id	sid	tid	token	lemma	pos
ejpam-5982	1	1	european	european	PROPN
ejpam-5982	1	2	journal	journal	PROPN
ejpam-5982	1	3	of	of	ADP
ejpam-5982	1	4	pure	pure	ADJ
ejpam-5982	1	5	and	and	CCONJ
ejpam-5982	1	6	applied	applied	ADJ
ejpam-5982	1	7	mathematics	mathematic	NOUN
ejpam-5982	1	8	2025	2025	NUM
ejpam-5982	1	9	,	,	PUNCT
ejpam-5982	1	10	vol	vol	NOUN
ejpam-5982	1	11	.	.	PROPN
ejpam-5982	1	12	18	18	NUM
ejpam-5982	1	13	,	,	PUNCT
ejpam-5982	1	14	issue	issue	NOUN
ejpam-5982	1	15	2	2	NUM
ejpam-5982	1	16	,	,	PUNCT
ejpam-5982	1	17	article	article	NOUN
ejpam-5982	1	18	number	number	NOUN
ejpam-5982	1	19	5982	5982	NUM
ejpam-5982	1	20	issn	issn	PROPN
ejpam-5982	1	21	1307	1307	NUM
ejpam-5982	1	22	-	-	SYM
ejpam-5982	1	23	5543	5543	NUM
ejpam-5982	1	24	–	–	PUNCT
ejpam-5982	1	25	ejpam.com	ejpam.com	X
ejpam-5982	1	26	published	publish	VERB
ejpam-5982	1	27	by	by	ADP
ejpam-5982	1	28	new	new	PROPN
ejpam-5982	1	29	york	york	PROPN
ejpam-5982	1	30	business	business	PROPN
ejpam-5982	1	31	global	global	ADJ
ejpam-5982	1	32	common	common	ADJ
ejpam-5982	1	33	fixed	fix	VERB
ejpam-5982	1	34	point	point	NOUN
ejpam-5982	1	35	theorems	theorem	NOUN
ejpam-5982	1	36	for	for	ADP
ejpam-5982	1	37	mappings	mapping	NOUN
ejpam-5982	1	38	satisfying	satisfy	VERB
ejpam-5982	1	39	implicit	implicit	ADJ
ejpam-5982	1	40	relation	relation	NOUN
ejpam-5982	1	41	in	in	ADP
ejpam-5982	1	42	bipolar	bipolar	ADJ
ejpam-5982	1	43	metric	metric	ADJ
ejpam-5982	1	44	space	space	NOUN
ejpam-5982	1	45	penumarthy	penumarthy	ADJ
ejpam-5982	1	46	parvateesam	parvateesam	NOUN
ejpam-5982	1	47	murthy1	murthy1	PROPN
ejpam-5982	1	48	,	,	PUNCT
ejpam-5982	1	49	chandra	chandra	PROPN
ejpam-5982	1	50	prakash	prakash	PROPN
ejpam-5982	1	51	dhuri1	dhuri1	PROPN
ejpam-5982	1	52	,	,	PUNCT
ejpam-5982	1	53	rajagopalan	rajagopalan	VERB
ejpam-5982	1	54	ramaswamy2,∗	ramaswamy2,∗	ADJ
ejpam-5982	1	55	,	,	PUNCT
ejpam-5982	1	56	khizar	khizar	PROPN
ejpam-5982	1	57	hayat	hayat	PROPN
ejpam-5982	1	58	khan2	khan2	PROPN
ejpam-5982	1	59	,	,	PUNCT
ejpam-5982	1	60	ola	ola	PROPN
ejpam-5982	1	61	ashour	ashour	PROPN
ejpam-5982	1	62	abdelnaby2,3	abdelnaby2,3	PROPN
ejpam-5982	1	63	1	1	NUM
ejpam-5982	1	64	department	department	NOUN
ejpam-5982	1	65	of	of	ADP
ejpam-5982	1	66	mathematics	mathematic	NOUN
ejpam-5982	1	67	,	,	PUNCT
ejpam-5982	1	68	guru	guru	NOUN
ejpam-5982	1	69	ghasidas	ghasidas	PROPN
ejpam-5982	1	70	vishwavidyalaya	vishwavidyalaya	PROPN
ejpam-5982	1	71	(	(	PUNCT
ejpam-5982	1	72	a	a	DET
ejpam-5982	1	73	central	central	ADJ
ejpam-5982	1	74	university	university	NOUN
ejpam-5982	1	75	)	)	PUNCT
ejpam-5982	1	76	,	,	PUNCT
ejpam-5982	1	77	bilaspur(cg	bilaspur(cg	PROPN
ejpam-5982	1	78	)	)	PUNCT
ejpam-5982	1	79	,	,	PUNCT
ejpam-5982	1	80	495	495	NUM
ejpam-5982	1	81	009	009	NUM
ejpam-5982	1	82	india	india	PROPN
ejpam-5982	1	83	2	2	NUM
ejpam-5982	1	84	department	department	NOUN
ejpam-5982	1	85	of	of	ADP
ejpam-5982	1	86	mathematics	mathematic	NOUN
ejpam-5982	1	87	,	,	PUNCT
ejpam-5982	1	88	college	college	NOUN
ejpam-5982	1	89	of	of	ADP
ejpam-5982	1	90	science	science	NOUN
ejpam-5982	1	91	and	and	CCONJ
ejpam-5982	1	92	humanities	humanity	NOUN
ejpam-5982	1	93	in	in	ADP
ejpam-5982	1	94	alkharj	alkharj	NOUN
ejpam-5982	1	95	,	,	PUNCT
ejpam-5982	1	96	prince	prince	PROPN
ejpam-5982	1	97	sattam	sattam	PROPN
ejpam-5982	1	98	bin	bin	PROPN
ejpam-5982	1	99	abdulaziz	abdulaziz	PROPN
ejpam-5982	1	100	university	university	PROPN
ejpam-5982	1	101	,	,	PUNCT
ejpam-5982	1	102	alkharj	alkharj	VERB
ejpam-5982	1	103	11942	11942	NUM
ejpam-5982	1	104	,	,	PUNCT
ejpam-5982	1	105	saudi	saudi	PROPN
ejpam-5982	1	106	arabia	arabia	PROPN
ejpam-5982	1	107	3	3	NUM
ejpam-5982	1	108	department	department	NOUN
ejpam-5982	1	109	of	of	ADP
ejpam-5982	1	110	mathematics	mathematic	NOUN
ejpam-5982	1	111	,	,	PUNCT
ejpam-5982	1	112	faculty	faculty	NOUN
ejpam-5982	1	113	of	of	ADP
ejpam-5982	1	114	science	science	NOUN
ejpam-5982	1	115	,	,	PUNCT
ejpam-5982	1	116	cairo	cairo	PROPN
ejpam-5982	1	117	university	university	PROPN
ejpam-5982	1	118	,	,	PUNCT
ejpam-5982	1	119	cairo	cairo	PROPN
ejpam-5982	1	120	,	,	PUNCT
ejpam-5982	1	121	egypt	egypt	PROPN
ejpam-5982	1	122	abstract	abstract	PROPN
ejpam-5982	1	123	.	.	PUNCT
ejpam-5982	2	1	in	in	ADP
ejpam-5982	2	2	this	this	DET
ejpam-5982	2	3	article	article	NOUN
ejpam-5982	2	4	,	,	PUNCT
ejpam-5982	2	5	we	we	PRON
ejpam-5982	2	6	introduce	introduce	VERB
ejpam-5982	2	7	the	the	DET
ejpam-5982	2	8	concept	concept	NOUN
ejpam-5982	2	9	of	of	ADP
ejpam-5982	2	10	compatible	compatible	ADJ
ejpam-5982	2	11	mappings	mapping	NOUN
ejpam-5982	2	12	of	of	ADP
ejpam-5982	2	13	type	type	NOUN
ejpam-5982	2	14	(	(	PUNCT
ejpam-5982	2	15	a	a	NOUN
ejpam-5982	2	16	)	)	PUNCT
ejpam-5982	2	17	and	and	CCONJ
ejpam-5982	2	18	weaken	weaken	VERB
ejpam-5982	2	19	the	the	DET
ejpam-5982	2	20	same	same	ADJ
ejpam-5982	2	21	in	in	ADP
ejpam-5982	2	22	the	the	DET
ejpam-5982	2	23	setting	setting	NOUN
ejpam-5982	2	24	of	of	ADP
ejpam-5982	2	25	bipolar	bipolar	ADJ
ejpam-5982	2	26	metric	metric	ADJ
ejpam-5982	2	27	spaces	space	NOUN
ejpam-5982	2	28	and	and	CCONJ
ejpam-5982	2	29	established	establish	VERB
ejpam-5982	2	30	fixed	fix	VERB
ejpam-5982	2	31	point	point	NOUN
ejpam-5982	2	32	results	result	NOUN
ejpam-5982	2	33	in	in	ADP
ejpam-5982	2	34	the	the	DET
ejpam-5982	2	35	setting	setting	NOUN
ejpam-5982	2	36	of	of	ADP
ejpam-5982	2	37	bipolar	bipolar	ADJ
ejpam-5982	2	38	metric	metric	ADJ
ejpam-5982	2	39	spaces	space	NOUN
ejpam-5982	2	40	,	,	PUNCT
ejpam-5982	2	41	using	use	VERB
ejpam-5982	2	42	implicit	implicit	ADJ
ejpam-5982	2	43	relation	relation	NOUN
ejpam-5982	2	44	function	function	NOUN
ejpam-5982	2	45	.	.	PUNCT
ejpam-5982	3	1	the	the	DET
ejpam-5982	3	2	derived	derive	VERB
ejpam-5982	3	3	results	result	NOUN
ejpam-5982	3	4	have	have	AUX
ejpam-5982	3	5	been	be	AUX
ejpam-5982	3	6	supplemented	supplement	VERB
ejpam-5982	3	7	with	with	ADP
ejpam-5982	3	8	suitable	suitable	ADJ
ejpam-5982	3	9	non	non	ADJ
ejpam-5982	3	10	-	-	ADJ
ejpam-5982	3	11	trivial	trivial	ADJ
ejpam-5982	3	12	examples	example	NOUN
ejpam-5982	3	13	.	.	PUNCT
ejpam-5982	4	1	our	our	PRON
ejpam-5982	4	2	results	result	NOUN
ejpam-5982	4	3	have	have	AUX
ejpam-5982	4	4	extended	extend	VERB
ejpam-5982	4	5	and	and	CCONJ
ejpam-5982	4	6	generalized	generalize	VERB
ejpam-5982	4	7	some	some	DET
ejpam-5982	4	8	results	result	NOUN
ejpam-5982	4	9	proven	prove	VERB
ejpam-5982	4	10	in	in	ADP
ejpam-5982	4	11	the	the	DET
ejpam-5982	4	12	past	past	NOUN
ejpam-5982	4	13	and	and	CCONJ
ejpam-5982	4	14	some	some	DET
ejpam-5982	4	15	open	open	ADJ
ejpam-5982	4	16	problems	problem	NOUN
ejpam-5982	4	17	for	for	ADP
ejpam-5982	4	18	future	future	ADJ
ejpam-5982	4	19	research	research	NOUN
ejpam-5982	4	20	has	have	AUX
ejpam-5982	4	21	been	be	AUX
ejpam-5982	4	22	given	give	VERB
ejpam-5982	4	23	.	.	PUNCT
ejpam-5982	5	1	2020	2020	NUM
ejpam-5982	5	2	mathematics	mathematic	NOUN
ejpam-5982	5	3	subject	subject	NOUN
ejpam-5982	5	4	classifications	classification	NOUN
ejpam-5982	5	5	:	:	PUNCT
ejpam-5982	5	6	47h10	47h10	NUM
ejpam-5982	5	7	,	,	PUNCT
ejpam-5982	5	8	54h25	54h25	NUM
ejpam-5982	5	9	key	key	ADJ
ejpam-5982	5	10	words	word	NOUN
ejpam-5982	5	11	and	and	CCONJ
ejpam-5982	5	12	phrases	phrase	NOUN
ejpam-5982	5	13	:	:	PUNCT
ejpam-5982	5	14	fixed	fix	VERB
ejpam-5982	5	15	points	point	NOUN
ejpam-5982	5	16	,	,	PUNCT
ejpam-5982	5	17	bipolar	bipolar	ADJ
ejpam-5982	5	18	metric	metric	ADJ
ejpam-5982	5	19	space	space	NOUN
ejpam-5982	5	20	,	,	PUNCT
ejpam-5982	5	21	covariant	covariant	PROPN
ejpam-5982	5	22	map	map	NOUN
ejpam-5982	5	23	,	,	PUNCT
ejpam-5982	5	24	contravariant	contravariant	PROPN
ejpam-5982	5	25	map	map	NOUN
ejpam-5982	5	26	,	,	PUNCT
ejpam-5982	5	27	compatible	compatible	ADJ
ejpam-5982	5	28	mapping	mapping	NOUN
ejpam-5982	5	29	of	of	ADP
ejpam-5982	5	30	type	type	NOUN
ejpam-5982	5	31	(	(	PUNCT
ejpam-5982	5	32	a	a	NOUN
ejpam-5982	5	33	)	)	PUNCT
ejpam-5982	5	34	,	,	PUNCT
ejpam-5982	5	35	property	property	NOUN
ejpam-5982	5	36	(	(	PUNCT
ejpam-5982	5	37	e.a	e.a	PROPN
ejpam-5982	5	38	.	.	PROPN
ejpam-5982	5	39	)	)	PUNCT
ejpam-5982	6	1	cauchy	cauchy	PROPN
ejpam-5982	6	2	bisequence	bisequence	NOUN
ejpam-5982	6	3	1	1	NUM
ejpam-5982	6	4	.	.	PUNCT
ejpam-5982	6	5	introduction	introduction	NOUN
ejpam-5982	6	6	it	it	PRON
ejpam-5982	6	7	would	would	AUX
ejpam-5982	6	8	be	be	AUX
ejpam-5982	6	9	fair	fair	ADJ
ejpam-5982	6	10	to	to	PART
ejpam-5982	6	11	say	say	VERB
ejpam-5982	6	12	that	that	SCONJ
ejpam-5982	6	13	the	the	DET
ejpam-5982	6	14	concept	concept	NOUN
ejpam-5982	6	15	of	of	ADP
ejpam-5982	6	16	metric	metric	ADJ
ejpam-5982	6	17	fixed	fix	VERB
ejpam-5982	6	18	point	point	NOUN
ejpam-5982	6	19	theory	theory	NOUN
ejpam-5982	6	20	started	start	VERB
ejpam-5982	6	21	with	with	ADP
ejpam-5982	6	22	the	the	DET
ejpam-5982	6	23	famous	famous	ADJ
ejpam-5982	6	24	contraction	contraction	NOUN
ejpam-5982	6	25	mapping	mapping	NOUN
ejpam-5982	6	26	theorem	theorem	NOUN
ejpam-5982	6	27	of	of	ADP
ejpam-5982	6	28	s.	s.	PROPN
ejpam-5982	6	29	banach	banach	PROPN
ejpam-5982	7	1	[	[	X
ejpam-5982	7	2	1	1	NUM
ejpam-5982	7	3	]	]	PUNCT
ejpam-5982	7	4	.	.	PUNCT
ejpam-5982	8	1	this	this	DET
ejpam-5982	8	2	theory	theory	NOUN
ejpam-5982	8	3	has	have	AUX
ejpam-5982	8	4	seen	see	VERB
ejpam-5982	8	5	rapid	rapid	ADJ
ejpam-5982	8	6	development	development	NOUN
ejpam-5982	8	7	in	in	ADP
ejpam-5982	8	8	the	the	DET
ejpam-5982	8	9	past	past	ADJ
ejpam-5982	8	10	nineteenth	nineteenth	ADJ
ejpam-5982	8	11	and	and	CCONJ
ejpam-5982	8	12	twentieth	twentieth	ADJ
ejpam-5982	8	13	centuries	century	NOUN
ejpam-5982	8	14	.	.	PUNCT
ejpam-5982	9	1	in	in	ADP
ejpam-5982	9	2	the	the	DET
ejpam-5982	9	3	overlaps	overlap	NOUN
ejpam-5982	9	4	made	make	VERB
ejpam-5982	9	5	in	in	ADP
ejpam-5982	9	6	these	these	DET
ejpam-5982	9	7	centuries	century	NOUN
ejpam-5982	9	8	,	,	PUNCT
ejpam-5982	9	9	while	while	SCONJ
ejpam-5982	9	10	metric	metric	ADJ
ejpam-5982	9	11	spaces	space	NOUN
ejpam-5982	9	12	and	and	CCONJ
ejpam-5982	9	13	normed	normed	ADJ
ejpam-5982	9	14	spaces	space	NOUN
ejpam-5982	9	15	developed	develop	VERB
ejpam-5982	9	16	,	,	PUNCT
ejpam-5982	9	17	the	the	DET
ejpam-5982	9	18	domains	domain	NOUN
ejpam-5982	9	19	were	be	AUX
ejpam-5982	9	20	only	only	ADV
ejpam-5982	9	21	taken	take	VERB
ejpam-5982	9	22	as	as	ADP
ejpam-5982	9	23	value	value	NOUN
ejpam-5982	9	24	regions	region	NOUN
ejpam-5982	9	25	with	with	ADP
ejpam-5982	9	26	single	single	ADJ
ejpam-5982	9	27	variables	variable	NOUN
ejpam-5982	9	28	and	and	CCONJ
ejpam-5982	9	29	real	real	ADJ
ejpam-5982	9	30	positive	positive	ADJ
ejpam-5982	9	31	numbers	number	NOUN
ejpam-5982	9	32	.	.	PUNCT
ejpam-5982	10	1	in	in	ADP
ejpam-5982	10	2	other	other	ADJ
ejpam-5982	10	3	words	word	NOUN
ejpam-5982	10	4	,	,	PUNCT
ejpam-5982	10	5	new	new	ADJ
ejpam-5982	10	6	metric	metric	ADJ
ejpam-5982	10	7	spaces	space	NOUN
ejpam-5982	10	8	are	be	AUX
ejpam-5982	10	9	produced	produce	VERB
ejpam-5982	10	10	by	by	ADP
ejpam-5982	10	11	taking	take	VERB
ejpam-5982	10	12	the	the	DET
ejpam-5982	10	13	domains	domain	NOUN
ejpam-5982	10	14	x	x	SYM
ejpam-5982	10	15	,	,	PUNCT
ejpam-5982	10	16	x2	x2	PROPN
ejpam-5982	10	17	,	,	PUNCT
ejpam-5982	10	18	and	and	CCONJ
ejpam-5982	10	19	x3	x3	ADJ
ejpam-5982	10	20	.	.	PUNCT
ejpam-5982	11	1	however	however	ADV
ejpam-5982	11	2	,	,	PUNCT
ejpam-5982	11	3	bipolar	bipolar	ADJ
ejpam-5982	11	4	metric	metric	ADJ
ejpam-5982	11	5	space	space	NOUN
ejpam-5982	11	6	is	be	AUX
ejpam-5982	11	7	defined	define	VERB
ejpam-5982	11	8	as	as	ADP
ejpam-5982	11	9	a	a	DET
ejpam-5982	11	10	new	new	ADJ
ejpam-5982	11	11	space	space	NOUN
ejpam-5982	11	12	by	by	ADP
ejpam-5982	11	13	going	go	VERB
ejpam-5982	11	14	beyond	beyond	ADP
ejpam-5982	11	15	the	the	DET
ejpam-5982	11	16	conventional	conventional	ADJ
ejpam-5982	11	17	definition	definition	NOUN
ejpam-5982	11	18	of	of	ADP
ejpam-5982	11	19	metric	metric	ADJ
ejpam-5982	11	20	spaces	space	NOUN
ejpam-5982	11	21	that	that	PRON
ejpam-5982	11	22	have	have	AUX
ejpam-5982	11	23	been	be	AUX
ejpam-5982	11	24	defined	define	VERB
ejpam-5982	11	25	for	for	ADP
ejpam-5982	11	26	years	year	NOUN
ejpam-5982	11	27	.	.	PUNCT
ejpam-5982	12	1	at	at	ADP
ejpam-5982	12	2	the	the	DET
ejpam-5982	12	3	same	same	ADJ
ejpam-5982	12	4	time	time	NOUN
ejpam-5982	12	5	,	,	PUNCT
ejpam-5982	12	6	this	this	DET
ejpam-5982	12	7	theory	theory	NOUN
ejpam-5982	12	8	has	have	AUX
ejpam-5982	12	9	been	be	AUX
ejpam-5982	12	10	applied	apply	VERB
ejpam-5982	12	11	to	to	ADP
ejpam-5982	12	12	real	real	ADJ
ejpam-5982	12	13	life	life	NOUN
ejpam-5982	12	14	and	and	CCONJ
ejpam-5982	12	15	various	various	ADJ
ejpam-5982	12	16	fields	field	NOUN
ejpam-5982	12	17	of	of	ADP
ejpam-5982	12	18	science	science	NOUN
ejpam-5982	12	19	,	,	PUNCT
ejpam-5982	12	20	namely	namely	ADV
ejpam-5982	12	21	engineering	engineering	NOUN
ejpam-5982	12	22	,	,	PUNCT
ejpam-5982	12	23	economics	economic	NOUN
ejpam-5982	12	24	,	,	PUNCT
ejpam-5982	12	25	medical	medical	ADJ
ejpam-5982	12	26	sciences	science	NOUN
ejpam-5982	12	27	,	,	PUNCT
ejpam-5982	12	28	∗corresponding	∗corresponde	VERB
ejpam-5982	12	29	author	author	NOUN
ejpam-5982	12	30	.	.	PUNCT
ejpam-5982	13	1	doi	doi	NOUN
ejpam-5982	13	2	:	:	PUNCT
ejpam-5982	13	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5982	https://doi.org/10.29020/nybg.ejpam.v18i2.5982	PROPN
ejpam-5982	13	4	email	email	NOUN
ejpam-5982	13	5	addresses	address	VERB
ejpam-5982	13	6	:	:	PUNCT
ejpam-5982	14	1	ppmurthy@gmail.com	ppmurthy@gmail.com	X
ejpam-5982	14	2	(	(	PUNCT
ejpam-5982	14	3	pp	pp	ADP
ejpam-5982	14	4	murthy	murthy	ADJ
ejpam-5982	14	5	)	)	PUNCT
ejpam-5982	14	6	,	,	PUNCT
ejpam-5982	14	7	cpdhuri@gmail.com	cpdhuri@gmail.com	X
ejpam-5982	15	1	(	(	PUNCT
ejpam-5982	15	2	cp	cp	PROPN
ejpam-5982	15	3	dhuri	dhuri	PROPN
ejpam-5982	15	4	)	)	PUNCT
ejpam-5982	15	5	,	,	PUNCT
ejpam-5982	16	1	r.gopalan@psau.edu.sa	r.gopalan@psau.edu.sa	NOUN
ejpam-5982	16	2	(	(	PUNCT
ejpam-5982	16	3	r	r	NOUN
ejpam-5982	16	4	ramaswamy	ramaswamy	ADJ
ejpam-5982	16	5	)	)	PUNCT
ejpam-5982	16	6	,	,	PUNCT
ejpam-5982	16	7	drkhizar@gmail.com	drkhizar@gmail.com	X
ejpam-5982	16	8	(	(	PUNCT
ejpam-5982	16	9	kh	kh	PROPN
ejpam-5982	16	10	khan	khan	PROPN
ejpam-5982	16	11	)	)	PUNCT
ejpam-5982	16	12	,	,	PUNCT
ejpam-5982	16	13	o.abdelnaby@psau.edu.sa	o.abdelnaby@psau.edu.sa	PROPN
ejpam-5982	16	14	(	(	PUNCT
ejpam-5982	16	15	ola	ola	PROPN
ejpam-5982	16	16	a.a	a.a	PROPN
ejpam-5982	16	17	)	)	PUNCT
ejpam-5982	16	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5982	16	19	1	1	NUM
ejpam-5982	16	20	copyright	copyright	NOUN
ejpam-5982	16	21	:	:	PUNCT
ejpam-5982	17	1	©	©	PROPN
ejpam-5982	17	2	2025	2025	NUM
ejpam-5982	17	3	the	the	DET
ejpam-5982	17	4	author(s	author(s	NOUN
ejpam-5982	17	5	)	)	PUNCT
ejpam-5982	17	6	.	.	PUNCT
ejpam-5982	18	1	(	(	PUNCT
ejpam-5982	18	2	cc	cc	NOUN
ejpam-5982	18	3	by	by	ADP
ejpam-5982	18	4	-	-	PUNCT
ejpam-5982	18	5	nc	nc	PROPN
ejpam-5982	18	6	4.0	4.0	NUM
ejpam-5982	18	7	)	)	PUNCT
ejpam-5982	19	1	p.	p.	NOUN
ejpam-5982	19	2	p.	p.	NOUN
ejpam-5982	20	1	murthy	murthy	ADJ
ejpam-5982	21	1	et	et	PROPN
ejpam-5982	21	2	al	al	PROPN
ejpam-5982	21	3	.	.	PUNCT
ejpam-5982	21	4	/	/	SYM
ejpam-5982	21	5	eur	eur	PROPN
ejpam-5982	21	6	.	.	PUNCT
ejpam-5982	22	1	j.	j.	PROPN
ejpam-5982	22	2	pure	pure	PROPN
ejpam-5982	22	3	appl	appl	PROPN
ejpam-5982	22	4	.	.	PROPN
ejpam-5982	22	5	math	math	PROPN
ejpam-5982	22	6	,	,	PUNCT
ejpam-5982	22	7	18	18	NUM
ejpam-5982	22	8	(	(	PUNCT
ejpam-5982	22	9	2	2	NUM
ejpam-5982	22	10	)	)	PUNCT
ejpam-5982	22	11	(	(	PUNCT
ejpam-5982	22	12	2025	2025	NUM
ejpam-5982	22	13	)	)	PUNCT
ejpam-5982	22	14	,	,	PUNCT
ejpam-5982	22	15	5982	5982	NUM
ejpam-5982	22	16	2	2	NUM
ejpam-5982	22	17	of	of	ADP
ejpam-5982	22	18	17	17	NUM
ejpam-5982	22	19	and	and	CCONJ
ejpam-5982	22	20	computer	computer	NOUN
ejpam-5982	22	21	,	,	PUNCT
ejpam-5982	22	22	etc	etc	X
ejpam-5982	22	23	.	.	X
ejpam-5982	23	1	in	in	ADP
ejpam-5982	23	2	the	the	DET
ejpam-5982	23	3	sequel	sequel	NOUN
ejpam-5982	23	4	of	of	ADP
ejpam-5982	23	5	generalisation	generalisation	NOUN
ejpam-5982	23	6	of	of	ADP
ejpam-5982	23	7	contractive	contractive	ADJ
ejpam-5982	23	8	conditions	condition	NOUN
ejpam-5982	23	9	,	,	PUNCT
ejpam-5982	23	10	jungck	jungck	PROPN
ejpam-5982	23	11	proposed	propose	VERB
ejpam-5982	23	12	a	a	DET
ejpam-5982	23	13	very	very	ADV
ejpam-5982	23	14	different	different	ADJ
ejpam-5982	23	15	type	type	NOUN
ejpam-5982	23	16	of	of	ADP
ejpam-5982	23	17	generalization	generalization	NOUN
ejpam-5982	23	18	of	of	ADP
ejpam-5982	23	19	the	the	DET
ejpam-5982	23	20	contraction	contraction	NOUN
ejpam-5982	23	21	condition	condition	NOUN
ejpam-5982	23	22	introduced	introduce	VERB
ejpam-5982	23	23	by	by	ADP
ejpam-5982	23	24	banach	banach	NOUN
ejpam-5982	23	25	for	for	ADP
ejpam-5982	23	26	a	a	DET
ejpam-5982	23	27	pair	pair	NOUN
ejpam-5982	23	28	of	of	ADP
ejpam-5982	23	29	compatible	compatible	ADJ
ejpam-5982	23	30	maps	map	NOUN
ejpam-5982	23	31	in	in	ADP
ejpam-5982	23	32	metric	metric	ADJ
ejpam-5982	23	33	spaces	space	NOUN
ejpam-5982	23	34	.	.	PUNCT
ejpam-5982	24	1	(	(	PUNCT
ejpam-5982	24	2	see	see	VERB
ejpam-5982	24	3	[	[	X
ejpam-5982	24	4	2	2	NUM
ejpam-5982	24	5	]	]	NUM
ejpam-5982	24	6	)	)	PUNCT
ejpam-5982	24	7	.	.	PUNCT
ejpam-5982	25	1	the	the	DET
ejpam-5982	25	2	concept	concept	NOUN
ejpam-5982	25	3	of	of	ADP
ejpam-5982	25	4	compatible	compatible	ADJ
ejpam-5982	25	5	mappings	mapping	NOUN
ejpam-5982	25	6	of	of	ADP
ejpam-5982	25	7	type	type	NOUN
ejpam-5982	25	8	(	(	PUNCT
ejpam-5982	25	9	a	a	NOUN
ejpam-5982	25	10	)	)	PUNCT
ejpam-5982	25	11	was	be	AUX
ejpam-5982	25	12	described	describe	VERB
ejpam-5982	25	13	in	in	ADP
ejpam-5982	25	14	[	[	X
ejpam-5982	25	15	3	3	X
ejpam-5982	25	16	]	]	PUNCT
ejpam-5982	25	17	on	on	ADP
ejpam-5982	25	18	complete	complete	ADJ
ejpam-5982	25	19	metric	metric	ADJ
ejpam-5982	25	20	spaces	space	NOUN
ejpam-5982	25	21	.	.	PUNCT
ejpam-5982	26	1	the	the	DET
ejpam-5982	26	2	common	common	ADJ
ejpam-5982	26	3	fixed	fix	VERB
ejpam-5982	26	4	point	point	NOUN
ejpam-5982	26	5	theorems	theorem	NOUN
ejpam-5982	26	6	have	have	AUX
ejpam-5982	26	7	been	be	AUX
ejpam-5982	26	8	proved	prove	VERB
ejpam-5982	26	9	for	for	ADP
ejpam-5982	26	10	the	the	DET
ejpam-5982	26	11	compatible	compatible	ADJ
ejpam-5982	26	12	mappings	mapping	NOUN
ejpam-5982	26	13	of	of	ADP
ejpam-5982	26	14	type	type	NOUN
ejpam-5982	26	15	(	(	PUNCT
ejpam-5982	26	16	a	a	NOUN
ejpam-5982	26	17	)	)	PUNCT
ejpam-5982	26	18	.	.	PUNCT
ejpam-5982	27	1	this	this	DET
ejpam-5982	27	2	type	type	NOUN
ejpam-5982	27	3	(	(	PUNCT
ejpam-5982	27	4	a	a	NOUN
ejpam-5982	27	5	)	)	PUNCT
ejpam-5982	27	6	was	be	AUX
ejpam-5982	27	7	shown	show	VERB
ejpam-5982	27	8	to	to	PART
ejpam-5982	27	9	be	be	AUX
ejpam-5982	27	10	equivalent	equivalent	ADJ
ejpam-5982	27	11	to	to	ADP
ejpam-5982	27	12	the	the	DET
ejpam-5982	27	13	context	context	NOUN
ejpam-5982	27	14	of	of	ADP
ejpam-5982	27	15	compatible	compatible	ADJ
ejpam-5982	27	16	mappings	mapping	NOUN
ejpam-5982	27	17	defined	define	VERB
ejpam-5982	27	18	by	by	ADP
ejpam-5982	27	19	jungck	jungck	NOUN
ejpam-5982	27	20	,	,	PUNCT
ejpam-5982	27	21	with	with	ADP
ejpam-5982	27	22	some	some	DET
ejpam-5982	27	23	restrictions	restriction	NOUN
ejpam-5982	27	24	(	(	PUNCT
ejpam-5982	27	25	see	see	VERB
ejpam-5982	27	26	[	[	X
ejpam-5982	27	27	3	3	NUM
ejpam-5982	27	28	]	]	NUM
ejpam-5982	27	29	)	)	PUNCT
ejpam-5982	27	30	.	.	PUNCT
ejpam-5982	28	1	valerie	valerie	PROPN
ejpam-5982	28	2	popa	popa	PROPN
ejpam-5982	28	3	demonstrated	demonstrate	VERB
ejpam-5982	28	4	fixed	fix	VERB
ejpam-5982	28	5	point	point	NOUN
ejpam-5982	28	6	theorems	theorem	NOUN
ejpam-5982	28	7	for	for	ADP
ejpam-5982	28	8	compatible	compatible	ADJ
ejpam-5982	28	9	mappings	mapping	NOUN
ejpam-5982	28	10	satisfying	satisfy	VERB
ejpam-5982	28	11	an	an	DET
ejpam-5982	28	12	implicit	implicit	ADJ
ejpam-5982	28	13	relation	relation	NOUN
ejpam-5982	28	14	in	in	ADP
ejpam-5982	28	15	[	[	X
ejpam-5982	28	16	4	4	NUM
ejpam-5982	28	17	]	]	PUNCT
ejpam-5982	28	18	.	.	PUNCT
ejpam-5982	29	1	in	in	ADP
ejpam-5982	29	2	[	[	X
ejpam-5982	29	3	5	5	NUM
ejpam-5982	29	4	]	]	PUNCT
ejpam-5982	29	5	,	,	PUNCT
ejpam-5982	29	6	the	the	DET
ejpam-5982	29	7	(	(	PUNCT
ejpam-5982	29	8	e.a	e.a	PROPN
ejpam-5982	29	9	.	.	PROPN
ejpam-5982	29	10	)	)	PUNCT
ejpam-5982	29	11	property	property	NOUN
ejpam-5982	29	12	in	in	ADP
ejpam-5982	29	13	metric	metric	ADJ
ejpam-5982	29	14	space	space	NOUN
ejpam-5982	29	15	was	be	AUX
ejpam-5982	29	16	defined	define	VERB
ejpam-5982	29	17	,	,	PUNCT
ejpam-5982	29	18	which	which	PRON
ejpam-5982	29	19	generalizes	generalize	VERB
ejpam-5982	29	20	the	the	DET
ejpam-5982	29	21	concept	concept	NOUN
ejpam-5982	29	22	of	of	ADP
ejpam-5982	29	23	non	non	ADJ
ejpam-5982	29	24	-	-	ADJ
ejpam-5982	29	25	compatible	compatible	ADJ
ejpam-5982	29	26	mappings	mapping	NOUN
ejpam-5982	29	27	,	,	PUNCT
ejpam-5982	29	28	and	and	CCONJ
ejpam-5982	29	29	some	some	DET
ejpam-5982	29	30	common	common	ADJ
ejpam-5982	29	31	fixed	fix	VERB
ejpam-5982	29	32	point	point	NOUN
ejpam-5982	29	33	theorems	theorem	NOUN
ejpam-5982	29	34	were	be	AUX
ejpam-5982	29	35	proved	prove	VERB
ejpam-5982	29	36	.	.	PUNCT
ejpam-5982	30	1	later	later	ADV
ejpam-5982	30	2	,	,	PUNCT
ejpam-5982	30	3	in	in	ADP
ejpam-5982	30	4	2016	2016	NUM
ejpam-5982	30	5	mutlu	mutlu	NOUN
ejpam-5982	30	6	and	and	CCONJ
ejpam-5982	30	7	gürdal	gürdal	VERB
ejpam-5982	30	8	[	[	X
ejpam-5982	30	9	6	6	NUM
ejpam-5982	30	10	]	]	PUNCT
ejpam-5982	30	11	introduced	introduce	VERB
ejpam-5982	30	12	the	the	DET
ejpam-5982	30	13	concept	concept	NOUN
ejpam-5982	30	14	of	of	ADP
ejpam-5982	30	15	a	a	DET
ejpam-5982	30	16	bipolar	bipolar	ADJ
ejpam-5982	30	17	metric	metric	ADJ
ejpam-5982	30	18	space	space	NOUN
ejpam-5982	30	19	which	which	PRON
ejpam-5982	30	20	is	be	AUX
ejpam-5982	30	21	the	the	DET
ejpam-5982	30	22	generalization	generalization	NOUN
ejpam-5982	30	23	of	of	ADP
ejpam-5982	30	24	a	a	DET
ejpam-5982	30	25	metric	metric	ADJ
ejpam-5982	30	26	space	space	NOUN
ejpam-5982	30	27	.	.	PUNCT
ejpam-5982	31	1	they	they	PRON
ejpam-5982	31	2	have	have	AUX
ejpam-5982	31	3	proved	prove	VERB
ejpam-5982	31	4	some	some	DET
ejpam-5982	31	5	generalizations	generalization	NOUN
ejpam-5982	31	6	of	of	ADP
ejpam-5982	31	7	banach	banach	ADV
ejpam-5982	31	8	fixed	fix	VERB
ejpam-5982	31	9	point	point	NOUN
ejpam-5982	31	10	theorem	theorem	VERB
ejpam-5982	31	11	[	[	X
ejpam-5982	31	12	1	1	NUM
ejpam-5982	31	13	]	]	PUNCT
ejpam-5982	31	14	in	in	ADP
ejpam-5982	31	15	bipolar	bipolar	ADJ
ejpam-5982	31	16	metric	metric	ADJ
ejpam-5982	31	17	spaces	space	NOUN
ejpam-5982	31	18	.	.	PUNCT
ejpam-5982	32	1	given	give	VERB
ejpam-5982	32	2	the	the	DET
ejpam-5982	32	3	theorem	theorem	NOUN
ejpam-5982	32	4	proved	prove	VERB
ejpam-5982	32	5	herein	herein	NOUN
ejpam-5982	32	6	,	,	PUNCT
ejpam-5982	32	7	it	it	PRON
ejpam-5982	32	8	is	be	AUX
ejpam-5982	32	9	highly	highly	ADV
ejpam-5982	32	10	demanded	demand	VERB
ejpam-5982	32	11	to	to	PART
ejpam-5982	32	12	recall	recall	VERB
ejpam-5982	32	13	the	the	DET
ejpam-5982	32	14	most	most	ADV
ejpam-5982	32	15	basic	basic	ADJ
ejpam-5982	32	16	definitions	definition	NOUN
ejpam-5982	32	17	and	and	CCONJ
ejpam-5982	32	18	properties	property	NOUN
ejpam-5982	32	19	in	in	ADP
ejpam-5982	32	20	bipolar	bipolar	ADJ
ejpam-5982	32	21	metric	metric	ADJ
ejpam-5982	32	22	spaces	space	NOUN
ejpam-5982	32	23	.	.	PUNCT
ejpam-5982	33	1	subsequently	subsequently	ADV
ejpam-5982	33	2	,	,	PUNCT
ejpam-5982	33	3	in	in	ADP
ejpam-5982	33	4	the	the	DET
ejpam-5982	33	5	recent	recent	ADJ
ejpam-5982	33	6	past	past	NOUN
ejpam-5982	33	7	,	,	PUNCT
ejpam-5982	33	8	various	various	ADJ
ejpam-5982	33	9	authors	author	NOUN
ejpam-5982	33	10	have	have	AUX
ejpam-5982	33	11	reported	report	VERB
ejpam-5982	33	12	fixed	fix	VERB
ejpam-5982	33	13	point	point	NOUN
ejpam-5982	33	14	results	result	NOUN
ejpam-5982	33	15	in	in	ADP
ejpam-5982	33	16	the	the	DET
ejpam-5982	33	17	setting	setting	NOUN
ejpam-5982	33	18	of	of	ADP
ejpam-5982	33	19	bipolar	bipolar	ADJ
ejpam-5982	33	20	metric	metric	ADJ
ejpam-5982	33	21	spaces	space	NOUN
ejpam-5982	33	22	using	use	VERB
ejpam-5982	33	23	various	various	ADJ
ejpam-5982	33	24	contractive	contractive	ADJ
ejpam-5982	33	25	conditions	condition	NOUN
ejpam-5982	33	26	.	.	PUNCT
ejpam-5982	34	1	for	for	SCONJ
ejpam-5982	34	2	more	more	ADJ
ejpam-5982	34	3	details	detail	NOUN
ejpam-5982	34	4	,	,	PUNCT
ejpam-5982	34	5	[	[	X
ejpam-5982	34	6	7–16	7–16	NOUN
ejpam-5982	34	7	]	]	PUNCT
ejpam-5982	34	8	.	.	PUNCT
ejpam-5982	34	9	inspired	inspire	VERB
ejpam-5982	34	10	,	,	PUNCT
ejpam-5982	34	11	in	in	ADP
ejpam-5982	34	12	this	this	DET
ejpam-5982	34	13	article	article	NOUN
ejpam-5982	34	14	we	we	PRON
ejpam-5982	34	15	aim	aim	VERB
ejpam-5982	34	16	to	to	PART
ejpam-5982	34	17	prove	prove	VERB
ejpam-5982	34	18	common	common	ADJ
ejpam-5982	34	19	fixed	fix	VERB
ejpam-5982	34	20	point	point	NOUN
ejpam-5982	34	21	theorems	theorem	NOUN
ejpam-5982	34	22	with	with	ADP
ejpam-5982	34	23	the	the	DET
ejpam-5982	34	24	concepts	concept	NOUN
ejpam-5982	34	25	of	of	ADP
ejpam-5982	34	26	maps	map	NOUN
ejpam-5982	34	27	of	of	ADP
ejpam-5982	34	28	type	type	NOUN
ejpam-5982	34	29	(	(	PUNCT
ejpam-5982	34	30	a	a	NOUN
ejpam-5982	34	31	)	)	PUNCT
ejpam-5982	34	32	and	and	CCONJ
ejpam-5982	34	33	property	property	NOUN
ejpam-5982	34	34	(	(	PUNCT
ejpam-5982	34	35	e.a	e.a	PROPN
ejpam-5982	34	36	)	)	PUNCT
ejpam-5982	34	37	,	,	PUNCT
ejpam-5982	34	38	and	and	CCONJ
ejpam-5982	34	39	implicit	implicit	ADJ
ejpam-5982	34	40	relations	relation	NOUN
ejpam-5982	34	41	in	in	ADP
ejpam-5982	34	42	bipolar	bipolar	ADJ
ejpam-5982	34	43	metric	metric	ADJ
ejpam-5982	34	44	spaces	space	NOUN
ejpam-5982	34	45	,	,	PUNCT
ejpam-5982	34	46	which	which	PRON
ejpam-5982	34	47	generalizes	generalize	VERB
ejpam-5982	34	48	some	some	DET
ejpam-5982	34	49	famous	famous	ADJ
ejpam-5982	34	50	well	well	ADV
ejpam-5982	34	51	-	-	PUNCT
ejpam-5982	34	52	known	know	VERB
ejpam-5982	34	53	results	result	NOUN
ejpam-5982	34	54	,	,	PUNCT
ejpam-5982	34	55	namely	namely	ADV
ejpam-5982	34	56	kannan	kannan	PROPN
ejpam-5982	35	1	[	[	X
ejpam-5982	35	2	17	17	NUM
ejpam-5982	35	3	]	]	PUNCT
ejpam-5982	35	4	,	,	PUNCT
ejpam-5982	35	5	reich	reich	PROPN
ejpam-5982	36	1	[	[	X
ejpam-5982	36	2	18	18	NUM
ejpam-5982	36	3	]	]	PUNCT
ejpam-5982	36	4	and	and	CCONJ
ejpam-5982	36	5	gaba	gaba	PROPN
ejpam-5982	37	1	[	[	X
ejpam-5982	37	2	16	16	NUM
ejpam-5982	37	3	]	]	PUNCT
ejpam-5982	37	4	.	.	PUNCT
ejpam-5982	38	1	the	the	DET
ejpam-5982	38	2	rest	rest	NOUN
ejpam-5982	38	3	of	of	ADP
ejpam-5982	38	4	the	the	DET
ejpam-5982	38	5	paper	paper	NOUN
ejpam-5982	38	6	is	be	AUX
ejpam-5982	38	7	organized	organize	VERB
ejpam-5982	38	8	as	as	SCONJ
ejpam-5982	38	9	follows	follow	VERB
ejpam-5982	38	10	:	:	PUNCT
ejpam-5982	38	11	in	in	ADP
ejpam-5982	38	12	section-2	section-2	NUM
ejpam-5982	38	13	,	,	PUNCT
ejpam-5982	38	14	we	we	PRON
ejpam-5982	38	15	review	review	VERB
ejpam-5982	38	16	some	some	DET
ejpam-5982	38	17	basic	basic	ADJ
ejpam-5982	38	18	preliminaries	preliminary	NOUN
ejpam-5982	38	19	and	and	CCONJ
ejpam-5982	38	20	monograph	monograph	NOUN
ejpam-5982	38	21	.	.	PUNCT
ejpam-5982	39	1	we	we	PRON
ejpam-5982	39	2	present	present	VERB
ejpam-5982	39	3	our	our	PRON
ejpam-5982	39	4	main	main	ADJ
ejpam-5982	39	5	results	result	NOUN
ejpam-5982	39	6	in	in	ADP
ejpam-5982	39	7	section-3	section-3	PROPN
ejpam-5982	39	8	,	,	PUNCT
ejpam-5982	39	9	establishing	establish	VERB
ejpam-5982	39	10	the	the	DET
ejpam-5982	39	11	fixed	fix	VERB
ejpam-5982	39	12	point	point	NOUN
ejpam-5982	39	13	results	result	NOUN
ejpam-5982	39	14	and	and	CCONJ
ejpam-5982	39	15	generalising	generalise	VERB
ejpam-5982	39	16	various	various	ADJ
ejpam-5982	39	17	fixed	fix	VERB
ejpam-5982	39	18	point	point	NOUN
ejpam-5982	39	19	results	result	NOUN
ejpam-5982	39	20	proven	prove	VERB
ejpam-5982	39	21	in	in	ADP
ejpam-5982	39	22	the	the	DET
ejpam-5982	39	23	past	past	NOUN
ejpam-5982	39	24	.	.	PUNCT
ejpam-5982	40	1	the	the	DET
ejpam-5982	40	2	derived	derive	VERB
ejpam-5982	40	3	results	result	NOUN
ejpam-5982	40	4	have	have	AUX
ejpam-5982	40	5	been	be	AUX
ejpam-5982	40	6	supported	support	VERB
ejpam-5982	40	7	with	with	ADP
ejpam-5982	40	8	suitable	suitable	ADJ
ejpam-5982	40	9	examples	example	NOUN
ejpam-5982	40	10	.	.	PUNCT
ejpam-5982	41	1	finally	finally	ADV
ejpam-5982	41	2	we	we	PRON
ejpam-5982	41	3	conclude	conclude	VERB
ejpam-5982	41	4	the	the	DET
ejpam-5982	41	5	manuscript	manuscript	NOUN
ejpam-5982	41	6	presenting	present	VERB
ejpam-5982	41	7	the	the	DET
ejpam-5982	41	8	scope	scope	NOUN
ejpam-5982	41	9	for	for	ADP
ejpam-5982	41	10	further	further	ADJ
ejpam-5982	41	11	research	research	NOUN
ejpam-5982	41	12	by	by	ADP
ejpam-5982	41	13	presenting	present	VERB
ejpam-5982	41	14	some	some	DET
ejpam-5982	41	15	open	open	ADJ
ejpam-5982	41	16	problems	problem	NOUN
ejpam-5982	41	17	for	for	ADP
ejpam-5982	41	18	future	future	ADJ
ejpam-5982	41	19	research	research	NOUN
ejpam-5982	41	20	.	.	PUNCT
ejpam-5982	42	1	2	2	X
ejpam-5982	42	2	.	.	NUM
ejpam-5982	42	3	preliminaries	preliminary	NOUN
ejpam-5982	42	4	the	the	DET
ejpam-5982	42	5	following	follow	VERB
ejpam-5982	42	6	are	be	AUX
ejpam-5982	42	7	required	require	VERB
ejpam-5982	42	8	in	in	ADP
ejpam-5982	42	9	the	the	DET
ejpam-5982	42	10	sequel	sequel	NOUN
ejpam-5982	42	11	.	.	PUNCT
ejpam-5982	43	1	definition	definition	NOUN
ejpam-5982	43	2	1	1	NUM
ejpam-5982	43	3	.	.	PUNCT
ejpam-5982	44	1	[	[	X
ejpam-5982	44	2	6	6	NUM
ejpam-5982	44	3	]	]	PUNCT
ejpam-5982	44	4	let	let	VERB
ejpam-5982	44	5	x	x	PRON
ejpam-5982	44	6	and	and	CCONJ
ejpam-5982	44	7	y	y	PROPN
ejpam-5982	44	8	be	be	AUX
ejpam-5982	44	9	two	two	NUM
ejpam-5982	44	10	non	non	ADJ
ejpam-5982	44	11	nonempty	nonempty	ADJ
ejpam-5982	44	12	sets	set	NOUN
ejpam-5982	44	13	and	and	CCONJ
ejpam-5982	44	14	d	d	NOUN
ejpam-5982	44	15	:	:	PUNCT
ejpam-5982	44	16	x	x	SYM
ejpam-5982	44	17	×	×	NOUN
ejpam-5982	44	18	y	y	X
ejpam-5982	44	19	→	→	PUNCT
ejpam-5982	44	20	[	[	X
ejpam-5982	44	21	0,+∞	0,+∞	NUM
ejpam-5982	44	22	)	)	PUNCT
ejpam-5982	44	23	be	be	AUX
ejpam-5982	44	24	a	a	DET
ejpam-5982	44	25	function	function	NOUN
ejpam-5982	44	26	.	.	PUNCT
ejpam-5982	45	1	then	then	ADV
ejpam-5982	45	2	the	the	DET
ejpam-5982	45	3	triplet	triplet	NOUN
ejpam-5982	45	4	(	(	PUNCT
ejpam-5982	45	5	x	x	NOUN
ejpam-5982	45	6	,	,	PUNCT
ejpam-5982	45	7	y	y	PROPN
ejpam-5982	45	8	,	,	PUNCT
ejpam-5982	45	9	d	d	NOUN
ejpam-5982	45	10	)	)	PUNCT
ejpam-5982	45	11	is	be	AUX
ejpam-5982	45	12	called	call	VERB
ejpam-5982	45	13	bipolar	bipolar	ADJ
ejpam-5982	45	14	metric	metric	ADJ
ejpam-5982	45	15	space	space	NOUN
ejpam-5982	45	16	and	and	CCONJ
ejpam-5982	45	17	d	d	NOUN
ejpam-5982	45	18	is	be	AUX
ejpam-5982	45	19	called	call	VERB
ejpam-5982	45	20	bipolar	bipolar	ADJ
ejpam-5982	46	1	p.	p.	NOUN
ejpam-5982	46	2	p.	p.	NOUN
ejpam-5982	46	3	murthy	murthy	PROPN
ejpam-5982	47	1	et	et	PROPN
ejpam-5982	47	2	al	al	PROPN
ejpam-5982	47	3	.	.	PUNCT
ejpam-5982	47	4	/	/	SYM
ejpam-5982	47	5	eur	eur	PROPN
ejpam-5982	47	6	.	.	PUNCT
ejpam-5982	48	1	j.	j.	PROPN
ejpam-5982	48	2	pure	pure	PROPN
ejpam-5982	48	3	appl	appl	PROPN
ejpam-5982	48	4	.	.	PROPN
ejpam-5982	48	5	math	math	PROPN
ejpam-5982	48	6	,	,	PUNCT
ejpam-5982	48	7	18	18	NUM
ejpam-5982	48	8	(	(	PUNCT
ejpam-5982	48	9	2	2	NUM
ejpam-5982	48	10	)	)	PUNCT
ejpam-5982	48	11	(	(	PUNCT
ejpam-5982	48	12	2025	2025	NUM
ejpam-5982	48	13	)	)	PUNCT
ejpam-5982	48	14	,	,	PUNCT
ejpam-5982	48	15	5982	5982	NUM
ejpam-5982	48	16	3	3	NUM
ejpam-5982	48	17	of	of	ADP
ejpam-5982	48	18	17	17	NUM
ejpam-5982	48	19	metric	metric	ADJ
ejpam-5982	48	20	on	on	ADP
ejpam-5982	48	21	(	(	PUNCT
ejpam-5982	48	22	x	x	X
ejpam-5982	48	23	,	,	PUNCT
ejpam-5982	48	24	y	y	PROPN
ejpam-5982	48	25	)	)	PUNCT
ejpam-5982	48	26	,	,	PUNCT
ejpam-5982	48	27	if	if	SCONJ
ejpam-5982	48	28	the	the	DET
ejpam-5982	48	29	following	follow	VERB
ejpam-5982	48	30	conditions	condition	NOUN
ejpam-5982	48	31	holds	hold	VERB
ejpam-5982	48	32	:	:	PUNCT
ejpam-5982	48	33	(	(	PUNCT
ejpam-5982	48	34	bp1	bp1	PROPN
ejpam-5982	48	35	)	)	PUNCT
ejpam-5982	48	36	d(x	d(x	PROPN
ejpam-5982	48	37	,	,	PUNCT
ejpam-5982	48	38	y	y	NOUN
ejpam-5982	48	39	)	)	PUNCT
ejpam-5982	48	40	=	=	SYM
ejpam-5982	48	41	0	0	PUNCT
ejpam-5982	49	1	if	if	SCONJ
ejpam-5982	49	2	and	and	CCONJ
ejpam-5982	49	3	only	only	ADV
ejpam-5982	49	4	if	if	SCONJ
ejpam-5982	49	5	x	x	X
ejpam-5982	49	6	=	=	PUNCT
ejpam-5982	49	7	y	y	PROPN
ejpam-5982	49	8	where	where	SCONJ
ejpam-5982	49	9	(	(	PUNCT
ejpam-5982	49	10	x	x	NOUN
ejpam-5982	49	11	,	,	PUNCT
ejpam-5982	49	12	y	y	NOUN
ejpam-5982	49	13	)	)	PUNCT
ejpam-5982	49	14	∈	∈	PROPN
ejpam-5982	49	15	x	x	SYM
ejpam-5982	49	16	×	×	PROPN
ejpam-5982	49	17	y	y	PROPN
ejpam-5982	49	18	,	,	PUNCT
ejpam-5982	49	19	(	(	PUNCT
ejpam-5982	49	20	bp2	bp2	PROPN
ejpam-5982	49	21	)	)	PUNCT
ejpam-5982	49	22	if	if	SCONJ
ejpam-5982	49	23	x	x	X
ejpam-5982	49	24	,	,	PUNCT
ejpam-5982	49	25	y	y	PROPN
ejpam-5982	49	26	∈	∈	PROPN
ejpam-5982	49	27	x	x	PUNCT
ejpam-5982	49	28	∩	∩	PROPN
ejpam-5982	49	29	y	y	PROPN
ejpam-5982	49	30	then	then	ADV
ejpam-5982	49	31	d(x	d(x	PROPN
ejpam-5982	49	32	,	,	PUNCT
ejpam-5982	49	33	y	y	NOUN
ejpam-5982	49	34	)	)	PUNCT
ejpam-5982	49	35	=	=	SYM
ejpam-5982	49	36	d(y	d(y	NOUN
ejpam-5982	49	37	,	,	PUNCT
ejpam-5982	49	38	x	x	NOUN
ejpam-5982	49	39	)	)	PUNCT
ejpam-5982	49	40	,	,	PUNCT
ejpam-5982	49	41	(	(	PUNCT
ejpam-5982	49	42	bp3	bp3	NOUN
ejpam-5982	49	43	)	)	PUNCT
ejpam-5982	49	44	d(x1	d(x1	NOUN
ejpam-5982	49	45	,	,	PUNCT
ejpam-5982	49	46	y2	y2	NOUN
ejpam-5982	49	47	)	)	PUNCT
ejpam-5982	49	48	≤	≤	NOUN
ejpam-5982	49	49	d(x1	d(x1	NOUN
ejpam-5982	49	50	,	,	PUNCT
ejpam-5982	49	51	y1	y1	NOUN
ejpam-5982	49	52	)	)	PUNCT
ejpam-5982	49	53	+	+	NUM
ejpam-5982	49	54	d(x2	d(x2	NOUN
ejpam-5982	49	55	,	,	PUNCT
ejpam-5982	49	56	y1	y1	PROPN
ejpam-5982	49	57	)	)	PUNCT
ejpam-5982	49	58	+	+	NUM
ejpam-5982	49	59	d(x2	d(x2	NOUN
ejpam-5982	49	60	,	,	PUNCT
ejpam-5982	49	61	y2	y2	PROPN
ejpam-5982	49	62	)	)	PUNCT
ejpam-5982	49	63	for	for	ADP
ejpam-5982	49	64	all	all	DET
ejpam-5982	49	65	x1	x1	PROPN
ejpam-5982	49	66	,	,	PUNCT
ejpam-5982	49	67	x2	x2	PROPN
ejpam-5982	49	68	∈	∈	PROPN
ejpam-5982	49	69	x	x	X
ejpam-5982	49	70	and	and	CCONJ
ejpam-5982	49	71	y1	y1	PROPN
ejpam-5982	49	72	,	,	PUNCT
ejpam-5982	49	73	y2	y2	PROPN
ejpam-5982	49	74	∈	∈	PROPN
ejpam-5982	49	75	y	y	PROPN
ejpam-5982	49	76	.	.	PUNCT
ejpam-5982	50	1	definition	definition	NOUN
ejpam-5982	50	2	2	2	NUM
ejpam-5982	50	3	.	.	PUNCT
ejpam-5982	51	1	[	[	X
ejpam-5982	51	2	6	6	NUM
ejpam-5982	51	3	]	]	PUNCT
ejpam-5982	51	4	let	let	VERB
ejpam-5982	51	5	(	(	PUNCT
ejpam-5982	51	6	x	x	NOUN
ejpam-5982	51	7	,	,	PUNCT
ejpam-5982	51	8	y	y	PROPN
ejpam-5982	51	9	,	,	PUNCT
ejpam-5982	51	10	d	d	NOUN
ejpam-5982	51	11	)	)	PUNCT
ejpam-5982	51	12	be	be	AUX
ejpam-5982	51	13	a	a	DET
ejpam-5982	51	14	bipolar	bipolar	ADJ
ejpam-5982	51	15	metric	metric	ADJ
ejpam-5982	51	16	space	space	NOUN
ejpam-5982	51	17	.	.	PUNCT
ejpam-5982	52	1	a	a	DET
ejpam-5982	52	2	sequence	sequence	NOUN
ejpam-5982	52	3	{	{	PUNCT
ejpam-5982	52	4	un	un	PROPN
ejpam-5982	52	5	}	}	PUNCT
ejpam-5982	52	6	is	be	AUX
ejpam-5982	52	7	said	say	VERB
ejpam-5982	52	8	to	to	PART
ejpam-5982	52	9	be	be	AUX
ejpam-5982	52	10	convergent	convergent	ADJ
ejpam-5982	52	11	to	to	ADP
ejpam-5982	52	12	a	a	DET
ejpam-5982	52	13	point	point	NOUN
ejpam-5982	52	14	t	t	NOUN
ejpam-5982	52	15	if	if	SCONJ
ejpam-5982	52	16	and	and	CCONJ
ejpam-5982	52	17	only	only	ADV
ejpam-5982	52	18	if	if	SCONJ
ejpam-5982	52	19	{	{	PUNCT
ejpam-5982	52	20	un	un	ADJ
ejpam-5982	52	21	}	}	PUNCT
ejpam-5982	52	22	is	be	AUX
ejpam-5982	52	23	a	a	DET
ejpam-5982	52	24	sequence	sequence	NOUN
ejpam-5982	52	25	in	in	ADP
ejpam-5982	52	26	x	x	PROPN
ejpam-5982	52	27	,	,	PUNCT
ejpam-5982	52	28	t	t	PROPN
ejpam-5982	52	29	is	be	AUX
ejpam-5982	52	30	a	a	DET
ejpam-5982	52	31	point	point	NOUN
ejpam-5982	52	32	in	in	ADP
ejpam-5982	52	33	y	y	PROPN
ejpam-5982	52	34	and	and	CCONJ
ejpam-5982	52	35	lim	lim	PROPN
ejpam-5982	52	36	n→+∞	n→+∞	PROPN
ejpam-5982	52	37	d(un	d(un	PROPN
ejpam-5982	52	38	,	,	PUNCT
ejpam-5982	52	39	t	t	PROPN
ejpam-5982	52	40	)	)	PUNCT
ejpam-5982	52	41	=	=	SYM
ejpam-5982	52	42	0	0	NUM
ejpam-5982	52	43	;	;	PUNCT
ejpam-5982	52	44	or	or	CCONJ
ejpam-5982	52	45	{	{	PUNCT
ejpam-5982	52	46	un	un	PROPN
ejpam-5982	52	47	}	}	PUNCT
ejpam-5982	52	48	is	be	AUX
