id	sid	tid	token	lemma	pos
cana-807	1	1	communications	communication	NOUN
cana-807	1	2	on	on	ADP
cana-807	1	3	applied	apply	VERB
cana-807	1	4	nonlinear	nonlinear	ADJ
cana-807	1	5	analysis	analysis	NOUN
cana-807	1	6	issn	issn	NOUN
cana-807	1	7	:	:	PUNCT
cana-807	1	8	1074	1074	NUM
cana-807	1	9	-	-	PUNCT
cana-807	1	10	133x	133x	NUM
cana-807	1	11	vol	vol	NOUN
cana-807	1	12	31	31	NUM
cana-807	1	13	no	no	NOUN
cana-807	1	14	.	.	PUNCT
cana-807	2	1	3s	3s	NUM
cana-807	2	2	(	(	PUNCT
cana-807	2	3	2024	2024	NUM
cana-807	2	4	)	)	PUNCT
cana-807	2	5	547	547	NUM
cana-807	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	2	7	neutrosophic	neutrosophic	ADJ
cana-807	2	8	pre	pre	VERB
cana-807	2	9	generalized	generalized	ADJ
cana-807	2	10	pre	pre	VERB
cana-807	2	11	regular	regular	ADJ
cana-807	2	12	star	star	NOUN
cana-807	2	13	weakly	weakly	ADJ
cana-807	2	14	closed	closed	ADJ
cana-807	2	15	sets	set	NOUN
cana-807	2	16	p.	p.	PROPN
cana-807	2	17	devi	devi	PROPN
cana-807	2	18	prabha1	prabha1	PROPN
cana-807	2	19	,	,	PUNCT
cana-807	2	20	r.	r.	PROPN
cana-807	2	21	asokan2	asokan2	PROPN
cana-807	3	1	1&2	1&2	PROPN
cana-807	3	2	department	department	PROPN
cana-807	3	3	of	of	ADP
cana-807	3	4	mathematics	mathematics	PROPN
cana-807	3	5	school	school	NOUN
cana-807	3	6	of	of	ADP
cana-807	3	7	mathematics	mathematics	PROPN
cana-807	3	8	madurai	madurai	PROPN
cana-807	3	9	kamaraj	kamaraj	ADJ
cana-807	3	10	university	university	NOUN
cana-807	3	11	madurai-625021	madurai-625021	NOUN
cana-807	3	12	,	,	PUNCT
cana-807	3	13	tamil	tamil	PROPN
cana-807	3	14	nadu	nadu	PROPN
cana-807	3	15	,	,	PUNCT
cana-807	3	16	india	india	PROPN
cana-807	3	17	email	email	NOUN
cana-807	3	18	:	:	PUNCT
cana-807	3	19	1deviprabhaponnusamy@gmail.com	1deviprabhaponnusamy@gmail.com	NUM
cana-807	3	20	,	,	PUNCT
cana-807	3	21	2	2	NUM
cana-807	3	22	asokan.maths@mkuniversity.org	asokan.maths@mkuniversity.org	NOUN
cana-807	3	23	article	article	NOUN
cana-807	3	24	history	history	NOUN
cana-807	3	25	:	:	PUNCT
cana-807	3	26	received	receive	VERB
cana-807	3	27	:	:	PUNCT
cana-807	3	28	10	10	NUM
cana-807	3	29	-	-	PUNCT
cana-807	3	30	04	04	NUM
cana-807	3	31	-	-	PUNCT
cana-807	3	32	2024	2024	NUM
cana-807	3	33	revised	revise	VERB
cana-807	3	34	:	:	PUNCT
cana-807	3	35	29	29	NUM
cana-807	3	36	-	-	SYM
cana-807	3	37	05	05	NUM
cana-807	3	38	-	-	PUNCT
cana-807	3	39	2024	2024	NUM
cana-807	3	40	accepted	accept	VERB
cana-807	3	41	:	:	PUNCT
cana-807	3	42	14	14	NUM
cana-807	3	43	-	-	SYM
cana-807	3	44	06	06	NUM
cana-807	3	45	-	-	PUNCT
cana-807	3	46	2024	2024	NUM
cana-807	3	47	abstract	abstract	NOUN
cana-807	3	48	the	the	DET
cana-807	3	49	objective	objective	NOUN
cana-807	3	50	of	of	ADP
cana-807	3	51	this	this	DET
cana-807	3	52	paper	paper	NOUN
cana-807	3	53	is	be	AUX
cana-807	3	54	to	to	PART
cana-807	3	55	emphasis	emphasis	VERB
cana-807	3	56	the	the	DET
cana-807	3	57	generalization	generalization	NOUN
cana-807	3	58	of	of	ADP
cana-807	3	59	pre	pre	ADJ
cana-807	3	60	generalized	generalized	ADJ
cana-807	3	61	pre	pre	NOUN
cana-807	3	62	regular	regular	ADJ
cana-807	3	63	weakly	weakly	ADJ
cana-807	3	64	closed	closed	ADJ
cana-807	3	65	sets	set	NOUN
cana-807	3	66	in	in	ADP
cana-807	3	67	neutrosophic	neutrosophic	ADJ
cana-807	3	68	environment	environment	NOUN
cana-807	4	1	namely	namely	ADV
cana-807	4	2	neutrosophic	neutrosophic	ADJ
cana-807	4	3	pre	pre	VERB
cana-807	4	4	generalized	generalized	ADJ
cana-807	4	5	pre	pre	VERB
cana-807	4	6	regular	regular	ADJ
cana-807	4	7	star	star	NOUN
cana-807	4	8	weakly	weakly	ADJ
cana-807	4	9	closed	closed	ADJ
cana-807	4	10	sets	set	NOUN
cana-807	4	11	.	.	PUNCT
cana-807	5	1	its	its	PRON
cana-807	5	2	properties	property	NOUN
cana-807	5	3	and	and	CCONJ
cana-807	5	4	characterizations	characterization	NOUN
cana-807	5	5	are	be	AUX
cana-807	5	6	defined	define	VERB
cana-807	5	7	and	and	CCONJ
cana-807	5	8	its	its	PRON
cana-807	5	9	relationships	relationship	NOUN
cana-807	5	10	with	with	ADP
cana-807	5	11	other	other	ADJ
cana-807	5	12	neutrosophic	neutrosophic	ADJ
cana-807	5	13	sets	set	NOUN
cana-807	5	14	are	be	AUX
cana-807	5	15	studied	study	VERB
cana-807	5	16	with	with	ADP
cana-807	5	17	suitable	suitable	ADJ
cana-807	5	18	examples	example	NOUN
cana-807	5	19	.	.	PUNCT
cana-807	6	1	keywords	keyword	NOUN
cana-807	6	2	:	:	PUNCT
cana-807	6	3	n	n	CCONJ
cana-807	6	4	-	-	PUNCT
cana-807	6	5	pgpr*wc(x	pgpr*wc(x	NOUN
cana-807	6	6	)	)	PUNCT
cana-807	6	7	,	,	PUNCT
cana-807	6	8	n	n	CCONJ
cana-807	6	9	-	-	PUNCT
cana-807	6	10	r*gαc(x	r*gαc(x	NOUN
cana-807	6	11	)	)	PUNCT
cana-807	6	12	,	,	PUNCT
cana-807	6	13	n	n	CCONJ
cana-807	6	14	-	-	PUNCT
cana-807	6	15	pgpr*gwo(x	pgpr*gwo(x	NOUN
cana-807	6	16	)	)	PUNCT
cana-807	6	17	,	,	PUNCT
cana-807	6	18	n	n	CCONJ
cana-807	6	19	-	-	PUNCT
cana-807	6	20	r*gαo(x	r*gαo(x	PROPN
cana-807	6	21	)	)	PUNCT
cana-807	6	22	.	.	PUNCT
cana-807	7	1	1	1	X
cana-807	7	2	.	.	X
cana-807	7	3	introduction	introduction	NOUN
cana-807	7	4	:	:	PUNCT
cana-807	7	5	.	.	PUNCT
cana-807	8	1	in	in	ADP
cana-807	8	2	recent	recent	ADJ
cana-807	8	3	times	time	NOUN
cana-807	8	4	many	many	ADJ
cana-807	8	5	researchers	researcher	NOUN
cana-807	8	6	undergoing	undergo	VERB
cana-807	8	7	their	their	PRON
cana-807	8	8	research	research	NOUN
cana-807	8	9	in	in	ADP
cana-807	8	10	the	the	DET
cana-807	8	11	area.of	area.of	PROPN
cana-807	8	12	g	g	NOUN
cana-807	8	13	clossd	clossd	NOUN
cana-807	8	14	sets	set	NOUN
cana-807	8	15	in	in	ADP
cana-807	8	16	1965	1965	NUM
cana-807	8	17	l.zadeh	l.zadeh	PUNCT
cana-807	9	1	[	[	X
cana-807	9	2	10	10	NUM
cana-807	9	3	]	]	PUNCT
cana-807	9	4	introduced	introduce	VERB
cana-807	9	5	the	the	DET
cana-807	9	6	concept	concept	NOUN
cana-807	9	7	of	of	ADP
cana-807	9	8	fuzzy	fuzzy	ADJ
cana-807	9	9	sets	set	NOUN
cana-807	9	10	which	which	PRON
cana-807	9	11	deals	deal	VERB
cana-807	9	12	with	with	ADP
cana-807	9	13	membership	membership	NOUN
cana-807	9	14	of	of	ADP
cana-807	9	15	a	a	DET
cana-807	9	16	set	set	NOUN
cana-807	9	17	.	.	PUNCT
cana-807	10	1	k.atanassova	k.atanassova	X
cana-807	11	1	[	[	X
cana-807	11	2	1	1	X
cana-807	11	3	]	]	PUNCT
cana-807	11	4	introduced	introduce	VERB
cana-807	11	5	intuitionistic	intuitionistic	ADJ
cana-807	11	6	fuzzy	fuzzy	ADJ
cana-807	11	7	sets	set	NOUN
cana-807	11	8	with	with	ADP
cana-807	11	9	membership	membership	NOUN
cana-807	11	10	and	and	CCONJ
cana-807	11	11	nonmembership	nonmembership	NOUN
cana-807	11	12	function	function	NOUN
cana-807	11	13	.	.	PUNCT
cana-807	12	1	f.	f.	PROPN
cana-807	12	2	smarandache	smarandache	PROPN
cana-807	13	1	[	[	X
cana-807	13	2	2	2	NUM
cana-807	13	3	]	]	PUNCT
cana-807	13	4	developed	develop	VERB
cana-807	13	5	his	his	PRON
cana-807	13	6	new	new	ADJ
cana-807	13	7	concept	concept	NOUN
cana-807	13	8	namely	namely	ADV
cana-807	13	9	neutrosophic	neutrosophic	ADJ
cana-807	13	10	set	set	NOUN
cana-807	13	11	that	that	SCONJ
cana-807	13	12	studies	study	NOUN
cana-807	13	13	membership	membership	NOUN
cana-807	13	14	,	,	PUNCT
cana-807	13	15	nonmembership	nonmembership	NOUN
cana-807	13	16	and	and	CCONJ
cana-807	13	17	indeterminacy	indeterminacy	NOUN
cana-807	13	18	.	.	PUNCT
cana-807	14	1	later	later	ADV
cana-807	14	2	on	on	ADP
cana-807	14	3	a.a.salama	a.a.salama	NOUN
cana-807	14	4	and	and	CCONJ
cana-807	14	5	s.a.alblow	s.a.alblow	VERB
cana-807	14	6	[	[	X
cana-807	14	7	6	6	NUM
cana-807	14	8	]	]	PUNCT
cana-807	14	9	introduced	introduce	VERB
cana-807	14	10	neutrosophic	neutrosophic	ADJ
cana-807	14	11	topological	topological	ADJ
cana-807	14	12	spaces	space	NOUN
cana-807	14	13	by	by	ADP
cana-807	14	14	using	use	VERB
cana-807	14	15	the	the	DET
cana-807	14	16	neutrosophic	neutrosophic	ADJ
cana-807	14	17	sets	set	NOUN
cana-807	14	18	.	.	PUNCT
cana-807	15	1	in	in	ADP
cana-807	15	2	the	the	DET
cana-807	15	3	year	year	NOUN
cana-807	15	4	2015	2015	NUM
cana-807	15	5	r.s.wali	r.s.wali	ADJ
cana-807	15	6	and	and	CCONJ
cana-807	15	7	vivekananda	vivekananda	NOUN
cana-807	15	8	dembre	dembre	VERB
cana-807	15	9	[	[	X
cana-807	15	10	9	9	NUM
cana-807	15	11	]	]	PUNCT
cana-807	15	12	introduced	introduce	VERB
cana-807	15	13	and	and	CCONJ
cana-807	15	14	studied	study	VERB
cana-807	15	15	pre	pre	ADJ
cana-807	15	16	generalized	generalized	ADJ
cana-807	15	17	pre	pre	NOUN
cana-807	15	18	regular	regular	ADJ
cana-807	15	19	weekly	weekly	ADJ
cana-807	15	20	closed	close	VERB
cana-807	15	21	and	and	CCONJ
cana-807	15	22	open	open	ADJ
cana-807	15	23	sets	set	NOUN
cana-807	15	24	respectively	respectively	ADV
cana-807	15	25	.	.	PUNCT
cana-807	16	1	in	in	ADP
cana-807	16	2	this	this	DET
cana-807	16	3	paper	paper	NOUN
cana-807	16	4	we	we	PRON
cana-807	16	5	apply	apply	VERB
cana-807	16	6	the	the	DET
cana-807	16	7	concept	concept	NOUN
cana-807	16	8	and	and	CCONJ
cana-807	16	9	properties	property	NOUN
cana-807	16	10	of	of	ADP
cana-807	16	11	neutrosophic	neutrosophic	ADJ
cana-807	16	12	sets	set	NOUN
cana-807	16	13	to	to	PART
cana-807	16	14	develop	develop	VERB
cana-807	16	15	a	a	DET
cana-807	16	16	new	new	ADJ
cana-807	16	17	class	class	NOUN
cana-807	16	18	of	of	ADP
cana-807	16	19	set	set	NOUN
cana-807	16	20	called	call	VERB
cana-807	16	21	neutrosophic	neutrosophic	ADJ
cana-807	16	22	pre	pre	ADJ
cana-807	16	23	-	-	ADJ
cana-807	16	24	generalized	generalized	ADJ
cana-807	16	25	pre	pre	ADJ
cana-807	16	26	regular	regular	ADJ
cana-807	16	27	star	star	NOUN
cana-807	16	28	weakly	weakly	ADV
cana-807	16	29	closed	close	VERB
cana-807	16	30	set	set	VERB
cana-807	16	31	in	in	ADP
cana-807	16	32	neutrosophic	neutrosophic	ADJ
cana-807	16	33	topological	topological	ADJ
cana-807	16	34	spaces	space	NOUN
cana-807	16	35	.	.	PUNCT
cana-807	17	1	in	in	ADP
cana-807	17	2	this	this	DET
cana-807	17	3	case	case	NOUN
cana-807	17	4	the	the	DET
cana-807	17	5	pair	pair	NOUN
cana-807	17	6	(	(	PUNCT
cana-807	17	7	x	x	NOUN
cana-807	17	8	,	,	PUNCT
cana-807	17	9			PROPN
cana-807	17	10	)	)	PUNCT
cana-807	17	11	is	be	AUX
cana-807	17	12	a	a	DET
cana-807	17	13	neutrosophic	neutrosophic	ADJ
cana-807	17	14	topological	topological	ADJ
cana-807	17	15	space	space	NOUN
cana-807	17	16	and	and	CCONJ
cana-807	17	17	any	any	DET
cana-807	17	18	neutrosophic	neutrosophic	ADJ
cana-807	17	19	set	set	NOUN
cana-807	17	20	in	in	ADP
cana-807	17	21	x	x	PROPN
cana-807	17	22	is	be	AUX
cana-807	17	23	known	know	VERB
cana-807	17	24	as	as	ADP
cana-807	17	25	a	a	DET
cana-807	17	26	neutrosopic	neutrosopic	ADJ
cana-807	17	27	open	open	ADJ
cana-807	17	28	set	set	NOUN
cana-807	17	29	(	(	PUNCT
cana-807	17	30	n	n	CCONJ
cana-807	17	31	-	-	PUNCT
cana-807	17	32	os	os	NOUN
cana-807	17	33	)	)	PUNCT
cana-807	17	34	in	in	ADP
cana-807	17	35	x.	x.	PROPN
cana-807	18	1	a	a	DET
cana-807	18	2	neutrosophic	neutrosophic	ADJ
cana-807	18	3	set	set	NOUN
cana-807	18	4	s	s	VERB
cana-807	18	5	is	be	AUX
cana-807	18	6	a	a	DET
cana-807	18	7	neutrosophic	neutrosophic	ADJ
cana-807	18	8	closed	close	VERB
cana-807	18	9	set	set	NOUN
cana-807	18	10	(	(	PUNCT
cana-807	18	11	n	n	CCONJ
cana-807	18	12	-	-	PUNCT
cana-807	18	13	cs	cs	ADJ
cana-807	18	14	)	)	PUNCT
cana-807	18	15	if	if	SCONJ
cana-807	18	16	and	and	CCONJ
cana-807	18	17	only	only	ADV
cana-807	18	18	if	if	SCONJ
cana-807	18	19	c(s	c(	NOUN
cana-807	18	20	)	)	PUNCT
cana-807	18	21	is	be	AUX
cana-807	18	22	a	a	DET
cana-807	18	23	neutrosophic	neutrosophic	ADJ
cana-807	18	24	open	open	ADJ
cana-807	18	25	set	set	VERB
cana-807	18	26	in	in	ADP
cana-807	18	27	x.	x.	NOUN
cana-807	18	28	here	here	ADV
cana-807	18	29	the	the	DET
cana-807	18	30	empty	empty	ADJ
cana-807	18	31	set	set	NOUN
cana-807	18	32	(	(	PUNCT
cana-807	18	33	on	on	ADP
cana-807	18	34	)	)	PUNCT
cana-807	18	35	and	and	CCONJ
cana-807	18	36	the	the	DET
cana-807	18	37	whole	whole	ADJ
cana-807	18	38	set	set	NOUN
cana-807	18	39	(	(	PUNCT
cana-807	18	40	in	in	ADP
cana-807	18	41	)	)	PUNCT
cana-807	18	42	may	may	AUX
cana-807	18	43	be	be	AUX
cana-807	18	44	defined	define	VERB
cana-807	18	45	as	as	SCONJ
cana-807	18	46	follows	follow	VERB
cana-807	18	47	we	we	PRON
cana-807	18	48	go	go	VERB
cana-807	18	49	through	through	ADP
cana-807	18	50	some	some	DET
cana-807	18	51	basic	basic	ADJ
cana-807	18	52	definitions	definition	NOUN
cana-807	18	53	in	in	ADP
cana-807	18	54	this	this	DET
cana-807	18	55	section	section	NOUN
cana-807	18	56	.	.	PUNCT
cana-807	19	1	definition	definition	NOUN
cana-807	19	2	2.1	2.1	NUM
cana-807	19	3	:	:	PUNCT
cana-807	20	1	[	[	X
cana-807	20	2	2	2	X
cana-807	20	3	]	]	PUNCT
cana-807	20	4	let	let	VERB
cana-807	20	5	x	x	PRON
cana-807	20	6	be	be	AUX
cana-807	20	7	a	a	DET
cana-807	20	8	non	non	ADJ
cana-807	20	9	-	-	ADJ
cana-807	20	10	empty	empty	ADJ
cana-807	20	11	fixed	fix	VERB
cana-807	20	12	set	set	NOUN
cana-807	20	13	.	.	PUNCT
cana-807	21	1	a	a	DET
cana-807	21	2	neutrosophic	neutrosophic	ADJ
cana-807	21	3	set	set	NOUN
cana-807	21	4	(	(	PUNCT
cana-807	21	5	n	n	CCONJ
cana-807	21	6	-	-	PUNCT
cana-807	21	7	s	s	NOUN
cana-807	21	8	)	)	PUNCT
cana-807	21	9	a	a	PRON
cana-807	21	10	is	be	AUX
cana-807	21	11	an	an	DET
cana-807	21	12	object	object	NOUN
cana-807	21	13	having	have	VERB
cana-807	21	14	the	the	DET
cana-807	21	15	form	form	NOUN
cana-807	21	16	a	a	PRON
cana-807	21	17	=	=	SYM
cana-807	21	18	{	{	PUNCT
cana-807	21	19	x	x	PROPN
cana-807	21	20	,	,	PUNCT
cana-807	21	21	a(x	a(x	NOUN
cana-807	21	22	)	)	PUNCT
cana-807	21	23	,	,	PUNCT
cana-807	21	24	a(x	a(x	PROPN
cana-807	21	25	)	)	PUNCT
cana-807	21	26	,	,	PUNCT
cana-807	21	27	va(x)	va(x)	NOUN
cana-807	21	28	:	:	PUNCT
cana-807	21	29	x	x	SYM
cana-807	21	30			NOUN
cana-807	21	31	x	x	X
cana-807	21	32	}	}	PUNCT
cana-807	21	33	where	where	SCONJ
cana-807	21	34	a(x	a(x	NOUN
cana-807	21	35	)	)	PUNCT
cana-807	21	36	,	,	PUNCT
cana-807	21	37	a(x	a(x	PROPN
cana-807	21	38	)	)	PUNCT
cana-807	21	39	,	,	PUNCT
cana-807	21	40	va(x	va(x	NOUN
cana-807	21	41	)	)	PUNCT
cana-807	21	42	represent	represent	VERB
cana-807	21	43	the	the	DET
cana-807	21	44	degree	degree	NOUN
cana-807	21	45	of	of	ADP
cana-807	21	46	membership	membership	NOUN
cana-807	21	47	,	,	PUNCT
cana-807	21	48	degree	degree	NOUN
cana-807	21	49	of	of	ADP
cana-807	21	50	indeterminacy	indeterminacy	NOUN
cana-807	21	51	and	and	CCONJ
cana-807	21	52	the	the	DET
cana-807	21	53	degree	degree	NOUN
cana-807	21	54	communications	communication	NOUN
cana-807	21	55	on	on	ADP
cana-807	21	56	applied	apply	VERB
cana-807	21	57	nonlinear	nonlinear	ADJ
cana-807	21	58	analysis	analysis	NOUN
cana-807	21	59	issn	issn	NOUN
cana-807	21	60	:	:	PUNCT
cana-807	21	61	1074	1074	NUM
cana-807	21	62	-	-	PUNCT
cana-807	21	63	133x	133x	NUM
cana-807	21	64	vol	vol	NOUN
cana-807	21	65	31	31	NUM
cana-807	21	66	no	no	NOUN
cana-807	21	67	.	.	PUNCT
cana-807	22	1	3s	3s	NUM
cana-807	22	2	(	(	PUNCT
cana-807	22	3	2024	2024	NUM
cana-807	22	4	)	)	PUNCT
cana-807	22	5	548	548	NUM
cana-807	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	22	7	of	of	ADP
cana-807	22	8	non	non	ADJ
cana-807	22	9	-	-	NOUN
cana-807	22	10	membership	membership	NOUN
cana-807	22	11	respectively	respectively	ADV
cana-807	22	12	of	of	ADP
cana-807	22	13	each	each	DET
cana-807	22	14	element	element	NOUN
cana-807	22	15	x	x	X
cana-807	22	16			NOUN
cana-807	22	17	x	x	X
cana-807	22	18	to	to	ADP
cana-807	22	19	the	the	DET
cana-807	22	20	set	set	NOUN
cana-807	22	21	a.	a.	NOUN
cana-807	22	22	a	a	DET
cana-807	22	23	neutrosophic	neutrosophic	ADJ
cana-807	22	24	set	set	VERB
cana-807	22	25	a	a	DET
cana-807	22	26	=	=	SYM
cana-807	22	27	{	{	PUNCT
cana-807	22	28	x	x	PROPN
cana-807	22	29	,	,	PUNCT
cana-807	22	30	a(x	a(x	NOUN
cana-807	22	31	)	)	PUNCT
cana-807	22	32	,	,	PUNCT
cana-807	22	33	a(x	a(x	PROPN
cana-807	22	34	)	)	PUNCT
cana-807	22	35	,	,	PUNCT
cana-807	22	36	va(x)	va(x)	NOUN
cana-807	22	37	:	:	PUNCT
cana-807	22	38	x	x	SYM
cana-807	22	39			NOUN
cana-807	22	40	x	x	VERB
cana-807	22	41	}	}	PUNCT
cana-807	22	42	can	can	AUX
cana-807	22	43	be	be	AUX
cana-807	22	44	identified	identify	VERB
cana-807	22	45	as	as	ADP
cana-807	22	46	an	an	DET
cana-807	22	47	ordered	ordered	ADJ
cana-807	22	48	triple	triple	ADJ
cana-807	22	49	a(x	a(x	NOUN
cana-807	22	50	)	)	PUNCT
cana-807	22	51	,	,	PUNCT
cana-807	22	52	a(x	a(x	PROPN
cana-807	22	53	)	)	PUNCT
cana-807	22	54	,	,	PUNCT
cana-807	22	55	va(x)	va(x)	NOUN
cana-807	22	56	in	in	ADP
cana-807	22	57	]	]	X
cana-807	22	58	0	0	NUM
cana-807	22	59	,	,	PUNCT
cana-807	22	60	1	1	NUM
cana-807	22	61	+	+	ADP
cana-807	22	62	[	[	PUNCT
cana-807	22	63	on	on	ADP
cana-807	22	64	x.	x.	NOUN
cana-807	22	65	definition	definition	NOUN
cana-807	22	66	2.2	2.2	NUM
cana-807	22	67	:	:	PUNCT
cana-807	23	1	[	[	X
cana-807	23	2	6	6	NUM
cana-807	23	3	]	]	PUNCT
cana-807	23	4	let	let	VERB
cana-807	23	5	a	a	DET
cana-807	23	6	=	=	SYM
cana-807	23	7	{	{	PUNCT
cana-807	23	8	x	x	PROPN
cana-807	23	9	,	,	PUNCT
cana-807	23	10	a(x	a(x	NOUN
cana-807	23	11	)	)	PUNCT
cana-807	23	12	,	,	PUNCT
cana-807	23	13	a(x	a(x	PROPN
cana-807	23	14	)	)	PUNCT
cana-807	23	15	,	,	PUNCT
cana-807	23	16	va(x)	va(x)	NOUN
cana-807	23	17	:	:	PUNCT
cana-807	23	18	x	x	SYM
cana-807	23	19			NOUN
cana-807	23	20	x	x	PART
cana-807	23	21	}	}	PUNCT
cana-807	23	22	be	be	AUX
cana-807	23	23	a	a	DET
cana-807	23	24	ns	ns	NOUN
cana-807	23	25	on	on	ADP
cana-807	23	26	x	x	NOUN
cana-807	23	27	,	,	PUNCT
cana-807	23	28	then	then	ADV
cana-807	23	29	the	the	DET
cana-807	23	30	complement	complement	NOUN
cana-807	23	31	c(a	c(a	ADV
cana-807	23	32	)	)	PUNCT
cana-807	23	33	may	may	AUX
cana-807	23	34	be	be	AUX
cana-807	23	35	defined	define	VERB
cana-807	23	36	as	as	ADP
cana-807	23	37	1	1	NUM
cana-807	23	38	.	.	PUNCT
cana-807	24	1	c(a	c(a	ADV
cana-807	24	2	)	)	PUNCT
cana-807	25	1	=	=	PRON
cana-807	25	2	{	{	PUNCT
cana-807	25	3	x	x	NOUN
cana-807	25	4	,	,	PUNCT
cana-807	25	5	1-a(x	1-a(x	NUM
cana-807	25	6	)	)	PUNCT
cana-807	25	7	,	,	PUNCT
cana-807	25	8	1	1	NUM
cana-807	25	9	-	-	PUNCT
cana-807	25	10	va(x)	va(x)	NOUN
cana-807	25	11	:	:	PUNCT
cana-807	25	12	x	x	SYM
cana-807	25	13			NOUN
cana-807	25	14	x	x	SYM
cana-807	25	15	}	}	PUNCT
cana-807	25	16	2	2	NUM
cana-807	25	17	.	.	PUNCT
cana-807	25	18	c(a	c(a	ADV
cana-807	25	19	)	)	PUNCT
cana-807	25	20	=	=	PRON
cana-807	25	21	{	{	PUNCT
cana-807	25	22	x	x	PROPN
cana-807	25	23	,	,	PUNCT
cana-807	25	24	a(x	a(x	NOUN
cana-807	25	25	)	)	PUNCT
cana-807	25	26	,	,	PUNCT
cana-807	25	27	a(x	a(x	PROPN
cana-807	25	28	)	)	PUNCT
cana-807	25	29	,	,	PUNCT
cana-807	25	30	va(x)	va(x)	NOUN
cana-807	25	31	:	:	PUNCT
cana-807	25	32	x	x	SYM
cana-807	25	33			NOUN
cana-807	25	34	x	x	SYM
cana-807	25	35	}	}	PUNCT
cana-807	25	36	3	3	NUM
cana-807	25	37	.	.	PUNCT
cana-807	25	38	c(a	c(a	ADV
cana-807	25	39	)	)	PUNCT
cana-807	25	40	=	=	PRON
cana-807	25	41	{	{	PUNCT
cana-807	25	42	x	x	PROPN
cana-807	25	43	,	,	PUNCT
cana-807	25	44	va(x	va(x	NOUN
cana-807	25	45	)	)	PUNCT
cana-807	25	46	,	,	PUNCT
cana-807	25	47	1-a(x	1-a(x	NUM
cana-807	25	48	)	)	PUNCT
cana-807	25	49	,	,	PUNCT
cana-807	25	50	a(x)	a(x)	PUNCT
cana-807	25	51	:	:	PUNCT
cana-807	25	52	x	x	SYM
cana-807	25	53			NOUN
cana-807	25	54	x	x	NUM
cana-807	25	55	}	}	PUNCT
cana-807	25	56	note	note	VERB
cana-807	25	57	that	that	SCONJ
cana-807	25	58	for	for	ADP
cana-807	25	59	any	any	DET
cana-807	25	60	two	two	NUM
cana-807	25	61	neutrosophic	neutrosophic	ADJ
cana-807	25	62	sets	set	NOUN
cana-807	25	63	a	a	DET
cana-807	25	64	and	and	CCONJ
cana-807	25	65	b	b	NOUN
cana-807	25	66	,	,	PUNCT
cana-807	25	67	4	4	NUM
cana-807	25	68	.	.	X
cana-807	25	69	c(a	c(a	PROPN
cana-807	25	70			NOUN
cana-807	25	71	b	b	NOUN
cana-807	25	72	)	)	PUNCT
cana-807	25	73	=	=	SYM
cana-807	25	74	c(a	c(a	NOUN
cana-807	25	75	)	)	PUNCT
cana-807	25	76			NOUN
cana-807	25	77	c(b	c(b	NOUN
cana-807	25	78	)	)	PUNCT
cana-807	25	79	5	5	NUM
cana-807	25	80	.	.	PUNCT
cana-807	25	81	c(a	c(a	PROPN
cana-807	25	82			NOUN
cana-807	25	83	b	b	NOUN
cana-807	25	84	)	)	PUNCT
cana-807	25	85	=	=	SYM
cana-807	25	86	c(a	c(a	PROPN
cana-807	25	87	)	)	PUNCT
cana-807	25	88			NOUN
cana-807	25	89	c(b	c(b	PROPN
cana-807	25	90	)	)	PUNCT
cana-807	25	91	definition	definition	NOUN
cana-807	25	92	2.3	2.3	NUM
cana-807	25	93	:	:	PUNCT
cana-807	26	1	[	[	X
cana-807	26	2	6	6	NUM
cana-807	26	3	]	]	PUNCT
cana-807	26	4	for	for	ADP
cana-807	26	5	any	any	DET
cana-807	26	6	two	two	NUM
cana-807	26	7	neutrosophic	neutrosophic	ADJ
cana-807	26	8	sets	set	VERB
cana-807	26	9	a	a	DET
cana-807	26	10	=	=	SYM
cana-807	26	11	{	{	PUNCT
cana-807	26	12	x	x	PROPN
cana-807	26	13	,	,	PUNCT
cana-807	26	14	a(x	a(x	NOUN
cana-807	26	15	)	)	PUNCT
cana-807	26	16	,	,	PUNCT
cana-807	26	17	a(x	a(x	PROPN
cana-807	26	18	)	)	PUNCT
cana-807	26	19	,	,	PUNCT
cana-807	26	20	va(x)	va(x)	NOUN
cana-807	26	21	:	:	PUNCT
cana-807	26	22	x	x	SYM
cana-807	26	23			NOUN
cana-807	26	24	x	x	X
cana-807	26	25	}	}	PUNCT
cana-807	26	26	and	and	CCONJ
cana-807	26	27	b	b	X
cana-807	26	28	=	=	SYM
cana-807	26	29	{	{	PUNCT
cana-807	26	30	x	x	PROPN
cana-807	26	31	,	,	PUNCT
cana-807	26	32	b(x	b(x	NOUN
cana-807	26	33	)	)	PUNCT
cana-807	26	34	,	,	PUNCT
cana-807	26	35	b(x	b(x	ADJ
cana-807	26	36	)	)	PUNCT
cana-807	26	37	,	,	PUNCT
cana-807	26	38	vb(x)	vb(x)	NOUN
cana-807	26	39	:	:	PUNCT
cana-807	26	40	x	x	SYM
cana-807	26	41			NOUN
cana-807	26	42	x	x	X
cana-807	26	43	}	}	PUNCT
cana-807	26	44	we	we	PRON
cana-807	26	45	may	may	AUX
cana-807	26	46	have	have	VERB
cana-807	26	47	.	.	PUNCT
cana-807	27	1	1	1	X
cana-807	27	2	.	.	X
cana-807	27	3	a	a	DET
cana-807	27	4			PROPN
cana-807	27	5	b	b	PROPN
cana-807	27	6			ADP
