id	sid	tid	token	lemma	pos
cana-5487	1	1	communications	communication	NOUN
cana-5487	1	2	on	on	ADP
cana-5487	1	3	applied	apply	VERB
cana-5487	1	4	nonlinear	nonlinear	ADJ
cana-5487	1	5	analysis	analysis	NOUN
cana-5487	1	6	issn	issn	NOUN
cana-5487	1	7	:	:	PUNCT
cana-5487	1	8	1074	1074	NUM
cana-5487	1	9	-	-	PUNCT
cana-5487	1	10	133x	133x	NUM
cana-5487	1	11	vol	vol	VERB
cana-5487	1	12	32	32	NUM
cana-5487	1	13	no	no	NOUN
cana-5487	1	14	.	.	PUNCT
cana-5487	2	1	10s	10	NOUN
cana-5487	2	2	(	(	PUNCT
cana-5487	2	3	2025	2025	NUM
cana-5487	2	4	)	)	PUNCT
cana-5487	2	5	2383	2383	NUM
cana-5487	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	2	7	reimagining	reimagine	VERB
cana-5487	2	8	fixed	fix	VERB
cana-5487	2	9	points	point	NOUN
cana-5487	2	10	:	:	PUNCT
cana-5487	2	11	exploring	explore	VERB
cana-5487	2	12	the	the	DET
cana-5487	2	13	role	role	NOUN
cana-5487	2	14	of	of	ADP
cana-5487	2	15	occasionally	occasionally	ADV
cana-5487	2	16	weakly	weakly	ADJ
cana-5487	2	17	compatible	compatible	ADJ
cana-5487	2	18	mappings	mapping	NOUN
cana-5487	2	19	in	in	ADP
cana-5487	2	20	fuzzy	fuzzy	ADJ
cana-5487	2	21	metrics	metric	NOUN
cana-5487	2	22	1priyanka	1priyanka	NUM
cana-5487	2	23	nigam	nigam	NOUN
cana-5487	2	24	,	,	PUNCT
cana-5487	2	25	2sandhya	2sandhya	NUM
cana-5487	2	26	shukla	shukla	NOUN
cana-5487	2	27	1presidency	1presidency	NUM
cana-5487	2	28	university	university	NOUN
cana-5487	2	29	,	,	PUNCT
cana-5487	2	30	bengaluru	bengaluru	PROPN
cana-5487	2	31	,	,	PUNCT
cana-5487	2	32	560089	560089	NUM
cana-5487	2	33	,	,	PUNCT
cana-5487	2	34	karnataka	karnataka	PROPN
cana-5487	2	35	,	,	PUNCT
cana-5487	2	36	india	india	PROPN
cana-5487	2	37	.	.	PUNCT
cana-5487	3	1	e	e	X
cana-5487	3	2	-	-	NOUN
cana-5487	3	3	mail	mail	NOUN
cana-5487	3	4	:	:	PUNCT
cana-5487	3	5	priyanka.nigam@presidencyuniversity.in	priyanka.nigam@presidencyuniversity.in	ADJ
cana-5487	3	6	,	,	PUNCT
cana-5487	3	7	priyanka_nigam01@yahoo.co.in	priyanka_nigam01@yahoo.co.in	PROPN
cana-5487	3	8	2university	2university	NUM
cana-5487	3	9	institute	institute	NOUN
cana-5487	3	10	of	of	ADP
cana-5487	3	11	technology	technology	PROPN
cana-5487	3	12	,	,	PUNCT
cana-5487	3	13	rgpv	rgpv	ADV
cana-5487	3	14	bhopal	bhopal	PROPN
cana-5487	3	15	,	,	PUNCT
cana-5487	3	16	462033	462033	NUM
cana-5487	3	17	,	,	PUNCT
cana-5487	3	18	madhya	madhya	PROPN
cana-5487	3	19	pradesh	pradesh	PROPN
cana-5487	3	20	,	,	PUNCT
cana-5487	3	21	india	india	PROPN
cana-5487	3	22	e	e	PROPN
cana-5487	3	23	-	-	NOUN
cana-5487	3	24	mail	mail	NOUN
cana-5487	3	25	:	:	PUNCT
cana-5487	3	26	maths.sandhyashukla@gmail.com	maths.sandhyashukla@gmail.com	X
cana-5487	3	27	article	article	NOUN
cana-5487	3	28	history	history	NOUN
cana-5487	3	29	:	:	PUNCT
cana-5487	3	30	received	receive	VERB
cana-5487	3	31	:	:	PUNCT
cana-5487	3	32	12	12	NUM
cana-5487	3	33	-	-	SYM
cana-5487	3	34	01	01	NUM
cana-5487	3	35	-	-	PUNCT
cana-5487	3	36	2025	2025	NUM
cana-5487	3	37	revised	revise	VERB
cana-5487	3	38	:	:	PUNCT
cana-5487	3	39	15	15	NUM
cana-5487	3	40	-	-	NUM
cana-5487	3	41	02	02	NUM
cana-5487	3	42	-	-	PUNCT
cana-5487	3	43	2025	2025	NUM
cana-5487	3	44	accepted	accept	VERB
cana-5487	3	45	:	:	PUNCT
cana-5487	3	46	01	01	NUM
cana-5487	3	47	-	-	SYM
cana-5487	3	48	03	03	NUM
cana-5487	3	49	-	-	PUNCT
cana-5487	3	50	2025	2025	NUM
cana-5487	3	51	abstract	abstract	NOUN
cana-5487	3	52	:	:	PUNCT
cana-5487	3	53	in	in	ADP
cana-5487	3	54	this	this	DET
cana-5487	3	55	work	work	NOUN
cana-5487	3	56	,	,	PUNCT
cana-5487	3	57	we	we	PRON
cana-5487	3	58	revisit	revisit	VERB
cana-5487	3	59	the	the	DET
cana-5487	3	60	concept	concept	NOUN
cana-5487	3	61	of	of	ADP
cana-5487	3	62	fixed	fix	VERB
cana-5487	3	63	points	point	NOUN
cana-5487	3	64	by	by	ADP
cana-5487	3	65	exploring	explore	VERB
cana-5487	3	66	the	the	DET
cana-5487	3	67	role	role	NOUN
cana-5487	3	68	of	of	ADP
cana-5487	3	69	occasionally	occasionally	ADV
cana-5487	3	70	weakly	weakly	ADV
cana-5487	3	71	compatible	compatible	ADJ
cana-5487	3	72	(	(	PUNCT
cana-5487	3	73	owc	owc	NOUN
cana-5487	3	74	)	)	PUNCT
cana-5487	3	75	mappings	mapping	NOUN
cana-5487	3	76	within	within	ADP
cana-5487	3	77	the	the	DET
cana-5487	3	78	structure	structure	NOUN
cana-5487	3	79	of	of	ADP
cana-5487	3	80	fuzzy	fuzzy	ADJ
cana-5487	3	81	metric	metric	ADJ
cana-5487	3	82	spaces	space	NOUN
cana-5487	3	83	.	.	PUNCT
cana-5487	4	1	building	build	VERB
cana-5487	4	2	on	on	ADP
cana-5487	4	3	the	the	DET
cana-5487	4	4	classical	classical	ADJ
cana-5487	4	5	fixed	fix	VERB
cana-5487	4	6	point	point	NOUN
cana-5487	4	7	theory	theory	NOUN
cana-5487	4	8	,	,	PUNCT
cana-5487	4	9	we	we	PRON
cana-5487	4	10	investigate	investigate	VERB
cana-5487	4	11	new	new	ADJ
cana-5487	4	12	conditions	condition	NOUN
cana-5487	4	13	under	under	ADP
cana-5487	4	14	which	which	PRON
cana-5487	4	15	tripled	triple	VERB
cana-5487	4	16	fixed	fix	VERB
cana-5487	4	17	points	point	NOUN
cana-5487	4	18	exist	exist	VERB
cana-5487	4	19	and	and	CCONJ
cana-5487	4	20	are	be	AUX
cana-5487	4	21	unique	unique	ADJ
cana-5487	4	22	.	.	PUNCT
cana-5487	5	1	by	by	ADP
cana-5487	5	2	employing	employ	VERB
cana-5487	5	3	the	the	DET
cana-5487	5	4	framework	framework	NOUN
cana-5487	5	5	of	of	ADP
cana-5487	5	6	fuzzy	fuzzy	ADJ
cana-5487	5	7	metrics	metric	NOUN
cana-5487	5	8	and	and	CCONJ
cana-5487	5	9	leveraging	leverage	VERB
cana-5487	5	10	the	the	DET
cana-5487	5	11	flexibility	flexibility	NOUN
cana-5487	5	12	of	of	ADP
cana-5487	5	13	owc	owc	PROPN
cana-5487	5	14	mappings	mapping	NOUN
cana-5487	5	15	,	,	PUNCT
cana-5487	5	16	we	we	PRON
cana-5487	5	17	establish	establish	VERB
cana-5487	5	18	a	a	DET
cana-5487	5	19	generalized	generalize	VERB
cana-5487	5	20	tripled	triple	VERB
cana-5487	5	21	fixed	fix	VERB
cana-5487	5	22	point	point	NOUN
cana-5487	5	23	theorem	theorem	NOUN
cana-5487	5	24	that	that	PRON
cana-5487	5	25	extends	extend	VERB
cana-5487	5	26	several	several	ADJ
cana-5487	5	27	known	know	VERB
cana-5487	5	28	results	result	NOUN
cana-5487	5	29	.	.	PUNCT
cana-5487	6	1	we	we	PRON
cana-5487	6	2	also	also	ADV
cana-5487	6	3	explore	explore	VERB
cana-5487	6	4	the	the	DET
cana-5487	6	5	concept	concept	NOUN
cana-5487	6	6	of	of	ADP
cana-5487	6	7	tripled	triple	VERB
cana-5487	6	8	fixed	fix	VERB
cana-5487	6	9	points	point	NOUN
cana-5487	6	10	for	for	ADP
cana-5487	6	11	occasionally	occasionally	ADV
cana-5487	6	12	weakly	weakly	ADJ
cana-5487	6	13	compatible	compatible	ADJ
cana-5487	6	14	mappings	mapping	NOUN
cana-5487	6	15	within	within	ADP
cana-5487	6	16	the	the	DET
cana-5487	6	17	framework	framework	NOUN
cana-5487	6	18	of	of	ADP
cana-5487	6	19	fuzzy	fuzzy	ADJ
cana-5487	6	20	metric	metric	ADJ
cana-5487	6	21	spaces	space	NOUN
cana-5487	6	22	.	.	PUNCT
cana-5487	7	1	we	we	PRON
cana-5487	7	2	establish	establish	VERB
cana-5487	7	3	several	several	ADJ
cana-5487	7	4	novel	novel	NOUN
cana-5487	7	5	tripled	triple	VERB
cana-5487	7	6	fixed	fix	VERB
cana-5487	7	7	-	-	PUNCT
cana-5487	7	8	point	point	NOUN
cana-5487	7	9	theorems	theorem	NOUN
cana-5487	7	10	that	that	PRON
cana-5487	7	11	extend	extend	VERB
cana-5487	7	12	existing	exist	VERB
cana-5487	7	13	results	result	NOUN
cana-5487	7	14	in	in	ADP
cana-5487	7	15	this	this	DET
cana-5487	7	16	area	area	NOUN
cana-5487	7	17	.	.	PUNCT
cana-5487	8	1	additionally	additionally	ADV
cana-5487	8	2	,	,	PUNCT
cana-5487	8	3	to	to	PART
cana-5487	8	4	validate	validate	VERB
cana-5487	8	5	the	the	DET
cana-5487	8	6	applicability	applicability	NOUN
cana-5487	8	7	of	of	ADP
cana-5487	8	8	our	our	PRON
cana-5487	8	9	theorems	theorem	NOUN
cana-5487	8	10	,	,	PUNCT
cana-5487	8	11	we	we	PRON
cana-5487	8	12	provide	provide	VERB
cana-5487	8	13	detailed	detailed	ADJ
cana-5487	8	14	illustrative	illustrative	ADJ
cana-5487	8	15	examples	example	NOUN
cana-5487	8	16	that	that	PRON
cana-5487	8	17	demonstrate	demonstrate	VERB
cana-5487	8	18	the	the	DET
cana-5487	8	19	effectiveness	effectiveness	NOUN
cana-5487	8	20	and	and	CCONJ
cana-5487	8	21	relevance	relevance	NOUN
cana-5487	8	22	of	of	ADP
cana-5487	8	23	the	the	DET
cana-5487	8	24	established	establish	VERB
cana-5487	8	25	results	result	NOUN
cana-5487	8	26	in	in	ADP
cana-5487	8	27	fuzzy	fuzzy	ADJ
cana-5487	8	28	metric	metric	ADJ
cana-5487	8	29	settings	setting	NOUN
cana-5487	8	30	.	.	PUNCT
cana-5487	9	1	these	these	DET
cana-5487	9	2	findings	finding	NOUN
cana-5487	9	3	contribute	contribute	VERB
cana-5487	9	4	to	to	ADP
cana-5487	9	5	the	the	DET
cana-5487	9	6	broader	broad	ADJ
cana-5487	9	7	understanding	understanding	NOUN
cana-5487	9	8	of	of	ADP
cana-5487	9	9	fixed	fix	VERB
cana-5487	9	10	-	-	PUNCT
cana-5487	9	11	point	point	NOUN
cana-5487	9	12	theory	theory	NOUN
cana-5487	9	13	in	in	ADP
cana-5487	9	14	fuzzy	fuzzy	ADJ
cana-5487	9	15	environments	environment	NOUN
cana-5487	9	16	and	and	CCONJ
cana-5487	9	17	open	open	VERB
cana-5487	9	18	new	new	ADJ
cana-5487	9	19	avenues	avenue	NOUN
cana-5487	9	20	for	for	ADP
cana-5487	9	21	future	future	ADJ
cana-5487	9	22	research	research	NOUN
cana-5487	9	23	in	in	ADP
cana-5487	9	24	generalized	generalized	ADJ
cana-5487	9	25	metric	metric	ADJ
cana-5487	9	26	spaces	space	NOUN
cana-5487	9	27	and	and	CCONJ
cana-5487	9	28	their	their	PRON
cana-5487	9	29	applications	application	NOUN
cana-5487	9	30	.	.	PUNCT
cana-5487	10	1	keywords	keyword	NOUN
cana-5487	10	2	:	:	PUNCT
cana-5487	10	3	occasionally	occasionally	ADV
cana-5487	10	4	weakly	weakly	ADJ
cana-5487	10	5	compatible	compatible	ADJ
cana-5487	10	6	mappings	mapping	NOUN
cana-5487	10	7	;	;	PUNCT
cana-5487	10	8	tripled	triple	VERB
cana-5487	10	9	fixed	fix	VERB
cana-5487	10	10	point	point	NOUN
cana-5487	10	11	;	;	PUNCT
cana-5487	10	12	fuzzy	fuzzy	ADJ
cana-5487	10	13	metric	metric	ADJ
cana-5487	10	14	space	space	NOUN
cana-5487	10	15	.	.	PUNCT
cana-5487	11	1	2000	2000	NUM
cana-5487	11	2	mathematics	mathematic	NOUN
cana-5487	11	3	subject	subject	ADJ
cana-5487	11	4	classification	classification	NOUN
cana-5487	11	5	:	:	PUNCT
cana-5487	11	6	47h10	47h10	NUM
cana-5487	11	7	;	;	PUNCT
cana-5487	11	8	54h25	54h25	NUM
cana-5487	11	9	.	.	X
cana-5487	12	1	1	1	NUM
cana-5487	12	2	introduction	introduction	NOUN
cana-5487	12	3	zadeh	zadeh	NOUN
cana-5487	12	4	[	[	X
cana-5487	12	5	14	14	NUM
cana-5487	12	6	]	]	X
cana-5487	12	7	defined	define	VERB
cana-5487	12	8	fuzzy	fuzzy	ADJ
cana-5487	12	9	sets	set	NOUN
cana-5487	12	10	.	.	PUNCT
cana-5487	13	1	kramosil	kramosil	NOUN
cana-5487	13	2	and	and	CCONJ
cana-5487	13	3	michalek	michalek	VERB
cana-5487	13	4	[	[	X
cana-5487	13	5	7	7	X
cana-5487	13	6	]	]	PUNCT
cana-5487	13	7	introduced	introduce	VERB
cana-5487	13	8	fuzzy	fuzzy	ADJ
cana-5487	13	9	metric	metric	ADJ
cana-5487	13	10	space	space	NOUN
cana-5487	13	11	,	,	PUNCT
cana-5487	13	12	george	george	PROPN
cana-5487	13	13	and	and	CCONJ
cana-5487	13	14	veermani	veermani	NOUN
cana-5487	14	1	[	[	X
cana-5487	14	2	3	3	NUM
cana-5487	14	3	]	]	X
cana-5487	14	4	modified	modify	VERB
cana-5487	14	5	the	the	DET
cana-5487	14	6	notion	notion	NOUN
cana-5487	14	7	and	and	CCONJ
cana-5487	14	8	gave	give	VERB
cana-5487	14	9	a	a	DET
cana-5487	14	10	new	new	ADJ
cana-5487	14	11	notion	notion	NOUN
cana-5487	14	12	with	with	ADP
cana-5487	14	13	the	the	DET
cana-5487	14	14	help	help	NOUN
cana-5487	14	15	of	of	ADP
cana-5487	14	16	continuous	continuous	ADJ
cana-5487	14	17	t	t	NOUN
cana-5487	14	18	-	-	PUNCT
cana-5487	14	19	norms	norm	NOUN
cana-5487	14	20	of	of	ADP
cana-5487	14	21	fuzzy	fuzzy	ADJ
cana-5487	14	22	metric	metric	ADJ
cana-5487	14	23	spaces	space	NOUN
cana-5487	14	24	.	.	PUNCT
cana-5487	15	1	many	many	ADJ
cana-5487	15	2	researchers	researcher	NOUN
cana-5487	15	3	have	have	AUX
cana-5487	15	4	obtained	obtain	VERB
cana-5487	15	5	common	common	ADJ
cana-5487	15	6	fixed	fix	VERB
cana-5487	15	7	point	point	NOUN
cana-5487	15	8	theorems	theorem	NOUN
cana-5487	15	9	for	for	ADP
cana-5487	15	10	mappings	mapping	NOUN
cana-5487	15	11	satisfying	satisfy	VERB
cana-5487	15	12	different	different	ADJ
cana-5487	15	13	types	type	NOUN
cana-5487	15	14	of	of	ADP
cana-5487	15	15	commutativity	commutativity	NOUN
cana-5487	15	16	conditions	condition	NOUN
cana-5487	15	17	.	.	PUNCT
cana-5487	16	1	fixed	fix	VERB
cana-5487	16	2	point	point	NOUN
cana-5487	16	3	theorems	theorem	NOUN
cana-5487	16	4	,	,	PUNCT
cana-5487	16	5	involving	involve	VERB
cana-5487	16	6	four	four	NUM
cana-5487	16	7	self	self	NOUN
cana-5487	16	8	-	-	PUNCT
cana-5487	16	9	maps	map	NOUN
cana-5487	16	10	,	,	PUNCT
cana-5487	16	11	began	begin	VERB
cana-5487	16	12	with	with	ADP
cana-5487	16	13	the	the	DET
cana-5487	16	14	assumption	assumption	NOUN
cana-5487	16	15	that	that	SCONJ
cana-5487	16	16	they	they	PRON
cana-5487	16	17	are	be	AUX
cana-5487	16	18	commuted	commute	VERB
cana-5487	16	19	.	.	PUNCT
cana-5487	17	1	sessa	sessa	PROPN
cana-5487	17	2	[	[	X
cana-5487	17	3	10	10	NUM
cana-5487	17	4	]	]	PUNCT
cana-5487	17	5	weakened	weaken	VERB
cana-5487	17	6	the	the	DET
cana-5487	17	7	condition	condition	NOUN
cana-5487	17	8	of	of	ADP
cana-5487	17	9	commutativity	commutativity	NOUN
cana-5487	17	10	to	to	ADP
cana-5487	17	11	that	that	PRON
cana-5487	17	12	of	of	ADP
cana-5487	17	13	pairwise	pairwise	NOUN
cana-5487	17	14	weakly	weakly	ADJ
cana-5487	17	15	commuting	commuting	NOUN
cana-5487	17	16	.	.	PUNCT
cana-5487	18	1	jungck	jungck	PROPN
cana-5487	18	2	generalized	generalize	VERB
cana-5487	18	3	the	the	DET
cana-5487	18	4	notion	notion	NOUN
cana-5487	18	5	of	of	ADP
cana-5487	18	6	weak	weak	ADJ
cana-5487	18	7	commutativity	commutativity	NOUN
cana-5487	18	8	to	to	ADP
cana-5487	18	9	that	that	PRON
cana-5487	18	10	of	of	ADP
cana-5487	18	11	pairwise	pairwise	NOUN
cana-5487	18	12	compatible	compatible	ADJ
cana-5487	19	1	[	[	X
cana-5487	19	2	4	4	NUM
cana-5487	19	3	]	]	PUNCT
cana-5487	19	4	and	and	CCONJ
cana-5487	19	5	then	then	ADV
cana-5487	19	6	pairwise	pairwise	VERB
cana-5487	19	7	weakly	weakly	ADV
cana-5487	19	8	compatible	compatible	ADJ
cana-5487	19	9	maps	map	NOUN
cana-5487	20	1	[	[	X
cana-5487	20	2	5	5	NUM
cana-5487	20	3	]	]	PUNCT
cana-5487	20	4	.	.	PUNCT
cana-5487	21	1	jungck	jungck	PROPN
cana-5487	21	2	and	and	CCONJ
cana-5487	21	3	rhoades	rhoade	NOUN
cana-5487	21	4	[	[	X
cana-5487	21	5	6	6	NUM
cana-5487	21	6	]	]	PUNCT
cana-5487	21	7	introduced	introduce	VERB
cana-5487	21	8	the	the	DET
cana-5487	21	9	concept	concept	NOUN
cana-5487	21	10	of	of	ADP
cana-5487	21	11	occasionally	occasionally	ADV
cana-5487	21	12	weakly	weakly	ADJ
cana-5487	21	13	compatible	compatible	ADJ
cana-5487	21	14	maps	map	NOUN
cana-5487	21	15	(	(	PUNCT
cana-5487	21	16	owc	owc	NOUN
cana-5487	21	17	)	)	PUNCT
cana-5487	21	18	.	.	PUNCT
cana-5487	22	1	some	some	PRON
cana-5487	22	2	of	of	ADP
cana-5487	22	3	the	the	DET
cana-5487	22	4	work	work	NOUN
cana-5487	22	5	cited	cite	VERB
cana-5487	22	6	in	in	ADP
cana-5487	22	7	references	reference	NOUN
cana-5487	22	8	[	[	X
cana-5487	22	9	1	1	NUM
cana-5487	22	10	]	]	PUNCT
cana-5487	22	11	,	,	PUNCT
cana-5487	22	12	[	[	X
cana-5487	22	13	2	2	NUM
cana-5487	22	14	]	]	PUNCT
cana-5487	22	15	,	,	PUNCT
cana-5487	22	16	[	[	X
cana-5487	22	17	8	8	NUM
cana-5487	22	18	]	]	PUNCT
cana-5487	22	19	,	,	PUNCT
cana-5487	22	20	[	[	X
cana-5487	22	21	9	9	NUM
cana-5487	22	22	]	]	PUNCT
cana-5487	22	23	,	,	PUNCT
cana-5487	22	24	[	[	X
cana-5487	22	25	11	11	NUM
cana-5487	22	26	]	]	PUNCT
cana-5487	22	27	,	,	PUNCT
cana-5487	22	28	[	[	X
cana-5487	22	29	12	12	NUM
cana-5487	22	30	]	]	PUNCT
cana-5487	22	31	and	and	CCONJ
cana-5487	22	32	[	[	X
cana-5487	22	33	13	13	NUM
cana-5487	22	34	]	]	PUNCT
cana-5487	22	35	is	be	AUX
cana-5487	22	36	also	also	ADV
cana-5487	22	37	significant	significant	ADJ
cana-5487	22	38	.	.	PUNCT
cana-5487	23	1	in	in	ADP
cana-5487	23	2	this	this	DET
cana-5487	23	3	work	work	NOUN
cana-5487	23	4	,	,	PUNCT
cana-5487	23	5	we	we	PRON
cana-5487	23	6	introduce	introduce	VERB
cana-5487	23	7	tripled	triple	VERB
cana-5487	23	8	fixed	fix	VERB
cana-5487	23	9	point	point	NOUN
cana-5487	23	10	for	for	ADP
cana-5487	23	11	occasionally	occasionally	ADV
cana-5487	23	12	weakly	weakly	ADJ
cana-5487	23	13	compatible	compatible	ADJ
cana-5487	23	14	mappings	mapping	NOUN
cana-5487	23	15	in	in	ADP
cana-5487	23	16	fuzzy	fuzzy	ADJ
cana-5487	23	17	metric	metric	ADJ
cana-5487	23	18	space	space	NOUN
cana-5487	23	19	and	and	CCONJ
cana-5487	23	20	also	also	ADV
cana-5487	23	21	proved	prove	VERB
cana-5487	23	22	some	some	DET
cana-5487	23	23	tripled	triple	VERB
cana-5487	23	24	fixed	fix	VERB
cana-5487	23	25	-	-	PUNCT
cana-5487	23	26	point	point	NOUN
cana-5487	23	27	theorem	theorem	NOUN
cana-5487	23	28	for	for	ADP
cana-5487	23	29	occasionally	occasionally	ADV
cana-5487	23	30	weakly	weakly	ADJ
cana-5487	23	31	compatible	compatible	ADJ
cana-5487	23	32	mailto:priyanka.nigam@presidencyuniversity.in	mailto:priyanka.nigam@presidencyuniversity.in	PROPN
cana-5487	23	33	mailto:priyanka_nigam01@yahoo.co.in	mailto:priyanka_nigam01@yahoo.co.in	PROPN
cana-5487	23	34	mailto:maths.sandhyashukla@gmail.com	mailto:maths.sandhyashukla@gmail.com	X
cana-5487	23	35	communications	communication	NOUN
cana-5487	23	36	on	on	ADP
cana-5487	23	37	applied	apply	VERB
cana-5487	23	38	nonlinear	nonlinear	ADJ
cana-5487	23	39	analysis	analysis	NOUN
