561	40	,	,	PUNCT
ejpam-3427	561	41	w2z2	w2z2	PRON
ejpam-3427	561	42	,	,	PUNCT
ejpam-3427	561	43	...	...	PUNCT
ejpam-3427	561	44	,	,	PUNCT
ejpam-3427	561	45	wnzn	wnzn	NOUN
ejpam-3427	561	46	)	)	PUNCT
ejpam-3427	561	47	}	}	PUNCT
ejpam-3427	561	48	=	=	SYM
ejpam-3427	561	49	µa∩b(w1z1	µa∩b(w1z1	PROPN
ejpam-3427	561	50	,	,	PUNCT
ejpam-3427	561	51	w2z2	w2z2	PRON
ejpam-3427	561	52	,	,	PUNCT
ejpam-3427	561	53	...	...	PUNCT
ejpam-3427	561	54	,	,	PUNCT
ejpam-3427	561	55	wnzn	wnzn	NOUN
ejpam-3427	561	56	)	)	PUNCT
ejpam-3427	561	57	=	=	SYM
ejpam-3427	561	58	µz((w1	µz((w1	PROPN
ejpam-3427	561	59	,	,	PUNCT
ejpam-3427	561	60	w2	w2	NOUN
ejpam-3427	561	61	,	,	PUNCT
ejpam-3427	561	62	...	...	PUNCT
ejpam-3427	561	63	,	,	PUNCT
ejpam-3427	561	64	wn	wn	PROPN
ejpam-3427	561	65	)	)	PUNCT
ejpam-3427	561	66	◦	◦	NOUN
ejpam-3427	561	67	(	(	PUNCT
ejpam-3427	561	68	z1	z1	PROPN
ejpam-3427	561	69	,	,	PUNCT
ejpam-3427	561	70	z2	z2	PROPN
ejpam-3427	561	71	,	,	PUNCT
ejpam-3427	561	72	...	...	PUNCT
ejpam-3427	561	73	,	,	PUNCT
ejpam-3427	561	74	zn	zn	PROPN
ejpam-3427	561	75	)	)	PUNCT
ejpam-3427	561	76	)	)	PUNCT
ejpam-3427	561	77	.	.	PUNCT
ejpam-3427	562	1	similarly	similarly	ADV
ejpam-3427	562	2	γz((z1	γz((z1	NOUN
ejpam-3427	562	3	,	,	PUNCT
ejpam-3427	562	4	z2	z2	PROPN
ejpam-3427	562	5	,	,	PUNCT
ejpam-3427	562	6	...	...	PUNCT
ejpam-3427	562	7	,	,	PUNCT
ejpam-3427	562	8	zn	zn	X
ejpam-3427	562	9	)	)	PUNCT
ejpam-3427	562	10	◦	◦	NOUN
ejpam-3427	562	11	(	(	PUNCT
ejpam-3427	562	12	w1	w1	NOUN
ejpam-3427	562	13	,	,	PUNCT
ejpam-3427	562	14	w2	w2	NOUN
ejpam-3427	562	15	,	,	PUNCT
ejpam-3427	562	16	...	...	PUNCT
ejpam-3427	562	17	,	,	PUNCT
ejpam-3427	562	18	wn	wn	PROPN
ejpam-3427	562	19	)	)	PUNCT
ejpam-3427	562	20	)	)	PUNCT
ejpam-3427	563	1	=	=	SYM
ejpam-3427	563	2	γz((w1	γz((w1	NOUN
ejpam-3427	563	3	,	,	PUNCT
ejpam-3427	563	4	w2	w2	NOUN
ejpam-3427	563	5	,	,	PUNCT
ejpam-3427	563	6	...	...	PUNCT
ejpam-3427	563	7	,	,	PUNCT
ejpam-3427	563	8	wn	wn	PROPN
ejpam-3427	563	9	)	)	PUNCT
ejpam-3427	563	10	◦	◦	NOUN
ejpam-3427	563	11	(	(	PUNCT
ejpam-3427	563	12	z1	z1	PROPN
ejpam-3427	563	13	,	,	PUNCT
ejpam-3427	563	14	z2	z2	PROPN
ejpam-3427	563	15	,	,	PUNCT
ejpam-3427	563	16	...	...	PUNCT
ejpam-3427	563	17	,	,	PUNCT
ejpam-3427	563	18	zn	zn	PROPN
ejpam-3427	563	19	)	)	PUNCT
ejpam-3427	563	20	)	)	PUNCT
ejpam-3427	563	21	.	.	PUNCT
ejpam-3427	564	1	hence	hence	ADV
ejpam-3427	564	2	z	z	NOUN
ejpam-3427	564	3	=	=	PUNCT
ejpam-3427	564	4	a	a	DET
ejpam-3427	564	5	∩	∩	ADJ
ejpam-3427	564	6	b	b	NOUN
ejpam-3427	564	7	is	be	AUX
ejpam-3427	564	8	an	an	DET
ejpam-3427	564	9	intuitionistic	intuitionistic	ADJ
ejpam-3427	564	10	anti	anti	ADJ
ejpam-3427	564	11	fuzzy	fuzzy	ADJ
ejpam-3427	564	12	normal	normal	ADJ
ejpam-3427	564	13	la	la	NOUN
ejpam-3427	564	14	-	-	PUNCT
ejpam-3427	564	15	subring	subring	NOUN
ejpam-3427	564	16	of	of	ADP
ejpam-3427	564	17	an	an	DET
ejpam-3427	564	18	la	la	ADJ
ejpam-3427	564	19	-	-	PUNCT
ejpam-3427	564	20	ring	ring	NOUN
ejpam-3427	564	21	r1	r1	NOUN
ejpam-3427	564	22	×r2	×r2	PROPN
ejpam-3427	564	23	×	×	NOUN
ejpam-3427	564	24	...	...	PUNCT
ejpam-3427	564	25	×rn	×rn	NOUN
ejpam-3427	564	26	.	.	PUNCT
ejpam-3427	565	1	corollary	corollary	ADJ
ejpam-3427	565	2	5	5	NUM
ejpam-3427	565	3	.	.	PUNCT
ejpam-3427	566	1	if	if	SCONJ
ejpam-3427	566	2	{	{	PUNCT
ejpam-3427	566	3	bi}i∈i	bi}i∈i	PRON
ejpam-3427	566	4	=	=	X
ejpam-3427	566	5	{	{	PUNCT
ejpam-3427	566	6	ai1	ai1	VERB
ejpam-3427	566	7	×ai2	×ai2	NOUN
ejpam-3427	566	8	×	×	PROPN
ejpam-3427	566	9	...	...	PUNCT
ejpam-3427	567	1	×ain}i∈i	×ain}i∈i	NOUN
ejpam-3427	567	2	is	be	AUX
ejpam-3427	567	3	a	a	DET
ejpam-3427	567	4	family	family	NOUN
ejpam-3427	567	5	of	of	ADP
ejpam-3427	567	6	intuitionistic	intuitionistic	ADJ
ejpam-3427	567	7	anti	anti	ADJ
ejpam-3427	567	8	fuzzy	fuzzy	ADJ
ejpam-3427	567	9	normal	normal	ADJ
ejpam-3427	567	10	la	la	NOUN
ejpam-3427	567	11	-	-	PUNCT
ejpam-3427	567	12	subrings	subring	NOUN
ejpam-3427	567	13	of	of	ADP
ejpam-3427	567	14	an	an	DET
ejpam-3427	567	15	la	la	ADJ
ejpam-3427	567	16	-	-	PUNCT
ejpam-3427	567	17	ring	ring	NOUN
ejpam-3427	567	18	r1	r1	NOUN
ejpam-3427	567	19	×	×	NOUN
ejpam-3427	567	20	r2	r2	PROPN
ejpam-3427	567	21	×	×	NOUN
ejpam-3427	567	22	...	...	PUNCT
ejpam-3427	567	23	×	×	PROPN
ejpam-3427	567	24	rn	rn	PROPN
ejpam-3427	567	25	,	,	PUNCT
ejpam-3427	567	26	then	then	ADV
ejpam-3427	567	27	b	b	X
ejpam-3427	567	28	=	=	NOUN
ejpam-3427	567	29	∩bi	∩bi	NOUN
ejpam-3427	567	30	is	be	AUX
ejpam-3427	567	31	also	also	ADV
ejpam-3427	567	32	an	an	DET
ejpam-3427	567	33	intuitionistic	intuitionistic	ADJ
ejpam-3427	567	34	anti	anti	ADJ
ejpam-3427	567	35	fuzzy	fuzzy	ADJ
ejpam-3427	567	36	normal	normal	ADJ
ejpam-3427	567	37	la	la	NOUN
ejpam-3427	567	38	-	-	PUNCT
ejpam-3427	567	39	subring	subring	NOUN
ejpam-3427	567	40	of	of	ADP
ejpam-3427	567	41	an	an	DET
ejpam-3427	567	42	la	la	ADJ
ejpam-3427	567	43	-	-	PUNCT
ejpam-3427	567	44	ring	ring	NOUN
ejpam-3427	567	45	r1	r1	NOUN
ejpam-3427	567	46	×r2	×r2	PROPN
ejpam-3427	567	47	×	×	NOUN
ejpam-3427	567	48	...	...	PUNCT
ejpam-3427	567	49	×rn	×rn	NOUN
ejpam-3427	567	50	.	.	PUNCT
ejpam-3427	568	1	proposition	proposition	NOUN
ejpam-3427	568	2	4	4	NUM
ejpam-3427	568	3	.	.	PUNCT
ejpam-3427	569	1	if	if	SCONJ
ejpam-3427	569	2	an	an	DET
ejpam-3427	569	3	ifs	ifs	PROPN
ejpam-3427	569	4	a	a	DET
ejpam-3427	569	5	=	=	NOUN
ejpam-3427	569	6	a1	a1	NOUN
ejpam-3427	569	7	×	×	PROPN
ejpam-3427	569	8	a2	a2	PROPN
ejpam-3427	569	9	×	×	NOUN
ejpam-3427	569	10	...	...	PUNCT
ejpam-3427	569	11	×	×	NOUN
ejpam-3427	569	12	an	an	PRON
ejpam-3427	569	13	is	be	AUX
ejpam-3427	569	14	an	an	DET
ejpam-3427	569	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	569	16	anti	anti	ADJ
ejpam-3427	569	17	fuzzy	fuzzy	ADJ
ejpam-3427	569	18	normal	normal	ADJ
ejpam-3427	569	19	la	la	NOUN
ejpam-3427	569	20	-	-	PUNCT
ejpam-3427	569	21	subring	subring	NOUN
ejpam-3427	569	22	of	of	ADP
ejpam-3427	569	23	an	an	DET
ejpam-3427	569	24	la	la	ADJ
ejpam-3427	569	25	-	-	PUNCT
ejpam-3427	569	26	ring	ring	NOUN
ejpam-3427	569	27	r1×r2×	r1×r2×	NOUN
ejpam-3427	569	28	...	...	PUNCT
ejpam-3427	569	29	×rn	×rn	PROPN
ejpam-3427	569	30	,	,	PUNCT
ejpam-3427	569	31	then	then	ADV
ejpam-3427	569	32	�	�	VERB
ejpam-3427	569	33	a	a	DET
ejpam-3427	569	34	=	=	PUNCT
ejpam-3427	569	35	(	(	PUNCT
ejpam-3427	569	36	µa	µa	PROPN
ejpam-3427	569	37	,	,	PUNCT
ejpam-3427	569	38	µa	µa	NOUN
ejpam-3427	569	39	)	)	PUNCT
ejpam-3427	569	40	(	(	PUNCT
ejpam-3427	569	41	resp	resp	NOUN
ejpam-3427	569	42	.	.	PUNCT
ejpam-3427	570	1	♦	♦	PROPN
ejpam-3427	570	2	a	a	PROPN
ejpam-3427	570	3	=	=	X
ejpam-3427	570	4	(	(	PUNCT
ejpam-3427	570	5	γa	γa	PROPN
ejpam-3427	570	6	,	,	PUNCT
ejpam-3427	570	7	γa	γa	PROPN
ejpam-3427	570	8	)	)	PUNCT
ejpam-3427	570	9	)	)	PUNCT
ejpam-3427	570	10	is	be	AUX
ejpam-3427	570	11	also	also	ADV
ejpam-3427	570	12	an	an	DET
ejpam-3427	570	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	570	14	anti	anti	ADJ
ejpam-3427	570	15	fuzzy	fuzzy	ADJ
ejpam-3427	570	16	normal	normal	ADJ
ejpam-3427	570	17	la	la	NOUN
ejpam-3427	570	18	-	-	PUNCT
ejpam-3427	570	19	subring	subring	NOUN
ejpam-3427	570	20	of	of	ADP
ejpam-3427	570	21	an	an	DET
ejpam-3427	570	22	la	la	ADJ
ejpam-3427	570	23	-	-	PUNCT
ejpam-3427	570	24	ring	ring	NOUN
ejpam-3427	570	25	r1	r1	NOUN
ejpam-3427	570	26	×r2	×r2	PROPN
ejpam-3427	570	27	×	×	NOUN
ejpam-3427	570	28	...	...	PUNCT
ejpam-3427	570	29	×rn	×rn	NOUN
ejpam-3427	570	30	.	.	PUNCT
ejpam-3427	571	1	proof	proof	NOUN
ejpam-3427	571	2	.	.	PUNCT
ejpam-3427	572	1	let	let	VERB
ejpam-3427	572	2	a1×a2×	a1×a2×	VERB
ejpam-3427	572	3	...	...	PUNCT
ejpam-3427	572	4	×an	×an	NOUN
ejpam-3427	572	5	be	be	VERB
ejpam-3427	572	6	an	an	DET
ejpam-3427	572	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	572	8	anti	anti	ADJ
ejpam-3427	572	9	fuzzy	fuzzy	ADJ
ejpam-3427	572	10	normal	normal	ADJ
ejpam-3427	572	11	la	la	NOUN
ejpam-3427	572	12	-	-	PUNCT
ejpam-3427	572	13	subring	subring	NOUN
ejpam-3427	572	14	of	of	ADP
ejpam-3427	572	15	an	an	DET
ejpam-3427	572	16	laringr1×r2×	laringr1×r2×	ADJ
ejpam-3427	572	17	...	...	PUNCT
ejpam-3427	572	18	×rn.we	×rn.we	NOUN
ejpam-3427	572	19	have	have	VERB
ejpam-3427	572	20	to	to	PART
ejpam-3427	572	21	show	show	VERB
ejpam-3427	572	22	that	that	SCONJ
ejpam-3427	572	23	�	�	NOUN
ejpam-3427	572	24	a1×a2×	a1×a2×	NOUN
ejpam-3427	572	25	...	...	PUNCT
ejpam-3427	572	26	×an	×an	NOUN
ejpam-3427	572	27	=	=	SYM
ejpam-3427	572	28	(	(	PUNCT
ejpam-3427	572	29	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	572	30	...	...	PUNCT
ejpam-3427	572	31	×an	×an	PROPN
ejpam-3427	572	32	,	,	PUNCT
ejpam-3427	572	33	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	572	34	...	...	PUNCT
ejpam-3427	572	35	×an	×an	PROPN
ejpam-3427	572	36	)	)	PUNCT
ejpam-3427	572	37	is	be	AUX
ejpam-3427	572	38	also	also	ADV
ejpam-3427	572	39	an	an	DET
ejpam-3427	572	40	intuitionistic	intuitionistic	ADJ
ejpam-3427	572	41	anti	anti	ADJ
ejpam-3427	572	42	fuzzy	fuzzy	ADJ
ejpam-3427	572	43	normal	normal	ADJ
ejpam-3427	572	44	la	la	NOUN
ejpam-3427	572	45	-	-	PUNCT
ejpam-3427	572	46	subring	subring	NOUN
ejpam-3427	572	47	of	of	ADP
ejpam-3427	572	48	an	an	DET
ejpam-3427	572	49	la	la	ADJ
ejpam-3427	572	50	-	-	PUNCT
ejpam-3427	572	51	ring	ring	NOUN
ejpam-3427	572	52	r1	r1	NOUN
ejpam-3427	572	53	×	×	NOUN
ejpam-3427	572	54	r2	r2	PROPN
ejpam-3427	572	55	×	×	NOUN
ejpam-3427	572	56	...	...	PUNCT
ejpam-3427	572	57	×	×	PROPN
ejpam-3427	572	58	rn	rn	PROPN
ejpam-3427	572	59	.	.	PUNCT
ejpam-3427	573	1	now	now	ADV
ejpam-3427	573	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	573	3	...	...	PUNCT
ejpam-3427	573	4	×an	×an	PROPN
ejpam-3427	573	5	(	(	PUNCT
ejpam-3427	573	6	(	(	PUNCT
ejpam-3427	573	7	x1	x1	PROPN
ejpam-3427	573	8	,	,	PUNCT
ejpam-3427	573	9	x2	x2	PROPN
ejpam-3427	573	10	,	,	PUNCT
ejpam-3427	573	11	...	...	PUNCT
ejpam-3427	573	12	,	,	PUNCT
ejpam-3427	573	13	xn)−	xn)−	X
ejpam-3427	573	14	(	(	PUNCT
ejpam-3427	573	15	y1	y1	INTJ
ejpam-3427	573	16	,	,	PUNCT
ejpam-3427	573	17	y2	y2	PROPN
ejpam-3427	573	18	,	,	PUNCT
ejpam-3427	573	19	...	...	PUNCT
ejpam-3427	573	20	,	,	PUNCT
ejpam-3427	573	21	yn	yn	PROPN
ejpam-3427	573	22	)	)	PUNCT
ejpam-3427	573	23	)	)	PUNCT
ejpam-3427	574	1	=	=	SYM
ejpam-3427	574	2	1−	1−	NUM
ejpam-3427	574	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	574	4	...	...	PUNCT
ejpam-3427	574	5	×an((x1	×an((x1	PROPN
ejpam-3427	574	6	,	,	PUNCT
ejpam-3427	574	7	x2	x2	PROPN
ejpam-3427	574	8	,	,	PUNCT
ejpam-3427	574	9	...	...	PUNCT
ejpam-3427	574	10	,	,	PUNCT
ejpam-3427	574	11	xn)−	xn)−	X
ejpam-3427	574	12	(	(	PUNCT
ejpam-3427	574	13	y1	y1	INTJ
ejpam-3427	574	14	,	,	PUNCT
ejpam-3427	574	15	y2	y2	PROPN
ejpam-3427	574	16	,	,	PUNCT
ejpam-3427	574	17	...	...	PUNCT
ejpam-3427	574	18	,	,	PUNCT
ejpam-3427	574	19	yn	yn	PROPN
ejpam-3427	574	20	)	)	PUNCT
ejpam-3427	574	21	)	)	PUNCT
ejpam-3427	574	22	k.	k.	PROPN
ejpam-3427	575	1	nasreen	nasreen	PROPN
ejpam-3427	575	2	/	/	SYM
ejpam-3427	575	3	eur	eur	PROPN
ejpam-3427	575	4	.	.	PUNCT
ejpam-3427	576	1	j.	j.	PROPN
ejpam-3427	576	2	pure	pure	PROPN
ejpam-3427	576	3	appl	appl	PROPN
ejpam-3427	576	4	.	.	PROPN
ejpam-3427	576	5	math	math	PROPN
ejpam-3427	576	6	,	,	PUNCT
ejpam-3427	576	7	12	12	NUM
ejpam-3427	576	8	(	(	PUNCT
ejpam-3427	576	9	2	2	NUM
ejpam-3427	576	10	)	)	PUNCT
ejpam-3427	576	11	(	(	PUNCT
ejpam-3427	576	12	2019	2019	NUM
ejpam-3427	576	13	)	)	PUNCT
ejpam-3427	576	14	,	,	PUNCT
ejpam-3427	576	15	622	622	NUM
ejpam-3427	576	16	-	-	SYM
ejpam-3427	576	17	648	648	NUM
ejpam-3427	576	18	642	642	NUM
ejpam-3427	576	19	≥	≥	NOUN
ejpam-3427	576	20	1−max	1−max	NUM
ejpam-3427	576	21	{	{	PUNCT
ejpam-3427	576	22	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	576	23	...	...	PUNCT
ejpam-3427	576	24	×an(x1	×an(x1	NOUN
ejpam-3427	576	25	,	,	PUNCT
ejpam-3427	576	26	x2	x2	PROPN
ejpam-3427	576	27	,	,	PUNCT
ejpam-3427	576	28	...	...	PUNCT
ejpam-3427	576	29	,	,	PUNCT
ejpam-3427	576	30	xn	xn	PROPN
ejpam-3427	576	31	)	)	PUNCT
ejpam-3427	576	32	,	,	PUNCT
ejpam-3427	576	33	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	576	34	...	...	PUNCT
ejpam-3427	576	35	×an(y1	×an(y1	NOUN
ejpam-3427	576	36	,	,	PUNCT
ejpam-3427	576	37	y2	y2	PROPN
ejpam-3427	576	38	,	,	PUNCT
ejpam-3427	576	39	...	...	PUNCT
ejpam-3427	576	40	,	,	PUNCT
ejpam-3427	576	41	yn	yn	PROPN
ejpam-3427	576	42	)	)	PUNCT
ejpam-3427	576	43	}	}	PUNCT
ejpam-3427	576	44	=	=	SYM
ejpam-3427	576	45	min	min	X
ejpam-3427	576	46	{	{	PUNCT
ejpam-3427	576	47	1−	1−	NUM
ejpam-3427	576	48	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	576	49	...	...	PUNCT
ejpam-3427	576	50	×an(x1	×an(x1	NOUN
ejpam-3427	576	51	,	,	PUNCT
ejpam-3427	576	52	x2	x2	PROPN
ejpam-3427	576	53	,	,	PUNCT
ejpam-3427	576	54	...	...	PUNCT
ejpam-3427	576	55	,	,	PUNCT
ejpam-3427	576	56	xn	xn	PROPN
ejpam-3427	576	57	)	)	PUNCT
ejpam-3427	576	58	,	,	PUNCT
ejpam-3427	576	59	1−	1−	NUM
ejpam-3427	576	60	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	576	61	...	...	PUNCT
ejpam-3427	576	62	×an(y1	×an(y1	NOUN
ejpam-3427	576	63	,	,	PUNCT
ejpam-3427	576	64	y2	y2	PROPN
ejpam-3427	576	65	,	,	PUNCT
ejpam-3427	576	66	...	...	PUNCT
ejpam-3427	576	67	,	,	PUNCT
ejpam-3427	576	68	yn	yn	PROPN
ejpam-3427	576	69	)	)	PUNCT
ejpam-3427	576	70	}	}	PUNCT
ejpam-3427	576	71	=	=	SYM
ejpam-3427	576	72	min{µa1×a2×	min{µa1×a2×	NOUN
ejpam-3427	576	73	...	...	PUNCT
ejpam-3427	576	74	×an	×an	NOUN
ejpam-3427	576	75	(	(	PUNCT
ejpam-3427	576	76	x1	x1	PROPN
ejpam-3427	576	77	,	,	PUNCT
ejpam-3427	576	78	x2	x2	PROPN
ejpam-3427	576	79	,	,	PUNCT
ejpam-3427	576	80	...	...	PUNCT
ejpam-3427	576	81	,	,	PUNCT
ejpam-3427	576	82	xn	xn	PROPN
ejpam-3427	576	83	)	)	PUNCT
ejpam-3427	576	84	,	,	PUNCT
ejpam-3427	576	85	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	576	86	...	...	PUNCT
ejpam-3427	576	87	×an	×an	PROPN
ejpam-3427	576	88	(	(	PUNCT
ejpam-3427	576	89	y1	y1	PROPN
ejpam-3427	576	90	,	,	PUNCT
ejpam-3427	576	91	y2	y2	PROPN
ejpam-3427	576	92	,	,	PUNCT
ejpam-3427	576	93	...	...	PUNCT
ejpam-3427	576	94	,	,	PUNCT
ejpam-3427	576	95	yn	yn	PROPN
ejpam-3427	576	96	)	)	PUNCT
ejpam-3427	576	97	}	}	PUNCT
ejpam-3427	576	98	and	and	CCONJ
ejpam-3427	576	99	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	576	100	...	...	PUNCT
ejpam-3427	576	101	×an	×an	PROPN
ejpam-3427	576	102	(	(	PUNCT
ejpam-3427	576	103	(	(	PUNCT
ejpam-3427	576	104	x1	x1	PROPN
ejpam-3427	576	105	,	,	PUNCT
ejpam-3427	576	106	x2	x2	PROPN
ejpam-3427	576	107	,	,	PUNCT
ejpam-3427	576	108	...	...	PUNCT
ejpam-3427	576	109	,	,	PUNCT
ejpam-3427	576	110	xn	xn	X
ejpam-3427	576	111	)	)	PUNCT
ejpam-3427	577	1	◦	◦	NOUN
ejpam-3427	577	2	(	(	PUNCT
ejpam-3427	577	3	y1	y1	INTJ
ejpam-3427	577	4	,	,	PUNCT
ejpam-3427	577	5	y2	y2	PROPN
ejpam-3427	577	6	,	,	PUNCT
ejpam-3427	577	7	...	...	PUNCT
ejpam-3427	577	8	,	,	PUNCT
ejpam-3427	577	9	yn	yn	PROPN
ejpam-3427	577	10	)	)	PUNCT
ejpam-3427	577	11	)	)	PUNCT
ejpam-3427	578	1	=	=	SYM
ejpam-3427	579	1	1−	1−	NUM
ejpam-3427	579	2	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	579	3	...	...	PUNCT
ejpam-3427	579	4	×an((x1	×an((x1	PROPN
ejpam-3427	579	5	,	,	PUNCT
ejpam-3427	579	6	x2	x2	PROPN
ejpam-3427	579	7	,	,	PUNCT
ejpam-3427	579	8	...	...	PUNCT
ejpam-3427	579	9	,	,	PUNCT
ejpam-3427	579	10	xn	xn	X
ejpam-3427	579	11	)	)	PUNCT
ejpam-3427	579	12	◦	◦	NOUN
ejpam-3427	579	13	(	(	PUNCT
ejpam-3427	579	14	y1	y1	INTJ
ejpam-3427	579	15	,	,	PUNCT
ejpam-3427	579	16	y2	y2	PROPN
ejpam-3427	579	17	,	,	PUNCT
ejpam-3427	579	18	...	...	PUNCT
ejpam-3427	579	19	,	,	PUNCT
ejpam-3427	579	20	yn	yn	PROPN
ejpam-3427	579	21	)	)	PUNCT
ejpam-3427	579	22	)	)	PUNCT
ejpam-3427	579	23	≥	≥	PROPN
ejpam-3427	580	1	1−max	1−max	NUM
ejpam-3427	580	2	{	{	PUNCT
ejpam-3427	580	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	580	4	...	...	PUNCT
ejpam-3427	580	5	×an(x1	×an(x1	NOUN
ejpam-3427	580	6	,	,	PUNCT
ejpam-3427	580	7	x2	x2	PROPN
ejpam-3427	580	8	,	,	PUNCT
ejpam-3427	580	9	...	...	PUNCT
ejpam-3427	580	10	,	,	PUNCT
ejpam-3427	580	11	xn	xn	PROPN
ejpam-3427	580	12	)	)	PUNCT
ejpam-3427	580	13	,	,	PUNCT
ejpam-3427	580	14	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	580	15	...	...	PUNCT
ejpam-3427	580	16	×an(y1	×an(y1	NOUN
ejpam-3427	580	17	,	,	PUNCT
ejpam-3427	580	18	y2	y2	PROPN
ejpam-3427	580	19	,	,	PUNCT
ejpam-3427	580	20	...	...	PUNCT
ejpam-3427	580	21	,	,	PUNCT
ejpam-3427	580	22	yn	yn	PROPN
ejpam-3427	580	23	)	)	PUNCT
ejpam-3427	580	24	}	}	PUNCT
ejpam-3427	580	25	=	=	SYM
ejpam-3427	580	26	min	min	X
ejpam-3427	580	27	{	{	PUNCT
ejpam-3427	580	28	1−	1−	NUM
ejpam-3427	580	29	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	580	30	...	...	PUNCT
ejpam-3427	580	31	×an(x1	×an(x1	NOUN
ejpam-3427	580	32	,	,	PUNCT
ejpam-3427	580	33	x2	x2	PROPN
ejpam-3427	580	34	,	,	PUNCT
ejpam-3427	580	35	...	...	PUNCT
ejpam-3427	580	36	,	,	PUNCT
ejpam-3427	580	37	xn	xn	PROPN
ejpam-3427	580	38	)	)	PUNCT
ejpam-3427	580	39	,	,	PUNCT
ejpam-3427	580	40	1−	1−	NUM
ejpam-3427	580	41	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	580	42	...	...	PUNCT
ejpam-3427	580	43	×an(y1	×an(y1	NOUN
ejpam-3427	580	44	,	,	PUNCT
ejpam-3427	580	45	y2	y2	PROPN
ejpam-3427	580	46	,	,	PUNCT
ejpam-3427	580	47	...	...	PUNCT
ejpam-3427	580	48	,	,	PUNCT
ejpam-3427	580	49	yn	yn	PROPN
ejpam-3427	580	50	)	)	PUNCT
ejpam-3427	580	51	}	}	PUNCT
ejpam-3427	580	52	=	=	SYM
ejpam-3427	580	53	min{µa1×a2×	min{µa1×a2×	NOUN
ejpam-3427	580	54	...	...	PUNCT
ejpam-3427	580	55	×an	×an	NOUN
ejpam-3427	580	56	(	(	PUNCT
ejpam-3427	580	57	x1	x1	PROPN
ejpam-3427	580	58	,	,	PUNCT
ejpam-3427	580	59	x2	x2	PROPN
ejpam-3427	580	60	,	,	PUNCT
ejpam-3427	580	61	...	...	PUNCT
ejpam-3427	580	62	,	,	PUNCT
ejpam-3427	580	63	xn	xn	PROPN
ejpam-3427	580	64	)	)	PUNCT
ejpam-3427	580	65	,	,	PUNCT
ejpam-3427	580	66	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	580	67	...	...	PUNCT
ejpam-3427	580	68	×an	×an	PROPN
ejpam-3427	580	69	(	(	PUNCT
ejpam-3427	580	70	y1	y1	PROPN
ejpam-3427	580	71	,	,	PUNCT
ejpam-3427	580	72	y2	y2	PROPN