cana-807	27	7	a	a	DET
cana-807	27	8	a(x	a(x	NOUN
cana-807	27	9	)	)	PUNCT
cana-807	27	10			NOUN
cana-807	27	11	b(x	b(x	NOUN
cana-807	27	12	)	)	PUNCT
cana-807	27	13	,	,	PUNCT
cana-807	27	14	a(x	a(x	PROPN
cana-807	27	15	)	)	PUNCT
cana-807	27	16			NOUN
cana-807	27	17	b(x	b(x	PUNCT
cana-807	27	18	)	)	PUNCT
cana-807	27	19	and	and	CCONJ
cana-807	27	20	va(x)	va(x)	PROPN
cana-807	27	21			NUM
cana-807	27	22	vb(x	vb(x	NUM
cana-807	27	23	)	)	PUNCT
cana-807	27	24			NOUN
cana-807	27	25	x	x	SYM
cana-807	27	26			NOUN
cana-807	27	27	x	x	X
cana-807	27	28	2	2	X
cana-807	27	29	.	.	PUNCT
cana-807	27	30	a	a	DET
cana-807	27	31			PROPN
cana-807	27	32	b	b	PROPN
cana-807	27	33			ADP
cana-807	27	34	a	a	DET
cana-807	27	35	a(x	a(x	NOUN
cana-807	27	36	)	)	PUNCT
cana-807	27	37			NOUN
cana-807	27	38	b(x	b(x	NOUN
cana-807	27	39	)	)	PUNCT
cana-807	27	40	,	,	PUNCT
cana-807	27	41	a(x	a(x	PROPN
cana-807	27	42	)	)	PUNCT
cana-807	27	43			NUM
cana-807	27	44	b(x	b(x	PUNCT
cana-807	27	45	)	)	PUNCT
cana-807	27	46	and	and	CCONJ
cana-807	27	47	va(x)	va(x)	PROPN
cana-807	27	48			NUM
cana-807	27	49	vb(x	vb(x	NUM
cana-807	27	50	)	)	PUNCT
cana-807	27	51			NOUN
cana-807	27	52	x	x	SYM
cana-807	27	53			NOUN
cana-807	27	54	x	x	X
cana-807	28	1	3	3	X
cana-807	28	2	.	.	PUNCT
cana-807	28	3	a	a	DET
cana-807	28	4			NOUN
cana-807	28	5	b	b	NOUN
cana-807	28	6	=	=	SYM
cana-807	28	7	x	x	PROPN
cana-807	28	8	,	,	PUNCT
cana-807	28	9	a(x	a(x	NOUN
cana-807	28	10	)	)	PUNCT
cana-807	28	11			PROPN
cana-807	28	12	b(x	b(x	NOUN
cana-807	28	13	)	)	PUNCT
cana-807	28	14	,	,	PUNCT
cana-807	28	15	a(x	a(x	PROPN
cana-807	28	16	)	)	PUNCT
cana-807	28	17			PROPN
cana-807	28	18	b(x	b(x	ADJ
cana-807	28	19	)	)	PUNCT
cana-807	28	20	,	,	PUNCT
cana-807	28	21	va(x)	va(x)	VERB
cana-807	28	22			ADJ
cana-807	28	23	vb(x)	vb(x)	NOUN
cana-807	28	24	4	4	NUM
cana-807	28	25	.	.	PUNCT
cana-807	29	1	a	a	DET
cana-807	29	2			NOUN
cana-807	29	3	b	b	NOUN
cana-807	29	4	=	=	SYM
cana-807	29	5	x	x	PROPN
cana-807	29	6	,	,	PUNCT
cana-807	29	7	a(x	a(x	NOUN
cana-807	29	8	)	)	PUNCT
cana-807	29	9			PROPN
cana-807	29	10	b(x	b(x	NOUN
cana-807	29	11	)	)	PUNCT
cana-807	29	12	,	,	PUNCT
cana-807	29	13	a(x	a(x	PROPN
cana-807	29	14	)	)	PUNCT
cana-807	29	15			NOUN
cana-807	29	16	b(x	b(x	ADJ
cana-807	29	17	)	)	PUNCT
cana-807	29	18	,	,	PUNCT
cana-807	29	19	va(x)	va(x)	VERB
cana-807	29	20			ADJ
cana-807	29	21	vb(x)	vb(x)	NOUN
cana-807	29	22	5	5	NUM
cana-807	29	23	.	.	PUNCT
cana-807	30	1	a	a	DET
cana-807	30	2			NOUN
cana-807	30	3	b	b	PROPN
cana-807	30	4	=	=	SYM
cana-807	30	5	x	x	PROPN
cana-807	30	6	,	,	PUNCT
cana-807	30	7	a(x	a(x	NOUN
cana-807	30	8	)	)	PUNCT
cana-807	30	9			NOUN
cana-807	30	10	b(x	b(x	NOUN
cana-807	30	11	)	)	PUNCT
cana-807	30	12	,	,	PUNCT
cana-807	30	13	a(x	a(x	PROPN
cana-807	30	14	)	)	PUNCT
cana-807	30	15			NOUN
cana-807	30	16	b(x	b(x	ADJ
cana-807	30	17	)	)	PUNCT
cana-807	30	18	,	,	PUNCT
cana-807	30	19	va(x)	va(x)	NOUN
cana-807	30	20			PROPN
cana-807	30	21	vb(x)	vb(x)	PROPN
cana-807	30	22	6	6	NUM
cana-807	30	23	.	.	PUNCT
cana-807	31	1	a	a	DET
cana-807	31	2			NOUN
cana-807	31	3	b	b	PROPN
cana-807	31	4	=	=	SYM
cana-807	31	5	x	x	PROPN
cana-807	31	6	,	,	PUNCT
cana-807	31	7	a(x	a(x	NOUN
cana-807	31	8	)	)	PUNCT
cana-807	31	9			NOUN
cana-807	31	10	b(x	b(x	NOUN
cana-807	31	11	)	)	PUNCT
cana-807	31	12	,	,	PUNCT
cana-807	31	13	a(x	a(x	PROPN
cana-807	31	14	)	)	PUNCT
cana-807	32	1			PROPN
cana-807	32	2	b(x	b(x	ADJ
cana-807	32	3	)	)	PUNCT
cana-807	32	4	,	,	PUNCT
cana-807	32	5	va(x)	va(x)	NOUN
cana-807	32	6			PROPN
cana-807	32	7	vb(x)	vb(x)	PROPN
cana-807	32	8	definition	definition	NOUN
cana-807	32	9	2.4	2.4	NUM
cana-807	32	10	:	:	PUNCT
cana-807	33	1	[	[	X
cana-807	33	2	6	6	NUM
cana-807	33	3	]	]	PUNCT
cana-807	33	4	a	a	DET
cana-807	33	5	neutrosophic	neutrosophic	ADJ
cana-807	33	6	topology	topology	NOUN
cana-807	33	7	on	on	ADP
cana-807	33	8	a	a	DET
cana-807	33	9	non	non	ADJ
cana-807	33	10	-	-	ADJ
cana-807	33	11	empty	empty	ADJ
cana-807	33	12	set	set	NOUN
cana-807	33	13	x	x	PUNCT
cana-807	33	14	is	be	AUX
cana-807	33	15	a	a	DET
cana-807	33	16	family	family	NOUN
cana-807	33	17			NOUN
cana-807	33	18	of	of	ADP
cana-807	33	19	neutrosophic	neutrosophic	ADJ
cana-807	33	20	subsets	subset	NOUN
cana-807	33	21	in	in	ADP
cana-807	33	22	x	x	X
cana-807	33	23	satisfies	satisfie	NOUN
cana-807	33	24	the	the	DET
cana-807	33	25	following	follow	VERB
cana-807	33	26	axioms	axiom	NOUN
cana-807	33	27	:	:	PUNCT
cana-807	33	28	(	(	PUNCT
cana-807	33	29	nt1	nt1	NOUN
cana-807	33	30	)	)	PUNCT
cana-807	33	31	(	(	PUNCT
cana-807	33	32	nt1	nt1	NOUN
cana-807	33	33	)	)	PUNCT
cana-807	33	34	0n	0n	NOUN
cana-807	33	35	,	,	PUNCT
cana-807	33	36	1n	1n	NUM
cana-807	33	37			NOUN
cana-807	33	38			PROPN
cana-807	33	39	(	(	PUNCT
cana-807	33	40			NOUN
cana-807	33	41	)	)	PUNCT
cana-807	33	42	g1	g1	PROPN
cana-807	33	43			PUNCT
cana-807	33	44	g2	g2	PROPN
cana-807	33	45			NOUN
cana-807	33	46			NOUN
cana-807	33	47	for	for	ADP
cana-807	33	48	any	any	DET
cana-807	33	49	g1	g1	NOUN
cana-807	33	50	,	,	PUNCT
cana-807	33	51	g2	g2	PROPN
cana-807	33	52			NOUN
cana-807	33	53			NOUN
cana-807	33	54	(	(	PUNCT
cana-807	33	55	nt3	nt3	NOUN
cana-807	33	56	)	)	PUNCT
cana-807	33	57			NOUN
cana-807	33	58	g1	g1	PROPN
cana-807	33	59			NOUN
cana-807	33	60			PROPN
cana-807	33	61			VERB
cana-807	33	62	{	{	PUNCT
cana-807	33	63	gi	gi	X
cana-807	33	64	:	:	PUNCT
cana-807	33	65	i	i	PRON
cana-807	33	66			PROPN
cana-807	33	67	j	j	PROPN
cana-807	33	68	}	}	PUNCT
cana-807	33	69			PROPN
cana-807	33	70			NOUN
cana-807	33	71	in	in	ADP
cana-807	33	72	this	this	DET
cana-807	33	73	case	case	NOUN
cana-807	33	74	the	the	DET
cana-807	33	75	pair	pair	NOUN
cana-807	33	76	(	(	PUNCT
cana-807	33	77	x	x	NOUN
cana-807	33	78	,	,	PUNCT
cana-807	33	79			PROPN
cana-807	33	80	)	)	PUNCT
cana-807	33	81	is	be	AUX
cana-807	33	82	a	a	DET
cana-807	33	83	neutrosophic	neutrosophic	ADJ
cana-807	33	84	topological	topological	ADJ
cana-807	33	85	space	space	NOUN
cana-807	33	86	and	and	CCONJ
cana-807	33	87	any	any	DET
cana-807	33	88	neutrosophic	neutrosophic	ADJ
cana-807	33	89	set	set	NOUN
cana-807	33	90	in	in	ADP
cana-807	33	91			NOUN
cana-807	33	92	is	be	AUX
cana-807	33	93	known	know	VERB
cana-807	33	94	as	as	ADP
cana-807	33	95	a	a	DET
cana-807	33	96	neutrosopic	neutrosopic	ADJ
cana-807	33	97	open	open	ADJ
cana-807	33	98	set	set	NOUN
cana-807	33	99	(	(	PUNCT
cana-807	33	100	nos	nos	NOUN
cana-807	33	101	)	)	PUNCT
cana-807	33	102	in	in	ADP
cana-807	33	103	x.	x.	PROPN
cana-807	33	104	a	a	DET
cana-807	33	105	neutrosophic	neutrosophic	PROPN
cana-807	33	106	set	set	NOUN
cana-807	33	107	a	a	PRON
cana-807	33	108	is	be	AUX
cana-807	33	109	a	a	DET
cana-807	33	110	neutrosophic	neutrosophic	ADJ
cana-807	33	111	closed	close	VERB
cana-807	33	112	set	set	NOUN
cana-807	33	113	(	(	PUNCT
cana-807	33	114	ncs	ncs	NOUN
cana-807	33	115	)	)	PUNCT
cana-807	34	1	if	if	SCONJ
cana-807	34	2	and	and	CCONJ
cana-807	34	3	only	only	ADV
cana-807	34	4	if	if	SCONJ
cana-807	34	5	its	its	PRON
cana-807	34	6	complement	complement	NOUN
cana-807	34	7	c(a	c(a	ADV
cana-807	34	8	)	)	PUNCT
cana-807	34	9	is	be	AUX
cana-807	34	10	a	a	DET
cana-807	34	11	neutrosophic	neutrosophic	ADJ
cana-807	34	12	open	open	ADJ
cana-807	34	13	set	set	VERB
cana-807	34	14	in	in	ADP
cana-807	34	15	x.	x.	NOUN
cana-807	34	16	here	here	ADV
cana-807	34	17	the	the	DET
cana-807	34	18	empty	empty	ADJ
cana-807	34	19	set	set	NOUN
cana-807	34	20	(	(	PUNCT
cana-807	34	21	os	os	NOUN
cana-807	34	22	)	)	PUNCT
cana-807	34	23	and	and	CCONJ
cana-807	34	24	the	the	DET
cana-807	34	25	whole	whole	ADJ
cana-807	34	26	set	set	NOUN
cana-807	34	27	(	(	PUNCT
cana-807	34	28	in	in	ADP
cana-807	34	29	)	)	PUNCT
cana-807	34	30	may	may	AUX
cana-807	34	31	be	be	AUX
cana-807	34	32	defined	define	VERB
cana-807	34	33	as	as	SCONJ
cana-807	34	34	follows	follow	VERB
cana-807	34	35	:	:	PUNCT
cana-807	34	36	(	(	PUNCT
cana-807	34	37	01	01	X
cana-807	34	38	)	)	PUNCT
cana-807	34	39	0n	0n	NOUN
cana-807	34	40	=	=	PUNCT
cana-807	34	41	{	{	PUNCT
cana-807	34	42	x	x	PROPN
cana-807	34	43	,	,	PUNCT
cana-807	34	44	0	0	NUM
cana-807	34	45	,	,	PUNCT
cana-807	34	46	0	0	NUM
cana-807	34	47	,	,	PUNCT
cana-807	34	48	1	1	PROPN
cana-807	34	49	:	:	PUNCT
cana-807	34	50	x	x	SYM
cana-807	34	51			NOUN
cana-807	34	52	x	x	NUM
cana-807	34	53	}	}	PUNCT
cana-807	34	54	(	(	PUNCT
cana-807	34	55	02	02	NUM
cana-807	34	56	)	)	PUNCT
cana-807	34	57	0n	0n	NOUN
cana-807	34	58	=	=	SYM
cana-807	34	59	{	{	PUNCT
cana-807	34	60	x	x	PROPN
cana-807	34	61	,	,	PUNCT
cana-807	34	62	0	0	NUM
cana-807	34	63	,	,	PUNCT
cana-807	34	64	1	1	NUM
cana-807	34	65	,	,	PUNCT
cana-807	34	66	1	1	PROPN
cana-807	34	67	:	:	PUNCT
cana-807	34	68	x	x	SYM
cana-807	34	69			NOUN
cana-807	34	70	x	x	NUM
cana-807	34	71	}	}	PUNCT
cana-807	34	72	(	(	PUNCT
cana-807	34	73	03	03	NUM
cana-807	34	74	)	)	PUNCT
cana-807	34	75	0n	0n	NOUN
cana-807	34	76	=	=	PUNCT
cana-807	34	77	{	{	PUNCT
cana-807	34	78	x	x	PROPN
cana-807	34	79	,	,	PUNCT
cana-807	34	80	0	0	NUM
cana-807	34	81	,	,	PUNCT
cana-807	34	82	1	1	NUM
cana-807	34	83	,	,	PUNCT
cana-807	34	84	0	0	NUM
cana-807	34	85	:	:	PUNCT
cana-807	34	86	x	x	SYM
cana-807	34	87			NOUN
cana-807	34	88	x	x	NUM
cana-807	34	89	}	}	PUNCT
cana-807	34	90	(	(	PUNCT
cana-807	34	91	04	04	NUM
cana-807	34	92	)	)	PUNCT
cana-807	34	93	0n	0n	NOUN
cana-807	34	94	=	=	PUNCT
cana-807	34	95	{	{	PUNCT
cana-807	34	96	x	x	PROPN
cana-807	34	97	,	,	PUNCT
cana-807	34	98	0	0	NUM
cana-807	34	99	,	,	PUNCT
cana-807	34	100	0	0	NUM
cana-807	34	101	,	,	PUNCT
cana-807	34	102	0	0	NUM
cana-807	34	103	:	:	PUNCT
cana-807	34	104	x	x	SYM
cana-807	34	105			NOUN
cana-807	34	106	x	x	NUM
cana-807	34	107	}	}	PUNCT
cana-807	34	108	communications	communication	NOUN
cana-807	34	109	on	on	ADP
cana-807	34	110	applied	apply	VERB
cana-807	34	111	nonlinear	nonlinear	ADJ
cana-807	34	112	analysis	analysis	NOUN
cana-807	34	113	issn	issn	NOUN
cana-807	34	114	:	:	PUNCT
cana-807	34	115	1074	1074	NUM
cana-807	34	116	-	-	PUNCT
cana-807	34	117	133x	133x	NUM
cana-807	34	118	vol	vol	NOUN
cana-807	34	119	31	31	NUM
cana-807	34	120	no	no	NOUN
cana-807	34	121	.	.	PUNCT
cana-807	35	1	3s	3s	NUM
cana-807	35	2	(	(	PUNCT
cana-807	35	3	2024	2024	NUM
cana-807	35	4	)	)	PUNCT
cana-807	35	5	549	549	NUM
cana-807	35	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	35	7	(	(	PUNCT
cana-807	35	8	11	11	NUM
cana-807	35	9	)	)	PUNCT
cana-807	35	10	1n	1n	NOUN
cana-807	35	11	=	=	SYM
cana-807	35	12	{	{	PUNCT
cana-807	35	13	x	x	NOUN
cana-807	35	14	,	,	PUNCT
cana-807	35	15	1	1	NUM
cana-807	35	16	,	,	PUNCT
cana-807	35	17	0	0	NUM
cana-807	35	18	,	,	PUNCT
cana-807	35	19	0	0	NUM
cana-807	35	20	:	:	PUNCT
cana-807	35	21	x	x	SYM
cana-807	35	22			NOUN
cana-807	35	23	x	x	NUM
cana-807	35	24	}	}	PUNCT
cana-807	35	25	(	(	PUNCT
cana-807	35	26	12	12	NUM
cana-807	35	27	)	)	PUNCT
cana-807	35	28	1n	1n	NOUN
cana-807	35	29	=	=	SYM
cana-807	35	30	{	{	PUNCT
cana-807	35	31	x	x	NOUN
cana-807	35	32	,	,	PUNCT
cana-807	35	33	1	1	NUM
cana-807	35	34	,	,	PUNCT
cana-807	35	35	0	0	NUM
cana-807	35	36	,	,	PUNCT
cana-807	35	37	1	1	PROPN
cana-807	35	38	:	:	PUNCT
cana-807	35	39	x	x	SYM
cana-807	35	40			NOUN
cana-807	35	41	x	x	NUM
cana-807	35	42	}	}	PUNCT
cana-807	35	43	(	(	PUNCT
cana-807	35	44	13	13	NUM
cana-807	35	45	)	)	PUNCT
cana-807	35	46	1n	1n	NOUN
cana-807	35	47	=	=	SYM
cana-807	35	48	{	{	PUNCT
cana-807	35	49	x	x	NOUN
cana-807	35	50	,	,	PUNCT
cana-807	35	51	1	1	NUM
cana-807	35	52	,	,	PUNCT
cana-807	35	53	1	1	NUM
cana-807	35	54	,	,	PUNCT
cana-807	35	55	0	0	NUM
cana-807	35	56	:	:	PUNCT
cana-807	35	57	x	x	SYM
cana-807	35	58			NOUN
cana-807	35	59	x	x	NUM
cana-807	35	60	}	}	PUNCT
cana-807	35	61	(	(	PUNCT
cana-807	35	62	14	14	NUM
cana-807	35	63	)	)	PUNCT
cana-807	35	64	1n	1n	NOUN
cana-807	35	65	=	=	SYM
cana-807	35	66	{	{	PUNCT
cana-807	35	67	x	x	NOUN
cana-807	35	68	,	,	PUNCT
cana-807	35	69	1	1	NUM
cana-807	35	70	,	,	PUNCT
cana-807	35	71	1	1	NUM
cana-807	35	72	,	,	PUNCT
cana-807	35	73	1	1	PROPN
cana-807	35	74	:	:	PUNCT
cana-807	35	75	x	x	SYM
cana-807	35	76			NOUN
cana-807	35	77	x	x	NUM
cana-807	35	78	}	}	PUNCT
cana-807	35	79	definition	definition	NOUN
cana-807	35	80	2.5	2.5	NUM
cana-807	35	81	:	:	PUNCT
cana-807	36	1	[	[	X
cana-807	36	2	6	6	NUM
cana-807	36	3	]	]	X
cana-807	36	4	let	let	VERB
cana-807	36	5	(	(	PUNCT
cana-807	36	6	x	x	NOUN
cana-807	36	7	,	,	PUNCT
cana-807	36	8			PROPN
cana-807	36	9	)	)	PUNCT
cana-807	36	10	be	be	VERB
cana-807	36	11	a	a	DET
cana-807	36	12	n	n	ADV
cana-807	36	13	-	-	PUNCT
cana-807	36	14	ts	ts	NOUN
cana-807	36	15	and	and	CCONJ
cana-807	36	16	a	a	DET
cana-807	36	17	=	=	X
cana-807	36	18	{	{	PUNCT
cana-807	36	19	x	x	PROPN
cana-807	36	20	,	,	PUNCT
cana-807	36	21	a(x	a(x	NOUN
cana-807	36	22	)	)	PUNCT
cana-807	36	23	,	,	PUNCT
cana-807	36	24	a(x	a(x	PROPN
cana-807	36	25	)	)	PUNCT
cana-807	36	26	,	,	PUNCT
cana-807	36	27	va(x)	va(x)	NOUN
cana-807	36	28	:	:	PUNCT
cana-807	36	29	x	x	SYM
cana-807	36	30			NOUN
cana-807	36	31	x	x	PART
cana-807	36	32	}	}	PUNCT
cana-807	36	33	be	be	AUX
cana-807	36	34	a	a	DET
cana-807	36	35	n	n	NOUN
cana-807	36	36	-	-	PUNCT
cana-807	36	37	s	s	NOUN
cana-807	36	38	in	in	ADP
cana-807	36	39	x.	x.	NOUN
cana-807	36	40	then	then	ADV
cana-807	36	41	the	the	DET
cana-807	36	42	neutrosophic	neutrosophic	ADJ
cana-807	36	43	interior	interior	NOUN
cana-807	36	44	and	and	CCONJ
cana-807	36	45	the	the	DET
cana-807	36	46	neutrosophic	neutrosophic	ADJ
cana-807	36	47	closure	closure	NOUN
cana-807	36	48	of	of	ADP
cana-807	36	49	a	a	PRON
cana-807	36	50	are	be	AUX
cana-807	36	51	defined	define	VERB
cana-807	36	52	by	by	ADP
cana-807	36	53	nint(a	nint(a	PROPN
cana-807	36	54	)	)	PUNCT
cana-807	36	55	=	=	SYM
cana-807	36	56			NOUN
cana-807	36	57	{	{	PUNCT
cana-807	36	58	g	g	NOUN
cana-807	36	59	:	:	PUNCT
cana-807	36	60	g	g	PROPN
cana-807	36	61	is	be	AUX
cana-807	36	62	an	an	DET
cana-807	36	63	nos	nos	X
cana-807	36	64	in	in	ADP
cana-807	36	65	x	x	X
cana-807	36	66	and	and	CCONJ
cana-807	36	67	g	g	PROPN
cana-807	36	68			PROPN
cana-807	36	69	a	a	DET
cana-807	36	70	}	}	PUNCT
cana-807	36	71	nint(a	nint(a	NOUN
cana-807	36	72	)	)	PUNCT
cana-807	36	73	=	=	VERB
cana-807	37	1			X
cana-807	37	2	{	{	PUNCT
cana-807	37	3	k	k	NOUN
cana-807	37	4	:	:	PUNCT
cana-807	37	5	k	k	X
cana-807	37	6	is	be	AUX
cana-807	37	7	an	an	DET
cana-807	37	8	nos	nos	X
cana-807	37	9	in	in	ADP
cana-807	37	10	x	x	X
cana-807	37	11	and	and	CCONJ
cana-807	37	12	a	a	DET
cana-807	37	13			PROPN
cana-807	37	14	k	k	PROPN
cana-807	37	15	}	}	PUNCT
cana-807	37	16	note	note	VERB
cana-807	37	17	that	that	SCONJ
cana-807	37	18	for	for	ADP
cana-807	37	19	any	any	DET
cana-807	37	20	ns	ns	NOUN
cana-807	37	21	a	a	PROPN
cana-807	37	22	,	,	PUNCT
cana-807	37	23	ncl(c(a	ncl(c(a	NOUN
cana-807	37	24	)	)	PUNCT
cana-807	37	25	)	)	PUNCT
cana-807	38	1	=	=	PUNCT
cana-807	38	2	c(nint(a	c(nint(a	NOUN
cana-807	38	3	)	)	PUNCT
cana-807	38	4	)	)	PUNCT
cana-807	38	5	and	and	CCONJ
cana-807	38	6	nint(c(a	nint(c(a	NUM
cana-807	38	7	)	)	PUNCT
cana-807	38	8	)	)	PUNCT
cana-807	39	1	=	=	PUNCT
cana-807	39	2	c(ncl(a	c(ncl(a	PROPN
cana-807	39	3	)	)	PUNCT
cana-807	39	4	)	)	PUNCT
cana-807	39	5	.	.	PUNCT
cana-807	40	1	definition	definition	NOUN
cana-807	40	2	2.6	2.6	NUM
cana-807	40	3	:	:	PUNCT
cana-807	41	1	[	[	X
cana-807	41	2	3	3	NUM
cana-807	41	3	]	]	SYM
cana-807	41	4	n	n	CCONJ
cana-807	41	5	-	-	SYM
cana-807	41	6	s	s	NOUN
cana-807	41	7	s	s	NOUN
cana-807	41	8	=	=	SYM
cana-807	41	9	{	{	PUNCT
cana-807	41	10	x	x	PROPN
cana-807	41	11	,	,	PUNCT
cana-807	41	12	a(x	a(x	NOUN
cana-807	41	13	)	)	PUNCT
cana-807	41	14	,	,	PUNCT
cana-807	41	15	a(x	a(x	PROPN
cana-807	41	16	)	)	PUNCT
cana-807	41	17	,	,	PUNCT
cana-807	41	18	va(x)	va(x)	NOUN
cana-807	41	19	:	:	PUNCT
cana-807	41	20	x	x	SYM
cana-807	41	21			NOUN
cana-807	41	22	x	x	PUNCT
cana-807	41	23	in	in	ADP
cana-807	41	24	a	a	DET
cana-807	41	25	n	n	ADV
cana-807	41	26	-	-	PUNCT
cana-807	41	27	ts	ts	NOUN
cana-807	41	28	the	the	DET
cana-807	41	29	n	n	NOUN
cana-807	41	30	-	-	PUNCT
cana-807	41	31	s	s	NOUN
cana-807	41	32	s	s	NOUN
cana-807	41	33	=	=	SYM
cana-807	41	34	{	{	PUNCT
cana-807	41	35	x	x	PROPN
cana-807	41	36	,	,	PUNCT
cana-807	41	37	a(x	a(x	NOUN
cana-807	41	38	)	)	PUNCT
cana-807	41	39	,	,	PUNCT
cana-807	41	40	a(x	a(x	PROPN
cana-807	41	41	)	)	PUNCT
cana-807	41	42	,	,	PUNCT
cana-807	41	43	va(x)	va(x)	NOUN
cana-807	41	44	:	:	PUNCT
cana-807	41	45	x	x	SYM
cana-807	41	46			NOUN
cana-807	41	47	x	x	X
cana-807	41	48	}	}	PUNCT
cana-807	41	49	.	.	PUNCT
cana-807	42	1	is	be	AUX
cana-807	42	2	said	say	VERB
cana-807	42	3	to	to	PART
cana-807	42	4	be	be	AUX
cana-807	42	5	neutrosophic	neutrosophic	ADJ
cana-807	42	6	•	•	NOUN
cana-807	42	7	(	(	PUNCT
cana-807	42	8	regular	regular	ADJ
cana-807	42	9	(	(	PUNCT
cana-807	42	10	n	n	CCONJ
cana-807	42	11	-	-	PUNCT
cana-807	42	12	rcs	rcs	NOUN
cana-807	42	13	)	)	PUNCT
cana-807	42	14	,	,	PUNCT
cana-807	42	15	semi(n	semi(n	NOUN
cana-807	42	16	-	-	PUNCT
cana-807	42	17	scs	scs	NOUN
cana-807	42	18	)	)	PUNCT
cana-807	42	19	,	,	PUNCT
cana-807	42	20	pre(n	pre(n	PROPN
cana-807	42	21	-	-	PUNCT
cana-807	42	22	pcs	pc	NOUN
cana-807	42	23	)	)	PUNCT
cana-807	42	24	,	,	PUNCT
cana-807	42	25	(	(	PUNCT
cana-807	42	26	n	n	CCONJ
cana-807	42	27	-	-	PUNCT
cana-807	42	28	αcs	αcs	NOUN
cana-807	42	29	)	)	PUNCT
cana-807	42	30	)	)	PUNCT
cana-807	43	1	if	if	SCONJ
cana-807	43	2	•	•	X
cana-807	43	3	(	(	PUNCT
cana-807	43	4	s	s	NOUN
cana-807	43	5	=	=	SYM
cana-807	43	6	n	n	CCONJ
cana-807	43	7	-	-	PUNCT
cana-807	43	8	cl(n	cl(n	NOUN
cana-807	43	9	-	-	PUNCT
cana-807	43	10	int(s	int(s	PROPN
cana-807	43	11	)	)	PUNCT
cana-807	43	12	)	)	PUNCT
cana-807	43	13	,	,	PUNCT
cana-807	43	14	n	n	CCONJ
cana-807	43	15	-	-	PUNCT
cana-807	43	16	int(n	int(n	NOUN
cana-807	43	17	-	-	PUNCT
cana-807	43	18	cl(s	cl(s	NOUN
cana-807	43	19	)	)	PUNCT
cana-807	43	20	)	)	PUNCT
cana-807	43	21			PROPN
cana-807	43	22	s	s	PROPN
cana-807	43	23	,	,	PUNCT
cana-807	43	24	n	n	CCONJ
cana-807	43	25	-	-	PUNCT
cana-807	43	26	cl(n	cl(n	NOUN
cana-807	43	27	-	-	PUNCT
cana-807	43	28	int(s	int(s	NUM
cana-807	43	29	)	)	PUNCT
cana-807	43	30	)	)	PUNCT
cana-807	43	31			PROPN
cana-807	43	32	s	s	PROPN
cana-807	43	33	,	,	PUNCT
cana-807	43	34	n	n	CCONJ
cana-807	43	35	-	-	PUNCT
cana-807	43	36	cl(n	cl(n	NOUN
cana-807	43	37	-	-	PUNCT
cana-807	43	38	int(n	int(n	NOUN
cana-807	43	39	-	-	PUNCT
cana-807	43	40	cl(s	cl(s	NOUN
cana-807	43	41	)	)	PUNCT
cana-807	43	42	)	)	PUNCT
cana-807	43	43	)	)	PUNCT
cana-807	43	44			PROPN
cana-807	43	45	s.	s.	PROPN
cana-807	43	46	)	)	PUNCT
cana-807	43	47	•	•	ADP
cana-807	43	48	the	the	DET
cana-807	43	49	complement	complement	NOUN
cana-807	43	50	of	of	ADP
cana-807	43	51	the	the	DET
cana-807	43	52	above	above	ADJ
cana-807	43	53	closed	closed	ADJ
cana-807	43	54	sets	set	NOUN
cana-807	43	55	are	be	AUX
cana-807	43	56	their	their	PRON
cana-807	43	57	respective	respective	ADJ
cana-807	43	58	open	open	ADJ
cana-807	43	59	sets	set	NOUN
cana-807	43	60	definition	definition	NOUN
cana-807	43	61	2.7	2.7	NUM
cana-807	43	62	:	:	PUNCT
cana-807	44	1	[	[	X
cana-807	44	2	8	8	X
cana-807	44	3	]	]	X
cana-807	44	4	the	the	DET
cana-807	44	5	n	n	NOUN
cana-807	44	6	-	-	PUNCT
cana-807	44	7	s	s	PRON
cana-807	44	8	a	a	PRON
cana-807	44	9	is	be	AUX
cana-807	44	10	said	say	VERB
cana-807	44	11	to	to	PART
cana-807	44	12	be	be	AUX
cana-807	44	13	neutrosophic	neutrosophic	ADJ
cana-807	44	14	generalized	generalize	VERB
cana-807	44	15	pre	pre	X
cana-807	44	16	closed	closed	ADJ
cana-807	44	17	set	set	NOUN
cana-807	44	18	(	(	PUNCT
cana-807	44	19	n	n	CCONJ
cana-807	44	20	-	-	PUNCT
cana-807	44	21	gpcs	gpc	NOUN
cana-807	44	22	)	)	PUNCT
cana-807	44	23	if	if	SCONJ
cana-807	44	24	npcl(a	npcl(a	NUM
cana-807	44	25	)	)	PUNCT
cana-807	44	26			PROPN
cana-807	44	27	u	u	PROPN
cana-807	44	28	whenever	whenever	SCONJ
cana-807	44	29	a	a	DET
cana-807	44	30			PROPN
cana-807	44	31	u	u	NOUN
cana-807	44	32	and	and	CCONJ
cana-807	44	33	u	u	NOUN
cana-807	44	34	is	be	AUX
cana-807	44	35	nos	nos	X
cana-807	44	36	in	in	ADV
cana-807	44	37	(	(	PUNCT
cana-807	44	38	x	x	NOUN
cana-807	44	39	,	,	PUNCT
cana-807	44	40	τ	τ	PROPN
cana-807	44	41	)	)	PUNCT
cana-807	44	42	.	.	PUNCT
cana-807	45	1	c	c	X
cana-807	45	2	(	(	PUNCT
cana-807	45	3	n	n	CCONJ
cana-807	45	4	-	-	PUNCT
cana-807	45	5	gpcs	gpc	NOUN
cana-807	45	6	)	)	PUNCT
cana-807	45	7	in	in	ADP
cana-807	45	8	(	(	PUNCT
cana-807	45	9	x	x	NOUN
cana-807	45	10	,	,	PUNCT
cana-807	45	11	τ	τ	X
cana-807	45	12	)	)	PUNCT
cana-807	45	13	is	be	AUX
cana-807	45	14	called	call	VERB
cana-807	45	15	neutrosophic	neutrosophic	ADJ
cana-807	45	16	generalized	generalize	VERB
cana-807	45	17	preopen	preopen	ADJ
cana-807	45	18	set	set	NOUN
cana-807	45	19	(	(	PUNCT
cana-807	45	20	n	n	CCONJ
cana-807	45	21	-	-	PUNCT
cana-807	45	22	gpos	gpos	NOUN
cana-807	45	23	shortly	shortly	ADV
cana-807	45	24	)	)	PUNCT
cana-807	45	25	.	.	PUNCT
cana-807	46	1	definition	definition	NOUN
cana-807	46	2	2.8	2.8	NUM
cana-807	46	3	:	:	PUNCT
cana-807	46	4	the	the	DET
cana-807	46	5	n	n	CCONJ
cana-807	46	6	-	-	PUNCT
cana-807	46	7	s	s	PRON
cana-807	46	8	a	a	PRON
cana-807	46	9	is	be	AUX
cana-807	46	10	said	say	VERB
cana-807	46	11	to	to	PART
cana-807	46	12	be	be	AUX
cana-807	46	13	neutrosophic	neutrosophic	ADJ
cana-807	46	14	generalized	generalize	VERB
cana-807	46	15	pre	pre	NOUN
cana-807	46	16	regular	regular	ADJ
cana-807	46	17	closed	closed	ADJ
