id	sid	tid	token	lemma	pos
cana-3840	1	1	communications	communication	NOUN
cana-3840	1	2	on	on	ADP
cana-3840	1	3	applied	apply	VERB
cana-3840	1	4	nonlinear	nonlinear	ADJ
cana-3840	1	5	analysis	analysis	NOUN
cana-3840	1	6	issn	issn	NOUN
cana-3840	1	7	:	:	PUNCT
cana-3840	1	8	1074	1074	NUM
cana-3840	1	9	-	-	PUNCT
cana-3840	1	10	133x	133x	NUM
cana-3840	1	11	vol	vol	NOUN
cana-3840	1	12	32	32	NUM
cana-3840	1	13	no	no	NOUN
cana-3840	1	14	.	.	PUNCT
cana-3840	2	1	9s	9s	NUM
cana-3840	2	2	(	(	PUNCT
cana-3840	2	3	2025	2025	NUM
cana-3840	2	4	)	)	PUNCT
cana-3840	2	5	88	88	NUM
cana-3840	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	3	2	topological	topological	ADJ
cana-3840	3	3	study	study	NOUN
cana-3840	3	4	on	on	ADP
cana-3840	3	5	revised	revise	VERB
cana-3840	3	6	fuzzy	fuzzy	ADJ
cana-3840	3	7	metric	metric	ADJ
cana-3840	3	8	spaces	space	NOUN
cana-3840	3	9	and	and	CCONJ
cana-3840	3	10	their	their	PRON
cana-3840	3	11	generalization	generalization	NOUN
cana-3840	3	12	1thangathamizh	1thangathamizh	NUM
cana-3840	3	13	r	r	NOUN
cana-3840	3	14	,	,	PUNCT
cana-3840	3	15	2abdul	2abdul	NUM
cana-3840	3	16	razak	razak	NOUN
cana-3840	3	17	,	,	PUNCT
cana-3840	3	18	3rajalakshmi	3rajalakshmi	PROPN
cana-3840	3	19	s	s	X
cana-3840	3	20	,	,	PUNCT
cana-3840	3	21	4gnanabala	4gnanabala	PROPN
cana-3840	3	22	k	k	NOUN
cana-3840	3	23	,	,	PUNCT
cana-3840	3	24	5thangammal	5thangammal	PROPN
cana-3840	3	25	r	r	NOUN
cana-3840	3	26	,	,	PUNCT
cana-3840	3	27	6shanmugavel	6shanmugavel	NUM
cana-3840	3	28	p	p	PRON
cana-3840	3	29	jeppiaar	jeppiaar	PROPN
cana-3840	3	30	institute	institute	PROPN
cana-3840	3	31	of	of	ADP
cana-3840	3	32	technology	technology	PROPN
cana-3840	3	33	,	,	PUNCT
cana-3840	3	34	sriperumpudhur	sriperumpudhur	NOUN
cana-3840	3	35	,	,	PUNCT
cana-3840	3	36	kanchipuram	kanchipuram	PROPN
cana-3840	3	37	,	,	PUNCT
cana-3840	3	38	email	email	NOUN
cana-3840	3	39	:	:	PUNCT
cana-3840	3	40	thamizh1418@gmail.com	thamizh1418@gmail.com	X
cana-3840	3	41	.	.	PROPN
cana-3840	3	42	,	,	PUNCT
cana-3840	3	43	k.ramakrishnan	k.ramakrishnan	PROPN
cana-3840	3	44	college	college	NOUN
cana-3840	3	45	of	of	ADP
cana-3840	3	46	engineering	engineering	NOUN
cana-3840	3	47	,	,	PUNCT
cana-3840	3	48	samayapuram	samayapuram	PROPN
cana-3840	3	49	,	,	PUNCT
cana-3840	3	50	trichy	trichy	NOUN
cana-3840	3	51	,	,	PUNCT
cana-3840	3	52	email	email	NOUN
cana-3840	3	53	:	:	PUNCT
cana-3840	3	54	arrazak76@gmail.com	arrazak76@gmail.com	X
cana-3840	3	55	.	.	PROPN
cana-3840	3	56	,	,	PUNCT
cana-3840	3	57	mahalashmi	mahalashmi	PROPN
cana-3840	3	58	women	woman	NOUN
cana-3840	3	59	’s	’s	PART
cana-3840	3	60	college	college	PROPN
cana-3840	3	61	of	of	ADP
cana-3840	3	62	arts	art	NOUN
cana-3840	3	63	and	and	CCONJ
cana-3840	3	64	science	science	NOUN
cana-3840	3	65	,	,	PUNCT
cana-3840	3	66	paruthipattu	paruthipattu	NOUN
cana-3840	3	67	,	,	PUNCT
cana-3840	3	68	avadi	avadi	NOUN
cana-3840	3	69	,	,	PUNCT
cana-3840	3	70	chennai	chennai	NOUN
cana-3840	3	71	,	,	PUNCT
cana-3840	3	72	email	email	NOUN
cana-3840	3	73	:	:	PUNCT
cana-3840	3	74	gbalamaths@gmail.com	gbalamaths@gmail.com	X
cana-3840	3	75	.	.	PROPN
cana-3840	3	76	,	,	PUNCT
cana-3840	3	77	chennai	chennai	PROPN
cana-3840	3	78	institute	institute	PROPN
cana-3840	3	79	of	of	ADP
cana-3840	3	80	technology	technology	PROPN
cana-3840	3	81	,	,	PUNCT
cana-3840	3	82	chennai	chennai	PROPN
cana-3840	3	83	,	,	PUNCT
cana-3840	3	84	email	email	NOUN
cana-3840	3	85	:	:	PUNCT
cana-3840	3	86	rajalakshmi301991@gmail.com	rajalakshmi301991@gmail.com	PROPN
cana-3840	3	87	.	.	PROPN
cana-3840	3	88	,	,	PUNCT
cana-3840	3	89	selvam	selvam	PROPN
cana-3840	3	90	college	college	PROPN
cana-3840	3	91	of	of	ADP
cana-3840	3	92	technology	technology	NOUN
cana-3840	3	93	(	(	PUNCT
cana-3840	3	94	autonomous	autonomous	ADJ
cana-3840	3	95	)	)	PUNCT
cana-3840	3	96	,	,	PUNCT
cana-3840	3	97	namakkal	namakkal	NOUN
cana-3840	3	98	,	,	PUNCT
cana-3840	3	99	email	email	NOUN
cana-3840	3	100	:	:	PUNCT
cana-3840	4	1	rthangam1981@gmail.com	rthangam1981@gmail.com	X
cana-3840	4	2	.	.	PUNCT
cana-3840	5	1	selvamm	selvamm	PROPN
cana-3840	5	2	arts	arts	PROPN
cana-3840	5	3	and	and	CCONJ
cana-3840	5	4	science	science	PROPN
cana-3840	5	5	college	college	PROPN
cana-3840	5	6	(	(	PUNCT
cana-3840	5	7	autonomous	autonomous	ADJ
cana-3840	5	8	)	)	PUNCT
cana-3840	5	9	,	,	PUNCT
cana-3840	5	10	namakkal	namakkal	NOUN
cana-3840	5	11	,	,	PUNCT
cana-3840	5	12	email	email	NOUN
cana-3840	5	13	:	:	PUNCT
cana-3840	5	14	p.sham1988@gmail.com	p.sham1988@gmail.com	X
cana-3840	5	15	.	.	PUNCT
cana-3840	6	1	article	article	NOUN
cana-3840	6	2	history	history	NOUN
cana-3840	6	3	:	:	PUNCT
cana-3840	6	4	received	receive	VERB
cana-3840	6	5	:	:	PUNCT
cana-3840	6	6	12	12	NUM
cana-3840	6	7	-	-	SYM
cana-3840	6	8	11	11	NUM
cana-3840	6	9	-	-	PUNCT
cana-3840	6	10	2024	2024	NUM
cana-3840	6	11	revised:24	revised:24	X
cana-3840	6	12	-	-	PUNCT
cana-3840	6	13	12	12	NUM
cana-3840	6	14	-	-	PUNCT
cana-3840	6	15	2024	2024	NUM
cana-3840	6	16	accepted:09	accepted:09	NOUN
cana-3840	6	17	-	-	PUNCT
cana-3840	6	18	01	01	NUM
cana-3840	6	19	-	-	PUNCT
cana-3840	6	20	2025	2025	NUM
cana-3840	6	21	abstract	abstract	NOUN
cana-3840	6	22	:	:	PUNCT
cana-3840	6	23	introduction	introduction	NOUN
cana-3840	6	24	in	in	ADP
cana-3840	6	25	this	this	DET
cana-3840	6	26	paper	paper	NOUN
cana-3840	6	27	,	,	PUNCT
cana-3840	6	28	we	we	PRON
cana-3840	6	29	explore	explore	VERB
cana-3840	6	30	the	the	DET
cana-3840	6	31	concept	concept	NOUN
cana-3840	6	32	of	of	ADP
cana-3840	6	33	metric	metric	ADJ
cana-3840	6	34	functions	function	NOUN
cana-3840	6	35	within	within	ADP
cana-3840	6	36	a	a	DET
cana-3840	6	37	revised	revise	VERB
cana-3840	6	38	fuzzy	fuzzy	ADJ
cana-3840	6	39	metric	metric	ADJ
cana-3840	6	40	space	space	NOUN
cana-3840	6	41	.	.	PUNCT
cana-3840	7	1	the	the	DET
cana-3840	7	2	study	study	NOUN
cana-3840	7	3	focuses	focus	VERB
cana-3840	7	4	on	on	ADP
cana-3840	7	5	understanding	understand	VERB
cana-3840	7	6	the	the	DET
cana-3840	7	7	relationships	relationship	NOUN
cana-3840	7	8	between	between	ADP
cana-3840	7	9	these	these	DET
cana-3840	7	10	metric	metric	ADJ
cana-3840	7	11	functions	function	NOUN
cana-3840	7	12	and	and	CCONJ
cana-3840	7	13	the	the	DET
cana-3840	7	14	topological	topological	ADJ
cana-3840	7	15	structures	structure	NOUN
cana-3840	7	16	they	they	PRON
cana-3840	7	17	generate	generate	VERB
cana-3840	7	18	.	.	PUNCT
cana-3840	8	1	specifically	specifically	ADV
cana-3840	8	2	,	,	PUNCT
cana-3840	8	3	we	we	PRON
cana-3840	8	4	introduce	introduce	VERB
cana-3840	8	5	the	the	DET
cana-3840	8	6	concept	concept	NOUN
cana-3840	8	7	of	of	ADP
cana-3840	8	8	a	a	DET
cana-3840	8	9	stratified	stratified	ADJ
cana-3840	8	10	function	function	NOUN
cana-3840	8	11	within	within	ADP
cana-3840	8	12	this	this	DET
cana-3840	8	13	framework	framework	NOUN
cana-3840	8	14	and	and	CCONJ
cana-3840	8	15	investigate	investigate	VERB
cana-3840	8	16	its	its	PRON
cana-3840	8	17	implications	implication	NOUN
cana-3840	8	18	.	.	PUNCT
cana-3840	9	1	objectives	objective	VERB
cana-3840	9	2	the	the	DET
cana-3840	9	3	main	main	ADJ
cana-3840	9	4	objectives	objective	NOUN
cana-3840	9	5	of	of	ADP
cana-3840	9	6	this	this	DET
cana-3840	9	7	paper	paper	NOUN
cana-3840	9	8	are	be	AUX
cana-3840	9	9	:	:	PUNCT
cana-3840	9	10	1	1	X
cana-3840	9	11	.	.	X
cana-3840	9	12	to	to	PART
cana-3840	9	13	define	define	VERB
cana-3840	9	14	and	and	CCONJ
cana-3840	9	15	analyze	analyze	VERB
cana-3840	9	16	stratified	stratified	ADJ
cana-3840	9	17	functions	function	NOUN
cana-3840	9	18	in	in	ADP
cana-3840	9	19	a	a	DET
cana-3840	9	20	revised	revise	VERB
cana-3840	9	21	fuzzy	fuzzy	ADJ
cana-3840	9	22	metric	metric	ADJ
cana-3840	9	23	space	space	NOUN
cana-3840	9	24	.	.	PUNCT
cana-3840	10	1	2	2	X
cana-3840	10	2	.	.	X
cana-3840	10	3	to	to	PART
cana-3840	10	4	demonstrate	demonstrate	VERB
cana-3840	10	5	that	that	SCONJ
cana-3840	10	6	the	the	DET
cana-3840	10	7	topology	topology	NOUN
cana-3840	10	8	generated	generate	VERB
cana-3840	10	9	by	by	ADP
cana-3840	10	10	the	the	DET
cana-3840	10	11	family	family	NOUN
cana-3840	10	12	of	of	ADP
cana-3840	10	13	stratified	stratified	ADJ
cana-3840	10	14	functions	function	NOUN
cana-3840	10	15	coincides	coincide	VERB
cana-3840	10	16	with	with	ADP
cana-3840	10	17	the	the	DET
cana-3840	10	18	topology	topology	NOUN
cana-3840	10	19	generated	generate	VERB
cana-3840	10	20	by	by	ADP
cana-3840	10	21	the	the	DET
cana-3840	10	22	revised	revise	VERB
cana-3840	10	23	fuzzy	fuzzy	ADJ
cana-3840	10	24	metric	metric	NOUN
cana-3840	10	25	.	.	PUNCT
cana-3840	11	1	3	3	X
cana-3840	11	2	.	.	X
cana-3840	11	3	to	to	PART
cana-3840	11	4	derive	derive	VERB
cana-3840	11	5	the	the	DET
cana-3840	11	6	concrete	concrete	ADJ
cana-3840	11	7	form	form	NOUN
cana-3840	11	8	of	of	ADP
cana-3840	11	9	the	the	DET
cana-3840	11	10	metric	metric	ADJ
cana-3840	11	11	function	function	NOUN
cana-3840	11	12	under	under	ADP
cana-3840	11	13	specific	specific	ADJ
cana-3840	11	14	conditions	condition	NOUN
cana-3840	11	15	.	.	PUNCT
cana-3840	12	1	methods	method	NOUN
cana-3840	12	2	we	we	PRON
cana-3840	12	3	approach	approach	VERB
cana-3840	12	4	these	these	DET
cana-3840	12	5	objectives	objective	NOUN
cana-3840	12	6	by	by	ADP
cana-3840	12	7	first	first	ADV
cana-3840	12	8	introducing	introduce	VERB
cana-3840	12	9	the	the	DET
cana-3840	12	10	notion	notion	NOUN
cana-3840	12	11	of	of	ADP
cana-3840	12	12	a	a	DET
cana-3840	12	13	stratified	stratified	ADJ
cana-3840	12	14	function	function	NOUN
cana-3840	12	15	in	in	ADP
cana-3840	12	16	a	a	DET
cana-3840	12	17	revised	revise	VERB
cana-3840	12	18	fuzzy	fuzzy	ADJ
cana-3840	12	19	metric	metric	ADJ
cana-3840	12	20	space	space	NOUN
cana-3840	12	21	.	.	PUNCT
cana-3840	13	1	using	use	VERB
cana-3840	13	2	this	this	DET
cana-3840	13	3	concept	concept	NOUN
cana-3840	13	4	,	,	PUNCT
cana-3840	13	5	we	we	PRON
cana-3840	13	6	prove	prove	VERB
cana-3840	13	7	that	that	SCONJ
cana-3840	13	8	the	the	DET
cana-3840	13	9	topology	topology	NOUN
cana-3840	13	10	generated	generate	VERB
cana-3840	13	11	by	by	ADP
cana-3840	13	12	the	the	DET
cana-3840	13	13	family	family	NOUN
cana-3840	13	14	of	of	ADP
cana-3840	13	15	stratified	stratified	ADJ
cana-3840	13	16	functions	function	NOUN
cana-3840	13	17	is	be	AUX
cana-3840	13	18	identical	identical	ADJ
cana-3840	13	19	to	to	ADP
cana-3840	13	20	the	the	DET
cana-3840	13	21	topology	topology	NOUN
cana-3840	13	22	generated	generate	VERB
cana-3840	13	23	by	by	ADP
cana-3840	13	24	the	the	DET
cana-3840	13	25	revised	revise	VERB
cana-3840	13	26	fuzzy	fuzzy	ADJ
cana-3840	13	27	metric	metric	NOUN
cana-3840	13	28	.	.	PUNCT
cana-3840	14	1	additionally	additionally	ADV
cana-3840	14	2	,	,	PUNCT
cana-3840	14	3	we	we	PRON
cana-3840	14	4	explore	explore	VERB
cana-3840	14	5	the	the	DET
cana-3840	14	6	conditions	condition	NOUN
cana-3840	14	7	under	under	ADP
cana-3840	14	8	which	which	PRON
cana-3840	14	9	a	a	DET
cana-3840	14	10	specific	specific	ADJ
cana-3840	14	11	form	form	NOUN
cana-3840	14	12	of	of	ADP
cana-3840	14	13	the	the	DET
cana-3840	14	14	metric	metric	ADJ
cana-3840	14	15	function	function	NOUN
cana-3840	14	16	can	can	AUX
cana-3840	14	17	be	be	AUX
cana-3840	14	18	determined	determine	VERB
cana-3840	14	19	.	.	PUNCT
cana-3840	15	1	results	result	VERB
cana-3840	15	2	our	our	PRON
cana-3840	15	3	findings	finding	NOUN
cana-3840	15	4	show	show	VERB
cana-3840	15	5	that	that	SCONJ
cana-3840	15	6	the	the	DET
cana-3840	15	7	topology	topology	NOUN
cana-3840	15	8	generated	generate	VERB
cana-3840	15	9	by	by	ADP
cana-3840	15	10	the	the	DET
cana-3840	15	11	stratified	stratified	ADJ
cana-3840	15	12	functions	function	NOUN
cana-3840	15	13	indeed	indeed	ADV
cana-3840	15	14	coincides	coincide	VERB
cana-3840	15	15	with	with	ADP
cana-3840	15	16	the	the	DET
cana-3840	15	17	topology	topology	NOUN
cana-3840	15	18	generated	generate	VERB
cana-3840	15	19	by	by	ADP
cana-3840	15	20	the	the	DET
cana-3840	15	21	revised	revise	VERB
cana-3840	15	22	fuzzy	fuzzy	ADJ
cana-3840	15	23	metric	metric	NOUN
cana-3840	15	24	.	.	PUNCT
cana-3840	16	1	moreover	moreover	ADV
cana-3840	16	2	,	,	PUNCT
cana-3840	16	3	under	under	ADP
cana-3840	16	4	certain	certain	ADJ
cana-3840	16	5	special	special	ADJ
cana-3840	16	6	conditions	condition	NOUN
cana-3840	16	7	,	,	PUNCT
cana-3840	16	8	we	we	PRON
cana-3840	16	9	can	can	AUX
cana-3840	16	10	obtain	obtain	VERB
cana-3840	16	11	a	a	DET
cana-3840	16	12	concrete	concrete	ADJ
cana-3840	16	13	representation	representation	NOUN
cana-3840	16	14	of	of	ADP
cana-3840	16	15	the	the	DET
cana-3840	16	16	metric	metric	ADJ
cana-3840	16	17	function	function	NOUN
cana-3840	16	18	.	.	PUNCT
cana-3840	17	1	conclusion	conclusion	NOUN
cana-3840	17	2	this	this	DET
cana-3840	17	3	paper	paper	NOUN
cana-3840	17	4	provides	provide	VERB
cana-3840	17	5	a	a	DET
cana-3840	17	6	deeper	deep	ADJ
cana-3840	17	7	understanding	understanding	NOUN
cana-3840	17	8	of	of	ADP
cana-3840	17	9	the	the	DET
cana-3840	17	10	structure	structure	NOUN
cana-3840	17	11	of	of	ADP
cana-3840	17	12	revised	revise	VERB
cana-3840	17	13	fuzzy	fuzzy	ADJ
cana-3840	17	14	metric	metric	ADJ
cana-3840	17	15	spaces	space	NOUN
cana-3840	17	16	.	.	PUNCT
cana-3840	18	1	the	the	DET
cana-3840	18	2	introduction	introduction	NOUN
cana-3840	18	3	of	of	ADP
cana-3840	18	4	stratified	stratified	ADJ
cana-3840	18	5	functions	function	NOUN
cana-3840	18	6	serves	serve	VERB
cana-3840	18	7	as	as	ADP
cana-3840	18	8	a	a	DET
cana-3840	18	9	key	key	ADJ
cana-3840	18	10	tool	tool	NOUN
cana-3840	18	11	for	for	ADP
cana-3840	18	12	analyzing	analyze	VERB
cana-3840	18	13	the	the	DET
cana-3840	18	14	topology	topology	NOUN
cana-3840	18	15	of	of	ADP
cana-3840	18	16	these	these	DET
cana-3840	18	17	spaces	space	NOUN
cana-3840	18	18	,	,	PUNCT
cana-3840	18	19	and	and	CCONJ
cana-3840	18	20	our	our	PRON
cana-3840	18	21	results	result	NOUN
cana-3840	18	22	offer	offer	VERB
cana-3840	18	23	a	a	DET
cana-3840	18	24	concrete	concrete	ADJ
cana-3840	18	25	form	form	NOUN
cana-3840	18	26	for	for	ADP
cana-3840	18	27	the	the	DET
cana-3840	18	28	metric	metric	ADJ
cana-3840	18	29	function	function	NOUN
cana-3840	18	30	under	under	ADP
cana-3840	18	31	specific	specific	ADJ
cana-3840	18	32	conditions	condition	NOUN
cana-3840	18	33	,	,	PUNCT
cana-3840	18	34	contributing	contribute	VERB
cana-3840	18	35	to	to	ADP
cana-3840	18	36	the	the	DET
cana-3840	18	37	broader	broad	ADJ
cana-3840	18	38	study	study	NOUN
cana-3840	18	39	of	of	ADP
cana-3840	18	40	fuzzy	fuzzy	ADJ
cana-3840	18	41	metric	metric	ADJ
cana-3840	18	42	spaces	space	NOUN
cana-3840	18	43	.	.	PUNCT
cana-3840	19	1	keywords	keyword	NOUN
cana-3840	19	2	:	:	PUNCT
cana-3840	19	3	t	t	NOUN
cana-3840	19	4	-	-	PUNCT
cana-3840	19	5	conorm	conorm	NOUN
cana-3840	19	6	,	,	PUNCT
cana-3840	19	7	revised	revise	VERB
cana-3840	19	8	fuzzy	fuzzy	ADJ
cana-3840	19	9	metric	metric	ADJ
cana-3840	19	10	2020	2020	NUM
cana-3840	19	11	mathematical	mathematical	ADJ
cana-3840	19	12	classification	classification	NOUN
cana-3840	19	13	:	:	PUNCT
cana-3840	19	14	37c25	37c25	NUM
cana-3840	19	15	,	,	PUNCT
cana-3840	19	16	46s40	46s40	NUM
cana-3840	19	17	,	,	PUNCT
cana-3840	19	18	46n20	46n20	NUM
cana-3840	19	19	,	,	PUNCT
cana-3840	19	20	47h10	47h10	NUM
cana-3840	19	21	1	1	NUM
cana-3840	19	22	.	.	PUNCT
cana-3840	20	1	introduction	introduction	NOUN
cana-3840	20	2	many	many	ADJ
cana-3840	20	3	scholars	scholar	NOUN
cana-3840	20	4	have	have	AUX
cana-3840	20	5	created	create	VERB
cana-3840	20	6	ideas	idea	NOUN
cana-3840	20	7	of	of	ADP
cana-3840	20	8	fuzzy	fuzzy	ADJ
cana-3840	20	9	metric	metric	ADJ
cana-3840	20	10	spaces	space	NOUN
cana-3840	20	11	and	and	CCONJ
cana-3840	20	12	examined	examine	VERB
cana-3840	20	13	their	their	PRON
cana-3840	20	14	characteristics	characteristic	NOUN
cana-3840	20	15	in	in	ADP
cana-3840	20	16	various	various	ADJ
cana-3840	20	17	ways	way	NOUN
cana-3840	20	18	since	since	SCONJ
cana-3840	20	19	zadeh	zadeh	PROPN
cana-3840	20	20	[	[	X
cana-3840	20	21	30	30	NUM
cana-3840	20	22	]	]	PUNCT
cana-3840	20	23	initially	initially	ADV
cana-3840	20	24	suggested	suggest	VERB
cana-3840	20	25	fuzzy	fuzzy	ADJ
cana-3840	20	26	set	set	NOUN
cana-3840	20	27	theory	theory	NOUN
cana-3840	20	28	in	in	ADP
cana-3840	20	29	1965	1965	NUM
cana-3840	20	30	.	.	PUNCT
cana-3840	21	1	in	in	ADP
cana-3840	21	2	1975	1975	NUM
cana-3840	21	3	,	,	PUNCT
cana-3840	21	4	kramosil	kramosil	NOUN
cana-3840	21	5	and	and	CCONJ
cana-3840	21	6	michalek	michalek	NOUN
cana-3840	21	7	mailto:thamizh1418@gmail.com	mailto:thamizh1418@gmail.com	X
cana-3840	21	8	https://cran.r-project.org/web/classifications/msc-2010.html#code:37c25	https://cran.r-project.org/web/classifications/msc-2010.html#code:37c25	NOUN
cana-3840	21	9	communications	communication	NOUN
cana-3840	21	10	on	on	ADP
cana-3840	21	11	applied	apply	VERB
cana-3840	21	12	nonlinear	nonlinear	ADJ
cana-3840	21	13	analysis	analysis	NOUN
cana-3840	21	14	issn	issn	NOUN
cana-3840	21	15	:	:	PUNCT
cana-3840	21	16	1074	1074	NUM
cana-3840	21	17	-	-	PUNCT
cana-3840	21	18	133x	133x	NUM
cana-3840	21	19	vol	vol	NOUN
cana-3840	21	20	32	32	NUM
cana-3840	21	21	no	no	NOUN
cana-3840	21	22	.	.	PUNCT
cana-3840	22	1	9s	9s	NUM
cana-3840	22	2	(	(	PUNCT
cana-3840	22	3	2025	2025	NUM
cana-3840	22	4	)	)	PUNCT
cana-3840	22	5	89	89	NUM
cana-3840	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	23	1	[	[	X
cana-3840	23	2	8	8	NUM
cana-3840	23	3	]	]	PUNCT
cana-3840	23	4	developed	develop	VERB
cana-3840	23	5	the	the	DET
cana-3840	23	6	concept	concept	NOUN
cana-3840	23	7	of	of	ADP
cana-3840	23	8	fuzzy	fuzzy	ADJ
cana-3840	23	9	metric	metric	NOUN
cana-3840	23	10	,	,	PUNCT
cana-3840	23	11	which	which	PRON
cana-3840	23	12	is	be	AUX
cana-3840	23	13	a	a	DET
cana-3840	23	14	fuzzy	fuzzy	ADJ
cana-3840	23	15	set	set	NOUN
cana-3840	23	16	in	in	ADP
cana-3840	23	17	the	the	DET
cana-3840	23	18	cartesian	cartesian	ADJ
cana-3840	23	19	product	product	NOUN
cana-3840	23	20	that	that	PRON
cana-3840	23	21	meets	meet	VERB
cana-3840	23	22	specific	specific	ADJ
cana-3840	23	23	requirements	requirement	NOUN
cana-3840	23	24	.	.	PUNCT
cana-3840	24	1	this	this	DET
cana-3840	24	2	concept	concept	NOUN
cana-3840	24	3	was	be	AUX
cana-3840	24	4	inspired	inspire	VERB
cana-3840	24	5	by	by	ADP
cana-3840	24	6	the	the	DET
cana-3840	24	7	idea	idea	NOUN
cana-3840	24	8	of	of	ADP
cana-3840	24	9	probabilistic	probabilistic	ADJ
cana-3840	24	10	metric	metric	ADJ
cana-3840	24	11	spaces	space	NOUN
cana-3840	24	12	.	.	PUNCT
cana-3840	25	1	subsequently	subsequently	ADV
cana-3840	25	2	,	,	PUNCT
cana-3840	25	3	george	george	PROPN
cana-3840	25	4	and	and	CCONJ
cana-3840	25	5	veeramani	veeramani	NOUN
cana-3840	26	1	[	[	X
cana-3840	26	2	2	2	NUM
cana-3840	26	3	]	]	PUNCT
cana-3840	26	4	modified	modify	VERB
cana-3840	26	5	this	this	DET
cana-3840	26	6	notion	notion	NOUN
cana-3840	26	7	of	of	ADP
cana-3840	26	8	fuzzy	fuzzy	ADJ
cana-3840	26	9	metric	metric	ADJ
cana-3840	26	10	space	space	NOUN
cana-3840	26	11	by	by	ADP
cana-3840	26	12	introducing	introduce	VERB
cana-3840	26	13	the	the	DET
cana-3840	26	14	idea	idea	NOUN
cana-3840	26	15	of	of	ADP
cana-3840	26	16	continuous	continuous	ADJ
cana-3840	26	17	t	t	NOUN
cana-3840	26	18	-	-	PUNCT
cana-3840	26	19	norms	norm	NOUN
cana-3840	26	20	and	and	CCONJ
cana-3840	26	21	shown	show	VERB
cana-3840	26	22	that	that	SCONJ
cana-3840	26	23	all	all	DET
cana-3840	26	24	fuzzy	fuzzy	ADJ
cana-3840	26	25	metric	metric	ADJ
cana-3840	26	26	spaces	space	NOUN
cana-3840	26	27	produce	produce	VERB
cana-3840	26	28	a	a	DET
cana-3840	26	29	hausdorff	hausdorff	NOUN
cana-3840	26	30	firstcountable	firstcountable	ADJ
cana-3840	26	31	topology	topology	NOUN
cana-3840	26	32	.	.	PUNCT
cana-3840	27	1	the	the	DET
cana-3840	27	2	theory	theory	NOUN
cana-3840	27	3	of	of	ADP
cana-3840	27	4	gv	gv	ADP
cana-3840	27	5	-	-	ADJ
cana-3840	27	6	fuzzy	fuzzy	ADJ
cana-3840	27	7	metric	metric	NOUN
cana-3840	27	8	has	have	AUX
cana-3840	27	9	been	be	AUX
cana-3840	27	10	established	establish	VERB
cana-3840	27	11	thus	thus	ADV
cana-3840	27	12	far	far	ADV
cana-3840	27	13	by	by	ADP
cana-3840	27	14	several	several	ADJ
cana-3840	27	15	academics	academic	NOUN
cana-3840	27	16	.	.	PUNCT
cana-3840	28	1	a	a	DET
cana-3840	28	2	great	great	ADJ
cana-3840	28	3	deal	deal	NOUN
cana-3840	28	4	of	of	ADP
cana-3840	28	5	knowledge	knowledge	NOUN
cana-3840	28	6	on	on	ADP
cana-3840	28	7	classical	classical	ADJ
cana-3840	28	8	metric	metric	ADJ
cana-3840	28	9	spaces	space	NOUN
cana-3840	28	10	was	be	AUX
cana-3840	28	11	extended	extend	VERB
cana-3840	28	12	to	to	ADP
cana-3840	28	13	fuzzy	fuzzy	ADJ
cana-3840	28	14	metric	metric	ADJ
cana-3840	28	15	spaces	space	NOUN
cana-3840	28	16	.	.	PUNCT
cana-3840	29	1	it	it	PRON
cana-3840	29	2	was	be	AUX
cana-3840	29	3	discovered	discover	VERB
cana-3840	29	4	throughout	throughout	ADP
cana-3840	29	5	this	this	DET
cana-3840	29	6	procedure	procedure	NOUN
cana-3840	29	7	that	that	PRON
cana-3840	29	8	the	the	DET
cana-3840	29	9	fuzzy	fuzzy	ADJ
cana-3840	29	10	metric	metric	ADJ
cana-3840	29	11	theory	theory	NOUN
cana-3840	29	12	differed	differ	VERB
cana-3840	29	13	greatly	greatly	ADV
cana-3840	29	14	from	from	ADP
cana-3840	29	15	the	the	DET
cana-3840	29	16	traditional	traditional	ADJ
cana-3840	29	17	theory	theory	NOUN
cana-3840	29	18	of	of	ADP
cana-3840	29	19	metric	metric	NOUN
cana-3840	29	20	.	.	PUNCT
cana-3840	30	1	as	as	ADP
cana-3840	30	2	an	an	DET
cana-3840	30	3	illustration	illustration	NOUN
cana-3840	30	4	,	,	PUNCT
cana-3840	30	5	gregori	gregori	PROPN
cana-3840	30	6	and	and	CCONJ
cana-3840	30	7	romaguera	romaguera	NOUN
cana-3840	30	8	[	[	X
cana-3840	30	9	3	3	X
cana-3840	30	10	]	]	PUNCT
cana-3840	30	11	demonstrated	demonstrate	VERB
cana-3840	30	12	the	the	DET
cana-3840	30	13	existence	existence	NOUN
cana-3840	30	14	of	of	ADP
cana-3840	30	15	an	an	DET
cana-3840	30	16	incompletable	incompletable	ADJ
cana-3840	30	17	gv	gv	ADP
cana-3840	30	18	fuzzy	fuzzy	ADJ
cana-3840	30	19	metric	metric	ADJ
cana-3840	30	20	space	space	NOUN
cana-3840	30	21	.	.	PUNCT
cana-3840	31	1	a	a	DET
cana-3840	31	2	classification	classification	NOUN
cana-3840	31	3	of	of	ADP
cana-3840	31	4	the	the	DET
cana-3840	31	5	class	class	NOUN
cana-3840	31	6	of	of	ADP
cana-3840	31	7	completable	completable	ADJ
cana-3840	31	8	strong	strong	ADJ
cana-3840	31	9	fuzzy	fuzzy	ADJ
cana-3840	31	10	metric	metric	ADJ
cana-3840	31	11	spaces	space	NOUN
cana-3840	31	12	was	be	AUX
cana-3840	31	13	provided	provide	VERB
cana-3840	31	14	by	by	ADP
cana-3840	31	15	[	[	X
cana-3840	31	16	2	2	NUM
cana-3840	31	17	,	,	PUNCT
cana-3840	31	18	4	4	NUM
cana-3840	31	19	-	-	SYM
cana-3840	31	20	7	7	NUM
cana-3840	31	21	,	,	PUNCT
cana-3840	31	22	12	12	NUM
cana-3840	31	23	-	-	SYM
cana-3840	31	24	13	13	NUM
cana-3840	31	25	,	,	PUNCT
cana-3840	31	26	23	23	NUM
cana-3840	31	27	,	,	PUNCT
cana-3840	31	28	28	28	NUM
cana-3840	31	29	-	-	SYM
cana-3840	31	30	29	29	NUM
cana-3840	31	31	]	]	PUNCT
cana-3840	31	32	.	.	PUNCT
cana-3840	32	1	jainrong	jainrong	PROPN
cana-3840	32	2	wu	wu	PROPN
cana-3840	32	3	and	and	CCONJ
cana-3840	32	4	hao	hao	PROPN
cana-3840	32	5	yang	yang	PROPN
cana-3840	33	1	[	[	X
cana-3840	33	2	11	11	NUM
cana-3840	33	3	]	]	PUNCT
cana-3840	33	4	found	find	VERB
cana-3840	33	5	a	a	DET
cana-3840	33	6	stronger	strong	ADJ
cana-3840	33	7	result	result	NOUN
cana-3840	33	8	in	in	ADP
cana-3840	33	9	2000	2000	NUM
cana-3840	33	10	,	,	PUNCT
cana-3840	33	11	which	which	PRON
cana-3840	33	12	was	be	AUX
cana-3840	33	13	a	a	DET
cana-3840	33	14	little	little	ADJ
cana-3840	33	15	unexpected	unexpected	ADJ
cana-3840	33	16	.	.	PUNCT
cana-3840	34	1	they	they	PRON
cana-3840	34	2	demonstrated	demonstrate	VERB
cana-3840	34	3	that	that	SCONJ
cana-3840	34	4	a	a	DET
cana-3840	34	5	metrizable	metrizable	ADJ
cana-3840	34	6	topology	topology	NOUN
cana-3840	34	7	is	be	AUX
cana-3840	34	8	produced	produce	VERB
cana-3840	34	9	by	by	ADP
cana-3840	34	10	each	each	DET
cana-3840	34	11	gv	gv	NOUN
cana-3840	34	12	-	-	ADJ
cana-3840	34	13	fuzzy	fuzzy	ADJ
cana-3840	34	14	metric	metric	NOUN
cana-3840	34	15	.	.	PUNCT
cana-3840	35	1	this	this	DET
cana-3840	35	2	crucial	crucial	ADJ
cana-3840	35	3	finding	finding	NOUN
cana-3840	35	4	establishes	establish	VERB
cana-3840	35	5	a	a	DET
cana-3840	35	6	link	link	NOUN
cana-3840	35	7	between	between	ADP
cana-3840	35	8	the	the	DET
cana-3840	35	9	classical	classical	ADJ
cana-3840	35	10	metric	metric	NOUN
cana-3840	35	11	and	and	CCONJ
cana-3840	35	12	the	the	DET
cana-3840	35	13	gv	gv	ADJ
cana-3840	35	14	-	-	ADJ
cana-3840	35	15	fuzzy	fuzzy	ADJ
cana-3840	35	16	metric	metric	NOUN
cana-3840	35	17	.	.	PUNCT
cana-3840	36	1	nonetheless	nonetheless	ADV
cana-3840	36	2	,	,	PUNCT
cana-3840	36	3	the	the	DET
cana-3840	36	4	metric	metric	ADJ
cana-3840	36	5	function	function	NOUN
cana-3840	36	6	's	's	PART
cana-3840	36	7	shape	shape	NOUN
cana-3840	36	8	has	have	AUX
cana-3840	36	9	n't	not	PART
cana-3840	36	10	been	be	AUX
cana-3840	36	11	examined	examine	VERB
cana-3840	36	12	in	in	ADP
cana-3840	36	13	any	any	PRON
cana-3840	36	14	of	of	ADP
cana-3840	36	15	the	the	DET
cana-3840	36	16	previous	previous	ADJ
cana-3840	36	17	research	research	NOUN
cana-3840	36	18	works	work	NOUN
cana-3840	36	19	.	.	PUNCT
cana-3840	37	1	that	that	PRON
cana-3840	37	2	is	be	AUX
cana-3840	37	3	only	only	ADV
cana-3840	37	4	the	the	DET
cana-3840	37	5	current	current	ADJ
cana-3840	37	6	paper	paper	NOUN
cana-3840	37	7	's	's	PART
cana-3840	37	8	primary	primary	ADJ
cana-3840	37	9	objective	objective	NOUN
cana-3840	37	10	.	.	PUNCT
cana-3840	38	1	most	most	ADV
cana-3840	38	2	interesting	interesting	ADJ
cana-3840	38	3	motivations	motivation	NOUN
cana-3840	38	4	is	be	AUX
cana-3840	38	5	the	the	DET
cana-3840	38	6	introduction	introduction	NOUN