ejpam-3427	580	73	,	,	PUNCT
ejpam-3427	580	74	...	...	PUNCT
ejpam-3427	580	75	,	,	PUNCT
ejpam-3427	580	76	yn	yn	PROPN
ejpam-3427	580	77	)	)	PUNCT
ejpam-3427	580	78	}	}	PUNCT
ejpam-3427	580	79	.	.	PUNCT
ejpam-3427	581	1	thus	thus	ADV
ejpam-3427	581	2	�	�	VERB
ejpam-3427	581	3	a1	a1	NOUN
ejpam-3427	581	4	×	×	PROPN
ejpam-3427	581	5	a2	a2	PROPN
ejpam-3427	581	6	×	×	NOUN
ejpam-3427	581	7	...	...	PUNCT
ejpam-3427	581	8	×	×	NOUN
ejpam-3427	581	9	an	an	PRON
ejpam-3427	581	10	=	=	X
ejpam-3427	581	11	(	(	PUNCT
ejpam-3427	581	12	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	581	13	...	...	PUNCT
ejpam-3427	581	14	×an	×an	PROPN
ejpam-3427	581	15	,	,	PUNCT
ejpam-3427	581	16	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	581	17	...	...	PUNCT
ejpam-3427	581	18	×an	×an	PROPN
ejpam-3427	581	19	)	)	PUNCT
ejpam-3427	581	20	is	be	AUX
ejpam-3427	581	21	an	an	DET
ejpam-3427	581	22	intuitionistic	intuitionistic	ADJ
ejpam-3427	581	23	anti	anti	ADJ
ejpam-3427	581	24	fuzzy	fuzzy	ADJ
ejpam-3427	581	25	la	la	NOUN
ejpam-3427	581	26	-	-	PUNCT
ejpam-3427	581	27	subring	subring	NOUN
ejpam-3427	581	28	of	of	ADP
ejpam-3427	581	29	an	an	DET
ejpam-3427	581	30	la	la	ADJ
ejpam-3427	581	31	-	-	PUNCT
ejpam-3427	581	32	ring	ring	NOUN
ejpam-3427	581	33	r1	r1	NOUN
ejpam-3427	581	34	×r2	×r2	PROPN
ejpam-3427	581	35	×	×	NOUN
ejpam-3427	581	36	...	...	PUNCT
ejpam-3427	581	37	×rn	×rn	NOUN
ejpam-3427	581	38	.	.	PUNCT
ejpam-3427	582	1	now	now	ADV
ejpam-3427	582	2	=	=	SYM
ejpam-3427	582	3	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	582	4	...	...	PUNCT
ejpam-3427	582	5	×an	×an	NOUN
ejpam-3427	582	6	(	(	PUNCT
ejpam-3427	582	7	(	(	PUNCT
ejpam-3427	582	8	x1	x1	PROPN
ejpam-3427	582	9	,	,	PUNCT
ejpam-3427	582	10	x2	x2	PROPN
ejpam-3427	582	11	,	,	PUNCT
ejpam-3427	582	12	...	...	PUNCT
ejpam-3427	582	13	,	,	PUNCT
ejpam-3427	582	14	xn	xn	X
ejpam-3427	582	15	)	)	PUNCT
ejpam-3427	582	16	◦	◦	NOUN
ejpam-3427	582	17	(	(	PUNCT
ejpam-3427	582	18	y1	y1	INTJ
ejpam-3427	582	19	,	,	PUNCT
ejpam-3427	582	20	y2	y2	PROPN
ejpam-3427	582	21	,	,	PUNCT
ejpam-3427	582	22	...	...	PUNCT
ejpam-3427	582	23	,	,	PUNCT
ejpam-3427	582	24	yn	yn	PROPN
ejpam-3427	582	25	)	)	PUNCT
ejpam-3427	582	26	)	)	PUNCT
ejpam-3427	583	1	=	=	SYM
ejpam-3427	583	2	1−	1−	NUM
ejpam-3427	583	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	583	4	...	...	PUNCT
ejpam-3427	583	5	×an((x1	×an((x1	PROPN
ejpam-3427	583	6	,	,	PUNCT
ejpam-3427	583	7	x2	x2	PROPN
ejpam-3427	583	8	,	,	PUNCT
ejpam-3427	583	9	...	...	PUNCT
ejpam-3427	583	10	,	,	PUNCT
ejpam-3427	583	11	xn	xn	X
ejpam-3427	583	12	)	)	PUNCT
ejpam-3427	583	13	◦	◦	NOUN
ejpam-3427	583	14	(	(	PUNCT
ejpam-3427	583	15	y1	y1	INTJ
ejpam-3427	583	16	,	,	PUNCT
ejpam-3427	583	17	y2	y2	PROPN
ejpam-3427	583	18	,	,	PUNCT
ejpam-3427	583	19	...	...	PUNCT
ejpam-3427	583	20	,	,	PUNCT
ejpam-3427	583	21	yn	yn	PROPN
ejpam-3427	583	22	)	)	PUNCT
ejpam-3427	583	23	)	)	PUNCT
ejpam-3427	583	24	=	=	SYM
ejpam-3427	584	1	1−	1−	NUM
ejpam-3427	584	2	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	584	3	...	...	PUNCT
ejpam-3427	584	4	×an((y1	×an((y1	PROPN
ejpam-3427	584	5	,	,	PUNCT
ejpam-3427	584	6	y2	y2	PROPN
ejpam-3427	584	7	,	,	PUNCT
ejpam-3427	584	8	...	...	PUNCT
ejpam-3427	584	9	,	,	PUNCT
ejpam-3427	584	10	yn	yn	PROPN
ejpam-3427	584	11	)	)	PUNCT
ejpam-3427	584	12	◦	◦	NOUN
ejpam-3427	584	13	(	(	PUNCT
ejpam-3427	584	14	x1	x1	PROPN
ejpam-3427	584	15	,	,	PUNCT
ejpam-3427	584	16	x2	x2	PROPN
ejpam-3427	584	17	,	,	PUNCT
ejpam-3427	584	18	...	...	PUNCT
ejpam-3427	584	19	,	,	PUNCT
ejpam-3427	584	20	xn	xn	PROPN
ejpam-3427	584	21	)	)	PUNCT
ejpam-3427	584	22	)	)	PUNCT
ejpam-3427	585	1	=	=	SYM
ejpam-3427	585	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	585	3	...	...	PUNCT
ejpam-3427	585	4	×an	×an	NOUN
ejpam-3427	585	5	(	(	PUNCT
ejpam-3427	585	6	(	(	PUNCT
ejpam-3427	585	7	y1	y1	INTJ
ejpam-3427	585	8	,	,	PUNCT
ejpam-3427	585	9	y2	y2	PROPN
ejpam-3427	585	10	,	,	PUNCT
ejpam-3427	585	11	...	...	PUNCT
ejpam-3427	585	12	,	,	PUNCT
ejpam-3427	585	13	yn	yn	PROPN
ejpam-3427	585	14	)	)	PUNCT
ejpam-3427	585	15	◦	◦	NOUN
ejpam-3427	585	16	(	(	PUNCT
ejpam-3427	585	17	x1	x1	PROPN
ejpam-3427	585	18	,	,	PUNCT
ejpam-3427	585	19	x2	x2	PROPN
ejpam-3427	585	20	,	,	PUNCT
ejpam-3427	585	21	...	...	PUNCT
ejpam-3427	585	22	,	,	PUNCT
ejpam-3427	585	23	xn	xn	PROPN
ejpam-3427	585	24	)	)	PUNCT
ejpam-3427	585	25	)	)	PUNCT
ejpam-3427	585	26	.	.	PUNCT
ejpam-3427	586	1	hence	hence	ADV
ejpam-3427	586	2	�	�	NOUN
ejpam-3427	586	3	a1	a1	PROPN
ejpam-3427	586	4	×	×	PROPN
ejpam-3427	586	5	a2	a2	PROPN
ejpam-3427	586	6	×	×	NOUN
ejpam-3427	586	7	...	...	PUNCT
ejpam-3427	586	8	×	×	NOUN
ejpam-3427	586	9	an	an	PRON
ejpam-3427	586	10	=	=	X
ejpam-3427	586	11	(	(	PUNCT
ejpam-3427	586	12	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	586	13	...	...	PUNCT
ejpam-3427	586	14	×an	×an	PROPN
ejpam-3427	586	15	,	,	PUNCT
ejpam-3427	586	16	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	586	17	...	...	PUNCT
ejpam-3427	586	18	×an	×an	PROPN
ejpam-3427	586	19	)	)	PUNCT
ejpam-3427	586	20	is	be	AUX
ejpam-3427	586	21	an	an	DET
ejpam-3427	586	22	intuitionistic	intuitionistic	ADJ
ejpam-3427	586	23	anti	anti	ADJ
ejpam-3427	586	24	fuzzy	fuzzy	ADJ
ejpam-3427	586	25	normal	normal	ADJ
ejpam-3427	586	26	la	la	NOUN
ejpam-3427	586	27	-	-	PUNCT
ejpam-3427	586	28	subring	subring	NOUN
ejpam-3427	586	29	of	of	ADP
ejpam-3427	586	30	an	an	DET
ejpam-3427	586	31	la	la	ADJ
ejpam-3427	586	32	-	-	PUNCT
ejpam-3427	586	33	ring	ring	NOUN
ejpam-3427	586	34	r1	r1	NOUN
ejpam-3427	586	35	×r2	×r2	PROPN
ejpam-3427	586	36	×	×	NOUN
ejpam-3427	586	37	...	...	PUNCT
ejpam-3427	586	38	×rn	×rn	NOUN
ejpam-3427	586	39	.	.	PUNCT
ejpam-3427	587	1	corollary	corollary	ADJ
ejpam-3427	587	2	6	6	NUM
ejpam-3427	587	3	.	.	PUNCT
ejpam-3427	588	1	an	an	DET
ejpam-3427	588	2	ifs	ifs	PROPN
ejpam-3427	588	3	a	a	DET
ejpam-3427	588	4	=	=	NOUN
ejpam-3427	588	5	a1	a1	NOUN
ejpam-3427	588	6	×	×	PROPN
ejpam-3427	588	7	a2	a2	PROPN
ejpam-3427	588	8	×	×	NOUN
ejpam-3427	588	9	...	...	PUNCT
ejpam-3427	588	10	×	×	PROPN
ejpam-3427	588	11	an	an	PRON
ejpam-3427	588	12	is	be	AUX
ejpam-3427	588	13	an	an	DET
ejpam-3427	588	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	588	15	anti	anti	ADJ
ejpam-3427	588	16	fuzzy	fuzzy	ADJ
ejpam-3427	588	17	normal	normal	ADJ
ejpam-3427	588	18	lasubring	lasubring	NOUN
ejpam-3427	588	19	of	of	ADP
ejpam-3427	588	20	an	an	DET
ejpam-3427	588	21	la	la	ADJ
ejpam-3427	588	22	-	-	PUNCT
ejpam-3427	588	23	ring	ring	NOUN
ejpam-3427	588	24	r1×r2×	r1×r2×	NOUN
ejpam-3427	588	25	...	...	PUNCT
ejpam-3427	588	26	×rn	×rn	VERB
ejpam-3427	588	27	if	if	SCONJ
ejpam-3427	588	28	and	and	CCONJ
ejpam-3427	588	29	only	only	ADV
ejpam-3427	588	30	if	if	SCONJ
ejpam-3427	588	31	�	�	NOUN
ejpam-3427	588	32	a	a	NOUN
ejpam-3427	588	33	=	=	X
ejpam-3427	588	34	(	(	PUNCT
ejpam-3427	588	35	µa	µa	PROPN
ejpam-3427	588	36	,	,	PUNCT
ejpam-3427	588	37	µa	µa	NOUN
ejpam-3427	588	38	)	)	PUNCT
ejpam-3427	588	39	(	(	PUNCT
ejpam-3427	588	40	resp	resp	NOUN
ejpam-3427	588	41	.	.	PUNCT
ejpam-3427	589	1	♦	♦	PROPN
ejpam-3427	589	2	a	a	PROPN
ejpam-3427	589	3	=	=	X
ejpam-3427	589	4	(	(	PUNCT
ejpam-3427	589	5	γa	γa	PROPN
ejpam-3427	589	6	,	,	PUNCT
ejpam-3427	589	7	γa	γa	PROPN
ejpam-3427	589	8	)	)	PUNCT
ejpam-3427	589	9	)	)	PUNCT
ejpam-3427	589	10	is	be	AUX
ejpam-3427	589	11	an	an	DET
ejpam-3427	589	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	589	13	anti	anti	ADJ
ejpam-3427	589	14	fuzzy	fuzzy	ADJ
ejpam-3427	589	15	normal	normal	ADJ
ejpam-3427	589	16	la	la	NOUN
ejpam-3427	589	17	-	-	PUNCT
ejpam-3427	589	18	subring	subring	NOUN
ejpam-3427	589	19	of	of	ADP
ejpam-3427	589	20	an	an	DET
ejpam-3427	589	21	la	la	ADJ
ejpam-3427	589	22	-	-	PUNCT
ejpam-3427	589	23	ring	ring	NOUN
ejpam-3427	589	24	r1	r1	NOUN
ejpam-3427	589	25	×r2	×r2	PROPN
ejpam-3427	589	26	×	×	NOUN
ejpam-3427	589	27	...	...	PUNCT
ejpam-3427	589	28	×rn	×rn	NOUN
ejpam-3427	589	29	.	.	PUNCT
ejpam-3427	590	1	theorem	theorem	VERB
ejpam-3427	590	2	8	8	NUM
ejpam-3427	590	3	.	.	PUNCT
ejpam-3427	591	1	an	an	DET
ejpam-3427	591	2	ifs	ifs	PROPN
ejpam-3427	591	3	a1×a2×	a1×a2×	VERB
ejpam-3427	591	4	...	...	PUNCT
ejpam-3427	591	5	×an	×an	NOUN
ejpam-3427	591	6	=	=	SYM
ejpam-3427	591	7	(	(	PUNCT
ejpam-3427	591	8	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	591	9	...	...	PUNCT
ejpam-3427	591	10	×an	×an	PROPN
ejpam-3427	591	11	,	,	PUNCT
ejpam-3427	591	12	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	591	13	...	...	PUNCT
ejpam-3427	591	14	×an	×an	PROPN
ejpam-3427	591	15	)	)	PUNCT
ejpam-3427	591	16	is	be	AUX
ejpam-3427	591	17	an	an	DET
ejpam-3427	591	18	intuitionistic	intuitionistic	ADJ
ejpam-3427	591	19	anti	anti	ADJ
ejpam-3427	591	20	fuzzy	fuzzy	ADJ
ejpam-3427	591	21	normal	normal	ADJ
ejpam-3427	591	22	la	la	NOUN
ejpam-3427	591	23	-	-	PUNCT
ejpam-3427	591	24	subring	subring	NOUN
ejpam-3427	591	25	of	of	ADP
ejpam-3427	591	26	an	an	DET
ejpam-3427	591	27	la	la	ADJ
ejpam-3427	591	28	-	-	PUNCT
ejpam-3427	591	29	ring	ring	NOUN
ejpam-3427	591	30	r1	r1	NOUN
ejpam-3427	591	31	×	×	NOUN
ejpam-3427	591	32	r2	r2	PROPN
ejpam-3427	591	33	×	×	NOUN
ejpam-3427	591	34	...	...	PUNCT
ejpam-3427	592	1	×	×	PROPN
ejpam-3427	592	2	rn	rn	NOUN
ejpam-3427	592	3	if	if	SCONJ
ejpam-3427	593	1	and	and	CCONJ
ejpam-3427	593	2	only	only	ADV
ejpam-3427	593	3	if	if	SCONJ
ejpam-3427	593	4	the	the	DET
ejpam-3427	593	5	fuzzy	fuzzy	ADJ
ejpam-3427	593	6	subsets	subset	NOUN
ejpam-3427	593	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	593	8	...	...	PUNCT
ejpam-3427	593	9	×an	×an	PROPN
ejpam-3427	593	10	and	and	CCONJ
ejpam-3427	593	11	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	593	12	...	...	PUNCT
ejpam-3427	594	1	×an	×an	PROPN
ejpam-3427	594	2	are	be	AUX
ejpam-3427	594	3	anti	anti	X
ejpam-3427	594	4	fuzzy	fuzzy	ADJ
ejpam-3427	594	5	normal	normal	ADJ
ejpam-3427	594	6	la	la	ADJ
ejpam-3427	594	7	-	-	PUNCT
ejpam-3427	594	8	subrings	subring	NOUN
ejpam-3427	594	9	of	of	ADP
ejpam-3427	594	10	an	an	DET
ejpam-3427	594	11	la	la	ADJ
ejpam-3427	594	12	-	-	PUNCT
ejpam-3427	594	13	ring	ring	NOUN
ejpam-3427	594	14	r1	r1	NOUN
ejpam-3427	594	15	×r2	×r2	PROPN
ejpam-3427	594	16	×	×	NOUN
ejpam-3427	594	17	...	...	PUNCT
ejpam-3427	594	18	×rn	×rn	NOUN
ejpam-3427	594	19	.	.	PUNCT
ejpam-3427	595	1	proof	proof	NOUN
ejpam-3427	595	2	.	.	PUNCT
ejpam-3427	596	1	let	let	VERB
ejpam-3427	596	2	a1×a2×	a1×a2×	VERB
ejpam-3427	596	3	...	...	PUNCT
ejpam-3427	596	4	×an	×an	PROPN
ejpam-3427	596	5	=	=	SYM
ejpam-3427	596	6	(	(	PUNCT
ejpam-3427	596	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	596	8	...	...	PUNCT
ejpam-3427	596	9	×an	×an	PROPN
ejpam-3427	596	10	,	,	PUNCT
ejpam-3427	596	11	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	596	12	...	...	PUNCT
ejpam-3427	596	13	×an	×an	PROPN
ejpam-3427	596	14	)	)	PUNCT
ejpam-3427	596	15	be	be	AUX
ejpam-3427	596	16	an	an	DET
ejpam-3427	596	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	596	18	anti	anti	ADJ
ejpam-3427	596	19	fuzzy	fuzzy	ADJ
ejpam-3427	596	20	normal	normal	ADJ
ejpam-3427	596	21	la	la	NOUN
ejpam-3427	596	22	-	-	PUNCT
ejpam-3427	596	23	subring	subring	NOUN
ejpam-3427	596	24	of	of	ADP
ejpam-3427	596	25	an	an	DET
ejpam-3427	596	26	la	la	ADJ
ejpam-3427	596	27	-	-	PUNCT
ejpam-3427	596	28	ring	ring	NOUN
ejpam-3427	596	29	r1×r2×	r1×r2×	NOUN
ejpam-3427	596	30	...	...	PUNCT
ejpam-3427	596	31	×rn	×rn	PROPN
ejpam-3427	596	32	.	.	PUNCT
ejpam-3427	597	1	this	this	PRON
ejpam-3427	597	2	implies	imply	VERB
ejpam-3427	597	3	that	that	SCONJ
ejpam-3427	597	4	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	597	5	...	...	PUNCT
ejpam-3427	597	6	×an	×an	PROPN
ejpam-3427	597	7	is	be	AUX
ejpam-3427	597	8	an	an	DET
ejpam-3427	597	9	anti	anti	ADJ
ejpam-3427	597	10	fuzzy	fuzzy	ADJ
ejpam-3427	597	11	normal	normal	ADJ
ejpam-3427	597	12	la	la	NOUN
ejpam-3427	597	13	-	-	PUNCT
ejpam-3427	597	14	subring	subring	NOUN
ejpam-3427	597	15	of	of	ADP
ejpam-3427	597	16	an	an	DET
ejpam-3427	597	17	la	la	ADJ
ejpam-3427	597	18	-	-	PUNCT
ejpam-3427	597	19	ring	ring	NOUN
ejpam-3427	597	20	r1×r2×	r1×r2×	NOUN
ejpam-3427	597	21	...	...	PUNCT
ejpam-3427	597	22	×rn	×rn	PROPN
ejpam-3427	597	23	.	.	PUNCT
ejpam-3427	598	1	we	we	PRON
ejpam-3427	598	2	have	have	VERB
ejpam-3427	598	3	to	to	PART
ejpam-3427	598	4	show	show	VERB
ejpam-3427	598	5	that	that	PRON
ejpam-3427	598	6	γa1×a2×	γa1×a2×	ADV
ejpam-3427	598	7	...	...	PUNCT
ejpam-3427	598	8	×an	×an	PROPN
ejpam-3427	598	9	is	be	AUX
ejpam-3427	598	10	also	also	ADV
ejpam-3427	598	11	an	an	DET
ejpam-3427	598	12	anti	anti	ADJ
ejpam-3427	598	13	fuzzy	fuzzy	ADJ
ejpam-3427	598	14	normal	normal	ADJ
ejpam-3427	598	15	la	la	NOUN
ejpam-3427	598	16	-	-	PUNCT
ejpam-3427	598	17	subring	subring	NOUN
ejpam-3427	598	18	of	of	ADP
ejpam-3427	598	19	an	an	DET
ejpam-3427	598	20	la	la	ADJ
ejpam-3427	598	21	-	-	PUNCT
ejpam-3427	598	22	ring	ring	NOUN
ejpam-3427	598	23	r1	r1	NOUN
ejpam-3427	598	24	×	×	NOUN
ejpam-3427	598	25	r2	r2	PROPN
ejpam-3427	598	26	×	×	NOUN
ejpam-3427	598	27	...	...	PUNCT
ejpam-3427	598	28	×	×	PROPN
ejpam-3427	598	29	rn	rn	PROPN
ejpam-3427	598	30	.	.	PUNCT
ejpam-3427	599	1	now	now	ADV
ejpam-3427	599	2	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	599	3	...	...	PUNCT
ejpam-3427	600	1	×an	×an	NOUN
ejpam-3427	600	2	(	(	PUNCT
ejpam-3427	600	3	(	(	PUNCT
ejpam-3427	600	4	x1	x1	PROPN
ejpam-3427	600	5	,	,	PUNCT
ejpam-3427	600	6	x2	x2	PROPN
ejpam-3427	600	7	,	,	PUNCT
ejpam-3427	600	8	...	...	PUNCT
ejpam-3427	600	9	,	,	PUNCT
ejpam-3427	600	10	xn)−	xn)−	X
ejpam-3427	600	11	(	(	PUNCT
ejpam-3427	600	12	y1	y1	INTJ
ejpam-3427	600	13	,	,	PUNCT
ejpam-3427	600	14	y2	y2	PROPN
ejpam-3427	600	15	,	,	PUNCT
ejpam-3427	600	16	...	...	PUNCT
ejpam-3427	600	17	,	,	PUNCT
ejpam-3427	600	18	yn	yn	PROPN
ejpam-3427	600	19	)	)	PUNCT
ejpam-3427	600	20	)	)	PUNCT
ejpam-3427	601	1	=	=	SYM
ejpam-3427	602	1	1−	1−	NUM
ejpam-3427	602	2	γa1×a2×	γa1×a2×	ADV
ejpam-3427	602	3	...	...	PUNCT
ejpam-3427	602	4	×an((x1	×an((x1	ADP
ejpam-3427	602	5	,	,	PUNCT
ejpam-3427	602	6	x2	x2	PROPN
ejpam-3427	602	7	,	,	PUNCT
ejpam-3427	602	8	...	...	PUNCT
ejpam-3427	602	9	,	,	PUNCT
ejpam-3427	602	10	xn)−	xn)−	X
ejpam-3427	602	11	(	(	PUNCT
ejpam-3427	602	12	y1	y1	INTJ
ejpam-3427	602	13	,	,	PUNCT
ejpam-3427	602	14	y2	y2	PROPN
ejpam-3427	602	15	,	,	PUNCT
ejpam-3427	602	16	...	...	PUNCT
ejpam-3427	602	17	,	,	PUNCT
ejpam-3427	602	18	yn	yn	PROPN
ejpam-3427	602	19	)	)	PUNCT
ejpam-3427	602	20	)	)	PUNCT
ejpam-3427	602	21	≤	≤	NOUN
ejpam-3427	602	22	1−min{γa1×a2×	1−min{γa1×a2×	NUM
ejpam-3427	602	23	...	...	PUNCT
ejpam-3427	602	24	×an(x1	×an(x1	NOUN
ejpam-3427	602	25	,	,	PUNCT
ejpam-3427	602	26	x2	x2	PROPN
ejpam-3427	602	27	,	,	PUNCT
ejpam-3427	602	28	...	...	PUNCT
ejpam-3427	602	29	,	,	PUNCT
ejpam-3427	602	30	xn	xn	PROPN
ejpam-3427	602	31	)	)	PUNCT
ejpam-3427	602	32	,	,	PUNCT
ejpam-3427	602	33	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	602	34	...	...	PUNCT
ejpam-3427	602	35	×an(y1	×an(y1	NOUN
ejpam-3427	602	36	,	,	PUNCT
ejpam-3427	602	37	y2	y2	PROPN
ejpam-3427	602	38	,	,	PUNCT
ejpam-3427	602	39	...	...	PUNCT
ejpam-3427	602	40	,	,	PUNCT
ejpam-3427	602	41	yn	yn	PROPN
ejpam-3427	602	42	)	)	PUNCT
ejpam-3427	602	43	}	}	PUNCT
ejpam-3427	602	44	=	=	SYM
ejpam-3427	603	1	max{1−	max{1−	PROPN
ejpam-3427	603	2	γa1×a2×	γa1×a2×	ADV
ejpam-3427	603	3	...	...	PUNCT
ejpam-3427	604	1	×an(x1	×an(x1	NOUN
ejpam-3427	604	2	,	,	PUNCT
ejpam-3427	604	3	x2	x2	PROPN
ejpam-3427	604	4	,	,	PUNCT
ejpam-3427	604	5	...	...	PUNCT
ejpam-3427	604	6	,	,	PUNCT
ejpam-3427	604	7	xn	xn	PROPN
ejpam-3427	604	8	)	)	PUNCT
ejpam-3427	604	9	,	,	PUNCT
ejpam-3427	604	10	1−	1−	NUM
ejpam-3427	604	11	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	604	12	...	...	PUNCT
ejpam-3427	604	13	×an(y1	×an(y1	NOUN
ejpam-3427	604	14	,	,	PUNCT
ejpam-3427	604	15	y2	y2	PROPN
ejpam-3427	604	16	,	,	PUNCT
ejpam-3427	604	17	...	...	PUNCT
ejpam-3427	604	18	,	,	PUNCT
ejpam-3427	604	19	yn	yn	PROPN
ejpam-3427	604	20	)	)	PUNCT
ejpam-3427	604	21	}	}	PUNCT
ejpam-3427	604	22	=	=	SYM
ejpam-3427	604	23	max{γa1×a2×	max{γa1×a2×	PROPN
ejpam-3427	604	24	...	...	PUNCT
ejpam-3427	604	25	×an	×an	PROPN
ejpam-3427	604	26	(	(	PUNCT
ejpam-3427	604	27	x1	x1	PROPN
ejpam-3427	604	28	,	,	PUNCT
ejpam-3427	604	29	x2	x2	PROPN
ejpam-3427	604	30	,	,	PUNCT
ejpam-3427	604	31	...	...	PUNCT
ejpam-3427	604	32	,	,	PUNCT
ejpam-3427	604	33	xn	xn	PROPN
ejpam-3427	604	34	)	)	PUNCT
ejpam-3427	604	35	,	,	PUNCT
ejpam-3427	604	36	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	604	37	...	...	PUNCT
ejpam-3427	605	1	×an	×an	PROPN
ejpam-3427	605	2	(	(	PUNCT
ejpam-3427	605	3	y1	y1	PROPN
ejpam-3427	605	4	,	,	PUNCT
ejpam-3427	605	5	y2	y2	PROPN
ejpam-3427	605	6	,	,	PUNCT
ejpam-3427	605	7	...	...	PUNCT
ejpam-3427	605	8	,	,	PUNCT
ejpam-3427	605	9	yn	yn	PROPN
ejpam-3427	605	10	)	)	PUNCT
ejpam-3427	605	11	}	}	PUNCT
ejpam-3427	605	12	.	.	PUNCT
ejpam-3427	606	1	and	and	CCONJ
ejpam-3427	606	2	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	606	3	...	...	PUNCT
ejpam-3427	607	1	×an	×an	PROPN
ejpam-3427	607	2	(	(	PUNCT
ejpam-3427	607	3	(	(	PUNCT
ejpam-3427	607	4	x1	x1	PROPN
ejpam-3427	607	5	,	,	PUNCT
ejpam-3427	607	6	x2	x2	PROPN
ejpam-3427	607	7	,	,	PUNCT
ejpam-3427	607	8	...	...	PUNCT
ejpam-3427	607	9	,	,	PUNCT
ejpam-3427	607	10	xn	xn	X
ejpam-3427	607	11	)	)	PUNCT
ejpam-3427	607	12	◦	◦	NOUN
ejpam-3427	607	13	(	(	PUNCT
ejpam-3427	607	14	y1	y1	INTJ
ejpam-3427	607	15	,	,	PUNCT
ejpam-3427	607	16	y2	y2	PROPN
ejpam-3427	607	17	,	,	PUNCT
ejpam-3427	607	18	...	...	PUNCT
ejpam-3427	607	19	,	,	PUNCT
ejpam-3427	607	20	yn	yn	PROPN
ejpam-3427	607	21	)	)	PUNCT
ejpam-3427	607	22	)	)	PUNCT
ejpam-3427	607	23	k.	k.	PROPN
ejpam-3427	608	1	nasreen	nasreen	PROPN
ejpam-3427	608	2	/	/	SYM
ejpam-3427	608	3	eur	eur	PROPN
ejpam-3427	608	4	.	.	PUNCT
ejpam-3427	609	1	j.	j.	PROPN
ejpam-3427	609	2	pure	pure	PROPN
ejpam-3427	609	3	appl	appl	PROPN
ejpam-3427	609	4	.	.	PROPN
