id	sid	tid	token	lemma	pos
cana-3296	1	1	communications	communication	NOUN
cana-3296	1	2	on	on	ADP
cana-3296	1	3	applied	apply	VERB
cana-3296	1	4	nonlinear	nonlinear	ADJ
cana-3296	1	5	analysis	analysis	NOUN
cana-3296	1	6	issn	issn	NOUN
cana-3296	1	7	:	:	PUNCT
cana-3296	1	8	1074	1074	NUM
cana-3296	1	9	-	-	PUNCT
cana-3296	1	10	133x	133x	NUM
cana-3296	1	11	vol	vol	NOUN
cana-3296	1	12	32	32	NUM
cana-3296	1	13	no	no	NOUN
cana-3296	1	14	.	.	PUNCT
cana-3296	2	1	6s	6s	NUM
cana-3296	2	2	(	(	PUNCT
cana-3296	2	3	2025	2025	NUM
cana-3296	2	4	)	)	PUNCT
cana-3296	2	5	295	295	NUM
cana-3296	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3296	3	2	new	new	ADJ
cana-3296	3	3	kinds	kind	NOUN
cana-3296	3	4	of	of	ADP
cana-3296	3	5	open	open	ADJ
cana-3296	3	6	sets	set	NOUN
cana-3296	3	7	in	in	ADP
cana-3296	3	8	intuitionistic	intuitionistic	ADJ
cana-3296	3	9	interval	interval	NOUN
cana-3296	3	10	-	-	PUNCT
cana-3296	3	11	valued	value	VERB
cana-3296	3	12	fuzzy	fuzzy	ADJ
cana-3296	3	13	topological	topological	ADJ
cana-3296	3	14	space	space	NOUN
cana-3296	3	15	s.sivaraja1	s.sivaraja1	PROPN
cana-3296	3	16	,	,	PUNCT
cana-3296	3	17	b.sudha2	b.sudha2	PROPN
cana-3296	3	18	,	,	PUNCT
cana-3296	3	19	k.	k.	PROPN
cana-3296	3	20	srinivasan3	srinivasan3	PROPN
cana-3296	3	21	,	,	PUNCT
cana-3296	3	22	k.bhuvaneswari4	k.bhuvaneswari4	PROPN
cana-3296	3	23	1assistant	1assistant	NUM
cana-3296	3	24	professor	professor	NOUN
cana-3296	3	25	,	,	PUNCT
cana-3296	3	26	department	department	NOUN
cana-3296	3	27	of	of	ADP
cana-3296	3	28	mathematics	mathematics	PROPN
cana-3296	3	29	,	,	PUNCT
cana-3296	3	30	k.ramakrishnan	k.ramakrishnan	ADP
cana-3296	3	31	college	college	NOUN
cana-3296	3	32	of	of	ADP
cana-3296	3	33	engineering(autonomous),trichy	engineering(autonomous),trichy	PROPN
cana-3296	3	34	.	.	PUNCT
cana-3296	4	1	2assistant	2assistant	NUM
cana-3296	4	2	professor	professor	NOUN
cana-3296	4	3	,	,	PUNCT
cana-3296	4	4	department	department	NOUN
cana-3296	4	5	of	of	ADP
cana-3296	4	6	mathematics	mathematics	PROPN
cana-3296	4	7	,	,	PUNCT
cana-3296	4	8	college	college	NOUN
cana-3296	4	9	of	of	ADP
cana-3296	4	10	engineering	engineering	NOUN
cana-3296	4	11	and	and	CCONJ
cana-3296	4	12	technology	technology	PROPN
cana-3296	4	13	srm	srm	PROPN
cana-3296	4	14	institute	institute	PROPN
cana-3296	4	15	of	of	ADP
cana-3296	4	16	science	science	NOUN
cana-3296	4	17	and	and	CCONJ
cana-3296	4	18	technology	technology	NOUN
cana-3296	4	19	,	,	PUNCT
cana-3296	4	20	kattankulathur	kattankulathur	PROPN
cana-3296	4	21	.	.	PUNCT
cana-3296	5	1	3associate	3associate	NUM
cana-3296	5	2	professor	professor	NOUN
cana-3296	5	3	,	,	PUNCT
cana-3296	5	4	department	department	NOUN
cana-3296	5	5	of	of	ADP
cana-3296	5	6	mathematics	mathematic	NOUN
cana-3296	5	7	,	,	PUNCT
cana-3296	5	8	r.m.k	r.m.k	VERB
cana-3296	5	9	engineering	engineering	PROPN
cana-3296	5	10	college	college	PROPN
cana-3296	5	11	,	,	PUNCT
cana-3296	5	12	kavaraipettai	kavaraipettai	NOUN
cana-3296	5	13	.	.	PUNCT
cana-3296	6	1	4assistant	4assistant	NUM
cana-3296	6	2	professor	professor	NOUN
cana-3296	6	3	,	,	PUNCT
cana-3296	6	4	department	department	NOUN
cana-3296	6	5	of	of	ADP
cana-3296	6	6	mathematics	mathematic	NOUN
cana-3296	6	7	,	,	PUNCT
cana-3296	6	8	sathyabama	sathyabama	PROPN
cana-3296	6	9	institute	institute	PROPN
cana-3296	6	10	of	of	ADP
cana-3296	6	11	science	science	NOUN
cana-3296	6	12	and	and	CCONJ
cana-3296	6	13	technology	technology	NOUN
cana-3296	6	14	,	,	PUNCT
cana-3296	6	15	chennai	chennai	NOUN
cana-3296	6	16	article	article	NOUN
cana-3296	6	17	history	history	NOUN
cana-3296	6	18	:	:	PUNCT
cana-3296	6	19	received	receive	VERB
cana-3296	6	20	:	:	PUNCT
cana-3296	6	21	19	19	NUM
cana-3296	6	22	-	-	SYM
cana-3296	6	23	10	10	NUM
cana-3296	6	24	-	-	PUNCT
cana-3296	6	25	2024	2024	NUM
cana-3296	6	26	revised	revise	VERB
cana-3296	6	27	:	:	PUNCT
cana-3296	6	28	03	03	NUM
cana-3296	6	29	-	-	SYM
cana-3296	6	30	12	12	NUM
cana-3296	6	31	-	-	PUNCT
cana-3296	6	32	2024	2024	NUM
cana-3296	6	33	accepted	accept	VERB
cana-3296	6	34	:	:	PUNCT
cana-3296	6	35	11	11	NUM
cana-3296	6	36	-	-	SYM
cana-3296	6	37	12	12	NUM
cana-3296	6	38	-	-	PUNCT
cana-3296	6	39	2024	2024	NUM
cana-3296	6	40	abstract	abstract	NOUN
cana-3296	6	41	:	:	PUNCT
cana-3296	6	42	in	in	ADP
cana-3296	6	43	an	an	DET
cana-3296	6	44	intuitionistic	intuitionistic	ADJ
cana-3296	6	45	interval	interval	NOUN
cana-3296	6	46	valued	value	VERB
cana-3296	6	47	fuzzy	fuzzy	ADJ
cana-3296	6	48	topological	topological	ADJ
cana-3296	6	49	space	space	NOUN
cana-3296	6	50	(	(	PUNCT
cana-3296	6	51	inshort	inshort	NOUN
cana-3296	6	52	iivfts	iivfts	PROPN
cana-3296	6	53	)	)	PUNCT
cana-3296	6	54	a	a	DET
cana-3296	6	55	new	new	ADJ
cana-3296	6	56	sort	sort	NOUN
cana-3296	6	57	of	of	ADP
cana-3296	6	58	open	open	ADJ
cana-3296	6	59	sets	set	NOUN
cana-3296	6	60	termed	term	VERB
cana-3296	6	61	intuitionistic	intuitionistic	ADJ
cana-3296	6	62	interval	interval	NOUN
cana-3296	6	63	valued	value	VERB
cana-3296	6	64	fuzzy	fuzzy	ADJ
cana-3296	6	65	minimum	minimum	ADJ
cana-3296	6	66	open(resp.maximal	open(resp.maximal	ADJ
cana-3296	6	67	open	open	NOUN
cana-3296	6	68	)	)	PUNCT
cana-3296	6	69	have	have	AUX
cana-3296	6	70	been	be	AUX
cana-3296	6	71	researched	research	VERB
cana-3296	6	72	.	.	PUNCT
cana-3296	7	1	keywords	keyword	NOUN
cana-3296	7	2	:	:	PUNCT
cana-3296	7	3	intuitionistic	intuitionistic	ADJ
cana-3296	7	4	interval	interval	NOUN
cana-3296	7	5	valued	value	VERB
cana-3296	7	6	fuzzy	fuzzy	ADJ
cana-3296	7	7	minimal	minimal	ADJ
cana-3296	7	8	open	open	ADJ
cana-3296	7	9	(	(	PUNCT
cana-3296	7	10	resp.minimal	resp.minimal	ADJ
cana-3296	7	11	open	open	NOUN
cana-3296	7	12	)	)	PUNCT
cana-3296	7	13	,	,	PUNCT
cana-3296	7	14	intuitionistic	intuitionistic	ADJ
cana-3296	7	15	interval	interval	NOUN
cana-3296	7	16	valued	value	VERB
cana-3296	7	17	fuzzy	fuzzy	ADJ
cana-3296	7	18	maximal	maximal	ADJ
cana-3296	7	19	closed	close	VERB
cana-3296	7	20	(	(	PUNCT
cana-3296	7	21	resp.minimal	resp.minimal	NOUN
cana-3296	7	22	closed	closed	ADJ
cana-3296	7	23	)	)	PUNCT
cana-3296	7	24	in	in	ADP
cana-3296	7	25	short	short	ADJ
cana-3296	7	26	iivfmio	iivfmio	NOUN
cana-3296	7	27	,	,	PUNCT
cana-3296	7	28	iivfmao	iivfmao	PROPN
cana-3296	7	29	,	,	PUNCT
cana-3296	7	30	iivfmac	iivfmac	NOUN
cana-3296	7	31	,	,	PUNCT
cana-3296	7	32	iivfmic	iivfmic	ADJ
cana-3296	7	33	.	.	PUNCT
cana-3296	8	1	1	1	X
cana-3296	8	2	.	.	X
cana-3296	8	3	introduction	introduction	NOUN
cana-3296	8	4	c.l	c.l	PROPN
cana-3296	8	5	.	.	PUNCT
cana-3296	9	1	chang[2	chang[2	PROPN
cana-3296	9	2	]	]	PUNCT
cana-3296	9	3	used	use	VERB
cana-3296	9	4	the	the	DET
cana-3296	9	5	fuzzy	fuzzy	ADJ
cana-3296	9	6	set	set	NOUN
cana-3296	9	7	,	,	PUNCT
cana-3296	9	8	which	which	PRON
cana-3296	9	9	zadeh[1	zadeh[1	AUX
cana-3296	9	10	]	]	PUNCT
cana-3296	9	11	had	have	AUX
cana-3296	9	12	established	establish	VERB
cana-3296	9	13	in	in	ADP
cana-3296	9	14	1965	1965	NUM
cana-3296	9	15	,	,	PUNCT
cana-3296	9	16	to	to	ADP
cana-3296	9	17	topology	topology	NOUN
cana-3296	9	18	in	in	ADP
cana-3296	9	19	1968	1968	NUM
cana-3296	9	20	.	.	PUNCT
cana-3296	10	1	once	once	ADV
cana-3296	10	2	more	more	ADV
cana-3296	10	3	,	,	PUNCT
cana-3296	10	4	mondal	mondal	NOUN
cana-3296	10	5	and	and	CCONJ
cana-3296	10	6	samanta	samanta	PROPN
cana-3296	10	7	[	[	X
cana-3296	10	8	4	4	X
cana-3296	10	9	]	]	PUNCT
cana-3296	10	10	used	use	VERB
cana-3296	10	11	the	the	DET
cana-3296	10	12	interval	interval	NOUN
cana-3296	10	13	-	-	PUNCT
cana-3296	10	14	valued	value	VERB
cana-3296	10	15	fuzzy	fuzzy	ADJ
cana-3296	10	16	set	set	NOUN
cana-3296	10	17	notion	notion	NOUN
cana-3296	10	18	used	use	VERB
cana-3296	10	19	by	by	ADP
cana-3296	10	20	zadeh	zadeh	PROPN
cana-3296	11	1	[	[	X
cana-3296	11	2	3	3	X
cana-3296	11	3	]	]	PUNCT
cana-3296	11	4	in	in	ADP
cana-3296	11	5	1975	1975	NUM
cana-3296	11	6	to	to	ADP
cana-3296	11	7	topology	topology	NOUN
cana-3296	11	8	in	in	ADP
cana-3296	11	9	1999	1999	NUM
cana-3296	11	10	.	.	PUNCT
cana-3296	12	1	additionally	additionally	ADV
cana-3296	12	2	,	,	PUNCT
cana-3296	12	3	in	in	ADP
cana-3296	12	4	1997	1997	NUM
cana-3296	12	5	,	,	PUNCT
cana-3296	12	6	coker[6	coker[6	PRON
cana-3296	12	7	]	]	PUNCT
cana-3296	12	8	applied	apply	VERB
cana-3296	12	9	the	the	DET
cana-3296	12	10	intuitionistic	intuitionistic	ADJ
cana-3296	12	11	fuzzy	fuzzy	ADJ
cana-3296	12	12	sets	set	NOUN
cana-3296	12	13	,	,	PUNCT
cana-3296	12	14	which	which	PRON
cana-3296	12	15	attanassov[5	attanassov[5	ADV
cana-3296	12	16	]	]	PUNCT
cana-3296	12	17	had	have	AUX
cana-3296	12	18	introduced	introduce	VERB
cana-3296	12	19	in	in	ADP
cana-3296	12	20	1986	1986	NUM
cana-3296	12	21	,	,	PUNCT
cana-3296	12	22	to	to	ADP
cana-3296	12	23	topology.in	topology.in	PROPN
cana-3296	12	24	2012	2012	NUM
cana-3296	12	25	,	,	PUNCT
cana-3296	12	26	pyung	pyung	PROPN
cana-3296	12	27	ki	ki	PROPN
cana-3296	12	28	lim	lim	PROPN
cana-3296	12	29	et	et	PROPN
cana-3296	12	30	al	al	PROPN
cana-3296	12	31	.	.	PUNCT
cana-3296	13	1	[	[	X
cana-3296	13	2	8	8	NUM
cana-3296	13	3	]	]	PUNCT
cana-3296	13	4	used	use	VERB
cana-3296	13	5	intuitionistic	intuitionistic	ADJ
cana-3296	13	6	interval	interval	NOUN
cana-3296	13	7	-	-	PUNCT
cana-3296	13	8	valued	value	VERB
cana-3296	13	9	fuzzy	fuzzy	ADJ
cana-3296	13	10	sets	set	NOUN
cana-3296	13	11	,	,	PUNCT
cana-3296	13	12	which	which	PRON
cana-3296	13	13	cheong	cheong	PROPN
cana-3296	13	14	and	and	CCONJ
cana-3296	13	15	hur	hur	PROPN
cana-3296	13	16	[	[	X
cana-3296	13	17	7	7	NUM
cana-3296	13	18	]	]	PUNCT
cana-3296	13	19	had	have	AUX
cana-3296	13	20	introduced	introduce	VERB
cana-3296	13	21	in	in	ADP
cana-3296	13	22	2010	2010	NUM
cana-3296	13	23	,	,	PUNCT
cana-3296	13	24	to	to	ADP
cana-3296	13	25	topology	topology	NOUN
cana-3296	13	26	.	.	PUNCT
cana-3296	14	1	in	in	ADP
cana-3296	14	2	fuzzy	fuzzy	ADJ
cana-3296	14	3	topological	topological	ADJ
cana-3296	14	4	space	space	NOUN
cana-3296	14	5	and	and	CCONJ
cana-3296	14	6	hesitant	hesitant	ADJ
cana-3296	14	7	fuzzy	fuzzy	ADJ
cana-3296	14	8	topological	topological	ADJ
cana-3296	14	9	space	space	NOUN
cana-3296	14	10	,	,	PUNCT
cana-3296	14	11	a.	a.	NOUN
cana-3296	14	12	swaminathan	swaminathan	NOUN
cana-3296	15	1	[	[	X
cana-3296	15	2	9,10	9,10	X
cana-3296	15	3	]	]	PUNCT
cana-3296	15	4	and	and	CCONJ
cana-3296	15	5	s.	s.	PROPN
cana-3296	15	6	sivaraja	sivaraja	PROPN
cana-3296	15	7	established	establish	VERB
cana-3296	15	8	and	and	CCONJ
cana-3296	15	9	expanded	expand	VERB
cana-3296	15	10	on	on	ADP
cana-3296	15	11	new	new	ADJ
cana-3296	15	12	types	type	NOUN
cana-3296	15	13	of	of	ADP
cana-3296	15	14	open	open	ADJ
cana-3296	15	15	and	and	CCONJ
cana-3296	15	16	closed	closed	ADJ
cana-3296	15	17	sets	set	NOUN
cana-3296	15	18	.	.	PUNCT
cana-3296	16	1	2	2	X
cana-3296	16	2	.	.	X
cana-3296	16	3	preliminaries	preliminary	NOUN
cana-3296	16	4	:	:	PUNCT
cana-3296	16	5	definition	definition	NOUN
cana-3296	16	6	2.1	2.1	NUM
