id	sid	tid	token	lemma	pos
ejpam-4953	1	1	european	european	PROPN
ejpam-4953	1	2	journal	journal	PROPN
ejpam-4953	1	3	of	of	ADP
ejpam-4953	1	4	pure	pure	ADJ
ejpam-4953	1	5	and	and	CCONJ
ejpam-4953	1	6	applied	apply	VERB
ejpam-4953	1	7	mathematics	mathematic	NOUN
ejpam-4953	1	8	vol	vol	NOUN
ejpam-4953	1	9	.	.	PUNCT
ejpam-4953	2	1	16	16	NUM
ejpam-4953	2	2	,	,	PUNCT
ejpam-4953	2	3	no	no	INTJ
ejpam-4953	2	4	.	.	NOUN
ejpam-4953	2	5	4	4	NUM
ejpam-4953	2	6	,	,	PUNCT
ejpam-4953	2	7	2023	2023	NUM
ejpam-4953	2	8	,	,	PUNCT
ejpam-4953	2	9	2405	2405	NUM
ejpam-4953	2	10	-	-	SYM
ejpam-4953	2	11	2418	2418	NUM
ejpam-4953	2	12	issn	issn	VERB
ejpam-4953	2	13	1307	1307	NUM
ejpam-4953	2	14	-	-	SYM
ejpam-4953	2	15	5543	5543	NUM
ejpam-4953	2	16	–	–	PUNCT
ejpam-4953	2	17	ejpam.com	ejpam.com	X
ejpam-4953	2	18	published	publish	VERB
ejpam-4953	2	19	by	by	ADP
ejpam-4953	2	20	new	new	PROPN
ejpam-4953	2	21	york	york	PROPN
ejpam-4953	2	22	business	business	PROPN
ejpam-4953	2	23	global	global	PROPN
ejpam-4953	2	24	metrical	metrical	ADJ
ejpam-4953	2	25	fixed	fix	VERB
ejpam-4953	2	26	point	point	NOUN
ejpam-4953	2	27	results	result	NOUN
ejpam-4953	2	28	on	on	ADP
ejpam-4953	2	29	b	b	NOUN
ejpam-4953	2	30	-	-	PUNCT
ejpam-4953	2	31	multiplicative	multiplicative	ADJ
ejpam-4953	2	32	metric	metric	ADJ
ejpam-4953	2	33	spaces	space	NOUN
ejpam-4953	2	34	employing	employ	VERB
ejpam-4953	2	35	binary	binary	ADJ
ejpam-4953	2	36	relaion	relaion	PROPN
ejpam-4953	2	37	ibtesam	ibtesam	PROPN
ejpam-4953	2	38	alshammari1,∗	alshammari1,∗	PROPN
ejpam-4953	2	39	,	,	PUNCT
ejpam-4953	2	40	shahbaz	shahbaz	PROPN
ejpam-4953	2	41	ali2	ali2	PROPN
ejpam-4953	2	42	,	,	PUNCT
ejpam-4953	2	43	qamrul	qamrul	NOUN
ejpam-4953	2	44	haque	haque	PROPN
ejpam-4953	2	45	khan2	khan2	PROPN
ejpam-4953	2	46	,	,	PUNCT
ejpam-4953	2	47	tawseef	tawseef	PROPN
ejpam-4953	2	48	rashid3	rashid3	PROPN
ejpam-4953	2	49	,	,	PUNCT
ejpam-4953	2	50	cenap	cenap	VERB
ejpam-4953	2	51	ozel4	ozel4	NOUN
ejpam-4953	2	52	1	1	NUM
ejpam-4953	2	53	department	department	NOUN
ejpam-4953	2	54	of	of	ADP
ejpam-4953	2	55	mathematics	mathematic	NOUN
ejpam-4953	2	56	,	,	PUNCT
ejpam-4953	2	57	university	university	NOUN
ejpam-4953	2	58	of	of	ADP
ejpam-4953	2	59	hafr	hafr	PROPN
ejpam-4953	2	60	al	al	PROPN
ejpam-4953	2	61	batin	batin	PROPN
ejpam-4953	2	62	,	,	PUNCT
ejpam-4953	2	63	hafr	hafr	PROPN
ejpam-4953	2	64	al	al	PROPN
ejpam-4953	2	65	batin	batin	PROPN
ejpam-4953	2	66	,	,	PUNCT
ejpam-4953	2	67	saudi	saudi	PROPN
ejpam-4953	2	68	arabia	arabia	PROPN
ejpam-4953	2	69	2	2	NUM
ejpam-4953	2	70	department	department	NOUN
ejpam-4953	2	71	of	of	ADP
ejpam-4953	2	72	mathematics	mathematics	PROPN
ejpam-4953	2	73	,	,	PUNCT
ejpam-4953	2	74	aligarh	aligarh	PROPN
ejpam-4953	2	75	muslim	muslim	PROPN
ejpam-4953	2	76	university	university	PROPN
ejpam-4953	2	77	,	,	PUNCT
ejpam-4953	2	78	aligarh	aligarh	PROPN
ejpam-4953	2	79	,	,	PUNCT
ejpam-4953	2	80	india	india	PROPN
ejpam-4953	2	81	3	3	NUM
ejpam-4953	2	82	department	department	PROPN
ejpam-4953	2	83	of	of	ADP
ejpam-4953	2	84	mathematical	mathematical	ADJ
ejpam-4953	2	85	sciences	science	NOUN
ejpam-4953	2	86	,	,	PUNCT
ejpam-4953	2	87	iust	iust	NOUN
ejpam-4953	2	88	,	,	PUNCT
ejpam-4953	2	89	awantipora	awantipora	PROPN
ejpam-4953	2	90	,	,	PUNCT
ejpam-4953	2	91	india	india	PROPN
ejpam-4953	2	92	4	4	NUM
ejpam-4953	2	93	department	department	NOUN
ejpam-4953	2	94	of	of	ADP
ejpam-4953	2	95	mathematics	mathematic	NOUN
ejpam-4953	2	96	,	,	PUNCT
ejpam-4953	2	97	king	king	PROPN
ejpam-4953	2	98	abdulaziz	abdulaziz	PROPN
ejpam-4953	2	99	university	university	PROPN
ejpam-4953	2	100	,	,	PUNCT
ejpam-4953	2	101	jeddah-21589	jeddah-21589	NOUN
ejpam-4953	2	102	,	,	PUNCT
ejpam-4953	2	103	saudi	saudi	PROPN
ejpam-4953	2	104	arabia	arabia	PROPN
ejpam-4953	2	105	abstract	abstract	NOUN
ejpam-4953	2	106	.	.	PUNCT
ejpam-4953	3	1	in	in	ADP
ejpam-4953	3	2	this	this	DET
ejpam-4953	3	3	manuscrit	manuscrit	NOUN
ejpam-4953	3	4	,	,	PUNCT
ejpam-4953	3	5	we	we	PRON
ejpam-4953	3	6	establish	establish	VERB
ejpam-4953	3	7	some	some	DET
ejpam-4953	3	8	results	result	NOUN
ejpam-4953	3	9	on	on	ADP
ejpam-4953	3	10	the	the	DET
ejpam-4953	3	11	existence	existence	NOUN
ejpam-4953	3	12	and	and	CCONJ
ejpam-4953	3	13	uniqueness	uniqueness	NOUN
ejpam-4953	3	14	of	of	ADP
ejpam-4953	3	15	fixed	fix	VERB
ejpam-4953	3	16	points	point	NOUN
ejpam-4953	3	17	by	by	ADP
ejpam-4953	3	18	using	use	VERB
ejpam-4953	3	19	b	b	NOUN
ejpam-4953	3	20	-	-	PUNCT
ejpam-4953	3	21	multiplicative	multiplicative	ADJ
ejpam-4953	3	22	metric	metric	ADJ
ejpam-4953	3	23	spaces(mms	spaces(mms	NOUN
ejpam-4953	3	24	)	)	PUNCT
ejpam-4953	3	25	endowed	endow	VERB
ejpam-4953	3	26	with	with	ADP
ejpam-4953	3	27	a	a	DET
ejpam-4953	3	28	binary	binary	ADJ
ejpam-4953	3	29	relation	relation	NOUN
ejpam-4953	3	30	.	.	PUNCT
ejpam-4953	4	1	we	we	PRON
ejpam-4953	4	2	also	also	ADV
ejpam-4953	4	3	find	find	VERB
ejpam-4953	4	4	result	result	NOUN
ejpam-4953	4	5	on	on	ADP
ejpam-4953	4	6	the	the	DET
ejpam-4953	4	7	coincidence	coincidence	NOUN
ejpam-4953	4	8	of	of	ADP
ejpam-4953	4	9	points	point	NOUN
ejpam-4953	4	10	involving	involve	VERB
ejpam-4953	4	11	a	a	DET
ejpam-4953	4	12	pair	pair	NOUN
ejpam-4953	4	13	of	of	ADP
ejpam-4953	4	14	mappings	mapping	NOUN
ejpam-4953	4	15	.	.	PUNCT
ejpam-4953	5	1	finally	finally	ADV
ejpam-4953	5	2	some	some	DET
ejpam-4953	5	3	examples	example	NOUN
ejpam-4953	5	4	are	be	AUX
ejpam-4953	5	5	presented	present	VERB
ejpam-4953	5	6	to	to	PART
ejpam-4953	5	7	illustrate	illustrate	VERB
ejpam-4953	5	8	the	the	DET
ejpam-4953	5	9	suitability	suitability	NOUN
ejpam-4953	5	10	of	of	ADP
ejpam-4953	5	11	our	our	PRON
ejpam-4953	5	12	results	result	NOUN
ejpam-4953	5	13	.	.	PUNCT
ejpam-4953	6	1	2020	2020	NUM
ejpam-4953	6	2	mathematics	mathematic	NOUN
ejpam-4953	6	3	subject	subject	NOUN
ejpam-4953	6	4	classifications	classification	NOUN
ejpam-4953	6	5	:	:	PUNCT
ejpam-4953	6	6	47h10	47h10	NUM
ejpam-4953	6	7	,	,	PUNCT
ejpam-4953	6	8	54h25,46j10	54h25,46j10	NUM
ejpam-4953	6	9	key	key	ADJ
ejpam-4953	6	10	words	word	NOUN
ejpam-4953	6	11	and	and	CCONJ
ejpam-4953	6	12	phrases	phrase	NOUN
ejpam-4953	6	13	:	:	PUNCT
ejpam-4953	6	14	multiplicative	multiplicative	ADJ
ejpam-4953	6	15	metric	metric	ADJ
ejpam-4953	6	16	space	space	NOUN
ejpam-4953	6	17	,	,	PUNCT
ejpam-4953	6	18	binary	binary	PROPN
ejpam-4953	6	19	relation	relation	PROPN
ejpam-4953	6	20	,	,	PUNCT
ejpam-4953	6	21	relation	relation	NOUN
ejpam-4953	6	22	theoretic	theoretic	ADJ
ejpam-4953	6	23	contractions	contraction	NOUN
ejpam-4953	6	24	,	,	PUNCT
ejpam-4953	6	25	coincidence	coincidence	NOUN
ejpam-4953	6	26	points	point	VERB
ejpam-4953	6	27	1	1	NUM
ejpam-4953	6	28	.	.	PUNCT
ejpam-4953	6	29	introduction	introduction	NOUN
ejpam-4953	6	30	and	and	CCONJ
ejpam-4953	6	31	prilimaries	prilimarie	NOUN
ejpam-4953	6	32	in	in	ADP
ejpam-4953	6	33	1922	1922	NUM
ejpam-4953	6	34	,	,	PUNCT
ejpam-4953	6	35	banach	banach	NOUN
ejpam-4953	6	36	[	[	X
ejpam-4953	6	37	1	1	NUM
ejpam-4953	6	38	]	]	PUNCT
ejpam-4953	6	39	laid	lay	VERB
ejpam-4953	6	40	the	the	DET
ejpam-4953	6	41	important	important	ADJ
ejpam-4953	6	42	result	result	NOUN
ejpam-4953	6	43	of	of	ADP
ejpam-4953	6	44	fixed	fix	VERB
ejpam-4953	6	45	point	point	NOUN
ejpam-4953	6	46	theory	theory	NOUN
ejpam-4953	6	47	in	in	ADP
ejpam-4953	6	48	metric	metric	ADJ
ejpam-4953	6	49	spaces	space	NOUN
ejpam-4953	6	50	.	.	PUNCT
ejpam-4953	7	1	later	later	ADV
ejpam-4953	7	2	on	on	ADV
ejpam-4953	7	3	,	,	PUNCT
ejpam-4953	7	4	several	several	ADJ
ejpam-4953	7	5	authors	author	NOUN
ejpam-4953	7	6	generalized	generalize	VERB
ejpam-4953	7	7	the	the	DET
ejpam-4953	7	8	banach	banach	NOUN
ejpam-4953	7	9	contraction	contraction	NOUN
ejpam-4953	7	10	principle	principle	NOUN
ejpam-4953	7	11	,	,	PUNCT
ejpam-4953	7	12	see[2–4	see[2–4	PROPN
ejpam-4953	7	13	]	]	PUNCT
ejpam-4953	7	14	.	.	PUNCT
ejpam-4953	8	1	inspired	inspire	VERB
ejpam-4953	8	2	by	by	ADP
ejpam-4953	8	3	turinici	turinici	NOUN
ejpam-4953	8	4	[	[	X
ejpam-4953	8	5	5	5	NUM
ejpam-4953	8	6	]	]	X
ejpam-4953	8	7	work	work	NOUN
ejpam-4953	8	8	,	,	PUNCT
ejpam-4953	8	9	ran	run	VERB
ejpam-4953	8	10	and	and	CCONJ
ejpam-4953	8	11	reurings	reuring	NOUN
ejpam-4953	9	1	[	[	X
ejpam-4953	9	2	6	6	NUM
ejpam-4953	9	3	]	]	PUNCT
ejpam-4953	9	4	in	in	ADP
ejpam-4953	9	5	2004	2004	NUM
ejpam-4953	9	6	worked	work	VERB
ejpam-4953	9	7	on	on	ADP
ejpam-4953	9	8	banach	banach	NOUN
ejpam-4953	9	9	contraction	contraction	NOUN
ejpam-4953	9	10	principle	principle	NOUN
ejpam-4953	9	11	in	in	ADP
ejpam-4953	9	12	ordered	order	VERB
ejpam-4953	9	13	metric	metric	ADJ
ejpam-4953	9	14	space	space	NOUN
ejpam-4953	9	15	and	and	CCONJ
ejpam-4953	9	16	assumed	assume	VERB
ejpam-4953	9	17	the	the	DET
ejpam-4953	9	18	contractive	contractive	ADJ
ejpam-4953	9	19	condition	condition	NOUN
ejpam-4953	9	20	only	only	ADV
ejpam-4953	9	21	to	to	PART
ejpam-4953	9	22	hold	hold	VERB
ejpam-4953	9	23	on	on	ADP
ejpam-4953	9	24	the	the	DET
ejpam-4953	9	25	comparable	comparable	ADJ
ejpam-4953	9	26	elements	element	NOUN
ejpam-4953	9	27	instead	instead	ADV
ejpam-4953	9	28	of	of	ADP
ejpam-4953	9	29	the	the	DET
ejpam-4953	9	30	whole	whole	ADJ
ejpam-4953	9	31	space	space	NOUN
ejpam-4953	9	32	.	.	PUNCT
ejpam-4953	10	1	fixed	fix	VERB
ejpam-4953	10	2	point	point	NOUN
ejpam-4953	10	3	in	in	ADP
ejpam-4953	10	4	ordered	order	VERB
ejpam-4953	10	5	metric	metric	ADJ
ejpam-4953	10	6	space	space	NOUN
ejpam-4953	10	7	has	have	AUX
ejpam-4953	10	8	been	be	AUX
ejpam-4953	10	9	extensively	extensively	ADV
ejpam-4953	10	10	studied	study	VERB
ejpam-4953	10	11	in	in	ADP
ejpam-4953	10	12	the	the	DET
ejpam-4953	10	13	literature	literature	NOUN
ejpam-4953	10	14	[	[	X
ejpam-4953	10	15	7–9	7–9	X
ejpam-4953	10	16	]	]	X
ejpam-4953	10	17	.	.	PUNCT
ejpam-4953	11	1	the	the	DET
ejpam-4953	11	2	idea	idea	NOUN
ejpam-4953	11	3	of	of	ADP
ejpam-4953	11	4	mms	mms	NOUN
ejpam-4953	11	5	,	,	PUNCT
ejpam-4953	11	6	which	which	PRON
ejpam-4953	11	7	is	be	AUX
ejpam-4953	11	8	a	a	DET
ejpam-4953	11	9	generalization	generalization	NOUN
ejpam-4953	11	10	of	of	ADP
ejpam-4953	11	11	metric	metric	ADJ
ejpam-4953	11	12	space	space	NOUN
ejpam-4953	11	13	,	,	PUNCT
ejpam-4953	11	14	was	be	AUX
ejpam-4953	11	15	first	first	ADV
ejpam-4953	11	16	introduced	introduce	VERB
ejpam-4953	11	17	by	by	ADP
ejpam-4953	11	18	bashirov	bashirov	PROPN
ejpam-4953	11	19	et	et	PROPN
ejpam-4953	11	20	al	al	PROPN
ejpam-4953	11	21	.	.	PUNCT
ejpam-4953	12	1	[	[	X
ejpam-4953	12	2	10	10	NUM
ejpam-4953	12	3	]	]	PUNCT
ejpam-4953	12	4	in	in	ADP
ejpam-4953	12	5	2008	2008	NUM
ejpam-4953	12	6	.	.	PUNCT
ejpam-4953	13	1	the	the	DET
ejpam-4953	13	2	main	main	ADJ
ejpam-4953	13	3	idea	idea	NOUN
ejpam-4953	13	4	behind	behind	ADP
ejpam-4953	13	5	introducing	introduce	VERB
ejpam-4953	13	6	mms	mms	NOUN
ejpam-4953	13	7	was	be	AUX
ejpam-4953	13	8	to	to	PART
ejpam-4953	13	9	replace	replace	VERB
ejpam-4953	13	10	usual	usual	ADJ
ejpam-4953	13	11	triangular	triangular	NOUN
ejpam-4953	13	12	inequality	inequality	NOUN
ejpam-4953	13	13	by	by	ADP
ejpam-4953	13	14	the	the	DET
ejpam-4953	13	15	multiplicative	multiplicative	ADJ
ejpam-4953	13	16	triangle	triangle	NOUN
ejpam-4953	13	17	inequality	inequality	NOUN
ejpam-4953	13	18	.	.	PUNCT
ejpam-4953	14	1	later	later	ADV
ejpam-4953	14	2	on	on	ADV
ejpam-4953	14	3	,	,	PUNCT
ejpam-4953	14	4	many	many	ADJ
ejpam-4953	14	5	research	research	NOUN
ejpam-4953	14	6	papers	paper	NOUN
ejpam-4953	14	7	were	be	AUX
ejpam-4953	14	8	written	write	VERB
ejpam-4953	14	9	on	on	ADP
ejpam-4953	14	10	fixed	fix	VERB
ejpam-4953	14	11	points	point	NOUN
ejpam-4953	14	12	in	in	ADP
ejpam-4953	14	13	mms	mms	NOUN
ejpam-4953	14	14	[	[	X
ejpam-4953	14	15	11–16	11–16	NUM
ejpam-4953	14	16	,	,	PUNCT
ejpam-4953	14	17	18–20	18–20	NUM
ejpam-4953	14	18	]	]	PUNCT
ejpam-4953	14	19	.	.	PUNCT
ejpam-4953	15	1	czerwik	czerwik	PROPN
ejpam-4953	16	1	[	[	X
ejpam-4953	16	2	17	17	NUM
ejpam-4953	16	3	]	]	PUNCT
ejpam-4953	16	4	introduced	introduce	VERB
ejpam-4953	16	5	the	the	DET
ejpam-4953	16	6	notion	notion	NOUN
ejpam-4953	16	7	of	of	ADP
ejpam-4953	16	8	b	b	NOUN
ejpam-4953	16	9	-	-	PUNCT
ejpam-4953	16	10	metric	metric	ADJ
ejpam-4953	16	11	space	space	NOUN
ejpam-4953	16	12	which	which	PRON
ejpam-4953	16	13	is	be	AUX
ejpam-4953	16	14	a	a	DET
ejpam-4953	16	15	generalization	generalization	NOUN
ejpam-4953	16	16	of	of	ADP
ejpam-4953	16	17	metric	metric	ADJ
ejpam-4953	16	18	space	space	NOUN
ejpam-4953	16	19	.	.	PUNCT
ejpam-4953	17	1	there	there	PRON
ejpam-4953	17	2	are	be	VERB
ejpam-4953	17	3	some	some	DET
ejpam-4953	17	4	fixed	fix	VERB
ejpam-4953	17	5	point	point	NOUN
ejpam-4953	17	6	results	result	NOUN
ejpam-4953	17	7	in	in	ADP
ejpam-4953	17	8	b	b	NOUN
ejpam-4953	17	9	-	-	PUNCT
ejpam-4953	17	10	metric	metric	ADJ
ejpam-4953	17	11	space	space	NOUN
ejpam-4953	17	12	.	.	PUNCT
ejpam-4953	18	1	later	later	ADV
ejpam-4953	18	2	on	on	ADV
ejpam-4953	18	3	,	,	PUNCT
ejpam-4953	18	4	muhammad	muhammad	PROPN
ejpam-4953	18	5	usman	usman	PROPN
ejpam-4953	18	6	et	et	PROPN
ejpam-4953	18	7	al	al	PROPN
ejpam-4953	18	8	.	.	PUNCT
ejpam-4953	19	1	[	[	X
ejpam-4953	19	2	21	21	NUM
ejpam-4953	19	3	]	]	PUNCT
ejpam-4953	19	4	introduce	introduce	VERB
ejpam-4953	19	5	the	the	DET
ejpam-4953	19	6	∗corresponding	∗corresponding	NOUN
ejpam-4953	19	7	author	author	NOUN
ejpam-4953	19	8	.	.	PUNCT
ejpam-4953	20	1	doi	doi	NOUN
ejpam-4953	20	2	:	:	PUNCT
ejpam-4953	20	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4953	https://doi.org/10.29020/nybg.ejpam.v16i4.4953	NUM
ejpam-4953	20	4	email	email	NOUN
ejpam-4953	20	5	addresses	address	NOUN
ejpam-4953	20	6	:	:	PUNCT
ejpam-4953	20	7	iealshamri@uhb.edu.sa	iealshamri@uhb.edu.sa	PROPN
ejpam-4953	20	8	(	(	PUNCT
ejpam-4953	20	9	i.	i.	PROPN
ejpam-4953	20	10	alshammari	alshammari	PROPN
ejpam-4953	20	11	)	)	PUNCT
ejpam-4953	20	12	,	,	PUNCT
ejpam-4953	20	13	shahbazali4786@gmail.com	shahbazali4786@gmail.com	X
ejpam-4953	20	14	(	(	PUNCT
ejpam-4953	20	15	s.	s.	PROPN
ejpam-4953	20	16	ali	ali	PROPN
ejpam-4953	20	17	)	)	PUNCT
ejpam-4953	20	18	,	,	PUNCT
ejpam-4953	20	19	qhkhan.ssitm@gmail.com	qhkhan.ssitm@gmail.com	X
ejpam-4953	20	20	(	(	PUNCT
ejpam-4953	20	21	q.h	q.h	PROPN
ejpam-4953	20	22	.	.	PROPN
ejpam-4953	20	23	khan	khan	PROPN
ejpam-4953	20	24	)	)	PUNCT
ejpam-4953	20	25	,	,	PUNCT
ejpam-4953	20	26	tawseefrashid123@gmail.com	tawseefrashid123@gmail.com	X
ejpam-4953	20	27	(	(	PUNCT
ejpam-4953	20	28	t.	t.	PROPN
ejpam-4953	20	29	rashid	rashid	PROPN
ejpam-4953	20	30	)	)	PUNCT
ejpam-4953	20	31	,	,	PUNCT
ejpam-4953	20	32	cenap.ozel@gmail.com	cenap.ozel@gmail.com	X
ejpam-4953	20	33	(	(	PUNCT
ejpam-4953	20	34	c.	c.	PROPN
ejpam-4953	20	35	ozel	ozel	PROPN
ejpam-4953	20	36	)	)	PUNCT
ejpam-4953	20	37	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4953	20	38	2405	2405	NUM
ejpam-4953	21	1	©	©	PROPN
ejpam-4953	21	2	2023	2023	NUM
ejpam-4953	21	3	ejpam	ejpam	NOUN
ejpam-4953	21	4	all	all	DET
ejpam-4953	21	5	rights	right	NOUN
ejpam-4953	21	6	reserved	reserve	VERB
ejpam-4953	21	7	.	.	PUNCT
ejpam-4953	22	1	i.	i.	PROPN
ejpam-4953	22	2	alshammari	alshammari	PROPN
ejpam-4953	22	3	et	et	PROPN
ejpam-4953	22	4	al	al	PROPN
ejpam-4953	22	5	.	.	PUNCT
ejpam-4953	22	6	/	/	SYM
ejpam-4953	22	7	eur	eur	PROPN
ejpam-4953	22	8	.	.	PUNCT
ejpam-4953	23	1	j.	j.	PROPN
ejpam-4953	23	2	pure	pure	PROPN
ejpam-4953	23	3	appl	appl	PROPN
ejpam-4953	23	4	.	.	PROPN
ejpam-4953	23	5	math	math	PROPN
ejpam-4953	23	6	,	,	PUNCT
ejpam-4953	23	7	16	16	NUM
ejpam-4953	23	8	(	(	PUNCT
ejpam-4953	23	9	4	4	NUM
ejpam-4953	23	10	)	)	PUNCT
ejpam-4953	23	11	(	(	PUNCT
ejpam-4953	23	12	2023	2023	NUM
ejpam-4953	23	13	)	)	PUNCT
ejpam-4953	23	14	,	,	PUNCT
ejpam-4953	23	15	2405	2405	NUM
ejpam-4953	23	16	-	-	SYM
ejpam-4953	23	17	2418	2418	NUM
ejpam-4953	23	18	2406	2406	NUM
ejpam-4953	23	19	new	new	ADJ
ejpam-4953	23	20	notion	notion	NOUN
ejpam-4953	23	21	of	of	ADP
ejpam-4953	23	22	b	b	NOUN
ejpam-4953	23	23	-	-	PUNCT
ejpam-4953	23	24	multiplicative	multiplicative	ADJ
ejpam-4953	23	25	metric	metric	ADJ
ejpam-4953	23	26	space	space	NOUN
ejpam-4953	23	27	and	and	CCONJ
ejpam-4953	23	28	proved	prove	VERB
ejpam-4953	23	29	fixed	fix	VERB
ejpam-4953	23	30	point	point	NOUN
ejpam-4953	23	31	theorems	theorem	NOUN
ejpam-4953	23	32	for	for	ADP
ejpam-4953	23	33	single	single	ADJ
ejpam-4953	23	34	and	and	CCONJ
ejpam-4953	23	35	multivalued	multivalued	ADJ
ejpam-4953	23	36	mapping	mapping	NOUN
ejpam-4953	23	37	on	on	ADP
ejpam-4953	23	38	b	b	X
ejpam-4953	23	39	-	-	PUNCT
ejpam-4953	23	40	multiplicative	multiplicative	ADJ
ejpam-4953	23	41	metric	metric	ADJ
ejpam-4953	23	42	spaces	space	NOUN
ejpam-4953	23	43	,	,	PUNCT
ejpam-4953	23	44	endowed	endow	VERB
ejpam-4953	23	45	with	with	ADP
ejpam-4953	23	46	a	a	DET
ejpam-4953	23	47	graph	graph	NOUN
ejpam-4953	23	48	.	.	PUNCT
ejpam-4953	24	1	in	in	ADP
ejpam-4953	24	2	this	this	DET
ejpam-4953	24	3	paper	paper	NOUN
ejpam-4953	24	4	we	we	PRON
ejpam-4953	24	5	prove	prove	VERB
ejpam-4953	24	6	fixed	fix	VERB
ejpam-4953	24	7	point	point	NOUN
ejpam-4953	24	8	theorems	theorem	NOUN
ejpam-4953	24	9	for	for	ADP
ejpam-4953	24	10	mapping	mapping	NOUN
ejpam-4953	24	11	on	on	ADP
ejpam-4953	24	12	b	b	NOUN
ejpam-4953	24	13	-	-	PUNCT
ejpam-4953	24	14	multiplicative	multiplicative	ADJ
ejpam-4953	24	15	metric	metric	ADJ
ejpam-4953	24	16	space	space	NOUN
ejpam-4953	24	17	endowed	endow	VERB
ejpam-4953	24	18	with	with	ADP
ejpam-4953	24	19	a	a	DET
ejpam-4953	24	20	binary	binary	ADJ
ejpam-4953	24	21	relation	relation	NOUN
ejpam-4953	24	22	and	and	CCONJ
ejpam-4953	24	23	also	also	ADV
ejpam-4953	24	24	prove	prove	VERB
ejpam-4953	24	25	a	a	DET
ejpam-4953	24	26	coincidence	coincidence	NOUN
ejpam-4953	24	27	of	of	ADP
ejpam-4953	24	28	points	point	NOUN
ejpam-4953	24	29	involving	involve	VERB
ejpam-4953	24	30	a	a	DET
ejpam-4953	24	31	pair	pair	NOUN
ejpam-4953	24	32	of	of	ADP
ejpam-4953	24	33	mapping	mapping	NOUN
ejpam-4953	24	34	and	and	CCONJ
ejpam-4953	24	35	provide	provide	VERB
ejpam-4953	24	36	some	some	DET
ejpam-4953	24	37	examples	example	NOUN
ejpam-4953	24	38	to	to	PART
ejpam-4953	24	39	demonstrate	demonstrate	VERB
ejpam-4953	24	40	our	our	PRON
ejpam-4953	24	41	results	result	NOUN
ejpam-4953	24	42	.	.	PUNCT
ejpam-4953	25	1	definition	definition	NOUN
ejpam-4953	25	2	1	1	NUM
ejpam-4953	25	3	.	.	PUNCT
ejpam-4953	26	1	[	[	X
ejpam-4953	26	2	21	21	NUM
ejpam-4953	26	3	]	]	PUNCT
ejpam-4953	26	4	.	.	PUNCT
ejpam-4953	27	1	let	let	VERB
ejpam-4953	27	2	ḧ	ḧ	PRON
ejpam-4953	27	3	be	be	AUX
ejpam-4953	27	4	a	a	DET
ejpam-4953	27	5	non	non	ADJ
ejpam-4953	27	6	-	-	ADJ
ejpam-4953	27	7	empty	empty	ADJ
ejpam-4953	27	8	set	set	NOUN
ejpam-4953	27	9	and	and	CCONJ
ejpam-4953	27	10	let	let	VERB
ejpam-4953	27	11	k	k	PROPN
ejpam-4953	27	12	≥	≥	NUM
ejpam-4953	27	13	1	1	NUM
ejpam-4953	27	14	be	be	AUX
ejpam-4953	27	15	a	a	DET
ejpam-4953	27	16	given	give	VERB
ejpam-4953	27	17	real	real	ADJ
ejpam-4953	27	18	number	number	NOUN
ejpam-4953	27	19	.	.	PUNCT
ejpam-4953	28	1	a	a	DET
ejpam-4953	28	2	mapping	mapping	NOUN
ejpam-4953	28	3	p	p	X
ejpam-4953	28	4	:	:	PUNCT
ejpam-4953	28	5	ḧ×ḧ	ḧ×ḧ	NOUN
ejpam-4953	28	6	→	→	SYM
ejpam-4953	28	7	r	r	NOUN
ejpam-4953	28	8	is	be	AUX
ejpam-4953	28	9	called	call	VERB
ejpam-4953	28	10	a	a	DET
ejpam-4953	28	11	b	b	NOUN
ejpam-4953	28	12	-	-	PUNCT
ejpam-4953	28	13	multiplicative	multiplicative	ADJ
ejpam-4953	28	14	metric	metric	NOUN
ejpam-4953	28	15	with	with	ADP
ejpam-4953	28	16	coefficient	coefficient	NOUN
ejpam-4953	28	17	k	k	NOUN
ejpam-4953	28	18	,	,	PUNCT
ejpam-4953	28	19	if	if	SCONJ
ejpam-4953	28	20	the	the	DET
ejpam-4953	28	21	following	follow	VERB
ejpam-4953	28	22	conditions	condition	NOUN
ejpam-4953	28	23	hold	hold	VERB
ejpam-4953	28	24	:	:	PUNCT
ejpam-4953	28	25	(	(	PUNCT
ejpam-4953	28	26	m1	m1	NOUN
ejpam-4953	28	27	)	)	PUNCT
ejpam-4953	28	28	p(ϖ	p(ϖ	PROPN
ejpam-4953	28	29	,	,	PUNCT
ejpam-4953	28	30	ρ	ρ	PROPN
ejpam-4953	28	31	)	)	PUNCT
ejpam-4953	28	32	≥	≥	NOUN
ejpam-4953	28	33	1	1	NUM
ejpam-4953	28	34	for	for	ADP
ejpam-4953	28	35	all	all	DET
ejpam-4953	28	36	ϖ	ϖ	PROPN
ejpam-4953	28	37	,	,	PUNCT
ejpam-4953	28	38	ρ	ρ	PROPN
ejpam-4953	28	39	∈	∈	PROPN
ejpam-4953	28	40	ḧ	ḧ	NOUN
ejpam-4953	28	41	and	and	CCONJ
ejpam-4953	28	42	p(ϖ	p(ϖ	PROPN
ejpam-4953	28	43	,	,	PUNCT
ejpam-4953	28	44	ρ	ρ	NOUN
ejpam-4953	28	45	)	)	PUNCT
ejpam-4953	28	46	=	=	NOUN
ejpam-4953	28	47	1	1	NUM
ejpam-4953	28	48	if	if	SCONJ
ejpam-4953	28	49	and	and	CCONJ
ejpam-4953	28	50	only	only	ADV
ejpam-4953	28	51	if	if	SCONJ
ejpam-4953	28	52	ϖ	ϖ	PROPN
ejpam-4953	28	53	=	=	SYM
ejpam-4953	28	54	ρ	ρ	PROPN
ejpam-4953	28	55	;	;	PUNCT
ejpam-4953	28	56	(	(	PUNCT
ejpam-4953	28	57	m2	m2	PROPN
ejpam-4953	28	58	)	)	PUNCT
ejpam-4953	28	59	p(ϖ	p(ϖ	PROPN
ejpam-4953	28	60	,	,	PUNCT
ejpam-4953	28	61	ρ	ρ	NOUN
ejpam-4953	28	62	)	)	PUNCT
ejpam-4953	28	63	=	=	SYM
ejpam-4953	28	64	p(ϖ	p(ϖ	PROPN
ejpam-4953	28	65	,	,	PUNCT
ejpam-4953	28	66	ρ	ρ	NOUN
ejpam-4953	28	67	)	)	PUNCT
ejpam-4953	28	68	for	for	ADP
ejpam-4953	28	69	all	all	DET
ejpam-4953	28	70	ϖ	ϖ	PROPN
ejpam-4953	28	71	,	,	PUNCT
ejpam-4953	28	72	ρ	ρ	PROPN
ejpam-4953	28	73	∈	∈	PROPN
ejpam-4953	28	74	ḧ	ḧ	NOUN
ejpam-4953	28	75	;	;	PUNCT
ejpam-4953	28	76	(	(	PUNCT
ejpam-4953	28	77	m3	m3	PROPN
ejpam-4953	28	78	)	)	PUNCT
ejpam-4953	28	79	p(ϖ	p(ϖ	PROPN
ejpam-4953	28	80	,	,	PUNCT
ejpam-4953	28	81	ρ	ρ	NOUN
ejpam-4953	28	82	)	)	PUNCT
ejpam-4953	28	83	≤	≤	NOUN
ejpam-4953	28	84	p(ϖ	p(ϖ	PROPN
ejpam-4953	28	85	,	,	PUNCT
ejpam-4953	28	86	z)k.p(z	z)k.p(z	NOUN
ejpam-4953	28	87	,	,	PUNCT
ejpam-4953	28	88	ρ)k	ρ)k	X
ejpam-4953	28	89	for	for	ADP
ejpam-4953	28	90	all	all	DET
ejpam-4953	28	91	ϖ	ϖ	PROPN
ejpam-4953	28	92	,	,	PUNCT
ejpam-4953	28	93	ρ	ρ	PROPN
ejpam-4953	28	94	,	,	PUNCT
ejpam-4953	28	95	z	z	PROPN
ejpam-4953	28	96	∈	∈	PROPN
ejpam-4953	28	97	ḧ.	ḧ.	VERB
ejpam-4953	28	98	the	the	DET
ejpam-4953	28	99	triplet	triplet	NOUN
ejpam-4953	28	100	(	(	PUNCT
ejpam-4953	28	101	ḧ,p	ḧ,p	PROPN
ejpam-4953	28	102	,	,	PUNCT
ejpam-4953	28	103	k	k	NOUN
ejpam-4953	28	104	)	)	PUNCT
ejpam-4953	28	105	is	be	AUX
ejpam-4953	28	106	called	call	VERB
ejpam-4953	28	107	a	a	DET
ejpam-4953	28	108	b	b	NOUN
ejpam-4953	28	109	-	-	PUNCT
ejpam-4953	28	110	multiplicative	multiplicative	ADJ
ejpam-4953	28	111	metric	metric	ADJ
ejpam-4953	28	112	space	space	NOUN
ejpam-4953	28	113	.	.	PUNCT
ejpam-4953	29	1	definition	definition	NOUN
ejpam-4953	29	2	2	2	NUM
ejpam-4953	29	3	.	.	PUNCT
ejpam-4953	30	1	[	[	X
ejpam-4953	30	2	4	4	NUM
ejpam-4953	30	3	]	]	PUNCT
ejpam-4953	30	4	.	.	PUNCT
ejpam-4953	31	1	let	let	AUX
ejpam-4953	31	2	(	(	PUNCT
ejpam-4953	31	3	ḧ	ḧ	NOUN
ejpam-4953	31	4	,	,	PUNCT
ejpam-4953	31	5	p	p	X
ejpam-4953	31	6	,	,	PUNCT
ejpam-4953	31	7	k	k	NOUN
ejpam-4953	31	8	)	)	PUNCT
ejpam-4953	31	9	be	be	VERB
ejpam-4953	31	10	any	any	DET
ejpam-4953	31	11	b	b	NOUN
ejpam-4953	31	12	-	-	PUNCT
ejpam-4953	31	13	mms	mms	ADJ
ejpam-4953	31	14	,	,	PUNCT
ejpam-4953	31	15	{	{	PUNCT
ejpam-4953	31	16	ϖn	ϖn	AUX
ejpam-4953	31	17	}	}	PUNCT
ejpam-4953	31	18	be	be	AUX
ejpam-4953	31	19	a	a	DET
ejpam-4953	31	20	sequence	sequence	NOUN
ejpam-4953	31	21	in	in	ADP
ejpam-4953	31	22	ḧ	ḧ	NOUN
ejpam-4953	31	23	and	and	CCONJ
ejpam-4953	31	24	ϖ	ϖ	X
ejpam-4953	31	25	∈	∈	NOUN
ejpam-4953	31	26	ḧ.	ḧ.	PROPN
ejpam-4953	31	27	if	if	SCONJ
ejpam-4953	31	28	for	for	ADP
ejpam-4953	31	29	every	every	DET
ejpam-4953	31	30	multiplicative	multiplicative	ADJ
ejpam-4953	31	31	open	open	ADJ
ejpam-4953	31	32	ball	ball	NOUN
ejpam-4953	31	33	bϵ(z	bϵ(z	NOUN
ejpam-4953	31	34	)	)	PUNCT
ejpam-4953	32	1	=	=	SYM
ejpam-4953	32	2	{	{	PUNCT
ejpam-4953	32	3	ρ	ρ	NOUN
ejpam-4953	32	4	:	:	PUNCT
ejpam-4953	32	5	p(ϖ	p(ϖ	PROPN
ejpam-4953	32	6	,	,	PUNCT
ejpam-4953	32	7	ρ	ρ	NOUN
ejpam-4953	32	8	)	)	PUNCT
ejpam-4953	32	9	<	<	X
ejpam-4953	32	10	ϵ	ϵ	X
ejpam-4953	32	11	}	}	PUNCT
ejpam-4953	32	12	,	,	PUNCT
ejpam-4953	32	13	ϵ	ϵ	X
ejpam-4953	32	14	>	>	X
ejpam-4953	32	15	1	1	NUM
ejpam-4953	32	16	,	,	PUNCT
ejpam-4953	32	17	there	there	PRON
ejpam-4953	32	18	exists	exist	VERB
ejpam-4953	32	19	a	a	DET
ejpam-4953	32	20	natural	natural	ADJ
ejpam-4953	32	21	number	number	NOUN
ejpam-4953	32	22	n	n	CCONJ
ejpam-4953	32	23	∈	∈	NOUN
ejpam-4953	32	24	n	n	PRON
ejpam-4953	32	25	such	such	ADJ
ejpam-4953	32	26	that	that	SCONJ
ejpam-4953	32	27	n	n	CCONJ
ejpam-4953	32	28	≥	≥	NOUN
ejpam-4953	32	29	n	n	ADV
ejpam-4953	32	30	and	and	CCONJ
ejpam-4953	32	31	ϖn	ϖn	ADP
ejpam-4953	32	32	∈	∈	PROPN
ejpam-4953	32	33	bϵ(ϖ	bϵ(ϖ	NOUN
ejpam-4953	32	34	)	)	PUNCT
ejpam-4953	32	35	.	.	PUNCT
ejpam-4953	33	1	then	then	ADV
ejpam-4953	33	2	the	the	DET
ejpam-4953	33	3	sequence	sequence	NOUN
ejpam-4953	33	4	{	{	PUNCT
ejpam-4953	33	5	ϖn	ϖn	NOUN
ejpam-4953	33	6	}	}	PUNCT
ejpam-4953	33	7	is	be	AUX
ejpam-4953	33	8	said	say	VERB
ejpam-4953	33	9	to	to	PART
ejpam-4953	33	10	be	be	AUX
ejpam-4953	33	11	multiplicative	multiplicative	ADJ
ejpam-4953	33	12	converging	converge	VERB
ejpam-4953	33	13	to	to	PART
ejpam-4953	33	14	ϖ.	ϖ.	VERB
ejpam-4953	33	15	we	we	PRON
ejpam-4953	33	16	denote	denote	VERB
ejpam-4953	33	17	as	as	ADP
ejpam-4953	33	18	ϖn	ϖn	NOUN
ejpam-4953	33	19	→	→	SYM
ejpam-4953	33	20	ϖ	ϖ	X
ejpam-4953	33	21	(	(	PUNCT
ejpam-4953	33	22	n	n	X
ejpam-4953	33	23	→	→	SYM
ejpam-4953	33	24	+	+	NOUN
ejpam-4953	33	25	∞	∞	NUM
ejpam-4953	33	26	)	)	PUNCT
ejpam-4953	33	27	.	.	PUNCT
ejpam-4953	34	1	lemma	lemma	PROPN
ejpam-4953	34	2	1	1	NUM
ejpam-4953	34	3	.	.	PUNCT
ejpam-4953	35	1	[	[	X
ejpam-4953	35	2	21	21	NUM
ejpam-4953	35	3	]	]	X
ejpam-4953	35	4	let	let	VERB
ejpam-4953	35	5	(	(	PUNCT
ejpam-4953	35	6	ḧ	ḧ	NOUN
ejpam-4953	35	7	,	,	PUNCT
ejpam-4953	35	8	p	p	X
ejpam-4953	35	9	,	,	PUNCT
ejpam-4953	35	10	k	k	NOUN
ejpam-4953	35	11	)	)	PUNCT
ejpam-4953	35	12	is	be	AUX
ejpam-4953	35	13	a	a	DET
ejpam-4953	35	14	b	b	NOUN
ejpam-4953	35	15	-	-	PUNCT
ejpam-4953	35	16	multiplicative	multiplicative	ADJ
ejpam-4953	35	17	metric	metric	ADJ
ejpam-4953	35	18	space	space	NOUN
ejpam-4953	35	19	.	.	PUNCT
ejpam-4953	36	1	if	if	SCONJ
ejpam-4953	36	2	a	a	DET
ejpam-4953	36	3	sequence	sequence	NOUN
ejpam-4953	36	4	{	{	PUNCT
ejpam-4953	36	5	ϖn	ϖn	NOUN
ejpam-4953	36	6	}	}	PUNCT
ejpam-4953	36	7	is	be	AUX
ejpam-4953	36	8	a	a	DET
ejpam-4953	36	9	multiplicative	multiplicative	ADJ
ejpam-4953	36	10	convergent	convergent	NOUN
ejpam-4953	36	11	,	,	PUNCT
ejpam-4953	36	12	then	then	ADV
