id	sid	tid	token	lemma	pos
cana-1482	1	1	communications	communication	NOUN
cana-1482	1	2	on	on	ADP
cana-1482	1	3	applied	apply	VERB
cana-1482	1	4	nonlinear	nonlinear	ADJ
cana-1482	1	5	analysis	analysis	NOUN
cana-1482	1	6	issn	issn	NOUN
cana-1482	1	7	:	:	PUNCT
cana-1482	1	8	1074	1074	NUM
cana-1482	1	9	-	-	PUNCT
cana-1482	1	10	133x	133x	NUM
cana-1482	1	11	vol	vol	NOUN
cana-1482	1	12	31	31	NUM
cana-1482	1	13	no	no	NOUN
cana-1482	1	14	.	.	PUNCT
cana-1482	2	1	8s	8s	PROPN
cana-1482	2	2	(	(	PUNCT
cana-1482	2	3	2024	2024	NUM
cana-1482	2	4	)	)	PUNCT
cana-1482	2	5	272	272	NUM
cana-1482	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	2	7	an	an	DET
cana-1482	2	8	in	in	ADP
cana-1482	2	9	-	-	PUNCT
cana-1482	2	10	depth	depth	NOUN
cana-1482	2	11	exploration	exploration	NOUN
cana-1482	2	12	of	of	ADP
cana-1482	2	13	the	the	DET
cana-1482	2	14	properties	property	NOUN
cana-1482	2	15	and	and	CCONJ
cana-1482	2	16	theoretical	theoretical	ADJ
cana-1482	2	17	foundations	foundation	NOUN
cana-1482	2	18	of	of	ADP
cana-1482	2	19	neutrosophic	neutrosophic	ADJ
cana-1482	2	20	generalized	generalize	VERB
cana-1482	2	21	semipreclosed	semipreclose	VERB
cana-1482	2	22	sets	set	NOUN
cana-1482	2	23	in	in	ADP
cana-1482	2	24	neutrosophic	neutrosophic	ADJ
cana-1482	2	25	topological	topological	ADJ
cana-1482	2	26	spaces	space	NOUN
cana-1482	2	27	1s	1s	NUM
cana-1482	2	28	.	.	PUNCT
cana-1482	3	1	sathishkumar	sathishkumar	PROPN
cana-1482	3	2	,	,	PUNCT
cana-1482	3	3	2s	2s	PROPN
cana-1482	3	4	.	.	PUNCT
cana-1482	4	1	vinoth	vinoth	ADJ
cana-1482	4	2	1,2	1,2	NUM
cana-1482	4	3	department	department	NOUN
cana-1482	4	4	of	of	ADP
cana-1482	4	5	mathematics	mathematic	NOUN
cana-1482	4	6	and	and	CCONJ
cana-1482	4	7	statistics	statistic	NOUN
cana-1482	4	8	,	,	PUNCT
cana-1482	4	9	vignan	vignan	NOUN
cana-1482	4	10	’s	’s	PART
cana-1482	4	11	foundation	foundation	PROPN
cana-1482	4	12	for	for	ADP
cana-1482	4	13	science	science	NOUN
cana-1482	4	14	,	,	PUNCT
cana-1482	4	15	technology	technology	NOUN
cana-1482	4	16	and	and	CCONJ
cana-1482	4	17	research	research	NOUN
cana-1482	4	18	,	,	PUNCT
cana-1482	4	19	guntur522213	guntur522213	PROPN
cana-1482	4	20	.	.	PUNCT
cana-1482	5	1	mail	mail	NOUN
cana-1482	6	1	i	i	NOUN
cana-1482	6	2	d	d	PROPN
cana-1482	6	3	:	:	PUNCT
cana-1482	6	4	1sathishmalar97@gmail.com	1sathishmalar97@gmail.com	NUM
cana-1482	6	5	,	,	PUNCT
cana-1482	6	6	2vinomaths6@gmail.com	2vinomaths6@gmail.com	NUM
cana-1482	6	7	.	.	PUNCT
cana-1482	7	1	article	article	NOUN
cana-1482	7	2	history	history	NOUN
cana-1482	7	3	:	:	PUNCT
cana-1482	7	4	received	receive	VERB
cana-1482	7	5	:	:	PUNCT
cana-1482	7	6	29	29	NUM
cana-1482	7	7	-	-	PUNCT
cana-1482	7	8	04	04	NUM
cana-1482	7	9	-	-	PUNCT
cana-1482	7	10	2024	2024	NUM
cana-1482	7	11	revised	revise	VERB
cana-1482	7	12	:	:	PUNCT
cana-1482	7	13	20	20	NUM
cana-1482	7	14	-	-	SYM
cana-1482	7	15	06	06	NUM
cana-1482	7	16	-	-	PUNCT
cana-1482	7	17	2024	2024	NUM
cana-1482	7	18	accepted	accept	VERB
cana-1482	7	19	:	:	PUNCT
cana-1482	7	20	01	01	NUM
cana-1482	7	21	-	-	SYM
cana-1482	7	22	07	07	NUM
cana-1482	7	23	-	-	PUNCT
cana-1482	7	24	2024	2024	NUM
cana-1482	7	25	abstract	abstract	NOUN
cana-1482	7	26	:	:	PUNCT
cana-1482	7	27	this	this	DET
cana-1482	7	28	paper	paper	NOUN
cana-1482	7	29	goes	go	VERB
cana-1482	7	30	deeply	deeply	ADV
cana-1482	7	31	into	into	ADP
cana-1482	7	32	various	various	ADJ
cana-1482	7	33	intrinsic	intrinsic	ADJ
cana-1482	7	34	characteristics	characteristic	NOUN
cana-1482	7	35	of	of	ADP
cana-1482	7	36	neutrosophic	neutrosophic	ADJ
cana-1482	7	37	generalized	generalize	VERB
cana-1482	7	38	semipre	semipre	NOUN
cana-1482	7	39	closed	close	VERB
cana-1482	7	40	sets	set	NOUN
cana-1482	7	41	.	.	PUNCT
cana-1482	8	1	by	by	ADP
cana-1482	8	2	meticulously	meticulously	ADV
cana-1482	8	3	investigating	investigate	VERB
cana-1482	8	4	these	these	DET
cana-1482	8	5	sets	set	NOUN
cana-1482	8	6	,	,	PUNCT
cana-1482	8	7	we	we	PRON
cana-1482	8	8	hope	hope	VERB
cana-1482	8	9	to	to	PART
cana-1482	8	10	identify	identify	VERB
cana-1482	8	11	their	their	PRON
cana-1482	8	12	essential	essential	ADJ
cana-1482	8	13	traits	trait	NOUN
cana-1482	8	14	and	and	CCONJ
cana-1482	8	15	behaviors	behavior	NOUN
cana-1482	8	16	within	within	ADP
cana-1482	8	17	the	the	DET
cana-1482	8	18	larger	large	ADJ
cana-1482	8	19	context	context	NOUN
cana-1482	8	20	of	of	ADP
cana-1482	8	21	neutrosophic	neutrosophic	ADJ
cana-1482	8	22	set	set	NOUN
cana-1482	8	23	theory	theory	NOUN
cana-1482	8	24	.	.	PUNCT
cana-1482	9	1	furthermore	furthermore	ADV
cana-1482	9	2	,	,	PUNCT
cana-1482	9	3	our	our	PRON
cana-1482	9	4	research	research	NOUN
cana-1482	9	5	looks	look	VERB
cana-1482	9	6	at	at	ADP
cana-1482	9	7	the	the	DET
cana-1482	9	8	complex	complex	ADJ
cana-1482	9	9	linkages	linkage	NOUN
cana-1482	9	10	and	and	CCONJ
cana-1482	9	11	interactions	interaction	NOUN
cana-1482	9	12	between	between	ADP
cana-1482	9	13	neutrosophic	neutrosophic	ADJ
cana-1482	9	14	generalized	generalize	VERB
cana-1482	9	15	semipre	semipre	NOUN
cana-1482	9	16	closed	close	VERB
cana-1482	9	17	sets	set	NOUN
cana-1482	9	18	and	and	CCONJ
cana-1482	9	19	other	other	ADJ
cana-1482	9	20	forms	form	NOUN
cana-1482	9	21	of	of	ADP
cana-1482	9	22	neutrosophic	neutrosophic	ADJ
cana-1482	9	23	sets	set	NOUN
cana-1482	9	24	.	.	PUNCT
cana-1482	10	1	through	through	ADP
cana-1482	10	2	this	this	DET
cana-1482	10	3	comparative	comparative	ADJ
cana-1482	10	4	research	research	NOUN
cana-1482	10	5	,	,	PUNCT
cana-1482	10	6	we	we	PRON
cana-1482	10	7	hope	hope	VERB
cana-1482	10	8	to	to	PART
cana-1482	10	9	highlight	highlight	VERB
cana-1482	10	10	the	the	DET
cana-1482	10	11	links	link	NOUN
cana-1482	10	12	and	and	CCONJ
cana-1482	10	13	distinctions	distinction	NOUN
cana-1482	10	14	that	that	PRON
cana-1482	10	15	exist	exist	VERB
cana-1482	10	16	across	across	ADP
cana-1482	10	17	these	these	DET
cana-1482	10	18	diverse	diverse	ADJ
cana-1482	10	19	types	type	NOUN
cana-1482	10	20	of	of	ADP
cana-1482	10	21	neutrosophic	neutrosophic	ADJ
cana-1482	10	22	sets	set	NOUN
cana-1482	10	23	,	,	PUNCT
cana-1482	10	24	thus	thus	ADV
cana-1482	10	25	contributing	contribute	VERB
cana-1482	10	26	to	to	ADP
cana-1482	10	27	a	a	DET
cana-1482	10	28	more	more	ADV
cana-1482	10	29	thorough	thorough	ADJ
cana-1482	10	30	understanding	understanding	NOUN
cana-1482	10	31	of	of	ADP
cana-1482	10	32	their	their	PRON
cana-1482	10	33	respective	respective	ADJ
cana-1482	10	34	functions	function	NOUN
cana-1482	10	35	and	and	CCONJ
cana-1482	10	36	applications	application	NOUN
cana-1482	10	37	in	in	ADP
cana-1482	10	38	the	the	DET
cana-1482	10	39	area	area	NOUN
cana-1482	10	40	.	.	PUNCT
cana-1482	11	1	keywords	keyword	NOUN
cana-1482	11	2	:	:	PUNCT
cana-1482	11	3	neutrosophic	neutrosophic	ADJ
cana-1482	11	4	subset	subset	NOUN
cana-1482	11	5	,	,	PUNCT
cana-1482	11	6	neutrosophic	neutrosophic	PROPN
cana-1482	11	7	topological	topological	ADJ
cana-1482	11	8	space	space	NOUN
cana-1482	11	9	,	,	PUNCT
cana-1482	11	10	neutrosophic	neutrosophic	ADJ
cana-1482	11	11	interior	interior	NOUN
cana-1482	11	12	,	,	PUNCT
cana-1482	11	13	neutrosophic	neutrosophic	ADJ
cana-1482	11	14	closure	closure	NOUN
cana-1482	11	15	.	.	PUNCT
cana-1482	12	1	2000	2000	NUM
cana-1482	12	2	ams	am	NOUN
cana-1482	12	3	subject	subject	ADJ
cana-1482	12	4	classification	classification	NOUN
cana-1482	12	5	:	:	PUNCT
cana-1482	12	6	54a4	54a4	NUM
cana-1482	12	7	,	,	PUNCT
cana-1482	12	8	08a72	08a72	NUM
cana-1482	12	9	.	.	PUNCT
cana-1482	13	1	a.	a.	NOUN
cana-1482	13	2	introduction	introduction	NOUN
cana-1482	13	3	:	:	PUNCT
cana-1482	13	4	the	the	DET
cana-1482	13	5	concept	concept	NOUN
cana-1482	13	6	of	of	ADP
cana-1482	13	7	generalized	generalized	ADJ
cana-1482	13	8	closed	close	VERB
cana-1482	13	9	sets	set	NOUN
cana-1482	13	10	in	in	ADP
cana-1482	13	11	topology	topology	NOUN
cana-1482	13	12	was	be	AUX
cana-1482	13	13	notably	notably	ADV
cana-1482	13	14	advanced	advance	VERB
cana-1482	13	15	by	by	ADP
cana-1482	13	16	levine	levine	PROPN
cana-1482	13	17	n.	n.	PROPN
cana-1482	13	18	,	,	PUNCT
cana-1482	13	19	while	while	SCONJ
cana-1482	13	20	palaniyappan	palaniyappan	PROPN
cana-1482	13	21	n.	n.	PROPN
cana-1482	13	22	and	and	CCONJ
cana-1482	13	23	rao	rao	PROPN
cana-1482	13	24	k.c	k.c	PROPN
cana-1482	13	25	.	.	PROPN
cana-1482	13	26	made	make	VERB
cana-1482	13	27	significant	significant	ADJ
cana-1482	13	28	contributions	contribution	NOUN
cana-1482	13	29	to	to	ADP
cana-1482	13	30	the	the	DET
cana-1482	13	31	understanding	understanding	NOUN
cana-1482	13	32	of	of	ADP
cana-1482	13	33	regular	regular	ADJ
cana-1482	13	34	generalized	generalize	VERB
cana-1482	13	35	closed	closed	ADJ
cana-1482	13	36	sets	set	NOUN
cana-1482	13	37	.	.	PUNCT
cana-1482	14	1	in	in	ADP
cana-1482	14	2	the	the	DET
cana-1482	14	3	realm	realm	NOUN
cana-1482	14	4	of	of	ADP
cana-1482	14	5	neutrosophic	neutrosophic	ADJ
cana-1482	14	6	sets	set	NOUN
cana-1482	14	7	and	and	CCONJ
cana-1482	14	8	neutrosophic	neutrosophic	ADJ
cana-1482	14	9	topological	topological	ADJ
cana-1482	14	10	spaces	space	NOUN
cana-1482	14	11	,	,	PUNCT
cana-1482	14	12	a.a	a.a	PROPN
cana-1482	14	13	.	.	PROPN
cana-1482	14	14	salama	salama	PROPN
cana-1482	14	15	and	and	CCONJ
cana-1482	14	16	s.a	s.a	PROPN
cana-1482	14	17	.	.	PROPN
cana-1482	14	18	alblowi	alblowi	PROPN
cana-1482	14	19	have	have	AUX
cana-1482	14	20	provided	provide	VERB
cana-1482	14	21	substantial	substantial	ADJ
cana-1482	14	22	insights	insight	NOUN
cana-1482	14	23	.	.	PUNCT
cana-1482	15	1	building	build	VERB
cana-1482	15	2	on	on	ADP
cana-1482	15	3	this	this	DET
cana-1482	15	4	extensive	extensive	ADJ
cana-1482	15	5	body	body	NOUN
cana-1482	15	6	of	of	ADP
cana-1482	15	7	work	work	NOUN
cana-1482	15	8	,	,	PUNCT
cana-1482	15	9	we	we	PRON
cana-1482	15	10	have	have	AUX
cana-1482	15	11	generalized	generalize	VERB
cana-1482	15	12	the	the	DET
cana-1482	15	13	concept	concept	NOUN
cana-1482	15	14	of	of	ADP
cana-1482	15	15	sets	set	NOUN
cana-1482	15	16	to	to	ADP
cana-1482	15	17	neutrosophic	neutrosophic	ADJ
cana-1482	15	18	topological	topological	ADJ
cana-1482	15	19	spaces	space	NOUN
cana-1482	15	20	.	.	PUNCT
cana-1482	16	1	in	in	ADP
cana-1482	16	2	this	this	DET
cana-1482	16	3	paper	paper	NOUN
cana-1482	16	4	,	,	PUNCT
cana-1482	16	5	we	we	PRON
cana-1482	16	6	present	present	VERB
cana-1482	16	7	several	several	ADJ
cana-1482	16	8	interesting	interesting	ADJ
cana-1482	16	9	theorems	theorem	NOUN
cana-1482	16	10	and	and	CCONJ
cana-1482	16	11	results	result	NOUN
cana-1482	16	12	on	on	ADP
cana-1482	16	13	neutrosophic	neutrosophic	ADJ
cana-1482	16	14	generalized	generalize	VERB
cana-1482	16	15	semipreclosed	semipreclose	VERB
cana-1482	16	16	sets	set	NOUN
cana-1482	16	17	,	,	PUNCT
cana-1482	16	18	contributing	contribute	VERB
cana-1482	16	19	to	to	ADP
cana-1482	16	20	the	the	DET
cana-1482	16	21	ongoing	ongoing	ADJ
cana-1482	16	22	development	development	NOUN
cana-1482	16	23	and	and	CCONJ
cana-1482	16	24	understanding	understanding	NOUN
cana-1482	16	25	of	of	ADP
cana-1482	16	26	neutrosophic	neutrosophic	ADJ
cana-1482	16	27	set	set	NOUN
cana-1482	16	28	theory	theory	NOUN
cana-1482	16	29	and	and	CCONJ
cana-1482	16	30	its	its	PRON
cana-1482	16	31	applications	application	NOUN
cana-1482	16	32	the	the	DET
cana-1482	16	33	concept	concept	NOUN
cana-1482	16	34	of	of	ADP
cana-1482	16	35	a	a	DET
cana-1482	16	36	fuzzy	fuzzy	ADJ
cana-1482	16	37	subset	subset	NOUN
cana-1482	16	38	was	be	AUX
cana-1482	16	39	first	first	ADV
cana-1482	16	40	introduced	introduce	VERB
cana-1482	16	41	and	and	CCONJ
cana-1482	16	42	thoroughly	thoroughly	ADV
cana-1482	16	43	studied	study	VERB
cana-1482	16	44	by	by	ADP
cana-1482	16	45	l.a	l.a	PROPN
cana-1482	16	46	.	.	PROPN
cana-1482	16	47	zadeh	zadeh	PROPN
cana-1482	17	1	[	[	X
cana-1482	17	2	15	15	NUM
cana-1482	17	3	]	]	X
cana-1482	17	4	in	in	ADP
cana-1482	17	5	1965	1965	NUM
cana-1482	17	6	,	,	PUNCT
cana-1482	17	7	marking	mark	VERB
cana-1482	17	8	a	a	DET
cana-1482	17	9	significant	significant	ADJ
cana-1482	17	10	milestone	milestone	NOUN
cana-1482	17	11	in	in	ADP
cana-1482	17	12	the	the	DET
cana-1482	17	13	field	field	NOUN
cana-1482	17	14	of	of	ADP
cana-1482	17	15	mathematical	mathematical	ADJ
cana-1482	17	16	sciences	science	NOUN
cana-1482	17	17	.	.	PUNCT
cana-1482	18	1	since	since	SCONJ
cana-1482	18	2	then	then	ADV
cana-1482	18	3	,	,	PUNCT
cana-1482	18	4	research	research	NOUN
cana-1482	18	5	activities	activity	NOUN
cana-1482	18	6	in	in	ADP
cana-1482	18	7	this	this	DET
cana-1482	18	8	area	area	NOUN
cana-1482	18	9	and	and	CCONJ
cana-1482	18	10	its	its	PRON
cana-1482	18	11	related	relate	VERB
cana-1482	18	12	domains	domain	NOUN
cana-1482	18	13	have	have	AUX
cana-1482	18	14	expanded	expand	VERB
cana-1482	18	15	,	,	PUNCT
cana-1482	18	16	finding	find	VERB
cana-1482	18	17	applications	application	NOUN
cana-1482	18	18	across	across	ADP
cana-1482	18	19	various	various	ADJ
cana-1482	18	20	branches	branch	NOUN
cana-1482	18	21	of	of	ADP
cana-1482	18	22	science	science	NOUN
cana-1482	18	23	and	and	CCONJ
cana-1482	18	24	engineering	engineering	NOUN
cana-1482	18	25	.	.	PUNCT
cana-1482	19	1	in	in	ADP
cana-1482	19	2	1986	1986	NUM
cana-1482	19	3	,	,	PUNCT
cana-1482	19	4	k.	k.	PROPN
cana-1482	19	5	atanassov	atanassov	PROPN
cana-1482	19	6	introduced	introduce	VERB
cana-1482	19	7	the	the	DET
cana-1482	19	8	concept	concept	NOUN
cana-1482	19	9	of	of	ADP
cana-1482	19	10	the	the	DET
cana-1482	19	11	intuitionistic	intuitionistic	ADJ
cana-1482	19	12	fuzzy	fuzzy	ADJ
cana-1482	19	13	set	set	NOUN
cana-1482	19	14	,	,	PUNCT
cana-1482	19	15	further	far	ADV
cana-1482	19	16	enriching	enrich	VERB
cana-1482	19	17	the	the	DET
cana-1482	19	18	fuzzy	fuzzy	ADJ
cana-1482	19	19	set	set	NOUN
cana-1482	19	20	theory	theory	NOUN
cana-1482	19	21	.	.	PUNCT
cana-1482	20	1	this	this	DET
cana-1482	20	2	foundational	foundational	ADJ
cana-1482	20	3	work	work	NOUN
cana-1482	20	4	has	have	AUX
cana-1482	20	5	since	since	ADV
cana-1482	20	6	been	be	AUX
cana-1482	20	7	extended	extend	VERB
cana-1482	20	8	by	by	ADP
cana-1482	20	9	numerous	numerous	ADJ
cana-1482	20	10	authors	author	NOUN
cana-1482	20	11	.	.	PUNCT
cana-1482	21	1	the	the	DET
cana-1482	21	2	concept	concept	NOUN
cana-1482	21	3	of	of	ADP
cana-1482	21	4	neutrosophic	neutrosophic	ADJ
cana-1482	21	5	set	set	NOUN
cana-1482	21	6	,	,	PUNCT
cana-1482	21	7	introduced	introduce	VERB
cana-1482	21	8	by	by	ADP
cana-1482	21	9	f.	f.	PROPN
cana-1482	21	10	smarandache	smarandache	PROPN
cana-1482	21	11	[	[	X
cana-1482	21	12	12	12	NUM
cana-1482	21	13	]	]	PUNCT
cana-1482	21	14	,	,	PUNCT
cana-1482	21	15	[	[	X
cana-1482	21	16	13	13	NUM
cana-1482	21	17	]	]	PUNCT
cana-1482	21	18	,	,	PUNCT
cana-1482	21	19	serves	serve	VERB
cana-1482	21	20	as	as	ADP
cana-1482	21	21	a	a	DET
cana-1482	21	22	mathematical	mathematical	ADJ
cana-1482	21	23	tool	tool	NOUN
cana-1482	21	24	for	for	ADP
cana-1482	21	25	handling	handle	VERB
cana-1482	21	26	problems	problem	NOUN
cana-1482	21	27	involving	involve	VERB
cana-1482	21	28	imprecise	imprecise	ADV
cana-1482	21	29	,	,	PUNCT
cana-1482	21	30	indeterminate	indeterminate	ADJ
cana-1482	21	31	,	,	PUNCT
cana-1482	21	32	and	and	CCONJ
cana-1482	21	33	inconsistent	inconsistent	ADJ
cana-1482	21	34	data	datum	NOUN
cana-1482	21	35	.	.	PUNCT
cana-1482	22	1	communications	communication	NOUN
cana-1482	22	2	on	on	ADP
cana-1482	22	3	applied	apply	VERB
cana-1482	22	4	nonlinear	nonlinear	ADJ
cana-1482	22	5	analysis	analysis	NOUN
cana-1482	22	6	issn	issn	NOUN
cana-1482	22	7	:	:	PUNCT
cana-1482	22	8	1074	1074	NUM
cana-1482	22	9	-	-	PUNCT
cana-1482	22	10	133x	133x	NUM
cana-1482	22	11	vol	vol	NOUN
cana-1482	22	12	31	31	NUM
cana-1482	22	13	no	no	NOUN
cana-1482	22	14	.	.	PUNCT
cana-1482	23	1	8s	8s	PROPN
cana-1482	23	2	(	(	PUNCT
cana-1482	23	3	2024	2024	NUM
cana-1482	23	4	)	)	PUNCT
cana-1482	23	5	273	273	NUM
cana-1482	23	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	23	7	our	our	PRON
cana-1482	23	8	research	research	NOUN
cana-1482	23	9	in	in	ADP
cana-1482	23	10	this	this	DET
cana-1482	23	11	paper	paper	NOUN
cana-1482	23	12	has	have	AUX
cana-1482	23	13	been	be	AUX
cana-1482	23	14	motivated	motivate	VERB
cana-1482	23	15	by	by	ADP
cana-1482	23	16	several	several	ADJ
cana-1482	23	17	key	key	ADJ
cana-1482	23	18	works	work	NOUN
cana-1482	23	19	in	in	ADP
cana-1482	23	20	the	the	DET
cana-1482	23	21	field	field	NOUN
cana-1482	23	22	.	.	PUNCT
cana-1482	24	1	in	in	ADP
cana-1482	24	2	1968	1968	NUM
cana-1482	24	3	,	,	PUNCT
cana-1482	24	4	c.l	c.l	PROPN
cana-1482	24	5	.	.	PROPN
cana-1482	24	6	chang	chang	PROPN
cana-1482	25	1	[	[	X
cana-1482	25	2	4	4	X
cana-1482	25	3	]	]	PUNCT
cana-1482	25	4	introduced	introduce	VERB
cana-1482	25	5	and	and	CCONJ
cana-1482	25	6	studied	study	VERB
cana-1482	25	7	fuzzy	fuzzy	ADJ
cana-1482	25	8	topological	topological	ADJ
cana-1482	25	9	spaces	space	NOUN
cana-1482	25	10	,	,	PUNCT
cana-1482	25	11	which	which	PRON
cana-1482	25	12	generalize	generalize	VERB
cana-1482	25	13	traditional	traditional	ADJ
cana-1482	25	14	topological	topological	ADJ
cana-1482	25	15	spaces	space	NOUN
cana-1482	25	16	.	.	PUNCT
cana-1482	26	1	this	this	DET
cana-1482	26	2	pioneering	pioneer	VERB
cana-1482	26	3	work	work	NOUN
cana-1482	26	4	has	have	AUX
cana-1482	26	5	inspired	inspire	VERB
cana-1482	26	6	many	many	ADJ
cana-1482	26	7	researchers	researcher	NOUN
cana-1482	26	8	to	to	PART
cana-1482	26	9	further	far	ADV
cana-1482	26	10	develop	develop	VERB
cana-1482	26	11	the	the	DET
cana-1482	26	12	theory	theory	NOUN
cana-1482	26	13	of	of	ADP
cana-1482	26	14	fuzzy	fuzzy	ADJ
cana-1482	26	15	topological	topological	ADJ
cana-1482	26	16	spaces	space	NOUN
cana-1482	26	17	.	.	PUNCT
cana-1482	27	1	among	among	ADP
cana-1482	27	2	them	they	PRON
cana-1482	27	3	,	,	PUNCT
cana-1482	27	4	andrijevic	andrijevic	VERB
cana-1482	27	5	[	[	X
cana-1482	27	6	1	1	X
cana-1482	27	7	]	]	PUNCT
cana-1482	27	8	introduced	introduce	VERB
cana-1482	27	9	semipreclosed	semipreclose	VERB
cana-1482	27	10	sets	set	NOUN
cana-1482	27	11	,	,	PUNCT
cana-1482	27	12	and	and	CCONJ
cana-1482	27	13	dontchev	dontchev	ADJ
cana-1482	28	1	[	[	X
cana-1482	28	2	5	5	NUM
cana-1482	28	3	]	]	PUNCT
cana-1482	28	4	extended	extend	VERB
cana-1482	28	5	this	this	DET
cana-1482	28	6	concept	concept	NOUN
cana-1482	28	7	to	to	ADP
cana-1482	28	8	generalized	generalize	VERB
cana-1482	28	9	semipreclosed	semipreclose	VERB
cana-1482	28	10	sets	set	NOUN
cana-1482	28	11	in	in	ADP
cana-1482	28	12	general	general	ADJ
cana-1482	28	13	topology	topology	NOUN
cana-1482	28	14	.	.	PUNCT
cana-1482	29	1	subsequently	subsequently	ADV
cana-1482	29	2	,	,	PUNCT
cana-1482	29	3	saraf	saraf	PROPN
cana-1482	29	4	and	and	CCONJ
cana-1482	29	5	khanna	khanna	PROPN
cana-1482	30	1	[	[	X
cana-1482	30	2	14	14	NUM
cana-1482	30	3	]	]	PUNCT
cana-1482	30	4	adapted	adapt	VERB
cana-1482	30	5	these	these	DET
cana-1482	30	6	sets	set	NOUN
cana-1482	30	7	to	to	ADP
cana-1482	30	8	fuzzy	fuzzy	ADJ
cana-1482	30	9	topological	topological	ADJ
cana-1482	30	10	spaces	space	NOUN
cana-1482	30	11	,	,	PUNCT
cana-1482	30	12	broadening	broaden	VERB
cana-1482	30	13	their	their	PRON
cana-1482	30	14	applicability	applicability	NOUN
cana-1482	30	15	.	.	PUNCT
cana-1482	31	1	further	further	ADJ
cana-1482	31	2	contributions	contribution	NOUN
cana-1482	31	3	include	include	VERB
cana-1482	31	4	the	the	DET
cana-1482	31	5	work	work	NOUN
cana-1482	31	6	of	of	ADP
cana-1482	31	7	tapas	tapas	NOUN
cana-1482	31	8	kumar	kumar	PROPN
cana-1482	31	9	mondal	mondal	PROPN
cana-1482	31	10	and	and	CCONJ
cana-1482	31	11	s.k	s.k	PROPN
cana-1482	31	12	.	.	PROPN
cana-1482	31	13	samantha	samantha	PROPN
cana-1482	32	1	[	[	X
cana-1482	32	2	9	9	NUM
cana-1482	32	3	]	]	PUNCT
cana-1482	32	4	,	,	PUNCT
cana-1482	32	5	who	who	PRON
cana-1482	32	6	introduced	introduce	VERB
cana-1482	32	7	the	the	DET
cana-1482	32	8	topology	topology	NOUN
cana-1482	32	9	of	of	ADP
cana-1482	32	10	interval	interval	NOUN
cana-1482	32	11	-	-	PUNCT
cana-1482	32	12	valued	value	VERB
cana-1482	32	13	fuzzy	fuzzy	ADJ
cana-1482	32	14	sets	set	NOUN
cana-1482	32	15	,	,	PUNCT
cana-1482	32	16	and	and	CCONJ
cana-1482	32	17	bhattacharya	bhattacharya	PROPN
cana-1482	32	18	b.	b.	PROPN
cana-1482	32	19	and	and	CCONJ
cana-1482	32	20	lahiri	lahiri	PROPN
cana-1482	32	21	b.k	b.k	PROPN
cana-1482	32	22	.	.	PUNCT
cana-1482	33	1	[	[	X
cana-1482	33	2	3	3	NUM
cana-1482	33	3	]	]	PUNCT
cana-1482	33	4	,	,	PUNCT
cana-1482	33	5	who	who	PRON
cana-1482	33	6	developed	develop	VERB
cana-1482	33	7	the	the	DET
cana-1482	33	8	concept	concept	NOUN
cana-1482	33	9	of	of	ADP
cana-1482	33	10	semigeneralized	semigeneralize	VERB
cana-1482	33	11	closed	closed	ADJ
cana-1482	33	12	sets	set	NOUN
cana-1482	33	13	in	in	ADP
cana-1482	33	14	topology	topology	NOUN
cana-1482	33	15	.	.	PUNCT
cana-1482	34	1	ganguly	ganguly	PROPN
cana-1482	34	2	s.	s.	PROPN
cana-1482	34	3	and	and	CCONJ
cana-1482	34	4	saha	saha	PROPN
cana-1482	34	5	s.	s.	PROPN
cana-1482	35	1	[	[	X
cana-1482	35	2	6	6	NUM
cana-1482	35	3	]	]	PUNCT
cana-1482	35	4	explored	explore	VERB
cana-1482	35	5	fuzzy	fuzzy	ADJ
cana-1482	35	6	semipreopen	semipreopen	ADJ
cana-1482	35	7	sets	set	NOUN
cana-1482	35	8	in	in	ADP
cana-1482	35	9	fuzzy	fuzzy	ADJ
cana-1482	35	10	topological	topological	ADJ
cana-1482	35	11	spaces	space	NOUN
cana-1482	35	12	,	,	PUNCT
cana-1482	35	13	while	while	SCONJ
cana-1482	35	14	indira	indira	PROPN
cana-1482	35	15	r.	r.	PROPN
cana-1482	35	16	et	et	PROPN
cana-1482	35	17	al	al	PROPN
cana-1482	35	18	.	.	PUNCT
cana-1482	36	1	[	[	X
cana-1482	36	2	7	7	NUM
cana-1482	36	3	]	]	PUNCT
cana-1482	36	4	investigated	investigate	VERB
cana-1482	36	5	intervalvalued	intervalvalue	VERB
cana-1482	36	6	fuzzy	fuzzy	ADJ
cana-1482	36	7	rw	rw	NOUN
cana-1482	36	8	-	-	PUNCT
cana-1482	36	9	closed	close	VERB
cana-1482	36	10	and	and	CCONJ
cana-1482	36	11	interval	interval	NOUN
cana-1482	36	12	-	-	PUNCT
cana-1482	36	13	valued	value	VERB
cana-1482	36	14	fuzzy	fuzzy	ADJ
cana-1482	36	15	rw	rw	NOUN
cana-1482	36	16	-	-	PUNCT
cana-1482	36	17	open	open	ADJ
cana-1482	36	18	sets	set	NOUN
cana-1482	36	19	in	in	ADP
cana-1482	36	20	interval	interval	NOUN
cana-1482	36	21	-	-	PUNCT
cana-1482	36	22	valued	value	VERB
cana-1482	36	23	fuzzy	fuzzy	ADJ
cana-1482	36	24	topological	topological	ADJ
cana-1482	36	25	spaces	space	NOUN
cana-1482	36	26	.	.	PUNCT
cana-1482	37	1	the	the	DET
cana-1482	37	2	concept	concept	NOUN
cana-1482	37	3	of	of	ADP
cana-1482	37	4	generalized	generalized	ADJ
cana-1482	37	5	closed	close	VERB
cana-1482	37	6	sets	set	NOUN
cana-1482	37	7	in	in	ADP
cana-1482	37	8	topology	topology	NOUN
cana-1482	37	9	was	be	AUX
cana-1482	37	10	notably	notably	ADV
cana-1482	37	11	advanced	advance	VERB
cana-1482	37	12	by	by	ADP
cana-1482	37	13	levine	levine	PROPN
cana-1482	37	14	n.	n.	PROPN
cana-1482	38	1	[	[	X
cana-1482	38	2	8	8	NUM
cana-1482	38	3	]	]	PUNCT
cana-1482	38	4	,	,	PUNCT
cana-1482	38	5	while	while	SCONJ
cana-1482	38	6	palaniyappan	palaniyappan	PROPN
cana-1482	38	7	n.	n.	NOUN
cana-1482	38	8	and	and	CCONJ
cana-1482	38	9	rao	rao	PROPN
cana-1482	38	10	k.c	k.c	PROPN
cana-1482	38	11	.	.	PUNCT
cana-1482	39	1	[	[	X
cana-1482	39	2	10	10	NUM
cana-1482	39	3	]	]	PUNCT
cana-1482	39	4	made	make	VERB
cana-1482	39	5	significant	significant	ADJ
cana-1482	39	6	contributions	contribution	NOUN
cana-1482	39	7	to	to	ADP
cana-1482	39	8	the	the	DET
cana-1482	39	9	understanding	understanding	NOUN
cana-1482	39	10	of	of	ADP
cana-1482	39	11	regular	regular	ADJ
cana-1482	39	12	generalized	generalize	VERB
cana-1482	39	13	closed	closed	ADJ
cana-1482	39	14	sets	set	NOUN
cana-1482	39	15	.	.	PUNCT
cana-1482	40	1	in	in	ADP
cana-1482	40	2	the	the	DET
cana-1482	40	3	realm	realm	NOUN
cana-1482	40	4	of	of	ADP
cana-1482	40	5	neutrosophic	neutrosophic	ADJ
cana-1482	40	6	sets	set	NOUN
cana-1482	40	7	and	and	CCONJ
cana-1482	40	8	neutrosophic	neutrosophic	ADJ
cana-1482	40	9	topological	topological	ADJ
cana-1482	40	10	spaces	space	NOUN
cana-1482	40	11	,	,	PUNCT
cana-1482	40	12	a.a	a.a	PROPN
cana-1482	40	13	.	.	PROPN
cana-1482	40	14	salama	salama	PROPN
cana-1482	40	15	and	and	CCONJ
cana-1482	40	16	s.a	s.a	PROPN
cana-1482	40	17	.	.	PROPN
cana-1482	40	18	alblowi	alblowi	PROPN
cana-1482	40	19	[	[	X
cana-1482	40	20	11	11	NUM
cana-1482	40	21	]	]	PUNCT
cana-1482	40	22	have	have	AUX
cana-1482	40	23	provided	provide	VERB
cana-1482	40	24	substantial	substantial	ADJ
cana-1482	40	25	insights	insight	NOUN
cana-1482	40	26	.	.	PUNCT
cana-1482	41	1	building	build	VERB
cana-1482	41	2	on	on	ADP
cana-1482	41	3	this	this	DET
cana-1482	41	4	extensive	extensive	ADJ
cana-1482	41	5	body	body	NOUN
cana-1482	41	6	of	of	ADP
cana-1482	41	7	work	work	NOUN
cana-1482	41	8	,	,	PUNCT
cana-1482	41	9	we	we	PRON
cana-1482	41	10	have	have	AUX
cana-1482	41	11	generalized	generalize	VERB
cana-1482	41	12	the	the	DET
cana-1482	41	13	concept	concept	NOUN
cana-1482	41	14	of	of	ADP
cana-1482	41	15	sets	set	NOUN
cana-1482	41	16	to	to	ADP
cana-1482	41	17	neutrosophic	neutrosophic	ADJ
cana-1482	41	18	topological	topological	ADJ
cana-1482	41	19	spaces	space	NOUN
cana-1482	41	20	.	.	PUNCT
cana-1482	42	1	in	in	ADP
cana-1482	42	2	this	this	DET
cana-1482	42	3	paper	paper	NOUN
cana-1482	42	4	,	,	PUNCT
cana-1482	42	5	we	we	PRON
cana-1482	42	6	present	present	VERB
cana-1482	42	7	several	several	ADJ
cana-1482	42	8	interesting	interesting	ADJ
cana-1482	42	9	theorems	theorem	NOUN
cana-1482	42	10	and	and	CCONJ
cana-1482	42	11	results	result	NOUN
cana-1482	42	12	on	on	ADP
cana-1482	42	13	neutrosophic	neutrosophic	ADJ
cana-1482	42	14	generalized	generalize	VERB
cana-1482	42	15	semipreclosed	semipreclose	VERB
cana-1482	42	16	sets	set	NOUN
cana-1482	42	17	,	,	PUNCT
cana-1482	42	18	contributing	contribute	VERB
cana-1482	42	19	to	to	ADP
cana-1482	42	20	the	the	DET
cana-1482	42	21	ongoing	ongoing	ADJ
cana-1482	42	22	development	development	NOUN
cana-1482	42	23	and	and	CCONJ
cana-1482	42	24	understanding	understanding	NOUN
cana-1482	42	25	of	of	ADP
cana-1482	42	26	neutrosophic	neutrosophic	ADJ
cana-1482	42	27	set	set	NOUN
cana-1482	42	28	theory	theory	NOUN
cana-1482	42	29	and	and	CCONJ
cana-1482	42	30	its	its	PRON
cana-1482	42	31	applications	application	NOUN
cana-1482	42	32	.	.	PUNCT
cana-1482	43	1	b.	b.	NOUN
cana-1482	43	2	motivation	motivation	NOUN
cana-1482	43	3	:	:	PUNCT
cana-1482	43	4	the	the	DET
cana-1482	43	5	motivation	motivation	NOUN
cana-1482	43	6	behind	behind	ADP
cana-1482	43	7	this	this	DET
cana-1482	43	8	paper	paper	NOUN
cana-1482	43	9	lies	lie	VERB
cana-1482	43	10	in	in	ADP
cana-1482	43	11	the	the	DET
cana-1482	43	12	progressive	progressive	ADJ
cana-1482	43	13	evolution	evolution	NOUN
cana-1482	43	14	of	of	ADP
cana-1482	43	15	mathematical	mathematical	ADJ
cana-1482	43	16	concepts	concept	NOUN
cana-1482	43	17	,	,	PUNCT
cana-1482	43	18	particularly	particularly	ADV
cana-1482	43	19	in	in	ADP
cana-1482	43	20	the	the	DET
cana-1482	43	21	realm	realm	NOUN
cana-1482	43	22	of	of	ADP
cana-1482	43	23	topology	topology	NOUN
cana-1482	43	24	and	and	CCONJ
cana-1482	43	25	set	set	VERB
cana-1482	43	26	theory	theory	NOUN
cana-1482	43	27	.	.	PUNCT
cana-1482	44	1	inspired	inspire	VERB