cana-3296	16	7	a	a	DET
cana-3296	16	8	function	function	NOUN
cana-3296	16	9	ϑ:x	ϑ:x	VERB
cana-3296	16	10	→	→	PUNCT
cana-3296	16	11	λ(i	λ(i	PROPN
cana-3296	16	12	⊕	⊕	PROPN
cana-3296	16	13	i	i	NOUN
cana-3296	16	14	)	)	PUNCT
cana-3296	16	15	is	be	AUX
cana-3296	16	16	an	an	DET
cana-3296	16	17	iivfs	iivfs	NOUN
cana-3296	16	18	for	for	ADP
cana-3296	16	19	any	any	DET
cana-3296	16	20	nonempty	nonempty	ADJ
cana-3296	16	21	x	x	NOUN
cana-3296	16	22	,	,	PUNCT
cana-3296	16	23	denoted	denote	VERB
cana-3296	16	24	by	by	ADP
cana-3296	16	25	ϑ	ϑ	PROPN
cana-3296	16	26	=	=	SYM
cana-3296	17	1	[	[	X
cana-3296	17	2	ϑl	ϑl	INTJ
cana-3296	17	3	,	,	PUNCT
cana-3296	17	4	ϑu	ϑu	X
cana-3296	17	5	]	]	PUNCT
cana-3296	17	6	=	=	PUNCT
cana-3296	18	1	[	[	X
cana-3296	18	2	(	(	PUNCT
cana-3296	18	3	αϑ	αϑ	PROPN
cana-3296	18	4	l	l	NOUN
cana-3296	18	5	,	,	PUNCT
cana-3296	18	6	β	β	X
cana-3296	18	7	ϑ	ϑ	X
cana-3296	18	8	l	l	NOUN
cana-3296	18	9	)	)	PUNCT
cana-3296	18	10	,	,	PUNCT
cana-3296	18	11	(	(	PUNCT
cana-3296	18	12	αϑ	αϑ	PROPN
cana-3296	18	13	u	u	PROPN
cana-3296	18	14	,	,	PUNCT
cana-3296	18	15	β	β	X
cana-3296	18	16	ϑ	ϑ	X
cana-3296	18	17	u	u	NOUN
cana-3296	18	18	)	)	PUNCT
cana-3296	18	19	]	]	PUNCT
cana-3296	18	20	.	.	PUNCT
cana-3296	19	1	the	the	DET
cana-3296	19	2	iivfs	iivfs	NOUN
cana-3296	19	3	whole	whole	ADJ
cana-3296	19	4	(	(	PUNCT
cana-3296	19	5	resp.empty	resp.empty	NOUN
cana-3296	19	6	)	)	PUNCT
cana-3296	19	7	fuzzy	fuzzy	ADJ
cana-3296	19	8	set	set	VERB
cana-3296	19	9	in	in	ADP
cana-3296	19	10	x	x	NOUN
cana-3296	19	11	,	,	PUNCT
cana-3296	19	12	is	be	AUX
cana-3296	19	13	given	give	VERB
cana-3296	19	14	by	by	ADP
cana-3296	19	15	1̃	1̃	NUM
cana-3296	19	16	=	=	PUNCT
cana-3296	20	1	[	[	X
cana-3296	20	2	(	(	PUNCT
cana-3296	20	3	1,0	1,0	NUM
cana-3296	20	4	)	)	PUNCT
cana-3296	20	5	,	,	PUNCT
cana-3296	20	6	(	(	PUNCT
cana-3296	20	7	1,0	1,0	NUM
cana-3296	20	8	)	)	PUNCT
cana-3296	20	9	]	]	PUNCT
cana-3296	20	10	(	(	PUNCT
cana-3296	20	11	resp	resp	NOUN
cana-3296	20	12	0̃	0̃	NOUN
cana-3296	20	13	=	=	SYM
cana-3296	21	1	[	[	X
cana-3296	21	2	(	(	PUNCT
cana-3296	21	3	0,1	0,1	NUM
cana-3296	21	4	)	)	PUNCT
cana-3296	21	5	,	,	PUNCT
cana-3296	21	6	(	(	PUNCT
cana-3296	21	7	0,1	0,1	NOUN
cana-3296	21	8	)	)	PUNCT
cana-3296	21	9	]	]	PUNCT
cana-3296	21	10	for	for	ADP
cana-3296	21	11	x	x	PROPN
cana-3296	21	12	∈	∈	PROPN
cana-3296	21	13	x.	x.	NOUN
cana-3296	21	14	definition	definition	NOUN
cana-3296	21	15	2.2	2.2	NUM
cana-3296	21	16	let	let	VERB
cana-3296	21	17	x	x	SYM
cana-3296	21	18	≠	≠	PROPN
cana-3296	21	19	ϕ	ϕ	NOUN
cana-3296	21	20	,	,	PUNCT
cana-3296	21	21	ζ	ζ	PROPN
cana-3296	21	22	∈	∈	PROPN
cana-3296	21	23	λ(i	λ(i	PROPN
cana-3296	21	24	⊕	⊕	PROPN
cana-3296	21	25	i)x	i)x	PROPN
cana-3296	21	26	then	then	ADV
cana-3296	21	27	ζ	ζ	NOUN
cana-3296	21	28	is	be	AUX
cana-3296	21	29	an	an	DET
cana-3296	21	30	intuitionistic	intuitionistic	ADJ
cana-3296	21	31	interval	interval	NOUN
cana-3296	21	32	valued	value	VERB
cana-3296	21	33	fuzzy	fuzzy	ADJ
cana-3296	21	34	topology	topology	NOUN
cana-3296	21	35	(	(	PUNCT
cana-3296	21	36	inshort	inshort	NOUN
cana-3296	21	37	iivft	iivft	NOUN
cana-3296	21	38	)	)	PUNCT
cana-3296	21	39	on	on	ADP
cana-3296	21	40	x	x	SYM
cana-3296	21	41	if	if	SCONJ
cana-3296	21	42	(	(	PUNCT
cana-3296	21	43	i	i	NOUN
cana-3296	21	44	)	)	PUNCT
cana-3296	21	45	0̃	0̃	PROPN
cana-3296	21	46	,	,	PUNCT
cana-3296	21	47	1̃	1̃	NUM
cana-3296	21	48	∈	∈	PROPN
cana-3296	21	49	ζ	ζ	NOUN
cana-3296	21	50	.	.	PUNCT
cana-3296	21	51	(	(	PUNCT
cana-3296	21	52	ii	ii	NOUN
cana-3296	21	53	)	)	PUNCT
cana-3296	21	54	ϑ1	ϑ1	NOUN
cana-3296	21	55	∩	∩	ADJ
cana-3296	21	56	ϑ2	ϑ2	PROPN
cana-3296	21	57	∈	∈	PROPN
cana-3296	21	58	ζ	ζ	PROPN
cana-3296	21	59	,	,	PUNCT
cana-3296	21	60	∀{ϑ1	∀{ϑ1	PROPN
cana-3296	21	61	,	,	PUNCT
cana-3296	21	62	ϑ2	ϑ2	PROPN
cana-3296	21	63	}	}	PUNCT
cana-3296	21	64	∈	∈	PROPN
cana-3296	21	65	ζ	ζ	NOUN
cana-3296	21	66	.	.	PUNCT
cana-3296	22	1	(	(	PUNCT
cana-3296	22	2	iii	iii	NOUN
cana-3296	22	3	)	)	PUNCT
cana-3296	22	4	⋃	⋃	NOUN
cana-3296	22	5	p∈k	p∈k	ADJ
cana-3296	22	6	ϑp	ϑp	NOUN
cana-3296	22	7	∈	∈	PROPN
cana-3296	22	8	ζ	ζ	NOUN
cana-3296	22	9	,	,	PUNCT
cana-3296	22	10	{	{	PUNCT
cana-3296	22	11	ϑp	ϑp	NOUN
cana-3296	22	12	}	}	PUNCT
cana-3296	22	13	p∈k	p∈k	VERB
cana-3296	22	14	⊂	⊂	PROPN
cana-3296	22	15	ζ	ζ	NOUN
cana-3296	22	16	.	.	PUNCT
cana-3296	23	1	the	the	DET
cana-3296	23	2	pair	pair	NOUN
cana-3296	23	3	(	(	PUNCT
cana-3296	23	4	x	x	NOUN
cana-3296	23	5	,	,	PUNCT
cana-3296	23	6	ζ	ζ	NOUN
cana-3296	23	7	)	)	PUNCT
cana-3296	23	8	is	be	AUX
cana-3296	23	9	iivfts	iivfts	NOUN
cana-3296	23	10	and	and	CCONJ
cana-3296	23	11	every	every	DET
cana-3296	23	12	ϑ	ϑ	PROPN
cana-3296	23	13	∈	∈	ADJ
cana-3296	23	14	ζ	ζ	NOUN
cana-3296	23	15	is	be	AUX
cana-3296	23	16	open	open	ADJ
cana-3296	23	17	and	and	CCONJ
cana-3296	23	18	ϑ	ϑ	X
cana-3296	23	19	is	be	AUX
cana-3296	23	20	said	say	VERB
cana-3296	23	21	to	to	PART
cana-3296	23	22	be	be	AUX
cana-3296	23	23	closed	close	VERB
cana-3296	23	24	in	in	ADP
cana-3296	23	25	x	x	PUNCT
cana-3296	23	26	if	if	SCONJ
cana-3296	23	27	ϑ	ϑ	PROPN
cana-3296	23	28	c	c	PROPN
cana-3296	23	29	∈	∈	PROPN
cana-3296	23	30	ζ	ζ	PROPN
cana-3296	23	31	.	.	PUNCT
cana-3296	23	32	communications	communication	NOUN
cana-3296	23	33	on	on	ADP
cana-3296	23	34	applied	apply	VERB
cana-3296	23	35	nonlinear	nonlinear	ADJ
cana-3296	23	36	analysis	analysis	NOUN
cana-3296	23	37	issn	issn	NOUN
cana-3296	23	38	:	:	PUNCT
cana-3296	23	39	1074	1074	NUM
cana-3296	23	40	-	-	PUNCT
cana-3296	23	41	133x	133x	NUM
cana-3296	23	42	vol	vol	NOUN
cana-3296	23	43	32	32	NUM
cana-3296	23	44	no	no	NOUN
cana-3296	23	45	.	.	PUNCT
cana-3296	24	1	6s	6s	NUM
cana-3296	24	2	(	(	PUNCT
cana-3296	24	3	2025	2025	NUM
cana-3296	24	4	)	)	PUNCT
cana-3296	24	5	296	296	NUM
cana-3296	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3296	24	7	3	3	NUM
cana-3296	24	8	intuitionistic	intuitionistic	ADJ
cana-3296	24	9	interval	interval	NOUN
cana-3296	24	10	valued	value	VERB
cana-3296	24	11	fuzzy	fuzzy	ADJ
cana-3296	24	12	minimal	minimal	ADJ
cana-3296	24	13	open	open	ADJ
cana-3296	24	14	sets	set	NOUN
cana-3296	24	15	definition	definition	NOUN
cana-3296	24	16	3.1	3.1	NUM
cana-3296	24	17	a	a	DET
cana-3296	24	18	proper	proper	ADJ
cana-3296	24	19	iivfo	iivfo	NOUN
cana-3296	24	20	set	set	VERB
cana-3296	24	21	ϑ	ϑ	PROPN
cana-3296	24	22	of	of	ADP
cana-3296	24	23	iivfts	iivfts	NOUN
cana-3296	24	24	(	(	PUNCT
cana-3296	24	25	x	x	NOUN
cana-3296	24	26	,	,	PUNCT
cana-3296	24	27	ζ	ζ	NOUN
cana-3296	24	28	)	)	PUNCT
cana-3296	24	29	is	be	AUX
cana-3296	24	30	said	say	VERB
cana-3296	24	31	to	to	PART
cana-3296	24	32	be	be	AUX
cana-3296	24	33	a	a	DET
cana-3296	24	34	iivfmio	iivfmio	ADJ
cana-3296	24	35	iff	iff	NOUN
cana-3296	24	36	any	any	DET
cana-3296	24	37	iivfo	iivfo	NOUN
cana-3296	24	38	which	which	PRON
cana-3296	24	39	is	be	AUX
cana-3296	24	40	contained	contain	VERB
cana-3296	24	41	in	in	ADP
cana-3296	24	42	ϑ	ϑ	PROPN
cana-3296	24	43	is	be	AUX
cana-3296	24	44	either	either	CCONJ
cana-3296	24	45	ϑ	ϑ	PROPN
cana-3296	24	46	or	or	CCONJ
cana-3296	24	47	0̃.	0̃.	NUM
cana-3296	24	48	lemma	lemma	PROPN
cana-3296	24	49	3.1	3.1	NUM
cana-3296	24	50	in	in	ADP
cana-3296	24	51	a	a	DET
cana-3296	24	52	iivfts	iivfts	NOUN
cana-3296	24	53	(	(	PUNCT
cana-3296	24	54	x	x	NOUN
cana-3296	24	55	,	,	PUNCT
cana-3296	24	56	ζ	ζ	NOUN
cana-3296	24	57	)	)	PUNCT
cana-3296	24	58	,	,	PUNCT
cana-3296	24	59	(	(	PUNCT
cana-3296	24	60	i	i	NOUN
cana-3296	24	61	)	)	PUNCT
cana-3296	24	62	if	if	SCONJ
cana-3296	24	63	ϑ1	ϑ1	PROPN
cana-3296	24	64	is	be	AUX
cana-3296	24	65	a	a	DET
cana-3296	24	66	iivfmio	iivfmio	NOUN
cana-3296	24	67	and	and	CCONJ
cana-3296	24	68	ϑ2	ϑ2	PROPN
cana-3296	24	69	is	be	AUX
cana-3296	24	70	a	a	DET
cana-3296	24	71	iivfo	iivfo	NOUN
cana-3296	24	72	set	set	VERB
cana-3296	24	73	in	in	ADP
cana-3296	24	74	x	x	PUNCT
cana-3296	24	75	then	then	ADV
cana-3296	24	76	,	,	PUNCT
cana-3296	24	77	ϑ1	ϑ1	NOUN
cana-3296	24	78	∩	∩	ADJ
cana-3296	24	79	ϑ2	ϑ2	NOUN
cana-3296	24	80	=	=	SYM
cana-3296	24	81	0̃	0̃	PROPN
cana-3296	24	82	or	or	CCONJ
cana-3296	24	83	ϑ1	ϑ1	NOUN
cana-3296	24	84	⊂	⊂	ADJ
cana-3296	24	85	ϑ2	ϑ2	PROPN
cana-3296	24	86	.	.	PUNCT
cana-3296	25	1	(	(	PUNCT
cana-3296	25	2	ii	ii	NOUN
cana-3296	25	3	)	)	PUNCT
cana-3296	25	4	if	if	SCONJ
cana-3296	25	5	ϑ1	ϑ1	PROPN
cana-3296	25	6	,	,	PUNCT
cana-3296	25	7	ϑ2	ϑ2	PROPN
cana-3296	25	8	are	be	AUX
cana-3296	25	9	iivfmio	iivfmio	ADJ
cana-3296	25	10	sets	set	NOUN
cana-3296	25	11	then	then	ADV
cana-3296	25	12	ϑ1	ϑ1	NOUN
cana-3296	25	13	∩	∩	ADJ
cana-3296	25	14	ϑ2	ϑ2	NOUN
cana-3296	25	15	=	=	SYM
cana-3296	25	16	0̃	0̃	PROPN
cana-3296	25	17	or	or	CCONJ
cana-3296	25	18	ϑ1	ϑ1	NOUN
cana-3296	25	19	=	=	SYM
cana-3296	25	20	ϑ2	ϑ2	PROPN
cana-3296	25	21	.	.	PUNCT
cana-3296	26	1	proof	proof	NOUN
cana-3296	26	2	.	.	PUNCT
cana-3296	27	1	(	(	PUNCT
cana-3296	27	2	i	i	NOUN
cana-3296	27	3	)	)	PUNCT
cana-3296	27	4	suppose	suppose	VERB
cana-3296	27	5	that	that	SCONJ
cana-3296	27	6	ϑ1	ϑ1	PROPN
cana-3296	27	7	∩	∩	ADJ
cana-3296	27	8	ϑ2	ϑ2	NOUN
cana-3296	27	9	≠	≠	PROPN
cana-3296	27	10	0̃	0̃	NOUN
cana-3296	27	11	for	for	ADP
cana-3296	27	12	any	any	DET
cana-3296	27	13	iivfo	iivfo	NOUN
cana-3296	27	14	set	set	VERB
cana-3296	27	15	ϑ2	ϑ2	PROPN
cana-3296	27	16	then	then	ADV
cana-3296	27	17	(	(	PUNCT
cana-3296	27	18	ϑ1	ϑ1	NOUN
cana-3296	27	19	∩	∩	ADJ
cana-3296	27	20	ϑ2	ϑ2	NOUN
cana-3296	27	21	)	)	PUNCT
cana-3296	27	22	⊂	⊂	PROPN
cana-3296	27	23	ϑ1	ϑ1	PROPN
cana-3296	27	24	,	,	PUNCT
cana-3296	27	25	a	a	DET
cana-3296	27	26	contradiction	contradiction	NOUN
cana-3296	27	27	to	to	ADP
cana-3296	27	28	minimality	minimality	NOUN
cana-3296	27	29	of	of	ADP
cana-3296	27	30	ϑ1	ϑ1	NOUN
cana-3296	27	31	,	,	PUNCT
cana-3296	27	32	then	then	ADV
cana-3296	27	33	(	(	PUNCT
cana-3296	27	34	ϑ1	ϑ1	NOUN
cana-3296	27	35	∩	∩	ADJ
cana-3296	27	36	ϑ2	ϑ2	NOUN
cana-3296	27	37	)	)	PUNCT
cana-3296	27	38	=	=	SYM
cana-3296	28	1	ϑ1	ϑ1	NOUN
cana-3296	28	2	implying	imply	VERB
cana-3296	28	3	that	that	SCONJ
cana-3296	28	4	ϑ1	ϑ1	PROPN
cana-3296	28	5	⊂	⊂	PROPN
cana-3296	28	6	ϑ2	ϑ2	PROPN
cana-3296	28	7	.	.	PUNCT
cana-3296	29	1	(	(	PUNCT
cana-3296	29	2	ii	ii	NOUN
cana-3296	29	3	)	)	PUNCT
cana-3296	29	4	as	as	ADP
cana-3296	29	5	ϑ1	ϑ1	PROPN
cana-3296	29	6	,	,	PUNCT
cana-3296	29	7	ϑ2	ϑ2	PROPN
cana-3296	29	8	are	be	AUX
cana-3296	29	9	iivfmio	iivfmio	ADJ
cana-3296	29	10	sets	set	NOUN
cana-3296	29	11	,	,	PUNCT
cana-3296	29	12	if	if	SCONJ
cana-3296	29	13	ϑ1	ϑ1	NOUN
cana-3296	29	14	∩	∩	ADJ
cana-3296	29	15	ϑ2	ϑ2	PROPN
cana-3296	29	16	≠	≠	PROPN
cana-3296	29	17	0̃	0̃	NOUN
cana-3296	29	18	then	then	ADV
cana-3296	29	19	ϑ1	ϑ1	NOUN
cana-3296	29	20	⊂	⊂	ADJ
cana-3296	29	21	ϑ2	ϑ2	PROPN
cana-3296	29	22	and	and	CCONJ
cana-3296	29	23	ϑ2	ϑ2	PROPN
cana-3296	29	24	⊂	⊂	PROPN
cana-3296	29	25	ϑ1	ϑ1	PROPN
cana-3296	29	26	,	,	PUNCT
cana-3296	29	27	implying	imply	VERB
cana-3296	29	28	that	that	SCONJ
cana-3296	29	29	ϑ1	ϑ1	NOUN
cana-3296	29	30	=	=	SYM