ejpam-5982	52	49	a	a	DET
ejpam-5982	52	50	sequence	sequence	NOUN
ejpam-5982	52	51	in	in	ADP
ejpam-5982	52	52	y	y	PROPN
ejpam-5982	52	53	,	,	PUNCT
ejpam-5982	52	54	t	t	PROPN
ejpam-5982	52	55	is	be	AUX
ejpam-5982	52	56	a	a	DET
ejpam-5982	52	57	point	point	NOUN
ejpam-5982	52	58	in	in	ADP
ejpam-5982	52	59	x	x	PUNCT
ejpam-5982	52	60	and	and	CCONJ
ejpam-5982	52	61	lim	lim	PROPN
ejpam-5982	52	62	n→+∞	n→+∞	PROPN
ejpam-5982	52	63	d(t	d(t	PROPN
ejpam-5982	52	64	,	,	PUNCT
ejpam-5982	52	65	un	un	PROPN
ejpam-5982	52	66	)	)	PUNCT
ejpam-5982	52	67	=	=	SYM
ejpam-5982	53	1	0	0	X
ejpam-5982	53	2	.	.	PUNCT
ejpam-5982	54	1	a	a	DET
ejpam-5982	54	2	sequence	sequence	NOUN
ejpam-5982	54	3	{	{	PUNCT
ejpam-5982	54	4	(	(	PUNCT
ejpam-5982	54	5	xn	xn	PROPN
ejpam-5982	54	6	,	,	PUNCT
ejpam-5982	54	7	yn	yn	PROPN
ejpam-5982	54	8	)	)	PUNCT
ejpam-5982	54	9	}	}	PUNCT
ejpam-5982	54	10	in	in	ADP
ejpam-5982	54	11	x	x	SYM
ejpam-5982	54	12	×	×	PROPN
ejpam-5982	54	13	y	y	PROPN
ejpam-5982	54	14	is	be	AUX
ejpam-5982	54	15	called	call	VERB
ejpam-5982	54	16	a	a	DET
ejpam-5982	54	17	bisequence	bisequence	NOUN
ejpam-5982	54	18	on	on	ADP
ejpam-5982	54	19	(	(	PUNCT
ejpam-5982	54	20	x	x	X
ejpam-5982	54	21	,	,	PUNCT
ejpam-5982	54	22	y	y	PROPN
ejpam-5982	54	23	)	)	PUNCT
ejpam-5982	54	24	.	.	PUNCT
ejpam-5982	55	1	this	this	DET
ejpam-5982	55	2	sequence	sequence	NOUN
ejpam-5982	55	3	is	be	AUX
ejpam-5982	55	4	simply	simply	ADV
ejpam-5982	55	5	denoted	denote	VERB
ejpam-5982	55	6	by	by	ADP
ejpam-5982	55	7	(	(	PUNCT
ejpam-5982	55	8	xn	xn	PROPN
ejpam-5982	55	9	,	,	PUNCT
ejpam-5982	55	10	yn	yn	PROPN
ejpam-5982	55	11	)	)	PUNCT
ejpam-5982	55	12	.	.	PUNCT
ejpam-5982	56	1	if	if	SCONJ
ejpam-5982	56	2	both	both	PRON
ejpam-5982	56	3	the	the	DET
ejpam-5982	56	4	sequences	sequence	NOUN
ejpam-5982	56	5	{	{	PUNCT
ejpam-5982	56	6	xn	xn	NUM
ejpam-5982	56	7	}	}	PUNCT
ejpam-5982	56	8	and	and	CCONJ
ejpam-5982	56	9	{	{	PUNCT
ejpam-5982	56	10	yn	yn	NOUN
ejpam-5982	56	11	}	}	PUNCT
ejpam-5982	56	12	converge	converge	VERB
ejpam-5982	56	13	,	,	PUNCT
ejpam-5982	56	14	then	then	ADV
ejpam-5982	56	15	the	the	DET
ejpam-5982	56	16	bisequence	bisequence	NOUN
ejpam-5982	56	17	(	(	PUNCT
ejpam-5982	56	18	xn	xn	PROPN
ejpam-5982	56	19	,	,	PUNCT
ejpam-5982	56	20	yn	yn	PROPN
ejpam-5982	56	21	)	)	PUNCT
ejpam-5982	56	22	is	be	AUX
ejpam-5982	56	23	said	say	VERB
ejpam-5982	56	24	to	to	PART
ejpam-5982	56	25	be	be	AUX
ejpam-5982	56	26	convergent	convergent	ADJ
ejpam-5982	56	27	.	.	PUNCT
ejpam-5982	57	1	if	if	SCONJ
ejpam-5982	57	2	both	both	PRON
ejpam-5982	57	3	the	the	DET
ejpam-5982	57	4	sequences	sequence	NOUN
ejpam-5982	57	5	{	{	PUNCT
ejpam-5982	57	6	xn	xn	NUM
ejpam-5982	57	7	}	}	PUNCT
ejpam-5982	57	8	and	and	CCONJ
ejpam-5982	57	9	{	{	PUNCT
ejpam-5982	57	10	yn	yn	NOUN
ejpam-5982	57	11	}	}	PUNCT
ejpam-5982	57	12	converge	converge	VERB
ejpam-5982	57	13	to	to	ADP
ejpam-5982	57	14	a	a	DET
ejpam-5982	57	15	same	same	ADJ
ejpam-5982	57	16	point	point	NOUN
ejpam-5982	57	17	u	u	NOUN
ejpam-5982	57	18	∈	∈	PROPN
ejpam-5982	57	19	x	x	SYM
ejpam-5982	57	20	∩	∩	PROPN
ejpam-5982	57	21	y	y	PROPN
ejpam-5982	57	22	then	then	ADV
ejpam-5982	57	23	(	(	PUNCT
ejpam-5982	57	24	xn	xn	PROPN
ejpam-5982	57	25	,	,	PUNCT
ejpam-5982	57	26	yn	yn	PROPN
ejpam-5982	57	27	)	)	PUNCT
ejpam-5982	57	28	is	be	AUX
ejpam-5982	57	29	called	call	VERB
ejpam-5982	57	30	biconvergent	biconvergent	NOUN
ejpam-5982	57	31	.	.	PUNCT
ejpam-5982	58	1	if	if	SCONJ
ejpam-5982	58	2	lim	lim	PROPN
ejpam-5982	58	3	n	n	CCONJ
ejpam-5982	58	4	,	,	PUNCT
ejpam-5982	58	5	m→+∞	m→+∞	PROPN
ejpam-5982	58	6	d(xn	d(xn	PROPN
ejpam-5982	58	7	,	,	PUNCT
ejpam-5982	58	8	ym	ym	NOUN
ejpam-5982	58	9	)	)	PUNCT
ejpam-5982	58	10	=	=	SYM
ejpam-5982	58	11	0	0	PUNCT
ejpam-5982	59	1	then	then	ADV
ejpam-5982	59	2	the	the	DET
ejpam-5982	59	3	bisequence	bisequence	NOUN
ejpam-5982	59	4	(	(	PUNCT
ejpam-5982	59	5	xn	xn	PROPN
ejpam-5982	59	6	,	,	PUNCT
ejpam-5982	59	7	yn	yn	PROPN
ejpam-5982	59	8	)	)	PUNCT
ejpam-5982	59	9	is	be	AUX
ejpam-5982	59	10	called	call	VERB
ejpam-5982	59	11	a	a	DET
ejpam-5982	59	12	cauchy	cauchy	ADJ
ejpam-5982	59	13	bisequence	bisequence	NOUN
ejpam-5982	59	14	.	.	PUNCT
ejpam-5982	60	1	in	in	ADP
ejpam-5982	60	2	a	a	DET
ejpam-5982	60	3	bipolar	bipolar	ADJ
ejpam-5982	60	4	metric	metric	ADJ
ejpam-5982	60	5	space	space	NOUN
ejpam-5982	60	6	,	,	PUNCT
ejpam-5982	60	7	every	every	DET
ejpam-5982	60	8	convergent	convergent	NOUN
ejpam-5982	60	9	cauchy	cauchy	ADJ
ejpam-5982	60	10	bisequence	bisequence	NOUN
ejpam-5982	60	11	is	be	AUX
ejpam-5982	60	12	biconvergent	biconvergent	NOUN
ejpam-5982	60	13	.	.	PUNCT
ejpam-5982	61	1	a	a	DET
ejpam-5982	61	2	bipolar	bipolar	ADJ
ejpam-5982	61	3	metric	metric	ADJ
ejpam-5982	61	4	space	space	NOUN
ejpam-5982	61	5	is	be	AUX
ejpam-5982	61	6	called	call	VERB
ejpam-5982	61	7	complete	complete	ADJ
ejpam-5982	61	8	,	,	PUNCT
ejpam-5982	61	9	if	if	SCONJ
ejpam-5982	61	10	every	every	DET
ejpam-5982	61	11	cauchy	cauchy	ADJ
ejpam-5982	61	12	bisequence	bisequence	NOUN
ejpam-5982	61	13	is	be	AUX
ejpam-5982	61	14	convergent	convergent	NOUN
ejpam-5982	61	15	,	,	PUNCT
ejpam-5982	61	16	hence	hence	ADV
ejpam-5982	61	17	biconvergent	biconvergent	NOUN
ejpam-5982	61	18	.	.	PUNCT
ejpam-5982	62	1	remark	remark	PROPN
ejpam-5982	62	2	1	1	NUM
ejpam-5982	62	3	.	.	PUNCT
ejpam-5982	63	1	limit	limit	NOUN
ejpam-5982	63	2	of	of	ADP
ejpam-5982	63	3	a	a	DET
ejpam-5982	63	4	convergent	convergent	NOUN
ejpam-5982	63	5	sequence	sequence	NOUN
ejpam-5982	63	6	in	in	ADP
ejpam-5982	63	7	a	a	DET
ejpam-5982	63	8	bipolar	bipolar	ADJ
ejpam-5982	63	9	metric	metric	ADJ
ejpam-5982	63	10	space	space	NOUN
ejpam-5982	63	11	need	need	AUX
ejpam-5982	63	12	not	not	PART
ejpam-5982	63	13	be	be	AUX
ejpam-5982	63	14	unique	unique	ADJ
ejpam-5982	63	15	,	,	PUNCT
ejpam-5982	63	16	but	but	CCONJ
ejpam-5982	63	17	if	if	SCONJ
ejpam-5982	63	18	a	a	DET
ejpam-5982	63	19	limit	limit	NOUN
ejpam-5982	63	20	is	be	AUX
ejpam-5982	63	21	in	in	ADP
ejpam-5982	63	22	x	x	X
ejpam-5982	63	23	∩	∩	ADJ
ejpam-5982	63	24	y	y	PROPN
ejpam-5982	63	25	,	,	PUNCT
ejpam-5982	63	26	then	then	ADV
ejpam-5982	63	27	it	it	PRON
ejpam-5982	63	28	is	be	AUX
ejpam-5982	63	29	the	the	DET
ejpam-5982	63	30	unique	unique	ADJ
ejpam-5982	63	31	limit	limit	NOUN
ejpam-5982	63	32	of	of	ADP
ejpam-5982	63	33	the	the	DET
ejpam-5982	63	34	sequence	sequence	NOUN
ejpam-5982	63	35	.	.	PUNCT
ejpam-5982	64	1	definition	definition	NOUN
ejpam-5982	64	2	3	3	NUM
ejpam-5982	64	3	.	.	PUNCT
ejpam-5982	65	1	[	[	X
ejpam-5982	65	2	6	6	NUM
ejpam-5982	65	3	]	]	PUNCT
ejpam-5982	65	4	let	let	VERB
ejpam-5982	65	5	x1	x1	PROPN
ejpam-5982	65	6	,	,	PUNCT
ejpam-5982	65	7	y1	y1	PROPN
ejpam-5982	65	8	,	,	PUNCT
ejpam-5982	65	9	x2	x2	PROPN
ejpam-5982	65	10	and	and	CCONJ
ejpam-5982	65	11	y2	y2	PROPN
ejpam-5982	65	12	be	be	VERB
ejpam-5982	65	13	four	four	NUM
ejpam-5982	65	14	sets	set	NOUN
ejpam-5982	65	15	.	.	PUNCT
ejpam-5982	66	1	a	a	DET
ejpam-5982	66	2	function	function	NOUN
ejpam-5982	66	3	f	f	NOUN
ejpam-5982	66	4	:	:	PUNCT
ejpam-5982	66	5	x1∪y1	x1∪y1	PROPN
ejpam-5982	66	6	→	→	SYM
ejpam-5982	66	7	x2∪y2	x2∪y2	PROPN
ejpam-5982	66	8	is	be	AUX
ejpam-5982	66	9	said	say	VERB
ejpam-5982	66	10	to	to	PART
ejpam-5982	66	11	be	be	AUX
ejpam-5982	66	12	a	a	DET
ejpam-5982	66	13	covariant	covariant	ADJ
ejpam-5982	66	14	map	map	NOUN
ejpam-5982	66	15	if	if	SCONJ
ejpam-5982	66	16	f(x1	f(x1	ADJ
ejpam-5982	66	17	)	)	PUNCT
ejpam-5982	66	18	⊆	⊆	NUM
ejpam-5982	66	19	x2	x2	NOUN
ejpam-5982	66	20	and	and	CCONJ
ejpam-5982	66	21	f(y1	f(y1	NOUN
ejpam-5982	66	22	)	)	PUNCT
ejpam-5982	66	23	⊆	⊆	NUM
ejpam-5982	66	24	y2	y2	NOUN
ejpam-5982	66	25	and	and	CCONJ
ejpam-5982	66	26	is	be	AUX
ejpam-5982	66	27	denoted	denote	VERB
ejpam-5982	66	28	as	as	ADP
ejpam-5982	66	29	f	f	PROPN
ejpam-5982	66	30	:	:	PUNCT
ejpam-5982	66	31	(	(	PUNCT
ejpam-5982	66	32	x1	x1	PROPN
ejpam-5982	66	33	,	,	PUNCT
ejpam-5982	66	34	y1	y1	NOUN
ejpam-5982	66	35	)	)	PUNCT
ejpam-5982	66	36	⇒	⇒	NOUN
ejpam-5982	66	37	(	(	PUNCT
ejpam-5982	66	38	x2	x2	PROPN
ejpam-5982	66	39	,	,	PUNCT
ejpam-5982	66	40	y2	y2	PROPN
ejpam-5982	66	41	)	)	PUNCT
ejpam-5982	66	42	.	.	PUNCT
ejpam-5982	67	1	in	in	ADP
ejpam-5982	67	2	particular	particular	ADJ
ejpam-5982	67	3	,	,	PUNCT
ejpam-5982	67	4	if	if	SCONJ
ejpam-5982	67	5	(	(	PUNCT
ejpam-5982	67	6	x1	x1	PROPN
ejpam-5982	67	7	,	,	PUNCT
ejpam-5982	67	8	y1	y1	NOUN
ejpam-5982	67	9	,	,	PUNCT
ejpam-5982	67	10	d1	d1	NOUN
ejpam-5982	67	11	)	)	PUNCT
ejpam-5982	67	12	and	and	CCONJ
ejpam-5982	67	13	(	(	PUNCT
ejpam-5982	67	14	x2	x2	PROPN
ejpam-5982	67	15	,	,	PUNCT
ejpam-5982	67	16	y2	y2	PROPN
ejpam-5982	67	17	,	,	PUNCT
ejpam-5982	67	18	d2	d2	PROPN
ejpam-5982	67	19	)	)	PUNCT
ejpam-5982	67	20	are	be	AUX
ejpam-5982	67	21	two	two	NUM
ejpam-5982	67	22	bipolar	bipolar	ADJ
ejpam-5982	67	23	metric	metric	ADJ
ejpam-5982	67	24	space	space	NOUN
ejpam-5982	67	25	then	then	ADV
ejpam-5982	67	26	we	we	PRON
ejpam-5982	67	27	use	use	VERB
ejpam-5982	67	28	the	the	DET
ejpam-5982	67	29	notaion	notaion	NOUN
ejpam-5982	67	30	f	f	NOUN
ejpam-5982	67	31	:	:	PUNCT
ejpam-5982	67	32	(	(	PUNCT
ejpam-5982	67	33	x1	x1	PROPN
ejpam-5982	67	34	,	,	PUNCT
ejpam-5982	67	35	y1	y1	NOUN
ejpam-5982	67	36	,	,	PUNCT
ejpam-5982	67	37	d1	d1	NOUN
ejpam-5982	67	38	)	)	PUNCT
ejpam-5982	67	39	⇒	⇒	NOUN
ejpam-5982	67	40	(	(	PUNCT
ejpam-5982	67	41	x2	x2	PROPN
ejpam-5982	67	42	,	,	PUNCT
ejpam-5982	67	43	y2	y2	PROPN
ejpam-5982	67	44	,	,	PUNCT
ejpam-5982	67	45	d2	d2	PROPN
ejpam-5982	67	46	)	)	PUNCT
ejpam-5982	67	47	for	for	ADP
ejpam-5982	67	48	covariant	covariant	ADJ
ejpam-5982	67	49	map	map	NOUN
ejpam-5982	67	50	f	f	PROPN
ejpam-5982	67	51	.	.	PUNCT
ejpam-5982	68	1	a	a	DET
ejpam-5982	68	2	function	function	NOUN
ejpam-5982	68	3	g	g	NOUN
ejpam-5982	68	4	:	:	PUNCT
ejpam-5982	68	5	x1	x1	PROPN
ejpam-5982	68	6	∪	∪	NOUN
ejpam-5982	68	7	y1	y1	NOUN
ejpam-5982	68	8	→	→	PUNCT
ejpam-5982	68	9	x2	x2	PROPN
ejpam-5982	68	10	∪	∪	NOUN
ejpam-5982	68	11	y2	y2	PROPN
ejpam-5982	68	12	is	be	AUX
ejpam-5982	68	13	said	say	VERB
ejpam-5982	68	14	to	to	PART
ejpam-5982	68	15	be	be	AUX
ejpam-5982	68	16	a	a	DET
ejpam-5982	68	17	contravariant	contravariant	ADJ
ejpam-5982	68	18	map	map	NOUN
ejpam-5982	68	19	if	if	SCONJ
ejpam-5982	68	20	g(x1	g(x1	NOUN
ejpam-5982	68	21	)	)	PUNCT
ejpam-5982	68	22	⊆	⊆	NUM
ejpam-5982	68	23	y2	y2	NOUN
ejpam-5982	68	24	and	and	CCONJ
ejpam-5982	68	25	g(y1	g(y1	NOUN
ejpam-5982	68	26	)	)	PUNCT
ejpam-5982	68	27	⊆	⊆	NUM
ejpam-5982	68	28	x2	x2	PROPN
ejpam-5982	68	29	and	and	CCONJ
ejpam-5982	68	30	is	be	AUX
ejpam-5982	68	31	denoted	denote	VERB
ejpam-5982	68	32	as	as	ADP
ejpam-5982	68	33	g	g	PROPN
ejpam-5982	68	34	:	:	PUNCT
ejpam-5982	68	35	(	(	PUNCT
ejpam-5982	68	36	x1	x1	PROPN
ejpam-5982	68	37	,	,	PUNCT
ejpam-5982	68	38	y1	y1	NOUN
ejpam-5982	68	39	)	)	PUNCT
ejpam-5982	68	40	⇄	⇄	PROPN
ejpam-5982	68	41	(	(	PUNCT
ejpam-5982	68	42	x2	x2	PROPN
ejpam-5982	68	43	,	,	PUNCT
ejpam-5982	68	44	y2	y2	PROPN
ejpam-5982	68	45	)	)	PUNCT
ejpam-5982	68	46	.	.	PUNCT
ejpam-5982	69	1	definition	definition	NOUN
ejpam-5982	69	2	4	4	NUM
ejpam-5982	69	3	.	.	PUNCT
ejpam-5982	70	1	let	let	VERB
ejpam-5982	70	2	(	(	PUNCT
ejpam-5982	70	3	x1	x1	ADJ
ejpam-5982	70	4	,	,	PUNCT
ejpam-5982	70	5	y1	y1	NOUN
ejpam-5982	70	6	,	,	PUNCT
ejpam-5982	70	7	d1	d1	NOUN
ejpam-5982	70	8	)	)	PUNCT
ejpam-5982	70	9	and	and	CCONJ
ejpam-5982	70	10	(	(	PUNCT
ejpam-5982	70	11	x2	x2	PROPN
ejpam-5982	70	12	,	,	PUNCT
ejpam-5982	70	13	y2	y2	PROPN
ejpam-5982	70	14	,	,	PUNCT
ejpam-5982	70	15	d2	d2	PROPN
ejpam-5982	70	16	)	)	PUNCT
ejpam-5982	70	17	be	be	VERB
ejpam-5982	70	18	two	two	NUM
ejpam-5982	70	19	bipolar	bipolar	ADJ
ejpam-5982	70	20	metric	metric	ADJ
ejpam-5982	70	21	spaces	space	NOUN
ejpam-5982	70	22	.	.	PUNCT
ejpam-5982	71	1	a	a	DET
ejpam-5982	71	2	covariant	covariant	ADJ
ejpam-5982	71	3	map	map	NOUN
ejpam-5982	71	4	f	f	X
ejpam-5982	71	5	:	:	PUNCT
ejpam-5982	71	6	(	(	PUNCT
ejpam-5982	71	7	x1	x1	PROPN
ejpam-5982	71	8	,	,	PUNCT
ejpam-5982	71	9	y1	y1	NOUN
ejpam-5982	71	10	)	)	PUNCT
ejpam-5982	71	11	⇒	⇒	NOUN
ejpam-5982	71	12	(	(	PUNCT
ejpam-5982	71	13	x2	x2	PROPN
ejpam-5982	71	14	,	,	PUNCT
ejpam-5982	71	15	y2	y2	PROPN
ejpam-5982	71	16	)	)	PUNCT
ejpam-5982	71	17	is	be	AUX
ejpam-5982	71	18	continuous	continuous	ADJ
ejpam-5982	71	19	at	at	ADP
ejpam-5982	71	20	v	v	NOUN
ejpam-5982	71	21	if	if	SCONJ
ejpam-5982	72	1	and	and	CCONJ
ejpam-5982	72	2	only	only	ADV
ejpam-5982	72	3	if	if	SCONJ
ejpam-5982	72	4	{	{	PUNCT
ejpam-5982	72	5	un	un	ADJ
ejpam-5982	72	6	}	}	PUNCT
ejpam-5982	72	7	converges	converge	VERB
ejpam-5982	72	8	to	to	ADP
ejpam-5982	72	9	v	v	NOUN
ejpam-5982	72	10	on	on	ADP
ejpam-5982	72	11	(	(	PUNCT
ejpam-5982	72	12	x1	x1	PROPN
ejpam-5982	72	13	,	,	PUNCT
ejpam-5982	72	14	y1	y1	NOUN
ejpam-5982	72	15	,	,	PUNCT
ejpam-5982	72	16	d1	d1	PROPN
ejpam-5982	72	17	)	)	PUNCT
ejpam-5982	72	18	implies	imply	VERB
ejpam-5982	72	19	{	{	PUNCT
ejpam-5982	72	20	f(un	f(un	NOUN
ejpam-5982	72	21	)	)	PUNCT
ejpam-5982	72	22	}	}	PUNCT
ejpam-5982	72	23	converges	converge	VERB
ejpam-5982	72	24	to	to	ADP
ejpam-5982	72	25	f(v	f(v	NOUN
ejpam-5982	72	26	)	)	PUNCT
ejpam-5982	72	27	on	on	ADP
ejpam-5982	72	28	(	(	PUNCT
ejpam-5982	72	29	x2	x2	PROPN
ejpam-5982	72	30	,	,	PUNCT
ejpam-5982	72	31	y2	y2	PROPN
ejpam-5982	72	32	,	,	PUNCT
ejpam-5982	72	33	d2	d2	PROPN
ejpam-5982	72	34	)	)	PUNCT
ejpam-5982	72	35	.	.	PUNCT
ejpam-5982	73	1	a	a	DET
ejpam-5982	73	2	contravariant	contravariant	PROPN
ejpam-5982	73	3	map	map	NOUN
ejpam-5982	73	4	g	g	NOUN
ejpam-5982	73	5	:	:	PUNCT
ejpam-5982	73	6	(	(	PUNCT
ejpam-5982	73	7	x1	x1	PROPN
ejpam-5982	73	8	,	,	PUNCT
ejpam-5982	73	9	y1	y1	NOUN
ejpam-5982	73	10	,	,	PUNCT
ejpam-5982	73	11	d1	d1	NOUN
ejpam-5982	73	12	)	)	PUNCT
ejpam-5982	73	13	⇄	⇄	PROPN
ejpam-5982	73	14	(	(	PUNCT
ejpam-5982	73	15	x2	x2	PROPN
ejpam-5982	73	16	,	,	PUNCT
ejpam-5982	73	17	y2	y2	PROPN
ejpam-5982	73	18	,	,	PUNCT
ejpam-5982	73	19	d2	d2	PROPN
ejpam-5982	73	20	)	)	PUNCT
ejpam-5982	73	21	is	be	AUX
ejpam-5982	73	22	continuous	continuous	ADJ
ejpam-5982	73	23	if	if	SCONJ
ejpam-5982	73	24	and	and	CCONJ
ejpam-5982	73	25	only	only	ADV
ejpam-5982	73	26	if	if	SCONJ
ejpam-5982	73	27	it	it	PRON
ejpam-5982	73	28	is	be	AUX
ejpam-5982	73	29	continuous	continuous	ADJ
ejpam-5982	73	30	as	as	ADP
ejpam-5982	73	31	a	a	DET
ejpam-5982	73	32	covariant	covariant	ADJ
ejpam-5982	73	33	map	map	NOUN
ejpam-5982	74	1	g	g	NOUN
ejpam-5982	74	2	:	:	PUNCT
ejpam-5982	74	3	(	(	PUNCT
ejpam-5982	74	4	x1	x1	PROPN
ejpam-5982	74	5	,	,	PUNCT
ejpam-5982	74	6	y1	y1	NOUN
ejpam-5982	74	7	,	,	PUNCT
ejpam-5982	74	8	d1	d1	NOUN
ejpam-5982	74	9	)	)	PUNCT
ejpam-5982	74	10	⇒	⇒	NOUN
ejpam-5982	74	11	(	(	PUNCT
ejpam-5982	74	12	y2	y2	INTJ
ejpam-5982	74	13	,	,	PUNCT
ejpam-5982	74	14	x2	x2	PROPN
ejpam-5982	74	15	,	,	PUNCT
ejpam-5982	74	16	d̄2	d̄2	PROPN
ejpam-5982	74	17	)	)	PUNCT
ejpam-5982	74	18	,	,	PUNCT
ejpam-5982	74	19	where	where	SCONJ
ejpam-5982	74	20	d̄2	d̄2	PROPN
ejpam-5982	74	21	is	be	AUX
ejpam-5982	74	22	defined	define	VERB
ejpam-5982	74	23	as	as	ADP
ejpam-5982	74	24	d̄2(y	d̄2(y	PROPN
ejpam-5982	74	25	,	,	PUNCT
ejpam-5982	74	26	x	x	NOUN
ejpam-5982	74	27	)	)	PUNCT
ejpam-5982	74	28	=	=	SYM
ejpam-5982	74	29	d2(x	d2(x	PROPN
ejpam-5982	74	30	,	,	PUNCT
ejpam-5982	74	31	y	y	NOUN
ejpam-5982	74	32	)	)	PUNCT
ejpam-5982	74	33	,	,	PUNCT
ejpam-5982	74	34	for	for	ADP
ejpam-5982	74	35	all	all	DET
ejpam-5982	74	36	(	(	PUNCT
ejpam-5982	74	37	y	y	PROPN
ejpam-5982	74	38	,	,	PUNCT
ejpam-5982	74	39	x	x	NOUN
ejpam-5982	74	40	)	)	PUNCT
ejpam-5982	74	41	∈	∈	NOUN
ejpam-5982	74	42	y2	y2	NOUN
ejpam-5982	74	43	×x2	×x2	NOUN
ejpam-5982	74	44	.	.	PUNCT
ejpam-5982	75	1	definition	definition	NOUN
ejpam-5982	75	2	5	5	NUM
ejpam-5982	75	3	.	.	PUNCT
ejpam-5982	76	1	[	[	X
ejpam-5982	76	2	7	7	X
ejpam-5982	76	3	]	]	X
ejpam-5982	76	4	if	if	SCONJ
ejpam-5982	76	5	s	s	PROPN
ejpam-5982	76	6	and	and	CCONJ
ejpam-5982	76	7	t	t	PROPN
ejpam-5982	76	8	are	be	AUX
ejpam-5982	76	9	covariant	covariant	ADJ
ejpam-5982	76	10	or	or	CCONJ
ejpam-5982	76	11	contravariant	contravariant	ADJ
ejpam-5982	76	12	maps	map	NOUN
ejpam-5982	76	13	on	on	ADP
ejpam-5982	76	14	x	x	PUNCT
ejpam-5982	76	15	∪	∪	PROPN
ejpam-5982	76	16	y	y	PROPN
ejpam-5982	76	17	,	,	PUNCT
ejpam-5982	76	18	then	then	ADV
ejpam-5982	76	19	(	(	PUNCT
ejpam-5982	76	20	i	i	NOUN
ejpam-5982	76	21	)	)	PUNCT
ejpam-5982	76	22	u	u	NOUN
ejpam-5982	76	23	∈	∈	PROPN
ejpam-5982	76	24	x	x	PUNCT
ejpam-5982	76	25	∪	∪	ADP
ejpam-5982	76	26	y	y	PROPN
ejpam-5982	76	27	is	be	AUX
ejpam-5982	76	28	called	call	VERB
ejpam-5982	76	29	fixed	fix	VERB
ejpam-5982	76	30	point	point	NOUN
ejpam-5982	76	31	of	of	ADP
ejpam-5982	76	32	t	t	PROPN
ejpam-5982	77	1	if	if	SCONJ
ejpam-5982	77	2	and	and	CCONJ
ejpam-5982	77	3	only	only	ADV
ejpam-5982	77	4	if	if	SCONJ
ejpam-5982	77	5	tu	tu	PROPN
ejpam-5982	77	6	=	=	SYM
ejpam-5982	77	7	u.	u.	PROPN
ejpam-5982	77	8	(	(	PUNCT
ejpam-5982	77	9	ii	ii	NOUN
ejpam-5982	77	10	)	)	PUNCT
ejpam-5982	77	11	u	u	NOUN
ejpam-5982	77	12	∈	∈	PROPN
ejpam-5982	77	13	x	x	PUNCT
ejpam-5982	77	14	∪	∪	ADP
ejpam-5982	77	15	y	y	PROPN
ejpam-5982	77	16	is	be	AUX
ejpam-5982	77	17	called	call	VERB
ejpam-5982	77	18	common	common	ADJ
ejpam-5982	77	19	fixed	fix	VERB
ejpam-5982	77	20	point	point	NOUN
ejpam-5982	77	21	of	of	ADP
ejpam-5982	77	22	s	s	PRON
ejpam-5982	77	23	and	and	CCONJ
ejpam-5982	77	24	t	t	PROPN
ejpam-5982	77	25	if	if	SCONJ
ejpam-5982	78	1	and	and	CCONJ
ejpam-5982	78	2	only	only	ADV
ejpam-5982	78	3	if	if	SCONJ
ejpam-5982	78	4	tu	tu	PROPN
ejpam-5982	78	5	=	=	PROPN
ejpam-5982	78	6	su	su	PROPN
ejpam-5982	78	7	=	=	PROPN
ejpam-5982	78	8	u.	u.	PROPN
ejpam-5982	78	9	(	(	PUNCT
ejpam-5982	78	10	iii	iii	NOUN
ejpam-5982	78	11	)	)	PUNCT
ejpam-5982	78	12	u	u	NOUN
ejpam-5982	78	13	∈	∈	PROPN
ejpam-5982	78	14	x	x	PUNCT
ejpam-5982	78	15	∪	∪	ADP
ejpam-5982	78	16	y	y	PROPN
ejpam-5982	78	17	is	be	AUX
ejpam-5982	78	18	called	call	VERB
ejpam-5982	78	19	coincidence	coincidence	NOUN
ejpam-5982	78	20	point	point	NOUN
ejpam-5982	78	21	of	of	ADP
ejpam-5982	78	22	s	s	PRON
ejpam-5982	78	23	and	and	CCONJ
ejpam-5982	78	24	t	t	PROPN
ejpam-5982	78	25	if	if	SCONJ
ejpam-5982	79	1	and	and	CCONJ
ejpam-5982	79	2	only	only	ADV
ejpam-5982	79	3	if	if	SCONJ
ejpam-5982	79	4	tu	tu	PROPN
ejpam-5982	79	5	=	=	PROPN
ejpam-5982	79	6	su	su	PROPN
ejpam-5982	79	7	.	.	PROPN
ejpam-5982	79	8	.	.	PUNCT
ejpam-5982	80	1	popa	popa	NOUN
ejpam-5982	81	1	[	[	X
ejpam-5982	81	2	4	4	X
ejpam-5982	81	3	]	]	PUNCT
ejpam-5982	81	4	studied	study	VERB
ejpam-5982	81	5	a	a	DET
ejpam-5982	81	6	new	new	ADJ
ejpam-5982	81	7	type	type	NOUN
ejpam-5982	81	8	of	of	ADP
ejpam-5982	81	9	contraction	contraction	NOUN
ejpam-5982	81	10	condition	condition	NOUN
ejpam-5982	81	11	by	by	ADP
ejpam-5982	81	12	employing	employ	VERB
ejpam-5982	81	13	the	the	DET
ejpam-5982	81	14	implicit	implicit	ADJ
ejpam-5982	81	15	function	function	NOUN
ejpam-5982	81	16	to	to	PART
ejpam-5982	81	17	obtain	obtain	VERB
ejpam-5982	81	18	fixed	fixed	ADJ
ejpam-5982	81	19	points	point	NOUN
ejpam-5982	81	20	.	.	PUNCT
ejpam-5982	82	1	in	in	ADP
ejpam-5982	82	2	the	the	DET
ejpam-5982	82	3	sequel	sequel	NOUN
ejpam-5982	82	4	,	,	PUNCT
ejpam-5982	82	5	we	we	PRON
ejpam-5982	82	6	are	be	AUX
ejpam-5982	82	7	also	also	ADV
ejpam-5982	82	8	going	go	VERB
ejpam-5982	82	9	to	to	PART
ejpam-5982	82	10	prove	prove	VERB
ejpam-5982	82	11	a	a	DET
ejpam-5982	82	12	few	few	ADJ
ejpam-5982	82	13	fixed	fix	VERB
ejpam-5982	82	14	-	-	PUNCT
ejpam-5982	82	15	point	point	NOUN
ejpam-5982	82	16	theorems	theorem	NOUN
ejpam-5982	82	17	by	by	ADP
ejpam-5982	82	18	employing	employ	VERB
ejpam-5982	82	19	implicit	implicit	ADJ
ejpam-5982	82	20	relations	relation	NOUN
ejpam-5982	82	21	in	in	ADP
ejpam-5982	82	22	a	a	DET
ejpam-5982	82	23	bipolar	bipolar	ADJ
ejpam-5982	82	24	metric	metric	ADJ
ejpam-5982	82	25	space	space	NOUN
ejpam-5982	82	26	.	.	PUNCT
ejpam-5982	83	1	p.	p.	NOUN
ejpam-5982	83	2	p.	p.	NOUN
ejpam-5982	84	1	murthy	murthy	ADJ
ejpam-5982	85	1	et	et	PROPN
ejpam-5982	85	2	al	al	PROPN
ejpam-5982	85	3	.	.	PUNCT
ejpam-5982	85	4	/	/	SYM
ejpam-5982	85	5	eur	eur	PROPN
ejpam-5982	85	6	.	.	PUNCT
ejpam-5982	86	1	j.	j.	PROPN
ejpam-5982	86	2	pure	pure	PROPN
ejpam-5982	86	3	appl	appl	PROPN
ejpam-5982	86	4	.	.	PROPN
ejpam-5982	86	5	math	math	PROPN
ejpam-5982	86	6	,	,	PUNCT
ejpam-5982	86	7	18	18	NUM
ejpam-5982	86	8	(	(	PUNCT
ejpam-5982	86	9	2	2	NUM
ejpam-5982	86	10	)	)	PUNCT
ejpam-5982	86	11	(	(	PUNCT
ejpam-5982	86	12	2025	2025	NUM
ejpam-5982	86	13	)	)	PUNCT
ejpam-5982	86	14	,	,	PUNCT
ejpam-5982	86	15	5982	5982	NUM
ejpam-5982	86	16	4	4	NUM
ejpam-5982	86	17	of	of	ADP
ejpam-5982	86	18	17	17	NUM
ejpam-5982	86	19	3	3	NUM
ejpam-5982	86	20	.	.	PUNCT
ejpam-5982	86	21	main	main	ADJ
ejpam-5982	86	22	results	result	NOUN
ejpam-5982	86	23	the	the	DET
ejpam-5982	86	24	concept	concept	NOUN
ejpam-5982	86	25	of	of	ADP
ejpam-5982	86	26	maps	map	NOUN
ejpam-5982	86	27	of	of	ADP
ejpam-5982	86	28	type	type	NOUN
ejpam-5982	86	29	(	(	PUNCT
ejpam-5982	86	30	a	a	NOUN
ejpam-5982	86	31	)	)	PUNCT
ejpam-5982	86	32	was	be	AUX
ejpam-5982	86	33	introduced	introduce	VERB
ejpam-5982	86	34	initially	initially	ADV
ejpam-5982	86	35	by	by	ADP
ejpam-5982	86	36	jungck	jungck	PROPN
ejpam-5982	86	37	,	,	PUNCT
ejpam-5982	86	38	murthy	murthy	ADJ
ejpam-5982	86	39	,	,	PUNCT
ejpam-5982	86	40	and	and	CCONJ
ejpam-5982	86	41	cho	cho	VERB
ejpam-5982	87	1	[	[	X
ejpam-5982	87	2	3	3	X
ejpam-5982	87	3	]	]	PUNCT
ejpam-5982	87	4	in	in	ADP
ejpam-5982	87	5	a	a	DET
ejpam-5982	87	6	metric	metric	ADJ
ejpam-5982	87	7	space	space	NOUN
ejpam-5982	87	8	.	.	PUNCT
ejpam-5982	88	1	now	now	ADV
ejpam-5982	88	2	we	we	PRON
ejpam-5982	88	3	are	be	AUX
ejpam-5982	88	4	ready	ready	ADJ
ejpam-5982	88	5	to	to	PART
ejpam-5982	88	6	introduce	introduce	VERB
ejpam-5982	88	7	the	the	DET
ejpam-5982	88	8	same	same	ADJ
ejpam-5982	88	9	concept	concept	NOUN
ejpam-5982	88	10	in	in	ADP
ejpam-5982	88	11	a	a	DET
ejpam-5982	88	12	bipolar	bipolar	ADJ
ejpam-5982	88	13	metric	metric	ADJ
ejpam-5982	88	14	space	space	NOUN
ejpam-5982	88	15	.	.	PUNCT
ejpam-5982	89	1	the	the	DET
ejpam-5982	89	2	definition	definition	NOUN
ejpam-5982	89	3	follows	follow	VERB
ejpam-5982	89	4	:	:	PUNCT
ejpam-5982	89	5	definition	definition	NOUN
ejpam-5982	89	6	6	6	NUM
ejpam-5982	89	7	.	.	PUNCT
ejpam-5982	90	1	let	let	VERB
ejpam-5982	90	2	(	(	PUNCT
ejpam-5982	90	3	x	x	X
ejpam-5982	90	4	,	,	PUNCT
ejpam-5982	90	5	y	y	PROPN
ejpam-5982	90	6	,	,	PUNCT
ejpam-5982	90	7	d	d	NOUN
ejpam-5982	90	8	)	)	PUNCT
ejpam-5982	90	9	be	be	AUX
ejpam-5982	90	10	a	a	DET
ejpam-5982	90	11	bipolar	bipolar	ADJ
ejpam-5982	90	12	metric	metric	ADJ
ejpam-5982	90	13	space	space	NOUN
ejpam-5982	90	14	.	.	PUNCT
ejpam-5982	91	1	also	also	ADV
ejpam-5982	91	2	,	,	PUNCT
ejpam-5982	91	3	let	let	VERB
ejpam-5982	91	4	t	t	NOUN
ejpam-5982	91	5	:	:	PUNCT
ejpam-5982	91	6	(	(	PUNCT
ejpam-5982	91	7	x	x	X
ejpam-5982	91	8	,	,	PUNCT
ejpam-5982	91	9	y	y	PROPN
ejpam-5982	91	10	,	,	PUNCT
ejpam-5982	91	11	d	d	NOUN
ejpam-5982	91	12	)	)	PUNCT
ejpam-5982	91	13	⇒	⇒	NOUN
ejpam-5982	91	14	(	(	PUNCT
ejpam-5982	91	15	x	x	X
ejpam-5982	91	16	,	,	PUNCT
ejpam-5982	91	17	y	y	PROPN
ejpam-5982	91	18	,	,	PUNCT
ejpam-5982	91	19	d	d	NOUN
ejpam-5982	91	20	)	)	PUNCT
ejpam-5982	91	21	be	be	AUX
ejpam-5982	91	22	a	a	DET
ejpam-5982	91	23	covariant	covariant	ADJ
ejpam-5982	91	24	map	map	NOUN
ejpam-5982	91	25	and	and	CCONJ
ejpam-5982	91	26	s	s	VERB
ejpam-5982	91	27	:	:	PUNCT
ejpam-5982	91	28	(	(	PUNCT
ejpam-5982	91	29	x	x	X
ejpam-5982	91	30	,	,	PUNCT
ejpam-5982	91	31	y	y	PROPN
ejpam-5982	91	32	,	,	PUNCT
ejpam-5982	91	33	d	d	NOUN
ejpam-5982	91	34	)	)	PUNCT
ejpam-5982	91	35	⇄	⇄	NOUN
ejpam-5982	91	36	(	(	PUNCT
ejpam-5982	91	37	x	x	X
ejpam-5982	91	38	,	,	PUNCT
ejpam-5982	91	39	y	y	PROPN
ejpam-5982	91	40	,	,	PUNCT
ejpam-5982	91	41	d	d	NOUN
ejpam-5982	91	42	)	)	PUNCT
ejpam-5982	91	43	be	be	AUX
ejpam-5982	91	44	a	a	DET
ejpam-5982	91	45	contravariant	contravariant	ADJ
ejpam-5982	91	46	map	map	NOUN
ejpam-5982	91	47	.	.	PUNCT
ejpam-5982	92	1	then	then	ADV
ejpam-5982	92	2	(	(	PUNCT
ejpam-5982	92	3	i	i	NOUN
ejpam-5982	92	4	)	)	PUNCT
ejpam-5982	92	5	s	s	PROPN
ejpam-5982	92	6	and	and	CCONJ
ejpam-5982	92	7	t	t	PROPN
ejpam-5982	92	8	are	be	AUX
ejpam-5982	92	9	said	say	VERB
ejpam-5982	92	10	to	to	PART
ejpam-5982	92	11	be	be	AUX
ejpam-5982	92	12	compatible	compatible	ADJ
ejpam-5982	92	13	mappings	mapping	NOUN
ejpam-5982	92	14	of	of	ADP
ejpam-5982	92	15	type	type	NOUN
ejpam-5982	92	16	(	(	PUNCT
ejpam-5982	92	17	a	a	NOUN
ejpam-5982	92	18	)	)	PUNCT
ejpam-5982	92	19	with	with	ADP
ejpam-5982	92	20	respect	respect	NOUN
ejpam-5982	92	21	to	to	ADP
ejpam-5982	92	22	x	x	PUNCT
ejpam-5982	92	23	if	if	SCONJ
ejpam-5982	93	1	and	and	CCONJ
ejpam-5982	93	2	only	only	ADV
ejpam-5982	93	3	if	if	SCONJ
ejpam-5982	93	4	lim	lim	PROPN
ejpam-5982	93	5	n→+∞	n→+∞	PROPN
ejpam-5982	93	6	d(ssun	d(ssun	PROPN
ejpam-5982	93	7	,	,	PUNCT
ejpam-5982	93	8	tsun	tsun	PROPN
ejpam-5982	93	9	)	)	PUNCT
ejpam-5982	93	10	=	=	SYM
ejpam-5982	93	11	0	0	NUM
ejpam-5982	93	12	or	or	CCONJ
ejpam-5982	93	13	lim	lim	PROPN
ejpam-5982	93	14	n→+∞	n→+∞	PROPN
ejpam-5982	93	15	d(ttun	d(ttun	PROPN
ejpam-5982	93	16	,	,	PUNCT
ejpam-5982	93	17	stun	stun	NOUN
ejpam-5982	93	18	)	)	PUNCT
ejpam-5982	93	19	=	=	SYM
ejpam-5982	94	1	0	0	NUM
ejpam-5982	94	2	,	,	PUNCT
ejpam-5982	94	3	whenever	whenever	SCONJ
ejpam-5982	94	4	{	{	PUNCT
ejpam-5982	94	5	un	un	AUX
ejpam-5982	94	6	}	}	PUNCT
ejpam-5982	94	7	be	be	AUX
ejpam-5982	94	8	a	a	DET
ejpam-5982	94	9	sequence	sequence	NOUN
ejpam-5982	94	10	in	in	ADP
ejpam-5982	94	11	x	x	INTJ
ejpam-5982	94	12	such	such	ADJ
ejpam-5982	94	13	that	that	SCONJ
ejpam-5982	94	14	lim	lim	PROPN
ejpam-5982	94	15	n→+∞	n→+∞	VERB
ejpam-5982	94	16	sun	sun	PROPN
ejpam-5982	94	17	=	=	SYM
ejpam-5982	94	18	lim	lim	PROPN
ejpam-5982	94	19	n→+∞	n→+∞	VERB
ejpam-5982	94	20	tun	tun	PROPN
ejpam-5982	95	1	=	=	SYM
ejpam-5982	95	2	t	t	PROPN
ejpam-5982	95	3	for	for	ADP
ejpam-5982	95	4	some	some	DET
ejpam-5982	95	5	t	t	NOUN
ejpam-5982	95	6	∈	∈	NOUN
ejpam-5982	95	7	x	x	PUNCT
ejpam-5982	95	8	∩	∩	PROPN
ejpam-5982	95	9	y	y	PROPN
ejpam-5982	95	10	.	.	PUNCT
ejpam-5982	96	1	(	(	PUNCT
ejpam-5982	96	2	ii	ii	NOUN
ejpam-5982	96	3	)	)	PUNCT
ejpam-5982	96	4	s	s	PROPN
ejpam-5982	96	5	and	and	CCONJ
ejpam-5982	96	6	t	t	PROPN
ejpam-5982	96	7	are	be	AUX
ejpam-5982	96	8	said	say	VERB
ejpam-5982	96	9	to	to	PART
ejpam-5982	96	10	be	be	AUX
ejpam-5982	96	11	compatible	compatible	ADJ
ejpam-5982	96	12	mappings	mapping	NOUN
ejpam-5982	96	13	of	of	ADP
ejpam-5982	96	14	type	type	NOUN
ejpam-5982	96	15	(	(	PUNCT
ejpam-5982	96	16	a	a	NOUN
ejpam-5982	96	17	)	)	PUNCT
ejpam-5982	96	18	with	with	ADP
ejpam-5982	96	19	respect	respect	NOUN
ejpam-5982	96	20	to	to	ADP
ejpam-5982	96	21	y	y	PRON
ejpam-5982	96	22	if	if	SCONJ
ejpam-5982	96	23	and	and	CCONJ
ejpam-5982	96	24	only	only	ADV
ejpam-5982	96	25	if	if	SCONJ
ejpam-5982	96	26	lim	lim	PROPN
ejpam-5982	96	27	n→+∞	n→+∞	PROPN
ejpam-5982	96	28	d(tsun	d(tsun	NOUN
ejpam-5982	96	29	,	,	PUNCT
ejpam-5982	96	30	ssun	ssun	NOUN
ejpam-5982	96	31	)	)	PUNCT
ejpam-5982	96	32	=	=	SYM
ejpam-5982	96	33	0	0	NUM
ejpam-5982	97	1	or	or	CCONJ
ejpam-5982	97	2	lim	lim	PROPN
ejpam-5982	97	3	n→+∞	n→+∞	PROPN
ejpam-5982	97	4	d(stun	d(stun	PROPN
ejpam-5982	97	5	,	,	PUNCT
ejpam-5982	97	6	ttun	ttun	ADJ
ejpam-5982	97	7	)	)	PUNCT
ejpam-5982	97	8	=	=	SYM
ejpam-5982	97	9	0	0	NUM
ejpam-5982	97	10	,	,	PUNCT
ejpam-5982	97	11	whenever	whenever	SCONJ
ejpam-5982	97	12	{	{	PUNCT
ejpam-5982	97	13	un	un	ADJ
ejpam-5982	97	14	}	}	PUNCT
ejpam-5982	97	15	is	be	AUX
ejpam-5982	97	16	a	a	DET
ejpam-5982	97	17	sequence	sequence	NOUN
ejpam-5982	97	18	in	in	ADP
ejpam-5982	97	19	y	y	PROPN
ejpam-5982	97	20	such	such	ADJ
ejpam-5982	97	21	that	that	SCONJ
ejpam-5982	97	22	lim	lim	PROPN
ejpam-5982	97	23	n→+∞	n→+∞	VERB
ejpam-5982	97	24	sun	sun	PROPN
ejpam-5982	97	25	=	=	SYM
ejpam-5982	97	26	lim	lim	PROPN
ejpam-5982	97	27	n→+∞	n→+∞	VERB
ejpam-5982	97	28	tun	tun	PROPN
ejpam-5982	98	1	=	=	SYM
ejpam-5982	98	2	t	t	PROPN
ejpam-5982	98	3	for	for	ADP
ejpam-5982	98	4	some	some	DET
ejpam-5982	98	5	t	t	NOUN
ejpam-5982	98	6	∈	∈	NOUN
ejpam-5982	98	7	x	x	PUNCT
ejpam-5982	98	8	∩	∩	PROPN
ejpam-5982	98	9	y	y	PROPN
ejpam-5982	98	10	.	.	PUNCT
ejpam-5982	99	1	(	(	PUNCT
ejpam-5982	99	2	iii	iii	X
ejpam-5982	99	3	)	)	PUNCT
ejpam-5982	99	4	s	s	PROPN
ejpam-5982	99	5	and	and	CCONJ
ejpam-5982	99	6	t	t	PROPN
ejpam-5982	99	7	are	be	AUX
ejpam-5982	99	8	said	say	VERB
ejpam-5982	99	9	to	to	PART
ejpam-5982	99	10	be	be	AUX
ejpam-5982	99	11	weak	weak	ADJ
ejpam-5982	99	12	compatible	compatible	ADJ
ejpam-5982	99	13	mappings	mapping	NOUN
ejpam-5982	99	14	of	of	ADP
ejpam-5982	99	15	type	type	NOUN
ejpam-5982	99	16	(	(	PUNCT
ejpam-5982	99	17	a	a	NOUN
ejpam-5982	99	18	)	)	PUNCT
ejpam-5982	99	19	if	if	SCONJ
ejpam-5982	100	1	and	and	CCONJ
ejpam-5982	100	2	only	only	ADV
ejpam-5982	100	3	if	if	SCONJ
ejpam-5982	100	4	tu	tu	PROPN
ejpam-5982	100	5	=	=	PROPN
ejpam-5982	100	6	su	su	PROPN
ejpam-5982	100	7	for	for	ADP
ejpam-5982	100	8	some	some	DET
ejpam-5982	100	9	u	u	NOUN
ejpam-5982	100	10	∈	∈	PROPN
ejpam-5982	100	11	x	x	SYM
ejpam-5982	100	12	∩	∩	PROPN
ejpam-5982	100	13	y	y	PROPN
ejpam-5982	100	14	,	,	PUNCT
ejpam-5982	100	15	then	then	ADV
ejpam-5982	100	16	tsu	tsu	NOUN
ejpam-5982	100	17	=	=	SYM
ejpam-5982	100	18	ssu	ssu	PROPN
ejpam-5982	100	19	(	(	PUNCT
ejpam-5982	100	20	or	or	CCONJ
ejpam-5982	100	21	equivalently	equivalently	ADV
ejpam-5982	100	22	,	,	PUNCT
ejpam-5982	100	23	stu	stu	PROPN
ejpam-5982	100	24	=	=	PUNCT
ejpam-5982	100	25	ttu	ttu	PROPN
ejpam-5982	100	26	.	.	PUNCT
ejpam-5982	100	27	)	)	PUNCT
ejpam-5982	100	28	example	example	NOUN
ejpam-5982	101	1	1	1	X
ejpam-5982	101	2	.	.	PUNCT
ejpam-5982	102	1	let	let	VERB
ejpam-5982	102	2	x	x	SYM
ejpam-5982	102	3	=	=	PUNCT
ejpam-5982	102	4	n∪	n∪	X
ejpam-5982	102	5	{	{	PUNCT
ejpam-5982	102	6	0	0	NUM
ejpam-5982	102	7	}	}	PUNCT
ejpam-5982	102	8	,	,	PUNCT
ejpam-5982	102	9	y	y	PROPN
ejpam-5982	102	10	=	=	PUNCT
ejpam-5982	103	1	[	[	X
ejpam-5982	103	2	0	0	NUM
ejpam-5982	103	3	,	,	PUNCT
ejpam-5982	103	4	1	1	NUM
ejpam-5982	103	5	]	]	PUNCT
ejpam-5982	103	6	and	and	CCONJ
ejpam-5982	103	7	the	the	DET
ejpam-5982	103	8	metric	metric	NOUN
ejpam-5982	103	9	d	d	PROPN
ejpam-5982	103	10	is	be	AUX
ejpam-5982	103	11	defined	define	VERB
ejpam-5982	103	12	by	by	ADP
ejpam-5982	103	13	d(x	d(x	PROPN
ejpam-5982	103	14	,	,	PUNCT
ejpam-5982	103	15	y	y	NOUN
ejpam-5982	103	16	)	)	PUNCT
ejpam-5982	103	17	=	=	PUNCT
ejpam-5982	103	18	|x−	|x−	NOUN
ejpam-5982	103	19	y|	y|	NOUN
ejpam-5982	103	20	,	,	PUNCT
ejpam-5982	103	21	where	where	SCONJ
ejpam-5982	103	22	n	n	PRON
ejpam-5982	103	23	is	be	AUX
ejpam-5982	103	24	the	the	DET
ejpam-5982	103	25	set	set	NOUN
ejpam-5982	103	26	of	of	ADP
ejpam-5982	103	27	positive	positive	ADJ
ejpam-5982	103	28	integers	integer	NOUN
ejpam-5982	103	29	.	.	PUNCT
ejpam-5982	104	1	then	then	ADV
ejpam-5982	104	2	(	(	PUNCT
ejpam-5982	104	3	x	x	X
ejpam-5982	104	4	,	,	PUNCT
ejpam-5982	104	5	y	y	PROPN
ejpam-5982	104	6	,	,	PUNCT
ejpam-5982	104	7	d	d	NOUN
ejpam-5982	104	8	)	)	PUNCT
ejpam-5982	104	9	is	be	AUX
ejpam-5982	104	10	a	a	DET
ejpam-5982	104	11	bipolar	bipolar	ADJ
ejpam-5982	104	12	metric	metric	ADJ
ejpam-5982	104	13	space	space	NOUN
ejpam-5982	104	14	.	.	PUNCT
ejpam-5982	105	1	let	let	VERB
ejpam-5982	105	2	t	t	PROPN
ejpam-5982	105	3	(	(	PUNCT
ejpam-5982	105	4	covariant	covariant	PROPN
ejpam-5982	105	5	)	)	PUNCT
ejpam-5982	105	6	and	and	CCONJ
ejpam-5982	105	7	s	s	PROPN
ejpam-5982	105	8	(	(	PUNCT
ejpam-5982	105	9	contravariant	contravariant	ADJ
ejpam-5982	105	10	map	map	NOUN
ejpam-5982	105	11	)	)	PUNCT
ejpam-5982	105	12	are	be	AUX
ejpam-5982	105	13	defined	define	VERB
ejpam-5982	105	14	as	as	ADP
ejpam-5982	105	15	s(n	s(n	NOUN
ejpam-5982	105	16	)	)	PUNCT
ejpam-5982	105	17	=	=	SYM
ejpam-5982	105	18	1	1	NUM
ejpam-5982	105	19	n	n	NOUN
ejpam-5982	105	20	,	,	PUNCT
ejpam-5982	105	21	for	for	ADP
ejpam-5982	105	22	all	all	DET
ejpam-5982	105	23	n	n	PRON
ejpam-5982	105	24	∈	∈	PROPN
ejpam-5982	105	25	n	n	CCONJ
ejpam-5982	105	26	,	,	PUNCT
ejpam-5982	105	27	s(y	s(y	PROPN
ejpam-5982	105	28	)	)	PUNCT
ejpam-5982	106	1	=	=	SYM
ejpam-5982	106	2	1	1	NUM
ejpam-5982	106	3	,	,	PUNCT
ejpam-5982	106	4	for	for	ADP
ejpam-5982	106	5	all	all	DET
ejpam-5982	106	6	y	y	PROPN
ejpam-5982	106	7	∈	∈	PROPN
ejpam-5982	106	8	y	y	PROPN
ejpam-5982	106	9	,	,	PUNCT
ejpam-5982	106	10	t	t	PROPN
ejpam-5982	106	11	(	(	PUNCT
ejpam-5982	106	12	n	n	CCONJ
ejpam-5982	106	13	)	)	PUNCT
ejpam-5982	106	14	=	=	SYM
ejpam-5982	106	15	2n	2n	NUM
ejpam-5982	106	16	,	,	PUNCT
ejpam-5982	106	17	for	for	ADP
ejpam-5982	106	18	all	all	DET
ejpam-5982	106	19	n	n	PRON
ejpam-5982	106	20	∈	∈	NOUN
ejpam-5982	106	21	x	x	INTJ
ejpam-5982	106	22	−	−	PROPN
ejpam-5982	106	23	{	{	PUNCT
ejpam-5982	106	24	1	1	NUM
ejpam-5982	106	25	}	}	PUNCT
ejpam-5982	106	26	,	,	PUNCT
ejpam-5982	107	1	t1	t1	NOUN
ejpam-5982	107	2	=	=	SYM
ejpam-5982	107	3	0	0	PUNCT
ejpam-5982	108	1	ty	ty	NOUN
ejpam-5982	108	2	=	=	PUNCT
ejpam-5982	109	1			PROPN
ejpam-5982	109	2	2y	2y	NUM
ejpam-5982	109	3	,	,	PUNCT
ejpam-5982	109	4	if	if	SCONJ
ejpam-5982	109	5	0	0	NUM
ejpam-5982	109	6	≤	≤	NUM
ejpam-5982	109	7	y	y	SYM
ejpam-5982	109	8	≤	≤	NUM
ejpam-5982	109	9	1	1	NUM
ejpam-5982	109	10	2	2	NUM
ejpam-5982	109	11	1	1	NUM
ejpam-5982	109	12	2	2	NUM
ejpam-5982	109	13	,	,	PUNCT
ejpam-5982	109	14	if	if	SCONJ
ejpam-5982	109	15	1	1	NUM
ejpam-5982	109	16	2	2	NUM
ejpam-5982	109	17	<	<	X
ejpam-5982	109	18	y	y	X
ejpam-5982	109	19	<	<	X
ejpam-5982	109	20	1	1	NUM
ejpam-5982	109	21	then	then	ADV
ejpam-5982	109	22	the	the	DET
ejpam-5982	109	23	maps	map	NOUN
ejpam-5982	109	24	s	s	PART
ejpam-5982	109	25	and	and	CCONJ
ejpam-5982	109	26	t	t	PROPN
ejpam-5982	109	27	are	be	AUX
ejpam-5982	109	28	compatible	compatible	ADJ
ejpam-5982	109	29	of	of	ADP
ejpam-5982	109	30	type	type	NOUN
ejpam-5982	109	31	(	(	PUNCT
ejpam-5982	109	32	a	a	NOUN
ejpam-5982	109	33	)	)	PUNCT
ejpam-5982	109	34	with	with	ADP
ejpam-5982	109	35	respect	respect	NOUN
ejpam-5982	109	36	to	to	ADP
ejpam-5982	109	37	x	x	PUNCT
ejpam-5982	109	38	vacuously	vacuously	ADV
ejpam-5982	109	39	as	as	SCONJ
ejpam-5982	109	40	there	there	PRON
ejpam-5982	109	41	is	be	VERB
ejpam-5982	109	42	no	no	DET
ejpam-5982	109	43	sequence	sequence	NOUN
ejpam-5982	109	44	{	{	PUNCT
ejpam-5982	109	45	xn	xn	NOUN
ejpam-5982	109	46	}	}	PUNCT
ejpam-5982	109	47	in	in	ADP
ejpam-5982	109	48	x	x	SYM
ejpam-5982	109	49	such	such	ADJ
ejpam-5982	109	50	that	that	SCONJ
ejpam-5982	109	51	lim	lim	PROPN
ejpam-5982	109	52	n→+∞	n→+∞	VERB
ejpam-5982	109	53	sxn	sxn	NOUN
ejpam-5982	110	1	=	=	SYM
ejpam-5982	110	2	lim	lim	PROPN
ejpam-5982	110	3	n→+∞	n→+∞	VERB
ejpam-5982	110	4	txn	txn	NOUN
ejpam-5982	110	5	=	=	SYM
ejpam-5982	110	6	1	1	NUM
ejpam-5982	110	7	∈	∈	NOUN
ejpam-5982	110	8	x	x	SYM
ejpam-5982	111	1	∩	∩	PROPN
ejpam-5982	111	2	y	y	PROPN
ejpam-5982	111	3	or	or	CCONJ
ejpam-5982	111	4	lim	lim	PROPN
ejpam-5982	111	5	n→+∞	n→+∞	VERB
ejpam-5982	111	6	sxn	sxn	NOUN
ejpam-5982	112	1	=	=	SYM
ejpam-5982	112	2	lim	lim	PROPN
ejpam-5982	112	3	n→+∞	n→+∞	VERB
ejpam-5982	112	4	txn	txn	PROPN
ejpam-5982	112	5	=	=	SYM
ejpam-5982	112	6	0	0	NUM
ejpam-5982	112	7	∈	∈	PROPN
ejpam-5982	112	8	x	x	X
ejpam-5982	112	9	∩	∩	PROPN
ejpam-5982	112	10	y	y	PROPN
ejpam-5982	112	11	,	,	PUNCT
ejpam-5982	112	12	but	but	CCONJ
ejpam-5982	112	13	it	it	PRON
ejpam-5982	112	14	is	be	AUX
ejpam-5982	112	15	not	not	PART
ejpam-5982	112	16	compatible	compatible	ADJ
ejpam-5982	112	17	of	of	ADP
ejpam-5982	112	18	type	type	NOUN
ejpam-5982	112	19	(	(	PUNCT
ejpam-5982	112	20	a	a	NOUN
ejpam-5982	112	21	)	)	PUNCT
ejpam-5982	112	22	with	with	ADP
ejpam-5982	112	23	respect	respect	NOUN
ejpam-5982	112	24	to	to	ADP
ejpam-5982	112	25	y	y	PROPN
ejpam-5982	112	26	as	as	ADP
ejpam-5982	112	27	the	the	DET
ejpam-5982	112	28	sequence	sequence	NOUN
ejpam-5982	112	29	{	{	PUNCT
ejpam-5982	112	30	1	1	NUM
ejpam-5982	112	31	2	2	NUM
ejpam-5982	112	32	−	−	NUM
ejpam-5982	112	33	1	1	NUM
ejpam-5982	112	34	2n	2n	NUM
ejpam-5982	112	35	}	}	PUNCT
ejpam-5982	112	36	in	in	ADP
ejpam-5982	112	37	y	y	PROPN
ejpam-5982	112	38	has	have	VERB
ejpam-5982	112	39	the	the	DET
ejpam-5982	112	40	following	follow	VERB
ejpam-5982	112	41	property	property	NOUN
ejpam-5982	112	42	lim	lim	PROPN
ejpam-5982	112	43	n→+∞	n→+∞	VERB
ejpam-5982	112	44	syn	syn	PROPN
ejpam-5982	112	45	=	=	PROPN
ejpam-5982	112	46	lim	lim	PROPN