cana-1482	44	2	by	by	ADP
cana-1482	44	3	seminal	seminal	ADJ
cana-1482	44	4	works	work	NOUN
cana-1482	44	5	by	by	ADP
cana-1482	44	6	levine	levine	PROPN
cana-1482	44	7	n.	n.	PROPN
cana-1482	44	8	,	,	PUNCT
cana-1482	44	9	palaniyappan	palaniyappan	NOUN
cana-1482	44	10	n.	n.	PROPN
cana-1482	44	11	,	,	PUNCT
cana-1482	44	12	rao	rao	PROPN
cana-1482	44	13	k.c	k.c	PROPN
cana-1482	44	14	.	.	PROPN
cana-1482	44	15	,	,	PUNCT
cana-1482	44	16	a.a	a.a	PROPN
cana-1482	44	17	.	.	PROPN
cana-1482	44	18	salama	salama	PROPN
cana-1482	44	19	,	,	PUNCT
cana-1482	44	20	and	and	CCONJ
cana-1482	44	21	s.a	s.a	PROPN
cana-1482	44	22	.	.	PROPN
cana-1482	44	23	alblowi	alblowi	PROPN
cana-1482	44	24	,	,	PUNCT
cana-1482	44	25	we	we	PRON
cana-1482	44	26	embark	embark	VERB
cana-1482	44	27	on	on	ADP
cana-1482	44	28	a	a	DET
cana-1482	44	29	journey	journey	NOUN
cana-1482	44	30	to	to	PART
cana-1482	44	31	generalize	generalize	VERB
cana-1482	44	32	the	the	DET
cana-1482	44	33	notion	notion	NOUN
cana-1482	44	34	of	of	ADP
cana-1482	44	35	sets	set	NOUN
cana-1482	44	36	to	to	ADP
cana-1482	44	37	neutrosophic	neutrosophic	ADJ
cana-1482	44	38	topological	topological	ADJ
cana-1482	44	39	spaces	space	NOUN
cana-1482	44	40	.	.	PUNCT
cana-1482	45	1	drawing	draw	VERB
cana-1482	45	2	from	from	ADP
cana-1482	45	3	the	the	DET
cana-1482	45	4	foundational	foundational	ADJ
cana-1482	45	5	research	research	NOUN
cana-1482	45	6	of	of	ADP
cana-1482	45	7	luminaries	luminary	NOUN
cana-1482	45	8	like	like	ADP
cana-1482	45	9	l.a	l.a	PROPN
cana-1482	45	10	.	.	PROPN
cana-1482	45	11	zadeh	zadeh	PROPN
cana-1482	45	12	and	and	CCONJ
cana-1482	45	13	k.	k.	PROPN
cana-1482	45	14	atanassov	atanassov	PROPN
cana-1482	45	15	,	,	PUNCT
cana-1482	45	16	who	who	PRON
cana-1482	45	17	introduced	introduce	VERB
cana-1482	45	18	fuzzy	fuzzy	ADJ
cana-1482	45	19	and	and	CCONJ
cana-1482	45	20	intuitionistic	intuitionistic	ADJ
cana-1482	45	21	fuzzy	fuzzy	ADJ
cana-1482	45	22	sets	set	NOUN
cana-1482	45	23	respectively	respectively	ADV
cana-1482	45	24	,	,	PUNCT
cana-1482	45	25	we	we	PRON
cana-1482	45	26	recognize	recognize	VERB
cana-1482	45	27	the	the	DET
cana-1482	45	28	significance	significance	NOUN
cana-1482	45	29	of	of	ADP
cana-1482	45	30	these	these	DET
cana-1482	45	31	frameworks	framework	NOUN
cana-1482	45	32	in	in	ADP
cana-1482	45	33	addressing	address	VERB
cana-1482	45	34	uncertainties	uncertainty	NOUN
cana-1482	45	35	inherent	inherent	ADJ
cana-1482	45	36	in	in	ADP
cana-1482	45	37	real	real	ADJ
cana-1482	45	38	-	-	PUNCT
cana-1482	45	39	world	world	NOUN
cana-1482	45	40	data	datum	NOUN
cana-1482	45	41	.	.	PUNCT
cana-1482	46	1	building	build	VERB
cana-1482	46	2	upon	upon	SCONJ
cana-1482	46	3	the	the	DET
cana-1482	46	4	pioneering	pioneer	VERB
cana-1482	46	5	efforts	effort	NOUN
cana-1482	46	6	of	of	ADP
cana-1482	46	7	c.l	c.l	PROPN
cana-1482	46	8	.	.	PROPN
cana-1482	46	9	chang	chang	PROPN
cana-1482	46	10	,	,	PUNCT
cana-1482	46	11	andrijevic	andrijevic	ADJ
cana-1482	46	12	,	,	PUNCT
cana-1482	46	13	dontchev	dontchev	PROPN
cana-1482	46	14	,	,	PUNCT
cana-1482	46	15	saraf	saraf	PROPN
cana-1482	46	16	,	,	PUNCT
cana-1482	46	17	khanna	khanna	PROPN
cana-1482	46	18	,	,	PUNCT
cana-1482	46	19	mondal	mondal	PROPN
cana-1482	46	20	,	,	PUNCT
cana-1482	46	21	samantha	samantha	PROPN
cana-1482	46	22	,	,	PUNCT
cana-1482	46	23	bhattacharya	bhattacharya	PROPN
cana-1482	46	24	,	,	PUNCT
cana-1482	46	25	lahiri	lahiri	PROPN
cana-1482	46	26	,	,	PUNCT
cana-1482	46	27	ganguly	ganguly	PROPN
cana-1482	46	28	,	,	PUNCT
cana-1482	46	29	saha	saha	PROPN
cana-1482	46	30	,	,	PUNCT
cana-1482	46	31	indira	indira	PROPN
cana-1482	46	32	,	,	PUNCT
cana-1482	46	33	and	and	CCONJ
cana-1482	46	34	others	other	NOUN
cana-1482	46	35	in	in	ADP
cana-1482	46	36	extending	extend	VERB
cana-1482	46	37	fuzzy	fuzzy	ADJ
cana-1482	46	38	set	set	NOUN
cana-1482	46	39	theory	theory	NOUN
cana-1482	46	40	to	to	ADP
cana-1482	46	41	various	various	ADJ
cana-1482	46	42	topological	topological	ADJ
cana-1482	46	43	spaces	space	NOUN
cana-1482	46	44	,	,	PUNCT
cana-1482	46	45	we	we	PRON
cana-1482	46	46	aspire	aspire	VERB
cana-1482	46	47	to	to	PART
cana-1482	46	48	expand	expand	VERB
cana-1482	46	49	the	the	DET
cana-1482	46	50	frontiers	frontier	NOUN
cana-1482	46	51	of	of	ADP
cana-1482	46	52	knowledge	knowledge	NOUN
cana-1482	46	53	in	in	ADP
cana-1482	46	54	neutrosophic	neutrosophic	ADJ
cana-1482	46	55	set	set	NOUN
cana-1482	46	56	theory	theory	NOUN
cana-1482	46	57	.	.	PUNCT
cana-1482	47	1	our	our	PRON
cana-1482	47	2	motivation	motivation	NOUN
cana-1482	47	3	is	be	AUX
cana-1482	47	4	to	to	PART
cana-1482	47	5	contribute	contribute	VERB
cana-1482	47	6	to	to	ADP
cana-1482	47	7	the	the	DET
cana-1482	47	8	ongoing	ongoing	ADJ
cana-1482	47	9	dialogue	dialogue	NOUN
cana-1482	47	10	surrounding	surround	VERB
cana-1482	47	11	the	the	DET
cana-1482	47	12	theoretical	theoretical	ADJ
cana-1482	47	13	foundations	foundation	NOUN
cana-1482	47	14	and	and	CCONJ
cana-1482	47	15	practical	practical	ADJ
cana-1482	47	16	applications	application	NOUN
cana-1482	47	17	of	of	ADP
cana-1482	47	18	neutrosophic	neutrosophic	ADJ
cana-1482	47	19	sets	set	NOUN
cana-1482	47	20	.	.	PUNCT
cana-1482	48	1	by	by	ADP
cana-1482	48	2	presenting	present	VERB
cana-1482	48	3	novel	novel	ADJ
cana-1482	48	4	theorems	theorem	NOUN
cana-1482	48	5	and	and	CCONJ
cana-1482	48	6	results	result	NOUN
cana-1482	48	7	on	on	ADP
cana-1482	48	8	neutrosophic	neutrosophic	ADJ
cana-1482	48	9	generalized	generalize	VERB
cana-1482	48	10	semipreclosed	semipreclose	VERB
cana-1482	48	11	sets	set	NOUN
cana-1482	48	12	,	,	PUNCT
cana-1482	48	13	we	we	PRON
cana-1482	48	14	aim	aim	VERB
cana-1482	48	15	to	to	PART
cana-1482	48	16	enrich	enrich	VERB
cana-1482	48	17	the	the	DET
cana-1482	48	18	understanding	understanding	NOUN
cana-1482	48	19	of	of	ADP
cana-1482	48	20	neutrosophic	neutrosophic	ADJ
cana-1482	48	21	set	set	NOUN
cana-1482	48	22	theory	theory	NOUN
cana-1482	48	23	and	and	CCONJ
cana-1482	48	24	its	its	PRON
cana-1482	48	25	potential	potential	ADJ
cana-1482	48	26	impact	impact	NOUN
cana-1482	48	27	across	across	ADP
cana-1482	48	28	diverse	diverse	ADJ
cana-1482	48	29	disciplines	discipline	NOUN
cana-1482	48	30	.	.	PUNCT
cana-1482	49	1	through	through	ADP
cana-1482	49	2	our	our	PRON
cana-1482	49	3	research	research	NOUN
cana-1482	49	4	,	,	PUNCT
cana-1482	49	5	we	we	PRON
cana-1482	49	6	hope	hope	VERB
cana-1482	49	7	to	to	PART
cana-1482	49	8	inspire	inspire	VERB
cana-1482	49	9	further	further	ADJ
cana-1482	49	10	exploration	exploration	NOUN
cana-1482	49	11	and	and	CCONJ
cana-1482	49	12	innovation	innovation	NOUN
cana-1482	49	13	in	in	ADP
cana-1482	49	14	the	the	DET
cana-1482	49	15	burgeoning	burgeon	VERB
cana-1482	49	16	field	field	NOUN
cana-1482	49	17	of	of	ADP
cana-1482	49	18	neutrosophic	neutrosophic	ADJ
cana-1482	49	19	topology	topology	NOUN
cana-1482	49	20	.	.	PUNCT
cana-1482	50	1	communications	communication	NOUN
cana-1482	50	2	on	on	ADP
cana-1482	50	3	applied	apply	VERB
cana-1482	50	4	nonlinear	nonlinear	ADJ
cana-1482	50	5	analysis	analysis	NOUN
cana-1482	50	6	issn	issn	NOUN
cana-1482	50	7	:	:	PUNCT
cana-1482	50	8	1074	1074	NUM
cana-1482	50	9	-	-	PUNCT
cana-1482	50	10	133x	133x	NUM
cana-1482	50	11	vol	vol	NOUN
cana-1482	50	12	31	31	NUM
cana-1482	50	13	no	no	NOUN
cana-1482	50	14	.	.	PUNCT
cana-1482	51	1	8s	8s	PROPN
cana-1482	51	2	(	(	PUNCT
cana-1482	51	3	2024	2024	NUM
cana-1482	51	4	)	)	PUNCT
cana-1482	51	5	274	274	NUM
cana-1482	51	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	51	7	1.preliminaries	1.preliminaries	NUM
cana-1482	51	8	:	:	PUNCT
cana-1482	51	9	1.1	1.1	NUM
cana-1482	51	10	definition:[8	definition:[8	NOUN
cana-1482	51	11	]	]	PUNCT
cana-1482	51	12	let	let	VERB
cana-1482	51	13	x	x	PRON
cana-1482	51	14	be	be	AUX
cana-1482	51	15	any	any	PRON
cana-1482	51	16	nonempty	nonempty	ADV
cana-1482	51	17	set	set	VERB
cana-1482	51	18	.	.	PUNCT
cana-1482	52	1	a	a	DET
cana-1482	52	2	mapping	mapping	NOUN
cana-1482	52	3	𝐴	𝐴	PROPN
cana-1482	52	4	:	:	PUNCT
cana-1482	52	5	𝑋	𝑋	NOUN
cana-1482	52	6	→	→	SYM
cana-1482	52	7	[	[	X
cana-1482	52	8	0,1	0,1	NUM
cana-1482	52	9	]	]	PUNCT
cana-1482	52	10	is	be	AUX
cana-1482	52	11	called	call	VERB
cana-1482	52	12	a	a	DET
cana-1482	52	13	fuzzy	fuzzy	ADJ
cana-1482	52	14	subset	subset	NOUN
cana-1482	52	15	(	(	PUNCT
cana-1482	52	16	briefly	briefly	ADV
cana-1482	52	17	,	,	PUNCT
cana-1482	52	18	fss	fss	PROPN
cana-1482	52	19	)	)	PUNCT
cana-1482	52	20	of	of	ADP
cana-1482	52	21	x.	x.	NOUN
cana-1482	52	22	1.2	1.2	NUM
cana-1482	52	23	definition	definition	NOUN
cana-1482	52	24	:	:	PUNCT
cana-1482	52	25	a	a	DET
cana-1482	52	26	intuitionistic	intuitionistic	ADJ
cana-1482	52	27	fuzzy	fuzzy	ADJ
cana-1482	52	28	subset	subset	NOUN
cana-1482	52	29	(	(	PUNCT
cana-1482	52	30	ifs	ifs	PROPN
cana-1482	52	31	)	)	PUNCT
cana-1482	52	32	a	a	PRON
cana-1482	52	33	of	of	ADP
cana-1482	52	34	an	an	DET
cana-1482	52	35	universal	universal	ADJ
cana-1482	52	36	set	set	NOUN
cana-1482	52	37	x	x	PUNCT
cana-1482	52	38	is	be	AUX
cana-1482	52	39	defined	define	VERB
cana-1482	52	40	as	as	ADP
cana-1482	52	41	an	an	DET
cana-1482	52	42	object	object	NOUN
cana-1482	52	43	of	of	ADP
cana-1482	52	44	the	the	DET
cana-1482	52	45	form	form	NOUN
cana-1482	53	1	a	a	NOUN
cana-1482	53	2	=	=	X
cana-1482	53	3	{	{	PUNCT
cana-1482	53	4			X
cana-1482	53	5	x	x	SYM
cana-1482	53	6	,	,	PUNCT
cana-1482	53	7	𝜇𝐴(𝑥	𝜇𝐴(𝑥	NUM
cana-1482	53	8	)	)	PUNCT
cana-1482	53	9	,	,	PUNCT
cana-1482	53	10	𝜗𝐴(𝑥	𝜗𝐴(𝑥	X
cana-1482	53	11	)	)	PUNCT
cana-1482	53	12			PROPN
cana-1482	53	13	/	/	SYM
cana-1482	53	14	xx	xx	PROPN
cana-1482	53	15	}	}	PUNCT
cana-1482	53	16	,	,	PUNCT
cana-1482	53	17	where	where	SCONJ
cana-1482	53	18	𝜇𝐴	𝜇𝐴	ADP
cana-1482	53	19	:	:	PUNCT
cana-1482	53	20	x→[0	x→[0	PROPN
cana-1482	53	21	,	,	PUNCT
cana-1482	53	22	1	1	NUM
cana-1482	53	23	]	]	PUNCT
cana-1482	53	24	and	and	CCONJ
cana-1482	53	25	𝜗𝐴	𝜗𝐴	ADJ
cana-1482	53	26	:	:	PUNCT
cana-1482	53	27	x	x	SYM
cana-1482	53	28	→[0	→[0	SYM
cana-1482	53	29	,	,	PUNCT
cana-1482	53	30	1	1	NUM
cana-1482	53	31	]	]	PUNCT
cana-1482	53	32	define	define	VERB
cana-1482	53	33	the	the	DET
cana-1482	53	34	degree	degree	NOUN
cana-1482	53	35	of	of	ADP
cana-1482	53	36	membership	membership	NOUN
cana-1482	53	37	and	and	CCONJ
cana-1482	53	38	the	the	DET
cana-1482	53	39	degree	degree	NOUN
cana-1482	53	40	of	of	ADP
cana-1482	53	41	non	non	ADJ
cana-1482	53	42	-	-	NOUN
cana-1482	53	43	membership	membership	NOUN
cana-1482	53	44	of	of	ADP
cana-1482	53	45	the	the	DET
cana-1482	53	46	element	element	NOUN
cana-1482	53	47	x	x	PUNCT
cana-1482	53	48	in	in	ADP
cana-1482	53	49	x	x	X
cana-1482	53	50	respectively	respectively	ADV
cana-1482	53	51	and	and	CCONJ
cana-1482	53	52	for	for	ADP
cana-1482	53	53	every	every	DET
cana-1482	53	54	x	x	NOUN
cana-1482	53	55	in	in	ADP
cana-1482	53	56	x	x	PUNCT
cana-1482	53	57	satisfying	satisfy	VERB
cana-1482	53	58	0	0	NUM
cana-1482	53	59			NUM
cana-1482	53	60	𝜇𝐴(𝑥	𝜇𝐴(𝑥	NOUN
cana-1482	53	61	)	)	PUNCT
cana-1482	54	1	+	+	PUNCT
cana-1482	54	2	𝜗𝐴(x	𝜗𝐴(x	X
cana-1482	54	3	)	)	PUNCT
cana-1482	54	4			PROPN
cana-1482	54	5	1	1	NUM
cana-1482	54	6	.	.	X
cana-1482	54	7	1.3	1.3	NUM
cana-1482	54	8	definition	definition	NOUN
cana-1482	54	9	:	:	PUNCT
cana-1482	54	10	a	a	DET
cana-1482	54	11	neutrosophic	neutrosophic	ADJ
cana-1482	54	12	subset	subset	NOUN
cana-1482	54	13	(	(	PUNCT
cana-1482	54	14	nss	nss	NOUN
cana-1482	54	15	)	)	PUNCT
cana-1482	54	16	�	�	PROPN
cana-1482	54	17	̅	̅	NOUN
cana-1482	54	18	�	�	NOUN
cana-1482	54	19	of	of	ADP
cana-1482	54	20	a	a	DET
cana-1482	54	21	set	set	NOUN
cana-1482	54	22	x	x	PUNCT
cana-1482	54	23	is	be	AUX
cana-1482	54	24	defined	define	VERB
cana-1482	54	25	as	as	ADP
cana-1482	54	26	an	an	DET
cana-1482	54	27	object	object	NOUN
cana-1482	54	28	of	of	ADP
cana-1482	54	29	the	the	DET
cana-1482	54	30	form	form	NOUN
cana-1482	54	31	�	�	NOUN
cana-1482	54	32	̅	̅	NOUN
cana-1482	54	33	�	�	NOUN
cana-1482	54	34	=	=	SYM
cana-1482	54	35	{	{	PUNCT
cana-1482	54	36			X
cana-1482	54	37	x	x	SYM
cana-1482	54	38	,	,	PUNCT
cana-1482	54	39	𝜇	𝜇	PART
cana-1482	54	40	�	�	NOUN
cana-1482	54	41	̅	̅	NOUN
cana-1482	54	42	�	�	NOUN
cana-1482	54	43	(𝑥	(𝑥	NOUN
cana-1482	54	44	)	)	PUNCT
cana-1482	54	45	,	,	PUNCT
cana-1482	54	46	𝜗	𝜗	PROPN
cana-1482	54	47	�	�	SYM
cana-1482	54	48	̅	̅	NOUN
cana-1482	54	49	�	�	NOUN
cana-1482	54	50	(𝑥	(𝑥	NOUN
cana-1482	54	51	)	)	PUNCT
cana-1482	54	52	,	,	PUNCT
cana-1482	54	53	γ	γ	PROPN
cana-1482	54	54	�	�	PROPN
cana-1482	54	55	̅	̅	NOUN
cana-1482	54	56	�	�	NOUN
cana-1482	54	57	(𝑥)	(𝑥)	NOUN
cana-1482	54	58	/	/	SYM
cana-1482	54	59	xx	xx	PROPN
cana-1482	54	60	}	}	PUNCT
cana-1482	54	61	,	,	PUNCT
cana-1482	54	62	where	where	SCONJ
cana-1482	54	63	𝜇	𝜇	ADP
cana-1482	54	64	�	�	PROPN
cana-1482	54	65	̅	̅	NOUN
cana-1482	54	66	�	�	NOUN
cana-1482	54	67	:	:	PUNCT
cana-1482	54	68	x→[0	x→[0	PROPN
cana-1482	54	69	,	,	PUNCT
cana-1482	54	70	1	1	NUM
cana-1482	54	71	]	]	PUNCT
cana-1482	54	72	and	and	CCONJ
cana-1482	54	73	𝜗	𝜗	PROPN
cana-1482	54	74	�	�	NOUN
cana-1482	54	75	̅	̅	NOUN
cana-1482	54	76	�	�	NOUN
cana-1482	54	77	:	:	PUNCT
cana-1482	54	78	x→[0	x→[0	PROPN
cana-1482	54	79	,	,	PUNCT
cana-1482	54	80	1	1	NUM
cana-1482	54	81	]	]	PUNCT
cana-1482	54	82	and	and	CCONJ
cana-1482	54	83	γ	γ	PROPN
cana-1482	54	84	�	�	PROPN
cana-1482	54	85	̅	̅	NOUN
cana-1482	54	86	�	�	PROPN
cana-1482	54	87	∶	∶	NOUN
cana-1482	54	88	x	x	SYM
cana-1482	54	89	→[0	→[0	PROPN
cana-1482	54	90	,	,	PUNCT
cana-1482	54	91	1	1	NUM
cana-1482	54	92	]	]	PUNCT
cana-1482	54	93	define	define	VERB
cana-1482	54	94	the	the	DET
cana-1482	54	95	degree	degree	NOUN
cana-1482	54	96	of	of	ADP
cana-1482	54	97	membership	membership	NOUN
cana-1482	54	98	,	,	PUNCT
cana-1482	54	99	degree	degree	NOUN
cana-1482	54	100	of	of	ADP
cana-1482	54	101	indeterminacy	indeterminacy	NOUN
cana-1482	54	102	and	and	CCONJ
cana-1482	54	103	the	the	DET
cana-1482	54	104	degree	degree	NOUN
cana-1482	54	105	of	of	ADP
cana-1482	54	106	non	non	ADJ
cana-1482	54	107	-	-	NOUN
cana-1482	54	108	membership	membership	NOUN
cana-1482	54	109	of	of	ADP
cana-1482	54	110	the	the	DET
cana-1482	54	111	element	element	NOUN
cana-1482	54	112	x	x	PUNCT
cana-1482	54	113	in	in	ADP
cana-1482	54	114	x	x	X
cana-1482	54	115	respectively	respectively	ADV
cana-1482	54	116	.	.	PUNCT
cana-1482	55	1	1.4	1.4	NUM
cana-1482	55	2	definition	definition	NOUN
cana-1482	55	3	:	:	PUNCT
cana-1482	55	4	let	let	VERB
cana-1482	55	5	�	�	PRON
cana-1482	55	6	̅	̅	VERB
cana-1482	55	7	�	�	PROPN
cana-1482	55	8	and	and	CCONJ
cana-1482	55	9	�	�	PROPN
cana-1482	55	10	̅	̅	NOUN
cana-1482	55	11	�	�	NOUN
cana-1482	55	12	be	be	VERB
cana-1482	55	13	any	any	DET
cana-1482	55	14	two	two	NUM
cana-1482	55	15	neutrosophic	neutrosophic	ADJ
cana-1482	55	16	subsets	subset	NOUN
cana-1482	55	17	of	of	ADP
cana-1482	55	18	a	a	DET
cana-1482	55	19	set	set	NOUN
cana-1482	55	20	x.	x.	NOUN
cana-1482	55	21	we	we	PRON
cana-1482	55	22	define	define	VERB
cana-1482	55	23	the	the	DET
cana-1482	55	24	following	follow	VERB
cana-1482	55	25	relations	relation	NOUN
cana-1482	55	26	and	and	CCONJ
cana-1482	55	27	operations	operation	NOUN
cana-1482	55	28	:	:	PUNCT
cana-1482	55	29	(	(	PUNCT
cana-1482	55	30	i	i	NOUN
cana-1482	55	31	)	)	PUNCT
cana-1482	55	32	�	�	PROPN
cana-1482	55	33	̅	̅	NOUN
cana-1482	55	34	�	�	PROPN
cana-1482	55	35			PROPN
cana-1482	55	36	�	�	PROPN
cana-1482	55	37	̅	̅	NOUN
cana-1482	55	38	�	�	PROPN
cana-1482	55	39	if	if	SCONJ
cana-1482	55	40	and	and	CCONJ
cana-1482	55	41	only	only	ADV
cana-1482	55	42	if	if	SCONJ
cana-1482	55	43	𝜇	𝜇	ADP
cana-1482	55	44	�	�	NOUN
cana-1482	55	45	̅	̅	NOUN
cana-1482	55	46	�	�	NOUN
cana-1482	55	47	(x	(x	NUM
cana-1482	55	48	)	)	PUNCT
cana-1482	55	49	≤	≤	NOUN
cana-1482	55	50	𝜇	𝜇	ADP
cana-1482	55	51	�	�	PROPN
cana-1482	55	52	̅	̅	NOUN
cana-1482	55	53	�	�	NOUN
cana-1482	55	54	(x	(x	NUM
cana-1482	55	55	)	)	PUNCT
cana-1482	55	56	and	and	CCONJ
cana-1482	55	57	𝜗	𝜗	NOUN
cana-1482	55	58	�	�	NOUN
cana-1482	55	59	̅	̅	NOUN
cana-1482	55	60	�	�	NOUN
cana-1482	55	61	(x	(x	NOUN
cana-1482	55	62	)	)	PUNCT
cana-1482	55	63	≤	≤	NOUN
cana-1482	55	64	𝜗	𝜗	SYM
cana-1482	55	65	�	�	SYM
cana-1482	55	66	̅	̅	NOUN
cana-1482	55	67	�	�	NOUN
cana-1482	55	68	(x	(x	NUM
cana-1482	55	69	)	)	PUNCT
cana-1482	55	70	and	and	CCONJ
cana-1482	55	71	γ	γ	PROPN
cana-1482	55	72	�	�	PROPN
cana-1482	55	73	̅	̅	NOUN
cana-1482	55	74	�	�	PROPN
cana-1482	55	75	(	(	PUNCT
cana-1482	55	76	x	x	NOUN
cana-1482	55	77	)	)	PUNCT
cana-1482	55	78	≤	≤	PROPN
cana-1482	55	79	γ	γ	PROPN
cana-1482	55	80	�	�	PROPN
cana-1482	55	81	̅	̅	NOUN
cana-1482	55	82	�	�	NOUN
cana-1482	55	83	(x	(x	NUM
cana-1482	55	84	)	)	PUNCT
cana-1482	55	85	for	for	ADP
cana-1482	55	86	all	all	DET
cana-1482	55	87	x	x	NOUN
cana-1482	55	88	in	in	ADP
cana-1482	55	89	x.	x.	PROPN
cana-1482	55	90	(	(	PUNCT
cana-1482	55	91	ii	ii	NOUN
cana-1482	55	92	)	)	PUNCT
cana-1482	55	93	�	�	PROPN
cana-1482	55	94	̅	̅	NOUN
cana-1482	55	95	�	�	PROPN
cana-1482	55	96	=	=	SYM
cana-1482	55	97	�	�	PROPN
cana-1482	55	98	̅	̅	NOUN
cana-1482	55	99	�	�	PROPN
cana-1482	55	100	if	if	SCONJ
cana-1482	55	101	and	and	CCONJ
cana-1482	55	102	only	only	ADV
cana-1482	55	103	if	if	SCONJ
cana-1482	55	104	𝜇	𝜇	ADP
cana-1482	55	105	�	�	NOUN
cana-1482	55	106	̅	̅	NOUN
cana-1482	55	107	�	�	NOUN
cana-1482	55	108	(x	(x	NUM
cana-1482	55	109	)	)	PUNCT
cana-1482	55	110	=	=	PUNCT
cana-1482	55	111	𝜇	𝜇	ADP
cana-1482	55	112	�	�	NOUN
cana-1482	55	113	̅	̅	NOUN
cana-1482	55	114	�	�	NOUN
cana-1482	55	115	(x	(x	NUM
cana-1482	55	116	)	)	PUNCT
cana-1482	55	117	and	and	CCONJ
cana-1482	55	118	𝜗	𝜗	NOUN
cana-1482	55	119	�	�	NOUN
cana-1482	55	120	̅	̅	NOUN
cana-1482	55	121	�	�	NOUN
cana-1482	55	122	(x	(x	NUM
cana-1482	55	123	)	)	PUNCT
cana-1482	55	124	=	=	SYM
cana-1482	56	1	𝜗	𝜗	NOUN
cana-1482	56	2	�	�	SYM
cana-1482	56	3	̅	̅	NOUN
cana-1482	56	4	�	�	NOUN
cana-1482	56	5	(x	(x	NUM
cana-1482	56	6	)	)	PUNCT
cana-1482	56	7	and	and	CCONJ
cana-1482	56	8	γ	γ	PROPN
cana-1482	56	9	�	�	PROPN
cana-1482	56	10	̅	̅	NOUN
cana-1482	56	11	�	�	NOUN
cana-1482	56	12	(	(	PUNCT
cana-1482	56	13	x	x	NOUN
cana-1482	56	14	)	)	PUNCT
cana-1482	56	15	=	=	SYM
cana-1482	56	16	γ	γ	X
cana-1482	56	17	�	�	PROPN
cana-1482	56	18	̅	̅	NOUN
cana-1482	56	19	�	�	NOUN
cana-1482	56	20	(x	(x	NUM
cana-1482	56	21	)	)	PUNCT
cana-1482	56	22	for	for	ADP
cana-1482	56	23	all	all	DET
cana-1482	56	24	x	x	NOUN
cana-1482	56	25	in	in	ADP
cana-1482	56	26	x.	x.	PROPN
cana-1482	56	27	(	(	PUNCT
cana-1482	56	28	iii	iii	NOUN
cana-1482	56	29	)	)	PUNCT
cana-1482	56	30	(	(	PUNCT
cana-1482	56	31	ā)c	ā)c	NOUN
cana-1482	56	32	=	=	PUNCT
cana-1482	56	33	{	{	PUNCT
cana-1482	56	34			X
cana-1482	56	35	x	x	X
cana-1482	56	36	,	,	PUNCT
cana-1482	56	37	γ	γ	PROPN
cana-1482	56	38	�	�	PROPN
cana-1482	56	39	̅	̅	NOUN
cana-1482	56	40	�	�	NOUN
cana-1482	56	41	(x	(x	NUM
cana-1482	56	42	)	)	PUNCT
cana-1482	56	43	,	,	PUNCT
cana-1482	56	44	1	1	NUM
cana-1482	56	45	−	−	NOUN
cana-1482	56	46	𝜗	𝜗	NOUN
cana-1482	56	47	�	�	SYM
cana-1482	56	48	̅	̅	NOUN
cana-1482	56	49	�	�	NOUN
cana-1482	56	50	(x	(x	NUM
cana-1482	56	51	)	)	PUNCT
cana-1482	56	52	,	,	PUNCT
cana-1482	56	53	𝜇	𝜇	ADP
cana-1482	56	54	�	�	NOUN
cana-1482	56	55	̅	̅	NOUN
cana-1482	56	56	�	�	NOUN
cana-1482	56	57	(x	(x	NOUN
cana-1482	56	58	)	)	PUNCT
cana-1482	56	59			PROPN
cana-1482	56	60	/	/	SYM
cana-1482	56	61	xx	xx	PROPN
cana-1482	56	62	}	}	PUNCT
cana-1482	56	63	.	.	PUNCT
cana-1482	57	1	(	(	PUNCT
cana-1482	57	2	iv	iv	X
cana-1482	57	3	)	)	PUNCT
cana-1482	57	4	�	�	PROPN
cana-1482	57	5	̅	̅	NOUN
cana-1482	57	6	�	�	PROPN
cana-1482	57	7			SYM
cana-1482	57	8	�	�	PROPN
cana-1482	57	9	̅	̅	NOUN
cana-1482	57	10	�	�	NOUN
cana-1482	57	11	=	=	SYM
cana-1482	57	12	{	{	PUNCT
cana-1482	57	13			X
cana-1482	57	14	x	x	SYM
cana-1482	57	15	,	,	PUNCT
cana-1482	57	16	min	min	NOUN
cana-1482	57	17	{	{	PUNCT
cana-1482	57	18	𝜇	𝜇	PART
cana-1482	57	19	�	�	NOUN
cana-1482	57	20	̅	̅	NOUN
cana-1482	57	21	�	�	NOUN
cana-1482	57	22	(x	(x	NUM
cana-1482	57	23	)	)	PUNCT
cana-1482	57	24	,	,	PUNCT
cana-1482	57	25	𝜇	𝜇	ADP
cana-1482	57	26	�	�	NOUN
cana-1482	57	27	̅	̅	NOUN
cana-1482	57	28	�	�	NOUN
cana-1482	57	29	(x	(x	NUM
cana-1482	57	30	)	)	PUNCT
cana-1482	57	31	}	}	PUNCT
cana-1482	57	32	,	,	PUNCT
cana-1482	57	33	min	min	NOUN
cana-1482	57	34	{	{	PUNCT
cana-1482	57	35	𝜗	𝜗	NOUN
cana-1482	57	36	�	�	NOUN
cana-1482	57	37	̅	̅	NOUN
cana-1482	57	38	�	�	NOUN
cana-1482	57	39	(x	(x	NUM
cana-1482	57	40	)	)	PUNCT
cana-1482	57	41	,	,	PUNCT
cana-1482	57	42	𝜗	𝜗	PROPN
cana-1482	57	43	�	�	SYM
cana-1482	57	44	̅	̅	NOUN
cana-1482	57	45	�	�	NOUN
cana-1482	57	46	(x	(x	NUM
cana-1482	57	47	)	)	PUNCT
cana-1482	57	48	}	}	PUNCT
cana-1482	57	49	,	,	PUNCT
cana-1482	57	50	max	max	PROPN
cana-1482	57	51	{	{	PUNCT
cana-1482	57	52	γ	γ	X
cana-1482	57	53	�	�	PROPN
cana-1482	57	54	̅	̅	NOUN
cana-1482	57	55	�	�	NOUN
cana-1482	57	56	(	(	PUNCT
cana-1482	57	57	x	x	NOUN
cana-1482	57	58	)	)	PUNCT
cana-1482	57	59	,	,	PUNCT
cana-1482	57	60	γ	γ	PROPN
cana-1482	57	61	�	�	PROPN
cana-1482	57	62	̅	̅	NOUN
cana-1482	57	63	�	�	NOUN
cana-1482	57	64	(x	(x	NUM
cana-1482	57	65	)	)	PUNCT
cana-1482	57	66	}	}	PUNCT
cana-1482	57	67			PROPN
cana-1482	57	68	/	/	SYM
cana-1482	57	69	xx	xx	PROPN
cana-1482	57	70	}	}	PUNCT
cana-1482	57	71	.	.	PUNCT
cana-1482	58	1	(	(	PUNCT
cana-1482	58	2	v	v	NOUN
cana-1482	58	3	)	)	PUNCT
cana-1482	58	4	�	�	NOUN
cana-1482	58	5	̅	̅	NOUN
cana-1482	58	6	�	�	PROPN
cana-1482	58	7			NOUN
cana-1482	58	8	�	�	PROPN
cana-1482	58	9	̅	̅	NOUN
cana-1482	58	10	�	�	NOUN
cana-1482	58	11	=	=	SYM
cana-1482	58	12	{	{	PUNCT
cana-1482	58	13			X
cana-1482	58	14	x	x	SYM
cana-1482	58	15	,	,	PUNCT
cana-1482	58	16	max	max	PROPN
cana-1482	58	17	{	{	PUNCT
cana-1482	58	18	𝜇	𝜇	NOUN
cana-1482	58	19	�	�	NOUN
cana-1482	58	20	̅	̅	NOUN
cana-1482	58	21	�	�	NOUN
cana-1482	58	22	(x	(x	NUM
cana-1482	58	23	)	)	PUNCT
cana-1482	58	24	,	,	PUNCT
cana-1482	58	25	𝜇	𝜇	ADP
cana-1482	58	26	�	�	NOUN
cana-1482	58	27	̅	̅	NOUN
cana-1482	58	28	�	�	NOUN
cana-1482	58	29	(x	(x	NUM
cana-1482	58	30	)	)	PUNCT
cana-1482	58	31	}	}	PUNCT
cana-1482	58	32	,	,	PUNCT
cana-1482	58	33	max	max	PROPN
cana-1482	58	34	{	{	PUNCT
cana-1482	58	35	𝜗	𝜗	PROPN
cana-1482	58	36	�	�	PROPN
cana-1482	58	37	̅	̅	NOUN
cana-1482	58	38	�	�	NOUN
cana-1482	58	39	(x	(x	NUM
cana-1482	58	40	)	)	PUNCT
cana-1482	58	41	,	,	PUNCT
cana-1482	58	42	𝜗	𝜗	PROPN
cana-1482	58	43	�	�	SYM
cana-1482	58	44	̅	̅	NOUN
cana-1482	58	45	�	�	NOUN
cana-1482	58	46	(x	(x	NUM
cana-1482	58	47	)	)	PUNCT
cana-1482	58	48	}	}	PUNCT
cana-1482	58	49	,	,	PUNCT
cana-1482	58	50	min	min	PROPN
cana-1482	58	51	{	{	PUNCT
cana-1482	58	52	γ	γ	X
cana-1482	58	53	�	�	PROPN
cana-1482	58	54	̅	̅	NOUN
cana-1482	58	55	�	�	NOUN
cana-1482	58	56	(	(	PUNCT
cana-1482	58	57	x	x	NOUN
cana-1482	58	58	)	)	PUNCT
cana-1482	58	59	,	,	PUNCT
cana-1482	58	60	γ	γ	PROPN
cana-1482	58	61	�	�	PROPN
cana-1482	58	62	̅	̅	NOUN
cana-1482	58	63	�	�	NOUN
cana-1482	58	64	(x	(x	NUM
cana-1482	58	65	)	)	PUNCT
cana-1482	58	66	}	}	PUNCT
cana-1482	58	67			PROPN
cana-1482	58	68	/	/	SYM
cana-1482	58	69	xx	xx	PROPN
cana-1482	58	70	}	}	PUNCT
cana-1482	58	71	.	.	PUNCT
cana-1482	59	1	(	(	PUNCT
cana-1482	59	2	vi	vi	X
cana-1482	59	3	)	)	PUNCT
cana-1482	59	4	0̅	0̅	NOUN
cana-1482	59	5	=	=	PUNCT
cana-1482	60	1	0	0	NUM
cana-1482	60	2	m	m	VERB
cana-1482	60	3	=	=	PUNCT
cana-1482	60	4	{	{	PUNCT
cana-1482	60	5	(	(	PUNCT
cana-1482	60	6	a	a	PRON
cana-1482	60	7	,	,	PUNCT
cana-1482	60	8	0	0	NUM
cana-1482	60	9	,	,	PUNCT
cana-1482	60	10	0	0	NUM
cana-1482	60	11	,	,	PUNCT
cana-1482	60	12	1	1	NUM
cana-1482	60	13	)	)	PUNCT
cana-1482	60	14	/	/	PUNCT
cana-1482	60	15	ak	ak	PUNCT
cana-1482	60	16	}	}	PUNCT
cana-1482	60	17	and	and	CCONJ
cana-1482	60	18	1̅	1̅	NUM
cana-1482	60	19	=	=	SYM
cana-1482	60	20	1	1	NUM
cana-1482	60	21	m	m	NOUN
cana-1482	60	22	=	=	PUNCT
cana-1482	60	23	{	{	PUNCT
cana-1482	60	24	(	(	PUNCT
cana-1482	60	25	a	a	DET
cana-1482	60	26	,	,	PUNCT
cana-1482	60	27	1	1	NUM
cana-1482	60	28	,	,	PUNCT
cana-1482	60	29	1	1	NUM
cana-1482	60	30	,	,	PUNCT
cana-1482	60	31	0	0	NUM
cana-1482	60	32	)	)	PUNCT
cana-1482	60	33	/	/	PUNCT
cana-1482	60	34	ak	ak	PUNCT
cana-1482	60	35	}	}	PUNCT
cana-1482	60	36	.	.	PUNCT
cana-1482	61	1	1.5	1.5	NUM
cana-1482	61	2	definition[8	definition[8	NOUN
cana-1482	61	3	]	]	PUNCT
cana-1482	61	4	:	:	PUNCT
cana-1482	61	5	let	let	VERB
cana-1482	61	6	x	x	PRON
cana-1482	61	7	be	be	AUX
cana-1482	61	8	a	a	DET
cana-1482	61	9	set	set	NOUN
cana-1482	61	10	and	and	CCONJ
cana-1482	61	11			NOUN
cana-1482	61	12	be	be	VERB
cana-1482	61	13	a	a	DET
cana-1482	61	14	family	family	NOUN
cana-1482	61	15	of	of	ADP
cana-1482	61	16	neutrosophic	neutrosophic	ADJ
cana-1482	61	17	subsets	subset	NOUN
cana-1482	61	18	of	of	ADP
cana-1482	61	19	x.	x.	NOUN
cana-1482	61	20	the	the	DET
cana-1482	61	21	family	family	NOUN
cana-1482	61	22			NOUN
cana-1482	61	23	is	be	AUX
cana-1482	61	24	called	call	VERB
cana-1482	61	25	an	an	DET
cana-1482	61	26	neutrosophic	neutrosophic	ADJ
cana-1482	61	27	topology	topology	NOUN
cana-1482	61	28	(	(	PUNCT
cana-1482	61	29	nst	nst	NOUN
cana-1482	61	30	)	)	PUNCT
cana-1482	61	31	on	on	ADP
cana-1482	61	32	x	x	SYM
cana-1482	61	33	if	if	SCONJ
cana-1482	61	34			NOUN
cana-1482	61	35	satisfies	satisfy	VERB
cana-1482	61	36	the	the	DET
cana-1482	61	37	following	follow	VERB
cana-1482	61	38	axioms	axiom	NOUN
cana-1482	61	39	(	(	PUNCT
cana-1482	61	40	i	i	NOUN
cana-1482	61	41	)	)	PUNCT
cana-1482	61	42	0̅	0̅	PROPN
cana-1482	61	43	,	,	PUNCT
cana-1482	61	44	1̅	1̅	PROPN
cana-1482	61	45	(	(	PUNCT
cana-1482	61	46	ii	ii	NOUN
cana-1482	61	47	)	)	PUNCT
cana-1482	61	48	if	if	SCONJ
cana-1482	61	49	{	{	PUNCT
cana-1482	61	50	�	�	NOUN
cana-1482	61	51	̅	̅	NOUN
cana-1482	61	52	�	�	NOUN
cana-1482	61	53	i	i	PRON
cana-1482	61	54	;	;	PUNCT
cana-1482	61	55	ii	ii	X
cana-1482	61	56	}	}	PUNCT
cana-1482	61	57			PROPN
cana-1482	61	58			NOUN
cana-1482	61	59	,	,	PUNCT
cana-1482	61	60	then	then	ADV
cana-1482	61	61	−	−	PROPN
cana-1482	61	62			PROPN
cana-1482	61	63			NOUN
cana-1482	61	64	i	i	PRON
cana-1482	61	65	ii	ii	VERB
cana-1482	61	66	a	a	DET
cana-1482	61	67			X
cana-1482	61	68	(	(	PUNCT
cana-1482	61	69	iii	iii	NOUN
cana-1482	61	70	)	)	PUNCT
cana-1482	61	71	if	if	SCONJ
cana-1482	61	72	�	�	NOUN
cana-1482	61	73	̅	̅	VERB
cana-1482	61	74	�	�	NOUN
cana-1482	61	75	1	1	NUM
cana-1482	61	76	,	,	PUNCT
cana-1482	61	77	�	�	NOUN
cana-1482	61	78	̅	̅	NOUN
cana-1482	61	79	�	�	NOUN
cana-1482	61	80	2	2	NUM
cana-1482	61	81	,	,	PUNCT
cana-1482	61	82	�	�	NOUN
cana-1482	61	83	̅	̅	NOUN
cana-1482	61	84	�	�	NOUN
cana-1482	61	85	3	3	NUM
cana-1482	61	86	,	,	PUNCT
cana-1482	61	87	…	…	PUNCT
cana-1482	61	88	..	..	PUNCT
cana-1482	61	89	�	�	SYM
cana-1482	61	90	̅	̅	NOUN
cana-1482	61	91	�	�	NOUN
cana-1482	61	92	n	n	NUM
cana-1482	61	93	,	,	PUNCT
cana-1482	61	94	then	then	ADV
cana-1482	61	95	−=	−=	VERB
cana-1482	61	96	=	=	SYM
cana-1482	61	97			NOUN
cana-1482	61	98	i	i	PRON
cana-1482	61	99	ni	ni	VERB
cana-1482	61	100	i	i	PRON
cana-1482	61	101	a	a	DET
cana-1482	61	102	1	1	NUM
cana-1482	61	103	.	.	PROPN
cana-1482	61	104	the	the	DET
cana-1482	61	105	pair	pair	NOUN
cana-1482	61	106	(	(	PUNCT
cana-1482	61	107	x	x	X
cana-1482	61	108	,	,	PUNCT
cana-1482	61	109			NOUN
cana-1482	61	110	)	)	PUNCT
cana-1482	61	111	is	be	AUX
cana-1482	61	112	called	call	VERB
