id	sid	tid	token	lemma	pos
ejpam-5279	1	1	european	european	PROPN
ejpam-5279	1	2	journal	journal	PROPN
ejpam-5279	1	3	of	of	ADP
ejpam-5279	1	4	pure	pure	ADJ
ejpam-5279	1	5	and	and	CCONJ
ejpam-5279	1	6	applied	apply	VERB
ejpam-5279	1	7	mathematics	mathematic	NOUN
ejpam-5279	1	8	vol	vol	NOUN
ejpam-5279	1	9	.	.	PROPN
ejpam-5279	2	1	17	17	NUM
ejpam-5279	2	2	,	,	PUNCT
ejpam-5279	2	3	no	no	INTJ
ejpam-5279	2	4	.	.	NOUN
ejpam-5279	2	5	3	3	NUM
ejpam-5279	2	6	,	,	PUNCT
ejpam-5279	2	7	2024	2024	NUM
ejpam-5279	2	8	,	,	PUNCT
ejpam-5279	2	9	1762	1762	NUM
ejpam-5279	2	10	-	-	SYM
ejpam-5279	2	11	1778	1778	NUM
ejpam-5279	2	12	issn	issn	PROPN
ejpam-5279	2	13	1307	1307	NUM
ejpam-5279	2	14	-	-	SYM
ejpam-5279	2	15	5543	5543	NUM
ejpam-5279	2	16	–	–	PUNCT
ejpam-5279	2	17	ejpam.com	ejpam.com	X
ejpam-5279	2	18	published	publish	VERB
ejpam-5279	2	19	by	by	ADP
ejpam-5279	2	20	new	new	PROPN
ejpam-5279	2	21	york	york	PROPN
ejpam-5279	2	22	business	business	PROPN
ejpam-5279	2	23	global	global	PROPN
ejpam-5279	2	24	on	on	ADP
ejpam-5279	2	25	the	the	DET
ejpam-5279	2	26	characterizations	characterization	NOUN
ejpam-5279	2	27	of	of	ADP
ejpam-5279	2	28	approach	approach	NOUN
ejpam-5279	2	29	groups	group	NOUN
ejpam-5279	2	30	t.m.g	t.m.g	PROPN
ejpam-5279	2	31	.	.	PUNCT
ejpam-5279	2	32	ahsanullah1,∗	ahsanullah1,∗	PROPN
ejpam-5279	2	33	,	,	PUNCT
ejpam-5279	2	34	fawzi	fawzi	PROPN
ejpam-5279	2	35	al	al	PROPN
ejpam-5279	2	36	-	-	PUNCT
ejpam-5279	2	37	thukair1	thukair1	PROPN
ejpam-5279	2	38	1	1	NUM
ejpam-5279	2	39	department	department	NOUN
ejpam-5279	2	40	of	of	ADP
ejpam-5279	2	41	mathematics	mathematic	NOUN
ejpam-5279	2	42	,	,	PUNCT
ejpam-5279	2	43	college	college	NOUN
ejpam-5279	2	44	of	of	ADP
ejpam-5279	2	45	science	science	NOUN
ejpam-5279	2	46	,	,	PUNCT
ejpam-5279	2	47	king	king	NOUN
ejpam-5279	2	48	saud	saud	PROPN
ejpam-5279	2	49	university	university	PROPN
ejpam-5279	2	50	,	,	PUNCT
ejpam-5279	2	51	riyadh	riyadh	PROPN
ejpam-5279	2	52	,	,	PUNCT
ejpam-5279	2	53	saudi	saudi	PROPN
ejpam-5279	2	54	arabia	arabia	PROPN
ejpam-5279	2	55	abstract	abstract	NOUN
ejpam-5279	2	56	.	.	PUNCT
ejpam-5279	3	1	in	in	ADP
ejpam-5279	3	2	this	this	DET
ejpam-5279	3	3	paper	paper	NOUN
ejpam-5279	3	4	,	,	PUNCT
ejpam-5279	3	5	we	we	PRON
ejpam-5279	3	6	present	present	VERB
ejpam-5279	3	7	several	several	ADJ
ejpam-5279	3	8	characterization	characterization	NOUN
ejpam-5279	3	9	theorems	theorem	NOUN
ejpam-5279	3	10	on	on	ADP
ejpam-5279	3	11	approach	approach	NOUN
ejpam-5279	3	12	groups	group	NOUN
ejpam-5279	3	13	,	,	PUNCT
ejpam-5279	3	14	and	and	CCONJ
ejpam-5279	3	15	ultra	ultra	ADJ
ejpam-5279	3	16	approach	approach	NOUN
ejpam-5279	3	17	groups	group	NOUN
ejpam-5279	3	18	.	.	PUNCT