ejpam-3427	609	5	math	math	PROPN
ejpam-3427	609	6	,	,	PUNCT
ejpam-3427	609	7	12	12	NUM
ejpam-3427	609	8	(	(	PUNCT
ejpam-3427	609	9	2	2	NUM
ejpam-3427	609	10	)	)	PUNCT
ejpam-3427	609	11	(	(	PUNCT
ejpam-3427	609	12	2019	2019	NUM
ejpam-3427	609	13	)	)	PUNCT
ejpam-3427	609	14	,	,	PUNCT
ejpam-3427	609	15	622	622	NUM
ejpam-3427	609	16	-	-	SYM
ejpam-3427	609	17	648	648	NUM
ejpam-3427	609	18	643	643	NUM
ejpam-3427	609	19	=	=	SYM
ejpam-3427	609	20	1−	1−	NUM
ejpam-3427	609	21	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	609	22	...	...	PUNCT
ejpam-3427	609	23	×an((x1	×an((x1	ADP
ejpam-3427	609	24	,	,	PUNCT
ejpam-3427	609	25	x2	x2	PROPN
ejpam-3427	609	26	,	,	PUNCT
ejpam-3427	609	27	...	...	PUNCT
ejpam-3427	609	28	,	,	PUNCT
ejpam-3427	609	29	xn	xn	X
ejpam-3427	609	30	)	)	PUNCT
ejpam-3427	610	1	◦	◦	NOUN
ejpam-3427	610	2	(	(	PUNCT
ejpam-3427	610	3	y1	y1	INTJ
ejpam-3427	610	4	,	,	PUNCT
ejpam-3427	610	5	y2	y2	PROPN
ejpam-3427	610	6	,	,	PUNCT
ejpam-3427	610	7	...	...	PUNCT
ejpam-3427	610	8	,	,	PUNCT
ejpam-3427	610	9	yn	yn	PROPN
ejpam-3427	610	10	)	)	PUNCT
ejpam-3427	610	11	)	)	PUNCT
ejpam-3427	610	12	≤	≤	NOUN
ejpam-3427	610	13	1−min{γa1×a2×	1−min{γa1×a2×	NUM
ejpam-3427	610	14	...	...	PUNCT
ejpam-3427	610	15	×an(x1	×an(x1	NOUN
ejpam-3427	610	16	,	,	PUNCT
ejpam-3427	610	17	x2	x2	PROPN
ejpam-3427	610	18	,	,	PUNCT
ejpam-3427	610	19	...	...	PUNCT
ejpam-3427	610	20	,	,	PUNCT
ejpam-3427	610	21	xn	xn	PROPN
ejpam-3427	610	22	)	)	PUNCT
ejpam-3427	610	23	,	,	PUNCT
ejpam-3427	610	24	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	610	25	...	...	PUNCT
ejpam-3427	610	26	×an(y1	×an(y1	NOUN
ejpam-3427	610	27	,	,	PUNCT
ejpam-3427	610	28	y2	y2	PROPN
ejpam-3427	610	29	,	,	PUNCT
ejpam-3427	610	30	...	...	PUNCT
ejpam-3427	610	31	,	,	PUNCT
ejpam-3427	610	32	yn	yn	PROPN
ejpam-3427	610	33	)	)	PUNCT
ejpam-3427	610	34	}	}	PUNCT
ejpam-3427	610	35	=	=	SYM
ejpam-3427	611	1	max{1−	max{1−	PROPN
ejpam-3427	611	2	γa1×a2×	γa1×a2×	ADV
ejpam-3427	611	3	...	...	PUNCT
ejpam-3427	612	1	×an(x1	×an(x1	NOUN
ejpam-3427	612	2	,	,	PUNCT
ejpam-3427	612	3	x2	x2	PROPN
ejpam-3427	612	4	,	,	PUNCT
ejpam-3427	612	5	...	...	PUNCT
ejpam-3427	612	6	,	,	PUNCT
ejpam-3427	612	7	xn	xn	PROPN
ejpam-3427	612	8	)	)	PUNCT
ejpam-3427	612	9	,	,	PUNCT
ejpam-3427	612	10	1−	1−	NUM
ejpam-3427	612	11	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	612	12	...	...	PUNCT
ejpam-3427	612	13	×an(y1	×an(y1	NOUN
ejpam-3427	612	14	,	,	PUNCT
ejpam-3427	612	15	y2	y2	PROPN
ejpam-3427	612	16	,	,	PUNCT
ejpam-3427	612	17	...	...	PUNCT
ejpam-3427	612	18	,	,	PUNCT
ejpam-3427	612	19	yn	yn	PROPN
ejpam-3427	612	20	)	)	PUNCT
ejpam-3427	612	21	}	}	PUNCT
ejpam-3427	612	22	=	=	SYM
ejpam-3427	612	23	max{γa1×a2×	max{γa1×a2×	PROPN
ejpam-3427	612	24	...	...	PUNCT
ejpam-3427	612	25	×an	×an	PROPN
ejpam-3427	612	26	(	(	PUNCT
ejpam-3427	612	27	x1	x1	PROPN
ejpam-3427	612	28	,	,	PUNCT
ejpam-3427	612	29	x2	x2	PROPN
ejpam-3427	612	30	,	,	PUNCT
ejpam-3427	612	31	...	...	PUNCT
ejpam-3427	612	32	,	,	PUNCT
ejpam-3427	612	33	xn	xn	PROPN
ejpam-3427	612	34	)	)	PUNCT
ejpam-3427	612	35	,	,	PUNCT
ejpam-3427	612	36	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	612	37	...	...	PUNCT
ejpam-3427	613	1	×an	×an	PROPN
ejpam-3427	613	2	(	(	PUNCT
ejpam-3427	613	3	y1	y1	PROPN
ejpam-3427	613	4	,	,	PUNCT
ejpam-3427	613	5	y2	y2	PROPN
ejpam-3427	613	6	,	,	PUNCT
ejpam-3427	613	7	...	...	PUNCT
ejpam-3427	613	8	,	,	PUNCT
ejpam-3427	613	9	yn	yn	PROPN
ejpam-3427	613	10	)	)	PUNCT
ejpam-3427	613	11	}	}	PUNCT
ejpam-3427	613	12	.	.	PUNCT
ejpam-3427	614	1	thus	thus	ADV
ejpam-3427	614	2	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	614	3	...	...	PUNCT
ejpam-3427	614	4	×an	×an	PROPN
ejpam-3427	614	5	is	be	AUX
ejpam-3427	614	6	an	an	DET
ejpam-3427	614	7	anti	anti	ADJ
ejpam-3427	614	8	fuzzy	fuzzy	ADJ
ejpam-3427	614	9	la	la	NOUN
ejpam-3427	614	10	-	-	PUNCT
ejpam-3427	614	11	subring	subring	NOUN
ejpam-3427	614	12	of	of	ADP
ejpam-3427	614	13	an	an	DET
ejpam-3427	614	14	la	la	ADJ
ejpam-3427	614	15	-	-	PUNCT
ejpam-3427	614	16	ring	ring	NOUN
ejpam-3427	614	17	r1×r2×	r1×r2×	NOUN
ejpam-3427	614	18	...	...	PUNCT
ejpam-3427	614	19	×rn	×rn	PROPN
ejpam-3427	614	20	.	.	PUNCT
ejpam-3427	615	1	now	now	ADV
ejpam-3427	615	2	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	615	3	...	...	PUNCT
ejpam-3427	616	1	×an	×an	NOUN
ejpam-3427	616	2	(	(	PUNCT
ejpam-3427	616	3	(	(	PUNCT
ejpam-3427	616	4	x1	x1	PROPN
ejpam-3427	616	5	,	,	PUNCT
ejpam-3427	616	6	x2	x2	PROPN
ejpam-3427	616	7	,	,	PUNCT
ejpam-3427	616	8	...	...	PUNCT
ejpam-3427	616	9	,	,	PUNCT
ejpam-3427	616	10	xn	xn	X
ejpam-3427	616	11	)	)	PUNCT
ejpam-3427	616	12	◦	◦	NOUN
ejpam-3427	616	13	(	(	PUNCT
ejpam-3427	616	14	y1	y1	INTJ
ejpam-3427	616	15	,	,	PUNCT
ejpam-3427	616	16	y2	y2	PROPN
ejpam-3427	616	17	,	,	PUNCT
ejpam-3427	616	18	...	...	PUNCT
ejpam-3427	616	19	,	,	PUNCT
ejpam-3427	616	20	yn	yn	PROPN
ejpam-3427	616	21	)	)	PUNCT
ejpam-3427	616	22	)	)	PUNCT
ejpam-3427	617	1	=	=	SYM
ejpam-3427	618	1	1−	1−	NUM
ejpam-3427	618	2	γa1×a2×	γa1×a2×	ADV
ejpam-3427	618	3	...	...	PUNCT
ejpam-3427	618	4	×an((x1	×an((x1	ADP
ejpam-3427	618	5	,	,	PUNCT
ejpam-3427	618	6	x2	x2	PROPN
ejpam-3427	618	7	,	,	PUNCT
ejpam-3427	618	8	...	...	PUNCT
ejpam-3427	618	9	,	,	PUNCT
ejpam-3427	618	10	xn	xn	X
ejpam-3427	618	11	)	)	PUNCT
ejpam-3427	619	1	◦	◦	NOUN
ejpam-3427	619	2	(	(	PUNCT
ejpam-3427	619	3	y1	y1	INTJ
ejpam-3427	619	4	,	,	PUNCT
ejpam-3427	619	5	y2	y2	PROPN
ejpam-3427	619	6	,	,	PUNCT
ejpam-3427	619	7	...	...	PUNCT
ejpam-3427	619	8	,	,	PUNCT
ejpam-3427	619	9	yn	yn	PROPN
ejpam-3427	619	10	)	)	PUNCT
ejpam-3427	619	11	)	)	PUNCT
ejpam-3427	620	1	=	=	SYM
ejpam-3427	621	1	1−	1−	NUM
ejpam-3427	621	2	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	621	3	...	...	PUNCT
ejpam-3427	621	4	×an((y1	×an((y1	ADJ
ejpam-3427	621	5	,	,	PUNCT
ejpam-3427	621	6	y2	y2	PROPN
ejpam-3427	621	7	,	,	PUNCT
ejpam-3427	621	8	...	...	PUNCT
ejpam-3427	621	9	,	,	PUNCT
ejpam-3427	621	10	yn	yn	PROPN
ejpam-3427	621	11	)	)	PUNCT
ejpam-3427	621	12	◦	◦	NOUN
ejpam-3427	621	13	(	(	PUNCT
ejpam-3427	621	14	x1	x1	PROPN
ejpam-3427	621	15	,	,	PUNCT
ejpam-3427	621	16	x2	x2	PROPN
ejpam-3427	621	17	,	,	PUNCT
ejpam-3427	621	18	...	...	PUNCT
ejpam-3427	621	19	,	,	PUNCT
ejpam-3427	621	20	xn	xn	PROPN
ejpam-3427	621	21	)	)	PUNCT
ejpam-3427	621	22	)	)	PUNCT
ejpam-3427	622	1	=	=	PUNCT
ejpam-3427	622	2	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	622	3	...	...	PUNCT
ejpam-3427	623	1	×an	×an	NOUN
ejpam-3427	623	2	(	(	PUNCT
ejpam-3427	623	3	(	(	PUNCT
ejpam-3427	623	4	y1	y1	INTJ
ejpam-3427	623	5	,	,	PUNCT
ejpam-3427	623	6	y2	y2	PROPN
ejpam-3427	623	7	,	,	PUNCT
ejpam-3427	623	8	...	...	PUNCT
ejpam-3427	623	9	,	,	PUNCT
ejpam-3427	623	10	yn	yn	PROPN
ejpam-3427	623	11	)	)	PUNCT
ejpam-3427	623	12	◦	◦	NOUN
ejpam-3427	623	13	(	(	PUNCT
ejpam-3427	623	14	x1	x1	PROPN
ejpam-3427	623	15	,	,	PUNCT
ejpam-3427	623	16	x2	x2	PROPN
ejpam-3427	623	17	,	,	PUNCT
ejpam-3427	623	18	...	...	PUNCT
ejpam-3427	623	19	,	,	PUNCT
ejpam-3427	623	20	xn	xn	PROPN
ejpam-3427	623	21	)	)	PUNCT
ejpam-3427	623	22	)	)	PUNCT
ejpam-3427	623	23	.	.	PUNCT
ejpam-3427	624	1	hence	hence	ADV
ejpam-3427	624	2	γa1×a2×	γa1×a2×	ADV
ejpam-3427	624	3	...	...	PUNCT
ejpam-3427	624	4	×an	×an	PROPN
ejpam-3427	624	5	is	be	AUX
ejpam-3427	624	6	an	an	DET
ejpam-3427	624	7	anti	anti	ADJ
ejpam-3427	624	8	fuzzy	fuzzy	ADJ
ejpam-3427	624	9	normal	normal	ADJ
ejpam-3427	624	10	la	la	NOUN
ejpam-3427	624	11	-	-	PUNCT
ejpam-3427	624	12	subring	subring	NOUN
ejpam-3427	624	13	of	of	ADP
ejpam-3427	624	14	an	an	DET
ejpam-3427	624	15	la	la	ADJ
ejpam-3427	624	16	-	-	PUNCT
ejpam-3427	624	17	ring	ring	NOUN
ejpam-3427	624	18	r1×r2×	r1×r2×	NOUN
ejpam-3427	624	19	...	...	PUNCT
ejpam-3427	624	20	×rn	×rn	NOUN
ejpam-3427	624	21	.	.	PUNCT
ejpam-3427	625	1	conversely	conversely	ADV
ejpam-3427	625	2	,	,	PUNCT
ejpam-3427	625	3	suppose	suppose	VERB
ejpam-3427	625	4	that	that	SCONJ
ejpam-3427	625	5	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	625	6	...	...	PUNCT
ejpam-3427	625	7	×an	×an	PROPN
ejpam-3427	625	8	and	and	CCONJ
ejpam-3427	625	9	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	625	10	...	...	PUNCT
ejpam-3427	625	11	×an	×an	PROPN
ejpam-3427	625	12	are	be	AUX
ejpam-3427	625	13	anti	anti	X
ejpam-3427	625	14	fuzzy	fuzzy	ADJ
ejpam-3427	625	15	normal	normal	ADJ
ejpam-3427	625	16	lasubrings	lasubring	NOUN
ejpam-3427	625	17	of	of	ADP
ejpam-3427	625	18	an	an	DET
ejpam-3427	625	19	la	la	ADJ
ejpam-3427	625	20	-	-	PUNCT
ejpam-3427	625	21	ring	ring	NOUN
ejpam-3427	625	22	r1	r1	NOUN
ejpam-3427	625	23	×	×	NOUN
ejpam-3427	625	24	r2	r2	PROPN
ejpam-3427	625	25	×	×	NOUN
ejpam-3427	625	26	...	...	PUNCT
ejpam-3427	625	27	×	×	PROPN
ejpam-3427	625	28	rn	rn	PROPN
ejpam-3427	625	29	.	.	PUNCT
ejpam-3427	626	1	we	we	PRON
ejpam-3427	626	2	have	have	VERB
ejpam-3427	626	3	to	to	PART
ejpam-3427	626	4	show	show	VERB
ejpam-3427	626	5	that	that	SCONJ
ejpam-3427	626	6	a1	a1	NOUN
ejpam-3427	626	7	×	×	PROPN
ejpam-3427	626	8	a2	a2	PROPN
ejpam-3427	626	9	×	×	NOUN
ejpam-3427	626	10	...	...	PUNCT
ejpam-3427	626	11	×	×	NOUN
ejpam-3427	626	12	an	an	DET
ejpam-3427	626	13	=	=	X
ejpam-3427	626	14	(	(	PUNCT
ejpam-3427	626	15	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	626	16	...	...	PUNCT
ejpam-3427	626	17	×an	×an	PROPN
ejpam-3427	626	18	,	,	PUNCT
ejpam-3427	626	19	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	626	20	...	...	PUNCT
ejpam-3427	626	21	×an	×an	PROPN
ejpam-3427	626	22	)	)	PUNCT
ejpam-3427	626	23	is	be	AUX
ejpam-3427	626	24	an	an	DET
ejpam-3427	626	25	intuitionistic	intuitionistic	ADJ
ejpam-3427	626	26	anti	anti	ADJ
ejpam-3427	626	27	fuzzy	fuzzy	ADJ
ejpam-3427	626	28	normal	normal	ADJ
ejpam-3427	626	29	la	la	NOUN
ejpam-3427	626	30	-	-	PUNCT
ejpam-3427	626	31	subring	subring	NOUN
ejpam-3427	626	32	of	of	ADP
ejpam-3427	626	33	an	an	DET
ejpam-3427	626	34	laring	laring	NOUN
ejpam-3427	626	35	r1	r1	PROPN
ejpam-3427	626	36	×r2	×r2	PROPN
ejpam-3427	626	37	×	×	NOUN
ejpam-3427	626	38	...	...	PUNCT
ejpam-3427	626	39	×rn	×rn	NOUN
ejpam-3427	626	40	.	.	PUNCT
ejpam-3427	627	1	now	now	ADV
ejpam-3427	627	2	1−	1−	NUM
ejpam-3427	627	3	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	627	4	...	...	PUNCT
ejpam-3427	628	1	×an((x1	×an((x1	ADP
ejpam-3427	628	2	,	,	PUNCT
ejpam-3427	628	3	x2	x2	PROPN
ejpam-3427	628	4	,	,	PUNCT
ejpam-3427	628	5	...	...	PUNCT
ejpam-3427	628	6	,	,	PUNCT
ejpam-3427	628	7	xn)−	xn)−	X
ejpam-3427	628	8	(	(	PUNCT
ejpam-3427	628	9	y1	y1	INTJ
ejpam-3427	628	10	,	,	PUNCT
ejpam-3427	628	11	y2	y2	PROPN
ejpam-3427	628	12	,	,	PUNCT
ejpam-3427	628	13	...	...	PUNCT
ejpam-3427	628	14	,	,	PUNCT
ejpam-3427	628	15	yn	yn	PROPN
ejpam-3427	628	16	)	)	PUNCT
ejpam-3427	628	17	)	)	PUNCT
ejpam-3427	629	1	=	=	PUNCT
ejpam-3427	629	2	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	629	3	...	...	PUNCT
ejpam-3427	630	1	×an	×an	NOUN
ejpam-3427	630	2	(	(	PUNCT
ejpam-3427	630	3	(	(	PUNCT
ejpam-3427	630	4	x1	x1	PROPN
ejpam-3427	630	5	,	,	PUNCT
ejpam-3427	630	6	x2	x2	PROPN
ejpam-3427	630	7	,	,	PUNCT
ejpam-3427	630	8	...	...	PUNCT
ejpam-3427	630	9	,	,	PUNCT
ejpam-3427	630	10	xn)−	xn)−	X
ejpam-3427	630	11	(	(	PUNCT
ejpam-3427	630	12	y1	y1	INTJ
ejpam-3427	630	13	,	,	PUNCT
ejpam-3427	630	14	y2	y2	PROPN
ejpam-3427	630	15	,	,	PUNCT
ejpam-3427	630	16	...	...	PUNCT
ejpam-3427	630	17	,	,	PUNCT
ejpam-3427	630	18	yn	yn	PROPN
ejpam-3427	630	19	)	)	PUNCT
ejpam-3427	630	20	)	)	PUNCT
ejpam-3427	630	21	≤	≤	PROPN
ejpam-3427	631	1	max{γa1×a2×	max{γa1×a2×	PROPN
ejpam-3427	631	2	...	...	PUNCT
ejpam-3427	631	3	×an	×an	PROPN
ejpam-3427	631	4	(	(	PUNCT
ejpam-3427	631	5	x1	x1	PROPN
ejpam-3427	631	6	,	,	PUNCT
ejpam-3427	631	7	x2	x2	PROPN
ejpam-3427	631	8	,	,	PUNCT
ejpam-3427	631	9	...	...	PUNCT
ejpam-3427	631	10	,	,	PUNCT
ejpam-3427	631	11	xn	xn	PROPN
ejpam-3427	631	12	)	)	PUNCT
ejpam-3427	631	13	,	,	PUNCT
ejpam-3427	631	14	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	631	15	...	...	PUNCT
ejpam-3427	632	1	×an	×an	PROPN
ejpam-3427	632	2	(	(	PUNCT
ejpam-3427	632	3	y1	y1	PROPN
ejpam-3427	632	4	,	,	PUNCT
ejpam-3427	632	5	y2	y2	PROPN
ejpam-3427	632	6	,	,	PUNCT
ejpam-3427	632	7	...	...	PUNCT
ejpam-3427	632	8	,	,	PUNCT
ejpam-3427	632	9	yn	yn	PROPN
ejpam-3427	632	10	)	)	PUNCT
ejpam-3427	632	11	}	}	PUNCT
ejpam-3427	632	12	=	=	SYM
ejpam-3427	633	1	max{1−	max{1−	PROPN
ejpam-3427	633	2	γa1×a2×	γa1×a2×	ADV
ejpam-3427	633	3	...	...	PUNCT
ejpam-3427	634	1	×an(x1	×an(x1	NOUN
ejpam-3427	634	2	,	,	PUNCT
ejpam-3427	634	3	x2	x2	PROPN
ejpam-3427	634	4	,	,	PUNCT
ejpam-3427	634	5	...	...	PUNCT
ejpam-3427	634	6	,	,	PUNCT
ejpam-3427	634	7	xn	xn	PROPN
ejpam-3427	634	8	)	)	PUNCT
ejpam-3427	634	9	,	,	PUNCT
ejpam-3427	634	10	1−	1−	NUM
ejpam-3427	634	11	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	634	12	...	...	PUNCT
ejpam-3427	634	13	×an(y1	×an(y1	NOUN
ejpam-3427	634	14	,	,	PUNCT
ejpam-3427	634	15	y2	y2	PROPN
ejpam-3427	634	16	,	,	PUNCT
ejpam-3427	634	17	...	...	PUNCT
ejpam-3427	634	18	,	,	PUNCT
ejpam-3427	634	19	yn	yn	PROPN
ejpam-3427	634	20	)	)	PUNCT
ejpam-3427	634	21	}	}	PUNCT
ejpam-3427	634	22	=	=	SYM
ejpam-3427	634	23	1−min{γa1×a2×	1−min{γa1×a2×	NUM
ejpam-3427	634	24	...	...	PUNCT
ejpam-3427	634	25	×an(x1	×an(x1	NOUN
ejpam-3427	634	26	,	,	PUNCT
ejpam-3427	634	27	x2	x2	PROPN
ejpam-3427	634	28	,	,	PUNCT
ejpam-3427	634	29	...	...	PUNCT
ejpam-3427	634	30	,	,	PUNCT
ejpam-3427	634	31	xn	xn	PROPN
ejpam-3427	634	32	)	)	PUNCT
ejpam-3427	634	33	,	,	PUNCT
ejpam-3427	634	34	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	634	35	...	...	PUNCT
ejpam-3427	634	36	×an(y1	×an(y1	NOUN
ejpam-3427	634	37	,	,	PUNCT
ejpam-3427	634	38	y2	y2	PROPN
ejpam-3427	634	39	,	,	PUNCT
ejpam-3427	634	40	...	...	PUNCT
ejpam-3427	634	41	,	,	PUNCT
ejpam-3427	634	42	yn	yn	PROPN
ejpam-3427	634	43	)	)	PUNCT
ejpam-3427	634	44	}	}	PUNCT
ejpam-3427	634	45	and	and	CCONJ
ejpam-3427	634	46	1−	1−	NUM
ejpam-3427	634	47	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	634	48	...	...	PUNCT
ejpam-3427	634	49	×an((x1	×an((x1	PROPN
ejpam-3427	634	50	,	,	PUNCT
ejpam-3427	634	51	x2	x2	PROPN
ejpam-3427	634	52	,	,	PUNCT
ejpam-3427	634	53	...	...	PUNCT
ejpam-3427	634	54	,	,	PUNCT
ejpam-3427	634	55	xn	xn	X
ejpam-3427	634	56	)	)	PUNCT
ejpam-3427	634	57	◦	◦	NOUN
ejpam-3427	634	58	(	(	PUNCT
ejpam-3427	634	59	y1	y1	INTJ
ejpam-3427	634	60	,	,	PUNCT
ejpam-3427	634	61	y2	y2	PROPN
ejpam-3427	634	62	,	,	PUNCT
ejpam-3427	634	63	...	...	PUNCT
ejpam-3427	634	64	,	,	PUNCT
ejpam-3427	634	65	yn	yn	PROPN
ejpam-3427	634	66	)	)	PUNCT
ejpam-3427	634	67	)	)	PUNCT
ejpam-3427	635	1	=	=	PUNCT
ejpam-3427	636	1	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	636	2	...	...	PUNCT
ejpam-3427	637	1	×an	×an	NOUN
ejpam-3427	637	2	(	(	PUNCT
ejpam-3427	637	3	(	(	PUNCT
ejpam-3427	637	4	x1	x1	PROPN
ejpam-3427	637	5	,	,	PUNCT
ejpam-3427	637	6	x2	x2	PROPN
ejpam-3427	637	7	,	,	PUNCT
ejpam-3427	637	8	...	...	PUNCT
ejpam-3427	637	9	,	,	PUNCT
ejpam-3427	637	10	xn	xn	X
ejpam-3427	637	11	)	)	PUNCT
ejpam-3427	637	12	◦	◦	NOUN
ejpam-3427	637	13	(	(	PUNCT
ejpam-3427	637	14	y1	y1	INTJ
ejpam-3427	637	15	,	,	PUNCT
ejpam-3427	637	16	y2	y2	PROPN
ejpam-3427	637	17	,	,	PUNCT
ejpam-3427	637	18	...	...	PUNCT
ejpam-3427	637	19	,	,	PUNCT
ejpam-3427	637	20	yn	yn	PROPN
ejpam-3427	637	21	)	)	PUNCT
ejpam-3427	637	22	)	)	PUNCT
ejpam-3427	637	23	≤	≤	PROPN
ejpam-3427	638	1	max{γa1×a2×	max{γa1×a2×	PROPN
ejpam-3427	638	2	...	...	PUNCT
ejpam-3427	638	3	×an	×an	PROPN
ejpam-3427	638	4	(	(	PUNCT
ejpam-3427	638	5	x1	x1	PROPN
ejpam-3427	638	6	,	,	PUNCT
ejpam-3427	638	7	x2	x2	PROPN
ejpam-3427	638	8	,	,	PUNCT
ejpam-3427	638	9	...	...	PUNCT
ejpam-3427	638	10	,	,	PUNCT
ejpam-3427	638	11	xn	xn	PROPN
ejpam-3427	638	12	)	)	PUNCT
ejpam-3427	638	13	,	,	PUNCT
ejpam-3427	638	14	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	638	15	...	...	PUNCT
ejpam-3427	639	1	×an	×an	PROPN
ejpam-3427	639	2	(	(	PUNCT
ejpam-3427	639	3	y1	y1	PROPN
ejpam-3427	639	4	,	,	PUNCT
ejpam-3427	639	5	y2	y2	PROPN
ejpam-3427	639	6	,	,	PUNCT
ejpam-3427	639	7	...	...	PUNCT
ejpam-3427	639	8	,	,	PUNCT
ejpam-3427	639	9	yn	yn	PROPN
ejpam-3427	639	10	)	)	PUNCT
ejpam-3427	639	11	}	}	PUNCT
ejpam-3427	639	12	=	=	SYM
ejpam-3427	640	1	max{1−	max{1−	PROPN
ejpam-3427	640	2	γa1×a2×	γa1×a2×	ADV
ejpam-3427	640	3	...	...	PUNCT
ejpam-3427	641	1	×an(x1	×an(x1	NOUN
ejpam-3427	641	2	,	,	PUNCT
ejpam-3427	641	3	x2	x2	PROPN
ejpam-3427	641	4	,	,	PUNCT
ejpam-3427	641	5	...	...	PUNCT
ejpam-3427	641	6	,	,	PUNCT
ejpam-3427	641	7	xn	xn	PROPN
ejpam-3427	641	8	)	)	PUNCT
ejpam-3427	641	9	,	,	PUNCT
ejpam-3427	641	10	1−	1−	NUM
ejpam-3427	641	11	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	641	12	...	...	PUNCT
ejpam-3427	641	13	×an(y1	×an(y1	NOUN
ejpam-3427	641	14	,	,	PUNCT
ejpam-3427	641	15	y2	y2	PROPN
ejpam-3427	641	16	,	,	PUNCT
ejpam-3427	641	17	...	...	PUNCT
ejpam-3427	641	18	,	,	PUNCT
ejpam-3427	641	19	yn	yn	PROPN
ejpam-3427	641	20	)	)	PUNCT
ejpam-3427	641	21	}	}	PUNCT
ejpam-3427	641	22	=	=	SYM
ejpam-3427	641	23	1−min{γa1×a2×	1−min{γa1×a2×	NUM
ejpam-3427	641	24	...	...	PUNCT
ejpam-3427	641	25	×an(x1	×an(x1	NOUN
ejpam-3427	641	26	,	,	PUNCT
ejpam-3427	641	27	x2	x2	PROPN
ejpam-3427	641	28	,	,	PUNCT
ejpam-3427	641	29	...	...	PUNCT
ejpam-3427	641	30	,	,	PUNCT
ejpam-3427	641	31	xn	xn	PROPN
ejpam-3427	641	32	)	)	PUNCT
ejpam-3427	641	33	,	,	PUNCT
ejpam-3427	641	34	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	641	35	...	...	PUNCT
ejpam-3427	641	36	×an(y1	×an(y1	NOUN
ejpam-3427	641	37	,	,	PUNCT
ejpam-3427	641	38	y2	y2	PROPN
ejpam-3427	641	39	,	,	PUNCT
ejpam-3427	641	40	...	...	PUNCT
ejpam-3427	641	41	,	,	PUNCT
ejpam-3427	641	42	yn	yn	PROPN
ejpam-3427	641	43	)	)	PUNCT
ejpam-3427	641	44	}	}	PUNCT
ejpam-3427	641	45	.	.	PUNCT
ejpam-3427	642	1	thus	thus	ADV
ejpam-3427	642	2	a1×a2×	a1×a2×	NOUN
ejpam-3427	642	3	...	...	PUNCT
ejpam-3427	642	4	×an	×an	NOUN
ejpam-3427	642	5	=	=	SYM
ejpam-3427	642	6	(	(	PUNCT