cana-1482	61	113	an	an	DET
cana-1482	61	114	neutrosophic	neutrosophic	ADJ
cana-1482	61	115	topological	topological	ADJ
cana-1482	61	116	space	space	NOUN
cana-1482	61	117	(	(	PUNCT
cana-1482	61	118	nsts	nst	NOUN
cana-1482	61	119	)	)	PUNCT
cana-1482	61	120	.	.	PUNCT
cana-1482	62	1	the	the	DET
cana-1482	62	2	members	member	NOUN
cana-1482	62	3	of	of	ADP
cana-1482	62	4			NOUN
cana-1482	62	5	are	be	AUX
cana-1482	62	6	called	call	VERB
cana-1482	62	7	neutrosophic	neutrosophic	ADJ
cana-1482	62	8	open	open	ADJ
cana-1482	62	9	sets	set	NOUN
cana-1482	62	10	(	(	PUNCT
cana-1482	62	11	nsos	nsos	NOUN
cana-1482	62	12	)	)	PUNCT
cana-1482	62	13	in	in	ADP
cana-1482	62	14	x.	x.	PROPN
cana-1482	62	15	a	a	DET
cana-1482	62	16	neutrosophic	neutrosophic	PROPN
cana-1482	62	17	subset	subset	NOUN
cana-1482	62	18	�	�	PROPN
cana-1482	62	19	̅	̅	NOUN
cana-1482	62	20	�	�	NOUN
cana-1482	62	21	in	in	ADP
cana-1482	62	22	x	x	PROPN
cana-1482	62	23	is	be	AUX
cana-1482	62	24	said	say	VERB
cana-1482	62	25	to	to	PART
cana-1482	62	26	be	be	AUX
cana-1482	62	27	neutrosophic	neutrosophic	ADJ
cana-1482	62	28	closed	close	VERB
cana-1482	62	29	set	set	NOUN
cana-1482	62	30	(	(	PUNCT
cana-1482	62	31	nscs	nscs	PROPN
cana-1482	62	32	)	)	PUNCT
cana-1482	62	33	in	in	ADP
cana-1482	62	34	x	x	PUNCT
cana-1482	62	35	if	if	SCONJ
cana-1482	62	36	and	and	CCONJ
cana-1482	62	37	only	only	ADV
cana-1482	62	38	if	if	SCONJ
cana-1482	62	39	(	(	PUNCT
cana-1482	62	40	�	�	NOUN
cana-1482	62	41	̅	̅	NOUN
cana-1482	62	42	�	�	NOUN
cana-1482	62	43	)c	)c	PUNCT
cana-1482	62	44	is	be	AUX
cana-1482	62	45	a	a	DET
cana-1482	62	46	nsos	nsos	NOUN
cana-1482	62	47	in	in	ADP
cana-1482	62	48	x.	x.	PROPN
cana-1482	62	49	1.6	1.6	NUM
cana-1482	62	50	definition	definition	NOUN
cana-1482	62	51	:	:	PUNCT
cana-1482	62	52	let	let	VERB
cana-1482	62	53	(	(	PUNCT
cana-1482	62	54	x	x	X
cana-1482	62	55	,	,	PUNCT
cana-1482	62	56			NOUN
cana-1482	62	57	)	)	PUNCT
cana-1482	62	58	be	be	AUX
cana-1482	62	59	an	an	DET
cana-1482	62	60	nsts	nst	NOUN
cana-1482	62	61	and	and	CCONJ
cana-1482	62	62	�	�	NOUN
cana-1482	62	63	̅	̅	NOUN
cana-1482	62	64	�	�	NOUN
cana-1482	62	65	be	be	AUX
cana-1482	62	66	an	an	DET
cana-1482	62	67	nss	nss	NOUN
cana-1482	62	68	in	in	ADP
cana-1482	62	69	x.	x.	NOUN
cana-1482	62	70	then	then	ADV
cana-1482	62	71	the	the	DET
cana-1482	62	72	neutrosophic	neutrosophic	ADJ
cana-1482	62	73	interior	interior	NOUN
cana-1482	62	74	and	and	CCONJ
cana-1482	62	75	neutrosophic	neutrosophic	ADJ
cana-1482	62	76	closure	closure	NOUN
cana-1482	62	77	are	be	AUX
cana-1482	62	78	defined	define	VERB
cana-1482	62	79	by	by	ADP
cana-1482	62	80	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	ADJ
cana-1482	62	81	�	�	PROPN
cana-1482	62	82	̅	̅	NOUN
cana-1482	62	83	�	�	NOUN
cana-1482	62	84	)	)	PUNCT
cana-1482	62	85	=	=	SYM
cana-1482	62	86	∪	∪	X
cana-1482	62	87	{	{	PUNCT
cana-1482	62	88	�	�	NOUN
cana-1482	62	89	̅	̅	NOUN
cana-1482	62	90	�	�	PROPN
cana-1482	62	91	∶	∶	PROPN
cana-1482	62	92	�	�	NOUN
cana-1482	62	93	̅	̅	NOUN
cana-1482	62	94	�	�	NOUN
cana-1482	62	95	is	be	AUX
cana-1482	62	96	an	an	DET
cana-1482	62	97	nsos	nsos	NOUN
cana-1482	62	98	in	in	ADP
cana-1482	62	99	x	x	PART
cana-1482	62	100	and	and	CCONJ
cana-1482	62	101	�	�	PROPN
cana-1482	62	102	̅	̅	NOUN
cana-1482	62	103	�	�	PROPN
cana-1482	62	104	⊆	⊆	NUM
cana-1482	62	105	�	�	NOUN
cana-1482	62	106	̅	̅	NOUN
cana-1482	62	107	�	�	PROPN
cana-1482	62	108	}	}	PUNCT
cana-1482	62	109	,	,	PUNCT
cana-1482	62	110	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	62	111	�	�	PROPN
cana-1482	62	112	̅	̅	NOUN
cana-1482	62	113	�	�	NOUN
cana-1482	62	114	)	)	PUNCT
cana-1482	62	115	=	=	SYM
cana-1482	62	116	∩	∩	NOUN
cana-1482	62	117	{	{	PUNCT
cana-1482	62	118	�	�	PROPN
cana-1482	62	119	̅	̅	NOUN
cana-1482	62	120	�	�	PROPN
cana-1482	62	121	∶	∶	PROPN
cana-1482	62	122	�	�	NOUN
cana-1482	62	123	̅	̅	NOUN
cana-1482	62	124	�	�	NOUN
cana-1482	62	125	is	be	AUX
cana-1482	62	126	an	an	DET
cana-1482	62	127	nscs	nscs	NOUN
cana-1482	62	128	in	in	ADP
cana-1482	62	129	x	x	PUNCT
cana-1482	62	130	and	and	CCONJ
cana-1482	62	131	�	�	PROPN
cana-1482	62	132	̅	̅	NOUN
cana-1482	62	133	�	�	PROPN
cana-1482	62	134	⊆	⊆	NUM
cana-1482	62	135	�	�	NOUN
cana-1482	62	136	̅	̅	NOUN
cana-1482	62	137	�	�	NOUN
cana-1482	62	138	}	}	PUNCT
cana-1482	62	139	.	.	PUNCT
cana-1482	63	1	for	for	ADP
cana-1482	63	2	any	any	DET
cana-1482	63	3	nss	nss	NOUN
cana-1482	63	4	�	�	NOUN
cana-1482	63	5	̅	̅	NOUN
cana-1482	63	6	�	�	NOUN
cana-1482	63	7	in	in	ADP
cana-1482	63	8	(	(	PUNCT
cana-1482	63	9	x	x	NOUN
cana-1482	63	10	,	,	PUNCT
cana-1482	63	11			NOUN
cana-1482	63	12	)	)	PUNCT
cana-1482	63	13	,	,	PUNCT
cana-1482	63	14	we	we	PRON
cana-1482	63	15	have	have	VERB
cana-1482	63	16	𝑛𝑠𝑐𝑙(a	𝑛𝑠𝑐𝑙(a	NOUN
cana-1482	63	17	𝑐	𝑐	NOUN
cana-1482	63	18	)	)	PUNCT
cana-1482	63	19	=	=	SYM
cana-1482	63	20	(	(	PUNCT
cana-1482	63	21	𝑛𝑠𝑖𝑛𝑡(a	𝑛𝑠𝑖𝑛𝑡(a	NOUN
cana-1482	63	22	)	)	PUNCT
cana-1482	63	23	)	)	PUNCT
cana-1482	63	24	𝑐	𝑐	PROPN
cana-1482	63	25	and	and	CCONJ
cana-1482	63	26	𝑛𝑠𝑖𝑛𝑡(a	𝑛𝑠𝑖𝑛𝑡(a	NOUN
cana-1482	63	27	𝑐	𝑐	NOUN
cana-1482	63	28	)	)	PUNCT
cana-1482	63	29	=	=	SYM
cana-1482	63	30	(	(	PUNCT
cana-1482	63	31	𝑛𝑠𝑐𝑙(a	𝑛𝑠𝑐𝑙(a	NOUN
cana-1482	63	32	)	)	PUNCT
cana-1482	63	33	)	)	PUNCT
cana-1482	64	1	𝑐.	𝑐.	NOUN
cana-1482	64	2	communications	communication	NOUN
cana-1482	64	3	on	on	ADP
cana-1482	64	4	applied	apply	VERB
cana-1482	64	5	nonlinear	nonlinear	ADJ
cana-1482	64	6	analysis	analysis	NOUN
cana-1482	64	7	issn	issn	NOUN
cana-1482	64	8	:	:	PUNCT
cana-1482	64	9	1074	1074	NUM
cana-1482	64	10	-	-	PUNCT
cana-1482	64	11	133x	133x	NUM
cana-1482	64	12	vol	vol	NOUN
cana-1482	64	13	31	31	NUM
cana-1482	64	14	no	no	NOUN
cana-1482	64	15	.	.	PUNCT
cana-1482	65	1	8s	8s	PROPN
cana-1482	65	2	(	(	PUNCT
cana-1482	65	3	2024	2024	NUM
cana-1482	65	4	)	)	PUNCT
cana-1482	65	5	275	275	NUM
cana-1482	65	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	65	7	1.7	1.7	NUM
cana-1482	65	8	definition	definition	NOUN
cana-1482	65	9	:	:	PUNCT
cana-1482	65	10	an	an	DET
cana-1482	65	11	nss	nss	NOUN
cana-1482	65	12	�	�	NOUN
cana-1482	65	13	̅	̅	NOUN
cana-1482	65	14	�	�	NOUN
cana-1482	65	15	of	of	ADP
cana-1482	65	16	an	an	DET
cana-1482	65	17	nsts	nst	NOUN
cana-1482	65	18	(	(	PUNCT
cana-1482	65	19	x	x	X
cana-1482	65	20	,	,	PUNCT
cana-1482	65	21			NOUN
cana-1482	65	22	)	)	PUNCT
cana-1482	65	23	is	be	AUX
cana-1482	65	24	said	say	VERB
cana-1482	65	25	to	to	PART
cana-1482	65	26	be	be	AUX
cana-1482	65	27	a	a	DET
cana-1482	65	28	(	(	PUNCT
cana-1482	65	29	i	i	NOUN
cana-1482	65	30	)	)	PUNCT
cana-1482	65	31	n	n	CCONJ
cana-1482	65	32	e	e	X
cana-1482	65	33	u	u	NOUN
cana-1482	65	34	t	t	NOUN
cana-1482	65	35	r	r	NOUN
cana-1482	65	36	o	o	X
cana-1482	65	37	s	s	X
cana-1482	65	38	o	o	X
cana-1482	65	39	p	p	NOUN
cana-1482	65	40	h	h	NOUN
cana-1482	66	1	i	i	NOUN
cana-1482	66	2	c	c	NOUN
cana-1482	67	1	r	r	NOUN
cana-1482	67	2	e	e	NOUN
cana-1482	67	3	g	g	NOUN
cana-1482	67	4	u	u	PROPN
cana-1482	67	5	l	l	NOUN
cana-1482	67	6	a	a	DET
cana-1482	67	7	r	r	NOUN
cana-1482	67	8	c	c	NOUN
cana-1482	67	9	l	l	NOUN
cana-1482	67	10	o	o	X
cana-1482	67	11	s	s	X
cana-1482	67	12	e	e	X
cana-1482	67	13	d	d	X
cana-1482	67	14	s	s	PROPN
cana-1482	67	15	e	e	X
cana-1482	67	16	t	t	PROPN
cana-1482	67	17	(	(	PUNCT
cana-1482	67	18	n	n	NOUN
cana-1482	67	19	s	s	NOUN
cana-1482	67	20	r	r	NOUN
cana-1482	67	21	c	c	NOUN
cana-1482	67	22	s	s	NOUN
cana-1482	67	23	f	f	NOUN
cana-1482	67	24	o	o	NOUN
cana-1482	67	25	r	r	NOUN
cana-1482	67	26	s	s	NOUN
cana-1482	67	27	h	h	NOUN
cana-1482	67	28	o	o	NOUN
cana-1482	67	29	r	r	NOUN
cana-1482	67	30	t	t	PROPN
cana-1482	67	31	)	)	PUNCT
cana-1482	68	1	i	i	PRON
cana-1482	68	2	f	f	PROPN
cana-1482	68	3	�	�	PROPN
cana-1482	68	4	̅	̅	NOUN
cana-1482	68	5	�	�	NOUN
cana-1482	68	6	=	=	SYM
cana-1482	68	7	𝑛𝑠𝑐𝑙	𝑛𝑠𝑐𝑙	PROPN
cana-1482	68	8	(	(	PUNCT
cana-1482	68	9	𝑛𝑠𝑖𝑛𝑡(a	𝑛𝑠𝑖𝑛𝑡(a	NOUN
cana-1482	68	10	)	)	PUNCT
cana-1482	68	11	)	)	PUNCT
cana-1482	68	12	(	(	PUNCT
cana-1482	68	13	ii	ii	NOUN
cana-1482	68	14	)	)	PUNCT
cana-1482	68	15	neutrosophic	neutrosophic	PROPN
cana-1482	68	16	semiclosed	semiclose	VERB
cana-1482	68	17	set	set	NOUN
cana-1482	68	18	(	(	PUNCT
cana-1482	68	19	nsscs	nssc	NOUN
cana-1482	68	20	for	for	ADP
cana-1482	68	21	short	short	ADJ
cana-1482	68	22	)	)	PUNCT
cana-1482	68	23	if	if	SCONJ
cana-1482	68	24	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(a̅	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(a̅	NUM
cana-1482	68	25	)	)	PUNCT
cana-1482	68	26	)	)	PUNCT
cana-1482	68	27	⊆	⊆	NUM
cana-1482	68	28	�	�	NOUN
cana-1482	68	29	̅	̅	NOUN
cana-1482	68	30	�	�	NOUN
cana-1482	68	31	(	(	PUNCT
cana-1482	68	32	i	i	PRON
cana-1482	68	33	i	i	VERB
cana-1482	68	34	i	i	VERB
cana-1482	68	35	)	)	PUNCT
cana-1482	68	36	n	n	CCONJ
cana-1482	68	37	e	e	X
cana-1482	68	38	u	u	NOUN
cana-1482	68	39	t	t	NOUN
cana-1482	68	40	r	r	NOUN
cana-1482	68	41	o	o	X
cana-1482	68	42	s	s	X
cana-1482	68	43	o	o	X
cana-1482	68	44	p	p	NOUN
cana-1482	68	45	h	h	NOUN
cana-1482	69	1	i	i	NOUN
cana-1482	69	2	c	c	VERB
cana-1482	69	3	p	p	NOUN
cana-1482	69	4	r	r	NOUN
cana-1482	69	5	e	e	NOUN
cana-1482	69	6	c	c	NOUN
cana-1482	69	7	l	l	NOUN
cana-1482	69	8	o	o	X
cana-1482	69	9	s	s	X
cana-1482	69	10	e	e	X
cana-1482	69	11	d	d	X
cana-1482	69	12	s	s	PROPN
cana-1482	69	13	e	e	X
cana-1482	69	14	t	t	PROPN
cana-1482	69	15	(	(	PUNCT
cana-1482	69	16	n	n	PROPN
cana-1482	69	17	s	s	VERB
cana-1482	69	18	p	p	X
cana-1482	69	19	c	c	NOUN
cana-1482	69	20	s	s	NOUN
cana-1482	70	1	f	f	NOUN
cana-1482	70	2	o	o	NOUN
cana-1482	70	3	r	r	NOUN
cana-1482	70	4	s	s	NOUN
cana-1482	70	5	h	h	NOUN
cana-1482	70	6	o	o	NOUN
cana-1482	70	7	r	r	NOUN
cana-1482	70	8	t	t	PROPN
cana-1482	70	9	)	)	PUNCT
cana-1482	71	1	i	i	PRON
cana-1482	71	2	f	f	PROPN
cana-1482	71	3	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(a̅	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(a̅	PROPN
cana-1482	71	4	)	)	PUNCT
cana-1482	71	5	)	)	PUNCT
cana-1482	72	1	⊆	⊆	NUM
cana-1482	72	2	�	�	NOUN
cana-1482	72	3	̅	̅	NOUN
cana-1482	72	4	�	�	PROPN
cana-1482	72	5	(	(	PUNCT
cana-1482	72	6	iv	iv	X
cana-1482	72	7	)	)	PUNCT
cana-1482	72	8	neutrosophic	neutrosophic	PROPN
cana-1482	72	9	α	α	PROPN
cana-1482	72	10	closed	close	VERB
cana-1482	72	11	set	set	NOUN
cana-1482	72	12	(	(	PUNCT
cana-1482	72	13	nsαcs	nsαcs	NOUN
cana-1482	72	14	for	for	ADP
cana-1482	72	15	short	short	ADJ
cana-1482	72	16	)	)	PUNCT
cana-1482	72	17	if	if	SCONJ
cana-1482	72	18	𝑛𝑠𝑐𝑙	𝑛𝑠𝑐𝑙	PROPN
cana-1482	72	19	(	(	PUNCT
cana-1482	72	20	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(a̅	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(a̅	NOUN
cana-1482	72	21	)	)	PUNCT
cana-1482	72	22	)	)	PUNCT
cana-1482	72	23	)	)	PUNCT
cana-1482	72	24	⊆	⊆	NUM
cana-1482	72	25	�	�	NOUN
cana-1482	72	26	̅	̅	NOUN
cana-1482	72	27	�	�	PROPN
cana-1482	72	28	(	(	PUNCT
cana-1482	72	29	v	v	NOUN
cana-1482	72	30	)	)	PUNCT
cana-1482	72	31	neutrosophic	neutrosophic	ADJ
cana-1482	72	32	β	β	X
cana-1482	72	33	closed	close	VERB
cana-1482	72	34	set	set	NOUN
cana-1482	72	35	(	(	PUNCT
cana-1482	72	36	nsβcs	nsβcs	NOUN
cana-1482	72	37	for	for	ADP
cana-1482	72	38	short	short	ADJ
cana-1482	72	39	)	)	PUNCT
cana-1482	72	40	if	if	SCONJ
cana-1482	72	41	𝑛𝑠𝑖𝑛𝑡	𝑛𝑠𝑖𝑛𝑡	VERB
cana-1482	72	42	(	(	PUNCT
cana-1482	72	43	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(a̅	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(a̅	NOUN
cana-1482	72	44	)	)	PUNCT
cana-1482	72	45	)	)	PUNCT
cana-1482	72	46	)	)	PUNCT
cana-1482	73	1	⊆	⊆	NUM
cana-1482	73	2	�	�	SYM
cana-1482	73	3	̅	̅	NOUN
cana-1482	73	4	�	�	NOUN
cana-1482	73	5	.	.	PUNCT
cana-1482	73	6	1.8	1.8	NUM
cana-1482	73	7	definition	definition	NOUN
cana-1482	73	8	:	:	PUNCT
cana-1482	73	9	an	an	DET
cana-1482	73	10	nss	nss	NOUN
cana-1482	73	11	�	�	NOUN
cana-1482	73	12	̅	̅	NOUN
cana-1482	73	13	�	�	NOUN
cana-1482	73	14	of	of	ADP
cana-1482	73	15	an	an	DET
cana-1482	73	16	nsts	nst	NOUN
cana-1482	73	17	(	(	PUNCT
cana-1482	73	18	x	x	X
cana-1482	73	19	,	,	PUNCT
cana-1482	73	20			NOUN
cana-1482	73	21	)	)	PUNCT
cana-1482	73	22	is	be	AUX
cana-1482	73	23	said	say	VERB
cana-1482	73	24	to	to	PART
cana-1482	73	25	be	be	AUX
cana-1482	73	26	an	an	DET
cana-1482	73	27	(	(	PUNCT
cana-1482	73	28	i	i	NOUN
cana-1482	73	29	)	)	PUNCT
cana-1482	73	30	n	n	CCONJ
cana-1482	73	31	e	e	X
cana-1482	73	32	u	u	NOUN
cana-1482	73	33	t	t	NOUN
cana-1482	73	34	r	r	NOUN
cana-1482	73	35	o	o	X
cana-1482	73	36	s	s	X
cana-1482	73	37	o	o	X
cana-1482	73	38	p	p	NOUN
cana-1482	73	39	h	h	NOUN
cana-1482	74	1	i	i	PRON
cana-1482	74	2	c	c	VERB
cana-1482	74	3	g	g	NOUN
cana-1482	74	4	e	e	PROPN
cana-1482	74	5	n	n	X
cana-1482	74	6	e	e	NOUN
cana-1482	74	7	r	r	NOUN
cana-1482	74	8	a	a	PROPN
cana-1482	74	9	l	l	NOUN
cana-1482	75	1	i	i	NOUN
cana-1482	75	2	z	z	NOUN
cana-1482	75	3	e	e	PUNCT
cana-1482	76	1	d	d	X
cana-1482	76	2	c	c	X
cana-1482	76	3	l	l	NOUN
cana-1482	76	4	o	o	X
cana-1482	76	5	s	s	X
cana-1482	76	6	e	e	X
cana-1482	76	7	d	d	X
cana-1482	76	8	s	s	PROPN
cana-1482	76	9	e	e	X
cana-1482	76	10	t	t	PROPN
cana-1482	76	11	(	(	PUNCT
cana-1482	76	12	n	n	NOUN
cana-1482	76	13	s	s	PART
cana-1482	76	14	g	g	NOUN
cana-1482	76	15	c	c	NOUN
cana-1482	76	16	s	s	NOUN
cana-1482	76	17	f	f	NOUN
cana-1482	76	18	o	o	NOUN
cana-1482	76	19	r	r	NOUN
cana-1482	76	20	s	s	NOUN
cana-1482	76	21	h	h	NOUN
cana-1482	76	22	o	o	NOUN
cana-1482	76	23	r	r	NOUN
cana-1482	76	24	t	t	PROPN
cana-1482	76	25	)	)	PUNCT
cana-1482	77	1	i	i	PRON
cana-1482	77	2	f	f	NOUN
cana-1482	77	3	𝑛𝑠𝑐𝑙(a̅	𝑛𝑠𝑐𝑙(a̅	NOUN
cana-1482	77	4	)	)	PUNCT
cana-1482	77	5	=	=	SYM
cana-1482	77	6	�	�	PROPN
cana-1482	77	7	̅	̅	NOUN
cana-1482	77	8	�	�	NOUN
cana-1482	77	9	,	,	PUNCT
cana-1482	77	10	whenever	whenever	SCONJ
cana-1482	77	11	�	�	PROPN
cana-1482	77	12	̅	̅	NOUN
cana-1482	77	13	�	�	PROPN
cana-1482	77	14	⊆	⊆	NUM
cana-1482	77	15	�	�	NOUN
cana-1482	77	16	̅	̅	NOUN
cana-1482	77	17	�	�	PROPN
cana-1482	77	18	and	and	CCONJ
cana-1482	77	19	�	�	PROPN
cana-1482	77	20	̅	̅	NOUN
cana-1482	77	21	�	�	NOUN
cana-1482	77	22	is	be	AUX
cana-1482	77	23	an	an	DET
cana-1482	77	24	nsos	nsos	ADJ
cana-1482	77	25	(	(	PUNCT
cana-1482	77	26	ii	ii	NOUN
cana-1482	77	27	)	)	PUNCT
cana-1482	77	28	neutrosophic	neutrosophic	ADJ
cana-1482	77	29	regular	regular	ADJ
cana-1482	77	30	generalized	generalize	VERB
cana-1482	77	31	closed	close	VERB
cana-1482	77	32	set	set	NOUN
cana-1482	77	33	(	(	PUNCT
cana-1482	77	34	nsrgcs	nsrgcs	NOUN
cana-1482	77	35	for	for	ADP
cana-1482	77	36	short	short	ADJ
cana-1482	77	37	)	)	PUNCT
cana-1482	77	38	if	if	SCONJ
cana-1482	77	39	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	77	40	�	�	PROPN
cana-1482	77	41	̅	̅	NOUN
cana-1482	77	42	�	�	NOUN
cana-1482	77	43	)	)	PUNCT
cana-1482	77	44	⊆	⊆	NUM
cana-1482	77	45	�	�	NOUN
cana-1482	77	46	̅	̅	NOUN
cana-1482	77	47	�	�	NOUN
cana-1482	77	48	,	,	PUNCT
cana-1482	77	49	whenever	whenever	SCONJ
cana-1482	77	50	�	�	PROPN
cana-1482	77	51	̅	̅	NOUN
cana-1482	77	52	�	�	PROPN
cana-1482	77	53	⊆	⊆	NUM
cana-1482	77	54	�	�	NOUN
cana-1482	77	55	̅	̅	NOUN
cana-1482	77	56	�	�	PROPN
cana-1482	77	57	and	and	CCONJ
cana-1482	77	58	�	�	PROPN
cana-1482	77	59	̅	̅	NOUN
cana-1482	77	60	�	�	NOUN
cana-1482	77	61	is	be	AUX
cana-1482	77	62	an	an	DET
cana-1482	77	63	nsros	nsro	NOUN
cana-1482	77	64	.	.	PUNCT
cana-1482	78	1	1.9	1.9	NUM
cana-1482	78	2	definition	definition	NOUN
cana-1482	78	3	:	:	PUNCT
cana-1482	78	4	a	a	DET
cana-1482	78	5	nss	nss	NOUN
cana-1482	78	6	�	�	NOUN
cana-1482	78	7	̅	̅	NOUN
cana-1482	78	8	�	�	NOUN
cana-1482	78	9	of	of	ADP
cana-1482	78	10	an	an	DET
cana-1482	78	11	nsts	nst	NOUN
cana-1482	78	12	(	(	PUNCT
cana-1482	78	13	x	x	X
cana-1482	78	14	,	,	PUNCT
cana-1482	78	15			NOUN
cana-1482	78	16	)	)	PUNCT
cana-1482	78	17	is	be	AUX
cana-1482	78	18	said	say	VERB
cana-1482	78	19	to	to	PART
cana-1482	78	20	be	be	AUX
cana-1482	78	21	an	an	DET
cana-1482	78	22	(	(	PUNCT
cana-1482	78	23	i	i	NOUN
cana-1482	78	24	)	)	PUNCT
cana-1482	78	25	neutrosophic	neutrosophic	PROPN
cana-1482	78	26	semipreclosed	semipreclose	VERB
cana-1482	78	27	set	set	NOUN
cana-1482	78	28	(	(	PUNCT
cana-1482	78	29	nsspcs	nsspc	NOUN
cana-1482	78	30	for	for	ADP
cana-1482	78	31	short	short	ADJ
cana-1482	78	32	)	)	PUNCT
cana-1482	78	33	if	if	SCONJ
cana-1482	78	34	there	there	PRON
cana-1482	78	35	exists	exist	VERB
cana-1482	78	36	an	an	DET
cana-1482	78	37	nspcs	nspcs	PROPN
cana-1482	78	38	�	�	NOUN
cana-1482	78	39	̅	̅	NOUN
cana-1482	78	40	�	�	NOUN
cana-1482	78	41	such	such	ADJ
cana-1482	78	42	that	that	SCONJ
cana-1482	78	43	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	ADJ
cana-1482	78	44	�	�	NOUN
cana-1482	78	45	̅	̅	NOUN
cana-1482	78	46	�	�	NOUN
cana-1482	78	47	)	)	PUNCT
cana-1482	78	48	⊆	⊆	NUM
cana-1482	78	49	�	�	NOUN
cana-1482	78	50	̅	̅	NOUN
cana-1482	78	51	�	�	PROPN
cana-1482	78	52	⊆	⊆	NUM
cana-1482	78	53	�	�	NOUN
cana-1482	78	54	̅	̅	NOUN
cana-1482	78	55	�	�	PROPN
cana-1482	78	56	(	(	PUNCT
cana-1482	78	57	ii	ii	NOUN
cana-1482	78	58	)	)	PUNCT
cana-1482	78	59	neutrosophic	neutrosophic	ADJ
cana-1482	78	60	semipreopen	semipreopen	NOUN
cana-1482	78	61	set	set	NOUN
cana-1482	78	62	(	(	PUNCT
cana-1482	78	63	nsspos	nsspos	X
cana-1482	78	64	for	for	ADP
cana-1482	78	65	short	short	ADJ
cana-1482	78	66	)	)	PUNCT
cana-1482	78	67	if	if	SCONJ
cana-1482	78	68	there	there	PRON
cana-1482	78	69	exists	exist	VERB
cana-1482	78	70	an	an	DET
cana-1482	78	71	nspos	nspos	NOUN
cana-1482	78	72	�	�	NOUN
cana-1482	78	73	̅	̅	NOUN
cana-1482	78	74	�	�	NOUN
cana-1482	78	75	such	such	ADJ
cana-1482	78	76	that	that	PRON
cana-1482	78	77	�	�	PROPN
cana-1482	78	78	̅	̅	NOUN
cana-1482	78	79	�	�	PROPN
cana-1482	78	80	⊆	⊆	NUM
cana-1482	78	81	�	�	NOUN
cana-1482	78	82	̅	̅	NOUN
cana-1482	78	83	�	�	NOUN
cana-1482	78	84	⊆	⊆	NUM
cana-1482	78	85	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	78	86	�	�	PROPN
cana-1482	78	87	̅	̅	NOUN
cana-1482	78	88	�	�	PROPN
cana-1482	78	89	)	)	PUNCT
cana-1482	78	90	.	.	PUNCT
cana-1482	79	1	1	1	NUM
cana-1482	79	2	.10	.10	NUM
cana-1482	79	3	de	de	X
cana-1482	79	4	f	f	PROPN
cana-1482	79	5	in	in	ADP
cana-1482	79	6	i	i	PRON
cana-1482	79	7	t	t	PROPN
cana-1482	79	8	ion	ion	NOUN
cana-1482	79	9	:	:	PUNCT
cana-1482	79	10	t	t	PROPN
cana-1482	79	11	wo	will	AUX
cana-1482	79	12	nsssa	nsssa	VERB
cana-1482	79	13	andb	andb	PROPN
cana-1482	79	14	ar	ar	PROPN
cana-1482	79	15	e	e	PROPN
cana-1482	79	16	s	s	PROPN
cana-1482	79	17	a	a	PRON
cana-1482	79	18	i	i	PROPN
cana-1482	79	19	d	d	PROPN
cana-1482	79	20	t	t	NOUN
cana-1482	79	21	o	o	NOUN
cana-1482	79	22	be	be	AUX
cana-1482	79	23	no	no	DET
cana-1482	79	24	t	t	NOUN
cana-1482	79	25	q	q	X
cana-1482	79	26	co	co	NOUN
cana-1482	79	27	in	in	ADP
cana-1482	79	28	c	c	PROPN
cana-1482	79	29	iden	iden	PROPN
cana-1482	80	1	t	t	PROPN
cana-1482	80	2	i	i	PRON
cana-1482	80	3	f	f	PROPN
cana-1482	80	4	and	and	CCONJ
cana-1482	80	5	only	only	ADV
cana-1482	80	6	ifa	ifa	PROPN
cana-1482	80	7	⊆	⊆	NUM
cana-1482	80	8	b	b	NOUN
cana-1482	80	9	𝑐.	𝑐.	NOUN
cana-1482	81	1	1.11	1.11	NUM
cana-1482	81	2	definition	definition	NOUN
cana-1482	81	3	:	:	PUNCT
cana-1482	81	4	let	let	VERB
cana-1482	81	5	�	�	PRON
cana-1482	81	6	̅	̅	NOUN
cana-1482	81	7	�	�	NOUN
cana-1482	81	8	be	be	AUX
cana-1482	81	9	an	an	DET
cana-1482	81	10	nss	nss	NOUN
cana-1482	81	11	in	in	ADP
cana-1482	81	12	an	an	DET
cana-1482	81	13	nsts	nst	NOUN
cana-1482	81	14	(	(	PUNCT
cana-1482	81	15	x	x	X
cana-1482	81	16	,	,	PUNCT
cana-1482	81	17			NOUN
cana-1482	81	18	)	)	PUNCT
cana-1482	81	19	.	.	PUNCT
cana-1482	82	1	then	then	ADV
cana-1482	82	2	the	the	DET
cana-1482	82	3	neutrosophic	neutrosophic	ADJ
cana-1482	82	4	semipre	semipre	ADJ
cana-1482	82	5	interior	interior	PROPN
cana-1482	82	6	of	of	ADP
cana-1482	82	7	�	�	PROPN
cana-1482	82	8	̅	̅	NOUN
cana-1482	82	9	�	�	PROPN
cana-1482	82	10	(	(	PUNCT
cana-1482	82	11	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	NOUN
cana-1482	82	12	�	�	SYM
cana-1482	82	13	̅	̅	NOUN
cana-1482	82	14	�	�	PROPN
cana-1482	82	15	)	)	PUNCT
cana-1482	82	16	for	for	ADP
cana-1482	82	17	short	short	ADJ
cana-1482	82	18	)	)	PUNCT
cana-1482	82	19	and	and	CCONJ
cana-1482	82	20	the	the	DET
cana-1482	82	21	neutrosophic	neutrosophic	ADJ
cana-1482	82	22	semipre	semipre	VERB
cana-1482	82	23	closure	closure	NOUN
cana-1482	82	24	of	of	ADP
cana-1482	82	25	�	�	NOUN
cana-1482	82	26	̅	̅	NOUN
cana-1482	82	27	�	�	PROPN
cana-1482	82	28	(	(	PUNCT
cana-1482	82	29	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	82	30	�	�	NOUN
cana-1482	82	31	̅	̅	NOUN
cana-1482	82	32	�	�	PROPN
cana-1482	82	33	)	)	PUNCT
cana-1482	82	34	for	for	ADP
cana-1482	82	35	short	short	ADJ
cana-1482	82	36	)	)	PUNCT
cana-1482	82	37	are	be	AUX
cana-1482	82	38	defined	define	VERB
cana-1482	82	39	by	by	ADP
cana-1482	82	40	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	ADJ
cana-1482	82	41	�	�	PROPN
cana-1482	82	42	̅	̅	NOUN
cana-1482	82	43	�	�	NOUN
cana-1482	82	44	)	)	PUNCT
cana-1482	82	45	=	=	SYM
cana-1482	82	46	∪	∪	ADP
cana-1482	82	47	{	{	PUNCT
cana-1482	82	48	�	�	NOUN
cana-1482	82	49	̅	̅	NOUN
cana-1482	82	50	�	�	PROPN
cana-1482	82	51	∶	∶	PROPN
cana-1482	82	52	�	�	NOUN
cana-1482	82	53	̅	̅	NOUN
cana-1482	82	54	�	�	NOUN
cana-1482	82	55	is	be	AUX
cana-1482	82	56	an	an	DET
cana-1482	82	57	nsspos	nsspos	NOUN
cana-1482	82	58	in	in	ADP
cana-1482	82	59	x	x	PUNCT
cana-1482	82	60	and	and	CCONJ
cana-1482	82	61	�	�	PROPN
cana-1482	82	62	̅	̅	NOUN
cana-1482	82	63	�	�	PROPN
cana-1482	82	64	⊆	⊆	NUM
cana-1482	82	65	�	�	NOUN
cana-1482	82	66	̅	̅	NOUN
cana-1482	82	67	�	�	NOUN
cana-1482	82	68	}	}	PUNCT
cana-1482	82	69	,	,	PUNCT
cana-1482	82	70	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	82	71	�	�	NOUN
cana-1482	82	72	̅	̅	NOUN
cana-1482	82	73	�	�	NOUN
cana-1482	82	74	)	)	PUNCT
cana-1482	83	1	=	=	NOUN
cana-1482	83	2	∩	∩	NOUN
cana-1482	83	3	{	{	PUNCT
cana-1482	83	4	�	�	NOUN
cana-1482	83	5	̅	̅	NOUN
cana-1482	83	6	�	�	PROPN
cana-1482	83	7	∶	∶	PROPN
cana-1482	83	8	�	�	NOUN
cana-1482	83	9	̅	̅	NOUN
cana-1482	83	10	�	�	NOUN
cana-1482	83	11	is	be	AUX
cana-1482	83	12	a	a	DET
cana-1482	83	13	nsspcs	nsspc	NOUN
cana-1482	83	14	in	in	ADP
cana-1482	83	15	x	x	PUNCT
cana-1482	83	16	and	and	CCONJ
cana-1482	83	17	�	�	PROPN
cana-1482	83	18	̅	̅	NOUN
cana-1482	83	19	�	�	PROPN
cana-1482	83	20	⊆	⊆	NUM
cana-1482	83	21	�	�	NOUN
cana-1482	83	22	̅	̅	NOUN
cana-1482	83	23	�	�	NOUN
cana-1482	83	24	}	}	PUNCT
cana-1482	83	25	.	.	PUNCT
cana-1482	84	1	for	for	ADP
cana-1482	84	2	any	any	DET
cana-1482	84	3	nss	nss	NOUN
cana-1482	84	4	�	�	NOUN
cana-1482	84	5	̅	̅	NOUN
cana-1482	84	6	�	�	PROPN
cana-1482	84	7	in	in	ADP
cana-1482	84	8	(	(	PUNCT
cana-1482	84	9	𝑋	𝑋	PROPN
cana-1482	84	10	,	,	PUNCT
cana-1482	84	11	ℑ	ℑ	PROPN
cana-1482	84	12	)	)	PUNCT
cana-1482	84	13	,	,	PUNCT
cana-1482	84	14	we	we	PRON
cana-1482	84	15	have	have	VERB
cana-1482	84	16	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	84	17	�	�	NOUN
cana-1482	84	18	̅	̅	NOUN
cana-1482	84	19	�	�	PROPN
cana-1482	84	20	𝑐	𝑐	NOUN
cana-1482	84	21	)	)	PUNCT
cana-1482	84	22	=	=	PRON
cana-1482	84	23	(	(	PUNCT
cana-1482	84	24	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	ADJ
cana-1482	84	25	�	�	SYM
cana-1482	84	26	̅	̅	NOUN
cana-1482	84	27	�	�	NOUN
cana-1482	84	28	)	)	PUNCT
cana-1482	84	29	)	)	PUNCT
cana-1482	84	30	𝑐	𝑐	PROPN
cana-1482	84	31	and	and	CCONJ
cana-1482	84	32	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	𝑛𝑠𝑠𝑝𝑖𝑛𝑡(	NOUN
cana-1482	84	33	�	�	NOUN
cana-1482	84	34	̅	̅	NOUN
cana-1482	84	35	�	�	PROPN
cana-1482	84	36	𝑐	𝑐	NOUN
cana-1482	84	37	)	)	PUNCT
cana-1482	84	38	=	=	PRON
cana-1482	84	39	(	(	PUNCT
cana-1482	84	40	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	84	41	�	�	NOUN
cana-1482	84	42	̅	̅	NOUN
cana-1482	84	43	�	�	NOUN
cana-1482	84	44	)	)	PUNCT
cana-1482	84	45	)	)	PUNCT
cana-1482	85	1	𝑐.	𝑐.	NOUN
cana-1482	85	2	1.12	1.12	NUM
cana-1482	85	3	definition	definition	NOUN
cana-1482	85	4	:	:	PUNCT
cana-1482	85	5	a	a	DET
cana-1482	85	6	nss	nss	NOUN
cana-1482	85	7	�	�	NOUN
cana-1482	85	8	̅	̅	NOUN
cana-1482	85	9	�	�	NOUN
cana-1482	85	10	in	in	ADP
cana-1482	85	11	nsts	nst	NOUN
cana-1482	85	12	(	(	PUNCT
cana-1482	85	13	x	x	X
cana-1482	85	14	,	,	PUNCT
cana-1482	85	15			NOUN
cana-1482	85	16	)	)	PUNCT
cana-1482	85	17	is	be	AUX
cana-1482	85	18	said	say	VERB
cana-1482	85	19	to	to	PART
cana-1482	85	20	be	be	AUX
cana-1482	85	21	a	a	DET
cana-1482	85	22	neutrosophic	neutrosophic	ADJ
cana-1482	85	23	generalized	generalize	VERB
cana-1482	85	24	semipreclosed	semipreclose	VERB
cana-1482	85	25	set	set	NOUN
cana-1482	85	26	(	(	PUNCT
cana-1482	85	27	nsgspcs	nsgspc	NOUN
cana-1482	85	28	for	for	ADP
cana-1482	85	29	short	short	ADJ
cana-1482	85	30	)	)	PUNCT
cana-1482	85	31	if	if	SCONJ
cana-1482	85	32	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	85	33	�	�	NOUN
cana-1482	85	34	̅	̅	NOUN
cana-1482	85	35	�	�	PROPN
cana-1482	85	36	)	)	PUNCT
cana-1482	85	37	⊆	⊆	NUM
cana-1482	85	38	�	�	NOUN
cana-1482	85	39	̅	̅	NOUN
cana-1482	85	40	�	�	NOUN
cana-1482	85	41	whenever	whenever	SCONJ
cana-1482	85	42	�	�	PROPN
cana-1482	85	43	̅	̅	NOUN
cana-1482	85	44	�	�	PROPN
cana-1482	85	45	⊆	⊆	NUM
cana-1482	85	46	�	�	NOUN
cana-1482	85	47	̅	̅	NOUN
cana-1482	85	48	�	�	PROPN
cana-1482	85	49	and	and	CCONJ
cana-1482	85	50	�	�	PROPN
cana-1482	85	51	̅	̅	NOUN
cana-1482	85	52	�	�	NOUN
cana-1482	85	53	is	be	AUX
cana-1482	85	54	an	an	DET
cana-1482	85	55	nsos	nsos	NOUN
cana-1482	85	56	in	in	ADP
cana-1482	85	57	(	(	PUNCT
cana-1482	85	58	x,	x,	X
cana-1482	85	59	)	)	PUNCT
cana-1482	85	60	.	.	PUNCT
cana-1482	86	1	1.13	1.13	NUM
cana-1482	86	2	example	example	NOUN
cana-1482	86	3	:	:	PUNCT
cana-1482	86	4	let	let	VERB
cana-1482	86	5	𝑋	𝑋	PROPN
cana-1482	86	6	=	=	SYM
cana-1482	86	7	{	{	PUNCT
cana-1482	86	8	𝑎	𝑎	NOUN
cana-1482	86	9	,	,	PUNCT
cana-1482	86	10	𝑏	𝑏	NOUN
cana-1482	86	11	}	}	PUNCT
cana-1482	86	12	and	and	CCONJ
cana-1482	86	13	�	�	PROPN
cana-1482	86	14	̅	̅	NOUN
cana-1482	86	15	�	�	NOUN
cana-1482	86	16	=	=	SYM
cana-1482	86	17	{	{	PUNCT
cana-1482	86	18	〈	〈	PROPN
cana-1482	86	19	𝑎	𝑎	NOUN
cana-1482	86	20	,	,	PUNCT
cana-1482	86	21	0.5	0.5	NUM
cana-1482	86	22	,	,	PUNCT
cana-1482	86	23	0.6	0.6	NUM
cana-1482	86	24	,	,	PUNCT
cana-1482	86	25	0.5	0.5	NUM
cana-1482	86	26	〉	〉	NOUN
cana-1482	86	27	,	,	PUNCT
cana-1482	86	28	〈	〈	PROPN
cana-1482	86	29	𝑏	𝑏	NOUN
cana-1482	86	30	,	,	PUNCT
cana-1482	86	31	0.4	0.4	NUM
cana-1482	86	32	,	,	PUNCT
cana-1482	86	33	0.5	0.5	NUM
cana-1482	86	34	,	,	PUNCT
cana-1482	86	35	0.6	0.6	NUM
cana-1482	86	36	〉	〉	NOUN
cana-1482	86	37	}	}	PUNCT
cana-1482	86	38	.	.	PUNCT
cana-1482	87	1	then	then	ADV
cana-1482	87	2			VERB
cana-1482	87	3	=	=	PUNCT
cana-1482	87	4	{	{	PUNCT
cana-1482	87	5	0̅	0̅	PROPN
cana-1482	87	6	,	,	PUNCT
cana-1482	87	7	�	�	PROPN
cana-1482	87	8	̅	̅	NOUN
cana-1482	87	9	�	�	PROPN
cana-1482	87	10	,	,	PUNCT
cana-1482	87	11	1̅	1̅	PROPN
cana-1482	87	12	}	}	PUNCT
cana-1482	87	13	is	be	AUX
cana-1482	87	14	an	an	DET
cana-1482	87	15	nst	nst	NOUN
cana-1482	87	16	on	on	ADP
cana-1482	87	17	x	x	PUNCT
cana-1482	87	18	and	and	CCONJ
cana-1482	87	19	the	the	DET
cana-1482	87	20	nss	nss	PROPN
cana-1482	87	21	�	�	NOUN
cana-1482	87	22	̅	̅	NOUN
cana-1482	87	23	�	�	NOUN
cana-1482	87	24	=	=	SYM
cana-1482	87	25	{	{	PUNCT
cana-1482	87	26	〈	〈	PROPN
cana-1482	87	27	𝑎	𝑎	NOUN
cana-1482	87	28	,	,	PUNCT
cana-1482	87	29	0.4	0.4	NUM
cana-1482	87	30	,	,	PUNCT
