id	sid	tid	token	lemma	pos
cana-4142	1	1	communications	communication	NOUN
cana-4142	1	2	on	on	ADP
cana-4142	1	3	applied	apply	VERB
cana-4142	1	4	nonlinear	nonlinear	ADJ
cana-4142	1	5	analysis	analysis	NOUN
cana-4142	1	6	issn	issn	NOUN
cana-4142	1	7	:	:	PUNCT
cana-4142	1	8	1074	1074	NUM
cana-4142	1	9	-	-	PUNCT
cana-4142	1	10	133x	133x	NUM
cana-4142	1	11	vol	vol	NOUN
cana-4142	1	12	x	x	NOUN
cana-4142	1	13	no	no	INTJ
cana-4142	1	14	.	.	PUNCT
cana-4142	2	1	y	y	PROPN
cana-4142	2	2	(	(	PUNCT
cana-4142	2	3	2025	2025	NUM
cana-4142	2	4	)	)	PUNCT
cana-4142	2	5	1319	1319	NUM
cana-4142	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	2	7	new	new	ADJ
cana-4142	2	8	contraction	contraction	NOUN
cana-4142	2	9	principle	principle	NOUN
cana-4142	2	10	in	in	ADP
cana-4142	2	11	revised	revise	VERB
cana-4142	2	12	fuzzy	fuzzy	ADJ
cana-4142	2	13	𝓴	𝓴	VERB
cana-4142	2	14	−metric	−metric	PROPN
cana-4142	2	15	spaces	space	VERB
cana-4142	2	16	1a	1a	PROPN
cana-4142	2	17	mohan	mohan	PROPN
cana-4142	2	18	,	,	PUNCT
cana-4142	2	19	2r	2r	NUM
cana-4142	2	20	thangathamizh	thangathamizh	PROPN
cana-4142	2	21	,	,	PUNCT
cana-4142	2	22	3a	3a	PROPN
cana-4142	2	23	muraliraj	muraliraj	VERB
cana-4142	2	24	1urumu	1urumu	NUM
cana-4142	2	25	dhanalakshmi	dhanalakshmi	PROPN
cana-4142	2	26	college	college	NOUN
cana-4142	2	27	,	,	PUNCT
cana-4142	2	28	bharathidasan	bharathidasan	ADJ
cana-4142	2	29	university	university	NOUN
cana-4142	2	30	,	,	PUNCT
cana-4142	2	31	trichy	trichy	PROPN
cana-4142	2	32	,	,	PUNCT
cana-4142	2	33	india	india	PROPN
cana-4142	2	34	.	.	PUNCT
cana-4142	3	1	email	email	NOUN
cana-4142	4	1	i	i	PROPN
cana-4142	4	2	d	d	PROPN
cana-4142	4	3	:	:	PUNCT
cana-4142	4	4	appavumohan@gmail.com	appavumohan@gmail.com	PROPN
cana-4142	5	1	2jeppiaar	2jeppiaar	NUM
cana-4142	5	2	institute	institute	NOUN
cana-4142	5	3	of	of	ADP
cana-4142	5	4	technology	technology	PROPN
cana-4142	5	5	,	,	PUNCT
cana-4142	5	6	sriperumbudur	sriperumbudur	NOUN
cana-4142	5	7	,	,	PUNCT
cana-4142	5	8	kanchipuram	kanchipuram	PROPN
cana-4142	5	9	,	,	PUNCT
cana-4142	5	10	india	india	PROPN
cana-4142	5	11	.	.	PUNCT
cana-4142	6	1	email	email	NOUN
cana-4142	7	1	i	i	PROPN
cana-4142	7	2	d	d	PROPN
cana-4142	7	3	:	:	PUNCT
cana-4142	7	4	thamizh1418@gmail.com	thamizh1418@gmail.com	X
cana-4142	8	1	3urumu	3urumu	NUM
cana-4142	8	2	dhanalakshmi	dhanalakshmi	NOUN
cana-4142	8	3	college	college	NOUN
cana-4142	8	4	,	,	PUNCT
cana-4142	8	5	bharathidasan	bharathidasan	ADJ
cana-4142	8	6	university	university	NOUN
cana-4142	8	7	,	,	PUNCT
cana-4142	8	8	trichy	trichy	PROPN
cana-4142	8	9	,	,	PUNCT
cana-4142	8	10	india	india	PROPN
cana-4142	8	11	.	.	PUNCT
cana-4142	9	1	email	email	NOUN
cana-4142	10	1	i	i	PROPN
cana-4142	10	2	d	d	PROPN
cana-4142	10	3	:	:	PUNCT
cana-4142	11	1	karguzali@gmail.com	karguzali@gmail.com	X
cana-4142	11	2	article	article	NOUN
cana-4142	11	3	history	history	NOUN
cana-4142	11	4	:	:	PUNCT
cana-4142	11	5	received	receive	VERB
cana-4142	11	6	:	:	PUNCT
cana-4142	11	7	12	12	NUM
cana-4142	11	8	-	-	SYM
cana-4142	11	9	01	01	NUM
cana-4142	11	10	-	-	PUNCT
cana-4142	11	11	2025	2025	NUM
cana-4142	11	12	revised	revise	VERB
cana-4142	11	13	:	:	PUNCT
cana-4142	11	14	15	15	NUM
cana-4142	11	15	-	-	NUM
cana-4142	11	16	02	02	NUM
cana-4142	11	17	-	-	PUNCT
cana-4142	11	18	2025	2025	NUM
cana-4142	11	19	accepted	accept	VERB
cana-4142	11	20	:	:	PUNCT
cana-4142	11	21	01	01	NUM
cana-4142	11	22	-	-	SYM
cana-4142	11	23	03	03	NUM
cana-4142	11	24	-	-	PUNCT
cana-4142	11	25	2025	2025	NUM
cana-4142	11	26	abstract	abstract	NOUN
cana-4142	11	27	:	:	PUNCT
cana-4142	11	28	introduction	introduction	NOUN
cana-4142	11	29	metric	metric	ADJ
cana-4142	11	30	spaces	space	NOUN
cana-4142	11	31	play	play	VERB
cana-4142	11	32	a	a	DET
cana-4142	11	33	crucial	crucial	ADJ
cana-4142	11	34	role	role	NOUN
cana-4142	11	35	in	in	ADP
cana-4142	11	36	mathematical	mathematical	ADJ
cana-4142	11	37	analysis	analysis	NOUN
cana-4142	11	38	and	and	CCONJ
cana-4142	11	39	topology	topology	NOUN
cana-4142	11	40	.	.	PUNCT
cana-4142	12	1	in	in	ADP
cana-4142	12	2	recent	recent	ADJ
cana-4142	12	3	years	year	NOUN
cana-4142	12	4	,	,	PUNCT
cana-4142	12	5	fuzzy	fuzzy	ADJ
cana-4142	12	6	metric	metric	ADJ
cana-4142	12	7	spaces	space	NOUN
cana-4142	12	8	have	have	AUX
cana-4142	12	9	been	be	AUX
cana-4142	12	10	widely	widely	ADV
cana-4142	12	11	studied	study	VERB
cana-4142	12	12	due	due	ADP
cana-4142	12	13	to	to	ADP
cana-4142	12	14	their	their	PRON
cana-4142	12	15	applications	application	NOUN
cana-4142	12	16	in	in	ADP
cana-4142	12	17	various	various	ADJ
cana-4142	12	18	fields	field	NOUN
cana-4142	12	19	.	.	PUNCT
cana-4142	13	1	alexander	alexander	PROPN
cana-4142	13	2	sostak	sostak	PROPN
cana-4142	13	3	introduced	introduce	VERB
cana-4142	13	4	the	the	DET
cana-4142	13	5	concept	concept	NOUN
cana-4142	13	6	of	of	ADP
cana-4142	13	7	revised	revise	VERB
cana-4142	13	8	fuzzy	fuzzy	ADJ
cana-4142	13	9	metric	metric	ADJ
cana-4142	13	10	spaces	space	NOUN
cana-4142	13	11	,	,	PUNCT
cana-4142	13	12	which	which	PRON
cana-4142	13	13	extends	extend	VERB
cana-4142	13	14	traditional	traditional	ADJ
cana-4142	13	15	fuzzy	fuzzy	ADJ
cana-4142	13	16	metric	metric	ADJ
cana-4142	13	17	spaces	space	NOUN
cana-4142	13	18	by	by	ADP
cana-4142	13	19	incorporating	incorporate	VERB
cana-4142	13	20	revised	revise	VERB
cana-4142	13	21	fuzzy	fuzzy	ADJ
cana-4142	13	22	sets	set	NOUN
cana-4142	13	23	.	.	PUNCT
cana-4142	14	1	in	in	ADP
cana-4142	14	2	this	this	DET
cana-4142	14	3	paper	paper	NOUN
cana-4142	14	4	,	,	PUNCT
cana-4142	14	5	we	we	PRON
cana-4142	14	6	introduce	introduce	VERB
cana-4142	14	7	a	a	DET
cana-4142	14	8	further	further	ADJ
cana-4142	14	9	generalization	generalization	NOUN
cana-4142	14	10	called	call	VERB
cana-4142	14	11	revised	revise	VERB
cana-4142	14	12	fuzzy	fuzzy	ADJ
cana-4142	14	13	𝓀	𝓀	PRON
cana-4142	14	14	−metric	−metric	ADJ
cana-4142	14	15	spaces	space	NOUN
cana-4142	14	16	,	,	PUNCT
cana-4142	14	17	which	which	PRON
cana-4142	14	18	allows	allow	VERB
cana-4142	14	19	for	for	ADP
cana-4142	14	20	the	the	DET
cana-4142	14	21	involvement	involvement	NOUN
cana-4142	14	22	of	of	ADP
cana-4142	14	23	multiple	multiple	ADJ
cana-4142	14	24	parameters	parameter	NOUN
cana-4142	14	25	(	(	PUNCT
cana-4142	14	26	𝓀	𝓀	X
cana-4142	14	27	)	)	PUNCT
cana-4142	14	28	,	,	PUNCT
cana-4142	14	29	thereby	thereby	ADV
cana-4142	14	30	enhancing	enhance	VERB
cana-4142	14	31	the	the	DET
cana-4142	14	32	flexibility	flexibility	NOUN
cana-4142	14	33	and	and	CCONJ
cana-4142	14	34	applicability	applicability	NOUN
cana-4142	14	35	of	of	ADP
cana-4142	14	36	the	the	DET
cana-4142	14	37	framework	framework	NOUN
cana-4142	14	38	.	.	PUNCT
cana-4142	15	1	objectives	objective	VERB
cana-4142	15	2	the	the	DET
cana-4142	15	3	primary	primary	ADJ
cana-4142	15	4	aim	aim	NOUN
cana-4142	15	5	of	of	ADP
cana-4142	15	6	this	this	DET
cana-4142	15	7	study	study	NOUN
cana-4142	15	8	is	be	AUX
cana-4142	15	9	to	to	PART
cana-4142	15	10	define	define	VERB
cana-4142	15	11	and	and	CCONJ
cana-4142	15	12	explore	explore	VERB
cana-4142	15	13	the	the	DET
cana-4142	15	14	fundamental	fundamental	ADJ
cana-4142	15	15	properties	property	NOUN
cana-4142	15	16	of	of	ADP
cana-4142	15	17	revised	revise	VERB
cana-4142	15	18	fuzzy	fuzzy	ADJ
cana-4142	15	19	𝓀	𝓀	PRON
cana-4142	15	20	−metric	−metric	ADJ
cana-4142	15	21	spaces	space	NOUN
cana-4142	15	22	.	.	PUNCT
cana-4142	16	1	we	we	PRON
cana-4142	16	2	investigate	investigate	VERB
cana-4142	16	3	their	their	PRON
cana-4142	16	4	topological	topological	ADJ
cana-4142	16	5	structure	structure	NOUN
cana-4142	16	6	and	and	CCONJ
cana-4142	16	7	establish	establish	VERB
cana-4142	16	8	significant	significant	ADJ
cana-4142	16	9	properties	property	NOUN
cana-4142	16	10	such	such	ADJ
cana-4142	16	11	as	as	ADP
cana-4142	16	12	first	first	ADJ
cana-4142	16	13	countability	countability	NOUN
cana-4142	16	14	and	and	CCONJ
cana-4142	16	15	the	the	DET
cana-4142	16	16	hausdorff	hausdorff	NOUN
cana-4142	16	17	condition	condition	NOUN
cana-4142	16	18	.	.	PUNCT
cana-4142	17	1	additionally	additionally	ADV
cana-4142	17	2	,	,	PUNCT
cana-4142	17	3	we	we	PRON
cana-4142	17	4	extend	extend	VERB
cana-4142	17	5	existing	exist	VERB
cana-4142	17	6	results	result	NOUN
cana-4142	17	7	in	in	ADP
cana-4142	17	8	the	the	DET
cana-4142	17	9	literature	literature	NOUN
cana-4142	17	10	by	by	ADP
cana-4142	17	11	proving	prove	VERB
cana-4142	17	12	a	a	DET
cana-4142	17	13	fixed	fix	VERB
cana-4142	17	14	-	-	PUNCT
cana-4142	17	15	point	point	NOUN
cana-4142	17	16	theorem	theorem	NOUN
cana-4142	17	17	in	in	ADP
cana-4142	17	18	this	this	DET
cana-4142	17	19	new	new	ADJ
cana-4142	17	20	setting	setting	NOUN
cana-4142	17	21	.	.	PUNCT
cana-4142	18	1	method	method	NOUN
cana-4142	18	2	we	we	PRON
cana-4142	18	3	begin	begin	VERB
cana-4142	18	4	by	by	ADP
cana-4142	18	5	formally	formally	ADV
cana-4142	18	6	defining	define	VERB
cana-4142	18	7	a	a	DET
cana-4142	18	8	revised	revise	VERB
cana-4142	18	9	fuzzy	fuzzy	ADJ
cana-4142	18	10	k	k	ADJ
cana-4142	18	11	-	-	ADJ
cana-4142	18	12	metric	metric	ADJ
cana-4142	18	13	space	space	NOUN
cana-4142	18	14	and	and	CCONJ
cana-4142	18	15	developing	develop	VERB
cana-4142	18	16	its	its	PRON
cana-4142	18	17	basic	basic	ADJ
cana-4142	18	18	properties	property	NOUN
cana-4142	18	19	.	.	PUNCT
cana-4142	19	1	using	use	VERB
cana-4142	19	2	topological	topological	ADJ
cana-4142	19	3	arguments	argument	NOUN
cana-4142	19	4	,	,	PUNCT
cana-4142	19	5	we	we	PRON
cana-4142	19	6	demonstrate	demonstrate	VERB
cana-4142	19	7	that	that	SCONJ
cana-4142	19	8	the	the	DET
cana-4142	19	9	topology	topology	NOUN
cana-4142	19	10	induced	induce	VERB
cana-4142	19	11	by	by	ADP
cana-4142	19	12	a	a	DET
cana-4142	19	13	revised	revise	VERB
cana-4142	19	14	fuzzy	fuzzy	ADJ
cana-4142	19	15	𝓀	𝓀	PRON
cana-4142	19	16	−metric	−metric	INTJ
cana-4142	19	17	is	be	AUX
cana-4142	19	18	first	first	ADV
cana-4142	19	19	countable	countable	ADJ
cana-4142	19	20	and	and	CCONJ
cana-4142	19	21	that	that	SCONJ
cana-4142	19	22	the	the	DET
cana-4142	19	23	space	space	NOUN
cana-4142	19	24	satisfies	satisfy	VERB
cana-4142	19	25	the	the	DET
cana-4142	19	26	hausdorff	hausdorff	NOUN
cana-4142	19	27	condition	condition	NOUN
cana-4142	19	28	.	.	PUNCT
cana-4142	20	1	finally	finally	ADV
cana-4142	20	2	,	,	PUNCT
cana-4142	20	3	we	we	PRON
cana-4142	20	4	extend	extend	VERB
cana-4142	20	5	the	the	DET
cana-4142	20	6	fixed	fix	VERB
cana-4142	20	7	-	-	PUNCT
cana-4142	20	8	point	point	NOUN
cana-4142	20	9	theorem	theorem	NOUN
cana-4142	20	10	established	establish	VERB
cana-4142	20	11	by	by	ADP
cana-4142	20	12	muraliraj	muraliraj	NOUN
cana-4142	20	13	and	and	CCONJ
cana-4142	20	14	thangathamizh	thangathamizh	ADJ
cana-4142	20	15	into	into	ADP
cana-4142	20	16	the	the	DET
cana-4142	20	17	context	context	NOUN
cana-4142	20	18	of	of	ADP
cana-4142	20	19	revised	revise	VERB
cana-4142	20	20	fuzzy	fuzzy	ADJ
cana-4142	20	21	𝓀	𝓀	PROPN
cana-4142	20	22	−metric	−metric	ADJ
cana-4142	20	23	spaces	space	NOUN
cana-4142	20	24	,	,	PUNCT
cana-4142	20	25	using	use	VERB
cana-4142	20	26	analytical	analytical	ADJ
cana-4142	20	27	and	and	CCONJ
cana-4142	20	28	set	set	ADJ
cana-4142	20	29	-	-	PUNCT
cana-4142	20	30	theoretic	theoretic	NOUN
cana-4142	20	31	techniques	technique	NOUN
cana-4142	20	32	.	.	PUNCT
cana-4142	21	1	result	result	VERB
cana-4142	21	2	our	our	PRON
cana-4142	21	3	findings	finding	NOUN
cana-4142	21	4	confirm	confirm	VERB
cana-4142	21	5	that	that	PRON
cana-4142	21	6	revised	revise	VERB
cana-4142	21	7	fuzzy	fuzzy	ADJ
cana-4142	21	8	k	k	ADJ
cana-4142	21	9	-	-	ADJ
cana-4142	21	10	metric	metric	ADJ
cana-4142	21	11	spaces	space	NOUN
cana-4142	21	12	preserve	preserve	VERB
cana-4142	21	13	essential	essential	ADJ
cana-4142	21	14	topological	topological	ADJ
cana-4142	21	15	characteristics	characteristic	NOUN
cana-4142	21	16	such	such	ADJ
cana-4142	21	17	as	as	ADP
cana-4142	21	18	first	first	ADJ
cana-4142	21	19	countability	countability	NOUN
cana-4142	21	20	and	and	CCONJ
cana-4142	21	21	hausdorff	hausdorff	NOUN
cana-4142	21	22	separation	separation	NOUN
cana-4142	21	23	.	.	PUNCT
cana-4142	22	1	furthermore	furthermore	ADV
cana-4142	22	2	,	,	PUNCT
cana-4142	22	3	the	the	DET
cana-4142	22	4	fixed	fix	VERB
cana-4142	22	5	-	-	PUNCT
cana-4142	22	6	point	point	NOUN
cana-4142	22	7	theorem	theorem	NOUN
cana-4142	22	8	proved	prove	VERB
cana-4142	22	9	in	in	ADP
cana-4142	22	10	this	this	DET
cana-4142	22	11	study	study	NOUN
cana-4142	22	12	generalizes	generalize	VERB
cana-4142	22	13	previous	previous	ADJ
cana-4142	22	14	results	result	NOUN
cana-4142	22	15	and	and	CCONJ
cana-4142	22	16	demonstrates	demonstrate	VERB
cana-4142	22	17	the	the	DET
cana-4142	22	18	broader	broad	ADJ
cana-4142	22	19	applicability	applicability	NOUN
cana-4142	22	20	of	of	ADP
cana-4142	22	21	revised	revise	VERB
cana-4142	22	22	fuzzy	fuzzy	ADJ
cana-4142	22	23	𝓀	𝓀	PRON
cana-4142	22	24	−metric	−metric	ADJ
cana-4142	22	25	spaces	space	NOUN
cana-4142	22	26	in	in	ADP
cana-4142	22	27	fixed	fix	VERB
cana-4142	22	28	-	-	PUNCT
cana-4142	22	29	point	point	NOUN
cana-4142	22	30	theory	theory	NOUN
cana-4142	22	31	.	.	PUNCT
cana-4142	23	1	conclusion	conclusion	NOUN
cana-4142	23	2	this	this	DET
cana-4142	23	3	study	study	NOUN
cana-4142	23	4	introduces	introduce	NOUN
cana-4142	23	5	revised	revise	VERB
cana-4142	23	6	fuzzy	fuzzy	ADJ
cana-4142	23	7	𝓀	𝓀	PRON
cana-4142	23	8	−metric	−metric	ADJ
cana-4142	23	9	spaces	space	NOUN
cana-4142	23	10	as	as	ADP
cana-4142	23	11	a	a	DET
cana-4142	23	12	generalization	generalization	NOUN
cana-4142	23	13	of	of	ADP
cana-4142	23	14	revised	revise	VERB
cana-4142	23	15	fuzzy	fuzzy	ADJ
cana-4142	23	16	metric	metric	ADJ
cana-4142	23	17	spaces	space	NOUN
cana-4142	23	18	,	,	PUNCT
cana-4142	23	19	providing	provide	VERB
cana-4142	23	20	a	a	DET
cana-4142	23	21	more	more	ADV
cana-4142	23	22	comprehensive	comprehensive	ADJ
cana-4142	23	23	framework	framework	NOUN
cana-4142	23	24	for	for	ADP
cana-4142	23	25	analyzing	analyze	VERB
cana-4142	23	26	metric	metric	ADJ
cana-4142	23	27	structures	structure	NOUN
cana-4142	23	28	with	with	ADP
cana-4142	23	29	multiple	multiple	ADJ
cana-4142	23	30	parameters	parameter	NOUN
cana-4142	23	31	.	.	PUNCT
cana-4142	24	1	the	the	DET
cana-4142	24	2	established	establish	VERB
cana-4142	24	3	topological	topological	ADJ
cana-4142	24	4	properties	property	NOUN
cana-4142	24	5	and	and	CCONJ
cana-4142	24	6	fixedcommunications	fixedcommunication	NOUN
cana-4142	24	7	on	on	ADP
cana-4142	24	8	applied	apply	VERB
cana-4142	24	9	nonlinear	nonlinear	ADJ
cana-4142	24	10	analysis	analysis	NOUN
cana-4142	24	11	issn	issn	NOUN
cana-4142	24	12	:	:	PUNCT
cana-4142	24	13	1074	1074	NUM
cana-4142	24	14	-	-	PUNCT
cana-4142	24	15	133x	133x	NUM
cana-4142	24	16	vol	vol	NOUN
cana-4142	24	17	x	x	NOUN
cana-4142	24	18	no	no	INTJ
cana-4142	24	19	.	.	PUNCT
cana-4142	25	1	y	y	PROPN
cana-4142	25	2	(	(	PUNCT
cana-4142	25	3	2025	2025	NUM
cana-4142	25	4	)	)	PUNCT
cana-4142	25	5	1320	1320	NUM
cana-4142	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	25	7	point	point	NOUN
cana-4142	25	8	theorem	theorem	VERB
cana-4142	25	9	contribute	contribute	NOUN
cana-4142	25	10	to	to	ADP
cana-4142	25	11	the	the	DET
cana-4142	25	12	further	further	ADJ
cana-4142	25	13	development	development	NOUN
cana-4142	25	14	of	of	ADP
cana-4142	25	15	fuzzy	fuzzy	ADJ
cana-4142	25	16	metric	metric	ADJ
cana-4142	25	17	theory	theory	NOUN
cana-4142	25	18	,	,	PUNCT
cana-4142	25	19	opening	open	VERB
cana-4142	25	20	new	new	ADJ
cana-4142	25	21	avenues	avenue	NOUN
cana-4142	25	22	for	for	ADP
cana-4142	25	23	future	future	ADJ
cana-4142	25	24	research	research	NOUN
cana-4142	25	25	in	in	ADP
cana-4142	25	26	mathematical	mathematical	ADJ
cana-4142	25	27	analysis	analysis	NOUN
cana-4142	25	28	and	and	CCONJ
cana-4142	25	29	its	its	PRON
cana-4142	25	30	applications	application	NOUN
cana-4142	25	31	.	.	PUNCT
cana-4142	26	1	keywords	keyword	NOUN
cana-4142	26	2	:	:	PUNCT
cana-4142	26	3	revised	revise	VERB
cana-4142	26	4	fuzzy	fuzzy	ADJ
cana-4142	26	5	𝓀	𝓀	PRON
cana-4142	26	6	−metric	−metric	ADJ
cana-4142	26	7	spaces	space	NOUN
cana-4142	26	8	,	,	PUNCT
cana-4142	26	9	revised	revise	VERB
cana-4142	26	10	fuzzy	fuzzy	ADJ
cana-4142	26	11	2	2	NUM
cana-4142	26	12	-	-	PUNCT
cana-4142	26	13	metric	metric	ADJ
cana-4142	26	14	spaces	space	NOUN
cana-4142	26	15	,	,	PUNCT
cana-4142	26	16	hausdorff	hausdorff	NOUN
cana-4142	26	17	spaces	space	NOUN
cana-4142	26	18	,	,	PUNCT
cana-4142	26	19	contractions	contraction	NOUN
cana-4142	26	20	,	,	PUNCT
cana-4142	26	21	fixed	fix	VERB
cana-4142	26	22	points	point	NOUN
cana-4142	26	23	1	1	NUM
cana-4142	26	24	.	.	X
cana-4142	27	1	introduction	introduction	NOUN
cana-4142	27	2	the	the	DET
cana-4142	27	3	idea	idea	NOUN
cana-4142	27	4	to	to	PART
cana-4142	27	5	revise	revise	VERB
cana-4142	27	6	the	the	DET
cana-4142	27	7	concept	concept	NOUN
cana-4142	27	8	of	of	ADP
cana-4142	27	9	a	a	DET
cana-4142	27	10	fuzzy	fuzzy	ADJ
cana-4142	27	11	metric	metric	NOUN
cana-4142	27	12	by	by	ADP
cana-4142	27	13	means	mean	NOUN
cana-4142	27	14	of	of	ADP
cana-4142	27	15	t	t	PROPN
cana-4142	27	16	-	-	PUNCT
cana-4142	27	17	conorms	conorm	NOUN
cana-4142	27	18	instead	instead	ADV
cana-4142	27	19	of	of	ADP
cana-4142	27	20	t	t	NOUN
cana-4142	27	21	-	-	PUNCT
cana-4142	27	22	norms	norm	NOUN
cana-4142	27	23	was	be	AUX
cana-4142	27	24	first	first	ADV
cana-4142	27	25	expressed	express	VERB
cana-4142	27	26	in	in	ADP
cana-4142	27	27	[	[	X
cana-4142	27	28	6	6	NUM
cana-4142	27	29	]	]	PUNCT
cana-4142	27	30	.	.	PUNCT
cana-4142	28	1	in	in	ADP
cana-4142	28	2	this	this	DET
cana-4142	28	3	paper	paper	NOUN
cana-4142	28	4	,	,	PUNCT
cana-4142	28	5	we	we	PRON
cana-4142	28	6	have	have	AUX
cana-4142	28	7	developed	develop	VERB
cana-4142	28	8	further	far	ADV
cana-4142	28	9	this	this	DET
cana-4142	28	10	approach	approach	NOUN
cana-4142	28	11	calling	call	VERB
cana-4142	28	12	fuzzy	fuzzy	ADJ
cana-4142	28	13	metrics	metric	NOUN
cana-4142	28	14	defined	define	VERB
cana-4142	28	15	on	on	ADP
cana-4142	28	16	the	the	DET
cana-4142	28	17	base	base	NOUN
cana-4142	28	18	of	of	ADP
cana-4142	28	19	a	a	DET
cana-4142	28	20	t	t	NOUN
cana-4142	28	21	-	-	PUNCT
cana-4142	28	22	conorm	conorm	NOUN
cana-4142	28	23	by	by	ADP
cana-4142	28	24	t	t	PROPN
cana-4142	28	25	-	-	PUNCT
cana-4142	28	26	conorm	conorm	NOUN
cana-4142	28	27	based	base	VERB
cana-4142	28	28	fuzzy	fuzzy	ADJ
cana-4142	28	29	metrics	metric	NOUN
cana-4142	28	30	or	or	CCONJ
cana-4142	28	31	by	by	ADP
cana-4142	28	32	cb	cb	PROPN
cana-4142	28	33	-	-	PUNCT
cana-4142	28	34	fuzzy	fuzzy	ADJ
cana-4142	28	35	metrics	metric	NOUN
cana-4142	28	36	for	for	ADP
cana-4142	28	37	short	short	ADJ
cana-4142	28	38	.	.	PUNCT
cana-4142	29	1	the	the	DET
cana-4142	29	2	three	three	NUM
cana-4142	29	3	main	main	ADJ
cana-4142	29	4	issues	issue	NOUN
cana-4142	29	5	considered	consider	VERB
cana-4142	29	6	in	in	ADP
cana-4142	29	7	the	the	DET
cana-4142	29	8	paper	paper	NOUN
cana-4142	29	9	are	be	AUX
cana-4142	29	10	the	the	DET
cana-4142	29	11	following	following	NOUN
cana-4142	29	12	.	.	PUNCT
cana-4142	30	1	construction	construction	NOUN
cana-4142	30	2	of	of	ADP
cana-4142	30	3	revised	revise	VERB
cana-4142	30	4	fuzzy	fuzzy	ADJ
cana-4142	30	5	𝓀	𝓀	PROPN
cana-4142	30	6	−metrics	−metric	NOUN
cana-4142	30	7	from	from	ADP
cana-4142	30	8	ordinary	ordinary	ADJ
cana-4142	30	9	metrics	metric	NOUN
cana-4142	30	10	(	(	PUNCT
cana-4142	30	11	section	section	NOUN
cana-4142	30	12	4	4	NUM
cana-4142	30	13	)	)	PUNCT
cana-4142	30	14	,	,	PUNCT
cana-4142	30	15	topological	topological	ADJ
cana-4142	30	16	structure	structure	NOUN
cana-4142	30	17	induced	induce	VERB
cana-4142	30	18	by	by	ADP
cana-4142	30	19	cb	cb	PROPN
cana-4142	30	20	-	-	PUNCT
cana-4142	30	21	fuzzy	fuzzy	ADJ
cana-4142	30	22	metrics	metric	NOUN
cana-4142	30	23	(	(	PUNCT
cana-4142	30	24	section	section	NOUN
cana-4142	30	25	5	5	NUM
cana-4142	30	26	)	)	PUNCT
cana-4142	30	27	,	,	PUNCT
cana-4142	30	28	and	and	CCONJ
cana-4142	30	29	interrelations	interrelation	NOUN
cana-4142	30	30	between	between	ADP
cana-4142	30	31	cb	cb	NOUN
cana-4142	30	32	-	-	PUNCT
cana-4142	30	33	fuzzy	fuzzy	ADJ
cana-4142	30	34	metrics	metric	NOUN
cana-4142	30	35	and	and	CCONJ
cana-4142	30	36	modular	modular	ADJ
cana-4142	30	37	metrics	metric	NOUN
cana-4142	30	38	(	(	PUNCT
cana-4142	30	39	section	section	NOUN
cana-4142	30	40	6	6	NUM
cana-4142	30	41	)	)	PUNCT
cana-4142	30	42	.	.	PUNCT
cana-4142	31	1	additionally	additionally	ADV
cana-4142	31	2	,	,	PUNCT
cana-4142	31	3	we	we	PRON
cana-4142	31	4	make	make	VERB
cana-4142	31	5	some	some	DET
cana-4142	31	6	comments	comment	NOUN
cana-4142	31	7	concerning	concern	VERB
cana-4142	31	8	the	the	DET
cana-4142	31	9	intuitionistic	intuitionistic	ADJ
cana-4142	31	10	counterpart	counterpart	NOUN
cana-4142	31	11	of	of	ADP
cana-4142	31	12	a	a	DET
cana-4142	31	13	cb	cb	NOUN
cana-4142	31	14	-	-	PUNCT
cana-4142	31	15	fuzzy	fuzzy	ADJ
cana-4142	31	16	metric	metric	NOUN
cana-4142	31	17	(	(	PUNCT
cana-4142	31	18	section	section	NOUN
cana-4142	31	19	7	7	NUM
cana-4142	31	20	)	)	PUNCT
cana-4142	31	21	.	.	PUNCT
cana-4142	32	1	concerning	concern	VERB
cana-4142	32	2	the	the	DET
cana-4142	32	3	construction	construction	NOUN
cana-4142	32	4	of	of	ADP
cana-4142	32	5	cb	cb	NOUN
cana-4142	32	6	-	-	PUNCT
cana-4142	32	7	fuzzy	fuzzy	ADJ
cana-4142	32	8	metrics	metric	NOUN
cana-4142	32	9	from	from	ADP
cana-4142	32	10	ordinary	ordinary	ADJ
cana-4142	32	11	metrics	metric	NOUN
cana-4142	32	12	we	we	PRON
cana-4142	32	13	mainly	mainly	ADV
cana-4142	32	14	restrict	restrict	VERB
cana-4142	32	15	the	the	DET
cana-4142	32	16	case	case	NOUN
cana-4142	32	17	of	of	ADP
cana-4142	32	18	fuzzy	fuzzy	ADJ
cana-4142	32	19	metrics	metric	NOUN
cana-4142	32	20	based	base	VERB
cana-4142	32	21	on	on	ADP
cana-4142	32	22	archimedean	archimedean	PROPN
cana-4142	32	23	t	t	PROPN
cana-4142	32	24	-	-	PUNCT
cana-4142	32	25	conorms	conorm	NOUN
cana-4142	32	26	.	.	PUNCT
cana-4142	33	1	just	just	ADV
cana-4142	33	2	in	in	ADP
cana-4142	33	3	this	this	DET
cana-4142	33	4	situation	situation	NOUN
cana-4142	33	5	we	we	PRON
cana-4142	33	6	can	can	AUX
cana-4142	33	7	effectively	effectively	ADV
cana-4142	33	8	use	use	VERB
cana-4142	33	9	the	the	DET
cana-4142	33	10	tools	tool	NOUN
cana-4142	33	11	provided	provide	VERB
cana-4142	33	12	by	by	ADP
cana-4142	33	13	additive	additive	ADJ
cana-4142	33	14	generators	generator	NOUN
cana-4142	33	15	of	of	ADP
cana-4142	33	16	t	t	PROPN
cana-4142	33	17	-	-	PUNCT
cana-4142	33	18	conorms	conorm	NOUN
cana-4142	33	19	.	.	PUNCT
cana-4142	34	1	by	by	ADP
cana-4142	34	2	using	use	VERB
cana-4142	34	3	additive	additive	ADJ
cana-4142	34	4	generators	generator	NOUN
cana-4142	34	5	for	for	ADP
cana-4142	34	6	such	such	ADJ
cana-4142	34	7	cb	cb	PROPN
cana-4142	34	8	-	-	PUNCT
cana-4142	34	9	fuzzy	fuzzy	ADJ
cana-4142	34	10	metrics	metric	NOUN
cana-4142	34	11	,	,	PUNCT
cana-4142	34	12	we	we	PRON
cana-4142	34	13	presented	present	VERB
cana-4142	34	14	a	a	DET
cana-4142	34	15	scheme	scheme	NOUN
cana-4142	34	16	for	for	ADP
cana-4142	34	17	construction	construction	NOUN
cana-4142	34	18	of	of	ADP
cana-4142	34	19	cb	cb	NOUN
cana-4142	34	20	-	-	PUNCT
cana-4142	34	21	fuzzy	fuzzy	ADJ
cana-4142	34	22	metrics	metric	NOUN
cana-4142	34	23	from	from	ADP
cana-4142	34	24	ordinary	ordinary	ADJ
cana-4142	34	25	metrics	metric	NOUN
cana-4142	34	26	and	and	CCONJ
cana-4142	34	27	illustrated	illustrate	VERB
cana-4142	34	28	it	it	PRON
cana-4142	34	29	with	with	ADP
cana-4142	34	30	examples	example	NOUN
cana-4142	34	31	for	for	ADP
cana-4142	34	32	some	some	DET
cana-4142	34	33	concrete	concrete	ADJ
cana-4142	34	34	t	t	NOUN
cana-4142	34	35	-	-	PUNCT
cana-4142	34	36	conorms	conorm	NOUN
cana-4142	34	37	.	.	PUNCT
cana-4142	35	1	we	we	PRON
cana-4142	35	2	guess	guess	VERB
cana-4142	35	3	that	that	SCONJ
cana-4142	35	4	the	the	DET
cana-4142	35	5	presented	present	VERB
cana-4142	35	6	construction	construction	NOUN
cana-4142	35	7	will	will	AUX
cana-4142	35	8	provide	provide	VERB
cana-4142	35	9	a	a	DET
cana-4142	35	10	scheme	scheme	NOUN
cana-4142	35	11	allowing	allow	VERB
cana-4142	35	12	to	to	PART
cana-4142	35	13	extend	extend	VERB
cana-4142	35	14	some	some	DET
cana-4142	35	15	results	result	NOUN
cana-4142	35	16	from	from	ADP
cana-4142	35	17	the	the	DET
cana-4142	35	18	theory	theory	NOUN
cana-4142	35	19	of	of	ADP
cana-4142	35	20	metric	metric	ADJ
cana-4142	35	21	spaces	space	NOUN
cana-4142	35	22	to	to	ADP
cana-4142	35	23	the	the	DET
cana-4142	35	24	corresponding	corresponding	ADJ
cana-4142	35	25	results	result	NOUN
cana-4142	35	26	for	for	ADP
cana-4142	35	27	cb	cb	NOUN
cana-4142	35	28	-	-	PUNCT
cana-4142	35	29	fuzzy	fuzzy	ADJ
cana-4142	35	30	metric	metric	ADJ
cana-4142	35	31	spaces	space	NOUN
cana-4142	35	32	.	.	PUNCT
cana-4142	36	1	specifically	specifically	ADV
cana-4142	36	2	,	,	PUNCT
cana-4142	36	3	this	this	PRON
cana-4142	36	4	can	can	AUX
cana-4142	36	5	concern	concern	VERB
cana-4142	36	6	the	the	DET
cana-4142	36	7	results	result	NOUN
cana-4142	36	8	in	in	ADP
cana-4142	36	9	the	the	DET
cana-4142	36	10	theory	theory	NOUN
cana-4142	36	11	of	of	ADP
cana-4142	36	12	fixed	fix	VERB
cana-4142	36	13	points	point	NOUN
cana-4142	36	14	.	.	PUNCT
cana-4142	37	1	the	the	DET
cana-4142	37	2	motivation	motivation	NOUN
cana-4142	37	3	in	in	ADP
cana-4142	37	4	this	this	DET
cana-4142	37	5	paper	paper	NOUN
cana-4142	37	6	for	for	ADP
cana-4142	37	7	inventing	invent	VERB
cana-4142	37	8	a	a	DET
cana-4142	37	9	new	new	ADJ
cana-4142	37	10	space	space	NOUN
cana-4142	37	11	,	,	PUNCT
cana-4142	37	12	which	which	PRON
cana-4142	37	13	is	be	AUX
cana-4142	37	14	more	more	ADV
cana-4142	37	15	general	general	ADJ
cana-4142	37	16	than	than	ADP
cana-4142	37	17	a	a	DET
cana-4142	37	18	revised	revise	VERB
cana-4142	37	19	fuzzy	fuzzy	ADJ
cana-4142	37	20	metric	metric	ADJ
cana-4142	37	21	space	space	NOUN
cana-4142	37	22	due	due	ADP
cana-4142	37	23	to	to	ADP
cana-4142	37	24	alexander	alexander	PROPN
cana-4142	37	25	sostack	sostack	PROPN
cana-4142	37	26	(	(	PUNCT
cana-4142	37	27	2018	2018	NUM
cana-4142	37	28	)	)	PUNCT
cana-4142	37	29	,	,	PUNCT
cana-4142	37	30	is	be	AUX
cana-4142	37	31	given	give	VERB
cana-4142	37	32	in	in	ADP
cana-4142	37	33	this	this	DET
cana-4142	37	34	paragraph	paragraph	NOUN
cana-4142	37	35	.	.	PUNCT
cana-4142	38	1	in	in	ADP
cana-4142	38	2	a	a	DET
cana-4142	38	3	revised	revise	VERB
cana-4142	38	4	fuzzy	fuzzy	ADJ
cana-4142	38	5	metric	metric	ADJ
cana-4142	38	6	space	space	NOUN
cana-4142	38	7	,	,	PUNCT
cana-4142	38	8	the	the	DET
cana-4142	38	9	fuzzy	fuzzy	ADJ
cana-4142	38	10	distance	distance	NOUN
cana-4142	38	11	of	of	ADP
cana-4142	38	12	two	two	NUM
cana-4142	38	13	points	point	NOUN
cana-4142	38	14	is	be	AUX
cana-4142	38	15	measured	measure	VERB
cana-4142	38	16	by	by	ADP
cana-4142	38	17	the	the	DET
cana-4142	38	18	degree	degree	NOUN
cana-4142	38	19	of	of	ADP
cana-4142	38	20	the	the	DET
cana-4142	38	21	nearness	nearness	NOUN
cana-4142	38	22	of	of	ADP
cana-4142	38	23	points	point	NOUN
cana-4142	38	24	with	with	ADP
cana-4142	38	25	respect	respect	NOUN
cana-4142	38	26	to	to	ADP
cana-4142	38	27	a	a	DET
cana-4142	38	28	parameter	parameter	NOUN
cana-4142	38	29	𝑡	𝑡	PROPN
cana-4142	38	30	∈	∈	PROPN
cana-4142	38	31	(	(	PUNCT
cana-4142	38	32	0	0	NUM
cana-4142	38	33	,	,	PUNCT
cana-4142	38	34	∞	∞	PROPN
cana-4142	38	35	)	)	PUNCT
cana-4142	38	36	.	.	PUNCT
cana-4142	39	1	for	for	ADP
cana-4142	39	2	instance	instance	NOUN
cana-4142	39	3	,	,	PUNCT
cana-4142	39	4	we	we	PRON
cana-4142	39	5	can	can	AUX
cana-4142	39	6	think	think	VERB
cana-4142	39	7	of	of	ADP
cana-4142	39	8	“	"	PUNCT
cana-4142	39	9	t	t	PROPN
cana-4142	39	10	”	"	PUNCT
cana-4142	39	11	as	as	ADP
cana-4142	39	12	the	the	DET
cana-4142	39	13	time	time	NOUN
cana-4142	39	14	required	require	VERB
cana-4142	39	15	to	to	PART
cana-4142	39	16	travel	travel	VERB
cana-4142	39	17	between	between	ADP
cana-4142	39	18	two	two	NUM
cana-4142	39	19	points	point	NOUN
cana-4142	39	20	𝑥	𝑥	NOUN
cana-4142	39	21	and	and	CCONJ
cana-4142	39	22	𝑦	𝑦	NOUN
cana-4142	39	23	in	in	ADP
cana-4142	39	24	a	a	DET
cana-4142	39	25	space	space	NOUN
cana-4142	39	26	.	.	PUNCT
cana-4142	40	1	there	there	PRON
cana-4142	40	2	is	be	VERB
cana-4142	40	3	an	an	DET
cana-4142	40	4	interesting	interesting	ADJ
cana-4142	40	5	situation	situation	NOUN
cana-4142	40	6	of	of	ADP
cana-4142	40	7	the	the	DET
cana-4142	40	8	degree	degree	NOUN
cana-4142	40	9	of	of	ADP
cana-4142	40	10	nearness	nearness	NOUN
cana-4142	40	11	when	when	SCONJ
cana-4142	40	12	we	we	PRON
cana-4142	40	13	measure	measure	VERB
cana-4142	40	14	this	this	DET
cana-4142	40	15	degree	degree	NOUN
cana-4142	40	16	with	with	ADP
cana-4142	40	17	respect	respect	NOUN
cana-4142	40	18	to	to	ADP
cana-4142	40	19	different	different	ADJ
cana-4142	40	20	(	(	PUNCT
cana-4142	40	21	more	more	ADJ
cana-4142	40	22	than	than	ADP
cana-4142	40	23	one	one	NUM
cana-4142	40	24	)	)	PUNCT
cana-4142	40	25	parameters	parameter	NOUN
cana-4142	40	26	.	.	PUNCT
cana-4142	41	1	for	for	ADP
cana-4142	41	2	instance	instance	NOUN
cana-4142	41	3	,	,	PUNCT
cana-4142	41	4	suppose	suppose	VERB
cana-4142	41	5	that	that	SCONJ
cana-4142	41	6	we	we	PRON
cana-4142	41	7	move	move	VERB
cana-4142	41	8	from	from	ADP
cana-4142	41	9	india	india	PROPN
cana-4142	41	10	,	,	PUNCT
cana-4142	41	11	represented	represent	VERB
cana-4142	41	12	by	by	ADP
cana-4142	41	13	𝑥	𝑥	PROPN
cana-4142	41	14	,	,	PUNCT
cana-4142	41	15	to	to	ADP
cana-4142	41	16	serbia	serbia	PROPN
cana-4142	41	17	,	,	PUNCT
cana-4142	41	18	represented	represent	VERB
cana-4142	41	19	by	by	ADP
cana-4142	41	20	𝑦	𝑦	PROPN
cana-4142	41	21	,	,	PUNCT
cana-4142	41	22	by	by	ADP
cana-4142	41	23	a	a	DET
cana-4142	41	24	plane	plane	NOUN
cana-4142	41	25	and	and	CCONJ
cana-4142	41	26	measure	measure	VERB
cana-4142	41	27	the	the	DET
cana-4142	41	28	degree	degree	NOUN
cana-4142	41	29	of	of	ADP
cana-4142	41	30	the	the	DET
cana-4142	41	31	nearness	nearness	NOUN
cana-4142	41	32	of	of	ADP
cana-4142	41	33	𝑥	𝑥	PROPN
cana-4142	41	34	and	and	CCONJ
cana-4142	41	35	𝑦	𝑦	NOUN
cana-4142	41	36	with	with	ADP
cana-4142	41	37	respect	respect	NOUN
cana-4142	41	38	to	to	ADP
cana-4142	41	39	time	time	NOUN
cana-4142	41	40	and	and	CCONJ
cana-4142	41	41	fuel	fuel	NOUN
cana-4142	41	42	consumption	consumption	NOUN
cana-4142	41	43	with	with	ADP
cana-4142	41	44	planes	plane	NOUN
cana-4142	41	45	of	of	ADP
cana-4142	41	46	different	different	ADJ
cana-4142	41	47	fuel	fuel	NOUN
cana-4142	41	48	efficiency	efficiency	NOUN
cana-4142	41	49	.	.	PUNCT
cana-4142	42	1	then	then	ADV
cana-4142	42	2	obviously	obviously	ADV
cana-4142	42	3	,	,	PUNCT
cana-4142	42	4	this	this	DET
cana-4142	42	5	degree	degree	NOUN
cana-4142	42	6	will	will	AUX
cana-4142	42	7	be	be	AUX
cana-4142	42	8	different	different	ADJ
cana-4142	42	9	for	for	ADP
cana-4142	42	10	distinct	distinct	ADJ
cana-4142	42	11	planes	plane	NOUN
cana-4142	42	12	even	even	ADV
cana-4142	42	13	for	for	ADP
cana-4142	42	14	the	the	DET
cana-4142	42	15	same	same	ADJ
cana-4142	42	16	time	time	NOUN
cana-4142	42	17	𝑡	𝑡	NOUN
cana-4142	42	18	,	,	PUNCT
cana-4142	42	19	as	as	ADV