ejpam-4953	36	13	the	the	DET
ejpam-4953	36	14	multiplicative	multiplicative	ADJ
ejpam-4953	36	15	limit	limit	NOUN
ejpam-4953	36	16	point	point	NOUN
ejpam-4953	36	17	is	be	AUX
ejpam-4953	36	18	unique	unique	ADJ
ejpam-4953	36	19	.	.	PUNCT
ejpam-4953	37	1	let	let	AUX
ejpam-4953	37	2	(	(	PUNCT
ejpam-4953	37	3	ḧ	ḧ	NOUN
ejpam-4953	37	4	,	,	PUNCT
ejpam-4953	37	5	p	p	X
ejpam-4953	37	6	)	)	PUNCT
ejpam-4953	37	7	be	be	AUX
ejpam-4953	37	8	a	a	DET
ejpam-4953	37	9	mms	mms	NOUN
ejpam-4953	37	10	,	,	PUNCT
ejpam-4953	37	11	{	{	PUNCT
ejpam-4953	37	12	ϖn	ϖn	AUX
ejpam-4953	37	13	}	}	PUNCT
ejpam-4953	37	14	be	be	AUX
ejpam-4953	37	15	a	a	DET
ejpam-4953	37	16	sequence	sequence	NOUN
ejpam-4953	37	17	in	in	ADP
ejpam-4953	37	18	ḧ	ḧ	NOUN
ejpam-4953	37	19	and	and	CCONJ
ejpam-4953	37	20	ϖ	ϖ	PROPN
ejpam-4953	37	21	∈	∈	PROPN
ejpam-4953	37	22	ḧ.	ḧ.	PROPN
ejpam-4953	37	23	then	then	ADV
ejpam-4953	37	24	ϖn	ϖn	ADP
ejpam-4953	37	25	→	→	SYM
ejpam-4953	37	26	ϖ(n	ϖ(n	X
ejpam-4953	37	27	→	→	SYM
ejpam-4953	37	28	+	+	NOUN
ejpam-4953	37	29	∞	∞	NOUN
ejpam-4953	37	30	)	)	PUNCT
ejpam-4953	37	31	⇔	⇔	PROPN
ejpam-4953	37	32	p(ϖn	p(ϖn	PROPN
ejpam-4953	37	33	,	,	PUNCT
ejpam-4953	37	34	ϖ	ϖ	NOUN
ejpam-4953	37	35	)	)	PUNCT
ejpam-4953	37	36	→	→	SYM
ejpam-4953	37	37	1(n	1(n	NUM
ejpam-4953	37	38	→	→	SYM
ejpam-4953	37	39	+	+	NOUN
ejpam-4953	37	40	∞	∞	NUM
ejpam-4953	37	41	)	)	PUNCT
ejpam-4953	37	42	.	.	PUNCT
ejpam-4953	38	1	definition	definition	NOUN
ejpam-4953	38	2	3	3	NUM
ejpam-4953	38	3	.	.	PUNCT
ejpam-4953	39	1	[	[	X
ejpam-4953	39	2	4	4	NUM
ejpam-4953	39	3	]	]	PUNCT
ejpam-4953	39	4	.	.	PUNCT
ejpam-4953	40	1	let	let	AUX
ejpam-4953	40	2	(	(	PUNCT
ejpam-4953	40	3	ḧ	ḧ	NOUN
ejpam-4953	40	4	,	,	PUNCT
ejpam-4953	40	5	p	p	X
ejpam-4953	40	6	)	)	PUNCT
ejpam-4953	40	7	be	be	AUX
ejpam-4953	40	8	a	a	DET
ejpam-4953	40	9	mms	mms	NOUN
ejpam-4953	40	10	and	and	CCONJ
ejpam-4953	40	11	{	{	PUNCT
ejpam-4953	40	12	ϖn	ϖn	NOUN
ejpam-4953	40	13	}	}	PUNCT
ejpam-4953	40	14	be	be	AUX
ejpam-4953	40	15	a	a	DET
ejpam-4953	40	16	sequence	sequence	NOUN
ejpam-4953	40	17	in	in	ADP
ejpam-4953	40	18	ḧ.	ḧ.	PROPN
ejpam-4953	40	19	•	•	ADV
ejpam-4953	40	20	then	then	ADV
ejpam-4953	40	21	{	{	PUNCT
ejpam-4953	40	22	ϖn	ϖn	NOUN
ejpam-4953	40	23	}	}	PUNCT
ejpam-4953	40	24	is	be	AUX
ejpam-4953	40	25	said	say	VERB
ejpam-4953	40	26	to	to	PART
ejpam-4953	40	27	be	be	AUX
ejpam-4953	40	28	multiplicative	multiplicative	ADJ
ejpam-4953	40	29	cauchy	cauchy	ADJ
ejpam-4953	40	30	sequence	sequence	NOUN
ejpam-4953	40	31	if	if	SCONJ
ejpam-4953	40	32	for	for	ADP
ejpam-4953	40	33	ϵ	ϵ	PROPN
ejpam-4953	40	34	>	>	X
ejpam-4953	40	35	1	1	NUM
ejpam-4953	40	36	,	,	PUNCT
ejpam-4953	40	37	there	there	PRON
ejpam-4953	40	38	exists	exist	VERB
ejpam-4953	40	39	a	a	DET
ejpam-4953	40	40	positive	positive	ADJ
ejpam-4953	40	41	integer	integer	NOUN
ejpam-4953	40	42	n	n	PRON
ejpam-4953	40	43	∈	∈	PROPN
ejpam-4953	40	44	n	n	PRON
ejpam-4953	40	45	such	such	ADJ
ejpam-4953	40	46	that	that	SCONJ
ejpam-4953	40	47	d(ϖm	d(ϖm	PROPN
ejpam-4953	40	48	,	,	PUNCT
ejpam-4953	40	49	ϖn	ϖn	NOUN
ejpam-4953	40	50	)	)	PUNCT
ejpam-4953	40	51	<	<	X
ejpam-4953	40	52	ϵ	ϵ	X
ejpam-4953	40	53	for	for	ADP
ejpam-4953	40	54	all	all	DET
ejpam-4953	40	55	n	n	CCONJ
ejpam-4953	40	56	,	,	PUNCT
ejpam-4953	40	57	m	m	PROPN
ejpam-4953	40	58	≥	≥	NOUN
ejpam-4953	40	59	n.	n.	NOUN
ejpam-4953	40	60	•	•	NOUN
ejpam-4953	40	61	then	then	ADV
ejpam-4953	40	62	{	{	PUNCT
ejpam-4953	40	63	ϖn	ϖn	NOUN
ejpam-4953	40	64	}	}	PUNCT
ejpam-4953	40	65	is	be	AUX
ejpam-4953	40	66	said	say	VERB
ejpam-4953	40	67	to	to	PART
ejpam-4953	40	68	be	be	AUX
ejpam-4953	40	69	multiplicative	multiplicative	ADJ
ejpam-4953	40	70	cauchy	cauchy	NOUN
ejpam-4953	40	71	if	if	SCONJ
ejpam-4953	40	72	and	and	CCONJ
ejpam-4953	40	73	only	only	ADV
ejpam-4953	40	74	if	if	SCONJ
ejpam-4953	40	75	p(ϖn	p(ϖn	NOUN
ejpam-4953	40	76	,	,	PUNCT
ejpam-4953	40	77	ϖm	ϖm	NOUN
ejpam-4953	40	78	)	)	PUNCT
ejpam-4953	40	79	→	→	SYM
ejpam-4953	40	80	1(n	1(n	NUM
ejpam-4953	40	81	,	,	PUNCT
ejpam-4953	40	82	m	m	PROPN
ejpam-4953	40	83	→	→	SYM
ejpam-4953	40	84	+	+	ADJ
ejpam-4953	40	85	∞	∞	NUM
ejpam-4953	40	86	)	)	PUNCT
ejpam-4953	40	87	.	.	PUNCT
ejpam-4953	41	1	definition	definition	NOUN
ejpam-4953	41	2	4	4	NUM
ejpam-4953	41	3	.	.	PUNCT
ejpam-4953	42	1	[	[	X
ejpam-4953	42	2	4	4	NUM
ejpam-4953	42	3	]	]	PUNCT
ejpam-4953	42	4	.	.	PUNCT
ejpam-4953	43	1	if	if	SCONJ
ejpam-4953	43	2	every	every	DET
ejpam-4953	43	3	multiplicative	multiplicative	ADJ
ejpam-4953	43	4	cauchy	cauchy	NOUN
ejpam-4953	43	5	sequence	sequence	NOUN
ejpam-4953	43	6	in	in	ADP
ejpam-4953	43	7	(	(	PUNCT
ejpam-4953	43	8	ḧ	ḧ	NOUN
ejpam-4953	43	9	,	,	PUNCT
ejpam-4953	43	10	p	p	NOUN
ejpam-4953	43	11	)	)	PUNCT
ejpam-4953	43	12	is	be	AUX
ejpam-4953	43	13	multiplicative	multiplicative	ADJ
ejpam-4953	43	14	convergent	convergent	NOUN
ejpam-4953	43	15	in	in	ADP
ejpam-4953	43	16	ḧ	ḧ	NOUN
ejpam-4953	43	17	,	,	PUNCT
ejpam-4953	43	18	then	then	ADV
ejpam-4953	43	19	mms	mms	NOUN
ejpam-4953	43	20	(	(	PUNCT
ejpam-4953	43	21	ḧ	ḧ	NOUN
ejpam-4953	43	22	,	,	PUNCT
ejpam-4953	43	23	p	p	NOUN
ejpam-4953	43	24	)	)	PUNCT
ejpam-4953	43	25	is	be	AUX
ejpam-4953	43	26	said	say	VERB
ejpam-4953	43	27	to	to	PART
ejpam-4953	43	28	be	be	AUX
ejpam-4953	43	29	multiplicative	multiplicative	ADJ
ejpam-4953	43	30	complete	complete	ADJ
ejpam-4953	43	31	definition	definition	NOUN
ejpam-4953	43	32	5	5	NUM
ejpam-4953	43	33	.	.	PUNCT
ejpam-4953	44	1	[	[	X
ejpam-4953	44	2	22	22	NUM
ejpam-4953	44	3	]	]	PUNCT
ejpam-4953	44	4	.	.	PUNCT
ejpam-4953	45	1	let	let	VERB
ejpam-4953	45	2	ḧ	ḧ	PRON
ejpam-4953	45	3	be	be	AUX
ejpam-4953	45	4	a	a	DET
ejpam-4953	45	5	nonempty	nonempty	ADJ
ejpam-4953	45	6	set	set	VERB
ejpam-4953	45	7	.	.	PUNCT
ejpam-4953	46	1	a	a	DET
ejpam-4953	46	2	subset	subset	NOUN
ejpam-4953	46	3	r̈	r̈	NOUN
ejpam-4953	46	4	of	of	ADP
ejpam-4953	46	5	ḧ2	ḧ2	NOUN
ejpam-4953	46	6	is	be	AUX
ejpam-4953	46	7	called	call	VERB
ejpam-4953	46	8	a	a	DET
ejpam-4953	46	9	binary	binary	ADJ
ejpam-4953	46	10	relation	relation	NOUN
ejpam-4953	46	11	on	on	ADP
ejpam-4953	46	12	ḧ.	ḧ.	PROPN
ejpam-4953	46	13	the	the	DET
ejpam-4953	46	14	subsets	subset	NOUN
ejpam-4953	46	15	,	,	PUNCT
ejpam-4953	46	16	ḧ2	ḧ2	NOUN
ejpam-4953	46	17	and	and	CCONJ
ejpam-4953	46	18	ϕ	ϕ	PROPN
ejpam-4953	46	19	of	of	ADP
ejpam-4953	46	20	ḧ2	ḧ2	NOUN
ejpam-4953	46	21	are	be	AUX
ejpam-4953	46	22	called	call	VERB
ejpam-4953	46	23	the	the	DET
ejpam-4953	46	24	universal	universal	ADJ
ejpam-4953	46	25	relation	relation	NOUN
ejpam-4953	46	26	and	and	CCONJ
ejpam-4953	46	27	empty	empty	ADJ
ejpam-4953	46	28	relation	relation	NOUN
ejpam-4953	46	29	respectively	respectively	ADV
ejpam-4953	46	30	.	.	PUNCT
ejpam-4953	47	1	definition	definition	NOUN
ejpam-4953	47	2	6	6	NUM
ejpam-4953	47	3	.	.	PUNCT
ejpam-4953	48	1	[	[	X
ejpam-4953	48	2	22	22	NUM
ejpam-4953	48	3	]	]	PUNCT
ejpam-4953	48	4	.	.	PUNCT
ejpam-4953	49	1	let	let	VERB
ejpam-4953	49	2	r̈	r̈	NOUN
ejpam-4953	49	3	be	be	AUX
ejpam-4953	49	4	a	a	DET
ejpam-4953	49	5	binary	binary	ADJ
ejpam-4953	49	6	relation	relation	NOUN
ejpam-4953	49	7	on	on	ADP
ejpam-4953	49	8	a	a	DET
ejpam-4953	49	9	nonempty	nonempty	ADV
ejpam-4953	49	10	set	set	VERB
ejpam-4953	49	11	ḧ.	ḧ.	PROPN
ejpam-4953	49	12	for	for	ADP
ejpam-4953	49	13	ϖ	ϖ	PROPN
ejpam-4953	49	14	,	,	PUNCT
ejpam-4953	49	15	ρ	ρ	PROPN
ejpam-4953	49	16	∈	∈	PROPN
ejpam-4953	49	17	ḧ	ḧ	NOUN
ejpam-4953	49	18	,	,	PUNCT
ejpam-4953	49	19	we	we	PRON
ejpam-4953	49	20	say	say	VERB
ejpam-4953	49	21	that	that	SCONJ
ejpam-4953	49	22	ϖ	ϖ	PROPN
ejpam-4953	49	23	and	and	CCONJ
ejpam-4953	49	24	ρ	ρ	PROPN
ejpam-4953	49	25	are	be	AUX
ejpam-4953	49	26	r̈-comparative	r̈-comparative	ADJ
ejpam-4953	49	27	if	if	SCONJ
ejpam-4953	49	28	either	either	CCONJ
ejpam-4953	49	29	(	(	PUNCT
ejpam-4953	49	30	ϖ	ϖ	NOUN
ejpam-4953	49	31	,	,	PUNCT
ejpam-4953	49	32	ρ	ρ	NOUN
ejpam-4953	49	33	)	)	PUNCT
ejpam-4953	49	34	∈	∈	PROPN
ejpam-4953	49	35	r̈	r̈	VERB
ejpam-4953	49	36	or	or	CCONJ
ejpam-4953	49	37	(	(	PUNCT
ejpam-4953	49	38	ρ,ϖ	ρ,ϖ	NOUN
ejpam-4953	49	39	)	)	PUNCT
ejpam-4953	49	40	∈	∈	NOUN
ejpam-4953	50	1	r̈.	r̈.	PROPN
ejpam-4953	50	2	we	we	PRON
ejpam-4953	50	3	denote	denote	VERB
ejpam-4953	50	4	it	it	PRON
ejpam-4953	50	5	by	by	ADP
ejpam-4953	50	6	[	[	X
ejpam-4953	50	7	ϖ	ϖ	X
ejpam-4953	50	8	,	,	PUNCT
ejpam-4953	50	9	ρ	ρ	NOUN
ejpam-4953	50	10	]	]	X
ejpam-4953	50	11	∈	∈	PROPN
ejpam-4953	50	12	r̈	r̈	PROPN
ejpam-4953	50	13	i.	i.	PROPN
ejpam-4953	50	14	alshammari	alshammari	PROPN
ejpam-4953	50	15	et	et	PROPN
ejpam-4953	50	16	al	al	PROPN
ejpam-4953	50	17	.	.	PUNCT
ejpam-4953	50	18	/	/	SYM
ejpam-4953	50	19	eur	eur	PROPN
ejpam-4953	50	20	.	.	PUNCT
ejpam-4953	51	1	j.	j.	PROPN
ejpam-4953	51	2	pure	pure	PROPN
ejpam-4953	51	3	appl	appl	PROPN
ejpam-4953	51	4	.	.	PROPN
ejpam-4953	51	5	math	math	PROPN
ejpam-4953	51	6	,	,	PUNCT
ejpam-4953	51	7	16	16	NUM
ejpam-4953	51	8	(	(	PUNCT
ejpam-4953	51	9	4	4	NUM
ejpam-4953	51	10	)	)	PUNCT
ejpam-4953	51	11	(	(	PUNCT
ejpam-4953	51	12	2023	2023	NUM
ejpam-4953	51	13	)	)	PUNCT
ejpam-4953	51	14	,	,	PUNCT
ejpam-4953	51	15	2405	2405	NUM
ejpam-4953	51	16	-	-	SYM
ejpam-4953	51	17	2418	2418	NUM
ejpam-4953	51	18	2407	2407	NUM
ejpam-4953	51	19	proposition	proposition	NOUN
ejpam-4953	51	20	1	1	NUM
ejpam-4953	51	21	.	.	PUNCT
ejpam-4953	52	1	if	if	SCONJ
ejpam-4953	52	2	(	(	PUNCT
ejpam-4953	52	3	ḧ	ḧ	NOUN
ejpam-4953	52	4	,	,	PUNCT
ejpam-4953	52	5	p	p	X
ejpam-4953	52	6	,	,	PUNCT
ejpam-4953	52	7	k	k	X
ejpam-4953	52	8	≥	≥	NUM
ejpam-4953	52	9	1	1	NUM
ejpam-4953	52	10	)	)	PUNCT
ejpam-4953	52	11	is	be	AUX
ejpam-4953	52	12	a	a	DET
ejpam-4953	52	13	b	b	NOUN
ejpam-4953	52	14	-	-	PUNCT
ejpam-4953	52	15	metric	metric	ADJ
ejpam-4953	52	16	space	space	NOUN
ejpam-4953	52	17	,	,	PUNCT
ejpam-4953	52	18	r̈	r̈	VERB
ejpam-4953	52	19	is	be	AUX
ejpam-4953	52	20	a	a	DET
ejpam-4953	52	21	binary	binary	ADJ
ejpam-4953	52	22	relation	relation	NOUN
ejpam-4953	52	23	on	on	ADP
ejpam-4953	52	24	ḧ	ḧ	NOUN
ejpam-4953	52	25	,	,	PUNCT
ejpam-4953	52	26	f̈	f̈	PROPN
ejpam-4953	52	27	is	be	AUX
ejpam-4953	52	28	a	a	DET
ejpam-4953	52	29	self	self	NOUN
ejpam-4953	52	30	-	-	PUNCT
ejpam-4953	52	31	mapping	mapping	NOUN
ejpam-4953	52	32	on	on	ADP
ejpam-4953	52	33	ḧ	ḧ	NOUN
ejpam-4953	52	34	and	and	CCONJ
ejpam-4953	52	35	λ	λ	X
ejpam-4953	52	36	∈	∈	PROPN
ejpam-4953	53	1	[	[	X
ejpam-4953	53	2	0	0	NUM
ejpam-4953	53	3	,	,	PUNCT
ejpam-4953	53	4	1k	1k	NUM
ejpam-4953	53	5	)	)	PUNCT
ejpam-4953	53	6	,	,	PUNCT
ejpam-4953	53	7	then	then	ADV
ejpam-4953	53	8	these	these	DET
ejpam-4953	53	9	conditions	condition	NOUN
ejpam-4953	53	10	are	be	AUX
ejpam-4953	53	11	equivalent	equivalent	ADJ
ejpam-4953	53	12	.	.	PUNCT
ejpam-4953	54	1	(	(	PUNCT
ejpam-4953	54	2	i	i	NOUN
ejpam-4953	54	3	)	)	PUNCT
ejpam-4953	54	4	p(f̈ϖ	p(f̈ϖ	PROPN
ejpam-4953	54	5	,	,	PUNCT
ejpam-4953	54	6	f̈ρ	f̈ρ	ADJ
ejpam-4953	54	7	)	)	PUNCT
ejpam-4953	54	8	≤	≤	NOUN
ejpam-4953	54	9	p(ϖ	p(ϖ	PROPN
ejpam-4953	54	10	,	,	PUNCT
ejpam-4953	54	11	ρ)λ	ρ)λ	ADJ
ejpam-4953	54	12	for	for	ADP
ejpam-4953	54	13	all	all	DET
ejpam-4953	54	14	ϖ	ϖ	PROPN
ejpam-4953	54	15	,	,	PUNCT
ejpam-4953	54	16	ρ	ρ	PROPN
ejpam-4953	54	17	∈	∈	PROPN
ejpam-4953	54	18	ḧ	ḧ	NOUN
ejpam-4953	54	19	with	with	ADP
ejpam-4953	54	20	(	(	PUNCT
ejpam-4953	54	21	ϖ	ϖ	NOUN
ejpam-4953	54	22	,	,	PUNCT
ejpam-4953	54	23	ρ	ρ	NOUN
ejpam-4953	54	24	)	)	PUNCT
ejpam-4953	54	25	∈	∈	PROPN
ejpam-4953	54	26	r̈	r̈	NOUN
ejpam-4953	54	27	,	,	PUNCT
ejpam-4953	54	28	(	(	PUNCT
ejpam-4953	54	29	ii	ii	NOUN
ejpam-4953	54	30	)	)	PUNCT
ejpam-4953	54	31	p(f̈ϖ	p(f̈ϖ	PROPN
ejpam-4953	54	32	,	,	PUNCT
ejpam-4953	54	33	f̈ρ	f̈ρ	ADJ
ejpam-4953	54	34	)	)	PUNCT
ejpam-4953	54	35	≤	≤	NOUN
ejpam-4953	54	36	p(ϖ	p(ϖ	PROPN
ejpam-4953	54	37	,	,	PUNCT
ejpam-4953	54	38	ρ)λ	ρ)λ	ADJ
ejpam-4953	54	39	for	for	ADP
ejpam-4953	54	40	all	all	DET
ejpam-4953	54	41	ϖ	ϖ	PROPN
ejpam-4953	54	42	,	,	PUNCT
ejpam-4953	54	43	ρ	ρ	PROPN
ejpam-4953	54	44	∈	∈	PROPN
ejpam-4953	54	45	ḧ	ḧ	NOUN
ejpam-4953	54	46	with	with	ADP
ejpam-4953	54	47	[	[	X
ejpam-4953	54	48	ϖ	ϖ	X
ejpam-4953	54	49	,	,	PUNCT
ejpam-4953	54	50	ρ	ρ	X
ejpam-4953	54	51	]	]	X
ejpam-4953	54	52	∈	∈	PROPN
ejpam-4953	54	53	r̈.	r̈.	PROPN
ejpam-4953	54	54	proof	proof	NOUN
ejpam-4953	54	55	.	.	PUNCT
ejpam-4953	55	1	the	the	DET
ejpam-4953	55	2	implication	implication	NOUN
ejpam-4953	55	3	(	(	PUNCT
ejpam-4953	55	4	ii	ii	NOUN
ejpam-4953	55	5	)	)	PUNCT
ejpam-4953	56	1	=	=	NOUN
ejpam-4953	56	2	⇒	⇒	NOUN
ejpam-4953	56	3	(	(	PUNCT
ejpam-4953	56	4	i	i	NOUN
ejpam-4953	56	5	)	)	PUNCT
ejpam-4953	56	6	is	be	AUX
ejpam-4953	56	7	trivial	trivial	ADJ
ejpam-4953	56	8	.	.	PUNCT
ejpam-4953	57	1	coversely	coversely	ADV
ejpam-4953	57	2	,	,	PUNCT
ejpam-4953	57	3	we	we	PRON
ejpam-4953	57	4	assume	assume	VERB
ejpam-4953	57	5	that	that	SCONJ
ejpam-4953	57	6	(	(	PUNCT
ejpam-4953	57	7	i	i	NOUN
ejpam-4953	57	8	)	)	PUNCT
ejpam-4953	57	9	holds	hold	VERB
ejpam-4953	57	10	.	.	PUNCT
ejpam-4953	58	1	take	take	VERB
ejpam-4953	58	2	ϖ	ϖ	NOUN
ejpam-4953	58	3	,	,	PUNCT
ejpam-4953	58	4	ρ	ρ	PROPN
ejpam-4953	58	5	∈	∈	PROPN
ejpam-4953	58	6	ḧ	ḧ	NOUN
ejpam-4953	58	7	with	with	ADP
ejpam-4953	58	8	[	[	X
ejpam-4953	58	9	ϖ	ϖ	X
ejpam-4953	58	10	,	,	PUNCT
ejpam-4953	58	11	ρ	ρ	X
ejpam-4953	58	12	]	]	X
ejpam-4953	58	13	∈	∈	PROPN
ejpam-4953	58	14	r̈.	r̈.	PROPN
ejpam-4953	59	1	if	if	SCONJ
ejpam-4953	59	2	(	(	PUNCT
ejpam-4953	59	3	ϖ	ϖ	NOUN
ejpam-4953	59	4	,	,	PUNCT
ejpam-4953	59	5	ρ	ρ	NOUN
ejpam-4953	59	6	)	)	PUNCT
ejpam-4953	59	7	∈	∈	PROPN
ejpam-4953	59	8	r̈	r̈	NOUN
ejpam-4953	59	9	,	,	PUNCT
ejpam-4953	59	10	then	then	ADV
ejpam-4953	59	11	(	(	PUNCT
ejpam-4953	59	12	ii	ii	NOUN
ejpam-4953	59	13	)	)	PUNCT
ejpam-4953	59	14	directly	directly	ADV
ejpam-4953	59	15	follows	follow	VERB
ejpam-4953	59	16	from	from	ADP
ejpam-4953	59	17	(	(	PUNCT
ejpam-4953	59	18	1	1	NUM
ejpam-4953	59	19	)	)	PUNCT
ejpam-4953	59	20	.	.	PUNCT
ejpam-4953	60	1	but	but	CCONJ
ejpam-4953	60	2	,	,	PUNCT
ejpam-4953	60	3	if	if	SCONJ
ejpam-4953	60	4	(	(	PUNCT
ejpam-4953	60	5	ρ,ϖ	ρ,ϖ	NOUN
ejpam-4953	60	6	)	)	PUNCT
ejpam-4953	60	7	∈	∈	PROPN
ejpam-4953	60	8	r̈	r̈	NOUN
ejpam-4953	60	9	,	,	PUNCT
ejpam-4953	60	10	then	then	ADV
ejpam-4953	60	11	using	use	VERB
ejpam-4953	60	12	the	the	DET
ejpam-4953	60	13	symmetry	symmetry	NOUN
ejpam-4953	60	14	of	of	ADP
ejpam-4953	60	15	p	p	PROPN
ejpam-4953	60	16	and	and	CCONJ
ejpam-4953	60	17	(	(	PUNCT
ejpam-4953	60	18	i	i	NOUN
ejpam-4953	60	19	)	)	PUNCT
ejpam-4953	60	20	,	,	PUNCT
ejpam-4953	60	21	we	we	PRON
ejpam-4953	60	22	obtain	obtain	VERB
ejpam-4953	60	23	p(f̈ϖ	p(f̈ϖ	PROPN
ejpam-4953	60	24	,	,	PUNCT
ejpam-4953	60	25	f̈ρ	f̈ρ	ADJ
ejpam-4953	60	26	)	)	PUNCT
ejpam-4953	60	27	=	=	SYM
ejpam-4953	61	1	p(f̈ρ	p(f̈ρ	ADJ
ejpam-4953	61	2	,	,	PUNCT
ejpam-4953	61	3	f̈ϖ	f̈ϖ	NOUN
ejpam-4953	61	4	)	)	PUNCT
ejpam-4953	61	5	≤	≤	NOUN
ejpam-4953	62	1	p(ρ,ϖ)λ	p(ρ,ϖ)λ	ADJ
ejpam-4953	62	2	=	=	PUNCT
ejpam-4953	62	3	p(ρ,ϖ)λ	p(ρ,ϖ)λ	ADJ
ejpam-4953	62	4	.	.	PUNCT
ejpam-4953	63	1	which	which	PRON
ejpam-4953	63	2	shows	show	VERB
ejpam-4953	63	3	that	that	SCONJ
ejpam-4953	63	4	(	(	PUNCT
ejpam-4953	63	5	i	i	NOUN
ejpam-4953	63	6	)	)	PUNCT
ejpam-4953	64	1	=	=	NOUN
ejpam-4953	64	2	⇒	⇒	NOUN
ejpam-4953	64	3	(	(	PUNCT
ejpam-4953	64	4	ii	ii	NOUN
ejpam-4953	64	5	)	)	PUNCT
ejpam-4953	64	6	.	.	PUNCT
ejpam-4953	65	1	proposition	proposition	NOUN
ejpam-4953	65	2	2	2	NUM
ejpam-4953	65	3	.	.	PUNCT
ejpam-4953	66	1	if	if	SCONJ
ejpam-4953	66	2	(	(	PUNCT
ejpam-4953	66	3	ḧ	ḧ	NOUN
ejpam-4953	66	4	,	,	PUNCT
ejpam-4953	66	5	p	p	X
ejpam-4953	66	6	,	,	PUNCT
ejpam-4953	66	7	k	k	X
ejpam-4953	66	8	≥	≥	NUM
ejpam-4953	66	9	1	1	NUM
ejpam-4953	66	10	)	)	PUNCT
ejpam-4953	66	11	is	be	AUX
ejpam-4953	66	12	a	a	DET
ejpam-4953	66	13	b	b	NOUN
ejpam-4953	66	14	-	-	PUNCT
ejpam-4953	66	15	metric	metric	ADJ
ejpam-4953	66	16	space	space	NOUN
ejpam-4953	66	17	,	,	PUNCT
ejpam-4953	66	18	r̈	r̈	VERB
ejpam-4953	66	19	is	be	AUX
ejpam-4953	66	20	a	a	DET
ejpam-4953	66	21	binary	binary	ADJ
ejpam-4953	66	22	relation	relation	NOUN
ejpam-4953	66	23	on	on	ADP
ejpam-4953	66	24	ḧ	ḧ	NOUN
ejpam-4953	66	25	,	,	PUNCT
ejpam-4953	66	26	f̈	f̈	PRON
ejpam-4953	66	27	and	and	CCONJ
ejpam-4953	66	28	s	s	NOUN
ejpam-4953	66	29	are	be	AUX
ejpam-4953	66	30	self	self	NOUN
ejpam-4953	66	31	-	-	PUNCT
ejpam-4953	66	32	mapping	mapping	NOUN
ejpam-4953	66	33	on	on	ADP
ejpam-4953	66	34	ḧ	ḧ	NOUN
ejpam-4953	66	35	and	and	CCONJ
ejpam-4953	66	36	λ	λ	X
ejpam-4953	66	37	∈	∈	PROPN
ejpam-4953	67	1	[	[	X
ejpam-4953	67	2	0	0	NUM
ejpam-4953	67	3	,	,	PUNCT
ejpam-4953	67	4	1k	1k	NUM
ejpam-4953	67	5	)	)	PUNCT
ejpam-4953	67	6	,	,	PUNCT
ejpam-4953	67	7	then	then	ADV
ejpam-4953	67	8	these	these	DET
ejpam-4953	67	9	conditions	condition	NOUN
ejpam-4953	67	10	are	be	AUX
ejpam-4953	67	11	equivalent	equivalent	ADJ
ejpam-4953	67	12	.	.	PUNCT
ejpam-4953	68	1	(	(	PUNCT
ejpam-4953	68	2	1	1	X
ejpam-4953	68	3	)	)	PUNCT
ejpam-4953	68	4	p(f̈ϖ	p(f̈ϖ	PROPN
ejpam-4953	68	5	,	,	PUNCT
ejpam-4953	68	6	f̈ρ	f̈ρ	ADJ
ejpam-4953	68	7	)	)	PUNCT
ejpam-4953	68	8	≤	≤	NOUN
ejpam-4953	68	9	p(sϖ	p(sϖ	NOUN
ejpam-4953	68	10	,	,	PUNCT
ejpam-4953	68	11	sρ)λ	sρ)λ	PROPN
ejpam-4953	68	12	for	for	ADP
ejpam-4953	68	13	all	all	DET
ejpam-4953	68	14	ϖ	ϖ	PROPN
ejpam-4953	68	15	,	,	PUNCT
ejpam-4953	68	16	ρ	ρ	PROPN
ejpam-4953	68	17	∈	∈	PROPN
ejpam-4953	68	18	ḧ	ḧ	NOUN
ejpam-4953	68	19	with	with	ADP
ejpam-4953	68	20	(	(	PUNCT
ejpam-4953	68	21	ϖ	ϖ	NOUN
ejpam-4953	68	22	,	,	PUNCT
ejpam-4953	68	23	ρ	ρ	NOUN
ejpam-4953	68	24	)	)	PUNCT
ejpam-4953	68	25	∈	∈	PROPN
ejpam-4953	68	26	r̈	r̈	NOUN
ejpam-4953	68	27	,	,	PUNCT
ejpam-4953	68	28	(	(	PUNCT
ejpam-4953	68	29	2	2	X
ejpam-4953	68	30	)	)	PUNCT
ejpam-4953	68	31	p(f̈ϖ	p(f̈ϖ	PROPN
ejpam-4953	68	32	,	,	PUNCT
ejpam-4953	68	33	f̈ρ	f̈ρ	ADJ
ejpam-4953	68	34	)	)	PUNCT
ejpam-4953	68	35	≤	≤	NOUN
ejpam-4953	68	36	p(sϖ	p(sϖ	NOUN
ejpam-4953	68	37	,	,	PUNCT
ejpam-4953	68	38	sρ)λ	sρ)λ	PROPN
ejpam-4953	68	39	for	for	ADP
ejpam-4953	68	40	all	all	DET
ejpam-4953	68	41	ϖ	ϖ	PROPN
ejpam-4953	68	42	,	,	PUNCT
ejpam-4953	68	43	ρ	ρ	PROPN
ejpam-4953	68	44	∈	∈	PROPN
ejpam-4953	68	45	ḧ	ḧ	NOUN
ejpam-4953	68	46	with	with	ADP
ejpam-4953	68	47	[	[	X
ejpam-4953	68	48	ϖ	ϖ	X
ejpam-4953	68	49	,	,	PUNCT
ejpam-4953	68	50	ρ	ρ	X
ejpam-4953	68	51	]	]	X
ejpam-4953	68	52	∈	∈	PROPN
ejpam-4953	68	53	r̈.	r̈.	PROPN
ejpam-4953	68	54	definition	definition	NOUN
ejpam-4953	68	55	7	7	NUM
ejpam-4953	68	56	.	.	PUNCT
ejpam-4953	69	1	[	[	X
ejpam-4953	69	2	23].“let	23].“let	NOUN
ejpam-4953	69	3	ḧ	ḧ	AUX
ejpam-4953	69	4	be	be	AUX
ejpam-4953	69	5	a	a	DET
ejpam-4953	69	6	non	non	ADJ
ejpam-4953	69	7	-	-	ADJ
ejpam-4953	69	8	empty	empty	ADJ
ejpam-4953	69	9	set	set	NOUN
ejpam-4953	69	10	and	and	CCONJ
ejpam-4953	69	11	r̈	r̈	VERB
ejpam-4953	69	12	be	be	AUX
ejpam-4953	69	13	a	a	DET
ejpam-4953	69	14	binary	binary	ADJ
ejpam-4953	69	15	relation	relation	NOUN
ejpam-4953	69	16	on	on	ADP
ejpam-4953	69	17	ḧ.	ḧ.	PROPN
ejpam-4953	69	18	(	(	PUNCT
ejpam-4953	69	19	1	1	X
ejpam-4953	69	20	)	)	PUNCT
ejpam-4953	69	21	the	the	DET
ejpam-4953	69	22	inverse	inverse	NOUN
ejpam-4953	69	23	,	,	PUNCT
ejpam-4953	69	24	transpose	transpose	NOUN
ejpam-4953	69	25	or	or	CCONJ
ejpam-4953	69	26	dual	dual	ADJ
ejpam-4953	69	27	relation	relation	NOUN
ejpam-4953	69	28	of	of	ADP
ejpam-4953	69	29	r̈	r̈	NOUN
ejpam-4953	69	30	,	,	PUNCT
ejpam-4953	69	31	denoted	denote	VERB
ejpam-4953	69	32	by	by	ADP
ejpam-4953	69	33	r̈−1	r̈−1	PROPN
ejpam-4953	69	34	is	be	AUX
ejpam-4953	69	35	defined	define	VERB
ejpam-4953	69	36	by	by	ADP
ejpam-4953	69	37	r̈−1	r̈−1	NOUN
ejpam-4953	69	38	=	=	PUNCT
ejpam-4953	69	39	{	{	PUNCT
ejpam-4953	69	40	(	(	PUNCT
ejpam-4953	69	41	ϖ	ϖ	NOUN
ejpam-4953	69	42	,	,	PUNCT
ejpam-4953	69	43	ρ	ρ	NOUN
ejpam-4953	69	44	)	)	PUNCT
ejpam-4953	69	45	∈	∈	PROPN
ejpam-4953	69	46	ḧ2	ḧ2	NOUN
ejpam-4953	69	47	:	:	PUNCT
ejpam-4953	69	48	(	(	PUNCT
ejpam-4953	69	49	ρ,ϖ	ρ,ϖ	NOUN
ejpam-4953	69	50	)	)	PUNCT
ejpam-4953	69	51	∈	∈	PROPN
ejpam-4953	69	52	r̈	r̈	NOUN
ejpam-4953	69	53	}	}	PUNCT
ejpam-4953	69	54	(	(	PUNCT
ejpam-4953	69	55	2	2	X
ejpam-4953	69	56	)	)	PUNCT
ejpam-4953	69	57	the	the	DET
ejpam-4953	69	58	reflexive	reflexive	ADJ
ejpam-4953	69	59	closure	closure	NOUN
ejpam-4953	69	60	of	of	ADP
ejpam-4953	69	61	r̈	r̈	NOUN
ejpam-4953	69	62	,	,	PUNCT
ejpam-4953	69	63	denoted	denote	VERB
ejpam-4953	69	64	by	by	ADP
ejpam-4953	69	65	r̈	r̈	NOUN
ejpam-4953	69	66	#	#	NUM
ejpam-4953	69	67	,	,	PUNCT
ejpam-4953	69	68	is	be	AUX
ejpam-4953	69	69	defined	define	VERB
ejpam-4953	69	70	to	to	PART
ejpam-4953	69	71	be	be	AUX
ejpam-4953	69	72	the	the	DET
ejpam-4953	69	73	set	set	NOUN
ejpam-4953	69	74	r̈	r̈	NOUN
ejpam-4953	69	75	∪	∪	ADJ
ejpam-4953	69	76	△	△	NOUN
ejpam-4953	69	77	ϖ	ϖ	X
ejpam-4953	69	78	(	(	PUNCT
ejpam-4953	69	79	i.e.	i.e.	X
ejpam-4953	69	80	,	,	PUNCT
ejpam-4953	69	81	r̈	r̈	NOUN
ejpam-4953	69	82	#	#	NOUN
ejpam-4953	69	83	:	:	PUNCT
ejpam-4953	69	84	=	=	NOUN
ejpam-4953	69	85	r̈	r̈	NOUN
ejpam-4953	69	86	∪	∪	VERB
ejpam-4953	69	87	△	△	NOUN
ejpam-4953	69	88	ϖ	ϖ	NOUN
ejpam-4953	69	89	)	)	PUNCT
ejpam-4953	69	90	.	.	PUNCT
ejpam-4953	70	1	(	(	PUNCT
ejpam-4953	70	2	3	3	X
ejpam-4953	70	3	)	)	PUNCT
ejpam-4953	70	4	the	the	DET
ejpam-4953	70	5	symmetric	symmetric	ADJ
ejpam-4953	70	6	closure	closure	NOUN
ejpam-4953	70	7	of	of	ADP
ejpam-4953	70	8	r̈	r̈	NOUN
ejpam-4953	70	9	,	,	PUNCT
ejpam-4953	70	10	denoted	denote	VERB
ejpam-4953	70	11	by	by	ADP
ejpam-4953	70	12	r̈s	r̈s	PROPN
ejpam-4953	70	13	,	,	PUNCT
ejpam-4953	70	14	is	be	AUX
ejpam-4953	70	15	defined	define	VERB
ejpam-4953	70	16	to	to	PART
ejpam-4953	70	17	be	be	AUX
ejpam-4953	70	18	the	the	DET
ejpam-4953	70	19	set	set	NOUN
ejpam-4953	70	20	r̈	r̈	NOUN
ejpam-4953	70	21	∪r̈−1	∪r̈−1	PUNCT
ejpam-4953	70	22	(	(	PUNCT
ejpam-4953	70	23	i.e.	i.e.	X
ejpam-4953	70	24	,	,	PUNCT
ejpam-4953	70	25	r̈	r̈	NOUN
ejpam-4953	70	26	#	#	NOUN
ejpam-4953	70	27	:	:	PUNCT
ejpam-4953	70	28	=	=	NOUN
ejpam-4953	70	29	r̈	r̈	NOUN
ejpam-4953	70	30	∪r̈−1	∪r̈−1	VERB
ejpam-4953	70	31	)	)	PUNCT
ejpam-4953	70	32	.	.	PUNCT
ejpam-4953	71	1	proposition	proposition	NOUN
ejpam-4953	71	2	3	3	NUM
ejpam-4953	71	3	.	.	PUNCT
ejpam-4953	72	1	[	[	X
ejpam-4953	72	2	24	24	NUM
ejpam-4953	72	3	]	]	PUNCT
ejpam-4953	72	4	for	for	ADP
ejpam-4953	72	5	a	a	DET
ejpam-4953	72	6	binary	binary	ADJ
ejpam-4953	72	7	relation	relation	NOUN
ejpam-4953	72	8	r̈	r̈	NOUN
ejpam-4953	72	9	defined	define	VERB
ejpam-4953	72	10	on	on	ADP
ejpam-4953	72	11	a	a	DET
ejpam-4953	72	12	nonempty	nonempty	ADV
ejpam-4953	72	13	set	set	VERB
ejpam-4953	72	14	ḧ	ḧ	NOUN
ejpam-4953	72	15	,	,	PUNCT
ejpam-4953	72	16	(	(	PUNCT
ejpam-4953	72	17	ϖ	ϖ	NOUN
ejpam-4953	72	18	,	,	PUNCT
ejpam-4953	72	19	ρ	ρ	NOUN
ejpam-4953	72	20	)	)	PUNCT
ejpam-4953	72	21	∈	∈	PROPN
ejpam-4953	72	22	r̈s	r̈	NOUN
ejpam-4953	72	23	⇐	⇐	ADJ
ejpam-4953	72	24	⇒	⇒	PROPN
ejpam-4953	72	25	[	[	X
ejpam-4953	72	26	ϖ	ϖ	X
ejpam-4953	72	27	,	,	PUNCT
ejpam-4953	72	28	ρ	ρ	X
ejpam-4953	72	29	]	]	X
ejpam-4953	72	30	∈	∈	PROPN
ejpam-4953	72	31	r̈.	r̈.	PROPN
ejpam-4953	72	32	definition	definition	NOUN
ejpam-4953	72	33	8	8	NUM
ejpam-4953	72	34	.	.	PUNCT
ejpam-4953	73	1	[	[	X
ejpam-4953	73	2	24	24	NUM
ejpam-4953	73	3	]	]	PUNCT
ejpam-4953	73	4	.	.	PUNCT
ejpam-4953	74	1	let	let	VERB
ejpam-4953	74	2	ḧ	ḧ	PRON
ejpam-4953	74	3	be	be	AUX
ejpam-4953	74	4	a	a	DET
ejpam-4953	74	5	non	non	ADJ
ejpam-4953	74	6	-	-	ADJ
ejpam-4953	74	7	empty	empty	ADJ
ejpam-4953	74	8	set	set	NOUN
ejpam-4953	74	9	and	and	CCONJ
ejpam-4953	74	10	r̈	r̈	VERB
ejpam-4953	74	11	a	a	DET
ejpam-4953	74	12	binary	binary	ADJ
ejpam-4953	74	13	relation	relation	NOUN
ejpam-4953	74	14	on	on	ADP
ejpam-4953	74	15	ḧ.	ḧ.	PROPN
ejpam-4953	74	16	a	a	DET
ejpam-4953	74	17	sequence	sequence	NOUN
ejpam-4953	74	18	ϖn	ϖn	ADP
ejpam-4953	74	19	⊂	⊂	PROPN
ejpam-4953	74	20	ḧ	ḧ	NOUN
ejpam-4953	74	21	is	be	AUX
ejpam-4953	74	22	called	call	VERB
ejpam-4953	74	23	r̈preserving	r̈preserve	VERB
ejpam-4953	74	24	if	if	SCONJ
ejpam-4953	74	25	(	(	PUNCT
ejpam-4953	74	26	ϖn	ϖn	NOUN
ejpam-4953	74	27	,	,	PUNCT
ejpam-4953	74	28	ϖn+1	ϖn+1	ADJ
ejpam-4953	74	29	)	)	PUNCT
ejpam-4953	74	30	∈	∈	PROPN
ejpam-4953	74	31	r̈	r̈	VERB
ejpam-4953	74	32	for	for	ADP
ejpam-4953	74	33	all	all	DET
ejpam-4953	74	34	n	n	PRON
ejpam-4953	74	35	∈	∈	PROPN
ejpam-4953	74	36	n0	n0	PROPN
ejpam-4953	74	37	.	.	PUNCT
ejpam-4953	75	1	definition	definition	NOUN
ejpam-4953	75	2	9	9	NUM
ejpam-4953	75	3	.	.	PUNCT
ejpam-4953	76	1	[	[	X
ejpam-4953	76	2	24	24	NUM
ejpam-4953	76	3	]	]	X
ejpam-4953	76	4	let	let	VERB
ejpam-4953	76	5	(	(	PUNCT
ejpam-4953	76	6	ḧ,p	ḧ,p	PROPN
ejpam-4953	76	7	)	)	PUNCT
ejpam-4953	76	8	be	be	AUX
ejpam-4953	76	9	a	a	DET
ejpam-4953	76	10	metric	metric	ADJ
ejpam-4953	76	11	space	space	NOUN
ejpam-4953	76	12	.	.	PUNCT
ejpam-4953	77	1	a	a	DET
ejpam-4953	77	2	binary	binary	ADJ
ejpam-4953	77	3	relation	relation	NOUN
ejpam-4953	77	4	r̈	r̈	NOUN
ejpam-4953	77	5	defined	define	VERB
ejpam-4953	77	6	on	on	ADP
ejpam-4953	77	7	ḧ	ḧ	NOUN
ejpam-4953	77	8	is	be	AUX
ejpam-4953	77	9	called	call	VERB
ejpam-4953	77	10	p	p	NOUN
ejpam-4953	77	11	-	-	PUNCT
ejpam-4953	77	12	selfclosed	selfclose	VERB
ejpam-4953	77	13	if	if	SCONJ
ejpam-4953	77	14	whenever	whenever	SCONJ
ejpam-4953	77	15	{	{	PUNCT
ejpam-4953	77	16	ϖn	ϖn	NOUN
ejpam-4953	77	17	}	}	PUNCT
ejpam-4953	77	18	is	be	AUX
ejpam-4953	77	19	an	an	DET
ejpam-4953	77	20	r̈-preserving	r̈-preserve	VERB
ejpam-4953	77	21	sequence	sequence	NOUN
ejpam-4953	77	22	and	and	CCONJ
ejpam-4953	77	23	ϖn	ϖn	NOUN
ejpam-4953	77	24	p−→	p−→	NOUN
ejpam-4953	77	25	ϖ	ϖ	INTJ
ejpam-4953	77	26	then	then	ADV
ejpam-4953	77	27	there	there	PRON
ejpam-4953	77	28	exists	exist	VERB
ejpam-4953	77	29	a	a	DET
ejpam-4953	77	30	subsequence	subsequence	NOUN
ejpam-4953	77	31	{	{	PUNCT
ejpam-4953	77	32	ϖnk	ϖnk	NOUN
ejpam-4953	77	33	}	}	PUNCT
ejpam-4953	77	34	of	of	ADP
ejpam-4953	77	35	{	{	PUNCT
ejpam-4953	77	36	ϖn	ϖn	NOUN
ejpam-4953	77	37	}	}	PUNCT
ejpam-4953	77	38	with	with	ADP
ejpam-4953	77	39	[	[	X
ejpam-4953	77	40	ϖnk	ϖnk	NOUN
ejpam-4953	77	41	,	,	PUNCT
ejpam-4953	77	42	ϖ	ϖ	X
ejpam-4953	77	43	]	]	X
ejpam-4953	77	44	∈	∈	PROPN
ejpam-4953	77	45	r̈	r̈	VERB
ejpam-4953	77	46	for	for	ADP
ejpam-4953	77	47	all	all	DET
ejpam-4953	77	48	k	k	PROPN
ejpam-4953	77	49	∈	∈	PROPN
ejpam-4953	77	50	n0	n0	PROPN
ejpam-4953	77	51	.	.	PUNCT
ejpam-4953	77	52	i.	i.	PROPN
ejpam-4953	77	53	alshammari	alshammari	PROPN
ejpam-4953	77	54	et	et	PROPN
ejpam-4953	77	55	al	al	PROPN