cana-1482	87	31	0.4	0.4	NUM
cana-1482	87	32	,	,	PUNCT
cana-1482	87	33	0.6	0.6	NUM
cana-1482	87	34	〉	〉	NOUN
cana-1482	87	35	,	,	PUNCT
cana-1482	87	36	〈	〈	PROPN
cana-1482	87	37	𝑏	𝑏	NOUN
cana-1482	87	38	,	,	PUNCT
cana-1482	87	39	0.2	0.2	NUM
cana-1482	87	40	,	,	PUNCT
cana-1482	87	41	0.3	0.3	NUM
cana-1482	87	42	,	,	PUNCT
cana-1482	87	43	0.7	0.7	NUM
cana-1482	87	44	〉	〉	NOUN
cana-1482	87	45	}	}	PUNCT
cana-1482	87	46	is	be	AUX
cana-1482	87	47	a	a	DET
cana-1482	87	48	nsgspcs	nsgspc	NOUN
cana-1482	87	49	in	in	ADP
cana-1482	87	50	(	(	PUNCT
cana-1482	87	51	x,	x,	X
cana-1482	87	52	)	)	PUNCT
cana-1482	87	53	.	.	PUNCT
cana-1482	88	1	communications	communication	NOUN
cana-1482	88	2	on	on	ADP
cana-1482	88	3	applied	apply	VERB
cana-1482	88	4	nonlinear	nonlinear	ADJ
cana-1482	88	5	analysis	analysis	NOUN
cana-1482	88	6	issn	issn	NOUN
cana-1482	88	7	:	:	PUNCT
cana-1482	88	8	1074	1074	NUM
cana-1482	88	9	-	-	PUNCT
cana-1482	88	10	133x	133x	NUM
cana-1482	88	11	vol	vol	NOUN
cana-1482	88	12	31	31	NUM
cana-1482	88	13	no	no	NOUN
cana-1482	88	14	.	.	PUNCT
cana-1482	89	1	8s	8s	PROPN
cana-1482	89	2	(	(	PUNCT
cana-1482	89	3	2024	2024	NUM
cana-1482	89	4	)	)	PUNCT
cana-1482	89	5	276	276	NUM
cana-1482	89	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	89	7	1.14	1.14	NUM
cana-1482	89	8	definition	definition	NOUN
cana-1482	89	9	:	:	PUNCT
cana-1482	89	10	let	let	VERB
cana-1482	89	11	𝛼	𝛼	X
cana-1482	89	12	,	,	PUNCT
cana-1482	89	13	𝛽	𝛽	NOUN
cana-1482	89	14	,	,	PUNCT
cana-1482	89	15			VERB
cana-1482	89	16	∈	∈	NOUN
cana-1482	90	1	[	[	X
cana-1482	90	2	0,1	0,1	NUM
cana-1482	90	3	]	]	PUNCT
cana-1482	90	4	.	.	PUNCT
cana-1482	91	1	a	a	DET
cana-1482	91	2	neutrosophic	neutrosophic	ADJ
cana-1482	91	3	point	point	NOUN
cana-1482	91	4	(	(	PUNCT
cana-1482	91	5	nsp	nsp	VERB
cana-1482	91	6	for	for	ADP
cana-1482	91	7	short	short	ADJ
cana-1482	91	8	)	)	PUNCT
cana-1482	91	9	,	,	PUNCT
cana-1482	91	10	written	write	VERB
cana-1482	91	11	as	as	ADP
cana-1482	91	12	�	�	PROPN
cana-1482	91	13	̅	̅	NOUN
cana-1482	91	14	�	�	PROPN
cana-1482	91	15	(	(	PUNCT
cana-1482	91	16	𝛼,𝛽,	𝛼,𝛽,	NOUN
cana-1482	91	17	)	)	PUNCT
cana-1482	91	18	is	be	AUX
cana-1482	91	19	defined	define	VERB
cana-1482	91	20	to	to	PART
cana-1482	91	21	be	be	AUX
cana-1482	91	22	an	an	DET
cana-1482	91	23	nss	nss	NOUN
cana-1482	91	24	of	of	ADP
cana-1482	91	25	x	x	VERB
cana-1482	91	26	is	be	AUX
cana-1482	91	27	given	give	VERB
cana-1482	91	28	by	by	ADP
cana-1482	91	29	�	�	NOUN
cana-1482	91	30	̅	̅	NOUN
cana-1482	91	31	�	�	PROPN
cana-1482	91	32	(	(	PUNCT
cana-1482	91	33	𝛼,𝛽,)(𝑥	𝛼,𝛽,)(𝑥	NUM
cana-1482	91	34	)	)	PUNCT
cana-1482	92	1	=	=	PRON
cana-1482	92	2	{	{	PUNCT
cana-1482	92	3	(	(	PUNCT
cana-1482	92	4	𝛼	𝛼	PROPN
cana-1482	92	5	,	,	PUNCT
cana-1482	92	6	𝛽	𝛽	NOUN
cana-1482	92	7	,	,	PUNCT
cana-1482	92	8			VERB
cana-1482	92	9	)	)	PUNCT
cana-1482	93	1	if	if	SCONJ
cana-1482	93	2	x	x	X
cana-1482	93	3	=	=	PRON
cana-1482	93	4	p	p	X
cana-1482	93	5	(	(	PUNCT
cana-1482	93	6	0,0,0	0,0,0	NOUN
cana-1482	93	7	)	)	PUNCT
cana-1482	93	8	otherwise	otherwise	ADV
cana-1482	93	9	.	.	PUNCT
cana-1482	94	1	2	2	X
cana-1482	94	2	.	.	X
cana-1482	95	1	some	some	DET
cana-1482	95	2	properties	property	NOUN
cana-1482	95	3	:	:	PUNCT
cana-1482	95	4	2.1	2.1	NUM
cana-1482	95	5	theorem	theorem	VERB
cana-1482	95	6	:	:	PUNCT
cana-1482	95	7	in	in	ADP
cana-1482	95	8	an	an	DET
cana-1482	95	9	nsts	nst	NOUN
cana-1482	95	10	(	(	PUNCT
cana-1482	95	11	x	x	X
cana-1482	95	12	,	,	PUNCT
cana-1482	95	13			NOUN
cana-1482	95	14	)	)	PUNCT
cana-1482	95	15	,	,	PUNCT
cana-1482	95	16	each	each	DET
cana-1482	95	17	nscs	nscs	NOUN
cana-1482	95	18	is	be	AUX
cana-1482	95	19	an	an	DET
cana-1482	95	20	nsgspcs	nsgspc	NOUN
cana-1482	95	21	in	in	ADP
cana-1482	95	22	(	(	PUNCT
cana-1482	95	23	x	x	NOUN
cana-1482	95	24	,	,	PUNCT
cana-1482	95	25			NOUN
cana-1482	95	26	)	)	PUNCT
cana-1482	95	27	.	.	PUNCT
cana-1482	96	1	proof	proof	NOUN
cana-1482	96	2	:	:	PUNCT
cana-1482	96	3	let	let	VERB
cana-1482	96	4	�	�	PRON
cana-1482	96	5	̅	̅	VERB
cana-1482	96	6	�	�	PROPN
cana-1482	96	7	in	in	ADP
cana-1482	96	8	(	(	PUNCT
cana-1482	96	9	x,	x,	X
cana-1482	96	10	)	)	PUNCT
cana-1482	96	11	be	be	VERB
cana-1482	96	12	an	an	DET
cana-1482	96	13	nscs	nscs	NOUN
cana-1482	96	14	.	.	PUNCT
cana-1482	97	1	let	let	VERB
cana-1482	97	2	us	we	PRON
cana-1482	97	3	assume	assume	VERB
cana-1482	97	4	that	that	SCONJ
cana-1482	97	5	in	in	ADP
cana-1482	97	6	(	(	PUNCT
cana-1482	97	7	x	x	NOUN
cana-1482	97	8	,	,	PUNCT
cana-1482	97	9			NOUN
cana-1482	97	10	)	)	PUNCT
cana-1482	97	11	�	�	PROPN
cana-1482	97	12	̅	̅	NOUN
cana-1482	97	13	�	�	PROPN
cana-1482	97	14	⊆	⊆	NUM
cana-1482	97	15	�	�	NOUN
cana-1482	97	16	̅	̅	NOUN
cana-1482	97	17	�	�	PROPN
cana-1482	97	18	and	and	CCONJ
cana-1482	97	19	�	�	PROPN
cana-1482	97	20	̅	̅	NOUN
cana-1482	97	21	�	�	NOUN
cana-1482	97	22	is	be	AUX
cana-1482	97	23	an	an	DET
cana-1482	97	24	nsos	nsos	NOUN
cana-1482	97	25	.	.	PUNCT
cana-1482	98	1	according	accord	VERB
cana-1482	98	2	to	to	ADP
cana-1482	98	3	hypothesis	hypothesis	NOUN
cana-1482	98	4	,	,	PUNCT
cana-1482	98	5	therefore	therefore	ADV
cana-1482	98	6	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	98	7	�	�	PROPN
cana-1482	98	8	̅	̅	NOUN
cana-1482	98	9	�	�	PROPN
cana-1482	98	10	)	)	PUNCT
cana-1482	98	11	⊆	⊆	NUM
cana-1482	98	12	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	98	13	�	�	PROPN
cana-1482	98	14	̅	̅	NOUN
cana-1482	98	15	�	�	NOUN
cana-1482	98	16	)	)	PUNCT
cana-1482	98	17	=	=	SYM
cana-1482	98	18	�	�	NOUN
cana-1482	98	19	̅	̅	NOUN
cana-1482	98	20	�	�	PROPN
cana-1482	98	21	⊆	⊆	NUM
cana-1482	98	22	�	�	NOUN
cana-1482	98	23	̅	̅	NOUN
cana-1482	98	24	�	�	NOUN
cana-1482	98	25	.	.	PUNCT
cana-1482	99	1	thus	thus	ADV
cana-1482	99	2	,	,	PUNCT
cana-1482	99	3	in	in	ADP
cana-1482	99	4	(	(	PUNCT
cana-1482	99	5	x	x	NOUN
cana-1482	99	6	,	,	PUNCT
cana-1482	99	7			NOUN
cana-1482	99	8	)	)	PUNCT
cana-1482	99	9	,	,	PUNCT
cana-1482	99	10	�	�	PROPN
cana-1482	99	11	̅	̅	NOUN
cana-1482	99	12	�	�	NOUN
cana-1482	99	13	is	be	AUX
cana-1482	99	14	an	an	DET
cana-1482	99	15	nsgspc	nsgspc	NOUN
cana-1482	99	16	2.2	2.2	NUM
cana-1482	99	17	remark	remark	NOUN
cana-1482	99	18	:	:	PUNCT
cana-1482	99	19	the	the	DET
cana-1482	99	20	following	follow	VERB
cana-1482	99	21	example	example	NOUN
cana-1482	99	22	shows	show	VERB
cana-1482	99	23	that	that	SCONJ
cana-1482	99	24	the	the	DET
cana-1482	99	25	converse	converse	NOUN
cana-1482	99	26	of	of	ADP
cana-1482	99	27	the	the	DET
cana-1482	99	28	preceding	precede	VERB
cana-1482	99	29	theorem	theorem	NOUN
cana-1482	99	30	need	need	AUX
cana-1482	99	31	not	not	PART
cana-1482	99	32	be	be	AUX
cana-1482	99	33	true	true	ADJ
cana-1482	99	34	.	.	PUNCT
cana-1482	100	1	llustration	llustration	NOUN
cana-1482	100	2	:	:	PUNCT
cana-1482	100	3	let	let	VERB
cana-1482	100	4	x=	x=	PUNCT
cana-1482	100	5	{	{	PUNCT
cana-1482	100	6	a	a	PRON
cana-1482	100	7	,	,	PUNCT
cana-1482	100	8	b	b	NOUN
cana-1482	100	9	}	}	PUNCT
cana-1482	100	10	and	and	CCONJ
cana-1482	100	11	�	�	PROPN
cana-1482	100	12	̅	̅	NOUN
cana-1482	100	13	�	�	NOUN
cana-1482	100	14	=	=	SYM
cana-1482	100	15	{	{	PUNCT
cana-1482	100	16	〈	〈	PROPN
cana-1482	100	17	𝑎	𝑎	NOUN
cana-1482	100	18	,	,	PUNCT
cana-1482	100	19	0.5	0.5	NUM
cana-1482	100	20	,	,	PUNCT
cana-1482	100	21	0.6	0.6	NUM
cana-1482	100	22	,	,	PUNCT
cana-1482	100	23	0.5	0.5	NUM
cana-1482	100	24	〉	〉	NOUN
cana-1482	100	25	,	,	PUNCT
cana-1482	100	26	〈	〈	PROPN
cana-1482	100	27	𝑏	𝑏	NOUN
cana-1482	100	28	,	,	PUNCT
cana-1482	100	29	0.4	0.4	NUM
cana-1482	100	30	,	,	PUNCT
cana-1482	100	31	0.5	0.5	NUM
cana-1482	100	32	,	,	PUNCT
cana-1482	100	33	0.6	0.6	NUM
cana-1482	100	34	〉	〉	NOUN
cana-1482	100	35	}	}	PUNCT
cana-1482	100	36	be	be	AUX
cana-1482	100	37	the	the	DET
cana-1482	100	38	example	example	NOUN
cana-1482	100	39	.	.	PUNCT
cana-1482	101	1	then	then	ADV
cana-1482	101	2	,	,	PUNCT
cana-1482	101	3	on	on	ADP
cana-1482	101	4	x	x	NOUN
cana-1482	101	5	,	,	PUNCT
cana-1482	101	6			NOUN
cana-1482	101	7	=	=	PUNCT
cana-1482	101	8	{	{	PUNCT
cana-1482	101	9	0̅	0̅	PROPN
cana-1482	101	10	,	,	PUNCT
cana-1482	101	11	�	�	PROPN
cana-1482	101	12	̅	̅	NOUN
cana-1482	101	13	�	�	PROPN
cana-1482	101	14	,	,	PUNCT
cana-1482	101	15	1̅	1̅	NUM
cana-1482	101	16	}	}	PUNCT
cana-1482	101	17	is	be	AUX
cana-1482	101	18	an	an	DET
cana-1482	101	19	nst	nst	NOUN
cana-1482	101	20	.	.	PUNCT
cana-1482	102	1	an	an	DET
cana-1482	102	2	nss	nss	NOUN
cana-1482	102	3	in	in	ADP
cana-1482	102	4	x	x	VERB
cana-1482	102	5	is	be	AUX
cana-1482	102	6	denoted	denote	VERB
cana-1482	102	7	by	by	ADP
cana-1482	102	8	�	�	NOUN
cana-1482	102	9	̅	̅	NOUN
cana-1482	102	10	�	�	NOUN
cana-1482	102	11	=	=	SYM
cana-1482	102	12	{	{	PUNCT
cana-1482	102	13	〈	〈	X
cana-1482	102	14	𝑎	𝑎	NOUN
cana-1482	102	15	,	,	PUNCT
cana-1482	102	16	0.4	0.4	NUM
cana-1482	102	17	,	,	PUNCT
cana-1482	102	18	0.5	0.5	NUM
cana-1482	102	19	,	,	PUNCT
cana-1482	102	20	0.6	0.6	NUM
cana-1482	102	21	〉	〉	NOUN
cana-1482	102	22	,	,	PUNCT
cana-1482	102	23	〈	〈	PROPN
cana-1482	102	24	𝑏	𝑏	NOUN
cana-1482	102	25	,	,	PUNCT
cana-1482	102	26	0.2	0.2	NUM
cana-1482	102	27	,	,	PUNCT
cana-1482	102	28	0.3,0.7	0.3,0.7	PROPN
cana-1482	102	29	,	,	PUNCT
cana-1482	102	30	〉	〉	NOUN
cana-1482	102	31	}	}	PUNCT
cana-1482	102	32	.	.	PUNCT
cana-1482	103	1	in	in	ADP
cana-1482	103	2	x	x	SYM
cana-1482	103	3	,	,	PUNCT
cana-1482	103	4	�	�	NOUN
cana-1482	103	5	̅	̅	NOUN
cana-1482	103	6	�	�	NOUN
cana-1482	103	7	is	be	AUX
cana-1482	103	8	not	not	PART
cana-1482	103	9	an	an	DET
cana-1482	103	10	nscs	nsc	NOUN
cana-1482	103	11	,	,	PUNCT
cana-1482	103	12	but	but	CCONJ
cana-1482	103	13	it	it	PRON
cana-1482	103	14	is	be	AUX
cana-1482	103	15	an	an	DET
cana-1482	103	16	nsgspcs	nsgspc	NOUN
cana-1482	103	17	.	.	PUNCT
cana-1482	104	1	2.3	2.3	NUM
cana-1482	104	2	theorem	theorem	VERB
cana-1482	104	3	:	:	PUNCT
cana-1482	104	4	each	each	DET
cana-1482	104	5	nsgspcs	nsgspc	NOUN
cana-1482	104	6	in	in	ADP
cana-1482	104	7	(	(	PUNCT
cana-1482	104	8	x	x	X
cana-1482	104	9	,	,	PUNCT
cana-1482	104	10			NOUN
cana-1482	104	11	)	)	PUNCT
cana-1482	104	12	is	be	AUX
cana-1482	104	13	an	an	DET
cana-1482	104	14	nsrcs	nsrcs	NOUN
cana-1482	104	15	in	in	ADP
cana-1482	104	16	the	the	DET
cana-1482	104	17	nsts	nst	NOUN
cana-1482	104	18	.	.	PUNCT
cana-1482	105	1	proof	proof	NOUN
cana-1482	105	2	:	:	PUNCT
cana-1482	105	3	theorem	theorem	VERB
cana-1482	105	4	2.1	2.1	NUM
cana-1482	105	5	makes	make	VERB
cana-1482	105	6	it	it	PRON
cana-1482	105	7	clear	clear	ADJ
cana-1482	105	8	that	that	SCONJ
cana-1482	105	9	every	every	DET
cana-1482	105	10	nsrcs	nsrcs	NOUN
cana-1482	105	11	is	be	AUX
cana-1482	105	12	an	an	DET
cana-1482	105	13	nscs	nscs	NOUN
cana-1482	105	14	.	.	PUNCT
cana-1482	106	1	2.4	2.4	NUM
cana-1482	106	2	remark	remark	NOUN
cana-1482	106	3	:	:	PUNCT
cana-1482	106	4	as	as	SCONJ
cana-1482	106	5	the	the	DET
cana-1482	106	6	following	follow	VERB
cana-1482	106	7	example	example	NOUN
cana-1482	106	8	illustrates	illustrate	VERB
cana-1482	106	9	,	,	PUNCT
cana-1482	106	10	the	the	DET
cana-1482	106	11	above	above	ADJ
cana-1482	106	12	theorem	theorem	NOUN
cana-1482	106	13	's	's	PART
cana-1482	106	14	converse	converse	NOUN
cana-1482	106	15	need	need	AUX
cana-1482	106	16	not	not	PART
cana-1482	106	17	be	be	AUX
cana-1482	106	18	true	true	ADJ
cana-1482	106	19	.	.	PUNCT
cana-1482	107	1	llustration	llustration	NOUN
cana-1482	107	2	:	:	PUNCT
cana-1482	107	3	for	for	ADP
cana-1482	107	4	illustration	illustration	NOUN
cana-1482	107	5	,	,	PUNCT
cana-1482	107	6	let	let	VERB
cana-1482	107	7	x=	x=	PUNCT
cana-1482	107	8	{	{	PUNCT
cana-1482	107	9	a	a	PRON
cana-1482	107	10	,	,	PUNCT
cana-1482	107	11	b	b	NOUN
cana-1482	107	12	}	}	PUNCT
cana-1482	107	13	and	and	CCONJ
cana-1482	107	14	�	�	PROPN
cana-1482	107	15	̅	̅	NOUN
cana-1482	107	16	�	�	NOUN
cana-1482	107	17	=	=	SYM
cana-1482	107	18	{	{	PUNCT
cana-1482	107	19	〈	〈	PROPN
cana-1482	107	20	𝑎	𝑎	NOUN
cana-1482	107	21	,	,	PUNCT
cana-1482	107	22	0.4	0.4	NUM
cana-1482	107	23	,	,	PUNCT
cana-1482	107	24	0.8,0	0.8,0	NOUN
cana-1482	107	25	〉	〉	NOUN
cana-1482	107	26	,	,	PUNCT
cana-1482	107	27	〈	〈	PROPN
cana-1482	107	28	𝑏	𝑏	NOUN
cana-1482	107	29	,	,	PUNCT
cana-1482	107	30	0.3	0.3	NUM
cana-1482	107	31	,	,	PUNCT
cana-1482	107	32	0.6,0	0.6,0	NOUN
cana-1482	107	33	〉	〉	NOUN
cana-1482	107	34	}	}	PUNCT
cana-1482	107	35	be	be	AUX
cana-1482	107	36	the	the	DET
cana-1482	107	37	example	example	NOUN
cana-1482	107	38	.	.	PUNCT
cana-1482	108	1	then	then	ADV
cana-1482	108	2	,	,	PUNCT
cana-1482	108	3	on	on	ADP
cana-1482	108	4	x	x	NOUN
cana-1482	108	5	,	,	PUNCT
cana-1482	108	6			NOUN
cana-1482	108	7	=	=	PUNCT
cana-1482	108	8	{	{	PUNCT
cana-1482	108	9	0̅	0̅	PROPN
cana-1482	108	10	,	,	PUNCT
cana-1482	108	11	�	�	PROPN
cana-1482	108	12	̅	̅	NOUN
cana-1482	108	13	�	�	PROPN
cana-1482	108	14	,	,	PUNCT
cana-1482	108	15	1̅	1̅	NUM
cana-1482	108	16	}	}	PUNCT
cana-1482	108	17	is	be	AUX
cana-1482	108	18	an	an	DET
cana-1482	108	19	nst	nst	PROPN
cana-1482	108	20	.	.	PROPN
cana-1482	108	21	assume	assume	VERB
cana-1482	108	22	that	that	SCONJ
cana-1482	108	23	an	an	DET
cana-1482	108	24	nss	nss	NOUN
cana-1482	108	25	in	in	ADP
cana-1482	108	26	x	x	VERB
cana-1482	108	27	is	be	AUX
cana-1482	108	28	�	�	NOUN
cana-1482	108	29	̅	̅	NOUN
cana-1482	108	30	�	�	NOUN
cana-1482	108	31	=	=	SYM
cana-1482	108	32	{	{	PUNCT
cana-1482	108	33	〈	〈	PROPN
cana-1482	108	34	𝑎	𝑎	NOUN
cana-1482	108	35	,	,	PUNCT
cana-1482	108	36	0.3	0.3	NUM
cana-1482	108	37	,	,	PUNCT
cana-1482	108	38	0.6,0	0.6,0	NOUN
cana-1482	108	39	〉	〉	NOUN
cana-1482	108	40	,	,	PUNCT
cana-1482	108	41	〈	〈	PROPN
cana-1482	108	42	𝑏	𝑏	NOUN
cana-1482	108	43	,	,	PUNCT
cana-1482	108	44	0.2	0.2	NUM
cana-1482	108	45	,	,	PUNCT
cana-1482	108	46	0.4,0	0.4,0	ADJ
cana-1482	108	47	〉	〉	NOUN
cana-1482	108	48	}	}	PUNCT
cana-1482	108	49	.	.	PUNCT
cana-1482	109	1	in	in	ADP
cana-1482	109	2	x	x	SYM
cana-1482	109	3	,	,	PUNCT
cana-1482	109	4	�	�	NOUN
cana-1482	109	5	̅	̅	NOUN
cana-1482	109	6	�	�	NOUN
cana-1482	109	7	is	be	AUX
cana-1482	109	8	therefore	therefore	ADV
cana-1482	109	9	an	an	DET
cana-1482	109	10	nsgspcs	nsgspc	NOUN
cana-1482	109	11	but	but	CCONJ
cana-1482	109	12	not	not	PART
cana-1482	109	13	an	an	DET
cana-1482	109	14	nsrcs	nsrcs	NOUN
cana-1482	109	15	.	.	PUNCT
cana-1482	110	1	2.5	2.5	NUM
cana-1482	110	2	theorem	theorem	VERB
cana-1482	110	3	:	:	PUNCT
cana-1482	110	4	a	a	DET
cana-1482	110	5	nsgspcs	nsgspc	NOUN
cana-1482	110	6	in	in	ADP
cana-1482	110	7	(	(	PUNCT
cana-1482	110	8	x	x	X
cana-1482	110	9	,	,	PUNCT
cana-1482	110	10			NOUN
cana-1482	110	11	)	)	PUNCT
cana-1482	110	12	is	be	AUX
cana-1482	110	13	an	an	DET
cana-1482	110	14	nsgcs	nsgcs	NOUN
cana-1482	110	15	for	for	ADP
cana-1482	110	16	every	every	DET
cana-1482	110	17	nsgcs	nsgcs	PROPN
cana-1482	110	18	in	in	ADP
cana-1482	110	19	an	an	DET
cana-1482	110	20	nsts	nst	NOUN
cana-1482	110	21	(	(	PUNCT
cana-1482	110	22	x	x	X
cana-1482	110	23	,	,	PUNCT
cana-1482	110	24			NOUN
cana-1482	110	25	)	)	PUNCT
cana-1482	110	26	.	.	PUNCT
cana-1482	111	1	proof	proof	NOUN
cana-1482	111	2	:	:	PUNCT
cana-1482	111	3	assume	assume	VERB
cana-1482	111	4	that	that	SCONJ
cana-1482	111	5	�	�	PROPN
cana-1482	111	6	̅	̅	NOUN
cana-1482	111	7	�	�	NOUN
cana-1482	111	8	is	be	AUX
cana-1482	111	9	an	an	DET
cana-1482	111	10	nsgcs	nsgcs	PROPN
cana-1482	111	11	in	in	ADP
cana-1482	111	12	nsts	nst	NOUN
cana-1482	111	13	(	(	PUNCT
cana-1482	111	14	x	x	X
cana-1482	111	15	,	,	PUNCT
cana-1482	111	16			NOUN
cana-1482	111	17	)	)	PUNCT
cana-1482	111	18	.	.	PUNCT
cana-1482	112	1	next	next	ADV
cana-1482	112	2	,	,	PUNCT
cana-1482	112	3	suppose	suppose	VERB
cana-1482	112	4	that	that	SCONJ
cana-1482	112	5	�	�	PROPN
cana-1482	112	6	̅	̅	NOUN
cana-1482	112	7	�	�	PROPN
cana-1482	112	8	⊆	⊆	NUM
cana-1482	112	9	�	�	NOUN
cana-1482	112	10	̅	̅	NOUN
cana-1482	112	11	�	�	PROPN
cana-1482	112	12	and	and	CCONJ
cana-1482	112	13	that	that	SCONJ
cana-1482	112	14	�	�	PROPN
cana-1482	112	15	̅	̅	NOUN
cana-1482	112	16	�	�	NOUN
cana-1482	112	17	is	be	AUX
cana-1482	112	18	an	an	DET
cana-1482	112	19	nso	nso	NOUN
cana-1482	112	20	in	in	ADP
cana-1482	112	21	(	(	PUNCT
cana-1482	112	22	x	x	NOUN
cana-1482	112	23	,	,	PUNCT
cana-1482	112	24			NOUN
cana-1482	112	25	)	)	PUNCT
cana-1482	112	26	.	.	PUNCT
cana-1482	113	1	by	by	ADP
cana-1482	113	2	hypothesis	hypothesis	NOUN
cana-1482	113	3	,	,	PUNCT
cana-1482	113	4	�	�	PROPN
cana-1482	113	5	̅	̅	NOUN
cana-1482	113	6	�	�	NOUN
cana-1482	113	7	is	be	AUX
cana-1482	113	8	an	an	DET
cana-1482	113	9	nsgspcs	nsgspc	NOUN
cana-1482	113	10	in	in	ADP
cana-1482	113	11	x	x	PUNCT
cana-1482	113	12	since	since	SCONJ
cana-1482	113	13	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	113	14	�	�	NOUN
cana-1482	113	15	̅	̅	NOUN
cana-1482	113	16	�	�	PROPN
cana-1482	113	17	)	)	PUNCT
cana-1482	113	18	⊆	⊆	NUM
cana-1482	113	19	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	113	20	�	�	PROPN
cana-1482	113	21	̅	̅	NOUN
cana-1482	113	22	�	�	PROPN
cana-1482	113	23	)	)	PUNCT
cana-1482	113	24	and	and	CCONJ
cana-1482	113	25	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	113	26	�	�	PROPN
cana-1482	113	27	̅	̅	NOUN
cana-1482	113	28	�	�	PROPN
cana-1482	113	29	)	)	PUNCT
cana-1482	113	30	⊆	⊆	NUM
cana-1482	113	31	�	�	NOUN
cana-1482	113	32	̅	̅	NOUN
cana-1482	113	33	�	�	PROPN
cana-1482	113	34	2.6	2.6	NUM
cana-1482	113	35	remark	remark	NOUN
cana-1482	113	36	:	:	PUNCT
cana-1482	113	37	the	the	DET
cana-1482	113	38	following	follow	VERB
cana-1482	113	39	example	example	NOUN
cana-1482	113	40	shows	show	VERB
cana-1482	113	41	that	that	SCONJ
cana-1482	113	42	the	the	DET
cana-1482	113	43	converse	converse	NOUN
cana-1482	113	44	of	of	ADP
cana-1482	113	45	the	the	DET
cana-1482	113	46	preceding	precede	VERB
cana-1482	113	47	theorem	theorem	NOUN
cana-1482	113	48	need	need	AUX
cana-1482	113	49	not	not	PART
cana-1482	113	50	be	be	AUX
cana-1482	113	51	true	true	ADJ
cana-1482	113	52	.	.	PUNCT
cana-1482	114	1	llustration	llustration	NOUN
cana-1482	114	2	:	:	PUNCT
cana-1482	114	3	let	let	VERB
cana-1482	114	4	x	x	PUNCT
cana-1482	114	5	=	=	PRON
cana-1482	114	6	{	{	PUNCT
cana-1482	114	7	a	a	PRON
cana-1482	114	8	,	,	PUNCT
cana-1482	114	9	b	b	NOUN
cana-1482	114	10	}	}	PUNCT
cana-1482	114	11	and	and	CCONJ
cana-1482	114	12	�	�	PROPN
cana-1482	114	13	̅	̅	NOUN
cana-1482	114	14	�	�	NOUN
cana-1482	114	15	=	=	SYM
cana-1482	114	16	{	{	PUNCT
cana-1482	114	17	〈	〈	PROPN
cana-1482	114	18	𝑎	𝑎	NOUN
cana-1482	114	19	,	,	PUNCT
cana-1482	114	20	0.5	0.5	NUM
cana-1482	114	21	,	,	PUNCT
cana-1482	114	22	0.6,0.5	0.6,0.5	NOUN
cana-1482	114	23	〉	〉	NOUN
cana-1482	114	24	,	,	PUNCT
cana-1482	114	25	〈	〈	PROPN
cana-1482	114	26	𝑏	𝑏	NOUN
cana-1482	114	27	,	,	PUNCT
cana-1482	114	28	0.4	0.4	NUM
cana-1482	114	29	,	,	PUNCT
cana-1482	114	30	0.5,0.6	0.5,0.6	PROPN
cana-1482	114	31	〉	〉	NOUN
cana-1482	114	32	}	}	PUNCT
cana-1482	114	33	be	be	AUX
cana-1482	114	34	the	the	DET
cana-1482	114	35	example	example	NOUN
cana-1482	114	36	.	.	PUNCT
cana-1482	115	1	then	then	ADV
cana-1482	115	2	,	,	PUNCT
cana-1482	115	3	on	on	ADP
cana-1482	115	4	x	x	NOUN
cana-1482	115	5	,	,	PUNCT
cana-1482	115	6			NOUN
cana-1482	115	7	=	=	PUNCT
cana-1482	115	8	{	{	PUNCT
cana-1482	115	9	0̅	0̅	PROPN
cana-1482	115	10	,	,	PUNCT
cana-1482	115	11	�	�	PROPN
cana-1482	115	12	̅	̅	NOUN
cana-1482	115	13	�	�	PROPN
cana-1482	115	14	,	,	PUNCT
cana-1482	115	15	1	1	NUM
cana-1482	115	16	̅	̅	NOUN
cana-1482	115	17	}	}	PUNCT
cana-1482	115	18	is	be	AUX
cana-1482	115	19	an	an	DET
cana-1482	115	20	ngt	ngt	PROPN
cana-1482	115	21	.	.	PUNCT
cana-1482	116	1	an	an	DET
cana-1482	116	2	nss	nss	NOUN
cana-1482	116	3	in	in	ADP
cana-1482	116	4	x	x	VERB
cana-1482	116	5	is	be	AUX
cana-1482	116	6	represented	represent	VERB
cana-1482	116	7	by	by	ADP
cana-1482	116	8	�	�	PROPN
cana-1482	116	9	̅	̅	NOUN
cana-1482	116	10	�	�	NOUN
cana-1482	116	11	=	=	SYM
cana-1482	116	12	{	{	PUNCT
cana-1482	116	13	〈	〈	PROPN
cana-1482	116	14	𝑎	𝑎	NOUN
cana-1482	116	15	,	,	PUNCT
cana-1482	116	16	0.4	0.4	NUM
cana-1482	116	17	,	,	PUNCT
cana-1482	116	18	0.5,0.6	0.5,0.6	PROPN
cana-1482	116	19	〉	〉	NOUN
cana-1482	116	20	,	,	PUNCT
cana-1482	116	21	〈	〈	PROPN
cana-1482	116	22	𝑏	𝑏	NOUN
cana-1482	116	23	,	,	PUNCT
cana-1482	116	24	0.2	0.2	NUM
cana-1482	116	25	,	,	PUNCT
cana-1482	116	26	0.3,0.7	0.3,0.7	NUM
cana-1482	116	27	〉	〉	NOUN
cana-1482	116	28	}	}	PUNCT
cana-1482	116	29	.	.	PUNCT
cana-1482	117	1	in	in	ADP
cana-1482	117	2	x	x	SYM
cana-1482	117	3	,	,	PUNCT
cana-1482	117	4	�	�	NOUN
cana-1482	117	5	̅	̅	NOUN
cana-1482	117	6	�	�	NOUN
cana-1482	117	7	is	be	AUX
cana-1482	117	8	therefore	therefore	ADV
cana-1482	117	9	an	an	DET
cana-1482	117	10	nsgspcs	nsgspc	NOUN
cana-1482	117	11	but	but	CCONJ
cana-1482	117	12	not	not	PART
cana-1482	117	13	an	an	DET
cana-1482	117	14	nggcs	nggcs	NOUN
cana-1482	117	15	.	.	PUNCT
cana-1482	118	1	2.7	2.7	NUM
cana-1482	118	2	theorem	theorem	VERB
cana-1482	118	3	:	:	PUNCT
cana-1482	118	4	states	state	VERB
cana-1482	118	5	that	that	SCONJ
cana-1482	118	6	each	each	DET
cana-1482	118	7	nsspcs	nsspc	NOUN
cana-1482	118	8	in	in	ADP
cana-1482	118	9	an	an	DET
cana-1482	118	10	nsts	nst	NOUN
cana-1482	118	11	(	(	PUNCT
cana-1482	118	12	x	x	X
cana-1482	118	13	,	,	PUNCT
cana-1482	118	14			NOUN
cana-1482	118	15	)	)	PUNCT
cana-1482	118	16	is	be	AUX
cana-1482	118	17	also	also	ADV
cana-1482	118	18	an	an	DET
cana-1482	118	19	nsgspcs	nsgspc	NOUN
cana-1482	118	20	in	in	ADP
cana-1482	118	21	(	(	PUNCT
cana-1482	118	22	x	x	NOUN
cana-1482	118	23	,	,	PUNCT
cana-1482	118	24			NOUN
cana-1482	118	25	)	)	PUNCT
cana-1482	118	26	.	.	PUNCT
cana-1482	119	1	proof	proof	NOUN
cana-1482	119	2	:	:	PUNCT
cana-1482	119	3	let	let	VERB
cana-1482	119	4	�	�	PRON
cana-1482	119	5	̅	̅	NOUN
cana-1482	119	6	�	�	NOUN
cana-1482	119	7	be	be	AUX
cana-1482	119	8	an	an	DET
cana-1482	119	9	nsspcs	nsspc	NOUN
cana-1482	119	10	in	in	ADP
cana-1482	119	11	x	x	PUNCT
cana-1482	119	12	as	as	ADP
cana-1482	119	13	proof	proof	NOUN
cana-1482	119	14	.	.	PUNCT
cana-1482	120	1	assume	assume	VERB
cana-1482	120	2	that	that	SCONJ
cana-1482	120	3	in	in	ADP
cana-1482	120	4	(	(	PUNCT
cana-1482	120	5	x	x	NOUN
cana-1482	120	6	,	,	PUNCT
cana-1482	120	7			NOUN
cana-1482	120	8	)	)	PUNCT
cana-1482	120	9	,	,	PUNCT
cana-1482	120	10	�	�	PROPN
cana-1482	120	11	̅	̅	NOUN
cana-1482	120	12	�	�	PROPN
cana-1482	120	13	⊆	⊆	NUM
cana-1482	120	14	�	�	NOUN
cana-1482	120	15	̅	̅	NOUN
cana-1482	120	16	�	�	PROPN
cana-1482	120	17	and	and	CCONJ
cana-1482	120	18	�	�	PROPN
cana-1482	120	19	̅	̅	NOUN
cana-1482	120	20	�	�	NOUN
cana-1482	120	21	is	be	AUX
cana-1482	120	22	an	an	DET
cana-1482	120	23	nsos	nsos	NOUN
cana-1482	120	24	.	.	PUNCT
cana-1482	121	1	we	we	PRON
cana-1482	121	2	then	then	ADV
cana-1482	121	3	have	have	VERB
cana-1482	121	4	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	121	5	�	�	NOUN
cana-1482	121	6	̅	̅	NOUN
cana-1482	121	7	�	�	PROPN
cana-1482	121	8	)	)	PUNCT
cana-1482	121	9	⊆	⊆	NUM
cana-1482	121	10	�	�	NOUN
cana-1482	121	11	̅	̅	NOUN
cana-1482	121	12	�	�	PROPN
cana-1482	121	13	since	since	SCONJ
cana-1482	121	14	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	121	15	�	�	NOUN
cana-1482	121	16	̅	̅	NOUN
cana-1482	121	17	�	�	NOUN
cana-1482	121	18	)	)	PUNCT
cana-1482	121	19	=	=	SYM
cana-1482	121	20	�	�	PROPN
cana-1482	121	21	̅	̅	NOUN
cana-1482	121	22	�	�	PROPN
cana-1482	121	23	.	.	PUNCT
cana-1482	122	1	thus	thus	ADV
cana-1482	122	2	,	,	PUNCT
cana-1482	122	3	in	in	ADP
cana-1482	122	4	(	(	PUNCT
cana-1482	122	5	x	x	NOUN
cana-1482	122	6	,	,	PUNCT
cana-1482	122	7			NOUN
cana-1482	122	8	)	)	PUNCT
cana-1482	122	9	,	,	PUNCT
cana-1482	122	10	a	a	DET
cana-1482	122	11	˅	˅	NOUN
cana-1482	122	12	is	be	AUX
cana-1482	122	13	an	an	DET
cana-1482	122	14	nsgspcs	nsgspc	NOUN
cana-1482	122	15	.	.	PUNCT
cana-1482	123	1	communications	communication	NOUN
cana-1482	123	2	on	on	ADP
cana-1482	123	3	applied	apply	VERB
cana-1482	123	4	nonlinear	nonlinear	ADJ
cana-1482	123	5	analysis	analysis	NOUN
cana-1482	123	6	issn	issn	NOUN
cana-1482	123	7	:	:	PUNCT
cana-1482	123	8	1074	1074	NUM
cana-1482	123	9	-	-	PUNCT
cana-1482	123	10	133x	133x	NUM
cana-1482	123	11	vol	vol	NOUN
cana-1482	123	12	31	31	NUM
cana-1482	123	13	no	no	NOUN
cana-1482	123	14	.	.	PUNCT
cana-1482	124	1	8s	8s	PROPN
cana-1482	124	2	(	(	PUNCT
cana-1482	124	3	2024	2024	NUM
cana-1482	124	4	)	)	PUNCT
cana-1482	124	5	277	277	NUM
cana-1482	124	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-1482	124	7	2.8	2.8	NUM
cana-1482	124	8	remark	remark	NOUN
cana-1482	124	9	:	:	PUNCT
cana-1482	124	10	as	as	SCONJ
cana-1482	124	11	the	the	DET
cana-1482	124	12	following	follow	VERB
cana-1482	124	13	example	example	NOUN
cana-1482	124	14	shows	show	VERB
cana-1482	124	15	,	,	PUNCT
cana-1482	124	16	the	the	DET
cana-1482	124	17	preceding	precede	VERB
cana-1482	124	18	theorem	theorem	NOUN
cana-1482	124	19	's	's	PART
cana-1482	124	20	converse	converse	NOUN
cana-1482	124	21	need	need	AUX
cana-1482	124	22	not	not	PART
cana-1482	124	23	hold	hold	VERB
cana-1482	124	24	true	true	ADJ
cana-1482	124	25	.	.	PUNCT
cana-1482	125	1	llustration	llustration	NOUN
cana-1482	125	2	:	:	PUNCT
cana-1482	125	3	let	let	VERB
cana-1482	125	4	x	x	PUNCT
cana-1482	125	5	=	=	PRON
cana-1482	125	6	{	{	PUNCT
cana-1482	125	7	a	a	PRON
cana-1482	125	8	,	,	PUNCT
cana-1482	125	9	b	b	NOUN
cana-1482	125	10	}	}	PUNCT
cana-1482	125	11	and	and	CCONJ
cana-1482	125	12	�	�	PROPN
cana-1482	125	13	̅	̅	NOUN
cana-1482	125	14	�	�	NOUN
cana-1482	125	15	=	=	SYM
cana-1482	125	16	{	{	PUNCT
cana-1482	125	17	〈	〈	PROPN
cana-1482	125	18	𝑎	𝑎	NOUN
cana-1482	125	19	,	,	PUNCT
cana-1482	125	20	0.5	0.5	NUM
cana-1482	125	21	,	,	PUNCT
cana-1482	125	22	0.6,0.5	0.6,0.5	NUM
cana-1482	125	23	〉	〉	NOUN
cana-1482	125	24	,	,	PUNCT
cana-1482	125	25	〈	〈	PROPN
cana-1482	125	26	𝑏	𝑏	NOUN
cana-1482	125	27	,	,	PUNCT
cana-1482	125	28	0.6	0.6	NUM
cana-1482	125	29	,	,	PUNCT
cana-1482	125	30	0.7,0.4	0.7,0.4	NOUN
cana-1482	125	31	〉	〉	NOUN
cana-1482	125	32	}	}	PUNCT
cana-1482	125	33	be	be	VERB
cana-1482	125	34	the	the	DET
cana-1482	125	35	example	example	NOUN
cana-1482	125	36	.	.	PUNCT
cana-1482	126	1	then	then	ADV
cana-1482	126	2	,	,	PUNCT
cana-1482	126	3	on	on	ADP
cana-1482	126	4	x	x	NOUN
cana-1482	126	5	,	,	PUNCT
cana-1482	126	6			NOUN
cana-1482	126	7	=	=	PUNCT
cana-1482	126	8	{	{	PUNCT
cana-1482	126	9	0̅	0̅	PROPN
cana-1482	126	10	,	,	PUNCT
cana-1482	126	11	�	�	PROPN
cana-1482	126	12	̅	̅	NOUN
cana-1482	126	13	�	�	PROPN
cana-1482	126	14	,	,	PUNCT
cana-1482	126	15	1̅	1̅	NUM
cana-1482	126	16	}	}	PUNCT
cana-1482	126	17	is	be	AUX
cana-1482	126	18	an	an	DET
cana-1482	126	19	nst	nst	NOUN
cana-1482	126	20	.	.	PUNCT
cana-1482	126	21	consider	consider	VERB
cana-1482	126	22	the	the	DET
cana-1482	126	23	following	follow	VERB
cana-1482	126	24	nss	nss	NOUN
cana-1482	126	25	in	in	ADP
cana-1482	126	26	x	x	NOUN
cana-1482	126	27	:	:	PUNCT
cana-1482	126	28	�	�	NOUN
cana-1482	126	29	̅	̅	NOUN
cana-1482	126	30	�	�	NOUN
cana-1482	126	31	=	=	SYM
cana-1482	126	32	{	{	PUNCT
cana-1482	126	33	〈	〈	PROPN
cana-1482	126	34	𝑎	𝑎	NOUN
cana-1482	126	35	,	,	PUNCT
cana-1482	126	36	0.5	0.5	NUM
cana-1482	126	37	,	,	PUNCT
cana-1482	126	38	0.6,0.5	0.6,0.5	NUM
cana-1482	126	39	〉	〉	NOUN
cana-1482	126	40	,	,	PUNCT
cana-1482	126	41	〈	〈	PROPN
cana-1482	126	42	𝑏	𝑏	NOUN