cana-4142	42	20	well	well	ADV
cana-4142	42	21	as	as	ADP
cana-4142	42	22	for	for	ADP
cana-4142	42	23	the	the	DET
cana-4142	42	24	same	same	ADJ
cana-4142	42	25	plane	plane	NOUN
cana-4142	42	26	but	but	CCONJ
cana-4142	42	27	for	for	ADP
cana-4142	42	28	different	different	ADJ
cana-4142	42	29	time	time	NOUN
cana-4142	42	30	intervals	interval	NOUN
cana-4142	42	31	.	.	PUNCT
cana-4142	43	1	the	the	DET
cana-4142	43	2	mentioned	mention	VERB
cana-4142	43	3	situation	situation	NOUN
cana-4142	43	4	in	in	ADP
cana-4142	43	5	the	the	DET
cana-4142	43	6	previous	previous	ADJ
cana-4142	43	7	paragraph	paragraph	NOUN
cana-4142	43	8	brings	bring	VERB
cana-4142	43	9	the	the	DET
cana-4142	43	10	inspiration	inspiration	NOUN
cana-4142	43	11	for	for	ADP
cana-4142	43	12	introducing	introduce	VERB
cana-4142	43	13	the	the	DET
cana-4142	43	14	notion	notion	NOUN
cana-4142	43	15	of	of	ADP
cana-4142	43	16	revised	revise	VERB
cana-4142	43	17	fuzzy	fuzzy	ADJ
cana-4142	43	18	𝓀	𝓀	PROPN
cana-4142	43	19	−metric	−metric	ADJ
cana-4142	43	20	spaces	space	NOUN
cana-4142	43	21	,	,	PUNCT
cana-4142	43	22	where	where	SCONJ
cana-4142	43	23	𝓀	𝓀	PROPN
cana-4142	43	24	∈	∈	PROPN
cana-4142	43	25	{	{	PUNCT
cana-4142	43	26	1	1	NUM
cana-4142	43	27	,	,	PUNCT
cana-4142	43	28	2	2	NUM
cana-4142	43	29	,	,	PUNCT
cana-4142	43	30	3	3	NUM
cana-4142	43	31	,	,	PUNCT
cana-4142	43	32	.	.	PUNCT
cana-4142	43	33	.	.	PUNCT
cana-4142	43	34	.	.	PUNCT
cana-4142	44	1	}	}	PUNCT
cana-4142	44	2	,	,	PUNCT
cana-4142	44	3	which	which	PRON
cana-4142	44	4	is	be	AUX
cana-4142	44	5	an	an	DET
cana-4142	44	6	extension	extension	NOUN
cana-4142	44	7	and	and	CCONJ
cana-4142	44	8	generalization	generalization	NOUN
cana-4142	44	9	of	of	ADP
cana-4142	44	10	the	the	DET
cana-4142	44	11	concept	concept	NOUN
cana-4142	44	12	of	of	ADP
cana-4142	44	13	fuzzy	fuzzy	ADJ
cana-4142	44	14	metric	metric	ADJ
cana-4142	44	15	spaces	space	NOUN
cana-4142	44	16	due	due	ADP
cana-4142	44	17	to	to	ADP
cana-4142	44	18	alexander	alexander	PROPN
cana-4142	44	19	sostack	sostack	PROPN
cana-4142	44	20	(	(	PUNCT
cana-4142	44	21	2018	2018	NUM
cana-4142	44	22	)	)	PUNCT
cana-4142	44	23	.	.	PUNCT
cana-4142	45	1	in	in	ADP
cana-4142	45	2	a	a	DET
cana-4142	45	3	revised	revise	VERB
cana-4142	45	4	fuzzy	fuzzy	ADJ
cana-4142	45	5	𝓀	𝓀	PROPN
cana-4142	45	6	−metric	−metric	ADJ
cana-4142	45	7	spaces	space	NOUN
cana-4142	45	8	,	,	PUNCT
cana-4142	45	9	the	the	DET
cana-4142	45	10	fuzzy	fuzzy	ADJ
cana-4142	45	11	distance	distance	NOUN
cana-4142	45	12	of	of	ADP
cana-4142	45	13	two	two	NUM
cana-4142	45	14	points	point	NOUN
cana-4142	45	15	is	be	AUX
cana-4142	45	16	measured	measure	VERB
cana-4142	45	17	by	by	ADP
cana-4142	45	18	the	the	DET
cana-4142	45	19	degree	degree	NOUN
cana-4142	45	20	of	of	ADP
cana-4142	45	21	nearness	nearness	NOUN
cana-4142	45	22	with	with	ADP
cana-4142	45	23	respect	respect	NOUN
cana-4142	45	24	to	to	ADP
cana-4142	45	25	𝓀	𝓀	PROPN
cana-4142	45	26	−parameter(s	−parameter(s	NUM
cana-4142	45	27	)	)	PUNCT
cana-4142	45	28	.	.	PUNCT
cana-4142	46	1	furthermore	furthermore	ADV
cana-4142	46	2	,	,	PUNCT
cana-4142	46	3	fixed	fix	VERB
cana-4142	46	4	point	point	NOUN
cana-4142	46	5	results	result	NOUN
cana-4142	46	6	for	for	ADP
cana-4142	46	7	contractive	contractive	ADJ
cana-4142	46	8	mappings	mapping	NOUN
cana-4142	46	9	in	in	ADP
cana-4142	46	10	revised	revise	VERB
cana-4142	46	11	fuzzy	fuzzy	ADJ
cana-4142	46	12	𝓀	𝓀	PRON
cana-4142	46	13	−metric	−metric	ADJ
cana-4142	46	14	spaces	space	NOUN
cana-4142	46	15	are	be	AUX
cana-4142	46	16	proved	prove	VERB
cana-4142	46	17	.	.	PUNCT
cana-4142	47	1	these	these	DET
cana-4142	47	2	results	result	NOUN
cana-4142	47	3	generalize	generalize	VERB
cana-4142	47	4	the	the	DET
cana-4142	47	5	fixed	fix	VERB
cana-4142	47	6	-	-	PUNCT
cana-4142	47	7	point	point	NOUN
cana-4142	47	8	results	result	NOUN
cana-4142	47	9	of	of	ADP
cana-4142	47	10	muraliraj	muraliraj	NOUN
cana-4142	47	11	and	and	CCONJ
cana-4142	47	12	thangathamizh	thangathamizh	PROPN
cana-4142	47	13	(	(	PUNCT
cana-4142	47	14	2022	2022	NUM
cana-4142	47	15	)	)	PUNCT
cana-4142	47	16	into	into	ADP
cana-4142	47	17	revised	revise	VERB
cana-4142	47	18	fuzzy	fuzzy	ADJ
cana-4142	47	19	𝓀	𝓀	PROPN
cana-4142	47	20	−metric	−metric	ADJ
cana-4142	47	21	spaces	space	NOUN
cana-4142	47	22	.	.	PUNCT
cana-4142	48	1	communications	communication	NOUN
cana-4142	48	2	on	on	ADP
cana-4142	48	3	applied	apply	VERB
cana-4142	48	4	nonlinear	nonlinear	ADJ
cana-4142	48	5	analysis	analysis	NOUN
cana-4142	48	6	issn	issn	NOUN
cana-4142	48	7	:	:	PUNCT
cana-4142	48	8	1074	1074	NUM
cana-4142	48	9	-	-	PUNCT
cana-4142	48	10	133x	133x	NUM
cana-4142	48	11	vol	vol	NOUN
cana-4142	48	12	x	x	NOUN
cana-4142	48	13	no	no	INTJ
cana-4142	48	14	.	.	PUNCT
cana-4142	49	1	y	y	PROPN
cana-4142	49	2	(	(	PUNCT
cana-4142	49	3	2025	2025	NUM
cana-4142	49	4	)	)	PUNCT
cana-4142	49	5	1321	1321	NUM
cana-4142	49	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	49	7	2	2	NUM
cana-4142	49	8	.	.	PUNCT
cana-4142	49	9	preliminaries	preliminary	NOUN
cana-4142	49	10	definition	definition	NOUN
cana-4142	49	11	1[22	1[22	NUM
cana-4142	49	12	]	]	PUNCT
cana-4142	49	13	(	(	PUNCT
cana-4142	49	14	schweizer	schweizer	PROPN
cana-4142	49	15	and	and	CCONJ
cana-4142	49	16	sklar	sklar	PROPN
cana-4142	49	17	(	(	PUNCT
cana-4142	49	18	1960	1960	NUM
cana-4142	49	19	)	)	PUNCT
cana-4142	49	20	a	a	DET
cana-4142	49	21	binary	binary	ADJ
cana-4142	49	22	operation	operation	NOUN
cana-4142	49	23	⨁	⨁	PROPN
cana-4142	49	24	:	:	PUNCT
cana-4142	50	1	[	[	X
cana-4142	50	2	0	0	NUM
cana-4142	50	3	,	,	PUNCT
cana-4142	50	4	1]2	1]2	NUM
cana-4142	50	5	→	→	PUNCT
cana-4142	50	6	[	[	X
cana-4142	50	7	0	0	NUM
cana-4142	50	8	,	,	PUNCT
cana-4142	50	9	1	1	NUM
cana-4142	50	10	]	]	PUNCT
cana-4142	50	11	is	be	AUX
cana-4142	50	12	called	call	VERB
cana-4142	50	13	a	a	DET
cana-4142	50	14	triangular	triangular	NOUN
cana-4142	50	15	conorm	conorm	NOUN
cana-4142	50	16	(	(	PUNCT
cana-4142	50	17	briefly	briefly	ADV
cana-4142	50	18	,	,	PUNCT
cana-4142	50	19	t	t	NOUN
cana-4142	50	20	-	-	PUNCT
cana-4142	50	21	conorm	conorm	NOUN
cana-4142	50	22	)	)	PUNCT
cana-4142	50	23	if	if	SCONJ
cana-4142	50	24	the	the	DET
cana-4142	50	25	following	follow	VERB
cana-4142	50	26	conditions	condition	NOUN
cana-4142	50	27	are	be	AUX
cana-4142	50	28	satisfied	satisfied	ADJ
cana-4142	50	29	for	for	ADP
cana-4142	50	30	all	all	DET
cana-4142	50	31	𝔭	𝔭	NOUN
cana-4142	50	32	,	,	PUNCT
cana-4142	50	33	𝔮	𝔮	PROPN
cana-4142	50	34	,	,	PUNCT
cana-4142	50	35	𝔯	𝔯	PROPN
cana-4142	50	36	,	,	PUNCT
cana-4142	50	37	𝔰	𝔰	PROPN
cana-4142	50	38	∈	∈	PROPN
cana-4142	51	1	[	[	X
cana-4142	51	2	0	0	NUM
cana-4142	51	3	,	,	PUNCT
cana-4142	51	4	1	1	NUM
cana-4142	51	5	]	]	SYM
cana-4142	51	6	:	:	PUNCT
cana-4142	52	1	1	1	X
cana-4142	52	2	.	.	X
cana-4142	52	3	⨁	⨁	PROPN
cana-4142	52	4	(	(	PUNCT
cana-4142	52	5	𝔭	𝔭	PROPN
cana-4142	52	6	,	,	PUNCT
cana-4142	52	7	𝔮	𝔮	NOUN
cana-4142	52	8	)	)	PUNCT
cana-4142	52	9	=	=	SYM
cana-4142	52	10	⨁	⨁	PROPN
cana-4142	52	11	(	(	PUNCT
cana-4142	52	12	𝔮	𝔮	PROPN
cana-4142	52	13	,	,	PUNCT
cana-4142	52	14	𝔭	𝔭	NOUN
cana-4142	52	15	)	)	PUNCT
cana-4142	52	16	;	;	PUNCT
cana-4142	53	1	2	2	X
cana-4142	53	2	.	.	X
cana-4142	53	3	if	if	SCONJ
cana-4142	53	4	𝔭	𝔭	ADP
cana-4142	53	5	≤	≤	ADV
cana-4142	53	6	𝔯	𝔯	PROPN
cana-4142	53	7	and	and	CCONJ
cana-4142	53	8	𝔮	𝔮	NOUN
cana-4142	53	9	≤	≤	NOUN
cana-4142	53	10	𝔰	𝔰	NOUN
cana-4142	53	11	,	,	PUNCT
cana-4142	53	12	then	then	ADV
cana-4142	53	13	⨁	⨁	PROPN
cana-4142	53	14	(	(	PUNCT
cana-4142	53	15	𝔭	𝔭	PROPN
cana-4142	53	16	,	,	PUNCT
cana-4142	53	17	𝔮	𝔮	NOUN
cana-4142	53	18	)	)	PUNCT
cana-4142	53	19	≤	≤	NUM
cana-4142	53	20	⨁	⨁	PROPN
cana-4142	53	21	(	(	PUNCT
cana-4142	53	22	𝔯	𝔯	PROPN
cana-4142	53	23	,	,	PUNCT
cana-4142	53	24	𝔰	𝔰	NOUN
cana-4142	53	25	)	)	PUNCT
cana-4142	53	26	;	;	PUNCT
cana-4142	53	27	3	3	X
cana-4142	53	28	.	.	X
cana-4142	53	29	⨁	⨁	PROPN
cana-4142	53	30	(	(	PUNCT
cana-4142	53	31	⨁	⨁	PROPN
cana-4142	53	32	(	(	PUNCT
cana-4142	53	33	𝔭	𝔭	PROPN
cana-4142	53	34	,	,	PUNCT
cana-4142	53	35	𝔮	𝔮	NOUN
cana-4142	53	36	)	)	PUNCT
cana-4142	53	37	,	,	PUNCT
cana-4142	53	38	𝔯	𝔯	PROPN
cana-4142	53	39	)	)	PUNCT
cana-4142	53	40	=	=	SYM
cana-4142	53	41	⨁	⨁	X
cana-4142	53	42	(	(	PUNCT
cana-4142	53	43	𝔭	𝔭	PROPN
cana-4142	53	44	,	,	PUNCT
cana-4142	53	45	⨁	⨁	PROPN
cana-4142	53	46	(	(	PUNCT
cana-4142	53	47	𝔮	𝔮	PROPN
cana-4142	53	48	,	,	PUNCT
cana-4142	53	49	𝔯	𝔯	PROPN
cana-4142	53	50	)	)	PUNCT
cana-4142	53	51	)	)	PUNCT
cana-4142	53	52	;	;	PUNCT
cana-4142	54	1	4	4	X
cana-4142	54	2	.	.	X
cana-4142	54	3	⨁	⨁	PROPN
cana-4142	54	4	(	(	PUNCT
cana-4142	54	5	𝔭	𝔭	NOUN
cana-4142	54	6	,	,	PUNCT
cana-4142	54	7	0	0	NUM
cana-4142	54	8	)	)	PUNCT
cana-4142	54	9	=	=	SYM
cana-4142	54	10	𝔭.	𝔭.	NOUN
cana-4142	54	11	if	if	SCONJ
cana-4142	54	12	⨁	⨁	PROPN
cana-4142	54	13	is	be	AUX
cana-4142	54	14	continuous	continuous	ADJ
cana-4142	54	15	,	,	PUNCT
cana-4142	54	16	it	it	PRON
cana-4142	54	17	is	be	AUX
cana-4142	54	18	called	call	VERB
cana-4142	54	19	a	a	DET
cana-4142	54	20	continuous	continuous	ADJ
cana-4142	54	21	t	t	NOUN
cana-4142	54	22	-	-	PUNCT
cana-4142	54	23	conorm	conorm	NOUN
cana-4142	54	24	.	.	PUNCT
cana-4142	55	1	for	for	ADP
cana-4142	55	2	each	each	DET
cana-4142	55	3	t	t	NOUN
cana-4142	55	4	-	-	PUNCT
cana-4142	55	5	conorm	conorm	NOUN
cana-4142	55	6	⨁	⨁	PROPN
cana-4142	55	7	:	:	PUNCT
cana-4142	55	8	[	[	X
cana-4142	55	9	0	0	NUM
cana-4142	55	10	,	,	PUNCT
cana-4142	55	11	1]2	1]2	NUM
cana-4142	55	12	→	→	PUNCT
cana-4142	55	13	[	[	X
cana-4142	55	14	0	0	NUM
cana-4142	55	15	,	,	PUNCT
cana-4142	55	16	1	1	NUM
cana-4142	55	17	]	]	PUNCT
cana-4142	55	18	and	and	CCONJ
cana-4142	55	19	𝔭	𝔭	NOUN
cana-4142	55	20	,	,	PUNCT
cana-4142	55	21	𝔮	𝔮	X
cana-4142	55	22	∈	∈	PROPN
cana-4142	56	1	[	[	X
cana-4142	56	2	0	0	NUM
cana-4142	56	3	,	,	PUNCT
cana-4142	56	4	1	1	NUM
cana-4142	56	5	]	]	PUNCT
cana-4142	56	6	,	,	PUNCT
cana-4142	56	7	instead	instead	ADV
cana-4142	56	8	of	of	ADP
cana-4142	56	9	⨁	⨁	PROPN
cana-4142	56	10	(	(	PUNCT
cana-4142	56	11	𝔭	𝔭	NOUN
cana-4142	56	12	,	,	PUNCT
cana-4142	56	13	𝔮	𝔮	X
cana-4142	56	14	)	)	PUNCT
cana-4142	56	15	we	we	PRON
cana-4142	56	16	will	will	AUX
cana-4142	56	17	use	use	VERB
cana-4142	56	18	the	the	DET
cana-4142	56	19	infix	infix	NOUN
cana-4142	56	20	notation	notation	NOUN
cana-4142	56	21	𝔭	𝔭	ADP
cana-4142	56	22	⨁	⨁	PROPN
cana-4142	56	23	𝔮.	𝔮.	NOUN
cana-4142	56	24	three	three	NUM
cana-4142	56	25	typical	typical	ADJ
cana-4142	56	26	examples	example	NOUN
cana-4142	56	27	of	of	ADP
cana-4142	56	28	continuous	continuous	ADJ
cana-4142	56	29	t	t	NOUN
cana-4142	56	30	-	-	PUNCT
cana-4142	56	31	norms	norm	NOUN
cana-4142	56	32	are	be	AUX
cana-4142	56	33	a	a	DET
cana-4142	56	34	product	product	NOUN
cana-4142	56	35	t	t	NOUN
cana-4142	56	36	-	-	PUNCT
cana-4142	56	37	conorm	conorm	NOUN
cana-4142	56	38	⨁1	⨁1	ADV
cana-4142	56	39	,	,	PUNCT
cana-4142	56	40	a	a	DET
cana-4142	56	41	minimum	minimum	NOUN
cana-4142	56	42	t	t	NOUN
cana-4142	56	43	-	-	PUNCT
cana-4142	56	44	conorm	conorm	NOUN
cana-4142	56	45	⨁2	⨁2	NOUN
cana-4142	56	46	and	and	CCONJ
cana-4142	56	47	a	a	DET
cana-4142	56	48	lukasiewicz	lukasiewicz	ADJ
cana-4142	56	49	t	t	NOUN
cana-4142	56	50	-	-	PUNCT
cana-4142	56	51	conorm	conorm	NOUN
cana-4142	56	52	⨁3	⨁3	NOUN
cana-4142	56	53	,	,	PUNCT
cana-4142	56	54	which	which	PRON
cana-4142	56	55	are	be	AUX
cana-4142	56	56	defined	define	VERB
cana-4142	56	57	for	for	ADP
cana-4142	56	58	each	each	DET
cana-4142	56	59	𝔭	𝔭	NOUN
cana-4142	56	60	,	,	PUNCT
cana-4142	56	61	𝔮	𝔮	X
cana-4142	56	62	∈	∈	PROPN
cana-4142	57	1	[	[	X
cana-4142	57	2	0	0	NUM
cana-4142	57	3	,	,	PUNCT
cana-4142	57	4	1	1	NUM
cana-4142	57	5	]	]	PUNCT
cana-4142	57	6	by	by	ADP
cana-4142	57	7	𝔭	𝔭	X
cana-4142	57	8	⨁1	⨁1	ADJ
cana-4142	57	9	𝔮	𝔮	X
cana-4142	57	10	=	=	SYM
cana-4142	57	11	𝑚𝑎𝑥{𝔭	𝑚𝑎𝑥{𝔭	PROPN
cana-4142	57	12	,	,	PUNCT
cana-4142	57	13	𝔮	𝔮	X
cana-4142	57	14	}	}	PUNCT
cana-4142	57	15	,	,	PUNCT
cana-4142	57	16	𝔭	𝔭	X
cana-4142	57	17	⨁2	⨁2	NOUN
cana-4142	57	18	𝔮	𝔮	X
cana-4142	57	19	=	=	SYM
cana-4142	57	20	𝔭	𝔭	X
cana-4142	57	21	+	+	CCONJ
cana-4142	57	22	𝔮	𝔮	NOUN
cana-4142	57	23	−	−	NOUN
cana-4142	57	24	𝔭𝔮	𝔭𝔮	NOUN
cana-4142	57	25	,	,	PUNCT
cana-4142	57	26	𝔭	𝔭	ADP
cana-4142	57	27	⨁3	⨁3	NOUN
cana-4142	57	28	𝔮	𝔮	X
cana-4142	57	29	=	=	SYM
cana-4142	57	30	𝑚𝑖𝑛{𝑎	𝑚𝑖𝑛{𝑎	PROPN
cana-4142	57	31	+	+	CCONJ
cana-4142	57	32	𝑏	𝑏	NOUN
cana-4142	57	33	,	,	PUNCT
cana-4142	57	34	1	1	NUM
cana-4142	57	35	}	}	PUNCT
cana-4142	57	36	.	.	PUNCT
cana-4142	58	1	remark	remark	NOUN
cana-4142	58	2	2	2	NUM
cana-4142	58	3	for	for	ADP
cana-4142	58	4	each	each	DET
cana-4142	58	5	t	t	NOUN
cana-4142	58	6	-	-	PUNCT
cana-4142	58	7	norm	norm	NOUN
cana-4142	58	8	⨁	⨁	PROPN
cana-4142	58	9	:	:	PUNCT
cana-4142	59	1	[	[	X
cana-4142	59	2	0	0	NUM
cana-4142	59	3	,	,	PUNCT
cana-4142	59	4	1]2	1]2	NUM
cana-4142	59	5	→	→	PUNCT
cana-4142	59	6	[	[	X
cana-4142	59	7	0	0	NUM
cana-4142	59	8	,	,	PUNCT
cana-4142	59	9	1	1	NUM
cana-4142	59	10	]	]	PUNCT
cana-4142	59	11	,	,	PUNCT
cana-4142	59	12	the	the	DET
cana-4142	59	13	following	follow	VERB
cana-4142	59	14	assertions	assertion	NOUN
cana-4142	59	15	hold	hold	VERB
cana-4142	59	16	:	:	PUNCT
cana-4142	59	17	1	1	X
cana-4142	59	18	.	.	X
cana-4142	60	1	for	for	ADP
cana-4142	60	2	each	each	DET
cana-4142	60	3	𝔭	𝔭	NOUN
cana-4142	60	4	,	,	PUNCT
cana-4142	60	5	𝔮	𝔮	X
cana-4142	60	6	∈	∈	PROPN
cana-4142	61	1	[	[	X
cana-4142	61	2	0	0	NUM
cana-4142	61	3	,	,	PUNCT
cana-4142	61	4	1	1	NUM
cana-4142	61	5	]	]	PUNCT
cana-4142	61	6	with	with	ADP
cana-4142	61	7	𝔭	𝔭	PRON
cana-4142	61	8	>	>	X
cana-4142	61	9	𝑞	𝑞	NOUN
cana-4142	61	10	,	,	PUNCT
cana-4142	61	11	there	there	PRON
cana-4142	61	12	is	be	VERB
cana-4142	61	13	𝔯	𝔯	PROPN
cana-4142	61	14	∈	∈	PROPN
cana-4142	61	15	(	(	PUNCT
cana-4142	61	16	0	0	NUM
cana-4142	61	17	,	,	PUNCT
cana-4142	61	18	1	1	NUM
cana-4142	61	19	)	)	PUNCT
cana-4142	61	20	such	such	ADJ
cana-4142	61	21	that	that	SCONJ
cana-4142	61	22	𝔭	𝔭	PROPN
cana-4142	61	23	⨁	⨁	PROPN
cana-4142	61	24	𝔯	𝔯	PROPN
cana-4142	61	25	≥	≥	NUM
cana-4142	61	26	𝔮	𝔮	PROPN
cana-4142	61	27	;	;	PUNCT
cana-4142	61	28	2	2	NUM
cana-4142	61	29	.	.	X
cana-4142	62	1	for	for	ADP
cana-4142	62	2	each	each	DET
cana-4142	62	3	𝔰	𝔰	PRON
cana-4142	62	4	∈	∈	PROPN
cana-4142	62	5	(	(	PUNCT
cana-4142	62	6	0	0	NUM
cana-4142	62	7	,	,	PUNCT
cana-4142	62	8	1	1	NUM
cana-4142	62	9	)	)	PUNCT
cana-4142	62	10	,	,	PUNCT
cana-4142	62	11	there	there	PRON
cana-4142	62	12	is	be	VERB
cana-4142	62	13	𝔱	𝔱	DET
cana-4142	62	14	∈	∈	PROPN
cana-4142	62	15	(	(	PUNCT
cana-4142	62	16	0	0	NUM
cana-4142	62	17	,	,	PUNCT
cana-4142	62	18	1	1	NUM
cana-4142	62	19	)	)	PUNCT
cana-4142	62	20	such	such	ADJ
cana-4142	62	21	that	that	SCONJ
cana-4142	62	22	𝔱	𝔱	PROPN
cana-4142	62	23	⨁	⨁	PROPN
cana-4142	62	24	𝔱	𝔱	PRON
cana-4142	62	25	≥	≥	NOUN
cana-4142	62	26	𝔰.	𝔰.	NOUN
cana-4142	62	27	definition	definition	NOUN
cana-4142	62	28	3[7	3[7	NUM
cana-4142	62	29	]	]	PUNCT
cana-4142	62	30	an	an	PRON
cana-4142	62	31	ordered	order	VERB
cana-4142	62	32	triple	triple	ADJ
cana-4142	62	33	(	(	PUNCT
cana-4142	62	34	𝔐	𝔐	PROPN
cana-4142	62	35	,	,	PUNCT
cana-4142	62	36	𝔑	𝔑	PROPN
cana-4142	62	37	,	,	PUNCT
cana-4142	62	38	⨁	⨁	PROPN
cana-4142	62	39	)	)	PUNCT
cana-4142	62	40	is	be	AUX
cana-4142	62	41	called	call	VERB
cana-4142	62	42	a	a	DET
cana-4142	62	43	revised	revise	VERB
cana-4142	62	44	fuzzy	fuzzy	ADJ
cana-4142	62	45	metric	metric	ADJ
cana-4142	62	46	space	space	NOUN
cana-4142	62	47	if	if	SCONJ
cana-4142	62	48	𝔐	𝔐	PRON
cana-4142	62	49	is	be	AUX
cana-4142	62	50	an	an	DET
cana-4142	62	51	arbitrary	arbitrary	ADJ
cana-4142	62	52	set	set	NOUN
cana-4142	62	53	,	,	PUNCT
cana-4142	62	54	⨁	⨁	PROPN
cana-4142	62	55	a	a	DET
cana-4142	62	56	continuous	continuous	ADJ
cana-4142	62	57	t	t	NOUN
cana-4142	62	58	-	-	PUNCT
cana-4142	62	59	conorm	conorm	NOUN
cana-4142	62	60	,	,	PUNCT
cana-4142	62	61	𝔐	𝔐	PROPN
cana-4142	62	62	is	be	AUX
cana-4142	62	63	a	a	DET
cana-4142	62	64	revised	revise	VERB
cana-4142	62	65	fuzzy	fuzzy	ADJ
cana-4142	62	66	set	set	NOUN
cana-4142	62	67	on	on	ADP
cana-4142	62	68	𝔐2	𝔐2	ADJ
cana-4142	62	69	×	×	NOUN
cana-4142	62	70	(	(	PUNCT
cana-4142	62	71	0	0	NUM
cana-4142	62	72	,	,	PUNCT
cana-4142	62	73	+	+	NOUN
cana-4142	62	74	∞	∞	NOUN
cana-4142	62	75	)	)	PUNCT
cana-4142	62	76	,	,	PUNCT
cana-4142	62	77	and	and	CCONJ
cana-4142	62	78	thefollowing	thefollowing	NOUN
cana-4142	62	79	conditions	condition	NOUN
cana-4142	62	80	are	be	AUX
cana-4142	62	81	satisfied	satisfied	ADJ
cana-4142	62	82	for	for	ADP
cana-4142	62	83	all	all	DET
cana-4142	62	84	𝔭	𝔭	NOUN
cana-4142	62	85	,	,	PUNCT
cana-4142	62	86	𝔮	𝔮	PROPN
cana-4142	62	87	∈	∈	PROPN
cana-4142	62	88	𝔐	𝔐	PROPN
cana-4142	62	89	,	,	PUNCT
cana-4142	62	90	𝔞	𝔞	PROPN
cana-4142	62	91	,	,	PUNCT
cana-4142	62	92	𝔟	𝔟	X
cana-4142	62	93	>	>	X
cana-4142	62	94	0	0	PUNCT
cana-4142	62	95	(	(	PUNCT
cana-4142	62	96	rf-1	rf-1	NOUN
cana-4142	62	97	)	)	PUNCT
cana-4142	62	98	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	62	99	,	,	PUNCT
cana-4142	62	100	𝔮	𝔮	PROPN
cana-4142	62	101	,	,	PUNCT
cana-4142	62	102	𝒶	𝒶	NOUN
cana-4142	62	103	)	)	PUNCT
cana-4142	62	104	<	<	X
cana-4142	62	105	1	1	NUM
cana-4142	62	106	;	;	PUNCT
cana-4142	62	107	(	(	PUNCT
cana-4142	62	108	rf-2	rf-2	X
cana-4142	62	109	)	)	PUNCT
cana-4142	62	110	𝔑(𝔭	𝔑(𝔭	X
cana-4142	62	111	,	,	PUNCT
cana-4142	62	112	𝔮	𝔮	PROPN
cana-4142	62	113	,	,	PUNCT
cana-4142	62	114	𝒶	𝒶	NOUN
cana-4142	62	115	)	)	PUNCT
cana-4142	62	116	=	=	SYM
cana-4142	62	117	0	0	PUNCT
cana-4142	63	1	if	if	SCONJ
cana-4142	63	2	and	and	CCONJ
cana-4142	63	3	only	only	ADV
cana-4142	63	4	if	if	SCONJ
cana-4142	63	5	𝔭	𝔭	ADP
cana-4142	63	6	=	=	SYM
cana-4142	63	7	𝔮	𝔮	PROPN
cana-4142	63	8	;	;	PUNCT
cana-4142	63	9	(	(	PUNCT
cana-4142	63	10	rf-3	rf-3	NOUN
cana-4142	63	11	)	)	PUNCT
cana-4142	63	12	𝔑(𝔭	𝔑(𝔭	X
cana-4142	63	13	,	,	PUNCT
cana-4142	63	14	𝔮	𝔮	PROPN
cana-4142	63	15	,	,	PUNCT
cana-4142	63	16	𝒶	𝒶	NOUN
cana-4142	63	17	)	)	PUNCT
cana-4142	63	18	=	=	SYM
cana-4142	63	19	𝔑(𝔮	𝔑(𝔮	PROPN
cana-4142	63	20	,	,	PUNCT
cana-4142	63	21	𝔭	𝔭	NOUN
cana-4142	63	22	,	,	PUNCT
cana-4142	63	23	𝒶	𝒶	NOUN
cana-4142	63	24	)	)	PUNCT
cana-4142	63	25	;	;	PUNCT
cana-4142	63	26	(	(	PUNCT
cana-4142	63	27	rf-4	rf-4	X
cana-4142	63	28	)	)	PUNCT
cana-4142	63	29	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	63	30	,	,	PUNCT
cana-4142	63	31	𝓇	𝓇	PROPN
cana-4142	63	32	,	,	PUNCT
cana-4142	63	33	𝒶	𝒶	NOUN
cana-4142	63	34	)	)	PUNCT
cana-4142	63	35	≤	≤	NOUN
cana-4142	63	36	𝔑(𝔭	𝔑(𝔭	PUNCT
cana-4142	63	37	,	,	PUNCT
cana-4142	63	38	𝔮	𝔮	PROPN
cana-4142	63	39	,	,	PUNCT
cana-4142	63	40	𝒶)⨁	𝒶)⨁	PROPN
cana-4142	63	41	𝔑(𝔮	𝔑(𝔮	PROPN
cana-4142	63	42	,	,	PUNCT
cana-4142	63	43	𝓇	𝓇	PROPN
cana-4142	63	44	,	,	PUNCT
cana-4142	63	45	𝒶	𝒶	NOUN
cana-4142	63	46	)	)	PUNCT
cana-4142	63	47	;	;	PUNCT
cana-4142	63	48	(	(	PUNCT
cana-4142	63	49	rf-5	rf-5	ADV
cana-4142	63	50	)	)	PUNCT
cana-4142	63	51	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	63	52	,	,	PUNCT
cana-4142	63	53	𝔮	𝔮	NOUN
cana-4142	63	54	,	,	PUNCT
cana-4142	63	55	−	−	PROPN
cana-4142	63	56	):	):	PUNCT
cana-4142	63	57	(	(	PUNCT
cana-4142	63	58	0	0	NUM
cana-4142	63	59	,	,	PUNCT
cana-4142	63	60	+	+	NOUN
cana-4142	63	61	∞)𝓀	∞)𝓀	NUM
cana-4142	63	62	→	→	SYM
cana-4142	63	63	[	[	X
cana-4142	63	64	0,1]is	0,1]is	X
cana-4142	63	65	a	a	DET
cana-4142	63	66	right	right	ADJ
cana-4142	63	67	continuous	continuous	ADJ
cana-4142	63	68	mapping	mapping	NOUN
cana-4142	63	69	.	.	PUNCT
cana-4142	64	1	example	example	NOUN
cana-4142	64	2	4[7	4[7	NUM
cana-4142	64	3	]	]	PUNCT
cana-4142	64	4	(	(	PUNCT
cana-4142	64	5	induced	induce	VERB
cana-4142	64	6	revised	revise	VERB
cana-4142	64	7	fuzzy	fuzzy	ADJ
cana-4142	64	8	metric	metric	NOUN
cana-4142	64	9	)	)	PUNCT
cana-4142	64	10	let	let	VERB
cana-4142	64	11	(	(	PUNCT
cana-4142	64	12	𝑋	𝑋	NOUN
cana-4142	64	13	,	,	PUNCT
cana-4142	64	14	𝑑	𝑑	NOUN
cana-4142	64	15	)	)	PUNCT
cana-4142	64	16	be	be	VERB
cana-4142	64	17	a	a	DET
cana-4142	64	18	metric	metric	ADJ
cana-4142	64	19	space	space	NOUN
cana-4142	64	20	and	and	CCONJ
cana-4142	64	21	⨁	⨁	PROPN
cana-4142	64	22	be	be	VERB
cana-4142	64	23	a	a	DET
cana-4142	64	24	product	product	NOUN
cana-4142	64	25	t	t	NOUN
cana-4142	64	26	-	-	PUNCT
cana-4142	64	27	conorm	conorm	NOUN
cana-4142	64	28	.	.	PUNCT
cana-4142	65	1	define	define	VERB
cana-4142	65	2	a	a	DET
cana-4142	65	3	revised	revise	VERB
cana-4142	65	4	fuzzy	fuzzy	ADJ
cana-4142	65	5	set	set	VERB
cana-4142	65	6	𝔑	𝔑	NOUN
cana-4142	65	7	on	on	ADP
cana-4142	65	8	𝔐2	𝔐2	ADJ
cana-4142	65	9	×	×	NOUN
cana-4142	65	10	(	(	PUNCT
cana-4142	65	11	0	0	NUM
cana-4142	65	12	,	,	PUNCT
cana-4142	65	13	+	+	NOUN
cana-4142	65	14	∞	∞	NOUN
cana-4142	65	15	)	)	PUNCT
cana-4142	65	16	by	by	ADP
cana-4142	65	17	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	65	18	,	,	PUNCT
cana-4142	65	19	𝔮	𝔮	PROPN
cana-4142	65	20	,	,	PUNCT
cana-4142	65	21	𝔞	𝔞	NOUN
cana-4142	65	22	)	)	PUNCT
cana-4142	65	23	=	=	PUNCT
cana-4142	65	24	𝕕(𝔭	𝕕(𝔭	X
cana-4142	65	25	,	,	PUNCT
cana-4142	65	26	𝔮	𝔮	NOUN
cana-4142	65	27	)	)	PUNCT
cana-4142	65	28	1	1	NUM
cana-4142	66	1	+	+	CCONJ
cana-4142	66	2	𝕕(𝔭	𝕕(𝔭	ADJ
cana-4142	66	3	,	,	PUNCT
cana-4142	66	4	𝔮	𝔮	NOUN
cana-4142	66	5	)	)	PUNCT
cana-4142	66	6	for	for	ADP
cana-4142	66	7	all	all	DET
cana-4142	66	8	𝔭	𝔭	NOUN
cana-4142	66	9	,	,	PUNCT
cana-4142	66	10	𝔮	𝔮	X
cana-4142	66	11	∈	∈	X
cana-4142	66	12	𝔐	𝔐	PROPN
cana-4142	66	13	and	and	CCONJ
cana-4142	66	14	𝔞	𝔞	ADJ
cana-4142	66	15	>	>	X
cana-4142	66	16	0	0	PROPN
cana-4142	66	17	,	,	PUNCT
cana-4142	66	18	where	where	SCONJ
cana-4142	66	19	𝑘	𝑘	X
cana-4142	66	20	,	,	PUNCT
cana-4142	66	21	𝑚	𝑚	PROPN
cana-4142	66	22	,	,	PUNCT
cana-4142	66	23	𝑛	𝑛	PROPN
cana-4142	66	24	>	>	X
cana-4142	66	25	0	0	X
cana-4142	66	26	.	.	PUNCT
cana-4142	67	1	then	then	ADV
cana-4142	67	2	,	,	PUNCT
cana-4142	67	3	(	(	PUNCT
cana-4142	67	4	𝔐	𝔐	X
cana-4142	67	5	,	,	PUNCT
cana-4142	67	6	𝔑	𝔑	PROPN
cana-4142	67	7	,	,	PUNCT
cana-4142	67	8	⨁	⨁	PROPN
cana-4142	67	9	)	)	PUNCT
cana-4142	67	10	is	be	AUX
cana-4142	67	11	a	a	DET
cana-4142	67	12	revised	revise	VERB
cana-4142	67	13	fuzzy	fuzzy	ADJ
cana-4142	67	14	metric	metric	ADJ
cana-4142	67	15	space	space	NOUN
cana-4142	67	16	called	call	VERB
cana-4142	67	17	the	the	DET
cana-4142	67	18	induced	induce	VERB
cana-4142	67	19	revised	revise	VERB
cana-4142	67	20	fuzzy	fuzzy	ADJ
cana-4142	67	21	metric	metric	NOUN
cana-4142	67	22	.	.	PUNCT
cana-4142	68	1	in	in	ADP
cana-4142	68	2	the	the	DET
cana-4142	68	3	above	above	ADJ
cana-4142	68	4	example	example	NOUN
cana-4142	68	5	,	,	PUNCT
cana-4142	68	6	note	note	VERB
cana-4142	68	7	that	that	SCONJ
cana-4142	68	8	communications	communication	NOUN
cana-4142	68	9	on	on	ADP
cana-4142	68	10	applied	apply	VERB
cana-4142	68	11	nonlinear	nonlinear	ADJ
cana-4142	68	12	analysis	analysis	NOUN
cana-4142	68	13	issn	issn	NOUN
cana-4142	68	14	:	:	PUNCT
cana-4142	68	15	1074	1074	NUM
cana-4142	68	16	-	-	PUNCT
cana-4142	68	17	133x	133x	NUM
cana-4142	68	18	vol	vol	NOUN
cana-4142	68	19	x	x	NOUN
cana-4142	68	20	no	no	INTJ
cana-4142	68	21	.	.	PUNCT
cana-4142	69	1	y	y	PROPN
cana-4142	69	2	(	(	PUNCT
cana-4142	69	3	2025	2025	NUM
cana-4142	69	4	)	)	PUNCT
cana-4142	69	5	1322	1322	NUM
cana-4142	69	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	70	1	lim	lim	PROPN
cana-4142	70	2	𝑡→+∞	𝑡→+∞	PROPN
cana-4142	70	3	𝔑(𝔭	𝔑(𝔭	X
cana-4142	70	4	,	,	PUNCT
cana-4142	70	5	𝔮	𝔮	PROPN
cana-4142	70	6	,	,	PUNCT
cana-4142	70	7	𝔞	𝔞	NOUN
cana-4142	70	8	)	)	PUNCT
cana-4142	70	9	=	=	SYM
cana-4142	70	10	0	0	NUM
cana-4142	70	11	for	for	ADP
cana-4142	70	12	all	all	DET
cana-4142	70	13	𝔭	𝔭	NOUN
cana-4142	70	14	,	,	PUNCT
cana-4142	70	15	𝔮	𝔮	X
cana-4142	70	16	∈	∈	PROPN
cana-4142	70	17	𝔐.	𝔐.	NOUN
cana-4142	70	18	(	(	PUNCT
cana-4142	70	19	1	1	NUM
cana-4142	70	20	)	)	PUNCT
cana-4142	70	21	as	as	ADP
cana-4142	70	22	(	(	PUNCT
cana-4142	70	23	𝔐	𝔐	PROPN
cana-4142	70	24	,	,	PUNCT
cana-4142	70	25	𝔑	𝔑	PROPN
cana-4142	70	26	,	,	PUNCT
cana-4142	70	27	⨁	⨁	PROPN
cana-4142	70	28	)	)	PUNCT
cana-4142	70	29	represents	represent	VERB
cana-4142	70	30	the	the	DET
cana-4142	70	31	degree	degree	NOUN
cana-4142	70	32	of	of	ADP
cana-4142	70	33	the	the	DET
cana-4142	70	34	nearness	nearness	NOUN
cana-4142	70	35	of	of	ADP
cana-4142	70	36	points	point	NOUN
cana-4142	70	37	𝔭	𝔭	NOUN
cana-4142	70	38	and	and	CCONJ
cana-4142	70	39	𝔮	𝔮	NOUN
cana-4142	70	40	with	with	ADP
cana-4142	70	41	respect	respect	NOUN
cana-4142	70	42	to	to	ADP
cana-4142	70	43	the	the	DET
cana-4142	70	44	parameter	parameter	NOUN
cana-4142	70	45	𝔞	𝔞	PROPN
cana-4142	70	46	and	and	CCONJ
cana-4142	70	47	it	it	PRON
cana-4142	70	48	is	be	AUX
cana-4142	70	49	a	a	DET
cana-4142	70	50	nondecreasing	nondecrease	VERB
cana-4142	70	51	function	function	NOUN
cana-4142	70	52	of	of	ADP
cana-4142	70	53	𝔞	𝔞	PROPN
cana-4142	70	54	for	for	ADP
cana-4142	70	55	all	all	DET
cana-4142	70	56	𝔭	𝔭	NOUN
cana-4142	70	57	,	,	PUNCT
cana-4142	70	58	𝔮	𝔮	X
cana-4142	70	59	∈	∈	PROPN
cana-4142	70	60	𝔐	𝔐	NOUN
cana-4142	70	61	;	;	PUNCT
cana-4142	70	62	therefore	therefore	ADV
cana-4142	70	63	,	,	PUNCT
cana-4142	70	64	condition	condition	NOUN
cana-4142	70	65	(	(	PUNCT
cana-4142	70	66	1	1	X
cana-4142	70	67	)	)	PUNCT
cana-4142	70	68	is	be	AUX
cana-4142	70	69	the	the	DET
cana-4142	70	70	most	most	ADV
cana-4142	70	71	natural	natural	ADJ
cana-4142	70	72	condition	condition	NOUN
cana-4142	70	73	for	for	ADP
cana-4142	70	74	the	the	DET
cana-4142	70	75	degree	degree	NOUN
cana-4142	70	76	of	of	ADP
cana-4142	70	77	the	the	DET
cana-4142	70	78	nearness	nearness	NOUN
cana-4142	70	79	to	to	PART
cana-4142	70	80	be	be	AUX
cana-4142	70	81	perfect	perfect	ADJ
cana-4142	70	82	(	(	PUNCT
cana-4142	70	83	that	that	PRON
cana-4142	70	84	is	is	ADV
cana-4142	70	85	,	,	PUNCT
cana-4142	70	86	unity	unity	NOUN
cana-4142	70	87	)	)	PUNCT
cana-4142	70	88	.	.	PUNCT
cana-4142	71	1	notice	notice	VERB
cana-4142	71	2	that	that	SCONJ
cana-4142	71	3	this	this	PRON
cana-4142	71	4	is	be	AUX
cana-4142	71	5	a	a	DET
cana-4142	71	6	specific	specific	ADJ
cana-4142	71	7	condition	condition	NOUN
cana-4142	71	8	and	and	CCONJ
cana-4142	71	9	may	may	AUX
cana-4142	71	10	not	not	PART
cana-4142	71	11	hold	hold	VERB
cana-4142	71	12	in	in	ADP
cana-4142	71	13	some	some	DET
cana-4142	71	14	fuzzy	fuzzy	ADJ
cana-4142	71	15	metric	metric	ADJ
cana-4142	71	16	spaces	space	NOUN
cana-4142	71	17	,	,	PUNCT
cana-4142	71	18	for	for	ADP
cana-4142	71	19	instance	instance	NOUN
cana-4142	71	20	,	,	PUNCT
cana-4142	71	21	in	in	ADP
cana-4142	71	22	stationary	stationary	ADJ
cana-4142	71	23	revised	revise	VERB
cana-4142	71	24	fuzzy	fuzzy	ADJ
cana-4142	71	25	metric	metric	ADJ
cana-4142	71	26	spaces	space	NOUN
cana-4142	71	27	.	.	PUNCT
cana-4142	72	1	this	this	PRON
cana-4142	72	2	brings	bring	VERB
cana-4142	72	3	to	to	ADP
cana-4142	72	4	the	the	DET
cana-4142	72	5	following	follow	VERB
cana-4142	72	6	definition	definition	NOUN
cana-4142	72	7	:	:	PUNCT
cana-4142	72	8	definition	definition	NOUN
cana-4142	72	9	5	5	NUM
cana-4142	72	10	a	a	DET
cana-4142	72	11	revised	revise	VERB
cana-4142	72	12	fuzzy	fuzzy	ADJ
cana-4142	72	13	metric	metric	ADJ
cana-4142	72	14	space	space	NOUN
cana-4142	72	15	(	(	PUNCT
cana-4142	72	16	𝔐	𝔐	PROPN
cana-4142	72	17	,	,	PUNCT
cana-4142	72	18	𝔑	𝔑	PROPN
cana-4142	72	19	,	,	PUNCT
cana-4142	72	20	⨁	⨁	PROPN
cana-4142	72	21	)	)	PUNCT
cana-4142	72	22	is	be	AUX
cana-4142	72	23	called	call	VERB
cana-4142	72	24	a	a	DET
cana-4142	72	25	natural	natural	ADJ
cana-4142	72	26	fuzzy	fuzzy	ADJ
cana-4142	72	27	metric	metric	ADJ
cana-4142	72	28	space	space	NOUN
cana-4142	72	29	if	if	SCONJ
cana-4142	72	30	and	and	CCONJ
cana-4142	72	31	only	only	ADV
cana-4142	73	1	if	if	SCONJ
cana-4142	73	2	lim	lim	PROPN
cana-4142	73	3	𝑡→+∞	𝑡→+∞	PROPN
cana-4142	73	4	𝔑(𝔭	𝔑(𝔭	X
cana-4142	73	5	,	,	PUNCT
cana-4142	73	6	𝔮	𝔮	PROPN
cana-4142	73	7	,	,	PUNCT
cana-4142	73	8	𝔞	𝔞	NOUN
cana-4142	73	9	)	)	PUNCT
cana-4142	73	10	=	=	SYM
cana-4142	73	11	0	0	NUM
cana-4142	73	12	for	for	ADP
cana-4142	73	13	all	all	DET
cana-4142	73	14	𝔭	𝔭	NOUN
cana-4142	73	15	,	,	PUNCT
cana-4142	73	16	𝔮	𝔮	X
cana-4142	73	17	∈	∈	PROPN
cana-4142	73	18	𝔐.	𝔐.	PROPN
cana-4142	73	19	definition	definition	NOUN
cana-4142	73	20	6	6	NUM
cana-4142	73	21	a	a	DET
cana-4142	73	22	3	3	NUM
cana-4142	73	23	-	-	PUNCT
cana-4142	73	24	tuple	tuple	NOUN
cana-4142	73	25	(	(	PUNCT
cana-4142	73	26	𝔐	𝔐	PROPN
cana-4142	73	27	,	,	PUNCT
cana-4142	73	28	𝔑	𝔑	PROPN
cana-4142	73	29	,	,	PUNCT
cana-4142	73	30	⨁	⨁	PROPN
cana-4142	73	31	)	)	PUNCT
cana-4142	73	32	is	be	AUX
cana-4142	73	33	said	say	VERB
cana-4142	73	34	to	to	PART
cana-4142	73	35	be	be	AUX
cana-4142	73	36	a	a	DET
cana-4142	73	37	revised	revise	VERB
cana-4142	73	38	fuzzy	fuzzy	ADJ
cana-4142	73	39	2	2	NUM
cana-4142	73	40	-	-	PUNCT
cana-4142	73	41	metric	metric	ADJ
cana-4142	73	42	space	space	NOUN
cana-4142	73	43	if	if	SCONJ
cana-4142	73	44	𝔐	𝔐	PRON
cana-4142	73	45	is	be	AUX
cana-4142	73	46	an	an	DET
cana-4142	73	47	arbitrary	arbitrary	ADJ
cana-4142	73	48	nonempty	nonempty	NOUN
cana-4142	73	49	set	set	NOUN
cana-4142	73	50	,	,	PUNCT
cana-4142	73	51	⨁is	⨁is	PROPN
cana-4142	73	52	a	a	DET
cana-4142	73	53	continuous	continuous	ADJ
cana-4142	73	54	t	t	NOUN
cana-4142	73	55	-	-	PUNCT
cana-4142	73	56	conorm	conorm	NOUN
cana-4142	73	57	,	,	PUNCT
cana-4142	73	58	and	and	CCONJ
cana-4142	73	59	𝔐	𝔐	PRON
cana-4142	73	60	is	be	AUX
cana-4142	73	61	a	a	DET
cana-4142	73	62	revised	revise	VERB
cana-4142	73	63	fuzzy	fuzzy	ADJ
cana-4142	73	64	set	set	VERB
cana-4142	73	65	on𝔐3	on𝔐3	NOUN
cana-4142	73	66	×	×	NOUN
cana-4142	73	67	(	(	PUNCT
cana-4142	73	68	0	0	NUM
cana-4142	73	69	,	,	PUNCT
cana-4142	73	70	+	+	NOUN
cana-4142	73	71	∞	∞	NOUN
cana-4142	73	72	)	)	PUNCT
cana-4142	73	73	satisfying	satisfy	VERB
cana-4142	73	74	the	the	DET
cana-4142	73	75	following	following	ADJ
cana-4142	73	76	conditions	condition	NOUN
cana-4142	73	77	:	:	PUNCT
cana-4142	73	78	for	for	ADP
cana-4142	73	79	all	all	DET
cana-4142	73	80	(	(	PUNCT
cana-4142	73	81	𝔭	𝔭	NUM
cana-4142	73	82	,	,	PUNCT
cana-4142	73	83	𝔮	𝔮	NOUN
cana-4142	73	84	,	,	PUNCT
cana-4142	73	85	𝓇	𝓇	X
cana-4142	73	86	∈	∈	PROPN
cana-4142	73	87	𝔐	𝔐	PROPN
cana-4142	73	88	,	,	PUNCT
cana-4142	73	89	𝒶	𝒶	NOUN
cana-4142	73	90	,	,	PUNCT
cana-4142	73	91	𝒶1	𝒶1	NOUN
cana-4142	73	92	,	,	PUNCT
cana-4142	73	93	𝒶2	𝒶2	PROPN
cana-4142	73	94	,	,	PUNCT
cana-4142	73	95	𝒶3	𝒶3	PROPN
cana-4142	73	96	∈	∈	PROPN
cana-4142	73	97	(	(	PUNCT
cana-4142	73	98	0	0	NUM
cana-4142	73	99	,	,	PUNCT
cana-4142	73	100	+	+	NOUN
cana-4142	73	101	∞	∞	NOUN
cana-4142	73	102	)	)	PUNCT
cana-4142	73	103	)	)	PUNCT
cana-4142	73	104	(	(	PUNCT
cana-4142	73	105	rf2m.1	rf2m.1	PROPN
cana-4142	73	106	)	)	PUNCT
cana-4142	73	107	given	give	VERB
cana-4142	73	108	distinct	distinct	ADJ
cana-4142	73	109	elements	element	NOUN
cana-4142	73	110	𝔭	𝔭	ADP
cana-4142	73	111	,	,	PUNCT
cana-4142	74	1	𝔮	𝔮	X
cana-4142	74	2	∈	∈	X
cana-4142	74	3	𝔐	𝔐	NOUN
cana-4142	74	4	there	there	PRON
cana-4142	74	5	is	be	VERB
cana-4142	74	6	an	an	DET
cana-4142	74	7	element𝔯	element𝔯	NOUN
cana-4142	74	8	∈	∈	NOUN
cana-4142	74	9	𝔐	𝔐	PRON
cana-4142	75	1	such	such	ADJ
cana-4142	75	2	that	that	SCONJ
cana-4142	75	3	𝔑(𝔭	𝔑(𝔭	ADP
cana-4142	75	4	,	,	PUNCT
cana-4142	75	5	𝔮	𝔮	PROPN
cana-4142	75	6	,	,	PUNCT
cana-4142	75	7	𝓇	𝓇	NOUN
cana-4142	75	8	,	,	PUNCT