cana-5487	23	40	issn	issn	NOUN
cana-5487	23	41	:	:	PUNCT
cana-5487	23	42	1074	1074	NUM
cana-5487	23	43	-	-	PUNCT
cana-5487	23	44	133x	133x	NUM
cana-5487	23	45	vol	vol	VERB
cana-5487	23	46	32	32	NUM
cana-5487	23	47	no	no	NOUN
cana-5487	23	48	.	.	PUNCT
cana-5487	24	1	10s	10	NOUN
cana-5487	24	2	(	(	PUNCT
cana-5487	24	3	2025	2025	NUM
cana-5487	24	4	)	)	PUNCT
cana-5487	24	5	2384	2384	NUM
cana-5487	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	24	7	mappings	mapping	NOUN
cana-5487	24	8	in	in	ADP
cana-5487	24	9	fuzzy	fuzzy	ADJ
cana-5487	24	10	metric	metric	ADJ
cana-5487	24	11	space	space	NOUN
cana-5487	24	12	.	.	PUNCT
cana-5487	25	1	our	our	PRON
cana-5487	25	2	results	result	NOUN
cana-5487	25	3	extend	extend	VERB
cana-5487	25	4	and	and	CCONJ
cana-5487	25	5	some	some	DET
cana-5487	25	6	recent	recent	ADJ
cana-5487	25	7	results	result	NOUN
cana-5487	25	8	in	in	ADP
cana-5487	25	9	literature	literature	NOUN
cana-5487	25	10	.	.	PUNCT
cana-5487	26	1	some	some	DET
cana-5487	26	2	illustrative	illustrative	ADJ
cana-5487	26	3	examples	example	NOUN
cana-5487	26	4	are	be	AUX
cana-5487	26	5	offered	offer	VERB
cana-5487	26	6	to	to	PART
cana-5487	26	7	support	support	VERB
cana-5487	26	8	our	our	PRON
cana-5487	26	9	theorems	theorem	NOUN
cana-5487	26	10	.	.	PUNCT
cana-5487	27	1	we	we	PRON
cana-5487	27	2	have	have	AUX
cana-5487	27	3	also	also	ADV
cana-5487	27	4	given	give	VERB
cana-5487	27	5	the	the	DET
cana-5487	27	6	diagram	diagram	NOUN
cana-5487	27	7	to	to	PART
cana-5487	27	8	demonstrate	demonstrate	VERB
cana-5487	27	9	the	the	DET
cana-5487	27	10	viability	viability	NOUN
cana-5487	27	11	and	and	CCONJ
cana-5487	27	12	applicability	applicability	NOUN
cana-5487	27	13	of	of	ADP
cana-5487	27	14	the	the	DET
cana-5487	27	15	result	result	NOUN
cana-5487	27	16	.	.	PUNCT
cana-5487	28	1	2	2	NUM
cana-5487	28	2	preliminary	preliminary	ADJ
cana-5487	28	3	notes	note	NOUN
cana-5487	28	4	definition	definition	NOUN
cana-5487	28	5	2.1	2.1	NUM
cana-5487	28	6	a	a	DET
cana-5487	28	7	fuzzy	fuzzy	ADJ
cana-5487	28	8	set	set	NOUN
cana-5487	28	9	a	a	PRON
cana-5487	28	10	in	in	NOUN
cana-5487	28	11	x	x	SYM
cana-5487	28	12	is	be	AUX
cana-5487	28	13	a	a	DET
cana-5487	28	14	function	function	NOUN
cana-5487	28	15	with	with	ADP
cana-5487	28	16	domain	domain	NOUN
cana-5487	28	17	x	x	PUNCT
cana-5487	28	18	and	and	CCONJ
cana-5487	28	19	values	value	NOUN
cana-5487	28	20	in	in	ADP
cana-5487	28	21	[	[	X
cana-5487	28	22	0	0	NUM
cana-5487	28	23	,	,	PUNCT
cana-5487	28	24	1	1	NUM
cana-5487	28	25	]	]	PUNCT
cana-5487	28	26	.	.	PUNCT
cana-5487	29	1	definition	definition	NOUN
cana-5487	29	2	2.2	2.2	NUM
cana-5487	29	3	a	a	DET
cana-5487	29	4	binary	binary	ADJ
cana-5487	29	5	operation	operation	NOUN
cana-5487	30	1	∗∶	∗∶	PROPN
cana-5487	31	1	[	[	X
cana-5487	31	2	0,1]	0,1]	X
cana-5487	32	1	[	[	X
cana-5487	32	2	0,1]→	0,1]→	NOUN
cana-5487	32	3	[	[	X
cana-5487	32	4	0,1	0,1	NUM
cana-5487	32	5	]	]	PUNCT
cana-5487	32	6	is	be	AUX
cana-5487	32	7	a	a	DET
cana-5487	32	8	continuous	continuous	ADJ
cana-5487	32	9	t	t	NOUN
cana-5487	32	10	-	-	PUNCT
cana-5487	32	11	norm	norm	NOUN
cana-5487	32	12	if	if	SCONJ
cana-5487	32	13	∗	∗	NOUN
cana-5487	32	14	is	be	AUX
cana-5487	32	15	satisfying	satisfy	VERB
cana-5487	32	16	conditions	condition	NOUN
cana-5487	32	17	:	:	PUNCT
cana-5487	32	18	(	(	PUNCT
cana-5487	32	19	i	i	NOUN
cana-5487	32	20	)	)	PUNCT
cana-5487	32	21	∗	∗	NOUN
cana-5487	32	22	is	be	AUX
cana-5487	32	23	an	an	DET
cana-5487	32	24	commutative	commutative	ADJ
cana-5487	32	25	and	and	CCONJ
cana-5487	32	26	associative	associative	ADJ
cana-5487	32	27	;	;	PUNCT
cana-5487	32	28	(	(	PUNCT
cana-5487	32	29	ii	ii	NOUN
cana-5487	32	30	)	)	PUNCT
cana-5487	33	1	∗is	∗is	ADJ
cana-5487	33	2	continuous	continuous	ADJ
cana-5487	33	3	;	;	PUNCT
cana-5487	33	4	(	(	PUNCT
cana-5487	33	5	iii	iii	X
cana-5487	33	6	)	)	PUNCT
cana-5487	33	7	𝑎	𝑎	NOUN
cana-5487	33	8	∗	∗	NOUN
cana-5487	33	9	1	1	NUM
cana-5487	33	10	=	=	SYM
cana-5487	33	11	𝑎	𝑎	NOUN
cana-5487	33	12	for	for	ADP
cana-5487	33	13	all	all	DET
cana-5487	33	14	a	a	PROPN
cana-5487	33	15	[	[	X
cana-5487	33	16	0,1	0,1	NUM
cana-5487	33	17	]	]	PUNCT
cana-5487	33	18	;	;	PUNCT
cana-5487	33	19	(	(	PUNCT
cana-5487	33	20	iv	iv	X
cana-5487	33	21	)	)	PUNCT
cana-5487	33	22	𝑎	𝑎	PRON
cana-5487	33	23	∗	∗	NOUN
cana-5487	33	24	𝑏𝑐	𝑏𝑐	NOUN
cana-5487	33	25	∗	∗	NOUN
cana-5487	33	26	𝑑	𝑑	NOUN
cana-5487	33	27	whenever	whenever	SCONJ
cana-5487	33	28	ca	can	AUX
cana-5487	33	29			NOUN
cana-5487	33	30	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5487	33	31	db	db	PROPN
cana-5487	33	32			PROPN
cana-5487	33	33	and	and	CCONJ
cana-5487	33	34	𝑎	𝑎	NOUN
cana-5487	33	35	,	,	PUNCT
cana-5487	33	36	𝑏	𝑏	NOUN
cana-5487	33	37	,	,	PUNCT
cana-5487	33	38	𝑐	𝑐	NOUN
cana-5487	33	39	,	,	PUNCT
cana-5487	33	40	𝑑[0,1	𝑑[0,1	NOUN
cana-5487	33	41	]	]	PUNCT
cana-5487	33	42	.	.	PUNCT
cana-5487	34	1	definition	definition	NOUN
cana-5487	34	2	2.3	2.3	NUM
cana-5487	34	3	a	a	DET
cana-5487	34	4	3	3	NUM
cana-5487	34	5	-	-	PUNCT
cana-5487	34	6	tuple	tuple	NOUN
cana-5487	34	7	(	(	PUNCT
cana-5487	34	8	𝑋,𝑀,∗	𝑋,𝑀,∗	NOUN
cana-5487	34	9	)	)	PUNCT
cana-5487	34	10	is	be	AUX
cana-5487	34	11	said	say	VERB
cana-5487	34	12	to	to	PART
cana-5487	34	13	be	be	AUX
cana-5487	34	14	a	a	DET
cana-5487	34	15	fuzzy	fuzzy	ADJ
cana-5487	34	16	metric	metric	ADJ
cana-5487	34	17	space	space	NOUN
cana-5487	34	18	if	if	SCONJ
cana-5487	34	19	x	x	PRON
cana-5487	34	20	is	be	AUX
cana-5487	34	21	an	an	DET
cana-5487	34	22	arbitrary	arbitrary	ADJ
cana-5487	34	23	set	set	NOUN
cana-5487	34	24	,	,	PUNCT
cana-5487	34	25	∗	∗	NOUN
cana-5487	34	26	is	be	AUX
cana-5487	34	27	a	a	DET
cana-5487	34	28	continuous	continuous	ADJ
cana-5487	34	29	𝑡	𝑡	NOUN
cana-5487	34	30	−	−	NOUN
cana-5487	34	31	𝑛𝑜𝑟𝑚	𝑛𝑜𝑟𝑚	NOUN
cana-5487	34	32	and	and	CCONJ
cana-5487	34	33	m	m	NOUN
cana-5487	34	34	is	be	AUX
cana-5487	34	35	a	a	DET
cana-5487	34	36	fuzzy	fuzzy	ADJ
cana-5487	34	37	set	set	NOUN
cana-5487	34	38	on	on	ADP
cana-5487	34	39	(	(	PUNCT
cana-5487	34	40	)	)	PUNCT
cana-5487	34	41			X
cana-5487	34	42	,	,	PUNCT
cana-5487	34	43	02x	02x	NOUN
cana-5487	34	44	satisfying	satisfy	VERB
cana-5487	34	45	the	the	DET
cana-5487	34	46	following	following	ADJ
cana-5487	34	47	conditions	condition	NOUN
cana-5487	34	48	,	,	PUNCT
cana-5487	34	49	for	for	ADP
cana-5487	34	50	all	all	DET
cana-5487	34	51	𝑥	𝑥	PROPN
cana-5487	34	52	,	,	PUNCT
cana-5487	34	53	𝑦	𝑦	NOUN
cana-5487	34	54	,	,	PUNCT
cana-5487	34	55	𝑧	𝑧	PRON
cana-5487	34	56			PROPN
cana-5487	34	57	𝑋	𝑋	PROPN
cana-5487	34	58	,	,	PUNCT
cana-5487	34	59	𝑠	𝑠	PROPN
cana-5487	34	60	,	,	PUNCT
cana-5487	34	61	𝑡	𝑡	X
cana-5487	34	62	>	>	X
cana-5487	34	63	0	0	NUM
cana-5487	34	64	,	,	PUNCT
cana-5487	34	65	(	(	PUNCT
cana-5487	34	66	𝑖	𝑖	X
cana-5487	34	67	)	)	PUNCT
cana-5487	34	68	0	0	NUM
cana-5487	34	69	)	)	PUNCT
cana-5487	34	70	,	,	PUNCT
cana-5487	34	71	,	,	PUNCT
cana-5487	34	72	(	(	PUNCT
cana-5487	34	73	tyxm	tyxm	ADV
cana-5487	34	74	;	;	PUNCT
cana-5487	34	75	(	(	PUNCT
cana-5487	34	76	𝑖𝑖	𝑖𝑖	NOUN
cana-5487	34	77	)	)	PUNCT
cana-5487	34	78	1	1	NUM
cana-5487	34	79	)	)	PUNCT
cana-5487	34	80	,	,	PUNCT
cana-5487	34	81	,	,	PUNCT
cana-5487	34	82	(	(	PUNCT
cana-5487	34	83	=	=	NOUN
cana-5487	34	84	tyxm	tyxm	NOUN
cana-5487	34	85	𝑖𝑓	𝑖𝑓	ADP
cana-5487	34	86	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-5487	34	87	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
cana-5487	34	88	𝑖𝑓	𝑖𝑓	ADP
cana-5487	34	89	yx	yx	PROPN
cana-5487	34	90	=	=	PUNCT
cana-5487	34	91	;	;	PUNCT
cana-5487	34	92	(	(	PUNCT
cana-5487	34	93	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
cana-5487	34	94	)	)	PUNCT
cana-5487	34	95	)	)	PUNCT
cana-5487	34	96	,	,	PUNCT
cana-5487	34	97	,	,	PUNCT
cana-5487	34	98	(	(	PUNCT
cana-5487	34	99	tyxm	tyxm	NOUN
cana-5487	34	100	=	=	PUNCT
cana-5487	34	101	)	)	PUNCT
cana-5487	34	102	,	,	PUNCT
cana-5487	34	103	,	,	PUNCT
cana-5487	34	104	(	(	PUNCT
cana-5487	34	105	txym	txym	INTJ
cana-5487	34	106	;	;	PUNCT
cana-5487	34	107	(	(	PUNCT
cana-5487	34	108	𝑖𝑣	𝑖𝑣	X
cana-5487	34	109	)	)	PUNCT
cana-5487	34	110	)	)	PUNCT
cana-5487	34	111	,	,	PUNCT
cana-5487	34	112	,	,	PUNCT
cana-5487	34	113	(	(	PUNCT
cana-5487	34	114	tyxm	tyxm	NOUN
cana-5487	34	115	∗	∗	NOUN
cana-5487	34	116	)	)	PUNCT
cana-5487	34	117	,	,	PUNCT
cana-5487	34	118	,	,	PUNCT
cana-5487	34	119	(	(	PUNCT
cana-5487	34	120	szym	szym	NOUN
cana-5487	34	121	)	)	PUNCT
cana-5487	34	122	,	,	PUNCT
cana-5487	34	123	,	,	PUNCT
cana-5487	34	124	(	(	PUNCT
cana-5487	34	125	stzxm	stzxm	ADJ
cana-5487	34	126	+	+	ADJ
cana-5487	34	127			NOUN
cana-5487	34	128	;	;	PUNCT
cana-5487	34	129	(	(	PUNCT
cana-5487	34	130	𝑣	𝑣	X
cana-5487	34	131	)	)	PUNCT
cana-5487	34	132	]	]	X
cana-5487	34	133	1,0(),0	1,0(),0	NUM
cana-5487	34	134	(:	(:	NOUN
cana-5487	34	135	)	)	PUNCT
cana-5487	34	136	,	,	PUNCT
cana-5487	34	137	,	,	PUNCT
cana-5487	34	138	(	(	PUNCT
cana-5487	34	139	→yxm	→yxm	NOUN
cana-5487	34	140	is	be	AUX
cana-5487	34	141	continuous	continuous	ADJ
cana-5487	34	142	.	.	PUNCT
cana-5487	35	1	then	then	ADV
cana-5487	35	2	m	m	PROPN
cana-5487	35	3	is	be	AUX
cana-5487	35	4	called	call	VERB
cana-5487	35	5	a	a	DET
cana-5487	35	6	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	NOUN
cana-5487	35	7	𝑚𝑒𝑡𝑟𝑖𝑐	𝑚𝑒𝑡𝑟𝑖𝑐	NOUN
cana-5487	35	8	on	on	ADP
cana-5487	35	9	x.	x.	PROPN
cana-5487	35	10	then	then	ADV
cana-5487	35	11	)	)	PUNCT
cana-5487	35	12	,	,	PUNCT
cana-5487	35	13	,	,	PUNCT
cana-5487	35	14	(	(	PUNCT
cana-5487	35	15	tyxm	tyxm	NOUN
cana-5487	35	16	denotes	denote	VERB
cana-5487	35	17	the	the	DET
cana-5487	35	18	degree	degree	NOUN
cana-5487	35	19	of	of	ADP
cana-5487	35	20	nearness	nearness	NOUN
cana-5487	35	21	between	between	ADP
cana-5487	35	22	x	x	PROPN
cana-5487	35	23	and	and	CCONJ
cana-5487	35	24	y	y	PROPN
cana-5487	35	25	with	with	ADP
cana-5487	35	26	respect	respect	NOUN
cana-5487	35	27	to	to	ADP
cana-5487	35	28	t.	t.	PROPN
cana-5487	35	29	example	example	NOUN
cana-5487	35	30	2.4	2.4	NUM
cana-5487	35	31	let	let	NOUN
cana-5487	35	32	)	)	PUNCT
cana-5487	35	33	,	,	PUNCT
cana-5487	35	34	(	(	PUNCT
cana-5487	35	35	dx	dx	PROPN
cana-5487	35	36	be	be	AUX
cana-5487	35	37	a	a	DET
cana-5487	35	38	metric	metric	ADJ
cana-5487	35	39	space	space	NOUN
cana-5487	35	40	.	.	PUNCT
cana-5487	36	1	denote	denote	VERB
cana-5487	36	2	𝑎	𝑎	NOUN
cana-5487	36	3	∗	∗	NOUN
cana-5487	36	4	𝑏	𝑏	NOUN
cana-5487	36	5	=	=	SYM
cana-5487	36	6	𝑎𝑏	𝑎𝑏	PROPN
cana-5487	36	7	for	for	ADP
cana-5487	36	8	all	all	DET
cana-5487	36	9			ADJ
cana-5487	36	10	1,0	1,0	PRON
cana-5487	36	11	,	,	PUNCT
cana-5487	36	12	ba	ba	VERB
cana-5487	36	13	and	and	CCONJ
cana-5487	36	14	let	let	VERB
cana-5487	36	15	dm	dm	PRON
cana-5487	36	16	be	be	AUX
cana-5487	36	17	fuzzy	fuzzy	ADJ
cana-5487	36	18	sets	set	NOUN
cana-5487	36	19	on	on	ADP
cana-5487	36	20	(	(	PUNCT
cana-5487	36	21	)	)	PUNCT
cana-5487	36	22			X
cana-5487	36	23	,	,	PUNCT
cana-5487	36	24	02x	02x	NOUN
cana-5487	36	25	defined	define	VERB
cana-5487	36	26	as	as	ADP
cana-5487	36	27	follows	follow	VERB
cana-5487	36	28	:	:	PUNCT
cana-5487	36	29	)	)	PUNCT
cana-5487	36	30	,	,	PUNCT
cana-5487	36	31	(	(	PUNCT
cana-5487	36	32	yxdt	yxdt	PROPN
cana-5487	36	33	t	t	PROPN
cana-5487	36	34	m	m	PROPN
cana-5487	36	35	d	d	NOUN
cana-5487	36	36	+	+	CCONJ
cana-5487	37	1	=	=	NOUN
cana-5487	37	2	.	.	PUNCT
cana-5487	38	1	then	then	ADV
cana-5487	38	2	(	(	PUNCT
cana-5487	38	3	𝑋	𝑋	PROPN
cana-5487	38	4	,	,	PUNCT
cana-5487	38	5	𝑀𝑑,∗	𝑀𝑑,∗	PROPN
cana-5487	38	6	)	)	PUNCT
cana-5487	38	7	is	be	AUX
cana-5487	38	8	a	a	DET
cana-5487	38	9	fuzzy	fuzzy	ADJ
cana-5487	38	10	metric	metric	ADJ
cana-5487	38	11	space	space	NOUN
cana-5487	38	12	.	.	PUNCT
cana-5487	39	1	lemma	lemma	PROPN
cana-5487	39	2	2.5let	2.5let	PROPN
cana-5487	39	3	(	(	PUNCT
cana-5487	39	4	x	x	X
cana-5487	39	5	,	,	PUNCT
cana-5487	39	6	m	m	PROPN
cana-5487	39	7	,	,	PUNCT
cana-5487	39	8	*	*	PUNCT
cana-5487	39	9	)	)	PUNCT
cana-5487	39	10	be	be	AUX
cana-5487	39	11	a	a	DET
cana-5487	39	12	fuzzy	fuzzy	ADJ
cana-5487	39	13	metric	metric	ADJ
cana-5487	39	14	space	space	NOUN
cana-5487	39	15	.	.	PUNCT
cana-5487	40	1	if	if	SCONJ
cana-5487	40	2	there	there	PRON
cana-5487	40	3	exists	exist	VERB
cana-5487	40	4	)	)	PUNCT
cana-5487	40	5	1,0(q	1,0(q	NUM
cana-5487	40	6	such	such	ADJ
cana-5487	40	7	that	that	SCONJ
cana-5487	40	8	m(x	m(x	PROPN
cana-5487	40	9	,	,	PUNCT
cana-5487	40	10	y	y	PROPN
cana-5487	40	11	,	,	PUNCT
cana-5487	40	12	qt	qt	NOUN
cana-5487	40	13	)	)	PUNCT
cana-5487	40	14			PROPN
cana-5487	40	15	m(x	m(x	PROPN
cana-5487	40	16	,	,	PUNCT
cana-5487	40	17	y	y	PROPN
cana-5487	40	18	,	,	PUNCT
cana-5487	40	19	t	t	PROPN
cana-5487	40	20	)	)	PUNCT
cana-5487	40	21	for	for	ADP
cana-5487	40	22	all	all	DET
cana-5487	40	23	x	x	NOUN
cana-5487	40	24	,	,	PUNCT
cana-5487	40	25	y	y	PROPN
cana-5487	40	26			NOUN
cana-5487	40	27	x	x	PUNCT
cana-5487	40	28	and	and	CCONJ
cana-5487	40	29	t>0	t>0	NOUN
cana-5487	40	30	,	,	PUNCT
cana-5487	40	31	then	then	ADV
cana-5487	40	32	x	x	X
cana-5487	40	33	=	=	PUNCT
cana-5487	40	34	y.	y.	NOUN
cana-5487	40	35	definition	definition	NOUN
cana-5487	40	36	2.6let	2.6let	NOUN
cana-5487	40	37	x	x	PUNCT
cana-5487	40	38	be	be	AUX
cana-5487	40	39	a	a	DET
cana-5487	40	40	non	non	ADJ
cana-5487	40	41	-	-	ADJ
cana-5487	40	42	empty	empty	ADJ
cana-5487	40	43	set	set	NOUN
cana-5487	40	44	.	.	PUNCT
cana-5487	41	1	an	an	DET
cana-5487	41	2	element	element	NOUN
cana-5487	41	3	(	(	PUNCT
cana-5487	41	4	𝑥	𝑥	PROPN
cana-5487	41	5	,	,	PUNCT
cana-5487	41	6	𝑦	𝑦	NOUN
cana-5487	41	7	,	,	PUNCT
cana-5487	41	8	𝑧	𝑧	NOUN
cana-5487	41	9	)	)	PUNCT
cana-5487	41	10	∈	∈	PROPN
cana-5487	42	1	𝑋	𝑋	NOUN
cana-5487	42	2	×	×	NOUN
cana-5487	42	3	𝑋	𝑋	NOUN
cana-5487	42	4	×	×	NOUN
cana-5487	42	5	𝑋	𝑋	PROPN
cana-5487	42	6	is	be	AUX
cana-5487	42	7	called	call	VERB
cana-5487	42	8	a	a	DET
cana-5487	42	9	tripled	triple	VERB
cana-5487	42	10	fixed	fix	VERB
cana-5487	42	11	point	point	NOUN
cana-5487	42	12	of	of	ADP
cana-5487	42	13	a	a	PRON
cana-5487	42	14	given	give	VERB
cana-5487	42	15	mapping	mapping	NOUN
cana-5487	42	16	𝑓	𝑓	X
cana-5487	42	17	:	:	PUNCT
cana-5487	42	18	𝑋	𝑋	PROPN
cana-5487	42	19	×	×	NOUN
cana-5487	42	20	𝑋	𝑋	PROPN
cana-5487	42	21	×	×	NOUN
cana-5487	42	22	𝑋	𝑋	PROPN
cana-5487	42	23	→	→	PUNCT
cana-5487	42	24	𝑋	𝑋	PROPN
cana-5487	42	25	if	if	SCONJ
cana-5487	42	26	𝑥	𝑥	PRON
cana-5487	42	27	=	=	SYM
cana-5487	42	28	𝑓(𝑥	𝑓(𝑥	NOUN
cana-5487	42	29	,	,	PUNCT
cana-5487	42	30	𝑦	𝑦	NOUN
cana-5487	42	31	,	,	PUNCT
cana-5487	42	32	𝑧	𝑧	PART
cana-5487	42	33	)	)	PUNCT
cana-5487	42	34	,	,	PUNCT
cana-5487	42	35	𝑦	𝑦	NOUN
cana-5487	42	36	=	=	PUNCT
cana-5487	42	37	𝑓(𝑦	𝑓(𝑦	PROPN
cana-5487	42	38	,	,	PUNCT
cana-5487	42	39	𝑧	𝑧	NOUN
cana-5487	42	40	,	,	PUNCT
cana-5487	42	41	𝑥	𝑥	NOUN
cana-5487	42	42	)	)	PUNCT
cana-5487	42	43	,	,	PUNCT
cana-5487	42	44	𝑧	𝑧	X
cana-5487	42	45	=	=	SYM
cana-5487	42	46	𝑓(𝑧	𝑓(𝑧	PROPN
cana-5487	42	47	,	,	PUNCT
cana-5487	42	48	𝑥	𝑥	NOUN
cana-5487	42	49	,	,	PUNCT
cana-5487	42	50	𝑦	𝑦	NOUN
cana-5487	42	51	)	)	PUNCT
cana-5487	42	52	.	.	PUNCT
cana-5487	43	1	example	example	NOUN
cana-5487	43	2	2.6.1let	2.6.1let	NUM
cana-5487	43	3	𝑋	𝑋	PROPN
cana-5487	43	4	=	=	SYM
cana-5487	43	5	𝑅	𝑅	PROPN
cana-5487	43	6	and	and	CCONJ
cana-5487	43	7	𝑆	𝑆	PROPN
cana-5487	43	8	:	:	PUNCT
cana-5487	43	9	𝑋	𝑋	PROPN
cana-5487	43	10	×	×	NOUN
cana-5487	43	11	𝑋	𝑋	PROPN
cana-5487	43	12	×	×	NOUN
cana-5487	43	13	𝑋	𝑋	PROPN
cana-5487	43	14	→	→	SYM
cana-5487	43	15	𝑋is	𝑋is	PROPN
cana-5487	43	16	defined	define	VERB
cana-5487	43	17	as	as	ADP
cana-5487	43	18	𝑆(𝑥	𝑆(𝑥	NOUN
cana-5487	43	19	,	,	PUNCT
cana-5487	43	20	𝑦	𝑦	NOUN
cana-5487	43	21	,	,	PUNCT
cana-5487	43	22	𝑧	𝑧	NOUN
cana-5487	43	23	)	)	PUNCT
cana-5487	43	24	=	=	SYM
cana-5487	44	1	𝑥	𝑥	PROPN
cana-5487	45	1	+	+	CCONJ
cana-5487	45	2	𝑥𝑦	𝑥𝑦	PROPN
cana-5487	45	3	+	+	ADJ
cana-5487	45	4	𝑥𝑧	𝑥𝑧	ADP
cana-5487	45	5	then	then	ADV
cana-5487	45	6	communications	communication	NOUN
cana-5487	45	7	on	on	ADP
cana-5487	45	8	applied	apply	VERB
cana-5487	45	9	nonlinear	nonlinear	ADJ
cana-5487	45	10	analysis	analysis	NOUN
cana-5487	45	11	issn	issn	NOUN
cana-5487	45	12	:	:	PUNCT
cana-5487	45	13	1074	1074	NUM
cana-5487	45	14	-	-	PUNCT
cana-5487	45	15	133x	133x	NUM
cana-5487	45	16	vol	vol	VERB
cana-5487	45	17	32	32	NUM
cana-5487	45	18	no	no	NOUN
cana-5487	45	19	.	.	PUNCT
cana-5487	46	1	10s	10	NOUN
cana-5487	46	2	(	(	PUNCT
cana-5487	46	3	2025	2025	NUM
cana-5487	46	4	)	)	PUNCT
cana-5487	46	5	2385	2385	NUM
cana-5487	46	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	46	7	𝑆(1,0,0	𝑆(1,0,0	NOUN
cana-5487	46	8	)	)	PUNCT
cana-5487	46	9	=	=	SYM
cana-5487	46	10	1	1	NUM
cana-5487	46	11	,	,	PUNCT
cana-5487	46	12	𝑆(0,1,0	𝑆(0,1,0	X
cana-5487	46	13	)	)	PUNCT
cana-5487	46	14	=	=	SYM
cana-5487	46	15	0	0	NUM
cana-5487	46	16	,	,	PUNCT
cana-5487	46	17	𝑆(0,0,1	𝑆(0,0,1	NUM
cana-5487	46	18	)	)	PUNCT
cana-5487	46	19	=	=	SYM
cana-5487	46	20	0	0	PUNCT
cana-5487	47	1	then	then	ADV
cana-5487	47	2	(	(	PUNCT
cana-5487	47	3	1,0,0),(0,1,0	1,0,0),(0,1,0	NUM
cana-5487	47	4	)	)	PUNCT
cana-5487	47	5	and	and	CCONJ
cana-5487	47	6	(	(	PUNCT
cana-5487	47	7	0,0,1	0,0,1	NOUN
cana-5487	47	8	)	)	PUNCT
cana-5487	47	9	are	be	AUX
cana-5487	47	10	tripled	triple	VERB