ejpam-4953	77	56	.	.	PUNCT
ejpam-4953	77	57	/	/	SYM
ejpam-4953	77	58	eur	eur	PROPN
ejpam-4953	77	59	.	.	PUNCT
ejpam-4953	78	1	j.	j.	PROPN
ejpam-4953	78	2	pure	pure	PROPN
ejpam-4953	78	3	appl	appl	PROPN
ejpam-4953	78	4	.	.	PROPN
ejpam-4953	78	5	math	math	PROPN
ejpam-4953	78	6	,	,	PUNCT
ejpam-4953	78	7	16	16	NUM
ejpam-4953	78	8	(	(	PUNCT
ejpam-4953	78	9	4	4	NUM
ejpam-4953	78	10	)	)	PUNCT
ejpam-4953	78	11	(	(	PUNCT
ejpam-4953	78	12	2023	2023	NUM
ejpam-4953	78	13	)	)	PUNCT
ejpam-4953	78	14	,	,	PUNCT
ejpam-4953	78	15	2405	2405	NUM
ejpam-4953	78	16	-	-	SYM
ejpam-4953	78	17	2418	2418	NUM
ejpam-4953	78	18	2408	2408	NUM
ejpam-4953	78	19	definition	definition	NOUN
ejpam-4953	78	20	10	10	NUM
ejpam-4953	78	21	.	.	PUNCT
ejpam-4953	79	1	[	[	X
ejpam-4953	79	2	24	24	NUM
ejpam-4953	79	3	]	]	PUNCT
ejpam-4953	79	4	let	let	VERB
ejpam-4953	79	5	ḧ	ḧ	PRON
ejpam-4953	79	6	be	be	AUX
ejpam-4953	79	7	a	a	DET
ejpam-4953	79	8	nonempty	nonempty	ADV
ejpam-4953	79	9	set	set	VERB
ejpam-4953	79	10	and	and	CCONJ
ejpam-4953	79	11	f̈	f̈	X
ejpam-4953	79	12	a	a	DET
ejpam-4953	79	13	self	self	NOUN
ejpam-4953	79	14	-	-	PUNCT
ejpam-4953	79	15	mapping	mapping	NOUN
ejpam-4953	79	16	on	on	ADP
ejpam-4953	79	17	ḧ.	ḧ.	PROPN
ejpam-4953	79	18	a	a	DET
ejpam-4953	79	19	binary	binary	ADJ
ejpam-4953	79	20	relation	relation	NOUN
ejpam-4953	79	21	r̈	r̈	NOUN
ejpam-4953	79	22	defined	define	VERB
ejpam-4953	79	23	on	on	ADP
ejpam-4953	79	24	ḧ	ḧ	NOUN
ejpam-4953	79	25	is	be	AUX
ejpam-4953	79	26	called	call	VERB
ejpam-4953	79	27	f̈-closed	f̈-close	VERB
ejpam-4953	79	28	if	if	SCONJ
ejpam-4953	79	29	for	for	ADP
ejpam-4953	79	30	any	any	DET
ejpam-4953	79	31	ϖ	ϖ	NOUN
ejpam-4953	79	32	,	,	PUNCT
ejpam-4953	79	33	ρ	ρ	PROPN
ejpam-4953	79	34	∈	∈	PROPN
ejpam-4953	79	35	ḧ	ḧ	X
ejpam-4953	79	36	(	(	PUNCT
ejpam-4953	79	37	ϖ	ϖ	NOUN
ejpam-4953	79	38	,	,	PUNCT
ejpam-4953	79	39	ρ	ρ	NOUN
ejpam-4953	79	40	)	)	PUNCT
ejpam-4953	79	41	∈	∈	PROPN
ejpam-4953	79	42	r̈	r̈	VERB
ejpam-4953	79	43	=	=	X
ejpam-4953	79	44	⇒	⇒	NOUN
ejpam-4953	79	45	(	(	PUNCT
ejpam-4953	79	46	f̈ϖ	f̈ϖ	NOUN
ejpam-4953	79	47	,	,	PUNCT
ejpam-4953	79	48	f̈ρ	f̈ρ	ADJ
ejpam-4953	79	49	)	)	PUNCT
ejpam-4953	79	50	∈	∈	PROPN
ejpam-4953	79	51	r̈.	r̈.	PROPN
ejpam-4953	79	52	proposition	proposition	NOUN
ejpam-4953	79	53	4	4	NUM
ejpam-4953	79	54	.	.	PUNCT
ejpam-4953	80	1	[	[	X
ejpam-4953	80	2	24	24	NUM
ejpam-4953	80	3	]	]	PUNCT
ejpam-4953	80	4	let	let	VERB
ejpam-4953	80	5	ḧ	ḧ	NOUN
ejpam-4953	80	6	,	,	PUNCT
ejpam-4953	80	7	f̈	f̈	PRON
ejpam-4953	80	8	and	and	CCONJ
ejpam-4953	80	9	r̈	r̈	VERB
ejpam-4953	80	10	be	be	AUX
ejpam-4953	80	11	same	same	ADJ
ejpam-4953	80	12	as	as	ADP
ejpam-4953	80	13	in	in	ADP
ejpam-4953	80	14	definition	definition	NOUN
ejpam-4953	80	15	1.10	1.10	NUM
ejpam-4953	80	16	.	.	PUNCT
ejpam-4953	81	1	r̈s	r̈s	PROPN
ejpam-4953	81	2	must	must	AUX
ejpam-4953	81	3	also	also	ADV
ejpam-4953	81	4	be	be	AUX
ejpam-4953	81	5	f̈-closed	f̈-close	VERB
ejpam-4953	81	6	if	if	SCONJ
ejpam-4953	81	7	r̈	r̈	NOUN
ejpam-4953	81	8	is	be	AUX
ejpam-4953	81	9	f̈-closed	f̈-close	VERB
ejpam-4953	81	10	.	.	PUNCT
ejpam-4953	82	1	definition	definition	NOUN
ejpam-4953	82	2	11	11	NUM
ejpam-4953	82	3	.	.	PUNCT
ejpam-4953	83	1	[	[	X
ejpam-4953	83	2	24	24	NUM
ejpam-4953	83	3	]	]	PUNCT
ejpam-4953	83	4	let	let	VERB
ejpam-4953	83	5	s	s	PRON
ejpam-4953	83	6	and	and	CCONJ
ejpam-4953	83	7	v	v	NOUN
ejpam-4953	83	8	are	be	AUX
ejpam-4953	83	9	self	self	NOUN
ejpam-4953	83	10	mappings	mapping	NOUN
ejpam-4953	83	11	on	on	ADP
ejpam-4953	83	12	a	a	DET
ejpam-4953	83	13	nonempty	nonempty	NOUN
ejpam-4953	83	14	set	set	VERB
ejpam-4953	83	15	ḧ.	ḧ.	PROPN
ejpam-4953	83	16	a	a	DET
ejpam-4953	83	17	binary	binary	ADJ
ejpam-4953	83	18	relation	relation	NOUN
ejpam-4953	83	19	r̈	r̈	VERB
ejpam-4953	83	20	on	on	ADP
ejpam-4953	83	21	ḧ	ḧ	NOUN
ejpam-4953	83	22	is	be	AUX
ejpam-4953	83	23	called	call	VERB
ejpam-4953	83	24	(	(	PUNCT
ejpam-4953	83	25	s	s	PROPN
ejpam-4953	83	26	,	,	PUNCT
ejpam-4953	83	27	v)-closed	v)-close	VERB
ejpam-4953	83	28	if	if	SCONJ
ejpam-4953	83	29	for	for	ADP
ejpam-4953	83	30	all	all	DET
ejpam-4953	83	31	ϖ	ϖ	PROPN
ejpam-4953	83	32	,	,	PUNCT
ejpam-4953	83	33	ρ	ρ	PROPN
ejpam-4953	83	34	∈	∈	PROPN
ejpam-4953	83	35	ḧ	ḧ	NOUN
ejpam-4953	83	36	,	,	PUNCT
ejpam-4953	83	37	(	(	PUNCT
ejpam-4953	83	38	vϖ	vϖ	INTJ
ejpam-4953	83	39	,	,	PUNCT
ejpam-4953	83	40	vρ	vρ	NOUN
ejpam-4953	83	41	)	)	PUNCT
ejpam-4953	83	42	∈	∈	PROPN
ejpam-4953	83	43	r̈	r̈	NOUN
ejpam-4953	83	44	yield	yield	VERB
ejpam-4953	83	45	that	that	SCONJ
ejpam-4953	83	46	(	(	PUNCT
ejpam-4953	83	47	sϖ	sϖ	PROPN
ejpam-4953	83	48	,	,	PUNCT
ejpam-4953	83	49	sρ	sρ	PRON
ejpam-4953	83	50	)	)	PUNCT
ejpam-4953	83	51	belong	belong	VERB
ejpam-4953	83	52	to	to	ADP
ejpam-4953	83	53	r̈.	r̈.	PROPN
ejpam-4953	83	54	if	if	SCONJ
ejpam-4953	83	55	we	we	PRON
ejpam-4953	83	56	take	take	VERB
ejpam-4953	83	57	v=	v=	NOUN
ejpam-4953	83	58	identity	identity	NOUN
ejpam-4953	83	59	mapping	mapping	NOUN
ejpam-4953	83	60	,	,	PUNCT
ejpam-4953	83	61	then	then	ADV
ejpam-4953	83	62	we	we	PRON
ejpam-4953	83	63	conclude	conclude	VERB
ejpam-4953	83	64	that	that	SCONJ
ejpam-4953	83	65	r̈	r̈	VERB
ejpam-4953	83	66	is	be	AUX
ejpam-4953	83	67	s	s	NOUN
ejpam-4953	83	68	-	-	PUNCT
ejpam-4953	83	69	closed	closed	ADJ
ejpam-4953	83	70	.	.	PUNCT
ejpam-4953	84	1	if	if	SCONJ
ejpam-4953	84	2	r̈	r̈	NOUN
ejpam-4953	84	3	is	be	AUX
ejpam-4953	84	4	s	s	NOUN
ejpam-4953	84	5	-	-	PUNCT
ejpam-4953	84	6	closed	closed	ADJ
ejpam-4953	84	7	,	,	PUNCT
ejpam-4953	84	8	then	then	ADV
ejpam-4953	84	9	r̈s	r̈s	NOUN
ejpam-4953	84	10	is	be	AUX
ejpam-4953	84	11	also	also	ADV
ejpam-4953	84	12	s	s	NOUN
ejpam-4953	84	13	-	-	PUNCT
ejpam-4953	84	14	closed	closed	ADJ
ejpam-4953	84	15	.	.	PUNCT
ejpam-4953	85	1	definition	definition	NOUN
ejpam-4953	85	2	12	12	NUM
ejpam-4953	85	3	.	.	PUNCT
ejpam-4953	86	1	[	[	X
ejpam-4953	86	2	25	25	NUM
ejpam-4953	86	3	]	]	X
ejpam-4953	86	4	let	let	VERB
ejpam-4953	86	5	(	(	PUNCT
ejpam-4953	86	6	ḧ	ḧ	NOUN
ejpam-4953	86	7	,	,	PUNCT
ejpam-4953	86	8	p	p	X
ejpam-4953	86	9	,	,	PUNCT
ejpam-4953	86	10	k	k	X
ejpam-4953	86	11	≥	≥	NUM
ejpam-4953	86	12	1	1	NUM
ejpam-4953	86	13	)	)	PUNCT
ejpam-4953	86	14	be	be	AUX
ejpam-4953	86	15	a	a	DET
ejpam-4953	86	16	b	b	NOUN
ejpam-4953	86	17	-	-	PUNCT
ejpam-4953	86	18	metric	metric	ADJ
ejpam-4953	86	19	space	space	NOUN
ejpam-4953	86	20	and	and	CCONJ
ejpam-4953	86	21	let	let	VERB
ejpam-4953	86	22	r̈	r̈	PROPN
ejpam-4953	86	23	a	a	DET
ejpam-4953	86	24	binary	binary	ADJ
ejpam-4953	86	25	relation	relation	NOUN
ejpam-4953	86	26	on	on	ADP
ejpam-4953	86	27	ḧ.	ḧ.	PROPN
ejpam-4953	86	28	(	(	PUNCT
ejpam-4953	86	29	i	i	NOUN
ejpam-4953	86	30	)	)	PUNCT
ejpam-4953	86	31	we	we	PRON
ejpam-4953	86	32	say	say	VERB
ejpam-4953	86	33	that	that	SCONJ
ejpam-4953	86	34	(	(	PUNCT
ejpam-4953	86	35	ϖ	ϖ	X
ejpam-4953	86	36	,	,	PUNCT
ejpam-4953	86	37	p	p	NOUN
ejpam-4953	86	38	)	)	PUNCT
ejpam-4953	86	39	is	be	AUX
ejpam-4953	86	40	r̈-complete	r̈-complete	ADJ
ejpam-4953	86	41	if	if	SCONJ
ejpam-4953	86	42	every	every	DET
ejpam-4953	86	43	r̈-preserving	r̈-preserve	VERB
ejpam-4953	86	44	b	b	X
ejpam-4953	86	45	-	-	PUNCT
ejpam-4953	86	46	cauchy	cauchy	ADJ
ejpam-4953	86	47	sequence	sequence	NOUN
ejpam-4953	86	48	in	in	ADP
ejpam-4953	86	49	ḧ	ḧ	NOUN
ejpam-4953	86	50	converges	converge	NOUN
ejpam-4953	86	51	.	.	PUNCT
ejpam-4953	87	1	(	(	PUNCT
ejpam-4953	87	2	ii	ii	NOUN
ejpam-4953	87	3	)	)	PUNCT
ejpam-4953	87	4	a	a	DET
ejpam-4953	87	5	subset	subset	NOUN
ejpam-4953	87	6	g	g	NOUN
ejpam-4953	87	7	of	of	ADP
ejpam-4953	87	8	ḧ	ḧ	NOUN
ejpam-4953	87	9	is	be	AUX
ejpam-4953	87	10	called	call	VERB
ejpam-4953	87	11	r̈-closed	r̈-close	VERB
ejpam-4953	87	12	if	if	SCONJ
ejpam-4953	87	13	every	every	DET
ejpam-4953	87	14	r̈-preserving	r̈-preserve	VERB
ejpam-4953	87	15	b	b	NOUN
ejpam-4953	87	16	-	-	PUNCT
ejpam-4953	87	17	convergent	convergent	ADJ
ejpam-4953	87	18	sequence	sequence	NOUN
ejpam-4953	87	19	in	in	ADP
ejpam-4953	87	20	g	g	PROPN
ejpam-4953	87	21	converges	converge	NOUN
ejpam-4953	87	22	to	to	ADP
ejpam-4953	87	23	a	a	DET
ejpam-4953	87	24	point	point	NOUN
ejpam-4953	87	25	of	of	ADP
ejpam-4953	87	26	g.	g.	PROPN
ejpam-4953	87	27	definition	definition	NOUN
ejpam-4953	87	28	13	13	NUM
ejpam-4953	87	29	.	.	PUNCT
ejpam-4953	88	1	[	[	X
ejpam-4953	88	2	25	25	NUM
ejpam-4953	88	3	]	]	X
ejpam-4953	88	4	let	let	VERB
ejpam-4953	88	5	(	(	PUNCT
ejpam-4953	88	6	ḧ	ḧ	NOUN
ejpam-4953	88	7	,	,	PUNCT
ejpam-4953	88	8	p	p	X
ejpam-4953	88	9	,	,	PUNCT
ejpam-4953	88	10	k	k	X
ejpam-4953	88	11	≥	≥	NUM
ejpam-4953	88	12	1	1	NUM
ejpam-4953	88	13	)	)	PUNCT
ejpam-4953	88	14	be	be	AUX
ejpam-4953	88	15	a	a	DET
ejpam-4953	88	16	b	b	NOUN
ejpam-4953	88	17	-	-	PUNCT
ejpam-4953	88	18	metric	metric	ADJ
ejpam-4953	88	19	space	space	NOUN
ejpam-4953	88	20	and	and	CCONJ
ejpam-4953	88	21	let	let	VERB
ejpam-4953	88	22	v	v	NOUN
ejpam-4953	88	23	:	:	PUNCT
ejpam-4953	88	24	ḧ	ḧ	X
ejpam-4953	88	25	→	→	SYM
ejpam-4953	88	26	ḧ.	ḧ.	PROPN
ejpam-4953	88	27	a	a	DET
ejpam-4953	88	28	binary	binary	ADJ
ejpam-4953	88	29	relation	relation	NOUN
ejpam-4953	88	30	r̈	r̈	NOUN
ejpam-4953	88	31	defined	define	VERB
ejpam-4953	88	32	on	on	ADP
ejpam-4953	88	33	ḧ	ḧ	NOUN
ejpam-4953	88	34	is	be	AUX
ejpam-4953	88	35	called	call	VERB
ejpam-4953	88	36	(	(	PUNCT
ejpam-4953	88	37	v	v	NOUN
ejpam-4953	88	38	,	,	PUNCT
ejpam-4953	88	39	bp)-self	bp)-self	PRON
ejpam-4953	88	40	closed	close	VERB
ejpam-4953	88	41	if	if	SCONJ
ejpam-4953	88	42	,	,	PUNCT
ejpam-4953	88	43	whenever	whenever	SCONJ
ejpam-4953	88	44	{	{	PUNCT
ejpam-4953	88	45	ϖn	ϖn	NOUN
ejpam-4953	88	46	}	}	PUNCT
ejpam-4953	88	47	is	be	AUX
ejpam-4953	88	48	an	an	DET
ejpam-4953	88	49	r̈-preserving	r̈-preserve	VERB
ejpam-4953	88	50	sequence	sequence	NOUN
ejpam-4953	88	51	and	and	CCONJ
ejpam-4953	88	52	ϖn	ϖn	ADP
ejpam-4953	88	53	→p	→p	PROPN
ejpam-4953	88	54	ϖ	ϖ	NOUN
ejpam-4953	88	55	,	,	PUNCT
ejpam-4953	88	56	there	there	PRON
ejpam-4953	88	57	exists	exist	VERB
ejpam-4953	88	58	a	a	DET
ejpam-4953	88	59	subsequence	subsequence	NOUN
ejpam-4953	88	60	{	{	PUNCT
ejpam-4953	88	61	ϖni	ϖni	NOUN
ejpam-4953	88	62	}	}	PUNCT
ejpam-4953	88	63	of	of	ADP
ejpam-4953	88	64	{	{	PUNCT
ejpam-4953	88	65	ϖn	ϖn	NOUN
ejpam-4953	88	66	}	}	PUNCT
ejpam-4953	88	67	with	with	ADP
ejpam-4953	88	68	[	[	X
ejpam-4953	88	69	v	v	X
ejpam-4953	88	70	ϖni	ϖni	NOUN
ejpam-4953	88	71	,	,	PUNCT
ejpam-4953	88	72	v	v	NOUN
ejpam-4953	88	73	ϖ	ϖ	X
ejpam-4953	88	74	]	]	X
ejpam-4953	88	75	∈	∈	PROPN
ejpam-4953	88	76	r̈	r̈	VERB
ejpam-4953	88	77	for	for	ADP
ejpam-4953	88	78	all	all	DET
ejpam-4953	88	79	i	i	PRON
ejpam-4953	88	80	∈	∈	PROPN
ejpam-4953	88	81	n.	n.	NOUN
ejpam-4953	88	82	if	if	SCONJ
ejpam-4953	88	83	v	v	NOUN
ejpam-4953	88	84	is	be	AUX
ejpam-4953	88	85	the	the	DET
ejpam-4953	88	86	identity	identity	NOUN
ejpam-4953	88	87	mapping	mapping	NOUN
ejpam-4953	88	88	,	,	PUNCT
ejpam-4953	88	89	then	then	ADV
ejpam-4953	88	90	we	we	PRON
ejpam-4953	88	91	get	get	VERB
ejpam-4953	88	92	the	the	DET
ejpam-4953	88	93	following	follow	VERB
ejpam-4953	88	94	definitions	definition	NOUN
ejpam-4953	88	95	:	:	PUNCT
ejpam-4953	88	96	definition	definition	NOUN
ejpam-4953	88	97	14	14	NUM
ejpam-4953	88	98	.	.	PUNCT
ejpam-4953	89	1	[	[	X
ejpam-4953	89	2	25	25	NUM
ejpam-4953	89	3	]	]	X
ejpam-4953	89	4	let	let	VERB
ejpam-4953	89	5	(	(	PUNCT
ejpam-4953	89	6	ḧ	ḧ	NOUN
ejpam-4953	89	7	,	,	PUNCT
ejpam-4953	89	8	p	p	X
ejpam-4953	89	9	,	,	PUNCT
ejpam-4953	89	10	k	k	X
ejpam-4953	89	11	≥	≥	NUM
ejpam-4953	89	12	1	1	NUM
ejpam-4953	89	13	)	)	PUNCT
ejpam-4953	89	14	be	be	AUX
ejpam-4953	89	15	a	a	DET
ejpam-4953	89	16	b	b	NOUN
ejpam-4953	89	17	-	-	PUNCT
ejpam-4953	89	18	metric	metric	ADJ
ejpam-4953	89	19	space	space	NOUN
ejpam-4953	89	20	.	.	PUNCT
ejpam-4953	90	1	a	a	DET
ejpam-4953	90	2	binary	binary	ADJ
ejpam-4953	90	3	relation	relation	NOUN
ejpam-4953	90	4	r̈	r̈	NOUN
ejpam-4953	90	5	defined	define	VERB
ejpam-4953	90	6	on	on	ADP
ejpam-4953	90	7	ḧ	ḧ	NOUN
ejpam-4953	90	8	is	be	AUX
ejpam-4953	90	9	called	call	VERB
ejpam-4953	90	10	bp	bp	PROPN
ejpam-4953	90	11	-	-	PUNCT
ejpam-4953	90	12	self	self	NOUN
ejpam-4953	90	13	closed	close	VERB
ejpam-4953	90	14	if	if	SCONJ
ejpam-4953	90	15	,	,	PUNCT
ejpam-4953	90	16	whenever	whenever	SCONJ
ejpam-4953	90	17	{	{	PUNCT
ejpam-4953	90	18	ϖn	ϖn	NOUN
ejpam-4953	90	19	}	}	PUNCT
ejpam-4953	90	20	is	be	AUX
ejpam-4953	90	21	an	an	DET
ejpam-4953	90	22	r̈-preserving	r̈-preserve	VERB
ejpam-4953	90	23	sequence	sequence	NOUN
ejpam-4953	90	24	and	and	CCONJ
ejpam-4953	90	25	ϖn	ϖn	ADP
ejpam-4953	90	26	→p	→p	PROPN
ejpam-4953	90	27	ϖ	ϖ	NOUN
ejpam-4953	90	28	,	,	PUNCT
ejpam-4953	90	29	there	there	PRON
ejpam-4953	90	30	exists	exist	VERB
ejpam-4953	90	31	a	a	DET
ejpam-4953	90	32	subsequence	subsequence	NOUN
ejpam-4953	90	33	{	{	PUNCT
ejpam-4953	90	34	ϖnj	ϖnj	NOUN
ejpam-4953	90	35	}	}	PUNCT
ejpam-4953	90	36	of	of	ADP
ejpam-4953	90	37	{	{	PUNCT
ejpam-4953	90	38	ϖn	ϖn	NOUN
ejpam-4953	90	39	}	}	PUNCT
ejpam-4953	90	40	with	with	ADP
ejpam-4953	90	41	(	(	PUNCT
ejpam-4953	90	42	ϖnj	ϖnj	INTJ
ejpam-4953	90	43	,	,	PUNCT
ejpam-4953	90	44	ϖ	ϖ	NOUN
ejpam-4953	90	45	)	)	PUNCT
ejpam-4953	90	46	∈	∈	PROPN
ejpam-4953	90	47	r̈	r̈	VERB
ejpam-4953	90	48	for	for	ADP
ejpam-4953	90	49	all	all	DET
ejpam-4953	90	50	j	j	PROPN
ejpam-4953	90	51	∈	∈	PROPN
ejpam-4953	90	52	n.	n.	PROPN
ejpam-4953	90	53	definition	definition	NOUN
ejpam-4953	90	54	15	15	NUM
ejpam-4953	90	55	.	.	PUNCT
ejpam-4953	91	1	[	[	X
ejpam-4953	91	2	26	26	NUM
ejpam-4953	91	3	]	]	PUNCT
ejpam-4953	91	4	let	let	VERB
ejpam-4953	91	5	ḧ	ḧ	PRON
ejpam-4953	91	6	be	be	AUX
ejpam-4953	91	7	a	a	DET
ejpam-4953	91	8	nonempty	nonempty	ADV
ejpam-4953	91	9	set	set	VERB
ejpam-4953	91	10	and	and	CCONJ
ejpam-4953	91	11	r̈	r̈	VERB
ejpam-4953	91	12	a	a	DET
ejpam-4953	91	13	binary	binary	ADJ
ejpam-4953	91	14	relation	relation	NOUN
ejpam-4953	91	15	on	on	ADP
ejpam-4953	91	16	ḧ.	ḧ.	PROPN
ejpam-4953	91	17	a	a	DET
ejpam-4953	91	18	subset	subset	NOUN
ejpam-4953	91	19	g	g	NOUN
ejpam-4953	91	20	of	of	ADP
ejpam-4953	91	21	ḧ	ḧ	NOUN
ejpam-4953	91	22	is	be	AUX
ejpam-4953	91	23	called	call	VERB
ejpam-4953	91	24	r̈-directed	r̈-directe	VERB
ejpam-4953	91	25	if	if	SCONJ
ejpam-4953	91	26	for	for	ADP
ejpam-4953	91	27	each	each	DET
ejpam-4953	91	28	ϖ	ϖ	NOUN
ejpam-4953	91	29	,	,	PUNCT
ejpam-4953	91	30	ρ	ρ	PROPN
ejpam-4953	91	31	∈	∈	PROPN
ejpam-4953	91	32	g	g	NOUN
ejpam-4953	91	33	,	,	PUNCT
ejpam-4953	91	34	there	there	PRON
ejpam-4953	91	35	exists	exist	VERB
ejpam-4953	91	36	z	z	NOUN
ejpam-4953	91	37	∈	∈	PROPN
ejpam-4953	91	38	ḧ	ḧ	NOUN
ejpam-4953	91	39	such	such	ADJ
ejpam-4953	91	40	that	that	SCONJ
ejpam-4953	91	41	(	(	PUNCT
ejpam-4953	91	42	ϖ	ϖ	NOUN
ejpam-4953	91	43	,	,	PUNCT
ejpam-4953	91	44	z	z	NOUN
ejpam-4953	91	45	)	)	PUNCT
ejpam-4953	91	46	∈	∈	PROPN
ejpam-4953	91	47	r̈	r̈	VERB
ejpam-4953	91	48	and	and	CCONJ
ejpam-4953	91	49	(	(	PUNCT
ejpam-4953	91	50	ρ	ρ	PROPN
ejpam-4953	91	51	,	,	PUNCT
ejpam-4953	91	52	z	z	NOUN
ejpam-4953	91	53	)	)	PUNCT
ejpam-4953	91	54	∈	∈	PROPN
ejpam-4953	91	55	r̈.	r̈.	PROPN
ejpam-4953	91	56	definition	definition	NOUN
ejpam-4953	91	57	16	16	NUM
ejpam-4953	91	58	.	.	PUNCT
ejpam-4953	92	1	[	[	X
ejpam-4953	92	2	27	27	NUM
ejpam-4953	92	3	]	]	PUNCT
ejpam-4953	92	4	let	let	VERB
ejpam-4953	92	5	ḧ	ḧ	PRON
ejpam-4953	92	6	be	be	AUX
ejpam-4953	92	7	a	a	DET
ejpam-4953	92	8	nonempty	nonempty	ADV
ejpam-4953	92	9	set	set	VERB
ejpam-4953	92	10	and	and	CCONJ
ejpam-4953	92	11	r̈	r̈	VERB
ejpam-4953	92	12	a	a	DET
ejpam-4953	92	13	binary	binary	ADJ
ejpam-4953	92	14	relation	relation	NOUN
ejpam-4953	92	15	on	on	ADP
ejpam-4953	92	16	ḧ.	ḧ.	PROPN
ejpam-4953	92	17	for	for	ADP
ejpam-4953	92	18	ϖ	ϖ	PROPN
ejpam-4953	92	19	,	,	PUNCT
ejpam-4953	92	20	ρ	ρ	PROPN
ejpam-4953	92	21	∈	∈	PROPN
ejpam-4953	92	22	ḧ	ḧ	NOUN
ejpam-4953	92	23	,	,	PUNCT
ejpam-4953	92	24	a	a	DET
ejpam-4953	92	25	path	path	NOUN
ejpam-4953	92	26	of	of	ADP
ejpam-4953	92	27	length	length	NOUN
ejpam-4953	92	28	k	k	PROPN
ejpam-4953	92	29	(	(	PUNCT
ejpam-4953	92	30	where	where	SCONJ
ejpam-4953	92	31	k	k	PROPN
ejpam-4953	92	32	is	be	AUX
ejpam-4953	92	33	a	a	DET
ejpam-4953	92	34	natural	natural	ADJ
ejpam-4953	92	35	number	number	NOUN
ejpam-4953	92	36	)	)	PUNCT
ejpam-4953	92	37	in	in	ADP
ejpam-4953	92	38	r̈	r̈	NOUN
ejpam-4953	92	39	from	from	ADP
ejpam-4953	92	40	ϖ	ϖ	PRON
ejpam-4953	92	41	to	to	ADP
ejpam-4953	92	42	ρ	ρ	PROPN
ejpam-4953	92	43	is	be	AUX
ejpam-4953	92	44	a	a	DET
ejpam-4953	92	45	finite	finite	ADJ
ejpam-4953	92	46	sequence	sequence	NOUN
ejpam-4953	92	47	{	{	PUNCT
ejpam-4953	92	48	t0	t0	PROPN
ejpam-4953	92	49	,	,	PUNCT
ejpam-4953	92	50	t1	t1	NOUN
ejpam-4953	92	51	,	,	PUNCT
ejpam-4953	92	52	t2	t2	NOUN
ejpam-4953	92	53	,	,	PUNCT
ejpam-4953	92	54	....	....	SYM
ejpam-4953	92	55	tk	tk	PROPN
ejpam-4953	92	56	}	}	PUNCT
ejpam-4953	92	57	⊂	⊂	X
ejpam-4953	92	58	ḧ	ḧ	AUX
ejpam-4953	92	59	satisfying	satisfy	VERB
ejpam-4953	92	60	the	the	DET
ejpam-4953	92	61	following	follow	VERB
ejpam-4953	92	62	conditions	condition	NOUN
ejpam-4953	92	63	:	:	PUNCT
ejpam-4953	92	64	(	(	PUNCT
ejpam-4953	92	65	i	i	NOUN
ejpam-4953	92	66	)	)	PUNCT
ejpam-4953	92	67	t0	t0	PROPN
ejpam-4953	92	68	=	=	SYM
ejpam-4953	93	1	ϖ	ϖ	PROPN
ejpam-4953	93	2	and	and	CCONJ
ejpam-4953	93	3	tk	tk	PROPN
ejpam-4953	93	4	=	=	PROPN
ejpam-4953	93	5	ρ	ρ	PROPN
ejpam-4953	93	6	(	(	PUNCT
ejpam-4953	93	7	ii	ii	PROPN
ejpam-4953	93	8	)	)	PUNCT
ejpam-4953	93	9	(	(	PUNCT
ejpam-4953	93	10	tj	tj	NOUN
ejpam-4953	93	11	,	,	PUNCT
ejpam-4953	93	12	tj+1	tj+1	PROPN
ejpam-4953	93	13	)	)	PUNCT
ejpam-4953	93	14	∈	∈	PROPN
ejpam-4953	93	15	r̈	r̈	NOUN
ejpam-4953	93	16	for	for	ADP
ejpam-4953	93	17	each	each	DET
ejpam-4953	93	18	j	j	PROPN
ejpam-4953	93	19	(	(	PUNCT
ejpam-4953	93	20	0	0	NUM
ejpam-4953	93	21	≤	≤	NUM
ejpam-4953	94	1	j	j	PROPN
ejpam-4953	94	2	≤	≤	PROPN
ejpam-4953	94	3	k	k	NOUN
ejpam-4953	95	1	−	−	NOUN
ejpam-4953	95	2	1	1	NUM
ejpam-4953	95	3	)	)	PUNCT
ejpam-4953	95	4	.	.	PUNCT
ejpam-4953	96	1	note	note	VERB
ejpam-4953	96	2	that	that	SCONJ
ejpam-4953	96	3	although	although	SCONJ
ejpam-4953	96	4	they	they	PRON
ejpam-4953	96	5	are	be	AUX
ejpam-4953	96	6	not	not	PART
ejpam-4953	96	7	necessarily	necessarily	ADV
ejpam-4953	96	8	distinct	distinct	ADJ
ejpam-4953	96	9	,	,	PUNCT
ejpam-4953	96	10	a	a	DET
ejpam-4953	96	11	path	path	NOUN
ejpam-4953	96	12	of	of	ADP
ejpam-4953	96	13	length	length	NOUN
ejpam-4953	96	14	k	k	PROPN
ejpam-4953	96	15	involves	involve	VERB
ejpam-4953	96	16	k+1	k+1	PRON
ejpam-4953	96	17	elements	element	NOUN
ejpam-4953	96	18	of	of	ADP
ejpam-4953	96	19	ḧ	ḧ	NOUN
ejpam-4953	96	20	”	"	PUNCT
ejpam-4953	96	21	.	.	PUNCT
ejpam-4953	97	1	i.	i.	PROPN
ejpam-4953	97	2	alshammari	alshammari	PROPN
ejpam-4953	97	3	et	et	PROPN
ejpam-4953	97	4	al	al	PROPN
ejpam-4953	97	5	.	.	PUNCT
ejpam-4953	97	6	/	/	SYM
ejpam-4953	97	7	eur	eur	PROPN
ejpam-4953	97	8	.	.	PUNCT
ejpam-4953	98	1	j.	j.	PROPN
ejpam-4953	98	2	pure	pure	PROPN
ejpam-4953	98	3	appl	appl	PROPN
ejpam-4953	98	4	.	.	PROPN
ejpam-4953	98	5	math	math	PROPN
ejpam-4953	98	6	,	,	PUNCT
ejpam-4953	98	7	16	16	NUM
ejpam-4953	98	8	(	(	PUNCT
ejpam-4953	98	9	4	4	NUM
ejpam-4953	98	10	)	)	PUNCT
ejpam-4953	98	11	(	(	PUNCT
ejpam-4953	98	12	2023	2023	NUM
ejpam-4953	98	13	)	)	PUNCT
ejpam-4953	98	14	,	,	PUNCT
ejpam-4953	98	15	2405	2405	NUM
ejpam-4953	98	16	-	-	SYM
ejpam-4953	98	17	2418	2418	NUM
ejpam-4953	98	18	2409	2409	NUM
ejpam-4953	98	19	definition	definition	NOUN
ejpam-4953	98	20	17	17	NUM
ejpam-4953	98	21	.	.	PUNCT
ejpam-4953	99	1	[	[	X
ejpam-4953	99	2	25	25	NUM
ejpam-4953	99	3	]	]	X
ejpam-4953	99	4	let	let	VERB
ejpam-4953	99	5	(	(	PUNCT
ejpam-4953	99	6	ḧ	ḧ	NOUN
ejpam-4953	99	7	,	,	PUNCT
ejpam-4953	99	8	p	p	X
ejpam-4953	99	9	,	,	PUNCT
ejpam-4953	99	10	k	k	X
ejpam-4953	99	11	≥	≥	NUM
ejpam-4953	99	12	1	1	NUM
ejpam-4953	99	13	)	)	PUNCT
ejpam-4953	99	14	be	be	AUX
ejpam-4953	99	15	a	a	DET
ejpam-4953	99	16	b	b	NOUN
ejpam-4953	99	17	-	-	PUNCT
ejpam-4953	99	18	metric	metric	ADJ
ejpam-4953	99	19	space	space	NOUN
ejpam-4953	99	20	,	,	PUNCT
ejpam-4953	99	21	let	let	VERB
ejpam-4953	99	22	r̈	r̈	NOUN
ejpam-4953	99	23	be	be	AUX
ejpam-4953	99	24	a	a	DET
ejpam-4953	99	25	binary	binary	ADJ
ejpam-4953	99	26	relation	relation	NOUN
ejpam-4953	99	27	on	on	ADP
ejpam-4953	99	28	ḧ	ḧ	NOUN
ejpam-4953	99	29	,	,	PUNCT
ejpam-4953	99	30	and	and	CCONJ
ejpam-4953	99	31	let	let	VERB
ejpam-4953	99	32	s	s	PRON
ejpam-4953	99	33	and	and	CCONJ
ejpam-4953	99	34	v	v	NOUN
ejpam-4953	99	35	be	be	AUX
ejpam-4953	99	36	two	two	NUM
ejpam-4953	99	37	self	self	NOUN
ejpam-4953	99	38	-	-	PUNCT
ejpam-4953	99	39	mappings	mapping	NOUN
ejpam-4953	99	40	on	on	ADP
ejpam-4953	99	41	ḧ.	ḧ.	PROPN
ejpam-4953	99	42	we	we	PRON
ejpam-4953	99	43	say	say	VERB
ejpam-4953	99	44	that	that	PRON
ejpam-4953	99	45	s	s	VERB
ejpam-4953	99	46	and	and	CCONJ
ejpam-4953	99	47	v	v	NOUN
ejpam-4953	99	48	are	be	AUX
ejpam-4953	99	49	r̈-compatible	r̈-compatible	ADJ
ejpam-4953	99	50	if	if	SCONJ
ejpam-4953	99	51	,	,	PUNCT
ejpam-4953	99	52	for	for	ADP
ejpam-4953	99	53	any	any	DET
ejpam-4953	99	54	sequence	sequence	NOUN
ejpam-4953	99	55	{	{	PUNCT
ejpam-4953	99	56	ϖn	ϖn	NOUN
ejpam-4953	99	57	}	}	PUNCT
ejpam-4953	99	58	∈	∈	NOUN
ejpam-4953	99	59	ḧ	ḧ	NOUN
ejpam-4953	99	60	such	such	ADJ
ejpam-4953	99	61	that	that	SCONJ
ejpam-4953	99	62	{	{	PUNCT
ejpam-4953	99	63	sϖn	sϖn	ADJ
ejpam-4953	99	64	}	}	PUNCT
ejpam-4953	99	65	and	and	CCONJ
ejpam-4953	99	66	{	{	PUNCT
ejpam-4953	99	67	v	v	AUX
ejpam-4953	99	68	ϖn	ϖn	NOUN
ejpam-4953	99	69	}	}	PUNCT
ejpam-4953	99	70	are	be	AUX
ejpam-4953	99	71	r̈-preserving	r̈-preserve	VERB
ejpam-4953	99	72	and	and	CCONJ
ejpam-4953	99	73	lim	lim	PROPN
ejpam-4953	99	74	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	99	75	v	v	PROPN
ejpam-4953	99	76	(	(	PUNCT
ejpam-4953	99	77	ϖn	ϖn	NOUN
ejpam-4953	99	78	)	)	PUNCT
ejpam-4953	100	1	=	=	SYM
ejpam-4953	100	2	lim	lim	PROPN
ejpam-4953	100	3	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	100	4	s(ϖn	s(ϖn	PROPN
ejpam-4953	100	5	)	)	PUNCT
ejpam-4953	100	6	,	,	PUNCT
ejpam-4953	100	7	we	we	PRON
ejpam-4953	100	8	have	have	VERB
ejpam-4953	100	9	lim	lim	PROPN
ejpam-4953	100	10	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	100	11	d(v	d(v	PROPN
ejpam-4953	100	12	p	p	NOUN
ejpam-4953	100	13	(	(	PUNCT
ejpam-4953	100	14	ϖn	ϖn	NOUN
ejpam-4953	100	15	)	)	PUNCT
ejpam-4953	100	16	,	,	PUNCT
ejpam-4953	100	17	pv	pv	ADP
ejpam-4953	100	18	ϖn	ϖn	NOUN
ejpam-4953	100	19	)	)	PUNCT
ejpam-4953	100	20	=	=	SYM
ejpam-4953	101	1	0	0	X
ejpam-4953	101	2	.	.	PUNCT
ejpam-4953	101	3	lemma	lemma	PROPN
ejpam-4953	101	4	2	2	NUM
ejpam-4953	101	5	.	.	PUNCT
ejpam-4953	102	1	[	[	X
ejpam-4953	102	2	28	28	NUM
ejpam-4953	102	3	]	]	PUNCT
ejpam-4953	102	4	let	let	VERB
ejpam-4953	102	5	ḧ	ḧ	PRON
ejpam-4953	102	6	be	be	AUX
ejpam-4953	102	7	a	a	DET
ejpam-4953	102	8	non	non	X
ejpam-4953	102	9	empty	empty	ADJ
ejpam-4953	102	10	set	set	NOUN
ejpam-4953	102	11	and	and	CCONJ
ejpam-4953	102	12	let	let	VERB
ejpam-4953	102	13	f̈	f̈	PRON
ejpam-4953	102	14	be	be	AUX
ejpam-4953	102	15	a	a	DET
ejpam-4953	102	16	self	self	NOUN
ejpam-4953	102	17	mapping	mapping	NOUN
ejpam-4953	102	18	on	on	ADP
ejpam-4953	102	19	ḧ.	ḧ.	PROPN
ejpam-4953	102	20	then	then	ADV
ejpam-4953	102	21	there	there	PRON
ejpam-4953	102	22	exists	exist	VERB
ejpam-4953	102	23	a	a	DET
ejpam-4953	102	24	subset	subset	NOUN
ejpam-4953	102	25	g	g	ADP
ejpam-4953	102	26	⊆	⊆	NUM
ejpam-4953	102	27	ḧ	ḧ	NOUN
ejpam-4953	102	28	such	such	DET
ejpam-4953	102	29	that	that	DET
ejpam-4953	102	30	f̈(g	f̈(g	NOUN
ejpam-4953	102	31	)	)	PUNCT
ejpam-4953	102	32	=	=	SYM
ejpam-4953	102	33	f̈(ḧ	f̈(ḧ	NOUN
ejpam-4953	102	34	)	)	PUNCT
ejpam-4953	102	35	and	and	CCONJ
ejpam-4953	102	36	f̈	f̈	NUM
ejpam-4953	102	37	:	:	PUNCT
ejpam-4953	102	38	g	g	X
ejpam-4953	102	39	→	→	SYM
ejpam-4953	102	40	ḧ	ḧ	NOUN
ejpam-4953	102	41	is	be	AUX
ejpam-4953	102	42	one	one	NUM
ejpam-4953	102	43	-	-	PUNCT
ejpam-4953	102	44	to	to	ADP
ejpam-4953	102	45	one	one	NUM
ejpam-4953	102	46	.	.	PUNCT
ejpam-4953	102	47	.	.	PUNCT
ejpam-4953	103	1	2	2	X
ejpam-4953	103	2	.	.	X
ejpam-4953	103	3	main	main	ADJ
ejpam-4953	103	4	result	result	NOUN
ejpam-4953	103	5	in	in	ADP
ejpam-4953	103	6	this	this	DET
ejpam-4953	103	7	manuscript	manuscript	NOUN
ejpam-4953	103	8	,	,	PUNCT
ejpam-4953	103	9	we	we	PRON
ejpam-4953	103	10	utilize	utilize	VERB
ejpam-4953	103	11	the	the	DET
ejpam-4953	103	12	following	following	ADJ
ejpam-4953	103	13	notations	notation	NOUN
ejpam-4953	103	14	:	:	PUNCT
ejpam-4953	103	15	(	(	PUNCT
ejpam-4953	103	16	i	i	NOUN
ejpam-4953	103	17	)	)	PUNCT
ejpam-4953	103	18	f(f̈	f(f̈	PROPN
ejpam-4953	103	19	)	)	PUNCT
ejpam-4953	103	20	=	=	NOUN
ejpam-4953	104	1	the	the	DET
ejpam-4953	104	2	set	set	NOUN
ejpam-4953	104	3	of	of	ADP
ejpam-4953	104	4	all	all	DET
ejpam-4953	104	5	fixed	fix	VERB
ejpam-4953	104	6	points	point	NOUN
ejpam-4953	104	7	of	of	ADP
ejpam-4953	104	8	f̈	f̈	PROPN
ejpam-4953	104	9	,	,	PUNCT
ejpam-4953	104	10	(	(	PUNCT
ejpam-4953	104	11	ii	ii	NOUN
ejpam-4953	104	12	)	)	PUNCT
ejpam-4953	104	13	ḧ(f̈	ḧ(f̈	NOUN
ejpam-4953	104	14	;	;	PUNCT
ejpam-4953	104	15	r̈	r̈	NOUN
ejpam-4953	104	16	)	)	PUNCT
ejpam-4953	104	17	:	:	PUNCT
ejpam-4953	104	18	=	=	SYM
ejpam-4953	104	19	{	{	PUNCT
ejpam-4953	104	20	ϖ	ϖ	X
ejpam-4953	104	21	∈	∈	PROPN
ejpam-4953	104	22	ḧ	ḧ	NOUN
ejpam-4953	104	23	:	:	PUNCT
ejpam-4953	104	24	(	(	PUNCT
ejpam-4953	104	25	ϖ	ϖ	NOUN
ejpam-4953	104	26	,	,	PUNCT
ejpam-4953	104	27	f̈ϖ	f̈ϖ	NOUN
ejpam-4953	104	28	)	)	PUNCT
ejpam-4953	104	29	∈	∈	PROPN
ejpam-4953	104	30	r̈	r̈	NOUN
ejpam-4953	104	31	}	}	PUNCT
ejpam-4953	104	32	,	,	PUNCT
ejpam-4953	104	33	(	(	PUNCT
ejpam-4953	104	34	iii	iii	X
ejpam-4953	104	35	)	)	PUNCT
ejpam-4953	104	36	γ	γ	NOUN
ejpam-4953	104	37	(	(	PUNCT
ejpam-4953	104	38	ϖ,ρ	ϖ,ρ	PROPN
ejpam-4953	104	39	,	,	PUNCT
ejpam-4953	104	40	r̈	r̈	NOUN
ejpam-4953	104	41	)	)	PUNCT
ejpam-4953	104	42	:	:	PUNCT
ejpam-4953	104	43	=	=	PUNCT
ejpam-4953	104	44	the	the	DET
ejpam-4953	104	45	class	class	NOUN
ejpam-4953	104	46	of	of	ADP
ejpam-4953	104	47	all	all	DET
ejpam-4953	104	48	paths	path	NOUN
ejpam-4953	104	49	in	in	ADP
ejpam-4953	104	50	r̈	r̈	NOUN
ejpam-4953	104	51	from	from	ADP
ejpam-4953	104	52	ϖ	ϖ	PRON
ejpam-4953	104	53	to	to	ADP
ejpam-4953	104	54	ρ	ρ	PROPN
ejpam-4953	104	55	theorem	theorem	ADJ
ejpam-4953	104	56	1	1	X
ejpam-4953	104	57	.	.	PUNCT
ejpam-4953	105	1	let	let	AUX