ejpam-3427	642	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	642	8	...	...	PUNCT
ejpam-3427	642	9	×an	×an	PROPN
ejpam-3427	642	10	,	,	PUNCT
ejpam-3427	642	11	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	642	12	...	...	PUNCT
ejpam-3427	642	13	×an	×an	PROPN
ejpam-3427	642	14	)	)	PUNCT
ejpam-3427	642	15	is	be	AUX
ejpam-3427	642	16	an	an	DET
ejpam-3427	642	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	642	18	anti	anti	ADJ
ejpam-3427	642	19	fuzzy	fuzzy	ADJ
ejpam-3427	642	20	la	la	NOUN
ejpam-3427	642	21	-	-	PUNCT
ejpam-3427	642	22	subring	subring	NOUN
ejpam-3427	642	23	of	of	ADP
ejpam-3427	642	24	an	an	DET
ejpam-3427	642	25	la	la	ADJ
ejpam-3427	642	26	-	-	PUNCT
ejpam-3427	642	27	ring	ring	NOUN
ejpam-3427	642	28	r1	r1	NOUN
ejpam-3427	642	29	×r2	×r2	PROPN
ejpam-3427	642	30	×	×	NOUN
ejpam-3427	642	31	...	...	PUNCT
ejpam-3427	642	32	×rn	×rn	NOUN
ejpam-3427	642	33	.	.	PUNCT
ejpam-3427	643	1	now	now	ADV
ejpam-3427	643	2	1−	1−	NUM
ejpam-3427	643	3	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	643	4	...	...	PUNCT
ejpam-3427	644	1	×an((x1	×an((x1	ADP
ejpam-3427	644	2	,	,	PUNCT
ejpam-3427	644	3	x2	x2	PROPN
ejpam-3427	644	4	,	,	PUNCT
ejpam-3427	644	5	...	...	PUNCT
ejpam-3427	644	6	,	,	PUNCT
ejpam-3427	644	7	xn	xn	X
ejpam-3427	644	8	)	)	PUNCT
ejpam-3427	644	9	◦	◦	NOUN
ejpam-3427	644	10	(	(	PUNCT
ejpam-3427	644	11	y1	y1	INTJ
ejpam-3427	644	12	,	,	PUNCT
ejpam-3427	644	13	y2	y2	PROPN
ejpam-3427	644	14	,	,	PUNCT
ejpam-3427	644	15	...	...	PUNCT
ejpam-3427	644	16	,	,	PUNCT
ejpam-3427	644	17	yn	yn	PROPN
ejpam-3427	644	18	)	)	PUNCT
ejpam-3427	644	19	)	)	PUNCT
ejpam-3427	645	1	=	=	PUNCT
ejpam-3427	645	2	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	645	3	...	...	PUNCT
ejpam-3427	646	1	×an	×an	NOUN
ejpam-3427	646	2	(	(	PUNCT
ejpam-3427	646	3	(	(	PUNCT
ejpam-3427	646	4	x1	x1	PROPN
ejpam-3427	646	5	,	,	PUNCT
ejpam-3427	646	6	x2	x2	PROPN
ejpam-3427	646	7	,	,	PUNCT
ejpam-3427	646	8	...	...	PUNCT
ejpam-3427	646	9	,	,	PUNCT
ejpam-3427	646	10	xn	xn	X
ejpam-3427	646	11	)	)	PUNCT
ejpam-3427	646	12	◦	◦	NOUN
ejpam-3427	646	13	(	(	PUNCT
ejpam-3427	646	14	y1	y1	INTJ
ejpam-3427	646	15	,	,	PUNCT
ejpam-3427	646	16	y2	y2	PROPN
ejpam-3427	646	17	,	,	PUNCT
ejpam-3427	646	18	...	...	PUNCT
ejpam-3427	646	19	,	,	PUNCT
ejpam-3427	646	20	yn	yn	PROPN
ejpam-3427	646	21	)	)	PUNCT
ejpam-3427	646	22	)	)	PUNCT
ejpam-3427	647	1	=	=	PUNCT
ejpam-3427	647	2	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	647	3	...	...	PUNCT
ejpam-3427	648	1	×an	×an	NOUN
ejpam-3427	648	2	(	(	PUNCT
ejpam-3427	648	3	(	(	PUNCT
ejpam-3427	648	4	y1	y1	INTJ
ejpam-3427	648	5	,	,	PUNCT
ejpam-3427	648	6	y2	y2	PROPN
ejpam-3427	648	7	,	,	PUNCT
ejpam-3427	648	8	...	...	PUNCT
ejpam-3427	648	9	,	,	PUNCT
ejpam-3427	648	10	yn	yn	PROPN
ejpam-3427	648	11	)	)	PUNCT
ejpam-3427	648	12	◦	◦	NOUN
ejpam-3427	648	13	(	(	PUNCT
ejpam-3427	648	14	x1	x1	PROPN
ejpam-3427	648	15	,	,	PUNCT
ejpam-3427	648	16	x2	x2	PROPN
ejpam-3427	648	17	,	,	PUNCT
ejpam-3427	648	18	...	...	PUNCT
ejpam-3427	648	19	,	,	PUNCT
ejpam-3427	648	20	xn	xn	PROPN
ejpam-3427	648	21	)	)	PUNCT
ejpam-3427	648	22	)	)	PUNCT
ejpam-3427	649	1	=	=	SYM
ejpam-3427	650	1	1−	1−	NUM
ejpam-3427	650	2	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	650	3	...	...	PUNCT
ejpam-3427	650	4	×an((y1	×an((y1	ADJ
ejpam-3427	650	5	,	,	PUNCT
ejpam-3427	650	6	y2	y2	PROPN
ejpam-3427	650	7	,	,	PUNCT
ejpam-3427	650	8	...	...	PUNCT
ejpam-3427	650	9	,	,	PUNCT
ejpam-3427	650	10	yn	yn	PROPN
ejpam-3427	650	11	)	)	PUNCT
ejpam-3427	650	12	◦	◦	NOUN
ejpam-3427	650	13	(	(	PUNCT
ejpam-3427	650	14	x1	x1	PROPN
ejpam-3427	650	15	,	,	PUNCT
ejpam-3427	650	16	x2	x2	PROPN
ejpam-3427	650	17	,	,	PUNCT
ejpam-3427	650	18	...	...	PUNCT
ejpam-3427	650	19	,	,	PUNCT
ejpam-3427	650	20	xn	xn	PROPN
ejpam-3427	650	21	)	)	PUNCT
ejpam-3427	650	22	)	)	PUNCT
ejpam-3427	650	23	.	.	PUNCT
ejpam-3427	651	1	hence	hence	ADV
ejpam-3427	651	2	a1×a2×	a1×a2×	PROPN
ejpam-3427	651	3	...	...	PUNCT
ejpam-3427	651	4	×an	×an	PROPN
ejpam-3427	651	5	=	=	SYM
ejpam-3427	651	6	(	(	PUNCT
ejpam-3427	651	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	651	8	...	...	PUNCT
ejpam-3427	651	9	×an	×an	PROPN
ejpam-3427	651	10	,	,	PUNCT
ejpam-3427	651	11	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	651	12	...	...	PUNCT
ejpam-3427	651	13	×an	×an	PROPN
ejpam-3427	651	14	)	)	PUNCT
ejpam-3427	651	15	is	be	AUX
ejpam-3427	651	16	an	an	DET
ejpam-3427	651	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	651	18	anti	anti	ADJ
ejpam-3427	651	19	fuzzy	fuzzy	ADJ
ejpam-3427	651	20	normal	normal	ADJ
ejpam-3427	651	21	la	la	NOUN
ejpam-3427	651	22	-	-	PUNCT
ejpam-3427	651	23	subring	subring	NOUN
ejpam-3427	651	24	of	of	ADP
ejpam-3427	651	25	an	an	DET
ejpam-3427	651	26	la	la	ADJ
ejpam-3427	651	27	-	-	PUNCT
ejpam-3427	651	28	ring	ring	NOUN
ejpam-3427	651	29	r1	r1	NOUN
ejpam-3427	651	30	×r2	×r2	PROPN
ejpam-3427	651	31	×	×	NOUN
ejpam-3427	651	32	...	...	PUNCT
ejpam-3427	651	33	×rn	×rn	PROPN
ejpam-3427	651	34	.	.	PUNCT
ejpam-3427	652	1	k.	k.	PROPN
ejpam-3427	652	2	nasreen	nasreen	PROPN
ejpam-3427	652	3	/	/	SYM
ejpam-3427	652	4	eur	eur	PROPN
ejpam-3427	652	5	.	.	PUNCT
ejpam-3427	653	1	j.	j.	PROPN
ejpam-3427	653	2	pure	pure	PROPN
ejpam-3427	653	3	appl	appl	PROPN
ejpam-3427	653	4	.	.	PROPN
ejpam-3427	653	5	math	math	PROPN
ejpam-3427	653	6	,	,	PUNCT
ejpam-3427	653	7	12	12	NUM
ejpam-3427	653	8	(	(	PUNCT
ejpam-3427	653	9	2	2	NUM
ejpam-3427	653	10	)	)	PUNCT
ejpam-3427	653	11	(	(	PUNCT
ejpam-3427	653	12	2019	2019	NUM
ejpam-3427	653	13	)	)	PUNCT
ejpam-3427	653	14	,	,	PUNCT
ejpam-3427	653	15	622	622	NUM
ejpam-3427	653	16	-	-	SYM
ejpam-3427	653	17	648	648	NUM
ejpam-3427	653	18	644	644	NUM
ejpam-3427	653	19	theorem	theorem	NOUN
ejpam-3427	653	20	9	9	NUM
ejpam-3427	653	21	.	.	PUNCT
ejpam-3427	654	1	an	an	DET
ejpam-3427	654	2	ifs	ifs	PROPN
ejpam-3427	654	3	a1×a2×	a1×a2×	VERB
ejpam-3427	654	4	...	...	PUNCT
ejpam-3427	654	5	×an	×an	NOUN
ejpam-3427	654	6	=	=	SYM
ejpam-3427	654	7	(	(	PUNCT
ejpam-3427	654	8	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	654	9	...	...	PUNCT
ejpam-3427	654	10	×an	×an	PROPN
ejpam-3427	654	11	,	,	PUNCT
ejpam-3427	654	12	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	654	13	...	...	PUNCT
ejpam-3427	654	14	×an	×an	PROPN
ejpam-3427	654	15	)	)	PUNCT
ejpam-3427	654	16	is	be	AUX
ejpam-3427	654	17	an	an	DET
ejpam-3427	654	18	intuitionistic	intuitionistic	ADJ
ejpam-3427	654	19	anti	anti	ADJ
ejpam-3427	654	20	fuzzy	fuzzy	ADJ
ejpam-3427	654	21	normal	normal	ADJ
ejpam-3427	654	22	la	la	NOUN
ejpam-3427	654	23	-	-	PUNCT
ejpam-3427	654	24	subring	subring	NOUN
ejpam-3427	654	25	of	of	ADP
ejpam-3427	654	26	an	an	DET
ejpam-3427	654	27	la	la	ADJ
ejpam-3427	654	28	-	-	PUNCT
ejpam-3427	654	29	ring	ring	NOUN
ejpam-3427	654	30	r1	r1	NOUN
ejpam-3427	654	31	×	×	NOUN
ejpam-3427	654	32	r2	r2	PROPN
ejpam-3427	654	33	×	×	NOUN
ejpam-3427	654	34	...	...	PUNCT
ejpam-3427	655	1	×	×	PROPN
ejpam-3427	655	2	rn	rn	NOUN
ejpam-3427	655	3	if	if	SCONJ
ejpam-3427	656	1	and	and	CCONJ
ejpam-3427	656	2	only	only	ADV
ejpam-3427	656	3	if	if	SCONJ
ejpam-3427	656	4	the	the	DET
ejpam-3427	656	5	fuzzy	fuzzy	ADJ
ejpam-3427	656	6	subsets	subset	NOUN
ejpam-3427	656	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	656	8	...	...	PUNCT
ejpam-3427	656	9	×an	×an	PROPN
ejpam-3427	656	10	and	and	CCONJ
ejpam-3427	656	11	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	656	12	...	...	PUNCT
ejpam-3427	657	1	×an	×an	PROPN
ejpam-3427	657	2	are	be	AUX
ejpam-3427	657	3	fuzzy	fuzzy	ADJ
ejpam-3427	657	4	normal	normal	ADJ
ejpam-3427	657	5	la	la	ADJ
ejpam-3427	657	6	-	-	PUNCT
ejpam-3427	657	7	subrings	subring	NOUN
ejpam-3427	657	8	of	of	ADP
ejpam-3427	657	9	an	an	DET
ejpam-3427	657	10	la	la	ADJ
ejpam-3427	657	11	-	-	PUNCT
ejpam-3427	657	12	ring	ring	NOUN
ejpam-3427	657	13	r1	r1	NOUN
ejpam-3427	657	14	×r2	×r2	PROPN
ejpam-3427	657	15	×	×	NOUN
ejpam-3427	657	16	...	...	PUNCT
ejpam-3427	657	17	×rn	×rn	NOUN
ejpam-3427	657	18	.	.	PUNCT
ejpam-3427	658	1	proof	proof	NOUN
ejpam-3427	658	2	.	.	PUNCT
ejpam-3427	659	1	let	let	VERB
ejpam-3427	659	2	a1×a2×	a1×a2×	VERB
ejpam-3427	659	3	...	...	PUNCT
ejpam-3427	659	4	×an	×an	PROPN
ejpam-3427	659	5	=	=	SYM
ejpam-3427	659	6	(	(	PUNCT
ejpam-3427	659	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	659	8	...	...	PUNCT
ejpam-3427	659	9	×an	×an	PROPN
ejpam-3427	659	10	,	,	PUNCT
ejpam-3427	659	11	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	659	12	...	...	PUNCT
ejpam-3427	659	13	×an	×an	PROPN
ejpam-3427	659	14	)	)	PUNCT
ejpam-3427	659	15	be	be	AUX
ejpam-3427	659	16	an	an	DET
ejpam-3427	659	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	659	18	anti	anti	ADJ
ejpam-3427	659	19	fuzzy	fuzzy	ADJ
ejpam-3427	659	20	normal	normal	ADJ
ejpam-3427	659	21	la	la	NOUN
ejpam-3427	659	22	-	-	PUNCT
ejpam-3427	659	23	subring	subring	NOUN
ejpam-3427	659	24	of	of	ADP
ejpam-3427	659	25	an	an	DET
ejpam-3427	659	26	la	la	ADJ
ejpam-3427	659	27	-	-	PUNCT
ejpam-3427	659	28	ring	ring	NOUN
ejpam-3427	659	29	r1×r2×	r1×r2×	NOUN
ejpam-3427	659	30	...	...	PUNCT
ejpam-3427	659	31	×rn	×rn	PROPN
ejpam-3427	659	32	.	.	PUNCT
ejpam-3427	660	1	this	this	PRON
ejpam-3427	660	2	means	mean	VERB
ejpam-3427	660	3	that	that	SCONJ
ejpam-3427	660	4	γa1×a2×	γa1×a2×	ADV
ejpam-3427	660	5	...	...	PUNCT
ejpam-3427	661	1	×an	×an	PROPN
ejpam-3427	661	2	is	be	AUX
ejpam-3427	661	3	a	a	DET
ejpam-3427	661	4	fuzzy	fuzzy	ADJ
ejpam-3427	661	5	normal	normal	ADJ
ejpam-3427	661	6	la	la	NOUN
ejpam-3427	661	7	-	-	PUNCT
ejpam-3427	661	8	subring	subring	NOUN
ejpam-3427	661	9	of	of	ADP
ejpam-3427	661	10	an	an	DET
ejpam-3427	661	11	la	la	ADJ
ejpam-3427	661	12	-	-	PUNCT
ejpam-3427	661	13	ring	ring	NOUN
ejpam-3427	661	14	r1	r1	NOUN
ejpam-3427	661	15	×	×	NOUN
ejpam-3427	661	16	r2	r2	PROPN
ejpam-3427	661	17	×	×	NOUN
ejpam-3427	661	18	...	...	PUNCT
ejpam-3427	661	19	×	×	PROPN
ejpam-3427	661	20	rn	rn	PROPN
ejpam-3427	661	21	.	.	PUNCT
ejpam-3427	662	1	we	we	PRON
ejpam-3427	662	2	have	have	VERB
ejpam-3427	662	3	to	to	PART
ejpam-3427	662	4	show	show	VERB
ejpam-3427	662	5	that	that	PRON
ejpam-3427	662	6	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	662	7	...	...	PUNCT
ejpam-3427	662	8	×an	×an	PROPN
ejpam-3427	662	9	is	be	AUX
ejpam-3427	662	10	also	also	ADV
ejpam-3427	662	11	a	a	DET
ejpam-3427	662	12	fuzzy	fuzzy	ADJ
ejpam-3427	662	13	normal	normal	ADJ
ejpam-3427	662	14	la	la	NOUN
ejpam-3427	662	15	-	-	PUNCT
ejpam-3427	662	16	subring	subring	NOUN
ejpam-3427	662	17	of	of	ADP
ejpam-3427	662	18	an	an	DET
ejpam-3427	662	19	la	la	ADJ
ejpam-3427	662	20	-	-	PUNCT
ejpam-3427	662	21	ring	ring	NOUN
ejpam-3427	662	22	r1	r1	NOUN
ejpam-3427	662	23	×r2	×r2	PROPN
ejpam-3427	662	24	×	×	NOUN
ejpam-3427	662	25	...	...	PUNCT
ejpam-3427	662	26	×rn	×rn	NOUN
ejpam-3427	662	27	.	.	PUNCT
ejpam-3427	663	1	now	now	ADV
ejpam-3427	663	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	663	3	...	...	PUNCT
ejpam-3427	663	4	×an	×an	PROPN
ejpam-3427	663	5	(	(	PUNCT
ejpam-3427	663	6	(	(	PUNCT
ejpam-3427	663	7	x1	x1	PROPN
ejpam-3427	663	8	,	,	PUNCT
ejpam-3427	663	9	x2	x2	PROPN
ejpam-3427	663	10	,	,	PUNCT
ejpam-3427	663	11	...	...	PUNCT
ejpam-3427	663	12	,	,	PUNCT
ejpam-3427	663	13	xn)−	xn)−	X
ejpam-3427	663	14	(	(	PUNCT
ejpam-3427	663	15	y1	y1	INTJ
ejpam-3427	663	16	,	,	PUNCT
ejpam-3427	663	17	y2	y2	PROPN
ejpam-3427	663	18	,	,	PUNCT
ejpam-3427	663	19	...	...	PUNCT
ejpam-3427	663	20	,	,	PUNCT
ejpam-3427	663	21	yn	yn	PROPN
ejpam-3427	663	22	)	)	PUNCT
ejpam-3427	663	23	)	)	PUNCT
ejpam-3427	664	1	=	=	SYM
ejpam-3427	664	2	1−	1−	NUM
ejpam-3427	664	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	664	4	...	...	PUNCT
ejpam-3427	664	5	×an((x1	×an((x1	PROPN
ejpam-3427	664	6	,	,	PUNCT
ejpam-3427	664	7	x2	x2	PROPN
ejpam-3427	664	8	,	,	PUNCT
ejpam-3427	664	9	...	...	PUNCT
ejpam-3427	664	10	,	,	PUNCT
ejpam-3427	664	11	xn)−	xn)−	X
ejpam-3427	664	12	(	(	PUNCT
ejpam-3427	664	13	y1	y1	INTJ
ejpam-3427	664	14	,	,	PUNCT
ejpam-3427	664	15	y2	y2	PROPN
ejpam-3427	664	16	,	,	PUNCT
ejpam-3427	664	17	...	...	PUNCT
ejpam-3427	664	18	,	,	PUNCT
ejpam-3427	664	19	yn	yn	PROPN
ejpam-3427	664	20	)	)	PUNCT
ejpam-3427	664	21	)	)	PUNCT
ejpam-3427	664	22	≥	≥	NOUN
ejpam-3427	664	23	1−max{µa1×a2×	1−max{µa1×a2×	NUM
ejpam-3427	664	24	...	...	PUNCT
ejpam-3427	665	1	×an(x1	×an(x1	NOUN
ejpam-3427	665	2	,	,	PUNCT
ejpam-3427	665	3	x2	x2	PROPN
ejpam-3427	665	4	,	,	PUNCT
ejpam-3427	665	5	...	...	PUNCT
ejpam-3427	665	6	,	,	PUNCT
ejpam-3427	665	7	xn	xn	PROPN
ejpam-3427	665	8	)	)	PUNCT
ejpam-3427	665	9	,	,	PUNCT
ejpam-3427	665	10	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	665	11	...	...	PUNCT
ejpam-3427	665	12	×an(y1	×an(y1	NOUN
ejpam-3427	665	13	,	,	PUNCT
ejpam-3427	665	14	y2	y2	PROPN
ejpam-3427	665	15	,	,	PUNCT
ejpam-3427	665	16	...	...	PUNCT
ejpam-3427	665	17	,	,	PUNCT
ejpam-3427	665	18	yn	yn	PROPN
ejpam-3427	665	19	)	)	PUNCT
ejpam-3427	665	20	}	}	PUNCT
ejpam-3427	665	21	=	=	PUNCT
ejpam-3427	666	1	min{1−	min{1−	VERB
ejpam-3427	666	2	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	666	3	...	...	PUNCT
ejpam-3427	666	4	×an(x1	×an(x1	NOUN
ejpam-3427	666	5	,	,	PUNCT
ejpam-3427	666	6	x2	x2	PROPN
ejpam-3427	666	7	,	,	PUNCT
ejpam-3427	666	8	...	...	PUNCT
ejpam-3427	666	9	,	,	PUNCT
ejpam-3427	666	10	xn	xn	PROPN
ejpam-3427	666	11	)	)	PUNCT
ejpam-3427	666	12	,	,	PUNCT
ejpam-3427	666	13	1−	1−	NUM
ejpam-3427	666	14	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	666	15	...	...	PUNCT
ejpam-3427	666	16	×an(y1	×an(y1	NOUN
ejpam-3427	666	17	,	,	PUNCT
ejpam-3427	666	18	y2	y2	PROPN
ejpam-3427	666	19	,	,	PUNCT
ejpam-3427	666	20	...	...	PUNCT
ejpam-3427	666	21	,	,	PUNCT
ejpam-3427	666	22	yn	yn	PROPN
ejpam-3427	666	23	)	)	PUNCT
ejpam-3427	666	24	}	}	PUNCT
ejpam-3427	666	25	=	=	SYM
ejpam-3427	666	26	min{µa1×a2×	min{µa1×a2×	NOUN
ejpam-3427	666	27	...	...	PUNCT
ejpam-3427	666	28	×an	×an	NOUN
ejpam-3427	666	29	(	(	PUNCT
ejpam-3427	666	30	x1	x1	PROPN
ejpam-3427	666	31	,	,	PUNCT
ejpam-3427	666	32	x2	x2	PROPN
ejpam-3427	666	33	,	,	PUNCT
ejpam-3427	666	34	...	...	PUNCT
ejpam-3427	666	35	,	,	PUNCT
ejpam-3427	666	36	xn	xn	PROPN
ejpam-3427	666	37	)	)	PUNCT
ejpam-3427	666	38	,	,	PUNCT
ejpam-3427	666	39	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	666	40	...	...	PUNCT
ejpam-3427	666	41	×an	×an	PROPN
ejpam-3427	666	42	(	(	PUNCT
ejpam-3427	666	43	y1	y1	PROPN
ejpam-3427	666	44	,	,	PUNCT
ejpam-3427	666	45	y2	y2	PROPN
ejpam-3427	666	46	,	,	PUNCT
ejpam-3427	666	47	...	...	PUNCT
ejpam-3427	666	48	,	,	PUNCT
ejpam-3427	666	49	yn	yn	PROPN
ejpam-3427	666	50	)	)	PUNCT
ejpam-3427	666	51	}	}	PUNCT
ejpam-3427	666	52	and	and	CCONJ
ejpam-3427	666	53	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	666	54	...	...	PUNCT
ejpam-3427	666	55	×an	×an	PROPN
ejpam-3427	666	56	(	(	PUNCT
ejpam-3427	666	57	(	(	PUNCT
ejpam-3427	666	58	x1	x1	PROPN
ejpam-3427	666	59	,	,	PUNCT
ejpam-3427	666	60	x2	x2	PROPN
ejpam-3427	666	61	,	,	PUNCT
ejpam-3427	666	62	...	...	PUNCT
ejpam-3427	666	63	,	,	PUNCT
ejpam-3427	666	64	xn	xn	X
ejpam-3427	666	65	)	)	PUNCT
ejpam-3427	666	66	◦	◦	NOUN
ejpam-3427	666	67	(	(	PUNCT
ejpam-3427	666	68	y1	y1	INTJ
ejpam-3427	666	69	,	,	PUNCT
ejpam-3427	666	70	y2	y2	PROPN
ejpam-3427	666	71	,	,	PUNCT
ejpam-3427	666	72	...	...	PUNCT
ejpam-3427	666	73	,	,	PUNCT
ejpam-3427	666	74	yn	yn	PROPN
ejpam-3427	666	75	)	)	PUNCT
ejpam-3427	666	76	)	)	PUNCT
ejpam-3427	667	1	=	=	SYM
ejpam-3427	668	1	1−	1−	NUM
ejpam-3427	668	2	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	668	3	...	...	PUNCT
ejpam-3427	668	4	×an((x1	×an((x1	PROPN
ejpam-3427	668	5	,	,	PUNCT
ejpam-3427	668	6	x2	x2	PROPN
ejpam-3427	668	7	,	,	PUNCT
ejpam-3427	668	8	...	...	PUNCT
ejpam-3427	668	9	,	,	PUNCT
ejpam-3427	668	10	xn	xn	X
ejpam-3427	668	11	)	)	PUNCT
ejpam-3427	668	12	◦	◦	NOUN
ejpam-3427	668	13	(	(	PUNCT
ejpam-3427	668	14	y1	y1	INTJ
ejpam-3427	668	15	,	,	PUNCT
ejpam-3427	668	16	y2	y2	PROPN
ejpam-3427	668	17	,	,	PUNCT
ejpam-3427	668	18	...	...	PUNCT
ejpam-3427	668	19	,	,	PUNCT
ejpam-3427	668	20	yn	yn	PROPN
ejpam-3427	668	21	)	)	PUNCT
ejpam-3427	668	22	)	)	PUNCT
ejpam-3427	668	23	≥	≥	NOUN
ejpam-3427	668	24	1−max{µa1×a2×	1−max{µa1×a2×	NUM
ejpam-3427	668	25	...	...	PUNCT
ejpam-3427	668	26	×an(x1	×an(x1	NOUN
ejpam-3427	668	27	,	,	PUNCT
ejpam-3427	668	28	x2	x2	PROPN
ejpam-3427	668	29	,	,	PUNCT
ejpam-3427	668	30	...	...	PUNCT
ejpam-3427	668	31	,	,	PUNCT
ejpam-3427	668	32	xn	xn	PROPN
ejpam-3427	668	33	)	)	PUNCT
ejpam-3427	668	34	,	,	PUNCT
ejpam-3427	668	35	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	668	36	...	...	PUNCT
ejpam-3427	668	37	×an(y1	×an(y1	NOUN
ejpam-3427	668	38	,	,	PUNCT
ejpam-3427	668	39	y2	y2	PROPN
ejpam-3427	668	40	,	,	PUNCT
ejpam-3427	668	41	...	...	PUNCT
ejpam-3427	668	42	,	,	PUNCT
ejpam-3427	668	43	yn	yn	PROPN
ejpam-3427	668	44	)	)	PUNCT
ejpam-3427	668	45	}	}	PUNCT
ejpam-3427	668	46	=	=	PUNCT
ejpam-3427	669	1	min{1−	min{1−	VERB
ejpam-3427	669	2	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	669	3	...	...	PUNCT
ejpam-3427	669	4	×an(x1	×an(x1	NOUN
ejpam-3427	669	5	,	,	PUNCT
ejpam-3427	669	6	x2	x2	PROPN
ejpam-3427	669	7	,	,	PUNCT
ejpam-3427	669	8	...	...	PUNCT
ejpam-3427	669	9	,	,	PUNCT
ejpam-3427	669	10	xn	xn	PROPN
ejpam-3427	669	11	)	)	PUNCT
ejpam-3427	669	12	,	,	PUNCT
ejpam-3427	669	13	1−	1−	NUM
ejpam-3427	669	14	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	669	15	...	...	PUNCT
ejpam-3427	669	16	×an(y1	×an(y1	NOUN
ejpam-3427	669	17	,	,	PUNCT
ejpam-3427	669	18	y2	y2	PROPN
ejpam-3427	669	19	,	,	PUNCT
ejpam-3427	669	20	...	...	PUNCT
ejpam-3427	669	21	,	,	PUNCT
ejpam-3427	669	22	yn	yn	PROPN
ejpam-3427	669	23	)	)	PUNCT
ejpam-3427	669	24	}	}	PUNCT
ejpam-3427	669	25	=	=	SYM
ejpam-3427	669	26	min{µa1×a2×	min{µa1×a2×	NOUN
ejpam-3427	669	27	...	...	PUNCT
ejpam-3427	669	28	×an	×an	NOUN
ejpam-3427	669	29	(	(	PUNCT
ejpam-3427	669	30	x1	x1	PROPN
ejpam-3427	669	31	,	,	PUNCT
ejpam-3427	669	32	x2	x2	PROPN
ejpam-3427	669	33	,	,	PUNCT