cana-3296	29	31	ϑ2	ϑ2	PROPN
cana-3296	29	32	.	.	PUNCT
cana-3296	30	1	theorem	theorem	VERB
cana-3296	30	2	3.2	3.2	NUM
cana-3296	30	3	if	if	SCONJ
cana-3296	30	4	ϑ	ϑ	NOUN
cana-3296	30	5	and	and	CCONJ
cana-3296	30	6	ϑp	ϑp	NOUN
cana-3296	30	7	are	be	AUX
cana-3296	30	8	iivfmio	iivfmio	ADJ
cana-3296	30	9	sets	set	NOUN
cana-3296	30	10	for	for	ADP
cana-3296	30	11	any	any	DET
cana-3296	30	12	p	p	PROPN
cana-3296	30	13	∈	∈	PROPN
cana-3296	30	14	k.	k.	NOUN
cana-3296	31	1	if	if	SCONJ
cana-3296	31	2	ϑ	ϑ	PROPN
cana-3296	31	3	⊆	⊆	NUM
cana-3296	31	4	⋃	⋃	ADV
cana-3296	31	5	p∈k	p∈k	ADJ
cana-3296	31	6	ϑp	ϑp	NOUN
cana-3296	31	7	then	then	ADV
cana-3296	31	8	∃	∃	PROPN
cana-3296	31	9	an	an	DET
cana-3296	31	10	element	element	NOUN
cana-3296	31	11	p	p	PROPN
cana-3296	31	12	∈	∈	PROPN
cana-3296	31	13	k	k	ADP
cana-3296	31	14	such	such	ADJ
cana-3296	31	15	that	that	SCONJ
cana-3296	31	16	ϑ	ϑ	X
cana-3296	31	17	=	=	X
cana-3296	31	18	ϑp	ϑp	NOUN
cana-3296	31	19	.	.	NOUN
cana-3296	31	20	proof	proof	NOUN
cana-3296	31	21	.	.	PUNCT
cana-3296	32	1	if	if	SCONJ
cana-3296	32	2	ϑ	ϑ	PRON
cana-3296	32	3	⊆	⊆	NUM
cana-3296	32	4	⋃	⋃	ADV
cana-3296	32	5	p∈k	p∈k	ADJ
cana-3296	32	6	ϑp	ϑp	NOUN
cana-3296	32	7	then	then	ADV
cana-3296	32	8	ϑ	ϑ	X
cana-3296	32	9	=	=	X
cana-3296	32	10	ϑ	ϑ	X
cana-3296	32	11	∩	∩	NOUN
cana-3296	32	12	(	(	PUNCT
cana-3296	32	13	⋃	⋃	ADV
cana-3296	32	14	p∈k	p∈k	ADJ
cana-3296	32	15	ϑp	ϑp	NOUN
cana-3296	32	16	)	)	PUNCT
cana-3296	32	17	=	=	SYM
cana-3296	32	18	⋃	⋃	NOUN
cana-3296	32	19	p∈k	p∈k	ADJ
cana-3296	32	20	(	(	PUNCT
cana-3296	32	21	ϑ	ϑ	X
cana-3296	32	22	∩	∩	ADJ
cana-3296	32	23	ϑp	ϑp	NOUN
cana-3296	32	24	)	)	PUNCT
cana-3296	32	25	.	.	PUNCT
cana-3296	33	1	clearly	clearly	ADV
cana-3296	33	2	(	(	PUNCT
cana-3296	33	3	ϑ	ϑ	X
cana-3296	33	4	∩	∩	ADJ
cana-3296	33	5	ϑp	ϑp	NOUN
cana-3296	33	6	)	)	PUNCT
cana-3296	33	7	=	=	SYM
cana-3296	33	8	0̃	0̃	NOUN
cana-3296	33	9	or	or	CCONJ
cana-3296	33	10	ϑ	ϑ	X
cana-3296	33	11	=	=	X
cana-3296	33	12	ϑp	ϑp	NOUN
cana-3296	33	13	as	as	ADP
cana-3296	33	14	ϑ	ϑ	NOUN
cana-3296	33	15	,	,	PUNCT
cana-3296	33	16	ϑp	ϑp	NOUN
cana-3296	33	17	are	be	AUX
cana-3296	33	18	iivfmio	iivfmio	ADJ
cana-3296	33	19	sets	set	NOUN
cana-3296	33	20	by	by	ADP
cana-3296	33	21	lemma-3.1(ii	lemma-3.1(ii	NOUN
cana-3296	33	22	)	)	PUNCT
cana-3296	33	23	.	.	PUNCT
cana-3296	34	1	if	if	SCONJ
cana-3296	34	2	(	(	PUNCT
cana-3296	34	3	ϑ	ϑ	X
cana-3296	34	4	∩	∩	ADJ
cana-3296	34	5	ϑp	ϑp	NOUN
cana-3296	34	6	)	)	PUNCT
cana-3296	34	7	=	=	SYM
cana-3296	34	8	0̃	0̃	NOUN
cana-3296	34	9	then	then	ADV
cana-3296	34	10	ϑ	ϑ	X
cana-3296	34	11	=	=	PROPN
cana-3296	34	12	0̃	0̃	PROPN
cana-3296	34	13	,	,	PUNCT
cana-3296	34	14	a	a	DET
cana-3296	34	15	contradiction	contradiction	NOUN
cana-3296	34	16	to	to	ADP
cana-3296	34	17	minimality	minimality	NOUN
cana-3296	34	18	of	of	ADP
cana-3296	34	19	ϑ.	ϑ.	NOUN
cana-3296	34	20	theorem	theorem	VERB
cana-3296	34	21	3.3	3.3	NUM
cana-3296	34	22	suppose	suppose	VERB
cana-3296	34	23	ϑ	ϑ	X
cana-3296	34	24	,	,	PUNCT
cana-3296	34	25	ϑp	ϑp	NOUN
cana-3296	34	26	are	be	AUX
cana-3296	34	27	iivfmio	iivfmio	ADJ
cana-3296	34	28	sets	set	NOUN
cana-3296	34	29	for	for	ADP
cana-3296	34	30	any	any	DET
cana-3296	34	31	p	p	PROPN
cana-3296	34	32	∈	∈	PROPN
cana-3296	34	33	k	k	NOUN
cana-3296	34	34	,	,	PUNCT
cana-3296	34	35	ϑ	ϑ	PROPN
cana-3296	34	36	≠	≠	PROPN
cana-3296	34	37	ϑp	ϑp	NOUN
cana-3296	34	38	for	for	ADP
cana-3296	34	39	any	any	DET
cana-3296	34	40	p	p	NOUN
cana-3296	34	41	∈	∈	PROPN
cana-3296	35	1	k	k	NOUN
cana-3296	35	2	then	then	ADV
cana-3296	35	3	ϑ	ϑ	AUX
cana-3296	35	4	∩	∩	NOUN
cana-3296	35	5	(	(	PUNCT
cana-3296	35	6	⋃	⋃	ADV
cana-3296	35	7	p∈k	p∈k	ADJ
cana-3296	35	8	ϑp	ϑp	NOUN
cana-3296	35	9	)	)	PUNCT
cana-3296	35	10	=	=	PUNCT
cana-3296	35	11	0̃.	0̃.	NUM
cana-3296	35	12	proof	proof	NOUN
cana-3296	35	13	.	.	PUNCT
cana-3296	35	14	suppose	suppose	VERB
cana-3296	35	15	that	that	SCONJ
cana-3296	35	16	ϑ	ϑ	PROPN
cana-3296	35	17	∩	∩	NOUN
cana-3296	35	18	(	(	PUNCT
cana-3296	35	19	⋃	⋃	ADV
cana-3296	35	20	p∈k	p∈k	ADJ
cana-3296	35	21	ϑp	ϑp	NOUN
cana-3296	35	22	)	)	PUNCT
cana-3296	35	23	≠	≠	PROPN
cana-3296	35	24	0̃	0̃	NOUN
cana-3296	35	25	,	,	PUNCT
cana-3296	35	26	then	then	ADV
cana-3296	35	27	∃	∃	PROPN
cana-3296	35	28	an	an	DET
cana-3296	35	29	element	element	NOUN
cana-3296	35	30	p	p	PROPN
cana-3296	35	31	∈	∈	PROPN
cana-3296	35	32	k	k	X
cana-3296	35	33	with	with	ADP
cana-3296	35	34	(	(	PUNCT
cana-3296	35	35	ϑ	ϑ	X
cana-3296	35	36	∩	∩	ADJ
cana-3296	35	37	ϑp	ϑp	NOUN
cana-3296	35	38	)	)	PUNCT
cana-3296	35	39	≠	≠	PROPN
cana-3296	35	40	0̃.	0̃.	NUM
cana-3296	35	41	by	by	ADP
cana-3296	35	42	lemma3.1(ii	lemma3.1(ii	PROPN
cana-3296	35	43	)	)	PUNCT
cana-3296	35	44	ϑ	ϑ	NOUN
cana-3296	35	45	=	=	SYM
cana-3296	35	46	ϑp	ϑp	NOUN
cana-3296	35	47	,	,	PUNCT
cana-3296	35	48	a	a	DET
cana-3296	35	49	contradiction	contradiction	NOUN
cana-3296	35	50	.	.	PUNCT
cana-3296	36	1	theorem	theorem	VERB
cana-3296	36	2	3.4	3.4	NUM
cana-3296	36	3	a	a	DET
cana-3296	36	4	iivfmio	iivfmio	NOUN
cana-3296	36	5	set	set	VERB
cana-3296	36	6	ϑp	ϑp	NOUN
cana-3296	36	7	for	for	ADP
cana-3296	36	8	any	any	DET
cana-3296	36	9	p	p	PROPN
cana-3296	36	10	∈	∈	PROPN
cana-3296	36	11	k	k	NOUN
cana-3296	36	12	;	;	PUNCT
cana-3296	36	13	|k|	|k|	NOUN
cana-3296	36	14	≥	≥	X
cana-3296	36	15	2	2	NUM
cana-3296	36	16	and	and	CCONJ
cana-3296	36	17	ϑl	ϑl	PROPN
cana-3296	36	18	≠	≠	PROPN
cana-3296	36	19	ϑp	ϑp	NOUN
cana-3296	36	20	for	for	ADP
cana-3296	36	21	any	any	DET
cana-3296	36	22	distinct	distinct	ADJ
cana-3296	36	23	l	l	NOUN
cana-3296	36	24	,	,	PUNCT
cana-3296	37	1	p	p	PROPN
cana-3296	37	2	∈	∈	PROPN
cana-3296	37	3	k	k	NOUN
cana-3296	37	4	,	,	PUNCT
cana-3296	37	5	then	then	ADV
cana-3296	37	6	for	for	ADP
cana-3296	37	7	any	any	DET
cana-3296	37	8	l	l	NOUN
cana-3296	37	9	∈	∈	PROPN
cana-3296	37	10	k	k	NOUN
cana-3296	37	11	,	,	PUNCT
cana-3296	37	12	ϑl	ϑl	PROPN
cana-3296	37	13	∩	∩	NOUN
cana-3296	37	14	(	(	PUNCT
cana-3296	37	15	⋃	⋃	ADV
cana-3296	37	16	p∈k	p∈k	ADJ
cana-3296	37	17	ϑp	ϑp	NOUN
cana-3296	37	18	)	)	PUNCT
cana-3296	37	19	=	=	PUNCT
cana-3296	37	20	0̃.	0̃.	NUM
cana-3296	37	21	proof	proof	NOUN
cana-3296	37	22	.	.	PUNCT
cana-3296	38	1	let	let	VERB
cana-3296	38	2	ϑl	ϑl	PROPN
cana-3296	38	3	∩	∩	NOUN
cana-3296	38	4	(	(	PUNCT
cana-3296	38	5	⋃	⋃	ADV
cana-3296	38	6	p∈k	p∈k	ADJ
cana-3296	38	7	/	/	SYM
cana-3296	38	8	l	l	NOUN
cana-3296	38	9	ϑp	ϑp	X
cana-3296	38	10	)	)	PUNCT
cana-3296	38	11	≠	≠	PROPN
cana-3296	38	12	0̃	0̃	NOUN
cana-3296	38	13	,	,	PUNCT
cana-3296	38	14	then	then	ADV
cana-3296	38	15	⋃	⋃	PUNCT
cana-3296	38	16	p∈k	p∈k	ADJ
cana-3296	38	17	/	/	SYM
cana-3296	38	18	l	l	NOUN
cana-3296	38	19	(	(	PUNCT
cana-3296	38	20	ϑl	ϑl	X
cana-3296	38	21	∩	∩	ADJ
cana-3296	38	22	ϑp	ϑp	NOUN
cana-3296	38	23	)	)	PUNCT
cana-3296	38	24	≠	≠	PROPN
cana-3296	38	25	0̃.	0̃.	NUM
cana-3296	38	26	by	by	ADP
cana-3296	38	27	lemma-3.1(ii	lemma-3.1(ii	NOUN
cana-3296	38	28	)	)	PUNCT
cana-3296	38	29	,	,	PUNCT
cana-3296	38	30	ϑl	ϑl	NOUN
cana-3296	38	31	=	=	NOUN
cana-3296	38	32	ϑp	ϑp	NOUN
cana-3296	38	33	,	,	PUNCT
cana-3296	38	34	a	a	DET
cana-3296	38	35	contradiction	contradiction	NOUN
cana-3296	38	36	.	.	PUNCT
cana-3296	39	1	theorem	theorem	VERB
cana-3296	39	2	3.5	3.5	NUM
cana-3296	39	3	if	if	SCONJ
cana-3296	39	4	ϑp	ϑp	NOUN
cana-3296	39	5	is	be	AUX
cana-3296	39	6	a	a	DET
cana-3296	39	7	iivfmio	iivfmio	NOUN
cana-3296	39	8	set	set	NOUN
cana-3296	39	9	for	for	ADP
cana-3296	39	10	any	any	DET
cana-3296	39	11	k	k	PROPN
cana-3296	39	12	∈	∈	PROPN
cana-3296	39	13	k	k	NOUN
cana-3296	39	14	;	;	PUNCT
cana-3296	39	15	|k|	|k|	NOUN
cana-3296	39	16	≥	≥	X
cana-3296	39	17	2	2	NUM
cana-3296	39	18	and	and	CCONJ
cana-3296	39	19	ϑl	ϑl	PROPN
cana-3296	39	20	≠	≠	PROPN
cana-3296	39	21	ϑp	ϑp	NOUN
cana-3296	39	22	for	for	ADP
cana-3296	39	23	any	any	DET
cana-3296	39	24	distinct	distinct	ADJ
cana-3296	39	25	l	l	NOUN
cana-3296	39	26	,	,	PUNCT
cana-3296	39	27	p	p	PROPN
cana-3296	39	28	∈	∈	PROPN
cana-3296	39	29	k.	k.	NOUN
cana-3296	40	1	if	if	SCONJ
cana-3296	40	2	α	α	PRON
cana-3296	40	3	is	be	AUX
cana-3296	40	4	a	a	DET
cana-3296	40	5	proper	proper	ADJ
cana-3296	40	6	iivf	iivf	NOUN
cana-3296	40	7	subset	subset	NOUN
cana-3296	40	8	of	of	ADP
cana-3296	40	9	k	k	NOUN
cana-3296	40	10	,	,	PUNCT
cana-3296	40	11	then	then	ADV
cana-3296	40	12	(	(	PUNCT
cana-3296	40	13	⋃	⋃	NOUN
cana-3296	40	14	k∈k	k∈k	NOUN
cana-3296	40	15	/	/	SYM
cana-3296	40	16	α	α	NOUN
cana-3296	40	17	ϑl	ϑl	NOUN
cana-3296	40	18	)	)	PUNCT
cana-3296	40	19	∩	∩	NOUN
cana-3296	40	20	(	(	PUNCT
cana-3296	40	21	⋃	⋃	PROPN
cana-3296	40	22	m∈α	m∈α	NOUN
cana-3296	40	23	ϑm	ϑm	NOUN
cana-3296	40	24	)	)	PUNCT
cana-3296	40	25	=	=	SYM
cana-3296	40	26	0̃.	0̃.	NUM
cana-3296	40	27	proof	proof	NOUN
cana-3296	40	28	.	.	PUNCT
cana-3296	41	1	by	by	ADP
cana-3296	41	2	assuming	assume	VERB
cana-3296	41	3	the	the	DET
cana-3296	41	4	contrary	contrary	NOUN
cana-3296	41	5	,	,	PUNCT
cana-3296	41	6	we	we	PRON
cana-3296	41	7	have	have	VERB
cana-3296	41	8	∪	∪	ADJ
cana-3296	41	9	(	(	PUNCT
cana-3296	41	10	ϑl	ϑl	INTJ
cana-3296	41	11	∩	∩	NOUN
cana-3296	41	12	ϑm	ϑm	ADP
cana-3296	41	13	)	)	PUNCT
cana-3296	41	14	≠	≠	PROPN
cana-3296	41	15	0̃	0̃	NOUN
cana-3296	41	16	for	for	ADP
cana-3296	41	17	k	k	PROPN
cana-3296	41	18	∈	∈	PROPN
cana-3296	41	19	k	k	PROPN
cana-3296	41	20	/	/	SYM
cana-3296	41	21	α	α	PROPN
cana-3296	41	22	and	and	CCONJ
cana-3296	41	23	m	m	PROPN
cana-3296	41	24	∈	∈	NOUN
cana-3296	41	25	α	α	NOUN
cana-3296	41	26	implying	imply	VERB
cana-3296	41	27	(	(	PUNCT
cana-3296	41	28	ϑl	ϑl	INTJ
cana-3296	41	29	∩	∩	NOUN
cana-3296	41	30	ϑm	ϑm	ADP
cana-3296	41	31	)	)	PUNCT
cana-3296	41	32	≠	≠	PROPN
cana-3296	41	33	0̃	0̃	NOUN
cana-3296	41	34	for	for	ADP
cana-3296	41	35	some	some	DET
cana-3296	41	36	p	p	NOUN
cana-3296	41	37	∈	∈	PROPN
cana-3296	41	38	k	k	NOUN
cana-3296	41	39	,	,	PUNCT
cana-3296	41	40	m	m	PROPN
cana-3296	41	41	∈	∈	PROPN
cana-3296	41	42	α.by	α.by	PROPN
cana-3296	41	43	lemma-0.1(ii	lemma-0.1(ii	PROPN
cana-3296	41	44	)	)	PUNCT
cana-3296	41	45	we	we	PRON
cana-3296	41	46	have	have	VERB
cana-3296	41	47	,	,	PUNCT
cana-3296	41	48	ϑl	ϑl	PROPN
cana-3296	41	49	=	=	X
cana-3296	41	50	ϑm	ϑm	PROPN
cana-3296	41	51	,	,	PUNCT
cana-3296	41	52	a	a	DET
cana-3296	41	53	contradiction	contradiction	NOUN
cana-3296	41	54	.	.	PUNCT
cana-3296	42	1	theorem	theorem	VERB
cana-3296	42	2	3.6	3.6	NUM
cana-3296	42	3	if	if	SCONJ
cana-3296	42	4	ϑp	ϑp	NOUN
cana-3296	42	5	is	be	AUX
cana-3296	42	6	a	a	DET
cana-3296	42	7	iivfmios	iivfmios	NOUN
cana-3296	42	8	for	for	ADP
cana-3296	42	9	any	any	DET
cana-3296	42	10	p	p	NOUN
cana-3296	42	11	∈	∈	PROPN
cana-3296	42	12	k	k	NOUN
cana-3296	42	13	with	with	ADP
cana-3296	42	14	ϑp	ϑp	NOUN
cana-3296	42	15	≠	≠	ADJ
cana-3296	42	16	ϑl	ϑl	NOUN
cana-3296	42	17	for	for	ADP
cana-3296	42	18	any	any	DET
cana-3296	42	19	distinct	distinct	ADJ
cana-3296	42	20	l	l	NOUN
cana-3296	42	21	,	,	PUNCT
cana-3296	43	1	p	p	PROPN
cana-3296	43	2	∈	∈	PROPN
cana-3296	43	3	k	k	NOUN
