id	sid	tid	token	lemma	pos
cana-1065	1	1	communications	communication	NOUN
cana-1065	1	2	on	on	ADP
cana-1065	1	3	applied	apply	VERB
cana-1065	1	4	nonlinear	nonlinear	ADJ
cana-1065	1	5	analysis	analysis	NOUN
cana-1065	1	6	issn	issn	NOUN
cana-1065	1	7	:	:	PUNCT
cana-1065	1	8	1074	1074	NUM
cana-1065	1	9	-	-	PUNCT
cana-1065	1	10	133x	133x	NUM
cana-1065	1	11	vol	vol	NOUN
cana-1065	1	12	31	31	NUM
cana-1065	1	13	no	no	NOUN
cana-1065	1	14	.	.	PUNCT
cana-1065	2	1	5s	5s	NUM
cana-1065	2	2	(	(	PUNCT
cana-1065	2	3	2024	2024	NUM
cana-1065	2	4	)	)	PUNCT
cana-1065	2	5	458	458	NUM
cana-1065	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	2	7	a	a	DET
cana-1065	2	8	common	common	ADJ
cana-1065	2	9	fixed	fix	VERB
cana-1065	2	10	point	point	NOUN
cana-1065	2	11	result	result	NOUN
cana-1065	2	12	in	in	ADP
cana-1065	2	13	menger	menger	PROPN
cana-1065	2	14	space	space	NOUN
cana-1065	2	15	*	*	PUNCT
cana-1065	2	16	1	1	NUM
cana-1065	2	17	ajay	ajay	PROPN
cana-1065	2	18	kumar	kumar	PROPN
cana-1065	2	19	chaudhary	chaudhary	PROPN
cana-1065	2	20	1department	1department	NUM
cana-1065	2	21	of	of	ADP
cana-1065	2	22	mathematics	mathematic	NOUN
cana-1065	2	23	,	,	PUNCT
cana-1065	2	24	trichandra	trichandra	VERB
cana-1065	2	25	multiple	multiple	ADJ
cana-1065	2	26	campus	campus	PROPN
cana-1065	2	27	tribhuvan	tribhuvan	PROPN
cana-1065	2	28	university	university	PROPN
cana-1065	2	29	,	,	PUNCT
cana-1065	2	30	kathmandu	kathmandu	NOUN
cana-1065	2	31	,	,	PUNCT
cana-1065	2	32	nepal	nepal	NOUN
cana-1065	2	33	*	*	PUNCT
cana-1065	2	34	corresponding	correspond	VERB
cana-1065	2	35	author	author	NOUN
cana-1065	2	36	e	e	NOUN
cana-1065	2	37	-	-	NOUN
cana-1065	2	38	mail	mail	NOUN
cana-1065	2	39	:	:	PUNCT
cana-1065	2	40	akcsaurya81@gmail.com	akcsaurya81@gmail.com	X
cana-1065	2	41	article	article	NOUN
cana-1065	2	42	history	history	NOUN
cana-1065	2	43	:	:	PUNCT
cana-1065	2	44	received	receive	VERB
cana-1065	2	45	:	:	PUNCT
cana-1065	2	46	13	13	NUM
cana-1065	2	47	-	-	SYM
cana-1065	2	48	05	05	NUM
cana-1065	2	49	-	-	PUNCT
cana-1065	2	50	2024	2024	NUM
cana-1065	2	51	revised	revise	VERB
cana-1065	2	52	:	:	PUNCT
cana-1065	2	53	23	23	NUM
cana-1065	2	54	-	-	SYM
cana-1065	2	55	06	06	NUM
cana-1065	2	56	-	-	PUNCT
cana-1065	2	57	2024	2024	NUM
cana-1065	2	58	accepted	accept	VERB
cana-1065	2	59	:	:	PUNCT
cana-1065	2	60	10	10	NUM
cana-1065	2	61	-	-	SYM
cana-1065	2	62	07	07	NUM
cana-1065	2	63	-	-	PUNCT
cana-1065	2	64	2024	2024	NUM
cana-1065	2	65	abstract	abstract	NOUN
cana-1065	2	66	:	:	PUNCT
cana-1065	2	67	by	by	ADP
cana-1065	2	68	using	use	VERB
cana-1065	2	69	compatibility	compatibility	NOUN
cana-1065	2	70	condition	condition	NOUN
cana-1065	2	71	type	type	NOUN
cana-1065	2	72	(	(	PUNCT
cana-1065	2	73	p	p	NOUN
cana-1065	2	74	)	)	PUNCT
cana-1065	2	75	in	in	ADP
cana-1065	2	76	probabilistic	probabilistic	ADJ
cana-1065	2	77	metric	metric	ADJ
cana-1065	2	78	space	space	NOUN
cana-1065	2	79	,	,	PUNCT
cana-1065	2	80	establish	establish	VERB
cana-1065	2	81	common	common	ADJ
cana-1065	2	82	fixed	fix	VERB
cana-1065	2	83	point	point	NOUN
cana-1065	2	84	results	result	NOUN
cana-1065	2	85	for	for	ADP
cana-1065	2	86	four	four	NUM
cana-1065	2	87	self	self	NOUN
cana-1065	2	88	-	-	PUNCT
cana-1065	2	89	mappings	mapping	NOUN
cana-1065	2	90	with	with	ADP
cana-1065	2	91	control	control	NOUN
cana-1065	2	92	function	function	NOUN
cana-1065	2	93	in	in	ADP
cana-1065	2	94	[	[	X
cana-1065	2	95	0,1	0,1	NUM
cana-1065	2	96	]	]	PUNCT
cana-1065	2	97	.	.	PUNCT
cana-1065	3	1	the	the	DET
cana-1065	3	2	result	result	NOUN
cana-1065	3	3	of	of	ADP
cana-1065	3	4	chaudhary	chaudhary	PROPN
cana-1065	3	5	et	et	PROPN
cana-1065	3	6	.	.	PUNCT
cana-1065	4	1	al	al	PROPN
cana-1065	5	1	[	[	X
cana-1065	5	2	5	5	NUM
cana-1065	5	3	]	]	PUNCT
cana-1065	5	4	is	be	AUX
cana-1065	5	5	a	a	DET
cana-1065	5	6	particular	particular	ADJ
cana-1065	5	7	case	case	NOUN
cana-1065	5	8	of	of	ADP
cana-1065	5	9	this	this	DET
cana-1065	5	10	new	new	ADJ
cana-1065	5	11	result	result	NOUN
cana-1065	5	12	and	and	CCONJ
cana-1065	5	13	it	it	PRON
cana-1065	5	14	extends	extend	VERB
cana-1065	5	15	and	and	CCONJ
cana-1065	5	16	generalizes	generalize	VERB
cana-1065	5	17	other	other	ADJ
cana-1065	5	18	similar	similar	ADJ
cana-1065	5	19	results	result	NOUN
cana-1065	5	20	in	in	ADP
cana-1065	5	21	the	the	DET
cana-1065	5	22	literature	literature	NOUN
cana-1065	5	23	.	.	PUNCT
cana-1065	6	1	keywords	keyword	NOUN
cana-1065	6	2	:	:	PUNCT
cana-1065	6	3	common	common	ADJ
cana-1065	6	4	fixed	fix	VERB
cana-1065	6	5	point	point	NOUN
cana-1065	6	6	,	,	PUNCT
cana-1065	6	7	menger	menger	PROPN
cana-1065	6	8	space	space	NOUN
cana-1065	6	9	,	,	PUNCT
cana-1065	6	10	compatible	compatible	ADJ
cana-1065	6	11	mappings	mapping	NOUN
cana-1065	6	12	,	,	PUNCT
cana-1065	6	13	compatible	compatible	ADJ
cana-1065	6	14	mappings	mapping	NOUN
cana-1065	6	15	of	of	ADP
cana-1065	6	16	type	type	NOUN
cana-1065	6	17	(	(	PUNCT
cana-1065	6	18	p	p	NOUN
cana-1065	6	19	)	)	PUNCT
cana-1065	6	20	.	.	PUNCT
cana-1065	7	1	mathematics	mathematic	NOUN
cana-1065	7	2	subject	subject	ADJ
cana-1065	7	3	classification	classification	NOUN
cana-1065	7	4	:	:	PUNCT
cana-1065	7	5	47h10	47h10	NUM
cana-1065	7	6	,	,	PUNCT
cana-1065	7	7	54h25	54h25	NUM
cana-1065	7	8	1	1	NUM
cana-1065	7	9	.	.	PUNCT
cana-1065	8	1	introduction	introduction	NOUN
cana-1065	8	2	:	:	PUNCT
cana-1065	8	3	probabilistic	probabilistic	ADJ
cana-1065	8	4	metric	metric	ADJ
cana-1065	8	5	space	space	NOUN
cana-1065	8	6	(	(	PUNCT
cana-1065	8	7	pm	pm	NOUN
cana-1065	8	8	space	space	NOUN
cana-1065	8	9	)	)	PUNCT
cana-1065	8	10	is	be	AUX
cana-1065	8	11	the	the	DET
cana-1065	8	12	idea	idea	NOUN
cana-1065	8	13	of	of	ADP
cana-1065	8	14	karl	karl	PROPN
cana-1065	8	15	’s	’s	PART
cana-1065	8	16	menger	menger	PROPN
cana-1065	9	1	[	[	X
cana-1065	9	2	11	11	NUM
cana-1065	9	3	]	]	PUNCT
cana-1065	9	4	,	,	PUNCT
cana-1065	9	5	a	a	DET
cana-1065	9	6	significant	significant	ADJ
cana-1065	9	7	generalization	generalization	NOUN
cana-1065	9	8	of	of	ADP
cana-1065	9	9	m.	m.	NOUN
cana-1065	9	10	frechet	frechet	PROPN
cana-1065	9	11	's	's	PART
cana-1065	9	12	[	[	X
cana-1065	9	13	3	3	NUM
cana-1065	9	14	]	]	X
cana-1065	9	15	metric	metric	ADJ
cana-1065	9	16	space	space	NOUN
cana-1065	9	17	.	.	PUNCT
cana-1065	10	1	if	if	SCONJ
cana-1065	10	2	pm	pm	NOUN
cana-1065	10	3	space	space	NOUN
cana-1065	10	4	includes	include	VERB
cana-1065	10	5	menger	menger	PROPN
cana-1065	10	6	inequality	inequality	NOUN
cana-1065	10	7	,	,	PUNCT
cana-1065	10	8	then	then	ADV
cana-1065	10	9	it	it	PRON
cana-1065	10	10	is	be	AUX
cana-1065	10	11	called	call	VERB
cana-1065	10	12	menger	menger	PROPN
cana-1065	10	13	space	space	NOUN
cana-1065	10	14	.	.	PUNCT
cana-1065	11	1	this	this	DET
cana-1065	11	2	space	space	NOUN
cana-1065	11	3	becomes	become	VERB
cana-1065	11	4	active	active	ADJ
cana-1065	11	5	after	after	ADP
cana-1065	11	6	the	the	DET
cana-1065	11	7	significant	significant	ADJ
cana-1065	11	8	work	work	NOUN
cana-1065	11	9	of	of	ADP
cana-1065	11	10	b.	b.	PROPN
cana-1065	11	11	schweizer	schweizer	PROPN
cana-1065	11	12	and	and	CCONJ
cana-1065	11	13	a.	a.	NOUN
cana-1065	11	14	skalar	skalar	PROPN
cana-1065	12	1	[	[	X
cana-1065	12	2	13	13	NUM
cana-1065	12	3	]	]	PUNCT
cana-1065	12	4	,	,	PUNCT
cana-1065	13	1	[	[	X
cana-1065	13	2	16	16	NUM
cana-1065	13	3	]	]	PUNCT
cana-1065	13	4	and	and	CCONJ
cana-1065	13	5	v.m	v.m	PROPN
cana-1065	13	6	.	.	PROPN
cana-1065	13	7	sehgal	sehgal	PROPN
cana-1065	13	8	and	and	CCONJ
cana-1065	13	9	a.t	a.t	PROPN
cana-1065	13	10	.	.	PROPN
cana-1065	13	11	barucha	barucha	PROPN
cana-1065	13	12	reid	reid	PROPN
cana-1065	14	1	[	[	X
cana-1065	14	2	14	14	NUM
cana-1065	14	3	]	]	PUNCT
cana-1065	14	4	.	.	PUNCT
cana-1065	15	1	in	in	ADP
cana-1065	15	2	1991	1991	NUM
cana-1065	15	3	,	,	PUNCT
cana-1065	15	4	s.	s.	PROPN
cana-1065	15	5	n.	n.	PROPN
cana-1065	15	6	mishra	mishra	PROPN
cana-1065	16	1	[	[	X
cana-1065	16	2	12	12	NUM
cana-1065	16	3	]	]	PUNCT
cana-1065	16	4	introduced	introduce	VERB
cana-1065	16	5	the	the	DET
cana-1065	16	6	notion	notion	NOUN
cana-1065	16	7	of	of	ADP
cana-1065	16	8	compatible	compatible	ADJ
cana-1065	16	9	mapping	mapping	NOUN
cana-1065	16	10	in	in	ADP
cana-1065	16	11	the	the	DET
cana-1065	16	12	menger	menger	PROPN
cana-1065	16	13	space	space	NOUN
cana-1065	16	14	and	and	CCONJ
cana-1065	16	15	then	then	ADV
cana-1065	16	16	so	so	ADV
cana-1065	16	17	many	many	ADJ
cana-1065	16	18	researchers	researcher	NOUN
cana-1065	16	19	worked	work	VERB
cana-1065	16	20	in	in	ADP
cana-1065	16	21	this	this	DET
cana-1065	16	22	space	space	NOUN
cana-1065	16	23	,	,	PUNCT
cana-1065	16	24	defining	define	VERB
cana-1065	16	25	weakly	weakly	ADJ
cana-1065	16	26	compatible	compatible	ADJ
cana-1065	16	27	mappings	mapping	NOUN
cana-1065	16	28	,	,	PUNCT
cana-1065	16	29	different	different	ADJ
cana-1065	16	30	compatible	compatible	ADJ
cana-1065	16	31	mappings	mapping	NOUN
cana-1065	16	32	types	type	NOUN
cana-1065	16	33	like	like	ADP
cana-1065	16	34	(	(	PUNCT
cana-1065	16	35	a	a	NOUN
cana-1065	16	36	)	)	PUNCT
cana-1065	16	37	,	,	PUNCT
cana-1065	16	38	(	(	PUNCT
cana-1065	16	39	k	k	NOUN
cana-1065	16	40	)	)	PUNCT
cana-1065	16	41	,	,	PUNCT
cana-1065	16	42	(	(	PUNCT
cana-1065	16	43	p	p	NOUN
cana-1065	16	44	)	)	PUNCT
cana-1065	16	45	etc	etc	X
cana-1065	16	46	.	.	X
cana-1065	16	47	see	see	VERB
cana-1065	16	48	references	reference	NOUN
cana-1065	16	49	[	[	X
cana-1065	16	50	[	[	X
cana-1065	16	51	2	2	NUM
cana-1065	16	52	]	]	PUNCT
cana-1065	16	53	,	,	PUNCT
cana-1065	16	54	[	[	X
cana-1065	16	55	6	6	NUM
cana-1065	16	56	]	]	PUNCT
cana-1065	16	57	,	,	PUNCT
cana-1065	16	58	[	[	X
cana-1065	16	59	7	7	NUM
cana-1065	16	60	]	]	PUNCT
cana-1065	16	61	,	,	PUNCT
cana-1065	16	62	[	[	X
cana-1065	16	63	8	8	NUM
cana-1065	16	64	]	]	PUNCT
cana-1065	16	65	,	,	PUNCT
cana-1065	17	1	[	[	X
cana-1065	17	2	9	9	NUM
cana-1065	17	3	]	]	PUNCT
cana-1065	17	4	,	,	PUNCT
cana-1065	17	5	[	[	X
cana-1065	17	6	10	10	NUM
cana-1065	17	7	]	]	PUNCT
cana-1065	17	8	,	,	PUNCT
cana-1065	17	9	[	[	X
cana-1065	17	10	14	14	NUM
cana-1065	17	11	]	]	PUNCT
cana-1065	17	12	,	,	PUNCT
cana-1065	18	1	[	[	X
cana-1065	18	2	16	16	NUM
cana-1065	18	3	]	]	X
cana-1065	18	4	]	]	PUNCT
cana-1065	18	5	.	.	PUNCT
cana-1065	19	1	recently	recently	ADV
cana-1065	19	2	,	,	PUNCT
cana-1065	19	3	chaudhary	chaudhary	PROPN
cana-1065	19	4	et	et	PROPN
cana-1065	19	5	.	.	PUNCT
cana-1065	20	1	al	al	PROPN
cana-1065	21	1	[	[	X
cana-1065	21	2	5	5	NUM
cana-1065	21	3	-	-	SYM
cana-1065	21	4	6	6	NUM
cana-1065	21	5	]	]	PUNCT
cana-1065	21	6	have	have	AUX
cana-1065	21	7	given	give	VERB
cana-1065	21	8	notions	notion	NOUN
cana-1065	21	9	of	of	ADP
cana-1065	21	10	compatible	compatible	ADJ
cana-1065	21	11	mapping	mapping	NOUN
cana-1065	21	12	of	of	ADP
cana-1065	21	13	type	type	NOUN
cana-1065	21	14	(	(	PUNCT
cana-1065	21	15	p	p	NOUN
cana-1065	21	16	)	)	PUNCT
cana-1065	21	17	and	and	CCONJ
cana-1065	21	18	weakly	weakly	ADJ
cana-1065	21	19	compatible	compatible	ADJ
cana-1065	21	20	mappings	mapping	NOUN
cana-1065	21	21	of	of	ADP
cana-1065	21	22	type	type	NOUN
cana-1065	21	23	(	(	PUNCT
cana-1065	21	24	p	p	NOUN
cana-1065	21	25	)	)	PUNCT
cana-1065	21	26	.	.	PUNCT
cana-1065	22	1	this	this	DET
cana-1065	22	2	paper	paper	NOUN
cana-1065	22	3	gives	give	VERB
cana-1065	22	4	the	the	DET
cana-1065	22	5	new	new	ADJ
cana-1065	22	6	results	result	NOUN
cana-1065	22	7	in	in	ADP
cana-1065	22	8	menger	menger	NOUN
cana-1065	22	9	space	space	NOUN
cana-1065	22	10	by	by	ADP
cana-1065	22	11	using	use	VERB
cana-1065	22	12	a	a	DET
cana-1065	22	13	control	control	NOUN
cana-1065	22	14	function	function	NOUN
cana-1065	22	15	φ	φ	NOUN
cana-1065	22	16	:	:	PUNCT
cana-1065	23	1	[	[	X
cana-1065	23	2	0,1	0,1	NUM
cana-1065	23	3	]	]	PUNCT
cana-1065	23	4	→	→	PUNCT
cana-1065	24	1	[	[	X
cana-1065	24	2	0,1	0,1	NUM
cana-1065	24	3	]	]	PUNCT
cana-1065	24	4	in	in	ADP
cana-1065	24	5	four	four	NUM
cana-1065	24	6	self	self	NOUN
cana-1065	24	7	-	-	PUNCT
cana-1065	24	8	mappings	mapping	NOUN
cana-1065	24	9	and	and	CCONJ
cana-1065	24	10	also	also	ADV
cana-1065	24	11	deduces	deduce	VERB
cana-1065	24	12	some	some	DET
cana-1065	24	13	consequences	consequence	NOUN
cana-1065	24	14	.	.	PUNCT
cana-1065	25	1	2	2	X
cana-1065	25	2	.	.	X
cana-1065	25	3	preliminaries	preliminary	NOUN
cana-1065	25	4	:	:	PUNCT
cana-1065	25	5	definition	definition	NOUN
cana-1065	25	6	2.1	2.1	NUM
cana-1065	26	1	[	[	X
cana-1065	26	2	16	16	NUM
cana-1065	26	3	]	]	X
cana-1065	26	4	:	:	PUNCT
cana-1065	26	5	if	if	SCONJ
cana-1065	26	6	a	a	DET
cana-1065	26	7	function	function	NOUN
cana-1065	26	8	𝑀	𝑀	NOUN
cana-1065	26	9	:	:	PUNCT
cana-1065	26	10	ℝ	ℝ	PROPN
cana-1065	26	11	→	→	SYM
cana-1065	26	12	ℝ+	ℝ+	PUNCT
cana-1065	26	13	is	be	AUX
cana-1065	26	14	(	(	PUNCT
cana-1065	26	15	i	i	NOUN
cana-1065	26	16	)	)	PUNCT
cana-1065	26	17	a	a	DET
cana-1065	26	18	non	non	ADJ
cana-1065	26	19	-	-	ADJ
cana-1065	26	20	decreasing	decrease	VERB
cana-1065	26	21	function	function	NOUN
cana-1065	26	22	,	,	PUNCT
cana-1065	26	23	(	(	PUNCT
cana-1065	26	24	ii	ii	NOUN
cana-1065	26	25	)	)	PUNCT
cana-1065	26	26	left	leave	VERB
cana-1065	26	27	continuous	continuous	ADJ
cana-1065	26	28	and	and	CCONJ
cana-1065	26	29	(	(	PUNCT
cana-1065	26	30	iii	iii	X
cana-1065	26	31	)	)	PUNCT
cana-1065	26	32	inf	inf	NOUN
cana-1065	26	33	{	{	PUNCT
cana-1065	26	34	𝐹	𝐹	PROPN
cana-1065	26	35	(	(	PUNCT
cana-1065	26	36	𝑥	𝑥	NOUN
cana-1065	26	37	):	):	PUNCT
cana-1065	26	38	𝑥	𝑥	PROPN
cana-1065	26	39	∈	∈	PROPN
cana-1065	26	40	ℝ	ℝ	PROPN
cana-1065	26	41	}	}	PUNCT
cana-1065	26	42	=	=	SYM
cana-1065	26	43	0	0	NUM
cana-1065	26	44	,	,	PUNCT
cana-1065	26	45	sup{𝐹	sup{𝐹	PROPN
cana-1065	26	46	(	(	PUNCT
cana-1065	26	47	𝑥	𝑥	NOUN
cana-1065	26	48	):	):	PUNCT
cana-1065	26	49	𝑥	𝑥	PROPN
cana-1065	26	50	∈	∈	PROPN
cana-1065	26	51	ℝ	ℝ	PROPN
cana-1065	26	52	}	}	PUNCT
cana-1065	26	53	=	=	SYM
cana-1065	26	54	1	1	NUM
cana-1065	26	55	then	then	ADV
cana-1065	26	56	𝑀	𝑀	PROPN
cana-1065	26	57	is	be	AUX
cana-1065	26	58	said	say	VERB
cana-1065	26	59	to	to	PART
cana-1065	26	60	be	be	AUX
cana-1065	26	61	a	a	DET
cana-1065	26	62	distribution	distribution	NOUN
cana-1065	26	63	function	function	NOUN
cana-1065	26	64	.	.	PUNCT
cana-1065	27	1	definition	definition	NOUN
cana-1065	27	2	2.2	2.2	NUM
cana-1065	28	1	[	[	X
cana-1065	28	2	4	4	NUM
cana-1065	28	3	]	]	PUNCT
cana-1065	28	4	:	:	PUNCT
cana-1065	28	5	let	let	VERB
cana-1065	28	6	𝑀	𝑀	PROPN
cana-1065	28	7	:	:	PUNCT
cana-1065	28	8	𝑌	𝑌	PROPN
cana-1065	28	9	×	×	NOUN
cana-1065	28	10	𝑌	𝑌	PROPN
cana-1065	28	11	→	→	SYM
cana-1065	28	12	𝐿	𝐿	PROPN
cana-1065	28	13	be	be	AUX
cana-1065	28	14	a	a	DET
cana-1065	28	15	distribution	distribution	NOUN
cana-1065	28	16	function	function	NOUN
cana-1065	28	17	,	,	PUNCT
cana-1065	28	18	𝐿	𝐿	PROPN
cana-1065	28	19	be	be	VERB
cana-1065	28	20	the	the	DET
cana-1065	28	21	set	set	NOUN
cana-1065	28	22	of	of	ADP
cana-1065	28	23	all	all	DET
cana-1065	28	24	distribution	distribution	NOUN
cana-1065	28	25	functions	function	NOUN
cana-1065	28	26	and	and	CCONJ
cana-1065	28	27	𝑌	𝑌	PROPN
cana-1065	28	28	be	be	VERB
cana-1065	28	29	a	a	DET
cana-1065	28	30	non	non	ADJ
cana-1065	28	31	-	-	ADJ
cana-1065	28	32	empty	empty	ADJ
cana-1065	28	33	set	set	NOUN
cana-1065	28	34	.	.	PUNCT
cana-1065	29	1	then	then	ADV
cana-1065	29	2	,	,	PUNCT
cana-1065	29	3	a	a	DET
cana-1065	29	4	pair	pair	NOUN
cana-1065	29	5	(	(	PUNCT
cana-1065	29	6	𝑌	𝑌	PROPN
cana-1065	29	7	,	,	PUNCT
cana-1065	29	8	𝑀	𝑀	PROPN
cana-1065	29	9	)	)	PUNCT
cana-1065	29	10	is	be	AUX
cana-1065	29	11	said	say	VERB
cana-1065	29	12	to	to	PART
cana-1065	29	13	be	be	AUX
cana-1065	29	14	probabilistic	probabilistic	ADJ
cana-1065	29	15	metric	metric	ADJ
cana-1065	29	16	space	space	NOUN
cana-1065	29	17	(	(	PUNCT
cana-1065	29	18	abbreviated	abbreviate	VERB
cana-1065	29	19	as	as	ADP
cana-1065	29	20	pm	pm	NOUN
cana-1065	29	21	-	-	PUNCT
cana-1065	29	22	space	space	NOUN
cana-1065	29	23	)	)	PUNCT
cana-1065	29	24	if	if	SCONJ
cana-1065	29	25	the	the	DET
cana-1065	29	26	distribution	distribution	NOUN
cana-1065	29	27	function	function	NOUN
cana-1065	29	28	𝑀	𝑀	PROPN
cana-1065	29	29	(	(	PUNCT
cana-1065	29	30	𝑝	𝑝	PROPN
cana-1065	29	31	,	,	PUNCT
cana-1065	29	32	𝑞	𝑞	NOUN
cana-1065	29	33	)	)	PUNCT
cana-1065	29	34	,	,	PUNCT
cana-1065	29	35	where	where	SCONJ
cana-1065	29	36	(	(	PUNCT
cana-1065	29	37	𝑝	𝑝	NOUN
cana-1065	29	38	,	,	PUNCT
cana-1065	29	39	𝑞	𝑞	NOUN
cana-1065	29	40	)	)	PUNCT
cana-1065	29	41	∈	∈	PROPN
cana-1065	29	42	𝑌	𝑌	PROPN
cana-1065	29	43	×	×	PROPN
cana-1065	29	44	𝑌	𝑌	PROPN
cana-1065	29	45	,	,	PUNCT
cana-1065	29	46	also	also	ADV
cana-1065	29	47	denoted	denote	VERB
cana-1065	29	48	by	by	ADP
cana-1065	29	49	𝑀𝑝,𝑞	𝑀𝑝,𝑞	PROPN
cana-1065	29	50	satisfies	satisfie	NOUN
cana-1065	29	51	following	follow	VERB
cana-1065	29	52	conditions	condition	NOUN
cana-1065	29	53	:	:	PUNCT
cana-1065	29	54	(	(	PUNCT
cana-1065	29	55	m1	m1	NOUN
cana-1065	29	56	)	)	PUNCT
cana-1065	29	57	𝑀𝑝,𝑞(𝑥	𝑀𝑝,𝑞(𝑥	ADV
cana-1065	29	58	)	)	PUNCT
cana-1065	29	59	=	=	SYM
cana-1065	29	60	1	1	NUM
cana-1065	29	61	for	for	ADP
cana-1065	29	62	every	every	DET
cana-1065	29	63	𝑥	𝑥	PROPN
cana-1065	29	64	>	>	X
cana-1065	29	65	0	0	PUNCT
cana-1065	30	1	if	if	SCONJ
cana-1065	30	2	and	and	CCONJ
cana-1065	30	3	only	only	ADV
cana-1065	30	4	if	if	SCONJ
cana-1065	30	5	𝑝	𝑝	PROPN
cana-1065	30	6	=	=	SYM
cana-1065	30	7	𝑞	𝑞	PROPN
cana-1065	30	8	,	,	PUNCT
cana-1065	30	9	communications	communication	NOUN
cana-1065	30	10	on	on	ADP
cana-1065	30	11	applied	apply	VERB
cana-1065	30	12	nonlinear	nonlinear	ADJ
cana-1065	30	13	analysis	analysis	NOUN
cana-1065	30	14	issn	issn	NOUN
cana-1065	30	15	:	:	PUNCT
cana-1065	30	16	1074	1074	NUM
cana-1065	30	17	-	-	PUNCT
cana-1065	30	18	133x	133x	NUM
cana-1065	30	19	vol	vol	NOUN
cana-1065	30	20	31	31	NUM
cana-1065	30	21	no	no	NOUN
cana-1065	30	22	.	.	PUNCT
cana-1065	31	1	5s	5s	NUM
cana-1065	31	2	(	(	PUNCT
cana-1065	31	3	2024	2024	NUM
cana-1065	31	4	)	)	PUNCT
cana-1065	31	5	459	459	NUM
cana-1065	31	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	31	7	(	(	PUNCT
cana-1065	31	8	m2	m2	PROPN
cana-1065	31	9	)	)	PUNCT
cana-1065	31	10	𝑀𝑝,𝑞(0	𝑀𝑝,𝑞(0	NOUN
cana-1065	31	11	)	)	PUNCT
cana-1065	31	12	=	=	SYM
cana-1065	31	13	0	0	NUM
cana-1065	31	14	for	for	ADP
cana-1065	31	15	every	every	DET
cana-1065	31	16	𝑝	𝑝	NOUN
cana-1065	31	17	,	,	PUNCT
cana-1065	31	18	𝑞	𝑞	PROPN
cana-1065	31	19	∈	∈	PROPN
cana-1065	31	20	𝐾	𝐾	PROPN
cana-1065	31	21	,	,	PUNCT
cana-1065	31	22	(	(	PUNCT
cana-1065	31	23	m3	m3	PROPN
cana-1065	31	24	)	)	PUNCT
cana-1065	31	25	𝑀𝑝,𝑞(𝑥	𝑀𝑝,𝑞(𝑥	ADV
cana-1065	31	26	)	)	PUNCT
cana-1065	31	27	=	=	SYM
cana-1065	32	1	𝑀𝑞,𝑝(𝑥	𝑀𝑞,𝑝(𝑥	NOUN
cana-1065	32	2	)	)	PUNCT
cana-1065	32	3	for	for	ADP
cana-1065	32	4	every	every	DET
cana-1065	32	5	𝑝	𝑝	NOUN
cana-1065	32	6	,	,	PUNCT
cana-1065	32	7	𝑞	𝑞	PROPN
cana-1065	32	8	∈	∈	PROPN
cana-1065	32	9	𝐾	𝐾	PROPN
cana-1065	32	10	,	,	PUNCT
cana-1065	32	11	and	and	CCONJ
cana-1065	32	12	(	(	PUNCT
cana-1065	32	13	m4	m4	PROPN
cana-1065	32	14	)	)	PUNCT
cana-1065	32	15	𝑀𝑝,𝑞(𝑥	𝑀𝑝,𝑞(𝑥	NOUN
cana-1065	32	16	+	+	CCONJ
cana-1065	32	17	𝑦	𝑦	X
cana-1065	32	18	)	)	PUNCT
cana-1065	32	19	=	=	SYM
cana-1065	32	20	1	1	NUM
cana-1065	32	21	if	if	SCONJ
cana-1065	32	22	and	and	CCONJ
cana-1065	32	23	only	only	ADV
cana-1065	32	24	if	if	SCONJ
cana-1065	32	25	𝑀𝑝,𝑟(𝑥	𝑀𝑝,𝑟(𝑥	NOUN
cana-1065	32	26	)	)	PUNCT
cana-1065	32	27	=	=	SYM
cana-1065	32	28	1	1	NUM
cana-1065	32	29	and	and	CCONJ
cana-1065	32	30	𝑀𝑟,𝑞(𝑦	𝑀𝑟,𝑞(𝑦	NUM
cana-1065	32	31	)	)	PUNCT
cana-1065	32	32	=	=	SYM
cana-1065	33	1	1	1	X
cana-1065	33	2	.	.	PUNCT
cana-1065	33	3	here	here	ADV
cana-1065	33	4	,	,	PUNCT
cana-1065	33	5	𝑀𝑝,𝑞(𝑥	𝑀𝑝,𝑞(𝑥	ADV
cana-1065	33	6	)	)	PUNCT
cana-1065	33	7	represents	represent	VERB
cana-1065	33	8	the	the	DET
cana-1065	33	9	value	value	NOUN
cana-1065	33	10	of	of	ADP
cana-1065	33	11	distribution	distribution	NOUN
cana-1065	33	12	function	function	NOUN
cana-1065	33	13	𝑀𝑝,𝑞	𝑀𝑝,𝑞	NOUN
cana-1065	33	14	𝑎𝑡	𝑎𝑡	ADP
cana-1065	33	15	𝑥	𝑥	DET
cana-1065	33	16	∈	∈	PROPN
cana-1065	33	17	ℝ.	ℝ.	PROPN
cana-1065	33	18	definition	definition	NOUN
cana-1065	33	19	2.3	2.3	NUM
cana-1065	34	1	[	[	X
cana-1065	34	2	4	4	NUM
cana-1065	34	3	]	]	X
cana-1065	34	4	:	:	PUNCT
cana-1065	34	5	a	a	DET
cana-1065	34	6	function	function	NOUN
cana-1065	34	7	𝑡	𝑡	X
cana-1065	34	8	∶	∶	NOUN
cana-1065	34	9	[	[	X
cana-1065	34	10	0	0	NUM
cana-1065	34	11	,	,	PUNCT
cana-1065	34	12	1	1	NUM
cana-1065	34	13	]	]	SYM
cana-1065	34	14	×	×	NOUN
cana-1065	34	15	[	[	X
cana-1065	34	16	0	0	NUM
cana-1065	34	17	,	,	PUNCT
cana-1065	34	18	1	1	NUM
cana-1065	34	19	]	]	PUNCT
cana-1065	34	20	→	→	PUNCT
cana-1065	34	21	[	[	X
cana-1065	34	22	0	0	NUM
cana-1065	34	23	,	,	PUNCT
cana-1065	34	24	1	1	NUM
cana-1065	34	25	]	]	PUNCT
cana-1065	34	26	is	be	AUX
cana-1065	34	27	referred	refer	VERB
cana-1065	34	28	to	to	ADP
cana-1065	34	29	as	as	ADP