ejpam-4953	105	2	(	(	PUNCT
ejpam-4953	105	3	ḧ,p	ḧ,p	PROPN
ejpam-4953	105	4	,	,	PUNCT
ejpam-4953	105	5	k	k	PROPN
ejpam-4953	105	6	≥	≥	NUM
ejpam-4953	105	7	1	1	NUM
ejpam-4953	105	8	)	)	PUNCT
ejpam-4953	105	9	be	be	AUX
ejpam-4953	105	10	a	a	DET
ejpam-4953	105	11	b	b	NOUN
ejpam-4953	105	12	-	-	PUNCT
ejpam-4953	105	13	complete	complete	ADJ
ejpam-4953	105	14	b	b	X
ejpam-4953	105	15	-	-	PUNCT
ejpam-4953	105	16	multiplicative	multiplicative	ADJ
ejpam-4953	105	17	metric	metric	ADJ
ejpam-4953	105	18	space	space	NOUN
ejpam-4953	105	19	and	and	CCONJ
ejpam-4953	105	20	r̈	r̈	VERB
ejpam-4953	105	21	a	a	DET
ejpam-4953	105	22	binary	binary	ADJ
ejpam-4953	105	23	relation	relation	NOUN
ejpam-4953	105	24	on	on	ADP
ejpam-4953	105	25	ḧ.	ḧ.	PROPN
ejpam-4953	105	26	f̈	f̈	NUM
ejpam-4953	105	27	:	:	PUNCT
ejpam-4953	105	28	ḧ	ḧ	NOUN
ejpam-4953	105	29	×	×	NOUN
ejpam-4953	105	30	ḧ	ḧ	NOUN
ejpam-4953	105	31	be	be	AUX
ejpam-4953	105	32	a	a	DET
ejpam-4953	105	33	self	self	NOUN
ejpam-4953	105	34	-	-	PUNCT
ejpam-4953	105	35	mapping	mapping	NOUN
ejpam-4953	105	36	satisfying	satisfy	VERB
ejpam-4953	105	37	the	the	DET
ejpam-4953	105	38	following	follow	VERB
ejpam-4953	105	39	conditions	condition	NOUN
ejpam-4953	105	40	given	give	VERB
ejpam-4953	105	41	below	below	ADV
ejpam-4953	105	42	.	.	PUNCT
ejpam-4953	106	1	(	(	PUNCT
ejpam-4953	106	2	i	i	NOUN
ejpam-4953	106	3	)	)	PUNCT
ejpam-4953	106	4	ḧ(f̈	ḧ(f̈	PROPN
ejpam-4953	106	5	;	;	PUNCT
ejpam-4953	106	6	r̈	r̈	NOUN
ejpam-4953	106	7	)	)	PUNCT
ejpam-4953	106	8	is	be	AUX
ejpam-4953	106	9	non	non	ADJ
ejpam-4953	106	10	-	-	ADJ
ejpam-4953	106	11	empty	empty	ADJ
ejpam-4953	106	12	.	.	PUNCT
ejpam-4953	107	1	(	(	PUNCT
ejpam-4953	107	2	ii	ii	NOUN
ejpam-4953	107	3	)	)	PUNCT
ejpam-4953	107	4	r̈	r̈	VERB
ejpam-4953	107	5	is	be	AUX
ejpam-4953	107	6	f̈-closed	f̈-close	VERB
ejpam-4953	107	7	.	.	PUNCT
ejpam-4953	108	1	(	(	PUNCT
ejpam-4953	108	2	iii	iii	NOUN
ejpam-4953	108	3	)	)	PUNCT
ejpam-4953	108	4	either	either	CCONJ
ejpam-4953	108	5	f̈	f̈	PRON
ejpam-4953	108	6	is	be	AUX
ejpam-4953	108	7	b	b	NOUN
ejpam-4953	108	8	-	-	PUNCT
ejpam-4953	108	9	continuous	continuous	ADJ
ejpam-4953	108	10	or	or	CCONJ
ejpam-4953	108	11	r̈	r̈	VERB
ejpam-4953	108	12	is	be	AUX
ejpam-4953	108	13	bp	bp	PROPN
ejpam-4953	108	14	-	-	PUNCT
ejpam-4953	108	15	self	self	NOUN
ejpam-4953	108	16	closed	closed	ADJ
ejpam-4953	108	17	.	.	PUNCT
ejpam-4953	109	1	(	(	PUNCT
ejpam-4953	109	2	iv	iv	X
ejpam-4953	109	3	)	)	PUNCT
ejpam-4953	109	4	there	there	PRON
ejpam-4953	109	5	exists	exist	VERB
ejpam-4953	109	6	λ	λ	X
ejpam-4953	109	7	∈	∈	PROPN
ejpam-4953	110	1	[	[	X
ejpam-4953	110	2	0	0	NUM
ejpam-4953	110	3	,	,	PUNCT
ejpam-4953	110	4	1k	1k	NUM
ejpam-4953	110	5	)	)	PUNCT
ejpam-4953	110	6	such	such	ADJ
ejpam-4953	110	7	that	that	PRON
ejpam-4953	110	8	.	.	PUNCT
ejpam-4953	111	1	p(f̈ϖ	p(f̈ϖ	NUM
ejpam-4953	111	2	,	,	PUNCT
ejpam-4953	111	3	f̈ρ	f̈ρ	ADJ
ejpam-4953	111	4	)	)	PUNCT
ejpam-4953	111	5	≤	≤	NOUN
ejpam-4953	111	6	p(ϖ	p(ϖ	PROPN
ejpam-4953	111	7	,	,	PUNCT
ejpam-4953	111	8	ρ)λ	ρ)λ	ADJ
ejpam-4953	111	9	then	then	ADV
ejpam-4953	111	10	f̈	f̈	PROPN
ejpam-4953	111	11	has	have	VERB
ejpam-4953	111	12	a	a	DET
ejpam-4953	111	13	fixed	fix	VERB
ejpam-4953	111	14	point	point	NOUN
ejpam-4953	111	15	.	.	PUNCT
ejpam-4953	112	1	i.e.	i.e.	X
ejpam-4953	112	2	,	,	PUNCT
ejpam-4953	112	3	there	there	PRON
ejpam-4953	112	4	exists	exist	VERB
ejpam-4953	112	5	ϖ∗	ϖ∗	PROPN
ejpam-4953	112	6	∈	∈	PROPN
ejpam-4953	112	7	ḧ	ḧ	NOUN
ejpam-4953	112	8	such	such	ADJ
ejpam-4953	112	9	that	that	DET
ejpam-4953	112	10	f̈ϖ∗	f̈ϖ∗	PROPN
ejpam-4953	112	11	=	=	PRON
ejpam-4953	112	12	ϖ∗.	ϖ∗.	PROPN
ejpam-4953	112	13	(	(	PUNCT
ejpam-4953	112	14	v	v	NOUN
ejpam-4953	112	15	)	)	PUNCT
ejpam-4953	112	16	γ(ϖ	γ(ϖ	NOUN
ejpam-4953	112	17	,	,	PUNCT
ejpam-4953	112	18	ρ	ρ	NOUN
ejpam-4953	112	19	,	,	PUNCT
ejpam-4953	112	20	r̈s	r̈s	NOUN
ejpam-4953	112	21	)	)	PUNCT
ejpam-4953	112	22	is	be	AUX
ejpam-4953	112	23	nonempty	nonempty	ADJ
ejpam-4953	112	24	,	,	PUNCT
ejpam-4953	112	25	for	for	ADP
ejpam-4953	112	26	each	each	DET
ejpam-4953	112	27	ϖ	ϖ	PROPN
ejpam-4953	112	28	,	,	PUNCT
ejpam-4953	112	29	ρ	ρ	PROPN
ejpam-4953	112	30	∈	∈	PROPN
ejpam-4953	113	1	ḧ	ḧ	NOUN
ejpam-4953	113	2	,	,	PUNCT
ejpam-4953	113	3	then	then	ADV
ejpam-4953	113	4	f̈	f̈	PROPN
ejpam-4953	113	5	has	have	VERB
ejpam-4953	113	6	a	a	DET
ejpam-4953	113	7	unique	unique	ADJ
ejpam-4953	113	8	fixed	fix	VERB
ejpam-4953	113	9	point	point	NOUN
ejpam-4953	113	10	.	.	PUNCT
ejpam-4953	114	1	proof	proof	NOUN
ejpam-4953	114	2	.	.	PUNCT
ejpam-4953	115	1	let	let	VERB
ejpam-4953	115	2	ϖ0	ϖ0	PRON
ejpam-4953	115	3	∈	∈	NOUN
ejpam-4953	115	4	(	(	PUNCT
ejpam-4953	115	5	f̈	f̈	NUM
ejpam-4953	115	6	;	;	PUNCT
ejpam-4953	115	7	r̈	r̈	NOUN
ejpam-4953	115	8	)	)	PUNCT
ejpam-4953	115	9	be	be	VERB
ejpam-4953	115	10	an	an	DET
ejpam-4953	115	11	arbitrary	arbitrary	ADJ
ejpam-4953	115	12	element	element	NOUN
ejpam-4953	115	13	.	.	PUNCT
ejpam-4953	116	1	now	now	ADV
ejpam-4953	116	2	we	we	PRON
ejpam-4953	116	3	define	define	VERB
ejpam-4953	116	4	the	the	DET
ejpam-4953	116	5	sequence	sequence	NOUN
ejpam-4953	116	6	ϖn	ϖn	NOUN
ejpam-4953	116	7	of	of	ADP
ejpam-4953	116	8	picard	picard	NOUN
ejpam-4953	116	9	iterates	iterate	NOUN
ejpam-4953	116	10	i.e.	i.e.	X
ejpam-4953	116	11	,	,	PUNCT
ejpam-4953	116	12	ϖn	ϖn	ADP
ejpam-4953	116	13	=	=	PUNCT
ejpam-4953	116	14	f̈ϖn−1	f̈ϖn−1	ADJ
ejpam-4953	117	1	=	=	PUNCT
ejpam-4953	117	2	f̈nϖ0	f̈nϖ0	NOUN
ejpam-4953	117	3	for	for	ADP
ejpam-4953	117	4	all	all	DET
ejpam-4953	117	5	n	n	PRON
ejpam-4953	117	6	∈	∈	PROPN
ejpam-4953	117	7	n.	n.	NOUN
ejpam-4953	117	8	as	as	ADP
ejpam-4953	117	9	(	(	PUNCT
ejpam-4953	117	10	ϖ0	ϖ0	NOUN
ejpam-4953	117	11	,	,	PUNCT
ejpam-4953	117	12	f̈ϖ0	f̈ϖ0	PROPN
ejpam-4953	117	13	)	)	PUNCT
ejpam-4953	117	14	∈	∈	PROPN
ejpam-4953	117	15	r̈	r̈	VERB
ejpam-4953	117	16	and	and	CCONJ
ejpam-4953	117	17	r̈	r̈	VERB
ejpam-4953	117	18	is	be	AUX
ejpam-4953	117	19	f̈-closed	f̈-close	VERB
ejpam-4953	117	20	,	,	PUNCT
ejpam-4953	117	21	we	we	PRON
ejpam-4953	117	22	get	get	VERB
ejpam-4953	117	23	.	.	PUNCT
ejpam-4953	118	1	(	(	PUNCT
ejpam-4953	118	2	f̈ϖ0	f̈ϖ0	PROPN
ejpam-4953	118	3	,	,	PUNCT
ejpam-4953	118	4	f̈	f̈	PROPN
ejpam-4953	118	5	2ϖ0	2ϖ0	NUM
ejpam-4953	118	6	)	)	PUNCT
ejpam-4953	118	7	,	,	PUNCT
ejpam-4953	118	8	(	(	PUNCT
ejpam-4953	118	9	f̈	f̈	X
ejpam-4953	118	10	2ϖ0	2ϖ0	NUM
ejpam-4953	118	11	,	,	PUNCT
ejpam-4953	118	12	f̈	f̈	X
ejpam-4953	118	13	3ϖ0	3ϖ0	NUM
ejpam-4953	118	14	)	)	PUNCT
ejpam-4953	118	15	,	,	PUNCT
ejpam-4953	118	16	.......	.......	PUNCT
ejpam-4953	118	17	,	,	PUNCT
ejpam-4953	118	18	(	(	PUNCT
ejpam-4953	118	19	f̈	f̈	X
ejpam-4953	118	20	nϖ0	nϖ0	NOUN
ejpam-4953	118	21	,	,	PUNCT
ejpam-4953	118	22	f̈	f̈	PROPN
ejpam-4953	118	23	n+1ϖ0	n+1ϖ0	PROPN
ejpam-4953	118	24	)	)	PUNCT
ejpam-4953	118	25	,	,	PUNCT
ejpam-4953	118	26	....	....	PUNCT
ejpam-4953	118	27	,	,	PUNCT
ejpam-4953	118	28	∈	∈	PROPN
ejpam-4953	118	29	r̈	r̈	PROPN
ejpam-4953	118	30	i.	i.	PROPN
ejpam-4953	118	31	alshammari	alshammari	PROPN
ejpam-4953	118	32	et	et	PROPN
ejpam-4953	118	33	al	al	PROPN
ejpam-4953	118	34	.	.	PUNCT
ejpam-4953	118	35	/	/	SYM
ejpam-4953	118	36	eur	eur	PROPN
ejpam-4953	118	37	.	.	PUNCT
ejpam-4953	119	1	j.	j.	PROPN
ejpam-4953	119	2	pure	pure	PROPN
ejpam-4953	119	3	appl	appl	PROPN
ejpam-4953	119	4	.	.	PROPN
ejpam-4953	119	5	math	math	PROPN
ejpam-4953	119	6	,	,	PUNCT
ejpam-4953	119	7	16	16	NUM
ejpam-4953	119	8	(	(	PUNCT
ejpam-4953	119	9	4	4	NUM
ejpam-4953	119	10	)	)	PUNCT
ejpam-4953	119	11	(	(	PUNCT
ejpam-4953	119	12	2023	2023	NUM
ejpam-4953	119	13	)	)	PUNCT
ejpam-4953	119	14	,	,	PUNCT
ejpam-4953	119	15	2405	2405	NUM
ejpam-4953	119	16	-	-	SYM
ejpam-4953	119	17	2418	2418	NUM
ejpam-4953	119	18	2410	2410	NUM
ejpam-4953	120	1	so	so	SCONJ
ejpam-4953	120	2	that	that	SCONJ
ejpam-4953	120	3	(	(	PUNCT
ejpam-4953	120	4	ϖn	ϖn	NOUN
ejpam-4953	120	5	,	,	PUNCT
ejpam-4953	120	6	ϖn+1	ϖn+1	ADJ
ejpam-4953	120	7	)	)	PUNCT
ejpam-4953	120	8	∈	∈	PROPN
ejpam-4953	120	9	r̈	r̈	NOUN
ejpam-4953	120	10	,	,	PUNCT
ejpam-4953	120	11	for	for	ADP
ejpam-4953	120	12	all	all	PRON
ejpam-4953	120	13	n	n	PRON
ejpam-4953	120	14	∈	∈	PROPN
ejpam-4953	120	15	n	n	CCONJ
ejpam-4953	120	16	(	(	PUNCT
ejpam-4953	120	17	1	1	NUM
ejpam-4953	120	18	)	)	PUNCT
ejpam-4953	120	19	therefore	therefore	ADV
ejpam-4953	120	20	the	the	DET
ejpam-4953	120	21	sequence	sequence	NOUN
ejpam-4953	120	22	ϖn	ϖn	NOUN
ejpam-4953	120	23	is	be	AUX
ejpam-4953	120	24	r̈-preserving	r̈-preserve	VERB
ejpam-4953	120	25	.	.	PUNCT
ejpam-4953	121	1	applying	apply	VERB
ejpam-4953	121	2	the	the	DET
ejpam-4953	121	3	contractivity	contractivity	NOUN
ejpam-4953	121	4	condition	condition	NOUN
ejpam-4953	121	5	(	(	PUNCT
ejpam-4953	121	6	iv	iv	X
ejpam-4953	121	7	)	)	PUNCT
ejpam-4953	121	8	to	to	ADP
ejpam-4953	121	9	(	(	PUNCT
ejpam-4953	121	10	1	1	NUM
ejpam-4953	121	11	)	)	PUNCT
ejpam-4953	121	12	.	.	PUNCT
ejpam-4953	122	1	we	we	PRON
ejpam-4953	122	2	deduce	deduce	VERB
ejpam-4953	122	3	,	,	PUNCT
ejpam-4953	122	4	for	for	ADP
ejpam-4953	122	5	all	all	DET
ejpam-4953	122	6	n	n	PRON
ejpam-4953	122	7	∈	∈	PROPN
ejpam-4953	122	8	n.	n.	NOUN
ejpam-4953	122	9	that	that	SCONJ
ejpam-4953	122	10	p(ϖn	p(ϖn	NOUN
ejpam-4953	122	11	,	,	PUNCT
ejpam-4953	122	12	ϖn+1	ϖn+1	ADJ
ejpam-4953	122	13	)	)	PUNCT
ejpam-4953	122	14	≤	≤	NOUN
ejpam-4953	122	15	p(ϖn−1	p(ϖn−1	NOUN
ejpam-4953	122	16	,	,	PUNCT
ejpam-4953	122	17	ϖn	ϖn	NOUN
ejpam-4953	122	18	)	)	PUNCT
ejpam-4953	122	19	λ	λ	NOUN
ejpam-4953	122	20	,	,	PUNCT
ejpam-4953	122	21	which	which	PRON
ejpam-4953	122	22	by	by	ADP
ejpam-4953	122	23	induction	induction	NOUN
ejpam-4953	122	24	yield	yield	NOUN
ejpam-4953	122	25	that	that	SCONJ
ejpam-4953	122	26	p(ϖn	p(ϖn	NOUN
ejpam-4953	122	27	,	,	PUNCT
ejpam-4953	122	28	ϖn+1	ϖn+1	ADJ
ejpam-4953	122	29	)	)	PUNCT
ejpam-4953	122	30	≤	≤	NOUN
ejpam-4953	122	31	p(ϖ0	p(ϖ0	NOUN
ejpam-4953	122	32	,	,	PUNCT
ejpam-4953	122	33	f̈ϖ0	f̈ϖ0	PROPN
ejpam-4953	122	34	)	)	PUNCT
ejpam-4953	122	35	λn	λn	NOUN
ejpam-4953	122	36	for	for	ADP
ejpam-4953	122	37	all	all	PRON
ejpam-4953	122	38	n	n	PRON
ejpam-4953	122	39	∈	∈	PROPN
ejpam-4953	122	40	n	n	CCONJ
ejpam-4953	122	41	(	(	PUNCT
ejpam-4953	122	42	2	2	NUM
ejpam-4953	122	43	)	)	PUNCT
ejpam-4953	122	44	by	by	ADP
ejpam-4953	122	45	using	use	VERB
ejpam-4953	122	46	(	(	PUNCT
ejpam-4953	122	47	2	2	NUM
ejpam-4953	122	48	)	)	PUNCT
ejpam-4953	122	49	and	and	CCONJ
ejpam-4953	122	50	multiplicative	multiplicative	ADJ
ejpam-4953	122	51	triangular	triangular	NOUN
ejpam-4953	122	52	inequality	inequality	NOUN
ejpam-4953	122	53	,	,	PUNCT
ejpam-4953	122	54	for	for	ADP
ejpam-4953	122	55	all	all	DET
ejpam-4953	122	56	n	n	PRON
ejpam-4953	122	57	∈	∈	PROPN
ejpam-4953	122	58	n	n	CCONJ
ejpam-4953	122	59	,	,	PUNCT
ejpam-4953	122	60	r	r	NOUN
ejpam-4953	122	61	∈	∈	PROPN
ejpam-4953	122	62	n	n	CCONJ
ejpam-4953	122	63	,	,	PUNCT
ejpam-4953	122	64	we	we	PRON
ejpam-4953	122	65	have	have	VERB
ejpam-4953	122	66	p(ϖn	p(ϖn	NOUN
ejpam-4953	122	67	,	,	PUNCT
ejpam-4953	122	68	ϖn+r	ϖn+r	NOUN
ejpam-4953	122	69	)	)	PUNCT
ejpam-4953	122	70	≤	≤	NOUN
ejpam-4953	122	71	p(ϖn	p(ϖn	PROPN
ejpam-4953	122	72	,	,	PUNCT
ejpam-4953	122	73	ϖn+1	ϖn+1	ADJ
ejpam-4953	122	74	)	)	PUNCT
ejpam-4953	123	1	kn	kn	PROPN
ejpam-4953	123	2	·	·	PUNCT
ejpam-4953	123	3	p(ϖn+1	p(ϖn+1	PROPN
ejpam-4953	123	4	,	,	PUNCT
ejpam-4953	123	5	ϖn+2	ϖn+2	X
ejpam-4953	123	6	)	)	PUNCT
ejpam-4953	123	7	kn+1	kn+1	PROPN
ejpam-4953	123	8	·	·	PUNCT
ejpam-4953	123	9	·	·	PUNCT
ejpam-4953	123	10	·	·	PUNCT
ejpam-4953	123	11	·	·	PUNCT
ejpam-4953	123	12	p(ϖn+r−1	p(ϖn+r−1	X
ejpam-4953	123	13	,	,	PUNCT
ejpam-4953	123	14	ϖn+r	ϖn+r	PROPN
ejpam-4953	123	15	)	)	PUNCT
ejpam-4953	123	16	kn+r−1	kn+r−1	ADJ
ejpam-4953	123	17	≤	≤	PROPN
ejpam-4953	123	18	p(ϖn	p(ϖn	PROPN
ejpam-4953	123	19	,	,	PUNCT
ejpam-4953	123	20	ϖn+1	ϖn+1	ADJ
ejpam-4953	123	21	)	)	PUNCT
ejpam-4953	123	22	λnkn	λnkn	PROPN
ejpam-4953	123	23	·	·	PUNCT
ejpam-4953	123	24	d(ϖn+1	d(ϖn+1	PROPN
ejpam-4953	123	25	,	,	PUNCT
ejpam-4953	123	26	ϖn+2	ϖn+2	NUM
ejpam-4953	123	27	)	)	PUNCT
ejpam-4953	123	28	λn+1kn+1	λn+1kn+1	VERB
ejpam-4953	123	29	·	·	PUNCT
ejpam-4953	123	30	·	·	PUNCT
ejpam-4953	123	31	·	·	PUNCT
ejpam-4953	123	32	·	·	PUNCT
ejpam-4953	123	33	p(ϖn+r−1	p(ϖn+r−1	X
ejpam-4953	123	34	,	,	PUNCT
ejpam-4953	123	35	ϖn+r	ϖn+r	PROPN
ejpam-4953	123	36	)	)	PUNCT
ejpam-4953	124	1	λn+r−1kn+r−1	λn+r−1kn+r−1	PROPN
ejpam-4953	124	2	≤	≤	ADJ
ejpam-4953	124	3	p(ϖ0	p(ϖ0	NOUN
ejpam-4953	124	4	,	,	PUNCT
ejpam-4953	124	5	f̈ϖ0	f̈ϖ0	PROPN
ejpam-4953	124	6	)	)	PUNCT
ejpam-4953	124	7	(	(	PUNCT
ejpam-4953	124	8	λk)n+(λk)n+1+···+(λk)n+r−1	λk)n+(λk)n+1+···+(λk)n+r−1	NOUN
ejpam-4953	124	9	≤	≤	ADJ
ejpam-4953	124	10	p(ϖ0	p(ϖ0	NOUN
ejpam-4953	124	11	,	,	PUNCT
ejpam-4953	124	12	f̈ϖ0	f̈ϖ0	PROPN
ejpam-4953	124	13	)	)	PUNCT
ejpam-4953	124	14	(	(	PUNCT
ejpam-4953	124	15	λk)n	λk)n	NOUN
ejpam-4953	124	16	1−(λk	1−(λk	NUM
ejpam-4953	124	17	)	)	PUNCT
ejpam-4953	124	18	.	.	PUNCT
ejpam-4953	125	1	this	this	PRON
ejpam-4953	125	2	implies	imply	VERB
ejpam-4953	125	3	that	that	SCONJ
ejpam-4953	125	4	p(ϖn	p(ϖn	NOUN
ejpam-4953	125	5	,	,	PUNCT
ejpam-4953	125	6	ϖn+r	ϖn+r	NOUN
ejpam-4953	125	7	)	)	PUNCT
ejpam-4953	125	8	→b	→b	NUM
ejpam-4953	125	9	1	1	NUM
ejpam-4953	125	10	,	,	PUNCT
ejpam-4953	125	11	(	(	PUNCT
ejpam-4953	125	12	as	as	ADP
ejpam-4953	125	13	n	n	NOUN
ejpam-4953	125	14	→	→	SYM
ejpam-4953	125	15	+	+	NOUN
ejpam-4953	125	16	∞	∞	NOUN
ejpam-4953	125	17	)	)	PUNCT
ejpam-4953	125	18	hence	hence	ADV
ejpam-4953	125	19	,	,	PUNCT
ejpam-4953	125	20	the	the	DET
ejpam-4953	125	21	sequence	sequence	NOUN
ejpam-4953	125	22	ϖn	ϖn	NOUN
ejpam-4953	125	23	is	be	AUX
ejpam-4953	125	24	multiplicative	multiplicative	ADJ
ejpam-4953	125	25	cauchy	cauchy	ADJ
ejpam-4953	125	26	sequence	sequence	NOUN
ejpam-4953	125	27	in	in	ADP
ejpam-4953	125	28	ḧ.	ḧ.	PROPN
ejpam-4953	125	29	as	as	ADP
ejpam-4953	125	30	(	(	PUNCT
ejpam-4953	125	31	ḧ	ḧ	NOUN
ejpam-4953	125	32	,	,	PUNCT
ejpam-4953	125	33	p	p	X
ejpam-4953	125	34	,	,	PUNCT
ejpam-4953	125	35	k	k	X
ejpam-4953	125	36	≥	≥	NUM
ejpam-4953	125	37	1	1	NUM
ejpam-4953	125	38	)	)	PUNCT
ejpam-4953	125	39	is	be	AUX
ejpam-4953	125	40	b	b	NOUN
ejpam-4953	125	41	-	-	PUNCT
ejpam-4953	125	42	complete	complete	ADJ
ejpam-4953	125	43	,	,	PUNCT
ejpam-4953	125	44	there	there	PRON
ejpam-4953	125	45	exists	exist	VERB
ejpam-4953	125	46	ϖ∗	ϖ∗	PROPN
ejpam-4953	125	47	∈	∈	PROPN
ejpam-4953	125	48	ḧ	ḧ	NOUN
ejpam-4953	126	1	such	such	ADJ
ejpam-4953	126	2	that	that	PRON
ejpam-4953	126	3	ϖn	ϖn	NOUN
ejpam-4953	126	4	−→	−→	NOUN
ejpam-4953	126	5	ϖ∗.	ϖ∗.	PUNCT
ejpam-4953	126	6	now	now	ADV
ejpam-4953	126	7	,	,	PUNCT
ejpam-4953	126	8	in	in	ADP
ejpam-4953	126	9	lieu	lieu	NOUN
ejpam-4953	126	10	of	of	ADP
ejpam-4953	126	11	(	(	PUNCT
ejpam-4953	126	12	iii	iii	NOUN
ejpam-4953	126	13	)	)	PUNCT
ejpam-4953	126	14	assume	assume	VERB
ejpam-4953	126	15	that	that	SCONJ
ejpam-4953	126	16	f̈	f̈	PROPN
ejpam-4953	126	17	is	be	AUX
ejpam-4953	126	18	b	b	NOUN
ejpam-4953	126	19	-	-	ADJ
ejpam-4953	126	20	continuous	continuous	ADJ
ejpam-4953	126	21	,	,	PUNCT
ejpam-4953	126	22	we	we	PRON
ejpam-4953	126	23	have	have	VERB
ejpam-4953	126	24	ϖn+1	ϖn+1	ADJ
ejpam-4953	126	25	=	=	SYM
ejpam-4953	126	26	f̈ϖn	f̈ϖn	NOUN
ejpam-4953	126	27	p−→	p−→	NOUN
ejpam-4953	127	1	f̈ϖ∗.	f̈ϖ∗.	PROPN
ejpam-4953	127	2	owing	owe	VERB
ejpam-4953	127	3	to	to	ADP
ejpam-4953	127	4	the	the	DET
ejpam-4953	127	5	uniqueness	uniqueness	NOUN
ejpam-4953	127	6	of	of	ADP
ejpam-4953	127	7	limit	limit	NOUN
ejpam-4953	127	8	,	,	PUNCT
ejpam-4953	127	9	we	we	PRON
ejpam-4953	127	10	obtain	obtain	VERB
ejpam-4953	127	11	f̈ϖ∗	f̈ϖ∗	PROPN
ejpam-4953	127	12	=	=	SYM
ejpam-4953	127	13	ϖ∗	ϖ∗	PROPN
ejpam-4953	127	14	i.e.	i.e.	X
ejpam-4953	127	15	,	,	PUNCT
ejpam-4953	127	16	ϖ∗	ϖ∗	PROPN
ejpam-4953	127	17	is	be	AUX
ejpam-4953	127	18	a	a	DET
ejpam-4953	127	19	fixed	fix	VERB
ejpam-4953	127	20	point	point	NOUN
ejpam-4953	127	21	of	of	ADP
ejpam-4953	127	22	f̈.	f̈.	NOUN
ejpam-4953	127	23	alternately	alternately	ADV
ejpam-4953	127	24	,	,	PUNCT
ejpam-4953	127	25	suppose	suppose	VERB
ejpam-4953	127	26	that	that	SCONJ
ejpam-4953	127	27	r̈	r̈	VERB
ejpam-4953	127	28	is	be	AUX
ejpam-4953	127	29	bp	bp	PROPN
ejpam-4953	127	30	−	−	PROPN
ejpam-4953	127	31	selfclosed	selfclose	VERB
ejpam-4953	127	32	.	.	PUNCT
ejpam-4953	128	1	since	since	SCONJ
ejpam-4953	128	2	ϖn	ϖn	NOUN
ejpam-4953	128	3	is	be	AUX
ejpam-4953	128	4	an	an	DET
ejpam-4953	128	5	r̈-preserving	r̈-preserve	VERB
ejpam-4953	128	6	sequence	sequence	NOUN
ejpam-4953	128	7	and	and	CCONJ
ejpam-4953	128	8	ϖn	ϖn	NOUN
ejpam-4953	128	9	p−→	p−→	NOUN
ejpam-4953	128	10	ϖ.	ϖ.	VERB
ejpam-4953	128	11	by	by	ADP
ejpam-4953	128	12	the	the	DET
ejpam-4953	128	13	bp	bp	PROPN
ejpam-4953	128	14	−	−	PROPN
ejpam-4953	128	15	selfcloseness	selfcloseness	NOUN
ejpam-4953	128	16	of	of	ADP
ejpam-4953	128	17	r̈	r̈	NOUN
ejpam-4953	128	18	,	,	PUNCT
ejpam-4953	128	19	there	there	PRON
ejpam-4953	128	20	exists	exist	VERB
ejpam-4953	128	21	a	a	DET
ejpam-4953	128	22	subsequence	subsequence	NOUN
ejpam-4953	128	23	{	{	PUNCT
ejpam-4953	128	24	ϖnj	ϖnj	NOUN
ejpam-4953	128	25	}	}	PUNCT
ejpam-4953	128	26	of	of	ADP
ejpam-4953	128	27	{	{	PUNCT
ejpam-4953	128	28	ϖn	ϖn	NOUN
ejpam-4953	128	29	}	}	PUNCT
ejpam-4953	128	30	with	with	ADP
ejpam-4953	128	31	[	[	X
ejpam-4953	128	32	ϖnj	ϖnj	NOUN
ejpam-4953	128	33	,	,	PUNCT
ejpam-4953	128	34	ϖ	ϖ	X
ejpam-4953	128	35	]	]	X
ejpam-4953	128	36	∈	∈	PROPN
ejpam-4953	128	37	r̈	r̈	VERB
ejpam-4953	128	38	for	for	ADP
ejpam-4953	128	39	all	all	DET
ejpam-4953	128	40	j	j	PROPN
ejpam-4953	128	41	∈	∈	PROPN
ejpam-4953	128	42	n	n	CCONJ
ejpam-4953	128	43	using	use	VERB
ejpam-4953	128	44	(	(	PUNCT
ejpam-4953	128	45	iv	iv	NUM
ejpam-4953	128	46	)	)	PUNCT
ejpam-4953	128	47	,	,	PUNCT
ejpam-4953	128	48	proposition	proposition	NOUN
ejpam-4953	128	49	(	(	PUNCT
ejpam-4953	128	50	1.1	1.1	NUM
ejpam-4953	128	51	)	)	PUNCT
ejpam-4953	128	52	,	,	PUNCT
ejpam-4953	128	53	we	we	PRON
ejpam-4953	128	54	obtain	obtain	VERB
ejpam-4953	128	55	p(ϖ∗	p(ϖ∗	NOUN
ejpam-4953	128	56	,	,	PUNCT
ejpam-4953	128	57	f̈ϖ∗	f̈ϖ∗	NOUN
ejpam-4953	128	58	)	)	PUNCT
ejpam-4953	128	59	≤	≤	NOUN
ejpam-4953	129	1	[	[	X
ejpam-4953	129	2	p(ϖ∗	p(ϖ∗	X
ejpam-4953	129	3	,	,	PUNCT
ejpam-4953	129	4	ϖn+1	ϖn+1	PROPN
ejpam-4953	129	5	)	)	PUNCT
ejpam-4953	129	6	·	·	PUNCT
ejpam-4953	129	7	p(ϖn+1	p(ϖn+1	PROPN
ejpam-4953	129	8	,	,	PUNCT
ejpam-4953	129	9	f̈ϖ	f̈ϖ	PROPN
ejpam-4953	129	10	∗)]k	∗)]k	NUM
ejpam-4953	129	11	i.	i.	NOUN
ejpam-4953	129	12	alshammari	alshammari	X
ejpam-4953	129	13	et	et	PROPN
ejpam-4953	129	14	al	al	PROPN
ejpam-4953	129	15	.	.	PUNCT
ejpam-4953	129	16	/	/	SYM
ejpam-4953	129	17	eur	eur	PROPN
ejpam-4953	129	18	.	.	PUNCT
ejpam-4953	130	1	j.	j.	PROPN
ejpam-4953	130	2	pure	pure	PROPN
ejpam-4953	130	3	appl	appl	PROPN
ejpam-4953	130	4	.	.	PROPN
ejpam-4953	130	5	math	math	PROPN
ejpam-4953	130	6	,	,	PUNCT
ejpam-4953	130	7	16	16	NUM
ejpam-4953	130	8	(	(	PUNCT
ejpam-4953	130	9	4	4	NUM
ejpam-4953	130	10	)	)	PUNCT
ejpam-4953	130	11	(	(	PUNCT
ejpam-4953	130	12	2023	2023	NUM
ejpam-4953	130	13	)	)	PUNCT
ejpam-4953	130	14	,	,	PUNCT
ejpam-4953	130	15	2405	2405	NUM
ejpam-4953	130	16	-	-	SYM
ejpam-4953	130	17	2418	2418	NUM
ejpam-4953	130	18	2411	2411	NUM
ejpam-4953	131	1	=	=	PUNCT
ejpam-4953	132	1	[	[	X
ejpam-4953	132	2	p(ϖ∗	p(ϖ∗	X
ejpam-4953	132	3	,	,	PUNCT
ejpam-4953	132	4	ϖn+1	ϖn+1	PROPN
ejpam-4953	132	5	)	)	PUNCT
ejpam-4953	132	6	·	·	PUNCT
ejpam-4953	132	7	p(ϖn+1	p(ϖn+1	PROPN
ejpam-4953	132	8	,	,	PUNCT
ejpam-4953	132	9	f̈ϖ	f̈ϖ	PROPN
ejpam-4953	132	10	∗)]k	∗)]k	PART
ejpam-4953	132	11	≤	≤	NOUN
ejpam-4953	132	12	[	[	X
ejpam-4953	132	13	p(ϖ∗	p(ϖ∗	X
ejpam-4953	132	14	,	,	PUNCT
ejpam-4953	132	15	ϖn+1	ϖn+1	PROPN
ejpam-4953	132	16	)	)	PUNCT
ejpam-4953	132	17	·	·	PUNCT
ejpam-4953	132	18	p(ϖn+1	p(ϖn+1	PROPN
ejpam-4953	132	19	,	,	PUNCT
ejpam-4953	132	20	f̈ϖ	f̈ϖ	PROPN
ejpam-4953	132	21	∗)λ]k	∗)λ]k	NOUN
ejpam-4953	132	22	→	→	SYM
ejpam-4953	132	23	1	1	NUM
ejpam-4953	132	24	as	as	ADP
ejpam-4953	132	25	n	n	PRON
ejpam-4953	132	26	→	→	PUNCT
ejpam-4953	132	27	+	+	PROPN
ejpam-4953	132	28	∞.	∞.	PROPN
ejpam-4953	132	29	hence	hence	ADV
ejpam-4953	132	30	,	,	PUNCT
ejpam-4953	132	31	f̈ϖ∗	f̈ϖ∗	PROPN
ejpam-4953	132	32	=	=	SYM
ejpam-4953	132	33	ϖ∗	ϖ∗	PROPN
ejpam-4953	132	34	and	and	CCONJ
ejpam-4953	132	35	ϖ∗	ϖ∗	PROPN
ejpam-4953	132	36	is	be	AUX
ejpam-4953	132	37	a	a	DET
ejpam-4953	132	38	fixed	fix	VERB
ejpam-4953	132	39	point	point	NOUN
ejpam-4953	132	40	of	of	ADP
ejpam-4953	132	41	f̈.	f̈.	PROPN
ejpam-4953	132	42	suppose	suppose	VERB
ejpam-4953	132	43	that	that	SCONJ
ejpam-4953	132	44	ρ∗	ρ∗	PROPN
ejpam-4953	132	45	is	be	AUX
ejpam-4953	132	46	another	another	DET
ejpam-4953	132	47	fixed	fix	VERB
ejpam-4953	132	48	point	point	NOUN
ejpam-4953	132	49	of	of	ADP
ejpam-4953	132	50	f̈.	f̈.	PROPN
ejpam-4953	132	51	by	by	ADP
ejpam-4953	132	52	assumption	assumption	NOUN
ejpam-4953	132	53	(	(	PUNCT
ejpam-4953	132	54	v	v	NOUN
ejpam-4953	132	55	)	)	PUNCT
ejpam-4953	132	56	,	,	PUNCT
ejpam-4953	132	57	there	there	PRON
ejpam-4953	132	58	exists	exist	VERB
ejpam-4953	132	59	a	a	DET
ejpam-4953	132	60	path	path	NOUN
ejpam-4953	132	61	(	(	PUNCT
ejpam-4953	132	62	say	say	INTJ
ejpam-4953	132	63	{	{	PUNCT
ejpam-4953	132	64	t0	t0	NOUN
ejpam-4953	132	65	,	,	PUNCT
ejpam-4953	132	66	t1	t1	NOUN
ejpam-4953	132	67	,	,	PUNCT
ejpam-4953	132	68	t2	t2	NOUN
ejpam-4953	132	69	,	,	PUNCT
ejpam-4953	132	70	.....	.....	PUNCT
ejpam-4953	132	71	tk	tk	PROPN
ejpam-4953	132	72	,	,	PUNCT
ejpam-4953	132	73	}	}	PUNCT
ejpam-4953	132	74	)	)	PUNCT
ejpam-4953	132	75	of	of	ADP
ejpam-4953	132	76	some	some	DET
ejpam-4953	132	77	finite	finite	ADJ
ejpam-4953	132	78	length	length	NOUN
ejpam-4953	132	79	k	k	PROPN
ejpam-4953	132	80	in	in	ADP
ejpam-4953	132	81	r̈s	r̈s	NOUN
ejpam-4953	132	82	from	from	ADP
ejpam-4953	132	83	ϖ	ϖ	PRON
ejpam-4953	132	84	to	to	ADP
ejpam-4953	132	85	ρ	ρ	NUM
ejpam-4953	132	86	so	so	SCONJ
ejpam-4953	132	87	that	that	SCONJ
ejpam-4953	132	88	t0	t0	AUX
ejpam-4953	132	89	=	=	SYM
ejpam-4953	133	1	ϖ	ϖ	PROPN
ejpam-4953	133	2	,	,	PUNCT
ejpam-4953	133	3	tk	tk	PROPN
ejpam-4953	133	4	=	=	PROPN
ejpam-4953	133	5	ρ	ρ	PROPN
ejpam-4953	133	6	,	,	PUNCT
ejpam-4953	133	7	[	[	X
ejpam-4953	133	8	tj	tj	X
ejpam-4953	133	9	,	,	PUNCT
ejpam-4953	133	10	tj+1	tj+1	PROPN
ejpam-4953	133	11	]	]	X
ejpam-4953	133	12	∈	∈	PROPN
ejpam-4953	133	13	r̈	r̈	VERB
ejpam-4953	133	14	for	for	ADP
ejpam-4953	133	15	each	each	DET
ejpam-4953	133	16	j	j	PROPN
ejpam-4953	133	17	(	(	PUNCT
ejpam-4953	133	18	0	0	NUM
ejpam-4953	133	19	≤	≤	NUM
ejpam-4953	134	1	j	j	PROPN
ejpam-4953	134	2	≤	≤	PROPN
ejpam-4953	134	3	k	k	NOUN
ejpam-4953	135	1	−	−	NOUN
ejpam-4953	135	2	1	1	NUM
ejpam-4953	135	3	)	)	PUNCT
ejpam-4953	135	4	.	.	PUNCT
ejpam-4953	136	1	(	(	PUNCT
ejpam-4953	136	2	3	3	X
ejpam-4953	136	3	)	)	PUNCT
ejpam-4953	136	4	as	as	SCONJ
ejpam-4953	136	5	r̈	r̈	NOUN
ejpam-4953	136	6	is	be	AUX
ejpam-4953	136	7	f̈-closed	f̈-close	VERB
ejpam-4953	136	8	,	,	PUNCT
ejpam-4953	136	9	by	by	ADP
ejpam-4953	136	10	using	use	VERB
ejpam-4953	136	11	proposition	proposition	NOUN
ejpam-4953	136	12	(	(	PUNCT
ejpam-4953	136	13	1.3	1.3	NUM
ejpam-4953	136	14	)	)	PUNCT
ejpam-4953	136	15	,	,	PUNCT
ejpam-4953	136	16	we	we	PRON
ejpam-4953	136	17	have	have	VERB
ejpam-4953	136	18	[	[	X
ejpam-4953	136	19	f̈ntj	f̈ntj	PROPN
ejpam-4953	136	20	,	,	PUNCT
ejpam-4953	136	21	f̈	f̈	ADP
ejpam-4953	136	22	ntj+1	ntj+1	PROPN
ejpam-4953	136	23	]	]	PUNCT
ejpam-4953	136	24	∈	∈	PROPN
ejpam-4953	136	25	r̈	r̈	VERB
ejpam-4953	136	26	for	for	ADP
ejpam-4953	136	27	each	each	DET
ejpam-4953	136	28	j	j	PROPN
ejpam-4953	136	29	(	(	PUNCT
ejpam-4953	136	30	0	0	NUM
ejpam-4953	136	31	≤	≤	NUM
ejpam-4953	137	1	j	j	PROPN
ejpam-4953	137	2	≤	≤	PROPN
ejpam-4953	137	3	k	k	NOUN
ejpam-4953	138	1	−	−	NOUN
ejpam-4953	138	2	1	1	NUM
ejpam-4953	138	3	)	)	PUNCT
ejpam-4953	138	4	and	and	CCONJ
ejpam-4953	138	5	for	for	ADP
ejpam-4953	138	6	each	each	DET
ejpam-4953	138	7	n	n	PRON
ejpam-4953	138	8	∈	∈	PROPN
ejpam-4953	138	9	n	n	CCONJ
ejpam-4953	138	10	(	(	PUNCT
ejpam-4953	138	11	4	4	X
ejpam-4953	138	12	)	)	PUNCT
ejpam-4953	138	13	making	make	VERB
ejpam-4953	138	14	use	use	NOUN
ejpam-4953	138	15	of	of	ADP
ejpam-4953	138	16	(	(	PUNCT
ejpam-4953	138	17	3	3	NUM
ejpam-4953	138	18	)	)	PUNCT
ejpam-4953	138	19	,	,	PUNCT
ejpam-4953	138	20	(	(	PUNCT
ejpam-4953	138	21	4	4	NUM
ejpam-4953	138	22	)	)	PUNCT
ejpam-4953	138	23	,	,	PUNCT
ejpam-4953	138	24	(	(	PUNCT
ejpam-4953	138	25	5	5	NUM
ejpam-4953	138	26	)	)	PUNCT
ejpam-4953	138	27	,	,	PUNCT
ejpam-4953	138	28	triangular	triangular	NOUN
ejpam-4953	138	29	inequality	inequality	NOUN
ejpam-4953	138	30	,	,	PUNCT
ejpam-4953	138	31	assumption	assumption	NOUN
ejpam-4953	138	32	(	(	PUNCT
ejpam-4953	138	33	iv	iv	NOUN
ejpam-4953	138	34	)	)	PUNCT
ejpam-4953	138	35	and	and	CCONJ
ejpam-4953	138	36	proposition	proposition	NOUN
ejpam-4953	138	37	(	(	PUNCT
ejpam-4953	138	38	1.1	1.1	NUM
ejpam-4953	138	39	)	)	PUNCT
ejpam-4953	138	40	,	,	PUNCT
ejpam-4953	138	41	we	we	PRON
ejpam-4953	138	42	obtain	obtain	VERB
ejpam-4953	138	43	p(ϖ	p(ϖ	PROPN
ejpam-4953	138	44	,	,	PUNCT
ejpam-4953	138	45	ρ	ρ	NOUN
ejpam-4953	138	46	)	)	PUNCT
ejpam-4953	138	47	=	=	SYM
ejpam-4953	139	1	p(f̈nt0	p(f̈nt0	NOUN
ejpam-4953	139	2	,	,	PUNCT
ejpam-4953	139	3	f̈	f̈	PROPN
ejpam-4953	139	4	ntk	ntk	PROPN
ejpam-4953	139	5	)	)	PUNCT
ejpam-4953	139	6	≤	≤	NOUN
ejpam-4953	140	1	k−1∏	k−1∏	PROPN
ejpam-4953	140	2	j=0	j=0	PROPN
ejpam-4953	140	3	(	(	PUNCT
ejpam-4953	140	4	f̈ntj	f̈ntj	PROPN
ejpam-4953	140	5	,	,	PUNCT
ejpam-4953	140	6	f̈	f̈	PRON
ejpam-4953	140	7	ntj+1	ntj+1	PROPN
ejpam-4953	140	8	)	)	PUNCT
ejpam-4953	140	9	≤	≤	NOUN
ejpam-4953	141	1	k−1∏	k−1∏	PROPN
ejpam-4953	141	2	j=0	j=0	PROPN
ejpam-4953	141	3	p(f̈n−1tj	p(f̈n−1tj	PROPN
ejpam-4953	141	4	,	,	PUNCT
ejpam-4953	141	5	f̈	f̈	PROPN
ejpam-4953	141	6	n−1tj+1	n−1tj+1	PRON
ejpam-4953	141	7	)	)	PUNCT
ejpam-4953	141	8	λ	λ	NOUN
ejpam-4953	141	9	≤	≤	NOUN
ejpam-4953	141	10	k−1∏	k−1∏	PROPN
ejpam-4953	141	11	j=0	j=0	PROPN
ejpam-4953	141	12	p(f̈n−2tj	p(f̈n−2tj	PRON
ejpam-4953	141	13	,	,	PUNCT
ejpam-4953	141	14	f̈	f̈	PROPN
ejpam-4953	141	15	n−2tj+1	n−2tj+1	NUM
ejpam-4953	141	16	)	)	PUNCT
ejpam-4953	141	17	λ2	λ2	NOUN
ejpam-4953	141	18	≤	≤	NOUN
ejpam-4953	141	19	·	·	PUNCT
ejpam-4953	141	20	·	·	PUNCT
ejpam-4953	141	21	·	·	PUNCT