cana-3296	43	4	,	,	PUNCT
cana-3296	43	5	then	then	ADV
cana-3296	43	6	[	[	X
cana-3296	43	7	⋃p∈k	⋃p∈k	PROPN
cana-3296	43	8	/	/	SYM
cana-3296	43	9	l	l	NOUN
cana-3296	43	10	ϑp	ϑp	X
cana-3296	43	11	]	]	X
cana-3296	43	12	∩	∩	NOUN
cana-3296	43	13	[	[	X
cana-3296	43	14	⋃l∈t	⋃l∈t	PROPN
cana-3296	43	15	ϑl	ϑl	X
cana-3296	43	16	]	]	X
cana-3296	43	17	=	=	SYM
cana-3296	43	18	0̃	0̃	NOUN
cana-3296	43	19	for	for	ADP
cana-3296	43	20	any	any	DET
cana-3296	43	21	proper	proper	ADJ
cana-3296	43	22	iivf	iivf	NOUN
cana-3296	43	23	subset	subset	PROPN
cana-3296	43	24	t	t	PROPN
cana-3296	43	25	of	of	ADP
cana-3296	43	26	k.	k.	PROPN
cana-3296	43	27	proof	proof	PROPN
cana-3296	43	28	.	.	PUNCT
cana-3296	44	1	by	by	ADP
cana-3296	44	2	assuming	assume	VERB
cana-3296	44	3	the	the	DET
cana-3296	44	4	contrary	contrary	NOUN
cana-3296	44	5	,	,	PUNCT
cana-3296	44	6	we	we	PRON
cana-3296	44	7	have	have	VERB
cana-3296	44	8	∪	∪	ADJ
cana-3296	44	9	[	[	PUNCT
cana-3296	44	10	ϑp	ϑp	NOUN
cana-3296	44	11	∩	∩	ADJ
cana-3296	44	12	ϑl	ϑl	ADP
cana-3296	44	13	]	]	PUNCT
cana-3296	44	14	≠	≠	PROPN
cana-3296	44	15	0̃	0̃	NOUN
cana-3296	44	16	,	,	PUNCT
cana-3296	44	17	∀p	∀p	NOUN
cana-3296	44	18	∈	∈	PROPN
cana-3296	44	19	k	k	NOUN
cana-3296	44	20	/	/	SYM
cana-3296	44	21	l	l	NOUN
cana-3296	44	22	,	,	PUNCT
cana-3296	44	23	l	l	PROPN
cana-3296	44	24	∈	∈	PROPN
cana-3296	44	25	t	t	NOUN
cana-3296	44	26	implying	imply	VERB
cana-3296	44	27	∪	∪	ADV
cana-3296	44	28	[	[	PUNCT
cana-3296	44	29	ϑp	ϑp	NOUN
cana-3296	44	30	∩	∩	ADJ
cana-3296	44	31	ϑl	ϑl	ADP
cana-3296	44	32	]	]	PUNCT
cana-3296	44	33	≠	≠	PROPN
cana-3296	44	34	0̃	0̃	NOUN
cana-3296	44	35	for	for	ADP
cana-3296	44	36	some	some	DET
cana-3296	44	37	p	p	NOUN
cana-3296	44	38	∈	∈	PROPN
cana-3296	44	39	k	k	NOUN
cana-3296	44	40	;	;	PUNCT
cana-3296	44	41	l	l	PROPN
cana-3296	44	42	∈	∈	PROPN
cana-3296	44	43	t.	t.	NOUN
cana-3296	44	44	by	by	ADP
cana-3296	44	45	lemma-3.1(ii	lemma-3.1(ii	NOUN
cana-3296	44	46	)	)	PUNCT
cana-3296	44	47	a	a	DET
cana-3296	44	48	contradiction	contradiction	NOUN
cana-3296	44	49	to	to	ADP
cana-3296	44	50	minimality	minimality	NOUN
cana-3296	44	51	of	of	ADP
cana-3296	44	52	ϑp	ϑp	PROPN
cana-3296	44	53	.	.	X
cana-3296	44	54	theorem	theorem	VERB
cana-3296	44	55	3.7	3.7	NUM
cana-3296	44	56	if	if	SCONJ
cana-3296	44	57	ϑp	ϑp	NOUN
cana-3296	44	58	,	,	PUNCT
cana-3296	44	59	ϑl	ϑl	NOUN
cana-3296	44	60	are	be	AUX
cana-3296	44	61	iivfmio	iivfmio	ADJ
cana-3296	44	62	sets	set	NOUN
cana-3296	44	63	for	for	ADP
cana-3296	44	64	any	any	DET
cana-3296	44	65	p	p	PROPN
cana-3296	44	66	∈	∈	PROPN
cana-3296	44	67	k	k	NOUN
cana-3296	44	68	;	;	PUNCT
cana-3296	44	69	l	l	PROPN
cana-3296	44	70	∈	∈	PROPN
cana-3296	44	71	t	t	NOUN
cana-3296	44	72	respectively	respectively	ADV
cana-3296	44	73	and	and	CCONJ
cana-3296	44	74	if	if	SCONJ
cana-3296	44	75	∃n	∃n	PROPN
cana-3296	44	76	∈	∈	PROPN
cana-3296	44	77	t	t	NOUN
cana-3296	44	78	such	such	ADJ
cana-3296	44	79	that	that	SCONJ
cana-3296	44	80	ϑp	ϑp	NOUN
cana-3296	44	81	≠	≠	PROPN
cana-3296	44	82	ϑn	ϑn	NOUN
cana-3296	44	83	,	,	PUNCT
cana-3296	44	84	for	for	ADP
cana-3296	44	85	any	any	DET
cana-3296	44	86	p	p	PROPN
cana-3296	44	87	∈	∈	PROPN
cana-3296	44	88	k	k	NOUN
cana-3296	44	89	,	,	PUNCT
cana-3296	44	90	then	then	ADV
cana-3296	44	91	[	[	X
cana-3296	44	92	⋃	⋃	NOUN
cana-3296	44	93	n∈m	n∈m	ADJ
cana-3296	44	94	ϑn][⋃	ϑn][⋃	NOUN
cana-3296	44	95	k∈k	k∈k	NOUN
cana-3296	44	96	ϑp	ϑp	NOUN
cana-3296	44	97	]	]	X
cana-3296	44	98	.	.	PUNCT
cana-3296	45	1	communications	communication	NOUN
cana-3296	45	2	on	on	ADP
cana-3296	45	3	applied	apply	VERB
cana-3296	45	4	nonlinear	nonlinear	ADJ
cana-3296	45	5	analysis	analysis	NOUN
cana-3296	45	6	issn	issn	NOUN
cana-3296	45	7	:	:	PUNCT
cana-3296	45	8	1074	1074	NUM
cana-3296	45	9	-	-	PUNCT
cana-3296	45	10	133x	133x	NUM
cana-3296	45	11	vol	vol	NOUN
cana-3296	45	12	32	32	NUM
cana-3296	45	13	no	no	NOUN
cana-3296	45	14	.	.	PUNCT
cana-3296	46	1	6s	6s	NUM
cana-3296	46	2	(	(	PUNCT
cana-3296	46	3	2025	2025	NUM
cana-3296	46	4	)	)	PUNCT
cana-3296	46	5	297	297	NUM
cana-3296	46	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3296	46	7	proof	proof	NOUN
cana-3296	46	8	.	.	PUNCT
cana-3296	47	1	by	by	ADP
cana-3296	47	2	assuming	assume	VERB
cana-3296	47	3	the	the	DET
cana-3296	47	4	contrary	contrary	NOUN
cana-3296	47	5	,	,	PUNCT
cana-3296	47	6	then	then	ADV
cana-3296	47	7	∃n	∃n	PROPN
cana-3296	47	8	∈	∈	PROPN
cana-3296	47	9	t	t	PROPN
cana-3296	47	10	with	with	ADP
cana-3296	47	11	ϑp	ϑp	PRON
cana-3296	47	12	≠	≠	ADJ
cana-3296	47	13	ϑn	ϑn	NOUN
cana-3296	47	14	for	for	ADP
cana-3296	47	15	any	any	DET
cana-3296	47	16	p	p	NOUN
cana-3296	47	17	∈	∈	PROPN
cana-3296	47	18	k	k	NOUN
cana-3296	47	19	,	,	PUNCT
cana-3296	47	20	then	then	ADV
cana-3296	47	21	[	[	X
cana-3296	47	22	⋃	⋃	ADJ
cana-3296	47	23	n∈m	n∈m	ADJ
cana-3296	47	24	ϑn	ϑn	NOUN
cana-3296	47	25	]	]	PUNCT
cana-3296	47	26	⊂	⊂	X
cana-3296	48	1	[	[	X
cana-3296	48	2	⋃	⋃	ADV
cana-3296	48	3	p∈k	p∈k	ADJ
cana-3296	48	4	ϑp	ϑp	NOUN
cana-3296	48	5	]	]	X
cana-3296	48	6	.	.	PUNCT
cana-3296	49	1	ϑn	ϑn	PROPN
cana-3296	49	2	⊂	⊂	PROPN
cana-3296	50	1	[	[	X
cana-3296	50	2	⋃	⋃	ADV
cana-3296	50	3	p∈k	p∈k	ADJ
cana-3296	50	4	ϑp	ϑp	NOUN
cana-3296	50	5	]	]	X
cana-3296	50	6	,	,	PUNCT
cana-3296	50	7	for	for	ADP
cana-3296	50	8	some	some	DET
cana-3296	50	9	n	n	PRON
cana-3296	50	10	∈	∈	NOUN
cana-3296	50	11	m.	m.	NOUN
cana-3296	50	12	hence	hence	ADV
cana-3296	50	13	,	,	PUNCT
cana-3296	50	14	ϑp	ϑp	NOUN
cana-3296	50	15	≠	≠	PROPN
cana-3296	50	16	ϑn	ϑn	NOUN
cana-3296	50	17	,	,	PUNCT
cana-3296	50	18	for	for	ADP
cana-3296	50	19	any	any	DET
cana-3296	50	20	p	p	PROPN
cana-3296	50	21	∈	∈	PROPN
cana-3296	50	22	k	k	NOUN
cana-3296	50	23	,	,	PUNCT
cana-3296	50	24	a	a	DET
cana-3296	50	25	contradiction	contradiction	NOUN
cana-3296	50	26	.	.	PUNCT
cana-3296	51	1	theorem	theorem	VERB
cana-3296	51	2	3.8	3.8	NUM
cana-3296	51	3	if	if	SCONJ
cana-3296	51	4	ϑp	ϑp	NOUN
cana-3296	51	5	is	be	AUX
cana-3296	51	6	a	a	DET
cana-3296	51	7	iivfmio	iivfmio	NOUN
cana-3296	51	8	for	for	ADP
cana-3296	51	9	any	any	DET
cana-3296	51	10	p	p	NOUN
cana-3296	51	11	∈	∈	PROPN
cana-3296	51	12	k	k	NOUN
cana-3296	51	13	with	with	ADP
cana-3296	51	14	ϑp	ϑp	DET
cana-3296	51	15	≠	≠	PROPN
cana-3296	51	16	ϑm	ϑm	ADP
cana-3296	51	17	for	for	ADP
cana-3296	51	18	any	any	DET
cana-3296	51	19	distinct	distinct	ADJ
cana-3296	51	20	p	p	NOUN
cana-3296	51	21	,	,	PUNCT
cana-3296	51	22	m	m	VERB
cana-3296	51	23	∈	∈	NOUN
cana-3296	52	1	k	k	NOUN
cana-3296	53	1	then	then	ADV
cana-3296	53	2	[	[	X
cana-3296	53	3	⋃	⋃	NOUN
cana-3296	53	4	p∈t	p∈t	NOUN
cana-3296	53	5	ϑp	ϑp	NOUN
cana-3296	53	6	]	]	X
cana-3296	53	7	⫋	⫋	NOUN
cana-3296	54	1	[	[	X
cana-3296	54	2	⋃	⋃	NOUN
cana-3296	54	3	m∈k	m∈k	NOUN
cana-3296	54	4	ϑm	ϑm	ADP
cana-3296	54	5	]	]	PUNCT
cana-3296	54	6	for	for	ADP
cana-3296	54	7	any	any	DET
cana-3296	54	8	proper	proper	ADJ
cana-3296	54	9	subset	subset	NOUN
cana-3296	54	10	t	t	PROPN
cana-3296	54	11	of	of	ADP
cana-3296	54	12	k.	k.	PROPN
cana-3296	54	13	proof	proof	PROPN
cana-3296	54	14	.	.	PUNCT
cana-3296	55	1	for	for	ADP
cana-3296	55	2	any	any	DET
cana-3296	55	3	p	p	NOUN
cana-3296	55	4	∈	∈	PROPN
cana-3296	55	5	k	k	PROPN
cana-3296	55	6	/	/	SYM
cana-3296	55	7	t	t	PROPN
cana-3296	55	8	,	,	PUNCT
cana-3296	55	9	ϑl	ϑl	PROPN
cana-3296	55	10	is	be	AUX
cana-3296	55	11	a	a	DET
cana-3296	55	12	iivfmio	iivfmio	NOUN
cana-3296	55	13	of	of	ADP
cana-3296	55	14	family	family	NOUN
cana-3296	55	15	{	{	PUNCT
cana-3296	55	16	ϑl|l	ϑl|l	PROPN
cana-3296	55	17	∈	∈	PROPN
cana-3296	55	18	k	k	PROPN
cana-3296	55	19	/	/	SYM
cana-3296	55	20	t	t	PROPN
cana-3296	55	21	}	}	PUNCT
cana-3296	55	22	of	of	ADP
cana-3296	55	23	iivfmio	iivfmio	ADJ
cana-3296	55	24	sets	set	NOUN
cana-3296	55	25	.	.	PUNCT
cana-3296	56	1	clearly	clearly	ADV
cana-3296	56	2	,	,	PUNCT
cana-3296	56	3	ϑl	ϑl	PRON
cana-3296	56	4	∩	∩	NOUN
cana-3296	56	5	[	[	X
cana-3296	56	6	⋃	⋃	ADP
cana-3296	56	7	p∈t	p∈t	NOUN
cana-3296	56	8	ϑp	ϑp	X
cana-3296	56	9	]	]	X
cana-3296	56	10	=	=	PUNCT
cana-3296	56	11	⋃	⋃	NOUN
cana-3296	56	12	p∈t	p∈t	NOUN
cana-3296	57	1	[	[	X
cana-3296	57	2	ϑl	ϑl	ADP
cana-3296	57	3	∩	∩	NOUN
cana-3296	57	4	ϑp	ϑp	X
cana-3296	57	5	]	]	X
cana-3296	57	6	=	=	SYM
cana-3296	57	7	0̃.	0̃.	NOUN
cana-3296	57	8	also	also	ADV
cana-3296	57	9	,	,	PUNCT
cana-3296	57	10	ϑl	ϑl	DET
cana-3296	57	11	∩	∩	NOUN
cana-3296	57	12	[	[	X
cana-3296	57	13	⋃	⋃	NOUN
cana-3296	57	14	m∈k	m∈k	NOUN
cana-3296	57	15	ϑm	ϑm	ADP
cana-3296	57	16	]	]	X
cana-3296	57	17	=	=	PUNCT
cana-3296	57	18	⋃	⋃	NOUN
cana-3296	57	19	m∈k	m∈k	NOUN
cana-3296	57	20	[	[	X
cana-3296	57	21	ϑl	ϑl	INTJ
cana-3296	57	22	∩	∩	NOUN
cana-3296	57	23	ϑm	ϑm	ADP
cana-3296	57	24	]	]	X
cana-3296	57	25	=	=	SYM
cana-3296	57	26	ϑl	ϑl	NOUN
cana-3296	57	27	.	.	PUNCT
cana-3296	58	1	in	in	ADP
cana-3296	58	2	case	case	NOUN
cana-3296	58	3	[	[	X
cana-3296	58	4	⋃l∈t	⋃l∈t	PROPN
cana-3296	58	5	ϑp	ϑp	X
cana-3296	58	6	]	]	X
cana-3296	58	7	=	=	SYM
cana-3296	59	1	[	[	X
cana-3296	59	2	⋃m∈k	⋃m∈k	NOUN
cana-3296	59	3	ϑm	ϑm	ADP
cana-3296	59	4	]	]	PUNCT
cana-3296	59	5	then	then	ADV
cana-3296	59	6	,	,	PUNCT
cana-3296	59	7	ϑl	ϑl	PROPN
cana-3296	59	8	=	=	PUNCT
cana-3296	59	9	0̃	0̃	PROPN
cana-3296	59	10	,	,	PUNCT
cana-3296	59	11	a	a	DET
cana-3296	59	12	contradiction	contradiction	NOUN
cana-3296	59	13	as	as	ADP
cana-3296	59	14	ϑl	ϑl	PROPN
cana-3296	59	15	is	be	AUX
cana-3296	59	16	a	a	DET
cana-3296	59	17	iivfmio	iivfmio	NOUN
cana-3296	59	18	.	.	PUNCT
cana-3296	60	1	hence	hence	ADV
cana-3296	60	2	proved	prove	VERB
cana-3296	60	3	.	.	PUNCT
cana-3296	61	1	theorem	theorem	VERB
cana-3296	61	2	3.9	3.9	NUM
cana-3296	61	3	if	if	SCONJ
cana-3296	61	4	ϑp	ϑp	NOUN
cana-3296	61	5	is	be	AUX
cana-3296	61	6	a	a	DET
cana-3296	61	7	iivfmio	iivfmio	NOUN
cana-3296	61	8	for	for	ADP
cana-3296	61	9	any	any	DET
cana-3296	61	10	p	p	PROPN
cana-3296	61	11	∈	∈	PROPN
cana-3296	61	12	t	t	NOUN
cana-3296	61	13	such	such	ADJ
cana-3296	61	14	that	that	SCONJ
cana-3296	61	15	ϑp	ϑp	NOUN
cana-3296	61	16	≠	≠	PROPN
cana-3296	61	17	ϑm	ϑm	ADP
cana-3296	61	18	for	for	ADP
cana-3296	61	19	any	any	DET
cana-3296	61	20	p	p	NOUN
cana-3296	61	21	,	,	PUNCT
cana-3296	61	22	m	m	PROPN
cana-3296	61	23	∈	∈	PROPN
cana-3296	61	24	t	t	PROPN
cana-3296	61	25	,	,	PUNCT
cana-3296	61	26	then	then	ADV
cana-3296	61	27	(	(	PUNCT
cana-3296	61	28	i	i	NOUN
cana-3296	61	29	)	)	PUNCT
cana-3296	61	30	ϑm	ϑm	ADP
cana-3296	61	31	⊂	⊂	PROPN
cana-3296	62	1	[	[	X
cana-3296	62	2	⋃l∈t	⋃l∈t	PROPN
cana-3296	62	3	/	/	SYM
cana-3296	62	4	m	m	VERB
cana-3296	62	5	ϑl	ϑl	NOUN
cana-3296	62	6	]	]	X
cana-3296	62	7	c	c	X
cana-3296	62	8	,	,	PUNCT
cana-3296	62	9	for	for	ADP
cana-3296	62	10	some	some	DET
cana-3296	62	11	m	m	NOUN
cana-3296	62	12	∈	∈	PROPN
cana-3296	62	13	k.	k.	PROPN
cana-3296	62	14	(	(	PUNCT
cana-3296	62	15	ii	ii	PROPN
cana-3296	62	16	)	)	PUNCT
cana-3296	62	17	⋃l∈k	⋃l∈k	NOUN
cana-3296	62	18	/	/	SYM
cana-3296	62	19	m	m	NOUN
cana-3296	62	20	ϑm	ϑm	ADP
cana-3296	62	21	≠	≠	PROPN