cana-1065	34	30	a	a	DET
cana-1065	34	31	triangular	triangular	NOUN
cana-1065	34	32	norm	norm	NOUN
cana-1065	34	33	(	(	PUNCT
cana-1065	34	34	shortly	shortly	ADV
cana-1065	34	35	t	t	NOUN
cana-1065	34	36	-	-	PUNCT
cana-1065	34	37	norm	norm	NOUN
cana-1065	34	38	)	)	PUNCT
cana-1065	34	39	if	if	SCONJ
cana-1065	34	40	it	it	PRON
cana-1065	34	41	satisfies	satisfy	VERB
cana-1065	34	42	the	the	DET
cana-1065	34	43	following	follow	VERB
cana-1065	34	44	conditions	condition	NOUN
cana-1065	34	45	:	:	PUNCT
cana-1065	34	46	t1	t1	NOUN
cana-1065	34	47	:	:	PUNCT
cana-1065	34	48	𝑡	𝑡	PROPN
cana-1065	34	49	(	(	PUNCT
cana-1065	34	50	0	0	NUM
cana-1065	34	51	,	,	PUNCT
cana-1065	34	52	0	0	NUM
cana-1065	34	53	)	)	PUNCT
cana-1065	34	54	=	=	SYM
cana-1065	34	55	0	0	NUM
cana-1065	34	56	,	,	PUNCT
cana-1065	34	57	t2	t2	NOUN
cana-1065	34	58	:	:	PUNCT
cana-1065	34	59	𝑡	𝑡	PROPN
cana-1065	34	60	(	(	PUNCT
cana-1065	34	61	𝑎	𝑎	NOUN
cana-1065	34	62	,	,	PUNCT
cana-1065	34	63	1	1	NUM
cana-1065	34	64	)	)	PUNCT
cana-1065	34	65	=	=	NOUN
cana-1065	35	1	𝑎	𝑎	NOUN
cana-1065	35	2	for	for	ADP
cana-1065	35	3	all	all	DET
cana-1065	35	4	𝑎	𝑎	PRON
cana-1065	35	5	∈	∈	NOUN
cana-1065	36	1	[	[	X
cana-1065	36	2	0	0	NUM
cana-1065	36	3	,	,	PUNCT
cana-1065	36	4	1	1	NUM
cana-1065	36	5	]	]	PUNCT
cana-1065	36	6	,	,	PUNCT
cana-1065	36	7	t3	t3	NOUN
cana-1065	36	8	:	:	PUNCT
cana-1065	36	9	𝑡	𝑡	PROPN
cana-1065	36	10	(	(	PUNCT
cana-1065	36	11	𝑎	𝑎	X
cana-1065	36	12	,	,	PUNCT
cana-1065	36	13	𝑏	𝑏	NOUN
cana-1065	36	14	)	)	PUNCT
cana-1065	36	15	=	=	SYM
cana-1065	36	16	𝑡	𝑡	PROPN
cana-1065	36	17	(	(	PUNCT
cana-1065	36	18	𝑏	𝑏	NOUN
cana-1065	36	19	,	,	PUNCT
cana-1065	36	20	𝑎	𝑎	NOUN
cana-1065	36	21	)	)	PUNCT
cana-1065	36	22	for	for	ADP
cana-1065	36	23	all	all	DET
cana-1065	36	24	𝑎	𝑎	NOUN
cana-1065	36	25	,	,	PUNCT
cana-1065	36	26	𝑏	𝑏	PROPN
cana-1065	36	27	∈	∈	PROPN
cana-1065	37	1	[	[	X
cana-1065	37	2	0	0	NUM
cana-1065	37	3	,	,	PUNCT
cana-1065	37	4	1	1	NUM
cana-1065	37	5	]	]	PUNCT
cana-1065	37	6	,	,	PUNCT
cana-1065	37	7	t4	t4	PROPN
cana-1065	37	8	:	:	PUNCT
cana-1065	37	9	𝑖𝑓	𝑖𝑓	NUM
cana-1065	37	10	𝑎	𝑎	X
cana-1065	37	11	≤	≤	NUM
cana-1065	37	12	𝑐	𝑐	NOUN
cana-1065	37	13	,	,	PUNCT
cana-1065	37	14	𝑏	𝑏	PROPN
cana-1065	37	15	≤	≤	NOUN
cana-1065	37	16	𝑑	𝑑	PROPN
cana-1065	37	17	then	then	ADV
cana-1065	37	18	𝑡	𝑡	X
cana-1065	37	19	(	(	PUNCT
cana-1065	37	20	𝑎	𝑎	PROPN
cana-1065	37	21	,	,	PUNCT
cana-1065	37	22	𝑏	𝑏	NOUN
cana-1065	37	23	)	)	PUNCT
cana-1065	37	24	≤	≤	NUM
cana-1065	37	25	𝑡	𝑡	PROPN
cana-1065	37	26	(	(	PUNCT
cana-1065	37	27	𝑐	𝑐	PROPN
cana-1065	37	28	,	,	PUNCT
cana-1065	37	29	𝑑	𝑑	NOUN
cana-1065	37	30	)	)	PUNCT
cana-1065	37	31	,	,	PUNCT
cana-1065	37	32	and	and	CCONJ
cana-1065	37	33	t5	t5	PROPN
cana-1065	37	34	:	:	PUNCT
cana-1065	37	35	𝑡	𝑡	PROPN
cana-1065	37	36	(	(	PUNCT
cana-1065	37	37	𝑡	𝑡	PROPN
cana-1065	37	38	(	(	PUNCT
cana-1065	37	39	𝑎	𝑎	PROPN
cana-1065	37	40	,	,	PUNCT
cana-1065	37	41	𝑏	𝑏	NOUN
cana-1065	37	42	)	)	PUNCT
cana-1065	37	43	,	,	PUNCT
cana-1065	37	44	𝑐	𝑐	NOUN
cana-1065	37	45	)	)	PUNCT
cana-1065	37	46	=	=	SYM
cana-1065	38	1	𝑡	𝑡	PROPN
cana-1065	38	2	(	(	PUNCT
cana-1065	38	3	𝑎	𝑎	X
cana-1065	38	4	,	,	PUNCT
cana-1065	38	5	𝑡	𝑡	X
cana-1065	38	6	(	(	PUNCT
cana-1065	38	7	𝑏	𝑏	NOUN
cana-1065	38	8	,	,	PUNCT
cana-1065	38	9	𝑐	𝑐	NOUN
cana-1065	38	10	)	)	PUNCT
cana-1065	38	11	)	)	PUNCT
cana-1065	38	12	,	,	PUNCT
cana-1065	38	13	where	where	SCONJ
cana-1065	38	14	𝑎	𝑎	X
cana-1065	38	15	,	,	PUNCT
cana-1065	38	16	𝑏	𝑏	NOUN
cana-1065	38	17	,	,	PUNCT
cana-1065	38	18	𝑐	𝑐	PROPN
cana-1065	38	19	,	,	PUNCT
cana-1065	38	20	𝑑	𝑑	PROPN
cana-1065	38	21	∈	∈	PROPN
cana-1065	39	1	[	[	X
cana-1065	39	2	0	0	NUM
cana-1065	39	3	,	,	PUNCT
cana-1065	39	4	1	1	NUM
cana-1065	39	5	]	]	PUNCT
cana-1065	39	6	.	.	PUNCT
cana-1065	40	1	definition	definition	NOUN
cana-1065	40	2	2.4	2.4	NUM
cana-1065	40	3	[	[	X
cana-1065	40	4	2	2	NUM
cana-1065	40	5	]	]	PUNCT
cana-1065	40	6	:	:	PUNCT
cana-1065	40	7	a	a	DET
cana-1065	40	8	probabilistic	probabilistic	ADJ
cana-1065	40	9	metric	metric	ADJ
cana-1065	40	10	space	space	NOUN
cana-1065	40	11	(	(	PUNCT
cana-1065	40	12	𝑌	𝑌	PROPN
cana-1065	40	13	,	,	PUNCT
cana-1065	40	14	𝑀	𝑀	PROPN
cana-1065	40	15	)	)	PUNCT
cana-1065	40	16	is	be	AUX
cana-1065	40	17	said	say	VERB
cana-1065	40	18	to	to	PART
cana-1065	40	19	be	be	AUX
cana-1065	40	20	menger	menger	NOUN
cana-1065	40	21	space	space	NOUN
cana-1065	40	22	(	(	PUNCT
cana-1065	40	23	𝑌	𝑌	PROPN
cana-1065	40	24	,	,	PUNCT
cana-1065	40	25	𝑀	𝑀	PROPN
cana-1065	40	26	,	,	PUNCT
cana-1065	40	27	𝑡	𝑡	PROPN
cana-1065	40	28	)	)	PUNCT
cana-1065	40	29	,	,	PUNCT
cana-1065	40	30	where	where	SCONJ
cana-1065	40	31	t	t	PROPN
cana-1065	40	32	is	be	AUX
cana-1065	40	33	a	a	DET
cana-1065	40	34	t	t	NOUN
cana-1065	40	35	-	-	PUNCT
cana-1065	40	36	norm	norm	NOUN
cana-1065	40	37	satisfying	satisfy	VERB
cana-1065	40	38	the	the	DET
cana-1065	40	39	following	follow	VERB
cana-1065	40	40	conditions	condition	NOUN
cana-1065	40	41	:	:	PUNCT
cana-1065	40	42	(	(	PUNCT
cana-1065	40	43	m5	m5	NOUN
cana-1065	40	44	)	)	PUNCT
cana-1065	40	45	𝑀𝑝,𝑞(𝑥	𝑀𝑝,𝑞(𝑥	NOUN
cana-1065	40	46	+	+	CCONJ
cana-1065	40	47	𝑦	𝑦	X
cana-1065	40	48	)	)	PUNCT
cana-1065	40	49	≥	≥	NOUN
cana-1065	40	50	𝑡	𝑡	PROPN
cana-1065	40	51	(	(	PUNCT
cana-1065	40	52	𝑀𝑝,𝑟(𝑥	𝑀𝑝,𝑟(𝑥	NOUN
cana-1065	40	53	)	)	PUNCT
cana-1065	40	54	,	,	PUNCT
cana-1065	40	55	𝑀𝑟,𝑞(𝑦	𝑀𝑟,𝑞(𝑦	NOUN
cana-1065	40	56	)	)	PUNCT
cana-1065	40	57	)	)	PUNCT
cana-1065	40	58	for	for	ADP
cana-1065	40	59	every	every	DET
cana-1065	40	60	𝑝	𝑝	PROPN
cana-1065	40	61	,	,	PUNCT
cana-1065	40	62	𝑞	𝑞	X
cana-1065	40	63	,	,	PUNCT
cana-1065	40	64	𝑟	𝑟	X
cana-1065	40	65	∈	∈	PROPN
cana-1065	40	66	𝑌	𝑌	PROPN
cana-1065	40	67	and	and	CCONJ
cana-1065	40	68	𝑥	𝑥	NOUN
cana-1065	40	69	,	,	PUNCT
cana-1065	40	70	𝑦	𝑦	NOUN
cana-1065	40	71	∈	∈	NOUN
cana-1065	40	72	ℝ	ℝ	NOUN
cana-1065	40	73	>	>	X
cana-1065	40	74	0	0	PROPN
cana-1065	40	75	.	.	PUNCT
cana-1065	41	1	definition	definition	NOUN
cana-1065	41	2	2.5	2.5	NUM
cana-1065	42	1	[	[	X
cana-1065	42	2	2	2	NUM
cana-1065	42	3	]	]	PUNCT
cana-1065	42	4	:	:	PUNCT
cana-1065	42	5	a	a	DET
cana-1065	42	6	mapping	mapping	NOUN
cana-1065	42	7	𝐴	𝐴	PROPN
cana-1065	42	8	:	:	PUNCT
cana-1065	42	9	𝑌	𝑌	PROPN
cana-1065	42	10	→	→	SYM
cana-1065	42	11	𝑌	𝑌	PROPN
cana-1065	42	12	in	in	ADP
cana-1065	42	13	menger	menger	PROPN
cana-1065	42	14	space	space	NOUN
cana-1065	42	15	(	(	PUNCT
cana-1065	42	16	𝐾	𝐾	PROPN
cana-1065	42	17	,	,	PUNCT
cana-1065	42	18	𝐹	𝐹	PROPN
cana-1065	42	19	,	,	PUNCT
cana-1065	42	20	𝑡	𝑡	PROPN
cana-1065	42	21	)	)	PUNCT
cana-1065	42	22	,	,	PUNCT
cana-1065	42	23	is	be	AUX
cana-1065	42	24	said	say	VERB
cana-1065	42	25	to	to	PART
cana-1065	42	26	be	be	AUX
cana-1065	42	27	continuous	continuous	ADJ
cana-1065	42	28	at	at	ADP
cana-1065	42	29	a	a	DET
cana-1065	42	30	point	point	NOUN
cana-1065	42	31	𝑝	𝑝	NOUN
cana-1065	42	32	∈	∈	NOUN
cana-1065	42	33	𝑌	𝑌	PROPN
cana-1065	42	34	if	if	SCONJ
cana-1065	42	35	for	for	ADP
cana-1065	42	36	every	every	DET
cana-1065	42	37			PROPN
cana-1065	42	38	>	>	X
cana-1065	42	39	0	0	PUNCT
cana-1065	42	40	and	and	CCONJ
cana-1065	42	41			X
cana-1065	42	42	>	>	X
cana-1065	42	43	0	0	NUM
cana-1065	42	44	,	,	PUNCT
cana-1065	42	45	there	there	PRON
cana-1065	42	46	exist	exist	VERB
cana-1065	42	47	1	1	PROPN
cana-1065	42	48	>	>	SYM
cana-1065	42	49	0	0	PUNCT
cana-1065	42	50	and	and	CCONJ
cana-1065	42	51	1	1	PROPN
cana-1065	42	52	>	>	X
cana-1065	42	53	0	0	NUM
cana-1065	43	1	such	such	ADJ
cana-1065	43	2	that	that	SCONJ
cana-1065	43	3	if	if	SCONJ
cana-1065	43	4	𝑀𝑝,𝑞	𝑀𝑝,𝑞	PROPN
cana-1065	43	5	(	(	PUNCT
cana-1065	43	6	1	1	PROPN
cana-1065	43	7	)	)	PUNCT
cana-1065	43	8	>	>	X
cana-1065	43	9	1	1	NUM
cana-1065	43	10	–	–	PUNCT
cana-1065	43	11	1	1	PUNCT
cana-1065	43	12	then	then	ADV
cana-1065	43	13	𝑀𝐴𝑝,𝐴𝑞(	𝑀𝐴𝑝,𝐴𝑞(	NUM
cana-1065	43	14	)	)	PUNCT
cana-1065	43	15	>	>	X
cana-1065	43	16	1	1	NUM
cana-1065	43	17	−	−	NUM
cana-1065	43	18	.	.	ADJ
cana-1065	43	19	definition	definition	NOUN
cana-1065	43	20	2.6	2.6	NUM
cana-1065	43	21	[	[	X
cana-1065	43	22	2	2	NUM
cana-1065	43	23	]	]	PUNCT
cana-1065	43	24	:	:	PUNCT
cana-1065	43	25	let	let	VERB
cana-1065	43	26	(	(	PUNCT
cana-1065	43	27	𝑌	𝑌	PROPN
cana-1065	43	28	,	,	PUNCT
cana-1065	43	29	𝑀	𝑀	PROPN
cana-1065	43	30	,	,	PUNCT
cana-1065	43	31	𝑡	𝑡	PROPN
cana-1065	43	32	)	)	PUNCT
cana-1065	43	33	be	be	VERB
cana-1065	43	34	a	a	DET
cana-1065	43	35	menger	menger	NOUN
cana-1065	43	36	space	space	NOUN
cana-1065	43	37	and	and	CCONJ
cana-1065	43	38	𝑡	𝑡	PROPN
cana-1065	43	39	be	be	VERB
cana-1065	43	40	a	a	DET
cana-1065	43	41	continuous	continuous	ADJ
cana-1065	43	42	t	t	NOUN
cana-1065	43	43	-	-	PUNCT
cana-1065	43	44	norm	norm	NOUN
cana-1065	43	45	.	.	PUNCT
cana-1065	44	1	then	then	ADV
cana-1065	44	2	,	,	PUNCT
cana-1065	44	3	(	(	PUNCT
cana-1065	44	4	a	a	X
cana-1065	44	5	)	)	PUNCT
cana-1065	44	6	a	a	DET
cana-1065	44	7	sequence	sequence	NOUN
cana-1065	44	8	{	{	PUNCT
cana-1065	44	9	𝑦𝑛	𝑦𝑛	NOUN
cana-1065	44	10	}	}	PUNCT
cana-1065	44	11	in	in	ADP
cana-1065	44	12	𝑌	𝑌	PROPN
cana-1065	44	13	is	be	AUX
cana-1065	44	14	said	say	VERB
cana-1065	44	15	to	to	PART
cana-1065	44	16	converge	converge	VERB
cana-1065	44	17	to	to	ADP
cana-1065	44	18	a	a	DET
cana-1065	44	19	point	point	NOUN
cana-1065	44	20	𝑦	𝑦	NOUN
cana-1065	44	21	in	in	ADP
cana-1065	44	22	𝑌	𝑌	PROPN
cana-1065	44	23	if	if	SCONJ
cana-1065	44	24	and	and	CCONJ
cana-1065	44	25	only	only	ADV
cana-1065	44	26	if	if	SCONJ
cana-1065	44	27	for	for	ADP
cana-1065	44	28	every	every	DET
cana-1065	44	29			PROPN
cana-1065	44	30	>	>	X
cana-1065	44	31	0	0	PUNCT
cana-1065	44	32	and	and	CCONJ
cana-1065	44	33			X
cana-1065	44	34	>	>	X
cana-1065	44	35	0	0	NUM
cana-1065	44	36	,	,	PUNCT
cana-1065	44	37	there	there	PRON
cana-1065	44	38	exist	exist	VERB
cana-1065	44	39	an	an	DET
cana-1065	44	40	integer	integer	NOUN
cana-1065	44	41	𝑁	𝑁	PROPN
cana-1065	44	42	=	=	PUNCT
cana-1065	44	43	𝑁	𝑁	PROPN
cana-1065	44	44	(	(	PUNCT
cana-1065	44	45			PROPN
cana-1065	44	46	,	,	PUNCT
cana-1065	44	47			X
cana-1065	44	48	)	)	PUNCT
cana-1065	44	49	such	such	ADJ
cana-1065	44	50	that	that	PRON
cana-1065	44	51	𝑀𝑦𝑛,𝑦(	𝑀𝑦𝑛,𝑦(	PROPN
cana-1065	44	52	)	)	PUNCT
cana-1065	44	53	>	>	X
cana-1065	44	54	1	1	NUM
cana-1065	44	55	−	−	NOUN
cana-1065	44	56			NOUN
cana-1065	44	57	for	for	SCONJ
cana-1065	44	58	all	all	DET
cana-1065	44	59	𝑛	𝑛	DET
cana-1065	44	60	≥	≥	NOUN
cana-1065	44	61	𝑁.	𝑁.	PROPN
cana-1065	44	62	in	in	ADP
cana-1065	44	63	this	this	DET
cana-1065	44	64	case	case	NOUN
cana-1065	44	65	,	,	PUNCT
cana-1065	44	66	we	we	PRON
cana-1065	44	67	write	write	VERB
cana-1065	44	68	,	,	PUNCT
cana-1065	44	69	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	44	70	𝑛→∞	𝑛→∞	NOUN
cana-1065	44	71	𝑦𝑛	𝑦𝑛	ADP
cana-1065	44	72	=	=	PUNCT
cana-1065	44	73	y.	y.	NOUN
cana-1065	44	74	(	(	PUNCT
cana-1065	44	75	b	b	X
cana-1065	44	76	)	)	PUNCT
cana-1065	44	77	a	a	DET
cana-1065	44	78	sequence	sequence	NOUN
cana-1065	44	79	{	{	PUNCT
cana-1065	44	80	𝑦𝑛	𝑦𝑛	NOUN
cana-1065	44	81	}	}	PUNCT
cana-1065	44	82	in	in	ADP
cana-1065	44	83	𝑌	𝑌	PROPN
cana-1065	44	84	is	be	AUX
cana-1065	44	85	said	say	VERB
cana-1065	44	86	to	to	PART
cana-1065	44	87	be	be	AUX
cana-1065	44	88	a	a	DET
cana-1065	44	89	cauchy	cauchy	ADJ
cana-1065	44	90	sequence	sequence	NOUN
cana-1065	44	91	if	if	SCONJ
cana-1065	44	92	for	for	ADP
cana-1065	44	93	every	every	DET
cana-1065	44	94			PROPN
cana-1065	44	95	>	>	X
cana-1065	44	96	0	0	PUNCT
cana-1065	44	97	and	and	CCONJ
cana-1065	44	98			X
cana-1065	44	99	>	>	X
cana-1065	44	100	0	0	NUM
cana-1065	44	101	,	,	PUNCT
cana-1065	44	102	there	there	PRON
cana-1065	44	103	exists	exist	VERB
cana-1065	44	104	an	an	DET
cana-1065	44	105	integer	integer	NOUN
cana-1065	44	106	𝑁	𝑁	PROPN
cana-1065	44	107	=	=	PUNCT
cana-1065	44	108	𝑁	𝑁	PROPN
cana-1065	44	109	(	(	PUNCT
cana-1065	44	110			PROPN
cana-1065	44	111	,	,	PUNCT
cana-1065	44	112			X
cana-1065	44	113	)	)	PUNCT
cana-1065	44	114	>	>	X
cana-1065	44	115	0	0	NUM
cana-1065	45	1	such	such	ADJ
cana-1065	45	2	that	that	SCONJ
cana-1065	45	3	𝑀𝑦𝑛,𝑦𝑚	𝑀𝑦𝑛,𝑦𝑚	PROPN
cana-1065	45	4	(	(	PUNCT
cana-1065	45	5			PROPN
cana-1065	45	6	)	)	PUNCT
cana-1065	45	7	>	>	X
cana-1065	45	8	1	1	NUM
cana-1065	45	9	−	−	NOUN
cana-1065	45	10			NOUN
cana-1065	45	11	for	for	ADP
cana-1065	45	12	all	all	DET
cana-1065	45	13	𝑚	𝑚	NOUN
cana-1065	45	14	,	,	PUNCT
cana-1065	45	15	𝑛	𝑛	DET
cana-1065	45	16	≥	≥	NOUN
cana-1065	45	17	𝑁.	𝑁.	PROPN
cana-1065	45	18	(	(	PUNCT
cana-1065	45	19	c	c	NOUN
cana-1065	45	20	)	)	PUNCT
cana-1065	45	21	a	a	DET
cana-1065	45	22	menger	menger	PROPN
cana-1065	45	23	space	space	NOUN
cana-1065	45	24	(	(	PUNCT
cana-1065	45	25	𝑌	𝑌	PROPN
cana-1065	45	26	,	,	PUNCT
cana-1065	45	27	𝑀	𝑀	PROPN
cana-1065	45	28	,	,	PUNCT
cana-1065	45	29	𝑡	𝑡	PROPN
cana-1065	45	30	)	)	PUNCT
cana-1065	45	31	is	be	AUX
cana-1065	45	32	said	say	VERB
cana-1065	45	33	to	to	PART
cana-1065	45	34	be	be	AUX
cana-1065	45	35	complete	complete	ADJ
cana-1065	45	36	if	if	SCONJ
cana-1065	45	37	every	every	DET
cana-1065	45	38	cauchy	cauchy	ADJ
cana-1065	45	39	sequence	sequence	NOUN
cana-1065	45	40	in	in	ADP
cana-1065	45	41	𝑌	𝑌	PROPN
cana-1065	45	42	converges	converge	VERB
cana-1065	45	43	to	to	ADP
cana-1065	45	44	a	a	DET
cana-1065	45	45	point	point	NOUN
cana-1065	45	46	in	in	ADP
cana-1065	45	47	𝑌.	𝑌.	PROPN
cana-1065	45	48	definition	definition	NOUN
cana-1065	45	49	2.7:[7	2.7:[7	NUM
cana-1065	45	50	]	]	PUNCT
cana-1065	45	51	common	common	ADJ
cana-1065	45	52	fixed	fix	VERB
cana-1065	45	53	point	point	NOUN
cana-1065	45	54	of	of	ADP
cana-1065	45	55	self	self	NOUN
cana-1065	45	56	-	-	PUNCT
cana-1065	45	57	mapping	mapping	NOUN
cana-1065	45	58	functions	function	NOUN
cana-1065	45	59	𝐴	𝐴	PROPN
cana-1065	45	60	,	,	PUNCT
cana-1065	45	61	𝐵	𝐵	NOUN
cana-1065	45	62	:	:	PUNCT
cana-1065	45	63	𝑌	𝑌	PROPN
cana-1065	45	64	→	→	SYM
cana-1065	45	65	𝑌	𝑌	PROPN
cana-1065	45	66	is	be	AUX
cana-1065	45	67	a	a	DET
cana-1065	45	68	point	point	NOUN
cana-1065	45	69	𝑦	𝑦	NOUN
cana-1065	45	70	î	î	PROPN
cana-1065	45	71	𝑌	𝑌	PROPN
cana-1065	45	72	if	if	SCONJ
cana-1065	45	73	𝐴(𝑦	𝐴(𝑦	NOUN
cana-1065	45	74	)	)	PUNCT
cana-1065	45	75	=	=	PUNCT
cana-1065	45	76	𝐵(𝑦	𝐵(𝑦	NOUN
cana-1065	45	77	)	)	PUNCT
cana-1065	45	78	=	=	PUNCT
cana-1065	45	79	𝑦.	𝑦.	PROPN
cana-1065	45	80	example	example	NOUN
cana-1065	45	81	2.1	2.1	NUM
cana-1065	45	82	:	:	PUNCT
cana-1065	45	83	let	let	VERB
cana-1065	45	84	𝐴	𝐴	PROPN
cana-1065	45	85	,	,	PUNCT
cana-1065	45	86	𝐵	𝐵	NOUN
cana-1065	45	87	:	:	PUNCT
cana-1065	45	88	ℝ	ℝ	PROPN
cana-1065	45	89	→	→	SYM
cana-1065	45	90	ℝ	ℝ	PROPN
cana-1065	45	91	be	be	AUX
cana-1065	45	92	functions	function	NOUN
cana-1065	45	93	such	such	ADJ
cana-1065	45	94	that	that	DET
cana-1065	45	95	𝐴(𝑦	𝐴(𝑦	NOUN
cana-1065	45	96	)	)	PUNCT
cana-1065	46	1	=	=	PUNCT
cana-1065	46	2	𝑦2	𝑦2	NOUN
cana-1065	46	3	4	4	NUM
cana-1065	46	4	and	and	CCONJ
cana-1065	46	5	𝐵(𝑦	𝐵(𝑦	NUM
cana-1065	46	6	)	)	PUNCT
cana-1065	46	7	=	=	SYM
cana-1065	46	8	2𝑦	2𝑦	NUM
cana-1065	46	9	−	−	PROPN
cana-1065	46	10	4	4	NUM
cana-1065	46	11	,	,	PUNCT
cana-1065	46	12	then	then	ADV
cana-1065	46	13	𝑦	𝑦	NOUN
cana-1065	46	14	=	=	SYM
cana-1065	46	15	4	4	NUM
cana-1065	46	16	is	be	AUX
cana-1065	46	17	a	a	DET
cana-1065	46	18	common	common	ADJ
cana-1065	46	19	fixed	fix	VERB
cana-1065	46	20	point	point	NOUN
cana-1065	46	21	of	of	ADP
cana-1065	46	22	𝐴	𝐴	PROPN
cana-1065	46	23	and	and	CCONJ
cana-1065	46	24	𝐵.	𝐵.	PROPN
cana-1065	46	25	communications	communication	NOUN
cana-1065	46	26	on	on	ADP
cana-1065	46	27	applied	apply	VERB
cana-1065	46	28	nonlinear	nonlinear	ADJ
cana-1065	46	29	analysis	analysis	NOUN
cana-1065	46	30	issn	issn	NOUN
cana-1065	46	31	:	:	PUNCT
cana-1065	46	32	1074	1074	NUM
cana-1065	46	33	-	-	PUNCT
cana-1065	46	34	133x	133x	NUM
cana-1065	46	35	vol	vol	NOUN
cana-1065	46	36	31	31	NUM
cana-1065	46	37	no	no	NOUN
cana-1065	46	38	.	.	PUNCT
cana-1065	47	1	5s	5s	NUM
cana-1065	47	2	(	(	PUNCT
cana-1065	47	3	2024	2024	NUM
cana-1065	47	4	)	)	PUNCT
cana-1065	47	5	460	460	NUM
cana-1065	48	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	48	2	definition	definition	NOUN
cana-1065	48	3	2.8:[12	2.8:[12	NUM
cana-1065	48	4	]	]	PUNCT
cana-1065	48	5	two	two	NUM
cana-1065	48	6	mappings	mapping	NOUN
cana-1065	48	7	𝐴	𝐴	PROPN
cana-1065	48	8	,	,	PUNCT
cana-1065	48	9	𝐵	𝐵	NOUN
cana-1065	48	10	:	:	PUNCT
cana-1065	48	11	𝑌	𝑌	PROPN
cana-1065	48	12	→	→	SYM
cana-1065	48	13	𝑌	𝑌	PROPN
cana-1065	48	14	are	be	AUX
cana-1065	48	15	said	say	VERB
cana-1065	48	16	to	to	PART
cana-1065	48	17	be	be	AUX
cana-1065	48	18	compatible	compatible	ADJ
cana-1065	48	19	mappings	mapping	NOUN
cana-1065	48	20	in	in	ADP
cana-1065	48	21	menger	menger	PROPN
cana-1065	48	22	space	space	NOUN
cana-1065	48	23	(	(	PUNCT
cana-1065	48	24	𝑌	𝑌	PROPN
cana-1065	48	25	,	,	PUNCT
cana-1065	48	26	𝑀	𝑀	PROPN
cana-1065	48	27	,	,	PUNCT
cana-1065	48	28	𝑡	𝑡	PROPN
cana-1065	48	29	)	)	PUNCT
cana-1065	48	30	iff	iff	PROPN
cana-1065	48	31	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	48	32	𝑛→∞	𝑛→∞	NUM
cana-1065	48	33	𝐹𝐴𝐵𝑥𝑛	𝐹𝐴𝐵𝑥𝑛	X
cana-1065	48	34	,	,	PUNCT
cana-1065	48	35	𝐵𝐴𝑥𝑛	𝐵𝐴𝑥𝑛	PROPN
cana-1065	48	36	(	(	PUNCT
cana-1065	48	37	𝑥	𝑥	NOUN
cana-1065	48	38	)	)	PUNCT
cana-1065	48	39	=	=	SYM
cana-1065	48	40	1	1	NUM
cana-1065	48	41	for	for	ADP
cana-1065	48	42	all	all	DET
cana-1065	48	43	𝑥	𝑥	PRON
cana-1065	48	44	>	>	X
cana-1065	48	45	0	0	NUM
cana-1065	48	46	,	,	PUNCT
cana-1065	48	47	whenever	whenever	SCONJ
cana-1065	48	48	sequence	sequence	NOUN
cana-1065	48	49	{	{	PUNCT
cana-1065	48	50	𝑥𝑛	𝑥𝑛	NOUN
cana-1065	48	51	}	}	PUNCT
cana-1065	48	52	in	in	ADP
cana-1065	48	53	y	y	PRON
cana-1065	48	54	such	such	ADJ
cana-1065	48	55	that	that	PRON
cana-1065	48	56	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	48	57	𝑛→∞	𝑛→∞	NOUN
cana-1065	49	1	𝐴𝑥𝑛	𝐴𝑥𝑛	NOUN
cana-1065	49	2	=	=	PRON
cana-1065	49	3	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	49	4	𝑛→∞	𝑛→∞	NUM
cana-1065	49	5	𝐵𝑥𝑛	𝐵𝑥𝑛	NOUN
cana-1065	49	6	=	=	SYM
cana-1065	49	7	𝑦	𝑦	NOUN
cana-1065	49	8	for	for	ADP
cana-1065	49	9	some	some	DET
cana-1065	49	10	𝑦	𝑦	NOUN
cana-1065	49	11	in	in	ADP
cana-1065	49	12	𝑌.	𝑌.	ADJ
cana-1065	49	13	definition	definition	NOUN
cana-1065	49	14	2.10	2.10	NUM
cana-1065	49	15	:	:	PUNCT
cana-1065	50	1	[	[	X
cana-1065	50	2	15	15	NUM
cana-1065	50	3	]	]	SYM
cana-1065	50	4	two	two	NUM
cana-1065	50	5	mappings	mapping	NOUN
cana-1065	50	6	𝐴	𝐴	PROPN
cana-1065	50	7	,	,	PUNCT
cana-1065	50	8	𝐵	𝐵	NOUN
cana-1065	50	9	:	:	PUNCT
cana-1065	50	10	𝑌	𝑌	PROPN
cana-1065	50	11	→	→	SYM
cana-1065	50	12	𝑌	𝑌	PROPN
cana-1065	50	13	are	be	AUX
cana-1065	50	14	said	say	VERB
cana-1065	50	15	to	to	PART
cana-1065	50	16	be	be	AUX
cana-1065	50	17	weakly	weakly	ADV
cana-1065	50	18	compatible	compatible	ADJ
cana-1065	50	19	(	(	PUNCT
cana-1065	50	20	or	or	CCONJ
cana-1065	50	21	coincidently	coincidently	ADV
cana-1065	50	22	commuting	commute	VERB
cana-1065	50	23	)	)	PUNCT
cana-1065	50	24	in	in	ADP
cana-1065	50	25	menger	menger	PROPN
cana-1065	50	26	space	space	NOUN
cana-1065	50	27	(	(	PUNCT
cana-1065	50	28	𝑌	𝑌	PROPN
cana-1065	50	29	,	,	PUNCT
cana-1065	50	30	𝐹	𝐹	PROPN
cana-1065	50	31	,	,	PUNCT
cana-1065	50	32	𝑡	𝑡	PROPN
cana-1065	50	33	)	)	PUNCT
cana-1065	50	34	if	if	SCONJ
cana-1065	50	35	they	they	PRON
cana-1065	50	36	commute	commute	VERB
cana-1065	50	37	at	at	ADP
cana-1065	50	38	their	their	PRON
cana-1065	50	39	coincidence	coincidence	NOUN
cana-1065	50	40	points	point	NOUN
cana-1065	50	41	,	,	PUNCT
cana-1065	50	42	that	that	ADV
cana-1065	50	43	is	is	ADV
cana-1065	50	44	,	,	PUNCT
cana-1065	50	45	if	if	SCONJ
cana-1065	50	46	𝐴𝑥	𝐴𝑥	PRON
cana-1065	50	47	=	=	PUNCT
cana-1065	50	48	𝐵𝑥	𝐵𝑥	PROPN
cana-1065	50	49	for	for	ADP
cana-1065	50	50	some	some	DET
cana-1065	50	51	𝑥	𝑥	PRON
cana-1065	50	52	∈	∈	PROPN
cana-1065	50	53	𝑌	𝑌	PROPN
cana-1065	50	54	then	then	ADV
cana-1065	50	55	𝐴𝐵𝑥	𝐴𝐵𝑥	X
cana-1065	50	56	=	=	SYM
cana-1065	50	57	𝐵𝐴𝑥.	𝐵𝐴𝑥.	PROPN
cana-1065	50	58	definition	definition	NOUN
cana-1065	50	59	2.11:[6	2.11:[6	NUM
cana-1065	50	60	]	]	PUNCT
cana-1065	50	61	two	two	NUM
cana-1065	50	62	mappings	mapping	NOUN
cana-1065	50	63	𝐴	𝐴	PROPN
cana-1065	50	64	,	,	PUNCT
cana-1065	50	65	𝐵	𝐵	NOUN
cana-1065	50	66	:	:	PUNCT
cana-1065	50	67	𝑌	𝑌	PROPN
cana-1065	50	68	→	→	SYM
cana-1065	50	69	𝑌	𝑌	PROPN
cana-1065	50	70	are	be	AUX
cana-1065	50	71	said	say	VERB
cana-1065	50	72	to	to	PART
cana-1065	50	73	be	be	AUX
cana-1065	50	74	compatible	compatible	ADJ
cana-1065	50	75	mappings	mapping	NOUN
cana-1065	50	76	of	of	ADP
cana-1065	50	77	type	type	NOUN
cana-1065	50	78	(	(	PUNCT
cana-1065	50	79	𝑷	𝑷	PROPN
cana-1065	50	80	)	)	PUNCT
cana-1065	50	81	in	in	ADP
cana-1065	50	82	menger	menger	PROPN
cana-1065	50	83	space	space	NOUN
cana-1065	50	84	(	(	PUNCT
cana-1065	50	85	𝑌	𝑌	PROPN
cana-1065	50	86	,	,	PUNCT
cana-1065	50	87	𝑀	𝑀	PROPN
cana-1065	50	88	,	,	PUNCT
cana-1065	50	89	𝑡	𝑡	PROPN
cana-1065	50	90	)	)	PUNCT
cana-1065	50	91	iff	iff	PROPN
cana-1065	50	92	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	50	93	𝑛→∞	𝑛→∞	PUNCT