ejpam-3427	669	34	...	...	PUNCT
ejpam-3427	669	35	,	,	PUNCT
ejpam-3427	669	36	xn	xn	PROPN
ejpam-3427	669	37	)	)	PUNCT
ejpam-3427	669	38	,	,	PUNCT
ejpam-3427	669	39	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	669	40	...	...	PUNCT
ejpam-3427	669	41	×an	×an	PROPN
ejpam-3427	669	42	(	(	PUNCT
ejpam-3427	669	43	y1	y1	PROPN
ejpam-3427	669	44	,	,	PUNCT
ejpam-3427	669	45	y2	y2	PROPN
ejpam-3427	669	46	,	,	PUNCT
ejpam-3427	669	47	...	...	PUNCT
ejpam-3427	669	48	,	,	PUNCT
ejpam-3427	669	49	yn	yn	PROPN
ejpam-3427	669	50	)	)	PUNCT
ejpam-3427	669	51	}	}	PUNCT
ejpam-3427	669	52	.	.	PUNCT
ejpam-3427	670	1	thus	thus	ADV
ejpam-3427	670	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	670	3	...	...	PUNCT
ejpam-3427	670	4	×an	×an	PROPN
ejpam-3427	670	5	is	be	AUX
ejpam-3427	670	6	a	a	DET
ejpam-3427	670	7	fuzzy	fuzzy	ADJ
ejpam-3427	670	8	la	la	NOUN
ejpam-3427	670	9	-	-	PUNCT
ejpam-3427	670	10	subring	subring	NOUN
ejpam-3427	670	11	of	of	ADP
ejpam-3427	670	12	an	an	DET
ejpam-3427	670	13	la	la	ADJ
ejpam-3427	670	14	-	-	PUNCT
ejpam-3427	670	15	ring	ring	NOUN
ejpam-3427	670	16	r1	r1	NOUN
ejpam-3427	670	17	×r2	×r2	PROPN
ejpam-3427	670	18	×	×	NOUN
ejpam-3427	670	19	...	...	PUNCT
ejpam-3427	670	20	×rn	×rn	NOUN
ejpam-3427	670	21	.	.	PUNCT
ejpam-3427	671	1	now	now	ADV
ejpam-3427	671	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	671	3	...	...	PUNCT
ejpam-3427	671	4	×an	×an	PROPN
ejpam-3427	671	5	(	(	PUNCT
ejpam-3427	671	6	(	(	PUNCT
ejpam-3427	671	7	x1	x1	PROPN
ejpam-3427	671	8	,	,	PUNCT
ejpam-3427	671	9	x2	x2	PROPN
ejpam-3427	671	10	,	,	PUNCT
ejpam-3427	671	11	...	...	PUNCT
ejpam-3427	671	12	,	,	PUNCT
ejpam-3427	671	13	xn	xn	X
ejpam-3427	671	14	)	)	PUNCT
ejpam-3427	671	15	◦	◦	NOUN
ejpam-3427	671	16	(	(	PUNCT
ejpam-3427	671	17	y1	y1	INTJ
ejpam-3427	671	18	,	,	PUNCT
ejpam-3427	671	19	y2	y2	PROPN
ejpam-3427	671	20	,	,	PUNCT
ejpam-3427	671	21	...	...	PUNCT
ejpam-3427	671	22	,	,	PUNCT
ejpam-3427	671	23	yn	yn	PROPN
ejpam-3427	671	24	)	)	PUNCT
ejpam-3427	671	25	)	)	PUNCT
ejpam-3427	672	1	=	=	SYM
ejpam-3427	672	2	1−	1−	NUM
ejpam-3427	672	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	672	4	...	...	PUNCT
ejpam-3427	672	5	×an((x1	×an((x1	PROPN
ejpam-3427	672	6	,	,	PUNCT
ejpam-3427	672	7	x2	x2	PROPN
ejpam-3427	672	8	,	,	PUNCT
ejpam-3427	672	9	...	...	PUNCT
ejpam-3427	672	10	,	,	PUNCT
ejpam-3427	672	11	xn	xn	X
ejpam-3427	672	12	)	)	PUNCT
ejpam-3427	672	13	◦	◦	NOUN
ejpam-3427	672	14	(	(	PUNCT
ejpam-3427	672	15	y1	y1	INTJ
ejpam-3427	672	16	,	,	PUNCT
ejpam-3427	672	17	y2	y2	PROPN
ejpam-3427	672	18	,	,	PUNCT
ejpam-3427	672	19	...	...	PUNCT
ejpam-3427	672	20	,	,	PUNCT
ejpam-3427	672	21	yn	yn	PROPN
ejpam-3427	672	22	)	)	PUNCT
ejpam-3427	672	23	)	)	PUNCT
ejpam-3427	672	24	=	=	SYM
ejpam-3427	673	1	1−	1−	NUM
ejpam-3427	673	2	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	673	3	...	...	PUNCT
ejpam-3427	673	4	×an((y1	×an((y1	PROPN
ejpam-3427	673	5	,	,	PUNCT
ejpam-3427	673	6	y2	y2	PROPN
ejpam-3427	673	7	,	,	PUNCT
ejpam-3427	673	8	...	...	PUNCT
ejpam-3427	673	9	,	,	PUNCT
ejpam-3427	673	10	yn	yn	PROPN
ejpam-3427	673	11	)	)	PUNCT
ejpam-3427	673	12	◦	◦	NOUN
ejpam-3427	673	13	(	(	PUNCT
ejpam-3427	673	14	x1	x1	PROPN
ejpam-3427	673	15	,	,	PUNCT
ejpam-3427	673	16	x2	x2	PROPN
ejpam-3427	673	17	,	,	PUNCT
ejpam-3427	673	18	...	...	PUNCT
ejpam-3427	673	19	,	,	PUNCT
ejpam-3427	673	20	xn	xn	PROPN
ejpam-3427	673	21	)	)	PUNCT
ejpam-3427	673	22	)	)	PUNCT
ejpam-3427	674	1	=	=	SYM
ejpam-3427	674	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	674	3	...	...	PUNCT
ejpam-3427	674	4	×an	×an	NOUN
ejpam-3427	674	5	(	(	PUNCT
ejpam-3427	674	6	(	(	PUNCT
ejpam-3427	674	7	y1	y1	INTJ
ejpam-3427	674	8	,	,	PUNCT
ejpam-3427	674	9	y2	y2	PROPN
ejpam-3427	674	10	,	,	PUNCT
ejpam-3427	674	11	...	...	PUNCT
ejpam-3427	674	12	,	,	PUNCT
ejpam-3427	674	13	yn	yn	PROPN
ejpam-3427	674	14	)	)	PUNCT
ejpam-3427	674	15	◦	◦	NOUN
ejpam-3427	674	16	(	(	PUNCT
ejpam-3427	674	17	x1	x1	PROPN
ejpam-3427	674	18	,	,	PUNCT
ejpam-3427	674	19	x2	x2	PROPN
ejpam-3427	674	20	,	,	PUNCT
ejpam-3427	674	21	...	...	PUNCT
ejpam-3427	674	22	,	,	PUNCT
ejpam-3427	674	23	xn	xn	PROPN
ejpam-3427	674	24	)	)	PUNCT
ejpam-3427	674	25	)	)	PUNCT
ejpam-3427	674	26	.	.	PUNCT
ejpam-3427	675	1	hence	hence	ADV
ejpam-3427	675	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	675	3	...	...	PUNCT
ejpam-3427	675	4	×an	×an	PROPN
ejpam-3427	675	5	is	be	AUX
ejpam-3427	675	6	a	a	DET
ejpam-3427	675	7	fuzzy	fuzzy	ADJ
ejpam-3427	675	8	normal	normal	ADJ
ejpam-3427	675	9	la	la	NOUN
ejpam-3427	675	10	-	-	PUNCT
ejpam-3427	675	11	subring	subring	NOUN
ejpam-3427	675	12	of	of	ADP
ejpam-3427	675	13	an	an	DET
ejpam-3427	675	14	la	la	ADJ
ejpam-3427	675	15	-	-	PUNCT
ejpam-3427	675	16	ring	ring	NOUN
ejpam-3427	675	17	r1	r1	NOUN
ejpam-3427	675	18	×r2	×r2	PROPN
ejpam-3427	675	19	×	×	NOUN
ejpam-3427	675	20	...	...	PUNCT
ejpam-3427	675	21	×rn	×rn	NOUN
ejpam-3427	675	22	.	.	PUNCT
ejpam-3427	676	1	conversely	conversely	ADV
ejpam-3427	676	2	,	,	PUNCT
ejpam-3427	676	3	assume	assume	VERB
ejpam-3427	676	4	that	that	SCONJ
ejpam-3427	676	5	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	676	6	...	...	PUNCT
ejpam-3427	676	7	×an	×an	PROPN
ejpam-3427	676	8	and	and	CCONJ
ejpam-3427	676	9	γa1×a2×	γa1×a2×	ADJ
ejpam-3427	676	10	...	...	PUNCT
ejpam-3427	676	11	×an	×an	PROPN
ejpam-3427	676	12	are	be	AUX
ejpam-3427	676	13	fuzzy	fuzzy	ADJ
ejpam-3427	676	14	normal	normal	ADJ
ejpam-3427	676	15	la	la	ADJ
ejpam-3427	676	16	-	-	PUNCT
ejpam-3427	676	17	subrings	subring	NOUN
ejpam-3427	676	18	of	of	ADP
ejpam-3427	676	19	an	an	DET
ejpam-3427	676	20	la	la	ADJ
ejpam-3427	676	21	-	-	PUNCT
ejpam-3427	676	22	ringr1×r2×	ringr1×r2×	NOUN
ejpam-3427	676	23	...	...	PUNCT
ejpam-3427	676	24	×rn.we	×rn.we	NOUN
ejpam-3427	676	25	have	have	VERB
ejpam-3427	676	26	to	to	PART
ejpam-3427	676	27	show	show	VERB
ejpam-3427	676	28	thata1×a2×	thata1×a2×	PROPN
ejpam-3427	676	29	...	...	PUNCT
ejpam-3427	676	30	×an	×an	PROPN
ejpam-3427	676	31	=	=	SYM
ejpam-3427	676	32	(	(	PUNCT
ejpam-3427	676	33	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	676	34	...	...	PUNCT
ejpam-3427	676	35	×an	×an	PROPN
ejpam-3427	676	36	,	,	PUNCT
ejpam-3427	676	37	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	676	38	...	...	PUNCT
ejpam-3427	676	39	×an	×an	PROPN
ejpam-3427	676	40	)	)	PUNCT
ejpam-3427	676	41	is	be	AUX
ejpam-3427	676	42	an	an	DET
ejpam-3427	676	43	intuitionistic	intuitionistic	ADJ
ejpam-3427	676	44	anti	anti	ADJ
ejpam-3427	676	45	fuzzy	fuzzy	ADJ
ejpam-3427	676	46	normal	normal	ADJ
ejpam-3427	676	47	la	la	NOUN
ejpam-3427	676	48	-	-	PUNCT
ejpam-3427	676	49	subring	subring	NOUN
ejpam-3427	676	50	of	of	ADP
ejpam-3427	676	51	an	an	DET
ejpam-3427	676	52	la	la	ADJ
ejpam-3427	676	53	-	-	PUNCT
ejpam-3427	676	54	ring	ring	NOUN
ejpam-3427	676	55	r1	r1	NOUN
ejpam-3427	676	56	×r2	×r2	PROPN
ejpam-3427	676	57	×	×	NOUN
ejpam-3427	676	58	...	...	PUNCT
ejpam-3427	676	59	×rn	×rn	NOUN
ejpam-3427	676	60	.	.	PUNCT
ejpam-3427	677	1	now	now	ADV
ejpam-3427	677	2	1−	1−	NUM
ejpam-3427	677	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	677	4	...	...	PUNCT
ejpam-3427	677	5	×an((x1	×an((x1	PROPN
ejpam-3427	677	6	,	,	PUNCT
ejpam-3427	677	7	x2	x2	PROPN
ejpam-3427	677	8	,	,	PUNCT
ejpam-3427	677	9	...	...	PUNCT
ejpam-3427	677	10	,	,	PUNCT
ejpam-3427	677	11	xn)−	xn)−	X
ejpam-3427	677	12	(	(	PUNCT
ejpam-3427	677	13	y1	y1	INTJ
ejpam-3427	677	14	,	,	PUNCT
ejpam-3427	677	15	y2	y2	PROPN
ejpam-3427	677	16	,	,	PUNCT
ejpam-3427	677	17	...	...	PUNCT
ejpam-3427	677	18	,	,	PUNCT
ejpam-3427	677	19	yn	yn	PROPN
ejpam-3427	677	20	)	)	PUNCT
ejpam-3427	677	21	)	)	PUNCT
ejpam-3427	678	1	=	=	SYM
ejpam-3427	678	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	678	3	...	...	PUNCT
ejpam-3427	678	4	×an	×an	NOUN
ejpam-3427	678	5	(	(	PUNCT
ejpam-3427	678	6	(	(	PUNCT
ejpam-3427	678	7	x1	x1	PROPN
ejpam-3427	678	8	,	,	PUNCT
ejpam-3427	678	9	x2	x2	PROPN
ejpam-3427	678	10	,	,	PUNCT
ejpam-3427	678	11	...	...	PUNCT
ejpam-3427	678	12	,	,	PUNCT
ejpam-3427	678	13	xn)−	xn)−	X
ejpam-3427	678	14	(	(	PUNCT
ejpam-3427	678	15	y1	y1	INTJ
ejpam-3427	678	16	,	,	PUNCT
ejpam-3427	678	17	y2	y2	PROPN
ejpam-3427	678	18	,	,	PUNCT
ejpam-3427	678	19	...	...	PUNCT
ejpam-3427	678	20	,	,	PUNCT
ejpam-3427	678	21	yn	yn	PROPN
ejpam-3427	678	22	)	)	PUNCT
ejpam-3427	678	23	)	)	PUNCT
ejpam-3427	678	24	≥	≥	PROPN
ejpam-3427	679	1	min{µa1×a2×	min{µa1×a2×	NOUN
ejpam-3427	679	2	...	...	PUNCT
ejpam-3427	679	3	×an	×an	PROPN
ejpam-3427	679	4	(	(	PUNCT
ejpam-3427	679	5	x1	x1	PROPN
ejpam-3427	679	6	,	,	PUNCT
ejpam-3427	679	7	x2	x2	PROPN
ejpam-3427	679	8	,	,	PUNCT
ejpam-3427	679	9	...	...	PUNCT
ejpam-3427	679	10	,	,	PUNCT
ejpam-3427	679	11	xn	xn	PROPN
ejpam-3427	679	12	)	)	PUNCT
ejpam-3427	679	13	,	,	PUNCT
ejpam-3427	679	14	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	679	15	...	...	PUNCT
ejpam-3427	679	16	×an	×an	PROPN
ejpam-3427	679	17	(	(	PUNCT
ejpam-3427	679	18	y1	y1	PROPN
ejpam-3427	679	19	,	,	PUNCT
ejpam-3427	679	20	y2	y2	PROPN
ejpam-3427	679	21	,	,	PUNCT
ejpam-3427	679	22	...	...	PUNCT
ejpam-3427	679	23	,	,	PUNCT
ejpam-3427	679	24	yn	yn	PROPN
ejpam-3427	679	25	)	)	PUNCT
ejpam-3427	679	26	}	}	PUNCT
ejpam-3427	679	27	=	=	PUNCT
ejpam-3427	680	1	min{1−	min{1−	VERB
ejpam-3427	680	2	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	680	3	...	...	PUNCT
ejpam-3427	680	4	×an(x1	×an(x1	NOUN
ejpam-3427	680	5	,	,	PUNCT
ejpam-3427	680	6	x2	x2	PROPN
ejpam-3427	680	7	,	,	PUNCT
ejpam-3427	680	8	...	...	PUNCT
ejpam-3427	680	9	,	,	PUNCT
ejpam-3427	680	10	xn	xn	PROPN
ejpam-3427	680	11	)	)	PUNCT
ejpam-3427	680	12	,	,	PUNCT
ejpam-3427	680	13	1−	1−	NUM
ejpam-3427	680	14	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	680	15	...	...	PUNCT
ejpam-3427	680	16	×an(y1	×an(y1	NOUN
ejpam-3427	680	17	,	,	PUNCT
ejpam-3427	680	18	y2	y2	PROPN
ejpam-3427	680	19	,	,	PUNCT
ejpam-3427	680	20	...	...	PUNCT
ejpam-3427	680	21	,	,	PUNCT
ejpam-3427	680	22	yn	yn	PROPN
ejpam-3427	680	23	)	)	PUNCT
ejpam-3427	680	24	}	}	PUNCT
ejpam-3427	680	25	=	=	SYM
ejpam-3427	680	26	1−max{µa1×a2×	1−max{µa1×a2×	NUM
ejpam-3427	680	27	...	...	PUNCT
ejpam-3427	680	28	×an(x1	×an(x1	NOUN
ejpam-3427	680	29	,	,	PUNCT
ejpam-3427	680	30	x2	x2	PROPN
ejpam-3427	680	31	,	,	PUNCT
ejpam-3427	680	32	...	...	PUNCT
ejpam-3427	680	33	,	,	PUNCT
ejpam-3427	680	34	xn	xn	PROPN
ejpam-3427	680	35	)	)	PUNCT
ejpam-3427	680	36	,	,	PUNCT
ejpam-3427	680	37	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	680	38	...	...	PUNCT
ejpam-3427	680	39	×an(y1	×an(y1	NOUN
ejpam-3427	680	40	,	,	PUNCT
ejpam-3427	680	41	y2	y2	PROPN
ejpam-3427	680	42	,	,	PUNCT
ejpam-3427	680	43	...	...	PUNCT
ejpam-3427	680	44	,	,	PUNCT
ejpam-3427	680	45	yn	yn	PROPN
ejpam-3427	680	46	)	)	PUNCT
ejpam-3427	680	47	}	}	PUNCT
ejpam-3427	680	48	and	and	CCONJ
ejpam-3427	680	49	1−	1−	NUM
ejpam-3427	680	50	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	680	51	...	...	PUNCT
ejpam-3427	680	52	×an((x1	×an((x1	PROPN
ejpam-3427	680	53	,	,	PUNCT
ejpam-3427	680	54	x2	x2	PROPN
ejpam-3427	680	55	,	,	PUNCT
ejpam-3427	680	56	...	...	PUNCT
ejpam-3427	680	57	,	,	PUNCT
ejpam-3427	680	58	xn	xn	X
ejpam-3427	680	59	)	)	PUNCT
ejpam-3427	680	60	◦	◦	NOUN
ejpam-3427	680	61	(	(	PUNCT
ejpam-3427	680	62	y1	y1	INTJ
ejpam-3427	680	63	,	,	PUNCT
ejpam-3427	680	64	y2	y2	PROPN
ejpam-3427	680	65	,	,	PUNCT
ejpam-3427	680	66	...	...	PUNCT
ejpam-3427	680	67	,	,	PUNCT
ejpam-3427	680	68	yn	yn	PROPN
ejpam-3427	680	69	)	)	PUNCT
ejpam-3427	680	70	)	)	PUNCT
ejpam-3427	681	1	=	=	SYM
ejpam-3427	681	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	681	3	...	...	PUNCT
ejpam-3427	681	4	×an	×an	NOUN
ejpam-3427	681	5	(	(	PUNCT
ejpam-3427	681	6	(	(	PUNCT
ejpam-3427	681	7	x1	x1	PROPN
ejpam-3427	681	8	,	,	PUNCT
ejpam-3427	681	9	x2	x2	PROPN
ejpam-3427	681	10	,	,	PUNCT
ejpam-3427	681	11	...	...	PUNCT
ejpam-3427	681	12	,	,	PUNCT
ejpam-3427	681	13	xn	xn	X
ejpam-3427	681	14	)	)	PUNCT
ejpam-3427	681	15	◦	◦	NOUN
ejpam-3427	681	16	(	(	PUNCT
ejpam-3427	681	17	y1	y1	INTJ
ejpam-3427	681	18	,	,	PUNCT
ejpam-3427	681	19	y2	y2	PROPN
ejpam-3427	681	20	,	,	PUNCT
ejpam-3427	681	21	...	...	PUNCT
ejpam-3427	681	22	,	,	PUNCT
ejpam-3427	681	23	yn	yn	PROPN
ejpam-3427	681	24	)	)	PUNCT
ejpam-3427	681	25	)	)	PUNCT
ejpam-3427	681	26	≥	≥	PROPN
ejpam-3427	682	1	min{µa1×a2×	min{µa1×a2×	NOUN
ejpam-3427	682	2	...	...	PUNCT
ejpam-3427	682	3	×an	×an	PROPN
ejpam-3427	682	4	(	(	PUNCT
ejpam-3427	682	5	x1	x1	PROPN
ejpam-3427	682	6	,	,	PUNCT
ejpam-3427	682	7	x2	x2	PROPN
ejpam-3427	682	8	,	,	PUNCT
ejpam-3427	682	9	...	...	PUNCT
ejpam-3427	682	10	,	,	PUNCT
ejpam-3427	682	11	xn	xn	PROPN
ejpam-3427	682	12	)	)	PUNCT
ejpam-3427	682	13	,	,	PUNCT
ejpam-3427	682	14	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	682	15	...	...	PUNCT
ejpam-3427	682	16	×an	×an	PROPN
ejpam-3427	682	17	(	(	PUNCT
ejpam-3427	682	18	y1	y1	PROPN
ejpam-3427	682	19	,	,	PUNCT
ejpam-3427	682	20	y2	y2	PROPN
ejpam-3427	682	21	,	,	PUNCT
ejpam-3427	682	22	...	...	PUNCT
ejpam-3427	682	23	,	,	PUNCT
ejpam-3427	682	24	yn	yn	PROPN
ejpam-3427	682	25	)	)	PUNCT
ejpam-3427	682	26	}	}	PUNCT
ejpam-3427	682	27	k.	k.	PROPN
ejpam-3427	683	1	nasreen	nasreen	PROPN
ejpam-3427	683	2	/	/	SYM
ejpam-3427	683	3	eur	eur	PROPN
ejpam-3427	683	4	.	.	PUNCT
ejpam-3427	684	1	j.	j.	PROPN
ejpam-3427	684	2	pure	pure	PROPN
ejpam-3427	684	3	appl	appl	PROPN
ejpam-3427	684	4	.	.	PROPN
ejpam-3427	684	5	math	math	PROPN
ejpam-3427	684	6	,	,	PUNCT
ejpam-3427	684	7	12	12	NUM
ejpam-3427	684	8	(	(	PUNCT
ejpam-3427	684	9	2	2	NUM
ejpam-3427	684	10	)	)	PUNCT
ejpam-3427	684	11	(	(	PUNCT
ejpam-3427	684	12	2019	2019	NUM
ejpam-3427	684	13	)	)	PUNCT
ejpam-3427	684	14	,	,	PUNCT
ejpam-3427	684	15	622	622	NUM
ejpam-3427	684	16	-	-	SYM
ejpam-3427	684	17	648	648	NUM
ejpam-3427	684	18	645	645	NUM
ejpam-3427	684	19	=	=	NOUN
ejpam-3427	684	20	min{1−	min{1−	VERB
ejpam-3427	684	21	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	684	22	...	...	PUNCT
ejpam-3427	684	23	×an(x1	×an(x1	NOUN
ejpam-3427	684	24	,	,	PUNCT
ejpam-3427	684	25	x2	x2	PROPN
ejpam-3427	684	26	,	,	PUNCT
ejpam-3427	684	27	...	...	PUNCT
ejpam-3427	684	28	,	,	PUNCT
ejpam-3427	684	29	xn	xn	PROPN
ejpam-3427	684	30	)	)	PUNCT
ejpam-3427	684	31	,	,	PUNCT
ejpam-3427	684	32	1−	1−	NUM
ejpam-3427	684	33	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	684	34	...	...	PUNCT
ejpam-3427	684	35	×an(y1	×an(y1	NOUN
ejpam-3427	684	36	,	,	PUNCT
ejpam-3427	684	37	y2	y2	PROPN
ejpam-3427	684	38	,	,	PUNCT
ejpam-3427	684	39	...	...	PUNCT
ejpam-3427	684	40	,	,	PUNCT
ejpam-3427	684	41	yn	yn	PROPN
ejpam-3427	684	42	)	)	PUNCT
ejpam-3427	684	43	}	}	PUNCT
ejpam-3427	684	44	=	=	SYM
ejpam-3427	684	45	1−max{µa1×a2×	1−max{µa1×a2×	NUM
ejpam-3427	684	46	...	...	PUNCT
ejpam-3427	684	47	×an(x1	×an(x1	NOUN
ejpam-3427	684	48	,	,	PUNCT
ejpam-3427	684	49	x2	x2	PROPN
ejpam-3427	684	50	,	,	PUNCT
ejpam-3427	684	51	...	...	PUNCT
ejpam-3427	684	52	,	,	PUNCT
ejpam-3427	684	53	xn	xn	PROPN
ejpam-3427	684	54	)	)	PUNCT
ejpam-3427	684	55	,	,	PUNCT
ejpam-3427	684	56	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	684	57	...	...	PUNCT
ejpam-3427	684	58	×an(y1	×an(y1	NOUN
ejpam-3427	684	59	,	,	PUNCT
ejpam-3427	684	60	y2	y2	PROPN
ejpam-3427	684	61	,	,	PUNCT
ejpam-3427	684	62	...	...	PUNCT
ejpam-3427	684	63	,	,	PUNCT
ejpam-3427	684	64	yn	yn	PROPN
ejpam-3427	684	65	)	)	PUNCT
ejpam-3427	684	66	}	}	PUNCT
ejpam-3427	684	67	.	.	PUNCT
ejpam-3427	685	1	thus	thus	ADV
ejpam-3427	685	2	a1×a2×	a1×a2×	NOUN
ejpam-3427	685	3	...	...	PUNCT
ejpam-3427	685	4	×an	×an	NOUN
ejpam-3427	685	5	=	=	SYM
ejpam-3427	685	6	(	(	PUNCT
ejpam-3427	685	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	685	8	...	...	PUNCT
ejpam-3427	685	9	×an	×an	PROPN
ejpam-3427	685	10	,	,	PUNCT
ejpam-3427	685	11	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	685	12	...	...	PUNCT
ejpam-3427	685	13	×an	×an	PROPN
ejpam-3427	685	14	)	)	PUNCT
ejpam-3427	685	15	is	be	AUX
ejpam-3427	685	16	an	an	DET
ejpam-3427	685	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	685	18	anti	anti	ADJ
ejpam-3427	685	19	fuzzy	fuzzy	ADJ
ejpam-3427	685	20	la	la	NOUN
ejpam-3427	685	21	-	-	PUNCT
ejpam-3427	685	22	subring	subring	NOUN
ejpam-3427	685	23	of	of	ADP
ejpam-3427	685	24	an	an	DET
ejpam-3427	685	25	la	la	ADJ
ejpam-3427	685	26	-	-	PUNCT
ejpam-3427	685	27	ring	ring	NOUN
ejpam-3427	685	28	r1	r1	NOUN
ejpam-3427	685	29	×r2	×r2	PROPN
ejpam-3427	685	30	×	×	NOUN
ejpam-3427	685	31	...	...	PUNCT
ejpam-3427	685	32	×rn	×rn	NOUN
ejpam-3427	685	33	.	.	PUNCT
ejpam-3427	686	1	now	now	ADV
ejpam-3427	686	2	1−	1−	NUM
ejpam-3427	686	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	686	4	...	...	PUNCT
ejpam-3427	686	5	×an((x1	×an((x1	PROPN
ejpam-3427	686	6	,	,	PUNCT
ejpam-3427	686	7	x2	x2	PROPN
ejpam-3427	686	8	,	,	PUNCT
ejpam-3427	686	9	...	...	PUNCT
ejpam-3427	686	10	,	,	PUNCT
ejpam-3427	686	11	xn	xn	X
ejpam-3427	686	12	)	)	PUNCT
ejpam-3427	686	13	◦	◦	NOUN
ejpam-3427	686	14	(	(	PUNCT
ejpam-3427	686	15	y1	y1	INTJ
ejpam-3427	686	16	,	,	PUNCT
ejpam-3427	686	17	y2	y2	PROPN
ejpam-3427	686	18	,	,	PUNCT
ejpam-3427	686	19	...	...	PUNCT
ejpam-3427	686	20	,	,	PUNCT
ejpam-3427	686	21	yn	yn	PROPN
ejpam-3427	686	22	)	)	PUNCT
ejpam-3427	686	23	)	)	PUNCT
ejpam-3427	687	1	=	=	SYM
ejpam-3427	687	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	687	3	...	...	PUNCT
ejpam-3427	687	4	×an	×an	NOUN
ejpam-3427	687	5	(	(	PUNCT
ejpam-3427	687	6	(	(	PUNCT
ejpam-3427	687	7	x1	x1	PROPN
ejpam-3427	687	8	,	,	PUNCT
ejpam-3427	687	9	x2	x2	PROPN
ejpam-3427	687	10	,	,	PUNCT
ejpam-3427	687	11	...	...	PUNCT
ejpam-3427	687	12	,	,	PUNCT
ejpam-3427	687	13	xn	xn	X
ejpam-3427	687	14	)	)	PUNCT
ejpam-3427	687	15	◦	◦	NOUN
ejpam-3427	687	16	(	(	PUNCT
ejpam-3427	687	17	y1	y1	INTJ
ejpam-3427	687	18	,	,	PUNCT
ejpam-3427	687	19	y2	y2	PROPN
ejpam-3427	687	20	,	,	PUNCT
ejpam-3427	687	21	...	...	PUNCT
ejpam-3427	687	22	,	,	PUNCT
ejpam-3427	687	23	yn	yn	PROPN
ejpam-3427	687	24	)	)	PUNCT