cana-4142	75	9	𝒶	𝒶	NOUN
cana-4142	75	10	)	)	PUNCT
cana-4142	75	11	<	<	X
cana-4142	75	12	1	1	NUM
cana-4142	75	13	for	for	ADP
cana-4142	75	14	each	each	PRON
cana-4142	75	15	𝔞	𝔞	PROPN
cana-4142	75	16	>	>	X
cana-4142	75	17	0	0	NUM
cana-4142	75	18	;	;	PUNCT
cana-4142	75	19	(	(	PUNCT
cana-4142	75	20	rf2m.2	rf2m.2	NOUN
cana-4142	75	21	)	)	PUNCT
cana-4142	75	22	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	75	23	,	,	PUNCT
cana-4142	75	24	𝔮	𝔮	PROPN
cana-4142	75	25	,	,	PUNCT
cana-4142	75	26	𝒶	𝒶	NOUN
cana-4142	75	27	)	)	PUNCT
cana-4142	75	28	=	=	SYM
cana-4142	75	29	0	0	PUNCT
cana-4142	75	30	if	if	SCONJ
cana-4142	75	31	at	at	ADV
cana-4142	75	32	least	least	ADV
cana-4142	75	33	two	two	NUM
cana-4142	75	34	of	of	ADP
cana-4142	75	35	𝔭	𝔭	NUM
cana-4142	75	36	,	,	PUNCT
cana-4142	75	37	𝔮	𝔮	NOUN
cana-4142	75	38	,	,	PUNCT
cana-4142	75	39	𝓇	𝓇	X
cana-4142	75	40	are	be	AUX
cana-4142	75	41	equal	equal	ADJ
cana-4142	75	42	.	.	PUNCT
cana-4142	76	1	(	(	PUNCT
cana-4142	76	2	rf2m.3	rf2m.3	NOUN
cana-4142	76	3	)	)	PUNCT
cana-4142	76	4	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	76	5	,	,	PUNCT
cana-4142	76	6	𝔮	𝔮	PROPN
cana-4142	76	7	,	,	PUNCT
cana-4142	76	8	𝔯	𝔯	PROPN
cana-4142	76	9	,	,	PUNCT
cana-4142	76	10	𝒶	𝒶	NOUN
cana-4142	76	11	)	)	PUNCT
cana-4142	76	12	=	=	SYM
cana-4142	76	13	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	76	14	,	,	PUNCT
cana-4142	76	15	𝔯	𝔯	PROPN
cana-4142	76	16	,	,	PUNCT
cana-4142	76	17	𝔮	𝔮	PROPN
cana-4142	76	18	,	,	PUNCT
cana-4142	76	19	𝒶	𝒶	NOUN
cana-4142	76	20	)	)	PUNCT
cana-4142	76	21	=	=	SYM
cana-4142	77	1	𝔑(𝔯	𝔑(𝔯	VERB
cana-4142	77	2	,	,	PUNCT
cana-4142	77	3	𝔭	𝔭	NOUN
cana-4142	77	4	,	,	PUNCT
cana-4142	77	5	𝔮	𝔮	NOUN
cana-4142	77	6	,	,	PUNCT
cana-4142	77	7	𝒶	𝒶	NOUN
cana-4142	77	8	)	)	PUNCT
cana-4142	77	9	for	for	ADP
cana-4142	77	10	all	all	DET
cana-4142	77	11	𝔭	𝔭	NOUN
cana-4142	77	12	,	,	PUNCT
cana-4142	77	13	𝔮	𝔮	NOUN
cana-4142	77	14	,	,	PUNCT
cana-4142	77	15	𝓇	𝓇	X
cana-4142	77	16	∈	∈	PROPN
cana-4142	77	17	𝔐	𝔐	NOUN
cana-4142	77	18	and	and	CCONJ
cana-4142	77	19	all	all	DET
cana-4142	77	20	𝔞	𝔞	PROPN
cana-4142	77	21	>	>	X
cana-4142	77	22	0	0	NUM
cana-4142	77	23	;	;	PUNCT
cana-4142	77	24	(	(	PUNCT
cana-4142	77	25	rf2m.4	rf2m.4	NOUN
cana-4142	77	26	)	)	PUNCT
cana-4142	77	27	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	77	28	,	,	PUNCT
cana-4142	77	29	𝔮	𝔮	PROPN
cana-4142	77	30	,	,	PUNCT
cana-4142	77	31	𝔯	𝔯	PROPN
cana-4142	77	32	,	,	PUNCT
cana-4142	77	33	𝒶1	𝒶1	NOUN
cana-4142	77	34	+	+	CCONJ
cana-4142	77	35	𝒶2	𝒶2	PROPN
cana-4142	77	36	+	+	CCONJ
cana-4142	77	37	𝒶3	𝒶3	NOUN
cana-4142	77	38	)	)	PUNCT
cana-4142	77	39	≤	≤	NOUN
cana-4142	77	40	𝔑(𝔭	𝔑(𝔭	PUNCT
cana-4142	77	41	,	,	PUNCT
cana-4142	77	42	𝔯	𝔯	PROPN
cana-4142	77	43	,	,	PUNCT
cana-4142	77	44	𝔰	𝔰	PROPN
cana-4142	77	45	,	,	PUNCT
cana-4142	77	46	𝒶1)⨁𝔑(𝔭	𝒶1)⨁𝔑(𝔭	NOUN
cana-4142	77	47	,	,	PUNCT
cana-4142	77	48	𝔰	𝔰	PROPN
cana-4142	77	49	,	,	PUNCT
cana-4142	77	50	𝔯	𝔯	PROPN
cana-4142	77	51	,	,	PUNCT
cana-4142	77	52	𝒶2)⨁𝔑(𝔰	𝒶2)⨁𝔑(𝔰	PROPN
cana-4142	77	53	,	,	PUNCT
cana-4142	77	54	𝔮	𝔮	PROPN
cana-4142	77	55	,	,	PUNCT
cana-4142	77	56	𝔯	𝔯	PROPN
cana-4142	77	57	,	,	PUNCT
cana-4142	77	58	𝒶3	𝒶3	PROPN
cana-4142	77	59	)	)	PUNCT
cana-4142	77	60	;	;	PUNCT
cana-4142	77	61	(	(	PUNCT
cana-4142	77	62	rf2m.5)𝔑(𝔭	rf2m.5)𝔑(𝔭	NOUN
cana-4142	77	63	,	,	PUNCT
cana-4142	77	64	𝔮	𝔮	PROPN
cana-4142	77	65	,	,	PUNCT
cana-4142	77	66	𝔯	𝔯	PROPN
cana-4142	77	67	,	,	PUNCT
cana-4142	77	68	−	−	NOUN
cana-4142	77	69	)	)	PUNCT
cana-4142	77	70	∶	∶	NOUN
cana-4142	77	71	(	(	PUNCT
cana-4142	77	72	0	0	NUM
cana-4142	77	73	,	,	PUNCT
cana-4142	77	74	∞	∞	PROPN
cana-4142	77	75	)	)	PUNCT
cana-4142	77	76	→	→	SYM
cana-4142	77	77	(	(	PUNCT
cana-4142	77	78	0,1	0,1	NUM
cana-4142	77	79	]	]	PUNCT
cana-4142	77	80	is	be	AUX
cana-4142	77	81	a	a	DET
cana-4142	77	82	continuous	continuous	ADJ
cana-4142	77	83	function	function	NOUN
cana-4142	77	84	.	.	PUNCT
cana-4142	78	1	the	the	DET
cana-4142	78	2	pair	pair	NOUN
cana-4142	78	3	(	(	PUNCT
cana-4142	78	4	𝔑	𝔑	PROPN
cana-4142	78	5	,	,	PUNCT
cana-4142	78	6	⨁	⨁	PROPN
cana-4142	78	7	)	)	PUNCT
cana-4142	78	8	(	(	PUNCT
cana-4142	78	9	or	or	CCONJ
cana-4142	78	10	only	only	ADV
cana-4142	78	11	𝔑	𝔑	PROPN
cana-4142	78	12	)	)	PUNCT
cana-4142	78	13	is	be	AUX
cana-4142	78	14	called	call	VERB
cana-4142	78	15	a	a	DET
cana-4142	78	16	revised	revise	VERB
cana-4142	78	17	fuzzy	fuzzy	ADJ
cana-4142	78	18	2	2	NUM
cana-4142	78	19	-	-	NOUN
cana-4142	78	20	metric	metric	ADJ
cana-4142	78	21	on	on	ADP
cana-4142	78	22	𝔐.	𝔐.	PROPN
cana-4142	78	23	3	3	NUM
cana-4142	78	24	.	.	PUNCT
cana-4142	78	25	revised	revise	VERB
cana-4142	78	26	fuzzy	fuzzy	ADJ
cana-4142	78	27	𝓴	𝓴	VERB
cana-4142	78	28	−metric	−metric	ADJ
cana-4142	78	29	spaces	space	NOUN
cana-4142	78	30	in	in	ADP
cana-4142	78	31	this	this	DET
cana-4142	78	32	section	section	NOUN
cana-4142	78	33	,	,	PUNCT
cana-4142	78	34	we	we	PRON
cana-4142	78	35	introduce	introduce	VERB
cana-4142	78	36	the	the	DET
cana-4142	78	37	idea	idea	NOUN
cana-4142	78	38	of	of	ADP
cana-4142	78	39	revised	revise	VERB
cana-4142	78	40	fuzzy	fuzzy	ADJ
cana-4142	78	41	𝓀	𝓀	PRON
cana-4142	78	42	−metric	−metric	ADJ
cana-4142	78	43	spaces	space	NOUN
cana-4142	78	44	and	and	CCONJ
cana-4142	78	45	investigate	investigate	VERB
cana-4142	78	46	the	the	DET
cana-4142	78	47	properties	property	NOUN
cana-4142	78	48	of	of	ADP
cana-4142	78	49	such	such	ADJ
cana-4142	78	50	spaces	space	NOUN
cana-4142	78	51	.	.	PUNCT
cana-4142	79	1	we	we	PRON
cana-4142	79	2	begin	begin	VERB
cana-4142	79	3	with	with	ADP
cana-4142	79	4	the	the	DET
cana-4142	79	5	following	follow	VERB
cana-4142	79	6	definition	definition	NOUN
cana-4142	79	7	definition	definition	NOUN
cana-4142	79	8	6	6	NUM
cana-4142	79	9	let	let	VERB
cana-4142	79	10	𝔐	𝔐	PRON
cana-4142	79	11	be	be	AUX
cana-4142	79	12	a	a	DET
cana-4142	79	13	nonempty	nonempty	ADJ
cana-4142	79	14	set	set	NOUN
cana-4142	79	15	,	,	PUNCT
cana-4142	79	16	⨁a	⨁a	PROPN
cana-4142	79	17	continuous	continuous	ADJ
cana-4142	79	18	t	t	PROPN
cana-4142	79	19	-	-	PUNCT
cana-4142	79	20	conorm	conorm	NOUN
cana-4142	79	21	,	,	PUNCT
cana-4142	79	22	𝓀a	𝓀a	ADP
cana-4142	79	23	positive	positive	ADJ
cana-4142	79	24	integer	integer	NOUN
cana-4142	79	25	and	and	CCONJ
cana-4142	79	26	𝔑be	𝔑be	PROPN
cana-4142	79	27	a	a	DET
cana-4142	79	28	revised	revise	VERB
cana-4142	79	29	fuzzy	fuzzy	ADJ
cana-4142	79	30	set	set	NOUN
cana-4142	79	31	on	on	ADP
cana-4142	79	32	𝔐2	𝔐2	ADJ
cana-4142	79	33	×	×	NOUN
cana-4142	79	34	(	(	PUNCT
cana-4142	79	35	0	0	NUM
cana-4142	79	36	,	,	PUNCT
cana-4142	80	1	+	+	ADJ
cana-4142	80	2	∞)𝓀.	∞)𝓀.	PROPN
cana-4142	80	3	an	an	DET
cana-4142	80	4	ordered	ordered	ADJ
cana-4142	80	5	triple	triple	ADJ
cana-4142	80	6	(	(	PUNCT
cana-4142	80	7	𝔐	𝔐	PROPN
cana-4142	80	8	,	,	PUNCT
cana-4142	80	9	𝔑	𝔑	PROPN
cana-4142	80	10	,	,	PUNCT
cana-4142	80	11	⨁)is	⨁)is	NOUN
cana-4142	80	12	called	call	VERB
cana-4142	80	13	a	a	DET
cana-4142	80	14	revised	revise	VERB
cana-4142	80	15	fuzzy𝓀	fuzzy𝓀	NOUN
cana-4142	80	16	−metric	−metric	ADJ
cana-4142	80	17	space	space	NOUN
cana-4142	80	18	if	if	SCONJ
cana-4142	80	19	the	the	DET
cana-4142	80	20	following	follow	VERB
cana-4142	80	21	conditions	condition	NOUN
cana-4142	80	22	are	be	AUX
cana-4142	80	23	satisfied	satisfied	ADJ
cana-4142	80	24	for	for	ADP
cana-4142	80	25	all	all	DET
cana-4142	80	26	𝔭	𝔭	NOUN
cana-4142	80	27	,	,	PUNCT
cana-4142	80	28	𝔮	𝔮	PROPN
cana-4142	80	29	,	,	PUNCT
cana-4142	80	30	𝔯	𝔯	PROPN
cana-4142	80	31	∈	∈	PROPN
cana-4142	80	32	𝔐	𝔐	PROPN
cana-4142	80	33	,	,	PUNCT
cana-4142	80	34	𝔞	𝔞	PROPN
cana-4142	80	35	,	,	PUNCT
cana-4142	80	36	𝔟	𝔟	X
cana-4142	80	37	>	>	X
cana-4142	80	38	0	0	NUM
cana-4142	80	39	and𝒶1	and𝒶1	PROPN
cana-4142	80	40	,	,	PUNCT
cana-4142	80	41	𝒶2	𝒶2	PROPN
cana-4142	80	42	,	,	PUNCT
cana-4142	80	43	.	.	PUNCT
cana-4142	80	44	.	.	PUNCT
cana-4142	81	1	.	.	PUNCT
cana-4142	82	1	,	,	PUNCT
cana-4142	82	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	82	3	>	>	X
cana-4142	82	4	0	0	NUM
cana-4142	82	5	:	:	PUNCT
cana-4142	82	6	(	(	PUNCT
cana-4142	82	7	rf	rf	NOUN
cana-4142	82	8	-	-	PUNCT
cana-4142	82	9	k1	k1	NOUN
cana-4142	82	10	)	)	PUNCT
cana-4142	82	11	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	82	12	,	,	PUNCT
cana-4142	82	13	𝔮	𝔮	PROPN
cana-4142	82	14	,	,	PUNCT
cana-4142	82	15	𝒶1	𝒶1	NOUN
cana-4142	82	16	,	,	PUNCT
cana-4142	82	17	𝒶2	𝒶2	PROPN
cana-4142	82	18	,	,	PUNCT
cana-4142	82	19	.	.	PUNCT
cana-4142	82	20	.	.	PUNCT
cana-4142	83	1	.	.	PUNCT
cana-4142	84	1	,	,	PUNCT
cana-4142	84	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	84	3	)	)	PUNCT
cana-4142	84	4	<	<	X
cana-4142	84	5	1	1	NUM
cana-4142	84	6	;	;	PUNCT
cana-4142	84	7	(	(	PUNCT
cana-4142	84	8	rf	rf	ADJ
cana-4142	84	9	-	-	PUNCT
cana-4142	84	10	k2	k2	NOUN
cana-4142	84	11	)	)	PUNCT
cana-4142	84	12	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	84	13	,	,	PUNCT
cana-4142	84	14	𝔮	𝔮	PROPN
cana-4142	84	15	,	,	PUNCT
cana-4142	84	16	𝒶1	𝒶1	NOUN
cana-4142	84	17	,	,	PUNCT
cana-4142	84	18	𝒶2	𝒶2	PROPN
cana-4142	84	19	,	,	PUNCT
cana-4142	84	20	.	.	PUNCT
cana-4142	84	21	.	.	PUNCT
cana-4142	85	1	.	.	PUNCT
cana-4142	86	1	,	,	PUNCT
cana-4142	86	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	86	3	)	)	PUNCT
cana-4142	86	4	=	=	SYM
cana-4142	86	5	0	0	PUNCT
cana-4142	87	1	if	if	SCONJ
cana-4142	87	2	and	and	CCONJ
cana-4142	87	3	only	only	ADV
cana-4142	87	4	if	if	SCONJ
cana-4142	87	5	𝔭	𝔭	ADP
cana-4142	87	6	=	=	SYM
cana-4142	87	7	𝔮	𝔮	PROPN
cana-4142	87	8	;	;	PUNCT
cana-4142	87	9	(	(	PUNCT
cana-4142	87	10	rf	rf	VERB
cana-4142	87	11	-	-	PUNCT
cana-4142	87	12	k3	k3	ADJ
cana-4142	87	13	)	)	PUNCT
cana-4142	87	14	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	87	15	,	,	PUNCT
cana-4142	87	16	𝔮	𝔮	PROPN
cana-4142	87	17	,	,	PUNCT
cana-4142	87	18	𝒶1	𝒶1	NOUN
cana-4142	87	19	,	,	PUNCT
cana-4142	87	20	𝒶2	𝒶2	PROPN
cana-4142	87	21	,	,	PUNCT
cana-4142	87	22	.	.	PUNCT
cana-4142	87	23	.	.	PUNCT
cana-4142	87	24	.	.	PUNCT
cana-4142	88	1	,	,	PUNCT
cana-4142	88	2	𝒶𝓀)is	𝒶𝓀)is	X
cana-4142	88	3	symmetric	symmetric	NOUN
cana-4142	88	4	.	.	PUNCT
cana-4142	89	1	(	(	PUNCT
cana-4142	89	2	rf	rf	NOUN
cana-4142	89	3	-	-	PUNCT
cana-4142	89	4	k4	k4	NOUN
cana-4142	89	5	)	)	PUNCT
cana-4142	89	6	for	for	ADP
cana-4142	89	7	any𝒿	any𝒿	PROPN
cana-4142	89	8	∈	∈	PROPN
cana-4142	89	9	{	{	PUNCT
cana-4142	89	10	1	1	NUM
cana-4142	89	11	,	,	PUNCT
cana-4142	89	12	2	2	NUM
cana-4142	89	13	,	,	PUNCT
cana-4142	89	14	3	3	NUM
cana-4142	89	15	,	,	PUNCT
cana-4142	89	16	.	.	PUNCT
cana-4142	89	17	.	.	PUNCT
cana-4142	90	1	.	.	PUNCT
cana-4142	91	1	,	,	PUNCT
cana-4142	91	2	𝑘	𝑘	X
cana-4142	91	3	}	}	PUNCT
cana-4142	91	4	,	,	PUNCT
cana-4142	91	5	we	we	PRON
cana-4142	91	6	have	have	VERB
cana-4142	91	7	𝔑(𝔭	𝔑(𝔭	X
cana-4142	91	8	,	,	PUNCT
cana-4142	91	9	𝓇	𝓇	NOUN
cana-4142	91	10	,	,	PUNCT
cana-4142	91	11	𝒶1	𝒶1	NOUN
cana-4142	91	12	,	,	PUNCT
cana-4142	91	13	𝒶2	𝒶2	PROPN
cana-4142	91	14	,	,	PUNCT
cana-4142	91	15	.	.	PUNCT
cana-4142	91	16	.	.	PUNCT
cana-4142	92	1	.	.	PUNCT
cana-4142	93	1	,	,	PUNCT
cana-4142	93	2	𝒶𝒿−1	𝒶𝒿−1	PROPN
cana-4142	93	3	,	,	PUNCT
cana-4142	93	4	𝒶	𝒶	PROPN
cana-4142	93	5	+	+	X
cana-4142	93	6	𝒷	𝒷	PROPN
cana-4142	93	7	,	,	PUNCT
cana-4142	93	8	𝒶𝒿+1	𝒶𝒿+1	X
cana-4142	93	9	,	,	PUNCT
cana-4142	93	10	…	…	PUNCT
cana-4142	93	11	,	,	PUNCT
cana-4142	93	12	𝒶𝓀	𝒶𝓀	NOUN
cana-4142	93	13	)	)	PUNCT
cana-4142	93	14	≤	≤	NOUN
cana-4142	93	15	{	{	PUNCT
cana-4142	93	16	𝔑(𝔮	𝔑(𝔮	NOUN
cana-4142	93	17	,	,	PUNCT
cana-4142	93	18	𝓇	𝓇	PROPN
cana-4142	93	19	,	,	PUNCT
cana-4142	93	20	𝒶1	𝒶1	NOUN
cana-4142	93	21	,	,	PUNCT
cana-4142	93	22	𝒶2	𝒶2	PROPN
cana-4142	93	23	,	,	PUNCT
cana-4142	93	24	.	.	PUNCT
cana-4142	93	25	.	.	PUNCT
cana-4142	94	1	.	.	PUNCT
cana-4142	95	1	,	,	PUNCT
cana-4142	95	2	𝒶𝒿−1	𝒶𝒿−1	PROPN
cana-4142	95	3	,	,	PUNCT
cana-4142	95	4	𝒶	𝒶	NOUN
cana-4142	95	5	,	,	PUNCT
cana-4142	95	6	𝒶𝒿+1	𝒶𝒿+1	X
cana-4142	95	7	,	,	PUNCT
cana-4142	95	8	…	…	PUNCT
cana-4142	95	9	,	,	PUNCT
cana-4142	95	10	𝒶𝒿−1	𝒶𝒿−1	PROPN
cana-4142	95	11	,	,	PUNCT
cana-4142	95	12	𝒶𝓀	𝒶𝓀	ADV
cana-4142	95	13	)	)	PUNCT
cana-4142	95	14	⨁	⨁	PROPN
cana-4142	95	15	𝔑(𝔮	𝔑(𝔮	PROPN
cana-4142	95	16	,	,	PUNCT
cana-4142	95	17	𝓇	𝓇	PROPN
cana-4142	95	18	,	,	PUNCT
cana-4142	95	19	𝒶1	𝒶1	NOUN
cana-4142	95	20	,	,	PUNCT
cana-4142	95	21	𝒶2	𝒶2	PROPN
cana-4142	95	22	,	,	PUNCT
cana-4142	95	23	.	.	PUNCT
cana-4142	95	24	.	.	PUNCT
cana-4142	96	1	.	.	PUNCT
cana-4142	97	1	,	,	PUNCT
cana-4142	97	2	𝒶𝒿−1	𝒶𝒿−1	PROPN
cana-4142	97	3	,	,	PUNCT
cana-4142	97	4	𝒷	𝒷	PROPN
cana-4142	97	5	,	,	PUNCT
cana-4142	97	6	𝒶𝒿+1	𝒶𝒿+1	X
cana-4142	97	7	,	,	PUNCT
cana-4142	97	8	…	…	PUNCT
cana-4142	97	9	,	,	PUNCT
cana-4142	97	10	𝒶𝒿−1	𝒶𝒿−1	PROPN
cana-4142	97	11	,	,	PUNCT
cana-4142	97	12	𝒶𝓀	𝒶𝓀	ADV
cana-4142	97	13	)	)	PUNCT
cana-4142	97	14	}	}	PUNCT
cana-4142	97	15	communications	communication	NOUN
cana-4142	97	16	on	on	ADP
cana-4142	97	17	applied	apply	VERB
cana-4142	97	18	nonlinear	nonlinear	ADJ
cana-4142	97	19	analysis	analysis	NOUN
cana-4142	97	20	issn	issn	NOUN
cana-4142	97	21	:	:	PUNCT
cana-4142	97	22	1074	1074	NUM
cana-4142	97	23	-	-	PUNCT
cana-4142	97	24	133x	133x	NUM
cana-4142	97	25	vol	vol	NOUN
cana-4142	97	26	x	x	NOUN
cana-4142	97	27	no	no	INTJ
cana-4142	97	28	.	.	PUNCT
cana-4142	98	1	y	y	PROPN
cana-4142	98	2	(	(	PUNCT
cana-4142	98	3	2025	2025	NUM
cana-4142	98	4	)	)	PUNCT
cana-4142	98	5	1323	1323	NUM
cana-4142	98	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	98	7	(	(	PUNCT
cana-4142	98	8	rf	rf	NOUN
cana-4142	98	9	-	-	PUNCT
cana-4142	98	10	k5	k5	NOUN
cana-4142	98	11	)	)	PUNCT
cana-4142	98	12	𝔑(𝔭	𝔑(𝔭	X
cana-4142	98	13	,	,	PUNCT
cana-4142	98	14	𝔮	𝔮	NOUN
cana-4142	98	15	,	,	PUNCT
cana-4142	98	16	−	−	PROPN
cana-4142	98	17	):	):	PUNCT
cana-4142	98	18	(	(	PUNCT
cana-4142	98	19	0	0	NUM
cana-4142	98	20	,	,	PUNCT
cana-4142	98	21	+	+	NOUN
cana-4142	98	22	∞)𝓀	∞)𝓀	NUM
cana-4142	98	23	→	→	SYM
cana-4142	98	24	[	[	X
cana-4142	98	25	0,1]is	0,1]is	X
cana-4142	98	26	a	a	DET
cana-4142	98	27	right	right	ADJ
cana-4142	98	28	continuous	continuous	ADJ
cana-4142	98	29	mapping	mapping	NOUN
cana-4142	98	30	.	.	PUNCT
cana-4142	99	1	remark	remark	NOUN
cana-4142	99	2	7for	7for	NUM
cana-4142	99	3	𝓀	𝓀	PROPN
cana-4142	99	4	=	=	SYM
cana-4142	99	5	1	1	NUM
cana-4142	99	6	,	,	PUNCT
cana-4142	99	7	the	the	DET
cana-4142	99	8	revised	revise	VERB
cana-4142	99	9	fuzzy𝓀	fuzzy𝓀	NOUN
cana-4142	99	10	−metric	−metric	ADJ
cana-4142	99	11	space	space	NOUN
cana-4142	99	12	reduces	reduce	VERB
cana-4142	99	13	into	into	ADP
cana-4142	99	14	the	the	DET
cana-4142	99	15	revised	revise	VERB
cana-4142	99	16	fuzzy	fuzzy	ADJ
cana-4142	99	17	metric	metric	ADJ
cana-4142	99	18	space	space	NOUN
cana-4142	99	19	in	in	ADP
cana-4142	99	20	the	the	DET
cana-4142	99	21	sense	sense	NOUN
cana-4142	99	22	of	of	ADP
cana-4142	99	23	alexander	alexander	PROPN
cana-4142	99	24	sostak	sostak	PROPN
cana-4142	99	25	.	.	PUNCT
cana-4142	100	1	example	example	NOUN
cana-4142	100	2	1	1	NUM
cana-4142	100	3	let	let	VERB
cana-4142	100	4	(	(	PUNCT
cana-4142	100	5	𝔐	𝔐	NOUN
cana-4142	100	6	,	,	PUNCT
cana-4142	100	7	𝕕)be	𝕕)be	PROPN
cana-4142	100	8	a	a	DET
cana-4142	100	9	metric	metric	ADJ
cana-4142	100	10	space	space	NOUN
cana-4142	100	11	,	,	PUNCT
cana-4142	100	12	⨁	⨁	PROPN
cana-4142	100	13	the	the	DET
cana-4142	100	14	product	product	NOUN
cana-4142	100	15	(	(	PUNCT
cana-4142	100	16	maximum	maximum	ADJ
cana-4142	100	17	)	)	PUNCT
cana-4142	100	18	t	t	NOUN
cana-4142	100	19	-	-	PUNCT
cana-4142	100	20	conorm	conorm	NOUN
cana-4142	100	21	,	,	PUNCT
cana-4142	100	22	𝔲	𝔲	PROPN
cana-4142	100	23	>	>	X
cana-4142	100	24	0	0	PUNCT
cana-4142	101	1	and	and	CCONJ
cana-4142	101	2	𝓀	𝓀	PROPN
cana-4142	101	3	be	be	VERB
cana-4142	101	4	a	a	DET
cana-4142	101	5	positive	positive	ADJ
cana-4142	101	6	integer	integer	NOUN
cana-4142	101	7	.	.	PUNCT
cana-4142	102	1	define	define	VERB
cana-4142	102	2	a	a	DET
cana-4142	102	3	revised	revise	VERB
cana-4142	102	4	fuzzy	fuzzy	ADJ
cana-4142	102	5	set	set	NOUN
cana-4142	102	6	𝔑on	𝔑on	PROPN
cana-4142	102	7	𝔐2	𝔐2	ADJ
cana-4142	102	8	×	×	NOUN
cana-4142	102	9	(	(	PUNCT
cana-4142	102	10	0	0	NUM
cana-4142	102	11	,	,	PUNCT
cana-4142	102	12	∞)𝓀by	∞)𝓀by	PROPN
cana-4142	102	13	𝔑(𝔭	𝔑(𝔭	X
cana-4142	102	14	,	,	PUNCT
cana-4142	102	15	𝔮	𝔮	PROPN
cana-4142	102	16	,	,	PUNCT
cana-4142	102	17	𝒶1	𝒶1	NOUN
cana-4142	102	18	,	,	PUNCT
cana-4142	102	19	𝒶2	𝒶2	PROPN
cana-4142	102	20	,	,	PUNCT
cana-4142	102	21	.	.	PUNCT
cana-4142	102	22	.	.	PUNCT
cana-4142	103	1	.	.	PUNCT
cana-4142	104	1	,	,	PUNCT
cana-4142	104	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	104	3	)	)	PUNCT
cana-4142	104	4	=	=	PUNCT
cana-4142	104	5	𝕕(𝔭	𝕕(𝔭	X
cana-4142	104	6	,	,	PUNCT
cana-4142	104	7	𝔮	𝔮	NOUN
cana-4142	104	8	)	)	PUNCT
cana-4142	104	9	𝔲(𝒶1	𝔲(𝒶1	NOUN
cana-4142	104	10	,	,	PUNCT
cana-4142	104	11	𝒶2	𝒶2	PROPN
cana-4142	104	12	,	,	PUNCT
cana-4142	104	13	.	.	PUNCT
cana-4142	104	14	.	.	PUNCT
cana-4142	105	1	.	.	PUNCT
cana-4142	106	1	,	,	PUNCT
cana-4142	106	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	106	3	)	)	PUNCT
cana-4142	106	4	+	+	CCONJ
cana-4142	106	5	𝕕(𝔭	𝕕(𝔭	ADJ
cana-4142	106	6	,	,	PUNCT
cana-4142	106	7	𝔮	𝔮	NOUN
cana-4142	106	8	)	)	PUNCT
cana-4142	106	9	for	for	ADP
cana-4142	106	10	all	all	DET
cana-4142	106	11	all	all	PRON
cana-4142	106	12	𝔭	𝔭	NOUN
cana-4142	106	13	,	,	PUNCT
cana-4142	106	14	𝔮	𝔮	PROPN
cana-4142	106	15	∈	∈	PROPN
cana-4142	106	16	𝔐	𝔐	PROPN
cana-4142	106	17	,	,	PUNCT
cana-4142	106	18	𝔞	𝔞	PROPN
cana-4142	106	19	,	,	PUNCT
cana-4142	106	20	𝔟	𝔟	X
cana-4142	106	21	>	>	X
cana-4142	106	22	0	0	PUNCT
cana-4142	106	23	and	and	CCONJ
cana-4142	106	24	𝒶1	𝒶1	PROPN
cana-4142	106	25	,	,	PUNCT
cana-4142	106	26	𝒶2	𝒶2	PROPN
cana-4142	106	27	,	,	PUNCT
cana-4142	106	28	.	.	PUNCT
cana-4142	106	29	.	.	PUNCT
cana-4142	106	30	.	.	PUNCT
cana-4142	107	1	,	,	PUNCT
cana-4142	107	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	107	3	>	>	X
cana-4142	107	4	0	0	X
cana-4142	107	5	.	.	PUNCT
cana-4142	108	1	then	then	ADV
cana-4142	108	2	,	,	PUNCT
cana-4142	108	3	(	(	PUNCT
cana-4142	108	4	𝔐	𝔐	X
cana-4142	108	5	,	,	PUNCT
cana-4142	108	6	𝔑	𝔑	PROPN
cana-4142	108	7	,	,	PUNCT
cana-4142	108	8	⨁	⨁	PROPN
cana-4142	108	9	)	)	PUNCT
cana-4142	108	10	is	be	AUX
cana-4142	108	11	a	a	DET
cana-4142	108	12	revised	revise	VERB
cana-4142	108	13	fuzzy	fuzzy	ADJ
cana-4142	108	14	𝓀	𝓀	X
cana-4142	108	15	−metric	−metric	ADJ
cana-4142	108	16	space	space	NOUN
cana-4142	108	17	.	.	PUNCT
cana-4142	109	1	from	from	ADP
cana-4142	109	2	the	the	DET
cana-4142	109	3	application	application	NOUN
cana-4142	109	4	point	point	NOUN
cana-4142	109	5	of	of	ADP
cana-4142	109	6	view	view	NOUN
cana-4142	109	7	,	,	PUNCT
cana-4142	109	8	one	one	PRON
cana-4142	109	9	should	should	AUX
cana-4142	109	10	define	define	VERB
cana-4142	109	11	the	the	DET
cana-4142	109	12	revised	revise	VERB
cana-4142	109	13	fuzzy	fuzzy	ADJ
cana-4142	109	14	𝓀	𝓀	PRON
cana-4142	109	15	−metric	−metric	ADJ
cana-4142	109	16	with	with	ADP
cana-4142	109	17	care	care	NOUN
cana-4142	109	18	to	to	ADP
cana-4142	109	19	the	the	DET
cana-4142	109	20	physical	physical	ADJ
cana-4142	109	21	nature	nature	NOUN
cana-4142	109	22	of	of	ADP
cana-4142	109	23	quantities	quantity	NOUN
cana-4142	109	24	.	.	PUNCT
cana-4142	110	1	for	for	ADP
cana-4142	110	2	instance	instance	NOUN
cana-4142	110	3	,	,	PUNCT
cana-4142	110	4	if	if	SCONJ
cana-4142	110	5	one	one	PRON
cana-4142	110	6	considers	consider	VERB
cana-4142	110	7	the	the	DET
cana-4142	110	8	degree	degree	NOUN
cana-4142	110	9	of	of	ADP
cana-4142	110	10	the	the	DET
cana-4142	110	11	nearness	nearness	NOUN
cana-4142	110	12	of	of	ADP
cana-4142	110	13	two	two	NUM
cana-4142	110	14	points	point	NOUN
cana-4142	110	15	𝔭	𝔭	ADP
cana-4142	110	16	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-4142	110	17	𝔮	𝔮	PROPN
cana-4142	110	18	in	in	ADP
cana-4142	110	19	a	a	DET
cana-4142	110	20	space	space	NOUN
cana-4142	110	21	with	with	ADP
cana-4142	110	22	respect	respect	NOUN
cana-4142	110	23	to	to	ADP
cana-4142	110	24	time	time	NOUN
cana-4142	110	25	and	and	CCONJ
cana-4142	110	26	fuel	fuel	NOUN
cana-4142	110	27	consumed	consume	VERB
cana-4142	110	28	in	in	ADP
cana-4142	110	29	moving	move	VERB
cana-4142	110	30	from	from	ADP
cana-4142	110	31	𝔭	𝔭	ADP
cana-4142	110	32	𝑡𝑜	𝑡𝑜	PROPN
cana-4142	110	33	𝔮	𝔮	PROPN
cana-4142	110	34	,	,	PUNCT
cana-4142	110	35	one	one	PRON
cana-4142	110	36	can	can	AUX
cana-4142	110	37	not	not	PART
cana-4142	110	38	use	use	VERB
cana-4142	110	39	the	the	DET
cana-4142	110	40	formulae	formulae	NOUN
cana-4142	110	41	for	for	ADP
cana-4142	110	42	the	the	DET
cana-4142	110	43	degree	degree	NOUN
cana-4142	110	44	of	of	ADP
cana-4142	110	45	the	the	DET
cana-4142	110	46	nearness	nearness	NOUN
cana-4142	110	47	as	as	SCONJ
cana-4142	110	48	given	give	VERB
cana-4142	110	49	in	in	ADP
cana-4142	110	50	the	the	DET
cana-4142	110	51	above	above	ADJ
cana-4142	110	52	examples	example	NOUN
cana-4142	110	53	due	due	ADP
cana-4142	110	54	to	to	ADP
cana-4142	110	55	the	the	DET
cana-4142	110	56	different	different	ADJ
cana-4142	110	57	dimensions	dimension	NOUN
cana-4142	110	58	of	of	ADP
cana-4142	110	59	these	these	DET
cana-4142	110	60	quantities	quantity	NOUN
cana-4142	110	61	.	.	PUNCT
cana-4142	111	1	in	in	ADP
cana-4142	111	2	the	the	DET
cana-4142	111	3	following	follow	VERB
cana-4142	111	4	example	example	NOUN
cana-4142	111	5	,	,	PUNCT
cana-4142	111	6	one	one	NUM
cana-4142	111	7	such	such	ADJ
cana-4142	111	8	case	case	NOUN
cana-4142	111	9	is	be	AUX
cana-4142	111	10	presented	present	VERB
cana-4142	111	11	.	.	PUNCT
cana-4142	112	1	example	example	NOUN
cana-4142	113	1	2	2	NUM
cana-4142	113	2	let	let	VERB
cana-4142	113	3	(	(	PUNCT
cana-4142	113	4	𝔐	𝔐	NOUN
cana-4142	113	5	,	,	PUNCT
cana-4142	113	6	𝕕	𝕕	X
cana-4142	113	7	)	)	PUNCT
cana-4142	113	8	be	be	AUX
cana-4142	113	9	a	a	DET
cana-4142	113	10	metric	metric	ADJ
cana-4142	113	11	space	space	NOUN
cana-4142	113	12	,	,	PUNCT
cana-4142	113	13	⨁	⨁	PROPN
cana-4142	113	14	the	the	DET
cana-4142	113	15	product	product	NOUN
cana-4142	113	16	(	(	PUNCT
cana-4142	113	17	maximum	maximum	ADJ
cana-4142	113	18	)	)	PUNCT
cana-4142	113	19	t	t	NOUN
cana-4142	113	20	-	-	PUNCT
cana-4142	113	21	conorm	conorm	NOUN
cana-4142	113	22	,	,	PUNCT
cana-4142	113	23	𝔲	𝔲	PROPN
cana-4142	113	24	>	>	X
cana-4142	113	25	0	0	PUNCT
cana-4142	113	26	and	and	CCONJ
cana-4142	113	27	𝓀	𝓀	PROPN
cana-4142	113	28	be	be	VERB
cana-4142	113	29	a	a	DET
cana-4142	113	30	positive	positive	ADJ
cana-4142	113	31	integer	integer	NOUN
cana-4142	113	32	.	.	PUNCT
cana-4142	114	1	define	define	VERB
cana-4142	114	2	a	a	DET
cana-4142	114	3	revised	revise	VERB
cana-4142	114	4	fuzzy	fuzzy	ADJ
cana-4142	114	5	set	set	VERB
cana-4142	114	6	𝔑	𝔑	NOUN
cana-4142	114	7	on	on	ADP
cana-4142	114	8	𝔐2	𝔐2	ADJ
cana-4142	114	9	×	×	NOUN
cana-4142	114	10	(	(	PUNCT
cana-4142	114	11	0	0	NUM
cana-4142	114	12	,	,	PUNCT
cana-4142	114	13	+	+	NOUN
cana-4142	114	14	∞)𝓀	∞)𝓀	NUM
cana-4142	114	15	by	by	ADP
cana-4142	114	16	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	114	17	,	,	PUNCT
cana-4142	114	18	𝔮	𝔮	PROPN
cana-4142	114	19	,	,	PUNCT
cana-4142	114	20	𝒶1	𝒶1	NOUN
cana-4142	114	21	,	,	PUNCT
cana-4142	114	22	𝒶2	𝒶2	PROPN
cana-4142	114	23	,	,	PUNCT
cana-4142	114	24	.	.	PUNCT
cana-4142	114	25	.	.	PUNCT
cana-4142	115	1	.	.	PUNCT
cana-4142	116	1	,	,	PUNCT
cana-4142	116	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	116	3	)	)	PUNCT
cana-4142	116	4	=	=	SYM
cana-4142	117	1	1	1	NUM
cana-4142	117	2	−	−	NOUN
cana-4142	118	1	𝔲	𝔲	NOUN
cana-4142	119	1	[	[	X
cana-4142	119	2	𝔲	𝔲	X
cana-4142	119	3	+	+	CCONJ
cana-4142	119	4	(	(	PUNCT
cana-4142	119	5	∑	∑	PROPN
cana-4142	119	6	1	1	NUM
cana-4142	119	7	𝒶𝒿	𝒶𝒿	NOUN
cana-4142	119	8	𝓀	𝓀	PROPN
cana-4142	119	9	𝒿=1	𝒿=1	X
cana-4142	119	10	)	)	PUNCT
cana-4142	119	11	𝕕(𝔭	𝕕(𝔭	PROPN
cana-4142	119	12	,	,	PUNCT
cana-4142	119	13	𝔮	𝔮	NOUN
cana-4142	119	14	)	)	PUNCT
cana-4142	119	15	]	]	PUNCT
cana-4142	119	16	−1	−1	NOUN
cana-4142	119	17	for	for	ADP
cana-4142	119	18	all	all	DET
cana-4142	119	19	all	all	PRON
cana-4142	119	20	𝔭	𝔭	NOUN
cana-4142	119	21	,	,	PUNCT
cana-4142	119	22	𝔮	𝔮	PROPN
cana-4142	119	23	∈	∈	PROPN
cana-4142	119	24	𝔐	𝔐	PROPN
cana-4142	119	25	,	,	PUNCT
cana-4142	119	26	𝔞	𝔞	PROPN
cana-4142	119	27	,	,	PUNCT
cana-4142	119	28	𝔟	𝔟	X
cana-4142	119	29	>	>	X
cana-4142	119	30	0	0	PUNCT
cana-4142	119	31	and	and	CCONJ
cana-4142	119	32	𝒶1	𝒶1	PROPN
cana-4142	119	33	,	,	PUNCT
cana-4142	119	34	𝒶2	𝒶2	PROPN
cana-4142	119	35	,	,	PUNCT
cana-4142	119	36	.	.	PUNCT
cana-4142	119	37	.	.	PUNCT
cana-4142	119	38	.	.	PUNCT
cana-4142	120	1	,	,	PUNCT
cana-4142	120	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	120	3	>	>	X
cana-4142	120	4	0	0	X
cana-4142	120	5	.	.	PUNCT
cana-4142	121	1	then	then	ADV
cana-4142	121	2	,	,	PUNCT
cana-4142	121	3	(	(	PUNCT
cana-4142	121	4	𝔐	𝔐	X
cana-4142	121	5	,	,	PUNCT
cana-4142	121	6	𝔑	𝔑	PROPN
cana-4142	121	7	,	,	PUNCT
cana-4142	121	8	⨁)is	⨁)is	ADJ
cana-4142	121	9	a	a	DET
cana-4142	121	10	revised	revise	VERB
cana-4142	121	11	fuzzy	fuzzy	ADJ
cana-4142	121	12	𝓀	𝓀	X
cana-4142	121	13	−metric	−metric	ADJ
cana-4142	121	14	space	space	NOUN
cana-4142	121	15	.	.	PUNCT
cana-4142	122	1	example	example	NOUN
cana-4142	122	2	3	3	NUM
cana-4142	122	3	let	let	VERB
cana-4142	122	4	𝔐	𝔐	PRON
cana-4142	122	5	=	=	SYM
cana-4142	122	6	ℛ𝓀	ℛ𝓀	PROPN
cana-4142	122	7	,	,	PUNCT
cana-4142	122	8	where	where	SCONJ
cana-4142	122	9	𝓀	𝓀	PROPN
cana-4142	122	10	is	be	AUX
cana-4142	122	11	a	a	DET
cana-4142	122	12	positive	positive	ADJ
cana-4142	122	13	integer	integer	NOUN
cana-4142	122	14	,	,	PUNCT
cana-4142	122	15	⨁	⨁	PROPN
cana-4142	122	16	the	the	DET
cana-4142	122	17	product	product	NOUN
cana-4142	122	18	t	t	NOUN
cana-4142	122	19	-	-	PUNCT
cana-4142	122	20	conorm	conorm	NOUN
cana-4142	122	21	.	.	PUNCT
cana-4142	123	1	define	define	VERB
cana-4142	123	2	a	a	DET
cana-4142	123	3	revised	revise	VERB
cana-4142	123	4	fuzzy	fuzzy	ADJ
cana-4142	123	5	set	set	VERB
cana-4142	123	6	𝔑	𝔑	NOUN
cana-4142	123	7	on	on	ADP
cana-4142	123	8	𝔐2	𝔐2	ADJ
cana-4142	123	9	×	×	NOUN
cana-4142	123	10	(	(	PUNCT
cana-4142	123	11	0	0	NUM
cana-4142	123	12	,	,	PUNCT
cana-4142	123	13	+	+	NOUN
cana-4142	123	14	∞)𝓀	∞)𝓀	NUM
cana-4142	123	15	by	by	ADP
cana-4142	123	16	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	123	17	,	,	PUNCT
cana-4142	123	18	𝔮	𝔮	PROPN
cana-4142	123	19	,	,	PUNCT
cana-4142	123	20	𝒶1	𝒶1	NOUN
cana-4142	123	21	,	,	PUNCT
cana-4142	123	22	𝒶2	𝒶2	PROPN
cana-4142	123	23	,	,	PUNCT
cana-4142	123	24	.	.	PUNCT
cana-4142	123	25	.	.	PUNCT
cana-4142	124	1	.	.	PUNCT
cana-4142	125	1	,	,	PUNCT
cana-4142	125	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	125	3	)	)	PUNCT
cana-4142	125	4	=	=	SYM
cana-4142	126	1	1	1	NUM
cana-4142	126	2	−	−	NOUN
cana-4142	127	1	𝔲	𝔲	NOUN
cana-4142	128	1	[	[	X
cana-4142	128	2	𝔲	𝔲	X
cana-4142	128	3	+	+	CCONJ
cana-4142	128	4	(	(	PUNCT
cana-4142	128	5	∑	∑	PART
cana-4142	128	6	|𝔮𝒿	|𝔮𝒿	NUM
cana-4142	128	7	−	−	PROPN
cana-4142	128	8	𝔭𝒿|	𝔭𝒿|	X
cana-4142	128	9	𝒶𝒿	𝒶𝒿	NOUN
cana-4142	128	10	𝓀	𝓀	PROPN
cana-4142	128	11	𝒿=1	𝒿=1	PROPN
cana-4142	128	12	)	)	PUNCT
cana-4142	128	13	]	]	PUNCT
cana-4142	128	14	−1	−1	NOUN
cana-4142	128	15	for	for	ADP
cana-4142	128	16	all	all	DET
cana-4142	128	17	𝔭	𝔭	NOUN
cana-4142	128	18	,	,	PUNCT
cana-4142	128	19	𝔮	𝔮	X
cana-4142	128	20	∈	∈	NOUN
cana-4142	128	21	𝔐	𝔐	PROPN
cana-4142	128	22	and	and	CCONJ
cana-4142	128	23	𝒶1	𝒶1	NOUN
cana-4142	128	24	,	,	PUNCT
cana-4142	128	25	𝒶2	𝒶2	PROPN
cana-4142	128	26	,	,	PUNCT
cana-4142	128	27	.	.	PUNCT
cana-4142	128	28	.	.	PUNCT
cana-4142	128	29	.	.	PUNCT
cana-4142	129	1	,	,	PUNCT
cana-4142	129	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	129	3	>	>	X
cana-4142	129	4	0	0	X
cana-4142	129	5	.	.	PUNCT
cana-4142	130	1	then	then	ADV
cana-4142	130	2	,	,	PUNCT
cana-4142	130	3	(	(	PUNCT
cana-4142	130	4	𝔐	𝔐	X
cana-4142	130	5	,	,	PUNCT
cana-4142	130	6	𝔑	𝔑	PROPN
cana-4142	130	7	,	,	PUNCT
cana-4142	130	8	⨁	⨁	PROPN
cana-4142	130	9	)	)	PUNCT
cana-4142	130	10	is	be	AUX
cana-4142	130	11	a	a	DET
cana-4142	130	12	revised	revise	VERB
cana-4142	130	13	fuzzy	fuzzy	ADJ
cana-4142	130	14	𝓀	𝓀	X
cana-4142	130	15	−metric	−metric	ADJ
cana-4142	130	16	space	space	NOUN
cana-4142	130	17	.	.	PUNCT
cana-4142	131	1	in	in	ADP
cana-4142	131	2	the	the	DET
cana-4142	131	3	present	present	ADJ
cana-4142	131	4	paper	paper	NOUN
cana-4142	131	5	,	,	PUNCT
cana-4142	131	6	we	we	PRON
cana-4142	131	7	restrict	restrict	VERB
cana-4142	131	8	ourselves	ourselves	PRON
cana-4142	131	9	to	to	ADP
cana-4142	131	10	only	only	ADV
cana-4142	131	11	mathematical	mathematical	ADJ
cana-4142	131	12	properties	property	NOUN
cana-4142	131	13	of	of	ADP
cana-4142	131	14	revised	revise	VERB
cana-4142	131	15	fuzzy	fuzzy	ADJ
cana-4142	131	16	𝓀	𝓀	X
cana-4142	131	17	−metric	−metric	ADJ
cana-4142	131	18	space	space	NOUN
cana-4142	131	19	.	.	PUNCT
cana-4142	132	1	definition	definition	NOUN
cana-4142	132	2	12	12	NUM
cana-4142	132	3	a	a	DET
cana-4142	132	4	revised	revise	VERB
cana-4142	132	5	fuzzy	fuzzy	ADJ
cana-4142	132	6	𝓀	𝓀	X
cana-4142	132	7	−metric	−metric	ADJ
cana-4142	132	8	space	space	NOUN
cana-4142	132	9	(	(	PUNCT
cana-4142	132	10	𝔐	𝔐	PROPN
cana-4142	132	11	,	,	PUNCT
cana-4142	132	12	𝔑	𝔑	PROPN
cana-4142	132	13	,	,	PUNCT
cana-4142	132	14	⨁	⨁	PROPN
cana-4142	132	15	)	)	PUNCT
cana-4142	132	16	is	be	AUX
cana-4142	132	17	called	call	VERB
cana-4142	132	18	𝒯	𝒯	PROPN
cana-4142	132	19	−natural	−natural	ADJ
cana-4142	132	20	revised	revise	VERB
cana-4142	132	21	fuzzy	fuzzy	ADJ
cana-4142	132	22	𝓀	𝓀	X
cana-4142	132	23	−metric	−metric	ADJ
cana-4142	132	24	space	space	NOUN
cana-4142	132	25	if	if	SCONJ
cana-4142	132	26	there	there	PRON
cana-4142	132	27	exists	exist	VERB
cana-4142	132	28	𝒯	𝒯	PROPN
cana-4142	132	29	∈	∈	PROPN
cana-4142	132	30	{	{	PUNCT
cana-4142	132	31	1	1	NUM
cana-4142	132	32	,	,	PUNCT
cana-4142	132	33	2	2	NUM
cana-4142	132	34	,	,	PUNCT
cana-4142	132	35	…	…	PUNCT
cana-4142	132	36	,	,	PUNCT
cana-4142	132	37	𝓀	𝓀	X
cana-4142	132	38	}	}	PUNCT
cana-4142	132	39	such	such	ADJ
cana-4142	132	40	that	that	SCONJ
cana-4142	132	41	communications	communication	NOUN
cana-4142	132	42	on	on	ADP
cana-4142	132	43	applied	apply	VERB
cana-4142	132	44	nonlinear	nonlinear	ADJ
cana-4142	132	45	analysis	analysis	NOUN
cana-4142	132	46	issn	issn	NOUN
cana-4142	132	47	:	:	PUNCT
cana-4142	132	48	1074	1074	NUM
cana-4142	132	49	-	-	PUNCT
cana-4142	132	50	133x	133x	NUM
cana-4142	132	51	vol	vol	NOUN
cana-4142	132	52	x	x	NOUN
cana-4142	133	1	no	no	INTJ
cana-4142	133	2	.	.	PUNCT
cana-4142	134	1	y	y	PROPN
cana-4142	134	2	(	(	PUNCT
cana-4142	134	3	2025	2025	NUM
cana-4142	134	4	)	)	PUNCT
cana-4142	134	5	1324	1324	NUM
cana-4142	134	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	135	1	lim	lim	PROPN
cana-4142	135	2	𝒶𝓀→+∞	𝒶𝓀→+∞	ADP
cana-4142	135	3	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	135	4	,	,	PUNCT
cana-4142	135	5	𝔮	𝔮	PROPN
cana-4142	135	6	,	,	PUNCT
cana-4142	135	7	𝒶1	𝒶1	NOUN
cana-4142	135	8	,	,	PUNCT
cana-4142	135	9	𝒶2	𝒶2	PROPN
cana-4142	135	10	,	,	PUNCT
cana-4142	135	11	.	.	PUNCT
cana-4142	135	12	.	.	PUNCT
cana-4142	135	13	.	.	PUNCT
cana-4142	136	1	,	,	PUNCT
cana-4142	136	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	136	3	)	)	PUNCT
cana-4142	136	4	=	=	SYM
cana-4142	136	5	0	0	NUM