cana-1065	50	94	𝑀𝐴𝐴𝑥𝑛	𝑀𝐴𝐴𝑥𝑛	PROPN
cana-1065	50	95	,	,	PUNCT
cana-1065	50	96	𝐵𝐵𝑥𝑛	𝐵𝐵𝑥𝑛	PROPN
cana-1065	50	97	(	(	PUNCT
cana-1065	50	98	𝑥	𝑥	NOUN
cana-1065	50	99	)	)	PUNCT
cana-1065	50	100	=	=	SYM
cana-1065	50	101	1	1	NUM
cana-1065	50	102	∀	∀	NOUN
cana-1065	51	1	𝑥	𝑥	X
cana-1065	51	2	>	>	X
cana-1065	51	3	0	0	PUNCT
cana-1065	52	1	whenever	whenever	SCONJ
cana-1065	52	2	{	{	PUNCT
cana-1065	52	3	𝑥𝑛	𝑥𝑛	NOUN
cana-1065	52	4	}	}	PUNCT
cana-1065	52	5	is	be	AUX
cana-1065	52	6	a	a	DET
cana-1065	52	7	sequence	sequence	NOUN
cana-1065	52	8	in	in	ADP
cana-1065	52	9	𝑌	𝑌	PROPN
cana-1065	52	10	such	such	DET
cana-1065	52	11	that	that	PRON
cana-1065	52	12	𝑙𝑖𝑚	𝑙𝑖𝑚	ADJ
cana-1065	52	13	𝑛→∞	𝑛→∞	NOUN
cana-1065	53	1	𝐴𝑥𝑛	𝐴𝑥𝑛	NOUN
cana-1065	53	2	=	=	PRON
cana-1065	53	3	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	53	4	𝑛→∞	𝑛→∞	NUM
cana-1065	53	5	𝐵𝑥𝑛	𝐵𝑥𝑛	NOUN
cana-1065	53	6	=	=	SYM
cana-1065	53	7	𝑦	𝑦	NOUN
cana-1065	53	8	for	for	ADP
cana-1065	53	9	some	some	DET
cana-1065	53	10	𝑦	𝑦	NOUN
cana-1065	53	11	𝑖𝑛	𝑖𝑛	PRON
cana-1065	53	12	𝑌.	𝑌.	ADJ
cana-1065	53	13	definition	definition	NOUN
cana-1065	53	14	2.12	2.12	NUM
cana-1065	53	15	:	:	PUNCT
cana-1065	54	1	[	[	X
cana-1065	54	2	5]two	5]two	NUM
cana-1065	54	3	mappings	mapping	NOUN
cana-1065	54	4	𝐴	𝐴	PROPN
cana-1065	54	5	,	,	PUNCT
cana-1065	54	6	𝐵	𝐵	NOUN
cana-1065	54	7	:	:	PUNCT
cana-1065	54	8	𝑌	𝑌	PROPN
cana-1065	54	9	→	→	SYM
cana-1065	54	10	𝑌	𝑌	PROPN
cana-1065	54	11	are	be	AUX
cana-1065	54	12	said	say	VERB
cana-1065	54	13	to	to	PART
cana-1065	54	14	be	be	AUX
cana-1065	54	15	weakly	weakly	ADV
cana-1065	54	16	compatible	compatible	ADJ
cana-1065	54	17	mapping	mapping	NOUN
cana-1065	54	18	of	of	ADP
cana-1065	54	19	type(𝑷	type(𝑷	NOUN
cana-1065	54	20	)	)	PUNCT
cana-1065	54	21	in	in	ADP
cana-1065	54	22	menger	menger	PROPN
cana-1065	54	23	space	space	NOUN
cana-1065	54	24	(	(	PUNCT
cana-1065	54	25	𝑌	𝑌	PROPN
cana-1065	54	26	,	,	PUNCT
cana-1065	54	27	𝑀	𝑀	PROPN
cana-1065	54	28	,	,	PUNCT
cana-1065	54	29	𝑡	𝑡	PROPN
cana-1065	54	30	)	)	PUNCT
cana-1065	54	31	iff	iff	PROPN
cana-1065	54	32	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	54	33	𝑛→∞	𝑛→∞	PUNCT
cana-1065	54	34	𝑀𝐴𝐴𝑥𝑛	𝑀𝐴𝐴𝑥𝑛	PROPN
cana-1065	54	35	,	,	PUNCT
cana-1065	54	36	𝐵𝐵𝑥𝑛	𝐵𝐵𝑥𝑛	PROPN
cana-1065	54	37	(	(	PUNCT
cana-1065	54	38	𝑥	𝑥	NOUN
cana-1065	54	39	)	)	PUNCT
cana-1065	54	40	≥	≥	NOUN
cana-1065	54	41	𝑀𝐴𝑥𝑛	𝑀𝐴𝑥𝑛	PROPN
cana-1065	54	42	,	,	PUNCT
cana-1065	54	43	𝐵𝑥𝑛	𝐵𝑥𝑛	PROPN
cana-1065	54	44	(	(	PUNCT
cana-1065	54	45	𝑥	𝑥	NOUN
cana-1065	54	46	)	)	PUNCT
cana-1065	54	47	∀	∀	X
cana-1065	55	1	𝑥	𝑥	X
cana-1065	55	2	>	>	X
cana-1065	55	3	0	0	NUM
cana-1065	55	4	,	,	PUNCT
cana-1065	55	5	whenever	whenever	SCONJ
cana-1065	55	6	{	{	PUNCT
cana-1065	55	7	𝑥𝑛	𝑥𝑛	NOUN
cana-1065	55	8	}	}	PUNCT
cana-1065	55	9	is	be	AUX
cana-1065	55	10	a	a	DET
cana-1065	55	11	sequence	sequence	NOUN
cana-1065	55	12	in	in	ADP
cana-1065	55	13	𝑌	𝑌	PROPN
cana-1065	56	1	such	such	DET
cana-1065	56	2	that	that	PRON
cana-1065	56	3	𝑙𝑖𝑚	𝑙𝑖𝑚	ADJ
cana-1065	56	4	𝑛→∞	𝑛→∞	NOUN
cana-1065	57	1	𝐴𝑥𝑛	𝐴𝑥𝑛	NOUN
cana-1065	57	2	=	=	PRON
cana-1065	57	3	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	57	4	𝑛→∞	𝑛→∞	NUM
cana-1065	57	5	𝐵𝑥𝑛	𝐵𝑥𝑛	NOUN
cana-1065	57	6	=	=	SYM
cana-1065	57	7	𝑦	𝑦	NOUN
cana-1065	57	8	for	for	ADP
cana-1065	57	9	some	some	DET
cana-1065	57	10	𝑦	𝑦	NOUN
cana-1065	57	11	𝑖𝑛	𝑖𝑛	DET
cana-1065	57	12	𝑌.	𝑌.	ADJ
cana-1065	57	13	example	example	NOUN
cana-1065	58	1	2.2	2.2	NUM
cana-1065	58	2	:	:	PUNCT
cana-1065	58	3	let	let	VERB
cana-1065	58	4	(	(	PUNCT
cana-1065	58	5	𝑌	𝑌	PROPN
cana-1065	58	6	,	,	PUNCT
cana-1065	58	7	𝑑	𝑑	NOUN
cana-1065	58	8	)	)	PUNCT
cana-1065	58	9	be	be	AUX
cana-1065	58	10	metric	metric	ADJ
cana-1065	58	11	space	space	NOUN
cana-1065	58	12	where	where	SCONJ
cana-1065	58	13	𝑌	𝑌	PROPN
cana-1065	58	14	=	=	PUNCT
cana-1065	59	1	[	[	X
cana-1065	59	2	0	0	NUM
cana-1065	59	3	,	,	PUNCT
cana-1065	59	4	2]with	2]with	NUM
cana-1065	59	5	usual	usual	ADJ
cana-1065	59	6	metric	metric	ADJ
cana-1065	59	7	𝑑(𝑥	𝑑(𝑥	PROPN
cana-1065	59	8	,	,	PUNCT
cana-1065	59	9	𝑦	𝑦	X
cana-1065	59	10	)	)	PUNCT
cana-1065	59	11	=	=	SYM
cana-1065	59	12	|𝑥	|𝑥	ADP
cana-1065	59	13	−	−	PROPN
cana-1065	59	14	𝑦|	𝑦|	PROPN
cana-1065	59	15	and	and	CCONJ
cana-1065	59	16	(	(	PUNCT
cana-1065	59	17	𝑌	𝑌	PROPN
cana-1065	59	18	,	,	PUNCT
cana-1065	59	19	𝑀	𝑀	PROPN
cana-1065	59	20	)	)	PUNCT
cana-1065	59	21	be	be	VERB
cana-1065	59	22	pm	pm	NOUN
cana-1065	59	23	space	space	NOUN
cana-1065	59	24	with	with	ADP
cana-1065	59	25	𝑀𝑥,𝑦(𝑡	𝑀𝑥,𝑦(𝑡	NUM
cana-1065	59	26	)	)	PUNCT
cana-1065	59	27	=	=	PRON
cana-1065	59	28	{	{	PUNCT
cana-1065	59	29	𝑒	𝑒	PROPN
cana-1065	59	30	𝑑(𝑥,𝑦	𝑑(𝑥,𝑦	NUM
cana-1065	59	31	)	)	PUNCT
cana-1065	59	32	𝑡	𝑡	PROPN
cana-1065	59	33	,	,	PUNCT
cana-1065	59	34	𝑖𝑓	𝑖𝑓	NUM
cana-1065	59	35	𝑡	𝑡	X
cana-1065	59	36	>	>	X
cana-1065	59	37	0	0	NUM
cana-1065	59	38	,	,	PUNCT
cana-1065	59	39	0	0	NUM
cana-1065	59	40	,	,	PUNCT
cana-1065	59	41	𝑖𝑓	𝑖𝑓	NUM
cana-1065	59	42	𝑡	𝑡	X
cana-1065	59	43	=	=	NOUN
cana-1065	59	44	0	0	X
cana-1065	59	45	.	.	PUNCT
cana-1065	60	1	for	for	ADP
cana-1065	60	2	all	all	DET
cana-1065	60	3	𝑥	𝑥	PROPN
cana-1065	60	4	,	,	PUNCT
cana-1065	60	5	𝑦	𝑦	NOUN
cana-1065	60	6	∈	∈	NOUN
cana-1065	60	7	𝑌.	𝑌.	NOUN
cana-1065	60	8	we	we	PRON
cana-1065	60	9	define	define	VERB
cana-1065	60	10	𝐴	𝐴	PROPN
cana-1065	60	11	and	and	CCONJ
cana-1065	60	12	𝐵	𝐵	NOUN
cana-1065	60	13	as	as	ADP
cana-1065	60	14	:	:	PUNCT
cana-1065	60	15	𝐴(𝑥	𝐴(𝑥	X
cana-1065	60	16	)	)	PUNCT
cana-1065	60	17	=	=	NOUN
cana-1065	60	18	{	{	PUNCT
cana-1065	61	1	1	1	NUM
cana-1065	61	2	−	−	PROPN
cana-1065	61	3	𝑥	𝑥	NOUN
cana-1065	61	4	,	,	PUNCT
cana-1065	61	5	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1065	61	6	𝑥	𝑥	PRON
cana-1065	61	7	∈	∈	PROPN
cana-1065	62	1	[	[	X
cana-1065	62	2	0	0	NUM
cana-1065	62	3	,	,	PUNCT
cana-1065	62	4	1/2	1/2	NUM
cana-1065	62	5	)	)	PUNCT
cana-1065	62	6	1	1	NUM
cana-1065	62	7	,	,	PUNCT
cana-1065	62	8	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1065	62	9	𝑥	𝑥	PRON
cana-1065	62	10	∈	∈	PROPN
cana-1065	62	11	[	[	PUNCT
cana-1065	62	12	1	1	NUM
cana-1065	62	13	2	2	NUM
cana-1065	62	14	,	,	PUNCT
cana-1065	62	15	2	2	NUM
cana-1065	62	16	]	]	PUNCT
cana-1065	62	17	and	and	CCONJ
cana-1065	62	18	𝐵(𝑥	𝐵(𝑥	NUM
cana-1065	62	19	)	)	PUNCT
cana-1065	62	20	=	=	PRON
cana-1065	62	21	{	{	PUNCT
cana-1065	62	22	𝑥	𝑥	NOUN
cana-1065	62	23	,	,	PUNCT
cana-1065	62	24	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1065	62	25	𝑥	𝑥	PRON
cana-1065	62	26	∈	∈	PROPN
cana-1065	63	1	[	[	X
cana-1065	63	2	0	0	NUM
cana-1065	63	3	,	,	PUNCT
cana-1065	63	4	1/2	1/2	NUM
cana-1065	63	5	)	)	PUNCT
cana-1065	63	6	1	1	NUM
cana-1065	63	7	,	,	PUNCT
cana-1065	63	8	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1065	63	9	𝑥	𝑥	PRON
cana-1065	63	10	∈	∈	PROPN
cana-1065	63	11	[	[	PUNCT
cana-1065	63	12	1	1	NUM
cana-1065	63	13	2	2	NUM
cana-1065	63	14	,	,	PUNCT
cana-1065	63	15	2	2	NUM
cana-1065	63	16	]	]	PUNCT
cana-1065	63	17	.	.	PUNCT
cana-1065	64	1	taking	take	VERB
cana-1065	64	2	sequence	sequence	NOUN
cana-1065	64	3	{	{	PUNCT
cana-1065	64	4	𝑥𝑛	𝑥𝑛	NOUN
cana-1065	64	5	}	}	PUNCT
cana-1065	64	6	in	in	ADP
cana-1065	64	7	𝑌	𝑌	PROPN
cana-1065	64	8	where	where	SCONJ
cana-1065	64	9	𝑥𝑛	𝑥𝑛	VERB
cana-1065	64	10	=	=	SYM
cana-1065	64	11	1	1	NUM
cana-1065	64	12	2	2	NUM
cana-1065	64	13	−	−	NUM
cana-1065	64	14	1	1	NUM
cana-1065	64	15	𝑛	𝑛	NOUN
cana-1065	64	16	,	,	PUNCT
cana-1065	64	17	𝑛	𝑛	DET
cana-1065	64	18	∈	∈	NOUN
cana-1065	64	19	𝑁.	𝑁.	PROPN
cana-1065	64	20	then	then	ADV
cana-1065	64	21	,	,	PUNCT
cana-1065	64	22	(	(	PUNCT
cana-1065	64	23	𝐴	𝐴	PROPN
cana-1065	64	24	,	,	PUNCT
cana-1065	64	25	𝐵	𝐵	PROPN
cana-1065	64	26	)	)	PUNCT
cana-1065	64	27	are	be	AUX
cana-1065	64	28	weakly	weakly	ADV
cana-1065	64	29	compatible	compatible	ADJ
cana-1065	64	30	mappings	mapping	NOUN
cana-1065	64	31	of	of	ADP
cana-1065	64	32	type	type	NOUN
cana-1065	64	33	(	(	PUNCT
cana-1065	64	34	𝑃	𝑃	NOUN
cana-1065	64	35	)	)	PUNCT
cana-1065	64	36	and	and	CCONJ
cana-1065	64	37	it	it	PRON
cana-1065	64	38	is	be	AUX
cana-1065	64	39	neither	neither	CCONJ
cana-1065	64	40	compatible	compatible	ADJ
cana-1065	64	41	mappings	mapping	NOUN
cana-1065	64	42	of	of	ADP
cana-1065	64	43	type	type	NOUN
cana-1065	64	44	(	(	PUNCT
cana-1065	64	45	𝑃	𝑃	NOUN
cana-1065	64	46	)	)	PUNCT
cana-1065	64	47	nor	nor	CCONJ
cana-1065	64	48	compatible	compatible	ADJ
cana-1065	64	49	mappings	mapping	NOUN
cana-1065	64	50	.	.	PUNCT
cana-1065	65	1	theorem	theorem	VERB
cana-1065	65	2	2.1[2	2.1[2	PROPN
cana-1065	65	3	]	]	PUNCT
cana-1065	65	4	:	:	PUNCT
cana-1065	65	5	let	let	VERB
cana-1065	66	1	(	(	PUNCT
cana-1065	66	2	𝑌	𝑌	PROPN
cana-1065	66	3	,	,	PUNCT
cana-1065	66	4	𝑀	𝑀	PROPN
cana-1065	66	5	,	,	PUNCT
cana-1065	66	6	𝑡	𝑡	PROPN
cana-1065	66	7	)	)	PUNCT
cana-1065	66	8	be	be	VERB
cana-1065	66	9	menger	menger	NOUN
cana-1065	66	10	space	space	NOUN
cana-1065	66	11	with	with	ADP
cana-1065	66	12	the	the	DET
cana-1065	66	13	continuous	continuous	ADJ
cana-1065	66	14	𝑡	𝑡	PROPN
cana-1065	66	15	−	−	PROPN
cana-1065	66	16	𝑛𝑜𝑟𝑚	𝑛𝑜𝑟𝑚	ADJ
cana-1065	66	17	𝑡	𝑡	PROPN
cana-1065	66	18	and	and	CCONJ
cana-1065	66	19	𝐴	𝐴	PROPN
cana-1065	66	20	:	:	PUNCT
cana-1065	66	21	𝑌	𝑌	PROPN
cana-1065	66	22	→	→	SYM
cana-1065	66	23	𝑌.	𝑌.	PROPN
cana-1065	66	24	then	then	ADV
cana-1065	66	25	,	,	PUNCT
cana-1065	66	26	𝐴	𝐴	PROPN
cana-1065	66	27	is	be	AUX
cana-1065	66	28	continuous	continuous	ADJ
cana-1065	66	29	at	at	ADP
cana-1065	66	30	a	a	DET
cana-1065	66	31	point	point	NOUN
cana-1065	66	32	𝑦	𝑦	NOUN
cana-1065	66	33	∈	∈	NOUN
cana-1065	66	34	𝑌	𝑌	PROPN
cana-1065	66	35	if	if	SCONJ
cana-1065	66	36	and	and	CCONJ
cana-1065	66	37	only	only	ADV
cana-1065	66	38	if	if	SCONJ
cana-1065	66	39	for	for	ADP
cana-1065	66	40	every	every	DET
cana-1065	66	41	sequence	sequence	NOUN
cana-1065	66	42	{	{	PUNCT
cana-1065	66	43	𝑦𝑛	𝑦𝑛	NOUN
cana-1065	66	44	}	}	PUNCT
cana-1065	66	45	in	in	ADP
cana-1065	66	46	𝑌	𝑌	PROPN
cana-1065	66	47	converging	converge	VERB
cana-1065	66	48	to	to	ADP
cana-1065	66	49	a	a	DET
cana-1065	66	50	point	point	NOUN
cana-1065	66	51	𝑦	𝑦	NOUN
cana-1065	66	52	,	,	PUNCT
cana-1065	66	53	then	then	ADV
cana-1065	66	54	sequence	sequence	NOUN
cana-1065	66	55	{	{	PUNCT
cana-1065	66	56	𝐴𝑦𝑛	𝐴𝑦𝑛	PROPN
cana-1065	66	57	}	}	PUNCT
cana-1065	66	58	converges	converge	NOUN
cana-1065	66	59	to	to	ADP
cana-1065	66	60	the	the	DET
cana-1065	66	61	point	point	NOUN
cana-1065	66	62	𝐴𝑦	𝐴𝑦	PROPN
cana-1065	66	63	,	,	PUNCT
cana-1065	66	64	i.e.	i.e.	X
cana-1065	66	65	if	if	SCONJ
cana-1065	66	66	{	{	PUNCT
cana-1065	66	67	𝑦𝑛	𝑦𝑛	NOUN
cana-1065	66	68	}	}	PUNCT
cana-1065	66	69	→	→	SYM
cana-1065	66	70	𝑦	𝑦	X
cana-1065	66	71	then	then	ADV
cana-1065	66	72	it	it	PRON
cana-1065	66	73	implies	imply	VERB
cana-1065	66	74	{	{	PUNCT
cana-1065	66	75	𝐴𝑦𝑛	𝐴𝑦𝑛	NOUN
cana-1065	66	76	}	}	PUNCT
cana-1065	66	77	→	→	PUNCT
cana-1065	66	78	𝐴𝑦.	𝐴𝑦.	NOUN
cana-1065	66	79	proposition	proposition	NOUN
cana-1065	66	80	2.1[9	2.1[9	NUM
cana-1065	66	81	]	]	X
cana-1065	66	82	:	:	PUNCT
cana-1065	66	83	in	in	ADP
cana-1065	66	84	menger	menger	PROPN
cana-1065	66	85	space(𝑌	space(𝑌	PROPN
cana-1065	66	86	,	,	PUNCT
cana-1065	66	87	𝑀	𝑀	PROPN
cana-1065	66	88	,	,	PUNCT
cana-1065	66	89	𝑡	𝑡	PROPN
cana-1065	66	90	)	)	PUNCT
cana-1065	66	91	,	,	PUNCT
cana-1065	66	92	if	if	SCONJ
cana-1065	66	93	𝑡	𝑡	PROPN
cana-1065	66	94	(	(	PUNCT
cana-1065	66	95	𝑘	𝑘	PROPN
cana-1065	66	96	,	,	PUNCT
cana-1065	66	97	𝑘	𝑘	NOUN
cana-1065	66	98	)	)	PUNCT
cana-1065	66	99	≥	≥	NOUN
cana-1065	66	100	𝑘	𝑘	NOUN
cana-1065	66	101	for	for	ADP
cana-1065	66	102	all	all	DET
cana-1065	66	103	𝑘	𝑘	DET
cana-1065	66	104	∈	∈	NOUN
cana-1065	67	1	[	[	X
cana-1065	67	2	0	0	NUM
cana-1065	67	3	,	,	PUNCT
cana-1065	67	4	1	1	NUM
cana-1065	67	5	]	]	PUNCT
cana-1065	67	6	then	then	ADV
cana-1065	67	7	𝑡(𝑎	𝑡(𝑎	PROPN
cana-1065	67	8	,	,	PUNCT
cana-1065	67	9	𝑏	𝑏	NOUN
cana-1065	67	10	)	)	PUNCT
cana-1065	67	11	=	=	SYM
cana-1065	67	12	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-1065	67	13	{	{	PUNCT
cana-1065	67	14	𝑎	𝑎	NOUN
cana-1065	67	15	,	,	PUNCT
cana-1065	67	16	𝑏	𝑏	NOUN
cana-1065	67	17	}	}	PUNCT
cana-1065	67	18	for	for	ADP
cana-1065	67	19	all	all	DET
cana-1065	67	20	𝑎	𝑎	NOUN
cana-1065	67	21	,	,	PUNCT
cana-1065	67	22	𝑏	𝑏	PROPN
cana-1065	67	23	∈	∈	PROPN
cana-1065	68	1	[	[	X
cana-1065	68	2	0	0	NUM
cana-1065	68	3	,	,	PUNCT
cana-1065	68	4	1	1	NUM
cana-1065	68	5	]	]	PUNCT
cana-1065	68	6	.	.	PUNCT
cana-1065	69	1	lemma	lemma	PROPN
cana-1065	69	2	2.1[15	2.1[15	PROPN
cana-1065	69	3	]	]	X
cana-1065	69	4	:	:	PUNCT
cana-1065	69	5	let	let	VERB
cana-1065	69	6	(	(	PUNCT
cana-1065	69	7	𝑌	𝑌	PROPN
cana-1065	69	8	,	,	PUNCT
cana-1065	69	9	𝑀	𝑀	PROPN
cana-1065	69	10	,	,	PUNCT
cana-1065	69	11	𝑡	𝑡	PROPN
cana-1065	69	12	)	)	PUNCT
cana-1065	69	13	be	be	VERB
cana-1065	69	14	a	a	DET
cana-1065	69	15	menger	menger	NOUN
cana-1065	69	16	space	space	NOUN
cana-1065	69	17	.	.	PUNCT
cana-1065	70	1	if	if	SCONJ
cana-1065	70	2	there	there	PRON
cana-1065	70	3	exists	exist	VERB
cana-1065	70	4	𝑘	𝑘	PRON
cana-1065	70	5	∈	∈	PROPN
cana-1065	70	6	(	(	PUNCT
cana-1065	70	7	0	0	NUM
cana-1065	70	8	,	,	PUNCT
cana-1065	70	9	1	1	NUM
cana-1065	70	10	)	)	PUNCT
cana-1065	70	11	such	such	ADJ
cana-1065	70	12	that	that	PRON
cana-1065	70	13	for	for	ADP
cana-1065	70	14	all	all	DET
cana-1065	70	15	𝑝	𝑝	NOUN
cana-1065	70	16	,	,	PUNCT
cana-1065	70	17	𝑞	𝑞	PROPN
cana-1065	70	18	∈	∈	PROPN
cana-1065	70	19	𝑌	𝑌	PROPN
cana-1065	70	20	,	,	PUNCT
cana-1065	70	21	𝑀𝑝,𝑞(𝑘𝑥	𝑀𝑝,𝑞(𝑘𝑥	NOUN
cana-1065	70	22	)	)	PUNCT
cana-1065	70	23	≥	≥	NOUN
cana-1065	70	24	𝑀𝑝,𝑞(𝑥	𝑀𝑝,𝑞(𝑥	ADV
cana-1065	70	25	)	)	PUNCT
cana-1065	70	26	then	then	ADV
cana-1065	70	27	𝑝	𝑝	X
cana-1065	70	28	=	=	SYM
cana-1065	70	29	𝑞.	𝑞.	ADJ
cana-1065	70	30	proposition	proposition	NOUN
cana-1065	70	31	2.2:[5	2.2:[5	NUM
cana-1065	70	32	]	]	PUNCT
cana-1065	70	33	let	let	ADJ
cana-1065	70	34	(	(	PUNCT
cana-1065	70	35	𝑌	𝑌	PROPN
cana-1065	70	36	,	,	PUNCT
cana-1065	70	37	𝑀	𝑀	PROPN
cana-1065	70	38	,	,	PUNCT
cana-1065	70	39	𝑡	𝑡	PROPN
cana-1065	70	40	)	)	PUNCT
cana-1065	70	41	be	be	VERB
cana-1065	70	42	a	a	DET
cana-1065	70	43	menger	menger	NOUN
cana-1065	70	44	space	space	NOUN
cana-1065	70	45	such	such	ADJ
cana-1065	70	46	that	that	SCONJ
cana-1065	70	47	the	the	DET
cana-1065	70	48	t	t	NOUN
cana-1065	70	49	-	-	PUNCT
cana-1065	70	50	norm	norm	NOUN
cana-1065	70	51	𝑡	𝑡	PROPN
cana-1065	70	52	is	be	AUX
cana-1065	70	53	continuous	continuous	ADJ
cana-1065	70	54	and	and	CCONJ
cana-1065	70	55	𝑡	𝑡	X
cana-1065	70	56	(	(	PUNCT
cana-1065	70	57	𝑥	𝑥	NOUN
cana-1065	70	58	,	,	PUNCT
cana-1065	70	59	𝑥	𝑥	NOUN
cana-1065	70	60	)	)	PUNCT
cana-1065	70	61	≥	≥	NOUN
cana-1065	70	62	𝑥	𝑥	NOUN
cana-1065	70	63	for	for	ADP
cana-1065	70	64	all	all	DET
cana-1065	70	65	𝑥	𝑥	DET
cana-1065	70	66	∈	∈	NOUN
cana-1065	71	1	[	[	X
cana-1065	71	2	0	0	NUM
cana-1065	71	3	,	,	PUNCT
cana-1065	71	4	1	1	NUM
cana-1065	71	5	]	]	PUNCT
cana-1065	71	6	and	and	CCONJ
cana-1065	71	7	𝐴	𝐴	PROPN
cana-1065	71	8	,	,	PUNCT
cana-1065	71	9	𝐵	𝐵	NOUN
cana-1065	71	10	:	:	PUNCT
cana-1065	71	11	𝑌	𝑌	PROPN
cana-1065	71	12	→	→	SYM
cana-1065	71	13	𝑌	𝑌	PROPN
cana-1065	71	14	be	be	VERB
cana-1065	71	15	continuous	continuous	ADJ
cana-1065	71	16	mappings	mapping	NOUN
cana-1065	71	17	.	.	PUNCT
cana-1065	72	1	then	then	ADV
cana-1065	72	2	,	,	PUNCT
cana-1065	72	3	𝐴	𝐴	PROPN
cana-1065	72	4	and	and	CCONJ
cana-1065	72	5	𝐵	𝐵	NOUN
cana-1065	72	6	are	be	AUX
cana-1065	72	7	weakly	weakly	ADV
cana-1065	72	8	compatible	compatible	ADJ
cana-1065	72	9	mappings	mapping	NOUN
cana-1065	72	10	of	of	ADP
cana-1065	72	11	type	type	NOUN
cana-1065	72	12	(	(	PUNCT
cana-1065	72	13	p	p	NOUN
cana-1065	72	14	)	)	PUNCT
cana-1065	72	15	if	if	SCONJ
cana-1065	72	16	they	they	PRON
cana-1065	72	17	are	be	AUX
cana-1065	72	18	compatible	compatible	ADJ
cana-1065	72	19	mappings	mapping	NOUN
cana-1065	72	20	of	of	ADP
cana-1065	72	21	type(𝑃	type(𝑃	PROPN
cana-1065	72	22	)	)	PUNCT
cana-1065	72	23	.	.	PUNCT
cana-1065	73	1	communications	communication	NOUN
cana-1065	73	2	on	on	ADP
cana-1065	73	3	applied	apply	VERB
cana-1065	73	4	nonlinear	nonlinear	ADJ
cana-1065	73	5	analysis	analysis	NOUN
cana-1065	73	6	issn	issn	NOUN
cana-1065	73	7	:	:	PUNCT
cana-1065	73	8	1074	1074	NUM
cana-1065	73	9	-	-	PUNCT
cana-1065	73	10	133x	133x	NUM
cana-1065	73	11	vol	vol	NOUN
cana-1065	73	12	31	31	NUM
cana-1065	73	13	no	no	NOUN
cana-1065	73	14	.	.	PUNCT
cana-1065	74	1	5s	5s	NUM
cana-1065	74	2	(	(	PUNCT
cana-1065	74	3	2024	2024	NUM
cana-1065	74	4	)	)	PUNCT
cana-1065	74	5	461	461	NUM
cana-1065	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	74	7	proposition	proposition	NOUN
cana-1065	74	8	2.3	2.3	NUM
cana-1065	74	9	:	:	PUNCT
cana-1065	75	1	[	[	X
cana-1065	75	2	5]let	5]let	NOUN
cana-1065	75	3	(	(	PUNCT
cana-1065	75	4	𝑌	𝑌	PROPN
cana-1065	75	5	,	,	PUNCT
cana-1065	75	6	𝑀	𝑀	PROPN
cana-1065	75	7	,	,	PUNCT
cana-1065	75	8	𝑡	𝑡	PROPN
cana-1065	75	9	)	)	PUNCT
cana-1065	75	10	be	be	VERB
cana-1065	75	11	a	a	DET
cana-1065	75	12	menger	menger	NOUN
cana-1065	75	13	space	space	NOUN
cana-1065	75	14	such	such	ADJ
cana-1065	75	15	that	that	SCONJ
cana-1065	75	16	the	the	DET
cana-1065	75	17	t	t	NOUN
cana-1065	75	18	-	-	PUNCT
cana-1065	75	19	norm	norm	NOUN
cana-1065	75	20	𝑡	𝑡	PROPN
cana-1065	75	21	is	be	AUX
cana-1065	75	22	continuous	continuous	ADJ
cana-1065	75	23	and	and	CCONJ
cana-1065	75	24	𝑡	𝑡	X
cana-1065	75	25	(	(	PUNCT
cana-1065	75	26	𝑥	𝑥	NOUN
cana-1065	75	27	,	,	PUNCT
cana-1065	75	28	𝑥	𝑥	NOUN
cana-1065	75	29	)	)	PUNCT
cana-1065	75	30	≥	≥	NOUN
cana-1065	75	31	𝑥	𝑥	NOUN
cana-1065	75	32	for	for	ADP
cana-1065	75	33	all	all	DET
cana-1065	75	34	𝑥	𝑥	DET
cana-1065	75	35	∈	∈	NOUN
cana-1065	76	1	[	[	X
cana-1065	76	2	0	0	NUM
cana-1065	76	3	,	,	PUNCT
cana-1065	76	4	1	1	NUM
cana-1065	76	5	]	]	PUNCT
cana-1065	76	6	and	and	CCONJ
cana-1065	76	7	𝐴	𝐴	PROPN
cana-1065	76	8	,	,	PUNCT
cana-1065	76	9	𝐵	𝐵	NOUN
cana-1065	76	10	:	:	PUNCT
cana-1065	76	11	𝑌	𝑌	PROPN
cana-1065	76	12	→	→	SYM
cana-1065	76	13	𝑌	𝑌	PROPN
cana-1065	76	14	be	be	VERB
cana-1065	76	15	continuous	continuous	ADJ
cana-1065	76	16	mappings	mapping	NOUN
cana-1065	76	17	.	.	PUNCT
cana-1065	77	1	then	then	ADV
cana-1065	77	2	,	,	PUNCT
cana-1065	77	3	a	a	PRON
cana-1065	77	4	and	and	CCONJ
cana-1065	77	5	b	b	NOUN
cana-1065	77	6	are	be	AUX
cana-1065	77	7	compatible	compatible	ADJ
cana-1065	77	8	mappings	mapping	NOUN
cana-1065	77	9	of	of	ADP
cana-1065	77	10	type	type	NOUN
cana-1065	77	11	(	(	PUNCT
cana-1065	77	12	𝑃	𝑃	NOUN
cana-1065	77	13	)	)	PUNCT
cana-1065	77	14	if	if	SCONJ
cana-1065	77	15	they	they	PRON
cana-1065	77	16	are	be	AUX
cana-1065	77	17	weakly	weakly	ADV
cana-1065	77	18	compatible	compatible	ADJ
cana-1065	77	19	mappings	mapping	NOUN
cana-1065	77	20	of	of	ADP
cana-1065	77	21	type	type	NOUN
cana-1065	77	22	(	(	PUNCT
cana-1065	77	23	𝑃	𝑃	NOUN
cana-1065	77	24	)	)	PUNCT
cana-1065	77	25	.	.	PUNCT
cana-1065	78	1	proposition	proposition	NOUN
cana-1065	78	2	2.4	2.4	NUM
cana-1065	78	3	:	:	PUNCT
cana-1065	79	1	[	[	X
cana-1065	79	2	5]let	5]let	NOUN
cana-1065	79	3	(	(	PUNCT
cana-1065	79	4	𝑌	𝑌	PROPN
cana-1065	79	5	,	,	PUNCT
cana-1065	79	6	𝑀	𝑀	PROPN
cana-1065	79	7	,	,	PUNCT
cana-1065	79	8	𝑡	𝑡	PROPN
cana-1065	79	9	)	)	PUNCT
cana-1065	79	10	be	be	VERB
cana-1065	79	11	a	a	DET
cana-1065	79	12	menger	menger	NOUN
cana-1065	79	13	space	space	NOUN
cana-1065	79	14	such	such	ADJ
cana-1065	79	15	that	that	SCONJ
cana-1065	79	16	the	the	DET
cana-1065	79	17	t	t	NOUN
cana-1065	79	18	-	-	PUNCT
cana-1065	79	19	norm	norm	NOUN
cana-1065	79	20	𝑡	𝑡	PROPN
cana-1065	79	21	is	be	AUX
cana-1065	79	22	continuous	continuous	ADJ
cana-1065	79	23	and	and	CCONJ
cana-1065	79	24	𝑡	𝑡	X
cana-1065	79	25	(	(	PUNCT
cana-1065	79	26	𝑥	𝑥	NOUN
cana-1065	79	27	,	,	PUNCT
cana-1065	79	28	𝑥	𝑥	NOUN
cana-1065	79	29	)	)	PUNCT
cana-1065	79	30	≥	≥	NOUN
cana-1065	79	31	𝑥	𝑥	NOUN
cana-1065	79	32	for	for	ADP
cana-1065	79	33	all	all	DET
cana-1065	79	34	𝑥	𝑥	DET
cana-1065	79	35	∈	∈	NOUN
cana-1065	80	1	[	[	X
cana-1065	80	2	0	0	NUM
cana-1065	80	3	,	,	PUNCT
cana-1065	80	4	1	1	NUM
cana-1065	80	5	]	]	PUNCT
cana-1065	80	6	and	and	CCONJ
cana-1065	80	7	𝐴	𝐴	PROPN
cana-1065	80	8	,	,	PUNCT
cana-1065	80	9	𝐵	𝐵	NOUN
cana-1065	80	10	:	:	PUNCT
cana-1065	80	11	𝑌	𝑌	PROPN
cana-1065	80	12	→	→	SYM
cana-1065	80	13	𝑌	𝑌	PROPN
cana-1065	80	14	be	be	VERB
cana-1065	80	15	mappings	mapping	NOUN
cana-1065	80	16	.	.	PUNCT
cana-1065	81	1	if	if	SCONJ
cana-1065	81	2	𝐴	𝐴	PROPN
cana-1065	81	3	and	and	CCONJ
cana-1065	81	4	𝐵	𝐵	NOUN
cana-1065	81	5	are	be	AUX
cana-1065	81	6	weakly	weakly	ADV
cana-1065	81	7	compatible	compatible	ADJ
cana-1065	81	8	mappings	mapping	NOUN
cana-1065	81	9	of	of	ADP
cana-1065	81	10	type	type	NOUN
cana-1065	81	11	(	(	PUNCT
cana-1065	81	12	𝑃	𝑃	NOUN
cana-1065	81	13	)	)	PUNCT
cana-1065	81	14	and	and	CCONJ
cana-1065	81	15	𝐴𝑘	𝐴𝑘	X
cana-1065	81	16	=	=	PUNCT
cana-1065	81	17	𝐵𝑘	𝐵𝑘	VERB
cana-1065	81	18	for	for	ADP
cana-1065	81	19	some	some	PRON