ejpam-5982	112	47	n→+∞	n→+∞	VERB
ejpam-5982	112	48	tyn	tyn	PROPN
ejpam-5982	113	1	=	=	SYM
ejpam-5982	113	2	1	1	NUM
ejpam-5982	114	1	but	but	CCONJ
ejpam-5982	114	2	lim	lim	PROPN
ejpam-5982	114	3	n→+∞	n→+∞	PROPN
ejpam-5982	114	4	d(tsyn	d(tsyn	PROPN
ejpam-5982	114	5	,	,	PUNCT
ejpam-5982	114	6	ssyn	ssyn	NOUN
ejpam-5982	114	7	)	)	PUNCT
ejpam-5982	114	8	=	=	PUNCT
ejpam-5982	115	1	d(0	d(0	NOUN
ejpam-5982	115	2	,	,	PUNCT
ejpam-5982	115	3	1	1	X
ejpam-5982	115	4	)	)	PUNCT
ejpam-5982	115	5	̸=	̸=	PROPN
ejpam-5982	115	6	0	0	NUM
ejpam-5982	115	7	and	and	CCONJ
ejpam-5982	115	8	lim	lim	PROPN
ejpam-5982	115	9	n→+∞	n→+∞	PROPN
ejpam-5982	115	10	d(styn	d(styn	PROPN
ejpam-5982	115	11	,	,	PUNCT
ejpam-5982	115	12	ttyn	ttyn	NOUN
ejpam-5982	115	13	)	)	PUNCT
ejpam-5982	115	14	=	=	PUNCT
ejpam-5982	116	1	d(1	d(1	PROPN
ejpam-5982	116	2	,	,	PUNCT
ejpam-5982	116	3	12	12	NUM
ejpam-5982	116	4	)	)	PUNCT
ejpam-5982	116	5	̸=	̸=	PROPN
ejpam-5982	116	6	0	0	NUM
ejpam-5982	116	7	.	.	PUNCT
ejpam-5982	117	1	p.	p.	NOUN
ejpam-5982	117	2	p.	p.	NOUN
ejpam-5982	118	1	murthy	murthy	PROPN
ejpam-5982	119	1	et	et	PROPN
ejpam-5982	119	2	al	al	PROPN
ejpam-5982	119	3	.	.	PUNCT
ejpam-5982	119	4	/	/	SYM
ejpam-5982	119	5	eur	eur	PROPN
ejpam-5982	119	6	.	.	PUNCT
ejpam-5982	120	1	j.	j.	PROPN
ejpam-5982	120	2	pure	pure	PROPN
ejpam-5982	120	3	appl	appl	PROPN
ejpam-5982	120	4	.	.	PROPN
ejpam-5982	120	5	math	math	PROPN
ejpam-5982	120	6	,	,	PUNCT
ejpam-5982	120	7	18	18	NUM
ejpam-5982	120	8	(	(	PUNCT
ejpam-5982	120	9	2	2	NUM
ejpam-5982	120	10	)	)	PUNCT
ejpam-5982	120	11	(	(	PUNCT
ejpam-5982	120	12	2025	2025	NUM
ejpam-5982	120	13	)	)	PUNCT
ejpam-5982	120	14	,	,	PUNCT
ejpam-5982	120	15	5982	5982	NUM
ejpam-5982	120	16	5	5	NUM
ejpam-5982	120	17	of	of	ADP
ejpam-5982	120	18	17	17	NUM
ejpam-5982	120	19	now	now	ADV
ejpam-5982	120	20	we	we	PRON
ejpam-5982	120	21	extend	extend	VERB
ejpam-5982	120	22	the	the	DET
ejpam-5982	120	23	definition	definition	NOUN
ejpam-5982	120	24	of	of	ADP
ejpam-5982	120	25	property	property	NOUN
ejpam-5982	120	26	(	(	PUNCT
ejpam-5982	120	27	e.a	e.a	PROPN
ejpam-5982	120	28	.	.	PROPN
ejpam-5982	120	29	)	)	PUNCT
ejpam-5982	121	1	[	[	X
ejpam-5982	121	2	5	5	X
ejpam-5982	121	3	]	]	PUNCT
ejpam-5982	121	4	to	to	ADP
ejpam-5982	121	5	bipolar	bipolar	ADJ
ejpam-5982	121	6	metric	metric	ADJ
ejpam-5982	121	7	space	space	NOUN
ejpam-5982	121	8	.	.	PUNCT
ejpam-5982	122	1	definition	definition	NOUN
ejpam-5982	122	2	7	7	NUM
ejpam-5982	122	3	.	.	PUNCT
ejpam-5982	123	1	let	let	VERB
ejpam-5982	123	2	(	(	PUNCT
ejpam-5982	123	3	x	x	X
ejpam-5982	123	4	,	,	PUNCT
ejpam-5982	123	5	y	y	PROPN
ejpam-5982	123	6	,	,	PUNCT
ejpam-5982	123	7	d	d	NOUN
ejpam-5982	123	8	)	)	PUNCT
ejpam-5982	123	9	be	be	AUX
ejpam-5982	123	10	a	a	DET
ejpam-5982	123	11	bipolar	bipolar	ADJ
ejpam-5982	123	12	metric	metric	ADJ
ejpam-5982	123	13	space	space	NOUN
ejpam-5982	123	14	and	and	CCONJ
ejpam-5982	123	15	let	let	VERB
ejpam-5982	123	16	t1	t1	NOUN
ejpam-5982	123	17	,	,	PUNCT
ejpam-5982	123	18	t2	t2	NOUN
ejpam-5982	123	19	:	:	PUNCT
ejpam-5982	123	20	(	(	PUNCT
ejpam-5982	123	21	x	x	X
ejpam-5982	123	22	,	,	PUNCT
ejpam-5982	123	23	y	y	PROPN
ejpam-5982	123	24	,	,	PUNCT
ejpam-5982	123	25	d	d	NOUN
ejpam-5982	123	26	)	)	PUNCT
ejpam-5982	123	27	⇒	⇒	NOUN
ejpam-5982	123	28	(	(	PUNCT
ejpam-5982	123	29	x	x	X
ejpam-5982	123	30	,	,	PUNCT
ejpam-5982	123	31	y	y	PROPN
ejpam-5982	123	32	,	,	PUNCT
ejpam-5982	123	33	d	d	NOUN
ejpam-5982	123	34	)	)	PUNCT
ejpam-5982	123	35	be	be	AUX
ejpam-5982	123	36	covariant	covariant	ADJ
ejpam-5982	123	37	maps	map	NOUN
ejpam-5982	123	38	and	and	CCONJ
ejpam-5982	123	39	s1	s1	NOUN
ejpam-5982	123	40	,	,	PUNCT
ejpam-5982	123	41	s2	s2	NOUN
ejpam-5982	123	42	:	:	PUNCT
ejpam-5982	123	43	(	(	PUNCT
ejpam-5982	123	44	x	x	X
ejpam-5982	123	45	,	,	PUNCT
ejpam-5982	123	46	y	y	PROPN
ejpam-5982	123	47	,	,	PUNCT
ejpam-5982	123	48	d	d	NOUN
ejpam-5982	123	49	)	)	PUNCT
ejpam-5982	123	50	⇄	⇄	NOUN
ejpam-5982	123	51	(	(	PUNCT
ejpam-5982	123	52	x	x	X
ejpam-5982	123	53	,	,	PUNCT
ejpam-5982	123	54	y	y	PROPN
ejpam-5982	123	55	,	,	PUNCT
ejpam-5982	123	56	d	d	NOUN
ejpam-5982	123	57	)	)	PUNCT
ejpam-5982	123	58	be	be	AUX
ejpam-5982	123	59	contravariant	contravariant	ADJ
ejpam-5982	123	60	maps	map	NOUN
ejpam-5982	123	61	.	.	PUNCT
ejpam-5982	124	1	we	we	PRON
ejpam-5982	124	2	say	say	VERB
ejpam-5982	124	3	that	that	SCONJ
ejpam-5982	124	4	(	(	PUNCT
ejpam-5982	124	5	i	i	NOUN
ejpam-5982	124	6	)	)	PUNCT
ejpam-5982	124	7	t1	t1	NOUN
ejpam-5982	124	8	and	and	CCONJ
ejpam-5982	124	9	s1	s1	NOUN
ejpam-5982	124	10	satisfy	satisfy	VERB
ejpam-5982	124	11	the	the	DET
ejpam-5982	124	12	property	property	NOUN
ejpam-5982	124	13	(	(	PUNCT
ejpam-5982	124	14	e.a	e.a	PROPN
ejpam-5982	124	15	.	.	PROPN
ejpam-5982	124	16	)	)	PUNCT
ejpam-5982	125	1	if	if	SCONJ
ejpam-5982	125	2	there	there	PRON
ejpam-5982	125	3	exists	exist	VERB
ejpam-5982	125	4	a	a	DET
ejpam-5982	125	5	sequence	sequence	NOUN
ejpam-5982	125	6	{	{	PUNCT
ejpam-5982	125	7	xn	xn	NOUN
ejpam-5982	125	8	}	}	PUNCT
ejpam-5982	125	9	in	in	ADP
ejpam-5982	125	10	x	x	X
ejpam-5982	125	11	and	and	CCONJ
ejpam-5982	125	12	a	a	DET
ejpam-5982	125	13	sequence	sequence	NOUN
ejpam-5982	125	14	{	{	PUNCT
ejpam-5982	125	15	yn	yn	NOUN
ejpam-5982	125	16	}	}	PUNCT
ejpam-5982	125	17	in	in	ADP
ejpam-5982	125	18	y	y	PRON
ejpam-5982	125	19	such	such	ADJ
ejpam-5982	125	20	that	that	PRON
ejpam-5982	125	21	limn→+∞	limn→+∞	VERB
ejpam-5982	125	22	s1xn	s1xn	PROPN
ejpam-5982	126	1	=	=	SYM
ejpam-5982	126	2	lim	lim	PROPN
ejpam-5982	126	3	n→+∞	n→+∞	VERB
ejpam-5982	126	4	t1xn	t1xn	PROPN
ejpam-5982	127	1	=	=	PRON
ejpam-5982	127	2	lim	lim	PROPN
ejpam-5982	127	3	n→+∞	n→+∞	VERB
ejpam-5982	127	4	s1yn	s1yn	NUM
ejpam-5982	128	1	=	=	NOUN
ejpam-5982	128	2	lim	lim	PROPN
ejpam-5982	128	3	n→+∞	n→+∞	VERB
ejpam-5982	128	4	t1yn	t1yn	PUNCT
ejpam-5982	129	1	=	=	SYM
ejpam-5982	129	2	t	t	PROPN
ejpam-5982	129	3	for	for	ADP
ejpam-5982	129	4	some	some	DET
ejpam-5982	129	5	t	t	NOUN
ejpam-5982	129	6	∈	∈	NOUN
ejpam-5982	129	7	x	x	SYM
ejpam-5982	129	8	∩	∩	PROPN
ejpam-5982	129	9	y	y	PROPN
ejpam-5982	129	10	(	(	PUNCT
ejpam-5982	129	11	ii	ii	PROPN
ejpam-5982	129	12	)	)	PUNCT
ejpam-5982	129	13	t1	t1	NOUN
ejpam-5982	129	14	and	and	CCONJ
ejpam-5982	129	15	s1	s1	NOUN
ejpam-5982	129	16	satisfy	satisfy	VERB
ejpam-5982	129	17	the	the	DET
ejpam-5982	129	18	weak	weak	ADJ
ejpam-5982	129	19	form	form	NOUN
ejpam-5982	129	20	of	of	ADP
ejpam-5982	129	21	property	property	NOUN
ejpam-5982	129	22	(	(	PUNCT
ejpam-5982	129	23	e.a	e.a	PROPN
ejpam-5982	129	24	.	.	PROPN
ejpam-5982	129	25	)	)	PUNCT
ejpam-5982	130	1	if	if	SCONJ
ejpam-5982	130	2	there	there	PRON
ejpam-5982	130	3	exists	exist	VERB
ejpam-5982	130	4	a	a	DET
ejpam-5982	130	5	sequence	sequence	NOUN
ejpam-5982	130	6	{	{	PUNCT
ejpam-5982	130	7	un	un	PROPN
ejpam-5982	130	8	}	}	PUNCT
ejpam-5982	130	9	in	in	ADP
ejpam-5982	130	10	x	x	INTJ
ejpam-5982	130	11	or	or	CCONJ
ejpam-5982	130	12	y	y	PRON
ejpam-5982	130	13	such	such	ADJ
ejpam-5982	130	14	that	that	SCONJ
ejpam-5982	130	15	lim	lim	PROPN
ejpam-5982	130	16	n→+∞	n→+∞	VERB
ejpam-5982	130	17	s1un	s1un	X
ejpam-5982	131	1	=	=	SYM
ejpam-5982	131	2	lim	lim	PROPN
ejpam-5982	131	3	n→+∞	n→+∞	VERB
ejpam-5982	131	4	t1un	t1un	PUNCT
ejpam-5982	131	5	=	=	SYM
ejpam-5982	131	6	t	t	PROPN
ejpam-5982	131	7	for	for	ADP
ejpam-5982	131	8	some	some	DET
ejpam-5982	131	9	t	t	NOUN
ejpam-5982	131	10	∈	∈	NOUN
ejpam-5982	131	11	x	x	SYM
ejpam-5982	131	12	∩	∩	PROPN
ejpam-5982	131	13	y	y	PROPN
ejpam-5982	131	14	(	(	PUNCT
ejpam-5982	131	15	iii	iii	NOUN
ejpam-5982	131	16	)	)	PUNCT
ejpam-5982	131	17	the	the	DET
ejpam-5982	131	18	quadruple	quadruple	NOUN
ejpam-5982	131	19	(	(	PUNCT
ejpam-5982	131	20	s1	s1	NOUN
ejpam-5982	131	21	,	,	PUNCT
ejpam-5982	131	22	t1	t1	NOUN
ejpam-5982	131	23	,	,	PUNCT
ejpam-5982	131	24	s2	s2	PROPN
ejpam-5982	131	25	,	,	PUNCT
ejpam-5982	131	26	t2	t2	NOUN
ejpam-5982	131	27	)	)	PUNCT
ejpam-5982	131	28	satisfies	satisfy	VERB
ejpam-5982	131	29	the	the	DET
ejpam-5982	131	30	property	property	NOUN
ejpam-5982	131	31	(	(	PUNCT
ejpam-5982	131	32	e.a	e.a	PROPN
ejpam-5982	131	33	.	.	PROPN
ejpam-5982	131	34	)	)	PUNCT
ejpam-5982	132	1	if	if	SCONJ
ejpam-5982	132	2	there	there	PRON
ejpam-5982	132	3	exists	exist	VERB
ejpam-5982	132	4	a	a	DET
ejpam-5982	132	5	sequence	sequence	NOUN
ejpam-5982	132	6	{	{	PUNCT
ejpam-5982	132	7	xn	xn	NOUN
ejpam-5982	132	8	}	}	PUNCT
ejpam-5982	132	9	in	in	ADP
ejpam-5982	132	10	x	x	X
ejpam-5982	132	11	and	and	CCONJ
ejpam-5982	132	12	a	a	DET
ejpam-5982	132	13	sequence	sequence	NOUN
ejpam-5982	132	14	{	{	PUNCT
ejpam-5982	132	15	yn	yn	NOUN
ejpam-5982	132	16	}	}	PUNCT
ejpam-5982	132	17	in	in	ADP
ejpam-5982	132	18	y	y	PRON
ejpam-5982	132	19	such	such	ADJ
ejpam-5982	132	20	that	that	SCONJ
ejpam-5982	132	21	lim	lim	PROPN
ejpam-5982	132	22	n→+∞	n→+∞	VERB
ejpam-5982	132	23	s1xn	s1xn	PUNCT
ejpam-5982	133	1	=	=	SYM
ejpam-5982	133	2	lim	lim	PROPN
ejpam-5982	133	3	n→+∞	n→+∞	VERB
ejpam-5982	133	4	t1xn	t1xn	PROPN
ejpam-5982	134	1	=	=	PRON
ejpam-5982	134	2	lim	lim	PROPN
ejpam-5982	134	3	n→+∞	n→+∞	VERB
ejpam-5982	134	4	s2yn	s2yn	PUNCT
ejpam-5982	134	5	=	=	PROPN
ejpam-5982	134	6	lim	lim	PROPN
ejpam-5982	134	7	n→+∞	n→+∞	VERB
ejpam-5982	134	8	t2yn	t2yn	NUM
ejpam-5982	134	9	=	=	SYM
ejpam-5982	134	10	t	t	PROPN
ejpam-5982	134	11	for	for	ADP
ejpam-5982	134	12	some	some	DET
ejpam-5982	134	13	t	t	NOUN
ejpam-5982	134	14	∈	∈	NOUN
ejpam-5982	134	15	x	x	SYM
ejpam-5982	134	16	∩	∩	PROPN
ejpam-5982	134	17	y	y	PROPN
ejpam-5982	134	18	the	the	DET
ejpam-5982	134	19	following	follow	VERB
ejpam-5982	134	20	proposition	proposition	NOUN
ejpam-5982	134	21	gives	give	VERB
ejpam-5982	134	22	the	the	DET
ejpam-5982	134	23	connection	connection	NOUN
ejpam-5982	134	24	between	between	ADP
ejpam-5982	134	25	compatible	compatible	ADJ
ejpam-5982	134	26	mappings	mapping	NOUN
ejpam-5982	134	27	of	of	ADP
ejpam-5982	134	28	type	type	NOUN
ejpam-5982	134	29	(	(	PUNCT
ejpam-5982	134	30	a	a	NOUN
ejpam-5982	134	31	)	)	PUNCT
ejpam-5982	134	32	and	and	CCONJ
ejpam-5982	134	33	a	a	DET
ejpam-5982	134	34	weak	weak	ADJ
ejpam-5982	134	35	compatible	compatible	ADJ
ejpam-5982	134	36	mappings	mapping	NOUN
ejpam-5982	134	37	of	of	ADP
ejpam-5982	134	38	type	type	NOUN
ejpam-5982	134	39	(	(	PUNCT
ejpam-5982	134	40	a	a	NOUN
ejpam-5982	134	41	)	)	PUNCT
ejpam-5982	134	42	.	.	PUNCT
ejpam-5982	135	1	proposition	proposition	NOUN
ejpam-5982	135	2	1	1	NUM
ejpam-5982	135	3	.	.	PUNCT
ejpam-5982	136	1	if	if	SCONJ
ejpam-5982	136	2	s	s	PRON
ejpam-5982	136	3	and	and	CCONJ
ejpam-5982	136	4	t	t	PROPN
ejpam-5982	136	5	are	be	AUX
ejpam-5982	136	6	mappings	mapping	NOUN
ejpam-5982	136	7	of	of	ADP
ejpam-5982	136	8	type	type	NOUN
ejpam-5982	136	9	(	(	PUNCT
ejpam-5982	136	10	a	a	NOUN
ejpam-5982	136	11	)	)	PUNCT
ejpam-5982	136	12	with	with	ADP
ejpam-5982	136	13	respect	respect	NOUN
ejpam-5982	136	14	to	to	ADP
ejpam-5982	136	15	x	x	PUNCT
ejpam-5982	136	16	or	or	CCONJ
ejpam-5982	136	17	y	y	PROPN
ejpam-5982	136	18	,	,	PUNCT
ejpam-5982	136	19	then	then	ADV
ejpam-5982	136	20	they	they	PRON
ejpam-5982	136	21	are	be	AUX
ejpam-5982	136	22	weak	weak	ADJ
ejpam-5982	136	23	compatible	compatible	ADJ
ejpam-5982	136	24	mappings	mapping	NOUN
ejpam-5982	136	25	of	of	ADP
ejpam-5982	136	26	type	type	NOUN
ejpam-5982	136	27	(	(	PUNCT
ejpam-5982	136	28	a	a	NOUN
ejpam-5982	136	29	)	)	PUNCT
ejpam-5982	136	30	.	.	PUNCT
ejpam-5982	137	1	proof	proof	NOUN
ejpam-5982	137	2	.	.	PUNCT
ejpam-5982	138	1	let	let	VERB
ejpam-5982	138	2	s	s	PRON
ejpam-5982	138	3	and	and	CCONJ
ejpam-5982	138	4	t	t	PROPN
ejpam-5982	138	5	are	be	AUX
ejpam-5982	138	6	compatible	compatible	ADJ
ejpam-5982	138	7	mappings	mapping	NOUN
ejpam-5982	138	8	of	of	ADP
ejpam-5982	138	9	type	type	NOUN
ejpam-5982	138	10	(	(	PUNCT
ejpam-5982	138	11	a	a	NOUN
ejpam-5982	138	12	)	)	PUNCT
ejpam-5982	138	13	with	with	ADP
ejpam-5982	138	14	respect	respect	NOUN
ejpam-5982	138	15	to	to	ADP
ejpam-5982	138	16	x	x	PUNCT
ejpam-5982	138	17	or	or	CCONJ
ejpam-5982	138	18	y	y	PROPN
ejpam-5982	138	19	with	with	ADP
ejpam-5982	138	20	su	su	PROPN
ejpam-5982	138	21	=	=	PROPN
ejpam-5982	138	22	tu	tu	PROPN
ejpam-5982	138	23	for	for	ADP
ejpam-5982	138	24	some	some	DET
ejpam-5982	138	25	u	u	NOUN
ejpam-5982	138	26	∈	∈	PROPN
ejpam-5982	138	27	x	x	SYM
ejpam-5982	138	28	∩	∩	PROPN
ejpam-5982	138	29	y	y	PROPN
ejpam-5982	138	30	,	,	PUNCT
ejpam-5982	138	31	then	then	ADV
ejpam-5982	138	32	the	the	DET
ejpam-5982	138	33	proposition	proposition	NOUN
ejpam-5982	138	34	can	can	AUX
ejpam-5982	138	35	be	be	AUX
ejpam-5982	138	36	proved	prove	VERB
ejpam-5982	138	37	easily	easily	ADV
ejpam-5982	138	38	by	by	ADP
ejpam-5982	138	39	taking	take	VERB
ejpam-5982	138	40	un	un	PROPN
ejpam-5982	138	41	=	=	PROPN
ejpam-5982	138	42	u	u	PROPN
ejpam-5982	138	43	in	in	ADP
ejpam-5982	138	44	the	the	DET
ejpam-5982	138	45	definitions	definition	NOUN
ejpam-5982	138	46	of	of	ADP
ejpam-5982	138	47	compatible	compatible	ADJ
ejpam-5982	138	48	mappings	mapping	NOUN
ejpam-5982	138	49	of	of	ADP
ejpam-5982	138	50	type	type	NOUN
ejpam-5982	138	51	(	(	PUNCT
ejpam-5982	138	52	a	a	NOUN
ejpam-5982	138	53	)	)	PUNCT
ejpam-5982	138	54	with	with	ADP
ejpam-5982	138	55	respect	respect	NOUN
ejpam-5982	138	56	to	to	ADP
ejpam-5982	138	57	x	x	PUNCT
ejpam-5982	138	58	and	and	CCONJ
ejpam-5982	138	59	y.	y.	NOUN
ejpam-5982	138	60	we	we	PRON
ejpam-5982	138	61	now	now	ADV
ejpam-5982	138	62	introduce	introduce	VERB
ejpam-5982	138	63	the	the	DET
ejpam-5982	138	64	following	follow	VERB
ejpam-5982	138	65	class	class	NOUN
ejpam-5982	138	66	of	of	ADP
ejpam-5982	138	67	implicit	implicit	ADJ
ejpam-5982	138	68	functions	function	NOUN
ejpam-5982	138	69	:	:	PUNCT
ejpam-5982	138	70	let	let	VERB
ejpam-5982	138	71	ψ	ψ	PART
ejpam-5982	138	72	be	be	AUX
ejpam-5982	138	73	the	the	DET
ejpam-5982	138	74	collection	collection	NOUN
ejpam-5982	138	75	of	of	ADP
ejpam-5982	138	76	all	all	DET
ejpam-5982	138	77	real	real	ADV
ejpam-5982	138	78	-	-	PUNCT
ejpam-5982	138	79	valued	value	VERB
ejpam-5982	138	80	function	function	NOUN
ejpam-5982	138	81	ψ	ψ	X
ejpam-5982	138	82	:	:	PUNCT
ejpam-5982	139	1	[	[	X
ejpam-5982	139	2	0,+∞)4	0,+∞)4	NOUN
ejpam-5982	139	3	→	→	SYM
ejpam-5982	139	4	r	r	NOUN
ejpam-5982	139	5	satisfying	satisfy	VERB
ejpam-5982	139	6	the	the	DET
ejpam-5982	139	7	following	follow	VERB
ejpam-5982	139	8	conditions	condition	NOUN
ejpam-5982	139	9	:	:	PUNCT
ejpam-5982	139	10	(	(	PUNCT
ejpam-5982	139	11	ψa	ψa	X
ejpam-5982	139	12	):	):	PUNCT
ejpam-5982	139	13	if	if	SCONJ
ejpam-5982	139	14	ψ(a	ψ(a	PROPN
ejpam-5982	139	15	,	,	PUNCT
ejpam-5982	139	16	b	b	PROPN
ejpam-5982	139	17	,	,	PUNCT
ejpam-5982	139	18	a	a	DET
ejpam-5982	139	19	,	,	PUNCT
ejpam-5982	139	20	b	b	NOUN
ejpam-5982	139	21	)	)	PUNCT
ejpam-5982	139	22	≤	≤	NOUN
ejpam-5982	139	23	0	0	NUM
ejpam-5982	139	24	or	or	CCONJ
ejpam-5982	139	25	ψ(a	ψ(a	PROPN
ejpam-5982	139	26	,	,	PUNCT
ejpam-5982	139	27	b	b	PROPN
ejpam-5982	139	28	,	,	PUNCT
ejpam-5982	139	29	b	b	PROPN
ejpam-5982	139	30	,	,	PUNCT
ejpam-5982	139	31	a	a	PRON
ejpam-5982	139	32	)	)	PUNCT
ejpam-5982	139	33	≤	≤	NOUN
ejpam-5982	139	34	0	0	PUNCT
ejpam-5982	139	35	then	then	ADV
ejpam-5982	139	36	there	there	PRON
ejpam-5982	139	37	exists	exist	VERB
ejpam-5982	139	38	k	k	PROPN
ejpam-5982	139	39	∈	∈	PROPN
ejpam-5982	140	1	[	[	X
ejpam-5982	140	2	0	0	NUM
ejpam-5982	140	3	,	,	PUNCT
ejpam-5982	140	4	1	1	NUM
ejpam-5982	140	5	)	)	PUNCT
ejpam-5982	140	6	such	such	ADJ
ejpam-5982	140	7	that	that	SCONJ
ejpam-5982	140	8	a	a	DET
ejpam-5982	140	9	≤	≤	PROPN
ejpam-5982	140	10	kb	kb	PROPN
ejpam-5982	140	11	.	.	PUNCT
ejpam-5982	141	1	(	(	PUNCT
ejpam-5982	141	2	ψb	ψb	ADV
ejpam-5982	141	3	):	):	PUNCT
ejpam-5982	141	4	if	if	SCONJ
ejpam-5982	141	5	ψ(a	ψ(a	PROPN
ejpam-5982	141	6	,	,	PUNCT
ejpam-5982	141	7	a	a	PRON
ejpam-5982	141	8	,	,	PUNCT
ejpam-5982	141	9	0	0	NUM
ejpam-5982	141	10	,	,	PUNCT
ejpam-5982	141	11	0	0	NUM
ejpam-5982	141	12	)	)	PUNCT
ejpam-5982	141	13	>	>	X
ejpam-5982	141	14	0	0	PUNCT
ejpam-5982	141	15	for	for	ADP
ejpam-5982	141	16	all	all	DET
ejpam-5982	141	17	a	a	DET
ejpam-5982	141	18	>	>	X
ejpam-5982	141	19	0	0	X
ejpam-5982	141	20	.	.	PUNCT
ejpam-5982	141	21	remark	remark	PROPN
ejpam-5982	141	22	2	2	NUM
ejpam-5982	141	23	.	.	PUNCT
ejpam-5982	141	24	by	by	ADP
ejpam-5982	141	25	definition	definition	NOUN
ejpam-5982	141	26	of	of	ADP
ejpam-5982	141	27	ψ	ψ	X
ejpam-5982	141	28	,	,	PUNCT
ejpam-5982	141	29	it	it	PRON
ejpam-5982	141	30	is	be	AUX
ejpam-5982	141	31	clear	clear	ADJ
ejpam-5982	141	32	that	that	SCONJ
ejpam-5982	141	33	the	the	DET
ejpam-5982	141	34	following	follow	VERB
ejpam-5982	141	35	implications	implication	NOUN
ejpam-5982	141	36	hold	hold	VERB
ejpam-5982	141	37	:	:	PUNCT
ejpam-5982	141	38	•	•	ADP
ejpam-5982	141	39	ψ(a	ψ(a	PROPN
ejpam-5982	141	40	,	,	PUNCT
ejpam-5982	141	41	a	a	PRON
ejpam-5982	141	42	,	,	PUNCT
ejpam-5982	141	43	0	0	NUM
ejpam-5982	141	44	,	,	PUNCT
ejpam-5982	141	45	0	0	NUM
ejpam-5982	141	46	)	)	PUNCT
ejpam-5982	141	47	≤	≤	NOUN
ejpam-5982	141	48	0	0	NUM
ejpam-5982	141	49	implies	imply	VERB
ejpam-5982	141	50	a	a	DET
ejpam-5982	141	51	=	=	SYM
ejpam-5982	141	52	0	0	NUM
ejpam-5982	141	53	.	.	NOUN
ejpam-5982	141	54	•	•	NUM
ejpam-5982	141	55	ψ(a	ψ(a	PROPN
ejpam-5982	141	56	,	,	PUNCT
ejpam-5982	141	57	0	0	NUM
ejpam-5982	141	58	,	,	PUNCT
ejpam-5982	141	59	0	0	NUM
ejpam-5982	141	60	,	,	PUNCT
ejpam-5982	141	61	a	a	DET
ejpam-5982	141	62	)	)	PUNCT
ejpam-5982	141	63	≤	≤	NOUN
ejpam-5982	141	64	0	0	NUM
ejpam-5982	141	65	implies	imply	VERB
ejpam-5982	141	66	a	a	DET
ejpam-5982	141	67	=	=	SYM
ejpam-5982	141	68	0	0	NUM
ejpam-5982	141	69	.	.	NOUN
ejpam-5982	141	70	•	•	NUM
ejpam-5982	141	71	ψ(a	ψ(a	PROPN
ejpam-5982	141	72	,	,	PUNCT
ejpam-5982	141	73	a	a	DET
ejpam-5982	141	74	,	,	PUNCT
ejpam-5982	141	75	a	a	DET
ejpam-5982	141	76	,	,	PUNCT
ejpam-5982	141	77	a	a	PRON
ejpam-5982	141	78	)	)	PUNCT
ejpam-5982	141	79	≤	≤	NOUN
ejpam-5982	141	80	0	0	NUM
ejpam-5982	141	81	implies	imply	VERB
ejpam-5982	141	82	a	a	DET
ejpam-5982	141	83	=	=	SYM
ejpam-5982	141	84	0	0	PROPN
ejpam-5982	141	85	.	.	NOUN
ejpam-5982	141	86	example	example	NOUN
ejpam-5982	142	1	2	2	NUM
ejpam-5982	142	2	.	.	X
ejpam-5982	142	3	the	the	DET
ejpam-5982	142	4	following	follow	VERB
ejpam-5982	142	5	functions	function	NOUN
ejpam-5982	142	6	are	be	AUX
ejpam-5982	142	7	members	member	NOUN
ejpam-5982	142	8	of	of	ADP
ejpam-5982	142	9	ψ	ψ	NOUN
ejpam-5982	142	10	.	.	NOUN
ejpam-5982	143	1	•	•	NUM
ejpam-5982	143	2	ψ1(a	ψ1(a	PROPN
ejpam-5982	143	3	,	,	PUNCT
ejpam-5982	143	4	b	b	PROPN
ejpam-5982	143	5	,	,	PUNCT
ejpam-5982	143	6	c	c	NOUN
ejpam-5982	143	7	,	,	PUNCT
ejpam-5982	143	8	d	d	NOUN
ejpam-5982	143	9	)	)	PUNCT
ejpam-5982	143	10	=	=	SYM
ejpam-5982	143	11	a−	a−	PROPN
ejpam-5982	143	12	k1b−	k1b−	PROPN
ejpam-5982	143	13	k2c−	k2c−	PROPN
ejpam-5982	143	14	k3d	k3d	PROPN
ejpam-5982	143	15	,	,	PUNCT
ejpam-5982	143	16	k1	k1	PROPN
ejpam-5982	143	17	,	,	PUNCT
ejpam-5982	143	18	k2	k2	NOUN
ejpam-5982	143	19	,	,	PUNCT
ejpam-5982	143	20	k3	k3	VERB
ejpam-5982	143	21	≥	≥	NOUN
ejpam-5982	143	22	0	0	NUM
ejpam-5982	143	23	,	,	PUNCT
ejpam-5982	143	24	k1	k1	X
ejpam-5982	143	25	+	+	CCONJ
ejpam-5982	143	26	k2	k2	NOUN
ejpam-5982	143	27	+	+	CCONJ
ejpam-5982	143	28	k3	k3	VERB
ejpam-5982	143	29	<	<	X
ejpam-5982	143	30	1	1	NUM
ejpam-5982	143	31	if	if	SCONJ
ejpam-5982	143	32	ψ1(a	ψ1(a	PROPN
ejpam-5982	143	33	,	,	PUNCT
ejpam-5982	143	34	b	b	PROPN
ejpam-5982	143	35	,	,	PUNCT
ejpam-5982	143	36	a	a	DET
ejpam-5982	143	37	,	,	PUNCT
ejpam-5982	143	38	b	b	NOUN
ejpam-5982	143	39	)	)	PUNCT
ejpam-5982	143	40	=	=	SYM
ejpam-5982	143	41	a−k1b−k2a−k3b	a−k1b−k2a−k3b	PROPN
ejpam-5982	143	42	≤	≤	NUM
ejpam-5982	143	43	0	0	PUNCT
ejpam-5982	144	1	then	then	ADV
ejpam-5982	144	2	a	a	DET
ejpam-5982	144	3	≤	≤	NOUN
ejpam-5982	144	4	(	(	PUNCT
ejpam-5982	144	5	k1	k1	NOUN
ejpam-5982	144	6	+	+	CCONJ
ejpam-5982	144	7	k3	k3	ADJ
ejpam-5982	144	8	1−	1−	PROPN
ejpam-5982	144	9	k2	k2	PROPN
ejpam-5982	144	10	)	)	PUNCT
ejpam-5982	144	11	b	b	NOUN
ejpam-5982	144	12	with	with	ADP
ejpam-5982	144	13	0	0	NUM
ejpam-5982	144	14	<	<	X
ejpam-5982	144	15	k1	k1	NOUN
ejpam-5982	144	16	+	+	CCONJ
ejpam-5982	144	17	k3	k3	PROPN
ejpam-5982	144	18	1−	1−	PROPN
ejpam-5982	144	19	k2	k2	X
ejpam-5982	144	20	<	<	X
ejpam-5982	144	21	1	1	NUM
ejpam-5982	144	22	.	.	PUNCT
ejpam-5982	145	1	if	if	SCONJ
ejpam-5982	145	2	ψ1(a	ψ1(a	PROPN
ejpam-5982	145	3	,	,	PUNCT
ejpam-5982	145	4	b	b	PROPN
ejpam-5982	145	5	,	,	PUNCT
ejpam-5982	145	6	b	b	PROPN
ejpam-5982	145	7	,	,	PUNCT
ejpam-5982	145	8	a	a	PRON
ejpam-5982	145	9	)	)	PUNCT
ejpam-5982	145	10	=	=	SYM
ejpam-5982	145	11	a−k1b−k2b−k3a	a−k1b−k2b−k3a	ADP
ejpam-5982	145	12	≤	≤	NOUN
ejpam-5982	145	13	0	0	NUM
ejpam-5982	146	1	then	then	ADV
ejpam-5982	146	2	a	a	DET
ejpam-5982	146	3	≤	≤	NOUN
ejpam-5982	146	4	(	(	PUNCT
ejpam-5982	146	5	k1	k1	NOUN
ejpam-5982	146	6	+	+	CCONJ
ejpam-5982	146	7	k2	k2	ADJ
ejpam-5982	146	8	1−	1−	NUM
ejpam-5982	146	9	k3	k3	ADJ
ejpam-5982	146	10	)	)	PUNCT
ejpam-5982	146	11	b	b	NOUN
ejpam-5982	146	12	with	with	ADP
ejpam-5982	146	13	0	0	NUM
ejpam-5982	146	14	<	<	X
ejpam-5982	146	15	k1	k1	PROPN
ejpam-5982	146	16	+	+	CCONJ
ejpam-5982	146	17	k2	k2	X
ejpam-5982	146	18	1−	1−	NUM
ejpam-5982	146	19	k3	k3	VERB
ejpam-5982	146	20	<	<	X
ejpam-5982	146	21	1	1	NUM
ejpam-5982	146	22	.	.	PUNCT
ejpam-5982	147	1	take	take	VERB
ejpam-5982	147	2	k	k	NOUN
ejpam-5982	147	3	=	=	X
ejpam-5982	147	4	max	max	PROPN
ejpam-5982	147	5	{	{	PUNCT
ejpam-5982	147	6	k1	k1	PROPN
ejpam-5982	147	7	+	+	CCONJ
ejpam-5982	147	8	k3	k3	ADJ
ejpam-5982	147	9	1−	1−	PROPN
ejpam-5982	147	10	k2	k2	NOUN
ejpam-5982	147	11	,	,	PUNCT
ejpam-5982	147	12	k1	k1	PROPN
ejpam-5982	147	13	+	+	CCONJ
ejpam-5982	147	14	k2	k2	ADJ
ejpam-5982	147	15	1−	1−	NUM
ejpam-5982	147	16	k3	k3	PROPN
ejpam-5982	147	17	}	}	PUNCT
ejpam-5982	148	1	p.	p.	NOUN
ejpam-5982	148	2	p.	p.	NOUN
ejpam-5982	149	1	murthy	murthy	ADJ
ejpam-5982	150	1	et	et	PROPN
ejpam-5982	150	2	al	al	PROPN
ejpam-5982	150	3	.	.	PUNCT
ejpam-5982	150	4	/	/	SYM
ejpam-5982	150	5	eur	eur	PROPN
ejpam-5982	150	6	.	.	PUNCT
ejpam-5982	151	1	j.	j.	PROPN
ejpam-5982	151	2	pure	pure	PROPN
ejpam-5982	151	3	appl	appl	PROPN
ejpam-5982	151	4	.	.	PROPN
ejpam-5982	151	5	math	math	PROPN
ejpam-5982	151	6	,	,	PUNCT
ejpam-5982	151	7	18	18	NUM
ejpam-5982	151	8	(	(	PUNCT
ejpam-5982	151	9	2	2	NUM
ejpam-5982	151	10	)	)	PUNCT
ejpam-5982	151	11	(	(	PUNCT
ejpam-5982	151	12	2025	2025	NUM
ejpam-5982	151	13	)	)	PUNCT
ejpam-5982	151	14	,	,	PUNCT
ejpam-5982	151	15	5982	5982	NUM
ejpam-5982	151	16	6	6	NUM
ejpam-5982	151	17	of	of	ADP
ejpam-5982	151	18	17	17	NUM
ejpam-5982	151	19	•	•	NUM
ejpam-5982	151	20	ψ2(a	ψ2(a	PROPN
ejpam-5982	151	21	,	,	PUNCT
ejpam-5982	151	22	b	b	NOUN
ejpam-5982	151	23	,	,	PUNCT
ejpam-5982	151	24	c	c	NOUN
ejpam-5982	151	25	,	,	PUNCT
ejpam-5982	151	26	d	d	NOUN
ejpam-5982	151	27	)	)	PUNCT
ejpam-5982	151	28	=	=	SYM
ejpam-5982	151	29	a−	a−	PROPN
ejpam-5982	151	30	k	k	PROPN
ejpam-5982	151	31	max{b	max{b	PROPN
ejpam-5982	151	32	,	,	PUNCT
ejpam-5982	151	33	c	c	X
ejpam-5982	151	34	,	,	PUNCT
ejpam-5982	151	35	d	d	NOUN
ejpam-5982	151	36	}	}	PUNCT
ejpam-5982	151	37	,	,	PUNCT
ejpam-5982	151	38	k	k	PROPN
ejpam-5982	151	39	∈	∈	PROPN
ejpam-5982	152	1	[	[	X
ejpam-5982	152	2	0	0	NUM
ejpam-5982	152	3	,	,	PUNCT
ejpam-5982	152	4	1	1	NUM
ejpam-5982	152	5	)	)	PUNCT
ejpam-5982	152	6	•	•	NUM
ejpam-5982	152	7	ψ3(a	ψ3(a	PROPN
ejpam-5982	152	8	,	,	PUNCT
ejpam-5982	152	9	b	b	PROPN
ejpam-5982	152	10	,	,	PUNCT
ejpam-5982	152	11	c	c	NOUN
ejpam-5982	152	12	,	,	PUNCT
ejpam-5982	152	13	d	d	NOUN
ejpam-5982	152	14	)	)	PUNCT
ejpam-5982	152	15	=	=	SYM
ejpam-5982	152	16	a−	a−	PROPN
ejpam-5982	152	17	k1b−	k1b−	PROPN
ejpam-5982	152	18	k2	k2	PROPN
ejpam-5982	152	19	max{c	max{c	PROPN
ejpam-5982	152	20	,	,	PUNCT
ejpam-5982	152	21	d	d	NOUN
ejpam-5982	152	22	}	}	PUNCT
ejpam-5982	152	23	k1	k1	NOUN
ejpam-5982	152	24	,	,	PUNCT
ejpam-5982	152	25	k2	k2	X
ejpam-5982	152	26	≥	≥	NOUN
ejpam-5982	152	27	0	0	NUM
ejpam-5982	152	28	,	,	PUNCT
ejpam-5982	152	29	k1	k1	NOUN
ejpam-5982	152	30	+	+	CCONJ
ejpam-5982	152	31	k2	k2	X
ejpam-5982	152	32	<	<	X
ejpam-5982	152	33	1	1	NUM
ejpam-5982	152	34	•	•	NUM
ejpam-5982	152	35	ψ4(a	ψ4(a	PROPN
ejpam-5982	152	36	,	,	PUNCT
ejpam-5982	152	37	b	b	NOUN
ejpam-5982	152	38	,	,	PUNCT
ejpam-5982	152	39	c	c	NOUN
ejpam-5982	152	40	,	,	PUNCT
ejpam-5982	152	41	d	d	NOUN
ejpam-5982	152	42	)	)	PUNCT
ejpam-5982	152	43	=	=	SYM
ejpam-5982	152	44	a−	a−	PROPN
ejpam-5982	152	45	k1	k1	PROPN
ejpam-5982	152	46	max{b	max{b	PROPN
ejpam-5982	152	47	,	,	PUNCT
ejpam-5982	152	48	c	c	NOUN
ejpam-5982	152	49	}	}	PUNCT
ejpam-5982	152	50	−	−	PROPN
ejpam-5982	152	51	k2d	k2d	PROPN
ejpam-5982	152	52	k1	k1	PROPN
ejpam-5982	152	53	,	,	PUNCT
ejpam-5982	152	54	k2	k2	X
ejpam-5982	152	55	≥	≥	NOUN
ejpam-5982	152	56	0	0	NUM
ejpam-5982	152	57	,	,	PUNCT
ejpam-5982	152	58	k1	k1	NOUN
ejpam-5982	152	59	+	+	CCONJ
ejpam-5982	152	60	k2	k2	X
ejpam-5982	152	61	<	<	X
ejpam-5982	152	62	1	1	NUM
ejpam-5982	152	63	•	•	NOUN
ejpam-5982	152	64	ψ(a	ψ(a	PROPN
ejpam-5982	152	65	,	,	PUNCT
ejpam-5982	152	66	b	b	NOUN
ejpam-5982	152	67	,	,	PUNCT
ejpam-5982	152	68	c	c	NOUN
ejpam-5982	152	69	,	,	PUNCT
ejpam-5982	152	70	d	d	NOUN
ejpam-5982	152	71	)	)	PUNCT
ejpam-5982	152	72	=	=	NOUN
ejpam-5982	152	73	a	a	DET
ejpam-5982	152	74	−	−	PROPN
ejpam-5982	152	75	kf	kf	PROPN
ejpam-5982	152	76	(	(	PUNCT
ejpam-5982	152	77	max{b	max{b	PROPN
ejpam-5982	152	78	,	,	PUNCT
ejpam-5982	152	79	c	c	X
ejpam-5982	152	80	,	,	PUNCT
ejpam-5982	152	81	d	d	NOUN
ejpam-5982	152	82	}	}	PUNCT
ejpam-5982	152	83	)	)	PUNCT
ejpam-5982	152	84	where	where	SCONJ
ejpam-5982	152	85	f	f	X
ejpam-5982	152	86	:	:	PUNCT
ejpam-5982	153	1	[	[	X
ejpam-5982	153	2	0,+∞	0,+∞	NUM
ejpam-5982	153	3	)	)	PUNCT
ejpam-5982	153	4	→	→	PUNCT
ejpam-5982	154	1	[	[	X
ejpam-5982	154	2	0,+∞	0,+∞	NUM
ejpam-5982	154	3	)	)	PUNCT
ejpam-5982	154	4	is	be	AUX
ejpam-5982	154	5	a	a	DET
ejpam-5982	154	6	function	function	NOUN
ejpam-5982	154	7	satisfying	satisfy	VERB
ejpam-5982	154	8	the	the	DET
ejpam-5982	154	9	condition	condition	NOUN
ejpam-5982	154	10	:	:	PUNCT
ejpam-5982	154	11	f	f	PROPN
ejpam-5982	154	12	(	(	PUNCT
ejpam-5982	154	13	t	t	PROPN
ejpam-5982	154	14	)	)	PUNCT
ejpam-5982	154	15	≤	≤	NOUN
ejpam-5982	154	16	t	t	PROPN
ejpam-5982	154	17	,	,	PUNCT
ejpam-5982	154	18	for	for	ADP
ejpam-5982	154	19	each	each	DET
ejpam-5982	154	20	t	t	NOUN
ejpam-5982	154	21	∈	∈	PROPN
ejpam-5982	154	22	(	(	PUNCT
ejpam-5982	154	23	0,+∞	0,+∞	NUM
ejpam-5982	154	24	)	)	PUNCT
ejpam-5982	154	25	and	and	CCONJ
ejpam-5982	154	26	k	k	PROPN
ejpam-5982	154	27	∈	∈	PROPN
ejpam-5982	155	1	[	[	X
ejpam-5982	155	2	0	0	NUM
ejpam-5982	155	3	,	,	PUNCT
ejpam-5982	155	4	1	1	NUM
ejpam-5982	155	5	)	)	PUNCT
ejpam-5982	155	6	.	.	PUNCT
ejpam-5982	156	1	now	now	ADV
ejpam-5982	156	2	we	we	PRON
ejpam-5982	156	3	consider	consider	VERB
ejpam-5982	156	4	a	a	DET
ejpam-5982	156	5	super	super	ADJ
ejpam-5982	156	6	class	class	NOUN
ejpam-5982	156	7	of	of	ADP
ejpam-5982	156	8	ψ	ψ	PROPN
ejpam-5982	156	9	which	which	PRON
ejpam-5982	156	10	will	will	AUX
ejpam-5982	156	11	be	be	AUX
ejpam-5982	156	12	denoted	denote	VERB
ejpam-5982	156	13	by	by	ADP
ejpam-5982	156	14	φ	φ	PROPN
ejpam-5982	156	15	and	and	CCONJ
ejpam-5982	156	16	it	it	PRON
ejpam-5982	156	17	is	be	AUX
ejpam-5982	156	18	the	the	DET
ejpam-5982	156	19	collection	collection	NOUN
ejpam-5982	156	20	of	of	ADP
ejpam-5982	156	21	all	all	DET
ejpam-5982	156	22	real	real	ADV
ejpam-5982	156	23	valued	value	VERB
ejpam-5982	156	24	functions	function	NOUN
ejpam-5982	156	25	ϕ	ϕ	NOUN
ejpam-5982	156	26	:	:	PUNCT
ejpam-5982	157	1	[	[	X
ejpam-5982	157	2	0,+∞)4	0,+∞)4	X
ejpam-5982	157	3	→	→	SYM
ejpam-5982	157	4	r	r	NOUN
ejpam-5982	157	5	satisfying	satisfy	VERB
ejpam-5982	157	6	the	the	DET
ejpam-5982	157	7	following	follow	VERB
ejpam-5982	157	8	condition	condition	NOUN
ejpam-5982	157	9	:	:	PUNCT
ejpam-5982	157	10	ϕ(a	ϕ(a	NOUN
ejpam-5982	157	11	,	,	PUNCT
ejpam-5982	157	12	a	a	PRON
ejpam-5982	157	13	,	,	PUNCT
ejpam-5982	157	14	0	0	NUM
ejpam-5982	157	15	,	,	PUNCT
ejpam-5982	157	16	0	0	NUM
ejpam-5982	157	17	)	)	PUNCT
ejpam-5982	157	18	>	>	X
ejpam-5982	157	19	0	0	NUM
ejpam-5982	157	20	,	,	PUNCT
ejpam-5982	157	21	for	for	ADP
ejpam-5982	157	22	all	all	DET
ejpam-5982	157	23	a	a	DET
ejpam-5982	157	24	>	>	X
ejpam-5982	157	25	0	0	X
ejpam-5982	157	26	.	.	PUNCT
ejpam-5982	158	1	now	now	ADV
ejpam-5982	158	2	we	we	PRON
ejpam-5982	158	3	prove	prove	VERB
ejpam-5982	158	4	a	a	DET
ejpam-5982	158	5	lemma	lemma	PROPN
ejpam-5982	158	6	that	that	PRON
ejpam-5982	158	7	will	will	AUX
ejpam-5982	158	8	be	be	AUX
ejpam-5982	158	9	used	use	VERB
ejpam-5982	158	10	in	in	ADP
ejpam-5982	158	11	proving	prove	VERB
ejpam-5982	158	12	our	our	PRON
ejpam-5982	158	13	theorems	theorem	NOUN
ejpam-5982	158	14	.	.	PUNCT
ejpam-5982	159	1	lemma	lemma	PROPN
ejpam-5982	159	2	1	1	X
ejpam-5982	159	3	.	.	PUNCT
ejpam-5982	160	1	let	let	VERB
ejpam-5982	160	2	(	(	PUNCT
ejpam-5982	160	3	x	x	X
ejpam-5982	160	4	,	,	PUNCT
ejpam-5982	160	5	y	y	PROPN
ejpam-5982	160	6	,	,	PUNCT
ejpam-5982	160	7	d	d	NOUN
ejpam-5982	160	8	)	)	PUNCT
ejpam-5982	160	9	be	be	AUX
ejpam-5982	160	10	a	a	DET
ejpam-5982	160	11	bipolar	bipolar	ADJ
ejpam-5982	160	12	metric	metric	ADJ
ejpam-5982	160	13	space	space	NOUN
ejpam-5982	160	14	and	and	CCONJ
ejpam-5982	160	15	(	(	PUNCT
ejpam-5982	160	16	xn	xn	PROPN
ejpam-5982	160	17	,	,	PUNCT
ejpam-5982	160	18	yn	yn	PROPN
ejpam-5982	160	19	)	)	PUNCT
ejpam-5982	160	20	is	be	AUX
ejpam-5982	160	21	a	a	DET
ejpam-5982	160	22	bisequence	bisequence	NOUN
ejpam-5982	160	23	in	in	ADP
ejpam-5982	160	24	x	x	SYM
ejpam-5982	160	25	×	×	NOUN
ejpam-5982	160	26	y	y	NOUN
ejpam-5982	160	27	satisfying	satisfy	VERB
ejpam-5982	160	28	the	the	DET
ejpam-5982	160	29	following	follow	VERB
ejpam-5982	160	30	condition	condition	NOUN
ejpam-5982	160	31	:	:	PUNCT
ejpam-5982	160	32	there	there	PRON
ejpam-5982	160	33	exists	exist	VERB
ejpam-5982	160	34	k	k	PROPN
ejpam-5982	160	35	∈	∈	PROPN
ejpam-5982	161	1	[	[	X
ejpam-5982	161	2	0	0	NUM
ejpam-5982	161	3	,	,	PUNCT
ejpam-5982	161	4	1	1	NUM
ejpam-5982	161	5	)	)	PUNCT
ejpam-5982	162	1	such	such	ADJ
ejpam-5982	162	2	that	that	SCONJ
ejpam-5982	162	3	d(xn+1	d(xn+1	PROPN
ejpam-5982	162	4	,	,	PUNCT
ejpam-5982	162	5	yn+1	yn+1	NOUN
ejpam-5982	162	6	)	)	PUNCT
ejpam-5982	162	7	≤	≤	NOUN
ejpam-5982	162	8	kd(xn+1	kd(xn+1	PROPN
ejpam-5982	162	9	,	,	PUNCT
ejpam-5982	162	10	yn	yn	PROPN
ejpam-5982	162	11	)	)	PUNCT
ejpam-5982	162	12	and	and	CCONJ
ejpam-5982	162	13	d(xn+1	d(xn+1	PROPN
ejpam-5982	162	14	,	,	PUNCT
ejpam-5982	162	15	yn	yn	NOUN
ejpam-5982	162	16	)	)	PUNCT
ejpam-5982	162	17	≤	≤	NUM
ejpam-5982	162	18	kd(xn	kd(xn	PROPN
ejpam-5982	162	19	,	,	PUNCT
ejpam-5982	162	20	yn	yn	PROPN
ejpam-5982	162	21	)	)	PUNCT
ejpam-5982	162	22	for	for	ADP
ejpam-5982	162	23	all	all	PRON
ejpam-5982	162	24	n	n	PRON
ejpam-5982	162	25	∈	∈	NOUN
ejpam-5982	162	26	n	n	NOUN
ejpam-5982	162	27	∪	∪	X
ejpam-5982	162	28	{	{	PUNCT
ejpam-5982	162	29	0	0	NUM
ejpam-5982	162	30	}	}	PUNCT
ejpam-5982	162	31	.	.	PUNCT
ejpam-5982	163	1	then	then	ADV
ejpam-5982	163	2	the	the	DET
ejpam-5982	163	3	bisequence	bisequence	NOUN
ejpam-5982	163	4	(	(	PUNCT
ejpam-5982	163	5	xn	xn	PROPN
ejpam-5982	163	6	,	,	PUNCT
ejpam-5982	163	7	yn	yn	PROPN
ejpam-5982	163	8	)	)	PUNCT
ejpam-5982	163	9	is	be	AUX
ejpam-5982	163	10	cauchy	cauchy	ADJ
ejpam-5982	163	11	bisequence	bisequence	NOUN
ejpam-5982	163	12	.	.	PUNCT
ejpam-5982	164	1	proof	proof	NOUN
ejpam-5982	164	2	.	.	PUNCT
ejpam-5982	165	1	first	first	ADV
ejpam-5982	165	2	,	,	PUNCT
ejpam-5982	165	3	we	we	PRON
ejpam-5982	165	4	observe	observe	VERB
ejpam-5982	165	5	that	that	SCONJ
ejpam-5982	165	6	the	the	DET
ejpam-5982	165	7	given	give	VERB
ejpam-5982	165	8	condition	condition	NOUN
ejpam-5982	165	9	implies	imply	VERB
ejpam-5982	165	10	the	the	DET
ejpam-5982	165	11	following	follow	VERB
ejpam-5982	165	12	condition	condition	NOUN
ejpam-5982	165	13	d(xn+1	d(xn+1	PROPN
ejpam-5982	165	14	,	,	PUNCT
ejpam-5982	165	15	yn+1	yn+1	NOUN
ejpam-5982	165	16	)	)	PUNCT
ejpam-5982	165	17	≤	≤	NOUN
ejpam-5982	166	1	k2d(xn	k2d(xn	PROPN
ejpam-5982	166	2	,	,	PUNCT
ejpam-5982	166	3	yn	yn	PROPN
ejpam-5982	166	4	)	)	PUNCT
ejpam-5982	166	5	for	for	ADP
ejpam-5982	166	6	all	all	PRON
ejpam-5982	166	7	n	n	PRON
ejpam-5982	166	8	∈	∈	NOUN
ejpam-5982	166	9	n	n	NOUN
ejpam-5982	166	10	∪	∪	X
ejpam-5982	166	11	{	{	PUNCT
ejpam-5982	166	12	0	0	NUM
ejpam-5982	166	13	}	}	SYM
ejpam-5982	166	14	d(xn+1	d(xn+1	PROPN
ejpam-5982	166	15	,	,	PUNCT
ejpam-5982	166	16	yn+1	yn+1	NOUN
ejpam-5982	166	17	)	)	PUNCT
ejpam-5982	166	18	≤	≤	PROPN
ejpam-5982	166	19	k2(n+1)d(x0	k2(n+1)d(x0	PROPN
ejpam-5982	166	20	,	,	PUNCT
ejpam-5982	166	21	y0	y0	PROPN
ejpam-5982	166	22	)	)	PUNCT
ejpam-5982	166	23	,	,	PUNCT
ejpam-5982	166	24	taking	take	VERB
ejpam-5982	166	25	limit	limit	NOUN
ejpam-5982	166	26	as	as	ADP
ejpam-5982	166	27	n→	n→	ADV
ejpam-5982	166	28	+	+	PROPN
ejpam-5982	166	29	∞	∞	PROPN
ejpam-5982	166	30	,	,	PUNCT
ejpam-5982	166	31	we	we	PRON
ejpam-5982	166	32	get	get	VERB
ejpam-5982	166	33	lim	lim	PROPN
ejpam-5982	166	34	n→+∞	n→+∞	VERB
ejpam-5982	166	35	d(xn	d(xn	PROPN
ejpam-5982	166	36	,	,	PUNCT
ejpam-5982	166	37	yn	yn	X
ejpam-5982	166	38	)	)	PUNCT
ejpam-5982	166	39	=	=	SYM
ejpam-5982	167	1	0	0	X
ejpam-5982	167	2	.	.	PUNCT
ejpam-5982	168	1	(	(	PUNCT
ejpam-5982	168	2	1	1	X
ejpam-5982	168	3	)	)	PUNCT
ejpam-5982	168	4	let	let	VERB
ejpam-5982	168	5	n	n	PRON
ejpam-5982	168	6	,	,	PUNCT
ejpam-5982	168	7	p	p	PROPN
ejpam-5982	168	8	∈	∈	PROPN
ejpam-5982	168	9	n	n	CCONJ
ejpam-5982	168	10	,	,	PUNCT
ejpam-5982	168	11	then	then	ADV
ejpam-5982	168	12	by	by	ADP
ejpam-5982	168	13	(	(	PUNCT
ejpam-5982	168	14	bp3	bp3	NOUN
ejpam-5982	168	15	)	)	PUNCT
ejpam-5982	168	16	and	and	CCONJ
ejpam-5982	168	17	given	give	VERB
ejpam-5982	168	18	condition	condition	NOUN
ejpam-5982	168	19	,	,	PUNCT
ejpam-5982	168	20	we	we	PRON
ejpam-5982	168	21	have	have	VERB
ejpam-5982	168	22	d(xn	d(xn	PROPN
ejpam-5982	168	23	,	,	PUNCT
ejpam-5982	168	24	yn+p	yn+p	NOUN
ejpam-5982	168	25	)	)	PUNCT
ejpam-5982	168	26	≤d(xn	≤d(xn	PROPN
ejpam-5982	168	27	,	,	PUNCT
ejpam-5982	168	28	yn	yn	PROPN
ejpam-5982	168	29	)	)	PUNCT
ejpam-5982	169	1	+	+	PUNCT
ejpam-5982	169	2	d(xn+1	d(xn+1	PROPN
ejpam-5982	169	3	,	,	PUNCT
ejpam-5982	169	4	yn	yn	PROPN
ejpam-5982	169	5	)	)	PUNCT
ejpam-5982	170	1	+	+	SYM
ejpam-5982	170	2	d(xn+1	d(xn+1	PROPN
ejpam-5982	170	3	,	,	PUNCT
ejpam-5982	170	4	yn+p	yn+p	NOUN
ejpam-5982	170	5	)	)	PUNCT
ejpam-5982	170	6	≤d(xn	≤d(xn	PROPN
ejpam-5982	170	7	,	,	PUNCT
ejpam-5982	170	8	yn	yn	PROPN
ejpam-5982	170	9	)	)	PUNCT
ejpam-5982	170	10	+	+	CCONJ
ejpam-5982	171	1	kd(xn	kd(xn	PROPN
ejpam-5982	171	2	,	,	PUNCT
ejpam-5982	171	3	yn	yn	PROPN
ejpam-5982	171	4	)	)	PUNCT
ejpam-5982	172	1	+	+	SYM
ejpam-5982	172	2	d(xn+1	d(xn+1	PROPN
ejpam-5982	172	3	,	,	PUNCT
ejpam-5982	172	4	yn+p	yn+p	NOUN
ejpam-5982	172	5	)	)	PUNCT
ejpam-5982	172	6	=(	=(	NOUN
ejpam-5982	172	7	1	1	NUM
ejpam-5982	172	8	+	+	NUM
ejpam-5982	172	9	k)d(xn	k)d(xn	PROPN
ejpam-5982	172	10	,	,	PUNCT
ejpam-5982	172	11	yn	yn	PROPN
ejpam-5982	172	12	)	)	PUNCT
ejpam-5982	173	1	+	+	SYM
ejpam-5982	173	2	d(xn+1	d(xn+1	PROPN
ejpam-5982	173	3	,	,	PUNCT
ejpam-5982	173	4	yn+p	yn+p	PROPN
ejpam-5982	173	5	)	)	PUNCT
ejpam-5982	174	1	≤(1	≤(1	PROPN
ejpam-5982	175	1	+	+	CCONJ
ejpam-5982	175	2	k)k2nd(x0	k)k2nd(x0	NOUN
ejpam-5982	175	3	,	,	PUNCT
ejpam-5982	175	4	y0	y0	PROPN
ejpam-5982	175	5	)	)	PUNCT
ejpam-5982	175	6	+	+	SYM
ejpam-5982	175	7	d(xn+1	d(xn+1	PROPN
ejpam-5982	175	8	,	,	PUNCT
ejpam-5982	175	9	yn+p	yn+p	PROPN
ejpam-5982	175	10	)	)	PUNCT
ejpam-5982	175	11	≤(1	≤(1	PROPN
ejpam-5982	176	1	+	+	NUM
ejpam-5982	176	2	k)(k2n	k)(k2n	NOUN
ejpam-5982	176	3	+	+	CCONJ
ejpam-5982	176	4	k2(n+1	k2(n+1	NOUN
ejpam-5982	176	5	)	)	PUNCT
ejpam-5982	176	6	+	+	NUM
ejpam-5982	176	7	k2(n+2	k2(n+2	NUM
ejpam-5982	176	8	)	)	PUNCT
ejpam-5982	177	1	+	+	CCONJ
ejpam-5982	177	2	·	·	PUNCT
ejpam-5982	177	3	·	·	PUNCT
ejpam-5982	177	4	·	·	PUNCT
ejpam-5982	177	5	+	+	NUM
ejpam-5982	177	6	k2(n+p−1))d(x0	k2(n+p−1))d(x0	ADJ
ejpam-5982	177	7	,	,	PUNCT
ejpam-5982	177	8	y0	y0	NOUN
ejpam-5982	177	9	)	)	PUNCT
ejpam-5982	177	10	+	+	CCONJ
ejpam-5982	177	11	d(xn+p	d(xn+p	PROPN
ejpam-5982	177	12	,	,	PUNCT
ejpam-5982	177	13	yn+p	yn+p	PROPN
ejpam-5982	177	14	)	)	PUNCT
ejpam-5982	177	15	≤(1	≤(1	PROPN
ejpam-5982	178	1	+	+	NUM