ejpam-5279	4	1	in	in	ADP
ejpam-5279	4	2	so	so	ADV
ejpam-5279	4	3	doing	do	VERB
ejpam-5279	4	4	,	,	PUNCT
ejpam-5279	4	5	we	we	PRON
ejpam-5279	4	6	first	first	ADV
ejpam-5279	4	7	give	give	VERB
ejpam-5279	4	8	necessary	necessary	ADJ
ejpam-5279	4	9	and	and	CCONJ
ejpam-5279	4	10	sufficient	sufficient	ADJ
ejpam-5279	4	11	conditions	condition	NOUN
ejpam-5279	4	12	for	for	ADP
ejpam-5279	4	13	an	an	DET
ejpam-5279	4	14	approach	approach	NOUN
ejpam-5279	4	15	structure	structure	NOUN
ejpam-5279	4	16	to	to	PART
ejpam-5279	4	17	be	be	AUX
ejpam-5279	4	18	compatible	compatible	ADJ
ejpam-5279	4	19	with	with	ADP
ejpam-5279	4	20	group	group	NOUN
ejpam-5279	4	21	structure	structure	NOUN
ejpam-5279	4	22	.	.	PUNCT
ejpam-5279	5	1	we	we	PRON
ejpam-5279	5	2	show	show	VERB
ejpam-5279	5	3	that	that	SCONJ
ejpam-5279	5	4	every	every	DET
ejpam-5279	5	5	ultra	ultra	ADJ
ejpam-5279	5	6	approach	approach	NOUN
ejpam-5279	5	7	group	group	NOUN
ejpam-5279	5	8	is	be	AUX
ejpam-5279	5	9	ultrauniformizable	ultrauniformizable	ADJ
ejpam-5279	5	10	.	.	PUNCT
ejpam-5279	6	1	secondly	secondly	ADV
ejpam-5279	6	2	,	,	PUNCT
ejpam-5279	6	3	starting	start	VERB
ejpam-5279	6	4	with	with	ADP
ejpam-5279	6	5	an	an	DET
ejpam-5279	6	6	approach	approach	NOUN
ejpam-5279	6	7	space	space	NOUN
ejpam-5279	6	8	,	,	PUNCT
ejpam-5279	6	9	and	and	CCONJ
ejpam-5279	6	10	its	its	PRON
ejpam-5279	6	11	natural	natural	ADJ
ejpam-5279	6	12	neighborhood	neighborhood	NOUN
ejpam-5279	6	13	system	system	NOUN
ejpam-5279	6	14	on	on	ADP
ejpam-5279	6	15	a	a	DET
ejpam-5279	6	16	group	group	NOUN
ejpam-5279	6	17	,	,	PUNCT
ejpam-5279	6	18	we	we	PRON
ejpam-5279	6	19	characterize	characterize	VERB
ejpam-5279	6	20	the	the	DET
ejpam-5279	6	21	resulting	result	VERB
ejpam-5279	6	22	neighborhood	neighborhood	NOUN
ejpam-5279	6	23	approach	approach	NOUN
ejpam-5279	6	24	group	group	NOUN
ejpam-5279	6	25	.	.	PUNCT
ejpam-5279	7	1	finally	finally	ADV
ejpam-5279	7	2	,	,	PUNCT
ejpam-5279	7	3	we	we	PRON
ejpam-5279	7	4	show	show	VERB
ejpam-5279	7	5	that	that	SCONJ
ejpam-5279	7	6	the	the	DET
ejpam-5279	7	7	category	category	NOUN
ejpam-5279	7	8	of	of	ADP
ejpam-5279	7	9	ultra	ultra	ADJ
ejpam-5279	7	10	approach	approach	NOUN
ejpam-5279	7	11	-	-	PUNCT
ejpam-5279	7	12	cauchy	cauchy	NOUN
ejpam-5279	7	13	group	group	NOUN
ejpam-5279	7	14	is	be	AUX
ejpam-5279	7	15	a	a	DET
ejpam-5279	7	16	topological	topological	ADJ
ejpam-5279	7	17	category	category	NOUN
ejpam-5279	7	18	,	,	PUNCT
ejpam-5279	7	19	and	and	CCONJ
ejpam-5279	7	20	more	more	ADV
ejpam-5279	7	21	importantly	importantly	ADV
ejpam-5279	7	22	,	,	PUNCT
ejpam-5279	7	23	we	we	PRON
ejpam-5279	7	24	show	show	VERB
ejpam-5279	7	25	that	that	SCONJ
ejpam-5279	7	26	the	the	DET