cana-1065	81	20	𝑘	𝑘	PRON
cana-1065	81	21	∈	∈	PROPN
cana-1065	81	22	𝐾	𝐾	PROPN
cana-1065	81	23	,	,	PUNCT
cana-1065	81	24	then	then	ADV
cana-1065	81	25	,	,	PUNCT
cana-1065	81	26	𝐴𝐴𝑘	𝐴𝐴𝑘	NOUN
cana-1065	81	27	=	=	SYM
cana-1065	81	28	𝐴𝐵𝑘	𝐴𝐵𝑘	NOUN
cana-1065	81	29	=	=	SYM
cana-1065	81	30	𝐵𝐴𝑘	𝐵𝐴𝑘	NOUN
cana-1065	81	31	=	=	SYM
cana-1065	81	32	𝐵𝐵𝑘.	𝐵𝐵𝑘.	NOUN
cana-1065	81	33	proposition	proposition	VERB
cana-1065	81	34	2.5:[5	2.5:[5	NOUN
cana-1065	81	35	]	]	X
cana-1065	81	36	let	let	ADJ
cana-1065	81	37	(	(	PUNCT
cana-1065	81	38	𝑌	𝑌	PROPN
cana-1065	81	39	,	,	PUNCT
cana-1065	81	40	𝑀	𝑀	PROPN
cana-1065	81	41	,	,	PUNCT
cana-1065	81	42	𝑡	𝑡	PROPN
cana-1065	81	43	)	)	PUNCT
cana-1065	81	44	be	be	VERB
cana-1065	81	45	a	a	DET
cana-1065	81	46	menger	menger	NOUN
cana-1065	81	47	space	space	NOUN
cana-1065	81	48	such	such	ADJ
cana-1065	81	49	that	that	SCONJ
cana-1065	81	50	the	the	DET
cana-1065	81	51	t	t	NOUN
cana-1065	81	52	-	-	PUNCT
cana-1065	81	53	norm	norm	NOUN
cana-1065	81	54	𝑡	𝑡	PROPN
cana-1065	81	55	is	be	AUX
cana-1065	81	56	continuous	continuous	ADJ
cana-1065	81	57	and	and	CCONJ
cana-1065	81	58	𝑡	𝑡	X
cana-1065	81	59	(	(	PUNCT
cana-1065	81	60	𝑥	𝑥	NOUN
cana-1065	81	61	,	,	PUNCT
cana-1065	81	62	𝑥	𝑥	NOUN
cana-1065	81	63	)	)	PUNCT
cana-1065	81	64	≥	≥	NOUN
cana-1065	81	65	𝑥	𝑥	NOUN
cana-1065	81	66	for	for	ADP
cana-1065	81	67	all	all	DET
cana-1065	81	68	𝑥	𝑥	DET
cana-1065	81	69	∈	∈	NOUN
cana-1065	82	1	[	[	X
cana-1065	82	2	0	0	NUM
cana-1065	82	3	,	,	PUNCT
cana-1065	82	4	1	1	NUM
cana-1065	82	5	]	]	PUNCT
cana-1065	82	6	and	and	CCONJ
cana-1065	82	7	𝐴	𝐴	PROPN
cana-1065	82	8	,	,	PUNCT
cana-1065	82	9	𝐵	𝐵	NOUN
cana-1065	82	10	:	:	PUNCT
cana-1065	82	11	𝑌	𝑌	PROPN
cana-1065	82	12	→	→	SYM
cana-1065	82	13	𝑌	𝑌	PROPN
cana-1065	82	14	be	be	VERB
cana-1065	82	15	mappings	mapping	NOUN
cana-1065	82	16	.	.	PUNCT
cana-1065	83	1	let	let	VERB
cana-1065	83	2	𝐴	𝐴	PROPN
cana-1065	83	3	and	and	CCONJ
cana-1065	83	4	𝐵	𝐵	NOUN
cana-1065	83	5	be	be	VERB
cana-1065	83	6	weakly	weakly	ADV
cana-1065	83	7	compatible	compatible	ADJ
cana-1065	83	8	mappings	mapping	NOUN
cana-1065	83	9	of	of	ADP
cana-1065	83	10	type	type	NOUN
cana-1065	83	11	(	(	PUNCT
cana-1065	83	12	𝑃	𝑃	NOUN
cana-1065	83	13	)	)	PUNCT
cana-1065	83	14	and	and	CCONJ
cana-1065	83	15	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	83	16	𝑛→∞	𝑛→∞	NUM
cana-1065	83	17	a𝑘𝑛	a𝑘𝑛	NOUN
cana-1065	83	18	=	=	PRON
cana-1065	83	19	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	83	20	𝑛→∞	𝑛→∞	NUM
cana-1065	83	21	b𝑘𝑛	b𝑘𝑛	NOUN
cana-1065	83	22	=	=	SYM
cana-1065	83	23	k	k	PROPN
cana-1065	83	24	for	for	ADP
cana-1065	83	25	some	some	PRON
cana-1065	83	26	𝑘	𝑘	PRON
cana-1065	83	27	∈	∈	NOUN
cana-1065	84	1	𝑌.	𝑌.	PROPN
cana-1065	84	2	then	then	ADV
cana-1065	84	3	we	we	PRON
cana-1065	84	4	have	have	AUX
cana-1065	84	5	,	,	PUNCT
cana-1065	84	6	(	(	PUNCT
cana-1065	84	7	𝑖	𝑖	X
cana-1065	84	8	)	)	PUNCT
cana-1065	84	9	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	84	10	𝑛→∞	𝑛→∞	NUM
cana-1065	84	11	bb𝑘𝑛	bb𝑘𝑛	NOUN
cana-1065	84	12	=	=	SYM
cana-1065	84	13	ak	ak	PROPN
cana-1065	84	14	𝑖𝑓	𝑖𝑓	PROPN
cana-1065	84	15	𝐴	𝐴	PROPN
cana-1065	85	1	𝑖𝑠	𝑖𝑠	PROPN
cana-1065	86	1	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	NOUN
cana-1065	86	2	𝑎𝑡	𝑎𝑡	ADP
cana-1065	86	3	𝑘	𝑘	PROPN
cana-1065	86	4	,	,	PUNCT
cana-1065	86	5	(	(	PUNCT
cana-1065	86	6	𝑖𝑖	𝑖𝑖	INTJ
cana-1065	86	7	)	)	PUNCT
cana-1065	86	8	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-1065	86	9	𝑛→∞	𝑛→∞	NUM
cana-1065	86	10	aa𝑘𝑛	aa𝑘𝑛	NOUN
cana-1065	87	1	=	=	PUNCT
cana-1065	87	2	bk	bk	NOUN
cana-1065	87	3	𝑖𝑓	𝑖𝑓	ADP
cana-1065	87	4	𝐵	𝐵	NOUN
cana-1065	87	5	𝑖𝑠	𝑖𝑠	PROPN
cana-1065	87	6	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	NOUN
cana-1065	87	7	𝑎𝑡	𝑎𝑡	ADP
cana-1065	87	8	𝑘	𝑘	PROPN
cana-1065	87	9	,	,	PUNCT
cana-1065	87	10	(	(	PUNCT
cana-1065	87	11	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
cana-1065	87	12	)	)	PUNCT
cana-1065	87	13	𝐴𝐵𝑘	𝐴𝐵𝑘	NOUN
cana-1065	87	14	=	=	SYM
cana-1065	87	15	𝐵𝐴𝑘	𝐵𝐴𝑘	NOUN
cana-1065	87	16	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
cana-1065	88	1	𝐴𝑘	𝐴𝑘	X
cana-1065	88	2	=	=	PUNCT
cana-1065	88	3	𝐵𝑘	𝐵𝑘	VERB
cana-1065	88	4	𝑖𝑓	𝑖𝑓	NOUN
cana-1065	88	5	𝐴	𝐴	PROPN
cana-1065	88	6	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1065	88	7	𝐵	𝐵	PROPN
cana-1065	88	8	𝑎𝑟𝑒	𝑎𝑟𝑒	NOUN
cana-1065	88	9	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	NOUN
cana-1065	88	10	𝑎𝑡	𝑎𝑡	ADP
cana-1065	88	11	𝑘.	𝑘.	NOUN
cana-1065	88	12	the	the	DET
cana-1065	88	13	following	follow	VERB
cana-1065	88	14	lemma	lemma	PROPN
cana-1065	88	15	needs	need	VERB
cana-1065	88	16	to	to	PART
cana-1065	88	17	prove	prove	VERB
cana-1065	88	18	the	the	DET
cana-1065	88	19	main	main	ADJ
cana-1065	88	20	theorem	theorem	NOUN
cana-1065	88	21	:	:	PUNCT
cana-1065	88	22	lemma	lemma	PROPN
cana-1065	88	23	2.2[15	2.2[15	NUM
cana-1065	88	24	]	]	X
cana-1065	88	25	:	:	PUNCT
cana-1065	88	26	let	let	VERB
cana-1065	88	27	{	{	PUNCT
cana-1065	88	28	𝑥𝑛	𝑥𝑛	AUX
cana-1065	88	29	}	}	PUNCT
cana-1065	88	30	be	be	AUX
cana-1065	88	31	a	a	DET
cana-1065	88	32	sequence	sequence	NOUN
cana-1065	88	33	in	in	ADP
cana-1065	88	34	menger	menger	PROPN
cana-1065	88	35	space	space	NOUN
cana-1065	88	36	(	(	PUNCT
cana-1065	88	37	𝑌	𝑌	PROPN
cana-1065	88	38	,	,	PUNCT
cana-1065	88	39	𝑀	𝑀	PROPN
cana-1065	88	40	,	,	PUNCT
cana-1065	88	41	𝑡	𝑡	PROPN
cana-1065	88	42	)	)	PUNCT
cana-1065	88	43	,	,	PUNCT
cana-1065	88	44	where	where	SCONJ
cana-1065	88	45	t	t	PROPN
cana-1065	88	46	is	be	AUX
cana-1065	88	47	continuous	continuous	ADJ
cana-1065	88	48	𝑡	𝑡	X
cana-1065	88	49	−norm	−norm	NOUN
cana-1065	88	50	and	and	CCONJ
cana-1065	88	51	𝑡	𝑡	PROPN
cana-1065	88	52	(	(	PUNCT
cana-1065	88	53	𝑥	𝑥	NOUN
cana-1065	88	54	,	,	PUNCT
cana-1065	88	55	𝑥	𝑥	NOUN
cana-1065	88	56	)	)	PUNCT
cana-1065	88	57	≥	≥	NOUN
cana-1065	88	58	𝑥	𝑥	NOUN
cana-1065	88	59	for	for	ADP
cana-1065	88	60	all	all	DET
cana-1065	88	61	𝑥	𝑥	DET
cana-1065	88	62	∈	∈	NOUN
cana-1065	89	1	[	[	X
cana-1065	89	2	0	0	NUM
cana-1065	89	3	,	,	PUNCT
cana-1065	89	4	1	1	NUM
cana-1065	89	5	]	]	PUNCT
cana-1065	89	6	.	.	PUNCT
cana-1065	90	1	if	if	SCONJ
cana-1065	90	2	there	there	PRON
cana-1065	90	3	exists	exist	VERB
cana-1065	90	4	a	a	DET
cana-1065	90	5	constant	constant	ADJ
cana-1065	90	6	𝑘	𝑘	PRON
cana-1065	90	7	∈	∈	PROPN
cana-1065	91	1	[	[	X
cana-1065	91	2	0	0	NUM
cana-1065	91	3	,	,	PUNCT
cana-1065	91	4	1	1	NUM
cana-1065	91	5	]	]	PUNCT
cana-1065	91	6	such	such	ADJ
cana-1065	91	7	that	that	SCONJ
cana-1065	91	8	𝑀𝑥𝑛	𝑀𝑥𝑛	PROPN
cana-1065	91	9	,	,	PUNCT
cana-1065	91	10	𝑥𝑛+1	𝑥𝑛+1	NUM
cana-1065	91	11	(	(	PUNCT
cana-1065	91	12	𝑘𝑥	𝑘𝑥	NOUN
cana-1065	91	13	)	)	PUNCT
cana-1065	91	14	≥	≥	NOUN
cana-1065	91	15	𝑀𝑥𝑛−1	𝑀𝑥𝑛−1	NOUN
cana-1065	91	16	,	,	PUNCT
cana-1065	91	17	𝑥𝑛	𝑥𝑛	PROPN
cana-1065	91	18	(	(	PUNCT
cana-1065	91	19	𝑥	𝑥	NOUN
cana-1065	91	20	)	)	PUNCT
cana-1065	91	21	for	for	ADP
cana-1065	91	22	all	all	DET
cana-1065	91	23	𝑥	𝑥	PRON
cana-1065	91	24	>	>	PUNCT
cana-1065	91	25	0	0	PUNCT
cana-1065	91	26	and	and	CCONJ
cana-1065	91	27	𝑛	𝑛	DET
cana-1065	91	28	∈	∈	PROPN
cana-1065	91	29	𝑁	𝑁	PROPN
cana-1065	91	30	,	,	PUNCT
cana-1065	91	31	then	then	ADV
cana-1065	91	32	{	{	PUNCT
cana-1065	91	33	𝑥𝑛	𝑥𝑛	NOUN
cana-1065	91	34	}	}	PUNCT
cana-1065	91	35	is	be	AUX
cana-1065	91	36	a	a	DET
cana-1065	91	37	cauchy	cauchy	ADJ
cana-1065	91	38	sequence	sequence	NOUN
cana-1065	91	39	in	in	ADP
cana-1065	91	40	𝑌.	𝑌.	PROPN
cana-1065	91	41	3	3	NUM
cana-1065	91	42	.	.	PUNCT
cana-1065	91	43	main	main	ADJ
cana-1065	91	44	theorem	theorem	NOUN
cana-1065	91	45	:	:	PUNCT
cana-1065	91	46	now	now	ADV
cana-1065	91	47	,	,	PUNCT
cana-1065	91	48	we	we	PRON
cana-1065	91	49	prove	prove	VERB
cana-1065	91	50	our	our	PRON
cana-1065	91	51	main	main	ADJ
cana-1065	91	52	theorem	theorem	NOUN
cana-1065	91	53	for	for	ADP
cana-1065	91	54	weakly	weakly	ADJ
cana-1065	91	55	compatible	compatible	ADJ
cana-1065	91	56	mappings	mapping	NOUN
cana-1065	91	57	of	of	ADP
cana-1065	91	58	type	type	NOUN
cana-1065	91	59	(	(	PUNCT
cana-1065	91	60	𝑃	𝑃	NOUN
cana-1065	91	61	)	)	PUNCT
cana-1065	91	62	in	in	ADP
cana-1065	91	63	complete	complete	ADJ
cana-1065	91	64	menger	menger	PROPN
cana-1065	91	65	space	space	NOUN
cana-1065	91	66	:	:	PUNCT
cana-1065	91	67	theorem	theorem	VERB
cana-1065	91	68	3.1	3.1	NUM
cana-1065	91	69	:	:	PUNCT
cana-1065	91	70	let	let	VERB
cana-1065	91	71	(	(	PUNCT
cana-1065	91	72	𝑌	𝑌	PROPN
cana-1065	91	73	,	,	PUNCT
cana-1065	91	74	𝑀	𝑀	PROPN
cana-1065	91	75	,	,	PUNCT
cana-1065	91	76	𝑡	𝑡	PROPN
cana-1065	91	77	)	)	PUNCT
cana-1065	91	78	be	be	VERB
cana-1065	91	79	a	a	DET
cana-1065	91	80	complete	complete	ADJ
cana-1065	91	81	menger	menger	NOUN
cana-1065	91	82	space	space	NOUN
cana-1065	91	83	with	with	ADP
cana-1065	91	84	𝑡	𝑡	PROPN
cana-1065	91	85	(	(	PUNCT
cana-1065	91	86	𝑥	𝑥	PROPN
cana-1065	91	87	,	,	PUNCT
cana-1065	91	88	𝑦	𝑦	NOUN
cana-1065	91	89	)	)	PUNCT
cana-1065	91	90	=	=	SYM
cana-1065	91	91	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-1065	91	92	{	{	PUNCT
cana-1065	91	93	𝑥	𝑥	NOUN
cana-1065	91	94	,	,	PUNCT
cana-1065	91	95	𝑦	𝑦	NOUN
cana-1065	91	96	}	}	PUNCT
cana-1065	91	97	for	for	ADP
cana-1065	91	98	all	all	DET
cana-1065	91	99	𝑥	𝑥	PROPN
cana-1065	91	100	,	,	PUNCT
cana-1065	91	101	𝑦	𝑦	PRON
cana-1065	91	102	∈	∈	NOUN
cana-1065	92	1	[	[	X
cana-1065	92	2	0	0	NUM
cana-1065	92	3	,	,	PUNCT
cana-1065	92	4	1	1	NUM
cana-1065	92	5	]	]	PUNCT
cana-1065	92	6	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1065	92	7	𝐴	𝐴	PROPN
cana-1065	92	8	,	,	PUNCT
cana-1065	92	9	𝐵	𝐵	PROPN
cana-1065	92	10	,	,	PUNCT
cana-1065	92	11	𝑆	𝑆	PROPN
cana-1065	92	12	,	,	PUNCT
cana-1065	92	13	𝑇	𝑇	PROPN
cana-1065	92	14	:	:	PUNCT
cana-1065	92	15	𝑌	𝑌	PROPN
cana-1065	92	16	→	→	SYM
cana-1065	92	17	𝑌	𝑌	PROPN
cana-1065	92	18	be	be	VERB
cana-1065	92	19	mappings	mapping	NOUN
cana-1065	92	20	such	such	ADJ
cana-1065	92	21	that	that	SCONJ
cana-1065	92	22	(	(	PUNCT
cana-1065	92	23	3.1.1	3.1.1	NUM
cana-1065	92	24	)	)	PUNCT
cana-1065	92	25	𝐴	𝐴	PROPN
cana-1065	92	26	(	(	PUNCT
cana-1065	92	27	𝑌	𝑌	PROPN
cana-1065	92	28	)	)	PUNCT
cana-1065	92	29	⊂	⊂	PROPN
cana-1065	92	30	𝑇	𝑇	PROPN
cana-1065	92	31	(	(	PUNCT
cana-1065	92	32	𝑌)𝑎𝑛𝑑	𝑌)𝑎𝑛𝑑	NUM
cana-1065	92	33	𝐵	𝐵	NOUN
cana-1065	92	34	(	(	PUNCT
cana-1065	92	35	𝑌	𝑌	PROPN
cana-1065	92	36	)	)	PUNCT
cana-1065	92	37	⊂	⊂	PROPN
cana-1065	92	38	𝑆	𝑆	PROPN
cana-1065	92	39	(	(	PUNCT
cana-1065	92	40	𝑌	𝑌	PROPN
cana-1065	92	41	)	)	PUNCT
cana-1065	92	42	,	,	PUNCT
cana-1065	92	43	(	(	PUNCT
cana-1065	92	44	3.1.2	3.1.2	X
cana-1065	92	45	)	)	PUNCT
cana-1065	92	46	the	the	DET
cana-1065	92	47	pairs	pair	NOUN
cana-1065	92	48	(	(	PUNCT
cana-1065	92	49	𝐴	𝐴	PROPN
cana-1065	92	50	,	,	PUNCT
cana-1065	92	51	𝑆	𝑆	PROPN
cana-1065	92	52	)	)	PUNCT
cana-1065	92	53	and	and	CCONJ
cana-1065	92	54	(	(	PUNCT
cana-1065	92	55	𝐵	𝐵	PROPN
cana-1065	92	56	,	,	PUNCT
cana-1065	92	57	𝑇	𝑇	PROPN
cana-1065	92	58	)	)	PUNCT
cana-1065	92	59	are	be	AUX
cana-1065	92	60	weakly	weakly	ADV
cana-1065	92	61	compatible	compatible	ADJ
cana-1065	92	62	mappings	mapping	NOUN
cana-1065	92	63	of	of	ADP
cana-1065	92	64	type	type	NOUN
cana-1065	92	65	(	(	PUNCT
cana-1065	92	66	𝑃	𝑃	NOUN
cana-1065	92	67	)	)	PUNCT
cana-1065	92	68	,	,	PUNCT
cana-1065	92	69	(	(	PUNCT
cana-1065	92	70	3.1.3	3.1.3	X
cana-1065	92	71	)	)	PUNCT
cana-1065	92	72	one	one	NUM
cana-1065	92	73	of	of	ADP
cana-1065	92	74	𝐴	𝐴	PROPN
cana-1065	92	75	,	,	PUNCT
cana-1065	92	76	𝑆	𝑆	PROPN
cana-1065	92	77	,	,	PUNCT
cana-1065	92	78	𝐵	𝐵	PROPN
cana-1065	92	79	,	,	PUNCT
cana-1065	92	80	𝑇	𝑇	PROPN
cana-1065	92	81	be	be	AUX
cana-1065	92	82	continuous	continuous	ADJ
cana-1065	92	83	,	,	PUNCT
cana-1065	92	84	and	and	CCONJ
cana-1065	92	85	(	(	PUNCT
cana-1065	92	86	3.1.4	3.1.4	NUM
cana-1065	92	87	)	)	PUNCT
cana-1065	92	88	there	there	PRON
cana-1065	92	89	exists	exist	VERB
cana-1065	92	90	a	a	DET
cana-1065	92	91	constant	constant	ADJ
cana-1065	92	92	ξ	ξ	X
cana-1065	92	93	∈	∈	PROPN
cana-1065	92	94	(	(	PUNCT
cana-1065	92	95	0	0	NUM
cana-1065	92	96	,	,	PUNCT
cana-1065	92	97	1	1	NUM
cana-1065	92	98	)	)	PUNCT
cana-1065	92	99	such	such	ADJ
cana-1065	92	100	that	that	PRON
cana-1065	92	101	𝑀(𝐴𝑥	𝑀(𝐴𝑥	PROPN
cana-1065	92	102	,	,	PUNCT
cana-1065	92	103	𝐵𝑦	𝐵𝑦	PROPN
cana-1065	92	104	,	,	PUNCT
cana-1065	92	105	ξ	ξ	PROPN
cana-1065	92	106	𝑞	𝑞	PROPN
cana-1065	92	107	)	)	PUNCT
cana-1065	92	108	≥	≥	NOUN
cana-1065	92	109	φ{min{𝑀(𝑆𝑥	φ{min{𝑀(𝑆𝑥	VERB
cana-1065	92	110	,	,	PUNCT
cana-1065	92	111	𝐴𝑥	𝐴𝑥	NOUN
cana-1065	92	112	,	,	PUNCT
cana-1065	92	113	𝑞	𝑞	NOUN
cana-1065	92	114	)	)	PUNCT
cana-1065	92	115	,	,	PUNCT
cana-1065	92	116	𝑀(𝑇𝑦	𝑀(𝑇𝑦	PROPN
cana-1065	92	117	,	,	PUNCT
cana-1065	92	118	𝐵𝑦	𝐵𝑦	PROPN
cana-1065	92	119	,	,	PUNCT
cana-1065	92	120	𝑞	𝑞	NOUN
cana-1065	92	121	)	)	PUNCT
cana-1065	92	122	,	,	PUNCT
cana-1065	92	123	𝑀	𝑀	PROPN
cana-1065	92	124	(	(	PUNCT
cana-1065	92	125	𝑇𝑦	𝑇𝑦	PROPN
cana-1065	92	126	,	,	PUNCT
cana-1065	92	127	𝐴𝑥	𝐴𝑥	NOUN
cana-1065	92	128	,	,	PUNCT
cana-1065	92	129	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	92	130	)	)	PUNCT
cana-1065	92	131	,	,	PUNCT
cana-1065	92	132	𝑀(𝑆𝑥	𝑀(𝑆𝑥	ADJ
cana-1065	92	133	,	,	PUNCT
cana-1065	92	134	𝐵𝑦(2	𝐵𝑦(2	ADJ
cana-1065	92	135	−	−	NOUN
cana-1065	92	136	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	92	137	,	,	PUNCT
cana-1065	92	138	𝑀(𝑆𝑥	𝑀(𝑆𝑥	ADJ
cana-1065	92	139	,	,	PUNCT
cana-1065	92	140	𝑇𝑦	𝑇𝑦	PROPN
cana-1065	92	141	,	,	PUNCT
cana-1065	92	142	𝑞	𝑞	NOUN
cana-1065	92	143	)	)	PUNCT
cana-1065	92	144	}	}	PUNCT
cana-1065	92	145	}	}	PUNCT
cana-1065	92	146	for	for	ADP
cana-1065	92	147	all	all	PRON
cana-1065	92	148	𝑥	𝑥	PROPN
cana-1065	92	149	,	,	PUNCT
cana-1065	92	150	𝑦	𝑦	NOUN
cana-1065	92	151	∈	∈	PROPN
cana-1065	92	152	𝑌	𝑌	PROPN
cana-1065	92	153	,	,	PUNCT
cana-1065	93	1	𝑟	𝑟	SYM
cana-1065	93	2	∈	∈	PROPN
cana-1065	93	3	(	(	PUNCT
cana-1065	93	4	0	0	NUM
cana-1065	93	5	,	,	PUNCT
cana-1065	93	6	2	2	NUM
cana-1065	93	7	)	)	PUNCT
cana-1065	93	8	and	and	CCONJ
cana-1065	93	9	𝑞	𝑞	X
cana-1065	93	10	>	>	X
cana-1065	93	11	0	0	PROPN
cana-1065	93	12	,	,	PUNCT
cana-1065	93	13	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1065	93	14	φ	φ	NOUN
cana-1065	93	15	:	:	PUNCT
cana-1065	94	1	[	[	X
cana-1065	94	2	0,1	0,1	NUM
cana-1065	94	3	]	]	PUNCT
cana-1065	94	4	→	→	X
cana-1065	94	5	[	[	X
cana-1065	94	6	0,1	0,1	NUM
cana-1065	94	7	]	]	X
cana-1065	94	8	satisfies	satisfie	NOUN
cana-1065	94	9	(	(	PUNCT
cana-1065	94	10	i	i	NOUN
cana-1065	94	11	)	)	PUNCT
cana-1065	94	12	φ	φ	PROPN
cana-1065	94	13	is	be	AUX
cana-1065	94	14	continuous	continuous	ADJ
cana-1065	94	15	and	and	CCONJ
cana-1065	94	16	non	non	ADJ
cana-1065	94	17	-	-	ADJ
cana-1065	94	18	decreasing	decrease	VERB
cana-1065	94	19	on	on	ADP
cana-1065	94	20	[	[	X
cana-1065	94	21	0,1	0,1	NUM
cana-1065	94	22	]	]	PUNCT
cana-1065	94	23	(	(	PUNCT
cana-1065	94	24	ii	ii	NOUN
cana-1065	94	25	)	)	PUNCT
cana-1065	94	26	φ(n	φ(n	NOUN
cana-1065	94	27	)	)	PUNCT
cana-1065	94	28	>	>	X
cana-1065	95	1	n	n	CCONJ
cana-1065	95	2	for	for	ADP
cana-1065	95	3	all	all	DET
cana-1065	95	4	n	n	NOUN
cana-1065	95	5	in	in	ADP
cana-1065	95	6	[	[	X
cana-1065	95	7	0,1	0,1	NUM
cana-1065	95	8	]	]	PUNCT
cana-1065	95	9	communications	communication	NOUN
cana-1065	95	10	on	on	ADP
cana-1065	95	11	applied	apply	VERB
cana-1065	95	12	nonlinear	nonlinear	ADJ
cana-1065	95	13	analysis	analysis	NOUN
cana-1065	95	14	issn	issn	NOUN
cana-1065	95	15	:	:	PUNCT
cana-1065	95	16	1074	1074	NUM
cana-1065	95	17	-	-	PUNCT
cana-1065	95	18	133x	133x	NUM
cana-1065	95	19	vol	vol	NOUN
cana-1065	95	20	31	31	NUM
cana-1065	95	21	no	no	NOUN
cana-1065	95	22	.	.	PUNCT
cana-1065	96	1	5s	5s	NUM
cana-1065	96	2	(	(	PUNCT
cana-1065	96	3	2024	2024	NUM
cana-1065	96	4	)	)	PUNCT
cana-1065	96	5	462	462	NUM
cana-1065	96	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	96	7	noting	note	VERB
cana-1065	96	8	that	that	SCONJ
cana-1065	96	9	if	if	SCONJ
cana-1065	96	10	φ	φ	PROPN
cana-1065	96	11	∈	∈	PROPN
cana-1065	96	12	φ	φ	PROPN
cana-1065	96	13	,	,	PUNCT
cana-1065	96	14	class	class	NOUN
cana-1065	96	15	of	of	ADP
cana-1065	96	16	all	all	DET
cana-1065	96	17	mappings	mapping	NOUN
cana-1065	96	18	φ	φ	NOUN
cana-1065	96	19	:	:	PUNCT
cana-1065	97	1	[	[	X
cana-1065	97	2	0,1	0,1	NUM
cana-1065	97	3	]	]	PUNCT
cana-1065	97	4	→	→	PUNCT
cana-1065	98	1	[	[	X
cana-1065	98	2	0,1	0,1	NUM
cana-1065	98	3	]	]	PUNCT
cana-1065	98	4	then	then	ADV
cana-1065	98	5	φ(0	φ(0	ADJ
cana-1065	98	6	)	)	PUNCT
cana-1065	98	7	=	=	SYM
cana-1065	98	8	0	0	NUM
cana-1065	98	9	,	,	PUNCT
cana-1065	98	10	φ(1	φ(1	PROPN
cana-1065	98	11	)	)	PUNCT
cana-1065	98	12	=	=	SYM
cana-1065	98	13	1	1	NUM
cana-1065	98	14	and	and	CCONJ
cana-1065	98	15	φ(n	φ(n	NOUN
cana-1065	98	16	)	)	PUNCT
cana-1065	98	17	≥	≥	NOUN
cana-1065	98	18	n	n	CCONJ
cana-1065	98	19	for	for	ADP
cana-1065	98	20	all	all	DET
cana-1065	98	21	n	n	NOUN
cana-1065	98	22	in	in	ADP
cana-1065	98	23	[	[	X
cana-1065	98	24	0,1	0,1	NUM
cana-1065	98	25	]	]	PUNCT
cana-1065	98	26	.	.	PUNCT
cana-1065	99	1	then	then	ADV
cana-1065	99	2	,	,	PUNCT
cana-1065	99	3	𝐴	𝐴	PROPN
cana-1065	99	4	,	,	PUNCT
cana-1065	99	5	𝐵	𝐵	PROPN
cana-1065	99	6	,	,	PUNCT
cana-1065	99	7	𝑆	𝑆	PROPN
cana-1065	99	8	,	,	PUNCT
cana-1065	99	9	𝑇	𝑇	PROPN
cana-1065	99	10	have	have	VERB
cana-1065	99	11	a	a	DET
cana-1065	99	12	unique	unique	ADJ
cana-1065	99	13	common	common	ADJ
cana-1065	99	14	fixed	fix	VERB
cana-1065	99	15	point	point	NOUN
cana-1065	99	16	in	in	ADP
cana-1065	99	17	𝑌.	𝑌.	PROPN
cana-1065	99	18	proof	proof	NOUN
cana-1065	99	19	:	:	PUNCT
cana-1065	99	20	consider	consider	VERB
cana-1065	99	21	𝑢0	𝑢0	PROPN
cana-1065	99	22	∈	∈	PROPN
cana-1065	99	23	𝑌.	𝑌.	PROPN
cana-1065	99	24	since	since	SCONJ
cana-1065	99	25	𝐴(𝑌	𝐴(𝑌	PROPN
cana-1065	99	26	)	)	PUNCT
cana-1065	99	27	⊂	⊂	PROPN
cana-1065	99	28	𝑇	𝑇	PROPN
cana-1065	99	29	(	(	PUNCT
cana-1065	99	30	𝑌	𝑌	PROPN
cana-1065	99	31	)	)	PUNCT
cana-1065	99	32	,	,	PUNCT
cana-1065	99	33	so	so	CCONJ
cana-1065	99	34	there	there	PRON
cana-1065	99	35	exists	exist	VERB
cana-1065	99	36	a	a	DET
cana-1065	99	37	point	point	NOUN
cana-1065	99	38	𝑢1𝑖𝑛	𝑢1𝑖𝑛	PUNCT
cana-1065	99	39	𝑌	𝑌	PROPN
cana-1065	99	40	such	such	ADJ
cana-1065	99	41	that	that	DET
cana-1065	99	42	𝐴𝑢0	𝐴𝑢0	NOUN
cana-1065	100	1	=	=	NOUN
cana-1065	100	2	𝑇𝑢	𝑇𝑢	ADP
cana-1065	100	3	1	1	NUM
cana-1065	100	4	=	=	SYM
cana-1065	100	5	𝑣0	𝑣0	NOUN
cana-1065	100	6	.	.	PUNCT
cana-1065	100	7	again	again	ADV
cana-1065	100	8	,	,	PUNCT
cana-1065	100	9	since	since	SCONJ
cana-1065	100	10	𝐵(𝑌	𝐵(𝑌	PRON
cana-1065	100	11	)	)	PUNCT
cana-1065	100	12	⊂	⊂	PROPN
cana-1065	100	13	𝑆	𝑆	PROPN
cana-1065	100	14	(	(	PUNCT
cana-1065	100	15	𝑌	𝑌	PROPN
cana-1065	100	16	)	)	PUNCT
cana-1065	100	17	,	,	PUNCT
cana-1065	100	18	so	so	ADV
cana-1065	100	19	for	for	ADP
cana-1065	100	20	𝑢1	𝑢1	NOUN
cana-1065	100	21	,	,	PUNCT
cana-1065	100	22	we	we	PRON
cana-1065	100	23	may	may	AUX
cana-1065	100	24	choose	choose	VERB
cana-1065	100	25	𝑢2	𝑢2	PROPN
cana-1065	100	26	in	in	ADP
cana-1065	100	27	𝑌	𝑌	PROPN
cana-1065	100	28	such	such	DET
cana-1065	100	29	that	that	DET
cana-1065	100	30	𝐵𝑢1	𝐵𝑢1	NOUN
cana-1065	100	31	=	=	NOUN
cana-1065	100	32	𝑆𝑢2	𝑆𝑢2	NOUN
cana-1065	100	33	=	=	SYM
cana-1065	100	34	𝑣1	𝑣1	NOUN
cana-1065	100	35	and	and	CCONJ
cana-1065	100	36	so	so	ADV
cana-1065	100	37	on	on	ADV
cana-1065	100	38	.	.	PUNCT
cana-1065	101	1	and	and	CCONJ
cana-1065	101	2	inductively	inductively	ADV
cana-1065	101	3	,	,	PUNCT
cana-1065	101	4	we	we	PRON
cana-1065	101	5	may	may	AUX
cana-1065	101	6	construct	construct	VERB
cana-1065	101	7	sequence	sequence	NOUN
cana-1065	101	8	{	{	PUNCT
cana-1065	101	9	𝑢𝑛	𝑢𝑛	NOUN
cana-1065	101	10	}	}	PUNCT
cana-1065	101	11	and	and	CCONJ
cana-1065	101	12	{	{	PUNCT
cana-1065	101	13	𝑣𝑛	𝑣𝑛	NOUN
cana-1065	101	14	}	}	PUNCT
cana-1065	101	15	in	in	ADP
cana-1065	101	16	𝑌	𝑌	PROPN
cana-1065	102	1	such	such	ADJ
cana-1065	102	2	that	that	PRON
cana-1065	102	3	𝐴𝑢2𝑛	𝐴𝑢2𝑛	ADJ
cana-1065	102	4	=	=	NOUN
cana-1065	102	5	𝑇𝑢	𝑇𝑢	ADP
cana-1065	102	6	2𝑛+1	2𝑛+1	NOUN
cana-1065	102	7	=	=	SYM
cana-1065	102	8	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	9	,	,	PUNCT
cana-1065	102	10	and	and	CCONJ
cana-1065	102	11	𝐵𝑢2𝑛+1	𝐵𝑢2𝑛+1	ADJ
cana-1065	102	12	=	=	PUNCT
cana-1065	102	13	𝑆𝑢	𝑆𝑢	PROPN
cana-1065	102	14	2𝑛+2	2𝑛+2	NOUN
cana-1065	102	15	=	=	SYM
cana-1065	102	16	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	17	,	,	PUNCT
cana-1065	102	18	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-1065	102	19	𝑛	𝑛	X
cana-1065	102	20	=	=	SYM
cana-1065	102	21	0	0	NUM
cana-1065	102	22	,	,	PUNCT
cana-1065	102	23	1	1	NUM
cana-1065	102	24	,	,	PUNCT
cana-1065	102	25	2	2	NUM
cana-1065	102	26	,	,	PUNCT
cana-1065	102	27	…	…	PUNCT
cana-1065	102	28	putting	put	VERB
cana-1065	102	29	𝑥	𝑥	X
cana-1065	102	30	=	=	PUNCT
cana-1065	102	31	𝑢2𝑛	𝑢2𝑛	X
cana-1065	102	32	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1065	102	33	𝑦	𝑦	PROPN
cana-1065	102	34	=	=	X
cana-1065	102	35	𝑢2𝑛+1	𝑢2𝑛+1	NOUN