ejpam-4953	141	22	·	·	PUNCT
ejpam-4953	141	23	≤	≤	NOUN
ejpam-4953	142	1	k−1∏	k−1∏	PROPN
ejpam-4953	142	2	j=0	j=0	PROPN
ejpam-4953	142	3	p(tj	p(tj	ADJ
ejpam-4953	142	4	,	,	PUNCT
ejpam-4953	142	5	tj+1	tj+1	X
ejpam-4953	142	6	)	)	PUNCT
ejpam-4953	142	7	λn	λn	NOUN
ejpam-4953	142	8	→	→	SYM
ejpam-4953	142	9	1	1	NUM
ejpam-4953	142	10	as	as	ADP
ejpam-4953	142	11	n	n	NOUN
ejpam-4953	142	12	→	→	SYM
ejpam-4953	142	13	+	+	ADJ
ejpam-4953	142	14	∞	∞	PROPN
ejpam-4953	142	15	(	(	PUNCT
ejpam-4953	142	16	5	5	NUM
ejpam-4953	142	17	)	)	PUNCT
ejpam-4953	142	18	so	so	ADV
ejpam-4953	142	19	,	,	PUNCT
ejpam-4953	142	20	that	that	SCONJ
ejpam-4953	142	21	ϖ=ρ	ϖ=ρ	PROPN
ejpam-4953	142	22	.	.	PUNCT
ejpam-4953	143	1	hence	hence	ADV
ejpam-4953	143	2	f̈	f̈	PROPN
ejpam-4953	143	3	has	have	VERB
ejpam-4953	143	4	a	a	DET
ejpam-4953	143	5	unique	unique	ADJ
ejpam-4953	143	6	fixed	fix	VERB
ejpam-4953	143	7	point	point	NOUN
ejpam-4953	143	8	.	.	PUNCT
ejpam-4953	144	1	theorem	theorem	NOUN
ejpam-4953	144	2	2	2	NUM
ejpam-4953	144	3	.	.	PUNCT
ejpam-4953	145	1	let	let	AUX
ejpam-4953	145	2	(	(	PUNCT
ejpam-4953	145	3	ḧ,d	ḧ,d	NOUN
ejpam-4953	145	4	,	,	PUNCT
ejpam-4953	145	5	k	k	PROPN
ejpam-4953	145	6	≥	≥	NUM
ejpam-4953	145	7	1	1	NUM
ejpam-4953	145	8	)	)	PUNCT
ejpam-4953	145	9	be	be	AUX
ejpam-4953	145	10	a	a	DET
ejpam-4953	145	11	b	b	NOUN
ejpam-4953	145	12	-	-	PUNCT
ejpam-4953	145	13	complete	complete	ADJ
ejpam-4953	145	14	b	b	X
ejpam-4953	145	15	-	-	PUNCT
ejpam-4953	145	16	multiplicative	multiplicative	ADJ
ejpam-4953	145	17	metric	metric	ADJ
ejpam-4953	145	18	space	space	NOUN
ejpam-4953	145	19	and	and	CCONJ
ejpam-4953	145	20	r̈	r̈	VERB
ejpam-4953	145	21	a	a	DET
ejpam-4953	145	22	binary	binary	ADJ
ejpam-4953	145	23	relation	relation	NOUN
ejpam-4953	145	24	on	on	ADP
ejpam-4953	145	25	ḧ.	ḧ.	PROPN
ejpam-4953	145	26	s	s	PROPN
ejpam-4953	145	27	,	,	PUNCT
ejpam-4953	145	28	v	v	NOUN
ejpam-4953	145	29	:	:	PUNCT
ejpam-4953	145	30	ḧ	ḧ	NOUN
ejpam-4953	145	31	→	→	SYM
ejpam-4953	145	32	ḧ	ḧ	VERB
ejpam-4953	145	33	be	be	AUX
ejpam-4953	145	34	a	a	DET
ejpam-4953	145	35	self	self	NOUN
ejpam-4953	145	36	-	-	PUNCT
ejpam-4953	145	37	mapping	mapping	NOUN
ejpam-4953	145	38	satisfying	satisfy	VERB
ejpam-4953	145	39	the	the	DET
ejpam-4953	145	40	following	follow	VERB
ejpam-4953	145	41	conditions	condition	NOUN
ejpam-4953	145	42	given	give	VERB
ejpam-4953	145	43	below	below	ADV
ejpam-4953	145	44	.	.	PUNCT
ejpam-4953	146	1	(	(	PUNCT
ejpam-4953	146	2	i	i	NOUN
ejpam-4953	146	3	)	)	PUNCT
ejpam-4953	146	4	ḧ(s	ḧ(s	PROPN
ejpam-4953	146	5	,	,	PUNCT
ejpam-4953	146	6	v	v	NOUN
ejpam-4953	146	7	;	;	PUNCT
ejpam-4953	146	8	r̈	r̈	NOUN
ejpam-4953	146	9	)	)	PUNCT
ejpam-4953	146	10	are	be	AUX
ejpam-4953	146	11	non	non	ADJ
ejpam-4953	146	12	-	-	ADJ
ejpam-4953	146	13	empty	empty	ADJ
ejpam-4953	146	14	and	and	CCONJ
ejpam-4953	146	15	s(ḧ	s(ḧ	NOUN
ejpam-4953	146	16	)	)	PUNCT
ejpam-4953	146	17	⊆	⊆	NUM
ejpam-4953	146	18	v	v	NOUN
ejpam-4953	146	19	(	(	PUNCT
ejpam-4953	146	20	ḧ	ḧ	NOUN
ejpam-4953	146	21	)	)	PUNCT
ejpam-4953	146	22	;	;	PUNCT
ejpam-4953	146	23	(	(	PUNCT
ejpam-4953	146	24	ii	ii	NOUN
ejpam-4953	146	25	)	)	PUNCT
ejpam-4953	146	26	r̈	r̈	VERB
ejpam-4953	146	27	is	be	AUX
ejpam-4953	146	28	(	(	PUNCT
ejpam-4953	146	29	s	s	X
ejpam-4953	146	30	,	,	PUNCT
ejpam-4953	146	31	v)-closed	v)-close	VERB
ejpam-4953	146	32	.	.	PUNCT
ejpam-4953	147	1	(	(	PUNCT
ejpam-4953	147	2	iii	iii	X
ejpam-4953	147	3	)	)	PUNCT
ejpam-4953	147	4	there	there	PRON
ejpam-4953	147	5	exists	exist	VERB
ejpam-4953	147	6	λ	λ	X
ejpam-4953	147	7	∈	∈	PROPN
ejpam-4953	148	1	[	[	X
ejpam-4953	148	2	0	0	NUM
ejpam-4953	148	3	,	,	PUNCT
ejpam-4953	148	4	1k	1k	NUM
ejpam-4953	148	5	)	)	PUNCT
ejpam-4953	149	1	such	such	ADJ
ejpam-4953	149	2	that	that	PRON
ejpam-4953	149	3	d(sϖ	d(sϖ	NOUN
ejpam-4953	149	4	,	,	PUNCT
ejpam-4953	149	5	sρ	sρ	PRON
ejpam-4953	149	6	)	)	PUNCT
ejpam-4953	149	7	≤	≤	NOUN
ejpam-4953	149	8	d(v	d(v	PROPN
ejpam-4953	149	9	ϖ	ϖ	X
ejpam-4953	149	10	,	,	PUNCT
ejpam-4953	149	11	v	v	ADP
ejpam-4953	149	12	ρ)λ	ρ)λ	ADJ
ejpam-4953	149	13	i.	i.	NOUN
ejpam-4953	149	14	alshammari	alshammari	PROPN
ejpam-4953	149	15	et	et	PROPN
ejpam-4953	149	16	al	al	PROPN
ejpam-4953	149	17	.	.	PUNCT
ejpam-4953	149	18	/	/	SYM
ejpam-4953	149	19	eur	eur	PROPN
ejpam-4953	149	20	.	.	PUNCT
ejpam-4953	150	1	j.	j.	PROPN
ejpam-4953	150	2	pure	pure	PROPN
ejpam-4953	150	3	appl	appl	PROPN
ejpam-4953	150	4	.	.	PROPN
ejpam-4953	150	5	math	math	PROPN
ejpam-4953	150	6	,	,	PUNCT
ejpam-4953	150	7	16	16	NUM
ejpam-4953	150	8	(	(	PUNCT
ejpam-4953	150	9	4	4	NUM
ejpam-4953	150	10	)	)	PUNCT
ejpam-4953	150	11	(	(	PUNCT
ejpam-4953	150	12	2023	2023	NUM
ejpam-4953	150	13	)	)	PUNCT
ejpam-4953	150	14	,	,	PUNCT
ejpam-4953	150	15	2405	2405	NUM
ejpam-4953	150	16	-	-	SYM
ejpam-4953	150	17	2418	2418	NUM
ejpam-4953	150	18	2412	2412	NUM
ejpam-4953	150	19	(	(	PUNCT
ejpam-4953	150	20	iv	iv	X
ejpam-4953	150	21	)	)	PUNCT
ejpam-4953	150	22	either	either	CCONJ
ejpam-4953	150	23	s	s	VERB
ejpam-4953	150	24	is	be	AUX
ejpam-4953	150	25	(	(	PUNCT
ejpam-4953	150	26	v	v	NOUN
ejpam-4953	150	27	,	,	PUNCT
ejpam-4953	150	28	r̈)-continuous	r̈)-continuous	PROPN
ejpam-4953	150	29	or	or	CCONJ
ejpam-4953	150	30	s	s	NOUN
ejpam-4953	150	31	and	and	CCONJ
ejpam-4953	150	32	v	v	NOUN
ejpam-4953	150	33	are	be	AUX
ejpam-4953	150	34	continuous	continuous	ADJ
ejpam-4953	150	35	.	.	PUNCT
ejpam-4953	151	1	or	or	CCONJ
ejpam-4953	151	2	(	(	PUNCT
ejpam-4953	151	3	iv	iv	X
ejpam-4953	151	4	’	'	PUNCT
ejpam-4953	151	5	)	)	PUNCT
ejpam-4953	151	6	s	s	PART
ejpam-4953	151	7	and	and	CCONJ
ejpam-4953	151	8	v	v	NOUN
ejpam-4953	151	9	are	be	AUX
ejpam-4953	151	10	r̈compatible	r̈compatible	PROPN
ejpam-4953	151	11	,	,	PUNCT
ejpam-4953	151	12	v	v	NOUN
ejpam-4953	151	13	is	be	AUX
ejpam-4953	151	14	r̈continuous	r̈continuous	PROPN
ejpam-4953	151	15	,	,	PUNCT
ejpam-4953	151	16	and	and	CCONJ
ejpam-4953	151	17	either	either	DET
ejpam-4953	151	18	s	s	VERB
ejpam-4953	151	19	is	be	AUX
ejpam-4953	151	20	r̈-continuous	r̈-continuous	ADJ
ejpam-4953	151	21	or	or	CCONJ
ejpam-4953	151	22	r̈	r̈	VERB
ejpam-4953	151	23	is	be	AUX
ejpam-4953	151	24	(	(	PUNCT
ejpam-4953	151	25	v	v	NOUN
ejpam-4953	151	26	,	,	PUNCT
ejpam-4953	151	27	bd)−	bd)−	ADJ
ejpam-4953	151	28	self	self	NOUN
ejpam-4953	151	29	−	−	NOUN
ejpam-4953	151	30	closed	closed	ADJ
ejpam-4953	151	31	,	,	PUNCT
ejpam-4953	151	32	then	then	ADV
ejpam-4953	151	33	s	s	X
ejpam-4953	151	34	and	and	CCONJ
ejpam-4953	151	35	v	v	AUX
ejpam-4953	151	36	have	have	VERB
ejpam-4953	151	37	a	a	DET
ejpam-4953	151	38	point	point	NOUN
ejpam-4953	151	39	of	of	ADP
ejpam-4953	151	40	coincidence	coincidence	NOUN
ejpam-4953	151	41	.	.	PUNCT
ejpam-4953	152	1	proof	proof	NOUN
ejpam-4953	152	2	.	.	PUNCT
ejpam-4953	153	1	let	let	VERB
ejpam-4953	153	2	ϖ0	ϖ0	NOUN
ejpam-4953	153	3	∈	∈	VERB
ejpam-4953	153	4	ḧ(s	ḧ(s	PROPN
ejpam-4953	153	5	,	,	PUNCT
ejpam-4953	153	6	v	v	NOUN
ejpam-4953	153	7	,	,	PUNCT
ejpam-4953	153	8	r̈	r̈	NOUN
ejpam-4953	153	9	)	)	PUNCT
ejpam-4953	153	10	be	be	VERB
ejpam-4953	153	11	an	an	DET
ejpam-4953	153	12	arbitrary	arbitrary	ADJ
ejpam-4953	153	13	element	element	NOUN
ejpam-4953	153	14	.	.	PUNCT
ejpam-4953	154	1	then	then	ADV
ejpam-4953	154	2	(	(	PUNCT
ejpam-4953	154	3	v	v	NOUN
ejpam-4953	154	4	ϖ0	ϖ0	NOUN
ejpam-4953	154	5	,	,	PUNCT
ejpam-4953	154	6	sϖ0	sϖ0	NOUN
ejpam-4953	154	7	)	)	PUNCT
ejpam-4953	154	8	∈	∈	PROPN
ejpam-4953	155	1	r̈.	r̈.	PROPN
ejpam-4953	155	2	if	if	SCONJ
ejpam-4953	155	3	v	v	X
ejpam-4953	155	4	(	(	PUNCT
ejpam-4953	155	5	ϖ0	ϖ0	NOUN
ejpam-4953	155	6	)	)	PUNCT
ejpam-4953	155	7	=	=	SYM
ejpam-4953	155	8	s(ϖ0	s(ϖ0	NOUN
ejpam-4953	155	9	)	)	PUNCT
ejpam-4953	155	10	,	,	PUNCT
ejpam-4953	155	11	then	then	ADV
ejpam-4953	155	12	ϖ0	ϖ0	NOUN
ejpam-4953	155	13	is	be	AUX
ejpam-4953	155	14	a	a	DET
ejpam-4953	155	15	coincidence	coincidence	NOUN
ejpam-4953	155	16	point	point	NOUN
ejpam-4953	155	17	of	of	ADP
ejpam-4953	155	18	s	s	PRON
ejpam-4953	155	19	and	and	CCONJ
ejpam-4953	155	20	v	v	NOUN
ejpam-4953	155	21	and	and	CCONJ
ejpam-4953	155	22	,	,	PUNCT
ejpam-4953	155	23	hence	hence	ADV
ejpam-4953	155	24	,	,	PUNCT
ejpam-4953	155	25	we	we	PRON
ejpam-4953	155	26	are	be	AUX
ejpam-4953	155	27	through	through	ADP
ejpam-4953	155	28	.	.	PUNCT
ejpam-4953	156	1	otherwise	otherwise	ADV
ejpam-4953	156	2	,	,	PUNCT
ejpam-4953	156	3	if	if	SCONJ
ejpam-4953	156	4	v	v	X
ejpam-4953	156	5	(	(	PUNCT
ejpam-4953	156	6	ϖ0	ϖ0	NOUN
ejpam-4953	156	7	)	)	PUNCT
ejpam-4953	156	8	̸=	̸=	PROPN
ejpam-4953	156	9	s(ϖ0	s(ϖ0	NOUN
ejpam-4953	156	10	)	)	PUNCT
ejpam-4953	156	11	,	,	PUNCT
ejpam-4953	156	12	then	then	ADV
ejpam-4953	156	13	,	,	PUNCT
ejpam-4953	156	14	in	in	ADP
ejpam-4953	156	15	view	view	NOUN
ejpam-4953	156	16	of	of	ADP
ejpam-4953	156	17	s(ḧ	s(ḧ	NOUN
ejpam-4953	156	18	)	)	PUNCT
ejpam-4953	156	19	⊆	⊆	NUM
ejpam-4953	156	20	v	v	NOUN
ejpam-4953	156	21	(	(	PUNCT
ejpam-4953	156	22	ḧ	ḧ	NOUN
ejpam-4953	156	23	)	)	PUNCT
ejpam-4953	156	24	,	,	PUNCT
ejpam-4953	156	25	we	we	PRON
ejpam-4953	156	26	can	can	AUX
ejpam-4953	156	27	choose	choose	VERB
ejpam-4953	156	28	ϖ1	ϖ1	PROPN
ejpam-4953	156	29	∈	∈	PROPN
ejpam-4953	156	30	ḧ	ḧ	NOUN
ejpam-4953	156	31	such	such	DET
ejpam-4953	156	32	that	that	DET
ejpam-4953	156	33	v	v	NOUN
ejpam-4953	156	34	(	(	PUNCT
ejpam-4953	156	35	ϖ1	ϖ1	PROPN
ejpam-4953	156	36	)	)	PUNCT
ejpam-4953	156	37	=	=	SYM
ejpam-4953	156	38	s(ϖ0	s(ϖ0	NOUN
ejpam-4953	156	39	)	)	PUNCT
ejpam-4953	156	40	.	.	PUNCT
ejpam-4953	157	1	again	again	ADV
ejpam-4953	157	2	from	from	ADP
ejpam-4953	157	3	s(ḧ	s(ḧ	NOUN
ejpam-4953	157	4	)	)	PUNCT
ejpam-4953	157	5	⊆	⊆	NUM
ejpam-4953	157	6	v	v	NOUN
ejpam-4953	157	7	(	(	PUNCT
ejpam-4953	157	8	ḧ	ḧ	NOUN
ejpam-4953	157	9	)	)	PUNCT
ejpam-4953	157	10	,	,	PUNCT
ejpam-4953	157	11	we	we	PRON
ejpam-4953	157	12	can	can	AUX
ejpam-4953	157	13	choose	choose	VERB
ejpam-4953	157	14	ϖ2	ϖ2	NOUN
ejpam-4953	157	15	∈	∈	PROPN
ejpam-4953	157	16	ḧ	ḧ	NOUN
ejpam-4953	157	17	such	such	DET
ejpam-4953	157	18	that	that	DET
ejpam-4953	157	19	v	v	NOUN
ejpam-4953	157	20	(	(	PUNCT
ejpam-4953	157	21	ϖ2	ϖ2	NOUN
ejpam-4953	157	22	)	)	PUNCT
ejpam-4953	157	23	=	=	PUNCT
ejpam-4953	157	24	s(ϖ1	s(ϖ1	NOUN
ejpam-4953	157	25	)	)	PUNCT
ejpam-4953	157	26	.	.	PUNCT
ejpam-4953	158	1	construct	construct	VERB
ejpam-4953	158	2	the	the	DET
ejpam-4953	158	3	sequence	sequence	NOUN
ejpam-4953	158	4	{	{	PUNCT
ejpam-4953	158	5	ϖn	ϖn	NOUN
ejpam-4953	158	6	}	}	PUNCT
ejpam-4953	158	7	⊂	⊂	X
ejpam-4953	158	8	ḧ	ḧ	VERB
ejpam-4953	158	9	such	such	ADJ
ejpam-4953	158	10	that	that	DET
ejpam-4953	158	11	v	v	NOUN
ejpam-4953	158	12	(	(	PUNCT
ejpam-4953	158	13	ϖn+1	ϖn+1	ADJ
ejpam-4953	158	14	)	)	PUNCT
ejpam-4953	158	15	=	=	SYM
ejpam-4953	158	16	s(ϖn	s(ϖn	PROPN
ejpam-4953	158	17	)	)	PUNCT
ejpam-4953	158	18	for	for	ADP
ejpam-4953	158	19	all	all	DET
ejpam-4953	158	20	n	n	PRON
ejpam-4953	158	21	∈	∈	PROPN
ejpam-4953	158	22	n	n	CCONJ
ejpam-4953	158	23	(	(	PUNCT
ejpam-4953	158	24	6	6	NUM
ejpam-4953	158	25	)	)	PUNCT
ejpam-4953	158	26	now	now	ADV
ejpam-4953	158	27	,	,	PUNCT
ejpam-4953	158	28	we	we	PRON
ejpam-4953	158	29	claim	claim	VERB
ejpam-4953	158	30	that	that	SCONJ
ejpam-4953	158	31	{	{	PUNCT
ejpam-4953	158	32	v	v	NOUN
ejpam-4953	158	33	ϖn	ϖn	NOUN
ejpam-4953	158	34	}	}	PUNCT
ejpam-4953	158	35	is	be	AUX
ejpam-4953	158	36	r̈preserving	r̈preserve	VERB
ejpam-4953	158	37	sequence	sequence	NOUN
ejpam-4953	158	38	,	,	PUNCT
ejpam-4953	158	39	i.e.	i.e.	X
ejpam-4953	158	40	,	,	PUNCT
ejpam-4953	158	41	(	(	PUNCT
ejpam-4953	158	42	v	v	NOUN
ejpam-4953	158	43	ϖn	ϖn	NOUN
ejpam-4953	158	44	,	,	PUNCT
ejpam-4953	158	45	v	v	X
ejpam-4953	158	46	ϖn+1	ϖn+1	ADJ
ejpam-4953	158	47	)	)	PUNCT
ejpam-4953	158	48	∈	∈	PROPN
ejpam-4953	158	49	r̈	r̈	VERB
ejpam-4953	158	50	for	for	ADP
ejpam-4953	158	51	all	all	PRON
ejpam-4953	158	52	n	n	PRON
ejpam-4953	158	53	∈	∈	PROPN
ejpam-4953	158	54	n	n	CCONJ
ejpam-4953	158	55	(	(	PUNCT
ejpam-4953	158	56	7	7	X
ejpam-4953	158	57	)	)	PUNCT
ejpam-4953	158	58	we	we	PRON
ejpam-4953	158	59	can	can	AUX
ejpam-4953	158	60	show	show	VERB
ejpam-4953	158	61	this	this	DET
ejpam-4953	158	62	fact	fact	NOUN
ejpam-4953	158	63	by	by	ADP
ejpam-4953	158	64	induction	induction	NOUN
ejpam-4953	158	65	.	.	PUNCT
ejpam-4953	159	1	by	by	ADP
ejpam-4953	159	2	equation	equation	NOUN
ejpam-4953	159	3	(	(	PUNCT
ejpam-4953	159	4	6	6	NUM
ejpam-4953	159	5	)	)	PUNCT
ejpam-4953	159	6	(	(	PUNCT
ejpam-4953	159	7	with	with	ADP
ejpam-4953	159	8	n=	n=	ADJ
ejpam-4953	159	9	0	0	NUM
ejpam-4953	159	10	)	)	PUNCT
ejpam-4953	159	11	and	and	CCONJ
ejpam-4953	159	12	fact	fact	NOUN
ejpam-4953	159	13	that	that	SCONJ
ejpam-4953	159	14	ϖ0	ϖ0	VERB
ejpam-4953	159	15	∈	∈	PROPN
ejpam-4953	159	16	ḧ(s	ḧ(s	PROPN
ejpam-4953	159	17	,	,	PUNCT
ejpam-4953	159	18	v	v	NOUN
ejpam-4953	159	19	,	,	PUNCT
ejpam-4953	159	20	r̈	r̈	NOUN
ejpam-4953	159	21	)	)	PUNCT
ejpam-4953	159	22	,	,	PUNCT
ejpam-4953	159	23	we	we	PRON
ejpam-4953	159	24	conclude	conclude	VERB
ejpam-4953	159	25	that	that	SCONJ
ejpam-4953	159	26	(	(	PUNCT
ejpam-4953	159	27	v	v	NUM
ejpam-4953	159	28	ϖ0	ϖ0	NOUN
ejpam-4953	159	29	,	,	PUNCT
ejpam-4953	159	30	v	v	NOUN
ejpam-4953	159	31	ϖ1	ϖ1	NOUN
ejpam-4953	159	32	)	)	PUNCT
ejpam-4953	159	33	∈	∈	PROPN
ejpam-4953	159	34	r̈.	r̈.	PROPN
ejpam-4953	159	35	which	which	PRON
ejpam-4953	159	36	means	mean	VERB
ejpam-4953	159	37	that	that	SCONJ
ejpam-4953	159	38	(	(	PUNCT
ejpam-4953	159	39	7	7	X
ejpam-4953	159	40	)	)	PUNCT
ejpam-4953	159	41	holds	hold	VERB
ejpam-4953	159	42	for	for	ADP
ejpam-4953	159	43	n=0	n=0	NUM
ejpam-4953	159	44	.	.	PUNCT
ejpam-4953	160	1	suppose	suppose	VERB
ejpam-4953	160	2	(	(	PUNCT
ejpam-4953	160	3	7	7	X
ejpam-4953	160	4	)	)	PUNCT
ejpam-4953	160	5	is	be	AUX
ejpam-4953	160	6	true	true	ADJ
ejpam-4953	160	7	for	for	ADP
ejpam-4953	160	8	n	n	NOUN
ejpam-4953	160	9	=	=	SYM
ejpam-4953	160	10	r	r	NOUN
ejpam-4953	160	11	≥	≥	NOUN
ejpam-4953	160	12	0	0	NUM
ejpam-4953	161	1	i.e.	i.e.	X
ejpam-4953	161	2	,	,	PUNCT
ejpam-4953	161	3	(	(	PUNCT
ejpam-4953	161	4	v	v	X
ejpam-4953	161	5	ϖr	ϖr	PROPN
ejpam-4953	161	6	,	,	PUNCT
ejpam-4953	161	7	v	v	NOUN
ejpam-4953	161	8	ϖr+1	ϖr+1	NOUN
ejpam-4953	161	9	)	)	PUNCT
ejpam-4953	161	10	∈	∈	PROPN
ejpam-4953	161	11	r̈.	r̈.	PROPN
ejpam-4953	161	12	as	as	SCONJ
ejpam-4953	161	13	r̈	r̈	NOUN
ejpam-4953	161	14	is	be	AUX
ejpam-4953	161	15	(	(	PUNCT
ejpam-4953	161	16	s	s	X
ejpam-4953	161	17	,	,	PUNCT
ejpam-4953	161	18	v)closed	v)close	VERB
ejpam-4953	161	19	,	,	PUNCT
ejpam-4953	161	20	we	we	PRON
ejpam-4953	161	21	get	get	VERB
ejpam-4953	161	22	(	(	PUNCT
ejpam-4953	161	23	sϖr	sϖr	NOUN
ejpam-4953	161	24	,	,	PUNCT
ejpam-4953	161	25	sϖr+1	sϖr+1	X
ejpam-4953	161	26	)	)	PUNCT
ejpam-4953	161	27	∈	∈	PROPN
ejpam-4953	161	28	r̈.	r̈.	PROPN
ejpam-4953	161	29	by	by	ADP
ejpam-4953	161	30	using	use	VERB
ejpam-4953	161	31	,	,	PUNCT
ejpam-4953	161	32	this	this	DET
ejpam-4953	161	33	yield	yield	NOUN
ejpam-4953	161	34	that	that	SCONJ
ejpam-4953	161	35	(	(	PUNCT
ejpam-4953	161	36	v	v	NOUN
ejpam-4953	161	37	ϖr+1	ϖr+1	PROPN
ejpam-4953	161	38	,	,	PUNCT
ejpam-4953	161	39	v	v	ADP
ejpam-4953	161	40	ϖr+2	ϖr+2	NOUN
ejpam-4953	161	41	)	)	PUNCT
ejpam-4953	161	42	∈	∈	PROPN
ejpam-4953	161	43	r̈	r̈	NOUN
ejpam-4953	161	44	,	,	PUNCT
ejpam-4953	161	45	i.e.	i.e.	X
ejpam-4953	161	46	,	,	PUNCT
ejpam-4953	161	47	inclusion	inclusion	NOUN
ejpam-4953	161	48	(	(	PUNCT
ejpam-4953	161	49	7	7	NUM
ejpam-4953	161	50	)	)	PUNCT
ejpam-4953	161	51	holds	hold	VERB
ejpam-4953	161	52	for	for	ADP
ejpam-4953	161	53	n	n	PRON
ejpam-4953	161	54	=	=	PROPN
ejpam-4953	161	55	r+1	r+1	PROPN
ejpam-4953	161	56	.	.	PUNCT
ejpam-4953	162	1	hence	hence	ADV
ejpam-4953	162	2	by	by	ADP
ejpam-4953	162	3	induction	induction	NOUN
ejpam-4953	162	4	,	,	PUNCT
ejpam-4953	162	5	inclusion	inclusion	NOUN
ejpam-4953	162	6	(	(	PUNCT
ejpam-4953	162	7	7	7	NUM
ejpam-4953	162	8	)	)	PUNCT
ejpam-4953	162	9	is	be	AUX
ejpam-4953	162	10	valid	valid	ADJ
ejpam-4953	162	11	for	for	ADP
ejpam-4953	162	12	all	all	PRON
ejpam-4953	162	13	n	n	DET
ejpam-4953	162	14	∈	∈	PROPN
ejpam-4953	162	15	n.	n.	NOUN
ejpam-4953	162	16	in	in	ADP
ejpam-4953	162	17	view	view	NOUN
ejpam-4953	162	18	of	of	ADP
ejpam-4953	162	19	(	(	PUNCT
ejpam-4953	162	20	6	6	NUM
ejpam-4953	162	21	)	)	PUNCT
ejpam-4953	162	22	and	and	CCONJ
ejpam-4953	162	23	(	(	PUNCT
ejpam-4953	162	24	7	7	NUM
ejpam-4953	162	25	)	)	PUNCT
ejpam-4953	162	26	,	,	PUNCT
ejpam-4953	162	27	the	the	DET
ejpam-4953	162	28	sequence	sequence	NOUN
ejpam-4953	162	29	{	{	PUNCT
ejpam-4953	162	30	sϖn	sϖn	ADJ
ejpam-4953	162	31	}	}	PUNCT
ejpam-4953	162	32	is	be	AUX
ejpam-4953	162	33	also	also	ADV
ejpam-4953	162	34	an	an	DET
ejpam-4953	162	35	r̈-preserving	r̈-preserve	VERB
ejpam-4953	162	36	,	,	PUNCT
ejpam-4953	162	37	i.e.	i.e.	X
ejpam-4953	162	38	,	,	PUNCT
ejpam-4953	162	39	(	(	PUNCT
ejpam-4953	162	40	sϖn	sϖn	ADJ
ejpam-4953	162	41	,	,	PUNCT
ejpam-4953	162	42	sϖn+1	sϖn+1	ADJ
ejpam-4953	162	43	)	)	PUNCT
ejpam-4953	162	44	∈	∈	PROPN
ejpam-4953	162	45	r̈	r̈	VERB
ejpam-4953	162	46	for	for	ADP
ejpam-4953	162	47	all	all	PRON
ejpam-4953	162	48	n	n	PRON
ejpam-4953	162	49	∈	∈	PRON
ejpam-4953	162	50	n	n	X
ejpam-4953	162	51	by	by	ADP
ejpam-4953	162	52	using	use	VERB
ejpam-4953	162	53	(	(	PUNCT
ejpam-4953	162	54	6	6	NUM
ejpam-4953	162	55	)	)	PUNCT
ejpam-4953	162	56	,	,	PUNCT
ejpam-4953	162	57	(	(	PUNCT
ejpam-4953	162	58	7)and	7)and	NUM
ejpam-4953	162	59	assumption	assumption	NOUN
ejpam-4953	162	60	(	(	PUNCT
ejpam-4953	162	61	iii	iii	NOUN
ejpam-4953	162	62	)	)	PUNCT
ejpam-4953	162	63	,	,	PUNCT
ejpam-4953	162	64	we	we	PRON
ejpam-4953	162	65	find	find	VERB
ejpam-4953	162	66	p(v	p(v	NOUN
ejpam-4953	162	67	ϖn	ϖn	NOUN
ejpam-4953	162	68	,	,	PUNCT
ejpam-4953	162	69	v	v	X
ejpam-4953	162	70	ϖn+1	ϖn+1	X
ejpam-4953	162	71	)	)	PUNCT
ejpam-4953	162	72	=	=	SYM
ejpam-4953	162	73	p(sϖn−1	p(sϖn−1	PROPN
ejpam-4953	162	74	,	,	PUNCT
ejpam-4953	162	75	sϖn	sϖn	ADJ
ejpam-4953	162	76	)	)	PUNCT
ejpam-4953	162	77	≤	≤	NOUN
ejpam-4953	162	78	p(v	p(v	PROPN
ejpam-4953	162	79	ϖn−1	ϖn−1	PROPN
ejpam-4953	162	80	,	,	PUNCT
ejpam-4953	162	81	v	v	NOUN
ejpam-4953	162	82	ϖn	ϖn	NOUN
ejpam-4953	162	83	)	)	PUNCT
ejpam-4953	162	84	λ	λ	NOUN
ejpam-4953	162	85	for	for	ADP
ejpam-4953	162	86	all	all	DET
ejpam-4953	162	87	n	n	PRON
ejpam-4953	162	88	∈	∈	PROPN
ejpam-4953	162	89	n	n	CCONJ
ejpam-4953	162	90	(	(	PUNCT
ejpam-4953	162	91	8)	8)	NUM
ejpam-4953	162	92	which	which	PRON
ejpam-4953	162	93	by	by	ADP
ejpam-4953	162	94	induction	induction	NOUN
ejpam-4953	162	95	yield	yield	NOUN
ejpam-4953	162	96	that	that	PRON
ejpam-4953	162	97	p(v	p(v	VERB
ejpam-4953	162	98	ϖn	ϖn	ADP
ejpam-4953	162	99	,	,	PUNCT
ejpam-4953	162	100	v	v	X
ejpam-4953	162	101	ϖn+1	ϖn+1	X
ejpam-4953	162	102	)	)	PUNCT
ejpam-4953	162	103	=	=	SYM
ejpam-4953	162	104	p(sϖn−1	p(sϖn−1	PROPN
ejpam-4953	162	105	,	,	PUNCT
ejpam-4953	162	106	sϖn	sϖn	ADJ
ejpam-4953	162	107	)	)	PUNCT
ejpam-4953	162	108	≤	≤	NOUN
ejpam-4953	162	109	p(v	p(v	PROPN
ejpam-4953	162	110	ϖn−1	ϖn−1	PROPN
ejpam-4953	162	111	,	,	PUNCT
ejpam-4953	162	112	v	v	NOUN
ejpam-4953	162	113	ϖn	ϖn	NOUN
ejpam-4953	162	114	)	)	PUNCT
ejpam-4953	162	115	λn	λn	NOUN
ejpam-4953	162	116	for	for	ADP
ejpam-4953	162	117	all	all	DET
ejpam-4953	162	118	n	n	PRON
ejpam-4953	162	119	∈	∈	PROPN
ejpam-4953	162	120	n	n	CCONJ
ejpam-4953	162	121	(	(	PUNCT
ejpam-4953	162	122	9	9	NUM
ejpam-4953	162	123	)	)	PUNCT
ejpam-4953	162	124	by	by	ADP
ejpam-4953	162	125	using	use	VERB
ejpam-4953	162	126	(	(	PUNCT
ejpam-4953	162	127	9	9	NUM
ejpam-4953	162	128	)	)	PUNCT
ejpam-4953	162	129	and	and	CCONJ
ejpam-4953	162	130	multiplicative	multiplicative	ADJ
ejpam-4953	162	131	triangular	triangular	NOUN
ejpam-4953	162	132	inequality	inequality	NOUN
ejpam-4953	162	133	,	,	PUNCT
ejpam-4953	162	134	for	for	ADP
ejpam-4953	162	135	all	all	DET
ejpam-4953	162	136	n	n	DET
ejpam-4953	162	137	∈	∈	PROPN
ejpam-4953	162	138	n	n	CCONJ
ejpam-4953	162	139	,	,	PUNCT
ejpam-4953	162	140	r	r	NOUN
ejpam-4953	162	141	∈	∈	PROPN
ejpam-4953	162	142	n	n	CCONJ
ejpam-4953	162	143	,	,	PUNCT
ejpam-4953	162	144	we	we	PRON
ejpam-4953	162	145	have	have	VERB
ejpam-4953	162	146	i.	i.	PROPN
ejpam-4953	162	147	alshammari	alshammari	PROPN
ejpam-4953	162	148	et	et	PROPN
ejpam-4953	162	149	al	al	PROPN
ejpam-4953	162	150	.	.	PUNCT
ejpam-4953	162	151	/	/	SYM
ejpam-4953	162	152	eur	eur	PROPN
ejpam-4953	162	153	.	.	PUNCT
ejpam-4953	163	1	j.	j.	PROPN
ejpam-4953	163	2	pure	pure	PROPN
ejpam-4953	163	3	appl	appl	PROPN
ejpam-4953	163	4	.	.	PROPN
ejpam-4953	163	5	math	math	PROPN
ejpam-4953	163	6	,	,	PUNCT
ejpam-4953	163	7	16	16	NUM
ejpam-4953	163	8	(	(	PUNCT
ejpam-4953	163	9	4	4	NUM
ejpam-4953	163	10	)	)	PUNCT
ejpam-4953	163	11	(	(	PUNCT
ejpam-4953	163	12	2023	2023	NUM
ejpam-4953	163	13	)	)	PUNCT
ejpam-4953	163	14	,	,	PUNCT
ejpam-4953	163	15	2405	2405	NUM
ejpam-4953	163	16	-	-	SYM
ejpam-4953	163	17	2418	2418	NUM
ejpam-4953	163	18	2413	2413	NUM
ejpam-4953	163	19	p(v	p(v	NOUN
ejpam-4953	163	20	ϖn	ϖn	ADP
ejpam-4953	163	21	,	,	PUNCT
ejpam-4953	163	22	v	v	X
ejpam-4953	163	23	ϖn+r	ϖn+r	NOUN
ejpam-4953	163	24	)	)	PUNCT
ejpam-4953	163	25	≤	≤	PUNCT
ejpam-4953	163	26	p(v	p(v	PROPN
ejpam-4953	163	27	ϖn	ϖn	ADP
ejpam-4953	163	28	,	,	PUNCT
ejpam-4953	163	29	v	v	X
ejpam-4953	163	30	ϖn+1	ϖn+1	ADJ
ejpam-4953	163	31	)	)	PUNCT
ejpam-4953	163	32	kn	kn	PROPN
ejpam-4953	163	33	·	·	PUNCT
ejpam-4953	163	34	p(v	p(v	PROPN
ejpam-4953	163	35	ϖn+1	ϖn+1	PROPN
ejpam-4953	163	36	,	,	PUNCT
ejpam-4953	163	37	v	v	ADP
ejpam-4953	163	38	ϖn+2	ϖn+2	PRON
ejpam-4953	163	39	)	)	PUNCT
ejpam-4953	163	40	kn+1	kn+1	PROPN
ejpam-4953	163	41	·	·	PUNCT
ejpam-4953	163	42	·	·	PUNCT
ejpam-4953	163	43	·	·	PUNCT
ejpam-4953	164	1	p(v	p(v	NOUN
ejpam-4953	164	2	ϖn+r−1	ϖn+r−1	PROPN
ejpam-4953	164	3	,	,	PUNCT
ejpam-4953	164	4	v	v	X
ejpam-4953	164	5	ϖn+r	ϖn+r	NOUN
ejpam-4953	164	6	)	)	PUNCT
ejpam-4953	164	7	kn+r−1	kn+r−1	PROPN
ejpam-4953	164	8	≤	≤	PUNCT
ejpam-4953	164	9	p(v	p(v	NOUN
ejpam-4953	164	10	ϖn	ϖn	ADP
ejpam-4953	164	11	,	,	PUNCT
ejpam-4953	164	12	v	v	X
ejpam-4953	164	13	ϖn+1	ϖn+1	ADJ
ejpam-4953	164	14	)	)	PUNCT
ejpam-4953	164	15	λnkn	λnkn	PROPN
ejpam-4953	164	16	·	·	PUNCT
ejpam-4953	164	17	p(v	p(v	PROPN
ejpam-4953	164	18	ϖn+1	ϖn+1	PROPN
ejpam-4953	164	19	,	,	PUNCT
ejpam-4953	164	20	v	v	ADP
ejpam-4953	164	21	ϖn+2	ϖn+2	NUM
ejpam-4953	164	22	)	)	PUNCT
ejpam-4953	164	23	λn+1kn+1	λn+1kn+1	VERB
ejpam-4953	164	24	·	·	PUNCT
ejpam-4953	164	25	·	·	PUNCT
ejpam-4953	164	26	·	·	PUNCT
ejpam-4953	164	27	·	·	PUNCT
ejpam-4953	165	1	p(v	p(v	NOUN
ejpam-4953	165	2	ϖn+r−1	ϖn+r−1	PROPN
ejpam-4953	165	3	,	,	PUNCT
ejpam-4953	165	4	v	v	X
ejpam-4953	165	5	ϖn+r	ϖn+r	PROPN
ejpam-4953	165	6	)	)	PUNCT
ejpam-4953	166	1	λn+r−1kn+r−1	λn+r−1kn+r−1	PROPN
ejpam-4953	166	2	≤	≤	ADJ
ejpam-4953	166	3	p(v	p(v	NOUN
ejpam-4953	166	4	ϖ0	ϖ0	NOUN
ejpam-4953	166	5	,	,	PUNCT
ejpam-4953	166	6	v	v	NOUN
ejpam-4953	166	7	ϖ1	ϖ1	NOUN
ejpam-4953	166	8	)	)	PUNCT
ejpam-4953	166	9	(	(	PUNCT
ejpam-4953	166	10	λk)n+(λk)n+1+···+(λk)n+r−1	λk)n+(λk)n+1+···+(λk)n+r−1	NOUN
ejpam-4953	166	11	≤	≤	ADJ
ejpam-4953	166	12	p(v	p(v	NOUN
ejpam-4953	166	13	ϖ0	ϖ0	NOUN
ejpam-4953	166	14	,	,	PUNCT
ejpam-4953	166	15	v	v	NOUN
ejpam-4953	166	16	ϖ1	ϖ1	NOUN
ejpam-4953	166	17	)	)	PUNCT
ejpam-4953	166	18	(	(	PUNCT
ejpam-4953	166	19	λk)n	λk)n	PROPN
ejpam-4953	166	20	1−(λk	1−(λk	NUM
ejpam-4953	166	21	)	)	PUNCT
ejpam-4953	166	22	.	.	PUNCT
ejpam-4953	167	1	this	this	PRON
ejpam-4953	167	2	implies	imply	VERB
ejpam-4953	167	3	that	that	SCONJ
ejpam-4953	167	4	p(v	p(v	NOUN
ejpam-4953	167	5	ϖn	ϖn	NOUN
ejpam-4953	167	6	,	,	PUNCT
ejpam-4953	167	7	v	v	ADP
ejpam-4953	167	8	ϖn+r	ϖn+r	NOUN
ejpam-4953	167	9	)	)	PUNCT
ejpam-4953	167	10	→b	→b	NUM
ejpam-4953	167	11	1	1	NUM
ejpam-4953	167	12	,	,	PUNCT
ejpam-4953	167	13	(	(	PUNCT
ejpam-4953	167	14	as	as	ADP
ejpam-4953	167	15	n	n	NOUN
ejpam-4953	167	16	→	→	SYM
ejpam-4953	167	17	+	+	NOUN
ejpam-4953	167	18	∞	∞	NOUN
ejpam-4953	167	19	)	)	PUNCT
ejpam-4953	167	20	hence	hence	ADV
ejpam-4953	167	21	,	,	PUNCT
ejpam-4953	167	22	the	the	DET
ejpam-4953	167	23	sequence	sequence	NOUN
ejpam-4953	167	24	v	v	NOUN
ejpam-4953	167	25	ϖn	ϖn	NOUN
ejpam-4953	167	26	is	be	AUX
ejpam-4953	167	27	multiplicative	multiplicative	ADJ
ejpam-4953	167	28	cauchy	cauchy	ADJ
ejpam-4953	167	29	sequence	sequence	NOUN
ejpam-4953	167	30	in	in	ADP
ejpam-4953	167	31	ḧ.	ḧ.	PROPN
ejpam-4953	167	32	by	by	ADP
ejpam-4953	167	33	using	use	VERB
ejpam-4953	167	34	(	(	PUNCT
ejpam-4953	167	35	3	3	NUM
ejpam-4953	167	36	)	)	PUNCT
ejpam-4953	167	37	,	,	PUNCT
ejpam-4953	167	38	we	we	PRON
ejpam-4953	167	39	have	have	VERB
ejpam-4953	167	40	v	v	NUM
ejpam-4953	167	41	ϖn	ϖn	ADP
ejpam-4953	167	42	⊆	⊆	NUM
ejpam-4953	167	43	s(ḧ	s(ḧ	NOUN
ejpam-4953	167	44	)	)	PUNCT
ejpam-4953	167	45	and	and	CCONJ
ejpam-4953	167	46	hence	hence	ADV
ejpam-4953	167	47	v	v	NOUN
ejpam-4953	167	48	ϖn	ϖn	NOUN
ejpam-4953	167	49	is	be	AUX
ejpam-4953	167	50	an	an	DET
ejpam-4953	167	51	r̈-preserving	r̈-preserving	ADJ
ejpam-4953	167	52	b	b	NOUN
ejpam-4953	167	53	-	-	PUNCT
ejpam-4953	167	54	multiplicative	multiplicative	ADJ
ejpam-4953	167	55	cauchy	cauchy	NOUN
ejpam-4953	167	56	sequence	sequence	NOUN
ejpam-4953	167	57	in	in	ADP
ejpam-4953	167	58	ḧ.	ḧ.	PROPN
ejpam-4953	167	59	as	as	ADP
ejpam-4953	167	60	(	(	PUNCT
ejpam-4953	167	61	ḧ	ḧ	NOUN
ejpam-4953	167	62	,	,	PUNCT
ejpam-4953	167	63	p	p	X
ejpam-4953	167	64	,	,	PUNCT
ejpam-4953	167	65	k	k	X
ejpam-4953	167	66	≥	≥	NUM
ejpam-4953	167	67	1	1	NUM
ejpam-4953	167	68	)	)	PUNCT
ejpam-4953	167	69	is	be	AUX
ejpam-4953	167	70	b	b	NOUN
ejpam-4953	167	71	-	-	PUNCT
ejpam-4953	167	72	complete	complete	ADJ
ejpam-4953	167	73	,	,	PUNCT
ejpam-4953	167	74	there	there	PRON
ejpam-4953	167	75	exists	exist	VERB
ejpam-4953	167	76	u	u	PROPN
ejpam-4953	167	77	∈	∈	PROPN
ejpam-4953	167	78	v	v	NOUN
ejpam-4953	167	79	(	(	PUNCT
ejpam-4953	167	80	ḧ	ḧ	NOUN
ejpam-4953	167	81	)	)	PUNCT
ejpam-4953	167	82	such	such	ADJ
ejpam-4953	167	83	that	that	SCONJ
ejpam-4953	167	84	lim	lim	PROPN
ejpam-4953	167	85	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	167	86	v	v	PROPN
ejpam-4953	167	87	(	(	PUNCT
ejpam-4953	167	88	ϖn	ϖn	NOUN
ejpam-4953	167	89	)	)	PUNCT
ejpam-4953	167	90	=	=	SYM
ejpam-4953	167	91	v	v	X
ejpam-4953	167	92	(	(	PUNCT
ejpam-4953	167	93	u	u	NOUN
ejpam-4953	167	94	)	)	PUNCT
ejpam-4953	167	95	(	(	PUNCT
ejpam-4953	167	96	10	10	NUM
ejpam-4953	167	97	)	)	PUNCT