ejpam-3427	687	25	)	)	PUNCT
ejpam-3427	688	1	=	=	SYM
ejpam-3427	688	2	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	688	3	...	...	PUNCT
ejpam-3427	688	4	×an	×an	NOUN
ejpam-3427	688	5	(	(	PUNCT
ejpam-3427	688	6	(	(	PUNCT
ejpam-3427	688	7	y1	y1	INTJ
ejpam-3427	688	8	,	,	PUNCT
ejpam-3427	688	9	y2	y2	PROPN
ejpam-3427	688	10	,	,	PUNCT
ejpam-3427	688	11	...	...	PUNCT
ejpam-3427	688	12	,	,	PUNCT
ejpam-3427	688	13	yn	yn	PROPN
ejpam-3427	688	14	)	)	PUNCT
ejpam-3427	688	15	◦	◦	NOUN
ejpam-3427	688	16	(	(	PUNCT
ejpam-3427	688	17	x1	x1	PROPN
ejpam-3427	688	18	,	,	PUNCT
ejpam-3427	688	19	x2	x2	PROPN
ejpam-3427	688	20	,	,	PUNCT
ejpam-3427	688	21	...	...	PUNCT
ejpam-3427	688	22	,	,	PUNCT
ejpam-3427	688	23	xn	xn	PROPN
ejpam-3427	688	24	)	)	PUNCT
ejpam-3427	688	25	)	)	PUNCT
ejpam-3427	689	1	=	=	SYM
ejpam-3427	689	2	1−	1−	NUM
ejpam-3427	689	3	µa1×a2×	µa1×a2×	NOUN
ejpam-3427	689	4	...	...	PUNCT
ejpam-3427	689	5	×an((y1	×an((y1	PROPN
ejpam-3427	689	6	,	,	PUNCT
ejpam-3427	689	7	y2	y2	PROPN
ejpam-3427	689	8	,	,	PUNCT
ejpam-3427	689	9	...	...	PUNCT
ejpam-3427	689	10	,	,	PUNCT
ejpam-3427	689	11	yn	yn	PROPN
ejpam-3427	689	12	)	)	PUNCT
ejpam-3427	689	13	◦	◦	NOUN
ejpam-3427	689	14	(	(	PUNCT
ejpam-3427	689	15	x1	x1	PROPN
ejpam-3427	689	16	,	,	PUNCT
ejpam-3427	689	17	x2	x2	PROPN
ejpam-3427	689	18	,	,	PUNCT
ejpam-3427	689	19	...	...	PUNCT
ejpam-3427	689	20	,	,	PUNCT
ejpam-3427	689	21	xn	xn	PROPN
ejpam-3427	689	22	)	)	PUNCT
ejpam-3427	689	23	)	)	PUNCT
ejpam-3427	689	24	.	.	PUNCT
ejpam-3427	690	1	hence	hence	ADV
ejpam-3427	690	2	a1×a2×	a1×a2×	PROPN
ejpam-3427	690	3	...	...	PUNCT
ejpam-3427	690	4	×an	×an	PROPN
ejpam-3427	690	5	=	=	SYM
ejpam-3427	690	6	(	(	PUNCT
ejpam-3427	690	7	µa1×a2×	µa1×a2×	PROPN
ejpam-3427	690	8	...	...	PUNCT
ejpam-3427	690	9	×an	×an	PROPN
ejpam-3427	690	10	,	,	PUNCT
ejpam-3427	690	11	γa1×a2×	γa1×a2×	PROPN
ejpam-3427	690	12	...	...	PUNCT
ejpam-3427	690	13	×an	×an	PROPN
ejpam-3427	690	14	)	)	PUNCT
ejpam-3427	690	15	is	be	AUX
ejpam-3427	690	16	an	an	DET
ejpam-3427	690	17	intuitionistic	intuitionistic	ADJ
ejpam-3427	690	18	anti	anti	ADJ
ejpam-3427	690	19	fuzzy	fuzzy	ADJ
ejpam-3427	690	20	normal	normal	ADJ
ejpam-3427	690	21	la	la	NOUN
ejpam-3427	690	22	-	-	PUNCT
ejpam-3427	690	23	subring	subring	NOUN
ejpam-3427	690	24	of	of	ADP
ejpam-3427	690	25	an	an	DET
ejpam-3427	690	26	la	la	ADJ
ejpam-3427	690	27	-	-	PUNCT
ejpam-3427	690	28	ring	ring	NOUN
ejpam-3427	690	29	r1	r1	NOUN
ejpam-3427	690	30	×r2	×r2	PROPN
ejpam-3427	690	31	×	×	NOUN
ejpam-3427	690	32	...	...	PUNCT
ejpam-3427	690	33	×rn	×rn	NOUN
ejpam-3427	690	34	.	.	PUNCT
ejpam-3427	691	1	proposition	proposition	NOUN
ejpam-3427	691	2	5	5	NUM
ejpam-3427	691	3	.	.	PUNCT
ejpam-3427	692	1	let	let	VERB
ejpam-3427	692	2	a	a	DET
ejpam-3427	692	3	=	=	PUNCT
ejpam-3427	692	4	a1	a1	NOUN
ejpam-3427	692	5	×a2	×a2	PROPN
ejpam-3427	692	6	×	×	PROPN
ejpam-3427	692	7	...	...	PUNCT
ejpam-3427	692	8	×an	×an	PROPN
ejpam-3427	692	9	and	and	CCONJ
ejpam-3427	692	10	b	b	X
ejpam-3427	692	11	=	=	NOUN
ejpam-3427	692	12	b1	b1	PROPN
ejpam-3427	692	13	×b2	×b2	PROPN
ejpam-3427	692	14	×	×	NOUN
ejpam-3427	692	15	...	...	PUNCT
ejpam-3427	692	16	×bn	×bn	AUX
ejpam-3427	692	17	be	be	AUX
ejpam-3427	692	18	intuitionistic	intuitionistic	ADJ
ejpam-3427	692	19	fuzzy	fuzzy	ADJ
ejpam-3427	692	20	sets	set	NOUN
ejpam-3427	692	21	of	of	ADP
ejpam-3427	692	22	la	la	NOUN
ejpam-3427	692	23	-	-	PUNCT
ejpam-3427	692	24	rings	ring	NOUN
ejpam-3427	692	25	r	r	NOUN
ejpam-3427	692	26	=	=	PUNCT
ejpam-3427	692	27	r1×r2×	r1×r2×	NOUN
ejpam-3427	692	28	...	...	PUNCT
ejpam-3427	692	29	×rn	×rn	ADJ
ejpam-3427	692	30	and	and	CCONJ
ejpam-3427	692	31	r′	r′	PROPN
ejpam-3427	692	32	=	=	NOUN
ejpam-3427	692	33	r′1×r′2×	r′1×r′2×	NOUN
ejpam-3427	692	34	...	...	PUNCT
ejpam-3427	692	35	×r′n	×r′n	PROPN
ejpam-3427	692	36	with	with	ADP
ejpam-3427	692	37	left	left	ADJ
ejpam-3427	692	38	identities	identity	NOUN
ejpam-3427	692	39	e	e	NOUN
ejpam-3427	692	40	=	=	SYM
ejpam-3427	692	41	(	(	PUNCT
ejpam-3427	692	42	e1	e1	PROPN
ejpam-3427	692	43	,	,	PUNCT
ejpam-3427	692	44	e2	e2	PROPN
ejpam-3427	692	45	,	,	PUNCT
ejpam-3427	692	46	...	...	PUNCT
ejpam-3427	692	47	,	,	PUNCT
ejpam-3427	692	48	en	en	X
ejpam-3427	692	49	)	)	PUNCT
ejpam-3427	692	50	and	and	CCONJ
ejpam-3427	692	51	e′	e′	NOUN
ejpam-3427	693	1	=	=	SYM
ejpam-3427	693	2	(	(	PUNCT
ejpam-3427	693	3	e1′	e1′	ADJ
ejpam-3427	693	4	,	,	PUNCT
ejpam-3427	693	5	e2′	e2′	NOUN
ejpam-3427	693	6	,	,	PUNCT
ejpam-3427	693	7	...	...	PUNCT
ejpam-3427	693	8	,	,	PUNCT
ejpam-3427	693	9	en′	en′	PROPN
ejpam-3427	693	10	)	)	PUNCT
ejpam-3427	693	11	,	,	PUNCT
ejpam-3427	693	12	respectively	respectively	ADV
ejpam-3427	693	13	.	.	PUNCT
ejpam-3427	694	1	if	if	SCONJ
ejpam-3427	694	2	a×b	a×b	PROPN
ejpam-3427	694	3	is	be	AUX
ejpam-3427	694	4	an	an	DET
ejpam-3427	694	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	694	6	anti	anti	ADJ
ejpam-3427	694	7	fuzzy	fuzzy	ADJ
ejpam-3427	694	8	la	la	NOUN
ejpam-3427	694	9	-	-	PUNCT
ejpam-3427	694	10	subring	subring	NOUN
ejpam-3427	694	11	of	of	ADP
ejpam-3427	694	12	an	an	DET
ejpam-3427	694	13	la	la	ADJ
ejpam-3427	694	14	-	-	PUNCT
ejpam-3427	694	15	ring	ring	NOUN
ejpam-3427	694	16	r×r′	r×r′	NOUN
ejpam-3427	694	17	,	,	PUNCT
ejpam-3427	694	18	then	then	ADV
ejpam-3427	694	19	at	at	ADP
ejpam-3427	694	20	least	least	ADJ
ejpam-3427	694	21	one	one	NUM
ejpam-3427	694	22	of	of	ADP
ejpam-3427	694	23	the	the	DET
ejpam-3427	694	24	following	follow	VERB
ejpam-3427	694	25	two	two	NUM
ejpam-3427	694	26	statements	statement	NOUN
ejpam-3427	694	27	must	must	AUX
ejpam-3427	694	28	hold	hold	VERB
ejpam-3427	694	29	.	.	PUNCT
ejpam-3427	695	1	1	1	X
ejpam-3427	695	2	.	.	X
ejpam-3427	695	3	µa	µa	NOUN
ejpam-3427	695	4	(	(	PUNCT
ejpam-3427	695	5	x	x	X
ejpam-3427	695	6	)	)	PUNCT
ejpam-3427	695	7	≥	≥	PROPN
ejpam-3427	695	8	µb	µb	PROPN
ejpam-3427	695	9	(	(	PUNCT
ejpam-3427	695	10	e′	e′	ADJ
ejpam-3427	695	11	)	)	PUNCT
ejpam-3427	695	12	and	and	CCONJ
ejpam-3427	695	13	γa	γa	PROPN
ejpam-3427	695	14	(	(	PUNCT
ejpam-3427	695	15	x	x	NOUN
ejpam-3427	695	16	)	)	PUNCT
ejpam-3427	695	17	≤	≤	NUM
ejpam-3427	695	18	γb	γb	NOUN
ejpam-3427	695	19	(	(	PUNCT
ejpam-3427	695	20	e′	e′	PROPN
ejpam-3427	695	21	)	)	PUNCT
ejpam-3427	695	22	,	,	PUNCT
ejpam-3427	695	23	for	for	ADP
ejpam-3427	695	24	all	all	DET
ejpam-3427	695	25	x	x	PROPN
ejpam-3427	695	26	∈	∈	PROPN
ejpam-3427	695	27	r.	r.	NOUN
ejpam-3427	695	28	2	2	NUM
ejpam-3427	695	29	.	.	PUNCT
ejpam-3427	696	1	µb	µb	PROPN
ejpam-3427	696	2	(	(	PUNCT
ejpam-3427	696	3	x	x	NOUN
ejpam-3427	696	4	)	)	PUNCT
ejpam-3427	696	5	≥	≥	NOUN
ejpam-3427	696	6	µa	µa	NOUN
ejpam-3427	696	7	(	(	PUNCT
ejpam-3427	696	8	e	e	NOUN
ejpam-3427	696	9	)	)	PUNCT
ejpam-3427	696	10	and	and	CCONJ
ejpam-3427	696	11	γb	γb	INTJ
ejpam-3427	696	12	(	(	PUNCT
ejpam-3427	696	13	x	x	NOUN
ejpam-3427	696	14	)	)	PUNCT
ejpam-3427	696	15	≤	≤	NOUN
ejpam-3427	696	16	γa	γa	NOUN
ejpam-3427	696	17	(	(	PUNCT
ejpam-3427	696	18	e	e	NOUN
ejpam-3427	696	19	)	)	PUNCT
ejpam-3427	696	20	,	,	PUNCT
ejpam-3427	696	21	for	for	ADP
ejpam-3427	696	22	all	all	PRON
ejpam-3427	696	23	x	x	SYM
ejpam-3427	696	24	∈	∈	NOUN
ejpam-3427	696	25	r′.	r′.	NOUN
ejpam-3427	696	26	proof	proof	NOUN
ejpam-3427	696	27	.	.	PUNCT
ejpam-3427	697	1	let	let	VERB
ejpam-3427	697	2	a×b	a×b	VERB
ejpam-3427	697	3	be	be	AUX
ejpam-3427	697	4	an	an	DET
ejpam-3427	697	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	697	6	anti	anti	ADJ
ejpam-3427	697	7	fuzzy	fuzzy	ADJ
ejpam-3427	697	8	la	la	NOUN
ejpam-3427	697	9	-	-	PUNCT
ejpam-3427	697	10	subring	subring	NOUN
ejpam-3427	697	11	of	of	ADP
ejpam-3427	697	12	an	an	DET
ejpam-3427	697	13	la	la	ADJ
ejpam-3427	697	14	-	-	PUNCT
ejpam-3427	697	15	ring	ring	NOUN
ejpam-3427	697	16	r×r′.	r×r′.	NOUN
ejpam-3427	697	17	by	by	ADP
ejpam-3427	697	18	contraposition	contraposition	NOUN
ejpam-3427	697	19	,	,	PUNCT
ejpam-3427	697	20	suppose	suppose	VERB
ejpam-3427	697	21	that	that	SCONJ
ejpam-3427	697	22	none	none	NOUN
ejpam-3427	697	23	of	of	ADP
ejpam-3427	697	24	the	the	DET
ejpam-3427	697	25	statements	statement	NOUN
ejpam-3427	697	26	(	(	PUNCT
ejpam-3427	697	27	i	i	NOUN
ejpam-3427	697	28	)	)	PUNCT
ejpam-3427	697	29	and	and	CCONJ
ejpam-3427	697	30	(	(	PUNCT
ejpam-3427	697	31	ii	ii	NOUN
ejpam-3427	697	32	)	)	PUNCT
ejpam-3427	697	33	holds	hold	VERB
ejpam-3427	697	34	.	.	PUNCT
ejpam-3427	698	1	then	then	ADV
ejpam-3427	698	2	we	we	PRON
ejpam-3427	698	3	can	can	AUX
ejpam-3427	698	4	find	find	VERB
ejpam-3427	698	5	a	a	PRON
ejpam-3427	698	6	and	and	CCONJ
ejpam-3427	698	7	b	b	NOUN
ejpam-3427	698	8	in	in	ADP
ejpam-3427	698	9	r	r	NOUN
ejpam-3427	698	10	and	and	CCONJ
ejpam-3427	698	11	r′	r′	PROPN
ejpam-3427	698	12	,	,	PUNCT
ejpam-3427	698	13	respectively	respectively	ADV
ejpam-3427	698	14	such	such	ADJ
ejpam-3427	698	15	that	that	SCONJ
ejpam-3427	698	16	µa	µa	NOUN
ejpam-3427	698	17	(	(	PUNCT
ejpam-3427	698	18	a	a	X
ejpam-3427	698	19	)	)	PUNCT
ejpam-3427	698	20	≤	≤	NUM
ejpam-3427	698	21	µb	µb	VERB
ejpam-3427	698	22	(	(	PUNCT
ejpam-3427	698	23	e′	e′	PROPN
ejpam-3427	698	24	)	)	PUNCT
ejpam-3427	698	25	and	and	CCONJ
ejpam-3427	698	26	γa	γa	PROPN
ejpam-3427	698	27	(	(	PUNCT
ejpam-3427	698	28	a	a	PRON
ejpam-3427	698	29	)	)	PUNCT
ejpam-3427	698	30	≥	≥	NOUN
ejpam-3427	698	31	γb	γb	PROPN
ejpam-3427	698	32	(	(	PUNCT
ejpam-3427	698	33	e′	e′	PROPN
ejpam-3427	698	34	)	)	PUNCT
ejpam-3427	698	35	.	.	PUNCT
ejpam-3427	699	1	µb	µb	VERB
ejpam-3427	699	2	(	(	PUNCT
ejpam-3427	699	3	b	b	NOUN
ejpam-3427	699	4	)	)	PUNCT
ejpam-3427	699	5	≤	≤	NOUN
ejpam-3427	699	6	µa	µa	NOUN
ejpam-3427	699	7	(	(	PUNCT
ejpam-3427	699	8	e	e	NOUN
ejpam-3427	699	9	)	)	PUNCT
ejpam-3427	699	10	and	and	CCONJ
ejpam-3427	699	11	γb	γb	INTJ
ejpam-3427	699	12	(	(	PUNCT
ejpam-3427	699	13	b	b	NOUN
ejpam-3427	699	14	)	)	PUNCT
ejpam-3427	699	15	≥	≥	NOUN
ejpam-3427	699	16	γa	γa	PROPN
ejpam-3427	699	17	(	(	PUNCT
ejpam-3427	699	18	e	e	NOUN
ejpam-3427	699	19	)	)	PUNCT
ejpam-3427	699	20	.	.	PUNCT
ejpam-3427	700	1	thus	thus	ADV
ejpam-3427	700	2	µa×b(a	µa×b(a	NOUN
ejpam-3427	700	3	,	,	PUNCT
ejpam-3427	700	4	b	b	NOUN
ejpam-3427	700	5	)	)	PUNCT
ejpam-3427	700	6	=	=	SYM
ejpam-3427	700	7	max{µa(a	max{µa(a	PROPN
ejpam-3427	700	8	)	)	PUNCT
ejpam-3427	700	9	,	,	PUNCT
ejpam-3427	700	10	µb(b	µb(b	NUM
ejpam-3427	700	11	)	)	PUNCT
ejpam-3427	700	12	}	}	PUNCT
ejpam-3427	700	13	≤	≤	NUM
ejpam-3427	700	14	max{µa(e	max{µa(e	NOUN
ejpam-3427	700	15	)	)	PUNCT
ejpam-3427	700	16	,	,	PUNCT
ejpam-3427	700	17	µb(e′	µb(e′	NOUN
ejpam-3427	700	18	)	)	PUNCT
ejpam-3427	700	19	}	}	PUNCT
ejpam-3427	700	20	=	=	SYM
ejpam-3427	700	21	µa×b(e	µa×b(e	NOUN
ejpam-3427	700	22	,	,	PUNCT
ejpam-3427	700	23	e′	e′	ADJ
ejpam-3427	700	24	)	)	PUNCT
ejpam-3427	700	25	and	and	CCONJ
ejpam-3427	700	26	γa×b(a	γa×b(a	ADJ
ejpam-3427	700	27	,	,	PUNCT
ejpam-3427	700	28	b	b	NOUN
ejpam-3427	700	29	)	)	PUNCT
ejpam-3427	700	30	=	=	SYM
ejpam-3427	700	31	min{γa(a	min{γa(a	PROPN
ejpam-3427	700	32	)	)	PUNCT
ejpam-3427	700	33	,	,	PUNCT
ejpam-3427	700	34	γb(b	γb(b	NOUN
ejpam-3427	700	35	)	)	PUNCT
ejpam-3427	700	36	}	}	PUNCT
ejpam-3427	700	37	≥	≥	PROPN
ejpam-3427	700	38	min{γa(e	min{γa(e	NOUN
ejpam-3427	700	39	)	)	PUNCT
ejpam-3427	700	40	,	,	PUNCT
ejpam-3427	700	41	γb(e′	γb(e′	PROPN
ejpam-3427	700	42	)	)	PUNCT
ejpam-3427	700	43	}	}	PUNCT
ejpam-3427	700	44	=	=	PUNCT
ejpam-3427	700	45	γa×b(e	γa×b(e	VERB
ejpam-3427	700	46	,	,	PUNCT
ejpam-3427	700	47	e′	e′	ADJ
ejpam-3427	700	48	)	)	PUNCT
ejpam-3427	700	49	.	.	PUNCT
ejpam-3427	701	1	therefore	therefore	ADV
ejpam-3427	701	2	a×	a×	PROPN
ejpam-3427	701	3	b	b	PROPN
ejpam-3427	701	4	is	be	AUX
ejpam-3427	701	5	not	not	PART
ejpam-3427	701	6	an	an	DET
ejpam-3427	701	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	701	8	anti	anti	ADJ
ejpam-3427	701	9	fuzzy	fuzzy	ADJ
ejpam-3427	701	10	la	la	NOUN
ejpam-3427	701	11	-	-	PUNCT
ejpam-3427	701	12	subring	subring	NOUN
ejpam-3427	701	13	of	of	ADP
ejpam-3427	701	14	an	an	DET
ejpam-3427	701	15	la	la	ADJ
ejpam-3427	701	16	-	-	PUNCT
ejpam-3427	701	17	ring	ring	NOUN
ejpam-3427	701	18	r	r	NOUN
ejpam-3427	701	19	×	×	NOUN
ejpam-3427	701	20	r′.	r′.	NOUN
ejpam-3427	701	21	hence	hence	ADV
ejpam-3427	701	22	either	either	CCONJ
ejpam-3427	701	23	µa	µa	INTJ
ejpam-3427	701	24	(	(	PUNCT
ejpam-3427	701	25	x	x	X
ejpam-3427	701	26	)	)	PUNCT
ejpam-3427	701	27	≥	≥	PROPN
ejpam-3427	701	28	µb	µb	PROPN
ejpam-3427	701	29	(	(	PUNCT
ejpam-3427	701	30	e′	e′	ADJ
ejpam-3427	701	31	)	)	PUNCT
ejpam-3427	701	32	and	and	CCONJ
ejpam-3427	701	33	γa	γa	PROPN
ejpam-3427	701	34	(	(	PUNCT
ejpam-3427	701	35	x	x	NOUN
ejpam-3427	701	36	)	)	PUNCT
ejpam-3427	701	37	≤	≤	NUM
ejpam-3427	701	38	γb	γb	NOUN
ejpam-3427	701	39	(	(	PUNCT
ejpam-3427	701	40	e′	e′	PROPN
ejpam-3427	701	41	)	)	PUNCT
ejpam-3427	701	42	,	,	PUNCT
ejpam-3427	701	43	for	for	ADP
ejpam-3427	701	44	all	all	DET
ejpam-3427	701	45	x	x	SYM
ejpam-3427	701	46	∈	∈	PROPN
ejpam-3427	701	47	r1	r1	NOUN
ejpam-3427	701	48	or	or	CCONJ
ejpam-3427	701	49	µb	µb	VERB
ejpam-3427	701	50	(	(	PUNCT
ejpam-3427	701	51	x	x	NOUN
ejpam-3427	701	52	)	)	PUNCT
ejpam-3427	701	53	≥	≥	NOUN
ejpam-3427	701	54	µa	µa	NOUN
ejpam-3427	701	55	(	(	PUNCT
ejpam-3427	701	56	e	e	NOUN
ejpam-3427	701	57	)	)	PUNCT
ejpam-3427	701	58	and	and	CCONJ
ejpam-3427	701	59	γb	γb	INTJ
ejpam-3427	701	60	(	(	PUNCT
ejpam-3427	701	61	x	x	NOUN
ejpam-3427	701	62	)	)	PUNCT
ejpam-3427	701	63	≤	≤	NOUN
ejpam-3427	701	64	γa	γa	NOUN
ejpam-3427	701	65	(	(	PUNCT
ejpam-3427	701	66	e	e	NOUN
ejpam-3427	701	67	)	)	PUNCT
ejpam-3427	701	68	,	,	PUNCT
ejpam-3427	701	69	for	for	ADP
ejpam-3427	701	70	all	all	DET
ejpam-3427	701	71	x	x	SYM
ejpam-3427	701	72	∈	∈	PROPN
ejpam-3427	701	73	r2	r2	NOUN
ejpam-3427	701	74	.	.	PUNCT
ejpam-3427	702	1	k.	k.	PROPN
ejpam-3427	702	2	nasreen	nasreen	PROPN
ejpam-3427	702	3	/	/	SYM
ejpam-3427	702	4	eur	eur	PROPN
ejpam-3427	702	5	.	.	PUNCT
ejpam-3427	703	1	j.	j.	PROPN
ejpam-3427	703	2	pure	pure	PROPN
ejpam-3427	703	3	appl	appl	PROPN
ejpam-3427	703	4	.	.	PROPN
ejpam-3427	703	5	math	math	PROPN
ejpam-3427	703	6	,	,	PUNCT
ejpam-3427	703	7	12	12	NUM
ejpam-3427	703	8	(	(	PUNCT
ejpam-3427	703	9	2	2	NUM
ejpam-3427	703	10	)	)	PUNCT
ejpam-3427	703	11	(	(	PUNCT
ejpam-3427	703	12	2019	2019	NUM
ejpam-3427	703	13	)	)	PUNCT
ejpam-3427	703	14	,	,	PUNCT
ejpam-3427	703	15	622	622	NUM
ejpam-3427	703	16	-	-	SYM
ejpam-3427	703	17	648	648	NUM
ejpam-3427	703	18	646	646	NUM
ejpam-3427	703	19	theorem	theorem	NOUN
ejpam-3427	703	20	10	10	NUM
ejpam-3427	703	21	.	.	PUNCT
ejpam-3427	704	1	let	let	VERB
ejpam-3427	704	2	a	a	DET
ejpam-3427	704	3	=	=	NOUN
ejpam-3427	704	4	a1×a2×	a1×a2×	NOUN
ejpam-3427	704	5	...	...	PUNCT
ejpam-3427	704	6	×an	×an	PROPN
ejpam-3427	704	7	and	and	CCONJ
ejpam-3427	704	8	b	b	X
ejpam-3427	704	9	=	=	PUNCT
ejpam-3427	704	10	b1×b2×	b1×b2×	ADJ
ejpam-3427	704	11	...	...	PUNCT
ejpam-3427	704	12	×bn	×bn	AUX
ejpam-3427	704	13	be	be	AUX
ejpam-3427	704	14	intuitionistic	intuitionistic	ADJ
ejpam-3427	704	15	fuzzy	fuzzy	ADJ
ejpam-3427	704	16	sets	set	NOUN
ejpam-3427	704	17	of	of	ADP
ejpam-3427	704	18	la	la	NOUN
ejpam-3427	704	19	-	-	PUNCT
ejpam-3427	704	20	rings	ring	NOUN
ejpam-3427	704	21	r	r	NOUN
ejpam-3427	704	22	=	=	SYM
ejpam-3427	704	23	r1	r1	NOUN
ejpam-3427	704	24	×	×	NOUN
ejpam-3427	704	25	r2	r2	PROPN
ejpam-3427	704	26	×	×	NOUN
ejpam-3427	704	27	...	...	PUNCT
ejpam-3427	704	28	×	×	PROPN
ejpam-3427	704	29	rn	rn	PROPN
ejpam-3427	704	30	and	and	CCONJ
ejpam-3427	704	31	r′	r′	PROPN
ejpam-3427	704	32	=	=	NOUN
ejpam-3427	704	33	r′1	r′1	NOUN
ejpam-3427	704	34	×	×	NOUN
ejpam-3427	704	35	r′2	r′2	NOUN
ejpam-3427	704	36	×	×	NOUN
ejpam-3427	704	37	...	...	PUNCT
ejpam-3427	704	38	×	×	NOUN
ejpam-3427	704	39	r′n	r′n	NOUN
ejpam-3427	704	40	with	with	ADP
ejpam-3427	704	41	left	left	ADJ
ejpam-3427	704	42	identities	identity	NOUN
ejpam-3427	704	43	e	e	NOUN
ejpam-3427	704	44	=	=	SYM
ejpam-3427	704	45	(	(	PUNCT
ejpam-3427	704	46	e1	e1	PROPN
ejpam-3427	704	47	,	,	PUNCT
ejpam-3427	704	48	e2	e2	PROPN
ejpam-3427	704	49	,	,	PUNCT
ejpam-3427	704	50	...	...	PUNCT
ejpam-3427	704	51	,	,	PUNCT
ejpam-3427	704	52	en	en	X
ejpam-3427	704	53	)	)	PUNCT
ejpam-3427	704	54	and	and	CCONJ
ejpam-3427	704	55	e′	e′	NOUN
ejpam-3427	704	56	=	=	SYM
ejpam-3427	704	57	(	(	PUNCT
ejpam-3427	704	58	e1′	e1′	ADJ
ejpam-3427	704	59	,	,	PUNCT
ejpam-3427	704	60	e2′	e2′	NOUN
ejpam-3427	704	61	,	,	PUNCT
ejpam-3427	704	62	...	...	PUNCT
ejpam-3427	704	63	,	,	PUNCT
ejpam-3427	704	64	en′	en′	PROPN
ejpam-3427	704	65	)	)	PUNCT
ejpam-3427	704	66	,	,	PUNCT
ejpam-3427	704	67	respectively	respectively	ADV
ejpam-3427	704	68	and	and	CCONJ
ejpam-3427	704	69	a×b	a×b	PROPN
ejpam-3427	704	70	is	be	AUX
ejpam-3427	704	71	an	an	DET
ejpam-3427	704	72	intuitionistic	intuitionistic	ADJ
ejpam-3427	704	73	anti	anti	ADJ
ejpam-3427	704	74	fuzzy	fuzzy	ADJ
ejpam-3427	704	75	normal	normal	ADJ
ejpam-3427	704	76	la	la	NOUN
ejpam-3427	704	77	-	-	PUNCT
ejpam-3427	704	78	subring	subring	NOUN
ejpam-3427	704	79	of	of	ADP
ejpam-3427	704	80	an	an	DET
ejpam-3427	704	81	la	la	ADJ
ejpam-3427	704	82	-	-	PUNCT
ejpam-3427	704	83	ring	ring	NOUN
ejpam-3427	704	84	r×r′.	r×r′.	NOUN
ejpam-3427	704	85	then	then	ADV
ejpam-3427	704	86	the	the	DET
ejpam-3427	704	87	following	follow	VERB
ejpam-3427	704	88	conditions	condition	NOUN
ejpam-3427	704	89	are	be	AUX
ejpam-3427	704	90	true	true	ADJ
ejpam-3427	704	91	.	.	PUNCT
ejpam-3427	705	1	1	1	X
ejpam-3427	705	2	.	.	X
ejpam-3427	706	1	if	if	SCONJ
ejpam-3427	706	2	µa	µa	PROPN
ejpam-3427	706	3	(	(	PUNCT
ejpam-3427	706	4	x	x	X
ejpam-3427	706	5	)	)	PUNCT
ejpam-3427	706	6	≥	≥	PROPN
ejpam-3427	706	7	µb	µb	PROPN
ejpam-3427	706	8	(	(	PUNCT
ejpam-3427	706	9	e′	e′	ADJ
ejpam-3427	706	10	)	)	PUNCT
ejpam-3427	706	11	and	and	CCONJ
ejpam-3427	706	12	γa	γa	PROPN
ejpam-3427	706	13	(	(	PUNCT
ejpam-3427	706	14	x	x	NOUN
ejpam-3427	706	15	)	)	PUNCT
ejpam-3427	706	16	≤	≤	NUM
ejpam-3427	706	17	γb	γb	NOUN
ejpam-3427	706	18	(	(	PUNCT
ejpam-3427	706	19	e′	e′	PROPN
ejpam-3427	706	20	)	)	PUNCT