cana-5487	47	11	fixed	fix	VERB
cana-5487	47	12	point	point	NOUN
cana-5487	47	13	.	.	PUNCT
cana-5487	48	1	definition	definition	NOUN
cana-5487	48	2	2.7	2.7	NUM
cana-5487	48	3	an	an	DET
cana-5487	48	4	element	element	NOUN
cana-5487	48	5	𝑥	𝑥	PRON
cana-5487	48	6	∈	∈	NOUN
cana-5487	48	7	𝑋	𝑋	NOUN
cana-5487	48	8	is	be	AUX
cana-5487	48	9	called	call	VERB
cana-5487	48	10	a	a	DET
cana-5487	48	11	common	common	ADJ
cana-5487	48	12	tripled	triple	VERB
cana-5487	48	13	fixed	fix	VERB
cana-5487	48	14	point	point	NOUN
cana-5487	48	15	of	of	ADP
cana-5487	48	16	the	the	DET
cana-5487	48	17	mappings	mapping	NOUN
cana-5487	48	18	𝑓	𝑓	X
cana-5487	48	19	:	:	PUNCT
cana-5487	48	20	𝑋	𝑋	NOUN
cana-5487	48	21	×	×	NOUN
cana-5487	48	22	𝑋	𝑋	PROPN
cana-5487	48	23	×	×	NOUN
cana-5487	48	24	𝑋	𝑋	PROPN
cana-5487	48	25	→	→	SYM
cana-5487	48	26	𝑋	𝑋	PROPN
cana-5487	48	27	and	and	CCONJ
cana-5487	48	28	𝑔	𝑔	NOUN
cana-5487	48	29	:	:	PUNCT
cana-5487	48	30	𝑋	𝑋	PROPN
cana-5487	48	31	→	→	SYM
cana-5487	48	32	𝑋	𝑋	PROPN
cana-5487	48	33	if	if	SCONJ
cana-5487	48	34	𝑥	𝑥	PRON
cana-5487	48	35	=	=	SYM
cana-5487	48	36	𝑓(𝑥	𝑓(𝑥	NOUN
cana-5487	48	37	,	,	PUNCT
cana-5487	48	38	𝑥	𝑥	NOUN
cana-5487	48	39	,	,	PUNCT
cana-5487	48	40	𝑥	𝑥	NOUN
cana-5487	48	41	)	)	PUNCT
cana-5487	48	42	=	=	SYM
cana-5487	48	43	𝑔(𝑥	𝑔(𝑥	PROPN
cana-5487	48	44	)	)	PUNCT
cana-5487	48	45	.	.	PUNCT
cana-5487	49	1	definition	definition	NOUN
cana-5487	49	2	2.8	2.8	NUM
cana-5487	49	3	an	an	DET
cana-5487	49	4	element	element	NOUN
cana-5487	49	5	(	(	PUNCT
cana-5487	49	6	𝑥	𝑥	PROPN
cana-5487	49	7	,	,	PUNCT
cana-5487	49	8	𝑦	𝑦	NOUN
cana-5487	49	9	,	,	PUNCT
cana-5487	49	10	𝑧	𝑧	NOUN
cana-5487	49	11	)	)	PUNCT
cana-5487	49	12	∈	∈	PROPN
cana-5487	49	13	𝑋	𝑋	NOUN
cana-5487	49	14	×	×	NOUN
cana-5487	49	15	𝑋	𝑋	NOUN
cana-5487	49	16	×	×	NOUN
cana-5487	49	17	𝑋	𝑋	PROPN
cana-5487	49	18	is	be	AUX
cana-5487	49	19	called	call	VERB
cana-5487	49	20	a	a	DET
cana-5487	49	21	tripled	triple	VERB
cana-5487	49	22	coincidence	coincidence	NOUN
cana-5487	49	23	point	point	NOUN
cana-5487	49	24	of	of	ADP
cana-5487	49	25	a	a	DET
cana-5487	49	26	mapping	mapping	NOUN
cana-5487	49	27	𝑓	𝑓	X
cana-5487	49	28	:	:	PUNCT
cana-5487	49	29	𝑋	𝑋	PROPN
cana-5487	49	30	×	×	NOUN
cana-5487	49	31	𝑋	𝑋	PROPN
cana-5487	49	32	×	×	NOUN
cana-5487	49	33	𝑋	𝑋	PROPN
cana-5487	49	34	→	→	SYM
cana-5487	49	35	𝑋	𝑋	PROPN
cana-5487	49	36	and	and	CCONJ
cana-5487	49	37	𝑔	𝑔	NOUN
cana-5487	49	38	:	:	PUNCT
cana-5487	49	39	𝑋	𝑋	PROPN
cana-5487	49	40	→	→	SYM
cana-5487	49	41	𝑋	𝑋	PROPN
cana-5487	49	42	if	if	SCONJ
cana-5487	49	43	𝑔𝑥	𝑔𝑥	PROPN
cana-5487	49	44	=	=	PUNCT
cana-5487	49	45	𝑓(𝑥	𝑓(𝑥	NOUN
cana-5487	49	46	,	,	PUNCT
cana-5487	49	47	𝑦	𝑦	NOUN
cana-5487	49	48	,	,	PUNCT
cana-5487	49	49	𝑧	𝑧	NOUN
cana-5487	49	50	)	)	PUNCT
cana-5487	49	51	,	,	PUNCT
cana-5487	49	52	𝑔𝑦	𝑔𝑦	ADV
cana-5487	49	53	=	=	PUNCT
cana-5487	49	54	𝑓(𝑦	𝑓(𝑦	PROPN
cana-5487	49	55	,	,	PUNCT
cana-5487	49	56	𝑧	𝑧	PRON
cana-5487	49	57	,	,	PUNCT
cana-5487	49	58	𝑥	𝑥	NOUN
cana-5487	49	59	)	)	PUNCT
cana-5487	49	60	,	,	PUNCT
cana-5487	49	61	𝑔𝑧	𝑔𝑧	ADP
cana-5487	49	62	=	=	SYM
cana-5487	49	63	𝑓(𝑧	𝑓(𝑧	PROPN
cana-5487	49	64	,	,	PUNCT
cana-5487	49	65	𝑥	𝑥	NOUN
cana-5487	49	66	,	,	PUNCT
cana-5487	49	67	𝑦	𝑦	NOUN
cana-5487	49	68	)	)	PUNCT
cana-5487	49	69	in	in	ADP
cana-5487	49	70	this	this	DET
cana-5487	49	71	case	case	NOUN
cana-5487	49	72	(	(	PUNCT
cana-5487	49	73	𝑔𝑥	𝑔𝑥	INTJ
cana-5487	49	74	,	,	PUNCT
cana-5487	49	75	𝑔𝑦	𝑔𝑦	PROPN
cana-5487	49	76	,	,	PUNCT
cana-5487	49	77	𝑔𝑧	𝑔𝑧	PROPN
cana-5487	49	78	)	)	PUNCT
cana-5487	49	79	is	be	AUX
cana-5487	49	80	called	call	VERB
cana-5487	49	81	a	a	DET
cana-5487	49	82	tripled	triple	VERB
cana-5487	49	83	point	point	NOUN
cana-5487	49	84	of	of	ADP
cana-5487	49	85	coincidence	coincidence	NOUN
cana-5487	49	86	.	.	PUNCT
cana-5487	50	1	definition	definition	NOUN
cana-5487	50	2	2.9	2.9	NUM
cana-5487	50	3	let	let	VERB
cana-5487	50	4	𝑓	𝑓	DET
cana-5487	50	5	∶	∶	NOUN
cana-5487	50	6	𝑋	𝑋	ADJ
cana-5487	50	7	×	×	NOUN
cana-5487	50	8	𝑋	𝑋	NOUN
cana-5487	50	9	×	×	NOUN
cana-5487	50	10	𝑋	𝑋	PROPN
cana-5487	50	11	→	→	SYM
cana-5487	50	12	𝑋	𝑋	PROPN
cana-5487	50	13	and	and	CCONJ
cana-5487	50	14	𝑔	𝑔	PROPN
cana-5487	50	15	∶	∶	NOUN
cana-5487	50	16	𝑋	𝑋	PROPN
cana-5487	50	17	→	→	PUNCT
cana-5487	50	18	𝑋	𝑋	NOUN
cana-5487	50	19	be	be	VERB
cana-5487	50	20	two	two	NUM
cana-5487	50	21	mappings	mapping	NOUN
cana-5487	50	22	.	.	PUNCT
cana-5487	51	1	𝑓	𝑓	DET
cana-5487	51	2	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-5487	51	3	𝑔	𝑔	PROPN
cana-5487	51	4	are	be	AUX
cana-5487	51	5	said	say	VERB
cana-5487	51	6	to	to	PART
cana-5487	51	7	be	be	AUX
cana-5487	51	8	weakly	weakly	ADV
cana-5487	51	9	compatible	compatible	ADJ
cana-5487	51	10	if	if	SCONJ
cana-5487	51	11	they	they	PRON
cana-5487	51	12	commute	commute	VERB
cana-5487	51	13	at	at	ADP
cana-5487	51	14	their	their	PRON
cana-5487	51	15	a	a	DET
cana-5487	51	16	tripled	triple	VERB
cana-5487	51	17	coincidence	coincidence	NOUN
cana-5487	51	18	point	point	NOUN
cana-5487	51	19	,	,	PUNCT
cana-5487	51	20	i.e.	i.e.	X
cana-5487	51	21	,	,	PUNCT
cana-5487	51	22	if	if	SCONJ
cana-5487	51	23	(	(	PUNCT
cana-5487	51	24	𝑥	𝑥	NOUN
cana-5487	51	25	,	,	PUNCT
cana-5487	51	26	𝑦	𝑦	NOUN
cana-5487	51	27	,	,	PUNCT
cana-5487	51	28	𝑧	𝑧	PART
cana-5487	51	29	)	)	PUNCT
cana-5487	51	30	is	be	AUX
cana-5487	51	31	a	a	DET
cana-5487	51	32	tripled	triple	VERB
cana-5487	51	33	coincidence	coincidence	NOUN
cana-5487	51	34	point	point	NOUN
cana-5487	51	35	of	of	ADP
cana-5487	51	36	𝑔	𝑔	PROPN
cana-5487	51	37	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5487	51	38	𝑓	𝑓	PROPN
cana-5487	51	39	,	,	PUNCT
cana-5487	51	40	then	then	ADV
cana-5487	51	41	𝑔𝑓(𝑥	𝑔𝑓(𝑥	NOUN
cana-5487	51	42	,	,	PUNCT
cana-5487	51	43	𝑦	𝑦	NOUN
cana-5487	51	44	,	,	PUNCT
cana-5487	51	45	𝑧	𝑧	NOUN
cana-5487	51	46	)	)	PUNCT
cana-5487	51	47	=	=	SYM
cana-5487	51	48	𝑓(𝑔(𝑥	𝑓(𝑔(𝑥	ADJ
cana-5487	51	49	)	)	PUNCT
cana-5487	51	50	,	,	PUNCT
cana-5487	51	51	𝑔(𝑦	𝑔(𝑦	PROPN
cana-5487	51	52	)	)	PUNCT
cana-5487	51	53	,	,	PUNCT
cana-5487	51	54	𝑔(𝑧	𝑔(𝑧	PROPN
cana-5487	51	55	)	)	PUNCT
cana-5487	51	56	)	)	PUNCT
cana-5487	51	57	.	.	PUNCT
cana-5487	52	1	example	example	NOUN
cana-5487	53	1	2.9.1let	2.9.1let	NUM
cana-5487	53	2	s	s	NOUN
cana-5487	53	3	:	:	PUNCT
cana-5487	53	4	x	x	SYM
cana-5487	53	5	×	×	NOUN
cana-5487	53	6	x	x	SYM
cana-5487	53	7	×	×	NOUN
cana-5487	53	8	x	x	INTJ
cana-5487	53	9	→	→	SYM
cana-5487	53	10	x	x	PROPN
cana-5487	53	11	&	&	CCONJ
cana-5487	53	12	𝑇	𝑇	PROPN
cana-5487	53	13	:	:	PUNCT
cana-5487	53	14	x	x	SYM
cana-5487	53	15	→	→	PUNCT
cana-5487	53	16	x	x	AUX
cana-5487	53	17	be	be	AUX
cana-5487	53	18	defined	define	VERB
cana-5487	53	19	by	by	ADP
cana-5487	53	20	s(x	s(x	PROPN
cana-5487	53	21	,	,	PUNCT
cana-5487	53	22	y	y	PROPN
cana-5487	53	23	,	,	PUNCT
cana-5487	53	24	z	z	NOUN
cana-5487	53	25	)	)	PUNCT
cana-5487	53	26	=	=	PUNCT
cana-5487	54	1	x	x	PUNCT
cana-5487	55	1	+	+	NUM
cana-5487	55	2	xy	xy	PROPN
cana-5487	56	1	+	+	PROPN
cana-5487	56	2	xz	xz	PROPN
cana-5487	56	3	𝑇(x	𝑇(x	NOUN
cana-5487	56	4	)	)	PUNCT
cana-5487	57	1	=	=	NOUN
cana-5487	57	2	{	{	PUNCT
cana-5487	57	3	0	0	NUM
cana-5487	57	4	,	,	PUNCT
cana-5487	57	5	if	if	SCONJ
cana-5487	57	6	x	x	PROPN
cana-5487	57	7	≠	≠	PROPN
cana-5487	57	8	1	1	NUM
cana-5487	57	9	;	;	PUNCT
cana-5487	57	10	1	1	NUM
cana-5487	57	11	,	,	PUNCT
cana-5487	57	12	ifx	ifx	PROPN
cana-5487	57	13	=	=	PROPN
cana-5487	57	14	1	1	X
cana-5487	57	15	.	.	PUNCT
cana-5487	57	16	here	here	ADV
cana-5487	57	17	,	,	PUNCT
cana-5487	57	18	(	(	PUNCT
cana-5487	57	19	1,0,0	1,0,0	NUM
cana-5487	57	20	)	)	PUNCT
cana-5487	57	21	,	,	PUNCT
cana-5487	57	22	(	(	PUNCT
cana-5487	57	23	0,1,0	0,1,0	NUM
cana-5487	57	24	)	)	PUNCT
cana-5487	57	25	and	and	CCONJ
cana-5487	57	26	(	(	PUNCT
cana-5487	57	27	0,0,1	0,0,1	NOUN
cana-5487	57	28	)	)	PUNCT
cana-5487	57	29	are	be	AUX
cana-5487	57	30	triple	triple	ADJ
cana-5487	57	31	coincidence	coincidence	NOUN
cana-5487	57	32	points	point	NOUN
cana-5487	57	33	of	of	ADP
cana-5487	57	34	s	s	PRON
cana-5487	57	35	and	and	CCONJ
cana-5487	57	36	t	t	PROPN
cana-5487	57	37	at	at	ADP
cana-5487	57	38	which	which	PRON
cana-5487	57	39	(	(	PUNCT
cana-5487	57	40	s	s	PROPN
cana-5487	57	41	,	,	PUNCT
cana-5487	57	42	t	t	PROPN
cana-5487	57	43	)	)	PUNCT
cana-5487	57	44	commute	commute	NOUN
cana-5487	57	45	.	.	PUNCT
cana-5487	58	1	so	so	ADV
cana-5487	58	2	,	,	PUNCT
cana-5487	58	3	s	s	X
cana-5487	58	4	and	and	CCONJ
cana-5487	58	5	t	t	PROPN
cana-5487	58	6	are	be	AUX
cana-5487	58	7	weakly	weakly	ADV
cana-5487	58	8	compatible	compatible	ADJ
cana-5487	58	9	.	.	PUNCT
cana-5487	59	1	example	example	NOUN
cana-5487	60	1	2.9.2let	2.9.2let	NUM
cana-5487	60	2	s	s	X
cana-5487	60	3	:	:	PUNCT
cana-5487	60	4	x	x	SYM
cana-5487	60	5	×	×	NOUN
cana-5487	60	6	x	x	SYM
cana-5487	60	7	×	×	NOUN
cana-5487	60	8	x	x	INTJ
cana-5487	60	9	→	→	SYM
cana-5487	60	10	x	x	PROPN
cana-5487	60	11	&	&	CCONJ
cana-5487	60	12	𝑇	𝑇	PROPN
cana-5487	60	13	:	:	PUNCT
cana-5487	60	14	x	x	SYM
cana-5487	60	15	→	→	PUNCT
cana-5487	60	16	x	x	AUX
cana-5487	60	17	be	be	AUX
cana-5487	60	18	defined	define	VERB
cana-5487	60	19	by	by	ADP
cana-5487	60	20	s(x	s(x	PROPN
cana-5487	60	21	,	,	PUNCT
cana-5487	60	22	y	y	PROPN
cana-5487	60	23	,	,	PUNCT
cana-5487	60	24	z	z	NOUN
cana-5487	60	25	)	)	PUNCT
cana-5487	60	26	=	=	SYM
cana-5487	60	27	xyz	xyz	PROPN
cana-5487	60	28	𝑇(x	𝑇(x	NOUN
cana-5487	60	29	)	)	PUNCT
cana-5487	60	30	=	=	NOUN
cana-5487	60	31	{	{	PUNCT
cana-5487	60	32	0	0	NUM
cana-5487	60	33	,	,	PUNCT
cana-5487	60	34	if	if	SCONJ
cana-5487	60	35	0	0	NUM
cana-5487	60	36	≤	≤	NUM
cana-5487	60	37	x	x	SYM
cana-5487	60	38	≤	≤	NUM
cana-5487	60	39	1	1	NUM
cana-5487	60	40	;	;	PUNCT
cana-5487	60	41	1	1	NUM
cana-5487	60	42	,	,	PUNCT
cana-5487	60	43	ifx	ifx	PROPN
cana-5487	60	44	≥	≥	PROPN
cana-5487	60	45	1	1	NUM
cana-5487	60	46	.	.	PUNCT
cana-5487	61	1	so	so	ADV
cana-5487	61	2	,	,	PUNCT
cana-5487	61	3	s	s	X
cana-5487	61	4	and	and	CCONJ
cana-5487	61	5	t	t	PROPN
cana-5487	61	6	are	be	AUX
cana-5487	61	7	weakly	weakly	ADV
cana-5487	61	8	compatible	compatible	ADJ
cana-5487	61	9	at	at	ADP
cana-5487	61	10	(	(	PUNCT
cana-5487	61	11	0	0	NUM
cana-5487	61	12	,	,	PUNCT
cana-5487	61	13	0	0	NUM
cana-5487	61	14	,	,	PUNCT
cana-5487	61	15	0	0	NUM
cana-5487	61	16	)	)	PUNCT
cana-5487	61	17	and	and	CCONJ
cana-5487	61	18	(	(	PUNCT
cana-5487	61	19	1	1	NUM
cana-5487	61	20	,	,	PUNCT
cana-5487	61	21	1	1	NUM
cana-5487	61	22	,	,	PUNCT
cana-5487	61	23	1	1	NUM
cana-5487	61	24	)	)	PUNCT
cana-5487	61	25	.	.	PUNCT
cana-5487	62	1	definition	definition	NOUN
cana-5487	62	2	2.10the	2.10the	PROPN
cana-5487	62	3	mappings𝑓	mappings𝑓	NOUN
cana-5487	62	4	:	:	PUNCT
cana-5487	62	5	𝑋	𝑋	NOUN
cana-5487	62	6	×	×	NOUN
cana-5487	62	7	𝑋	𝑋	PROPN
cana-5487	62	8	×	×	NOUN
cana-5487	62	9	𝑋	𝑋	PROPN
cana-5487	62	10	→	→	SYM
cana-5487	62	11	𝑋	𝑋	PROPN
cana-5487	62	12	and	and	CCONJ
cana-5487	62	13	𝑔:𝑋	𝑔:𝑋	PROPN
cana-5487	62	14	→	→	SYM
cana-5487	62	15	𝑋	𝑋	NOUN
cana-5487	62	16	of	of	ADP
cana-5487	62	17	a	a	DET
cana-5487	62	18	set	set	NOUN
cana-5487	62	19	x	x	VERB
cana-5487	62	20	are	be	AUX
cana-5487	62	21	occasionally	occasionally	ADV
cana-5487	62	22	weakly	weakly	ADJ
cana-5487	62	23	compatible(𝑜𝑤𝑐)iff	compatible(𝑜𝑤𝑐)iff	NOUN
cana-5487	62	24	there	there	PRON
cana-5487	62	25	is	be	VERB
cana-5487	62	26	a	a	DET
cana-5487	62	27	point	point	NOUN
cana-5487	62	28	(	(	PUNCT
cana-5487	62	29	𝑥	𝑥	NOUN
cana-5487	62	30	,	,	PUNCT
cana-5487	62	31	𝑦	𝑦	NOUN
cana-5487	62	32	,	,	PUNCT
cana-5487	62	33	𝑧	𝑧	NOUN
cana-5487	62	34	)	)	PUNCT
cana-5487	62	35	∈	∈	PROPN
cana-5487	62	36	𝑋	𝑋	NOUN
cana-5487	62	37	×	×	NOUN
cana-5487	62	38	𝑋	𝑋	PROPN
cana-5487	62	39	×	×	NOUN
cana-5487	62	40	𝑋	𝑋	NOUN
cana-5487	62	41	which	which	PRON
cana-5487	62	42	is	be	AUX
cana-5487	62	43	a	a	DET
cana-5487	62	44	coincidence	coincidence	NOUN
cana-5487	62	45	point	point	NOUN
cana-5487	62	46	of	of	ADP
cana-5487	62	47	f	f	PROPN
cana-5487	62	48	and	and	CCONJ
cana-5487	62	49	g	g	PROPN
cana-5487	62	50	at	at	ADP
cana-5487	62	51	which	which	PRON
cana-5487	62	52	f	f	PROPN
cana-5487	62	53	and	and	CCONJ
cana-5487	62	54	g	g	PROPN
cana-5487	62	55	commute	commute	NOUN
cana-5487	62	56	i.e.	i.e.	X
cana-5487	62	57	(	(	PUNCT
cana-5487	62	58	𝑓	𝑓	PROPN
cana-5487	62	59	,	,	PUNCT
cana-5487	62	60	𝑔	𝑔	NOUN
cana-5487	62	61	)	)	PUNCT
cana-5487	62	62	are	be	AUX
cana-5487	62	63	occasionally	occasionally	ADV
cana-5487	62	64	weakly	weakly	ADJ
cana-5487	62	65	compatible	compatible	ADJ
cana-5487	62	66	maps	map	NOUN
cana-5487	62	67	iff	iff	VERB
cana-5487	62	68	𝑓(𝑥	𝑓(𝑥	NOUN
cana-5487	62	69	,	,	PUNCT
cana-5487	62	70	𝑦	𝑦	NOUN
cana-5487	62	71	,	,	PUNCT
cana-5487	62	72	𝑧	𝑧	NOUN
cana-5487	62	73	)	)	PUNCT
cana-5487	62	74	=	=	SYM
cana-5487	62	75	𝑔(𝑥	𝑔(𝑥	PROPN
cana-5487	62	76	)	)	PUNCT
cana-5487	62	77	,	,	PUNCT
cana-5487	62	78	𝑓(𝑦	𝑓(𝑦	PROPN
cana-5487	62	79	,	,	PUNCT
cana-5487	62	80	𝑧	𝑧	PRON
cana-5487	62	81	,	,	PUNCT
cana-5487	62	82	𝑥	𝑥	NOUN
cana-5487	62	83	)	)	PUNCT
cana-5487	62	84	=	=	SYM
cana-5487	62	85	𝑔(𝑦	𝑔(𝑦	PROPN
cana-5487	62	86	)	)	PUNCT
cana-5487	62	87	,	,	PUNCT
cana-5487	62	88	𝑓(𝑧	𝑓(𝑧	PROPN
cana-5487	62	89	,	,	PUNCT
cana-5487	62	90	𝑥	𝑥	NOUN
cana-5487	62	91	,	,	PUNCT
cana-5487	62	92	𝑦	𝑦	NOUN
cana-5487	62	93	)	)	PUNCT
cana-5487	62	94	=	=	SYM
cana-5487	62	95	𝑔𝑧	𝑔𝑧	PROPN
cana-5487	62	96	implies	imply	VERB
cana-5487	62	97	𝑔𝑓(𝑥	𝑔𝑓(𝑥	NOUN
cana-5487	62	98	,	,	PUNCT
cana-5487	62	99	𝑦	𝑦	NOUN
cana-5487	62	100	,	,	PUNCT
cana-5487	62	101	𝑧	𝑧	NOUN
cana-5487	62	102	)	)	PUNCT
cana-5487	62	103	=	=	SYM
cana-5487	62	104	𝑓(𝑔𝑥	𝑓(𝑔𝑥	PROPN
cana-5487	62	105	,	,	PUNCT
cana-5487	62	106	𝑔𝑦	𝑔𝑦	PROPN
cana-5487	62	107	,	,	PUNCT
cana-5487	62	108	𝑔𝑧	𝑔𝑧	PROPN
cana-5487	62	109	)	)	PUNCT
cana-5487	62	110	,	,	PUNCT
cana-5487	62	111	𝑔𝑓(𝑦	𝑔𝑓(𝑦	X
cana-5487	62	112	,	,	PUNCT
cana-5487	62	113	𝑧	𝑧	PROPN
cana-5487	62	114	,	,	PUNCT
cana-5487	62	115	𝑥	𝑥	NOUN
cana-5487	62	116	)	)	PUNCT
cana-5487	62	117	=	=	SYM
cana-5487	62	118	𝑓(𝑔𝑦	𝑓(𝑔𝑦	ADJ
cana-5487	62	119	,	,	PUNCT
cana-5487	62	120	𝑔𝑧	𝑔𝑧	SYM
cana-5487	62	121	,	,	PUNCT
cana-5487	62	122	𝑔𝑥	𝑔𝑥	PROPN
cana-5487	62	123	)	)	PUNCT
cana-5487	62	124	,	,	PUNCT
cana-5487	62	125	𝑔𝑓(𝑧	𝑔𝑓(𝑧	NUM
cana-5487	62	126	,	,	PUNCT
cana-5487	62	127	𝑥	𝑥	X
cana-5487	62	128	,	,	PUNCT
cana-5487	62	129	𝑦	𝑦	NOUN
cana-5487	62	130	)	)	PUNCT
cana-5487	62	131	=	=	SYM
cana-5487	63	1	𝑓(𝑔𝑧	𝑓(𝑔𝑧	PROPN
cana-5487	63	2	,	,	PUNCT
cana-5487	63	3	𝑔𝑥	𝑔𝑥	PROPN
cana-5487	63	4	,	,	PUNCT
cana-5487	63	5	𝑔𝑦	𝑔𝑦	PROPN
cana-5487	63	6	)	)	PUNCT
cana-5487	63	7	for	for	ADP
cana-5487	63	8	(	(	PUNCT
cana-5487	63	9	𝑥	𝑥	PROPN
cana-5487	63	10	,	,	PUNCT
cana-5487	63	11	𝑦	𝑦	NOUN
cana-5487	63	12	,	,	PUNCT
cana-5487	63	13	𝑧	𝑧	NOUN
cana-5487	63	14	)	)	PUNCT
cana-5487	63	15	∈	∈	PROPN
cana-5487	63	16	𝑋	𝑋	NOUN
cana-5487	63	17	×	×	NOUN
cana-5487	63	18	𝑋	𝑋	PROPN
cana-5487	63	19	×	×	PROPN
cana-5487	63	20	𝑋.	𝑋.	PROPN
cana-5487	63	21	example	example	NOUN
cana-5487	63	22	2.10.1	2.10.1	NUM
cana-5487	63	23	let	let	VERB
cana-5487	63	24	(	(	PUNCT
cana-5487	63	25	x	x	NOUN
cana-5487	63	26	,	,	PUNCT
cana-5487	63	27	ℱ,∗	ℱ,∗	NUM
cana-5487	63	28	)	)	PUNCT
cana-5487	63	29	be	be	VERB
cana-5487	63	30	a	a	DET
cana-5487	63	31	fuzzy	fuzzy	ADJ
cana-5487	63	32	metric	metric	ADJ
cana-5487	63	33	space	space	NOUN
cana-5487	63	34	,	,	PUNCT
cana-5487	63	35	where	where	SCONJ
cana-5487	63	36	x	x	X
cana-5487	63	37	=	=	PUNCT
cana-5487	64	1	[	[	X
cana-5487	64	2	0,1	0,1	NUM
cana-5487	64	3	]	]	PUNCT
cana-5487	64	4	with	with	ADP
cana-5487	64	5	𝑎	𝑎	NOUN
cana-5487	64	6	∗	∗	NOUN
cana-5487	64	7	𝑏	𝑏	NOUN
cana-5487	64	8	=	=	SYM
cana-5487	64	9	min{𝑎	min{𝑎	PROPN
cana-5487	64	10	,	,	PUNCT
cana-5487	64	11	𝑏}and	𝑏}and	PROPN
cana-5487	64	12	m(x	m(x	PROPN
cana-5487	64	13	,	,	PUNCT
cana-5487	64	14	y	y	PROPN
cana-5487	64	15	,	,	PUNCT
cana-5487	64	16	t	t	PROPN
cana-5487	64	17	)	)	PUNCT
cana-5487	64	18	=	=	PRON
cana-5487	64	19	{	{	PUNCT
cana-5487	64	20	t	t	NOUN
cana-5487	64	21	t	t	PROPN
cana-5487	64	22	+	+	CCONJ
cana-5487	64	23	|x	|x	NOUN
cana-5487	65	1	−	−	VERB
cana-5487	65	2	y|	y|	NOUN
cana-5487	65	3	,	,	PUNCT
cana-5487	65	4	if	if	SCONJ
cana-5487	65	5	t	t	PROPN
cana-5487	65	6	>	>	X
cana-5487	65	7	0	0	NUM
cana-5487	65	8	;	;	PUNCT
cana-5487	65	9	0	0	NUM
cana-5487	65	10	,	,	PUNCT
cana-5487	65	11	if	if	SCONJ
cana-5487	65	12	t	t	NOUN
cana-5487	65	13	=	=	SYM
cana-5487	65	14	0	0	X
cana-5487	65	15	.	.	PUNCT
cana-5487	66	1	let	let	VERB
cana-5487	66	2	f	f	X
cana-5487	66	3	:	:	PUNCT
cana-5487	66	4	x	x	SYM
cana-5487	66	5	×	×	NOUN
cana-5487	66	6	x	x	SYM
cana-5487	66	7	×	×	NOUN
cana-5487	66	8	x	x	INTJ
cana-5487	66	9	→	→	SYM
cana-5487	66	10	x	x	PROPN
cana-5487	66	11	&	&	CCONJ
cana-5487	66	12	𝑔	𝑔	ADJ
cana-5487	67	1	:	:	PUNCT
cana-5487	67	2	x	x	SYM
cana-5487	67	3	→	→	PUNCT
cana-5487	67	4	x	x	AUX
cana-5487	67	5	be	be	AUX
cana-5487	67	6	defined	define	VERB
cana-5487	67	7	by	by	ADP
cana-5487	67	8	f(x	f(x	PROPN
cana-5487	67	9	,	,	PUNCT
cana-5487	67	10	y	y	PROPN
cana-5487	67	11	,	,	PUNCT
cana-5487	67	12	z	z	NOUN
cana-5487	67	13	)	)	PUNCT
cana-5487	67	14	=	=	SYM
cana-5487	67	15	2x	2x	NOUN
cana-5487	67	16	+	+	CCONJ
cana-5487	67	17	2y	2y	PROPN
cana-5487	67	18	+	+	CCONJ
cana-5487	67	19	z	z	NOUN
cana-5487	67	20	2	2	NUM
cana-5487	67	21	communications	communication	NOUN