ejpam-5279	7	27	category	category	NOUN
ejpam-5279	7	28	of	of	ADP
ejpam-5279	7	29	ultra	ultra	ADJ
ejpam-5279	7	30	approach	approach	NOUN
ejpam-5279	7	31	-	-	PUNCT
ejpam-5279	7	32	cauchy	cauchy	NOUN
ejpam-5279	7	33	groups	group	NOUN
ejpam-5279	7	34	and	and	CCONJ
ejpam-5279	7	35	the	the	DET
ejpam-5279	7	36	category	category	NOUN
ejpam-5279	7	37	of	of	ADP
ejpam-5279	7	38	strongly	strongly	ADV
ejpam-5279	7	39	normal	normal	ADJ
ejpam-5279	7	40	ultra	ultra	ADJ
ejpam-5279	7	41	approach	approach	NOUN
ejpam-5279	7	42	-	-	PUNCT
ejpam-5279	7	43	limit	limit	NOUN
ejpam-5279	7	44	groups	group	NOUN
ejpam-5279	7	45	are	be	AUX
ejpam-5279	7	46	isomorphic	isomorphic	ADJ
ejpam-5279	7	47	.	.	PUNCT
ejpam-5279	8	1	2020	2020	NUM
ejpam-5279	8	2	mathematics	mathematic	NOUN
ejpam-5279	8	3	subject	subject	NOUN
ejpam-5279	8	4	classifications	classification	NOUN
ejpam-5279	8	5	:	:	PUNCT
ejpam-5279	8	6	54a20	54a20	NUM
ejpam-5279	8	7	,	,	PUNCT
ejpam-5279	8	8	54e70	54e70	NUM
ejpam-5279	8	9	,	,	PUNCT
ejpam-5279	8	10	54e90	54e90	NUM
ejpam-5279	8	11	,	,	PUNCT
ejpam-5279	8	12	54h11	54h11	NUM
ejpam-5279	8	13	key	key	ADJ
ejpam-5279	8	14	words	word	NOUN
ejpam-5279	8	15	and	and	CCONJ
ejpam-5279	8	16	phrases	phrase	NOUN
ejpam-5279	8	17	:	:	PUNCT
ejpam-5279	8	18	approach	approach	NOUN
ejpam-5279	8	19	space	space	NOUN
ejpam-5279	8	20	,	,	PUNCT
ejpam-5279	8	21	approach	approach	NOUN
ejpam-5279	8	22	group	group	NOUN
ejpam-5279	8	23	,	,	PUNCT
ejpam-5279	8	24	ultra	ultra	ADJ
ejpam-5279	8	25	approach	approach	NOUN
ejpam-5279	8	26	group	group	NOUN
ejpam-5279	8	27	,	,	PUNCT
ejpam-5279	8	28	approach	approach	NOUN
ejpam-5279	8	29	neighborhood	neighborhood	NOUN
ejpam-5279	8	30	system	system	NOUN
ejpam-5279	8	31	,	,	PUNCT
ejpam-5279	8	32	approach	approach	NOUN
ejpam-5279	8	33	limit	limit	NOUN
ejpam-5279	8	34	space	space	NOUN
ejpam-5279	8	35	,	,	PUNCT
ejpam-5279	8	36	approach	approach	NOUN
ejpam-5279	8	37	limit	limit	NOUN
ejpam-5279	8	38	group	group	NOUN
ejpam-5279	8	39	,	,	PUNCT
ejpam-5279	8	40	approach	approach	NOUN
ejpam-5279	8	41	uniformity	uniformity	NOUN
ejpam-5279	8	42	,	,	PUNCT
ejpam-5279	8	43	ultra	ultra	ADJ
ejpam-5279	8	44	approach	approach	NOUN
ejpam-5279	8	45	-	-	PUNCT
ejpam-5279	8	46	cauchy	cauchy	NOUN
ejpam-5279	8	47	group	group	NOUN
ejpam-5279	8	48	,	,	PUNCT
ejpam-5279	8	49	strongly	strongly	ADV
ejpam-5279	8	50	normal	normal	ADJ
ejpam-5279	8	51	ultra	ultra	ADJ
ejpam-5279	8	52	approach	approach	NOUN
ejpam-5279	8	53	-	-	PUNCT
ejpam-5279	8	54	cauchy	cauchy	NOUN
ejpam-5279	8	55	group	group	NOUN
ejpam-5279	8	56	,	,	PUNCT
ejpam-5279	8	57	category	category	NOUN
ejpam-5279	8	58	1	1	NUM
ejpam-5279	8	59	.	.	PUNCT
ejpam-5279	9	1	introduction	introduction	NOUN