cana-3840	38	7	of	of	ADP
cana-3840	38	8	revised	revise	VERB
cana-3840	38	9	fuzzy	fuzzy	ADJ
cana-3840	38	10	metric	metric	ADJ
cana-3840	38	11	spaces	space	NOUN
cana-3840	38	12	by	by	ADP
cana-3840	38	13	alexandar	alexandar	NOUN
cana-3840	38	14	sostak	sostak	NOUN
cana-3840	38	15	[	[	X
cana-3840	38	16	1	1	NUM
cana-3840	38	17	]	]	PUNCT
cana-3840	38	18	.	.	PUNCT
cana-3840	39	1	later	later	ADV
cana-3840	39	2	on	on	ADV
cana-3840	39	3	,	,	PUNCT
cana-3840	39	4	olga	olga	PROPN
cana-3840	39	5	grigorenko	grigorenko	PROPN
cana-3840	40	1	[	[	X
cana-3840	40	2	17	17	NUM
cana-3840	40	3	]	]	PUNCT
cana-3840	40	4	,	,	PUNCT
cana-3840	40	5	juan	juan	PROPN
cana-3840	40	6	jose	jose	PROPN
cana-3840	40	7	minana	minana	PROPN
cana-3840	40	8	,	,	PUNCT
cana-3840	40	9	alexander	alexander	PROPN
cana-3840	40	10	sostak	sostak	PROPN
cana-3840	40	11	,	,	PUNCT
cana-3840	40	12	oscar	oscar	PROPN
cana-3840	40	13	valero	valero	PROPN
cana-3840	40	14	introduced	introduce	VERB
cana-3840	40	15	“	"	PUNCT
cana-3840	40	16	on	on	ADP
cana-3840	40	17	t	t	PROPN
cana-3840	40	18	-	-	PUNCT
cana-3840	40	19	conorm	conorm	NOUN
cana-3840	40	20	based	base	VERB
cana-3840	40	21	fuzzy	fuzzy	ADJ
cana-3840	40	22	(	(	PUNCT
cana-3840	40	23	pseudo	pseudo	NOUN
cana-3840	40	24	)	)	PUNCT
cana-3840	40	25	metrics	metric	NOUN
cana-3840	40	26	”	"	PUNCT
cana-3840	40	27	,	,	PUNCT
cana-3840	40	28	they	they	PRON
cana-3840	40	29	develop	develop	VERB
cana-3840	40	30	the	the	DET
cana-3840	40	31	basics	basic	NOUN
cana-3840	40	32	of	of	ADP
cana-3840	40	33	the	the	DET
cana-3840	40	34	theory	theory	NOUN
cana-3840	40	35	of	of	ADP
cana-3840	40	36	cb	cb	PROPN
cana-3840	40	37	-	-	ADJ
cana-3840	40	38	fuzzy	fuzzy	ADJ
cana-3840	40	39	(	(	PUNCT
cana-3840	40	40	pseudo	pseudo	NOUN
cana-3840	40	41	)	)	PUNCT
cana-3840	40	42	metrics	metric	NOUN
cana-3840	40	43	and	and	CCONJ
cana-3840	40	44	compare	compare	VERB
cana-3840	40	45	them	they	PRON
cana-3840	40	46	with	with	ADP
cana-3840	40	47	“	"	PUNCT
cana-3840	40	48	classic	classic	ADJ
cana-3840	40	49	”	"	PUNCT
cana-3840	40	50	fuzzy	fuzzy	ADJ
cana-3840	40	51	(	(	PUNCT
cana-3840	40	52	pseudo	pseudo	NOUN
cana-3840	40	53	)	)	PUNCT
cana-3840	40	54	metrics	metric	NOUN
cana-3840	40	55	[	[	X
cana-3840	40	56	2020	2020	NUM
cana-3840	40	57	]	]	PUNCT
cana-3840	40	58	.	.	PUNCT
cana-3840	41	1	after	after	ADP
cana-3840	41	2	that	that	DET
cana-3840	41	3	muraliraj	muraliraj	NOUN
cana-3840	41	4	and	and	CCONJ
cana-3840	41	5	thangathamizh	thangathamizh	ADJ
cana-3840	41	6	[	[	X
cana-3840	41	7	21	21	NUM
cana-3840	41	8	]	]	PUNCT
cana-3840	41	9	proved	prove	VERB
cana-3840	41	10	the	the	DET
cana-3840	41	11	existence	existence	NOUN
cana-3840	41	12	of	of	ADP
cana-3840	41	13	fixed	fix	VERB
cana-3840	41	14	points	point	NOUN
cana-3840	41	15	in	in	ADP
cana-3840	41	16	revised	revise	VERB
cana-3840	41	17	fuzzy	fuzzy	ADJ
cana-3840	41	18	metric	metric	ADJ
cana-3840	41	19	space	space	NOUN
cana-3840	41	20	.	.	PUNCT
cana-3840	42	1	good	good	ADJ
cana-3840	42	2	,	,	PUNCT
cana-3840	42	3	related	related	ADJ
cana-3840	42	4	results	result	NOUN
cana-3840	42	5	about	about	ADP
cana-3840	42	6	fixed	fix	VERB
cana-3840	42	7	point	point	NOUN
cana-3840	42	8	in	in	ADP
cana-3840	42	9	fuzzy	fuzzy	ADJ
cana-3840	42	10	metric	metric	ADJ
cana-3840	42	11	spaces	space	NOUN
cana-3840	42	12	were	be	AUX
cana-3840	42	13	introduced	introduce	VERB
cana-3840	42	14	recently	recently	ADV
cana-3840	42	15	[	[	X
cana-3840	42	16	14	14	NUM
cana-3840	42	17	,	,	PUNCT
cana-3840	42	18	16	16	NUM
cana-3840	42	19	,	,	PUNCT
cana-3840	42	20	26	26	NUM
cana-3840	42	21	-	-	SYM
cana-3840	42	22	27	27	NUM
cana-3840	42	23	]	]	PUNCT
cana-3840	42	24	in	in	ADP
cana-3840	42	25	this	this	DET
cana-3840	42	26	study	study	NOUN
cana-3840	42	27	,	,	PUNCT
cana-3840	42	28	we	we	PRON
cana-3840	42	29	first	first	ADV
cana-3840	42	30	present	present	VERB
cana-3840	42	31	the	the	DET
cana-3840	42	32	idea	idea	NOUN
cana-3840	42	33	of	of	ADP
cana-3840	42	34	a	a	DET
cana-3840	42	35	stratified	stratified	ADJ
cana-3840	42	36	function	function	NOUN
cana-3840	42	37	in	in	ADP
cana-3840	42	38	a	a	DET
cana-3840	42	39	revised	revise	VERB
cana-3840	42	40	fuzzy	fuzzy	ADJ
cana-3840	42	41	metric	metric	ADJ
cana-3840	42	42	space	space	NOUN
cana-3840	42	43	,	,	PUNCT
cana-3840	42	44	which	which	PRON
cana-3840	42	45	differs	differ	VERB
cana-3840	42	46	somewhat	somewhat	ADV
cana-3840	42	47	from	from	ADP
cana-3840	42	48	the	the	DET
cana-3840	42	49	rgv	rgv	PROPN
cana-3840	42	50	-	-	PUNCT
cana-3840	42	51	fuzzy	fuzzy	ADJ
cana-3840	42	52	metric	metric	ADJ
cana-3840	42	53	space	space	NOUN
cana-3840	42	54	.	.	PUNCT
cana-3840	43	1	we	we	PRON
cana-3840	43	2	next	next	ADV
cana-3840	43	3	demonstrate	demonstrate	VERB
cana-3840	43	4	that	that	SCONJ
cana-3840	43	5	the	the	DET
cana-3840	43	6	metrizable	metrizable	ADJ
cana-3840	43	7	topology	topology	NOUN
cana-3840	43	8	may	may	AUX
cana-3840	43	9	coexist	coexist	VERB
cana-3840	43	10	with	with	ADP
cana-3840	43	11	the	the	DET
cana-3840	43	12	topology	topology	NOUN
cana-3840	43	13	produced	produce	VERB
cana-3840	43	14	by	by	ADP
cana-3840	43	15	the	the	DET
cana-3840	43	16	family	family	NOUN
cana-3840	43	17	of	of	ADP
cana-3840	43	18	stratified	stratified	ADJ
cana-3840	43	19	functions	function	NOUN
cana-3840	43	20	.	.	PUNCT
cana-3840	44	1	the	the	DET
cana-3840	44	2	concrete	concrete	ADJ
cana-3840	44	3	metric	metric	ADJ
cana-3840	44	4	function	function	NOUN
cana-3840	44	5	whose	whose	DET
cana-3840	44	6	topology	topology	NOUN
cana-3840	44	7	agrees	agree	VERB
cana-3840	44	8	with	with	ADP
cana-3840	44	9	the	the	DET
cana-3840	44	10	metrizable	metrizable	ADJ
cana-3840	44	11	topology	topology	NOUN
cana-3840	44	12	is	be	AUX
cana-3840	44	13	then	then	ADV
cana-3840	44	14	provided	provide	VERB
cana-3840	44	15	,	,	PUNCT
cana-3840	44	16	subject	subject	ADJ
cana-3840	44	17	to	to	ADP
cana-3840	44	18	certain	certain	ADJ
cana-3840	44	19	restrictions	restriction	NOUN
cana-3840	44	20	.	.	PUNCT
cana-3840	45	1	the	the	DET
cana-3840	45	2	paper	paper	NOUN
cana-3840	45	3	is	be	AUX
cana-3840	45	4	organized	organize	VERB
cana-3840	45	5	as	as	SCONJ
cana-3840	45	6	follows	follow	VERB
cana-3840	45	7	.	.	PUNCT
cana-3840	46	1	we	we	PRON
cana-3840	46	2	address	address	VERB
cana-3840	46	3	the	the	DET
cana-3840	46	4	early	early	ADJ
cana-3840	46	5	ideas	idea	NOUN
cana-3840	46	6	on	on	ADP
cana-3840	46	7	revised	revise	VERB
cana-3840	46	8	fuzzy	fuzzy	ADJ
cana-3840	46	9	metrics	metric	NOUN
cana-3840	46	10	in	in	ADP
cana-3840	46	11	the	the	DET
cana-3840	46	12	next	next	ADJ
cana-3840	46	13	section	section	NOUN
cana-3840	46	14	.	.	PUNCT
cana-3840	47	1	section	section	NOUN
cana-3840	47	2	3	3	NUM
cana-3840	47	3	presents	present	VERB
cana-3840	47	4	our	our	PRON
cana-3840	47	5	primary	primary	ADJ
cana-3840	47	6	findings	finding	NOUN
cana-3840	47	7	.	.	PUNCT
cana-3840	48	1	lastly	lastly	ADV
cana-3840	48	2	,	,	PUNCT
cana-3840	48	3	we	we	PRON
cana-3840	48	4	conclude	conclude	VERB
cana-3840	48	5	in	in	ADP
cana-3840	48	6	section	section	NOUN
cana-3840	48	7	4	4	NUM
cana-3840	48	8	with	with	ADP
cana-3840	48	9	some	some	DET
cana-3840	48	10	last	last	ADJ
cana-3840	48	11	thoughts	thought	NOUN
cana-3840	48	12	.	.	PUNCT
cana-3840	49	1	2	2	X
cana-3840	49	2	.	.	X
cana-3840	49	3	preliminaries	preliminary	NOUN
cana-3840	49	4	in	in	ADP
cana-3840	49	5	this	this	DET
cana-3840	49	6	section	section	NOUN
cana-3840	49	7	,	,	PUNCT
cana-3840	49	8	we	we	PRON
cana-3840	49	9	first	first	ADV
cana-3840	49	10	introduce	introduce	VERB
cana-3840	49	11	some	some	DET
cana-3840	49	12	basic	basic	ADJ
cana-3840	49	13	concepts	concept	NOUN
cana-3840	49	14	and	and	CCONJ
cana-3840	49	15	properties	property	NOUN
cana-3840	49	16	of	of	ADP
cana-3840	49	17	revised	revise	VERB
cana-3840	49	18	fuzzy	fuzzy	ADJ
cana-3840	49	19	metric	metric	ADJ
cana-3840	49	20	spaces	space	NOUN
cana-3840	49	21	.	.	PUNCT
cana-3840	50	1	definition	definition	NOUN
cana-3840	50	2	1[30	1[30	NUM
cana-3840	50	3	]	]	PUNCT
cana-3840	50	4	.	.	PUNCT
cana-3840	51	1	a	a	DET
cana-3840	51	2	binary	binary	ADJ
cana-3840	51	3	operation	operation	NOUN
cana-3840	51	4	⨁	⨁	PROPN
cana-3840	51	5	:	:	PUNCT
cana-3840	52	1	[	[	X
cana-3840	52	2	0	0	NUM
cana-3840	52	3	,	,	PUNCT
cana-3840	52	4	1]2	1]2	NUM
cana-3840	52	5	⟶	⟶	NOUN
cana-3840	52	6	[	[	X
cana-3840	52	7	0	0	NUM
cana-3840	52	8	,	,	PUNCT
cana-3840	52	9	1	1	NUM
cana-3840	52	10	]	]	PUNCT
cana-3840	52	11	is	be	AUX
cana-3840	52	12	a	a	DET
cana-3840	52	13	continuous	continuous	ADJ
cana-3840	52	14	t	t	NOUN
cana-3840	52	15	-	-	PUNCT
cana-3840	52	16	conorm	conorm	NOUN
cana-3840	52	17	if	if	SCONJ
cana-3840	52	18	it	it	PRON
cana-3840	52	19	satisfies	satisfy	VERB
cana-3840	52	20	the	the	DET
cana-3840	52	21	following	follow	VERB
cana-3840	52	22	conditions	condition	NOUN
cana-3840	52	23	:	:	PUNCT
cana-3840	52	24	(	(	PUNCT
cana-3840	52	25	1	1	X
cana-3840	52	26	)	)	PUNCT
cana-3840	52	27	⨁	⨁	PROPN
cana-3840	52	28	is	be	AUX
cana-3840	52	29	associative	associative	ADJ
cana-3840	52	30	and	and	CCONJ
cana-3840	52	31	commutative	commutative	ADJ
cana-3840	52	32	(	(	PUNCT
cana-3840	52	33	2	2	NUM
cana-3840	52	34	)	)	PUNCT
cana-3840	52	35	⨁	⨁	PROPN
cana-3840	52	36	is	be	AUX
cana-3840	52	37	continuous	continuous	ADJ
cana-3840	52	38	(	(	PUNCT
cana-3840	52	39	3	3	NUM
cana-3840	52	40	)	)	PUNCT
cana-3840	52	41	0	0	NUM
cana-3840	52	42	⨁	⨁	PROPN
cana-3840	52	43	𝕡	𝕡	PROPN
cana-3840	52	44	=	=	SYM
cana-3840	52	45	𝕡	𝕡	PROPN
cana-3840	52	46	for	for	ADP
cana-3840	52	47	each	each	DET
cana-3840	52	48	𝕡	𝕡	PRON
cana-3840	52	49	∈	∈	PROPN
cana-3840	53	1	[	[	X
cana-3840	53	2	0,1	0,1	NUM
cana-3840	53	3	]	]	PUNCT
cana-3840	53	4	(	(	PUNCT
cana-3840	53	5	4	4	X
cana-3840	53	6	)	)	PUNCT
cana-3840	53	7	𝕡	𝕡	DET
cana-3840	53	8	⨁	⨁	PROPN
cana-3840	53	9	𝕢	𝕢	DET
cana-3840	53	10	≤	≤	PROPN
cana-3840	53	11	𝕣	𝕣	X
cana-3840	53	12	⨁	⨁	PROPN
cana-3840	53	13	𝕤	𝕤	NOUN
cana-3840	53	14	whenever	whenever	SCONJ
cana-3840	53	15	𝕡	𝕡	DET
cana-3840	53	16	≤	≤	X
cana-3840	53	17	𝕣	𝕣	PRON
cana-3840	53	18	and	and	CCONJ
cana-3840	53	19	𝕢	𝕢	DET
cana-3840	53	20	≤	≤	NUM
cana-3840	53	21	𝕤	𝕤	NOUN
cana-3840	53	22	with	with	ADP
cana-3840	53	23	𝕡	𝕡	PROPN
cana-3840	53	24	,	,	PUNCT
cana-3840	53	25	𝕢	𝕢	PROPN
cana-3840	53	26	,	,	PUNCT
cana-3840	53	27	𝕣	𝕣	PROPN
cana-3840	53	28	,	,	PUNCT
cana-3840	53	29	𝕤	𝕤	DET
cana-3840	53	30	∈	∈	PROPN
cana-3840	54	1	[	[	X
cana-3840	54	2	0	0	NUM
cana-3840	54	3	,	,	PUNCT
cana-3840	54	4	1	1	NUM
cana-3840	54	5	]	]	PUNCT
cana-3840	54	6	.	.	PUNCT
cana-3840	55	1	the	the	DET
cana-3840	55	2	following	follow	VERB
cana-3840	55	3	continuous	continuous	ADJ
cana-3840	55	4	t	t	PROPN
cana-3840	55	5	-	-	PUNCT
cana-3840	55	6	conorms	conorm	NOUN
cana-3840	55	7	are	be	AUX
cana-3840	55	8	used	use	VERB
cana-3840	55	9	in	in	ADP
cana-3840	55	10	this	this	DET
cana-3840	55	11	paper	paper	NOUN
cana-3840	55	12	:	:	PUNCT
cana-3840	55	13	communications	communication	NOUN
cana-3840	55	14	on	on	ADP
cana-3840	55	15	applied	apply	VERB
cana-3840	55	16	nonlinear	nonlinear	ADJ
cana-3840	55	17	analysis	analysis	NOUN
cana-3840	55	18	issn	issn	NOUN
cana-3840	55	19	:	:	PUNCT
cana-3840	55	20	1074	1074	NUM
cana-3840	55	21	-	-	PUNCT
cana-3840	55	22	133x	133x	NUM
cana-3840	55	23	vol	vol	NOUN
cana-3840	55	24	32	32	NUM
cana-3840	56	1	no	no	NOUN
cana-3840	56	2	.	.	PUNCT
cana-3840	57	1	9s	9s	NUM
cana-3840	57	2	(	(	PUNCT
cana-3840	57	3	2025	2025	NUM
cana-3840	57	4	)	)	PUNCT
cana-3840	58	1	90	90	NUM
cana-3840	58	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	58	3	𝕡	𝕡	X
cana-3840	58	4	⋁1	⋁1	NOUN
cana-3840	58	5	𝕢	𝕢	X
cana-3840	58	6	=	=	PUNCT
cana-3840	58	7	𝑚𝑎𝑥{𝕡	𝑚𝑎𝑥{𝕡	ADV
cana-3840	58	8	,	,	PUNCT
cana-3840	58	9	𝕢	𝕢	X
cana-3840	58	10	}	}	PUNCT
cana-3840	58	11	,	,	PUNCT
cana-3840	58	12	𝕡	𝕡	X
cana-3840	58	13	⋁2	⋁2	NOUN
cana-3840	58	14	𝕢	𝕢	X
cana-3840	58	15	=	=	SYM
cana-3840	58	16	𝕡	𝕡	PROPN
cana-3840	59	1	+	+	CCONJ
cana-3840	59	2	𝕢	𝕢	DET
cana-3840	59	3	−	−	PROPN
cana-3840	59	4	𝑎𝕢	𝑎𝕢	PROPN
cana-3840	59	5	,	,	PUNCT
cana-3840	59	6	𝕡	𝕡	DET
cana-3840	59	7	⋁3	⋁3	NOUN
cana-3840	59	8	𝕢	𝕢	X
cana-3840	59	9	=	=	PUNCT
cana-3840	59	10	𝑚𝑖𝑛{𝕡	𝑚𝑖𝑛{𝕡	PROPN
cana-3840	59	11	+	+	CCONJ
cana-3840	59	12	𝕢	𝕢	PROPN
cana-3840	59	13	,	,	PUNCT
cana-3840	59	14	1	1	NUM
cana-3840	59	15	}	}	PUNCT
cana-3840	59	16	.	.	PUNCT
cana-3840	60	1	(	(	PUNCT
cana-3840	60	2	1	1	X
cana-3840	60	3	)	)	PUNCT
cana-3840	60	4	in	in	ADP
cana-3840	60	5	the	the	DET
cana-3840	60	6	sense	sense	NOUN
cana-3840	60	7	of	of	ADP
cana-3840	60	8	alexander	alexander	PROPN
cana-3840	60	9	sostack	sostack	PROPN
cana-3840	60	10	,	,	PUNCT
cana-3840	60	11	a	a	DET
cana-3840	60	12	gv	gv	NOUN
cana-3840	60	13	-	-	ADJ
cana-3840	60	14	fuzzy	fuzzy	ADJ
cana-3840	60	15	metric	metric	NOUN
cana-3840	60	16	is	be	AUX
cana-3840	60	17	defined	define	VERB
cana-3840	60	18	by	by	ADP
cana-3840	60	19	the	the	DET
cana-3840	60	20	follows	follow	NOUN
cana-3840	60	21	.	.	PUNCT
cana-3840	61	1	definition	definition	NOUN
cana-3840	61	2	2[1	2[1	NUM
cana-3840	61	3	]	]	PUNCT
cana-3840	61	4	.	.	PUNCT
cana-3840	62	1	let	let	VERB
cana-3840	62	2	𝔘	𝔘	PRON
cana-3840	62	3	be	be	AUX
cana-3840	62	4	a	a	DET
cana-3840	62	5	nonempty	nonempty	ADV
cana-3840	62	6	set	set	VERB
cana-3840	62	7	and	and	CCONJ
cana-3840	62	8	⨁	⨁	PROPN
cana-3840	62	9	be	be	VERB
cana-3840	62	10	a	a	DET
cana-3840	62	11	continuous	continuous	ADJ
cana-3840	62	12	t	t	NOUN
cana-3840	62	13	-	-	PUNCT
cana-3840	62	14	conorm	conorm	NOUN
cana-3840	62	15	.	.	PUNCT
cana-3840	63	1	a	a	DET
cana-3840	63	2	revised	revise	VERB
cana-3840	63	3	fuzzy	fuzzy	ADJ
cana-3840	63	4	metric	metric	ADJ
cana-3840	63	5	𝕎	𝕎	PROPN
cana-3840	63	6	on	on	ADP
cana-3840	63	7	the	the	DET
cana-3840	63	8	set	set	ADJ
cana-3840	63	9	𝔘	𝔘	PROPN
cana-3840	63	10	is	be	AUX
cana-3840	63	11	a	a	DET
cana-3840	63	12	mapping	mapping	NOUN
cana-3840	63	13	𝕎	𝕎	NOUN
cana-3840	63	14	:	:	PUNCT
cana-3840	63	15	𝔘2	𝔘2	PROPN
cana-3840	63	16	×	×	PROPN
cana-3840	63	17	(	(	PUNCT
cana-3840	63	18	0	0	NUM
cana-3840	63	19	,	,	PUNCT
cana-3840	63	20	∞	∞	NUM
cana-3840	63	21	)	)	PUNCT
cana-3840	63	22	⟶	⟶	NOUN
cana-3840	63	23	(	(	PUNCT
cana-3840	63	24	0	0	NUM
cana-3840	63	25	,	,	PUNCT
cana-3840	63	26	1	1	NUM
cana-3840	63	27	]	]	PUNCT
cana-3840	63	28	satisfying	satisfy	VERB
cana-3840	63	29	the	the	DET
cana-3840	63	30	following	follow	VERB
cana-3840	63	31	conditions	condition	NOUN
cana-3840	63	32	:	:	PUNCT
cana-3840	63	33	for	for	ADP
cana-3840	63	34	all	all	PRON
cana-3840	63	35	𝕒	𝕒	PROPN
cana-3840	63	36	,	,	PUNCT
cana-3840	63	37	𝕓	𝕓	PROPN
cana-3840	63	38	,	,	PUNCT
cana-3840	63	39	𝕔	𝕔	DET
cana-3840	63	40	∈	∈	PROPN
cana-3840	63	41	𝔘	𝔘	PROPN
cana-3840	63	42	,	,	PUNCT
cana-3840	63	43	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-3840	63	44	𝓉	𝓉	PROPN
cana-3840	63	45	,	,	PUNCT
cana-3840	63	46	𝓈	𝓈	X
cana-3840	63	47	>	>	X
cana-3840	63	48	0	0	NUM
cana-3840	63	49	:	:	PUNCT
cana-3840	63	50	(	(	PUNCT
cana-3840	63	51	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	63	52	𝟏	𝟏	NUM
cana-3840	63	53	)	)	PUNCT
cana-3840	63	54	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	63	55	,	,	PUNCT
cana-3840	63	56	𝕓	𝕓	X
cana-3840	63	57	,	,	PUNCT
cana-3840	63	58	𝓉	𝓉	PRON
cana-3840	63	59	)	)	PUNCT
cana-3840	63	60	<	<	X
cana-3840	63	61	1	1	NUM
cana-3840	63	62	,	,	PUNCT
cana-3840	63	63	(	(	PUNCT
cana-3840	63	64	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	63	65	𝟐	𝟐	NUM
cana-3840	63	66	)	)	PUNCT
cana-3840	63	67	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	63	68	,	,	PUNCT
cana-3840	63	69	𝕓	𝕓	X
cana-3840	63	70	,	,	PUNCT
cana-3840	63	71	𝓉	𝓉	PRON
cana-3840	63	72	)	)	PUNCT
cana-3840	64	1	=	=	SYM
cana-3840	64	2	0	0	PUNCT
cana-3840	65	1	if	if	SCONJ
cana-3840	65	2	and	and	CCONJ
cana-3840	65	3	only	only	ADV
cana-3840	65	4	if	if	SCONJ
cana-3840	65	5	𝕒	𝕒	X
cana-3840	65	6	=	=	SYM
cana-3840	65	7	𝕓	𝕓	PROPN
cana-3840	65	8	,	,	PUNCT
cana-3840	65	9	(	(	PUNCT
cana-3840	65	10	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	65	11	𝟑	𝟑	X
cana-3840	65	12	)	)	PUNCT
cana-3840	65	13	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	65	14	,	,	PUNCT
cana-3840	65	15	𝕓	𝕓	X
cana-3840	65	16	,	,	PUNCT
cana-3840	65	17	𝓉	𝓉	PRON
cana-3840	65	18	)	)	PUNCT
cana-3840	65	19	=	=	SYM
cana-3840	66	1	𝕎(𝕓	𝕎(𝕓	PROPN
cana-3840	66	2	,	,	PUNCT
cana-3840	66	3	𝕒	𝕒	X
cana-3840	66	4	,	,	PUNCT
cana-3840	66	5	𝓉	𝓉	PROPN
cana-3840	66	6	)	)	PUNCT
cana-3840	66	7	,	,	PUNCT
cana-3840	66	8	(	(	PUNCT
cana-3840	66	9	2	2	X
cana-3840	66	10	)	)	PUNCT
cana-3840	66	11	(	(	PUNCT
cana-3840	66	12	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	66	13	𝟒	𝟒	NUM
cana-3840	66	14	)	)	PUNCT
cana-3840	66	15	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	66	16	,	,	PUNCT
cana-3840	66	17	𝕓	𝕓	X
cana-3840	66	18	,	,	PUNCT
cana-3840	66	19	𝓉)⨁	𝓉)⨁	VERB
cana-3840	66	20	𝕎(𝕓	𝕎(𝕓	PROPN
cana-3840	66	21	,	,	PUNCT
cana-3840	66	22	𝕔	𝕔	NOUN
cana-3840	66	23	,	,	PUNCT
cana-3840	66	24	𝓈	𝓈	NOUN
cana-3840	66	25	)	)	PUNCT
cana-3840	66	26	≥	≥	NOUN
cana-3840	66	27	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	66	28	,	,	PUNCT
cana-3840	66	29	𝕓	𝕓	X
cana-3840	66	30	,	,	PUNCT
cana-3840	66	31	𝓉	𝓉	PROPN
cana-3840	66	32	)	)	PUNCT
cana-3840	66	33	,	,	PUNCT
cana-3840	66	34	(	(	PUNCT
cana-3840	66	35	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	66	36	𝟓	𝟓	NUM
cana-3840	66	37	)	)	PUNCT
cana-3840	66	38	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	66	39	,	,	PUNCT
cana-3840	66	40	𝕓	𝕓	X
cana-3840	66	41	,	,	PUNCT
cana-3840	66	42	−	−	PROPN
cana-3840	66	43	):	):	PUNCT
cana-3840	66	44	(	(	PUNCT
cana-3840	66	45	0	0	NUM
cana-3840	66	46	,	,	PUNCT
cana-3840	66	47	∞	∞	NUM
cana-3840	66	48	)	)	PUNCT
cana-3840	66	49	⟶	⟶	NOUN
cana-3840	66	50	(	(	PUNCT
cana-3840	66	51	0,1	0,1	NUM
cana-3840	66	52	]	]	PUNCT
cana-3840	66	53	is	be	AUX
cana-3840	66	54	continuous	continuous	ADJ
cana-3840	66	55	.	.	PUNCT
cana-3840	67	1	if	if	SCONJ
cana-3840	67	2	𝕎	𝕎	PROPN
cana-3840	67	3	is	be	AUX
cana-3840	67	4	a	a	DET
cana-3840	67	5	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	67	6	–	–	PUNCT
cana-3840	67	7	revised	revise	VERB
cana-3840	67	8	fuzzy	fuzzy	ADJ
cana-3840	67	9	metric	metric	ADJ
cana-3840	67	10	on	on	ADP
cana-3840	67	11	𝔘	𝔘	PROPN
cana-3840	67	12	,	,	PUNCT
cana-3840	67	13	then	then	ADV
cana-3840	67	14	the	the	DET
cana-3840	67	15	3	3	NUM
cana-3840	67	16	-	-	PUNCT
cana-3840	67	17	tuple	tuple	NOUN
cana-3840	67	18	(	(	PUNCT
cana-3840	67	19	𝔘	𝔘	PROPN
cana-3840	67	20	,	,	PUNCT
cana-3840	67	21	𝕎	𝕎	PROPN
cana-3840	67	22	,	,	PUNCT
cana-3840	67	23	⨁	⨁	PROPN
cana-3840	67	24	)	)	PUNCT
cana-3840	67	25	is	be	AUX
cana-3840	67	26	said	say	VERB
cana-3840	67	27	to	to	PART
cana-3840	67	28	be	be	AUX
cana-3840	67	29	a	a	DET
cana-3840	67	30	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	67	31	–	–	PUNCT
cana-3840	67	32	revised	revise	VERB
cana-3840	67	33	fuzzy	fuzzy	ADJ
cana-3840	67	34	metric	metric	ADJ
cana-3840	67	35	space	space	NOUN
cana-3840	67	36	.	.	PUNCT
cana-3840	68	1	in	in	ADP
cana-3840	68	2	that	that	DET
cana-3840	68	3	case	case	NOUN
cana-3840	68	4	,	,	PUNCT
cana-3840	68	5	if	if	SCONJ
cana-3840	68	6	confusion	confusion	NOUN
cana-3840	68	7	is	be	AUX
cana-3840	68	8	not	not	PART
cana-3840	68	9	possible	possible	ADJ
cana-3840	68	10	,	,	PUNCT
cana-3840	68	11	we	we	PRON
cana-3840	68	12	call	call	VERB
cana-3840	68	13	𝔘	𝔘	PROPN
cana-3840	68	14	a	a	DET
cana-3840	68	15	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	68	16	–	–	PUNCT
cana-3840	68	17	revised	revise	VERB
cana-3840	68	18	fuzzy	fuzzy	ADJ
cana-3840	68	19	metric	metric	ADJ
cana-3840	68	20	space	space	NOUN
cana-3840	68	21	for	for	ADP
cana-3840	68	22	short	short	ADJ
cana-3840	68	23	.	.	PUNCT
cana-3840	69	1	the	the	DET
cana-3840	69	2	following	follow	VERB
cana-3840	69	3	is	be	AUX
cana-3840	69	4	a	a	DET
cana-3840	69	5	well	well	ADV
cana-3840	69	6	-	-	PUNCT
cana-3840	69	7	known	know	VERB
cana-3840	69	8	result	result	NOUN
cana-3840	69	9	.	.	PUNCT
cana-3840	70	1	lemma	lemma	PROPN
cana-3840	70	2	1	1	X
cana-3840	70	3	.	.	PUNCT
cana-3840	71	1	let	let	VERB
cana-3840	71	2	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	71	3	,	,	PUNCT
cana-3840	71	4	𝕓	𝕓	X
cana-3840	71	5	,	,	PUNCT
cana-3840	71	6	−	−	NOUN
cana-3840	71	7	)	)	PUNCT
cana-3840	71	8	is	be	AUX
cana-3840	71	9	non	non	ADJ
cana-3840	71	10	-	-	ADJ
cana-3840	71	11	increasing	increase	VERB
cana-3840	71	12	for	for	ADP
cana-3840	71	13	all	all	DET
cana-3840	71	14	𝕒	𝕒	PROPN
cana-3840	71	15	,	,	PUNCT
cana-3840	71	16	𝕓	𝕓	PROPN
cana-3840	71	17	∈	∈	PROPN
cana-3840	71	18	𝔘.	𝔘.	PROPN
cana-3840	71	19	alexander	alexander	PROPN
cana-3840	71	20	sostack	sostack	NOUN
cana-3840	71	21	in	in	ADP
cana-3840	71	22	that	that	SCONJ
cana-3840	71	23	every	every	DET
cana-3840	71	24	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	71	25	–	–	PUNCT
cana-3840	71	26	revised	revise	VERB
cana-3840	71	27	fuzzy	fuzzy	ADJ
cana-3840	71	28	metric	metric	ADJ
cana-3840	71	29	𝕎	𝕎	PROPN
cana-3840	71	30	on	on	ADP
cana-3840	71	31	𝔘	𝔘	PROPN
cana-3840	71	32	generates	generate	VERB
cana-3840	71	33	a	a	DET
cana-3840	71	34	topology	topology	NOUN
cana-3840	71	35	𝜏𝕎	𝜏𝕎	NOUN
cana-3840	71	36	which	which	PRON
cana-3840	71	37	has	have	VERB
cana-3840	71	38	as	as	ADP
cana-3840	71	39	a	a	DET
cana-3840	71	40	base	base	NOUN
cana-3840	71	41	{	{	PUNCT
cana-3840	71	42	𝔅𝕎(𝕒	𝔅𝕎(𝕒	ADJ
cana-3840	71	43	,	,	PUNCT
cana-3840	71	44	𝕣	𝕣	PROPN
cana-3840	71	45	,	,	PUNCT
cana-3840	71	46	𝓉	𝓉	PROPN
cana-3840	71	47	):	):	PUNCT
cana-3840	71	48	𝕒	𝕒	PROPN
cana-3840	71	49	∈	∈	PROPN
cana-3840	71	50	𝔘	𝔘	PROPN
cana-3840	71	51	,	,	PUNCT
cana-3840	71	52	𝕣(0,1	𝕣(0,1	NOUN
cana-3840	71	53	)	)	PUNCT
cana-3840	71	54	,	,	PUNCT
cana-3840	71	55	𝓉	𝓉	PROPN
cana-3840	71	56	>	>	X
cana-3840	71	57	0	0	NUM
cana-3840	71	58	}	}	PUNCT
cana-3840	71	59	(	(	PUNCT
cana-3840	71	60	3	3	X
cana-3840	71	61	)	)	PUNCT
cana-3840	71	62	were	be	AUX
cana-3840	71	63	𝔅𝕎(𝕒	𝔅𝕎(𝕒	ADJ
cana-3840	71	64	,	,	PUNCT
cana-3840	71	65	𝕣	𝕣	PROPN
cana-3840	71	66	,	,	PUNCT
cana-3840	71	67	𝓉	𝓉	PROPN
cana-3840	71	68	)	)	PUNCT
cana-3840	71	69	=	=	PRON
cana-3840	71	70	{	{	PUNCT
cana-3840	71	71	𝕓	𝕓	PROPN
cana-3840	71	72	∈	∈	PROPN
cana-3840	71	73	𝔘	𝔘	PROPN
cana-3840	71	74	:	:	PUNCT
cana-3840	71	75	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	71	76	,	,	PUNCT
cana-3840	71	77	𝕓	𝕓	X
cana-3840	71	78	,	,	PUNCT
cana-3840	71	79	𝓉	𝓉	PROPN
cana-3840	71	80	)	)	PUNCT
cana-3840	71	81	<	<	X
cana-3840	71	82	𝕣	𝕣	X
cana-3840	71	83	}	}	PUNCT
cana-3840	71	84	,	,	PUNCT
cana-3840	71	85	for	for	ADP
cana-3840	71	86	all	all	DET
cana-3840	71	87	𝕒	𝕒	PRON
cana-3840	71	88	∈	∈	PROPN
cana-3840	71	89	𝔘	𝔘	PROPN
cana-3840	71	90	,	,	PUNCT
cana-3840	71	91	𝕣	𝕣	PRON
cana-3840	71	92	∈	∈	PROPN
cana-3840	71	93	(	(	PUNCT
cana-3840	71	94	0,1	0,1	NOUN
cana-3840	71	95	)	)	PUNCT
cana-3840	71	96	,	,	PUNCT
cana-3840	71	97	and	and	CCONJ
cana-3840	71	98	𝓉	𝓉	PROPN
cana-3840	71	99	>	>	X
cana-3840	71	100	0	0	NUM
cana-3840	71	101	.	.	PUNCT
cana-3840	72	1	(	(	PUNCT
cana-3840	72	2	4	4	X
cana-3840	72	3	)	)	PUNCT
cana-3840	72	4	they	they	PRON
cana-3840	72	5	proved	prove	VERB
cana-3840	72	6	that	that	SCONJ
cana-3840	72	7	for	for	ADP
cana-3840	72	8	each	each	DET
cana-3840	72	9	𝕒	𝕒	PROPN
cana-3840	72	10	∈	∈	PROPN
cana-3840	72	11	𝔘	𝔘	PROPN
cana-3840	72	12	,	,	PUNCT
cana-3840	72	13	the	the	DET
cana-3840	72	14	family	family	NOUN
cana-3840	72	15	{	{	PUNCT
cana-3840	72	16	𝔅𝕎	𝔅𝕎	PROPN
cana-3840	72	17	(	(	PUNCT
cana-3840	72	18	𝕒	𝕒	PROPN
cana-3840	72	19	,	,	PUNCT
cana-3840	72	20	(	(	PUNCT
cana-3840	72	21	1	1	NUM
cana-3840	72	22	𝕟	𝕟	PROPN
cana-3840	72	23	)	)	PUNCT
cana-3840	72	24	,	,	PUNCT
cana-3840	72	25	(	(	PUNCT
cana-3840	72	26	1	1	NUM
cana-3840	72	27	𝕟	𝕟	PROPN
cana-3840	72	28	)	)	PUNCT
cana-3840	72	29	)	)	PUNCT
cana-3840	72	30	:	:	PUNCT
cana-3840	72	31	𝕟	𝕟	PROPN
cana-3840	72	32	∈	∈	PROPN
cana-3840	72	33	ℕ	ℕ	PROPN
cana-3840	72	34	}	}	PUNCT
cana-3840	72	35	is	be	AUX
cana-3840	72	36	a	a	DET
cana-3840	72	37	local	local	ADJ
cana-3840	72	38	base	base	NOUN
cana-3840	72	39	at	at	ADP
cana-3840	72	40	𝕒.	𝕒.	NOUN
cana-3840	72	41	a	a	DET
cana-3840	72	42	sequence	sequence	NOUN
cana-3840	72	43	{	{	PUNCT
cana-3840	72	44	𝕒𝕟	𝕒𝕟	X
cana-3840	72	45	}	}	PUNCT
cana-3840	72	46	in	in	ADP
cana-3840	72	47	{	{	PUNCT
cana-3840	72	48	𝔘	𝔘	PROPN
cana-3840	72	49	,	,	PUNCT
cana-3840	72	50	𝜏𝕎	𝜏𝕎	NOUN
cana-3840	72	51	)	)	PUNCT
cana-3840	72	52	}	}	PUNCT
cana-3840	72	53	converges	converge	VERB
cana-3840	72	54	to	to	ADP
cana-3840	72	55	𝕒	𝕒	SYM
cana-3840	72	56	∈	∈	PROPN
cana-3840	72	57	𝔘	𝔘	NOUN
cana-3840	72	58	if	if	SCONJ
cana-3840	72	59	and	and	CCONJ
cana-3840	72	60	only	only	ADV
cana-3840	72	61	if	if	SCONJ
cana-3840	72	62	log	log	VERB
cana-3840	72	63	𝑛	𝑛	PRON
cana-3840	72	64	⟶	⟶	NOUN
cana-3840	72	65	∞	∞	NUM
cana-3840	72	66	𝕎(𝕒𝕟	𝕎(𝕒𝕟	PROPN
cana-3840	72	67	,	,	PUNCT
cana-3840	72	68	𝕒	𝕒	X
cana-3840	72	69	,	,	PUNCT
cana-3840	72	70	𝓉	𝓉	PROPN
cana-3840	72	71	)	)	PUNCT
cana-3840	72	72	=	=	SYM
cana-3840	72	73	0	0	NUM
cana-3840	72	74	for	for	ADP
cana-3840	72	75	all	all	DET
cana-3840	72	76	𝓉	𝓉	PROPN
cana-3840	72	77	>	>	X
cana-3840	72	78	0	0	X
cana-3840	72	79	.	.	PUNCT
cana-3840	73	1	also	also	ADV
cana-3840	73	2	,	,	PUNCT
cana-3840	73	3	by	by	ADP
cana-3840	73	4	using	use	VERB
cana-3840	73	5	kelley	kelley	PROPN
cana-3840	73	6	metrization	metrization	PROPN
cana-3840	73	7	lemma	lemma	PROPN
cana-3840	74	1	[	[	X
cana-3840	74	2	21	21	NUM
cana-3840	74	3	]	]	PUNCT
cana-3840	74	4	,	,	PUNCT
cana-3840	74	5	they	they	PRON
cana-3840	74	6	also	also	ADV
cana-3840	74	7	proved	prove	VERB
cana-3840	74	8	that	that	SCONJ
cana-3840	74	9	𝜏𝕎	𝜏𝕎	NOUN
cana-3840	74	10	is	be	AUX
cana-3840	74	11	a	a	DET
cana-3840	74	12	metrizable	metrizable	ADJ
cana-3840	74	13	topology	topology	NOUN
cana-3840	74	14	.	.	PUNCT
cana-3840	75	1	3	3	X
cana-3840	75	2	.	.	X
cana-3840	75	3	main	main	ADJ