ejpam-5982	178	2	k)(k2n+k2(n+1)+k2(n+2	k)(k2n+k2(n+1)+k2(n+2	X
ejpam-5982	178	3	)	)	PUNCT
ejpam-5982	178	4	+	+	CCONJ
ejpam-5982	178	5	·	·	PUNCT
ejpam-5982	178	6	·	·	PUNCT
ejpam-5982	178	7	·	·	PUNCT
ejpam-5982	178	8	)	)	PUNCT
ejpam-5982	178	9	d(x0	d(x0	NOUN
ejpam-5982	178	10	,	,	PUNCT
ejpam-5982	178	11	y0)+d(xn+p	y0)+d(xn+p	PROPN
ejpam-5982	178	12	,	,	PUNCT
ejpam-5982	178	13	yn+p	yn+p	PROPN
ejpam-5982	178	14	)	)	PUNCT
ejpam-5982	178	15	≤(1	≤(1	PROPN
ejpam-5982	179	1	+	+	CCONJ
ejpam-5982	179	2	k)k2n(1	k)k2n(1	NOUN
ejpam-5982	179	3	+	+	CCONJ
ejpam-5982	179	4	k2	k2	ADJ
ejpam-5982	179	5	+	+	CCONJ
ejpam-5982	179	6	k4	k4	NOUN
ejpam-5982	179	7	+	+	X
ejpam-5982	179	8	·	·	PUNCT
ejpam-5982	179	9	·	·	PUNCT
ejpam-5982	179	10	·	·	PUNCT
ejpam-5982	179	11	)	)	PUNCT
ejpam-5982	179	12	d(x0	d(x0	NOUN
ejpam-5982	179	13	,	,	PUNCT
ejpam-5982	179	14	y0	y0	PROPN
ejpam-5982	179	15	)	)	PUNCT
ejpam-5982	179	16	+	+	CCONJ
ejpam-5982	179	17	k2(n+p)d(x0	k2(n+p)d(x0	ADJ
ejpam-5982	179	18	,	,	PUNCT
ejpam-5982	179	19	y0	y0	NOUN
ejpam-5982	179	20	)	)	PUNCT
ejpam-5982	179	21	=	=	PUNCT
ejpam-5982	179	22	(	(	PUNCT
ejpam-5982	179	23	1	1	NUM
ejpam-5982	179	24	+	+	CCONJ
ejpam-5982	179	25	k)k2n	k)k2n	PROPN
ejpam-5982	179	26	1−	1−	PROPN
ejpam-5982	179	27	k2	k2	PROPN
ejpam-5982	179	28	d(x0	d(x0	NOUN
ejpam-5982	179	29	,	,	PUNCT
ejpam-5982	179	30	y0	y0	PROPN
ejpam-5982	179	31	)	)	PUNCT
ejpam-5982	179	32	+	+	CCONJ
ejpam-5982	179	33	k(2n+2p)d(x0	k(2n+2p)d(x0	PROPN
ejpam-5982	179	34	,	,	PUNCT
ejpam-5982	179	35	y0	y0	PROPN
ejpam-5982	179	36	)	)	PUNCT
ejpam-5982	179	37	.	.	PUNCT
ejpam-5982	180	1	p.	p.	NOUN
ejpam-5982	180	2	p.	p.	NOUN
ejpam-5982	181	1	murthy	murthy	ADJ
ejpam-5982	182	1	et	et	PROPN
ejpam-5982	182	2	al	al	PROPN
ejpam-5982	182	3	.	.	PUNCT
ejpam-5982	182	4	/	/	SYM
ejpam-5982	182	5	eur	eur	PROPN
ejpam-5982	182	6	.	.	PUNCT
ejpam-5982	183	1	j.	j.	PROPN
ejpam-5982	183	2	pure	pure	PROPN
ejpam-5982	183	3	appl	appl	PROPN
ejpam-5982	183	4	.	.	PROPN
ejpam-5982	183	5	math	math	PROPN
ejpam-5982	183	6	,	,	PUNCT
ejpam-5982	183	7	18	18	NUM
ejpam-5982	183	8	(	(	PUNCT
ejpam-5982	183	9	2	2	NUM
ejpam-5982	183	10	)	)	PUNCT
ejpam-5982	183	11	(	(	PUNCT
ejpam-5982	183	12	2025	2025	NUM
ejpam-5982	183	13	)	)	PUNCT
ejpam-5982	183	14	,	,	PUNCT
ejpam-5982	183	15	5982	5982	NUM
ejpam-5982	183	16	7	7	NUM
ejpam-5982	183	17	of	of	ADP
ejpam-5982	183	18	17	17	NUM
ejpam-5982	183	19	this	this	PRON
ejpam-5982	183	20	implies	imply	VERB
ejpam-5982	183	21	lim	lim	PROPN
ejpam-5982	183	22	n→+∞	n→+∞	VERB
ejpam-5982	183	23	d(xn	d(xn	PROPN
ejpam-5982	183	24	,	,	PUNCT
ejpam-5982	183	25	yn+p	yn+p	PROPN
ejpam-5982	183	26	)	)	PUNCT
ejpam-5982	183	27	=	=	NOUN
ejpam-5982	184	1	0	0	X
ejpam-5982	184	2	.	.	PUNCT
ejpam-5982	185	1	(	(	PUNCT
ejpam-5982	185	2	2	2	X
ejpam-5982	185	3	)	)	PUNCT
ejpam-5982	185	4	now	now	ADV
ejpam-5982	185	5	to	to	PART
ejpam-5982	185	6	prove	prove	VERB
ejpam-5982	185	7	that	that	SCONJ
ejpam-5982	185	8	lim	lim	PROPN
ejpam-5982	185	9	n→+∞	n→+∞	PROPN
ejpam-5982	185	10	d(xn+p	d(xn+p	PROPN
ejpam-5982	185	11	,	,	PUNCT
ejpam-5982	185	12	yn	yn	PROPN
ejpam-5982	185	13	)	)	PUNCT
ejpam-5982	185	14	=	=	SYM
ejpam-5982	185	15	0	0	NUM
ejpam-5982	185	16	,	,	PUNCT
ejpam-5982	185	17	consider	consider	VERB
ejpam-5982	185	18	the	the	DET
ejpam-5982	185	19	inequality	inequality	NOUN
ejpam-5982	185	20	(	(	PUNCT
ejpam-5982	185	21	by	by	ADP
ejpam-5982	185	22	property	property	NOUN
ejpam-5982	185	23	(	(	PUNCT
ejpam-5982	185	24	bp3	bp3	NOUN
ejpam-5982	185	25	)	)	PUNCT
ejpam-5982	185	26	)	)	PUNCT
ejpam-5982	185	27	d(xn+p	d(xn+p	PROPN
ejpam-5982	185	28	,	,	PUNCT
ejpam-5982	185	29	yn	yn	PROPN
ejpam-5982	185	30	)	)	PUNCT
ejpam-5982	185	31	≤	≤	NOUN
ejpam-5982	185	32	d(xn+p	d(xn+p	PROPN
ejpam-5982	185	33	,	,	PUNCT
ejpam-5982	185	34	yn+p	yn+p	PROPN
ejpam-5982	185	35	)	)	PUNCT
ejpam-5982	186	1	+	+	CCONJ
ejpam-5982	187	1	d(xn	d(xn	PROPN
ejpam-5982	187	2	,	,	PUNCT
ejpam-5982	187	3	yn+p	yn+p	PROPN
ejpam-5982	187	4	)	)	PUNCT
ejpam-5982	187	5	+	+	CCONJ
ejpam-5982	187	6	d(xn	d(xn	PROPN
ejpam-5982	187	7	,	,	PUNCT
ejpam-5982	187	8	yn	yn	NOUN
ejpam-5982	187	9	)	)	PUNCT
ejpam-5982	188	1	and	and	CCONJ
ejpam-5982	188	2	take	take	VERB
ejpam-5982	188	3	the	the	DET
ejpam-5982	188	4	limit	limit	NOUN
ejpam-5982	188	5	as	as	ADP
ejpam-5982	188	6	n→	n→	ADV
ejpam-5982	188	7	+	+	PROPN
ejpam-5982	188	8	∞	∞	NOUN
ejpam-5982	188	9	and	and	CCONJ
ejpam-5982	188	10	use	use	NOUN
ejpam-5982	188	11	(	(	PUNCT
ejpam-5982	188	12	1	1	NUM
ejpam-5982	188	13	)	)	PUNCT
ejpam-5982	188	14	and	and	CCONJ
ejpam-5982	188	15	(	(	PUNCT
ejpam-5982	188	16	2	2	NUM
ejpam-5982	188	17	)	)	PUNCT
ejpam-5982	188	18	.	.	PUNCT
ejpam-5982	189	1	hence	hence	ADV
ejpam-5982	189	2	(	(	PUNCT
ejpam-5982	189	3	xn	xn	PROPN
ejpam-5982	189	4	,	,	PUNCT
ejpam-5982	189	5	yn	yn	PROPN
ejpam-5982	189	6	)	)	PUNCT
ejpam-5982	189	7	is	be	AUX
ejpam-5982	189	8	a	a	DET
ejpam-5982	189	9	cauchy	cauchy	ADJ
ejpam-5982	189	10	sequence	sequence	NOUN
ejpam-5982	189	11	.	.	PUNCT
ejpam-5982	190	1	our	our	PRON
ejpam-5982	190	2	first	first	ADJ
ejpam-5982	190	3	main	main	ADJ
ejpam-5982	190	4	result	result	NOUN
ejpam-5982	190	5	is	be	AUX
ejpam-5982	190	6	the	the	DET
ejpam-5982	190	7	following	following	NOUN
ejpam-5982	190	8	.	.	PUNCT
ejpam-5982	191	1	theorem	theorem	NOUN
ejpam-5982	191	2	1	1	NUM
ejpam-5982	191	3	.	.	PUNCT
ejpam-5982	192	1	let	let	AUX
ejpam-5982	192	2	(	(	PUNCT
ejpam-5982	192	3	x	x	X
ejpam-5982	192	4	,	,	PUNCT
ejpam-5982	192	5	y	y	PROPN
ejpam-5982	192	6	,	,	PUNCT
ejpam-5982	192	7	d	d	NOUN
ejpam-5982	192	8	)	)	PUNCT
ejpam-5982	192	9	be	be	AUX
ejpam-5982	192	10	a	a	DET
ejpam-5982	192	11	complete	complete	ADJ
ejpam-5982	192	12	bipolar	bipolar	ADJ
ejpam-5982	192	13	metric	metric	ADJ
ejpam-5982	192	14	space	space	NOUN
ejpam-5982	192	15	and	and	CCONJ
ejpam-5982	192	16	let	let	VERB
ejpam-5982	192	17	t1	t1	NOUN
ejpam-5982	192	18	,	,	PUNCT
ejpam-5982	192	19	t2	t2	NOUN
ejpam-5982	192	20	:	:	PUNCT
ejpam-5982	192	21	(	(	PUNCT
ejpam-5982	192	22	x	x	X
ejpam-5982	192	23	,	,	PUNCT
ejpam-5982	192	24	y	y	PROPN
ejpam-5982	192	25	,	,	PUNCT
ejpam-5982	192	26	d	d	NOUN
ejpam-5982	192	27	)	)	PUNCT
ejpam-5982	192	28	⇒	⇒	NOUN
ejpam-5982	192	29	(	(	PUNCT
ejpam-5982	192	30	x	x	X
ejpam-5982	192	31	,	,	PUNCT
ejpam-5982	192	32	y	y	PROPN
ejpam-5982	192	33	,	,	PUNCT
ejpam-5982	192	34	d	d	NOUN
ejpam-5982	192	35	)	)	PUNCT
ejpam-5982	192	36	be	be	AUX
ejpam-5982	192	37	two	two	NUM
ejpam-5982	192	38	covariant	covariant	ADJ
ejpam-5982	192	39	maps	map	NOUN
ejpam-5982	192	40	and	and	CCONJ
ejpam-5982	192	41	s1	s1	NOUN
ejpam-5982	192	42	,	,	PUNCT
ejpam-5982	192	43	s2	s2	NOUN
ejpam-5982	192	44	:	:	PUNCT
ejpam-5982	192	45	(	(	PUNCT
ejpam-5982	192	46	x	x	X
ejpam-5982	192	47	,	,	PUNCT
ejpam-5982	192	48	y	y	PROPN
ejpam-5982	192	49	,	,	PUNCT
ejpam-5982	192	50	d	d	NOUN
ejpam-5982	192	51	)	)	PUNCT
ejpam-5982	192	52	⇄	⇄	NOUN
ejpam-5982	192	53	(	(	PUNCT
ejpam-5982	192	54	x	x	X
ejpam-5982	192	55	,	,	PUNCT
ejpam-5982	192	56	y	y	PROPN
ejpam-5982	192	57	,	,	PUNCT
ejpam-5982	192	58	d	d	NOUN
ejpam-5982	192	59	)	)	PUNCT
ejpam-5982	192	60	be	be	AUX
ejpam-5982	192	61	two	two	NUM
ejpam-5982	192	62	contravariant	contravariant	ADJ
ejpam-5982	192	63	maps	map	NOUN
ejpam-5982	192	64	satisfying	satisfy	VERB
ejpam-5982	192	65	the	the	DET
ejpam-5982	192	66	following	follow	VERB
ejpam-5982	192	67	conditions	condition	NOUN
ejpam-5982	192	68	:	:	PUNCT
ejpam-5982	192	69	(	(	PUNCT
ejpam-5982	192	70	i	i	NOUN
ejpam-5982	192	71	)	)	PUNCT
ejpam-5982	192	72	the	the	DET
ejpam-5982	192	73	mappings	mapping	NOUN
ejpam-5982	192	74	s2	s2	PROPN
ejpam-5982	192	75	,	,	PUNCT
ejpam-5982	192	76	t1	t1	NUM
ejpam-5982	192	77	are	be	AUX
ejpam-5982	192	78	compatible	compatible	ADJ
ejpam-5982	192	79	of	of	ADP
ejpam-5982	192	80	type	type	NOUN
ejpam-5982	192	81	(	(	PUNCT
ejpam-5982	192	82	a	a	NOUN
ejpam-5982	192	83	)	)	PUNCT
ejpam-5982	192	84	with	with	ADP
ejpam-5982	192	85	respect	respect	NOUN
ejpam-5982	192	86	to	to	ADP
ejpam-5982	192	87	y.	y.	PROPN
ejpam-5982	192	88	(	(	PUNCT
ejpam-5982	192	89	ii	ii	PROPN
ejpam-5982	192	90	)	)	PUNCT
ejpam-5982	192	91	the	the	DET
ejpam-5982	192	92	mappings	mapping	NOUN
ejpam-5982	192	93	s1	s1	NOUN
ejpam-5982	192	94	,	,	PUNCT
ejpam-5982	192	95	t2	t2	NOUN
ejpam-5982	192	96	are	be	AUX
ejpam-5982	192	97	compatible	compatible	ADJ
ejpam-5982	192	98	of	of	ADP
ejpam-5982	192	99	type	type	NOUN
ejpam-5982	192	100	(	(	PUNCT
ejpam-5982	192	101	a	a	NOUN
ejpam-5982	192	102	)	)	PUNCT
ejpam-5982	192	103	with	with	ADP
ejpam-5982	192	104	respect	respect	NOUN
ejpam-5982	192	105	to	to	ADP
ejpam-5982	192	106	x.	x.	NOUN
ejpam-5982	192	107	(	(	PUNCT
ejpam-5982	192	108	iii	iii	NOUN
ejpam-5982	192	109	)	)	PUNCT
ejpam-5982	192	110	s1(x	s1(x	NOUN
ejpam-5982	192	111	∪	∪	PROPN
ejpam-5982	192	112	y	y	PROPN
ejpam-5982	192	113	)	)	PUNCT
ejpam-5982	193	1	⊆	⊆	NUM
ejpam-5982	193	2	t1(x	t1(x	NOUN
ejpam-5982	193	3	∪	∪	PROPN
ejpam-5982	193	4	y	y	PROPN
ejpam-5982	193	5	)	)	PUNCT
ejpam-5982	193	6	and	and	CCONJ
ejpam-5982	193	7	s2(x	s2(x	PROPN
ejpam-5982	193	8	∪	∪	PROPN
ejpam-5982	193	9	y	y	PROPN
ejpam-5982	193	10	)	)	PUNCT
ejpam-5982	193	11	⊆	⊆	PROPN
ejpam-5982	193	12	t2(x	t2(x	SYM
ejpam-5982	193	13	∪	∪	PROPN
ejpam-5982	193	14	y	y	PROPN
ejpam-5982	193	15	)	)	PUNCT
ejpam-5982	193	16	.	.	PUNCT
ejpam-5982	194	1	(	(	PUNCT
ejpam-5982	194	2	iv	iv	X
ejpam-5982	194	3	)	)	PUNCT
ejpam-5982	194	4	all	all	DET
ejpam-5982	194	5	the	the	DET
ejpam-5982	194	6	four	four	NUM
ejpam-5982	194	7	mappings	mapping	NOUN
ejpam-5982	194	8	s1	s1	NOUN
ejpam-5982	194	9	,	,	PUNCT
ejpam-5982	194	10	s2	s2	PROPN
ejpam-5982	194	11	,	,	PUNCT
ejpam-5982	194	12	t1	t1	NOUN
ejpam-5982	194	13	and	and	CCONJ
ejpam-5982	194	14	t2	t2	NOUN
ejpam-5982	194	15	are	be	AUX
ejpam-5982	194	16	continuous	continuous	ADJ
ejpam-5982	194	17	.	.	PUNCT
ejpam-5982	195	1	(	(	PUNCT
ejpam-5982	195	2	v	v	NOUN
ejpam-5982	195	3	)	)	PUNCT
ejpam-5982	195	4	there	there	PRON
ejpam-5982	195	5	exists	exist	VERB
ejpam-5982	195	6	ψ	ψ	ADP
ejpam-5982	195	7	∈	∈	NOUN
ejpam-5982	195	8	ψ	ψ	ADP
ejpam-5982	195	9	such	such	ADJ
ejpam-5982	195	10	that	that	DET
ejpam-5982	195	11	ψ(d(s2y	ψ(d(s2y	PROPN
ejpam-5982	195	12	,	,	PUNCT
ejpam-5982	195	13	s1x	s1x	PROPN
ejpam-5982	195	14	)	)	PUNCT
ejpam-5982	195	15	,	,	PUNCT
ejpam-5982	195	16	d(t2x	d(t2x	VERB
ejpam-5982	195	17	,	,	PUNCT
ejpam-5982	195	18	t1y	t1y	NOUN
ejpam-5982	195	19	)	)	PUNCT
ejpam-5982	195	20	,	,	PUNCT
ejpam-5982	195	21	d(t2x	d(t2x	NOUN
ejpam-5982	195	22	,	,	PUNCT
ejpam-5982	195	23	s1x	s1x	PROPN
ejpam-5982	195	24	)	)	PUNCT
ejpam-5982	195	25	,	,	PUNCT
ejpam-5982	195	26	d(s2y	d(s2y	NOUN
ejpam-5982	195	27	,	,	PUNCT
ejpam-5982	195	28	t1y	t1y	NOUN
ejpam-5982	195	29	)	)	PUNCT
ejpam-5982	195	30	)	)	PUNCT
ejpam-5982	196	1	≤	≤	ADV
ejpam-5982	196	2	0	0	NUM
ejpam-5982	196	3	,	,	PUNCT
ejpam-5982	196	4	(	(	PUNCT
ejpam-5982	196	5	3	3	X
ejpam-5982	196	6	)	)	PUNCT
ejpam-5982	196	7	for	for	ADP
ejpam-5982	196	8	all	all	DET
ejpam-5982	196	9	(	(	PUNCT
ejpam-5982	196	10	x	x	NOUN
ejpam-5982	196	11	,	,	PUNCT
ejpam-5982	196	12	y	y	NOUN
ejpam-5982	196	13	)	)	PUNCT
ejpam-5982	196	14	∈	∈	PROPN
ejpam-5982	196	15	x	x	PUNCT
ejpam-5982	196	16	×	×	NOUN
ejpam-5982	196	17	y	y	PROPN
ejpam-5982	196	18	.	.	PUNCT
ejpam-5982	197	1	then	then	ADV
ejpam-5982	197	2	the	the	DET
ejpam-5982	197	3	functions	function	NOUN
ejpam-5982	197	4	s1	s1	NOUN
ejpam-5982	197	5	,	,	PUNCT
ejpam-5982	197	6	s2	s2	PROPN
ejpam-5982	197	7	,	,	PUNCT
ejpam-5982	197	8	t1	t1	NOUN
ejpam-5982	197	9	and	and	CCONJ
ejpam-5982	197	10	t2	t2	PROPN
ejpam-5982	197	11	have	have	VERB
ejpam-5982	197	12	a	a	DET
ejpam-5982	197	13	unique	unique	ADJ
ejpam-5982	197	14	common	common	ADJ
ejpam-5982	197	15	fixed	fix	VERB
ejpam-5982	197	16	point	point	NOUN
ejpam-5982	197	17	.	.	PUNCT
ejpam-5982	198	1	proof	proof	NOUN
ejpam-5982	198	2	.	.	PUNCT
ejpam-5982	199	1	let	let	VERB
ejpam-5982	199	2	x0	x0	PROPN
ejpam-5982	199	3	∈	∈	PROPN
ejpam-5982	199	4	x	x	PUNCT
ejpam-5982	199	5	and	and	CCONJ
ejpam-5982	199	6	choose	choose	VERB
ejpam-5982	199	7	x1	x1	PROPN
ejpam-5982	199	8	∈	∈	PROPN
ejpam-5982	199	9	x	x	X
ejpam-5982	199	10	and	and	CCONJ
ejpam-5982	199	11	y1	y1	PROPN
ejpam-5982	199	12	∈	∈	PROPN
ejpam-5982	199	13	y	y	PROPN
ejpam-5982	199	14	such	such	ADJ
ejpam-5982	199	15	that	that	SCONJ
ejpam-5982	199	16	s1x0	s1x0	NOUN
ejpam-5982	199	17	=	=	SYM
ejpam-5982	199	18	t1y1	t1y1	SYM
ejpam-5982	199	19	=	=	SYM
ejpam-5982	199	20	v0	v0	NOUN
ejpam-5982	199	21	and	and	CCONJ
ejpam-5982	199	22	s2y1	s2y1	NOUN
ejpam-5982	199	23	=	=	PUNCT
ejpam-5982	199	24	t2x1	t2x1	X
ejpam-5982	199	25	=	=	SYM
ejpam-5982	199	26	u1	u1	NOUN
ejpam-5982	199	27	.	.	PUNCT
ejpam-5982	200	1	this	this	PRON
ejpam-5982	200	2	can	can	AUX
ejpam-5982	200	3	be	be	AUX
ejpam-5982	200	4	done	do	VERB
ejpam-5982	200	5	since	since	SCONJ
ejpam-5982	200	6	s1(x	s1(x	PROPN
ejpam-5982	200	7	∪	∪	VERB
ejpam-5982	200	8	y	y	PROPN
ejpam-5982	200	9	)	)	PUNCT
ejpam-5982	201	1	⊆	⊆	NUM
ejpam-5982	201	2	t1(x	t1(x	NOUN
ejpam-5982	201	3	∪	∪	PROPN
ejpam-5982	201	4	y	y	PROPN
ejpam-5982	201	5	)	)	PUNCT
ejpam-5982	201	6	and	and	CCONJ
ejpam-5982	201	7	s2(x	s2(x	PROPN
ejpam-5982	201	8	∪	∪	PROPN
ejpam-5982	201	9	y	y	PROPN
ejpam-5982	201	10	)	)	PUNCT
ejpam-5982	201	11	⊆	⊆	NUM
ejpam-5982	201	12	t2(x	t2(x	NOUN
ejpam-5982	201	13	∪y	∪y	NUM
ejpam-5982	201	14	)	)	PUNCT
ejpam-5982	201	15	.	.	PUNCT
ejpam-5982	202	1	in	in	ADP
ejpam-5982	202	2	general	general	ADJ
ejpam-5982	202	3	we	we	PRON
ejpam-5982	202	4	can	can	AUX
ejpam-5982	202	5	choose	choose	VERB
ejpam-5982	202	6	(	(	PUNCT
ejpam-5982	202	7	xn	xn	PROPN
ejpam-5982	202	8	,	,	PUNCT
ejpam-5982	202	9	yn	yn	NOUN
ejpam-5982	202	10	)	)	PUNCT
ejpam-5982	202	11	∈	∈	PROPN
ejpam-5982	202	12	x	x	PUNCT
ejpam-5982	202	13	×y	×y	NOUN
ejpam-5982	203	1	such	such	ADJ
ejpam-5982	203	2	that	that	SCONJ
ejpam-5982	203	3	s1xn	s1xn	NUM
ejpam-5982	203	4	=	=	SYM
ejpam-5982	203	5	t1yn+1	t1yn+1	NOUN
ejpam-5982	203	6	=	=	PUNCT
ejpam-5982	203	7	vn	vn	PROPN
ejpam-5982	203	8	and	and	CCONJ
ejpam-5982	203	9	s2yn+1	s2yn+1	ADJ
ejpam-5982	203	10	=	=	SYM
ejpam-5982	203	11	t2xn+1	t2xn+1	NOUN
ejpam-5982	203	12	=	=	SYM
ejpam-5982	203	13	un+1	un+1	NOUN
ejpam-5982	203	14	for	for	ADP
ejpam-5982	203	15	all	all	PRON
ejpam-5982	203	16	n	n	PRON
ejpam-5982	203	17	∈	∈	NOUN
ejpam-5982	203	18	n	n	NOUN
ejpam-5982	203	19	∪	∪	X
ejpam-5982	203	20	{	{	PUNCT
ejpam-5982	203	21	0	0	NUM
ejpam-5982	203	22	}	}	PUNCT
ejpam-5982	203	23	.	.	PUNCT
ejpam-5982	204	1	now	now	ADV
ejpam-5982	204	2	putting	put	VERB
ejpam-5982	204	3	x	x	X
ejpam-5982	204	4	=	=	SYM
ejpam-5982	204	5	xn+1	xn+1	PROPN
ejpam-5982	204	6	and	and	CCONJ
ejpam-5982	204	7	y	y	PROPN
ejpam-5982	204	8	=	=	SYM
ejpam-5982	204	9	yn+1	yn+1	PROPN
ejpam-5982	204	10	in	in	ADP
ejpam-5982	204	11	(	(	PUNCT
ejpam-5982	204	12	3	3	NUM
ejpam-5982	204	13	)	)	PUNCT
ejpam-5982	204	14	,	,	PUNCT
ejpam-5982	204	15	we	we	PRON
ejpam-5982	204	16	get	get	VERB
ejpam-5982	204	17	ψ(d(s2yn+1	ψ(d(s2yn+1	NOUN
ejpam-5982	204	18	,	,	PUNCT
ejpam-5982	204	19	s1xn+1	s1xn+1	NOUN
ejpam-5982	204	20	)	)	PUNCT
ejpam-5982	204	21	,	,	PUNCT
ejpam-5982	204	22	d(t2xn+1	d(t2xn+1	PROPN
ejpam-5982	204	23	,	,	PUNCT
ejpam-5982	204	24	t1yn+1	t1yn+1	NOUN
ejpam-5982	204	25	)	)	PUNCT
ejpam-5982	204	26	,	,	PUNCT
ejpam-5982	204	27	d(t2xn+1	d(t2xn+1	PROPN
ejpam-5982	204	28	,	,	PUNCT
ejpam-5982	204	29	s1xn+1	s1xn+1	NOUN
ejpam-5982	204	30	)	)	PUNCT
ejpam-5982	204	31	,	,	PUNCT
ejpam-5982	204	32	d(s2yn+1	d(s2yn+1	PROPN
ejpam-5982	204	33	,	,	PUNCT
ejpam-5982	204	34	t1yn+1	t1yn+1	NOUN
ejpam-5982	204	35	)	)	PUNCT
ejpam-5982	204	36	)	)	PUNCT
ejpam-5982	205	1	≤	≤	NOUN
ejpam-5982	205	2	0	0	X
ejpam-5982	206	1	ψ(d(un+1	ψ(d(un+1	ADJ
ejpam-5982	206	2	,	,	PUNCT
ejpam-5982	206	3	vn+1	vn+1	PROPN
ejpam-5982	206	4	)	)	PUNCT
ejpam-5982	206	5	,	,	PUNCT
ejpam-5982	206	6	d(un+1	d(un+1	PROPN
ejpam-5982	206	7	,	,	PUNCT
ejpam-5982	206	8	vn	vn	NOUN
ejpam-5982	206	9	)	)	PUNCT
ejpam-5982	206	10	,	,	PUNCT
ejpam-5982	206	11	d(un+1	d(un+1	PROPN
ejpam-5982	206	12	,	,	PUNCT
ejpam-5982	206	13	vn+1	vn+1	NOUN
ejpam-5982	206	14	)	)	PUNCT
ejpam-5982	206	15	,	,	PUNCT
ejpam-5982	206	16	d(un+1	d(un+1	PROPN
ejpam-5982	206	17	,	,	PUNCT
ejpam-5982	206	18	vn	vn	NOUN
ejpam-5982	206	19	)	)	PUNCT
ejpam-5982	206	20	)	)	PUNCT
ejpam-5982	206	21	≤	≤	ADV
ejpam-5982	206	22	0	0	X
ejpam-5982	206	23	.	.	PUNCT
ejpam-5982	207	1	so	so	ADV
ejpam-5982	207	2	by	by	ADP
ejpam-5982	207	3	property	property	NOUN
ejpam-5982	207	4	of	of	ADP
ejpam-5982	207	5	ψ	ψ	NOUN
ejpam-5982	207	6	,	,	PUNCT
ejpam-5982	207	7	there	there	PRON
ejpam-5982	207	8	exists	exist	VERB
ejpam-5982	207	9	k	k	PROPN
ejpam-5982	207	10	∈	∈	PROPN
ejpam-5982	208	1	[	[	X
ejpam-5982	208	2	0	0	NUM
ejpam-5982	208	3	,	,	PUNCT
ejpam-5982	208	4	1	1	NUM
ejpam-5982	208	5	)	)	PUNCT
ejpam-5982	208	6	such	such	ADJ
ejpam-5982	208	7	that	that	PRON
ejpam-5982	208	8	d(un+1	d(un+1	PROPN
ejpam-5982	208	9	,	,	PUNCT
ejpam-5982	208	10	vn+1	vn+1	NOUN
ejpam-5982	208	11	)	)	PUNCT
ejpam-5982	208	12	≤	≤	NUM
ejpam-5982	208	13	kd(un+1	kd(un+1	NOUN
ejpam-5982	208	14	,	,	PUNCT
ejpam-5982	208	15	vn	vn	NOUN
ejpam-5982	208	16	)	)	PUNCT
ejpam-5982	208	17	.	.	PUNCT
ejpam-5982	209	1	(	(	PUNCT
ejpam-5982	209	2	4	4	X
ejpam-5982	209	3	)	)	PUNCT
ejpam-5982	209	4	again	again	ADV
ejpam-5982	209	5	putting	put	VERB
ejpam-5982	209	6	x	x	X
ejpam-5982	209	7	=	=	PUNCT
ejpam-5982	209	8	xn	xn	PROPN
ejpam-5982	209	9	and	and	CCONJ
ejpam-5982	209	10	y	y	PROPN
ejpam-5982	209	11	=	=	SYM
ejpam-5982	209	12	yn+1	yn+1	PROPN
ejpam-5982	209	13	in	in	ADP
ejpam-5982	209	14	(	(	PUNCT
ejpam-5982	209	15	3	3	NUM
ejpam-5982	209	16	)	)	PUNCT
ejpam-5982	209	17	,	,	PUNCT
ejpam-5982	209	18	we	we	PRON
ejpam-5982	209	19	get	get	VERB
ejpam-5982	209	20	ψ((d(s2yn+1	ψ((d(s2yn+1	NUM
ejpam-5982	209	21	,	,	PUNCT
ejpam-5982	209	22	s1xn	s1xn	NUM
ejpam-5982	209	23	)	)	PUNCT
ejpam-5982	209	24	,	,	PUNCT
ejpam-5982	209	25	d(t2xn	d(t2xn	NUM
ejpam-5982	209	26	,	,	PUNCT
ejpam-5982	209	27	t1yn+1	t1yn+1	NOUN
ejpam-5982	209	28	)	)	PUNCT
ejpam-5982	209	29	,	,	PUNCT
ejpam-5982	209	30	d(t2xn	d(t2xn	PRON
ejpam-5982	209	31	,	,	PUNCT
ejpam-5982	209	32	s1xn	s1xn	NOUN
ejpam-5982	209	33	)	)	PUNCT
ejpam-5982	209	34	,	,	PUNCT
ejpam-5982	209	35	d(s2yn+1	d(s2yn+1	PROPN
ejpam-5982	209	36	,	,	PUNCT
ejpam-5982	209	37	t1yn+1	t1yn+1	NOUN
ejpam-5982	209	38	)	)	PUNCT
ejpam-5982	209	39	)	)	PUNCT
ejpam-5982	210	1	≤	≤	NOUN
ejpam-5982	210	2	0	0	X
ejpam-5982	211	1	ψ(d(un+1	ψ(d(un+1	ADJ
ejpam-5982	211	2	,	,	PUNCT
ejpam-5982	211	3	vn	vn	NOUN
ejpam-5982	211	4	)	)	PUNCT
ejpam-5982	211	5	,	,	PUNCT
ejpam-5982	211	6	d(un	d(un	PROPN
ejpam-5982	211	7	,	,	PUNCT
ejpam-5982	211	8	vn	vn	NOUN
ejpam-5982	211	9	)	)	PUNCT
ejpam-5982	211	10	,	,	PUNCT
ejpam-5982	211	11	d(un	d(un	PROPN
ejpam-5982	211	12	,	,	PUNCT
ejpam-5982	211	13	vn	vn	NOUN
ejpam-5982	211	14	)	)	PUNCT
ejpam-5982	211	15	,	,	PUNCT
ejpam-5982	211	16	d(un+1	d(un+1	PROPN
ejpam-5982	211	17	,	,	PUNCT
ejpam-5982	211	18	vn	vn	NOUN
ejpam-5982	211	19	)	)	PUNCT
ejpam-5982	211	20	)	)	PUNCT
ejpam-5982	211	21	≤	≤	ADV
ejpam-5982	211	22	0	0	NUM
ejpam-5982	211	23	.	.	PUNCT
ejpam-5982	212	1	p.	p.	NOUN
ejpam-5982	212	2	p.	p.	NOUN
ejpam-5982	213	1	murthy	murthy	PROPN
ejpam-5982	214	1	et	et	PROPN
ejpam-5982	214	2	al	al	PROPN
ejpam-5982	214	3	.	.	PUNCT
ejpam-5982	214	4	/	/	SYM
ejpam-5982	214	5	eur	eur	PROPN
ejpam-5982	214	6	.	.	PUNCT
ejpam-5982	215	1	j.	j.	PROPN
ejpam-5982	215	2	pure	pure	PROPN
ejpam-5982	215	3	appl	appl	PROPN
ejpam-5982	215	4	.	.	PROPN
ejpam-5982	215	5	math	math	PROPN
ejpam-5982	215	6	,	,	PUNCT
ejpam-5982	215	7	18	18	NUM
ejpam-5982	215	8	(	(	PUNCT
ejpam-5982	215	9	2	2	NUM
ejpam-5982	215	10	)	)	PUNCT
ejpam-5982	215	11	(	(	PUNCT
ejpam-5982	215	12	2025	2025	NUM
ejpam-5982	215	13	)	)	PUNCT
ejpam-5982	215	14	,	,	PUNCT
ejpam-5982	215	15	5982	5982	NUM
ejpam-5982	215	16	8	8	NUM
ejpam-5982	215	17	of	of	ADP
ejpam-5982	215	18	17	17	NUM
ejpam-5982	215	19	so	so	ADV
ejpam-5982	215	20	by	by	ADP
ejpam-5982	215	21	property	property	NOUN
ejpam-5982	215	22	of	of	ADP
ejpam-5982	215	23	ψ	ψ	NOUN
ejpam-5982	215	24	,	,	PUNCT
ejpam-5982	215	25	we	we	PRON
ejpam-5982	215	26	have	have	AUX
ejpam-5982	215	27	d(un+1	d(un+1	NUM
ejpam-5982	215	28	,	,	PUNCT
ejpam-5982	215	29	vn	vn	NOUN
ejpam-5982	215	30	)	)	PUNCT
ejpam-5982	215	31	≤	≤	NOUN
ejpam-5982	216	1	kd(un	kd(un	PROPN
ejpam-5982	216	2	,	,	PUNCT
ejpam-5982	216	3	vn	vn	PROPN
ejpam-5982	216	4	)	)	PUNCT
ejpam-5982	216	5	.	.	PUNCT
ejpam-5982	217	1	(	(	PUNCT
ejpam-5982	217	2	5	5	NUM
ejpam-5982	217	3	)	)	PUNCT
ejpam-5982	217	4	by	by	ADP
ejpam-5982	217	5	(	(	PUNCT
ejpam-5982	217	6	4	4	NUM
ejpam-5982	217	7	)	)	PUNCT
ejpam-5982	217	8	,	,	PUNCT
ejpam-5982	217	9	(	(	PUNCT
ejpam-5982	217	10	5	5	NUM
ejpam-5982	217	11	)	)	PUNCT
ejpam-5982	217	12	and	and	CCONJ
ejpam-5982	217	13	lemma	lemma	PROPN
ejpam-5982	217	14	1	1	NUM
ejpam-5982	217	15	,	,	PUNCT
ejpam-5982	217	16	the	the	DET
ejpam-5982	217	17	sequence	sequence	NOUN
ejpam-5982	217	18	(	(	PUNCT
ejpam-5982	217	19	un	un	PROPN
ejpam-5982	217	20	,	,	PUNCT
ejpam-5982	217	21	vn	vn	NOUN
ejpam-5982	217	22	)	)	PUNCT
ejpam-5982	217	23	is	be	AUX
ejpam-5982	217	24	cauchy	cauchy	ADJ
ejpam-5982	217	25	bisequence	bisequence	NOUN
ejpam-5982	217	26	and	and	CCONJ
ejpam-5982	217	27	since	since	SCONJ
ejpam-5982	217	28	given	give	VERB
ejpam-5982	217	29	bipolar	bipolar	ADJ
ejpam-5982	217	30	metric	metric	ADJ
ejpam-5982	217	31	space	space	NOUN
ejpam-5982	217	32	is	be	AUX
ejpam-5982	217	33	complete	complete	ADJ
ejpam-5982	217	34	,	,	PUNCT
ejpam-5982	217	35	hence	hence	ADV
ejpam-5982	217	36	the	the	DET
ejpam-5982	217	37	sequence	sequence	NOUN
ejpam-5982	217	38	(	(	PUNCT
ejpam-5982	217	39	un	un	PROPN
ejpam-5982	217	40	,	,	PUNCT
ejpam-5982	217	41	vn	vn	NOUN
ejpam-5982	217	42	)	)	PUNCT
ejpam-5982	217	43	biconverges	biconverge	NOUN
ejpam-5982	217	44	to	to	ADP
ejpam-5982	217	45	a	a	DET
ejpam-5982	217	46	point	point	NOUN
ejpam-5982	217	47	t	t	X
ejpam-5982	217	48	∈	∈	NOUN
ejpam-5982	217	49	x	x	PUNCT
ejpam-5982	217	50	∩	∩	PROPN
ejpam-5982	217	51	y	y	PROPN
ejpam-5982	217	52	.	.	PUNCT
ejpam-5982	218	1	so	so	ADV
ejpam-5982	218	2	lim	lim	PROPN
ejpam-5982	218	3	n→+∞	n→+∞	VERB
ejpam-5982	218	4	s1xn	s1xn	PUNCT
ejpam-5982	219	1	=	=	SYM
ejpam-5982	219	2	lim	lim	PROPN
ejpam-5982	219	3	n→+∞	n→+∞	VERB
ejpam-5982	219	4	t1yn	t1yn	PUNCT
ejpam-5982	220	1	=	=	SYM
ejpam-5982	220	2	lim	lim	PROPN
ejpam-5982	220	3	n→+∞	n→+∞	VERB
ejpam-5982	220	4	s2yn	s2yn	PUNCT
ejpam-5982	220	5	=	=	PROPN
ejpam-5982	220	6	lim	lim	PROPN
ejpam-5982	220	7	n→+∞	n→+∞	VERB
ejpam-5982	220	8	t2xn	t2xn	PUNCT
ejpam-5982	220	9	=	=	PUNCT
ejpam-5982	220	10	t.	t.	NOUN
ejpam-5982	220	11	(	(	PUNCT
ejpam-5982	220	12	6	6	NUM
ejpam-5982	220	13	)	)	PUNCT
ejpam-5982	220	14	by	by	ADP
ejpam-5982	220	15	using	use	VERB
ejpam-5982	220	16	compatibility	compatibility	NOUN
ejpam-5982	220	17	of	of	ADP
ejpam-5982	220	18	type	type	NOUN
ejpam-5982	220	19	(	(	PUNCT
ejpam-5982	220	20	a	a	NOUN
ejpam-5982	220	21	)	)	PUNCT
ejpam-5982	220	22	with	with	ADP
ejpam-5982	220	23	respect	respect	NOUN
ejpam-5982	220	24	to	to	ADP
ejpam-5982	220	25	y	y	PROPN
ejpam-5982	220	26	of	of	ADP
ejpam-5982	220	27	mappings	mapping	NOUN
ejpam-5982	220	28	s2	s2	NOUN
ejpam-5982	220	29	and	and	CCONJ
ejpam-5982	220	30	t1	t1	NOUN
ejpam-5982	220	31	and	and	CCONJ
ejpam-5982	220	32	(	(	PUNCT
ejpam-5982	220	33	6	6	NUM
ejpam-5982	220	34	)	)	PUNCT
ejpam-5982	220	35	,	,	PUNCT
ejpam-5982	220	36	we	we	PRON
ejpam-5982	220	37	get	get	VERB
ejpam-5982	220	38	lim	lim	PROPN
ejpam-5982	220	39	n→+∞	n→+∞	PROPN
ejpam-5982	220	40	d(t1s2yn	d(t1s2yn	PROPN
ejpam-5982	220	41	,	,	PUNCT
ejpam-5982	220	42	s2s2yn	s2s2yn	ADJ
ejpam-5982	220	43	)	)	PUNCT
ejpam-5982	220	44	=	=	SYM
ejpam-5982	220	45	0	0	NUM
ejpam-5982	220	46	or	or	CCONJ
ejpam-5982	220	47	lim	lim	PROPN
ejpam-5982	220	48	n→+∞	n→+∞	PROPN
ejpam-5982	220	49	d(s2t1yn	d(s2t1yn	PROPN
ejpam-5982	220	50	,	,	PUNCT
ejpam-5982	220	51	t1t1yn	t1t1yn	PRON
ejpam-5982	220	52	)	)	PUNCT
ejpam-5982	220	53	=	=	SYM
ejpam-5982	221	1	0	0	X
ejpam-5982	221	2	.	.	PUNCT
ejpam-5982	222	1	by	by	ADP
ejpam-5982	222	2	continuity	continuity	NOUN
ejpam-5982	222	3	of	of	ADP
ejpam-5982	222	4	mappings	mapping	NOUN
ejpam-5982	222	5	s2	s2	NOUN
ejpam-5982	222	6	and	and	CCONJ
ejpam-5982	222	7	t1	t1	VERB
ejpam-5982	222	8	,	,	PUNCT
ejpam-5982	222	9	we	we	PRON
ejpam-5982	222	10	have	have	VERB
ejpam-5982	222	11	d(t1	d(t1	PROPN
ejpam-5982	222	12	t	t	PROPN
ejpam-5982	222	13	,	,	PUNCT
ejpam-5982	222	14	s2	s2	PROPN
ejpam-5982	222	15	t	t	PROPN
ejpam-5982	222	16	)	)	PUNCT
ejpam-5982	222	17	=	=	SYM
ejpam-5982	223	1	0	0	NUM
ejpam-5982	223	2	t1	t1	NUM
ejpam-5982	223	3	t	t	PROPN
ejpam-5982	223	4	=	=	SYM
ejpam-5982	223	5	s2	s2	PROPN
ejpam-5982	223	6	t.	t.	NOUN
ejpam-5982	223	7	(	(	PUNCT
ejpam-5982	223	8	7	7	NUM
ejpam-5982	223	9	)	)	PUNCT
ejpam-5982	223	10	similarly	similarly	ADV
ejpam-5982	223	11	by	by	ADP
ejpam-5982	223	12	using	use	VERB
ejpam-5982	223	13	compatibility	compatibility	NOUN
ejpam-5982	223	14	of	of	ADP
ejpam-5982	223	15	type	type	NOUN
ejpam-5982	223	16	(	(	PUNCT
ejpam-5982	223	17	a	a	NOUN
ejpam-5982	223	18	)	)	PUNCT
ejpam-5982	223	19	with	with	ADP
ejpam-5982	223	20	respect	respect	NOUN
ejpam-5982	223	21	tox	tox	NOUN
ejpam-5982	223	22	and	and	CCONJ
ejpam-5982	223	23	continuity	continuity	NOUN
ejpam-5982	223	24	of	of	ADP
ejpam-5982	223	25	mappings	mapping	NOUN
ejpam-5982	223	26	s1	s1	NOUN
ejpam-5982	223	27	and	and	CCONJ
ejpam-5982	223	28	t2	t2	PROPN
ejpam-5982	223	29	and	and	CCONJ
ejpam-5982	223	30	(	(	PUNCT
ejpam-5982	223	31	6	6	NUM
ejpam-5982	223	32	)	)	PUNCT
ejpam-5982	223	33	,	,	PUNCT
ejpam-5982	223	34	we	we	PRON
ejpam-5982	223	35	get	get	VERB
ejpam-5982	223	36	t2	t2	NOUN
ejpam-5982	223	37	t	t	NOUN
ejpam-5982	223	38	=	=	SYM
ejpam-5982	223	39	s1	s1	PROPN
ejpam-5982	223	40	t.	t.	PROPN
ejpam-5982	223	41	(	(	PUNCT
ejpam-5982	223	42	8)	8)	NUM
ejpam-5982	223	43	now	now	ADV
ejpam-5982	223	44	putting	put	VERB
ejpam-5982	223	45	x	x	PUNCT
ejpam-5982	223	46	=	=	PUNCT
ejpam-5982	223	47	y	y	PROPN
ejpam-5982	223	48	=	=	SYM
ejpam-5982	223	49	t	t	PROPN
ejpam-5982	223	50	in	in	ADP
ejpam-5982	223	51	(	(	PUNCT
ejpam-5982	223	52	3	3	NUM
ejpam-5982	223	53	)	)	PUNCT
ejpam-5982	223	54	and	and	CCONJ
ejpam-5982	223	55	using	use	VERB
ejpam-5982	223	56	(	(	PUNCT
ejpam-5982	223	57	7	7	NUM
ejpam-5982	223	58	)	)	PUNCT
ejpam-5982	223	59	and	and	CCONJ
ejpam-5982	223	60	(	(	PUNCT
ejpam-5982	223	61	8)	8)	NUM
ejpam-5982	223	62	,	,	PUNCT
ejpam-5982	223	63	we	we	PRON
ejpam-5982	223	64	get	get	VERB
ejpam-5982	223	65	ψ(d(s2	ψ(d(s2	PROPN
ejpam-5982	223	66	t	t	PROPN
ejpam-5982	223	67	,	,	PUNCT
ejpam-5982	223	68	s1	s1	PROPN
ejpam-5982	223	69	t	t	PROPN
ejpam-5982	223	70	)	)	PUNCT
ejpam-5982	223	71	,	,	PUNCT
ejpam-5982	223	72	d(t2	d(t2	NOUN
ejpam-5982	223	73	t	t	PROPN
ejpam-5982	223	74	,	,	PUNCT
ejpam-5982	223	75	t1	t1	PROPN
ejpam-5982	223	76	t	t	PROPN
ejpam-5982	223	77	)	)	PUNCT
ejpam-5982	223	78	,	,	PUNCT
ejpam-5982	223	79	d(t2	d(t2	NOUN
ejpam-5982	223	80	t	t	PROPN
ejpam-5982	223	81	,	,	PUNCT
ejpam-5982	223	82	s1	s1	PROPN
ejpam-5982	223	83	t	t	PROPN
ejpam-5982	223	84	)	)	PUNCT
ejpam-5982	223	85	,	,	PUNCT
ejpam-5982	223	86	d(s2	d(s2	NOUN
ejpam-5982	223	87	t	t	PROPN
ejpam-5982	223	88	,	,	PUNCT
ejpam-5982	223	89	t1	t1	PROPN
ejpam-5982	223	90	t	t	PROPN
ejpam-5982	223	91	)	)	PUNCT
ejpam-5982	223	92	)	)	PUNCT
ejpam-5982	224	1	≤	≤	NOUN
ejpam-5982	224	2	0	0	NUM
ejpam-5982	224	3	ψ(d(s2	ψ(d(s2	PROPN
ejpam-5982	224	4	t	t	PROPN
ejpam-5982	224	5	,	,	PUNCT
ejpam-5982	224	6	s1	s1	PROPN
ejpam-5982	224	7	t	t	PROPN
ejpam-5982	224	8	)	)	PUNCT
ejpam-5982	224	9	,	,	PUNCT
ejpam-5982	224	10	d(s1	d(s1	PROPN
ejpam-5982	224	11	t	t	PROPN
ejpam-5982	224	12	,	,	PUNCT
ejpam-5982	224	13	s2	s2	PROPN
ejpam-5982	224	14	t	t	PROPN
ejpam-5982	224	15	)	)	PUNCT
ejpam-5982	224	16	,	,	PUNCT
ejpam-5982	224	17	0	0	NUM
ejpam-5982	224	18	,	,	PUNCT
ejpam-5982	224	19	0	0	NUM
ejpam-5982	224	20	)	)	PUNCT
ejpam-5982	224	21	≤	≤	NOUN
ejpam-5982	224	22	0	0	NUM
ejpam-5982	224	23	.	.	PUNCT
ejpam-5982	225	1	so	so	ADV
ejpam-5982	225	2	by	by	ADP
ejpam-5982	225	3	property	property	NOUN
ejpam-5982	225	4	of	of	ADP
ejpam-5982	225	5	ψ	ψ	NOUN
ejpam-5982	225	6	,	,	PUNCT
ejpam-5982	225	7	this	this	PRON
ejpam-5982	225	8	implies	imply	VERB
ejpam-5982	225	9	that	that	SCONJ
ejpam-5982	225	10	s2	s2	PROPN
ejpam-5982	225	11	t	t	NOUN
ejpam-5982	225	12	=	=	SYM
ejpam-5982	225	13	s1	s1	PROPN
ejpam-5982	225	14	t.	t.	NOUN
ejpam-5982	225	15	hence	hence	ADV
ejpam-5982	225	16	we	we	PRON
ejpam-5982	225	17	get	get	VERB
ejpam-5982	225	18	s2	s2	PROPN
ejpam-5982	225	19	t	t	NOUN
ejpam-5982	225	20	=	=	SYM
ejpam-5982	225	21	s1	s1	PROPN
ejpam-5982	225	22	t	t	NOUN
ejpam-5982	225	23	=	=	SYM
ejpam-5982	225	24	t2	t2	PROPN
ejpam-5982	225	25	t	t	NOUN
ejpam-5982	225	26	=	=	SYM
ejpam-5982	225	27	t1	t1	PROPN
ejpam-5982	225	28	t	t	PROPN
ejpam-5982	225	29	=	=	SYM
ejpam-5982	225	30	u	u	PROPN
ejpam-5982	225	31	(	(	PUNCT
ejpam-5982	225	32	say	say	INTJ
ejpam-5982	225	33	)	)	PUNCT
ejpam-5982	225	34	that	that	PRON
ejpam-5982	225	35	is	be	AUX
ejpam-5982	225	36	t	t	NOUN
ejpam-5982	225	37	is	be	AUX
ejpam-5982	225	38	a	a	DET
ejpam-5982	225	39	coincidence	coincidence	NOUN
ejpam-5982	225	40	point	point	NOUN
ejpam-5982	225	41	of	of	ADP
ejpam-5982	225	42	s1	s1	NOUN
ejpam-5982	225	43	,	,	PUNCT
ejpam-5982	225	44	s2	s2	PROPN
ejpam-5982	225	45	,	,	PUNCT
ejpam-5982	225	46	t1	t1	NOUN
ejpam-5982	225	47	and	and	CCONJ
ejpam-5982	225	48	t2	t2	NOUN
ejpam-5982	225	49	.	.	PUNCT
ejpam-5982	226	1	by	by	ADP
ejpam-5982	226	2	proposition	proposition	NOUN
ejpam-5982	226	3	1	1	NUM
ejpam-5982	226	4	the	the	DET
ejpam-5982	226	5	pairs	pair	NOUN
ejpam-5982	226	6	(	(	PUNCT
ejpam-5982	226	7	s2	s2	PROPN
ejpam-5982	226	8	,	,	PUNCT
ejpam-5982	226	9	t1	t1	NOUN
ejpam-5982	226	10	)	)	PUNCT
ejpam-5982	226	11	and	and	CCONJ
ejpam-5982	226	12	(	(	PUNCT
ejpam-5982	226	13	s1	s1	NOUN
ejpam-5982	226	14	,	,	PUNCT
ejpam-5982	226	15	t2	t2	NOUN
ejpam-5982	226	16	)	)	PUNCT
ejpam-5982	226	17	are	be	AUX
ejpam-5982	226	18	weak	weak	ADJ
ejpam-5982	226	19	compatibility	compatibility	NOUN
ejpam-5982	226	20	of	of	ADP
ejpam-5982	226	21	type	type	NOUN
ejpam-5982	226	22	(	(	PUNCT
ejpam-5982	226	23	a	a	NOUN
ejpam-5982	226	24	)	)	PUNCT
ejpam-5982	226	25	.	.	PUNCT
ejpam-5982	227	1	so	so	ADV
ejpam-5982	227	2	we	we	PRON
ejpam-5982	227	3	get	get	VERB
ejpam-5982	227	4	t1s2	t1s2	NOUN
ejpam-5982	227	5	t	t	NOUN
ejpam-5982	227	6	=	=	PUNCT
ejpam-5982	227	7	s2s2	s2s2	NOUN
ejpam-5982	227	8	t	t	PROPN
ejpam-5982	227	9	and	and	CCONJ
ejpam-5982	227	10	t2s1	t2s1	PROPN
ejpam-5982	227	11	t	t	NOUN
ejpam-5982	227	12	=	=	PUNCT
ejpam-5982	228	1	s1s1	s1s1	PROPN
ejpam-5982	228	2	t.	t.	NOUN
ejpam-5982	228	3	this	this	PRON
ejpam-5982	228	4	implies	imply	VERB
ejpam-5982	228	5	t1u	t1u	VERB
ejpam-5982	228	6	=	=	SYM
ejpam-5982	228	7	s2u	s2u	NOUN
ejpam-5982	228	8	and	and	CCONJ
ejpam-5982	228	9	t2u	t2u	NOUN
ejpam-5982	228	10	=	=	SYM
ejpam-5982	228	11	s1u	s1u	NOUN
ejpam-5982	228	12	.	.	PUNCT
ejpam-5982	229	1	(	(	PUNCT
ejpam-5982	229	2	9	9	X
ejpam-5982	229	3	)	)	PUNCT
ejpam-5982	229	4	now	now	ADV
ejpam-5982	229	5	putting	put	VERB
ejpam-5982	229	6	x	x	PUNCT
ejpam-5982	229	7	=	=	PUNCT
ejpam-5982	229	8	y	y	PROPN
ejpam-5982	229	9	=	=	SYM
ejpam-5982	229	10	u	u	PROPN
ejpam-5982	229	11	in	in	ADP
ejpam-5982	229	12	(	(	PUNCT
ejpam-5982	229	13	3	3	NUM
ejpam-5982	229	14	)	)	PUNCT
ejpam-5982	229	15	,	,	PUNCT
ejpam-5982	229	16	we	we	PRON
ejpam-5982	229	17	get	get	VERB
ejpam-5982	229	18	ψ(d(s2u	ψ(d(s2u	NOUN
ejpam-5982	229	19	,	,	PUNCT
ejpam-5982	229	20	s1u	s1u	NOUN
ejpam-5982	229	21	)	)	PUNCT
ejpam-5982	229	22	,	,	PUNCT
ejpam-5982	229	23	d(t2u	d(t2u	PRON
ejpam-5982	229	24	,	,	PUNCT
ejpam-5982	229	25	t1u	t1u	NOUN
ejpam-5982	229	26	)	)	PUNCT
ejpam-5982	229	27	,	,	PUNCT
ejpam-5982	229	28	d(t2u	d(t2u	NOUN
ejpam-5982	229	29	,	,	PUNCT
ejpam-5982	229	30	s1u	s1u	NOUN
ejpam-5982	229	31	)	)	PUNCT
ejpam-5982	229	32	,	,	PUNCT
ejpam-5982	229	33	d(s2u	d(s2u	NOUN
ejpam-5982	229	34	,	,	PUNCT
ejpam-5982	229	35	t1u	t1u	NOUN
ejpam-5982	229	36	)	)	PUNCT
ejpam-5982	229	37	)	)	PUNCT
ejpam-5982	230	1	≤	≤	ADV
ejpam-5982	230	2	0	0	X
ejpam-5982	231	1	ψ(d(s2u	ψ(d(s2u	NOUN
ejpam-5982	231	2	,	,	PUNCT
ejpam-5982	231	3	s1u	s1u	NOUN
ejpam-5982	231	4	)	)	PUNCT
ejpam-5982	231	5	,	,	PUNCT
ejpam-5982	231	6	d(s1u	d(s1u	NOUN
ejpam-5982	231	7	,	,	PUNCT
ejpam-5982	231	8	s2u	s2u	NOUN
ejpam-5982	231	9	)	)	PUNCT
ejpam-5982	231	10	,	,	PUNCT
ejpam-5982	231	11	0	0	NUM
ejpam-5982	231	12	,	,	PUNCT
ejpam-5982	231	13	0	0	NUM
ejpam-5982	231	14	)	)	PUNCT
ejpam-5982	231	15	≤	≤	NOUN
ejpam-5982	231	16	0	0	NUM
ejpam-5982	232	1	d(s2u	d(s2u	NOUN
ejpam-5982	232	2	,	,	PUNCT
ejpam-5982	232	3	s1u	s1u	NOUN
ejpam-5982	232	4	)	)	PUNCT
ejpam-5982	232	5	=	=	SYM
ejpam-5982	232	6	0	0	PUNCT
ejpam-5982	232	7	(	(	PUNCT
ejpam-5982	232	8	by	by	ADP
ejpam-5982	232	9	property	property	NOUN
ejpam-5982	232	10	of	of	ADP
ejpam-5982	232	11	ψ	ψ	NOUN
ejpam-5982	232	12	)	)	PUNCT
ejpam-5982	232	13	s2u	s2u	NOUN
ejpam-5982	232	14	=	=	SYM
ejpam-5982	232	15	s1u	s1u	NOUN
ejpam-5982	232	16	.	.	PUNCT
ejpam-5982	233	1	(	(	PUNCT
ejpam-5982	233	2	10	10	NUM
ejpam-5982	233	3	)	)	PUNCT
ejpam-5982	234	1	p.	p.	NOUN
ejpam-5982	234	2	p.	p.	NOUN
ejpam-5982	235	1	murthy	murthy	ADJ
ejpam-5982	236	1	et	et	PROPN
ejpam-5982	236	2	al	al	PROPN
ejpam-5982	236	3	.	.	PUNCT
ejpam-5982	236	4	/	/	SYM
ejpam-5982	236	5	eur	eur	PROPN
ejpam-5982	236	6	.	.	PUNCT
ejpam-5982	237	1	j.	j.	PROPN
ejpam-5982	237	2	pure	pure	PROPN
ejpam-5982	237	3	appl	appl	PROPN
ejpam-5982	237	4	.	.	PROPN
ejpam-5982	237	5	math	math	PROPN
ejpam-5982	237	6	,	,	PUNCT
ejpam-5982	237	7	18	18	NUM
ejpam-5982	237	8	(	(	PUNCT
ejpam-5982	237	9	2	2	NUM
ejpam-5982	237	10	)	)	PUNCT
ejpam-5982	237	11	(	(	PUNCT
ejpam-5982	237	12	2025	2025	NUM
ejpam-5982	237	13	)	)	PUNCT
ejpam-5982	237	14	,	,	PUNCT
ejpam-5982	237	15	5982	5982	NUM
ejpam-5982	237	16	9	9	NUM
ejpam-5982	237	17	of	of	ADP
ejpam-5982	237	18	17	17	NUM
ejpam-5982	237	19	by	by	ADP
ejpam-5982	237	20	(	(	PUNCT
ejpam-5982	237	21	9	9	NUM
ejpam-5982	237	22	)	)	PUNCT
ejpam-5982	237	23	and	and	CCONJ
ejpam-5982	237	24	(	(	PUNCT
ejpam-5982	237	25	10	10	NUM
ejpam-5982	237	26	)	)	PUNCT
ejpam-5982	237	27	,	,	PUNCT
ejpam-5982	237	28	we	we	PRON
ejpam-5982	237	29	get	get	AUX
ejpam-5982	237	30	t1u	t1u	VERB
ejpam-5982	237	31	=	=	PUNCT
ejpam-5982	237	32	t2u	t2u	VERB
ejpam-5982	237	33	=	=	PUNCT
ejpam-5982	238	1	s1u	s1u	NOUN
ejpam-5982	238	2	=	=	SYM
ejpam-5982	238	3	s2u	s2u	NOUN
ejpam-5982	238	4	.	.	PUNCT
ejpam-5982	239	1	so	so	ADV
ejpam-5982	239	2	u	u	NOUN
ejpam-5982	239	3	is	be	AUX
ejpam-5982	239	4	also	also	ADV
ejpam-5982	239	5	a	a	DET
ejpam-5982	239	6	coincidence	coincidence	NOUN
ejpam-5982	239	7	point	point	NOUN
ejpam-5982	239	8	of	of	ADP
ejpam-5982	239	9	s1	s1	NOUN
ejpam-5982	239	10	,	,	PUNCT
ejpam-5982	239	11	s2	s2	PROPN
ejpam-5982	239	12	,	,	PUNCT
ejpam-5982	239	13	t1	t1	NOUN
ejpam-5982	239	14	and	and	CCONJ
ejpam-5982	239	15	t2	t2	NOUN
ejpam-5982	239	16	.	.	PUNCT
ejpam-5982	240	1	now	now	ADV
ejpam-5982	240	2	putting	put	VERB
ejpam-5982	240	3	y	y	NOUN
ejpam-5982	240	4	=	=	PUNCT
ejpam-5982	240	5	u	u	PROPN
ejpam-5982	240	6	and	and	CCONJ
ejpam-5982	240	7	x	x	X
ejpam-5982	240	8	=	=	SYM
ejpam-5982	240	9	t	t	PROPN
ejpam-5982	240	10	in	in	ADP
ejpam-5982	240	11	(	(	PUNCT
ejpam-5982	240	12	3	3	NUM
ejpam-5982	240	13	)	)	PUNCT
ejpam-5982	240	14	,	,	PUNCT
ejpam-5982	240	15	we	we	PRON
ejpam-5982	240	16	get	get	VERB
ejpam-5982	240	17	ψ(d(s2u	ψ(d(s2u	NOUN
ejpam-5982	240	18	,	,	PUNCT
ejpam-5982	240	19	s1	s1	PROPN
ejpam-5982	240	20	t	t	PROPN
ejpam-5982	240	21	)	)	PUNCT
ejpam-5982	240	22	,	,	PUNCT
ejpam-5982	240	23	d(t2	d(t2	NOUN
ejpam-5982	240	24	t	t	NOUN
ejpam-5982	240	25	,	,	PUNCT
ejpam-5982	240	26	t1u	t1u	NOUN
ejpam-5982	240	27	)	)	PUNCT
ejpam-5982	240	28	,	,	PUNCT
ejpam-5982	240	29	d(t2	d(t2	NOUN
ejpam-5982	240	30	t	t	PROPN
ejpam-5982	240	31	,	,	PUNCT
ejpam-5982	240	32	s1	s1	PROPN
ejpam-5982	240	33	t	t	PROPN
ejpam-5982	240	34	)	)	PUNCT
ejpam-5982	240	35	,	,	PUNCT
ejpam-5982	240	36	d(s2u	d(s2u	NOUN
ejpam-5982	240	37	,	,	PUNCT
ejpam-5982	240	38	t1u	t1u	NOUN
ejpam-5982	240	39	)	)	PUNCT
ejpam-5982	240	40	)	)	PUNCT
ejpam-5982	241	1	≤	≤	ADV
ejpam-5982	241	2	0	0	X
ejpam-5982	242	1	ψ(d(s2u	ψ(d(s2u	NOUN
ejpam-5982	242	2	,	,	PUNCT
ejpam-5982	242	3	u	u	NOUN
ejpam-5982	242	4	)	)	PUNCT
ejpam-5982	242	5	,	,	PUNCT
ejpam-5982	242	6	d(u	d(u	PROPN