cana-807	46	18	set	set	NOUN
cana-807	46	19	(	(	PUNCT
cana-807	46	20	n	n	CCONJ
cana-807	46	21	-	-	PUNCT
cana-807	46	22	gprcs	gprcs	NOUN
cana-807	46	23	shortly	shortly	ADV
cana-807	46	24	)	)	PUNCT
cana-807	46	25	if	if	SCONJ
cana-807	46	26	n	n	CCONJ
cana-807	46	27	-	-	PUNCT
cana-807	46	28	pcl(a	pcl(a	ADJ
cana-807	46	29	)	)	PUNCT
cana-807	46	30			PROPN
cana-807	46	31	u	u	PROPN
cana-807	46	32	whenever	whenever	SCONJ
cana-807	46	33	a	a	DET
cana-807	46	34			PROPN
cana-807	46	35	u	u	NOUN
cana-807	46	36	and	and	CCONJ
cana-807	46	37	u	u	NOUN
cana-807	46	38	is	be	AUX
cana-807	46	39	n	n	PRON
cana-807	46	40	-	-	PUNCT
cana-807	46	41	ros	ros	NOUN
cana-807	46	42	in	in	ADP
cana-807	46	43	(	(	PUNCT
cana-807	46	44	x	x	NOUN
cana-807	46	45	,	,	PUNCT
cana-807	46	46	τ	τ	PROPN
cana-807	46	47	)	)	PUNCT
cana-807	46	48	.	.	PUNCT
cana-807	46	49	.	.	PUNCT
cana-807	47	1	definition	definition	NOUN
cana-807	47	2	2.9	2.9	NUM
cana-807	47	3	:	:	PUNCT
cana-807	48	1	[	[	X
cana-807	48	2	3	3	X
cana-807	48	3	]	]	PUNCT
cana-807	48	4	the	the	DET
cana-807	48	5	n	n	NOUN
cana-807	48	6	-	-	PUNCT
cana-807	48	7	s	s	PRON
cana-807	48	8	a	a	PRON
cana-807	48	9	is	be	AUX
cana-807	48	10	said	say	VERB
cana-807	48	11	to	to	PART
cana-807	48	12	be	be	AUX
cana-807	48	13	neutrosophic	neutrosophic	ADJ
cana-807	48	14	regular	regular	ADJ
cana-807	48	15	generalized	generalize	VERB
cana-807	48	16	closed	close	VERB
cana-807	48	17	set	set	NOUN
cana-807	48	18	(	(	PUNCT
cana-807	48	19	n	n	CCONJ
cana-807	48	20	-	-	PUNCT
cana-807	48	21	rgcs	rgcs	NOUN
cana-807	48	22	shortly	shortly	ADV
cana-807	48	23	)	)	PUNCT
cana-807	48	24	if	if	SCONJ
cana-807	48	25	n	n	X
cana-807	48	26	-	-	PUNCT
cana-807	48	27	cl(a	cl(a	NUM
cana-807	48	28	)	)	PUNCT
cana-807	48	29			PROPN
cana-807	48	30	u	u	PROPN
cana-807	48	31	whenever	whenever	SCONJ
cana-807	48	32	a	a	DET
cana-807	48	33			PROPN
cana-807	48	34	u	u	NOUN
cana-807	48	35	and	and	CCONJ
cana-807	48	36	u	u	NOUN
cana-807	48	37	is	be	AUX
cana-807	48	38	n	n	PRON
cana-807	48	39	-	-	PUNCT
cana-807	48	40	ros	ros	NOUN
cana-807	48	41	in	in	ADP
cana-807	48	42	(	(	PUNCT
cana-807	48	43	x	x	NOUN
cana-807	48	44	,	,	PUNCT
cana-807	48	45	τ	τ	PROPN
cana-807	48	46	)	)	PUNCT
cana-807	48	47	.	.	PUNCT
cana-807	48	48	.	.	PUNCT
cana-807	49	1	definition	definition	NOUN
cana-807	49	2	2.10	2.10	NUM
cana-807	49	3	:	:	PUNCT
cana-807	49	4	the	the	DET
cana-807	49	5	n	n	CCONJ
cana-807	49	6	-	-	PUNCT
cana-807	49	7	s	s	PRON
cana-807	49	8	a	a	PRON
cana-807	49	9	is	be	AUX
cana-807	49	10	said	say	VERB
cana-807	49	11	to	to	PART
cana-807	49	12	be	be	AUX
cana-807	49	13	neutrosophic	neutrosophic	ADJ
cana-807	49	14	generalized	generalize	VERB
cana-807	49	15	pre	pre	NOUN
cana-807	49	16	regular	regular	ADJ
cana-807	49	17	weakly	weakly	ADJ
cana-807	49	18	closed	closed	ADJ
cana-807	49	19	set	set	NOUN
cana-807	49	20	(	(	PUNCT
cana-807	49	21	ngprwcs	ngprwcs	ADV
cana-807	49	22	shortly	shortly	ADV
cana-807	49	23	)	)	PUNCT
cana-807	50	1	if	if	SCONJ
cana-807	50	2	n	n	CCONJ
cana-807	50	3	-	-	PUNCT
cana-807	50	4	pcl(a	pcl(a	ADJ
cana-807	50	5	)	)	PUNCT
cana-807	50	6			PROPN
cana-807	50	7	u	u	PROPN
cana-807	50	8	whenever	whenever	SCONJ
cana-807	50	9	a	a	DET
cana-807	50	10			PROPN
cana-807	50	11	u	u	NOUN
cana-807	50	12	and	and	CCONJ
cana-807	50	13	u	u	NOUN
cana-807	50	14	is	be	AUX
cana-807	50	15	n	n	ADV
cana-807	50	16	-	-	PUNCT
cana-807	50	17	rsos	rsos	NOUN
cana-807	50	18	in	in	ADP
cana-807	50	19	(	(	PUNCT
cana-807	50	20	x	x	X
cana-807	50	21	,	,	PUNCT
cana-807	50	22	τ	τ	PROPN
cana-807	50	23	)	)	PUNCT
cana-807	50	24	.	.	PUNCT
cana-807	51	1	.	.	PUNCT
cana-807	52	1	definition	definition	NOUN
cana-807	52	2	2.11	2.11	NUM
cana-807	52	3	:	:	PUNCT
cana-807	52	4	the	the	DET
cana-807	52	5	n	n	CCONJ
cana-807	52	6	-	-	PUNCT
cana-807	52	7	s	s	PRON
cana-807	52	8	a	a	PRON
cana-807	52	9	is	be	AUX
cana-807	52	10	said	say	VERB
cana-807	52	11	to	to	PART
cana-807	52	12	be	be	AUX
cana-807	52	13	n	n	PRON
cana-807	52	14	-	-	PUNCT
cana-807	52	15	r*α	r*α	ADJ
cana-807	52	16	-	-	PUNCT
cana-807	52	17	closed	close	VERB
cana-807	52	18	set	set	NOUN
cana-807	52	19	(	(	PUNCT
cana-807	52	20	n	n	CCONJ
cana-807	52	21	-	-	PUNCT
cana-807	52	22	r*αcs	r*αcs	NOUN
cana-807	52	23	shortly	shortly	ADV
cana-807	52	24	)	)	PUNCT
cana-807	52	25	if	if	SCONJ
cana-807	52	26	u	u	NOUN
cana-807	52	27	is	be	AUX
cana-807	52	28	n	n	CCONJ
cana-807	52	29	-	-	PUNCT
cana-807	52	30	rcs	rcs	NOUN
cana-807	52	31	such	such	ADJ
cana-807	52	32	that	that	SCONJ
cana-807	52	33	a	a	DET
cana-807	52	34			PROPN
cana-807	52	35	u	u	NOUN
cana-807	52	36			NOUN
cana-807	52	37	n	n	CCONJ
cana-807	52	38	=	=	NOUN
cana-807	52	39	αint(u	αint(u	NUM
cana-807	52	40	)	)	PUNCT
cana-807	52	41	.	.	PUNCT
cana-807	53	1	definition	definition	NOUN
cana-807	53	2	2.12	2.12	NUM
cana-807	53	3	:	:	PUNCT
cana-807	53	4	the	the	DET
cana-807	53	5	n	n	CCONJ
cana-807	53	6	-	-	PUNCT
cana-807	53	7	s	s	PRON
cana-807	53	8	a	a	PRON
cana-807	53	9	is	be	AUX
cana-807	53	10	said	say	VERB
cana-807	53	11	to	to	PART
cana-807	53	12	be	be	AUX
cana-807	53	13	neutrosophic	neutrosophic	ADJ
cana-807	53	14	r*gα	r*gα	PROPN
cana-807	53	15	-	-	PUNCT
cana-807	53	16	closed	close	VERB
cana-807	53	17	set	set	NOUN
cana-807	53	18	(	(	PUNCT
cana-807	53	19	n	n	CCONJ
cana-807	53	20	-	-	PUNCT
cana-807	53	21	r*gαcs	r*gαcs	NOUN
cana-807	53	22	shortly	shortly	ADV
cana-807	53	23	)	)	PUNCT
cana-807	53	24	if	if	SCONJ
cana-807	53	25	nαcl(a	nαcl(a	NOUN
cana-807	53	26	)	)	PUNCT
cana-807	53	27			PROPN
cana-807	53	28	u	u	PROPN
cana-807	53	29	whenever	whenever	SCONJ
cana-807	53	30	a	a	DET
cana-807	53	31			PROPN
cana-807	53	32	u	u	NOUN
cana-807	53	33	and	and	CCONJ
cana-807	53	34	u	u	NOUN
cana-807	53	35	is	be	AUX
cana-807	53	36	n	n	PRON
cana-807	53	37	-	-	PUNCT
cana-807	53	38	r*αos	r*αo	NOUN
cana-807	53	39	in	in	ADP
cana-807	53	40	(	(	PUNCT
cana-807	53	41	x	x	NOUN
cana-807	53	42	,	,	PUNCT
cana-807	53	43	τ	τ	PROPN
cana-807	53	44	)	)	PUNCT
cana-807	53	45	.	.	PUNCT
cana-807	54	1	the	the	DET
cana-807	54	2	complement	complement	NOUN
cana-807	54	3	of	of	ADP
cana-807	54	4	the	the	DET
cana-807	54	5	above	above	ADJ
cana-807	54	6	closed	closed	ADJ
cana-807	54	7	sets	set	NOUN
cana-807	54	8	are	be	AUX
cana-807	54	9	their	their	PRON
cana-807	54	10	respective	respective	ADJ
cana-807	54	11	open	open	ADJ
cana-807	54	12	sets	set	NOUN
cana-807	54	13	3	3	NUM
cana-807	54	14	.	.	X
cana-807	55	1	neutrosophic	neutrosophic	PROPN
cana-807	55	2	pre	pre	VERB
cana-807	55	3	generalized	generalized	ADJ
cana-807	55	4	pre	pre	VERB
cana-807	55	5	regular	regular	ADJ
cana-807	55	6	star	star	NOUN
cana-807	55	7	weakly	weakly	ADJ
cana-807	55	8	closed	closed	ADJ
cana-807	55	9	sets	set	NOUN
cana-807	55	10	definition	definition	NOUN
cana-807	55	11	3.1	3.1	NUM
cana-807	55	12	:	:	PUNCT
cana-807	55	13	the	the	DET
cana-807	55	14	n	n	NOUN
cana-807	55	15	-	-	PUNCT
cana-807	55	16	s	s	NOUN
cana-807	55	17	s	s	NOUN
cana-807	55	18	is	be	AUX
cana-807	55	19	nutrosophic	nutrosophic	ADV
cana-807	55	20	pre	pre	ADJ
cana-807	55	21	generalised	generalised	ADJ
cana-807	55	22	pre	pre	VERB
cana-807	55	23	regular	regular	ADJ
cana-807	55	24	star	star	NOUN
cana-807	55	25	weakly	weakly	ADV
cana-807	55	26	closed	closed	ADJ
cana-807	55	27	(	(	PUNCT
cana-807	55	28	n	n	X
cana-807	55	29	pgpr*wcs	pgpr*wcs	NOUN
cana-807	55	30	)	)	PUNCT
cana-807	56	1	if	if	SCONJ
cana-807	56	2	n	n	CCONJ
cana-807	56	3	-	-	PUNCT
cana-807	56	4	pcl(s	pcl(s	NOUN
cana-807	56	5	)	)	PUNCT
cana-807	56	6			PROPN
cana-807	56	7	m	m	VERB
cana-807	56	8	whenever	whenever	SCONJ
cana-807	56	9	s	s	VERB
cana-807	56	10			PROPN
cana-807	56	11	m	m	PROPN
cana-807	56	12	and	and	CCONJ
cana-807	56	13	m	m	PROPN
cana-807	56	14	is	be	AUX
cana-807	56	15	n	n	PRON
cana-807	56	16	-	-	PUNCT
cana-807	56	17	r*gαos	r*gαo	NOUN
cana-807	56	18	in	in	ADP
cana-807	56	19	x.	x.	NOUN
cana-807	56	20	c(n	c(n	NOUN
cana-807	56	21	-	-	PUNCT
cana-807	56	22	pgpr*wcs	pgpr*wcs	NOUN
cana-807	56	23	)	)	PUNCT
cana-807	56	24	is	be	AUX
cana-807	56	25	communications	communication	NOUN
cana-807	56	26	on	on	ADP
cana-807	56	27	applied	apply	VERB
cana-807	56	28	nonlinear	nonlinear	ADJ
cana-807	56	29	analysis	analysis	NOUN
cana-807	56	30	issn	issn	NOUN
cana-807	56	31	:	:	PUNCT
cana-807	56	32	1074	1074	NUM
cana-807	56	33	-	-	PUNCT
cana-807	56	34	133x	133x	NUM
cana-807	56	35	vol	vol	NOUN
cana-807	56	36	31	31	NUM
cana-807	56	37	no	no	NOUN
cana-807	56	38	.	.	PUNCT
cana-807	57	1	3s	3s	NUM
cana-807	57	2	(	(	PUNCT
cana-807	57	3	2024	2024	NUM
cana-807	57	4	)	)	PUNCT
cana-807	57	5	550	550	NUM
cana-807	57	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	57	7	neutrosophic	neutrosophic	ADJ
cana-807	57	8	pre	pre	VERB
cana-807	57	9	generalized	generalized	ADJ
cana-807	57	10	pre	pre	VERB
cana-807	57	11	regular	regular	ADJ
cana-807	57	12	star	star	NOUN
cana-807	57	13	weakly	weakly	ADJ
cana-807	57	14	opensets	openset	NOUN
cana-807	57	15	(	(	PUNCT
cana-807	57	16	n	n	CCONJ
cana-807	57	17	-	-	PUNCT
cana-807	57	18	pgpr*wos	pgpr*wos	NOUN
cana-807	57	19	)	)	PUNCT
cana-807	57	20	.	.	PUNCT
cana-807	58	1	the	the	DET
cana-807	58	2	collection	collection	NOUN
cana-807	58	3	of	of	ADP
cana-807	58	4	all	all	DET
cana-807	58	5	n	n	NOUN
cana-807	58	6	-	-	PUNCT
cana-807	58	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	58	8	of	of	ADP
cana-807	58	9	n	n	X
cana-807	58	10	-	-	PUNCT
cana-807	58	11	ts	ts	NOUN
cana-807	58	12	(	(	PUNCT
cana-807	58	13	x	x	X
cana-807	58	14	,	,	PUNCT
cana-807	58	15	τ	τ	X
cana-807	58	16	)	)	PUNCT
cana-807	58	17	is	be	AUX
cana-807	58	18	denoted	denote	VERB
cana-807	58	19	by	by	ADP
cana-807	58	20	n	n	CCONJ
cana-807	58	21	-	-	PUNCT
cana-807	58	22	pgpr*wc	pgpr*wc	NOUN
cana-807	58	23	-	-	NOUN
cana-807	58	24	x.	x.	NOUN
cana-807	58	25	the	the	DET
cana-807	58	26	n	n	ADV
cana-807	58	27	-	-	PUNCT
cana-807	58	28	ts	ts	NOUN
cana-807	58	29	(	(	PUNCT
cana-807	58	30	x	x	X
cana-807	58	31	,	,	PUNCT
cana-807	58	32	τ	τ	X
cana-807	58	33	)	)	PUNCT
cana-807	58	34	is	be	AUX
cana-807	58	35	denoted	denote	VERB
cana-807	58	36	by	by	ADP
cana-807	58	37	n	n	X
cana-807	58	38	-	-	PUNCT
cana-807	58	39	ts	ts	NOUN
cana-807	58	40	.	.	PUNCT
cana-807	59	1	x	x	PUNCT
cana-807	59	2	or	or	CCONJ
cana-807	59	3	x	x	ADJ
cana-807	59	4	example	example	NOUN
cana-807	59	5	3.2	3.2	NUM
cana-807	59	6	let	let	VERB
cana-807	59	7	x	x	PUNCT
cana-807	59	8	=	=	PRON
cana-807	59	9	{	{	PUNCT
cana-807	59	10	a	a	PROPN
cana-807	59	11	,	,	PUNCT
cana-807	59	12	b	b	NOUN
cana-807	59	13	}	}	PUNCT
cana-807	59	14	and	and	CCONJ
cana-807	59	15	τ	τ	PROPN
cana-807	59	16	=	=	PUNCT
cana-807	59	17	{	{	PUNCT
cana-807	59	18	0n	0n	NUM
cana-807	59	19	,	,	PUNCT
cana-807	59	20	s	s	X
cana-807	59	21	,	,	PUNCT
cana-807	59	22	m	m	PROPN
cana-807	59	23	,	,	PUNCT
cana-807	59	24	1n	1n	NUM
cana-807	59	25	}	}	PUNCT
cana-807	59	26	.	.	PUNCT
cana-807	60	1	where	where	SCONJ
cana-807	60	2	m	m	VERB
cana-807	60	3	=	=	SYM
cana-807	60	4	⟨(0.2	⟨(0.2	PROPN
cana-807	60	5	,	,	PUNCT
cana-807	60	6	0.2	0.2	NUM
cana-807	60	7	,	,	PUNCT
cana-807	60	8	0.7	0.7	NUM
cana-807	60	9	)	)	PUNCT
cana-807	60	10	,	,	PUNCT
cana-807	60	11	(	(	PUNCT
cana-807	60	12	0.3	0.3	NUM
cana-807	60	13	,	,	PUNCT
cana-807	60	14	0.1	0.1	NUM
cana-807	60	15	,	,	PUNCT
cana-807	60	16	0.7)⟩	0.7)⟩	NOUN
cana-807	60	17	v	v	ADP
cana-807	60	18	=	=	SYM
cana-807	60	19	⟨(0.8	⟨(0.8	ADJ
cana-807	60	20	,	,	PUNCT
cana-807	60	21	0.2	0.2	NUM
cana-807	60	22	,	,	PUNCT
cana-807	60	23	0.2	0.2	NUM
cana-807	60	24	)	)	PUNCT
cana-807	60	25	,	,	PUNCT
cana-807	60	26	(	(	PUNCT
cana-807	60	27	0.7	0.7	NUM
cana-807	60	28	,	,	PUNCT
cana-807	60	29	0.2	0.2	NUM
cana-807	60	30	,	,	PUNCT
cana-807	60	31	0.2)⟩	0.2)⟩	ADJ
cana-807	60	32	then	then	ADV
cana-807	60	33	x	x	PUNCT
cana-807	60	34	is	be	AUX
cana-807	60	35	a	a	DET
cana-807	60	36	n	n	ADV
cana-807	60	37	-	-	PUNCT
cana-807	60	38	ts	ts	NOUN
cana-807	60	39	.	.	PUNCT
cana-807	61	1	here	here	ADV
cana-807	61	2	the	the	DET
cana-807	61	3	n	n	CCONJ
cana-807	61	4	-	-	PUNCT
cana-807	61	5	s	s	NOUN
cana-807	61	6	s=	s=	NOUN
cana-807	61	7	⟨(0.1	⟨(0.1	PROPN
cana-807	61	8	,	,	PUNCT
cana-807	61	9	0.2	0.2	NUM
cana-807	61	10	,	,	PUNCT
cana-807	61	11	0.7	0.7	NUM
cana-807	61	12	)	)	PUNCT
cana-807	61	13	,	,	PUNCT
cana-807	61	14	(	(	PUNCT
cana-807	61	15	0.2	0.2	NUM
cana-807	61	16	,	,	PUNCT
cana-807	61	17	0.2	0.2	NUM
cana-807	61	18	,	,	PUNCT
cana-807	61	19	0.8)⟩	0.8)⟩	NOUN
cana-807	61	20	is	be	AUX
cana-807	61	21	a	a	DET
cana-807	61	22	n	n	NOUN
cana-807	61	23	-	-	PUNCT
cana-807	61	24	pgpr*wcs	pgpr*wcs	NOUN
cana-807	61	25	in	in	ADP
cana-807	61	26	x.	x.	NOUN
cana-807	61	27	since	since	SCONJ
cana-807	61	28	s	s	PROPN
cana-807	61	29			PROPN
cana-807	61	30	m	m	PROPN
cana-807	61	31	and	and	CCONJ
cana-807	61	32	v	v	NOUN
cana-807	61	33	is	be	AUX
cana-807	61	34	nr*gαos	nr*gαo	NOUN
cana-807	61	35	since	since	SCONJ
cana-807	61	36	n	n	NOUN
cana-807	61	37	-	-	PUNCT
cana-807	61	38	pcl(s	pcl(s	NOUN
cana-807	61	39	)	)	PUNCT
cana-807	62	1	=	=	SYM
cana-807	62	2	s	s	PART
cana-807	62	3			PROPN
cana-807	62	4	m.	m.	NOUN
cana-807	62	5	proposition	proposition	NOUN
cana-807	62	6	3.3	3.3	NUM
cana-807	62	7	in	in	ADP
cana-807	62	8	(	(	PUNCT
cana-807	62	9	x	x	X
cana-807	62	10	,	,	PUNCT
cana-807	62	11	τ	τ	X
cana-807	62	12	)	)	PUNCT
cana-807	62	13	let	let	VERB
cana-807	62	14	s	s	PRON
cana-807	62	15	be	be	AUX
cana-807	62	16	a	a	DET
cana-807	62	17	n	n	CCONJ
cana-807	62	18	-	-	PUNCT
cana-807	62	19	s	s	NOUN
cana-807	62	20	satisfying	satisfy	VERB
cana-807	62	21	the	the	DET
cana-807	62	22	following	follow	VERB
cana-807	62	23	properties	property	NOUN
cana-807	62	24	every	every	DET
cana-807	62	25	1	1	NUM
cana-807	62	26	.	.	PUNCT
cana-807	62	27	n	n	CCONJ
cana-807	62	28	-	-	PUNCT
cana-807	62	29	pcs	pc	NOUN
cana-807	62	30	is	be	AUX
cana-807	62	31	n	n	PRON
cana-807	62	32	-	-	PUNCT
cana-807	62	33	pgpr*wcs	pgpr*wcs	NOUN
cana-807	62	34	2	2	NUM
cana-807	62	35	.	.	NUM
cana-807	62	36	n	n	CCONJ
cana-807	62	37	-	-	PUNCT
cana-807	62	38	cs	cs	PROPN
cana-807	62	39	is	be	AUX
cana-807	62	40	n	n	PRON
cana-807	62	41	-	-	PUNCT
cana-807	62	42	pgpr*wcs	pgpr*wcs	NOUN
cana-807	62	43	3	3	NUM
cana-807	62	44	.	.	PUNCT
cana-807	62	45	n	n	CCONJ
cana-807	62	46	-	-	PUNCT
cana-807	62	47	rcs	rcs	PROPN
cana-807	62	48	is	be	AUX
cana-807	62	49	n	n	PRON
cana-807	62	50	-	-	PUNCT
cana-807	62	51	pgpr*wcs	pgpr*wcs	NOUN
cana-807	62	52	4	4	NUM
cana-807	62	53	.	.	PUNCT
cana-807	63	1	n	n	CCONJ
cana-807	63	2	-	-	PUNCT
cana-807	63	3	gcs	gcs	PROPN
cana-807	63	4	is	be	AUX
cana-807	63	5	n	n	PRON
cana-807	63	6	-	-	PUNCT
cana-807	63	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	63	8	5	5	NUM
cana-807	63	9	.	.	PUNCT
cana-807	63	10	n	n	CCONJ
cana-807	63	11	-	-	PUNCT
cana-807	63	12	alphacs	alphacs	PROPN
cana-807	63	13	is	be	AUX
cana-807	63	14	n	n	PRON
cana-807	63	15	-	-	PUNCT
cana-807	63	16	pgpr*wcs	pgpr*wcs	NOUN
cana-807	63	17	6	6	NUM
cana-807	63	18	.	.	PUNCT
cana-807	64	1	n	n	CCONJ
cana-807	64	2	-	-	PUNCT
cana-807	64	3	wcs	wcs	NOUN
cana-807	64	4	is	be	AUX
cana-807	64	5	n	n	PRON
cana-807	64	6	-	-	PUNCT
cana-807	64	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	64	8	7	7	NUM
cana-807	64	9	.	.	NUM
cana-807	64	10	n	n	CCONJ
cana-807	64	11	-	-	PUNCT
cana-807	64	12	pgpr*wcs	pgpr*wcs	NOUN
cana-807	64	13	is	be	AUX
cana-807	64	14	n	n	PRON
cana-807	64	15	-	-	PUNCT
cana-807	64	16	gpcs	gpcs	NOUN
cana-807	64	17	8	8	NUM
cana-807	64	18	.	.	PUNCT
cana-807	65	1	n	n	CCONJ
cana-807	65	2	-	-	PUNCT
cana-807	65	3	pgpr*wcs	pgpr*wcs	NOUN
cana-807	65	4	is	be	AUX
cana-807	65	5	n	n	CCONJ
cana-807	65	6	-	-	PUNCT
cana-807	65	7	gprcs	gprcs	NOUN
cana-807	65	8	proof	proof	NOUN
cana-807	65	9	.	.	PUNCT
cana-807	66	1	1	1	X
cana-807	66	2	.	.	X
cana-807	66	3	let	let	VERB
cana-807	66	4	m	m	PRON
cana-807	66	5	be	be	AUX
cana-807	66	6	a	a	DET
cana-807	66	7	n	n	NOUN
cana-807	66	8	-	-	PUNCT
cana-807	66	9	r*gαcs	r*gαcs	NOUN
cana-807	66	10	in	in	ADP
cana-807	66	11	x	x	PUNCT
cana-807	66	12	as	as	SCONJ
cana-807	66	13	s	s	PROPN
cana-807	66	14	is	be	AUX
cana-807	66	15	npcs	npc	NOUN
cana-807	66	16	,	,	PUNCT
cana-807	66	17	then	then	ADV
cana-807	66	18	n	n	CCONJ
cana-807	66	19	-	-	PUNCT
cana-807	66	20	cl(n	cl(n	NOUN
cana-807	66	21	-	-	PUNCT
cana-807	66	22	int	int	NOUN
cana-807	66	23	(	(	PUNCT
cana-807	66	24	s	s	NOUN
cana-807	66	25	)	)	PUNCT
cana-807	66	26	)	)	PUNCT
cana-807	67	1	⊆	⊆	NUM
cana-807	67	2	s.	s.	PROPN
cana-807	67	3	so	so	CCONJ
cana-807	67	4	n	n	CCONJ
cana-807	67	5	-	-	PUNCT
cana-807	67	6	pcl	pcl	PROPN
cana-807	67	7	(	(	PUNCT
cana-807	67	8	s	s	PROPN
cana-807	67	9	)	)	PUNCT
cana-807	67	10	=	=	SYM
cana-807	67	11	s	s	X
cana-807	67	12	∪	∪	VERB
cana-807	67	13	ncl(n	ncl(n	ADJ
cana-807	67	14	-	-	PUNCT
cana-807	67	15	int	int	NOUN
cana-807	67	16	(	(	PUNCT
cana-807	67	17	s	s	NOUN
cana-807	67	18	)	)	PUNCT
cana-807	67	19	)	)	PUNCT
cana-807	68	1	⊆	⊆	NUM
cana-807	68	2	s∪	s∪	NOUN
cana-807	68	3	s	s	PART
cana-807	68	4	=	=	SYM
cana-807	68	5	s	s	PROPN
cana-807	68	6	⊆	⊆	NUM
cana-807	68	7	m	m	VERB
cana-807	68	8	hence	hence	ADV
cana-807	68	9	s	s	NOUN
cana-807	68	10	is	be	AUX
cana-807	68	11	n	n	PRON
cana-807	68	12	-	-	PUNCT
cana-807	68	13	pgpr*wcs	pgpr*wcs	NOUN
cana-807	68	14	in	in	ADP
cana-807	68	15	x.	x.	NOUN
cana-807	68	16	2	2	X
cana-807	68	17	.	.	PUNCT
cana-807	69	1	let	let	VERB
cana-807	69	2	s	s	PRON
cana-807	69	3	⊆	⊆	NUM
cana-807	69	4	m	m	NOUN
cana-807	69	5	and	and	CCONJ
cana-807	69	6	,	,	PUNCT
cana-807	69	7	m	m	AUX
cana-807	69	8	be	be	VERB
cana-807	69	9	a	a	DET
cana-807	69	10	n	n	NOUN
cana-807	69	11	-	-	PUNCT
cana-807	69	12	r*gαcs	r*gαcs	NOUN
cana-807	69	13	in	in	ADP
cana-807	69	14	x	x	PUNCT
cana-807	69	15	as	as	SCONJ
cana-807	69	16	s	s	PROPN
cana-807	69	17	is	be	AUX
cana-807	69	18	n	n	PRON
cana-807	69	19	-	-	PUNCT
cana-807	69	20	cs	cs	NOUN
cana-807	69	21	in	in	ADP
cana-807	69	22	x	x	PROPN
cana-807	69	23	,	,	PUNCT
cana-807	69	24	n	n	CCONJ
cana-807	69	25	-	-	PUNCT
cana-807	69	26	cl	cl	NOUN
cana-807	69	27	(	(	PUNCT
cana-807	69	28	s	s	X
cana-807	69	29	)	)	PUNCT
cana-807	69	30	=	=	PUNCT
cana-807	69	31	s.	s.	PROPN
cana-807	69	32	implies	imply	VERB
cana-807	69	33	n	n	CCONJ
cana-807	69	34	-	-	PUNCT
cana-807	69	35	pcl	pcl	PROPN
cana-807	69	36	(	(	PUNCT
cana-807	69	37	s	s	PROPN
cana-807	69	38	)	)	PUNCT
cana-807	69	39	⊆	⊆	NUM
cana-807	69	40	n	n	CCONJ
cana-807	69	41	-	-	PUNCT
cana-807	69	42	cl(s	cl(s	NOUN
cana-807	69	43	)	)	PUNCT
cana-807	69	44	=	=	SYM
cana-807	69	45	s	s	PROPN
cana-807	69	46	⊆	⊆	NUM
cana-807	69	47	m	m	NOUN
cana-807	69	48	by	by	ADP
cana-807	69	49	hypothesis	hypothesis	NOUN
cana-807	69	50	.	.	PUNCT
cana-807	70	1	s	s	PART
cana-807	70	2	is	be	AUX
cana-807	70	3	a	a	DET
cana-807	70	4	npgpr*wcs	npgpr*wcs	NOUN
cana-807	70	5	in	in	ADP
cana-807	70	6	x.	x.	NOUN
cana-807	70	7	3	3	X
cana-807	70	8	.	.	PUNCT
cana-807	71	1	let	let	VERB
cana-807	71	2	s	s	PRON
cana-807	71	3	⊆	⊆	NUM
cana-807	71	4	m	m	NOUN
cana-807	71	5	and	and	CCONJ
cana-807	71	6	m	m	AUX
cana-807	71	7	be	be	AUX
cana-807	71	8	a	a	DET
cana-807	71	9	n	n	NOUN
cana-807	71	10	-	-	PUNCT
cana-807	71	11	r*gαos	r*gαo	NOUN
cana-807	71	12	as	as	SCONJ
cana-807	71	13	every	every	DET
cana-807	71	14	n	n	NUM
cana-807	71	15	-	-	PUNCT
cana-807	71	16	rcs	rcs	NOUN
cana-807	71	17	is	be	AUX
cana-807	71	18	n	n	NOUN
cana-807	71	19	-	-	PUNCT
cana-807	71	20	cs	cs	PROPN
cana-807	71	21	then	then	ADV
cana-807	71	22	n	n	CCONJ
cana-807	71	23	-	-	PUNCT
cana-807	71	24	cl(s	cl(s	NOUN
cana-807	71	25	)	)	PUNCT
cana-807	71	26	=	=	VERB
cana-807	72	1	s.	s.	PROPN
cana-807	72	2	by	by	ADP
cana-807	72	3	hypothesis	hypothesis	NOUN
cana-807	72	4	npcl	npcl	NOUN
cana-807	72	5	(	(	PUNCT
cana-807	72	6	s	s	X
cana-807	72	7	)	)	PUNCT
cana-807	72	8	⊆	⊆	NUM
cana-807	72	9	n	n	CCONJ
cana-807	72	10	-	-	PUNCT
cana-807	72	11	cl	cl	NOUN
cana-807	72	12	(	(	PUNCT
cana-807	72	13	s	s	NOUN
cana-807	72	14	)	)	PUNCT
cana-807	72	15	⊆	⊆	NUM
cana-807	72	16	m	m	NOUN
cana-807	72	17	and	and	CCONJ
cana-807	72	18	n	n	CCONJ
cana-807	72	19	-	-	PUNCT
cana-807	72	20	pcl(s	pcl(s	PROPN
cana-807	72	21	)	)	PUNCT
cana-807	72	22	⊆	⊆	NUM
cana-807	72	23	m	m	VERB
cana-807	72	24	thus	thus	ADV
cana-807	72	25	s	s	VERB
cana-807	72	26	is	be	AUX
cana-807	72	27	n	n	PRON
cana-807	72	28	-	-	PUNCT
cana-807	72	29	pgpr*wcs	pgpr*wcs	NOUN
cana-807	72	30	4	4	NUM
cana-807	72	31	.	.	PUNCT
cana-807	73	1	let	let	VERB
cana-807	73	2	m	m	PRON
cana-807	73	3	be	be	AUX
cana-807	73	4	a	a	DET
cana-807	73	5	n	n	NOUN
cana-807	73	6	-	-	PUNCT
cana-807	73	7	r*gαos	r*gαo	NOUN
cana-807	73	8	such	such	ADJ
cana-807	73	9	that	that	PRON
cana-807	73	10	s	s	VERB
cana-807	73	11	⊆	⊆	NUM
cana-807	73	12	m.	m.	NOUN
cana-807	73	13	since	since	SCONJ
cana-807	73	14	every	every	DET
cana-807	73	15	n	n	CCONJ
cana-807	73	16	-	-	PUNCT
cana-807	73	17	gcs	gcs	PROPN
cana-807	73	18	is	be	AUX
cana-807	73	19	a	a	DET
cana-807	73	20	n	n	NOUN
cana-807	73	21	-	-	PUNCT
cana-807	73	22	cs	cs	PROPN
cana-807	73	23	.	.	PUNCT
cana-807	74	1	we	we	PRON
cana-807	74	2	have	have	VERB
cana-807	74	3	n	n	NOUN
cana-807	74	4	-	-	PUNCT
cana-807	74	5	cl(s	cl(s	NOUN
cana-807	74	6	)	)	PUNCT
cana-807	74	7	=	=	VERB
cana-807	75	1	s.	s.	PROPN
cana-807	75	2	by	by	ADP
cana-807	75	3	hypothesis	hypothesis	NOUN
cana-807	75	4	,	,	PUNCT
cana-807	75	5	n	n	CCONJ
cana-807	75	6	-	-	PUNCT
cana-807	75	7	pcl(s	pcl(s	PROPN
cana-807	75	8	)	)	PUNCT
cana-807	75	9	⊆	⊆	NUM
cana-807	75	10	n	n	CCONJ
cana-807	75	11	-	-	PUNCT
cana-807	75	12	cl(s	cl(s	NOUN
cana-807	75	13	)	)	PUNCT