cana-3840	75	4	results	result	NOUN
cana-3840	75	5	first	first	ADV
cana-3840	75	6	,	,	PUNCT
cana-3840	75	7	we	we	PRON
cana-3840	75	8	introduce	introduce	VERB
cana-3840	75	9	the	the	DET
cana-3840	75	10	concept	concept	NOUN
cana-3840	75	11	of	of	ADP
cana-3840	75	12	a	a	DET
cana-3840	75	13	stratified	stratified	ADJ
cana-3840	75	14	function	function	NOUN
cana-3840	75	15	in	in	ADP
cana-3840	75	16	a	a	DET
cana-3840	75	17	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	75	18	–	–	PUNCT
cana-3840	75	19	revised	revise	VERB
cana-3840	75	20	fuzzy	fuzzy	ADJ
cana-3840	75	21	metric	metric	ADJ
cana-3840	75	22	space	space	NOUN
cana-3840	75	23	.	.	PUNCT
cana-3840	76	1	definition	definition	NOUN
cana-3840	76	2	3	3	X
cana-3840	76	3	.	.	PUNCT
cana-3840	77	1	let	let	VERB
cana-3840	77	2	(	(	PUNCT
cana-3840	77	3	𝔘	𝔘	PROPN
cana-3840	77	4	,	,	PUNCT
cana-3840	77	5	𝕎	𝕎	PROPN
cana-3840	77	6	,	,	PUNCT
cana-3840	77	7	⨁	⨁	PROPN
cana-3840	77	8	)	)	PUNCT
cana-3840	77	9	is	be	AUX
cana-3840	77	10	a	a	DET
cana-3840	77	11	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	77	12	–	–	PUNCT
cana-3840	77	13	revised	revise	VERB
cana-3840	77	14	fuzzy	fuzzy	ADJ
cana-3840	77	15	metric	metric	ADJ
cana-3840	77	16	space	space	NOUN
cana-3840	77	17	.	.	PUNCT
cana-3840	78	1	let	let	VERB
cana-3840	78	2	𝕣	𝕣	PRON
cana-3840	78	3	∈	∈	PROPN
cana-3840	78	4	(	(	PUNCT
cana-3840	78	5	0,1	0,1	NUM
cana-3840	78	6	)	)	PUNCT
cana-3840	78	7	and	and	CCONJ
cana-3840	78	8	𝕒	𝕒	X
cana-3840	78	9	,	,	PUNCT
cana-3840	78	10	𝕓	𝕓	PROPN
cana-3840	78	11	∈	∈	PROPN
cana-3840	78	12	𝔘	𝔘	PROPN
cana-3840	78	13	;	;	PUNCT
cana-3840	78	14	set	set	VERB
cana-3840	78	15	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	78	16	,	,	PUNCT
cana-3840	78	17	𝕓	𝕓	X
cana-3840	78	18	)	)	PUNCT
cana-3840	78	19	=	=	SYM
cana-3840	78	20	𝑠𝑢𝑝{𝓉	𝑠𝑢𝑝{𝓉	VERB
cana-3840	78	21	>	>	X
cana-3840	78	22	0	0	NUM
cana-3840	78	23	:	:	PUNCT
cana-3840	78	24	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	78	25	,	,	PUNCT
cana-3840	78	26	𝕓	𝕓	X
cana-3840	78	27	,	,	PUNCT
cana-3840	78	28	𝓉	𝓉	PROPN
cana-3840	78	29	)	)	PUNCT
cana-3840	78	30	<	<	X
cana-3840	78	31	𝕣	𝕣	X
cana-3840	78	32	}	}	PUNCT
cana-3840	78	33	.	.	PUNCT
cana-3840	79	1	(	(	PUNCT
cana-3840	79	2	5	5	NUM
cana-3840	79	3	)	)	PUNCT
cana-3840	79	4	then	then	ADV
cana-3840	79	5	,	,	PUNCT
cana-3840	79	6	𝕕𝕣	𝕕𝕣	ADJ
cana-3840	79	7	is	be	AUX
cana-3840	79	8	called	call	VERB
cana-3840	79	9	a	a	DET
cana-3840	79	10	𝕣	𝕣	NUM
cana-3840	79	11	-stratified	-stratified	ADJ
cana-3840	79	12	function	function	NOUN
cana-3840	79	13	with	with	ADP
cana-3840	79	14	respect	respect	NOUN
cana-3840	79	15	to	to	ADP
cana-3840	79	16	(	(	PUNCT
cana-3840	79	17	𝔘	𝔘	PROPN
cana-3840	79	18	,	,	PUNCT
cana-3840	79	19	𝕎	𝕎	PROPN
cana-3840	79	20	,	,	PUNCT
cana-3840	79	21	⨁	⨁	PROPN
cana-3840	79	22	)	)	PUNCT
cana-3840	79	23	,	,	PUNCT
cana-3840	79	24	{	{	PUNCT
cana-3840	79	25	𝕕𝕣	𝕕𝕣	ADP
cana-3840	79	26	:	:	SYM
cana-3840	79	27	0	0	NUM
cana-3840	79	28	<	<	X
cana-3840	79	29	𝕣	𝕣	X
cana-3840	79	30	<	<	X
cana-3840	79	31	1	1	NUM
cana-3840	79	32	}	}	PUNCT
cana-3840	79	33	,	,	PUNCT
cana-3840	79	34	the	the	DET
cana-3840	79	35	family	family	NOUN
cana-3840	79	36	of	of	ADP
cana-3840	79	37	stratified	stratified	ADJ
cana-3840	79	38	functions	function	NOUN
cana-3840	79	39	.	.	PUNCT
cana-3840	80	1	to	to	PART
cana-3840	80	2	avoid	avoid	VERB
cana-3840	80	3	the	the	DET
cana-3840	80	4	occurrence	occurrence	NOUN
cana-3840	80	5	of	of	ADP
cana-3840	80	6	the	the	DET
cana-3840	80	7	empty	empty	ADJ
cana-3840	80	8	set	set	NOUN
cana-3840	80	9	,	,	PUNCT
cana-3840	80	10	by	by	ADP
cana-3840	80	11	a	a	DET
cana-3840	80	12	revised	revise	VERB
cana-3840	80	13	fuzzy	fuzzy	ADJ
cana-3840	80	14	metric	metric	NOUN
cana-3840	80	15	in	in	ADP
cana-3840	80	16	the	the	DET
cana-3840	80	17	rest	rest	NOUN
cana-3840	80	18	of	of	ADP
cana-3840	80	19	this	this	DET
cana-3840	80	20	paper	paper	NOUN
cana-3840	80	21	,	,	PUNCT
cana-3840	80	22	we	we	PRON
cana-3840	80	23	mean	mean	VERB
cana-3840	80	24	an	an	DET
cana-3840	80	25	rgv	rgv	NOUN
cana-3840	80	26	-	-	PUNCT
cana-3840	80	27	fuzzy	fuzzy	ADJ
cana-3840	80	28	metric	metric	ADJ
cana-3840	80	29	satisfying	satisfying	NOUN
cana-3840	80	30	(	(	PUNCT
cana-3840	80	31	𝓡𝓖𝓥	𝓡𝓖𝓥	PROPN
cana-3840	80	32	𝟔	𝟔	NUM
cana-3840	80	33	)	)	PUNCT
cana-3840	80	34	log	log	NOUN
cana-3840	80	35	𝑛	𝑛	PRON
cana-3840	80	36	⟶	⟶	NOUN
cana-3840	80	37	∞	∞	PROPN
cana-3840	80	38	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	80	39	,	,	PUNCT
cana-3840	80	40	𝕓	𝕓	X
cana-3840	80	41	,	,	PUNCT
cana-3840	80	42	𝓉	𝓉	PRON
cana-3840	80	43	)	)	PUNCT
cana-3840	80	44	=	=	SYM
cana-3840	80	45	0	0	NUM
cana-3840	80	46	,	,	PUNCT
cana-3840	80	47	∀	∀	X
cana-3840	81	1	𝕒	𝕒	X
cana-3840	81	2	,	,	PUNCT
cana-3840	81	3	𝕓	𝕓	PROPN
cana-3840	81	4	∈	∈	PROPN
cana-3840	81	5	𝔘.	𝔘.	PROPN
cana-3840	81	6	(	(	PUNCT
cana-3840	81	7	6	6	NUM
cana-3840	81	8	)	)	PUNCT
cana-3840	81	9	lemma	lemma	PROPN
cana-3840	81	10	2	2	X
cana-3840	81	11	.	.	PUNCT
cana-3840	82	1	let	let	AUX
cana-3840	82	2	(	(	PUNCT
cana-3840	82	3	𝔘	𝔘	PROPN
cana-3840	82	4	,	,	PUNCT
cana-3840	82	5	𝕎	𝕎	PROPN
cana-3840	82	6	,	,	PUNCT
cana-3840	82	7	⨁	⨁	PROPN
cana-3840	82	8	)	)	PUNCT
cana-3840	82	9	be	be	VERB
cana-3840	82	10	a	a	DET
cana-3840	82	11	revised	revise	VERB
cana-3840	82	12	fuzzy	fuzzy	ADJ
cana-3840	82	13	metric	metric	ADJ
cana-3840	82	14	space	space	NOUN
cana-3840	82	15	,	,	PUNCT
cana-3840	82	16	𝕣	𝕣	PRON
cana-3840	82	17	∈	∈	PROPN
cana-3840	82	18	(	(	PUNCT
cana-3840	82	19	0,1	0,1	NUM
cana-3840	82	20	)	)	PUNCT
cana-3840	82	21	,	,	PUNCT
cana-3840	82	22	𝓉	𝓉	PROPN
cana-3840	82	23	>	>	X
cana-3840	82	24	0	0	PROPN
cana-3840	82	25	,	,	PUNCT
cana-3840	82	26	𝕒	𝕒	X
cana-3840	82	27	,	,	PUNCT
cana-3840	82	28	𝕓	𝕓	X
cana-3840	82	29	∈	∈	PROPN
cana-3840	82	30	𝔘.	𝔘.	PROPN
cana-3840	82	31	then	then	ADV
cana-3840	82	32	,	,	PUNCT
cana-3840	82	33	communications	communication	NOUN
cana-3840	82	34	on	on	ADP
cana-3840	82	35	applied	apply	VERB
cana-3840	82	36	nonlinear	nonlinear	ADJ
cana-3840	82	37	analysis	analysis	NOUN
cana-3840	82	38	issn	issn	NOUN
cana-3840	82	39	:	:	PUNCT
cana-3840	82	40	1074	1074	NUM
cana-3840	82	41	-	-	PUNCT
cana-3840	82	42	133x	133x	NUM
cana-3840	82	43	vol	vol	NOUN
cana-3840	82	44	32	32	NUM
cana-3840	82	45	no	no	NOUN
cana-3840	82	46	.	.	PUNCT
cana-3840	83	1	9s	9s	NUM
cana-3840	83	2	(	(	PUNCT
cana-3840	83	3	2025	2025	NUM
cana-3840	83	4	)	)	PUNCT
cana-3840	83	5	91	91	NUM
cana-3840	83	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	83	7	(	(	PUNCT
cana-3840	83	8	1	1	NUM
cana-3840	83	9	)	)	PUNCT
cana-3840	83	10	for	for	ADP
cana-3840	83	11	any	any	DET
cana-3840	83	12	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	83	13	,	,	PUNCT
cana-3840	83	14	𝕓	𝕓	X
cana-3840	83	15	)	)	PUNCT
cana-3840	83	16	<	<	X
cana-3840	83	17	𝜆	𝜆	X
cana-3840	83	18	,	,	PUNCT
cana-3840	83	19	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	83	20	,	,	PUNCT
cana-3840	83	21	𝕓	𝕓	X
cana-3840	83	22	,	,	PUNCT
cana-3840	83	23	𝜆	𝜆	NOUN
cana-3840	83	24	)	)	PUNCT
cana-3840	83	25	≤	≤	NUM
cana-3840	83	26	𝕣	𝕣	PRON
cana-3840	83	27	(	(	PUNCT
cana-3840	83	28	2	2	NUM
cana-3840	83	29	)	)	PUNCT
cana-3840	83	30	the	the	DET
cana-3840	83	31	function	function	NOUN
cana-3840	83	32	𝕕𝕣	𝕕𝕣	ADP
cana-3840	83	33	is	be	AUX
cana-3840	83	34	non	non	ADJ
cana-3840	83	35	-	-	ADJ
cana-3840	83	36	decreasing	decrease	VERB
cana-3840	83	37	with	with	ADP
cana-3840	83	38	respect	respect	NOUN
cana-3840	83	39	to	to	ADP
cana-3840	83	40	𝕣	𝕣	PRON
cana-3840	83	41	∈	∈	PROPN
cana-3840	83	42	(	(	PUNCT
cana-3840	83	43	0,1	0,1	NUM
cana-3840	83	44	)	)	PUNCT
cana-3840	83	45	(	(	PUNCT
cana-3840	83	46	3	3	NUM
cana-3840	83	47	)	)	PUNCT
cana-3840	83	48	𝔅𝕎(𝕒	𝔅𝕎(𝕒	ADJ
cana-3840	83	49	,	,	PUNCT
cana-3840	83	50	𝕣	𝕣	PROPN
cana-3840	83	51	,	,	PUNCT
cana-3840	83	52	𝓉	𝓉	PROPN
cana-3840	83	53	)	)	PUNCT
cana-3840	83	54	=	=	SYM
cana-3840	83	55	ℕ𝕎(𝕒	ℕ𝕎(𝕒	ADJ
cana-3840	83	56	,	,	PUNCT
cana-3840	83	57	𝓉	𝓉	PROPN
cana-3840	83	58	)	)	PUNCT
cana-3840	83	59	,	,	PUNCT
cana-3840	83	60	were	be	AUX
cana-3840	83	61	ℕ𝕣(𝕒	ℕ𝕣(𝕒	NOUN
cana-3840	83	62	,	,	PUNCT
cana-3840	83	63	𝓉	𝓉	PRON
cana-3840	83	64	)	)	PUNCT
cana-3840	83	65	=	=	PUNCT
cana-3840	83	66	𝕓	𝕓	PUNCT
cana-3840	83	67	∈	∈	PROPN
cana-3840	83	68	𝔘	𝔘	PROPN
cana-3840	83	69	:	:	PUNCT
cana-3840	83	70	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	83	71	,	,	PUNCT
cana-3840	83	72	𝕓	𝕓	X
cana-3840	83	73	)	)	PUNCT
cana-3840	83	74	<	<	X
cana-3840	83	75	𝑡.	𝑡.	NOUN
cana-3840	83	76	(	(	PUNCT
cana-3840	83	77	7	7	NUM
cana-3840	83	78	)	)	PUNCT
cana-3840	83	79	(	(	PUNCT
cana-3840	83	80	4	4	X
cana-3840	83	81	)	)	PUNCT
cana-3840	83	82	the	the	DET
cana-3840	83	83	function	function	NOUN
cana-3840	83	84	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	83	85	,	,	PUNCT
cana-3840	83	86	𝕓	𝕓	X
cana-3840	83	87	,	,	PUNCT
cana-3840	83	88	−	−	NOUN
cana-3840	83	89	)	)	PUNCT
cana-3840	83	90	is	be	AUX
cana-3840	83	91	strictly	strictly	ADV
cana-3840	83	92	non	non	ADJ
cana-3840	83	93	-	-	ADJ
cana-3840	83	94	increasing	increase	VERB
cana-3840	83	95	for	for	ADP
cana-3840	83	96	the	the	DET
cana-3840	83	97	fixed	fix	VERB
cana-3840	83	98	points	point	NOUN
cana-3840	83	99	𝕒	𝕒	NOUN
cana-3840	83	100	,	,	PUNCT
cana-3840	83	101	𝕓	𝕓	PROPN
cana-3840	83	102	∈	∈	PROPN
cana-3840	83	103	𝔘	𝔘	PROPN
cana-3840	83	104	,	,	PUNCT
cana-3840	83	105	if	if	SCONJ
cana-3840	83	106	and	and	CCONJ
cana-3840	83	107	only	only	ADV
cana-3840	83	108	if	if	SCONJ
cana-3840	83	109	for	for	ADP
cana-3840	83	110	any	any	DET
cana-3840	83	111	𝕣	𝕣	PRON
cana-3840	83	112	∈	∈	PROPN
cana-3840	83	113	(	(	PUNCT
cana-3840	83	114	0,1	0,1	NOUN
cana-3840	83	115	)	)	PUNCT
cana-3840	83	116	,	,	PUNCT
cana-3840	83	117	𝕕𝕣(𝕒	𝕕𝕣(𝕒	X
cana-3840	83	118	,	,	PUNCT
cana-3840	83	119	𝕓	𝕓	X
cana-3840	83	120	)	)	PUNCT
cana-3840	83	121	=	=	SYM
cana-3840	83	122	𝑠𝑢𝑝{𝓉	𝑠𝑢𝑝{𝓉	VERB
cana-3840	83	123	>	>	X
cana-3840	83	124	0	0	NUM
cana-3840	83	125	:	:	PUNCT
cana-3840	83	126	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	83	127	,	,	PUNCT
cana-3840	83	128	𝕓	𝕓	X
cana-3840	83	129	,	,	PUNCT
cana-3840	83	130	𝓉	𝓉	PROPN
cana-3840	83	131	)	)	PUNCT
cana-3840	83	132	<	<	X
cana-3840	83	133	𝕣	𝕣	X
cana-3840	83	134	}	}	PUNCT
cana-3840	83	135	.	.	PUNCT
cana-3840	84	1	(	(	PUNCT
cana-3840	84	2	8)	8)	NUM
cana-3840	84	3	proof	proof	NOUN
cana-3840	84	4	(	(	PUNCT
cana-3840	84	5	1	1	X
cana-3840	84	6	)	)	PUNCT
cana-3840	84	7	let	let	VERB
cana-3840	84	8	𝜆	𝜆	PRON
cana-3840	84	9	>	>	X
cana-3840	84	10	𝕕𝕣(𝕒	𝕕𝕣(𝕒	SYM
cana-3840	84	11	,	,	PUNCT
cana-3840	84	12	𝕓	𝕓	X
cana-3840	84	13	)	)	PUNCT
cana-3840	84	14	.	.	PUNCT
cana-3840	85	1	from	from	ADP
cana-3840	85	2	definition	definition	NOUN
cana-3840	85	3	3	3	NUM
cana-3840	85	4	,	,	PUNCT
cana-3840	85	5	there	there	PRON
cana-3840	85	6	exists	exist	VERB
cana-3840	85	7	0	0	PUNCT
cana-3840	85	8	<	<	X
cana-3840	86	1	𝓉	𝓉	PROPN
cana-3840	86	2	<	<	X
cana-3840	86	3	𝜆	𝜆	X
cana-3840	86	4	such	such	ADJ
cana-3840	86	5	that	that	DET
cana-3840	86	6	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	86	7	,	,	PUNCT
cana-3840	86	8	𝕓	𝕓	X
cana-3840	86	9	,	,	PUNCT
cana-3840	86	10	𝓉	𝓉	PROPN
cana-3840	86	11	)	)	PUNCT
cana-3840	86	12	≤	≤	NUM
cana-3840	87	1	𝕣.	𝕣.	NOUN
cana-3840	88	1	so	so	ADV
cana-3840	88	2	,	,	PUNCT
cana-3840	88	3	𝕎(𝕒	𝕎(𝕒	X
cana-3840	88	4	,	,	PUNCT
cana-3840	88	5	𝕓	𝕓	X
cana-3840	88	6	,	,	PUNCT
cana-3840	88	7	𝜆	𝜆	NOUN
cana-3840	88	8	)	)	PUNCT
cana-3840	88	9	≤	≤	ADJ
cana-3840	88	10	𝕣.	𝕣.	NOUN
cana-3840	88	11	(	(	PUNCT
cana-3840	88	12	2	2	X
cana-3840	88	13	)	)	PUNCT
cana-3840	88	14	it	it	PRON
cana-3840	88	15	follows	follow	VERB
cana-3840	88	16	from	from	ADP
cana-3840	88	17	lemma	lemma	PROPN
cana-3840	88	18	1	1	NUM
cana-3840	88	19	directly	directly	ADV
cana-3840	88	20	.	.	PUNCT
cana-3840	89	1	(	(	PUNCT
cana-3840	89	2	3	3	X
cana-3840	89	3	)	)	PUNCT
cana-3840	89	4	let	let	VERB
cana-3840	89	5	𝕓	𝕓	PRON
cana-3840	89	6	∈	∈	PROPN
cana-3840	89	7	𝔅𝕎(𝕒	𝔅𝕎(𝕒	ADJ
cana-3840	89	8	,	,	PUNCT
cana-3840	89	9	𝕣	𝕣	PROPN
cana-3840	89	10	,	,	PUNCT
cana-3840	89	11	𝓉	𝓉	PROPN
cana-3840	89	12	)	)	PUNCT
cana-3840	89	13	,	,	PUNCT
cana-3840	89	14	that	that	ADV
cana-3840	89	15	is	is	ADV
cana-3840	89	16	,	,	PUNCT
cana-3840	89	17	𝕎(𝕒	𝕎(𝕒	X
cana-3840	89	18	,	,	PUNCT
cana-3840	89	19	𝕓	𝕓	X
cana-3840	89	20	,	,	PUNCT
cana-3840	89	21	𝓉	𝓉	PROPN
cana-3840	89	22	)	)	PUNCT
cana-3840	89	23	≤	≤	NOUN
cana-3840	89	24	𝕣.	𝕣.	NOUN
cana-3840	89	25	since	since	SCONJ
cana-3840	89	26	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	89	27	,	,	PUNCT
cana-3840	89	28	𝕓	𝕓	X
cana-3840	89	29	,	,	PUNCT
cana-3840	89	30	−	−	NOUN
cana-3840	89	31	)	)	PUNCT
cana-3840	89	32	is	be	AUX
cana-3840	89	33	continuous	continuous	ADJ
cana-3840	89	34	and	and	CCONJ
cana-3840	89	35	non	non	ADJ
cana-3840	89	36	-	-	ADJ
cana-3840	89	37	increasing	increase	VERB
cana-3840	89	38	,	,	PUNCT
cana-3840	89	39	there	there	PRON
cana-3840	89	40	exists	exist	VERB
cana-3840	89	41	0	0	PUNCT
cana-3840	89	42	<	<	X
cana-3840	89	43	𝓉1	𝓉1	X
cana-3840	89	44	<	<	X
cana-3840	89	45	𝓉	𝓉	PROPN
cana-3840	89	46	such	such	ADJ
cana-3840	89	47	that	that	DET
cana-3840	89	48	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	89	49	,	,	PUNCT
cana-3840	89	50	𝕓	𝕓	X
cana-3840	89	51	,	,	PUNCT
cana-3840	89	52	𝓉1	𝓉1	INTJ
cana-3840	89	53	)	)	PUNCT
cana-3840	89	54	≤	≤	NOUN
cana-3840	89	55	𝕣.	𝕣.	NOUN
cana-3840	89	56	from	from	ADP
cana-3840	89	57	(	(	PUNCT
cana-3840	89	58	5	5	NUM
cana-3840	89	59	)	)	PUNCT
cana-3840	89	60	,	,	PUNCT
cana-3840	89	61	we	we	PRON
cana-3840	89	62	know	know	VERB
cana-3840	89	63	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	89	64	,	,	PUNCT
cana-3840	89	65	𝕓	𝕓	X
cana-3840	89	66	)	)	PUNCT
cana-3840	89	67	≤	≤	NUM
cana-3840	89	68	𝓉1	𝓉1	VERB
cana-3840	89	69	<	<	X
cana-3840	89	70	𝑡.	𝑡.	NOUN
cana-3840	89	71	so	so	ADV
cana-3840	89	72	,	,	PUNCT
cana-3840	89	73	𝕓	𝕓	X
cana-3840	89	74	∈	∈	PROPN
cana-3840	89	75	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	89	76	,	,	PUNCT
cana-3840	89	77	𝓉	𝓉	PROPN
cana-3840	89	78	)	)	PUNCT
cana-3840	89	79	.	.	PUNCT
cana-3840	90	1	from	from	ADP
cana-3840	90	2	the	the	DET
cana-3840	90	3	arbitrariness	arbitrariness	NOUN
cana-3840	90	4	of	of	ADP
cana-3840	90	5	𝕓	𝕓	NUM
cana-3840	90	6	,	,	PUNCT
cana-3840	90	7	we	we	PRON
cana-3840	90	8	know	know	VERB
cana-3840	90	9	that	that	SCONJ
cana-3840	90	10	𝔅𝕎(𝕒	𝔅𝕎(𝕒	ADJ
cana-3840	90	11	,	,	PUNCT
cana-3840	90	12	𝕣	𝕣	PROPN
cana-3840	90	13	,	,	PUNCT
cana-3840	90	14	𝓉	𝓉	PROPN
cana-3840	90	15	)	)	PUNCT
cana-3840	90	16	⊆	⊆	NUM
cana-3840	90	17	ℕ𝕣(𝕒	ℕ𝕣(𝕒	NOUN
cana-3840	90	18	,	,	PUNCT
cana-3840	90	19	𝓉	𝓉	PROPN
cana-3840	90	20	)	)	PUNCT
cana-3840	90	21	.	.	PUNCT
cana-3840	91	1	(	(	PUNCT
cana-3840	91	2	9	9	X
cana-3840	91	3	)	)	PUNCT
cana-3840	91	4	let	let	VERB
cana-3840	91	5	𝕔	𝕔	DET
cana-3840	91	6	∈	∈	NOUN
cana-3840	91	7	ℕ𝕣(𝕒	ℕ𝕣(𝕒	NOUN
cana-3840	91	8	,	,	PUNCT
cana-3840	91	9	𝓉	𝓉	PROPN
cana-3840	91	10	)	)	PUNCT
cana-3840	91	11	;	;	PUNCT
cana-3840	91	12	then	then	ADV
cana-3840	91	13	,	,	PUNCT
cana-3840	91	14	𝕕𝕣(𝕒	𝕕𝕣(𝕒	X
cana-3840	91	15	,	,	PUNCT
cana-3840	91	16	𝕔	𝕔	NOUN
cana-3840	91	17	)	)	PUNCT
cana-3840	91	18	<	<	X
cana-3840	91	19	𝓉.	𝓉.	NOUN
cana-3840	91	20	from	from	ADP
cana-3840	91	21	(	(	PUNCT
cana-3840	91	22	5	5	NUM
cana-3840	91	23	)	)	PUNCT
cana-3840	91	24	,	,	PUNCT
cana-3840	91	25	there	there	PRON
cana-3840	91	26	exists	exist	VERB
cana-3840	91	27	0	0	PUNCT
cana-3840	91	28	<	<	X
cana-3840	91	29	𝓉2	𝓉2	X
cana-3840	91	30	<	<	X
cana-3840	91	31	𝓉	𝓉	PROPN
cana-3840	91	32	such	such	ADJ
cana-3840	91	33	that	that	PRON
cana-3840	91	34	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	91	35	,	,	PUNCT
cana-3840	91	36	𝕓	𝕓	X
cana-3840	91	37	,	,	PUNCT
cana-3840	91	38	𝓉2	𝓉2	NOUN
cana-3840	91	39	)	)	PUNCT
cana-3840	91	40	≤	≤	NOUN
cana-3840	92	1	𝕣.	𝕣.	NOUN
cana-3840	93	1	therefore	therefore	ADV
cana-3840	93	2	,	,	PUNCT
cana-3840	93	3	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	93	4	,	,	PUNCT
cana-3840	93	5	𝕔	𝕔	PROPN
cana-3840	93	6	,	,	PUNCT
cana-3840	93	7	𝓉	𝓉	PROPN
cana-3840	93	8	)	)	PUNCT
cana-3840	93	9	≤	≤	NOUN
cana-3840	93	10	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	93	11	,	,	PUNCT
cana-3840	93	12	𝕓	𝕓	X
cana-3840	93	13	,	,	PUNCT
cana-3840	93	14	𝓉2	𝓉2	NOUN
cana-3840	93	15	)	)	PUNCT
cana-3840	93	16	≤	≤	NOUN
cana-3840	94	1	𝕣.	𝕣.	NOUN
cana-3840	95	1	so	so	ADV
cana-3840	95	2	,	,	PUNCT
cana-3840	95	3	𝕔	𝕔	DET
cana-3840	95	4	∈	∈	PROPN
cana-3840	95	5	𝔅𝕎(𝕒	𝔅𝕎(𝕒	ADJ
cana-3840	95	6	,	,	PUNCT
cana-3840	95	7	𝕣	𝕣	PROPN
cana-3840	95	8	,	,	PUNCT
cana-3840	95	9	𝓉	𝓉	PROPN
cana-3840	95	10	)	)	PUNCT
cana-3840	95	11	.	.	PUNCT
cana-3840	96	1	from	from	ADP
cana-3840	96	2	the	the	DET
cana-3840	96	3	arbitrariness	arbitrariness	NOUN
cana-3840	96	4	of	of	ADP
cana-3840	96	5	𝕔	𝕔	PROPN
cana-3840	96	6	,	,	PUNCT
cana-3840	96	7	we	we	PRON
cana-3840	96	8	know	know	VERB
cana-3840	96	9	that	that	SCONJ
cana-3840	96	10	ℕ𝕣(𝕒	ℕ𝕣(𝕒	NOUN
cana-3840	96	11	,	,	PUNCT
cana-3840	96	12	𝓉	𝓉	PROPN
cana-3840	96	13	)	)	PUNCT
cana-3840	97	1	⊆	⊆	NUM
cana-3840	97	2	𝔅𝕎(𝕒	𝔅𝕎(𝕒	ADJ
cana-3840	97	3	,	,	PUNCT
cana-3840	97	4	𝕣	𝕣	PROPN
cana-3840	97	5	,	,	PUNCT
cana-3840	97	6	𝓉	𝓉	PROPN
cana-3840	97	7	)	)	PUNCT
cana-3840	97	8	.	.	PUNCT
cana-3840	98	1	(	(	PUNCT
cana-3840	98	2	4	4	X
cana-3840	98	3	)	)	PUNCT
cana-3840	98	4	suppose	suppose	VERB
cana-3840	98	5	that	that	SCONJ
cana-3840	98	6	(	(	PUNCT
cana-3840	98	7	8)	8)	NUM
cana-3840	98	8	holds	hold	VERB
cana-3840	98	9	;	;	PUNCT
cana-3840	98	10	however	however	ADV
cana-3840	98	11	,	,	PUNCT
cana-3840	98	12	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	98	13	,	,	PUNCT
cana-3840	98	14	𝕓	𝕓	X
cana-3840	98	15	,	,	PUNCT
cana-3840	98	16	−	−	NOUN
cana-3840	98	17	)	)	PUNCT
cana-3840	98	18	is	be	AUX
cana-3840	98	19	not	not	PART
cana-3840	98	20	strictly	strictly	ADV
cana-3840	98	21	decreasing	decrease	VERB
cana-3840	98	22	.	.	PUNCT
cana-3840	99	1	then	then	ADV
cana-3840	99	2	,	,	PUNCT
cana-3840	99	3	there	there	PRON
cana-3840	99	4	exist	exist	VERB
cana-3840	99	5	𝓉1	𝓉1	NOUN
cana-3840	99	6	,	,	PUNCT
cana-3840	99	7	𝓉2	𝓉2	NOUN
cana-3840	99	8	∈	∈	PROPN
cana-3840	99	9	{	{	PUNCT
cana-3840	99	10	𝓉	𝓉	PROPN
cana-3840	99	11	>	>	X
cana-3840	99	12	0	0	NUM
cana-3840	99	13	:	:	SYM
cana-3840	99	14	0	0	NUM
cana-3840	99	15	<	<	X
cana-3840	99	16	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	99	17	,	,	PUNCT
cana-3840	99	18	𝕓	𝕓	X
cana-3840	99	19	,	,	PUNCT
cana-3840	99	20	𝓉	𝓉	PRON
cana-3840	99	21	)	)	PUNCT
cana-3840	99	22	<	<	X
cana-3840	99	23	1	1	X
cana-3840	99	24	}	}	PUNCT
cana-3840	99	25	such	such	ADJ
cana-3840	99	26	that	that	SCONJ
cana-3840	99	27	𝓉1	𝓉1	VERB
cana-3840	99	28	<	<	X
cana-3840	99	29	𝓉2	𝓉2	NOUN
cana-3840	99	30	and	and	CCONJ
cana-3840	99	31	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	99	32	,	,	PUNCT
cana-3840	99	33	𝕓	𝕓	X
cana-3840	99	34	,	,	PUNCT
cana-3840	99	35	−	−	NOUN
cana-3840	99	36	)	)	PUNCT
cana-3840	99	37	≡	≡	PROPN
cana-3840	99	38	𝕣0	𝕣0	VERB
cana-3840	99	39	on	on	ADP
cana-3840	99	40	[	[	X
cana-3840	99	41	𝓉1	𝓉1	X
cana-3840	99	42	,	,	PUNCT
cana-3840	99	43	𝓉2	𝓉2	NOUN
cana-3840	99	44	]	]	PUNCT
cana-3840	99	45	.	.	PUNCT
cana-3840	100	1	thus	thus	ADV
cana-3840	100	2	,	,	PUNCT
cana-3840	100	3	𝑠𝑢𝑝{𝓉	𝑠𝑢𝑝{𝓉	VERB
cana-3840	100	4	>	>	X
cana-3840	100	5	0	0	NUM
cana-3840	100	6	:	:	PUNCT
cana-3840	100	7	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	100	8	,	,	PUNCT
cana-3840	100	9	𝕓	𝕓	X
cana-3840	100	10	,	,	PUNCT
cana-3840	100	11	𝓉	𝓉	PROPN
cana-3840	100	12	)	)	PUNCT
cana-3840	100	13	≥	≥	NOUN
cana-3840	100	14	𝕣0	𝕣0	PROPN
cana-3840	100	15	}	}	PUNCT
cana-3840	100	16	≥	≥	PROPN
cana-3840	100	17	𝓉2	𝓉2	PROPN
cana-3840	100	18	>	>	X
cana-3840	100	19	𝓉1	𝓉1	X
cana-3840	100	20	≥	≥	X
cana-3840	100	21	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	X
cana-3840	100	22	>	>	X
cana-3840	100	23	0	0	NUM
cana-3840	100	24	:	:	PUNCT
cana-3840	100	25	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	100	26	,	,	PUNCT
cana-3840	100	27	𝕓	𝕓	X
cana-3840	100	28	,	,	PUNCT
cana-3840	100	29	𝓉	𝓉	PROPN
cana-3840	100	30	)	)	PUNCT
cana-3840	100	31	≤	≤	NOUN
cana-3840	100	32	𝕣0	𝕣0	PROPN
cana-3840	100	33	}	}	PUNCT
cana-3840	100	34	.	.	PUNCT
cana-3840	101	1	(	(	PUNCT
cana-3840	101	2	10	10	NUM
cana-3840	101	3	)	)	PUNCT
cana-3840	101	4	it	it	PRON
cana-3840	101	5	is	be	AUX
cana-3840	101	6	easy	easy	ADJ
cana-3840	101	7	to	to	PART
cana-3840	101	8	see	see	VERB
cana-3840	101	9	that	that	SCONJ
cana-3840	101	10	𝑠𝑢𝑝{𝓉	𝑠𝑢𝑝{𝓉	VERB
cana-3840	101	11	>	>	X
cana-3840	101	12	0	0	NUM
cana-3840	101	13	:	:	PUNCT
cana-3840	101	14	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	101	15	,	,	PUNCT
cana-3840	101	16	𝕓	𝕓	X
cana-3840	101	17	,	,	PUNCT
cana-3840	101	18	𝓉	𝓉	PROPN
cana-3840	101	19	)	)	PUNCT
cana-3840	101	20	≥	≥	NOUN
cana-3840	101	21	𝕣0	𝕣0	ADJ
cana-3840	101	22	}	}	PUNCT
cana-3840	101	23	=	=	PUNCT
cana-3840	101	24	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	X
cana-3840	101	25	>	>	X
cana-3840	101	26	0	0	NUM
cana-3840	101	27	:	:	PUNCT
cana-3840	101	28	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	101	29	,	,	PUNCT
cana-3840	101	30	𝕓	𝕓	X
cana-3840	101	31	,	,	PUNCT
cana-3840	101	32	𝓉	𝓉	PRON
cana-3840	101	33	)	)	PUNCT
cana-3840	101	34	<	<	X
cana-3840	101	35	𝕣0	𝕣0	PROPN
cana-3840	101	36	}	}	PUNCT
cana-3840	101	37	.	.	PUNCT
cana-3840	102	1	(	(	PUNCT
cana-3840	102	2	11	11	NUM
cana-3840	102	3	)	)	PUNCT
cana-3840	102	4	therefore	therefore	ADV
cana-3840	102	5	,	,	PUNCT
cana-3840	102	6	𝕕𝕣0	𝕕𝕣0	X
cana-3840	102	7	(	(	PUNCT
cana-3840	102	8	𝕒	𝕒	X
cana-3840	102	9	,	,	PUNCT
cana-3840	102	10	𝕓	𝕓	X
cana-3840	102	11	)	)	PUNCT
cana-3840	102	12	>	>	X
cana-3840	102	13	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	PROPN
cana-3840	102	14	>	>	X
cana-3840	102	15	0	0	NUM
cana-3840	102	16	:	:	PUNCT
cana-3840	102	17	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	102	18	,	,	PUNCT
cana-3840	102	19	𝕓	𝕓	X
cana-3840	102	20	,	,	PUNCT
cana-3840	102	21	𝓉	𝓉	PRON
cana-3840	102	22	)	)	PUNCT
cana-3840	102	23	<	<	X
cana-3840	102	24	𝕣0	𝕣0	PROPN
cana-3840	102	25	}	}	PUNCT
cana-3840	102	26	,	,	PUNCT
cana-3840	102	27	which	which	PRON
cana-3840	102	28	conflicts	conflict	VERB
cana-3840	102	29	with	with	ADP
cana-3840	102	30	(	(	PUNCT
cana-3840	102	31	8)	8)	NUM
cana-3840	102	32	.	.	PUNCT
cana-3840	103	1	conversely	conversely	ADV
cana-3840	103	2	,	,	PUNCT
cana-3840	103	3	suppose	suppose	VERB
cana-3840	103	4	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	103	5	,	,	PUNCT
cana-3840	103	6	𝕓	𝕓	X
cana-3840	103	7	,	,	PUNCT
cana-3840	103	8	−	−	NOUN
cana-3840	103	9	)	)	PUNCT
cana-3840	103	10	is	be	AUX
cana-3840	103	11	strictly	strictly	ADV
cana-3840	103	12	decreasing	decrease	VERB
cana-3840	103	13	.	.	PUNCT
cana-3840	104	1	let	let	VERB
cana-3840	104	2	,	,	PUNCT
cana-3840	104	3	𝓉0	𝓉0	PROPN
cana-3840	104	4	=	=	SYM
cana-3840	104	5	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	104	6	,	,	PUNCT
cana-3840	104	7	𝕓	𝕓	X
cana-3840	104	8	)	)	PUNCT
cana-3840	104	9	=	=	PUNCT
cana-3840	104	10	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	X
cana-3840	104	11	>	>	X
cana-3840	104	12	0	0	NUM
cana-3840	104	13	:	:	PUNCT
cana-3840	104	14	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	104	15	,	,	PUNCT
cana-3840	104	16	𝕓	𝕓	X
cana-3840	104	17	,	,	PUNCT
cana-3840	104	18	𝓉	𝓉	PROPN
cana-3840	104	19	)	)	PUNCT
cana-3840	104	20	<	<	X
cana-3840	104	21	𝑟	𝑟	NOUN
cana-3840	104	22	}	}	PUNCT
cana-3840	104	23	.	.	PUNCT
cana-3840	105	1	(	(	PUNCT
cana-3840	105	2	12	12	NUM
cana-3840	105	3	)	)	PUNCT
cana-3840	105	4	obviously	obviously	ADV
cana-3840	105	5	,	,	PUNCT
cana-3840	105	6	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	X
cana-3840	105	7	>	>	X
cana-3840	105	8	0	0	NUM
cana-3840	105	9	:	:	PUNCT
cana-3840	105	10	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	105	11	,	,	PUNCT
cana-3840	105	12	𝕓	𝕓	X
cana-3840	105	13	,	,	PUNCT
cana-3840	105	14	𝓉	𝓉	PROPN
cana-3840	105	15	)	)	PUNCT
cana-3840	105	16	≤	≤	NOUN
cana-3840	105	17	𝕣	𝕣	X
cana-3840	105	18	}	}	PUNCT
cana-3840	105	19	≤	≤	NUM
cana-3840	105	20	𝓉0	𝓉0	ADJ
cana-3840	105	21	.	.	PUNCT
cana-3840	106	1	if	if	SCONJ
cana-3840	106	2	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	PROPN
cana-3840	106	3	>	>	X
cana-3840	106	4	0	0	NUM
cana-3840	106	5	:	:	PUNCT
cana-3840	106	6	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	106	7	,	,	PUNCT
cana-3840	106	8	𝕓	𝕓	X
cana-3840	106	9	,	,	PUNCT
cana-3840	106	10	𝓉	𝓉	PROPN
cana-3840	106	11	)	)	PUNCT
cana-3840	106	12	<	<	X
cana-3840	106	13	𝑟	𝑟	X
cana-3840	106	14	}	}	PUNCT
cana-3840	106	15	<	<	X
cana-3840	106	16	𝓉0	𝓉0	ADJ
cana-3840	106	17	,	,	PUNCT
cana-3840	106	18	then	then	ADV
cana-3840	106	19	there	there	PRON
cana-3840	106	20	is	be	VERB
cana-3840	106	21	0	0	NUM
cana-3840	106	22	<	<	X
cana-3840	106	23	𝓉2	𝓉2	NOUN
cana-3840	106	24	<	<	X
cana-3840	106	25	𝓉1	𝓉1	PRON
cana-3840	106	26	such	such	ADJ
cana-3840	106	27	that	that	DET
cana-3840	106	28	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	106	29	,	,	PUNCT
cana-3840	106	30	𝕓	𝕓	X
cana-3840	106	31	,	,	PUNCT
cana-3840	106	32	𝓉2	𝓉2	NOUN
cana-3840	106	33	)	)	PUNCT
cana-3840	106	34	≤	≤	NOUN
cana-3840	106	35	𝕣	𝕣	NOUN
cana-3840	106	36	,	,	PUNCT
cana-3840	106	37	so	so	ADV
cana-3840	106	38	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	106	39	,	,	PUNCT
cana-3840	106	40	𝕓	𝕓	X
cana-3840	106	41	,	,	PUNCT
cana-3840	106	42	𝓉2	𝓉2	NOUN
cana-3840	106	43	)	)	PUNCT
cana-3840	106	44	<	<	X
cana-3840	106	45	𝑟.	𝑟.	NOUN
cana-3840	106	46	since	since	SCONJ
cana-3840	106	47	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	106	48	,	,	PUNCT
cana-3840	106	49	𝕓	𝕓	X
cana-3840	106	50	,	,	PUNCT
cana-3840	106	51	−	−	NOUN
cana-3840	106	52	)	)	PUNCT
cana-3840	106	53	is	be	AUX
cana-3840	106	54	right	right	ADV
cana-3840	106	55	continuous	continuous	ADJ
cana-3840	106	56	at	at	ADP
cana-3840	106	57	𝓉0	𝓉0	ADJ
cana-3840	106	58	,	,	PUNCT