ejpam-5982	242	7	,	,	PUNCT
ejpam-5982	242	8	s2u	s2u	NOUN
ejpam-5982	242	9	)	)	PUNCT
ejpam-5982	242	10	,	,	PUNCT
ejpam-5982	242	11	0	0	NUM
ejpam-5982	242	12	,	,	PUNCT
ejpam-5982	242	13	0	0	NUM
ejpam-5982	242	14	)	)	PUNCT
ejpam-5982	242	15	≤	≤	NOUN
ejpam-5982	242	16	0	0	PUNCT
ejpam-5982	243	1	ψ(d(s2u	ψ(d(s2u	NOUN
ejpam-5982	243	2	,	,	PUNCT
ejpam-5982	243	3	u	u	NOUN
ejpam-5982	243	4	)	)	PUNCT
ejpam-5982	243	5	,	,	PUNCT
ejpam-5982	243	6	d(s2u	d(s2u	PROPN
ejpam-5982	243	7	,	,	PUNCT
ejpam-5982	243	8	u	u	NOUN
ejpam-5982	243	9	)	)	PUNCT
ejpam-5982	243	10	,	,	PUNCT
ejpam-5982	243	11	0	0	NUM
ejpam-5982	243	12	,	,	PUNCT
ejpam-5982	243	13	0	0	NUM
ejpam-5982	243	14	)	)	PUNCT
ejpam-5982	243	15	≤	≤	NOUN
ejpam-5982	243	16	0	0	NUM
ejpam-5982	244	1	d(s2u	d(s2u	NOUN
ejpam-5982	244	2	,	,	PUNCT
ejpam-5982	244	3	u	u	NOUN
ejpam-5982	244	4	)	)	PUNCT
ejpam-5982	244	5	=	=	SYM
ejpam-5982	244	6	0	0	NUM
ejpam-5982	244	7	s2u	s2u	NOUN
ejpam-5982	244	8	=	=	PUNCT
ejpam-5982	244	9	u.	u.	NOUN
ejpam-5982	244	10	so	so	ADV
ejpam-5982	244	11	u	u	NOUN
ejpam-5982	244	12	is	be	AUX
ejpam-5982	244	13	a	a	DET
ejpam-5982	244	14	common	common	ADJ
ejpam-5982	244	15	fixed	fix	VERB
ejpam-5982	244	16	point	point	NOUN
ejpam-5982	244	17	of	of	ADP
ejpam-5982	244	18	all	all	DET
ejpam-5982	244	19	the	the	DET
ejpam-5982	244	20	given	give	VERB
ejpam-5982	244	21	four	four	NUM
ejpam-5982	244	22	mappings	mapping	NOUN
ejpam-5982	244	23	.	.	PUNCT
ejpam-5982	245	1	now	now	ADV
ejpam-5982	245	2	we	we	PRON
ejpam-5982	245	3	prove	prove	VERB
ejpam-5982	245	4	that	that	SCONJ
ejpam-5982	245	5	the	the	DET
ejpam-5982	245	6	fixed	fix	VERB
ejpam-5982	245	7	point	point	NOUN
ejpam-5982	245	8	is	be	AUX
ejpam-5982	245	9	unique	unique	ADJ
ejpam-5982	245	10	.	.	PUNCT
ejpam-5982	246	1	for	for	ADP
ejpam-5982	246	2	this	this	PRON
ejpam-5982	246	3	let	let	VERB
ejpam-5982	246	4	us	we	PRON
ejpam-5982	246	5	assume	assume	VERB
ejpam-5982	246	6	that	that	SCONJ
ejpam-5982	246	7	u1	u1	NOUN
ejpam-5982	246	8	is	be	AUX
ejpam-5982	246	9	another	another	DET
ejpam-5982	246	10	common	common	ADJ
ejpam-5982	246	11	fixed	fix	VERB
ejpam-5982	246	12	point	point	NOUN
ejpam-5982	246	13	then	then	ADV
ejpam-5982	246	14	putting	put	VERB
ejpam-5982	246	15	y	y	PROPN
ejpam-5982	246	16	=	=	PUNCT
ejpam-5982	246	17	u	u	PROPN
ejpam-5982	246	18	and	and	CCONJ
ejpam-5982	246	19	x	x	NOUN
ejpam-5982	246	20	=	=	SYM
ejpam-5982	246	21	u1	u1	NOUN
ejpam-5982	246	22	in	in	ADP
ejpam-5982	246	23	(	(	PUNCT
ejpam-5982	246	24	3	3	NUM
ejpam-5982	246	25	)	)	PUNCT
ejpam-5982	246	26	,	,	PUNCT
ejpam-5982	246	27	we	we	PRON
ejpam-5982	246	28	get	get	VERB
ejpam-5982	246	29	ψ(d(s2u	ψ(d(s2u	NOUN
ejpam-5982	246	30	,	,	PUNCT
ejpam-5982	246	31	s1u1	s1u1	NOUN
ejpam-5982	246	32	)	)	PUNCT
ejpam-5982	246	33	,	,	PUNCT
ejpam-5982	246	34	d(t2u1	d(t2u1	NOUN
ejpam-5982	246	35	,	,	PUNCT
ejpam-5982	246	36	t1u	t1u	NOUN
ejpam-5982	246	37	)	)	PUNCT
ejpam-5982	246	38	,	,	PUNCT
ejpam-5982	246	39	d(t2u1	d(t2u1	PROPN
ejpam-5982	246	40	,	,	PUNCT
ejpam-5982	246	41	s1u1	s1u1	NOUN
ejpam-5982	246	42	)	)	PUNCT
ejpam-5982	246	43	,	,	PUNCT
ejpam-5982	246	44	d(s2u	d(s2u	NOUN
ejpam-5982	246	45	,	,	PUNCT
ejpam-5982	246	46	t1u	t1u	NOUN
ejpam-5982	246	47	)	)	PUNCT
ejpam-5982	246	48	)	)	PUNCT
ejpam-5982	247	1	≤	≤	NOUN
ejpam-5982	247	2	0	0	NUM
ejpam-5982	248	1	ψ(d(u	ψ(d(u	NOUN
ejpam-5982	248	2	,	,	PUNCT
ejpam-5982	248	3	u1	u1	NOUN
ejpam-5982	248	4	)	)	PUNCT
ejpam-5982	248	5	,	,	PUNCT
ejpam-5982	248	6	d(u1	d(u1	NOUN
ejpam-5982	248	7	,	,	PUNCT
ejpam-5982	248	8	u	u	NOUN
ejpam-5982	248	9	)	)	PUNCT
ejpam-5982	248	10	,	,	PUNCT
ejpam-5982	248	11	0	0	NUM
ejpam-5982	248	12	,	,	PUNCT
ejpam-5982	248	13	0	0	NUM
ejpam-5982	248	14	)	)	PUNCT
ejpam-5982	248	15	≤	≤	NOUN
ejpam-5982	248	16	0	0	NUM
ejpam-5982	248	17	.	.	PUNCT
ejpam-5982	249	1	by	by	ADP
ejpam-5982	249	2	remark	remark	NOUN
ejpam-5982	249	3	2	2	NUM
ejpam-5982	249	4	,	,	PUNCT
ejpam-5982	249	5	this	this	PRON
ejpam-5982	249	6	implies	imply	VERB
ejpam-5982	249	7	d(u	d(u	PROPN
ejpam-5982	249	8	,	,	PUNCT
ejpam-5982	249	9	u1	u1	NOUN
ejpam-5982	249	10	)	)	PUNCT
ejpam-5982	249	11	=	=	SYM
ejpam-5982	250	1	0	0	NUM
ejpam-5982	250	2	u	u	NOUN
ejpam-5982	250	3	=	=	NOUN
ejpam-5982	250	4	u1	u1	NOUN
ejpam-5982	250	5	.	.	PUNCT
ejpam-5982	251	1	so	so	ADV
ejpam-5982	251	2	u	u	NOUN
ejpam-5982	251	3	is	be	AUX
ejpam-5982	251	4	the	the	DET
ejpam-5982	251	5	unique	unique	ADJ
ejpam-5982	251	6	common	common	ADJ
ejpam-5982	251	7	fixed	fix	VERB
ejpam-5982	251	8	point	point	NOUN
ejpam-5982	251	9	.	.	PUNCT
ejpam-5982	252	1	now	now	ADV
ejpam-5982	252	2	we	we	PRON
ejpam-5982	252	3	prove	prove	VERB
ejpam-5982	252	4	some	some	DET
ejpam-5982	252	5	corollaries	corollary	NOUN
ejpam-5982	252	6	derived	derive	VERB
ejpam-5982	252	7	from	from	ADP
ejpam-5982	252	8	theorem	theorem	ADJ
ejpam-5982	252	9	1	1	NUM
ejpam-5982	252	10	corollary	corollary	NOUN
ejpam-5982	252	11	1	1	NUM
ejpam-5982	252	12	.	.	PUNCT
ejpam-5982	253	1	let	let	VERB
ejpam-5982	253	2	(	(	PUNCT
ejpam-5982	253	3	x	x	X
ejpam-5982	253	4	,	,	PUNCT
ejpam-5982	253	5	y	y	PROPN
ejpam-5982	253	6	,	,	PUNCT
ejpam-5982	253	7	d	d	NOUN
ejpam-5982	253	8	)	)	PUNCT
ejpam-5982	253	9	be	be	AUX
ejpam-5982	253	10	a	a	DET
ejpam-5982	253	11	complete	complete	ADJ
ejpam-5982	253	12	bipolar	bipolar	ADJ
ejpam-5982	253	13	metric	metric	ADJ
ejpam-5982	253	14	space	space	NOUN
ejpam-5982	253	15	and	and	CCONJ
ejpam-5982	253	16	let	let	VERB
ejpam-5982	253	17	t	t	NOUN
ejpam-5982	253	18	:	:	PUNCT
ejpam-5982	253	19	(	(	PUNCT
ejpam-5982	253	20	x	x	X
ejpam-5982	253	21	,	,	PUNCT
ejpam-5982	253	22	y	y	PROPN
ejpam-5982	253	23	,	,	PUNCT
ejpam-5982	253	24	d	d	NOUN
ejpam-5982	253	25	)	)	PUNCT
ejpam-5982	253	26	⇒	⇒	NOUN
ejpam-5982	253	27	(	(	PUNCT
ejpam-5982	253	28	x	x	X
ejpam-5982	253	29	,	,	PUNCT
ejpam-5982	253	30	y	y	PROPN
ejpam-5982	253	31	,	,	PUNCT
ejpam-5982	253	32	d	d	NOUN
ejpam-5982	253	33	)	)	PUNCT
ejpam-5982	253	34	be	be	AUX
ejpam-5982	253	35	a	a	DET
ejpam-5982	253	36	covariant	covariant	ADJ
ejpam-5982	253	37	map	map	NOUN
ejpam-5982	253	38	and	and	CCONJ
ejpam-5982	253	39	s1	s1	NOUN
ejpam-5982	253	40	,	,	PUNCT
ejpam-5982	253	41	s2	s2	NOUN
ejpam-5982	253	42	:	:	PUNCT
ejpam-5982	253	43	(	(	PUNCT
ejpam-5982	253	44	x	x	X
ejpam-5982	253	45	,	,	PUNCT
ejpam-5982	253	46	y	y	PROPN
ejpam-5982	253	47	,	,	PUNCT
ejpam-5982	253	48	d	d	NOUN
ejpam-5982	253	49	)	)	PUNCT
ejpam-5982	253	50	⇄	⇄	NOUN
ejpam-5982	253	51	(	(	PUNCT
ejpam-5982	253	52	x	x	X
ejpam-5982	253	53	,	,	PUNCT
ejpam-5982	253	54	y	y	PROPN
ejpam-5982	253	55	,	,	PUNCT
ejpam-5982	253	56	d	d	NOUN
ejpam-5982	253	57	)	)	PUNCT
ejpam-5982	253	58	be	be	AUX
ejpam-5982	253	59	two	two	NUM
ejpam-5982	253	60	contravariant	contravariant	ADJ
ejpam-5982	253	61	maps	map	NOUN
ejpam-5982	253	62	satisfying	satisfy	VERB
ejpam-5982	253	63	the	the	DET
ejpam-5982	253	64	following	follow	VERB
ejpam-5982	253	65	conditions	condition	NOUN
ejpam-5982	253	66	(	(	PUNCT
ejpam-5982	253	67	i	i	NOUN
ejpam-5982	253	68	)	)	PUNCT
ejpam-5982	253	69	s2	s2	NOUN
ejpam-5982	253	70	and	and	CCONJ
ejpam-5982	253	71	t	t	PROPN
ejpam-5982	253	72	are	be	AUX
ejpam-5982	253	73	compatible	compatible	ADJ
ejpam-5982	253	74	of	of	ADP
ejpam-5982	253	75	type	type	NOUN
ejpam-5982	253	76	(	(	PUNCT
ejpam-5982	253	77	a	a	NOUN
ejpam-5982	253	78	)	)	PUNCT
ejpam-5982	253	79	with	with	ADP
ejpam-5982	253	80	respect	respect	NOUN
ejpam-5982	253	81	to	to	ADP
ejpam-5982	253	82	y.	y.	PROPN
ejpam-5982	253	83	(	(	PUNCT
ejpam-5982	253	84	ii	ii	PROPN
ejpam-5982	253	85	)	)	PUNCT
ejpam-5982	253	86	s1	s1	PROPN
ejpam-5982	253	87	and	and	CCONJ
ejpam-5982	253	88	t	t	NOUN
ejpam-5982	253	89	are	be	AUX
ejpam-5982	253	90	compatible	compatible	ADJ
ejpam-5982	253	91	of	of	ADP
ejpam-5982	253	92	type	type	NOUN
ejpam-5982	253	93	(	(	PUNCT
ejpam-5982	253	94	a	a	NOUN
ejpam-5982	253	95	)	)	PUNCT
ejpam-5982	253	96	with	with	ADP
ejpam-5982	253	97	respect	respect	NOUN
ejpam-5982	253	98	to	to	ADP
ejpam-5982	253	99	x.	x.	NOUN
ejpam-5982	253	100	(	(	PUNCT
ejpam-5982	253	101	iii	iii	NOUN
ejpam-5982	253	102	)	)	PUNCT
ejpam-5982	253	103	s1(x	s1(x	NOUN
ejpam-5982	253	104	∪	∪	PROPN
ejpam-5982	253	105	y	y	PROPN
ejpam-5982	253	106	)	)	PUNCT
ejpam-5982	253	107	⊆	⊆	NUM
ejpam-5982	253	108	t	t	NOUN
ejpam-5982	253	109	(	(	PUNCT
ejpam-5982	253	110	x	x	X
ejpam-5982	253	111	∪	∪	PROPN
ejpam-5982	253	112	y	y	PROPN
ejpam-5982	253	113	)	)	PUNCT
ejpam-5982	253	114	and	and	CCONJ
ejpam-5982	253	115	s2(x	s2(x	PROPN
ejpam-5982	253	116	∪	∪	PROPN
ejpam-5982	253	117	y	y	PROPN
ejpam-5982	253	118	)	)	PUNCT
ejpam-5982	253	119	⊆	⊆	NUM
ejpam-5982	253	120	t	t	NOUN
ejpam-5982	253	121	(	(	PUNCT
ejpam-5982	253	122	x	x	X
ejpam-5982	253	123	∪	∪	PROPN
ejpam-5982	253	124	y	y	PROPN
ejpam-5982	253	125	)	)	PUNCT
ejpam-5982	253	126	.	.	PUNCT
ejpam-5982	254	1	(	(	PUNCT
ejpam-5982	254	2	iv	iv	X
ejpam-5982	254	3	)	)	PUNCT
ejpam-5982	254	4	all	all	DET
ejpam-5982	254	5	the	the	DET
ejpam-5982	254	6	three	three	NUM
ejpam-5982	254	7	mappings	mapping	NOUN
ejpam-5982	254	8	s1	s1	NOUN
ejpam-5982	254	9	,	,	PUNCT
ejpam-5982	254	10	s2	s2	NOUN
ejpam-5982	254	11	and	and	CCONJ
ejpam-5982	254	12	t	t	PROPN
ejpam-5982	254	13	are	be	AUX
ejpam-5982	254	14	continuous	continuous	ADJ
ejpam-5982	254	15	.	.	PUNCT
ejpam-5982	255	1	(	(	PUNCT
ejpam-5982	255	2	v	v	NOUN
ejpam-5982	255	3	)	)	PUNCT
ejpam-5982	255	4	there	there	PRON
ejpam-5982	255	5	exists	exist	VERB
ejpam-5982	255	6	ψ	ψ	ADP
ejpam-5982	255	7	∈	∈	NOUN
ejpam-5982	255	8	ψ	ψ	ADP
ejpam-5982	255	9	such	such	ADJ
ejpam-5982	255	10	that	that	DET
ejpam-5982	255	11	ψ(d(s2y	ψ(d(s2y	PROPN
ejpam-5982	255	12	,	,	PUNCT
ejpam-5982	255	13	s1x	s1x	PROPN
ejpam-5982	255	14	)	)	PUNCT
ejpam-5982	255	15	,	,	PUNCT
ejpam-5982	255	16	d(tx	d(tx	PROPN
ejpam-5982	255	17	,	,	PUNCT
ejpam-5982	255	18	ty	ty	NOUN
ejpam-5982	255	19	)	)	PUNCT
ejpam-5982	255	20	,	,	PUNCT
ejpam-5982	255	21	d(tx	d(tx	PROPN
ejpam-5982	255	22	,	,	PUNCT
ejpam-5982	255	23	s1x	s1x	PROPN
ejpam-5982	255	24	)	)	PUNCT
ejpam-5982	255	25	,	,	PUNCT
ejpam-5982	255	26	d(s2y	d(s2y	PROPN
ejpam-5982	255	27	,	,	PUNCT
ejpam-5982	255	28	ty	ty	NOUN
ejpam-5982	255	29	)	)	PUNCT
ejpam-5982	255	30	)	)	PUNCT
ejpam-5982	255	31	≤	≤	ADV
ejpam-5982	255	32	0	0	NUM
ejpam-5982	255	33	,	,	PUNCT
ejpam-5982	255	34	(	(	PUNCT
ejpam-5982	255	35	11	11	NUM
ejpam-5982	255	36	)	)	PUNCT
ejpam-5982	255	37	for	for	ADP
ejpam-5982	255	38	all	all	DET
ejpam-5982	255	39	(	(	PUNCT
ejpam-5982	255	40	x	x	NOUN
ejpam-5982	255	41	,	,	PUNCT
ejpam-5982	255	42	y	y	NOUN
ejpam-5982	255	43	)	)	PUNCT
ejpam-5982	255	44	∈	∈	PROPN
ejpam-5982	255	45	x	x	PUNCT
ejpam-5982	255	46	×	×	NOUN
ejpam-5982	255	47	y	y	PROPN
ejpam-5982	255	48	.	.	PUNCT
ejpam-5982	256	1	then	then	ADV
ejpam-5982	256	2	the	the	DET
ejpam-5982	256	3	functions	function	NOUN
ejpam-5982	256	4	s1	s1	NOUN
ejpam-5982	256	5	,	,	PUNCT
ejpam-5982	256	6	s2	s2	NOUN
ejpam-5982	256	7	,	,	PUNCT
ejpam-5982	256	8	and	and	CCONJ
ejpam-5982	256	9	t	t	PROPN
ejpam-5982	256	10	have	have	VERB
ejpam-5982	256	11	a	a	DET
ejpam-5982	256	12	unique	unique	ADJ
ejpam-5982	256	13	common	common	ADJ
ejpam-5982	256	14	fixed	fix	VERB
ejpam-5982	256	15	point	point	NOUN
ejpam-5982	256	16	.	.	PUNCT
ejpam-5982	257	1	p.	p.	NOUN
ejpam-5982	257	2	p.	p.	NOUN
ejpam-5982	258	1	murthy	murthy	PROPN
ejpam-5982	259	1	et	et	PROPN
ejpam-5982	259	2	al	al	PROPN
ejpam-5982	259	3	.	.	PUNCT
ejpam-5982	259	4	/	/	SYM
ejpam-5982	259	5	eur	eur	PROPN
ejpam-5982	259	6	.	.	PUNCT
ejpam-5982	260	1	j.	j.	PROPN
ejpam-5982	260	2	pure	pure	PROPN
ejpam-5982	260	3	appl	appl	PROPN
ejpam-5982	260	4	.	.	PROPN
ejpam-5982	260	5	math	math	PROPN
ejpam-5982	260	6	,	,	PUNCT
ejpam-5982	260	7	18	18	NUM
ejpam-5982	260	8	(	(	PUNCT
ejpam-5982	260	9	2	2	NUM
ejpam-5982	260	10	)	)	PUNCT
ejpam-5982	260	11	(	(	PUNCT
ejpam-5982	260	12	2025	2025	NUM
ejpam-5982	260	13	)	)	PUNCT
ejpam-5982	260	14	,	,	PUNCT
ejpam-5982	260	15	5982	5982	NUM
ejpam-5982	260	16	10	10	NUM
ejpam-5982	260	17	of	of	ADP
ejpam-5982	260	18	17	17	NUM
ejpam-5982	260	19	proof	proof	NOUN
ejpam-5982	260	20	.	.	PUNCT
ejpam-5982	261	1	take	take	VERB
ejpam-5982	261	2	t1	t1	NOUN
ejpam-5982	261	3	=	=	SYM
ejpam-5982	261	4	t2	t2	PROPN
ejpam-5982	261	5	=	=	SYM
ejpam-5982	261	6	t	t	PROPN
ejpam-5982	261	7	in	in	ADP
ejpam-5982	261	8	theorem	theorem	NOUN
ejpam-5982	261	9	1	1	NUM
ejpam-5982	261	10	.	.	PUNCT
ejpam-5982	261	11	corollary	corollary	ADJ
ejpam-5982	261	12	2	2	NUM
ejpam-5982	261	13	.	.	PUNCT
ejpam-5982	262	1	let	let	VERB
ejpam-5982	262	2	(	(	PUNCT
ejpam-5982	262	3	x	x	X
ejpam-5982	262	4	,	,	PUNCT
ejpam-5982	262	5	y	y	PROPN
ejpam-5982	262	6	,	,	PUNCT
ejpam-5982	262	7	d	d	NOUN
ejpam-5982	262	8	)	)	PUNCT
ejpam-5982	262	9	be	be	AUX
ejpam-5982	262	10	a	a	DET
ejpam-5982	262	11	complete	complete	ADJ
ejpam-5982	262	12	bipolar	bipolar	ADJ
ejpam-5982	262	13	metric	metric	ADJ
ejpam-5982	262	14	space	space	NOUN
ejpam-5982	262	15	and	and	CCONJ
ejpam-5982	262	16	let	let	VERB
ejpam-5982	262	17	t	t	NOUN
ejpam-5982	262	18	:	:	PUNCT
ejpam-5982	262	19	(	(	PUNCT
ejpam-5982	262	20	x	x	X
ejpam-5982	262	21	,	,	PUNCT
ejpam-5982	262	22	y	y	PROPN
ejpam-5982	262	23	,	,	PUNCT
ejpam-5982	262	24	d	d	NOUN
ejpam-5982	262	25	)	)	PUNCT
ejpam-5982	262	26	⇒	⇒	NOUN
ejpam-5982	262	27	(	(	PUNCT
ejpam-5982	262	28	x	x	X
ejpam-5982	262	29	,	,	PUNCT
ejpam-5982	262	30	y	y	PROPN
ejpam-5982	262	31	,	,	PUNCT
ejpam-5982	262	32	d	d	NOUN
ejpam-5982	262	33	)	)	PUNCT
ejpam-5982	262	34	be	be	AUX
ejpam-5982	262	35	a	a	DET
ejpam-5982	262	36	covariant	covariant	ADJ
ejpam-5982	262	37	map	map	NOUN
ejpam-5982	262	38	and	and	CCONJ
ejpam-5982	262	39	s	s	VERB
ejpam-5982	262	40	:	:	PUNCT
ejpam-5982	262	41	(	(	PUNCT
ejpam-5982	262	42	x	x	X
ejpam-5982	262	43	,	,	PUNCT
ejpam-5982	262	44	y	y	PROPN
ejpam-5982	262	45	,	,	PUNCT
ejpam-5982	262	46	d	d	NOUN
ejpam-5982	262	47	)	)	PUNCT
ejpam-5982	262	48	⇄	⇄	NOUN
ejpam-5982	262	49	(	(	PUNCT
ejpam-5982	262	50	x	x	X
ejpam-5982	262	51	,	,	PUNCT
ejpam-5982	262	52	y	y	PROPN
ejpam-5982	262	53	,	,	PUNCT
ejpam-5982	262	54	d	d	NOUN
ejpam-5982	262	55	)	)	PUNCT
ejpam-5982	262	56	be	be	AUX
ejpam-5982	262	57	a	a	DET
ejpam-5982	262	58	contravariant	contravariant	ADJ
ejpam-5982	262	59	map	map	NOUN
ejpam-5982	262	60	satisfying	satisfy	VERB
ejpam-5982	262	61	the	the	DET
ejpam-5982	262	62	following	follow	VERB
ejpam-5982	262	63	conditions	condition	NOUN
ejpam-5982	262	64	:	:	PUNCT
ejpam-5982	262	65	(	(	PUNCT
ejpam-5982	262	66	i	i	NOUN
ejpam-5982	262	67	)	)	PUNCT
ejpam-5982	262	68	s	s	PROPN
ejpam-5982	262	69	and	and	CCONJ
ejpam-5982	262	70	t	t	PROPN
ejpam-5982	262	71	are	be	AUX
ejpam-5982	262	72	compatible	compatible	ADJ
ejpam-5982	262	73	of	of	ADP
ejpam-5982	262	74	type	type	NOUN
ejpam-5982	262	75	(	(	PUNCT
ejpam-5982	262	76	a	a	NOUN
ejpam-5982	262	77	)	)	PUNCT
ejpam-5982	262	78	with	with	ADP
ejpam-5982	262	79	respect	respect	NOUN
ejpam-5982	262	80	to	to	ADP
ejpam-5982	262	81	x	x	PUNCT
ejpam-5982	262	82	or	or	CCONJ
ejpam-5982	262	83	y.	y.	PROPN
ejpam-5982	262	84	(	(	PUNCT
ejpam-5982	262	85	ii	ii	PROPN
ejpam-5982	262	86	)	)	PUNCT
ejpam-5982	262	87	s(x	s(x	PROPN
ejpam-5982	262	88	∪	∪	PROPN
ejpam-5982	262	89	y	y	PROPN
ejpam-5982	262	90	)	)	PUNCT
ejpam-5982	262	91	⊆	⊆	NUM
ejpam-5982	262	92	t	t	NOUN
ejpam-5982	262	93	(	(	PUNCT
ejpam-5982	262	94	x	x	X
ejpam-5982	262	95	∪	∪	PROPN
ejpam-5982	262	96	y	y	PROPN
ejpam-5982	262	97	)	)	PUNCT
ejpam-5982	262	98	.	.	PUNCT
ejpam-5982	263	1	(	(	PUNCT
ejpam-5982	263	2	iii	iii	X
ejpam-5982	263	3	)	)	PUNCT
ejpam-5982	263	4	s	s	PROPN
ejpam-5982	263	5	and	and	CCONJ
ejpam-5982	263	6	t	t	PROPN
ejpam-5982	263	7	are	be	AUX
ejpam-5982	263	8	continuous	continuous	ADJ
ejpam-5982	263	9	.	.	PUNCT
ejpam-5982	264	1	(	(	PUNCT
ejpam-5982	264	2	iv	iv	X
ejpam-5982	264	3	)	)	PUNCT
ejpam-5982	264	4	there	there	PRON
ejpam-5982	264	5	exists	exist	VERB
ejpam-5982	264	6	ψ	ψ	ADP
ejpam-5982	264	7	∈	∈	NOUN
ejpam-5982	264	8	ψ	ψ	ADP
ejpam-5982	264	9	such	such	ADJ
ejpam-5982	264	10	that	that	DET
ejpam-5982	264	11	ψ(d(sy	ψ(d(sy	ADJ
ejpam-5982	264	12	,	,	PUNCT
ejpam-5982	264	13	sx	sx	PROPN
ejpam-5982	264	14	)	)	PUNCT
ejpam-5982	264	15	,	,	PUNCT
ejpam-5982	264	16	d(tx	d(tx	PROPN
ejpam-5982	264	17	,	,	PUNCT
ejpam-5982	264	18	ty	ty	NOUN
ejpam-5982	264	19	)	)	PUNCT
ejpam-5982	264	20	,	,	PUNCT
ejpam-5982	264	21	d(tx	d(tx	PROPN
ejpam-5982	264	22	,	,	PUNCT
ejpam-5982	264	23	sx	sx	PROPN
ejpam-5982	264	24	)	)	PUNCT
ejpam-5982	264	25	,	,	PUNCT
ejpam-5982	264	26	d(sy	d(sy	PROPN
ejpam-5982	264	27	,	,	PUNCT
ejpam-5982	264	28	ty	ty	NOUN
ejpam-5982	264	29	)	)	PUNCT
ejpam-5982	264	30	)	)	PUNCT
ejpam-5982	264	31	≤	≤	ADV
ejpam-5982	264	32	0	0	NUM
ejpam-5982	264	33	,	,	PUNCT
ejpam-5982	264	34	(	(	PUNCT
ejpam-5982	264	35	12	12	NUM
ejpam-5982	264	36	)	)	PUNCT
ejpam-5982	264	37	for	for	ADP
ejpam-5982	264	38	all	all	DET
ejpam-5982	264	39	(	(	PUNCT
ejpam-5982	264	40	x	x	NOUN
ejpam-5982	264	41	,	,	PUNCT
ejpam-5982	264	42	y	y	NOUN
ejpam-5982	264	43	)	)	PUNCT
ejpam-5982	264	44	∈	∈	PROPN
ejpam-5982	264	45	x	x	PUNCT
ejpam-5982	264	46	×	×	NOUN
ejpam-5982	264	47	y	y	PROPN
ejpam-5982	264	48	.	.	PUNCT
ejpam-5982	265	1	then	then	ADV
ejpam-5982	265	2	the	the	DET
ejpam-5982	265	3	functions	function	NOUN
ejpam-5982	265	4	s	s	PART
ejpam-5982	265	5	and	and	CCONJ
ejpam-5982	265	6	t	t	PROPN
ejpam-5982	265	7	have	have	VERB
ejpam-5982	265	8	a	a	DET
ejpam-5982	265	9	unique	unique	ADJ
ejpam-5982	265	10	common	common	ADJ
ejpam-5982	265	11	fixed	fix	VERB
ejpam-5982	265	12	point	point	NOUN
ejpam-5982	265	13	.	.	PUNCT
ejpam-5982	266	1	proof	proof	NOUN
ejpam-5982	266	2	.	.	PUNCT
ejpam-5982	267	1	take	take	VERB
ejpam-5982	267	2	t1	t1	NOUN
ejpam-5982	267	3	=	=	SYM
ejpam-5982	267	4	t2	t2	PROPN
ejpam-5982	267	5	=	=	SYM
ejpam-5982	267	6	t	t	PROPN
ejpam-5982	267	7	and	and	CCONJ
ejpam-5982	267	8	s1	s1	PROPN
ejpam-5982	267	9	=	=	SYM
ejpam-5982	267	10	s2	s2	PROPN
ejpam-5982	267	11	=	=	SYM
ejpam-5982	267	12	s	s	PROPN
ejpam-5982	267	13	in	in	ADP
ejpam-5982	267	14	theorem	theorem	NOUN
ejpam-5982	267	15	1	1	NUM
ejpam-5982	267	16	.	.	PUNCT
ejpam-5982	267	17	corollary	corollary	ADJ
ejpam-5982	267	18	3	3	X
ejpam-5982	267	19	.	.	PUNCT
ejpam-5982	268	1	let	let	AUX
ejpam-5982	268	2	(	(	PUNCT
ejpam-5982	268	3	x	x	X
ejpam-5982	268	4	,	,	PUNCT
ejpam-5982	268	5	y	y	PROPN
ejpam-5982	268	6	,	,	PUNCT
ejpam-5982	268	7	d	d	NOUN
ejpam-5982	268	8	)	)	PUNCT
ejpam-5982	268	9	be	be	AUX
ejpam-5982	268	10	a	a	DET
ejpam-5982	268	11	complete	complete	ADJ
ejpam-5982	268	12	bipolar	bipolar	ADJ
ejpam-5982	268	13	metric	metric	ADJ
ejpam-5982	268	14	space	space	NOUN
ejpam-5982	268	15	and	and	CCONJ
ejpam-5982	268	16	let	let	VERB
ejpam-5982	268	17	s	s	PRON
ejpam-5982	268	18	:	:	PUNCT
ejpam-5982	268	19	(	(	PUNCT
ejpam-5982	268	20	x	x	X
ejpam-5982	268	21	,	,	PUNCT
ejpam-5982	268	22	y	y	PROPN
ejpam-5982	268	23	,	,	PUNCT
ejpam-5982	268	24	d	d	NOUN
ejpam-5982	268	25	)	)	PUNCT
ejpam-5982	268	26	⇄	⇄	NOUN
ejpam-5982	268	27	(	(	PUNCT
ejpam-5982	268	28	x	x	X
ejpam-5982	268	29	,	,	PUNCT
ejpam-5982	268	30	y	y	PROPN
ejpam-5982	268	31	,	,	PUNCT
ejpam-5982	268	32	d	d	NOUN
ejpam-5982	268	33	)	)	PUNCT
ejpam-5982	268	34	be	be	AUX
ejpam-5982	268	35	a	a	DET
ejpam-5982	268	36	contravariant	contravariant	ADJ
ejpam-5982	268	37	map	map	NOUN
ejpam-5982	268	38	satisfying	satisfy	VERB
ejpam-5982	268	39	the	the	DET
ejpam-5982	268	40	following	follow	VERB
ejpam-5982	268	41	conditions	condition	NOUN
ejpam-5982	268	42	:	:	PUNCT
ejpam-5982	268	43	(	(	PUNCT
ejpam-5982	268	44	i	i	NOUN
ejpam-5982	268	45	)	)	PUNCT
ejpam-5982	268	46	s	s	AUX
ejpam-5982	268	47	is	be	AUX
ejpam-5982	268	48	continuous	continuous	ADJ
ejpam-5982	268	49	.	.	PUNCT
ejpam-5982	269	1	(	(	PUNCT
ejpam-5982	269	2	ii	ii	NOUN
ejpam-5982	269	3	)	)	PUNCT
ejpam-5982	269	4	there	there	PRON
ejpam-5982	269	5	exists	exist	VERB
ejpam-5982	269	6	ψ	ψ	ADP
ejpam-5982	269	7	∈	∈	NOUN
ejpam-5982	269	8	ψ	ψ	ADP
ejpam-5982	269	9	such	such	ADJ
ejpam-5982	269	10	that	that	DET
ejpam-5982	269	11	ψ(d(sy	ψ(d(sy	ADJ
ejpam-5982	269	12	,	,	PUNCT
ejpam-5982	269	13	sx	sx	PROPN
ejpam-5982	269	14	)	)	PUNCT
ejpam-5982	269	15	,	,	PUNCT
ejpam-5982	269	16	d(x	d(x	PROPN
ejpam-5982	269	17	,	,	PUNCT
ejpam-5982	269	18	y	y	NOUN
ejpam-5982	269	19	)	)	PUNCT
ejpam-5982	269	20	,	,	PUNCT
ejpam-5982	269	21	d(x	d(x	PROPN
ejpam-5982	269	22	,	,	PUNCT
ejpam-5982	269	23	sx	sx	PROPN
ejpam-5982	269	24	)	)	PUNCT
ejpam-5982	269	25	,	,	PUNCT
ejpam-5982	269	26	d(sy	d(sy	X
ejpam-5982	269	27	,	,	PUNCT
ejpam-5982	269	28	y	y	NOUN
ejpam-5982	269	29	)	)	PUNCT
ejpam-5982	269	30	)	)	PUNCT
ejpam-5982	269	31	≤	≤	ADV
ejpam-5982	269	32	0	0	NUM
ejpam-5982	269	33	,	,	PUNCT
ejpam-5982	269	34	(	(	PUNCT
ejpam-5982	269	35	13	13	NUM
ejpam-5982	269	36	)	)	PUNCT
ejpam-5982	269	37	for	for	ADP
ejpam-5982	269	38	all	all	DET
ejpam-5982	269	39	(	(	PUNCT
ejpam-5982	269	40	x	x	NOUN
ejpam-5982	269	41	,	,	PUNCT
ejpam-5982	269	42	y	y	NOUN
ejpam-5982	269	43	)	)	PUNCT
ejpam-5982	269	44	∈	∈	PROPN
ejpam-5982	269	45	x	x	PUNCT
ejpam-5982	269	46	×	×	NOUN
ejpam-5982	269	47	y	y	PROPN
ejpam-5982	269	48	.	.	PUNCT
ejpam-5982	270	1	then	then	ADV
ejpam-5982	270	2	the	the	DET
ejpam-5982	270	3	function	function	NOUN
ejpam-5982	270	4	s	s	PART
ejpam-5982	270	5	has	have	VERB
ejpam-5982	270	6	a	a	DET
ejpam-5982	270	7	unique	unique	ADJ
ejpam-5982	270	8	fixed	fix	VERB
ejpam-5982	270	9	point	point	NOUN
ejpam-5982	270	10	.	.	PUNCT
ejpam-5982	271	1	proof	proof	NOUN
ejpam-5982	271	2	.	.	PUNCT
ejpam-5982	272	1	take	take	VERB
ejpam-5982	272	2	t1	t1	NOUN
ejpam-5982	272	3	=	=	SYM
ejpam-5982	272	4	t2	t2	NOUN
ejpam-5982	272	5	=	=	PUNCT
ejpam-5982	272	6	i	i	NOUN
ejpam-5982	272	7	and	and	CCONJ
ejpam-5982	272	8	s1	s1	PROPN
ejpam-5982	272	9	=	=	SYM
ejpam-5982	272	10	s2	s2	PROPN
ejpam-5982	272	11	=	=	SYM
ejpam-5982	272	12	s	s	PROPN
ejpam-5982	272	13	in	in	ADP
ejpam-5982	272	14	theorem	theorem	NOUN
ejpam-5982	272	15	1	1	NUM
ejpam-5982	272	16	,	,	PUNCT
ejpam-5982	272	17	where	where	SCONJ
ejpam-5982	272	18	i	i	PRON
ejpam-5982	272	19	is	be	AUX
ejpam-5982	272	20	the	the	DET
ejpam-5982	272	21	identity	identity	NOUN
ejpam-5982	272	22	mapping	mapping	NOUN
ejpam-5982	272	23	.	.	PUNCT
ejpam-5982	273	1	in	in	ADP
ejpam-5982	273	2	the	the	DET
ejpam-5982	273	3	following	follow	VERB
ejpam-5982	273	4	corollary	corollary	NOUN
ejpam-5982	273	5	,	,	PUNCT
ejpam-5982	273	6	we	we	PRON
ejpam-5982	273	7	take	take	VERB
ejpam-5982	273	8	ψ	ψ	PRON
ejpam-5982	273	9	as	as	ADP
ejpam-5982	273	10	a	a	DET
ejpam-5982	273	11	continuous	continuous	ADJ
ejpam-5982	273	12	function	function	NOUN
ejpam-5982	273	13	and	and	CCONJ
ejpam-5982	273	14	s	s	AUX
ejpam-5982	273	15	need	need	AUX
ejpam-5982	273	16	not	not	PART
ejpam-5982	273	17	be	be	AUX
ejpam-5982	273	18	continuous	continuous	ADJ
ejpam-5982	273	19	.	.	PUNCT
ejpam-5982	274	1	corollary	corollary	ADJ
ejpam-5982	274	2	4	4	NUM
ejpam-5982	274	3	.	.	PUNCT
ejpam-5982	275	1	let	let	AUX
ejpam-5982	275	2	(	(	PUNCT
ejpam-5982	275	3	x	x	X
ejpam-5982	275	4	,	,	PUNCT
ejpam-5982	275	5	y	y	PROPN
ejpam-5982	275	6	,	,	PUNCT
ejpam-5982	275	7	d	d	NOUN
ejpam-5982	275	8	)	)	PUNCT
ejpam-5982	275	9	be	be	AUX
ejpam-5982	275	10	a	a	DET
ejpam-5982	275	11	complete	complete	ADJ
ejpam-5982	275	12	bipolar	bipolar	ADJ
ejpam-5982	275	13	metric	metric	ADJ
ejpam-5982	275	14	space	space	NOUN
ejpam-5982	275	15	and	and	CCONJ
ejpam-5982	275	16	let	let	VERB
ejpam-5982	275	17	s	s	PRON
ejpam-5982	275	18	:	:	PUNCT
ejpam-5982	275	19	(	(	PUNCT
ejpam-5982	275	20	x	x	X
ejpam-5982	275	21	,	,	PUNCT
ejpam-5982	275	22	y	y	PROPN
ejpam-5982	275	23	,	,	PUNCT
ejpam-5982	275	24	d	d	NOUN
ejpam-5982	275	25	)	)	PUNCT
ejpam-5982	275	26	⇄	⇄	NOUN
ejpam-5982	275	27	(	(	PUNCT
ejpam-5982	275	28	x	x	X
ejpam-5982	275	29	,	,	PUNCT
ejpam-5982	275	30	y	y	PROPN
ejpam-5982	275	31	,	,	PUNCT
ejpam-5982	275	32	d	d	NOUN
ejpam-5982	275	33	)	)	PUNCT
ejpam-5982	275	34	be	be	AUX
ejpam-5982	275	35	a	a	DET
ejpam-5982	275	36	contravariant	contravariant	ADJ
ejpam-5982	275	37	map	map	NOUN
ejpam-5982	275	38	satisfying	satisfy	VERB
ejpam-5982	275	39	the	the	DET
ejpam-5982	275	40	following	follow	VERB
ejpam-5982	275	41	condition	condition	NOUN
ejpam-5982	275	42	:	:	PUNCT
ejpam-5982	275	43	ψ(d(sy	ψ(d(sy	ADJ
ejpam-5982	275	44	,	,	PUNCT
ejpam-5982	275	45	sx	sx	PROPN
ejpam-5982	275	46	)	)	PUNCT
ejpam-5982	275	47	,	,	PUNCT
ejpam-5982	275	48	d(x	d(x	PROPN
ejpam-5982	275	49	,	,	PUNCT
ejpam-5982	275	50	y	y	NOUN
ejpam-5982	275	51	)	)	PUNCT
ejpam-5982	275	52	,	,	PUNCT
ejpam-5982	275	53	d(x	d(x	PROPN
ejpam-5982	275	54	,	,	PUNCT
ejpam-5982	275	55	sx	sx	PROPN
ejpam-5982	275	56	)	)	PUNCT
ejpam-5982	275	57	,	,	PUNCT
ejpam-5982	275	58	d(sy	d(sy	X
ejpam-5982	275	59	,	,	PUNCT
ejpam-5982	275	60	y	y	NOUN
ejpam-5982	275	61	)	)	PUNCT
ejpam-5982	275	62	)	)	PUNCT
ejpam-5982	276	1	≤	≤	ADV
ejpam-5982	276	2	0	0	NUM
ejpam-5982	276	3	,	,	PUNCT
ejpam-5982	276	4	for	for	ADP
ejpam-5982	276	5	some	some	DET
ejpam-5982	276	6	continuous	continuous	ADJ
ejpam-5982	276	7	function	function	NOUN
ejpam-5982	276	8	ψ	ψ	ADP
ejpam-5982	276	9	∈	∈	NOUN
ejpam-5982	276	10	ψ	ψ	NOUN
ejpam-5982	276	11	and	and	CCONJ
ejpam-5982	276	12	for	for	ADP
ejpam-5982	276	13	all	all	DET
ejpam-5982	276	14	(	(	PUNCT
ejpam-5982	276	15	x	x	NOUN
ejpam-5982	276	16	,	,	PUNCT
ejpam-5982	276	17	y	y	NOUN
ejpam-5982	276	18	)	)	PUNCT
ejpam-5982	276	19	∈	∈	PROPN
ejpam-5982	276	20	x	x	PUNCT
ejpam-5982	276	21	×	×	NOUN
ejpam-5982	276	22	y	y	PROPN
ejpam-5982	276	23	.	.	PUNCT
ejpam-5982	277	1	then	then	ADV
ejpam-5982	277	2	the	the	DET
ejpam-5982	277	3	function	function	NOUN
ejpam-5982	277	4	s	s	PART
ejpam-5982	277	5	has	have	VERB
ejpam-5982	277	6	a	a	DET
ejpam-5982	277	7	unique	unique	ADJ
ejpam-5982	277	8	fixed	fix	VERB
ejpam-5982	277	9	point	point	NOUN
ejpam-5982	277	10	.	.	PUNCT
ejpam-5982	278	1	p.	p.	NOUN
ejpam-5982	278	2	p.	p.	NOUN
ejpam-5982	279	1	murthy	murthy	PROPN
ejpam-5982	280	1	et	et	PROPN
ejpam-5982	280	2	al	al	PROPN
ejpam-5982	280	3	.	.	PUNCT
ejpam-5982	280	4	/	/	SYM
ejpam-5982	280	5	eur	eur	PROPN
ejpam-5982	280	6	.	.	PUNCT
ejpam-5982	281	1	j.	j.	PROPN
ejpam-5982	281	2	pure	pure	PROPN
ejpam-5982	281	3	appl	appl	PROPN
ejpam-5982	281	4	.	.	PROPN
ejpam-5982	281	5	math	math	PROPN
ejpam-5982	281	6	,	,	PUNCT
ejpam-5982	281	7	18	18	NUM
ejpam-5982	281	8	(	(	PUNCT
ejpam-5982	281	9	2	2	NUM
ejpam-5982	281	10	)	)	PUNCT
ejpam-5982	281	11	(	(	PUNCT
ejpam-5982	281	12	2025	2025	NUM
ejpam-5982	281	13	)	)	PUNCT
ejpam-5982	281	14	,	,	PUNCT
ejpam-5982	281	15	5982	5982	NUM
ejpam-5982	281	16	11	11	NUM
ejpam-5982	281	17	of	of	ADP
ejpam-5982	281	18	17	17	NUM
ejpam-5982	281	19	proof	proof	NOUN
ejpam-5982	281	20	.	.	PUNCT
ejpam-5982	282	1	as	as	ADP
ejpam-5982	282	2	in	in	ADP
ejpam-5982	282	3	the	the	DET
ejpam-5982	282	4	proof	proof	NOUN
ejpam-5982	282	5	of	of	ADP
ejpam-5982	282	6	the	the	DET
ejpam-5982	282	7	previous	previous	ADJ
ejpam-5982	282	8	corollary	corollary	NOUN
ejpam-5982	282	9	,	,	PUNCT
ejpam-5982	282	10	we	we	PRON
ejpam-5982	282	11	obtain	obtain	VERB
ejpam-5982	282	12	a	a	DET
ejpam-5982	282	13	bisequence	bisequence	NOUN
ejpam-5982	282	14	(	(	PUNCT
ejpam-5982	282	15	syn	syn	PROPN
ejpam-5982	282	16	,	,	PUNCT
ejpam-5982	282	17	sxn	sxn	PROPN
ejpam-5982	282	18	)	)	PUNCT
ejpam-5982	282	19	biconverging	biconverge	VERB
ejpam-5982	282	20	to	to	ADP
ejpam-5982	282	21	a	a	DET
ejpam-5982	282	22	point	point	NOUN
ejpam-5982	282	23	t	t	X
ejpam-5982	282	24	∈	∈	NOUN
ejpam-5982	282	25	x	x	PUNCT
ejpam-5982	282	26	∩	∩	PROPN
ejpam-5982	282	27	y	y	PROPN
ejpam-5982	282	28	,	,	PUNCT
ejpam-5982	282	29	where	where	SCONJ
ejpam-5982	282	30	syn	syn	PROPN
ejpam-5982	282	31	=	=	PROPN
ejpam-5982	282	32	xn	xn	PROPN
ejpam-5982	282	33	and	and	CCONJ
ejpam-5982	282	34	sxn	sxn	NOUN
ejpam-5982	282	35	=	=	SYM
ejpam-5982	282	36	yn+1	yn+1	PROPN
ejpam-5982	282	37	.	.	PROPN
ejpam-5982	282	38	in	in	ADP
ejpam-5982	282	39	the	the	DET
ejpam-5982	282	40	given	give	VERB
ejpam-5982	282	41	condition	condition	NOUN
ejpam-5982	282	42	,	,	PUNCT
ejpam-5982	282	43	if	if	SCONJ
ejpam-5982	282	44	we	we	PRON
ejpam-5982	282	45	take	take	VERB
ejpam-5982	282	46	y	y	NOUN
ejpam-5982	282	47	=	=	PUNCT
ejpam-5982	282	48	yn	yn	PROPN
ejpam-5982	282	49	and	and	CCONJ
ejpam-5982	282	50	x	x	X
ejpam-5982	282	51	=	=	SYM
ejpam-5982	282	52	t	t	PROPN
ejpam-5982	282	53	,	,	PUNCT
ejpam-5982	282	54	then	then	ADV
ejpam-5982	282	55	we	we	PRON
ejpam-5982	282	56	get	get	VERB
ejpam-5982	282	57	,	,	PUNCT
ejpam-5982	282	58	ψ(d(syn	ψ(d(syn	PROPN
ejpam-5982	282	59	,	,	PUNCT
ejpam-5982	282	60	st	st	PROPN
ejpam-5982	282	61	)	)	PUNCT
ejpam-5982	282	62	,	,	PUNCT
ejpam-5982	282	63	d(t	d(t	PROPN
ejpam-5982	282	64	,	,	PUNCT
ejpam-5982	282	65	yn	yn	PROPN
ejpam-5982	282	66	)	)	PUNCT
ejpam-5982	282	67	,	,	PUNCT
ejpam-5982	282	68	d(t	d(t	PROPN
ejpam-5982	282	69	,	,	PUNCT
ejpam-5982	282	70	st	st	PROPN
ejpam-5982	282	71	)	)	PUNCT
ejpam-5982	282	72	,	,	PUNCT
ejpam-5982	282	73	d(syn	d(syn	PROPN
ejpam-5982	282	74	,	,	PUNCT
ejpam-5982	282	75	yn	yn	PROPN
ejpam-5982	282	76	)	)	PUNCT
ejpam-5982	282	77	)	)	PUNCT
ejpam-5982	283	1	≤	≤	ADV
ejpam-5982	283	2	0	0	NUM
ejpam-5982	283	3	,	,	PUNCT
ejpam-5982	283	4	taking	take	VERB
ejpam-5982	283	5	limit	limit	NOUN
ejpam-5982	283	6	as	as	ADP
ejpam-5982	283	7	n→	n→	ADV
ejpam-5982	283	8	+	+	PROPN
ejpam-5982	283	9	∞	∞	PROPN
ejpam-5982	283	10	,	,	PUNCT
ejpam-5982	283	11	we	we	PRON
ejpam-5982	283	12	get	get	VERB
ejpam-5982	283	13	ψ(d(t	ψ(d(t	PROPN
ejpam-5982	283	14	,	,	PUNCT
ejpam-5982	283	15	st	st	PROPN
ejpam-5982	283	16	)	)	PUNCT
ejpam-5982	283	17	,	,	PUNCT
ejpam-5982	283	18	0	0	NUM
ejpam-5982	283	19	,	,	PUNCT
ejpam-5982	283	20	d(t	d(t	PROPN
ejpam-5982	283	21	,	,	PUNCT
ejpam-5982	283	22	st	st	PROPN
ejpam-5982	283	23	)	)	PUNCT
ejpam-5982	283	24	,	,	PUNCT
ejpam-5982	283	25	0	0	X
ejpam-5982	283	26	)	)	PUNCT
ejpam-5982	283	27	≤	≤	NOUN
ejpam-5982	283	28	0	0	PUNCT
ejpam-5982	284	1	so	so	CCONJ
ejpam-5982	284	2	by	by	ADP
ejpam-5982	284	3	property	property	NOUN
ejpam-5982	284	4	of	of	ADP
ejpam-5982	284	5	ψ	ψ	NOUN
ejpam-5982	284	6	,	,	PUNCT
ejpam-5982	284	7	we	we	PRON
ejpam-5982	284	8	get	get	VERB
ejpam-5982	284	9	d(t	d(t	PROPN
ejpam-5982	284	10	,	,	PUNCT
ejpam-5982	284	11	st	st	PROPN
ejpam-5982	284	12	)	)	PUNCT
ejpam-5982	284	13	=	=	NOUN
ejpam-5982	285	1	0	0	X
ejpam-5982	285	2	.	.	PUNCT
ejpam-5982	286	1	this	this	PRON
ejpam-5982	286	2	implies	imply	VERB
ejpam-5982	286	3	that	that	SCONJ
ejpam-5982	286	4	t	t	PROPN
ejpam-5982	286	5	is	be	AUX
ejpam-5982	286	6	a	a	DET
ejpam-5982	286	7	fixed	fix	VERB
ejpam-5982	286	8	point	point	NOUN
ejpam-5982	286	9	of	of	ADP
ejpam-5982	286	10	s.	s.	PROPN
ejpam-5982	286	11	uniqueness	uniqueness	PROPN
ejpam-5982	286	12	can	can	AUX
ejpam-5982	286	13	be	be	AUX
ejpam-5982	286	14	proved	prove	VERB
ejpam-5982	286	15	as	as	SCONJ
ejpam-5982	286	16	given	give	VERB
ejpam-5982	286	17	in	in	ADP
ejpam-5982	286	18	the	the	DET
ejpam-5982	286	19	previous	previous	ADJ
ejpam-5982	286	20	corollary	corollary	NOUN
ejpam-5982	286	21	.	.	PUNCT
ejpam-5982	287	1	the	the	DET
ejpam-5982	287	2	contraction	contraction	NOUN
ejpam-5982	287	3	used	use	VERB
ejpam-5982	287	4	in	in	ADP
ejpam-5982	287	5	the	the	DET
ejpam-5982	287	6	following	follow	VERB
ejpam-5982	287	7	corollary	corollary	NOUN
ejpam-5982	287	8	is	be	AUX
ejpam-5982	287	9	reich	reich	NOUN
ejpam-5982	287	10	-	-	PUNCT
ejpam-5982	287	11	type	type	NOUN
ejpam-5982	287	12	contraction	contraction	NOUN
ejpam-5982	287	13	(	(	PUNCT
ejpam-5982	287	14	see	see	VERB
ejpam-5982	287	15	[	[	X
ejpam-5982	287	16	18	18	NUM
ejpam-5982	287	17	]	]	PUNCT
ejpam-5982	287	18	,	,	PUNCT
ejpam-5982	287	19	[	[	X
ejpam-5982	287	20	16	16	NUM
ejpam-5982	287	21	]	]	SYM
ejpam-5982	287	22	)	)	PUNCT
ejpam-5982	287	23	.	.	PUNCT
ejpam-5982	288	1	corollary	corollary	ADJ
ejpam-5982	288	2	5	5	NUM
ejpam-5982	288	3	.	.	PUNCT
ejpam-5982	289	1	let	let	VERB
ejpam-5982	289	2	(	(	PUNCT
ejpam-5982	289	3	x	x	X
ejpam-5982	289	4	,	,	PUNCT
ejpam-5982	289	5	y	y	PROPN
ejpam-5982	289	6	,	,	PUNCT
ejpam-5982	289	7	d	d	NOUN
ejpam-5982	289	8	)	)	PUNCT
ejpam-5982	289	9	be	be	AUX
ejpam-5982	289	10	a	a	DET
ejpam-5982	289	11	complete	complete	ADJ
ejpam-5982	289	12	bipolar	bipolar	ADJ
ejpam-5982	289	13	metric	metric	ADJ
ejpam-5982	289	14	space	space	NOUN
ejpam-5982	289	15	and	and	CCONJ
ejpam-5982	289	16	let	let	VERB
ejpam-5982	289	17	s	s	PRON
ejpam-5982	289	18	:	:	PUNCT
ejpam-5982	289	19	(	(	PUNCT
ejpam-5982	289	20	x	x	X
ejpam-5982	289	21	,	,	PUNCT
ejpam-5982	289	22	y	y	PROPN
ejpam-5982	289	23	,	,	PUNCT
ejpam-5982	289	24	d	d	NOUN
ejpam-5982	289	25	)	)	PUNCT
ejpam-5982	289	26	⇄	⇄	NOUN
ejpam-5982	289	27	(	(	PUNCT
ejpam-5982	289	28	x	x	X
ejpam-5982	289	29	,	,	PUNCT
ejpam-5982	289	30	y	y	PROPN
ejpam-5982	289	31	,	,	PUNCT
ejpam-5982	289	32	d	d	NOUN
ejpam-5982	289	33	)	)	PUNCT
ejpam-5982	289	34	be	be	AUX
ejpam-5982	289	35	a	a	DET
ejpam-5982	289	36	continuous	continuous	ADJ
ejpam-5982	289	37	contravariant	contravariant	ADJ
ejpam-5982	289	38	map	map	NOUN
ejpam-5982	289	39	satisfying	satisfy	VERB
ejpam-5982	289	40	the	the	DET
ejpam-5982	289	41	following	follow	VERB
ejpam-5982	289	42	condition	condition	NOUN
ejpam-5982	289	43	:	:	PUNCT
ejpam-5982	289	44	d(sy	d(sy	PROPN
ejpam-5982	289	45	,	,	PUNCT
ejpam-5982	289	46	sx	sx	NOUN
ejpam-5982	289	47	)	)	PUNCT
ejpam-5982	289	48	≤	≤	NOUN
ejpam-5982	289	49	k1d(x	k1d(x	PROPN
ejpam-5982	289	50	,	,	PUNCT
ejpam-5982	289	51	y	y	NOUN
ejpam-5982	289	52	)	)	PUNCT
ejpam-5982	290	1	+	+	CCONJ
ejpam-5982	290	2	k2d(x	k2d(x	PROPN
ejpam-5982	290	3	,	,	PUNCT
ejpam-5982	290	4	sx	sx	PROPN
ejpam-5982	290	5	)	)	PUNCT
ejpam-5982	290	6	+	+	CCONJ
ejpam-5982	290	7	k3d(sy	k3d(sy	PROPN
ejpam-5982	290	8	,	,	PUNCT
ejpam-5982	290	9	y	y	NOUN
ejpam-5982	290	10	)	)	PUNCT
ejpam-5982	290	11	)	)	PUNCT
ejpam-5982	290	12	,	,	PUNCT
ejpam-5982	290	13	for	for	ADP
ejpam-5982	290	14	all	all	PRON
ejpam-5982	290	15	(	(	PUNCT
ejpam-5982	290	16	x	x	NOUN
ejpam-5982	290	17	,	,	PUNCT
ejpam-5982	290	18	y	y	NOUN
ejpam-5982	290	19	)	)	PUNCT
ejpam-5982	290	20	∈	∈	PROPN
ejpam-5982	290	21	x	x	SYM
ejpam-5982	290	22	×	×	PROPN
ejpam-5982	290	23	y	y	PROPN
ejpam-5982	290	24	,	,	PUNCT
ejpam-5982	290	25	where	where	SCONJ
ejpam-5982	290	26	k1	k1	NOUN
ejpam-5982	290	27	+	+	CCONJ
ejpam-5982	290	28	k2	k2	PROPN
ejpam-5982	290	29	+	+	CCONJ
ejpam-5982	290	30	k3	k3	VERB
ejpam-5982	290	31	<	<	X
ejpam-5982	290	32	1	1	NUM
ejpam-5982	290	33	.	.	PUNCT
ejpam-5982	290	34	then	then	ADV
ejpam-5982	290	35	the	the	DET
ejpam-5982	290	36	function	function	NOUN
ejpam-5982	290	37	s	s	PART
ejpam-5982	290	38	has	have	VERB
ejpam-5982	290	39	a	a	DET
ejpam-5982	290	40	unique	unique	ADJ
ejpam-5982	290	41	fixed	fix	VERB
ejpam-5982	290	42	point	point	NOUN
ejpam-5982	290	43	.	.	PUNCT
ejpam-5982	291	1	proof	proof	NOUN
ejpam-5982	291	2	.	.	PUNCT
ejpam-5982	292	1	in	in	ADP
ejpam-5982	292	2	corollary	corollary	ADJ
ejpam-5982	292	3	4	4	NUM
ejpam-5982	292	4	,	,	PUNCT
ejpam-5982	292	5	take	take	VERB
ejpam-5982	292	6	ψ(a	ψ(a	PROPN
ejpam-5982	292	7	,	,	PUNCT
ejpam-5982	292	8	b	b	NOUN
ejpam-5982	292	9	,	,	PUNCT
ejpam-5982	292	10	c	c	NOUN
ejpam-5982	292	11	,	,	PUNCT
ejpam-5982	292	12	d	d	NOUN
ejpam-5982	292	13	)	)	PUNCT
ejpam-5982	292	14	=	=	SYM
ejpam-5982	292	15	a−	a−	PROPN
ejpam-5982	292	16	(	(	PUNCT
ejpam-5982	292	17	k1b+	k1b+	PROPN
ejpam-5982	292	18	k2c+	k2c+	PROPN
ejpam-5982	292	19	k3d	k3d	PROPN
ejpam-5982	292	20	)	)	PUNCT
ejpam-5982	292	21	.	.	PUNCT
ejpam-5982	293	1	corollary	corollary	ADJ
ejpam-5982	293	2	6	6	NUM
ejpam-5982	293	3	.	.	PUNCT
ejpam-5982	294	1	let	let	VERB
ejpam-5982	294	2	(	(	PUNCT
ejpam-5982	294	3	x	x	X
ejpam-5982	294	4	,	,	PUNCT
ejpam-5982	294	5	y	y	PROPN
ejpam-5982	294	6	,	,	PUNCT
ejpam-5982	294	7	d	d	NOUN
ejpam-5982	294	8	)	)	PUNCT
ejpam-5982	294	9	be	be	AUX
ejpam-5982	294	10	a	a	DET
ejpam-5982	294	11	complete	complete	ADJ
ejpam-5982	294	12	bipolar	bipolar	ADJ
ejpam-5982	294	13	metric	metric	ADJ
ejpam-5982	294	14	space	space	NOUN
ejpam-5982	294	15	and	and	CCONJ
ejpam-5982	294	16	let	let	VERB
ejpam-5982	294	17	s	s	PRON
ejpam-5982	294	18	:	:	PUNCT
ejpam-5982	294	19	(	(	PUNCT
ejpam-5982	294	20	x	x	X
ejpam-5982	294	21	,	,	PUNCT
ejpam-5982	294	22	y	y	PROPN
ejpam-5982	294	23	,	,	PUNCT
ejpam-5982	294	24	d	d	NOUN
ejpam-5982	294	25	)	)	PUNCT
ejpam-5982	294	26	⇄	⇄	NOUN
ejpam-5982	294	27	(	(	PUNCT
ejpam-5982	294	28	x	x	X
ejpam-5982	294	29	,	,	PUNCT
ejpam-5982	294	30	y	y	PROPN
ejpam-5982	294	31	,	,	PUNCT
ejpam-5982	294	32	d	d	NOUN
ejpam-5982	294	33	)	)	PUNCT
ejpam-5982	294	34	be	be	AUX