cana-1065	102	36	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1065	102	37	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1065	102	38	𝑞	𝑞	X
cana-1065	102	39	>	>	X
cana-1065	102	40	0	0	NUM
cana-1065	102	41	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1065	102	42	𝑟	𝑟	NOUN
cana-1065	102	43	=	=	SYM
cana-1065	102	44	1	1	NUM
cana-1065	102	45	−	−	PROPN
cana-1065	102	46	𝑝	𝑝	NOUN
cana-1065	102	47	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-1065	102	48	𝑝	𝑝	PROPN
cana-1065	102	49	𝜖	𝜖	X
cana-1065	102	50	(	(	PUNCT
cana-1065	102	51	0	0	NUM
cana-1065	102	52	,	,	PUNCT
cana-1065	102	53	1	1	NUM
cana-1065	102	54	)	)	PUNCT
cana-1065	102	55	𝑖𝑛	𝑖𝑛	NOUN
cana-1065	102	56	(	(	PUNCT
cana-1065	102	57	3	3	NUM
cana-1065	102	58	.	.	NOUN
cana-1065	102	59	1.4	1.4	NUM
cana-1065	102	60	)	)	PUNCT
cana-1065	102	61	,	,	PUNCT
cana-1065	102	62	we	we	PRON
cana-1065	102	63	get	get	VERB
cana-1065	102	64	𝑀(𝐴𝑢2𝑛	𝑀(𝐴𝑢2𝑛	NOUN
cana-1065	102	65	,	,	PUNCT
cana-1065	102	66	𝐵𝑢2𝑛+1	𝐵𝑢2𝑛+1	NUM
cana-1065	102	67	,	,	PUNCT
cana-1065	102	68	ξ	ξ	PROPN
cana-1065	102	69	𝑞	𝑞	PROPN
cana-1065	102	70	)	)	PUNCT
cana-1065	102	71	≥	≥	NOUN
cana-1065	102	72	φ{min{𝑀(𝑆𝑢2𝑛	φ{min{𝑀(𝑆𝑢2𝑛	NOUN
cana-1065	102	73	,	,	PUNCT
cana-1065	102	74	𝐴𝑢2𝑛	𝐴𝑢2𝑛	ADJ
cana-1065	102	75	,	,	PUNCT
cana-1065	102	76	𝑞	𝑞	NOUN
cana-1065	102	77	)	)	PUNCT
cana-1065	102	78	,	,	PUNCT
cana-1065	102	79	𝑀(𝑇𝑢2𝑛+1	𝑀(𝑇𝑢2𝑛+1	PROPN
cana-1065	102	80	,	,	PUNCT
cana-1065	102	81	𝐵𝑢2𝑛+1	𝐵𝑢2𝑛+1	PROPN
cana-1065	102	82	,	,	PUNCT
cana-1065	102	83	𝑞	𝑞	NOUN
cana-1065	102	84	)	)	PUNCT
cana-1065	102	85	,	,	PUNCT
cana-1065	102	86	𝑀(𝑇𝑢2𝑛+1	𝑀(𝑇𝑢2𝑛+1	PROPN
cana-1065	102	87	,	,	PUNCT
cana-1065	102	88	𝐴𝑢2𝑛	𝐴𝑢2𝑛	ADJ
cana-1065	102	89	,	,	PUNCT
cana-1065	102	90	(	(	PUNCT
cana-1065	102	91	(	(	PUNCT
cana-1065	102	92	1	1	NUM
cana-1065	102	93	−	−	NUM
cana-1065	102	94	𝑝))𝑞	𝑝))𝑞	ADJ
cana-1065	102	95	)	)	PUNCT
cana-1065	102	96	,	,	PUNCT
cana-1065	102	97	𝑀(𝑆𝑢2𝑛	𝑀(𝑆𝑢2𝑛	PROPN
cana-1065	102	98	,	,	PUNCT
cana-1065	102	99	𝐵𝑢2𝑛+1,((1	𝐵𝑢2𝑛+1,((1	PROPN
cana-1065	102	100	+	+	ADJ
cana-1065	102	101	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	102	102	)	)	PUNCT
cana-1065	102	103	,	,	PUNCT
cana-1065	102	104	𝑀(𝑆𝑢2𝑛	𝑀(𝑆𝑢2𝑛	PROPN
cana-1065	102	105	,	,	PUNCT
cana-1065	102	106	𝑇𝑢2𝑛+1	𝑇𝑢2𝑛+1	NUM
cana-1065	102	107	,	,	PUNCT
cana-1065	102	108	𝑞	𝑞	NOUN
cana-1065	102	109	)	)	PUNCT
cana-1065	102	110	}	}	PUNCT
cana-1065	102	111	}	}	PUNCT
cana-1065	102	112	or	or	CCONJ
cana-1065	102	113	,	,	PUNCT
cana-1065	102	114	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	ADJ
cana-1065	102	115	,	,	PUNCT
cana-1065	102	116	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	117	,	,	PUNCT
cana-1065	102	118	ξ	ξ	PROPN
cana-1065	102	119	𝑞	𝑞	PROPN
cana-1065	102	120	)	)	PUNCT
cana-1065	102	121	≥	≥	NOUN
cana-1065	102	122	φ{min{𝑀(𝑣2𝑛−1	φ{min{𝑀(𝑣2𝑛−1	PROPN
cana-1065	102	123	,	,	PUNCT
cana-1065	102	124	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	125	,	,	PUNCT
cana-1065	102	126	𝑞	𝑞	NOUN
cana-1065	102	127	)	)	PUNCT
cana-1065	102	128	,	,	PUNCT
cana-1065	102	129	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	130	,	,	PUNCT
cana-1065	102	131	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	132	,	,	PUNCT
cana-1065	102	133	𝑞	𝑞	NOUN
cana-1065	102	134	)	)	PUNCT
cana-1065	102	135	,	,	PUNCT
cana-1065	102	136	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	137	,	,	PUNCT
cana-1065	102	138	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	139	,	,	PUNCT
cana-1065	102	140	(	(	PUNCT
cana-1065	102	141	(	(	PUNCT
cana-1065	102	142	1	1	NUM
cana-1065	102	143	−	−	NUM
cana-1065	102	144	𝑝))𝑞	𝑝))𝑞	ADJ
cana-1065	102	145	)	)	PUNCT
cana-1065	102	146	,	,	PUNCT
cana-1065	102	147	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	148	,	,	PUNCT
cana-1065	102	149	𝑣2𝑛+1,((1	𝑣2𝑛+1,((1	NOUN
cana-1065	102	150	+	+	CCONJ
cana-1065	102	151	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	102	152	)	)	PUNCT
cana-1065	102	153	,	,	PUNCT
cana-1065	102	154	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	155	,	,	PUNCT
cana-1065	102	156	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	157	,	,	PUNCT
cana-1065	102	158	𝑞	𝑞	NOUN
cana-1065	102	159	)	)	PUNCT
cana-1065	102	160	}	}	PUNCT
cana-1065	102	161	}	}	PUNCT
cana-1065	102	162	≥	≥	PROPN
cana-1065	102	163	φ{min{𝑀(𝑣2𝑛−1	φ{min{𝑀(𝑣2𝑛−1	PROPN
cana-1065	102	164	,	,	PUNCT
cana-1065	102	165	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	166	,	,	PUNCT
cana-1065	102	167	𝑞	𝑞	NOUN
cana-1065	102	168	)	)	PUNCT
cana-1065	102	169	,	,	PUNCT
cana-1065	102	170	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	171	,	,	PUNCT
cana-1065	102	172	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	173	,	,	PUNCT
cana-1065	102	174	𝑞	𝑞	NOUN
cana-1065	102	175	)	)	PUNCT
cana-1065	102	176	,	,	PUNCT
cana-1065	102	177	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	178	,	,	PUNCT
cana-1065	102	179	𝑣2𝑛+1,((1	𝑣2𝑛+1,((1	NOUN
cana-1065	102	180	+	+	CCONJ
cana-1065	102	181	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	102	182	)	)	PUNCT
cana-1065	102	183	,	,	PUNCT
cana-1065	102	184	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	185	,	,	PUNCT
cana-1065	102	186	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	187	,	,	PUNCT
cana-1065	102	188	𝑞	𝑞	NOUN
cana-1065	102	189	)	)	PUNCT
cana-1065	102	190	}	}	PUNCT
cana-1065	102	191	}	}	PUNCT
cana-1065	102	192	≥	≥	PROPN
cana-1065	102	193	φ{min{𝑀(𝑣2𝑛−1	φ{min{𝑀(𝑣2𝑛−1	PROPN
cana-1065	102	194	,	,	PUNCT
cana-1065	102	195	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	196	,	,	PUNCT
cana-1065	102	197	𝑞	𝑞	NOUN
cana-1065	102	198	)	)	PUNCT
cana-1065	102	199	,	,	PUNCT
cana-1065	102	200	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	201	,	,	PUNCT
cana-1065	102	202	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	203	,	,	PUNCT
cana-1065	102	204	𝑞	𝑞	NOUN
cana-1065	102	205	)	)	PUNCT
cana-1065	102	206	,	,	PUNCT
cana-1065	102	207	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	208	,	,	PUNCT
cana-1065	102	209	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	210	,	,	PUNCT
cana-1065	102	211	𝑞	𝑞	NOUN
cana-1065	102	212	)	)	PUNCT
cana-1065	102	213	,	,	PUNCT
cana-1065	102	214	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	215	,	,	PUNCT
cana-1065	102	216	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	217	,	,	PUNCT
cana-1065	102	218	𝑝𝑞	𝑝𝑞	NOUN
cana-1065	102	219	)	)	PUNCT
cana-1065	102	220	,	,	PUNCT
cana-1065	102	221	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	222	,	,	PUNCT
cana-1065	102	223	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	224	,	,	PUNCT
cana-1065	102	225	𝑞	𝑞	NOUN
cana-1065	102	226	)	)	PUNCT
cana-1065	102	227	}	}	PUNCT
cana-1065	102	228	}	}	PUNCT
cana-1065	102	229	≥	≥	PROPN
cana-1065	102	230	φ{min{𝑀(𝑣2𝑛−1	φ{min{𝑀(𝑣2𝑛−1	PROPN
cana-1065	102	231	,	,	PUNCT
cana-1065	102	232	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	233	,	,	PUNCT
cana-1065	102	234	𝑞	𝑞	NOUN
cana-1065	102	235	)	)	PUNCT
cana-1065	102	236	,	,	PUNCT
cana-1065	102	237	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	238	,	,	PUNCT
cana-1065	102	239	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	240	,	,	PUNCT
cana-1065	102	241	𝑞	𝑞	NOUN
cana-1065	102	242	)	)	PUNCT
cana-1065	102	243	,	,	PUNCT
cana-1065	102	244	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	245	,	,	PUNCT
cana-1065	102	246	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	247	,	,	PUNCT
cana-1065	102	248	𝑝𝑞	𝑝𝑞	NOUN
cana-1065	102	249	)	)	PUNCT
cana-1065	102	250	}	}	PUNCT
cana-1065	102	251	}	}	PUNCT
cana-1065	102	252	as	as	ADP
cana-1065	102	253	𝑝	𝑝	NOUN
cana-1065	102	254	→	→	SYM
cana-1065	102	255	1	1	NUM
cana-1065	102	256	,	,	PUNCT
cana-1065	102	257	we	we	PRON
cana-1065	102	258	obtain	obtain	VERB
cana-1065	102	259	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	ADJ
cana-1065	102	260	,	,	PUNCT
cana-1065	102	261	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	262	,	,	PUNCT
cana-1065	102	263	ξ	ξ	PROPN
cana-1065	102	264	𝑞	𝑞	PROPN
cana-1065	102	265	)	)	PUNCT
cana-1065	102	266	≥	≥	NOUN
cana-1065	102	267	φ{min{𝑀(𝑣2𝑛−1	φ{min{𝑀(𝑣2𝑛−1	PROPN
cana-1065	102	268	,	,	PUNCT
cana-1065	102	269	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	270	,	,	PUNCT
cana-1065	102	271	𝑞	𝑞	NOUN
cana-1065	102	272	)	)	PUNCT
cana-1065	102	273	,	,	PUNCT
cana-1065	102	274	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	275	,	,	PUNCT
cana-1065	102	276	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	277	,	,	PUNCT
cana-1065	102	278	𝑞	𝑞	NOUN
cana-1065	102	279	)	)	PUNCT
cana-1065	102	280	,	,	PUNCT
cana-1065	102	281	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	282	,	,	PUNCT
cana-1065	102	283	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	284	,	,	PUNCT
cana-1065	102	285	𝑞	𝑞	NOUN
cana-1065	102	286	)	)	PUNCT
cana-1065	102	287	}	}	PUNCT
cana-1065	102	288	}	}	PUNCT
cana-1065	102	289	≥	≥	PROPN
cana-1065	102	290	φ{min{𝑀(𝑣2𝑛−1	φ{min{𝑀(𝑣2𝑛−1	PROPN
cana-1065	102	291	,	,	PUNCT
cana-1065	102	292	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	293	,	,	PUNCT
cana-1065	102	294	𝑞	𝑞	NOUN
cana-1065	102	295	)	)	PUNCT
cana-1065	102	296	,	,	PUNCT
cana-1065	102	297	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	NOUN
cana-1065	102	298	,	,	PUNCT
cana-1065	102	299	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	300	,	,	PUNCT
cana-1065	102	301	𝑞	𝑞	NOUN
cana-1065	102	302	)	)	PUNCT
cana-1065	102	303	}	}	PUNCT
cana-1065	102	304	or	or	CCONJ
cana-1065	102	305	,	,	PUNCT
cana-1065	102	306	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	ADJ
cana-1065	102	307	,	,	PUNCT
cana-1065	102	308	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	309	,	,	PUNCT
cana-1065	102	310	ξ	ξ	PROPN
cana-1065	102	311	𝑞	𝑞	PROPN
cana-1065	102	312	)	)	PUNCT
cana-1065	102	313	≥	≥	PROPN
cana-1065	102	314	φ{𝑀(𝑣2𝑛−1	φ{𝑀(𝑣2𝑛−1	PROPN
cana-1065	102	315	,	,	PUNCT
cana-1065	102	316	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	317	,	,	PUNCT
cana-1065	102	318	𝑞	𝑞	NOUN
cana-1065	102	319	)	)	PUNCT
cana-1065	102	320	}	}	PUNCT
cana-1065	102	321	>	>	X
cana-1065	102	322	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	323	,	,	PUNCT
cana-1065	102	324	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	325	,	,	PUNCT
cana-1065	102	326	𝑞	𝑞	NOUN
cana-1065	102	327	)	)	PUNCT
cana-1065	102	328	,	,	PUNCT
cana-1065	102	329	by	by	ADP
cana-1065	102	330	property	property	NOUN
cana-1065	102	331	of	of	ADP
cana-1065	102	332	φ	φ	PROPN
cana-1065	102	333	hence	hence	ADV
cana-1065	102	334	,	,	PUNCT
cana-1065	102	335	we	we	PRON
cana-1065	102	336	get	get	VERB
cana-1065	102	337	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	ADJ
cana-1065	102	338	,	,	PUNCT
cana-1065	102	339	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	340	,	,	PUNCT
cana-1065	102	341	ξ	ξ	PROPN
cana-1065	102	342	𝑞	𝑞	PROPN
cana-1065	102	343	)	)	PUNCT
cana-1065	102	344	≥	≥	NOUN
cana-1065	102	345	𝑀(𝑣2𝑛−1	𝑀(𝑣2𝑛−1	NOUN
cana-1065	102	346	,	,	PUNCT
cana-1065	102	347	𝑣2𝑛	𝑣2𝑛	PROPN
cana-1065	102	348	,	,	PUNCT
cana-1065	102	349	𝑞	𝑞	NOUN
cana-1065	102	350	)	)	PUNCT
cana-1065	102	351	similarly	similarly	ADV
cana-1065	102	352	,	,	PUNCT
cana-1065	102	353	we	we	PRON
cana-1065	102	354	obtain	obtain	VERB
cana-1065	102	355	𝑀(𝑣2𝑛+1	𝑀(𝑣2𝑛+1	PROPN
cana-1065	102	356	,	,	PUNCT
cana-1065	102	357	𝑣2𝑛+2	𝑣2𝑛+2	PROPN
cana-1065	102	358	,	,	PUNCT
cana-1065	102	359	ξ	ξ	PROPN
cana-1065	102	360	𝑞	𝑞	PROPN
cana-1065	102	361	)	)	PUNCT
cana-1065	102	362	≥	≥	PROPN
cana-1065	102	363	𝑀(𝑣2𝑛	𝑀(𝑣2𝑛	PROPN
cana-1065	102	364	,	,	PUNCT
cana-1065	102	365	𝑣2𝑛+1	𝑣2𝑛+1	PROPN
cana-1065	102	366	,	,	PUNCT
cana-1065	102	367	𝑞	𝑞	NOUN
cana-1065	102	368	)	)	PUNCT
cana-1065	102	369	therefore	therefore	ADV
cana-1065	102	370	,	,	PUNCT
cana-1065	102	371	for	for	ADP
cana-1065	102	372	every	every	DET
cana-1065	102	373	𝑛	𝑛	PRON
cana-1065	102	374	∈	∈	PROPN
cana-1065	102	375	𝑁	𝑁	PROPN
cana-1065	102	376	,	,	PUNCT
cana-1065	102	377	𝑀(𝑣𝑛	𝑀(𝑣𝑛	PROPN
cana-1065	102	378	,	,	PUNCT
cana-1065	102	379	𝑣𝑛+1	𝑣𝑛+1	PROPN
cana-1065	102	380	,	,	PUNCT
cana-1065	102	381	ξ	ξ	X
cana-1065	102	382	𝑞	𝑞	PROPN
cana-1065	102	383	)	)	PUNCT
cana-1065	102	384	≥	≥	PROPN
cana-1065	102	385	𝑀(𝑣𝑛−1	𝑀(𝑣𝑛−1	PROPN
cana-1065	102	386	,	,	PUNCT
cana-1065	102	387	𝑣𝑛	𝑣𝑛	PROPN
cana-1065	102	388	,	,	PUNCT
cana-1065	102	389	𝑞	𝑞	NOUN
cana-1065	102	390	)	)	PUNCT
cana-1065	102	391	so	so	ADV
cana-1065	102	392	,	,	PUNCT
cana-1065	102	393	using	use	VERB
cana-1065	102	394	lemma	lemma	PROPN
cana-1065	102	395	(	(	PUNCT
cana-1065	102	396	2.2	2.2	NUM
cana-1065	102	397	)	)	PUNCT
cana-1065	102	398	,	,	PUNCT
cana-1065	102	399	{	{	PUNCT
cana-1065	102	400	𝑣𝑛	𝑣𝑛	NOUN
cana-1065	102	401	}	}	PUNCT
cana-1065	102	402	is	be	AUX
cana-1065	102	403	a	a	DET
cana-1065	102	404	cauchy	cauchy	ADJ
cana-1065	102	405	sequence	sequence	NOUN
cana-1065	102	406	in	in	ADP
cana-1065	102	407	𝐾.	𝐾.	PROPN
cana-1065	102	408	since	since	SCONJ
cana-1065	102	409	the	the	DET
cana-1065	102	410	menger	menger	PROPN
cana-1065	102	411	space	space	NOUN
cana-1065	102	412	(	(	PUNCT
cana-1065	102	413	𝑌	𝑌	PROPN
cana-1065	102	414	,	,	PUNCT
cana-1065	102	415	𝑀	𝑀	PROPN
cana-1065	102	416	,	,	PUNCT
cana-1065	102	417	𝑡	𝑡	PROPN
cana-1065	102	418	)	)	PUNCT
cana-1065	102	419	is	be	AUX
cana-1065	102	420	complete	complete	ADJ
cana-1065	102	421	,	,	PUNCT
cana-1065	102	422	so	so	CCONJ
cana-1065	102	423	{	{	PUNCT
cana-1065	102	424	𝑣𝑛	𝑣𝑛	NOUN
cana-1065	102	425	}	}	PUNCT
cana-1065	102	426	converges	converge	NOUN
cana-1065	102	427	to	to	ADP
cana-1065	102	428	a	a	DET
cana-1065	102	429	point	point	NOUN
cana-1065	102	430	𝑧	𝑧	VERB
cana-1065	102	431	in	in	ADP
cana-1065	102	432	𝑌	𝑌	PROPN
cana-1065	102	433	and	and	CCONJ
cana-1065	102	434	consequently	consequently	ADV
cana-1065	102	435	the	the	DET
cana-1065	102	436	subsequences	subsequence	NOUN
cana-1065	102	437	{	{	PUNCT
cana-1065	102	438	𝐴𝑢2𝑛	𝐴𝑢2𝑛	PROPN
cana-1065	102	439	}	}	PUNCT
cana-1065	102	440	,	,	PUNCT
cana-1065	102	441	{	{	PUNCT
cana-1065	102	442	𝐵𝑢2𝑛+1	𝐵𝑢2𝑛+1	X
cana-1065	102	443	}	}	PUNCT
cana-1065	102	444	,	,	PUNCT
cana-1065	102	445	{	{	PUNCT
cana-1065	102	446	𝑆𝑢2𝑛	𝑆𝑢2𝑛	PROPN
cana-1065	102	447	}	}	PUNCT
cana-1065	102	448	,	,	PUNCT
cana-1065	102	449	{	{	PUNCT
cana-1065	102	450	𝑇𝑢2𝑛+1	𝑇𝑢2𝑛+1	NOUN
cana-1065	102	451	}	}	PUNCT
cana-1065	102	452	of	of	ADP
cana-1065	102	453	{	{	PUNCT
cana-1065	102	454	𝑣𝑛	𝑣𝑛	NOUN
cana-1065	102	455	}	}	PUNCT
cana-1065	102	456	also	also	ADV
cana-1065	102	457	converges	converge	VERB
cana-1065	102	458	to	to	ADP
cana-1065	102	459	𝑧.	𝑧.	NOUN
cana-1065	102	460	communications	communication	NOUN
cana-1065	102	461	on	on	ADP
cana-1065	102	462	applied	apply	VERB
cana-1065	102	463	nonlinear	nonlinear	ADJ
cana-1065	102	464	analysis	analysis	NOUN
cana-1065	102	465	issn	issn	NOUN
cana-1065	102	466	:	:	PUNCT
cana-1065	102	467	1074	1074	NUM
cana-1065	102	468	-	-	PUNCT
cana-1065	102	469	133x	133x	NUM
cana-1065	102	470	vol	vol	NOUN
cana-1065	102	471	31	31	NUM
cana-1065	102	472	no	no	NOUN
cana-1065	102	473	.	.	PUNCT
cana-1065	103	1	5s	5s	NUM
cana-1065	103	2	(	(	PUNCT
cana-1065	103	3	2024	2024	NUM
cana-1065	103	4	)	)	PUNCT
cana-1065	103	5	463	463	NUM
cana-1065	103	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	103	7	now	now	ADV
cana-1065	103	8	,	,	PUNCT
cana-1065	103	9	suppose	suppose	VERB
cana-1065	103	10	that	that	SCONJ
cana-1065	103	11	𝑇	𝑇	PROPN
cana-1065	103	12	is	be	AUX
cana-1065	103	13	continuous	continuous	ADJ
cana-1065	103	14	.	.	PUNCT
cana-1065	104	1	then	then	ADV
cana-1065	104	2	,	,	PUNCT
cana-1065	104	3	since	since	SCONJ
cana-1065	104	4	𝐵	𝐵	PROPN
cana-1065	104	5	&	&	CCONJ
cana-1065	104	6	𝑇	𝑇	PROPN
cana-1065	104	7	are	be	AUX
cana-1065	104	8	weakly	weakly	ADV
cana-1065	104	9	compatible	compatible	ADJ
cana-1065	104	10	mappings	mapping	NOUN
cana-1065	104	11	of	of	ADP
cana-1065	104	12	type	type	NOUN
cana-1065	104	13	(	(	PUNCT
cana-1065	104	14	𝑃	𝑃	NOUN
cana-1065	104	15	)	)	PUNCT
cana-1065	104	16	then	then	ADV
cana-1065	104	17	by	by	ADP
cana-1065	104	18	proposition	proposition	NOUN
cana-1065	104	19	2.5	2.5	NUM
cana-1065	104	20	,	,	PUNCT
cana-1065	104	21	𝐵𝐵𝑢2𝑛+1	𝐵𝐵𝑢2𝑛+1	NOUN
cana-1065	104	22	,	,	PUNCT
cana-1065	104	23	𝑇𝐵𝑢2𝑛+1	𝑇𝐵𝑢2𝑛+1	PROPN
cana-1065	104	24	→	→	PUNCT
cana-1065	105	1	𝑇𝑧	𝑇𝑧	ADP
cana-1065	105	2	𝑎𝑠	𝑎𝑠	PROPN
cana-1065	105	3	𝑛	𝑛	PROPN
cana-1065	105	4	→	→	PUNCT
cana-1065	105	5	∞.	∞.	PROPN
cana-1065	105	6	putting	put	VERB
cana-1065	105	7	𝑥	𝑥	X
cana-1065	105	8	=	=	PUNCT
cana-1065	105	9	𝑢2𝑛	𝑢2𝑛	PROPN
cana-1065	105	10	and	and	CCONJ
cana-1065	105	11	𝑦	𝑦	NOUN
cana-1065	105	12	=	=	X
cana-1065	105	13	𝐵𝑢2𝑛+1	𝐵𝑢2𝑛+1	X
cana-1065	105	14	in	in	ADP
cana-1065	105	15	relation	relation	NOUN
cana-1065	105	16	(	(	PUNCT
cana-1065	105	17	3.1.4	3.1.4	NUM
cana-1065	105	18	)	)	PUNCT
cana-1065	105	19	,	,	PUNCT
cana-1065	105	20	we	we	PRON
cana-1065	105	21	get	get	VERB
cana-1065	105	22	𝑀(𝐴𝑢2𝑛	𝑀(𝐴𝑢2𝑛	NOUN
cana-1065	105	23	,	,	PUNCT
cana-1065	105	24	𝐵𝐵𝑢2𝑛+1	𝐵𝐵𝑢2𝑛+1	PROPN
cana-1065	105	25	,	,	PUNCT
cana-1065	105	26	ξ	ξ	PROPN
cana-1065	105	27	𝑞	𝑞	PROPN
cana-1065	105	28	)	)	PUNCT
cana-1065	105	29	≥	≥	NOUN
cana-1065	105	30	φ{min{𝑀(𝑆𝑢2𝑛	φ{min{𝑀(𝑆𝑢2𝑛	NOUN
cana-1065	105	31	,	,	PUNCT
cana-1065	105	32	𝐴𝑢2𝑛	𝐴𝑢2𝑛	ADJ
cana-1065	105	33	,	,	PUNCT
cana-1065	105	34	𝑞	𝑞	NOUN
cana-1065	105	35	)	)	PUNCT
cana-1065	105	36	,	,	PUNCT
cana-1065	105	37	𝑀(𝑇𝐵𝑢2𝑛+1	𝑀(𝑇𝐵𝑢2𝑛+1	NOUN
cana-1065	105	38	,	,	PUNCT
cana-1065	105	39	𝐵𝐵𝑢2𝑛+1	𝐵𝐵𝑢2𝑛+1	PROPN
cana-1065	105	40	,	,	PUNCT
cana-1065	105	41	𝑞	𝑞	NOUN
cana-1065	105	42	)	)	PUNCT
cana-1065	105	43	,	,	PUNCT
cana-1065	105	44	𝑀(𝑇𝐵𝑢2𝑛+1	𝑀(𝑇𝐵𝑢2𝑛+1	ADJ
cana-1065	105	45	,	,	PUNCT
cana-1065	105	46	𝐴𝑢2𝑛	𝐴𝑢2𝑛	ADJ
cana-1065	105	47	,	,	PUNCT
cana-1065	105	48	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	105	49	)	)	PUNCT
cana-1065	105	50	,	,	PUNCT
cana-1065	105	51	𝑀(𝑆𝑢2𝑛	𝑀(𝑆𝑢2𝑛	PROPN
cana-1065	105	52	,	,	PUNCT
cana-1065	105	53	𝐵𝐵𝑢2𝑛+1,(2	𝐵𝐵𝑢2𝑛+1,(2	ADP
cana-1065	105	54	−	−	PROPN
cana-1065	105	55	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	105	56	)	)	PUNCT
cana-1065	105	57	,	,	PUNCT
cana-1065	105	58	𝑀(𝑆𝑢2𝑛	𝑀(𝑆𝑢2𝑛	PROPN
cana-1065	105	59	,	,	PUNCT
cana-1065	105	60	𝑇𝐵𝑢2𝑛+1	𝑇𝐵𝑢2𝑛+1	PROPN
cana-1065	105	61	,	,	PUNCT
cana-1065	105	62	𝑞	𝑞	NOUN
cana-1065	105	63	)	)	PUNCT
cana-1065	105	64	}	}	PUNCT
cana-1065	105	65	}	}	PUNCT
cana-1065	105	66	taking	take	VERB
cana-1065	105	67	𝑛	𝑛	PRON
cana-1065	105	68	→	→	SYM
cana-1065	105	69	∞	∞	PROPN
cana-1065	105	70	,	,	PUNCT
cana-1065	105	71	we	we	PRON
cana-1065	105	72	have	have	VERB
cana-1065	105	73	𝑀(𝑧	𝑀(𝑧	NUM
cana-1065	105	74	,	,	PUNCT
cana-1065	105	75	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	105	76	,	,	PUNCT
cana-1065	105	77	ξ	ξ	PROPN
cana-1065	105	78	𝑞	𝑞	PROPN
cana-1065	105	79	)	)	PUNCT
cana-1065	105	80	≥	≥	NOUN
cana-1065	105	81	φ{min{𝑀(𝑧	φ{min{𝑀(𝑧	PROPN
cana-1065	105	82	,	,	PUNCT
cana-1065	105	83	𝑧	𝑧	PROPN
cana-1065	105	84	,	,	PUNCT
cana-1065	105	85	𝑞	𝑞	NOUN
cana-1065	105	86	)	)	PUNCT
cana-1065	105	87	,	,	PUNCT
cana-1065	105	88	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	105	89	,	,	PUNCT
cana-1065	105	90	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	105	91	,	,	PUNCT
cana-1065	105	92	𝑞	𝑞	NOUN
cana-1065	105	93	)	)	PUNCT
cana-1065	105	94	,	,	PUNCT
cana-1065	105	95	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	105	96	,	,	PUNCT
cana-1065	105	97	𝑧	𝑧	NOUN
cana-1065	105	98	,	,	PUNCT
cana-1065	105	99	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	105	100	)	)	PUNCT
cana-1065	105	101	,	,	PUNCT
cana-1065	105	102	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	105	103	,	,	PUNCT
cana-1065	105	104	𝑇𝑧(2	𝑇𝑧(2	NOUN
cana-1065	105	105	−	−	NOUN
cana-1065	105	106	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	105	107	)	)	PUNCT
cana-1065	105	108	,	,	PUNCT
cana-1065	105	109	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	105	110	,	,	PUNCT
cana-1065	105	111	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	105	112	,	,	PUNCT
cana-1065	105	113	𝑞	𝑞	NOUN
cana-1065	105	114	)	)	PUNCT
cana-1065	105	115	}	}	PUNCT
cana-1065	105	116	}	}	PUNCT
cana-1065	105	117	letting	let	VERB
cana-1065	105	118	𝑟	𝑟	NOUN
cana-1065	105	119	=	=	SYM
cana-1065	105	120	1	1	NUM
cana-1065	105	121	−	−	PROPN
cana-1065	105	122	𝑝	𝑝	NOUN
cana-1065	105	123	with	with	ADP
cana-1065	105	124	𝑝	𝑝	PROPN
cana-1065	105	125	∈	∈	PROPN
cana-1065	105	126	(	(	PUNCT
cana-1065	105	127	0	0	NUM
cana-1065	105	128	,	,	PUNCT
cana-1065	105	129	1	1	NUM
cana-1065	105	130	)	)	PUNCT
cana-1065	105	131	then	then	ADV
cana-1065	105	132	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	105	133	,	,	PUNCT
cana-1065	105	134	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	105	135	,	,	PUNCT
cana-1065	105	136	ξ	ξ	PROPN
cana-1065	105	137	𝑞	𝑞	PROPN
cana-1065	105	138	)	)	PUNCT
cana-1065	105	139	≥	≥	NOUN
cana-1065	105	140	φ{min	φ{min	NOUN
cana-1065	105	141	{	{	PUNCT
cana-1065	105	142	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	105	143	,	,	PUNCT