cana-3840	106	59	there	there	PRON
cana-3840	106	60	is	be	VERB
cana-3840	106	61	𝛿	𝛿	PROPN
cana-3840	106	62	>	>	X
cana-3840	106	63	0	0	NUM
cana-3840	106	64	such	such	ADJ
cana-3840	106	65	that	that	DET
cana-3840	106	66	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	106	67	,	,	PUNCT
cana-3840	106	68	𝕓	𝕓	X
cana-3840	106	69	,	,	PUNCT
cana-3840	106	70	𝓉0	𝓉0	ADJ
cana-3840	106	71	−	−	PROPN
cana-3840	106	72	𝛿	𝛿	NOUN
cana-3840	106	73	)	)	PUNCT
cana-3840	106	74	<	<	X
cana-3840	106	75	𝑟	𝑟	NOUN
cana-3840	106	76	,	,	PUNCT
cana-3840	106	77	which	which	PRON
cana-3840	106	78	conflicts	conflict	VERB
cana-3840	106	79	with	with	ADP
cana-3840	106	80	the	the	DET
cana-3840	106	81	definition	definition	NOUN
cana-3840	106	82	of	of	ADP
cana-3840	106	83	𝓉0	𝓉0	PROPN
cana-3840	106	84	.	.	PUNCT
cana-3840	107	1	thus	thus	ADV
cana-3840	107	2	,	,	PUNCT
cana-3840	107	3	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	X
cana-3840	107	4	>	>	X
cana-3840	107	5	0	0	NUM
cana-3840	107	6	:	:	PUNCT
cana-3840	107	7	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	107	8	,	,	PUNCT
cana-3840	107	9	𝕓	𝕓	X
cana-3840	107	10	,	,	PUNCT
cana-3840	107	11	𝓉	𝓉	PROPN
cana-3840	107	12	)	)	PUNCT
cana-3840	107	13	≤	≤	NUM
cana-3840	107	14	𝕣	𝕣	X
cana-3840	107	15	}	}	PUNCT
cana-3840	107	16	≥	≥	NOUN
cana-3840	107	17	𝓉0	𝓉0	ADJ
cana-3840	107	18	.	.	PUNCT
cana-3840	108	1	so	so	ADV
cana-3840	108	2	,	,	PUNCT
cana-3840	108	3	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	108	4	,	,	PUNCT
cana-3840	108	5	𝕓	𝕓	X
cana-3840	108	6	)	)	PUNCT
cana-3840	108	7	=	=	SYM
cana-3840	108	8	𝓉0	𝓉0	ADJ
cana-3840	108	9	=	=	SYM
cana-3840	108	10	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	X
cana-3840	108	11	>	>	X
cana-3840	108	12	0	0	NUM
cana-3840	108	13	:	:	PUNCT
cana-3840	108	14	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	108	15	,	,	PUNCT
cana-3840	108	16	𝕓	𝕓	X
cana-3840	108	17	,	,	PUNCT
cana-3840	108	18	𝓉	𝓉	PROPN
cana-3840	108	19	)	)	PUNCT
cana-3840	108	20	≤	≤	NOUN
cana-3840	108	21	𝕣	𝕣	X
cana-3840	108	22	}	}	PUNCT
cana-3840	108	23	.	.	PUNCT
cana-3840	109	1	(	(	PUNCT
cana-3840	109	2	13	13	NUM
cana-3840	109	3	)	)	PUNCT
cana-3840	109	4	communications	communication	NOUN
cana-3840	109	5	on	on	ADP
cana-3840	109	6	applied	apply	VERB
cana-3840	109	7	nonlinear	nonlinear	ADJ
cana-3840	109	8	analysis	analysis	NOUN
cana-3840	109	9	issn	issn	NOUN
cana-3840	109	10	:	:	PUNCT
cana-3840	109	11	1074	1074	NUM
cana-3840	109	12	-	-	PUNCT
cana-3840	109	13	133x	133x	NUM
cana-3840	109	14	vol	vol	NOUN
cana-3840	109	15	32	32	NUM
cana-3840	109	16	no	no	NOUN
cana-3840	109	17	.	.	PUNCT
cana-3840	110	1	9s	9s	NUM
cana-3840	110	2	(	(	PUNCT
cana-3840	110	3	2025	2025	NUM
cana-3840	110	4	)	)	PUNCT
cana-3840	110	5	92	92	NUM
cana-3840	110	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	110	7	now	now	ADV
cana-3840	110	8	,	,	PUNCT
cana-3840	110	9	from	from	ADP
cana-3840	110	10	lemma	lemma	PROPN
cana-3840	110	11	2	2	NUM
cana-3840	110	12	(	(	PUNCT
cana-3840	110	13	3	3	NUM
cana-3840	110	14	)	)	PUNCT
cana-3840	110	15	,	,	PUNCT
cana-3840	110	16	it	it	PRON
cana-3840	110	17	is	be	AUX
cana-3840	110	18	easy	easy	ADJ
cana-3840	110	19	to	to	PART
cana-3840	110	20	see	see	VERB
cana-3840	110	21	that	that	SCONJ
cana-3840	110	22	the	the	DET
cana-3840	110	23	topology	topology	NOUN
cana-3840	110	24	𝜏𝕎	𝜏𝕎	NOUN
cana-3840	110	25	can	can	AUX
cana-3840	110	26	be	be	AUX
cana-3840	110	27	induced	induce	VERB
cana-3840	110	28	by	by	ADP
cana-3840	110	29	the	the	DET
cana-3840	110	30	family	family	NOUN
cana-3840	110	31	of	of	ADP
cana-3840	110	32	stratified	stratified	ADJ
cana-3840	110	33	functions	function	NOUN
cana-3840	110	34	.	.	PUNCT
cana-3840	111	1	that	that	PRON
cana-3840	111	2	is	is	ADV
cana-3840	111	3	,	,	PUNCT
cana-3840	111	4	we	we	PRON
cana-3840	111	5	obtain	obtain	VERB
cana-3840	111	6	the	the	DET
cana-3840	111	7	following	follow	VERB
cana-3840	111	8	theorem	theorem	VERB
cana-3840	111	9	.	.	PUNCT
cana-3840	111	10	theorem	theorem	NOUN
cana-3840	111	11	1	1	NUM
cana-3840	111	12	.	.	PUNCT
cana-3840	112	1	let	let	VERB
cana-3840	112	2	𝔐	𝔐	PRON
cana-3840	112	3	=	=	PUNCT
cana-3840	112	4	{	{	PUNCT
cana-3840	112	5	𝕕𝕣	𝕕𝕣	SCONJ
cana-3840	112	6	:	:	SYM
cana-3840	112	7	0	0	NUM
cana-3840	112	8	<	<	X
cana-3840	112	9	𝕣	𝕣	X
cana-3840	112	10	<	<	X
cana-3840	112	11	1	1	NUM
cana-3840	112	12	}	}	PUNCT
cana-3840	112	13	be	be	AUX
cana-3840	112	14	the	the	DET
cana-3840	112	15	family	family	NOUN
cana-3840	112	16	of	of	ADP
cana-3840	112	17	stratified	stratified	ADJ
cana-3840	112	18	functions	function	NOUN
cana-3840	112	19	with	with	ADP
cana-3840	112	20	respect	respect	NOUN
cana-3840	112	21	to	to	ADP
cana-3840	112	22	a	a	DET
cana-3840	112	23	revised	revise	VERB
cana-3840	112	24	fuzzy	fuzzy	ADJ
cana-3840	112	25	metric	metric	ADJ
cana-3840	112	26	space	space	NOUN
cana-3840	112	27	(	(	PUNCT
cana-3840	112	28	𝔘	𝔘	PROPN
cana-3840	112	29	,	,	PUNCT
cana-3840	112	30	𝕎	𝕎	PROPN
cana-3840	112	31	,	,	PUNCT
cana-3840	112	32	⨁	⨁	PROPN
cana-3840	112	33	)	)	PUNCT
cana-3840	112	34	,	,	PUNCT
cana-3840	112	35	ℕ𝕣(𝕒	ℕ𝕣(𝕒	NOUN
cana-3840	112	36	,	,	PUNCT
cana-3840	112	37	𝓉	𝓉	PRON
cana-3840	112	38	)	)	PUNCT
cana-3840	112	39	be	be	AUX
cana-3840	112	40	defined	define	VERB
cana-3840	112	41	by	by	ADP
cana-3840	112	42	(	(	PUNCT
cana-3840	112	43	8)	8)	NUM
cana-3840	112	44	,	,	PUNCT
cana-3840	112	45	and	and	CCONJ
cana-3840	112	46	𝔅𝕒	𝔅𝕒	NOUN
cana-3840	112	47	=	=	PUNCT
cana-3840	112	48	{	{	PUNCT
cana-3840	112	49	ℕ𝕣(𝕒	ℕ𝕣(𝕒	NOUN
cana-3840	112	50	,	,	PUNCT
cana-3840	112	51	𝓉	𝓉	PRON
cana-3840	112	52	):	):	PUNCT
cana-3840	112	53	𝕣	𝕣	PROPN
cana-3840	112	54	∈	∈	PROPN
cana-3840	112	55	(	(	PUNCT
cana-3840	112	56	0	0	NUM
cana-3840	112	57	,	,	PUNCT
cana-3840	112	58	1	1	NUM
cana-3840	112	59	)	)	PUNCT
cana-3840	112	60	,	,	PUNCT
cana-3840	112	61	𝓉	𝓉	PROPN
cana-3840	112	62	>	>	X
cana-3840	112	63	0	0	NUM
cana-3840	112	64	}	}	PUNCT
cana-3840	112	65	.	.	PUNCT
cana-3840	113	1	(	(	PUNCT
cana-3840	113	2	14	14	NUM
cana-3840	113	3	)	)	PUNCT
cana-3840	113	4	then	then	ADV
cana-3840	113	5	,	,	PUNCT
cana-3840	113	6	(	(	PUNCT
cana-3840	113	7	1	1	X
cana-3840	113	8	)	)	PUNCT
cana-3840	113	9	𝔅𝕒	𝔅𝕒	NOUN
cana-3840	113	10	is	be	AUX
cana-3840	113	11	a	a	DET
cana-3840	113	12	base	base	NOUN
cana-3840	113	13	of	of	ADP
cana-3840	113	14	neighborhoods	neighborhood	NOUN
cana-3840	113	15	at	at	ADP
cana-3840	113	16	𝕒	𝕒	PROPN
cana-3840	113	17	∈	∈	PROPN
cana-3840	113	18	𝔸.	𝔸.	PROPN
cana-3840	113	19	(	(	PUNCT
cana-3840	113	20	2	2	X
cana-3840	113	21	)	)	PUNCT
cana-3840	113	22	the	the	DET
cana-3840	113	23	topology	topology	NOUN
cana-3840	113	24	𝜏𝕎	𝜏𝕎	NOUN
cana-3840	113	25	generated	generate	VERB
cana-3840	113	26	by	by	ADP
cana-3840	113	27	{	{	PUNCT
cana-3840	113	28	𝔅𝕒	𝔅𝕒	NOUN
cana-3840	113	29	:	:	PUNCT
cana-3840	113	30	𝕒	𝕒	PROPN
cana-3840	113	31	∈	∈	PROPN
cana-3840	113	32	𝔸	𝔸	PROPN
cana-3840	113	33	}	}	PUNCT
cana-3840	113	34	coincides	coincide	VERB
cana-3840	113	35	with	with	ADP
cana-3840	113	36	the	the	DET
cana-3840	113	37	topology	topology	NOUN
cana-3840	113	38	𝜏𝕎.	𝜏𝕎.	PROPN
cana-3840	113	39	generally	generally	ADV
cana-3840	113	40	,	,	PUNCT
cana-3840	113	41	a	a	DET
cana-3840	113	42	stratified	stratified	ADJ
cana-3840	113	43	function	function	NOUN
cana-3840	113	44	is	be	AUX
cana-3840	113	45	not	not	PART
cana-3840	113	46	a	a	DET
cana-3840	113	47	pseudo	pseudo	NOUN
cana-3840	113	48	metric	metric	NOUN
cana-3840	113	49	.	.	PUNCT
cana-3840	114	1	in	in	ADP
cana-3840	114	2	fact	fact	NOUN
cana-3840	114	3	,	,	PUNCT
cana-3840	114	4	we	we	PRON
cana-3840	114	5	have	have	VERB
cana-3840	114	6	the	the	DET
cana-3840	114	7	following	follow	VERB
cana-3840	114	8	result	result	NOUN
cana-3840	114	9	.	.	PUNCT
cana-3840	115	1	theorem	theorem	NOUN
cana-3840	115	2	2	2	NUM
cana-3840	115	3	.	.	PUNCT
cana-3840	116	1	let	let	AUX
cana-3840	116	2	(	(	PUNCT
cana-3840	116	3	𝔘	𝔘	PROPN
cana-3840	116	4	,	,	PUNCT
cana-3840	116	5	𝕎	𝕎	PROPN
cana-3840	116	6	,	,	PUNCT
cana-3840	116	7	⨁	⨁	PROPN
cana-3840	116	8	)	)	PUNCT
cana-3840	116	9	be	be	VERB
cana-3840	116	10	a	a	DET
cana-3840	116	11	revised	revise	VERB
cana-3840	116	12	fuzzy	fuzzy	ADJ
cana-3840	116	13	metric	metric	ADJ
cana-3840	116	14	space	space	NOUN
cana-3840	116	15	.	.	PUNCT
cana-3840	117	1	a	a	DET
cana-3840	117	2	stratified	stratified	ADJ
cana-3840	117	3	function	function	NOUN
cana-3840	117	4	(	(	PUNCT
cana-3840	117	5	𝕕𝕣(𝕣	𝕕𝕣(𝕣	PUNCT
cana-3840	117	6	∈	∈	NOUN
cana-3840	117	7	(	(	PUNCT
cana-3840	117	8	0	0	NUM
cana-3840	117	9	,	,	PUNCT
cana-3840	117	10	1	1	NUM
cana-3840	117	11	)	)	PUNCT
cana-3840	117	12	)	)	PUNCT
cana-3840	117	13	)	)	PUNCT
cana-3840	117	14	is	be	AUX
cana-3840	117	15	a	a	DET
cana-3840	117	16	pseudometric	pseudometric	NOUN
cana-3840	117	17	on	on	ADP
cana-3840	117	18	𝔸	𝔸	PROPN
cana-3840	117	19	if	if	SCONJ
cana-3840	118	1	and	and	CCONJ
cana-3840	118	2	only	only	ADV
cana-3840	118	3	if	if	SCONJ
cana-3840	118	4	𝕎	𝕎	PROPN
cana-3840	118	5	satisfies	satisfy	VERB
cana-3840	118	6	the	the	DET
cana-3840	118	7	following	follow	VERB
cana-3840	118	8	condition	condition	NOUN
cana-3840	118	9	:	:	PUNCT
cana-3840	118	10	for	for	ADP
cana-3840	118	11	any	any	DET
cana-3840	118	12	𝕒	𝕒	PROPN
cana-3840	118	13	,	,	PUNCT
cana-3840	118	14	𝕓	𝕓	PROPN
cana-3840	118	15	,	,	PUNCT
cana-3840	118	16	𝕔	𝕔	DET
cana-3840	118	17	∈	∈	PROPN
cana-3840	118	18	𝔸	𝔸	PROPN
cana-3840	118	19	,	,	PUNCT
cana-3840	118	20	𝓉1	𝓉1	PROPN
cana-3840	118	21	,	,	PUNCT
cana-3840	118	22	𝓉2	𝓉2	NOUN
cana-3840	118	23	>	>	X
cana-3840	118	24	0	0	NUM
cana-3840	118	25	,	,	PUNCT
cana-3840	118	26	if	if	SCONJ
cana-3840	118	27	𝕎(𝕒	𝕎(𝕒	ADJ
cana-3840	118	28	,	,	PUNCT
cana-3840	118	29	𝕔	𝕔	PROPN
cana-3840	118	30	,	,	PUNCT
cana-3840	118	31	𝓉1	𝓉1	INTJ
cana-3840	118	32	)	)	PUNCT
cana-3840	118	33	<	<	X
cana-3840	118	34	𝑟	𝑟	X
cana-3840	118	35	,	,	PUNCT
cana-3840	118	36	𝕎(𝕔	𝕎(𝕔	PROPN
cana-3840	118	37	,	,	PUNCT
cana-3840	118	38	𝕓	𝕓	X
cana-3840	118	39	,	,	PUNCT
cana-3840	118	40	𝓉2	𝓉2	NOUN
cana-3840	118	41	)	)	PUNCT
cana-3840	118	42	<	<	X
cana-3840	119	1	𝑟	𝑟	X
cana-3840	119	2	,	,	PUNCT
cana-3840	119	3	then	then	ADV
cana-3840	119	4	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	119	5	,	,	PUNCT
cana-3840	119	6	𝕓	𝕓	PRON
cana-3840	119	7	,	,	PUNCT
cana-3840	119	8	𝓉1	𝓉1	VERB
cana-3840	119	9	+	+	CCONJ
cana-3840	119	10	𝓉2	𝓉2	NOUN
cana-3840	119	11	)	)	PUNCT
cana-3840	119	12	<	<	X
cana-3840	120	1	𝑟.	𝑟.	X
cana-3840	120	2	(	(	PUNCT
cana-3840	120	3	15	15	NUM
cana-3840	120	4	)	)	PUNCT
cana-3840	120	5	proof	proof	NOUN
cana-3840	120	6	.	.	PUNCT
cana-3840	121	1	for	for	ADP
cana-3840	121	2	any	any	DET
cana-3840	121	3	𝕣	𝕣	PRON
cana-3840	121	4	∈	∈	PROPN
cana-3840	121	5	(	(	PUNCT
cana-3840	121	6	0	0	NUM
cana-3840	121	7	,	,	PUNCT
cana-3840	121	8	1	1	NUM
cana-3840	121	9	)	)	PUNCT
cana-3840	121	10	,	,	PUNCT
cana-3840	121	11	it	it	PRON
cana-3840	121	12	is	be	AUX
cana-3840	121	13	obvious	obvious	ADJ
cana-3840	121	14	that	that	SCONJ
cana-3840	121	15	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	121	16	,	,	PUNCT
cana-3840	121	17	𝕓	𝕓	X
cana-3840	121	18	)	)	PUNCT
cana-3840	121	19	≥	≥	NOUN
cana-3840	121	20	0	0	NUM
cana-3840	121	21	,	,	PUNCT
cana-3840	121	22	𝕕𝕣(𝕒	𝕕𝕣(𝕒	X
cana-3840	121	23	,	,	PUNCT
cana-3840	121	24	𝕓	𝕓	X
cana-3840	121	25	)	)	PUNCT
cana-3840	121	26	=	=	SYM
cana-3840	121	27	𝕕𝕣(𝕓	𝕕𝕣(𝕓	NOUN
cana-3840	121	28	,	,	PUNCT
cana-3840	121	29	𝕒	𝕒	NOUN
cana-3840	121	30	)	)	PUNCT
cana-3840	121	31	,	,	PUNCT
cana-3840	121	32	and	and	CCONJ
cana-3840	121	33	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	121	34	,	,	PUNCT
cana-3840	121	35	𝕓	𝕓	X
cana-3840	121	36	)	)	PUNCT
cana-3840	121	37	=	=	SYM
cana-3840	122	1	0	0	NUM
cana-3840	122	2	when	when	SCONJ
cana-3840	122	3	𝕒	𝕒	X
cana-3840	122	4	=	=	PROPN
cana-3840	122	5	𝕓.	𝕓.	PROPN
cana-3840	122	6	thus	thus	ADV
cana-3840	122	7	,	,	PUNCT
cana-3840	122	8	to	to	PART
cana-3840	122	9	complete	complete	VERB
cana-3840	122	10	the	the	DET
cana-3840	122	11	proof	proof	NOUN
cana-3840	122	12	,	,	PUNCT
cana-3840	122	13	we	we	PRON
cana-3840	122	14	only	only	ADV
cana-3840	122	15	must	must	AUX
cana-3840	122	16	prove	prove	VERB
cana-3840	122	17	that	that	SCONJ
cana-3840	122	18	𝕕𝕣(𝕒	𝕕𝕣(𝕒	VERB
cana-3840	122	19	,	,	PUNCT
cana-3840	122	20	𝕓	𝕓	X
cana-3840	122	21	)	)	PUNCT
cana-3840	122	22	≤	≤	NOUN
cana-3840	122	23	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	122	24	,	,	PUNCT
cana-3840	122	25	𝕔	𝕔	NOUN
cana-3840	122	26	)	)	PUNCT
cana-3840	122	27	+	+	CCONJ
cana-3840	122	28	𝕕𝕣(𝕔	𝕕𝕣(𝕔	NOUN
cana-3840	122	29	,	,	PUNCT
cana-3840	122	30	𝕓	𝕓	X
cana-3840	122	31	)	)	PUNCT
cana-3840	122	32	if	if	SCONJ
cana-3840	122	33	and	and	CCONJ
cana-3840	122	34	only	only	ADV
cana-3840	122	35	if	if	SCONJ
cana-3840	122	36	𝕎	𝕎	PROPN
cana-3840	122	37	satisfies	satisfy	VERB
cana-3840	122	38	condition	condition	NOUN
cana-3840	122	39	(	(	PUNCT
cana-3840	122	40	15	15	NUM
cana-3840	122	41	)	)	PUNCT
cana-3840	122	42	.	.	PUNCT
cana-3840	123	1	sufficiency	sufficiency	PROPN
cana-3840	123	2	.	.	PUNCT
cana-3840	124	1	for	for	ADP
cana-3840	124	2	any	any	DET
cana-3840	124	3	휀	휀	NOUN
cana-3840	124	4	>	>	X
cana-3840	124	5	0	0	NUM
cana-3840	124	6	,	,	PUNCT
cana-3840	124	7	from	from	ADP
cana-3840	124	8	lemma	lemma	PROPN
cana-3840	124	9	2	2	NUM
cana-3840	124	10	(	(	PUNCT
cana-3840	124	11	1	1	NUM
cana-3840	124	12	)	)	PUNCT
cana-3840	124	13	,	,	PUNCT
cana-3840	124	14	we	we	PRON
cana-3840	124	15	obtain	obtain	VERB
cana-3840	124	16	𝕎	𝕎	PROPN
cana-3840	124	17	(	(	PUNCT
cana-3840	124	18	𝕒	𝕒	PROPN
cana-3840	124	19	,	,	PUNCT
cana-3840	124	20	𝕔	𝕔	NOUN
cana-3840	124	21	,	,	PUNCT
cana-3840	124	22	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	124	23	,	,	PUNCT
cana-3840	124	24	𝕔	𝕔	NOUN
cana-3840	124	25	)	)	PUNCT
cana-3840	124	26	+	+	NUM
cana-3840	124	27	𝜀	𝜀	X
cana-3840	124	28	2	2	NUM
cana-3840	124	29	)	)	PUNCT
cana-3840	124	30	<	<	X
cana-3840	125	1	𝑟	𝑟	X
cana-3840	125	2	,	,	PUNCT
cana-3840	125	3	𝕎	𝕎	PROPN
cana-3840	125	4	(	(	PUNCT
cana-3840	125	5	𝕔	𝕔	PROPN
cana-3840	125	6	,	,	PUNCT
cana-3840	125	7	𝕓	𝕓	X
cana-3840	125	8	,	,	PUNCT
cana-3840	125	9	𝕕𝕣(𝕔	𝕕𝕣(𝕔	NOUN
cana-3840	125	10	,	,	PUNCT
cana-3840	125	11	𝕓	𝕓	X
cana-3840	125	12	)	)	PUNCT
cana-3840	125	13	+	+	NUM
cana-3840	125	14	𝜀	𝜀	X
cana-3840	125	15	2	2	NUM
cana-3840	125	16	)	)	PUNCT
cana-3840	125	17	<	<	X
cana-3840	125	18	𝑟.	𝑟.	X
cana-3840	125	19	(	(	PUNCT
cana-3840	125	20	16	16	NUM
cana-3840	125	21	)	)	PUNCT
cana-3840	125	22	from	from	ADP
cana-3840	125	23	(	(	PUNCT
cana-3840	125	24	15	15	NUM
cana-3840	125	25	)	)	PUNCT
cana-3840	125	26	,	,	PUNCT
cana-3840	125	27	we	we	PRON
cana-3840	125	28	have	have	VERB
cana-3840	125	29	𝕎	𝕎	PROPN
cana-3840	125	30	(	(	PUNCT
cana-3840	125	31	𝕒	𝕒	PROPN
cana-3840	125	32	,	,	PUNCT
cana-3840	125	33	𝕓	𝕓	X
cana-3840	125	34	,	,	PUNCT
cana-3840	125	35	𝕕𝕣(𝕒	𝕕𝕣(𝕒	X
cana-3840	125	36	,	,	PUNCT
cana-3840	125	37	𝕔	𝕔	NOUN
cana-3840	125	38	)	)	PUNCT
cana-3840	125	39	+	+	CCONJ
cana-3840	125	40	𝕕𝕣(𝕔	𝕕𝕣(𝕔	NOUN
cana-3840	125	41	,	,	PUNCT
cana-3840	125	42	𝕓	𝕓	X
cana-3840	125	43	)	)	PUNCT
cana-3840	126	1	+	+	NUM
cana-3840	126	2	𝜀	𝜀	X
cana-3840	126	3	2	2	NUM
cana-3840	126	4	)	)	PUNCT
cana-3840	126	5	<	<	X
cana-3840	126	6	𝑟.	𝑟.	X
cana-3840	126	7	(	(	PUNCT
cana-3840	126	8	17	17	NUM
cana-3840	126	9	)	)	PUNCT
cana-3840	126	10	therefore	therefore	ADV
cana-3840	126	11	,	,	PUNCT
cana-3840	126	12	𝕕𝕣(𝕒	𝕕𝕣(𝕒	X
cana-3840	126	13	,	,	PUNCT
cana-3840	126	14	𝕓	𝕓	NOUN
cana-3840	126	15	)	)	PUNCT
cana-3840	126	16	≤	≤	NOUN
cana-3840	126	17	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	126	18	,	,	PUNCT
cana-3840	126	19	𝕔	𝕔	NOUN
cana-3840	126	20	)	)	PUNCT
cana-3840	126	21	+	+	CCONJ
cana-3840	126	22	𝕕𝕣(𝕔	𝕕𝕣(𝕔	NOUN
cana-3840	126	23	,	,	PUNCT
cana-3840	126	24	𝕓	𝕓	X
cana-3840	126	25	)	)	PUNCT
cana-3840	126	26	+	+	X
cana-3840	126	27	휀	휀	X
cana-3840	126	28	.	.	PUNCT
cana-3840	126	29	(	(	PUNCT
cana-3840	126	30	18	18	NUM
cana-3840	126	31	)	)	PUNCT
cana-3840	126	32	from	from	ADP
cana-3840	126	33	the	the	DET
cana-3840	126	34	arbitrariness	arbitrariness	NOUN
cana-3840	126	35	of	of	ADP
cana-3840	126	36	휀	휀	NOUN
cana-3840	126	37	>	>	X
cana-3840	126	38	0	0	NUM
cana-3840	126	39	,	,	PUNCT
cana-3840	126	40	we	we	PRON
cana-3840	126	41	know	know	VERB
cana-3840	126	42	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	126	43	,	,	PUNCT
cana-3840	126	44	𝕓	𝕓	NOUN
cana-3840	126	45	)	)	PUNCT
cana-3840	126	46	≤	≤	NOUN
cana-3840	126	47	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	126	48	,	,	PUNCT
cana-3840	126	49	𝕔	𝕔	NOUN
cana-3840	126	50	)	)	PUNCT
cana-3840	127	1	+	+	CCONJ
cana-3840	127	2	𝕕𝕣(𝕔	𝕕𝕣(𝕔	NOUN
cana-3840	127	3	,	,	PUNCT
cana-3840	127	4	𝕓	𝕓	X
cana-3840	127	5	)	)	PUNCT
cana-3840	127	6	.	.	PUNCT
cana-3840	128	1	necessity	necessity	NOUN
cana-3840	128	2	.	.	PUNCT
cana-3840	129	1	suppose	suppose	VERB
cana-3840	129	2	that	that	SCONJ
cana-3840	129	3	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	129	4	,	,	PUNCT
cana-3840	129	5	𝕔	𝕔	PROPN
cana-3840	129	6	,	,	PUNCT
cana-3840	129	7	𝓉1	𝓉1	INTJ
cana-3840	129	8	)	)	PUNCT
cana-3840	129	9	<	<	X
cana-3840	130	1	𝑟	𝑟	X
cana-3840	130	2	,	,	PUNCT
cana-3840	130	3	𝕎(𝕔	𝕎(𝕔	PROPN
cana-3840	130	4	,	,	PUNCT
cana-3840	130	5	𝕓	𝕓	X
cana-3840	130	6	,	,	PUNCT
cana-3840	130	7	𝓉2	𝓉2	NOUN
cana-3840	130	8	)	)	PUNCT
cana-3840	130	9	<	<	X
cana-3840	130	10	𝑟.	𝑟.	NOUN
cana-3840	130	11	by	by	ADP
cana-3840	130	12	(	(	PUNCT
cana-3840	130	13	rgv5	rgv5	PROPN
cana-3840	130	14	)	)	PUNCT
cana-3840	130	15	,	,	PUNCT
cana-3840	130	16	there	there	PRON
cana-3840	130	17	exists	exist	VERB
cana-3840	130	18	𝛿	𝛿	PROPN
cana-3840	130	19	>	>	X
cana-3840	130	20	0	0	NUM
cana-3840	130	21	such	such	ADJ
cana-3840	130	22	that	that	PRON
cana-3840	130	23	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	130	24	,	,	PUNCT
cana-3840	130	25	𝕔	𝕔	PROPN
cana-3840	130	26	,	,	PUNCT
cana-3840	130	27	𝓉1	𝓉1	VERB
cana-3840	130	28	−	−	PROPN
cana-3840	130	29	δ	δ	PROPN
cana-3840	130	30	)	)	PUNCT
cana-3840	130	31	<	<	X
cana-3840	130	32	𝑟	𝑟	X
cana-3840	130	33	,	,	PUNCT
cana-3840	130	34	𝕎(𝕔	𝕎(𝕔	PROPN
cana-3840	130	35	,	,	PUNCT
cana-3840	130	36	𝕓	𝕓	PRON
cana-3840	130	37	,	,	PUNCT
cana-3840	130	38	𝓉2	𝓉2	NOUN
cana-3840	130	39	−	−	PROPN
cana-3840	130	40	δ	δ	PROPN
cana-3840	130	41	)	)	PUNCT
cana-3840	130	42	<	<	X
cana-3840	130	43	𝑟	𝑟	X
cana-3840	130	44	,	,	PUNCT
cana-3840	130	45	(	(	PUNCT
cana-3840	130	46	19	19	NUM
cana-3840	130	47	)	)	PUNCT
cana-3840	130	48	so	so	ADV
cana-3840	130	49	,	,	PUNCT
cana-3840	130	50	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	130	51	,	,	PUNCT
cana-3840	130	52	𝕔	𝕔	NOUN
cana-3840	130	53	)	)	PUNCT
cana-3840	130	54	≤	≤	NUM
cana-3840	130	55	𝓉1	𝓉1	VERB
cana-3840	130	56	−	−	PROPN
cana-3840	130	57	𝛿	𝛿	ADJ
cana-3840	130	58	and	and	CCONJ
cana-3840	130	59	𝕕𝕣(𝕔	𝕕𝕣(𝕔	NOUN
cana-3840	130	60	,	,	PUNCT
cana-3840	130	61	𝕓	𝕓	NOUN
cana-3840	130	62	)	)	PUNCT
cana-3840	130	63	≤	≤	NOUN
cana-3840	130	64	𝓉2	𝓉2	NOUN
cana-3840	130	65	−	−	NOUN
cana-3840	130	66	𝛿.	𝛿.	ADJ
cana-3840	130	67	since	since	SCONJ
cana-3840	130	68	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	130	69	,	,	PUNCT
cana-3840	130	70	𝕓	𝕓	NOUN
cana-3840	130	71	)	)	PUNCT
cana-3840	130	72	≤	≤	NOUN
cana-3840	130	73	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	130	74	,	,	PUNCT
cana-3840	130	75	𝕔	𝕔	NOUN
cana-3840	130	76	)	)	PUNCT
cana-3840	130	77	+	+	CCONJ
cana-3840	130	78	𝕕𝕣(𝕔	𝕕𝕣(𝕔	NOUN
cana-3840	130	79	,	,	PUNCT
cana-3840	130	80	𝕓	𝕓	X
cana-3840	130	81	)	)	PUNCT
cana-3840	130	82	,	,	PUNCT
cana-3840	130	83	we	we	PRON
cana-3840	130	84	have	have	VERB
cana-3840	130	85	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	130	86	,	,	PUNCT
cana-3840	130	87	𝕓	𝕓	X
cana-3840	130	88	)	)	PUNCT
cana-3840	130	89	≤	≤	PUNCT
cana-3840	130	90	𝓉1	𝓉1	VERB
cana-3840	130	91	+	+	CCONJ
cana-3840	130	92	𝓉2	𝓉2	NOUN
cana-3840	130	93	−	−	NOUN
cana-3840	130	94	2𝛿	2𝛿	PROPN
cana-3840	130	95	<	<	X
cana-3840	130	96	𝓉1	𝓉1	NOUN
cana-3840	130	97	+	+	X
cana-3840	130	98	𝓉2	𝓉2	NOUN
cana-3840	130	99	.	.	PUNCT
cana-3840	131	1	by	by	ADP
cana-3840	131	2	the	the	DET
cana-3840	131	3	definition	definition	NOUN
cana-3840	131	4	of	of	ADP
cana-3840	131	5	𝕕𝕣(𝕒	𝕕𝕣(𝕒	NOUN
cana-3840	131	6	,	,	PUNCT
cana-3840	131	7	𝕓	𝕓	X
cana-3840	131	8	)	)	PUNCT
cana-3840	131	9	,	,	PUNCT
cana-3840	131	10	there	there	PRON
cana-3840	131	11	exists	exist	VERB
cana-3840	131	12	𝓉0	𝓉0	PROPN
cana-3840	131	13	<	<	X
cana-3840	131	14	𝓉1	𝓉1	PROPN
cana-3840	132	1	+	+	X
cana-3840	132	2	𝓉2	𝓉2	NOUN
cana-3840	132	3	such	such	ADJ
cana-3840	132	4	that	that	DET
cana-3840	132	5	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	132	6	,	,	PUNCT
cana-3840	132	7	𝕓	𝕓	X
cana-3840	132	8	,	,	PUNCT
cana-3840	132	9	𝓉0	𝓉0	ADJ
cana-3840	132	10	)	)	PUNCT
cana-3840	132	11	<	<	X
cana-3840	132	12	𝑟.	𝑟.	NOUN
cana-3840	132	13	thus	thus	ADV
cana-3840	132	14	,	,	PUNCT
cana-3840	132	15	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	132	16	,	,	PUNCT
cana-3840	132	17	𝕓	𝕓	PRON
cana-3840	132	18	,	,	PUNCT
cana-3840	132	19	𝓉1	𝓉1	VERB
cana-3840	132	20	+	+	CCONJ
cana-3840	132	21	𝓉2	𝓉2	NOUN
cana-3840	132	22	)	)	PUNCT
cana-3840	132	23	<	<	X
cana-3840	132	24	𝑟.	𝑟.	NOUN
cana-3840	132	25	remark	remark	VERB
cana-3840	132	26	1	1	NUM
cana-3840	132	27	.	.	PUNCT
cana-3840	133	1	let	let	VERB
cana-3840	133	2	𝕎	𝕎	PROPN
cana-3840	133	3	satisfies	satisfie	NOUN
cana-3840	133	4	(	(	PUNCT
cana-3840	133	5	15	15	NUM
cana-3840	133	6	)	)	PUNCT
cana-3840	133	7	if	if	SCONJ
cana-3840	133	8	⨁	⨁	PROPN
cana-3840	133	9	=	=	SYM
cana-3840	133	10	⋁3	⋁3	PROPN
cana-3840	133	11	.	.	PUNCT
cana-3840	134	1	now	now	ADV
cana-3840	134	2	,	,	PUNCT
cana-3840	134	3	we	we	PRON
cana-3840	134	4	explore	explore	VERB
cana-3840	134	5	the	the	DET
cana-3840	134	6	metric	metric	NOUN
cana-3840	134	7	which	which	PRON
cana-3840	134	8	induces	induce	VERB
cana-3840	134	9	the	the	DET
cana-3840	134	10	topology	topology	NOUN
cana-3840	134	11	𝜏𝕎.	𝜏𝕎.	PROPN
cana-3840	134	12	communications	communication	NOUN
cana-3840	134	13	on	on	ADP
cana-3840	134	14	applied	apply	VERB
cana-3840	134	15	nonlinear	nonlinear	ADJ
cana-3840	134	16	analysis	analysis	NOUN
cana-3840	134	17	issn	issn	NOUN
cana-3840	134	18	:	:	PUNCT
cana-3840	134	19	1074	1074	NUM
cana-3840	134	20	-	-	PUNCT
cana-3840	134	21	133x	133x	NUM
cana-3840	134	22	vol	vol	NOUN
cana-3840	134	23	32	32	NUM
cana-3840	134	24	no	no	NOUN
cana-3840	134	25	.	.	PUNCT
cana-3840	135	1	9s	9s	NUM
cana-3840	135	2	(	(	PUNCT
cana-3840	135	3	2025	2025	NUM
cana-3840	135	4	)	)	PUNCT
cana-3840	135	5	93	93	NUM
cana-3840	135	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	135	7	definition	definition	NOUN
cana-3840	135	8	4	4	X
cana-3840	135	9	.	.	PUNCT
cana-3840	136	1	let	let	VERB
cana-3840	136	2	ℝ+	ℝ+	PUNCT
cana-3840	136	3	=	=	PUNCT
cana-3840	137	1	[	[	X
cana-3840	137	2	0	0	NUM
cana-3840	137	3	,	,	PUNCT
cana-3840	137	4	∞	∞	PROPN
cana-3840	137	5	)	)	PUNCT
cana-3840	137	6	.	.	PUNCT
cana-3840	138	1	we	we	PRON
cana-3840	138	2	call	call	VERB
cana-3840	138	3	function	function	NOUN
cana-3840	138	4	𝒢	𝒢	PROPN
cana-3840	138	5	:	:	PUNCT
cana-3840	138	6	ℝ+	ℝ+	PUNCT
cana-3840	138	7	⟶	⟶	NOUN
cana-3840	138	8	ℝ+	ℝ+	PUNCT
cana-3840	138	9	satisfies	satisfie	NOUN
cana-3840	138	10	(	(	PUNCT
cana-3840	138	11	i	i	NOUN
cana-3840	138	12	)	)	PUNCT
cana-3840	138	13	the	the	DET
cana-3840	138	14	condition	condition	NOUN
cana-3840	138	15	ℭ1	ℭ1	PROPN
cana-3840	138	16	,	,	PUNCT
cana-3840	138	17	if	if	SCONJ
cana-3840	138	18	𝒢(0	𝒢(0	VERB
cana-3840	138	19	)	)	PUNCT
cana-3840	138	20	=	=	SYM
cana-3840	138	21	0	0	NUM
cana-3840	138	22	,	,	PUNCT
cana-3840	138	23	𝒢(𝓉	𝒢(𝓉	X
cana-3840	138	24	)	)	PUNCT
cana-3840	138	25	≡	≡	PROPN
cana-3840	138	26	0	0	NUM
cana-3840	138	27	,	,	PUNCT
cana-3840	138	28	and	and	CCONJ
cana-3840	138	29	𝒢	𝒢	PROPN
cana-3840	138	30	is	be	AUX
cana-3840	138	31	non	non	ADJ
cana-3840	138	32	increasing	increase	VERB
cana-3840	138	33	and	and	CCONJ
cana-3840	138	34	continuous	continuous	ADJ
cana-3840	138	35	at	at	ADP
cana-3840	138	36	0	0	NUM
cana-3840	138	37	.	.	PUNCT
cana-3840	139	1	(	(	PUNCT
cana-3840	139	2	ii	ii	NOUN
cana-3840	139	3	)	)	PUNCT
cana-3840	139	4	the	the	DET
cana-3840	139	5	condition	condition	NOUN
cana-3840	139	6	ℭ2	ℭ2	ADV
cana-3840	139	7	,	,	PUNCT
cana-3840	139	8	if	if	SCONJ
cana-3840	139	9	𝒢(0	𝒢(0	VERB
cana-3840	139	10	)	)	PUNCT
cana-3840	139	11	=	=	SYM
cana-3840	139	12	0	0	NUM
cana-3840	139	13	,	,	PUNCT
cana-3840	139	14	𝒢(0	𝒢(0	NOUN
cana-3840	139	15	)	)	PUNCT
cana-3840	139	16	>	>	X
cana-3840	139	17	0	0	PUNCT
cana-3840	140	1	as	as	ADP
cana-3840	140	2	𝓉	𝓉	PROPN
cana-3840	140	3	>	>	X
cana-3840	140	4	0	0	PROPN
cana-3840	140	5	,	,	PUNCT
cana-3840	140	6	lim	lim	PROPN
cana-3840	140	7	𝓉	𝓉	PROPN
cana-3840	140	8	⟶	⟶	ADJ
cana-3840	140	9	+	+	SYM
cana-3840	140	10	∞	∞	NUM
cana-3840	140	11	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	140	12	)	)	PUNCT
cana-3840	140	13	=	=	PUNCT
cana-3840	141	1	+	+	NUM
cana-3840	141	2	∞	∞	NOUN
cana-3840	141	3	,	,	PUNCT
cana-3840	141	4	𝒢(𝓉1	𝒢(𝓉1	PROPN
cana-3840	141	5	+	+	CCONJ
cana-3840	141	6	𝓉2	𝓉2	NOUN
cana-3840	141	7	)	)	PUNCT
cana-3840	141	8	≤	≤	NOUN
cana-3840	141	9	𝒢(𝓉1	𝒢(𝓉1	PROPN
cana-3840	141	10	)	)	PUNCT
cana-3840	141	11	+	+	NUM
cana-3840	141	12	𝒢(𝓉2	𝒢(𝓉2	NOUN
cana-3840	141	13	)	)	PUNCT
cana-3840	141	14	for	for	ADP
cana-3840	141	15	any	any	DET
cana-3840	141	16	𝓉1	𝓉1	NOUN
cana-3840	141	17	+	+	CCONJ
cana-3840	141	18	𝓉2	𝓉2	NOUN
cana-3840	141	19	∈	∈	NOUN
cana-3840	141	20	ℝ+	ℝ+	NOUN
cana-3840	141	21	,	,	PUNCT
cana-3840	141	22	and	and	CCONJ
cana-3840	141	23	𝒢	𝒢	NOUN
cana-3840	141	24	is	be	AUX
cana-3840	141	25	right	right	ADV
cana-3840	141	26	continuous	continuous	ADJ
cana-3840	141	27	and	and	CCONJ
cana-3840	141	28	non	non	ADJ
cana-3840	141	29	-	-	ADJ
cana-3840	141	30	increasing	increasing	ADJ
cana-3840	141	31	.	.	PUNCT
cana-3840	142	1	theorem	theorem	NOUN
cana-3840	142	2	3	3	X
cana-3840	142	3	.	.	PUNCT
cana-3840	143	1	let	let	VERB