ejpam-5982	294	35	a	a	DET
ejpam-5982	294	36	contravariant	contravariant	ADJ
ejpam-5982	294	37	map	map	NOUN
ejpam-5982	294	38	satisfying	satisfy	VERB
ejpam-5982	294	39	the	the	DET
ejpam-5982	294	40	following	follow	VERB
ejpam-5982	294	41	condition	condition	NOUN
ejpam-5982	294	42	:	:	PUNCT
ejpam-5982	294	43	d(sy	d(sy	PROPN
ejpam-5982	294	44	,	,	PUNCT
ejpam-5982	294	45	sx	sx	NOUN
ejpam-5982	294	46	)	)	PUNCT
ejpam-5982	294	47	≤	≤	NOUN
ejpam-5982	294	48	k(d(x	k(d(x	PROPN
ejpam-5982	294	49	,	,	PUNCT
ejpam-5982	294	50	y	y	PROPN
ejpam-5982	294	51	)	)	PUNCT
ejpam-5982	295	1	+	+	CCONJ
ejpam-5982	295	2	d(x	d(x	PROPN
ejpam-5982	295	3	,	,	PUNCT
ejpam-5982	295	4	sx	sx	PROPN
ejpam-5982	295	5	)	)	PUNCT
ejpam-5982	296	1	+	+	CCONJ
ejpam-5982	296	2	d(sy	d(sy	PROPN
ejpam-5982	296	3	,	,	PUNCT
ejpam-5982	296	4	y	y	NOUN
ejpam-5982	296	5	)	)	PUNCT
ejpam-5982	296	6	)	)	PUNCT
ejpam-5982	296	7	,	,	PUNCT
ejpam-5982	296	8	for	for	ADP
ejpam-5982	296	9	all	all	PRON
ejpam-5982	296	10	(	(	PUNCT
ejpam-5982	296	11	x	x	NOUN
ejpam-5982	296	12	,	,	PUNCT
ejpam-5982	296	13	y	y	NOUN
ejpam-5982	296	14	)	)	PUNCT
ejpam-5982	296	15	∈	∈	PROPN
ejpam-5982	296	16	x	x	SYM
ejpam-5982	296	17	×	×	PROPN
ejpam-5982	296	18	y	y	PROPN
ejpam-5982	296	19	,	,	PUNCT
ejpam-5982	296	20	where	where	SCONJ
ejpam-5982	296	21	k	k	PROPN
ejpam-5982	296	22	<	<	X
ejpam-5982	296	23	1	1	NUM
ejpam-5982	296	24	3	3	NUM
ejpam-5982	296	25	.	.	PUNCT
ejpam-5982	297	1	then	then	ADV
ejpam-5982	297	2	the	the	DET
ejpam-5982	297	3	function	function	NOUN
ejpam-5982	297	4	s	s	PART
ejpam-5982	297	5	has	have	VERB
ejpam-5982	297	6	a	a	DET
ejpam-5982	297	7	unique	unique	ADJ
ejpam-5982	297	8	fixed	fix	VERB
ejpam-5982	297	9	point	point	NOUN
ejpam-5982	297	10	.	.	PUNCT
ejpam-5982	298	1	proof	proof	NOUN
ejpam-5982	298	2	.	.	PUNCT
ejpam-5982	299	1	in	in	ADP
ejpam-5982	299	2	corollary	corollary	ADJ
ejpam-5982	299	3	5	5	NUM
ejpam-5982	299	4	,	,	PUNCT
ejpam-5982	299	5	take	take	VERB
ejpam-5982	299	6	k1	k1	NOUN
ejpam-5982	299	7	=	=	SYM
ejpam-5982	299	8	k2	k2	NOUN
ejpam-5982	299	9	=	=	PUNCT
ejpam-5982	299	10	k3	k3	PROPN
ejpam-5982	299	11	=	=	PROPN
ejpam-5982	299	12	k.	k.	PROPN
ejpam-5982	299	13	in	in	ADP
ejpam-5982	299	14	the	the	DET
ejpam-5982	299	15	following	follow	VERB
ejpam-5982	299	16	corollary	corollary	NOUN
ejpam-5982	299	17	,	,	PUNCT
ejpam-5982	299	18	kannan	kannan	PROPN
ejpam-5982	299	19	-	-	PUNCT
ejpam-5982	299	20	type	type	NOUN
ejpam-5982	299	21	contraction	contraction	NOUN
ejpam-5982	299	22	(	(	PUNCT
ejpam-5982	299	23	see	see	VERB
ejpam-5982	299	24	[	[	X
ejpam-5982	299	25	16	16	NUM
ejpam-5982	299	26	,	,	PUNCT
ejpam-5982	299	27	17	17	NUM
ejpam-5982	299	28	]	]	PUNCT
ejpam-5982	299	29	)	)	PUNCT
ejpam-5982	299	30	is	be	AUX
ejpam-5982	299	31	used	use	VERB
ejpam-5982	299	32	.	.	PUNCT
ejpam-5982	300	1	corollary	corollary	ADJ
ejpam-5982	300	2	7	7	NUM
ejpam-5982	300	3	.	.	PUNCT
ejpam-5982	301	1	let	let	VERB
ejpam-5982	301	2	(	(	PUNCT
ejpam-5982	301	3	x	x	X
ejpam-5982	301	4	,	,	PUNCT
ejpam-5982	301	5	y	y	PROPN
ejpam-5982	301	6	,	,	PUNCT
ejpam-5982	301	7	d	d	NOUN
ejpam-5982	301	8	)	)	PUNCT
ejpam-5982	301	9	be	be	AUX
ejpam-5982	301	10	a	a	DET
ejpam-5982	301	11	complete	complete	ADJ
ejpam-5982	301	12	bipolar	bipolar	ADJ
ejpam-5982	301	13	metric	metric	ADJ
ejpam-5982	301	14	space	space	NOUN
ejpam-5982	301	15	and	and	CCONJ
ejpam-5982	301	16	let	let	VERB
ejpam-5982	301	17	s	s	PRON
ejpam-5982	301	18	:	:	PUNCT
ejpam-5982	301	19	(	(	PUNCT
ejpam-5982	301	20	x	x	X
ejpam-5982	301	21	,	,	PUNCT
ejpam-5982	301	22	y	y	PROPN
ejpam-5982	301	23	,	,	PUNCT
ejpam-5982	301	24	d	d	NOUN
ejpam-5982	301	25	)	)	PUNCT
ejpam-5982	301	26	⇄	⇄	NOUN
ejpam-5982	301	27	(	(	PUNCT
ejpam-5982	301	28	x	x	X
ejpam-5982	301	29	,	,	PUNCT
ejpam-5982	301	30	y	y	PROPN
ejpam-5982	301	31	,	,	PUNCT
ejpam-5982	301	32	d	d	NOUN
ejpam-5982	301	33	)	)	PUNCT
ejpam-5982	301	34	be	be	AUX
ejpam-5982	301	35	a	a	DET
ejpam-5982	301	36	contravariant	contravariant	ADJ
ejpam-5982	301	37	map	map	NOUN
ejpam-5982	301	38	satisfying	satisfy	VERB
ejpam-5982	301	39	the	the	DET
ejpam-5982	301	40	following	follow	VERB
ejpam-5982	301	41	condition	condition	NOUN
ejpam-5982	301	42	:	:	PUNCT
ejpam-5982	301	43	d(sy	d(sy	PROPN
ejpam-5982	301	44	,	,	PUNCT
ejpam-5982	301	45	sx	sx	NOUN
ejpam-5982	301	46	)	)	PUNCT
ejpam-5982	301	47	≤	≤	NOUN
ejpam-5982	302	1	k(d(x	k(d(x	PROPN
ejpam-5982	302	2	,	,	PUNCT
ejpam-5982	302	3	sx	sx	PROPN
ejpam-5982	302	4	)	)	PUNCT
ejpam-5982	302	5	+	+	CCONJ
ejpam-5982	302	6	d(sy	d(sy	PROPN
ejpam-5982	302	7	,	,	PUNCT
ejpam-5982	302	8	y	y	NOUN
ejpam-5982	302	9	)	)	PUNCT
ejpam-5982	302	10	)	)	PUNCT
ejpam-5982	302	11	,	,	PUNCT
ejpam-5982	302	12	for	for	ADP
ejpam-5982	302	13	all	all	PRON
ejpam-5982	302	14	(	(	PUNCT
ejpam-5982	302	15	x	x	NOUN
ejpam-5982	302	16	,	,	PUNCT
ejpam-5982	302	17	y	y	NOUN
ejpam-5982	302	18	)	)	PUNCT
ejpam-5982	302	19	∈	∈	PROPN
ejpam-5982	302	20	x	x	SYM
ejpam-5982	302	21	×	×	PROPN
ejpam-5982	302	22	y	y	PROPN
ejpam-5982	302	23	,	,	PUNCT
ejpam-5982	302	24	where	where	SCONJ
ejpam-5982	302	25	k	k	PROPN
ejpam-5982	302	26	<	<	X
ejpam-5982	302	27	1	1	NUM
ejpam-5982	302	28	2	2	NUM
ejpam-5982	302	29	.	.	PUNCT
ejpam-5982	303	1	then	then	ADV
ejpam-5982	303	2	the	the	DET
ejpam-5982	303	3	function	function	NOUN
ejpam-5982	303	4	s	s	PART
ejpam-5982	303	5	has	have	VERB
ejpam-5982	303	6	a	a	DET
ejpam-5982	303	7	unique	unique	ADJ
ejpam-5982	303	8	fixed	fix	VERB
ejpam-5982	303	9	point	point	NOUN
ejpam-5982	303	10	.	.	PUNCT
ejpam-5982	304	1	proof	proof	NOUN
ejpam-5982	304	2	.	.	PUNCT
ejpam-5982	305	1	in	in	ADP
ejpam-5982	305	2	corollary	corollary	ADJ
ejpam-5982	305	3	5	5	NUM
ejpam-5982	305	4	,	,	PUNCT
ejpam-5982	305	5	take	take	VERB
ejpam-5982	305	6	k1	k1	NOUN
ejpam-5982	305	7	=	=	SYM
ejpam-5982	305	8	0	0	NUM
ejpam-5982	305	9	,	,	PUNCT
ejpam-5982	305	10	k2	k2	NOUN
ejpam-5982	305	11	=	=	SYM
ejpam-5982	305	12	k3	k3	PROPN
ejpam-5982	305	13	=	=	PROPN
ejpam-5982	305	14	k.	k.	PROPN
ejpam-5982	305	15	in	in	ADP
ejpam-5982	305	16	our	our	PRON
ejpam-5982	305	17	next	next	ADJ
ejpam-5982	305	18	theorem	theorem	NOUN
ejpam-5982	305	19	,	,	PUNCT
ejpam-5982	305	20	we	we	PRON
ejpam-5982	305	21	do	do	AUX
ejpam-5982	305	22	not	not	PART
ejpam-5982	305	23	require	require	VERB
ejpam-5982	305	24	the	the	DET
ejpam-5982	305	25	continuity	continuity	NOUN
ejpam-5982	305	26	of	of	ADP
ejpam-5982	305	27	maps	map	NOUN
ejpam-5982	305	28	.	.	PUNCT
ejpam-5982	306	1	theorem	theorem	NOUN
ejpam-5982	306	2	2	2	NUM
ejpam-5982	306	3	.	.	X
ejpam-5982	307	1	let	let	VERB
ejpam-5982	307	2	(	(	PUNCT
ejpam-5982	307	3	x	x	X
ejpam-5982	307	4	,	,	PUNCT
ejpam-5982	307	5	y	y	PROPN
ejpam-5982	307	6	,	,	PUNCT
ejpam-5982	307	7	d	d	NOUN
ejpam-5982	307	8	)	)	PUNCT
ejpam-5982	307	9	be	be	AUX
ejpam-5982	307	10	a	a	DET
ejpam-5982	307	11	bipolar	bipolar	ADJ
ejpam-5982	307	12	metric	metric	ADJ
ejpam-5982	307	13	space	space	NOUN
ejpam-5982	307	14	and	and	CCONJ
ejpam-5982	307	15	let	let	VERB
ejpam-5982	307	16	t1	t1	NOUN
ejpam-5982	307	17	,	,	PUNCT
ejpam-5982	307	18	t2	t2	NOUN
ejpam-5982	307	19	:	:	PUNCT
ejpam-5982	307	20	(	(	PUNCT
ejpam-5982	307	21	x	x	X
ejpam-5982	307	22	,	,	PUNCT
ejpam-5982	307	23	y	y	PROPN
ejpam-5982	307	24	,	,	PUNCT
ejpam-5982	307	25	d	d	NOUN
ejpam-5982	307	26	)	)	PUNCT
ejpam-5982	307	27	⇒	⇒	NOUN
ejpam-5982	307	28	(	(	PUNCT
ejpam-5982	307	29	x	x	X
ejpam-5982	307	30	,	,	PUNCT
ejpam-5982	307	31	y	y	PROPN
ejpam-5982	307	32	,	,	PUNCT
ejpam-5982	307	33	d	d	NOUN
ejpam-5982	307	34	)	)	PUNCT
ejpam-5982	307	35	be	be	AUX
ejpam-5982	307	36	two	two	NUM
ejpam-5982	307	37	covariant	covariant	ADJ
ejpam-5982	307	38	maps	map	NOUN
ejpam-5982	307	39	and	and	CCONJ
ejpam-5982	307	40	s1	s1	NOUN
ejpam-5982	307	41	,	,	PUNCT
ejpam-5982	307	42	s2	s2	NOUN
ejpam-5982	307	43	:	:	PUNCT
ejpam-5982	307	44	(	(	PUNCT
ejpam-5982	307	45	x	x	X
ejpam-5982	307	46	,	,	PUNCT
ejpam-5982	307	47	y	y	PROPN
ejpam-5982	307	48	,	,	PUNCT
ejpam-5982	307	49	d	d	NOUN
ejpam-5982	307	50	)	)	PUNCT
ejpam-5982	307	51	⇄	⇄	NOUN
ejpam-5982	307	52	(	(	PUNCT
ejpam-5982	307	53	x	x	X
ejpam-5982	307	54	,	,	PUNCT
ejpam-5982	307	55	y	y	PROPN
ejpam-5982	307	56	,	,	PUNCT
ejpam-5982	307	57	d	d	NOUN
ejpam-5982	307	58	)	)	PUNCT
ejpam-5982	307	59	be	be	AUX
ejpam-5982	307	60	two	two	NUM
ejpam-5982	307	61	contravariant	contravariant	ADJ
ejpam-5982	307	62	maps	map	NOUN
ejpam-5982	307	63	satisfying	satisfy	VERB
ejpam-5982	307	64	the	the	DET
ejpam-5982	307	65	following	follow	VERB
ejpam-5982	307	66	conditions	condition	NOUN
ejpam-5982	307	67	:	:	PUNCT
ejpam-5982	308	1	p.	p.	NOUN
ejpam-5982	308	2	p.	p.	NOUN
ejpam-5982	308	3	murthy	murthy	ADJ
ejpam-5982	309	1	et	et	PROPN
ejpam-5982	309	2	al	al	PROPN
ejpam-5982	309	3	.	.	PUNCT
ejpam-5982	309	4	/	/	SYM
ejpam-5982	309	5	eur	eur	PROPN
ejpam-5982	309	6	.	.	PUNCT
ejpam-5982	310	1	j.	j.	PROPN
ejpam-5982	310	2	pure	pure	PROPN
ejpam-5982	310	3	appl	appl	PROPN
ejpam-5982	310	4	.	.	PROPN
ejpam-5982	310	5	math	math	PROPN
ejpam-5982	310	6	,	,	PUNCT
ejpam-5982	310	7	18	18	NUM
ejpam-5982	310	8	(	(	PUNCT
ejpam-5982	310	9	2	2	NUM
ejpam-5982	310	10	)	)	PUNCT
ejpam-5982	310	11	(	(	PUNCT
ejpam-5982	310	12	2025	2025	NUM
ejpam-5982	310	13	)	)	PUNCT
ejpam-5982	310	14	,	,	PUNCT
ejpam-5982	310	15	5982	5982	NUM
ejpam-5982	310	16	12	12	NUM
ejpam-5982	310	17	of	of	ADP
ejpam-5982	310	18	17	17	NUM
ejpam-5982	310	19	(	(	PUNCT
ejpam-5982	310	20	i	i	NOUN
ejpam-5982	310	21	)	)	PUNCT
ejpam-5982	310	22	the	the	DET
ejpam-5982	310	23	pairs	pair	NOUN
ejpam-5982	310	24	(	(	PUNCT
ejpam-5982	310	25	s2	s2	PROPN
ejpam-5982	310	26	,	,	PUNCT
ejpam-5982	310	27	t1	t1	NOUN
ejpam-5982	310	28	)	)	PUNCT
ejpam-5982	310	29	and	and	CCONJ
ejpam-5982	310	30	(	(	PUNCT
ejpam-5982	310	31	s1	s1	NOUN
ejpam-5982	310	32	,	,	PUNCT
ejpam-5982	310	33	t2	t2	NOUN
ejpam-5982	310	34	)	)	PUNCT
ejpam-5982	310	35	are	be	AUX
ejpam-5982	310	36	weak	weak	ADJ
ejpam-5982	310	37	compatible	compatible	ADJ
ejpam-5982	310	38	of	of	ADP
ejpam-5982	310	39	type(a	type(a	NOUN
ejpam-5982	310	40	)	)	PUNCT
ejpam-5982	310	41	.	.	PUNCT
ejpam-5982	311	1	(	(	PUNCT
ejpam-5982	311	2	ii	ii	NOUN
ejpam-5982	311	3	)	)	PUNCT
ejpam-5982	311	4	s1(x	s1(x	NOUN
ejpam-5982	311	5	)	)	PUNCT
ejpam-5982	312	1	⊆	⊆	NUM
ejpam-5982	312	2	t1(y	t1(y	NUM
ejpam-5982	312	3	)	)	PUNCT
ejpam-5982	312	4	or	or	CCONJ
ejpam-5982	312	5	s2(y	s2(y	NUM
ejpam-5982	312	6	)	)	PUNCT
ejpam-5982	312	7	⊆	⊆	NUM
ejpam-5982	312	8	t2(x	t2(x	NOUN
ejpam-5982	312	9	)	)	PUNCT
ejpam-5982	312	10	.	.	PUNCT
ejpam-5982	313	1	(	(	PUNCT
ejpam-5982	313	2	iii	iii	X
ejpam-5982	313	3	)	)	PUNCT
ejpam-5982	313	4	(	(	PUNCT
ejpam-5982	313	5	t2(x	t2(x	NOUN
ejpam-5982	313	6	)	)	PUNCT
ejpam-5982	313	7	,	,	PUNCT
ejpam-5982	313	8	t1(y	t1(y	NUM
ejpam-5982	313	9	)	)	PUNCT
ejpam-5982	313	10	,	,	PUNCT
ejpam-5982	313	11	d	d	X
ejpam-5982	313	12	)	)	PUNCT
ejpam-5982	313	13	or	or	CCONJ
ejpam-5982	313	14	(	(	PUNCT
ejpam-5982	313	15	s2(y	s2(y	PROPN
ejpam-5982	313	16	)	)	PUNCT
ejpam-5982	313	17	,	,	PUNCT
ejpam-5982	313	18	s1(x	s1(x	NOUN
ejpam-5982	313	19	)	)	PUNCT
ejpam-5982	313	20	,	,	PUNCT
ejpam-5982	313	21	d	d	X
ejpam-5982	313	22	)	)	PUNCT
ejpam-5982	313	23	is	be	AUX
ejpam-5982	313	24	complete	complete	ADJ
ejpam-5982	313	25	.	.	PUNCT
ejpam-5982	314	1	(	(	PUNCT
ejpam-5982	314	2	iv	iv	X
ejpam-5982	314	3	)	)	PUNCT
ejpam-5982	314	4	there	there	PRON
ejpam-5982	314	5	exists	exist	VERB
ejpam-5982	314	6	continuous	continuous	ADJ
ejpam-5982	314	7	function	function	NOUN
ejpam-5982	314	8	ψ	ψ	X
ejpam-5982	314	9	∈	∈	NOUN
ejpam-5982	314	10	ψ	ψ	ADP
ejpam-5982	314	11	such	such	ADJ
ejpam-5982	314	12	that	that	DET
ejpam-5982	314	13	ψ(d(s2y	ψ(d(s2y	PROPN
ejpam-5982	314	14	,	,	PUNCT
ejpam-5982	314	15	s1x	s1x	PROPN
ejpam-5982	314	16	)	)	PUNCT
ejpam-5982	314	17	,	,	PUNCT
ejpam-5982	314	18	d(t2x	d(t2x	VERB
ejpam-5982	314	19	,	,	PUNCT
ejpam-5982	314	20	t1y	t1y	NOUN
ejpam-5982	314	21	)	)	PUNCT
ejpam-5982	314	22	,	,	PUNCT
ejpam-5982	314	23	d(t2x	d(t2x	NOUN
ejpam-5982	314	24	,	,	PUNCT
ejpam-5982	314	25	s1x	s1x	PROPN
ejpam-5982	314	26	)	)	PUNCT
ejpam-5982	314	27	,	,	PUNCT
ejpam-5982	314	28	d(s2y	d(s2y	NOUN
ejpam-5982	314	29	,	,	PUNCT
ejpam-5982	314	30	t1y	t1y	NOUN
ejpam-5982	314	31	)	)	PUNCT
ejpam-5982	314	32	)	)	PUNCT
ejpam-5982	314	33	≤	≤	ADV
ejpam-5982	314	34	0	0	NUM
ejpam-5982	314	35	,	,	PUNCT
ejpam-5982	314	36	(	(	PUNCT
ejpam-5982	314	37	14	14	NUM
ejpam-5982	314	38	)	)	PUNCT
ejpam-5982	314	39	for	for	ADP
ejpam-5982	314	40	all	all	DET
ejpam-5982	314	41	(	(	PUNCT
ejpam-5982	314	42	x	x	NOUN
ejpam-5982	314	43	,	,	PUNCT
ejpam-5982	314	44	y	y	NOUN
ejpam-5982	314	45	)	)	PUNCT
ejpam-5982	314	46	∈	∈	PROPN
ejpam-5982	314	47	x	x	PUNCT
ejpam-5982	314	48	×	×	NOUN
ejpam-5982	314	49	y	y	PROPN
ejpam-5982	314	50	.	.	PUNCT
ejpam-5982	315	1	then	then	ADV
ejpam-5982	315	2	the	the	DET
ejpam-5982	315	3	functions	function	NOUN
ejpam-5982	315	4	s1	s1	NOUN
ejpam-5982	315	5	,	,	PUNCT
ejpam-5982	315	6	s2	s2	PROPN
ejpam-5982	315	7	,	,	PUNCT
ejpam-5982	315	8	t1	t1	NOUN
ejpam-5982	315	9	and	and	CCONJ
ejpam-5982	315	10	t2	t2	PROPN
ejpam-5982	315	11	have	have	VERB
ejpam-5982	315	12	a	a	DET
ejpam-5982	315	13	unique	unique	ADJ
ejpam-5982	315	14	common	common	ADJ
ejpam-5982	315	15	fixed	fix	VERB
ejpam-5982	315	16	point	point	NOUN
ejpam-5982	315	17	.	.	PUNCT
ejpam-5982	316	1	proof	proof	NOUN
ejpam-5982	316	2	.	.	PUNCT
ejpam-5982	317	1	let	let	VERB
ejpam-5982	317	2	the	the	DET
ejpam-5982	317	3	bisequence	bisequence	NOUN
ejpam-5982	317	4	(	(	PUNCT
ejpam-5982	317	5	xn	xn	PROPN
ejpam-5982	317	6	,	,	PUNCT
ejpam-5982	317	7	yn	yn	PROPN
ejpam-5982	317	8	)	)	PUNCT
ejpam-5982	317	9	and	and	CCONJ
ejpam-5982	317	10	(	(	PUNCT
ejpam-5982	317	11	un	un	PROPN
ejpam-5982	317	12	,	,	PUNCT
ejpam-5982	317	13	vn	vn	NOUN
ejpam-5982	317	14	)	)	PUNCT
ejpam-5982	317	15	be	be	AUX
ejpam-5982	317	16	defined	define	VERB
ejpam-5982	317	17	as	as	ADP
ejpam-5982	317	18	in	in	ADP
ejpam-5982	317	19	theorem	theorem	NOUN
ejpam-5982	317	20	1	1	NUM
ejpam-5982	317	21	.	.	PUNCT
ejpam-5982	318	1	by	by	ADP
ejpam-5982	318	2	the	the	DET
ejpam-5982	318	3	same	same	ADJ
ejpam-5982	318	4	argument	argument	NOUN
ejpam-5982	318	5	as	as	SCONJ
ejpam-5982	318	6	given	give	VERB
ejpam-5982	318	7	in	in	ADP
ejpam-5982	318	8	the	the	DET
ejpam-5982	318	9	same	same	ADJ
ejpam-5982	318	10	theorem	theorem	NOUN
ejpam-5982	318	11	,	,	PUNCT
ejpam-5982	318	12	(	(	PUNCT
ejpam-5982	318	13	un	un	PROPN
ejpam-5982	318	14	,	,	PUNCT
ejpam-5982	318	15	vn	vn	NOUN
ejpam-5982	318	16	)	)	PUNCT
ejpam-5982	318	17	is	be	AUX
ejpam-5982	318	18	cauchy	cauchy	ADJ
ejpam-5982	318	19	bisequence	bisequence	NOUN
ejpam-5982	318	20	in	in	ADP
ejpam-5982	318	21	(	(	PUNCT
ejpam-5982	318	22	t2(x	t2(x	NOUN
ejpam-5982	318	23	)	)	PUNCT
ejpam-5982	318	24	,	,	PUNCT
ejpam-5982	318	25	t1(y	t1(y	NUM
ejpam-5982	318	26	)	)	PUNCT
ejpam-5982	318	27	,	,	PUNCT
ejpam-5982	318	28	d	d	X
ejpam-5982	318	29	)	)	PUNCT
ejpam-5982	318	30	and	and	CCONJ
ejpam-5982	318	31	(	(	PUNCT
ejpam-5982	318	32	s2(y	s2(y	PROPN
ejpam-5982	318	33	)	)	PUNCT
ejpam-5982	318	34	,	,	PUNCT
ejpam-5982	318	35	s1(x	s1(x	NOUN
ejpam-5982	318	36	)	)	PUNCT
ejpam-5982	318	37	,	,	PUNCT
ejpam-5982	318	38	d	d	NOUN
ejpam-5982	318	39	)	)	PUNCT
ejpam-5982	318	40	.	.	PUNCT
ejpam-5982	319	1	the	the	DET
ejpam-5982	319	2	following	follow	VERB
ejpam-5982	319	3	two	two	NUM
ejpam-5982	319	4	cases	case	NOUN
ejpam-5982	319	5	arise	arise	VERB
ejpam-5982	319	6	case	case	NOUN
ejpam-5982	320	1	i	i	PRON
ejpam-5982	320	2	:	:	PUNCT
ejpam-5982	320	3	if	if	SCONJ
ejpam-5982	320	4	(	(	PUNCT
ejpam-5982	320	5	t2(x	t2(x	NOUN
ejpam-5982	320	6	)	)	PUNCT
ejpam-5982	320	7	,	,	PUNCT
ejpam-5982	320	8	t1(y	t1(y	NUM
ejpam-5982	320	9	)	)	PUNCT
ejpam-5982	320	10	,	,	PUNCT
ejpam-5982	320	11	d	d	X
ejpam-5982	320	12	)	)	PUNCT
ejpam-5982	320	13	is	be	AUX
ejpam-5982	320	14	complete	complete	ADJ
ejpam-5982	320	15	,	,	PUNCT
ejpam-5982	320	16	then	then	ADV
ejpam-5982	320	17	the	the	DET
ejpam-5982	320	18	sequence	sequence	NOUN
ejpam-5982	320	19	(	(	PUNCT
ejpam-5982	320	20	un	un	PROPN
ejpam-5982	320	21	,	,	PUNCT
ejpam-5982	320	22	vn	vn	NOUN
ejpam-5982	320	23	)	)	PUNCT
ejpam-5982	320	24	biconverges	biconverge	NOUN
ejpam-5982	320	25	to	to	ADP
ejpam-5982	320	26	some	some	DET
ejpam-5982	320	27	point	point	NOUN
ejpam-5982	320	28	in	in	ADP
ejpam-5982	320	29	t2(x	t2(x	NOUN
ejpam-5982	320	30	)	)	PUNCT
ejpam-5982	320	31	∩	∩	NOUN
ejpam-5982	320	32	t1(y	t1(y	PRON
ejpam-5982	320	33	)	)	PUNCT
ejpam-5982	320	34	.	.	PUNCT
ejpam-5982	321	1	case	case	NOUN
ejpam-5982	321	2	ii	ii	NOUN
ejpam-5982	321	3	:	:	PUNCT
ejpam-5982	321	4	if	if	SCONJ
ejpam-5982	321	5	(	(	PUNCT
ejpam-5982	321	6	s2(y	s2(y	NUM
ejpam-5982	321	7	)	)	PUNCT
ejpam-5982	321	8	,	,	PUNCT
ejpam-5982	321	9	s1(x	s1(x	NOUN
ejpam-5982	321	10	)	)	PUNCT
ejpam-5982	321	11	,	,	PUNCT
ejpam-5982	321	12	d	d	X
ejpam-5982	321	13	)	)	PUNCT
ejpam-5982	321	14	is	be	AUX
ejpam-5982	321	15	complete	complete	ADJ
ejpam-5982	321	16	,	,	PUNCT
ejpam-5982	321	17	then	then	ADV
ejpam-5982	321	18	the	the	DET
ejpam-5982	321	19	sequence	sequence	NOUN
ejpam-5982	321	20	(	(	PUNCT
ejpam-5982	321	21	un	un	PROPN
ejpam-5982	321	22	,	,	PUNCT
ejpam-5982	321	23	vn	vn	NOUN
ejpam-5982	321	24	)	)	PUNCT
ejpam-5982	321	25	biconverges	biconverge	NOUN
ejpam-5982	321	26	to	to	ADP
ejpam-5982	321	27	a	a	DET
ejpam-5982	321	28	point	point	NOUN
ejpam-5982	321	29	in	in	ADP
ejpam-5982	321	30	s2(y	s2(y	NUM
ejpam-5982	321	31	)	)	PUNCT
ejpam-5982	321	32	∩	∩	NOUN
ejpam-5982	321	33	s1(x	s1(x	NOUN
ejpam-5982	321	34	)	)	PUNCT
ejpam-5982	321	35	.	.	PUNCT
ejpam-5982	322	1	this	this	PRON
ejpam-5982	322	2	implies	imply	VERB
ejpam-5982	322	3	that	that	SCONJ
ejpam-5982	322	4	(	(	PUNCT
ejpam-5982	322	5	un	un	PROPN
ejpam-5982	322	6	,	,	PUNCT
ejpam-5982	322	7	vn	vn	NOUN
ejpam-5982	322	8	)	)	PUNCT
ejpam-5982	322	9	biconverges	biconverge	NOUN
ejpam-5982	322	10	to	to	ADP
ejpam-5982	322	11	a	a	DET
ejpam-5982	322	12	point	point	NOUN
ejpam-5982	322	13	in	in	ADP
ejpam-5982	322	14	t2(x	t2(x	NOUN
ejpam-5982	322	15	)	)	PUNCT
ejpam-5982	322	16	∩	∩	NOUN
ejpam-5982	322	17	t1(y	t1(y	NUM
ejpam-5982	322	18	)	)	PUNCT
ejpam-5982	322	19	as	as	ADP
ejpam-5982	322	20	s2(y	s2(y	NUM
ejpam-5982	322	21	)	)	PUNCT
ejpam-5982	322	22	∩	∩	NOUN
ejpam-5982	322	23	s1(x	s1(x	NOUN
ejpam-5982	322	24	)	)	PUNCT
ejpam-5982	322	25	⊂	⊂	PROPN
ejpam-5982	322	26	t2(x	t2(x	NOUN
ejpam-5982	322	27	)	)	PUNCT
ejpam-5982	322	28	∩	∩	NOUN
ejpam-5982	322	29	t1(y	t1(y	PRON
ejpam-5982	322	30	)	)	PUNCT
ejpam-5982	322	31	.	.	PUNCT
ejpam-5982	323	1	so	so	ADV
ejpam-5982	323	2	in	in	ADP
ejpam-5982	323	3	both	both	CCONJ
ejpam-5982	323	4	the	the	DET
ejpam-5982	323	5	cases	case	NOUN
ejpam-5982	323	6	,	,	PUNCT
ejpam-5982	323	7	it	it	PRON
ejpam-5982	323	8	converges	converge	VERB
ejpam-5982	323	9	to	to	ADP
ejpam-5982	323	10	a	a	DET
ejpam-5982	323	11	point	point	NOUN
ejpam-5982	323	12	t	t	NOUN
ejpam-5982	323	13	(	(	PUNCT
ejpam-5982	323	14	say	say	INTJ
ejpam-5982	323	15	)	)	PUNCT
ejpam-5982	323	16	in	in	ADP
ejpam-5982	323	17	t2(x	t2(x	NOUN
ejpam-5982	323	18	)	)	PUNCT
ejpam-5982	323	19	∩	∩	NOUN
ejpam-5982	323	20	t1(y	t1(y	PRON
ejpam-5982	323	21	)	)	PUNCT
ejpam-5982	323	22	.	.	PUNCT
ejpam-5982	324	1	hence	hence	ADV
ejpam-5982	324	2	,	,	PUNCT
ejpam-5982	324	3	there	there	PRON
ejpam-5982	324	4	exist	exist	VERB
ejpam-5982	324	5	p	p	PROPN
ejpam-5982	324	6	∈	∈	PROPN
ejpam-5982	324	7	b	b	PROPN
ejpam-5982	324	8	and	and	CCONJ
ejpam-5982	324	9	q	q	PROPN
ejpam-5982	324	10	∈	∈	PROPN
ejpam-5982	324	11	a	a	DET
ejpam-5982	324	12	such	such	ADJ
ejpam-5982	324	13	that	that	DET
ejpam-5982	324	14	t	t	NOUN
ejpam-5982	324	15	=	=	SYM
ejpam-5982	324	16	t1p	t1p	NOUN
ejpam-5982	324	17	=	=	PUNCT
ejpam-5982	324	18	t2q	t2q	PROPN
ejpam-5982	324	19	.	.	PUNCT
ejpam-5982	325	1	(	(	PUNCT
ejpam-5982	325	2	15	15	NUM
ejpam-5982	325	3	)	)	PUNCT
ejpam-5982	325	4	now	now	ADV
ejpam-5982	325	5	putting	put	VERB
ejpam-5982	325	6	y	y	NOUN
ejpam-5982	325	7	=	=	PUNCT
ejpam-5982	325	8	yn	yn	PROPN
ejpam-5982	325	9	and	and	CCONJ
ejpam-5982	325	10	x	x	X
ejpam-5982	325	11	=	=	PUNCT
ejpam-5982	325	12	q	q	X
ejpam-5982	325	13	in	in	ADP
ejpam-5982	325	14	(	(	PUNCT
ejpam-5982	325	15	14	14	NUM
ejpam-5982	325	16	)	)	PUNCT
ejpam-5982	325	17	,	,	PUNCT
ejpam-5982	325	18	we	we	PRON
ejpam-5982	325	19	get	get	VERB
ejpam-5982	325	20	ψ(d(s2yn	ψ(d(s2yn	ADJ
ejpam-5982	325	21	,	,	PUNCT
ejpam-5982	325	22	s1q	s1q	NOUN
ejpam-5982	325	23	)	)	PUNCT
ejpam-5982	325	24	,	,	PUNCT
ejpam-5982	325	25	d(t2q	d(t2q	NOUN
ejpam-5982	325	26	,	,	PUNCT
ejpam-5982	325	27	t1yn	t1yn	PROPN
ejpam-5982	325	28	)	)	PUNCT
ejpam-5982	325	29	,	,	PUNCT
ejpam-5982	325	30	d(t2q	d(t2q	NOUN
ejpam-5982	325	31	,	,	PUNCT
ejpam-5982	325	32	s1q	s1q	NOUN
ejpam-5982	325	33	)	)	PUNCT
ejpam-5982	325	34	,	,	PUNCT
ejpam-5982	325	35	d(s2yn	d(s2yn	ADJ
ejpam-5982	325	36	,	,	PUNCT
ejpam-5982	325	37	t1yn	t1yn	PUNCT
ejpam-5982	325	38	)	)	PUNCT
ejpam-5982	325	39	)	)	PUNCT
ejpam-5982	326	1	≤	≤	ADV
ejpam-5982	326	2	0	0	NUM
ejpam-5982	326	3	,	,	PUNCT
ejpam-5982	326	4	taking	take	VERB
ejpam-5982	326	5	limit	limit	NOUN
ejpam-5982	326	6	as	as	ADP
ejpam-5982	326	7	n→	n→	ADV
ejpam-5982	326	8	+	+	PROPN
ejpam-5982	326	9	∞	∞	PROPN
ejpam-5982	326	10	,	,	PUNCT
ejpam-5982	326	11	we	we	PRON
ejpam-5982	326	12	get	get	VERB
ejpam-5982	326	13	ψ(d(t	ψ(d(t	ADJ
ejpam-5982	326	14	,	,	PUNCT
ejpam-5982	326	15	s1q	s1q	NOUN
ejpam-5982	326	16	)	)	PUNCT
ejpam-5982	326	17	,	,	PUNCT
ejpam-5982	326	18	d(t	d(t	PROPN
ejpam-5982	326	19	,	,	PUNCT
ejpam-5982	326	20	t	t	PROPN
ejpam-5982	326	21	)	)	PUNCT
ejpam-5982	326	22	,	,	PUNCT
ejpam-5982	326	23	d(t	d(t	PROPN
ejpam-5982	326	24	,	,	PUNCT
ejpam-5982	326	25	s1q	s1q	NOUN
ejpam-5982	326	26	)	)	PUNCT
ejpam-5982	326	27	,	,	PUNCT
ejpam-5982	326	28	0	0	X
ejpam-5982	326	29	)	)	PUNCT
ejpam-5982	326	30	≤	≤	NOUN
ejpam-5982	326	31	0	0	NUM
ejpam-5982	327	1	ψ(d(t	ψ(d(t	ADJ
ejpam-5982	327	2	,	,	PUNCT
ejpam-5982	327	3	s1q	s1q	NOUN
ejpam-5982	327	4	)	)	PUNCT
ejpam-5982	327	5	,	,	PUNCT
ejpam-5982	327	6	0	0	NUM
ejpam-5982	327	7	,	,	PUNCT
ejpam-5982	327	8	d(t	d(t	NOUN
ejpam-5982	327	9	,	,	PUNCT
ejpam-5982	327	10	s1q	s1q	NOUN
ejpam-5982	327	11	)	)	PUNCT
ejpam-5982	327	12	,	,	PUNCT
ejpam-5982	327	13	0	0	X
ejpam-5982	327	14	)	)	PUNCT
ejpam-5982	327	15	≤	≤	NOUN
ejpam-5982	327	16	0	0	NUM
ejpam-5982	327	17	s1q	s1q	NOUN
ejpam-5982	327	18	=	=	PROPN
ejpam-5982	327	19	t.	t.	NOUN
ejpam-5982	327	20	(	(	PUNCT
ejpam-5982	327	21	16	16	NUM
ejpam-5982	327	22	)	)	PUNCT
ejpam-5982	327	23	again	again	ADV
ejpam-5982	327	24	putting	put	VERB
ejpam-5982	327	25	y	y	NOUN
ejpam-5982	327	26	=	=	PUNCT
ejpam-5982	327	27	p	p	PROPN
ejpam-5982	327	28	and	and	CCONJ
ejpam-5982	327	29	x	x	SYM
ejpam-5982	327	30	=	=	SYM
ejpam-5982	327	31	xn	xn	PROPN
ejpam-5982	327	32	in	in	ADP
ejpam-5982	327	33	(	(	PUNCT
ejpam-5982	327	34	14	14	NUM
ejpam-5982	327	35	)	)	PUNCT
ejpam-5982	327	36	,	,	PUNCT
ejpam-5982	327	37	we	we	PRON
ejpam-5982	327	38	get	get	VERB
ejpam-5982	327	39	ψ(d(s2p	ψ(d(s2p	NOUN
ejpam-5982	327	40	,	,	PUNCT
ejpam-5982	327	41	s1xn	s1xn	NUM
ejpam-5982	327	42	)	)	PUNCT
ejpam-5982	327	43	,	,	PUNCT
ejpam-5982	327	44	d(t2xn	d(t2xn	PRON
ejpam-5982	327	45	,	,	PUNCT
ejpam-5982	327	46	t1p	t1p	NOUN
ejpam-5982	327	47	)	)	PUNCT
ejpam-5982	327	48	,	,	PUNCT
ejpam-5982	327	49	d(t2xn	d(t2xn	PRON
ejpam-5982	327	50	,	,	PUNCT
ejpam-5982	327	51	s1xn	s1xn	NOUN
ejpam-5982	327	52	)	)	PUNCT
ejpam-5982	327	53	,	,	PUNCT
ejpam-5982	327	54	d(s2p	d(s2p	NOUN
ejpam-5982	327	55	,	,	PUNCT
ejpam-5982	327	56	t1p	t1p	NOUN
ejpam-5982	327	57	)	)	PUNCT
ejpam-5982	327	58	)	)	PUNCT
ejpam-5982	327	59	≤	≤	ADV
ejpam-5982	327	60	0	0	NUM
ejpam-5982	327	61	,	,	PUNCT
ejpam-5982	327	62	taking	take	VERB
ejpam-5982	327	63	limit	limit	NOUN
ejpam-5982	327	64	as	as	ADP
ejpam-5982	327	65	n→	n→	ADV
ejpam-5982	327	66	+	+	PROPN
ejpam-5982	327	67	∞	∞	PROPN
ejpam-5982	327	68	,	,	PUNCT
ejpam-5982	327	69	we	we	PRON
ejpam-5982	327	70	get	get	VERB
ejpam-5982	327	71	ψ(d(s2p	ψ(d(s2p	NOUN
ejpam-5982	327	72	,	,	PUNCT
ejpam-5982	327	73	t	t	PROPN
ejpam-5982	327	74	)	)	PUNCT
ejpam-5982	327	75	,	,	PUNCT
ejpam-5982	327	76	d(t	d(t	PROPN
ejpam-5982	327	77	,	,	PUNCT
ejpam-5982	327	78	t	t	PROPN
ejpam-5982	327	79	)	)	PUNCT
ejpam-5982	327	80	,	,	PUNCT
ejpam-5982	327	81	d(t	d(t	PROPN
ejpam-5982	327	82	,	,	PUNCT
ejpam-5982	327	83	t	t	PROPN
ejpam-5982	327	84	)	)	PUNCT
ejpam-5982	327	85	,	,	PUNCT
ejpam-5982	327	86	d(s2p	d(s2p	NOUN
ejpam-5982	327	87	,	,	PUNCT
ejpam-5982	327	88	t	t	PROPN
ejpam-5982	327	89	)	)	PUNCT
ejpam-5982	327	90	)	)	PUNCT
ejpam-5982	328	1	≤	≤	NOUN
ejpam-5982	328	2	0	0	NUM
ejpam-5982	329	1	ψ(d(s2p	ψ(d(s2p	NOUN
ejpam-5982	329	2	,	,	PUNCT
ejpam-5982	329	3	t	t	PROPN
ejpam-5982	329	4	)	)	PUNCT
ejpam-5982	329	5	,	,	PUNCT
ejpam-5982	329	6	0	0	NUM
ejpam-5982	329	7	,	,	PUNCT
ejpam-5982	329	8	0	0	NUM
ejpam-5982	329	9	,	,	PUNCT
ejpam-5982	329	10	d(s2p	d(s2p	NOUN
ejpam-5982	329	11	,	,	PUNCT
ejpam-5982	329	12	t	t	PROPN
ejpam-5982	329	13	)	)	PUNCT
ejpam-5982	329	14	)	)	PUNCT
ejpam-5982	330	1	≤	≤	NOUN
ejpam-5982	330	2	0	0	NUM
ejpam-5982	331	1	p.	p.	NOUN
ejpam-5982	331	2	p.	p.	NOUN
ejpam-5982	331	3	murthy	murthy	ADJ
ejpam-5982	332	1	et	et	PROPN
ejpam-5982	332	2	al	al	PROPN
ejpam-5982	332	3	.	.	PUNCT
ejpam-5982	332	4	/	/	SYM
ejpam-5982	332	5	eur	eur	PROPN
ejpam-5982	332	6	.	.	PUNCT
ejpam-5982	333	1	j.	j.	PROPN
ejpam-5982	333	2	pure	pure	PROPN
ejpam-5982	333	3	appl	appl	PROPN
ejpam-5982	333	4	.	.	PROPN
ejpam-5982	333	5	math	math	PROPN
ejpam-5982	333	6	,	,	PUNCT
ejpam-5982	333	7	18	18	NUM
ejpam-5982	333	8	(	(	PUNCT
ejpam-5982	333	9	2	2	NUM
ejpam-5982	333	10	)	)	PUNCT
ejpam-5982	333	11	(	(	PUNCT
ejpam-5982	333	12	2025	2025	NUM
ejpam-5982	333	13	)	)	PUNCT
ejpam-5982	333	14	,	,	PUNCT
ejpam-5982	333	15	5982	5982	NUM
ejpam-5982	333	16	13	13	NUM
ejpam-5982	333	17	of	of	ADP
ejpam-5982	333	18	17	17	NUM
ejpam-5982	333	19	s2p	s2p	NOUN
ejpam-5982	333	20	=	=	SYM
ejpam-5982	333	21	t.	t.	NOUN
ejpam-5982	333	22	(	(	PUNCT
ejpam-5982	333	23	17	17	NUM
ejpam-5982	333	24	)	)	PUNCT
ejpam-5982	333	25	from	from	ADP
ejpam-5982	333	26	(	(	PUNCT
ejpam-5982	333	27	15	15	NUM
ejpam-5982	333	28	)	)	PUNCT
ejpam-5982	333	29	,	,	PUNCT
ejpam-5982	333	30	(	(	PUNCT
ejpam-5982	333	31	16	16	NUM
ejpam-5982	333	32	)	)	PUNCT
ejpam-5982	333	33	and	and	CCONJ
ejpam-5982	333	34	(	(	PUNCT
ejpam-5982	333	35	17	17	NUM
ejpam-5982	333	36	)	)	PUNCT
ejpam-5982	333	37	,	,	PUNCT
ejpam-5982	333	38	we	we	PRON
ejpam-5982	333	39	get	get	VERB
ejpam-5982	333	40	t1p	t1p	PRON
ejpam-5982	333	41	=	=	PUNCT
ejpam-5982	333	42	t2q	t2q	NOUN
ejpam-5982	333	43	=	=	PUNCT
ejpam-5982	333	44	s1q	s1q	NOUN
ejpam-5982	333	45	=	=	PUNCT
ejpam-5982	333	46	s2p	s2p	PROPN
ejpam-5982	333	47	=	=	SYM
ejpam-5982	333	48	t.	t.	NOUN
ejpam-5982	333	49	(	(	PUNCT
ejpam-5982	333	50	18	18	NUM
ejpam-5982	333	51	)	)	PUNCT
ejpam-5982	333	52	since	since	SCONJ
ejpam-5982	333	53	the	the	DET
ejpam-5982	333	54	pairs	pair	NOUN
ejpam-5982	333	55	(	(	PUNCT
ejpam-5982	333	56	s2	s2	PROPN
ejpam-5982	333	57	,	,	PUNCT
ejpam-5982	333	58	t1	t1	NOUN
ejpam-5982	333	59	)	)	PUNCT
ejpam-5982	333	60	and	and	CCONJ
ejpam-5982	333	61	(	(	PUNCT
ejpam-5982	333	62	s1	s1	NOUN
ejpam-5982	333	63	,	,	PUNCT
ejpam-5982	333	64	t2	t2	NOUN
ejpam-5982	333	65	)	)	PUNCT
ejpam-5982	333	66	are	be	AUX
ejpam-5982	333	67	weak	weak	ADJ
ejpam-5982	333	68	compatible	compatible	ADJ
ejpam-5982	333	69	of	of	ADP
ejpam-5982	333	70	type(a	type(a	NOUN
ejpam-5982	333	71	)	)	PUNCT
ejpam-5982	333	72	,	,	PUNCT
ejpam-5982	333	73	equations	equation	NOUN
ejpam-5982	333	74	(	(	PUNCT
ejpam-5982	333	75	18	18	NUM
ejpam-5982	333	76	)	)	PUNCT
ejpam-5982	333	77	imply	imply	VERB
ejpam-5982	333	78	that	that	SCONJ
ejpam-5982	333	79	t1s2p	t1s2p	PUNCT
ejpam-5982	333	80	=	=	NOUN
ejpam-5982	333	81	s2s2p	s2s2p	PUNCT
ejpam-5982	333	82	or	or	CCONJ
ejpam-5982	333	83	s2t1p	s2t1p	PUNCT
ejpam-5982	333	84	=	=	NOUN
ejpam-5982	333	85	t1t1p	t1t1p	NUM
ejpam-5982	333	86	;	;	PUNCT
ejpam-5982	333	87	and	and	CCONJ
ejpam-5982	333	88	t2s1q	t2s1q	PUNCT
ejpam-5982	333	89	=	=	SYM
ejpam-5982	333	90	s1s1q	s1s1q	PUNCT
ejpam-5982	333	91	or	or	CCONJ
ejpam-5982	333	92	s1t2q	s1t2q	PUNCT
ejpam-5982	333	93	=	=	SYM
ejpam-5982	333	94	t2t2q	t2t2q	X
ejpam-5982	333	95	.	.	PUNCT
ejpam-5982	334	1	so	so	ADV
ejpam-5982	334	2	t1	t1	PROPN
ejpam-5982	334	3	t	t	PROPN
ejpam-5982	334	4	=	=	SYM
ejpam-5982	334	5	s2	s2	PROPN
ejpam-5982	334	6	t	t	PROPN
ejpam-5982	334	7	,	,	PUNCT
ejpam-5982	334	8	t2	t2	NOUN
ejpam-5982	334	9	t	t	NOUN
ejpam-5982	334	10	=	=	SYM
ejpam-5982	334	11	s1	s1	PROPN
ejpam-5982	334	12	t.	t.	NOUN
ejpam-5982	334	13	now	now	ADV
ejpam-5982	334	14	putting	put	VERB
ejpam-5982	334	15	x	x	PUNCT
ejpam-5982	334	16	=	=	PUNCT
ejpam-5982	334	17	y	y	PROPN
ejpam-5982	334	18	=	=	SYM
ejpam-5982	334	19	t	t	PROPN
ejpam-5982	334	20	in	in	ADP
ejpam-5982	334	21	(	(	PUNCT
ejpam-5982	334	22	14	14	NUM
ejpam-5982	334	23	)	)	PUNCT
ejpam-5982	334	24	,	,	PUNCT
ejpam-5982	334	25	we	we	PRON
ejpam-5982	334	26	get	get	VERB
ejpam-5982	334	27	ψ(d(s2	ψ(d(s2	PROPN
ejpam-5982	334	28	t	t	PROPN
ejpam-5982	334	29	,	,	PUNCT
ejpam-5982	334	30	s1	s1	PROPN
ejpam-5982	334	31	t	t	PROPN
ejpam-5982	334	32	)	)	PUNCT
ejpam-5982	334	33	,	,	PUNCT
ejpam-5982	334	34	d(t2	d(t2	NOUN
ejpam-5982	334	35	t	t	PROPN
ejpam-5982	334	36	,	,	PUNCT
ejpam-5982	334	37	t1	t1	PROPN
ejpam-5982	334	38	t	t	PROPN
ejpam-5982	334	39	)	)	PUNCT
ejpam-5982	334	40	,	,	PUNCT
ejpam-5982	334	41	d(t2	d(t2	NOUN
ejpam-5982	334	42	t	t	PROPN
ejpam-5982	334	43	,	,	PUNCT
ejpam-5982	334	44	s1	s1	PROPN
ejpam-5982	334	45	t	t	PROPN
ejpam-5982	334	46	)	)	PUNCT
ejpam-5982	334	47	,	,	PUNCT
ejpam-5982	334	48	d(s2	d(s2	NOUN
ejpam-5982	334	49	t	t	PROPN
ejpam-5982	334	50	,	,	PUNCT
ejpam-5982	334	51	t1	t1	PROPN
ejpam-5982	334	52	t	t	PROPN
ejpam-5982	334	53	)	)	PUNCT
ejpam-5982	334	54	)	)	PUNCT
ejpam-5982	335	1	≤	≤	NOUN
ejpam-5982	335	2	0	0	NUM
ejpam-5982	336	1	ψ(d(s2	ψ(d(s2	PROPN
ejpam-5982	336	2	t	t	PROPN
ejpam-5982	336	3	,	,	PUNCT
ejpam-5982	336	4	s1	s1	PROPN
ejpam-5982	336	5	t	t	PROPN
ejpam-5982	336	6	)	)	PUNCT
ejpam-5982	336	7	,	,	PUNCT
ejpam-5982	336	8	d(s1	d(s1	PROPN
ejpam-5982	336	9	t	t	PROPN
ejpam-5982	336	10	,	,	PUNCT
ejpam-5982	336	11	s2	s2	PROPN
ejpam-5982	336	12	t	t	PROPN
ejpam-5982	336	13	)	)	PUNCT
ejpam-5982	336	14	,	,	PUNCT
ejpam-5982	336	15	0	0	NUM
ejpam-5982	336	16	,	,	PUNCT
ejpam-5982	336	17	0	0	NUM
ejpam-5982	336	18	)	)	PUNCT
ejpam-5982	336	19	≤	≤	NOUN
ejpam-5982	336	20	0	0	NUM
ejpam-5982	336	21	d(s2	d(s2	PROPN
ejpam-5982	336	22	t	t	PROPN
ejpam-5982	336	23	,	,	PUNCT
ejpam-5982	336	24	s1	s1	PROPN
ejpam-5982	336	25	t	t	PROPN
ejpam-5982	336	26	)	)	PUNCT
ejpam-5982	336	27	=	=	SYM
ejpam-5982	336	28	0	0	NUM
ejpam-5982	336	29	s2	s2	PROPN
ejpam-5982	336	30	t	t	NOUN
ejpam-5982	336	31	=	=	SYM
ejpam-5982	336	32	s1	s1	PROPN
ejpam-5982	336	33	t.	t.	NOUN
ejpam-5982	337	1	so	so	ADV
ejpam-5982	337	2	we	we	PRON
ejpam-5982	337	3	get	get	VERB
ejpam-5982	337	4	t1	t1	NOUN
ejpam-5982	337	5	t	t	NOUN
ejpam-5982	337	6	=	=	SYM
ejpam-5982	337	7	s2	s2	PROPN
ejpam-5982	337	8	t	t	NOUN
ejpam-5982	337	9	=	=	SYM
ejpam-5982	337	10	t2	t2	PROPN
ejpam-5982	337	11	t	t	NOUN
ejpam-5982	337	12	=	=	SYM
ejpam-5982	337	13	s1	s1	PROPN
ejpam-5982	337	14	t.	t.	NOUN
ejpam-5982	337	15	(	(	PUNCT
ejpam-5982	337	16	19	19	NUM
ejpam-5982	337	17	)	)	PUNCT
ejpam-5982	337	18	that	that	PRON
ejpam-5982	337	19	is	be	AUX
ejpam-5982	337	20	,	,	PUNCT
ejpam-5982	337	21	t	t	PROPN
ejpam-5982	337	22	is	be	AUX
ejpam-5982	337	23	a	a	DET
ejpam-5982	337	24	coincidence	coincidence	NOUN
ejpam-5982	337	25	point	point	NOUN
ejpam-5982	337	26	of	of	ADP
ejpam-5982	337	27	t1	t1	PROPN
ejpam-5982	337	28	,	,	PUNCT
ejpam-5982	337	29	s2	s2	PROPN
ejpam-5982	337	30	,	,	PUNCT
ejpam-5982	337	31	t2	t2	NOUN
ejpam-5982	337	32	and	and	CCONJ
ejpam-5982	337	33	s1	s1	NOUN
ejpam-5982	337	34	.	.	PUNCT
ejpam-5982	338	1	now	now	ADV
ejpam-5982	338	2	we	we	PRON
ejpam-5982	338	3	show	show	VERB
ejpam-5982	338	4	that	that	SCONJ
ejpam-5982	338	5	t	t	PROPN
ejpam-5982	338	6	is	be	AUX
ejpam-5982	338	7	a	a	DET
ejpam-5982	338	8	common	common	ADJ
ejpam-5982	338	9	fixed	fix	VERB
ejpam-5982	338	10	point	point	NOUN
ejpam-5982	338	11	of	of	ADP
ejpam-5982	338	12	these	these	DET
ejpam-5982	338	13	four	four	NUM
ejpam-5982	338	14	mappings	mapping	NOUN
ejpam-5982	338	15	.	.	PUNCT
ejpam-5982	339	1	for	for	ADP
ejpam-5982	339	2	this	this	PRON
ejpam-5982	339	3	,	,	PUNCT
ejpam-5982	339	4	substituting	substitute	VERB
ejpam-5982	339	5	x	x	X
ejpam-5982	339	6	=	=	SYM
ejpam-5982	339	7	t	t	PROPN
ejpam-5982	339	8	and	and	CCONJ
ejpam-5982	339	9	y	y	PROPN
ejpam-5982	339	10	=	=	PROPN
ejpam-5982	339	11	p	p	NOUN
ejpam-5982	339	12	in	in	ADP
ejpam-5982	339	13	(	(	PUNCT
ejpam-5982	339	14	14	14	NUM
ejpam-5982	339	15	)	)	PUNCT
ejpam-5982	339	16	and	and	CCONJ
ejpam-5982	339	17	using	use	VERB
ejpam-5982	339	18	(	(	PUNCT
ejpam-5982	339	19	18	18	NUM
ejpam-5982	339	20	)	)	PUNCT
ejpam-5982	339	21	and	and	CCONJ
ejpam-5982	339	22	(	(	PUNCT
ejpam-5982	339	23	19	19	NUM
ejpam-5982	339	24	)	)	PUNCT
ejpam-5982	339	25	,	,	PUNCT
ejpam-5982	339	26	we	we	PRON
ejpam-5982	339	27	get	get	VERB
ejpam-5982	339	28	ψ(d(s2p	ψ(d(s2p	NOUN
ejpam-5982	339	29	,	,	PUNCT
ejpam-5982	339	30	s1	s1	PROPN
ejpam-5982	339	31	t	t	PROPN
ejpam-5982	339	32	)	)	PUNCT
ejpam-5982	339	33	,	,	PUNCT
ejpam-5982	339	34	d(t2	d(t2	NOUN
ejpam-5982	339	35	t	t	NOUN
ejpam-5982	339	36	,	,	PUNCT
ejpam-5982	339	37	t1p	t1p	NOUN
ejpam-5982	339	38	)	)	PUNCT
ejpam-5982	339	39	,	,	PUNCT
ejpam-5982	339	40	d(t2	d(t2	NOUN
ejpam-5982	339	41	t	t	PROPN
ejpam-5982	339	42	,	,	PUNCT
ejpam-5982	339	43	s1	s1	PROPN
ejpam-5982	339	44	t	t	PROPN
ejpam-5982	339	45	)	)	PUNCT
ejpam-5982	339	46	,	,	PUNCT
ejpam-5982	339	47	d(s2p	d(s2p	NOUN
ejpam-5982	339	48	,	,	PUNCT
ejpam-5982	339	49	t1p	t1p	NOUN
ejpam-5982	339	50	)	)	PUNCT
ejpam-5982	339	51	)	)	PUNCT
ejpam-5982	340	1	≤	≤	NOUN
ejpam-5982	340	2	0	0	NUM
ejpam-5982	341	1	ψ(d(t	ψ(d(t	ADJ
ejpam-5982	341	2	,	,	PUNCT
ejpam-5982	341	3	s1	s1	PROPN
ejpam-5982	341	4	t	t	PROPN
ejpam-5982	341	5	)	)	PUNCT
ejpam-5982	341	6	,	,	PUNCT
ejpam-5982	341	7	d(s1	d(s1	PROPN
ejpam-5982	341	8	t	t	PROPN
ejpam-5982	341	9	,	,	PUNCT
ejpam-5982	341	10	t	t	PROPN
ejpam-5982	341	11	)	)	PUNCT
ejpam-5982	341	12	,	,	PUNCT
ejpam-5982	341	13	0	0	NUM
ejpam-5982	341	14	,	,	PUNCT
ejpam-5982	341	15	0	0	NUM
ejpam-5982	341	16	)	)	PUNCT
ejpam-5982	341	17	≤	≤	NOUN
ejpam-5982	341	18	0	0	NUM
ejpam-5982	341	19	s1	s1	NOUN
ejpam-5982	341	20	t	t	NOUN
ejpam-5982	341	21	=	=	PUNCT
ejpam-5982	341	22	t.	t.	NOUN
ejpam-5982	342	1	so	so	PROPN
ejpam-5982	342	2	t	t	PROPN
ejpam-5982	342	3	is	be	AUX
ejpam-5982	342	4	a	a	DET
ejpam-5982	342	5	common	common	ADJ
ejpam-5982	342	6	fixed	fix	VERB
ejpam-5982	342	7	point	point	NOUN
ejpam-5982	342	8	of	of	ADP
ejpam-5982	342	9	given	give	VERB
ejpam-5982	342	10	four	four	NUM
ejpam-5982	342	11	mappings	mapping	NOUN
ejpam-5982	342	12	.	.	PUNCT
ejpam-5982	343	1	the	the	DET
ejpam-5982	343	2	uniqueness	uniqueness	NOUN
ejpam-5982	343	3	of	of	ADP
ejpam-5982	343	4	a	a	DET
ejpam-5982	343	5	common	common	ADJ
ejpam-5982	343	6	fixed	fix	VERB
ejpam-5982	343	7	point	point	NOUN
ejpam-5982	343	8	can	can	AUX
ejpam-5982	343	9	be	be	AUX
ejpam-5982	343	10	proved	prove	VERB
ejpam-5982	343	11	as	as	ADP
ejpam-5982	343	12	in	in	ADP
ejpam-5982	343	13	theorem	theorem	NOUN
ejpam-5982	343	14	1	1	X
ejpam-5982	343	15	.	.	PUNCT
ejpam-5982	344	1	our	our	PRON
ejpam-5982	344	2	next	next	ADJ
ejpam-5982	344	3	theorem	theorem	NOUN
ejpam-5982	344	4	is	be	AUX
ejpam-5982	344	5	about	about	ADP
ejpam-5982	344	6	the	the	DET
ejpam-5982	344	7	common	common	ADJ
ejpam-5982	344	8	fixed	fix	VERB
ejpam-5982	344	9	point	point	NOUN
ejpam-5982	344	10	of	of	ADP
ejpam-5982	344	11	four	four	NUM
ejpam-5982	344	12	mappings	mapping	NOUN
ejpam-5982	344	13	and	and	CCONJ
ejpam-5982	344	14	is	be	AUX
ejpam-5982	344	15	a	a	DET
ejpam-5982	344	16	generalization	generalization	NOUN
ejpam-5982	344	17	of	of	ADP
ejpam-5982	344	18	the	the	DET
ejpam-5982	344	19	theorem	theorem	ADJ
ejpam-5982	344	20	1	1	NUM
ejpam-5982	344	21	.	.	PUNCT
ejpam-5982	344	22	theorem	theorem	NOUN
ejpam-5982	344	23	3	3	X
ejpam-5982	344	24	.	.	PUNCT
ejpam-5982	345	1	let	let	AUX
ejpam-5982	345	2	(	(	PUNCT
ejpam-5982	345	3	x	x	X
ejpam-5982	345	4	,	,	PUNCT
ejpam-5982	345	5	y	y	PROPN
ejpam-5982	345	6	,	,	PUNCT
ejpam-5982	345	7	d	d	NOUN
ejpam-5982	345	8	)	)	PUNCT
ejpam-5982	345	9	be	be	AUX
ejpam-5982	345	10	a	a	DET
ejpam-5982	345	11	complete	complete	ADJ