cana-3296	62	22	1̃	1̃	NUM
cana-3296	62	23	,	,	PUNCT
cana-3296	62	24	∀m	∀m	PROPN
cana-3296	62	25	∈	∈	PROPN
cana-3296	62	26	k.	k.	NOUN
cana-3296	62	27	proof	proof	NOUN
cana-3296	62	28	.	.	PUNCT
cana-3296	63	1	(	(	PUNCT
cana-3296	63	2	i	i	NOUN
cana-3296	63	3	)	)	PUNCT
cana-3296	63	4	given	give	VERB
cana-3296	63	5	that	that	SCONJ
cana-3296	63	6	ϑp	ϑp	NOUN
cana-3296	63	7	≠	≠	PROPN
cana-3296	63	8	ϑm	ϑm	ADP
cana-3296	63	9	for	for	ADP
cana-3296	63	10	any	any	DET
cana-3296	63	11	p	p	NOUN
cana-3296	63	12	,	,	PUNCT
cana-3296	63	13	m	m	PROPN
cana-3296	63	14	∈	∈	PROPN
cana-3296	63	15	t	t	PROPN
cana-3296	63	16	,	,	PUNCT
cana-3296	63	17	=	=	PRON
cana-3296	63	18	>	>	X
cana-3296	63	19	⋃p∈t	⋃p∈t	PROPN
cana-3296	64	1	[	[	X
cana-3296	64	2	ϑp	ϑp	X
cana-3296	64	3	]	]	X
cana-3296	64	4	∩	∩	NOUN
cana-3296	64	5	ϑm	ϑm	ADP
cana-3296	64	6	=	=	SYM
cana-3296	64	7	0̃.	0̃.	NOUN
cana-3296	65	1	=	=	SYM
cana-3296	65	2	>	>	X
cana-3296	65	3	⋃p∈t	⋃p∈t	PROPN
cana-3296	66	1	[	[	X
cana-3296	66	2	ϑp	ϑp	NOUN
cana-3296	66	3	∩	∩	NOUN
cana-3296	66	4	ϑm	ϑm	ADP
cana-3296	66	5	]	]	PUNCT
cana-3296	66	6	=	=	PUNCT
cana-3296	66	7	0̃.	0̃.	NOUN
cana-3296	67	1	=	=	SYM
cana-3296	67	2	>	>	X
cana-3296	68	1	[	[	X
cana-3296	68	2	ϑp	ϑp	NOUN
cana-3296	68	3	∩	∩	NOUN
cana-3296	68	4	ϑm	ϑm	ADP
cana-3296	68	5	]	]	PUNCT
cana-3296	68	6	=	=	PUNCT
cana-3296	68	7	0̃.	0̃.	NOUN
cana-3296	69	1	=	=	SYM
cana-3296	69	2	>	>	X
cana-3296	69	3	ϑp	ϑp	X
cana-3296	69	4	⊂	⊂	PROPN
cana-3296	69	5	ϑm	ϑm	ADP
cana-3296	69	6	c	c	NOUN
cana-3296	69	7	.	.	PUNCT
cana-3296	70	1	=	=	PRON
cana-3296	70	2	>	>	X
cana-3296	70	3	ϑm	ϑm	ADP
cana-3296	70	4	⊂	⊂	PROPN
cana-3296	71	1	[	[	X
cana-3296	71	2	⋃l∈t	⋃l∈t	PROPN
cana-3296	71	3	/	/	SYM
cana-3296	71	4	m	m	VERB
cana-3296	71	5	ϑl	ϑl	NOUN
cana-3296	71	6	]	]	X
cana-3296	71	7	c.	c.	PROPN
cana-3296	71	8	(	(	PUNCT
cana-3296	71	9	ii	ii	PROPN
cana-3296	71	10	)	)	PUNCT
cana-3296	71	11	suppose	suppose	VERB
cana-3296	71	12	that	that	SCONJ
cana-3296	71	13	,	,	PUNCT
cana-3296	71	14	⋃l∈t	⋃l∈t	PROPN
cana-3296	71	15	/	/	SYM
cana-3296	71	16	m	m	PROPN
cana-3296	71	17	ϑl	ϑl	NOUN
cana-3296	71	18	=	=	SYM
cana-3296	71	19	1̃.	1̃.	NUM
cana-3296	71	20	=	=	NOUN
cana-3296	71	21	>	>	X
cana-3296	71	22	ϑl	ϑl	NOUN
cana-3296	71	23	=	=	PROPN
cana-3296	71	24	0̃	0̃	PROPN
cana-3296	71	25	,	,	PUNCT
cana-3296	71	26	a	a	DET
cana-3296	71	27	contradiction	contradiction	NOUN
cana-3296	71	28	for	for	ADP
cana-3296	71	29	minimality	minimality	NOUN
cana-3296	71	30	of	of	ADP
cana-3296	71	31	ϑl	ϑl	PROPN
cana-3296	71	32	.	.	PUNCT
cana-3296	72	1	this	this	PRON
cana-3296	72	2	completes	complete	VERB
cana-3296	72	3	the	the	DET
cana-3296	72	4	proof	proof	NOUN
cana-3296	72	5	.	.	PUNCT
cana-3296	73	1	corollary	corollary	ADJ
cana-3296	73	2	3.10	3.10	NUM
cana-3296	73	3	if	if	SCONJ
cana-3296	73	4	ϑp	ϑp	NOUN
cana-3296	73	5	is	be	AUX
cana-3296	73	6	a	a	DET
cana-3296	73	7	iivfmio	iivfmio	NOUN
cana-3296	73	8	for	for	ADP
cana-3296	73	9	any	any	DET
cana-3296	73	10	p	p	PROPN
cana-3296	73	11	∈	∈	PROPN
cana-3296	73	12	t	t	NOUN
cana-3296	73	13	such	such	ADJ
cana-3296	73	14	that	that	SCONJ
cana-3296	73	15	ϑp	ϑp	NOUN
cana-3296	73	16	≠	≠	PROPN
cana-3296	73	17	ϑm	ϑm	ADP
cana-3296	73	18	for	for	ADP
cana-3296	73	19	any	any	DET
cana-3296	73	20	p	p	NOUN
cana-3296	73	21	,	,	PUNCT
cana-3296	73	22	m	m	PROPN
cana-3296	73	23	∈	∈	PROPN
cana-3296	73	24	t	t	PROPN
cana-3296	74	1	,	,	PUNCT
cana-3296	74	2	then	then	ADV
cana-3296	74	3	ϑp	ϑp	NOUN
cana-3296	74	4	∪	∪	ADJ
cana-3296	74	5	ϑm	ϑm	ADP
cana-3296	74	6	≠	≠	ADJ
cana-3296	74	7	1̃.	1̃.	NUM
cana-3296	74	8	proof	proof	NOUN
cana-3296	74	9	.	.	PUNCT
cana-3296	75	1	similar	similar	ADJ
cana-3296	75	2	to	to	ADP
cana-3296	75	3	the	the	DET
cana-3296	75	4	previous	previous	ADJ
cana-3296	75	5	theorem	theorem	PROPN
cana-3296	75	6	.	.	PUNCT
cana-3296	75	7	theorem	theorem	VERB
cana-3296	75	8	3.11	3.11	NUM
cana-3296	75	9	if	if	SCONJ
cana-3296	75	10	ϑp	ϑp	NOUN
cana-3296	75	11	is	be	AUX
cana-3296	75	12	a	a	DET
cana-3296	75	13	iivfmio	iivfmio	NOUN
cana-3296	75	14	for	for	ADP
cana-3296	75	15	any	any	DET
cana-3296	75	16	p	p	PROPN
cana-3296	75	17	∈	∈	PROPN
cana-3296	75	18	t	t	NOUN
cana-3296	75	19	such	such	ADJ
cana-3296	75	20	that	that	SCONJ
cana-3296	75	21	ϑp	ϑp	NOUN
cana-3296	75	22	≠	≠	PROPN
cana-3296	75	23	ϑm	ϑm	ADP
cana-3296	75	24	for	for	ADP
cana-3296	75	25	any	any	DET
cana-3296	75	26	p	p	NOUN
cana-3296	75	27	,	,	PUNCT
cana-3296	75	28	m	m	PROPN
cana-3296	75	29	∈	∈	PROPN
cana-3296	75	30	t	t	PROPN
cana-3296	75	31	,	,	PUNCT
cana-3296	75	32	then	then	ADV
cana-3296	75	33	ϑm	ϑm	ADV
cana-3296	75	34	=	=	PUNCT
cana-3296	76	1	[	[	X
cana-3296	76	2	⋃	⋃	NOUN
cana-3296	76	3	p∈t	p∈t	NOUN
cana-3296	76	4	ϑp	ϑp	X
cana-3296	76	5	]	]	X
cana-3296	76	6	∩	∩	NOUN
cana-3296	76	7	[	[	X
cana-3296	76	8	⋃	⋃	NOUN
cana-3296	76	9	p∈t	p∈t	NOUN
cana-3296	76	10	/	/	SYM
cana-3296	76	11	m	m	PROPN
cana-3296	76	12	ϑp]c	ϑp]c	PROPN
cana-3296	76	13	,	,	PUNCT
cana-3296	76	14	for	for	ADP
cana-3296	76	15	any	any	DET
cana-3296	76	16	m	m	NOUN
cana-3296	76	17	∈	∈	NOUN
cana-3296	76	18	t.	t.	NOUN
cana-3296	76	19	proof	proof	NOUN
cana-3296	76	20	.	.	PUNCT
cana-3296	77	1	=	=	PUNCT
cana-3296	77	2	>	>	X
cana-3296	78	1	[	[	X
cana-3296	78	2	⋃	⋃	NOUN
cana-3296	78	3	p∈t	p∈t	NOUN
cana-3296	78	4	ϑp	ϑp	X
cana-3296	78	5	]	]	X
cana-3296	78	6	∩	∩	NOUN
cana-3296	78	7	[	[	X
cana-3296	78	8	⋃	⋃	NOUN
cana-3296	78	9	p∈t	p∈t	NOUN
cana-3296	78	10	/	/	SYM
cana-3296	78	11	m	m	PROPN
cana-3296	78	12	ϑp]c	ϑp]c	NOUN
cana-3296	78	13	=	=	PUNCT
cana-3296	79	1	[	[	X
cana-3296	79	2	⋃	⋃	NOUN
cana-3296	79	3	p∈t	p∈t	NOUN
cana-3296	79	4	/	/	SYM
cana-3296	79	5	m	m	VERB
cana-3296	79	6	ϑp	ϑp	NOUN
cana-3296	79	7	∪	∪	NOUN
cana-3296	79	8	ϑm	ϑm	ADP
cana-3296	79	9	]	]	PUNCT
cana-3296	79	10	∩	∩	NOUN
cana-3296	79	11	[	[	X
cana-3296	79	12	⋃	⋃	NOUN
cana-3296	79	13	p∈t	p∈t	NOUN
cana-3296	79	14	/	/	SYM
cana-3296	79	15	m	m	NOUN
cana-3296	79	16	ϑp]c	ϑp]c	PROPN
cana-3296	79	17	.	.	PUNCT
cana-3296	80	1	=	=	PRON
cana-3296	80	2	(	(	PUNCT
cana-3296	80	3	⋃	⋃	NOUN
cana-3296	80	4	p∈t	p∈t	NOUN
cana-3296	80	5	/	/	SYM
cana-3296	80	6	m	m	VERB
cana-3296	80	7	ϑp	ϑp	NOUN
cana-3296	80	8	∩	∩	NOUN
cana-3296	80	9	[	[	X
cana-3296	80	10	⋃	⋃	NOUN
cana-3296	80	11	p∈t	p∈t	NOUN
cana-3296	80	12	/	/	SYM
cana-3296	80	13	m	m	PROPN
cana-3296	80	14	ϑp]c	ϑp]c	PROPN
cana-3296	80	15	)	)	PUNCT
cana-3296	80	16	.	.	PUNCT
cana-3296	81	1	communications	communication	NOUN
cana-3296	81	2	on	on	ADP
cana-3296	81	3	applied	apply	VERB
cana-3296	81	4	nonlinear	nonlinear	ADJ
cana-3296	81	5	analysis	analysis	NOUN
cana-3296	81	6	issn	issn	NOUN
cana-3296	81	7	:	:	PUNCT
cana-3296	81	8	1074	1074	NUM
cana-3296	81	9	-	-	PUNCT
cana-3296	81	10	133x	133x	NUM
cana-3296	81	11	vol	vol	NOUN
cana-3296	81	12	32	32	NUM
cana-3296	81	13	no	no	NOUN
cana-3296	81	14	.	.	PUNCT
cana-3296	82	1	6s	6s	NUM
cana-3296	82	2	(	(	PUNCT
cana-3296	82	3	2025	2025	NUM
cana-3296	82	4	)	)	PUNCT
cana-3296	82	5	298	298	NUM
cana-3296	83	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3296	83	2	=	=	SYM
cana-3296	83	3	∪	∪	X
cana-3296	83	4	(	(	PUNCT
cana-3296	83	5	ϑm	ϑm	ADP
cana-3296	83	6	∩	∩	NOUN
cana-3296	83	7	[	[	X
cana-3296	83	8	⋃	⋃	NOUN
cana-3296	83	9	p∈t	p∈t	NOUN
cana-3296	83	10	/	/	SYM
cana-3296	83	11	m	m	PROPN
cana-3296	83	12	ϑp]c	ϑp]c	PROPN
cana-3296	83	13	)	)	PUNCT
cana-3296	83	14	.	.	PUNCT
cana-3296	84	1	=	=	PUNCT
cana-3296	84	2	0̃	0̃	NOUN
cana-3296	84	3	∪	∪	ADJ
cana-3296	84	4	ϑm=ϑm	ϑm=ϑm	PROPN
cana-3296	84	5	.	.	PROPN
cana-3296	84	6	4	4	NUM
cana-3296	84	7	.	.	PUNCT
cana-3296	84	8	intuitionistic	intuitionistic	ADJ
cana-3296	84	9	interval	interval	NOUN
cana-3296	84	10	valued	value	VERB
cana-3296	84	11	fuzzy	fuzzy	ADJ
cana-3296	84	12	maximal	maximal	ADJ
cana-3296	84	13	open	open	ADJ
cana-3296	84	14	sets	set	NOUN
cana-3296	84	15	definition	definition	NOUN
cana-3296	84	16	4.1	4.1	NUM
cana-3296	84	17	a	a	DET
cana-3296	84	18	proper	proper	ADJ
cana-3296	84	19	iivfo	iivfo	ADJ
cana-3296	84	20	sets	set	NOUN
cana-3296	84	21	ϑ	ϑ	PROPN
cana-3296	84	22	of	of	ADP
cana-3296	84	23	a	a	DET
cana-3296	84	24	iivfts	iivfts	NOUN
cana-3296	84	25	(	(	PUNCT
cana-3296	84	26	x	x	NOUN
cana-3296	84	27	,	,	PUNCT
cana-3296	84	28	ζ	ζ	NOUN
cana-3296	84	29	)	)	PUNCT
cana-3296	84	30	is	be	AUX
cana-3296	84	31	said	say	VERB
cana-3296	84	32	to	to	PART
cana-3296	84	33	be	be	AUX
cana-3296	84	34	iivfmao	iivfmao	ADJ
cana-3296	84	35	set	set	VERB
cana-3296	84	36	if	if	SCONJ
cana-3296	84	37	any	any	DET
cana-3296	84	38	iivfo	iivfo	NOUN
cana-3296	84	39	set	set	NOUN
cana-3296	84	40	containing	contain	VERB
cana-3296	84	41	ϑ	ϑ	PROPN
cana-3296	84	42	is	be	AUX
cana-3296	84	43	either	either	CCONJ
cana-3296	84	44	1̃	1̃	NUM
cana-3296	84	45	or	or	CCONJ
cana-3296	84	46	itself	itself	PRON
cana-3296	84	47	.	.	PUNCT
cana-3296	85	1	example	example	NOUN
cana-3296	85	2	4.2	4.2	NUM
cana-3296	85	3	let	let	VERB
cana-3296	85	4	x	x	PUNCT
cana-3296	85	5	=	=	PRON
cana-3296	85	6	{	{	PUNCT
cana-3296	85	7	l	l	NOUN
cana-3296	85	8	,	,	PUNCT
cana-3296	85	9	m	m	PROPN
cana-3296	85	10	,	,	PUNCT
cana-3296	85	11	n	n	CCONJ
cana-3296	85	12	}	}	PUNCT
cana-3296	85	13	and	and	CCONJ
cana-3296	85	14	(	(	PUNCT
cana-3296	85	15	x	x	NOUN
cana-3296	85	16	,	,	PUNCT
cana-3296	85	17	ζ	ζ	NOUN
cana-3296	85	18	)	)	PUNCT
cana-3296	85	19	is	be	AUX
cana-3296	85	20	iivfts	iivfts	NOUN
cana-3296	85	21	with	with	ADP
cana-3296	85	22	iivfs	iivfs	NOUN
cana-3296	85	23	ϑ	ϑ	AUX
cana-3296	85	24	given	give	VERB
cana-3296	85	25	by	by	ADP
cana-3296	85	26	:	:	PUNCT
cana-3296	85	27	ϑ(l	ϑ(l	NUM
cana-3296	85	28	)	)	PUNCT
cana-3296	85	29	=	=	PUNCT
cana-3296	86	1	[	[	X
cana-3296	86	2	(	(	PUNCT
cana-3296	86	3	0.55,0.45	0.55,0.45	NUM
cana-3296	86	4	)	)	PUNCT
cana-3296	86	5	,	,	PUNCT
cana-3296	86	6	(	(	PUNCT
cana-3296	86	7	0.65,0.25	0.65,0.25	NOUN
cana-3296	86	8	)	)	PUNCT
cana-3296	86	9	]	]	X
cana-3296	86	10	,	,	PUNCT
cana-3296	86	11	ϑ(m	ϑ(m	NOUN
cana-3296	86	12	)	)	PUNCT
cana-3296	86	13	=	=	SYM
cana-3296	87	1	[	[	X
cana-3296	87	2	(	(	PUNCT
cana-3296	87	3	0.45,0.35	0.45,0.35	NOUN
cana-3296	87	4	)	)	PUNCT
cana-3296	87	5	,	,	PUNCT
cana-3296	87	6	(	(	PUNCT
cana-3296	87	7	0.55,0.35	0.55,0.35	NOUN
cana-3296	87	8	)	)	PUNCT
cana-3296	87	9	]	]	PUNCT
cana-3296	87	10	,	,	PUNCT
cana-3296	87	11	ϑ(n	ϑ(n	ADV
cana-3296	87	12	)	)	PUNCT
cana-3296	87	13	=	=	PUNCT
cana-3296	88	1	[	[	X
cana-3296	88	2	(	(	PUNCT
cana-3296	88	3	0.25,0.55	0.25,0.55	NOUN
cana-3296	88	4	)	)	PUNCT
cana-3296	88	5	,	,	PUNCT
cana-3296	88	6	(	(	PUNCT
cana-3296	88	7	0.45,0.35	0.45,0.35	NOUN
cana-3296	88	8	)	)	PUNCT
cana-3296	88	9	]	]	PUNCT
cana-3296	88	10	and	and	CCONJ
cana-3296	88	11	ζ	ζ	NOUN
cana-3296	88	12	=	=	SYM
cana-3296	88	13	{	{	PUNCT
cana-3296	88	14	0̃	0̃	PROPN
cana-3296	88	15	,	,	PUNCT
cana-3296	88	16	ϑ	ϑ	NOUN
cana-3296	88	17	,	,	PUNCT
cana-3296	88	18	1̃	1̃	NUM
cana-3296	88	19	}	}	PUNCT
cana-3296	88	20	then	then	ADV
cana-3296	88	21	ϑ	ϑ	X
cana-3296	88	22	is	be	AUX
cana-3296	88	23	both	both	PRON
cana-3296	88	24	iivfmio	iivfmio	ADJ