ejpam-5279	9	2	it	it	PRON
ejpam-5279	9	3	is	be	AUX
ejpam-5279	9	4	observed	observe	VERB
ejpam-5279	9	5	in	in	ADP
ejpam-5279	9	6	[	[	X
ejpam-5279	9	7	15	15	NUM
ejpam-5279	9	8	]	]	PUNCT
ejpam-5279	9	9	,	,	PUNCT
ejpam-5279	9	10	and	and	CCONJ
ejpam-5279	9	11	elsewhere	elsewhere	ADV
ejpam-5279	9	12	that	that	DET
ejpam-5279	9	13	top	top	NOUN
ejpam-5279	9	14	,	,	PUNCT
ejpam-5279	9	15	the	the	DET
ejpam-5279	9	16	category	category	NOUN
ejpam-5279	9	17	of	of	ADP
ejpam-5279	9	18	topological	topological	ADJ
ejpam-5279	9	19	spaces	space	NOUN
ejpam-5279	9	20	,	,	PUNCT
ejpam-5279	9	21	is	be	AUX
ejpam-5279	9	22	simultaneously	simultaneously	ADV
ejpam-5279	9	23	bireflectively	bireflectively	ADV
ejpam-5279	9	24	and	and	CCONJ
ejpam-5279	9	25	bicoreflectively	bicoreflectively	ADV
ejpam-5279	9	26	embedded	embed	VERB
ejpam-5279	9	27	in	in	ADP
ejpam-5279	9	28	ap	ap	PROPN
ejpam-5279	9	29	,	,	PUNCT
ejpam-5279	9	30	the	the	DET
ejpam-5279	9	31	category	category	NOUN
ejpam-5279	9	32	of	of	ADP
ejpam-5279	9	33	approach	approach	NOUN
ejpam-5279	9	34	spaces	space	NOUN
ejpam-5279	9	35	.	.	PUNCT
ejpam-5279	10	1	this	this	DET
ejpam-5279	10	2	shows	show	NOUN
ejpam-5279	10	3	,	,	PUNCT
ejpam-5279	10	4	however	however	ADV
ejpam-5279	10	5	,	,	PUNCT
ejpam-5279	10	6	that	that	SCONJ
ejpam-5279	10	7	it	it	PRON
ejpam-5279	10	8	makes	make	VERB
ejpam-5279	10	9	not	not	PART
ejpam-5279	10	10	much	much	ADJ
ejpam-5279	10	11	difference	difference	NOUN
ejpam-5279	10	12	notions	notion	NOUN
ejpam-5279	10	13	like	like	ADP
ejpam-5279	10	14	limits	limit	NOUN
ejpam-5279	10	15	,	,	PUNCT
ejpam-5279	10	16	colimits	colimit	NOUN
ejpam-5279	10	17	,	,	PUNCT
ejpam-5279	10	18	initial	initial	ADJ
ejpam-5279	10	19	structure	structure	NOUN
ejpam-5279	10	20	that	that	PRON
ejpam-5279	10	21	we	we	PRON
ejpam-5279	10	22	may	may	AUX
ejpam-5279	10	23	consider	consider	VERB
ejpam-5279	10	24	either	either	CCONJ
ejpam-5279	10	25	in	in	ADP
ejpam-5279	10	26	top	top	NOUN
ejpam-5279	10	27	or	or	CCONJ
ejpam-5279	10	28	in	in	ADP
ejpam-5279	10	29	ap	ap	PROPN
ejpam-5279	10	30	.	.	PUNCT
ejpam-5279	11	1	but	but	CCONJ
ejpam-5279	11	2	it	it	PRON
ejpam-5279	11	3	does	do	AUX
ejpam-5279	11	4	make	make	VERB
ejpam-5279	11	5	difference	difference	NOUN
ejpam-5279	11	6	whether	whether	SCONJ
ejpam-5279	11	7	we	we	PRON
ejpam-5279	11	8	make	make	VERB
ejpam-5279	11	9	initial	initial	ADJ
ejpam-5279	11	10	structures	structure	NOUN
ejpam-5279	11	11	of	of	ADP
ejpam-5279	11	12	∞pq	∞pq	NOUN
ejpam-5279	11	13	-	-	PUNCT
ejpam-5279	11	14	metric	metric	ADJ
ejpam-5279	11	15	approach	approach	NOUN
ejpam-5279	11	16	spaces	space	NOUN
ejpam-5279	11	17	in	in	ADP
ejpam-5279	11	18	pqmet∞	pqmet∞	NOUN
ejpam-5279	11	19	,	,	PUNCT