cana-1482	126	43	,	,	PUNCT
cana-1482	126	44	0.7	0.7	NUM
cana-1482	126	45	,	,	PUNCT
cana-1482	126	46	0.8,0.3	0.8,0.3	NUM
cana-1482	126	47	〉	〉	NOUN
cana-1482	126	48	}	}	PUNCT
cana-1482	126	49	.	.	PUNCT
cana-1482	127	1	in	in	ADP
cana-1482	127	2	x	x	SYM
cana-1482	127	3	,	,	PUNCT
cana-1482	127	4	�	�	NOUN
cana-1482	127	5	̅	̅	NOUN
cana-1482	127	6	�	�	NOUN
cana-1482	127	7	is	be	AUX
cana-1482	127	8	not	not	PART
cana-1482	127	9	an	an	DET
cana-1482	127	10	nsspcs	nsspc	NOUN
cana-1482	127	11	,	,	PUNCT
cana-1482	127	12	but	but	CCONJ
cana-1482	127	13	it	it	PRON
cana-1482	127	14	is	be	AUX
cana-1482	127	15	an	an	DET
cana-1482	127	16	nsgspcs	nsgspc	NOUN
cana-1482	127	17	.	.	PUNCT
cana-1482	128	1	2.9	2.9	NUM
cana-1482	128	2	theorem	theorem	VERB
cana-1482	128	3	:	:	PUNCT
cana-1482	128	4	each	each	DET
cana-1482	128	5	nsαcs	nsαcs	NOUN
cana-1482	128	6	within	within	ADP
cana-1482	128	7	an	an	DET
cana-1482	128	8	nsts	nst	NOUN
cana-1482	128	9	(	(	PUNCT
cana-1482	128	10	x	x	X
cana-1482	128	11	,	,	PUNCT
cana-1482	128	12			NOUN
cana-1482	128	13	)	)	PUNCT
cana-1482	128	14	corresponds	correspond	VERB
cana-1482	128	15	to	to	ADP
cana-1482	128	16	an	an	DET
cana-1482	128	17	nsgspcs	nsgspc	NOUN
cana-1482	128	18	within	within	ADP
cana-1482	128	19	(	(	PUNCT
cana-1482	128	20	x	x	NOUN
cana-1482	128	21	,	,	PUNCT
cana-1482	128	22			NOUN
cana-1482	128	23	)	)	PUNCT
cana-1482	128	24	.	.	PUNCT
cana-1482	129	1	proof	proof	NOUN
cana-1482	129	2	:	:	PUNCT
cana-1482	129	3	theorem	theorem	VERB
cana-1482	129	4	2.7	2.7	NUM
cana-1482	129	5	makes	make	VERB
cana-1482	129	6	it	it	PRON
cana-1482	129	7	clear	clear	ADJ
cana-1482	129	8	that	that	SCONJ
cana-1482	129	9	every	every	DET
cana-1482	129	10	nsαcs	nsαcs	NOUN
cana-1482	129	11	is	be	AUX
cana-1482	129	12	also	also	ADV
cana-1482	129	13	an	an	DET
cana-1482	129	14	nsspcs	nsspc	NOUN
cana-1482	129	15	.	.	PUNCT
cana-1482	130	1	2.10	2.10	NUM
cana-1482	130	2	remark	remark	NOUN
cana-1482	130	3	:	:	PUNCT
cana-1482	130	4	the	the	DET
cana-1482	130	5	following	follow	VERB
cana-1482	130	6	example	example	NOUN
cana-1482	130	7	shows	show	VERB
cana-1482	130	8	that	that	SCONJ
cana-1482	130	9	the	the	DET
cana-1482	130	10	converse	converse	NOUN
cana-1482	130	11	of	of	ADP
cana-1482	130	12	the	the	DET
cana-1482	130	13	preceding	precede	VERB
cana-1482	130	14	theorem	theorem	NOUN
cana-1482	130	15	need	need	AUX
cana-1482	130	16	not	not	PART
cana-1482	130	17	be	be	AUX
cana-1482	130	18	true	true	ADJ
cana-1482	130	19	.	.	PUNCT
cana-1482	131	1	llustration	llustration	NOUN
cana-1482	131	2	:	:	PUNCT
cana-1482	131	3	let	let	VERB
cana-1482	131	4	x	x	PUNCT
cana-1482	131	5	=	=	PRON
cana-1482	131	6	{	{	PUNCT
cana-1482	131	7	a	a	PRON
cana-1482	131	8	,	,	PUNCT
cana-1482	131	9	b	b	NOUN
cana-1482	131	10	}	}	PUNCT
cana-1482	131	11	and	and	CCONJ
cana-1482	131	12	�	�	PROPN
cana-1482	131	13	̅	̅	NOUN
cana-1482	131	14	�	�	NOUN
cana-1482	131	15	=	=	SYM
cana-1482	131	16	{	{	PUNCT
cana-1482	131	17	〈	〈	PROPN
cana-1482	131	18	𝑎	𝑎	NOUN
cana-1482	131	19	,	,	PUNCT
cana-1482	131	20	0.5	0.5	NUM
cana-1482	131	21	,	,	PUNCT
cana-1482	131	22	0.6,0.5	0.6,0.5	NUM
cana-1482	131	23	〉	〉	NOUN
cana-1482	131	24	,	,	PUNCT
cana-1482	131	25	〈	〈	PROPN
cana-1482	131	26	𝑏	𝑏	NOUN
cana-1482	131	27	,	,	PUNCT
cana-1482	131	28	0.6,0.5,0.4	0.6,0.5,0.4	NUM
cana-1482	131	29	〉	〉	NOUN
cana-1482	131	30	}	}	PUNCT
cana-1482	131	31	be	be	AUX
cana-1482	131	32	the	the	DET
cana-1482	131	33	example	example	NOUN
cana-1482	131	34	.	.	PUNCT
cana-1482	132	1	then	then	ADV
cana-1482	132	2	,	,	PUNCT
cana-1482	132	3	on	on	ADP
cana-1482	132	4	x	x	NOUN
cana-1482	132	5	,	,	PUNCT
cana-1482	132	6			NOUN
cana-1482	132	7	=	=	PUNCT
cana-1482	132	8	{	{	PUNCT
cana-1482	132	9	0̅	0̅	PROPN
cana-1482	132	10	,	,	PUNCT
cana-1482	132	11	�	�	PROPN
cana-1482	132	12	̅	̅	NOUN
cana-1482	132	13	�	�	NOUN
cana-1482	132	14	,	,	PUNCT
cana-1482	132	15	1	1	NUM
cana-1482	132	16	̅	̅	NOUN
cana-1482	132	17	}	}	PUNCT
cana-1482	132	18	is	be	AUX
cana-1482	132	19	an	an	DET
cana-1482	132	20	nst	nst	PROPN
cana-1482	132	21	.	.	PROPN
cana-1482	132	22	assume	assume	VERB
cana-1482	132	23	that	that	SCONJ
cana-1482	132	24	an	an	DET
cana-1482	132	25	nss	nss	NOUN
cana-1482	132	26	in	in	ADP
cana-1482	132	27	x	x	VERB
cana-1482	132	28	is	be	AUX
cana-1482	132	29	�	�	NOUN
cana-1482	132	30	̅	̅	NOUN
cana-1482	132	31	�	�	NOUN
cana-1482	132	32	=	=	SYM
cana-1482	132	33	{	{	PUNCT
cana-1482	132	34	〈	〈	PROPN
cana-1482	132	35	𝑎	𝑎	NOUN
cana-1482	132	36	,	,	PUNCT
cana-1482	132	37	0.5,0.6,0.5	0.5,0.6,0.5	NUM
cana-1482	132	38	〉	〉	NOUN
cana-1482	132	39	,	,	PUNCT
cana-1482	132	40	〈	〈	NOUN
cana-1482	132	41	𝑏	𝑏	NOUN
cana-1482	132	42	,	,	PUNCT
cana-1482	132	43	0.7	0.7	NUM
cana-1482	132	44	,	,	PUNCT
cana-1482	132	45	0.5,0.3	0.5,0.3	NUM
cana-1482	132	46	〉	〉	NOUN
cana-1482	132	47	}	}	PUNCT
cana-1482	132	48	.	.	PUNCT
cana-1482	133	1	in	in	ADP
cana-1482	133	2	(	(	PUNCT
cana-1482	133	3	x	x	X
cana-1482	133	4	,	,	PUNCT
cana-1482	133	5			NOUN
cana-1482	133	6	)	)	PUNCT
cana-1482	133	7	,	,	PUNCT
cana-1482	133	8	�	�	PROPN
cana-1482	133	9	̅	̅	NOUN
cana-1482	133	10	�	�	NOUN
cana-1482	133	11	is	be	AUX
cana-1482	133	12	therefore	therefore	ADV
cana-1482	133	13	an	an	DET
cana-1482	133	14	nsgspcs	nsgspc	NOUN
cana-1482	133	15	but	but	CCONJ
cana-1482	133	16	not	not	PART
cana-1482	133	17	an	an	DET
cana-1482	133	18	nsαcs	nsαcs	NOUN
cana-1482	133	19	.	.	PUNCT
cana-1482	134	1	2.11	2.11	NUM
cana-1482	134	2	theorem	theorem	VERB
cana-1482	134	3	:	:	PUNCT
cana-1482	134	4	states	state	VERB
cana-1482	134	5	that	that	SCONJ
cana-1482	134	6	any	any	DET
cana-1482	134	7	nsβcs	nsβcs	NOUN
cana-1482	134	8	in	in	ADP
cana-1482	134	9	an	an	DET
cana-1482	134	10	nsts	nst	NOUN
cana-1482	134	11	(	(	PUNCT
cana-1482	134	12	x	x	X
cana-1482	134	13	,	,	PUNCT
cana-1482	134	14			NOUN
cana-1482	134	15	)	)	PUNCT
cana-1482	134	16	is	be	AUX
cana-1482	134	17	an	an	DET
cana-1482	134	18	nsgspcs	nsgspc	NOUN
cana-1482	134	19	in	in	ADP
cana-1482	134	20	(	(	PUNCT
cana-1482	134	21	x	x	NOUN
cana-1482	134	22	,	,	PUNCT
cana-1482	134	23			NOUN
cana-1482	134	24	)	)	PUNCT
cana-1482	134	25	.	.	PUNCT
cana-1482	135	1	proof	proof	NOUN
cana-1482	135	2	:	:	PUNCT
cana-1482	135	3	let	let	VERB
cana-1482	135	4	�	�	PRON
cana-1482	135	5	̅	̅	NOUN
cana-1482	135	6	�	�	NOUN
cana-1482	135	7	be	be	AUX
cana-1482	135	8	an	an	DET
cana-1482	135	9	nsβcs	nsβcs	NOUN
cana-1482	135	10	in	in	ADP
cana-1482	135	11	x	x	X
cana-1482	135	12	as	as	ADP
cana-1482	135	13	proof	proof	NOUN
cana-1482	135	14	.	.	PUNCT
cana-1482	136	1	suppose	suppose	VERB
cana-1482	136	2	that	that	SCONJ
cana-1482	136	3	in	in	ADP
cana-1482	136	4	(	(	PUNCT
cana-1482	136	5	x	x	X
cana-1482	136	6	,	,	PUNCT
cana-1482	136	7			NOUN
cana-1482	136	8	)	)	PUNCT
cana-1482	136	9	,	,	PUNCT
cana-1482	136	10	�	�	PROPN
cana-1482	136	11	̅	̅	NOUN
cana-1482	136	12	�	�	PROPN
cana-1482	136	13	⊆	⊆	NUM
cana-1482	136	14	�	�	NOUN
cana-1482	136	15	̅	̅	NOUN
cana-1482	136	16	�	�	PROPN
cana-1482	136	17	and	and	CCONJ
cana-1482	136	18	�	�	PROPN
cana-1482	136	19	̅	̅	NOUN
cana-1482	136	20	�	�	NOUN
cana-1482	136	21	is	be	AUX
cana-1482	136	22	an	an	DET
cana-1482	136	23	nsos	nsos	NOUN
cana-1482	136	24	.	.	PUNCT
cana-1482	137	1	we	we	PRON
cana-1482	137	2	have	have	VERB
cana-1482	137	3	𝑛𝑠𝛽𝑐𝑙(	𝑛𝑠𝛽𝑐𝑙(	NOUN
cana-1482	137	4	�	�	NOUN
cana-1482	137	5	̅	̅	NOUN
cana-1482	137	6	�	�	PROPN
cana-1482	137	7	)	)	PUNCT
cana-1482	137	8	⊆	⊆	NUM
cana-1482	137	9	�	�	NOUN
cana-1482	137	10	̅	̅	NOUN
cana-1482	137	11	�	�	NOUN
cana-1482	137	12	since	since	SCONJ
cana-1482	137	13	𝑛𝑠𝛽𝑐𝑙(	𝑛𝑠𝛽𝑐𝑙(	NOUN
cana-1482	137	14	�	�	NOUN
cana-1482	137	15	̅	̅	NOUN
cana-1482	137	16	�	�	NOUN
cana-1482	137	17	)	)	PUNCT
cana-1482	137	18	=	=	SYM
cana-1482	137	19	�	�	PROPN
cana-1482	137	20	̅	̅	NOUN
cana-1482	137	21	�	�	PROPN
cana-1482	137	22	.	.	PUNCT
cana-1482	138	1	thus	thus	ADV
cana-1482	138	2	,	,	PUNCT
cana-1482	138	3	in	in	ADP
cana-1482	138	4	(	(	PUNCT
cana-1482	138	5	x	x	NOUN
cana-1482	138	6	,	,	PUNCT
cana-1482	138	7			NOUN
cana-1482	138	8	)	)	PUNCT
cana-1482	138	9	,	,	PUNCT
cana-1482	138	10	�	�	PROPN
cana-1482	138	11	̅	̅	NOUN
cana-1482	138	12	�	�	NOUN
cana-1482	138	13	is	be	AUX
cana-1482	138	14	an	an	DET
cana-1482	138	15	nsgspcs	nsgspc	NOUN
cana-1482	138	16	.	.	PUNCT
cana-1482	139	1	2.12	2.12	NUM
cana-1482	139	2	remark	remark	NOUN
cana-1482	139	3	:	:	PUNCT
cana-1482	139	4	as	as	SCONJ
cana-1482	139	5	the	the	DET
cana-1482	139	6	following	follow	VERB
cana-1482	139	7	example	example	NOUN
cana-1482	139	8	illustrates	illustrate	VERB
cana-1482	139	9	,	,	PUNCT
cana-1482	139	10	the	the	DET
cana-1482	139	11	above	above	ADJ
cana-1482	139	12	theorem	theorem	NOUN
cana-1482	139	13	's	's	PART
cana-1482	139	14	converse	converse	NOUN
cana-1482	139	15	need	need	AUX
cana-1482	139	16	not	not	PART
cana-1482	139	17	be	be	AUX
cana-1482	139	18	true	true	ADJ
cana-1482	139	19	.	.	PUNCT
cana-1482	140	1	llustration	llustration	NOUN
cana-1482	140	2	:	:	PUNCT
cana-1482	140	3	let	let	VERB
cana-1482	140	4	x	x	PUNCT
cana-1482	140	5	=	=	PRON
cana-1482	140	6	{	{	PUNCT
cana-1482	140	7	a	a	PRON
cana-1482	140	8	,	,	PUNCT
cana-1482	140	9	b	b	NOUN
cana-1482	140	10	}	}	PUNCT
cana-1482	140	11	and	and	CCONJ
cana-1482	140	12	�	�	PROPN
cana-1482	140	13	̅	̅	NOUN
cana-1482	140	14	�	�	NOUN
cana-1482	140	15	=	=	SYM
cana-1482	140	16	{	{	PUNCT
cana-1482	140	17	〈	〈	PROPN
cana-1482	140	18	𝑎	𝑎	NOUN
cana-1482	140	19	,	,	PUNCT
cana-1482	140	20	0.5	0.5	NUM
cana-1482	140	21	,	,	PUNCT
cana-1482	140	22	0.6,0.5	0.6,0.5	NUM
cana-1482	140	23	〉	〉	NOUN
cana-1482	140	24	,	,	PUNCT
cana-1482	140	25	〈	〈	PROPN
cana-1482	140	26	𝑏	𝑏	NOUN
cana-1482	140	27	,	,	PUNCT
cana-1482	140	28	0.6	0.6	NUM
cana-1482	140	29	,	,	PUNCT
cana-1482	140	30	0.5,0.4	0.5,0.4	X
cana-1482	140	31	〉	〉	NOUN
cana-1482	140	32	}	}	PUNCT
cana-1482	140	33	be	be	AUX
cana-1482	140	34	the	the	DET
cana-1482	140	35	example	example	NOUN
cana-1482	140	36	.	.	PUNCT
cana-1482	141	1	then	then	ADV
cana-1482	141	2	,	,	PUNCT
cana-1482	141	3	on	on	ADP
cana-1482	141	4	x	x	NOUN
cana-1482	141	5	,	,	PUNCT
cana-1482	141	6			NOUN
cana-1482	141	7	=	=	PUNCT
cana-1482	141	8	{	{	PUNCT
cana-1482	141	9	0̅	0̅	PROPN
cana-1482	141	10	,	,	PUNCT
cana-1482	141	11	�	�	PROPN
cana-1482	141	12	̅	̅	NOUN
cana-1482	141	13	�	�	PROPN
cana-1482	141	14	,	,	PUNCT
cana-1482	141	15	1̅	1̅	PROPN
cana-1482	141	16	}	}	PUNCT
cana-1482	141	17	is	be	AUX
cana-1482	141	18	an	an	DET
cana-1482	141	19	nst	nst	PROPN
cana-1482	141	20	.	.	PROPN
cana-1482	141	21	assume	assume	VERB
cana-1482	141	22	that	that	SCONJ
cana-1482	141	23	an	an	DET
cana-1482	141	24	nss	nss	NOUN
cana-1482	141	25	in	in	ADP
cana-1482	141	26	x	x	VERB
cana-1482	141	27	is	be	AUX
cana-1482	141	28	�	�	NOUN
cana-1482	141	29	̅	̅	NOUN
cana-1482	141	30	�	�	NOUN
cana-1482	141	31	=	=	SYM
cana-1482	141	32	{	{	PUNCT
cana-1482	141	33	〈	〈	PROPN
cana-1482	141	34	𝑎	𝑎	NOUN
cana-1482	141	35	,	,	PUNCT
cana-1482	141	36	0.5	0.5	NUM
cana-1482	141	37	,	,	PUNCT
cana-1482	141	38	0.6,0.5	0.6,0.5	NOUN
cana-1482	141	39	〉	〉	NOUN
cana-1482	141	40	,	,	PUNCT
cana-1482	141	41	〈	〈	PROPN
cana-1482	141	42	𝑏	𝑏	NOUN
cana-1482	141	43	,	,	PUNCT
cana-1482	141	44	0.7	0.7	NUM
cana-1482	141	45	,	,	PUNCT
cana-1482	141	46	0.5,0.3	0.5,0.3	NUM
cana-1482	141	47	〉	〉	NOUN
cana-1482	141	48	}	}	PUNCT
cana-1482	141	49	.	.	PUNCT
cana-1482	142	1	in	in	ADP
cana-1482	142	2	such	such	ADJ
cana-1482	142	3	case	case	NOUN
cana-1482	142	4	,	,	PUNCT
cana-1482	142	5	�	�	PROPN
cana-1482	142	6	̅	̅	NOUN
cana-1482	142	7	�	�	NOUN
cana-1482	142	8	in	in	ADP
cana-1482	142	9	x	x	PROPN
cana-1482	142	10	is	be	AUX
cana-1482	142	11	an	an	DET
cana-1482	142	12	nsgspcs	nsgspc	NOUN
cana-1482	142	13	but	but	CCONJ
cana-1482	142	14	not	not	PART
cana-1482	142	15	an	an	DET
cana-1482	142	16	nsβcs	nsβcs	NOUN
cana-1482	142	17	.	.	PUNCT
cana-1482	143	1	2.13	2.13	NUM
cana-1482	143	2	theorem	theorem	VERB
cana-1482	143	3	:	:	PUNCT
cana-1482	143	4	each	each	DET
cana-1482	143	5	nsgspcs	nsgspc	NOUN
cana-1482	143	6	in	in	ADP
cana-1482	143	7	(	(	PUNCT
cana-1482	143	8	x	x	X
cana-1482	143	9	,	,	PUNCT
cana-1482	143	10			NOUN
cana-1482	143	11	)	)	PUNCT
cana-1482	143	12	is	be	AUX
cana-1482	143	13	an	an	DET
cana-1482	143	14	nsscs	nsscs	NOUN
cana-1482	143	15	in	in	ADP
cana-1482	143	16	the	the	DET
cana-1482	143	17	nsts	nst	NOUN
cana-1482	143	18	(	(	PUNCT
cana-1482	143	19	x	x	X
cana-1482	143	20	,	,	PUNCT
cana-1482	143	21			NOUN
cana-1482	143	22	)	)	PUNCT
cana-1482	143	23	.	.	PUNCT
cana-1482	144	1	proof	proof	NOUN
cana-1482	144	2	:	:	PUNCT
cana-1482	144	3	let	let	VERB
cana-1482	144	4	a	a	DET
cana-1482	144	5	˅	˅	NOUN
cana-1482	144	6	in	in	ADP
cana-1482	144	7	(	(	PUNCT
cana-1482	144	8	x	x	X
cana-1482	144	9	,	,	PUNCT
cana-1482	144	10			NOUN
cana-1482	144	11	)	)	PUNCT
cana-1482	144	12	be	be	AUX
cana-1482	144	13	an	an	DET
cana-1482	144	14	nsscs	nssc	NOUN
cana-1482	144	15	.	.	PUNCT
cana-1482	145	1	according	accord	VERB
cana-1482	145	2	to	to	ADP
cana-1482	145	3	theorem	theorem	ADJ
cana-1482	145	4	2.7	2.7	NUM
cana-1482	145	5	,	,	PUNCT
cana-1482	145	6	�	�	NOUN
cana-1482	145	7	̅	̅	NOUN
cana-1482	145	8	�	�	NOUN
cana-1482	145	9	is	be	AUX
cana-1482	145	10	an	an	DET
cana-1482	145	11	nsgspcs	nsgspc	NOUN
cana-1482	145	12	in	in	ADP
cana-1482	145	13	(	(	PUNCT
cana-1482	145	14	x	x	X
cana-1482	145	15	,	,	PUNCT
cana-1482	145	16			NOUN
cana-1482	145	17	)	)	PUNCT
cana-1482	145	18	since	since	SCONJ
cana-1482	145	19	every	every	DET
cana-1482	145	20	nsscs	nsscs	NOUN
cana-1482	145	21	is	be	AUX
cana-1482	145	22	an	an	DET
cana-1482	145	23	nsspcs	nsspc	NOUN
cana-1482	145	24	.	.	PUNCT
cana-1482	146	1	2.14	2.14	NUM
cana-1482	146	2	remark	remark	NOUN
cana-1482	146	3	:	:	PUNCT
cana-1482	146	4	as	as	SCONJ
cana-1482	146	5	the	the	DET
cana-1482	146	6	following	follow	VERB
cana-1482	146	7	example	example	NOUN
cana-1482	146	8	illustrates	illustrate	VERB
cana-1482	146	9	,	,	PUNCT
cana-1482	146	10	the	the	DET
cana-1482	146	11	above	above	ADJ
cana-1482	146	12	theorem	theorem	NOUN
cana-1482	146	13	's	's	PART
cana-1482	146	14	converse	converse	NOUN
cana-1482	146	15	need	need	AUX
cana-1482	146	16	not	not	PART
cana-1482	146	17	be	be	AUX
cana-1482	146	18	true	true	ADJ
cana-1482	146	19	.	.	PUNCT
cana-1482	147	1	llustration	llustration	NOUN
cana-1482	147	2	:	:	PUNCT
cana-1482	147	3	let	let	VERB
cana-1482	147	4	x	x	PUNCT
cana-1482	147	5	=	=	PRON
cana-1482	147	6	{	{	PUNCT
cana-1482	147	7	a	a	PRON
cana-1482	147	8	,	,	PUNCT
cana-1482	147	9	b	b	NOUN
cana-1482	147	10	}	}	PUNCT
cana-1482	147	11	and	and	CCONJ
cana-1482	147	12	�	�	PROPN
cana-1482	147	13	̅	̅	NOUN
cana-1482	147	14	�	�	NOUN
cana-1482	147	15	=	=	SYM
cana-1482	147	16	{	{	PUNCT
cana-1482	147	17	〈	〈	PROPN
cana-1482	147	18	𝑎	𝑎	NOUN
cana-1482	147	19	,	,	PUNCT
cana-1482	147	20	0.5	0.5	NUM
cana-1482	147	21	,	,	PUNCT
cana-1482	147	22	0.6,0.5	0.6,0.5	NOUN
cana-1482	147	23	〉	〉	NOUN
cana-1482	147	24	,	,	PUNCT
cana-1482	147	25	〈	〈	PROPN
cana-1482	147	26	𝑏	𝑏	NOUN
cana-1482	147	27	,	,	PUNCT
cana-1482	147	28	0.6	0.6	NUM
cana-1482	147	29	,	,	PUNCT
cana-1482	147	30	0.5,0.4	0.5,0.4	X
cana-1482	147	31	〉	〉	NOUN
cana-1482	147	32	}	}	PUNCT
cana-1482	147	33	be	be	AUX
cana-1482	147	34	the	the	DET
cana-1482	147	35	example	example	NOUN
cana-1482	147	36	.	.	PUNCT
cana-1482	148	1	then	then	ADV
cana-1482	148	2	,	,	PUNCT
cana-1482	148	3	on	on	ADP
cana-1482	148	4	x	x	NOUN
cana-1482	148	5	,	,	PUNCT
cana-1482	148	6			NOUN
cana-1482	148	7	=	=	PUNCT
cana-1482	148	8	{	{	PUNCT
cana-1482	148	9	0̅	0̅	PROPN
cana-1482	148	10	,	,	PUNCT
cana-1482	148	11	�	�	PROPN
cana-1482	148	12	̅	̅	NOUN
cana-1482	148	13	�	�	PROPN
cana-1482	148	14	,	,	PUNCT
cana-1482	148	15	1	1	NUM
cana-1482	148	16	̅	̅	NOUN
cana-1482	148	17	}	}	PUNCT
cana-1482	148	18	is	be	AUX
cana-1482	148	19	an	an	DET
cana-1482	148	20	nst	nst	PROPN
cana-1482	148	21	.	.	PROPN
cana-1482	148	22	assume	assume	VERB
cana-1482	148	23	that	that	SCONJ
cana-1482	148	24	the	the	DET
cana-1482	148	25	nss	nss	NOUN
cana-1482	148	26	in	in	ADP
cana-1482	148	27	x	x	VERB
cana-1482	148	28	is	be	AUX
cana-1482	148	29	�	�	NOUN
cana-1482	148	30	̅	̅	NOUN
cana-1482	148	31	�	�	NOUN
cana-1482	148	32	=	=	SYM
cana-1482	148	33	{	{	PUNCT
cana-1482	148	34	〈	〈	PROPN
cana-1482	148	35	𝑎	𝑎	NOUN
cana-1482	148	36	,	,	PUNCT
cana-1482	148	37	0.5	0.5	NUM
cana-1482	148	38	,	,	PUNCT
cana-1482	148	39	0.6,0.5	0.6,0.5	NUM
cana-1482	148	40	〉	〉	NOUN
cana-1482	148	41	,	,	PUNCT
cana-1482	148	42	〈	〈	PROPN
cana-1482	148	43	𝑏	𝑏	NOUN
cana-1482	148	44	,	,	PUNCT
cana-1482	148	45	0.7	0.7	NUM
cana-1482	148	46	,	,	PUNCT
cana-1482	148	47	0.5	0.5	NUM
cana-1482	148	48	,	,	PUNCT
cana-1482	148	49	0.3	0.3	NUM
cana-1482	148	50	〉	〉	NOUN
cana-1482	148	51	}	}	PUNCT
cana-1482	148	52	.	.	PUNCT
cana-1482	149	1	in	in	ADP
cana-1482	149	2	x	x	SYM
cana-1482	149	3	,	,	PUNCT
cana-1482	149	4	�	�	NOUN
cana-1482	149	5	̅	̅	NOUN
cana-1482	149	6	�	�	NOUN
cana-1482	149	7	is	be	AUX
cana-1482	149	8	an	an	DET
cana-1482	149	9	nsgspcs	nsgspc	NOUN
cana-1482	149	10	,	,	PUNCT
cana-1482	149	11	but	but	CCONJ
cana-1482	149	12	it	it	PRON
cana-1482	149	13	is	be	AUX
cana-1482	149	14	not	not	PART
cana-1482	149	15	an	an	DET
cana-1482	149	16	nsscs	nssc	NOUN
cana-1482	149	17	.	.	PUNCT
cana-1482	150	1	2.15	2.15	NUM
cana-1482	150	2	theorem	theorem	NOUN
cana-1482	150	3	:	:	PUNCT
cana-1482	150	4	each	each	DET
cana-1482	150	5	nsgspcs	nsgspc	NOUN
cana-1482	150	6	in	in	ADP
cana-1482	150	7	(	(	PUNCT
cana-1482	150	8	x	x	X
cana-1482	150	9	,	,	PUNCT
cana-1482	150	10			NOUN
cana-1482	150	11	)	)	PUNCT
cana-1482	150	12	is	be	AUX
cana-1482	150	13	an	an	DET
cana-1482	150	14	nspcs	nspc	NOUN
cana-1482	150	15	in	in	ADP
cana-1482	150	16	an	an	DET
cana-1482	150	17	nsts	nst	NOUN
cana-1482	150	18	.	.	PUNCT
cana-1482	151	1	proof	proof	NOUN
cana-1482	151	2	:	:	PUNCT
cana-1482	151	3	theorem	theorem	VERB
cana-1482	151	4	2.7	2.7	NUM
cana-1482	151	5	makes	make	VERB
cana-1482	151	6	it	it	PRON
cana-1482	151	7	clear	clear	ADJ
cana-1482	151	8	that	that	SCONJ
cana-1482	151	9	every	every	DET
cana-1482	151	10	nspcs	nspc	NOUN
cana-1482	151	11	is	be	AUX
cana-1482	151	12	an	an	DET
cana-1482	151	13	nsspcs	nsspc	NOUN
cana-1482	151	14	.	.	PUNCT
cana-1482	152	1	2.16	2.16	NUM
cana-1482	152	2	remark	remark	NOUN
cana-1482	152	3	:	:	PUNCT
cana-1482	152	4	as	as	SCONJ
cana-1482	152	5	the	the	DET
cana-1482	152	6	following	follow	VERB
cana-1482	152	7	example	example	NOUN
cana-1482	152	8	illustrates	illustrate	VERB
cana-1482	152	9	,	,	PUNCT
cana-1482	152	10	the	the	DET
cana-1482	152	11	above	above	ADJ
cana-1482	152	12	theorem	theorem	NOUN
cana-1482	152	13	's	's	PART
cana-1482	152	14	converse	converse	NOUN
cana-1482	152	15	need	need	AUX
cana-1482	152	16	not	not	PART
cana-1482	152	17	be	be	AUX
cana-1482	152	18	true	true	ADJ
cana-1482	152	19	.	.	PUNCT
cana-1482	153	1	llustration	llustration	NOUN
cana-1482	153	2	:	:	PUNCT
cana-1482	153	3	let	let	VERB
cana-1482	153	4	x	x	PUNCT
cana-1482	153	5	=	=	PRON
cana-1482	153	6	{	{	PUNCT
cana-1482	153	7	a	a	PRON
cana-1482	153	8	,	,	PUNCT
cana-1482	153	9	b	b	NOUN
cana-1482	153	10	}	}	PUNCT
cana-1482	153	11	and	and	CCONJ
cana-1482	153	12	�	�	PROPN
cana-1482	153	13	̅	̅	NOUN
cana-1482	153	14	�	�	NOUN
cana-1482	153	15	=	=	SYM
cana-1482	153	16	{	{	PUNCT
cana-1482	153	17	〈	〈	PROPN
cana-1482	153	18	𝑎	𝑎	NOUN
cana-1482	153	19	,	,	PUNCT
cana-1482	153	20	0.5	0.5	NUM
cana-1482	153	21	,	,	PUNCT
cana-1482	153	22	0.6,0.5	0.6,0.5	NOUN
cana-1482	153	23	〉	〉	NOUN
cana-1482	153	24	,	,	PUNCT
cana-1482	153	25	〈	〈	NOUN
cana-1482	153	26	𝑏	𝑏	NOUN
cana-1482	153	27	,	,	PUNCT
cana-1482	153	28	0.6	0.6	NUM
cana-1482	153	29	,	,	PUNCT
cana-1482	153	30	0.5,0.4	0.5,0.4	X
cana-1482	153	31	〉	〉	NOUN
cana-1482	153	32	}	}	PUNCT
cana-1482	153	33	be	be	AUX
cana-1482	153	34	the	the	DET
cana-1482	153	35	example	example	NOUN
cana-1482	153	36	.	.	PUNCT
cana-1482	154	1	hence	hence	ADV
cana-1482	154	2	,	,	PUNCT
cana-1482	154	3	an	an	DET
cana-1482	154	4	nst	nst	NOUN
cana-1482	154	5	on	on	ADP
cana-1482	154	6	x	x	AUX
cana-1482	154	7	is	be	AUX
cana-1482	154	8	given	give	VERB
cana-1482	154	9	by	by	ADP
cana-1482	154	10	=	=	PROPN
cana-1482	154	11	{	{	PUNCT
cana-1482	154	12	0̅	0̅	PROPN
cana-1482	154	13	,	,	PUNCT
cana-1482	154	14	�	�	PROPN
cana-1482	154	15	̅	̅	NOUN
cana-1482	154	16	�	�	NOUN
cana-1482	154	17	,	,	PUNCT
cana-1482	154	18	1̅	1̅	NUM
cana-1482	154	19	}	}	PUNCT
cana-1482	154	20	.	.	PUNCT
cana-1482	155	1	consider	consider	VERB
cana-1482	155	2	the	the	DET
cana-1482	155	3	following	follow	VERB
cana-1482	155	4	nss	nss	NOUN
cana-1482	155	5	in	in	ADP
cana-1482	155	6	x.	x.	NOUN
cana-1482	155	7	a̅	a̅	PROPN
cana-1482	156	1	=	=	PUNCT
cana-1482	156	2	{	{	PUNCT
cana-1482	156	3	〈	〈	PROPN
cana-1482	156	4	𝑎	𝑎	NOUN
cana-1482	156	5	,	,	PUNCT
cana-1482	156	6	0.5	0.5	NUM
cana-1482	156	7	,	,	PUNCT
cana-1482	156	8	0.6,0.5	0.6,0.5	NUM
cana-1482	156	9	〉	〉	NOUN
cana-1482	156	10	,	,	PUNCT
cana-1482	156	11	〈	〈	PROPN
cana-1482	156	12	𝑏	𝑏	NOUN
cana-1482	156	13	,	,	PUNCT
cana-1482	156	14	0.7	0.7	NUM
cana-1482	156	15	,	,	PUNCT
cana-1482	156	16	0.5,0.3	0.5,0.3	PROPN
cana-1482	156	17	〉	〉	NOUN
cana-1482	156	18	}	}	PUNCT
cana-1482	156	19	.	.	PUNCT
cana-1482	157	1	in	in	ADP
cana-1482	157	2	x	x	SYM
cana-1482	157	3	,	,	PUNCT
cana-1482	157	4	𝐴	𝐴	PROPN
cana-1482	157	5	is	be	AUX
cana-1482	157	6	an	an	DET
cana-1482	157	7	nsgspcs	nsgspc	NOUN
cana-1482	157	8	,	,	PUNCT
cana-1482	157	9	but	but	CCONJ
cana-1482	157	10	it	it	PRON
cana-1482	157	11	is	be	AUX
cana-1482	157	12	not	not	PART
cana-1482	157	13	an	an	DET
cana-1482	157	14	nspcs	nspc	NOUN
cana-1482	157	15	.	.	PUNCT
cana-1482	158	1	2.17	2.17	NUM
cana-1482	158	2	remark	remark	NOUN
cana-1482	158	3	:	:	PUNCT
cana-1482	158	4	in	in	ADP
cana-1482	158	5	an	an	DET
cana-1482	158	6	nsts	nst	NOUN
cana-1482	158	7	(	(	PUNCT
cana-1482	158	8	x,	x,	X
cana-1482	158	9	)	)	PUNCT
cana-1482	158	10	,	,	PUNCT
cana-1482	158	11	the	the	DET
cana-1482	158	12	union	union	NOUN
cana-1482	158	13	of	of	ADP
cana-1482	158	14	any	any	DET
cana-1482	158	15	two	two	NUM
cana-1482	158	16	nsgspcs	nsgspc	NOUN
cana-1482	158	17	is	be	AUX
cana-1482	158	18	not	not	PART
cana-1482	158	19	an	an	DET
cana-1482	158	20	nsgspcs	nsgspc	NOUN
cana-1482	158	21	in	in	ADP
cana-1482	158	22	(	(	PUNCT
cana-1482	158	23	x,	x,	X
cana-1482	158	24	)	)	PUNCT
cana-1482	158	25	.	.	PUNCT
cana-1482	159	1	communications	communication	NOUN
cana-1482	159	2	on	on	ADP
cana-1482	159	3	applied	apply	VERB
cana-1482	159	4	nonlinear	nonlinear	ADJ
cana-1482	159	5	analysis	analysis	NOUN
cana-1482	159	6	issn	issn	NOUN
cana-1482	159	7	:	:	PUNCT
cana-1482	159	8	1074	1074	NUM
cana-1482	159	9	-	-	PUNCT
cana-1482	159	10	133x	133x	NUM
cana-1482	159	11	vol	vol	NOUN
cana-1482	159	12	31	31	NUM
cana-1482	159	13	no	no	NOUN
cana-1482	159	14	.	.	PUNCT
cana-1482	160	1	8s	8s	PROPN
cana-1482	160	2	(	(	PUNCT
cana-1482	160	3	2024	2024	NUM
cana-1482	160	4	)	)	PUNCT
cana-1482	160	5	278	278	NUM
cana-1482	160	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	160	7	llustration	llustration	NOUN
cana-1482	160	8	:	:	PUNCT
cana-1482	160	9	x	x	SYM
cana-1482	160	10	=	=	X
cana-1482	160	11	{	{	PUNCT
cana-1482	160	12	a	a	PRON
cana-1482	160	13	,	,	PUNCT
cana-1482	160	14	b	b	NOUN
cana-1482	160	15	}	}	PUNCT
cana-1482	160	16	and	and	CCONJ
cana-1482	160	17	�	�	PROPN
cana-1482	160	18	̅	̅	NOUN
cana-1482	160	19	�	�	NOUN
cana-1482	160	20	1	1	NUM
cana-1482	160	21	=	=	SYM
cana-1482	160	22	{	{	PUNCT
cana-1482	160	23	〈	〈	PROPN
cana-1482	160	24	𝑎	𝑎	NOUN
cana-1482	160	25	,	,	PUNCT
cana-1482	160	26	0.7	0.7	NUM
cana-1482	160	27	,	,	PUNCT
cana-1482	160	28	0.5,0.3	0.5,0.3	PROPN
cana-1482	160	29	〉	〉	NOUN
cana-1482	160	30	,	,	PUNCT
cana-1482	160	31	〈	〈	PROPN
cana-1482	160	32	𝑏	𝑏	NOUN
cana-1482	160	33	,	,	PUNCT
cana-1482	160	34	0.8	0.8	NUM
cana-1482	160	35	,	,	PUNCT
cana-1482	160	36	0.5,0.2	0.5,0.2	PROPN
cana-1482	160	37	〉	〉	NOUN
cana-1482	160	38	}	}	PUNCT
cana-1482	160	39	and	and	CCONJ
cana-1482	160	40	�	�	PROPN
cana-1482	160	41	̅	̅	NOUN
cana-1482	160	42	�	�	NOUN
cana-1482	160	43	2	2	NUM
cana-1482	160	44	=	=	NOUN
cana-1482	160	45	{	{	PUNCT
cana-1482	160	46	〈	〈	PROPN
cana-1482	160	47	𝑎	𝑎	NOUN
cana-1482	160	48	,	,	PUNCT
cana-1482	160	49	0.6	0.6	NUM
cana-1482	160	50	,	,	PUNCT
cana-1482	160	51	0.5,0.4	0.5,0.4	NOUN
cana-1482	160	52	〉	〉	NOUN
cana-1482	160	53	,	,	PUNCT
cana-1482	160	54	〈	〈	PROPN
cana-1482	160	55	𝑏	𝑏	NOUN
cana-1482	160	56	,	,	PUNCT
cana-1482	160	57	0.7	0.7	NUM
cana-1482	160	58	,	,	PUNCT
cana-1482	160	59	0.5,0.3	0.5,0.3	PROPN
cana-1482	160	60	〉	〉	NOUN
cana-1482	160	61	}	}	PUNCT
cana-1482	160	62	.	.	PUNCT
cana-1482	161	1	should	should	AUX
cana-1482	161	2	be	be	AUX
cana-1482	161	3	taken	take	VERB
cana-1482	161	4	into	into	ADP
cana-1482	161	5	consideration	consideration	NOUN
cana-1482	161	6	.	.	PUNCT
cana-1482	162	1	hence	hence	ADV
cana-1482	162	2	,	,	PUNCT
cana-1482	162	3	an	an	DET
cana-1482	162	4	nst	nst	NOUN
cana-1482	162	5	on	on	ADP
cana-1482	162	6	x	x	AUX
cana-1482	162	7	is	be	AUX
cana-1482	162	8	given	give	VERB
cana-1482	162	9	by	by	ADP
cana-1482	162	10	=	=	PROPN
cana-1482	162	11	{	{	PUNCT
cana-1482	162	12	0̅	0̅	PROPN
cana-1482	162	13	,	,	PUNCT
cana-1482	162	14	�	�	PROPN
cana-1482	162	15	̅	̅	NOUN
cana-1482	162	16	�	�	NOUN
cana-1482	162	17	1	1	NUM
cana-1482	162	18	,	,	PUNCT
cana-1482	162	19	�	�	NOUN
cana-1482	162	20	̅	̅	NOUN
cana-1482	162	21	�	�	NOUN
cana-1482	162	22	2	2	NUM
cana-1482	162	23	,	,	PUNCT
cana-1482	162	24	1	1	NUM
cana-1482	162	25	̅	̅	NOUN
cana-1482	162	26	}	}	PUNCT
cana-1482	162	27	.	.	PUNCT
cana-1482	163	1	assume	assume	VERB
cana-1482	163	2	that	that	SCONJ
cana-1482	163	3	there	there	PRON
cana-1482	163	4	are	be	VERB
cana-1482	163	5	two	two	NUM
cana-1482	163	6	nsss	nsss	NOUN
cana-1482	163	7	in	in	ADP
cana-1482	163	8	x	x	PROPN
cana-1482	163	9	,	,	PUNCT
cana-1482	163	10	�	�	NOUN
cana-1482	163	11	̅	̅	NOUN
cana-1482	163	12	�	�	NOUN
cana-1482	163	13	=	=	SYM
cana-1482	163	14	{	{	PUNCT
cana-1482	163	15	〈	〈	PROPN
cana-1482	163	16	𝑎	𝑎	NOUN
cana-1482	163	17	,	,	PUNCT
cana-1482	163	18	0.6	0.6	NUM
cana-1482	163	19	,	,	PUNCT
cana-1482	163	20	0.5,0.4	0.5,0.4	X
cana-1482	163	21	〉	〉	NOUN
cana-1482	163	22	,	,	PUNCT
cana-1482	163	23	〈	〈	PROPN
cana-1482	163	24	𝑏	𝑏	NOUN
cana-1482	163	25	,	,	PUNCT
cana-1482	163	26	0.4	0.4	NUM
cana-1482	163	27	,	,	PUNCT
cana-1482	163	28	0.5,0.3	0.5,0.3	PROPN
cana-1482	163	29	〉	〉	NOUN
cana-1482	163	30	}	}	PUNCT
cana-1482	163	31	and	and	CCONJ
cana-1482	163	32	�	�	PROPN
cana-1482	163	33	̅	̅	NOUN
cana-1482	163	34	�	�	NOUN
cana-1482	163	35	=	=	SYM
cana-1482	163	36	{	{	PUNCT
cana-1482	163	37	〈	〈	PROPN
cana-1482	163	38	𝑎	𝑎	NOUN
cana-1482	163	39	,	,	PUNCT
cana-1482	163	40	0.4	0.4	NUM
cana-1482	163	41	,	,	PUNCT
cana-1482	163	42	0.5,0.4	0.5,0.4	NUM
cana-1482	163	43	〉	〉	NOUN
cana-1482	163	44	,	,	PUNCT
cana-1482	163	45	〈	〈	PROPN
cana-1482	163	46	𝑏	𝑏	NOUN
cana-1482	163	47	,	,	PUNCT
cana-1482	163	48	0.8	0.8	NUM
cana-1482	163	49	,	,	PUNCT
cana-1482	163	50	0.5,0.2	0.5,0.2	PROPN
cana-1482	163	51	〉	〉	NOUN
cana-1482	163	52	}	}	PUNCT
cana-1482	163	53	since	since	SCONJ
cana-1482	163	54	�	�	NOUN
cana-1482	163	55	̅	̅	NOUN
cana-1482	163	56	�	�	PROPN
cana-1482	163	57	∪	∪	ADP
cana-1482	163	58	�	�	PROPN
cana-1482	163	59	̅	̅	NOUN
cana-1482	163	60	�	�	NOUN
cana-1482	163	61	=	=	SYM
cana-1482	163	62	{	{	PUNCT
cana-1482	163	63	〈	〈	PROPN