cana-5487	67	22	on	on	ADP
cana-5487	67	23	applied	apply	VERB
cana-5487	67	24	nonlinear	nonlinear	ADJ
cana-5487	67	25	analysis	analysis	NOUN
cana-5487	67	26	issn	issn	NOUN
cana-5487	67	27	:	:	PUNCT
cana-5487	67	28	1074	1074	NUM
cana-5487	67	29	-	-	PUNCT
cana-5487	67	30	133x	133x	NUM
cana-5487	67	31	vol	vol	VERB
cana-5487	67	32	32	32	NUM
cana-5487	67	33	no	no	NOUN
cana-5487	67	34	.	.	PUNCT
cana-5487	68	1	10s	10	NOUN
cana-5487	68	2	(	(	PUNCT
cana-5487	68	3	2025	2025	NUM
cana-5487	68	4	)	)	PUNCT
cana-5487	68	5	2386	2386	NUM
cana-5487	68	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	68	7	g(x	g(x	NOUN
cana-5487	68	8	)	)	PUNCT
cana-5487	68	9	=	=	PRON
cana-5487	68	10	{	{	PUNCT
cana-5487	68	11	x	x	X
cana-5487	68	12	,	,	PUNCT
cana-5487	68	13	if	if	SCONJ
cana-5487	68	14	0	0	NUM
cana-5487	68	15	≤	≤	NUM
cana-5487	68	16	x	x	X
cana-5487	68	17	<	<	X
cana-5487	68	18	1	1	NUM
cana-5487	68	19	;	;	PUNCT
cana-5487	68	20	5	5	NUM
cana-5487	68	21	2	2	NUM
cana-5487	68	22	,	,	PUNCT
cana-5487	68	23	ifx	ifx	PROPN
cana-5487	68	24	≥	≥	PROPN
cana-5487	68	25	1	1	NUM
cana-5487	68	26	.	.	PUNCT
cana-5487	69	1	here	here	ADV
cana-5487	69	2	,	,	PUNCT
cana-5487	69	3	(	(	PUNCT
cana-5487	69	4	0,0,0	0,0,0	NOUN
cana-5487	69	5	)	)	PUNCT
cana-5487	69	6	and	and	CCONJ
cana-5487	69	7	(	(	PUNCT
cana-5487	69	8	1,1,1	1,1,1	NUM
cana-5487	69	9	)	)	PUNCT
cana-5487	69	10	are	be	AUX
cana-5487	69	11	two	two	NUM
cana-5487	69	12	coincidence	coincidence	NOUN
cana-5487	69	13	points	point	NOUN
cana-5487	69	14	of	of	ADP
cana-5487	69	15	f	f	PROPN
cana-5487	69	16	and	and	CCONJ
cana-5487	69	17	g.	g.	PROPN
cana-5487	70	1	that	that	PRON
cana-5487	70	2	isf(0,0,0	isf(0,0,0	NOUN
cana-5487	70	3	)	)	PUNCT
cana-5487	70	4	=	=	SYM
cana-5487	70	5	0	0	PUNCT
cana-5487	70	6	=	=	SYM
cana-5487	70	7	g(0	g(0	PROPN
cana-5487	70	8	)	)	PUNCT
cana-5487	70	9	,	,	PUNCT
cana-5487	70	10	f(1,1,1	f(1,1,1	PROPN
cana-5487	70	11	)	)	PUNCT
cana-5487	70	12	=	=	SYM
cana-5487	70	13	1	1	NUM
cana-5487	70	14	=	=	SYM
cana-5487	70	15	g(1)butgf(0,0,0	g(1)butgf(0,0,0	NOUN
cana-5487	70	16	)	)	PUNCT
cana-5487	70	17	=	=	SYM
cana-5487	70	18	0	0	NUM
cana-5487	71	1	=	=	SYM
cana-5487	71	2	f(g0	f(g0	NOUN
cana-5487	71	3	,	,	PUNCT
cana-5487	71	4	g0	g0	PROPN
cana-5487	71	5	,	,	PUNCT
cana-5487	71	6	g0	g0	NOUN
cana-5487	71	7	)	)	PUNCT
cana-5487	71	8	,	,	PUNCT
cana-5487	71	9	gf(1,1,1	gf(1,1,1	NOUN
cana-5487	71	10	)	)	PUNCT
cana-5487	71	11	≠	≠	PROPN
cana-5487	71	12	f(g1	f(g1	NOUN
cana-5487	71	13	,	,	PUNCT
cana-5487	71	14	g1	g1	NOUN
cana-5487	71	15	,	,	PUNCT
cana-5487	71	16	g1	g1	NOUN
cana-5487	71	17	)	)	PUNCT
cana-5487	71	18	.	.	PUNCT
cana-5487	72	1	thus	thus	ADV
cana-5487	72	2	f	f	PROPN
cana-5487	72	3	and	and	CCONJ
cana-5487	72	4	g	g	PROPN
cana-5487	72	5	are	be	AUX
cana-5487	72	6	owc	owc	NOUN
cana-5487	72	7	but	but	CCONJ
cana-5487	72	8	not	not	PART
cana-5487	72	9	weakly	weakly	ADV
cana-5487	72	10	compatible	compatible	ADJ
cana-5487	72	11	.	.	PUNCT
cana-5487	73	1	the	the	DET
cana-5487	73	2	mesh	mesh	NOUN
cana-5487	73	3	diagram	diagram	NOUN
cana-5487	73	4	for	for	ADP
cana-5487	73	5	the	the	DET
cana-5487	73	6	given	give	VERB
cana-5487	73	7	example	example	NOUN
cana-5487	73	8	is	be	AUX
cana-5487	73	9	shown	show	VERB
cana-5487	73	10	in	in	ADP
cana-5487	73	11	fig	fig	NOUN
cana-5487	73	12	[	[	X
cana-5487	73	13	2.1	2.1	NUM
cana-5487	73	14	]	]	PUNCT
cana-5487	73	15	.	.	PUNCT
cana-5487	74	1	fig	fig	NOUN
cana-5487	75	1	[	[	X
cana-5487	75	2	2.1	2.1	NUM
cana-5487	75	3	]	]	SYM
cana-5487	75	4	3	3	NUM
cana-5487	75	5	main	main	ADJ
cana-5487	75	6	results	result	NOUN
cana-5487	75	7	theorem	theorem	VERB
cana-5487	75	8	:	:	PUNCT
cana-5487	75	9	3.1let	3.1let	NUM
cana-5487	75	10	(	(	PUNCT
cana-5487	75	11	𝑋	𝑋	PROPN
cana-5487	75	12	,	,	PUNCT
cana-5487	75	13	𝑀	𝑀	PROPN
cana-5487	75	14	,	,	PUNCT
cana-5487	75	15			PROPN
cana-5487	75	16	)	)	PUNCT
cana-5487	75	17	be	be	AUX
cana-5487	75	18	a	a	DET
cana-5487	75	19	fuzzy	fuzzy	ADJ
cana-5487	75	20	metric	metric	ADJ
cana-5487	75	21	space	space	NOUN
cana-5487	75	22	with	with	ADP
cana-5487	75	23	𝑡	𝑡	PROPN
cana-5487	75	24	∗	∗	NOUN
cana-5487	75	25	𝑡	𝑡	X
cana-5487	75	26	=	=	PUNCT
cana-5487	75	27	𝑡	𝑡	PROPN
cana-5487	75	28	for	for	ADP
cana-5487	75	29	all	all	DET
cana-5487	75	30	𝑡	𝑡	ADP
cana-5487	75	31	∈	∈	PROPN
cana-5487	76	1	[	[	X
cana-5487	76	2	0,1	0,1	NUM
cana-5487	76	3	]	]	PUNCT
cana-5487	76	4	.	.	PUNCT
cana-5487	77	1	let	let	VERB
cana-5487	77	2	𝐴	𝐴	PROPN
cana-5487	77	3	,	,	PUNCT
cana-5487	77	4	𝐵	𝐵	PROPN
cana-5487	77	5	:	:	PUNCT
cana-5487	77	6	𝑋	𝑋	NOUN
cana-5487	77	7	×	×	NOUN
cana-5487	77	8	𝑋	𝑋	PROPN
cana-5487	77	9	×	×	NOUN
cana-5487	77	10	𝑋	𝑋	PROPN
cana-5487	77	11	→	→	SYM
cana-5487	77	12	𝑋	𝑋	PROPN
cana-5487	77	13	and	and	CCONJ
cana-5487	77	14	𝑆	𝑆	PROPN
cana-5487	77	15	,	,	PUNCT
cana-5487	77	16	𝑇	𝑇	PROPN
cana-5487	77	17	:	:	PUNCT
cana-5487	77	18	𝑋	𝑋	PROPN
cana-5487	77	19	→	→	SYM
cana-5487	77	20	𝑋	𝑋	PROPN
cana-5487	77	21	be	be	VERB
cana-5487	77	22	four	four	NUM
cana-5487	77	23	self	self	NOUN
cana-5487	77	24	-	-	PUNCT
cana-5487	77	25	mappings	mapping	NOUN
cana-5487	77	26	satisfying	satisfy	VERB
cana-5487	77	27	the	the	DET
cana-5487	77	28	following	follow	VERB
cana-5487	77	29	conditions	condition	NOUN
cana-5487	77	30	:	:	PUNCT
cana-5487	77	31	(	(	PUNCT
cana-5487	77	32	i	i	NOUN
cana-5487	77	33	)	)	PUNCT
cana-5487	77	34	𝑀𝑃(𝐴(𝑥	𝑀𝑃(𝐴(𝑥	NUM
cana-5487	77	35	,	,	PUNCT
cana-5487	77	36	𝑦	𝑦	NOUN
cana-5487	77	37	,	,	PUNCT
cana-5487	77	38	𝑧	𝑧	NOUN
cana-5487	77	39	)	)	PUNCT
cana-5487	77	40	,	,	PUNCT
cana-5487	77	41	𝐵(𝑢	𝐵(𝑢	PROPN
cana-5487	77	42	,	,	PUNCT
cana-5487	77	43	𝑣	𝑣	NOUN
cana-5487	77	44	,	,	PUNCT
cana-5487	77	45	𝑤	𝑤	ADP
cana-5487	77	46	)	)	PUNCT
cana-5487	77	47	,	,	PUNCT
cana-5487	77	48	𝑞𝑡	𝑞𝑡	PRON
cana-5487	77	49	)	)	PUNCT
cana-5487	77	50	≥	≥	NOUN
cana-5487	78	1	𝜑	𝜑	NOUN
cana-5487	79	1	[	[	PUNCT
cana-5487	79	2	𝑎	𝑎	PROPN
cana-5487	79	3	𝑀𝑝(𝑆𝑥	𝑀𝑝(𝑆𝑥	NOUN
cana-5487	79	4	,	,	PUNCT
cana-5487	79	5	𝑇𝑢	𝑇𝑢	PROPN
cana-5487	79	6	,	,	PUNCT
cana-5487	79	7	𝑡	𝑡	NOUN
cana-5487	79	8	)	)	PUNCT
cana-5487	79	9	+	+	CCONJ
cana-5487	79	10	(	(	PUNCT
cana-5487	79	11	1	1	NUM
cana-5487	79	12	−	−	NUM
cana-5487	79	13	𝑎	𝑎	NOUN
cana-5487	79	14	)	)	PUNCT
cana-5487	79	15	min	min	NOUN
cana-5487	79	16	{	{	PUNCT
cana-5487	79	17	𝑀𝑝(𝐴(𝑥	𝑀𝑝(𝐴(𝑥	NOUN
cana-5487	79	18	,	,	PUNCT
cana-5487	79	19	𝑦	𝑦	NOUN
cana-5487	79	20	,	,	PUNCT
cana-5487	79	21	𝑧	𝑧	NOUN
cana-5487	79	22	)	)	PUNCT
cana-5487	79	23	,	,	PUNCT
cana-5487	79	24	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	79	25	,	,	PUNCT
cana-5487	79	26	𝑡	𝑡	PROPN
cana-5487	79	27	)	)	PUNCT
cana-5487	79	28	,	,	PUNCT
cana-5487	79	29	𝑀𝑝(𝐵(𝑢	𝑀𝑝(𝐵(𝑢	PROPN
cana-5487	79	30	,	,	PUNCT
cana-5487	79	31	𝑣	𝑣	NOUN
cana-5487	79	32	,	,	PUNCT
cana-5487	79	33	𝑤	𝑤	ADP
cana-5487	79	34	)	)	PUNCT
cana-5487	79	35	,	,	PUNCT
cana-5487	80	1	𝑇𝑢	𝑇𝑢	PROPN
cana-5487	80	2	,	,	PUNCT
cana-5487	80	3	𝑡	𝑡	PROPN
cana-5487	80	4	)	)	PUNCT
cana-5487	80	5	,	,	PUNCT
cana-5487	80	6	𝑀	𝑀	PROPN
cana-5487	80	7	𝑝	𝑝	PROPN
cana-5487	80	8	2(𝐴(𝑥	2(𝐴(𝑥	NUM
cana-5487	80	9	,	,	PUNCT
cana-5487	80	10	𝑦	𝑦	NOUN
cana-5487	80	11	,	,	PUNCT
cana-5487	80	12	𝑧	𝑧	NOUN
cana-5487	80	13	)	)	PUNCT
cana-5487	80	14	,	,	PUNCT
cana-5487	80	15	𝑇𝑢	𝑇𝑢	PROPN
cana-5487	80	16	,	,	PUNCT
cana-5487	80	17	𝑡	𝑡	PROPN
cana-5487	80	18	)	)	PUNCT
cana-5487	80	19	.	.	PUNCT
cana-5487	81	1	𝑀	𝑀	PROPN
cana-5487	81	2	𝑝	𝑝	PROPN
cana-5487	81	3	2(𝐵(𝑢	2(𝐵(𝑢	PROPN
cana-5487	81	4	,	,	PUNCT
cana-5487	81	5	𝑣	𝑣	NOUN
cana-5487	81	6	,	,	PUNCT
cana-5487	81	7	𝑤	𝑤	ADP
cana-5487	81	8	)	)	PUNCT
cana-5487	81	9	,	,	PUNCT
cana-5487	82	1	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	82	2	,	,	PUNCT
cana-5487	82	3	𝑡	𝑡	PROPN
cana-5487	82	4	)	)	PUNCT
cana-5487	82	5	,	,	PUNCT
cana-5487	82	6	1	1	NUM
cana-5487	82	7	2	2	NUM
cana-5487	82	8	[	[	X
cana-5487	82	9	𝑀𝑝(𝐴(𝑥	𝑀𝑝(𝐴(𝑥	NUM
cana-5487	82	10	,	,	PUNCT
cana-5487	82	11	𝑦	𝑦	NOUN
cana-5487	82	12	,	,	PUNCT
cana-5487	82	13	𝑧	𝑧	NOUN
cana-5487	82	14	)	)	PUNCT
cana-5487	82	15	,	,	PUNCT
cana-5487	82	16	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	82	17	,	,	PUNCT
cana-5487	82	18	𝑡	𝑡	PROPN
cana-5487	82	19	)	)	PUNCT
cana-5487	82	20	+	+	X
cana-5487	82	21	𝑀𝑝(𝐵(𝑢	𝑀𝑝(𝐵(𝑢	PROPN
cana-5487	82	22	,	,	PUNCT
cana-5487	82	23	𝑣	𝑣	NOUN
cana-5487	82	24	,	,	PUNCT
cana-5487	82	25	𝑤	𝑤	ADP
cana-5487	82	26	)	)	PUNCT
cana-5487	82	27	,	,	PUNCT
cana-5487	82	28	𝑇𝑢	𝑇𝑢	PROPN
cana-5487	82	29	,	,	PUNCT
cana-5487	82	30	𝑡	𝑡	PROPN
cana-5487	82	31	)	)	PUNCT
cana-5487	82	32	]	]	PUNCT
cana-5487	82	33	}	}	PUNCT
cana-5487	82	34	]	]	PUNCT
cana-5487	82	35	for	for	ADP
cana-5487	82	36	all	all	PRON
cana-5487	82	37	𝑥	𝑥	PROPN
cana-5487	82	38	,	,	PUNCT
cana-5487	82	39	𝑦	𝑦	NOUN
cana-5487	82	40	,	,	PUNCT
cana-5487	82	41	𝑧	𝑧	NOUN
cana-5487	82	42	,	,	PUNCT
cana-5487	82	43	𝑢	𝑢	PROPN
cana-5487	82	44	,	,	PUNCT
cana-5487	82	45	𝑣	𝑣	NOUN
cana-5487	82	46	,	,	PUNCT
cana-5487	82	47	𝑤	𝑤	ADP
cana-5487	82	48	∈	∈	PROPN
cana-5487	82	49	𝑋	𝑋	PROPN
cana-5487	82	50	,	,	PUNCT
cana-5487	82	51	0	0	NUM
cana-5487	82	52	≤	≤	NUM
cana-5487	83	1	𝑎	𝑎	PRON
cana-5487	83	2	≤	≤	NUM
cana-5487	83	3	1	1	NUM
cana-5487	83	4	,	,	PUNCT
cana-5487	83	5	𝑝	𝑝	PRON
cana-5487	83	6	≥	≥	NOUN
cana-5487	83	7	1	1	NUM
cana-5487	83	8	and	and	CCONJ
cana-5487	83	9	𝜑	𝜑	NOUN
cana-5487	83	10	:	:	PUNCT
cana-5487	83	11	𝑅+	𝑅+	PROPN
cana-5487	83	12	→	→	SYM
cana-5487	83	13	𝑅+	𝑅+	ADP
cana-5487	83	14	such	such	ADJ
cana-5487	83	15	that	that	SCONJ
cana-5487	83	16	𝜑	𝜑	PROPN
cana-5487	83	17	is	be	AUX
cana-5487	83	18	upper	upper	ADJ
cana-5487	83	19	semi	semi	ADV
cana-5487	83	20	continuous	continuous	ADJ
cana-5487	83	21	,	,	PUNCT
cana-5487	83	22	nonincreasing	nonincreasing	NOUN
cana-5487	83	23	and	and	CCONJ
cana-5487	83	24	𝜑(𝑡	𝜑(𝑡	PROPN
cana-5487	83	25	)	)	PUNCT
cana-5487	83	26	>	>	PUNCT
cana-5487	83	27	𝑡	𝑡	PROPN
cana-5487	83	28	for	for	ADP
cana-5487	83	29	any	any	DET
cana-5487	83	30	t>0	t>0	NOUN
cana-5487	83	31	.	.	PUNCT
cana-5487	84	1	(	(	PUNCT
cana-5487	84	2	ii	ii	NOUN
cana-5487	84	3	)	)	PUNCT
cana-5487	84	4	𝑦	𝑦	NOUN
cana-5487	84	5	=	=	SYM
cana-5487	84	6	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	84	7	,	,	PUNCT
cana-5487	84	8	𝑦	𝑦	NOUN
cana-5487	84	9	,	,	PUNCT
cana-5487	84	10	𝑧	𝑧	NOUN
cana-5487	84	11	)	)	PUNCT
cana-5487	84	12	communications	communication	NOUN
cana-5487	84	13	on	on	ADP
cana-5487	84	14	applied	apply	VERB
cana-5487	84	15	nonlinear	nonlinear	ADJ
cana-5487	84	16	analysis	analysis	NOUN
cana-5487	84	17	issn	issn	NOUN
cana-5487	84	18	:	:	PUNCT
cana-5487	84	19	1074	1074	NUM
cana-5487	84	20	-	-	PUNCT
cana-5487	84	21	133x	133x	NUM
cana-5487	84	22	vol	vol	VERB
cana-5487	84	23	32	32	NUM
cana-5487	84	24	no	no	NOUN
cana-5487	84	25	.	.	PUNCT
cana-5487	85	1	10s	10	NOUN
cana-5487	85	2	(	(	PUNCT
cana-5487	85	3	2025	2025	NUM
cana-5487	85	4	)	)	PUNCT
cana-5487	85	5	2387	2387	NUM
cana-5487	85	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	86	1	moreover	moreover	ADV
cana-5487	86	2	if	if	SCONJ
cana-5487	86	3	the	the	DET
cana-5487	86	4	pairs	pair	NOUN
cana-5487	86	5	(	(	PUNCT
cana-5487	86	6	𝐴	𝐴	PROPN
cana-5487	86	7	,	,	PUNCT
cana-5487	86	8	𝑆	𝑆	PROPN
cana-5487	86	9	)	)	PUNCT
cana-5487	86	10	and	and	CCONJ
cana-5487	86	11	(	(	PUNCT
cana-5487	86	12	𝐵	𝐵	PROPN
cana-5487	86	13	,	,	PUNCT
cana-5487	86	14	𝑇	𝑇	PROPN
cana-5487	86	15	)	)	PUNCT
cana-5487	86	16	are	be	AUX
cana-5487	86	17	owc	owc	NUM
cana-5487	86	18	,	,	PUNCT
cana-5487	86	19	then	then	ADV
cana-5487	86	20	there	there	PRON
cana-5487	86	21	exists	exist	VERB
cana-5487	86	22	a	a	DET
cana-5487	86	23	unique	unique	ADJ
cana-5487	86	24	point	point	NOUN
cana-5487	86	25	𝑥	𝑥	NOUN
cana-5487	86	26	in	in	ADP
cana-5487	86	27	𝑋	𝑋	PROPN
cana-5487	86	28	such	such	ADJ
cana-5487	86	29	that𝐴(𝑥	that𝐴(𝑥	NOUN
cana-5487	86	30	,	,	PUNCT
cana-5487	86	31	𝑥	𝑥	NOUN
cana-5487	86	32	,	,	PUNCT
cana-5487	86	33	𝑥	𝑥	NOUN
cana-5487	86	34	)	)	PUNCT
cana-5487	86	35	=	=	SYM
cana-5487	87	1	𝑇(𝑥	𝑇(𝑥	X
cana-5487	87	2	)	)	PUNCT
cana-5487	87	3	=	=	SYM
cana-5487	88	1	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	88	2	,	,	PUNCT
cana-5487	88	3	𝑥	𝑥	NOUN
cana-5487	88	4	,	,	PUNCT
cana-5487	88	5	𝑥	𝑥	NOUN
cana-5487	88	6	)	)	PUNCT
cana-5487	88	7	=	=	SYM
cana-5487	88	8	𝑆(𝑥	𝑆(𝑥	X
cana-5487	88	9	)	)	PUNCT
cana-5487	88	10	=	=	NOUN
cana-5487	89	1	𝑥.	𝑥.	ADJ
cana-5487	89	2	proof	proof	NOUN
cana-5487	89	3	:	:	PUNCT
cana-5487	89	4	since	since	SCONJ
cana-5487	89	5	the	the	DET
cana-5487	89	6	pairs	pair	NOUN
cana-5487	89	7	(	(	PUNCT
cana-5487	89	8	a	a	DET
cana-5487	89	9	,	,	PUNCT
cana-5487	89	10	s	s	PART
cana-5487	89	11	)	)	PUNCT
cana-5487	89	12	and	and	CCONJ
cana-5487	89	13	(	(	PUNCT
cana-5487	89	14	b	b	NOUN
cana-5487	89	15	,	,	PUNCT
cana-5487	89	16	t	t	PROPN
cana-5487	89	17	)	)	PUNCT
cana-5487	89	18	are	be	AUX
cana-5487	89	19	owc	owc	NUM
cana-5487	89	20	so	so	SCONJ
cana-5487	89	21	there	there	PRON
cana-5487	89	22	are	be	VERB
cana-5487	89	23	points	point	NOUN
cana-5487	89	24	𝑎	𝑎	NOUN
cana-5487	89	25	,	,	PUNCT
cana-5487	89	26	𝑏	𝑏	NOUN
cana-5487	89	27	,	,	PUNCT
cana-5487	89	28	𝑐	𝑐	NOUN
cana-5487	89	29	,	,	PUNCT
cana-5487	89	30	𝑎′	𝑎′	NUM
cana-5487	89	31	,	,	PUNCT
cana-5487	89	32	𝑏′	𝑏′	PROPN
cana-5487	89	33	,	,	PUNCT
cana-5487	89	34	𝑐′	𝑐′	NOUN
cana-5487	89	35	in	in	ADP
cana-5487	89	36	x	x	PROPN
cana-5487	89	37	such	such	ADJ
cana-5487	89	38	that	that	SCONJ
cana-5487	89	39	𝐴(𝑎	𝐴(𝑎	PROPN
cana-5487	89	40	,	,	PUNCT
cana-5487	89	41	𝑏	𝑏	NOUN
cana-5487	89	42	,	,	PUNCT
cana-5487	89	43	𝑐	𝑐	NOUN
cana-5487	89	44	)	)	PUNCT
cana-5487	89	45	=	=	SYM
cana-5487	89	46	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	89	47	,	,	PUNCT
cana-5487	89	48	𝐴(𝑏	𝐴(𝑏	PROPN
cana-5487	89	49	,	,	PUNCT
cana-5487	89	50	𝑐	𝑐	NOUN
cana-5487	89	51	,	,	PUNCT
cana-5487	89	52	𝑎	𝑎	NOUN
cana-5487	89	53	)	)	PUNCT
cana-5487	89	54	=	=	SYM
cana-5487	90	1	𝑆𝑏	𝑆𝑏	PROPN
cana-5487	90	2	,	,	PUNCT
cana-5487	90	3	𝐴(𝑐	𝐴(𝑐	PROPN
cana-5487	90	4	,	,	PUNCT
cana-5487	90	5	𝑎	𝑎	NOUN
cana-5487	90	6	,	,	PUNCT
cana-5487	90	7	𝑏	𝑏	NOUN
cana-5487	90	8	)	)	PUNCT
cana-5487	90	9	=	=	SYM
cana-5487	90	10	𝑆𝑐and	𝑆𝑐and	PROPN
cana-5487	90	11	𝐵(𝑎′	𝐵(𝑎′	PROPN
cana-5487	90	12	,	,	PUNCT
cana-5487	90	13	𝑏′	𝑏′	PROPN
cana-5487	90	14	,	,	PUNCT
cana-5487	90	15	𝑐′	𝑐′	NUM
cana-5487	90	16	)	)	PUNCT
cana-5487	91	1	=	=	SYM
cana-5487	91	2	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	91	3	,	,	PUNCT
cana-5487	91	4	𝐵(𝑏′	𝐵(𝑏′	NOUN
cana-5487	91	5	,	,	PUNCT
cana-5487	91	6	𝑐′	𝑐′	NUM
cana-5487	91	7	,	,	PUNCT
cana-5487	91	8	𝑎′	𝑎′	NUM
cana-5487	91	9	)	)	PUNCT
cana-5487	91	10	=	=	SYM
cana-5487	91	11	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	91	12	,	,	PUNCT
cana-5487	91	13	𝐵(𝑐′	𝐵(𝑐′	PROPN
cana-5487	91	14	,	,	PUNCT
cana-5487	91	15	𝑎′	𝑎′	PROPN
cana-5487	91	16	,	,	PUNCT
cana-5487	91	17	𝑏′	𝑏′	NUM
cana-5487	91	18	)	)	PUNCT
cana-5487	91	19	=	=	VERB
cana-5487	92	1	𝑇𝑐′	𝑇𝑐′	NOUN
cana-5487	92	2	we	we	PRON
cana-5487	92	3	claim	claim	VERB
cana-5487	92	4	that	that	SCONJ
cana-5487	92	5	𝑆𝑎	𝑆𝑎	NOUN
cana-5487	92	6	=	=	NOUN
cana-5487	92	7	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	92	8	.	.	PUNCT
cana-5487	93	1	if	if	SCONJ
cana-5487	93	2	not	not	PART
cana-5487	93	3	,	,	PUNCT
cana-5487	93	4	by	by	ADP
cana-5487	93	5	inequality	inequality	NOUN
cana-5487	93	6	(	(	PUNCT
cana-5487	93	7	𝑖	𝑖	X
cana-5487	93	8	)	)	PUNCT
cana-5487	93	9	we	we	PRON
cana-5487	93	10	get	get	VERB
cana-5487	93	11	𝑀𝑃(𝐴(𝑎	𝑀𝑃(𝐴(𝑎	ADP
cana-5487	93	12	,	,	PUNCT
cana-5487	93	13	𝑏	𝑏	NOUN
cana-5487	93	14	,	,	PUNCT
cana-5487	93	15	𝑐	𝑐	NOUN
cana-5487	93	16	)	)	PUNCT
cana-5487	93	17	,	,	PUNCT
cana-5487	93	18	𝐵(𝑎′	𝐵(𝑎′	PROPN
cana-5487	93	19	,	,	PUNCT
cana-5487	93	20	𝑏′	𝑏′	PROPN
cana-5487	93	21	,	,	PUNCT
cana-5487	93	22	𝑐′	𝑐′	NUM
cana-5487	93	23	)	)	PUNCT
cana-5487	93	24	,	,	PUNCT
cana-5487	93	25	𝑞𝑡	𝑞𝑡	PRON
cana-5487	93	26	)	)	PUNCT
cana-5487	93	27	≥	≥	NOUN
cana-5487	93	28	𝜑	𝜑	NOUN
cana-5487	93	29	[	[	PUNCT
cana-5487	93	30	𝑎	𝑎	PROPN
cana-5487	93	31	𝑀𝑝(𝑆𝑎	𝑀𝑝(𝑆𝑎	ADP
cana-5487	93	32	,	,	PUNCT
cana-5487	93	33	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	93	34	,	,	PUNCT
cana-5487	93	35	𝑡	𝑡	PROPN
cana-5487	93	36	)	)	PUNCT
cana-5487	94	1	+	+	CCONJ
cana-5487	94	2	(	(	PUNCT
cana-5487	94	3	1	1	NUM
cana-5487	94	4	−	−	NUM
cana-5487	94	5	𝑎	𝑎	NOUN
cana-5487	94	6	)	)	PUNCT
cana-5487	94	7	min	min	NOUN
cana-5487	94	8	{	{	PUNCT
cana-5487	94	9	𝑀𝑝(𝐴(𝑎	𝑀𝑝(𝐴(𝑎	PROPN
cana-5487	94	10	,	,	PUNCT
cana-5487	94	11	𝑏	𝑏	NOUN
cana-5487	94	12	,	,	PUNCT
cana-5487	94	13	𝑐	𝑐	NOUN
cana-5487	94	14	)	)	PUNCT
cana-5487	94	15	,	,	PUNCT
cana-5487	94	16	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	94	17	,	,	PUNCT
cana-5487	94	18	𝑡	𝑡	NOUN
cana-5487	94	19	)	)	PUNCT
cana-5487	94	20	,	,	PUNCT
cana-5487	94	21	𝑀𝑝(𝐵(𝑎′	𝑀𝑝(𝐵(𝑎′	PROPN
cana-5487	94	22	,	,	PUNCT
cana-5487	94	23	𝑏′	𝑏′	PROPN
cana-5487	94	24	,	,	PUNCT
cana-5487	94	25	𝑐′	𝑐′	NUM
cana-5487	94	26	)	)	PUNCT
cana-5487	94	27	,	,	PUNCT
cana-5487	94	28	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	29	,	,	PUNCT
cana-5487	94	30	𝑡	𝑡	PROPN
cana-5487	94	31	)	)	PUNCT
cana-5487	94	32	,	,	PUNCT
cana-5487	94	33	𝑀	𝑀	PROPN
cana-5487	94	34	𝑝	𝑝	PROPN
cana-5487	94	35	2(𝐴(𝑎	2(𝐴(𝑎	NUM
cana-5487	94	36	,	,	PUNCT
cana-5487	94	37	𝑏	𝑏	NOUN
cana-5487	94	38	,	,	PUNCT
cana-5487	94	39	𝑐	𝑐	NOUN
cana-5487	94	40	)	)	PUNCT
cana-5487	94	41	,	,	PUNCT