cana-3840	143	2	(	(	PUNCT
cana-3840	143	3	𝔘	𝔘	PROPN
cana-3840	143	4	,	,	PUNCT
cana-3840	143	5	𝕎	𝕎	PROPN
cana-3840	143	6	,	,	PUNCT
cana-3840	143	7	⨁	⨁	PROPN
cana-3840	143	8	)	)	PUNCT
cana-3840	143	9	is	be	AUX
cana-3840	143	10	revised	revise	VERB
cana-3840	143	11	fuzzy	fuzzy	ADJ
cana-3840	143	12	metric	metric	ADJ
cana-3840	143	13	space	space	NOUN
cana-3840	143	14	;	;	PUNCT
cana-3840	143	15	the	the	DET
cana-3840	143	16	functions	function	NOUN
cana-3840	143	17	𝒦	𝒦	PROPN
cana-3840	143	18	and	and	CCONJ
cana-3840	143	19	𝒢	𝒢	NOUN
cana-3840	143	20	satisfy	satisfy	VERB
cana-3840	143	21	the	the	DET
cana-3840	143	22	conditions	condition	NOUN
cana-3840	143	23	ℭ1	ℭ1	ADV
cana-3840	143	24	and	and	CCONJ
cana-3840	143	25	ℭ2	ℭ2	PROPN
cana-3840	143	26	,	,	PUNCT
cana-3840	143	27	respectively	respectively	ADV
cana-3840	143	28	.	.	PUNCT
cana-3840	144	1	define	define	VERB
cana-3840	144	2	a	a	DET
cana-3840	144	3	function	function	NOUN
cana-3840	144	4	𝕕	𝕕	NOUN
cana-3840	144	5	on	on	ADP
cana-3840	144	6	𝔘2	𝔘2	PROPN
cana-3840	144	7	as	as	ADP
cana-3840	144	8	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	144	9	,	,	PUNCT
cana-3840	144	10	𝕓	𝕓	X
cana-3840	144	11	)	)	PUNCT
cana-3840	144	12	=	=	SYM
cana-3840	145	1	𝑖𝑛𝑓{𝓈	𝑖𝑛𝑓{𝓈	PROPN
cana-3840	145	2	>	>	X
cana-3840	146	1	0	0	NUM
cana-3840	146	2	:	:	PUNCT
cana-3840	146	3	if	if	SCONJ
cana-3840	146	4	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	146	5	)	)	PUNCT
cana-3840	146	6	>	>	X
cana-3840	146	7	𝓈	𝓈	PROPN
cana-3840	146	8	,	,	PUNCT
cana-3840	146	9	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
cana-3840	146	10	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	146	11	,	,	PUNCT
cana-3840	146	12	𝕓	𝕓	X
cana-3840	146	13	,	,	PUNCT
cana-3840	146	14	𝓉	𝓉	PRON
cana-3840	146	15	)	)	PUNCT
cana-3840	146	16	+	+	PUNCT
cana-3840	146	17	𝒦(𝓉	𝒦(𝓉	X
cana-3840	146	18	)	)	PUNCT
cana-3840	146	19	≤	≤	NOUN
cana-3840	146	20	0	0	NUM
cana-3840	146	21	,	,	PUNCT
cana-3840	146	22	∀𝕒	∀𝕒	PROPN
cana-3840	146	23	,	,	PUNCT
cana-3840	146	24	𝕓	𝕓	PROPN
cana-3840	146	25	∈	∈	PROPN
cana-3840	146	26	𝔘	𝔘	PROPN
cana-3840	146	27	}	}	PUNCT
cana-3840	146	28	.	.	PUNCT
cana-3840	147	1	(	(	PUNCT
cana-3840	147	2	20	20	NUM
cana-3840	147	3	)	)	PUNCT
cana-3840	147	4	if	if	SCONJ
cana-3840	147	5	one	one	NUM
cana-3840	147	6	of	of	ADP
cana-3840	147	7	the	the	DET
cana-3840	147	8	following	follow	VERB
cana-3840	147	9	conditions	condition	NOUN
cana-3840	147	10	is	be	AUX
cana-3840	147	11	satisfied	satisfied	ADJ
cana-3840	147	12	:	:	PUNCT
cana-3840	147	13	(	(	PUNCT
cana-3840	147	14	i	i	NOUN
cana-3840	147	15	)	)	PUNCT
cana-3840	147	16	𝕎	𝕎	PROPN
cana-3840	147	17	satisfies	satisfy	VERB
cana-3840	147	18	condition	condition	NOUN
cana-3840	147	19	(	(	PUNCT
cana-3840	147	20	15	15	NUM
cana-3840	147	21	)	)	PUNCT
cana-3840	147	22	(	(	PUNCT
cana-3840	147	23	ii	ii	NOUN
cana-3840	147	24	)	)	PUNCT
cana-3840	147	25	⨁	⨁	PROPN
cana-3840	147	26	≥	≥	NUM
cana-3840	147	27	𝛥1	𝛥1	PROPN
cana-3840	147	28	,	,	PUNCT
cana-3840	147	29	∀𝓇1	∀𝓇1	ADP
cana-3840	147	30	,	,	PUNCT
cana-3840	147	31	𝓇2	𝓇2	NOUN
cana-3840	147	32	∈	∈	PROPN
cana-3840	148	1	[	[	X
cana-3840	148	2	0	0	NUM
cana-3840	148	3	,	,	PUNCT
cana-3840	148	4	∞	∞	NUM
cana-3840	148	5	)	)	PUNCT
cana-3840	148	6	𝒦(𝓇1	𝒦(𝓇1	NOUN
cana-3840	148	7	)	)	PUNCT
cana-3840	148	8	+	+	SYM
cana-3840	148	9	𝒦(𝓇2	𝒦(𝓇2	SYM
cana-3840	148	10	)	)	PUNCT
cana-3840	148	11	≤	≤	NUM
cana-3840	148	12	𝒦(𝓇1	𝒦(𝓇1	NOUN
cana-3840	148	13	)	)	PUNCT
cana-3840	149	1	+	+	CCONJ
cana-3840	149	2	𝓇2	𝓇2	NOUN
cana-3840	149	3	,	,	PUNCT
cana-3840	149	4	(	(	PUNCT
cana-3840	149	5	21	21	NUM
cana-3840	149	6	)	)	PUNCT
cana-3840	149	7	then	then	ADV
cana-3840	149	8	𝕕	𝕕	PRON
cana-3840	149	9	is	be	AUX
cana-3840	149	10	a	a	DET
cana-3840	149	11	metric	metric	NOUN
cana-3840	149	12	on	on	ADP
cana-3840	149	13	𝔘.	𝔘.	PROPN
cana-3840	149	14	proof	proof	NOUN
cana-3840	149	15	.	.	PUNCT
cana-3840	150	1	first	first	ADV
cana-3840	150	2	,	,	PUNCT
cana-3840	150	3	we	we	PRON
cana-3840	150	4	prove	prove	VERB
cana-3840	150	5	the	the	DET
cana-3840	150	6	following	follow	VERB
cana-3840	150	7	fact	fact	NOUN
cana-3840	150	8	:	:	PUNCT
cana-3840	150	9	if	if	SCONJ
cana-3840	150	10	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	150	11	)	)	PUNCT
cana-3840	150	12	<	<	X
cana-3840	150	13	𝓇	𝓇	X
cana-3840	150	14	<	<	X
cana-3840	150	15	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	150	16	,	,	PUNCT
cana-3840	150	17	𝕓	𝕓	X
cana-3840	150	18	)	)	PUNCT
cana-3840	150	19	,	,	PUNCT
cana-3840	150	20	(	(	PUNCT
cana-3840	150	21	22	22	NUM
cana-3840	150	22	)	)	PUNCT
cana-3840	150	23	then	then	ADV
cana-3840	150	24	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	150	25	,	,	PUNCT
cana-3840	150	26	𝕓	𝕓	X
cana-3840	150	27	,	,	PUNCT
cana-3840	150	28	𝓉	𝓉	PROPN
cana-3840	150	29	)	)	PUNCT
cana-3840	150	30	<	<	X
cana-3840	150	31	𝓇	𝓇	X
cana-3840	150	32	≤	≤	NUM
cana-3840	150	33	𝒦(𝓇	𝒦(𝓇	PUNCT
cana-3840	150	34	)	)	PUNCT
cana-3840	150	35	.	.	PUNCT
cana-3840	151	1	(	(	PUNCT
cana-3840	151	2	23	23	NUM
cana-3840	151	3	)	)	PUNCT
cana-3840	151	4	in	in	ADP
cana-3840	151	5	fact	fact	NOUN
cana-3840	151	6	,	,	PUNCT
cana-3840	151	7	if	if	SCONJ
cana-3840	151	8	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	151	9	)	)	PUNCT
cana-3840	151	10	<	<	X
cana-3840	151	11	𝓇	𝓇	X
cana-3840	151	12	<	<	X
cana-3840	151	13	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	151	14	,	,	PUNCT
cana-3840	151	15	𝕓	𝕓	X
cana-3840	151	16	)	)	PUNCT
cana-3840	151	17	,	,	PUNCT
cana-3840	151	18	from	from	ADP
cana-3840	151	19	the	the	DET
cana-3840	151	20	definition	definition	NOUN
cana-3840	151	21	of	of	ADP
cana-3840	151	22	𝕕	𝕕	X
cana-3840	151	23	,	,	PUNCT
cana-3840	151	24	we	we	PRON
cana-3840	151	25	obtain	obtain	VERB
cana-3840	151	26	there	there	ADV
cana-3840	151	27	exists	exist	VERB
cana-3840	151	28	0	0	PUNCT
cana-3840	151	29	<	<	X
cana-3840	151	30	𝓈	𝓈	X
cana-3840	151	31	<	<	X
cana-3840	151	32	𝓇	𝓇	X
cana-3840	151	33	such	such	ADJ
cana-3840	151	34	that	that	SCONJ
cana-3840	151	35	if	if	SCONJ
cana-3840	151	36	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	151	37	)	)	PUNCT
cana-3840	151	38	>	>	X
cana-3840	151	39	𝓈	𝓈	PROPN
cana-3840	151	40	,	,	PUNCT
cana-3840	151	41	then	then	ADV
cana-3840	151	42	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	151	43	,	,	PUNCT
cana-3840	151	44	𝕓	𝕓	X
cana-3840	151	45	,	,	PUNCT
cana-3840	151	46	𝓉	𝓉	PRON
cana-3840	151	47	)	)	PUNCT
cana-3840	151	48	+	+	NUM
cana-3840	151	49	𝒦(𝓇	𝒦(𝓇	NOUN
cana-3840	151	50	)	)	PUNCT
cana-3840	151	51	≤	≤	NUM
cana-3840	151	52	0	0	NUM
cana-3840	151	53	.	.	PUNCT
cana-3840	152	1	therefore	therefore	ADV
cana-3840	152	2	,	,	PUNCT
cana-3840	152	3	if	if	SCONJ
cana-3840	152	4	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	152	5	)	)	PUNCT
cana-3840	152	6	>	>	X
cana-3840	152	7	𝓇	𝓇	PROPN
cana-3840	152	8	,	,	PUNCT
cana-3840	152	9	then	then	ADV
cana-3840	152	10	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	152	11	)	)	PUNCT
cana-3840	152	12	>	>	X
cana-3840	152	13	𝓈	𝓈	PROPN
cana-3840	152	14	,	,	PUNCT
cana-3840	152	15	and	and	CCONJ
cana-3840	152	16	hence	hence	ADV
cana-3840	152	17	,	,	PUNCT
cana-3840	152	18	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	152	19	,	,	PUNCT
cana-3840	152	20	𝕓	𝕓	X
cana-3840	152	21	,	,	PUNCT
cana-3840	152	22	𝓉	𝓉	PRON
cana-3840	152	23	)	)	PUNCT
cana-3840	152	24	+	+	CCONJ
cana-3840	152	25	𝒦(𝑟	𝒦(𝑟	X
cana-3840	152	26	)	)	PUNCT
cana-3840	152	27	≤	≤	NOUN
cana-3840	152	28	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	152	29	,	,	PUNCT
cana-3840	152	30	𝕓	𝕓	X
cana-3840	152	31	,	,	PUNCT
cana-3840	152	32	𝓉	𝓉	PRON
cana-3840	152	33	)	)	PUNCT
cana-3840	152	34	+	+	CCONJ
cana-3840	152	35	𝒦(𝓈	𝒦(𝓈	X
cana-3840	152	36	)	)	PUNCT
cana-3840	152	37	≤	≤	NUM
cana-3840	152	38	0	0	NUM
cana-3840	152	39	.	.	PUNCT
cana-3840	153	1	that	that	PRON
cana-3840	153	2	is	be	AUX
cana-3840	153	3	,	,	PUNCT
cana-3840	153	4	𝕎(𝕒	𝕎(𝕒	X
cana-3840	153	5	,	,	PUNCT
cana-3840	153	6	𝕓	𝕓	X
cana-3840	153	7	,	,	PUNCT
cana-3840	153	8	𝓉	𝓉	PROPN
cana-3840	153	9	)	)	PUNCT
cana-3840	153	10	≤	≤	NOUN
cana-3840	154	1	𝒦(𝑟	𝒦(𝑟	NOUN
cana-3840	154	2	)	)	PUNCT
cana-3840	154	3	.	.	PUNCT
cana-3840	155	1	next	next	ADV
cana-3840	155	2	,	,	PUNCT
cana-3840	155	3	we	we	PRON
cana-3840	155	4	prove	prove	VERB
cana-3840	155	5	𝕕	𝕕	PRON
cana-3840	155	6	is	be	AUX
cana-3840	155	7	a	a	DET
cana-3840	155	8	metric	metric	NOUN
cana-3840	155	9	on	on	ADP
cana-3840	155	10	𝔘	𝔘	PROPN
cana-3840	155	11	,	,	PUNCT
cana-3840	155	12	that	that	ADV
cana-3840	155	13	is	is	ADV
cana-3840	155	14	,	,	PUNCT
cana-3840	155	15	𝕕	𝕕	PRON
cana-3840	155	16	satisfies	satisfy	VERB
cana-3840	155	17	the	the	DET
cana-3840	155	18	following	follow	VERB
cana-3840	155	19	properties	property	NOUN
cana-3840	155	20	:	:	PUNCT
cana-3840	155	21	for	for	ADP
cana-3840	155	22	any	any	DET
cana-3840	155	23	𝕒	𝕒	PROPN
cana-3840	155	24	,	,	PUNCT
cana-3840	155	25	𝕓	𝕓	PROPN
cana-3840	155	26	,	,	PUNCT
cana-3840	155	27	𝕔	𝕔	PRON
cana-3840	155	28	∈	∈	PROPN
cana-3840	155	29	𝔘	𝔘	PROPN
cana-3840	155	30	,	,	PUNCT
cana-3840	155	31	(	(	PUNCT
cana-3840	155	32	m1	m1	NOUN
cana-3840	155	33	)	)	PUNCT
cana-3840	155	34	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	155	35	,	,	PUNCT
cana-3840	155	36	𝕓	𝕓	X
cana-3840	155	37	)	)	PUNCT
cana-3840	155	38	≥	≥	NOUN
cana-3840	155	39	0	0	NUM
cana-3840	155	40	,	,	PUNCT
cana-3840	155	41	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	155	42	,	,	PUNCT
cana-3840	155	43	𝕓	𝕓	X
cana-3840	155	44	)	)	PUNCT
cana-3840	155	45	=	=	SYM
cana-3840	155	46	𝕕(𝕓	𝕕(𝕓	PROPN
cana-3840	155	47	,	,	PUNCT
cana-3840	155	48	𝕒	𝕒	NOUN
cana-3840	155	49	)	)	PUNCT
cana-3840	155	50	,	,	PUNCT
cana-3840	155	51	(	(	PUNCT
cana-3840	155	52	m2	m2	PROPN
cana-3840	155	53	)	)	PUNCT
cana-3840	155	54	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	155	55	,	,	PUNCT
cana-3840	155	56	𝕓	𝕓	X
cana-3840	155	57	)	)	PUNCT
cana-3840	155	58	=	=	SYM
cana-3840	155	59	0	0	PUNCT
cana-3840	156	1	if	if	SCONJ
cana-3840	156	2	and	and	CCONJ
cana-3840	156	3	only	only	ADV
cana-3840	156	4	if	if	SCONJ
cana-3840	156	5	𝕒	𝕒	X
cana-3840	156	6	=	=	SYM
cana-3840	156	7	𝕓	𝕓	PROPN
cana-3840	156	8	,	,	PUNCT
cana-3840	156	9	(	(	PUNCT
cana-3840	156	10	m3	m3	PROPN
cana-3840	156	11	)	)	PUNCT
cana-3840	156	12	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	156	13	,	,	PUNCT
cana-3840	156	14	𝕓	𝕓	X
cana-3840	156	15	)	)	PUNCT
cana-3840	156	16	≤	≤	NOUN
cana-3840	156	17	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	156	18	,	,	PUNCT
cana-3840	156	19	𝕔	𝕔	NOUN
cana-3840	156	20	)	)	PUNCT
cana-3840	156	21	+	+	CCONJ
cana-3840	156	22	𝕕(𝕔	𝕕(𝕔	PROPN
cana-3840	156	23	,	,	PUNCT
cana-3840	156	24	𝕓	𝕓	X
cana-3840	156	25	)	)	PUNCT
cana-3840	156	26	.	.	PUNCT
cana-3840	157	1	the	the	DET
cana-3840	157	2	conclusion	conclusion	NOUN
cana-3840	157	3	(	(	PUNCT
cana-3840	157	4	m1	m1	NOUN
cana-3840	157	5	)	)	PUNCT
cana-3840	157	6	is	be	AUX
cana-3840	157	7	obvious	obvious	ADJ
cana-3840	157	8	.	.	PUNCT
cana-3840	158	1	for	for	ADP
cana-3840	158	2	the	the	DET
cana-3840	158	3	conclusion	conclusion	NOUN
cana-3840	158	4	(	(	PUNCT
cana-3840	158	5	m2	m2	PROPN
cana-3840	158	6	)	)	PUNCT
cana-3840	158	7	,	,	PUNCT
cana-3840	158	8	it	it	PRON
cana-3840	158	9	is	be	AUX
cana-3840	158	10	easy	easy	ADJ
cana-3840	158	11	to	to	PART
cana-3840	158	12	see	see	VERB
cana-3840	158	13	that	that	SCONJ
cana-3840	158	14	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	158	15	,	,	PUNCT
cana-3840	158	16	𝕓	𝕓	X
cana-3840	158	17	)	)	PUNCT
cana-3840	158	18	=	=	SYM
cana-3840	159	1	0	0	PUNCT
cana-3840	160	1	if	if	SCONJ
cana-3840	160	2	𝕒	𝕒	X
cana-3840	160	3	=	=	PUNCT
cana-3840	160	4	𝕓.	𝕓.	PROPN
cana-3840	160	5	now	now	ADV
cana-3840	160	6	,	,	PUNCT
cana-3840	160	7	we	we	PRON
cana-3840	160	8	suppose	suppose	VERB
cana-3840	160	9	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	160	10	,	,	PUNCT
cana-3840	160	11	𝕓	𝕓	X
cana-3840	160	12	)	)	PUNCT
cana-3840	160	13	≥	≥	NOUN
cana-3840	160	14	0	0	NUM
cana-3840	160	15	;	;	PUNCT
cana-3840	160	16	however	however	ADV
cana-3840	160	17	,	,	PUNCT
cana-3840	160	18	𝕒	𝕒	PROPN
cana-3840	160	19	≠	≠	PROPN
cana-3840	160	20	𝕓.	𝕓.	NOUN
cana-3840	160	21	then	then	ADV
cana-3840	160	22	,	,	PUNCT
cana-3840	160	23	there	there	PRON
cana-3840	160	24	exists	exist	VERB
cana-3840	160	25	𝓉0	𝓉0	PROPN
cana-3840	160	26	>	>	X
cana-3840	160	27	0	0	NUM
cana-3840	160	28	such	such	ADJ
cana-3840	160	29	that	that	PRON
cana-3840	160	30	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	160	31	,	,	PUNCT
cana-3840	160	32	𝕓	𝕓	X
cana-3840	160	33	,	,	PUNCT
cana-3840	160	34	𝓉0	𝓉0	ADJ
cana-3840	160	35	)	)	PUNCT
cana-3840	160	36	≠	≠	PROPN
cana-3840	160	37	0	0	NUM
cana-3840	160	38	,	,	PUNCT
cana-3840	160	39	that	that	ADV
cana-3840	160	40	is	is	ADV
cana-3840	160	41	,	,	PUNCT
cana-3840	160	42	𝕎(𝕒	𝕎(𝕒	X
cana-3840	160	43	,	,	PUNCT
cana-3840	160	44	𝕓	𝕓	X
cana-3840	160	45	,	,	PUNCT
cana-3840	160	46	𝓉0	𝓉0	ADJ
cana-3840	160	47	)	)	PUNCT
cana-3840	160	48	≥	≥	NOUN
cana-3840	160	49	0	0	NUM
cana-3840	160	50	.	.	PUNCT
cana-3840	161	1	since	since	SCONJ
cana-3840	161	2	𝒦	𝒦	PROPN
cana-3840	161	3	is	be	AUX
cana-3840	161	4	continuous	continuous	ADJ
cana-3840	161	5	at	at	ADP
cana-3840	161	6	0	0	NUM
cana-3840	161	7	,	,	PUNCT
cana-3840	161	8	there	there	PRON
cana-3840	161	9	exists	exist	VERB
cana-3840	161	10	0	0	PUNCT
cana-3840	161	11	<	<	X
cana-3840	161	12	𝓇	𝓇	X
cana-3840	161	13	<	<	X
cana-3840	161	14	𝒢(𝓉	𝒢(𝓉	X
cana-3840	161	15	)	)	PUNCT
cana-3840	161	16	such	such	ADJ
cana-3840	161	17	that	that	DET
cana-3840	161	18	𝒦(𝓇0	𝒦(𝓇0	NOUN
cana-3840	161	19	)	)	PUNCT
cana-3840	161	20	>	>	X
cana-3840	162	1	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	162	2	,	,	PUNCT
cana-3840	162	3	𝕓	𝕓	X
cana-3840	162	4	,	,	PUNCT
cana-3840	162	5	𝓉0	𝓉0	ADJ
cana-3840	162	6	)	)	PUNCT
cana-3840	162	7	.	.	PUNCT
cana-3840	163	1	this	this	PRON
cana-3840	163	2	is	be	AUX
cana-3840	163	3	indirect	indirect	ADJ
cana-3840	163	4	contradiction	contradiction	NOUN
cana-3840	163	5	to	to	ADP
cana-3840	163	6	(	(	PUNCT
cana-3840	163	7	23	23	NUM
cana-3840	163	8	)	)	PUNCT
cana-3840	163	9	.	.	PUNCT
cana-3840	164	1	thus	thus	ADV
cana-3840	164	2	,	,	PUNCT
cana-3840	164	3	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	164	4	,	,	PUNCT
cana-3840	164	5	𝕓	𝕓	X
cana-3840	164	6	)	)	PUNCT
cana-3840	164	7	=	=	SYM
cana-3840	164	8	0	0	NUM
cana-3840	164	9	implies	imply	VERB
cana-3840	164	10	that	that	SCONJ
cana-3840	164	11	𝕒	𝕒	X
cana-3840	164	12	=	=	PUNCT
cana-3840	164	13	𝕓.	𝕓.	NOUN
cana-3840	164	14	to	to	PART
cana-3840	164	15	prove	prove	VERB
cana-3840	164	16	(	(	PUNCT
cana-3840	164	17	m3	m3	PROPN
cana-3840	164	18	)	)	PUNCT
cana-3840	164	19	,	,	PUNCT
cana-3840	164	20	we	we	PRON
cana-3840	164	21	take	take	VERB
cana-3840	164	22	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	164	23	,	,	PUNCT
cana-3840	164	24	𝕔	𝕔	NOUN
cana-3840	164	25	)	)	PUNCT
cana-3840	164	26	<	<	X
cana-3840	164	27	𝓇1	𝓇1	PROPN
cana-3840	164	28	and	and	CCONJ
cana-3840	164	29	𝕕(𝕔	𝕕(𝕔	PROPN
cana-3840	164	30	,	,	PUNCT
cana-3840	164	31	𝕓	𝕓	X
cana-3840	164	32	)	)	PUNCT
cana-3840	164	33	<	<	X
cana-3840	164	34	𝓇2	𝓇2	NOUN
cana-3840	164	35	arbitrarily	arbitrarily	ADV
cana-3840	164	36	.	.	PUNCT
cana-3840	165	1	from	from	ADP
cana-3840	165	2	(	(	PUNCT
cana-3840	165	3	23	23	NUM
cana-3840	165	4	)	)	PUNCT
cana-3840	165	5	,	,	PUNCT
cana-3840	165	6	we	we	PRON
cana-3840	165	7	know	know	VERB
cana-3840	165	8	(	(	PUNCT
cana-3840	165	9	i	i	NOUN
cana-3840	165	10	)	)	PUNCT
cana-3840	165	11	if	if	SCONJ
cana-3840	165	12	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	165	13	)	)	PUNCT
cana-3840	165	14	>	>	X
cana-3840	166	1	𝓇1	𝓇1	PROPN
cana-3840	166	2	,	,	PUNCT
cana-3840	166	3	then	then	ADV
cana-3840	166	4	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	166	5	,	,	PUNCT
cana-3840	166	6	𝕔	𝕔	PROPN
cana-3840	166	7	,	,	PUNCT
cana-3840	166	8	𝓉	𝓉	PROPN
cana-3840	166	9	)	)	PUNCT
cana-3840	166	10	≤	≤	NUM
cana-3840	166	11	𝒦(𝓇1	𝒦(𝓇1	NOUN
cana-3840	166	12	)	)	PUNCT
cana-3840	166	13	≤	≤	NUM
cana-3840	166	14	𝒦(𝓇1	𝒦(𝓇1	NOUN
cana-3840	166	15	)	)	PUNCT
cana-3840	167	1	+	+	CCONJ
cana-3840	167	2	𝓇2	𝓇2	NOUN
cana-3840	167	3	.	.	PUNCT
cana-3840	168	1	(	(	PUNCT
cana-3840	168	2	24	24	NUM
cana-3840	168	3	)	)	PUNCT
cana-3840	168	4	communications	communication	NOUN
cana-3840	168	5	on	on	ADP
cana-3840	168	6	applied	apply	VERB
cana-3840	168	7	nonlinear	nonlinear	ADJ
cana-3840	168	8	analysis	analysis	NOUN
cana-3840	168	9	issn	issn	NOUN
cana-3840	168	10	:	:	PUNCT
cana-3840	168	11	1074	1074	NUM
cana-3840	168	12	-	-	PUNCT
cana-3840	168	13	133x	133x	NUM
cana-3840	168	14	vol	vol	NOUN
cana-3840	168	15	32	32	NUM
cana-3840	168	16	no	no	NOUN
cana-3840	168	17	.	.	PUNCT
cana-3840	169	1	9s	9s	NUM
cana-3840	169	2	(	(	PUNCT
cana-3840	169	3	2025	2025	NUM
cana-3840	169	4	)	)	PUNCT
cana-3840	169	5	94	94	NUM
cana-3840	170	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	170	2	(	(	PUNCT
cana-3840	170	3	ii	ii	NOUN
cana-3840	170	4	)	)	PUNCT
cana-3840	170	5	if	if	SCONJ
cana-3840	170	6	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	170	7	)	)	PUNCT
cana-3840	170	8	>	>	X
cana-3840	170	9	𝓇2	𝓇2	NOUN
cana-3840	170	10	,	,	PUNCT
cana-3840	170	11	then	then	ADV
cana-3840	170	12	𝕎(𝕔	𝕎(𝕔	PROPN
cana-3840	170	13	,	,	PUNCT
cana-3840	170	14	𝕓	𝕓	X
cana-3840	170	15	,	,	PUNCT
cana-3840	170	16	𝓉	𝓉	PROPN
cana-3840	170	17	)	)	PUNCT
cana-3840	170	18	≤	≤	NUM
cana-3840	170	19	𝒦(𝓇2	𝒦(𝓇2	NOUN
cana-3840	170	20	)	)	PUNCT
cana-3840	170	21	≤	≤	NUM
cana-3840	170	22	𝒦(𝓇1	𝒦(𝓇1	NOUN
cana-3840	170	23	)	)	PUNCT
cana-3840	171	1	+	+	CCONJ
cana-3840	171	2	𝓇2	𝓇2	NOUN
cana-3840	171	3	.	.	PUNCT
cana-3840	172	1	(	(	PUNCT
cana-3840	172	2	25	25	NUM
cana-3840	172	3	)	)	PUNCT
cana-3840	172	4	now	now	ADV
cana-3840	172	5	,	,	PUNCT
cana-3840	172	6	suppose	suppose	VERB
cana-3840	172	7	that	that	SCONJ
cana-3840	172	8	𝓇1	𝓇1	PROPN
cana-3840	172	9	+	+	CCONJ
cana-3840	172	10	𝓇2	𝓇2	NOUN
cana-3840	172	11	<	<	X
cana-3840	172	12	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	172	13	)	)	PUNCT
cana-3840	172	14	.	.	PUNCT
cana-3840	173	1	let	let	VERB
cana-3840	173	2	𝓉0	𝓉0	ADJ
cana-3840	173	3	=	=	PUNCT
cana-3840	173	4	𝑠𝑢𝑝{𝓈	𝑠𝑢𝑝{𝓈	NOUN
cana-3840	173	5	:	:	PUNCT
cana-3840	173	6	0	0	NUM
cana-3840	173	7	<	<	X
cana-3840	173	8	𝓈	𝓈	X
cana-3840	173	9	≤	≤	PUNCT
cana-3840	173	10	𝓉	𝓉	PROPN
cana-3840	173	11	,	,	PUNCT
cana-3840	173	12	𝒢(𝓈	𝒢(𝓈	X
cana-3840	173	13	)	)	PUNCT
cana-3840	173	14	>	>	X
cana-3840	173	15	𝓇1	𝓇1	PROPN
cana-3840	173	16	}	}	PUNCT
cana-3840	173	17	.	.	PUNCT
cana-3840	174	1	(	(	PUNCT
cana-3840	174	2	26	26	NUM
cana-3840	174	3	)	)	PUNCT
cana-3840	174	4	obviously	obviously	ADV
cana-3840	174	5	,	,	PUNCT
cana-3840	174	6	𝓉0	𝓉0	PROPN
cana-3840	174	7	≤	≤	PROPN
cana-3840	174	8	𝓉.	𝓉.	NOUN
cana-3840	174	9	it	it	PRON
cana-3840	174	10	is	be	AUX
cana-3840	174	11	easy	easy	ADJ
cana-3840	174	12	to	to	PART
cana-3840	174	13	prove	prove	VERB
cana-3840	174	14	that	that	SCONJ
cana-3840	174	15	𝒢(𝓉0	𝒢(𝓉0	NOUN
cana-3840	174	16	)	)	PUNCT
cana-3840	174	17	≤	≤	NUM
cana-3840	174	18	𝓇1	𝓇1	PROPN
cana-3840	174	19	.	.	PUNCT
cana-3840	175	1	in	in	ADP
cana-3840	175	2	fact	fact	NOUN
cana-3840	175	3	,	,	PUNCT
cana-3840	175	4	if	if	SCONJ
cana-3840	175	5	𝒢(𝓉0	𝒢(𝓉0	PRON
cana-3840	175	6	)	)	PUNCT
cana-3840	175	7	≤	≤	NUM
cana-3840	175	8	𝓇1	𝓇1	PROPN
cana-3840	175	9	,	,	PUNCT
cana-3840	175	10	from	from	ADP
cana-3840	175	11	the	the	DET
cana-3840	175	12	right	right	ADJ
cana-3840	175	13	continuity	continuity	NOUN
cana-3840	175	14	of	of	ADP
cana-3840	175	15	𝒢	𝒢	PROPN
cana-3840	175	16	,	,	PUNCT
cana-3840	175	17	there	there	PRON
cana-3840	175	18	exists	exist	VERB
cana-3840	175	19	η	η	PROPN
cana-3840	175	20	>	>	X
cana-3840	175	21	0	0	PROPN
cana-3840	175	22	,	,	PUNCT
cana-3840	175	23	such	such	ADJ
cana-3840	175	24	that	that	SCONJ
cana-3840	175	25	𝒢(𝓉0	𝒢(𝓉0	AUX
cana-3840	175	26	−	−	NUM
cana-3840	175	27	𝜂	𝜂	NOUN
cana-3840	175	28	)	)	PUNCT
cana-3840	175	29	≤	≤	NUM
cana-3840	175	30	𝓇1	𝓇1	PROPN
cana-3840	175	31	.	.	PUNCT
cana-3840	176	1	by	by	ADP
cana-3840	176	2	the	the	DET
cana-3840	176	3	definition	definition	NOUN
cana-3840	176	4	of	of	ADP
cana-3840	176	5	𝓉0	𝓉0	PROPN
cana-3840	176	6	,	,	PUNCT
cana-3840	176	7	we	we	PRON
cana-3840	176	8	conclude	conclude	VERB
cana-3840	176	9	that	that	SCONJ
cana-3840	176	10	𝓉0	𝓉0	PROPN
cana-3840	176	11	≤	≤	PROPN
cana-3840	176	12	𝓉0	𝓉0	PROPN
cana-3840	176	13	−	−	PROPN
cana-3840	176	14	𝜂	𝜂	PROPN
cana-3840	176	15	,	,	PUNCT
cana-3840	176	16	a	a	DET
cana-3840	176	17	contradiction	contradiction	NOUN
cana-3840	176	18	.	.	PUNCT
cana-3840	177	1	the	the	DET
cana-3840	177	2	fact	fact	NOUN
cana-3840	177	3	𝒢(𝓉0	𝒢(𝓉0	NOUN
cana-3840	177	4	)	)	PUNCT
cana-3840	177	5	≤	≤	ADJ
cana-3840	177	6	𝓇1	𝓇1	PROPN
cana-3840	177	7	implies	imply	VERB
cana-3840	177	8	that	that	SCONJ
cana-3840	177	9	𝒢(𝓉	𝒢(𝓉	PUNCT
cana-3840	177	10	−	−	PUNCT
cana-3840	177	11	𝓉0	𝓉0	ADJ
cana-3840	177	12	)	)	PUNCT
cana-3840	177	13	≤	≤	NOUN
cana-3840	177	14	(	(	PUNCT
cana-3840	177	15	𝒢𝓉	𝒢𝓉	ADP
cana-3840	177	16	−	−	NOUN
cana-3840	177	17	𝒢𝓉0	𝒢𝓉0	NOUN
cana-3840	177	18	)	)	PUNCT
cana-3840	177	19	≤	≤	NOUN
cana-3840	177	20	𝓇2	𝓇2	NOUN
cana-3840	177	21	by	by	ADP
cana-3840	177	22	the	the	DET
cana-3840	177	23	right	right	ADJ
cana-3840	177	24	continuity	continuity	NOUN
cana-3840	177	25	of	of	ADP
cana-3840	177	26	𝒢	𝒢	PROPN
cana-3840	177	27	again	again	ADV
cana-3840	177	28	,	,	PUNCT
cana-3840	177	29	we	we	PRON
cana-3840	177	30	know	know	VERB
cana-3840	177	31	there	there	PRON
cana-3840	177	32	exists	exist	VERB
cana-3840	177	33	휀	휀	PRON
cana-3840	177	34	>	>	X
cana-3840	177	35	0	0	NUM
cana-3840	177	36	such	such	ADJ
cana-3840	177	37	that	that	SCONJ
cana-3840	177	38	𝒢(𝓉	𝒢(𝓉	PUNCT
cana-3840	177	39	−	−	PROPN
cana-3840	177	40	(	(	PUNCT
cana-3840	177	41	𝓉0	𝓉0	PROPN
cana-3840	177	42	+	+	CCONJ
cana-3840	177	43	휀	휀	NOUN
cana-3840	177	44	)	)	PUNCT
cana-3840	177	45	)	)	PUNCT
cana-3840	177	46	≤	≤	NOUN
cana-3840	177	47	𝒢(𝓉	𝒢(𝓉	PUNCT
cana-3840	177	48	−	−	PROPN
cana-3840	177	49	(	(	PUNCT
cana-3840	177	50	𝓉0	𝓉0	PROPN
cana-3840	177	51	−	−	PROPN
cana-3840	177	52	휀	휀	NOUN
cana-3840	177	53	)	)	PUNCT
cana-3840	177	54	)	)	PUNCT
cana-3840	178	1	<	<	X
cana-3840	178	2	𝓇2	𝓇2	NOUN
cana-3840	178	3	.	.	PUNCT
cana-3840	179	1	by	by	ADP
cana-3840	179	2	the	the	DET
cana-3840	179	3	definition	definition	NOUN
cana-3840	179	4	of	of	ADP
cana-3840	179	5	𝓉0	𝓉0	PROPN
cana-3840	179	6	,	,	PUNCT
cana-3840	179	7	there	there	PRON
cana-3840	179	8	exists	exist	VERB
cana-3840	179	9	0	0	PUNCT
cana-3840	179	10	<	<	X
cana-3840	179	11	𝓈1	𝓈1	NOUN
cana-3840	179	12	≤	≤	NUM
cana-3840	179	13	𝑡	𝑡	NOUN
cana-3840	179	14	such	such	ADJ
cana-3840	179	15	that	that	DET
cana-3840	179	16	𝒢(𝓈1	𝒢(𝓈1	NOUN
cana-3840	179	17	)	)	PUNCT
cana-3840	179	18	≤	≤	NOUN
cana-3840	179	19	𝓇1	𝓇1	PROPN
cana-3840	179	20	and	and	CCONJ
cana-3840	179	21	𝓈1	𝓈1	NOUN
cana-3840	179	22	≤	≤	NOUN
cana-3840	179	23	𝓉0	𝓉0	PROPN
cana-3840	180	1	+	+	CCONJ
cana-3840	180	2	휀	휀	AUX
cana-3840	180	3	.	.	PUNCT
cana-3840	180	4	noting	note	VERB
cana-3840	180	5	that	that	SCONJ
cana-3840	180	6	g	g	PROPN
cana-3840	180	7	is	be	AUX
cana-3840	180	8	increasing	increase	VERB
cana-3840	180	9	,	,	PUNCT
cana-3840	180	10	we	we	PRON
cana-3840	180	11	obtain	obtain	VERB
cana-3840	180	12	that	that	SCONJ
cana-3840	180	13	𝒢(𝓉	𝒢(𝓉	PUNCT
cana-3840	180	14	−	−	DET
cana-3840	180	15	휀	휀	NOUN
cana-3840	180	16	)	)	PUNCT
cana-3840	180	17	)	)	PUNCT
cana-3840	180	18	≤	≤	NOUN
cana-3840	181	1	𝒢(𝓈1	𝒢(𝓈1	NOUN
cana-3840	181	2	)	)	PUNCT
cana-3840	182	1	<	<	X
cana-3840	182	2	𝓇1	𝓇1	PROPN
cana-3840	182	3	.	.	PUNCT
cana-3840	183	1	from	from	ADP
cana-3840	183	2	(	(	PUNCT
cana-3840	183	3	i	i	NOUN
cana-3840	183	4	)	)	PUNCT
cana-3840	183	5	and	and	CCONJ
cana-3840	183	6	(	(	PUNCT
cana-3840	183	7	ii	ii	NOUN
cana-3840	183	8	)	)	PUNCT
cana-3840	183	9	,	,	PUNCT
cana-3840	183	10	we	we	PRON
cana-3840	183	11	have	have	VERB
cana-3840	183	12	𝕎(𝕔	𝕎(𝕔	PROPN
cana-3840	183	13	,	,	PUNCT
cana-3840	183	14	𝕓	𝕓	PRON
cana-3840	183	15	,	,	PUNCT
cana-3840	183	16	(	(	PUNCT
cana-3840	183	17	𝓉0	𝓉0	ADJ
cana-3840	183	18	+	+	CCONJ
cana-3840	183	19	휀	휀	NOUN
cana-3840	183	20	)	)	PUNCT
cana-3840	183	21	)	)	PUNCT
cana-3840	183	22	≤	≤	NUM
cana-3840	184	1	𝒦(𝓇1	𝒦(𝓇1	NOUN
cana-3840	184	2	)	)	PUNCT
cana-3840	184	3	≤	≤	PUNCT
cana-3840	185	1	𝒦(𝓇1	𝒦(𝓇1	PROPN
cana-3840	185	2	+	+	X
cana-3840	185	3	𝓇2	𝓇2	NOUN
cana-3840	185	4	)	)	PUNCT
cana-3840	185	5	𝕎(𝕔	𝕎(𝕔	PROPN
cana-3840	185	6	,	,	PUNCT
cana-3840	185	7	𝕓	𝕓	PRON
cana-3840	185	8	,	,	PUNCT
cana-3840	185	9	𝓉	𝓉	PROPN
cana-3840	185	10	−	−	PROPN
cana-3840	185	11	(	(	PUNCT
cana-3840	185	12	𝓉0	𝓉0	PROPN
cana-3840	185	13	+	+	CCONJ
cana-3840	185	14	휀	휀	NOUN
cana-3840	185	15	)	)	PUNCT
cana-3840	185	16	)	)	PUNCT
cana-3840	185	17	≤	≤	NUM
cana-3840	185	18	𝒦(𝓇2	𝒦(𝓇2	NOUN
cana-3840	185	19	)	)	PUNCT
cana-3840	185	20	≤	≤	NUM
cana-3840	185	21	𝒦(𝓇1	𝒦(𝓇1	NOUN
cana-3840	185	22	)	)	PUNCT
cana-3840	186	1	+	+	CCONJ
cana-3840	186	2	𝓇2	𝓇2	NOUN
cana-3840	186	3	.	.	PUNCT
cana-3840	187	1	(	(	PUNCT
cana-3840	187	2	27	27	NUM
cana-3840	187	3	)	)	PUNCT
cana-3840	187	4	combining	combine	VERB
cana-3840	187	5	conditions	condition	NOUN
cana-3840	187	6	(	(	PUNCT
cana-3840	187	7	i	i	NOUN
cana-3840	187	8	)	)	PUNCT
cana-3840	187	9	and	and	CCONJ
cana-3840	187	10	(	(	PUNCT
cana-3840	187	11	ii	ii	NOUN
cana-3840	187	12	)	)	PUNCT
cana-3840	187	13	,	,	PUNCT
cana-3840	187	14	we	we	PRON
cana-3840	187	15	get	get	VERB
cana-3840	187	16	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	187	17	,	,	PUNCT
cana-3840	187	18	𝕓	𝕓	X
cana-3840	187	19	,	,	PUNCT
cana-3840	187	20	𝓉	𝓉	PROPN
cana-3840	187	21	)	)	PUNCT
cana-3840	187	22	≤	≤	PUNCT
cana-3840	188	1	𝒦(𝓇1	𝒦(𝓇1	PROPN
cana-3840	188	2	+	+	X
cana-3840	188	3	𝓇2	𝓇2	NOUN
cana-3840	188	4	)	)	PUNCT
cana-3840	188	5	.	.	PUNCT
cana-3840	189	1	by	by	ADP
cana-3840	189	2	the	the	DET