cana-1065	105	144	𝑧	𝑧	PRON
cana-1065	105	145	,	,	PUNCT
cana-1065	105	146	(	(	PUNCT
cana-1065	105	147	1	1	NUM
cana-1065	105	148	−	−	NOUN
cana-1065	105	149	𝑝)𝑞	𝑝)𝑞	NOUN
cana-1065	105	150	)	)	PUNCT
cana-1065	105	151	,	,	PUNCT
cana-1065	105	152	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	105	153	,	,	PUNCT
cana-1065	106	1	𝑇𝑧(2	𝑇𝑧(2	NOUN
cana-1065	106	2	−	−	PROPN
cana-1065	107	1	(	(	PUNCT
cana-1065	107	2	1	1	NUM
cana-1065	107	3	−	−	NOUN
cana-1065	107	4	𝑝)𝑞	𝑝)𝑞	NOUN
cana-1065	107	5	)	)	PUNCT
cana-1065	107	6	,	,	PUNCT
cana-1065	107	7	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	8	,	,	PUNCT
cana-1065	107	9	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	10	,	,	PUNCT
cana-1065	107	11	𝑞	𝑞	NOUN
cana-1065	107	12	)	)	PUNCT
cana-1065	107	13	}	}	PUNCT
cana-1065	107	14	}	}	PUNCT
cana-1065	107	15	or	or	CCONJ
cana-1065	107	16	,	,	PUNCT
cana-1065	107	17	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	18	,	,	PUNCT
cana-1065	107	19	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	20	,	,	PUNCT
cana-1065	107	21	ξ	ξ	PROPN
cana-1065	107	22	𝑞	𝑞	PROPN
cana-1065	107	23	)	)	PUNCT
cana-1065	107	24	≥	≥	NOUN
cana-1065	107	25	φ{min	φ{min	NOUN
cana-1065	107	26	{	{	PUNCT
cana-1065	107	27	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	107	28	,	,	PUNCT
cana-1065	107	29	𝑧	𝑧	PRON
cana-1065	107	30	,	,	PUNCT
cana-1065	107	31	(	(	PUNCT
cana-1065	107	32	1	1	NUM
cana-1065	107	33	−	−	NOUN
cana-1065	107	34	𝑝)𝑞	𝑝)𝑞	NOUN
cana-1065	107	35	)	)	PUNCT
cana-1065	107	36	,	,	PUNCT
cana-1065	107	37	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	38	,	,	PUNCT
cana-1065	107	39	𝑇𝑧(1	𝑇𝑧(1	NOUN
cana-1065	107	40	+	+	CCONJ
cana-1065	107	41	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	107	42	)	)	PUNCT
cana-1065	107	43	,	,	PUNCT
cana-1065	107	44	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	45	,	,	PUNCT
cana-1065	107	46	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	47	,	,	PUNCT
cana-1065	107	48	𝑞	𝑞	NOUN
cana-1065	107	49	)	)	PUNCT
cana-1065	107	50	}	}	PUNCT
cana-1065	107	51	}	}	PUNCT
cana-1065	107	52	≥	≥	X
cana-1065	107	53	φ{min	φ{min	NOUN
cana-1065	107	54	{	{	PUNCT
cana-1065	107	55	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	107	56	,	,	PUNCT
cana-1065	107	57	𝑧	𝑧	PRON
cana-1065	107	58	,	,	PUNCT
cana-1065	107	59	(	(	PUNCT
cana-1065	107	60	1	1	NUM
cana-1065	107	61	−	−	PROPN
cana-1065	107	62	𝑝	𝑝	NOUN
cana-1065	107	63	+	+	CCONJ
cana-1065	107	64	1	1	NUM
cana-1065	107	65	+	+	CCONJ
cana-1065	107	66	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	107	67	)	)	PUNCT
cana-1065	107	68	,	,	PUNCT
cana-1065	107	69	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	70	,	,	PUNCT
cana-1065	107	71	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	72	,	,	PUNCT
cana-1065	107	73	𝑞	𝑞	NOUN
cana-1065	107	74	)	)	PUNCT
cana-1065	107	75	}	}	PUNCT
cana-1065	107	76	}	}	PUNCT
cana-1065	107	77	≥	≥	X
cana-1065	107	78	φ{min	φ{min	NOUN
cana-1065	107	79	{	{	PUNCT
cana-1065	107	80	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	107	81	,	,	PUNCT
cana-1065	107	82	𝑧	𝑧	NOUN
cana-1065	107	83	,	,	PUNCT
cana-1065	107	84	2𝑞	2𝑞	NOUN
cana-1065	107	85	)	)	PUNCT
cana-1065	107	86	,	,	PUNCT
cana-1065	107	87	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	88	,	,	PUNCT
cana-1065	107	89	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	90	,	,	PUNCT
cana-1065	107	91	𝑞	𝑞	NOUN
cana-1065	107	92	)	)	PUNCT
cana-1065	107	93	}	}	PUNCT
cana-1065	107	94	}	}	PUNCT
cana-1065	107	95	≥	≥	X
cana-1065	107	96	φ{min	φ{min	PROPN
cana-1065	107	97	{	{	PUNCT
cana-1065	107	98	𝑀(𝑧	𝑀(𝑧	NUM
cana-1065	107	99	,	,	PUNCT
cana-1065	107	100	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	101	,	,	PUNCT
cana-1065	107	102	𝑞	𝑞	NOUN
cana-1065	107	103	)	)	PUNCT
cana-1065	107	104	}	}	PUNCT
cana-1065	107	105	}	}	PUNCT
cana-1065	107	106	therefore	therefore	ADV
cana-1065	107	107	,	,	PUNCT
cana-1065	107	108	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	109	,	,	PUNCT
cana-1065	107	110	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	111	,	,	PUNCT
cana-1065	107	112	ξ	ξ	PROPN
cana-1065	107	113	𝑞	𝑞	PROPN
cana-1065	107	114	)	)	PUNCT
cana-1065	107	115	≥	≥	NOUN
cana-1065	107	116	φ{𝑀(𝑧	φ{𝑀(𝑧	NOUN
cana-1065	107	117	,	,	PUNCT
cana-1065	107	118	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	119	,	,	PUNCT
cana-1065	107	120	𝑞	𝑞	NOUN
cana-1065	107	121	)	)	PUNCT
cana-1065	107	122	}	}	PUNCT
cana-1065	107	123	or	or	CCONJ
cana-1065	107	124	,	,	PUNCT
cana-1065	107	125	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	107	126	,	,	PUNCT
cana-1065	107	127	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	128	,	,	PUNCT
cana-1065	107	129	ξ	ξ	PROPN
cana-1065	107	130	𝑞	𝑞	PROPN
cana-1065	107	131	)	)	PUNCT
cana-1065	107	132	≥	≥	NOUN
cana-1065	107	133	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	107	134	,	,	PUNCT
cana-1065	107	135	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	107	136	,	,	PUNCT
cana-1065	107	137	𝑞	𝑞	NOUN
cana-1065	107	138	)	)	PUNCT
cana-1065	107	139	,	,	PUNCT
cana-1065	107	140	by	by	ADP
cana-1065	107	141	property	property	NOUN
cana-1065	107	142	of	of	ADP
cana-1065	107	143	φ	φ	NUM
cana-1065	107	144	which	which	PRON
cana-1065	107	145	implies	imply	VERB
cana-1065	107	146	𝑧	𝑧	X
cana-1065	107	147	=	=	PUNCT
cana-1065	108	1	𝑇𝑧	𝑇𝑧	ADP
cana-1065	108	2	by	by	ADP
cana-1065	108	3	lemma	lemma	PROPN
cana-1065	108	4	2.1	2.1	NUM
cana-1065	108	5	.	.	PUNCT
cana-1065	109	1	similarly	similarly	ADV
cana-1065	109	2	,	,	PUNCT
cana-1065	109	3	replacing	replace	VERB
cana-1065	109	4	𝑥	𝑥	PRON
cana-1065	109	5	by	by	ADP
cana-1065	109	6	𝑢2𝑛	𝑢2𝑛	PROPN
cana-1065	109	7	and	and	CCONJ
cana-1065	109	8	𝑦	𝑦	NOUN
cana-1065	109	9	𝑏𝑦	𝑏𝑦	NOUN
cana-1065	109	10	𝑧	𝑧	VERB
cana-1065	109	11	in	in	ADP
cana-1065	109	12	relation	relation	NOUN
cana-1065	109	13	(	(	PUNCT
cana-1065	109	14	3.1.4	3.1.4	NUM
cana-1065	109	15	)	)	PUNCT
cana-1065	109	16	,	,	PUNCT
cana-1065	109	17	we	we	PRON
cana-1065	109	18	have	have	VERB
cana-1065	109	19	𝑀(𝐴𝑢2𝑛	𝑀(𝐴𝑢2𝑛	NOUN
cana-1065	109	20	,	,	PUNCT
cana-1065	109	21	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	109	22	,	,	PUNCT
cana-1065	109	23	ξ	ξ	PROPN
cana-1065	109	24	𝑞	𝑞	PROPN
cana-1065	109	25	)	)	PUNCT
cana-1065	109	26	≥	≥	NOUN
cana-1065	109	27	φ{min{𝑀(𝑆𝑢2𝑛	φ{min{𝑀(𝑆𝑢2𝑛	NOUN
cana-1065	109	28	,	,	PUNCT
cana-1065	109	29	𝐴𝑢2𝑛	𝐴𝑢2𝑛	ADJ
cana-1065	109	30	,	,	PUNCT
cana-1065	109	31	𝑞	𝑞	NOUN
cana-1065	109	32	)	)	PUNCT
cana-1065	109	33	,	,	PUNCT
cana-1065	109	34	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	109	35	,	,	PUNCT
cana-1065	109	36	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	109	37	,	,	PUNCT
cana-1065	109	38	𝑞	𝑞	NOUN
cana-1065	109	39	)	)	PUNCT
cana-1065	109	40	,	,	PUNCT
cana-1065	109	41	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	109	42	,	,	PUNCT
cana-1065	109	43	𝐴𝑢2𝑛	𝐴𝑢2𝑛	ADJ
cana-1065	109	44	,	,	PUNCT
cana-1065	109	45	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	109	46	)	)	PUNCT
cana-1065	109	47	,	,	PUNCT
cana-1065	109	48	𝑀(𝑆𝑢2𝑛	𝑀(𝑆𝑢2𝑛	PROPN
cana-1065	109	49	,	,	PUNCT
cana-1065	109	50	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	109	51	,	,	PUNCT
cana-1065	109	52	(	(	PUNCT
cana-1065	109	53	2	2	NUM
cana-1065	109	54	−	−	NOUN
cana-1065	109	55	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	109	56	)	)	PUNCT
cana-1065	109	57	,	,	PUNCT
cana-1065	109	58	𝑀(𝑆𝑢2𝑛	𝑀(𝑆𝑢2𝑛	PROPN
cana-1065	109	59	,	,	PUNCT
cana-1065	109	60	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	109	61	,	,	PUNCT
cana-1065	109	62	𝑞	𝑞	NOUN
cana-1065	109	63	)	)	PUNCT
cana-1065	109	64	}	}	PUNCT
cana-1065	109	65	}	}	PUNCT
cana-1065	109	66	taking	take	VERB
cana-1065	109	67	𝑛	𝑛	PRON
cana-1065	109	68	→	→	SYM
cana-1065	109	69	∞	∞	PROPN
cana-1065	109	70	,	,	PUNCT
cana-1065	109	71	we	we	PRON
cana-1065	109	72	get	get	VERB
cana-1065	109	73	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	109	74	,	,	PUNCT
cana-1065	109	75	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	109	76	,	,	PUNCT
cana-1065	109	77	ξ	ξ	PROPN
cana-1065	109	78	𝑞	𝑞	PROPN
cana-1065	109	79	)	)	PUNCT
cana-1065	109	80	≥	≥	NOUN
cana-1065	109	81	φ{min{𝑀(𝑧	φ{min{𝑀(𝑧	PROPN
cana-1065	109	82	,	,	PUNCT
cana-1065	109	83	𝑧	𝑧	PROPN
cana-1065	109	84	,	,	PUNCT
cana-1065	109	85	𝑞	𝑞	NOUN
cana-1065	109	86	)	)	PUNCT
cana-1065	109	87	,	,	PUNCT
cana-1065	109	88	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	109	89	,	,	PUNCT
cana-1065	109	90	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	109	91	,	,	PUNCT
cana-1065	109	92	𝑞	𝑞	NOUN
cana-1065	109	93	)	)	PUNCT
cana-1065	109	94	,	,	PUNCT
cana-1065	109	95	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	109	96	,	,	PUNCT
cana-1065	109	97	𝑧	𝑧	VERB
cana-1065	109	98	,	,	PUNCT
cana-1065	109	99	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	109	100	)	)	PUNCT
cana-1065	109	101	,	,	PUNCT
cana-1065	109	102	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	109	103	,	,	PUNCT
cana-1065	109	104	𝐵𝑧(2	𝐵𝑧(2	NOUN
cana-1065	109	105	−	−	NOUN
cana-1065	109	106	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	109	107	)	)	PUNCT
cana-1065	109	108	,	,	PUNCT
cana-1065	109	109	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	109	110	,	,	PUNCT
cana-1065	109	111	𝑧	𝑧	PROPN
cana-1065	109	112	,	,	PUNCT
cana-1065	109	113	𝑞	𝑞	NOUN
cana-1065	109	114	)	)	PUNCT
cana-1065	109	115	}	}	PUNCT
cana-1065	109	116	}	}	PUNCT
cana-1065	109	117	≥	≥	X
cana-1065	109	118	φ{min	φ{min	PROPN
cana-1065	109	119	{	{	PUNCT
cana-1065	109	120	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	109	121	,	,	PUNCT
cana-1065	109	122	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	109	123	,	,	PUNCT
cana-1065	109	124	𝑞	𝑞	NOUN
cana-1065	109	125	)	)	PUNCT
cana-1065	109	126	,	,	PUNCT
cana-1065	109	127	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	109	128	,	,	PUNCT
cana-1065	109	129	𝐵𝑧(2	𝐵𝑧(2	NOUN
cana-1065	109	130	−	−	PROPN
cana-1065	110	1	(	(	PUNCT
cana-1065	110	2	1	1	NUM
cana-1065	110	3	−	−	NOUN
cana-1065	110	4	𝑝))𝑞	𝑝))𝑞	ADJ
cana-1065	110	5	)	)	PUNCT
cana-1065	110	6	}	}	PUNCT
cana-1065	110	7	}	}	PUNCT
cana-1065	110	8	≥	≥	X
cana-1065	110	9	φ{min	φ{min	PROPN
cana-1065	110	10	{	{	PUNCT
cana-1065	110	11	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	110	12	,	,	PUNCT
cana-1065	110	13	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	14	,	,	PUNCT
cana-1065	110	15	𝑞	𝑞	NOUN
cana-1065	110	16	)	)	PUNCT
cana-1065	110	17	,	,	PUNCT
cana-1065	110	18	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	110	19	,	,	PUNCT
cana-1065	110	20	𝐵𝑧(1	𝐵𝑧(1	NOUN
cana-1065	110	21	+	+	CCONJ
cana-1065	110	22	𝑝))𝑞	𝑝))𝑞	ADJ
cana-1065	110	23	)	)	PUNCT
cana-1065	110	24	}	}	PUNCT
cana-1065	110	25	}	}	PUNCT
cana-1065	110	26	≥	≥	X
cana-1065	110	27	φ{min	φ{min	PROPN
cana-1065	110	28	{	{	PUNCT
cana-1065	110	29	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	110	30	,	,	PUNCT
cana-1065	110	31	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	32	,	,	PUNCT
cana-1065	110	33	𝑞	𝑞	NOUN
cana-1065	110	34	)	)	PUNCT
cana-1065	110	35	,	,	PUNCT
cana-1065	110	36	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	110	37	,	,	PUNCT
cana-1065	110	38	𝑧	𝑧	NOUN
cana-1065	110	39	,	,	PUNCT
cana-1065	110	40	𝑞	𝑞	NOUN
cana-1065	110	41	)	)	PUNCT
cana-1065	110	42	,	,	PUNCT
cana-1065	110	43	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	110	44	,	,	PUNCT
cana-1065	110	45	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	46	,	,	PUNCT
cana-1065	110	47	𝑝𝑞	𝑝𝑞	NOUN
cana-1065	110	48	)	)	PUNCT
cana-1065	110	49	}	}	PUNCT
cana-1065	110	50	}	}	PUNCT
cana-1065	110	51	≥	≥	X
cana-1065	110	52	φ{min	φ{min	PROPN
cana-1065	110	53	{	{	PUNCT
cana-1065	110	54	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	110	55	,	,	PUNCT
cana-1065	110	56	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	57	,	,	PUNCT
cana-1065	110	58	𝑞	𝑞	NOUN
cana-1065	110	59	)	)	PUNCT
cana-1065	110	60	,	,	PUNCT
cana-1065	110	61	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	110	62	,	,	PUNCT
cana-1065	110	63	𝑧	𝑧	NOUN
cana-1065	110	64	,	,	PUNCT
cana-1065	110	65	𝑞	𝑞	NOUN
cana-1065	110	66	)	)	PUNCT
cana-1065	110	67	,	,	PUNCT
cana-1065	110	68	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	110	69	,	,	PUNCT
cana-1065	110	70	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	71	,	,	PUNCT
cana-1065	110	72	𝑝𝑞	𝑝𝑞	NOUN
cana-1065	110	73	)	)	PUNCT
cana-1065	110	74	}	}	PUNCT
cana-1065	110	75	}	}	PUNCT
cana-1065	110	76	≥	≥	X
cana-1065	110	77	φ{min	φ{min	PROPN
cana-1065	110	78	{	{	PUNCT
cana-1065	110	79	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	110	80	,	,	PUNCT
cana-1065	110	81	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	82	,	,	PUNCT
cana-1065	110	83	𝑞	𝑞	NOUN
cana-1065	110	84	)	)	PUNCT
cana-1065	110	85	,	,	PUNCT
cana-1065	110	86	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	110	87	,	,	PUNCT
cana-1065	110	88	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	89	,	,	PUNCT
cana-1065	110	90	𝑞	𝑞	NOUN
cana-1065	110	91	)	)	PUNCT
cana-1065	110	92	}	}	PUNCT
cana-1065	110	93	}	}	PUNCT
cana-1065	110	94	,	,	PUNCT
cana-1065	110	95	as	as	ADP
cana-1065	110	96	𝑝	𝑝	NOUN
cana-1065	110	97	→	→	SYM
cana-1065	110	98	1	1	NUM
cana-1065	110	99	so	so	SCONJ
cana-1065	110	100	that	that	SCONJ
cana-1065	110	101	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	110	102	,	,	PUNCT
cana-1065	110	103	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	104	,	,	PUNCT
cana-1065	110	105	ξ	ξ	PROPN
cana-1065	110	106	𝑞	𝑞	PROPN
cana-1065	110	107	)	)	PUNCT
cana-1065	110	108	≥	≥	NOUN
cana-1065	110	109	φ{𝑀(𝑧	φ{𝑀(𝑧	PROPN
cana-1065	110	110	,	,	PUNCT
cana-1065	110	111	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	112	,	,	PUNCT
cana-1065	110	113	𝑞	𝑞	NOUN
cana-1065	110	114	)	)	PUNCT
cana-1065	110	115	}	}	PUNCT
cana-1065	110	116	or	or	CCONJ
cana-1065	110	117	,	,	PUNCT
cana-1065	110	118	𝑀(𝑧	𝑀(𝑧	X
cana-1065	110	119	,	,	PUNCT
cana-1065	110	120	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	121	,	,	PUNCT
cana-1065	110	122	ξ	ξ	PROPN
cana-1065	110	123	𝑞	𝑞	PROPN
cana-1065	110	124	)	)	PUNCT
cana-1065	110	125	≥	≥	NOUN
cana-1065	110	126	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	110	127	,	,	PUNCT
cana-1065	110	128	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	129	,	,	PUNCT
cana-1065	110	130	𝑞	𝑞	NOUN
cana-1065	110	131	)	)	PUNCT
cana-1065	110	132	,	,	PUNCT
cana-1065	110	133	by	by	ADP
cana-1065	110	134	property	property	NOUN
cana-1065	110	135	of	of	ADP
cana-1065	110	136	φ	φ	NUM
cana-1065	110	137	which	which	PRON
cana-1065	110	138	implies	imply	VERB
cana-1065	110	139	𝑧	𝑧	X
cana-1065	110	140	=	=	SYM
cana-1065	110	141	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	110	142	by	by	ADP
cana-1065	110	143	lemma	lemma	PROPN
cana-1065	110	144	2.1	2.1	NUM
cana-1065	110	145	.	.	PUNCT
cana-1065	111	1	since	since	SCONJ
cana-1065	111	2	,	,	PUNCT
cana-1065	111	3	𝐵(𝑌	𝐵(𝑌	PROPN
cana-1065	111	4	)	)	PUNCT
cana-1065	111	5	⊂	⊂	PROPN
cana-1065	111	6	𝑆	𝑆	PROPN
cana-1065	111	7	(	(	PUNCT
cana-1065	111	8	𝑌	𝑌	PROPN
cana-1065	111	9	)	)	PUNCT
cana-1065	111	10	,	,	PUNCT
cana-1065	111	11	so	so	CCONJ
cana-1065	111	12	there	there	PRON
cana-1065	111	13	exists	exist	VERB
cana-1065	111	14	a	a	DET
cana-1065	111	15	point	point	NOUN
cana-1065	111	16	𝑤	𝑤	ADP
cana-1065	111	17	in	in	ADP
cana-1065	111	18	𝑌	𝑌	PROPN
cana-1065	111	19	such	such	ADJ
cana-1065	111	20	that	that	SCONJ
cana-1065	111	21	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	111	22	=	=	SYM
cana-1065	111	23	𝑆𝑤	𝑆𝑤	PROPN
cana-1065	111	24	=	=	PUNCT
cana-1065	111	25	𝑧.	𝑧.	NOUN
cana-1065	111	26	by	by	ADP
cana-1065	111	27	using	use	VERB
cana-1065	111	28	relation	relation	NOUN
cana-1065	111	29	(	(	PUNCT
cana-1065	111	30	3.1.4	3.1.4	NUM
cana-1065	111	31	)	)	PUNCT
cana-1065	111	32	with	with	ADP
cana-1065	111	33	𝑥	𝑥	PROPN
cana-1065	111	34	=	=	SYM
cana-1065	111	35	𝑤	𝑤	PROPN
cana-1065	111	36	,	,	PUNCT
cana-1065	111	37	𝑦	𝑦	NOUN
cana-1065	111	38	=	=	SYM
cana-1065	111	39	𝑧	𝑧	PROPN
cana-1065	111	40	,	,	PUNCT
cana-1065	111	41	we	we	PRON
cana-1065	111	42	have	have	VERB
cana-1065	111	43	communications	communication	NOUN
cana-1065	111	44	on	on	ADP
cana-1065	111	45	applied	apply	VERB
cana-1065	111	46	nonlinear	nonlinear	ADJ
cana-1065	111	47	analysis	analysis	NOUN
cana-1065	111	48	issn	issn	NOUN
cana-1065	111	49	:	:	PUNCT
cana-1065	111	50	1074	1074	NUM
cana-1065	111	51	-	-	PUNCT
cana-1065	111	52	133x	133x	NUM
cana-1065	111	53	vol	vol	NOUN
cana-1065	111	54	31	31	NUM
cana-1065	111	55	no	no	NOUN
cana-1065	111	56	.	.	PUNCT
cana-1065	112	1	5s	5s	NUM
cana-1065	112	2	(	(	PUNCT
cana-1065	112	3	2024	2024	NUM
cana-1065	112	4	)	)	PUNCT
cana-1065	112	5	464	464	NUM
cana-1065	113	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	113	2	𝑀(𝐴𝑤	𝑀(𝐴𝑤	PROPN
cana-1065	113	3	,	,	PUNCT
cana-1065	113	4	𝑧	𝑧	PROPN
cana-1065	113	5	,	,	PUNCT
cana-1065	113	6	ξ	ξ	PROPN
cana-1065	113	7	𝑞	𝑞	PROPN
cana-1065	113	8	)	)	PUNCT
cana-1065	113	9	≥	≥	PROPN
cana-1065	113	10	φ{min{𝑀(𝑆𝑤	φ{min{𝑀(𝑆𝑤	PROPN
cana-1065	113	11	,	,	PUNCT
cana-1065	113	12	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	13	,	,	PUNCT
cana-1065	113	14	𝑞	𝑞	NOUN
cana-1065	113	15	)	)	PUNCT
cana-1065	113	16	,	,	PUNCT
cana-1065	113	17	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	113	18	,	,	PUNCT
cana-1065	113	19	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	113	20	,	,	PUNCT
cana-1065	113	21	𝑞	𝑞	NOUN
cana-1065	113	22	)	)	PUNCT
cana-1065	113	23	,	,	PUNCT
cana-1065	113	24	𝑀	𝑀	PROPN
cana-1065	113	25	(	(	PUNCT
cana-1065	113	26	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	113	27	,	,	PUNCT
cana-1065	113	28	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	113	29	,	,	PUNCT
cana-1065	113	30	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	113	31	)	)	PUNCT
cana-1065	113	32	,	,	PUNCT
cana-1065	113	33	𝑀(𝑆𝑤	𝑀(𝑆𝑤	NOUN
cana-1065	113	34	,	,	PUNCT
cana-1065	113	35	𝐵𝑧(2	𝐵𝑧(2	NOUN
cana-1065	113	36	−	−	NOUN
cana-1065	113	37	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	113	38	,	,	PUNCT
cana-1065	113	39	𝑀(𝑆𝑤	𝑀(𝑆𝑤	NOUN
cana-1065	113	40	,	,	PUNCT
cana-1065	113	41	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	113	42	,	,	PUNCT
cana-1065	113	43	𝑞	𝑞	NOUN
cana-1065	113	44	)	)	PUNCT
cana-1065	113	45	}	}	PUNCT
cana-1065	113	46	}	}	PUNCT
cana-1065	113	47	≥	≥	PROPN
cana-1065	113	48	φ{min{𝑀(𝑧	φ{min{𝑀(𝑧	PROPN
cana-1065	113	49	,	,	PUNCT
cana-1065	113	50	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	51	,	,	PUNCT
cana-1065	113	52	𝑞	𝑞	NOUN
cana-1065	113	53	)	)	PUNCT
cana-1065	113	54	,	,	PUNCT
cana-1065	113	55	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	113	56	,	,	PUNCT
cana-1065	113	57	𝑧	𝑧	PROPN
cana-1065	113	58	,	,	PUNCT
cana-1065	113	59	𝑞	𝑞	NOUN
cana-1065	113	60	)	)	PUNCT
cana-1065	113	61	,	,	PUNCT
cana-1065	113	62	𝑀	𝑀	PROPN
cana-1065	113	63	(	(	PUNCT
cana-1065	113	64	𝑧	𝑧	PROPN
cana-1065	113	65	,	,	PUNCT
cana-1065	113	66	𝐴𝑤	𝐴𝑤	NOUN
cana-1065	113	67	,	,	PUNCT
cana-1065	113	68	(	(	PUNCT
cana-1065	113	69	1	1	NUM
cana-1065	113	70	−	−	NOUN
cana-1065	113	71	𝑝)𝑞	𝑝)𝑞	NOUN
cana-1065	113	72	)	)	PUNCT
cana-1065	113	73	,	,	PUNCT
cana-1065	113	74	𝑀(𝑆𝑤	𝑀(𝑆𝑤	NOUN
cana-1065	113	75	,	,	PUNCT
cana-1065	113	76	𝑧(1	𝑧(1	PROPN
cana-1065	113	77	+	+	CCONJ
cana-1065	113	78	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	113	79	,	,	PUNCT
cana-1065	113	80	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	113	81	,	,	PUNCT
cana-1065	113	82	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	113	83	,	,	PUNCT
cana-1065	113	84	𝑞	𝑞	NOUN
cana-1065	113	85	)	)	PUNCT
cana-1065	113	86	}	}	PUNCT
cana-1065	113	87	}	}	PUNCT
cana-1065	113	88	≥	≥	PROPN
cana-1065	113	89	φ{min{𝑀(𝑧	φ{min{𝑀(𝑧	PROPN
cana-1065	113	90	,	,	PUNCT
cana-1065	113	91	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	92	,	,	PUNCT
cana-1065	113	93	𝑞	𝑞	NOUN
cana-1065	113	94	)	)	PUNCT
cana-1065	113	95	,	,	PUNCT
cana-1065	113	96	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	113	97	,	,	PUNCT
cana-1065	113	98	𝑧	𝑧	PROPN
cana-1065	113	99	,	,	PUNCT
cana-1065	113	100	𝑞	𝑞	NOUN
cana-1065	113	101	)	)	PUNCT
cana-1065	113	102	,	,	PUNCT
cana-1065	113	103	𝑀	𝑀	PROPN
cana-1065	113	104	(	(	PUNCT
cana-1065	113	105	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	106	,	,	PUNCT
cana-1065	113	107	𝑧	𝑧	PRON
cana-1065	113	108	,	,	PUNCT
cana-1065	113	109	(	(	PUNCT
cana-1065	113	110	1	1	NUM
cana-1065	113	111	−	−	NOUN
cana-1065	113	112	𝑝)𝑞	𝑝)𝑞	NOUN
cana-1065	113	113	)	)	PUNCT
cana-1065	113	114	,	,	PUNCT
cana-1065	113	115	𝑀(𝑆𝑤	𝑀(𝑆𝑤	NOUN
cana-1065	113	116	,	,	PUNCT
cana-1065	113	117	𝑧(1	𝑧(1	PROPN
cana-1065	113	118	+	+	CCONJ
cana-1065	113	119	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	113	120	,	,	PUNCT
cana-1065	113	121	𝑀(𝑧	𝑀(𝑧	PRON
cana-1065	113	122	,	,	PUNCT