cana-807	75	14	⊆	⊆	NUM
cana-807	75	15	m	m	NOUN
cana-807	75	16	hence	hence	ADV
cana-807	75	17	n	n	CCONJ
cana-807	75	18	-	-	PUNCT
cana-807	75	19	pcl(s	pcl(s	PROPN
cana-807	75	20	)	)	PUNCT
cana-807	75	21	⊆	⊆	NUM
cana-807	75	22	m.	m.	NOUN
cana-807	75	23	thus	thus	ADV
cana-807	75	24	s	s	VERB
cana-807	75	25	is	be	AUX
cana-807	75	26	n	n	PRON
cana-807	75	27	-	-	PUNCT
cana-807	75	28	pgpr*wcs	pgpr*wcs	NOUN
cana-807	75	29	.	.	PUNCT
cana-807	76	1	5	5	X
cana-807	76	2	.	.	X
cana-807	76	3	let	let	VERB
cana-807	76	4	s	s	PRON
cana-807	76	5	⊆	⊆	NUM
cana-807	76	6	m	m	NOUN
cana-807	76	7	and	and	CCONJ
cana-807	76	8	m	m	AUX
cana-807	76	9	be	be	AUX
cana-807	76	10	a	a	DET
cana-807	76	11	n	n	NOUN
cana-807	76	12	-	-	PUNCT
cana-807	76	13	r*gαos	r*gαo	NOUN
cana-807	76	14	in	in	SCONJ
cana-807	76	15	x	x	PUNCT
cana-807	76	16	as	as	SCONJ
cana-807	76	17	s	s	PROPN
cana-807	76	18	is	be	AUX
cana-807	76	19	n	n	PRON
cana-807	76	20	-	-	PUNCT
cana-807	76	21	αcs	αcs	NOUN
cana-807	76	22	n	n	NOUN
cana-807	76	23	-	-	PUNCT
cana-807	76	24	cl	cl	NOUN
cana-807	76	25	(	(	PUNCT
cana-807	76	26	n	n	CCONJ
cana-807	76	27	-	-	PUNCT
cana-807	76	28	int	int	NOUN
cana-807	76	29	(	(	PUNCT
cana-807	76	30	n	n	CCONJ
cana-807	76	31	-	-	PUNCT
cana-807	76	32	cl	cl	NOUN
cana-807	76	33	(	(	PUNCT
cana-807	76	34	s	s	NOUN
cana-807	76	35	)	)	PUNCT
cana-807	76	36	)	)	PUNCT
cana-807	77	1	⊆	⊆	X
cana-807	77	2	s.	s.	PROPN
cana-807	77	3	also	also	ADV
cana-807	77	4	s	s	VERB
cana-807	77	5	⊆	⊆	NUM
cana-807	77	6	n	n	CCONJ
cana-807	77	7	-	-	PUNCT
cana-807	77	8	cl	cl	NOUN
cana-807	77	9	(	(	PUNCT
cana-807	77	10	s	s	NOUN
cana-807	77	11	)	)	PUNCT
cana-807	77	12	implies	imply	VERB
cana-807	77	13	n	n	CCONJ
cana-807	77	14	-	-	PUNCT
cana-807	77	15	cl	cl	NOUN
cana-807	77	16	(	(	PUNCT
cana-807	77	17	n	n	CCONJ
cana-807	77	18	-	-	PUNCT
cana-807	77	19	int(s	int(s	PROPN
cana-807	77	20	)	)	PUNCT
cana-807	77	21	)	)	PUNCT
cana-807	78	1	⊆	⊆	NUM
cana-807	78	2	n	n	CCONJ
cana-807	78	3	-	-	PUNCT
cana-807	78	4	cl(n	cl(n	NOUN
cana-807	78	5	-	-	PUNCT
cana-807	78	6	int	int	NOUN
cana-807	78	7	(	(	PUNCT
cana-807	78	8	n	n	CCONJ
cana-807	78	9	-	-	PUNCT
cana-807	78	10	cl(s	cl(s	NOUN
cana-807	78	11	)	)	PUNCT
cana-807	78	12	)	)	PUNCT
cana-807	79	1	⊆	⊆	NUM
cana-807	79	2	s.	s.	PROPN
cana-807	79	3	implies	imply	VERB
cana-807	79	4	n	n	CCONJ
cana-807	79	5	-	-	PUNCT
cana-807	79	6	pcl(s	pcl(s	PROPN
cana-807	79	7	)	)	PUNCT
cana-807	79	8	=	=	PUNCT
cana-807	79	9	s∪n	s∪n	NOUN
cana-807	79	10	-	-	PUNCT
cana-807	79	11	cl(nint(s	cl(nint(s	X
cana-807	79	12	)	)	PUNCT
cana-807	79	13	)	)	PUNCT
cana-807	80	1	⊆	⊆	NUM
cana-807	80	2	s∪s	s∪s	NOUN
cana-807	80	3	=	=	SYM
cana-807	80	4	s⊆	s⊆	PROPN
cana-807	80	5	m.	m.	NOUN
cana-807	80	6	therefore	therefore	ADV
cana-807	80	7	s	s	VERB
cana-807	80	8	is	be	AUX
cana-807	80	9	n	n	PRON
cana-807	80	10	-	-	PUNCT
cana-807	80	11	pgpr*wcs	pgpr*wcs	NOUN
cana-807	80	12	in	in	ADP
cana-807	80	13	x	x	PROPN
cana-807	80	14	communications	communication	NOUN
cana-807	80	15	on	on	ADP
cana-807	80	16	applied	apply	VERB
cana-807	80	17	nonlinear	nonlinear	ADJ
cana-807	80	18	analysis	analysis	NOUN
cana-807	80	19	issn	issn	NOUN
cana-807	80	20	:	:	PUNCT
cana-807	80	21	1074	1074	NUM
cana-807	80	22	-	-	PUNCT
cana-807	80	23	133x	133x	NUM
cana-807	80	24	vol	vol	NOUN
cana-807	80	25	31	31	NUM
cana-807	80	26	no	no	NOUN
cana-807	80	27	.	.	PUNCT
cana-807	81	1	3s	3s	NUM
cana-807	81	2	(	(	PUNCT
cana-807	81	3	2024	2024	NUM
cana-807	81	4	)	)	PUNCT
cana-807	81	5	551	551	NUM
cana-807	81	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	81	7	6	6	NUM
cana-807	81	8	.	.	PUNCT
cana-807	82	1	let	let	VERB
cana-807	82	2	s	s	PRON
cana-807	82	3	⊆	⊆	NUM
cana-807	82	4	m	m	NOUN
cana-807	82	5	,	,	PUNCT
cana-807	82	6	and	and	CCONJ
cana-807	82	7	m	m	AUX
cana-807	82	8	be	be	AUX
cana-807	82	9	a	a	DET
cana-807	82	10	n	n	NOUN
cana-807	82	11	-	-	PUNCT
cana-807	82	12	r*gαos	r*gαo	NOUN
cana-807	82	13	in	in	ADP
cana-807	82	14	x.as	x.as	NOUN
cana-807	82	15	every	every	DET
cana-807	82	16	n	n	NUM
cana-807	82	17	-	-	PUNCT
cana-807	82	18	wcs	wcs	NOUN
cana-807	82	19	is	be	AUX
cana-807	82	20	n	n	X
cana-807	82	21	-	-	PUNCT
cana-807	82	22	cs	cs	PROPN
cana-807	82	23	.	.	PROPN
cana-807	82	24	implies	imply	VERB
cana-807	82	25	n	n	CCONJ
cana-807	82	26	-	-	PUNCT
cana-807	82	27	cl	cl	NOUN
cana-807	82	28	(	(	PUNCT
cana-807	82	29	s	s	X
cana-807	82	30	)	)	PUNCT
cana-807	82	31	=	=	VERB
cana-807	82	32	s.	s.	PROPN
cana-807	82	33	by	by	ADP
cana-807	82	34	hypothesis	hypothesis	NOUN
cana-807	82	35	,	,	PUNCT
cana-807	82	36	n	n	CCONJ
cana-807	82	37	-	-	PUNCT
cana-807	82	38	pcl(s	pcl(s	PROPN
cana-807	82	39	)	)	PUNCT
cana-807	82	40	⊆	⊆	NUM
cana-807	82	41	n	n	CCONJ
cana-807	82	42	-	-	PUNCT
cana-807	82	43	cl(s	cl(s	NOUN
cana-807	82	44	)	)	PUNCT
cana-807	82	45	⊆	⊆	NUM
cana-807	82	46	m	m	NOUN
cana-807	82	47	then	then	ADV
cana-807	82	48	n	n	CCONJ
cana-807	82	49	-	-	PUNCT
cana-807	82	50	pcl(s	pcl(s	PROPN
cana-807	82	51	)	)	PUNCT
cana-807	82	52	⊆	⊆	NUM
cana-807	82	53	s.	s.	PROPN
cana-807	82	54	since	since	SCONJ
cana-807	82	55	s	s	PROPN
cana-807	82	56	is	be	AUX
cana-807	82	57	n	n	PRON
cana-807	82	58	-	-	PUNCT
cana-807	82	59	os	os	NOUN
cana-807	82	60	.	.	PUNCT
cana-807	83	1	we	we	PRON
cana-807	83	2	have	have	VERB
cana-807	83	3	n	n	NOUN
cana-807	83	4	-	-	PUNCT
cana-807	83	5	cl	cl	NOUN
cana-807	83	6	(	(	PUNCT
cana-807	83	7	s	s	X
cana-807	83	8	)	)	PUNCT
cana-807	83	9	=	=	SYM
cana-807	83	10	s	s	AUX
cana-807	83	11	therefore	therefore	ADV
cana-807	83	12	s	s	X
cana-807	83	13	is	be	AUX
cana-807	83	14	n	n	PRON
cana-807	83	15	-	-	PUNCT
cana-807	83	16	pgpr*wcs	pgpr*wcs	NOUN
cana-807	83	17	in	in	ADP
cana-807	83	18	x	x	PROPN
cana-807	83	19	7	7	X
cana-807	83	20	.	.	PUNCT
cana-807	84	1	let	let	AUX
cana-807	84	2	m	m	PRON
cana-807	84	3	be	be	AUX
cana-807	84	4	n	n	ADV
cana-807	84	5	-	-	ADJ
cana-807	84	6	os	os	NOUN
cana-807	84	7	in	in	ADP
cana-807	84	8	x	x	PUNCT
cana-807	84	9	such	such	ADJ
cana-807	84	10	that	that	DET
cana-807	84	11	s	s	NOUN
cana-807	84	12	⊆	⊆	NUM
cana-807	84	13	m.	m.	NOUN
cana-807	84	14	since	since	SCONJ
cana-807	84	15	s	s	PROPN
cana-807	84	16	is	be	AUX
cana-807	84	17	n	n	PRON
cana-807	84	18	-	-	PUNCT
cana-807	84	19	pgpr*wcs	pgpr*wcs	NOUN
cana-807	84	20	in	in	ADP
cana-807	84	21	x	x	NOUN
cana-807	84	22	,	,	PUNCT
cana-807	84	23	and	and	CCONJ
cana-807	84	24	every	every	DET
cana-807	84	25	n	n	CCONJ
cana-807	84	26	-	-	PUNCT
cana-807	84	27	os	os	NOUN
cana-807	84	28	is	be	AUX
cana-807	84	29	nr*gαos	nr*gαo	NOUN
cana-807	84	30	in	in	ADP
cana-807	84	31	x.we	x.we	PROPN
cana-807	84	32	have	have	AUX
cana-807	84	33	m	m	PROPN
cana-807	84	34	is	be	AUX
cana-807	84	35	n	n	PRON
cana-807	84	36	-	-	PUNCT
cana-807	84	37	r*gαos	r*gαo	NOUN
cana-807	84	38	in	in	ADP
cana-807	84	39	x	x	SYM
cana-807	84	40	therefore	therefore	ADV
cana-807	84	41	,	,	PUNCT
cana-807	84	42	n	n	CCONJ
cana-807	84	43	-	-	PUNCT
cana-807	84	44	pcl(s	pcl(s	PROPN
cana-807	84	45	)	)	PUNCT
cana-807	84	46	⊆	⊆	NUM
cana-807	84	47	m.	m.	NOUN
cana-807	84	48	hence	hence	ADV
cana-807	84	49	s	s	VERB
cana-807	84	50	is	be	AUX
cana-807	84	51	n	n	PRON
cana-807	84	52	-	-	PUNCT
cana-807	84	53	gpcs	gpc	NOUN
cana-807	84	54	in	in	ADP
cana-807	84	55	x	x	SYM
cana-807	84	56	8	8	NUM
cana-807	84	57	let	let	VERB
cana-807	84	58	s	s	PRON
cana-807	84	59	be	be	AUX
cana-807	84	60	n	n	X
cana-807	84	61	-	-	PUNCT
cana-807	84	62	pgpr*wcs	pgpr*wcs	NOUN
cana-807	84	63	and	and	CCONJ
cana-807	84	64	m	m	AUX
cana-807	84	65	be	be	VERB
cana-807	84	66	n	n	PRON
cana-807	84	67	-	-	PUNCT
cana-807	84	68	pos	pos	NOUN
cana-807	84	69	such	such	ADJ
cana-807	84	70	that	that	PRON
cana-807	84	71	s	s	VERB
cana-807	84	72	⊆	⊆	NUM
cana-807	84	73	m.	m.	NOUN
cana-807	84	74	as	as	SCONJ
cana-807	84	75	every	every	DET
cana-807	84	76	n	n	NUM
cana-807	84	77	-	-	PUNCT
cana-807	84	78	ros	ros	PROPN
cana-807	84	79	is	be	AUX
cana-807	84	80	n	n	PRON
cana-807	84	81	-	-	PUNCT
cana-807	84	82	r*gαos	r*gαo	NOUN
cana-807	84	83	m	m	NOUN
cana-807	84	84	is	be	AUX
cana-807	84	85	n	n	PRON
cana-807	84	86	-	-	PUNCT
cana-807	84	87	r*gαos	r*gαo	NOUN
cana-807	84	88	in	in	SCONJ
cana-807	84	89	x	x	PUNCT
cana-807	84	90	since	since	SCONJ
cana-807	84	91	s	s	PROPN
cana-807	84	92	is	be	AUX
cana-807	84	93	n	n	PRON
cana-807	84	94	-	-	PUNCT
cana-807	84	95	pgpr*wcs	pgpr*wcs	NOUN
cana-807	84	96	.	.	PUNCT
cana-807	85	1	then	then	ADV
cana-807	85	2	n	n	CCONJ
cana-807	85	3	-	-	PUNCT
cana-807	85	4	pcl(s	pcl(s	PROPN
cana-807	85	5	)	)	PUNCT
cana-807	86	1	⊆	⊆	NUM
cana-807	86	2	u.	u.	NOUN
cana-807	86	3	therefore	therefore	ADV
cana-807	86	4	s	s	VERB
cana-807	86	5	is	be	AUX
cana-807	86	6	n	n	PRON
cana-807	86	7	-	-	PUNCT
cana-807	86	8	gprcs	gprcs	NOUN
cana-807	86	9	.	.	PUNCT
cana-807	87	1	remark	remark	NOUN
cana-807	87	2	:	:	PUNCT
cana-807	87	3	3.4	3.4	NUM
cana-807	87	4	the	the	DET
cana-807	87	5	converse	converse	NOUN
cana-807	87	6	of	of	ADP
cana-807	87	7	subdivision	subdivision	NOUN
cana-807	87	8	1	1	NUM
cana-807	87	9	to	to	PART
cana-807	87	10	5	5	NUM
cana-807	87	11	of	of	ADP
cana-807	87	12	theorem	theorem	ADJ
cana-807	87	13	3.3	3.3	NUM
cana-807	87	14	n	n	NOUN
cana-807	87	15	e	e	NOUN
cana-807	87	16	e	e	X
cana-807	87	17	d	d	PROPN
cana-807	87	18	n	n	X
cana-807	87	19	o	o	X
cana-807	87	20	t	t	PROPN
cana-807	87	21	b	b	PROPN
cana-807	87	22	e	e	PROPN
cana-807	87	23	t	t	PROPN
cana-807	87	24	r	r	NOUN
cana-807	87	25	u	u	NOUN
cana-807	87	26	e	e	PROPN
cana-807	87	27	c	c	PROPN
cana-807	87	28	a	a	PRON
cana-807	87	29	n	n	NOUN
cana-807	87	30	b	b	NOUN
cana-807	87	31	e	e	NOUN
cana-807	87	32	proved	prove	VERB
cana-807	87	33	by	by	ADP
cana-807	87	34	the	the	DET
cana-807	87	35	following	following	ADJ
cana-807	87	36	example	example	NOUN
cana-807	87	37	example	example	NOUN
cana-807	87	38	3.5	3.5	NUM
cana-807	87	39	as	as	ADP
cana-807	87	40	x	x	PROPN
cana-807	87	41	,	,	PUNCT
cana-807	87	42	τ	τ	PROPN
cana-807	87	43	,	,	PUNCT
cana-807	87	44	u	u	PROPN
cana-807	87	45	,	,	PUNCT
cana-807	87	46	m	m	AUX
cana-807	87	47	defined	define	VERB
cana-807	87	48	in	in	ADP
cana-807	87	49	ex	ex	NUM
cana-807	87	50	3.2	3.2	NUM
cana-807	87	51	n	n	CCONJ
cana-807	87	52	-	-	PUNCT
cana-807	87	53	s	s	NOUN
cana-807	87	54	s=	s=	NOUN
cana-807	87	55	⟨(0.8	⟨(0.8	PROPN
cana-807	87	56	,	,	PUNCT
cana-807	87	57	0.2	0.2	NUM
cana-807	87	58	,	,	PUNCT
cana-807	87	59	0.1	0.1	NUM
cana-807	87	60	)	)	PUNCT
cana-807	87	61	,	,	PUNCT
cana-807	87	62	(	(	PUNCT
cana-807	87	63	0.8	0.8	NUM
cana-807	87	64	,	,	PUNCT
cana-807	87	65	0.2	0.2	NUM
cana-807	87	66	,	,	PUNCT
cana-807	87	67	0.1)⟩	0.1)⟩	PRON
cana-807	87	68	is	be	AUX
cana-807	87	69	a	a	DET
cana-807	87	70	n	n	NOUN
cana-807	87	71	-	-	PUNCT
cana-807	87	72	pgpr*wcs	pgpr*wcs	NOUN
cana-807	87	73	in	in	ADP
cana-807	87	74	x	x	X
cana-807	87	75	s	s	VERB
cana-807	87	76	is	be	AUX
cana-807	87	77	not	not	PART
cana-807	87	78	n	n	CCONJ
cana-807	87	79	-	-	PUNCT
cana-807	87	80	cs	cs	ADJ
cana-807	87	81	,	,	PUNCT
cana-807	87	82	n	n	CCONJ
cana-807	87	83	-	-	PUNCT
cana-807	87	84	pcs	pc	NOUN
cana-807	87	85	,	,	PUNCT
cana-807	87	86	−cs	−cs	NOUN
cana-807	87	87	,	,	PUNCT
cana-807	87	88	n	n	CCONJ
cana-807	87	89	-	-	PUNCT
cana-807	87	90	rcs	rcs	NOUN
cana-807	87	91	,	,	PUNCT
cana-807	87	92	n	n	CCONJ
cana-807	87	93	-	-	PUNCT
cana-807	87	94	wcs	wcs	NOUN
cana-807	87	95	,	,	PUNCT
cana-807	87	96	n	n	CCONJ
cana-807	87	97	-	-	PUNCT
cana-807	87	98	gcs	gcs	PROPN
cana-807	87	99	remark	remark	NOUN
cana-807	87	100	:	:	PUNCT
cana-807	87	101	3.6	3.6	NUM
cana-807	87	102	the	the	DET
cana-807	87	103	converse	converse	NOUN
cana-807	87	104	of	of	ADP
cana-807	87	105	subdivision	subdivision	NOUN
cana-807	87	106	6	6	NUM
cana-807	87	107	and	and	CCONJ
cana-807	87	108	7	7	NUM
cana-807	87	109	of	of	ADP
cana-807	87	110	theorem	theorem	ADJ
cana-807	87	111	3.3	3.3	NUM
cana-807	87	112	proved	prove	VERB
cana-807	87	113	by	by	ADP
cana-807	87	114	the	the	DET
cana-807	87	115	following	following	ADJ
cana-807	87	116	example	example	NOUN
cana-807	87	117	example	example	NOUN
cana-807	87	118	3.7	3.7	NUM
cana-807	87	119	as	as	ADP
cana-807	87	120	x	x	PROPN
cana-807	87	121	,	,	PUNCT
cana-807	87	122	τ	τ	PROPN
cana-807	87	123	,	,	PUNCT
cana-807	87	124	defined	define	VERB
cana-807	87	125	in	in	ADP
cana-807	87	126	ex	ex	NUM
cana-807	87	127	3.2	3.2	NUM
cana-807	87	128	m	m	NOUN
cana-807	87	129	=	=	SYM
cana-807	87	130	{	{	PUNCT
cana-807	87	131	(	(	PUNCT
cana-807	87	132	0.5	0.5	NUM
cana-807	87	133	,	,	PUNCT
cana-807	87	134	0.2	0.2	NUM
cana-807	87	135	,	,	PUNCT
cana-807	87	136	0.5	0.5	NUM
cana-807	87	137	)	)	PUNCT
cana-807	87	138	,	,	PUNCT
cana-807	87	139	(	(	PUNCT
cana-807	87	140	0.2	0.2	NUM
cana-807	87	141	,	,	PUNCT
cana-807	87	142	02	02	NUM
cana-807	87	143	,	,	PUNCT
cana-807	87	144	0.8	0.8	NUM
cana-807	87	145	)	)	PUNCT
cana-807	87	146	}	}	PUNCT
cana-807	87	147	here	here	ADV
cana-807	87	148	the	the	DET
cana-807	87	149	n	n	NOUN
cana-807	87	150	-	-	PUNCT
cana-807	87	151	s	s	NOUN
cana-807	87	152	s	s	NOUN
cana-807	87	153	=	=	NOUN
cana-807	87	154	⟨	⟨	X
cana-807	87	155	(	(	PUNCT
cana-807	87	156	0.2,0.2	0.2,0.2	PROPN
cana-807	87	157	,	,	PUNCT
cana-807	87	158	0.6	0.6	NUM
cana-807	87	159	)	)	PUNCT
cana-807	87	160	(	(	PUNCT
cana-807	87	161	0.2	0.2	NUM
cana-807	87	162	0.2	0.2	NUM
cana-807	87	163	,	,	PUNCT
cana-807	87	164	0.8	0.8	NUM
cana-807	87	165	)	)	PUNCT
cana-807	87	166	⟩	⟩	NOUN
cana-807	87	167	s	s	NOUN
cana-807	87	168	is	be	AUX
cana-807	87	169	n	n	PRON
cana-807	87	170	-	-	PUNCT
cana-807	87	171	gpcs	gpc	NOUN
cana-807	87	172	as	as	ADP
cana-807	87	173	s	s	NOUN
cana-807	87	174	⊆	⊆	NUM
cana-807	87	175	m	m	NOUN
cana-807	87	176	,	,	PUNCT
cana-807	87	177	m	m	VERB
cana-807	87	178	is	be	AUX
cana-807	87	179	n	n	PRON
cana-807	87	180	-	-	PUNCT
cana-807	87	181	r*gαos	r*gαo	NOUN
cana-807	87	182	also	also	ADV
cana-807	87	183	n	n	CCONJ
cana-807	87	184	-	-	PUNCT
cana-807	87	185	pcl(s	pcl(s	NOUN
cana-807	87	186	)	)	PUNCT
cana-807	87	187	=	=	PUNCT
cana-807	88	1	mc	mc	PROPN
cana-807	88	2	⊆	⊆	NUM
cana-807	88	3	m	m	PROPN
cana-807	88	4	,	,	PUNCT
cana-807	88	5	.	.	PUNCT
cana-807	89	1	n	n	CCONJ
cana-807	89	2	-	-	PUNCT
cana-807	89	3	cl(s	cl(s	NOUN
cana-807	89	4	)	)	PUNCT
cana-807	90	1	=	=	PUNCT
cana-807	90	2	mc	mc	PROPN
cana-807	90	3	≠	≠	PROPN
cana-807	90	4	s.	s.	PROPN
cana-807	90	5	implies	imply	VERB
cana-807	90	6	s	s	PART
cana-807	90	7	is	be	AUX
cana-807	90	8	not	not	PART
cana-807	90	9	n	n	CCONJ
cana-807	90	10	-	-	PUNCT
cana-807	90	11	pgpr*wcs	pgpr*wcs	NOUN
cana-807	90	12	example	example	NOUN
cana-807	90	13	3.8	3.8	NUM
cana-807	90	14	from	from	ADP
cana-807	90	15	ex	ex	NUM
cana-807	90	16	3.7	3.7	NUM
cana-807	90	17	s	s	NOUN
cana-807	90	18	is	be	AUX
cana-807	90	19	n	n	PRON
cana-807	90	20	-	-	PUNCT
cana-807	90	21	gprcs	gprcs	NOUN
cana-807	90	22	the	the	DET
cana-807	90	23	n	n	NOUN
cana-807	90	24	-	-	PUNCT
cana-807	90	25	s	s	NOUN
cana-807	90	26	s	s	NOUN
cana-807	90	27	=	=	SYM
cana-807	90	28	⟨(0.5	⟨(0.5	PROPN
cana-807	90	29	,	,	PUNCT
cana-807	90	30	0.2	0.2	NUM
cana-807	90	31	,	,	PUNCT
cana-807	90	32	0.5	0.5	NUM
cana-807	90	33	)	)	PUNCT
cana-807	90	34	,	,	PUNCT
cana-807	90	35	(	(	PUNCT
cana-807	90	36	0.2	0.2	NUM
cana-807	90	37	,	,	PUNCT
cana-807	90	38	0.2	0.2	NUM
cana-807	90	39	,	,	PUNCT
cana-807	90	40	0.8)⟩	0.8)⟩	NOUN
cana-807	90	41	is	be	AUX
cana-807	90	42	n	n	CCONJ
cana-807	90	43	-	-	PUNCT
cana-807	90	44	gprcs	gprcs	NOUN
cana-807	90	45	since	since	SCONJ
cana-807	90	46	n	n	NOUN
cana-807	90	47	-	-	PUNCT
cana-807	90	48	pcl(s	pcl(s	NOUN
cana-807	90	49	)	)	PUNCT
cana-807	90	50	=	=	SYM
cana-807	90	51	s	s	PROPN
cana-807	90	52	⊆	⊆	NUM
cana-807	90	53	m	m	NOUN
cana-807	90	54	whenever	whenever	SCONJ
cana-807	90	55	s	s	VERB
cana-807	90	56	⊆	⊆	NUM
cana-807	90	57	m	m	NOUN
cana-807	90	58	where	where	SCONJ
cana-807	90	59	m	m	PRON
cana-807	90	60	is	be	AUX
cana-807	90	61	n	n	PRON
cana-807	90	62	-	-	PUNCT
cana-807	90	63	ros	ros	PROPN
cana-807	90	64	.	.	PUNCT
cana-807	91	1	but	but	CCONJ
cana-807	91	2	n	n	CCONJ
cana-807	91	3	-	-	PUNCT
cana-807	91	4	α	α	NOUN
cana-807	91	5	pcl(s	pcl(s	PROPN
cana-807	91	6	)	)	PUNCT
cana-807	92	1	=	=	SYM
cana-807	92	2	mc	mc	PROPN
cana-807	92	3	⊈	⊈	PROPN
cana-807	92	4	m.	m.	NOUN
cana-807	92	5	the	the	DET
cana-807	92	6	n	n	NOUN
cana-807	92	7	-	-	PUNCT
cana-807	92	8	s	s	NOUN
cana-807	92	9	s	s	NOUN
cana-807	92	10	is	be	AUX
cana-807	92	11	not	not	PART
cana-807	92	12	n	n	CCONJ
cana-807	92	13	-	-	PUNCT
cana-807	92	14	pgpr*wcs	pgpr*wcs	NOUN
cana-807	92	15	.	.	PUNCT
cana-807	93	1	theorem	theorem	VERB
cana-807	93	2	3.9	3.9	NUM
cana-807	93	3	:	:	PUNCT
cana-807	93	4	if	if	SCONJ
cana-807	93	5	s	s	NOUN
cana-807	93	6	is	be	AUX
cana-807	93	7	n	n	PRON
cana-807	93	8	-	-	PUNCT
cana-807	93	9	ros	ros	PROPN
cana-807	93	10	and	and	CCONJ
cana-807	93	11	n	n	CCONJ
cana-807	93	12	-	-	PUNCT
cana-807	93	13	pgpr*wcs	pgpr*wcs	NOUN
cana-807	93	14	in	in	ADP
cana-807	93	15	x	x	PUNCT
cana-807	94	1	then	then	ADV
cana-807	94	2	s	s	VERB
cana-807	94	3	is	be	AUX
cana-807	94	4	n	n	PRON
cana-807	94	5	-	-	PUNCT
cana-807	94	6	pcs	pc	NOUN
cana-807	94	7	proof	proof	NOUN
cana-807	94	8	:	:	PUNCT
cana-807	94	9	let	let	VERB
cana-807	94	10	s	s	PRON
cana-807	94	11	be	be	AUX
cana-807	94	12	n	n	X
cana-807	94	13	-	-	PUNCT
cana-807	94	14	ros	ros	PROPN
cana-807	94	15	&	&	CCONJ
cana-807	94	16	n	n	CCONJ
cana-807	94	17	-	-	PUNCT
cana-807	94	18	pgpr*wcs	pgpr*wcs	NOUN
cana-807	94	19	as	as	SCONJ
cana-807	94	20	every	every	DET
cana-807	94	21	n	n	NUM
cana-807	94	22	-	-	PUNCT
cana-807	94	23	ros	ros	PROPN
cana-807	94	24	is	be	AUX
cana-807	94	25	n	n	PRON
cana-807	94	26	-	-	PUNCT
cana-807	94	27	r*gαos	r*gαo	NOUN
cana-807	94	28	.	.	PUNCT
cana-807	95	1	since	since	SCONJ
cana-807	95	2	s	s	X
cana-807	95	3			PROPN
cana-807	95	4	s	s	PROPN
cana-807	95	5	and	and	CCONJ
cana-807	95	6	s	s	NOUN
cana-807	95	7	is	be	AUX
cana-807	95	8	n	n	PRON
cana-807	95	9	communications	communication	NOUN
cana-807	95	10	on	on	ADP
cana-807	95	11	applied	apply	VERB
cana-807	95	12	nonlinear	nonlinear	ADJ
cana-807	95	13	analysis	analysis	NOUN
cana-807	95	14	issn	issn	NOUN
cana-807	95	15	:	:	PUNCT
cana-807	95	16	1074	1074	NUM
cana-807	95	17	-	-	PUNCT
cana-807	95	18	133x	133x	NUM
cana-807	95	19	vol	vol	NOUN
cana-807	95	20	31	31	NUM
cana-807	95	21	no	no	NOUN
cana-807	95	22	.	.	PUNCT
cana-807	96	1	3s	3s	NUM
cana-807	96	2	(	(	PUNCT
cana-807	96	3	2024	2024	NUM
cana-807	96	4	)	)	PUNCT
cana-807	96	5	552	552	NUM
cana-807	96	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	96	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	96	8	.	.	PUNCT
cana-807	97	1	we	we	PRON
cana-807	97	2	have	have	VERB
cana-807	97	3	n	n	NUM
cana-807	97	4	-	-	PUNCT
cana-807	97	5	pcl(s	pcl(s	PROPN
cana-807	97	6	)	)	PUNCT
cana-807	97	7			PROPN
cana-807	97	8	s	s	PROPN
cana-807	97	9	and	and	CCONJ
cana-807	97	10	also	also	ADV
cana-807	97	11	s	s	PART
cana-807	97	12			PROPN
cana-807	97	13	n	n	CCONJ
cana-807	97	14	-	-	PUNCT
cana-807	97	15	pcl(s	pcl(s	PROPN
cana-807	97	16	)	)	PUNCT
cana-807	97	17	n	n	CCONJ
cana-807	97	18	-	-	PUNCT
cana-807	97	19	pcl(s	pcl(s	PROPN
cana-807	97	20	)	)	PUNCT
cana-807	97	21	=	=	SYM
cana-807	98	1	s.	s.	PROPN
cana-807	98	2	hence	hence	ADV
cana-807	98	3	s	s	PART
cana-807	98	4	is	be	AUX
cana-807	98	5	n	n	PRON
cana-807	98	6	-	-	PUNCT
cana-807	98	7	pcs	pc	NOUN
cana-807	98	8	.	.	PUNCT
cana-807	99	1	theorem	theorem	VERB
cana-807	99	2	3.10	3.10	NUM
cana-807	99	3	:	:	PUNCT
cana-807	99	4	if	if	SCONJ
cana-807	99	5	s	s	NOUN
cana-807	99	6	is	be	AUX
cana-807	99	7	n	n	PRON
cana-807	99	8	-	-	PUNCT
cana-807	99	9	os	os	NOUN
cana-807	99	10	and	and	CCONJ
cana-807	99	11	n	n	CCONJ
cana-807	99	12	-	-	PUNCT
cana-807	99	13	gpcs	gpc	NOUN
cana-807	99	14	then	then	ADV
cana-807	99	15	s	s	VERB
cana-807	99	16	is	be	AUX
cana-807	99	17	n	n	PRON
cana-807	99	18	-	-	PUNCT
cana-807	99	19	pgr*wcs	pgr*wcs	NOUN
cana-807	99	20	.	.	PUNCT
cana-807	100	1	proof	proof	NOUN
cana-807	100	2	:	:	PUNCT
cana-807	100	3	let	let	VERB
cana-807	100	4	s	s	PRON
cana-807	100	5	be	be	AUX
cana-807	100	6	n	n	X
cana-807	100	7	-	-	PUNCT
cana-807	100	8	os	os	NOUN
cana-807	100	9	and	and	CCONJ
cana-807	100	10	n	n	CCONJ
cana-807	100	11	-	-	PUNCT
cana-807	100	12	gpcs	gpc	NOUN
cana-807	100	13	.	.	PUNCT
cana-807	101	1	let	let	VERB
cana-807	101	2	m	m	PRON
cana-807	101	3	be	be	AUX
cana-807	101	4	n	n	PRON
cana-807	101	5	-	-	PUNCT
cana-807	101	6	r*gαos	r*gαo	NOUN
cana-807	101	7	such	such	ADJ
cana-807	101	8	that	that	DET
cana-807	101	9	s	s	PART
cana-807	101	10			PROPN
cana-807	101	11	m.	m.	NOUN
cana-807	101	12	since	since	SCONJ
cana-807	101	13	s	s	PROPN
cana-807	101	14	is	be	AUX
cana-807	101	15	n	n	PRON
cana-807	101	16	-	-	PUNCT
cana-807	101	17	os	os	NOUN
cana-807	101	18	and	and	CCONJ
cana-807	101	19	n	n	CCONJ
cana-807	101	20	-	-	PUNCT
cana-807	101	21	gpcs	gpc	NOUN
cana-807	101	22	.	.	PUNCT
cana-807	102	1	we	we	PRON
cana-807	102	2	have	have	VERB
cana-807	102	3	n	n	NUM
cana-807	102	4	-	-	PUNCT
cana-807	102	5	pcl(s	pcl(s	ADJ
cana-807	102	6	)	)	PUNCT
cana-807	102	7			PROPN
cana-807	102	8	s	s	PART
cana-807	102	9			PROPN
cana-807	102	10	m.	m.	NOUN
cana-807	102	11	so	so	ADV
cana-807	102	12	,	,	PUNCT
cana-807	102	13	n	n	CCONJ
cana-807	102	14	-	-	PUNCT
cana-807	102	15	pcl(s	pcl(s	PROPN
cana-807	102	16	)	)	PUNCT
cana-807	102	17			PROPN
cana-807	102	18	m	m	VERB
cana-807	102	19	whenever	whenever	SCONJ
cana-807	102	20	s	s	VERB
cana-807	102	21			PROPN
cana-807	102	22	m.	m.	NOUN
cana-807	102	23	therefore	therefore	ADV