cana-3840	189	3	definition	definition	NOUN
cana-3840	189	4	of	of	ADP
cana-3840	189	5	𝕕	𝕕	X
cana-3840	189	6	,	,	PUNCT
cana-3840	189	7	we	we	PRON
cana-3840	189	8	know	know	VERB
cana-3840	189	9	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	189	10	,	,	PUNCT
cana-3840	189	11	𝕓	𝕓	X
cana-3840	189	12	)	)	PUNCT
cana-3840	189	13	≤	≤	NUM
cana-3840	189	14	𝓇1	𝓇1	PROPN
cana-3840	189	15	+	+	CCONJ
cana-3840	189	16	𝓇2	𝓇2	NOUN
cana-3840	189	17	.	.	PUNCT
cana-3840	190	1	since	since	SCONJ
cana-3840	190	2	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	190	3	,	,	PUNCT
cana-3840	190	4	𝕔	𝕔	NOUN
cana-3840	190	5	)	)	PUNCT
cana-3840	190	6	<	<	X
cana-3840	190	7	𝓇1	𝓇1	PROPN
cana-3840	190	8	and	and	CCONJ
cana-3840	190	9	𝕕(𝕔	𝕕(𝕔	PROPN
cana-3840	190	10	,	,	PUNCT
cana-3840	190	11	𝕓	𝕓	X
cana-3840	190	12	)	)	PUNCT
cana-3840	190	13	<	<	X
cana-3840	190	14	𝓇2	𝓇2	NOUN
cana-3840	190	15	,	,	PUNCT
cana-3840	190	16	we	we	PRON
cana-3840	190	17	obtain	obtain	VERB
cana-3840	190	18	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	190	19	,	,	PUNCT
cana-3840	190	20	𝕓	𝕓	X
cana-3840	190	21	)	)	PUNCT
cana-3840	190	22	≤	≤	NOUN
cana-3840	190	23	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	190	24	,	,	PUNCT
cana-3840	190	25	𝕔	𝕔	NOUN
cana-3840	190	26	)	)	PUNCT
cana-3840	191	1	+	+	CCONJ
cana-3840	191	2	𝕕(𝕔	𝕕(𝕔	PROPN
cana-3840	191	3	,	,	PUNCT
cana-3840	191	4	𝕓	𝕓	X
cana-3840	191	5	)	)	PUNCT
cana-3840	191	6	.	.	PUNCT
cana-3840	192	1	(	(	PUNCT
cana-3840	192	2	28	28	NUM
cana-3840	192	3	)	)	PUNCT
cana-3840	192	4	directly	directly	ADV
cana-3840	192	5	.	.	PUNCT
cana-3840	193	1	theorem	theorem	ADJ
cana-3840	193	2	4	4	NUM
cana-3840	193	3	.	.	PUNCT
cana-3840	193	4	(	(	PUNCT
cana-3840	193	5	𝔘	𝔘	PROPN
cana-3840	193	6	,	,	PUNCT
cana-3840	193	7	𝕎	𝕎	PROPN
cana-3840	193	8	,	,	PUNCT
cana-3840	193	9	⨁	⨁	PROPN
cana-3840	193	10	)	)	PUNCT
cana-3840	193	11	is	be	AUX
cana-3840	193	12	a	a	DET
cana-3840	193	13	revised	revise	VERB
cana-3840	193	14	fuzzy	fuzzy	ADJ
cana-3840	193	15	metric	metric	ADJ
cana-3840	193	16	space	space	NOUN
cana-3840	193	17	.	.	PUNCT
cana-3840	194	1	𝕎	𝕎	NOUN
cana-3840	194	2	satisfies	satisfy	VERB
cana-3840	194	3	condition	condition	NOUN
cana-3840	194	4	(	(	PUNCT
cana-3840	194	5	15	15	NUM
cana-3840	194	6	)	)	PUNCT
cana-3840	194	7	or	or	CCONJ
cana-3840	194	8	⨁	⨁	PROPN
cana-3840	194	9	≥	≥	NUM
cana-3840	194	10	𝛥1	𝛥1	PROPN
cana-3840	194	11	.	.	PUNCT
cana-3840	195	1	let	let	VERB
cana-3840	195	2	𝜌(𝕒	𝜌(𝕒	PROPN
cana-3840	195	3	,	,	PUNCT
cana-3840	195	4	𝕓	𝕓	X
cana-3840	195	5	)	)	PUNCT
cana-3840	195	6	=	=	PUNCT
cana-3840	196	1	𝑠𝑢𝑝{𝕩	𝑠𝑢𝑝{𝕩	X
cana-3840	196	2	>	>	X
cana-3840	196	3	0	0	NUM
cana-3840	196	4	:	:	PUNCT
cana-3840	196	5	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	196	6	,	,	PUNCT
cana-3840	196	7	𝕓	𝕓	X
cana-3840	196	8	,	,	PUNCT
cana-3840	196	9	𝕩	𝕩	NOUN
cana-3840	196	10	)	)	PUNCT
cana-3840	197	1	+	+	CCONJ
cana-3840	197	2	𝕩	𝕩	X
cana-3840	197	3	≤	≤	NUM
cana-3840	197	4	0	0	NUM
cana-3840	197	5	}	}	PUNCT
cana-3840	197	6	,	,	PUNCT
cana-3840	197	7	∀𝕒	∀𝕒	PROPN
cana-3840	197	8	,	,	PUNCT
cana-3840	197	9	𝕓	𝕓	X
cana-3840	197	10	∈	∈	PROPN
cana-3840	197	11	𝔘.	𝔘.	PROPN
cana-3840	197	12	(	(	PUNCT
cana-3840	197	13	29	29	NUM
cana-3840	197	14	)	)	PUNCT
cana-3840	197	15	then	then	ADV
cana-3840	197	16	,	,	PUNCT
cana-3840	197	17	𝜌	𝜌	X
cana-3840	197	18	is	be	AUX
cana-3840	197	19	a	a	DET
cana-3840	197	20	metric	metric	NOUN
cana-3840	197	21	on	on	ADP
cana-3840	197	22	𝔘	𝔘	NOUN
cana-3840	197	23	and	and	CCONJ
cana-3840	197	24	the	the	DET
cana-3840	197	25	topology	topology	NOUN
cana-3840	197	26	𝜏𝜌	𝜏𝜌	ADV
cana-3840	197	27	induced	induce	VERB
cana-3840	197	28	by	by	ADP
cana-3840	197	29	𝜌	𝜌	X
cana-3840	197	30	coincides	coincide	NOUN
cana-3840	197	31	with	with	ADP
cana-3840	197	32	the	the	DET
cana-3840	197	33	topology	topology	NOUN
cana-3840	197	34	𝜏𝕎.	𝜏𝕎.	ADJ
cana-3840	197	35	proof	proof	NOUN
cana-3840	197	36	.	.	PUNCT
cana-3840	198	1	let	let	VERB
cana-3840	198	2	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	198	3	)	)	PUNCT
cana-3840	198	4	=	=	SYM
cana-3840	198	5	𝓉	𝓉	PROPN
cana-3840	198	6	,	,	PUNCT
cana-3840	198	7	𝒦(𝓉	𝒦(𝓉	X
cana-3840	198	8	)	)	PUNCT
cana-3840	198	9	=	=	SYM
cana-3840	198	10	𝓉	𝓉	PROPN
cana-3840	198	11	,	,	PUNCT
cana-3840	198	12	∀t	∀t	PROPN
cana-3840	198	13	≥	≥	NOUN
cana-3840	198	14	0	0	NUM
cana-3840	198	15	.	.	PUNCT
cana-3840	199	1	then	then	ADV
cana-3840	199	2	,	,	PUNCT
cana-3840	199	3	𝒦	𝒦	PROPN
cana-3840	199	4	and	and	CCONJ
cana-3840	199	5	𝒢	𝒢	PROPN
cana-3840	199	6	satisfy	satisfy	VERB
cana-3840	199	7	the	the	DET
cana-3840	199	8	conditions	condition	NOUN
cana-3840	199	9	ℭ1	ℭ1	ADV
cana-3840	199	10	and	and	CCONJ
cana-3840	199	11	ℭ2	ℭ2	PROPN
cana-3840	199	12	,	,	PUNCT
cana-3840	199	13	respectively	respectively	ADV
cana-3840	199	14	.	.	PUNCT
cana-3840	200	1	besides	besides	ADV
cana-3840	200	2	,	,	PUNCT
cana-3840	200	3	𝒦	𝒦	PROPN
cana-3840	200	4	satisfies	satisfy	VERB
cana-3840	200	5	condition	condition	NOUN
cana-3840	200	6	(	(	PUNCT
cana-3840	200	7	21	21	NUM
cana-3840	200	8	)	)	PUNCT
cana-3840	200	9	.	.	PUNCT
cana-3840	201	1	from	from	ADP
cana-3840	201	2	theorem	theorem	ADJ
cana-3840	201	3	3	3	NUM
cana-3840	201	4	,	,	PUNCT
cana-3840	201	5	we	we	PRON
cana-3840	201	6	know	know	VERB
cana-3840	201	7	that	that	SCONJ
cana-3840	201	8	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	201	9	,	,	PUNCT
cana-3840	201	10	𝕓	𝕓	X
cana-3840	201	11	)	)	PUNCT
cana-3840	201	12	=	=	SYM
cana-3840	201	13	𝑠𝑢𝑝{𝓈	𝑠𝑢𝑝{𝓈	NOUN
cana-3840	201	14	>	>	NOUN
cana-3840	201	15	0	0	NUM
cana-3840	201	16	:	:	PUNCT
cana-3840	202	1	𝑖𝑓	𝑖𝑓	ADP
cana-3840	202	2	𝓉	𝓉	PROPN
cana-3840	202	3	>	>	PUNCT
cana-3840	202	4	𝓈	𝓈	PROPN
cana-3840	202	5	,	,	PUNCT
cana-3840	202	6	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	202	7	,	,	PUNCT
cana-3840	202	8	𝕓	𝕓	X
cana-3840	202	9	,	,	PUNCT
cana-3840	202	10	𝓉	𝓉	PRON
cana-3840	202	11	)	)	PUNCT
cana-3840	202	12	+	+	CCONJ
cana-3840	202	13	𝓈	𝓈	X
cana-3840	202	14	≤	≤	NUM
cana-3840	202	15	0	0	NUM
cana-3840	202	16	}	}	PUNCT
cana-3840	202	17	,	,	PUNCT
cana-3840	202	18	∀𝕒	∀𝕒	PROPN
cana-3840	202	19	,	,	PUNCT
cana-3840	202	20	𝕓	𝕓	X
cana-3840	202	21	∈	∈	PROPN
cana-3840	202	22	𝔘.	𝔘.	PROPN
cana-3840	202	23	(	(	PUNCT
cana-3840	202	24	30	30	NUM
cana-3840	202	25	)	)	PUNCT
cana-3840	202	26	is	be	AUX
cana-3840	202	27	a	a	DET
cana-3840	202	28	metric	metric	NOUN
cana-3840	202	29	on	on	ADP
cana-3840	202	30	𝔘.	𝔘.	PROPN
cana-3840	202	31	thus	thus	ADV
cana-3840	202	32	,	,	PUNCT
cana-3840	202	33	to	to	PART
cana-3840	202	34	show	show	VERB
cana-3840	202	35	that	that	SCONJ
cana-3840	202	36	𝜌	𝜌	PRON
cana-3840	202	37	is	be	AUX
cana-3840	202	38	a	a	DET
cana-3840	202	39	metric	metric	NOUN
cana-3840	202	40	,	,	PUNCT
cana-3840	202	41	we	we	PRON
cana-3840	202	42	only	only	ADV
cana-3840	202	43	need	need	VERB
cana-3840	202	44	to	to	PART
cana-3840	202	45	show	show	VERB
cana-3840	202	46	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	202	47	,	,	PUNCT
cana-3840	202	48	𝕓	𝕓	X
cana-3840	202	49	)	)	PUNCT
cana-3840	202	50	≤	≤	NOUN
cana-3840	202	51	𝜌(𝕒	𝜌(𝕒	PROPN
cana-3840	202	52	,	,	PUNCT
cana-3840	202	53	𝕔	𝕔	NOUN
cana-3840	202	54	)	)	PUNCT
cana-3840	202	55	,	,	PUNCT
cana-3840	202	56	∀𝕒	∀𝕒	PROPN
cana-3840	202	57	,	,	PUNCT
cana-3840	202	58	𝕓	𝕓	X
cana-3840	202	59	∈	∈	PROPN
cana-3840	202	60	𝔘.	𝔘.	PROPN
cana-3840	202	61	in	in	ADP
cana-3840	202	62	fact	fact	NOUN
cana-3840	202	63	,	,	PUNCT
cana-3840	202	64	if	if	SCONJ
cana-3840	202	65	𝕩	𝕩	PROPN
cana-3840	202	66	>	>	X
cana-3840	202	67	0	0	NUM
cana-3840	202	68	,	,	PUNCT
cana-3840	202	69	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	202	70	,	,	PUNCT
cana-3840	202	71	𝕓	𝕓	X
cana-3840	202	72	,	,	PUNCT
cana-3840	202	73	𝕩	𝕩	NOUN
cana-3840	202	74	)	)	PUNCT
cana-3840	203	1	+	+	CCONJ
cana-3840	203	2	𝕩	𝕩	PROPN
cana-3840	203	3	≤	≤	NOUN
cana-3840	203	4	0	0	NUM
cana-3840	203	5	and	and	CCONJ
cana-3840	203	6	𝓉	𝓉	PROPN
cana-3840	203	7	>	>	X
cana-3840	203	8	𝕩	𝕩	PROPN
cana-3840	203	9	,	,	PUNCT
cana-3840	203	10	then	then	ADV
cana-3840	203	11	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	203	12	,	,	PUNCT
cana-3840	203	13	𝕓	𝕓	X
cana-3840	203	14	,	,	PUNCT
cana-3840	203	15	𝓉	𝓉	PRON
cana-3840	203	16	)	)	PUNCT
cana-3840	204	1	+	+	CCONJ
cana-3840	204	2	𝕩	𝕩	PROPN
cana-3840	204	3	≤	≤	PROPN
cana-3840	204	4	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	204	5	,	,	PUNCT
cana-3840	204	6	𝕓	𝕓	X
cana-3840	204	7	,	,	PUNCT
cana-3840	204	8	𝕩	𝕩	NOUN
cana-3840	204	9	)	)	PUNCT
cana-3840	204	10	+	+	CCONJ
cana-3840	204	11	𝕩	𝕩	PROPN
cana-3840	204	12	≤	≤	PROPN
cana-3840	204	13	0	0	NUM
cana-3840	204	14	.	.	PUNCT
cana-3840	205	1	therefore	therefore	ADV
cana-3840	205	2	,	,	PUNCT
cana-3840	205	3	{	{	PUNCT
cana-3840	205	4	𝕩	𝕩	X
cana-3840	205	5	>	>	X
cana-3840	205	6	0	0	NUM
cana-3840	205	7	:	:	PUNCT
cana-3840	205	8	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	205	9	,	,	PUNCT
cana-3840	205	10	𝕓	𝕓	X
cana-3840	205	11	,	,	PUNCT
cana-3840	205	12	𝕩	𝕩	NOUN
cana-3840	205	13	)	)	PUNCT
cana-3840	206	1	+	+	CCONJ
cana-3840	206	2	𝕩	𝕩	PROPN
cana-3840	206	3	≤	≤	ADV
cana-3840	206	4	0	0	NUM
cana-3840	206	5	}	}	PUNCT
cana-3840	206	6	⊆	⊆	NUM
cana-3840	206	7	{	{	PUNCT
cana-3840	206	8	𝓈	𝓈	X
cana-3840	206	9	>	>	X
cana-3840	206	10	0	0	NUM
cana-3840	206	11	:	:	PUNCT
cana-3840	206	12	𝑖𝑓	𝑖𝑓	ADP
cana-3840	206	13	𝓉	𝓉	PROPN
cana-3840	206	14	>	>	PUNCT
cana-3840	206	15	𝓈	𝓈	PROPN
cana-3840	206	16	,	,	PUNCT
cana-3840	206	17	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
cana-3840	206	18	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	206	19	,	,	PUNCT
cana-3840	206	20	𝕓	𝕓	X
cana-3840	206	21	,	,	PUNCT
cana-3840	206	22	𝓉	𝓉	PRON
cana-3840	206	23	)	)	PUNCT
cana-3840	207	1	+	+	CCONJ
cana-3840	207	2	𝓈	𝓈	X
cana-3840	207	3	≤	≤	NUM
cana-3840	207	4	0	0	NUM
cana-3840	207	5	}	}	PUNCT
cana-3840	207	6	.	.	PUNCT
cana-3840	208	1	(	(	PUNCT
cana-3840	208	2	31	31	NUM
cana-3840	208	3	)	)	PUNCT
cana-3840	208	4	from	from	ADP
cana-3840	208	5	(	(	PUNCT
cana-3840	208	6	29	29	NUM
cana-3840	208	7	)	)	PUNCT
cana-3840	208	8	and	and	CCONJ
cana-3840	208	9	(	(	PUNCT
cana-3840	208	10	30	30	NUM
cana-3840	208	11	)	)	PUNCT
cana-3840	208	12	,	,	PUNCT
cana-3840	208	13	we	we	PRON
cana-3840	208	14	get	get	VERB
cana-3840	208	15	𝜌(𝕒	𝜌(𝕒	NOUN
cana-3840	208	16	,	,	PUNCT
cana-3840	208	17	𝕓	𝕓	X
cana-3840	208	18	)	)	PUNCT
cana-3840	208	19	≥	≥	NOUN
cana-3840	208	20	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	208	21	,	,	PUNCT
cana-3840	208	22	𝕓	𝕓	X
cana-3840	208	23	)	)	PUNCT
cana-3840	208	24	.	.	PUNCT
cana-3840	209	1	on	on	ADP
cana-3840	209	2	the	the	DET
cana-3840	209	3	other	other	ADJ
cana-3840	209	4	hand	hand	NOUN
cana-3840	209	5	,	,	PUNCT
cana-3840	209	6	from	from	ADP
cana-3840	209	7	(	(	PUNCT
cana-3840	209	8	30	30	NUM
cana-3840	209	9	)	)	PUNCT
cana-3840	209	10	,	,	PUNCT
cana-3840	209	11	for	for	ADP
cana-3840	209	12	any	any	PRON
cana-3840	209	13	휀	휀	NOUN
cana-3840	209	14	>	>	X
cana-3840	209	15	0	0	NUM
cana-3840	209	16	,	,	PUNCT
cana-3840	209	17	there	there	PRON
cana-3840	209	18	exists	exist	VERB
cana-3840	209	19	𝓈	𝓈	PROPN
cana-3840	209	20	>	>	X
cana-3840	209	21	0	0	NUM
cana-3840	209	22	such	such	ADJ
cana-3840	210	1	that	that	SCONJ
cana-3840	210	2	𝓈	𝓈	X
cana-3840	210	3	>	>	X
cana-3840	210	4	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	210	5	,	,	PUNCT
cana-3840	210	6	𝕓	𝕓	X
cana-3840	210	7	)	)	PUNCT
cana-3840	210	8	+	+	CCONJ
cana-3840	210	9	휀	휀	NOUN
cana-3840	210	10	and	and	CCONJ
cana-3840	210	11	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	210	12	,	,	PUNCT
cana-3840	210	13	𝕓	𝕓	X
cana-3840	210	14	,	,	PUNCT
cana-3840	210	15	𝓉	𝓉	PRON
cana-3840	210	16	)	)	PUNCT
cana-3840	210	17	+	+	CCONJ
cana-3840	210	18	𝓈	𝓈	X
cana-3840	210	19	≤	≤	NUM
cana-3840	210	20	0	0	NUM
cana-3840	211	1	when	when	SCONJ
cana-3840	211	2	𝓉	𝓉	PROPN
cana-3840	211	3	>	>	X
cana-3840	211	4	𝓈.	𝓈.	PROPN
cana-3840	211	5	thus	thus	ADV
cana-3840	211	6	,	,	PUNCT
cana-3840	211	7	communications	communication	NOUN
cana-3840	211	8	on	on	ADP
cana-3840	211	9	applied	apply	VERB
cana-3840	211	10	nonlinear	nonlinear	ADJ
cana-3840	211	11	analysis	analysis	NOUN
cana-3840	211	12	issn	issn	NOUN
cana-3840	211	13	:	:	PUNCT
cana-3840	211	14	1074	1074	NUM
cana-3840	211	15	-	-	PUNCT
cana-3840	211	16	133x	133x	NUM
cana-3840	211	17	vol	vol	NOUN
cana-3840	211	18	32	32	NUM
cana-3840	211	19	no	no	NOUN
cana-3840	212	1	.	.	PUNCT
cana-3840	213	1	9s	9s	NUM
cana-3840	213	2	(	(	PUNCT
cana-3840	213	3	2025	2025	NUM
cana-3840	213	4	)	)	PUNCT
cana-3840	213	5	95	95	NUM
cana-3840	214	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	214	2	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	214	3	,	,	PUNCT
cana-3840	214	4	𝕓	𝕓	X
cana-3840	214	5	,	,	PUNCT
cana-3840	214	6	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	214	7	,	,	PUNCT
cana-3840	214	8	𝕓	𝕓	X
cana-3840	214	9	)	)	PUNCT
cana-3840	214	10	+	+	SYM
cana-3840	214	11	휀	휀	X
cana-3840	214	12	)	)	PUNCT
cana-3840	214	13	+	+	CCONJ
cana-3840	214	14	𝓈	𝓈	X
cana-3840	214	15	≤	≤	NUM
cana-3840	214	16	0	0	NUM
cana-3840	214	17	.	.	PUNCT
cana-3840	215	1	(	(	PUNCT
cana-3840	215	2	32	32	NUM
cana-3840	215	3	)	)	PUNCT
cana-3840	215	4	and	and	CCONJ
cana-3840	215	5	hence	hence	ADV
cana-3840	215	6	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	215	7	,	,	PUNCT
cana-3840	215	8	𝕓	𝕓	X
cana-3840	215	9	,	,	PUNCT
cana-3840	215	10	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	215	11	,	,	PUNCT
cana-3840	215	12	𝕓	𝕓	X
cana-3840	215	13	)	)	PUNCT
cana-3840	215	14	+	+	SYM
cana-3840	215	15	휀	휀	X
cana-3840	215	16	)	)	PUNCT
cana-3840	215	17	+	+	CCONJ
cana-3840	215	18	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	215	19	,	,	PUNCT
cana-3840	215	20	𝕓	𝕓	X
cana-3840	215	21	)	)	PUNCT
cana-3840	215	22	+	+	SYM
cana-3840	216	1	휀	휀	X
cana-3840	216	2	)	)	PUNCT
cana-3840	216	3	<	<	X
cana-3840	216	4	0	0	NUM
cana-3840	216	5	.	.	PUNCT
cana-3840	217	1	(	(	PUNCT
cana-3840	217	2	33	33	NUM
cana-3840	217	3	)	)	PUNCT
cana-3840	217	4	from	from	ADP
cana-3840	217	5	(	(	PUNCT
cana-3840	217	6	29	29	NUM
cana-3840	217	7	)	)	PUNCT
cana-3840	217	8	,	,	PUNCT
cana-3840	217	9	we	we	PRON
cana-3840	217	10	get	get	VERB
cana-3840	217	11	𝜌(𝕒	𝜌(𝕒	NOUN
cana-3840	217	12	,	,	PUNCT
cana-3840	217	13	𝕓	𝕓	X
cana-3840	217	14	)	)	PUNCT
cana-3840	217	15	≤	≤	NOUN
cana-3840	217	16	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	217	17	,	,	PUNCT
cana-3840	217	18	𝕓	𝕓	X
cana-3840	217	19	)	)	PUNCT
cana-3840	218	1	+	+	X
cana-3840	219	1	휀	휀	X
cana-3840	219	2	.	.	NOUN
cana-3840	219	3	from	from	ADP
cana-3840	219	4	the	the	DET
cana-3840	219	5	arbitrariness	arbitrariness	NOUN
cana-3840	219	6	of	of	ADP
cana-3840	219	7	ε	ε	PROPN
cana-3840	219	8	>	>	X
cana-3840	219	9	0	0	PROPN
cana-3840	219	10	,	,	PUNCT
cana-3840	219	11	we	we	PRON
cana-3840	219	12	have	have	VERB
cana-3840	219	13	𝜌(𝕒	𝜌(𝕒	PROPN
cana-3840	219	14	,	,	PUNCT
cana-3840	219	15	𝕓	𝕓	X
cana-3840	219	16	)	)	PUNCT
cana-3840	219	17	≤	≤	NOUN
cana-3840	219	18	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	219	19	,	,	PUNCT
cana-3840	219	20	𝕓	𝕓	X
cana-3840	219	21	)	)	PUNCT
cana-3840	219	22	.	.	PUNCT
cana-3840	220	1	thus	thus	ADV
cana-3840	220	2	,	,	PUNCT
cana-3840	220	3	𝜌(𝕒	𝜌(𝕒	PROPN
cana-3840	220	4	,	,	PUNCT
cana-3840	220	5	𝕓	𝕓	X
cana-3840	220	6	)	)	PUNCT
cana-3840	220	7	=	=	SYM
cana-3840	220	8	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	220	9	,	,	PUNCT
cana-3840	220	10	𝕓	𝕓	X
cana-3840	220	11	)	)	PUNCT
cana-3840	220	12	.	.	PUNCT
cana-3840	221	1	let	let	VERB
cana-3840	221	2	𝓇	𝓇	X
cana-3840	221	3	∈	∈	PROPN
cana-3840	221	4	(	(	PUNCT
cana-3840	221	5	0,1	0,1	NOUN
cana-3840	221	6	)	)	PUNCT
cana-3840	221	7	,	,	PUNCT
cana-3840	221	8	𝕒	𝕒	PROPN
cana-3840	221	9	∈	∈	PROPN
cana-3840	221	10	𝔘.	𝔘.	NOUN
cana-3840	221	11	we	we	PRON
cana-3840	221	12	put	put	VERB
cana-3840	221	13	𝒰(𝕒	𝒰(𝕒	ADP
cana-3840	221	14	,	,	PUNCT
cana-3840	221	15	𝓇	𝓇	NOUN
cana-3840	221	16	)	)	PUNCT
cana-3840	221	17	=	=	NOUN
cana-3840	221	18	{	{	PUNCT
cana-3840	221	19	𝕓	𝕓	PROPN
cana-3840	221	20	∈	∈	PROPN
cana-3840	221	21	𝔘	𝔘	PROPN
cana-3840	221	22	:	:	PUNCT
cana-3840	221	23	𝜌(𝕒	𝜌(𝕒	PROPN
cana-3840	221	24	,	,	PUNCT
cana-3840	221	25	𝕓	𝕓	X
cana-3840	221	26	)	)	PUNCT
cana-3840	221	27	<	<	X
cana-3840	221	28	𝓇	𝓇	NOUN
cana-3840	221	29	}	}	PUNCT
cana-3840	221	30	.	.	PUNCT
cana-3840	222	1	it	it	PRON
cana-3840	222	2	is	be	AUX
cana-3840	222	3	easy	easy	ADJ
cana-3840	222	4	to	to	PART
cana-3840	222	5	show	show	VERB
cana-3840	222	6	that	that	SCONJ
cana-3840	222	7	𝒰(𝕒	𝒰(𝕒	ADP
cana-3840	222	8	,	,	PUNCT
cana-3840	222	9	𝓇	𝓇	NOUN
cana-3840	222	10	)	)	PUNCT
cana-3840	222	11	⊆	⊆	NUM
cana-3840	222	12	ℬ𝕎(𝕒	ℬ𝕎(𝕒	NOUN
cana-3840	222	13	,	,	PUNCT
cana-3840	222	14	𝓇	𝓇	NOUN
cana-3840	222	15	,	,	PUNCT
cana-3840	222	16	𝓇	𝓇	NOUN
cana-3840	222	17	)	)	PUNCT
cana-3840	222	18	⊆	⊆	NUM
cana-3840	222	19	𝒰(𝕒	𝒰(𝕒	NOUN
cana-3840	222	20	,	,	PUNCT
cana-3840	222	21	2𝓇	2𝓇	NOUN
cana-3840	222	22	)	)	PUNCT
cana-3840	222	23	.	.	PUNCT
cana-3840	223	1	(	(	PUNCT
cana-3840	223	2	34	34	NUM
cana-3840	223	3	)	)	PUNCT
cana-3840	223	4	in	in	ADP
cana-3840	223	5	fact	fact	NOUN
cana-3840	223	6	,	,	PUNCT
cana-3840	223	7	for	for	ADP
cana-3840	223	8	any	any	DET
cana-3840	223	9	𝕓	𝕓	PROPN
cana-3840	223	10	∈	∈	PROPN
cana-3840	223	11	𝒰(𝕒	𝒰(𝕒	NOUN
cana-3840	223	12	,	,	PUNCT
cana-3840	223	13	𝓇	𝓇	NOUN
cana-3840	223	14	)	)	PUNCT
cana-3840	223	15	,	,	PUNCT
cana-3840	223	16	𝑠𝑢𝑝{𝕩	𝑠𝑢𝑝{𝕩	VERB
cana-3840	223	17	>	>	X
cana-3840	223	18	0	0	NUM
cana-3840	223	19	:	:	PUNCT
cana-3840	223	20	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	223	21	,	,	PUNCT
cana-3840	223	22	𝕓	𝕓	X
cana-3840	223	23	,	,	PUNCT
cana-3840	223	24	𝕩	𝕩	NOUN
cana-3840	223	25	)	)	PUNCT
cana-3840	224	1	+	+	CCONJ
cana-3840	224	2	𝕩	𝕩	X
cana-3840	224	3	≤	≤	ADV
cana-3840	224	4	0	0	NUM
cana-3840	224	5	}	}	PUNCT
cana-3840	224	6	<	<	X
cana-3840	224	7	𝓇.	𝓇.	NOUN
cana-3840	224	8	thus	thus	ADV
cana-3840	224	9	,	,	PUNCT
cana-3840	224	10	there	there	PRON
cana-3840	224	11	is	be	VERB
cana-3840	224	12	𝕩	𝕩	PROPN
cana-3840	224	13	>	>	X
cana-3840	224	14	0	0	NUM
cana-3840	224	15	such	such	ADJ
cana-3840	224	16	that	that	SCONJ
cana-3840	224	17	𝕩	𝕩	PROPN
cana-3840	224	18	<	<	X
cana-3840	224	19	𝓇	𝓇	X
cana-3840	224	20	and	and	CCONJ
cana-3840	224	21	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	224	22	,	,	PUNCT
cana-3840	224	23	𝕓	𝕓	X
cana-3840	224	24	,	,	PUNCT
cana-3840	224	25	𝕩	𝕩	NOUN
cana-3840	224	26	)	)	PUNCT
cana-3840	225	1	+	+	CCONJ
cana-3840	225	2	𝕩	𝕩	PROPN
cana-3840	225	3	≤	≤	ADJ
cana-3840	225	4	0	0	NUM
cana-3840	225	5	.	.	PUNCT
cana-3840	226	1	so	so	ADV
cana-3840	226	2	,	,	PUNCT
cana-3840	226	3	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	226	4	,	,	PUNCT
cana-3840	226	5	𝕓	𝕓	X
cana-3840	226	6	,	,	PUNCT
cana-3840	226	7	𝓇	𝓇	NOUN
cana-3840	226	8	)	)	PUNCT
cana-3840	226	9	+	+	CCONJ
cana-3840	226	10	𝓇	𝓇	X
cana-3840	226	11	≤	≤	NUM
cana-3840	226	12	0	0	NUM
cana-3840	226	13	,	,	PUNCT
cana-3840	226	14	that	that	ADV
cana-3840	226	15	is	is	ADV
cana-3840	226	16	,	,	PUNCT
cana-3840	226	17	𝕓	𝕓	PROPN
cana-3840	226	18	∈	∈	PROPN
cana-3840	226	19	ℬ𝕎(𝕒	ℬ𝕎(𝕒	NOUN
cana-3840	226	20	,	,	PUNCT
cana-3840	226	21	𝓇	𝓇	NOUN
cana-3840	226	22	,	,	PUNCT
cana-3840	226	23	𝓇	𝓇	NOUN
cana-3840	226	24	)	)	PUNCT
cana-3840	226	25	.	.	PUNCT
cana-3840	227	1	hence	hence	ADV
cana-3840	227	2	,	,	PUNCT
cana-3840	227	3	𝒰(𝕒	𝒰(𝕒	X
cana-3840	227	4	,	,	PUNCT
cana-3840	227	5	𝓇	𝓇	NOUN
cana-3840	227	6	)	)	PUNCT
cana-3840	227	7	⊆	⊆	NUM
cana-3840	227	8	ℬ𝕎(𝕒	ℬ𝕎(𝕒	NOUN
cana-3840	227	9	,	,	PUNCT
cana-3840	227	10	𝓇	𝓇	NOUN
cana-3840	227	11	,	,	PUNCT
cana-3840	227	12	𝓇	𝓇	NOUN
cana-3840	227	13	)	)	PUNCT
cana-3840	227	14	.	.	PUNCT
cana-3840	228	1	(	(	PUNCT
cana-3840	228	2	35	35	NUM
cana-3840	228	3	)	)	PUNCT
cana-3840	228	4	on	on	ADP
cana-3840	228	5	the	the	DET
cana-3840	228	6	other	other	ADJ
cana-3840	228	7	hand	hand	NOUN
cana-3840	228	8	,	,	PUNCT
cana-3840	228	9	for	for	ADP
cana-3840	228	10	any	any	DET
cana-3840	228	11	𝕔	𝕔	DET
cana-3840	228	12	∈	∈	PROPN
cana-3840	228	13	ℬ𝕎(𝕒	ℬ𝕎(𝕒	NOUN
cana-3840	228	14	,	,	PUNCT
cana-3840	228	15	𝓇	𝓇	NOUN
cana-3840	228	16	,	,	PUNCT
cana-3840	228	17	𝓇	𝓇	NOUN
cana-3840	228	18	)	)	PUNCT
cana-3840	228	19	,	,	PUNCT
cana-3840	228	20	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	228	21	,	,	PUNCT
cana-3840	228	22	𝕔	𝕔	NOUN
cana-3840	228	23	,	,	PUNCT
cana-3840	228	24	𝓇	𝓇	NOUN
cana-3840	228	25	)	)	PUNCT
cana-3840	228	26	+	+	CCONJ
cana-3840	228	27	𝓇	𝓇	X
cana-3840	228	28	<	<	X
cana-3840	228	29	0	0	NUM
cana-3840	228	30	.	.	PUNCT
cana-3840	229	1	from	from	ADP
cana-3840	229	2	(	(	PUNCT
cana-3840	229	3	29	29	NUM
cana-3840	229	4	)	)	PUNCT
cana-3840	229	5	,	,	PUNCT
cana-3840	229	6	we	we	PRON
cana-3840	229	7	get	get	VERB
cana-3840	229	8	𝜌(𝕒	𝜌(𝕒	NOUN
cana-3840	229	9	,	,	PUNCT
cana-3840	229	10	𝕔	𝕔	NOUN
cana-3840	229	11	)	)	PUNCT
cana-3840	229	12	≤	≤	NOUN
cana-3840	230	1	𝓇	𝓇	ADP
cana-3840	230	2	<	<	X
cana-3840	230	3	𝓇.	𝓇.	NOUN
cana-3840	231	1	so	so	ADV
cana-3840	231	2	,	,	PUNCT
cana-3840	231	3	𝕔	𝕔	PRON
cana-3840	231	4	∈	∈	PROPN
cana-3840	231	5	𝒰(𝕒	𝒰(𝕒	NOUN
cana-3840	231	6	,	,	PUNCT
cana-3840	231	7	2𝓇	2𝓇	NOUN
cana-3840	231	8	)	)	PUNCT
cana-3840	231	9	.	.	PUNCT
cana-3840	232	1	hence	hence	ADV
cana-3840	232	2	,	,	PUNCT
cana-3840	232	3	ℬ𝕎(𝕒	ℬ𝕎(𝕒	ADJ
cana-3840	232	4	,	,	PUNCT
cana-3840	232	5	𝓇	𝓇	NOUN
cana-3840	232	6	,	,	PUNCT
cana-3840	232	7	𝓇	𝓇	NOUN
cana-3840	232	8	)	)	PUNCT
cana-3840	232	9	⊆	⊆	NUM
cana-3840	232	10	𝒰(𝕒	𝒰(𝕒	NOUN
cana-3840	232	11	,	,	PUNCT
cana-3840	232	12	𝓇	𝓇	NOUN
cana-3840	232	13	)	)	PUNCT
cana-3840	232	14	.	.	PUNCT
cana-3840	233	1	(	(	PUNCT
cana-3840	233	2	36	36	NUM
cana-3840	233	3	)	)	PUNCT
cana-3840	233	4	and	and	CCONJ
cana-3840	233	5	(	(	PUNCT
cana-3840	233	6	34	34	NUM
cana-3840	233	7	)	)	PUNCT
cana-3840	233	8	holds	hold	NOUN
cana-3840	233	9	,	,	PUNCT
cana-3840	233	10	which	which	PRON
cana-3840	233	11	implies	imply	VERB
cana-3840	233	12	that	that	SCONJ
cana-3840	233	13	𝜏𝜌	𝜏𝜌	ADP
cana-3840	233	14	=	=	PUNCT
cana-3840	233	15	𝜏𝕎	𝜏𝕎	NOUN
cana-3840	233	16	directly	directly	ADV
cana-3840	233	17	.	.	PUNCT
cana-3840	234	1	lemma	lemma	PROPN
cana-3840	234	2	3	3	X
cana-3840	234	3	.	.	PUNCT
cana-3840	235	1	let	let	VERB
cana-3840	235	2	(	(	PUNCT
cana-3840	235	3	𝔘	𝔘	PROPN
cana-3840	235	4	,	,	PUNCT
cana-3840	235	5	𝕎	𝕎	PROPN
cana-3840	235	6	,	,	PUNCT
cana-3840	235	7	⨁	⨁	PROPN
cana-3840	235	8	)	)	PUNCT
cana-3840	235	9	is	be	AUX
cana-3840	235	10	revised	revise	VERB
cana-3840	235	11	fuzzy	fuzzy	ADJ
cana-3840	235	12	metric	metric	ADJ
cana-3840	235	13	space	space	NOUN
cana-3840	235	14	;	;	PUNCT
cana-3840	235	15	the	the	DET
cana-3840	235	16	functions	function	NOUN
cana-3840	235	17	𝒦	𝒦	PROPN
cana-3840	235	18	and	and	CCONJ
cana-3840	235	19	𝒢	𝒢	NOUN
cana-3840	235	20	satisfy	satisfy	VERB
cana-3840	235	21	the	the	DET
cana-3840	235	22	conditions	condition	NOUN
cana-3840	235	23	ℭ1	ℭ1	ADV
cana-3840	235	24	and	and	CCONJ
cana-3840	235	25	ℭ2	ℭ2	PROPN
cana-3840	235	26	,	,	PUNCT
cana-3840	235	27	respectively	respectively	ADV
cana-3840	235	28	.	.	PUNCT
cana-3840	236	1	then	then	ADV
cana-3840	236	2	,	,	PUNCT
cana-3840	236	3	the	the	DET
cana-3840	236	4	function	function	NOUN
cana-3840	236	5	d	d	NOUN
cana-3840	236	6	defined	define	VERB
cana-3840	236	7	by	by	ADP
cana-3840	236	8	(	(	PUNCT
cana-3840	236	9	20	20	NUM
cana-3840	236	10	)	)	PUNCT
cana-3840	236	11	can	can	AUX
cana-3840	236	12	be	be	AUX
cana-3840	236	13	represented	represent	VERB
cana-3840	236	14	as	as	ADP
cana-3840	236	15	follows	follow	VERB
cana-3840	236	16	,	,	PUNCT
cana-3840	236	17	∀	∀	X
cana-3840	236	18	𝕒	𝕒	X
cana-3840	236	19	,	,	PUNCT
cana-3840	236	20	𝕓	𝕓	PROPN
cana-3840	236	21	∈	∈	PROPN
cana-3840	236	22	𝔘	𝔘	PROPN
cana-3840	236	23	:	:	PUNCT
cana-3840	236	24	𝕕(𝕒	𝕕(𝕒	PROPN
cana-3840	236	25	,	,	PUNCT
cana-3840	236	26	𝕓	𝕓	X
cana-3840	236	27	)	)	PUNCT
cana-3840	236	28	=	=	SYM
cana-3840	237	1	𝑖𝑛𝑓{𝓈	𝑖𝑛𝑓{𝓈	X
cana-3840	237	2	>	>	X
cana-3840	238	1	0	0	NUM
cana-3840	238	2	:	:	PUNCT
cana-3840	238	3	𝑖𝑓	𝑖𝑓	NUM
cana-3840	238	4	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	238	5	)	)	PUNCT
cana-3840	238	6	>	>	X
cana-3840	238	7	𝓈	𝓈	PROPN
cana-3840	238	8	,	,	PUNCT
cana-3840	238	9	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
cana-3840	238	10	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	238	11	,	,	PUNCT
cana-3840	238	12	𝕓	𝕓	X
cana-3840	238	13	,	,	PUNCT
cana-3840	238	14	𝓉	𝓉	PRON
cana-3840	238	15	)	)	PUNCT
cana-3840	239	1	+	+	CCONJ
cana-3840	239	2	𝒦(𝓈	𝒦(𝓈	X
cana-3840	239	3	)	)	PUNCT
cana-3840	239	4	<	<	X
cana-3840	239	5	0	0	NUM
cana-3840	239	6	}	}	PUNCT
cana-3840	239	7	.	.	PUNCT
cana-3840	240	1	(	(	PUNCT
cana-3840	240	2	37	37	NUM
cana-3840	240	3	)	)	PUNCT
cana-3840	240	4	proof	proof	NOUN
cana-3840	240	5	.	.	PUNCT
cana-3840	241	1	in	in	ADP
cana-3840	241	2	fact	fact	NOUN
cana-3840	241	3	,	,	PUNCT
cana-3840	241	4	we	we	PRON
cana-3840	241	5	only	only	ADV
cana-3840	241	6	need	need	VERB
cana-3840	241	7	to	to	PART
cana-3840	241	8	show	show	VERB
cana-3840	241	9	that	that	SCONJ
cana-3840	241	10	𝑖𝑛𝑓𝔸	𝑖𝑛𝑓𝔸	NOUN
cana-3840	241	11	=	=	SYM
cana-3840	241	12	𝑠𝑢𝑝𝔹	𝑠𝑢𝑝𝔹	NOUN
cana-3840	241	13	,	,	PUNCT
cana-3840	241	14	where	where	SCONJ
cana-3840	241	15	,	,	PUNCT
cana-3840	241	16	𝔸	𝔸	PROPN
cana-3840	241	17	=	=	SYM
cana-3840	241	18	{	{	PUNCT
cana-3840	241	19	𝕩	𝕩	X
cana-3840	241	20	>	>	X
cana-3840	241	21	0	0	PROPN
cana-3840	241	22	:	:	PUNCT
cana-3840	241	23	𝑖𝑓	𝑖𝑓	NUM
cana-3840	241	24	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	241	25	)	)	PUNCT
cana-3840	241	26	>	>	X
cana-3840	241	27	𝕩	𝕩	PROPN
cana-3840	241	28	,	,	PUNCT
cana-3840	241	29	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3840	241	30	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	241	31	,	,	PUNCT
cana-3840	241	32	𝕓	𝕓	X
cana-3840	241	33	,	,	PUNCT
cana-3840	241	34	𝕩	𝕩	NOUN
cana-3840	241	35	)	)	PUNCT
cana-3840	241	36	+	+	PUNCT