cana-5487	94	42	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	43	,	,	PUNCT
cana-5487	94	44	𝑡).𝑀	𝑡).𝑀	PROPN
cana-5487	94	45	𝑝	𝑝	ADP
cana-5487	94	46	2(𝐵(𝑎′	2(𝐵(𝑎′	NUM
cana-5487	94	47	,	,	PUNCT
cana-5487	94	48	𝑏′	𝑏′	PROPN
cana-5487	94	49	,	,	PUNCT
cana-5487	94	50	𝑐′	𝑐′	NUM
cana-5487	94	51	)	)	PUNCT
cana-5487	94	52	,	,	PUNCT
cana-5487	94	53	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	94	54	,	,	PUNCT
cana-5487	94	55	𝑡	𝑡	NOUN
cana-5487	94	56	)	)	PUNCT
cana-5487	94	57	,	,	PUNCT
cana-5487	94	58	1	1	NUM
cana-5487	94	59	2	2	NUM
cana-5487	94	60	[	[	X
cana-5487	94	61	𝑀𝑝(𝐴(𝑎	𝑀𝑝(𝐴(𝑎	NOUN
cana-5487	94	62	,	,	PUNCT
cana-5487	94	63	𝑏	𝑏	NOUN
cana-5487	94	64	,	,	PUNCT
cana-5487	94	65	𝑐	𝑐	NOUN
cana-5487	94	66	)	)	PUNCT
cana-5487	94	67	,	,	PUNCT
cana-5487	94	68	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	94	69	,	,	PUNCT
cana-5487	94	70	𝑡	𝑡	NOUN
cana-5487	94	71	)	)	PUNCT
cana-5487	94	72	+	+	CCONJ
cana-5487	94	73	𝑀𝑝(𝐵(𝑎′	𝑀𝑝(𝐵(𝑎′	PROPN
cana-5487	94	74	,	,	PUNCT
cana-5487	94	75	𝑏′	𝑏′	PROPN
cana-5487	94	76	,	,	PUNCT
cana-5487	94	77	𝑐′	𝑐′	NUM
cana-5487	94	78	)	)	PUNCT
cana-5487	94	79	,	,	PUNCT
cana-5487	94	80	𝑇𝑎′	𝑇𝑎′	PROPN
cana-5487	94	81	,	,	PUNCT
cana-5487	94	82	𝑡	𝑡	PROPN
cana-5487	94	83	)	)	PUNCT
cana-5487	94	84	]	]	PUNCT
cana-5487	94	85	}	}	PUNCT
cana-5487	94	86	]	]	PUNCT
cana-5487	94	87	=	=	PUNCT
cana-5487	94	88	𝜑(𝑎	𝜑(𝑎	PROPN
cana-5487	94	89	𝑀𝑝(𝑆𝑎	𝑀𝑝(𝑆𝑎	PROPN
cana-5487	94	90	,	,	PUNCT
cana-5487	94	91	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	92	,	,	PUNCT
cana-5487	94	93	𝑡	𝑡	PROPN
cana-5487	94	94	)	)	PUNCT
cana-5487	94	95	+	+	CCONJ
cana-5487	94	96	(	(	PUNCT
cana-5487	94	97	1	1	NUM
cana-5487	94	98	−	−	PROPN
cana-5487	94	99	𝑎)min	𝑎)min	PROPN
cana-5487	94	100	{	{	PUNCT
cana-5487	94	101	1,1	1,1	NUM
cana-5487	94	102	,	,	PUNCT
cana-5487	94	103	𝑀𝑝(𝑆𝑎	𝑀𝑝(𝑆𝑎	ADP
cana-5487	94	104	,	,	PUNCT
cana-5487	94	105	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	106	,	,	PUNCT
cana-5487	94	107	𝑡	𝑡	PROPN
cana-5487	94	108	)	)	PUNCT
cana-5487	94	109	,	,	PUNCT
cana-5487	94	110	1	1	X
cana-5487	94	111	}	}	PUNCT
cana-5487	94	112	=	=	SYM
cana-5487	94	113	𝜑(𝑎	𝜑(𝑎	PROPN
cana-5487	94	114	𝑀𝑝(𝑆𝑎	𝑀𝑝(𝑆𝑎	PROPN
cana-5487	94	115	,	,	PUNCT
cana-5487	94	116	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	117	,	,	PUNCT
cana-5487	94	118	𝑡	𝑡	PROPN
cana-5487	94	119	)	)	PUNCT
cana-5487	94	120	+	+	CCONJ
cana-5487	94	121	(	(	PUNCT
cana-5487	94	122	1	1	NUM
cana-5487	94	123	−	−	PROPN
cana-5487	94	124	𝑎)min	𝑎)min	PROPN
cana-5487	94	125	𝑀𝑝(𝑆𝑎	𝑀𝑝(𝑆𝑎	PROPN
cana-5487	94	126	,	,	PUNCT
cana-5487	94	127	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	128	,	,	PUNCT
cana-5487	94	129	𝑡	𝑡	PROPN
cana-5487	94	130	)	)	PUNCT
cana-5487	94	131	>	>	X
cana-5487	94	132	𝑀(𝑆𝑎	𝑀(𝑆𝑎	PROPN
cana-5487	94	133	,	,	PUNCT
cana-5487	94	134	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	135	,	,	PUNCT
cana-5487	94	136	𝑡	𝑡	NOUN
cana-5487	94	137	)	)	PUNCT
cana-5487	94	138	⇒	⇒	VERB
cana-5487	94	139	𝑆𝑎	𝑆𝑎	NOUN
cana-5487	94	140	=	=	SYM
cana-5487	94	141	𝑇𝑎′	𝑇𝑎′	PROPN
cana-5487	94	142	therefore	therefore	ADV
cana-5487	94	143	𝐴(𝑎	𝐴(𝑎	PROPN
cana-5487	94	144	,	,	PUNCT
cana-5487	94	145	𝑏	𝑏	NOUN
cana-5487	94	146	,	,	PUNCT
cana-5487	94	147	𝑐	𝑐	NOUN
cana-5487	94	148	)	)	PUNCT
cana-5487	94	149	=	=	SYM
cana-5487	94	150	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	94	151	=	=	PUNCT
cana-5487	94	152	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	94	153	=	=	PUNCT
cana-5487	94	154	𝐵(𝑎′	𝐵(𝑎′	PROPN
cana-5487	94	155	,	,	PUNCT
cana-5487	94	156	𝑏′	𝑏′	PROPN
cana-5487	94	157	,	,	PUNCT
cana-5487	94	158	𝑐′	𝑐′	NUM
cana-5487	94	159	)	)	PUNCT
cana-5487	94	160	similarly	similarly	ADV
cana-5487	94	161	𝐴(𝑏	𝐴(𝑏	PROPN
cana-5487	94	162	,	,	PUNCT
cana-5487	94	163	𝑐	𝑐	NOUN
cana-5487	94	164	,	,	PUNCT
cana-5487	94	165	𝑎	𝑎	NOUN
cana-5487	94	166	)	)	PUNCT
cana-5487	94	167	=	=	SYM
cana-5487	94	168	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	95	1	=	=	PUNCT
cana-5487	95	2	𝑆𝑏	𝑆𝑏	INTJ
cana-5487	95	3	=	=	NOUN
cana-5487	95	4	𝐵(𝑏′	𝐵(𝑏′	PROPN
cana-5487	95	5	,	,	PUNCT
cana-5487	95	6	𝑐′	𝑐′	NUM
cana-5487	95	7	,	,	PUNCT
cana-5487	95	8	𝑎′	𝑎′	NUM
cana-5487	95	9	)	)	PUNCT
cana-5487	95	10	𝐴(𝑐	𝐴(𝑐	NOUN
cana-5487	95	11	,	,	PUNCT
cana-5487	95	12	𝑎	𝑎	NOUN
cana-5487	95	13	,	,	PUNCT
cana-5487	95	14	𝑏	𝑏	NOUN
cana-5487	95	15	)	)	PUNCT
cana-5487	95	16	=	=	SYM
cana-5487	95	17	𝑇𝑐′	𝑇𝑐′	NOUN
cana-5487	95	18	=	=	PUNCT
cana-5487	95	19	𝑆𝑐	𝑆𝑐	PROPN
cana-5487	95	20	=	=	SYM
cana-5487	95	21	𝐵(𝑐′	𝐵(𝑐′	PROPN
cana-5487	95	22	,	,	PUNCT
cana-5487	95	23	𝑎′	𝑎′	NUM
cana-5487	95	24	,	,	PUNCT
cana-5487	95	25	𝑏′	𝑏′	NUM
cana-5487	95	26	)	)	PUNCT
cana-5487	95	27	thus	thus	ADV
cana-5487	95	28	the	the	DET
cana-5487	95	29	pairs	pair	NOUN
cana-5487	95	30	(	(	PUNCT
cana-5487	95	31	𝐴	𝐴	PROPN
cana-5487	95	32	,	,	PUNCT
cana-5487	95	33	𝑆)and(𝐵	𝑆)and(𝐵	PROPN
cana-5487	95	34	,	,	PUNCT
cana-5487	95	35	𝑇)have	𝑇)have	VERB
cana-5487	95	36	common	common	ADJ
cana-5487	95	37	coincidence	coincidence	NOUN
cana-5487	95	38	points	point	NOUN
cana-5487	95	39	.	.	PUNCT
cana-5487	96	1	let	let	VERB
cana-5487	96	2	𝐴(𝑎	𝐴(𝑎	PRON
cana-5487	96	3	,	,	PUNCT
cana-5487	96	4	𝑏	𝑏	NOUN
cana-5487	96	5	,	,	PUNCT
cana-5487	96	6	𝑐	𝑐	NOUN
cana-5487	96	7	)	)	PUNCT
cana-5487	96	8	=	=	SYM
cana-5487	97	1	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	97	2	=	=	PUNCT
cana-5487	97	3	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	97	4	=	=	SYM
cana-5487	97	5	𝐵(𝑎′	𝐵(𝑎′	PROPN
cana-5487	97	6	,	,	PUNCT
cana-5487	97	7	𝑏′	𝑏′	PROPN
cana-5487	97	8	,	,	PUNCT
cana-5487	97	9	𝑐′	𝑐′	NUM
cana-5487	97	10	)	)	PUNCT
cana-5487	97	11	=	=	SYM
cana-5487	98	1	𝑥	𝑥	PROPN
cana-5487	98	2	and	and	CCONJ
cana-5487	98	3	𝐴(𝑏	𝐴(𝑏	PROPN
cana-5487	98	4	,	,	PUNCT
cana-5487	98	5	𝑐	𝑐	PROPN
cana-5487	98	6	,	,	PUNCT
cana-5487	98	7	𝑎	𝑎	NOUN
cana-5487	98	8	)	)	PUNCT
cana-5487	98	9	=	=	SYM
cana-5487	99	1	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	100	1	=	=	PUNCT
cana-5487	100	2	𝑆𝑏	𝑆𝑏	INTJ
cana-5487	100	3	=	=	NOUN
cana-5487	100	4	𝐵(𝑏′	𝐵(𝑏′	PROPN
cana-5487	100	5	,	,	PUNCT
cana-5487	100	6	𝑐′	𝑐′	NUM
cana-5487	100	7	,	,	PUNCT
cana-5487	100	8	𝑎′	𝑎′	NUM
cana-5487	100	9	)	)	PUNCT
cana-5487	100	10	=	=	SYM
cana-5487	100	11	𝑦	𝑦	PROPN
cana-5487	100	12	𝐴(𝑐	𝐴(𝑐	PROPN
cana-5487	100	13	,	,	PUNCT
cana-5487	100	14	𝑎	𝑎	NOUN
cana-5487	100	15	,	,	PUNCT
cana-5487	100	16	𝑏	𝑏	NOUN
cana-5487	100	17	)	)	PUNCT
cana-5487	100	18	=	=	SYM
cana-5487	100	19	𝑇𝑐′	𝑇𝑐′	NOUN
cana-5487	100	20	=	=	PUNCT
cana-5487	100	21	𝑆𝑐	𝑆𝑐	PROPN
cana-5487	100	22	=	=	SYM
cana-5487	100	23	𝐵(𝑐′	𝐵(𝑐′	PROPN
cana-5487	100	24	,	,	PUNCT
cana-5487	100	25	𝑎′	𝑎′	NUM
cana-5487	100	26	,	,	PUNCT
cana-5487	100	27	𝑏′	𝑏′	NUM
cana-5487	100	28	)	)	PUNCT
cana-5487	100	29	=	=	PUNCT
cana-5487	101	1	𝑧	𝑧	DET
cana-5487	101	2	since(𝐴	since(𝐴	PROPN
cana-5487	101	3	,	,	PUNCT
cana-5487	101	4	𝑆	𝑆	PROPN
cana-5487	101	5	)	)	PUNCT
cana-5487	101	6	and	and	CCONJ
cana-5487	101	7	(	(	PUNCT
cana-5487	101	8	𝐵	𝐵	PROPN
cana-5487	101	9	,	,	PUNCT
cana-5487	101	10	𝑇	𝑇	PROPN
cana-5487	101	11	)	)	PUNCT
cana-5487	101	12	are	be	AUX
cana-5487	101	13	owc	owc	NOUN
cana-5487	101	14	so	so	ADV
cana-5487	101	15	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	101	16	=	=	PUNCT
cana-5487	101	17	𝑆𝐴(𝑎	𝑆𝐴(𝑎	NOUN
cana-5487	101	18	,	,	PUNCT
cana-5487	101	19	𝑏	𝑏	NOUN
cana-5487	101	20	,	,	PUNCT
cana-5487	101	21	𝑐	𝑐	NOUN
cana-5487	101	22	)	)	PUNCT
cana-5487	102	1	=	=	NOUN
cana-5487	102	2	𝐴(𝑆𝑎	𝐴(𝑆𝑎	NOUN
cana-5487	102	3	,	,	PUNCT
cana-5487	102	4	𝑆𝑏	𝑆𝑏	PROPN
cana-5487	102	5	,	,	PUNCT
cana-5487	102	6	𝑆𝑐	𝑆𝑐	PROPN
cana-5487	102	7	)	)	PUNCT
cana-5487	102	8	=	=	PUNCT
cana-5487	102	9	𝐴(𝑥	𝐴(𝑥	NOUN
cana-5487	102	10	,	,	PUNCT
cana-5487	102	11	𝑦	𝑦	NOUN
cana-5487	102	12	,	,	PUNCT
cana-5487	102	13	𝑧	𝑧	NOUN
cana-5487	102	14	)	)	PUNCT
cana-5487	102	15	and	and	CCONJ
cana-5487	102	16	𝑆𝑦	𝑆𝑦	PROPN
cana-5487	102	17	=	=	PUNCT
cana-5487	102	18	𝑆𝐴(𝑏	𝑆𝐴(𝑏	PROPN
cana-5487	102	19	,	,	PUNCT
cana-5487	102	20	𝑐	𝑐	NOUN
cana-5487	102	21	,	,	PUNCT
cana-5487	102	22	𝑎	𝑎	NOUN
cana-5487	102	23	)	)	PUNCT
cana-5487	102	24	=	=	SYM
cana-5487	102	25	𝐴(𝑆𝑏	𝐴(𝑆𝑏	NOUN
cana-5487	102	26	,	,	PUNCT
cana-5487	102	27	𝑆𝑐	𝑆𝑐	PROPN
cana-5487	102	28	,	,	PUNCT
cana-5487	102	29	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	102	30	)	)	PUNCT
cana-5487	102	31	=	=	SYM
cana-5487	102	32	𝐴(𝑦	𝐴(𝑦	X
cana-5487	102	33	,	,	PUNCT
cana-5487	102	34	𝑧	𝑧	NOUN
cana-5487	102	35	,	,	PUNCT
cana-5487	102	36	𝑥	𝑥	NOUN
cana-5487	102	37	)	)	PUNCT
cana-5487	102	38	𝑆𝑧	𝑆𝑧	NOUN
cana-5487	102	39	=	=	PUNCT
cana-5487	102	40	𝑆𝐴(𝑐	𝑆𝐴(𝑐	PROPN
cana-5487	102	41	,	,	PUNCT
cana-5487	102	42	𝑎	𝑎	NOUN
cana-5487	102	43	,	,	PUNCT
cana-5487	102	44	𝑏	𝑏	NOUN
cana-5487	102	45	)	)	PUNCT
cana-5487	102	46	=	=	SYM
cana-5487	102	47	𝐴(𝑆𝑐	𝐴(𝑆𝑐	PROPN
cana-5487	102	48	,	,	PUNCT
cana-5487	102	49	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	102	50	,	,	PUNCT
cana-5487	102	51	𝑆𝑏	𝑆𝑏	PROPN
cana-5487	102	52	)	)	PUNCT
cana-5487	102	53	=	=	SYM
cana-5487	102	54	𝐴(𝑧	𝐴(𝑧	X
cana-5487	102	55	,	,	PUNCT
cana-5487	102	56	𝑥	𝑥	NOUN
cana-5487	102	57	,	,	PUNCT
cana-5487	102	58	𝑦	𝑦	NOUN
cana-5487	102	59	)	)	PUNCT
cana-5487	102	60	also	also	ADV
cana-5487	102	61	𝑇𝑥	𝑇𝑥	PROPN
cana-5487	102	62	=	=	SYM
cana-5487	102	63	𝑇𝐵(𝑎′	𝑇𝐵(𝑎′	PROPN
cana-5487	102	64	,	,	PUNCT
cana-5487	102	65	𝑏′	𝑏′	PROPN
cana-5487	102	66	,	,	PUNCT
cana-5487	102	67	𝑐′	𝑐′	NUM
cana-5487	102	68	)	)	PUNCT
cana-5487	103	1	=	=	SYM
cana-5487	103	2	𝐵(𝑇𝑎′	𝐵(𝑇𝑎′	NOUN
cana-5487	103	3	,	,	PUNCT
cana-5487	103	4	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	103	5	,	,	PUNCT
cana-5487	103	6	𝑇𝑐′	𝑇𝑐′	ADJ
cana-5487	103	7	)	)	PUNCT
cana-5487	103	8	=	=	SYM
cana-5487	103	9	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	103	10	,	,	PUNCT
cana-5487	103	11	𝑦	𝑦	NOUN
cana-5487	103	12	,	,	PUNCT
cana-5487	103	13	𝑧	𝑧	NOUN
cana-5487	103	14	)	)	PUNCT
cana-5487	104	1	𝑇𝑧	𝑇𝑧	PROPN
cana-5487	104	2	=	=	SYM
cana-5487	104	3	𝑇𝐵(𝑐′	𝑇𝐵(𝑐′	PROPN
cana-5487	104	4	,	,	PUNCT
cana-5487	104	5	𝑎′	𝑎′	PROPN
cana-5487	104	6	,	,	PUNCT
cana-5487	104	7	𝑏′	𝑏′	NUM
cana-5487	104	8	)	)	PUNCT
cana-5487	104	9	=	=	SYM
cana-5487	105	1	𝐵(𝑇𝑐′	𝐵(𝑇𝑐′	PROPN
cana-5487	105	2	,	,	PUNCT
cana-5487	105	3	𝑇𝑎′	𝑇𝑎′	NOUN
cana-5487	105	4	,	,	PUNCT
cana-5487	105	5	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	105	6	)	)	PUNCT
cana-5487	105	7	=	=	SYM
cana-5487	106	1	𝐵(𝑧	𝐵(𝑧	NUM
cana-5487	106	2	,	,	PUNCT
cana-5487	106	3	𝑥	𝑥	NOUN
cana-5487	106	4	,	,	PUNCT
cana-5487	106	5	𝑦	𝑦	NOUN
cana-5487	106	6	)	)	PUNCT
cana-5487	106	7	next	next	ADV
cana-5487	106	8	we	we	PRON
cana-5487	106	9	show	show	VERB
cana-5487	106	10	that	that	SCONJ
cana-5487	106	11	𝑥	𝑥	NOUN
cana-5487	106	12	=	=	PUNCT
cana-5487	106	13	𝑦	𝑦	SYM
cana-5487	106	14	=	=	SYM
cana-5487	106	15	𝑧	𝑧	ADJ
cana-5487	106	16	,	,	PUNCT
cana-5487	106	17	for	for	ADP
cana-5487	106	18	this	this	DET
cana-5487	106	19	putting	put	VERB
cana-5487	106	20	𝑥	𝑥	X
cana-5487	106	21	=	=	SYM
cana-5487	106	22	𝑎	𝑎	NOUN
cana-5487	106	23	,	,	PUNCT
cana-5487	106	24	𝑦	𝑦	NOUN
cana-5487	106	25	=	=	SYM
cana-5487	106	26	𝑏	𝑏	PROPN
cana-5487	106	27	,	,	PUNCT
cana-5487	106	28	𝑧	𝑧	PROPN
cana-5487	106	29	=	=	ADJ
cana-5487	106	30	𝑐	𝑐	PROPN
cana-5487	106	31	,	,	PUNCT
cana-5487	106	32	𝑢	𝑢	X
cana-5487	106	33	=	=	SYM
cana-5487	106	34	𝑏′	𝑏′	PROPN
cana-5487	106	35	,	,	PUNCT
cana-5487	106	36	𝑣	𝑣	X
cana-5487	106	37	=	=	PUNCT
cana-5487	106	38	𝑐′,𝑤	𝑐′,𝑤	NOUN
cana-5487	106	39	=	=	PUNCT
cana-5487	106	40	𝑎′	𝑎′	X
cana-5487	106	41	in	in	ADP
cana-5487	106	42	(	(	PUNCT
cana-5487	106	43	i	i	NOUN
cana-5487	106	44	)	)	PUNCT
cana-5487	106	45	,	,	PUNCT
cana-5487	106	46	communications	communication	NOUN
cana-5487	106	47	on	on	ADP
cana-5487	106	48	applied	apply	VERB
cana-5487	106	49	nonlinear	nonlinear	ADJ
cana-5487	106	50	analysis	analysis	NOUN
cana-5487	106	51	issn	issn	NOUN
cana-5487	106	52	:	:	PUNCT
cana-5487	106	53	1074	1074	NUM
cana-5487	106	54	-	-	PUNCT
cana-5487	106	55	133x	133x	NUM
cana-5487	106	56	vol	vol	VERB
cana-5487	106	57	32	32	NUM
cana-5487	106	58	no	no	NOUN
cana-5487	106	59	.	.	PUNCT
cana-5487	107	1	10s	10	NOUN
cana-5487	107	2	(	(	PUNCT
cana-5487	107	3	2025	2025	NUM
cana-5487	107	4	)	)	PUNCT
cana-5487	107	5	2388	2388	NUM
cana-5487	107	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	107	7	𝑀𝑃(𝐴(𝑎	𝑀𝑃(𝐴(𝑎	X
cana-5487	107	8	,	,	PUNCT
cana-5487	107	9	𝑏	𝑏	NOUN
cana-5487	107	10	,	,	PUNCT
cana-5487	107	11	𝑐	𝑐	NOUN
cana-5487	107	12	)	)	PUNCT
cana-5487	107	13	,	,	PUNCT
cana-5487	107	14	𝐵(𝑏′	𝐵(𝑏′	NOUN
cana-5487	107	15	,	,	PUNCT
cana-5487	107	16	𝑐′	𝑐′	NUM
cana-5487	107	17	,	,	PUNCT
cana-5487	107	18	𝑎′	𝑎′	NUM
cana-5487	107	19	)	)	PUNCT
cana-5487	107	20	,	,	PUNCT
cana-5487	107	21	𝑞𝑡	𝑞𝑡	PRON
cana-5487	107	22	)	)	PUNCT
cana-5487	107	23	≥	≥	NOUN
cana-5487	107	24	𝜑	𝜑	NOUN
cana-5487	107	25	[	[	PUNCT
cana-5487	107	26	𝑎	𝑎	PROPN
cana-5487	107	27	𝑀𝑝(𝑆𝑎	𝑀𝑝(𝑆𝑎	ADP
cana-5487	107	28	,	,	PUNCT
cana-5487	107	29	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	107	30	,	,	PUNCT
cana-5487	107	31	𝑡	𝑡	X
cana-5487	107	32	)	)	PUNCT
cana-5487	107	33	+	+	CCONJ
cana-5487	107	34	(	(	PUNCT
cana-5487	107	35	1	1	NUM
cana-5487	107	36	−	−	NUM
cana-5487	107	37	𝑎	𝑎	NOUN
cana-5487	107	38	)	)	PUNCT
cana-5487	107	39	min	min	NOUN
cana-5487	107	40	{	{	PUNCT
cana-5487	107	41	𝑀𝑝(𝐴(𝑎	𝑀𝑝(𝐴(𝑎	PROPN
cana-5487	107	42	,	,	PUNCT
cana-5487	107	43	𝑏	𝑏	NOUN
cana-5487	107	44	,	,	PUNCT
cana-5487	107	45	𝑐	𝑐	NOUN
cana-5487	107	46	)	)	PUNCT
cana-5487	107	47	,	,	PUNCT
cana-5487	107	48	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	107	49	,	,	PUNCT
cana-5487	107	50	𝑡	𝑡	NOUN
cana-5487	107	51	)	)	PUNCT
cana-5487	107	52	,	,	PUNCT
cana-5487	107	53	𝑀𝑝(𝐵(𝑏′	𝑀𝑝(𝐵(𝑏′	PROPN
cana-5487	107	54	,	,	PUNCT
cana-5487	107	55	𝑐′	𝑐′	NUM
cana-5487	107	56	,	,	PUNCT
cana-5487	107	57	𝑎′	𝑎′	NUM
cana-5487	107	58	)	)	PUNCT
cana-5487	107	59	,	,	PUNCT
cana-5487	107	60	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	107	61	,	,	PUNCT
cana-5487	107	62	𝑡),𝑀	𝑡),𝑀	PUNCT
cana-5487	107	63	𝑝	𝑝	NOUN
cana-5487	107	64	2(𝐴(𝑎	2(𝐴(𝑎	NUM
cana-5487	107	65	,	,	PUNCT
cana-5487	107	66	𝑏	𝑏	NOUN
cana-5487	107	67	,	,	PUNCT
cana-5487	107	68	𝑐	𝑐	NOUN
cana-5487	107	69	)	)	PUNCT
cana-5487	107	70	,	,	PUNCT
cana-5487	107	71	𝑇𝑏′	𝑇𝑏′	PROPN
cana-5487	107	72	,	,	PUNCT
cana-5487	107	73	𝑡).𝑀	𝑡).𝑀	PROPN
cana-5487	107	74	𝑝	𝑝	NOUN
cana-5487	107	75	2(𝐵(𝑏′	2(𝐵(𝑏′	NUM
cana-5487	107	76	,	,	PUNCT
cana-5487	107	77	𝑐′	𝑐′	NUM
cana-5487	107	78	,	,	PUNCT
cana-5487	107	79	𝑎′	𝑎′	NUM
cana-5487	107	80	)	)	PUNCT
cana-5487	107	81	,	,	PUNCT
cana-5487	107	82	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	107	83	,	,	PUNCT
cana-5487	107	84	𝑡	𝑡	NOUN
cana-5487	107	85	)	)	PUNCT
cana-5487	107	86	,	,	PUNCT
cana-5487	107	87	1	1	NUM
cana-5487	107	88	2	2	NUM
cana-5487	107	89	[	[	X
cana-5487	107	90	𝑀𝑝(𝐴(𝑎	𝑀𝑝(𝐴(𝑎	NOUN
cana-5487	107	91	,	,	PUNCT
cana-5487	107	92	𝑏	𝑏	NOUN
cana-5487	107	93	,	,	PUNCT
cana-5487	107	94	𝑐	𝑐	NOUN
cana-5487	107	95	)	)	PUNCT
cana-5487	107	96	,	,	PUNCT
cana-5487	107	97	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	107	98	,	,	PUNCT
cana-5487	107	99	𝑡	𝑡	NOUN
cana-5487	107	100	)	)	PUNCT
cana-5487	107	101	+	+	CCONJ
cana-5487	107	102	𝑀𝑝(𝐵(𝑏′	𝑀𝑝(𝐵(𝑏′	PROPN
cana-5487	107	103	,	,	PUNCT
cana-5487	107	104	𝑐′	𝑐′	NUM
cana-5487	107	105	,	,	PUNCT
cana-5487	107	106	𝑎′	𝑎′	NUM
cana-5487	107	107	)	)	PUNCT
cana-5487	107	108	,	,	PUNCT
cana-5487	107	109	𝑇𝑏′	𝑇𝑏′	NOUN
cana-5487	107	110	,	,	PUNCT
cana-5487	107	111	𝑡	𝑡	NOUN
cana-5487	107	112	)	)	PUNCT
cana-5487	107	113	]	]	PUNCT
cana-5487	107	114	}	}	PUNCT
cana-5487	107	115	]	]	PUNCT
cana-5487	107	116	𝑀𝑝(𝑥	𝑀𝑝(𝑥	NOUN
cana-5487	107	117	,	,	PUNCT
cana-5487	107	118	𝑦	𝑦	NOUN
cana-5487	107	119	,	,	PUNCT
cana-5487	107	120	𝑞𝑡	𝑞𝑡	PRON
cana-5487	107	121	)	)	PUNCT
cana-5487	107	122	≥	≥	NOUN
cana-5487	107	123	𝜑(𝑎	𝜑(𝑎	PROPN
cana-5487	107	124	𝑀𝑝(𝑥	𝑀𝑝(𝑥	NOUN
cana-5487	107	125	,	,	PUNCT
cana-5487	107	126	𝑦	𝑦	NOUN
cana-5487	107	127	,	,	PUNCT
cana-5487	107	128	𝑡	𝑡	NOUN
cana-5487	107	129	)	)	PUNCT
cana-5487	107	130	+	+	CCONJ
cana-5487	107	131	(	(	PUNCT
cana-5487	107	132	1	1	NUM
cana-5487	107	133	−	−	NUM
cana-5487	107	134	𝑎)𝑚𝑖𝑛{1,1,𝑀𝑝(𝑥	𝑎)𝑚𝑖𝑛{1,1,𝑀𝑝(𝑥	NOUN
cana-5487	107	135	,	,	PUNCT
cana-5487	107	136	𝑦	𝑦	NOUN
cana-5487	107	137	,	,	PUNCT
cana-5487	107	138	𝑡	𝑡	NOUN
cana-5487	107	139	)	)	PUNCT
cana-5487	107	140	,	,	PUNCT
cana-5487	107	141	1	1	X
cana-5487	107	142	}	}	PUNCT
cana-5487	107	143	=	=	PUNCT
cana-5487	107	144	𝜑𝑀𝑝(𝑥	𝜑𝑀𝑝(𝑥	ADP
cana-5487	107	145	,	,	PUNCT
cana-5487	107	146	𝑦	𝑦	NOUN
cana-5487	107	147	,	,	PUNCT
cana-5487	107	148	𝑡	𝑡	NOUN
cana-5487	107	149	)	)	PUNCT
cana-5487	107	150	>	>	X
cana-5487	108	1	𝑀(𝑥	𝑀(𝑥	PROPN
cana-5487	108	2	,	,	PUNCT
cana-5487	108	3	𝑦	𝑦	NOUN
cana-5487	108	4	,	,	PUNCT
cana-5487	108	5	𝑡	𝑡	PROPN
cana-5487	108	6	)	)	PUNCT
cana-5487	108	7	⟹	⟹	AUX
cana-5487	109	1	𝑥	𝑥	NOUN
cana-5487	109	2	=	=	SYM
cana-5487	109	3	𝑦	𝑦	NOUN
cana-5487	109	4	again	again	ADV
cana-5487	109	5	putting	put	VERB
cana-5487	109	6	𝑥	𝑥	NOUN
cana-5487	109	7	=	=	SYM
cana-5487	109	8	𝑎	𝑎	NOUN
cana-5487	109	9	,	,	PUNCT
cana-5487	109	10	𝑦	𝑦	NOUN
cana-5487	109	11	=	=	SYM
cana-5487	109	12	𝑏	𝑏	PROPN
cana-5487	109	13	,	,	PUNCT
cana-5487	109	14	𝑧	𝑧	PROPN
cana-5487	109	15	=	=	ADJ
cana-5487	109	16	𝑐	𝑐	PROPN
cana-5487	109	17	,	,	PUNCT
cana-5487	109	18	𝑢	𝑢	X