cana-1482	163	64	𝑎	𝑎	NOUN
cana-1482	163	65	,	,	PUNCT
cana-1482	163	66	0.6	0.6	NUM
cana-1482	163	67	,	,	PUNCT
cana-1482	163	68	0.5,0.4	0.5,0.4	X
cana-1482	163	69	〉	〉	NOUN
cana-1482	163	70	,	,	PUNCT
cana-1482	163	71	〈	〈	PROPN
cana-1482	163	72	𝑏	𝑏	NOUN
cana-1482	163	73	,	,	PUNCT
cana-1482	163	74	0.8	0.8	NUM
cana-1482	163	75	,	,	PUNCT
cana-1482	163	76	0.5,0.2	0.5,0.2	PROPN
cana-1482	163	77	〉	〉	NOUN
cana-1482	163	78	}	}	PUNCT
cana-1482	163	79	⊆	⊆	NUM
cana-1482	163	80	�	�	NOUN
cana-1482	163	81	̅	̅	NOUN
cana-1482	163	82	�	�	NOUN
cana-1482	163	83	1	1	NUM
cana-1482	163	84	,	,	PUNCT
cana-1482	163	85	then	then	ADV
cana-1482	163	86	�	�	PROPN
cana-1482	163	87	̅	̅	NOUN
cana-1482	163	88	�	�	PROPN
cana-1482	163	89	and	and	CCONJ
cana-1482	163	90	�	�	PROPN
cana-1482	163	91	̅	̅	NOUN
cana-1482	163	92	�	�	NOUN
cana-1482	163	93	are	be	AUX
cana-1482	163	94	nsgspcs	nsgspc	NOUN
cana-1482	163	95	,	,	PUNCT
cana-1482	163	96	but	but	CCONJ
cana-1482	163	97	�	�	PROPN
cana-1482	163	98	̅	̅	NOUN
cana-1482	163	99	�	�	PROPN
cana-1482	163	100	∪	∪	ADP
cana-1482	163	101	�	�	PROPN
cana-1482	163	102	̅	̅	NOUN
cana-1482	163	103	�	�	NOUN
cana-1482	163	104	is	be	AUX
cana-1482	163	105	not	not	PART
cana-1482	163	106	in	in	ADP
cana-1482	163	107	x.	x.	PROPN
cana-1482	163	108	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	163	109	�	�	PROPN
cana-1482	163	110	̅	̅	NOUN
cana-1482	163	111	�	�	PROPN
cana-1482	163	112	∪	∪	ADP
cana-1482	163	113	�	�	PROPN
cana-1482	163	114	̅	̅	NOUN
cana-1482	163	115	�	�	NOUN
cana-1482	163	116	)	)	PUNCT
cana-1482	163	117	=	=	SYM
cana-1482	164	1	1̅	1̅	NUM
cana-1482	164	2	⊄	⊄	NOUN
cana-1482	164	3	�	�	NOUN
cana-1482	164	4	̅	̅	NOUN
cana-1482	164	5	�	�	NOUN
cana-1482	164	6	1	1	NUM
cana-1482	164	7	.	.	PUNCT
cana-1482	164	8	2.18	2.18	NUM
cana-1482	164	9	remark	remark	NOUN
cana-1482	164	10	:	:	PUNCT
cana-1482	164	11	an	an	DET
cana-1482	164	12	nsgspcs	nsgspc	NOUN
cana-1482	164	13	in	in	ADP
cana-1482	164	14	(	(	PUNCT
cana-1482	164	15	x	x	X
cana-1482	164	16	,	,	PUNCT
cana-1482	164	17			NOUN
cana-1482	164	18	)	)	PUNCT
cana-1482	164	19	is	be	AUX
cana-1482	164	20	not	not	PART
cana-1482	164	21	the	the	DET
cana-1482	164	22	intersection	intersection	NOUN
cana-1482	164	23	of	of	ADP
cana-1482	164	24	two	two	NUM
cana-1482	164	25	nsgspcs	nsgspc	NOUN
cana-1482	164	26	in	in	ADP
cana-1482	164	27	an	an	DET
cana-1482	164	28	nsts	nst	NOUN
cana-1482	164	29	.	.	PUNCT
cana-1482	165	1	llustration	llustration	NOUN
cana-1482	165	2	:	:	PUNCT
cana-1482	165	3	consider	consider	VERB
cana-1482	165	4	the	the	DET
cana-1482	165	5	example	example	NOUN
cana-1482	165	6	,	,	PUNCT
cana-1482	165	7	let	let	VERB
cana-1482	165	8	𝑋	𝑋	PROPN
cana-1482	165	9	=	=	SYM
cana-1482	165	10	{	{	PUNCT
cana-1482	165	11	𝑎	𝑎	NOUN
cana-1482	165	12	,	,	PUNCT
cana-1482	165	13	𝑏	𝑏	NOUN
cana-1482	165	14	}	}	PUNCT
cana-1482	165	15	and	and	CCONJ
cana-1482	165	16	�	�	PROPN
cana-1482	165	17	̅	̅	NOUN
cana-1482	165	18	�	�	NOUN
cana-1482	165	19	=	=	SYM
cana-1482	165	20	{	{	PUNCT
cana-1482	165	21	〈	〈	PROPN
cana-1482	165	22	𝑎	𝑎	NOUN
cana-1482	165	23	,	,	PUNCT
cana-1482	165	24	0.5	0.5	NUM
cana-1482	165	25	,	,	PUNCT
cana-1482	165	26	0.6,0.5	0.6,0.5	NUM
cana-1482	165	27	〉	〉	NOUN
cana-1482	165	28	,	,	PUNCT
cana-1482	165	29	〈	〈	PROPN
cana-1482	165	30	𝑏	𝑏	NOUN
cana-1482	165	31	,	,	PUNCT
cana-1482	165	32	0.6,0.5,0.4	0.6,0.5,0.4	NUM
cana-1482	165	33	〉	〉	NOUN
cana-1482	165	34	}	}	PUNCT
cana-1482	165	35	.	.	PUNCT
cana-1482	166	1	then	then	ADV
cana-1482	166	2			VERB
cana-1482	166	3	=	=	PUNCT
cana-1482	166	4	{	{	PUNCT
cana-1482	166	5	0̅	0̅	PROPN
cana-1482	166	6	,	,	PUNCT
cana-1482	166	7	�	�	PROPN
cana-1482	166	8	̅	̅	NOUN
cana-1482	166	9	�	�	NOUN
cana-1482	166	10	,	,	PUNCT
cana-1482	166	11	1	1	NUM
cana-1482	166	12	̅	̅	NOUN
cana-1482	166	13	}	}	PUNCT
cana-1482	166	14	is	be	AUX
cana-1482	166	15	an	an	DET
cana-1482	166	16	nst	nst	NOUN
cana-1482	166	17	on	on	ADP
cana-1482	166	18	x.	x.	PROPN
cana-1482	166	19	let	let	VERB
cana-1482	166	20	�	�	PRON
cana-1482	166	21	̅	̅	VERB
cana-1482	166	22	�	�	NOUN
cana-1482	166	23	=	=	SYM
cana-1482	166	24	{	{	PUNCT
cana-1482	166	25	〈	〈	PROPN
cana-1482	166	26	𝑎	𝑎	NOUN
cana-1482	166	27	,	,	PUNCT
cana-1482	166	28	0.5	0.5	NUM
cana-1482	166	29	,	,	PUNCT
cana-1482	166	30	0.6,0.5	0.6,0.5	NUM
cana-1482	166	31	〉	〉	NOUN
cana-1482	166	32	,	,	PUNCT
cana-1482	166	33	〈	〈	PROPN
cana-1482	166	34	𝑏	𝑏	NOUN
cana-1482	166	35	,	,	PUNCT
cana-1482	166	36	0.7	0.7	NUM
cana-1482	166	37	,	,	PUNCT
cana-1482	166	38	0.5,0.3	0.5,0.3	PROPN
cana-1482	166	39	〉	〉	NOUN
cana-1482	166	40	}	}	PUNCT
cana-1482	166	41	and	and	CCONJ
cana-1482	166	42	�	�	PROPN
cana-1482	166	43	̅	̅	NOUN
cana-1482	166	44	�	�	NOUN
cana-1482	166	45	=	=	SYM
cana-1482	166	46	{	{	PUNCT
cana-1482	166	47	〈	〈	PROPN
cana-1482	166	48	𝑎	𝑎	NOUN
cana-1482	166	49	,	,	PUNCT
cana-1482	166	50	0.6	0.6	NUM
cana-1482	166	51	,	,	PUNCT
cana-1482	166	52	0.5,0.4	0.5,0.4	NOUN
cana-1482	166	53	〉	〉	NOUN
cana-1482	166	54	,	,	PUNCT
cana-1482	166	55	〈	〈	PROPN
cana-1482	166	56	𝑏	𝑏	NOUN
cana-1482	166	57	,	,	PUNCT
cana-1482	166	58	0.6	0.6	NUM
cana-1482	166	59	,	,	PUNCT
cana-1482	166	60	0.5,0.4	0.5,0.4	X
cana-1482	166	61	〉	〉	NOUN
cana-1482	166	62	}	}	PUNCT
cana-1482	166	63	be	be	VERB
cana-1482	166	64	nss	nss	NOUN
cana-1482	166	65	in	in	ADP
cana-1482	166	66	x.	x.	NOUN
cana-1482	166	67	then	then	ADV
cana-1482	166	68	�	�	PROPN
cana-1482	166	69	̅	̅	NOUN
cana-1482	166	70	�	�	PROPN
cana-1482	166	71	and	and	CCONJ
cana-1482	166	72	�	�	PROPN
cana-1482	166	73	̅	̅	NOUN
cana-1482	166	74	�	�	NOUN
cana-1482	166	75	are	be	AUX
cana-1482	166	76	nsgspcs	nsgspc	NOUN
cana-1482	166	77	but	but	CCONJ
cana-1482	166	78	�	�	NOUN
cana-1482	166	79	̅	̅	NOUN
cana-1482	166	80	�	�	PROPN
cana-1482	166	81	∩	∩	ADJ
cana-1482	166	82	�	�	PROPN
cana-1482	166	83	̅	̅	NOUN
cana-1482	166	84	�	�	NOUN
cana-1482	166	85	is	be	AUX
cana-1482	166	86	not	not	PART
cana-1482	166	87	an	an	DET
cana-1482	166	88	nsgspcs	nsgspc	NOUN
cana-1482	166	89	in	in	ADP
cana-1482	166	90	x	x	NOUN
cana-1482	166	91	,	,	PUNCT
cana-1482	166	92	since	since	SCONJ
cana-1482	166	93	�	�	NOUN
cana-1482	166	94	̅	̅	NOUN
cana-1482	166	95	�	�	PROPN
cana-1482	166	96	∩	∩	NOUN
cana-1482	166	97	�	�	PROPN
cana-1482	166	98	̅	̅	NOUN
cana-1482	166	99	�	�	NOUN
cana-1482	166	100	=	=	SYM
cana-1482	166	101	{	{	PUNCT
cana-1482	166	102	〈	〈	PROPN
cana-1482	166	103	𝑎	𝑎	NOUN
cana-1482	166	104	,	,	PUNCT
cana-1482	166	105	0.5	0.5	NUM
cana-1482	166	106	,	,	PUNCT
cana-1482	166	107	0.5,0.5	0.5,0.5	PROPN
cana-1482	166	108	〉	〉	NOUN
cana-1482	166	109	,	,	PUNCT
cana-1482	166	110	〈	〈	PROPN
cana-1482	166	111	𝑏	𝑏	NOUN
cana-1482	166	112	,	,	PUNCT
cana-1482	166	113	0.6	0.6	NUM
cana-1482	166	114	,	,	PUNCT
cana-1482	167	1	0.5,0.4	0.5,0.4	X
cana-1482	167	2	〉	〉	NOUN
cana-1482	167	3	}	}	PUNCT
cana-1482	167	4	⊆	⊆	NUM
cana-1482	167	5	�	�	NOUN
cana-1482	167	6	̅	̅	NOUN
cana-1482	167	7	�	�	PROPN
cana-1482	167	8	but	but	CCONJ
cana-1482	167	9	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	167	10	�	�	PROPN
cana-1482	167	11	̅	̅	NOUN
cana-1482	167	12	�	�	PROPN
cana-1482	167	13	∩	∩	ADJ
cana-1482	167	14	�	�	PROPN
cana-1482	167	15	̅	̅	NOUN
cana-1482	167	16	�	�	NOUN
cana-1482	167	17	)	)	PUNCT
cana-1482	168	1	=	=	PUNCT
cana-1482	168	2	1̅	1̅	NUM
cana-1482	168	3	⊄	⊄	NOUN
cana-1482	168	4	�	�	NOUN
cana-1482	168	5	̅	̅	NOUN
cana-1482	168	6	�	�	PROPN
cana-1482	168	7	.	.	PUNCT
cana-1482	169	1	2.19	2.19	NUM
cana-1482	169	2	theorem	theorem	VERB
cana-1482	169	3	:	:	PUNCT
cana-1482	169	4	let	let	VERB
cana-1482	169	5	(	(	PUNCT
cana-1482	169	6	x,	x,	X
cana-1482	169	7	)	)	PUNCT
cana-1482	169	8	be	be	VERB
cana-1482	169	9	an	an	DET
cana-1482	169	10	nsts	nst	NOUN
cana-1482	169	11	.	.	PUNCT
cana-1482	170	1	then	then	ADV
cana-1482	170	2	for	for	ADP
cana-1482	170	3	every	every	DET
cana-1482	170	4	�	�	NOUN
cana-1482	170	5	̅	̅	NOUN
cana-1482	170	6	�	�	PROPN
cana-1482	170	7	∈	∈	PROPN
cana-1482	170	8	nsgspc(x	nsgspc(x	PROPN
cana-1482	170	9	)	)	PUNCT
cana-1482	170	10	and	and	CCONJ
cana-1482	170	11	for	for	ADP
cana-1482	170	12	every	every	DET
cana-1482	170	13	�	�	NOUN
cana-1482	170	14	̅	̅	NOUN
cana-1482	170	15	�	�	NOUN
cana-1482	170	16	∈nss(x	∈nss(x	PROPN
cana-1482	170	17	)	)	PUNCT
cana-1482	170	18	,	,	PUNCT
cana-1482	170	19	�	�	PROPN
cana-1482	170	20	̅	̅	NOUN
cana-1482	170	21	�	�	PROPN
cana-1482	170	22	⊆	⊆	NUM
cana-1482	170	23	�	�	NOUN
cana-1482	170	24	̅	̅	NOUN
cana-1482	170	25	�	�	NOUN
cana-1482	170	26	⊆	⊆	NUM
cana-1482	170	27	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	170	28	�	�	NOUN
cana-1482	170	29	̅	̅	NOUN
cana-1482	170	30	�	�	NOUN
cana-1482	170	31	)	)	PUNCT
cana-1482	170	32	implies	imply	VERB
cana-1482	170	33	�	�	NOUN
cana-1482	170	34	̅	̅	NOUN
cana-1482	170	35	�	�	PROPN
cana-1482	170	36	∈	∈	PROPN
cana-1482	170	37	nsgspc(x	nsgspc(x	PROPN
cana-1482	170	38	)	)	PUNCT
cana-1482	170	39	.	.	PUNCT
cana-1482	171	1	proof	proof	NOUN
cana-1482	171	2	:	:	PUNCT
cana-1482	171	3	let	let	VERB
cana-1482	171	4	�	�	PRON
cana-1482	171	5	̅	̅	VERB
cana-1482	171	6	�	�	PROPN
cana-1482	171	7	⊆	⊆	NUM
cana-1482	171	8	�	�	NOUN
cana-1482	171	9	̅	̅	NOUN
cana-1482	171	10	�	�	PROPN
cana-1482	171	11	and	and	CCONJ
cana-1482	171	12	�	�	PROPN
cana-1482	171	13	̅	̅	NOUN
cana-1482	171	14	�	�	NOUN
cana-1482	171	15	be	be	AUX
cana-1482	171	16	a	a	DET
cana-1482	171	17	nsos	nsos	NOUN
cana-1482	171	18	in	in	ADP
cana-1482	171	19	(	(	PUNCT
cana-1482	171	20	x,	x,	X
cana-1482	171	21	)	)	PUNCT
cana-1482	171	22	.	.	PUNCT
cana-1482	172	1	then	then	ADV
cana-1482	172	2	since	since	SCONJ
cana-1482	172	3	�	�	PROPN
cana-1482	172	4	̅	̅	NOUN
cana-1482	172	5	�	�	NOUN
cana-1482	172	6	⊆	⊆	NUM
cana-1482	172	7	𝐵,̅	𝐵,̅	PROPN
cana-1482	172	8	�	�	PROPN
cana-1482	172	9	̅	̅	NOUN
cana-1482	172	10	�	�	PROPN
cana-1482	172	11	⊆	⊆	NUM
cana-1482	172	12	�	�	NOUN
cana-1482	172	13	̅	̅	NOUN
cana-1482	172	14	�	�	NOUN
cana-1482	172	15	.	.	PUNCT
cana-1482	172	16	by	by	ADP
cana-1482	172	17	hypothesis	hypothesis	NOUN
cana-1482	172	18	,	,	PUNCT
cana-1482	172	19	�	�	PROPN
cana-1482	172	20	̅	̅	NOUN
cana-1482	172	21	�	�	NOUN
cana-1482	172	22	⊆	⊆	NUM
cana-1482	172	23	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	172	24	�	�	NOUN
cana-1482	172	25	̅	̅	NOUN
cana-1482	172	26	�	�	NUM
cana-1482	172	27	)	)	PUNCT
cana-1482	172	28	.therefore	.therefore	ADP
cana-1482	172	29	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	172	30	�	�	PROPN
cana-1482	172	31	̅	̅	NOUN
cana-1482	172	32	�	�	NOUN
cana-1482	172	33	)	)	PUNCT
cana-1482	172	34	⊆	⊆	NUM
cana-1482	172	35	𝑛𝑠𝑠𝑝𝑐𝑙(𝑛𝑠𝑠𝑝𝑐𝑙(𝐴	𝑛𝑠𝑠𝑝𝑐𝑙(𝑛𝑠𝑠𝑝𝑐𝑙(𝐴	NOUN
cana-1482	172	36	̅	̅	NOUN
cana-1482	172	37	)	)	PUNCT
cana-1482	172	38	)	)	PUNCT
cana-1482	173	1	=	=	PUNCT
cana-1482	173	2	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	173	3	�	�	PROPN
cana-1482	173	4	̅	̅	NOUN
cana-1482	173	5	�	�	PROPN
cana-1482	173	6	)	)	PUNCT
cana-1482	173	7	⊆	⊆	NUM
cana-1482	173	8	�	�	NOUN
cana-1482	173	9	̅	̅	NOUN
cana-1482	173	10	�	�	NOUN
cana-1482	173	11	,	,	PUNCT
cana-1482	173	12	since	since	SCONJ
cana-1482	173	13	�	�	NOUN
cana-1482	173	14	̅	̅	NOUN
cana-1482	173	15	�	�	NOUN
cana-1482	173	16	is	be	AUX
cana-1482	173	17	a	a	DET
cana-1482	173	18	nsgspcs	nsgspc	NOUN
cana-1482	173	19	in	in	ADP
cana-1482	173	20	(	(	PUNCT
cana-1482	173	21	x,	x,	X
cana-1482	173	22	)	)	PUNCT
cana-1482	173	23	.	.	PUNCT
cana-1482	174	1	hence	hence	ADV
cana-1482	174	2	�	�	NOUN
cana-1482	174	3	̅	̅	NOUN
cana-1482	174	4	�	�	PROPN
cana-1482	174	5	∈	∈	PROPN
cana-1482	174	6	nsgspc(x	nsgspc(x	PROPN
cana-1482	174	7	)	)	PUNCT
cana-1482	174	8	.	.	PUNCT
cana-1482	175	1	2.20	2.20	NUM
cana-1482	175	2	theorem	theorem	VERB
cana-1482	175	3	:	:	PUNCT
cana-1482	175	4	if	if	SCONJ
cana-1482	175	5	and	and	CCONJ
cana-1482	175	6	only	only	ADV
cana-1482	175	7	if	if	SCONJ
cana-1482	175	8	�	�	NOUN
cana-1482	175	9	̅	̅	NOUN
cana-1482	175	10	�	�	NOUN
cana-1482	175	11	is	be	AUX
cana-1482	175	12	not	not	PART
cana-1482	175	13	qcoincident	qcoincident	ADJ
cana-1482	175	14	�	�	PROPN
cana-1482	175	15	̅	̅	NOUN
cana-1482	175	16	�	�	PROPN
cana-1482	175	17	⟹	⟹	NUM
cana-1482	175	18	𝑛𝑠𝑠𝑝𝑐𝑙(𝐴	𝑛𝑠𝑠𝑝𝑐𝑙(𝐴	PROPN
cana-1482	175	19	̅	̅	NOUN
cana-1482	175	20	)	)	PUNCT
cana-1482	175	21	not	not	PART
cana-1482	175	22	q	q	ADJ
cana-1482	175	23	-	-	NOUN
cana-1482	175	24	coincident	coincident	ADJ
cana-1482	175	25	𝐹,̅for	𝐹,̅for	ADP
cana-1482	175	26	each	each	DET
cana-1482	175	27	nscs	nscs	ADJ
cana-1482	175	28	𝐹,̅	𝐹,̅	NOUN
cana-1482	175	29	of	of	ADP
cana-1482	175	30	x	x	PROPN
cana-1482	175	31	,	,	PUNCT
cana-1482	175	32	then	then	ADV
cana-1482	175	33	an	an	DET
cana-1482	175	34	nss	nss	NOUN
cana-1482	175	35	�	�	NOUN
cana-1482	175	36	̅	̅	NOUN
cana-1482	175	37	�	�	NOUN
cana-1482	175	38	of	of	ADP
cana-1482	175	39	an	an	DET
cana-1482	175	40	nsts	nst	NOUN
cana-1482	175	41	(	(	PUNCT
cana-1482	175	42	x,	x,	X
cana-1482	175	43	)	)	PUNCT
cana-1482	175	44	is	be	AUX
cana-1482	175	45	an	an	DET
cana-1482	175	46	nsgspcs	nsgspc	NOUN
cana-1482	175	47	in	in	ADP
cana-1482	175	48	(	(	PUNCT
cana-1482	175	49	x,	x,	X
cana-1482	175	50	)	)	PUNCT
cana-1482	175	51	.	.	PUNCT
cana-1482	176	1	proof	proof	NOUN
cana-1482	176	2	:	:	PUNCT
cana-1482	176	3	necessity	necessity	NOUN
cana-1482	176	4	:	:	PUNCT
cana-1482	176	5	assume	assume	VERB
cana-1482	176	6	that	that	SCONJ
cana-1482	176	7	a	a	DET
cana-1482	176	8	̅	̅	NOUN
cana-1482	176	9	is	be	AUX
cana-1482	176	10	not	not	PART
cana-1482	176	11	q	q	ADJ
cana-1482	176	12	-	-	ADJ
cana-1482	176	13	coincident	coincident	ADJ
cana-1482	176	14	𝐹,̅	𝐹,̅	NOUN
cana-1482	176	15	and	and	CCONJ
cana-1482	176	16	that	that	SCONJ
cana-1482	176	17	𝐹,̅	𝐹,̅	PROPN
cana-1482	176	18	is	be	AUX
cana-1482	176	19	an	an	DET
cana-1482	176	20	nscs	nscs	NOUN
cana-1482	176	21	in	in	ADP
cana-1482	176	22	(	(	PUNCT
cana-1482	176	23	x,	x,	X
cana-1482	176	24	)	)	PUNCT
cana-1482	176	25	.	.	PUNCT
cana-1482	177	1	by	by	ADP
cana-1482	177	2	definition	definition	NOUN
cana-1482	177	3	1.10	1.10	NUM
cana-1482	177	4	�	�	NOUN
cana-1482	177	5	̅	̅	NOUN
cana-1482	177	6	�	�	PROPN
cana-1482	177	7	⊆	⊆	NUM
cana-1482	177	8	�	�	NOUN
cana-1482	177	9	̅	̅	NOUN
cana-1482	177	10	�	�	NOUN
cana-1482	177	11	𝑐	𝑐	NOUN
cana-1482	177	12	,	,	PUNCT
cana-1482	177	13	where	where	SCONJ
cana-1482	177	14	an	an	DET
cana-1482	177	15	nsos	nsos	NOUN
cana-1482	177	16	in	in	ADP
cana-1482	177	17	(	(	PUNCT
cana-1482	177	18	x,	x,	X
cana-1482	177	19	)	)	PUNCT
cana-1482	177	20	is	be	AUX
cana-1482	177	21	represented	represent	VERB
cana-1482	177	22	by	by	ADP
cana-1482	177	23	𝐹	𝐹	PROPN
cana-1482	177	24	̅̅	̅̅	PROPN
cana-1482	177	25	̅𝑐.	̅𝑐.	PUNCT
cana-1482	178	1	then	then	ADV
cana-1482	178	2	,	,	PUNCT
cana-1482	178	3	theoretically	theoretically	ADV
cana-1482	178	4	,	,	PUNCT
cana-1482	178	5	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	178	6	�	�	NOUN
cana-1482	178	7	̅	̅	NOUN
cana-1482	178	8	�	�	NOUN
cana-1482	178	9	)	)	PUNCT
cana-1482	178	10	⊆	⊆	NUM
cana-1482	178	11	𝐹	𝐹	PROPN
cana-1482	178	12	̅𝑐.	̅𝑐.	PUNCT
cana-1482	178	13	therefore	therefore	ADV
cana-1482	178	14	,	,	PUNCT
cana-1482	178	15	according	accord	VERB
cana-1482	178	16	to	to	ADP
cana-1482	178	17	definition	definition	NOUN
cana-1482	178	18	1.10	1.10	NUM
cana-1482	178	19	once	once	ADV
cana-1482	178	20	more	more	ADJ
cana-1482	178	21	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	178	22	�	�	NOUN
cana-1482	178	23	̅	̅	NOUN
cana-1482	178	24	�	�	PROPN
cana-1482	178	25	)	)	PUNCT
cana-1482	178	26	is	be	AUX
cana-1482	178	27	not	not	PART
cana-1482	178	28	q	q	ADJ
cana-1482	178	29	-	-	ADJ
cana-1482	178	30	coincident	coincident	ADJ
cana-1482	178	31	𝐹.̅	𝐹.̅	DET
cana-1482	178	32	sufficiency	sufficiency	NOUN
cana-1482	178	33	:	:	PUNCT
cana-1482	178	34	assume	assume	VERB
cana-1482	178	35	that	that	SCONJ
cana-1482	178	36	�	�	PROPN
cana-1482	178	37	̅	̅	NOUN
cana-1482	178	38	�	�	PROPN
cana-1482	178	39	⊆	⊆	NUM
cana-1482	178	40	�	�	NOUN
cana-1482	178	41	̅	̅	NOUN
cana-1482	178	42	�	�	PROPN
cana-1482	178	43	and	and	CCONJ
cana-1482	178	44	that	that	SCONJ
cana-1482	178	45	�	�	PROPN
cana-1482	178	46	̅	̅	NOUN
cana-1482	178	47	�	�	NOUN
cana-1482	178	48	is	be	AUX
cana-1482	178	49	an	an	DET
cana-1482	178	50	nsos	nsos	NOUN
cana-1482	178	51	in	in	ADP
cana-1482	178	52	(	(	PUNCT
cana-1482	178	53	x,	x,	X
cana-1482	178	54	)	)	PUNCT
cana-1482	178	55	.	.	PUNCT
cana-1482	179	1	therefore	therefore	ADV
cana-1482	179	2	,	,	PUNCT
cana-1482	179	3	�	�	PROPN
cana-1482	179	4	̅	̅	NOUN
cana-1482	179	5	�	�	NOUN
cana-1482	179	6	⊆	⊆	NUM
cana-1482	179	7	(	(	PUNCT
cana-1482	179	8	𝑈	𝑈	PROPN
cana-1482	179	9	̅𝑐)𝑐and	̅𝑐)𝑐and	PROPN
cana-1482	179	10	�	�	PROPN
cana-1482	179	11	̅	̅	NOUN
cana-1482	179	12	�	�	NOUN
cana-1482	179	13	𝑐	𝑐	NOUN
cana-1482	179	14	are	be	AUX
cana-1482	179	15	nscss	nscss	ADJ
cana-1482	179	16	in	in	ADP
cana-1482	179	17	(	(	PUNCT
cana-1482	179	18	x,	x,	X
cana-1482	179	19	)	)	PUNCT
cana-1482	179	20	.	.	PUNCT
cana-1482	180	1	according	accord	VERB
cana-1482	180	2	to	to	ADP
cana-1482	180	3	the	the	DET
cana-1482	180	4	theory	theory	NOUN
cana-1482	180	5	,	,	PUNCT
cana-1482	180	6	�	�	PROPN
cana-1482	180	7	̅	̅	NOUN
cana-1482	180	8	�	�	NOUN
cana-1482	180	9	is	be	AUX
cana-1482	180	10	not	not	PART
cana-1482	180	11	q	q	ADJ
cana-1482	180	12	-	-	ADJ
cana-1482	180	13	coincident	coincident	ADJ
cana-1482	180	14	𝑈	𝑈	NOUN
cana-1482	180	15	̅𝑐.	̅𝑐.	PUNCT
cana-1482	180	16	not	not	PART
cana-1482	180	17	q	q	ADJ
cana-1482	180	18	-	-	ADJ
cana-1482	180	19	coincident	coincident	ADJ
cana-1482	180	20	𝑈	𝑈	PROPN
cana-1482	180	21	̅𝑐	̅𝑐	ADJ
cana-1482	180	22	,	,	PUNCT
cana-1482	180	23	but	but	CCONJ
cana-1482	180	24	rather⟹	rather⟹	NOUN
cana-1482	180	25	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	180	26	�	�	PROPN
cana-1482	180	27	̅	̅	NOUN
cana-1482	180	28	�	�	PROPN
cana-1482	180	29	)	)	PUNCT
cana-1482	180	30	.	.	PUNCT
cana-1482	181	1	therefore	therefore	ADV
cana-1482	181	2	,	,	PUNCT
cana-1482	181	3	𝑖𝑣𝑖𝑓𝑠𝑝𝑐𝑙(	𝑖𝑣𝑖𝑓𝑠𝑝𝑐𝑙(	PROPN
cana-1482	181	4	�	�	SYM
cana-1482	181	5	̅	̅	NOUN
cana-1482	181	6	�	�	PROPN
cana-1482	181	7	)	)	PUNCT
cana-1482	182	1	⊆	⊆	NUM
cana-1482	182	2	(	(	PUNCT
cana-1482	182	3	𝑈	𝑈	NOUN
cana-1482	182	4	̅𝑐)𝑐	̅𝑐)𝑐	ADJ
cana-1482	182	5	=	=	SYM
cana-1482	182	6	𝑈	𝑈	PROPN
cana-1482	182	7	̅	̅	NOUN
cana-1482	182	8	by	by	ADP
cana-1482	182	9	definition	definition	NOUN
cana-1482	182	10	1.10	1.10	NUM
cana-1482	182	11	.	.	PUNCT
cana-1482	183	1	consequently	consequently	ADV
cana-1482	183	2	,	,	PUNCT
cana-1482	183	3	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	183	4	�	�	NOUN
cana-1482	183	5	̅	̅	NOUN
cana-1482	183	6	�	�	NOUN
cana-1482	183	7	)	)	PUNCT
cana-1482	183	8	⊆	⊆	X
cana-1482	183	9	𝑈	𝑈	PROPN
cana-1482	183	10	̅.	̅.	PROPN
cana-1482	183	11	therefore	therefore	ADV
cana-1482	183	12	�	�	PROPN
cana-1482	183	13	̅	̅	NOUN
cana-1482	183	14	�	�	PROPN
cana-1482	183	15	in	in	ADP
cana-1482	183	16	(	(	PUNCT
cana-1482	183	17	x	x	X
cana-1482	183	18	,	,	PUNCT
cana-1482	183	19			NOUN
cana-1482	183	20	)	)	PUNCT
cana-1482	183	21	is	be	AUX
cana-1482	183	22	an	an	DET
cana-1482	183	23	nsgspcs	nsgspc	NOUN
cana-1482	183	24	.	.	PUNCT
cana-1482	184	1	2.21	2.21	NUM
cana-1482	184	2	theorem	theorem	VERB
cana-1482	184	3	:	:	PUNCT
cana-1482	184	4	let	let	VERB
cana-1482	184	5	(	(	PUNCT
cana-1482	184	6	𝑋,	𝑋,	PROPN
cana-1482	184	7	)	)	PUNCT
cana-1482	184	8	be	be	AUX
cana-1482	184	9	a	a	DET
cana-1482	184	10	nsts	nst	NOUN
cana-1482	184	11	.	.	PUNCT
cana-1482	185	1	then	then	ADV
cana-1482	185	2	every	every	DET
cana-1482	185	3	nss	nss	NOUN
cana-1482	185	4	in	in	ADP
cana-1482	185	5	(	(	PUNCT
cana-1482	185	6	𝑋,	𝑋,	PROPN
cana-1482	185	7	)	)	PUNCT
cana-1482	185	8	is	be	AUX
cana-1482	185	9	a	a	DET
cana-1482	185	10	nsgspcs	nsgspc	NOUN
cana-1482	185	11	in	in	ADP
cana-1482	185	12	(	(	PUNCT
cana-1482	185	13	𝑋,	𝑋,	PROPN
cana-1482	185	14	)	)	PUNCT
cana-1482	185	15	if	if	SCONJ
cana-1482	185	16	and	and	CCONJ
cana-1482	185	17	only	only	ADV
cana-1482	185	18	if	if	SCONJ
cana-1482	185	19	nsspo(x	nsspo(x	ADJ
cana-1482	185	20	)	)	PUNCT
cana-1482	185	21	=	=	NOUN
cana-1482	185	22	nsspc(x	nsspc(x	NOUN
cana-1482	185	23	)	)	PUNCT
cana-1482	185	24	.	.	PUNCT
cana-1482	186	1	proof	proof	NOUN
cana-1482	186	2	:	:	PUNCT
cana-1482	186	3	necessity	necessity	NOUN
cana-1482	186	4	:	:	PUNCT
cana-1482	186	5	suppose	suppose	VERB
cana-1482	186	6	that	that	SCONJ
cana-1482	186	7	every	every	DET
cana-1482	186	8	nss	nss	NOUN
cana-1482	186	9	in	in	ADP
cana-1482	186	10	(	(	PUNCT
cana-1482	186	11	𝑋,	𝑋,	PROPN
cana-1482	186	12	)	)	PUNCT
cana-1482	186	13	is	be	AUX
cana-1482	186	14	a	a	DET
cana-1482	186	15	nsgspcs	nsgspc	NOUN
cana-1482	186	16	in	in	ADP
cana-1482	186	17	(	(	PUNCT
cana-1482	186	18	𝑋,	𝑋,	PROPN
cana-1482	186	19	)	)	PUNCT
cana-1482	186	20	.	.	PUNCT
cana-1482	187	1	let	let	VERB
cana-1482	187	2	𝑈	𝑈	NOUN
cana-1482	187	3	̅	̅	VERB
cana-1482	187	4	∈	∈	NOUN
cana-1482	187	5	𝑁𝑆𝑂(𝑋	𝑁𝑆𝑂(𝑋	NOUN
cana-1482	187	6	)	)	PUNCT
cana-1482	187	7	.	.	PUNCT
cana-1482	188	1	then	then	ADV
cana-1482	188	2	𝑈	𝑈	PROPN
cana-1482	188	3	̅	̅	NOUN
cana-1482	188	4	∈	∈	NOUN
cana-1482	188	5	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	NOUN
cana-1482	188	6	)	)	PUNCT
cana-1482	188	7	and	and	CCONJ
cana-1482	188	8	by	by	ADP
cana-1482	188	9	hypothesis	hypothesis	NOUN
cana-1482	188	10	,	,	PUNCT
cana-1482	188	11	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	NOUN
cana-1482	188	12	̅	̅	NOUN
cana-1482	188	13	)	)	PUNCT
cana-1482	188	14	⊆	⊆	NUM
cana-1482	188	15	𝑈	𝑈	PROPN
cana-1482	188	16	̅	̅	NOUN
cana-1482	188	17	⊆	⊆	NUM
cana-1482	188	18	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	NOUN
cana-1482	188	19	̅	̅	NOUN
cana-1482	188	20	)	)	PUNCT
cana-1482	188	21	.	.	PUNCT
cana-1482	189	1	this	this	PRON
cana-1482	189	2	implies	imply	VERB
cana-1482	189	3	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	NOUN
cana-1482	189	4	̅	̅	NOUN
cana-1482	189	5	)	)	PUNCT
cana-1482	189	6	=	=	VERB
cana-1482	190	1	𝑈.̅	𝑈.̅	NOUN
cana-1482	190	2	therefore	therefore	ADV
cana-1482	190	3	𝑈	𝑈	PROPN
cana-1482	190	4	̅	̅	NOUN
cana-1482	190	5	∈	∈	NOUN
cana-1482	190	6	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	190	7	)	)	PUNCT
cana-1482	190	8	.	.	PUNCT
cana-1482	191	1	hence	hence	ADV
cana-1482	191	2	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	NOUN
cana-1482	191	3	)	)	PUNCT
cana-1482	192	1	⊆	⊆	NUM
cana-1482	192	2	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	192	3	)	)	PUNCT
cana-1482	192	4	.	.	PUNCT
cana-1482	193	1	let	let	VERB
cana-1482	193	2	�	�	PRON
cana-1482	193	3	̅	̅	VERB
cana-1482	193	4	�	�	PROPN
cana-1482	193	5	∈	∈	PROPN
cana-1482	193	6	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	193	7	)	)	PUNCT
cana-1482	193	8	.	.	PUNCT
cana-1482	194	1	then	then	ADV
cana-1482	194	2	�	�	PROPN
cana-1482	194	3	̅	̅	NOUN
cana-1482	194	4	�	�	PROPN
cana-1482	194	5	𝑐	𝑐	PROPN
cana-1482	194	6	∈	∈	PROPN
cana-1482	194	7	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	PROPN
cana-1482	194	8	)	)	PUNCT
cana-1482	195	1	⊆	⊆	NUM
cana-1482	195	2	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	195	3	)	)	PUNCT
cana-1482	195	4	.	.	PUNCT
cana-1482	196	1	that	that	PRON
cana-1482	196	2	is	be	AUX
cana-1482	196	3	�	�	PROPN
cana-1482	196	4	̅	̅	NOUN
cana-1482	196	5	�	�	PROPN
cana-1482	196	6	𝑐	𝑐	PROPN
cana-1482	196	7	∈	∈	PROPN
cana-1482	196	8	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	196	9	)	)	PUNCT
cana-1482	196	10	.	.	PUNCT
cana-1482	197	1	therefore	therefore	ADV
cana-1482	197	2	�	�	PROPN
cana-1482	197	3	̅	̅	NOUN
cana-1482	197	4	�	�	PROPN
cana-1482	197	5	∈	∈	PROPN
cana-1482	197	6	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	NOUN
cana-1482	197	7	)	)	PUNCT
cana-1482	197	8	.	.	PUNCT
cana-1482	198	1	hence	hence	ADV
cana-1482	198	2	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	198	3	)	)	PUNCT
cana-1482	198	4	⊆	⊆	NUM
cana-1482	198	5	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	NOUN
cana-1482	198	6	)	)	PUNCT
cana-1482	198	7	.	.	PUNCT
cana-1482	199	1	thus	thus	ADV
cana-1482	199	2	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	VERB
cana-1482	199	3	)	)	PUNCT
cana-1482	199	4	=	=	PUNCT
cana-1482	200	1	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	200	2	)	)	PUNCT
cana-1482	200	3	.	.	PUNCT
cana-1482	201	1	communications	communication	NOUN
cana-1482	201	2	on	on	ADP
cana-1482	201	3	applied	apply	VERB
cana-1482	201	4	nonlinear	nonlinear	ADJ
cana-1482	201	5	analysis	analysis	NOUN
cana-1482	201	6	issn	issn	NOUN
cana-1482	201	7	:	:	PUNCT
cana-1482	201	8	1074	1074	NUM
cana-1482	201	9	-	-	PUNCT
cana-1482	201	10	133x	133x	NUM
cana-1482	201	11	vol	vol	NOUN
cana-1482	201	12	31	31	NUM
cana-1482	201	13	no	no	NOUN
cana-1482	201	14	.	.	PUNCT
cana-1482	202	1	8s	8s	PROPN
cana-1482	202	2	(	(	PUNCT
cana-1482	202	3	2024	2024	NUM
cana-1482	202	4	)	)	PUNCT
cana-1482	202	5	279	279	NUM
cana-1482	202	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	202	7	sufficiency	sufficiency	NOUN
cana-1482	202	8	:	:	PUNCT
cana-1482	202	9	suppose	suppose	VERB
cana-1482	202	10	that	that	SCONJ
cana-1482	202	11	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	NOUN
cana-1482	202	12	)	)	PUNCT
cana-1482	202	13	=	=	PUNCT
cana-1482	203	1	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	203	2	)	)	PUNCT
cana-1482	203	3	.	.	PUNCT
cana-1482	204	1	let	let	VERB
cana-1482	204	2	�	�	PRON
cana-1482	204	3	̅	̅	VERB
cana-1482	204	4	�	�	PROPN
cana-1482	204	5	⊆	⊆	NUM
cana-1482	204	6	𝑈	𝑈	PROPN
cana-1482	204	7	̅	̅	NOUN
cana-1482	204	8	and	and	CCONJ
cana-1482	204	9	𝑈	𝑈	NOUN
cana-1482	204	10	̅	̅	NOUN
cana-1482	204	11	be	be	VERB
cana-1482	204	12	a	a	DET
cana-1482	204	13	nsos	nsos	NOUN
cana-1482	204	14	in	in	ADP
cana-1482	204	15	(	(	PUNCT
cana-1482	204	16	𝑋,	𝑋,	PROPN
cana-1482	204	17	)	)	PUNCT
cana-1482	204	18	.	.	PUNCT
cana-1482	205	1	then	then	ADV
cana-1482	205	2	𝑈	𝑈	PROPN
cana-1482	205	3	̅	̅	NOUN
cana-1482	205	4	∈	∈	NOUN
cana-1482	205	5	𝑁𝑆𝑆𝑃𝑂(𝑋	𝑁𝑆𝑆𝑃𝑂(𝑋	NOUN
cana-1482	205	6	)	)	PUNCT
cana-1482	205	7	and	and	CCONJ
cana-1482	205	8	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	205	9	�	�	NOUN
cana-1482	205	10	̅	̅	NOUN
cana-1482	205	11	�	�	PROPN
cana-1482	205	12	)	)	PUNCT
cana-1482	205	13	⊆	⊆	NUM
cana-1482	205	14	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	𝑛𝑠𝑠𝑝𝑐𝑙(𝑈	NOUN
cana-1482	205	15	̅	̅	NOUN
cana-1482	205	16	)	)	PUNCT
cana-1482	205	17	=	=	SYM
cana-1482	205	18	𝑈	𝑈	NOUN
cana-1482	205	19	̅	̅	NOUN
cana-1482	205	20	,	,	PUNCT
cana-1482	205	21	since	since	SCONJ
cana-1482	205	22	𝑈	𝑈	PROPN
cana-1482	205	23	̅	̅	NOUN
cana-1482	205	24	∈	∈	PROPN
cana-1482	205	25	nsspc(x	nsspc(x	NOUN
cana-1482	205	26	)	)	PUNCT
cana-1482	205	27	,	,	PUNCT
cana-1482	205	28	by	by	ADP
cana-1482	205	29	hypothesis	hypothesis	NOUN
cana-1482	205	30	.	.	PUNCT
cana-1482	206	1	therefore	therefore	ADV
cana-1482	206	2	�	�	PROPN
cana-1482	206	3	̅	̅	NOUN
cana-1482	206	4	�	�	NOUN
cana-1482	206	5	is	be	AUX
cana-1482	206	6	an	an	DET
cana-1482	206	7	nsgspcs	nsgspc	NOUN
cana-1482	206	8	in	in	ADP
cana-1482	206	9	x.	x.	PROPN
cana-1482	206	10	2.22	2.22	NUM
cana-1482	206	11	theorem	theorem	VERB
cana-1482	206	12	:	:	PUNCT
cana-1482	206	13	if	if	SCONJ
cana-1482	206	14	�	�	NOUN
cana-1482	206	15	̅	̅	NOUN
cana-1482	206	16	�	�	NOUN
cana-1482	206	17	is	be	AUX
cana-1482	206	18	a	a	DET
cana-1482	206	19	nsos	nsos	NOUN
cana-1482	206	20	and	and	CCONJ
cana-1482	206	21	a	a	DET
cana-1482	206	22	nsgspcs	nsgspc	NOUN