ejpam-4953	167	98	by	by	ADP
ejpam-4953	167	99	using	use	VERB
ejpam-4953	167	100	(	(	PUNCT
ejpam-4953	167	101	6	6	NUM
ejpam-4953	167	102	)	)	PUNCT
ejpam-4953	167	103	and	and	CCONJ
ejpam-4953	167	104	(	(	PUNCT
ejpam-4953	167	105	10	10	NUM
ejpam-4953	167	106	)	)	PUNCT
ejpam-4953	167	107	,	,	PUNCT
ejpam-4953	167	108	we	we	PRON
ejpam-4953	167	109	get	get	VERB
ejpam-4953	167	110	lim	lim	PROPN
ejpam-4953	167	111	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	167	112	s(ϖn	s(ϖn	PROPN
ejpam-4953	167	113	)	)	PUNCT
ejpam-4953	168	1	=	=	SYM
ejpam-4953	168	2	v	v	X
ejpam-4953	168	3	(	(	PUNCT
ejpam-4953	168	4	u	u	NOUN
ejpam-4953	168	5	)	)	PUNCT
ejpam-4953	168	6	(	(	PUNCT
ejpam-4953	168	7	11	11	NUM
ejpam-4953	168	8	)	)	PUNCT
ejpam-4953	168	9	now	now	ADV
ejpam-4953	168	10	we	we	PRON
ejpam-4953	168	11	show	show	VERB
ejpam-4953	168	12	that	that	SCONJ
ejpam-4953	168	13	u	u	PRON
ejpam-4953	168	14	is	be	AUX
ejpam-4953	168	15	a	a	DET
ejpam-4953	168	16	coincidence	coincidence	NOUN
ejpam-4953	168	17	point	point	NOUN
ejpam-4953	168	18	of	of	ADP
ejpam-4953	168	19	s	s	PRON
ejpam-4953	168	20	and	and	CCONJ
ejpam-4953	168	21	v.	v.	ADP
ejpam-4953	168	22	now	now	ADV
ejpam-4953	168	23	,	,	PUNCT
ejpam-4953	168	24	in	in	ADP
ejpam-4953	168	25	lieu	lieu	NOUN
ejpam-4953	168	26	of	of	ADP
ejpam-4953	168	27	(	(	PUNCT
ejpam-4953	168	28	iv	iv	X
ejpam-4953	168	29	)	)	PUNCT
ejpam-4953	168	30	consider	consider	VERB
ejpam-4953	168	31	that	that	SCONJ
ejpam-4953	168	32	p	p	NOUN
ejpam-4953	168	33	is	be	AUX
ejpam-4953	168	34	(	(	PUNCT
ejpam-4953	168	35	v	v	NOUN
ejpam-4953	168	36	,	,	PUNCT
ejpam-4953	168	37	r̈)-continuous	r̈)-continuous	NUM
ejpam-4953	168	38	,	,	PUNCT
ejpam-4953	168	39	thus	thus	ADV
ejpam-4953	168	40	utilizing	utilize	VERB
ejpam-4953	168	41	(	(	PUNCT
ejpam-4953	168	42	7	7	NUM
ejpam-4953	168	43	)	)	PUNCT
ejpam-4953	168	44	and	and	CCONJ
ejpam-4953	168	45	(	(	PUNCT
ejpam-4953	168	46	10	10	NUM
ejpam-4953	168	47	)	)	PUNCT
ejpam-4953	168	48	we	we	PRON
ejpam-4953	168	49	obtain	obtain	VERB
ejpam-4953	168	50	lim	lim	PROPN
ejpam-4953	168	51	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	168	52	s(ϖn	s(ϖn	PROPN
ejpam-4953	168	53	)	)	PUNCT
ejpam-4953	168	54	=	=	SYM
ejpam-4953	168	55	s(u	s(u	PROPN
ejpam-4953	168	56	)	)	PUNCT
ejpam-4953	168	57	(	(	PUNCT
ejpam-4953	168	58	12	12	NUM
ejpam-4953	168	59	)	)	PUNCT
ejpam-4953	168	60	in	in	ADP
ejpam-4953	168	61	view	view	NOUN
ejpam-4953	168	62	of	of	ADP
ejpam-4953	168	63	(	(	PUNCT
ejpam-4953	168	64	11	11	NUM
ejpam-4953	168	65	)	)	PUNCT
ejpam-4953	168	66	and	and	CCONJ
ejpam-4953	168	67	(	(	PUNCT
ejpam-4953	168	68	12	12	NUM
ejpam-4953	168	69	)	)	PUNCT
ejpam-4953	168	70	,	,	PUNCT
ejpam-4953	168	71	we	we	PRON
ejpam-4953	168	72	obtain	obtain	VERB
ejpam-4953	168	73	v(u)=	v(u)=	ADJ
ejpam-4953	168	74	s(u	s(u	NOUN
ejpam-4953	168	75	)	)	PUNCT
ejpam-4953	168	76	.	.	PUNCT
ejpam-4953	169	1	hence	hence	ADV
ejpam-4953	169	2	,	,	PUNCT
ejpam-4953	169	3	we	we	PRON
ejpam-4953	169	4	are	be	AUX
ejpam-4953	169	5	completed	complete	VERB
ejpam-4953	169	6	.	.	PUNCT
ejpam-4953	170	1	second	second	ADV
ejpam-4953	170	2	,	,	PUNCT
ejpam-4953	170	3	we	we	PRON
ejpam-4953	170	4	assume	assume	VERB
ejpam-4953	170	5	that	that	SCONJ
ejpam-4953	170	6	s	s	VERB
ejpam-4953	170	7	and	and	CCONJ
ejpam-4953	170	8	v	v	NOUN
ejpam-4953	170	9	are	be	AUX
ejpam-4953	170	10	continuous	continuous	ADJ
ejpam-4953	170	11	and	and	CCONJ
ejpam-4953	170	12	owing	owe	VERB
ejpam-4953	170	13	to	to	ADP
ejpam-4953	170	14	the	the	DET
ejpam-4953	170	15	lemma	lemma	PROPN
ejpam-4953	170	16	1.1	1.1	NUM
ejpam-4953	170	17	,	,	PUNCT
ejpam-4953	170	18	there	there	PRON
ejpam-4953	170	19	exists	exist	VERB
ejpam-4953	170	20	a	a	DET
ejpam-4953	170	21	subset	subset	NOUN
ejpam-4953	170	22	g	g	ADP
ejpam-4953	170	23	⊆	⊆	NUM
ejpam-4953	170	24	ḧ	ḧ	NOUN
ejpam-4953	170	25	such	such	DET
ejpam-4953	170	26	that	that	PRON
ejpam-4953	170	27	v	v	NOUN
ejpam-4953	170	28	(	(	PUNCT
ejpam-4953	170	29	g	g	NOUN
ejpam-4953	170	30	)	)	PUNCT
ejpam-4953	170	31	=	=	NOUN
ejpam-4953	170	32	v	v	NOUN
ejpam-4953	170	33	(	(	PUNCT
ejpam-4953	170	34	ḧ	ḧ	NOUN
ejpam-4953	170	35	)	)	PUNCT
ejpam-4953	170	36	and	and	CCONJ
ejpam-4953	170	37	v	v	X
ejpam-4953	170	38	:	:	PUNCT
ejpam-4953	170	39	g	g	NOUN
ejpam-4953	170	40	→	→	SYM
ejpam-4953	170	41	ḧ	ḧ	NOUN
ejpam-4953	170	42	is	be	AUX
ejpam-4953	170	43	one	one	NUM
ejpam-4953	170	44	to	to	ADP
ejpam-4953	170	45	one	one	NUM
ejpam-4953	170	46	.	.	PUNCT
ejpam-4953	171	1	now	now	ADV
ejpam-4953	171	2	we	we	PRON
ejpam-4953	171	3	define	define	VERB
ejpam-4953	171	4	f̈	f̈	NUM
ejpam-4953	171	5	:	:	PUNCT
ejpam-4953	171	6	v	v	NOUN
ejpam-4953	171	7	(	(	PUNCT
ejpam-4953	171	8	g	g	NOUN
ejpam-4953	171	9	)	)	PUNCT
ejpam-4953	171	10	→	→	SYM
ejpam-4953	171	11	v	v	X
ejpam-4953	171	12	(	(	PUNCT
ejpam-4953	171	13	ḧ	ḧ	NOUN
ejpam-4953	171	14	)	)	PUNCT
ejpam-4953	171	15	by	by	ADP
ejpam-4953	171	16	f̈(v	f̈(v	PROPN
ejpam-4953	171	17	a	a	X
ejpam-4953	171	18	)	)	PUNCT
ejpam-4953	171	19	=	=	SYM
ejpam-4953	171	20	s(a	s(a	PROPN
ejpam-4953	171	21	)	)	PUNCT
ejpam-4953	171	22	for	for	ADP
ejpam-4953	171	23	all	all	PRON
ejpam-4953	171	24	v	v	NOUN
ejpam-4953	171	25	(	(	PUNCT
ejpam-4953	171	26	a	a	PRON
ejpam-4953	171	27	)	)	PUNCT
ejpam-4953	171	28	∈	∈	NOUN
ejpam-4953	171	29	v	v	NOUN
ejpam-4953	171	30	(	(	PUNCT
ejpam-4953	171	31	g	g	NOUN
ejpam-4953	171	32	)	)	PUNCT
ejpam-4953	171	33	where	where	SCONJ
ejpam-4953	171	34	a	a	DET
ejpam-4953	171	35	∈	∈	PROPN
ejpam-4953	171	36	g	g	NOUN
ejpam-4953	171	37	as	as	ADP
ejpam-4953	171	38	v	v	NOUN
ejpam-4953	171	39	:	:	PUNCT
ejpam-4953	171	40	g	g	NOUN
ejpam-4953	171	41	→	→	SYM
ejpam-4953	171	42	ḧ	ḧ	NOUN
ejpam-4953	171	43	is	be	AUX
ejpam-4953	171	44	injective	injective	ADJ
ejpam-4953	171	45	and	and	CCONJ
ejpam-4953	171	46	s(ḧ	s(ḧ	NOUN
ejpam-4953	171	47	)	)	PUNCT
ejpam-4953	171	48	⊆	⊆	NUM
ejpam-4953	171	49	v	v	NOUN
ejpam-4953	171	50	(	(	PUNCT
ejpam-4953	171	51	ḧ	ḧ	NOUN
ejpam-4953	171	52	)	)	PUNCT
ejpam-4953	171	53	,	,	PUNCT
ejpam-4953	171	54	we	we	PRON
ejpam-4953	171	55	get	get	VERB
ejpam-4953	171	56	to	to	ADP
ejpam-4953	171	57	the	the	DET
ejpam-4953	171	58	conclusion	conclusion	NOUN
ejpam-4953	171	59	that	that	SCONJ
ejpam-4953	171	60	f̈	f̈	PROPN
ejpam-4953	171	61	is	be	AUX
ejpam-4953	171	62	well	well	ADV
ejpam-4953	171	63	defined	define	VERB
ejpam-4953	171	64	.	.	PUNCT
ejpam-4953	172	1	additionally	additionally	ADV
ejpam-4953	172	2	,	,	PUNCT
ejpam-4953	172	3	f̈	f̈	PROPN
ejpam-4953	172	4	is	be	AUX
ejpam-4953	172	5	continuous	continuous	ADJ
ejpam-4953	172	6	because	because	SCONJ
ejpam-4953	172	7	s	s	NOUN
ejpam-4953	172	8	and	and	CCONJ
ejpam-4953	172	9	v	v	NOUN
ejpam-4953	172	10	are	be	AUX
ejpam-4953	172	11	continuous	continuous	ADJ
ejpam-4953	172	12	.	.	PUNCT
ejpam-4953	173	1	as	as	ADP
ejpam-4953	173	2	v	v	NOUN
ejpam-4953	173	3	(	(	PUNCT
ejpam-4953	173	4	ḧ	ḧ	NOUN
ejpam-4953	173	5	)	)	PUNCT
ejpam-4953	173	6	=	=	SYM
ejpam-4953	173	7	v	v	X
ejpam-4953	173	8	(	(	PUNCT
ejpam-4953	173	9	g	g	NOUN
ejpam-4953	173	10	)	)	PUNCT
ejpam-4953	173	11	and	and	CCONJ
ejpam-4953	173	12	s(ḧ	s(ḧ	NOUN
ejpam-4953	173	13	)	)	PUNCT
ejpam-4953	173	14	∈	∈	PROPN
ejpam-4953	173	15	v	v	NOUN
ejpam-4953	173	16	(	(	PUNCT
ejpam-4953	173	17	ḧ	ḧ	NOUN
ejpam-4953	173	18	)	)	PUNCT
ejpam-4953	173	19	,	,	PUNCT
ejpam-4953	173	20	we	we	PRON
ejpam-4953	173	21	get	get	VERB
ejpam-4953	173	22	s(ḧ	s(ḧ	NOUN
ejpam-4953	173	23	)	)	PUNCT
ejpam-4953	173	24	∈	∈	PROPN
ejpam-4953	173	25	v	v	NOUN
ejpam-4953	173	26	(	(	PUNCT
ejpam-4953	173	27	g	g	NOUN
ejpam-4953	173	28	)	)	PUNCT
ejpam-4953	173	29	.	.	PUNCT
ejpam-4953	174	1	this	this	PRON
ejpam-4953	174	2	means	mean	VERB
ejpam-4953	174	3	that	that	SCONJ
ejpam-4953	174	4	,	,	PUNCT
ejpam-4953	174	5	it	it	PRON
ejpam-4953	174	6	is	be	AUX
ejpam-4953	174	7	possible	possible	ADJ
ejpam-4953	174	8	to	to	PART
ejpam-4953	174	9	construct	construct	VERB
ejpam-4953	174	10	{	{	PUNCT
ejpam-4953	174	11	ϖn	ϖn	NOUN
ejpam-4953	174	12	}	}	PUNCT
ejpam-4953	174	13	∈	∈	NOUN
ejpam-4953	174	14	g	g	NOUN
ejpam-4953	174	15	satisfying	satisfying	NOUN
ejpam-4953	174	16	relation	relation	NOUN
ejpam-4953	174	17	(	(	PUNCT
ejpam-4953	174	18	6	6	NUM
ejpam-4953	174	19	)	)	PUNCT
ejpam-4953	174	20	and	and	CCONJ
ejpam-4953	174	21	we	we	PRON
ejpam-4953	174	22	choose	choose	VERB
ejpam-4953	174	23	u	u	PRON
ejpam-4953	174	24	∈	∈	PROPN
ejpam-4953	174	25	g.	g.	NOUN
ejpam-4953	174	26	utilizing	utilize	VERB
ejpam-4953	174	27	equation	equation	NOUN
ejpam-4953	174	28	(	(	PUNCT
ejpam-4953	174	29	10	10	NUM
ejpam-4953	174	30	)	)	PUNCT
ejpam-4953	174	31	and	and	CCONJ
ejpam-4953	174	32	(	(	PUNCT
ejpam-4953	174	33	11	11	NUM
ejpam-4953	174	34	)	)	PUNCT
ejpam-4953	174	35	and	and	CCONJ
ejpam-4953	174	36	the	the	DET
ejpam-4953	174	37	continuity	continuity	NOUN
ejpam-4953	174	38	of	of	ADP
ejpam-4953	174	39	f̈	f̈	PROPN
ejpam-4953	174	40	,	,	PUNCT
ejpam-4953	174	41	we	we	PRON
ejpam-4953	174	42	find	find	VERB
ejpam-4953	174	43	s(u	s(u	NOUN
ejpam-4953	174	44	)	)	PUNCT
ejpam-4953	174	45	=	=	SYM
ejpam-4953	174	46	f̈(v	f̈(v	NUM
ejpam-4953	174	47	u	u	NOUN
ejpam-4953	174	48	)	)	PUNCT
ejpam-4953	174	49	=	=	SYM
ejpam-4953	174	50	f̈	f̈	PROPN
ejpam-4953	174	51	(	(	PUNCT
ejpam-4953	174	52	lim	lim	PROPN
ejpam-4953	174	53	n→+∞	n→+∞	VERB
ejpam-4953	174	54	v	v	NOUN
ejpam-4953	174	55	ϖn	ϖn	NOUN
ejpam-4953	174	56	)	)	PUNCT
ejpam-4953	175	1	=	=	SYM
ejpam-4953	175	2	lim	lim	PROPN
ejpam-4953	175	3	n→+∞	n→+∞	VERB
ejpam-4953	175	4	f̈(v	f̈(v	PROPN
ejpam-4953	175	5	ϖn	ϖn	NOUN
ejpam-4953	175	6	)	)	PUNCT
ejpam-4953	176	1	=	=	SYM
ejpam-4953	177	1	lim	lim	PROPN
ejpam-4953	177	2	n→+∞	n→+∞	PROPN
ejpam-4953	177	3	s(ϖn	s(ϖn	PROPN
ejpam-4953	177	4	)	)	PUNCT
ejpam-4953	178	1	=	=	SYM
ejpam-4953	178	2	v	v	X
ejpam-4953	178	3	(	(	PUNCT
ejpam-4953	178	4	u	u	NOUN
ejpam-4953	178	5	)	)	PUNCT
ejpam-4953	178	6	hence	hence	ADV
ejpam-4953	178	7	,	,	PUNCT
ejpam-4953	178	8	u	u	PROPN
ejpam-4953	178	9	∈	∈	PROPN
ejpam-4953	178	10	ḧ	ḧ	NOUN
ejpam-4953	178	11	is	be	AUX
ejpam-4953	178	12	a	a	DET
ejpam-4953	178	13	point	point	NOUN
ejpam-4953	178	14	of	of	ADP
ejpam-4953	178	15	coincidence	coincidence	NOUN
ejpam-4953	178	16	of	of	ADP
ejpam-4953	178	17	a	a	DET
ejpam-4953	178	18	pair	pair	NOUN
ejpam-4953	178	19	of	of	ADP
ejpam-4953	178	20	maps	map	NOUN
ejpam-4953	178	21	.	.	PUNCT
ejpam-4953	179	1	this	this	PRON
ejpam-4953	179	2	end	end	VERB
ejpam-4953	179	3	the	the	DET
ejpam-4953	179	4	proof	proof	NOUN
ejpam-4953	179	5	.	.	PUNCT
ejpam-4953	180	1	owing	owe	VERB
ejpam-4953	180	2	to	to	ADP
ejpam-4953	180	3	(	(	PUNCT
ejpam-4953	180	4	6	6	NUM
ejpam-4953	180	5	)	)	PUNCT
ejpam-4953	180	6	,	,	PUNCT
ejpam-4953	180	7	we	we	PRON
ejpam-4953	180	8	have	have	VERB
ejpam-4953	180	9	{	{	PUNCT
ejpam-4953	180	10	v	v	NOUN
ejpam-4953	180	11	ϖn	ϖn	NOUN
ejpam-4953	180	12	}	}	PUNCT
ejpam-4953	180	13	⊆	⊆	NUM
ejpam-4953	180	14	s(ḧ	s(ḧ	NOUN
ejpam-4953	180	15	)	)	PUNCT
ejpam-4953	180	16	and	and	CCONJ
ejpam-4953	180	17	hence	hence	ADV
ejpam-4953	180	18	,	,	PUNCT
ejpam-4953	180	19	{	{	PUNCT
ejpam-4953	180	20	v	v	NOUN
ejpam-4953	180	21	ϖn	ϖn	NOUN
ejpam-4953	180	22	}	}	PUNCT
ejpam-4953	180	23	is	be	AUX
ejpam-4953	180	24	b	b	NOUN
ejpam-4953	180	25	-	-	PUNCT
ejpam-4953	180	26	multiplicative	multiplicative	ADJ
ejpam-4953	180	27	cauchy	cauchy	NOUN
ejpam-4953	180	28	sequence	sequence	NOUN
ejpam-4953	180	29	in	in	ADP
ejpam-4953	180	30	ḧ.	ḧ.	PROPN
ejpam-4953	180	31	as	as	SCONJ
ejpam-4953	180	32	ḧ	ḧ	PRON
ejpam-4953	180	33	is	be	AUX
ejpam-4953	180	34	b	b	NOUN
ejpam-4953	180	35	-	-	PUNCT
ejpam-4953	180	36	complete	complete	ADJ
ejpam-4953	180	37	,	,	PUNCT
ejpam-4953	180	38	there	there	PRON
ejpam-4953	180	39	exists	exist	VERB
ejpam-4953	180	40	u	u	PROPN
ejpam-4953	180	41	∈	∈	PROPN
ejpam-4953	180	42	v	v	NOUN
ejpam-4953	180	43	(	(	PUNCT
ejpam-4953	180	44	ḧ	ḧ	NOUN
ejpam-4953	180	45	)	)	PUNCT
ejpam-4953	180	46	such	such	ADJ
ejpam-4953	180	47	that	that	DET
ejpam-4953	180	48	i.	i.	PROPN
ejpam-4953	180	49	alshammari	alshammari	PROPN
ejpam-4953	180	50	et	et	PROPN
ejpam-4953	180	51	al	al	PROPN
ejpam-4953	180	52	.	.	PUNCT
ejpam-4953	180	53	/	/	SYM
ejpam-4953	180	54	eur	eur	PROPN
ejpam-4953	180	55	.	.	PUNCT
ejpam-4953	181	1	j.	j.	PROPN
ejpam-4953	181	2	pure	pure	PROPN
ejpam-4953	181	3	appl	appl	PROPN
ejpam-4953	181	4	.	.	PROPN
ejpam-4953	181	5	math	math	PROPN
ejpam-4953	181	6	,	,	PUNCT
ejpam-4953	181	7	16	16	NUM
ejpam-4953	181	8	(	(	PUNCT
ejpam-4953	181	9	4	4	NUM
ejpam-4953	181	10	)	)	PUNCT
ejpam-4953	181	11	(	(	PUNCT
ejpam-4953	181	12	2023	2023	NUM
ejpam-4953	181	13	)	)	PUNCT
ejpam-4953	181	14	,	,	PUNCT
ejpam-4953	181	15	2405	2405	NUM
ejpam-4953	181	16	-	-	SYM
ejpam-4953	181	17	2418	2418	NUM
ejpam-4953	181	18	2414	2414	NUM
ejpam-4953	181	19	lim	lim	NOUN
ejpam-4953	181	20	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	181	21	v	v	PROPN
ejpam-4953	181	22	(	(	PUNCT
ejpam-4953	181	23	ϖn	ϖn	NOUN
ejpam-4953	181	24	)	)	PUNCT
ejpam-4953	181	25	=	=	SYM
ejpam-4953	181	26	v	v	X
ejpam-4953	181	27	(	(	PUNCT
ejpam-4953	181	28	u	u	NOUN
ejpam-4953	181	29	)	)	PUNCT
ejpam-4953	181	30	.	.	PUNCT
ejpam-4953	182	1	(	(	PUNCT
ejpam-4953	182	2	13	13	NUM
ejpam-4953	182	3	)	)	PUNCT
ejpam-4953	182	4	by	by	ADP
ejpam-4953	182	5	using	use	VERB
ejpam-4953	182	6	(	(	PUNCT
ejpam-4953	182	7	6	6	NUM
ejpam-4953	182	8	)	)	PUNCT
ejpam-4953	182	9	and	and	CCONJ
ejpam-4953	182	10	(	(	PUNCT
ejpam-4953	182	11	13	13	NUM
ejpam-4953	182	12	)	)	PUNCT
ejpam-4953	182	13	,	,	PUNCT
ejpam-4953	182	14	we	we	PRON
ejpam-4953	182	15	get	get	VERB
ejpam-4953	182	16	lim	lim	PROPN
ejpam-4953	182	17	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	182	18	s(ϖn	s(ϖn	PROPN
ejpam-4953	182	19	)	)	PUNCT
ejpam-4953	183	1	=	=	SYM
ejpam-4953	183	2	v	v	X
ejpam-4953	183	3	(	(	PUNCT
ejpam-4953	183	4	u	u	NOUN
ejpam-4953	183	5	)	)	PUNCT
ejpam-4953	183	6	.	.	PUNCT
ejpam-4953	184	1	(	(	PUNCT
ejpam-4953	184	2	14	14	NUM
ejpam-4953	184	3	)	)	PUNCT
ejpam-4953	184	4	as	as	ADP
ejpam-4953	184	5	v	v	NUM
ejpam-4953	184	6	is	be	AUX
ejpam-4953	184	7	r̈-continuous	r̈-continuous	ADJ
ejpam-4953	184	8	,	,	PUNCT
ejpam-4953	184	9	we	we	PRON
ejpam-4953	184	10	find	find	VERB
ejpam-4953	184	11	lim	lim	PROPN
ejpam-4953	184	12	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	184	13	v	v	PROPN
ejpam-4953	184	14	(	(	PUNCT
ejpam-4953	184	15	v	v	NOUN
ejpam-4953	184	16	ϖn	ϖn	NOUN
ejpam-4953	184	17	)	)	PUNCT
ejpam-4953	184	18	=	=	SYM
ejpam-4953	184	19	v	v	X
ejpam-4953	184	20	(	(	PUNCT
ejpam-4953	184	21	lim	lim	PROPN
ejpam-4953	184	22	n→+∞	n→+∞	VERB
ejpam-4953	184	23	v	v	PROPN
ejpam-4953	184	24	(	(	PUNCT
ejpam-4953	184	25	ϖn	ϖn	NOUN
ejpam-4953	184	26	)	)	PUNCT
ejpam-4953	184	27	)	)	PUNCT
ejpam-4953	185	1	=	=	SYM
ejpam-4953	185	2	v	v	X
ejpam-4953	185	3	(	(	PUNCT
ejpam-4953	185	4	v	v	NOUN
ejpam-4953	185	5	(	(	PUNCT
ejpam-4953	185	6	u	u	NOUN
ejpam-4953	185	7	)	)	PUNCT
ejpam-4953	185	8	)	)	PUNCT
ejpam-4953	185	9	(	(	PUNCT
ejpam-4953	185	10	15	15	NUM
ejpam-4953	185	11	)	)	PUNCT
ejpam-4953	185	12	moreover	moreover	ADV
ejpam-4953	185	13	,	,	PUNCT
ejpam-4953	185	14	we	we	PRON
ejpam-4953	185	15	get	get	VERB
ejpam-4953	185	16	lim	lim	PROPN
ejpam-4953	185	17	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	185	18	v	v	PROPN
ejpam-4953	185	19	(	(	PUNCT
ejpam-4953	185	20	sϖn	sϖn	ADJ
ejpam-4953	185	21	)	)	PUNCT
ejpam-4953	185	22	=	=	SYM
ejpam-4953	185	23	v	v	X
ejpam-4953	185	24	(	(	PUNCT
ejpam-4953	185	25	lim	lim	PROPN
ejpam-4953	185	26	n→+∞	n→+∞	PROPN
ejpam-4953	185	27	s(ϖn	s(ϖn	PROPN
ejpam-4953	185	28	)	)	PUNCT
ejpam-4953	185	29	)	)	PUNCT
ejpam-4953	186	1	=	=	SYM
ejpam-4953	186	2	v	v	X
ejpam-4953	186	3	(	(	PUNCT
ejpam-4953	186	4	v	v	NOUN
ejpam-4953	186	5	(	(	PUNCT
ejpam-4953	186	6	u	u	NOUN
ejpam-4953	186	7	)	)	PUNCT
ejpam-4953	186	8	)	)	PUNCT
ejpam-4953	186	9	(	(	PUNCT
ejpam-4953	186	10	16	16	NUM
ejpam-4953	186	11	)	)	PUNCT
ejpam-4953	186	12	since	since	SCONJ
ejpam-4953	186	13	{	{	PUNCT
ejpam-4953	186	14	sϖn	sϖn	ADJ
ejpam-4953	186	15	}	}	PUNCT
ejpam-4953	186	16	and	and	CCONJ
ejpam-4953	186	17	{	{	PUNCT
ejpam-4953	186	18	v	v	NOUN
ejpam-4953	186	19	ϖn	ϖn	NOUN
ejpam-4953	186	20	}	}	PUNCT
ejpam-4953	186	21	are	be	AUX
ejpam-4953	186	22	r̈-preserving	r̈-preserve	VERB
ejpam-4953	186	23	lim	lim	PROPN
ejpam-4953	186	24	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	186	25	s(ϖn	s(ϖn	PROPN
ejpam-4953	186	26	)	)	PUNCT
ejpam-4953	187	1	=	=	SYM
ejpam-4953	187	2	v	v	X
ejpam-4953	187	3	(	(	PUNCT
ejpam-4953	187	4	u	u	NOUN
ejpam-4953	187	5	)	)	PUNCT
ejpam-4953	187	6	=	=	SYM
ejpam-4953	187	7	lim	lim	PROPN
ejpam-4953	187	8	n→+∞	n→+∞	VERB
ejpam-4953	187	9	v	v	PROPN
ejpam-4953	187	10	(	(	PUNCT
ejpam-4953	187	11	ϖn	ϖn	NOUN
ejpam-4953	187	12	)	)	PUNCT
ejpam-4953	187	13	(	(	PUNCT
ejpam-4953	187	14	17	17	NUM
ejpam-4953	187	15	)	)	PUNCT
ejpam-4953	187	16	and	and	CCONJ
ejpam-4953	187	17	s	s	PROPN
ejpam-4953	187	18	and	and	CCONJ
ejpam-4953	187	19	v	v	NOUN
ejpam-4953	187	20	are	be	AUX
ejpam-4953	187	21	r̈-compatible	r̈-compatible	ADJ
ejpam-4953	187	22	,	,	PUNCT
ejpam-4953	187	23	we	we	PRON
ejpam-4953	187	24	obtain	obtain	VERB
ejpam-4953	187	25	lim	lim	PROPN
ejpam-4953	187	26	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	187	27	p(v	p(v	PROPN
ejpam-4953	187	28	s(ϖn	s(ϖn	PROPN
ejpam-4953	187	29	)	)	PUNCT
ejpam-4953	187	30	,	,	PUNCT
ejpam-4953	187	31	sv	sv	INTJ
ejpam-4953	187	32	(	(	PUNCT
ejpam-4953	187	33	ϖn	ϖn	NOUN
ejpam-4953	187	34	)	)	PUNCT
ejpam-4953	187	35	)	)	PUNCT
ejpam-4953	188	1	=	=	PUNCT
ejpam-4953	188	2	0	0	X
ejpam-4953	188	3	.	.	PUNCT
ejpam-4953	189	1	(	(	PUNCT
ejpam-4953	189	2	18	18	NUM
ejpam-4953	189	3	)	)	PUNCT
ejpam-4953	189	4	now	now	ADV
ejpam-4953	189	5	,	,	PUNCT
ejpam-4953	189	6	we	we	PRON
ejpam-4953	189	7	demonstrate	demonstrate	VERB
ejpam-4953	189	8	that	that	PRON
ejpam-4953	189	9	v(u	v(u	NOUN
ejpam-4953	189	10	)	)	PUNCT
ejpam-4953	189	11	is	be	AUX
ejpam-4953	189	12	a	a	DET
ejpam-4953	189	13	coincidence	coincidence	NOUN
ejpam-4953	189	14	point	point	NOUN
ejpam-4953	189	15	of	of	ADP
ejpam-4953	189	16	s	s	PRON
ejpam-4953	189	17	and	and	CCONJ
ejpam-4953	189	18	v.	v.	CCONJ
ejpam-4953	189	19	we	we	PRON
ejpam-4953	189	20	assume	assume	VERB
ejpam-4953	189	21	that	that	SCONJ
ejpam-4953	189	22	s	s	VERB
ejpam-4953	189	23	is	be	AUX
ejpam-4953	189	24	r̈-continuous	r̈-continuous	ADJ
ejpam-4953	189	25	.	.	PUNCT
ejpam-4953	190	1	by	by	ADP
ejpam-4953	190	2	using	use	VERB
ejpam-4953	190	3	(	(	PUNCT
ejpam-4953	190	4	7	7	NUM
ejpam-4953	190	5	)	)	PUNCT
ejpam-4953	190	6	,	,	PUNCT
ejpam-4953	190	7	we	we	PRON
ejpam-4953	190	8	get	get	VERB
ejpam-4953	190	9	lim	lim	PROPN
ejpam-4953	190	10	ϖ→+∞	ϖ→+∞	PROPN
ejpam-4953	190	11	s(v	s(v	PROPN
ejpam-4953	190	12	ϖn	ϖn	NOUN
ejpam-4953	190	13	)	)	PUNCT
ejpam-4953	190	14	=	=	SYM
ejpam-4953	190	15	s	s	PART
ejpam-4953	190	16	lim	lim	NOUN
ejpam-4953	190	17	n→+∞	n→+∞	VERB
ejpam-4953	190	18	v	v	PROPN
ejpam-4953	190	19	(	(	PUNCT
ejpam-4953	190	20	ϖn	ϖn	NOUN
ejpam-4953	190	21	)	)	PUNCT
ejpam-4953	191	1	=	=	SYM
ejpam-4953	191	2	=	=	SYM
ejpam-4953	191	3	s(v	s(v	PROPN
ejpam-4953	191	4	(	(	PUNCT
ejpam-4953	191	5	u	u	NOUN
ejpam-4953	191	6	)	)	PUNCT
ejpam-4953	191	7	)	)	PUNCT
ejpam-4953	191	8	(	(	PUNCT
ejpam-4953	191	9	19	19	NUM
ejpam-4953	191	10	)	)	PUNCT
ejpam-4953	191	11	suppose	suppose	VERB
ejpam-4953	191	12	that	that	SCONJ
ejpam-4953	191	13	v(u	v(u	NOUN
ejpam-4953	191	14	)	)	PUNCT
ejpam-4953	192	1	=	=	SYM
ejpam-4953	192	2	z	z	NOUN
ejpam-4953	192	3	,	,	PUNCT
ejpam-4953	192	4	utilizing	utilize	VERB
ejpam-4953	192	5	triangle	triangle	NOUN
ejpam-4953	192	6	inequality	inequality	NOUN
ejpam-4953	192	7	,	,	PUNCT
ejpam-4953	192	8	we	we	PRON
ejpam-4953	192	9	get	get	VERB
ejpam-4953	192	10	p(v	p(v	PROPN
ejpam-4953	192	11	z	z	PROPN
ejpam-4953	192	12	,	,	PUNCT
ejpam-4953	192	13	sz	sz	NOUN
ejpam-4953	192	14	)	)	PUNCT
ejpam-4953	192	15	≤	≤	NOUN
ejpam-4953	193	1	[	[	X
ejpam-4953	193	2	p(v	p(v	NOUN
ejpam-4953	193	3	z	z	PROPN
ejpam-4953	193	4	,	,	PUNCT
ejpam-4953	193	5	v	v	X
ejpam-4953	193	6	(	(	PUNCT
ejpam-4953	193	7	sϖn	sϖn	ADJ
ejpam-4953	193	8	)	)	PUNCT
ejpam-4953	193	9	)	)	PUNCT
ejpam-4953	194	1	·	·	PUNCT
ejpam-4953	194	2	p(v	p(v	NOUN
ejpam-4953	194	3	(	(	PUNCT
ejpam-4953	194	4	sϖn	sϖn	ADJ
ejpam-4953	194	5	)	)	PUNCT
ejpam-4953	194	6	,	,	PUNCT
ejpam-4953	194	7	sz	sz	PROPN
ejpam-4953	194	8	)	)	PUNCT
ejpam-4953	194	9	]	]	PUNCT
ejpam-4953	195	1	k	k	PROPN
ejpam-4953	195	2	≤	≤	PROPN
ejpam-4953	195	3	p(v	p(v	PROPN
ejpam-4953	195	4	z	z	PROPN
ejpam-4953	195	5	,	,	PUNCT
ejpam-4953	195	6	v	v	PROPN
ejpam-4953	195	7	(	(	PUNCT
ejpam-4953	195	8	sϖn	sϖn	ADJ
ejpam-4953	195	9	)	)	PUNCT
ejpam-4953	195	10	)	)	PUNCT
ejpam-4953	196	1	k	k	X
ejpam-4953	196	2	·	·	PUNCT
ejpam-4953	197	1	[	[	X
ejpam-4953	197	2	p(v	p(v	NOUN
ejpam-4953	197	3	(	(	PUNCT
ejpam-4953	197	4	sϖn	sϖn	ADJ
ejpam-4953	197	5	)	)	PUNCT
ejpam-4953	197	6	,	,	PUNCT
ejpam-4953	197	7	s(v	s(v	PROPN
ejpam-4953	197	8	ϖn	ϖn	NOUN
ejpam-4953	197	9	)	)	PUNCT
ejpam-4953	197	10	·	·	PUNCT
ejpam-4953	197	11	p(s(v	p(s(v	NOUN
ejpam-4953	197	12	ϖn	ϖn	NOUN
ejpam-4953	197	13	)	)	PUNCT
ejpam-4953	197	14	,	,	PUNCT
ejpam-4953	197	15	sz	sz	NOUN
ejpam-4953	197	16	)	)	PUNCT
ejpam-4953	197	17	k2	k2	PROPN
ejpam-4953	197	18	making	making	NOUN
ejpam-4953	197	19	n	n	X
ejpam-4953	197	20	→	→	SYM
ejpam-4953	197	21	+	+	PROPN
ejpam-4953	197	22	∞	∞	PROPN
ejpam-4953	197	23	,	,	PUNCT
ejpam-4953	197	24	we	we	PRON
ejpam-4953	197	25	get	get	VERB
ejpam-4953	197	26	p(vz	p(vz	NOUN
ejpam-4953	197	27	,	,	PUNCT
ejpam-4953	197	28	sz	sz	NOUN
ejpam-4953	197	29	)	)	PUNCT
ejpam-4953	197	30	=	=	SYM
ejpam-4953	198	1	1	1	NUM
ejpam-4953	198	2	,	,	PUNCT
ejpam-4953	198	3	which	which	PRON
ejpam-4953	198	4	implies	imply	VERB
ejpam-4953	198	5	vz	vz	X
ejpam-4953	198	6	=	=	SYM
ejpam-4953	198	7	sz	sz	PROPN
ejpam-4953	198	8	,	,	PUNCT
ejpam-4953	198	9	i.e.	i.e.	X
ejpam-4953	198	10	,	,	PUNCT
ejpam-4953	198	11	z	z	NOUN
ejpam-4953	198	12	=	=	SYM
ejpam-4953	198	13	v(u	v(u	PROPN
ejpam-4953	198	14	)	)	PUNCT
ejpam-4953	198	15	is	be	AUX
ejpam-4953	198	16	coincidence	coincidence	NOUN
ejpam-4953	198	17	point	point	NOUN
ejpam-4953	198	18	of	of	ADP
ejpam-4953	198	19	s	s	PRON
ejpam-4953	198	20	and	and	CCONJ
ejpam-4953	198	21	v.	v.	ADP
ejpam-4953	198	22	alternatively	alternatively	ADV
ejpam-4953	198	23	,	,	PUNCT
ejpam-4953	198	24	assume	assume	VERB
ejpam-4953	198	25	that	that	SCONJ
ejpam-4953	198	26	r̈	r̈	VERB
ejpam-4953	198	27	is	be	AUX
ejpam-4953	198	28	(	(	PUNCT
ejpam-4953	198	29	v	v	NOUN
ejpam-4953	198	30	,	,	PUNCT
ejpam-4953	198	31	bp)-self	bp)-self	NOUN
ejpam-4953	198	32	closed	close	VERB
ejpam-4953	198	33	.	.	PUNCT
ejpam-4953	199	1	since	since	SCONJ
ejpam-4953	199	2	{	{	PUNCT
ejpam-4953	199	3	v	v	NOUN
ejpam-4953	199	4	ϖn	ϖn	NOUN
ejpam-4953	199	5	}	}	PUNCT
ejpam-4953	199	6	is	be	AUX
ejpam-4953	199	7	r̈-preserving	r̈-preserve	VERB
ejpam-4953	199	8	and	and	CCONJ
ejpam-4953	199	9	v	v	ADP
ejpam-4953	199	10	ϖn	ϖn	NOUN
ejpam-4953	199	11	→	→	SYM
ejpam-4953	199	12	v	v	NUM
ejpam-4953	199	13	u	u	NOUN
ejpam-4953	199	14	,	,	PUNCT
ejpam-4953	199	15	in	in	ADP
ejpam-4953	199	16	view	view	NOUN
ejpam-4953	199	17	of	of	ADP
ejpam-4953	199	18	the	the	DET
ejpam-4953	199	19	(	(	PUNCT
ejpam-4953	199	20	v	v	NOUN
ejpam-4953	199	21	,	,	PUNCT
ejpam-4953	199	22	bp)-self	bp)-self	NOUN
ejpam-4953	199	23	closeness	closeness	NOUN
ejpam-4953	199	24	of	of	ADP
ejpam-4953	199	25	r̈	r̈	NOUN
ejpam-4953	199	26	,	,	PUNCT
ejpam-4953	199	27	there	there	PRON
ejpam-4953	199	28	exists	exist	VERB
ejpam-4953	199	29	a	a	DET
ejpam-4953	199	30	subsequence	subsequence	NOUN
ejpam-4953	199	31	{	{	PUNCT
ejpam-4953	199	32	v	v	NOUN
ejpam-4953	199	33	ϖni	ϖni	NOUN
ejpam-4953	199	34	}	}	PUNCT
ejpam-4953	199	35	of	of	ADP
ejpam-4953	199	36	{	{	PUNCT
ejpam-4953	199	37	v	v	NOUN
ejpam-4953	199	38	ϖn	ϖn	NOUN
ejpam-4953	199	39	}	}	PUNCT
ejpam-4953	199	40	such	such	ADJ
ejpam-4953	199	41	that	that	SCONJ
ejpam-4953	199	42	[	[	X
ejpam-4953	199	43	v	v	NUM
ejpam-4953	199	44	v	v	NOUN
ejpam-4953	199	45	ϖni	ϖni	NOUN
ejpam-4953	199	46	,	,	PUNCT
ejpam-4953	199	47	v	v	NOUN
ejpam-4953	199	48	v	v	NUM
ejpam-4953	199	49	u	u	NOUN
ejpam-4953	199	50	]	]	PUNCT
ejpam-4953	199	51	belongs	belong	VERB
ejpam-4953	199	52	to	to	ADP
ejpam-4953	199	53	r̈	r̈	NOUN
ejpam-4953	199	54	for	for	SCONJ
ejpam-4953	199	55	all	all	PRON
ejpam-4953	199	56	i	i	PRON
ejpam-4953	199	57	∈	∈	VERB
ejpam-4953	199	58	n	n	PART
ejpam-4953	199	59	∪	∪	X
ejpam-4953	199	60	{	{	PUNCT
ejpam-4953	199	61	0	0	NUM
ejpam-4953	199	62	}	}	PUNCT
ejpam-4953	199	63	.	.	PUNCT
ejpam-4953	200	1	since	since	SCONJ
ejpam-4953	200	2	v	v	NUM
ejpam-4953	200	3	ϖni	ϖni	NOUN
ejpam-4953	200	4	→	→	SYM
ejpam-4953	200	5	v	v	NOUN
ejpam-4953	200	6	u	u	NOUN
ejpam-4953	200	7	,	,	PUNCT
ejpam-4953	200	8	in	in	ADP
ejpam-4953	200	9	the	the	DET
ejpam-4953	200	10	view	view	NOUN
ejpam-4953	200	11	of	of	ADP
ejpam-4953	200	12	proposition	proposition	NOUN
ejpam-4953	200	13	1.3	1.3	NUM
ejpam-4953	200	14	,	,	PUNCT
ejpam-4953	200	15	we	we	PRON
ejpam-4953	200	16	get	get	VERB
ejpam-4953	200	17	p(sv	p(sv	NOUN
ejpam-4953	200	18	ϖni	ϖni	NOUN
ejpam-4953	200	19	,	,	PUNCT
ejpam-4953	200	20	sv	sv	NOUN
ejpam-4953	200	21	u	u	NOUN
ejpam-4953	200	22	)	)	PUNCT
ejpam-4953	200	23	≤	≤	PUNCT
ejpam-4953	201	1	p(v	p(v	NOUN
ejpam-4953	201	2	v	v	ADP
ejpam-4953	201	3	ϖni	ϖni	NOUN
ejpam-4953	201	4	,	,	PUNCT
ejpam-4953	201	5	v	v	NOUN
ejpam-4953	201	6	v	v	NOUN
ejpam-4953	201	7	u)λ	u)λ	NOUN
ejpam-4953	201	8	for	for	ADP
ejpam-4953	201	9	all	all	DET
ejpam-4953	201	10	i	i	PRON
ejpam-4953	201	11	∈	∈	VERB
ejpam-4953	201	12	n	n	PART
ejpam-4953	201	13	∪	∪	X
ejpam-4953	201	14	{	{	PUNCT
ejpam-4953	201	15	0	0	NUM
ejpam-4953	201	16	}	}	PUNCT
ejpam-4953	201	17	i.	i.	NOUN
ejpam-4953	201	18	alshammari	alshammari	PROPN
ejpam-4953	201	19	et	et	PROPN
ejpam-4953	201	20	al	al	PROPN
ejpam-4953	201	21	.	.	PUNCT
ejpam-4953	201	22	/	/	SYM
ejpam-4953	201	23	eur	eur	PROPN
ejpam-4953	201	24	.	.	PUNCT
ejpam-4953	202	1	j.	j.	PROPN
ejpam-4953	202	2	pure	pure	PROPN
ejpam-4953	202	3	appl	appl	PROPN
ejpam-4953	202	4	.	.	PROPN
ejpam-4953	202	5	math	math	PROPN
ejpam-4953	202	6	,	,	PUNCT
ejpam-4953	202	7	16	16	NUM
ejpam-4953	202	8	(	(	PUNCT
ejpam-4953	202	9	4	4	NUM
ejpam-4953	202	10	)	)	PUNCT
ejpam-4953	202	11	(	(	PUNCT
ejpam-4953	202	12	2023	2023	NUM
ejpam-4953	202	13	)	)	PUNCT
ejpam-4953	202	14	,	,	PUNCT