ejpam-5982	345	12	bipolar	bipolar	ADJ
ejpam-5982	345	13	metric	metric	ADJ
ejpam-5982	345	14	space	space	NOUN
ejpam-5982	345	15	and	and	CCONJ
ejpam-5982	345	16	let	let	VERB
ejpam-5982	345	17	t1	t1	NOUN
ejpam-5982	345	18	,	,	PUNCT
ejpam-5982	345	19	t2	t2	NOUN
ejpam-5982	345	20	:	:	PUNCT
ejpam-5982	345	21	(	(	PUNCT
ejpam-5982	345	22	x	x	X
ejpam-5982	345	23	,	,	PUNCT
ejpam-5982	345	24	y	y	PROPN
ejpam-5982	345	25	,	,	PUNCT
ejpam-5982	345	26	d	d	NOUN
ejpam-5982	345	27	)	)	PUNCT
ejpam-5982	345	28	⇒	⇒	NOUN
ejpam-5982	345	29	(	(	PUNCT
ejpam-5982	345	30	x	x	X
ejpam-5982	345	31	,	,	PUNCT
ejpam-5982	345	32	y	y	PROPN
ejpam-5982	345	33	,	,	PUNCT
ejpam-5982	345	34	d	d	NOUN
ejpam-5982	345	35	)	)	PUNCT
ejpam-5982	345	36	be	be	AUX
ejpam-5982	345	37	two	two	NUM
ejpam-5982	345	38	covariant	covariant	ADJ
ejpam-5982	345	39	maps	map	NOUN
ejpam-5982	345	40	and	and	CCONJ
ejpam-5982	345	41	s1	s1	NOUN
ejpam-5982	345	42	,	,	PUNCT
ejpam-5982	345	43	s2	s2	NOUN
ejpam-5982	345	44	:	:	PUNCT
ejpam-5982	345	45	(	(	PUNCT
ejpam-5982	345	46	x	x	X
ejpam-5982	345	47	,	,	PUNCT
ejpam-5982	345	48	y	y	PROPN
ejpam-5982	345	49	,	,	PUNCT
ejpam-5982	345	50	d	d	NOUN
ejpam-5982	345	51	)	)	PUNCT
ejpam-5982	345	52	⇄	⇄	NOUN
ejpam-5982	345	53	(	(	PUNCT
ejpam-5982	345	54	x	x	X
ejpam-5982	345	55	,	,	PUNCT
ejpam-5982	345	56	y	y	PROPN
ejpam-5982	345	57	,	,	PUNCT
ejpam-5982	345	58	d	d	NOUN
ejpam-5982	345	59	)	)	PUNCT
ejpam-5982	345	60	be	be	AUX
ejpam-5982	345	61	two	two	NUM
ejpam-5982	345	62	contravariant	contravariant	ADJ
ejpam-5982	345	63	maps	map	NOUN
ejpam-5982	345	64	satisfying	satisfy	VERB
ejpam-5982	345	65	the	the	DET
ejpam-5982	345	66	following	follow	VERB
ejpam-5982	345	67	conditions	condition	NOUN
ejpam-5982	345	68	:	:	PUNCT
ejpam-5982	345	69	(	(	PUNCT
ejpam-5982	345	70	i	i	NOUN
ejpam-5982	345	71	)	)	PUNCT
ejpam-5982	345	72	s2	s2	NOUN
ejpam-5982	345	73	and	and	CCONJ
ejpam-5982	345	74	t1	t1	NOUN
ejpam-5982	345	75	are	be	AUX
ejpam-5982	345	76	compatible	compatible	ADJ
ejpam-5982	345	77	of	of	ADP
ejpam-5982	345	78	type	type	NOUN
ejpam-5982	345	79	(	(	PUNCT
ejpam-5982	345	80	a	a	NOUN
ejpam-5982	345	81	)	)	PUNCT
ejpam-5982	345	82	with	with	ADP
ejpam-5982	345	83	respect	respect	NOUN
ejpam-5982	345	84	to	to	ADP
ejpam-5982	345	85	y.	y.	PROPN
ejpam-5982	345	86	(	(	PUNCT
ejpam-5982	345	87	ii	ii	PROPN
ejpam-5982	345	88	)	)	PUNCT
ejpam-5982	345	89	s1	s1	NOUN
ejpam-5982	345	90	and	and	CCONJ
ejpam-5982	345	91	t2	t2	NOUN
ejpam-5982	345	92	are	be	AUX
ejpam-5982	345	93	compatible	compatible	ADJ
ejpam-5982	345	94	of	of	ADP
ejpam-5982	345	95	type	type	NOUN
ejpam-5982	345	96	(	(	PUNCT
ejpam-5982	345	97	a	a	NOUN
ejpam-5982	345	98	)	)	PUNCT
ejpam-5982	345	99	with	with	ADP
ejpam-5982	345	100	respect	respect	NOUN
ejpam-5982	345	101	to	to	ADP
ejpam-5982	345	102	x.	x.	PROPN
ejpam-5982	345	103	(	(	PUNCT
ejpam-5982	345	104	iii	iii	X
ejpam-5982	345	105	)	)	PUNCT
ejpam-5982	345	106	the	the	DET
ejpam-5982	345	107	quadruple	quadruple	NOUN
ejpam-5982	345	108	(	(	PUNCT
ejpam-5982	345	109	s1	s1	NOUN
ejpam-5982	345	110	,	,	PUNCT
ejpam-5982	345	111	t2	t2	NOUN
ejpam-5982	345	112	,	,	PUNCT
ejpam-5982	345	113	s2	s2	PROPN
ejpam-5982	345	114	,	,	PUNCT
ejpam-5982	345	115	t1	t1	PROPN
ejpam-5982	345	116	)	)	PUNCT
ejpam-5982	345	117	satisfies	satisfy	VERB
ejpam-5982	345	118	the	the	DET
ejpam-5982	345	119	property	property	NOUN
ejpam-5982	345	120	(	(	PUNCT
ejpam-5982	345	121	e.a	e.a	PROPN
ejpam-5982	345	122	.	.	PROPN
ejpam-5982	345	123	)	)	PUNCT
ejpam-5982	345	124	.	.	PUNCT
ejpam-5982	346	1	(	(	PUNCT
ejpam-5982	346	2	iv	iv	X
ejpam-5982	346	3	)	)	PUNCT
ejpam-5982	346	4	all	all	DET
ejpam-5982	346	5	the	the	DET
ejpam-5982	346	6	four	four	NUM
ejpam-5982	346	7	mappings	mapping	NOUN
ejpam-5982	346	8	s1	s1	NOUN
ejpam-5982	346	9	,	,	PUNCT
ejpam-5982	346	10	s2	s2	PROPN
ejpam-5982	346	11	,	,	PUNCT
ejpam-5982	346	12	t1	t1	NOUN
ejpam-5982	346	13	and	and	CCONJ
ejpam-5982	346	14	t2	t2	NOUN
ejpam-5982	346	15	are	be	AUX
ejpam-5982	346	16	continuous	continuous	ADJ
ejpam-5982	346	17	.	.	PUNCT
ejpam-5982	347	1	(	(	PUNCT
ejpam-5982	347	2	v	v	NOUN
ejpam-5982	347	3	)	)	PUNCT
ejpam-5982	347	4	there	there	PRON
ejpam-5982	347	5	exists	exist	VERB
ejpam-5982	347	6	ϕ	ϕ	PROPN
ejpam-5982	347	7	∈	∈	PROPN
ejpam-5982	347	8	φ	φ	NUM
ejpam-5982	348	1	such	such	ADJ
ejpam-5982	348	2	that	that	DET
ejpam-5982	348	3	p.	p.	NOUN
ejpam-5982	348	4	p.	p.	NOUN
ejpam-5982	349	1	murthy	murthy	PROPN
ejpam-5982	350	1	et	et	PROPN
ejpam-5982	350	2	al	al	PROPN
ejpam-5982	350	3	.	.	PUNCT
ejpam-5982	350	4	/	/	SYM
ejpam-5982	350	5	eur	eur	PROPN
ejpam-5982	350	6	.	.	PUNCT
ejpam-5982	351	1	j.	j.	PROPN
ejpam-5982	351	2	pure	pure	PROPN
ejpam-5982	351	3	appl	appl	PROPN
ejpam-5982	351	4	.	.	PROPN
ejpam-5982	351	5	math	math	PROPN
ejpam-5982	351	6	,	,	PUNCT
ejpam-5982	351	7	18	18	NUM
ejpam-5982	351	8	(	(	PUNCT
ejpam-5982	351	9	2	2	NUM
ejpam-5982	351	10	)	)	PUNCT
ejpam-5982	351	11	(	(	PUNCT
ejpam-5982	351	12	2025	2025	NUM
ejpam-5982	351	13	)	)	PUNCT
ejpam-5982	351	14	,	,	PUNCT
ejpam-5982	351	15	5982	5982	NUM
ejpam-5982	351	16	14	14	NUM
ejpam-5982	351	17	of	of	ADP
ejpam-5982	351	18	17	17	NUM
ejpam-5982	351	19	ϕ(d(s2y	ϕ(d(s2y	PROPN
ejpam-5982	351	20	,	,	PUNCT
ejpam-5982	351	21	s1x	s1x	PROPN
ejpam-5982	351	22	)	)	PUNCT
ejpam-5982	351	23	,	,	PUNCT
ejpam-5982	351	24	d(t2x	d(t2x	VERB
ejpam-5982	351	25	,	,	PUNCT
ejpam-5982	351	26	t1y	t1y	NOUN
ejpam-5982	351	27	)	)	PUNCT
ejpam-5982	351	28	,	,	PUNCT
ejpam-5982	351	29	d(t2x	d(t2x	NOUN
ejpam-5982	351	30	,	,	PUNCT
ejpam-5982	351	31	s1x	s1x	PROPN
ejpam-5982	351	32	)	)	PUNCT
ejpam-5982	351	33	,	,	PUNCT
ejpam-5982	351	34	d(s2y	d(s2y	NOUN
ejpam-5982	351	35	,	,	PUNCT
ejpam-5982	351	36	t1y	t1y	NOUN
ejpam-5982	351	37	)	)	PUNCT
ejpam-5982	351	38	)	)	PUNCT
ejpam-5982	352	1	≤	≤	ADV
ejpam-5982	352	2	0	0	NUM
ejpam-5982	352	3	,	,	PUNCT
ejpam-5982	352	4	(	(	PUNCT
ejpam-5982	352	5	20	20	NUM
ejpam-5982	352	6	)	)	PUNCT
ejpam-5982	352	7	for	for	ADP
ejpam-5982	352	8	all	all	DET
ejpam-5982	352	9	(	(	PUNCT
ejpam-5982	352	10	x	x	NOUN
ejpam-5982	352	11	,	,	PUNCT
ejpam-5982	352	12	y	y	NOUN
ejpam-5982	352	13	)	)	PUNCT
ejpam-5982	352	14	∈	∈	PROPN
ejpam-5982	352	15	x	x	PUNCT
ejpam-5982	352	16	×	×	NOUN
ejpam-5982	352	17	y	y	PROPN
ejpam-5982	352	18	.	.	PUNCT
ejpam-5982	353	1	then	then	ADV
ejpam-5982	353	2	the	the	DET
ejpam-5982	353	3	functions	function	NOUN
ejpam-5982	353	4	s1	s1	NOUN
ejpam-5982	353	5	,	,	PUNCT
ejpam-5982	353	6	s2	s2	PROPN
ejpam-5982	353	7	,	,	PUNCT
ejpam-5982	353	8	t1	t1	NOUN
ejpam-5982	353	9	and	and	CCONJ
ejpam-5982	353	10	t2	t2	PROPN
ejpam-5982	353	11	have	have	VERB
ejpam-5982	353	12	a	a	DET
ejpam-5982	353	13	unique	unique	ADJ
ejpam-5982	353	14	common	common	ADJ
ejpam-5982	353	15	fixed	fix	VERB
ejpam-5982	353	16	point	point	NOUN
ejpam-5982	353	17	.	.	PUNCT
ejpam-5982	354	1	proof	proof	NOUN
ejpam-5982	354	2	.	.	PUNCT
ejpam-5982	355	1	since	since	SCONJ
ejpam-5982	355	2	the	the	DET
ejpam-5982	355	3	quadruple	quadruple	NOUN
ejpam-5982	355	4	(	(	PUNCT
ejpam-5982	355	5	s1	s1	NOUN
ejpam-5982	355	6	,	,	PUNCT
ejpam-5982	355	7	t2	t2	NOUN
ejpam-5982	355	8	,	,	PUNCT
ejpam-5982	355	9	s2	s2	PROPN
ejpam-5982	355	10	,	,	PUNCT
ejpam-5982	355	11	t1	t1	PROPN
ejpam-5982	355	12	)	)	PUNCT
ejpam-5982	355	13	satisfies	satisfy	VERB
ejpam-5982	355	14	the	the	DET
ejpam-5982	355	15	property	property	NOUN
ejpam-5982	355	16	(	(	PUNCT
ejpam-5982	355	17	e.a	e.a	PROPN
ejpam-5982	355	18	.	.	PROPN
ejpam-5982	355	19	)	)	PUNCT
ejpam-5982	356	1	,	,	PUNCT
ejpam-5982	356	2	so	so	CCONJ
ejpam-5982	356	3	there	there	PRON
ejpam-5982	356	4	exists	exist	VERB
ejpam-5982	356	5	a	a	DET
ejpam-5982	356	6	sequence	sequence	NOUN
ejpam-5982	356	7	{	{	PUNCT
ejpam-5982	356	8	(	(	PUNCT
ejpam-5982	356	9	xn	xn	PROPN
ejpam-5982	356	10	,	,	PUNCT
ejpam-5982	356	11	yn	yn	PROPN
ejpam-5982	356	12	)	)	PUNCT
ejpam-5982	356	13	}	}	PUNCT
ejpam-5982	356	14	in	in	ADP
ejpam-5982	356	15	x	x	X
ejpam-5982	356	16	×	×	NOUN
ejpam-5982	356	17	y	y	PROPN
ejpam-5982	356	18	such	such	ADJ
ejpam-5982	356	19	that	that	SCONJ
ejpam-5982	356	20	lim	lim	PROPN
ejpam-5982	356	21	n→+∞	n→+∞	VERB
ejpam-5982	356	22	s1xn	s1xn	PUNCT
ejpam-5982	357	1	=	=	SYM
ejpam-5982	357	2	lim	lim	PROPN
ejpam-5982	357	3	n→+∞	n→+∞	VERB
ejpam-5982	357	4	t2xn	t2xn	PUNCT
ejpam-5982	357	5	=	=	SYM
ejpam-5982	358	1	lim	lim	PROPN
ejpam-5982	358	2	n→+∞	n→+∞	VERB
ejpam-5982	358	3	s2yn	s2yn	PUNCT
ejpam-5982	358	4	=	=	PROPN
ejpam-5982	358	5	lim	lim	PROPN
ejpam-5982	358	6	n→+∞	n→+∞	VERB
ejpam-5982	358	7	t1yn	t1yn	PUNCT
ejpam-5982	359	1	=	=	SYM
ejpam-5982	359	2	t.	t.	NOUN
ejpam-5982	359	3	(	(	PUNCT
ejpam-5982	359	4	21	21	NUM
ejpam-5982	359	5	)	)	PUNCT
ejpam-5982	359	6	this	this	PRON
ejpam-5982	359	7	is	be	AUX
ejpam-5982	359	8	the	the	DET
ejpam-5982	359	9	equation	equation	NOUN
ejpam-5982	359	10	(	(	PUNCT
ejpam-5982	359	11	6	6	NUM
ejpam-5982	359	12	)	)	PUNCT
ejpam-5982	359	13	in	in	ADP
ejpam-5982	359	14	theorem	theorem	NOUN
ejpam-5982	359	15	1	1	NUM
ejpam-5982	359	16	.	.	PUNCT
ejpam-5982	360	1	the	the	DET
ejpam-5982	360	2	remaining	remain	VERB
ejpam-5982	360	3	proof	proof	NOUN
ejpam-5982	360	4	of	of	ADP
ejpam-5982	360	5	the	the	DET
ejpam-5982	360	6	theorem	theorem	NOUN
ejpam-5982	360	7	is	be	AUX
ejpam-5982	360	8	the	the	DET
ejpam-5982	360	9	same	same	ADJ
ejpam-5982	360	10	as	as	ADP
ejpam-5982	360	11	the	the	DET
ejpam-5982	360	12	proof	proof	NOUN
ejpam-5982	360	13	of	of	ADP
ejpam-5982	360	14	the	the	DET
ejpam-5982	360	15	theorem	theorem	NOUN
ejpam-5982	360	16	1	1	NUM
ejpam-5982	360	17	with	with	ADP
ejpam-5982	360	18	ψ	ψ	PRON
ejpam-5982	360	19	replaced	replace	VERB
ejpam-5982	360	20	by	by	ADP
ejpam-5982	360	21	ϕ.	ϕ.	PROPN
ejpam-5982	360	22	like	like	ADP
ejpam-5982	360	23	theorem	theorem	NOUN
ejpam-5982	360	24	1	1	NUM
ejpam-5982	360	25	,	,	PUNCT
ejpam-5982	360	26	many	many	ADJ
ejpam-5982	360	27	corollaries	corollary	NOUN
ejpam-5982	360	28	can	can	AUX
ejpam-5982	360	29	be	be	AUX
ejpam-5982	360	30	derived	derive	VERB
ejpam-5982	360	31	here	here	ADV
ejpam-5982	360	32	also	also	ADV
ejpam-5982	360	33	.	.	PUNCT
ejpam-5982	361	1	one	one	NUM
ejpam-5982	361	2	of	of	ADP
ejpam-5982	361	3	the	the	DET
ejpam-5982	361	4	corollaries	corollary	NOUN
ejpam-5982	361	5	is	be	AUX
ejpam-5982	361	6	given	give	VERB
ejpam-5982	361	7	below	below	ADP
ejpam-5982	361	8	:	:	PUNCT
ejpam-5982	361	9	corollary	corollary	ADJ
ejpam-5982	361	10	8	8	NUM
ejpam-5982	361	11	.	.	PUNCT
ejpam-5982	362	1	let	let	VERB
ejpam-5982	362	2	(	(	PUNCT
ejpam-5982	362	3	x	x	X
ejpam-5982	362	4	,	,	PUNCT
ejpam-5982	362	5	y	y	PROPN
ejpam-5982	362	6	,	,	PUNCT
ejpam-5982	362	7	d	d	NOUN
ejpam-5982	362	8	)	)	PUNCT
ejpam-5982	362	9	be	be	AUX
ejpam-5982	362	10	a	a	DET
ejpam-5982	362	11	complete	complete	ADJ
ejpam-5982	362	12	bipolar	bipolar	ADJ
ejpam-5982	362	13	metric	metric	ADJ
ejpam-5982	362	14	space	space	NOUN
ejpam-5982	362	15	and	and	CCONJ
ejpam-5982	362	16	let	let	VERB
ejpam-5982	362	17	t	t	NOUN
ejpam-5982	362	18	:	:	PUNCT
ejpam-5982	362	19	(	(	PUNCT
ejpam-5982	362	20	x	x	X
ejpam-5982	362	21	,	,	PUNCT
ejpam-5982	362	22	y	y	PROPN
ejpam-5982	362	23	,	,	PUNCT
ejpam-5982	362	24	d	d	NOUN
ejpam-5982	362	25	)	)	PUNCT
ejpam-5982	362	26	⇒	⇒	NOUN
ejpam-5982	362	27	(	(	PUNCT
ejpam-5982	362	28	x	x	X
ejpam-5982	362	29	,	,	PUNCT
ejpam-5982	362	30	y	y	PROPN
ejpam-5982	362	31	,	,	PUNCT
ejpam-5982	362	32	d	d	NOUN
ejpam-5982	362	33	)	)	PUNCT
ejpam-5982	362	34	be	be	AUX
ejpam-5982	362	35	a	a	DET
ejpam-5982	362	36	covariant	covariant	ADJ
ejpam-5982	362	37	map	map	NOUN
ejpam-5982	362	38	and	and	CCONJ
ejpam-5982	362	39	s	s	VERB
ejpam-5982	362	40	:	:	PUNCT
ejpam-5982	362	41	(	(	PUNCT
ejpam-5982	362	42	x	x	X
ejpam-5982	362	43	,	,	PUNCT
ejpam-5982	362	44	y	y	PROPN
ejpam-5982	362	45	,	,	PUNCT
ejpam-5982	362	46	d	d	NOUN
ejpam-5982	362	47	)	)	PUNCT
ejpam-5982	362	48	⇄	⇄	NOUN
ejpam-5982	362	49	(	(	PUNCT
ejpam-5982	362	50	x	x	X
ejpam-5982	362	51	,	,	PUNCT
ejpam-5982	362	52	y	y	PROPN
ejpam-5982	362	53	,	,	PUNCT
ejpam-5982	362	54	d	d	NOUN
ejpam-5982	362	55	)	)	PUNCT
ejpam-5982	362	56	be	be	AUX
ejpam-5982	362	57	a	a	DET
ejpam-5982	362	58	contravariant	contravariant	ADJ
ejpam-5982	362	59	map	map	NOUN
ejpam-5982	362	60	satisfying	satisfy	VERB
ejpam-5982	362	61	the	the	DET
ejpam-5982	362	62	following	follow	VERB
ejpam-5982	362	63	conditions	condition	NOUN
ejpam-5982	362	64	:	:	PUNCT
ejpam-5982	362	65	(	(	PUNCT
ejpam-5982	362	66	i	i	NOUN
ejpam-5982	362	67	)	)	PUNCT
ejpam-5982	362	68	s	s	PROPN
ejpam-5982	362	69	and	and	CCONJ
ejpam-5982	362	70	t	t	PROPN
ejpam-5982	362	71	are	be	AUX
ejpam-5982	362	72	compatible	compatible	ADJ
ejpam-5982	362	73	of	of	ADP
ejpam-5982	362	74	type	type	NOUN
ejpam-5982	362	75	(	(	PUNCT
ejpam-5982	362	76	a	a	NOUN
ejpam-5982	362	77	)	)	PUNCT
ejpam-5982	362	78	with	with	ADP
ejpam-5982	362	79	respect	respect	NOUN
ejpam-5982	362	80	to	to	ADP
ejpam-5982	362	81	x	x	PUNCT
ejpam-5982	362	82	or	or	CCONJ
ejpam-5982	362	83	y.	y.	PROPN
ejpam-5982	362	84	(	(	PUNCT
ejpam-5982	362	85	ii	ii	PROPN
ejpam-5982	362	86	)	)	PUNCT
ejpam-5982	362	87	t	t	PROPN
ejpam-5982	362	88	and	and	CCONJ
ejpam-5982	362	89	s	s	AUX
ejpam-5982	362	90	satisfy	satisfy	VERB
ejpam-5982	362	91	the	the	DET
ejpam-5982	362	92	weak	weak	ADJ
ejpam-5982	362	93	form	form	NOUN
ejpam-5982	362	94	of	of	ADP
ejpam-5982	362	95	property	property	NOUN
ejpam-5982	362	96	(	(	PUNCT
ejpam-5982	362	97	e.a	e.a	PROPN
ejpam-5982	362	98	.	.	PROPN
ejpam-5982	362	99	)	)	PUNCT
ejpam-5982	362	100	.	.	PUNCT
ejpam-5982	363	1	(	(	PUNCT
ejpam-5982	363	2	iii	iii	X
ejpam-5982	363	3	)	)	PUNCT
ejpam-5982	363	4	s	s	PROPN
ejpam-5982	363	5	and	and	CCONJ
ejpam-5982	363	6	t	t	PROPN
ejpam-5982	363	7	are	be	AUX
ejpam-5982	363	8	continuous	continuous	ADJ
ejpam-5982	363	9	.	.	PUNCT
ejpam-5982	364	1	(	(	PUNCT
ejpam-5982	364	2	iv	iv	X
ejpam-5982	364	3	)	)	PUNCT
ejpam-5982	364	4	there	there	PRON
ejpam-5982	364	5	exists	exist	VERB
ejpam-5982	364	6	ϕ	ϕ	PROPN
ejpam-5982	364	7	∈	∈	PROPN
ejpam-5982	364	8	φ	φ	NUM
ejpam-5982	364	9	such	such	ADJ
ejpam-5982	364	10	that	that	SCONJ
ejpam-5982	364	11	ϕ(d(sy	ϕ(d(sy	PROPN
ejpam-5982	364	12	,	,	PUNCT
ejpam-5982	364	13	sx	sx	PROPN
ejpam-5982	364	14	)	)	PUNCT
ejpam-5982	364	15	,	,	PUNCT
ejpam-5982	364	16	d(tx	d(tx	PROPN
ejpam-5982	364	17	,	,	PUNCT
ejpam-5982	364	18	ty	ty	NOUN
ejpam-5982	364	19	)	)	PUNCT
ejpam-5982	364	20	,	,	PUNCT
ejpam-5982	364	21	d(tx	d(tx	PROPN
ejpam-5982	364	22	,	,	PUNCT
ejpam-5982	364	23	sx	sx	PROPN
ejpam-5982	364	24	)	)	PUNCT
ejpam-5982	364	25	,	,	PUNCT
ejpam-5982	364	26	d(sy	d(sy	PROPN
ejpam-5982	364	27	,	,	PUNCT
ejpam-5982	364	28	ty	ty	NOUN
ejpam-5982	364	29	)	)	PUNCT
ejpam-5982	364	30	)	)	PUNCT
ejpam-5982	364	31	≤	≤	NUM
ejpam-5982	364	32	0	0	NUM
ejpam-5982	364	33	for	for	ADP
ejpam-5982	364	34	all	all	DET
ejpam-5982	364	35	(	(	PUNCT
ejpam-5982	364	36	x	x	NOUN
ejpam-5982	364	37	,	,	PUNCT
ejpam-5982	364	38	y	y	NOUN
ejpam-5982	364	39	)	)	PUNCT
ejpam-5982	364	40	∈	∈	PROPN
ejpam-5982	364	41	x	x	PUNCT
ejpam-5982	364	42	×	×	NOUN
ejpam-5982	364	43	y	y	PROPN
ejpam-5982	364	44	.	.	PUNCT
ejpam-5982	365	1	then	then	ADV
ejpam-5982	365	2	the	the	DET
ejpam-5982	365	3	functions	function	NOUN
ejpam-5982	365	4	s	s	PART
ejpam-5982	365	5	and	and	CCONJ
ejpam-5982	365	6	t	t	PROPN
ejpam-5982	365	7	have	have	VERB
ejpam-5982	365	8	a	a	DET
ejpam-5982	365	9	unique	unique	ADJ
ejpam-5982	365	10	common	common	ADJ
ejpam-5982	365	11	fixed	fix	VERB
ejpam-5982	365	12	point	point	NOUN
ejpam-5982	365	13	.	.	PUNCT
ejpam-5982	366	1	example	example	NOUN
ejpam-5982	367	1	3	3	X
ejpam-5982	367	2	.	.	PUNCT
ejpam-5982	367	3	let	let	VERB
ejpam-5982	367	4	a	a	PRON
ejpam-5982	367	5	,	,	PUNCT
ejpam-5982	367	6	d	d	PROPN
ejpam-5982	367	7	∈	∈	PROPN
ejpam-5982	367	8	r	r	NOUN
ejpam-5982	367	9	,	,	PUNCT
ejpam-5982	367	10	0	0	NUM
ejpam-5982	368	1	̸=	̸=	PROPN
ejpam-5982	368	2	b	b	PROPN
ejpam-5982	368	3	∈	∈	PROPN
ejpam-5982	368	4	c	c	NOUN
ejpam-5982	368	5	with	with	ADP
ejpam-5982	368	6	bb̄	bb̄	NOUN
ejpam-5982	368	7	−	−	PROPN
ejpam-5982	368	8	ad	ad	NOUN
ejpam-5982	368	9	>	>	X
ejpam-5982	368	10	0	0	PUNCT
ejpam-5982	368	11	,	,	PUNCT
ejpam-5982	368	12	where	where	SCONJ
ejpam-5982	368	13	c	c	PROPN
ejpam-5982	368	14	is	be	AUX
ejpam-5982	368	15	the	the	DET
ejpam-5982	368	16	set	set	NOUN
ejpam-5982	368	17	of	of	ADP
ejpam-5982	368	18	complex	complex	ADJ
ejpam-5982	368	19	numbers	number	NOUN
ejpam-5982	368	20	.	.	PUNCT
ejpam-5982	369	1	let	let	VERB
ejpam-5982	369	2	us	we	PRON
ejpam-5982	369	3	define	define	VERB
ejpam-5982	369	4	two	two	NUM
ejpam-5982	369	5	sets	set	NOUN
ejpam-5982	369	6	c(a	c(a	PROPN
ejpam-5982	369	7	,	,	PUNCT
ejpam-5982	369	8	b	b	NOUN
ejpam-5982	369	9	,	,	PUNCT
ejpam-5982	369	10	d	d	NOUN
ejpam-5982	369	11	)	)	PUNCT
ejpam-5982	369	12	=	=	PRON
ejpam-5982	369	13	{	{	PUNCT
ejpam-5982	369	14	z	z	NOUN
ejpam-5982	369	15	∈	∈	PROPN
ejpam-5982	369	16	c	c	NOUN
ejpam-5982	369	17	:	:	PUNCT
ejpam-5982	369	18	azz̄	azz̄	X
ejpam-5982	370	1	+	+	CCONJ
ejpam-5982	370	2	bz̄	bz̄	VERB
ejpam-5982	370	3	+	+	CCONJ
ejpam-5982	370	4	b̄z	b̄z	NOUN
ejpam-5982	370	5	+	+	PUNCT
ejpam-5982	370	6	d	d	NOUN
ejpam-5982	370	7	=	=	SYM
ejpam-5982	370	8	0	0	NUM
ejpam-5982	370	9	}	}	PUNCT
ejpam-5982	370	10	and	and	CCONJ
ejpam-5982	370	11	l(b	l(b	PROPN
ejpam-5982	370	12	,	,	PUNCT
ejpam-5982	370	13	d	d	NOUN
ejpam-5982	370	14	)	)	PUNCT
ejpam-5982	370	15	=	=	SYM
ejpam-5982	370	16	{	{	PUNCT
ejpam-5982	370	17	z	z	NOUN
ejpam-5982	370	18	∈	∈	PROPN
ejpam-5982	370	19	c	c	NOUN
ejpam-5982	370	20	:	:	PUNCT
ejpam-5982	370	21	bz̄+b̄z+d	bz̄+b̄z+d	PROPN
ejpam-5982	370	22	=	=	PUNCT
ejpam-5982	370	23	0	0	NUM
ejpam-5982	370	24	}	}	PUNCT
ejpam-5982	370	25	.	.	PUNCT
ejpam-5982	371	1	it	it	PRON
ejpam-5982	371	2	is	be	AUX
ejpam-5982	371	3	clear	clear	ADJ
ejpam-5982	371	4	that	that	SCONJ
ejpam-5982	371	5	c(a.b	c(a.b	PROPN
ejpam-5982	371	6	,	,	PUNCT
ejpam-5982	371	7	d	d	NOUN
ejpam-5982	371	8	)	)	PUNCT
ejpam-5982	371	9	and	and	CCONJ
ejpam-5982	371	10	l(b	l(b	PROPN
ejpam-5982	371	11	,	,	PUNCT
ejpam-5982	371	12	d	d	X
ejpam-5982	371	13	)	)	PUNCT
ejpam-5982	371	14	represent	represent	VERB
ejpam-5982	371	15	a	a	DET
ejpam-5982	371	16	circle(if	circle(if	NOUN
ejpam-5982	371	17	a	a	DET
ejpam-5982	371	18	̸=	̸=	PROPN
ejpam-5982	371	19	0	0	NUM
ejpam-5982	371	20	)	)	PUNCT
ejpam-5982	371	21	and	and	CCONJ
ejpam-5982	371	22	a	a	DET
ejpam-5982	371	23	straight	straight	ADJ
ejpam-5982	371	24	line	line	NOUN
ejpam-5982	371	25	respectively	respectively	ADV
ejpam-5982	371	26	in	in	ADP
ejpam-5982	371	27	a	a	DET
ejpam-5982	371	28	complex	complex	ADJ
ejpam-5982	371	29	plane	plane	NOUN
ejpam-5982	371	30	.	.	PUNCT
ejpam-5982	372	1	let	let	VERB
ejpam-5982	372	2	x	x	PUNCT
ejpam-5982	372	3	=	=	PRON
ejpam-5982	372	4	{	{	PUNCT
ejpam-5982	372	5	c(a	c(a	PROPN
ejpam-5982	372	6	,	,	PUNCT
ejpam-5982	372	7	b	b	NOUN
ejpam-5982	372	8	,	,	PUNCT
ejpam-5982	372	9	d	d	PROPN
ejpam-5982	372	10	)	)	PUNCT
ejpam-5982	372	11	:	:	PUNCT
ejpam-5982	372	12	a	a	X
ejpam-5982	372	13	,	,	PUNCT
ejpam-5982	372	14	d	d	X
ejpam-5982	372	15	∈	∈	NOUN
ejpam-5982	372	16	r	r	NOUN
ejpam-5982	372	17	}	}	PUNCT
ejpam-5982	372	18	and	and	CCONJ
ejpam-5982	372	19	y	y	PROPN
ejpam-5982	372	20	=	=	SYM
ejpam-5982	372	21	{	{	PUNCT
ejpam-5982	372	22	l(b	l(b	PROPN
ejpam-5982	372	23	,	,	PUNCT
ejpam-5982	372	24	d	d	NOUN
ejpam-5982	372	25	)	)	PUNCT
ejpam-5982	372	26	:	:	PUNCT
ejpam-5982	373	1	d	d	X
ejpam-5982	373	2	∈	∈	PROPN
ejpam-5982	373	3	r	r	NOUN
ejpam-5982	373	4	}	}	PUNCT
ejpam-5982	373	5	.	.	PUNCT
ejpam-5982	374	1	hence	hence	ADV
ejpam-5982	374	2	y	y	PROPN
ejpam-5982	374	3	⊆	⊆	NUM
ejpam-5982	374	4	x.	x.	NOUN
ejpam-5982	374	5	let	let	VERB
ejpam-5982	374	6	ρ	ρ	NOUN
ejpam-5982	374	7	:	:	PUNCT
ejpam-5982	374	8	x	x	SYM
ejpam-5982	374	9	×	×	NOUN
ejpam-5982	374	10	y	y	X
ejpam-5982	374	11	→	→	PUNCT
ejpam-5982	375	1	[	[	X
ejpam-5982	375	2	0,+∞	0,+∞	NUM
ejpam-5982	375	3	)	)	PUNCT
ejpam-5982	375	4	is	be	AUX
ejpam-5982	375	5	defined	define	VERB
ejpam-5982	375	6	as	as	ADP
ejpam-5982	375	7	ρ(c(a	ρ(c(a	PROPN
ejpam-5982	375	8	,	,	PUNCT
ejpam-5982	375	9	b	b	PROPN
ejpam-5982	375	10	,	,	PUNCT
ejpam-5982	375	11	d	d	NOUN
ejpam-5982	375	12	)	)	PUNCT
ejpam-5982	375	13	,	,	PUNCT
ejpam-5982	375	14	l(b	l(b	PROPN
ejpam-5982	375	15	,	,	PUNCT
ejpam-5982	375	16	d1	d1	NOUN
ejpam-5982	375	17	)	)	PUNCT
ejpam-5982	375	18	)	)	PUNCT
ejpam-5982	376	1	=	=	SYM
ejpam-5982	376	2	|a|	|a|	PROPN
ejpam-5982	376	3	+	+	NOUN
ejpam-5982	376	4	|d−	|d−	NOUN
ejpam-5982	376	5	d1|	d1|	NOUN
ejpam-5982	376	6	for	for	ADP
ejpam-5982	376	7	all	all	DET
ejpam-5982	376	8	c(a	c(a	PROPN
ejpam-5982	376	9	,	,	PUNCT
ejpam-5982	376	10	b	b	NOUN
ejpam-5982	376	11	,	,	PUNCT
ejpam-5982	376	12	d	d	NOUN
ejpam-5982	376	13	)	)	PUNCT
ejpam-5982	376	14	∈	∈	PROPN
ejpam-5982	376	15	x	x	SYM
ejpam-5982	376	16	,	,	PUNCT
ejpam-5982	376	17	l(b	l(b	PROPN
ejpam-5982	376	18	,	,	PUNCT
ejpam-5982	376	19	d1	d1	NOUN
ejpam-5982	376	20	)	)	PUNCT
ejpam-5982	376	21	∈	∈	PROPN
ejpam-5982	376	22	y	y	PROPN
ejpam-5982	376	23	.	.	PUNCT
ejpam-5982	377	1	then	then	ADV
ejpam-5982	377	2	(	(	PUNCT
ejpam-5982	377	3	x	x	X
ejpam-5982	377	4	,	,	PUNCT
ejpam-5982	377	5	y	y	PROPN
ejpam-5982	377	6	,	,	PUNCT
ejpam-5982	377	7	ρ	ρ	PROPN
ejpam-5982	377	8	)	)	PUNCT
ejpam-5982	377	9	is	be	AUX
ejpam-5982	377	10	a	a	DET
ejpam-5982	377	11	complete	complete	ADJ
ejpam-5982	377	12	bipolar	bipolar	ADJ
ejpam-5982	377	13	metric	metric	ADJ
ejpam-5982	377	14	space	space	NOUN
ejpam-5982	377	15	.	.	PUNCT
ejpam-5982	378	1	let	let	VERB
ejpam-5982	378	2	t	t	NOUN
ejpam-5982	378	3	:	:	PUNCT
ejpam-5982	378	4	(	(	PUNCT
ejpam-5982	378	5	x	x	X
ejpam-5982	378	6	,	,	PUNCT
ejpam-5982	378	7	y	y	PROPN
ejpam-5982	378	8	,	,	PUNCT
ejpam-5982	378	9	ρ	ρ	PROPN
ejpam-5982	378	10	)	)	PUNCT
ejpam-5982	378	11	⇒	⇒	NOUN
ejpam-5982	378	12	(	(	PUNCT
ejpam-5982	378	13	x	x	X
ejpam-5982	378	14	,	,	PUNCT
ejpam-5982	378	15	y	y	PROPN
ejpam-5982	378	16	,	,	PUNCT
ejpam-5982	378	17	ρ	ρ	PROPN
ejpam-5982	378	18	)	)	PUNCT
ejpam-5982	378	19	be	be	VERB
ejpam-5982	378	20	a	a	DET
ejpam-5982	378	21	covariant	covariant	ADJ
ejpam-5982	378	22	map	map	NOUN
ejpam-5982	378	23	and	and	CCONJ
ejpam-5982	378	24	s	s	AUX
ejpam-5982	378	25	:	:	PUNCT
ejpam-5982	378	26	(	(	PUNCT
ejpam-5982	378	27	x	x	X
ejpam-5982	378	28	,	,	PUNCT
ejpam-5982	378	29	y	y	PROPN
ejpam-5982	378	30	,	,	PUNCT
ejpam-5982	378	31	ρ	ρ	PROPN
ejpam-5982	378	32	)	)	PUNCT
ejpam-5982	378	33	⇄	⇄	NOUN
ejpam-5982	378	34	(	(	PUNCT
ejpam-5982	378	35	x	x	X
ejpam-5982	378	36	,	,	PUNCT
ejpam-5982	378	37	y	y	PROPN
ejpam-5982	378	38	,	,	PUNCT
ejpam-5982	378	39	ρ	ρ	PROPN
ejpam-5982	378	40	)	)	PUNCT
ejpam-5982	378	41	be	be	VERB
ejpam-5982	378	42	a	a	DET
ejpam-5982	378	43	contravariant	contravariant	ADJ
ejpam-5982	378	44	map	map	NOUN
ejpam-5982	378	45	defined	define	VERB
ejpam-5982	378	46	as	as	ADP
ejpam-5982	378	47	follows	follow	VERB
ejpam-5982	378	48	:	:	PUNCT
ejpam-5982	378	49	s(c(a	s(c(a	PROPN
ejpam-5982	378	50	,	,	PUNCT
ejpam-5982	378	51	b	b	PROPN
ejpam-5982	378	52	,	,	PUNCT
ejpam-5982	378	53	d	d	NOUN
ejpam-5982	378	54	)	)	PUNCT
ejpam-5982	378	55	)	)	PUNCT
ejpam-5982	379	1	=	=	SYM
ejpam-5982	379	2	l(b	l(b	PROPN
ejpam-5982	379	3	,	,	PUNCT
ejpam-5982	379	4	d8	d8	PROPN
ejpam-5982	379	5	)	)	PUNCT
ejpam-5982	379	6	,	,	PUNCT
ejpam-5982	379	7	s(l(b	s(l(b	PROPN
ejpam-5982	379	8	,	,	PUNCT
ejpam-5982	379	9	d	d	NOUN
ejpam-5982	379	10	)	)	PUNCT
ejpam-5982	379	11	)	)	PUNCT
ejpam-5982	380	1	=	=	SYM
ejpam-5982	380	2	l(b	l(b	PROPN
ejpam-5982	380	3	,	,	PUNCT
ejpam-5982	380	4	d	d	NOUN
ejpam-5982	380	5	8	8	NUM
ejpam-5982	380	6	)	)	PUNCT
ejpam-5982	380	7	,	,	PUNCT
ejpam-5982	380	8	t	t	PROPN
ejpam-5982	380	9	(	(	PUNCT
ejpam-5982	380	10	c(a	c(a	PROPN
ejpam-5982	380	11	,	,	PUNCT
ejpam-5982	380	12	b	b	NOUN
ejpam-5982	380	13	,	,	PUNCT
ejpam-5982	380	14	d	d	NOUN
ejpam-5982	380	15	)	)	PUNCT
ejpam-5982	380	16	)	)	PUNCT
ejpam-5982	381	1	=	=	PUNCT
ejpam-5982	381	2	c	c	X
ejpam-5982	381	3	(	(	PUNCT
ejpam-5982	381	4	a	a	DET
ejpam-5982	381	5	2	2	NUM
ejpam-5982	381	6	,	,	PUNCT
ejpam-5982	381	7	b	b	NOUN
ejpam-5982	381	8	,	,	PUNCT
ejpam-5982	381	9	d	d	PROPN
ejpam-5982	381	10	2	2	NUM
ejpam-5982	381	11	)	)	PUNCT
ejpam-5982	381	12	,	,	PUNCT
ejpam-5982	381	13	t	t	PROPN
ejpam-5982	381	14	(	(	PUNCT
ejpam-5982	381	15	l(b	l(b	PROPN
ejpam-5982	381	16	,	,	PUNCT
ejpam-5982	381	17	d	d	NOUN
ejpam-5982	381	18	)	)	PUNCT
ejpam-5982	381	19	)	)	PUNCT
ejpam-5982	382	1	=	=	PUNCT
ejpam-5982	382	2	l	l	NOUN
ejpam-5982	382	3	(	(	PUNCT
ejpam-5982	382	4	b	b	X
ejpam-5982	382	5	,	,	PUNCT
ejpam-5982	382	6	d	d	NOUN
ejpam-5982	382	7	2	2	NUM
ejpam-5982	382	8	)	)	PUNCT
ejpam-5982	383	1	p.	p.	NOUN
ejpam-5982	383	2	p.	p.	NOUN
ejpam-5982	384	1	murthy	murthy	ADJ
ejpam-5982	385	1	et	et	PROPN
ejpam-5982	385	2	al	al	PROPN
ejpam-5982	385	3	.	.	PUNCT
ejpam-5982	385	4	/	/	SYM
ejpam-5982	385	5	eur	eur	PROPN
ejpam-5982	385	6	.	.	PUNCT
ejpam-5982	386	1	j.	j.	PROPN
ejpam-5982	386	2	pure	pure	PROPN
ejpam-5982	386	3	appl	appl	PROPN
ejpam-5982	386	4	.	.	PROPN
ejpam-5982	386	5	math	math	PROPN
ejpam-5982	386	6	,	,	PUNCT
ejpam-5982	386	7	18	18	NUM
ejpam-5982	386	8	(	(	PUNCT
ejpam-5982	386	9	2	2	NUM
ejpam-5982	386	10	)	)	PUNCT
ejpam-5982	386	11	(	(	PUNCT
ejpam-5982	386	12	2025	2025	NUM
ejpam-5982	386	13	)	)	PUNCT
ejpam-5982	386	14	,	,	PUNCT
ejpam-5982	386	15	5982	5982	NUM
ejpam-5982	386	16	15	15	NUM
ejpam-5982	386	17	of	of	ADP
ejpam-5982	386	18	17	17	NUM
ejpam-5982	386	19	then	then	ADV
ejpam-5982	386	20	s	s	PRON
ejpam-5982	386	21	and	and	CCONJ
ejpam-5982	386	22	t	t	PROPN
ejpam-5982	386	23	are	be	AUX
ejpam-5982	386	24	continuous	continuous	ADJ
ejpam-5982	386	25	mappings	mapping	NOUN
ejpam-5982	386	26	,	,	PUNCT
ejpam-5982	386	27	s(x	s(x	PROPN
ejpam-5982	386	28	∪	∪	PROPN
ejpam-5982	386	29	y	y	PROPN
ejpam-5982	386	30	)	)	PUNCT
ejpam-5982	387	1	=	=	SYM
ejpam-5982	388	1	y	y	PROPN
ejpam-5982	388	2	⊆	⊆	NUM
ejpam-5982	388	3	x	x	SYM
ejpam-5982	388	4	∪	∪	ADP
ejpam-5982	388	5	y	y	PROPN
ejpam-5982	388	6	=	=	SYM
ejpam-5982	388	7	t	t	PROPN
ejpam-5982	388	8	(	(	PUNCT
ejpam-5982	388	9	x	x	X
ejpam-5982	388	10	∪	∪	PROPN
ejpam-5982	388	11	y	y	PROPN
ejpam-5982	388	12	)	)	PUNCT
ejpam-5982	388	13	.	.	PUNCT
ejpam-5982	389	1	the	the	DET
ejpam-5982	389	2	mappings	mapping	NOUN
ejpam-5982	389	3	s	s	PART
ejpam-5982	389	4	and	and	CCONJ
ejpam-5982	389	5	t	t	PROPN
ejpam-5982	389	6	are	be	AUX
ejpam-5982	389	7	compatible	compatible	ADJ
ejpam-5982	389	8	of	of	ADP
ejpam-5982	389	9	type	type	NOUN
ejpam-5982	389	10	(	(	PUNCT
ejpam-5982	389	11	a	a	NOUN
ejpam-5982	389	12	)	)	PUNCT
ejpam-5982	389	13	with	with	ADP
ejpam-5982	389	14	respect	respect	NOUN
ejpam-5982	389	15	to	to	ADP
ejpam-5982	389	16	y	y	PROPN
ejpam-5982	389	17	,	,	PUNCT
ejpam-5982	389	18	for	for	ADP
ejpam-5982	389	19	let	let	VERB
ejpam-5982	389	20	the	the	DET
ejpam-5982	389	21	sequence	sequence	NOUN
ejpam-5982	389	22	{	{	PUNCT
ejpam-5982	389	23	l(b	l(b	PROPN
ejpam-5982	389	24	,	,	PUNCT
ejpam-5982	389	25	dn	dn	NOUN
ejpam-5982	389	26	)	)	PUNCT
ejpam-5982	389	27	}	}	PUNCT
ejpam-5982	389	28	in	in	ADP
ejpam-5982	389	29	y	y	PROPN
ejpam-5982	389	30	satisfies	satisfy	VERB
ejpam-5982	389	31	the	the	DET
ejpam-5982	389	32	condition	condition	NOUN
ejpam-5982	389	33	:	:	PUNCT
ejpam-5982	389	34	lim	lim	PROPN
ejpam-5982	389	35	n→+∞	n→+∞	VERB
ejpam-5982	389	36	s(l(b	s(l(b	PROPN
ejpam-5982	389	37	,	,	PUNCT
ejpam-5982	389	38	dn	dn	PROPN
ejpam-5982	389	39	)	)	PUNCT
ejpam-5982	389	40	)	)	PUNCT
ejpam-5982	390	1	=	=	VERB
ejpam-5982	391	1	lim	lim	PROPN
ejpam-5982	391	2	n→+∞	n→+∞	PROPN
ejpam-5982	391	3	t	t	PROPN
ejpam-5982	391	4	(	(	PUNCT
ejpam-5982	391	5	l(b	l(b	PROPN
ejpam-5982	391	6	,	,	PUNCT
ejpam-5982	391	7	dn	dn	NOUN
ejpam-5982	391	8	)	)	PUNCT
ejpam-5982	391	9	)	)	PUNCT
ejpam-5982	392	1	=	=	SYM
ejpam-5982	392	2	l(b	l(b	PROPN
ejpam-5982	392	3	,	,	PUNCT
ejpam-5982	392	4	d0	d0	NOUN
ejpam-5982	392	5	)	)	PUNCT
ejpam-5982	392	6	∈	∈	PROPN
ejpam-5982	392	7	x	x	SYM
ejpam-5982	392	8	∩	∩	PROPN
ejpam-5982	392	9	y	y	PROPN
ejpam-5982	392	10	,	,	PUNCT
ejpam-5982	392	11	then	then	ADV
ejpam-5982	392	12	dn	dn	PROPN
ejpam-5982	392	13	8	8	NUM
ejpam-5982	392	14	→	→	SYM
ejpam-5982	392	15	d0	d0	NOUN
ejpam-5982	392	16	and	and	CCONJ
ejpam-5982	392	17	dn	dn	PROPN
ejpam-5982	392	18	2	2	NUM
ejpam-5982	392	19	→	→	SYM
ejpam-5982	392	20	d0	d0	NOUN
ejpam-5982	392	21	,	,	PUNCT
ejpam-5982	392	22	so	so	ADV
ejpam-5982	392	23	d0	d0	NOUN
ejpam-5982	392	24	=	=	SYM
ejpam-5982	392	25	0	0	NUM
ejpam-5982	392	26	,	,	PUNCT
ejpam-5982	392	27	this	this	PRON
ejpam-5982	392	28	implies	imply	VERB
ejpam-5982	392	29	that	that	SCONJ
ejpam-5982	392	30	ρ(ts(l(b	ρ(ts(l(b	NOUN
ejpam-5982	392	31	,	,	PUNCT
ejpam-5982	392	32	dn	dn	NOUN
ejpam-5982	392	33	)	)	PUNCT
ejpam-5982	392	34	)	)	PUNCT
ejpam-5982	392	35	,	,	PUNCT
ejpam-5982	392	36	ss(l(b	ss(l(b	NOUN
ejpam-5982	392	37	,	,	PUNCT
ejpam-5982	392	38	dn	dn	NOUN
ejpam-5982	392	39	)	)	PUNCT
ejpam-5982	392	40	)	)	PUNCT
ejpam-5982	392	41	)	)	PUNCT
ejpam-5982	392	42	→	→	SYM
ejpam-5982	392	43	0	0	NUM
ejpam-5982	392	44	and	and	CCONJ
ejpam-5982	392	45	ρ(st	ρ(st	NOUN
ejpam-5982	392	46	(	(	PUNCT
ejpam-5982	392	47	l(b	l(b	PROPN
ejpam-5982	392	48	,	,	PUNCT
ejpam-5982	392	49	dn	dn	NOUN
ejpam-5982	392	50	)	)	PUNCT
ejpam-5982	392	51	)	)	PUNCT
ejpam-5982	392	52	,	,	PUNCT
ejpam-5982	392	53	tt	tt	PROPN
ejpam-5982	392	54	(	(	PUNCT
ejpam-5982	392	55	l(b	l(b	PROPN
ejpam-5982	392	56	,	,	PUNCT
ejpam-5982	392	57	dn	dn	NOUN
ejpam-5982	392	58	)	)	PUNCT
ejpam-5982	392	59	)	)	PUNCT
ejpam-5982	392	60	)	)	PUNCT
ejpam-5982	393	1	→	→	SYM
ejpam-5982	393	2	0	0	X
ejpam-5982	393	3	.	.	X
ejpam-5982	393	4	s	s	PART
ejpam-5982	393	5	and	and	CCONJ
ejpam-5982	393	6	t	t	PROPN
ejpam-5982	393	7	satisfy	satisfy	VERB
ejpam-5982	393	8	the	the	DET
ejpam-5982	393	9	following	follow	VERB
ejpam-5982	393	10	condition	condition	NOUN
ejpam-5982	393	11	ψ(d(sy	ψ(d(sy	NOUN
ejpam-5982	393	12	,	,	PUNCT
ejpam-5982	393	13	sx	sx	PROPN
ejpam-5982	393	14	)	)	PUNCT
ejpam-5982	393	15	,	,	PUNCT
ejpam-5982	393	16	d(tx	d(tx	PROPN
ejpam-5982	393	17	,	,	PUNCT
ejpam-5982	393	18	ty	ty	NOUN
ejpam-5982	393	19	)	)	PUNCT
ejpam-5982	393	20	,	,	PUNCT
ejpam-5982	393	21	d(tx	d(tx	PROPN
ejpam-5982	393	22	,	,	PUNCT
ejpam-5982	393	23	sx	sx	PROPN
ejpam-5982	393	24	)	)	PUNCT
ejpam-5982	393	25	,	,	PUNCT
ejpam-5982	393	26	d(sy	d(sy	PROPN
ejpam-5982	393	27	,	,	PUNCT
ejpam-5982	393	28	ty	ty	NOUN
ejpam-5982	393	29	)	)	PUNCT
ejpam-5982	393	30	)	)	PUNCT
ejpam-5982	393	31	≤	≤	ADV
ejpam-5982	393	32	0	0	NUM
ejpam-5982	393	33	,	,	PUNCT
ejpam-5982	393	34	for	for	ADP
ejpam-5982	393	35	all	all	PRON
ejpam-5982	393	36	(	(	PUNCT
ejpam-5982	393	37	x	x	NOUN
ejpam-5982	393	38	,	,	PUNCT
ejpam-5982	393	39	y	y	NOUN
ejpam-5982	393	40	)	)	PUNCT
ejpam-5982	393	41	∈	∈	PROPN
ejpam-5982	393	42	x	x	SYM
ejpam-5982	393	43	×	×	PROPN
ejpam-5982	393	44	y	y	PROPN
ejpam-5982	393	45	,	,	PUNCT
ejpam-5982	393	46	where	where	SCONJ
ejpam-5982	393	47	ψ(a	ψ(a	PROPN
ejpam-5982	393	48	,	,	PUNCT
ejpam-5982	393	49	b	b	PROPN
ejpam-5982	393	50	,	,	PUNCT
ejpam-5982	393	51	c	c	NOUN
ejpam-5982	393	52	,	,	PUNCT
ejpam-5982	393	53	d	d	NOUN
ejpam-5982	393	54	)	)	PUNCT
ejpam-5982	393	55	=	=	NOUN
ejpam-5982	393	56	a	a	DET
ejpam-5982	393	57	−	−	PROPN
ejpam-5982	393	58	1	1	NUM
ejpam-5982	393	59	4	4	NUM
ejpam-5982	393	60	(	(	PUNCT
ejpam-5982	393	61	b	b	NOUN
ejpam-5982	393	62	+	+	CCONJ
ejpam-5982	393	63	c	c	NOUN
ejpam-5982	393	64	+	+	CCONJ
ejpam-5982	393	65	d	d	NOUN
ejpam-5982	393	66	)	)	PUNCT
ejpam-5982	393	67	.	.	PUNCT
ejpam-5982	394	1	so	so	ADV
ejpam-5982	394	2	all	all	DET
ejpam-5982	394	3	the	the	DET
ejpam-5982	394	4	conditions	condition	NOUN
ejpam-5982	394	5	of	of	ADP
ejpam-5982	394	6	corollary	corollary	ADJ
ejpam-5982	394	7	2	2	NUM
ejpam-5982	394	8	are	be	AUX
ejpam-5982	394	9	satisfied	satisfied	ADJ
ejpam-5982	394	10	,	,	PUNCT
ejpam-5982	394	11	so	so	SCONJ
ejpam-5982	394	12	s	s	PROPN
ejpam-5982	394	13	and	and	CCONJ
ejpam-5982	394	14	t	t	PROPN
ejpam-5982	394	15	have	have	VERB
ejpam-5982	394	16	unique	unique	ADJ
ejpam-5982	394	17	common	common	ADJ
ejpam-5982	394	18	fixed	fix	VERB
ejpam-5982	394	19	point	point	NOUN
ejpam-5982	394	20	.	.	PUNCT
ejpam-5982	395	1	in	in	ADP
ejpam-5982	395	2	fact	fact	NOUN
ejpam-5982	395	3	,	,	PUNCT
ejpam-5982	395	4	l(b	l(b	PROPN
ejpam-5982	395	5	,	,	PUNCT
ejpam-5982	395	6	0	0	NUM
ejpam-5982	395	7	)	)	PUNCT
ejpam-5982	395	8	,	,	PUNCT
ejpam-5982	395	9	(	(	PUNCT
ejpam-5982	395	10	thatisbz̄	thatisbz̄	VERB
ejpam-5982	395	11	+	+	CCONJ
ejpam-5982	395	12	b̄z	b̄z	NOUN
ejpam-5982	395	13	=	=	SYM
ejpam-5982	395	14	0	0	NUM
ejpam-5982	395	15	)	)	PUNCT
ejpam-5982	395	16	is	be	AUX
ejpam-5982	395	17	the	the	DET
ejpam-5982	395	18	unique	unique	ADJ
ejpam-5982	395	19	common	common	ADJ
ejpam-5982	395	20	fixed	fix	VERB
ejpam-5982	395	21	point	point	NOUN
ejpam-5982	395	22	of	of	ADP
ejpam-5982	395	23	s	s	PRON
ejpam-5982	395	24	and	and	CCONJ
ejpam-5982	395	25	t.	t.	PROPN
ejpam-5982	395	26	example	example	NOUN
ejpam-5982	396	1	4	4	X
ejpam-5982	396	2	.	.	PUNCT
ejpam-5982	397	1	let	let	VERB
ejpam-5982	397	2	x	x	PUNCT
ejpam-5982	397	3	=	=	SYM
ejpam-5982	397	4	(	(	PUNCT
ejpam-5982	397	5	−∞	−∞	NOUN
ejpam-5982	397	6	,	,	PUNCT
ejpam-5982	397	7	0	0	NUM
ejpam-5982	397	8	]	]	PUNCT
ejpam-5982	397	9	and	and	CCONJ
ejpam-5982	397	10	y	y	PROPN
ejpam-5982	397	11	=	=	PUNCT
ejpam-5982	398	1	[	[	X
ejpam-5982	398	2	0,+∞	0,+∞	NUM
ejpam-5982	398	3	)	)	PUNCT
ejpam-5982	398	4	,	,	PUNCT
ejpam-5982	398	5	then	then	ADV
ejpam-5982	398	6	(	(	PUNCT
ejpam-5982	398	7	x	x	X
ejpam-5982	398	8	,	,	PUNCT
ejpam-5982	398	9	y	y	PROPN
ejpam-5982	398	10	,	,	PUNCT
ejpam-5982	398	11	d	d	NOUN
ejpam-5982	398	12	)	)	PUNCT
ejpam-5982	398	13	is	be	AUX
ejpam-5982	398	14	a	a	DET
ejpam-5982	398	15	complete	complete	ADJ
ejpam-5982	398	16	bipolar	bipolar	ADJ
ejpam-5982	398	17	metric	metric	ADJ
ejpam-5982	398	18	space	space	NOUN
ejpam-5982	398	19	where	where	SCONJ
ejpam-5982	398	20	d	d	NOUN
ejpam-5982	398	21	is	be	AUX
ejpam-5982	398	22	defined	define	VERB
ejpam-5982	398	23	as	as	ADP
ejpam-5982	398	24	d(x	d(x	PROPN
ejpam-5982	398	25	,	,	PUNCT
ejpam-5982	398	26	y	y	NOUN
ejpam-5982	398	27	)	)	PUNCT
ejpam-5982	398	28	=	=	SYM
ejpam-5982	398	29	|x−	|x−	NOUN
ejpam-5982	398	30	y|	y|	NOUN
ejpam-5982	398	31	.	.	PUNCT
ejpam-5982	399	1	let	let	VERB
ejpam-5982	399	2	maps	map	NOUN
ejpam-5982	399	3	t1	t1	VERB
ejpam-5982	399	4	,	,	PUNCT
ejpam-5982	399	5	t2	t2	NOUN
ejpam-5982	399	6	,	,	PUNCT
ejpam-5982	399	7	s1	s1	PROPN
ejpam-5982	399	8	and	and	CCONJ
ejpam-5982	399	9	s2	s2	NOUN
ejpam-5982	399	10	be	be	AUX
ejpam-5982	399	11	defined	define	VERB
ejpam-5982	399	12	as	as	ADP
ejpam-5982	399	13	s1(x	s1(x	NOUN
ejpam-5982	399	14	)	)	PUNCT
ejpam-5982	400	1	=	=	PUNCT
ejpam-5982	400	2	−x	−x	NOUN
ejpam-5982	400	3	12	12	NUM
ejpam-5982	400	4	,	,	PUNCT
ejpam-5982	400	5	s2(x	s2(x	X
ejpam-5982	400	6	)	)	PUNCT
ejpam-5982	400	7	=	=	SYM
ejpam-5982	401	1	−x	−x	NOUN
ejpam-5982	401	2	6	6	NUM
ejpam-5982	401	3	,	,	PUNCT
ejpam-5982	401	4	t1(x	t1(x	NOUN
ejpam-5982	401	5	)	)	PUNCT
ejpam-5982	401	6	=	=	NOUN
ejpam-5982	401	7	x	x	SYM
ejpam-5982	401	8	2	2	NUM
ejpam-5982	401	9	and	and	CCONJ
ejpam-5982	401	10	t2(x	t2(x	NOUN
ejpam-5982	401	11	)	)	PUNCT
ejpam-5982	401	12	=	=	SYM
ejpam-5982	402	1	x	x	SYM
ejpam-5982	402	2	4	4	NUM
ejpam-5982	402	3	,	,	PUNCT
ejpam-5982	402	4	for	for	ADP
ejpam-5982	402	5	all	all	DET
ejpam-5982	402	6	x	x	SYM
ejpam-5982	402	7	∈	∈	NOUN
ejpam-5982	402	8	x	x	SYM
ejpam-5982	402	9	∪	∪	ADP
ejpam-5982	402	10	y.	y.	PROPN
ejpam-5982	402	11	then	then	ADV
ejpam-5982	402	12	s1	s1	PROPN
ejpam-5982	402	13	,	,	PUNCT
ejpam-5982	402	14	s2	s2	PROPN
ejpam-5982	402	15	are	be	AUX
ejpam-5982	402	16	two	two	NUM
ejpam-5982	402	17	continuous	continuous	ADJ
ejpam-5982	402	18	contravariant	contravariant	ADJ
ejpam-5982	402	19	maps	map	NOUN
ejpam-5982	402	20	and	and	CCONJ
ejpam-5982	402	21	t1	t1	NOUN
ejpam-5982	402	22	,	,	PUNCT
ejpam-5982	402	23	t2	t2	PROPN
ejpam-5982	402	24	are	be	AUX
ejpam-5982	402	25	continuous	continuous	ADJ
ejpam-5982	402	26	covariant	covariant	ADJ
ejpam-5982	402	27	maps	map	NOUN
ejpam-5982	402	28	.	.	PUNCT
ejpam-5982	403	1	all	all	DET
ejpam-5982	403	2	these	these	DET
ejpam-5982	403	3	maps	map	NOUN
ejpam-5982	403	4	satisfy	satisfy	VERB
ejpam-5982	403	5	the	the	DET
ejpam-5982	403	6	condition	condition	NOUN
ejpam-5982	403	7	ψ(d(s2y	ψ(d(s2y	ADP
ejpam-5982	403	8	,	,	PUNCT
ejpam-5982	403	9	s1x	s1x	PROPN
ejpam-5982	403	10	)	)	PUNCT
ejpam-5982	403	11	,	,	PUNCT
ejpam-5982	403	12	d(t2x	d(t2x	VERB