ejpam-5279	11	20	the	the	DET
ejpam-5279	11	21	set	set	NOUN
ejpam-5279	11	22	of	of	ADP
ejpam-5279	11	23	all	all	DET
ejpam-5279	11	24	∞pq	∞pq	NOUN
ejpam-5279	11	25	-	-	PUNCT
ejpam-5279	11	26	metrics	metric	NOUN
ejpam-5279	11	27	or	or	CCONJ
ejpam-5279	11	28	in	in	ADP
ejpam-5279	11	29	ap	ap	PROPN
ejpam-5279	11	30	.	.	PUNCT
ejpam-5279	12	1	this	this	PRON
ejpam-5279	12	2	is	be	AUX
ejpam-5279	12	3	so	so	ADV
ejpam-5279	12	4	,	,	PUNCT
ejpam-5279	12	5	because	because	SCONJ
ejpam-5279	12	6	of	of	ADP
ejpam-5279	12	7	the	the	DET
ejpam-5279	12	8	facts	fact	NOUN
ejpam-5279	12	9	that	that	SCONJ
ejpam-5279	12	10	in	in	ADP
ejpam-5279	12	11	the	the	DET
ejpam-5279	12	12	first	first	ADJ
ejpam-5279	12	13	place	place	NOUN
ejpam-5279	12	14	,	,	PUNCT
ejpam-5279	12	15	the	the	DET
ejpam-5279	12	16	domain	domain	NOUN
ejpam-5279	12	17	of	of	ADP
ejpam-5279	12	18	the	the	DET
ejpam-5279	12	19	ordinary	ordinary	ADJ
ejpam-5279	12	20	metric	metric	ADJ
ejpam-5279	12	21	space	space	NOUN
ejpam-5279	12	22	object	object	NOUN
ejpam-5279	12	23	(	(	PUNCT
ejpam-5279	12	24	x	x	X
ejpam-5279	12	25	,	,	PUNCT
ejpam-5279	12	26	d	d	X
ejpam-5279	12	27	:	:	PUNCT
ejpam-5279	12	28	x	x	PROPN
ejpam-5279	12	29	×x	×x	X
ejpam-5279	12	30	→	→	X
ejpam-5279	12	31	[	[	X
ejpam-5279	12	32	0,∞	0,∞	NOUN
ejpam-5279	12	33	]	]	PUNCT
ejpam-5279	12	34	)	)	PUNCT
ejpam-5279	12	35	belonging	belong	VERB
ejpam-5279	12	36	to	to	ADP
ejpam-5279	12	37	the	the	DET
ejpam-5279	12	38	category	category	NOUN
ejpam-5279	12	39	pqmet∞	pqmet∞	NOUN
ejpam-5279	12	40	and	and	CCONJ
ejpam-5279	12	41	the	the	DET
ejpam-5279	12	42	distance	distance	NOUN
ejpam-5279	12	43	space	space	NOUN
ejpam-5279	12	44	object	object	NOUN
ejpam-5279	12	45	(	(	PUNCT
ejpam-5279	12	46	x,∆d	x,∆d	NUM
ejpam-5279	12	47	:	:	PUNCT
ejpam-5279	12	48	x	x	SYM
ejpam-5279	12	49	×	×	NOUN
ejpam-5279	12	50	2x	2x	NUM
ejpam-5279	12	51	→	→	PUNCT
ejpam-5279	12	52	[	[	X
ejpam-5279	12	53	0,∞	0,∞	X
ejpam-5279	12	54	]	]	PUNCT
ejpam-5279	12	55	)	)	PUNCT
ejpam-5279	12	56	known	know	VERB
ejpam-5279	12	57	as	as	ADP
ejpam-5279	12	58	metric	metric	ADJ
ejpam-5279	12	59	distance	distance	NOUN
ejpam-5279	12	60	space	space	NOUN
ejpam-5279	12	61	,	,	PUNCT
ejpam-5279	12	62	being	be	AUX
ejpam-5279	12	63	member	member	NOUN
ejpam-5279	12	64	of	of	ADP
ejpam-5279	12	65	the	the	DET
ejpam-5279	12	66	category	category	NOUN
ejpam-5279	12	67	ap	ap	PROPN
ejpam-5279	12	68	,	,	PUNCT
ejpam-5279	12	69	are	be	AUX
ejpam-5279	12	70	essentially	essentially	ADV
ejpam-5279	12	71	different	different	ADJ
ejpam-5279	12	72	;	;	PUNCT
ejpam-5279	12	73	and	and	CCONJ
ejpam-5279	12	74	,	,	PUNCT
ejpam-5279	12	75	in	in	ADP
ejpam-5279	12	76	the	the	DET
ejpam-5279	12	77	second	second	ADJ