cana-807	102	24	s	s	VERB
cana-807	102	25	is	be	AUX
cana-807	102	26	n	n	PRON
cana-807	102	27	-	-	PUNCT
cana-807	102	28	pgpr*wcs	pgpr*wcs	NOUN
cana-807	102	29	.	.	PUNCT
cana-807	103	1	theorem	theorem	VERB
cana-807	103	2	3.11	3.11	NUM
cana-807	103	3	:	:	PUNCT
cana-807	103	4	the	the	DET
cana-807	103	5	union	union	NOUN
cana-807	103	6	of	of	ADP
cana-807	103	7	two	two	NUM
cana-807	103	8	n	n	CCONJ
cana-807	103	9	-	-	PUNCT
cana-807	103	10	pgpr*wcs	pgpr*wcs	NOUN
cana-807	103	11	of	of	ADP
cana-807	103	12	x	x	PROPN
cana-807	103	13	is	be	AUX
cana-807	103	14	n	n	PRON
cana-807	103	15	-	-	PUNCT
cana-807	103	16	pgpr*wcs	pgpr*wcs	NOUN
cana-807	103	17	.	.	PUNCT
cana-807	104	1	proof	proof	NOUN
cana-807	104	2	:	:	PUNCT
cana-807	104	3	let	let	VERB
cana-807	104	4	c	c	NOUN
cana-807	104	5	and	and	CCONJ
cana-807	104	6	d	d	X
cana-807	104	7	be	be	VERB
cana-807	104	8	n	n	PRON
cana-807	104	9	-	-	PUNCT
cana-807	104	10	pgpr*wcs	pgpr*wcs	NOUN
cana-807	104	11	in	in	ADP
cana-807	104	12	x.by	x.by	PROPN
cana-807	104	13	definition	definition	NOUN
cana-807	104	14	c	c	NOUN
cana-807	104	15	,	,	PUNCT
cana-807	104	16	d	d	PROPN
cana-807	104	17			PROPN
cana-807	104	18	m	m	PROPN
cana-807	104	19	and	and	CCONJ
cana-807	104	20	m	m	AUX
cana-807	104	21	be	be	VERB
cana-807	104	22	a	a	DET
cana-807	104	23	n	n	NOUN
cana-807	104	24	-	-	PUNCT
cana-807	104	25	r*gαos	r*gαo	NOUN
cana-807	104	26	in	in	ADP
cana-807	104	27	x	x	SYM
cana-807	104	28	where	where	SCONJ
cana-807	104	29	c	c	NOUN
cana-807	104	30			PROPN
cana-807	104	31	m	m	PROPN
cana-807	104	32	and	and	CCONJ
cana-807	104	33	d	d	PROPN
cana-807	104	34			PROPN
cana-807	104	35	m.	m.	NOUN
cana-807	104	36	then	then	ADV
cana-807	104	37	n	n	CCONJ
cana-807	104	38	-	-	PUNCT
cana-807	104	39	pcl(c∪d	pcl(c∪d	NOUN
cana-807	104	40	)	)	PUNCT
cana-807	104	41	=	=	SYM
cana-807	104	42	n	n	CCONJ
cana-807	104	43	-	-	PUNCT
cana-807	104	44	pcl(a)∪	pcl(a)∪	NUM
cana-807	104	45	n	n	CCONJ
cana-807	104	46	-	-	PUNCT
cana-807	104	47	pcl(s	pcl(s	PROPN
cana-807	104	48	)	)	PUNCT
cana-807	104	49			PROPN
cana-807	104	50	m	m	VERB
cana-807	104	51	by	by	ADP
cana-807	104	52	hypothesis	hypothesis	NOUN
cana-807	104	53	.	.	PUNCT
cana-807	105	1	hence	hence	ADV
cana-807	105	2	c∪d	c∪d	NOUN
cana-807	105	3	is	be	AUX
cana-807	105	4	also	also	ADV
cana-807	105	5	n	n	CCONJ
cana-807	105	6	-	-	PUNCT
cana-807	105	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	105	8	in	in	ADP
cana-807	105	9	x.	x.	PROPN
cana-807	105	10	theorem	theorem	VERB
cana-807	105	11	3.12	3.12	NUM
cana-807	105	12	:	:	PUNCT
cana-807	105	13	the	the	DET
cana-807	105	14	intersection	intersection	NOUN
cana-807	105	15	of	of	ADP
cana-807	105	16	two	two	NUM
cana-807	105	17	n	n	CCONJ
cana-807	105	18	-	-	PUNCT
cana-807	105	19	pgpr*wcs	pgpr*wcs	NOUN
cana-807	105	20	in	in	ADP
cana-807	105	21	x	x	PROPN
cana-807	105	22	is	be	AUX
cana-807	105	23	generally	generally	ADV
cana-807	105	24	not	not	PART
cana-807	105	25	an	an	DET
cana-807	105	26	n	n	NOUN
cana-807	105	27	-	-	PUNCT
cana-807	105	28	pgpr*wcs	pgpr*wcs	NOUN
cana-807	105	29	in	in	ADP
cana-807	105	30	x.	x.	PROPN
cana-807	105	31	theorem	theorem	VERB
cana-807	105	32	3.13	3.13	NUM
cana-807	105	33	:	:	PUNCT
cana-807	105	34	if	if	SCONJ
cana-807	105	35	s	s	NOUN
cana-807	105	36	is	be	AUX
cana-807	105	37	n	n	PRON
cana-807	105	38	-	-	PUNCT
cana-807	105	39	pgpr*wcs	pgpr*wcs	NOUN
cana-807	105	40	and	and	CCONJ
cana-807	105	41	s	s	NOUN
cana-807	105	42			PROPN
cana-807	105	43	c	c	PROPN
cana-807	105	44			PROPN
cana-807	105	45	n	n	CCONJ
cana-807	105	46	-	-	PUNCT
cana-807	105	47	pcl(s	pcl(s	PROPN
cana-807	105	48	)	)	PUNCT
cana-807	105	49	.	.	PUNCT
cana-807	106	1	then	then	ADV
cana-807	106	2	c	c	PROPN
cana-807	106	3	is	be	AUX
cana-807	106	4	also	also	ADV
cana-807	106	5	npgpr*wcs	npgpr*wcs	NOUN
cana-807	106	6	in	in	ADP
cana-807	106	7	x.	x.	NOUN
cana-807	106	8	proof	proof	NOUN
cana-807	106	9	:	:	PUNCT
cana-807	106	10	let	let	VERB
cana-807	106	11	s	s	PRON
cana-807	106	12	be	be	AUX
cana-807	106	13	npgpr*wcs	npgpr*wcs	NOUN
cana-807	106	14	in	in	ADP
cana-807	106	15	x.	x.	NOUN
cana-807	106	16	to	to	PART
cana-807	106	17	prove	prove	VERB
cana-807	106	18	c	c	PROPN
cana-807	106	19	is	be	AUX
cana-807	106	20	npgpr*wcs	npgpr*wcs	NOUN
cana-807	106	21	in	in	ADP
cana-807	106	22	x.	x.	NOUN
cana-807	106	23	let	let	VERB
cana-807	106	24	m	m	PRON
cana-807	106	25	be	be	AUX
cana-807	106	26	an	an	DET
cana-807	106	27	n	n	NOUN
cana-807	106	28	-	-	PUNCT
cana-807	106	29	r*gαos	r*gαo	NOUN
cana-807	106	30	in	in	ADP
cana-807	106	31	x	x	SYM
cana-807	106	32	such	such	ADJ
cana-807	106	33	that	that	SCONJ
cana-807	106	34	c	c	PROPN
cana-807	106	35			PROPN
cana-807	106	36	m.	m.	NOUN
cana-807	106	37	since	since	SCONJ
cana-807	106	38	s	s	PROPN
cana-807	106	39	is	be	AUX
cana-807	106	40	n	n	PRON
cana-807	106	41	-	-	PUNCT
cana-807	106	42	pgpr*wcs	pgpr*wcs	NOUN
cana-807	106	43	and	and	CCONJ
cana-807	106	44	s	s	NOUN
cana-807	106	45			PROPN
cana-807	106	46	c.	c.	PROPN
cana-807	106	47	we	we	PRON
cana-807	106	48	have	have	VERB
cana-807	106	49	npcl(s	npcl(s	NOUN
cana-807	106	50	)	)	PUNCT
cana-807	106	51			PROPN
cana-807	106	52	m	m	PROPN
cana-807	106	53	and	and	CCONJ
cana-807	106	54	s	s	PROPN
cana-807	106	55			PROPN
cana-807	106	56	m.	m.	NOUN
cana-807	106	57	now	now	ADV
cana-807	106	58	c	c	PROPN
cana-807	106	59			PROPN
cana-807	106	60	n	n	CCONJ
cana-807	106	61	-	-	PUNCT
cana-807	106	62	pcl(s)∪	pcl(s)∪	NOUN
cana-807	106	63	n	n	CCONJ
cana-807	106	64	-	-	PUNCT
cana-807	106	65	pcl(c	pcl(c	NOUN
cana-807	106	66	)	)	PUNCT
cana-807	106	67			PROPN
cana-807	106	68	n	n	CCONJ
cana-807	106	69	-	-	PUNCT
cana-807	106	70	pcl(n	pcl(n	PROPN
cana-807	106	71	-	-	PUNCT
cana-807	106	72	pcl(s	pcl(s	PROPN
cana-807	106	73	)	)	PUNCT
cana-807	106	74	)	)	PUNCT
cana-807	107	1			PROPN
cana-807	107	2	n	n	CCONJ
cana-807	107	3	-	-	PUNCT
cana-807	107	4	pcl(s	pcl(s	PROPN
cana-807	107	5	)	)	PUNCT
cana-807	107	6			PROPN
cana-807	107	7	m.	m.	NOUN
cana-807	107	8	therefore	therefore	ADV
cana-807	107	9	n	n	CCONJ
cana-807	107	10	-	-	PUNCT
cana-807	107	11	pcl(s	pcl(s	PROPN
cana-807	107	12	)	)	PUNCT
cana-807	107	13			PROPN
cana-807	107	14	m.	m.	NOUN
cana-807	107	15	hence	hence	ADV
cana-807	107	16	c	c	PROPN
cana-807	107	17	is	be	AUX
cana-807	107	18	n	n	CCONJ
cana-807	107	19	-	-	PUNCT
cana-807	107	20	pgpr*wcs	pgpr*wcs	NOUN
cana-807	107	21	in	in	ADP
cana-807	107	22	x.	x.	PROPN
cana-807	107	23	theorem	theorem	VERB
cana-807	107	24	3.14	3.14	NUM
cana-807	107	25	:	:	PUNCT
cana-807	107	26	if	if	SCONJ
cana-807	107	27	a	a	DET
cana-807	107	28	subset	subset	NOUN
cana-807	107	29	s	s	X
cana-807	107	30	is	be	AUX
cana-807	107	31	both	both	PRON
cana-807	107	32	n	n	CCONJ
cana-807	107	33	-	-	PUNCT
cana-807	107	34	sos	sos	NOUN
cana-807	107	35	and	and	CCONJ
cana-807	107	36	n	n	CCONJ
cana-807	107	37	-	-	PUNCT
cana-807	107	38	wcs	wcs	NOUN
cana-807	108	1	then	then	ADV
cana-807	108	2	s	s	VERB
cana-807	108	3	is	be	AUX
cana-807	108	4	n	n	PRON
cana-807	108	5	-	-	PUNCT
cana-807	108	6	pgpr*wcs	pgpr*wcs	NOUN
cana-807	108	7	in	in	ADP
cana-807	108	8	x	x	PROPN
cana-807	108	9	..	..	PUNCT
cana-807	108	10	proof	proof	NOUN
cana-807	108	11	:	:	PUNCT
cana-807	108	12	let	let	VERB
cana-807	108	13	s	s	PRON
cana-807	108	14	be	be	AUX
cana-807	108	15	n	n	PRON
cana-807	108	16	-	-	PUNCT
cana-807	108	17	s	s	NOUN
cana-807	108	18	open	open	ADJ
cana-807	108	19	and	and	CCONJ
cana-807	108	20	n	n	CCONJ
cana-807	108	21	-	-	PUNCT
cana-807	108	22	wcs	wcs	NOUN
cana-807	108	23	in	in	ADP
cana-807	108	24	x.	x.	NOUN
cana-807	108	25	:	:	PUNCT
cana-807	108	26	let	let	VERB
cana-807	108	27	s	s	PRON
cana-807	108	28			PROPN
cana-807	108	29	m	m	PROPN
cana-807	108	30	and	and	CCONJ
cana-807	108	31	m	m	AUX
cana-807	108	32	be	be	VERB
cana-807	108	33	n	n	PRON
cana-807	108	34	-	-	PUNCT
cana-807	108	35	r*gαos	r*gαo	NOUN
cana-807	108	36	in	in	ADP
cana-807	108	37	x.	x.	NOUN
cana-807	108	38	now	now	ADV
cana-807	108	39	s	s	VERB
cana-807	108	40			PROPN
cana-807	108	41	s.	s.	PROPN
cana-807	108	42	by	by	ADP
cana-807	108	43	hypothesis	hypothesis	NOUN
cana-807	108	44	,	,	PUNCT
cana-807	108	45	n	n	CCONJ
cana-807	108	46	-	-	PUNCT
cana-807	108	47	ci(s	ci(s	NUM
cana-807	108	48	)	)	PUNCT
cana-807	109	1			PROPN
cana-807	109	2	s.	s.	PROPN
cana-807	109	3	therefore	therefore	ADV
cana-807	109	4	n	n	CCONJ
cana-807	109	5	-	-	PUNCT
cana-807	109	6	pcl(s	pcl(s	PROPN
cana-807	109	7	)	)	PUNCT
cana-807	109	8			PROPN
cana-807	109	9	n	n	CCONJ
cana-807	109	10	-	-	PUNCT
cana-807	109	11	cl(s	cl(s	NOUN
cana-807	109	12	)	)	PUNCT
cana-807	109	13			PROPN
cana-807	109	14	s	s	PART
cana-807	109	15			PROPN
cana-807	109	16	m.	m.	NOUN
cana-807	109	17	hence	hence	ADV
cana-807	109	18	s	s	VERB
cana-807	109	19	is	be	AUX
cana-807	109	20	n	n	CCONJ
cana-807	109	21	-	-	PUNCT
cana-807	109	22	pgpr*wcs	pgpr*wcs	NOUN
cana-807	109	23	in	in	ADP
cana-807	109	24	x.	x.	NOUN
cana-807	109	25	communications	communication	NOUN
cana-807	109	26	on	on	ADP
cana-807	109	27	applied	apply	VERB
cana-807	109	28	nonlinear	nonlinear	ADJ
cana-807	109	29	analysis	analysis	NOUN
cana-807	109	30	issn	issn	NOUN
cana-807	109	31	:	:	PUNCT
cana-807	109	32	1074	1074	NUM
cana-807	109	33	-	-	PUNCT
cana-807	109	34	133x	133x	NUM
cana-807	109	35	vol	vol	NOUN
cana-807	109	36	31	31	NUM
cana-807	109	37	no	no	NOUN
cana-807	109	38	.	.	PUNCT
cana-807	110	1	3s	3s	NUM
cana-807	110	2	(	(	PUNCT
cana-807	110	3	2024	2024	NUM
cana-807	110	4	)	)	PUNCT
cana-807	110	5	553	553	NUM
cana-807	110	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	110	7	theorem	theorem	VERB
cana-807	110	8	3.15	3.15	NUM
cana-807	110	9	:	:	PUNCT
cana-807	110	10	if	if	SCONJ
cana-807	110	11	s	s	NOUN
cana-807	110	12	is	be	AUX
cana-807	110	13	both	both	PRON
cana-807	110	14	n	n	NOUN
cana-807	110	15	-	-	PUNCT
cana-807	110	16	ros	ros	PROPN
cana-807	110	17	and	and	CCONJ
cana-807	110	18	n	n	CCONJ
cana-807	110	19	-	-	PUNCT
cana-807	110	20	rgcs	rgcs	NOUN
cana-807	110	21	then	then	ADV
cana-807	110	22	s	s	VERB
cana-807	110	23	is	be	AUX
cana-807	110	24	n	n	PRON
cana-807	110	25	-	-	PUNCT
cana-807	110	26	pgpr*wcs	pgpr*wcs	NOUN
cana-807	110	27	proof	proof	NOUN
cana-807	110	28	:	:	PUNCT
cana-807	110	29	let	let	VERB
cana-807	110	30	s	s	PRON
cana-807	110	31			PROPN
cana-807	110	32	m	m	VERB
cana-807	110	33	where	where	SCONJ
cana-807	110	34	m	m	PRON
cana-807	110	35	is	be	AUX
cana-807	110	36	n	n	PRON
cana-807	110	37	-	-	PUNCT
cana-807	110	38	r*gαos	r*gαo	NOUN
cana-807	110	39	since	since	SCONJ
cana-807	110	40	s	s	PROPN
cana-807	110	41	is	be	AUX
cana-807	110	42	n	n	PRON
cana-807	110	43	-	-	PUNCT
cana-807	110	44	ros	ros	PROPN
cana-807	110	45	and	and	CCONJ
cana-807	110	46	n	n	CCONJ
cana-807	110	47	-	-	PUNCT
cana-807	110	48	rgcs	rgcs	NOUN
cana-807	110	49	.	.	PUNCT
cana-807	111	1	we	we	PRON
cana-807	111	2	have	have	VERB
cana-807	111	3	s	s	PROPN
cana-807	111	4			PROPN
cana-807	111	5	s.	s.	PROPN
cana-807	111	6	so	so	ADV
cana-807	111	7	n	n	CCONJ
cana-807	111	8	-	-	PUNCT
cana-807	111	9	cl(s	cl(s	NOUN
cana-807	111	10	)	)	PUNCT
cana-807	111	11			PROPN
cana-807	111	12	s	s	PROPN
cana-807	111	13	then	then	ADV
cana-807	111	14	n	n	CCONJ
cana-807	111	15	-	-	PUNCT
cana-807	111	16	pcl(s	pcl(s	PROPN
cana-807	111	17	)	)	PUNCT
cana-807	111	18			PROPN
cana-807	111	19	n	n	CCONJ
cana-807	111	20	-	-	PUNCT
cana-807	111	21	cl(s	cl(s	NOUN
cana-807	111	22	)	)	PUNCT
cana-807	111	23	.	.	PUNCT
cana-807	112	1	therefore	therefore	ADV
cana-807	112	2	n	n	CCONJ
cana-807	112	3	-	-	PUNCT
cana-807	112	4	pcl(s	pcl(s	PROPN
cana-807	112	5	)	)	PUNCT
cana-807	112	6			PROPN
cana-807	112	7	s	s	PROPN
cana-807	112	8	,	,	PUNCT
cana-807	112	9	whenever	whenever	SCONJ
cana-807	112	10	s	s	AUX
cana-807	112	11			PROPN
cana-807	112	12	m.	m.	NOUN
cana-807	112	13	proves	prove	VERB
cana-807	112	14	.	.	PUNCT
cana-807	113	1	s	s	PART
cana-807	113	2	is	be	AUX
cana-807	113	3	n	n	PRON
cana-807	113	4	-	-	PUNCT
cana-807	113	5	pgpr*wcs	pgpr*wcs	NOUN
cana-807	113	6	theorem	theorem	VERB
cana-807	113	7	3.16	3.16	NUM
cana-807	113	8	:	:	PUNCT
cana-807	113	9	if	if	SCONJ
cana-807	113	10	s	s	NOUN
cana-807	113	11	is	be	AUX
cana-807	113	12	both	both	PRON
cana-807	113	13	n	n	CCONJ
cana-807	113	14	-	-	PUNCT
cana-807	113	15	rsos	rsos	ADJ
cana-807	113	16	and	and	CCONJ
cana-807	113	17	n	n	CCONJ
cana-807	113	18	-	-	PUNCT
cana-807	113	19	gprwcs	gprwcs	NOUN
cana-807	113	20	then	then	ADV
cana-807	113	21	it	it	PRON
cana-807	113	22	is	be	AUX
cana-807	113	23	n	n	CCONJ
cana-807	113	24	-	-	PUNCT
cana-807	113	25	pgpr*wcs	pgpr*wcs	NOUN
cana-807	113	26	.	.	PUNCT
cana-807	114	1	proof	proof	NOUN
cana-807	114	2	:	:	PUNCT
cana-807	114	3	let	let	VERB
cana-807	114	4	s	s	PRON
cana-807	114	5	be	be	AUX
cana-807	114	6	n	n	X
cana-807	114	7	-	-	PUNCT
cana-807	114	8	rsos	rsos	ADJ
cana-807	114	9	and	and	CCONJ
cana-807	114	10	n	n	CCONJ
cana-807	114	11	-	-	PUNCT
cana-807	114	12	gprwcs	gprwcs	NOUN
cana-807	114	13	,	,	PUNCT
cana-807	114	14	m	m	VERB
cana-807	114	15	be	be	VERB
cana-807	114	16	n	n	PRON
cana-807	114	17	-	-	PUNCT
cana-807	114	18	r*gαos	r*gαo	NOUN
cana-807	114	19	in	in	ADP
cana-807	114	20	x	x	SYM
cana-807	114	21	such	such	ADJ
cana-807	114	22	that	that	DET
cana-807	114	23	s	s	X
cana-807	114	24			PROPN
cana-807	114	25	m	m	VERB
cana-807	114	26	by	by	ADP
cana-807	114	27	hypothesis	hypothesis	NOUN
cana-807	114	28	,	,	PUNCT
cana-807	114	29	s	s	PART
cana-807	114	30			PROPN
cana-807	114	31	s.	s.	PROPN
cana-807	114	32	therefore	therefore	ADV
cana-807	114	33	n	n	CCONJ
cana-807	114	34	-	-	PUNCT
cana-807	114	35	pcl(s	pcl(s	PROPN
cana-807	114	36	)	)	PUNCT
cana-807	115	1			PROPN
cana-807	115	2	s	s	PART
cana-807	115	3			PROPN
cana-807	115	4	m.	m.	NOUN
cana-807	115	5	so	so	CCONJ
cana-807	115	6	n	n	CCONJ
cana-807	115	7	-	-	PUNCT
cana-807	115	8	pcl(s	pcl(s	ADJ
cana-807	115	9	)	)	PUNCT
cana-807	115	10			PROPN
cana-807	115	11	m.	m.	NOUN
cana-807	115	12	hence	hence	ADV
cana-807	115	13	s	s	VERB
cana-807	115	14	is	be	AUX
cana-807	115	15	a	a	DET
cana-807	115	16	n	n	NOUN
cana-807	115	17	-	-	PUNCT
cana-807	115	18	pgpr*wcs	pgpr*wcs	NOUN
cana-807	115	19	.	.	PUNCT
cana-807	116	1	remark	remark	VERB
cana-807	116	2	3.17	3.17	NUM
cana-807	116	3	:	:	PUNCT
cana-807	116	4	to	to	PART
cana-807	116	5	prove	prove	VERB
cana-807	116	6	the	the	DET
cana-807	116	7	converse	converse	NOUN
cana-807	116	8	need	need	AUX
cana-807	116	9	not	not	PART
cana-807	116	10	be	be	AUX
cana-807	116	11	true	true	ADJ
cana-807	116	12	as	as	SCONJ
cana-807	116	13	x	x	X
cana-807	116	14	,	,	PUNCT
cana-807	116	15	τ	τ	PROPN
cana-807	116	16	,	,	PUNCT
cana-807	116	17	defined	define	VERB
cana-807	116	18	in	in	ADP
cana-807	116	19	ex	ex	PRON
cana-807	116	20	3.2	3.2	NUM
cana-807	116	21	where	where	SCONJ
cana-807	116	22	m	m	VERB
cana-807	116	23	=	=	PUNCT
cana-807	116	24	⟨x	⟨x	VERB
cana-807	116	25	,	,	PUNCT
cana-807	116	26	(	(	PUNCT
cana-807	116	27	0.7,0.2	0.7,0.2	PROPN
cana-807	116	28	,	,	PUNCT
cana-807	116	29	0.5),(0.8,0.2	0.5),(0.8,0.2	NUM
cana-807	116	30	,	,	PUNCT
cana-807	116	31	0	0	NUM
cana-807	116	32	..	..	SYM
cana-807	116	33	3)⟩	3)⟩	NUM
cana-807	116	34	u	u	NOUN
cana-807	116	35	=	=	PUNCT
cana-807	116	36	⟨x	⟨x	VERB
cana-807	116	37	,	,	PUNCT
cana-807	116	38	(	(	PUNCT
cana-807	116	39	0.2,0.2,0.9	0.2,0.2,0.9	INTJ
cana-807	116	40	)	)	PUNCT
cana-807	116	41	(	(	PUNCT
cana-807	116	42	0.3	0.3	NUM
cana-807	116	43	0.2	0.2	NUM
cana-807	116	44	,	,	PUNCT
cana-807	116	45	0.9)⟩	0.9)⟩	NOUN
cana-807	116	46	a	a	DET
cana-807	116	47	=	=	PUNCT
cana-807	116	48	⟨x	⟨x	VERB
cana-807	116	49	,	,	PUNCT
cana-807	116	50	(	(	PUNCT
cana-807	116	51	0.3,.0.2,0.7	0.3,.0.2,0.7	NUM
cana-807	116	52	)	)	PUNCT
cana-807	116	53	(	(	PUNCT
cana-807	116	54	0.3,0.2	0.3,0.2	NOUN
cana-807	116	55	,	,	PUNCT
cana-807	116	56	0.8)⟩	0.8)⟩	NOUN
cana-807	116	57	n	n	CCONJ
cana-807	116	58	-	-	SYM
cana-807	116	59	s	s	NOUN
cana-807	116	60	s	s	NOUN
cana-807	116	61	is	be	AUX
cana-807	116	62	n	n	PRON
cana-807	116	63	-	-	PUNCT
cana-807	116	64	pgpr*wcs	pgpr*wcs	NOUN
cana-807	116	65	when	when	SCONJ
cana-807	116	66	m	m	PROPN
cana-807	116	67	is	be	AUX
cana-807	116	68	n	n	PRON
cana-807	116	69	-	-	PUNCT
cana-807	116	70	r*gαos	r*gαo	NOUN
cana-807	116	71	.	.	PUNCT
cana-807	117	1	as	as	ADP
cana-807	117	2	n	n	NOUN
cana-807	117	3	-	-	PUNCT
cana-807	117	4	pcl(a	pcl(a	NOUN
cana-807	117	5	)	)	PUNCT
cana-807	117	6	=	=	NOUN
cana-807	117	7	mc	mc	PROPN
cana-807	117	8			PROPN
cana-807	117	9	m	m	VERB
cana-807	117	10	whenever	whenever	SCONJ
cana-807	117	11	s	s	VERB
cana-807	117	12			PROPN
cana-807	117	13	m.	m.	PROPN
cana-807	117	14	sut	sut	PROPN
cana-807	117	15	since	since	SCONJ
cana-807	117	16	s⊈	s⊈	ADJ
cana-807	117	17	n	n	CCONJ
cana-807	117	18	-	-	PUNCT
cana-807	117	19	cl(ints	cl(int	NOUN
cana-807	117	20	)	)	PUNCT
cana-807	117	21	i.e.	i.e.	X
cana-807	117	22	s⊈	s⊈	ADJ
cana-807	117	23	mc	mc	PROPN
cana-807	117	24	.	.	PUNCT
cana-807	118	1	s	s	PART
cana-807	118	2	is	be	AUX
cana-807	118	3	not	not	PART
cana-807	118	4	n	n	ADV
cana-807	118	5	-	-	PUNCT
cana-807	118	6	rsos	rsos	NOUN
cana-807	118	7	and	and	CCONJ
cana-807	118	8	thus	thus	ADV
cana-807	118	9	s	s	X
cana-807	118	10	is	be	AUX
cana-807	118	11	not	not	PART
cana-807	118	12	n	n	CCONJ
cana-807	118	13	-	-	PUNCT
cana-807	118	14	gprwcs	gprwcs	NOUN
cana-807	118	15	.	.	PUNCT
cana-807	119	1	theorem	theorem	VERB
cana-807	119	2	3.18	3.18	NUM
cana-807	119	3	:	:	PUNCT
cana-807	119	4	if	if	SCONJ
cana-807	119	5	a	a	PRON
cana-807	119	6	is	be	AUX
cana-807	119	7	both	both	PRON
cana-807	119	8	n	n	CCONJ
cana-807	119	9	-	-	PUNCT
cana-807	119	10	os	os	NOUN
cana-807	119	11	and	and	CCONJ
cana-807	119	12	n	n	CCONJ
cana-807	119	13	-	-	PUNCT
cana-807	119	14	gcs	gcs	PROPN
cana-807	119	15	then	then	ADV
cana-807	119	16	s	s	VERB
cana-807	119	17	is	be	AUX
cana-807	119	18	n	n	PRON
cana-807	119	19	-	-	PUNCT
cana-807	119	20	pgpr*wcs	pgpr*wcs	NOUN
cana-807	119	21	.	.	PUNCT
cana-807	120	1	proof	proof	NOUN
cana-807	120	2	:	:	PUNCT
cana-807	120	3	given	give	VERB
cana-807	120	4	s	s	PROPN
cana-807	120	5	is	be	AUX
cana-807	120	6	n	n	PRON
cana-807	120	7	-	-	PUNCT
cana-807	120	8	os	os	NOUN
cana-807	120	9	and	and	CCONJ
cana-807	120	10	n	n	CCONJ
cana-807	120	11	-	-	PUNCT
cana-807	120	12	gcs	gcs	PROPN
cana-807	120	13	.	.	PUNCT
cana-807	121	1	let	let	VERB
cana-807	121	2	m	m	PRON
cana-807	121	3	be	be	AUX
cana-807	121	4	any	any	DET
cana-807	121	5	n	n	NOUN
cana-807	121	6	-	-	PUNCT
cana-807	121	7	r*gαos	r*gαo	NOUN
cana-807	121	8	such	such	ADJ
cana-807	121	9	that	that	DET
cana-807	121	10	s	s	PART
cana-807	121	11			PROPN
cana-807	121	12	m.	m.	NOUN
cana-807	121	13	since	since	SCONJ
cana-807	121	14	s	s	PROPN
cana-807	121	15			PROPN
cana-807	121	16	s	s	PROPN
cana-807	121	17	and	and	CCONJ
cana-807	121	18	n	n	CCONJ
cana-807	121	19	-	-	PUNCT
cana-807	121	20	os	os	NOUN
cana-807	121	21	and	and	CCONJ
cana-807	121	22	s	s	NOUN
cana-807	121	23	is	be	AUX
cana-807	121	24	n	n	PRON
cana-807	121	25	-	-	PUNCT
cana-807	121	26	gcl(s	gcl(s	NOUN
cana-807	121	27	)	)	PUNCT
cana-807	121	28	.	.	PUNCT
cana-807	122	1	hence	hence	ADV
cana-807	122	2	n	n	CCONJ
cana-807	122	3	-	-	PUNCT
cana-807	122	4	cl(s	cl(s	NOUN
cana-807	122	5	)	)	PUNCT
cana-807	122	6			PROPN
cana-807	122	7	s	s	PROPN
cana-807	122	8	and	and	CCONJ
cana-807	122	9	n	n	CCONJ
cana-807	122	10	-	-	PUNCT
cana-807	122	11	pcl(s	pcl(s	PROPN
cana-807	122	12	)	)	PUNCT
cana-807	122	13			PROPN
cana-807	122	14	n	n	CCONJ
cana-807	122	15	-	-	PUNCT
cana-807	122	16	cl(s	cl(s	NOUN
cana-807	122	17	)	)	PUNCT
cana-807	122	18			PROPN
cana-807	122	19	s	s	PART
cana-807	122	20			PROPN
cana-807	122	21	m.	m.	NOUN
cana-807	122	22	thus	thus	ADV
cana-807	122	23	n	n	CCONJ
cana-807	122	24	-	-	PUNCT
cana-807	122	25	pcl(s	pcl(s	PROPN
cana-807	122	26	)	)	PUNCT
cana-807	122	27			PROPN
cana-807	122	28	m	m	VERB
cana-807	122	29	whenever	whenever	SCONJ
cana-807	122	30	s	s	VERB
cana-807	122	31			PROPN
cana-807	122	32	m	m	PROPN
cana-807	122	33	and	and	CCONJ
cana-807	122	34	m	m	PROPN
cana-807	122	35	is	be	AUX
cana-807	122	36	n	n	PRON
cana-807	122	37	-	-	PUNCT
cana-807	122	38	r*gαos	r*gαo	NOUN
cana-807	122	39	in	in	ADP
cana-807	122	40	x	x	NOUN
cana-807	122	41	,	,	PUNCT
cana-807	122	42	.	.	PUNCT
cana-807	123	1	therefore	therefore	ADV
cana-807	123	2	s	s	X
cana-807	123	3	is	be	AUX
cana-807	123	4	npgpr*wcs	npgpr*wcs	NOUN
cana-807	123	5	.	.	PUNCT
cana-807	124	1	theorem	theorem	VERB
cana-807	124	2	3.19	3.19	NUM
cana-807	124	3	:	:	PUNCT
cana-807	124	4	if	if	SCONJ
cana-807	124	5	s	s	NOUN
cana-807	124	6	is	be	AUX
cana-807	124	7	n	n	PRON
cana-807	124	8	-	-	PUNCT
cana-807	124	9	ros	ros	PROPN
cana-807	124	10	and	and	CCONJ
cana-807	124	11	n	n	CCONJ
cana-807	124	12	-	-	PUNCT
cana-807	124	13	gprcs	gprcs	NOUN
cana-807	124	14	then	then	ADV
cana-807	124	15	it	it	PRON
cana-807	124	16	is	be	AUX
cana-807	124	17	n	n	CCONJ
cana-807	124	18	-	-	PUNCT
cana-807	124	19	pgpr*wcs	pgpr*wcs	NOUN
cana-807	124	20	.	.	PUNCT
cana-807	125	1	proof	proof	NOUN
cana-807	125	2	:	:	PUNCT
cana-807	125	3	if	if	SCONJ
cana-807	125	4	a	a	PRON
cana-807	125	5	is	be	AUX
cana-807	125	6	n	n	PRON
cana-807	125	7	-	-	PUNCT
cana-807	125	8	ros	ros	PROPN
cana-807	125	9	and	and	CCONJ
cana-807	125	10	n	n	CCONJ
cana-807	125	11	-	-	PUNCT
cana-807	125	12	gprcs	gprcs	NOUN
cana-807	125	13	.	.	PUNCT
cana-807	126	1	let	let	VERB
cana-807	126	2	m	m	PRON
cana-807	126	3	be	be	AUX
cana-807	126	4	any	any	DET
cana-807	126	5	n	n	NOUN
cana-807	126	6	-	-	PUNCT
cana-807	126	7	r*gαos	r*gαo	NOUN