cana-5487	109	19	=	=	X
cana-5487	109	20	𝑐′	𝑐′	NOUN
cana-5487	109	21	,	,	PUNCT
cana-5487	109	22	𝑣	𝑣	NOUN
cana-5487	109	23	=	=	NOUN
cana-5487	109	24	𝑎′,𝑤	𝑎′,𝑤	NOUN
cana-5487	109	25	=	=	SYM
cana-5487	109	26	𝑏′	𝑏′	PROPN
cana-5487	109	27	in	in	ADP
cana-5487	109	28	(	(	PUNCT
cana-5487	109	29	i	i	NOUN
cana-5487	109	30	)	)	PUNCT
cana-5487	109	31	,	,	PUNCT
cana-5487	109	32	𝑀𝑃(𝐴(𝑎	𝑀𝑃(𝐴(𝑎	ADV
cana-5487	109	33	,	,	PUNCT
cana-5487	109	34	𝑏	𝑏	NOUN
cana-5487	109	35	,	,	PUNCT
cana-5487	109	36	𝑐	𝑐	NOUN
cana-5487	109	37	)	)	PUNCT
cana-5487	109	38	,	,	PUNCT
cana-5487	109	39	𝐵(𝑐′	𝐵(𝑐′	PROPN
cana-5487	109	40	,	,	PUNCT
cana-5487	109	41	𝑎′	𝑎′	NUM
cana-5487	109	42	,	,	PUNCT
cana-5487	109	43	𝑏′	𝑏′	NUM
cana-5487	109	44	)	)	PUNCT
cana-5487	109	45	,	,	PUNCT
cana-5487	109	46	𝑞𝑡	𝑞𝑡	PRON
cana-5487	109	47	)	)	PUNCT
cana-5487	109	48	≥	≥	NOUN
cana-5487	109	49	𝜑	𝜑	NOUN
cana-5487	109	50	[	[	PUNCT
cana-5487	109	51	𝑎	𝑎	PROPN
cana-5487	109	52	𝑀𝑝(𝑆𝑎	𝑀𝑝(𝑆𝑎	ADP
cana-5487	109	53	,	,	PUNCT
cana-5487	109	54	𝑇𝑐′	𝑇𝑐′	ADJ
cana-5487	109	55	,	,	PUNCT
cana-5487	109	56	𝑡	𝑡	NOUN
cana-5487	109	57	)	)	PUNCT
cana-5487	109	58	+	+	CCONJ
cana-5487	109	59	(	(	PUNCT
cana-5487	109	60	1	1	NUM
cana-5487	109	61	−	−	NUM
cana-5487	109	62	𝑎	𝑎	NOUN
cana-5487	109	63	)	)	PUNCT
cana-5487	109	64	min	min	NOUN
cana-5487	109	65	{	{	PUNCT
cana-5487	109	66	𝑀𝑝(𝐴(𝑎	𝑀𝑝(𝐴(𝑎	PROPN
cana-5487	109	67	,	,	PUNCT
cana-5487	109	68	𝑏	𝑏	NOUN
cana-5487	109	69	,	,	PUNCT
cana-5487	109	70	𝑐	𝑐	NOUN
cana-5487	109	71	)	)	PUNCT
cana-5487	109	72	,	,	PUNCT
cana-5487	109	73	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	109	74	,	,	PUNCT
cana-5487	109	75	𝑡	𝑡	NOUN
cana-5487	109	76	)	)	PUNCT
cana-5487	109	77	,	,	PUNCT
cana-5487	109	78	𝑀𝑝(𝐵(𝑐′	𝑀𝑝(𝐵(𝑐′	PROPN
cana-5487	109	79	,	,	PUNCT
cana-5487	109	80	𝑎′	𝑎′	NUM
cana-5487	109	81	,	,	PUNCT
cana-5487	109	82	𝑏′	𝑏′	NUM
cana-5487	109	83	)	)	PUNCT
cana-5487	109	84	,	,	PUNCT
cana-5487	109	85	𝑇𝑐′	𝑇𝑐′	NOUN
cana-5487	109	86	,	,	PUNCT
cana-5487	109	87	𝑡),𝑀	𝑡),𝑀	PUNCT
cana-5487	109	88	𝑝	𝑝	NOUN
cana-5487	109	89	2(𝐴(𝑎	2(𝐴(𝑎	NUM
cana-5487	109	90	,	,	PUNCT
cana-5487	109	91	𝑏	𝑏	NOUN
cana-5487	109	92	,	,	PUNCT
cana-5487	109	93	𝑐	𝑐	NOUN
cana-5487	109	94	)	)	PUNCT
cana-5487	109	95	,	,	PUNCT
cana-5487	109	96	𝑇𝑐′	𝑇𝑐′	NOUN
cana-5487	109	97	,	,	PUNCT
cana-5487	109	98	𝑡).𝑀	𝑡).𝑀	PROPN
cana-5487	109	99	𝑝	𝑝	PROPN
cana-5487	109	100	2(𝐵(𝑐′	2(𝐵(𝑐′	NUM
cana-5487	109	101	,	,	PUNCT
cana-5487	109	102	𝑎′	𝑎′	NUM
cana-5487	109	103	,	,	PUNCT
cana-5487	109	104	𝑏′	𝑏′	NUM
cana-5487	109	105	)	)	PUNCT
cana-5487	109	106	,	,	PUNCT
cana-5487	109	107	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	109	108	,	,	PUNCT
cana-5487	109	109	𝑡	𝑡	NOUN
cana-5487	109	110	)	)	PUNCT
cana-5487	109	111	,	,	PUNCT
cana-5487	109	112	1	1	NUM
cana-5487	109	113	2	2	NUM
cana-5487	109	114	[	[	X
cana-5487	109	115	𝑀𝑝(𝐴(𝑎	𝑀𝑝(𝐴(𝑎	NOUN
cana-5487	109	116	,	,	PUNCT
cana-5487	109	117	𝑏	𝑏	NOUN
cana-5487	109	118	,	,	PUNCT
cana-5487	109	119	𝑐	𝑐	NOUN
cana-5487	109	120	)	)	PUNCT
cana-5487	109	121	,	,	PUNCT
cana-5487	109	122	𝑆𝑎	𝑆𝑎	PROPN
cana-5487	109	123	,	,	PUNCT
cana-5487	109	124	𝑡	𝑡	NOUN
cana-5487	109	125	)	)	PUNCT
cana-5487	109	126	+	+	CCONJ
cana-5487	109	127	𝑀𝑝(𝐵(𝑐′	𝑀𝑝(𝐵(𝑐′	PROPN
cana-5487	109	128	,	,	PUNCT
cana-5487	109	129	𝑎′	𝑎′	PRON
cana-5487	109	130	,	,	PUNCT
cana-5487	109	131	𝑏′	𝑏′	NUM
cana-5487	109	132	)	)	PUNCT
cana-5487	109	133	,	,	PUNCT
cana-5487	109	134	𝑇𝑐′	𝑇𝑐′	NOUN
cana-5487	109	135	,	,	PUNCT
cana-5487	109	136	𝑡	𝑡	PROPN
cana-5487	109	137	)	)	PUNCT
cana-5487	109	138	]	]	PUNCT
cana-5487	109	139	}	}	PUNCT
cana-5487	109	140	]	]	PUNCT
cana-5487	109	141	𝑀𝑝(𝑥	𝑀𝑝(𝑥	NOUN
cana-5487	109	142	,	,	PUNCT
cana-5487	109	143	𝑧	𝑧	NOUN
cana-5487	109	144	,	,	PUNCT
cana-5487	109	145	𝑞𝑡	𝑞𝑡	PRON
cana-5487	109	146	)	)	PUNCT
cana-5487	109	147	≥	≥	NOUN
cana-5487	109	148	𝜑(𝑎	𝜑(𝑎	PROPN
cana-5487	109	149	𝑀𝑝(𝑥	𝑀𝑝(𝑥	NOUN
cana-5487	109	150	,	,	PUNCT
cana-5487	109	151	𝑧	𝑧	NOUN
cana-5487	109	152	,	,	PUNCT
cana-5487	109	153	𝑡	𝑡	NOUN
cana-5487	109	154	)	)	PUNCT
cana-5487	109	155	+	+	CCONJ
cana-5487	109	156	(	(	PUNCT
cana-5487	109	157	1	1	NUM
cana-5487	109	158	−	−	NOUN
cana-5487	109	159	𝑎)𝑚𝑖𝑛{1,1	𝑎)𝑚𝑖𝑛{1,1	NOUN
cana-5487	109	160	,	,	PUNCT
cana-5487	109	161	𝑀𝑝(𝑥	𝑀𝑝(𝑥	NOUN
cana-5487	109	162	,	,	PUNCT
cana-5487	109	163	𝑧	𝑧	NOUN
cana-5487	109	164	,	,	PUNCT
cana-5487	109	165	𝑡	𝑡	NOUN
cana-5487	109	166	)	)	PUNCT
cana-5487	109	167	,	,	PUNCT
cana-5487	109	168	1	1	X
cana-5487	109	169	}	}	PUNCT
cana-5487	109	170	=	=	PUNCT
cana-5487	109	171	𝜑𝑀𝑝(𝑥	𝜑𝑀𝑝(𝑥	ADP
cana-5487	109	172	,	,	PUNCT
cana-5487	109	173	𝑧	𝑧	NOUN
cana-5487	109	174	,	,	PUNCT
cana-5487	109	175	𝑡	𝑡	PROPN
cana-5487	109	176	)	)	PUNCT
cana-5487	109	177	>	>	X
cana-5487	109	178	𝑀𝑝(𝑥	𝑀𝑝(𝑥	PROPN
cana-5487	109	179	,	,	PUNCT
cana-5487	109	180	𝑧	𝑧	NOUN
cana-5487	109	181	,	,	PUNCT
cana-5487	109	182	𝑡	𝑡	PROPN
cana-5487	109	183	)	)	PUNCT
cana-5487	109	184	⟹	⟹	PUNCT
cana-5487	110	1	𝑥	𝑥	NOUN
cana-5487	110	2	=	=	PUNCT
cana-5487	110	3	𝑧	𝑧	PROPN
cana-5487	110	4	⇒	⇒	NOUN
cana-5487	111	1	𝑥	𝑥	X
cana-5487	111	2	=	=	SYM
cana-5487	111	3	𝑦	𝑦	SYM
cana-5487	111	4	=	=	X
cana-5487	111	5	𝑧	𝑧	VERB
cana-5487	111	6	now	now	ADV
cana-5487	111	7	we	we	PRON
cana-5487	111	8	prove	prove	VERB
cana-5487	111	9	that	that	SCONJ
cana-5487	111	10	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	111	11	=	=	PUNCT
cana-5487	111	12	𝑇𝑥	𝑇𝑥	PROPN
cana-5487	111	13	𝑀𝑃(𝐴(𝑥	𝑀𝑃(𝐴(𝑥	SYM
cana-5487	111	14	,	,	PUNCT
cana-5487	111	15	𝑦	𝑦	NOUN
cana-5487	111	16	,	,	PUNCT
cana-5487	111	17	𝑧	𝑧	NOUN
cana-5487	111	18	)	)	PUNCT
cana-5487	111	19	,	,	PUNCT
cana-5487	111	20	𝐵(𝑦	𝐵(𝑦	ADV
cana-5487	111	21	,	,	PUNCT
cana-5487	111	22	𝑧	𝑧	NOUN
cana-5487	111	23	,	,	PUNCT
cana-5487	111	24	𝑥	𝑥	NOUN
cana-5487	111	25	)	)	PUNCT
cana-5487	111	26	,	,	PUNCT
cana-5487	111	27	𝑞𝑡	𝑞𝑡	PRON
cana-5487	111	28	)	)	PUNCT
cana-5487	111	29	≥	≥	NOUN
cana-5487	111	30	𝜑	𝜑	NOUN
cana-5487	111	31	[	[	PUNCT
cana-5487	111	32	𝑎	𝑎	PROPN
cana-5487	111	33	𝑀𝑝(𝑆𝑥	𝑀𝑝(𝑆𝑥	NOUN
cana-5487	111	34	,	,	PUNCT
cana-5487	111	35	𝑇𝑦	𝑇𝑦	PROPN
cana-5487	111	36	,	,	PUNCT
cana-5487	111	37	𝑡	𝑡	NOUN
cana-5487	111	38	)	)	PUNCT
cana-5487	112	1	+	+	CCONJ
cana-5487	112	2	(	(	PUNCT
cana-5487	112	3	1	1	NUM
cana-5487	112	4	−	−	NUM
cana-5487	112	5	𝑎	𝑎	NOUN
cana-5487	112	6	)	)	PUNCT
cana-5487	112	7	min	min	NOUN
cana-5487	112	8	{	{	PUNCT
cana-5487	112	9	𝑀𝑝(𝐴(𝑥	𝑀𝑝(𝐴(𝑥	NOUN
cana-5487	112	10	,	,	PUNCT
cana-5487	112	11	𝑦	𝑦	NOUN
cana-5487	112	12	,	,	PUNCT
cana-5487	112	13	𝑧	𝑧	NOUN
cana-5487	112	14	)	)	PUNCT
cana-5487	112	15	,	,	PUNCT
cana-5487	112	16	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	112	17	,	,	PUNCT
cana-5487	112	18	𝑡	𝑡	NOUN
cana-5487	112	19	)	)	PUNCT
cana-5487	112	20	,	,	PUNCT
cana-5487	112	21	𝑀𝑝(𝐵(𝑦	𝑀𝑝(𝐵(𝑦	PROPN
cana-5487	112	22	,	,	PUNCT
cana-5487	112	23	𝑧	𝑧	NOUN
cana-5487	112	24	,	,	PUNCT
cana-5487	112	25	𝑥	𝑥	NOUN
cana-5487	112	26	)	)	PUNCT
cana-5487	112	27	,	,	PUNCT
cana-5487	112	28	𝑇𝑦	𝑇𝑦	PROPN
cana-5487	112	29	,	,	PUNCT
cana-5487	112	30	𝑡),𝑀	𝑡),𝑀	PUNCT
cana-5487	112	31	𝑝	𝑝	NOUN
cana-5487	112	32	2(𝐴(𝑥	2(𝐴(𝑥	NUM
cana-5487	112	33	,	,	PUNCT
cana-5487	112	34	𝑦	𝑦	NOUN
cana-5487	112	35	,	,	PUNCT
cana-5487	112	36	𝑧	𝑧	NOUN
cana-5487	112	37	)	)	PUNCT
cana-5487	112	38	,	,	PUNCT
cana-5487	112	39	𝑇𝑦	𝑇𝑦	PROPN
cana-5487	112	40	,	,	PUNCT
cana-5487	112	41	𝑡).𝑀	𝑡).𝑀	PRON
cana-5487	112	42	𝑝	𝑝	NOUN
cana-5487	112	43	2(𝐵(𝑦	2(𝐵(𝑦	NUM
cana-5487	112	44	,	,	PUNCT
cana-5487	112	45	𝑧	𝑧	PROPN
cana-5487	112	46	,	,	PUNCT
cana-5487	112	47	𝑥	𝑥	NOUN
cana-5487	112	48	)	)	PUNCT
cana-5487	112	49	,	,	PUNCT
cana-5487	112	50	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	112	51	,	,	PUNCT
cana-5487	112	52	𝑡	𝑡	PROPN
cana-5487	112	53	)	)	PUNCT
cana-5487	112	54	,	,	PUNCT
cana-5487	112	55	1	1	NUM
cana-5487	112	56	2	2	NUM
cana-5487	112	57	[	[	X
cana-5487	112	58	𝑀𝑝(𝐴(𝑥	𝑀𝑝(𝐴(𝑥	NUM
cana-5487	112	59	,	,	PUNCT
cana-5487	112	60	𝑦	𝑦	NOUN
cana-5487	112	61	,	,	PUNCT
cana-5487	112	62	𝑧	𝑧	NOUN
cana-5487	112	63	)	)	PUNCT
cana-5487	112	64	,	,	PUNCT
cana-5487	112	65	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	112	66	,	,	PUNCT
cana-5487	112	67	𝑡	𝑡	NOUN
cana-5487	112	68	)	)	PUNCT
cana-5487	112	69	+	+	CCONJ
cana-5487	112	70	𝑀𝑝(𝐵(𝑦	𝑀𝑝(𝐵(𝑦	PROPN
cana-5487	112	71	,	,	PUNCT
cana-5487	112	72	𝑧	𝑧	NOUN
cana-5487	112	73	,	,	PUNCT
cana-5487	112	74	𝑥	𝑥	NOUN
cana-5487	112	75	)	)	PUNCT
cana-5487	112	76	,	,	PUNCT
cana-5487	112	77	𝑇𝑦	𝑇𝑦	PROPN
cana-5487	112	78	,	,	PUNCT
cana-5487	112	79	𝑡	𝑡	NOUN
cana-5487	112	80	)	)	PUNCT
cana-5487	112	81	]	]	PUNCT
cana-5487	112	82	}	}	PUNCT
cana-5487	112	83	]	]	PUNCT
cana-5487	112	84	𝑀𝑝(𝑆𝑥	𝑀𝑝(𝑆𝑥	X
cana-5487	112	85	,	,	PUNCT
cana-5487	112	86	𝑇𝑦	𝑇𝑦	PROPN
cana-5487	112	87	,	,	PUNCT
cana-5487	112	88	𝑞𝑡	𝑞𝑡	PRON
cana-5487	112	89	)	)	PUNCT
cana-5487	112	90	≥	≥	NOUN
cana-5487	112	91	𝜑(𝑎	𝜑(𝑎	PROPN
cana-5487	112	92	𝑀𝑝(𝑆𝑥	𝑀𝑝(𝑆𝑥	NUM
cana-5487	112	93	,	,	PUNCT
cana-5487	112	94	𝑇𝑦	𝑇𝑦	NOUN
cana-5487	112	95	,	,	PUNCT
cana-5487	112	96	𝑡	𝑡	NOUN
cana-5487	112	97	)	)	PUNCT
cana-5487	112	98	+	+	CCONJ
cana-5487	112	99	(	(	PUNCT
cana-5487	112	100	1	1	NUM
cana-5487	112	101	−	−	NOUN
cana-5487	112	102	𝑎)𝑚𝑖𝑛{1,1	𝑎)𝑚𝑖𝑛{1,1	NOUN
cana-5487	112	103	,	,	PUNCT
cana-5487	112	104	𝑀𝑝(𝑆𝑥	𝑀𝑝(𝑆𝑥	ADV
cana-5487	112	105	,	,	PUNCT
cana-5487	112	106	𝑇𝑦	𝑇𝑦	NOUN
cana-5487	112	107	,	,	PUNCT
cana-5487	112	108	𝑡	𝑡	NOUN
cana-5487	112	109	)	)	PUNCT
cana-5487	112	110	,	,	PUNCT
cana-5487	112	111	1	1	X
cana-5487	112	112	}	}	PUNCT
cana-5487	112	113	=	=	PUNCT
cana-5487	112	114	𝜑𝑀𝑝(𝑆𝑥	𝜑𝑀𝑝(𝑆𝑥	ADJ
cana-5487	112	115	,	,	PUNCT
cana-5487	112	116	𝑇𝑦	𝑇𝑦	NOUN
cana-5487	112	117	,	,	PUNCT
cana-5487	112	118	𝑡	𝑡	NOUN
cana-5487	112	119	)	)	PUNCT
cana-5487	112	120	>	>	X
cana-5487	112	121	𝑀𝑝(𝑆𝑥	𝑀𝑝(𝑆𝑥	NOUN
cana-5487	112	122	,	,	PUNCT
cana-5487	112	123	𝑇𝑦	𝑇𝑦	PROPN
cana-5487	112	124	,	,	PUNCT
cana-5487	112	125	𝑡	𝑡	NOUN
cana-5487	112	126	)	)	PUNCT
cana-5487	112	127	⟹	⟹	PUNCT
cana-5487	113	1	𝑆𝑥	𝑆𝑥	NOUN
cana-5487	113	2	=	=	PUNCT
cana-5487	113	3	𝑇𝑥	𝑇𝑥	PROPN
cana-5487	113	4	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	113	5	=	=	PUNCT
cana-5487	113	6	𝑇𝑥	𝑇𝑥	PROPN
cana-5487	113	7	=	=	SYM
cana-5487	113	8	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	113	9	,	,	PUNCT
cana-5487	113	10	𝑦	𝑦	NOUN
cana-5487	113	11	,	,	PUNCT
cana-5487	113	12	𝑧	𝑧	NOUN
cana-5487	113	13	)	)	PUNCT
cana-5487	113	14	=	=	SYM
cana-5487	113	15	𝐴(𝑥	𝐴(𝑥	NOUN
cana-5487	113	16	,	,	PUNCT
cana-5487	113	17	𝑦	𝑦	NOUN
cana-5487	113	18	,	,	PUNCT
cana-5487	113	19	𝑧	𝑧	NOUN
cana-5487	113	20	)	)	PUNCT
cana-5487	113	21	or	or	CCONJ
cana-5487	113	22	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	113	23	=	=	SYM
cana-5487	113	24	𝑇𝑥	𝑇𝑥	PROPN
cana-5487	113	25	=	=	SYM
cana-5487	113	26	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	113	27	,	,	PUNCT
cana-5487	113	28	𝑥	𝑥	NOUN
cana-5487	113	29	,	,	PUNCT
cana-5487	113	30	𝑥	𝑥	NOUN
cana-5487	113	31	)	)	PUNCT
cana-5487	113	32	=	=	PUNCT
cana-5487	114	1	𝐴(𝑥	𝐴(𝑥	NOUN
cana-5487	114	2	,	,	PUNCT
cana-5487	114	3	𝑥	𝑥	PRON
cana-5487	114	4	,	,	PUNCT
cana-5487	114	5	𝑥	𝑥	NOUN
cana-5487	114	6	)	)	PUNCT
cana-5487	114	7	communications	communication	NOUN
cana-5487	114	8	on	on	ADP
cana-5487	114	9	applied	apply	VERB
cana-5487	114	10	nonlinear	nonlinear	ADJ
cana-5487	114	11	analysis	analysis	NOUN
cana-5487	114	12	issn	issn	NOUN
cana-5487	114	13	:	:	PUNCT
cana-5487	114	14	1074	1074	NUM
cana-5487	114	15	-	-	PUNCT
cana-5487	114	16	133x	133x	NUM
cana-5487	114	17	vol	vol	VERB
cana-5487	114	18	32	32	NUM
cana-5487	114	19	no	no	NOUN
cana-5487	114	20	.	.	PUNCT
cana-5487	115	1	10s	10	NOUN
cana-5487	115	2	(	(	PUNCT
cana-5487	115	3	2025	2025	NUM
cana-5487	115	4	)	)	PUNCT
cana-5487	115	5	2389	2389	NUM
cana-5487	115	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	115	7	also	also	ADV
cana-5487	115	8	by	by	ADP
cana-5487	115	9	condition	condition	NOUN
cana-5487	115	10	(	(	PUNCT
cana-5487	115	11	ii	ii	NOUN
cana-5487	115	12	)	)	PUNCT
cana-5487	115	13	we	we	PRON
cana-5487	115	14	have	have	VERB
cana-5487	115	15	,	,	PUNCT
cana-5487	115	16	𝑥	𝑥	X
cana-5487	115	17	=	=	SYM
cana-5487	115	18	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	115	19	,	,	PUNCT
cana-5487	115	20	𝑥	𝑥	NOUN
cana-5487	115	21	,	,	PUNCT
cana-5487	115	22	𝑥	𝑥	NOUN
cana-5487	115	23	)	)	PUNCT
cana-5487	115	24	thus	thus	ADV
cana-5487	115	25	𝐴(𝑥	𝐴(𝑥	ADP
cana-5487	115	26	,	,	PUNCT
cana-5487	115	27	𝑥	𝑥	PRON
cana-5487	115	28	,	,	PUNCT
cana-5487	115	29	𝑥	𝑥	NOUN
cana-5487	115	30	)	)	PUNCT
cana-5487	115	31	=	=	SYM
cana-5487	115	32	𝑇(𝑥	𝑇(𝑥	X
cana-5487	115	33	)	)	PUNCT
cana-5487	115	34	=	=	SYM
cana-5487	116	1	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	116	2	,	,	PUNCT
cana-5487	116	3	𝑥	𝑥	NOUN
cana-5487	116	4	,	,	PUNCT
cana-5487	116	5	𝑥	𝑥	NOUN
cana-5487	116	6	)	)	PUNCT
cana-5487	116	7	=	=	SYM
cana-5487	116	8	𝑆(𝑥	𝑆(𝑥	X
cana-5487	116	9	)	)	PUNCT
cana-5487	116	10	=	=	PUNCT
cana-5487	116	11	𝑥	𝑥	PRON
cana-5487	117	1	∎	∎	PROPN
cana-5487	117	2	example	example	NOUN
cana-5487	117	3	3.1.1	3.1.1	NUM
cana-5487	117	4	let𝑋	let𝑋	NOUN
cana-5487	118	1	=	=	PUNCT
cana-5487	119	1	[	[	X
cana-5487	119	2	0,1	0,1	NUM
cana-5487	119	3	]	]	PUNCT
cana-5487	119	4	with	with	ADP
cana-5487	119	5	the	the	DET
cana-5487	119	6	metric	metric	NOUN
cana-5487	119	7	𝑑	𝑑	AUX
cana-5487	119	8	defined	define	VERB
cana-5487	119	9	by	by	ADP
cana-5487	119	10	𝑑(𝑥	𝑑(𝑥	PROPN
cana-5487	119	11	,	,	PUNCT
cana-5487	119	12	𝑦	𝑦	X
cana-5487	119	13	)	)	PUNCT
cana-5487	119	14	=	=	SYM
cana-5487	119	15	|𝑥	|𝑥	ADP
cana-5487	119	16	−	−	PROPN
cana-5487	119	17	𝑦|	𝑦|	PROPN
cana-5487	119	18	and	and	CCONJ
cana-5487	119	19	for	for	ADP
cana-5487	119	20	each	each	DET
cana-5487	119	21	𝑡	𝑡	PROPN
cana-5487	119	22	∈	∈	PROPN
cana-5487	120	1	[	[	X
cana-5487	120	2	0,1	0,1	NUM
cana-5487	120	3	]	]	PUNCT
cana-5487	120	4	,	,	PUNCT
cana-5487	120	5	define	define	VERB
cana-5487	120	6	m(x	m(x	PROPN
cana-5487	120	7	,	,	PUNCT
cana-5487	120	8	y	y	PROPN
cana-5487	120	9	,	,	PUNCT
cana-5487	120	10	t	t	PROPN
cana-5487	120	11	)	)	PUNCT
cana-5487	120	12	=	=	PRON
cana-5487	120	13	{	{	PUNCT
cana-5487	120	14	t	t	NOUN
cana-5487	120	15	t	t	PROPN
cana-5487	120	16	+	+	CCONJ
cana-5487	120	17	|x	|x	NOUN
cana-5487	120	18	−	−	VERB
cana-5487	120	19	y|	y|	NOUN
cana-5487	120	20	,	,	PUNCT
cana-5487	120	21	if	if	SCONJ
cana-5487	120	22	t	t	PROPN
cana-5487	120	23	>	>	X
cana-5487	120	24	0	0	NUM
cana-5487	120	25	;	;	PUNCT
cana-5487	120	26	0	0	NUM
cana-5487	120	27	,	,	PUNCT
cana-5487	120	28	if	if	SCONJ
cana-5487	120	29	t	t	PROPN
cana-5487	120	30	=	=	SYM
cana-5487	120	31	0	0	NUM
cana-5487	120	32	for	for	ADP
cana-5487	120	33	all	all	PRON
cana-5487	120	34	𝑥	𝑥	PROPN
cana-5487	120	35	,	,	PUNCT
cana-5487	120	36	𝑦	𝑦	PRON
cana-5487	120	37	∈	∈	PROPN
cana-5487	120	38	𝑋.	𝑋.	PROPN
cana-5487	120	39	clearly	clearly	ADV
cana-5487	120	40	(	(	PUNCT
cana-5487	120	41	x	x	X
cana-5487	120	42	,	,	PUNCT
cana-5487	120	43	ℱ,∗	ℱ,∗	NUM
cana-5487	120	44	)	)	PUNCT
cana-5487	120	45	be	be	VERB
cana-5487	120	46	a	a	DET
cana-5487	120	47	fuzzy	fuzzy	ADJ
cana-5487	120	48	metric	metric	ADJ
cana-5487	120	49	space	space	NOUN
cana-5487	120	50	,	,	PUNCT
cana-5487	120	51	with	with	ADP
cana-5487	120	52	𝑎	𝑎	DET
cana-5487	120	53	∗	∗	NOUN
cana-5487	120	54	𝑏	𝑏	NOUN
cana-5487	120	55	=	=	SYM
cana-5487	120	56	min{𝑎	min{𝑎	PROPN
cana-5487	120	57	,	,	PUNCT
cana-5487	120	58	𝑏	𝑏	NOUN
cana-5487	120	59	}	}	PUNCT
cana-5487	120	60	.	.	PUNCT
cana-5487	121	1	let	let	VERB
cana-5487	121	2	𝑆	𝑆	PROPN
cana-5487	121	3	,	,	PUNCT
cana-5487	121	4	𝑇	𝑇	PROPN
cana-5487	121	5	:	:	PUNCT
cana-5487	121	6	𝑋	𝑋	PROPN
cana-5487	121	7	→	→	SYM
cana-5487	121	8	𝑋and	𝑋and	PROPN
cana-5487	121	9	𝐴	𝐴	PROPN
cana-5487	121	10	,	,	PUNCT
cana-5487	121	11	𝐵:𝑋	𝐵:𝑋	PROPN
cana-5487	121	12	×	×	NOUN
cana-5487	121	13	𝑋	𝑋	PROPN
cana-5487	121	14	×	×	NOUN
cana-5487	121	15	𝑋	𝑋	PROPN
cana-5487	121	16	→	→	SYM
cana-5487	121	17	𝑋	𝑋	PROPN
cana-5487	121	18	defined	define	VERB
cana-5487	121	19	by	by	ADP
cana-5487	121	20	a(x	a(x	NOUN
cana-5487	121	21	,	,	PUNCT
cana-5487	121	22	y	y	PROPN
cana-5487	121	23	,	,	PUNCT
cana-5487	121	24	z	z	NOUN
cana-5487	121	25	)	)	PUNCT
cana-5487	121	26	=	=	SYM
cana-5487	122	1	2x	2x	NOUN
cana-5487	123	1	+	+	CCONJ
cana-5487	123	2	y	y	PROPN
cana-5487	123	3	+	+	CCONJ
cana-5487	123	4	z	z	PROPN
cana-5487	123	5	2	2	NUM
cana-5487	123	6	s(x	s(x	PROPN
cana-5487	123	7	)	)	PUNCT
cana-5487	123	8	=	=	PRON
cana-5487	123	9	{	{	PUNCT
cana-5487	123	10	x	x	NOUN
cana-5487	123	11	,	,	PUNCT
cana-5487	123	12	if0	if0	VERB
cana-5487	123	13	≤	≤	NUM
cana-5487	123	14	x	x	PUNCT
cana-5487	123	15	<	<	X
cana-5487	123	16	1	1	NUM
cana-5487	123	17	;	;	PUNCT
cana-5487	123	18	5	5	NUM
cana-5487	123	19	2	2	NUM
cana-5487	123	20	,	,	PUNCT
cana-5487	123	21	ifx	ifx	PROPN
cana-5487	123	22	≥	≥	PROPN
cana-5487	123	23	1	1	NUM
cana-5487	123	24	.	.	PUNCT
cana-5487	124	1	b(x	b(x	NOUN
cana-5487	124	2	,	,	PUNCT
cana-5487	124	3	y	y	PROPN
cana-5487	124	4	,	,	PUNCT
cana-5487	124	5	z	z	NOUN
cana-5487	124	6	)	)	PUNCT
cana-5487	124	7	=	=	SYM
cana-5487	124	8	yt(x	yt(x	NOUN
cana-5487	124	9	)	)	PUNCT
cana-5487	124	10	=	=	PRON
cana-5487	124	11	{	{	PUNCT
cana-5487	124	12	x	x	NOUN
cana-5487	124	13	,	,	PUNCT
cana-5487	124	14	if0	if0	VERB
cana-5487	124	15	≤	≤	NUM
cana-5487	124	16	x	x	PUNCT
cana-5487	124	17	<	<	X
cana-5487	124	18	1	1	NUM
cana-5487	124	19	;	;	PUNCT