cana-4142	136	6	,	,	PUNCT
cana-4142	136	7	𝔭	𝔭	NOUN
cana-4142	136	8	,	,	PUNCT
cana-4142	136	9	𝔮	𝔮	PROPN
cana-4142	136	10	∈	∈	NOUN
cana-4142	136	11	𝔐	𝔐	PROPN
cana-4142	136	12	and	and	CCONJ
cana-4142	136	13	𝒶1	𝒶1	NOUN
cana-4142	136	14	,	,	PUNCT
cana-4142	136	15	𝒶2	𝒶2	PROPN
cana-4142	136	16	,	,	PUNCT
cana-4142	136	17	.	.	PUNCT
cana-4142	136	18	.	.	PUNCT
cana-4142	137	1	.	.	PUNCT
cana-4142	138	1	,	,	PUNCT
cana-4142	138	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	138	3	>	>	X
cana-4142	138	4	0	0	X
cana-4142	138	5	.	.	PUNCT
cana-4142	139	1	for	for	ADP
cana-4142	139	2	the	the	DET
cana-4142	139	3	rest	rest	NOUN
cana-4142	139	4	of	of	ADP
cana-4142	139	5	this	this	DET
cana-4142	139	6	paper	paper	NOUN
cana-4142	139	7	,	,	PUNCT
cana-4142	139	8	for	for	ADP
cana-4142	139	9	a	a	DET
cana-4142	139	10	given	give	VERB
cana-4142	139	11	revised	revise	VERB
cana-4142	139	12	fuzzy	fuzzy	ADJ
cana-4142	139	13	𝓀	𝓀	PROPN
cana-4142	139	14	−metric	−metric	PROPN
cana-4142	139	15	space(𝔐	space(𝔐	PROPN
cana-4142	139	16	,	,	PUNCT
cana-4142	139	17	𝔑	𝔑	PROPN
cana-4142	139	18	,	,	PUNCT
cana-4142	139	19	⨁	⨁	PROPN
cana-4142	139	20	)	)	PUNCT
cana-4142	139	21	,	,	PUNCT
cana-4142	139	22	𝔭	𝔭	NOUN
cana-4142	139	23	,	,	PUNCT
cana-4142	139	24	𝔮	𝔮	PROPN
cana-4142	139	25	∈	∈	NOUN
cana-4142	139	26	𝔐	𝔐	PROPN
cana-4142	139	27	and	and	CCONJ
cana-4142	139	28	𝒶1	𝒶1	NOUN
cana-4142	139	29	,	,	PUNCT
cana-4142	139	30	𝒶2	𝒶2	PROPN
cana-4142	139	31	,	,	PUNCT
cana-4142	139	32	.	.	PUNCT
cana-4142	139	33	.	.	PUNCT
cana-4142	139	34	.	.	PUNCT
cana-4142	140	1	,	,	PUNCT
cana-4142	140	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	140	3	>	>	X
cana-4142	140	4	0	0	NUM
cana-4142	140	5	,	,	PUNCT
cana-4142	140	6	for	for	ADP
cana-4142	140	7	simplicity	simplicity	NOUN
cana-4142	140	8	,	,	PUNCT
cana-4142	140	9	we	we	PRON
cana-4142	140	10	write	write	VERB
cana-4142	140	11	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	140	12	,	,	PUNCT
cana-4142	140	13	𝔮	𝔮	PROPN
cana-4142	140	14	,	,	PUNCT
cana-4142	140	15	𝒶1	𝒶1	PROPN
cana-4142	140	16	𝓀	𝓀	PROPN
cana-4142	140	17	)	)	PUNCT
cana-4142	140	18	instead	instead	ADV
cana-4142	140	19	𝔑(𝔭	𝔑(𝔭	X
cana-4142	140	20	,	,	PUNCT
cana-4142	140	21	𝔮	𝔮	PROPN
cana-4142	140	22	,	,	PUNCT
cana-4142	140	23	𝒶1	𝒶1	NOUN
cana-4142	140	24	,	,	PUNCT
cana-4142	140	25	𝒶2	𝒶2	PROPN
cana-4142	140	26	,	,	PUNCT
cana-4142	140	27	.	.	PUNCT
cana-4142	140	28	.	.	PUNCT
cana-4142	141	1	.	.	PUNCT
cana-4142	142	1	,	,	PUNCT
cana-4142	142	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	142	3	)	)	PUNCT
cana-4142	142	4	.	.	PUNCT
cana-4142	143	1	next	next	ADV
cana-4142	143	2	,	,	PUNCT
cana-4142	143	3	we	we	PRON
cana-4142	143	4	discuss	discuss	VERB
cana-4142	143	5	some	some	DET
cana-4142	143	6	properties	property	NOUN
cana-4142	143	7	revised	revise	VERB
cana-4142	143	8	fuzzy	fuzzy	ADJ
cana-4142	143	9	𝓀	𝓀	X
cana-4142	143	10	−metric	−metric	ADJ
cana-4142	143	11	space	space	NOUN
cana-4142	143	12	and	and	CCONJ
cana-4142	143	13	establish	establish	VERB
cana-4142	143	14	the	the	DET
cana-4142	143	15	topology	topology	NOUN
cana-4142	143	16	of	of	ADP
cana-4142	143	17	such	such	ADJ
cana-4142	143	18	spaces	space	NOUN
cana-4142	143	19	.	.	PUNCT
cana-4142	144	1	proposition	proposition	NOUN
cana-4142	144	2	13	13	NUM
cana-4142	144	3	let	let	VERB
cana-4142	144	4	(	(	PUNCT
cana-4142	144	5	𝔐	𝔐	X
cana-4142	144	6	,	,	PUNCT
cana-4142	144	7	𝔑	𝔑	PROPN
cana-4142	144	8	,	,	PUNCT
cana-4142	144	9	⨁	⨁	PROPN
cana-4142	144	10	)	)	PUNCT
cana-4142	144	11	be	be	VERB
cana-4142	144	12	a	a	DET
cana-4142	144	13	revised	revise	VERB
cana-4142	144	14	fuzzy	fuzzy	ADJ
cana-4142	144	15	𝓀	𝓀	X
cana-4142	144	16	−metric	−metric	ADJ
cana-4142	144	17	space	space	NOUN
cana-4142	144	18	,	,	PUNCT
cana-4142	144	19	𝒶	𝒶	NOUN
cana-4142	144	20	,	,	PUNCT
cana-4142	144	21	𝒶1	𝒶1	NOUN
cana-4142	144	22	,	,	PUNCT
cana-4142	144	23	𝒶2	𝒶2	PROPN
cana-4142	144	24	,	,	PUNCT
cana-4142	144	25	.	.	PUNCT
cana-4142	144	26	.	.	PUNCT
cana-4142	144	27	.	.	PUNCT
cana-4142	145	1	,	,	PUNCT
cana-4142	145	2	𝒶𝓀.	𝒶𝓀.	NOUN
cana-4142	145	3	suppose	suppose	VERB
cana-4142	145	4	that	that	SCONJ
cana-4142	145	5	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	145	6	<	<	X
cana-4142	145	7	𝑎	𝑎	NOUN
cana-4142	145	8	for	for	ADP
cana-4142	145	9	some	some	DET
cana-4142	145	10	𝒯	𝒯	PROPN
cana-4142	145	11	∈	∈	PROPN
cana-4142	145	12	{	{	PUNCT
cana-4142	145	13	1	1	NUM
cana-4142	145	14	,	,	PUNCT
cana-4142	145	15	2	2	NUM
cana-4142	145	16	,	,	PUNCT
cana-4142	145	17	…	…	PUNCT
cana-4142	145	18	,	,	PUNCT
cana-4142	145	19	𝓀	𝓀	X
cana-4142	145	20	}	}	PUNCT
cana-4142	145	21	.	.	PUNCT
cana-4142	146	1	then	then	ADV
cana-4142	146	2	,	,	PUNCT
cana-4142	146	3	𝔑(𝔭	𝔑(𝔭	X
cana-4142	146	4	,	,	PUNCT
cana-4142	146	5	𝔮	𝔮	PROPN
cana-4142	146	6	,	,	PUNCT
cana-4142	146	7	𝒶1	𝒶1	PROPN
cana-4142	146	8	𝓀	𝓀	PROPN
cana-4142	146	9	)	)	PUNCT
cana-4142	146	10	≥	≥	PROPN
cana-4142	146	11	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	146	12	,	,	PUNCT
cana-4142	146	13	𝔮	𝔮	PROPN
cana-4142	146	14	,	,	PUNCT
cana-4142	146	15	𝒶1	𝒶1	NOUN
cana-4142	146	16	,	,	PUNCT
cana-4142	146	17	𝒶2	𝒶2	PROPN
cana-4142	146	18	,	,	PUNCT
cana-4142	146	19	.	.	PUNCT
cana-4142	146	20	.	.	PUNCT
cana-4142	147	1	,	,	PUNCT
cana-4142	147	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	147	3	,	,	PUNCT
cana-4142	147	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	147	5	,	,	PUNCT
cana-4142	147	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	147	7	,	,	PUNCT
cana-4142	147	8	.	.	PUNCT
cana-4142	147	9	.	.	PUNCT
cana-4142	148	1	,	,	PUNCT
cana-4142	148	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	148	3	)	)	PUNCT
cana-4142	148	4	for	for	ADP
cana-4142	148	5	all	all	DET
cana-4142	148	6	𝔭	𝔭	NOUN
cana-4142	148	7	,	,	PUNCT
cana-4142	148	8	𝔮	𝔮	X
cana-4142	148	9	∈	∈	PROPN
cana-4142	148	10	𝔐.	𝔐.	NOUN
cana-4142	148	11	remark	remark	NOUN
cana-4142	148	12	14	14	NUM
cana-4142	148	13	in	in	ADP
cana-4142	148	14	a	a	DET
cana-4142	148	15	revised	revise	VERB
cana-4142	148	16	fuzzy	fuzzy	ADJ
cana-4142	148	17	𝓀	𝓀	PROPN
cana-4142	148	18	−metric	−metric	PROPN
cana-4142	148	19	space(𝔐	space(𝔐	PROPN
cana-4142	148	20	,	,	PUNCT
cana-4142	148	21	𝔑	𝔑	PROPN
cana-4142	148	22	,	,	PUNCT
cana-4142	148	23	⨁	⨁	PROPN
cana-4142	148	24	)	)	PUNCT
cana-4142	148	25	,	,	PUNCT
cana-4142	148	26	if	if	SCONJ
cana-4142	148	27	𝔑(𝔭	𝔑(𝔭	X
cana-4142	148	28	,	,	PUNCT
cana-4142	148	29	𝔮	𝔮	PROPN
cana-4142	148	30	,	,	PUNCT
cana-4142	148	31	𝒶1	𝒶1	PROPN
cana-4142	148	32	𝓀	𝓀	X
cana-4142	148	33	)	)	PUNCT
cana-4142	148	34	<	<	X
cana-4142	149	1	휀	휀	X
cana-4142	149	2	,	,	PUNCT
cana-4142	149	3	where	where	SCONJ
cana-4142	149	4	𝔭	𝔭	NOUN
cana-4142	149	5	,	,	PUNCT
cana-4142	149	6	𝔮	𝔮	PROPN
cana-4142	149	7	∈	∈	PROPN
cana-4142	149	8	𝔐	𝔐	PROPN
cana-4142	149	9	,	,	PUNCT
cana-4142	149	10	𝒶1	𝒶1	NOUN
cana-4142	149	11	,	,	PUNCT
cana-4142	149	12	𝒶2	𝒶2	PROPN
cana-4142	149	13	,	,	PUNCT
cana-4142	149	14	.	.	PUNCT
cana-4142	149	15	.	.	PUNCT
cana-4142	149	16	.	.	PUNCT
cana-4142	150	1	,	,	PUNCT
cana-4142	150	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	150	3	>	>	X
cana-4142	150	4	0	0	PUNCT
cana-4142	150	5	and	and	CCONJ
cana-4142	150	6	0	0	NUM
cana-4142	150	7	<	<	X
cana-4142	150	8	휀	휀	X
cana-4142	150	9	<	<	X
cana-4142	150	10	1	1	NUM
cana-4142	150	11	,	,	PUNCT
cana-4142	150	12	then	then	ADV
cana-4142	150	13	for	for	ADP
cana-4142	150	14	each	each	DET
cana-4142	150	15	𝒯	𝒯	PROPN
cana-4142	150	16	∈	∈	PROPN
cana-4142	150	17	{	{	PUNCT
cana-4142	150	18	1	1	NUM
cana-4142	150	19	,	,	PUNCT
cana-4142	150	20	2	2	NUM
cana-4142	150	21	,	,	PUNCT
cana-4142	150	22	…	…	PUNCT
cana-4142	150	23	,	,	PUNCT
cana-4142	150	24	𝓀	𝓀	X
cana-4142	150	25	}	}	PUNCT
cana-4142	150	26	,	,	PUNCT
cana-4142	150	27	we	we	PRON
cana-4142	150	28	can	can	AUX
cana-4142	150	29	find	find	VERB
cana-4142	150	30	𝒶	𝒶	PRON
cana-4142	150	31	∈	∈	PROPN
cana-4142	150	32	(	(	PUNCT
cana-4142	150	33	0	0	NUM
cana-4142	150	34	,	,	PUNCT
cana-4142	150	35	𝒶𝓀	𝒶𝓀	ADV
cana-4142	150	36	)	)	PUNCT
cana-4142	150	37	such	such	ADJ
cana-4142	150	38	that	that	SCONJ
cana-4142	150	39	𝔑(𝔭	𝔑(𝔭	ADP
cana-4142	150	40	,	,	PUNCT
cana-4142	150	41	𝔮	𝔮	NOUN
cana-4142	150	42	,	,	PUNCT
cana-4142	150	43	𝒶1	𝒶1	NOUN
cana-4142	150	44	,	,	PUNCT
cana-4142	150	45	𝒶2	𝒶2	PROPN
cana-4142	150	46	,	,	PUNCT
cana-4142	150	47	.	.	PUNCT
cana-4142	150	48	.	.	PUNCT
cana-4142	151	1	,	,	PUNCT
cana-4142	151	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	151	3	,	,	PUNCT
cana-4142	151	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	151	5	,	,	PUNCT
cana-4142	151	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	151	7	,	,	PUNCT
cana-4142	151	8	.	.	PUNCT
cana-4142	151	9	.	.	PUNCT
cana-4142	152	1	,	,	PUNCT
cana-4142	152	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	152	3	)	)	PUNCT
cana-4142	152	4	<	<	X
cana-4142	152	5	휀	휀	X
cana-4142	152	6	.	.	PUNCT
cana-4142	152	7	definition	definition	NOUN
cana-4142	152	8	15	15	NUM
cana-4142	152	9	let	let	VERB
cana-4142	152	10	(	(	PUNCT
cana-4142	152	11	𝔐	𝔐	X
cana-4142	152	12	,	,	PUNCT
cana-4142	152	13	𝔑	𝔑	PROPN
cana-4142	152	14	,	,	PUNCT
cana-4142	152	15	⨁	⨁	PROPN
cana-4142	152	16	)	)	PUNCT
cana-4142	152	17	be	be	VERB
cana-4142	152	18	a	a	DET
cana-4142	152	19	revised	revise	VERB
cana-4142	152	20	fuzzy	fuzzy	ADJ
cana-4142	152	21	𝓀	𝓀	X
cana-4142	152	22	−metric	−metric	ADJ
cana-4142	152	23	space	space	NOUN
cana-4142	152	24	.	.	PUNCT
cana-4142	153	1	an	an	DET
cana-4142	153	2	open	open	ADJ
cana-4142	153	3	ball	ball	NOUN
cana-4142	153	4	with	with	ADP
cana-4142	153	5	center	center	NOUN
cana-4142	153	6	𝔭	𝔭	ADP
cana-4142	153	7	∈	∈	NOUN
cana-4142	153	8	𝔐	𝔐	NOUN
cana-4142	153	9	and	and	CCONJ
cana-4142	153	10	radius	radius	NOUN
cana-4142	153	11	휀	휀	PROPN
cana-4142	153	12	∈	∈	PROPN
cana-4142	153	13	(	(	PUNCT
cana-4142	153	14	0,1	0,1	NOUN
cana-4142	153	15	)	)	PUNCT
cana-4142	153	16	with	with	ADP
cana-4142	153	17	respect	respect	NOUN
cana-4142	153	18	to	to	ADP
cana-4142	153	19	parameters𝒶1	parameters𝒶1	NOUN
cana-4142	153	20	,	,	PUNCT
cana-4142	153	21	𝒶2	𝒶2	PROPN
cana-4142	153	22	,	,	PUNCT
cana-4142	153	23	.	.	PUNCT
cana-4142	153	24	.	.	PUNCT
cana-4142	153	25	.	.	PUNCT
cana-4142	154	1	,	,	PUNCT
cana-4142	154	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	154	3	>	>	X
cana-4142	154	4	0	0	NUM
cana-4142	154	5	,	,	PUNCT
cana-4142	154	6	denoted	denote	VERB
cana-4142	154	7	by	by	ADP
cana-4142	154	8	𝔅(𝔭	𝔅(𝔭	PROPN
cana-4142	154	9	,	,	PUNCT
cana-4142	154	10	휀	휀	NOUN
cana-4142	154	11	;	;	PUNCT
cana-4142	154	12	𝒶1	𝒶1	NOUN
cana-4142	154	13	,	,	PUNCT
cana-4142	154	14	𝒶2	𝒶2	PROPN
cana-4142	154	15	,	,	PUNCT
cana-4142	154	16	.	.	PUNCT
cana-4142	154	17	.	.	PUNCT
cana-4142	155	1	.	.	PUNCT
cana-4142	156	1	,	,	PUNCT
cana-4142	156	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	156	3	)	)	PUNCT
cana-4142	156	4	,	,	PUNCT
cana-4142	156	5	is	be	AUX
cana-4142	156	6	defined	define	VERB
cana-4142	156	7	by	by	ADP
cana-4142	156	8	𝔅(𝔭	𝔅(𝔭	PROPN
cana-4142	156	9	,	,	PUNCT
cana-4142	156	10	휀	휀	NOUN
cana-4142	156	11	;	;	PUNCT
cana-4142	156	12	𝒶1	𝒶1	NOUN
cana-4142	156	13	,	,	PUNCT
cana-4142	156	14	𝒶2	𝒶2	PROPN
cana-4142	156	15	,	,	PUNCT
cana-4142	156	16	.	.	PUNCT
cana-4142	156	17	.	.	PUNCT
cana-4142	157	1	.	.	PUNCT
cana-4142	158	1	,	,	PUNCT
cana-4142	158	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	158	3	)	)	PUNCT
cana-4142	158	4	=	=	SYM
cana-4142	158	5	{	{	PUNCT
cana-4142	158	6	𝔮	𝔮	NOUN
cana-4142	158	7	∈	∈	PROPN
cana-4142	158	8	𝔐	𝔐	NOUN
cana-4142	158	9	:	:	PUNCT
cana-4142	158	10	𝔑(𝔭	𝔑(𝔭	X
cana-4142	158	11	,	,	PUNCT
cana-4142	158	12	𝔮	𝔮	PROPN
cana-4142	158	13	,	,	PUNCT
cana-4142	158	14	𝒶1	𝒶1	PROPN
cana-4142	158	15	𝓀	𝓀	X
cana-4142	158	16	)	)	PUNCT
cana-4142	158	17	<	<	X
cana-4142	159	1	휀	휀	X
cana-4142	159	2	}	}	PUNCT
cana-4142	159	3	.	.	PUNCT
cana-4142	160	1	definition	definition	NOUN
cana-4142	160	2	16	16	NUM
cana-4142	160	3	let	let	VERB
cana-4142	160	4	(	(	PUNCT
cana-4142	160	5	𝔐	𝔐	X
cana-4142	160	6	,	,	PUNCT
cana-4142	160	7	𝔑	𝔑	PROPN
cana-4142	160	8	,	,	PUNCT
cana-4142	160	9	⨁	⨁	PROPN
cana-4142	160	10	)	)	PUNCT
cana-4142	160	11	be	be	VERB
cana-4142	160	12	a	a	DET
cana-4142	160	13	revised	revise	VERB
cana-4142	160	14	fuzzy	fuzzy	ADJ
cana-4142	160	15	𝓀	𝓀	X
cana-4142	160	16	−metric	−metric	ADJ
cana-4142	160	17	space	space	NOUN
cana-4142	160	18	.	.	PUNCT
cana-4142	161	1	a	a	DET
cana-4142	161	2	subset	subset	NOUN
cana-4142	161	3	𝒳	𝒳	PROPN
cana-4142	161	4	of	of	ADP
cana-4142	161	5	𝔐	𝔐	PROPN
cana-4142	161	6	is	be	AUX
cana-4142	161	7	called	call	VERB
cana-4142	161	8	an	an	DET
cana-4142	161	9	open	open	ADJ
cana-4142	161	10	set	set	NOUN
cana-4142	161	11	if	if	SCONJ
cana-4142	161	12	and	and	CCONJ
cana-4142	161	13	only	only	ADV
cana-4142	161	14	if	if	SCONJ
cana-4142	161	15	there	there	PRON
cana-4142	161	16	is	be	VERB
cana-4142	161	17	an	an	DET
cana-4142	161	18	open	open	ADJ
cana-4142	161	19	ball	ball	NOUN
cana-4142	161	20	𝔅	𝔅	NOUN
cana-4142	161	21	such	such	ADJ
cana-4142	161	22	that	that	PRON
cana-4142	161	23	𝔅	𝔅	PROPN
cana-4142	161	24	⊆	⊆	NUM
cana-4142	161	25	𝒳.	𝒳.	NOUN
cana-4142	161	26	a	a	DET
cana-4142	161	27	subset	subset	ADJ
cana-4142	161	28	𝒴	𝒴	NOUN
cana-4142	161	29	of	of	ADP
cana-4142	161	30	𝔐	𝔐	PROPN
cana-4142	161	31	is	be	AUX
cana-4142	161	32	called	call	VERB
cana-4142	161	33	a	a	DET
cana-4142	161	34	closed	closed	ADJ
cana-4142	161	35	set	set	NOUN
cana-4142	161	36	if	if	SCONJ
cana-4142	161	37	and	and	CCONJ
cana-4142	161	38	only	only	ADV
cana-4142	161	39	if	if	SCONJ
cana-4142	161	40	its	its	PRON
cana-4142	161	41	complement	complement	NOUN
cana-4142	161	42	is	be	AUX
cana-4142	161	43	an	an	DET
cana-4142	161	44	open	open	ADJ
cana-4142	161	45	set	set	NOUN
cana-4142	161	46	.	.	PUNCT
cana-4142	162	1	theorem	theorem	VERB
cana-4142	162	2	17	17	NUM
cana-4142	162	3	every	every	DET
cana-4142	162	4	open	open	ADJ
cana-4142	162	5	ball	ball	NOUN
cana-4142	162	6	in	in	ADP
cana-4142	162	7	a	a	DET
cana-4142	162	8	revised	revise	VERB
cana-4142	162	9	fuzzy	fuzzy	ADJ
cana-4142	162	10	𝓀	𝓀	PROPN
cana-4142	162	11	−metric	−metric	ADJ
cana-4142	162	12	space	space	NOUN
cana-4142	162	13	is	be	AUX
cana-4142	162	14	an	an	DET
cana-4142	162	15	open	open	ADJ
cana-4142	162	16	set	set	NOUN
cana-4142	162	17	.	.	PUNCT
cana-4142	163	1	proof	proof	NOUN
cana-4142	163	2	let	let	VERB
cana-4142	163	3	(	(	PUNCT
cana-4142	163	4	𝔐	𝔐	X
cana-4142	163	5	,	,	PUNCT
cana-4142	163	6	𝔑	𝔑	PROPN
cana-4142	163	7	,	,	PUNCT
cana-4142	163	8	⨁	⨁	PROPN
cana-4142	163	9	)	)	PUNCT
cana-4142	163	10	be	be	VERB
cana-4142	163	11	a	a	DET
cana-4142	163	12	revised	revise	VERB
cana-4142	163	13	fuzzy	fuzzy	ADJ
cana-4142	163	14	𝓀	𝓀	PROPN
cana-4142	163	15	−metric	−metric	ADJ
cana-4142	163	16	space	space	NOUN
cana-4142	163	17	,	,	PUNCT
cana-4142	163	18	𝔭	𝔭	NOUN
cana-4142	163	19	∈	∈	PROPN
cana-4142	163	20	𝔐	𝔐	PROPN
cana-4142	163	21	,	,	PUNCT
cana-4142	163	22	𝒶1	𝒶1	NOUN
cana-4142	163	23	,	,	PUNCT
cana-4142	163	24	𝒶2	𝒶2	PROPN
cana-4142	163	25	,	,	PUNCT
cana-4142	163	26	.	.	PUNCT
cana-4142	163	27	.	.	PUNCT
cana-4142	163	28	.	.	PUNCT
cana-4142	164	1	,	,	PUNCT
cana-4142	164	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	164	3	>	>	X
cana-4142	164	4	0	0	PUNCT
cana-4142	165	1	and	and	CCONJ
cana-4142	165	2	휀	휀	PRON
cana-4142	165	3	∈	∈	PROPN
cana-4142	165	4	(	(	PUNCT
cana-4142	165	5	0,1	0,1	NUM
cana-4142	165	6	)	)	PUNCT
cana-4142	165	7	.	.	PUNCT
cana-4142	166	1	assume	assume	VERB
cana-4142	166	2	that	that	SCONJ
cana-4142	166	3	𝔮	𝔮	X
cana-4142	166	4	∈	∈	PROPN
cana-4142	166	5	𝔅(𝔭	𝔅(𝔭	NOUN
cana-4142	166	6	,	,	PUNCT
cana-4142	166	7	휀	휀	NOUN
cana-4142	166	8	;	;	PUNCT
cana-4142	166	9	𝒶1	𝒶1	NOUN
cana-4142	166	10	,	,	PUNCT
cana-4142	166	11	𝒶2	𝒶2	PROPN
cana-4142	166	12	,	,	PUNCT
cana-4142	166	13	.	.	PUNCT
cana-4142	166	14	.	.	PUNCT
cana-4142	166	15	.	.	PUNCT
cana-4142	167	1	,	,	PUNCT
cana-4142	167	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	167	3	)	)	PUNCT
cana-4142	167	4	.	.	PUNCT
cana-4142	168	1	then	then	ADV
cana-4142	168	2	,	,	PUNCT
cana-4142	168	3	we	we	PRON
cana-4142	168	4	have	have	VERB
cana-4142	168	5	𝔑(𝔭	𝔑(𝔭	X
cana-4142	168	6	,	,	PUNCT
cana-4142	168	7	𝔮	𝔮	PROPN
cana-4142	168	8	,	,	PUNCT
cana-4142	168	9	𝒶1	𝒶1	PROPN
cana-4142	168	10	𝓀	𝓀	X
cana-4142	168	11	)	)	PUNCT
cana-4142	168	12	<	<	X
cana-4142	169	1	휀	휀	X
cana-4142	169	2	.	.	PUNCT
cana-4142	169	3	therefore	therefore	ADV
cana-4142	169	4	,	,	PUNCT
cana-4142	169	5	we	we	PRON
cana-4142	169	6	can	can	AUX
cana-4142	169	7	find	find	VERB
cana-4142	169	8	𝒯	𝒯	PROPN
cana-4142	169	9	∈	∈	PROPN
cana-4142	169	10	{	{	PUNCT
cana-4142	169	11	1	1	NUM
cana-4142	169	12	,	,	PUNCT
cana-4142	169	13	2	2	NUM
cana-4142	169	14	,	,	PUNCT
cana-4142	169	15	…	…	PUNCT
cana-4142	169	16	,	,	PUNCT
cana-4142	169	17	𝓀	𝓀	X
cana-4142	169	18	}	}	PUNCT
cana-4142	169	19	and	and	CCONJ
cana-4142	169	20	𝒶	𝒶	PRON
cana-4142	169	21	∈	∈	PROPN
cana-4142	169	22	(	(	PUNCT
cana-4142	169	23	0	0	NUM
cana-4142	169	24	,	,	PUNCT
cana-4142	169	25	𝒶𝓀)such	𝒶𝓀)such	NOUN
cana-4142	169	26	that	that	PRON
cana-4142	169	27	휀0	휀0	NOUN
cana-4142	169	28	≔	≔	NOUN
cana-4142	169	29	𝔑(𝔭	𝔑(𝔭	NOUN
cana-4142	169	30	,	,	PUNCT
cana-4142	169	31	𝔮	𝔮	PROPN
cana-4142	169	32	,	,	PUNCT
cana-4142	169	33	𝒶1	𝒶1	NOUN
cana-4142	169	34	,	,	PUNCT
cana-4142	169	35	𝒶2	𝒶2	PROPN
cana-4142	169	36	,	,	PUNCT
cana-4142	169	37	.	.	PUNCT
cana-4142	169	38	.	.	PUNCT
cana-4142	170	1	,	,	PUNCT
cana-4142	170	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	170	3	,	,	PUNCT
cana-4142	170	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	170	5	,	,	PUNCT
cana-4142	170	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	170	7	,	,	PUNCT
cana-4142	170	8	.	.	PUNCT
cana-4142	170	9	.	.	PUNCT
cana-4142	171	1	,	,	PUNCT
cana-4142	171	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	171	3	)	)	PUNCT
cana-4142	171	4	<	<	X
cana-4142	172	1	휀	휀	X
cana-4142	172	2	.	.	PUNCT
cana-4142	172	3	then	then	ADV
cana-4142	172	4	,	,	PUNCT
cana-4142	172	5	we	we	PRON
cana-4142	172	6	can	can	AUX
cana-4142	172	7	find	find	VERB
cana-4142	172	8	𝛿	𝛿	DET
cana-4142	172	9	∈	∈	PROPN
cana-4142	172	10	(	(	PUNCT
cana-4142	172	11	0	0	NUM
cana-4142	172	12	,	,	PUNCT
cana-4142	172	13	1	1	NUM
cana-4142	172	14	)	)	PUNCT
cana-4142	172	15	such	such	ADJ
cana-4142	172	16	that	that	SCONJ
cana-4142	172	17	휀0	휀0	NOUN
cana-4142	172	18	<	<	X
cana-4142	172	19	𝛿	𝛿	X
cana-4142	172	20	<	<	X
cana-4142	172	21	휀	휀	NOUN
cana-4142	172	22	.	.	PUNCT
cana-4142	172	23	by	by	ADP
cana-4142	172	24	remark	remark	NOUN
cana-4142	172	25	2	2	NUM
cana-4142	172	26	,	,	PUNCT
cana-4142	172	27	there	there	PRON
cana-4142	172	28	is	be	VERB
cana-4142	172	29	휀1	휀1	NOUN
cana-4142	172	30	∈	∈	PROPN
cana-4142	172	31	(	(	PUNCT
cana-4142	172	32	0	0	NUM
cana-4142	172	33	,	,	PUNCT
cana-4142	172	34	1	1	NUM
cana-4142	172	35	)	)	PUNCT
cana-4142	172	36	such	such	ADJ
cana-4142	172	37	that	that	DET
cana-4142	172	38	휀0⨁휀1	휀0⨁휀1	NOUN
cana-4142	172	39	≤	≤	NOUN
cana-4142	172	40	𝛿.	𝛿.	ADV
cana-4142	173	1	now	now	ADV
cana-4142	173	2	,	,	PUNCT
cana-4142	173	3	we	we	PRON
cana-4142	173	4	will	will	AUX
cana-4142	173	5	claim	claim	VERB
cana-4142	173	6	that	that	SCONJ
cana-4142	173	7	𝔅(𝔭	𝔅(𝔭	NOUN
cana-4142	173	8	,	,	PUNCT
cana-4142	173	9	휀	휀	NOUN
cana-4142	173	10	;	;	PUNCT
cana-4142	173	11	𝒶1	𝒶1	NOUN
cana-4142	173	12	,	,	PUNCT
cana-4142	173	13	𝒶2	𝒶2	PROPN
cana-4142	173	14	,	,	PUNCT
cana-4142	173	15	.	.	PUNCT
cana-4142	173	16	.	.	PUNCT
cana-4142	173	17	.	.	PUNCT
cana-4142	174	1	,	,	PUNCT
cana-4142	174	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	174	3	)	)	PUNCT
cana-4142	174	4	⊆	⊆	NUM
cana-4142	174	5	𝔅(𝔮	𝔅(𝔮	NUM
cana-4142	174	6	,	,	PUNCT
cana-4142	174	7	1	1	NUM
cana-4142	174	8	−	−	NOUN
cana-4142	174	9	휀1	휀1	NOUN
cana-4142	174	10	;	;	PUNCT
cana-4142	174	11	𝒶1	𝒶1	NOUN
cana-4142	174	12	,	,	PUNCT
cana-4142	174	13	𝒶2	𝒶2	PROPN
cana-4142	174	14	,	,	PUNCT
cana-4142	174	15	.	.	PUNCT
cana-4142	174	16	.	.	PUNCT
cana-4142	175	1	,	,	PUNCT
cana-4142	175	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	175	3	,	,	PUNCT
cana-4142	175	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	175	5	,	,	PUNCT
cana-4142	175	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	175	7	,	,	PUNCT
cana-4142	175	8	.	.	PUNCT
cana-4142	175	9	.	.	PUNCT
cana-4142	176	1	,	,	PUNCT
cana-4142	176	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	176	3	)	)	PUNCT
cana-4142	176	4	.	.	PUNCT
cana-4142	177	1	assume	assume	VERB
cana-4142	177	2	that	that	SCONJ
cana-4142	177	3	𝔯	𝔯	PROPN
cana-4142	177	4	∈	∈	PROPN
cana-4142	177	5	𝔅(𝔮	𝔅(𝔮	PRON
cana-4142	177	6	,	,	PUNCT
cana-4142	177	7	1	1	NUM
cana-4142	177	8	−	−	NOUN
cana-4142	177	9	휀1	휀1	NOUN
cana-4142	177	10	;	;	PUNCT
cana-4142	177	11	𝒶1	𝒶1	NOUN
cana-4142	177	12	,	,	PUNCT
cana-4142	177	13	𝒶2	𝒶2	PROPN
cana-4142	177	14	,	,	PUNCT
cana-4142	177	15	.	.	PUNCT
cana-4142	177	16	.	.	PUNCT
cana-4142	178	1	,	,	PUNCT
cana-4142	178	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	178	3	,	,	PUNCT
cana-4142	178	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	178	5	,	,	PUNCT
cana-4142	178	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	178	7	,	,	PUNCT
cana-4142	178	8	.	.	PUNCT
cana-4142	178	9	.	.	PUNCT
cana-4142	179	1	,	,	PUNCT
cana-4142	179	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	179	3	)	)	PUNCT
cana-4142	179	4	.	.	PUNCT
cana-4142	180	1	then	then	ADV
cana-4142	180	2	,	,	PUNCT
cana-4142	180	3	𝔑(𝔮	𝔑(𝔮	PROPN
cana-4142	180	4	,	,	PUNCT
cana-4142	180	5	𝔯	𝔯	PROPN
cana-4142	180	6	,	,	PUNCT
cana-4142	180	7	𝒶1	𝒶1	NOUN
cana-4142	180	8	,	,	PUNCT
cana-4142	180	9	𝒶2	𝒶2	PROPN
cana-4142	180	10	,	,	PUNCT
cana-4142	180	11	.	.	PUNCT
cana-4142	180	12	.	.	PUNCT
cana-4142	181	1	,	,	PUNCT
cana-4142	181	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	181	3	,	,	PUNCT
cana-4142	181	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	181	5	,	,	PUNCT
cana-4142	181	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	181	7	,	,	PUNCT
cana-4142	181	8	.	.	PUNCT
cana-4142	181	9	.	.	PUNCT
cana-4142	182	1	,	,	PUNCT
cana-4142	182	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	182	3	)	)	PUNCT
cana-4142	182	4	<	<	X
cana-4142	182	5	1	1	NUM
cana-4142	182	6	−	−	NOUN
cana-4142	182	7	휀1	휀1	NOUN
cana-4142	182	8	.	.	PUNCT
cana-4142	183	1	communications	communication	NOUN
cana-4142	183	2	on	on	ADP
cana-4142	183	3	applied	apply	VERB
cana-4142	183	4	nonlinear	nonlinear	ADJ
cana-4142	183	5	analysis	analysis	NOUN
cana-4142	183	6	issn	issn	NOUN
cana-4142	183	7	:	:	PUNCT
cana-4142	183	8	1074	1074	NUM
cana-4142	183	9	-	-	PUNCT
cana-4142	183	10	133x	133x	NUM
cana-4142	183	11	vol	vol	NOUN
cana-4142	183	12	x	x	NOUN
cana-4142	183	13	no	no	INTJ
cana-4142	183	14	.	.	PUNCT
cana-4142	184	1	y	y	PROPN
cana-4142	184	2	(	(	PUNCT
cana-4142	184	3	2025	2025	NUM
cana-4142	184	4	)	)	PUNCT
cana-4142	184	5	1325	1325	NUM
cana-4142	184	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	184	7	𝔑(𝔭	𝔑(𝔭	X
cana-4142	184	8	,	,	PUNCT
cana-4142	184	9	𝔯	𝔯	PROPN
cana-4142	184	10	,	,	PUNCT
cana-4142	184	11	𝒶1	𝒶1	PROPN
cana-4142	184	12	𝓀	𝓀	PROPN
cana-4142	184	13	)	)	PUNCT
cana-4142	184	14	≤	≤	NOUN
cana-4142	184	15	𝔑(𝔭	𝔑(𝔭	PUNCT
cana-4142	184	16	,	,	PUNCT
cana-4142	184	17	𝔮	𝔮	PROPN
cana-4142	184	18	,	,	PUNCT
cana-4142	184	19	𝒶1	𝒶1	NOUN
cana-4142	184	20	,	,	PUNCT
cana-4142	184	21	𝒶2	𝒶2	PROPN
cana-4142	184	22	,	,	PUNCT
cana-4142	184	23	.	.	PUNCT
cana-4142	184	24	.	.	PUNCT
cana-4142	185	1	,	,	PUNCT
cana-4142	185	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	185	3	,	,	PUNCT
cana-4142	185	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	185	5	,	,	PUNCT
cana-4142	185	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	185	7	,	,	PUNCT
cana-4142	185	8	.	.	PUNCT
cana-4142	185	9	.	.	PUNCT
cana-4142	186	1	,	,	PUNCT
cana-4142	186	2	𝒶𝓀)⨁	𝒶𝓀)⨁	PROPN
cana-4142	186	3	𝔑(𝔮	𝔑(𝔮	PROPN
cana-4142	186	4	,	,	PUNCT
cana-4142	186	5	𝔯	𝔯	PROPN
cana-4142	186	6	,	,	PUNCT
cana-4142	186	7	𝒶1	𝒶1	NOUN
cana-4142	186	8	,	,	PUNCT
cana-4142	186	9	𝒶2	𝒶2	PROPN
cana-4142	186	10	,	,	PUNCT
cana-4142	186	11	.	.	PUNCT
cana-4142	186	12	.	.	PUNCT
cana-4142	187	1	,	,	PUNCT
cana-4142	187	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	187	3	,	,	PUNCT
cana-4142	187	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	187	5	,	,	PUNCT
cana-4142	187	6	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	187	7	,	,	PUNCT
cana-4142	187	8	.	.	PUNCT
cana-4142	187	9	.	.	PUNCT
cana-4142	188	1	,	,	PUNCT
cana-4142	188	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	188	3	)	)	PUNCT
cana-4142	188	4	≤	≤	PUNCT
cana-4142	188	5	휀0⨁휀1	휀0⨁휀1	ADJ
cana-4142	188	6	≤	≤	ADJ
cana-4142	188	7	𝛿	𝛿	PRON
cana-4142	188	8	<	<	X
cana-4142	188	9	휀	휀	NOUN
cana-4142	188	10	,	,	PUNCT
cana-4142	188	11	which	which	PRON
cana-4142	188	12	proves	prove	VERB
cana-4142	188	13	the	the	DET
cana-4142	188	14	result	result	NOUN
cana-4142	188	15	.	.	PUNCT
cana-4142	189	1	from	from	ADP
cana-4142	189	2	the	the	DET
cana-4142	189	3	above	above	ADJ
cana-4142	189	4	theorem	theorem	NOUN
cana-4142	189	5	,	,	PUNCT
cana-4142	189	6	we	we	PRON
cana-4142	189	7	can	can	AUX
cana-4142	189	8	directly	directly	ADV
cana-4142	189	9	get	get	VERB
cana-4142	189	10	the	the	DET
cana-4142	189	11	following	follow	VERB
cana-4142	189	12	result	result	NOUN
cana-4142	189	13	:	:	PUNCT
cana-4142	189	14	theorem	theorem	VERB
cana-4142	189	15	18	18	NUM
cana-4142	189	16	let	let	NOUN
cana-4142	189	17	(	(	PUNCT
cana-4142	189	18	𝔐	𝔐	X
cana-4142	189	19	,	,	PUNCT
cana-4142	189	20	𝔑	𝔑	PROPN
cana-4142	189	21	,	,	PUNCT
cana-4142	189	22	⨁	⨁	PROPN
cana-4142	189	23	)	)	PUNCT
cana-4142	189	24	be	be	VERB
cana-4142	189	25	a	a	DET
cana-4142	189	26	revised	revise	VERB
cana-4142	189	27	fuzzy	fuzzy	ADJ
cana-4142	189	28	𝓀	𝓀	X
cana-4142	189	29	−metric	−metric	ADJ
cana-4142	189	30	space	space	NOUN
cana-4142	189	31	and	and	CCONJ
cana-4142	189	32	𝜏	𝜏	NOUN
cana-4142	189	33	=	=	PUNCT
cana-4142	189	34	{	{	PUNCT
cana-4142	189	35	𝒳	𝒳	PROPN
cana-4142	189	36	⊆	⊆	NUM
cana-4142	189	37	𝔐	𝔐	NOUN
cana-4142	189	38	∶	∶	NOUN
cana-4142	189	39	𝒶	𝒶	X
cana-4142	189	40	∈	∈	NOUN
cana-4142	189	41	𝔐	𝔐	NOUN
cana-4142	190	1	if	if	SCONJ
cana-4142	191	1	and	and	CCONJ
cana-4142	191	2	only	only	ADV
cana-4142	191	3	if	if	SCONJ
cana-4142	191	4	there	there	PRON
cana-4142	191	5	exist	exist	VERB
cana-4142	191	6	𝒶1	𝒶1	NOUN
cana-4142	191	7	,	,	PUNCT
cana-4142	191	8	𝒶2	𝒶2	PROPN
cana-4142	191	9	,	,	PUNCT
cana-4142	191	10	.	.	PUNCT
cana-4142	191	11	.	.	PUNCT
cana-4142	192	1	.	.	PUNCT
cana-4142	193	1	,	,	PUNCT
cana-4142	193	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	193	3	>	>	X
cana-4142	193	4	0	0	PUNCT
cana-4142	194	1	and	and	CCONJ
cana-4142	194	2	휀	휀	PRON
cana-4142	194	3	∈	∈	PROPN
cana-4142	194	4	(	(	PUNCT
cana-4142	194	5	0,1	0,1	NOUN
cana-4142	194	6	)	)	PUNCT
cana-4142	194	7	such	such	ADJ
cana-4142	194	8	that	that	SCONJ
cana-4142	194	9	𝔅(𝔭	𝔅(𝔭	NOUN
cana-4142	194	10	,	,	PUNCT
cana-4142	194	11	휀	휀	NOUN
cana-4142	194	12	;	;	PUNCT
cana-4142	194	13	𝒶1	𝒶1	NOUN
cana-4142	194	14	,	,	PUNCT
cana-4142	194	15	𝒶2	𝒶2	PROPN
cana-4142	194	16	,	,	PUNCT
cana-4142	194	17	.	.	PUNCT
cana-4142	194	18	.	.	PUNCT
cana-4142	195	1	.	.	PUNCT
cana-4142	196	1	,	,	PUNCT
cana-4142	196	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	196	3	)	)	PUNCT
cana-4142	196	4	⊆	⊆	NUM
cana-4142	196	5	𝒳	𝒳	PROPN
cana-4142	196	6	}	}	PUNCT
cana-4142	196	7	.	.	PUNCT
cana-4142	197	1	then	then	ADV
cana-4142	197	2	,	,	PUNCT
cana-4142	197	3	𝜏	𝜏	PROPN
cana-4142	197	4	is	be	AUX
cana-4142	197	5	a	a	DET
cana-4142	197	6	topology	topology	NOUN
cana-4142	197	7	on	on	ADP
cana-4142	197	8	𝔐.	𝔐.	PROPN
cana-4142	197	9	remark	remark	NOUN
cana-4142	197	10	19	19	NUM
cana-4142	197	11	let	let	VERB
cana-4142	197	12	(	(	PUNCT
cana-4142	197	13	𝔐	𝔐	X
cana-4142	197	14	,	,	PUNCT
cana-4142	197	15	𝔑	𝔑	PROPN
cana-4142	197	16	,	,	PUNCT
cana-4142	197	17	⨁	⨁	PROPN
cana-4142	197	18	)	)	PUNCT
cana-4142	197	19	be	be	VERB
cana-4142	197	20	a	a	DET
cana-4142	197	21	revised	revise	VERB
cana-4142	197	22	fuzzy	fuzzy	ADJ
cana-4142	197	23	𝓀	𝓀	X
cana-4142	197	24	−metric	−metric	ADJ
cana-4142	197	25	space	space	NOUN
cana-4142	197	26	and	and	CCONJ
cana-4142	197	27	𝒶	𝒶	PRON
cana-4142	197	28	∈	∈	PROPN
cana-4142	197	29	𝔐.	𝔐.	PROPN
cana-4142	197	30	since	since	SCONJ
cana-4142	197	31	𝔅𝔭	𝔅𝔭	PROPN
cana-4142	197	32	=	=	PUNCT
cana-4142	197	33	{	{	PUNCT
cana-4142	197	34	𝔅	𝔅	PROPN
cana-4142	197	35	(	(	PUNCT
cana-4142	197	36	𝔭	𝔭	NOUN
cana-4142	197	37	,	,	PUNCT
cana-4142	197	38	1	1	NUM
cana-4142	197	39	𝑛	𝑛	NOUN
cana-4142	197	40	;	;	PUNCT
cana-4142	197	41	𝒶1	𝒶1	NOUN
cana-4142	197	42	,	,	PUNCT
cana-4142	197	43	𝒶2	𝒶2	PROPN
cana-4142	197	44	,	,	PUNCT
cana-4142	197	45	.	.	PUNCT
cana-4142	197	46	.	.	PUNCT
cana-4142	198	1	.	.	PUNCT
cana-4142	199	1	,	,	PUNCT
cana-4142	199	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	199	3	)	)	PUNCT
cana-4142	199	4	:	:	PUNCT
cana-4142	200	1	𝓃	𝓃	X
cana-4142	200	2	∈	∈	PROPN
cana-4142	200	3	𝒩	𝒩	PROPN
cana-4142	200	4	}	}	PUNCT
cana-4142	200	5	where	where	SCONJ
cana-4142	200	6	𝒶1	𝒶1	NOUN
cana-4142	200	7	=	=	SYM
cana-4142	200	8	𝒶2	𝒶2	PROPN
cana-4142	200	9	=	=	X
cana-4142	200	10	.	.	PUNCT
cana-4142	200	11	.	.	PUNCT
cana-4142	200	12	.	.	PUNCT
cana-4142	201	1	=	=	PUNCT
cana-4142	202	1	𝒶𝓀	𝒶𝓀	NOUN
cana-4142	202	2	=	=	SYM
cana-4142	202	3	1	1	NUM
cana-4142	202	4	𝓃	𝓃	NOUN
cana-4142	202	5	,	,	PUNCT
cana-4142	202	6	is	be	AUX
cana-4142	202	7	a	a	DET
cana-4142	202	8	local	local	ADJ
cana-4142	202	9	base	base	NOUN
cana-4142	202	10	at	at	ADP
cana-4142	202	11	a	a	DET
cana-4142	202	12	point	point	NOUN
cana-4142	202	13	𝒶	𝒶	NOUN
cana-4142	202	14	,	,	PUNCT
cana-4142	202	15	the	the	DET
cana-4142	202	16	topology	topology	NOUN
cana-4142	202	17	𝜏	𝜏	NOUN
cana-4142	202	18	given	give	VERB
cana-4142	202	19	in	in	ADP
cana-4142	202	20	theorem	theorem	ADJ
cana-4142	202	21	18	18	NUM
cana-4142	202	22	is	be	AUX
cana-4142	202	23	first	first	ADV
cana-4142	202	24	countable	countable	ADJ
cana-4142	202	25	.	.	PUNCT
cana-4142	203	1	theorem	theorem	VERB
cana-4142	203	2	20	20	NUM
cana-4142	203	3	every	every	DET
cana-4142	203	4	revised	revise	VERB
cana-4142	203	5	fuzzy	fuzzy	ADJ
cana-4142	203	6	𝓀	𝓀	X
cana-4142	203	7	−metric	−metric	ADJ
cana-4142	203	8	space	space	NOUN
cana-4142	203	9	is	be	AUX
cana-4142	203	10	hausdorff	hausdorff	NOUN
cana-4142	203	11	.	.	PUNCT
cana-4142	204	1	definition	definition	NOUN
cana-4142	204	2	21	21	NUM
cana-4142	204	3	let	let	VERB
cana-4142	204	4	(	(	PUNCT
cana-4142	204	5	𝔐	𝔐	X
cana-4142	204	6	,	,	PUNCT
cana-4142	204	7	𝔑	𝔑	PROPN
cana-4142	204	8	,	,	PUNCT
cana-4142	204	9	⨁	⨁	PROPN
cana-4142	204	10	)	)	PUNCT
cana-4142	204	11	be	be	VERB
cana-4142	204	12	a	a	DET
cana-4142	204	13	revised	revise	VERB
cana-4142	204	14	fuzzy	fuzzy	ADJ
cana-4142	204	15	𝓀	𝓀	X
cana-4142	204	16	−metric	−metric	ADJ
cana-4142	204	17	space	space	NOUN
cana-4142	204	18	.	.	PUNCT
cana-4142	205	1	a	a	DET
cana-4142	205	2	sequence	sequence	NOUN
cana-4142	205	3	{	{	PUNCT
cana-4142	205	4	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	205	5	}	}	PUNCT
cana-4142	205	6	in	in	ADP
cana-4142	205	7	𝔐	𝔐	PROPN
cana-4142	205	8	is	be	AUX
cana-4142	205	9	said	say	VERB
cana-4142	205	10	to	to	PART
cana-4142	205	11	be	be	AUX
cana-4142	205	12	convergent	convergent	ADJ
cana-4142	205	13	and	and	CCONJ