ejpam-4953	202	15	2405	2405	NUM
ejpam-4953	202	16	-	-	SYM
ejpam-4953	202	17	2418	2418	NUM
ejpam-4953	202	18	2415	2415	NUM
ejpam-4953	202	19	we	we	PRON
ejpam-4953	202	20	choose	choose	VERB
ejpam-4953	202	21	vu	vu	X
ejpam-4953	202	22	=	=	PUNCT
ejpam-4953	202	23	z.	z.	PROPN
ejpam-4953	202	24	by	by	ADP
ejpam-4953	202	25	the	the	DET
ejpam-4953	202	26	triangle	triangle	NOUN
ejpam-4953	202	27	inequality	inequality	NOUN
ejpam-4953	202	28	,	,	PUNCT
ejpam-4953	202	29	we	we	PRON
ejpam-4953	202	30	get	get	VERB
ejpam-4953	202	31	p(v	p(v	PROPN
ejpam-4953	202	32	z	z	PROPN
ejpam-4953	202	33	,	,	PUNCT
ejpam-4953	202	34	sz	sz	NOUN
ejpam-4953	202	35	)	)	PUNCT
ejpam-4953	202	36	≤	≤	NOUN
ejpam-4953	203	1	[	[	X
ejpam-4953	203	2	p(v	p(v	NOUN
ejpam-4953	203	3	z	z	PROPN
ejpam-4953	203	4	,	,	PUNCT
ejpam-4953	203	5	v	v	ADJ
ejpam-4953	203	6	(	(	PUNCT
ejpam-4953	203	7	sϖni	sϖni	NOUN
ejpam-4953	203	8	)	)	PUNCT
ejpam-4953	203	9	)	)	PUNCT
ejpam-4953	203	10	·	·	PUNCT
ejpam-4953	204	1	p(v	p(v	NOUN
ejpam-4953	204	2	(	(	PUNCT
ejpam-4953	204	3	sϖni	sϖni	NOUN
ejpam-4953	204	4	)	)	PUNCT
ejpam-4953	204	5	,	,	PUNCT
ejpam-4953	204	6	sz	sz	PROPN
ejpam-4953	204	7	)	)	PUNCT
ejpam-4953	204	8	]	]	PUNCT
ejpam-4953	205	1	k	k	PROPN
ejpam-4953	205	2	≤	≤	PROPN
ejpam-4953	205	3	p(v	p(v	PROPN
ejpam-4953	205	4	z	z	PROPN
ejpam-4953	205	5	,	,	PUNCT
ejpam-4953	205	6	v	v	ADJ
ejpam-4953	205	7	(	(	PUNCT
ejpam-4953	205	8	sϖni	sϖni	NOUN
ejpam-4953	205	9	)	)	PUNCT
ejpam-4953	205	10	)	)	PUNCT
ejpam-4953	206	1	k	k	X
ejpam-4953	206	2	·	·	PUNCT
ejpam-4953	207	1	[	[	X
ejpam-4953	207	2	p(v	p(v	NOUN
ejpam-4953	207	3	(	(	PUNCT
ejpam-4953	207	4	sϖni	sϖni	NOUN
ejpam-4953	207	5	)	)	PUNCT
ejpam-4953	207	6	,	,	PUNCT
ejpam-4953	207	7	s(v	s(v	PROPN
ejpam-4953	207	8	ϖni	ϖni	NOUN
ejpam-4953	207	9	)	)	PUNCT
ejpam-4953	207	10	·	·	PUNCT
ejpam-4953	207	11	p(s(v	p(s(v	NOUN
ejpam-4953	207	12	ϖni	ϖni	NOUN
ejpam-4953	207	13	)	)	PUNCT
ejpam-4953	207	14	,	,	PUNCT
ejpam-4953	207	15	sz	sz	PROPN
ejpam-4953	207	16	)	)	PUNCT
ejpam-4953	207	17	]	]	PUNCT
ejpam-4953	208	1	k2	k2	PROPN
ejpam-4953	208	2	≤	≤	PUNCT
ejpam-4953	208	3	p(v	p(v	PROPN
ejpam-4953	208	4	z	z	PROPN
ejpam-4953	208	5	,	,	PUNCT
ejpam-4953	208	6	v	v	ADJ
ejpam-4953	208	7	(	(	PUNCT
ejpam-4953	208	8	sϖni	sϖni	NOUN
ejpam-4953	208	9	)	)	PUNCT
ejpam-4953	208	10	)	)	PUNCT
ejpam-4953	209	1	k	k	X
ejpam-4953	209	2	·	·	PUNCT
ejpam-4953	209	3	p(v	p(v	PROPN
ejpam-4953	209	4	(	(	PUNCT
ejpam-4953	209	5	sϖni	sϖni	NOUN
ejpam-4953	209	6	)	)	PUNCT
ejpam-4953	209	7	,	,	PUNCT
ejpam-4953	209	8	s(v	s(v	PROPN
ejpam-4953	209	9	ϖni	ϖni	NOUN
ejpam-4953	209	10	)	)	PUNCT
ejpam-4953	209	11	k2	k2	PROPN
ejpam-4953	209	12	·	·	PUNCT
ejpam-4953	209	13	p(s(v	p(s(v	PROPN
ejpam-4953	209	14	ϖni	ϖni	NOUN
ejpam-4953	209	15	)	)	PUNCT
ejpam-4953	209	16	,	,	PUNCT
ejpam-4953	209	17	sz	sz	PROPN
ejpam-4953	209	18	)	)	PUNCT
ejpam-4953	209	19	.	.	PUNCT
ejpam-4953	210	1	λk2	λk2	PROPN
ejpam-4953	210	2	making	make	VERB
ejpam-4953	210	3	i	i	PRON
ejpam-4953	210	4	→	→	PUNCT
ejpam-4953	210	5	+	+	NOUN
ejpam-4953	210	6	∞	∞	PROPN
ejpam-4953	210	7	,	,	PUNCT
ejpam-4953	210	8	we	we	PRON
ejpam-4953	210	9	get	get	VERB
ejpam-4953	210	10	p(vz	p(vz	NOUN
ejpam-4953	210	11	,	,	PUNCT
ejpam-4953	210	12	sz	sz	NOUN
ejpam-4953	210	13	)	)	PUNCT
ejpam-4953	210	14	=	=	SYM
ejpam-4953	211	1	1	1	NUM
ejpam-4953	211	2	,	,	PUNCT
ejpam-4953	211	3	which	which	PRON
ejpam-4953	211	4	implies	imply	VERB
ejpam-4953	211	5	vz	vz	X
ejpam-4953	211	6	=	=	PUNCT
ejpam-4953	211	7	sz	sz	PROPN
ejpam-4953	211	8	,	,	PUNCT
ejpam-4953	211	9	that	that	ADV
ejpam-4953	211	10	is	be	AUX
ejpam-4953	211	11	,	,	PUNCT
ejpam-4953	211	12	z	z	PROPN
ejpam-4953	211	13	=	=	SYM
ejpam-4953	211	14	v(u	v(u	PROPN
ejpam-4953	211	15	)	)	PUNCT
ejpam-4953	211	16	is	be	AUX
ejpam-4953	211	17	a	a	DET
ejpam-4953	211	18	coincidence	coincidence	NOUN
ejpam-4953	211	19	point	point	NOUN
ejpam-4953	211	20	of	of	ADP
ejpam-4953	211	21	s	s	PRON
ejpam-4953	211	22	and	and	CCONJ
ejpam-4953	211	23	v.	v.	ADP
ejpam-4953	211	24	now	now	ADV
ejpam-4953	211	25	we	we	PRON
ejpam-4953	211	26	can	can	AUX
ejpam-4953	211	27	give	give	VERB
ejpam-4953	211	28	examples	example	NOUN
ejpam-4953	211	29	in	in	ADP
ejpam-4953	211	30	support	support	NOUN
ejpam-4953	211	31	of	of	ADP
ejpam-4953	211	32	theorem	theorem	NOUN
ejpam-4953	211	33	1	1	NUM
ejpam-4953	211	34	.	.	NOUN
ejpam-4953	211	35	example	example	NOUN
ejpam-4953	212	1	1	1	NUM
ejpam-4953	212	2	.	.	PUNCT
ejpam-4953	213	1	let	let	VERB
ejpam-4953	213	2	ḧ	ḧ	NOUN
ejpam-4953	213	3	=	=	SYM
ejpam-4953	213	4	r+	r+	NOUN
ejpam-4953	213	5	and	and	CCONJ
ejpam-4953	213	6	p	p	NOUN
ejpam-4953	213	7	=	=	NOUN
ejpam-4953	214	1	|ϖρ	|ϖρ	X
ejpam-4953	214	2	|	|	ADV
ejpam-4953	214	3	,	,	PUNCT
ejpam-4953	214	4	then	then	ADV
ejpam-4953	214	5	(	(	PUNCT
ejpam-4953	214	6	ḧ,p	ḧ,p	PROPN
ejpam-4953	214	7	)	)	PUNCT
ejpam-4953	214	8	is	be	AUX
ejpam-4953	214	9	a	a	DET
ejpam-4953	214	10	complete	complete	ADJ
ejpam-4953	214	11	multiplicative	multiplicative	ADJ
ejpam-4953	214	12	metric	metric	ADJ
ejpam-4953	214	13	space	space	NOUN
ejpam-4953	214	14	.	.	PUNCT
ejpam-4953	215	1	define	define	VERB
ejpam-4953	215	2	binary	binary	ADJ
ejpam-4953	215	3	relation	relation	NOUN
ejpam-4953	215	4	r̈	r̈	VERB
ejpam-4953	215	5	=	=	PRON
ejpam-4953	215	6	{	{	PUNCT
ejpam-4953	215	7	(	(	PUNCT
ejpam-4953	215	8	ϖ	ϖ	NOUN
ejpam-4953	215	9	,	,	PUNCT
ejpam-4953	215	10	ρ	ρ	NOUN
ejpam-4953	215	11	)	)	PUNCT
ejpam-4953	215	12	∈	∈	PROPN
ejpam-4953	215	13	r2	r2	NOUN
ejpam-4953	216	1	+	+	PROPN
ejpam-4953	216	2	:	:	PUNCT
ejpam-4953	216	3	ϖ	ϖ	X
ejpam-4953	216	4	ρ	ρ	X
ejpam-4953	216	5	≥	≥	NOUN
ejpam-4953	216	6	1	1	NUM
ejpam-4953	216	7	,	,	PUNCT
ejpam-4953	216	8	ϖ	ϖ	PROPN
ejpam-4953	216	9	,	,	PUNCT
ejpam-4953	216	10	ρ	ρ	PROPN
ejpam-4953	216	11	∈	∈	PROPN
ejpam-4953	216	12	r+	r+	PUNCT
ejpam-4953	216	13	}	}	PUNCT
ejpam-4953	216	14	on	on	ADP
ejpam-4953	216	15	ḧ.	ḧ.	PROPN
ejpam-4953	216	16	consider	consider	VERB
ejpam-4953	216	17	mapping	mapping	NOUN
ejpam-4953	216	18	f̈:ḧ	f̈:ḧ	NOUN
ejpam-4953	216	19	→	→	PUNCT
ejpam-4953	217	1	ḧ	ḧ	NOUN
ejpam-4953	217	2	defined	define	VERB
ejpam-4953	217	3	by	by	ADP
ejpam-4953	217	4	f̈(ϖ	f̈(ϖ	NUM
ejpam-4953	217	5	)	)	PUNCT
ejpam-4953	217	6	=	=	PUNCT
ejpam-4953	218	1	ϖ	ϖ	X
ejpam-4953	218	2	2	2	NUM
ejpam-4953	218	3	3	3	NUM
ejpam-4953	218	4	obviously	obviously	ADV
ejpam-4953	218	5	,	,	PUNCT
ejpam-4953	218	6	r̈	r̈	VERB
ejpam-4953	218	7	is	be	AUX
ejpam-4953	218	8	f̈	f̈	NUM
ejpam-4953	218	9	closed	close	VERB
ejpam-4953	218	10	and	and	CCONJ
ejpam-4953	218	11	f̈	f̈	PRON
ejpam-4953	218	12	is	be	AUX
ejpam-4953	218	13	continuous	continuous	ADJ
ejpam-4953	218	14	.	.	PUNCT
ejpam-4953	219	1	now	now	ADV
ejpam-4953	219	2	,	,	PUNCT
ejpam-4953	219	3	for	for	ADP
ejpam-4953	219	4	ϖ	ϖ	PROPN
ejpam-4953	219	5	,	,	PUNCT
ejpam-4953	219	6	ρ	ρ	PROPN
ejpam-4953	219	7	∈	∈	PROPN
ejpam-4953	219	8	ḧ	ḧ	NOUN
ejpam-4953	219	9	with	with	ADP
ejpam-4953	219	10	(	(	PUNCT
ejpam-4953	219	11	ϖ	ϖ	NOUN
ejpam-4953	219	12	,	,	PUNCT
ejpam-4953	219	13	ρ	ρ	NOUN
ejpam-4953	219	14	)	)	PUNCT
ejpam-4953	219	15	∈	∈	PROPN
ejpam-4953	219	16	r+	r+	NOUN
ejpam-4953	219	17	.	.	PUNCT
ejpam-4953	220	1	we	we	PRON
ejpam-4953	220	2	have	have	VERB
ejpam-4953	220	3	p(f̈ϖ	p(f̈ϖ	NUM
ejpam-4953	220	4	,	,	PUNCT
ejpam-4953	220	5	f̈ρ	f̈ρ	ADJ
ejpam-4953	220	6	)	)	PUNCT
ejpam-4953	220	7	=	=	PUNCT
ejpam-4953	220	8	∣∣∣∣ϖ	∣∣∣∣ϖ	NOUN
ejpam-4953	220	9	2	2	NUM
ejpam-4953	220	10	3	3	NUM
ejpam-4953	220	11	ρ	ρ	NUM
ejpam-4953	220	12	2	2	NUM
ejpam-4953	220	13	3	3	NUM
ejpam-4953	220	14	∣∣∣∣	∣∣∣∣	NOUN
ejpam-4953	220	15	=	=	SYM
ejpam-4953	220	16	∣∣∣∣ϖρ	∣∣∣∣ϖρ	PROPN
ejpam-4953	220	17	∣∣∣∣	∣∣∣∣	PROPN
ejpam-4953	220	18	23	23	NUM
ejpam-4953	220	19	=	=	SYM
ejpam-4953	220	20	p(ϖ	p(ϖ	PROPN
ejpam-4953	220	21	,	,	PUNCT
ejpam-4953	220	22	ρ	ρ	NOUN
ejpam-4953	220	23	)	)	PUNCT
ejpam-4953	220	24	2	2	NUM
ejpam-4953	220	25	3	3	NUM
ejpam-4953	220	26	<	<	X
ejpam-4953	220	27	p(ϖ	p(ϖ	PROPN
ejpam-4953	220	28	,	,	PUNCT
ejpam-4953	220	29	ρ	ρ	NOUN
ejpam-4953	220	30	)	)	PUNCT
ejpam-4953	220	31	3	3	NUM
ejpam-4953	220	32	4	4	NUM
ejpam-4953	220	33	i.e.	i.e.	X
ejpam-4953	220	34	,	,	PUNCT
ejpam-4953	220	35	f̈	f̈	PROPN
ejpam-4953	220	36	satisfies	satisfy	VERB
ejpam-4953	220	37	assumption	assumption	NOUN
ejpam-4953	220	38	(	(	PUNCT
ejpam-4953	220	39	iv	iv	X
ejpam-4953	220	40	)	)	PUNCT
ejpam-4953	220	41	of	of	ADP
ejpam-4953	220	42	theorem	theorem	NOUN
ejpam-4953	220	43	(	(	PUNCT
ejpam-4953	220	44	2.1	2.1	NUM
ejpam-4953	220	45	)	)	PUNCT
ejpam-4953	220	46	for	for	ADP
ejpam-4953	220	47	λ	λ	NOUN
ejpam-4953	220	48	=	=	NOUN
ejpam-4953	220	49	3	3	NUM
ejpam-4953	220	50	4	4	NUM
ejpam-4953	220	51	.	.	PUNCT
ejpam-4953	221	1	consequently	consequently	ADV
ejpam-4953	221	2	,	,	PUNCT
ejpam-4953	221	3	every	every	DET
ejpam-4953	221	4	conditions	condition	NOUN
ejpam-4953	221	5	(	(	PUNCT
ejpam-4953	221	6	i)-(iv	i)-(iv	X
ejpam-4953	221	7	)	)	PUNCT
ejpam-4953	221	8	of	of	ADP
ejpam-4953	221	9	theorem	theorem	NOUN
ejpam-4953	221	10	(	(	PUNCT
ejpam-4953	221	11	2.1	2.1	NUM
ejpam-4953	221	12	)	)	PUNCT
ejpam-4953	221	13	also	also	ADV
ejpam-4953	221	14	holds	hold	VERB
ejpam-4953	221	15	and	and	CCONJ
ejpam-4953	221	16	therefore	therefore	ADV
ejpam-4953	221	17	,	,	PUNCT
ejpam-4953	221	18	f̈	f̈	PROPN
ejpam-4953	221	19	has	have	VERB
ejpam-4953	221	20	a	a	DET
ejpam-4953	221	21	unique	unique	ADJ
ejpam-4953	221	22	fixed	fix	VERB
ejpam-4953	221	23	point	point	NOUN
ejpam-4953	221	24	(	(	PUNCT
ejpam-4953	221	25	for	for	ADP
ejpam-4953	221	26	ϖ	ϖ	NOUN
ejpam-4953	221	27	=	=	SYM
ejpam-4953	221	28	1	1	NUM
ejpam-4953	221	29	)	)	PUNCT
ejpam-4953	221	30	.	.	PUNCT
ejpam-4953	222	1	example	example	NOUN
ejpam-4953	223	1	2	2	NUM
ejpam-4953	223	2	.	.	PUNCT
ejpam-4953	223	3	let	let	VERB
ejpam-4953	223	4	ḧ	ḧ	NOUN
ejpam-4953	223	5	=	=	PUNCT
ejpam-4953	224	1	[	[	X
ejpam-4953	224	2	0.1	0.1	NUM
ejpam-4953	224	3	,	,	PUNCT
ejpam-4953	224	4	1	1	NUM
ejpam-4953	224	5	]	]	PUNCT
ejpam-4953	224	6	and	and	CCONJ
ejpam-4953	224	7	p	p	NOUN
ejpam-4953	224	8	=	=	NOUN
ejpam-4953	225	1	|ϖρ	|ϖρ	X
ejpam-4953	225	2	|	|	ADV
ejpam-4953	225	3	,	,	PUNCT
ejpam-4953	225	4	then	then	ADV
ejpam-4953	225	5	(	(	PUNCT
ejpam-4953	225	6	ḧ,p	ḧ,p	PROPN
ejpam-4953	225	7	)	)	PUNCT
ejpam-4953	225	8	is	be	AUX
ejpam-4953	225	9	complete	complete	ADJ
ejpam-4953	225	10	b	b	X
ejpam-4953	225	11	-	-	PUNCT
ejpam-4953	225	12	mms	mms	ADJ
ejpam-4953	225	13	.	.	PUNCT
ejpam-4953	226	1	define	define	VERB
ejpam-4953	226	2	binary	binary	ADJ
ejpam-4953	226	3	relation	relation	NOUN
ejpam-4953	226	4	r̈	r̈	VERB
ejpam-4953	226	5	=	=	PRON
ejpam-4953	226	6	{	{	PUNCT
ejpam-4953	226	7	(	(	PUNCT
ejpam-4953	226	8	ϖ	ϖ	NOUN
ejpam-4953	226	9	,	,	PUNCT
ejpam-4953	226	10	ρ	ρ	NOUN
ejpam-4953	226	11	)	)	PUNCT
ejpam-4953	226	12	∈	∈	PROPN
ejpam-4953	227	1	[	[	X
ejpam-4953	227	2	0.1	0.1	NUM
ejpam-4953	227	3	,	,	PUNCT
ejpam-4953	227	4	1]2	1]2	NUM
ejpam-4953	227	5	:	:	PUNCT
ejpam-4953	227	6	ϖ	ϖ	X
ejpam-4953	227	7	ρ	ρ	X
ejpam-4953	227	8	≥	≥	NOUN
ejpam-4953	227	9	1	1	NUM
ejpam-4953	227	10	,	,	PUNCT
ejpam-4953	227	11	ϖ	ϖ	PROPN
ejpam-4953	227	12	,	,	PUNCT
ejpam-4953	227	13	ρ	ρ	PROPN
ejpam-4953	227	14	∈	∈	PROPN
ejpam-4953	227	15	r+	r+	PUNCT
ejpam-4953	227	16	}	}	PUNCT
ejpam-4953	227	17	on	on	ADP
ejpam-4953	227	18	ḧ.	ḧ.	PROPN
ejpam-4953	227	19	consider	consider	VERB
ejpam-4953	227	20	mapping	mapping	NOUN
ejpam-4953	227	21	f̈	f̈	X
ejpam-4953	227	22	:	:	PUNCT
ejpam-4953	227	23	ḧ	ḧ	X
ejpam-4953	227	24	→	→	SYM
ejpam-4953	227	25	ḧ	ḧ	NOUN
ejpam-4953	227	26	defined	define	VERB
ejpam-4953	227	27	by	by	ADP
ejpam-4953	227	28	f̈(ϖ	f̈(ϖ	NUM
ejpam-4953	227	29	)	)	PUNCT
ejpam-4953	227	30	=	=	SYM
ejpam-4953	227	31	eϖ−1−ϖ3	eϖ−1−ϖ3	NOUN
ejpam-4953	227	32	10	10	NUM
ejpam-4953	227	33	obviously	obviously	ADV
ejpam-4953	227	34	,	,	PUNCT
ejpam-4953	227	35	r̈	r̈	VERB
ejpam-4953	227	36	is	be	AUX
ejpam-4953	227	37	f̈-closed	f̈-close	VERB
ejpam-4953	227	38	and	and	CCONJ
ejpam-4953	227	39	f̈	f̈	PROPN
ejpam-4953	227	40	is	be	AUX
ejpam-4953	227	41	continuous	continuous	ADJ
ejpam-4953	227	42	.	.	PUNCT
ejpam-4953	228	1	now	now	ADV
ejpam-4953	228	2	,	,	PUNCT
ejpam-4953	228	3	for	for	ADP
ejpam-4953	228	4	ϖ	ϖ	X
ejpam-4953	228	5	,	,	PUNCT
ejpam-4953	228	6	ρ	ρ	PROPN
ejpam-4953	228	7	∈	∈	PROPN
ejpam-4953	229	1	[	[	X
ejpam-4953	229	2	0.1	0.1	NUM
ejpam-4953	229	3	,	,	PUNCT
ejpam-4953	229	4	1	1	NUM
ejpam-4953	229	5	]	]	PUNCT
ejpam-4953	229	6	.	.	PUNCT
ejpam-4953	230	1	we	we	PRON
ejpam-4953	230	2	have	have	VERB
ejpam-4953	230	3	p(f̈ϖ	p(f̈ϖ	NUM
ejpam-4953	230	4	,	,	PUNCT
ejpam-4953	230	5	f̈ρ	f̈ρ	ADJ
ejpam-4953	230	6	)	)	PUNCT
ejpam-4953	230	7	=	=	SYM
ejpam-4953	230	8	∣∣∣∣	∣∣∣∣	NOUN
ejpam-4953	230	9	f̈ϖf̈ρ	f̈ϖf̈ρ	NOUN
ejpam-4953	230	10	∣∣∣∣	∣∣∣∣	NOUN
ejpam-4953	230	11	≤	≤	ADJ
ejpam-4953	230	12	∣∣∣∣ϖρ	∣∣∣∣ϖρ	NOUN
ejpam-4953	230	13	∣∣∣∣λ	∣∣∣∣λ	NOUN
ejpam-4953	230	14	=	=	PUNCT
ejpam-4953	231	1	p(ϖ	p(ϖ	PROPN
ejpam-4953	231	2	,	,	PUNCT
ejpam-4953	231	3	ρ)λ	ρ)λ	ADJ
ejpam-4953	231	4	for	for	ADP
ejpam-4953	231	5	all	all	DET
ejpam-4953	231	6	ϖ	ϖ	PROPN
ejpam-4953	231	7	,	,	PUNCT
ejpam-4953	231	8	ρ	ρ	PROPN
ejpam-4953	231	9	∈	∈	PROPN
ejpam-4953	231	10	x	x	SYM
ejpam-4953	231	11	where	where	SCONJ
ejpam-4953	231	12	,	,	PUNCT
ejpam-4953	231	13	λ	λ	X
ejpam-4953	231	14	=	=	SYM
ejpam-4953	231	15	0.997	0.997	NUM
ejpam-4953	231	16	,	,	PUNCT
ejpam-4953	231	17	finally	finally	ADV
ejpam-4953	231	18	,	,	PUNCT
ejpam-4953	231	19	we	we	PRON
ejpam-4953	231	20	can	can	AUX
ejpam-4953	231	21	say	say	VERB
ejpam-4953	231	22	that	that	SCONJ
ejpam-4953	231	23	f̈	f̈	PROPN
ejpam-4953	231	24	has	have	VERB
ejpam-4953	231	25	a	a	DET
ejpam-4953	231	26	unique	unique	ADJ
ejpam-4953	231	27	fixed	fix	VERB
ejpam-4953	231	28	point	point	NOUN
ejpam-4953	231	29	0.7411317711	0.7411317711	NUM
ejpam-4953	231	30	∈	∈	NOUN
ejpam-4953	231	31	x.	x.	NOUN
ejpam-4953	231	32	references	reference	NOUN
ejpam-4953	231	33	2416	2416	NUM
ejpam-4953	231	34	example	example	NOUN
ejpam-4953	231	35	3	3	NUM
ejpam-4953	231	36	.	.	PUNCT
ejpam-4953	231	37	let	let	VERB
ejpam-4953	231	38	ḧ	ḧ	NOUN
ejpam-4953	231	39	=	=	PUNCT
ejpam-4953	232	1	[	[	X
ejpam-4953	232	2	1	1	NUM
ejpam-4953	232	3	,	,	PUNCT
ejpam-4953	232	4	3	3	NUM
ejpam-4953	232	5	]	]	PUNCT
ejpam-4953	232	6	and	and	CCONJ
ejpam-4953	232	7	p	p	NOUN
ejpam-4953	232	8	=	=	NOUN
ejpam-4953	233	1	|ϖρ	|ϖρ	X
ejpam-4953	233	2	|	|	ADV
ejpam-4953	233	3	,	,	PUNCT
ejpam-4953	233	4	then	then	ADV
ejpam-4953	233	5	(	(	PUNCT
ejpam-4953	233	6	ḧ	ḧ	NOUN
ejpam-4953	233	7	,	,	PUNCT
ejpam-4953	233	8	p	p	NOUN
ejpam-4953	233	9	)	)	PUNCT
ejpam-4953	233	10	is	be	AUX
ejpam-4953	233	11	complete	complete	ADJ
ejpam-4953	233	12	b	b	X
ejpam-4953	233	13	-	-	PUNCT
ejpam-4953	233	14	mms	mms	ADJ
ejpam-4953	233	15	.	.	PUNCT
ejpam-4953	234	1	define	define	VERB
ejpam-4953	234	2	binary	binary	ADJ
ejpam-4953	234	3	relation	relation	NOUN
ejpam-4953	234	4	r̈={(1,1),(2,1),(2,2),(3,1),(3,2	r̈={(1,1),(2,1),(2,2),(3,1),(3,2	PROPN
ejpam-4953	234	5	)	)	PUNCT
ejpam-4953	234	6	}	}	PUNCT
ejpam-4953	234	7	on	on	ADP
ejpam-4953	234	8	ḧ	ḧ	NOUN
ejpam-4953	234	9	and	and	CCONJ
ejpam-4953	234	10	a	a	DET
ejpam-4953	234	11	mapping	mapping	NOUN
ejpam-4953	234	12	f̈:ḧ	f̈:ḧ	NOUN
ejpam-4953	234	13	→	→	PUNCT
ejpam-4953	235	1	ḧ	ḧ	NOUN
ejpam-4953	235	2	defined	define	VERB
ejpam-4953	235	3	by	by	ADP
ejpam-4953	235	4	f̈(ϖ	f̈(ϖ	NUM
ejpam-4953	235	5	)	)	PUNCT
ejpam-4953	235	6	=	=	NOUN
ejpam-4953	235	7	{	{	PUNCT
ejpam-4953	235	8	1	1	NUM
ejpam-4953	235	9	,	,	PUNCT
ejpam-4953	235	10	if	if	SCONJ
ejpam-4953	235	11	1	1	NUM
ejpam-4953	235	12	≤	≤	NUM
ejpam-4953	235	13	ϖ	ϖ	NOUN
ejpam-4953	235	14	≤	≤	NUM
ejpam-4953	235	15	2	2	NUM
ejpam-4953	235	16	2	2	NUM
ejpam-4953	235	17	if	if	SCONJ
ejpam-4953	235	18	2	2	NUM
ejpam-4953	235	19	<	<	X
ejpam-4953	235	20	ϖ	ϖ	X
ejpam-4953	235	21	≤	≤	NUM
ejpam-4953	235	22	3	3	NUM
ejpam-4953	235	23	,	,	PUNCT
ejpam-4953	235	24	obviously	obviously	ADV
ejpam-4953	235	25	,	,	PUNCT
ejpam-4953	235	26	r̈	r̈	VERB
ejpam-4953	235	27	is	be	AUX
ejpam-4953	235	28	f̈	f̈	NUM
ejpam-4953	235	29	closed	close	VERB
ejpam-4953	235	30	but	but	CCONJ
ejpam-4953	235	31	f̈	f̈	NUM
ejpam-4953	235	32	is	be	AUX
ejpam-4953	235	33	not	not	PART
ejpam-4953	235	34	continuous	continuous	ADJ
ejpam-4953	235	35	.	.	PUNCT
ejpam-4953	236	1	take	take	VERB
ejpam-4953	236	2	an	an	DET
ejpam-4953	236	3	r̈-preserving	r̈-preserve	VERB
ejpam-4953	236	4	sequence	sequence	NOUN
ejpam-4953	236	5	{	{	PUNCT
ejpam-4953	236	6	ϖn	ϖn	NOUN
ejpam-4953	236	7	}	}	PUNCT
ejpam-4953	236	8	such	such	ADJ
ejpam-4953	236	9	that	that	DET
ejpam-4953	236	10	ϖn	ϖn	NOUN
ejpam-4953	236	11	p−→	p−→	NOUN
ejpam-4953	237	1	ϖ	ϖ	INTJ
ejpam-4953	238	1	so	so	ADV
ejpam-4953	238	2	that	that	SCONJ
ejpam-4953	238	3	(	(	PUNCT
ejpam-4953	238	4	ϖn	ϖn	NOUN
ejpam-4953	238	5	,	,	PUNCT
ejpam-4953	238	6	ϖn+1	ϖn+1	ADJ
ejpam-4953	238	7	)	)	PUNCT
ejpam-4953	238	8	∈	∈	PROPN
ejpam-4953	238	9	r̈	r̈	VERB
ejpam-4953	238	10	for	for	ADP
ejpam-4953	238	11	all	all	PRON
ejpam-4953	238	12	n	n	DET
ejpam-4953	238	13	∈	∈	PROPN
ejpam-4953	238	14	n.	n.	NOUN
ejpam-4953	238	15	here	here	ADV
ejpam-4953	238	16	,	,	PUNCT
ejpam-4953	238	17	one	one	PRON
ejpam-4953	238	18	can	can	AUX
ejpam-4953	238	19	observe	observe	VERB
ejpam-4953	238	20	that	that	SCONJ
ejpam-4953	238	21	(	(	PUNCT
ejpam-4953	238	22	ϖn	ϖn	NOUN
ejpam-4953	238	23	,	,	PUNCT
ejpam-4953	238	24	ϖn+1	ϖn+1	ADJ
ejpam-4953	238	25	)	)	PUNCT
ejpam-4953	238	26	/∈	/∈	PUNCT
ejpam-4953	239	1	{	{	PUNCT
ejpam-4953	239	2	(	(	PUNCT
ejpam-4953	239	3	3	3	NUM
ejpam-4953	239	4	,	,	PUNCT
ejpam-4953	239	5	1	1	NUM
ejpam-4953	239	6	)	)	PUNCT
ejpam-4953	239	7	,	,	PUNCT
ejpam-4953	239	8	(	(	PUNCT
ejpam-4953	239	9	3	3	NUM
ejpam-4953	239	10	,	,	PUNCT
ejpam-4953	239	11	2	2	NUM
ejpam-4953	239	12	)	)	PUNCT
ejpam-4953	239	13	}	}	PUNCT
ejpam-4953	239	14	so	so	SCONJ
ejpam-4953	239	15	that	that	SCONJ
ejpam-4953	239	16	(	(	PUNCT
ejpam-4953	239	17	ϖn	ϖn	NOUN
ejpam-4953	239	18	,	,	PUNCT
ejpam-4953	239	19	ϖn+1	ϖn+1	ADJ
ejpam-4953	239	20	)	)	PUNCT
ejpam-4953	239	21	∈	∈	NOUN
ejpam-4953	239	22	{	{	PUNCT
ejpam-4953	239	23	(	(	PUNCT
ejpam-4953	239	24	1	1	NUM
ejpam-4953	239	25	,	,	PUNCT
ejpam-4953	239	26	1	1	NUM
ejpam-4953	239	27	)	)	PUNCT
ejpam-4953	239	28	,	,	PUNCT
ejpam-4953	239	29	(	(	PUNCT
ejpam-4953	239	30	2	2	NUM
ejpam-4953	239	31	,	,	PUNCT
ejpam-4953	239	32	1	1	NUM
ejpam-4953	239	33	)	)	PUNCT
ejpam-4953	239	34	,	,	PUNCT
ejpam-4953	239	35	(	(	PUNCT
ejpam-4953	239	36	2	2	NUM
ejpam-4953	239	37	,	,	PUNCT
ejpam-4953	239	38	2	2	NUM
ejpam-4953	239	39	)	)	PUNCT
ejpam-4953	239	40	}	}	PUNCT
ejpam-4953	239	41	which	which	PRON
ejpam-4953	239	42	gives	give	VERB
ejpam-4953	239	43	rise	rise	NOUN
ejpam-4953	239	44	to	to	ADP
ejpam-4953	239	45	{	{	PUNCT
ejpam-4953	239	46	ϖn	ϖn	NOUN
ejpam-4953	239	47	}	}	PUNCT
ejpam-4953	239	48	⊂	⊂	PRON
ejpam-4953	239	49	{	{	PUNCT
ejpam-4953	239	50	1	1	NUM
ejpam-4953	239	51	,	,	PUNCT
ejpam-4953	239	52	2}.{1	2}.{1	NUM
ejpam-4953	239	53	,	,	PUNCT
ejpam-4953	239	54	2	2	NUM
ejpam-4953	239	55	}	}	PUNCT
ejpam-4953	239	56	is	be	AUX
ejpam-4953	239	57	closed	closed	ADJ
ejpam-4953	239	58	,	,	PUNCT
ejpam-4953	239	59	we	we	PRON
ejpam-4953	239	60	have	have	VERB
ejpam-4953	239	61	[	[	X
ejpam-4953	239	62	ϖn	ϖn	NOUN
ejpam-4953	239	63	,	,	PUNCT
ejpam-4953	239	64	ϖ	ϖ	X
ejpam-4953	239	65	]	]	X
ejpam-4953	239	66	∈	∈	PROPN
ejpam-4953	240	1	r̈.	r̈.	PROPN
ejpam-4953	240	2	therefore	therefore	ADV
ejpam-4953	240	3	,	,	PUNCT
ejpam-4953	240	4	r̈	r̈	VERB
ejpam-4953	240	5	is	be	AUX
ejpam-4953	240	6	p	p	NOUN
ejpam-4953	240	7	-	-	PUNCT
ejpam-4953	240	8	closed	closed	ADJ
ejpam-4953	240	9	.	.	PUNCT
ejpam-4953	241	1	assumption	assumption	NOUN
ejpam-4953	241	2	can	can	AUX
ejpam-4953	241	3	be	be	AUX
ejpam-4953	241	4	verified	verify	VERB
ejpam-4953	241	5	(	(	PUNCT
ejpam-4953	241	6	iv	iv	X
ejpam-4953	241	7	)	)	PUNCT
ejpam-4953	241	8	of	of	ADP
ejpam-4953	241	9	theorem	theorem	ADJ
ejpam-4953	241	10	2.1	2.1	NUM
ejpam-4953	241	11	with	with	ADP
ejpam-4953	241	12	λ	λ	NOUN
ejpam-4953	241	13	=	=	NOUN
ejpam-4953	241	14	1	1	NUM
ejpam-4953	241	15	2	2	NUM
ejpam-4953	241	16	.	.	PUNCT
ejpam-4953	242	1	thus	thus	ADV
ejpam-4953	242	2	,	,	PUNCT
ejpam-4953	242	3	all	all	DET
ejpam-4953	242	4	the	the	DET
ejpam-4953	242	5	condition	condition	NOUN
ejpam-4953	242	6	(	(	PUNCT
ejpam-4953	242	7	i)(iv	i)(iv	NUM
ejpam-4953	242	8	)	)	PUNCT
ejpam-4953	242	9	of	of	ADP
ejpam-4953	242	10	theorem	theorem	ADJ
ejpam-4953	242	11	2.1	2.1	NUM
ejpam-4953	242	12	are	be	AUX
ejpam-4953	242	13	satisfies	satisfie	NOUN
ejpam-4953	242	14	and	and	CCONJ
ejpam-4953	242	15	f̈	f̈	PRON
ejpam-4953	242	16	has	have	VERB
ejpam-4953	242	17	a	a	DET
ejpam-4953	242	18	fixed	fix	VERB
ejpam-4953	242	19	point	point	NOUN
ejpam-4953	242	20	in	in	ADP
ejpam-4953	242	21	ḧ	ḧ	NOUN
ejpam-4953	242	22	(	(	PUNCT
ejpam-4953	242	23	for	for	ADP
ejpam-4953	242	24	ϖ=1	ϖ=1	PROPN
ejpam-4953	242	25	)	)	PUNCT
ejpam-4953	242	26	.	.	PUNCT
ejpam-4953	243	1	references	reference	NOUN
ejpam-4953	243	2	[	[	X
ejpam-4953	243	3	1	1	X
ejpam-4953	243	4	]	]	PUNCT
ejpam-4953	243	5	s.	s.	PROPN
ejpam-4953	243	6	banach	banach	PROPN
ejpam-4953	243	7	,	,	PUNCT
ejpam-4953	243	8	sure	sure	SCONJ
ejpam-4953	243	9	operations	operation	NOUN
ejpam-4953	243	10	dans	dan	NOUN
ejpam-4953	243	11	les	le	NOUN
ejpam-4953	243	12	ensembles	ensemble	NOUN
ejpam-4953	243	13	abstraits	abstrait	NOUN
ejpam-4953	243	14	et	et	PROPN
ejpam-4953	243	15	leur	leur	X
ejpam-4953	243	16	application	application	PROPN
ejpam-4953	243	17	aux	aux	PROPN
ejpam-4953	243	18	equations	equations	PROPN
ejpam-4953	243	19	integrals	integral	NOUN
ejpam-4953	243	20	,	,	PUNCT
ejpam-4953	243	21	fund	fund	NOUN
ejpam-4953	243	22	.	.	PUNCT
ejpam-4953	244	1	maths	math	NOUN
ejpam-4953	244	2	.	.	PUNCT
ejpam-4953	245	1	3(1922	3(1922	NUM
ejpam-4953	245	2	)	)	PUNCT
ejpam-4953	245	3	,	,	PUNCT
ejpam-4953	245	4	133	133	NUM
ejpam-4953	245	5	-	-	SYM
ejpam-4953	245	6	181	181	NUM
ejpam-4953	245	7	.	.	PUNCT
ejpam-4953	246	1	[	[	X
ejpam-4953	246	2	2	2	NUM
ejpam-4953	246	3	]	]	X
ejpam-4953	246	4	d.w.boyd	d.w.boyd	NOUN
ejpam-4953	246	5	,	,	PUNCT
ejpam-4953	246	6	j.s.wong	j.s.wong	PROPN
ejpam-4953	246	7	,	,	PUNCT
ejpam-4953	246	8	on	on	ADP
ejpam-4953	246	9	nonlinear	nonlinear	ADJ
ejpam-4953	246	10	contractions	contraction	NOUN
ejpam-4953	246	11	,	,	PUNCT
ejpam-4953	246	12	proc.am.soc	proc.am.soc	NOUN
ejpam-4953	246	13	.	.	NOUN
ejpam-4953	246	14	20(1969	20(1969	NUM
ejpam-4953	246	15	)	)	PUNCT
ejpam-4953	246	16	,	,	PUNCT
ejpam-4953	246	17	458	458	NUM
ejpam-4953	246	18	-	-	SYM
ejpam-4953	246	19	464	464	NUM
ejpam-4953	246	20	.	.	PUNCT
ejpam-4953	247	1	[	[	X
ejpam-4953	247	2	3	3	NUM
ejpam-4953	247	3	]	]	PUNCT
ejpam-4953	247	4	a.	a.	NOUN
ejpam-4953	247	5	stouti	stouti	PROPN
ejpam-4953	247	6	and	and	CCONJ
ejpam-4953	247	7	a.	a.	NOUN
ejpam-4953	247	8	maaden	maaden	PROPN
ejpam-4953	247	9	,	,	PUNCT
ejpam-4953	247	10	fixed	fix	VERB
ejpam-4953	247	11	points	point	NOUN
ejpam-4953	247	12	and	and	CCONJ
ejpam-4953	247	13	common	common	ADJ
ejpam-4953	247	14	fixed	fix	VERB
ejpam-4953	247	15	points	point	NOUN
ejpam-4953	247	16	theorems	theorem	NOUN
ejpam-4953	247	17	in	in	ADP
ejpam-4953	247	18	pseudoordered	pseudoordere	VERB
ejpam-4953	247	19	sets.proyecciones	sets.proyeccione	NOUN
ejpam-4953	247	20	32	32	NUM
ejpam-4953	247	21	(	(	PUNCT
ejpam-4953	247	22	2013	2013	NUM
ejpam-4953	247	23	)	)	PUNCT
ejpam-4953	247	24	,	,	PUNCT
ejpam-4953	247	25	409	409	NUM
ejpam-4953	247	26	-	-	SYM
ejpam-4953	247	27	418	418	NUM
ejpam-4953	247	28	.	.	PUNCT
ejpam-4953	248	1	[	[	X
ejpam-4953	248	2	4	4	X
ejpam-4953	248	3	]	]	PUNCT
ejpam-4953	248	4	m.	m.	NOUN
ejpam-4953	248	5	özaşvar	özaşvar	PROPN
ejpam-4953	248	6	,	,	PUNCT
ejpam-4953	248	7	a.c	a.c	PROPN
ejpam-4953	248	8	.	.	PROPN
ejpam-4953	248	9	çevikel	çevikel	PROPN
ejpam-4953	248	10	,	,	PUNCT
ejpam-4953	248	11	fixed	fix	VERB
ejpam-4953	248	12	points	point	NOUN
ejpam-4953	248	13	of	of	ADP
ejpam-4953	248	14	multiplicative	multiplicative	ADJ
ejpam-4953	248	15	contraction	contraction	NOUN
ejpam-4953	248	16	mappings	mapping	NOUN
ejpam-4953	248	17	on	on	ADP
ejpam-4953	248	18	multiplicative	multiplicative	ADJ
ejpam-4953	248	19	metric	metric	ADJ
ejpam-4953	248	20	spaces	space	NOUN
ejpam-4953	248	21	,	,	PUNCT
ejpam-4953	248	22	arxiv	arxiv	NOUN
ejpam-4953	248	23	:	:	PUNCT
ejpam-4953	248	24	1205.5131v1	1205.5131v1	NUM
ejpam-4953	249	1	[	[	X
ejpam-4953	249	2	matn.gn](2012	matn.gn](2012	NOUN
ejpam-4953	249	3	)	)	PUNCT
ejpam-4953	249	4	.	.	PUNCT
ejpam-4953	250	1	[	[	X
ejpam-4953	250	2	5	5	NUM
ejpam-4953	250	3	]	]	PUNCT
ejpam-4953	250	4	m.	m.	NOUN
ejpam-4953	250	5	turinici	turinici	PROPN
ejpam-4953	250	6	,	,	PUNCT
ejpam-4953	250	7	abstract	abstract	ADJ
ejpam-4953	250	8	comparison	comparison	NOUN
ejpam-4953	250	9	principles	principle	NOUN
ejpam-4953	250	10	and	and	CCONJ
ejpam-4953	250	11	multivariable	multivariable	ADJ
ejpam-4953	250	12	gronwall	gronwall	ADJ
ejpam-4953	250	13	-	-	PUNCT
ejpam-4953	250	14	bellman	bellman	NOUN
ejpam-4953	250	15	inequalities	inequality	NOUN
ejpam-4953	250	16	,	,	PUNCT
ejpam-4953	250	17	j.	j.	PROPN
ejpam-4953	250	18	math	math	PROPN
ejpam-4953	250	19	.	.	PUNCT
ejpam-4953	251	1	anal	anal	PROPN
ejpam-4953	251	2	.	.	PUNCT
ejpam-4953	252	1	appl	appl	PROPN
ejpam-4953	252	2	.	.	PROPN
ejpam-4953	253	1	117	117	NUM
ejpam-4953	253	2	(	(	PUNCT
ejpam-4953	253	3	1	1	NUM
ejpam-4953	253	4	)	)	PUNCT
ejpam-4953	253	5	(	(	PUNCT
ejpam-4953	253	6	1986	1986	NUM
ejpam-4953	253	7	)	)	PUNCT
ejpam-4953	253	8	,	,	PUNCT
ejpam-4953	253	9	100	100	NUM
ejpam-4953	253	10	-	-	SYM
ejpam-4953	253	11	127	127	NUM
ejpam-4953	253	12	.	.	PUNCT
ejpam-4953	254	1	[	[	X
ejpam-4953	254	2	6	6	NUM
ejpam-4953	254	3	]	]	X
ejpam-4953	254	4	a.c.m	a.c.m	X
ejpam-4953	254	5	.	.	PROPN
ejpam-4953	254	6	ran	ran	NOUN
ejpam-4953	254	7	,	,	PUNCT
ejpam-4953	254	8	m.c.b	m.c.b	NOUN
ejpam-4953	254	9	.	.	PUNCT
ejpam-4953	255	1	reurings	reuring	NOUN
ejpam-4953	255	2	,	,	PUNCT
ejpam-4953	255	3	a	a	DET
ejpam-4953	255	4	fixed	fix	VERB
ejpam-4953	255	5	point	point	NOUN
ejpam-4953	255	6	theorem	theorem	VERB
ejpam-4953	255	7	in	in	ADP
ejpam-4953	255	8	partial	partial	ADJ
ejpam-4953	255	9	ordered	order	VERB
ejpam-4953	255	10	sets	set	NOUN
ejpam-4953	255	11	and	and	CCONJ
ejpam-4953	255	12	some	some	DET
ejpam-4953	255	13	applications	application	NOUN
ejpam-4953	255	14	to	to	PART
ejpam-4953	255	15	matrix	matrix	VERB