ejpam-5279	12	78	place	place	NOUN
ejpam-5279	12	79	,	,	PUNCT
ejpam-5279	12	80	they	they	PRON
ejpam-5279	12	81	are	be	AUX
ejpam-5279	12	82	also	also	ADV
ejpam-5279	12	83	different	different	ADJ
ejpam-5279	12	84	from	from	ADP
ejpam-5279	12	85	categorical	categorical	ADJ
ejpam-5279	12	86	viewpoint	viewpoint	NOUN
ejpam-5279	12	87	.	.	PUNCT
ejpam-5279	13	1	given	give	VERB
ejpam-5279	13	2	the	the	DET
ejpam-5279	13	3	importance	importance	NOUN
ejpam-5279	13	4	of	of	ADP
ejpam-5279	13	5	the	the	DET
ejpam-5279	13	6	preceding	precede	VERB
ejpam-5279	13	7	paragraph	paragraph	NOUN
ejpam-5279	13	8	,	,	PUNCT
ejpam-5279	13	9	a	a	DET
ejpam-5279	13	10	vast	vast	ADJ
ejpam-5279	13	11	scale	scale	NOUN
ejpam-5279	13	12	of	of	ADP
ejpam-5279	13	13	research	research	NOUN
ejpam-5279	13	14	articles	article	NOUN
ejpam-5279	13	15	appeared	appear	VERB
ejpam-5279	13	16	over	over	ADP
ejpam-5279	13	17	the	the	DET
ejpam-5279	13	18	years	year	NOUN
ejpam-5279	13	19	on	on	ADP
ejpam-5279	13	20	studying	study	VERB
ejpam-5279	13	21	various	various	ADJ
ejpam-5279	13	22	aspects	aspect	NOUN
ejpam-5279	13	23	of	of	ADP
ejpam-5279	13	24	approach	approach	NOUN
ejpam-5279	13	25	spaces	space	NOUN
ejpam-5279	13	26	,	,	PUNCT
ejpam-5279	13	27	and	and	CCONJ
ejpam-5279	13	28	their	their	PRON
ejpam-5279	13	29	equivalence	equivalence	NOUN
ejpam-5279	13	30	structures	structure	NOUN
ejpam-5279	13	31	.	.	PUNCT
ejpam-5279	14	1	a	a	DET
ejpam-5279	14	2	tiny	tiny	ADJ
ejpam-5279	14	3	part	part	NOUN
ejpam-5279	14	4	of	of	ADP
ejpam-5279	14	5	the	the	DET
ejpam-5279	14	6	work	work	NOUN
ejpam-5279	14	7	cited	cite	VERB
ejpam-5279	14	8	in	in	ADP
ejpam-5279	14	9	this	this	DET
ejpam-5279	14	10	paper	paper	NOUN
ejpam-5279	14	11	cf	cf	NOUN
ejpam-5279	14	12	.	.	PUNCT
ejpam-5279	15	1	[	[	X
ejpam-5279	15	2	4	4	NUM
ejpam-5279	15	3	,	,	PUNCT
ejpam-5279	15	4	6–8	6–8	NOUN
ejpam-5279	15	5	,	,	PUNCT
ejpam-5279	15	6	13	13	NUM
ejpam-5279	15	7	,	,	PUNCT
ejpam-5279	15	8	14	14	NUM
ejpam-5279	15	9	,	,	PUNCT
ejpam-5279	15	10	17	17	NUM
ejpam-5279	15	11	,	,	PUNCT
ejpam-5279	15	12	18	18	NUM
ejpam-5279	15	13	,	,	PUNCT
ejpam-5279	15	14	20	20	NUM
ejpam-5279	15	15	]	]	PUNCT
ejpam-5279	15	16	,	,	PUNCT
ejpam-5279	15	17	whereas	whereas	SCONJ
ejpam-5279	15	18	∗corresponding	∗corresponde	VERB
ejpam-5279	15	19	author	author	NOUN
ejpam-5279	15	20	.	.	PUNCT
ejpam-5279	16	1	doi	doi	NOUN
ejpam-5279	16	2	:	:	PUNCT
ejpam-5279	16	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5279	https://doi.org/10.29020/nybg.ejpam.v17i3.5279	VERB
ejpam-5279	16	4	email	email	NOUN
ejpam-5279	16	5	addresses	address	NOUN
ejpam-5279	16	6	:	:	PUNCT
ejpam-5279	16	7	tmga1@ksu.edu.sa	tmga1@ksu.edu.sa	PROPN
ejpam-5279	16	8	(	(	PUNCT
ejpam-5279	16	9	t.m.g	t.m.g	X