cana-4142	205	14	converges	converge	NOUN
cana-4142	205	15	to	to	ADP
cana-4142	205	16	𝔭	𝔭	SYM
cana-4142	205	17	∈	∈	NOUN
cana-4142	205	18	𝔐	𝔐	NOUN
cana-4142	205	19	if	if	SCONJ
cana-4142	205	20	and	and	CCONJ
cana-4142	205	21	only	only	ADV
cana-4142	205	22	if	if	SCONJ
cana-4142	205	23	for	for	ADP
cana-4142	205	24	every	every	DET
cana-4142	205	25	real	real	ADJ
cana-4142	205	26	𝜖	𝜖	PROPN
cana-4142	205	27	∈	∈	PROPN
cana-4142	205	28	(	(	PUNCT
cana-4142	205	29	0	0	NUM
cana-4142	205	30	,	,	PUNCT
cana-4142	205	31	1	1	NUM
cana-4142	205	32	)	)	PUNCT
cana-4142	205	33	,	,	PUNCT
cana-4142	205	34	there	there	PRON
cana-4142	205	35	exists	exist	VERB
cana-4142	205	36	𝑛0	𝑛0	VERB
cana-4142	205	37	∈	∈	PROPN
cana-4142	205	38	𝒩	𝒩	NOUN
cana-4142	205	39	such	such	ADJ
cana-4142	205	40	that	that	DET
cana-4142	205	41	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	205	42	,	,	PUNCT
cana-4142	205	43	𝔭	𝔭	NOUN
cana-4142	205	44	,	,	PUNCT
cana-4142	205	45	𝒶1	𝒶1	NOUN
cana-4142	205	46	𝓀	𝓀	X
cana-4142	205	47	)	)	PUNCT
cana-4142	205	48	<	<	X
cana-4142	205	49	𝜖	𝜖	PROPN
cana-4142	205	50	for	for	ADP
cana-4142	205	51	all	all	DET
cana-4142	205	52	𝑛	𝑛	PRON
cana-4142	205	53	∈	∈	NOUN
cana-4142	205	54	𝑛0	𝑛0	VERB
cana-4142	205	55	and	and	CCONJ
cana-4142	205	56	𝒶1	𝒶1	NOUN
cana-4142	205	57	,	,	PUNCT
cana-4142	205	58	𝒶2	𝒶2	PROPN
cana-4142	205	59	,	,	PUNCT
cana-4142	205	60	.	.	PUNCT
cana-4142	205	61	.	.	PUNCT
cana-4142	205	62	.	.	PUNCT
cana-4142	206	1	,	,	PUNCT
cana-4142	206	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	206	3	>	>	X
cana-4142	206	4	0	0	X
cana-4142	206	5	.	.	PUNCT
cana-4142	207	1	the	the	DET
cana-4142	207	2	proof	proof	NOUN
cana-4142	207	3	of	of	ADP
cana-4142	207	4	the	the	DET
cana-4142	207	5	following	follow	VERB
cana-4142	207	6	lemma	lemma	PROPN
cana-4142	207	7	is	be	AUX
cana-4142	207	8	straightforward	straightforward	ADJ
cana-4142	207	9	,	,	PUNCT
cana-4142	207	10	so	so	SCONJ
cana-4142	207	11	we	we	PRON
cana-4142	207	12	will	will	AUX
cana-4142	207	13	omit	omit	VERB
cana-4142	207	14	the	the	DET
cana-4142	207	15	proof	proof	NOUN
cana-4142	207	16	.	.	PUNCT
cana-4142	208	1	lemma	lemma	PROPN
cana-4142	208	2	22	22	NUM
cana-4142	208	3	let	let	VERB
cana-4142	208	4	(	(	PUNCT
cana-4142	208	5	𝔐	𝔐	X
cana-4142	208	6	,	,	PUNCT
cana-4142	208	7	𝔑	𝔑	PROPN
cana-4142	208	8	,	,	PUNCT
cana-4142	208	9	⨁	⨁	PROPN
cana-4142	208	10	)	)	PUNCT
cana-4142	208	11	be	be	VERB
cana-4142	208	12	a	a	DET
cana-4142	208	13	revised	revise	VERB
cana-4142	208	14	fuzzy	fuzzy	ADJ
cana-4142	208	15	𝓀	𝓀	X
cana-4142	208	16	−metric	−metric	ADJ
cana-4142	208	17	space	space	NOUN
cana-4142	208	18	.	.	PUNCT
cana-4142	209	1	a	a	DET
cana-4142	209	2	sequence	sequence	NOUN
cana-4142	209	3	{	{	PUNCT
cana-4142	209	4	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	209	5	}	}	PUNCT
cana-4142	209	6	in	in	ADP
cana-4142	209	7	𝔐	𝔐	PROPN
cana-4142	209	8	converges	converge	VERB
cana-4142	209	9	to	to	ADP
cana-4142	209	10	𝔭	𝔭	SYM
cana-4142	209	11	∈	∈	NOUN
cana-4142	209	12	𝔐	𝔐	NOUN
cana-4142	209	13	if	if	SCONJ
cana-4142	209	14	and	and	CCONJ
cana-4142	209	15	only	only	ADV
cana-4142	209	16	if	if	SCONJ
cana-4142	209	17	lim	lim	PROPN
cana-4142	209	18	𝑛→+∞	𝑛→+∞	PROPN
cana-4142	209	19	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	209	20	,	,	PUNCT
cana-4142	209	21	𝔭	𝔭	NOUN
cana-4142	209	22	,	,	PUNCT
cana-4142	209	23	𝒶1	𝒶1	NOUN
cana-4142	209	24	𝓀	𝓀	PRON
cana-4142	209	25	)	)	PUNCT
cana-4142	209	26	=	=	SYM
cana-4142	209	27	0	0	NUM
cana-4142	209	28	for	for	ADP
cana-4142	209	29	all	all	DET
cana-4142	209	30	𝒶1	𝒶1	NOUN
cana-4142	209	31	,	,	PUNCT
cana-4142	209	32	𝒶2	𝒶2	PROPN
cana-4142	209	33	,	,	PUNCT
cana-4142	209	34	.	.	PUNCT
cana-4142	209	35	.	.	PUNCT
cana-4142	209	36	.	.	PUNCT
cana-4142	210	1	,	,	PUNCT
cana-4142	210	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	210	3	>	>	X
cana-4142	210	4	0	0	X
cana-4142	210	5	.	.	PUNCT
cana-4142	211	1	definition	definition	NOUN
cana-4142	211	2	23	23	NUM
cana-4142	211	3	let	let	VERB
cana-4142	211	4	(	(	PUNCT
cana-4142	211	5	𝔐	𝔐	X
cana-4142	211	6	,	,	PUNCT
cana-4142	211	7	𝔑	𝔑	PROPN
cana-4142	211	8	,	,	PUNCT
cana-4142	211	9	⨁	⨁	PROPN
cana-4142	211	10	)	)	PUNCT
cana-4142	211	11	be	be	VERB
cana-4142	211	12	a	a	DET
cana-4142	211	13	revised	revise	VERB
cana-4142	211	14	fuzzy	fuzzy	ADJ
cana-4142	211	15	𝓀	𝓀	X
cana-4142	211	16	−metric	−metric	ADJ
cana-4142	211	17	space	space	NOUN
cana-4142	211	18	and	and	CCONJ
cana-4142	211	19	{	{	PUNCT
cana-4142	211	20	𝔭𝓃	𝔭𝓃	AUX
cana-4142	211	21	}	}	PUNCT
cana-4142	211	22	be	be	AUX
cana-4142	211	23	a	a	DET
cana-4142	211	24	sequence	sequence	NOUN
cana-4142	211	25	in	in	ADP
cana-4142	211	26	𝔐.	𝔐.	PROPN
cana-4142	211	27	1	1	PROPN
cana-4142	211	28	.	.	PUNCT
cana-4142	212	1	{	{	PUNCT
cana-4142	212	2	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	212	3	}	}	PUNCT
cana-4142	212	4	is	be	AUX
cana-4142	212	5	called	call	VERB
cana-4142	212	6	an	an	DET
cana-4142	212	7	𝔑	𝔑	PROPN
cana-4142	212	8	−cauchy	−cauchy	ADJ
cana-4142	212	9	sequence	sequence	NOUN
cana-4142	212	10	if	if	SCONJ
cana-4142	212	11	for	for	ADP
cana-4142	212	12	every	every	DET
cana-4142	212	13	𝜖	𝜖	PROPN
cana-4142	212	14	∈	∈	PROPN
cana-4142	212	15	(	(	PUNCT
cana-4142	212	16	0	0	NUM
cana-4142	212	17	,	,	PUNCT
cana-4142	212	18	1	1	NUM
cana-4142	212	19	)	)	PUNCT
cana-4142	212	20	,	,	PUNCT
cana-4142	212	21	there	there	PRON
cana-4142	212	22	exists	exist	VERB
cana-4142	212	23	𝑛0	𝑛0	VERB
cana-4142	212	24	∈	∈	PROPN
cana-4142	212	25	𝒩	𝒩	NOUN
cana-4142	212	26	such	such	ADJ
cana-4142	212	27	that	that	DET
cana-4142	212	28	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	212	29	,	,	PUNCT
cana-4142	212	30	𝔭𝑚	𝔭𝑚	PROPN
cana-4142	212	31	,	,	PUNCT
cana-4142	212	32	𝒶1	𝒶1	PROPN
cana-4142	212	33	𝓀	𝓀	PRON
cana-4142	212	34	)	)	PUNCT
cana-4142	212	35	<	<	X
cana-4142	212	36	𝜖	𝜖	PROPN
cana-4142	212	37	for	for	ADP
cana-4142	212	38	all	all	DET
cana-4142	212	39	𝑛	𝑛	PROPN
cana-4142	212	40	,	,	PUNCT
cana-4142	212	41	𝑚	𝑚	X
cana-4142	212	42	>	>	X
cana-4142	212	43	𝑛0	𝑛0	VERB
cana-4142	212	44	and	and	CCONJ
cana-4142	212	45	𝒶1	𝒶1	NOUN
cana-4142	212	46	,	,	PUNCT
cana-4142	212	47	𝒶2	𝒶2	PROPN
cana-4142	212	48	,	,	PUNCT
cana-4142	212	49	.	.	PUNCT
cana-4142	212	50	.	.	PUNCT
cana-4142	213	1	.	.	PUNCT
cana-4142	214	1	,	,	PUNCT
cana-4142	214	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	214	3	>	>	X
cana-4142	214	4	0	0	X
cana-4142	214	5	.	.	NOUN
cana-4142	215	1	2	2	NUM
cana-4142	215	2	.	.	PUNCT
cana-4142	215	3	{	{	PUNCT
cana-4142	215	4	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	215	5	}	}	PUNCT
cana-4142	215	6	is	be	AUX
cana-4142	215	7	called	call	VERB
cana-4142	215	8	a	a	DET
cana-4142	215	9	𝔾	𝔾	ADJ
cana-4142	215	10	−cauchy	−cauchy	ADJ
cana-4142	215	11	sequence	sequence	NOUN
cana-4142	215	12	if	if	SCONJ
cana-4142	215	13	lim	lim	PROPN
cana-4142	215	14	𝑛→+∞	𝑛→+∞	PROPN
cana-4142	215	15	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	215	16	,	,	PUNCT
cana-4142	215	17	𝔭𝓃+𝓍	𝔭𝓃+𝓍	NUM
cana-4142	215	18	,	,	PUNCT
cana-4142	215	19	𝒶1	𝒶1	NOUN
cana-4142	215	20	𝓀	𝓀	X
cana-4142	215	21	)	)	PUNCT
cana-4142	215	22	=	=	SYM
cana-4142	215	23	0	0	NUM
cana-4142	215	24	for	for	ADP
cana-4142	215	25	all	all	DET
cana-4142	215	26	𝒶1	𝒶1	NOUN
cana-4142	215	27	,	,	PUNCT
cana-4142	215	28	𝒶2	𝒶2	PROPN
cana-4142	215	29	,	,	PUNCT
cana-4142	215	30	.	.	PUNCT
cana-4142	215	31	.	.	PUNCT
cana-4142	216	1	.	.	PUNCT
cana-4142	217	1	,	,	PUNCT
cana-4142	217	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	217	3	>	>	X
cana-4142	217	4	0	0	PUNCT
cana-4142	217	5	and	and	CCONJ
cana-4142	217	6	𝓍	𝓍	ADJ
cana-4142	217	7	>	>	X
cana-4142	217	8	0	0	X
cana-4142	217	9	.	.	PUNCT
cana-4142	217	10	note	note	VERB
cana-4142	217	11	that	that	SCONJ
cana-4142	217	12	the	the	DET
cana-4142	217	13	above	above	ADJ
cana-4142	217	14	definitions	definition	NOUN
cana-4142	217	15	of	of	ADP
cana-4142	217	16	cauchy	cauchy	ADJ
cana-4142	217	17	sequences	sequence	NOUN
cana-4142	217	18	are	be	AUX
cana-4142	217	19	different	different	ADJ
cana-4142	217	20	(	(	PUNCT
cana-4142	217	21	for	for	ADP
cana-4142	217	22	the	the	DET
cana-4142	217	23	case	case	NOUN
cana-4142	218	1	𝓀	𝓀	X
cana-4142	218	2	=	=	NOUN
cana-4142	218	3	1	1	X
cana-4142	218	4	.	.	PUNCT
cana-4142	218	5	definition	definition	NOUN
cana-4142	218	6	24	24	NUM
cana-4142	218	7	let	let	VERB
cana-4142	218	8	(	(	PUNCT
cana-4142	218	9	𝔐	𝔐	X
cana-4142	218	10	,	,	PUNCT
cana-4142	218	11	𝔑	𝔑	PROPN
cana-4142	218	12	,	,	PUNCT
cana-4142	218	13	⨁	⨁	PROPN
cana-4142	218	14	)	)	PUNCT
cana-4142	218	15	be	be	VERB
cana-4142	218	16	a	a	DET
cana-4142	218	17	revised	revise	VERB
cana-4142	218	18	fuzzy	fuzzy	ADJ
cana-4142	218	19	𝓀	𝓀	X
cana-4142	218	20	−metric	−metric	ADJ
cana-4142	218	21	space	space	NOUN
cana-4142	218	22	.	.	PUNCT
cana-4142	219	1	communications	communication	NOUN
cana-4142	219	2	on	on	ADP
cana-4142	219	3	applied	apply	VERB
cana-4142	219	4	nonlinear	nonlinear	ADJ
cana-4142	219	5	analysis	analysis	NOUN
cana-4142	219	6	issn	issn	NOUN
cana-4142	219	7	:	:	PUNCT
cana-4142	219	8	1074	1074	NUM
cana-4142	219	9	-	-	PUNCT
cana-4142	219	10	133x	133x	NUM
cana-4142	219	11	vol	vol	NOUN
cana-4142	219	12	x	x	NOUN
cana-4142	219	13	no	no	INTJ
cana-4142	219	14	.	.	PUNCT
cana-4142	220	1	y	y	PROPN
cana-4142	220	2	(	(	PUNCT
cana-4142	220	3	2025	2025	NUM
cana-4142	220	4	)	)	PUNCT
cana-4142	220	5	1326	1326	NUM
cana-4142	220	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	220	7	1	1	NUM
cana-4142	220	8	.	.	PUNCT
cana-4142	220	9	(	(	PUNCT
cana-4142	220	10	𝔐	𝔐	PROPN
cana-4142	220	11	,	,	PUNCT
cana-4142	220	12	𝔑	𝔑	PROPN
cana-4142	220	13	,	,	PUNCT
cana-4142	220	14	⨁	⨁	PROPN
cana-4142	220	15	)	)	PUNCT
cana-4142	220	16	is	be	AUX
cana-4142	220	17	said	say	VERB
cana-4142	220	18	to	to	PART
cana-4142	220	19	be	be	AUX
cana-4142	220	20	𝔑	𝔑	PROPN
cana-4142	220	21	−complete	−complete	PROPN
cana-4142	220	22	if	if	SCONJ
cana-4142	220	23	every	every	DET
cana-4142	220	24	𝔑	𝔑	PROPN
cana-4142	220	25	−cauchy	−cauchy	ADJ
cana-4142	220	26	sequence	sequence	NOUN
cana-4142	220	27	in	in	ADP
cana-4142	220	28	𝔐	𝔐	PROPN
cana-4142	220	29	converges	converge	VERB
cana-4142	220	30	to	to	ADP
cana-4142	220	31	some	some	PRON
cana-4142	220	32	𝔭	𝔭	ADP
cana-4142	220	33	∈	∈	PROPN
cana-4142	220	34	𝔐.	𝔐.	PROPN
cana-4142	220	35	2	2	NUM
cana-4142	220	36	.	.	PUNCT
cana-4142	221	1	(	(	PUNCT
cana-4142	221	2	𝔐	𝔐	PROPN
cana-4142	221	3	,	,	PUNCT
cana-4142	221	4	𝔑	𝔑	PROPN
cana-4142	221	5	,	,	PUNCT
cana-4142	221	6	⨁	⨁	PROPN
cana-4142	221	7	)	)	PUNCT
cana-4142	221	8	is	be	AUX
cana-4142	221	9	said	say	VERB
cana-4142	221	10	to	to	PART
cana-4142	221	11	be	be	AUX
cana-4142	221	12	𝔾	𝔾	PROPN
cana-4142	221	13	−complete	−complete	NOUN
cana-4142	221	14	if	if	SCONJ
cana-4142	221	15	every	every	DET
cana-4142	221	16	g	g	NOUN
cana-4142	221	17	-	-	PUNCT
cana-4142	221	18	cauchy	cauchy	ADJ
cana-4142	221	19	sequence	sequence	NOUN
cana-4142	221	20	in	in	ADP
cana-4142	221	21	𝔐	𝔐	PROPN
cana-4142	221	22	converges	converge	VERB
cana-4142	221	23	to	to	ADP
cana-4142	221	24	some	some	PRON
cana-4142	221	25	𝔭	𝔭	ADP
cana-4142	221	26	∈	∈	PROPN
cana-4142	221	27	𝔐.	𝔐.	PROPN
cana-4142	221	28	4	4	NUM
cana-4142	221	29	.	.	PUNCT
cana-4142	221	30	fixed	fix	VERB
cana-4142	221	31	point	point	NOUN
cana-4142	221	32	theorems	theorem	NOUN
cana-4142	221	33	in	in	ADP
cana-4142	221	34	this	this	DET
cana-4142	221	35	section	section	NOUN
cana-4142	221	36	,	,	PUNCT
cana-4142	221	37	we	we	PRON
cana-4142	221	38	prove	prove	VERB
cana-4142	221	39	many	many	ADJ
cana-4142	221	40	fixed	fix	VERB
cana-4142	221	41	-	-	PUNCT
cana-4142	221	42	point	point	NOUN
cana-4142	221	43	results	result	NOUN
cana-4142	221	44	in	in	ADP
cana-4142	221	45	revised	revise	VERB
cana-4142	221	46	fuzzy	fuzzy	ADJ
cana-4142	221	47	𝓀	𝓀	X
cana-4142	221	48	−metric	−metric	ADJ
cana-4142	221	49	space	space	NOUN
cana-4142	221	50	.	.	PUNCT
cana-4142	222	1	for	for	ADP
cana-4142	222	2	simplicity	simplicity	NOUN
cana-4142	222	3	,	,	PUNCT
cana-4142	222	4	for	for	ADP
cana-4142	222	5	a	a	DET
cana-4142	222	6	given	give	VERB
cana-4142	222	7	revised	revise	VERB
cana-4142	222	8	fuzzy	fuzzy	ADJ
cana-4142	222	9	𝓀	𝓀	PROPN
cana-4142	222	10	−metric	−metric	PROPN
cana-4142	222	11	space(𝔐	space(𝔐	PROPN
cana-4142	222	12	,	,	PUNCT
cana-4142	222	13	𝔑	𝔑	PROPN
cana-4142	222	14	,	,	PUNCT
cana-4142	222	15	⨁	⨁	PROPN
cana-4142	222	16	)	)	PUNCT
cana-4142	222	17	,	,	PUNCT
cana-4142	222	18	𝒯	𝒯	PROPN
cana-4142	222	19	∈	∈	PROPN
cana-4142	222	20	{	{	PUNCT
cana-4142	222	21	1	1	NUM
cana-4142	222	22	,	,	PUNCT
cana-4142	222	23	2	2	NUM
cana-4142	222	24	,	,	PUNCT
cana-4142	222	25	…	…	PUNCT
cana-4142	222	26	,	,	PUNCT
cana-4142	222	27	𝓀	𝓀	X
cana-4142	222	28	}	}	PUNCT
cana-4142	222	29	,	,	PUNCT
cana-4142	222	30	𝒷	𝒷	X
cana-4142	222	31	>	>	X
cana-4142	222	32	0	0	NUM
cana-4142	222	33	,	,	PUNCT
cana-4142	222	34	𝔭	𝔭	NOUN
cana-4142	222	35	,	,	PUNCT
cana-4142	222	36	𝔮	𝔮	PROPN
cana-4142	222	37	∈	∈	NOUN
cana-4142	222	38	𝔐	𝔐	PROPN
cana-4142	222	39	and	and	CCONJ
cana-4142	222	40	𝒶1	𝒶1	NOUN
cana-4142	222	41	,	,	PUNCT
cana-4142	222	42	𝒶2	𝒶2	PROPN
cana-4142	222	43	,	,	PUNCT
cana-4142	222	44	.	.	PUNCT
cana-4142	222	45	.	.	PUNCT
cana-4142	222	46	.	.	PUNCT
cana-4142	223	1	,	,	PUNCT
cana-4142	223	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	223	3	>	>	X
cana-4142	223	4	0	0	NUM
cana-4142	223	5	,	,	PUNCT
cana-4142	223	6	we	we	PRON
cana-4142	223	7	write	write	VERB
cana-4142	223	8	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	223	9	𝒷(𝔭	𝒷(𝔭	PROPN
cana-4142	223	10	,	,	PUNCT
cana-4142	223	11	𝔮	𝔮	PROPN
cana-4142	223	12	,	,	PUNCT
cana-4142	223	13	𝒶1	𝒶1	PROPN
cana-4142	223	14	𝓀	𝓀	PROPN
cana-4142	223	15	)	)	PUNCT
cana-4142	223	16	instead	instead	ADV
cana-4142	223	17	𝔑	𝔑	PROPN
cana-4142	223	18	(	(	PUNCT
cana-4142	223	19	𝔭	𝔭	PROPN
cana-4142	223	20	,	,	PUNCT
cana-4142	223	21	𝔮	𝔮	NOUN
cana-4142	223	22	,	,	PUNCT
cana-4142	223	23	𝒶1	𝒶1	NOUN
cana-4142	223	24	,	,	PUNCT
cana-4142	223	25	𝒶2	𝒶2	PROPN
cana-4142	223	26	,	,	PUNCT
cana-4142	223	27	.	.	PUNCT
cana-4142	223	28	.	.	PUNCT
cana-4142	224	1	,	,	PUNCT
cana-4142	224	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	224	3	,	,	PUNCT
cana-4142	224	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	224	5	𝒷	𝒷	PROPN
cana-4142	224	6	,	,	PUNCT
cana-4142	224	7	𝒶𝒯+1	𝒶𝒯+1	X
cana-4142	224	8	,	,	PUNCT
cana-4142	224	9	.	.	PUNCT
cana-4142	224	10	.	.	PUNCT
cana-4142	225	1	,	,	PUNCT
cana-4142	225	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	225	3	)	)	PUNCT
cana-4142	225	4	.	.	PUNCT
cana-4142	226	1	theorem	theorem	ADJ
cana-4142	226	2	26	26	NUM
cana-4142	226	3	let	let	VERB
cana-4142	226	4	(	(	PUNCT
cana-4142	226	5	𝔐	𝔐	X
cana-4142	226	6	,	,	PUNCT
cana-4142	226	7	𝔑	𝔑	PROPN
cana-4142	226	8	,	,	PUNCT
cana-4142	226	9	⨁	⨁	PROPN
cana-4142	226	10	)	)	PUNCT
cana-4142	226	11	be	be	VERB
cana-4142	226	12	a	a	DET
cana-4142	226	13	𝔾	𝔾	PROPN
cana-4142	226	14	−complete	−complete	PROPN
cana-4142	226	15	revised	revise	VERB
cana-4142	226	16	fuzzy	fuzzy	ADJ
cana-4142	226	17	𝓀	𝓀	X
cana-4142	226	18	−metric	−metric	ADJ
cana-4142	226	19	space	space	NOUN
cana-4142	226	20	and	and	CCONJ
cana-4142	226	21	𝔗	𝔗	NOUN
cana-4142	226	22	:	:	PUNCT
cana-4142	226	23	𝔐	𝔐	PROPN
cana-4142	226	24	→	→	SYM
cana-4142	226	25	𝔐	𝔐	PRON
cana-4142	226	26	be	be	AUX
cana-4142	226	27	a	a	DET
cana-4142	226	28	mapping	mapping	NOUN
cana-4142	226	29	satisfying	satisfy	VERB
cana-4142	226	30	the	the	DET
cana-4142	226	31	following	follow	VERB
cana-4142	226	32	condition	condition	NOUN
cana-4142	226	33	:	:	PUNCT
cana-4142	226	34	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	226	35	1	1	NUM
cana-4142	226	36	𝜆	𝜆	PRON
cana-4142	226	37	(	(	PUNCT
cana-4142	226	38	𝔗𝔭	𝔗𝔭	PROPN
cana-4142	226	39	,	,	PUNCT
cana-4142	226	40	𝔗𝔮	𝔗𝔮	PROPN
cana-4142	226	41	,	,	PUNCT
cana-4142	226	42	𝒶1	𝒶1	NOUN
cana-4142	226	43	𝓀	𝓀	X
cana-4142	226	44	)	)	PUNCT
cana-4142	226	45	≤	≤	NOUN
cana-4142	226	46	𝔑(𝔭	𝔑(𝔭	PUNCT
cana-4142	226	47	,	,	PUNCT
cana-4142	226	48	𝔮	𝔮	PROPN
cana-4142	226	49	,	,	PUNCT
cana-4142	226	50	𝒶1	𝒶1	PROPN
cana-4142	226	51	𝓀	𝓀	PROPN
cana-4142	226	52	)	)	PUNCT
cana-4142	226	53	(	(	PUNCT
cana-4142	226	54	2	2	X
cana-4142	226	55	)	)	PUNCT
cana-4142	226	56	for	for	ADP
cana-4142	226	57	all	all	DET
cana-4142	226	58	𝔭	𝔭	NOUN
cana-4142	226	59	,	,	PUNCT
cana-4142	226	60	𝔮	𝔮	X
cana-4142	226	61	∈	∈	NOUN
cana-4142	226	62	𝔐	𝔐	PROPN
cana-4142	226	63	and	and	CCONJ
cana-4142	226	64	𝒶1	𝒶1	NOUN
cana-4142	226	65	,	,	PUNCT
cana-4142	226	66	𝒶2	𝒶2	PROPN
cana-4142	226	67	,	,	PUNCT
cana-4142	226	68	.	.	PUNCT
cana-4142	226	69	.	.	PUNCT
cana-4142	227	1	.	.	PUNCT
cana-4142	228	1	,	,	PUNCT
cana-4142	228	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	228	3	>	>	X
cana-4142	228	4	0	0	PROPN
cana-4142	228	5	,	,	PUNCT
cana-4142	228	6	where	where	SCONJ
cana-4142	228	7	𝒯	𝒯	PROPN
cana-4142	228	8	∈	∈	PROPN
cana-4142	228	9	{	{	PUNCT
cana-4142	228	10	1	1	NUM
cana-4142	228	11	,	,	PUNCT
cana-4142	228	12	2	2	NUM
cana-4142	228	13	,	,	PUNCT
cana-4142	228	14	…	…	PUNCT
cana-4142	228	15	,	,	PUNCT
cana-4142	228	16	𝓀	𝓀	X
cana-4142	228	17	}	}	PUNCT
cana-4142	228	18	and	and	CCONJ
cana-4142	228	19	𝜆	𝜆	PRON
cana-4142	228	20	∈	∈	PROPN
cana-4142	228	21	(	(	PUNCT
cana-4142	228	22	0,1	0,1	NOUN
cana-4142	228	23	)	)	PUNCT
cana-4142	228	24	is	be	AUX
cana-4142	228	25	a	a	DET
cana-4142	228	26	constant	constant	ADJ
cana-4142	228	27	.	.	PUNCT
cana-4142	228	28	suppose	suppose	VERB
cana-4142	228	29	that	that	SCONJ
cana-4142	228	30	(	(	PUNCT
cana-4142	228	31	𝔐	𝔐	PROPN
cana-4142	228	32	,	,	PUNCT
cana-4142	228	33	𝔑	𝔑	PROPN
cana-4142	228	34	,	,	PUNCT
cana-4142	228	35	⨁	⨁	PROPN
cana-4142	228	36	)	)	PUNCT
cana-4142	228	37	is	be	AUX
cana-4142	228	38	an	an	DET
cana-4142	228	39	𝒯	𝒯	PROPN
cana-4142	228	40	−natural	−natural	NOUN
cana-4142	228	41	revised	revise	VERB
cana-4142	228	42	fuzzy	fuzzy	ADJ
cana-4142	228	43	𝓀	𝓀	X
cana-4142	228	44	−metric	−metric	ADJ
cana-4142	228	45	space	space	NOUN
cana-4142	228	46	.	.	PUNCT
cana-4142	229	1	then	then	ADV
cana-4142	229	2	,	,	PUNCT
cana-4142	229	3	𝔗	𝔗	PROPN
cana-4142	229	4	has	have	VERB
cana-4142	229	5	a	a	DET
cana-4142	229	6	unique	unique	ADJ
cana-4142	229	7	fixed	fix	VERB
cana-4142	229	8	point	point	NOUN
cana-4142	229	9	.	.	PUNCT
cana-4142	230	1	proof	proof	NOUN
cana-4142	230	2	first	first	ADV
cana-4142	230	3	,	,	PUNCT
cana-4142	230	4	we	we	PRON
cana-4142	230	5	will	will	AUX
cana-4142	230	6	show	show	VERB
cana-4142	230	7	that	that	SCONJ
cana-4142	230	8	if	if	SCONJ
cana-4142	230	9	a	a	DET
cana-4142	230	10	fixed	fixed	ADJ
cana-4142	230	11	point	point	NOUN
cana-4142	230	12	of	of	ADP
cana-4142	230	13	𝔗	𝔗	PROPN
cana-4142	230	14	exists	exist	VERB
cana-4142	230	15	,	,	PUNCT
cana-4142	230	16	then	then	ADV
cana-4142	230	17	it	it	PRON
cana-4142	230	18	is	be	AUX
cana-4142	230	19	unique	unique	ADJ
cana-4142	230	20	.	.	PUNCT
cana-4142	230	21	suppose	suppose	VERB
cana-4142	230	22	that	that	SCONJ
cana-4142	230	23	𝔵	𝔵	PROPN
cana-4142	230	24	and	and	CCONJ
cana-4142	230	25	𝔶	𝔶	NOUN
cana-4142	230	26	are	be	AUX
cana-4142	230	27	fixed	fix	VERB
cana-4142	230	28	points	point	NOUN
cana-4142	230	29	of	of	ADP
cana-4142	230	30	𝔗.	𝔗.	PROPN
cana-4142	230	31	by	by	ADP
cana-4142	230	32	(	(	PUNCT
cana-4142	230	33	2	2	NUM
cana-4142	230	34	)	)	PUNCT
cana-4142	230	35	,	,	PUNCT
cana-4142	230	36	we	we	PRON
cana-4142	230	37	have	have	VERB
cana-4142	230	38	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	230	39	,	,	PUNCT
cana-4142	230	40	𝔶	𝔶	ADP
cana-4142	230	41	,	,	PUNCT
cana-4142	230	42	𝒶1	𝒶1	NOUN
cana-4142	230	43	𝓀	𝓀	PRON
cana-4142	230	44	)	)	PUNCT
cana-4142	230	45	=	=	SYM
cana-4142	231	1	𝔑(𝔗𝔵	𝔑(𝔗𝔵	PROPN
cana-4142	231	2	,	,	PUNCT
cana-4142	231	3	𝔗𝔶	𝔗𝔶	PROPN
cana-4142	231	4	,	,	PUNCT
cana-4142	231	5	𝒶1	𝒶1	PROPN
cana-4142	231	6	𝓀	𝓀	X
cana-4142	231	7	)	)	PUNCT
cana-4142	231	8	≤	≤	NOUN
cana-4142	231	9	𝔑	𝔑	PROPN
cana-4142	231	10	(	(	PUNCT
cana-4142	231	11	𝔵	𝔵	PROPN
cana-4142	231	12	,	,	PUNCT
cana-4142	231	13	𝔶	𝔶	ADJ
cana-4142	231	14	,	,	PUNCT
cana-4142	231	15	𝒶1	𝒶1	NOUN
cana-4142	231	16	,	,	PUNCT
cana-4142	231	17	𝒶2	𝒶2	PROPN
cana-4142	231	18	,	,	PUNCT
cana-4142	231	19	.	.	PUNCT
cana-4142	231	20	.	.	PUNCT
cana-4142	232	1	,	,	PUNCT
cana-4142	232	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	232	3	,	,	PUNCT
cana-4142	232	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	232	5	𝜆	𝜆	ADP
cana-4142	232	6	,	,	PUNCT
cana-4142	232	7	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	232	8	,	,	PUNCT
cana-4142	232	9	.	.	PUNCT
cana-4142	232	10	.	.	PUNCT
cana-4142	233	1	,	,	PUNCT
cana-4142	233	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	233	3	)	)	PUNCT
cana-4142	234	1	=	=	SYM
cana-4142	234	2	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	234	3	𝜆	𝜆	PRON
cana-4142	234	4	(	(	PUNCT
cana-4142	234	5	𝔵	𝔵	NOUN
cana-4142	234	6	,	,	PUNCT
cana-4142	234	7	𝔶	𝔶	NOUN
cana-4142	234	8	,	,	PUNCT
cana-4142	234	9	𝒶1	𝒶1	PROPN
cana-4142	234	10	𝓀	𝓀	PROPN
cana-4142	234	11	)	)	PUNCT
cana-4142	234	12	by	by	ADP
cana-4142	234	13	repeating	repeat	VERB
cana-4142	234	14	this	this	DET
cana-4142	234	15	process	process	NOUN
cana-4142	234	16	,	,	PUNCT
cana-4142	234	17	we	we	PRON
cana-4142	234	18	obtain	obtain	VERB
cana-4142	234	19	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	234	20	,	,	PUNCT
cana-4142	234	21	𝔶	𝔶	ADP
cana-4142	234	22	,	,	PUNCT
cana-4142	234	23	𝒶1	𝒶1	PROPN
cana-4142	234	24	𝓀	𝓀	X
cana-4142	234	25	)	)	PUNCT
cana-4142	234	26	≤	≤	NOUN
cana-4142	235	1	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	235	2	𝜆𝑛	𝜆𝑛	PROPN
cana-4142	235	3	(	(	PUNCT
cana-4142	235	4	𝔵	𝔵	NOUN
cana-4142	235	5	,	,	PUNCT
cana-4142	235	6	𝔶	𝔶	NOUN
cana-4142	235	7	,	,	PUNCT
cana-4142	235	8	𝒶1	𝒶1	PROPN
cana-4142	235	9	𝓀	𝓀	PROPN
cana-4142	235	10	)	)	PUNCT
cana-4142	235	11	(	(	PUNCT
cana-4142	235	12	3	3	X
cana-4142	235	13	)	)	PUNCT
cana-4142	235	14	for	for	ADP
cana-4142	235	15	all	all	DET
cana-4142	235	16	𝓃	𝓃	NOUN
cana-4142	235	17	∈	∈	PROPN
cana-4142	235	18	𝒩.	𝒩.	PROPN
cana-4142	235	19	note	note	NOUN
cana-4142	235	20	that	that	SCONJ
cana-4142	235	21	,	,	PUNCT
cana-4142	235	22	if	if	SCONJ
cana-4142	235	23	{	{	PUNCT
cana-4142	235	24	𝔭𝓃	𝔭𝓃	PART
cana-4142	235	25	}	}	PUNCT
cana-4142	235	26	be	be	AUX
cana-4142	235	27	any	any	DET
cana-4142	235	28	sequence	sequence	NOUN
cana-4142	235	29	such	such	ADJ
cana-4142	235	30	that	that	PRON
cana-4142	235	31	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	235	32	>	>	X
cana-4142	235	33	0	0	PUNCT
cana-4142	235	34	and	and	CCONJ
cana-4142	235	35	lim	lim	PROPN
cana-4142	235	36	𝑛→∞	𝑛→∞	NUM
cana-4142	235	37	𝔭𝓃	𝔭𝓃	VERB
cana-4142	235	38	=	=	NOUN
cana-4142	235	39	0	0	NUM
cana-4142	235	40	,	,	PUNCT
cana-4142	235	41	then	then	ADV
cana-4142	235	42	since	since	SCONJ
cana-4142	235	43	(	(	PUNCT
cana-4142	235	44	𝔐	𝔐	PROPN
cana-4142	235	45	,	,	PUNCT
cana-4142	235	46	𝔑	𝔑	PROPN
cana-4142	235	47	,	,	PUNCT
cana-4142	235	48	⨁	⨁	PROPN
cana-4142	235	49	)	)	PUNCT
cana-4142	235	50	is	be	AUX
cana-4142	235	51	𝒯	𝒯	PROPN
cana-4142	235	52	−natural	−natural	NOUN
cana-4142	235	53	,	,	PUNCT
cana-4142	235	54	we	we	PRON
cana-4142	235	55	have	have	VERB
cana-4142	235	56	lim	lim	PROPN
cana-4142	235	57	𝑛→+∞	𝑛→+∞	PROPN
cana-4142	235	58	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	235	59	𝒷𝓃(𝔭	𝒷𝓃(𝔭	X
cana-4142	235	60	,	,	PUNCT
cana-4142	235	61	𝔮	𝔮	PROPN
cana-4142	235	62	,	,	PUNCT
cana-4142	235	63	𝒶1	𝒶1	NOUN
cana-4142	235	64	𝓀	𝓀	PRON
cana-4142	235	65	)	)	PUNCT
cana-4142	235	66	=	=	SYM
cana-4142	235	67	0	0	NUM
cana-4142	235	68	for	for	ADP
cana-4142	235	69	all	all	DET
cana-4142	235	70	𝒶1	𝒶1	NOUN
cana-4142	235	71	,	,	PUNCT
cana-4142	235	72	𝒶2	𝒶2	PROPN
cana-4142	235	73	,	,	PUNCT
cana-4142	235	74	.	.	PUNCT
cana-4142	235	75	.	.	PUNCT
cana-4142	236	1	.	.	PUNCT
cana-4142	237	1	,	,	PUNCT
cana-4142	237	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	237	3	>	>	X
cana-4142	237	4	0	0	X
cana-4142	237	5	.	.	PUNCT
cana-4142	238	1	using	use	VERB
cana-4142	238	2	this	this	DET
cana-4142	238	3	fact	fact	NOUN
cana-4142	238	4	in	in	ADP
cana-4142	238	5	(	(	PUNCT
cana-4142	238	6	3	3	NUM
cana-4142	238	7	)	)	PUNCT
cana-4142	238	8	,	,	PUNCT
cana-4142	238	9	we	we	PRON
cana-4142	238	10	obtain	obtain	VERB
cana-4142	238	11	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	238	12	,	,	PUNCT
cana-4142	238	13	𝔶	𝔶	ADP
cana-4142	238	14	,	,	PUNCT
cana-4142	238	15	𝒶1	𝒶1	NOUN
cana-4142	238	16	𝓀	𝓀	PRON
cana-4142	238	17	)	)	PUNCT
cana-4142	238	18	=	=	SYM
cana-4142	238	19	0	0	NUM
cana-4142	238	20	for	for	ADP
cana-4142	238	21	all	all	DET
cana-4142	238	22	𝒶1	𝒶1	NOUN
cana-4142	238	23	,	,	PUNCT
cana-4142	238	24	𝒶2	𝒶2	PROPN
cana-4142	238	25	,	,	PUNCT
cana-4142	238	26	.	.	PUNCT
cana-4142	238	27	.	.	PUNCT
cana-4142	239	1	.	.	PUNCT
cana-4142	240	1	,	,	PUNCT
cana-4142	240	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	240	3	>	>	X
cana-4142	240	4	0	0	NUM
cana-4142	240	5	,	,	PUNCT
cana-4142	240	6	that	that	ADV
cana-4142	240	7	is	is	ADV
cana-4142	240	8	,	,	PUNCT
cana-4142	240	9	𝔵	𝔵	X
cana-4142	240	10	=	=	NOUN
cana-4142	240	11	𝔶.	𝔶.	NOUN
cana-4142	240	12	therefore	therefore	ADV
cana-4142	240	13	,	,	PUNCT
cana-4142	240	14	the	the	DET
cana-4142	240	15	fixed	fix	VERB
cana-4142	240	16	point	point	NOUN
cana-4142	240	17	of	of	ADP
cana-4142	240	18	𝔗	𝔗	PROPN
cana-4142	240	19	is	be	AUX
cana-4142	240	20	unique	unique	ADJ
cana-4142	240	21	.	.	PUNCT
cana-4142	241	1	for	for	ADP
cana-4142	241	2	the	the	DET
cana-4142	241	3	existence	existence	NOUN
cana-4142	241	4	of	of	ADP
cana-4142	241	5	a	a	DET
cana-4142	241	6	fixed	fix	VERB
cana-4142	241	7	point	point	NOUN
cana-4142	241	8	of	of	ADP
cana-4142	241	9	𝔗	𝔗	PROPN
cana-4142	241	10	,	,	PUNCT
cana-4142	241	11	we	we	PRON
cana-4142	241	12	choose	choose	VERB
cana-4142	241	13	𝔭0	𝔭0	PROPN
cana-4142	241	14	∈	∈	PROPN
cana-4142	241	15	𝔐	𝔐	PROPN
cana-4142	241	16	and	and	CCONJ
cana-4142	241	17	define	define	VERB
cana-4142	241	18	an	an	DET
cana-4142	241	19	iterative	iterative	NOUN
cana-4142	241	20	sequence	sequence	NOUN
cana-4142	241	21	{	{	PUNCT
cana-4142	241	22	𝔭𝓃	𝔭𝓃	VERB
cana-4142	241	23	}	}	PUNCT
cana-4142	241	24	by	by	ADP
cana-4142	241	25	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	241	26	=	=	PUNCT
cana-4142	241	27	𝔗𝔭𝓃−1for	𝔗𝔭𝓃−1for	ADP
cana-4142	241	28	all	all	DET
cana-4142	241	29	𝓃	𝓃	NOUN
cana-4142	241	30	∈	∈	NOUN
cana-4142	241	31	𝒩.	𝒩.	PROPN
cana-4142	241	32	if	if	SCONJ
cana-4142	241	33	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	241	34	=	=	PUNCT
cana-4142	241	35	𝔭𝓃−1	𝔭𝓃−1	NOUN
cana-4142	241	36	for	for	ADP
cana-4142	241	37	some	some	DET
cana-4142	241	38	𝓃	𝓃	NOUN
cana-4142	241	39	∈	∈	PROPN
cana-4142	241	40	𝒩	𝒩	PROPN
cana-4142	241	41	,	,	PUNCT
cana-4142	241	42	then	then	ADV
cana-4142	241	43	𝔭𝓃	𝔭𝓃	VERB
cana-4142	241	44	is	be	AUX
cana-4142	241	45	the	the	DET
cana-4142	241	46	unique	unique	ADJ
cana-4142	241	47	fixed	fix	VERB
cana-4142	241	48	point	point	NOUN
cana-4142	241	49	of	of	ADP
cana-4142	241	50	𝔗.	𝔗.	PROPN
cana-4142	241	51	therefore	therefore	ADV
cana-4142	241	52	,	,	PUNCT
cana-4142	241	53	we	we	PRON
cana-4142	241	54	may	may	AUX
cana-4142	241	55	assume	assume	VERB
cana-4142	241	56	that	that	SCONJ
cana-4142	241	57	𝔭𝓃	𝔭𝓃	VERB
cana-4142	241	58	≠	≠	NOUN
cana-4142	241	59	𝔭𝓃−1	𝔭𝓃−1	NOUN
cana-4142	241	60	for	for	ADP
cana-4142	241	61	all	all	DET
cana-4142	241	62	𝓃	𝓃	NOUN
cana-4142	241	63	∈	∈	NOUN
cana-4142	241	64	𝒩.	𝒩.	PROPN
cana-4142	241	65	for	for	ADP
cana-4142	241	66	any	any	DET
cana-4142	241	67	𝓃	𝓃	NOUN
cana-4142	241	68	∈	∈	PROPN
cana-4142	241	69	𝒩	𝒩	PROPN
cana-4142	241	70	and	and	CCONJ
cana-4142	241	71	𝒶1	𝒶1	PROPN
cana-4142	241	72	,	,	PUNCT
cana-4142	241	73	𝒶2	𝒶2	PROPN
cana-4142	241	74	,	,	PUNCT
cana-4142	241	75	.	.	PUNCT
cana-4142	241	76	.	.	PUNCT
cana-4142	242	1	.	.	PUNCT
cana-4142	243	1	,	,	PUNCT
cana-4142	243	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	243	3	>	>	X
cana-4142	243	4	0	0	NUM
cana-4142	243	5	,	,	PUNCT
cana-4142	243	6	we	we	PRON
cana-4142	243	7	have	have	VERB
cana-4142	243	8	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	243	9	,	,	PUNCT
cana-4142	243	10	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	243	11	,	,	PUNCT
cana-4142	243	12	𝒶1	𝒶1	NOUN
cana-4142	243	13	𝓀	𝓀	X
cana-4142	243	14	)	)	PUNCT
cana-4142	243	15	=	=	SYM
cana-4142	243	16	𝔑(𝔗𝔭𝓃−1	𝔑(𝔗𝔭𝓃−1	PROPN
cana-4142	243	17	,	,	PUNCT
cana-4142	243	18	𝔗𝔭𝓃	𝔗𝔭𝓃	PROPN
cana-4142	243	19	,	,	PUNCT
cana-4142	243	20	𝒶1	𝒶1	NOUN
cana-4142	243	21	𝓀	𝓀	X
cana-4142	243	22	)	)	PUNCT
cana-4142	243	23	≤	≤	NOUN
cana-4142	244	1	𝔑	𝔑	PROPN
cana-4142	244	2	(	(	PUNCT
cana-4142	244	3	𝔵	𝔵	PROPN
cana-4142	244	4	,	,	PUNCT
cana-4142	244	5	𝔶	𝔶	ADJ
cana-4142	244	6	,	,	PUNCT
cana-4142	244	7	𝒶1	𝒶1	NOUN
cana-4142	244	8	,	,	PUNCT
cana-4142	244	9	𝒶2	𝒶2	PROPN
cana-4142	244	10	,	,	PUNCT
cana-4142	244	11	.	.	PUNCT
cana-4142	244	12	.	.	PUNCT
cana-4142	245	1	,	,	PUNCT
cana-4142	245	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	245	3	,	,	PUNCT
cana-4142	245	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	245	5	𝜆	𝜆	ADP
cana-4142	245	6	,	,	PUNCT
cana-4142	245	7	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	245	8	,	,	PUNCT
cana-4142	245	9	.	.	PUNCT
cana-4142	245	10	.	.	PUNCT
cana-4142	246	1	,	,	PUNCT
cana-4142	246	2	𝒶𝓀	𝒶𝓀	PROPN
cana-4142	246	3	)	)	PUNCT
cana-4142	247	1	=	=	SYM
cana-4142	247	2	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	247	3	𝜆	𝜆	PRON
cana-4142	247	4	(	(	PUNCT
cana-4142	247	5	𝔭𝓃−1	𝔭𝓃−1	NOUN
cana-4142	247	6	,	,	PUNCT
cana-4142	247	7	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	247	8	,	,	PUNCT
cana-4142	247	9	𝒶1	𝒶1	PROPN
cana-4142	247	10	𝓀	𝓀	PROPN
cana-4142	247	11	)	)	PUNCT
cana-4142	247	12	communications	communication	NOUN
cana-4142	247	13	on	on	ADP
cana-4142	247	14	applied	apply	VERB
cana-4142	247	15	nonlinear	nonlinear	ADJ
cana-4142	247	16	analysis	analysis	NOUN
cana-4142	247	17	issn	issn	NOUN
cana-4142	247	18	:	:	PUNCT
cana-4142	247	19	1074	1074	NUM
cana-4142	247	20	-	-	PUNCT
cana-4142	247	21	133x	133x	NUM
cana-4142	247	22	vol	vol	NOUN
cana-4142	247	23	x	x	NOUN
cana-4142	247	24	no	no	INTJ
cana-4142	247	25	.	.	PUNCT
cana-4142	248	1	y	y	PROPN
cana-4142	248	2	(	(	PUNCT
cana-4142	248	3	2025	2025	NUM
cana-4142	248	4	)	)	PUNCT
cana-4142	248	5	1327	1327	NUM
cana-4142	248	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	248	7	by	by	ADP
cana-4142	248	8	repeating	repeat	VERB
cana-4142	248	9	this	this	DET
cana-4142	248	10	process	process	NOUN
cana-4142	248	11	,	,	PUNCT
cana-4142	248	12	we	we	PRON
cana-4142	248	13	obtain	obtain	VERB
cana-4142	248	14	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	248	15	,	,	PUNCT
cana-4142	248	16	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	248	17	,	,	PUNCT
cana-4142	248	18	𝒶1	𝒶1	NOUN
cana-4142	248	19	𝓀	𝓀	X
cana-4142	248	20	)	)	PUNCT
cana-4142	248	21	≤	≤	NOUN
cana-4142	249	1	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	249	2	𝜆𝑛	𝜆𝑛	PROPN
cana-4142	249	3	(	(	PUNCT
cana-4142	249	4	𝔭0	𝔭0	ADJ
cana-4142	249	5	,	,	PUNCT
cana-4142	249	6	𝔭1	𝔭1	ADJ
cana-4142	249	7	,	,	PUNCT
cana-4142	249	8	𝒶1	𝒶1	PROPN
cana-4142	249	9	𝓀	𝓀	PROPN
cana-4142	249	10	)	)	PUNCT
cana-4142	249	11	(	(	PUNCT
cana-4142	249	12	4	4	X
cana-4142	249	13	)	)	PUNCT
cana-4142	249	14	for	for	ADP
cana-4142	249	15	all	all	DET
cana-4142	249	16	𝓃	𝓃	NOUN