cana-3840	241	37	𝒦(𝕩	𝒦(𝕩	X
cana-3840	241	38	)	)	PUNCT
cana-3840	241	39	≤	≤	NOUN
cana-3840	241	40	0	0	NUM
cana-3840	241	41	}	}	PUNCT
cana-3840	241	42	,	,	PUNCT
cana-3840	241	43	𝔹	𝔹	VERB
cana-3840	241	44	=	=	SYM
cana-3840	241	45	{	{	PUNCT
cana-3840	241	46	𝕪	𝕪	X
cana-3840	241	47	>	>	X
cana-3840	241	48	0	0	NUM
cana-3840	241	49	:	:	PUNCT
cana-3840	241	50	𝑖𝑓	𝑖𝑓	NUM
cana-3840	241	51	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	241	52	)	)	PUNCT
cana-3840	241	53	>	>	X
cana-3840	242	1	𝕪	𝕪	PROPN
cana-3840	242	2	,	,	PUNCT
cana-3840	242	3	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
cana-3840	242	4	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	242	5	,	,	PUNCT
cana-3840	242	6	𝕓	𝕓	X
cana-3840	242	7	,	,	PUNCT
cana-3840	242	8	𝕩	𝕩	NOUN
cana-3840	242	9	)	)	PUNCT
cana-3840	242	10	+	+	CCONJ
cana-3840	242	11	𝒦(𝕪	𝒦(𝕪	X
cana-3840	242	12	)	)	PUNCT
cana-3840	242	13	≤	≤	NOUN
cana-3840	242	14	0	0	NUM
cana-3840	242	15	}	}	PUNCT
cana-3840	242	16	.	.	PUNCT
cana-3840	243	1	(	(	PUNCT
cana-3840	243	2	38	38	NUM
cana-3840	243	3	)	)	PUNCT
cana-3840	243	4	take	take	VERB
cana-3840	243	5	𝕩	𝕩	PROPN
cana-3840	243	6	∈	∈	PROPN
cana-3840	243	7	𝔸	𝔸	PROPN
cana-3840	243	8	,	,	PUNCT
cana-3840	243	9	𝕪	𝕪	PROPN
cana-3840	243	10	∈	∈	PROPN
cana-3840	243	11	𝔹	𝔹	NOUN
cana-3840	243	12	arbitrarily	arbitrarily	ADV
cana-3840	243	13	.	.	PUNCT
cana-3840	244	1	suppose	suppose	VERB
cana-3840	244	2	𝕪	𝕪	X
cana-3840	244	3	>	>	X
cana-3840	244	4	𝕩.	𝕩.	NOUN
cana-3840	244	5	if	if	SCONJ
cana-3840	244	6	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	244	7	)	)	PUNCT
cana-3840	244	8	>	>	X
cana-3840	245	1	𝕪	𝕪	PROPN
cana-3840	245	2	,	,	PUNCT
cana-3840	245	3	then	then	ADV
cana-3840	245	4	𝒢(𝓉	𝒢(𝓉	X
cana-3840	245	5	)	)	PUNCT
cana-3840	245	6	>	>	X
cana-3840	245	7	𝕩.	𝕩.	NOUN
cana-3840	245	8	from	from	ADP
cana-3840	245	9	the	the	DET
cana-3840	245	10	definitions	definition	NOUN
cana-3840	245	11	of	of	ADP
cana-3840	245	12	𝔸	𝔸	PROPN
cana-3840	245	13	and	and	CCONJ
cana-3840	245	14	𝔹	𝔹	PROPN
cana-3840	245	15	,	,	PUNCT
cana-3840	245	16	we	we	PRON
cana-3840	245	17	obtain	obtain	VERB
cana-3840	245	18	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	245	19	,	,	PUNCT
cana-3840	245	20	𝕓	𝕓	X
cana-3840	245	21	,	,	PUNCT
cana-3840	245	22	𝓉	𝓉	PRON
cana-3840	245	23	)	)	PUNCT
cana-3840	245	24	+	+	PUNCT
cana-3840	245	25	𝒦(𝕩	𝒦(𝕩	X
cana-3840	245	26	)	)	PUNCT
cana-3840	245	27	≤	≤	NOUN
cana-3840	245	28	0	0	NUM
cana-3840	245	29	>	>	X
cana-3840	245	30	𝕎(𝕒	𝕎(𝕒	PROPN
cana-3840	245	31	,	,	PUNCT
cana-3840	245	32	𝕓	𝕓	X
cana-3840	245	33	,	,	PUNCT
cana-3840	245	34	𝓉	𝓉	PRON
cana-3840	245	35	)	)	PUNCT
cana-3840	245	36	+	+	CCONJ
cana-3840	245	37	𝒦(𝕪	𝒦(𝕪	X
cana-3840	245	38	)	)	PUNCT
cana-3840	245	39	.	.	PUNCT
cana-3840	246	1	(	(	PUNCT
cana-3840	246	2	39	39	NUM
cana-3840	246	3	)	)	PUNCT
cana-3840	246	4	this	this	PRON
cana-3840	246	5	is	be	AUX
cana-3840	246	6	in	in	ADP
cana-3840	246	7	direct	direct	ADJ
cana-3840	246	8	contradiction	contradiction	NOUN
cana-3840	246	9	to	to	ADP
cana-3840	246	10	the	the	DET
cana-3840	246	11	condition	condition	NOUN
cana-3840	246	12	that	that	SCONJ
cana-3840	246	13	𝒦	𝒦	PROPN
cana-3840	246	14	is	be	AUX
cana-3840	246	15	non	non	PRON
cana-3840	246	16	increasing	increase	VERB
cana-3840	246	17	.	.	PUNCT
cana-3840	247	1	so	so	ADV
cana-3840	247	2	,	,	PUNCT
cana-3840	247	3	𝕩	𝕩	PROPN
cana-3840	247	4	≥	≥	X
cana-3840	247	5	𝕪	𝕪	ADP
cana-3840	247	6	,	,	PUNCT
cana-3840	247	7	and	and	CCONJ
cana-3840	247	8	hence	hence	ADV
cana-3840	247	9	,	,	PUNCT
cana-3840	247	10	𝑖𝑛𝑓𝔸	𝑖𝑛𝑓𝔸	X
cana-3840	247	11	≥	≥	NOUN
cana-3840	247	12	𝑠𝑢𝑝𝔹.	𝑠𝑢𝑝𝔹.	ADP
cana-3840	247	13	now	now	ADV
cana-3840	247	14	,	,	PUNCT
cana-3840	247	15	let	let	VERB
cana-3840	247	16	𝑖𝑛𝑓𝔸	𝑖𝑛𝑓𝔸	PRON
cana-3840	247	17	=	=	SYM
cana-3840	247	18	𝛼	𝛼	X
cana-3840	247	19	,	,	PUNCT
cana-3840	247	20	𝑠𝑢𝑝𝔹	𝑠𝑢𝑝𝔹	NOUN
cana-3840	247	21	=	=	SYM
cana-3840	247	22	𝛽.	𝛽.	NOUN
cana-3840	247	23	for	for	ADP
cana-3840	247	24	any	any	DET
cana-3840	247	25	𝛿	𝛿	PROPN
cana-3840	247	26	>	>	X
cana-3840	247	27	0	0	NUM
cana-3840	247	28	,	,	PUNCT
cana-3840	247	29	we	we	PRON
cana-3840	247	30	know	know	VERB
cana-3840	247	31	𝛽	𝛽	NOUN
cana-3840	247	32	+	+	ADP
cana-3840	247	33	𝛿	𝛿	PRON
cana-3840	247	34	∉	∉	PROPN
cana-3840	247	35	𝔹	𝔹	PROPN
cana-3840	247	36	,	,	PUNCT
cana-3840	247	37	that	that	ADV
cana-3840	247	38	is	is	ADV
cana-3840	247	39	,	,	PUNCT
cana-3840	247	40	𝛽	𝛽	NOUN
cana-3840	247	41	+	+	ADP
cana-3840	247	42	𝛿	𝛿	DET
cana-3840	247	43	∈	∈	PROPN
cana-3840	247	44	𝔸	𝔸	PROPN
cana-3840	247	45	,	,	PUNCT
cana-3840	247	46	which	which	PRON
cana-3840	247	47	implies	imply	VERB
cana-3840	247	48	that	that	DET
cana-3840	247	49	inf	inf	NOUN
cana-3840	247	50	𝔸	𝔸	PROPN
cana-3840	247	51	≤	≤	PROPN
cana-3840	247	52	𝛽	𝛽	NOUN
cana-3840	247	53	+	+	CCONJ
cana-3840	247	54	𝛿.	𝛿.	ADJ
cana-3840	247	55	by	by	ADP
cana-3840	247	56	the	the	DET
cana-3840	247	57	arbitrariness	arbitrariness	NOUN
cana-3840	247	58	of	of	ADP
cana-3840	247	59	𝛿	𝛿	ADJ
cana-3840	247	60	,	,	PUNCT
cana-3840	247	61	we	we	PRON
cana-3840	247	62	know	know	VERB
cana-3840	247	63	that	that	SCONJ
cana-3840	247	64	𝑖𝑛𝑓𝔸	𝑖𝑛𝑓𝔸	NOUN
cana-3840	247	65	≤	≤	X
cana-3840	247	66	𝛽	𝛽	NOUN
cana-3840	247	67	=	=	PUNCT
cana-3840	247	68	𝑠𝑢𝑝𝔹.	𝑠𝑢𝑝𝔹.	NOUN
cana-3840	247	69	this	this	PRON
cana-3840	247	70	completes	complete	VERB
cana-3840	247	71	the	the	DET
cana-3840	247	72	proof	proof	NOUN
cana-3840	247	73	.	.	PUNCT
cana-3840	248	1	communications	communication	NOUN
cana-3840	248	2	on	on	ADP
cana-3840	248	3	applied	apply	VERB
cana-3840	248	4	nonlinear	nonlinear	ADJ
cana-3840	248	5	analysis	analysis	NOUN
cana-3840	248	6	issn	issn	NOUN
cana-3840	248	7	:	:	PUNCT
cana-3840	248	8	1074	1074	NUM
cana-3840	248	9	-	-	PUNCT
cana-3840	248	10	133x	133x	NUM
cana-3840	248	11	vol	vol	NOUN
cana-3840	248	12	32	32	NUM
cana-3840	248	13	no	no	NOUN
cana-3840	248	14	.	.	PUNCT
cana-3840	249	1	9s	9s	NUM
cana-3840	249	2	(	(	PUNCT
cana-3840	249	3	2025	2025	NUM
cana-3840	249	4	)	)	PUNCT
cana-3840	249	5	96	96	NUM
cana-3840	250	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	250	2	theorem	theorem	VERB
cana-3840	250	3	5	5	NUM
cana-3840	250	4	.	.	PUNCT
cana-3840	251	1	let	let	VERB
cana-3840	251	2	(	(	PUNCT
cana-3840	251	3	𝔘	𝔘	PROPN
cana-3840	251	4	,	,	PUNCT
cana-3840	251	5	𝕎	𝕎	PROPN
cana-3840	251	6	,	,	PUNCT
cana-3840	251	7	⨁	⨁	PROPN
cana-3840	251	8	)	)	PUNCT
cana-3840	251	9	is	be	AUX
cana-3840	251	10	a	a	DET
cana-3840	251	11	revised	revise	VERB
cana-3840	251	12	fuzzy	fuzzy	ADJ
cana-3840	251	13	metric	metric	ADJ
cana-3840	251	14	space	space	NOUN
cana-3840	251	15	.	.	PUNCT
cana-3840	252	1	if	if	SCONJ
cana-3840	252	2	𝕎	𝕎	PROPN
cana-3840	252	3	satisfies	satisfy	VERB
cana-3840	252	4	condition	condition	NOUN
cana-3840	252	5	(	(	PUNCT
cana-3840	252	6	15	15	NUM
cana-3840	252	7	)	)	PUNCT
cana-3840	252	8	or	or	CCONJ
cana-3840	252	9	⨁	⨁	PROPN
cana-3840	252	10	≥	≥	NUM
cana-3840	252	11	𝛥1	𝛥1	NOUN
cana-3840	252	12	,	,	PUNCT
cana-3840	252	13	then	then	ADV
cana-3840	252	14	∀𝕒	∀𝕒	PROPN
cana-3840	252	15	,	,	PUNCT
cana-3840	252	16	𝕓	𝕓	PROPN
cana-3840	252	17	∈	∈	PROPN
cana-3840	252	18	𝔘	𝔘	PROPN
cana-3840	252	19	:	:	PUNCT
cana-3840	252	20	𝜌(𝕒	𝜌(𝕒	PROPN
cana-3840	252	21	,	,	PUNCT
cana-3840	252	22	𝕓	𝕓	X
cana-3840	252	23	)	)	PUNCT
cana-3840	252	24	=	=	PUNCT
cana-3840	252	25	𝑖𝑛𝑓{𝓉	𝑖𝑛𝑓{𝓉	X
cana-3840	252	26	>	>	X
cana-3840	252	27	0	0	NUM
cana-3840	252	28	:	:	PUNCT
cana-3840	252	29	𝕎(𝕒	𝕎(𝕒	NUM
cana-3840	252	30	,	,	PUNCT
cana-3840	252	31	𝕓	𝕓	X
cana-3840	252	32	,	,	PUNCT
cana-3840	252	33	𝑡	𝑡	X
cana-3840	252	34	)	)	PUNCT
cana-3840	252	35	+	+	CCONJ
cana-3840	252	36	𝓉	𝓉	PROPN
cana-3840	252	37	<	<	X
cana-3840	252	38	0	0	NUM
cana-3840	252	39	}	}	PUNCT
cana-3840	252	40	,	,	PUNCT
cana-3840	252	41	(	(	PUNCT
cana-3840	252	42	40	40	NUM
cana-3840	252	43	)	)	PUNCT
cana-3840	252	44	where	where	SCONJ
cana-3840	252	45	ρ	ρ	PROPN
cana-3840	252	46	is	be	AUX
cana-3840	252	47	defined	define	VERB
cana-3840	252	48	by	by	ADP
cana-3840	252	49	(	(	PUNCT
cana-3840	252	50	29	29	NUM
cana-3840	252	51	)	)	PUNCT
cana-3840	252	52	.	.	PUNCT
cana-3840	253	1	proof	proof	NOUN
cana-3840	253	2	.	.	PUNCT
cana-3840	254	1	let	let	VERB
cana-3840	254	2	𝒢(𝓉	𝒢(𝓉	NOUN
cana-3840	254	3	)	)	PUNCT
cana-3840	254	4	=	=	SYM
cana-3840	254	5	𝓉	𝓉	PROPN
cana-3840	254	6	,	,	PUNCT
cana-3840	254	7	𝒦(𝓉	𝒦(𝓉	X
cana-3840	254	8	)	)	PUNCT
cana-3840	254	9	=	=	SYM
cana-3840	254	10	𝓉	𝓉	PROPN
cana-3840	254	11	,	,	PUNCT
cana-3840	254	12	∀t	∀t	PROPN
cana-3840	254	13	≥	≥	NOUN
cana-3840	254	14	0	0	NUM
cana-3840	254	15	.	.	PUNCT
cana-3840	255	1	then	then	ADV
cana-3840	255	2	,	,	PUNCT
cana-3840	255	3	(	(	PUNCT
cana-3840	255	4	40	40	NUM
cana-3840	255	5	)	)	PUNCT
cana-3840	255	6	follows	follow	VERB
cana-3840	255	7	from	from	ADP
cana-3840	255	8	lemma	lemma	PROPN
cana-3840	255	9	3	3	NUM
cana-3840	255	10	directly	directly	ADV
cana-3840	255	11	.	.	PUNCT
cana-3840	256	1	4	4	X
cana-3840	256	2	.	.	X
cana-3840	256	3	conclusions	conclusion	NOUN
cana-3840	256	4	we	we	PRON
cana-3840	256	5	study	study	VERB
cana-3840	256	6	several	several	ADJ
cana-3840	256	7	metric	metric	ADJ
cana-3840	256	8	structures	structure	NOUN
cana-3840	256	9	in	in	ADP
cana-3840	256	10	a	a	DET
cana-3840	256	11	revised	revise	VERB
cana-3840	256	12	fuzzy	fuzzy	ADJ
cana-3840	256	13	metric	metric	ADJ
cana-3840	256	14	space	space	NOUN
cana-3840	256	15	in	in	ADP
cana-3840	256	16	current	current	ADJ
cana-3840	256	17	research	research	NOUN
cana-3840	256	18	.	.	PUNCT
cana-3840	257	1	we	we	PRON
cana-3840	257	2	provide	provide	VERB
cana-3840	257	3	the	the	DET
cana-3840	257	4	explicit	explicit	ADJ
cana-3840	257	5	form	form	NOUN
cana-3840	257	6	of	of	ADP
cana-3840	257	7	the	the	DET
cana-3840	257	8	metric	metric	ADJ
cana-3840	257	9	function	function	NOUN
cana-3840	257	10	about	about	ADP
cana-3840	257	11	the	the	DET
cana-3840	257	12	metrizable	metrizable	ADJ
cana-3840	257	13	topology	topology	NOUN
cana-3840	257	14	for	for	ADP
cana-3840	257	15	a	a	DET
cana-3840	257	16	revised	revise	VERB
cana-3840	257	17	fuzzy	fuzzy	ADJ
cana-3840	257	18	metric	metric	NOUN
cana-3840	257	19	in	in	ADP
cana-3840	257	20	two	two	NUM
cana-3840	257	21	exceptional	exceptional	ADJ
cana-3840	257	22	scenarios	scenario	NOUN
cana-3840	257	23	.	.	PUNCT
cana-3840	258	1	interesting	interesting	ADJ
cana-3840	258	2	future	future	ADJ
cana-3840	258	3	research	research	NOUN
cana-3840	258	4	investigations	investigation	NOUN
cana-3840	258	5	concerning	concern	VERB
cana-3840	258	6	linked	link	VERB
cana-3840	258	7	themes	theme	NOUN
cana-3840	258	8	may	may	AUX
cana-3840	258	9	be	be	AUX
cana-3840	258	10	prospective	prospective	ADJ
cana-3840	258	11	,	,	PUNCT
cana-3840	258	12	based	base	VERB
cana-3840	258	13	on	on	ADP
cana-3840	258	14	the	the	DET
cana-3840	258	15	paper	paper	NOUN
cana-3840	258	16	's	's	PART
cana-3840	258	17	conclusions	conclusion	NOUN
cana-3840	258	18	.	.	PUNCT
cana-3840	259	1	furthermore	furthermore	ADV
cana-3840	259	2	,	,	PUNCT
cana-3840	259	3	this	this	DET
cana-3840	259	4	paper	paper	NOUN
cana-3840	259	5	's	's	PART
cana-3840	259	6	methodology	methodology	NOUN
cana-3840	259	7	suggests	suggest	VERB
cana-3840	259	8	discussing	discuss	VERB
cana-3840	259	9	the	the	DET
cana-3840	259	10	relevant	relevant	ADJ
cana-3840	259	11	issue	issue	NOUN
cana-3840	259	12	in	in	ADP
cana-3840	259	13	a	a	DET
cana-3840	259	14	broader	broad	ADJ
cana-3840	259	15	context	context	NOUN
cana-3840	259	16	.	.	PUNCT
cana-3840	260	1	5	5	X
cana-3840	260	2	.	.	NUM
cana-3840	260	3	references	reference	NOUN
cana-3840	260	4	[	[	X
cana-3840	260	5	1	1	NUM
cana-3840	260	6	]	]	X
cana-3840	260	7	alexander	alexander	PROPN
cana-3840	260	8	sostak	sostak	PROPN
cana-3840	260	9	“	"	PUNCT
cana-3840	260	10	george	george	NOUN
cana-3840	260	11	-	-	PUNCT
cana-3840	260	12	veeramani	veeramani	NOUN
cana-3840	260	13	fuzzy	fuzzy	ADJ
cana-3840	260	14	metrics	metric	NOUN
cana-3840	260	15	revised	revise	VERB
cana-3840	260	16	”	"	PUNCT
cana-3840	260	17	,	,	PUNCT
cana-3840	260	18	axioms	axiom	VERB
cana-3840	260	19	2018,7,60	2018,7,60	NUM
cana-3840	260	20	.	.	PUNCT
cana-3840	261	1	[	[	X
cana-3840	261	2	2	2	X
cana-3840	261	3	]	]	X
cana-3840	261	4	george.a	george.a	PROPN
cana-3840	261	5	and	and	CCONJ
cana-3840	261	6	veeramani	veeramani	NOUN
cana-3840	261	7	.	.	PUNCT
cana-3840	262	1	p	p	X
cana-3840	262	2	,	,	PUNCT
cana-3840	262	3	“	"	PUNCT
cana-3840	262	4	on	on	ADP
cana-3840	262	5	some	some	DET
cana-3840	262	6	results	result	NOUN
cana-3840	262	7	in	in	ADP
cana-3840	262	8	fuzzy	fuzzy	ADJ
cana-3840	262	9	metric	metric	ADJ
cana-3840	262	10	spaces	space	NOUN
cana-3840	262	11	,	,	PUNCT
cana-3840	262	12	”	"	PUNCT
cana-3840	262	13	fuzzy	fuzzy	ADJ
cana-3840	262	14	sets	set	NOUN
cana-3840	262	15	and	and	CCONJ
cana-3840	262	16	systems	system	NOUN
cana-3840	262	17	,	,	PUNCT
cana-3840	262	18	vol	vol	NOUN
cana-3840	262	19	.	.	PROPN
cana-3840	263	1	64	64	NUM
cana-3840	263	2	,	,	PUNCT
cana-3840	263	3	no	no	INTJ
cana-3840	263	4	.	.	NOUN
cana-3840	263	5	3	3	NUM
cana-3840	263	6	,	,	PUNCT
cana-3840	263	7	pp	pp	ADJ
cana-3840	263	8	.	.	PUNCT
cana-3840	264	1	395–399	395–399	NUM
cana-3840	264	2	,	,	PUNCT
cana-3840	264	3	1994	1994	NUM
cana-3840	264	4	.	.	PUNCT
cana-3840	265	1	[	[	X
cana-3840	265	2	3	3	NUM
cana-3840	265	3	]	]	X
cana-3840	265	4	gregori	gregori	X
cana-3840	265	5	.	.	PUNCT
cana-3840	266	1	v	v	NOUN
cana-3840	266	2	and	and	CCONJ
cana-3840	266	3	romaguera	romaguera	NOUN
cana-3840	266	4	.	.	PUNCT
cana-3840	267	1	s	s	X
cana-3840	267	2	,	,	PUNCT
cana-3840	267	3	“	"	PUNCT
cana-3840	267	4	on	on	ADP
cana-3840	267	5	completion	completion	NOUN
cana-3840	267	6	of	of	ADP
cana-3840	267	7	fuzzy	fuzzy	ADJ
cana-3840	267	8	metric	metric	ADJ
cana-3840	267	9	spaces	space	NOUN
cana-3840	267	10	,	,	PUNCT
cana-3840	267	11	”	"	PUNCT
cana-3840	267	12	fuzzy	fuzzy	ADJ
cana-3840	267	13	sets	set	NOUN
cana-3840	267	14	and	and	CCONJ
cana-3840	267	15	systems	system	NOUN
cana-3840	267	16	,	,	PUNCT
cana-3840	267	17	vol	vol	NOUN
cana-3840	267	18	.	.	PROPN
cana-3840	267	19	130	130	NUM
cana-3840	267	20	,	,	PUNCT
cana-3840	267	21	no	no	INTJ
cana-3840	267	22	.	.	NOUN
cana-3840	267	23	3	3	NUM
cana-3840	267	24	,	,	PUNCT
cana-3840	267	25	pp	pp	ADJ
cana-3840	267	26	.	.	PUNCT
cana-3840	268	1	399–404	399–404	NUM
cana-3840	268	2	,	,	PUNCT
cana-3840	268	3	2002	2002	NUM
cana-3840	268	4	.	.	PUNCT
cana-3840	269	1	[	[	X
cana-3840	269	2	4	4	NUM
cana-3840	269	3	]	]	X
cana-3840	269	4	gregori	gregori	X
cana-3840	269	5	.	.	PUNCT
cana-3840	270	1	v	v	NOUN
cana-3840	270	2	,	,	PUNCT
cana-3840	270	3	morillas	morilla	NOUN
cana-3840	270	4	.	.	PUNCT
cana-3840	271	1	s	s	X
cana-3840	271	2	,	,	PUNCT
cana-3840	271	3	and	and	CCONJ
cana-3840	271	4	sapena	sapena	NOUN
cana-3840	271	5	.	.	PUNCT
cana-3840	272	1	a	a	DET
cana-3840	272	2	,	,	PUNCT
cana-3840	272	3	“	"	PUNCT
cana-3840	272	4	examples	example	NOUN
cana-3840	272	5	of	of	ADP
cana-3840	272	6	fuzzy	fuzzy	ADJ
cana-3840	272	7	metrics	metric	NOUN
cana-3840	272	8	and	and	CCONJ
cana-3840	272	9	applications	application	NOUN
cana-3840	272	10	,	,	PUNCT
cana-3840	272	11	”	"	PUNCT
cana-3840	272	12	fuzzy	fuzzy	ADJ
cana-3840	272	13	sets	set	NOUN
cana-3840	272	14	and	and	CCONJ
cana-3840	272	15	systems	system	NOUN
cana-3840	272	16	,	,	PUNCT
cana-3840	272	17	vol	vol	NOUN
cana-3840	272	18	.	.	PROPN
cana-3840	272	19	170	170	NUM
cana-3840	272	20	,	,	PUNCT
cana-3840	272	21	no	no	INTJ
cana-3840	272	22	.	.	NOUN
cana-3840	272	23	1	1	NUM
cana-3840	272	24	,	,	PUNCT
cana-3840	272	25	pp	pp	ADJ
cana-3840	272	26	.	.	PUNCT
cana-3840	273	1	95–111	95–111	NUM
cana-3840	273	2	,	,	PUNCT
cana-3840	273	3	2011	2011	NUM
cana-3840	273	4	.	.	PUNCT
cana-3840	274	1	[	[	X
cana-3840	274	2	5	5	NUM
cana-3840	274	3	]	]	X
cana-3840	274	4	grigorenko	grigorenko	PROPN
cana-3840	274	5	,	,	PUNCT
cana-3840	274	6	o	o	NOUN
cana-3840	274	7	,	,	PUNCT
cana-3840	274	8	miñana	miñana	PROPN
cana-3840	274	9	,	,	PUNCT
cana-3840	274	10	j.j	j.j	PROPN
cana-3840	274	11	,	,	PUNCT
cana-3840	274	12	šostak	šostak	NOUN
cana-3840	274	13	,	,	PUNCT
cana-3840	274	14	a.	a.	NOUN
cana-3840	274	15	,	,	PUNCT
cana-3840	274	16	valero	valero	PROPN
cana-3840	274	17	,	,	PUNCT
cana-3840	274	18	o.	o.	PROPN
cana-3840	274	19	on	on	ADP
cana-3840	274	20	t	t	PROPN
cana-3840	274	21	-	-	PUNCT
cana-3840	274	22	conorm	conorm	NOUN
cana-3840	274	23	based	base	VERB
cana-3840	274	24	fuzzy	fuzzy	ADJ
cana-3840	274	25	(	(	PUNCT
cana-3840	274	26	pseudo	pseudo	NOUN
cana-3840	274	27	)	)	PUNCT
cana-3840	274	28	metrics	metric	NOUN
cana-3840	274	29	.	.	PUNCT
cana-3840	275	1	axioms	axiom	VERB
cana-3840	275	2	2020	2020	NUM
cana-3840	275	3	,	,	PUNCT
cana-3840	275	4	9	9	NUM
cana-3840	275	5	,	,	PUNCT
cana-3840	275	6	78	78	NUM
cana-3840	275	7	.	.	PUNCT
cana-3840	276	1	https://doi.org/10.3390/axioms9030078	https://doi.org/10.3390/axioms9030078	PROPN
cana-3840	276	2	.	.	PUNCT
cana-3840	277	1	[	[	X
cana-3840	277	2	6	6	NUM
cana-3840	277	3	]	]	X
cana-3840	277	4	gupta	gupta	PROPN
cana-3840	277	5	.	.	PUNCT
cana-3840	278	1	v	v	NOUN
cana-3840	278	2	,	,	PUNCT
cana-3840	278	3	kaushik	kaushik	PROPN
cana-3840	278	4	.	.	PUNCT
cana-3840	279	1	a	a	PRON
cana-3840	279	2	,	,	PUNCT
cana-3840	279	3	and	and	CCONJ
cana-3840	279	4	m.	m.	PROPN
cana-3840	279	5	verma	verma	PROPN
cana-3840	279	6	.	.	PUNCT
cana-3840	280	1	m	m	PROPN
cana-3840	280	2	,	,	PUNCT
cana-3840	280	3	“	"	PUNCT
cana-3840	280	4	some	some	DET
cana-3840	280	5	new	new	ADJ
cana-3840	280	6	fixed	fix	VERB
cana-3840	280	7	-	-	PUNCT
cana-3840	280	8	point	point	NOUN
cana-3840	280	9	results	result	NOUN
cana-3840	280	10	on	on	ADP
cana-3840	280	11	v	v	NOUN
cana-3840	280	12	-	-	PUNCT
cana-3840	280	13	ψ	ψ	NOUN
cana-3840	280	14	-	-	ADJ
cana-3840	280	15	fuzzy	fuzzy	ADJ
cana-3840	280	16	contraction	contraction	NOUN
cana-3840	280	17	endowed	endow	VERB
cana-3840	280	18	with	with	ADP
cana-3840	280	19	graph	graph	NOUN
cana-3840	280	20	,	,	PUNCT
cana-3840	280	21	journal	journal	NOUN
cana-3840	280	22	of	of	ADP
cana-3840	280	23	intelligent	intelligent	ADJ
cana-3840	280	24	&	&	CCONJ
cana-3840	280	25	fuzzy	fuzzy	ADJ
cana-3840	280	26	systems	system	NOUN
cana-3840	280	27	,	,	PUNCT
cana-3840	280	28	vol	vol	NOUN
cana-3840	280	29	.	.	PROPN
cana-3840	281	1	36	36	NUM
cana-3840	281	2	,	,	PUNCT
cana-3840	281	3	no	no	INTJ
cana-3840	281	4	.	.	NOUN
cana-3840	281	5	6	6	NUM
cana-3840	281	6	,	,	PUNCT
cana-3840	281	7	pp	pp	ADJ
cana-3840	281	8	.	.	PUNCT
cana-3840	282	1	6549–6554	6549–6554	NUM
cana-3840	282	2	,	,	PUNCT
cana-3840	282	3	2019	2019	NUM
cana-3840	282	4	.	.	PUNCT
cana-3840	283	1	[	[	X
cana-3840	283	2	7	7	NUM
cana-3840	283	3	]	]	X
cana-3840	283	4	kelley	kelley	PROPN
cana-3840	283	5	.	.	PUNCT
cana-3840	284	1	j.	j.	PROPN
cana-3840	284	2	l	l	PROPN
cana-3840	284	3	,	,	PUNCT
cana-3840	284	4	general	general	ADJ
cana-3840	284	5	topology	topology	NOUN
cana-3840	284	6	,	,	PUNCT
cana-3840	284	7	van	van	PROPN
cana-3840	284	8	nostrand	nostrand	PROPN
cana-3840	284	9	,	,	PUNCT
cana-3840	284	10	new	new	PROPN
cana-3840	284	11	york	york	PROPN
cana-3840	284	12	,	,	PUNCT
cana-3840	284	13	ny	ny	PROPN
cana-3840	284	14	,	,	PUNCT
cana-3840	284	15	usa	usa	PROPN
cana-3840	284	16	,	,	PUNCT
cana-3840	284	17	1955	1955	NUM
cana-3840	284	18	.	.	PUNCT
cana-3840	285	1	[	[	X
cana-3840	285	2	8	8	NUM
cana-3840	285	3	]	]	SYM
cana-3840	285	4	kramosil	kramosil	NOUN
cana-3840	285	5	.	.	PUNCT
cana-3840	286	1	i	i	PRON
cana-3840	286	2	and	and	CCONJ
cana-3840	286	3	michalek	michalek	NOUN
cana-3840	286	4	.	.	PUNCT
cana-3840	287	1	j	j	NOUN
cana-3840	287	2	,	,	PUNCT
cana-3840	287	3	“	"	PUNCT
cana-3840	287	4	fuzzy	fuzzy	ADJ
cana-3840	287	5	metrics	metric	NOUN
cana-3840	287	6	and	and	CCONJ
cana-3840	287	7	statistical	statistical	ADJ
cana-3840	287	8	metric	metric	ADJ
cana-3840	287	9	spaces	space	NOUN
cana-3840	287	10	,	,	PUNCT
cana-3840	287	11	”	"	PUNCT
cana-3840	287	12	kybernetika	kybernetika	NOUN
cana-3840	287	13	,	,	PUNCT
cana-3840	287	14	vol	vol	NOUN
cana-3840	287	15	.	.	PROPN
cana-3840	287	16	11	11	NUM
cana-3840	287	17	,	,	PUNCT
cana-3840	287	18	pp	pp	ADJ
cana-3840	287	19	.	.	PUNCT
cana-3840	288	1	336–344	336–344	NUM
cana-3840	288	2	,	,	PUNCT
cana-3840	288	3	1975	1975	NUM
cana-3840	288	4	.	.	PUNCT
cana-3840	289	1	[	[	X
cana-3840	289	2	9	9	NUM
cana-3840	289	3	]	]	X
cana-3840	289	4	jehad	jehad	PROPN
cana-3840	289	5	r	r	NOUN
cana-3840	289	6	,	,	PUNCT
cana-3840	289	7	madhan	madhan	NOUN
cana-3840	289	8	kider	kider	NOUN
cana-3840	289	9	“	"	PUNCT
cana-3840	289	10	some	some	DET
cana-3840	289	11	properties	property	NOUN
cana-3840	289	12	of	of	ADP
cana-3840	289	13	algebra	algebra	NOUN
cana-3840	289	14	fuzzy	fuzzy	ADJ
cana-3840	289	15	metric	metric	ADJ
cana-3840	289	16	space	space	NOUN
cana-3840	289	17	”	"	PUNCT
cana-3840	289	18	,	,	PUNCT
cana-3840	289	19	journal	journal	NOUN
cana-3840	289	20	of	of	ADP
cana-3840	289	21	alqadisiyah	alqadisiyah	NOUN
cana-3840	289	22	for	for	ADP
cana-3840	289	23	computer	computer	NOUN
cana-3840	289	24	science	science	NOUN
cana-3840	289	25	and	and	CCONJ
cana-3840	289	26	mathematics	mathematic	NOUN
cana-3840	289	27	vol.12(2	vol.12(2	ADV
cana-3840	289	28	)	)	PUNCT
cana-3840	289	29	2020	2020	NUM
cana-3840	289	30	,	,	PUNCT
cana-3840	289	31	pp	pp	ADP
cana-3840	289	32	math	math	NOUN
cana-3840	289	33	43–56	43–56	PROPN
cana-3840	289	34	43	43	NUM
cana-3840	289	35	.	.	PUNCT
cana-3840	290	1	[	[	X
cana-3840	290	2	10	10	NUM
cana-3840	290	3	]	]	X
cana-3840	290	4	jehad	jehad	PROPN
cana-3840	290	5	r	r	PROPN
cana-3840	290	6	,	,	PUNCT
cana-3840	290	7	madhan	madhan	NOUN
cana-3840	290	8	kider	kider	NOUN
cana-3840	290	9	“	"	PUNCT
cana-3840	290	10	application	application	NOUN
cana-3840	290	11	of	of	ADP
cana-3840	290	12	fixed	fix	VERB
cana-3840	290	13	points	point	NOUN
cana-3840	290	14	in	in	ADP
cana-3840	290	15	algebra	algebra	PROPN
cana-3840	290	16	fuzzy	fuzzy	ADJ
cana-3840	290	17	normed	norme	VERB
cana-3840	290	18	spaces	space	NOUN
cana-3840	290	19	”	"	PUNCT
cana-3840	290	20	,	,	PUNCT
cana-3840	290	21	journal	journal	NOUN
cana-3840	290	22	of	of	ADP
cana-3840	290	23	physics	physics	PROPN
cana-3840	290	24	:	:	PUNCT
cana-3840	290	25	conference	conference	NOUN
cana-3840	290	26	series	series	NOUN
cana-3840	290	27	1879	1879	NUM
cana-3840	290	28	(	(	PUNCT
cana-3840	290	29	2021	2021	NUM
cana-3840	290	30	)	)	PUNCT
cana-3840	290	31	022099	022099	NUM
cana-3840	290	32	,	,	PUNCT
cana-3840	290	33	iop	iop	NOUN
cana-3840	290	34	publishing	publishing	NOUN
cana-3840	290	35	.	.	PUNCT
cana-3840	291	1	[	[	X
cana-3840	291	2	11	11	NUM
cana-3840	291	3	]	]	X
cana-3840	291	4	jianrong	jianrong	PROPN
cana-3840	291	5	wu	wu	PROPN
cana-3840	291	6	and	and	CCONJ
cana-3840	291	7	hao	hao	PROPN
cana-3840	291	8	yang	yang	PROPN
cana-3840	291	9	,	,	PUNCT
cana-3840	291	10	“	"	PUNCT
cana-3840	291	11	on	on	ADP
cana-3840	291	12	metrization	metrization	NOUN
cana-3840	291	13	of	of	ADP
cana-3840	291	14	the	the	DET
cana-3840	291	15	topologies	topology	NOUN
cana-3840	291	16	induced	induce	VERB
cana-3840	291	17	by	by	ADP
cana-3840	291	18	fuzzy	fuzzy	ADJ
cana-3840	291	19	metrics	metric	NOUN
cana-3840	291	20	”	"	PUNCT
cana-3840	291	21	,	,	PUNCT
cana-3840	291	22	journal	journal	NOUN
cana-3840	291	23	of	of	ADP
cana-3840	291	24	functiom	functiom	ADJ
cana-3840	291	25	spaces	space	NOUN
cana-3840	291	26	,	,	PUNCT
cana-3840	291	27	volume	volume	NOUN
cana-3840	291	28	2020	2020	NUM
cana-3840	291	29	,	,	PUNCT
cana-3840	291	30	article	article	NOUN
cana-3840	291	31	i	i	PROPN
cana-3840	291	32	d	d	PROPN
cana-3840	291	33	4731357	4731357	NUM
cana-3840	291	34	,	,	PUNCT
cana-3840	291	35	5	5	NUM
cana-3840	291	36	pages	page	NOUN
cana-3840	291	37	.	.	PUNCT
cana-3840	292	1	[	[	X
cana-3840	292	2	12	12	NUM
cana-3840	292	3	]	]	PUNCT
cana-3840	292	4	morillas	morilla	NOUN
cana-3840	292	5	.	.	PUNCT
cana-3840	293	1	s	s	X
cana-3840	293	2	,	,	PUNCT
cana-3840	293	3	gregori	gregori	PROPN
cana-3840	293	4	.	.	PUNCT
cana-3840	294	1	v	v	X
cana-3840	294	2	,	,	PUNCT
cana-3840	294	3	peris	peris	PROPN
cana-3840	294	4	-	-	PUNCT
cana-3840	294	5	fajarn´es	fajarn´es	NOUN
cana-3840	294	6	.	.	PUNCT
cana-3840	295	1	g	g	NOUN
cana-3840	295	2	,	,	PUNCT
cana-3840	295	3	and	and	CCONJ
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cana-3840	295	5	.	.	PUNCT
cana-3840	296	1	p	p	X
cana-3840	296	2	,	,	PUNCT
cana-3840	296	3	“	"	PUNCT
cana-3840	296	4	a	a	DET
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cana-3840	296	6	vector	vector	NOUN
cana-3840	296	7	median	median	NOUN
cana-3840	296	8	filter	filter	NOUN
cana-3840	296	9	based	base	VERB
cana-3840	296	10	on	on	ADP
cana-3840	296	11	fuzzy	fuzzy	ADJ
cana-3840	296	12	metrics	metric	NOUN
cana-3840	296	13	,	,	PUNCT
cana-3840	296	14	”	"	PUNCT
cana-3840	296	15	in	in	ADP
cana-3840	296	16	lecture	lecture	NOUN
cana-3840	296	17	notes	note	NOUN
cana-3840	296	18	in	in	ADP
cana-3840	296	19	computer	computer	NOUN
cana-3840	296	20	science	science	NOUN
cana-3840	296	21	,	,	PUNCT
cana-3840	296	22	vol	vol	NOUN
cana-3840	296	23	.	.	PROPN
cana-3840	296	24	3656	3656	NUM
cana-3840	296	25	,	,	PUNCT
cana-3840	296	26	pp	pp	ADP
cana-3840	296	27	.	.	PUNCT
cana-3840	297	1	81–90	81–90	NUM
cana-3840	297	2	,	,	PUNCT
cana-3840	297	3	springer	springer	NOUN
cana-3840	297	4	,	,	PUNCT
cana-3840	297	5	berlin	berlin	PROPN
cana-3840	297	6	,	,	PUNCT
cana-3840	297	7	germany	germany	PROPN
cana-3840	297	8	,	,	PUNCT
cana-3840	297	9	2005	2005	NUM
cana-3840	297	10	.	.	PUNCT
cana-3840	298	1	[	[	X
cana-3840	298	2	13	13	NUM
cana-3840	298	3	]	]	SYM
cana-3840	298	4	morillas	morilla	NOUN
cana-3840	298	5	.	.	PUNCT
cana-3840	299	1	s	s	X
cana-3840	299	2	,	,	PUNCT
cana-3840	299	3	gregori	gregori	PROPN
cana-3840	299	4	.	.	PUNCT
cana-3840	300	1	v	v	X
cana-3840	300	2	,	,	PUNCT
cana-3840	300	3	peris	peris	PROPN
cana-3840	300	4	-	-	PUNCT
cana-3840	300	5	fajarn´es	fajarn´es	NOUN
cana-3840	300	6	.	.	PUNCT