ejpam-5279	16	10	.	.	PUNCT
ejpam-5279	17	1	ahsanullah	ahsanullah	NOUN
ejpam-5279	17	2	)	)	PUNCT
ejpam-5279	17	3	,	,	PUNCT
ejpam-5279	17	4	thukair@ksu.edu.sa	thukair@ksu.edu.sa	PROPN
ejpam-5279	17	5	(	(	PUNCT
ejpam-5279	17	6	fawzi	fawzi	PROPN
ejpam-5279	17	7	al	al	PROPN
ejpam-5279	17	8	-	-	PUNCT
ejpam-5279	17	9	thukair	thukair	NOUN
ejpam-5279	17	10	)	)	PUNCT
ejpam-5279	17	11	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5279	17	12	1762	1762	NUM
ejpam-5279	17	13	©	©	ADP
ejpam-5279	17	14	2024	2024	NUM
ejpam-5279	17	15	ejpam	ejpam	NOUN
ejpam-5279	17	16	all	all	DET
ejpam-5279	17	17	rights	right	NOUN
ejpam-5279	17	18	reserved	reserve	VERB
ejpam-5279	17	19	.	.	PUNCT
ejpam-5279	18	1	t.m.g	t.m.g	ADJ
ejpam-5279	18	2	.	.	PUNCT
ejpam-5279	19	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	19	2	,	,	PUNCT
ejpam-5279	19	3	fawzi	fawzi	PROPN
ejpam-5279	19	4	al	al	PROPN
ejpam-5279	19	5	-	-	PUNCT
ejpam-5279	19	6	thukair	thukair	NOUN
ejpam-5279	19	7	/	/	SYM
ejpam-5279	19	8	eur	eur	NOUN
ejpam-5279	19	9	.	.	PUNCT
ejpam-5279	20	1	j.	j.	PROPN
ejpam-5279	20	2	pure	pure	PROPN
ejpam-5279	20	3	appl	appl	PROPN
ejpam-5279	20	4	.	.	PROPN
ejpam-5279	20	5	math	math	PROPN
ejpam-5279	20	6	,	,	PUNCT
ejpam-5279	20	7	17	17	NUM
ejpam-5279	20	8	(	(	PUNCT
ejpam-5279	20	9	3	3	NUM
ejpam-5279	20	10	)	)	PUNCT
ejpam-5279	20	11	(	(	PUNCT
ejpam-5279	20	12	2024	2024	NUM
ejpam-5279	20	13	)	)	PUNCT
ejpam-5279	20	14	,	,	PUNCT
ejpam-5279	20	15	1762	1762	NUM
ejpam-5279	20	16	-	-	SYM
ejpam-5279	20	17	1778	1778	NUM
ejpam-5279	20	18	1763	1763	NUM
ejpam-5279	20	19	the	the	DET
ejpam-5279	20	20	vast	vast	ADJ
ejpam-5279	20	21	majority	majority	NOUN
ejpam-5279	20	22	of	of	ADP
ejpam-5279	20	23	research	research	NOUN
ejpam-5279	20	24	carried	carry	VERB
ejpam-5279	20	25	out	out	ADP
ejpam-5279	20	26	in	in	ADP
ejpam-5279	20	27	these	these	DET
ejpam-5279	20	28	ever	ever	ADV
ejpam-5279	20	29	-	-	PUNCT
ejpam-5279	20	30	growing	grow	VERB
ejpam-5279	20	31	area	area	NOUN
ejpam-5279	20	32	are	be	AUX
ejpam-5279	20	33	not	not	PART
ejpam-5279	20	34	mentioned	mention	VERB
ejpam-5279	20	35	here	here	ADV
ejpam-5279	20	36	just	just	ADV
ejpam-5279	20	37	because	because	SCONJ
ejpam-5279	20	38	the	the	DET
ejpam-5279	20	39	present	present	ADJ
ejpam-5279	20	40	paper	paper	NOUN
ejpam-5279	20	41	is	be	AUX
ejpam-5279	20	42	not	not	PART
ejpam-5279	20	43	directly	directly	ADV
ejpam-5279	20	44	linked	link	VERB
ejpam-5279	20	45	to	to	ADP
ejpam-5279	20	46	those	those	DET
ejpam-5279	20	47	works	work	NOUN
ejpam-5279	20	48	.	.	PUNCT
ejpam-5279	21	1	it	it	PRON
ejpam-5279	21	2	is	be	AUX
ejpam-5279	21	3	pointed	point	VERB
ejpam-5279	21	4	out	out	ADP
ejpam-5279	21	5	in	in	ADP