cana-3296	88	25	and	and	CCONJ
cana-3296	88	26	iivfmao	iivfmao	NOUN
cana-3296	88	27	in	in	ADP
cana-3296	88	28	(	(	PUNCT
cana-3296	88	29	x	x	NOUN
cana-3296	88	30	,	,	PUNCT
cana-3296	88	31	ζ	ζ	NOUN
cana-3296	88	32	)	)	PUNCT
cana-3296	88	33	.	.	PUNCT
cana-3296	89	1	lemma	lemma	PROPN
cana-3296	89	2	4.2	4.2	NUM
cana-3296	89	3	let	let	VERB
cana-3296	89	4	(	(	PUNCT
cana-3296	89	5	x	x	NOUN
cana-3296	89	6	,	,	PUNCT
cana-3296	89	7	ζ	ζ	NOUN
cana-3296	89	8	)	)	PUNCT
cana-3296	89	9	be	be	VERB
cana-3296	89	10	a	a	DET
cana-3296	89	11	iivfts	iivfts	NOUN
cana-3296	89	12	then	then	ADV
cana-3296	89	13	,	,	PUNCT
cana-3296	89	14	(	(	PUNCT
cana-3296	89	15	i	i	NOUN
cana-3296	89	16	)	)	PUNCT
cana-3296	89	17	if	if	SCONJ
cana-3296	89	18	ϑ1	ϑ1	PROPN
cana-3296	89	19	is	be	AUX
cana-3296	89	20	iivfmao	iivfmao	ADJ
cana-3296	89	21	and	and	CCONJ
cana-3296	89	22	ϑ2	ϑ2	PROPN
cana-3296	89	23	is	be	AUX
cana-3296	89	24	a	a	DET
cana-3296	89	25	iivfo	iivfo	NOUN
cana-3296	89	26	set	set	VERB
cana-3296	89	27	in	in	ADP
cana-3296	89	28	x	x	NOUN
cana-3296	89	29	,	,	PUNCT
cana-3296	89	30	then	then	ADV
cana-3296	89	31	either	either	CCONJ
cana-3296	89	32	ϑ2	ϑ2	PROPN
cana-3296	89	33	⊂	⊂	PROPN
cana-3296	89	34	ϑ1	ϑ1	PROPN
cana-3296	89	35	or	or	CCONJ
cana-3296	89	36	(	(	PUNCT
cana-3296	89	37	ϑ1	ϑ1	PROPN
cana-3296	89	38	∪	∪	ADJ
cana-3296	89	39	ϑ2	ϑ2	NOUN
cana-3296	89	40	)	)	PUNCT
cana-3296	89	41	=	=	SYM
cana-3296	90	1	1̃.	1̃.	NUM
cana-3296	90	2	(	(	PUNCT
cana-3296	90	3	ii	ii	NOUN
cana-3296	90	4	)	)	PUNCT
cana-3296	90	5	if	if	SCONJ
cana-3296	90	6	ϑ1	ϑ1	PROPN
cana-3296	90	7	,	,	PUNCT
cana-3296	90	8	ϑ2	ϑ2	PROPN
cana-3296	90	9	are	be	AUX
cana-3296	90	10	iivfmaos	iivfmaos	NOUN
cana-3296	90	11	in	in	ADP
cana-3296	90	12	x	x	NOUN
cana-3296	90	13	,	,	PUNCT
cana-3296	90	14	then	then	ADV
cana-3296	90	15	eithet	eithet	VERB
cana-3296	90	16	ϑ1	ϑ1	NOUN
cana-3296	90	17	=	=	SYM
cana-3296	90	18	ϑ2	ϑ2	PROPN
cana-3296	90	19	or	or	CCONJ
cana-3296	90	20	(	(	PUNCT
cana-3296	90	21	ϑ1	ϑ1	PROPN
cana-3296	90	22	∪	∪	ADJ
cana-3296	90	23	ϑ2	ϑ2	NOUN
cana-3296	90	24	)	)	PUNCT
cana-3296	90	25	=	=	SYM
cana-3296	90	26	1̃.	1̃.	NUM
cana-3296	90	27	proof	proof	NOUN
cana-3296	90	28	.	.	PUNCT
cana-3296	91	1	(	(	PUNCT
cana-3296	91	2	i	i	NOUN
cana-3296	91	3	)	)	PUNCT
cana-3296	91	4	suppose	suppose	VERB
cana-3296	91	5	that	that	SCONJ
cana-3296	91	6	(	(	PUNCT
cana-3296	91	7	ϑ1	ϑ1	NOUN
cana-3296	91	8	∪	∪	ADJ
cana-3296	91	9	ϑ2	ϑ2	NOUN
cana-3296	91	10	)	)	PUNCT
cana-3296	91	11	≠	≠	PROPN
cana-3296	91	12	1̃	1̃	NUM
cana-3296	91	13	for	for	ADP
cana-3296	91	14	any	any	DET
cana-3296	91	15	iivfmao	iivfmao	NOUN
cana-3296	91	16	set	set	VERB
cana-3296	91	17	ϑ1	ϑ1	NOUN
cana-3296	91	18	and	and	CCONJ
cana-3296	91	19	iivfo	iivfo	ADJ
cana-3296	91	20	ϑ2	ϑ2	PROPN
cana-3296	91	21	,	,	PUNCT
cana-3296	91	22	if	if	SCONJ
cana-3296	91	23	ϑ2ϑ1	ϑ2ϑ1	PRON
cana-3296	91	24	then	then	ADV
cana-3296	91	25	ϑ1	ϑ1	PROPN
cana-3296	91	26	⊂	⊂	PROPN
cana-3296	91	27	(	(	PUNCT
cana-3296	91	28	ϑ1	ϑ1	PROPN
cana-3296	91	29	∪	∪	ADJ
cana-3296	91	30	ϑ2	ϑ2	PROPN
cana-3296	91	31	)	)	PUNCT
cana-3296	91	32	,	,	PUNCT
cana-3296	91	33	a	a	DET
cana-3296	91	34	contradiction	contradiction	NOUN
cana-3296	91	35	.	.	PUNCT
cana-3296	92	1	hence	hence	ADV
cana-3296	92	2	,	,	PUNCT
cana-3296	92	3	ϑ2	ϑ2	PROPN
cana-3296	92	4	⊂	⊂	PROPN
cana-3296	92	5	ϑ1	ϑ1	PROPN
cana-3296	92	6	.	.	PUNCT
cana-3296	93	1	(	(	PUNCT
cana-3296	93	2	ii	ii	NOUN
cana-3296	93	3	)	)	PUNCT
cana-3296	93	4	suppose	suppose	VERB
cana-3296	93	5	that	that	SCONJ
cana-3296	93	6	(	(	PUNCT
cana-3296	93	7	ϑ1	ϑ1	NOUN
cana-3296	93	8	∪	∪	ADJ
cana-3296	93	9	ϑ2	ϑ2	NOUN
cana-3296	93	10	)	)	PUNCT
cana-3296	93	11	≠	≠	PROPN
cana-3296	93	12	1̃	1̃	NUM
cana-3296	93	13	for	for	ADP
cana-3296	93	14	any	any	DET
cana-3296	93	15	iivfmaos	iivfmaos	NOUN
cana-3296	93	16	ϑ1	ϑ1	NOUN
cana-3296	93	17	,	,	PUNCT
cana-3296	93	18	ϑ2	ϑ2	PROPN
cana-3296	93	19	,	,	PUNCT
cana-3296	93	20	then	then	ADV
cana-3296	93	21	ϑ1	ϑ1	NOUN
cana-3296	93	22	⊂	⊂	ADJ
cana-3296	93	23	ϑ2	ϑ2	PROPN
cana-3296	93	24	and	and	CCONJ
cana-3296	93	25	ϑ2	ϑ2	PROPN
cana-3296	93	26	⊂	⊂	PROPN
cana-3296	93	27	ϑ1	ϑ1	PROPN
cana-3296	93	28	then	then	ADV
cana-3296	93	29	ϑ1	ϑ1	NOUN
cana-3296	93	30	=	=	PUNCT
cana-3296	93	31	ϑ2	ϑ2	PROPN
cana-3296	93	32	.	.	PUNCT
cana-3296	93	33	theorem	theorem	VERB
cana-3296	93	34	4.3	4.3	NUM
cana-3296	93	35	if	if	SCONJ
cana-3296	93	36	ϑl	ϑl	NOUN
cana-3296	93	37	,	,	PUNCT
cana-3296	93	38	ϑm	ϑm	ADV
cana-3296	93	39	,	,	PUNCT
cana-3296	93	40	ϑp	ϑp	NOUN
cana-3296	93	41	are	be	AUX
cana-3296	93	42	iivfmao	iivfmao	ADJ
cana-3296	93	43	sets	set	NOUN
cana-3296	93	44	such	such	ADJ
cana-3296	93	45	that	that	SCONJ
cana-3296	93	46	ϑl	ϑl	PROPN
cana-3296	93	47	≠	≠	PROPN
cana-3296	93	48	ϑm	ϑm	ADV
cana-3296	93	49	and	and	CCONJ
cana-3296	93	50	(	(	PUNCT
cana-3296	93	51	ϑl	ϑl	INTJ
cana-3296	93	52	∩	∩	NOUN
cana-3296	93	53	ϑm	ϑm	ADP
cana-3296	93	54	)	)	PUNCT
cana-3296	93	55	⊂	⊂	PROPN
cana-3296	93	56	ϑp	ϑp	NOUN
cana-3296	93	57	,	,	PUNCT
cana-3296	93	58	then	then	ADV
cana-3296	93	59	either	either	CCONJ
cana-3296	93	60	ϑl	ϑl	X
cana-3296	93	61	=	=	NOUN
cana-3296	93	62	ϑp	ϑp	NOUN
cana-3296	93	63	or	or	CCONJ
cana-3296	93	64	ϑm	ϑm	ADP
cana-3296	93	65	=	=	PUNCT
cana-3296	93	66	ϑp	ϑp	NOUN
cana-3296	93	67	.	.	NOUN
cana-3296	93	68	proof	proof	NOUN
cana-3296	93	69	.	.	PUNCT
cana-3296	94	1	suppose	suppose	VERB
cana-3296	94	2	that	that	SCONJ
cana-3296	94	3	(	(	PUNCT
cana-3296	94	4	ϑl	ϑl	INTJ
cana-3296	94	5	∩	∩	NOUN
cana-3296	94	6	ϑm	ϑm	ADP
cana-3296	94	7	)	)	PUNCT
cana-3296	94	8	⊂	⊂	PROPN
cana-3296	94	9	ϑp	ϑp	NOUN
cana-3296	94	10	and	and	CCONJ
cana-3296	94	11	ϑl	ϑl	PROPN
cana-3296	94	12	≠	≠	PROPN
cana-3296	94	13	ϑp	ϑp	NOUN
cana-3296	94	14	then	then	ADV
cana-3296	94	15	,	,	PUNCT
cana-3296	94	16	(	(	PUNCT
cana-3296	94	17	ϑm	ϑm	ADP
cana-3296	94	18	∩	∩	ADJ
cana-3296	94	19	ϑp	ϑp	NOUN
cana-3296	94	20	)	)	PUNCT
cana-3296	94	21	=	=	SYM
cana-3296	94	22	(	(	PUNCT
cana-3296	94	23	ϑm	ϑm	NOUN
cana-3296	94	24	)	)	PUNCT
cana-3296	94	25	∩	∩	NOUN
cana-3296	94	26	(	(	PUNCT
cana-3296	94	27	ϑp	ϑp	NOUN
cana-3296	94	28	∩	∩	ADJ
cana-3296	94	29	1̃	1̃	NUM
cana-3296	94	30	)	)	PUNCT
cana-3296	94	31	=(	=(	PROPN
cana-3296	94	32	ϑm	ϑm	NOUN
cana-3296	94	33	)	)	PUNCT
cana-3296	94	34	∩	∩	NOUN
cana-3296	94	35	[	[	X
cana-3296	94	36	(	(	PUNCT
cana-3296	94	37	ϑp	ϑp	NOUN
cana-3296	94	38	)	)	PUNCT
cana-3296	94	39	∩	∩	NOUN
cana-3296	94	40	(	(	PUNCT
cana-3296	94	41	ϑl	ϑl	INTJ
cana-3296	94	42	∪	∪	NOUN
cana-3296	94	43	ϑm	ϑm	NOUN
cana-3296	94	44	)	)	PUNCT
cana-3296	94	45	]	]	PUNCT
cana-3296	95	1	=	=	X
cana-3296	95	2	(	(	PUNCT
cana-3296	95	3	ϑm	ϑm	NOUN
cana-3296	95	4	)	)	PUNCT
cana-3296	95	5	∩	∩	NOUN
cana-3296	95	6	[	[	X
cana-3296	95	7	(	(	PUNCT
cana-3296	95	8	ϑp	ϑp	NOUN
cana-3296	95	9	)	)	PUNCT
cana-3296	95	10	∩	∩	ADJ
cana-3296	95	11	ϑl	ϑl	NOUN
cana-3296	95	12	)	)	PUNCT
cana-3296	95	13	∪	∪	NOUN
cana-3296	95	14	(	(	PUNCT
cana-3296	95	15	ϑp	ϑp	NOUN
cana-3296	95	16	)	)	PUNCT
cana-3296	95	17	∩	∩	NOUN
cana-3296	95	18	ϑm	ϑm	ADP
cana-3296	95	19	)	)	PUNCT
cana-3296	95	20	]	]	PUNCT
cana-3296	96	1	=	=	X
cana-3296	96	2	(	(	PUNCT
cana-3296	96	3	ϑm	ϑm	ADP
cana-3296	96	4	∩	∩	ADJ
cana-3296	96	5	ϑp	ϑp	ADP
cana-3296	96	6	∩	∩	ADJ
cana-3296	96	7	ϑl	ϑl	NOUN
cana-3296	96	8	)	)	PUNCT
cana-3296	96	9	∪	∪	NOUN
cana-3296	96	10	(	(	PUNCT
cana-3296	96	11	ϑm	ϑm	ADP
cana-3296	96	12	∩	∩	ADJ
cana-3296	96	13	ϑp	ϑp	NOUN
cana-3296	96	14	∩	∩	NOUN
cana-3296	96	15	ϑm	ϑm	ADP
cana-3296	96	16	)	)	PUNCT
cana-3296	96	17	]	]	PUNCT
cana-3296	97	1	=	=	X
cana-3296	97	2	(	(	PUNCT
cana-3296	97	3	ϑm	ϑm	ADP
cana-3296	97	4	∩	∩	ADJ
cana-3296	97	5	ϑl	ϑl	NOUN
cana-3296	97	6	)	)	PUNCT
cana-3296	97	7	∪	∪	NOUN
cana-3296	97	8	(	(	PUNCT
cana-3296	97	9	ϑm	ϑm	ADP
cana-3296	97	10	∩	∩	ADJ
cana-3296	97	11	ϑp	ϑp	NOUN
cana-3296	97	12	)	)	PUNCT
cana-3296	97	13	=	=	NOUN
cana-3296	97	14	ϑm	ϑm	ADP
cana-3296	97	15	∩	∩	NOUN
cana-3296	97	16	(	(	PUNCT
cana-3296	97	17	ϑl	ϑl	ADP
cana-3296	97	18	∪	∪	ADJ
cana-3296	97	19	ϑp	ϑp	NOUN
cana-3296	97	20	)	)	PUNCT
cana-3296	97	21	=	=	NOUN
cana-3296	97	22	ϑm	ϑm	ADP
cana-3296	97	23	∩	∩	ADJ
cana-3296	97	24	1̃	1̃	NUM
cana-3296	97	25	=	=	SYM
cana-3296	97	26	ϑm	ϑm	NOUN
cana-3296	97	27	.	.	PUNCT
cana-3296	98	1	(	(	PUNCT
cana-3296	98	2	ϑm	ϑm	ADP
cana-3296	98	3	∩	∩	ADJ
cana-3296	98	4	ϑ3	ϑ3	NOUN
cana-3296	98	5	)	)	PUNCT
cana-3296	98	6	=	=	PUNCT
cana-3296	99	1	ϑm	ϑm	ADP
cana-3296	99	2	=	=	NOUN
cana-3296	99	3	>	>	X
cana-3296	99	4	ϑm	ϑm	ADP
cana-3296	99	5	⊂	⊂	PROPN
cana-3296	99	6	ϑp	ϑp	NOUN
cana-3296	99	7	.	.	NOUN
cana-3296	100	1	as	as	ADP
cana-3296	100	2	ϑm	ϑm	ADV
cana-3296	100	3	is	be	AUX
cana-3296	100	4	a	a	DET
cana-3296	100	5	iivfmao	iivfmao	NOUN
cana-3296	100	6	set	set	NOUN
cana-3296	100	7	ϑp	ϑp	ADP
cana-3296	100	8	⊂	⊂	PROPN
cana-3296	100	9	ϑm	ϑm	ADP
cana-3296	100	10	this	this	PRON
cana-3296	100	11	implies	imply	VERB
cana-3296	100	12	ϑm	ϑm	ADP
cana-3296	100	13	=	=	SYM
cana-3296	100	14	ϑp	ϑp	PROPN
cana-3296	100	15	.	.	X
cana-3296	100	16	theorem	theorem	VERB
cana-3296	100	17	4.4	4.4	NUM
cana-3296	101	1	[	[	X
cana-3296	101	2	ϑ1	ϑ1	NOUN
cana-3296	101	3	∩	∩	ADJ
cana-3296	101	4	ϑ2][ϑ1	ϑ2][ϑ1	NOUN
cana-3296	101	5	∩	∩	ADJ
cana-3296	101	6	ϑ3	ϑ3	PROPN
cana-3296	101	7	]	]	PUNCT
cana-3296	101	8	for	for	ADP
cana-3296	101	9	any	any	DET
cana-3296	101	10	distinct	distinct	ADJ
cana-3296	101	11	iivfmao	iivfmao	ADJ
cana-3296	101	12	sets	set	NOUN
cana-3296	101	13	ϑ1	ϑ1	PROPN
cana-3296	101	14	,	,	PUNCT
cana-3296	101	15	ϑ2	ϑ2	PROPN
cana-3296	101	16	,	,	PUNCT
cana-3296	101	17	ϑ3	ϑ3	NOUN
cana-3296	101	18	.	.	PUNCT
cana-3296	102	1	proof	proof	NOUN
cana-3296	102	2	.	.	PUNCT
cana-3296	103	1	let	let	VERB
cana-3296	103	2	us	we	PRON
cana-3296	103	3	assume	assume	VERB
cana-3296	103	4	the	the	DET
cana-3296	103	5	contrary	contrary	NOUN
cana-3296	103	6	,	,	PUNCT
cana-3296	103	7	[	[	X
cana-3296	103	8	ϑ1	ϑ1	NOUN
cana-3296	103	9	∩	∩	ADJ
cana-3296	103	10	ϑ2	ϑ2	NOUN
cana-3296	103	11	]	]	PUNCT
cana-3296	103	12	⊂	⊂	PROPN
cana-3296	104	1	[	[	X
cana-3296	104	2	ϑ1	ϑ1	PROPN
cana-3296	104	3	∩	∩	ADJ
cana-3296	104	4	ϑ3	ϑ3	NOUN
cana-3296	104	5	]	]	PUNCT
cana-3296	104	6	for	for	ADP
cana-3296	104	7	any	any	DET
cana-3296	104	8	distinct	distinct	ADJ
cana-3296	104	9	iivfmao	iivfmao	NOUN
cana-3296	104	10	sets	set	NOUN
cana-3296	104	11	then	then	ADV
cana-3296	104	12	,	,	PUNCT
cana-3296	104	13	[	[	X
cana-3296	104	14	ϑ1	ϑ1	NOUN
cana-3296	104	15	∩	∩	ADJ
cana-3296	104	16	ϑ2	ϑ2	NOUN
cana-3296	104	17	]	]	PUNCT
cana-3296	104	18	∪	∪	ADP
cana-3296	104	19	[	[	PUNCT
cana-3296	104	20	ϑ2	ϑ2	PROPN
cana-3296	104	21	∩	∩	ADJ
cana-3296	104	22	ϑ3	ϑ3	NOUN
cana-3296	104	23	]	]	PUNCT
cana-3296	104	24	⊂	⊂	PROPN
cana-3296	105	1	[	[	X
cana-3296	105	2	ϑ1	ϑ1	PROPN
cana-3296	105	3	∩	∩	ADJ
cana-3296	105	4	ϑ3	ϑ3	NOUN
cana-3296	105	5	]	]	PUNCT
cana-3296	105	6	∪	∪	ADP
cana-3296	105	7	[	[	PUNCT
cana-3296	105	8	ϑ2	ϑ2	PROPN
cana-3296	105	9	∩	∩	ADJ
cana-3296	105	10	ϑ3	ϑ3	NOUN
cana-3296	105	11	]	]	PUNCT
cana-3296	105	12	.	.	PUNCT
cana-3296	106	1	=	=	PUNCT