ejpam-3427	706	21	,	,	PUNCT
ejpam-3427	706	22	for	for	ADP
ejpam-3427	706	23	all	all	DET
ejpam-3427	706	24	x	x	SYM
ejpam-3427	706	25	∈	∈	PROPN
ejpam-3427	706	26	r	r	NOUN
ejpam-3427	706	27	,	,	PUNCT
ejpam-3427	706	28	then	then	ADV
ejpam-3427	706	29	a	a	PRON
ejpam-3427	706	30	is	be	AUX
ejpam-3427	706	31	an	an	DET
ejpam-3427	706	32	intuitionistic	intuitionistic	ADJ
ejpam-3427	706	33	anti	anti	ADJ
ejpam-3427	706	34	fuzzy	fuzzy	ADJ
ejpam-3427	706	35	normal	normal	ADJ
ejpam-3427	706	36	la	la	NOUN
ejpam-3427	706	37	-	-	PUNCT
ejpam-3427	706	38	subring	subring	NOUN
ejpam-3427	706	39	of	of	ADP
ejpam-3427	706	40	r.	r.	PROPN
ejpam-3427	706	41	2	2	NUM
ejpam-3427	706	42	.	.	PUNCT
ejpam-3427	707	1	if	if	SCONJ
ejpam-3427	707	2	µb	µb	VERB
ejpam-3427	707	3	(	(	PUNCT
ejpam-3427	707	4	x′	x′	NUM
ejpam-3427	707	5	)	)	PUNCT
ejpam-3427	707	6	≥	≥	NOUN
ejpam-3427	707	7	µa	µa	NOUN
ejpam-3427	707	8	(	(	PUNCT
ejpam-3427	707	9	e	e	NOUN
ejpam-3427	707	10	)	)	PUNCT
ejpam-3427	707	11	and	and	CCONJ
ejpam-3427	707	12	γb	γb	X
ejpam-3427	707	13	(	(	PUNCT
ejpam-3427	707	14	x′	x′	NUM
ejpam-3427	707	15	)	)	PUNCT
ejpam-3427	707	16	≤	≤	NOUN
ejpam-3427	707	17	γa	γa	NOUN
ejpam-3427	707	18	(	(	PUNCT
ejpam-3427	707	19	e	e	NOUN
ejpam-3427	707	20	)	)	PUNCT
ejpam-3427	707	21	,	,	PUNCT
ejpam-3427	707	22	for	for	ADP
ejpam-3427	707	23	all	all	PRON
ejpam-3427	707	24	x′	x′	PROPN
ejpam-3427	707	25	∈	∈	PROPN
ejpam-3427	707	26	r′	r′	PROPN
ejpam-3427	707	27	,	,	PUNCT
ejpam-3427	707	28	then	then	ADV
ejpam-3427	707	29	b	b	PROPN
ejpam-3427	707	30	is	be	AUX
ejpam-3427	707	31	an	an	DET
ejpam-3427	707	32	intuitionistic	intuitionistic	ADJ
ejpam-3427	707	33	anti	anti	ADJ
ejpam-3427	707	34	fuzzy	fuzzy	ADJ
ejpam-3427	707	35	normal	normal	ADJ
ejpam-3427	707	36	la	la	NOUN
ejpam-3427	707	37	-	-	PUNCT
ejpam-3427	707	38	subring	subring	NOUN
ejpam-3427	707	39	of	of	ADP
ejpam-3427	707	40	r′.	r′.	NOUN
ejpam-3427	707	41	proof	proof	NOUN
ejpam-3427	707	42	.	.	PUNCT
ejpam-3427	708	1	1	1	X
ejpam-3427	708	2	.	.	X
ejpam-3427	708	3	let	let	VERB
ejpam-3427	708	4	µa	µa	INTJ
ejpam-3427	708	5	(	(	PUNCT
ejpam-3427	708	6	x	x	X
ejpam-3427	708	7	)	)	PUNCT
ejpam-3427	708	8	≥	≥	PROPN
ejpam-3427	708	9	µb	µb	PROPN
ejpam-3427	708	10	(	(	PUNCT
ejpam-3427	708	11	e′	e′	ADJ
ejpam-3427	708	12	)	)	PUNCT
ejpam-3427	708	13	and	and	CCONJ
ejpam-3427	708	14	γa	γa	PROPN
ejpam-3427	708	15	(	(	PUNCT
ejpam-3427	708	16	x	x	NOUN
ejpam-3427	708	17	)	)	PUNCT
ejpam-3427	708	18	≤	≤	NUM
ejpam-3427	708	19	γb	γb	NOUN
ejpam-3427	708	20	(	(	PUNCT
ejpam-3427	708	21	e′	e′	NOUN
ejpam-3427	708	22	)	)	PUNCT
ejpam-3427	708	23	for	for	ADP
ejpam-3427	708	24	all	all	DET
ejpam-3427	708	25	x	x	SYM
ejpam-3427	708	26	∈	∈	PROPN
ejpam-3427	708	27	r	r	NOUN
ejpam-3427	708	28	,	,	PUNCT
ejpam-3427	708	29	and	and	CCONJ
ejpam-3427	708	30	y	y	PROPN
ejpam-3427	708	31	∈	∈	PROPN
ejpam-3427	708	32	r.	r.	NOUN
ejpam-3427	708	33	we	we	PRON
ejpam-3427	708	34	have	have	VERB
ejpam-3427	708	35	to	to	PART
ejpam-3427	708	36	show	show	VERB
ejpam-3427	708	37	that	that	SCONJ
ejpam-3427	708	38	a	a	PRON
ejpam-3427	708	39	is	be	AUX
ejpam-3427	708	40	an	an	DET
ejpam-3427	708	41	intuitionistic	intuitionistic	ADJ
ejpam-3427	708	42	anti	anti	ADJ
ejpam-3427	708	43	fuzzy	fuzzy	ADJ
ejpam-3427	708	44	normal	normal	ADJ
ejpam-3427	708	45	la	la	NOUN
ejpam-3427	708	46	-	-	PUNCT
ejpam-3427	708	47	subring	subring	NOUN
ejpam-3427	708	48	of	of	ADP
ejpam-3427	708	49	r.	r.	PROPN
ejpam-3427	708	50	now	now	ADV
ejpam-3427	708	51	µa(x−	µa(x−	NUM
ejpam-3427	708	52	y	y	NOUN
ejpam-3427	708	53	)	)	PUNCT
ejpam-3427	708	54	=	=	SYM
ejpam-3427	709	1	µa(x+	µa(x+	PROPN
ejpam-3427	709	2	(	(	PUNCT
ejpam-3427	709	3	−y	−y	NOUN
ejpam-3427	709	4	)	)	PUNCT
ejpam-3427	709	5	)	)	PUNCT
ejpam-3427	710	1	=	=	SYM
ejpam-3427	710	2	max{µa(x+	max{µa(x+	PROPN
ejpam-3427	710	3	(	(	PUNCT
ejpam-3427	710	4	−y	−y	NOUN
ejpam-3427	710	5	)	)	PUNCT
ejpam-3427	710	6	)	)	PUNCT
ejpam-3427	710	7	,	,	PUNCT
ejpam-3427	710	8	µb(e′	µb(e′	PROPN
ejpam-3427	710	9	+	+	CCONJ
ejpam-3427	710	10	(	(	PUNCT
ejpam-3427	710	11	−e′	−e′	NOUN
ejpam-3427	710	12	)	)	PUNCT
ejpam-3427	710	13	)	)	PUNCT
ejpam-3427	710	14	}	}	PUNCT
ejpam-3427	710	15	=	=	SYM
ejpam-3427	710	16	µa×b(x+	µa×b(x+	X
ejpam-3427	710	17	(	(	PUNCT
ejpam-3427	710	18	−y	−y	NOUN
ejpam-3427	710	19	)	)	PUNCT
ejpam-3427	710	20	,	,	PUNCT
ejpam-3427	710	21	e′	e′	X
ejpam-3427	710	22	+	+	CCONJ
ejpam-3427	710	23	(	(	PUNCT
ejpam-3427	710	24	−e′	−e′	NOUN
ejpam-3427	710	25	)	)	PUNCT
ejpam-3427	710	26	)	)	PUNCT
ejpam-3427	711	1	=	=	PUNCT
ejpam-3427	711	2	µa×b((x	µa×b((x	VERB
ejpam-3427	711	3	,	,	PUNCT
ejpam-3427	711	4	e′	e′	ADJ
ejpam-3427	711	5	)	)	PUNCT
ejpam-3427	712	1	+	+	CCONJ
ejpam-3427	712	2	(	(	PUNCT
ejpam-3427	712	3	−y,−e′	−y,−e′	INTJ
ejpam-3427	712	4	)	)	PUNCT
ejpam-3427	712	5	)	)	PUNCT
ejpam-3427	713	1	=	=	PUNCT
ejpam-3427	713	2	µa×b((x	µa×b((x	VERB
ejpam-3427	713	3	,	,	PUNCT
ejpam-3427	713	4	e′)−	e′)−	PROPN
ejpam-3427	713	5	(	(	PUNCT
ejpam-3427	713	6	y	y	NOUN
ejpam-3427	713	7	,	,	PUNCT
ejpam-3427	713	8	e′	e′	ADJ
ejpam-3427	713	9	)	)	PUNCT
ejpam-3427	713	10	)	)	PUNCT
ejpam-3427	713	11	≤	≤	NUM
ejpam-3427	713	12	µa×b(x	µa×b(x	ADJ
ejpam-3427	713	13	,	,	PUNCT
ejpam-3427	713	14	e′	e′	ADJ
ejpam-3427	713	15	)	)	PUNCT
ejpam-3427	713	16	∨	∨	NOUN
ejpam-3427	713	17	µa×b(y	µa×b(y	ADJ
ejpam-3427	713	18	,	,	PUNCT
ejpam-3427	713	19	e′	e′	ADJ
ejpam-3427	713	20	)	)	PUNCT
ejpam-3427	713	21	=	=	SYM
ejpam-3427	713	22	max{max{µa(x	max{max{µa(x	NOUN
ejpam-3427	713	23	)	)	PUNCT
ejpam-3427	713	24	,	,	PUNCT
ejpam-3427	713	25	µb(e′)},max{µa(y	µb(e′)},max{µa(y	NOUN
ejpam-3427	713	26	)	)	PUNCT
ejpam-3427	713	27	,	,	PUNCT
ejpam-3427	713	28	µb(e′	µb(e′	NOUN
ejpam-3427	713	29	)	)	PUNCT
ejpam-3427	713	30	}	}	PUNCT
ejpam-3427	713	31	}	}	PUNCT
ejpam-3427	713	32	=	=	SYM
ejpam-3427	713	33	µa(x	µa(x	NOUN
ejpam-3427	713	34	)	)	PUNCT
ejpam-3427	713	35	∨	∨	NUM
ejpam-3427	713	36	µa(y	µa(y	NOUN
ejpam-3427	713	37	)	)	PUNCT
ejpam-3427	713	38	and	and	CCONJ
ejpam-3427	713	39	µa(xy	µa(xy	NOUN
ejpam-3427	713	40	)	)	PUNCT
ejpam-3427	713	41	=	=	SYM
ejpam-3427	713	42	max{µa(xy	max{µa(xy	PROPN
ejpam-3427	713	43	)	)	PUNCT
ejpam-3427	713	44	,	,	PUNCT
ejpam-3427	713	45	µb(e′e′	µb(e′e′	VERB
ejpam-3427	713	46	)	)	PUNCT
ejpam-3427	713	47	}	}	PUNCT
ejpam-3427	713	48	=	=	SYM
ejpam-3427	713	49	µa×b(xy	µa×b(xy	PROPN
ejpam-3427	713	50	,	,	PUNCT
ejpam-3427	713	51	e′e′	e′e′	NOUN
ejpam-3427	713	52	)	)	PUNCT
ejpam-3427	713	53	=	=	PUNCT
ejpam-3427	713	54	µa×b((x	µa×b((x	VERB
ejpam-3427	713	55	,	,	PUNCT
ejpam-3427	713	56	e′	e′	ADJ
ejpam-3427	713	57	)	)	PUNCT
ejpam-3427	714	1	◦	◦	NOUN
ejpam-3427	714	2	(	(	PUNCT
ejpam-3427	714	3	y	y	NOUN
ejpam-3427	714	4	,	,	PUNCT
ejpam-3427	714	5	e′	e′	ADJ
ejpam-3427	714	6	)	)	PUNCT
ejpam-3427	714	7	)	)	PUNCT
ejpam-3427	714	8	≤	≤	NUM
ejpam-3427	714	9	µa×b(x	µa×b(x	ADJ
ejpam-3427	714	10	,	,	PUNCT
ejpam-3427	714	11	e′	e′	ADJ
ejpam-3427	714	12	)	)	PUNCT
ejpam-3427	714	13	∨	∨	NOUN
ejpam-3427	714	14	µa×b(y	µa×b(y	ADJ
ejpam-3427	714	15	,	,	PUNCT
ejpam-3427	714	16	e′	e′	ADJ
ejpam-3427	714	17	)	)	PUNCT
ejpam-3427	714	18	=	=	SYM
ejpam-3427	714	19	max{max{µa(x	max{max{µa(x	NOUN
ejpam-3427	714	20	)	)	PUNCT
ejpam-3427	714	21	,	,	PUNCT
ejpam-3427	714	22	µb(e′)},max{µa(y	µb(e′)},max{µa(y	NOUN
ejpam-3427	714	23	)	)	PUNCT
ejpam-3427	714	24	,	,	PUNCT
ejpam-3427	714	25	µb(e′	µb(e′	NOUN
ejpam-3427	714	26	)	)	PUNCT
ejpam-3427	714	27	}	}	PUNCT
ejpam-3427	714	28	}	}	PUNCT
ejpam-3427	714	29	=	=	SYM
ejpam-3427	714	30	µa(x	µa(x	NOUN
ejpam-3427	714	31	)	)	PUNCT
ejpam-3427	714	32	∨	∨	NUM
ejpam-3427	714	33	µa(y	µa(y	NOUN
ejpam-3427	714	34	)	)	PUNCT
ejpam-3427	714	35	.	.	PUNCT
ejpam-3427	715	1	similarly	similarly	ADV
ejpam-3427	715	2	,	,	PUNCT
ejpam-3427	715	3	we	we	PRON
ejpam-3427	715	4	have	have	VERB
ejpam-3427	715	5	γa(x−	γa(x−	PROPN
ejpam-3427	715	6	y	y	PROPN
ejpam-3427	715	7	)	)	PUNCT
ejpam-3427	715	8	≥	≥	PROPN
ejpam-3427	715	9	min{γa(x	min{γa(x	NOUN
ejpam-3427	715	10	)	)	PUNCT
ejpam-3427	715	11	,	,	PUNCT
ejpam-3427	715	12	γa(y	γa(y	NOUN
ejpam-3427	715	13	)	)	PUNCT
ejpam-3427	715	14	}	}	PUNCT
ejpam-3427	715	15	and	and	CCONJ
ejpam-3427	715	16	γa(xy	γa(xy	PROPN
ejpam-3427	715	17	)	)	PUNCT
ejpam-3427	715	18	≥	≥	PROPN
ejpam-3427	715	19	min{γa(x	min{γa(x	NOUN
ejpam-3427	715	20	)	)	PUNCT
ejpam-3427	715	21	,	,	PUNCT
ejpam-3427	715	22	γa(y	γa(y	NOUN
ejpam-3427	715	23	)	)	PUNCT
ejpam-3427	715	24	}	}	PUNCT
ejpam-3427	715	25	.	.	PUNCT
ejpam-3427	716	1	thus	thus	ADV
ejpam-3427	716	2	a	a	PRON
ejpam-3427	716	3	is	be	AUX
ejpam-3427	716	4	an	an	DET
ejpam-3427	716	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	716	6	anti	anti	ADJ
ejpam-3427	716	7	fuzzy	fuzzy	ADJ
ejpam-3427	716	8	la	la	NOUN
ejpam-3427	716	9	-	-	PUNCT
ejpam-3427	716	10	subring	subring	NOUN
ejpam-3427	716	11	of	of	ADP
ejpam-3427	716	12	r.	r.	PROPN
ejpam-3427	716	13	now	now	ADV
ejpam-3427	716	14	µa(xy	µa(xy	PROPN
ejpam-3427	716	15	)	)	PUNCT
ejpam-3427	717	1	=	=	SYM
ejpam-3427	717	2	max{µa(xy	max{µa(xy	PROPN
ejpam-3427	717	3	)	)	PUNCT
ejpam-3427	717	4	,	,	PUNCT
ejpam-3427	717	5	µb(e′e′	µb(e′e′	VERB
ejpam-3427	717	6	)	)	PUNCT
ejpam-3427	717	7	}	}	PUNCT
ejpam-3427	717	8	=	=	SYM
ejpam-3427	717	9	µa×b	µa×b	PROPN
ejpam-3427	717	10	(	(	PUNCT
ejpam-3427	717	11	xy	xy	PROPN
ejpam-3427	717	12	,	,	PUNCT
ejpam-3427	717	13	e′e′	e′e′	VERB
ejpam-3427	717	14	)	)	PUNCT
ejpam-3427	718	1	=	=	SYM
ejpam-3427	718	2	µa×b	µa×b	PROPN
ejpam-3427	718	3	(	(	PUNCT
ejpam-3427	718	4	(	(	PUNCT
ejpam-3427	718	5	x	x	NOUN
ejpam-3427	718	6	,	,	PUNCT
ejpam-3427	718	7	e′	e′	X
ejpam-3427	718	8	)	)	PUNCT
ejpam-3427	718	9	◦	◦	NOUN
ejpam-3427	718	10	(	(	PUNCT
ejpam-3427	718	11	y	y	NOUN
ejpam-3427	718	12	,	,	PUNCT
ejpam-3427	718	13	e′	e′	NOUN
ejpam-3427	718	14	)	)	PUNCT
ejpam-3427	718	15	)	)	PUNCT
ejpam-3427	719	1	=	=	SYM
ejpam-3427	719	2	µa×b	µa×b	PROPN
ejpam-3427	719	3	(	(	PUNCT
ejpam-3427	719	4	(	(	PUNCT
ejpam-3427	719	5	y	y	NOUN
ejpam-3427	719	6	,	,	PUNCT
ejpam-3427	719	7	e′	e′	NOUN
ejpam-3427	719	8	)	)	PUNCT
ejpam-3427	719	9	◦	◦	NOUN
ejpam-3427	719	10	(	(	PUNCT
ejpam-3427	719	11	x	x	NOUN
ejpam-3427	719	12	,	,	PUNCT
ejpam-3427	719	13	e′	e′	NOUN
ejpam-3427	719	14	)	)	PUNCT
ejpam-3427	719	15	)	)	PUNCT
ejpam-3427	720	1	=	=	SYM
ejpam-3427	720	2	µa×b(yx	µa×b(yx	X
ejpam-3427	720	3	,	,	PUNCT
ejpam-3427	720	4	e′e′	e′e′	NOUN
ejpam-3427	720	5	)	)	PUNCT
ejpam-3427	720	6	=	=	SYM
ejpam-3427	721	1	max{µa(yx	max{µa(yx	NOUN
ejpam-3427	721	2	)	)	PUNCT
ejpam-3427	721	3	,	,	PUNCT
ejpam-3427	721	4	µb(e′e′	µb(e′e′	VERB
ejpam-3427	721	5	)	)	PUNCT
ejpam-3427	721	6	}	}	PUNCT
ejpam-3427	721	7	references	reference	VERB
ejpam-3427	721	8	647	647	NUM
ejpam-3427	721	9	=	=	SYM
ejpam-3427	721	10	µa(yx	µa(yx	NOUN
ejpam-3427	721	11	)	)	PUNCT
ejpam-3427	721	12	.	.	PUNCT
ejpam-3427	722	1	similarly	similarly	ADV
ejpam-3427	722	2	,	,	PUNCT
ejpam-3427	722	3	γb(xy	γb(xy	NOUN
ejpam-3427	722	4	)	)	PUNCT
ejpam-3427	722	5	=	=	SYM
ejpam-3427	722	6	γb(yx	γb(yx	PROPN
ejpam-3427	722	7	)	)	PUNCT
ejpam-3427	722	8	.	.	PUNCT
ejpam-3427	723	1	hence	hence	ADV
ejpam-3427	723	2	a	a	PRON
ejpam-3427	723	3	is	be	AUX
ejpam-3427	723	4	an	an	DET
ejpam-3427	723	5	intuitionistic	intuitionistic	ADJ
ejpam-3427	723	6	anti	anti	ADJ
ejpam-3427	723	7	fuzzy	fuzzy	ADJ
ejpam-3427	723	8	normal	normal	ADJ
ejpam-3427	723	9	la	la	NOUN
ejpam-3427	723	10	-	-	PUNCT
ejpam-3427	723	11	subring	subring	NOUN
ejpam-3427	723	12	of	of	ADP
ejpam-3427	723	13	r.	r.	PROPN
ejpam-3427	723	14	2	2	NUM
ejpam-3427	723	15	.	.	PUNCT
ejpam-3427	723	16	is	be	AUX
ejpam-3427	723	17	same	same	ADJ
ejpam-3427	723	18	as	as	ADP
ejpam-3427	723	19	1	1	NUM
ejpam-3427	723	20	.	.	PUNCT
ejpam-3427	723	21	conclusion	conclusion	NOUN
ejpam-3427	723	22	1	1	NUM
ejpam-3427	723	23	.	.	PUNCT
ejpam-3427	724	1	our	our	PRON
ejpam-3427	724	2	aim	aim	NOUN
ejpam-3427	724	3	is	be	AUX
ejpam-3427	724	4	to	to	PART
ejpam-3427	724	5	encourage	encourage	VERB
ejpam-3427	724	6	the	the	DET
ejpam-3427	724	7	research	research	NOUN
ejpam-3427	724	8	of	of	ADP
ejpam-3427	724	9	associative	associative	ADJ
ejpam-3427	724	10	algebraic	algebraic	ADJ
ejpam-3427	724	11	structure	structure	NOUN
ejpam-3427	724	12	by	by	ADP
ejpam-3427	724	13	studying	study	VERB
ejpam-3427	724	14	a	a	DET
ejpam-3427	724	15	class	class	NOUN
ejpam-3427	724	16	of	of	ADP
ejpam-3427	724	17	non	non	ADJ
ejpam-3427	724	18	-	-	ADJ
ejpam-3427	724	19	associative	associative	ADJ
ejpam-3427	724	20	and	and	CCONJ
ejpam-3427	724	21	non	non	ADJ
ejpam-3427	724	22	-	-	ADJ
ejpam-3427	724	23	commutative	commutative	ADJ
ejpam-3427	724	24	algebraic	algebraic	ADJ
ejpam-3427	724	25	structure	structure	NOUN
ejpam-3427	724	26	means	mean	VERB
ejpam-3427	724	27	laring	lare	VERB
ejpam-3427	724	28	and	and	CCONJ
ejpam-3427	724	29	explored	explore	VERB
ejpam-3427	724	30	new	new	ADJ
ejpam-3427	724	31	methodological	methodological	ADJ
ejpam-3427	724	32	developments	development	NOUN
ejpam-3427	724	33	on	on	ADP
ejpam-3427	724	34	la	la	ADJ
ejpam-3427	724	35	-	-	PUNCT
ejpam-3427	724	36	ring	ring	NOUN
ejpam-3427	724	37	,	,	PUNCT
ejpam-3427	724	38	which	which	PRON
ejpam-3427	724	39	will	will	AUX
ejpam-3427	724	40	be	be	AUX
ejpam-3427	724	41	helpful	helpful	ADJ
ejpam-3427	724	42	in	in	ADP
ejpam-3427	724	43	future	future	NOUN
ejpam-3427	724	44	.	.	PUNCT
ejpam-3427	725	1	the	the	DET
ejpam-3427	725	2	objective	objective	NOUN
ejpam-3427	725	3	of	of	ADP
ejpam-3427	725	4	this	this	DET
ejpam-3427	725	5	paper	paper	NOUN
ejpam-3427	725	6	is	be	AUX
ejpam-3427	725	7	to	to	PART
ejpam-3427	725	8	initiate	initiate	VERB
ejpam-3427	725	9	the	the	DET
ejpam-3427	725	10	notion	notion	NOUN
ejpam-3427	725	11	of	of	ADP
ejpam-3427	725	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	725	13	anti	anti	ADJ
ejpam-3427	725	14	fuzzy	fuzzy	ADJ
ejpam-3427	725	15	normal	normal	ADJ
ejpam-3427	725	16	subrings	subring	NOUN
ejpam-3427	725	17	on	on	ADP
ejpam-3427	725	18	la	la	ADJ
ejpam-3427	725	19	-	-	PUNCT
ejpam-3427	725	20	ring	ring	NOUN
ejpam-3427	725	21	and	and	CCONJ
ejpam-3427	725	22	established	establish	VERB
ejpam-3427	725	23	some	some	DET
ejpam-3427	725	24	imperative	imperative	ADJ
ejpam-3427	725	25	properties	property	NOUN
ejpam-3427	725	26	of	of	ADP
ejpam-3427	725	27	such	such	ADJ
ejpam-3427	725	28	subrings	subring	NOUN
ejpam-3427	725	29	.	.	PUNCT
ejpam-3427	726	1	we	we	PRON
ejpam-3427	726	2	hope	hope	VERB
ejpam-3427	726	3	that	that	SCONJ
ejpam-3427	726	4	in	in	ADP
ejpam-3427	726	5	future	future	NOUN
ejpam-3427	726	6	,	,	PUNCT
ejpam-3427	726	7	this	this	DET
ejpam-3427	726	8	concept	concept	NOUN
ejpam-3427	726	9	would	would	AUX
ejpam-3427	726	10	be	be	AUX
ejpam-3427	726	11	a	a	DET
ejpam-3427	726	12	useful	useful	ADJ
ejpam-3427	726	13	contribution	contribution	NOUN
ejpam-3427	726	14	in	in	ADP
ejpam-3427	726	15	the	the	DET
ejpam-3427	726	16	theory	theory	NOUN
ejpam-3427	726	17	of	of	ADP
ejpam-3427	726	18	nonassociative	nonassociative	ADJ
ejpam-3427	726	19	algebraic	algebraic	ADJ
ejpam-3427	726	20	structures	structure	NOUN
ejpam-3427	726	21	.	.	PUNCT
ejpam-3427	727	1	references	reference	NOUN
ejpam-3427	727	2	[	[	X
ejpam-3427	727	3	1	1	NUM
ejpam-3427	727	4	]	]	PUNCT
ejpam-3427	727	5	k.	k.	PROPN
ejpam-3427	727	6	atanassov	atanassov	PROPN
ejpam-3427	727	7	,	,	PUNCT
ejpam-3427	727	8	intuitionistic	intuitionistic	ADJ
ejpam-3427	727	9	fuzzy	fuzzy	ADJ
ejpam-3427	727	10	sets	set	NOUN
ejpam-3427	727	11	,	,	PUNCT
ejpam-3427	727	12	fuzzy	fuzzy	ADJ
ejpam-3427	727	13	sets	set	NOUN
ejpam-3427	727	14	and	and	CCONJ
ejpam-3427	727	15	systems	system	NOUN
ejpam-3427	727	16	20(1986	20(1986	NUM
ejpam-3427	727	17	)	)	PUNCT
ejpam-3427	727	18	87	87	NUM
ejpam-3427	727	19	-	-	SYM
ejpam-3427	727	20	96	96	NUM
ejpam-3427	727	21	.	.	PUNCT
ejpam-3427	728	1	[	[	X
ejpam-3427	728	2	2	2	NUM
ejpam-3427	728	3	]	]	PUNCT
ejpam-3427	728	4	k.	k.	PROPN
ejpam-3427	728	5	atanassov	atanassov	PROPN
ejpam-3427	728	6	,	,	PUNCT
ejpam-3427	728	7	new	new	ADJ
ejpam-3427	728	8	operations	operation	NOUN
ejpam-3427	728	9	defined	define	VERB
ejpam-3427	728	10	over	over	ADP
ejpam-3427	728	11	the	the	DET
ejpam-3427	728	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	728	13	fuzzy	fuzzy	ADJ
ejpam-3427	728	14	sets	set	NOUN
ejpam-3427	728	15	,	,	PUNCT
ejpam-3427	728	16	fuzzy	fuzzy	ADJ
ejpam-3427	728	17	sets	set	NOUN
ejpam-3427	728	18	and	and	CCONJ
ejpam-3427	728	19	systems	system	NOUN
ejpam-3427	728	20	,	,	PUNCT
ejpam-3427	728	21	61(1994	61(1994	NUM
ejpam-3427	728	22	)	)	PUNCT
ejpam-3427	728	23	137	137	NUM
ejpam-3427	728	24	-	-	SYM
ejpam-3427	728	25	142	142	NUM
ejpam-3427	728	26	.	.	PUNCT
ejpam-3427	729	1	[	[	X
ejpam-3427	729	2	3	3	X
ejpam-3427	729	3	]	]	X
ejpam-3427	729	4	b.	b.	PROPN
ejpam-3427	729	5	banerjee	banerjee	PROPN
ejpam-3427	729	6	and	and	CCONJ
ejpam-3427	729	7	d.	d.	PROPN
ejpam-3427	729	8	k.	k.	PROPN
ejpam-3427	729	9	basnet	basnet	PROPN
ejpam-3427	729	10	,	,	PUNCT
ejpam-3427	729	11	intuitionistic	intuitionistic	ADJ
ejpam-3427	729	12	fuzzy	fuzzy	ADJ
ejpam-3427	729	13	subrings	subring	NOUN
ejpam-3427	729	14	and	and	CCONJ
ejpam-3427	729	15	ideals	ideal	NOUN
ejpam-3427	729	16	,	,	PUNCT
ejpam-3427	729	17	j.	j.	PROPN
ejpam-3427	729	18	fuzzy	fuzzy	PROPN
ejpam-3427	729	19	math	math	PROPN
ejpam-3427	729	20	.	.	PUNCT
ejpam-3427	729	21	,	,	PUNCT
ejpam-3427	730	1	11(2003	11(2003	NUM
ejpam-3427	730	2	)	)	PUNCT
ejpam-3427	730	3	139	139	NUM
ejpam-3427	730	4	-	-	SYM
ejpam-3427	730	5	155	155	NUM
ejpam-3427	730	6	.	.	PUNCT
ejpam-3427	731	1	[	[	X
ejpam-3427	731	2	4	4	NUM
ejpam-3427	731	3	]	]	X
ejpam-3427	731	4	r.	r.	PROPN
ejpam-3427	731	5	biswas	biswas	PROPN
ejpam-3427	731	6	,	,	PUNCT
ejpam-3427	731	7	intuitionistic	intuitionistic	ADJ
ejpam-3427	731	8	fuzzy	fuzzy	ADJ
ejpam-3427	731	9	subgroups	subgroup	NOUN
ejpam-3427	731	10	,	,	PUNCT
ejpam-3427	731	11	math	math	NOUN
ejpam-3427	731	12	.	.	PUNCT
ejpam-3427	732	1	forum	forum	PROPN
ejpam-3427	732	2	,	,	PUNCT
ejpam-3427	732	3	x(1989	x(1989	PROPN
ejpam-3427	732	4	)	)	PUNCT
ejpam-3427	732	5	37	37	NUM
ejpam-3427	732	6	-	-	SYM
ejpam-3427	732	7	46	46	NUM