ejpam-5982	403	13	,	,	PUNCT
ejpam-5982	403	14	t1y	t1y	NOUN
ejpam-5982	403	15	)	)	PUNCT
ejpam-5982	403	16	,	,	PUNCT
ejpam-5982	403	17	d(t2x	d(t2x	NOUN
ejpam-5982	403	18	,	,	PUNCT
ejpam-5982	403	19	s1x	s1x	PROPN
ejpam-5982	403	20	)	)	PUNCT
ejpam-5982	403	21	,	,	PUNCT
ejpam-5982	403	22	d(s2y	d(s2y	NOUN
ejpam-5982	403	23	,	,	PUNCT
ejpam-5982	403	24	t1y	t1y	NOUN
ejpam-5982	403	25	)	)	PUNCT
ejpam-5982	403	26	)	)	PUNCT
ejpam-5982	403	27	≤	≤	ADV
ejpam-5982	403	28	0	0	NUM
ejpam-5982	403	29	,	,	PUNCT
ejpam-5982	403	30	for	for	ADP
ejpam-5982	403	31	all	all	PRON
ejpam-5982	403	32	(	(	PUNCT
ejpam-5982	403	33	x	x	NOUN
ejpam-5982	403	34	,	,	PUNCT
ejpam-5982	403	35	y	y	NOUN
ejpam-5982	403	36	)	)	PUNCT
ejpam-5982	403	37	∈	∈	PROPN
ejpam-5982	403	38	x×y	x×y	PROPN
ejpam-5982	403	39	where	where	SCONJ
ejpam-5982	403	40	ψ(a	ψ(a	PROPN
ejpam-5982	403	41	,	,	PUNCT
ejpam-5982	403	42	b	b	PROPN
ejpam-5982	403	43	,	,	PUNCT
ejpam-5982	403	44	c	c	NOUN
ejpam-5982	403	45	,	,	PUNCT
ejpam-5982	403	46	d	d	NOUN
ejpam-5982	403	47	)	)	PUNCT
ejpam-5982	403	48	=	=	SYM
ejpam-5982	403	49	a−k1b−k2c−k3d	a−k1b−k2c−k3d	VERB
ejpam-5982	403	50	with	with	ADP
ejpam-5982	403	51	k1	k1	NOUN
ejpam-5982	403	52	=	=	SYM
ejpam-5982	403	53	2	2	NUM
ejpam-5982	403	54	3	3	NUM
ejpam-5982	403	55	,	,	PUNCT
ejpam-5982	403	56	k2	k2	NOUN
ejpam-5982	403	57	=	=	SYM
ejpam-5982	403	58	1	1	NUM
ejpam-5982	403	59	12	12	NUM
ejpam-5982	403	60	,	,	PUNCT
ejpam-5982	403	61	k3	k3	VERB
ejpam-5982	403	62	=	=	NOUN
ejpam-5982	403	63	1	1	NUM
ejpam-5982	403	64	12	12	NUM
ejpam-5982	403	65	.	.	PUNCT
ejpam-5982	404	1	all	all	DET
ejpam-5982	404	2	the	the	DET
ejpam-5982	404	3	other	other	ADJ
ejpam-5982	404	4	conditions	condition	NOUN
ejpam-5982	404	5	of	of	ADP
ejpam-5982	404	6	theorem	theorem	NOUN
ejpam-5982	404	7	1	1	NUM
ejpam-5982	404	8	are	be	AUX
ejpam-5982	404	9	also	also	ADV
ejpam-5982	404	10	satisfied	satisfied	ADJ
ejpam-5982	404	11	,	,	PUNCT
ejpam-5982	404	12	so	so	ADV
ejpam-5982	404	13	s1	s1	NOUN
ejpam-5982	404	14	,	,	PUNCT
ejpam-5982	404	15	s2	s2	PROPN
ejpam-5982	404	16	,	,	PUNCT
ejpam-5982	404	17	t1	t1	NOUN
ejpam-5982	404	18	and	and	CCONJ
ejpam-5982	404	19	t2	t2	PROPN
ejpam-5982	404	20	have	have	VERB
ejpam-5982	404	21	a	a	DET
ejpam-5982	404	22	unique	unique	ADJ
ejpam-5982	404	23	common	common	ADJ
ejpam-5982	404	24	fixed	fix	VERB
ejpam-5982	404	25	point	point	NOUN
ejpam-5982	404	26	.	.	PUNCT
ejpam-5982	405	1	example	example	NOUN
ejpam-5982	406	1	5	5	NUM
ejpam-5982	406	2	.	.	PUNCT
ejpam-5982	406	3	let	let	VERB
ejpam-5982	406	4	x	x	PUNCT
ejpam-5982	406	5	=	=	SYM
ejpam-5982	406	6	(	(	PUNCT
ejpam-5982	406	7	−∞	−∞	NOUN
ejpam-5982	406	8	,	,	PUNCT
ejpam-5982	406	9	0	0	NUM
ejpam-5982	406	10	]	]	PUNCT
ejpam-5982	406	11	and	and	CCONJ
ejpam-5982	406	12	y	y	PROPN
ejpam-5982	406	13	=	=	PUNCT
ejpam-5982	407	1	[	[	X
ejpam-5982	407	2	0,+∞	0,+∞	NUM
ejpam-5982	407	3	)	)	PUNCT
ejpam-5982	407	4	,	,	PUNCT
ejpam-5982	407	5	then	then	ADV
ejpam-5982	407	6	(	(	PUNCT
ejpam-5982	407	7	x	x	X
ejpam-5982	407	8	,	,	PUNCT
ejpam-5982	407	9	y	y	PROPN
ejpam-5982	407	10	,	,	PUNCT
ejpam-5982	407	11	d	d	NOUN
ejpam-5982	407	12	)	)	PUNCT
ejpam-5982	407	13	is	be	AUX
ejpam-5982	407	14	a	a	DET
ejpam-5982	407	15	complete	complete	ADJ
ejpam-5982	407	16	bipolar	bipolar	ADJ
ejpam-5982	407	17	metric	metric	ADJ
ejpam-5982	407	18	space	space	NOUN
ejpam-5982	407	19	where	where	SCONJ
ejpam-5982	407	20	d	d	NOUN
ejpam-5982	407	21	is	be	AUX
ejpam-5982	407	22	defined	define	VERB
ejpam-5982	407	23	as	as	ADP
ejpam-5982	407	24	d(x	d(x	PROPN
ejpam-5982	407	25	,	,	PUNCT
ejpam-5982	407	26	y	y	NOUN
ejpam-5982	407	27	)	)	PUNCT
ejpam-5982	407	28	=	=	SYM
ejpam-5982	407	29	|x−	|x−	NOUN
ejpam-5982	407	30	y|	y|	NOUN
ejpam-5982	407	31	.	.	PUNCT
ejpam-5982	408	1	let	let	VERB
ejpam-5982	408	2	t	t	PROPN
ejpam-5982	408	3	(	(	PUNCT
ejpam-5982	408	4	covariant	covariant	PROPN
ejpam-5982	408	5	)	)	PUNCT
ejpam-5982	408	6	and	and	CCONJ
ejpam-5982	408	7	s(contravariant	s(contravariant	ADJ
ejpam-5982	408	8	map	map	NOUN
ejpam-5982	408	9	)	)	PUNCT
ejpam-5982	408	10	are	be	AUX
ejpam-5982	408	11	defined	define	VERB
ejpam-5982	408	12	as	as	ADP
ejpam-5982	408	13	s(x	s(x	NOUN
ejpam-5982	408	14	)	)	PUNCT
ejpam-5982	409	1	=	=	PUNCT
ejpam-5982	409	2	−x	−x	NOUN
ejpam-5982	409	3	3	3	NUM
ejpam-5982	409	4	and	and	CCONJ
ejpam-5982	409	5	t	t	PROPN
ejpam-5982	409	6	(	(	PUNCT
ejpam-5982	409	7	x	x	X
ejpam-5982	409	8	)	)	PUNCT
ejpam-5982	409	9	=	=	SYM
ejpam-5982	409	10	x	x	SYM
ejpam-5982	409	11	2	2	NUM
ejpam-5982	409	12	,	,	PUNCT
ejpam-5982	409	13	for	for	ADP
ejpam-5982	409	14	all	all	DET
ejpam-5982	409	15	x	x	SYM
ejpam-5982	409	16	∈	∈	NOUN
ejpam-5982	409	17	x	x	SYM
ejpam-5982	409	18	∪	∪	ADP
ejpam-5982	409	19	y.	y.	PROPN
ejpam-5982	409	20	then	then	ADV
ejpam-5982	409	21	s	s	VERB
ejpam-5982	409	22	and	and	CCONJ
ejpam-5982	409	23	t	t	PROPN
ejpam-5982	409	24	are	be	AUX
ejpam-5982	409	25	continuous	continuous	ADJ
ejpam-5982	409	26	functions	function	NOUN
ejpam-5982	409	27	,	,	PUNCT
ejpam-5982	409	28	compatible	compatible	ADJ
ejpam-5982	409	29	of	of	ADP
ejpam-5982	409	30	type	type	NOUN
ejpam-5982	409	31	(	(	PUNCT
ejpam-5982	409	32	a	a	NOUN
ejpam-5982	409	33	)	)	PUNCT
ejpam-5982	409	34	with	with	ADP
ejpam-5982	409	35	respect	respect	NOUN
ejpam-5982	409	36	to	to	ADP
ejpam-5982	409	37	x	x	PUNCT
ejpam-5982	409	38	and	and	CCONJ
ejpam-5982	409	39	y	y	PROPN
ejpam-5982	409	40	both	both	PRON
ejpam-5982	409	41	,	,	PUNCT
ejpam-5982	409	42	s(x	s(x	PROPN
ejpam-5982	409	43	∪	∪	PROPN
ejpam-5982	409	44	y	y	PROPN
ejpam-5982	409	45	)	)	PUNCT
ejpam-5982	409	46	⊆	⊆	NUM
ejpam-5982	409	47	t	t	NOUN
ejpam-5982	409	48	(	(	PUNCT
ejpam-5982	409	49	x	x	X
ejpam-5982	409	50	∪	∪	PROPN
ejpam-5982	409	51	y	y	PROPN
ejpam-5982	409	52	)	)	PUNCT
ejpam-5982	409	53	,	,	PUNCT
ejpam-5982	409	54	and	and	CCONJ
ejpam-5982	409	55	satisfy	satisfy	VERB
ejpam-5982	409	56	the	the	DET
ejpam-5982	409	57	condition	condition	NOUN
ejpam-5982	409	58	ψ(d(sy	ψ(d(sy	NOUN
ejpam-5982	409	59	,	,	PUNCT
ejpam-5982	409	60	sx	sx	PROPN
ejpam-5982	409	61	)	)	PUNCT
ejpam-5982	409	62	,	,	PUNCT
ejpam-5982	409	63	d(tx	d(tx	PROPN
ejpam-5982	409	64	,	,	PUNCT
ejpam-5982	409	65	ty	ty	NOUN
ejpam-5982	409	66	)	)	PUNCT
ejpam-5982	409	67	,	,	PUNCT
ejpam-5982	409	68	d(tx	d(tx	PROPN
ejpam-5982	409	69	,	,	PUNCT
ejpam-5982	409	70	sx	sx	PROPN
ejpam-5982	409	71	)	)	PUNCT
ejpam-5982	409	72	,	,	PUNCT
ejpam-5982	409	73	d(sy	d(sy	PROPN
ejpam-5982	409	74	,	,	PUNCT
ejpam-5982	409	75	ty	ty	NOUN
ejpam-5982	409	76	)	)	PUNCT
ejpam-5982	409	77	)	)	PUNCT
ejpam-5982	409	78	≤	≤	ADV
ejpam-5982	409	79	0	0	NUM
ejpam-5982	409	80	,	,	PUNCT
ejpam-5982	409	81	for	for	ADP
ejpam-5982	409	82	all	all	DET
ejpam-5982	409	83	(	(	PUNCT
ejpam-5982	409	84	x	x	NOUN
ejpam-5982	409	85	,	,	PUNCT
ejpam-5982	409	86	y	y	NOUN
ejpam-5982	409	87	)	)	PUNCT
ejpam-5982	409	88	∈	∈	PROPN
ejpam-5982	409	89	x×y	x×y	PROPN
ejpam-5982	409	90	where	where	SCONJ
ejpam-5982	409	91	ψ(a	ψ(a	PROPN
ejpam-5982	409	92	,	,	PUNCT
ejpam-5982	409	93	b	b	PROPN
ejpam-5982	409	94	,	,	PUNCT
ejpam-5982	409	95	c	c	NOUN
ejpam-5982	409	96	,	,	PUNCT
ejpam-5982	409	97	d	d	NOUN
ejpam-5982	409	98	)	)	PUNCT
ejpam-5982	409	99	=	=	SYM
ejpam-5982	409	100	a−k1b−k2c−k3d	a−k1b−k2c−k3d	VERB
ejpam-5982	409	101	with	with	ADP
ejpam-5982	409	102	k1	k1	NOUN
ejpam-5982	409	103	=	=	SYM
ejpam-5982	409	104	2	2	NUM
ejpam-5982	409	105	3	3	NUM
ejpam-5982	409	106	,	,	PUNCT
ejpam-5982	409	107	k2	k2	NOUN
ejpam-5982	409	108	=	=	SYM
ejpam-5982	409	109	1	1	NUM
ejpam-5982	409	110	12	12	NUM
ejpam-5982	409	111	,	,	PUNCT
ejpam-5982	409	112	k3	k3	VERB
ejpam-5982	409	113	=	=	NOUN
ejpam-5982	409	114	1	1	NUM
ejpam-5982	409	115	12	12	NUM
ejpam-5982	409	116	.	.	PUNCT
ejpam-5982	410	1	so	so	ADV
ejpam-5982	410	2	all	all	DET
ejpam-5982	410	3	the	the	DET
ejpam-5982	410	4	conditions	condition	NOUN
ejpam-5982	410	5	of	of	ADP
ejpam-5982	410	6	corollary	corollary	ADJ
ejpam-5982	410	7	2	2	NUM
ejpam-5982	410	8	are	be	AUX
ejpam-5982	410	9	satisfied	satisfied	ADJ
ejpam-5982	410	10	,	,	PUNCT
ejpam-5982	410	11	so	so	SCONJ
ejpam-5982	410	12	s	s	PROPN
ejpam-5982	410	13	and	and	CCONJ
ejpam-5982	410	14	t	t	PROPN
ejpam-5982	410	15	have	have	VERB
ejpam-5982	410	16	a	a	DET
ejpam-5982	410	17	unique	unique	ADJ
ejpam-5982	410	18	common	common	ADJ
ejpam-5982	410	19	fixed	fix	VERB
ejpam-5982	410	20	point	point	NOUN
ejpam-5982	410	21	.	.	PUNCT
ejpam-5982	411	1	p.	p.	NOUN
ejpam-5982	411	2	p.	p.	NOUN
ejpam-5982	412	1	murthy	murthy	PROPN
ejpam-5982	413	1	et	et	PROPN
ejpam-5982	413	2	al	al	PROPN
ejpam-5982	413	3	.	.	PUNCT
ejpam-5982	413	4	/	/	SYM
ejpam-5982	413	5	eur	eur	PROPN
ejpam-5982	413	6	.	.	PUNCT
ejpam-5982	414	1	j.	j.	PROPN
ejpam-5982	414	2	pure	pure	PROPN
ejpam-5982	414	3	appl	appl	PROPN
ejpam-5982	414	4	.	.	PROPN
ejpam-5982	414	5	math	math	PROPN
ejpam-5982	414	6	,	,	PUNCT
ejpam-5982	414	7	18	18	NUM
ejpam-5982	414	8	(	(	PUNCT
ejpam-5982	414	9	2	2	NUM
ejpam-5982	414	10	)	)	PUNCT
ejpam-5982	414	11	(	(	PUNCT
ejpam-5982	414	12	2025	2025	NUM
ejpam-5982	414	13	)	)	PUNCT
ejpam-5982	414	14	,	,	PUNCT
ejpam-5982	414	15	5982	5982	NUM
ejpam-5982	414	16	16	16	NUM
ejpam-5982	414	17	of	of	ADP
ejpam-5982	414	18	17	17	NUM
ejpam-5982	414	19	4	4	NUM
ejpam-5982	414	20	.	.	PUNCT
ejpam-5982	414	21	conclusion	conclusion	NOUN
ejpam-5982	414	22	in	in	ADP
ejpam-5982	414	23	the	the	DET
ejpam-5982	414	24	article	article	NOUN
ejpam-5982	414	25	,	,	PUNCT
ejpam-5982	414	26	fixed	fix	VERB
ejpam-5982	414	27	point	point	NOUN
ejpam-5982	414	28	results	result	NOUN
ejpam-5982	414	29	in	in	ADP
ejpam-5982	414	30	the	the	DET
ejpam-5982	414	31	setting	setting	NOUN
ejpam-5982	414	32	of	of	ADP
ejpam-5982	414	33	bipolar	bipolar	ADJ
ejpam-5982	414	34	metric	metric	ADJ
ejpam-5982	414	35	space	space	NOUN
ejpam-5982	414	36	generalising	generalise	VERB
ejpam-5982	414	37	some	some	DET
ejpam-5982	414	38	famous	famous	ADJ
ejpam-5982	414	39	results	result	NOUN
ejpam-5982	414	40	of	of	ADP
ejpam-5982	414	41	kannan	kannan	PROPN
ejpam-5982	414	42	[	[	X
ejpam-5982	414	43	17	17	NUM
ejpam-5982	414	44	]	]	PUNCT
ejpam-5982	414	45	,	,	PUNCT
ejpam-5982	414	46	reich	reich	PROPN
ejpam-5982	415	1	[	[	X
ejpam-5982	415	2	18	18	NUM
ejpam-5982	415	3	]	]	PUNCT
ejpam-5982	415	4	and	and	CCONJ
ejpam-5982	415	5	gaba	gaba	PROPN
ejpam-5982	415	6	[	[	X
ejpam-5982	415	7	16	16	NUM
ejpam-5982	415	8	]	]	PUNCT
ejpam-5982	415	9	have	have	AUX
ejpam-5982	415	10	been	be	AUX
ejpam-5982	415	11	proven	prove	VERB
ejpam-5982	415	12	.	.	PUNCT
ejpam-5982	416	1	suitable	suitable	ADJ
ejpam-5982	416	2	non	non	ADJ
ejpam-5982	416	3	-	-	ADJ
ejpam-5982	416	4	trivial	trivial	ADJ
ejpam-5982	416	5	examples	example	NOUN
ejpam-5982	416	6	have	have	AUX
ejpam-5982	416	7	been	be	AUX
ejpam-5982	416	8	provided	provide	VERB
ejpam-5982	416	9	in	in	ADP
ejpam-5982	416	10	support	support	NOUN
ejpam-5982	416	11	of	of	ADP
ejpam-5982	416	12	the	the	DET
ejpam-5982	416	13	derived	derive	VERB
ejpam-5982	416	14	results	result	NOUN
ejpam-5982	416	15	.	.	PUNCT
ejpam-5982	417	1	it	it	PRON
ejpam-5982	417	2	will	will	AUX
ejpam-5982	417	3	be	be	AUX
ejpam-5982	417	4	an	an	DET
ejpam-5982	417	5	open	open	ADJ
ejpam-5982	417	6	problem	problem	NOUN
ejpam-5982	417	7	to	to	PART
ejpam-5982	417	8	find	find	VERB
ejpam-5982	417	9	some	some	DET
ejpam-5982	417	10	applications	application	NOUN
ejpam-5982	417	11	to	to	PART
ejpam-5982	417	12	examine	examine	VERB
ejpam-5982	417	13	the	the	DET
ejpam-5982	417	14	existence	existence	NOUN
ejpam-5982	417	15	and	and	CCONJ
ejpam-5982	417	16	uniqueness	uniqueness	NOUN
ejpam-5982	417	17	of	of	ADP
ejpam-5982	417	18	solutions	solution	NOUN
ejpam-5982	417	19	to	to	PART
ejpam-5982	417	20	differential	differential	ADJ
ejpam-5982	417	21	equations	equation	NOUN
ejpam-5982	417	22	,	,	PUNCT
ejpam-5982	417	23	integral	integral	ADJ
ejpam-5982	417	24	equations	equation	NOUN
ejpam-5982	417	25	and	and	CCONJ
ejpam-5982	417	26	also	also	ADV
ejpam-5982	417	27	extend	extend	VERB
ejpam-5982	417	28	the	the	DET
ejpam-5982	417	29	proven	prove	VERB
ejpam-5982	417	30	results	result	NOUN
ejpam-5982	417	31	using	use	VERB
ejpam-5982	417	32	generalised	generalise	VERB
ejpam-5982	417	33	contractive	contractive	ADJ
ejpam-5982	417	34	conditions	condition	NOUN
ejpam-5982	417	35	.	.	PUNCT
ejpam-5982	418	1	acknowledgements	acknowledgement	NOUN
ejpam-5982	418	2	this	this	DET
ejpam-5982	418	3	study	study	NOUN
ejpam-5982	418	4	is	be	AUX
ejpam-5982	418	5	supported	support	VERB
ejpam-5982	418	6	via	via	ADP
ejpam-5982	418	7	funding	funding	NOUN
ejpam-5982	418	8	from	from	ADP
ejpam-5982	418	9	prince	prince	PROPN
ejpam-5982	418	10	sattam	sattam	PROPN
ejpam-5982	418	11	bin	bin	PROPN
ejpam-5982	418	12	abdulaziz	abdulaziz	PROPN
ejpam-5982	418	13	university	university	PROPN
ejpam-5982	418	14	project	project	NOUN
ejpam-5982	418	15	number	number	NOUN
ejpam-5982	418	16	(	(	PUNCT
ejpam-5982	418	17	psau/2025	psau/2025	NOUN
ejpam-5982	418	18	/	/	SYM
ejpam-5982	418	19	r/1446	r/1446	PROPN
ejpam-5982	418	20	)	)	PUNCT
ejpam-5982	418	21	.	.	PUNCT
ejpam-5982	419	1	conflict	conflict	NOUN
ejpam-5982	419	2	of	of	ADP
ejpam-5982	419	3	interest	interest	NOUN
ejpam-5982	419	4	the	the	DET
ejpam-5982	419	5	authors	author	NOUN
ejpam-5982	419	6	declare	declare	VERB
ejpam-5982	419	7	no	no	DET
ejpam-5982	419	8	conflict	conflict	NOUN
ejpam-5982	419	9	of	of	ADP
ejpam-5982	419	10	interest	interest	NOUN
ejpam-5982	419	11	.	.	PUNCT
ejpam-5982	420	1	references	reference	NOUN
ejpam-5982	420	2	[	[	X
ejpam-5982	420	3	1	1	X
ejpam-5982	420	4	]	]	PUNCT
ejpam-5982	420	5	s.	s.	PROPN
ejpam-5982	420	6	banach	banach	PROPN
ejpam-5982	420	7	.	.	PUNCT
ejpam-5982	421	1	sur	sur	PROPN
ejpam-5982	421	2	les	les	X
ejpam-5982	421	3	opérations	opération	NOUN
ejpam-5982	421	4	dans	dan	NOUN
ejpam-5982	421	5	les	les	X
ejpam-5982	421	6	ensembles	ensemble	NOUN
ejpam-5982	421	7	abstraits	abstrait	NOUN
ejpam-5982	421	8	et	et	PROPN
ejpam-5982	421	9	leur	leur	X
ejpam-5982	421	10	application	application	PROPN
ejpam-5982	421	11	aux	aux	PROPN
ejpam-5982	421	12	équations	équations	PROPN
ejpam-5982	421	13	intégrales	intégrale	NOUN
ejpam-5982	421	14	.	.	PUNCT
ejpam-5982	422	1	fund	fund	PROPN
ejpam-5982	422	2	.	.	PUNCT
ejpam-5982	423	1	math	math	NOUN
ejpam-5982	423	2	.	.	PUNCT
ejpam-5982	423	3	,	,	PUNCT
ejpam-5982	424	1	3:133–181	3:133–181	NUM
ejpam-5982	424	2	,	,	PUNCT
ejpam-5982	424	3	1922	1922	NUM
ejpam-5982	424	4	.	.	PUNCT
ejpam-5982	425	1	[	[	X
ejpam-5982	425	2	2	2	X
ejpam-5982	425	3	]	]	X
ejpam-5982	425	4	g.	g.	PROPN
ejpam-5982	425	5	jungck	jungck	PROPN
ejpam-5982	425	6	.	.	PUNCT
ejpam-5982	426	1	compatible	compatible	ADJ
ejpam-5982	426	2	mappings	mapping	NOUN
ejpam-5982	426	3	and	and	CCONJ
ejpam-5982	426	4	common	common	ADJ
ejpam-5982	426	5	fixed	fix	VERB
ejpam-5982	426	6	points	point	NOUN
ejpam-5982	426	7	.	.	PUNCT
ejpam-5982	427	1	int	int	NOUN
ejpam-5982	427	2	.	.	PUNCT
ejpam-5982	428	1	j.	j.	PROPN
ejpam-5982	428	2	math	math	PROPN
ejpam-5982	428	3	.	.	PUNCT
ejpam-5982	429	1	math	math	NOUN
ejpam-5982	429	2	.	.	PUNCT
ejpam-5982	430	1	sci	sci	PROPN
ejpam-5982	430	2	.	.	PROPN
ejpam-5982	430	3	,	,	PUNCT
ejpam-5982	430	4	9(4):771–779	9(4):771–779	NUM
ejpam-5982	430	5	,	,	PUNCT
ejpam-5982	430	6	1986	1986	NUM
ejpam-5982	430	7	.	.	PUNCT
ejpam-5982	431	1	[	[	X
ejpam-5982	431	2	3	3	X
ejpam-5982	431	3	]	]	X
ejpam-5982	431	4	g.	g.	PROPN
ejpam-5982	431	5	jungck	jungck	PROPN
ejpam-5982	431	6	.	.	PUNCT
ejpam-5982	432	1	compatible	compatible	ADJ
ejpam-5982	432	2	mappings	mapping	NOUN
ejpam-5982	432	3	and	and	CCONJ
ejpam-5982	432	4	common	common	ADJ
ejpam-5982	432	5	fixed	fix	VERB
ejpam-5982	432	6	points	point	NOUN
ejpam-5982	432	7	.	.	PUNCT
ejpam-5982	433	1	int	int	NOUN
ejpam-5982	433	2	.	.	PUNCT
ejpam-5982	434	1	j.	j.	PROPN
ejpam-5982	434	2	math	math	PROPN
ejpam-5982	434	3	.	.	PUNCT
ejpam-5982	435	1	math	math	NOUN
ejpam-5982	435	2	.	.	PUNCT
ejpam-5982	436	1	sci	sci	PROPN
ejpam-5982	436	2	.	.	PROPN
ejpam-5982	436	3	,	,	PUNCT
ejpam-5982	436	4	9(4):771–779	9(4):771–779	NUM
ejpam-5982	436	5	,	,	PUNCT
ejpam-5982	436	6	1986	1986	NUM
ejpam-5982	436	7	.	.	PUNCT
ejpam-5982	437	1	[	[	X
ejpam-5982	437	2	4	4	X
ejpam-5982	437	3	]	]	PUNCT
ejpam-5982	437	4	v.	v.	CCONJ
ejpam-5982	437	5	popa	popa	NOUN
ejpam-5982	437	6	.	.	PUNCT
ejpam-5982	438	1	common	common	ADJ
ejpam-5982	438	2	fixed	fix	VERB
ejpam-5982	438	3	point	point	NOUN
ejpam-5982	438	4	theorems	theorem	NOUN
ejpam-5982	438	5	for	for	ADP
ejpam-5982	438	6	compatible	compatible	ADJ
ejpam-5982	438	7	mappings	mapping	NOUN
ejpam-5982	438	8	of	of	ADP
ejpam-5982	438	9	type	type	NOUN
ejpam-5982	438	10	(	(	PUNCT
ejpam-5982	438	11	a	a	X
ejpam-5982	438	12	)	)	PUNCT
ejpam-5982	438	13	satisfying	satisfy	VERB
ejpam-5982	438	14	an	an	DET
ejpam-5982	438	15	implicit	implicit	ADJ
ejpam-5982	438	16	relation	relation	NOUN
ejpam-5982	438	17	.	.	PUNCT
ejpam-5982	439	1	stud	stud	PROPN
ejpam-5982	439	2	.	.	PUNCT
ejpam-5982	440	1	cercet	cercet	PROPN
ejpam-5982	440	2	.	.	PUNCT
ejpam-5982	441	1	stiint	stiint	PROPN
ejpam-5982	441	2	.	.	PUNCT
ejpam-5982	442	1	ser	ser	PROPN
ejpam-5982	442	2	.	.	PROPN
ejpam-5982	442	3	mat	mat	PROPN
ejpam-5982	442	4	.	.	PROPN
ejpam-5982	442	5	univ	univ	PROPN
ejpam-5982	442	6	.	.	PUNCT
ejpam-5982	442	7	bacau	bacau	PROPN
ejpam-5982	442	8	,	,	PUNCT
ejpam-5982	442	9	9:165–172	9:165–172	NUM
ejpam-5982	442	10	,	,	PUNCT
ejpam-5982	442	11	1999	1999	NUM
ejpam-5982	442	12	.	.	PUNCT
ejpam-5982	443	1	[	[	X
ejpam-5982	443	2	5	5	X
ejpam-5982	443	3	]	]	PUNCT
ejpam-5982	443	4	m.	m.	NOUN
ejpam-5982	443	5	aamri	aamri	PROPN
ejpam-5982	443	6	and	and	CCONJ
ejpam-5982	443	7	d.	d.	PROPN
ejpam-5982	443	8	el	el	PROPN
ejpam-5982	443	9	moutawakil	moutawakil	PROPN
ejpam-5982	443	10	.	.	PUNCT
ejpam-5982	444	1	some	some	DET
ejpam-5982	444	2	new	new	ADJ
ejpam-5982	444	3	common	common	ADJ
ejpam-5982	444	4	fixed	fix	VERB
ejpam-5982	444	5	point	point	NOUN
ejpam-5982	444	6	theorems	theorem	NOUN
ejpam-5982	444	7	under	under	ADP
ejpam-5982	444	8	strict	strict	ADJ
ejpam-5982	444	9	contractive	contractive	ADJ
ejpam-5982	444	10	conditions	condition	NOUN
ejpam-5982	444	11	.	.	PUNCT
ejpam-5982	445	1	j.	j.	PROPN
ejpam-5982	445	2	math	math	PROPN
ejpam-5982	445	3	.	.	PUNCT
ejpam-5982	446	1	anal	anal	PROPN
ejpam-5982	446	2	.	.	PUNCT
ejpam-5982	447	1	appl	appl	PROPN
ejpam-5982	447	2	.	.	PROPN
ejpam-5982	447	3	,	,	PUNCT
ejpam-5982	448	1	27:181–188	27:181–188	NUM
ejpam-5982	448	2	,	,	PUNCT
ejpam-5982	448	3	2002	2002	NUM
ejpam-5982	448	4	.	.	PUNCT
ejpam-5982	449	1	[	[	X
ejpam-5982	449	2	6	6	NUM
ejpam-5982	449	3	]	]	PUNCT
ejpam-5982	449	4	a.	a.	NOUN
ejpam-5982	449	5	mutulu	mutulu	PROPN
ejpam-5982	449	6	and	and	CCONJ
ejpam-5982	449	7	u.	u.	PROPN
ejpam-5982	449	8	gürdal	gürdal	PROPN
ejpam-5982	449	9	.	.	PUNCT
ejpam-5982	450	1	bipoloar	bipoloar	PROPN
ejpam-5982	450	2	metric	metric	ADJ
ejpam-5982	450	3	spaces	space	NOUN
ejpam-5982	450	4	and	and	CCONJ
ejpam-5982	450	5	some	some	DET
ejpam-5982	450	6	fixed	fix	VERB
ejpam-5982	450	7	point	point	NOUN
ejpam-5982	450	8	problems	problem	NOUN
ejpam-5982	450	9	.	.	PUNCT
ejpam-5982	451	1	j.	j.	PROPN
ejpam-5982	451	2	nonlinear	nonlinear	PROPN
ejpam-5982	451	3	sci	sci	PROPN
ejpam-5982	451	4	.	.	PUNCT
ejpam-5982	451	5	appl	appl	PROPN
ejpam-5982	451	6	.	.	PROPN
ejpam-5982	451	7	,	,	PUNCT
ejpam-5982	451	8	9(9):5362–5373	9(9):5362–5373	NUM
ejpam-5982	451	9	,	,	PUNCT
ejpam-5982	451	10	2016	2016	NUM
ejpam-5982	451	11	.	.	PUNCT
ejpam-5982	452	1	[	[	X
ejpam-5982	452	2	7	7	X
ejpam-5982	452	3	]	]	X
ejpam-5982	452	4	b.	b.	PROPN
ejpam-5982	452	5	s.	s.	PROPN
ejpam-5982	452	6	rao	rao	PROPN
ejpam-5982	452	7	and	and	CCONJ
ejpam-5982	452	8	g.	g.	PROPN
ejpam-5982	452	9	n.	n.	PROPN
ejpam-5982	452	10	v.	v.	PROPN
ejpam-5982	452	11	kishore	kishore	PROPN
ejpam-5982	452	12	.	.	PUNCT
ejpam-5982	453	1	common	common	ADJ
ejpam-5982	453	2	fixed	fix	VERB
ejpam-5982	453	3	point	point	NOUN
ejpam-5982	453	4	theorems	theorem	NOUN
ejpam-5982	453	5	in	in	ADP
ejpam-5982	453	6	bipolar	bipolar	ADJ
ejpam-5982	453	7	metric	metric	ADJ
ejpam-5982	453	8	spaces	space	NOUN
ejpam-5982	453	9	with	with	ADP
ejpam-5982	453	10	applications	application	NOUN
ejpam-5982	453	11	to	to	ADP
ejpam-5982	453	12	integral	integral	ADJ
ejpam-5982	453	13	equations	equation	NOUN
ejpam-5982	453	14	.	.	PUNCT
ejpam-5982	454	1	int	int	NOUN
ejpam-5982	454	2	.	.	PUNCT
ejpam-5982	455	1	j.	j.	PROPN
ejpam-5982	455	2	eng	eng	PROPN
ejpam-5982	455	3	.	.	PROPN
ejpam-5982	456	1	technol	technol	PROPN
ejpam-5982	456	2	.	.	PROPN
ejpam-5982	456	3	,	,	PUNCT
ejpam-5982	456	4	7:1022–1026	7:1022–1026	PROPN
ejpam-5982	456	5	,	,	PUNCT
ejpam-5982	456	6	2018	2018	NUM
ejpam-5982	456	7	.	.	PUNCT
ejpam-5982	457	1	[	[	X
ejpam-5982	457	2	8	8	NUM
ejpam-5982	457	3	]	]	X
ejpam-5982	457	4	g.	g.	PROPN
ejpam-5982	457	5	mani	mani	PROPN
ejpam-5982	457	6	,	,	PUNCT
ejpam-5982	457	7	r.	r.	PROPN
ejpam-5982	457	8	ramaswamy	ramaswamy	PROPN
ejpam-5982	457	9	,	,	PUNCT
ejpam-5982	457	10	a.	a.	PROPN
ejpam-5982	457	11	j.	j.	PROPN
ejpam-5982	457	12	gnanaprakasam	gnanaprakasam	PROPN
ejpam-5982	457	13	,	,	PUNCT
ejpam-5982	457	14	a.	a.	NOUN
ejpam-5982	457	15	elsonbaty	elsonbaty	NOUN
ejpam-5982	457	16	,	,	PUNCT
ejpam-5982	457	17	o.	o.	PROPN
ejpam-5982	457	18	a.	a.	PROPN
ejpam-5982	457	19	a.	a.	PROPN
ejpam-5982	457	20	abdelnaby	abdelnaby	PROPN
ejpam-5982	457	21	,	,	PUNCT
ejpam-5982	457	22	and	and	CCONJ
ejpam-5982	457	23	s.	s.	PROPN
ejpam-5982	458	1	radenović.	radenović.	PROPN
ejpam-5982	458	2	application	application	NOUN
ejpam-5982	458	3	of	of	ADP
ejpam-5982	458	4	fixed	fix	VERB
ejpam-5982	458	5	points	point	NOUN
ejpam-5982	458	6	in	in	ADP
ejpam-5982	458	7	bipolar	bipolar	ADJ
ejpam-5982	458	8	controlled	control	VERB
ejpam-5982	458	9	metric	metric	ADJ
ejpam-5982	458	10	space	space	NOUN
ejpam-5982	458	11	to	to	PART
ejpam-5982	458	12	solve	solve	VERB
ejpam-5982	458	13	fractional	fractional	ADJ
ejpam-5982	458	14	differential	differential	NOUN
ejpam-5982	458	15	equation	equation	NOUN
ejpam-5982	458	16	.	.	PUNCT
ejpam-5982	459	1	fractal	fractal	ADJ
ejpam-5982	459	2	and	and	CCONJ
ejpam-5982	459	3	fractional	fractional	ADJ
ejpam-5982	459	4	,	,	PUNCT
ejpam-5982	459	5	7(3):doi.org/10.3390	7(3):doi.org/10.3390	NUM
ejpam-5982	459	6	/	/	SYM
ejpam-5982	459	7	fractalfract7030242	fractalfract7030242	PROPN
ejpam-5982	459	8	,	,	PUNCT
ejpam-5982	459	9	2023	2023	NUM
ejpam-5982	459	10	.	.	PUNCT
ejpam-5982	460	1	[	[	X
ejpam-5982	460	2	9	9	NUM
ejpam-5982	460	3	]	]	PUNCT
ejpam-5982	460	4	k.	k.	PROPN
ejpam-5982	460	5	özkan	özkan	PROPN
ejpam-5982	460	6	,	,	PUNCT
ejpam-5982	460	7	u.	u.	PROPN
ejpam-5982	460	8	gürdal	gürdal	NOUN
ejpam-5982	460	9	,	,	PUNCT
ejpam-5982	460	10	and	and	CCONJ
ejpam-5982	460	11	a.	a.	NOUN
ejpam-5982	460	12	mutlu	mutlu	PROPN
ejpam-5982	460	13	.	.	PUNCT
ejpam-5982	461	1	caristi	caristi	PROPN
ejpam-5982	461	2	’s	’s	PART
ejpam-5982	461	3	and	and	CCONJ
ejpam-5982	461	4	downing	downing	NOUN
ejpam-5982	461	5	-	-	PUNCT
ejpam-5982	461	6	kirk	kirk	NOUN
ejpam-5982	461	7	’s	’s	PART
ejpam-5982	461	8	fixed	fix	VERB
ejpam-5982	461	9	point	point	NOUN
ejpam-5982	461	10	theorems	theorem	NOUN
ejpam-5982	461	11	on	on	ADP
ejpam-5982	461	12	bipolar	bipolar	ADJ
ejpam-5982	461	13	metric	metric	ADJ
ejpam-5982	461	14	spaces	space	NOUN
ejpam-5982	461	15	.	.	PUNCT
ejpam-5982	462	1	fixed	fix	VERB
ejpam-5982	462	2	point	point	NOUN
ejpam-5982	462	3	theory	theory	NOUN
ejpam-5982	462	4	,	,	PUNCT
ejpam-5982	462	5	22(2):785–794	22(2):785–794	PROPN
ejpam-5982	462	6	,	,	PUNCT
ejpam-5982	462	7	2021	2021	NUM
ejpam-5982	462	8	.	.	PUNCT
ejpam-5982	463	1	[	[	X
ejpam-5982	463	2	10	10	NUM
ejpam-5982	463	3	]	]	X
ejpam-5982	463	4	g.	g.	PROPN
ejpam-5982	463	5	mani	mani	PROPN
ejpam-5982	463	6	,	,	PUNCT
ejpam-5982	463	7	r.	r.	PROPN
ejpam-5982	463	8	ramaswamy	ramaswamy	PROPN
ejpam-5982	463	9	,	,	PUNCT
ejpam-5982	463	10	a.	a.	PROPN
ejpam-5982	463	11	j.	j.	PROPN
ejpam-5982	463	12	gnanaprakasam	gnanaprakasam	PROPN
ejpam-5982	463	13	,	,	PUNCT
ejpam-5982	463	14	v.	v.	ADP
ejpam-5982	463	15	stojiljković	stojiljković	NOUN
ejpam-5982	463	16	,	,	PUNCT
ejpam-5982	463	17	z.	z.	PROPN
ejpam-5982	463	18	m.	m.	PROPN
ejpam-5982	463	19	fadail	fadail	PROPN
ejpam-5982	463	20	,	,	PUNCT
ejpam-5982	463	21	and	and	CCONJ
ejpam-5982	463	22	s.	s.	PROPN
ejpam-5982	464	1	radenović.	radenović.	PROPN
ejpam-5982	464	2	application	application	NOUN
ejpam-5982	464	3	of	of	ADP
ejpam-5982	464	4	fixed	fix	VERB
ejpam-5982	464	5	point	point	NOUN
ejpam-5982	464	6	results	result	NOUN
ejpam-5982	464	7	in	in	ADP
ejpam-5982	464	8	the	the	DET
ejpam-5982	464	9	setting	setting	NOUN
ejpam-5982	464	10	of	of	ADP
ejpam-5982	464	11	f	f	NOUN
ejpam-5982	464	12	-	-	PUNCT
ejpam-5982	464	13	contraction	contraction	NOUN
ejpam-5982	464	14	and	and	CCONJ
ejpam-5982	464	15	simp	simp	ADJ
ejpam-5982	464	16	.	.	PUNCT
ejpam-5982	465	1	p.	p.	NOUN
ejpam-5982	465	2	murthy	murthy	PROPN
ejpam-5982	466	1	et	et	PROPN
ejpam-5982	466	2	al	al	PROPN
ejpam-5982	466	3	.	.	PUNCT
ejpam-5982	466	4	/	/	SYM
ejpam-5982	466	5	eur	eur	PROPN
ejpam-5982	466	6	.	.	PUNCT
ejpam-5982	467	1	j.	j.	PROPN
ejpam-5982	467	2	pure	pure	PROPN
ejpam-5982	467	3	appl	appl	PROPN
ejpam-5982	467	4	.	.	PROPN
ejpam-5982	467	5	math	math	PROPN
ejpam-5982	467	6	,	,	PUNCT
ejpam-5982	467	7	18	18	NUM
ejpam-5982	467	8	(	(	PUNCT
ejpam-5982	467	9	2	2	NUM
ejpam-5982	467	10	)	)	PUNCT
ejpam-5982	467	11	(	(	PUNCT
ejpam-5982	467	12	2025	2025	NUM
ejpam-5982	467	13	)	)	PUNCT
ejpam-5982	467	14	,	,	PUNCT
ejpam-5982	467	15	5982	5982	NUM
ejpam-5982	467	16	17	17	NUM
ejpam-5982	467	17	of	of	ADP
ejpam-5982	467	18	17	17	NUM
ejpam-5982	467	19	ulation	ulation	NOUN
ejpam-5982	467	20	function	function	NOUN
ejpam-5982	467	21	in	in	ADP
ejpam-5982	467	22	the	the	DET
ejpam-5982	467	23	setting	setting	NOUN
ejpam-5982	467	24	of	of	ADP
ejpam-5982	467	25	bipolar	bipolar	ADJ
ejpam-5982	467	26	metric	metric	ADJ
ejpam-5982	467	27	space	space	NOUN
ejpam-5982	467	28	.	.	PUNCT
ejpam-5982	468	1	aims	aim	VERB
ejpam-5982	468	2	mathematics	mathematic	NOUN
ejpam-5982	468	3	,	,	PUNCT
ejpam-5982	468	4	8(2):3269	8(2):3269	NUM
ejpam-5982	468	5	–	–	PUNCT
ejpam-5982	468	6	3285	3285	NUM
ejpam-5982	468	7	,	,	PUNCT
ejpam-5982	468	8	2023	2023	NUM
ejpam-5982	468	9	.	.	PUNCT
ejpam-5982	469	1	[	[	X
ejpam-5982	469	2	11	11	NUM
ejpam-5982	469	3	]	]	X
ejpam-5982	469	4	g.	g.	PROPN
ejpam-5982	469	5	mani	mani	PROPN
ejpam-5982	469	6	,	,	PUNCT
ejpam-5982	469	7	s.	s.	PROPN
ejpam-5982	469	8	s.	s.	PROPN
ejpam-5982	469	9	ramulu	ramulu	PROPN
ejpam-5982	469	10	,	,	PUNCT
ejpam-5982	469	11	s.	s.	PROPN
ejpam-5982	469	12	aljohani	aljohani	PROPN
ejpam-5982	469	13	,	,	PUNCT
ejpam-5982	469	14	z.	z.	PROPN
ejpam-5982	469	15	d.	d.	PROPN
ejpam-5982	469	16	mitrović	mitrović	PROPN
ejpam-5982	469	17	,	,	PUNCT
ejpam-5982	469	18	and	and	CCONJ
ejpam-5982	469	19	n.	n.	PROPN
ejpam-5982	469	20	mlaiki	mlaiki	PROPN
ejpam-5982	469	21	.	.	PUNCT
ejpam-5982	470	1	results	result	NOUN
ejpam-5982	470	2	on	on	ADP
ejpam-5982	470	3	fixed	fix	VERB
ejpam-5982	470	4	points	point	NOUN
ejpam-5982	470	5	and	and	CCONJ
ejpam-5982	470	6	common	common	ADJ
ejpam-5982	470	7	fixed	fix	VERB
ejpam-5982	470	8	points	point	NOUN
ejpam-5982	470	9	on	on	ADP
ejpam-5982	470	10	bipolar	bipolar	ADJ
ejpam-5982	470	11	b	b	NOUN
ejpam-5982	470	12	-	-	PUNCT
ejpam-5982	470	13	metric	metric	ADJ
ejpam-5982	470	14	space	space	NOUN
ejpam-5982	470	15	with	with	ADP
ejpam-5982	470	16	applications	application	NOUN
ejpam-5982	470	17	.	.	PUNCT
ejpam-5982	471	1	j.	j.	PROPN
ejpam-5982	471	2	math	math	PROPN
ejpam-5982	471	3	.	.	PUNCT
ejpam-5982	472	1	computer	computer	PROPN
ejpam-5982	472	2	sci	sci	PROPN
ejpam-5982	472	3	.	.	PROPN
ejpam-5982	472	4	,	,	PUNCT
ejpam-5982	472	5	37:274–2865	37:274–2865	NUM
ejpam-5982	472	6	,	,	PUNCT
ejpam-5982	472	7	2025	2025	NUM
ejpam-5982	472	8	.	.	PUNCT
ejpam-5982	473	1	[	[	X
ejpam-5982	473	2	12	12	NUM
ejpam-5982	473	3	]	]	PUNCT
ejpam-5982	473	4	m.	m.	NOUN
ejpam-5982	473	5	kumar	kumar	PROPN
ejpam-5982	473	6	,	,	PUNCT
ejpam-5982	473	7	p.	p.	PROPN
ejpam-5982	473	8	kumar	kumar	PROPN
ejpam-5982	473	9	,	,	PUNCT
ejpam-5982	473	10	a.	a.	PROPN
ejpam-5982	473	11	mutlu	mutlu	PROPN
ejpam-5982	473	12	,	,	PUNCT
ejpam-5982	473	13	r.	r.	PROPN
ejpam-5982	473	14	ramaswamy	ramaswamy	PROPN
ejpam-5982	473	15	,	,	PUNCT
ejpam-5982	473	16	o.	o.	PROPN
ejpam-5982	473	17	a.	a.	PROPN
ejpam-5982	473	18	a.	a.	PROPN
ejpam-5982	473	19	abdelnaby	abdelnaby	PROPN
ejpam-5982	473	20	,	,	PUNCT
ejpam-5982	473	21	and	and	CCONJ
ejpam-5982	473	22	s.	s.	PROPN
ejpam-5982	474	1	radenović.	radenović.	PROPN
ejpam-5982	474	2	ulam	ulam	NOUN
ejpam-5982	474	3	-	-	PUNCT
ejpam-5982	474	4	hyers	hyer	NOUN
ejpam-5982	474	5	stability	stability	NOUN
ejpam-5982	474	6	and	and	CCONJ
ejpam-5982	474	7	well	well	ADV
ejpam-5982	474	8	-	-	PUNCT
ejpam-5982	474	9	posedness	posedness	NOUN
ejpam-5982	474	10	of	of	ADP
ejpam-5982	474	11	fixed	fix	VERB
ejpam-5982	474	12	point	point	NOUN
ejpam-5982	474	13	problems	problem	NOUN
ejpam-5982	474	14	in	in	ADP
ejpam-5982	474	15	c*-algebra	c*-algebra	PROPN
ejpam-5982	474	16	valued	value	VERB
ejpam-5982	474	17	bipolar	bipolar	ADJ
ejpam-5982	474	18	b	b	NOUN
ejpam-5982	474	19	-	-	PUNCT
ejpam-5982	474	20	metric	metric	ADJ
ejpam-5982	474	21	spaces	space	NOUN
ejpam-5982	474	22	.	.	PUNCT
ejpam-5982	475	1	mathematics	mathematic	NOUN
ejpam-5982	475	2	,	,	PUNCT
ejpam-5982	475	3	11(10):doi.org/10.3390	11(10):doi.org/10.3390	NUM
ejpam-5982	475	4	/	/	SYM
ejpam-5982	475	5	math11102323	math11102323	PROPN
ejpam-5982	475	6	,	,	PUNCT
ejpam-5982	475	7	2023	2023	NUM
ejpam-5982	475	8	.	.	PUNCT
ejpam-5982	476	1	[	[	X
ejpam-5982	476	2	13	13	NUM
ejpam-5982	476	3	]	]	PUNCT
ejpam-5982	476	4	m.	m.	NOUN
ejpam-5982	476	5	kumar	kumar	PROPN
ejpam-5982	476	6	,	,	PUNCT
ejpam-5982	476	7	p.	p.	PROPN
ejpam-5982	476	8	kumar	kumar	PROPN
ejpam-5982	476	9	,	,	PUNCT
ejpam-5982	476	10	r.	r.	PROPN
ejpam-5982	476	11	ramaswamy	ramaswamy	PROPN
ejpam-5982	476	12	,	,	PUNCT
ejpam-5982	476	13	o.	o.	PROPN
ejpam-5982	476	14	a.	a.	PROPN
ejpam-5982	476	15	a.	a.	PROPN
ejpam-5982	476	16	abdelnaby	abdelnaby	PROPN
ejpam-5982	476	17	,	,	PUNCT
ejpam-5982	476	18	and	and	CCONJ
ejpam-5982	476	19	s.	s.	PROPN
ejpam-5982	476	20	radenović.	radenović.	PROPN
ejpam-5982	476	21	(	(	PUNCT
ejpam-5982	476	22	α	α	NOUN
ejpam-5982	476	23	−	−	NOUN
ejpam-5982	476	24	ψ	ψ	NOUN
ejpam-5982	476	25	)	)	PUNCT
ejpam-5982	476	26	meir	meir	PROPN
ejpam-5982	476	27	–	–	PUNCT
ejpam-5982	476	28	keeler	keeler	NOUN
ejpam-5982	476	29	contractions	contraction	NOUN
ejpam-5982	476	30	in	in	ADP
ejpam-5982	476	31	bipolar	bipolar	ADJ
ejpam-5982	476	32	metric	metric	ADJ
ejpam-5982	476	33	spaces	space	NOUN
ejpam-5982	476	34	.	.	PUNCT
ejpam-5982	477	1	mathematics	mathematic	NOUN
ejpam-5982	477	2	,	,	PUNCT
ejpam-5982	477	3	11(6):doi.org/10.3390	11(6):doi.org/10.3390	NUM
ejpam-5982	477	4	/	/	SYM
ejpam-5982	477	5	math11061310	math11061310	PROPN
ejpam-5982	477	6	,	,	PUNCT
ejpam-5982	477	7	2023	2023	NUM
ejpam-5982	477	8	.	.	PUNCT
ejpam-5982	478	1	[	[	X
ejpam-5982	478	2	14	14	NUM
ejpam-5982	478	3	]	]	PUNCT
ejpam-5982	479	1	p.	p.	NOUN
ejpam-5982	479	2	p.	p.	NOUN
ejpam-5982	480	1	murthy	murthy	PROPN
ejpam-5982	480	2	,	,	PUNCT
ejpam-5982	480	3	z.	z.	PROPN
ejpam-5982	480	4	mitrović	mitrović	PROPN
ejpam-5982	480	5	,	,	PUNCT
ejpam-5982	480	6	c.	c.	PROPN
ejpam-5982	480	7	p.	p.	PROPN
ejpam-5982	480	8	dhuri	dhuri	PROPN
ejpam-5982	480	9	,	,	PUNCT
ejpam-5982	480	10	and	and	CCONJ
ejpam-5982	480	11	s.	s.	PROPN
ejpam-5982	481	1	radenović.	radenović.	PROPN
ejpam-5982	481	2	the	the	DET
ejpam-5982	481	3	common	common	ADJ
ejpam-5982	481	4	fixed	fix	VERB
ejpam-5982	481	5	point	point	NOUN
ejpam-5982	481	6	theorems	theorem	NOUN
ejpam-5982	481	7	in	in	ADP
ejpam-5982	481	8	bipolar	bipolar	ADJ
ejpam-5982	481	9	metric	metric	ADJ
ejpam-5982	481	10	space	space	NOUN
ejpam-5982	481	11	.	.	PUNCT
ejpam-5982	482	1	gulf	gulf	PROPN
ejpam-5982	482	2	journal	journal	PROPN
ejpam-5982	482	3	of	of	ADP
ejpam-5982	482	4	mathematics	mathematic	NOUN
ejpam-5982	482	5	,	,	PUNCT
ejpam-5982	482	6	12(2):31–38	12(2):31–38	NUM
ejpam-5982	482	7	,	,	PUNCT
ejpam-5982	482	8	2022	2022	NUM
ejpam-5982	482	9	.	.	PUNCT
ejpam-5982	483	1	[	[	X
ejpam-5982	483	2	15	15	NUM
ejpam-5982	483	3	]	]	X
ejpam-5982	483	4	r.	r.	PROPN
ejpam-5982	483	5	ramaswamy	ramaswamy	PROPN
ejpam-5982	483	6	,	,	PUNCT
ejpam-5982	483	7	g.	g.	PROPN
ejpam-5982	483	8	mani	mani	PROPN
ejpam-5982	483	9	,	,	PUNCT
ejpam-5982	483	10	a.	a.	PROPN
ejpam-5982	483	11	j.	j.	PROPN
ejpam-5982	483	12	gnanaprakasam	gnanaprakasam	PROPN
ejpam-5982	483	13	,	,	PUNCT
ejpam-5982	483	14	o.	o.	PROPN
ejpam-5982	483	15	a.	a.	PROPN
ejpam-5982	483	16	a.	a.	PROPN
ejpam-5982	483	17	abdelnaby	abdelnaby	PROPN
ejpam-5982	483	18	,	,	PUNCT
ejpam-5982	483	19	v.	v.	ADP
ejpam-5982	483	20	stojiljković	stojiljković	PROPN
ejpam-5982	483	21	,	,	PUNCT
ejpam-5982	483	22	s.	s.	PROPN
ejpam-5982	483	23	radojević	radojević	PROPN
ejpam-5982	483	24	,	,	PUNCT
ejpam-5982	483	25	and	and	CCONJ
ejpam-5982	483	26	s.	s.	PROPN
ejpam-5982	484	1	radenović.	radenović.	PROPN
ejpam-5982	484	2	fixed	fix	VERB
ejpam-5982	484	3	points	point	NOUN
ejpam-5982	484	4	on	on	ADP
ejpam-5982	484	5	covariant	covariant	NOUN
ejpam-5982	484	6	and	and	CCONJ
ejpam-5982	484	7	contravariant	contravariant	PROPN
ejpam-5982	484	8	maps	map	NOUN
ejpam-5982	484	9	with	with	ADP
ejpam-5982	484	10	an	an	DET
ejpam-5982	484	11	application	application	NOUN
ejpam-5982	484	12	.	.	PUNCT
ejpam-5982	485	1	mathematics	mathematic	NOUN
ejpam-5982	485	2	,	,	PUNCT
ejpam-5982	485	3	10(22):https://doi.org/10.3390	10(22):https://doi.org/10.3390	NUM
ejpam-5982	485	4	/	/	SYM
ejpam-5982	485	5	math10224385	math10224385	PROPN
ejpam-5982	485	6	,	,	PUNCT
ejpam-5982	485	7	2022	2022	NUM
ejpam-5982	485	8	.	.	PUNCT
ejpam-5982	486	1	[	[	X
ejpam-5982	486	2	16	16	X
ejpam-5982	486	3	]	]	X
ejpam-5982	486	4	y.	y.	PROPN
ejpam-5982	486	5	u.	u.	PROPN
ejpam-5982	486	6	gaba	gaba	PROPN
ejpam-5982	486	7	,	,	PUNCT
ejpam-5982	486	8	m.	m.	NOUN
ejpam-5982	486	9	aphane	aphane	PROPN
ejpam-5982	486	10	,	,	PUNCT
ejpam-5982	486	11	and	and	CCONJ
ejpam-5982	486	12	h.	h.	PROPN
ejpam-5982	486	13	aydi	aydi	VERB
ejpam-5982	486	14	.	.	PUNCT
ejpam-5982	487	1	(	(	PUNCT
ejpam-5982	487	2	α	α	NOUN
ejpam-5982	487	3	,	,	PUNCT
ejpam-5982	487	4	bk)-contraction	bk)-contraction	NOUN
ejpam-5982	487	5	in	in	ADP
ejpam-5982	487	6	bipolar	bipolar	ADJ
ejpam-5982	487	7	metric	metric	ADJ
ejpam-5982	487	8	spaces	space	NOUN
ejpam-5982	487	9	.	.	PUNCT
ejpam-5982	488	1	journal	journal	NOUN
ejpam-5982	488	2	of	of	ADP
ejpam-5982	488	3	mathematics	mathematic	NOUN
ejpam-5982	488	4	,	,	PUNCT
ejpam-5982	488	5	2021	2021	NUM
ejpam-5982	488	6	:	:	PUNCT
ejpam-5982	488	7	doi.org/10.1155/2021/5562651	doi.org/10.1155/2021/5562651	PROPN
ejpam-5982	488	8	,	,	PUNCT
ejpam-5982	488	9	2021	2021	NUM
ejpam-5982	488	10	.	.	PUNCT
ejpam-5982	489	1	[	[	X
ejpam-5982	489	2	17	17	NUM
ejpam-5982	489	3	]	]	X
ejpam-5982	489	4	r.	r.	PROPN
ejpam-5982	489	5	kannan	kannan	PROPN
ejpam-5982	489	6	.	.	PUNCT
ejpam-5982	490	1	some	some	DET
ejpam-5982	490	2	results	result	NOUN
ejpam-5982	490	3	on	on	ADP
ejpam-5982	490	4	fixed	fix	VERB
ejpam-5982	490	5	points	point	NOUN
ejpam-5982	490	6	.	.	PUNCT
ejpam-5982	491	1	bull	bull	NOUN
ejpam-5982	491	2	.	.	PUNCT
ejpam-5982	492	1	calcutta	calcutta	PROPN
ejpam-5982	492	2	math	math	PROPN
ejpam-5982	492	3	.	.	PUNCT
ejpam-5982	493	1	soc	soc	PROPN
ejpam-5982	493	2	.	.	PUNCT
ejpam-5982	493	3	,	,	PUNCT
ejpam-5982	493	4	60:71–76	60:71–76	NUM
ejpam-5982	493	5	,	,	PUNCT
ejpam-5982	493	6	1968	1968	NUM
ejpam-5982	493	7	.	.	PUNCT
ejpam-5982	494	1	[	[	X
ejpam-5982	494	2	18	18	NUM
ejpam-5982	494	3	]	]	X
ejpam-5982	494	4	s.	s.	PROPN
ejpam-5982	494	5	reich	reich	PROPN
ejpam-5982	494	6	.	.	PUNCT
ejpam-5982	495	1	kannan	kannan	PROPN
ejpam-5982	495	2	’s	’s	PART
ejpam-5982	495	3	fixed	fix	VERB
ejpam-5982	495	4	point	point	NOUN
ejpam-5982	495	5	theorem	theorem	VERB
ejpam-5982	495	6	.	.	PROPN
ejpam-5982	495	7	boll	boll	PROPN
ejpam-5982	495	8	.	.	PUNCT
ejpam-5982	496	1	un	un	PROPN
ejpam-5982	496	2	.	.	PROPN
ejpam-5982	496	3	mat	mat	PROPN
ejpam-5982	496	4	.	.	PUNCT
ejpam-5982	496	5	ital	ital	PROPN
ejpam-5982	496	6	.	.	PROPN
ejpam-5982	496	7	,	,	PUNCT
ejpam-5982	496	8	4(4):1–11	4(4):1–11	NUM
ejpam-5982	496	9	,	,	PUNCT
ejpam-5982	496	10	1971	1971	NUM
ejpam-5982	496	11	.	.	PUNCT