cana-1065	113	123	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	113	124	,	,	PUNCT
cana-1065	113	125	𝑞	𝑞	NOUN
cana-1065	113	126	)	)	PUNCT
cana-1065	113	127	}	}	PUNCT
cana-1065	113	128	}	}	PUNCT
cana-1065	113	129	≥	≥	PROPN
cana-1065	113	130	φ{min{𝑀(𝑧	φ{min{𝑀(𝑧	PROPN
cana-1065	113	131	,	,	PUNCT
cana-1065	113	132	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	133	,	,	PUNCT
cana-1065	113	134	𝑞	𝑞	NOUN
cana-1065	113	135	)	)	PUNCT
cana-1065	113	136	,	,	PUNCT
cana-1065	113	137	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	113	138	,	,	PUNCT
cana-1065	113	139	𝑧	𝑧	NOUN
cana-1065	113	140	,	,	PUNCT
cana-1065	113	141	𝑞	𝑞	NOUN
cana-1065	113	142	)	)	PUNCT
cana-1065	113	143	,	,	PUNCT
cana-1065	113	144	𝑀	𝑀	PROPN
cana-1065	113	145	(	(	PUNCT
cana-1065	113	146	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	147	,	,	PUNCT
cana-1065	113	148	𝑆𝑤	𝑆𝑤	PROPN
cana-1065	113	149	,	,	PUNCT
cana-1065	113	150	(	(	PUNCT
cana-1065	113	151	1	1	NUM
cana-1065	113	152	−	−	PROPN
cana-1065	113	153	𝑝	𝑝	NOUN
cana-1065	113	154	+	+	CCONJ
cana-1065	113	155	1	1	NUM
cana-1065	113	156	+	+	CCONJ
cana-1065	113	157	𝑝)𝑞	𝑝)𝑞	ADJ
cana-1065	113	158	)	)	PUNCT
cana-1065	113	159	}	}	PUNCT
cana-1065	113	160	}	}	PUNCT
cana-1065	113	161	≥	≥	PROPN
cana-1065	113	162	φ{min{𝑀(𝑧	φ{min{𝑀(𝑧	PROPN
cana-1065	113	163	,	,	PUNCT
cana-1065	113	164	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	165	,	,	PUNCT
cana-1065	113	166	𝑞	𝑞	NOUN
cana-1065	113	167	)	)	PUNCT
cana-1065	113	168	,	,	PUNCT
cana-1065	113	169	𝑀	𝑀	PROPN
cana-1065	113	170	(	(	PUNCT
cana-1065	113	171	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	172	,	,	PUNCT
cana-1065	113	173	𝑧	𝑧	NOUN
cana-1065	113	174	,	,	PUNCT
cana-1065	113	175	2𝑞	2𝑞	NOUN
cana-1065	113	176	)	)	PUNCT
cana-1065	113	177	}	}	PUNCT
cana-1065	113	178	}	}	PUNCT
cana-1065	113	179	therefore	therefore	ADV
cana-1065	113	180	,	,	PUNCT
cana-1065	113	181	𝑀(𝐴𝑤	𝑀(𝐴𝑤	PROPN
cana-1065	113	182	,	,	PUNCT
cana-1065	113	183	𝑧	𝑧	PROPN
cana-1065	113	184	,	,	PUNCT
cana-1065	113	185	ξ	ξ	PROPN
cana-1065	113	186	𝑞	𝑞	PROPN
cana-1065	113	187	)	)	PUNCT
cana-1065	113	188	≥	≥	NOUN
cana-1065	113	189	φ{𝑀(𝑧	φ{𝑀(𝑧	PROPN
cana-1065	113	190	,	,	PUNCT
cana-1065	113	191	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	192	,	,	PUNCT
cana-1065	113	193	𝑞	𝑞	NOUN
cana-1065	113	194	)	)	PUNCT
cana-1065	113	195	}	}	PUNCT
cana-1065	113	196	or	or	CCONJ
cana-1065	113	197	,	,	PUNCT
cana-1065	113	198	𝑀(𝐴𝑤	𝑀(𝐴𝑤	PROPN
cana-1065	113	199	,	,	PUNCT
cana-1065	113	200	𝑧	𝑧	PROPN
cana-1065	113	201	,	,	PUNCT
cana-1065	113	202	ξ	ξ	PROPN
cana-1065	113	203	𝑞	𝑞	PROPN
cana-1065	113	204	)	)	PUNCT
cana-1065	113	205	≥	≥	NOUN
cana-1065	113	206	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	113	207	,	,	PUNCT
cana-1065	113	208	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	209	,	,	PUNCT
cana-1065	113	210	𝑞	𝑞	NOUN
cana-1065	113	211	)	)	PUNCT
cana-1065	113	212	,	,	PUNCT
cana-1065	113	213	by	by	ADP
cana-1065	113	214	property	property	NOUN
cana-1065	113	215	of	of	ADP
cana-1065	113	216	φ	φ	NUM
cana-1065	113	217	which	which	PRON
cana-1065	113	218	implies	imply	VERB
cana-1065	113	219	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	113	220	=	=	SYM
cana-1065	113	221	𝑧	𝑧	NOUN
cana-1065	113	222	by	by	ADP
cana-1065	113	223	lemma	lemma	PROPN
cana-1065	113	224	2.1	2.1	NUM
cana-1065	113	225	.	.	PUNCT
cana-1065	114	1	again	again	ADV
cana-1065	114	2	,	,	PUNCT
cana-1065	114	3	since	since	SCONJ
cana-1065	114	4	𝐴	𝐴	PROPN
cana-1065	114	5	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1065	114	6	𝑆	𝑆	PROPN
cana-1065	114	7	are	be	AUX
cana-1065	114	8	weakly	weakly	ADV
cana-1065	114	9	compatible	compatible	ADJ
cana-1065	114	10	mappings	mapping	NOUN
cana-1065	114	11	of	of	ADP
cana-1065	114	12	type	type	NOUN
cana-1065	114	13	(	(	PUNCT
cana-1065	114	14	𝑃	𝑃	NOUN
cana-1065	114	15	)	)	PUNCT
cana-1065	114	16	and	and	CCONJ
cana-1065	114	17	𝐴𝑤	𝐴𝑤	PROPN
cana-1065	114	18	=	=	SYM
cana-1065	114	19	𝑆𝑤	𝑆𝑤	PROPN
cana-1065	114	20	=	=	PUNCT
cana-1065	114	21	𝑧	𝑧	PROPN
cana-1065	114	22	,	,	PUNCT
cana-1065	114	23	by	by	ADP
cana-1065	114	24	proposition	proposition	NOUN
cana-1065	114	25	2.4	2.4	NUM
cana-1065	114	26	,	,	PUNCT
cana-1065	114	27	we	we	PRON
cana-1065	114	28	have	have	VERB
cana-1065	114	29	for	for	ADP
cana-1065	114	30	every	every	DET
cana-1065	114	31			PROPN
cana-1065	114	32	>	>	X
cana-1065	114	33	0	0	NUM
cana-1065	114	34	1	1	NUM
cana-1065	114	35	=	=	SYM
cana-1065	114	36	𝑀(𝐴𝐴𝑤	𝑀(𝐴𝐴𝑤	PROPN
cana-1065	114	37	,	,	PUNCT
cana-1065	114	38	𝑆𝑆𝑤	𝑆𝑆𝑤	PROPN
cana-1065	114	39	,	,	PUNCT
cana-1065	114	40	𝜖	𝜖	PROPN
cana-1065	114	41	)	)	PUNCT
cana-1065	114	42	≥	≥	NOUN
cana-1065	114	43	𝑀(𝐴𝑤	𝑀(𝐴𝑤	NOUN
cana-1065	114	44	,	,	PUNCT
cana-1065	114	45	𝑆𝑤	𝑆𝑤	PROPN
cana-1065	114	46	,	,	PUNCT
cana-1065	114	47	𝜖	𝜖	NOUN
cana-1065	114	48	)	)	PUNCT
cana-1065	114	49	hence	hence	ADV
cana-1065	114	50	𝐴𝑤	𝐴𝑤	NOUN
cana-1065	114	51	=	=	SYM
cana-1065	114	52	𝐴𝐴𝑤	𝐴𝐴𝑤	NOUN
cana-1065	114	53	=	=	NOUN
cana-1065	114	54	𝑆𝑆𝑤	𝑆𝑆𝑤	NOUN
cana-1065	114	55	=	=	PUNCT
cana-1065	115	1	𝑆𝑤	𝑆𝑤	PROPN
cana-1065	115	2	finally	finally	ADV
cana-1065	115	3	,	,	PUNCT
cana-1065	115	4	by	by	ADP
cana-1065	115	5	relation	relation	NOUN
cana-1065	115	6	(	(	PUNCT
cana-1065	115	7	3.1.4	3.1.4	NUM
cana-1065	115	8	)	)	PUNCT
cana-1065	115	9	with	with	ADP
cana-1065	115	10	𝑥	𝑥	PROPN
cana-1065	115	11	=	=	PUNCT
cana-1065	115	12	𝑧	𝑧	PROPN
cana-1065	115	13	,	,	PUNCT
cana-1065	115	14	𝑦	𝑦	NOUN
cana-1065	115	15	=	=	SYM
cana-1065	115	16	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	115	17	=	=	SYM
cana-1065	115	18	𝑧	𝑧	X
cana-1065	115	19	,	,	PUNCT
cana-1065	115	20	we	we	PRON
cana-1065	115	21	have	have	VERB
cana-1065	115	22	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	23	,	,	PUNCT
cana-1065	115	24	𝑧	𝑧	PROPN
cana-1065	115	25	,	,	PUNCT
cana-1065	115	26	ξ	ξ	PROPN
cana-1065	115	27	𝑞	𝑞	NOUN
cana-1065	115	28	)	)	PUNCT
cana-1065	115	29	=	=	SYM
cana-1065	115	30	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	31	,	,	PUNCT
cana-1065	115	32	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	115	33	,	,	PUNCT
cana-1065	115	34	ξ	ξ	PROPN
cana-1065	115	35	𝑞	𝑞	PROPN
cana-1065	115	36	)	)	PUNCT
cana-1065	115	37	≥	≥	NOUN
cana-1065	115	38	φ{min{𝑀(𝑆𝑧	φ{min{𝑀(𝑆𝑧	PROPN
cana-1065	115	39	,	,	PUNCT
cana-1065	115	40	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	115	41	,	,	PUNCT
cana-1065	115	42	𝑞	𝑞	NOUN
cana-1065	115	43	)	)	PUNCT
cana-1065	115	44	,	,	PUNCT
cana-1065	115	45	𝑀(𝑇𝑧	𝑀(𝑇𝑧	NUM
cana-1065	115	46	,	,	PUNCT
cana-1065	115	47	𝑧	𝑧	PROPN
cana-1065	115	48	,	,	PUNCT
cana-1065	115	49	𝑞	𝑞	NOUN
cana-1065	115	50	)	)	PUNCT
cana-1065	115	51	,	,	PUNCT
cana-1065	115	52	𝑀	𝑀	PROPN
cana-1065	115	53	(	(	PUNCT
cana-1065	115	54	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	115	55	,	,	PUNCT
cana-1065	115	56	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	115	57	,	,	PUNCT
cana-1065	115	58	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	115	59	)	)	PUNCT
cana-1065	115	60	,	,	PUNCT
cana-1065	115	61	𝑀(𝑆𝑧	𝑀(𝑆𝑧	NOUN
cana-1065	115	62	,	,	PUNCT
cana-1065	115	63	𝑧(2	𝑧(2	PROPN
cana-1065	115	64	−	−	NOUN
cana-1065	115	65	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	115	66	,	,	PUNCT
cana-1065	115	67	𝑀(𝑆𝑧	𝑀(𝑆𝑧	NOUN
cana-1065	115	68	,	,	PUNCT
cana-1065	115	69	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	115	70	,	,	PUNCT
cana-1065	115	71	𝑞	𝑞	NOUN
cana-1065	115	72	)	)	PUNCT
cana-1065	115	73	}	}	PUNCT
cana-1065	115	74	≥	≥	NOUN
cana-1065	115	75	φ{min{𝑀(𝐴𝑧	φ{min{𝑀(𝐴𝑧	PROPN
cana-1065	115	76	,	,	PUNCT
cana-1065	115	77	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	115	78	,	,	PUNCT
cana-1065	115	79	𝑞	𝑞	NOUN
cana-1065	115	80	)	)	PUNCT
cana-1065	115	81	,	,	PUNCT
cana-1065	115	82	𝑀(𝑧	𝑀(𝑧	PROPN
cana-1065	115	83	,	,	PUNCT
cana-1065	115	84	𝑧	𝑧	NOUN
cana-1065	115	85	,	,	PUNCT
cana-1065	115	86	𝑞	𝑞	NOUN
cana-1065	115	87	)	)	PUNCT
cana-1065	115	88	,	,	PUNCT
cana-1065	115	89	𝑀	𝑀	PROPN
cana-1065	115	90	(	(	PUNCT
cana-1065	115	91	𝑧	𝑧	PROPN
cana-1065	115	92	,	,	PUNCT
cana-1065	115	93	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	115	94	,	,	PUNCT
cana-1065	115	95	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	115	96	)	)	PUNCT
cana-1065	115	97	,	,	PUNCT
cana-1065	115	98	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	99	,	,	PUNCT
cana-1065	115	100	𝑧(2	𝑧(2	PROPN
cana-1065	115	101	−	−	NOUN
cana-1065	115	102	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	115	103	,	,	PUNCT
cana-1065	115	104	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	105	,	,	PUNCT
cana-1065	115	106	𝑧	𝑧	PROPN
cana-1065	115	107	,	,	PUNCT
cana-1065	115	108	𝑞	𝑞	NOUN
cana-1065	115	109	)	)	PUNCT
cana-1065	115	110	}	}	PUNCT
cana-1065	115	111	≥	≥	X
cana-1065	115	112	φ{min	φ{min	X
cana-1065	115	113	{	{	PUNCT
cana-1065	115	114	𝑀	𝑀	PROPN
cana-1065	115	115	(	(	PUNCT
cana-1065	115	116	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	115	117	,	,	PUNCT
cana-1065	115	118	𝑧	𝑧	VERB
cana-1065	115	119	,	,	PUNCT
cana-1065	115	120	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	115	121	)	)	PUNCT
cana-1065	115	122	,	,	PUNCT
cana-1065	115	123	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	115	124	,	,	PUNCT
cana-1065	115	125	𝐴𝑧(2	𝐴𝑧(2	NOUN
cana-1065	115	126	−	−	NOUN
cana-1065	115	127	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	115	128	,	,	PUNCT
cana-1065	115	129	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	130	,	,	PUNCT
cana-1065	115	131	𝑧	𝑧	PROPN
cana-1065	115	132	,	,	PUNCT
cana-1065	115	133	𝑞	𝑞	NOUN
cana-1065	115	134	)	)	PUNCT
cana-1065	115	135	}	}	PUNCT
cana-1065	115	136	≥	≥	X
cana-1065	115	137	φ{min	φ{min	NOUN
cana-1065	115	138	{	{	PUNCT
cana-1065	115	139	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	140	,	,	PUNCT
cana-1065	115	141	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	115	142	,	,	PUNCT
cana-1065	115	143	𝑟𝑞	𝑟𝑞	NOUN
cana-1065	115	144	+	+	CCONJ
cana-1065	115	145	(	(	PUNCT
cana-1065	115	146	2	2	NUM
cana-1065	115	147	−	−	NOUN
cana-1065	115	148	𝑟)𝑞	𝑟)𝑞	NOUN
cana-1065	115	149	,	,	PUNCT
cana-1065	115	150	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	151	,	,	PUNCT
cana-1065	115	152	𝑧	𝑧	PROPN
cana-1065	115	153	,	,	PUNCT
cana-1065	115	154	𝑞	𝑞	NOUN
cana-1065	115	155	)	)	PUNCT
cana-1065	115	156	}	}	PUNCT
cana-1065	115	157	≥	≥	X
cana-1065	115	158	φ{min	φ{min	NOUN
cana-1065	115	159	{	{	PUNCT
cana-1065	115	160	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	161	,	,	PUNCT
cana-1065	115	162	𝑧	𝑧	PROPN
cana-1065	115	163	,	,	PUNCT
cana-1065	115	164	𝑞	𝑞	NOUN
cana-1065	115	165	)	)	PUNCT
cana-1065	115	166	}	}	PUNCT
cana-1065	115	167	≥	≥	X
cana-1065	115	168	φ{𝑀(𝐴𝑧	φ{𝑀(𝐴𝑧	PROPN
cana-1065	115	169	,	,	PUNCT
cana-1065	115	170	𝑧	𝑧	PROPN
cana-1065	115	171	,	,	PUNCT
cana-1065	115	172	𝑞	𝑞	NOUN
cana-1065	115	173	)	)	PUNCT
cana-1065	115	174	}	}	PUNCT
cana-1065	115	175	or	or	CCONJ
cana-1065	115	176	,	,	PUNCT
cana-1065	115	177	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	178	,	,	PUNCT
cana-1065	115	179	𝑧	𝑧	PROPN
cana-1065	115	180	,	,	PUNCT
cana-1065	115	181	ξ	ξ	PROPN
cana-1065	115	182	𝑞	𝑞	PROPN
cana-1065	115	183	)	)	PUNCT
cana-1065	115	184	≥	≥	PROPN
cana-1065	115	185	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	115	186	,	,	PUNCT
cana-1065	115	187	𝑧	𝑧	PROPN
cana-1065	115	188	,	,	PUNCT
cana-1065	115	189	𝑞	𝑞	NOUN
cana-1065	115	190	)	)	PUNCT
cana-1065	115	191	,	,	PUNCT
cana-1065	115	192	by	by	ADP
cana-1065	115	193	property	property	NOUN
cana-1065	115	194	of	of	ADP
cana-1065	115	195	φ	φ	PROPN
cana-1065	115	196			PROPN
cana-1065	116	1	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	116	2	=	=	PUNCT
cana-1065	116	3	𝑧	𝑧	PROPN
cana-1065	116	4	,	,	PUNCT
cana-1065	116	5	by	by	ADP
cana-1065	116	6	lemma	lemma	PROPN
cana-1065	116	7	2.1	2.1	NUM
cana-1065	116	8	.	.	PUNCT
cana-1065	117	1	hence	hence	ADV
cana-1065	117	2	,	,	PUNCT
cana-1065	117	3	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	117	4	=	=	SYM
cana-1065	117	5	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	117	6	=	=	PUNCT
cana-1065	117	7	𝑆𝑧	𝑆𝑧	PROPN
cana-1065	117	8	=	=	PUNCT
cana-1065	118	1	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	118	2	=	=	PUNCT
cana-1065	118	3	𝑧	𝑧	VERB
cana-1065	118	4	.	.	PUNCT
cana-1065	119	1	that	that	PRON
cana-1065	119	2	is	be	AUX
cana-1065	119	3	,	,	PUNCT
cana-1065	119	4	𝑧	𝑧	PRON
cana-1065	119	5	is	be	AUX
cana-1065	119	6	a	a	DET
cana-1065	119	7	common	common	ADJ
cana-1065	119	8	fixed	fix	VERB
cana-1065	119	9	point	point	NOUN
cana-1065	119	10	of	of	ADP
cana-1065	119	11	given	give	VERB
cana-1065	119	12	mappings	mapping	NOUN
cana-1065	119	13	𝐴	𝐴	PROPN
cana-1065	119	14	,	,	PUNCT
cana-1065	119	15	𝐵	𝐵	PROPN
cana-1065	119	16	,	,	PUNCT
cana-1065	119	17	𝑆	𝑆	PROPN
cana-1065	119	18	&	&	CCONJ
cana-1065	119	19	𝑇.	𝑇.	PROPN
cana-1065	119	20	uniqueness	uniqueness	NOUN
cana-1065	119	21	:	:	PUNCT
cana-1065	119	22	suppose	suppose	VERB
cana-1065	119	23	𝑧1	𝑧1	NOUN
cana-1065	119	24	is	be	AUX
cana-1065	119	25	another	another	DET
cana-1065	119	26	point	point	NOUN
cana-1065	119	27	in	in	ADP
cana-1065	119	28	𝑌	𝑌	PROPN
cana-1065	119	29	such	such	DET
cana-1065	119	30	that	that	SCONJ
cana-1065	119	31	𝑧1	𝑧1	NOUN
cana-1065	119	32	=	=	NOUN
cana-1065	119	33	𝐴𝑧1	𝐴𝑧1	NOUN
cana-1065	119	34	=	=	PUNCT
cana-1065	119	35	𝐵𝑧1	𝐵𝑧1	ADJ
cana-1065	119	36	=	=	NOUN
cana-1065	119	37	𝑆𝑧1	𝑆𝑧1	NOUN
cana-1065	119	38	=	=	PUNCT
cana-1065	119	39	𝑇𝑧1	𝑇𝑧1	ADJ
cana-1065	119	40	.	.	PUNCT
cana-1065	120	1	then	then	ADV
cana-1065	120	2	,	,	PUNCT
cana-1065	120	3	putting	put	VERB
cana-1065	120	4	𝑥	𝑥	X
cana-1065	120	5	=	=	PUNCT
cana-1065	120	6	𝑧	𝑧	PROPN
cana-1065	120	7	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1065	120	8	𝑦	𝑦	NOUN
cana-1065	120	9	=	=	SYM
cana-1065	120	10	𝑧1	𝑧1	NOUN
cana-1065	120	11	,	,	PUNCT
cana-1065	120	12	𝑟	𝑟	NOUN
cana-1065	120	13	=	=	SYM
cana-1065	120	14	1	1	NUM
cana-1065	120	15	in	in	ADP
cana-1065	120	16	(	(	PUNCT
cana-1065	120	17	3.1.4	3.1.4	NUM
cana-1065	120	18	)	)	PUNCT
cana-1065	120	19	,	,	PUNCT
cana-1065	120	20	we	we	PRON
cana-1065	120	21	get	get	VERB
cana-1065	120	22	𝑀(𝐴𝑧	𝑀(𝐴𝑧	NOUN
cana-1065	120	23	,	,	PUNCT
cana-1065	120	24	𝐵𝑧1	𝐵𝑧1	ADJ
cana-1065	120	25	,	,	PUNCT
cana-1065	120	26	ξ	ξ	PROPN
cana-1065	120	27	𝑞	𝑞	PROPN
cana-1065	120	28	)	)	PUNCT
cana-1065	120	29	=	=	SYM
cana-1065	120	30	𝑀(𝑧	𝑀(𝑧	NUM
cana-1065	120	31	,	,	PUNCT
cana-1065	120	32	𝑧1	𝑧1	NOUN
cana-1065	120	33	,	,	PUNCT
cana-1065	120	34	ξ	ξ	PROPN
cana-1065	120	35	𝑞	𝑞	PROPN
cana-1065	120	36	)	)	PUNCT
cana-1065	120	37	≥	≥	NOUN
cana-1065	120	38	φ{min{𝑀(𝑆𝑧	φ{min{𝑀(𝑆𝑧	PROPN
cana-1065	120	39	,	,	PUNCT
cana-1065	120	40	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	120	41	,	,	PUNCT
cana-1065	120	42	𝑞	𝑞	NOUN
cana-1065	120	43	)	)	PUNCT
cana-1065	120	44	,	,	PUNCT
cana-1065	120	45	𝑀(𝑇𝑧1	𝑀(𝑇𝑧1	PROPN
cana-1065	120	46	,	,	PUNCT
cana-1065	120	47	𝐵𝑧1	𝐵𝑧1	ADJ
cana-1065	120	48	,	,	PUNCT
cana-1065	120	49	𝑞	𝑞	PROPN
cana-1065	120	50	)	)	PUNCT
cana-1065	120	51	,	,	PUNCT
cana-1065	120	52	𝑀	𝑀	PROPN
cana-1065	120	53	(	(	PUNCT
cana-1065	120	54	𝑇𝑧1	𝑇𝑧1	ADJ
cana-1065	120	55	,	,	PUNCT
cana-1065	121	1	𝐴𝑧	𝐴𝑧	PROPN
cana-1065	121	2	,	,	PUNCT
cana-1065	121	3	𝑞	𝑞	NOUN
cana-1065	121	4	)	)	PUNCT
cana-1065	121	5	,	,	PUNCT
cana-1065	121	6	𝑀(𝑆𝑧	𝑀(𝑆𝑧	NOUN
cana-1065	121	7	,	,	PUNCT
cana-1065	121	8	𝑇𝑧1	𝑇𝑧1	NOUN
cana-1065	121	9	,	,	PUNCT
cana-1065	121	10	𝑞	𝑞	NOUN
cana-1065	121	11	)	)	PUNCT
cana-1065	121	12	}	}	PUNCT
cana-1065	121	13	communications	communication	NOUN
cana-1065	121	14	on	on	ADP
cana-1065	121	15	applied	apply	VERB
cana-1065	121	16	nonlinear	nonlinear	ADJ
cana-1065	121	17	analysis	analysis	NOUN
cana-1065	121	18	issn	issn	NOUN
cana-1065	121	19	:	:	PUNCT
cana-1065	121	20	1074	1074	NUM
cana-1065	121	21	-	-	PUNCT
cana-1065	121	22	133x	133x	NUM
cana-1065	121	23	vol	vol	NOUN
cana-1065	121	24	31	31	NUM
cana-1065	121	25	no	no	NOUN
cana-1065	121	26	.	.	PUNCT
cana-1065	122	1	5s	5s	NUM
cana-1065	122	2	(	(	PUNCT
cana-1065	122	3	2024	2024	NUM
cana-1065	122	4	)	)	PUNCT
cana-1065	122	5	465	465	NUM
cana-1065	122	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1065	122	7	or	or	CCONJ
cana-1065	122	8	,	,	PUNCT
cana-1065	122	9	𝑀(𝑧	𝑀(𝑧	NUM
cana-1065	122	10	,	,	PUNCT
cana-1065	122	11	𝑧1	𝑧1	NOUN
cana-1065	122	12	,	,	PUNCT
cana-1065	122	13	ξ	ξ	PROPN
cana-1065	122	14	𝑞	𝑞	PROPN
cana-1065	122	15	)	)	PUNCT
cana-1065	122	16	≥	≥	NOUN
cana-1065	122	17	φ{min{𝑀(𝑧	φ{min{𝑀(𝑧	PROPN
cana-1065	122	18	,	,	PUNCT
cana-1065	122	19	𝑧1	𝑧1	PROPN
cana-1065	122	20	,	,	PUNCT
cana-1065	122	21	𝑞	𝑞	NOUN
cana-1065	122	22	)	)	PUNCT
cana-1065	122	23	,	,	PUNCT
cana-1065	122	24	𝑀	𝑀	PROPN
cana-1065	122	25	(	(	PUNCT
cana-1065	122	26	𝑧	𝑧	PROPN
cana-1065	122	27	,	,	PUNCT
cana-1065	122	28	𝑧	𝑧	PROPN
cana-1065	122	29	,	,	PUNCT
cana-1065	122	30	𝑞	𝑞	NOUN
cana-1065	122	31	)	)	PUNCT
cana-1065	122	32	}	}	PUNCT
cana-1065	122	33	or	or	CCONJ
cana-1065	122	34	,	,	PUNCT
cana-1065	122	35	𝑀(𝑧	𝑀(𝑧	NUM
cana-1065	122	36	,	,	PUNCT
cana-1065	122	37	𝑧1	𝑧1	NOUN
cana-1065	122	38	,	,	PUNCT
cana-1065	122	39	ξ	ξ	PROPN
cana-1065	122	40	𝑞	𝑞	PROPN
cana-1065	122	41	)	)	PUNCT
cana-1065	122	42	≥	≥	NOUN
cana-1065	122	43	φ{𝑀(𝑧	φ{𝑀(𝑧	NOUN
cana-1065	122	44	,	,	PUNCT
cana-1065	122	45	𝑧1	𝑧1	PROPN
cana-1065	122	46	,	,	PUNCT
cana-1065	122	47	𝑞	𝑞	NOUN
cana-1065	122	48	)	)	PUNCT
cana-1065	122	49	}	}	PUNCT
cana-1065	122	50	𝑀(𝑧	𝑀(𝑧	NUM
cana-1065	122	51	,	,	PUNCT
cana-1065	122	52	𝑧1	𝑧1	NOUN
cana-1065	122	53	,	,	PUNCT
cana-1065	122	54	ξ	ξ	PROPN
cana-1065	122	55	𝑞	𝑞	PROPN
cana-1065	122	56	)	)	PUNCT
cana-1065	122	57	≥	≥	NOUN
cana-1065	122	58	𝑀(𝑧	𝑀(𝑧	NOUN
cana-1065	122	59	,	,	PUNCT
cana-1065	122	60	𝑧1	𝑧1	NOUN
cana-1065	122	61	,	,	PUNCT
cana-1065	122	62	𝑞	𝑞	NOUN
cana-1065	122	63	)	)	PUNCT
cana-1065	122	64	,	,	PUNCT
cana-1065	122	65	by	by	ADP
cana-1065	122	66	property	property	NOUN
cana-1065	122	67	of	of	ADP
cana-1065	122	68	φ	φ	PROPN
cana-1065	122	69			PROPN
cana-1065	123	1	𝑧	𝑧	PROPN
cana-1065	123	2	=	=	X
cana-1065	123	3	𝑧1	𝑧1	NOUN
cana-1065	123	4	,	,	PUNCT
cana-1065	123	5	by	by	ADP
cana-1065	123	6	lemma	lemma	PROPN
cana-1065	123	7	2.1	2.1	NUM
cana-1065	123	8	.	.	PUNCT
cana-1065	124	1	hence	hence	ADV
cana-1065	124	2	,	,	PUNCT
cana-1065	124	3	𝑧	𝑧	PROPN
cana-1065	124	4	=	=	PUNCT
cana-1065	124	5	𝐴𝑧	𝐴𝑧	X
cana-1065	124	6	=	=	SYM
cana-1065	124	7	𝐵𝑧	𝐵𝑧	PROPN
cana-1065	124	8	=	=	PUNCT
cana-1065	124	9	𝑆𝑧	𝑆𝑧	PROPN
cana-1065	124	10	=	=	SYM
cana-1065	124	11	𝑇𝑧	𝑇𝑧	PROPN
cana-1065	124	12	,	,	PUNCT
cana-1065	124	13	and	and	CCONJ
cana-1065	124	14	𝑧	𝑧	PRON
cana-1065	124	15	is	be	AUX
cana-1065	124	16	a	a	DET
cana-1065	124	17	unique	unique	ADJ
cana-1065	124	18	common	common	ADJ
cana-1065	124	19	fixed	fix	VERB
cana-1065	124	20	point	point	NOUN
cana-1065	124	21	for	for	ADP
cana-1065	124	22	a	a	DET
cana-1065	124	23	,	,	PUNCT
cana-1065	124	24	b	b	PROPN
cana-1065	124	25	,	,	PUNCT
cana-1065	124	26	s	s	PROPN
cana-1065	124	27	,	,	PUNCT
cana-1065	124	28	and	and	CCONJ
cana-1065	124	29	t	t	X
cana-1065	124	30	in	in	ADP
cana-1065	124	31	𝑌.	𝑌.	PROPN
cana-1065	124	32	this	this	PRON
cana-1065	124	33	completes	complete	VERB
cana-1065	124	34	the	the	DET
cana-1065	124	35	proof	proof	NOUN
cana-1065	124	36	.	.	PUNCT
cana-1065	125	1	4	4	X
cana-1065	125	2	.	.	X
cana-1065	125	3	conclusion	conclusion	NOUN
cana-1065	125	4	:	:	PUNCT
cana-1065	125	5	in	in	ADP
cana-1065	125	6	conclusion	conclusion	NOUN
cana-1065	125	7	,	,	PUNCT
cana-1065	125	8	the	the	DET
cana-1065	125	9	result	result	NOUN
cana-1065	125	10	of	of	ADP
cana-1065	125	11	chaudhary	chaudhary	PROPN
cana-1065	125	12	et	et	PROPN
cana-1065	125	13	.	.	PUNCT
cana-1065	126	1	al	al	PROPN
cana-1065	126	2	.	.	PUNCT
cana-1065	127	1	[	[	X
cana-1065	127	2	5	5	NUM
cana-1065	127	3	]	]	PUNCT
cana-1065	127	4	is	be	AUX
cana-1065	127	5	a	a	DET
cana-1065	127	6	particular	particular	ADJ
cana-1065	127	7	case	case	NOUN
cana-1065	127	8	of	of	ADP
cana-1065	127	9	this	this	DET
cana-1065	127	10	theorem	theorem	NOUN
cana-1065	127	11	.	.	PUNCT
cana-1065	128	1	also	also	ADV
cana-1065	128	2	,	,	PUNCT
cana-1065	128	3	this	this	DET
cana-1065	128	4	theorem	theorem	NOUN
cana-1065	128	5	may	may	AUX
cana-1065	128	6	apply	apply	VERB
cana-1065	128	7	to	to	ADP
cana-1065	128	8	consequences	consequence	NOUN
cana-1065	128	9	results	result	NOUN
cana-1065	128	10	in	in	ADP
cana-1065	128	11	metric	metric	ADJ
cana-1065	128	12	space	space	NOUN
cana-1065	128	13	in	in	ADP
cana-1065	128	14	four	four	NUM
cana-1065	128	15	self	self	NOUN
cana-1065	128	16	-	-	PUNCT
cana-1065	128	17	mappings	mapping	NOUN
cana-1065	128	18	and	and	CCONJ
cana-1065	128	19	generalizes	generalize	VERB
cana-1065	128	20	and	and	CCONJ
cana-1065	128	21	improves	improve	VERB
cana-1065	128	22	other	other	ADJ
cana-1065	128	23	similar	similar	ADJ
cana-1065	128	24	results	result	NOUN
cana-1065	128	25	in	in	ADP
cana-1065	128	26	the	the	DET
cana-1065	128	27	literature	literature	NOUN
cana-1065	128	28	.	.	PUNCT