ejpam-3427	732	8	.	.	PUNCT
ejpam-3427	733	1	[	[	X
ejpam-3427	733	2	5	5	NUM
ejpam-3427	733	3	]	]	PUNCT
ejpam-3427	733	4	r.	r.	PROPN
ejpam-3427	733	5	j.	j.	PROPN
ejpam-3427	733	6	cho	cho	PROPN
ejpam-3427	733	7	,	,	PUNCT
ejpam-3427	733	8	j.	j.	PROPN
ejpam-3427	733	9	jezek	jezek	PROPN
ejpam-3427	733	10	and	and	CCONJ
ejpam-3427	733	11	t.	t.	PROPN
ejpam-3427	733	12	kepka	kepka	NOUN
ejpam-3427	733	13	,	,	PUNCT
ejpam-3427	733	14	paramedial	paramedial	ADJ
ejpam-3427	733	15	groupoids	groupoid	NOUN
ejpam-3427	733	16	,	,	PUNCT
ejpam-3427	733	17	czechoslovak	czechoslovak	ADJ
ejpam-3427	733	18	math	math	NOUN
ejpam-3427	733	19	.	.	PUNCT
ejpam-3427	734	1	j.	j.	PROPN
ejpam-3427	734	2	,	,	PUNCT
ejpam-3427	734	3	49(1999	49(1999	PROPN
ejpam-3427	734	4	)	)	PUNCT
ejpam-3427	734	5	277	277	NUM
ejpam-3427	734	6	-	-	SYM
ejpam-3427	734	7	290	290	NUM
ejpam-3427	734	8	.	.	PUNCT
ejpam-3427	735	1	[	[	X
ejpam-3427	735	2	6	6	NUM
ejpam-3427	735	3	]	]	PUNCT
ejpam-3427	735	4	k.	k.	PROPN
ejpam-3427	735	5	hur	hur	PROPN
ejpam-3427	735	6	,	,	PUNCT
ejpam-3427	735	7	h.	h.	PROPN
ejpam-3427	735	8	w.	w.	PROPN
ejpam-3427	735	9	kang	kang	PROPN
ejpam-3427	735	10	and	and	CCONJ
ejpam-3427	735	11	h.	h.	PROPN
ejpam-3427	735	12	k.	k.	PROPN
ejpam-3427	735	13	song	song	PROPN
ejpam-3427	735	14	,	,	PUNCT
ejpam-3427	735	15	intuitionistic	intuitionistic	ADJ
ejpam-3427	735	16	fuzzy	fuzzy	ADJ
ejpam-3427	735	17	subgroups	subgroup	NOUN
ejpam-3427	735	18	and	and	CCONJ
ejpam-3427	735	19	subrings	subring	NOUN
ejpam-3427	735	20	,	,	PUNCT
ejpam-3427	735	21	honam	honam	PROPN
ejpam-3427	735	22	math	math	PROPN
ejpam-3427	735	23	.	.	PUNCT
ejpam-3427	736	1	j.	j.	PROPN
ejpam-3427	736	2	,	,	PUNCT
ejpam-3427	736	3	25(2003	25(2003	NUM
ejpam-3427	736	4	)	)	PUNCT
ejpam-3427	736	5	19	19	NUM
ejpam-3427	736	6	-	-	SYM
ejpam-3427	736	7	41	41	NUM
ejpam-3427	736	8	.	.	PUNCT
ejpam-3427	737	1	[	[	X
ejpam-3427	737	2	7	7	X
ejpam-3427	737	3	]	]	X
ejpam-3427	737	4	j.	j.	PROPN
ejpam-3427	737	5	jezek	jezek	PROPN
ejpam-3427	737	6	and	and	CCONJ
ejpam-3427	737	7	t.	t.	PROPN
ejpam-3427	737	8	kepka	kepka	NOUN
ejpam-3427	737	9	,	,	PUNCT
ejpam-3427	737	10	medial	medial	ADJ
ejpam-3427	737	11	groupoids	groupoid	NOUN
ejpam-3427	737	12	,	,	PUNCT
ejpam-3427	737	13	rozpravy	rozpravy	PROPN
ejpam-3427	737	14	csav	csav	PROPN
ejpam-3427	737	15	rada	rada	PROPN
ejpam-3427	737	16	mat	mat	PROPN
ejpam-3427	737	17	.	.	PUNCT
ejpam-3427	738	1	a	a	DET
ejpam-3427	738	2	prir	prir	NOUN
ejpam-3427	738	3	.	.	PUNCT
ejpam-3427	739	1	ved	ve	VERB
ejpam-3427	739	2	.	.	PROPN
ejpam-3427	739	3	,	,	PUNCT
ejpam-3427	739	4	93/2	93/2	NUM
ejpam-3427	739	5	,	,	PUNCT
ejpam-3427	739	6	1983	1983	NUM
ejpam-3427	739	7	,	,	PUNCT
ejpam-3427	739	8	93	93	NUM
ejpam-3427	739	9	pp	pp	NOUN
ejpam-3427	739	10	.	.	PUNCT
ejpam-3427	740	1	[	[	X
ejpam-3427	740	2	8	8	NUM
ejpam-3427	740	3	]	]	PUNCT
ejpam-3427	740	4	m.	m.	NOUN
ejpam-3427	740	5	s.	s.	PROPN
ejpam-3427	740	6	kamran	kamran	PROPN
ejpam-3427	740	7	,	,	PUNCT
ejpam-3427	740	8	conditions	condition	NOUN
ejpam-3427	740	9	for	for	ADP
ejpam-3427	740	10	la	la	NOUN
ejpam-3427	740	11	-	-	PUNCT
ejpam-3427	740	12	semigroups	semigroup	NOUN
ejpam-3427	740	13	to	to	PART
ejpam-3427	740	14	resemble	resemble	VERB
ejpam-3427	740	15	associative	associative	ADJ
ejpam-3427	740	16	structures	structure	NOUN
ejpam-3427	740	17	,	,	PUNCT
ejpam-3427	740	18	ph.d	ph.d	PROPN
ejpam-3427	740	19	.	.	PUNCT
ejpam-3427	741	1	thesis	thesis	NOUN
ejpam-3427	741	2	,	,	PUNCT
ejpam-3427	741	3	quaid	quaid	PROPN
ejpam-3427	741	4	-	-	PUNCT
ejpam-3427	741	5	i	i	PROPN
ejpam-3427	741	6	-	-	PUNCT
ejpam-3427	741	7	azam	azam	PROPN
ejpam-3427	741	8	university	university	PROPN
ejpam-3427	741	9	,	,	PUNCT
ejpam-3427	741	10	islamabad	islamabad	PROPN
ejpam-3427	741	11	1993	1993	NUM
ejpam-3427	741	12	.	.	PUNCT
ejpam-3427	742	1	[	[	X
ejpam-3427	742	2	9	9	NUM
ejpam-3427	742	3	]	]	PUNCT
ejpam-3427	742	4	m.	m.	NOUN
ejpam-3427	742	5	a.	a.	PROPN
ejpam-3427	742	6	kazim	kazim	PROPN
ejpam-3427	742	7	and	and	CCONJ
ejpam-3427	742	8	m.	m.	PROPN
ejpam-3427	742	9	naseerudin	naseerudin	PROPN
ejpam-3427	742	10	,	,	PUNCT
ejpam-3427	742	11	on	on	ADP
ejpam-3427	742	12	almost	almost	ADV
ejpam-3427	742	13	semigroups	semigroup	NOUN
ejpam-3427	742	14	,	,	PUNCT
ejpam-3427	742	15	alig	alig	PROPN
ejpam-3427	742	16	.	.	PUNCT
ejpam-3427	743	1	bull	bull	PROPN
ejpam-3427	743	2	.	.	PUNCT
ejpam-3427	744	1	math	math	NOUN
ejpam-3427	744	2	.	.	PUNCT
ejpam-3427	744	3	,	,	PUNCT
ejpam-3427	744	4	2(1972	2(1972	X
ejpam-3427	744	5	)	)	PUNCT
ejpam-3427	744	6	1	1	NUM
ejpam-3427	744	7	-	-	SYM
ejpam-3427	744	8	7	7	NUM
ejpam-3427	744	9	.	.	PUNCT
ejpam-3427	745	1	[	[	X
ejpam-3427	745	2	10	10	NUM
ejpam-3427	745	3	]	]	PUNCT
ejpam-3427	745	4	m.	m.	PROPN
ejpam-3427	745	5	f.	f.	PROPN
ejpam-3427	745	6	marashdeh	marashdeh	PROPN
ejpam-3427	745	7	and	and	CCONJ
ejpam-3427	745	8	a.	a.	PROPN
ejpam-3427	745	9	r.	r.	PROPN
ejpam-3427	745	10	salleh	salleh	PROPN
ejpam-3427	745	11	,	,	PUNCT
ejpam-3427	745	12	intuitionistic	intuitionistic	ADJ
ejpam-3427	745	13	fuzzy	fuzzy	ADJ
ejpam-3427	745	14	rings	ring	NOUN
ejpam-3427	745	15	,	,	PUNCT
ejpam-3427	745	16	int	int	NOUN
ejpam-3427	745	17	.	.	PUNCT
ejpam-3427	746	1	j.	j.	PROPN
ejpam-3427	746	2	algebra	algebra	PROPN
ejpam-3427	746	3	,	,	PUNCT
ejpam-3427	746	4	5(2011	5(2011	NUM
ejpam-3427	746	5	)	)	PUNCT
ejpam-3427	746	6	37	37	NUM
ejpam-3427	746	7	-	-	SYM
ejpam-3427	746	8	47	47	NUM
ejpam-3427	746	9	.	.	PUNCT
ejpam-3427	747	1	[	[	X
ejpam-3427	747	2	11	11	NUM
ejpam-3427	747	3	]	]	PUNCT
ejpam-3427	747	4	m.	m.	NOUN
ejpam-3427	747	5	t.	t.	PROPN
ejpam-3427	747	6	a.	a.	PROPN
ejpam-3427	747	7	osman	osman	PROPN
ejpam-3427	747	8	,	,	PUNCT
ejpam-3427	747	9	on	on	ADP
ejpam-3427	747	10	the	the	DET
ejpam-3427	747	11	direct	direct	ADJ
ejpam-3427	747	12	product	product	NOUN
ejpam-3427	747	13	of	of	ADP
ejpam-3427	747	14	fuzzy	fuzzy	ADJ
ejpam-3427	747	15	subgroups	subgroup	NOUN
ejpam-3427	747	16	,	,	PUNCT
ejpam-3427	747	17	fuzzy	fuzzy	ADJ
ejpam-3427	747	18	sets	set	NOUN
ejpam-3427	747	19	and	and	CCONJ
ejpam-3427	747	20	systems	system	NOUN
ejpam-3427	747	21	,	,	PUNCT
ejpam-3427	747	22	12(1984	12(1984	NUM
ejpam-3427	747	23	)	)	PUNCT
ejpam-3427	747	24	87	87	NUM
ejpam-3427	747	25	-	-	SYM
ejpam-3427	747	26	91	91	NUM
ejpam-3427	747	27	.	.	PUNCT
ejpam-3427	748	1	[	[	X
ejpam-3427	748	2	12	12	NUM
ejpam-3427	748	3	]	]	PUNCT
ejpam-3427	748	4	m.	m.	NOUN
ejpam-3427	748	5	t.	t.	PROPN
ejpam-3427	748	6	a.	a.	PROPN
ejpam-3427	748	7	osman	osman	PROPN
ejpam-3427	748	8	,	,	PUNCT
ejpam-3427	748	9	on	on	ADP
ejpam-3427	748	10	some	some	DET
ejpam-3427	748	11	product	product	NOUN
ejpam-3427	748	12	of	of	ADP
ejpam-3427	748	13	fuzzy	fuzzy	ADJ
ejpam-3427	748	14	subgroups	subgroup	NOUN
ejpam-3427	748	15	,	,	PUNCT
ejpam-3427	748	16	fuzzy	fuzzy	ADJ
ejpam-3427	748	17	sets	set	NOUN
ejpam-3427	748	18	and	and	CCONJ
ejpam-3427	748	19	systems	system	NOUN
ejpam-3427	748	20	,	,	PUNCT
ejpam-3427	748	21	24(1987	24(1987	NUM
ejpam-3427	748	22	)	)	PUNCT
ejpam-3427	748	23	79	79	NUM
ejpam-3427	748	24	-	-	SYM
ejpam-3427	748	25	86	86	NUM
ejpam-3427	748	26	.	.	PUNCT
ejpam-3427	749	1	references	reference	NOUN
ejpam-3427	749	2	648	648	NUM
ejpam-3427	749	3	[	[	SYM
ejpam-3427	749	4	13	13	NUM
ejpam-3427	749	5	]	]	X
ejpam-3427	749	6	n.	n.	PROPN
ejpam-3427	749	7	palaniappan	palaniappan	PROPN
ejpam-3427	749	8	,	,	PUNCT
ejpam-3427	749	9	k.	k.	PROPN
ejpam-3427	749	10	arjunan	arjunan	PROPN
ejpam-3427	749	11	and	and	CCONJ
ejpam-3427	749	12	v.	v.	ADP
ejpam-3427	749	13	veeramani	veeramani	PROPN
ejpam-3427	749	14	,	,	PUNCT
ejpam-3427	749	15	the	the	DET
ejpam-3427	749	16	homomorphism	homomorphism	NOUN
ejpam-3427	749	17	,	,	PUNCT
ejpam-3427	749	18	antihomomorphism	antihomomorphism	NOUN
ejpam-3427	749	19	of	of	ADP
ejpam-3427	749	20	intuitionistic	intuitionistic	ADJ
ejpam-3427	749	21	fuzzy	fuzzy	ADJ
ejpam-3427	749	22	normal	normal	ADJ
ejpam-3427	749	23	subrings	subring	NOUN
ejpam-3427	749	24	,	,	PUNCT
ejpam-3427	749	25	acta	acta	PROPN
ejpam-3427	749	26	ciencia	ciencia	PROPN
ejpam-3427	749	27	indica	indica	PROPN
ejpam-3427	749	28	math	math	PROPN
ejpam-3427	749	29	.	.	PUNCT
ejpam-3427	749	30	,	,	PUNCT
ejpam-3427	749	31	xxxiii(2007	xxxiii(2007	PROPN
ejpam-3427	749	32	)	)	PUNCT
ejpam-3427	749	33	219	219	NUM
ejpam-3427	749	34	-	-	SYM
ejpam-3427	749	35	224	224	NUM
ejpam-3427	749	36	.	.	PUNCT
ejpam-3427	750	1	[	[	X
ejpam-3427	750	2	14	14	NUM
ejpam-3427	750	3	]	]	X
ejpam-3427	750	4	n.	n.	PROPN
ejpam-3427	750	5	palaniappan	palaniappan	PROPN
ejpam-3427	750	6	,	,	PUNCT
ejpam-3427	750	7	k.	k.	PROPN
ejpam-3427	750	8	arjunan	arjunan	PROPN
ejpam-3427	750	9	and	and	CCONJ
ejpam-3427	750	10	v.	v.	ADP
ejpam-3427	750	11	veeramani	veeramani	PROPN
ejpam-3427	750	12	,	,	PUNCT
ejpam-3427	750	13	some	some	DET
ejpam-3427	750	14	properties	property	NOUN
ejpam-3427	750	15	of	of	ADP
ejpam-3427	750	16	intuitionistic	intuitionistic	ADJ
ejpam-3427	750	17	fuzzy	fuzzy	ADJ
ejpam-3427	750	18	normal	normal	ADJ
ejpam-3427	750	19	subrings	subring	NOUN
ejpam-3427	750	20	,	,	PUNCT
ejpam-3427	750	21	applied	apply	VERB
ejpam-3427	750	22	math	math	NOUN
ejpam-3427	750	23	.	.	PUNCT
ejpam-3427	751	1	sci	sci	PROPN
ejpam-3427	751	2	.	.	PROPN
ejpam-3427	751	3	,	,	PUNCT
ejpam-3427	751	4	4(2010	4(2010	NOUN
ejpam-3427	751	5	)	)	PUNCT
ejpam-3427	751	6	2119	2119	NUM
ejpam-3427	751	7	-	-	SYM
ejpam-3427	751	8	2124	2124	NUM
ejpam-3427	751	9	.	.	PUNCT
ejpam-3427	752	1	[	[	X
ejpam-3427	752	2	15	15	NUM
ejpam-3427	752	3	]	]	X
ejpam-3427	752	4	p.	p.	NOUN
ejpam-3427	752	5	v.	v.	ADP
ejpam-3427	752	6	protic	protic	PROPN
ejpam-3427	752	7	and	and	CCONJ
ejpam-3427	752	8	n.	n.	PROPN
ejpam-3427	752	9	stevanovic	stevanovic	PROPN
ejpam-3427	752	10	,	,	PUNCT
ejpam-3427	752	11	ag	ag	NOUN
ejpam-3427	752	12	-	-	PUNCT
ejpam-3427	752	13	test	test	NOUN
ejpam-3427	752	14	and	and	CCONJ
ejpam-3427	752	15	some	some	DET
ejpam-3427	752	16	general	general	ADJ
ejpam-3427	752	17	properties	property	NOUN
ejpam-3427	752	18	of	of	ADP
ejpam-3427	752	19	abelgrassmann	abelgrassmann	PROPN
ejpam-3427	752	20	’s	’s	PART
ejpam-3427	752	21	groupoids	groupoid	NOUN
ejpam-3427	752	22	,	,	PUNCT
ejpam-3427	752	23	pure	pure	ADJ
ejpam-3427	752	24	math	math	NOUN
ejpam-3427	752	25	.	.	PUNCT
ejpam-3427	753	1	appli	appli	PROPN
ejpam-3427	753	2	.	.	PROPN
ejpam-3427	753	3	,	,	PUNCT
ejpam-3427	754	1	6(1995	6(1995	NUM
ejpam-3427	754	2	)	)	PUNCT
ejpam-3427	754	3	371	371	NUM
ejpam-3427	754	4	-	-	SYM
ejpam-3427	754	5	383	383	NUM
ejpam-3427	754	6	.	.	PUNCT
ejpam-3427	755	1	[	[	X
ejpam-3427	755	2	16	16	NUM
ejpam-3427	755	3	]	]	PUNCT
ejpam-3427	755	4	a.	a.	PROPN
ejpam-3427	755	5	k.	k.	PROPN
ejpam-3427	755	6	ray	ray	PROPN
ejpam-3427	755	7	,	,	PUNCT
ejpam-3427	755	8	on	on	ADP
ejpam-3427	755	9	product	product	NOUN
ejpam-3427	755	10	of	of	ADP
ejpam-3427	755	11	fuzzy	fuzzy	ADJ
ejpam-3427	755	12	subgroups	subgroup	NOUN
ejpam-3427	755	13	,	,	PUNCT
ejpam-3427	755	14	fuzzy	fuzzy	ADJ
ejpam-3427	755	15	sets	set	NOUN
ejpam-3427	755	16	and	and	CCONJ
ejpam-3427	755	17	systems	system	NOUN
ejpam-3427	755	18	,	,	PUNCT
ejpam-3427	755	19	11(1983	11(1983	NUM
ejpam-3427	755	20	)	)	PUNCT
ejpam-3427	755	21	79	79	NUM
ejpam-3427	755	22	-	-	SYM
ejpam-3427	755	23	89	89	NUM
ejpam-3427	755	24	.	.	PUNCT
ejpam-3427	756	1	[	[	X
ejpam-3427	756	2	17	17	NUM
ejpam-3427	756	3	]	]	PUNCT
ejpam-3427	756	4	t.	t.	NOUN
ejpam-3427	756	5	shah	shah	NOUN
ejpam-3427	756	6	,	,	PUNCT
ejpam-3427	756	7	n.	n.	PROPN
ejpam-3427	756	8	kausar	kausar	PROPN
ejpam-3427	756	9	and	and	CCONJ
ejpam-3427	756	10	i.	i.	PROPN
ejpam-3427	756	11	rehman	rehman	PROPN
ejpam-3427	756	12	,	,	PUNCT
ejpam-3427	756	13	intuitionistic	intuitionistic	ADJ
ejpam-3427	756	14	fuzzy	fuzzy	ADJ
ejpam-3427	756	15	normal	normal	ADJ
ejpam-3427	756	16	subrings	subring	NOUN
ejpam-3427	756	17	over	over	ADP
ejpam-3427	756	18	a	a	DET
ejpam-3427	756	19	nonassociative	nonassociative	ADJ
ejpam-3427	756	20	ring	ring	NOUN
ejpam-3427	756	21	,	,	PUNCT
ejpam-3427	756	22	an	an	PROPN
ejpam-3427	756	23	.	.	PUNCT
ejpam-3427	756	24	st	st	PROPN
ejpam-3427	756	25	.	.	PROPN
ejpam-3427	756	26	univ	univ	PROPN
ejpam-3427	756	27	.	.	PUNCT
ejpam-3427	757	1	ovidius	ovidius	PROPN
ejpam-3427	757	2	constanta	constanta	PROPN
ejpam-3427	757	3	,	,	PUNCT
ejpam-3427	757	4	1(2012	1(2012	NUM
ejpam-3427	757	5	)	)	PUNCT
ejpam-3427	757	6	369	369	NUM
ejpam-3427	757	7	-	-	SYM
ejpam-3427	757	8	386	386	NUM
ejpam-3427	757	9	.	.	PUNCT
ejpam-3427	758	1	[	[	X
ejpam-3427	758	2	18	18	NUM
ejpam-3427	758	3	]	]	PUNCT
ejpam-3427	758	4	t.	t.	NOUN
ejpam-3427	758	5	shah	shah	PROPN
ejpam-3427	758	6	and	and	CCONJ
ejpam-3427	758	7	i.	i.	PROPN
ejpam-3427	758	8	rehman	rehman	PROPN
ejpam-3427	758	9	,	,	PUNCT
ejpam-3427	758	10	on	on	ADP
ejpam-3427	758	11	la	la	NOUN
ejpam-3427	758	12	-	-	PUNCT
ejpam-3427	758	13	rings	ring	NOUN
ejpam-3427	758	14	of	of	ADP
ejpam-3427	758	15	finitely	finitely	ADJ
ejpam-3427	758	16	non	non	ADJ
ejpam-3427	758	17	-	-	ADJ
ejpam-3427	758	18	zero	zero	NUM
ejpam-3427	758	19	functions	function	NOUN
ejpam-3427	758	20	,	,	PUNCT
ejpam-3427	758	21	int	int	NOUN
ejpam-3427	758	22	.	.	PUNCT
ejpam-3427	759	1	j.	j.	PROPN
ejpam-3427	759	2	contemp	contemp	PROPN
ejpam-3427	759	3	.	.	PUNCT
ejpam-3427	760	1	math	math	NOUN
ejpam-3427	760	2	.	.	PUNCT
ejpam-3427	761	1	sci	sci	PROPN
ejpam-3427	761	2	.	.	PROPN
ejpam-3427	761	3	,	,	PUNCT
ejpam-3427	761	4	5(2010	5(2010	NUM
ejpam-3427	761	5	)	)	PUNCT
ejpam-3427	761	6	209	209	NUM
ejpam-3427	761	7	-	-	SYM
ejpam-3427	761	8	222	222	NUM
ejpam-3427	761	9	.	.	PUNCT
ejpam-3427	762	1	[	[	X
ejpam-3427	762	2	19	19	NUM
ejpam-3427	762	3	]	]	X
ejpam-3427	762	4	kausar	kausar	PROPN
ejpam-3427	762	5	,	,	PUNCT
ejpam-3427	762	6	waqar	waqar	PROPN
ejpam-3427	762	7	,	,	PUNCT
ejpam-3427	762	8	characterizations	characterization	NOUN
ejpam-3427	762	9	of	of	ADP
ejpam-3427	762	10	non	non	ADJ
ejpam-3427	762	11	-	-	ADJ
ejpam-3427	762	12	associative	associative	ADJ
ejpam-3427	762	13	rings	ring	NOUN
ejpam-3427	762	14	by	by	ADP
ejpam-3427	762	15	their	their	PRON
ejpam-3427	762	16	intuitionistic	intuitionistic	ADJ
ejpam-3427	762	17	fuzzy	fuzzy	ADJ
ejpam-3427	762	18	bi	bi	NOUN
ejpam-3427	762	19	-	-	NOUN
ejpam-3427	762	20	ideals	ideal	NOUN
ejpam-3427	762	21	,	,	PUNCT
ejpam-3427	762	22	european	european	ADJ
ejpam-3427	762	23	journal	journal	PROPN
ejpam-3427	762	24	of	of	ADP
ejpam-3427	762	25	pure	pure	ADJ
ejpam-3427	762	26	and	and	CCONJ
ejpam-3427	762	27	applied	applied	ADJ
ejpam-3427	762	28	mathematics	mathematic	NOUN
ejpam-3427	762	29	,	,	PUNCT
ejpam-3427	762	30	vol	vol	NOUN
ejpam-3427	762	31	.	.	PUNCT
ejpam-3427	762	32	12(2019	12(2019	NUM
ejpam-3427	762	33	)	)	PUNCT
ejpam-3427	762	34	226	226	NUM
ejpam-3427	762	35	-	-	SYM
ejpam-3427	762	36	250	250	NUM
ejpam-3427	762	37	.	.	PUNCT
ejpam-3427	763	1	[	[	X
ejpam-3427	763	2	20	20	NUM
ejpam-3427	763	3	]	]	PUNCT
ejpam-3427	763	4	h.	h.	PROPN
ejpam-3427	763	5	sherwood	sherwood	PROPN
ejpam-3427	763	6	,	,	PUNCT
ejpam-3427	763	7	product	product	NOUN
ejpam-3427	763	8	of	of	ADP
ejpam-3427	763	9	fuzzy	fuzzy	ADJ
ejpam-3427	763	10	subgroups	subgroup	NOUN
ejpam-3427	763	11	,	,	PUNCT
ejpam-3427	763	12	fuzzy	fuzzy	ADJ
ejpam-3427	763	13	sets	set	NOUN
ejpam-3427	763	14	and	and	CCONJ
ejpam-3427	763	15	systems	system	NOUN
ejpam-3427	763	16	,	,	PUNCT
ejpam-3427	763	17	105(1999	105(1999	NUM
ejpam-3427	763	18	)	)	PUNCT
ejpam-3427	763	19	181183	181183	NUM
ejpam-3427	763	20	.	.	PUNCT
ejpam-3427	764	1	[	[	X
ejpam-3427	764	2	21	21	NUM
ejpam-3427	764	3	]	]	X
ejpam-3427	764	4	l.	l.	PROPN
ejpam-3427	764	5	m.	m.	PROPN
ejpam-3427	764	6	yan	yan	PROPN
ejpam-3427	764	7	,	,	PUNCT
ejpam-3427	764	8	intuitionistic	intuitionistic	ADJ
ejpam-3427	764	9	fuzzy	fuzzy	ADJ
ejpam-3427	764	10	ring	ring	NOUN
ejpam-3427	764	11	and	and	CCONJ
ejpam-3427	764	12	its	its	PRON
ejpam-3427	764	13	homomorphism	homomorphism	NOUN
ejpam-3427	764	14	image	image	NOUN
ejpam-3427	764	15	,	,	PUNCT
ejpam-3427	764	16	int	int	NOUN
ejpam-3427	764	17	.	.	PUNCT
ejpam-3427	765	1	seminar	seminar	NOUN
ejpam-3427	765	2	on	on	ADP
ejpam-3427	765	3	future	future	ADJ
ejpam-3427	765	4	biomedical	biomedical	ADJ
ejpam-3427	765	5	inform	inform	NOUN
ejpam-3427	765	6	.	.	PUNCT
ejpam-3427	766	1	eng	eng	PROPN
ejpam-3427	766	2	.	.	PROPN
ejpam-3427	766	3	,	,	PUNCT
ejpam-3427	766	4	(	(	PUNCT
ejpam-3427	766	5	2008	2008	NUM
ejpam-3427	766	6	)	)	PUNCT
ejpam-3427	766	7	75	75	NUM
ejpam-3427	766	8	-	-	SYM
ejpam-3427	766	9	77	77	NUM
ejpam-3427	766	10	.	.	PUNCT
ejpam-3427	767	1	[	[	X
ejpam-3427	767	2	22	22	NUM
ejpam-3427	767	3	]	]	X
ejpam-3427	767	4	l.	l.	PROPN
ejpam-3427	767	5	a.	a.	PROPN
ejpam-3427	767	6	zadeh	zadeh	PROPN
ejpam-3427	767	7	,	,	PUNCT
ejpam-3427	767	8	fuzzy	fuzzy	ADJ
ejpam-3427	767	9	sets	set	NOUN
ejpam-3427	767	10	,	,	PUNCT
ejpam-3427	767	11	information	information	NOUN
ejpam-3427	767	12	and	and	CCONJ
ejpam-3427	767	13	control	control	NOUN
ejpam-3427	767	14	,	,	PUNCT
ejpam-3427	767	15	8(1965	8(1965	NUM
ejpam-3427	767	16	)	)	PUNCT
ejpam-3427	767	17	338	338	NUM
ejpam-3427	767	18	-	-	SYM
ejpam-3427	767	19	353	353	NUM
ejpam-3427	767	20	.	.	PUNCT
ejpam-3427	768	1	[	[	X
ejpam-3427	768	2	23	23	NUM
ejpam-3427	768	3	]	]	PUNCT
ejpam-3427	768	4	s.	s.	PROPN
ejpam-3427	768	5	a.	a.	PROPN
ejpam-3427	768	6	zaid	zaid	PROPN
ejpam-3427	768	7	,	,	PUNCT
ejpam-3427	768	8	on	on	ADP
ejpam-3427	768	9	normal	normal	ADJ
ejpam-3427	768	10	fuzzy	fuzzy	ADJ
ejpam-3427	768	11	subgroups	subgroup	NOUN
ejpam-3427	768	12	,	,	PUNCT
ejpam-3427	768	13	j.	j.	PROPN
ejpam-3427	768	14	fac	fac	PROPN
ejpam-3427	768	15	.	.	PROPN
ejpam-3427	768	16	educ	educ	PROPN
ejpam-3427	768	17	.	.	PUNCT
ejpam-3427	769	1	ain	ain	PROPN
ejpam-3427	769	2	shams	sham	NOUN
ejpam-3427	769	3	univ	univ	PROPN
ejpam-3427	769	4	,	,	PUNCT
ejpam-3427	769	5	cairo	cairo	PROPN
ejpam-3427	769	6	,	,	PUNCT
ejpam-3427	769	7	13(1988	13(1988	NUM
ejpam-3427	769	8	)	)	PUNCT
ejpam-3427	769	9	115	115	NUM
ejpam-3427	769	10	-	-	SYM
ejpam-3427	769	11	125	125	NUM
ejpam-3427	769	12	.	.	PUNCT