cana-3840	301	1	g	g	NOUN
cana-3840	301	2	,	,	PUNCT
cana-3840	301	3	and	and	CCONJ
cana-3840	301	4	latorre	latorre	NOUN
cana-3840	301	5	.	.	PUNCT
cana-3840	302	1	p	p	X
cana-3840	302	2	,	,	PUNCT
cana-3840	302	3	“	"	PUNCT
cana-3840	302	4	a	a	DET
cana-3840	302	5	fast	fast	ADJ
cana-3840	302	6	impulsive	impulsive	ADJ
cana-3840	302	7	noise	noise	NOUN
cana-3840	302	8	color	color	NOUN
cana-3840	302	9	image	image	NOUN
cana-3840	302	10	filter	filter	NOUN
cana-3840	302	11	using	use	VERB
cana-3840	302	12	fuzzy	fuzzy	ADJ
cana-3840	302	13	metrics	metric	NOUN
cana-3840	302	14	,	,	PUNCT
cana-3840	302	15	real	real	ADJ
cana-3840	302	16	-	-	PUNCT
cana-3840	302	17	time	time	NOUN
cana-3840	302	18	imaging	imaging	NOUN
cana-3840	302	19	,	,	PUNCT
cana-3840	302	20	vol	vol	NOUN
cana-3840	302	21	.	.	PROPN
cana-3840	303	1	11	11	NUM
cana-3840	303	2	,	,	PUNCT
cana-3840	303	3	no	no	INTJ
cana-3840	303	4	.	.	NOUN
cana-3840	303	5	5	5	NUM
cana-3840	303	6	-	-	SYM
cana-3840	303	7	6	6	NUM
cana-3840	303	8	,	,	PUNCT
cana-3840	303	9	pp	pp	ADJ
cana-3840	303	10	.	.	PUNCT
cana-3840	304	1	417–428	417–428	NUM
cana-3840	304	2	,	,	PUNCT
cana-3840	304	3	2005	2005	NUM
cana-3840	304	4	.	.	PUNCT
cana-3840	305	1	communications	communication	NOUN
cana-3840	305	2	on	on	ADP
cana-3840	305	3	applied	apply	VERB
cana-3840	305	4	nonlinear	nonlinear	ADJ
cana-3840	305	5	analysis	analysis	NOUN
cana-3840	305	6	issn	issn	NOUN
cana-3840	305	7	:	:	PUNCT
cana-3840	305	8	1074	1074	NUM
cana-3840	305	9	-	-	PUNCT
cana-3840	305	10	133x	133x	NUM
cana-3840	305	11	vol	vol	NOUN
cana-3840	305	12	32	32	NUM
cana-3840	305	13	no	no	NOUN
cana-3840	305	14	.	.	PUNCT
cana-3840	306	1	9s	9s	NUM
cana-3840	306	2	(	(	PUNCT
cana-3840	306	3	2025	2025	NUM
cana-3840	306	4	)	)	PUNCT
cana-3840	306	5	97	97	NUM
cana-3840	306	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	307	1	[	[	X
cana-3840	307	2	14	14	NUM
cana-3840	307	3	]	]	X
cana-3840	307	4	muraliraj	muraliraj	VERB
cana-3840	307	5	a	a	DET
cana-3840	307	6	,	,	PUNCT
cana-3840	307	7	thangathamizh	thangathamizh	ADJ
cana-3840	307	8	r	r	NOUN
cana-3840	307	9	,	,	PUNCT
cana-3840	307	10	popovic	popovic	NOUN
cana-3840	307	11	n	n	CCONJ
cana-3840	307	12	,	,	PUNCT
cana-3840	307	13	savic	savic	PROPN
cana-3840	307	14	a	a	X
cana-3840	307	15	,	,	PUNCT
cana-3840	307	16	radenovic	radenovic	PROPN
cana-3840	307	17	s.	s.	PROPN
cana-3840	307	18	the	the	DET
cana-3840	307	19	first	first	ADJ
cana-3840	307	20	rational	rational	ADJ
cana-3840	307	21	type	type	NOUN
cana-3840	307	22	revisd	revisd	NOUN
cana-3840	307	23	fuzzy	fuzzy	ADJ
cana-3840	307	24	-	-	PUNCT
cana-3840	307	25	contractions	contraction	NOUN
cana-3840	307	26	in	in	ADP
cana-3840	307	27	revisd	revisd	NOUN
cana-3840	307	28	fuzzy	fuzzy	ADJ
cana-3840	307	29	metric	metric	ADJ
cana-3840	307	30	spaces	space	NOUN
cana-3840	307	31	with	with	ADP
cana-3840	307	32	an	an	DET
cana-3840	307	33	applications	application	NOUN
cana-3840	307	34	.	.	PUNCT
cana-3840	308	1	mathematics	mathematic	NOUN
cana-3840	308	2	.	.	PUNCT
cana-3840	309	1	2023;11(10):2244	2023;11(10):2244	X
cana-3840	309	2	.	.	PUNCT
cana-3840	310	1	[	[	X
cana-3840	310	2	15	15	NUM
cana-3840	310	3	]	]	PUNCT
cana-3840	310	4	muraliraj.a	muraliraj.a	PROPN
cana-3840	310	5	and	and	CCONJ
cana-3840	310	6	thangathamizh	thangathamizh	ADJ
cana-3840	310	7	.	.	PUNCT
cana-3840	311	1	r	r	X
cana-3840	311	2	,	,	PUNCT
cana-3840	311	3	“	"	PUNCT
cana-3840	311	4	some	some	DET
cana-3840	311	5	topological	topological	ADJ
cana-3840	311	6	properties	property	NOUN
cana-3840	311	7	of	of	ADP
cana-3840	311	8	revised	revise	VERB
cana-3840	311	9	fuzzy	fuzzy	ADJ
cana-3840	311	10	cone	cone	NOUN
cana-3840	311	11	metric	metric	ADJ
cana-3840	311	12	space	space	NOUN
cana-3840	311	13	”	"	PUNCT
cana-3840	311	14	,	,	PUNCT
cana-3840	311	15	ratio	ratio	PROPN
cana-3840	311	16	mathematica	mathematica	PROPN
cana-3840	311	17	,	,	PUNCT
cana-3840	311	18	volume	volume	NOUN
cana-3840	311	19	26	26	NUM
cana-3840	311	20	,	,	PUNCT
cana-3840	311	21	number	number	NOUN
cana-3840	311	22	2	2	NUM
cana-3840	311	23	,	,	PUNCT
cana-3840	311	24	(	(	PUNCT
cana-3840	311	25	2023	2023	NUM
cana-3840	311	26	)	)	PUNCT
cana-3840	311	27	.	.	PUNCT
cana-3840	312	1	[	[	X
cana-3840	312	2	16	16	NUM
cana-3840	312	3	]	]	X
cana-3840	312	4	muraliraj	muraliraj	VERB
cana-3840	312	5	a	a	DET
cana-3840	312	6	and	and	CCONJ
cana-3840	312	7	thangathamizh	thangathamizh	ADJ
cana-3840	312	8	r	r	NOUN
cana-3840	312	9	,	,	PUNCT
cana-3840	312	10	“	"	PUNCT
cana-3840	312	11	new	new	ADJ
cana-3840	312	12	relation	relation	NOUN
cana-3840	312	13	-	-	PUNCT
cana-3840	312	14	theoretic	theoretic	ADJ
cana-3840	312	15	fixed	fix	VERB
cana-3840	312	16	-	-	PUNCT
cana-3840	312	17	point	point	NOUN
cana-3840	312	18	theorems	theorem	NOUN
cana-3840	312	19	in	in	ADP
cana-3840	312	20	revised	revise	VERB
cana-3840	312	21	fuzzy	fuzzy	ADJ
cana-3840	312	22	metric	metric	ADJ
cana-3840	312	23	spaces	space	NOUN
cana-3840	312	24	with	with	ADP
cana-3840	312	25	an	an	DET
cana-3840	312	26	application	application	NOUN
cana-3840	312	27	to	to	ADP
cana-3840	312	28	fractional	fractional	ADJ
cana-3840	312	29	differential	differential	ADJ
cana-3840	312	30	equations	equation	NOUN
cana-3840	312	31	”	"	PUNCT
cana-3840	312	32	,	,	PUNCT
cana-3840	312	33	communications	communication	NOUN
cana-3840	312	34	in	in	ADP
cana-3840	312	35	mathematics	mathematic	NOUN
cana-3840	312	36	and	and	CCONJ
cana-3840	312	37	applications	application	NOUN
cana-3840	312	38	,	,	PUNCT
cana-3840	312	39	vol.12	vol.12	NOUN
cana-3840	312	40	,	,	PUNCT
cana-3840	312	41	no	no	DET
cana-3840	312	42	22	22	NUM
cana-3840	312	43	.	.	PUNCT
cana-3840	312	44	(	(	PUNCT
cana-3840	312	45	2023	2023	NUM
cana-3840	312	46	)	)	PUNCT
cana-3840	312	47	.	.	PUNCT
cana-3840	313	1	[	[	X
cana-3840	313	2	17	17	NUM
cana-3840	313	3	]	]	PUNCT
cana-3840	313	4	muraliraj.a	muraliraj.a	PROPN
cana-3840	313	5	and	and	CCONJ
cana-3840	313	6	thangathamizh	thangathamizh	ADJ
cana-3840	313	7	.	.	PUNCT
cana-3840	314	1	r	r	X
cana-3840	314	2	,	,	PUNCT
cana-3840	314	3	“	"	PUNCT
cana-3840	314	4	fixed	fix	VERB
cana-3840	314	5	point	point	NOUN
cana-3840	314	6	theorems	theorem	NOUN
cana-3840	314	7	in	in	ADP
cana-3840	314	8	revised	revise	VERB
cana-3840	314	9	fuzzy	fuzzy	ADJ
cana-3840	314	10	metric	metric	ADJ
cana-3840	314	11	space	space	NOUN
cana-3840	314	12	”	"	PUNCT
cana-3840	314	13	,	,	PUNCT
cana-3840	314	14	advances	advance	NOUN
cana-3840	314	15	in	in	ADP
cana-3840	314	16	fuzzy	fuzzy	ADJ
cana-3840	314	17	sets	set	NOUN
cana-3840	314	18	and	and	CCONJ
cana-3840	314	19	systems	system	NOUN
cana-3840	314	20	,	,	PUNCT
cana-3840	314	21	volume	volume	NOUN
cana-3840	314	22	26	26	NUM
cana-3840	314	23	,	,	PUNCT
cana-3840	314	24	number	number	NOUN
cana-3840	314	25	2	2	NUM
cana-3840	314	26	,	,	PUNCT
cana-3840	314	27	2021	2021	NUM
cana-3840	314	28	.	.	PUNCT
cana-3840	315	1	[	[	X
cana-3840	315	2	18	18	NUM
cana-3840	315	3	]	]	X
cana-3840	315	4	muraliraj	muraliraj	NOUN
cana-3840	315	5	.	.	PUNCT
cana-3840	316	1	m	m	PROPN
cana-3840	316	2	and	and	CCONJ
cana-3840	316	3	thangathamizh	thangathamizh	ADJ
cana-3840	316	4	.	.	PUNCT
cana-3840	317	1	r	r	X
cana-3840	317	2	,	,	PUNCT
cana-3840	317	3	“	"	PUNCT
cana-3840	317	4	introduction	introduction	NOUN
cana-3840	317	5	on	on	ADP
cana-3840	317	6	revised	revise	VERB
cana-3840	317	7	fuzzy	fuzzy	ADJ
cana-3840	317	8	modular	modular	ADJ
cana-3840	317	9	spaces	space	NOUN
cana-3840	317	10	”	"	PUNCT
cana-3840	317	11	,	,	PUNCT
cana-3840	317	12	global	global	ADJ
cana-3840	317	13	journal	journal	NOUN
cana-3840	317	14	of	of	ADP
cana-3840	317	15	pure	pure	ADJ
cana-3840	317	16	and	and	CCONJ
cana-3840	317	17	applied	applied	ADJ
cana-3840	317	18	mathematics	mathematic	NOUN
cana-3840	317	19	.	.	PUNCT
cana-3840	318	1	volume	volume	NOUN
cana-3840	318	2	17	17	NUM
cana-3840	318	3	,	,	PUNCT
cana-3840	318	4	number	number	NOUN
cana-3840	318	5	2	2	NUM
cana-3840	318	6	(	(	PUNCT
cana-3840	318	7	2021	2021	NUM
cana-3840	318	8	)	)	PUNCT
cana-3840	318	9	,	,	PUNCT
cana-3840	318	10	pp	pp	ADP
cana-3840	318	11	.	.	PUNCT
cana-3840	319	1	303	303	NUM
cana-3840	319	2	-	-	SYM
cana-3840	319	3	317	317	NUM
cana-3840	319	4	.	.	PUNCT
cana-3840	320	1	[	[	X
cana-3840	320	2	19	19	NUM
cana-3840	320	3	]	]	X
cana-3840	320	4	muraliraj	muraliraj	VERB
cana-3840	320	5	a	a	PRON
cana-3840	320	6	and	and	CCONJ
cana-3840	320	7	shanmugavel	shanmugavel	NOUN
cana-3840	320	8	p	p	X
cana-3840	320	9	,	,	PUNCT
cana-3840	320	10	thangathamizh	thangathamizh	ADJ
cana-3840	320	11	r	r	NOUN
cana-3840	320	12	“	"	PUNCT
cana-3840	320	13	existence	existence	NOUN
cana-3840	320	14	of	of	ADP
cana-3840	320	15	fixed	fix	VERB
cana-3840	320	16	point	point	NOUN
cana-3840	320	17	theorems	theorem	NOUN
cana-3840	320	18	in	in	ADP
cana-3840	320	19	revised	revise	VERB
cana-3840	320	20	fuzzy	fuzzy	ADJ
cana-3840	320	21	modular	modular	ADJ
cana-3840	320	22	spaces	space	NOUN
cana-3840	320	23	”	"	PUNCT
cana-3840	320	24	,	,	PUNCT
cana-3840	320	25	advances	advance	NOUN
cana-3840	320	26	in	in	ADP
cana-3840	320	27	nonlinear	nonlinear	ADJ
cana-3840	320	28	variational	variational	ADJ
cana-3840	320	29	inequalities	inequality	NOUN
cana-3840	320	30	,	,	PUNCT
cana-3840	320	31	vol	vol	NOUN
cana-3840	320	32	.	.	PROPN
cana-3840	320	33	27	27	NUM
cana-3840	320	34	no	no	NOUN
cana-3840	320	35	.	.	NOUN
cana-3840	320	36	2	2	NUM
cana-3840	320	37	(	(	PUNCT
cana-3840	320	38	2024	2024	NUM
cana-3840	320	39	)	)	PUNCT
cana-3840	320	40	.	.	PUNCT
cana-3840	321	1	[	[	X
cana-3840	321	2	20	20	NUM
cana-3840	321	3	]	]	X
cana-3840	321	4	muraliraj	muraliraj	VERB
cana-3840	321	5	a	a	PRON
cana-3840	321	6	and	and	CCONJ
cana-3840	321	7	shanmugavel	shanmugavel	NOUN
cana-3840	321	8	p	p	X
cana-3840	321	9	,	,	PUNCT
cana-3840	321	10	thangathamizh	thangathamizh	ADJ
cana-3840	321	11	r	r	NOUN
cana-3840	321	12	,	,	PUNCT
cana-3840	321	13	“	"	PUNCT
cana-3840	321	14	fixed	fix	VERB
cana-3840	321	15	point	point	NOUN
cana-3840	321	16	theorems	theorem	NOUN
cana-3840	321	17	in	in	ADP
cana-3840	321	18	modular	modular	ADJ
cana-3840	321	19	revised	revise	VERB
cana-3840	321	20	fuzzy	fuzzy	ADJ
cana-3840	321	21	metric	metric	ADJ
cana-3840	321	22	spaces	space	NOUN
cana-3840	321	23	”	"	PUNCT
cana-3840	321	24	,	,	PUNCT
cana-3840	321	25	communications	communication	NOUN
cana-3840	321	26	on	on	ADP
cana-3840	321	27	applied	apply	VERB
cana-3840	321	28	nonlinear	nonlinear	ADJ
cana-3840	321	29	analysis	analysis	NOUN
cana-3840	321	30	,	,	PUNCT
cana-3840	321	31	vol	vol	NOUN
cana-3840	321	32	31	31	NUM
cana-3840	321	33	,	,	PUNCT
cana-3840	321	34	no	no	INTJ
cana-3840	321	35	.	.	PUNCT
cana-3840	321	36	3s	3s	NUM
cana-3840	321	37	,	,	PUNCT
cana-3840	321	38	2024	2024	NUM
cana-3840	321	39	.	.	PUNCT
cana-3840	322	1	doi.10.52783	doi.10.52783	NOUN
cana-3840	322	2	/	/	SYM
cana-3840	322	3	cana	cana	PROPN
cana-3840	322	4	.	.	PUNCT
cana-3840	323	1	v31.760	v31.760	PROPN
cana-3840	323	2	.	.	PUNCT
cana-3840	324	1	[	[	X
cana-3840	324	2	21	21	NUM
cana-3840	324	3	]	]	X
cana-3840	324	4	olga	olga	PROPN
cana-3840	324	5	grigorenko	grigorenko	PROPN
cana-3840	324	6	,	,	PUNCT
cana-3840	324	7	juan	juan	PROPN
cana-3840	324	8	jose	jose	PROPN
cana-3840	324	9	minana	minana	PROPN
cana-3840	324	10	,	,	PUNCT
cana-3840	324	11	alexander	alexander	PROPN
cana-3840	324	12	sostak	sostak	PROPN
cana-3840	324	13	“	"	PUNCT
cana-3840	324	14	on	on	ADP
cana-3840	324	15	t	t	PROPN
cana-3840	324	16	-	-	PUNCT
cana-3840	324	17	conorm	conorm	NOUN
cana-3840	324	18	based	base	VERB
cana-3840	324	19	fuzzy	fuzzy	ADJ
cana-3840	324	20	(	(	PUNCT
cana-3840	324	21	pseudo	pseudo	NOUN
cana-3840	324	22	)	)	PUNCT
cana-3840	324	23	metrics	metric	NOUN
cana-3840	324	24	”	"	PUNCT
cana-3840	324	25	,	,	PUNCT
cana-3840	324	26	axioms	axiom	VERB
cana-3840	324	27	2020	2020	NUM
cana-3840	324	28	,	,	PUNCT
cana-3840	324	29	9	9	NUM
cana-3840	324	30	,	,	PUNCT
cana-3840	324	31	78	78	NUM
cana-3840	324	32	.	.	PUNCT
cana-3840	325	1	[	[	X
cana-3840	325	2	22	22	NUM
cana-3840	325	3	]	]	PUNCT
cana-3840	325	4	parakath	parakath	PROPN
cana-3840	325	5	nisha	nisha	PROPN
cana-3840	325	6	bagam	bagam	PROPN
cana-3840	326	1	p	p	PROPN
cana-3840	326	2	,	,	PUNCT
cana-3840	326	3	sandhya	sandhya	PROPN
cana-3840	326	4	p	p	PROPN
cana-3840	326	5	,	,	PUNCT
cana-3840	326	6	thangathamizh	thangathamizh	ADJ
cana-3840	326	7	r	r	NOUN
cana-3840	326	8	,	,	PUNCT
cana-3840	326	9	shanmugavel	shanmugavel	NOUN
cana-3840	326	10	p	p	NOUN
cana-3840	326	11	,	,	PUNCT
cana-3840	326	12	sarathbabu	sarathbabu	PROPN
cana-3840	326	13	k	k	NOUN
cana-3840	326	14	,	,	PUNCT
cana-3840	326	15	anusuya	anusuya	PROPN
cana-3840	326	16	r	r	NOUN
cana-3840	326	17	,	,	PUNCT
cana-3840	326	18	“	"	PUNCT
cana-3840	326	19	fixed	fix	VERB
cana-3840	326	20	point	point	NOUN
cana-3840	326	21	theorems	theorem	NOUN
cana-3840	326	22	in	in	ADP
cana-3840	326	23	revised	revise	VERB
cana-3840	326	24	fuzzy	fuzzy	ADJ
cana-3840	326	25	metric	metric	ADJ
cana-3840	326	26	space	space	NOUN
cana-3840	326	27	via	via	ADP
cana-3840	326	28	𝑅𝐹	𝑅𝐹	PROPN
cana-3840	326	29	−contraction	−contraction	PROPN
cana-3840	326	30	”	"	PUNCT
cana-3840	326	31	,	,	PUNCT
cana-3840	326	32	communications	communication	NOUN
cana-3840	326	33	on	on	ADP
cana-3840	326	34	applied	apply	VERB
cana-3840	326	35	nonlinear	nonlinear	ADJ
cana-3840	326	36	analysis	analysis	NOUN
cana-3840	326	37	,	,	PUNCT
cana-3840	326	38	vol	vol	NOUN
cana-3840	326	39	31	31	NUM
cana-3840	326	40	,	,	PUNCT
cana-3840	326	41	no	no	INTJ
cana-3840	326	42	.	.	PUNCT
cana-3840	326	43	3s	3s	NUM
cana-3840	326	44	,	,	PUNCT
cana-3840	326	45	2024	2024	NUM
cana-3840	326	46	.	.	PUNCT
cana-3840	327	1	doi.10.52783	doi.10.52783	NOUN
cana-3840	327	2	/	/	SYM
cana-3840	327	3	cana	cana	PROPN
cana-3840	327	4	.	.	PUNCT
cana-3840	328	1	v31.761	v31.761	VERB
cana-3840	328	2	.	.	PUNCT
cana-3840	329	1	[	[	X
cana-3840	329	2	23	23	NUM
cana-3840	329	3	]	]	PUNCT
cana-3840	329	4	romaguera	romaguera	NOUN
cana-3840	329	5	.	.	PUNCT
cana-3840	330	1	s	s	X
cana-3840	330	2	,	,	PUNCT
cana-3840	330	3	sapena	sapena	NOUN
cana-3840	330	4	.	.	PUNCT
cana-3840	331	1	a	a	PRON
cana-3840	331	2	,	,	PUNCT
cana-3840	331	3	and	and	CCONJ
cana-3840	331	4	tirado	tirado	NOUN
cana-3840	331	5	.	.	PUNCT
cana-3840	332	1	p	p	X
cana-3840	332	2	,	,	PUNCT
cana-3840	332	3	the	the	DET
cana-3840	332	4	banach	banach	ADV
cana-3840	332	5	fixed	fix	VERB
cana-3840	332	6	point	point	NOUN
cana-3840	332	7	theorem	theorem	VERB
cana-3840	332	8	in	in	ADP
cana-3840	332	9	fuzzy	fuzzy	ADJ
cana-3840	332	10	quasi	quasi	ADJ
cana-3840	332	11	-	-	ADJ
cana-3840	332	12	metric	metric	ADJ
cana-3840	332	13	spaces	space	NOUN
cana-3840	332	14	with	with	ADP
cana-3840	332	15	application	application	NOUN
cana-3840	332	16	to	to	ADP
cana-3840	332	17	the	the	DET
cana-3840	332	18	domain	domain	NOUN
cana-3840	332	19	of	of	ADP
cana-3840	332	20	words	word	NOUN
cana-3840	332	21	,	,	PUNCT
cana-3840	332	22	”	"	PUNCT
cana-3840	332	23	topology	topology	NOUN
cana-3840	332	24	and	and	CCONJ
cana-3840	332	25	its	its	PRON
cana-3840	332	26	applications	application	NOUN
cana-3840	332	27	,	,	PUNCT
cana-3840	332	28	vol	vol	NOUN
cana-3840	332	29	.	.	PROPN
cana-3840	333	1	154	154	NUM
cana-3840	333	2	,	,	PUNCT
cana-3840	333	3	no	no	INTJ
cana-3840	333	4	.	.	NOUN
cana-3840	333	5	10	10	NUM
cana-3840	333	6	,	,	PUNCT
cana-3840	333	7	pp	pp	ADJ
cana-3840	333	8	.	.	PUNCT
cana-3840	334	1	2196–2203	2196–2203	NOUN
cana-3840	334	2	,	,	PUNCT
cana-3840	334	3	2007	2007	NUM
cana-3840	334	4	.	.	PUNCT
cana-3840	335	1	[	[	X
cana-3840	335	2	24	24	NUM
cana-3840	335	3	]	]	X
cana-3840	335	4	tarkan	tarkan	PROPN
cana-3840	335	5	oner	oner	PROPN
cana-3840	335	6	,	,	PUNCT
cana-3840	335	7	alexander	alexander	PROPN
cana-3840	335	8	sostak	sostak	PROPN
cana-3840	335	9	,	,	PUNCT
cana-3840	335	10	“	"	PUNCT
cana-3840	335	11	on	on	ADP
cana-3840	335	12	metric	metric	ADJ
cana-3840	335	13	-	-	PUNCT
cana-3840	335	14	type	type	NOUN
cana-3840	335	15	spaces	space	NOUN
cana-3840	335	16	based	base	VERB
cana-3840	335	17	on	on	ADP
cana-3840	335	18	extended	extend	VERB
cana-3840	335	19	t	t	PROPN
cana-3840	335	20	-	-	PUNCT
cana-3840	335	21	conorms	conorm	NOUN
cana-3840	335	22	”	"	PUNCT
cana-3840	335	23	mathematics	mathematic	NOUN
cana-3840	335	24	2020	2020	NUM
cana-3840	335	25	,	,	PUNCT
cana-3840	335	26	8	8	NUM
cana-3840	335	27	,	,	PUNCT
cana-3840	335	28	1097	1097	NUM
cana-3840	335	29	.	.	PUNCT
cana-3840	336	1	[	[	X
cana-3840	336	2	25	25	NUM
cana-3840	336	3	]	]	PUNCT
cana-3840	336	4	thangathamizh	thangathamizh	PROPN
cana-3840	336	5	r	r	NOUN
cana-3840	336	6	,	,	PUNCT
cana-3840	336	7	muraliraj	muraliraj	VERB
cana-3840	336	8	a	a	PRON
cana-3840	336	9	and	and	CCONJ
cana-3840	336	10	shanmugavel	shanmugavel	NOUN
cana-3840	336	11	p	p	X
cana-3840	336	12	“	"	PUNCT
cana-3840	336	13	new	new	ADJ
cana-3840	336	14	approach	approach	NOUN
cana-3840	336	15	of	of	ADP
cana-3840	336	16	lebesgue	lebesgue	PROPN
cana-3840	336	17	integral	integral	ADJ
cana-3840	336	18	in	in	ADP
cana-3840	336	19	revised	revise	VERB
cana-3840	336	20	fuzzy	fuzzy	ADJ
cana-3840	336	21	cone	cone	NOUN
cana-3840	336	22	metric	metric	ADJ
cana-3840	336	23	spaces	space	NOUN
cana-3840	336	24	vie	vie	X
cana-3840	336	25	unique	unique	ADJ
cana-3840	336	26	coupled	couple	VERB
cana-3840	336	27	fixed	fix	VERB
cana-3840	336	28	point	point	NOUN
cana-3840	336	29	theorems	theorem	NOUN
cana-3840	336	30	”	"	PUNCT
cana-3840	336	31	,	,	PUNCT
cana-3840	336	32	military	military	ADJ
cana-3840	336	33	technical	technical	ADJ
cana-3840	336	34	courier	courier	NOUN
cana-3840	336	35	,	,	PUNCT
cana-3840	336	36	no	no	DET
cana-3840	336	37	3	3	NUM
cana-3840	336	38	,	,	PUNCT
cana-3840	336	39	2024	2024	NUM
cana-3840	336	40	.	.	PUNCT
cana-3840	337	1	doi.10.5937	doi.10.5937	NOUN
cana-3840	337	2	/	/	SYM
cana-3840	337	3	vojtehg72	vojtehg72	NOUN
cana-3840	337	4	-	-	PUNCT
cana-3840	337	5	48816	48816	NUM
cana-3840	337	6	.	.	PUNCT
cana-3840	338	1	[	[	X
cana-3840	338	2	26	26	NUM
cana-3840	338	3	]	]	PUNCT
cana-3840	338	4	thangathamizh	thangathamizh	PROPN
cana-3840	338	5	r	r	NOUN
cana-3840	338	6	,	,	PUNCT
cana-3840	338	7	balamurugan	balamurugan	VERB
cana-3840	338	8	k	k	PROPN
cana-3840	338	9	,	,	PUNCT
cana-3840	338	10	karnan	karnan	PROPN
cana-3840	338	11	c	c	NOUN
cana-3840	338	12	,	,	PUNCT
cana-3840	338	13	shanmugavel	shanmugavel	NOUN
cana-3840	338	14	p	p	NOUN
cana-3840	338	15	,	,	PUNCT
cana-3840	338	16	and	and	CCONJ
cana-3840	338	17	balraj	balraj	X
cana-3840	338	18	d	d	NOUN
cana-3840	338	19	,	,	PUNCT
cana-3840	338	20	“	"	PUNCT
cana-3840	338	21	revised	revise	VERB
cana-3840	338	22	fuzzy	fuzzy	ADJ
cana-3840	338	23	differential	differential	ADJ
cana-3840	338	24	equations	equation	NOUN
cana-3840	338	25	using	use	VERB
cana-3840	338	26	weakly	weakly	ADJ
cana-3840	338	27	compatible	compatible	ADJ
cana-3840	338	28	self	self	NOUN
cana-3840	338	29	-	-	PUNCT
cana-3840	338	30	mappings	mapping	NOUN
cana-3840	338	31	in	in	ADP
cana-3840	338	32	revised	revise	VERB
cana-3840	338	33	fuzzy	fuzzy	ADJ
cana-3840	338	34	metric	metric	ADJ
cana-3840	338	35	spaces	space	NOUN
cana-3840	338	36	”	"	PUNCT
cana-3840	338	37	,	,	PUNCT
cana-3840	338	38	advances	advance	NOUN
cana-3840	338	39	in	in	ADP
cana-3840	338	40	nonlinear	nonlinear	ADJ
cana-3840	338	41	variational	variational	ADJ
cana-3840	338	42	inequalities	inequality	NOUN
cana-3840	338	43	,	,	PUNCT
cana-3840	338	44	vol	vol	NOUN
cana-3840	338	45	.	.	PROPN
cana-3840	339	1	27	27	NUM
cana-3840	339	2	no	no	NOUN
cana-3840	339	3	.	.	NOUN
cana-3840	339	4	2	2	NUM
cana-3840	339	5	(	(	PUNCT
cana-3840	339	6	2024	2024	NUM
cana-3840	339	7	)	)	PUNCT
cana-3840	339	8	.	.	PUNCT
cana-3840	340	1	[	[	X
cana-3840	340	2	27	27	NUM
cana-3840	340	3	]	]	PUNCT
cana-3840	340	4	thangathamizh	thangathamizh	PROPN
cana-3840	340	5	r	r	NOUN
cana-3840	340	6	,	,	PUNCT
cana-3840	340	7	abdelhamid	abdelhamid	ADP
cana-3840	340	8	moussaoui	moussaoui	NOUN
cana-3840	340	9	,	,	PUNCT
cana-3840	340	10	tatjana	tatjana	PROPN
cana-3840	340	11	dosenovic	dosenovic	PROPN
cana-3840	340	12	,	,	PUNCT
cana-3840	340	13	stojan	stojan	ADP
cana-3840	340	14	radenovic	radenovic	PROPN
cana-3840	340	15	“	"	PUNCT
cana-3840	340	16	fixed	fix	VERB
cana-3840	340	17	point	point	NOUN
cana-3840	340	18	results	result	NOUN
cana-3840	340	19	in	in	ADP
cana-3840	340	20	controlled	control	VERB
cana-3840	340	21	revised	revise	VERB
cana-3840	340	22	fuzzy	fuzzy	ADJ
cana-3840	340	23	metric	metric	ADJ
cana-3840	340	24	spaces	space	NOUN
cana-3840	340	25	with	with	ADP
cana-3840	340	26	an	an	DET
cana-3840	340	27	application	application	NOUN
cana-3840	340	28	to	to	ADP
cana-3840	340	29	the	the	DET
cana-3840	340	30	transformation	transformation	NOUN
cana-3840	340	31	of	of	ADP
cana-3840	340	32	solar	solar	ADJ
cana-3840	340	33	energy	energy	NOUN
cana-3840	340	34	to	to	ADP
cana-3840	340	35	electric	electric	ADJ
cana-3840	340	36	power	power	NOUN
cana-3840	340	37	”	"	PUNCT
cana-3840	340	38	,	,	PUNCT
cana-3840	340	39	military	military	ADJ
cana-3840	340	40	technical	technical	ADJ
cana-3840	340	41	courier	courier	NOUN
cana-3840	340	42	,	,	PUNCT
cana-3840	340	43	no	no	DET
cana-3840	340	44	4	4	NUM
cana-3840	340	45	,	,	PUNCT
cana-3840	340	46	2024	2024	NUM
cana-3840	340	47	.	.	PUNCT
cana-3840	341	1	doi.10.5937	doi.10.5937	NOUN
cana-3840	341	2	/	/	SYM
cana-3840	341	3	vojtehg7249064	vojtehg7249064	PROPN
cana-3840	341	4	.	.	PUNCT
cana-3840	342	1	[	[	X
cana-3840	342	2	28	28	NUM
cana-3840	342	3	]	]	X
cana-3840	342	4	j.-r	j.-r	PROPN
cana-3840	342	5	.	.	PUNCT
cana-3840	343	1	wuand	wuand	PROPN
cana-3840	343	2	z.-y	z.-y	NOUN
cana-3840	343	3	.	.	PUNCT
cana-3840	344	1	jin	jin	NOUN
cana-3840	344	2	,	,	PUNCT
cana-3840	344	3	“	"	PUNCT
cana-3840	344	4	a	a	DET
cana-3840	344	5	note	note	NOUN
cana-3840	344	6	on	on	ADP
cana-3840	344	7	ulam	ulam	PROPN
cana-3840	344	8	stability	stability	NOUN
cana-3840	344	9	of	of	ADP
cana-3840	344	10	some	some	DET
cana-3840	344	11	fuzzy	fuzzy	ADJ
cana-3840	344	12	number	number	NOUN
cana-3840	344	13	-	-	PUNCT
cana-3840	344	14	valued	value	VERB
cana-3840	344	15	functional	functional	ADJ
cana-3840	344	16	equations	equation	NOUN
cana-3840	344	17	,	,	PUNCT
cana-3840	344	18	”	"	PUNCT
cana-3840	344	19	fuzzy	fuzzy	ADJ
cana-3840	344	20	sets	set	NOUN
cana-3840	344	21	and	and	CCONJ
cana-3840	344	22	systems	system	NOUN
cana-3840	344	23	,	,	PUNCT
cana-3840	344	24	vol	vol	NOUN
cana-3840	344	25	.	.	PROPN
cana-3840	344	26	375	375	NUM
cana-3840	344	27	,	,	PUNCT
cana-3840	344	28	pp	pp	ADJ
cana-3840	344	29	.	.	PUNCT
cana-3840	345	1	191–195	191–195	NUM
cana-3840	345	2	,	,	PUNCT
cana-3840	345	3	2019	2019	NUM
cana-3840	345	4	.	.	PUNCT
cana-3840	346	1	[	[	X
cana-3840	346	2	29	29	NUM
cana-3840	346	3	]	]	X
cana-3840	346	4	j.-r	j.-r	PROPN
cana-3840	346	5	.	.	PUNCT
cana-3840	347	1	wu	wu	PROPN
cana-3840	347	2	,	,	PUNCT
cana-3840	347	3	x.-w	x.-w	PROPN
cana-3840	347	4	.	.	PUNCT
cana-3840	348	1	kai	kai	PROPN
cana-3840	348	2	,	,	PUNCT
cana-3840	348	3	and	and	CCONJ
cana-3840	348	4	j.-j	j.-j	PROPN
cana-3840	348	5	.	.	PUNCT
cana-3840	349	1	li	li	PROPN
cana-3840	349	2	,	,	PUNCT
cana-3840	349	3	“	"	PUNCT
cana-3840	349	4	atoms	atom	NOUN
cana-3840	349	5	of	of	ADP
cana-3840	349	6	monotone	monotone	ADJ
cana-3840	349	7	set	set	NOUN
cana-3840	349	8	valued	value	VERB
cana-3840	349	9	measures	measure	NOUN
cana-3840	349	10	and	and	CCONJ
cana-3840	349	11	integrals	integral	NOUN
cana-3840	349	12	,	,	PUNCT
cana-3840	349	13	”	"	PUNCT
cana-3840	349	14	fuzzy	fuzzy	ADJ
cana-3840	349	15	sets	set	NOUN
cana-3840	349	16	and	and	CCONJ
cana-3840	349	17	systems	system	NOUN
cana-3840	349	18	,	,	PUNCT
cana-3840	349	19	vol	vol	NOUN
cana-3840	349	20	.	.	PROPN
cana-3840	350	1	304	304	NUM
cana-3840	350	2	,	,	PUNCT
cana-3840	350	3	no	no	INTJ
cana-3840	350	4	.	.	NOUN
cana-3840	350	5	1	1	NUM
cana-3840	350	6	,	,	PUNCT
cana-3840	350	7	pp	pp	ADJ
cana-3840	350	8	.	.	PUNCT
cana-3840	351	1	131–139	131–139	NUM
cana-3840	351	2	,	,	PUNCT
cana-3840	351	3	2016	2016	NUM
cana-3840	351	4	.	.	PUNCT
cana-3840	352	1	https://internationalpubls.com/index.php/anvi/issue/view/54	https://internationalpubls.com/index.php/anvi/issue/view/54	PROPN
cana-3840	352	2	https://internationalpubls.com/index.php/anvi/issue/view/54	https://internationalpubls.com/index.php/anvi/issue/view/54	PROPN
cana-3840	352	3	communications	communication	NOUN
cana-3840	352	4	on	on	ADP
cana-3840	352	5	applied	apply	VERB
cana-3840	352	6	nonlinear	nonlinear	ADJ
cana-3840	352	7	analysis	analysis	NOUN
cana-3840	352	8	issn	issn	NOUN
cana-3840	352	9	:	:	PUNCT
cana-3840	352	10	1074	1074	NUM
cana-3840	352	11	-	-	PUNCT
cana-3840	352	12	133x	133x	NUM
cana-3840	352	13	vol	vol	NOUN
cana-3840	352	14	32	32	NUM
cana-3840	352	15	no	no	NOUN
cana-3840	352	16	.	.	PUNCT
cana-3840	353	1	9s	9s	NUM
cana-3840	353	2	(	(	PUNCT
cana-3840	353	3	2025	2025	NUM
cana-3840	353	4	)	)	PUNCT
cana-3840	353	5	98	98	NUM
cana-3840	353	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3840	354	1	[	[	X
cana-3840	354	2	30	30	NUM
cana-3840	354	3	]	]	X
cana-3840	354	4	l.	l.	PROPN
cana-3840	354	5	a.	a.	PROPN
cana-3840	354	6	zadeh	zadeh	PROPN
cana-3840	354	7	,	,	PUNCT
cana-3840	354	8	“	"	PUNCT
cana-3840	354	9	fuzzy	fuzzy	ADJ
cana-3840	354	10	sets	set	NOUN
cana-3840	354	11	,	,	PUNCT
cana-3840	354	12	”	"	PUNCT
cana-3840	354	13	information	information	NOUN
cana-3840	354	14	and	and	CCONJ
cana-3840	354	15	control	control	NOUN
cana-3840	354	16	,	,	PUNCT
cana-3840	354	17	vol	vol	NOUN
cana-3840	354	18	.	.	PROPN
cana-3840	354	19	8	8	NUM
cana-3840	354	20	,	,	PUNCT
cana-3840	354	21	no	no	INTJ
cana-3840	354	22	.	.	NOUN
cana-3840	354	23	3	3	NUM
cana-3840	354	24	,	,	PUNCT
cana-3840	354	25	pp	pp	ADJ
cana-3840	354	26	.	.	PUNCT
cana-3840	355	1	338–353	338–353	NUM
cana-3840	355	2	,	,	PUNCT
cana-3840	355	3	1965	1965	NUM
cana-3840	355	4	.	.	PUNCT