cana-1065	129	1	5	5	X
cana-1065	129	2	.	.	X
cana-1065	129	3	acknowledgments	acknowledgment	NOUN
cana-1065	129	4	:	:	PUNCT
cana-1065	130	1	i	i	PRON
cana-1065	130	2	am	be	AUX
cana-1065	130	3	very	very	ADV
cana-1065	130	4	thankful	thankful	ADJ
cana-1065	130	5	to	to	ADP
cana-1065	130	6	the	the	DET
cana-1065	130	7	editor	editor	NOUN
cana-1065	130	8	and	and	CCONJ
cana-1065	130	9	to	to	ADP
cana-1065	130	10	the	the	DET
cana-1065	130	11	anonymous	anonymous	ADJ
cana-1065	130	12	reviewers	reviewer	NOUN
cana-1065	130	13	for	for	ADP
cana-1065	130	14	their	their	PRON
cana-1065	130	15	careful	careful	ADJ
cana-1065	130	16	suggestions	suggestion	NOUN
cana-1065	130	17	,	,	PUNCT
cana-1065	130	18	and	and	CCONJ
cana-1065	130	19	also	also	ADV
cana-1065	130	20	thanks	thank	NOUN
cana-1065	130	21	to	to	ADP
cana-1065	130	22	the	the	DET
cana-1065	130	23	university	university	NOUN
cana-1065	130	24	grant	grant	PROPN
cana-1065	130	25	commission	commission	PROPN
cana-1065	130	26	,	,	PUNCT
cana-1065	130	27	nepal	nepal	NOUN
cana-1065	130	28	for	for	ADP
cana-1065	130	29	their	their	PRON
cana-1065	130	30	financial	financial	ADJ
cana-1065	130	31	contribution	contribution	NOUN
cana-1065	130	32	for	for	ADP
cana-1065	130	33	my	my	PRON
cana-1065	130	34	research	research	NOUN
cana-1065	130	35	publication	publication	NOUN
cana-1065	130	36	.	.	PUNCT
cana-1065	131	1	references	reference	NOUN
cana-1065	131	2	[	[	X
cana-1065	131	3	1	1	NUM
cana-1065	131	4	]	]	PUNCT
cana-1065	131	5	banach	banach	NOUN
cana-1065	131	6	,	,	PUNCT
cana-1065	131	7	s.	s.	PROPN
cana-1065	131	8	,	,	PUNCT
cana-1065	131	9	sur	sur	PROPN
cana-1065	131	10	les	les	PROPN
cana-1065	131	11	operations	operation	NOUN
cana-1065	131	12	dans	dan	NOUN
cana-1065	131	13	les	le	NOUN
cana-1065	131	14	ensembles	ensemble	NOUN
cana-1065	131	15	abstraits	abstrait	NOUN
cana-1065	131	16	et	et	PROPN
cana-1065	131	17	leur	leur	PROPN
cana-1065	131	18	applications	applications	PROPN
cana-1065	131	19	aux	aux	PROPN
cana-1065	131	20	equations	equation	NOUN
cana-1065	131	21	integral	integral	ADJ
cana-1065	131	22	,	,	PUNCT
cana-1065	131	23	fund	fund	NOUN
cana-1065	131	24	math	math	NOUN
cana-1065	131	25	.	.	PUNCT
cana-1065	132	1	3(1922	3(1922	NUM
cana-1065	132	2	)	)	PUNCT
cana-1065	132	3	,	,	PUNCT
cana-1065	132	4	133	133	NUM
cana-1065	132	5	-	-	SYM
cana-1065	132	6	181	181	NUM
cana-1065	132	7	.	.	PUNCT
cana-1065	133	1	[	[	X
cana-1065	133	2	2	2	X
cana-1065	133	3	]	]	X
cana-1065	133	4	cho	cho	PROPN
cana-1065	133	5	y.	y.	PROPN
cana-1065	133	6	j.	j.	PROPN
cana-1065	133	7	,	,	PUNCT
cana-1065	133	8	murthy	murthy	PROPN
cana-1065	133	9	p.	p.	PROPN
cana-1065	133	10	p.	p.	NOUN
cana-1065	133	11	,	,	PUNCT
cana-1065	133	12	and	and	CCONJ
cana-1065	133	13	stojakovic	stojakovic	ADJ
cana-1065	133	14	m.	m.	NOUN
cana-1065	133	15	,	,	PUNCT
cana-1065	133	16	compatible	compatible	ADJ
cana-1065	133	17	mappings	mapping	NOUN
cana-1065	133	18	of	of	ADP
cana-1065	133	19	type	type	NOUN
cana-1065	133	20	(	(	PUNCT
cana-1065	133	21	a	a	NOUN
cana-1065	133	22	)	)	PUNCT
cana-1065	133	23	and	and	CCONJ
cana-1065	133	24	common	common	ADJ
cana-1065	133	25	fixed	fix	VERB
cana-1065	133	26	point	point	NOUN
cana-1065	133	27	in	in	ADP
cana-1065	133	28	menger	menger	PROPN
cana-1065	133	29	space	space	NOUN
cana-1065	133	30	,	,	PUNCT
cana-1065	133	31	comm	comm	NOUN
cana-1065	133	32	.	.	PUNCT
cana-1065	134	1	korean	korean	ADJ
cana-1065	134	2	math	math	PROPN
cana-1065	134	3	.	.	PUNCT
cana-1065	135	1	soc	soc	PROPN
cana-1065	135	2	.	.	PUNCT
cana-1065	136	1	7(2	7(2	PUNCT
cana-1065	136	2	)	)	PUNCT
cana-1065	137	1	(	(	PUNCT
cana-1065	137	2	1992	1992	NUM
cana-1065	137	3	)	)	PUNCT
cana-1065	137	4	,	,	PUNCT
cana-1065	137	5	325	325	NUM
cana-1065	137	6	-	-	SYM
cana-1065	137	7	339	339	NUM
cana-1065	137	8	.	.	PUNCT
cana-1065	138	1	[	[	X
cana-1065	138	2	3	3	NUM
cana-1065	138	3	]	]	X
cana-1065	138	4	frechet	frechet	NOUN
cana-1065	138	5	,	,	PUNCT
cana-1065	138	6	m.	m.	NOUN
cana-1065	138	7	,	,	PUNCT
cana-1065	138	8	sur	sur	PROPN
cana-1065	138	9	quelques	quelques	PROPN
cana-1065	138	10	points	point	NOUN
cana-1065	138	11	du	du	PROPN
cana-1065	138	12	calcul	calcul	PROPN
cana-1065	138	13	fonctionnel	fonctionnel	PROPN
cana-1065	138	14	,	,	PUNCT
cana-1065	138	15	rendic	rendic	ADJ
cana-1065	138	16	.	.	PUNCT
cana-1065	139	1	circ	circ	PROPN
cana-1065	139	2	.	.	PUNCT
cana-1065	140	1	mat	mat	PROPN
cana-1065	140	2	.	.	PUNCT
cana-1065	140	3	palermo	palermo	PROPN
cana-1065	140	4	(	(	PUNCT
cana-1065	140	5	1906),1	1906),1	NUM
cana-1065	140	6	-	-	SYM
cana-1065	140	7	74	74	NUM
cana-1065	141	1	[	[	X
cana-1065	141	2	4	4	X
cana-1065	141	3	]	]	X
cana-1065	141	4	hadzic	hadzic	PROPN
cana-1065	141	5	o	o	X
cana-1065	141	6	and	and	CCONJ
cana-1065	141	7	pap	pap	ADP
cana-1065	141	8	e	e	NOUN
cana-1065	141	9	,	,	PUNCT
cana-1065	141	10	probabilistic	probabilistic	ADJ
cana-1065	141	11	fixed	fix	VERB
cana-1065	141	12	-	-	PUNCT
cana-1065	141	13	point	point	NOUN
cana-1065	141	14	theory	theory	NOUN
cana-1065	141	15	in	in	ADP
cana-1065	141	16	probabilistic	probabilistic	ADJ
cana-1065	141	17	metric	metric	ADJ
cana-1065	141	18	space	space	NOUN
cana-1065	141	19	,	,	PUNCT
cana-1065	141	20	kluwer	kluwer	NOUN
cana-1065	141	21	academic	academic	PROPN
cana-1065	141	22	publisher	publisher	NOUN
cana-1065	141	23	,	,	PUNCT
cana-1065	141	24	london	london	PROPN
cana-1065	141	25	.	.	PUNCT
cana-1065	142	1	536	536	NUM
cana-1065	142	2	,	,	PUNCT
cana-1065	142	3	2010	2010	NUM
cana-1065	142	4	.	.	PUNCT
cana-1065	143	1	[	[	X
cana-1065	143	2	5	5	NUM
cana-1065	143	3	]	]	X
cana-1065	143	4	chaudhary	chaudhary	PROPN
cana-1065	143	5	a.k	a.k	PROPN
cana-1065	143	6	.	.	PROPN
cana-1065	143	7	,	,	PUNCT
cana-1065	143	8	manandhar	manandhar	VERB
cana-1065	143	9	k.b	k.b	PROPN
cana-1065	143	10	.	.	PROPN
cana-1065	143	11	,	,	PUNCT
cana-1065	143	12	jha	jha	PROPN
cana-1065	143	13	k.	k.	PROPN
cana-1065	143	14	,	,	PUNCT
cana-1065	143	15	and	and	CCONJ
cana-1065	143	16	pathak	pathak	PROPN
cana-1065	143	17	,	,	PUNCT
cana-1065	143	18	h.	h.	PROPN
cana-1065	143	19	k.	k.	PROPN
cana-1065	143	20	,	,	PUNCT
cana-1065	143	21	a	a	DET
cana-1065	143	22	common	common	ADJ
cana-1065	143	23	fixed	fix	VERB
cana-1065	143	24	point	point	NOUN
cana-1065	143	25	theorem	theorem	VERB
cana-1065	143	26	in	in	ADP
cana-1065	143	27	menger	menger	PROPN
cana-1065	143	28	space	space	NOUN
cana-1065	143	29	with	with	ADP
cana-1065	143	30	weakly	weakly	ADJ
cana-1065	143	31	compatible	compatible	ADJ
cana-1065	143	32	mapping	mapping	NOUN
cana-1065	143	33	of	of	ADP
cana-1065	143	34	type	type	NOUN
cana-1065	143	35	(	(	PUNCT
cana-1065	143	36	p	p	NOUN
cana-1065	143	37	)	)	PUNCT
cana-1065	143	38	,	,	PUNCT
cana-1065	143	39	advances	advance	NOUN
cana-1065	143	40	in	in	ADP
cana-1065	143	41	mathematics	mathematic	NOUN
cana-1065	143	42	:	:	PUNCT
cana-1065	143	43	scientific	scientific	ADJ
cana-1065	143	44	journal	journal	NOUN
cana-1065	143	45	,	,	PUNCT
cana-1065	143	46	11(11	11(11	NUM
cana-1065	143	47	)	)	PUNCT
cana-1065	143	48	(	(	PUNCT
cana-1065	143	49	2022	2022	NUM
cana-1065	143	50	)	)	PUNCT
cana-1065	143	51	,	,	PUNCT
cana-1065	143	52	10191031	10191031	NUM
cana-1065	143	53	.	.	PUNCT
cana-1065	144	1	[	[	X
cana-1065	144	2	6	6	NUM
cana-1065	144	3	]	]	X
cana-1065	144	4	chaudhary	chaudhary	PROPN
cana-1065	144	5	a.k	a.k	PROPN
cana-1065	144	6	.	.	PROPN
cana-1065	144	7	,	,	PUNCT
cana-1065	144	8	manandhar	manandhar	VERB
cana-1065	144	9	k.b	k.b	PROPN
cana-1065	144	10	.	.	PROPN
cana-1065	144	11	,	,	PUNCT
cana-1065	144	12	and	and	CCONJ
cana-1065	144	13	jha	jha	PROPN
cana-1065	144	14	k.	k.	PROPN
cana-1065	144	15	,	,	PUNCT
cana-1065	144	16	a	a	DET
cana-1065	144	17	common	common	ADJ
cana-1065	144	18	fixed	fix	VERB
cana-1065	144	19	point	point	NOUN
cana-1065	144	20	theorem	theorem	VERB
cana-1065	144	21	in	in	ADP
cana-1065	144	22	menger	menger	PROPN
cana-1065	144	23	space	space	NOUN
cana-1065	144	24	with	with	ADP
cana-1065	144	25	compatible	compatible	ADJ
cana-1065	144	26	mapping	mapping	NOUN
cana-1065	144	27	of	of	ADP
cana-1065	144	28	type	type	NOUN
cana-1065	144	29	(	(	PUNCT
cana-1065	144	30	p	p	NOUN
cana-1065	144	31	)	)	PUNCT
cana-1065	144	32	,	,	PUNCT
cana-1065	144	33	international	international	ADJ
cana-1065	144	34	journal	journal	NOUN
cana-1065	144	35	of	of	ADP
cana-1065	144	36	math	math	NOUN
cana-1065	144	37	.	.	PUNCT
cana-1065	145	1	sci	sci	PROPN
cana-1065	145	2	.	.	PROPN
cana-1065	145	3	&	&	CCONJ
cana-1065	145	4	engg	engg	PROPN
cana-1065	145	5	.	.	PUNCT
cana-1065	146	1	appls	appls	PROPN
cana-1065	146	2	.	.	PUNCT
cana-1065	146	3	,	,	PUNCT
cana-1065	146	4	15(2	15(2	NUM
cana-1065	146	5	)	)	PUNCT
cana-1065	146	6	(	(	PUNCT
cana-1065	146	7	2021	2021	NUM
cana-1065	146	8	)	)	PUNCT
cana-1065	146	9	,	,	PUNCT
cana-1065	146	10	59	59	NUM
cana-1065	146	11	-	-	SYM
cana-1065	146	12	70	70	NUM
cana-1065	146	13	.	.	PUNCT
cana-1065	147	1	[	[	X
cana-1065	147	2	7	7	X
cana-1065	147	3	]	]	X
cana-1065	147	4	chaudhary	chaudhary	PROPN
cana-1065	147	5	a.k	a.k	PROPN
cana-1065	147	6	.	.	PROPN
cana-1065	147	7	,	,	PUNCT
cana-1065	147	8	manandhar	manandhar	VERB
cana-1065	147	9	k.b	k.b	PROPN
cana-1065	147	10	.	.	PROPN
cana-1065	147	11	,	,	PUNCT
cana-1065	147	12	and	and	CCONJ
cana-1065	147	13	jha	jha	PROPN
cana-1065	147	14	k.	k.	PROPN
cana-1065	147	15	,	,	PUNCT
cana-1065	147	16	a	a	DET
cana-1065	147	17	common	common	ADJ
cana-1065	147	18	fixed	fix	VERB
cana-1065	147	19	point	point	NOUN
cana-1065	147	20	theorem	theorem	VERB
cana-1065	147	21	in	in	ADP
cana-1065	147	22	menger	menger	PROPN
cana-1065	147	23	space	space	NOUN
cana-1065	147	24	with	with	ADP
cana-1065	147	25	compatible	compatible	ADJ
cana-1065	147	26	mapping	mapping	NOUN
cana-1065	147	27	of	of	ADP
cana-1065	147	28	type	type	NOUN
cana-1065	147	29	(	(	PUNCT
cana-1065	147	30	k	k	NOUN
cana-1065	147	31	)	)	PUNCT
cana-1065	147	32	,	,	PUNCT
cana-1065	147	33	advances	advance	NOUN
cana-1065	147	34	in	in	ADP
cana-1065	147	35	mathematics	mathematic	NOUN
cana-1065	147	36	:	:	PUNCT
cana-1065	147	37	scientific	scientific	ADJ
cana-1065	147	38	journal	journal	NOUN
cana-1065	147	39	,	,	PUNCT
cana-1065	147	40	11(10	11(10	NUM
cana-1065	147	41	)	)	PUNCT
cana-1065	147	42	(	(	PUNCT
cana-1065	147	43	2022	2022	NUM
cana-1065	147	44	)	)	PUNCT
cana-1065	147	45	,	,	PUNCT
cana-1065	147	46	883	883	NUM
cana-1065	147	47	-	-	SYM
cana-1065	147	48	892	892	NUM
cana-1065	147	49	.	.	PUNCT
cana-1065	148	1	[	[	X
cana-1065	148	2	8	8	NUM
cana-1065	148	3	]	]	X
cana-1065	148	4	jungck	jungck	NOUN
cana-1065	148	5	,	,	PUNCT
cana-1065	148	6	g.	g.	PROPN
cana-1065	148	7	,	,	PUNCT
cana-1065	148	8	compatible	compatible	ADJ
cana-1065	148	9	mapping	mapping	NOUN
cana-1065	148	10	and	and	CCONJ
cana-1065	148	11	common	common	ADJ
cana-1065	148	12	fixed	fix	VERB
cana-1065	148	13	points	point	NOUN
cana-1065	148	14	,	,	PUNCT
cana-1065	148	15	internat	internat	PROPN
cana-1065	148	16	.	.	PUNCT
cana-1065	149	1	j.	j.	PROPN
cana-1065	149	2	math	math	PROPN
cana-1065	149	3	.	.	PUNCT
cana-1065	150	1	sci	sci	PROPN
cana-1065	150	2	.	.	PROPN
cana-1065	150	3	,	,	PUNCT
cana-1065	150	4	9(4	9(4	NUM
cana-1065	150	5	)	)	PUNCT
cana-1065	150	6	(	(	PUNCT
cana-1065	150	7	1986),771	1986),771	NUM
cana-1065	150	8	-	-	SYM
cana-1065	150	9	779	779	NUM
cana-1065	150	10	.	.	PUNCT
cana-1065	151	1	[	[	X
cana-1065	151	2	9	9	NUM
cana-1065	151	3	]	]	SYM
cana-1065	151	4	jungck	jungck	NOUN
cana-1065	151	5	,	,	PUNCT
cana-1065	151	6	g.	g.	PROPN
cana-1065	151	7	,	,	PUNCT
cana-1065	151	8	murthy	murthy	PROPN
cana-1065	151	9	p.	p.	PROPN
cana-1065	151	10	p.	p.	NOUN
cana-1065	151	11	,	,	PUNCT
cana-1065	151	12	and	and	CCONJ
cana-1065	151	13	cho	cho	PROPN
cana-1065	151	14	y.	y.	PROPN
cana-1065	151	15	j.	j.	PROPN
cana-1065	151	16	,	,	PUNCT
cana-1065	151	17	compatible	compatible	ADJ
cana-1065	151	18	mappings	mapping	NOUN
cana-1065	151	19	of	of	ADP
cana-1065	151	20	type	type	NOUN
cana-1065	151	21	(	(	PUNCT
cana-1065	151	22	a	a	NOUN
cana-1065	151	23	)	)	PUNCT
cana-1065	151	24	and	and	CCONJ
cana-1065	151	25	common	common	ADJ
cana-1065	151	26	fixed	fix	VERB
cana-1065	151	27	points	point	NOUN
cana-1065	151	28	,	,	PUNCT
cana-1065	151	29	math	math	NOUN
cana-1065	151	30	.	.	PUNCT
cana-1065	152	1	japonica	japonica	PROPN
cana-1065	152	2	,	,	PUNCT
cana-1065	152	3	38(1993	38(1993	NUM
cana-1065	152	4	)	)	PUNCT
cana-1065	152	5	,	,	PUNCT
cana-1065	152	6	381–390	381–390	NUM
cana-1065	152	7	.	.	PUNCT
cana-1065	153	1	[	[	X
cana-1065	153	2	10	10	NUM
cana-1065	153	3	]	]	X
cana-1065	153	4	jha	jha	PROPN
cana-1065	153	5	k.	k.	PROPN
cana-1065	153	6	,	,	PUNCT
cana-1065	153	7	popa	popa	NOUN
cana-1065	153	8	v.	v.	ADV
cana-1065	153	9	,	,	PUNCT
cana-1065	153	10	and	and	CCONJ
cana-1065	153	11	manandhar	manandhar	VERB
cana-1065	153	12	k.b	k.b	PROPN
cana-1065	153	13	.	.	PROPN
cana-1065	153	14	,	,	PUNCT
cana-1065	153	15	a	a	DET
cana-1065	153	16	common	common	ADJ
cana-1065	153	17	fixed	fix	VERB
cana-1065	153	18	point	point	NOUN
cana-1065	153	19	theorem	theorem	NOUN
cana-1065	153	20	for	for	ADP
cana-1065	153	21	compatible	compatible	ADJ
cana-1065	153	22	mappings	mapping	NOUN
cana-1065	153	23	of	of	ADP
cana-1065	153	24	type	type	NOUN
cana-1065	153	25	(	(	PUNCT
cana-1065	153	26	k	k	NOUN
cana-1065	153	27	)	)	PUNCT
cana-1065	153	28	in	in	ADP
cana-1065	153	29	metric	metric	ADJ
cana-1065	153	30	space	space	NOUN
cana-1065	153	31	,	,	PUNCT
cana-1065	153	32	international	international	ADJ
cana-1065	153	33	journal	journal	NOUN
cana-1065	153	34	of	of	ADP
cana-1065	153	35	math	math	NOUN
cana-1065	153	36	.	.	PUNCT
cana-1065	154	1	sci	sci	PROPN
cana-1065	154	2	.	.	PROPN
cana-1065	154	3	&	&	CCONJ
cana-1065	154	4	engg	engg	PROPN
cana-1065	154	5	.	.	PUNCT
cana-1065	155	1	appls	appls	PROPN
cana-1065	155	2	.	.	PUNCT
cana-1065	155	3	,	,	PUNCT
cana-1065	155	4	8(1	8(1	NUM
cana-1065	155	5	)	)	PUNCT
cana-1065	155	6	(	(	PUNCT
cana-1065	155	7	2014),383	2014),383	NUM
cana-1065	155	8	-	-	SYM
cana-1065	155	9	391	391	NUM
cana-1065	155	10	.	.	PUNCT
cana-1065	156	1	[	[	X
cana-1065	156	2	11	11	NUM
cana-1065	156	3	]	]	X
cana-1065	156	4	menger	menger	PROPN
cana-1065	156	5	,	,	PUNCT
cana-1065	156	6	k.	k.	PROPN
cana-1065	156	7	,	,	PUNCT
cana-1065	156	8	statistical	statistical	ADJ
cana-1065	156	9	matrices	matrix	NOUN
cana-1065	156	10	,	,	PUNCT
cana-1065	156	11	proceedings	proceeding	NOUN
cana-1065	156	12	of	of	ADP
cana-1065	156	13	national	national	PROPN
cana-1065	156	14	academy	academy	PROPN
cana-1065	156	15	of	of	ADP
cana-1065	156	16	sciences	sciences	PROPN
cana-1065	156	17	of	of	ADP
cana-1065	156	18	usa	usa	PROPN
cana-1065	156	19	,	,	PUNCT
cana-1065	156	20	28	28	NUM
cana-1065	156	21	(	(	PUNCT
cana-1065	156	22	1942	1942	NUM
cana-1065	156	23	)	)	PUNCT
cana-1065	156	24	,	,	PUNCT
cana-1065	156	25	535	535	NUM
cana-1065	156	26	-	-	SYM
cana-1065	156	27	537	537	NUM
cana-1065	156	28	.	.	PUNCT
cana-1065	157	1	[	[	X
cana-1065	157	2	12	12	NUM
cana-1065	157	3	]	]	X
cana-1065	157	4	mishra	mishra	PROPN
cana-1065	157	5	s.n	s.n	PROPN
cana-1065	157	6	.	.	PROPN
cana-1065	157	7	,	,	PUNCT
cana-1065	157	8	common	common	ADJ
cana-1065	157	9	fixed	fix	VERB
cana-1065	157	10	points	point	NOUN
cana-1065	157	11	of	of	ADP
cana-1065	157	12	compatible	compatible	ADJ
cana-1065	157	13	mappings	mapping	NOUN
cana-1065	157	14	in	in	ADP
cana-1065	157	15	probabilistic	probabilistic	ADJ
cana-1065	157	16	metric	metric	ADJ
cana-1065	157	17	space	space	NOUN
cana-1065	157	18	,	,	PUNCT
cana-1065	157	19	math	math	NOUN
cana-1065	157	20	.	.	PUNCT
cana-1065	158	1	japon	japon	PROPN
cana-1065	158	2	.	.	PROPN
cana-1065	158	3	,36	,36	PUNCT
cana-1065	158	4	(	(	PUNCT
cana-1065	158	5	1991	1991	NUM
cana-1065	158	6	)	)	PUNCT
cana-1065	158	7	283	283	NUM
cana-1065	158	8	-	-	SYM
cana-1065	158	9	289	289	NUM
cana-1065	158	10	.	.	PUNCT
cana-1065	159	1	[	[	X
cana-1065	159	2	13	13	NUM
cana-1065	159	3	]	]	PUNCT
cana-1065	159	4	schweizer	schweizer	PROPN
cana-1065	159	5	b.	b.	PROPN
cana-1065	159	6	,	,	PUNCT
cana-1065	159	7	and	and	CCONJ
cana-1065	159	8	sklar	sklar	ADJ
cana-1065	159	9	a.	a.	NOUN
cana-1065	159	10	statistical	statistical	ADJ
cana-1065	159	11	metric	metric	ADJ
cana-1065	159	12	space	space	NOUN
cana-1065	159	13	,	,	PUNCT
cana-1065	159	14	pacific	pacific	PROPN
cana-1065	159	15	j.	j.	PROPN
cana-1065	159	16	of	of	ADP
cana-1065	159	17	math	math	PROPN
cana-1065	159	18	.	.	PUNCT
cana-1065	160	1	,	,	PUNCT
cana-1065	160	2	10	10	NUM
cana-1065	160	3	(	(	PUNCT
cana-1065	160	4	1960	1960	NUM
cana-1065	160	5	)	)	PUNCT
cana-1065	160	6	314	314	NUM
cana-1065	160	7	-	-	SYM
cana-1065	160	8	334	334	NUM
cana-1065	160	9	.	.	PUNCT
cana-1065	161	1	[	[	X
cana-1065	161	2	14	14	NUM
cana-1065	161	3	]	]	X
cana-1065	161	4	sehgal	sehgal	PROPN
cana-1065	161	5	,	,	PUNCT
cana-1065	161	6	v.m	v.m	PROPN
cana-1065	161	7	.	.	PROPN
cana-1065	161	8	and	and	CCONJ
cana-1065	161	9	bharucha	bharucha	ADV
cana-1065	161	10	-	-	PUNCT
cana-1065	161	11	reid	reid	PROPN
cana-1065	161	12	a.t	a.t	PROPN
cana-1065	161	13	.	.	PROPN
cana-1065	161	14	,	,	PUNCT
cana-1065	161	15	fixed	fix	VERB
cana-1065	161	16	point	point	NOUN
cana-1065	161	17	contraction	contraction	NOUN
cana-1065	161	18	mapping	mapping	NOUN
cana-1065	161	19	in	in	ADP
cana-1065	161	20	probabilistic	probabilistic	ADJ
cana-1065	161	21	metric	metric	ADJ
cana-1065	161	22	space	space	NOUN
cana-1065	161	23	.	.	PUNCT
cana-1065	162	1	math	math	NOUN
cana-1065	162	2	system	system	NOUN
cana-1065	162	3	theory	theory	NOUN
cana-1065	162	4	,	,	PUNCT
cana-1065	162	5	6	6	NUM
cana-1065	162	6	(	(	PUNCT
cana-1065	162	7	1972	1972	NUM
cana-1065	162	8	)	)	PUNCT
cana-1065	162	9	,	,	PUNCT
cana-1065	162	10	97	97	NUM
cana-1065	162	11	-	-	SYM
cana-1065	162	12	102	102	NUM
cana-1065	162	13	.	.	PUNCT
cana-1065	163	1	[	[	X
cana-1065	163	2	15	15	NUM
cana-1065	163	3	]	]	X
cana-1065	163	4	singh	singh	PROPN
cana-1065	163	5	b.	b.	PROPN
cana-1065	163	6	and	and	CCONJ
cana-1065	163	7	jain	jain	PROPN
cana-1065	163	8	s.	s.	PROPN
cana-1065	163	9	common	common	ADJ
cana-1065	163	10	fixed	fix	VERB
cana-1065	163	11	point	point	NOUN
cana-1065	163	12	theorem	theorem	VERB
cana-1065	163	13	in	in	ADP
cana-1065	163	14	menger	menger	PROPN
cana-1065	163	15	space	space	NOUN
cana-1065	163	16	through	through	ADP
cana-1065	163	17	weak	weak	ADJ
cana-1065	163	18	compatibility	compatibility	NOUN
cana-1065	163	19	,	,	PUNCT
cana-1065	163	20	j.	j.	PROPN
cana-1065	163	21	math	math	PROPN
cana-1065	163	22	.	.	PUNCT
cana-1065	164	1	anal	anal	PROPN
cana-1065	164	2	.	.	PUNCT
cana-1065	164	3	appl	appl	PROPN
cana-1065	164	4	.	.	PROPN
cana-1065	164	5	,	,	PUNCT
cana-1065	164	6	301(2005	301(2005	NUM
cana-1065	164	7	)	)	PUNCT
cana-1065	164	8	,	,	PUNCT
cana-1065	164	9	439	439	NUM
cana-1065	164	10	-	-	SYM
cana-1065	164	11	448	448	NUM
cana-1065	164	12	.	.	PUNCT
cana-1065	165	1	[	[	X
cana-1065	165	2	16	16	NUM
cana-1065	165	3	]	]	X
cana-1065	165	4	sklar	sklar	ADJ
cana-1065	165	5	a.	a.	PROPN
cana-1065	165	6	and	and	CCONJ
cana-1065	165	7	schweizer	schweizer	PROPN
cana-1065	165	8	b.	b.	PROPN
cana-1065	165	9	probabilistic	probabilistic	PROPN
cana-1065	165	10	metric	metric	ADJ
cana-1065	165	11	space	space	NOUN
cana-1065	165	12	.	.	PUNCT
cana-1065	166	1	dover	dover	PROPN
cana-1065	166	2	publications	publications	PROPN
cana-1065	166	3	,	,	PUNCT
cana-1065	166	4	inc	inc	PROPN
cana-1065	166	5	,	,	PUNCT
cana-1065	166	6	mineola	mineola	PROPN
cana-1065	166	7	,	,	PUNCT
cana-1065	166	8	new	new	PROPN
cana-1065	166	9	york	york	PROPN
cana-1065	166	10	(	(	PUNCT
cana-1065	166	11	2005	2005	NUM
cana-1065	166	12	)	)	PUNCT
cana-1065	166	13	.	.	PUNCT
cana-1065	167	1	[	[	X
cana-1065	167	2	17	17	NUM
cana-1065	167	3	]	]	X
cana-1065	167	4	sessa	sessa	PROPN
cana-1065	167	5	,	,	PUNCT
cana-1065	167	6	s.	s.	PROPN
cana-1065	167	7	,	,	PUNCT
cana-1065	167	8	on	on	ADP
cana-1065	167	9	a	a	DET
cana-1065	167	10	weak	weak	ADJ
cana-1065	167	11	commutativity	commutativity	NOUN
cana-1065	167	12	condition	condition	NOUN
cana-1065	167	13	of	of	ADP
cana-1065	167	14	mappings	mapping	NOUN
cana-1065	167	15	in	in	ADP
cana-1065	167	16	fixed	fix	VERB
cana-1065	167	17	point	point	NOUN
cana-1065	167	18	considerations	consideration	NOUN
cana-1065	167	19	,	,	PUNCT
cana-1065	167	20	publ	publ	NOUN
cana-1065	167	21	.	.	PUNCT
cana-1065	168	1	inst	inst	PROPN
cana-1065	168	2	.	.	PUNCT
cana-1065	168	3	math	math	NOUN
cana-1065	168	4	.	.	PUNCT
cana-1065	169	1	(	(	PUNCT
cana-1065	169	2	beograd	beograd	PROPN
cana-1065	169	3	)	)	PUNCT
cana-1065	169	4	(	(	PUNCT
cana-1065	169	5	n.s	n.s	PROPN
cana-1065	169	6	.	.	PROPN
cana-1065	169	7	)	)	PUNCT
cana-1065	169	8	,	,	PUNCT
cana-1065	169	9	32(46	32(46	NOUN
cana-1065	169	10	)	)	PUNCT
cana-1065	169	11	(	(	PUNCT
cana-1065	169	12	1982	1982	NUM
cana-1065	169	13	)	)	PUNCT
cana-1065	169	14	,	,	PUNCT
cana-1065	169	15	149–153	149–153	NUM
cana-1065	169	16	.	.	PUNCT