cana-4142	249	17	∈	∈	NOUN
cana-4142	249	18	𝒩.	𝒩.	PROPN
cana-4142	249	19	for	for	ADP
cana-4142	249	20	each	each	DET
cana-4142	249	21	𝓃	𝓃	NOUN
cana-4142	249	22	∈	∈	PROPN
cana-4142	249	23	𝒩	𝒩	PROPN
cana-4142	249	24	and	and	CCONJ
cana-4142	249	25	𝒶1	𝒶1	PROPN
cana-4142	249	26	,	,	PUNCT
cana-4142	249	27	𝒶2	𝒶2	PROPN
cana-4142	249	28	,	,	PUNCT
cana-4142	249	29	.	.	PUNCT
cana-4142	249	30	.	.	PUNCT
cana-4142	249	31	.	.	PUNCT
cana-4142	250	1	,	,	PUNCT
cana-4142	250	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	250	3	>	>	X
cana-4142	250	4	0	0	PUNCT
cana-4142	250	5	and	and	CCONJ
cana-4142	250	6	𝓍	𝓍	ADJ
cana-4142	250	7	>	>	X
cana-4142	250	8	0	0	NUM
cana-4142	250	9	,	,	PUNCT
cana-4142	250	10	we	we	PRON
cana-4142	250	11	have	have	VERB
cana-4142	250	12	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	250	13	,	,	PUNCT
cana-4142	250	14	𝔭𝓃+𝓍	𝔭𝓃+𝓍	NUM
cana-4142	250	15	,	,	PUNCT
cana-4142	250	16	𝒶1	𝒶1	NOUN
cana-4142	250	17	𝓀	𝓀	PROPN
cana-4142	250	18	)	)	PUNCT
cana-4142	250	19	≤	≤	NOUN
cana-4142	250	20	{	{	PUNCT
cana-4142	250	21	𝔑	𝔑	PROPN
cana-4142	250	22	(	(	PUNCT
cana-4142	250	23	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	250	24	,	,	PUNCT
cana-4142	250	25	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	250	26	,	,	PUNCT
cana-4142	250	27	𝒶1	𝒶1	NOUN
cana-4142	250	28	,	,	PUNCT
cana-4142	250	29	𝒶2	𝒶2	PROPN
cana-4142	250	30	,	,	PUNCT
cana-4142	250	31	.	.	PUNCT
cana-4142	250	32	.	.	PUNCT
cana-4142	251	1	,	,	PUNCT
cana-4142	251	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	251	3	,	,	PUNCT
cana-4142	251	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	251	5	2	2	NUM
cana-4142	251	6	,	,	PUNCT
cana-4142	251	7	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	251	8	,	,	PUNCT
cana-4142	251	9	.	.	PUNCT
cana-4142	251	10	.	.	PUNCT
cana-4142	252	1	,	,	PUNCT
cana-4142	252	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	252	3	)	)	PUNCT
cana-4142	252	4	⨁𝔑	⨁𝔑	NOUN
cana-4142	252	5	(	(	PUNCT
cana-4142	252	6	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	252	7	,	,	PUNCT
cana-4142	252	8	𝔭𝓃+𝓍	𝔭𝓃+𝓍	NUM
cana-4142	252	9	,	,	PUNCT
cana-4142	252	10	𝒶1	𝒶1	NOUN
cana-4142	252	11	,	,	PUNCT
cana-4142	252	12	𝒶2	𝒶2	PROPN
cana-4142	252	13	,	,	PUNCT
cana-4142	252	14	.	.	PUNCT
cana-4142	252	15	.	.	PUNCT
cana-4142	253	1	,	,	PUNCT
cana-4142	253	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	253	3	,	,	PUNCT
cana-4142	253	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	253	5	2	2	NUM
cana-4142	253	6	,	,	PUNCT
cana-4142	253	7	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	253	8	,	,	PUNCT
cana-4142	253	9	.	.	PUNCT
cana-4142	253	10	.	.	PUNCT
cana-4142	254	1	,	,	PUNCT
cana-4142	254	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	254	3	)	)	PUNCT
cana-4142	254	4	}	}	PUNCT
cana-4142	254	5	≤	≤	NOUN
cana-4142	254	6	{	{	PUNCT
cana-4142	254	7	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	254	8	2	2	NUM
cana-4142	254	9	(	(	PUNCT
cana-4142	254	10	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	254	11	,	,	PUNCT
cana-4142	254	12	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	254	13	,	,	PUNCT
cana-4142	254	14	𝒶1	𝒶1	NOUN
cana-4142	254	15	𝓀)⨁𝔑	𝓀)⨁𝔑	PROPN
cana-4142	254	16	(	(	PUNCT
cana-4142	254	17	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	254	18	,	,	PUNCT
cana-4142	254	19	𝔭𝓃+2	𝔭𝓃+2	NUM
cana-4142	254	20	,	,	PUNCT
cana-4142	254	21	𝒶1	𝒶1	NOUN
cana-4142	254	22	,	,	PUNCT
cana-4142	254	23	𝒶2	𝒶2	PROPN
cana-4142	254	24	,	,	PUNCT
cana-4142	254	25	.	.	PUNCT
cana-4142	254	26	.	.	PUNCT
cana-4142	255	1	,	,	PUNCT
cana-4142	255	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	255	3	,	,	PUNCT
cana-4142	255	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	255	5	22	22	NUM
cana-4142	255	6	,	,	PUNCT
cana-4142	255	7	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	255	8	,	,	PUNCT
cana-4142	255	9	.	.	PUNCT
cana-4142	255	10	.	.	PUNCT
cana-4142	256	1	,	,	PUNCT
cana-4142	256	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	256	3	)	)	PUNCT
cana-4142	256	4	⨁𝔑	⨁𝔑	NOUN
cana-4142	256	5	(	(	PUNCT
cana-4142	256	6	𝔭𝓃+2	𝔭𝓃+2	NUM
cana-4142	256	7	,	,	PUNCT
cana-4142	256	8	𝔭𝓃+𝓍	𝔭𝓃+𝓍	NUM
cana-4142	256	9	,	,	PUNCT
cana-4142	256	10	𝒶1	𝒶1	NOUN
cana-4142	256	11	,	,	PUNCT
cana-4142	256	12	𝒶2	𝒶2	PROPN
cana-4142	256	13	,	,	PUNCT
cana-4142	256	14	.	.	PUNCT
cana-4142	256	15	.	.	PUNCT
cana-4142	257	1	,	,	PUNCT
cana-4142	257	2	𝒶𝒯−1	𝒶𝒯−1	PROPN
cana-4142	257	3	,	,	PUNCT
cana-4142	257	4	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	257	5	22	22	NUM
cana-4142	257	6	,	,	PUNCT
cana-4142	257	7	𝒶𝒯+1	𝒶𝒯+1	NOUN
cana-4142	257	8	,	,	PUNCT
cana-4142	257	9	.	.	PUNCT
cana-4142	257	10	.	.	PUNCT
cana-4142	258	1	,	,	PUNCT
cana-4142	258	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	258	3	)	)	PUNCT
cana-4142	258	4	}	}	PUNCT
cana-4142	258	5	≤	≤	NOUN
cana-4142	258	6	{	{	PUNCT
cana-4142	258	7	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	258	8	2	2	NUM
cana-4142	258	9	(	(	PUNCT
cana-4142	258	10	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	258	11	,	,	PUNCT
cana-4142	258	12	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	258	13	,	,	PUNCT
cana-4142	258	14	𝒶1	𝒶1	NOUN
cana-4142	258	15	𝓀)⨁𝔑𝒯	𝓀)⨁𝔑𝒯	NOUN
cana-4142	258	16	22	22	NUM
cana-4142	258	17	(	(	PUNCT
cana-4142	258	18	𝔭𝓃+1	𝔭𝓃+1	NUM
cana-4142	258	19	,	,	PUNCT
cana-4142	258	20	𝔭𝓃+2	𝔭𝓃+2	NUM
cana-4142	258	21	,	,	PUNCT
cana-4142	258	22	𝒶1	𝒶1	NOUN
cana-4142	258	23	𝓀)⨁	𝓀)⨁	PROPN
cana-4142	258	24	…	…	PUNCT
cana-4142	258	25	⨁	⨁	PROPN
cana-4142	258	26	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	258	27	2𝓍−1(𝔭𝓃+𝓍−2	2𝓍−1(𝔭𝓃+𝓍−2	NUM
cana-4142	258	28	,	,	PUNCT
cana-4142	258	29	𝔭𝓃+𝓍+1	𝔭𝓃+𝓍+1	NUM
cana-4142	258	30	,	,	PUNCT
cana-4142	258	31	𝒶1	𝒶1	NOUN
cana-4142	258	32	𝓀)⨁𝔑2(𝔭𝓃+𝓍−1	𝓀)⨁𝔑2(𝔭𝓃+𝓍−1	PROPN
cana-4142	258	33	,	,	PUNCT
cana-4142	258	34	𝔭𝓃+𝓍	𝔭𝓃+𝓍	NUM
cana-4142	258	35	,	,	PUNCT
cana-4142	258	36	𝒶1	𝒶1	NOUN
cana-4142	258	37	𝓀	𝓀	PROPN
cana-4142	258	38	)	)	PUNCT
cana-4142	258	39	}	}	PUNCT
cana-4142	258	40	by	by	ADP
cana-4142	258	41	using	use	VERB
cana-4142	258	42	(	(	PUNCT
cana-4142	258	43	4	4	NUM
cana-4142	258	44	)	)	PUNCT
cana-4142	258	45	,	,	PUNCT
cana-4142	258	46	we	we	PRON
cana-4142	258	47	obtain	obtain	VERB
cana-4142	258	48	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	258	49	,	,	PUNCT
cana-4142	258	50	𝔭𝓃+𝓍	𝔭𝓃+𝓍	NUM
cana-4142	258	51	,	,	PUNCT
cana-4142	258	52	𝒶1	𝒶1	NOUN
cana-4142	258	53	𝓀	𝓀	X
cana-4142	258	54	)	)	PUNCT
cana-4142	258	55	≤	≤	NOUN
cana-4142	259	1	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	259	2	2𝜆𝑛	2𝜆𝑛	NOUN
cana-4142	259	3	(	(	PUNCT
cana-4142	259	4	𝔭0	𝔭0	ADJ
cana-4142	259	5	,	,	PUNCT
cana-4142	259	6	𝔭1	𝔭1	ADJ
cana-4142	259	7	,	,	PUNCT
cana-4142	259	8	𝒶1	𝒶1	PROPN
cana-4142	259	9	𝓀	𝓀	X
cana-4142	259	10	)	)	PUNCT
cana-4142	259	11	⨁	⨁	PROPN
cana-4142	259	12	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	259	13	22𝜆𝑛+1	22𝜆𝑛+1	NUM
cana-4142	259	14	(	(	PUNCT
cana-4142	259	15	𝔭0	𝔭0	ADJ
cana-4142	259	16	,	,	PUNCT
cana-4142	259	17	𝔭1	𝔭1	ADJ
cana-4142	259	18	,	,	PUNCT
cana-4142	259	19	𝒶1	𝒶1	NOUN
cana-4142	259	20	𝓀)⨁	𝓀)⨁	NUM
cana-4142	259	21	…	…	PUNCT
cana-4142	259	22	⨁𝔑𝒯	⨁𝔑𝒯	ADJ
cana-4142	259	23	2𝓍−1𝜆𝑛+𝓍−1	2𝓍−1𝜆𝑛+𝓍−1	NOUN
cana-4142	259	24	(	(	PUNCT
cana-4142	259	25	𝔭0	𝔭0	ADJ
cana-4142	259	26	,	,	PUNCT
cana-4142	259	27	𝔭1	𝔭1	ADJ
cana-4142	259	28	,	,	PUNCT
cana-4142	259	29	𝒶1	𝒶1	NOUN
cana-4142	259	30	𝓀	𝓀	PROPN
cana-4142	259	31	)	)	PUNCT
cana-4142	259	32	since	since	SCONJ
cana-4142	259	33	(	(	PUNCT
cana-4142	259	34	𝔐	𝔐	PROPN
cana-4142	259	35	,	,	PUNCT
cana-4142	259	36	𝔑	𝔑	PROPN
cana-4142	259	37	,	,	PUNCT
cana-4142	259	38	⨁	⨁	PROPN
cana-4142	259	39	)	)	PUNCT
cana-4142	259	40	is	be	AUX
cana-4142	259	41	𝒯	𝒯	PROPN
cana-4142	259	42	−natural	−natural	NOUN
cana-4142	259	43	,	,	PUNCT
cana-4142	259	44	it	it	PRON
cana-4142	259	45	follows	follow	VERB
cana-4142	259	46	from	from	ADP
cana-4142	259	47	the	the	DET
cana-4142	259	48	above	above	ADJ
cana-4142	259	49	inequality	inequality	NOUN
cana-4142	259	50	that	that	PRON
cana-4142	259	51	lim	lim	PROPN
cana-4142	259	52	𝑛→+∞	𝑛→+∞	PROPN
cana-4142	259	53	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	259	54	,	,	PUNCT
cana-4142	259	55	𝔭𝓃+𝓍	𝔭𝓃+𝓍	NUM
cana-4142	259	56	,	,	PUNCT
cana-4142	259	57	𝒶1	𝒶1	NOUN
cana-4142	259	58	𝓀	𝓀	X
cana-4142	259	59	)	)	PUNCT
cana-4142	259	60	=	=	SYM
cana-4142	260	1	0	0	X
cana-4142	260	2	.	.	PUNCT
cana-4142	261	1	therefore	therefore	ADV
cana-4142	261	2	,	,	PUNCT
cana-4142	261	3	{	{	PUNCT
cana-4142	261	4	𝔭𝓃	𝔭𝓃	PART
cana-4142	261	5	}	}	PUNCT
cana-4142	261	6	is	be	AUX
cana-4142	261	7	a	a	DET
cana-4142	261	8	𝔾	𝔾	ADJ
cana-4142	261	9	−cauchy	−cauchy	ADJ
cana-4142	261	10	sequence	sequence	NOUN
cana-4142	261	11	.	.	PUNCT
cana-4142	262	1	by	by	ADP
cana-4142	262	2	the	the	DET
cana-4142	262	3	𝔾	𝔾	PROPN
cana-4142	262	4	−completeness	−completeness	PROPN
cana-4142	262	5	of	of	ADP
cana-4142	262	6	(	(	PUNCT
cana-4142	262	7	𝔐	𝔐	PROPN
cana-4142	262	8	,	,	PUNCT
cana-4142	262	9	𝔑	𝔑	PROPN
cana-4142	262	10	,	,	PUNCT
cana-4142	262	11	⨁	⨁	PROPN
cana-4142	262	12	)	)	PUNCT
cana-4142	262	13	,	,	PUNCT
cana-4142	262	14	there	there	PRON
cana-4142	262	15	exists	exist	VERB
cana-4142	262	16	𝔵	𝔵	DET
cana-4142	262	17	∈	∈	NOUN
cana-4142	262	18	𝔐	𝔐	NOUN
cana-4142	262	19	such	such	ADJ
cana-4142	262	20	that	that	SCONJ
cana-4142	262	21	lim	lim	PROPN
cana-4142	262	22	𝑛→+∞	𝑛→+∞	PROPN
cana-4142	262	23	𝔑(𝔭𝓃	𝔑(𝔭𝓃	ADJ
cana-4142	262	24	,	,	PUNCT
cana-4142	262	25	𝔵	𝔵	NOUN
cana-4142	262	26	,	,	PUNCT
cana-4142	262	27	𝒶1	𝒶1	NOUN
cana-4142	262	28	𝓀	𝓀	X
cana-4142	262	29	)	)	PUNCT
cana-4142	262	30	=	=	SYM
cana-4142	262	31	0	0	NUM
cana-4142	262	32	,	,	PUNCT
cana-4142	262	33	for	for	ADP
cana-4142	262	34	all	all	DET
cana-4142	262	35	𝒶1	𝒶1	NOUN
cana-4142	262	36	,	,	PUNCT
cana-4142	262	37	𝒶2	𝒶2	PROPN
cana-4142	262	38	,	,	PUNCT
cana-4142	262	39	.	.	PUNCT
cana-4142	262	40	.	.	PUNCT
cana-4142	263	1	.	.	PUNCT
cana-4142	264	1	,	,	PUNCT
cana-4142	264	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	264	3	>	>	X
cana-4142	264	4	0	0	X
cana-4142	264	5	.	.	PUNCT
cana-4142	265	1	we	we	PRON
cana-4142	265	2	will	will	AUX
cana-4142	265	3	show	show	VERB
cana-4142	265	4	that	that	SCONJ
cana-4142	265	5	𝔵	𝔵	NOUN
cana-4142	265	6	is	be	AUX
cana-4142	265	7	a	a	DET
cana-4142	265	8	fixed	fix	VERB
cana-4142	265	9	point	point	NOUN
cana-4142	265	10	of	of	ADP
cana-4142	265	11	𝔗.	𝔗.	PROPN
cana-4142	265	12	for	for	ADP
cana-4142	265	13	each	each	DET
cana-4142	265	14	𝒶1	𝒶1	NOUN
cana-4142	265	15	,	,	PUNCT
cana-4142	265	16	𝒶2	𝒶2	PROPN
cana-4142	265	17	,	,	PUNCT
cana-4142	265	18	.	.	PUNCT
cana-4142	265	19	.	.	PUNCT
cana-4142	266	1	.	.	PUNCT
cana-4142	267	1	,	,	PUNCT
cana-4142	267	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	267	3	>	>	X
cana-4142	267	4	0	0	NUM
cana-4142	267	5	,	,	PUNCT
cana-4142	267	6	we	we	PRON
cana-4142	267	7	have	have	VERB
cana-4142	267	8	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	267	9	,	,	PUNCT
cana-4142	267	10	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	267	11	,	,	PUNCT
cana-4142	267	12	𝒶1	𝒶1	NOUN
cana-4142	267	13	𝓀	𝓀	X
cana-4142	267	14	)	)	PUNCT
cana-4142	267	15	≤	≤	NUM
cana-4142	267	16	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	267	17	2	2	NUM
cana-4142	267	18	(	(	PUNCT
cana-4142	267	19	𝔵	𝔵	NOUN
cana-4142	267	20	,	,	PUNCT
cana-4142	267	21	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	267	22	,	,	PUNCT
cana-4142	267	23	𝒶1	𝒶1	NOUN
cana-4142	267	24	𝓀)⨁	𝓀)⨁	NUM
cana-4142	267	25	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	267	26	2	2	NUM
cana-4142	267	27	(	(	PUNCT
cana-4142	267	28	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	267	29	,	,	PUNCT
cana-4142	267	30	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	267	31	,	,	PUNCT
cana-4142	267	32	𝒶1	𝒶1	NOUN
cana-4142	267	33	𝓀	𝓀	PRON
cana-4142	267	34	)	)	PUNCT
cana-4142	267	35	=	=	SYM
cana-4142	267	36	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	267	37	2	2	NUM
cana-4142	267	38	(	(	PUNCT
cana-4142	267	39	𝔵	𝔵	NOUN
cana-4142	267	40	,	,	PUNCT
cana-4142	267	41	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	267	42	,	,	PUNCT
cana-4142	267	43	𝒶1	𝒶1	NOUN
cana-4142	267	44	𝓀)⨁	𝓀)⨁	NUM
cana-4142	267	45	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	267	46	2	2	NUM
cana-4142	267	47	(	(	PUNCT
cana-4142	267	48	𝔗𝔭𝓃−1	𝔗𝔭𝓃−1	NOUN
cana-4142	267	49	,	,	PUNCT
cana-4142	267	50	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	267	51	,	,	PUNCT
cana-4142	267	52	𝒶1	𝒶1	NOUN
cana-4142	267	53	𝓀	𝓀	X
cana-4142	267	54	)	)	PUNCT
cana-4142	267	55	≤	≤	NUM
cana-4142	267	56	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	267	57	2	2	NUM
cana-4142	267	58	(	(	PUNCT
cana-4142	267	59	𝔵	𝔵	NOUN
cana-4142	267	60	,	,	PUNCT
cana-4142	267	61	𝔭𝓃	𝔭𝓃	NOUN
cana-4142	267	62	,	,	PUNCT
cana-4142	267	63	𝒶1	𝒶1	NOUN
cana-4142	267	64	𝓀)⨁	𝓀)⨁	PROPN
cana-4142	267	65	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	267	66	2𝜆(𝔭𝓃−1	2𝜆(𝔭𝓃−1	PROPN
cana-4142	267	67	,	,	PUNCT
cana-4142	267	68	𝔵	𝔵	NOUN
cana-4142	267	69	,	,	PUNCT
cana-4142	267	70	𝒶1	𝒶1	NOUN
cana-4142	267	71	𝓀	𝓀	PROPN
cana-4142	267	72	)	)	PUNCT
cana-4142	267	73	by	by	ADP
cana-4142	267	74	using	use	VERB
cana-4142	267	75	(	(	PUNCT
cana-4142	267	76	5	5	NUM
cana-4142	267	77	)	)	PUNCT
cana-4142	267	78	in	in	ADP
cana-4142	267	79	the	the	DET
cana-4142	267	80	above	above	ADJ
cana-4142	267	81	inequality	inequality	NOUN
cana-4142	267	82	,	,	PUNCT
cana-4142	267	83	we	we	PRON
cana-4142	267	84	obtain	obtain	VERB
cana-4142	267	85	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	267	86	,	,	PUNCT
cana-4142	267	87	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	267	88	,	,	PUNCT
cana-4142	267	89	𝒶1	𝒶1	NOUN
cana-4142	267	90	𝓀	𝓀	X
cana-4142	267	91	)	)	PUNCT
cana-4142	267	92	=	=	SYM
cana-4142	267	93	0	0	NUM
cana-4142	267	94	for	for	ADP
cana-4142	267	95	all	all	DET
cana-4142	267	96	𝒶1	𝒶1	NOUN
cana-4142	267	97	,	,	PUNCT
cana-4142	267	98	𝒶2	𝒶2	PROPN
cana-4142	267	99	,	,	PUNCT
cana-4142	267	100	.	.	PUNCT
cana-4142	267	101	.	.	PUNCT
cana-4142	267	102	.	.	PUNCT
cana-4142	268	1	,	,	PUNCT
cana-4142	268	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	268	3	>	>	X
cana-4142	268	4	0	0	NUM
cana-4142	268	5	,	,	PUNCT
cana-4142	268	6	that	that	ADV
cana-4142	268	7	is	is	ADV
cana-4142	268	8	,	,	PUNCT
cana-4142	268	9	𝔵	𝔵	X
cana-4142	268	10	=	=	SYM
cana-4142	268	11	𝔗𝔵.	𝔗𝔵.	PROPN
cana-4142	268	12	thus	thus	ADV
cana-4142	268	13	,	,	PUNCT
cana-4142	268	14	𝔵	𝔵	PRON
cana-4142	268	15	is	be	AUX
cana-4142	268	16	the	the	DET
cana-4142	268	17	unique	unique	ADJ
cana-4142	268	18	fixed	fix	VERB
cana-4142	268	19	point	point	NOUN
cana-4142	268	20	of	of	ADP
cana-4142	268	21	𝔗.	𝔗.	PROPN
cana-4142	268	22	for	for	ADP
cana-4142	268	23	𝓀	𝓀	PROPN
cana-4142	268	24	=	=	SYM
cana-4142	268	25	1	1	NUM
cana-4142	268	26	,	,	PUNCT
cana-4142	268	27	the	the	DET
cana-4142	268	28	above	above	ADJ
cana-4142	268	29	theorem	theorem	NOUN
cana-4142	268	30	reduces	reduce	VERB
cana-4142	268	31	to	to	ADP
cana-4142	268	32	the	the	DET
cana-4142	268	33	following	following	ADJ
cana-4142	268	34	result	result	NOUN
cana-4142	268	35	of	of	ADP
cana-4142	268	36	grabiec	grabiec	PROPN
cana-4142	268	37	(	(	PUNCT
cana-4142	268	38	1988	1988	NUM
cana-4142	268	39	)	)	PUNCT
cana-4142	268	40	.	.	PUNCT
cana-4142	269	1	corollary	corollary	ADJ
cana-4142	269	2	27	27	NUM
cana-4142	269	3	let	let	VERB
cana-4142	269	4	(	(	PUNCT
cana-4142	269	5	𝔐	𝔐	X
cana-4142	269	6	,	,	PUNCT
cana-4142	269	7	𝔑	𝔑	PROPN
cana-4142	269	8	,	,	PUNCT
cana-4142	269	9	⨁	⨁	PROPN
cana-4142	269	10	)	)	PUNCT
cana-4142	269	11	be	be	VERB
cana-4142	269	12	a	a	DET
cana-4142	269	13	𝔾	𝔾	PROPN
cana-4142	269	14	−complete	−complete	PROPN
cana-4142	269	15	revised	revise	VERB
cana-4142	269	16	fuzzy	fuzzy	ADJ
cana-4142	269	17	metric	metric	ADJ
cana-4142	269	18	space	space	NOUN
cana-4142	269	19	such	such	ADJ
cana-4142	269	20	that	that	SCONJ
cana-4142	269	21	lim	lim	PROPN
cana-4142	269	22	𝑡→+∞	𝑡→+∞	PROPN
cana-4142	269	23	𝔑(𝔭	𝔑(𝔭	X
cana-4142	269	24	,	,	PUNCT
cana-4142	269	25	𝔮	𝔮	PROPN
cana-4142	269	26	,	,	PUNCT
cana-4142	269	27	𝔞	𝔞	NOUN
cana-4142	269	28	)	)	PUNCT
cana-4142	269	29	=	=	SYM
cana-4142	269	30	0	0	NUM
cana-4142	269	31	for	for	ADP
cana-4142	269	32	all	all	DET
cana-4142	269	33	𝔭	𝔭	NOUN
cana-4142	269	34	,	,	PUNCT
cana-4142	269	35	𝔮	𝔮	X
cana-4142	269	36	∈	∈	X
cana-4142	269	37	𝔐	𝔐	NOUN
cana-4142	269	38	(	(	PUNCT
cana-4142	269	39	6	6	NUM
cana-4142	269	40	)	)	PUNCT
cana-4142	269	41	and	and	CCONJ
cana-4142	269	42	𝔗	𝔗	ADJ
cana-4142	269	43	:	:	PUNCT
cana-4142	269	44	𝔐	𝔐	PROPN
cana-4142	269	45	→	→	SYM
cana-4142	269	46	𝔐be	𝔐be	VERB
cana-4142	269	47	a	a	DET
cana-4142	269	48	mapping	mapping	NOUN
cana-4142	269	49	.	.	PUNCT
cana-4142	270	1	suppose	suppose	VERB
cana-4142	270	2	that	that	SCONJ
cana-4142	270	3	there	there	PRON
cana-4142	270	4	exists	exist	VERB
cana-4142	270	5	𝜆	𝜆	DET
cana-4142	270	6	∈	∈	PROPN
cana-4142	270	7	(	(	PUNCT
cana-4142	270	8	0	0	NUM
cana-4142	270	9	,	,	PUNCT
cana-4142	270	10	1	1	NUM
cana-4142	270	11	)	)	PUNCT
cana-4142	270	12	such	such	ADJ
cana-4142	270	13	that	that	SCONJ
cana-4142	270	14	𝔑(𝔗𝔭	𝔑(𝔗𝔭	NOUN
cana-4142	270	15	,	,	PUNCT
cana-4142	270	16	𝔗𝔮	𝔗𝔮	PROPN
cana-4142	270	17	,	,	PUNCT
cana-4142	270	18	𝔞	𝔞	NOUN
cana-4142	270	19	)	)	PUNCT
cana-4142	270	20	≤	≤	NOUN
cana-4142	270	21	𝔑(𝔭	𝔑(𝔭	PUNCT
cana-4142	270	22	,	,	PUNCT
cana-4142	270	23	𝔮	𝔮	PROPN
cana-4142	270	24	,	,	PUNCT
cana-4142	270	25	𝔞	𝔞	NOUN
cana-4142	270	26	)	)	PUNCT
cana-4142	270	27	(	(	PUNCT
cana-4142	270	28	7	7	X
cana-4142	270	29	)	)	PUNCT
cana-4142	270	30	for	for	ADP
cana-4142	270	31	all	all	PRON
cana-4142	270	32	𝔵	𝔵	PROPN
cana-4142	270	33	,	,	PUNCT
cana-4142	270	34	𝔶	𝔶	PRON
cana-4142	270	35	∈	∈	PROPN
cana-4142	270	36	𝔐.	𝔐.	PROPN
cana-4142	270	37	then	then	ADV
cana-4142	270	38	,	,	PUNCT
cana-4142	270	39	𝔗	𝔗	PROPN
cana-4142	270	40	has	have	VERB
cana-4142	270	41	a	a	DET
cana-4142	270	42	unique	unique	ADJ
cana-4142	270	43	fixed	fix	VERB
cana-4142	270	44	point	point	NOUN
cana-4142	270	45	.	.	PUNCT
cana-4142	271	1	remark	remark	PROPN
cana-4142	271	2	28	28	NUM
cana-4142	271	3	let	let	VERB
cana-4142	271	4	(	(	PUNCT
cana-4142	271	5	𝔐	𝔐	X
cana-4142	271	6	,	,	PUNCT
cana-4142	271	7	𝔑	𝔑	PROPN
cana-4142	271	8	,	,	PUNCT
cana-4142	271	9	⨁	⨁	PROPN
cana-4142	271	10	)	)	PUNCT
cana-4142	271	11	be	be	VERB
cana-4142	271	12	a	a	DET
cana-4142	271	13	revised	revise	VERB
cana-4142	271	14	fuzzy	fuzzy	ADJ
cana-4142	271	15	metric	metric	ADJ
cana-4142	271	16	space	space	NOUN
cana-4142	271	17	and	and	CCONJ
cana-4142	271	18	𝔗	𝔗	NOUN
cana-4142	271	19	:	:	PUNCT
cana-4142	271	20	𝔐	𝔐	PROPN
cana-4142	272	1	→	→	SYM
cana-4142	272	2	𝔐	𝔐	PRON
cana-4142	272	3	be	be	AUX
cana-4142	272	4	a	a	DET
cana-4142	272	5	mapping	mapping	NOUN
cana-4142	272	6	.	.	PUNCT
cana-4142	273	1	the	the	DET
cana-4142	273	2	contractive	contractive	ADJ
cana-4142	273	3	communications	communication	NOUN
cana-4142	273	4	on	on	ADP
cana-4142	273	5	applied	apply	VERB
cana-4142	273	6	nonlinear	nonlinear	ADJ
cana-4142	273	7	analysis	analysis	NOUN
cana-4142	273	8	issn	issn	NOUN
cana-4142	273	9	:	:	PUNCT
cana-4142	273	10	1074	1074	NUM
cana-4142	273	11	-	-	PUNCT
cana-4142	273	12	133x	133x	NUM
cana-4142	273	13	vol	vol	NOUN
cana-4142	273	14	x	x	NOUN
cana-4142	273	15	no	no	INTJ
cana-4142	273	16	.	.	PUNCT
cana-4142	274	1	y	y	PROPN
cana-4142	274	2	(	(	PUNCT
cana-4142	274	3	2025	2025	NUM
cana-4142	274	4	)	)	PUNCT
cana-4142	274	5	1328	1328	NUM
cana-4142	274	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	274	7	condition	condition	NOUN
cana-4142	274	8	(	(	PUNCT
cana-4142	274	9	7	7	X
cana-4142	274	10	)	)	PUNCT
cana-4142	274	11	tells	tell	VERB
cana-4142	274	12	that	that	SCONJ
cana-4142	274	13	the	the	DET
cana-4142	274	14	mapping	mapping	NOUN
cana-4142	274	15	𝔗	𝔗	PROPN
cana-4142	274	16	contract	contract	NOUN
cana-4142	274	17	the	the	DET
cana-4142	274	18	space	space	NOUN
cana-4142	274	19	with	with	ADP
cana-4142	274	20	respect	respect	NOUN
cana-4142	274	21	to	to	ADP
cana-4142	274	22	the	the	DET
cana-4142	274	23	parameter	parameter	NOUN
cana-4142	274	24	t	t	PROPN
cana-4142	274	25	in	in	ADP
cana-4142	274	26	the	the	DET
cana-4142	274	27	sense	sense	NOUN
cana-4142	274	28	that	that	SCONJ
cana-4142	274	29	the	the	DET
cana-4142	274	30	degree	degree	NOUN
cana-4142	274	31	of	of	ADP
cana-4142	274	32	the	the	DET
cana-4142	274	33	nearness	nearness	NOUN
cana-4142	274	34	of	of	ADP
cana-4142	274	35	images	image	NOUN
cana-4142	274	36	of	of	ADP
cana-4142	274	37	any	any	DET
cana-4142	274	38	two	two	NUM
cana-4142	274	39	points	point	NOUN
cana-4142	274	40	under	under	ADP
cana-4142	274	41	𝔗	𝔗	PROPN
cana-4142	274	42	is	be	AUX
cana-4142	274	43	not	not	PART
cana-4142	274	44	less	less	ADJ
cana-4142	274	45	than	than	ADP
cana-4142	274	46	the	the	DET
cana-4142	274	47	degree	degree	NOUN
cana-4142	274	48	of	of	ADP
cana-4142	274	49	the	the	DET
cana-4142	274	50	nearness	nearness	NOUN
cana-4142	274	51	of	of	ADP
cana-4142	274	52	corresponding	correspond	VERB
cana-4142	274	53	points	point	NOUN
cana-4142	274	54	(	(	PUNCT
cana-4142	274	55	obviously	obviously	ADV
cana-4142	274	56	in	in	ADP
cana-4142	274	57	case	case	NOUN
cana-4142	274	58	of	of	ADP
cana-4142	274	59	stationery	stationery	NOUN
cana-4142	274	60	revised	revise	VERB
cana-4142	274	61	fuzzy	fuzzy	ADJ
cana-4142	274	62	metric	metric	ADJ
cana-4142	274	63	spaces	space	NOUN
cana-4142	274	64	(	(	PUNCT
cana-4142	274	65	see	see	VERB
cana-4142	274	66	gregori	gregori	PROPN
cana-4142	274	67	and	and	CCONJ
cana-4142	274	68	romaguera	romaguera	NOUN
cana-4142	274	69	2004	2004	NUM
cana-4142	274	70	)	)	PUNCT
cana-4142	275	1	it	it	PRON
cana-4142	275	2	is	be	AUX
cana-4142	275	3	not	not	PART
cana-4142	275	4	applicable	applicable	ADJ
cana-4142	275	5	)	)	PUNCT
cana-4142	275	6	.	.	PUNCT
cana-4142	276	1	in	in	ADP
cana-4142	276	2	theorem	theorem	NOUN
cana-4142	276	3	26	26	NUM
cana-4142	276	4	,	,	PUNCT
cana-4142	276	5	the	the	DET
cana-4142	276	6	mapping	mapping	NOUN
cana-4142	276	7	contracts	contract	VERB
cana-4142	276	8	the	the	DET
cana-4142	276	9	space	space	NOUN
cana-4142	276	10	with	with	ADP
cana-4142	276	11	respect	respect	NOUN
cana-4142	276	12	to	to	ADP
cana-4142	276	13	only	only	ADV
cana-4142	276	14	parameter	parameter	NOUN
cana-4142	276	15	𝒶𝒯	𝒶𝒯	PROPN
cana-4142	276	16	for	for	ADP
cana-4142	276	17	some	some	DET
cana-4142	276	18	𝒯	𝒯	PROPN
cana-4142	276	19	∈	∈	PROPN
cana-4142	276	20	{	{	PUNCT
cana-4142	276	21	1	1	NUM
cana-4142	276	22	,	,	PUNCT
cana-4142	276	23	2	2	NUM
cana-4142	276	24	,	,	PUNCT
cana-4142	276	25	…	…	PUNCT
cana-4142	276	26	,	,	PUNCT
cana-4142	276	27	𝓀	𝓀	X
cana-4142	276	28	}	}	PUNCT
cana-4142	276	29	and	and	CCONJ
cana-4142	276	30	it	it	PRON
cana-4142	276	31	may	may	AUX
cana-4142	276	32	not	not	PART
cana-4142	276	33	be	be	AUX
cana-4142	276	34	contractive	contractive	ADJ
cana-4142	276	35	with	with	ADP
cana-4142	276	36	respect	respect	NOUN
cana-4142	276	37	to	to	ADP
cana-4142	276	38	other	other	ADJ
cana-4142	276	39	parameters	parameter	NOUN
cana-4142	276	40	.	.	PUNCT
cana-4142	277	1	similarly	similarly	ADV
cana-4142	277	2	,	,	PUNCT
cana-4142	277	3	(	(	PUNCT
cana-4142	277	4	𝔐	𝔐	X
cana-4142	277	5	,	,	PUNCT
cana-4142	277	6	𝔑	𝔑	PROPN
cana-4142	277	7	,	,	PUNCT
cana-4142	277	8	⨁	⨁	PROPN
cana-4142	277	9	)	)	PUNCT
cana-4142	277	10	is	be	AUX
cana-4142	277	11	assumed	assume	VERB
cana-4142	277	12	l	l	ADJ
cana-4142	277	13	-	-	ADJ
cana-4142	277	14	natural	natural	ADJ
cana-4142	277	15	k	k	ADJ
cana-4142	277	16	-	-	ADJ
cana-4142	277	17	fuzzy	fuzzy	ADJ
cana-4142	277	18	metric	metric	ADJ
cana-4142	277	19	space	space	NOUN
cana-4142	277	20	for	for	ADP
cana-4142	277	21	at	at	ADV
cana-4142	277	22	least	least	ADV
cana-4142	277	23	one	one	NUM
cana-4142	277	24	𝒯	𝒯	PROPN
cana-4142	277	25	∈	∈	PROPN
cana-4142	277	26	{	{	PUNCT
cana-4142	277	27	1	1	NUM
cana-4142	277	28	,	,	PUNCT
cana-4142	277	29	2	2	NUM
cana-4142	277	30	,	,	PUNCT
cana-4142	277	31	…	…	PUNCT
cana-4142	277	32	,	,	PUNCT
cana-4142	277	33	𝓀	𝓀	X
cana-4142	277	34	}	}	PUNCT
cana-4142	277	35	only	only	ADV
cana-4142	277	36	.	.	PUNCT
cana-4142	278	1	the	the	DET
cana-4142	278	2	following	follow	VERB
cana-4142	278	3	example	example	NOUN
cana-4142	278	4	verifies	verifie	NOUN
cana-4142	278	5	the	the	DET
cana-4142	278	6	above	above	ADJ
cana-4142	278	7	remark	remark	NOUN
cana-4142	278	8	.	.	PUNCT
cana-4142	279	1	example	example	NOUN
cana-4142	279	2	29	29	NUM
cana-4142	279	3	let	let	VERB
cana-4142	279	4	𝔐	𝔐	PRON
cana-4142	279	5	=	=	PUNCT
cana-4142	280	1	[	[	X
cana-4142	280	2	0	0	NUM
cana-4142	280	3	,	,	PUNCT
cana-4142	280	4	1	1	NUM
cana-4142	280	5	]	]	SYM
cana-4142	280	6	×	×	NOUN
cana-4142	281	1	[	[	X
cana-4142	281	2	0	0	NUM
cana-4142	281	3	,	,	PUNCT
cana-4142	281	4	1	1	NUM
cana-4142	281	5	]	]	PUNCT
cana-4142	281	6	and	and	CCONJ
cana-4142	281	7	⨁	⨁	PROPN
cana-4142	281	8	be	be	VERB
cana-4142	281	9	the	the	DET
cana-4142	281	10	product	product	NOUN
cana-4142	281	11	t	t	NOUN
cana-4142	281	12	-	-	PUNCT
cana-4142	281	13	conorm	conorm	NOUN
cana-4142	281	14	and	and	CCONJ
cana-4142	281	15	the	the	DET
cana-4142	281	16	revised	revise	VERB
cana-4142	281	17	fuzzy	fuzzy	ADJ
cana-4142	281	18	set	set	VERB
cana-4142	281	19	𝔑	𝔑	NOUN
cana-4142	281	20	on	on	ADP
cana-4142	281	21	𝔐2	𝔐2	ADJ
cana-4142	281	22	×	×	NOUN
cana-4142	281	23	(	(	PUNCT
cana-4142	281	24	0	0	NUM
cana-4142	281	25	,	,	PUNCT
cana-4142	281	26	∞)2	∞)2	PROPN
cana-4142	281	27	be	be	AUX
cana-4142	281	28	defined	define	VERB
cana-4142	281	29	by	by	ADP
cana-4142	281	30	𝔑(𝔭	𝔑(𝔭	X
cana-4142	281	31	,	,	PUNCT
cana-4142	281	32	𝔮	𝔮	NOUN
cana-4142	281	33	,	,	PUNCT
cana-4142	281	34	𝔞1	𝔞1	NOUN
cana-4142	281	35	,	,	PUNCT
cana-4142	281	36	𝔞2	𝔞2	NUM
cana-4142	281	37	)	)	PUNCT
cana-4142	281	38	=	=	SYM
cana-4142	282	1	1	1	NUM
cana-4142	282	2	−	−	NOUN
cana-4142	282	3	[	[	X
cana-4142	282	4	1	1	NUM
cana-4142	282	5	+	+	NUM
cana-4142	282	6	|𝔮1	|𝔮1	NOUN
cana-4142	282	7	−	−	PROPN
cana-4142	282	8	𝔭1|	𝔭1|	PROPN
cana-4142	283	1	+	+	CCONJ
cana-4142	283	2	|𝔮2	|𝔮2	NUM
cana-4142	283	3	−	−	PROPN
cana-4142	283	4	𝔭2|	𝔭2|	NUM
cana-4142	283	5	𝔞1	𝔞1	NOUN
cana-4142	283	6	]	]	PUNCT
cana-4142	283	7	−1	−1	NOUN
cana-4142	283	8	for	for	ADP
cana-4142	283	9	all	all	PRON
cana-4142	283	10	𝔭	𝔭	NOUN
cana-4142	283	11	=	=	PUNCT
cana-4142	283	12	(	(	PUNCT
cana-4142	283	13	𝔭1	𝔭1	PROPN
cana-4142	283	14	,	,	PUNCT
cana-4142	283	15	𝔭2	𝔭2	PROPN
cana-4142	283	16	)	)	PUNCT
cana-4142	283	17	,	,	PUNCT
cana-4142	283	18	𝔮	𝔮	X
cana-4142	283	19	=	=	SYM
cana-4142	283	20	(	(	PUNCT
cana-4142	283	21	𝔮1	𝔮1	PROPN
cana-4142	283	22	,	,	PUNCT
cana-4142	283	23	𝔮2	𝔮2	ADJ
cana-4142	283	24	)	)	PUNCT
cana-4142	283	25	∈	∈	PROPN
cana-4142	283	26	𝔐	𝔐	PROPN
cana-4142	283	27	and	and	CCONJ
cana-4142	283	28	𝔞1	𝔞1	NOUN
cana-4142	283	29	,	,	PUNCT
cana-4142	283	30	𝔞2	𝔞2	PROPN
cana-4142	283	31	>	>	X
cana-4142	283	32	0	0	PROPN
cana-4142	283	33	.	.	PUNCT
cana-4142	284	1	then	then	ADV
cana-4142	284	2	,	,	PUNCT
cana-4142	284	3	(	(	PUNCT
cana-4142	284	4	𝔐	𝔐	X
cana-4142	284	5	,	,	PUNCT
cana-4142	284	6	𝔑	𝔑	PROPN
cana-4142	284	7	,	,	PUNCT
cana-4142	284	8	⨁	⨁	PROPN
cana-4142	284	9	)	)	PUNCT
cana-4142	284	10	is	be	AUX
cana-4142	284	11	a	a	DET
cana-4142	284	12	𝔾	𝔾	PROPN
cana-4142	284	13	−complete	−complete	PROPN
cana-4142	284	14	revised	revise	VERB
cana-4142	284	15	fuzzy	fuzzy	ADJ
cana-4142	284	16	2	2	NUM
cana-4142	284	17	-	-	PUNCT
cana-4142	284	18	metric	metric	ADJ
cana-4142	284	19	space	space	NOUN
cana-4142	284	20	(	(	PUNCT
cana-4142	284	21	𝓀	𝓀	X
cana-4142	284	22	=	=	NOUN
cana-4142	284	23	2	2	NUM
cana-4142	284	24	)	)	PUNCT
cana-4142	284	25	.	.	PUNCT
cana-4142	285	1	moreover	moreover	ADV
cana-4142	285	2	,	,	PUNCT
cana-4142	285	3	lim	lim	PROPN
cana-4142	285	4	𝔞1→+∞	𝔞1→+∞	PROPN
cana-4142	285	5	𝔑(𝔭	𝔑(𝔭	X
cana-4142	285	6	,	,	PUNCT
cana-4142	285	7	𝔮	𝔮	NOUN
cana-4142	285	8	,	,	PUNCT
cana-4142	285	9	𝔞1	𝔞1	NOUN
cana-4142	285	10	,	,	PUNCT
cana-4142	285	11	𝔞2	𝔞2	NUM
cana-4142	285	12	)	)	PUNCT
cana-4142	285	13	=	=	SYM
cana-4142	285	14	0	0	NUM
cana-4142	285	15	for	for	ADP
cana-4142	285	16	all	all	DET
cana-4142	285	17	𝔭	𝔭	NOUN
cana-4142	285	18	,	,	PUNCT
cana-4142	285	19	𝔮	𝔮	X
cana-4142	285	20	∈	∈	PROPN
cana-4142	285	21	𝔐	𝔐	PROPN
cana-4142	285	22	,	,	PUNCT
cana-4142	285	23	𝔞2	𝔞2	PROPN
cana-4142	285	24	>	>	PUNCT
cana-4142	285	25	0,that	0,that	PRON
cana-4142	285	26	is	be	AUX
cana-4142	285	27	,	,	PUNCT
cana-4142	285	28	(	(	PUNCT
cana-4142	285	29	𝔐	𝔐	X
cana-4142	285	30	,	,	PUNCT
cana-4142	285	31	𝔑	𝔑	PROPN
cana-4142	285	32	,	,	PUNCT
cana-4142	285	33	⨁	⨁	PROPN
cana-4142	285	34	)	)	PUNCT
cana-4142	285	35	is	be	AUX
cana-4142	285	36	a	a	DET
cana-4142	285	37	1	1	NUM
cana-4142	285	38	−natural	−natural	ADJ
cana-4142	285	39	revised	revise	VERB
cana-4142	285	40	fuzzy	fuzzy	ADJ
cana-4142	285	41	2	2	NUM
cana-4142	285	42	-	-	PUNCT
cana-4142	285	43	metric	metric	ADJ
cana-4142	285	44	space	space	NOUN
cana-4142	285	45	.	.	PUNCT
cana-4142	286	1	define	define	VERB
cana-4142	286	2	a	a	DET
cana-4142	286	3	mapping	mapping	NOUN
cana-4142	286	4	𝔗	𝔗	NOUN
cana-4142	286	5	:	:	PUNCT
cana-4142	286	6	𝔐	𝔐	PROPN
cana-4142	286	7	→	→	SYM
cana-4142	286	8	𝔐	𝔐	NOUN
cana-4142	286	9	by	by	ADP
cana-4142	286	10	𝔑(𝔗𝔭	𝔑(𝔗𝔭	NOUN
cana-4142	286	11	,	,	PUNCT
cana-4142	286	12	𝔗𝔮	𝔗𝔮	PROPN
cana-4142	286	13	,	,	PUNCT
cana-4142	286	14	𝜆𝔞1	𝜆𝔞1	PROPN
cana-4142	286	15	,	,	PUNCT
cana-4142	286	16	𝔞2	𝔞2	PROPN
cana-4142	286	17	)	)	PUNCT
cana-4142	286	18	=	=	SYM
cana-4142	287	1	1	1	NUM
cana-4142	287	2	−	−	NOUN
cana-4142	288	1	[	[	X
cana-4142	288	2	1	1	NUM
cana-4142	288	3	+	+	NUM
cana-4142	288	4	|𝔮1−𝔭1|+|𝔮2−𝔭2|	|𝔮1−𝔭1|+|𝔮2−𝔭2|	PROPN
cana-4142	288	5	2𝜆𝔞1	2𝜆𝔞1	NUM
cana-4142	288	6	]	]	PUNCT
cana-4142	288	7	−1	−1	NOUN
cana-4142	288	8	≤	≤	ADV
cana-4142	288	9	1	1	NUM
cana-4142	288	10	−	−	NOUN
cana-4142	289	1	[	[	X
cana-4142	289	2	1	1	NUM
cana-4142	289	3	+	+	NUM
cana-4142	289	4	|𝔮1−𝔭1|+|𝔮2−𝔭2|	|𝔮1−𝔭1|+|𝔮2−𝔭2|	PROPN
cana-4142	289	5	𝔞1	𝔞1	NOUN
cana-4142	289	6	]	]	PUNCT
cana-4142	289	7	−1	−1	NOUN
cana-4142	289	8	=	=	SYM
cana-4142	289	9	𝔑(𝔭	𝔑(𝔭	X
cana-4142	289	10	,	,	PUNCT
cana-4142	289	11	𝔮	𝔮	NOUN
cana-4142	289	12	,	,	PUNCT
cana-4142	289	13	𝔞1	𝔞1	NOUN
cana-4142	289	14	,	,	PUNCT
cana-4142	289	15	𝔞2	𝔞2	PROPN
cana-4142	289	16	)	)	PUNCT
cana-4142	289	17	for	for	ADP
cana-4142	289	18	𝜆	𝜆	DET
cana-4142	289	19	∈	∈	PROPN
cana-4142	290	1	[	[	X
cana-4142	290	2	1/2	1/2	NUM
cana-4142	290	3	,	,	PUNCT
cana-4142	290	4	1	1	NUM
cana-4142	290	5	)	)	PUNCT
cana-4142	290	6	.	.	PUNCT
cana-4142	291	1	by	by	ADP
cana-4142	291	2	theorem	theorem	NOUN
cana-4142	291	3	26	26	NUM
cana-4142	291	4	,	,	PUNCT
cana-4142	291	5	𝔗	𝔗	PROPN
cana-4142	291	6	has	have	VERB
cana-4142	291	7	a	a	DET
cana-4142	291	8	unique	unique	ADJ
cana-4142	291	9	fixed	fix	VERB
cana-4142	291	10	point	point	NOUN
cana-4142	291	11	.	.	PUNCT
cana-4142	292	1	in	in	ADP
cana-4142	292	2	this	this	DET
cana-4142	292	3	case	case	NOUN
cana-4142	292	4	,	,	PUNCT
cana-4142	292	5	a	a	DET
cana-4142	292	6	point	point	NOUN
cana-4142	292	7	(	(	PUNCT
cana-4142	292	8	0,0	0,0	NOUN
cana-4142	292	9	)	)	PUNCT
cana-4142	292	10	∈	∈	NOUN
cana-4142	292	11	𝔐	𝔐	PROPN
cana-4142	292	12	is	be	AUX
cana-4142	292	13	a	a	DET
cana-4142	292	14	fixed	fixed	ADJ
cana-4142	292	15	point	point	NOUN
cana-4142	292	16	of	of	ADP