cana-1482	206	23	in	in	ADP
cana-1482	206	24	(	(	PUNCT
cana-1482	206	25	𝑋,	𝑋,	PROPN
cana-1482	206	26	)	)	PUNCT
cana-1482	206	27	,	,	PUNCT
cana-1482	206	28	then	then	ADV
cana-1482	206	29	�	�	PROPN
cana-1482	206	30	̅	̅	NOUN
cana-1482	206	31	�	�	NOUN
cana-1482	206	32	is	be	AUX
cana-1482	206	33	a	a	DET
cana-1482	206	34	nsspcs	nsspc	NOUN
cana-1482	206	35	in	in	ADP
cana-1482	206	36	(	(	PUNCT
cana-1482	206	37	𝑋,	𝑋,	PROPN
cana-1482	206	38	)	)	PUNCT
cana-1482	206	39	.	.	PUNCT
cana-1482	207	1	proof	proof	NOUN
cana-1482	207	2	:	:	PUNCT
cana-1482	207	3	since	since	SCONJ
cana-1482	207	4	�	�	PROPN
cana-1482	207	5	̅	̅	NOUN
cana-1482	207	6	�	�	PROPN
cana-1482	207	7	⊆	⊆	NUM
cana-1482	207	8	�	�	NOUN
cana-1482	207	9	̅	̅	NOUN
cana-1482	207	10	�	�	PROPN
cana-1482	207	11	and	and	CCONJ
cana-1482	207	12	�	�	PROPN
cana-1482	207	13	̅	̅	NOUN
cana-1482	207	14	�	�	NOUN
cana-1482	207	15	is	be	AUX
cana-1482	207	16	a	a	DET
cana-1482	207	17	nsos	nsos	NOUN
cana-1482	207	18	in	in	ADP
cana-1482	207	19	(	(	PUNCT
cana-1482	207	20	𝑋,	𝑋,	PROPN
cana-1482	207	21	)	)	PUNCT
cana-1482	207	22	,	,	PUNCT
cana-1482	207	23	by	by	ADP
cana-1482	207	24	hypothesis	hypothesis	NOUN
cana-1482	207	25	,	,	PUNCT
cana-1482	207	26	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	207	27	�	�	NOUN
cana-1482	207	28	̅	̅	NOUN
cana-1482	207	29	�	�	PROPN
cana-1482	207	30	)	)	PUNCT
cana-1482	207	31	⊆	⊆	NUM
cana-1482	207	32	�	�	NOUN
cana-1482	207	33	̅	̅	NOUN
cana-1482	207	34	�	�	PROPN
cana-1482	207	35	.	.	PUNCT
cana-1482	208	1	but	but	CCONJ
cana-1482	208	2	�	�	PROPN
cana-1482	208	3	̅	̅	NOUN
cana-1482	208	4	�	�	NOUN
cana-1482	208	5	⊆	⊆	NUM
cana-1482	208	6	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	208	7	�	�	NOUN
cana-1482	208	8	̅	̅	NOUN
cana-1482	208	9	�	�	PROPN
cana-1482	208	10	)	)	PUNCT
cana-1482	208	11	.	.	PUNCT
cana-1482	209	1	therefore	therefore	ADV
cana-1482	209	2	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	209	3	�	�	PROPN
cana-1482	209	4	̅	̅	NOUN
cana-1482	209	5	�	�	NOUN
cana-1482	209	6	)	)	PUNCT
cana-1482	209	7	=	=	SYM
cana-1482	209	8	�	�	PROPN
cana-1482	209	9	̅	̅	NOUN
cana-1482	209	10	�	�	NOUN
cana-1482	209	11	.	.	PUNCT
cana-1482	209	12	hence	hence	ADV
cana-1482	209	13	�	�	PROPN
cana-1482	209	14	̅	̅	NOUN
cana-1482	209	15	�	�	NOUN
cana-1482	209	16	is	be	AUX
cana-1482	209	17	a	a	DET
cana-1482	209	18	nsspcs	nsspc	NOUN
cana-1482	209	19	in	in	ADP
cana-1482	209	20	(	(	PUNCT
cana-1482	209	21	𝑋,	𝑋,	PROPN
cana-1482	209	22	)	)	PUNCT
cana-1482	209	23	.	.	PUNCT
cana-1482	210	1	2.23	2.23	NUM
cana-1482	210	2	theorem	theorem	VERB
cana-1482	210	3	:	:	PUNCT
cana-1482	210	4	let	let	VERB
cana-1482	210	5	�	�	PRON
cana-1482	210	6	̅	̅	NOUN
cana-1482	210	7	�	�	NOUN
cana-1482	210	8	be	be	AUX
cana-1482	210	9	a	a	DET
cana-1482	210	10	nsgspcs	nsgspc	NOUN
cana-1482	210	11	in	in	ADP
cana-1482	210	12	(	(	PUNCT
cana-1482	210	13	𝑋,	𝑋,	PROPN
cana-1482	210	14	)	)	PUNCT
cana-1482	210	15	and	and	CCONJ
cana-1482	210	16	�	�	PROPN
cana-1482	210	17	̅	̅	NOUN
cana-1482	210	18	�	�	PROPN
cana-1482	210	19	(	(	PUNCT
cana-1482	210	20	𝛼,𝛽,	𝛼,𝛽,	NOUN
cana-1482	210	21	)	)	PUNCT
cana-1482	210	22	be	be	AUX
cana-1482	210	23	an	an	DET
cana-1482	210	24	nsp	nsp	NOUN
cana-1482	210	25	in	in	ADP
cana-1482	210	26	x	x	SYM
cana-1482	210	27	such	such	ADJ
cana-1482	210	28	that	that	PRON
cana-1482	210	29	�	�	PROPN
cana-1482	210	30	̅	̅	NOUN
cana-1482	210	31	�	�	PROPN
cana-1482	210	32	(	(	PUNCT
cana-1482	210	33	α	α	PROPN
cana-1482	210	34	,	,	PUNCT
cana-1482	210	35	β,)𝑞	β,)𝑞	SYM
cana-1482	210	36	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	210	37	�	�	PROPN
cana-1482	210	38	̅	̅	NOUN
cana-1482	210	39	�	�	PROPN
cana-1482	210	40	)	)	PUNCT
cana-1482	210	41	.	.	PUNCT
cana-1482	211	1	then	then	ADV
cana-1482	211	2	ns𝑐𝑙(	ns𝑐𝑙(	PROPN
cana-1482	211	3	�	�	PROPN
cana-1482	211	4	̅	̅	NOUN
cana-1482	211	5	�	�	PROPN
cana-1482	211	6	(	(	PUNCT
cana-1482	211	7	α	α	NOUN
cana-1482	211	8	,	,	PUNCT
cana-1482	211	9	β,	β,	PROPN
cana-1482	211	10	)	)	PUNCT
cana-1482	211	11	)	)	PUNCT
cana-1482	211	12	𝑞	𝑞	PROPN
cana-1482	211	13	�	�	PROPN
cana-1482	211	14	̅	̅	NOUN
cana-1482	211	15	�	�	PROPN
cana-1482	211	16	.	.	PUNCT
cana-1482	212	1	proof	proof	NOUN
cana-1482	212	2	:	:	PUNCT
cana-1482	212	3	let	let	VERB
cana-1482	212	4	�	�	PRON
cana-1482	212	5	̅	̅	NOUN
cana-1482	212	6	�	�	NOUN
cana-1482	212	7	be	be	AUX
cana-1482	212	8	an	an	DET
cana-1482	212	9	nsgspcs	nsgspc	NOUN
cana-1482	212	10	in	in	ADP
cana-1482	212	11	(	(	PUNCT
cana-1482	212	12	𝑋,	𝑋,	PROPN
cana-1482	212	13	)	)	PUNCT
cana-1482	212	14	and	and	CCONJ
cana-1482	212	15	let	let	VERB
cana-1482	212	16	�	�	PRON
cana-1482	212	17	̅	̅	VERB
cana-1482	212	18	�	�	PROPN
cana-1482	212	19	(	(	PUNCT
cana-1482	212	20	𝛼,𝛽,)𝑞	𝛼,𝛽,)𝑞	PROPN
cana-1482	212	21	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	212	22	�	�	PROPN
cana-1482	212	23	̅	̅	NOUN
cana-1482	212	24	�	�	PROPN
cana-1482	212	25	)	)	PUNCT
cana-1482	212	26	.	.	PUNCT
cana-1482	213	1	if	if	SCONJ
cana-1482	213	2	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	213	3	�	�	PROPN
cana-1482	213	4	̅	̅	NOUN
cana-1482	213	5	�	�	PROPN
cana-1482	213	6	(	(	PUNCT
cana-1482	213	7	𝛼,𝛽,	𝛼,𝛽,	NOUN
cana-1482	213	8	)	)	PUNCT
cana-1482	213	9	)	)	PUNCT
cana-1482	213	10	not	not	PART
cana-1482	213	11	qcoincident	qcoincident	VERB
cana-1482	213	12	�	�	PROPN
cana-1482	213	13	̅	̅	NOUN
cana-1482	213	14	�	�	NOUN
cana-1482	213	15	,	,	PUNCT
cana-1482	213	16	then	then	ADV
cana-1482	213	17	by	by	ADP
cana-1482	213	18	definition	definition	NOUN
cana-1482	213	19	1.10	1.10	NUM
cana-1482	213	20	,	,	PUNCT
cana-1482	213	21	�	�	NOUN
cana-1482	213	22	̅	̅	NOUN
cana-1482	213	23	�	�	NOUN
cana-1482	213	24	⊆	⊆	NUM
cana-1482	213	25	(	(	PUNCT
cana-1482	213	26	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	213	27	�	�	PROPN
cana-1482	213	28	̅	̅	NOUN
cana-1482	213	29	�	�	NOUN
cana-1482	213	30	(𝛼,𝛽,	(𝛼,𝛽,	NOUN
cana-1482	213	31	)	)	PUNCT
cana-1482	213	32	)	)	PUNCT
cana-1482	213	33	)	)	PUNCT
cana-1482	213	34	𝑐	𝑐	NOUN
cana-1482	213	35	,	,	PUNCT
cana-1482	213	36	where	where	SCONJ
cana-1482	213	37	(	(	PUNCT
cana-1482	213	38	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	213	39	�	�	PROPN
cana-1482	213	40	̅	̅	NOUN
cana-1482	213	41	�	�	NOUN
cana-1482	213	42	(𝛼,𝛽,	(𝛼,𝛽,	NOUN
cana-1482	213	43	)	)	PUNCT
cana-1482	213	44	)	)	PUNCT
cana-1482	213	45	)	)	PUNCT
cana-1482	214	1	𝑐	𝑐	PROPN
cana-1482	214	2	is	be	AUX
cana-1482	214	3	a	a	DET
cana-1482	214	4	nsos	nsos	NOUN
cana-1482	214	5	in	in	ADP
cana-1482	214	6	(	(	PUNCT
cana-1482	214	7	𝑋,	𝑋,	PROPN
cana-1482	214	8	)	)	PUNCT
cana-1482	214	9	.	.	PUNCT
cana-1482	215	1	then	then	ADV
cana-1482	215	2	by	by	ADP
cana-1482	215	3	hypothesis	hypothesis	NOUN
cana-1482	215	4	,	,	PUNCT
cana-1482	215	5	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	215	6	�	�	NOUN
cana-1482	215	7	̅	̅	NOUN
cana-1482	215	8	�	�	NOUN
cana-1482	215	9	)	)	PUNCT
cana-1482	215	10	⊆	⊆	NUM
cana-1482	215	11	(	(	PUNCT
cana-1482	215	12	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	215	13	�	�	PROPN
cana-1482	215	14	̅	̅	NOUN
cana-1482	215	15	�	�	PROPN
cana-1482	215	16	(	(	PUNCT
cana-1482	215	17	𝛼,𝛽,	𝛼,𝛽,	PROPN
cana-1482	215	18	)	)	PUNCT
cana-1482	215	19	)	)	PUNCT
cana-1482	215	20	)	)	PUNCT
cana-1482	216	1	𝑐	𝑐	PROPN
cana-1482	216	2	⊆	⊆	NUM
cana-1482	216	3	(	(	PUNCT
cana-1482	216	4	�	�	NOUN
cana-1482	216	5	̅	̅	NOUN
cana-1482	216	6	�	�	PROPN
cana-1482	216	7	(	(	PUNCT
cana-1482	216	8	𝛼,𝛽,	𝛼,𝛽,	NOUN
cana-1482	216	9	)	)	PUNCT
cana-1482	216	10	)	)	PUNCT
cana-1482	216	11	𝑐	𝑐	PROPN
cana-1482	216	12	.	.	PUNCT
cana-1482	217	1	therefore	therefore	ADV
cana-1482	217	2	by	by	ADP
cana-1482	217	3	definition	definition	NOUN
cana-1482	217	4	1.10	1.10	NUM
cana-1482	217	5	,	,	PUNCT
cana-1482	217	6	�	�	NOUN
cana-1482	217	7	̅	̅	NOUN
cana-1482	217	8	�	�	NOUN
cana-1482	217	9	(𝛼,𝛽,	(𝛼,𝛽,	NOUN
cana-1482	217	10	)	)	PUNCT
cana-1482	217	11	not	not	PART
cana-1482	217	12	q	q	ADJ
cana-1482	217	13	-	-	ADJ
cana-1482	217	14	coincident	coincident	ADJ
cana-1482	217	15	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	217	16	�	�	NOUN
cana-1482	217	17	̅	̅	NOUN
cana-1482	217	18	�	�	PROPN
cana-1482	217	19	)	)	PUNCT
cana-1482	217	20	,	,	PUNCT
cana-1482	217	21	which	which	PRON
cana-1482	217	22	is	be	AUX
cana-1482	217	23	a	a	DET
cana-1482	217	24	contradiction	contradiction	NOUN
cana-1482	217	25	to	to	ADP
cana-1482	217	26	the	the	DET
cana-1482	217	27	hypothesis	hypothesis	NOUN
cana-1482	217	28	.	.	PUNCT
cana-1482	218	1	hence	hence	ADV
cana-1482	218	2	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	VERB
cana-1482	218	3	�	�	PROPN
cana-1482	218	4	̅	̅	NOUN
cana-1482	218	5	�	�	PROPN
cana-1482	218	6	(	(	PUNCT
cana-1482	218	7	𝛼,𝛽,	𝛼,𝛽,	NOUN
cana-1482	218	8	)	)	PUNCT
cana-1482	218	9	)	)	PUNCT
cana-1482	219	1	𝑞	𝑞	PROPN
cana-1482	219	2	�	�	PROPN
cana-1482	219	3	̅	̅	NOUN
cana-1482	219	4	�	�	NOUN
cana-1482	219	5	.	.	PUNCT
cana-1482	219	6	2.24	2.24	NUM
cana-1482	219	7	theorem	theorem	VERB
cana-1482	219	8	:	:	PUNCT
cana-1482	219	9	for	for	ADP
cana-1482	219	10	any	any	DET
cana-1482	219	11	nss	nss	NOUN
cana-1482	219	12	�	�	NOUN
cana-1482	219	13	̅	̅	NOUN
cana-1482	219	14	�	�	NOUN
cana-1482	219	15	in	in	ADP
cana-1482	219	16	a	a	DET
cana-1482	219	17	nsts	nst	NOUN
cana-1482	219	18	(	(	PUNCT
cana-1482	219	19	𝑋,	𝑋,	PROPN
cana-1482	219	20	)	)	PUNCT
cana-1482	219	21	,	,	PUNCT
cana-1482	219	22	the	the	DET
cana-1482	219	23	following	follow	VERB
cana-1482	219	24	conditions	condition	NOUN
cana-1482	219	25	are	be	AUX
cana-1482	219	26	equivalent	equivalent	ADJ
cana-1482	219	27	:	:	PUNCT
cana-1482	219	28	(	(	PUNCT
cana-1482	219	29	i	i	NOUN
cana-1482	219	30	)	)	PUNCT
cana-1482	219	31	�	�	PROPN
cana-1482	219	32	̅	̅	NOUN
cana-1482	219	33	�	�	NOUN
cana-1482	219	34	is	be	AUX
cana-1482	219	35	a	a	DET
cana-1482	219	36	nsos	nsos	NOUN
cana-1482	219	37	and	and	CCONJ
cana-1482	219	38	a	a	DET
cana-1482	219	39	nsgspcs	nsgspc	NOUN
cana-1482	219	40	in	in	ADP
cana-1482	219	41	(	(	PUNCT
cana-1482	219	42	𝑋,	𝑋,	PROPN
cana-1482	219	43	)	)	PUNCT
cana-1482	219	44	(	(	PUNCT
cana-1482	219	45	ii	ii	NOUN
cana-1482	219	46	)	)	PUNCT
cana-1482	219	47	�	�	PROPN
cana-1482	219	48	̅	̅	NOUN
cana-1482	219	49	�	�	NOUN
cana-1482	219	50	is	be	AUX
cana-1482	219	51	a	a	DET
cana-1482	219	52	nsros	nsro	NOUN
cana-1482	219	53	in	in	ADP
cana-1482	219	54	(	(	PUNCT
cana-1482	219	55	𝑋,	𝑋,	PROPN
cana-1482	219	56	)	)	PUNCT
cana-1482	219	57	.	.	PUNCT
cana-1482	220	1	proof	proof	NOUN
cana-1482	220	2	:	:	PUNCT
cana-1482	220	3	(	(	PUNCT
cana-1482	220	4	i	i	NOUN
cana-1482	220	5	)	)	PUNCT
cana-1482	220	6	⇒	⇒	NOUN
cana-1482	220	7	(	(	PUNCT
cana-1482	220	8	𝑖𝑖	𝑖𝑖	NOUN
cana-1482	220	9	)	)	PUNCT
cana-1482	220	10	let	let	VERB
cana-1482	220	11	�	�	PRON
cana-1482	220	12	̅	̅	NOUN
cana-1482	220	13	�	�	NOUN
cana-1482	220	14	be	be	AUX
cana-1482	220	15	a	a	DET
cana-1482	220	16	nsos	nsos	NOUN
cana-1482	220	17	and	and	CCONJ
cana-1482	220	18	a	a	DET
cana-1482	220	19	nsgspcs	nsgspc	NOUN
cana-1482	220	20	in	in	ADP
cana-1482	220	21	a	a	DET
cana-1482	220	22	nsts	nst	NOUN
cana-1482	220	23	(	(	PUNCT
cana-1482	220	24	𝑋,	𝑋,	PROPN
cana-1482	220	25	)	)	PUNCT
cana-1482	220	26	.	.	PUNCT
cana-1482	221	1	then	then	ADV
cana-1482	221	2	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	221	3	�	�	PROPN
cana-1482	221	4	̅	̅	NOUN
cana-1482	221	5	�	�	PROPN
cana-1482	221	6	)	)	PUNCT
cana-1482	221	7	⊆	⊆	NUM
cana-1482	221	8	�	�	NOUN
cana-1482	221	9	̅	̅	NOUN
cana-1482	221	10	�	�	NOUN
cana-1482	221	11	.	.	PUNCT
cana-1482	222	1	since	since	SCONJ
cana-1482	222	2	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	222	3	�	�	NOUN
cana-1482	222	4	̅	̅	NOUN
cana-1482	222	5	�	�	PROPN
cana-1482	222	6	)	)	PUNCT
cana-1482	222	7	is	be	AUX
cana-1482	222	8	nsspcs	nsspc	NOUN
cana-1482	222	9	,	,	PUNCT
cana-1482	222	10	by	by	ADP
cana-1482	222	11	definition	definition	NOUN
cana-1482	222	12	1.7	1.7	NUM
cana-1482	222	13	,	,	PUNCT
cana-1482	222	14	there	there	PRON
cana-1482	222	15	exists	exist	VERB
cana-1482	222	16	a	a	DET
cana-1482	222	17	nspcs	nspcs	PROPN
cana-1482	222	18	�	�	NOUN
cana-1482	222	19	̅	̅	NOUN
cana-1482	222	20	�	�	NOUN
cana-1482	222	21	such	such	ADJ
cana-1482	222	22	that	that	SCONJ
cana-1482	222	23	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	ADJ
cana-1482	222	24	�	�	NOUN
cana-1482	222	25	̅	̅	NOUN
cana-1482	222	26	�	�	NOUN
cana-1482	222	27	)	)	PUNCT
cana-1482	222	28	⊆	⊆	NUM
cana-1482	222	29	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	222	30	�	�	NOUN
cana-1482	222	31	̅	̅	NOUN
cana-1482	222	32	�	�	PROPN
cana-1482	222	33	)	)	PUNCT
cana-1482	222	34	⊆	⊆	NUM
cana-1482	222	35	�	�	NOUN
cana-1482	222	36	̅	̅	NOUN
cana-1482	222	37	�	�	PROPN
cana-1482	222	38	and	and	CCONJ
cana-1482	222	39	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	222	40	�	�	NOUN
cana-1482	222	41	̅	̅	NOUN
cana-1482	222	42	�	�	NOUN
cana-1482	222	43	)	)	PUNCT
cana-1482	222	44	)	)	PUNCT
cana-1482	223	1	⊆	⊆	NUM
cana-1482	223	2	�	�	SYM
cana-1482	223	3	̅	̅	NOUN
cana-1482	223	4	�	�	PROPN
cana-1482	223	5	.	.	PUNCT
cana-1482	223	6	now	now	ADV
cana-1482	223	7	𝑛𝑠𝑖𝑛𝑡	𝑛𝑠𝑖𝑛𝑡	VERB
cana-1482	223	8	(	(	PUNCT
cana-1482	223	9	𝑛𝑠𝑐𝑙	𝑛𝑠𝑐𝑙	PROPN
cana-1482	223	10	(	(	PUNCT
cana-1482	223	11	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	223	12	�	�	SYM
cana-1482	223	13	̅	̅	NOUN
cana-1482	223	14	�	�	NOUN
cana-1482	223	15	)	)	PUNCT
cana-1482	223	16	)	)	PUNCT
cana-1482	223	17	)	)	PUNCT
cana-1482	223	18	)	)	PUNCT
cana-1482	224	1	⊆	⊆	NUM
cana-1482	224	2	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	224	3	�	�	NOUN
cana-1482	224	4	̅	̅	NOUN
cana-1482	224	5	�	�	NOUN
cana-1482	224	6	)	)	PUNCT
cana-1482	224	7	)	)	PUNCT
cana-1482	224	8	)	)	PUNCT
cana-1482	225	1	⊆	⊆	NUM
cana-1482	225	2	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	ADJ
cana-1482	225	3	�	�	NOUN
cana-1482	225	4	̅	̅	NOUN
cana-1482	225	5	�	�	NOUN
cana-1482	225	6	)	)	PUNCT
cana-1482	225	7	⊆	⊆	NUM
cana-1482	225	8	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	PROPN
cana-1482	225	9	�	�	NOUN
cana-1482	225	10	̅	̅	NOUN
cana-1482	225	11	�	�	PROPN
cana-1482	225	12	)	)	PUNCT
cana-1482	225	13	.	.	PUNCT
cana-1482	226	1	now	now	ADV
cana-1482	226	2	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	226	3	�	�	NOUN
cana-1482	226	4	̅	̅	NOUN
cana-1482	226	5	�	�	NOUN
cana-1482	226	6	)	)	PUNCT
cana-1482	226	7	)	)	PUNCT
cana-1482	226	8	)	)	PUNCT
cana-1482	227	1	⊆	⊆	X
cana-1482	227	2	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	227	3	�	�	NOUN
cana-1482	227	4	̅	̅	NOUN
cana-1482	227	5	�	�	NOUN
cana-1482	227	6	)	)	PUNCT
cana-1482	227	7	)	)	PUNCT
cana-1482	227	8	)	)	PUNCT
cana-1482	227	9	)	)	PUNCT
cana-1482	228	1	⊆	⊆	NUM
cana-1482	228	2	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	228	3	�	�	NOUN
cana-1482	228	4	̅	̅	NOUN
cana-1482	228	5	�	�	PROPN
cana-1482	228	6	)	)	PUNCT
cana-1482	228	7	.	.	PUNCT
cana-1482	229	1	therefore	therefore	ADV
cana-1482	229	2	�	�	PROPN
cana-1482	229	3	̅	̅	NOUN
cana-1482	229	4	�	�	NOUN
cana-1482	229	5	∪	∪	ADP
cana-1482	229	6	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	229	7	�	�	NOUN
cana-1482	229	8	̅	̅	NOUN
cana-1482	229	9	�	�	NOUN
cana-1482	229	10	)	)	PUNCT
cana-1482	229	11	)	)	PUNCT
cana-1482	229	12	)	)	PUNCT
cana-1482	230	1	⊆	⊆	NUM
cana-1482	230	2	𝑛𝑠𝑠𝑝𝑐𝑙(	𝑛𝑠𝑠𝑝𝑐𝑙(	NOUN
cana-1482	230	3	�	�	NOUN
cana-1482	230	4	̅	̅	NOUN
cana-1482	230	5	�	�	PROPN
cana-1482	230	6	)	)	PUNCT
cana-1482	230	7	⊆	⊆	NUM
cana-1482	230	8	�	�	NOUN
cana-1482	230	9	̅	̅	NOUN
cana-1482	230	10	�	�	PROPN
cana-1482	230	11	.	.	PUNCT
cana-1482	231	1	this	this	PRON
cana-1482	231	2	implies	imply	VERB
cana-1482	231	3	that	that	SCONJ
cana-1482	231	4	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	PROPN
cana-1482	231	5	�	�	PROPN
cana-1482	231	6	̅	̅	NOUN
cana-1482	231	7	�	�	NOUN
cana-1482	231	8	)	)	PUNCT
cana-1482	231	9	)	)	PUNCT
cana-1482	231	10	)	)	PUNCT
cana-1482	231	11	⊆	⊆	NUM
cana-1482	231	12	�	�	NOUN
cana-1482	231	13	̅	̅	NOUN
cana-1482	231	14	�	�	PROPN
cana-1482	231	15	.	.	PUNCT
cana-1482	232	1	since	since	SCONJ
cana-1482	232	2	�	�	PROPN
cana-1482	232	3	̅	̅	NOUN
cana-1482	232	4	�	�	NOUN
cana-1482	232	5	is	be	AUX
cana-1482	232	6	a	a	DET
cana-1482	232	7	nsos	nsos	ADJ
cana-1482	232	8	,	,	PUNCT
cana-1482	232	9	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	ADJ
cana-1482	232	10	�	�	NOUN
cana-1482	232	11	̅	̅	NOUN
cana-1482	232	12	�	�	NOUN
cana-1482	232	13	)	)	PUNCT
cana-1482	232	14	=	=	SYM
cana-1482	232	15	�	�	NOUN
cana-1482	232	16	̅	̅	NOUN
cana-1482	232	17	�	�	PROPN
cana-1482	232	18	.	.	PUNCT
cana-1482	233	1	therefore	therefore	ADV
cana-1482	233	2	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	PROPN
cana-1482	233	3	�	�	PROPN
cana-1482	233	4	̅	̅	NOUN
cana-1482	233	5	�	�	NOUN
cana-1482	233	6	)	)	PUNCT
cana-1482	233	7	⊆	⊆	NUM
cana-1482	233	8	�	�	NOUN
cana-1482	233	9	̅	̅	NOUN
cana-1482	233	10	�	�	PROPN
cana-1482	233	11	.	.	PUNCT
cana-1482	234	1	since	since	SCONJ
cana-1482	234	2	�	�	PROPN
cana-1482	234	3	̅	̅	NOUN
cana-1482	234	4	�	�	NOUN
cana-1482	234	5	is	be	AUX
cana-1482	234	6	an	an	DET
cana-1482	234	7	nsos	nsos	NOUN
cana-1482	234	8	,	,	PUNCT
cana-1482	234	9	it	it	PRON
cana-1482	234	10	is	be	AUX
cana-1482	234	11	a	a	DET
cana-1482	234	12	nspos	nspos	NOUN
cana-1482	234	13	.	.	PUNCT
cana-1482	235	1	hence	hence	ADV
cana-1482	235	2	�	�	NOUN
cana-1482	235	3	̅	̅	NOUN
cana-1482	235	4	�	�	PROPN
cana-1482	235	5	⊆	⊆	NUM
cana-1482	235	6	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	NOUN
cana-1482	235	7	�	�	PROPN
cana-1482	235	8	̅	̅	NOUN
cana-1482	235	9	�	�	NOUN
cana-1482	235	10	)	)	PUNCT
cana-1482	235	11	)	)	PUNCT
cana-1482	235	12	.	.	PUNCT
cana-1482	236	1	therefore	therefore	ADV
cana-1482	236	2	�	�	PROPN
cana-1482	236	3	̅	̅	NOUN
cana-1482	236	4	�	�	NOUN
cana-1482	236	5	=	=	SYM
cana-1482	236	6	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	NOUN
cana-1482	236	7	�	�	PROPN
cana-1482	236	8	̅	̅	NOUN
cana-1482	236	9	�	�	NUM
cana-1482	236	10	)	)	PUNCT
cana-1482	236	11	.	.	PUNCT
cana-1482	237	1	hence	hence	ADV
cana-1482	237	2	�	�	NOUN
cana-1482	237	3	̅	̅	NOUN
cana-1482	237	4	�	�	NOUN
cana-1482	237	5	is	be	AUX
cana-1482	237	6	a	a	DET
cana-1482	237	7	nsros	nsro	NOUN
cana-1482	237	8	in	in	ADP
cana-1482	237	9	(	(	PUNCT
cana-1482	237	10	𝑋,	𝑋,	PROPN
cana-1482	237	11	)	)	PUNCT
cana-1482	237	12	.	.	PUNCT
cana-1482	238	1	(	(	PUNCT
cana-1482	238	2	ii)⇒	ii)⇒	PROPN
cana-1482	238	3	(	(	PUNCT
cana-1482	238	4	ii	ii	NOUN
cana-1482	238	5	)	)	PUNCT
cana-1482	238	6	let	let	VERB
cana-1482	238	7	�	�	PRON
cana-1482	238	8	̅	̅	NOUN
cana-1482	238	9	�	�	NOUN
cana-1482	238	10	be	be	AUX
cana-1482	238	11	a	a	DET
cana-1482	238	12	nsros	nsro	NOUN
cana-1482	238	13	in	in	ADP
cana-1482	238	14	(	(	PUNCT
cana-1482	238	15	𝑋,	𝑋,	PROPN
cana-1482	238	16	)	)	PUNCT
cana-1482	238	17	.	.	PUNCT
cana-1482	239	1	therefore	therefore	ADV
cana-1482	239	2	�	�	PROPN
cana-1482	239	3	̅	̅	NOUN
cana-1482	239	4	�	�	NOUN
cana-1482	239	5	=	=	SYM
cana-1482	239	6	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	NOUN
cana-1482	239	7	�	�	PROPN
cana-1482	239	8	̅	̅	NOUN
cana-1482	239	9	�	�	NOUN
cana-1482	239	10	)	)	PUNCT
cana-1482	239	11	)	)	PUNCT
cana-1482	239	12	.	.	PUNCT
cana-1482	240	1	since	since	SCONJ
cana-1482	240	2	every	every	DET
cana-1482	240	3	nsros	nsro	NOUN
cana-1482	240	4	is	be	AUX
cana-1482	240	5	a	a	DET
cana-1482	240	6	nsos	nsos	ADJ
cana-1482	240	7	,	,	PUNCT
cana-1482	240	8	�	�	NOUN
cana-1482	240	9	̅	̅	NOUN
cana-1482	240	10	�	�	NOUN
cana-1482	240	11	is	be	AUX
cana-1482	240	12	a	a	DET
cana-1482	240	13	nsos	nsos	ADJ
cana-1482	240	14	and	and	CCONJ
cana-1482	240	15	�	�	PROPN
cana-1482	240	16	̅	̅	NOUN
cana-1482	240	17	�	�	PROPN
cana-1482	240	18	⊆	⊆	NUM
cana-1482	240	19	�	�	NOUN
cana-1482	240	20	̅	̅	NOUN
cana-1482	240	21	�	�	NOUN
cana-1482	240	22	.	.	PUNCT
cana-1482	241	1	this	this	PRON
cana-1482	241	2	implies	imply	VERB
cana-1482	241	3	that	that	PRON
cana-1482	241	4	𝑛𝑠𝑖𝑛𝑡	𝑛𝑠𝑖𝑛𝑡	NOUN
cana-1482	241	5	(	(	PUNCT
cana-1482	241	6	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	241	7	�	�	PROPN
cana-1482	241	8	̅	̅	NOUN
cana-1482	241	9	�	�	NOUN
cana-1482	241	10	)	)	PUNCT
cana-1482	241	11	)	)	PUNCT
cana-1482	241	12	⊆	⊆	NUM
cana-1482	241	13	�	�	NOUN
cana-1482	241	14	̅	̅	NOUN
cana-1482	241	15	�	�	NOUN
cana-1482	241	16	.	.	PUNCT
cana-1482	242	1	that	that	PRON
cana-1482	242	2	is	be	AUX
cana-1482	242	3	𝑛𝑠𝑖𝑛𝑡	𝑛𝑠𝑖𝑛𝑡	X
cana-1482	242	4	(	(	PUNCT
cana-1482	242	5	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	242	6	�	�	NOUN
cana-1482	242	7	̅	̅	NOUN
cana-1482	242	8	�	�	NOUN
cana-1482	242	9	)	)	PUNCT
cana-1482	242	10	)	)	PUNCT
cana-1482	242	11	)	)	PUNCT
cana-1482	243	1	=	=	PUNCT
cana-1482	244	1	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	NOUN
cana-1482	244	2	�	�	PROPN
cana-1482	244	3	̅	̅	NOUN
cana-1482	244	4	�	�	NOUN
cana-1482	244	5	)	)	PUNCT
cana-1482	244	6	)	)	PUNCT
cana-1482	245	1	⊆	⊆	NUM
cana-1482	245	2	�	�	SYM
cana-1482	245	3	̅	̅	NOUN
cana-1482	245	4	�	�	NOUN
cana-1482	245	5	.	.	PUNCT
cana-1482	246	1	thus	thus	ADV
cana-1482	246	2	�	�	NOUN
cana-1482	246	3	̅	̅	NOUN
cana-1482	246	4	�	�	NOUN
cana-1482	246	5	is	be	AUX
cana-1482	246	6	a	a	DET
cana-1482	246	7	nsβcs	nsβcs	NOUN
cana-1482	246	8	.	.	PUNCT
cana-1482	247	1	hence	hence	ADV
cana-1482	247	2	by	by	ADP
cana-1482	247	3	theorem	theorem	ADJ
cana-1482	247	4	2.11	2.11	NUM
cana-1482	247	5	,	,	PUNCT
cana-1482	247	6	�	�	NOUN
cana-1482	247	7	̅	̅	NOUN
cana-1482	247	8	�	�	NOUN
cana-1482	247	9	is	be	AUX
cana-1482	247	10	a	a	DET
cana-1482	247	11	nsgspcs	nsgspc	NOUN
cana-1482	247	12	in	in	ADP
cana-1482	247	13	(	(	PUNCT
cana-1482	247	14	𝑋,	𝑋,	PROPN
cana-1482	247	15	)	)	PUNCT
cana-1482	247	16	.	.	PUNCT
cana-1482	248	1	2.25	2.25	NUM
cana-1482	248	2	theorem	theorem	VERB
cana-1482	248	3	:	:	PUNCT
cana-1482	248	4	for	for	ADP
cana-1482	248	5	a	a	DET
cana-1482	248	6	nsos	nsos	PROPN
cana-1482	248	7	�	�	PROPN
cana-1482	248	8	̅	̅	NOUN
cana-1482	248	9	�	�	PROPN
cana-1482	248	10	in	in	ADP
cana-1482	248	11	(	(	PUNCT
cana-1482	248	12	𝑋,	𝑋,	PROPN
cana-1482	248	13	)	)	PUNCT
cana-1482	248	14	,	,	PUNCT
cana-1482	248	15	the	the	DET
cana-1482	248	16	following	follow	VERB
cana-1482	248	17	conditions	condition	NOUN
cana-1482	248	18	are	be	AUX
cana-1482	248	19	equivalent	equivalent	ADJ
cana-1482	248	20	:	:	PUNCT
cana-1482	248	21	(	(	PUNCT
cana-1482	248	22	i	i	NOUN
cana-1482	248	23	)	)	PUNCT
cana-1482	248	24	�	�	PROPN
cana-1482	248	25	̅	̅	NOUN
cana-1482	248	26	�	�	NOUN
cana-1482	248	27	is	be	AUX
cana-1482	248	28	a	a	DET
cana-1482	248	29	nscs	nscs	NOUN
cana-1482	248	30	in	in	ADP
cana-1482	248	31	(	(	PUNCT
cana-1482	248	32	𝑋,	𝑋,	PROPN
cana-1482	248	33	)	)	PUNCT
cana-1482	248	34	,	,	PUNCT
cana-1482	248	35	(	(	PUNCT
cana-1482	248	36	ii	ii	NOUN
cana-1482	248	37	)	)	PUNCT
cana-1482	248	38	�	�	PROPN
cana-1482	248	39	̅	̅	NOUN
cana-1482	248	40	�	�	NOUN
cana-1482	248	41	is	be	AUX
cana-1482	248	42	a	a	DET
cana-1482	248	43	nsgspcs	nsgspc	NOUN
cana-1482	248	44	and	and	CCONJ
cana-1482	248	45	a	a	DET
cana-1482	248	46	nsq	nsq	NOUN
cana-1482	248	47	-	-	PUNCT
cana-1482	248	48	set	set	VERB
cana-1482	248	49	in	in	ADP
cana-1482	248	50	(	(	PUNCT
cana-1482	248	51	𝑋,	𝑋,	PROPN
cana-1482	248	52	)	)	PUNCT
cana-1482	248	53	.	.	PUNCT
cana-1482	249	1	communications	communication	NOUN
cana-1482	249	2	on	on	ADP
cana-1482	249	3	applied	apply	VERB
cana-1482	249	4	nonlinear	nonlinear	ADJ
cana-1482	249	5	analysis	analysis	NOUN
cana-1482	249	6	issn	issn	NOUN
cana-1482	249	7	:	:	PUNCT
cana-1482	249	8	1074	1074	NUM
cana-1482	249	9	-	-	PUNCT
cana-1482	249	10	133x	133x	NUM
cana-1482	249	11	vol	vol	NOUN
cana-1482	249	12	31	31	NUM
cana-1482	249	13	no	no	NOUN
cana-1482	249	14	.	.	PUNCT
cana-1482	250	1	8s	8s	PROPN
cana-1482	250	2	(	(	PUNCT
cana-1482	250	3	2024	2024	NUM
cana-1482	250	4	)	)	PUNCT
cana-1482	250	5	280	280	NUM
cana-1482	250	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	250	7	proof	proof	NOUN
cana-1482	250	8	:	:	PUNCT
cana-1482	250	9	(	(	PUNCT
cana-1482	250	10	𝑖	𝑖	X
cana-1482	250	11	)	)	PUNCT
cana-1482	250	12	⟹	⟹	X
cana-1482	250	13	(	(	PUNCT
cana-1482	250	14	𝑖𝑖	𝑖𝑖	NOUN
cana-1482	250	15	)	)	PUNCT
cana-1482	250	16	since	since	SCONJ
cana-1482	250	17	�	�	NOUN
cana-1482	250	18	̅	̅	NOUN
cana-1482	250	19	�	�	NOUN
cana-1482	250	20	is	be	AUX
cana-1482	250	21	a	a	DET
cana-1482	250	22	nscs	nsc	NOUN
cana-1482	250	23	,	,	PUNCT
cana-1482	250	24	it	it	PRON
cana-1482	250	25	is	be	AUX
cana-1482	250	26	a	a	DET
cana-1482	250	27	nsgspcs	nsgspc	NOUN
cana-1482	250	28	in	in	ADP
cana-1482	250	29	(	(	PUNCT
cana-1482	250	30	𝑋,	𝑋,	PROPN
cana-1482	250	31	)	)	PUNCT
cana-1482	250	32	.	.	PUNCT
cana-1482	251	1	now	now	ADV
cana-1482	251	2	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	NOUN
cana-1482	251	3	�	�	PROPN
cana-1482	251	4	̅	̅	NOUN
cana-1482	251	5	�	�	NOUN
cana-1482	251	6	)	)	PUNCT
cana-1482	251	7	)	)	PUNCT
cana-1482	252	1	=	=	PUNCT
cana-1482	252	2	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	252	3	�	�	PROPN
cana-1482	252	4	̅	̅	NOUN
cana-1482	252	5	�	�	NOUN
cana-1482	252	6	)	)	PUNCT
cana-1482	252	7	=	=	SYM
cana-1482	252	8	�	�	NOUN
cana-1482	252	9	̅	̅	NOUN
cana-1482	252	10	�	�	NOUN
cana-1482	252	11	=	=	SYM
cana-1482	252	12	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	252	13	�	�	PROPN
cana-1482	252	14	̅	̅	NOUN
cana-1482	252	15	�	�	NOUN
cana-1482	252	16	)	)	PUNCT
cana-1482	252	17	=	=	PUNCT
cana-1482	252	18	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	252	19	�	�	NOUN
cana-1482	252	20	̅	̅	NOUN
cana-1482	252	21	�	�	NOUN
cana-1482	252	22	)	)	PUNCT
cana-1482	252	23	)	)	PUNCT
cana-1482	252	24	,	,	PUNCT
cana-1482	252	25	by	by	ADP
cana-1482	252	26	hyp	hyp	PROPN
cana-1482	252	27	-	-	PUNCT
cana-1482	252	28	othesis	othesis	NOUN
cana-1482	252	29	.	.	PUNCT
cana-1482	253	1	hence	hence	ADV
cana-1482	253	2	�	�	NOUN
cana-1482	253	3	̅	̅	NOUN
cana-1482	253	4	�	�	NOUN
cana-1482	253	5	is	be	AUX
cana-1482	253	6	a	a	DET
cana-1482	253	7	nsq	nsq	NOUN
cana-1482	253	8	-	-	PUNCT
cana-1482	253	9	set	set	VERB
cana-1482	253	10	in	in	ADP
cana-1482	253	11	(	(	PUNCT
cana-1482	253	12	𝑋,	𝑋,	PROPN
cana-1482	253	13	)	)	PUNCT
cana-1482	253	14	.	.	PUNCT
cana-1482	254	1	(	(	PUNCT
cana-1482	254	2	𝑖𝑖	𝑖𝑖	NOUN
cana-1482	254	3	)	)	PUNCT
cana-1482	254	4	⟹	⟹	X
cana-1482	254	5	(	(	PUNCT
cana-1482	254	6	𝑖	𝑖	X
cana-1482	254	7	)	)	PUNCT
cana-1482	254	8	since	since	SCONJ
cana-1482	254	9	�	�	NOUN
cana-1482	254	10	̅	̅	NOUN
cana-1482	254	11	�	�	NOUN
cana-1482	254	12	is	be	AUX
cana-1482	254	13	a	a	DET
cana-1482	254	14	nsos	nsos	NOUN
cana-1482	254	15	and	and	CCONJ
cana-1482	254	16	a	a	DET
cana-1482	254	17	nsgspcs	nsgspc	NOUN
cana-1482	254	18	in	in	ADP
cana-1482	254	19	(	(	PUNCT
cana-1482	254	20	𝑋,	𝑋,	PROPN
cana-1482	254	21	)	)	PUNCT
cana-1482	254	22	,	,	PUNCT
cana-1482	254	23	by	by	ADP
cana-1482	254	24	theorem	theorem	NOUN
cana-1482	254	25	2.24	2.24	NUM
cana-1482	254	26	,	,	PUNCT
cana-1482	254	27	�	�	NOUN
cana-1482	254	28	̅	̅	NOUN
cana-1482	254	29	�	�	NOUN
cana-1482	254	30	is	be	AUX
cana-1482	254	31	a	a	DET
cana-1482	254	32	nsros	nsro	NOUN
cana-1482	254	33	in	in	ADP
cana-1482	254	34	(	(	PUNCT
cana-1482	254	35	𝑋,	𝑋,	PROPN
cana-1482	254	36	)	)	PUNCT
cana-1482	254	37	.	.	PUNCT
cana-1482	255	1	therefore	therefore	ADV
cana-1482	255	2	�	�	PROPN
cana-1482	255	3	̅	̅	NOUN
cana-1482	255	4	�	�	NOUN