cana-807	126	8	such	such	ADJ
cana-807	126	9	that	that	DET
cana-807	126	10	s	s	PART
cana-807	126	11			PROPN
cana-807	126	12	m.	m.	NOUN
cana-807	126	13	thus	thus	ADV
cana-807	126	14	n	n	CCONJ
cana-807	126	15	-	-	PUNCT
cana-807	126	16	pcl(s	pcl(s	PROPN
cana-807	126	17	)	)	PUNCT
cana-807	126	18			PROPN
cana-807	126	19	s	s	PROPN
cana-807	126	20	and	and	CCONJ
cana-807	126	21	n	n	CCONJ
cana-807	126	22	-	-	PUNCT
cana-807	126	23	pcl(s	pcl(s	PROPN
cana-807	126	24	)	)	PUNCT
cana-807	126	25			PROPN
cana-807	126	26	m.	m.	NOUN
cana-807	126	27	therefore	therefore	ADV
cana-807	126	28	npcl(s	npcl(s	ADJ
cana-807	126	29	)	)	PUNCT
cana-807	126	30			PROPN
cana-807	126	31	m	m	VERB
cana-807	126	32	whenever	whenever	SCONJ
cana-807	126	33	s	s	VERB
cana-807	126	34			PROPN
cana-807	126	35	m	m	PROPN
cana-807	126	36	and	and	CCONJ
cana-807	126	37	m	m	PROPN
cana-807	126	38	is	be	AUX
cana-807	126	39	n	n	PRON
cana-807	126	40	-	-	PUNCT
cana-807	126	41	r*gαos	r*gαo	NOUN
cana-807	126	42	in	in	ADP
cana-807	126	43	x.	x.	NOUN
cana-807	126	44	therefore	therefore	ADV
cana-807	126	45	s	s	VERB
cana-807	126	46	is	be	AUX
cana-807	126	47	n	n	PRON
cana-807	126	48	communications	communication	NOUN
cana-807	126	49	on	on	ADP
cana-807	126	50	applied	apply	VERB
cana-807	126	51	nonlinear	nonlinear	ADJ
cana-807	126	52	analysis	analysis	NOUN
cana-807	126	53	issn	issn	NOUN
cana-807	126	54	:	:	PUNCT
cana-807	126	55	1074	1074	NUM
cana-807	126	56	-	-	PUNCT
cana-807	126	57	133x	133x	NUM
cana-807	126	58	vol	vol	NOUN
cana-807	126	59	31	31	NUM
cana-807	126	60	no	no	NOUN
cana-807	126	61	.	.	PUNCT
cana-807	127	1	3s	3s	NUM
cana-807	127	2	(	(	PUNCT
cana-807	127	3	2024	2024	NUM
cana-807	127	4	)	)	PUNCT
cana-807	127	5	554	554	NUM
cana-807	127	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	127	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	127	8	.	.	PUNCT
cana-807	127	9	theorem	theorem	VERB
cana-807	127	10	3.20	3.20	NUM
cana-807	127	11	:	:	PUNCT
cana-807	127	12	if	if	SCONJ
cana-807	127	13	s	s	NOUN
cana-807	127	14	is	be	AUX
cana-807	127	15	both	both	PRON
cana-807	127	16	n	n	CCONJ
cana-807	127	17	-	-	PUNCT
cana-807	127	18	rsos	rsos	ADJ
cana-807	127	19	and	and	CCONJ
cana-807	127	20	n	n	CCONJ
cana-807	127	21	-	-	PUNCT
cana-807	127	22	gprwos	gprwos	NOUN
cana-807	127	23	then	then	ADV
cana-807	127	24	it	it	PRON
cana-807	127	25	is	be	AUX
cana-807	127	26	n	n	ADV
cana-807	127	27	-	-	PUNCT
cana-807	127	28	pgpr*w	pgpr*w	NOUN
cana-807	127	29	closed	closed	ADJ
cana-807	127	30	.	.	PUNCT
cana-807	128	1	proof	proof	NOUN
cana-807	128	2	:	:	PUNCT
cana-807	128	3	subset	subset	VERB
cana-807	128	4	let	let	VERB
cana-807	128	5	s	s	PRON
cana-807	128	6	be	be	AUX
cana-807	128	7	n	n	ADV
cana-807	128	8	-	-	PUNCT
cana-807	128	9	rs	rs	NOUN
cana-807	128	10	open	open	ADJ
cana-807	128	11	and	and	CCONJ
cana-807	128	12	n	n	CCONJ
cana-807	128	13	-	-	PUNCT
cana-807	128	14	gprw	gprw	VERB
cana-807	128	15	closed	close	VERB
cana-807	128	16	.	.	PUNCT
cana-807	129	1	let	let	VERB
cana-807	129	2	m	m	PRON
cana-807	129	3	be	be	AUX
cana-807	129	4	n	n	PRON
cana-807	129	5	-	-	PUNCT
cana-807	129	6	r*gαos	r*gαo	NOUN
cana-807	129	7	open	open	ADJ
cana-807	129	8	in	in	ADP
cana-807	129	9	x	x	PROPN
cana-807	129	10	such	such	ADJ
cana-807	129	11	that	that	DET
cana-807	129	12	s	s	NUM
cana-807	129	13	m	m	VERB
cana-807	129	14	by	by	ADP
cana-807	129	15	hypothesis	hypothesis	NOUN
cana-807	129	16	,	,	PUNCT
cana-807	129	17	s	s	PROPN
cana-807	129	18			PROPN
cana-807	129	19	s.	s.	PROPN
cana-807	129	20	therefore	therefore	ADV
cana-807	129	21	,	,	PUNCT
cana-807	129	22	n	n	CCONJ
cana-807	129	23	-	-	PUNCT
cana-807	129	24	pcl(s	pcl(s	PROPN
cana-807	129	25	)	)	PUNCT
cana-807	129	26			PROPN
cana-807	129	27	s	s	PART
cana-807	129	28			PROPN
cana-807	129	29	m	m	VERB
cana-807	129	30	so	so	ADV
cana-807	129	31	,	,	PUNCT
cana-807	129	32	n	n	CCONJ
cana-807	129	33	-	-	PUNCT
cana-807	129	34	pcl(s	pcl(s	PROPN
cana-807	129	35	)	)	PUNCT
cana-807	129	36			PROPN
cana-807	129	37	m.	m.	NOUN
cana-807	129	38	thus	thus	ADV
cana-807	129	39	,	,	PUNCT
cana-807	129	40	s	s	X
cana-807	129	41	is	be	AUX
cana-807	129	42	n	n	PRON
cana-807	129	43	-	-	PUNCT
cana-807	129	44	pgpr*wcs	pgpr*wcs	NOUN
cana-807	129	45	.	.	PUNCT
cana-807	130	1	theorem	theorem	VERB
cana-807	130	2	3.21	3.21	NUM
cana-807	130	3	:	:	PUNCT
cana-807	130	4	if	if	SCONJ
cana-807	130	5	s	s	VERB
cana-807	130	6	in	in	ADP
cana-807	130	7	x	x	INTJ
cana-807	130	8	such	such	ADJ
cana-807	130	9	that	that	SCONJ
cana-807	130	10	k	k	PROPN
cana-807	130	11			PROPN
cana-807	130	12	n	n	CCONJ
cana-807	130	13	-	-	PUNCT
cana-807	130	14	pcl(s)-s	pcl(s)-	NOUN
cana-807	130	15	then	then	ADV
cana-807	130	16	k	k	PROPN
cana-807	130	17	=	=	SYM
cana-807	130	18	𝛟	𝛟	PROPN
cana-807	130	19	.where	.where	X
cana-807	131	1	k	k	PROPN
cana-807	131	2	is	be	AUX
cana-807	131	3	a	a	DET
cana-807	131	4	non	non	ADJ
cana-807	131	5	-	-	ADJ
cana-807	131	6	empty	empty	ADJ
cana-807	131	7	nr*gαos	nr*gαo	NOUN
cana-807	131	8	of	of	ADP
cana-807	131	9	n	n	CCONJ
cana-807	131	10	-	-	PUNCT
cana-807	131	11	pcl(s)-s	pcl(s)-	NOUN
cana-807	131	12	proof	proof	NOUN
cana-807	131	13	:	:	PUNCT
cana-807	131	14	let	let	VERB
cana-807	131	15	s	s	PRON
cana-807	131	16	be	be	AUX
cana-807	131	17	n	n	X
cana-807	131	18	-	-	PUNCT
cana-807	131	19	pgpr*wcs	pgpr*wcs	NOUN
cana-807	131	20	in	in	ADP
cana-807	131	21	x.	x.	NOUN
cana-807	131	22	given	give	VERB
cana-807	131	23	k	k	PROPN
cana-807	131	24			PROPN
cana-807	131	25	n	n	CCONJ
cana-807	131	26	-	-	PUNCT
cana-807	131	27	pcl(s)-s	pcl(s)-	NOUN
cana-807	131	28	k	k	X
cana-807	131	29			PROPN
cana-807	131	30	n	n	CCONJ
cana-807	131	31	-	-	PUNCT
cana-807	131	32	pcl(s)-s	pcl(s)-	NOUN
cana-807	131	33	implies	imply	VERB
cana-807	131	34	k	k	PROPN
cana-807	131	35			PROPN
cana-807	131	36	n	n	CCONJ
cana-807	131	37	-	-	PUNCT
cana-807	131	38	pcl(s)∩	pcl(s)∩	NOUN
cana-807	131	39	s	s	PART
cana-807	131	40	k	k	PROPN
cana-807	131	41			PROPN
cana-807	131	42	n	n	CCONJ
cana-807	131	43	-	-	PUNCT
cana-807	131	44	pcl(s	pcl(s	NOUN
cana-807	131	45	)	)	PUNCT
cana-807	131	46	(	(	PUNCT
cana-807	131	47	1	1	X
cana-807	131	48	)	)	PUNCT
cana-807	131	49	k	k	NOUN
cana-807	131	50			PROPN
cana-807	131	51	x	x	X
cana-807	131	52	-	-	PUNCT
cana-807	131	53	s	s	X
cana-807	131	54	and	and	CCONJ
cana-807	131	55	s	s	NOUN
cana-807	131	56			PROPN
cana-807	131	57	x	x	X
cana-807	131	58	-	-	PUNCT
cana-807	131	59	k	k	ADP
cana-807	131	60	we	we	PRON
cana-807	131	61	have	have	VERB
cana-807	131	62	x	x	SYM
cana-807	131	63	k	k	PROPN
cana-807	131	64	is	be	AUX
cana-807	131	65	n	n	PRON
cana-807	131	66	-	-	PUNCT
cana-807	131	67	r*gαos	r*gαo	NOUN
cana-807	131	68	and	and	CCONJ
cana-807	131	69	s	s	VERB
cana-807	131	70	is	be	AUX
cana-807	131	71	npgpr*wcs	npgpr*wcs	PROPN
cana-807	131	72	.implies	.implie	NOUN
cana-807	131	73	n	n	CCONJ
cana-807	131	74	-	-	PUNCT
cana-807	131	75	pcl(s	pcl(s	PROPN
cana-807	131	76	)	)	PUNCT
cana-807	131	77			PROPN
cana-807	131	78	x	x	INTJ
cana-807	131	79	-	-	PROPN
cana-807	131	80	k	k	X
cana-807	131	81	..	..	PUNCT
cana-807	131	82	therefore	therefore	ADV
cana-807	131	83	,	,	PUNCT
cana-807	131	84	k	k	PROPN
cana-807	131	85			PROPN
cana-807	131	86	x	x	PROPN
cana-807	131	87	-	-	ADJ
cana-807	131	88	n	n	CCONJ
cana-807	131	89	-	-	PUNCT
cana-807	131	90	pcl(s	pcl(s	NOUN
cana-807	131	91	)	)	PUNCT
cana-807	131	92	(	(	PUNCT
cana-807	131	93	2	2	NUM
cana-807	131	94	)	)	PUNCT
cana-807	131	95	from	from	ADP
cana-807	131	96	(	(	PUNCT
cana-807	131	97	1	1	NUM
cana-807	131	98	)	)	PUNCT
cana-807	131	99	and	and	CCONJ
cana-807	131	100	(	(	PUNCT
cana-807	131	101	2	2	NUM
cana-807	131	102	)	)	PUNCT
cana-807	131	103	,	,	PUNCT
cana-807	131	104	k	k	PROPN
cana-807	131	105			PROPN
cana-807	131	106	n	n	CCONJ
cana-807	131	107	-	-	PUNCT
cana-807	131	108	pcl(s)intersection(x	pcl(s)intersection(x	NOUN
cana-807	131	109	-	-	PUNCT
cana-807	131	110	n	n	CCONJ
cana-807	131	111	-	-	PUNCT
cana-807	131	112	pcl(s	pcl(s	NOUN
cana-807	131	113	)	)	PUNCT
cana-807	131	114	)	)	PUNCT
cana-807	131	115	=	=	SYM
cana-807	132	1	𝛟	𝛟	PROPN
cana-807	132	2	implies	imply	VERB
cana-807	132	3	k	k	X
cana-807	132	4	=	=	PUNCT
cana-807	132	5	𝛟.	𝛟.	NOUN
cana-807	132	6	thus	thus	ADV
cana-807	132	7	,	,	PUNCT
cana-807	132	8	n	n	CCONJ
cana-807	132	9	-	-	PUNCT
cana-807	132	10	pcl(s)-s	pcl(s)-	NOUN
cana-807	132	11	does	do	AUX
cana-807	132	12	not	not	PART
cana-807	132	13	contain	contain	VERB
cana-807	132	14	any	any	DET
cana-807	132	15	non	non	ADJ
cana-807	132	16	empty	empty	ADJ
cana-807	132	17	n	n	CCONJ
cana-807	132	18	-	-	PUNCT
cana-807	132	19	r*gα	r*gα	ADJ
cana-807	132	20	cs	cs	PROPN
cana-807	132	21	.	.	PROPN
cana-807	132	22	theorem	theorem	VERB
cana-807	132	23	3.22	3.22	NUM
cana-807	132	24	:	:	PUNCT
cana-807	132	25	let	let	VERB
cana-807	132	26	s	s	PRON
cana-807	132	27	be	be	AUX
cana-807	132	28	n	n	X
cana-807	132	29	-	-	PUNCT
cana-807	132	30	pgpr*wcs	pgpr*wcs	NOUN
cana-807	132	31	in	in	ADP
cana-807	132	32	x.	x.	NOUN
cana-807	132	33	thus	thus	ADV
cana-807	132	34	s	s	VERB
cana-807	132	35	is	be	AUX
cana-807	132	36	n	n	PRON
cana-807	132	37	-	-	PUNCT
cana-807	132	38	pcs	pc	NOUN
cana-807	132	39	iff	iff	PROPN
cana-807	132	40	n	n	CCONJ
cana-807	132	41	-	-	PUNCT
cana-807	132	42	pcl(s	pcl(s	PROPN
cana-807	132	43	)	)	PUNCT
cana-807	132	44	-s	-s	PROPN
cana-807	132	45	is	be	AUX
cana-807	132	46	n	n	PRON
cana-807	132	47	-	-	PUNCT
cana-807	132	48	rcs	rcs	NOUN
cana-807	132	49	.	.	PUNCT
cana-807	133	1	proof	proof	NOUN
cana-807	133	2	:	:	PUNCT
cana-807	133	3	suppose	suppose	VERB
cana-807	133	4	s	s	PRON
cana-807	133	5	is	be	AUX
cana-807	133	6	n	n	PRON
cana-807	133	7	-	-	PUNCT
cana-807	133	8	pcs	pc	NOUN
cana-807	133	9	.	.	PUNCT
cana-807	134	1	then	then	ADV
cana-807	134	2	n	n	CCONJ
cana-807	134	3	-	-	PUNCT
cana-807	134	4	pcl(s	pcl(s	NOUN
cana-807	134	5	)	)	PUNCT
cana-807	134	6	=	=	PUNCT
cana-807	135	1	s.	s.	PROPN
cana-807	135	2	so	so	ADV
cana-807	135	3	,	,	PUNCT
cana-807	135	4	n	n	CCONJ
cana-807	135	5	-	-	PUNCT
cana-807	135	6	pcl(s)-s	pcl(s)-	NOUN
cana-807	135	7	=	=	SYM
cana-807	135	8	𝛟	𝛟	X
cana-807	135	9	which	which	PRON
cana-807	135	10	is	be	AUX
cana-807	135	11	n	n	PRON
cana-807	135	12	-	-	PUNCT
cana-807	135	13	rcs	rcs	NOUN
cana-807	135	14	.	.	PUNCT
cana-807	136	1	conversely	conversely	ADV
cana-807	136	2	,	,	PUNCT
cana-807	136	3	suppose	suppose	VERB
cana-807	136	4	s	s	NOUN
cana-807	136	5	is	be	AUX
cana-807	136	6	n	n	PRON
cana-807	136	7	-	-	PUNCT
cana-807	136	8	pgpr*wcs	pgpr*wcs	NOUN
cana-807	136	9	and	and	CCONJ
cana-807	136	10	n	n	CCONJ
cana-807	136	11	-	-	PUNCT
cana-807	136	12	pcl(s	pcl(s	NOUN
cana-807	136	13	)	)	PUNCT
cana-807	136	14	=	=	SYM
cana-807	136	15	s	s	PROPN
cana-807	136	16	is	be	AUX
cana-807	136	17	n	n	PRON
cana-807	136	18	-	-	PUNCT
cana-807	136	19	rcs	rcs	NOUN
cana-807	136	20	.	.	PUNCT
cana-807	137	1	by	by	ADP
cana-807	137	2	the	the	DET
cana-807	137	3	theorem	theorem	PROPN
cana-807	137	4	,	,	PUNCT
cana-807	137	5	n	n	CCONJ
cana-807	137	6	-	-	PUNCT
cana-807	137	7	pcl(s	pcl(s	PROPN
cana-807	137	8	)	)	PUNCT
cana-807	137	9	s	s	PART
cana-807	137	10	implies	imply	VERB
cana-807	137	11	n	n	CCONJ
cana-807	137	12	-	-	PUNCT
cana-807	137	13	pcl(s	pcl(s	NOUN
cana-807	137	14	)	)	PUNCT
cana-807	137	15	=	=	PUNCT
cana-807	137	16	s.	s.	PROPN
cana-807	137	17	therefore	therefore	ADV
cana-807	137	18	,	,	PUNCT
cana-807	137	19	s	s	X
cana-807	137	20	is	be	AUX
cana-807	137	21	n	n	PRON
cana-807	137	22	-	-	PUNCT
cana-807	137	23	pcs	pc	NOUN
cana-807	137	24	4	4	NUM
cana-807	137	25	.	.	NOUN
cana-807	138	1	neutrosophic	neutrosophic	PROPN
cana-807	138	2	pre	pre	VERB
cana-807	138	3	generalized	generalized	ADJ
cana-807	138	4	pre	pre	VERB
cana-807	138	5	regular	regular	ADJ
cana-807	138	6	star	star	NOUN
cana-807	138	7	weakly	weakly	ADV
cana-807	138	8	open	open	ADJ
cana-807	138	9	sets	set	NOUN
cana-807	138	10	we	we	PRON
cana-807	138	11	go	go	VERB
cana-807	138	12	through	through	ADP
cana-807	138	13	some	some	DET
cana-807	138	14	basic	basic	ADJ
cana-807	138	15	definitions	definition	NOUN
cana-807	138	16	in	in	ADP
cana-807	138	17	this	this	DET
cana-807	138	18	section	section	NOUN
cana-807	138	19	.	.	PUNCT
cana-807	139	1	definition	definition	NOUN
cana-807	139	2	4.1	4.1	NUM
cana-807	139	3	:	:	PUNCT
cana-807	139	4	the	the	DET
cana-807	139	5	ns	ns	NOUN
cana-807	139	6	a	a	PRON
cana-807	139	7	is	be	AUX
cana-807	139	8	said	say	VERB
cana-807	139	9	to	to	PART
cana-807	139	10	be	be	AUX
cana-807	139	11	neutrosophic	neutrosophic	ADJ
cana-807	139	12	pre	pre	X
cana-807	139	13	generalised	generalised	ADJ
cana-807	139	14	pre	pre	NOUN
cana-807	140	1	regular	regular	ADJ
cana-807	140	2	weakly	weakly	ADV
cana-807	140	3	open	open	ADJ
cana-807	140	4	set	set	NOUN
cana-807	140	5	(	(	PUNCT
cana-807	140	6	npgpr*wos	npgpr*wos	NOUN
cana-807	140	7	shortly	shortly	ADV
cana-807	140	8	)	)	PUNCT
cana-807	140	9	if	if	SCONJ
cana-807	140	10	n	n	NOUN
cana-807	140	11	-	-	PUNCT
cana-807	140	12	pint(s	pint(s	NOUN
cana-807	140	13	)	)	PUNCT
cana-807	140	14	⸧m	⸧m	NOUN
cana-807	140	15	whenever	whenever	SCONJ
cana-807	140	16	s⸧m	s⸧m	NOUN
cana-807	140	17	and	and	CCONJ
cana-807	140	18	m	m	PROPN
cana-807	140	19	is	be	AUX
cana-807	140	20	n	n	PRON
cana-807	140	21	-	-	PUNCT
cana-807	140	22	r*gαcs	r*gαcs	NOUN
cana-807	140	23	in	in	ADP
cana-807	140	24	x.	x.	NOUN
cana-807	140	25	the	the	DET
cana-807	140	26	family	family	NOUN
cana-807	140	27	of	of	ADP
cana-807	140	28	all	all	DET
cana-807	140	29	n'pgpr*wos	n'pgpr*wos	NOUN
cana-807	140	30	of	of	ADP
cana-807	140	31	n	n	CCONJ
cana-807	140	32	-	-	PUNCT
cana-807	140	33	ts	ts	NOUN
cana-807	140	34	(	(	PUNCT
cana-807	140	35	x	x	X
cana-807	140	36	,	,	PUNCT
cana-807	140	37	τ	τ	X
cana-807	140	38	)	)	PUNCT
cana-807	140	39	is	be	AUX
cana-807	140	40	denoted	denote	VERB
cana-807	140	41	by	by	ADP
cana-807	140	42	n	n	CCONJ
cana-807	140	43	-	-	PUNCT
cana-807	140	44	pgpr*wo	pgpr*wo	NOUN
cana-807	140	45	-	-	PUNCT
cana-807	140	46	x.	x.	NOUN
cana-807	140	47	communications	communication	NOUN
cana-807	140	48	on	on	ADP
cana-807	140	49	applied	apply	VERB
cana-807	140	50	nonlinear	nonlinear	ADJ
cana-807	140	51	analysis	analysis	NOUN
cana-807	140	52	issn	issn	NOUN
cana-807	140	53	:	:	PUNCT
cana-807	140	54	1074	1074	NUM
cana-807	140	55	-	-	PUNCT
cana-807	140	56	133x	133x	NUM
cana-807	140	57	vol	vol	NOUN
cana-807	140	58	31	31	NUM
cana-807	140	59	no	no	NOUN
cana-807	140	60	.	.	PUNCT
cana-807	141	1	3s	3s	NUM
cana-807	141	2	(	(	PUNCT
cana-807	141	3	2024	2024	NUM
cana-807	141	4	)	)	PUNCT
cana-807	141	5	555	555	NUM
cana-807	141	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	141	7	example	example	NOUN
cana-807	141	8	4.2	4.2	NUM
cana-807	141	9	:	:	PUNCT
cana-807	141	10	let	let	VERB
cana-807	141	11	x	x	PUNCT
cana-807	141	12	=	=	PRON
cana-807	141	13	{	{	PUNCT
cana-807	141	14	a	a	DET
cana-807	141	15	,	,	PUNCT
cana-807	141	16	b	b	NOUN
cana-807	141	17	}	}	PUNCT
cana-807	141	18	and	and	CCONJ
cana-807	141	19	τ	τ	PROPN
cana-807	141	20	=	=	PUNCT
cana-807	141	21	{	{	PUNCT
cana-807	141	22	0n	0n	NUM
cana-807	141	23	,	,	PUNCT
cana-807	141	24	l	l	PROPN
cana-807	141	25	,	,	PUNCT
cana-807	141	26	m	m	PROPN
cana-807	141	27	,	,	PUNCT
cana-807	141	28	1n	1n	NUM
cana-807	141	29	}	}	PUNCT
cana-807	141	30	.	.	PUNCT
cana-807	142	1	where	where	SCONJ
cana-807	142	2	m	m	VERB
cana-807	142	3	=	=	SYM
cana-807	142	4	⟨(0.5	⟨(0.5	PROPN
cana-807	142	5	,	,	PUNCT
cana-807	142	6	0.3	0.3	NUM
cana-807	142	7	,	,	PUNCT
cana-807	142	8	0.6	0.6	NUM
cana-807	142	9	)	)	PUNCT
cana-807	142	10	,	,	PUNCT
cana-807	142	11	(	(	PUNCT
cana-807	142	12	0.4	0.4	NUM
cana-807	142	13	,	,	PUNCT
cana-807	142	14	0.4	0.4	NUM
cana-807	142	15	,	,	PUNCT
cana-807	142	16	0.7)⟩	0.7)⟩	NOUN
cana-807	142	17	l	l	NOUN
cana-807	142	18	=	=	SYM
cana-807	142	19	⟨(0.7	⟨(0.7	PROPN
cana-807	142	20	,	,	PUNCT
cana-807	142	21	0.5	0.5	NUM
cana-807	142	22	,	,	PUNCT
cana-807	142	23	0	0	NUM
cana-807	142	24	.3	.3	NUM
cana-807	142	25	)	)	PUNCT
cana-807	142	26	,	,	PUNCT
cana-807	142	27	(	(	PUNCT
cana-807	142	28	0.7	0.7	NUM
cana-807	142	29	,	,	PUNCT
cana-807	142	30	0.5	0.5	NUM
cana-807	142	31	,	,	PUNCT
cana-807	142	32	0.2)⟩	0.2)⟩	ADJ
cana-807	142	33	then	then	ADV
cana-807	142	34	x	x	PUNCT
cana-807	142	35	is	be	AUX
cana-807	142	36	a	a	DET
cana-807	142	37	n	n	NOUN
cana-807	142	38	-	-	PUNCT
cana-807	142	39	cs	cs	PROPN
cana-807	142	40	.	.	PUNCT
cana-807	143	1	here	here	ADV
cana-807	143	2	the	the	DET
cana-807	143	3	n	n	NOUN
cana-807	143	4	-	-	PUNCT
cana-807	143	5	s	s	NOUN
cana-807	143	6	s	s	NOUN
cana-807	143	7	=	=	SYM
cana-807	143	8	⟨(0.8	⟨(0.8	ADJ
cana-807	143	9	,	,	PUNCT
cana-807	143	10	0.9	0.9	NUM
cana-807	143	11	,	,	PUNCT
cana-807	143	12	0.2	0.2	NUM
cana-807	143	13	)	)	PUNCT
cana-807	143	14	,	,	PUNCT
cana-807	143	15	(	(	PUNCT
cana-807	143	16	0.9	0.9	NUM
cana-807	143	17	,	,	PUNCT
cana-807	143	18	0.6	0.6	NUM
cana-807	143	19	,	,	PUNCT
cana-807	143	20	0.1)⟩	0.1)⟩	PRON
cana-807	143	21	is	be	AUX
cana-807	143	22	a	a	DET
cana-807	143	23	n	n	ADV
cana-807	143	24	-	-	PUNCT
cana-807	143	25	pgpr*wos	pgpr*wos	NOUN
cana-807	143	26	in	in	ADP
cana-807	143	27	x	x	X
cana-807	143	28	,	,	PUNCT
cana-807	143	29	.	.	PUNCT
cana-807	144	1	since	since	SCONJ
cana-807	144	2	s	s	PRON
cana-807	144	3	⸧mc	⸧mc	ADJ
cana-807	144	4	and	and	CCONJ
cana-807	144	5	mc	mc	PROPN
cana-807	144	6	is	be	AUX
cana-807	144	7	a	a	DET
cana-807	144	8	n	n	NOUN
cana-807	144	9	-	-	PUNCT
cana-807	144	10	r*gαcs	r*gαcs	NOUN
cana-807	144	11	,	,	PUNCT
cana-807	144	12	as	as	SCONJ
cana-807	144	13	we	we	PRON
cana-807	144	14	have	have	VERB
cana-807	144	15	n	n	NOUN
cana-807	144	16	-	-	PUNCT
cana-807	144	17	pint(s	pint(s	NOUN
cana-807	144	18	)	)	PUNCT
cana-807	144	19	=	=	SYM
cana-807	144	20	s	s	PART
cana-807	144	21	⸧mc	⸧mc	NOUN
cana-807	144	22	.	.	PUNCT
cana-807	145	1	theorem	theorem	VERB
cana-807	145	2	4.3	4.3	NUM
cana-807	145	3	:	:	PUNCT
cana-807	145	4	every	every	DET
cana-807	145	5	n	n	X
cana-807	145	6	-	-	PUNCT
cana-807	145	7	os	os	NOUN
cana-807	145	8	is	be	AUX
cana-807	145	9	n	n	PRON
cana-807	145	10	-	-	PUNCT
cana-807	145	11	pgpr*wos	pgpr*wos	NOUN
cana-807	145	12	.	.	PUNCT
cana-807	146	1	but	but	CCONJ
cana-807	146	2	the	the	DET
cana-807	146	3	converse	converse	NOUN
cana-807	146	4	may	may	AUX
cana-807	146	5	not	not	PART
cana-807	146	6	be	be	AUX
cana-807	146	7	true	true	ADJ
cana-807	146	8	.	.	PUNCT
cana-807	147	1	proof	proof	NOUN
cana-807	147	2	:	:	PUNCT
cana-807	147	3	let	let	VERB
cana-807	147	4	m	m	PRON
cana-807	147	5	be	be	AUX
cana-807	147	6	n	n	PRON
cana-807	147	7	-	-	PUNCT
cana-807	147	8	r*gαcs	r*gαcs	NOUN
cana-807	147	9	in	in	ADP
cana-807	147	10	x	x	INTJ
cana-807	147	11	such	such	ADJ
cana-807	147	12	that	that	PRON
cana-807	147	13	s	s	NOUN
cana-807	147	14	⸧m	⸧m	NOUN
cana-807	147	15	.	.	PUNCT
cana-807	148	1	since	since	SCONJ
cana-807	148	2	s	s	PROPN
cana-807	148	3	is	be	AUX
cana-807	148	4	n	n	PRON
cana-807	148	5	-	-	PUNCT
cana-807	148	6	os	os	NOUN
cana-807	148	7	,	,	PUNCT
cana-807	148	8	n	n	CCONJ
cana-807	148	9	-	-	PUNCT
cana-807	148	10	pint(s	pint(s	NOUN
cana-807	148	11	)	)	PUNCT
cana-807	149	1	=	=	SYM
cana-807	149	2	s	s	X
cana-807	149	3	,	,	PUNCT
cana-807	149	4	by	by	ADP
cana-807	149	5	hypothesis	hypothesis	NOUN
cana-807	149	6	,	,	PUNCT
cana-807	149	7	.	.	PUNCT
cana-807	150	1	n	n	CCONJ
cana-807	150	2	-	-	PUNCT
cana-807	150	3	pint(s	pint(s	NOUN
cana-807	150	4	)	)	PUNCT
cana-807	150	5	=	=	SYM
cana-807	150	6	s	s	NOUN
cana-807	150	7	∩	∩	NOUN
cana-807	150	8	n	n	CCONJ
cana-807	150	9	-	-	PUNCT
cana-807	150	10	int(n	int(n	NOUN
cana-807	150	11	-	-	PUNCT
cana-807	150	12	cl(s	cl(s	NOUN
cana-807	150	13	)	)	PUNCT
cana-807	150	14	)	)	PUNCT
cana-807	151	1	=	=	SYM
cana-807	151	2	s	s	NOUN
cana-807	151	3	∩	∩	NOUN
cana-807	151	4	n	n	CCONJ
cana-807	151	5	-	-	PUNCT
cana-807	151	6	cl(s	cl(s	NOUN
cana-807	151	7	)	)	PUNCT
cana-807	151	8	⊃	⊃	PROPN
cana-807	151	9	s∩s	s∩s	PROPN
cana-807	151	10	=	=	NOUN
cana-807	151	11	scontainsm	scontainsm	NOUN
cana-807	151	12	therefore	therefore	ADV
cana-807	151	13	s	s	VERB
cana-807	151	14	is	be	AUX
cana-807	151	15	npgpr*wos	npgpr*wos	ADJ
cana-807	151	16	in	in	ADP
cana-807	151	17	x.	x.	PROPN
cana-807	151	18	example	example	NOUN
cana-807	151	19	4.4	4.4	NUM
cana-807	151	20	in	in	ADP
cana-807	151	21	example	example	NOUN
cana-807	151	22	4.2	4.2	NUM
cana-807	151	23	the	the	DET
cana-807	151	24	n	n	NUM
cana-807	151	25	-	-	PUNCT
cana-807	151	26	s	s	NOUN
cana-807	151	27	s	s	NOUN
cana-807	151	28	=	=	SYM
cana-807	151	29	⟨(0.8	⟨(0.8	ADJ
cana-807	151	30	,	,	PUNCT
cana-807	151	31	0.9	0.9	NUM
cana-807	151	32	,	,	PUNCT
cana-807	151	33	0.2	0.2	NUM
cana-807	151	34	)	)	PUNCT
cana-807	151	35	,	,	PUNCT
cana-807	151	36	(	(	PUNCT
cana-807	151	37	0.9	0.9	NUM
cana-807	151	38	,	,	PUNCT
cana-807	151	39	0.6	0.6	NUM
cana-807	151	40	,	,	PUNCT
cana-807	151	41	0.1)⟩	0.1)⟩	PRON
cana-807	151	42	is	be	AUX
cana-807	151	43	an	an	DET
cana-807	151	44	n	n	ADV
cana-807	151	45	-	-	PUNCT
cana-807	151	46	pgpr*wos	pgpr*wos	NOUN
cana-807	151	47	in	in	ADP
cana-807	151	48	x	x	NOUN
cana-807	151	49	,	,	PUNCT
cana-807	151	50	but	but	CCONJ
cana-807	151	51	not	not	PART
cana-807	151	52	a	a	DET
cana-807	151	53	n	n	NOUN
cana-807	151	54	-	-	PUNCT
cana-807	151	55	os	os	NOUN
cana-807	151	56	in	in	ADP
cana-807	151	57	x.	x.	PROPN
cana-807	151	58	theorem	theorem	VERB
cana-807	151	59	4.5	4.5	NUM
cana-807	151	60	:	:	PUNCT
cana-807	151	61	for	for	ADP
cana-807	151	62	any	any	DET