cana-4142	292	17	𝔗.	𝔗.	PROPN
cana-4142	292	18	in	in	ADP
cana-4142	292	19	theorem	theorem	PROPN
cana-4142	292	20	26	26	NUM
cana-4142	292	21	,	,	PUNCT
cana-4142	292	22	corresponding	correspond	VERB
cana-4142	292	23	to	to	ADP
cana-4142	292	24	condition	condition	NOUN
cana-4142	292	25	(	(	PUNCT
cana-4142	292	26	2	2	NUM
cana-4142	292	27	)	)	PUNCT
cana-4142	292	28	,	,	PUNCT
cana-4142	292	29	we	we	PRON
cana-4142	292	30	assume	assume	VERB
cana-4142	292	31	that	that	SCONJ
cana-4142	292	32	the	the	DET
cana-4142	292	33	space	space	NOUN
cana-4142	292	34	(	(	PUNCT
cana-4142	292	35	𝔐	𝔐	PROPN
cana-4142	292	36	,	,	PUNCT
cana-4142	292	37	𝔑	𝔑	PROPN
cana-4142	292	38	,	,	PUNCT
cana-4142	292	39	⨁	⨁	PROPN
cana-4142	292	40	)	)	PUNCT
cana-4142	292	41	is	be	AUX
cana-4142	292	42	𝒯	𝒯	PROPN
cana-4142	292	43	−natural	−natural	PROPN
cana-4142	292	44	.	.	PUNCT
cana-4142	293	1	notice	notice	VERB
cana-4142	293	2	that	that	SCONJ
cana-4142	293	3	,	,	PUNCT
cana-4142	293	4	for	for	ADP
cana-4142	293	5	the	the	DET
cana-4142	293	6	existence	existence	NOUN
cana-4142	293	7	of	of	ADP
cana-4142	293	8	a	a	DET
cana-4142	293	9	fixed	fix	VERB
cana-4142	293	10	point	point	NOUN
cana-4142	293	11	,	,	PUNCT
cana-4142	293	12	the	the	DET
cana-4142	293	13	𝒯	𝒯	PROPN
cana-4142	293	14	−naturalness	−naturalness	NOUN
cana-4142	293	15	can	can	AUX
cana-4142	293	16	not	not	PART
cana-4142	293	17	be	be	AUX
cana-4142	293	18	replaced	replace	VERB
cana-4142	293	19	by	by	ADP
cana-4142	293	20	the	the	DET
cana-4142	293	21	𝔪	𝔪	NOUN
cana-4142	293	22	−naturalness	−naturalness	NOUN
cana-4142	293	23	with	with	ADP
cana-4142	293	24	𝔪	𝔪	DET
cana-4142	293	25	≠	≠	PROPN
cana-4142	293	26	𝒯.	𝒯.	PROPN
cana-4142	293	27	the	the	DET
cana-4142	293	28	following	follow	VERB
cana-4142	293	29	example	example	NOUN
cana-4142	293	30	verifies	verifie	NOUN
cana-4142	293	31	this	this	DET
cana-4142	293	32	fact	fact	NOUN
cana-4142	293	33	.	.	PUNCT
cana-4142	294	1	example	example	NOUN
cana-4142	294	2	30	30	NUM
cana-4142	294	3	let	let	VERB
cana-4142	294	4	𝔐	𝔐	PRON
cana-4142	294	5	=	=	PUNCT
cana-4142	295	1	[	[	X
cana-4142	295	2	0	0	NUM
cana-4142	295	3	,	,	PUNCT
cana-4142	295	4	1	1	NUM
cana-4142	295	5	]	]	SYM
cana-4142	295	6	×	×	NOUN
cana-4142	296	1	[	[	X
cana-4142	296	2	0	0	NUM
cana-4142	296	3	,	,	PUNCT
cana-4142	296	4	1	1	NUM
cana-4142	296	5	]	]	PUNCT
cana-4142	296	6	and	and	CCONJ
cana-4142	296	7	⨁	⨁	PROPN
cana-4142	296	8	be	be	VERB
cana-4142	296	9	the	the	DET
cana-4142	296	10	product	product	NOUN
cana-4142	296	11	t	t	NOUN
cana-4142	296	12	-	-	PUNCT
cana-4142	296	13	conorm	conorm	NOUN
cana-4142	296	14	and	and	CCONJ
cana-4142	296	15	the	the	DET
cana-4142	296	16	revised	revise	VERB
cana-4142	296	17	fuzzy	fuzzy	ADJ
cana-4142	296	18	set	set	VERB
cana-4142	296	19	𝔑	𝔑	NOUN
cana-4142	296	20	on	on	ADP
cana-4142	296	21	𝔐2	𝔐2	ADJ
cana-4142	296	22	×	×	NOUN
cana-4142	296	23	(	(	PUNCT
cana-4142	296	24	0	0	NUM
cana-4142	296	25	,	,	PUNCT
cana-4142	296	26	∞)2	∞)2	PROPN
cana-4142	296	27	be	be	AUX
cana-4142	296	28	defined	define	VERB
cana-4142	296	29	by	by	ADP
cana-4142	296	30	𝔑(𝔭	𝔑(𝔭	X
cana-4142	296	31	,	,	PUNCT
cana-4142	296	32	𝔮	𝔮	NOUN
cana-4142	296	33	,	,	PUNCT
cana-4142	296	34	𝔞1	𝔞1	NOUN
cana-4142	296	35	,	,	PUNCT
cana-4142	296	36	𝔞2	𝔞2	NUM
cana-4142	296	37	)	)	PUNCT
cana-4142	296	38	=	=	SYM
cana-4142	297	1	1	1	NUM
cana-4142	297	2	−	−	NOUN
cana-4142	297	3	[	[	X
cana-4142	297	4	1	1	NUM
cana-4142	297	5	+	+	NUM
cana-4142	297	6	|𝔮1	|𝔮1	NOUN
cana-4142	297	7	−	−	PROPN
cana-4142	297	8	𝔭1|	𝔭1|	PROPN
cana-4142	298	1	+	+	CCONJ
cana-4142	298	2	|𝔮2	|𝔮2	NUM
cana-4142	298	3	−	−	PROPN
cana-4142	298	4	𝔭2|	𝔭2|	NUM
cana-4142	298	5	𝔞2	𝔞2	PROPN
cana-4142	298	6	]	]	PUNCT
cana-4142	298	7	−1	−1	NOUN
cana-4142	298	8	for	for	ADP
cana-4142	298	9	all	all	PRON
cana-4142	298	10	𝔭	𝔭	NOUN
cana-4142	298	11	=	=	PUNCT
cana-4142	298	12	(	(	PUNCT
cana-4142	298	13	𝔭1	𝔭1	PROPN
cana-4142	298	14	,	,	PUNCT
cana-4142	298	15	𝔭2	𝔭2	PROPN
cana-4142	298	16	)	)	PUNCT
cana-4142	298	17	,	,	PUNCT
cana-4142	298	18	𝔮	𝔮	X
cana-4142	298	19	=	=	SYM
cana-4142	298	20	(	(	PUNCT
cana-4142	298	21	𝔮1	𝔮1	PROPN
cana-4142	298	22	,	,	PUNCT
cana-4142	298	23	𝔮2	𝔮2	ADJ
cana-4142	298	24	)	)	PUNCT
cana-4142	298	25	∈	∈	PROPN
cana-4142	298	26	𝔐	𝔐	PROPN
cana-4142	298	27	and	and	CCONJ
cana-4142	298	28	𝔞1	𝔞1	NOUN
cana-4142	298	29	,	,	PUNCT
cana-4142	298	30	𝔞2	𝔞2	PROPN
cana-4142	298	31	>	>	X
cana-4142	298	32	0	0	PROPN
cana-4142	298	33	.	.	PUNCT
cana-4142	299	1	then	then	ADV
cana-4142	299	2	,	,	PUNCT
cana-4142	299	3	(	(	PUNCT
cana-4142	299	4	𝔐	𝔐	X
cana-4142	299	5	,	,	PUNCT
cana-4142	299	6	𝔑	𝔑	PROPN
cana-4142	299	7	,	,	PUNCT
cana-4142	299	8	⨁	⨁	PROPN
cana-4142	299	9	)	)	PUNCT
cana-4142	299	10	is	be	AUX
cana-4142	299	11	a	a	DET
cana-4142	299	12	𝔾	𝔾	PROPN
cana-4142	299	13	−complete	−complete	PROPN
cana-4142	299	14	revised	revise	VERB
cana-4142	299	15	fuzzy	fuzzy	ADJ
cana-4142	299	16	2	2	NUM
cana-4142	299	17	-	-	PUNCT
cana-4142	299	18	metric	metric	ADJ
cana-4142	299	19	space	space	NOUN
cana-4142	299	20	(	(	PUNCT
cana-4142	299	21	𝓀	𝓀	X
cana-4142	299	22	=	=	NOUN
cana-4142	299	23	2	2	NUM
cana-4142	299	24	)	)	PUNCT
cana-4142	299	25	.	.	PUNCT
cana-4142	300	1	moreover	moreover	ADV
cana-4142	300	2	,	,	PUNCT
cana-4142	300	3	lim	lim	PROPN
cana-4142	300	4	𝔞1→+∞	𝔞1→+∞	PROPN
cana-4142	300	5	𝔑(𝔭	𝔑(𝔭	X
cana-4142	300	6	,	,	PUNCT
cana-4142	300	7	𝔮	𝔮	NOUN
cana-4142	300	8	,	,	PUNCT
cana-4142	300	9	𝔞1	𝔞1	NOUN
cana-4142	300	10	,	,	PUNCT
cana-4142	300	11	𝔞2	𝔞2	NUM
cana-4142	300	12	)	)	PUNCT
cana-4142	300	13	=	=	SYM
cana-4142	300	14	0	0	NUM
cana-4142	300	15	for	for	ADP
cana-4142	300	16	all	all	DET
cana-4142	300	17	𝔭	𝔭	NOUN
cana-4142	300	18	,	,	PUNCT
cana-4142	300	19	𝔮	𝔮	PROPN
cana-4142	300	20	∈	∈	PROPN
cana-4142	300	21	𝔐	𝔐	PROPN
cana-4142	300	22	,	,	PUNCT
cana-4142	300	23	𝔞1	𝔞1	PROPN
cana-4142	300	24	>	>	X
cana-4142	300	25	0	0	NUM
cana-4142	300	26	,	,	PUNCT
cana-4142	300	27	that	that	ADV
cana-4142	300	28	is	is	ADV
cana-4142	300	29	,	,	PUNCT
cana-4142	300	30	(	(	PUNCT
cana-4142	300	31	𝔐	𝔐	X
cana-4142	300	32	,	,	PUNCT
cana-4142	300	33	𝔑	𝔑	PROPN
cana-4142	300	34	,	,	PUNCT
cana-4142	300	35	⨁	⨁	PROPN
cana-4142	300	36	)	)	PUNCT
cana-4142	300	37	is	be	AUX
cana-4142	300	38	a	a	DET
cana-4142	300	39	2	2	NUM
cana-4142	300	40	−natural	−natural	NOUN
cana-4142	300	41	revised	revise	VERB
cana-4142	300	42	fuzzy	fuzzy	ADJ
cana-4142	300	43	2	2	NUM
cana-4142	300	44	-	-	PUNCT
cana-4142	300	45	metric	metric	ADJ
cana-4142	300	46	space	space	NOUN
cana-4142	300	47	.	.	PUNCT
cana-4142	301	1	define	define	VERB
cana-4142	301	2	a	a	DET
cana-4142	301	3	mapping	mapping	NOUN
cana-4142	301	4	𝔗	𝔗	NOUN
cana-4142	301	5	:	:	PUNCT
cana-4142	301	6	𝔐	𝔐	PROPN
cana-4142	301	7	→	→	SYM
cana-4142	301	8	𝔐	𝔐	NOUN
cana-4142	301	9	by	by	ADP
cana-4142	301	10	𝔗(𝔭1	𝔗(𝔭1	PROPN
cana-4142	301	11	,	,	PUNCT
cana-4142	301	12	𝔭2	𝔭2	PROPN
cana-4142	301	13	)	)	PUNCT
cana-4142	302	1	=	=	SYM
cana-4142	302	2	(	(	PUNCT
cana-4142	302	3	𝔭1	𝔭1	PROPN
cana-4142	302	4	,	,	PUNCT
cana-4142	302	5	𝔭2	𝔭2	PROPN
cana-4142	302	6	)	)	PUNCT
cana-4142	302	7	for	for	ADP
cana-4142	302	8	all	all	PRON
cana-4142	302	9	(	(	PUNCT
cana-4142	302	10	𝔭1	𝔭1	ADJ
cana-4142	302	11	,	,	PUNCT
cana-4142	302	12	𝔭2	𝔭2	PROPN
cana-4142	302	13	)	)	PUNCT
cana-4142	302	14	∈	∈	PROPN
cana-4142	302	15	𝔐.	𝔐.	PROPN
cana-4142	302	16	communications	communication	NOUN
cana-4142	302	17	on	on	ADP
cana-4142	302	18	applied	apply	VERB
cana-4142	302	19	nonlinear	nonlinear	ADJ
cana-4142	302	20	analysis	analysis	NOUN
cana-4142	302	21	issn	issn	NOUN
cana-4142	302	22	:	:	PUNCT
cana-4142	302	23	1074	1074	NUM
cana-4142	302	24	-	-	PUNCT
cana-4142	302	25	133x	133x	NUM
cana-4142	302	26	vol	vol	NOUN
cana-4142	302	27	x	x	NOUN
cana-4142	302	28	no	no	INTJ
cana-4142	302	29	.	.	PUNCT
cana-4142	303	1	y	y	PROPN
cana-4142	303	2	(	(	PUNCT
cana-4142	303	3	2025	2025	NUM
cana-4142	303	4	)	)	PUNCT
cana-4142	303	5	1329	1329	NUM
cana-4142	303	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	303	7	notice	notice	VERB
cana-4142	303	8	that	that	SCONJ
cana-4142	303	9	,	,	PUNCT
cana-4142	303	10	for	for	ADP
cana-4142	303	11	any	any	DET
cana-4142	303	12	arbitrary	arbitrary	ADJ
cana-4142	303	13	𝜆	𝜆	ADP
cana-4142	303	14	∈	∈	PROPN
cana-4142	303	15	(	(	PUNCT
cana-4142	303	16	0	0	NUM
cana-4142	303	17	,	,	PUNCT
cana-4142	303	18	1	1	NUM
cana-4142	303	19	)	)	PUNCT
cana-4142	303	20	𝔑(𝔗𝔭	𝔑(𝔗𝔭	NOUN
cana-4142	303	21	,	,	PUNCT
cana-4142	303	22	𝔗𝔮	𝔗𝔮	PROPN
cana-4142	303	23	,	,	PUNCT
cana-4142	303	24	𝜆𝔞1	𝜆𝔞1	PROPN
cana-4142	303	25	,	,	PUNCT
cana-4142	303	26	𝔞2	𝔞2	PROPN
cana-4142	303	27	)	)	PUNCT
cana-4142	303	28	≤	≤	NOUN
cana-4142	303	29	𝔑(𝔭	𝔑(𝔭	PUNCT
cana-4142	303	30	,	,	PUNCT
cana-4142	303	31	𝔮	𝔮	NOUN
cana-4142	303	32	,	,	PUNCT
cana-4142	303	33	𝔞1	𝔞1	NOUN
cana-4142	303	34	,	,	PUNCT
cana-4142	303	35	𝔞2	𝔞2	PROPN
cana-4142	303	36	)	)	PUNCT
cana-4142	303	37	but	but	CCONJ
cana-4142	303	38	the	the	DET
cana-4142	303	39	fixed	fix	VERB
cana-4142	303	40	point	point	NOUN
cana-4142	303	41	of	of	ADP
cana-4142	303	42	𝔗	𝔗	PROPN
cana-4142	303	43	is	be	AUX
cana-4142	303	44	not	not	PART
cana-4142	303	45	unique	unique	ADJ
cana-4142	303	46	.	.	PUNCT
cana-4142	304	1	indeed	indeed	ADV
cana-4142	304	2	,	,	PUNCT
cana-4142	304	3	every	every	DET
cana-4142	304	4	point	point	NOUN
cana-4142	304	5	(	(	PUNCT
cana-4142	304	6	𝔭1	𝔭1	ADJ
cana-4142	304	7	,	,	PUNCT
cana-4142	304	8	𝔭2	𝔭2	PROPN
cana-4142	304	9	)	)	PUNCT
cana-4142	304	10	∈	∈	PROPN
cana-4142	304	11	𝔐	𝔐	PROPN
cana-4142	304	12	is	be	AUX
cana-4142	304	13	a	a	DET
cana-4142	304	14	fixed	fix	VERB
cana-4142	304	15	point	point	NOUN
cana-4142	304	16	of	of	ADP
cana-4142	304	17	𝔗.	𝔗.	PROPN
cana-4142	304	18	finally	finally	ADV
cana-4142	304	19	,	,	PUNCT
cana-4142	304	20	we	we	PRON
cana-4142	304	21	will	will	AUX
cana-4142	304	22	prove	prove	VERB
cana-4142	304	23	a	a	DET
cana-4142	304	24	fixed	fix	VERB
cana-4142	304	25	-	-	PUNCT
cana-4142	304	26	point	point	NOUN
cana-4142	304	27	result	result	NOUN
cana-4142	304	28	for	for	ADP
cana-4142	304	29	a	a	DET
cana-4142	304	30	revised	revise	VERB
cana-4142	304	31	fuzzy	fuzzy	ADJ
cana-4142	304	32	𝓀	𝓀	PROPN
cana-4142	304	33	−contraction	−contraction	NUM
cana-4142	304	34	mapping	mapping	NOUN
cana-4142	304	35	.	.	PUNCT
cana-4142	305	1	we	we	PRON
cana-4142	305	2	begin	begin	VERB
cana-4142	305	3	with	with	ADP
cana-4142	305	4	the	the	DET
cana-4142	305	5	definition	definition	NOUN
cana-4142	305	6	of	of	ADP
cana-4142	305	7	a	a	DET
cana-4142	305	8	revised	revise	VERB
cana-4142	305	9	fuzzy	fuzzy	ADJ
cana-4142	305	10	𝓀	𝓀	PROPN
cana-4142	305	11	−contraction	−contraction	NOUN
cana-4142	305	12	mapping	mapping	NOUN
cana-4142	305	13	as	as	SCONJ
cana-4142	305	14	follows	follow	VERB
cana-4142	305	15	:	:	PUNCT
cana-4142	305	16	definition	definition	NOUN
cana-4142	305	17	31	31	NUM
cana-4142	305	18	let	let	VERB
cana-4142	305	19	(	(	PUNCT
cana-4142	305	20	𝔐	𝔐	X
cana-4142	305	21	,	,	PUNCT
cana-4142	305	22	𝔑	𝔑	PROPN
cana-4142	305	23	,	,	PUNCT
cana-4142	305	24	⨁	⨁	PROPN
cana-4142	305	25	)	)	PUNCT
cana-4142	305	26	be	be	VERB
cana-4142	305	27	a	a	DET
cana-4142	305	28	revised	revise	VERB
cana-4142	305	29	fuzzy	fuzzy	ADJ
cana-4142	305	30	𝓀	𝓀	X
cana-4142	305	31	−metric	−metric	ADJ
cana-4142	305	32	space	space	NOUN
cana-4142	305	33	.	.	PUNCT
cana-4142	306	1	a	a	DET
cana-4142	306	2	mapping	mapping	NOUN
cana-4142	306	3	𝔗	𝔗	NOUN
cana-4142	306	4	:	:	PUNCT
cana-4142	306	5	𝔐	𝔐	PROPN
cana-4142	306	6	→	→	SYM
cana-4142	306	7	𝔐	𝔐	PROPN
cana-4142	306	8	is	be	AUX
cana-4142	306	9	called	call	VERB
cana-4142	306	10	a	a	DET
cana-4142	306	11	revised	revise	VERB
cana-4142	306	12	fuzzy	fuzzy	ADJ
cana-4142	306	13	𝓀	𝓀	PROPN
cana-4142	306	14	−contraction	−contraction	NOUN
cana-4142	306	15	mapping	mapping	NOUN
cana-4142	306	16	if	if	SCONJ
cana-4142	306	17	𝔑(𝔗𝔭	𝔑(𝔗𝔭	NUM
cana-4142	306	18	,	,	PUNCT
cana-4142	306	19	𝔗𝔮	𝔗𝔮	PROPN
cana-4142	306	20	,	,	PUNCT
cana-4142	306	21	𝒶1	𝒶1	NOUN
cana-4142	306	22	𝓀	𝓀	PROPN
cana-4142	306	23	)	)	PUNCT
cana-4142	306	24	≤	≤	NUM
cana-4142	306	25	𝜆{𝔑(𝔭	𝜆{𝔑(𝔭	NOUN
cana-4142	306	26	,	,	PUNCT
cana-4142	306	27	𝔮	𝔮	PROPN
cana-4142	306	28	,	,	PUNCT
cana-4142	306	29	𝒶1	𝒶1	PROPN
cana-4142	306	30	𝓀	𝓀	PROPN
cana-4142	306	31	)	)	PUNCT
cana-4142	306	32	}	}	PUNCT
cana-4142	306	33	(	(	PUNCT
cana-4142	306	34	8)	8)	NUM
cana-4142	306	35	for	for	ADP
cana-4142	306	36	all	all	DET
cana-4142	306	37	(	(	PUNCT
cana-4142	306	38	𝔭1	𝔭1	ADJ
cana-4142	306	39	,	,	PUNCT
cana-4142	306	40	𝔭2	𝔭2	PROPN
cana-4142	306	41	)	)	PUNCT
cana-4142	306	42	∈	∈	PROPN
cana-4142	306	43	𝔐and𝒶1	𝔐and𝒶1	PROPN
cana-4142	306	44	,	,	PUNCT
cana-4142	306	45	𝒶2	𝒶2	PROPN
cana-4142	306	46	,	,	PUNCT
cana-4142	306	47	.	.	PUNCT
cana-4142	306	48	.	.	PUNCT
cana-4142	307	1	.	.	PUNCT
cana-4142	308	1	,	,	PUNCT
cana-4142	308	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	308	3	>	>	X
cana-4142	308	4	0	0	NUM
cana-4142	308	5	,	,	PUNCT
cana-4142	309	1	where	where	SCONJ
cana-4142	309	2	𝜆	𝜆	DET
cana-4142	309	3	∈	∈	PROPN
cana-4142	309	4	[	[	X
cana-4142	309	5	0	0	NUM
cana-4142	309	6	,	,	PUNCT
cana-4142	309	7	1	1	NUM
cana-4142	309	8	)	)	PUNCT
cana-4142	309	9	is	be	AUX
cana-4142	309	10	a	a	DET
cana-4142	309	11	constant	constant	ADJ
cana-4142	309	12	.	.	PUNCT
cana-4142	310	1	theorem	theorem	ADJ
cana-4142	310	2	32	32	NUM
cana-4142	310	3	let	let	VERB
cana-4142	310	4	(	(	PUNCT
cana-4142	310	5	𝔐	𝔐	X
cana-4142	310	6	,	,	PUNCT
cana-4142	310	7	𝔑	𝔑	PROPN
cana-4142	310	8	,	,	PUNCT
cana-4142	310	9	⨁	⨁	PROPN
cana-4142	310	10	)	)	PUNCT
cana-4142	310	11	be	be	VERB
cana-4142	310	12	a	a	DET
cana-4142	310	13	𝔾	𝔾	PROPN
cana-4142	310	14	−complete	−complete	PROPN
cana-4142	310	15	revised	revise	VERB
cana-4142	310	16	fuzzy	fuzzy	ADJ
cana-4142	310	17	𝓀	𝓀	X
cana-4142	310	18	−metric	−metric	ADJ
cana-4142	310	19	space	space	NOUN
cana-4142	310	20	and	and	CCONJ
cana-4142	310	21	𝔗	𝔗	NOUN
cana-4142	310	22	:	:	PUNCT
cana-4142	310	23	𝔐	𝔐	PROPN
cana-4142	310	24	→	→	SYM
cana-4142	310	25	𝔐	𝔐	PROPN
cana-4142	310	26	is	be	AUX
cana-4142	310	27	called	call	VERB
cana-4142	310	28	a	a	DET
cana-4142	310	29	revised	revise	VERB
cana-4142	310	30	fuzzy	fuzzy	ADJ
cana-4142	310	31	𝓀	𝓀	PROPN
cana-4142	310	32	−contraction	−contraction	NUM
cana-4142	310	33	mapping	mapping	NOUN
cana-4142	310	34	.	.	PUNCT
cana-4142	311	1	then	then	ADV
cana-4142	311	2	,	,	PUNCT
cana-4142	311	3	𝔗	𝔗	PROPN
cana-4142	311	4	has	have	VERB
cana-4142	311	5	a	a	DET
cana-4142	311	6	unique	unique	ADJ
cana-4142	311	7	fixed	fix	VERB
cana-4142	311	8	point	point	NOUN
cana-4142	311	9	.	.	PUNCT
cana-4142	312	1	proof	proof	NOUN
cana-4142	312	2	let	let	VERB
cana-4142	312	3	𝔭0	𝔭0	PROPN
cana-4142	312	4	∈	∈	PROPN
cana-4142	312	5	𝔐	𝔐	PROPN
cana-4142	312	6	and	and	CCONJ
cana-4142	312	7	define	define	VERB
cana-4142	312	8	a	a	DET
cana-4142	312	9	sequence	sequence	NOUN
cana-4142	312	10	{	{	PUNCT
cana-4142	312	11	𝔭𝑛	𝔭𝑛	NOUN
cana-4142	312	12	}	}	PUNCT
cana-4142	312	13	by	by	ADP
cana-4142	312	14	𝔭𝑛	𝔭𝑛	NUM
cana-4142	312	15	=	=	NOUN
cana-4142	312	16	𝔗𝔭𝑛−1	𝔗𝔭𝑛−1	NOUN
cana-4142	312	17	for	for	ADP
cana-4142	312	18	all	all	DET
cana-4142	312	19	𝑛	𝑛	PRON
cana-4142	312	20	∈	∈	PROPN
cana-4142	312	21	𝒩.	𝒩.	NOUN
cana-4142	312	22	we	we	PRON
cana-4142	312	23	will	will	AUX
cana-4142	312	24	show	show	VERB
cana-4142	312	25	that	that	SCONJ
cana-4142	312	26	this	this	DET
cana-4142	312	27	sequence	sequence	NOUN
cana-4142	312	28	is	be	AUX
cana-4142	312	29	a	a	DET
cana-4142	312	30	𝔾	𝔾	ADJ
cana-4142	312	31	−cauchy	−cauchy	ADJ
cana-4142	312	32	sequence	sequence	NOUN
cana-4142	312	33	.	.	PUNCT
cana-4142	313	1	for	for	ADP
cana-4142	313	2	any	any	DET
cana-4142	313	3	𝑛	𝑛	PRON
cana-4142	313	4	∈	∈	PROPN
cana-4142	313	5	𝒩	𝒩	PROPN
cana-4142	313	6	,	,	PUNCT
cana-4142	313	7	we	we	PRON
cana-4142	313	8	have	have	VERB
cana-4142	313	9	𝔑(𝔭𝑛	𝔑(𝔭𝑛	NOUN
cana-4142	313	10	,	,	PUNCT
cana-4142	313	11	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	313	12	,	,	PUNCT
cana-4142	313	13	𝒶1	𝒶1	NOUN
cana-4142	313	14	𝓀	𝓀	PRON
cana-4142	313	15	)	)	PUNCT
cana-4142	313	16	=	=	SYM
cana-4142	313	17	𝔑(𝔗𝔭𝑛−1	𝔑(𝔗𝔭𝑛−1	PROPN
cana-4142	313	18	,	,	PUNCT
cana-4142	313	19	𝔗𝔭𝑛	𝔗𝔭𝑛	PROPN
cana-4142	313	20	,	,	PUNCT
cana-4142	313	21	𝒶1	𝒶1	NOUN
cana-4142	313	22	𝓀	𝓀	PROPN
cana-4142	313	23	)	)	PUNCT
cana-4142	313	24	≤	≤	NUM
cana-4142	313	25	𝜆{𝔑(𝔭𝑛−1	𝜆{𝔑(𝔭𝑛−1	PROPN
cana-4142	313	26	,	,	PUNCT
cana-4142	313	27	𝔭𝑛	𝔭𝑛	PRON
cana-4142	313	28	,	,	PUNCT
cana-4142	313	29	𝒶1	𝒶1	NOUN
cana-4142	313	30	𝓀	𝓀	PROPN
cana-4142	313	31	)	)	PUNCT
cana-4142	313	32	}	}	PUNCT
cana-4142	313	33	by	by	ADP
cana-4142	313	34	repeating	repeat	VERB
cana-4142	313	35	in	in	ADP
cana-4142	313	36	this	this	DET
cana-4142	313	37	manner	manner	NOUN
cana-4142	313	38	,	,	PUNCT
cana-4142	313	39	we	we	PRON
cana-4142	313	40	obtain	obtain	VERB
cana-4142	313	41	𝔑(𝔭𝑛	𝔑(𝔭𝑛	NOUN
cana-4142	313	42	,	,	PUNCT
cana-4142	313	43	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	313	44	,	,	PUNCT
cana-4142	313	45	𝒶1	𝒶1	NOUN
cana-4142	313	46	𝓀	𝓀	X
cana-4142	313	47	)	)	PUNCT
cana-4142	313	48	≤	≤	NOUN
cana-4142	313	49	𝜆𝑛{𝔑(𝔭𝑛−1	𝜆𝑛{𝔑(𝔭𝑛−1	NOUN
cana-4142	313	50	,	,	PUNCT
cana-4142	313	51	𝔭𝑛	𝔭𝑛	INTJ
cana-4142	313	52	,	,	PUNCT
cana-4142	313	53	𝒶1	𝒶1	NOUN
cana-4142	313	54	𝓀	𝓀	PROPN
cana-4142	313	55	)	)	PUNCT
cana-4142	313	56	}	}	PUNCT
cana-4142	313	57	(	(	PUNCT
cana-4142	313	58	9	9	X
cana-4142	313	59	)	)	PUNCT
cana-4142	313	60	for	for	ADP
cana-4142	313	61	all	all	DET
cana-4142	313	62	𝑛	𝑛	DET
cana-4142	313	63	∈	∈	PROPN
cana-4142	313	64	𝒩.	𝒩.	NOUN
cana-4142	313	65	since	since	SCONJ
cana-4142	313	66	𝜆	𝜆	DET
cana-4142	313	67	∈	∈	PROPN
cana-4142	314	1	[	[	X
cana-4142	314	2	0	0	NUM
cana-4142	314	3	,	,	PUNCT
cana-4142	314	4	1	1	NUM
cana-4142	314	5	)	)	PUNCT
cana-4142	314	6	,	,	PUNCT
cana-4142	314	7	we	we	PRON
cana-4142	314	8	conclude	conclude	VERB
cana-4142	314	9	from	from	ADP
cana-4142	314	10	(	(	PUNCT
cana-4142	314	11	9	9	NUM
cana-4142	314	12	)	)	PUNCT
cana-4142	314	13	that	that	PRON
cana-4142	314	14	lim	lim	PROPN
cana-4142	314	15	𝓃→+∞	𝓃→+∞	PROPN
cana-4142	314	16	{	{	PUNCT
cana-4142	314	17	𝔑(𝔭𝑛	𝔑(𝔭𝑛	PROPN
cana-4142	314	18	,	,	PUNCT
cana-4142	314	19	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	314	20	,	,	PUNCT
cana-4142	314	21	𝒶1	𝒶1	NOUN
cana-4142	314	22	𝓀	𝓀	PROPN
cana-4142	314	23	)	)	PUNCT
cana-4142	314	24	}	}	PUNCT
cana-4142	314	25	≥	≥	NOUN
cana-4142	314	26	1	1	NUM
cana-4142	314	27	,	,	PUNCT
cana-4142	314	28	that	that	ADV
cana-4142	314	29	is	is	ADV
cana-4142	314	30	,	,	PUNCT
cana-4142	314	31	lim	lim	PROPN
cana-4142	314	32	𝓃→+∞	𝓃→+∞	PROPN
cana-4142	314	33	𝔑(𝔭𝑛	𝔑(𝔭𝑛	PROPN
cana-4142	314	34	,	,	PUNCT
cana-4142	314	35	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	314	36	,	,	PUNCT
cana-4142	314	37	𝒶1	𝒶1	NOUN
cana-4142	314	38	𝓀	𝓀	PRON
cana-4142	314	39	)	)	PUNCT
cana-4142	314	40	=	=	SYM
cana-4142	314	41	0	0	NUM
cana-4142	314	42	,	,	PUNCT
cana-4142	314	43	(	(	PUNCT
cana-4142	314	44	10	10	NUM
cana-4142	314	45	)	)	PUNCT
cana-4142	314	46	for	for	ADP
cana-4142	314	47	all	all	DET
cana-4142	314	48	𝒶1	𝒶1	NOUN
cana-4142	314	49	,	,	PUNCT
cana-4142	314	50	𝒶2	𝒶2	PROPN
cana-4142	314	51	,	,	PUNCT
cana-4142	314	52	.	.	PUNCT
cana-4142	314	53	.	.	PUNCT
cana-4142	315	1	.	.	PUNCT
cana-4142	316	1	,	,	PUNCT
cana-4142	316	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	316	3	>	>	X
cana-4142	316	4	0	0	X
cana-4142	316	5	.	.	PUNCT
cana-4142	317	1	for	for	ADP
cana-4142	317	2	each	each	DET
cana-4142	317	3	𝑛	𝑛	PRON
cana-4142	317	4	∈	∈	PROPN
cana-4142	317	5	𝒩	𝒩	PROPN
cana-4142	317	6	,	,	PUNCT
cana-4142	317	7	𝓍	𝓍	X
cana-4142	317	8	>	>	X
cana-4142	317	9	0	0	PUNCT
cana-4142	317	10	and	and	CCONJ
cana-4142	317	11	𝒶1	𝒶1	PROPN
cana-4142	317	12	,	,	PUNCT
cana-4142	317	13	𝒶2	𝒶2	PROPN
cana-4142	317	14	,	,	PUNCT
cana-4142	317	15	.	.	PUNCT
cana-4142	317	16	.	.	PUNCT
cana-4142	317	17	.	.	PUNCT
cana-4142	318	1	,	,	PUNCT
cana-4142	318	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	318	3	>	>	X
cana-4142	318	4	0	0	NUM
cana-4142	318	5	,	,	PUNCT
cana-4142	318	6	we	we	PRON
cana-4142	318	7	have	have	VERB
cana-4142	318	8	𝔑(𝔭𝑛	𝔑(𝔭𝑛	NOUN
cana-4142	318	9	,	,	PUNCT
cana-4142	318	10	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	318	11	,	,	PUNCT
cana-4142	318	12	𝒶1	𝒶1	NOUN
cana-4142	318	13	𝓀	𝓀	X
cana-4142	318	14	)	)	PUNCT
cana-4142	318	15	≤	≤	NUM
cana-4142	318	16	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	318	17	2	2	NUM
cana-4142	318	18	(	(	PUNCT
cana-4142	318	19	𝔭𝑛	𝔭𝑛	INTJ
cana-4142	318	20	,	,	PUNCT
cana-4142	318	21	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	318	22	,	,	PUNCT
cana-4142	318	23	𝒶1	𝒶1	NOUN
cana-4142	318	24	𝓀)⨁𝔑𝒯	𝓀)⨁𝔑𝒯	NOUN
cana-4142	318	25	2	2	NUM
cana-4142	318	26	(	(	PUNCT
cana-4142	318	27	𝔭𝑛	𝔭𝑛	INTJ
cana-4142	318	28	,	,	PUNCT
cana-4142	318	29	𝔭𝑛+𝓍	𝔭𝑛+𝓍	NUM
cana-4142	318	30	,	,	PUNCT
cana-4142	318	31	𝒶1	𝒶1	NOUN
cana-4142	318	32	𝓀	𝓀	PROPN
cana-4142	318	33	)	)	PUNCT
cana-4142	318	34	≤	≤	NOUN
cana-4142	318	35	{	{	PUNCT
cana-4142	318	36	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	318	37	2	2	NUM
cana-4142	318	38	(	(	PUNCT
cana-4142	318	39	𝔭𝑛	𝔭𝑛	INTJ
cana-4142	318	40	,	,	PUNCT
cana-4142	318	41	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	318	42	,	,	PUNCT
cana-4142	318	43	𝒶1	𝒶1	NOUN
cana-4142	318	44	𝓀)⨁𝔑𝒯	𝓀)⨁𝔑𝒯	NOUN
cana-4142	318	45	22	22	NUM
cana-4142	318	46	(	(	PUNCT
cana-4142	318	47	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	318	48	,	,	PUNCT
cana-4142	318	49	𝔭𝑛+2	𝔭𝑛+2	NUM
cana-4142	318	50	,	,	PUNCT
cana-4142	318	51	𝒶1	𝒶1	NOUN
cana-4142	318	52	𝓀	𝓀	PROPN
cana-4142	318	53	)	)	PUNCT
cana-4142	318	54	⨁	⨁	PROPN
cana-4142	318	55	…	…	SYM
cana-4142	318	56	⨁	⨁	PROPN
cana-4142	318	57	𝔑𝒯	𝔑𝒯	PROPN
cana-4142	318	58	2𝓍−1	2𝓍−1	NUM
cana-4142	318	59	(	(	PUNCT
cana-4142	318	60	𝔭𝑛+𝓍−2	𝔭𝑛+𝓍−2	PROPN
cana-4142	318	61	,	,	PUNCT
cana-4142	318	62	𝔭𝑛+𝓍−1	𝔭𝑛+𝓍−1	PROPN
cana-4142	318	63	,	,	PUNCT
cana-4142	318	64	𝒶1	𝒶1	NOUN
cana-4142	318	65	𝓀)⨁𝔑𝒯	𝓀)⨁𝔑𝒯	NOUN
cana-4142	318	66	2𝓍−1	2𝓍−1	NUM
cana-4142	318	67	(	(	PUNCT
cana-4142	318	68	𝔭𝑛+𝓍−1	𝔭𝑛+𝓍−1	PROPN
cana-4142	318	69	,	,	PUNCT
cana-4142	318	70	𝔭𝑛+𝓍	𝔭𝑛+𝓍	PROPN
cana-4142	318	71	,	,	PUNCT
cana-4142	318	72	𝒶1	𝒶1	PROPN
cana-4142	318	73	𝓀	𝓀	PROPN
cana-4142	318	74	)	)	PUNCT
cana-4142	318	75	}	}	PUNCT
cana-4142	318	76	(	(	PUNCT
cana-4142	318	77	11	11	NUM
cana-4142	318	78	)	)	PUNCT
cana-4142	318	79	from	from	ADP
cana-4142	318	80	(	(	PUNCT
cana-4142	318	81	10	10	NUM
cana-4142	318	82	)	)	PUNCT
cana-4142	318	83	,	,	PUNCT
cana-4142	318	84	we	we	PRON
cana-4142	318	85	have	have	VERB
cana-4142	318	86	,	,	PUNCT
cana-4142	318	87	lim	lim	PROPN
cana-4142	318	88	𝓃→+∞	𝓃→+∞	PROPN
cana-4142	318	89	𝔑𝓍	𝔑𝓍	PROPN
cana-4142	318	90	𝒯(𝔭𝑛	𝒯(𝔭𝑛	NOUN
cana-4142	318	91	,	,	PUNCT
cana-4142	318	92	𝔭𝑛+1	𝔭𝑛+1	NUM
cana-4142	318	93	,	,	PUNCT
cana-4142	318	94	𝒶1	𝒶1	NOUN
cana-4142	318	95	𝓀	𝓀	PRON
cana-4142	318	96	)	)	PUNCT
cana-4142	318	97	=	=	SYM
cana-4142	318	98	0	0	NUM
cana-4142	318	99	,	,	PUNCT
cana-4142	318	100	for	for	ADP
cana-4142	318	101	all	all	DET
cana-4142	318	102	𝒶1	𝒶1	NOUN
cana-4142	318	103	,	,	PUNCT
cana-4142	318	104	𝒶2	𝒶2	PROPN
cana-4142	318	105	,	,	PUNCT
cana-4142	318	106	.	.	PUNCT
cana-4142	318	107	.	.	PUNCT
cana-4142	319	1	.	.	PUNCT
cana-4142	320	1	,	,	PUNCT
cana-4142	320	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	320	3	>	>	X
cana-4142	320	4	0	0	PUNCT
cana-4142	320	5	and	and	CCONJ
cana-4142	320	6	𝓍	𝓍	ADJ
cana-4142	320	7	>	>	X
cana-4142	320	8	0,which	0,which	X
cana-4142	320	9	together	together	ADV
cana-4142	320	10	with	with	ADP
cana-4142	320	11	inequality	inequality	NOUN
cana-4142	320	12	(	(	PUNCT
cana-4142	320	13	11	11	NUM
cana-4142	320	14	)	)	PUNCT
cana-4142	320	15	yields	yield	NOUN
cana-4142	320	16	,	,	PUNCT
cana-4142	320	17	lim	lim	PROPN
cana-4142	320	18	𝓃→+∞	𝓃→+∞	PROPN
cana-4142	320	19	𝔑(𝔭𝑛	𝔑(𝔭𝑛	PROPN
cana-4142	320	20	,	,	PUNCT
cana-4142	320	21	𝔭𝑛+𝓍	𝔭𝑛+𝓍	NUM
cana-4142	320	22	,	,	PUNCT
cana-4142	320	23	𝒶1	𝒶1	NOUN
cana-4142	320	24	𝓀	𝓀	X
cana-4142	320	25	)	)	PUNCT
cana-4142	320	26	≤	≤	NOUN
cana-4142	320	27	0	0	NUM
cana-4142	320	28	⨁	⨁	PROPN
cana-4142	320	29	0	0	NUM
cana-4142	320	30	⨁	⨁	PROPN
cana-4142	320	31	…	…	SYM
cana-4142	320	32	⨁	⨁	PROPN
cana-4142	320	33	0	0	NUM
cana-4142	320	34	=	=	SYM
cana-4142	320	35	0	0	PROPN
cana-4142	320	36	,	,	PUNCT
cana-4142	320	37	for	for	ADP
cana-4142	320	38	all	all	DET
cana-4142	320	39	𝒶1	𝒶1	NOUN
cana-4142	320	40	,	,	PUNCT
cana-4142	320	41	𝒶2	𝒶2	PROPN
cana-4142	320	42	,	,	PUNCT
cana-4142	320	43	.	.	PUNCT
cana-4142	320	44	.	.	PUNCT
cana-4142	321	1	.	.	PUNCT
cana-4142	322	1	,	,	PUNCT
cana-4142	322	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	322	3	>	>	X
cana-4142	322	4	0	0	PUNCT
cana-4142	322	5	and	and	CCONJ
cana-4142	322	6	𝓍	𝓍	ADJ
cana-4142	322	7	>	>	X
cana-4142	322	8	0	0	X
cana-4142	322	9	.	.	PUNCT
cana-4142	323	1	communications	communication	NOUN
cana-4142	323	2	on	on	ADP
cana-4142	323	3	applied	apply	VERB
cana-4142	323	4	nonlinear	nonlinear	ADJ
cana-4142	323	5	analysis	analysis	NOUN
cana-4142	323	6	issn	issn	NOUN
cana-4142	323	7	:	:	PUNCT
cana-4142	323	8	1074	1074	NUM
cana-4142	323	9	-	-	PUNCT
cana-4142	323	10	133x	133x	NUM
cana-4142	323	11	vol	vol	NOUN
cana-4142	323	12	x	x	NOUN
cana-4142	323	13	no	no	INTJ
cana-4142	323	14	.	.	PUNCT
cana-4142	324	1	y	y	PROPN
cana-4142	324	2	(	(	PUNCT
cana-4142	324	3	2025	2025	NUM
cana-4142	324	4	)	)	PUNCT
cana-4142	324	5	1330	1330	NUM
cana-4142	324	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	324	7	therefore	therefore	ADV
cana-4142	324	8	,	,	PUNCT
cana-4142	324	9	the	the	DET
cana-4142	324	10	sequence	sequence	NOUN
cana-4142	324	11	{	{	PUNCT
cana-4142	324	12	𝔭𝑛	𝔭𝑛	NOUN
cana-4142	324	13	}	}	PUNCT
cana-4142	324	14	is	be	AUX
cana-4142	324	15	a	a	DET
cana-4142	324	16	𝔾	𝔾	ADJ
cana-4142	324	17	−cauchy	−cauchy	ADJ
cana-4142	324	18	sequence	sequence	NOUN
cana-4142	324	19	in	in	ADP
cana-4142	324	20	𝔐.	𝔐.	PROPN
cana-4142	324	21	by	by	ADP
cana-4142	324	22	the	the	DET
cana-4142	324	23	𝔾	𝔾	PROPN
cana-4142	324	24	−completeness	−completeness	NOUN
cana-4142	324	25	of	of	ADP
cana-4142	324	26	𝔐	𝔐	PROPN
cana-4142	324	27	,	,	PUNCT
cana-4142	324	28	there	there	PRON
cana-4142	324	29	exists	exist	VERB
cana-4142	324	30	𝔵	𝔵	DET
cana-4142	324	31	∈	∈	NOUN
cana-4142	324	32	𝔐	𝔐	NOUN
cana-4142	324	33	such	such	ADJ
cana-4142	324	34	that	that	SCONJ
cana-4142	324	35	the	the	DET
cana-4142	324	36	sequence	sequence	NOUN
cana-4142	324	37	{	{	PUNCT
cana-4142	324	38	𝔭𝑛	𝔭𝑛	PROPN
cana-4142	324	39	}	}	PUNCT
cana-4142	324	40	converges	converge	NOUN
cana-4142	324	41	to	to	ADP
cana-4142	324	42	𝔵	𝔵	PRON
cana-4142	324	43	,	,	PUNCT
cana-4142	324	44	that	that	ADV
cana-4142	324	45	is	is	ADV
cana-4142	324	46	,	,	PUNCT
cana-4142	324	47	lim	lim	PROPN
cana-4142	324	48	𝓃→+∞	𝓃→+∞	PROPN
cana-4142	324	49	𝔑(𝔭𝑛	𝔑(𝔭𝑛	PROPN
cana-4142	324	50	,	,	PUNCT
cana-4142	324	51	𝔵	𝔵	NOUN
cana-4142	324	52	,	,	PUNCT
cana-4142	324	53	𝒶1	𝒶1	NOUN
cana-4142	324	54	𝓀	𝓀	X
cana-4142	324	55	)	)	PUNCT
cana-4142	324	56	=	=	SYM
cana-4142	324	57	0	0	NUM
cana-4142	324	58	,	,	PUNCT
cana-4142	324	59	(	(	PUNCT
cana-4142	324	60	12	12	NUM
cana-4142	324	61	)	)	PUNCT
cana-4142	324	62	for	for	ADP
cana-4142	324	63	all	all	DET
cana-4142	324	64	𝒶1	𝒶1	NOUN
cana-4142	324	65	,	,	PUNCT
cana-4142	324	66	𝒶2	𝒶2	PROPN
cana-4142	324	67	,	,	PUNCT
cana-4142	324	68	.	.	PUNCT
cana-4142	324	69	.	.	PUNCT
cana-4142	325	1	.	.	PUNCT
cana-4142	326	1	,	,	PUNCT
cana-4142	326	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	326	3	>	>	X
cana-4142	326	4	0	0	X
cana-4142	326	5	.	.	PUNCT
cana-4142	327	1	now	now	ADV
cana-4142	327	2	,	,	PUNCT
cana-4142	327	3	we	we	PRON
cana-4142	327	4	will	will	AUX
cana-4142	327	5	show	show	VERB
cana-4142	327	6	that	that	SCONJ
cana-4142	327	7	𝔵	𝔵	NOUN
cana-4142	327	8	is	be	AUX
cana-4142	327	9	a	a	DET
cana-4142	327	10	fixed	fix	VERB
cana-4142	327	11	point	point	NOUN
cana-4142	327	12	of	of	ADP
cana-4142	327	13	𝔗.	𝔗.	PROPN
cana-4142	327	14	for	for	ADP
cana-4142	327	15	each	each	DET
cana-4142	327	16	𝑛	𝑛	PRON
cana-4142	327	17	∈	∈	PROPN
cana-4142	327	18	𝒩	𝒩	PROPN
cana-4142	327	19	,	,	PUNCT
cana-4142	327	20	we	we	PRON
cana-4142	327	21	have	have	VERB
cana-4142	327	22	𝔑(𝔭𝑛+1	𝔑(𝔭𝑛+1	PROPN
cana-4142	327	23	,	,	PUNCT
cana-4142	327	24	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	327	25	,	,	PUNCT
cana-4142	327	26	𝒶1	𝒶1	NOUN
cana-4142	327	27	𝓀	𝓀	PRON
cana-4142	327	28	)	)	PUNCT
cana-4142	327	29	=	=	SYM
cana-4142	327	30	𝔑(𝔗𝔭𝑛	𝔑(𝔗𝔭𝑛	NOUN
cana-4142	327	31	,	,	PUNCT
cana-4142	327	32	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	327	33	,	,	PUNCT
cana-4142	327	34	𝒶1	𝒶1	NOUN
cana-4142	327	35	𝓀	𝓀	PROPN
cana-4142	327	36	)	)	PUNCT
cana-4142	327	37	≤	≤	NOUN
cana-4142	327	38	𝜆{𝔑(𝔭𝑛	𝜆{𝔑(𝔭𝑛	NOUN
cana-4142	327	39	,	,	PUNCT
cana-4142	327	40	𝔵	𝔵	NOUN
cana-4142	327	41	,	,	PUNCT
cana-4142	327	42	𝒶1	𝒶1	NOUN
cana-4142	327	43	𝓀	𝓀	PROPN
cana-4142	327	44	)	)	PUNCT
cana-4142	327	45	}	}	PUNCT
cana-4142	327	46	by	by	ADP
cana-4142	327	47	using	use	VERB
cana-4142	327	48	(	(	PUNCT
cana-4142	327	49	12	12	NUM
cana-4142	327	50	)	)	PUNCT
cana-4142	327	51	,	,	PUNCT
cana-4142	327	52	we	we	PRON
cana-4142	327	53	have	have	VERB
cana-4142	327	54	,	,	PUNCT
cana-4142	327	55	lim	lim	PROPN
cana-4142	327	56	𝓃→+∞	𝓃→+∞	PROPN
cana-4142	327	57	{	{	PUNCT
cana-4142	327	58	𝔑(𝔭𝑛+1	𝔑(𝔭𝑛+1	PROPN
cana-4142	327	59	,	,	PUNCT
cana-4142	327	60	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	327	61	,	,	PUNCT
cana-4142	327	62	𝒶1	𝒶1	NOUN
cana-4142	327	63	𝓀	𝓀	PROPN
cana-4142	327	64	)	)	PUNCT
cana-4142	327	65	}	}	PUNCT
cana-4142	328	1	=	=	SYM
cana-4142	328	2	0,that	0,that	PRON
cana-4142	328	3	is	be	AUX
cana-4142	328	4	,	,	PUNCT
cana-4142	328	5	lim	lim	PROPN
cana-4142	328	6	𝓃→+∞	𝓃→+∞	PROPN
cana-4142	328	7	𝔑(𝔭𝑛+1	𝔑(𝔭𝑛+1	PROPN
cana-4142	328	8	,	,	PUNCT
cana-4142	328	9	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	328	10	,	,	PUNCT
cana-4142	328	11	𝒶1	𝒶1	NOUN
cana-4142	328	12	𝓀	𝓀	X
cana-4142	328	13	)	)	PUNCT
cana-4142	328	14	=	=	SYM
cana-4142	328	15	0	0	NUM
cana-4142	328	16	(	(	PUNCT
cana-4142	328	17	13	13	NUM
cana-4142	328	18	)	)	PUNCT
cana-4142	328	19	for	for	ADP
cana-4142	328	20	all	all	DET
cana-4142	328	21	𝒶1	𝒶1	NOUN
cana-4142	328	22	,	,	PUNCT
cana-4142	328	23	𝒶2	𝒶2	PROPN
cana-4142	328	24	,	,	PUNCT
cana-4142	328	25	.	.	PUNCT
cana-4142	328	26	.	.	PUNCT
cana-4142	328	27	.	.	PUNCT
cana-4142	329	1	,	,	PUNCT
cana-4142	329	2	𝒶𝓀	𝒶𝓀	ADV
cana-4142	329	3	>	>	X
cana-4142	329	4	0	0	X
cana-4142	329	5	.	.	PUNCT
cana-4142	330	1	for	for	ADP
cana-4142	330	2	any	any	DET
cana-4142	330	3	𝑛	𝑛	PRON
cana-4142	330	4	∈	∈	PROPN
cana-4142	330	5	𝒩	𝒩	PROPN
cana-4142	330	6	,	,	PUNCT
cana-4142	330	7	we	we	PRON
cana-4142	330	8	have	have	VERB
cana-4142	330	9	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	330	10	,	,	PUNCT
cana-4142	330	11	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	330	12	,	,	PUNCT
cana-4142	330	13	𝒶1	𝒶1	NOUN
cana-4142	330	14	𝓀	𝓀	PROPN
cana-4142	330	15	)	)	PUNCT
cana-4142	330	16	≤	≤	NUM
cana-4142	330	17	𝔑2	𝔑2	PROPN
cana-4142	330	18	𝒯(𝔵	𝒯(𝔵	PROPN
cana-4142	330	19	,	,	PUNCT
cana-4142	330	20	𝔵𝑛+1	𝔵𝑛+1	NUM
cana-4142	330	21	,	,	PUNCT
cana-4142	330	22	𝒶1	𝒶1	VERB
cana-4142	330	23	𝓀)⨁𝔑2	𝓀)⨁𝔑2	PROPN
cana-4142	330	24	𝓀(𝔭𝑛+1	𝓀(𝔭𝑛+1	PROPN
cana-4142	330	25	,	,	PUNCT
cana-4142	330	26	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	330	27	,	,	PUNCT
cana-4142	330	28	𝒶1	𝒶1	NOUN
cana-4142	330	29	𝓀	𝓀	PROPN