cana-3296	106	2	>	>	X
cana-3296	107	1	[	[	X
cana-3296	107	2	ϑ1	ϑ1	PROPN
cana-3296	107	3	∪	∪	PROPN
cana-3296	107	4	ϑ3	ϑ3	NOUN
cana-3296	107	5	]	]	PUNCT
cana-3296	107	6	∩	∩	ADJ
cana-3296	107	7	ϑ2	ϑ2	PROPN
cana-3296	107	8	⊂	⊂	PROPN
cana-3296	108	1	[	[	X
cana-3296	108	2	ϑ1	ϑ1	PROPN
cana-3296	108	3	∪	∪	ADJ
cana-3296	108	4	ϑ2	ϑ2	NOUN
cana-3296	108	5	]	]	PUNCT
cana-3296	108	6	∩	∩	PROPN
cana-3296	108	7	ϑ3	ϑ3	NOUN
cana-3296	108	8	=	=	PROPN
cana-3296	108	9	>	>	X
cana-3296	108	10	1̃	1̃	NUM
cana-3296	108	11	∩	∩	ADJ
cana-3296	108	12	ϑ2	ϑ2	PROPN
cana-3296	108	13	⊂	⊂	PROPN
cana-3296	108	14	1̃	1̃	NUM
cana-3296	108	15	∩	∩	PROPN
cana-3296	108	16	ϑ3	ϑ3	NOUN
cana-3296	108	17	=	=	PROPN
cana-3296	108	18	>	>	X
cana-3296	108	19	ϑ2	ϑ2	PROPN
cana-3296	108	20	⊂	⊂	PROPN
cana-3296	108	21	ϑ3	ϑ3	PROPN
cana-3296	108	22	,	,	PUNCT
cana-3296	108	23	a	a	DET
cana-3296	108	24	contradiction	contradiction	NOUN
cana-3296	108	25	.	.	PUNCT
cana-3296	109	1	communications	communication	NOUN
cana-3296	109	2	on	on	ADP
cana-3296	109	3	applied	apply	VERB
cana-3296	109	4	nonlinear	nonlinear	ADJ
cana-3296	109	5	analysis	analysis	NOUN
cana-3296	109	6	issn	issn	NOUN
cana-3296	109	7	:	:	PUNCT
cana-3296	109	8	1074	1074	NUM
cana-3296	109	9	-	-	PUNCT
cana-3296	109	10	133x	133x	NUM
cana-3296	109	11	vol	vol	NOUN
cana-3296	109	12	32	32	NUM
cana-3296	109	13	no	no	NOUN
cana-3296	109	14	.	.	PUNCT
cana-3296	110	1	6s	6s	NUM
cana-3296	110	2	(	(	PUNCT
cana-3296	110	3	2025	2025	NUM
cana-3296	110	4	)	)	PUNCT
cana-3296	110	5	299	299	NUM
cana-3296	110	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3296	110	7	remark	remark	NOUN
cana-3296	110	8	4.5	4.5	NUM
cana-3296	110	9	these	these	DET
cana-3296	110	10	proofs	proof	NOUN
cana-3296	110	11	are	be	AUX
cana-3296	110	12	left	leave	VERB
cana-3296	110	13	out	out	ADP
cana-3296	110	14	because	because	SCONJ
cana-3296	110	15	theorem	theorem	ADJ
cana-3296	110	16	4.5	4.5	NUM
cana-3296	110	17	,	,	PUNCT
cana-3296	110	18	corollary	corollary	ADJ
cana-3296	110	19	4.6	4.6	NUM
cana-3296	110	20	,	,	PUNCT
cana-3296	110	21	theorem	theorem	VERB
cana-3296	110	22	4.7	4.7	NUM
cana-3296	110	23	,	,	PUNCT
cana-3296	110	24	and	and	CCONJ
cana-3296	110	25	theorem	theorem	VERB
cana-3296	110	26	4.8	4.8	NUM
cana-3296	110	27	are	be	AUX
cana-3296	110	28	comparable	comparable	ADJ
cana-3296	110	29	to	to	ADP
cana-3296	110	30	theorem	theorem	VERB
cana-3296	110	31	3.9	3.9	NUM
cana-3296	110	32	,	,	PUNCT
cana-3296	110	33	corollary	corollary	ADJ
cana-3296	110	34	3.10	3.10	NUM
cana-3296	110	35	,	,	PUNCT
cana-3296	110	36	theorem	theorem	VERB
cana-3296	110	37	3.11	3.11	NUM
cana-3296	110	38	,	,	PUNCT
cana-3296	110	39	and	and	CCONJ
cana-3296	110	40	theorem	theorem	VERB
cana-3296	110	41	3.8	3.8	NUM
cana-3296	110	42	.	.	PUNCT
cana-3296	111	1	the	the	DET
cana-3296	111	2	4.6	4.6	NUM
cana-3296	111	3	theorem	theorem	NOUN
cana-3296	111	4	.	.	PUNCT
cana-3296	111	5	theorem	theorem	VERB
cana-3296	111	6	4.5	4.5	NUM
cana-3296	111	7	if	if	SCONJ
cana-3296	111	8	ϑp	ϑp	NOUN
cana-3296	111	9	is	be	AUX
cana-3296	111	10	a	a	DET
cana-3296	111	11	iivfmao	iivfmao	NOUN
cana-3296	111	12	for	for	ADP
cana-3296	111	13	any	any	DET
cana-3296	111	14	p	p	PROPN
cana-3296	111	15	∈	∈	PROPN
cana-3296	111	16	k	k	NOUN
cana-3296	111	17	,	,	PUNCT
cana-3296	111	18	a	a	DET
cana-3296	111	19	finite	finite	NOUN
cana-3296	111	20	set	set	NOUN
cana-3296	111	21	and	and	CCONJ
cana-3296	111	22	ϑp	ϑp	NOUN
cana-3296	111	23	≠	≠	PROPN
cana-3296	111	24	ϑm	ϑm	ADP
cana-3296	111	25	for	for	ADP
cana-3296	111	26	any	any	DET
cana-3296	111	27	distinct	distinct	ADJ
cana-3296	111	28	m	m	NOUN
cana-3296	111	29	,	,	PUNCT
cana-3296	111	30	p	p	PROPN
cana-3296	111	31	∈	∈	PROPN
cana-3296	112	1	k	k	NOUN
cana-3296	112	2	then	then	ADV
cana-3296	112	3	(	(	PUNCT
cana-3296	112	4	i	i	NOUN
cana-3296	112	5	)	)	PUNCT
cana-3296	113	1	[	[	X
cana-3296	113	2	⋂k	⋂k	X
cana-3296	113	3	/	/	SYM
cana-3296	113	4	m	m	VERB
cana-3296	113	5	ϑp]c	ϑp]c	PROPN
cana-3296	113	6	⊂	⊂	PROPN
cana-3296	113	7	ϑm	ϑm	ADP
cana-3296	113	8	for	for	ADP
cana-3296	113	9	any	any	DET
cana-3296	113	10	m	m	PROPN
cana-3296	113	11	∈	∈	PROPN
cana-3296	113	12	k.	k.	PROPN
cana-3296	113	13	(	(	PUNCT
cana-3296	113	14	ii	ii	PROPN
cana-3296	113	15	)	)	PUNCT
cana-3296	114	1	[	[	X
cana-3296	114	2	⋂k	⋂k	X
cana-3296	114	3	/	/	SYM
cana-3296	114	4	m	m	VERB
cana-3296	114	5	ϑp	ϑp	X
cana-3296	114	6	]	]	PUNCT
cana-3296	114	7	≠	≠	PROPN
cana-3296	114	8	0̃	0̃	NOUN
cana-3296	114	9	for	for	ADP
cana-3296	114	10	any	any	DET
cana-3296	114	11	m	m	PROPN
cana-3296	114	12	∈	∈	PROPN
cana-3296	114	13	k.	k.	NOUN
cana-3296	114	14	corollary	corollary	NOUN
cana-3296	114	15	4.6	4.6	NUM
cana-3296	114	16	if	if	SCONJ
cana-3296	114	17	ϑp	ϑp	NOUN
cana-3296	114	18	is	be	AUX
cana-3296	114	19	a	a	DET
cana-3296	114	20	iivfmao	iivfmao	NOUN
cana-3296	114	21	for	for	ADP
cana-3296	114	22	any	any	DET
cana-3296	114	23	p	p	PROPN
cana-3296	114	24	∈	∈	PROPN
cana-3296	114	25	k	k	NOUN
cana-3296	114	26	,	,	PUNCT
cana-3296	114	27	a	a	DET
cana-3296	114	28	finite	finite	NOUN
cana-3296	114	29	set	set	NOUN
cana-3296	114	30	and	and	CCONJ
cana-3296	114	31	ϑp	ϑp	NOUN
cana-3296	114	32	≠	≠	PROPN
cana-3296	114	33	ϑm	ϑm	ADP
cana-3296	114	34	for	for	ADP
cana-3296	114	35	any	any	DET
cana-3296	114	36	distinct	distinct	ADJ
cana-3296	114	37	m	m	NOUN
cana-3296	114	38	,	,	PUNCT
cana-3296	114	39	p	p	PROPN
cana-3296	114	40	∈	∈	PROPN
cana-3296	115	1	k	k	X
cana-3296	115	2	then	then	ADV
cana-3296	115	3	[	[	X
cana-3296	115	4	ϑp	ϑp	NOUN
cana-3296	115	5	∩	∩	ADJ
cana-3296	115	6	ϑm	ϑm	ADP
cana-3296	115	7	]	]	PUNCT
cana-3296	115	8	≠	≠	PROPN
cana-3296	115	9	0̃.	0̃.	NOUN
cana-3296	115	10	theorem	theorem	VERB
cana-3296	115	11	4.7	4.7	NUM
cana-3296	115	12	if	if	SCONJ
cana-3296	115	13	ϑp	ϑp	NOUN
cana-3296	115	14	is	be	AUX
cana-3296	115	15	a	a	DET
cana-3296	115	16	iivfmao	iivfmao	NOUN
cana-3296	115	17	for	for	ADP
cana-3296	115	18	any	any	DET
cana-3296	115	19	p	p	PROPN
cana-3296	115	20	∈	∈	PROPN
cana-3296	115	21	k	k	NOUN
cana-3296	115	22	,	,	PUNCT
cana-3296	115	23	a	a	DET
cana-3296	115	24	finite	finite	NOUN
cana-3296	115	25	set	set	NOUN
cana-3296	115	26	and	and	CCONJ
cana-3296	115	27	ϑp	ϑp	NOUN
cana-3296	115	28	≠	≠	PROPN
cana-3296	115	29	ϑm	ϑm	ADP
cana-3296	115	30	for	for	ADP
cana-3296	115	31	any	any	DET
cana-3296	115	32	distinct	distinct	ADJ
cana-3296	115	33	m	m	NOUN
cana-3296	115	34	,	,	PUNCT
cana-3296	115	35	p	p	PROPN
cana-3296	115	36	∈	∈	PROPN
cana-3296	115	37	k	k	NOUN
cana-3296	115	38	,	,	PUNCT
cana-3296	115	39	then	then	ADV
cana-3296	115	40	ϑm	ϑm	ADV
cana-3296	115	41	=	=	PUNCT
cana-3296	116	1	[	[	X
cana-3296	116	2	⋂p∈k	⋂p∈k	X
cana-3296	116	3	ϑp	ϑp	X
cana-3296	116	4	]	]	X
cana-3296	116	5	∪	∪	ADP
cana-3296	116	6	[	[	X
cana-3296	116	7	⋂p∈k	⋂p∈k	NUM
cana-3296	116	8	/	/	SYM
cana-3296	116	9	m	m	NOUN
cana-3296	116	10	ϑp]c	ϑp]c	NOUN
cana-3296	116	11	for	for	ADP
cana-3296	116	12	any	any	DET
cana-3296	116	13	m	m	PROPN
cana-3296	116	14	∈	∈	PROPN
cana-3296	116	15	k.	k.	NOUN
cana-3296	116	16	theorem	theorem	VERB
cana-3296	116	17	4.8	4.8	NUM
cana-3296	116	18	if	if	SCONJ
cana-3296	116	19	ϑp	ϑp	NOUN
cana-3296	116	20	is	be	AUX
cana-3296	116	21	a	a	DET
cana-3296	116	22	iivfmao	iivfmao	NOUN
cana-3296	116	23	for	for	ADP
cana-3296	116	24	any	any	DET
cana-3296	116	25	p	p	PROPN
cana-3296	116	26	∈	∈	PROPN
cana-3296	116	27	k	k	NOUN
cana-3296	116	28	,	,	PUNCT
cana-3296	116	29	a	a	DET
cana-3296	116	30	finite	finite	NOUN
cana-3296	116	31	set	set	NOUN
cana-3296	116	32	and	and	CCONJ
cana-3296	116	33	ϑp	ϑp	NOUN
cana-3296	116	34	≠	≠	PROPN
cana-3296	116	35	ϑm	ϑm	ADP
cana-3296	116	36	for	for	ADP
cana-3296	116	37	any	any	DET
cana-3296	116	38	distinct	distinct	ADJ
cana-3296	116	39	m	m	NOUN
cana-3296	116	40	,	,	PUNCT
cana-3296	116	41	p	p	PROPN
cana-3296	116	42	∈	∈	PROPN
cana-3296	116	43	k	k	NOUN
cana-3296	116	44	,	,	PUNCT
cana-3296	116	45	and	and	CCONJ
cana-3296	116	46	if	if	SCONJ
cana-3296	116	47	t	t	PROPN
cana-3296	116	48	is	be	AUX
cana-3296	116	49	a	a	DET
cana-3296	116	50	proper	proper	ADJ
cana-3296	116	51	nonempty	nonempty	NOUN
cana-3296	116	52	subset	subset	NOUN
cana-3296	116	53	of	of	ADP
cana-3296	116	54	k	k	NOUN
cana-3296	116	55	,	,	PUNCT
cana-3296	116	56	then	then	ADV
cana-3296	116	57	⋂p∈k	⋂p∈k	ADV
cana-3296	116	58	ϑp	ϑp	PROPN
cana-3296	116	59	⊂	⊂	PROPN
cana-3296	116	60	⋂t∈t	⋂t∈t	PROPN
cana-3296	116	61	ϑt	ϑt	PROPN
cana-3296	116	62	.	.	PUNCT
cana-3296	117	1	theorem	theorem	VERB
cana-3296	117	2	4.9	4.9	NUM
cana-3296	117	3	if	if	SCONJ
cana-3296	117	4	ϑp	ϑp	NOUN
cana-3296	117	5	is	be	AUX
cana-3296	117	6	a	a	DET
cana-3296	117	7	iivfmao	iivfmao	NOUN
cana-3296	117	8	for	for	ADP
cana-3296	117	9	any	any	DET
cana-3296	117	10	p	p	PROPN
cana-3296	117	11	∈	∈	PROPN
cana-3296	117	12	k	k	NOUN
cana-3296	117	13	,	,	PUNCT
cana-3296	117	14	a	a	DET
cana-3296	117	15	finite	finite	NOUN
cana-3296	117	16	set	set	NOUN
cana-3296	117	17	and	and	CCONJ
cana-3296	117	18	ϑp	ϑp	NOUN
cana-3296	117	19	≠	≠	PROPN
cana-3296	117	20	ϑm	ϑm	ADP
cana-3296	117	21	for	for	ADP
cana-3296	117	22	any	any	DET
cana-3296	117	23	distinct	distinct	ADJ
cana-3296	117	24	m	m	NOUN
cana-3296	117	25	,	,	PUNCT
cana-3296	117	26	p	p	PROPN
cana-3296	117	27	∈	∈	PROPN
cana-3296	117	28	k	k	NOUN
cana-3296	117	29	and	and	CCONJ
cana-3296	117	30	if	if	SCONJ
cana-3296	117	31	⋂p∈k	⋂p∈k	NUM
cana-3296	117	32	ϑp	ϑp	NOUN
cana-3296	117	33	is	be	AUX
cana-3296	117	34	a	a	DET
cana-3296	117	35	iivfc	iivfc	NOUN
cana-3296	117	36	set	set	NOUN
cana-3296	117	37	,	,	PUNCT
cana-3296	117	38	then	then	ADV
cana-3296	117	39	ϑm	ϑm	ADV
cana-3296	117	40	is	be	AUX
cana-3296	117	41	a	a	DET
cana-3296	117	42	iivfc	iivfc	NOUN
cana-3296	117	43	set	set	VERB
cana-3296	117	44	for	for	ADP
cana-3296	117	45	any	any	DET
cana-3296	117	46	j	j	PROPN
cana-3296	117	47	∈	∈	PROPN
cana-3296	117	48	k.	k.	PROPN
cana-3296	117	49	proof	proof	NOUN
cana-3296	117	50	.	.	PUNCT
cana-3296	118	1	by	by	ADP
cana-3296	118	2	theorem-4.7	theorem-4.7	NOUN
cana-3296	118	3	,	,	PUNCT
cana-3296	118	4	we	we	PRON
cana-3296	118	5	have	have	VERB
cana-3296	118	6	ϑm	ϑm	ADP
cana-3296	118	7	=	=	PUNCT
cana-3296	119	1	[	[	X
cana-3296	119	2	⋂p∈k	⋂p∈k	X
cana-3296	119	3	ϑp	ϑp	X
cana-3296	119	4	]	]	X
cana-3296	119	5	∪	∪	ADP
cana-3296	119	6	[	[	X
cana-3296	119	7	⋂p∈k	⋂p∈k	NUM
cana-3296	119	8	/	/	SYM
cana-3296	119	9	m	m	NOUN
cana-3296	119	10	ϑp]c	ϑp]c	NOUN
cana-3296	119	11	for	for	ADP
cana-3296	119	12	any	any	DET
cana-3296	119	13	m	m	NOUN
cana-3296	119	14	∈	∈	NOUN
cana-3296	119	15	k	k	NOUN
cana-3296	120	1	=	=	X
cana-3296	120	2	>	>	X
cana-3296	120	3	ϑm	ϑm	ADP
cana-3296	120	4	=	=	X
cana-3296	121	1	[	[	X
cana-3296	121	2	⋂p∈k	⋂p∈k	X
cana-3296	121	3	ϑp	ϑp	X
cana-3296	121	4	]	]	X
cana-3296	121	5	∪	∪	ADP
cana-3296	121	6	[	[	X
cana-3296	121	7	⋂p∈k	⋂p∈k	NUM
cana-3296	121	8	/	/	SYM
cana-3296	121	9	m	m	PROPN
cana-3296	121	10	(	(	PUNCT
cana-3296	121	11	ϑp)c	ϑp)c	PROPN
cana-3296	121	12	]	]	PUNCT
cana-3296	121	13	.	.	PUNCT
cana-3296	122	1	as	as	SCONJ