cana-5487	124	20	5	5	NUM
cana-5487	124	21	,	,	PUNCT
cana-5487	124	22	ifx	ifx	PROPN
cana-5487	124	23	≥	≥	NUM
cana-5487	124	24	1	1	NUM
cana-5487	124	25	.	.	PUNCT
cana-5487	124	26	also	also	ADV
cana-5487	124	27	the	the	DET
cana-5487	124	28	pairs	pair	NOUN
cana-5487	124	29	(	(	PUNCT
cana-5487	124	30	𝐴	𝐴	PROPN
cana-5487	124	31	,	,	PUNCT
cana-5487	124	32	𝑆	𝑆	PROPN
cana-5487	124	33	)	)	PUNCT
cana-5487	124	34	and	and	CCONJ
cana-5487	124	35	(	(	PUNCT
cana-5487	124	36	𝐵	𝐵	PROPN
cana-5487	124	37	,	,	PUNCT
cana-5487	124	38	𝑇	𝑇	PROPN
cana-5487	124	39	)	)	PUNCT
cana-5487	124	40	are	be	AUX
cana-5487	124	41	owc	owc	NOUN
cana-5487	124	42	.	.	PUNCT
cana-5487	125	1	clearly	clearly	ADV
cana-5487	125	2	all	all	DET
cana-5487	125	3	the	the	DET
cana-5487	125	4	conditions	condition	NOUN
cana-5487	125	5	of	of	ADP
cana-5487	125	6	the	the	DET
cana-5487	125	7	above	above	ADJ
cana-5487	125	8	theorem	theorem	NOUN
cana-5487	125	9	are	be	AUX
cana-5487	125	10	satisfied	satisfied	ADJ
cana-5487	125	11	.	.	PUNCT
cana-5487	126	1	also	also	ADV
cana-5487	126	2	𝑆𝐴(0,0,0	𝑆𝐴(0,0,0	NOUN
cana-5487	126	3	)	)	PUNCT
cana-5487	126	4	=	=	SYM
cana-5487	126	5	𝐴(𝑆0	𝐴(𝑆0	PROPN
cana-5487	126	6	,	,	PUNCT
cana-5487	126	7	𝑆0	𝑆0	NOUN
cana-5487	126	8	,	,	PUNCT
cana-5487	126	9	𝑆0	𝑆0	NOUN
cana-5487	126	10	)	)	PUNCT
cana-5487	126	11	and	and	CCONJ
cana-5487	126	12	𝑇𝐵(0,0,0	𝑇𝐵(0,0,0	NOUN
cana-5487	126	13	)	)	PUNCT
cana-5487	126	14	=	=	SYM
cana-5487	126	15	𝐵(𝑇0	𝐵(𝑇0	PROPN
cana-5487	126	16	,	,	PUNCT
cana-5487	126	17	𝑇0	𝑇0	NOUN
cana-5487	126	18	,	,	PUNCT
cana-5487	126	19	𝑇0	𝑇0	NOUN
cana-5487	126	20	)	)	PUNCT
cana-5487	127	1	so	so	ADV
cana-5487	127	2	,	,	PUNCT
cana-5487	127	3	(	(	PUNCT
cana-5487	127	4	a	a	PRON
cana-5487	127	5	,	,	PUNCT
cana-5487	127	6	s	s	NOUN
cana-5487	127	7	)	)	PUNCT
cana-5487	127	8	and	and	CCONJ
cana-5487	127	9	(	(	PUNCT
cana-5487	127	10	b	b	PROPN
cana-5487	127	11	,	,	PUNCT
cana-5487	127	12	t	t	PROPN
cana-5487	127	13	)	)	PUNCT
cana-5487	127	14	are	be	AUX
cana-5487	127	15	owc	owc	PROPN
cana-5487	127	16	maps	map	NOUN
cana-5487	127	17	and(0	and(0	NOUN
cana-5487	127	18	,	,	PUNCT
cana-5487	127	19	0	0	NUM
cana-5487	127	20	,	,	PUNCT
cana-5487	127	21	0	0	NUM
cana-5487	127	22	)	)	PUNCT
cana-5487	128	1	is	be	AUX
cana-5487	128	2	the	the	DET
cana-5487	128	3	common	common	ADJ
cana-5487	128	4	tripled	triple	VERB
cana-5487	128	5	fixed	fix	VERB
cana-5487	128	6	point	point	NOUN
cana-5487	128	7	of	of	ADP
cana-5487	128	8	a	a	DET
cana-5487	128	9	,	,	PUNCT
cana-5487	128	10	b	b	NOUN
cana-5487	128	11	,	,	PUNCT
cana-5487	128	12	s	s	PART
cana-5487	128	13	and	and	CCONJ
cana-5487	128	14	t.	t.	NOUN
cana-5487	128	15	the	the	DET
cana-5487	128	16	mesh	mesh	NOUN
cana-5487	128	17	diagram	diagram	NOUN
cana-5487	128	18	for	for	ADP
cana-5487	128	19	the	the	DET
cana-5487	128	20	given	give	VERB
cana-5487	128	21	example	example	NOUN
cana-5487	128	22	is	be	AUX
cana-5487	128	23	shown	show	VERB
cana-5487	128	24	in	in	ADP
cana-5487	128	25	fig	fig	NOUN
cana-5487	129	1	[	[	X
cana-5487	129	2	3.1	3.1	NUM
cana-5487	129	3	]	]	PUNCT
cana-5487	129	4	.	.	PUNCT
cana-5487	130	1	fig	fig	NOUN
cana-5487	131	1	[	[	X
cana-5487	131	2	3.1	3.1	NUM
cana-5487	131	3	]	]	PUNCT
cana-5487	131	4	theorem	theorem	NOUN
cana-5487	131	5	:	:	PUNCT
cana-5487	131	6	3.2	3.2	NUM
cana-5487	131	7	let	let	VERB
cana-5487	131	8	(	(	PUNCT
cana-5487	131	9	𝑋	𝑋	PROPN
cana-5487	131	10	,	,	PUNCT
cana-5487	131	11	𝑀	𝑀	PROPN
cana-5487	131	12	,	,	PUNCT
cana-5487	131	13			PROPN
cana-5487	131	14	)	)	PUNCT
cana-5487	131	15	be	be	AUX
cana-5487	131	16	a	a	DET
cana-5487	131	17	fuzzy	fuzzy	ADJ
cana-5487	131	18	metric	metric	ADJ
cana-5487	131	19	space	space	NOUN
cana-5487	131	20	with	with	ADP
cana-5487	131	21	𝑡	𝑡	PROPN
cana-5487	131	22	∗	∗	NOUN
cana-5487	131	23	𝑡	𝑡	X
cana-5487	131	24	=	=	PUNCT
cana-5487	131	25	𝑡	𝑡	PROPN
cana-5487	131	26	for	for	ADP
cana-5487	131	27	all	all	DET
cana-5487	131	28	𝑡	𝑡	ADP
cana-5487	131	29	∈	∈	PROPN
cana-5487	132	1	[	[	X
cana-5487	132	2	0,1	0,1	NUM
cana-5487	132	3	]	]	PUNCT
cana-5487	132	4	.	.	PUNCT
cana-5487	133	1	let	let	VERB
cana-5487	133	2	𝐴	𝐴	PROPN
cana-5487	133	3	,	,	PUNCT
cana-5487	133	4	𝐵	𝐵	PROPN
cana-5487	133	5	:	:	PUNCT
cana-5487	133	6	𝑋	𝑋	NOUN
cana-5487	133	7	×	×	NOUN
cana-5487	133	8	𝑋	𝑋	PROPN
cana-5487	133	9	×	×	NOUN
cana-5487	133	10	𝑋	𝑋	PROPN
cana-5487	133	11	→	→	SYM
cana-5487	133	12	𝑋	𝑋	PROPN
cana-5487	133	13	and	and	CCONJ
cana-5487	133	14	𝑆	𝑆	PROPN
cana-5487	133	15	,	,	PUNCT
cana-5487	133	16	𝑇	𝑇	PROPN
cana-5487	133	17	:	:	PUNCT
cana-5487	133	18	𝑋	𝑋	PROPN
cana-5487	133	19	→	→	SYM
cana-5487	133	20	𝑋	𝑋	PROPN
cana-5487	133	21	be	be	VERB
cana-5487	133	22	four	four	NUM
cana-5487	133	23	self	self	NOUN
cana-5487	133	24	-	-	PUNCT
cana-5487	133	25	mappings	mapping	NOUN
cana-5487	133	26	satisfying	satisfy	VERB
cana-5487	133	27	the	the	DET
cana-5487	133	28	following	follow	VERB
cana-5487	133	29	conditions	condition	NOUN
cana-5487	133	30	:	:	PUNCT
cana-5487	133	31	communications	communication	NOUN
cana-5487	133	32	on	on	ADP
cana-5487	133	33	applied	apply	VERB
cana-5487	133	34	nonlinear	nonlinear	ADJ
cana-5487	133	35	analysis	analysis	NOUN
cana-5487	133	36	issn	issn	NOUN
cana-5487	133	37	:	:	PUNCT
cana-5487	133	38	1074	1074	NUM
cana-5487	133	39	-	-	PUNCT
cana-5487	133	40	133x	133x	NUM
cana-5487	133	41	vol	vol	VERB
cana-5487	133	42	32	32	NUM
cana-5487	133	43	no	no	NOUN
cana-5487	133	44	.	.	PUNCT
cana-5487	134	1	10s	10	NOUN
cana-5487	134	2	(	(	PUNCT
cana-5487	134	3	2025	2025	NUM
cana-5487	134	4	)	)	PUNCT
cana-5487	134	5	2390	2390	NUM
cana-5487	134	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	134	7	(	(	PUNCT
cana-5487	134	8	i	i	NOUN
cana-5487	134	9	)	)	PUNCT
cana-5487	134	10	𝑀𝑃(𝐴(𝑥	𝑀𝑃(𝐴(𝑥	NUM
cana-5487	134	11	,	,	PUNCT
cana-5487	134	12	𝑦	𝑦	NOUN
cana-5487	134	13	,	,	PUNCT
cana-5487	134	14	𝑧	𝑧	NOUN
cana-5487	134	15	)	)	PUNCT
cana-5487	134	16	,	,	PUNCT
cana-5487	134	17	𝐵(𝑢	𝐵(𝑢	PROPN
cana-5487	134	18	,	,	PUNCT
cana-5487	134	19	𝑣	𝑣	NOUN
cana-5487	134	20	,	,	PUNCT
cana-5487	134	21	𝑤	𝑤	ADP
cana-5487	134	22	)	)	PUNCT
cana-5487	134	23	,	,	PUNCT
cana-5487	134	24	𝑞𝑡	𝑞𝑡	PRON
cana-5487	134	25	)	)	PUNCT
cana-5487	134	26	≥	≥	PROPN
cana-5487	134	27	min	min	NOUN
cana-5487	134	28	{	{	PUNCT
cana-5487	134	29	𝑀𝑝(𝑆𝑥	𝑀𝑝(𝑆𝑥	NOUN
cana-5487	134	30	,	,	PUNCT
cana-5487	134	31	𝑇𝑢	𝑇𝑢	NOUN
cana-5487	134	32	,	,	PUNCT
cana-5487	134	33	𝑡),𝑀(𝐴(𝑥	𝑡),𝑀(𝐴(𝑥	NOUN
cana-5487	134	34	,	,	PUNCT
cana-5487	134	35	𝑦	𝑦	NOUN
cana-5487	134	36	,	,	PUNCT
cana-5487	134	37	𝑧	𝑧	NOUN
cana-5487	134	38	)	)	PUNCT
cana-5487	134	39	,	,	PUNCT
cana-5487	135	1	𝑇𝑢	𝑇𝑢	PROPN
cana-5487	135	2	,	,	PUNCT
cana-5487	135	3	𝑡).𝑀𝑝−1(𝐵(𝑢	𝑡).𝑀𝑝−1(𝐵(𝑢	PROPN
cana-5487	135	4	,	,	PUNCT
cana-5487	135	5	𝑣	𝑣	NOUN
cana-5487	135	6	,	,	PUNCT
cana-5487	135	7	𝑤	𝑤	ADP
cana-5487	135	8	)	)	PUNCT
cana-5487	135	9	,	,	PUNCT
cana-5487	135	10	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	135	11	,	,	PUNCT
cana-5487	135	12	𝑡	𝑡	NOUN
cana-5487	135	13	)	)	PUNCT
cana-5487	135	14	}	}	PUNCT
cana-5487	135	15	for	for	ADP
cana-5487	135	16	all	all	PRON
cana-5487	135	17	𝑥	𝑥	PROPN
cana-5487	135	18	,	,	PUNCT
cana-5487	135	19	𝑦	𝑦	NOUN
cana-5487	135	20	,	,	PUNCT
cana-5487	135	21	𝑧	𝑧	NOUN
cana-5487	135	22	,	,	PUNCT
cana-5487	135	23	𝑢	𝑢	PROPN
cana-5487	135	24	,	,	PUNCT
cana-5487	135	25	𝑣	𝑣	NOUN
cana-5487	135	26	,	,	PUNCT
cana-5487	135	27	𝑤	𝑤	ADP
cana-5487	135	28	∈	∈	PROPN
cana-5487	135	29	𝑋	𝑋	PROPN
cana-5487	135	30	,	,	PUNCT
cana-5487	135	31	0	0	NUM
cana-5487	135	32	≤	≤	NUM
cana-5487	135	33	𝑎	𝑎	PRON
cana-5487	135	34	≤	≤	NUM
cana-5487	135	35	1	1	NUM
cana-5487	135	36	,	,	PUNCT
cana-5487	135	37	𝑝	𝑝	PRON
cana-5487	135	38	≥	≥	NOUN
cana-5487	135	39	1	1	NUM
cana-5487	135	40	.	.	PUNCT
cana-5487	135	41	(	(	PUNCT
cana-5487	135	42	ii	ii	NOUN
cana-5487	135	43	)	)	PUNCT
cana-5487	135	44	𝑦	𝑦	NOUN
cana-5487	135	45	=	=	SYM
cana-5487	135	46	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	135	47	,	,	PUNCT
cana-5487	135	48	𝑦	𝑦	NOUN
cana-5487	135	49	,	,	PUNCT
cana-5487	135	50	𝑧	𝑧	PART
cana-5487	135	51	)	)	PUNCT
cana-5487	135	52	moreover	moreover	ADV
cana-5487	135	53	if	if	SCONJ
cana-5487	135	54	the	the	DET
cana-5487	135	55	pairs	pair	NOUN
cana-5487	135	56	(	(	PUNCT
cana-5487	135	57	𝐴	𝐴	PROPN
cana-5487	135	58	,	,	PUNCT
cana-5487	135	59	𝑆	𝑆	PROPN
cana-5487	135	60	)	)	PUNCT
cana-5487	135	61	and	and	CCONJ
cana-5487	135	62	(	(	PUNCT
cana-5487	135	63	𝐵	𝐵	PROPN
cana-5487	135	64	,	,	PUNCT
cana-5487	135	65	𝑇	𝑇	PROPN
cana-5487	135	66	)	)	PUNCT
cana-5487	135	67	are	be	AUX
cana-5487	135	68	owc	owc	NUM
cana-5487	135	69	,	,	PUNCT
cana-5487	135	70	then	then	ADV
cana-5487	135	71	there	there	PRON
cana-5487	135	72	exists	exist	VERB
cana-5487	135	73	a	a	DET
cana-5487	135	74	unique	unique	ADJ
cana-5487	135	75	point	point	NOUN
cana-5487	135	76	𝑥	𝑥	NOUN
cana-5487	135	77	in	in	ADP
cana-5487	135	78	𝑋	𝑋	PROPN
cana-5487	135	79	such	such	ADJ
cana-5487	135	80	that𝐴(𝑥	that𝐴(𝑥	NOUN
cana-5487	135	81	,	,	PUNCT
cana-5487	135	82	𝑥	𝑥	NOUN
cana-5487	135	83	,	,	PUNCT
cana-5487	135	84	𝑥	𝑥	NOUN
cana-5487	135	85	)	)	PUNCT
cana-5487	135	86	=	=	SYM
cana-5487	135	87	𝑇(𝑥	𝑇(𝑥	X
cana-5487	135	88	)	)	PUNCT
cana-5487	135	89	=	=	SYM
cana-5487	136	1	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	136	2	,	,	PUNCT
cana-5487	136	3	𝑥	𝑥	NOUN
cana-5487	136	4	,	,	PUNCT
cana-5487	136	5	𝑥	𝑥	NOUN
cana-5487	136	6	)	)	PUNCT
cana-5487	136	7	=	=	SYM
cana-5487	136	8	𝑆(𝑥	𝑆(𝑥	X
cana-5487	136	9	)	)	PUNCT
cana-5487	136	10	=	=	PUNCT
cana-5487	136	11	𝑥.	𝑥.	NOUN
cana-5487	136	12	theorem	theorem	VERB
cana-5487	136	13	:	:	PUNCT
cana-5487	136	14	3.3let	3.3let	NUM
cana-5487	136	15	(	(	PUNCT
cana-5487	136	16	𝑋	𝑋	PROPN
cana-5487	136	17	,	,	PUNCT
cana-5487	136	18	𝑀	𝑀	PROPN
cana-5487	136	19	,	,	PUNCT
cana-5487	136	20			PROPN
cana-5487	136	21	)	)	PUNCT
cana-5487	136	22	be	be	AUX
cana-5487	136	23	a	a	DET
cana-5487	136	24	fuzzy	fuzzy	ADJ
cana-5487	136	25	metric	metric	ADJ
cana-5487	136	26	space	space	NOUN
cana-5487	136	27	with	with	ADP
cana-5487	136	28	𝑡	𝑡	PROPN
cana-5487	136	29	∗	∗	NOUN
cana-5487	136	30	𝑡	𝑡	X
cana-5487	136	31	=	=	PUNCT
cana-5487	136	32	𝑡	𝑡	PROPN
cana-5487	136	33	for	for	ADP
cana-5487	136	34	all	all	DET
cana-5487	136	35	𝑡	𝑡	ADP
cana-5487	136	36	∈	∈	PROPN
cana-5487	137	1	[	[	X
cana-5487	137	2	0,1	0,1	NUM
cana-5487	137	3	]	]	PUNCT
cana-5487	137	4	.	.	PUNCT
cana-5487	138	1	let	let	VERB
cana-5487	138	2	𝐴	𝐴	PROPN
cana-5487	138	3	,	,	PUNCT
cana-5487	138	4	𝐵	𝐵	PROPN
cana-5487	138	5	:	:	PUNCT
cana-5487	138	6	𝑋	𝑋	NOUN
cana-5487	138	7	×	×	NOUN
cana-5487	138	8	𝑋	𝑋	PROPN
cana-5487	138	9	×	×	NOUN
cana-5487	138	10	𝑋	𝑋	PROPN
cana-5487	138	11	→	→	SYM
cana-5487	138	12	𝑋	𝑋	PROPN
cana-5487	138	13	and	and	CCONJ
cana-5487	138	14	𝑆	𝑆	PROPN
cana-5487	138	15	,	,	PUNCT
cana-5487	138	16	𝑇	𝑇	PROPN
cana-5487	138	17	:	:	PUNCT
cana-5487	138	18	𝑋	𝑋	PROPN
cana-5487	138	19	→	→	SYM
cana-5487	138	20	𝑋	𝑋	PROPN
cana-5487	138	21	be	be	VERB
cana-5487	138	22	four	four	NUM
cana-5487	138	23	self	self	NOUN
cana-5487	138	24	-	-	PUNCT
cana-5487	138	25	mappings	mapping	NOUN
cana-5487	138	26	satisfying	satisfy	VERB
cana-5487	138	27	the	the	DET
cana-5487	138	28	following	following	ADJ
cana-5487	138	29	conditions	condition	NOUN
cana-5487	138	30	:	:	PUNCT
cana-5487	138	31	𝑀(𝐴(𝑥	𝑀(𝐴(𝑥	NOUN
cana-5487	138	32	,	,	PUNCT
cana-5487	138	33	𝑦	𝑦	NOUN
cana-5487	138	34	,	,	PUNCT
cana-5487	138	35	𝑧	𝑧	NOUN
cana-5487	138	36	)	)	PUNCT
cana-5487	138	37	,	,	PUNCT
cana-5487	138	38	𝐵(𝑢	𝐵(𝑢	PROPN
cana-5487	138	39	,	,	PUNCT
cana-5487	138	40	𝑣	𝑣	NOUN
cana-5487	138	41	,	,	PUNCT
cana-5487	138	42	𝑤	𝑤	ADP
cana-5487	138	43	)	)	PUNCT
cana-5487	138	44	,	,	PUNCT
cana-5487	138	45	𝑞𝑡	𝑞𝑡	PRON
cana-5487	138	46	)	)	PUNCT
cana-5487	138	47	≥	≥	PROPN
cana-5487	138	48	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-5487	138	49	{	{	PUNCT
cana-5487	138	50	𝑀(𝑆𝑥	𝑀(𝑆𝑥	ADJ
cana-5487	138	51	,	,	PUNCT
cana-5487	138	52	𝑇𝑢	𝑇𝑢	PROPN
cana-5487	138	53	,	,	PUNCT
cana-5487	138	54	𝑡	𝑡	NOUN
cana-5487	138	55	)	)	PUNCT
cana-5487	138	56	+	+	CCONJ
cana-5487	138	57	1	1	NUM
cana-5487	138	58	2	2	NUM
cana-5487	138	59	(	(	PUNCT
cana-5487	138	60	1	1	NUM
cana-5487	138	61	+	+	NUM
cana-5487	138	62	𝑀(𝐴(𝑥	𝑀(𝐴(𝑥	NOUN
cana-5487	138	63	,	,	PUNCT
cana-5487	138	64	𝑦	𝑦	NOUN
cana-5487	138	65	,	,	PUNCT
cana-5487	138	66	𝑧	𝑧	NOUN
cana-5487	138	67	)	)	PUNCT
cana-5487	138	68	,	,	PUNCT
cana-5487	138	69	𝑆𝑥	𝑆𝑥	PROPN
cana-5487	138	70	,	,	PUNCT
cana-5487	138	71	𝑡	𝑡	NOUN
cana-5487	138	72	)	)	PUNCT
cana-5487	138	73	𝑀(𝐵(𝑢	𝑀(𝐵(𝑢	NOUN
cana-5487	138	74	,	,	PUNCT
cana-5487	138	75	𝑣	𝑣	X
cana-5487	138	76	,	,	PUNCT
cana-5487	138	77	𝑤	𝑤	ADP
cana-5487	138	78	)	)	PUNCT
cana-5487	138	79	,	,	PUNCT
cana-5487	139	1	𝑇𝑢	𝑇𝑢	PROPN
cana-5487	139	2	,	,	PUNCT
cana-5487	139	3	𝑡	𝑡	NOUN
cana-5487	139	4	)	)	PUNCT
cana-5487	139	5	)	)	PUNCT
cana-5487	139	6	}	}	PUNCT
cana-5487	139	7	for	for	ADP
cana-5487	139	8	all	all	DET
cana-5487	139	9	𝑥	𝑥	PROPN
cana-5487	139	10	,	,	PUNCT
cana-5487	139	11	𝑦	𝑦	NOUN
cana-5487	139	12	,	,	PUNCT
cana-5487	139	13	𝑢	𝑢	X
cana-5487	139	14	,	,	PUNCT
cana-5487	139	15	𝑣	𝑣	PRON
cana-5487	139	16	∈	∈	PROPN
cana-5487	139	17	𝑋	𝑋	PROPN
cana-5487	139	18	(	(	PUNCT
cana-5487	139	19	i	i	NOUN
cana-5487	139	20	)	)	PUNCT
cana-5487	139	21	𝑦	𝑦	NOUN
cana-5487	139	22	=	=	SYM
cana-5487	139	23	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	139	24	,	,	PUNCT
cana-5487	139	25	𝑦	𝑦	NOUN
cana-5487	139	26	)	)	PUNCT
cana-5487	139	27	moreover	moreover	ADV
cana-5487	139	28	if	if	SCONJ
cana-5487	139	29	the	the	DET
cana-5487	139	30	pairs	pair	NOUN
cana-5487	139	31	(	(	PUNCT
cana-5487	139	32	𝐴	𝐴	PROPN
cana-5487	139	33	,	,	PUNCT
cana-5487	139	34	𝑆	𝑆	PROPN
cana-5487	139	35	)	)	PUNCT
cana-5487	139	36	and	and	CCONJ
cana-5487	139	37	(	(	PUNCT
cana-5487	139	38	𝐵	𝐵	PROPN
cana-5487	139	39	,	,	PUNCT
cana-5487	139	40	𝑇	𝑇	PROPN
cana-5487	139	41	)	)	PUNCT
cana-5487	139	42	are	be	AUX
cana-5487	139	43	owc	owc	NUM
cana-5487	139	44	,	,	PUNCT
cana-5487	139	45	then	then	ADV
cana-5487	139	46	there	there	PRON
cana-5487	139	47	exists	exist	VERB
cana-5487	139	48	a	a	DET
cana-5487	139	49	unique	unique	ADJ
cana-5487	139	50	point	point	NOUN
cana-5487	139	51	𝑥	𝑥	NOUN
cana-5487	139	52	in	in	ADP
cana-5487	139	53	𝑋	𝑋	NOUN
cana-5487	140	1	such	such	ADJ
cana-5487	140	2	that	that	SCONJ
cana-5487	140	3	𝐴(𝑥	𝐴(𝑥	NOUN
cana-5487	140	4	,	,	PUNCT
cana-5487	140	5	𝑥	𝑥	PRON
cana-5487	140	6	,	,	PUNCT
cana-5487	140	7	𝑥	𝑥	NOUN
cana-5487	140	8	)	)	PUNCT
cana-5487	140	9	=	=	SYM
cana-5487	140	10	𝑇(𝑥	𝑇(𝑥	X
cana-5487	140	11	)	)	PUNCT
cana-5487	140	12	=	=	SYM
cana-5487	140	13	𝐵(𝑥	𝐵(𝑥	PROPN
cana-5487	140	14	,	,	PUNCT
cana-5487	140	15	𝑥	𝑥	NOUN
cana-5487	140	16	,	,	PUNCT
cana-5487	140	17	𝑥	𝑥	NOUN
cana-5487	140	18	)	)	PUNCT
cana-5487	140	19	=	=	SYM
cana-5487	140	20	𝑆(𝑥	𝑆(𝑥	X
cana-5487	140	21	)	)	PUNCT
cana-5487	140	22	=	=	PUNCT
cana-5487	141	1	𝑥.	𝑥.	ADJ
cana-5487	141	2	references	reference	NOUN
cana-5487	141	3	[	[	X
cana-5487	141	4	1	1	X
cana-5487	141	5	]	]	PUNCT
cana-5487	141	6	t.	t.	NOUN
cana-5487	141	7	g.	g.	PROPN
cana-5487	141	8	bhaskar	bhaskar	PROPN
cana-5487	141	9	and	and	CCONJ
cana-5487	141	10	v.	v.	ADP
cana-5487	141	11	lakshmikantham	lakshmikantham	NOUN
cana-5487	141	12	,	,	PUNCT
cana-5487	141	13	"	"	PUNCT
cana-5487	141	14	fixed	fixed	ADJ
cana-5487	141	15	point	point	NOUN
cana-5487	141	16	theorems	theorem	NOUN
cana-5487	141	17	in	in	ADP
cana-5487	141	18	partially	partially	ADV
cana-5487	141	19	ordered	order	VERB
cana-5487	141	20	metric	metric	ADJ
cana-5487	141	21	spaces	space	NOUN
cana-5487	141	22	and	and	CCONJ
cana-5487	141	23	applications	application	NOUN
cana-5487	141	24	,	,	PUNCT
cana-5487	141	25	"	"	PUNCT
cana-5487	141	26	*	*	PUNCT
cana-5487	141	27	nonlinear	nonlinear	ADJ
cana-5487	141	28	analysis	analysis	NOUN
cana-5487	141	29	:	:	PUNCT
cana-5487	141	30	theory	theory	NOUN
cana-5487	141	31	,	,	PUNCT
cana-5487	141	32	methods	method	NOUN
cana-5487	141	33	&	&	CCONJ
cana-5487	141	34	applications	application	NOUN
cana-5487	141	35	*	*	PUNCT
cana-5487	141	36	,	,	PUNCT
cana-5487	141	37	vol	vol	NOUN
cana-5487	141	38	.	.	PROPN
cana-5487	141	39	65	65	NUM
cana-5487	141	40	,	,	PUNCT
cana-5487	141	41	no	no	INTJ
cana-5487	141	42	.	.	NOUN
cana-5487	141	43	7	7	NUM
cana-5487	141	44	,	,	PUNCT
cana-5487	141	45	pp	pp	ADJ
cana-5487	141	46	.	.	PUNCT
cana-5487	141	47	1379–1393	1379–1393	NUM
cana-5487	141	48	,	,	PUNCT
cana-5487	141	49	2006	2006	NUM
cana-5487	141	50	.	.	PUNCT
cana-5487	142	1	[	[	X
cana-5487	142	2	2	2	X
cana-5487	142	3	]	]	PUNCT
cana-5487	142	4	j.	j.	PROPN
cana-5487	142	5	x.	x.	PROPN
cana-5487	142	6	fang	fang	PROPN
cana-5487	142	7	,	,	PUNCT
cana-5487	142	8	"	"	PUNCT
cana-5487	142	9	common	common	ADJ
cana-5487	142	10	fixed	fix	VERB
cana-5487	142	11	point	point	NOUN
cana-5487	142	12	theorems	theorem	NOUN
cana-5487	142	13	of	of	ADP
cana-5487	142	14	compatible	compatible	ADJ
cana-5487	142	15	and	and	CCONJ
cana-5487	142	16	weakly	weakly	ADJ
cana-5487	142	17	compatible	compatible	ADJ
cana-5487	142	18	maps	map	NOUN
cana-5487	142	19	in	in	ADP
cana-5487	142	20	menger	menger	PROPN
cana-5487	142	21	spaces	space	NOUN
cana-5487	142	22	,	,	PUNCT
cana-5487	142	23	"	"	PUNCT
cana-5487	142	24	*	*	PUNCT
cana-5487	142	25	nonlinear	nonlinear	ADJ