cana-807	151	63	nuetrosophic	nuetrosophic	ADJ
cana-807	151	64	topological	topological	ADJ
cana-807	151	65	space	space	NOUN
cana-807	151	66	x	x	NOUN
cana-807	151	67	,	,	PUNCT
cana-807	151	68	.	.	PUNCT
cana-807	152	1	we	we	PRON
cana-807	152	2	have	have	VERB
cana-807	152	3	the	the	DET
cana-807	152	4	following	follow	VERB
cana-807	152	5	1	1	NUM
cana-807	152	6	.	.	PUNCT
cana-807	153	1	every	every	DET
cana-807	153	2	n	n	NOUN
cana-807	153	3	-	-	PUNCT
cana-807	153	4	ros	ros	PROPN
cana-807	153	5	,	,	PUNCT
cana-807	153	6	n	n	CCONJ
cana-807	153	7	-	-	PUNCT
cana-807	153	8	os	os	NOUN
cana-807	153	9	,	,	PUNCT
cana-807	153	10	n	n	CCONJ
cana-807	153	11	-	-	PUNCT
cana-807	153	12	wos	wos	NOUN
cana-807	153	13	,	,	PUNCT
cana-807	153	14	n	n	CCONJ
cana-807	153	15	-	-	PUNCT
cana-807	153	16	pos	pos	NOUN
cana-807	153	17	,	,	PUNCT
cana-807	153	18	n	n	CCONJ
cana-807	153	19	-	-	PUNCT
cana-807	153	20	gos	go	NOUN
cana-807	153	21	is	be	AUX
cana-807	153	22	a	a	DET
cana-807	153	23	n	n	ADV
cana-807	153	24	-	-	PUNCT
cana-807	153	25	pgpr*wos	pgpr*wos	NOUN
cana-807	153	26	but	but	CCONJ
cana-807	153	27	the	the	DET
cana-807	153	28	converse	converse	NOUN
cana-807	153	29	need	need	VERB
cana-807	153	30	not	not	PART
cana-807	153	31	true	true	ADJ
cana-807	153	32	.	.	PUNCT
cana-807	154	1	2	2	X
cana-807	154	2	.	.	X
cana-807	154	3	every	every	DET
cana-807	154	4	n	n	NOUN
cana-807	154	5	-	-	PUNCT
cana-807	154	6	gpos	gpos	NOUN
cana-807	154	7	,	,	PUNCT
cana-807	154	8	n	n	CCONJ
cana-807	154	9	-	-	PUNCT
cana-807	154	10	gpros	gpro	NOUN
cana-807	154	11	is	be	AUX
cana-807	154	12	n	n	ADV
cana-807	154	13	-	-	PUNCT
cana-807	154	14	pgpr*wos	pgpr*wos	NOUN
cana-807	154	15	but	but	CCONJ
cana-807	154	16	the	the	DET
cana-807	154	17	converse	converse	NOUN
cana-807	154	18	need	need	VERB
cana-807	154	19	not	not	PART
cana-807	154	20	true	true	ADJ
cana-807	154	21	in	in	ADP
cana-807	154	22	general	general	ADJ
cana-807	154	23	.	.	PUNCT
cana-807	155	1	remark	remark	VERB
cana-807	155	2	4.6	4.6	NUM
cana-807	155	3	converse	converse	NOUN
cana-807	155	4	of	of	ADP
cana-807	155	5	theorem	theorem	NOUN
cana-807	155	6	4.5	4.5	NUM
cana-807	155	7	can	can	AUX
cana-807	155	8	be	be	AUX
cana-807	155	9	proved	prove	VERB
cana-807	155	10	by	by	ADP
cana-807	155	11	the	the	DET
cana-807	155	12	following	following	ADJ
cana-807	155	13	example	example	NOUN
cana-807	155	14	to	to	PART
cana-807	155	15	show	show	VERB
cana-807	155	16	it	it	PRON
cana-807	155	17	is	be	AUX
cana-807	155	18	not	not	PART
cana-807	155	19	true	true	ADJ
cana-807	155	20	1	1	NUM
cana-807	155	21	.	.	PUNCT
cana-807	156	1	m	m	VERB
cana-807	156	2	=	=	SYM
cana-807	156	3	⟨(0.5	⟨(0.5	PROPN
cana-807	156	4	,	,	PUNCT
cana-807	156	5	0.3	0.3	NUM
cana-807	156	6	,	,	PUNCT
cana-807	156	7	0.4	0.4	NUM
cana-807	156	8	)	)	PUNCT
cana-807	156	9	,	,	PUNCT
cana-807	156	10	(	(	PUNCT
cana-807	156	11	0.9	0.9	NUM
cana-807	156	12	,	,	PUNCT
cana-807	156	13	0.7	0.7	NUM
cana-807	156	14	,	,	PUNCT
cana-807	156	15	0.8)⟩.	0.8)⟩.	NOUN
cana-807	156	16	then	then	ADV
cana-807	156	17	(	(	PUNCT
cana-807	156	18	x	x	NOUN
cana-807	156	19	,	,	PUNCT
cana-807	156	20	)	)	PUNCT
cana-807	156	21	is	be	AUX
cana-807	156	22	a	a	DET
cana-807	156	23	n	n	NOUN
cana-807	156	24	-	-	PUNCT
cana-807	156	25	cs	cs	PROPN
cana-807	156	26	.	.	PUNCT
cana-807	157	1	here	here	ADV
cana-807	157	2	the	the	DET
cana-807	157	3	n	n	NOUN
cana-807	157	4	-	-	PUNCT
cana-807	157	5	s	s	NOUN
cana-807	157	6	s	s	X
cana-807	157	7	=	=	SYM
cana-807	157	8	⟨(0.3	⟨(0.3	PROPN
cana-807	157	9	,	,	PUNCT
cana-807	157	10	0.9	0.9	NUM
cana-807	157	11	,	,	PUNCT
cana-807	157	12	0	0	NUM
cana-807	157	13	.7	.7	NUM
cana-807	157	14	)	)	PUNCT
cana-807	157	15	,	,	PUNCT
cana-807	157	16	(	(	PUNCT
cana-807	157	17	0.7	0.7	NUM
cana-807	157	18	,	,	PUNCT
cana-807	157	19	0.5	0.5	NUM
cana-807	157	20	,	,	PUNCT
cana-807	157	21	0.9)⟩	0.9)⟩	NOUN
cana-807	157	22	is	be	AUX
cana-807	157	23	a	a	DET
cana-807	157	24	n	n	NOUN
cana-807	157	25	-	-	PUNCT
cana-807	157	26	pgpr*wcs	pgpr*wcs	NOUN
cana-807	157	27	in	in	ADP
cana-807	157	28	x	x	PROPN
cana-807	157	29	,	,	PUNCT
cana-807	157	30	.	.	PUNCT
cana-807	158	1	since	since	SCONJ
cana-807	158	2	s	s	PROPN
cana-807	158	3	⸧	⸧	NOUN
cana-807	158	4	m	m	VERB
cana-807	158	5	we	we	PRON
cana-807	158	6	have	have	VERB
cana-807	158	7	n	n	ADV
cana-807	158	8	-	-	PUNCT
cana-807	158	9	pint(s	pint(s	NOUN
cana-807	158	10	)	)	PUNCT
cana-807	159	1	=	=	SYM
cana-807	159	2	1n	1n	NUM
cana-807	159	3	⸧	⸧	NOUN
cana-807	159	4	1n	1n	NUM
cana-807	159	5	,	,	PUNCT
cana-807	159	6	but	but	CCONJ
cana-807	159	7	since	since	SCONJ
cana-807	159	8	n	n	CCONJ
cana-807	159	9	-	-	PUNCT
cana-807	159	10	int(n	int(n	NOUN
cana-807	159	11	-	-	PUNCT
cana-807	159	12	cl(s	cl(s	NOUN
cana-807	159	13	)	)	PUNCT
cana-807	159	14	)	)	PUNCT
cana-807	160	1	=	=	SYM
cana-807	161	1	1n	1n	NUM
cana-807	161	2	≠	≠	PROPN
cana-807	161	3	s	s	X
cana-807	161	4	,	,	PUNCT
cana-807	161	5	s	s	X
cana-807	161	6	is	be	AUX
cana-807	161	7	not	not	PART
cana-807	161	8	a	a	DET
cana-807	161	9	n	n	CCONJ
cana-807	161	10	-	-	PUNCT
cana-807	161	11	ros	ros	PROPN
cana-807	161	12	,	,	PUNCT
cana-807	161	13	similarly	similarly	ADV
cana-807	161	14	s	s	X
cana-807	161	15	is	be	AUX
cana-807	161	16	not	not	PART
cana-807	161	17	n	n	CCONJ
cana-807	161	18	-	-	PUNCT
cana-807	161	19	os	os	NOUN
cana-807	161	20	,	,	PUNCT
cana-807	161	21	n	n	CCONJ
cana-807	161	22	-	-	PUNCT
cana-807	161	23	wos	wos	NOUN
cana-807	161	24	,	,	PUNCT
cana-807	161	25	n	n	CCONJ
cana-807	161	26	-	-	PUNCT
cana-807	161	27	pos	pos	NOUN
cana-807	161	28	and	and	CCONJ
cana-807	161	29	n	n	CCONJ
cana-807	161	30	-	-	PUNCT
cana-807	161	31	gos	go	NOUN
cana-807	161	32	in	in	ADP
cana-807	161	33	(	(	PUNCT
cana-807	161	34	x	x	NOUN
cana-807	161	35	,	,	PUNCT
cana-807	161	36	)	)	PUNCT
cana-807	161	37	2	2	X
cana-807	161	38	.	.	X
cana-807	162	1	let	let	VERB
cana-807	162	2	x	x	PUNCT
cana-807	162	3	=	=	PRON
cana-807	162	4	{	{	PUNCT
cana-807	162	5	a	a	DET
cana-807	162	6	,	,	PUNCT
cana-807	162	7	b	b	NOUN
cana-807	162	8	}	}	PUNCT
cana-807	162	9	and	and	CCONJ
cana-807	162	10	=	=	SYM
cana-807	162	11	{	{	PUNCT
cana-807	162	12	0n	0n	NUM
cana-807	162	13	,	,	PUNCT
cana-807	162	14	m	m	PROPN
cana-807	162	15	,	,	PUNCT
cana-807	162	16	1n	1n	NUM
cana-807	162	17	}	}	PUNCT
cana-807	162	18	where	where	SCONJ
cana-807	162	19	m	m	VERB
cana-807	162	20	=	=	SYM
cana-807	162	21	⟨(0.4	⟨(0.4	PROPN
cana-807	162	22	,	,	PUNCT
cana-807	162	23	0.2	0.2	NUM
cana-807	162	24	,	,	PUNCT
cana-807	162	25	0.3	0.3	NUM
cana-807	162	26	)	)	PUNCT
cana-807	162	27	,	,	PUNCT
cana-807	162	28	(	(	PUNCT
cana-807	162	29	0.8	0.8	NUM
cana-807	162	30	,	,	PUNCT
cana-807	162	31	0.6	0.6	NUM
cana-807	162	32	,	,	PUNCT
cana-807	162	33	0.7)⟩.	0.7)⟩.	NOUN
cana-807	162	34	then	then	ADV
cana-807	162	35	x	x	X
cana-807	162	36	,	,	PUNCT
cana-807	162	37	is	be	AUX
cana-807	162	38	a	a	DET
cana-807	162	39	n	n	NOUN
cana-807	162	40	-	-	PUNCT
cana-807	162	41	cs	cs	PROPN
cana-807	162	42	.	.	PROPN
cana-807	162	43	m=	m=	X
cana-807	162	44	⟨(0.2	⟨(0.2	PROPN
cana-807	162	45	,	,	PUNCT
cana-807	162	46	0.1	0.1	NUM
cana-807	162	47	,	,	PUNCT
cana-807	162	48	0.8	0.8	NUM
cana-807	162	49	)	)	PUNCT
cana-807	162	50	,	,	PUNCT
cana-807	162	51	(	(	PUNCT
cana-807	162	52	0.2	0.2	NUM
cana-807	162	53	,	,	PUNCT
cana-807	162	54	0.1	0.1	NUM
cana-807	162	55	,	,	PUNCT
cana-807	162	56	0.8)⟩	0.8)⟩	NOUN
cana-807	162	57	here	here	ADV
cana-807	162	58	the	the	DET
cana-807	162	59	n	n	NOUN
cana-807	162	60	-	-	PUNCT
cana-807	162	61	s	s	NOUN
cana-807	162	62	s	s	NOUN
cana-807	162	63	=	=	SYM
cana-807	162	64	⟨(0.5	⟨(0.5	PROPN
cana-807	162	65	,	,	PUNCT
cana-807	162	66	0.9	0.9	NUM
cana-807	162	67	,	,	PUNCT
cana-807	162	68	0.5	0.5	NUM
cana-807	162	69	)	)	PUNCT
cana-807	162	70	,	,	PUNCT
cana-807	162	71	(	(	PUNCT
cana-807	162	72	0.8	0.8	NUM
cana-807	162	73	,	,	PUNCT
cana-807	162	74	0.9	0.9	NUM
cana-807	162	75	,	,	PUNCT
cana-807	162	76	0.2)⟩	0.2)⟩	NUM
cana-807	162	77	is	be	AUX
cana-807	162	78	a	a	DET
cana-807	162	79	n	n	NOUN
cana-807	162	80	-	-	PUNCT
cana-807	162	81	pgpr*wcs	pgpr*wcs	NOUN
cana-807	162	82	in	in	ADP
cana-807	162	83	x	x	PROPN
cana-807	162	84	,	,	PUNCT
cana-807	162	85	.	.	PUNCT
cana-807	163	1	since	since	SCONJ
cana-807	163	2	s	s	PROPN
cana-807	163	3	⸧	⸧	VERB
cana-807	163	4	0n	0n	NUM
cana-807	163	5	,	,	PUNCT
cana-807	163	6	we	we	PRON
cana-807	163	7	have	have	VERB
cana-807	163	8	n	n	ADV
cana-807	163	9	-	-	PUNCT
cana-807	163	10	pint(s	pint(s	NOUN
cana-807	163	11	)	)	PUNCT
cana-807	164	1	=	=	PUNCT
cana-807	164	2	0n	0n	NUM
cana-807	164	3	⸧	⸧	ADJ
cana-807	164	4	0n	0n	NUM
cana-807	164	5	,	,	PUNCT
cana-807	164	6	so	so	CCONJ
cana-807	164	7	s	s	NOUN
cana-807	164	8	is	be	AUX
cana-807	164	9	not	not	PART
cana-807	164	10	a	a	PRON
cana-807	164	11	n	n	CCONJ
cana-807	164	12	-	-	PUNCT
cana-807	164	13	gpos	gpos	NOUN
cana-807	164	14	,	,	PUNCT
cana-807	164	15	n	n	CCONJ
cana-807	164	16	-	-	PUNCT
cana-807	164	17	gpros	gpro	NOUN
cana-807	164	18	in	in	ADP
cana-807	164	19	x	x	X
cana-807	164	20	,	,	PUNCT
cana-807	164	21	.	.	PUNCT
cana-807	165	1	communications	communication	NOUN
cana-807	165	2	on	on	ADP
cana-807	165	3	applied	apply	VERB
cana-807	165	4	nonlinear	nonlinear	ADJ
cana-807	165	5	analysis	analysis	NOUN
cana-807	165	6	issn	issn	NOUN
cana-807	165	7	:	:	PUNCT
cana-807	165	8	1074	1074	NUM
cana-807	165	9	-	-	PUNCT
cana-807	165	10	133x	133x	NUM
cana-807	165	11	vol	vol	NOUN
cana-807	165	12	31	31	NUM
cana-807	165	13	no	no	NOUN
cana-807	165	14	.	.	PUNCT
cana-807	166	1	3s	3s	NUM
cana-807	166	2	(	(	PUNCT
cana-807	166	3	2024	2024	NUM
cana-807	166	4	)	)	PUNCT
cana-807	166	5	556	556	NUM
cana-807	166	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	166	7	theorem	theorem	VERB
cana-807	166	8	4.7	4.7	NUM
cana-807	166	9	:	:	PUNCT
cana-807	166	10	the	the	DET
cana-807	166	11	intersection	intersection	NOUN
cana-807	166	12	of	of	ADP
cana-807	166	13	two	two	NUM
cana-807	166	14	n	n	CCONJ
cana-807	166	15	-	-	PUNCT
cana-807	166	16	pgpr*wosis	pgpr*wosis	NOUN
cana-807	166	17	n	n	CCONJ
cana-807	166	18	-	-	PUNCT
cana-807	166	19	pgpr*wos	pgpr*wos	NOUN
cana-807	166	20	.	.	PUNCT
cana-807	167	1	proof	proof	NOUN
cana-807	167	2	:	:	PUNCT
cana-807	167	3	let	let	VERB
cana-807	167	4	c	c	NOUN
cana-807	168	1	and	and	CCONJ
cana-807	168	2	d	d	NOUN
cana-807	168	3	be	be	AUX
cana-807	168	4	the	the	DET
cana-807	168	5	n	n	CCONJ
cana-807	168	6	-	-	PUNCT
cana-807	168	7	pgpr*wosin	pgpr*wosin	NOUN
cana-807	168	8	x	x	NOUN
cana-807	168	9	let	let	VERB
cana-807	168	10	c∩	c∩	VERB
cana-807	168	11	d⊃m	d⊃m	PUNCT
cana-807	168	12	and	and	CCONJ
cana-807	168	13	m	m	AUX
cana-807	168	14	be	be	VERB
cana-807	168	15	n	n	PRON
cana-807	168	16	-	-	PUNCT
cana-807	168	17	r*gαos	r*gαo	NOUN
cana-807	168	18	in	in	ADP
cana-807	168	19	x	x	SYM
cana-807	168	20	where	where	SCONJ
cana-807	168	21	c⊃	c⊃	NOUN
cana-807	168	22	m	m	VERB
cana-807	168	23	and	and	CCONJ
cana-807	168	24	d⊃	d⊃	VERB
cana-807	168	25	m	m	PROPN
cana-807	168	26	then	then	ADV
cana-807	168	27	n	n	CCONJ
cana-807	168	28	-pint(c∩d)=	-pint(c∩d)=	ADV
cana-807	168	29	(	(	PUNCT
cana-807	168	30	c∩d)∩npint(n	c∩d)∩npint(n	PROPN
cana-807	168	31	-	-	PUNCT
cana-807	168	32	pcl(n	pcl(n	NOUN
cana-807	168	33	-	-	PUNCT
cana-807	168	34	pint(c∩d	pint(c∩d	NOUN
cana-807	168	35	)	)	PUNCT
cana-807	168	36	)	)	PUNCT
cana-807	168	37	)	)	PUNCT
cana-807	169	1	(	(	PUNCT
cana-807	169	2	c∩	c∩	NOUN
cana-807	169	3	d	d	NOUN
cana-807	169	4	)	)	PUNCT
cana-807	169	5	∩	∩	NOUN
cana-807	169	6	n	n	CCONJ
cana-807	169	7	-	-	SYM
cana-807	169	8	pint(c∩	pint(c∩	NUM
cana-807	169	9	d	d	NOUN
cana-807	169	10	)	)	PUNCT
cana-807	169	11	=	=	SYM
cana-807	169	12	(	(	PUNCT
cana-807	169	13	c∩	c∩	PROPN
cana-807	169	14	d)∩	d)∩	NUM
cana-807	169	15	n	n	CCONJ
cana-807	169	16	-	-	PUNCT
cana-807	169	17	pint(c))∩	pint(c))∩	NOUN
cana-807	169	18	n	n	CCONJ
cana-807	169	19	-	-	PUNCT
cana-807	169	20	pint(d)⊃	pint(d)⊃	NOUN
cana-807	169	21	m	m	VERB
cana-807	169	22	by	by	ADP
cana-807	169	23	hypothesis	hypothesis	NOUN
cana-807	169	24	hence	hence	ADV
cana-807	169	25	c∩	c∩	PROPN
cana-807	169	26	d	d	PROPN
cana-807	169	27	is	be	AUX
cana-807	169	28	also	also	ADV
cana-807	169	29	n	n	CCONJ
cana-807	169	30	-	-	PUNCT
cana-807	169	31	pgpr*wos	pgpr*wos	NOUN
cana-807	169	32	in	in	ADP
cana-807	169	33	x.	x.	NOUN
cana-807	169	34	theorem	theorem	VERB
cana-807	169	35	4.8	4.8	NUM
cana-807	169	36	:	:	PUNCT
cana-807	169	37	the	the	DET
cana-807	169	38	n	n	CCONJ
cana-807	169	39	-	-	PUNCT
cana-807	169	40	s	s	NOUN
cana-807	169	41	s	s	X
cana-807	169	42	of	of	ADP
cana-807	169	43	ncs	ncs	PROPN
cana-807	169	44	x	x	X
cana-807	169	45	,	,	PUNCT
cana-807	169	46	is	be	AUX
cana-807	169	47	a	a	DET
cana-807	169	48	n	n	CCONJ
cana-807	169	49	-	-	PUNCT
cana-807	169	50	pgpr*wosin	pgpr*wosin	NOUN
cana-807	169	51	x	x	NOUN
cana-807	169	52	,	,	PUNCT
cana-807	169	53	iff	iff	PROPN
cana-807	169	54	m	m	PROPN
cana-807	169	55			PROPN
cana-807	169	56	n	n	CCONJ
cana-807	169	57	-	-	PUNCT
cana-807	169	58	pint(s	pint(s	NOUN
cana-807	169	59	)	)	PUNCT
cana-807	169	60	whenever	whenever	SCONJ
cana-807	169	61	m	m	VERB
cana-807	169	62	is	be	AUX
cana-807	169	63	a	a	DET
cana-807	169	64	nr*gαcs	nr*gαcs	NOUN
cana-807	169	65	in	in	ADP
cana-807	169	66	x	x	NOUN
cana-807	169	67	,	,	PUNCT
cana-807	169	68	and	and	CCONJ
cana-807	169	69	m	m	PROPN
cana-807	169	70			PROPN
cana-807	169	71	s	s	PART
cana-807	169	72	proof	proof	NOUN
cana-807	169	73	:	:	PUNCT
cana-807	169	74	let	let	VERB
cana-807	169	75	s	s	PRON
cana-807	169	76	be	be	AUX
cana-807	169	77	n	n	X
cana-807	169	78	-	-	PUNCT
cana-807	169	79	pgpr*wosin	pgpr*wosin	NOUN
cana-807	169	80	x	x	NOUN
cana-807	169	81	and	and	CCONJ
cana-807	169	82	m	m	PROPN
cana-807	169	83	is	be	AUX
cana-807	169	84	n	n	PRON
cana-807	169	85	-	-	PUNCT
cana-807	169	86	r*gαcs	r*gαcs	NOUN
cana-807	169	87	in	in	ADP
cana-807	169	88	x	x	NOUN
cana-807	169	89	,	,	PUNCT
cana-807	169	90	such	such	ADJ
cana-807	169	91	that	that	DET
cana-807	169	92	s⸧m	s⸧m	NOUN
cana-807	169	93	.	.	PUNCT
cana-807	170	1	then	then	ADV
cana-807	170	2	x	x	X
cana-807	170	3	-	-	PUNCT
cana-807	170	4	s	s	X
cana-807	170	5	is	be	AUX
cana-807	170	6	n	n	PRON
cana-807	170	7	-	-	PUNCT
cana-807	170	8	pgpr*w	pgpr*w	ADV
cana-807	170	9	-	-	PUNCT
cana-807	170	10	closed	closed	ADJ
cana-807	170	11	in	in	ADP
cana-807	170	12	x.	x.	NOUN
cana-807	170	13	also	also	ADV
cana-807	170	14	x	x	PROPN
cana-807	170	15	-	-	PUNCT
cana-807	170	16	s	s	X
cana-807	170	17			PROPN
cana-807	170	18	x	x	PROPN
cana-807	170	19	-	-	NOUN
cana-807	170	20	m	m	PRON
cana-807	170	21	and	and	CCONJ
cana-807	170	22	x	x	NOUN
cana-807	170	23	-	-	NOUN
cana-807	170	24	m	m	VERB
cana-807	170	25	is	be	AUX
cana-807	170	26	n	n	PRON
cana-807	170	27	-	-	PUNCT
cana-807	170	28	r*gαos	r*gαo	NOUN
cana-807	170	29	in	in	ADP
cana-807	170	30	x.	x.	NOUN
cana-807	170	31	hence	hence	ADV
cana-807	170	32	n	n	CCONJ
cana-807	170	33	-	-	PUNCT
cana-807	170	34	pcl(x	pcl(x	PROPN
cana-807	170	35	-	-	PUNCT
cana-807	170	36	s	s	NOUN
cana-807	170	37	)	)	PUNCT
cana-807	170	38			PROPN
cana-807	170	39	x	x	NOUN
cana-807	170	40	-	-	NOUN
cana-807	170	41	m	m	VERB
cana-807	170	42	we	we	PRON
cana-807	170	43	know	know	VERB
cana-807	170	44	that	that	SCONJ
cana-807	170	45	n	n	CCONJ
cana-807	170	46	-	-	PUNCT
cana-807	170	47	pcl(x	pcl(x	PROPN
cana-807	170	48	-	-	PUNCT
cana-807	170	49	s	s	NOUN
cana-807	170	50	)	)	PUNCT
cana-807	170	51	=	=	SYM
cana-807	170	52	x	x	SYM
cana-807	170	53	n	n	CCONJ
cana-807	170	54	-	-	PUNCT
cana-807	170	55	pint(s	pint(s	NOUN
cana-807	170	56	)	)	PUNCT
cana-807	170	57	x	x	NOUN
cana-807	170	58	-	-	PUNCT
cana-807	170	59	n	n	CCONJ
cana-807	170	60	-	-	PUNCT
cana-807	170	61	pint(s	pint(s	NOUN
cana-807	170	62	)	)	PUNCT
cana-807	170	63			PROPN
cana-807	170	64	x	x	INTJ
cana-807	170	65	-	-	NOUN
cana-807	170	66	m.	m.	NOUN
cana-807	171	1	so	so	ADV
cana-807	171	2	,	,	PUNCT
cana-807	171	3	m	m	VERB
cana-807	171	4			ADJ
cana-807	171	5	n	n	CCONJ
cana-807	171	6	-	-	PUNCT
cana-807	171	7	pint(s	pint(s	NOUN
cana-807	171	8	)	)	PUNCT
cana-807	171	9	.	.	PUNCT
cana-807	172	1	conversely	conversely	ADV
cana-807	172	2	,	,	PUNCT
cana-807	172	3	suppose	suppose	VERB
cana-807	172	4	m	m	PROPN
cana-807	172	5			ADJ
cana-807	172	6	n	n	CCONJ
cana-807	172	7	-	-	PUNCT
cana-807	172	8	pint(s	pint(s	NOUN
cana-807	172	9	)	)	PUNCT
cana-807	172	10	whenever	whenever	SCONJ
cana-807	172	11	m	m	PROPN
cana-807	172	12	is	be	AUX
cana-807	172	13	n	n	PRON
cana-807	172	14	-	-	PUNCT
cana-807	172	15	r*gαcs	r*gαcs	NOUN
cana-807	172	16	and	and	CCONJ
cana-807	172	17	s⸧m	s⸧m	NOUN
cana-807	172	18	.	.	PUNCT
cana-807	173	1	to	to	PART
cana-807	173	2	prove	prove	VERB
cana-807	173	3	s	s	NOUN
cana-807	173	4	is	be	AUX
cana-807	173	5	n	n	PRON
cana-807	173	6	-	-	PUNCT
cana-807	173	7	pgpr*wos	pgpr*wos	NOUN
cana-807	173	8	.	.	PUNCT
cana-807	174	1	let	let	VERB
cana-807	174	2	f	f	PRON
cana-807	174	3	be	be	AUX
cana-807	174	4	n	n	PRON
cana-807	174	5	-	-	PUNCT
cana-807	174	6	r*gαcs	r*gαcs	NOUN
cana-807	174	7	of	of	ADP
cana-807	174	8	x	x	INTJ
cana-807	174	9	such	such	ADJ
cana-807	174	10	that	that	SCONJ
cana-807	174	11	x	x	VERB
cana-807	174	12	-	-	PUNCT
cana-807	174	13	s	s	X
cana-807	174	14			PROPN
cana-807	174	15	f	f	PROPN
cana-807	174	16	then	then	ADV
cana-807	174	17	x	x	PROPN
cana-807	174	18	-	-	PUNCT
cana-807	174	19	f	f	PROPN
cana-807	174	20			PROPN
cana-807	174	21	s.	s.	PROPN
cana-807	174	22	now	now	ADV
cana-807	174	23	x	x	VERB
cana-807	174	24	-	-	PUNCT
cana-807	174	25	f	f	PROPN
cana-807	174	26	is	be	AUX
cana-807	174	27	n	n	PRON
cana-807	174	28	-	-	PUNCT
cana-807	174	29	r*gαcs	r*gαcs	NOUN
cana-807	174	30	contained	contain	VERB
cana-807	174	31	in	in	ADP
cana-807	174	32	s.	s.	PROPN
cana-807	174	33	so	so	ADV
cana-807	174	34	x	x	PROPN
cana-807	174	35	-	-	PUNCT
cana-807	174	36	f	f	PROPN
cana-807	174	37			PROPN
cana-807	174	38	n	n	CCONJ
cana-807	174	39	-	-	PUNCT
cana-807	174	40	pint(s	pint(s	NOUN
cana-807	174	41	)	)	PUNCT
cana-807	174	42	implies	imply	VERB
cana-807	174	43	x	x	NOUN
cana-807	174	44	-	-	PUNCT
cana-807	174	45	n	n	CCONJ
cana-807	174	46	-	-	PUNCT
cana-807	174	47	pint(s	pint(s	NOUN
cana-807	174	48	)	)	PUNCT
cana-807	174	49			PROPN
cana-807	174	50	f.	f.	PROPN
cana-807	174	51	but	but	CCONJ
cana-807	174	52	n	n	CCONJ
cana-807	174	53	-	-	PUNCT
cana-807	174	54	pcl(x	pcl(x	PROPN
cana-807	174	55	-	-	PUNCT
cana-807	174	56	s	s	NOUN
cana-807	174	57	)	)	PUNCT
cana-807	174	58	=	=	SYM
cana-807	174	59	x	x	X
cana-807	174	60	-	-	PUNCT
cana-807	174	61	n	n	CCONJ
cana-807	174	62	-	-	PUNCT
cana-807	174	63	pints	pint	NOUN
cana-807	174	64			PROPN
cana-807	174	65	f.	f.	PROPN
cana-807	174	66	therefore	therefore	ADV
cana-807	174	67	x	x	VERB
cana-807	174	68	-	-	PUNCT
cana-807	174	69	s	s	X
cana-807	174	70	is	be	AUX
cana-807	174	71	n	n	PRON
cana-807	174	72	-	-	PUNCT
cana-807	174	73	pgpr*wcs	pgpr*wcs	NOUN
cana-807	174	74	.	.	PUNCT
cana-807	175	1	hence	hence	ADV
cana-807	175	2	s	s	X
cana-807	175	3	is	be	AUX
cana-807	175	4	n	n	PRON
cana-807	175	5	-	-	PUNCT
cana-807	175	6	pgpr*wos	pgpr*wos	NOUN
cana-807	175	7	.	.	PUNCT
cana-807	176	1	theorem	theorem	VERB
cana-807	176	2	4.9	4.9	NUM
cana-807	176	3	:	:	PUNCT
cana-807	176	4	if	if	SCONJ
cana-807	176	5	s	s	VERB
cana-807	176	6			PROPN
cana-807	176	7	x	x	X
cana-807	176	8	is	be	AUX
cana-807	176	9	n	n	CCONJ
cana-807	176	10	-	-	PUNCT
cana-807	176	11	pgpr*wcs	pgpr*wcs	NOUN
cana-807	176	12	in	in	ADP
cana-807	176	13	x	x	PROPN
cana-807	176	14	and	and	CCONJ
cana-807	176	15	f	f	PROPN
cana-807	176	16	=	=	SYM
cana-807	176	17	𝛟	𝛟	ADP
cana-807	176	18	then	then	ADV
cana-807	176	19	n	n	CCONJ
cana-807	176	20	-	-	PUNCT
cana-807	176	21	pcl(s)-s	pcl(s)-	NOUN
cana-807	176	22	is	be	AUX
cana-807	176	23	n	n	CCONJ
cana-807	176	24	-	-	PUNCT
cana-807	176	25	pgpr*wos	pgpr*wos	NOUN
cana-807	176	26	.	.	PUNCT
cana-807	177	1	proof	proof	NOUN
cana-807	177	2	:	:	PUNCT
cana-807	177	3	let	let	VERB
cana-807	177	4	s	s	PRON
cana-807	177	5	be	be	AUX
cana-807	177	6	n	n	X
cana-807	177	7	-	-	PUNCT
cana-807	177	8	pgpr*wcs	pgpr*wcs	NOUN
cana-807	177	9	and	and	CCONJ
cana-807	177	10	f	f	NOUN
cana-807	177	11	=	=	SYM
cana-807	177	12	𝛟.	𝛟.	NOUN
cana-807	177	13	let	let	VERB
cana-807	177	14	s	s	PRON
cana-807	177	15			PROPN
cana-807	177	16	n	n	CCONJ
cana-807	177	17	-	-	PUNCT
cana-807	177	18	pcl(a)-a	pcl(a)-a	PROPN
cana-807	177	19	where	where	SCONJ
cana-807	177	20	f	f	PROPN
cana-807	177	21	is	be	AUX
cana-807	177	22	n	n	PRON
cana-807	177	23	-	-	PUNCT
cana-807	177	24	r*gαcs	r*gαcs	NOUN
cana-807	177	25	.	.	PUNCT
cana-807	178	1	then	then	ADV
cana-807	178	2	by	by	ADP
cana-807	178	3	the	the	DET
cana-807	178	4	theorem	theorem	NOUN
cana-807	178	5	n	n	CCONJ
cana-807	178	6	-	-	PUNCT
cana-807	178	7	pcl(s)-s	pcl(s)-	NOUN
cana-807	178	8	does	do	AUX
cana-807	178	9	not	not	PART
cana-807	178	10	contain	contain	VERB
cana-807	178	11	n	n	PRON
cana-807	178	12	-	-	PUNCT
cana-807	178	13	r*gαcs	r*gαcs	NOUN
cana-807	178	14	in	in	ADP
cana-807	178	15	x.	x.	NOUN
cana-807	179	1	i.e.	i.e.	X
cana-807	179	2	f	f	X
cana-807	179	3	=	=	PUNCT
cana-807	179	4	𝛟.	𝛟.	NOUN
cana-807	179	5	therefore	therefore	ADV
cana-807	179	6	f	f	PROPN
cana-807	179	7			PROPN
cana-807	179	8	n	n	CCONJ
cana-807	179	9	-	-	PUNCT
cana-807	179	10	pint(n	pint(n	NOUN
cana-807	179	11	-	-	PUNCT
cana-807	179	12	pcl(s)-s	pcl(s)-	NOUN
cana-807	179	13	)	)	PUNCT
cana-807	179	14	.	.	PUNCT
cana-807	180	1	n	n	CCONJ
cana-807	180	2	-	-	PUNCT
cana-807	180	3	pcl(s)-s	pcl(s)-	NOUN