cana-3296	122	2	k	k	PROPN
cana-3296	122	3	is	be	AUX
cana-3296	122	4	finite	finite	ADJ
cana-3296	122	5	,	,	PUNCT
cana-3296	122	6	[	[	X
cana-3296	122	7	⋂p∈k	⋂p∈k	X
cana-3296	122	8	/	/	SYM
cana-3296	122	9	m	m	PROPN
cana-3296	122	10	(	(	PUNCT
cana-3296	122	11	ϑp)c	ϑp)c	PROPN
cana-3296	122	12	]	]	X
cana-3296	122	13	is	be	AUX
cana-3296	122	14	iivfc	iivfc	NOUN
cana-3296	122	15	.	.	PUNCT
cana-3296	123	1	hence	hence	ADV
cana-3296	123	2	,	,	PUNCT
cana-3296	123	3	ϑm	ϑm	ADP
cana-3296	123	4	is	be	AUX
cana-3296	123	5	iivfc	iivfc	NOUN
cana-3296	123	6	.	.	PUNCT
cana-3296	124	1	theorem	theorem	VERB
cana-3296	124	2	4.10	4.10	NUM
cana-3296	124	3	if	if	SCONJ
cana-3296	124	4	ϑp	ϑp	NOUN
cana-3296	124	5	is	be	AUX
cana-3296	124	6	a	a	DET
cana-3296	124	7	iivfmao	iivfmao	NOUN
cana-3296	124	8	for	for	ADP
cana-3296	124	9	any	any	DET
cana-3296	124	10	p	p	PROPN
cana-3296	124	11	∈	∈	PROPN
cana-3296	124	12	k	k	NOUN
cana-3296	124	13	,	,	PUNCT
cana-3296	124	14	a	a	DET
cana-3296	124	15	finite	finite	NOUN
cana-3296	124	16	set	set	NOUN
cana-3296	124	17	and	and	CCONJ
cana-3296	124	18	ϑp	ϑp	NOUN
cana-3296	124	19	≠	≠	PROPN
cana-3296	124	20	ϑm	ϑm	ADP
cana-3296	124	21	for	for	ADP
cana-3296	124	22	any	any	DET
cana-3296	124	23	distinct	distinct	ADJ
cana-3296	124	24	m	m	NOUN
cana-3296	124	25	,	,	PUNCT
cana-3296	124	26	p	p	PROPN
cana-3296	124	27	∈	∈	PROPN
cana-3296	124	28	k.	k.	PROPN
cana-3296	125	1	if	if	SCONJ
cana-3296	125	2	⋂p∈k	⋂p∈k	NUM
cana-3296	125	3	ϑp	ϑp	NOUN
cana-3296	125	4	=	=	NOUN
cana-3296	125	5	0̃	0̃	PROPN
cana-3296	125	6	,	,	PUNCT
cana-3296	125	7	then	then	ADV
cana-3296	125	8	{	{	PUNCT
cana-3296	125	9	ϑp|p	ϑp|p	NOUN
cana-3296	125	10	∈	∈	PROPN
cana-3296	125	11	k	k	PROPN
cana-3296	125	12	}	}	PUNCT
cana-3296	125	13	is	be	AUX
cana-3296	125	14	a	a	DET
cana-3296	125	15	set	set	NOUN
cana-3296	125	16	of	of	ADP
cana-3296	125	17	all	all	DET
cana-3296	125	18	iivfmao	iivfmao	ADJ
cana-3296	125	19	sets	set	NOUN
cana-3296	125	20	of	of	ADP
cana-3296	125	21	x.	x.	NOUN
cana-3296	125	22	proof	proof	NOUN
cana-3296	125	23	.	.	PUNCT
cana-3296	126	1	suppose	suppose	VERB
cana-3296	126	2	∃ϑn	∃ϑn	NOUN
cana-3296	126	3	another	another	DET
cana-3296	126	4	iivfmao	iivfmao	ADJ
cana-3296	126	5	set	set	NOUN
cana-3296	126	6	of	of	ADP
cana-3296	126	7	x	x	PUNCT
cana-3296	126	8	such	such	ADJ
cana-3296	126	9	that	that	DET
cana-3296	126	10	ϑn	ϑn	NOUN
cana-3296	126	11	≠	≠	PROPN
cana-3296	126	12	ϑp∀p	ϑp∀p	PROPN
cana-3296	126	13	∈	∈	PROPN
cana-3296	126	14	k.	k.	PROPN
cana-3296	127	1	clearly	clearly	ADV
cana-3296	127	2	,	,	PUNCT
cana-3296	127	3	0̃	0̃	NOUN
cana-3296	127	4	=	=	SYM
cana-3296	127	5	⋂p∈k	⋂p∈k	NUM
cana-3296	127	6	ϑp	ϑp	NOUN
cana-3296	127	7	=	=	SYM
cana-3296	127	8	⋂p∈(k∪n)/n	⋂p∈(k∪n)/n	PROPN
cana-3296	127	9	ϑp	ϑp	NOUN
cana-3296	127	10	≠	≠	PROPN
cana-3296	127	11	0̃	0̃	NOUN
cana-3296	127	12	,	,	PUNCT
cana-3296	127	13	a	a	DET
cana-3296	127	14	contradiction	contradiction	NOUN
cana-3296	127	15	.	.	PUNCT
cana-3296	128	1	hence	hence	ADV
cana-3296	128	2	,	,	PUNCT
cana-3296	128	3	proved	prove	VERB
cana-3296	128	4	.	.	PUNCT
cana-3296	129	1	references	reference	NOUN
cana-3296	129	2	[	[	X
cana-3296	129	3	1	1	NUM
cana-3296	129	4	]	]	PUNCT
cana-3296	129	5	l.a.zadeh	l.a.zadeh	NOUN
cana-3296	129	6	,	,	PUNCT
cana-3296	129	7	fuzzy	fuzzy	ADJ
cana-3296	129	8	sets	set	NOUN
cana-3296	129	9	,	,	PUNCT
cana-3296	129	10	inf.control.8	inf.control.8	PROPN
cana-3296	129	11	(	(	PUNCT
cana-3296	129	12	1965	1965	NUM
cana-3296	129	13	)	)	PUNCT
cana-3296	129	14	,	,	PUNCT
cana-3296	129	15	338	338	NUM
cana-3296	129	16	-	-	SYM
cana-3296	129	17	353	353	NUM
cana-3296	129	18	.	.	PUNCT
cana-3296	130	1	[	[	X
cana-3296	130	2	2	2	NUM
cana-3296	130	3	]	]	X
cana-3296	130	4	c.l.chang	c.l.chang	NOUN
cana-3296	130	5	,	,	PUNCT
cana-3296	130	6	fuzzy	fuzzy	ADJ
cana-3296	130	7	topological	topological	ADJ
cana-3296	130	8	spaces	space	NOUN
cana-3296	130	9	.	.	PUNCT
cana-3296	131	1	j.math	j.math	NOUN
cana-3296	131	2	.	.	PUNCT
cana-3296	132	1	anal	anal	PROPN
cana-3296	132	2	.	.	PUNCT
cana-3296	132	3	appl	appl	PROPN
cana-3296	132	4	.	.	PROPN
cana-3296	133	1	24	24	NUM
cana-3296	133	2	(	(	PUNCT
cana-3296	133	3	1968	1968	NUM
cana-3296	133	4	)	)	PUNCT
cana-3296	133	5	,	,	PUNCT
cana-3296	133	6	182	182	NUM
cana-3296	133	7	-	-	SYM
cana-3296	133	8	190	190	NUM
cana-3296	133	9	.	.	PUNCT
cana-3296	134	1	[	[	X
cana-3296	134	2	3	3	NUM
cana-3296	134	3	]	]	PUNCT
cana-3296	134	4	l.a.zadeh	l.a.zadeh	NOUN
cana-3296	134	5	,	,	PUNCT
cana-3296	134	6	the	the	DET
cana-3296	134	7	concept	concept	NOUN
cana-3296	134	8	of	of	ADP
cana-3296	134	9	a	a	DET
cana-3296	134	10	linguistic	linguistic	ADJ
cana-3296	134	11	variable	variable	NOUN
cana-3296	134	12	and	and	CCONJ
cana-3296	134	13	its	its	PRON
cana-3296	134	14	application	application	NOUN
cana-3296	134	15	to	to	PART
cana-3296	134	16	approximate	approximate	ADJ
cana-3296	134	17	reasoning	reason	VERB
cana-3296	134	18	i	i	PRON
cana-3296	134	19	,	,	PUNCT
cana-3296	134	20	inform.sci	inform.sci	X
cana-3296	134	21	.	.	NOUN
cana-3296	134	22	8	8	NUM
cana-3296	134	23	(	(	PUNCT
cana-3296	134	24	1975),199	1975),199	NUM
cana-3296	134	25	-	-	SYM
cana-3296	134	26	249	249	NUM
cana-3296	134	27	.	.	PUNCT
cana-3296	135	1	[	[	X
cana-3296	135	2	4	4	NUM
cana-3296	135	3	]	]	X
cana-3296	135	4	t.k.mondal	t.k.mondal	ADJ
cana-3296	135	5	and	and	CCONJ
cana-3296	135	6	s.k.samanta	s.k.samanta	NOUN
cana-3296	135	7	,	,	PUNCT
cana-3296	135	8	topology	topology	NOUN
cana-3296	135	9	of	of	ADP
cana-3296	135	10	interval	interval	NOUN
cana-3296	135	11	valued	value	VERB
cana-3296	135	12	fuzzy	fuzzy	ADJ
cana-3296	135	13	sets	set	NOUN
cana-3296	135	14	,	,	PUNCT
cana-3296	135	15	indian	indian	ADJ
cana-3296	135	16	j.pure	j.pure	NOUN
cana-3296	135	17	appl.math	appl.math	PROPN
cana-3296	135	18	.	.	PUNCT
cana-3296	136	1	(	(	PUNCT
cana-3296	136	2	january	january	PROPN
cana-3296	136	3	1999	1999	NUM
cana-3296	136	4	)	)	PUNCT
cana-3296	136	5	.	.	PUNCT
cana-3296	137	1	[	[	X
cana-3296	137	2	5	5	NUM
cana-3296	137	3	]	]	SYM
cana-3296	137	4	k.attanassov	k.attanassov	NOUN
cana-3296	137	5	,	,	PUNCT
cana-3296	137	6	intuitionistic	intuitionistic	ADJ
cana-3296	137	7	fuzzy	fuzzy	ADJ
cana-3296	137	8	sets	set	NOUN
cana-3296	137	9	,	,	PUNCT
cana-3296	137	10	fuzzy	fuzzy	ADJ
cana-3296	137	11	sets	set	NOUN
cana-3296	137	12	and	and	CCONJ
cana-3296	137	13	systems	system	NOUN
cana-3296	137	14	20	20	NUM
cana-3296	137	15	(	(	PUNCT
cana-3296	137	16	1986	1986	NUM
cana-3296	137	17	)	)	PUNCT
cana-3296	137	18	,	,	PUNCT
cana-3296	137	19	87	87	NUM
cana-3296	137	20	-	-	SYM
cana-3296	137	21	96	96	NUM
cana-3296	137	22	.	.	PUNCT
cana-3296	138	1	[	[	X
cana-3296	138	2	6	6	NUM
cana-3296	138	3	]	]	PUNCT
cana-3296	138	4	d.coker	d.coker	NOUN
cana-3296	138	5	,	,	PUNCT
cana-3296	138	6	an	an	DET
cana-3296	138	7	introduction	introduction	NOUN
cana-3296	138	8	to	to	ADP
cana-3296	138	9	intuitionistic	intuitionistic	ADJ
cana-3296	138	10	fuzzy	fuzzy	ADJ
cana-3296	138	11	topological	topological	ADJ
cana-3296	138	12	spaces	space	NOUN
cana-3296	138	13	,	,	PUNCT
cana-3296	138	14	fuzzy	fuzzy	ADJ
cana-3296	138	15	sets	set	NOUN
cana-3296	138	16	and	and	CCONJ
cana-3296	138	17	systems	system	NOUN
cana-3296	138	18	88	88	NUM
cana-3296	138	19	(	(	PUNCT
cana-3296	138	20	1987	1987	NUM
cana-3296	138	21	)	)	PUNCT
cana-3296	138	22	,	,	PUNCT
cana-3296	138	23	81	81	NUM
cana-3296	138	24	-	-	SYM
cana-3296	138	25	89	89	NUM
cana-3296	138	26	.	.	PUNCT
cana-3296	139	1	[	[	X
cana-3296	139	2	7	7	NUM
cana-3296	139	3	]	]	SYM
cana-3296	139	4	m.s.cheong	m.s.cheong	NOUN
cana-3296	139	5	and	and	CCONJ
cana-3296	139	6	k.hur	k.hur	PROPN
cana-3296	139	7	,	,	PUNCT
cana-3296	139	8	instutionistic	instutionistic	ADJ
cana-3296	139	9	interval	interval	NOUN
cana-3296	139	10	valued	value	VERB
cana-3296	139	11	fuzzy	fuzzy	ADJ
cana-3296	139	12	sets	set	NOUN
cana-3296	139	13	,	,	PUNCT
cana-3296	139	14	j.korean	j.korean	ADJ
cana-3296	139	15	institute	institute	NOUN
cana-3296	139	16	of	of	ADP
cana-3296	139	17	intelligent	intelligent	ADJ
cana-3296	139	18	systems	system	NOUN
cana-3296	139	19	20	20	NUM
cana-3296	139	20	(	(	PUNCT
cana-3296	139	21	6	6	NUM
cana-3296	139	22	)	)	PUNCT
cana-3296	139	23	(	(	PUNCT
cana-3296	139	24	2010	2010	NUM
cana-3296	139	25	)	)	PUNCT
cana-3296	139	26	,	,	PUNCT
cana-3296	139	27	864	864	NUM
cana-3296	139	28	-	-	SYM
cana-3296	139	29	874	874	NUM
cana-3296	139	30	.	.	PUNCT
cana-3296	140	1	[	[	X
cana-3296	140	2	8	8	NUM
cana-3296	140	3	]	]	X
cana-3296	140	4	pyung	pyung	PROPN
cana-3296	140	5	ki	ki	PROPN
cana-3296	140	6	lim	lim	PROPN
cana-3296	140	7	et.all	et.all	PROPN
cana-3296	140	8	,	,	PUNCT
cana-3296	140	9	instutionistic	instutionistic	ADJ
cana-3296	140	10	interval	interval	NOUN
cana-3296	140	11	valued	value	VERB
cana-3296	140	12	fuzzy	fuzzy	ADJ
cana-3296	140	13	topological	topological	ADJ
cana-3296	140	14	spaces	space	NOUN
cana-3296	140	15	,	,	PUNCT
cana-3296	140	16	jkiis	jkiis	NOUN
cana-3296	140	17	(	(	PUNCT
cana-3296	140	18	22	22	NUM
cana-3296	140	19	)	)	PUNCT
cana-3296	140	20	(	(	PUNCT
cana-3296	140	21	2012	2012	NUM
cana-3296	140	22	)	)	PUNCT
cana-3296	140	23	,	,	PUNCT
cana-3296	140	24	126	126	NUM
cana-3296	140	25	-	-	SYM
cana-3296	140	26	134	134	NUM
cana-3296	140	27	.	.	PUNCT
cana-3296	141	1	[	[	X
cana-3296	141	2	9	9	NUM
cana-3296	141	3	]	]	PUNCT
cana-3296	141	4	a.swaminathan	a.swaminathan	NOUN
cana-3296	141	5	and	and	CCONJ
cana-3296	141	6	s.sivaraja	s.sivaraja	NOUN
cana-3296	141	7	,	,	PUNCT
cana-3296	141	8	fuzzy	fuzzy	ADJ
cana-3296	141	9	maximal	maximal	ADJ
cana-3296	141	10	,	,	PUNCT
cana-3296	141	11	minimal	minimal	ADJ
cana-3296	141	12	open	open	ADJ
cana-3296	141	13	and	and	CCONJ
cana-3296	141	14	closed	closed	ADJ
cana-3296	141	15	sets	set	NOUN
cana-3296	141	16	,	,	PUNCT
cana-3296	141	17	journal	journal	NOUN
cana-3296	141	18	of	of	ADP
cana-3296	141	19	advances	advance	NOUN
cana-3296	141	20	in	in	ADP
cana-3296	141	21	mathematics	mathematic	NOUN
cana-3296	141	22	9(10),2020	9(10),2020	NUM
cana-3296	141	23	7741	7741	NUM
cana-3296	141	24	-	-	SYM
cana-3296	141	25	7747	7747	NUM
cana-3296	141	26	.	.	PUNCT
cana-3296	142	1	[	[	X
cana-3296	142	2	10	10	NUM
cana-3296	142	3	]	]	PUNCT
cana-3296	142	4	a.swaminathan	a.swaminathan	NOUN
cana-3296	142	5	and	and	CCONJ
cana-3296	142	6	s.sivaraja	s.sivaraja	NOUN
cana-3296	142	7	,	,	PUNCT
cana-3296	142	8	hesitant	hesitant	ADJ
cana-3296	142	9	fuzzy	fuzzy	ADJ
cana-3296	142	10	minimal	minimal	ADJ
cana-3296	142	11	and	and	CCONJ
cana-3296	142	12	maximal	maximal	ADJ
cana-3296	142	13	open	open	ADJ
cana-3296	142	14	sets	set	NOUN
cana-3296	142	15	,	,	PUNCT
cana-3296	142	16	journal	journal	NOUN
cana-3296	142	17	of	of	ADP
cana-3296	142	18	appl.and	appl.and	X
cana-3296	142	19	pure	pure	ADJ
cana-3296	142	20	mathematics	mathematic	NOUN
cana-3296	142	21	5(2),2023	5(2),2023	NUM
cana-3296	142	22	121	121	NUM
cana-3296	142	23	-	-	SYM
cana-3296	142	24	128	128	NUM
cana-3296	142	25	.	.	PUNCT