cana-5487	142	26	analysis	analysis	NOUN
cana-5487	142	27	:	:	PUNCT
cana-5487	142	28	theory	theory	NOUN
cana-5487	142	29	,	,	PUNCT
cana-5487	142	30	methods	method	NOUN
cana-5487	142	31	&	&	CCONJ
cana-5487	142	32	applications	application	NOUN
cana-5487	142	33	*	*	PUNCT
cana-5487	142	34	,	,	PUNCT
cana-5487	142	35	vol	vol	NOUN
cana-5487	142	36	.	.	PROPN
cana-5487	142	37	71	71	NUM
cana-5487	142	38	,	,	PUNCT
cana-5487	142	39	no	no	INTJ
cana-5487	142	40	.	.	PUNCT
cana-5487	143	1	5–6	5–6	NUM
cana-5487	143	2	,	,	PUNCT
cana-5487	143	3	pp	pp	ADJ
cana-5487	143	4	.	.	PUNCT
cana-5487	144	1	1833–1843	1833–1843	NUM
cana-5487	144	2	,	,	PUNCT
cana-5487	144	3	2009	2009	NUM
cana-5487	144	4	.	.	PUNCT
cana-5487	145	1	[	[	X
cana-5487	145	2	3	3	NUM
cana-5487	145	3	]	]	PUNCT
cana-5487	145	4	a.	a.	NOUN
cana-5487	145	5	george	george	PROPN
cana-5487	145	6	and	and	CCONJ
cana-5487	145	7	p.	p.	PROPN
cana-5487	145	8	veeramani	veeramani	PROPN
cana-5487	145	9	,	,	PUNCT
cana-5487	145	10	"	"	PUNCT
cana-5487	145	11	on	on	ADP
cana-5487	145	12	some	some	DET
cana-5487	145	13	results	result	NOUN
cana-5487	145	14	in	in	ADP
cana-5487	145	15	fuzzy	fuzzy	ADJ
cana-5487	145	16	metric	metric	ADJ
cana-5487	145	17	spaces	space	NOUN
cana-5487	145	18	,	,	PUNCT
cana-5487	145	19	"	"	PUNCT
cana-5487	145	20	*	*	PUNCT
cana-5487	145	21	fuzzy	fuzzy	ADJ
cana-5487	145	22	sets	set	NOUN
cana-5487	145	23	and	and	CCONJ
cana-5487	145	24	systems	system	NOUN
cana-5487	145	25	*	*	PUNCT
cana-5487	145	26	,	,	PUNCT
cana-5487	145	27	vol	vol	NOUN
cana-5487	145	28	.	.	PROPN
cana-5487	145	29	64	64	NUM
cana-5487	145	30	,	,	PUNCT
cana-5487	145	31	pp	pp	ADJ
cana-5487	145	32	.	.	PUNCT
cana-5487	146	1	395–399	395–399	NUM
cana-5487	146	2	,	,	PUNCT
cana-5487	146	3	1994	1994	NUM
cana-5487	146	4	.	.	PUNCT
cana-5487	147	1	[	[	X
cana-5487	147	2	4	4	X
cana-5487	147	3	]	]	X
cana-5487	147	4	g.	g.	PROPN
cana-5487	147	5	jungck	jungck	PROPN
cana-5487	147	6	,	,	PUNCT
cana-5487	147	7	"	"	PUNCT
cana-5487	147	8	compatible	compatible	ADJ
cana-5487	147	9	mappings	mapping	NOUN
cana-5487	147	10	and	and	CCONJ
cana-5487	147	11	common	common	ADJ
cana-5487	147	12	fixed	fix	VERB
cana-5487	147	13	points	point	NOUN
cana-5487	147	14	,	,	PUNCT
cana-5487	147	15	"	"	PUNCT
cana-5487	147	16	*	*	PUNCT
cana-5487	147	17	international	international	ADJ
cana-5487	147	18	journal	journal	NOUN
cana-5487	147	19	of	of	ADP
cana-5487	147	20	mathematics	mathematics	PROPN
cana-5487	147	21	and	and	CCONJ
cana-5487	147	22	mathematical	mathematical	ADJ
cana-5487	147	23	sciences	science	NOUN
cana-5487	147	24	*	*	NOUN
cana-5487	147	25	,	,	PUNCT
cana-5487	147	26	vol	vol	NOUN
cana-5487	147	27	.	.	NOUN
cana-5487	147	28	9	9	NUM
cana-5487	147	29	,	,	PUNCT
cana-5487	147	30	no	no	INTJ
cana-5487	147	31	.	.	NOUN
cana-5487	147	32	4	4	NUM
cana-5487	147	33	,	,	PUNCT
cana-5487	147	34	pp	pp	ADJ
cana-5487	147	35	.	.	PUNCT
cana-5487	148	1	771–779	771–779	NUM
cana-5487	148	2	,	,	PUNCT
cana-5487	148	3	1986	1986	NUM
cana-5487	148	4	.	.	PUNCT
cana-5487	149	1	[	[	X
cana-5487	149	2	5	5	X
cana-5487	149	3	]	]	PUNCT
cana-5487	149	4	g.	g.	PROPN
cana-5487	149	5	jungck	jungck	PROPN
cana-5487	149	6	,	,	PUNCT
cana-5487	149	7	"	"	PUNCT
cana-5487	149	8	common	common	ADJ
cana-5487	149	9	fixed	fix	VERB
cana-5487	149	10	points	point	NOUN
cana-5487	149	11	for	for	ADP
cana-5487	149	12	non	non	ADJ
cana-5487	149	13	-	-	ADJ
cana-5487	149	14	continuous	continuous	ADJ
cana-5487	149	15	non	non	ADJ
cana-5487	149	16	-	-	ADJ
cana-5487	149	17	self	self	NOUN
cana-5487	149	18	-	-	PUNCT
cana-5487	149	19	maps	map	NOUN
cana-5487	149	20	on	on	ADP
cana-5487	149	21	nonmetric	nonmetric	ADJ
cana-5487	149	22	spaces	space	NOUN
cana-5487	149	23	,	,	PUNCT
cana-5487	149	24	"	"	PUNCT
cana-5487	149	25	*	*	PUNCT
cana-5487	149	26	far	far	PROPN
cana-5487	149	27	east	east	NOUN
cana-5487	149	28	journal	journal	PROPN
cana-5487	149	29	of	of	ADP
cana-5487	149	30	mathematical	mathematical	ADJ
cana-5487	149	31	sciences	science	NOUN
cana-5487	149	32	*	*	NOUN
cana-5487	149	33	,	,	PUNCT
cana-5487	149	34	vol	vol	NOUN
cana-5487	149	35	.	.	PROPN
cana-5487	149	36	4	4	NUM
cana-5487	149	37	,	,	PUNCT
cana-5487	149	38	no	no	INTJ
cana-5487	149	39	.	.	NOUN
cana-5487	149	40	2	2	NUM
cana-5487	149	41	,	,	PUNCT
cana-5487	149	42	pp	pp	ADJ
cana-5487	149	43	.	.	PUNCT
cana-5487	150	1	199–215	199–215	NUM
cana-5487	150	2	,	,	PUNCT
cana-5487	150	3	1996	1996	NUM
cana-5487	150	4	.	.	PUNCT
cana-5487	151	1	[	[	X
cana-5487	151	2	6	6	NUM
cana-5487	151	3	]	]	PUNCT
cana-5487	151	4	g.	g.	PROPN
cana-5487	151	5	jungck	jungck	PROPN
cana-5487	151	6	and	and	CCONJ
cana-5487	151	7	b.	b.	PROPN
cana-5487	151	8	e.	e.	PROPN
cana-5487	151	9	rhoades	rhoades	PROPN
cana-5487	151	10	,	,	PUNCT
cana-5487	151	11	"	"	PUNCT
cana-5487	151	12	fixed	fix	VERB
cana-5487	151	13	point	point	NOUN
cana-5487	151	14	theorems	theorem	NOUN
cana-5487	151	15	for	for	ADP
cana-5487	151	16	occasionally	occasionally	ADV
cana-5487	151	17	weakly	weakly	ADJ
cana-5487	151	18	compatible	compatible	ADJ
cana-5487	151	19	mappings	mapping	NOUN
cana-5487	151	20	,	,	PUNCT
cana-5487	151	21	"	"	PUNCT
cana-5487	151	22	*	*	PUNCT
cana-5487	151	23	fixed	fix	VERB
cana-5487	151	24	point	point	NOUN
cana-5487	151	25	theory	theory	NOUN
cana-5487	151	26	*	*	NOUN
cana-5487	151	27	,	,	PUNCT
cana-5487	151	28	vol	vol	NOUN
cana-5487	151	29	.	.	PROPN
cana-5487	151	30	7	7	NUM
cana-5487	151	31	,	,	PUNCT
cana-5487	151	32	no	no	INTJ
cana-5487	151	33	.	.	NOUN
cana-5487	151	34	2	2	NUM
cana-5487	151	35	,	,	PUNCT
cana-5487	151	36	pp	pp	ADJ
cana-5487	151	37	.	.	PUNCT
cana-5487	152	1	287–296	287–296	NUM
cana-5487	152	2	,	,	PUNCT
cana-5487	152	3	2006	2006	NUM
cana-5487	152	4	.	.	PUNCT
cana-5487	153	1	[	[	X
cana-5487	153	2	7	7	X
cana-5487	153	3	]	]	X
cana-5487	153	4	o.	o.	NOUN
cana-5487	153	5	kramosil	kramosil	PROPN
cana-5487	153	6	and	and	CCONJ
cana-5487	153	7	j.	j.	PROPN
cana-5487	153	8	michalek	michalek	PROPN
cana-5487	153	9	,	,	PUNCT
cana-5487	153	10	"	"	PUNCT
cana-5487	153	11	fuzzy	fuzzy	ADJ
cana-5487	153	12	metric	metric	ADJ
cana-5487	153	13	and	and	CCONJ
cana-5487	153	14	statistical	statistical	ADJ
cana-5487	153	15	metric	metric	ADJ
cana-5487	153	16	spaces	space	NOUN
cana-5487	153	17	,	,	PUNCT
cana-5487	153	18	"	"	PUNCT
cana-5487	154	1	*	*	NOUN
cana-5487	154	2	kybernetika	kybernetika	NOUN
cana-5487	154	3	*	*	NOUN
cana-5487	154	4	,	,	PUNCT
cana-5487	154	5	vol	vol	NOUN
cana-5487	154	6	.	.	PROPN
cana-5487	154	7	11	11	NUM
cana-5487	154	8	,	,	PUNCT
cana-5487	154	9	pp	pp	ADJ
cana-5487	154	10	.	.	PUNCT
cana-5487	155	1	326–334	326–334	NUM
cana-5487	155	2	,	,	PUNCT
cana-5487	155	3	1975	1975	NUM
cana-5487	155	4	.	.	PUNCT
cana-5487	156	1	[	[	X
cana-5487	156	2	8	8	X
cana-5487	156	3	]	]	PUNCT
cana-5487	156	4	p.	p.	NOUN
cana-5487	156	5	nigam	nigam	PROPN
cana-5487	156	6	and	and	CCONJ
cana-5487	156	7	n.	n.	PROPN
cana-5487	156	8	malviya	malviya	PROPN
cana-5487	156	9	,	,	PUNCT
cana-5487	156	10	"	"	PUNCT
cana-5487	156	11	some	some	DET
cana-5487	156	12	fixed	fix	VERB
cana-5487	156	13	point	point	NOUN
cana-5487	156	14	theorems	theorem	NOUN
cana-5487	156	15	for	for	ADP
cana-5487	156	16	occasionally	occasionally	ADV
cana-5487	156	17	weakly	weakly	ADJ
cana-5487	156	18	compatible	compatible	ADJ
cana-5487	156	19	mappings	mapping	NOUN
cana-5487	156	20	in	in	ADP
cana-5487	156	21	fuzzy	fuzzy	ADJ
cana-5487	156	22	2	2	NUM
cana-5487	156	23	-	-	PUNCT
cana-5487	156	24	metric	metric	ADJ
cana-5487	156	25	space	space	NOUN
cana-5487	156	26	,	,	PUNCT
cana-5487	156	27	"	"	PUNCT
cana-5487	156	28	*	*	PUNCT
cana-5487	156	29	international	international	ADJ
cana-5487	156	30	journal	journal	NOUN
cana-5487	156	31	of	of	ADP
cana-5487	156	32	fuzzy	fuzzy	ADJ
cana-5487	156	33	mathematics	mathematics	PROPN
cana-5487	156	34	&	&	CCONJ
cana-5487	156	35	systems	system	NOUN
cana-5487	156	36	*	*	PUNCT
cana-5487	156	37	,	,	PUNCT
cana-5487	156	38	vol	vol	NOUN
cana-5487	156	39	.	.	PROPN
cana-5487	156	40	1	1	NUM
cana-5487	156	41	,	,	PUNCT
cana-5487	156	42	no	no	INTJ
cana-5487	156	43	.	.	NOUN
cana-5487	156	44	1	1	NUM
cana-5487	156	45	,	,	PUNCT
cana-5487	156	46	pp	pp	ADJ
cana-5487	156	47	.	.	PUNCT
cana-5487	157	1	81–92	81–92	NUM
cana-5487	157	2	,	,	PUNCT
cana-5487	157	3	2011	2011	NUM
cana-5487	157	4	.	.	PUNCT
cana-5487	158	1	communications	communication	NOUN
cana-5487	158	2	on	on	ADP
cana-5487	158	3	applied	apply	VERB
cana-5487	158	4	nonlinear	nonlinear	ADJ
cana-5487	158	5	analysis	analysis	NOUN
cana-5487	158	6	issn	issn	NOUN
cana-5487	158	7	:	:	PUNCT
cana-5487	158	8	1074	1074	NUM
cana-5487	158	9	-	-	PUNCT
cana-5487	158	10	133x	133x	NUM
cana-5487	158	11	vol	vol	VERB
cana-5487	158	12	32	32	NUM
cana-5487	158	13	no	no	NOUN
cana-5487	158	14	.	.	PUNCT
cana-5487	159	1	10s	10	NOUN
cana-5487	159	2	(	(	PUNCT
cana-5487	159	3	2025	2025	NUM
cana-5487	159	4	)	)	PUNCT
cana-5487	159	5	2391	2391	NUM
cana-5487	159	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5487	160	1	[	[	X
cana-5487	160	2	9	9	NUM
cana-5487	160	3	]	]	PUNCT
cana-5487	160	4	p.	p.	NOUN
cana-5487	160	5	nigam	nigam	PROPN
cana-5487	160	6	and	and	CCONJ
cana-5487	160	7	s.	s.	PROPN
cana-5487	160	8	s.	s.	PROPN
cana-5487	160	9	pagey	pagey	PROPN
cana-5487	160	10	,	,	PUNCT
cana-5487	160	11	"	"	PUNCT
cana-5487	160	12	fixed	fix	VERB
cana-5487	160	13	point	point	NOUN
cana-5487	160	14	results	result	NOUN
cana-5487	160	15	for	for	ADP
cana-5487	160	16	hybrid	hybrid	ADJ
cana-5487	160	17	pairs	pair	NOUN
cana-5487	160	18	of	of	ADP
cana-5487	160	19	occasionally	occasionally	ADV
cana-5487	160	20	weakly	weakly	ADJ
cana-5487	160	21	compatible	compatible	ADJ
cana-5487	160	22	mappings	mapping	NOUN
cana-5487	160	23	defined	define	VERB
cana-5487	160	24	on	on	ADP
cana-5487	160	25	fuzzy	fuzzy	ADJ
cana-5487	160	26	metric	metric	ADJ
cana-5487	160	27	space	space	NOUN
cana-5487	160	28	,	,	PUNCT
cana-5487	160	29	"	"	PUNCT
cana-5487	160	30	*	*	PUNCT
cana-5487	160	31	annals	annal	NOUN
cana-5487	160	32	of	of	ADP
cana-5487	160	33	fuzzy	fuzzy	ADJ
cana-5487	160	34	mathematics	mathematic	NOUN
cana-5487	160	35	and	and	CCONJ
cana-5487	160	36	informatics	informatic	NOUN
cana-5487	160	37	*	*	PUNCT
cana-5487	160	38	,	,	PUNCT
cana-5487	160	39	vol	vol	NOUN
cana-5487	160	40	.	.	NOUN
cana-5487	161	1	9	9	NUM
cana-5487	161	2	,	,	PUNCT
cana-5487	161	3	no	no	INTJ
cana-5487	161	4	.	.	NOUN
cana-5487	161	5	6	6	NUM
cana-5487	161	6	,	,	PUNCT
cana-5487	161	7	pp	pp	ADJ
cana-5487	161	8	.	.	PUNCT
cana-5487	162	1	891–899	891–899	NUM
cana-5487	162	2	,	,	PUNCT
cana-5487	162	3	2015	2015	NUM
cana-5487	162	4	.	.	PUNCT
cana-5487	163	1	[	[	X
cana-5487	163	2	10	10	NUM
cana-5487	163	3	]	]	X
cana-5487	163	4	s.	s.	PROPN
cana-5487	163	5	sessa	sessa	PROPN
cana-5487	163	6	,	,	PUNCT
cana-5487	163	7	"	"	PUNCT
cana-5487	163	8	on	on	ADP
cana-5487	163	9	a	a	DET
cana-5487	163	10	weak	weak	ADJ
cana-5487	163	11	commutativity	commutativity	NOUN
cana-5487	163	12	condition	condition	NOUN
cana-5487	163	13	of	of	ADP
cana-5487	163	14	mappings	mapping	NOUN
cana-5487	163	15	in	in	ADP
cana-5487	163	16	fixed	fix	VERB
cana-5487	163	17	point	point	NOUN
cana-5487	163	18	considerations	consideration	NOUN
cana-5487	163	19	,	,	PUNCT
cana-5487	163	20	"	"	PUNCT
cana-5487	163	21	*	*	PUNCT
cana-5487	163	22	publications	publication	NOUN
cana-5487	163	23	de	de	X
cana-5487	163	24	l’institute	l’institute	X
cana-5487	163	25	mathématique	mathématique	NOUN
cana-5487	163	26	*	*	PROPN
cana-5487	163	27	,	,	PUNCT
cana-5487	163	28	vol	vol	NOUN
cana-5487	163	29	.	.	PROPN
cana-5487	163	30	32	32	NUM
cana-5487	163	31	,	,	PUNCT
cana-5487	163	32	no	no	INTJ
cana-5487	163	33	.	.	NOUN
cana-5487	163	34	46	46	NUM
cana-5487	163	35	,	,	PUNCT
cana-5487	163	36	pp	pp	ADJ
cana-5487	163	37	.	.	PUNCT
cana-5487	164	1	149–153	149–153	NUM
cana-5487	164	2	,	,	PUNCT
cana-5487	164	3	1982	1982	NUM
cana-5487	164	4	.	.	PUNCT
cana-5487	165	1	[	[	X
cana-5487	165	2	11	11	NUM
cana-5487	165	3	]	]	X
cana-5487	165	4	s.	s.	PROPN
cana-5487	165	5	sedghi	sedghi	PROPN
cana-5487	165	6	,	,	PUNCT
cana-5487	165	7	i.	i.	NOUN
cana-5487	165	8	altun	altun	PROPN
cana-5487	165	9	,	,	PUNCT
cana-5487	165	10	and	and	CCONJ
cana-5487	165	11	n.	n.	PROPN
cana-5487	165	12	shobe	shobe	PROPN
cana-5487	165	13	,	,	PUNCT
cana-5487	165	14	"	"	PUNCT
cana-5487	165	15	coupled	couple	VERB
cana-5487	165	16	fixed	fix	VERB
cana-5487	165	17	point	point	NOUN
cana-5487	165	18	theorems	theorem	NOUN
cana-5487	165	19	for	for	ADP
cana-5487	165	20	contractions	contraction	NOUN
cana-5487	165	21	in	in	ADP
cana-5487	165	22	fuzzy	fuzzy	ADJ
cana-5487	165	23	metric	metric	ADJ
cana-5487	165	24	spaces	space	NOUN
cana-5487	165	25	,	,	PUNCT
cana-5487	165	26	"	"	PUNCT
cana-5487	165	27	*	*	PUNCT
cana-5487	165	28	nonlinear	nonlinear	ADJ
cana-5487	165	29	analysis	analysis	NOUN
cana-5487	165	30	:	:	PUNCT
cana-5487	165	31	theory	theory	NOUN
cana-5487	165	32	,	,	PUNCT
cana-5487	165	33	methods	method	NOUN
cana-5487	165	34	&	&	CCONJ
cana-5487	165	35	applications	application	NOUN
cana-5487	165	36	*	*	PUNCT
cana-5487	165	37	,	,	PUNCT
cana-5487	165	38	vol	vol	NOUN
cana-5487	165	39	.	.	PROPN
cana-5487	165	40	72	72	NUM
cana-5487	165	41	,	,	PUNCT
cana-5487	165	42	no	no	INTJ
cana-5487	165	43	.	.	PUNCT
cana-5487	166	1	3–4	3–4	NUM
cana-5487	166	2	,	,	PUNCT
cana-5487	166	3	pp	pp	ADJ
cana-5487	166	4	.	.	PUNCT
cana-5487	167	1	4341–4349	4341–4349	NUM
cana-5487	167	2	,	,	PUNCT
cana-5487	167	3	2010	2010	NUM
cana-5487	167	4	.	.	PUNCT
cana-5487	168	1	[	[	X
cana-5487	168	2	12	12	NUM
cana-5487	168	3	]	]	X
cana-5487	168	4	s.	s.	PROPN
cana-5487	168	5	shukla	shukla	PROPN
cana-5487	168	6	and	and	CCONJ
cana-5487	168	7	p.	p.	PROPN
cana-5487	168	8	nigam	nigam	PROPN
cana-5487	168	9	,	,	PUNCT
cana-5487	168	10	"	"	PUNCT
cana-5487	168	11	coupled	couple	VERB
cana-5487	168	12	fixed	fix	VERB
cana-5487	168	13	point	point	NOUN
cana-5487	168	14	theorems	theorem	NOUN
cana-5487	168	15	for	for	ADP
cana-5487	168	16	occasionally	occasionally	ADV
cana-5487	168	17	weakly	weakly	ADJ
cana-5487	168	18	compatible	compatible	ADJ
cana-5487	168	19	mappings	mapping	NOUN
cana-5487	168	20	in	in	ADP
cana-5487	168	21	fuzzy	fuzzy	ADJ
cana-5487	168	22	metric	metric	ADJ
cana-5487	168	23	space	space	NOUN
cana-5487	168	24	,	,	PUNCT
cana-5487	168	25	"	"	PUNCT
cana-5487	168	26	*	*	PUNCT
cana-5487	168	27	advances	advance	NOUN
cana-5487	168	28	in	in	ADP
cana-5487	168	29	inequalities	inequality	NOUN
cana-5487	168	30	and	and	CCONJ
cana-5487	168	31	applications	application	NOUN
cana-5487	168	32	*	*	PUNCT
cana-5487	168	33	,	,	PUNCT
cana-5487	168	34	vol	vol	NOUN
cana-5487	168	35	.	.	NOUN
cana-5487	168	36	2019	2019	NUM
cana-5487	168	37	,	,	PUNCT
cana-5487	168	38	article	article	NOUN
cana-5487	168	39	i	i	PROPN
cana-5487	168	40	d	d	PROPN
cana-5487	168	41	2	2	NUM
cana-5487	168	42	,	,	PUNCT
cana-5487	168	43	14	14	NUM
cana-5487	168	44	pages	page	NOUN
cana-5487	168	45	,	,	PUNCT
cana-5487	168	46	2019	2019	NUM
cana-5487	168	47	.	.	PUNCT
cana-5487	169	1	[	[	X
cana-5487	169	2	13	13	NUM
cana-5487	169	3	]	]	SYM
cana-5487	169	4	x.-q	x.-q	PROPN
cana-5487	169	5	.	.	PUNCT
cana-5487	170	1	hu	hu	PROPN
cana-5487	170	2	,	,	PUNCT
cana-5487	170	3	"	"	PUNCT
cana-5487	170	4	common	common	ADJ
cana-5487	170	5	coupled	couple	VERB
cana-5487	170	6	fixed	fix	VERB
cana-5487	170	7	point	point	NOUN
cana-5487	170	8	theorems	theorem	NOUN
cana-5487	170	9	for	for	ADP
cana-5487	170	10	contractive	contractive	ADJ
cana-5487	170	11	mappings	mapping	NOUN
cana-5487	170	12	in	in	ADP
cana-5487	170	13	fuzzy	fuzzy	ADJ
cana-5487	170	14	metric	metric	ADJ
cana-5487	170	15	spaces	space	NOUN
cana-5487	170	16	,	,	PUNCT
cana-5487	170	17	"	"	PUNCT
cana-5487	170	18	*	*	VERB
cana-5487	170	19	fixed	fix	VERB
cana-5487	170	20	point	point	NOUN
cana-5487	170	21	theory	theory	NOUN
cana-5487	170	22	and	and	CCONJ
cana-5487	170	23	applications	application	NOUN
cana-5487	170	24	*	*	PUNCT
cana-5487	170	25	,	,	PUNCT
cana-5487	170	26	vol	vol	NOUN
cana-5487	170	27	.	.	NOUN
cana-5487	170	28	2011	2011	NUM
cana-5487	170	29	,	,	PUNCT
cana-5487	170	30	article	article	NOUN
cana-5487	170	31	i	i	PROPN
cana-5487	170	32	d	d	PROPN
cana-5487	170	33	363716	363716	NUM
cana-5487	170	34	,	,	PUNCT
cana-5487	170	35	14	14	NUM
cana-5487	170	36	pages	page	NOUN
cana-5487	170	37	,	,	PUNCT
cana-5487	170	38	2011	2011	NUM
cana-5487	170	39	.	.	PUNCT
cana-5487	171	1	[	[	X
cana-5487	171	2	14	14	NUM
cana-5487	171	3	]	]	X
cana-5487	171	4	l.	l.	PROPN
cana-5487	171	5	a.	a.	PROPN
cana-5487	171	6	zadeh	zadeh	PROPN
cana-5487	171	7	,	,	PUNCT
cana-5487	171	8	"	"	PUNCT
cana-5487	171	9	fuzzy	fuzzy	ADJ
cana-5487	171	10	sets	set	NOUN
cana-5487	171	11	,	,	PUNCT
cana-5487	171	12	"	"	PUNCT
cana-5487	171	13	*	*	PUNCT
cana-5487	171	14	information	information	NOUN
cana-5487	171	15	and	and	CCONJ
cana-5487	171	16	control	control	NOUN
cana-5487	171	17	*	*	NOUN
cana-5487	171	18	,	,	PUNCT
cana-5487	171	19	vol	vol	NOUN
cana-5487	171	20	.	.	NOUN
cana-5487	171	21	8	8	NUM
cana-5487	171	22	,	,	PUNCT
cana-5487	171	23	pp	pp	ADJ
cana-5487	171	24	.	.	PUNCT
cana-5487	172	1	338–353	338–353	NUM
cana-5487	172	2	,	,	PUNCT
cana-5487	172	3	1965	1965	NUM
cana-5487	172	4	.	.	PUNCT