cana-807	180	4	is	be	AUX
cana-807	180	5	n	n	CCONJ
cana-807	180	6	-	-	PUNCT
cana-807	180	7	pgpr*wos	pgpr*wos	NOUN
cana-807	180	8	theorem	theorem	VERB
cana-807	180	9	4.10	4.10	NUM
cana-807	180	10	:	:	PUNCT
cana-807	180	11	if	if	SCONJ
cana-807	180	12	s	s	PRON
cana-807	180	13	be	be	AUX
cana-807	180	14	n	n	PRON
cana-807	180	15	-	-	PUNCT
cana-807	180	16	s	s	NOUN
cana-807	180	17	is	be	AUX
cana-807	180	18	n	n	ADV
cana-807	180	19	-	-	PUNCT
cana-807	180	20	pgpr*wos	pgpr*wos	NOUN
cana-807	180	21	in	in	ADP
cana-807	180	22	x	x	PUNCT
cana-807	180	23	iff	iff	PROPN
cana-807	180	24	m	m	NOUN
cana-807	180	25	=	=	ADJ
cana-807	180	26	x	x	INTJ
cana-807	180	27	whenever	whenever	SCONJ
cana-807	180	28	m	m	VERB
cana-807	180	29	is	be	AUX
cana-807	180	30	n	n	PRON
cana-807	180	31	-	-	PUNCT
cana-807	180	32	r*gαcs	r*gαcs	NOUN
cana-807	180	33	and	and	CCONJ
cana-807	180	34	mc	mc	PROPN
cana-807	180	35			PROPN
cana-807	180	36	communications	communication	NOUN
cana-807	180	37	on	on	ADP
cana-807	180	38	applied	apply	VERB
cana-807	180	39	nonlinear	nonlinear	ADJ
cana-807	180	40	analysis	analysis	NOUN
cana-807	180	41	issn	issn	NOUN
cana-807	180	42	:	:	PUNCT
cana-807	180	43	1074	1074	NUM
cana-807	180	44	-	-	PUNCT
cana-807	180	45	133x	133x	NUM
cana-807	180	46	vol	vol	NOUN
cana-807	180	47	31	31	NUM
cana-807	180	48	no	no	NOUN
cana-807	180	49	.	.	PUNCT
cana-807	181	1	3s	3s	NUM
cana-807	181	2	(	(	PUNCT
cana-807	181	3	2024	2024	NUM
cana-807	181	4	)	)	PUNCT
cana-807	181	5	557	557	NUM
cana-807	181	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-807	181	7	(	(	PUNCT
cana-807	181	8	(	(	PUNCT
cana-807	181	9	n	n	CCONJ
cana-807	181	10	-	-	PUNCT
cana-807	181	11	pint(s	pint(s	NOUN
cana-807	181	12	)	)	PUNCT
cana-807	181	13	)	)	PUNCT
cana-807	182	1	sc)c	sc)c	NOUN
cana-807	182	2	.	.	PUNCT
cana-807	183	1	proof	proof	NOUN
cana-807	183	2	:	:	PUNCT
cana-807	183	3	suppos	suppos	NOUN
cana-807	183	4	s	s	PART
cana-807	183	5	is	be	AUX
cana-807	183	6	n	n	ADV
cana-807	183	7	-	-	PUNCT
cana-807	183	8	pgpr*wos	pgpr*wos	NOUN
cana-807	183	9	in	in	ADP
cana-807	183	10	x.	x.	NOUN
cana-807	183	11	let	let	VERB
cana-807	183	12	m	m	PRON
cana-807	183	13	be	be	AUX
cana-807	183	14	n	n	PRON
cana-807	183	15	-	-	PUNCT
cana-807	183	16	r*gαcs	r*gαcs	NOUN
cana-807	183	17	and	and	CCONJ
cana-807	183	18	mc	mc	PROPN
cana-807	183	19			PROPN
cana-807	183	20	(	(	PUNCT
cana-807	183	21	(	(	PUNCT
cana-807	183	22	n	n	CCONJ
cana-807	183	23	-	-	PUNCT
cana-807	183	24	pints	pint	NOUN
cana-807	183	25	)	)	PUNCT
cana-807	183	26	sc	sc	PROPN
cana-807	183	27	)	)	PUNCT
cana-807	183	28	c	c	PROPN
cana-807	183	29	mc	mc	PROPN
cana-807	183	30	⊆	⊆	NUM
cana-807	183	31	(	(	PUNCT
cana-807	183	32	n	n	CCONJ
cana-807	183	33	-	-	PUNCT
cana-807	183	34	pint	pint	NOUN
cana-807	183	35	(	(	PUNCT
cana-807	183	36	s))c	s))c	PROPN
cana-807	183	37	∩	∩	PROPN
cana-807	183	38	s	s	PART
cana-807	183	39	mc	mc	PROPN
cana-807	183	40			PROPN
cana-807	183	41	(	(	PUNCT
cana-807	183	42	n	n	CCONJ
cana-807	183	43	-	-	PUNCT
cana-807	183	44	pints	pint	NOUN
cana-807	183	45	)	)	PUNCT
cana-807	183	46	c	c	NOUN
cana-807	184	1	−	−	PROPN
cana-807	184	2	s	s	PART
cana-807	184	3	.	.	PUNCT
cana-807	185	1	mc	mc	PROPN
cana-807	185	2			PROPN
cana-807	185	3	n	n	CCONJ
cana-807	185	4	-	-	PUNCT
cana-807	185	5	pcl	pcl	PROPN
cana-807	185	6	(	(	PUNCT
cana-807	185	7	sc	sc	PROPN
cana-807	185	8	)	)	PUNCT
cana-807	185	9	−	−	PROPN
cana-807	185	10	sc	sc	PROPN
cana-807	185	11	m'c	m'c	PRON
cana-807	185	12	is	be	AUX
cana-807	185	13	n	n	PRON
cana-807	185	14	-	-	PUNCT
cana-807	185	15	r*gαcs	r*gαcs	NOUN
cana-807	185	16	mc	mc	PROPN
cana-807	185	17	=	=	SYM
cana-807	185	18	𝛟	𝛟	PROPN
cana-807	185	19	and	and	CCONJ
cana-807	185	20	therefore	therefore	ADV
cana-807	185	21	m	m	VERB
cana-807	185	22	=	=	NOUN
cana-807	185	23	x	x	SYM
cana-807	185	24	theorem	theorem	VERB
cana-807	185	25	4.11	4.11	NUM
cana-807	185	26	:	:	PUNCT
cana-807	185	27	let	let	VERB
cana-807	185	28	x	x	PRON
cana-807	185	29	be	be	AUX
cana-807	185	30	a	a	DET
cana-807	185	31	n	n	ADV
cana-807	185	32	-	-	PUNCT
cana-807	185	33	ts	ts	NOUN
cana-807	185	34	.	.	PUNCT
cana-807	186	1	and	and	CCONJ
cana-807	186	2	s	s	AUX
cana-807	186	3	be	be	AUX
cana-807	186	4	a	a	DET
cana-807	186	5	n	n	ADV
cana-807	186	6	-	-	PUNCT
cana-807	186	7	pgpr*wos	pgpr*wos	NOUN
cana-807	186	8	in	in	ADP
cana-807	186	9	x.	x.	NOUN
cana-807	186	10	then	then	ADV
cana-807	186	11	x	x	X
cana-807	187	1	=	=	PUNCT
cana-807	187	2	s.	s.	PROPN
cana-807	187	3	we	we	PRON
cana-807	187	4	know	know	VERB
cana-807	187	5	that	that	SCONJ
cana-807	187	6	n	n	CCONJ
cana-807	187	7	-	-	PUNCT
cana-807	187	8	pcl(x	pcl(x	PROPN
cana-807	187	9	-	-	PUNCT
cana-807	187	10	s	s	NOUN
cana-807	187	11	)	)	PUNCT
cana-807	187	12	=	=	SYM
cana-807	188	1	x	x	X
cana-807	188	2	-	-	PUNCT
cana-807	188	3	n	n	CCONJ
cana-807	188	4	-	-	PUNCT
cana-807	188	5	pint(s	pint(s	NOUN
cana-807	188	6	)	)	PUNCT
cana-807	188	7	n	n	CCONJ
cana-807	188	8	-	-	PUNCT
cana-807	188	9	pcl(x	pcl(x	PROPN
cana-807	188	10	-	-	PUNCT
cana-807	188	11	s	s	NOUN
cana-807	188	12	)	)	PUNCT
cana-807	188	13	+	+	CCONJ
cana-807	188	14	n	n	CCONJ
cana-807	188	15	-	-	PUNCT
cana-807	188	16	pint(s	pint(s	NOUN
cana-807	188	17	)	)	PUNCT
cana-807	188	18	=	=	SYM
cana-807	188	19	s	s	NOUN
cana-807	188	20	n	n	ADV
cana-807	188	21	-	-	PUNCT
cana-807	188	22	pcl(s)c	pcl(s)c	ADJ
cana-807	188	23	+	+	CCONJ
cana-807	188	24	n	n	CCONJ
cana-807	188	25	-	-	PUNCT
cana-807	188	26	pint(s	pint(s	NOUN
cana-807	188	27	)	)	PUNCT
cana-807	189	1	=	=	SYM
cana-807	189	2	s	s	X
cana-807	189	3	(	(	PUNCT
cana-807	189	4	n	n	CCONJ
cana-807	189	5	-	-	PUNCT
cana-807	189	6	pints)c	pints)c	ADJ
cana-807	189	7	+	+	CCONJ
cana-807	189	8	n	n	CCONJ
cana-807	189	9	-	-	PUNCT
cana-807	189	10	pints	pint	NOUN
cana-807	189	11	=	=	SYM
cana-807	189	12	s	s	NOUN
cana-807	189	13	theorem	theorem	NOUN
cana-807	189	14	4.12	4.12	NUM
cana-807	189	15	:	:	PUNCT
cana-807	189	16	if	if	SCONJ
cana-807	189	17	s	s	X
cana-807	189	18	and	and	CCONJ
cana-807	189	19	c	c	PROPN
cana-807	189	20	are	be	AUX
cana-807	189	21	neutrosophic	neutrosophic	ADJ
cana-807	189	22	sets	set	NOUN
cana-807	189	23	of	of	ADP
cana-807	189	24	x.	x.	NOUN
cana-807	189	25	if	if	SCONJ
cana-807	189	26	c	c	PROPN
cana-807	189	27	is	be	AUX
cana-807	189	28	n	n	ADV
cana-807	189	29	-	-	PUNCT
cana-807	189	30	pgpr*wos	pgpr*wos	NOUN
cana-807	189	31	and	and	CCONJ
cana-807	189	32	n	n	CCONJ
cana-807	189	33	-	-	PUNCT
cana-807	189	34	pint(s	pint(s	NOUN
cana-807	189	35	)	)	PUNCT
cana-807	189	36			PROPN
cana-807	189	37	s	s	PART
cana-807	189	38	then	then	ADV
cana-807	189	39	cc	cc	PROPN
cana-807	189	40			PROPN
cana-807	189	41	n	n	CCONJ
cana-807	189	42	-	-	PUNCT
cana-807	189	43	pcl	pcl	PROPN
cana-807	189	44	(	(	PUNCT
cana-807	189	45	cc	cc	PROPN
cana-807	189	46	)	)	PUNCT
cana-807	189	47			PROPN
cana-807	189	48	sc	sc	PROPN
cana-807	189	49	.	.	PUNCT
cana-807	189	50	proof	proof	NOUN
cana-807	189	51	:	:	PUNCT
cana-807	189	52	n	n	NUM
cana-807	189	53	-	-	PUNCT
cana-807	189	54	pint(c	pint(c	ADJ
cana-807	189	55	)	)	PUNCT
cana-807	189	56			PROPN
cana-807	189	57	s	s	PROPN
cana-807	189	58	and	and	CCONJ
cana-807	189	59	c	c	PROPN
cana-807	189	60			PROPN
cana-807	189	61	n	n	CCONJ
cana-807	189	62	-	-	PUNCT
cana-807	189	63	pint(c	pint(c	ADJ
cana-807	189	64	)	)	PUNCT
cana-807	189	65	c	c	NOUN
cana-807	189	66			PROPN
cana-807	189	67	n	n	CCONJ
cana-807	189	68	-	-	PUNCT
cana-807	189	69	pint(c	pint(c	ADJ
cana-807	189	70	)	)	PUNCT
cana-807	189	71			PROPN
cana-807	189	72	s	s	PART
cana-807	189	73	x	x	X
cana-807	189	74	–	–	PUNCT
cana-807	189	75	c	c	X
cana-807	189	76			PROPN
cana-807	189	77	x	x	INTJ
cana-807	189	78	–	–	PUNCT
cana-807	189	79	n	n	CCONJ
cana-807	189	80	-	-	PUNCT
cana-807	189	81	pint(c	pint(c	ADJ
cana-807	189	82	)	)	PUNCT
cana-807	189	83			PROPN
cana-807	189	84	x	x	VERB
cana-807	189	85	-	-	PUNCT
cana-807	189	86	s	s	X
cana-807	189	87	cc	cc	NOUN
cana-807	189	88			PROPN
cana-807	189	89	n	n	CCONJ
cana-807	189	90	-	-	PUNCT
cana-807	189	91	pcl(x	pcl(x	PROPN
cana-807	189	92	-	-	PUNCT
cana-807	189	93	c	c	NOUN
cana-807	189	94	)	)	PUNCT
cana-807	189	95			PROPN
cana-807	189	96	sc	sc	PROPN
cana-807	189	97	cc	cc	PROPN
cana-807	189	98			PROPN
cana-807	189	99	n	n	CCONJ
cana-807	189	100	-	-	PUNCT
cana-807	189	101	pcl(cc	pcl(cc	ADJ
cana-807	189	102	)	)	PUNCT
cana-807	189	103			PROPN
cana-807	189	104	sc	sc	PROPN
cana-807	189	105	.	.	PUNCT
cana-807	189	106	:	:	PUNCT
cana-807	190	1	theorem	theorem	VERB
cana-807	190	2	4.13	4.13	NUM
cana-807	190	3	:	:	PUNCT
cana-807	190	4	let	let	VERB
cana-807	190	5	s	s	PRON
cana-807	190	6	be	be	AUX
cana-807	190	7	a	a	DET
cana-807	190	8	n	n	ADV
cana-807	190	9	-	-	PUNCT
cana-807	190	10	pgpr*wos	pgpr*wos	NOUN
cana-807	190	11	in	in	ADP
cana-807	190	12	x	x	PUNCT
cana-807	190	13	and	and	CCONJ
cana-807	190	14	s	s	PROPN
cana-807	190	15	⊇	⊇	PROPN
cana-807	190	16	c	c	PROPN
cana-807	190	17	⊇	⊇	PROPN
cana-807	190	18	n	n	CCONJ
cana-807	190	19	-	-	PUNCT
cana-807	190	20	pint(s	pint(s	NOUN
cana-807	190	21	)	)	PUNCT
cana-807	190	22	,	,	PUNCT
cana-807	190	23	then	then	ADV
cana-807	190	24	c	c	PROPN
cana-807	190	25	is	be	AUX
cana-807	190	26	n	n	ADV
cana-807	190	27	-	-	PUNCT
cana-807	190	28	pgpr*wos	pgpr*wos	NOUN
cana-807	190	29	in	in	ADP
cana-807	190	30	x	x	NOUN
cana-807	190	31	proof	proof	NOUN
cana-807	190	32	:	:	PUNCT
cana-807	190	33	let	let	VERB
cana-807	190	34	s	s	PRON
cana-807	190	35	be	be	AUX
cana-807	190	36	aa	aa	NOUN
cana-807	190	37	n	n	ADV
cana-807	190	38	-	-	PUNCT
cana-807	190	39	pgpr*wos	pgpr*wos	NOUN
cana-807	190	40	in	in	ADP
cana-807	190	41	x	x	PUNCT
cana-807	190	42	and	and	CCONJ
cana-807	190	43	c	c	PROPN
cana-807	190	44	be	be	AUX
cana-807	190	45	a	a	DET
cana-807	190	46	neutrosophic	neutrosophic	ADJ
cana-807	190	47	set	set	NOUN
cana-807	190	48	in	in	ADP
cana-807	190	49	x.	x.	NOUN
cana-807	190	50	let	let	VERB
cana-807	190	51	s⊇	s⊇	PROPN
cana-807	190	52	c	c	PROPN
cana-807	190	53	⊇	⊇	PROPN
cana-807	190	54	n	n	CCONJ
cana-807	190	55	-	-	PUNCT
cana-807	190	56	pint(s	pint(s	NOUN
cana-807	190	57	)	)	PUNCT
cana-807	190	58	.	.	PUNCT
cana-807	191	1	then	then	ADV
cana-807	191	2	sc	sc	PROPN
cana-807	191	3	is	be	AUX
cana-807	191	4	a	a	DET
cana-807	191	5	n	n	NOUN
cana-807	191	6	-	-	PUNCT
cana-807	191	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	191	8	in	in	ADP
cana-807	191	9	x	x	PUNCT
cana-807	191	10	and	and	CCONJ
cana-807	191	11	sc	sc	PROPN
cana-807	191	12	⊆	⊆	NUM
cana-807	191	13	cc	cc	ADP
cana-807	191	14	⊆	⊆	NUM
cana-807	191	15	n	n	CCONJ
cana-807	191	16	-	-	PUNCT
cana-807	191	17	pcl(sc	pcl(sc	NUM
cana-807	191	18	)	)	PUNCT
cana-807	191	19	.	.	PUNCT
cana-807	192	1	then	then	ADV
cana-807	192	2	cc	cc	PROPN
cana-807	192	3	is	be	AUX
cana-807	192	4	a	a	DET
cana-807	192	5	n	n	CCONJ
cana-807	192	6	-	-	PUNCT
cana-807	192	7	pgpr*wcs	pgpr*wcs	NOUN
cana-807	192	8	set	set	VERB
cana-807	192	9	in	in	ADP
cana-807	192	10	x	x	X
cana-807	192	11	.	.	PUNCT
cana-807	193	1	hence	hence	ADV
cana-807	193	2	c	c	PROPN
cana-807	193	3	is	be	AUX
cana-807	193	4	a	a	DET
cana-807	193	5	n	n	ADV
cana-807	193	6	-	-	PUNCT
cana-807	193	7	pgpr*wos	pgpr*wos	NOUN
cana-807	193	8	in	in	ADP
cana-807	193	9	x.	x.	NOUN
cana-807	193	10	references	reference	NOUN
cana-807	193	11	:	:	PUNCT
cana-807	194	1	[	[	X
cana-807	194	2	1	1	NUM
cana-807	194	3	]	]	X
cana-807	194	4	atanassom	atanassom	PROPN
cana-807	194	5	k.c	k.c	PROPN
cana-807	194	6	.	.	PROPN
cana-807	194	7	,	,	PUNCT
cana-807	194	8	intuitionistic	intuitionistic	ADJ
cana-807	194	9	fuzzy	fuzzy	ADJ
cana-807	194	10	sets	set	NOUN
cana-807	194	11	,	,	PUNCT
cana-807	194	12	fuzzy	fuzzy	ADJ
cana-807	194	13	sets	set	NOUN
cana-807	194	14	and	and	CCONJ
cana-807	194	15	systems	system	NOUN
cana-807	194	16	,	,	PUNCT
cana-807	194	17	1986	1986	NUM
cana-807	194	18	,	,	PUNCT
cana-807	194	19	20	20	NUM
cana-807	194	20	,	,	PUNCT
cana-807	194	21	87	87	NUM
cana-807	194	22	-	-	SYM
cana-807	194	23	96	96	NUM
cana-807	194	24	.	.	PUNCT
cana-807	195	1	[	[	X
cana-807	195	2	2	2	NUM
cana-807	195	3	]	]	PUNCT
cana-807	195	4	floretin	floretin	ADJ
cana-807	195	5	smarandache	smarandache	NOUN
cana-807	195	6	,	,	PUNCT
cana-807	195	7	neutrosophic	neutrosophic	ADJ
cana-807	195	8	set	set	NOUN
cana-807	195	9	:	:	PUNCT
cana-807	195	10	a	a	DET
cana-807	195	11	generalization	generalization	NOUN
cana-807	195	12	of	of	ADP
cana-807	195	13	intuitional	intuitional	ADJ
cana-807	195	14	fuzzy	fuzzy	ADJ
cana-807	195	15	set	set	NOUN
cana-807	195	16	,	,	PUNCT
cana-807	195	17	journal	journal	NOUN
cana-807	195	18	of	of	ADP
cana-807	195	19	defense	defense	NOUN
cana-807	195	20	,	,	PUNCT
cana-807	195	21	2010	2010	NUM
cana-807	195	22	,	,	PUNCT
cana-807	195	23	1	1	NUM
cana-807	195	24	,	,	PUNCT
cana-807	195	25	107	107	NUM
cana-807	195	26	-	-	SYM
cana-807	195	27	116	116	NUM
cana-807	195	28	.	.	PUNCT
cana-807	196	1	[	[	X
cana-807	196	2	3	3	NUM
cana-807	196	3	]	]	X
cana-807	196	4	harshitha	harshitha	PROPN
cana-807	196	5	,	,	PUNCT
cana-807	196	6	a.	a.	NOUN
cana-807	196	7	and	and	CCONJ
cana-807	196	8	jayanthi	jayanthi	PROPN
cana-807	196	9	,	,	PUNCT
cana-807	196	10	d.	d.	PROPN
cana-807	196	11	,	,	PUNCT
cana-807	196	12	regular	regular	ADP
cana-807	196	13	a	a	DET
cana-807	196	14	generalized	generalize	VERB
cana-807	196	15	closed	closed	ADJ
cana-807	196	16	sets	set	NOUN
cana-807	196	17	in	in	ADP
cana-807	196	18	neutrosophic	neutrosophic	ADJ
cana-807	196	19	topological	topological	ADJ
cana-807	196	20	spaces	space	NOUN
cana-807	196	21	,	,	PUNCT
cana-807	196	22	iosr	iosr	ADJ
cana-807	196	23	journal	journal	NOUN
cana-807	196	24	of	of	ADP
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cana-807	196	26	,	,	PUNCT
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cana-807	196	28	,	,	PUNCT
cana-807	196	29	15(02	15(02	NUM
cana-807	196	30	)	)	PUNCT
cana-807	196	31	,	,	PUNCT
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cana-807	196	33	-	-	SYM
cana-807	196	34	18	18	NUM
cana-807	196	35	.	.	PUNCT
cana-807	197	1	[	[	X
cana-807	197	2	4	4	NUM
cana-807	197	3	]	]	X
cana-807	197	4	jayanthi	jayanthi	PROPN
cana-807	197	5	,	,	PUNCT
cana-807	197	6	d.	d.	PROPN
cana-807	197	7	,	,	PUNCT
cana-807	197	8	on	on	ADP
cana-807	197	9	generalized	generalized	ADJ
cana-807	197	10	closed	closed	ADJ
cana-807	197	11	sets	set	NOUN
cana-807	197	12	in	in	ADP
cana-807	197	13	neutrosophic	neutrosophic	ADJ
cana-807	197	14	spaces	space	NOUN
cana-807	197	15	,	,	PUNCT
cana-807	197	16	international	international	ADJ
cana-807	197	17	conference	conference	NOUN
cana-807	197	18	on	on	ADP
cana-807	197	19	recent	recent	ADJ
cana-807	197	20	crends	crend	NOUN
cana-807	197	21	in	in	ADP
cana-807	197	22	mathematics	mathematic	NOUN
cana-807	197	23	and	and	CCONJ
cana-807	197	24	information	information	NOUN
cana-807	197	25	cechnology	cechnology	NOUN
cana-807	197	26	,	,	PUNCT
cana-807	197	27	2018	2018	NUM
cana-807	197	28	,	,	PUNCT
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cana-807	197	30	,	,	PUNCT
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cana-807	197	32	-	-	SYM
cana-807	197	33	91	91	NUM
cana-807	197	34	.	.	PUNCT
cana-807	198	1	[	[	X
cana-807	198	2	5	5	NUM
cana-807	198	3	]	]	X
cana-807	198	4	pushpalatha	pushpalatha	PROPN
cana-807	198	5	,	,	PUNCT
cana-807	198	6	a.	a.	NOUN
cana-807	198	7	and	and	CCONJ
cana-807	198	8	nandhini	nandhini	PROPN
cana-807	198	9	,	,	PUNCT
cana-807	198	10	c.	c.	NOUN
cana-807	198	11	,	,	PUNCT
cana-807	198	12	generalized	generalize	VERB
cana-807	198	13	closed	close	VERB
cana-807	198	14	sets	set	NOUN
cana-807	198	15	mia	mia	PROPN
cana-807	198	16	neutrosophic	neutrosophic	PROPN
cana-807	198	17	topological	topological	ADJ
cana-807	198	18	spaces	space	NOUN
cana-807	198	19	,	,	PUNCT
cana-807	198	20	malaya	malaya	PROPN
cana-807	198	21	journal	journal	PROPN
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cana-807	198	26	,	,	PUNCT
cana-807	198	27	7(1	7(1	NUM
cana-807	198	28	)	)	PUNCT
cana-807	198	29	,	,	PUNCT
cana-807	198	30	50	50	NUM
cana-807	198	31	-	-	SYM
cana-807	198	32	54	54	NUM
cana-807	198	33	.	.	PUNCT
cana-807	199	1	[	[	X
cana-807	199	2	6	6	NUM
cana-807	199	3	]	]	PUNCT
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cana-807	199	5	,	,	PUNCT
cana-807	199	6	a.a	a.a	PROPN
cana-807	199	7	.	.	PROPN
cana-807	199	8	and	and	CCONJ
cana-807	199	9	alslowi	alslowi	PROPN
cana-807	199	10	,	,	PUNCT
cana-807	199	11	s.a	s.a	PROPN
cana-807	199	12	.	.	PROPN
cana-807	199	13	,	,	PUNCT
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cana-807	199	15	set	set	NOUN
cana-807	199	16	and	and	CCONJ
cana-807	199	17	nuetrosophic	nuetrosophic	ADJ
cana-807	199	18	topological	topological	ADJ
cana-807	199	19	space	space	NOUN
cana-807	199	20	,	,	PUNCT
cana-807	199	21	iosr	iosr	ADJ
cana-807	199	22	journal	journal	NOUN
cana-807	199	23	of	of	ADP
cana-807	199	24	communications	communication	NOUN
cana-807	199	25	on	on	ADP
cana-807	199	26	applied	apply	VERB
cana-807	199	27	nonlinear	nonlinear	ADJ
cana-807	199	28	analysis	analysis	NOUN
cana-807	199	29	issn	issn	NOUN
cana-807	199	30	:	:	PUNCT
cana-807	199	31	1074	1074	NUM
cana-807	199	32	-	-	PUNCT
cana-807	199	33	133x	133x	NUM
cana-807	199	34	vol	vol	NOUN
cana-807	199	35	31	31	NUM
cana-807	199	36	no	no	NOUN
cana-807	199	37	.	.	PUNCT
cana-807	200	1	3s	3s	NUM
cana-807	200	2	(	(	PUNCT
cana-807	200	3	2024	2024	NUM
cana-807	200	4	)	)	PUNCT
cana-807	200	5	558	558	NUM
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cana-807	200	7	mathematics	mathematic	NOUN
cana-807	200	8	,	,	PUNCT
cana-807	200	9	2012	2012	NUM
cana-807	200	10	,	,	PUNCT
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cana-807	200	12	-	-	SYM
cana-807	200	13	35	35	NUM
cana-807	200	14	.	.	PUNCT
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cana-807	201	3	]	]	X
cana-807	201	4	santhi	santhi	ADJ
cana-807	201	5	,	,	PUNCT
cana-807	201	6	r.	r.	PROPN
cana-807	201	7	and	and	CCONJ
cana-807	201	8	udhayarani	udhayarani	PROPN
cana-807	201	9	,	,	PUNCT
cana-807	201	10	n.	n.	NOUN
cana-807	201	11	,	,	PUNCT
cana-807	201	12	n	n	CCONJ
cana-807	201	13	-	-	PUNCT
cana-807	201	14	closed	close	VERB
cana-807	201	15	set	set	NOUN
cana-807	201	16	in	in	ADP
cana-807	201	17	neutrosophic	neutrosophic	ADJ
cana-807	201	18	copological	copological	ADJ
cana-807	201	19	spaces	space	NOUN
cana-807	201	20	,	,	PUNCT
cana-807	201	21	neutrosophic	neutrosophic	ADJ
cana-807	201	22	sets	set	NOUN
cana-807	201	23	and	and	CCONJ
cana-807	201	24	systems	system	NOUN
cana-807	201	25	,	,	PUNCT
cana-807	201	26	2016	2016	NUM
cana-807	201	27	,	,	PUNCT
cana-807	201	28	12	12	NUM
cana-807	201	29	,	,	PUNCT
cana-807	201	30	114	114	NUM
cana-807	201	31	-	-	SYM
cana-807	201	32	117	117	NUM
cana-807	201	33	.	.	PUNCT
cana-807	202	1	[	[	X
cana-807	202	2	8	8	NUM
cana-807	202	3	]	]	PUNCT
cana-807	202	4	wadei	wadei	VERB
cana-807	202	5	al	al	PROPN
cana-807	202	6	-	-	PUNCT
cana-807	202	7	omeri	omeri	PROPN
cana-807	202	8	and	and	CCONJ
cana-807	202	9	saeid	saeid	PROPN
cana-807	202	10	jafari	jafari	PROPN
cana-807	202	11	,	,	PUNCT
cana-807	202	12	on	on	ADP
cana-807	202	13	generalized	generalized	ADJ
cana-807	202	14	closed	closed	ADJ
cana-807	202	15	sets	set	NOUN
cana-807	202	16	and	and	CCONJ
cana-807	202	17	generalized	generalized	ADJ
cana-807	202	18	pre	pre	VERB
cana-807	202	19	-	-	VERB
cana-807	202	20	closed	closed	ADJ
cana-807	202	21	in	in	ADP
cana-807	202	22	neutrosophic	neutrosophic	ADJ
cana-807	202	23	copological	copological	ADJ
cana-807	202	24	spaces	space	NOUN
cana-807	202	25	,	,	PUNCT
cana-807	202	26	mathematics	mathematics	PROPN
cana-807	202	27	mdpi	mdpi	PROPN
cana-807	202	28	,	,	PUNCT
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cana-807	202	30	,	,	PUNCT
cana-807	202	31	7(1	7(1	NUM
cana-807	202	32	)	)	PUNCT
cana-807	202	33	,	,	PUNCT
cana-807	202	34	01	01	NUM
cana-807	202	35	-	-	SYM
cana-807	202	36	12	12	NUM
cana-807	202	37	.	.	PUNCT
cana-807	203	1	[	[	X
cana-807	203	2	9	9	NUM
cana-807	203	3	]	]	SYM
cana-807	203	4	wali	wali	PROPN
cana-807	203	5	,	,	PUNCT
cana-807	203	6	r.s	r.s	PROPN
cana-807	203	7	.	.	PROPN
cana-807	203	8	and	and	CCONJ
cana-807	203	9	mimekananda	mimekananda	PROPN
cana-807	203	10	demsre	demsre	NOUN
cana-807	203	11	,	,	PUNCT
cana-807	203	12	on	on	ADP
cana-807	203	13	pre	pre	ADJ
cana-807	203	14	generalized	generalized	ADJ
cana-807	203	15	pre	pre	NOUN
cana-807	203	16	regular	regular	ADJ
cana-807	203	17	weakly	weakly	ADJ
cana-807	203	18	closed	closed	ADJ
cana-807	203	19	sets	set	NOUN
cana-807	203	20	in	in	ADP
cana-807	203	21	copological	copological	ADJ
cana-807	203	22	spaces	space	NOUN
cana-807	203	23	,	,	PUNCT
cana-807	203	24	journal	journal	NOUN
cana-807	203	25	of	of	ADP
cana-807	203	26	computer	computer	NOUN
cana-807	203	27	and	and	CCONJ
cana-807	203	28	mathematical	mathematical	ADJ
cana-807	203	29	sciences	science	NOUN
cana-807	203	30	,	,	PUNCT
cana-807	203	31	2015	2015	NUM
cana-807	203	32	,	,	PUNCT
cana-807	203	33	6(2	6(2	NUM
cana-807	203	34	)	)	PUNCT
cana-807	203	35	,	,	PUNCT
cana-807	203	36	113	113	NUM
cana-807	203	37	-	-	SYM
cana-807	203	38	125	125	NUM
cana-807	203	39	zadeh	zadeh	PROPN
cana-807	203	40	,	,	PUNCT
cana-807	203	41	l.a	l.a	PROPN
cana-807	203	42	.	.	PROPN
cana-807	203	43	,	,	PUNCT
cana-807	203	44	fuzzy	fuzzy	ADJ
cana-807	203	45	sets	set	NOUN
cana-807	203	46	,	,	PUNCT
cana-807	203	47	information	information	NOUN
cana-807	203	48	and	and	CCONJ