ejpam-4953	255	16	equations	equation	NOUN
ejpam-4953	255	17	,	,	PUNCT
ejpam-4953	255	18	proc	proc	NOUN
ejpam-4953	255	19	.	.	PUNCT
ejpam-4953	255	20	am	be	AUX
ejpam-4953	255	21	.	.	PUNCT
ejpam-4953	256	1	math	math	NOUN
ejpam-4953	256	2	.	.	PUNCT
ejpam-4953	257	1	soc	soc	PROPN
ejpam-4953	257	2	.	.	PUNCT
ejpam-4953	258	1	132(5	132(5	NUM
ejpam-4953	258	2	)	)	PUNCT
ejpam-4953	258	3	(	(	PUNCT
ejpam-4953	258	4	2004	2004	NUM
ejpam-4953	258	5	)	)	PUNCT
ejpam-4953	258	6	,	,	PUNCT
ejpam-4953	258	7	1435	1435	NUM
ejpam-4953	258	8	-	-	SYM
ejpam-4953	258	9	1443	1443	NUM
ejpam-4953	258	10	.	.	PUNCT
ejpam-4953	259	1	[	[	X
ejpam-4953	259	2	7	7	X
ejpam-4953	259	3	]	]	X
ejpam-4953	259	4	h.	h.	PROPN
ejpam-4953	259	5	aydi	aydi	PROPN
ejpam-4953	259	6	,	,	PUNCT
ejpam-4953	259	7	e.	e.	PROPN
ejpam-4953	259	8	karapinar	karapinar	PROPN
ejpam-4953	259	9	,	,	PUNCT
ejpam-4953	259	10	w.	w.	PROPN
ejpam-4953	259	11	shatanawi	shatanawi	PROPN
ejpam-4953	259	12	,	,	PUNCT
ejpam-4953	259	13	coupled	couple	VERB
ejpam-4953	259	14	coincidence	coincidence	NOUN
ejpam-4953	259	15	points	point	NOUN
ejpam-4953	259	16	in	in	ADP
ejpam-4953	259	17	partially	partially	ADV
ejpam-4953	259	18	ordered	order	VERB
ejpam-4953	259	19	cone	cone	NOUN
ejpam-4953	259	20	metric	metric	ADJ
ejpam-4953	259	21	spaces	space	NOUN
ejpam-4953	259	22	with	with	ADP
ejpam-4953	259	23	a	a	DET
ejpam-4953	259	24	c	c	NOUN
ejpam-4953	259	25	-	-	PUNCT
ejpam-4953	259	26	distance	distance	NOUN
ejpam-4953	259	27	,	,	PUNCT
ejpam-4953	259	28	journal	journal	NOUN
ejpam-4953	259	29	of	of	ADP
ejpam-4953	259	30	applied	apply	VERB
ejpam-4953	259	31	mathematics	mathematic	NOUN
ejpam-4953	259	32	,	,	PUNCT
ejpam-4953	259	33	volume	volume	NOUN
ejpam-4953	259	34	2012	2012	NUM
ejpam-4953	259	35	,	,	PUNCT
ejpam-4953	259	36	article	article	NOUN
ejpam-4953	259	37	i	i	PROPN
ejpam-4953	259	38	d	d	PROPN
ejpam-4953	259	39	312078	312078	NUM
ejpam-4953	259	40	,	,	PUNCT
ejpam-4953	259	41	15	15	NUM
ejpam-4953	259	42	pages	page	NOUN
ejpam-4953	259	43	.	.	PUNCT
ejpam-4953	260	1	references	reference	NOUN
ejpam-4953	260	2	2417	2417	NUM
ejpam-4953	260	3	[	[	X
ejpam-4953	260	4	8	8	NUM
ejpam-4953	260	5	]	]	PUNCT
ejpam-4953	260	6	m.	m.	NOUN
ejpam-4953	260	7	abbas	abbas	PROPN
ejpam-4953	260	8	,	,	PUNCT
ejpam-4953	260	9	b.	b.	PROPN
ejpam-4953	260	10	ali	ali	PROPN
ejpam-4953	260	11	,	,	PUNCT
ejpam-4953	260	12	y.i	y.i	PROPN
ejpam-4953	260	13	.	.	PROPN
ejpam-4953	260	14	suleiman	suleiman	PROPN
ejpam-4953	260	15	,	,	PUNCT
ejpam-4953	260	16	generalized	generalize	VERB
ejpam-4953	260	17	coupled	couple	VERB
ejpam-4953	260	18	common	common	ADJ
ejpam-4953	260	19	fixed	fix	VERB
ejpam-4953	260	20	point	point	NOUN
ejpam-4953	260	21	results	result	NOUN
ejpam-4953	260	22	in	in	ADP
ejpam-4953	260	23	partially	partially	ADV
ejpam-4953	260	24	ordered	order	VERB
ejpam-4953	260	25	a	a	PRON
ejpam-4953	260	26	-	-	PUNCT
ejpam-4953	260	27	metric	metric	ADJ
ejpam-4953	260	28	spaces	space	NOUN
ejpam-4953	260	29	,	,	PUNCT
ejpam-4953	260	30	fixed	fix	VERB
ejpam-4953	260	31	point	point	NOUN
ejpam-4953	260	32	theory	theory	NOUN
ejpam-4953	260	33	appl	appl	PROPN
ejpam-4953	260	34	.	.	PUNCT
ejpam-4953	261	1	(	(	PUNCT
ejpam-4953	261	2	2015	2015	NUM
ejpam-4953	261	3	)	)	PUNCT
ejpam-4953	261	4	2015:64	2015:64	NUM
ejpam-4953	261	5	,	,	PUNCT
ejpam-4953	261	6	24	24	NUM
ejpam-4953	261	7	pages	page	NOUN
ejpam-4953	261	8	.	.	PUNCT
ejpam-4953	262	1	[	[	X
ejpam-4953	262	2	9	9	NUM
ejpam-4953	262	3	]	]	X
ejpam-4953	262	4	n.v	n.v	PROPN
ejpam-4953	262	5	.	.	PROPN
ejpam-4953	262	6	luong	luong	PROPN
ejpam-4953	262	7	,	,	PUNCT
ejpam-4953	262	8	n.x	n.x	PROPN
ejpam-4953	262	9	.	.	PROPN
ejpam-4953	262	10	thuan	thuan	PROPN
ejpam-4953	262	11	,	,	PUNCT
ejpam-4953	262	12	coupled	couple	VERB
ejpam-4953	262	13	fixed	fix	VERB
ejpam-4953	262	14	point	point	NOUN
ejpam-4953	262	15	theorems	theorem	NOUN
ejpam-4953	262	16	in	in	ADP
ejpam-4953	262	17	partially	partially	ADV
ejpam-4953	262	18	ordered	order	VERB
ejpam-4953	262	19	metric	metric	ADJ
ejpam-4953	262	20	spacesd	spacesd	NOUN
ejpam-4953	262	21	epended	epende	VERB
ejpam-4953	262	22	on	on	ADP
ejpam-4953	262	23	another	another	DET
ejpam-4953	262	24	function	function	NOUN
ejpam-4953	262	25	,	,	PUNCT
ejpam-4953	262	26	bull	bull	NOUN
ejpam-4953	262	27	.	.	PUNCT
ejpam-4953	262	28	math	math	NOUN
ejpam-4953	262	29	.	.	PUNCT
ejpam-4953	263	1	anal	anal	ADJ
ejpam-4953	263	2	.	.	PUNCT
ejpam-4953	264	1	3	3	NUM
ejpam-4953	264	2	(	(	PUNCT
ejpam-4953	264	3	2011	2011	NUM
ejpam-4953	264	4	)	)	PUNCT
ejpam-4953	264	5	,	,	PUNCT
ejpam-4953	264	6	129	129	NUM
ejpam-4953	264	7	-	-	SYM
ejpam-4953	264	8	140	140	NUM
ejpam-4953	264	9	.	.	PUNCT
ejpam-4953	265	1	[	[	X
ejpam-4953	265	2	10	10	NUM
ejpam-4953	265	3	]	]	X
ejpam-4953	265	4	a.e	a.e	PROPN
ejpam-4953	265	5	.	.	PROPN
ejpam-4953	265	6	bashirov	bashirov	PROPN
ejpam-4953	265	7	,	,	PUNCT
ejpam-4953	265	8	e.m	e.m	PROPN
ejpam-4953	265	9	.	.	PROPN
ejpam-4953	265	10	kurpinar	kurpinar	PROPN
ejpam-4953	265	11	,	,	PUNCT
ejpam-4953	265	12	a.	a.	PROPN
ejpam-4953	265	13	özyapici	özyapici	PROPN
ejpam-4953	265	14	,	,	PUNCT
ejpam-4953	265	15	multiplicative	multiplicative	ADJ
ejpam-4953	265	16	calculus	calculus	NOUN
ejpam-4953	265	17	and	and	CCONJ
ejpam-4953	265	18	its	its	PRON
ejpam-4953	265	19	applications	application	NOUN
ejpam-4953	265	20	,	,	PUNCT
ejpam-4953	265	21	j.	j.	PROPN
ejpam-4953	265	22	math	math	PROPN
ejpam-4953	265	23	.	.	PUNCT
ejpam-4953	266	1	anal	anal	PROPN
ejpam-4953	266	2	.	.	PUNCT
ejpam-4953	266	3	appl	appl	PROPN
ejpam-4953	266	4	.	.	PUNCT
ejpam-4953	267	1	337(2008	337(2008	NOUN
ejpam-4953	267	2	)	)	PUNCT
ejpam-4953	267	3	,	,	PUNCT
ejpam-4953	267	4	36	36	NUM
ejpam-4953	267	5	-	-	SYM
ejpam-4953	267	6	48	48	NUM
ejpam-4953	267	7	.	.	PUNCT
ejpam-4953	268	1	[	[	X
ejpam-4953	268	2	11	11	NUM
ejpam-4953	268	3	]	]	PUNCT
ejpam-4953	268	4	m.	m.	NOUN
ejpam-4953	268	5	abbas	abbas	PROPN
ejpam-4953	268	6	,	,	PUNCT
ejpam-4953	268	7	b.	b.	PROPN
ejpam-4953	268	8	ali	ali	PROPN
ejpam-4953	268	9	,	,	PUNCT
ejpam-4953	268	10	y.	y.	PROPN
ejpam-4953	268	11	suleiman	suleiman	PROPN
ejpam-4953	268	12	,	,	PUNCT
ejpam-4953	268	13	common	common	ADJ
ejpam-4953	268	14	fixed	fix	VERB
ejpam-4953	268	15	points	point	NOUN
ejpam-4953	268	16	of	of	ADP
ejpam-4953	268	17	locally	locally	ADV
ejpam-4953	268	18	contractive	contractive	ADJ
ejpam-4953	268	19	mappings	mapping	NOUN
ejpam-4953	268	20	in	in	ADP
ejpam-4953	268	21	multiplicative	multiplicative	ADJ
ejpam-4953	268	22	metric	metric	ADJ
ejpam-4953	268	23	spaces	space	NOUN
ejpam-4953	268	24	with	with	ADP
ejpam-4953	268	25	applications	application	NOUN
ejpam-4953	268	26	,	,	PUNCT
ejpam-4953	268	27	int	int	NOUN
ejpam-4953	268	28	.	.	PUNCT
ejpam-4953	269	1	j.	j.	PROPN
ejpam-4953	269	2	math	math	PROPN
ejpam-4953	269	3	.	.	PUNCT
ejpam-4953	270	1	math	math	NOUN
ejpam-4953	270	2	.	.	PUNCT
ejpam-4953	271	1	sci	sci	PROPN
ejpam-4953	271	2	.	.	PROPN
ejpam-4953	271	3	,	,	PUNCT
ejpam-4953	271	4	(	(	PUNCT
ejpam-4953	271	5	2015	2015	NUM
ejpam-4953	271	6	)	)	PUNCT
ejpam-4953	271	7	(	(	PUNCT
ejpam-4953	271	8	2015	2015	NUM
ejpam-4953	271	9	)	)	PUNCT
ejpam-4953	271	10	,	,	PUNCT
ejpam-4953	271	11	article	article	NOUN
ejpam-4953	271	12	i	i	PROPN
ejpam-4953	271	13	d	d	PROPN
ejpam-4953	271	14	218683	218683	NUM
ejpam-4953	271	15	.	.	PUNCT
ejpam-4953	272	1	[	[	X
ejpam-4953	272	2	12	12	NUM
ejpam-4953	272	3	]	]	PUNCT
ejpam-4953	272	4	m.	m.	NOUN
ejpam-4953	272	5	abbas	abbas	PROPN
ejpam-4953	272	6	,	,	PUNCT
ejpam-4953	272	7	m.de	m.de	PROPN
ejpam-4953	272	8	la	la	PROPN
ejpam-4953	272	9	sen	sen	PROPN
ejpam-4953	272	10	,	,	PUNCT
ejpam-4953	272	11	t.	t.	PROPN
ejpam-4953	272	12	nazir	nazir	PROPN
ejpam-4953	272	13	,	,	PUNCT
ejpam-4953	272	14	common	common	ADJ
ejpam-4953	272	15	fixed	fix	VERB
ejpam-4953	272	16	points	point	NOUN
ejpam-4953	272	17	of	of	ADP
ejpam-4953	272	18	generalized	generalized	ADJ
ejpam-4953	272	19	rational	rational	ADJ
ejpam-4953	272	20	type	type	NOUN
ejpam-4953	272	21	cocyclic	cocyclic	NOUN
ejpam-4953	272	22	mappings	mapping	NOUN
ejpam-4953	272	23	in	in	ADP
ejpam-4953	272	24	multiplicative	multiplicative	ADJ
ejpam-4953	272	25	metric	metric	ADJ
ejpam-4953	272	26	spaces	space	NOUN
ejpam-4953	272	27	,	,	PUNCT
ejpam-4953	272	28	discrete	discrete	ADJ
ejpam-4953	272	29	dyn	dyn	NOUN
ejpam-4953	272	30	.	.	PUNCT
ejpam-4953	273	1	nat	nat	PROPN
ejpam-4953	273	2	.	.	PUNCT
ejpam-4953	274	1	soc	soc	PROPN
ejpam-4953	274	2	.	.	PUNCT
ejpam-4953	274	3	,	,	PUNCT
ejpam-4953	274	4	(	(	PUNCT
ejpam-4953	274	5	2015	2015	NUM
ejpam-4953	274	6	)	)	PUNCT
ejpam-4953	274	7	2015	2015	NUM
ejpam-4953	274	8	,	,	PUNCT
ejpam-4953	274	9	article	article	NOUN
ejpam-4953	274	10	i	i	PROPN
ejpam-4953	274	11	d	d	PROPN
ejpam-4953	274	12	532725	532725	NUM
ejpam-4953	274	13	.	.	PUNCT
ejpam-4953	275	1	[	[	X
ejpam-4953	275	2	13	13	NUM
ejpam-4953	275	3	]	]	X
ejpam-4953	275	4	a.a.n	a.a.n	NOUN
ejpam-4953	275	5	.	.	PUNCT
ejpam-4953	276	1	abdou	abdou	PROPN
ejpam-4953	276	2	,	,	PUNCT
ejpam-4953	276	3	fixed	fix	VERB
ejpam-4953	276	4	point	point	NOUN
ejpam-4953	276	5	theorems	theorem	NOUN
ejpam-4953	276	6	for	for	ADP
ejpam-4953	276	7	generalized	generalized	ADJ
ejpam-4953	276	8	contraction	contraction	NOUN
ejpam-4953	276	9	mappings	mapping	NOUN
ejpam-4953	276	10	in	in	ADP
ejpam-4953	276	11	multiplicative	multiplicative	ADJ
ejpam-4953	276	12	metric	metric	ADJ
ejpam-4953	276	13	spaces	space	NOUN
ejpam-4953	276	14	,	,	PUNCT
ejpam-4953	276	15	j.	j.	PROPN
ejpam-4953	276	16	nonlinear	nonlinear	PROPN
ejpam-4953	276	17	sci	sci	PROPN
ejpam-4953	276	18	.	.	PUNCT
ejpam-4953	276	19	appl	appl	PROPN
ejpam-4953	276	20	.	.	PROPN
ejpam-4953	276	21	,	,	PUNCT
ejpam-4953	276	22	9(2016	9(2016	NUM
ejpam-4953	276	23	)	)	PUNCT
ejpam-4953	276	24	,	,	PUNCT
ejpam-4953	276	25	2347	2347	NUM
ejpam-4953	276	26	-	-	SYM
ejpam-4953	276	27	2363	2363	NUM
ejpam-4953	276	28	.	.	PUNCT
ejpam-4953	277	1	[	[	X
ejpam-4953	277	2	14	14	NUM
ejpam-4953	277	3	]	]	X
ejpam-4953	277	4	q.h	q.h	PROPN
ejpam-4953	277	5	.	.	PROPN
ejpam-4953	277	6	khan	khan	PROPN
ejpam-4953	277	7	,	,	PUNCT
ejpam-4953	277	8	m.	m.	PROPN
ejpam-4953	277	9	imdad	imdad	PROPN
ejpam-4953	277	10	,	,	PUNCT
ejpam-4953	277	11	coincidence	coincidence	NOUN
ejpam-4953	277	12	and	and	CCONJ
ejpam-4953	277	13	common	common	ADJ
ejpam-4953	277	14	fixed	fix	VERB
ejpam-4953	277	15	point	point	NOUN
ejpam-4953	277	16	theorems	theorem	NOUN
ejpam-4953	277	17	in	in	ADP
ejpam-4953	277	18	a	a	DET
ejpam-4953	277	19	multiplicative	multiplicative	ADJ
ejpam-4953	277	20	metric	metric	ADJ
ejpam-4953	277	21	space	space	NOUN
ejpam-4953	277	22	,	,	PUNCT
ejpam-4953	277	23	adv	adv	PROPN
ejpam-4953	277	24	.	.	PUNCT
ejpam-4953	277	25	fixed	fix	VERB
ejpam-4953	277	26	point	point	NOUN
ejpam-4953	277	27	theory	theory	NOUN
ejpam-4953	277	28	,	,	PUNCT
ejpam-4953	277	29	6(2016	6(2016	NUM
ejpam-4953	277	30	)	)	PUNCT
ejpam-4953	277	31	,	,	PUNCT
ejpam-4953	277	32	no	no	INTJ
ejpam-4953	277	33	.	.	NOUN
ejpam-4953	277	34	1	1	NUM
ejpam-4953	277	35	,	,	PUNCT
ejpam-4953	277	36	1	1	NUM
ejpam-4953	277	37	-	-	SYM
ejpam-4953	277	38	9	9	NUM
ejpam-4953	277	39	.	.	PUNCT
ejpam-4953	278	1	[	[	X
ejpam-4953	278	2	15	15	NUM
ejpam-4953	278	3	]	]	X
ejpam-4953	278	4	s.	s.	PROPN
ejpam-4953	278	5	laishram	laishram	PROPN
ejpam-4953	278	6	,	,	PUNCT
ejpam-4953	278	7	r.	r.	PROPN
ejpam-4953	278	8	yumnam	yumnam	PROPN
ejpam-4953	278	9	,	,	PUNCT
ejpam-4953	278	10	tripled	triple	VERB
ejpam-4953	278	11	fixed	fix	VERB
ejpam-4953	278	12	point	point	NOUN
ejpam-4953	278	13	in	in	ADP
ejpam-4953	278	14	ordered	order	VERB
ejpam-4953	278	15	multiplicative	multiplicative	ADJ
ejpam-4953	278	16	metric	metric	ADJ
ejpam-4953	278	17	spaces	space	NOUN
ejpam-4953	278	18	,	,	PUNCT
ejpam-4953	278	19	journal	journal	NOUN
ejpam-4953	278	20	nonlinear	nonlinear	ADJ
ejpam-4953	278	21	analysis	analysis	NOUN
ejpam-4953	278	22	application	application	NOUN
ejpam-4953	278	23	,	,	PUNCT
ejpam-4953	278	24	1(2017	1(2017	NUM
ejpam-4953	278	25	)	)	PUNCT
ejpam-4953	278	26	,	,	PUNCT
ejpam-4953	278	27	56	56	NUM
ejpam-4953	278	28	-	-	SYM
ejpam-4953	278	29	65	65	NUM
ejpam-4953	278	30	.	.	PUNCT
ejpam-4953	279	1	[	[	X
ejpam-4953	279	2	16	16	NUM
ejpam-4953	279	3	]	]	X
ejpam-4953	279	4	x.h	x.h	PROPN
ejpam-4953	279	5	.	.	PROPN
ejpam-4953	279	6	song	song	PROPN
ejpam-4953	279	7	,	,	PUNCT
ejpam-4953	279	8	m.d	m.d	PROPN
ejpam-4953	279	9	.	.	PROPN
ejpam-4953	279	10	chen	chen	PROPN
ejpam-4953	279	11	,	,	PUNCT
ejpam-4953	279	12	common	common	ADJ
ejpam-4953	279	13	fixed	fix	VERB
ejpam-4953	279	14	points	point	NOUN
ejpam-4953	279	15	for	for	ADP
ejpam-4953	279	16	weak	weak	ADJ
ejpam-4953	279	17	commutative	commutative	ADJ
ejpam-4953	279	18	mappings	mapping	NOUN
ejpam-4953	279	19	on	on	ADP
ejpam-4953	279	20	a	a	DET
ejpam-4953	279	21	multiplicative	multiplicative	ADJ
ejpam-4953	279	22	metric	metric	ADJ
ejpam-4953	279	23	space	space	NOUN
ejpam-4953	279	24	,	,	PUNCT
ejpam-4953	279	25	fixed	fix	VERB
ejpam-4953	279	26	point	point	NOUN
ejpam-4953	279	27	theory	theory	NOUN
ejpam-4953	279	28	appl	appl	NOUN
ejpam-4953	279	29	.	.	PUNCT
ejpam-4953	280	1	48	48	NUM
ejpam-4953	280	2	2014	2014	NUM
ejpam-4953	280	3	,	,	PUNCT
ejpam-4953	280	4	2014	2014	NUM
ejpam-4953	280	5	.	.	PUNCT
ejpam-4953	281	1	[	[	X
ejpam-4953	281	2	17	17	NUM
ejpam-4953	281	3	]	]	X
ejpam-4953	281	4	s.czerwik	s.czerwik	PROPN
ejpam-4953	281	5	,	,	PUNCT
ejpam-4953	281	6	contraction	contraction	NOUN
ejpam-4953	281	7	mapping	mapping	NOUN
ejpam-4953	281	8	in	in	ADP
ejpam-4953	281	9	b	b	NOUN
ejpam-4953	281	10	-	-	ADJ
ejpam-4953	281	11	metric	metric	ADJ
ejpam-4953	281	12	spaces	space	NOUN
ejpam-4953	281	13	,	,	PUNCT
ejpam-4953	281	14	acta	acta	PROPN
ejpam-4953	281	15	math	math	PROPN
ejpam-4953	281	16	.	.	PUNCT
ejpam-4953	282	1	inf.univ.ostraviensis	inf.univ.ostraviensis	NUM
ejpam-4953	282	2	.	.	PUNCT
ejpam-4953	283	1	1(1993	1(1993	NUM
ejpam-4953	283	2	)	)	PUNCT
ejpam-4953	284	1	,	,	PUNCT
ejpam-4953	284	2	5	5	NUM
ejpam-4953	284	3	-	-	SYM
ejpam-4953	284	4	11	11	NUM
ejpam-4953	284	5	.	.	PUNCT
ejpam-4953	285	1	[	[	X
ejpam-4953	285	2	18	18	NUM
ejpam-4953	285	3	]	]	X
ejpam-4953	285	4	i.m	i.m	PROPN
ejpam-4953	285	5	.	.	PROPN
ejpam-4953	285	6	alanazi	alanazi	PROPN
ejpam-4953	285	7	,	,	PUNCT
ejpam-4953	285	8	q.h	q.h	PROPN
ejpam-4953	285	9	.	.	PROPN
ejpam-4953	285	10	khan	khan	PROPN
ejpam-4953	285	11	,	,	PUNCT
ejpam-4953	285	12	s.	s.	PROPN
ejpam-4953	285	13	ali	ali	PROPN
ejpam-4953	285	14	,	,	PUNCT
ejpam-4953	285	15	t.	t.	PROPN
ejpam-4953	285	16	rashid	rashid	PROPN
ejpam-4953	285	17	,	,	PUNCT
ejpam-4953	285	18	f.a	f.a	PROPN
ejpam-4953	285	19	.	.	PROPN
ejpam-4953	285	20	khan	khan	PROPN
ejpam-4953	285	21	,	,	PUNCT
ejpam-4953	285	22	on	on	ADP
ejpam-4953	285	23	coupled	couple	VERB
ejpam-4953	285	24	coincidence	coincidence	NOUN
ejpam-4953	285	25	points	point	NOUN
ejpam-4953	285	26	in	in	ADP
ejpam-4953	285	27	multiplicative	multiplicative	ADJ
ejpam-4953	285	28	metric	metric	ADJ
ejpam-4953	285	29	spaces	space	NOUN
ejpam-4953	285	30	with	with	ADP
ejpam-4953	285	31	an	an	DET
ejpam-4953	285	32	application	application	NOUN
ejpam-4953	285	33	,	,	PUNCT
ejpam-4953	285	34	nonlinear	nonlinear	ADJ
ejpam-4953	285	35	functional	functional	ADJ
ejpam-4953	285	36	analysis	analysis	NOUN
ejpam-4953	285	37	and	and	CCONJ
ejpam-4953	285	38	application	application	NOUN
ejpam-4953	285	39	,	,	PUNCT
ejpam-4953	285	40	28(3	28(3	NUM
ejpam-4953	285	41	)	)	PUNCT
ejpam-4953	285	42	(	(	PUNCT
ejpam-4953	285	43	2023	2023	NUM
ejpam-4953	285	44	)	)	PUNCT
ejpam-4953	285	45	,	,	PUNCT
ejpam-4953	285	46	775	775	NUM
ejpam-4953	285	47	-	-	SYM
ejpam-4953	285	48	791	791	NUM
ejpam-4953	285	49	.	.	PUNCT
ejpam-4953	286	1	[	[	X
ejpam-4953	286	2	19	19	NUM
ejpam-4953	286	3	]	]	SYM
ejpam-4953	286	4	t	t	PROPN
ejpam-4953	286	5	,	,	PUNCT
ejpam-4953	286	6	dosenovic	dosenovic	ADJ
ejpam-4953	286	7	,	,	PUNCT
ejpam-4953	286	8	m.	m.	NOUN
ejpam-4953	286	9	postolache	postolache	PROPN
ejpam-4953	286	10	and	and	CCONJ
ejpam-4953	286	11	s.	s.	PROPN
ejpam-4953	286	12	radenovic	radenovic	PROPN
ejpam-4953	286	13	,	,	PUNCT
ejpam-4953	286	14	on	on	ADP
ejpam-4953	286	15	multiplicative	multiplicative	ADJ
ejpam-4953	286	16	metric	metric	ADJ
ejpam-4953	286	17	spaces	space	NOUN
ejpam-4953	286	18	:	:	PUNCT
ejpam-4953	286	19	survey	survey	NOUN
ejpam-4953	286	20	,	,	PUNCT
ejpam-4953	286	21	fixed	fix	VERB
ejpam-4953	286	22	point	point	NOUN
ejpam-4953	286	23	theory	theory	NOUN
ejpam-4953	286	24	and	and	CCONJ
ejpam-4953	286	25	application	application	NOUN
ejpam-4953	286	26	,	,	PUNCT
ejpam-4953	286	27	92	92	NUM
ejpam-4953	286	28	(	(	PUNCT
ejpam-4953	286	29	2016	2016	NUM
ejpam-4953	286	30	)	)	PUNCT
ejpam-4953	286	31	,	,	PUNCT
ejpam-4953	286	32	2016	2016	NUM
ejpam-4953	286	33	.	.	PUNCT
ejpam-4953	287	1	[	[	X
ejpam-4953	287	2	20	20	NUM
ejpam-4953	287	3	]	]	PUNCT
ejpam-4953	287	4	a.	a.	NOUN
ejpam-4953	287	5	shoaib	shoaib	PROPN
ejpam-4953	287	6	,	,	PUNCT
ejpam-4953	287	7	common	common	ADJ
ejpam-4953	287	8	fixed	fix	VERB
ejpam-4953	287	9	point	point	NOUN
ejpam-4953	287	10	for	for	ADP
ejpam-4953	287	11	generalized	generalized	ADJ
ejpam-4953	287	12	contraction	contraction	NOUN
ejpam-4953	287	13	in	in	ADP
ejpam-4953	287	14	b	b	NOUN
ejpam-4953	287	15	-	-	PUNCT
ejpam-4953	287	16	multiplicatice	multiplicatice	NOUN
ejpam-4953	287	17	metric	metric	ADJ
ejpam-4953	287	18	spaces	space	NOUN
ejpam-4953	287	19	with	with	ADP
ejpam-4953	287	20	applications	application	NOUN
ejpam-4953	287	21	,	,	PUNCT
ejpam-4953	287	22	bulletin	bulletin	NOUN
ejpam-4953	287	23	of	of	ADP
ejpam-4953	287	24	mathematical	mathematical	ADJ
ejpam-4953	287	25	analysis	analysis	NOUN
ejpam-4953	287	26	and	and	CCONJ
ejpam-4953	287	27	applications	application	NOUN
ejpam-4953	287	28	,	,	PUNCT
ejpam-4953	287	29	12(3	12(3	NUM
ejpam-4953	287	30	)	)	PUNCT
ejpam-4953	287	31	(	(	PUNCT
ejpam-4953	287	32	2020	2020	NUM
ejpam-4953	287	33	)	)	PUNCT
ejpam-4953	287	34	,	,	PUNCT
ejpam-4953	287	35	46	46	NUM
ejpam-4953	287	36	-	-	SYM
ejpam-4953	287	37	49	49	NUM
ejpam-4953	287	38	.	.	PUNCT
ejpam-4953	288	1	[	[	X
ejpam-4953	288	2	21	21	NUM
ejpam-4953	288	3	]	]	X
ejpam-4953	288	4	m.u.ali	m.u.ali	PROPN
ejpam-4953	288	5	,	,	PUNCT
ejpam-4953	288	6	t.kamran	t.kamran	NOUN
ejpam-4953	288	7	,	,	PUNCT
ejpam-4953	288	8	a.kurdi	a.kurdi	PRON
ejpam-4953	288	9	,	,	PUNCT
ejpam-4953	288	10	fixed	fix	VERB
ejpam-4953	288	11	point	point	NOUN
ejpam-4953	288	12	theorem	theorem	VERB
ejpam-4953	288	13	in	in	ADP
ejpam-4953	288	14	b	b	NOUN
ejpam-4953	288	15	-	-	PUNCT
ejpam-4953	288	16	multiplicative	multiplicative	ADJ
ejpam-4953	288	17	metric	metric	ADJ
ejpam-4953	288	18	spaces	space	NOUN
ejpam-4953	288	19	,	,	PUNCT
ejpam-4953	288	20	u.p.b.sci.bull	u.p.b.sci.bull	NOUN
ejpam-4953	288	21	.	.	PROPN
ejpam-4953	288	22	,	,	PUNCT
ejpam-4953	288	23	79(3)(2017	79(3)(2017	NUM
ejpam-4953	288	24	)	)	PUNCT
ejpam-4953	288	25	,	,	PUNCT
ejpam-4953	288	26	107	107	NUM
ejpam-4953	288	27	-	-	SYM
ejpam-4953	288	28	116	116	NUM
ejpam-4953	288	29	.	.	PUNCT
ejpam-4953	289	1	[	[	X
ejpam-4953	289	2	22	22	NUM
ejpam-4953	289	3	]	]	X
ejpam-4953	289	4	t.g	t.g	PROPN
ejpam-4953	289	5	.	.	PROPN
ejpam-4953	289	6	bhaskar	bhaskar	NOUN
ejpam-4953	289	7	,	,	PUNCT
ejpam-4953	289	8	v.	v.	ADP
ejpam-4953	289	9	lakshmikantham	lakshmikantham	ADJ
ejpam-4953	289	10	,	,	PUNCT
ejpam-4953	289	11	fixed	fix	VERB
ejpam-4953	289	12	point	point	NOUN
ejpam-4953	289	13	theorems	theorem	NOUN
ejpam-4953	289	14	in	in	ADP
ejpam-4953	289	15	partially	partially	ADV
ejpam-4953	289	16	ordered	order	VERB
ejpam-4953	289	17	metric	metric	ADJ
ejpam-4953	289	18	spaces	space	NOUN
ejpam-4953	289	19	and	and	CCONJ
ejpam-4953	289	20	applications	application	NOUN
ejpam-4953	289	21	,	,	PUNCT
ejpam-4953	289	22	nonlin	nonlin	NOUN
ejpam-4953	289	23	.	.	PUNCT
ejpam-4953	290	1	anal	anal	PROPN
ejpam-4953	290	2	.	.	PUNCT
ejpam-4953	290	3	,	,	PUNCT
ejpam-4953	290	4	65(2006	65(2006	NUM
ejpam-4953	290	5	)	)	PUNCT
ejpam-4953	290	6	,	,	PUNCT
ejpam-4953	290	7	1379	1379	NUM
ejpam-4953	290	8	-	-	SYM
ejpam-4953	290	9	1393	1393	NUM
ejpam-4953	290	10	.	.	PUNCT
ejpam-4953	291	1	references	reference	NOUN
ejpam-4953	291	2	2418	2418	NUM
ejpam-4953	291	3	[	[	X
ejpam-4953	291	4	23	23	NUM
ejpam-4953	291	5	]	]	PUNCT
ejpam-4953	291	6	s.lipschutz	s.lipschutz	VERB
ejpam-4953	291	7	,	,	PUNCT
ejpam-4953	291	8	schaum	schaum	PROPN
ejpam-4953	291	9	’s	’s	PART
ejpam-4953	291	10	outlines	outline	NOUN
ejpam-4953	291	11	of	of	ADP
ejpam-4953	291	12	theory	theory	NOUN
ejpam-4953	291	13	and	and	CCONJ
ejpam-4953	291	14	problems	problem	NOUN
ejpam-4953	291	15	of	of	ADP
ejpam-4953	291	16	set	set	ADJ
ejpam-4953	291	17	theory	theory	NOUN
ejpam-4953	291	18	ans	an	NOUN
ejpam-4953	291	19	related	relate	VERB
ejpam-4953	291	20	topics	topic	NOUN
ejpam-4953	291	21	,	,	PUNCT
ejpam-4953	291	22	mcgraw	mcgraw	NOUN
ejpam-4953	291	23	-	-	PUNCT
ejpam-4953	291	24	hill	hill	PROPN
ejpam-4953	291	25	,	,	PUNCT
ejpam-4953	291	26	new	new	PROPN
ejpam-4953	291	27	york	york	PROPN
ejpam-4953	291	28	,	,	PUNCT
ejpam-4953	291	29	1964	1964	NUM
ejpam-4953	291	30	.	.	PUNCT
ejpam-4953	292	1	[	[	X
ejpam-4953	292	2	24	24	NUM
ejpam-4953	292	3	]	]	PUNCT
ejpam-4953	292	4	a.alam	a.alam	PROPN
ejpam-4953	292	5	and	and	CCONJ
ejpam-4953	292	6	m.imdad	m.imdad	NOUN
ejpam-4953	292	7	,	,	PUNCT
ejpam-4953	292	8	relation	relation	NOUN
ejpam-4953	292	9	-	-	PUNCT
ejpam-4953	292	10	theoretic	theoretic	NOUN
ejpam-4953	292	11	contraction	contraction	NOUN
ejpam-4953	292	12	principle	principle	NOUN
ejpam-4953	292	13	,	,	PUNCT
ejpam-4953	292	14	j.fixed	j.fixed	ADJ
ejpam-4953	292	15	point	point	NOUN
ejpam-4953	292	16	theory	theory	NOUN
ejpam-4953	292	17	appl	appl	PROPN
ejpam-4953	292	18	.	.	PROPN
ejpam-4953	292	19	,	,	PUNCT
ejpam-4953	292	20	17	17	NUM
ejpam-4953	292	21	(	(	PUNCT
ejpam-4953	292	22	2015	2015	NUM
ejpam-4953	292	23	)	)	PUNCT
ejpam-4953	292	24	,	,	PUNCT
ejpam-4953	292	25	693	693	NUM
ejpam-4953	292	26	-	-	SYM
ejpam-4953	292	27	702	702	NUM
ejpam-4953	292	28	.	.	PUNCT
ejpam-4953	293	1	[	[	X
ejpam-4953	293	2	25	25	NUM
ejpam-4953	293	3	]	]	PUNCT
ejpam-4953	293	4	s.chandok	s.chandok	NOUN
ejpam-4953	293	5	,	,	PUNCT
ejpam-4953	293	6	arbitrary	arbitrary	ADJ
ejpam-4953	293	7	binary	binary	ADJ
ejpam-4953	293	8	relations	relation	NOUN
ejpam-4953	293	9	,	,	PUNCT
ejpam-4953	293	10	contraction	contraction	NOUN
ejpam-4953	293	11	mappings	mapping	NOUN
ejpam-4953	293	12	,	,	PUNCT
ejpam-4953	293	13	and	and	CCONJ
ejpam-4953	293	14	b	b	X
ejpam-4953	293	15	-	-	PUNCT
ejpam-4953	293	16	metric	metric	ADJ
ejpam-4953	293	17	spaces	space	NOUN
ejpam-4953	293	18	,	,	PUNCT
ejpam-4953	293	19	ukrainian	ukrainian	ADJ
ejpam-4953	293	20	mathematical	mathematical	ADJ
ejpam-4953	293	21	journal,70(4	journal,70(4	PROPN
ejpam-4953	293	22	)	)	PUNCT
ejpam-4953	293	23	(	(	PUNCT
ejpam-4953	293	24	2020),651	2020),651	NUM
ejpam-4953	293	25	-	-	SYM
ejpam-4953	293	26	662	662	NUM
ejpam-4953	293	27	.	.	PUNCT
ejpam-4953	294	1	[	[	X
ejpam-4953	294	2	26	26	NUM
ejpam-4953	294	3	]	]	PUNCT
ejpam-4953	294	4	b.samet	b.samet	PUNCT
ejpam-4953	294	5	and	and	CCONJ
ejpam-4953	294	6	m.turinici	m.turinici	NOUN
ejpam-4953	294	7	,	,	PUNCT
ejpam-4953	294	8	fixed	fix	VERB
ejpam-4953	294	9	point	point	NOUN
ejpam-4953	294	10	theorems	theorem	NOUN
ejpam-4953	294	11	on	on	ADP
ejpam-4953	294	12	a	a	DET
ejpam-4953	294	13	metric	metric	ADJ
ejpam-4953	294	14	space	space	NOUN
ejpam-4953	294	15	endowed	endow	VERB
ejpam-4953	294	16	with	with	ADP
ejpam-4953	294	17	an	an	DET
ejpam-4953	294	18	arbitrary	arbitrary	ADJ
ejpam-4953	294	19	binary	binary	ADJ
ejpam-4953	294	20	relation	relation	NOUN
ejpam-4953	294	21	and	and	CCONJ
ejpam-4953	294	22	applications	application	NOUN
ejpam-4953	294	23	.	.	PUNCT
ejpam-4953	295	1	commun	commun	PROPN
ejpam-4953	295	2	.	.	PUNCT
ejpam-4953	296	1	math	math	PROPN
ejpam-4953	296	2	.	.	PUNCT
ejpam-4953	297	1	anal	anal	PROPN
ejpam-4953	297	2	.	.	PUNCT
ejpam-4953	298	1	13	13	NUM
ejpam-4953	298	2	(	(	PUNCT
ejpam-4953	298	3	2012	2012	NUM
ejpam-4953	298	4	)	)	PUNCT
ejpam-4953	298	5	,	,	PUNCT
ejpam-4953	298	6	82	82	NUM
ejpam-4953	298	7	-	-	SYM
ejpam-4953	298	8	97	97	NUM
ejpam-4953	298	9	.	.	PUNCT
ejpam-4953	299	1	[	[	X
ejpam-4953	299	2	27	27	NUM
ejpam-4953	299	3	]	]	SYM
ejpam-4953	299	4	b.kolman	b.kolman	PROPN
ejpam-4953	299	5	,	,	PUNCT
ejpam-4953	299	6	r.c	r.c	PROPN
ejpam-4953	299	7	.	.	PROPN
ejpam-4953	299	8	busby	busby	PROPN
ejpam-4953	299	9	and	and	CCONJ
ejpam-4953	299	10	s.ross	s.ros	NOUN
ejpam-4953	299	11	,	,	PUNCT
ejpam-4953	299	12	discrete	discrete	ADJ
ejpam-4953	299	13	mathematical	mathematical	ADJ
ejpam-4953	299	14	structures	structure	NOUN
ejpam-4953	299	15	,	,	PUNCT
ejpam-4953	299	16	3rd	3rd	ADJ
ejpam-4953	299	17	ed	ed	NOUN
ejpam-4953	299	18	.	.	PROPN
ejpam-4953	299	19	,	,	PUNCT
ejpam-4953	299	20	phi	phi	PROPN
ejpam-4953	299	21	pvt	pvt	PROPN
ejpam-4953	299	22	.	.	PROPN
ejpam-4953	299	23	ltd	ltd	PROPN
ejpam-4953	299	24	.	.	PROPN
ejpam-4953	299	25	,	,	PUNCT
ejpam-4953	299	26	new	new	PROPN
ejpam-4953	299	27	delhi	delhi	PROPN
ejpam-4953	299	28	,	,	PUNCT
ejpam-4953	299	29	2000	2000	NUM
ejpam-4953	299	30	.	.	PUNCT
ejpam-4953	300	1	[	[	X
ejpam-4953	300	2	28	28	NUM
ejpam-4953	300	3	]	]	X
ejpam-4953	300	4	r.	r.	PROPN
ejpam-4953	300	5	h.	h.	PROPN
ejpam-4953	300	6	haghi	haghi	PROPN
ejpam-4953	300	7	,	,	PUNCT
ejpam-4953	300	8	sh	sh	PROPN
ejpam-4953	300	9	.	.	PROPN
ejpam-4953	300	10	rezapour	rezapour	PROPN
ejpam-4953	300	11	,	,	PUNCT
ejpam-4953	300	12	and	and	CCONJ
ejpam-4953	300	13	n.	n.	PROPN
ejpam-4953	300	14	shahzad	shahzad	PROPN
ejpam-4953	300	15	,	,	PUNCT
ejpam-4953	300	16	some	some	DET
ejpam-4953	300	17	fixed	fix	VERB
ejpam-4953	300	18	point	point	NOUN
ejpam-4953	300	19	generalizations	generalization	NOUN
ejpam-4953	300	20	are	be	AUX
ejpam-4953	300	21	not	not	PART
ejpam-4953	300	22	real	real	ADJ
ejpam-4953	300	23	generalizations	generalization	NOUN
ejpam-4953	300	24	,	,	PUNCT
ejpam-4953	300	25	nonlin	nonlin	PROPN
ejpam-4953	300	26	.	.	PUNCT
ejpam-4953	301	1	anal	anal	PROPN
ejpam-4953	301	2	.	.	PROPN
ejpam-4953	301	3	,	,	PUNCT
ejpam-4953	301	4	74	74	NUM
ejpam-4953	301	5	(	(	PUNCT
ejpam-4953	301	6	2011	2011	NUM
ejpam-4953	301	7	)	)	PUNCT
ejpam-4953	301	8	,	,	PUNCT
ejpam-4953	301	9	1799–1803	1799–1803	NUM
ejpam-4953	301	10	.	.	PUNCT