cana-1482	255	5	=	=	SYM
cana-1482	255	6	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	𝑛𝑠𝑖𝑛𝑡(𝑛𝑠𝑐𝑙(	NOUN
cana-1482	255	7	�	�	PROPN
cana-1482	255	8	̅	̅	NOUN
cana-1482	255	9	�	�	NOUN
cana-1482	255	10	)	)	PUNCT
cana-1482	255	11	)	)	PUNCT
cana-1482	256	1	=	=	PUNCT
cana-1482	256	2	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑐𝑙(𝑛𝑠𝑖𝑛𝑡(	NOUN
cana-1482	256	3	�	�	NOUN
cana-1482	256	4	̅	̅	NOUN
cana-1482	256	5	�	�	NOUN
cana-1482	256	6	)	)	PUNCT
cana-1482	256	7	)	)	PUNCT
cana-1482	257	1	=	=	PUNCT
cana-1482	257	2	𝑛𝑠𝑐𝑙(	𝑛𝑠𝑐𝑙(	PROPN
cana-1482	257	3	�	�	PROPN
cana-1482	257	4	̅	̅	NOUN
cana-1482	257	5	�	�	PROPN
cana-1482	257	6	)	)	PUNCT
cana-1482	257	7	,	,	PUNCT
cana-1482	257	8	by	by	ADP
cana-1482	257	9	hypothesis	hypothesis	NOUN
cana-1482	257	10	.	.	PUNCT
cana-1482	258	1	hence	hence	ADV
cana-1482	258	2	�	�	NOUN
cana-1482	258	3	̅	̅	NOUN
cana-1482	258	4	�	�	NOUN
cana-1482	258	5	is	be	AUX
cana-1482	258	6	a	a	DET
cana-1482	258	7	nscs	nscs	NOUN
cana-1482	258	8	in	in	ADP
cana-1482	258	9	(	(	PUNCT
cana-1482	258	10	𝑋,	𝑋,	PROPN
cana-1482	258	11	)	)	PUNCT
cana-1482	258	12	.	.	PUNCT
cana-1482	259	1	2.26	2.26	NUM
cana-1482	259	2	theorem	theorem	NOUN
cana-1482	259	3	:	:	PUNCT
cana-1482	259	4	let	let	VERB
cana-1482	259	5	(	(	PUNCT
cana-1482	259	6	𝑋,	𝑋,	PROPN
cana-1482	259	7	)	)	PUNCT
cana-1482	259	8	be	be	AUX
cana-1482	259	9	a	a	DET
cana-1482	259	10	nsts	nst	NOUN
cana-1482	259	11	.	.	PUNCT
cana-1482	260	1	then	then	ADV
cana-1482	260	2	for	for	ADP
cana-1482	260	3	every	every	DET
cana-1482	260	4	�	�	NOUN
cana-1482	260	5	̅	̅	NOUN
cana-1482	260	6	�	�	PROPN
cana-1482	260	7	∈nsspc(x	∈nsspc(x	NOUN
cana-1482	260	8	)	)	PUNCT
cana-1482	260	9	and	and	CCONJ
cana-1482	260	10	for	for	ADP
cana-1482	260	11	every	every	DET
cana-1482	260	12	nss	nss	PROPN
cana-1482	260	13	�	�	NOUN
cana-1482	260	14	̅	̅	NOUN
cana-1482	260	15	�	�	NOUN
cana-1482	260	16	in	in	ADP
cana-1482	260	17	x	x	PROPN
cana-1482	260	18	,	,	PUNCT
cana-1482	260	19	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	ADJ
cana-1482	260	20	�	�	NOUN
cana-1482	260	21	̅	̅	NOUN
cana-1482	260	22	�	�	NOUN
cana-1482	260	23	)	)	PUNCT
cana-1482	260	24	⊆	⊆	NUM
cana-1482	260	25	�	�	NOUN
cana-1482	260	26	̅	̅	NOUN
cana-1482	260	27	�	�	PROPN
cana-1482	260	28	⊆	⊆	NUM
cana-1482	260	29	�	�	NOUN
cana-1482	260	30	̅	̅	NOUN
cana-1482	260	31	�	�	NOUN
cana-1482	260	32	implies	imply	VERB
cana-1482	260	33	�	�	NOUN
cana-1482	260	34	̅	̅	NOUN
cana-1482	260	35	�	�	PROPN
cana-1482	260	36	∈	∈	PROPN
cana-1482	260	37	𝑁𝑆𝐺𝑆𝑃𝐶(𝑋	𝑁𝑆𝐺𝑆𝑃𝐶(𝑋	NOUN
cana-1482	260	38	)	)	PUNCT
cana-1482	260	39	.	.	PUNCT
cana-1482	261	1	proof	proof	NOUN
cana-1482	261	2	:	:	PUNCT
cana-1482	261	3	let	let	VERB
cana-1482	261	4	�	�	PRON
cana-1482	261	5	̅	̅	NOUN
cana-1482	261	6	�	�	NOUN
cana-1482	261	7	be	be	AUX
cana-1482	261	8	a	a	DET
cana-1482	261	9	nsspcs	nsspc	NOUN
cana-1482	261	10	in	in	ADP
cana-1482	261	11	x.	x.	NOUN
cana-1482	261	12	then	then	ADV
cana-1482	261	13	by	by	ADP
cana-1482	261	14	definition	definition	NOUN
cana-1482	261	15	1.7	1.7	NUM
cana-1482	261	16	,	,	PUNCT
cana-1482	261	17	there	there	PRON
cana-1482	261	18	exists	exist	VERB
cana-1482	261	19	a	a	DET
cana-1482	261	20	nspcs	nspc	NOUN
cana-1482	261	21	,	,	PUNCT
cana-1482	261	22	say	say	VERB
cana-1482	261	23	𝐶̅	𝐶̅	NOUN
cana-1482	261	24	such	such	ADJ
cana-1482	261	25	that	that	SCONJ
cana-1482	261	26	𝑛𝑠𝑖𝑛𝑡(𝐶̅	𝑛𝑠𝑖𝑛𝑡(𝐶̅	NOUN
cana-1482	261	27	)	)	PUNCT
cana-1482	261	28	⊆	⊆	NUM
cana-1482	261	29	�	�	NOUN
cana-1482	261	30	̅	̅	NOUN
cana-1482	261	31	�	�	NOUN
cana-1482	261	32	⊆	⊆	NUM
cana-1482	261	33	𝐶̅.	𝐶̅.	VERB
cana-1482	261	34	by	by	ADP
cana-1482	261	35	hypothesis	hypothesis	NOUN
cana-1482	261	36	,	,	PUNCT
cana-1482	261	37	�	�	PROPN
cana-1482	261	38	̅	̅	NOUN
cana-1482	261	39	�	�	NOUN
cana-1482	261	40	⊆	⊆	NUM
cana-1482	261	41	𝐴.̅	𝐴.̅	PROPN
cana-1482	261	42	therefore	therefore	ADV
cana-1482	261	43	�	�	PROPN
cana-1482	261	44	̅	̅	NOUN
cana-1482	261	45	�	�	NOUN
cana-1482	261	46	⊆	⊆	NUM
cana-1482	261	47	𝐶̅.	𝐶̅.	PROPN
cana-1482	261	48	since	since	SCONJ
cana-1482	261	49	𝑛𝑠𝑖𝑛𝑡(𝐶̅	𝑛𝑠𝑖𝑛𝑡(𝐶̅	NOUN
cana-1482	261	50	)	)	PUNCT
cana-1482	261	51	⊆	⊆	NUM
cana-1482	261	52	�	�	NOUN
cana-1482	261	53	̅	̅	NOUN
cana-1482	261	54	�	�	PROPN
cana-1482	261	55	,	,	PUNCT
cana-1482	261	56	𝑛𝑠𝑖𝑛𝑡(𝐶̅	𝑛𝑠𝑖𝑛𝑡(𝐶̅	NOUN
cana-1482	261	57	)	)	PUNCT
cana-1482	261	58	⊆	⊆	NUM
cana-1482	261	59	𝑛𝑠𝑖𝑛𝑡(	𝑛𝑠𝑖𝑛𝑡(	ADJ
cana-1482	261	60	�	�	NOUN
cana-1482	261	61	̅	̅	NOUN
cana-1482	261	62	�	�	NOUN
cana-1482	261	63	)	)	PUNCT
cana-1482	261	64	and	and	CCONJ
cana-1482	261	65	𝑛𝑠𝑖𝑛𝑡(𝐶̅	𝑛𝑠𝑖𝑛𝑡(𝐶̅	NOUN
cana-1482	261	66	)	)	PUNCT
cana-1482	261	67	⊆	⊆	NUM
cana-1482	261	68	𝐵.̅	𝐵.̅	NOUN
cana-1482	261	69	thus	thus	ADV
cana-1482	261	70	𝑛𝑠𝑖𝑛𝑡(𝐶̅	𝑛𝑠𝑖𝑛𝑡(𝐶̅	NOUN
cana-1482	261	71	)	)	PUNCT
cana-1482	262	1	⊆	⊆	NUM
cana-1482	262	2	�	�	NOUN
cana-1482	262	3	̅	̅	NOUN
cana-1482	262	4	�	�	NOUN
cana-1482	262	5	⊆	⊆	NUM
cana-1482	262	6	𝐶̅	𝐶̅	NOUN
cana-1482	262	7	and	and	CCONJ
cana-1482	262	8	by	by	ADP
cana-1482	262	9	definition	definition	NOUN
cana-1482	262	10	1.7	1.7	NUM
cana-1482	262	11	,	,	PUNCT
cana-1482	262	12	�	�	NOUN
cana-1482	262	13	̅	̅	NOUN
cana-1482	262	14	�	�	PROPN
cana-1482	262	15	∈	∈	PROPN
cana-1482	262	16	𝑁𝑆𝑆𝑃𝐶(𝑋	𝑁𝑆𝑆𝑃𝐶(𝑋	NUM
cana-1482	262	17	)	)	PUNCT
cana-1482	262	18	.	.	PUNCT
cana-1482	263	1	hence	hence	ADV
cana-1482	263	2	by	by	ADP
cana-1482	263	3	theorem	theorem	ADJ
cana-1482	263	4	2.7	2.7	NUM
cana-1482	263	5	,	,	PUNCT
cana-1482	263	6	�	�	NOUN
cana-1482	263	7	̅	̅	NOUN
cana-1482	263	8	�	�	PROPN
cana-1482	263	9	∈	∈	PROPN
cana-1482	263	10	𝑁𝑆𝐺𝑆𝑃𝐶(𝑋	𝑁𝑆𝐺𝑆𝑃𝐶(𝑋	NOUN
cana-1482	263	11	)	)	PUNCT
cana-1482	263	12	.	.	PUNCT
cana-1482	264	1	conclusion	conclusion	NOUN
cana-1482	264	2	:	:	PUNCT
cana-1482	264	3	this	this	DET
cana-1482	264	4	study	study	NOUN
cana-1482	264	5	examines	examine	VERB
cana-1482	264	6	the	the	DET
cana-1482	264	7	features	feature	NOUN
cana-1482	264	8	of	of	ADP
cana-1482	264	9	neutrosophic	neutrosophic	ADJ
cana-1482	264	10	generalized	generalize	VERB
cana-1482	264	11	semipreclosed	semipreclose	VERB
cana-1482	264	12	sets	set	NOUN
cana-1482	264	13	in	in	ADP
cana-1482	264	14	neutrosophic	neutrosophic	ADJ
cana-1482	264	15	topological	topological	ADJ
cana-1482	264	16	space	space	NOUN
cana-1482	264	17	.	.	PUNCT
cana-1482	265	1	by	by	ADP
cana-1482	265	2	diving	diving	NOUN
cana-1482	265	3	into	into	ADP
cana-1482	265	4	these	these	DET
cana-1482	265	5	sets	set	NOUN
cana-1482	265	6	'	'	PART
cana-1482	265	7	traits	trait	NOUN
cana-1482	265	8	and	and	CCONJ
cana-1482	265	9	behaviors	behavior	NOUN
cana-1482	265	10	,	,	PUNCT
cana-1482	265	11	the	the	DET
cana-1482	265	12	study	study	NOUN
cana-1482	265	13	hopes	hope	VERB
cana-1482	265	14	to	to	PART
cana-1482	265	15	provide	provide	VERB
cana-1482	265	16	the	the	DET
cana-1482	265	17	groundwork	groundwork	NOUN
cana-1482	265	18	for	for	ADP
cana-1482	265	19	future	future	ADJ
cana-1482	265	20	research	research	NOUN
cana-1482	265	21	and	and	CCONJ
cana-1482	265	22	advancement	advancement	NOUN
cana-1482	265	23	in	in	ADP
cana-1482	265	24	this	this	DET
cana-1482	265	25	sector	sector	NOUN
cana-1482	265	26	.	.	PUNCT
cana-1482	266	1	the	the	DET
cana-1482	266	2	theorems	theorem	NOUN
cana-1482	266	3	offered	offer	VERB
cana-1482	266	4	herein	herein	NOUN
cana-1482	266	5	are	be	AUX
cana-1482	266	6	not	not	PART
cana-1482	266	7	only	only	ADV
cana-1482	266	8	useful	useful	ADJ
cana-1482	266	9	for	for	ADP
cana-1482	266	10	advancing	advance	VERB
cana-1482	266	11	theoretical	theoretical	ADJ
cana-1482	266	12	understanding	understanding	NOUN
cana-1482	266	13	,	,	PUNCT
cana-1482	266	14	but	but	CCONJ
cana-1482	266	15	they	they	PRON
cana-1482	266	16	also	also	ADV
cana-1482	266	17	serve	serve	VERB
cana-1482	266	18	as	as	ADP
cana-1482	266	19	the	the	DET
cana-1482	266	20	foundation	foundation	NOUN
cana-1482	266	21	for	for	ADP
cana-1482	266	22	practical	practical	ADJ
cana-1482	266	23	applications	application	NOUN
cana-1482	266	24	.	.	PUNCT
cana-1482	267	1	the	the	DET
cana-1482	267	2	detailed	detailed	ADJ
cana-1482	267	3	analysis	analysis	NOUN
cana-1482	267	4	reveals	reveal	VERB
cana-1482	267	5	how	how	SCONJ
cana-1482	267	6	neutrosophic	neutrosophic	ADJ
cana-1482	267	7	generalized	generalize	VERB
cana-1482	267	8	semipreclosed	semipreclose	VERB
cana-1482	267	9	sets	set	NOUN
cana-1482	267	10	can	can	AUX
cana-1482	267	11	be	be	AUX
cana-1482	267	12	leveraged	leverage	VERB
cana-1482	267	13	to	to	PART
cana-1482	267	14	extend	extend	VERB
cana-1482	267	15	various	various	ADJ
cana-1482	267	16	aspects	aspect	NOUN
cana-1482	267	17	of	of	ADP
cana-1482	267	18	topological	topological	ADJ
cana-1482	267	19	theory	theory	NOUN
cana-1482	267	20	.	.	PUNCT
cana-1482	268	1	specifically	specifically	ADV
cana-1482	268	2	,	,	PUNCT
cana-1482	268	3	the	the	DET
cana-1482	268	4	study	study	NOUN
cana-1482	268	5	explores	explore	VERB
cana-1482	268	6	the	the	DET
cana-1482	268	7	potential	potential	NOUN
cana-1482	268	8	to	to	PART
cana-1482	268	9	apply	apply	VERB
cana-1482	268	10	these	these	DET
cana-1482	268	11	sets	set	NOUN
cana-1482	268	12	to	to	ADP
cana-1482	268	13	functions	function	NOUN
cana-1482	268	14	,	,	PUNCT
cana-1482	268	15	including	include	VERB
cana-1482	268	16	open	open	ADJ
cana-1482	268	17	maps	map	NOUN
cana-1482	268	18	,	,	PUNCT
cana-1482	268	19	closed	closed	ADJ
cana-1482	268	20	maps	map	NOUN
cana-1482	268	21	,	,	PUNCT
cana-1482	268	22	and	and	CCONJ
cana-1482	268	23	homeomorphisms	homeomorphisms	PROPN
cana-1482	268	24	.	.	PUNCT
cana-1482	269	1	this	this	DET
cana-1482	269	2	extension	extension	NOUN
cana-1482	269	3	of	of	ADP
cana-1482	269	4	the	the	DET
cana-1482	269	5	theory	theory	NOUN
cana-1482	269	6	offers	offer	VERB
cana-1482	269	7	a	a	DET
cana-1482	269	8	robust	robust	ADJ
cana-1482	269	9	framework	framework	NOUN
cana-1482	269	10	for	for	ADP
cana-1482	269	11	investigating	investigate	VERB
cana-1482	269	12	the	the	DET
cana-1482	269	13	continuity	continuity	NOUN
cana-1482	269	14	and	and	CCONJ
cana-1482	269	15	compatibility	compatibility	NOUN
cana-1482	269	16	of	of	ADP
cana-1482	269	17	functions	function	NOUN
cana-1482	269	18	within	within	ADP
cana-1482	269	19	neutrosophic	neutrosophic	ADJ
cana-1482	269	20	topological	topological	ADJ
cana-1482	269	21	spaces	space	NOUN
cana-1482	269	22	.	.	PUNCT
cana-1482	270	1	through	through	ADP
cana-1482	270	2	this	this	DET
cana-1482	270	3	work	work	NOUN
cana-1482	270	4	,	,	PUNCT
cana-1482	270	5	the	the	DET
cana-1482	270	6	utility	utility	NOUN
cana-1482	270	7	of	of	ADP
cana-1482	270	8	neutrosophic	neutrosophic	ADJ
cana-1482	270	9	generalized	generalize	VERB
cana-1482	270	10	semipreclosed	semipreclose	VERB
cana-1482	270	11	sets	set	NOUN
cana-1482	270	12	in	in	ADP
cana-1482	270	13	broadening	broaden	VERB
cana-1482	270	14	the	the	DET
cana-1482	270	15	scope	scope	NOUN
cana-1482	270	16	of	of	ADP
cana-1482	270	17	topological	topological	ADJ
cana-1482	270	18	studies	study	NOUN
cana-1482	270	19	is	be	AUX
cana-1482	270	20	demonstrated	demonstrate	VERB
cana-1482	270	21	,	,	PUNCT
cana-1482	270	22	paving	pave	VERB
cana-1482	270	23	the	the	DET
cana-1482	270	24	way	way	NOUN
cana-1482	270	25	for	for	ADP
cana-1482	270	26	new	new	ADJ
cana-1482	270	27	discoveries	discovery	NOUN
cana-1482	270	28	and	and	CCONJ
cana-1482	270	29	applications	application	NOUN
cana-1482	270	30	in	in	ADP
cana-1482	270	31	the	the	DET
cana-1482	270	32	realm	realm	NOUN
cana-1482	270	33	of	of	ADP
cana-1482	270	34	neutrosophic	neutrosophic	ADJ
cana-1482	270	35	topology	topology	NOUN
cana-1482	270	36	.	.	PUNCT
cana-1482	271	1	by	by	ADP
cana-1482	271	2	establishing	establish	VERB
cana-1482	271	3	these	these	DET
cana-1482	271	4	foundational	foundational	ADJ
cana-1482	271	5	principles	principle	NOUN
cana-1482	271	6	,	,	PUNCT
cana-1482	271	7	the	the	DET
cana-1482	271	8	aim	aim	NOUN
cana-1482	271	9	is	be	AUX
cana-1482	271	10	to	to	PART
cana-1482	271	11	inspire	inspire	VERB
cana-1482	271	12	further	further	ADJ
cana-1482	271	13	research	research	NOUN
cana-1482	271	14	that	that	PRON
cana-1482	271	15	will	will	AUX
cana-1482	271	16	continue	continue	VERB
cana-1482	271	17	to	to	PART
cana-1482	271	18	expand	expand	VERB
cana-1482	271	19	and	and	CCONJ
cana-1482	271	20	refine	refine	VERB
cana-1482	271	21	the	the	DET
cana-1482	271	22	understanding	understanding	NOUN
cana-1482	271	23	of	of	ADP
cana-1482	271	24	neutrosophic	neutrosophic	ADJ
cana-1482	271	25	topological	topological	ADJ
cana-1482	271	26	structures	structure	NOUN
cana-1482	271	27	and	and	CCONJ
cana-1482	271	28	their	their	PRON
cana-1482	271	29	practical	practical	ADJ
cana-1482	271	30	implications	implication	NOUN
cana-1482	271	31	.	.	PUNCT
cana-1482	272	1	references	reference	NOUN
cana-1482	272	2	:	:	PUNCT
cana-1482	273	1	[	[	X
cana-1482	273	2	1	1	NUM
cana-1482	273	3	]	]	PUNCT
cana-1482	273	4	andrijevic.d	andrijevic.d	ADP
cana-1482	273	5	,	,	PUNCT
cana-1482	273	6	semipreopen	semipreopen	ADJ
cana-1482	273	7	sets	set	NOUN
cana-1482	273	8	,	,	PUNCT
cana-1482	273	9	mat.vesnic	mat.vesnic	NOUN
cana-1482	273	10	,	,	PUNCT
cana-1482	273	11	38	38	NUM
cana-1482	273	12	,	,	PUNCT
cana-1482	273	13	(	(	PUNCT
cana-1482	273	14	1986	1986	NUM
cana-1482	273	15	)	)	PUNCT
cana-1482	273	16	,	,	PUNCT
cana-1482	273	17	24	24	NUM
cana-1482	273	18	-	-	SYM
cana-1482	273	19	32	32	NUM
cana-1482	273	20	.	.	PUNCT
cana-1482	274	1	[	[	X
cana-1482	274	2	2	2	NUM
cana-1482	274	3	]	]	PUNCT
cana-1482	274	4	k.	k.	PROPN
cana-1482	274	5	atanassov	atanassov	PROPN
cana-1482	274	6	,	,	PUNCT
cana-1482	274	7	intuitionistic	intuitionistic	ADJ
cana-1482	274	8	fuzzy	fuzzy	ADJ
cana-1482	274	9	sets	set	NOUN
cana-1482	274	10	,	,	PUNCT
cana-1482	274	11	fuzzy	fuzzy	ADJ
cana-1482	274	12	sets	set	NOUN
cana-1482	274	13	and	and	CCONJ
cana-1482	274	14	systems	system	NOUN
cana-1482	274	15	20	20	NUM
cana-1482	274	16	(	(	PUNCT
cana-1482	274	17	1986	1986	NUM
cana-1482	274	18	)	)	PUNCT
cana-1482	274	19	,	,	PUNCT
cana-1482	274	20	87	87	NUM
cana-1482	274	21	-	-	SYM
cana-1482	274	22	96	96	NUM
cana-1482	274	23	.	.	PUNCT
cana-1482	275	1	[	[	X
cana-1482	275	2	3	3	NUM
cana-1482	275	3	]	]	PUNCT
cana-1482	275	4	bhattacarya.b	bhattacarya.b	NUM
cana-1482	275	5	.	.	PROPN
cana-1482	275	6	,	,	PUNCT
cana-1482	275	7	and	and	CCONJ
cana-1482	275	8	lahiri.b.k	lahiri.b.k	NOUN
cana-1482	275	9	.	.	PUNCT
cana-1482	275	10	,	,	PUNCT
cana-1482	275	11	semi	semi	ADJ
cana-1482	275	12	-	-	ADJ
cana-1482	275	13	generalized	generalized	ADJ
cana-1482	275	14	closed	close	VERB
cana-1482	275	15	set	set	VERB
cana-1482	275	16	in	in	ADP
cana-1482	275	17	topology	topology	NOUN
cana-1482	275	18	,	,	PUNCT
cana-1482	275	19	indian	indian	PROPN
cana-1482	275	20	jour.math	jour.math	PROPN
cana-1482	275	21	.	.	PROPN
cana-1482	275	22	,29	,29	PROPN
cana-1482	275	23	(	(	PUNCT
cana-1482	275	24	1987	1987	NUM
cana-1482	275	25	)	)	PUNCT
cana-1482	275	26	,	,	PUNCT
cana-1482	275	27	375	375	NUM
cana-1482	275	28	-	-	SYM
cana-1482	275	29	382	382	NUM
cana-1482	275	30	.	.	PUNCT
cana-1482	276	1	[	[	X
cana-1482	276	2	4	4	NUM
cana-1482	276	3	]	]	SYM
cana-1482	276	4	chang.c.l	chang.c.l	NOUN
cana-1482	276	5	.	.	PUNCT
cana-1482	276	6	,	,	PUNCT
cana-1482	276	7	ftss	ftss	NOUN
cana-1482	276	8	.	.	PUNCT
cana-1482	277	1	ji	ji	PROPN
cana-1482	277	2	.	.	PROPN
cana-1482	277	3	math	math	PROPN
cana-1482	277	4	.	.	PUNCT
cana-1482	278	1	anal	anal	PROPN
cana-1482	278	2	.	.	PUNCT
cana-1482	278	3	appl	appl	PROPN
cana-1482	278	4	.	.	PROPN
cana-1482	278	5	,	,	PUNCT
cana-1482	278	6	24(1968	24(1968	NUM
cana-1482	278	7	)	)	PUNCT
cana-1482	278	8	,	,	PUNCT
cana-1482	278	9	182	182	NUM
cana-1482	278	10	-	-	SYM
cana-1482	278	11	190	190	NUM
cana-1482	278	12	.	.	PUNCT
cana-1482	279	1	[	[	X
cana-1482	279	2	5	5	NUM
cana-1482	279	3	]	]	PUNCT
cana-1482	279	4	dontchev.j	dontchev.j	NOUN
cana-1482	279	5	.	.	PROPN
cana-1482	279	6	,	,	PUNCT
cana-1482	279	7	on	on	ADP
cana-1482	279	8	generalizing	generalize	VERB
cana-1482	279	9	semipreopen	semipreopen	ADJ
cana-1482	279	10	sets	set	NOUN
cana-1482	279	11	,	,	PUNCT
cana-1482	279	12	mem	mem	PROPN
cana-1482	279	13	.	.	PUNCT
cana-1482	279	14	fac	fac	PROPN
cana-1482	279	15	.	.	PUNCT
cana-1482	279	16	sci	sci	PROPN
cana-1482	279	17	.	.	PROPN
cana-1482	279	18	kochi	kochi	PROPN
cana-1482	279	19	.	.	PUNCT
cana-1482	280	1	univ	univ	PROPN
cana-1482	280	2	.	.	PUNCT
cana-1482	280	3	ser	ser	PROPN
cana-1482	280	4	.	.	PUNCT
cana-1482	281	1	a	a	DET
cana-1482	281	2	,	,	PUNCT
cana-1482	281	3	math	math	NOUN
cana-1482	281	4	.	.	PUNCT
cana-1482	281	5	,16	,16	PUNCT
cana-1482	281	6	,	,	PUNCT
cana-1482	281	7	(	(	PUNCT
cana-1482	281	8	1995	1995	NUM
cana-1482	281	9	)	)	PUNCT
cana-1482	281	10	,	,	PUNCT
cana-1482	281	11	35	35	NUM
cana-1482	281	12	-	-	SYM
cana-1482	281	13	48	48	NUM
cana-1482	281	14	.	.	PUNCT
cana-1482	282	1	[	[	X
cana-1482	282	2	6	6	NUM
cana-1482	282	3	]	]	PUNCT
cana-1482	282	4	ganguly.s	ganguly.s	PROPN
cana-1482	282	5	and	and	CCONJ
cana-1482	282	6	saha.s	saha.s	PROPN
cana-1482	282	7	,	,	PUNCT
cana-1482	282	8	a	a	DET
cana-1482	282	9	note	note	NOUN
cana-1482	282	10	on	on	ADP
cana-1482	282	11	fuzzy	fuzzy	ADJ
cana-1482	282	12	semipreopen	semipreopen	ADJ
cana-1482	282	13	sets	set	NOUN
cana-1482	282	14	in	in	ADP
cana-1482	282	15	fuzzy	fuzzy	ADJ
cana-1482	282	16	topological	topological	ADJ
cana-1482	282	17	spaces	space	NOUN
cana-1482	282	18	,	,	PUNCT
cana-1482	282	19	fuzzy	fuzzy	ADJ
cana-1482	282	20	sets	set	NOUN
cana-1482	282	21	and	and	CCONJ
cana-1482	282	22	system	system	NOUN
cana-1482	282	23	,	,	PUNCT
cana-1482	282	24	18	18	NUM
cana-1482	282	25	,	,	PUNCT
cana-1482	282	26	(	(	PUNCT
cana-1482	282	27	1986	1986	NUM
cana-1482	282	28	)	)	PUNCT
cana-1482	282	29	,	,	PUNCT
cana-1482	282	30	83	83	NUM
cana-1482	282	31	-	-	SYM
cana-1482	282	32	96	96	NUM
cana-1482	282	33	.	.	PUNCT
cana-1482	283	1	communications	communication	NOUN
cana-1482	283	2	on	on	ADP
cana-1482	283	3	applied	apply	VERB
cana-1482	283	4	nonlinear	nonlinear	ADJ
cana-1482	283	5	analysis	analysis	NOUN
cana-1482	283	6	issn	issn	NOUN
cana-1482	283	7	:	:	PUNCT
cana-1482	283	8	1074	1074	NUM
cana-1482	283	9	-	-	PUNCT
cana-1482	283	10	133x	133x	NUM
cana-1482	283	11	vol	vol	NOUN
cana-1482	283	12	31	31	NUM
cana-1482	283	13	no	no	NOUN
cana-1482	283	14	.	.	PUNCT
cana-1482	284	1	8s	8s	PROPN
cana-1482	284	2	(	(	PUNCT
cana-1482	284	3	2024	2024	NUM
cana-1482	284	4	)	)	PUNCT
cana-1482	284	5	281	281	NUM
cana-1482	284	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1482	285	1	[	[	X
cana-1482	285	2	7	7	NUM
cana-1482	285	3	]	]	SYM
cana-1482	285	4	indira.r	indira.r	NUM
cana-1482	285	5	,	,	PUNCT
cana-1482	285	6	arjunan.k	arjunan.k	PUNCT
cana-1482	285	7	and	and	CCONJ
cana-1482	285	8	palaniappan.n	palaniappan.n	PROPN
cana-1482	285	9	,	,	PUNCT
cana-1482	285	10	notes	note	VERB
cana-1482	285	11	on	on	ADP
cana-1482	285	12	interval	interval	NOUN
cana-1482	285	13	valued	value	VERB
cana-1482	285	14	fuzzy	fuzzy	ADJ
cana-1482	285	15	rw	rw	NOUN
cana-1482	285	16	-	-	PUNCT
cana-1482	285	17	closed	closed	ADJ
cana-1482	285	18	,	,	PUNCT
cana-1482	285	19	interval	interval	NOUN
cana-1482	285	20	valued	value	VERB
cana-1482	285	21	fuzzy	fuzzy	ADJ
cana-1482	285	22	rw	rw	ADJ
cana-1482	285	23	-	-	PUNCT
cana-1482	285	24	open	open	ADJ
cana-1482	285	25	sets	set	NOUN
cana-1482	285	26	in	in	ADP
cana-1482	285	27	interval	interval	NOUN
cana-1482	285	28	valued	value	VERB
cana-1482	285	29	fuzzy	fuzzy	ADJ
cana-1482	285	30	topological	topological	ADJ
cana-1482	285	31	space	space	NOUN
cana-1482	285	32	,	,	PUNCT
cana-1482	285	33	international	international	ADJ
cana-1482	285	34	journal	journal	NOUN
cana-1482	285	35	of	of	ADP
cana-1482	285	36	computational	computational	ADJ
cana-1482	285	37	and	and	CCONJ
cana-1482	285	38	applied	applied	ADJ
cana-1482	285	39	mathematics	mathematic	NOUN
cana-1482	285	40	.	.	PUNCT
cana-1482	285	41	,vol	,vol	PUNCT
cana-1482	285	42	.3,no.1(2013	.3,no.1(2013	PROPN
cana-1482	285	43	)	)	PUNCT
cana-1482	285	44	,	,	PUNCT
cana-1482	285	45	23	23	NUM
cana-1482	285	46	-	-	SYM
cana-1482	285	47	38	38	NUM
cana-1482	286	1	[	[	SYM
cana-1482	286	2	8	8	NUM
cana-1482	286	3	]	]	SYM
cana-1482	286	4	levine.n	levine.n	PROPN
cana-1482	286	5	,	,	PUNCT
cana-1482	286	6	generalized	generalize	VERB
cana-1482	286	7	closed	closed	ADJ
cana-1482	286	8	sets	set	NOUN
cana-1482	286	9	in	in	ADP
cana-1482	286	10	topology	topology	NOUN
cana-1482	286	11	,	,	PUNCT
cana-1482	286	12	rend	rend	VERB
cana-1482	286	13	.	.	PUNCT
cana-1482	287	1	circ	circ	PROPN
cana-1482	287	2	.	.	PUNCT
cana-1482	288	1	math	math	NOUN
cana-1482	288	2	.	.	PUNCT
cana-1482	289	1	palermo,19,(1970),89	palermo,19,(1970),89	ADJ
cana-1482	289	2	-	-	PUNCT
cana-1482	289	3	96	96	NUM
cana-1482	289	4	.	.	PUNCT
cana-1482	290	1	[	[	X
cana-1482	290	2	9	9	NUM
cana-1482	290	3	]	]	PUNCT
cana-1482	290	4	mondal.t.k	mondal.t.k	NOUN
cana-1482	290	5	.	.	PUNCT
cana-1482	290	6	,	,	PUNCT
cana-1482	290	7	topology	topology	NOUN
cana-1482	290	8	of	of	ADP
cana-1482	290	9	interval	interval	NOUN
cana-1482	290	10	valued	value	VERB
cana-1482	290	11	fuzzy	fuzzy	ADJ
cana-1482	290	12	sets	set	NOUN
cana-1482	290	13	,	,	PUNCT
cana-1482	290	14	indian	indian	ADJ
cana-1482	290	15	j.	j.	PROPN
cana-1482	290	16	pure	pure	PROPN
cana-1482	290	17	appl.math.30	appl.math.30	PROPN
cana-1482	290	18	(	(	PUNCT
cana-1482	290	19	1999	1999	NUM
cana-1482	290	20	)	)	PUNCT
cana-1482	290	21	,	,	PUNCT
cana-1482	290	22	no.1	no.1	NUM
cana-1482	290	23	,	,	PUNCT
cana-1482	290	24	23	23	NUM
cana-1482	290	25	-	-	SYM
cana-1482	290	26	38	38	NUM
cana-1482	290	27	.	.	PUNCT
cana-1482	291	1	[	[	X
cana-1482	291	2	10	10	NUM
cana-1482	291	3	]	]	SYM
cana-1482	291	4	palaniyappan.n	palaniyappan.n	NOUN
cana-1482	291	5	and	and	CCONJ
cana-1482	291	6	rao	rao	PROPN
cana-1482	291	7	.k.c	.k.c	PROPN
cana-1482	291	8	.	.	PROPN
cana-1482	291	9	,	,	PUNCT
cana-1482	291	10	regular	regular	ADJ
cana-1482	291	11	generalized	generalize	VERB
cana-1482	291	12	closed	close	VERB
cana-1482	291	13	sets	set	NOUN
cana-1482	291	14	,	,	PUNCT
cana-1482	291	15	kyunpook	kyunpook	VERB
cana-1482	291	16	math	math	PROPN
cana-1482	291	17	.	.	PUNCT
cana-1482	292	1	jour	jour	PROPN
cana-1482	292	2	.	.	PROPN
cana-1482	292	3	,	,	PUNCT
cana-1482	292	4	33	33	NUM
cana-1482	292	5	,	,	PUNCT
cana-1482	292	6	(	(	PUNCT
cana-1482	292	7	1993	1993	NUM
cana-1482	292	8	)	)	PUNCT
cana-1482	292	9	,	,	PUNCT
cana-1482	292	10	211219	211219	NUM
cana-1482	292	11	.	.	PUNCT
cana-1482	293	1	[	[	X
cana-1482	293	2	11	11	NUM
cana-1482	293	3	]	]	PUNCT
cana-1482	293	4	a.a.salama	a.a.salama	NOUN
cana-1482	293	5	and	and	CCONJ
cana-1482	293	6	s.a.alblowi	s.a.alblowi	NOUN
cana-1482	293	7	,	,	PUNCT
cana-1482	293	8	neutrosophic	neutrosophic	ADJ
cana-1482	293	9	set	set	NOUN
cana-1482	293	10	and	and	CCONJ
cana-1482	293	11	neutrosophic	neutrosophic	ADJ
cana-1482	293	12	topological	topological	ADJ
cana-1482	293	13	space	space	NOUN
cana-1482	293	14	,	,	PUNCT
cana-1482	293	15	isor	isor	PROPN
cana-1482	293	16	j.	j.	PROPN
cana-1482	293	17	mathematics	mathematics	PROPN
cana-1482	293	18	,	,	PUNCT
cana-1482	293	19	vol.(3	vol.(3	NOUN
cana-1482	293	20	)	)	PUNCT
cana-1482	293	21	,	,	PUNCT
cana-1482	293	22	issue(4	issue(4	PROPN
cana-1482	293	23	)	)	PUNCT
cana-1482	293	24	,	,	PUNCT
cana-1482	293	25	(	(	PUNCT
cana-1482	293	26	2012	2012	NUM
cana-1482	293	27	)	)	PUNCT
cana-1482	293	28	.	.	PUNCT
cana-1482	294	1	pp-31	pp-31	NOUN
cana-1482	294	2	-	-	PUNCT
cana-1482	294	3	35	35	NUM
cana-1482	294	4	.	.	PUNCT
cana-1482	295	1	[	[	X
cana-1482	295	2	12	12	NUM
cana-1482	295	3	]	]	SYM
cana-1482	295	4	smarandache.f	smarandache.f	NOUN
cana-1482	295	5	,	,	PUNCT
cana-1482	295	6	“	"	PUNCT
cana-1482	295	7	neutrosophy	neutrosophy	NOUN
cana-1482	295	8	and	and	CCONJ
cana-1482	295	9	neutrosophic	neutrosophic	ADJ
cana-1482	295	10	logic	logic	NOUN
cana-1482	295	11	”	"	PUNCT
cana-1482	295	12	,	,	PUNCT
cana-1482	295	13	first	first	ADJ
cana-1482	295	14	international	international	ADJ
cana-1482	295	15	conference	conference	NOUN
cana-1482	295	16	on	on	ADP
cana-1482	295	17	neutrosophy	neutrosophy	NOUN
cana-1482	295	18	,	,	PUNCT
cana-1482	295	19	neutrosophic	neutrosophic	ADJ
cana-1482	295	20	logic	logic	NOUN
cana-1482	295	21	,	,	PUNCT
cana-1482	295	22	set	set	NOUN
cana-1482	295	23	,	,	PUNCT
cana-1482	295	24	probability	probability	NOUN
cana-1482	295	25	and	and	CCONJ
cana-1482	295	26	statistics	statistics	PROPN
cana-1482	295	27	university	university	PROPN
cana-1482	295	28	of	of	ADP
cana-1482	295	29	new	new	PROPN
cana-1482	295	30	mexico	mexico	PROPN
cana-1482	295	31	,	,	PUNCT
cana-1482	295	32	gallup	gallup	PROPN
cana-1482	295	33	,	,	PUNCT
cana-1482	295	34	nm	nm	PROPN
cana-1482	295	35	87301	87301	NUM
cana-1482	295	36	,	,	PUNCT
cana-1482	295	37	usa	usa	PROPN
cana-1482	295	38	(	(	PUNCT
cana-1482	295	39	2002	2002	NUM
cana-1482	295	40	)	)	PUNCT
cana-1482	295	41	.	.	PUNCT
cana-1482	296	1	[	[	X
cana-1482	296	2	13	13	NUM
cana-1482	296	3	]	]	SYM
cana-1482	296	4	smarandache.f	smarandache.f	NOUN
cana-1482	296	5	,	,	PUNCT
cana-1482	296	6	“	"	PUNCT
cana-1482	296	7	neutrosophic	neutrosophic	ADJ
cana-1482	296	8	set	set	NOUN
cana-1482	296	9	,	,	PUNCT
cana-1482	296	10	a	a	DET
cana-1482	296	11	generalisation	generalisation	NOUN
cana-1482	296	12	of	of	ADP
cana-1482	296	13	the	the	DET
cana-1482	296	14	intuitionistic	intuitionistic	ADJ
cana-1482	296	15	sets	set	NOUN
cana-1482	296	16	”	"	PUNCT
cana-1482	296	17	,	,	PUNCT
cana-1482	296	18	int	int	NOUN
cana-1482	296	19	.	.	PUNCT
cana-1482	297	1	j.	j.	PROPN
cana-1482	297	2	pure	pure	PROPN
cana-1482	297	3	appl.math	appl.math	PROPN
cana-1482	297	4	.	.	PROPN
cana-1482	297	5	24	24	NUM
cana-1482	297	6	(	(	PUNCT
cana-1482	297	7	2005	2005	NUM
cana-1482	297	8	)	)	PUNCT
cana-1482	297	9	287	287	NUM
cana-1482	297	10	-	-	SYM
cana-1482	297	11	297	297	NUM
cana-1482	297	12	.	.	PUNCT
cana-1482	298	1	[	[	X
cana-1482	298	2	14	14	NUM
cana-1482	298	3	]	]	X
cana-1482	298	4	saraf.r.k	saraf.r.k	NOUN
cana-1482	298	5	and	and	CCONJ
cana-1482	298	6	khanna.k	khanna.k	PROPN
cana-1482	298	7	.	.	PROPN
cana-1482	298	8	,	,	PUNCT
cana-1482	298	9	fuzzy	fuzzy	ADJ
cana-1482	298	10	generalized	generalize	VERB
cana-1482	298	11	semipreclosed	semipreclose	VERB
cana-1482	298	12	sets	set	NOUN
cana-1482	298	13	,	,	PUNCT
cana-1482	298	14	jour.tripura	jour.tripura	NOUN
cana-1482	298	15	.	.	PUNCT
cana-1482	299	1	math.soc	math.soc	NOUN
cana-1482	299	2	.	.	NOUN
cana-1482	299	3	,3	,3	PROPN
cana-1482	299	4	,	,	PUNCT
cana-1482	299	5	(	(	PUNCT
cana-1482	299	6	2001	2001	NUM
cana-1482	299	7	)	)	PUNCT
cana-1482	299	8	,	,	PUNCT
cana-1482	299	9	5968	5968	NUM
cana-1482	299	10	.	.	PUNCT
cana-1482	300	1	[	[	X
cana-1482	300	2	15	15	NUM
cana-1482	300	3	]	]	PUNCT
cana-1482	300	4	zadeh.l.a	zadeh.l.a	NOUN
cana-1482	300	5	.	.	PUNCT
cana-1482	300	6	,	,	PUNCT
cana-1482	300	7	fuzzy	fuzzy	ADJ
cana-1482	300	8	sets	set	NOUN
cana-1482	300	9	,	,	PUNCT
cana-1482	300	10	information	information	NOUN
cana-1482	300	11	and	and	CCONJ
cana-1482	300	12	control	control	NOUN
cana-1482	300	13	,	,	PUNCT
cana-1482	300	14	vol.8	vol.8	PROPN
cana-1482	300	15	(	(	PUNCT
cana-1482	300	16	1965	1965	NUM
cana-1482	300	17	)	)	PUNCT
cana-1482	300	18	,	,	PUNCT
cana-1482	300	19	338	338	NUM
cana-1482	300	20	-	-	SYM
cana-1482	300	21	353	353	NUM
cana-1482	300	22	.	.	PUNCT