cana-4142	330	30	)	)	PUNCT
cana-4142	330	31	,	,	PUNCT
cana-4142	330	32	which	which	PRON
cana-4142	330	33	together	together	ADV
cana-4142	330	34	with	with	ADP
cana-4142	330	35	(	(	PUNCT
cana-4142	330	36	12	12	NUM
cana-4142	330	37	)	)	PUNCT
cana-4142	330	38	and	and	CCONJ
cana-4142	330	39	(	(	PUNCT
cana-4142	330	40	13	13	NUM
cana-4142	330	41	)	)	PUNCT
cana-4142	330	42	yields	yield	NOUN
cana-4142	330	43	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	330	44	,	,	PUNCT
cana-4142	330	45	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	330	46	,	,	PUNCT
cana-4142	330	47	𝒶1	𝒶1	NOUN
cana-4142	330	48	𝓀	𝓀	X
cana-4142	330	49	)	)	PUNCT
cana-4142	331	1	=	=	SYM
cana-4142	331	2	0	0	NUM
cana-4142	331	3	for	for	ADP
cana-4142	331	4	all	all	DET
cana-4142	331	5	𝓌1	𝓌1	NOUN
cana-4142	331	6	,	,	PUNCT
cana-4142	331	7	𝓌2	𝓌2	NOUN
cana-4142	331	8	,	,	PUNCT
cana-4142	331	9	.	.	PUNCT
cana-4142	331	10	.	.	PUNCT
cana-4142	331	11	.	.	PUNCT
cana-4142	332	1	,	,	PUNCT
cana-4142	332	2	𝓌𝓀	𝓌𝓀	X
cana-4142	332	3	>	>	X
cana-4142	332	4	0	0	X
cana-4142	332	5	.	.	PUNCT
cana-4142	333	1	that	that	PRON
cana-4142	333	2	is	be	AUX
cana-4142	333	3	,	,	PUNCT
cana-4142	333	4	𝔗𝔵	𝔗𝔵	PROPN
cana-4142	333	5	=	=	SYM
cana-4142	333	6	𝔵.	𝔵.	NOUN
cana-4142	333	7	thus	thus	ADV
cana-4142	333	8	,	,	PUNCT
cana-4142	333	9	𝔵	𝔵	PRON
cana-4142	333	10	is	be	AUX
cana-4142	333	11	a	a	DET
cana-4142	333	12	fixed	fix	VERB
cana-4142	333	13	point	point	NOUN
cana-4142	333	14	of	of	ADP
cana-4142	333	15	𝔗.	𝔗.	PROPN
cana-4142	333	16	𝔑(𝔵	𝔑(𝔵	SYM
cana-4142	333	17	,	,	PUNCT
cana-4142	333	18	𝔶	𝔶	ADP
cana-4142	333	19	,	,	PUNCT
cana-4142	333	20	𝓌1	𝓌1	PROPN
cana-4142	333	21	𝓀	𝓀	PROPN
cana-4142	333	22	)	)	PUNCT
cana-4142	333	23	>	>	X
cana-4142	333	24	0	0	NUM
cana-4142	333	25	,	,	PUNCT
cana-4142	333	26	that	that	ADV
cana-4142	333	27	is	is	ADV
cana-4142	333	28	,	,	PUNCT
cana-4142	333	29	𝔑(𝔵	𝔑(𝔵	X
cana-4142	333	30	,	,	PUNCT
cana-4142	333	31	𝔶	𝔶	ADP
cana-4142	333	32	,	,	PUNCT
cana-4142	333	33	𝓌1	𝓌1	PROPN
cana-4142	333	34	𝓀	𝓀	X
cana-4142	333	35	)	)	PUNCT
cana-4142	333	36	<	<	X
cana-4142	334	1	1	1	X
cana-4142	334	2	.	.	PUNCT
cana-4142	334	3	now	now	ADV
cana-4142	334	4	,	,	PUNCT
cana-4142	334	5	we	we	PRON
cana-4142	334	6	have	have	VERB
cana-4142	334	7	𝔑(𝔵	𝔑(𝔵	NUM
cana-4142	334	8	,	,	PUNCT
cana-4142	334	9	𝔶	𝔶	ADP
cana-4142	334	10	,	,	PUNCT
cana-4142	334	11	𝓌1	𝓌1	PROPN
cana-4142	334	12	𝓀	𝓀	X
cana-4142	334	13	)	)	PUNCT
cana-4142	334	14	=	=	SYM
cana-4142	335	1	𝔑(𝔗𝔵	𝔑(𝔗𝔵	PROPN
cana-4142	335	2	,	,	PUNCT
cana-4142	335	3	𝔗𝔶	𝔗𝔶	PROPN
cana-4142	335	4	,	,	PUNCT
cana-4142	335	5	𝓌1	𝓌1	PROPN
cana-4142	335	6	𝓀	𝓀	PROPN
cana-4142	335	7	)	)	PUNCT
cana-4142	335	8	≤	≤	PROPN
cana-4142	336	1	𝜆{𝔑(𝔵	𝜆{𝔑(𝔵	PROPN
cana-4142	336	2	,	,	PUNCT
cana-4142	336	3	𝔶	𝔶	ADP
cana-4142	336	4	,	,	PUNCT
cana-4142	336	5	𝓌1	𝓌1	PROPN
cana-4142	336	6	𝓀	𝓀	PROPN
cana-4142	336	7	)	)	PUNCT
cana-4142	336	8	}	}	PUNCT
cana-4142	336	9	since	since	SCONJ
cana-4142	336	10	𝜆	𝜆	PRON
cana-4142	336	11	<	<	X
cana-4142	336	12	1	1	NUM
cana-4142	336	13	,	,	PUNCT
cana-4142	336	14	the	the	DET
cana-4142	336	15	above	above	ADJ
cana-4142	336	16	inequality	inequality	NOUN
cana-4142	336	17	yields	yield	VERB
cana-4142	336	18	a	a	DET
cana-4142	336	19	contradiction	contradiction	NOUN
cana-4142	336	20	.	.	PUNCT
cana-4142	337	1	therefore	therefore	ADV
cana-4142	337	2	,	,	PUNCT
cana-4142	337	3	we	we	PRON
cana-4142	337	4	must	must	AUX
cana-4142	337	5	have	have	VERB
cana-4142	337	6	,	,	PUNCT
cana-4142	337	7	𝔵	𝔵	X
cana-4142	337	8	=	=	X
cana-4142	337	9	𝔶.	𝔶.	NOUN
cana-4142	337	10	thus	thus	ADV
cana-4142	337	11	,	,	PUNCT
cana-4142	337	12	the	the	DET
cana-4142	337	13	fixed	fix	VERB
cana-4142	337	14	point	point	NOUN
cana-4142	337	15	of	of	ADP
cana-4142	337	16	𝑇	𝑇	PROPN
cana-4142	337	17	is	be	AUX
cana-4142	337	18	unique	unique	ADJ
cana-4142	337	19	.	.	PUNCT
cana-4142	338	1	references	reference	NOUN
cana-4142	338	2	1	1	NUM
cana-4142	338	3	.	.	PUNCT
cana-4142	338	4	muraliraj	muraliraj	PROPN
cana-4142	338	5	a	a	PRON
cana-4142	338	6	and	and	CCONJ
cana-4142	338	7	thangathamizh	thangathamizh	ADJ
cana-4142	338	8	,	,	PUNCT
cana-4142	338	9	“	"	PUNCT
cana-4142	338	10	the	the	DET
cana-4142	338	11	first	first	ADJ
cana-4142	338	12	rational	rational	ADJ
cana-4142	338	13	type	type	NOUN
cana-4142	338	14	revised	revise	VERB
cana-4142	338	15	fuzzy	fuzzy	ADJ
cana-4142	338	16	-	-	PUNCT
cana-4142	338	17	contractions	contraction	NOUN
cana-4142	338	18	in	in	ADP
cana-4142	338	19	revised	revise	VERB
cana-4142	338	20	fuzzy	fuzzy	ADJ
cana-4142	338	21	metric	metric	ADJ
cana-4142	338	22	spaces	space	NOUN
cana-4142	338	23	with	with	ADP
cana-4142	338	24	an	an	DET
cana-4142	338	25	applications	application	NOUN
cana-4142	338	26	”	"	PUNCT
cana-4142	338	27	,	,	PUNCT
cana-4142	338	28	mathematics,11	mathematics,11	NOUN
cana-4142	338	29	,	,	PUNCT
cana-4142	338	30	2244	2244	NUM
cana-4142	338	31	,	,	PUNCT
cana-4142	338	32	2023	2023	NUM
cana-4142	338	33	.	.	PUNCT
cana-4142	339	1	2	2	NUM
cana-4142	339	2	.	.	PUNCT
cana-4142	339	3	a.	a.	NOUN
cana-4142	339	4	moussaoui	moussaoui	PROPN
cana-4142	339	5	,	,	PUNCT
cana-4142	339	6	v.	v.	ADP
cana-4142	339	7	todorˇcevi´c	todorˇcevi´c	NOUN
cana-4142	339	8	,	,	PUNCT
cana-4142	339	9	mirjana	mirjana	PROPN
cana-4142	339	10	pantovi´c	pantovi´c	PROPN
cana-4142	339	11	,	,	PUNCT
cana-4142	339	12	s.	s.	PROPN
cana-4142	339	13	radenovi´c	radenovi´c	PROPN
cana-4142	339	14	,	,	PUNCT
cana-4142	339	15	s.	s.	PROPN
cana-4142	339	16	mellian	mellian	PROPN
cana-4142	339	17	,	,	PUNCT
cana-4142	339	18	“	"	PUNCT
cana-4142	339	19	fixed	fix	VERB
cana-4142	339	20	point	point	NOUN
cana-4142	339	21	results	result	NOUN
cana-4142	339	22	via	via	ADP
cana-4142	339	23	g	g	NOUN
cana-4142	339	24	-	-	PUNCT
cana-4142	339	25	transitive	transitive	ADJ
cana-4142	339	26	binary	binary	ADJ
cana-4142	339	27	relation	relation	NOUN
cana-4142	339	28	and	and	CCONJ
cana-4142	339	29	fuzzy	fuzzy	ADJ
cana-4142	339	30	l	l	NOUN
cana-4142	339	31	-	-	PUNCT
cana-4142	339	32	r	r	NOUN
cana-4142	339	33	-	-	PUNCT
cana-4142	339	34	contraction	contraction	NOUN
cana-4142	339	35	”	"	PUNCT
cana-4142	339	36	,	,	PUNCT
cana-4142	339	37	mathematics	mathematics	NOUN
cana-4142	339	38	2023	2023	NUM
cana-4142	339	39	,	,	PUNCT
cana-4142	339	40	11	11	NUM
cana-4142	339	41	,	,	PUNCT
cana-4142	339	42	1768	1768	NUM
cana-4142	339	43	.	.	PUNCT
cana-4142	340	1	3	3	X
cana-4142	340	2	.	.	PUNCT
cana-4142	340	3	a.	a.	NOUN
cana-4142	340	4	moussaoui	moussaoui	PROPN
cana-4142	340	5	,	,	PUNCT
cana-4142	340	6	s.	s.	PROPN
cana-4142	340	7	radenovi´c	radenovi´c	PROPN
cana-4142	340	8	,	,	PUNCT
cana-4142	340	9	s.	s.	PROPN
cana-4142	340	10	mellian	mellian	PROPN
cana-4142	340	11	,	,	PUNCT
cana-4142	340	12	“	"	PUNCT
cana-4142	340	13	new	new	ADJ
cana-4142	340	14	fixed	fix	VERB
cana-4142	340	15	-	-	PUNCT
cana-4142	340	16	point	point	NOUN
cana-4142	340	17	results	result	NOUN
cana-4142	340	18	for	for	ADP
cana-4142	340	19	𝛼	𝛼	NOUN
cana-4142	340	20	−	−	PROPN
cana-4142	340	21	𝜂	𝜂	NOUN
cana-4142	340	22	−	−	X
cana-4142	340	23	𝛩𝑓	𝛩𝑓	PROPN
cana-4142	340	24	−type	−type	NOUN
cana-4142	340	25	fuzzy	fuzzy	ADJ
cana-4142	340	26	contraction	contraction	NOUN
cana-4142	340	27	”	"	PUNCT
cana-4142	340	28	,	,	PUNCT
cana-4142	340	29	commun	commun	PROPN
cana-4142	340	30	.	.	PUNCT
cana-4142	341	1	optim	optim	PROPN
cana-4142	341	2	.	.	PUNCT
cana-4142	341	3	theory	theory	NOUN
cana-4142	341	4	2023	2023	NUM
cana-4142	341	5	(	(	PUNCT
cana-4142	341	6	2023	2023	NUM
cana-4142	341	7	)	)	PUNCT
cana-4142	341	8	15	15	NUM
cana-4142	341	9	,	,	PUNCT
cana-4142	341	10	https://doi.org/10.23952/cot.2023.15	https://doi.org/10.23952/cot.2023.15	PROPN
cana-4142	341	11	4	4	NUM
cana-4142	341	12	.	.	PUNCT
cana-4142	342	1	t.	t.	PROPN
cana-4142	342	2	došenovi´c	došenovi´c	PROPN
cana-4142	342	3	,	,	PUNCT
cana-4142	342	4	d.	d.	PROPN
cana-4142	342	5	raki´c	raki´c	PROPN
cana-4142	342	6	,	,	PUNCT
cana-4142	342	7	s.	s.	PROPN
cana-4142	342	8	radenovi´c	radenovi´c	PROPN
cana-4142	342	9	,	,	PUNCT
cana-4142	342	10	b.	b.	PROPN
cana-4142	342	11	cari´c	cari´c	PROPN
cana-4142	342	12	,	,	PUNCT
cana-4142	342	13	“	"	PUNCT
cana-4142	342	14	ciric	ciric	ADJ
cana-4142	342	15	type	type	NOUN
cana-4142	342	16	nonunique	nonunique	ADJ
cana-4142	342	17	fixed	fix	VERB
cana-4142	342	18	point	point	NOUN
cana-4142	342	19	theorems	theorem	NOUN
cana-4142	342	20	in	in	ADP
cana-4142	342	21	the	the	DET
cana-4142	342	22	frame	frame	NOUN
cana-4142	342	23	of	of	ADP
cana-4142	342	24	fuzzy	fuzzy	ADJ
cana-4142	342	25	metric	metric	ADJ
cana-4142	342	26	spaces	space	NOUN
cana-4142	342	27	”	"	PUNCT
cana-4142	342	28	,	,	PUNCT
cana-4142	342	29	aims	aim	VERB
cana-4142	342	30	mathematics	mathematic	NOUN
cana-4142	342	31	,	,	PUNCT
cana-4142	342	32	8	8	NUM
cana-4142	342	33	(	(	PUNCT
cana-4142	342	34	1	1	NUM
cana-4142	342	35	):	):	PUNCT
cana-4142	342	36	2154	2154	NUM
cana-4142	342	37	-	-	SYM
cana-4142	342	38	2167	2167	NUM
cana-4142	342	39	,	,	PUNCT
cana-4142	342	40	doi	doi	NOUN
cana-4142	342	41	:	:	PUNCT
cana-4142	342	42	10.3934	10.3934	NUM
cana-4142	342	43	/	/	SYM
cana-4142	343	1	math.2023111	math.2023111	ADJ
cana-4142	343	2	.	.	NOUN
cana-4142	343	3	5	5	NUM
cana-4142	343	4	.	.	X
cana-4142	343	5	u.d.patel	u.d.patel	PROPN
cana-4142	343	6	,	,	PUNCT
cana-4142	343	7	s.	s.	PROPN
cana-4142	343	8	radenovi´c	radenovi´c	PROPN
cana-4142	343	9	,	,	PUNCT
cana-4142	343	10	“	"	PUNCT
cana-4142	343	11	an	an	DET
cana-4142	343	12	application	application	NOUN
cana-4142	343	13	to	to	ADP
cana-4142	343	14	nonlinear	nonlinear	ADJ
cana-4142	343	15	fractional	fractional	ADJ
cana-4142	343	16	differential	differential	NOUN
cana-4142	343	17	equation	equation	NOUN
cana-4142	343	18	via𝛼	via𝛼	PROPN
cana-4142	344	1	−	−	PROPN
cana-4142	345	1	𝛤𝐹	𝛤𝐹	PROPN
cana-4142	345	2	−fuzzy	−fuzzy	NUM
cana-4142	345	3	contractive	contractive	ADJ
cana-4142	345	4	mappings	mapping	NOUN
cana-4142	345	5	in	in	ADP
cana-4142	345	6	a	a	DET
cana-4142	345	7	fuzzy	fuzzy	ADJ
cana-4142	345	8	metric	metric	ADJ
cana-4142	345	9	space	space	NOUN
cana-4142	345	10	”	"	PUNCT
cana-4142	345	11	,	,	PUNCT
cana-4142	345	12	mathematics	mathematic	NOUN
cana-4142	345	13	,	,	PUNCT
cana-4142	345	14	2022	2022	NUM
cana-4142	345	15	,	,	PUNCT
cana-4142	345	16	10	10	NUM
cana-4142	345	17	,	,	PUNCT
cana-4142	345	18	2831	2831	NUM
cana-4142	345	19	.	.	PUNCT
cana-4142	346	1	6	6	NUM
cana-4142	346	2	.	.	X
cana-4142	347	1	d.raki´c	d.raki´c	NOUN
cana-4142	347	2	,	,	PUNCT
cana-4142	347	3	a.	a.	NOUN
cana-4142	347	4	mukheimer	mukheimer	PROPN
cana-4142	347	5	,	,	PUNCT
cana-4142	347	6	t.	t.	NOUN
cana-4142	347	7	došenovi´c	došenovi´c	PROPN
cana-4142	347	8	,	,	PUNCT
cana-4142	347	9	z.	z.	PROPN
cana-4142	347	10	d.	d.	PROPN
cana-4142	347	11	mitrovi´c	mitrovi´c	PROPN
cana-4142	347	12	,	,	PUNCT
cana-4142	347	13	s.	s.	PROPN
cana-4142	347	14	radenovi´c	radenovi´c	PROPN
cana-4142	347	15	,	,	PUNCT
cana-4142	347	16	“	"	PUNCT
cana-4142	347	17	some	some	DET
cana-4142	347	18	new	new	ADJ
cana-4142	347	19	fixed	fix	VERB
cana-4142	347	20	-	-	PUNCT
cana-4142	347	21	point	point	NOUN
cana-4142	347	22	results	result	NOUN
cana-4142	347	23	in	in	ADP
cana-4142	347	24	b	b	NOUN
cana-4142	347	25	-	-	PUNCT
cana-4142	347	26	fuzzy	fuzzy	ADJ
cana-4142	347	27	metric	metric	ADJ
cana-4142	347	28	spaces	space	NOUN
cana-4142	347	29	”	"	PUNCT
cana-4142	347	30	,	,	PUNCT
cana-4142	347	31	j.	j.	PROPN
cana-4142	347	32	inequalities	inequalities	PROPN
cana-4142	347	33	appl	appl	PROPN
cana-4142	347	34	.	.	PROPN
cana-4142	347	35	,	,	PUNCT
cana-4142	347	36	(	(	PUNCT
cana-4142	347	37	2020	2020	NUM
cana-4142	347	38	)	)	PUNCT
cana-4142	347	39	2020:99	2020:99	NUM
cana-4142	347	40	.	.	PUNCT
cana-4142	348	1	7	7	X
cana-4142	348	2	.	.	X
cana-4142	348	3	muraliraj	muraliraj	PROPN
cana-4142	348	4	a	a	PRON
cana-4142	348	5	and	and	CCONJ
cana-4142	348	6	thangathamizh	thangathamizh	ADJ
cana-4142	348	7	r	r	NOUN
cana-4142	348	8	,	,	PUNCT
cana-4142	348	9	“	"	PUNCT
cana-4142	348	10	new	new	ADJ
cana-4142	348	11	relation	relation	NOUN
cana-4142	348	12	-	-	PUNCT
cana-4142	348	13	theoretic	theoretic	ADJ
cana-4142	348	14	fixed	fix	VERB
cana-4142	348	15	-	-	PUNCT
cana-4142	348	16	point	point	NOUN
cana-4142	348	17	theorems	theorem	NOUN
cana-4142	348	18	in	in	ADP
cana-4142	348	19	revised	revise	VERB
cana-4142	348	20	fuzzy	fuzzy	ADJ
cana-4142	348	21	metric	metric	ADJ
cana-4142	348	22	spaces	space	NOUN
cana-4142	348	23	with	with	ADP
cana-4142	348	24	an	an	DET
cana-4142	348	25	application	application	NOUN
cana-4142	348	26	to	to	ADP
cana-4142	348	27	fractional	fractional	ADJ
cana-4142	348	28	differential	differential	ADJ
cana-4142	348	29	equations	equation	NOUN
cana-4142	348	30	”	"	PUNCT
cana-4142	348	31	,	,	PUNCT
cana-4142	348	32	communications	communication	NOUN
cana-4142	348	33	communications	communication	NOUN
cana-4142	348	34	on	on	ADP
cana-4142	348	35	applied	apply	VERB
cana-4142	348	36	nonlinear	nonlinear	ADJ
cana-4142	348	37	analysis	analysis	NOUN
cana-4142	348	38	issn	issn	NOUN
cana-4142	348	39	:	:	PUNCT
cana-4142	348	40	1074	1074	NUM
cana-4142	348	41	-	-	PUNCT
cana-4142	348	42	133x	133x	NUM
cana-4142	348	43	vol	vol	NOUN
cana-4142	348	44	x	x	NOUN
cana-4142	348	45	no	no	INTJ
cana-4142	348	46	.	.	PUNCT
cana-4142	349	1	y	y	PROPN
cana-4142	349	2	(	(	PUNCT
cana-4142	349	3	2025	2025	NUM
cana-4142	349	4	)	)	PUNCT
cana-4142	349	5	1331	1331	NUM
cana-4142	349	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4142	349	7	in	in	ADP
cana-4142	349	8	mathematics	mathematic	NOUN
cana-4142	349	9	and	and	CCONJ
cana-4142	349	10	applications	application	NOUN
cana-4142	349	11	,	,	PUNCT
cana-4142	349	12	vol.12	vol.12	NOUN
cana-4142	349	13	,	,	PUNCT
cana-4142	349	14	no	no	DET
cana-4142	349	15	22	22	NUM
cana-4142	349	16	,	,	PUNCT
cana-4142	349	17	2023	2023	NUM
cana-4142	349	18	.	.	PUNCT
cana-4142	350	1	8	8	NUM
cana-4142	350	2	.	.	X
cana-4142	350	3	muraliraj	muraliraj	PROPN
cana-4142	350	4	.	.	PUNCT
cana-4142	351	1	a	a	PRON
cana-4142	351	2	and	and	CCONJ
cana-4142	351	3	thangathamizh	thangathamizh	ADJ
cana-4142	351	4	.	.	PUNCT
cana-4142	352	1	r	r	X
cana-4142	352	2	,	,	PUNCT
cana-4142	352	3	“	"	PUNCT
cana-4142	352	4	fixed	fix	VERB
cana-4142	352	5	point	point	NOUN
cana-4142	352	6	theorems	theorem	NOUN
cana-4142	352	7	in	in	ADP
cana-4142	352	8	revised	revise	VERB
cana-4142	352	9	fuzzy	fuzzy	ADJ
cana-4142	352	10	metric	metric	ADJ
cana-4142	352	11	space	space	NOUN
cana-4142	352	12	”	"	PUNCT
cana-4142	352	13	,	,	PUNCT
cana-4142	352	14	advances	advance	NOUN
cana-4142	352	15	in	in	ADP
cana-4142	352	16	fuzzy	fuzzy	ADJ
cana-4142	352	17	sets	set	NOUN
cana-4142	352	18	and	and	CCONJ
cana-4142	352	19	systems	system	NOUN
cana-4142	352	20	,	,	PUNCT
cana-4142	352	21	volume	volume	NOUN
cana-4142	352	22	26	26	NUM
cana-4142	352	23	,	,	PUNCT
cana-4142	352	24	number	number	NOUN
cana-4142	352	25	2	2	NUM
cana-4142	352	26	,	,	PUNCT
cana-4142	352	27	2021	2021	NUM
cana-4142	352	28	.	.	PUNCT
cana-4142	353	1	9	9	X
cana-4142	353	2	.	.	X
cana-4142	353	3	muraliraj	muraliraj	PROPN
cana-4142	353	4	.	.	PUNCT
cana-4142	354	1	a	a	PRON
cana-4142	354	2	and	and	CCONJ
cana-4142	354	3	thangathamizh	thangathamizh	ADJ
cana-4142	354	4	.	.	PUNCT
cana-4142	355	1	r	r	X
cana-4142	355	2	,	,	PUNCT
cana-4142	355	3	“	"	PUNCT
cana-4142	355	4	introduction	introduction	NOUN
cana-4142	355	5	on	on	ADP
cana-4142	355	6	revised	revise	VERB
cana-4142	355	7	fuzzy	fuzzy	ADJ
cana-4142	355	8	modular	modular	ADJ
cana-4142	355	9	spaces	space	NOUN
cana-4142	355	10	”	"	PUNCT
cana-4142	355	11	,	,	PUNCT
cana-4142	355	12	issn	issn	PROPN
cana-4142	355	13	09731768	09731768	NUM
cana-4142	355	14	,	,	PUNCT
cana-4142	355	15	volume	volume	NOUN
cana-4142	355	16	17	17	NUM
cana-4142	355	17	,	,	PUNCT
cana-4142	355	18	number	number	NOUN
cana-4142	355	19	2	2	NUM
cana-4142	355	20	(	(	PUNCT
cana-4142	355	21	2021	2021	NUM
cana-4142	355	22	)	)	PUNCT
cana-4142	355	23	,	,	PUNCT
cana-4142	355	24	pp	pp	ADP
cana-4142	355	25	.	.	PUNCT
cana-4142	356	1	303	303	NUM
cana-4142	356	2	-	-	SYM
cana-4142	356	3	317	317	NUM
cana-4142	356	4	10	10	NUM
cana-4142	356	5	.	.	PUNCT
cana-4142	357	1	muraliraj.a	muraliraj.a	PROPN
cana-4142	357	2	and	and	CCONJ
cana-4142	357	3	thangathamizh.r	thangathamizh.r	NUM
cana-4142	357	4	,	,	PUNCT
cana-4142	357	5	“	"	PUNCT
cana-4142	357	6	relation	relation	NOUN
cana-4142	357	7	–	–	PUNCT
cana-4142	357	8	theoretic	theoretic	NOUN
cana-4142	357	9	revised	revise	VERB
cana-4142	357	10	fuzzy	fuzzy	ADJ
cana-4142	357	11	banach	banach	NOUN
cana-4142	357	12	contraction	contraction	NOUN
cana-4142	357	13	principle	principle	NOUN
cana-4142	357	14	and	and	CCONJ
cana-4142	357	15	revised	revise	VERB
cana-4142	357	16	fuzzy	fuzzy	ADJ
cana-4142	357	17	eldestein	eldestein	NOUN
cana-4142	357	18	contraction	contraction	NOUN
cana-4142	357	19	theorem	theorem	VERB
cana-4142	357	20	”	"	PUNCT
cana-4142	357	21	,	,	PUNCT
cana-4142	357	22	jmscm	jmscm	PROPN
cana-4142	357	23	,	,	PUNCT
cana-4142	357	24	vol.3	vol.3	PROPN
cana-4142	357	25	,	,	PUNCT
cana-4142	357	26	no.2	no.2	PROPN
cana-4142	357	27	,	,	PUNCT
cana-4142	357	28	january	january	PROPN
cana-4142	357	29	2022	2022	NUM
cana-4142	357	30	.	.	PUNCT
cana-4142	358	1	11	11	NUM
cana-4142	358	2	.	.	PUNCT
cana-4142	359	1	olga	olga	PROPN
cana-4142	359	2	grigorenko	grigorenko	PROPN
cana-4142	359	3	,	,	PUNCT
cana-4142	359	4	juan	juan	PROPN
cana-4142	359	5	jose	jose	PROPN
cana-4142	359	6	minana	minana	PROPN
cana-4142	359	7	,	,	PUNCT
cana-4142	359	8	alexander	alexander	PROPN
cana-4142	359	9	sostak	sostak	PROPN
cana-4142	359	10	“	"	PUNCT
cana-4142	359	11	on	on	ADP
cana-4142	359	12	t	t	PROPN
cana-4142	359	13	-	-	PUNCT
cana-4142	359	14	conorm	conorm	NOUN
cana-4142	359	15	based	base	VERB
cana-4142	359	16	fuzzy	fuzzy	ADJ
cana-4142	359	17	(	(	PUNCT
cana-4142	359	18	pseudo	pseudo	NOUN
cana-4142	359	19	)	)	PUNCT
cana-4142	359	20	metrics	metric	NOUN
cana-4142	359	21	”	"	PUNCT
cana-4142	359	22	,	,	PUNCT
cana-4142	359	23	axioms	axiom	VERB
cana-4142	359	24	2020	2020	NUM
cana-4142	359	25	,	,	PUNCT
cana-4142	359	26	9	9	NUM
cana-4142	359	27	,	,	PUNCT
cana-4142	359	28	78	78	NUM
cana-4142	359	29	.	.	PUNCT
cana-4142	360	1	12	12	NUM
cana-4142	360	2	.	.	PUNCT
cana-4142	361	1	tarkan	tarkan	PROPN
cana-4142	361	2	oner	oner	PROPN
cana-4142	361	3	,	,	PUNCT
cana-4142	361	4	alexander	alexander	PROPN
cana-4142	361	5	sostak	sostak	PROPN
cana-4142	361	6	,	,	PUNCT
cana-4142	361	7	“	"	PUNCT
cana-4142	361	8	on	on	ADP
cana-4142	361	9	metric	metric	ADJ
cana-4142	361	10	-	-	PUNCT
cana-4142	361	11	type	type	NOUN
cana-4142	361	12	spaces	space	NOUN
cana-4142	361	13	based	base	VERB
cana-4142	361	14	on	on	ADP
cana-4142	361	15	extended	extend	VERB
cana-4142	361	16	t	t	PROPN
cana-4142	361	17	-	-	PUNCT
cana-4142	361	18	conorms	conorm	NOUN
cana-4142	361	19	”	"	PUNCT
cana-4142	361	20	mathematics	mathematic	NOUN
cana-4142	361	21	2020	2020	NUM
cana-4142	361	22	,	,	PUNCT
cana-4142	361	23	8	8	NUM
cana-4142	361	24	,	,	PUNCT
cana-4142	361	25	1097	1097	NUM
cana-4142	361	26	.	.	PUNCT
cana-4142	362	1	13	13	NUM
cana-4142	362	2	.	.	PUNCT
cana-4142	362	3	dhananjay	dhananjay	PROPN
cana-4142	362	4	gopal	gopal	PROPN
cana-4142	362	5	,	,	PUNCT
cana-4142	362	6	wutiphol	wutiphol	NOUN
cana-4142	362	7	sintunavarat	sintunavarat	NOUN
cana-4142	362	8	,	,	PUNCT
cana-4142	362	9	abhay	abhay	PROPN
cana-4142	362	10	s.	s.	PROPN
cana-4142	362	11	ranadive	ranadive	PROPN
cana-4142	362	12	,	,	PUNCT
cana-4142	362	13	satish	satish	ADJ
cana-4142	362	14	shukla	shukla	NOUN
cana-4142	362	15	,	,	PUNCT
cana-4142	362	16	“	"	PUNCT
cana-4142	362	17	the	the	DET
cana-4142	362	18	investigation	investigation	NOUN
cana-4142	362	19	of	of	ADP
cana-4142	362	20	k	k	ADJ
cana-4142	362	21	-	-	ADJ
cana-4142	362	22	fuzzy	fuzzy	ADJ
cana-4142	362	23	metric	metric	ADJ
cana-4142	362	24	spaces	space	NOUN
cana-4142	362	25	with	with	ADP
cana-4142	362	26	the	the	DET
cana-4142	362	27	first	first	ADJ
cana-4142	362	28	contraction	contraction	NOUN
cana-4142	362	29	principle	principle	NOUN
cana-4142	362	30	in	in	ADP
cana-4142	362	31	such	such	ADJ
cana-4142	362	32	spaces	space	NOUN
cana-4142	362	33	,	,	PUNCT
cana-4142	362	34	”	"	PUNCT
cana-4142	362	35	soft	soft	ADJ
cana-4142	362	36	computing	computing	NOUN
cana-4142	362	37	,	,	PUNCT
cana-4142	362	38	3	3	NUM
cana-4142	362	39	march	march	NOUN
cana-4142	362	40	,	,	PUNCT
cana-4142	362	41	2023	2023	NUM
cana-4142	362	42	.	.	PUNCT
cana-4142	363	1	14	14	NUM
cana-4142	363	2	.	.	PUNCT
cana-4142	364	1	thangathamizh	thangathamizh	PROPN
cana-4142	364	2	,	,	PUNCT
cana-4142	364	3	r.	r.	PROPN
cana-4142	364	4	,	,	PUNCT
cana-4142	364	5	muraliraj	muraliraj	PROPN
cana-4142	364	6	,	,	PUNCT
cana-4142	364	7	a.	a.	NOUN
cana-4142	364	8	&	&	CCONJ
cana-4142	364	9	shanmugavel	shanmugavel	PROPN
cana-4142	364	10	,	,	PUNCT
cana-4142	364	11	p.	p.	NOUN
cana-4142	364	12	2024	2024	NUM
cana-4142	364	13	.	.	PUNCT
cana-4142	365	1	new	new	ADJ
cana-4142	365	2	approach	approach	NOUN
cana-4142	365	3	of	of	ADP
cana-4142	365	4	lebesgue	lebesgue	PROPN
cana-4142	365	5	integral	integral	ADJ
cana-4142	365	6	in	in	ADP
cana-4142	365	7	revised	revise	VERB
cana-4142	365	8	fuzzy	fuzzy	ADJ
cana-4142	365	9	cone	cone	NOUN
cana-4142	365	10	metric	metric	ADJ
cana-4142	365	11	spaces	space	NOUN
cana-4142	365	12	via	via	ADP
cana-4142	365	13	unique	unique	ADJ
cana-4142	365	14	coupled	couple	VERB
cana-4142	365	15	fixed	fix	VERB
cana-4142	365	16	point	point	NOUN
cana-4142	365	17	theorems	theorem	NOUN
cana-4142	365	18	.	.	PUNCT
cana-4142	366	1	vojnotehnički	vojnotehnički	PROPN
cana-4142	366	2	glasnik	glasnik	PROPN
cana-4142	366	3	/	/	SYM
cana-4142	366	4	military	military	ADJ
cana-4142	366	5	technical	technical	ADJ
cana-4142	366	6	courier	courier	NOUN
cana-4142	366	7	,	,	PUNCT
cana-4142	366	8	72(3	72(3	NUM
cana-4142	366	9	)	)	PUNCT
cana-4142	366	10	,	,	PUNCT
cana-4142	366	11	pp.10291045	pp.10291045	PROPN
cana-4142	366	12	.	.	PUNCT
cana-4142	366	13	available	available	ADJ
cana-4142	366	14	at	at	ADP
cana-4142	366	15	:	:	PUNCT
cana-4142	366	16	https://doi.org/10.5937/vojtehg72-48816	https://doi.org/10.5937/vojtehg72-48816	PROPN
cana-4142	366	17	.	.	PUNCT
cana-4142	367	1	15	15	NUM
cana-4142	367	2	.	.	X
cana-4142	367	3	parakath	parakath	PROPN
cana-4142	367	4	nisha	nisha	PROPN
cana-4142	367	5	bagam	bagam	PROPN
cana-4142	367	6	p	p	PROPN
cana-4142	367	7	,	,	PUNCT
cana-4142	367	8	sandhya	sandhya	PROPN
cana-4142	367	9	p	p	PROPN
cana-4142	367	10	,	,	PUNCT
cana-4142	367	11	thangathamizh	thangathamizh	ADJ
cana-4142	367	12	r	r	NOUN
cana-4142	367	13	,	,	PUNCT
cana-4142	367	14	shanmugavel	shanmugavel	NOUN
cana-4142	367	15	p	p	NOUN
cana-4142	367	16	,	,	PUNCT
cana-4142	367	17	sarathbabu	sarathbabu	PROPN
cana-4142	367	18	k	k	NOUN
cana-4142	367	19	,	,	PUNCT
cana-4142	367	20	anusuya	anusuya	PROPN
cana-4142	367	21	r	r	NOUN
cana-4142	367	22	,	,	PUNCT
cana-4142	367	23	“	"	PUNCT
cana-4142	367	24	fixed	fix	VERB
cana-4142	367	25	point	point	NOUN
cana-4142	367	26	theorems	theorem	NOUN
cana-4142	367	27	in	in	ADP
cana-4142	367	28	revised	revise	VERB
cana-4142	367	29	fuzzy	fuzzy	ADJ
cana-4142	367	30	metric	metric	ADJ
cana-4142	367	31	space	space	NOUN
cana-4142	367	32	via	via	ADP
cana-4142	367	33	𝑅𝐹	𝑅𝐹	PROPN
cana-4142	367	34	−contraction	−contraction	PROPN
cana-4142	367	35	”	"	PUNCT
cana-4142	367	36	,	,	PUNCT
cana-4142	367	37	communications	communication	NOUN
cana-4142	367	38	on	on	ADP
cana-4142	367	39	applied	apply	VERB
cana-4142	367	40	nonlinear	nonlinear	ADJ
cana-4142	367	41	analysis	analysis	NOUN
cana-4142	367	42	,	,	PUNCT
cana-4142	367	43	vol	vol	NOUN
cana-4142	367	44	.	.	PROPN
cana-4142	367	45	31	31	NUM
cana-4142	368	1	no	no	NOUN
cana-4142	368	2	.	.	PUNCT
cana-4142	369	1	3s	3s	NUM
cana-4142	369	2	(	(	PUNCT
cana-4142	369	3	2024	2024	NUM
cana-4142	369	4	)	)	PUNCT
cana-4142	369	5	.	.	PUNCT
cana-4142	370	1	16	16	NUM
cana-4142	370	2	.	.	PUNCT
cana-4142	370	3	a.	a.	PROPN
cana-4142	370	4	muraliraj	muraliraj	PROPN
cana-4142	370	5	,	,	PUNCT
cana-4142	370	6	p.	p.	PROPN
cana-4142	370	7	shanmugavel	shanmugavel	PROPN
cana-4142	370	8	,	,	PUNCT
cana-4142	370	9	r.	r.	PROPN
cana-4142	370	10	thangathamizh	thangathamizh	PROPN
cana-4142	370	11	,	,	PUNCT
cana-4142	370	12	“	"	PUNCT
cana-4142	370	13	existence	existence	NOUN
cana-4142	370	14	of	of	ADP
cana-4142	370	15	fixed	fix	VERB
cana-4142	370	16	-	-	PUNCT
cana-4142	370	17	point	point	NOUN
cana-4142	370	18	theorems	theorem	NOUN
cana-4142	370	19	in	in	ADP
cana-4142	370	20	revised	revise	VERB
cana-4142	370	21	fuzzy	fuzzy	ADJ
cana-4142	370	22	modular	modular	ADJ
cana-4142	370	23	spaces	space	NOUN
cana-4142	370	24	”	"	PUNCT
cana-4142	370	25	,	,	PUNCT
cana-4142	370	26	advances	advance	NOUN
cana-4142	370	27	in	in	ADP
cana-4142	370	28	nonlinear	nonlinear	ADJ
cana-4142	370	29	variational	variational	ADJ
cana-4142	370	30	inequalities	inequality	NOUN
cana-4142	370	31	,	,	PUNCT
cana-4142	370	32	vol	vol	NOUN
cana-4142	370	33	24	24	NUM
cana-4142	370	34	no	no	DET
cana-4142	370	35	2	2	NUM
cana-4142	370	36	.	.	PUNCT
cana-4142	370	37	(	(	PUNCT
cana-4142	370	38	2024	2024	NUM
cana-4142	370	39	)	)	PUNCT
cana-4142	370	40	.	.	PUNCT
cana-4142	371	1	17	17	NUM
cana-4142	371	2	.	.	PUNCT
cana-4142	371	3	a.	a.	PROPN
cana-4142	371	4	muraliraj	muraliraj	PROPN
cana-4142	371	5	,	,	PUNCT
cana-4142	371	6	p.	p.	PROPN
cana-4142	371	7	shanmugavel	shanmugavel	PROPN
cana-4142	371	8	,	,	PUNCT
cana-4142	371	9	r.	r.	PROPN
cana-4142	371	10	thangathamizh	thangathamizh	PROPN
cana-4142	371	11	,	,	PUNCT
cana-4142	371	12	“	"	PUNCT
cana-4142	371	13	fixed	fix	VERB
cana-4142	371	14	point	point	NOUN
cana-4142	371	15	theorems	theorem	NOUN
cana-4142	371	16	on	on	ADP
cana-4142	371	17	modular	modular	ADJ
cana-4142	371	18	revised	revise	VERB
cana-4142	371	19	fuzzy	fuzzy	ADJ
cana-4142	371	20	metric	metric	ADJ
cana-4142	371	21	spaces	space	NOUN
cana-4142	371	22	”	"	PUNCT
cana-4142	371	23	,	,	PUNCT
cana-4142	371	24	communications	communication	NOUN
cana-4142	371	25	on	on	ADP
cana-4142	371	26	applied	apply	VERB
cana-4142	371	27	nonlinear	nonlinear	ADJ
cana-4142	371	28	analysis	analysis	NOUN
cana-4142	371	29	,	,	PUNCT
cana-4142	371	30	vol	vol	NOUN
cana-4142	371	31	.	.	PROPN
cana-4142	371	32	31	31	NUM
cana-4142	372	1	no	no	NOUN
cana-4142	372	2	.	.	PUNCT
cana-4142	373	1	3s	3s	NUM
cana-4142	373	2	(	(	PUNCT
cana-4142	373	3	2024	2024	NUM
cana-4142	373	4	)	)	PUNCT
cana-4142	373	5	18	18	NUM
cana-4142	373	6	.	.	PUNCT
cana-4142	374	1	r.	r.	PROPN
cana-4142	374	2	thangathamizh	thangathamizh	PROPN
cana-4142	374	3	,	,	PUNCT
cana-4142	374	4	k.	k.	PROPN
cana-4142	374	5	balamurugan	balamurugan	PROPN
cana-4142	374	6	,	,	PUNCT
cana-4142	374	7	c.	c.	PROPN
cana-4142	374	8	karnan	karnan	PROPN
cana-4142	374	9	,	,	PUNCT
cana-4142	374	10	p.	p.	NOUN
cana-4142	374	11	shanmugavel	shanmugavel	PROPN
cana-4142	374	12	,	,	PUNCT
cana-4142	374	13	d.	d.	PROPN
cana-4142	374	14	balraj	balraj	PROPN
cana-4142	374	15	,	,	PUNCT
cana-4142	374	16	“	"	PUNCT
cana-4142	374	17	a	a	DET
cana-4142	374	18	revised	revise	VERB
cana-4142	374	19	fuzzy	fuzzy	ADJ
cana-4142	374	20	differential	differential	ADJ
cana-4142	374	21	equations	equation	NOUN
cana-4142	374	22	using	use	VERB
cana-4142	374	23	weakly	weakly	ADJ
cana-4142	374	24	compatible	compatible	ADJ
cana-4142	374	25	self	self	NOUN
cana-4142	374	26	-	-	PUNCT
cana-4142	374	27	mappings	mapping	NOUN
cana-4142	374	28	in	in	ADP
cana-4142	374	29	revised	revise	VERB
cana-4142	374	30	fuzzy	fuzzy	ADJ
cana-4142	374	31	metric	metric	ADJ
cana-4142	374	32	spaces	space	NOUN
cana-4142	374	33	”	"	PUNCT
cana-4142	374	34	advances	advance	NOUN
cana-4142	374	35	in	in	ADP
cana-4142	374	36	nonlinear	nonlinear	ADJ
cana-4142	374	37	variational	variational	ADJ
cana-4142	374	38	inequalities	inequality	NOUN
cana-4142	374	39	,	,	PUNCT
cana-4142	374	40	vol	vol	NOUN
cana-4142	374	41	24	24	NUM
cana-4142	374	42	no	no	DET
cana-4142	374	43	2	2	NUM
cana-4142	374	44	.	.	PUNCT
cana-4142	374	45	(	(	PUNCT
cana-4142	374	46	2024	2024	NUM
cana-4142	374	47	)	)	PUNCT
cana-4142	374	48	.	.	PUNCT
cana-4142	375	1	19	19	NUM
cana-4142	375	2	.	.	X
cana-4142	375	3	r.	r.	PROPN
cana-4142	375	4	thangathamizh	thangathamizh	PROPN
cana-4142	375	5	,	,	PUNCT
cana-4142	375	6	a.	a.	NOUN
cana-4142	375	7	muraliraj	muraliraj	PROPN
cana-4142	375	8	,	,	PUNCT
cana-4142	375	9	p.	p.	NOUN
cana-4142	375	10	shanmugavel	shanmugavel	NOUN
cana-4142	375	11	,	,	PUNCT
cana-4142	375	12	“	"	PUNCT
cana-4142	375	13	new	new	ADJ
cana-4142	375	14	approach	approach	NOUN
cana-4142	375	15	of	of	ADP
cana-4142	375	16	lebesgue	lebesgue	PROPN
cana-4142	375	17	integral	integral	ADJ
cana-4142	375	18	in	in	ADP
cana-4142	375	19	revised	revise	VERB
cana-4142	375	20	fuzzy	fuzzy	ADJ
cana-4142	375	21	cone	cone	NOUN
cana-4142	375	22	metric	metric	ADJ
cana-4142	375	23	spaces	space	NOUN
cana-4142	375	24	vie	vie	X
cana-4142	375	25	unique	unique	ADJ
cana-4142	375	26	coupled	couple	VERB
cana-4142	375	27	fixed	fix	VERB
cana-4142	375	28	-	-	PUNCT
cana-4142	375	29	point	point	NOUN
cana-4142	375	30	theorems	theorem	NOUN
cana-4142	375	31	”	"	PUNCT
cana-4142	375	32	,	,	PUNCT
cana-4142	375	33	military	military	ADJ
cana-4142	375	34	technical	technical	ADJ
cana-4142	375	35	courier	courier	NOUN
cana-4142	375	36	,	,	PUNCT
cana-4142	375	37	http://dx.doi.org/10.5937/vojtehg72-48816	http://dx.doi.org/10.5937/vojtehg72-48816	PROPN
cana-4142	375	38	.	.	PROPN
cana-4142	375	39	20	20	NUM
cana-4142	375	40	.	.	PUNCT
cana-4142	376	1	ravichandran	ravichandran	PROPN
cana-4142	376	2	thangathamizh	thangathamizh	ADJ
cana-4142	376	3	,	,	PUNCT
cana-4142	376	4	abdelhamid	abdelhamid	ADP
cana-4142	376	5	moussaoui	moussaoui	NOUN
cana-4142	376	6	,	,	PUNCT
cana-4142	376	7	tatjana	tatjana	PROPN
cana-4142	376	8	došenović	došenović	PROPN
cana-4142	376	9	,	,	PUNCT
cana-4142	376	10	stojan	stojan	PROPN
cana-4142	376	11	radenović	radenović	NOUN
cana-4142	376	12	,	,	PUNCT
cana-4142	376	13	“	"	PUNCT
cana-4142	376	14	fixed	fix	VERB
cana-4142	376	15	point	point	NOUN
cana-4142	376	16	results	result	NOUN
cana-4142	376	17	in	in	ADP
cana-4142	376	18	controlled	control	VERB
cana-4142	376	19	revised	revise	VERB
cana-4142	376	20	fuzzy	fuzzy	ADJ
cana-4142	376	21	metric	metric	ADJ
cana-4142	376	22	spaces	space	NOUN
cana-4142	376	23	with	with	ADP
cana-4142	376	24	an	an	DET
cana-4142	376	25	application	application	NOUN
cana-4142	376	26	to	to	ADP
cana-4142	376	27	solar	solar	ADJ
cana-4142	376	28	energy	energy	NOUN
cana-4142	376	29	to	to	ADP
cana-4142	376	30	electric	electric	ADJ
cana-4142	376	31	power	power	NOUN
cana-4142	376	32	”	"	PUNCT
cana-4142	376	33	,	,	PUNCT
cana-4142	376	34	https://doi.org/10.5937/vojtehg72-49064	https://doi.org/10.5937/vojtehg72-49064	PROPN
cana-4142	376	35	.	.	PUNCT
cana-4142	377	1	https://doi.org/10.5937/vojtehg72-48816	https://doi.org/10.5937/vojtehg72-48816	PROPN
cana-4142	377	2	https://internationalpubls.com/index.php/cana/issue/view/58	https://internationalpubls.com/index.php/cana/issue/view/58	PROPN
cana-4142	378	1	https://internationalpubls.com/index.php/cana/issue/view/58	https://internationalpubls.com/index.php/cana/issue/view/58	PROPN
cana-4142	378	2	http://dx.doi.org/10.5937/vojtehg72-48816	http://dx.doi.org/10.5937/vojtehg72-48816	PROPN