ejpam-5279	21	6	[	[	X
ejpam-5279	21	7	9	9	NUM
ejpam-5279	21	8	,	,	PUNCT
ejpam-5279	21	9	14	14	NUM
ejpam-5279	21	10	,	,	PUNCT
ejpam-5279	21	11	19	19	NUM
ejpam-5279	21	12	,	,	PUNCT
ejpam-5279	21	13	21	21	NUM
ejpam-5279	21	14	]	]	PUNCT
ejpam-5279	21	15	,	,	PUNCT
ejpam-5279	21	16	the	the	DET
ejpam-5279	21	17	importance	importance	NOUN
ejpam-5279	21	18	of	of	ADP
ejpam-5279	21	19	non	non	ADJ
ejpam-5279	21	20	-	-	ADJ
ejpam-5279	21	21	archimedean	archimedean	ADJ
ejpam-5279	21	22	approach	approach	NOUN
ejpam-5279	21	23	structures	structure	NOUN
ejpam-5279	21	24	or	or	CCONJ
ejpam-5279	21	25	ultra	ultra	ADJ
ejpam-5279	21	26	approach	approach	NOUN
ejpam-5279	21	27	structures	structure	NOUN
ejpam-5279	21	28	.	.	PUNCT
ejpam-5279	22	1	we	we	PRON
ejpam-5279	22	2	find	find	VERB
ejpam-5279	22	3	it	it	PRON
ejpam-5279	22	4	interesting	interesting	ADJ
ejpam-5279	22	5	to	to	PART
ejpam-5279	22	6	study	study	VERB
ejpam-5279	22	7	the	the	DET
ejpam-5279	22	8	compatibility	compatibility	NOUN
ejpam-5279	22	9	of	of	ADP
ejpam-5279	22	10	the	the	DET
ejpam-5279	22	11	non	non	ADJ
ejpam-5279	22	12	-	-	ADJ
ejpam-5279	22	13	archimedean	archimedean	ADJ
ejpam-5279	22	14	approach	approach	NOUN
ejpam-5279	22	15	structures	structure	NOUN
ejpam-5279	22	16	with	with	ADP
ejpam-5279	22	17	group	group	NOUN
ejpam-5279	22	18	structures	structure	NOUN
ejpam-5279	22	19	,	,	PUNCT
ejpam-5279	22	20	particularly	particularly	ADV
ejpam-5279	22	21	,	,	PUNCT
ejpam-5279	22	22	ultra	ultra	ADJ
ejpam-5279	22	23	approach	approach	NOUN
ejpam-5279	22	24	-	-	PUNCT
ejpam-5279	22	25	cauchy	cauchy	NOUN
ejpam-5279	22	26	structures	structure	NOUN
ejpam-5279	22	27	,	,	PUNCT
ejpam-5279	22	28	ultra	ultra	ADJ
ejpam-5279	22	29	approach	approach	NOUN
ejpam-5279	22	30	limit	limit	NOUN
ejpam-5279	22	31	structures	structure	NOUN
ejpam-5279	22	32	,	,	PUNCT
ejpam-5279	22	33	and	and	CCONJ
ejpam-5279	22	34	so	so	ADV
ejpam-5279	22	35	on	on	ADV
ejpam-5279	22	36	;	;	PUNCT
ejpam-5279	22	37	although	although	SCONJ
ejpam-5279	22	38	we	we	PRON
ejpam-5279	22	39	do	do	AUX
ejpam-5279	22	40	not	not	PART
ejpam-5279	22	41	intend	intend	VERB
ejpam-5279	22	42	to	to	PART
ejpam-5279	22	43	study	study	VERB
ejpam-5279	22	44	nonarchimedean	nonarchimedean	ADJ
ejpam-5279	22	45	metric	metric	ADJ
ejpam-5279	22	46	group	group	NOUN
ejpam-5279	22	47	or	or	CCONJ
ejpam-5279	22	48	ultra	ultra	ADJ
ejpam-5279	22	49	metric	metric	ADJ
ejpam-5279	22	50	group	group	NOUN
ejpam-5279	22	51	structures	structure	NOUN
ejpam-5279	22	52	here	here	ADV
ejpam-5279	22	53	in	in	ADP
ejpam-5279	22	54	this	this	DET
ejpam-5279	22	55	paper	paper	NOUN
ejpam-5279	22	56	.	.	PUNCT
ejpam-5279	23	1	the	the	DET
ejpam-5279	23	2	idea	idea	NOUN
ejpam-5279	23	3	of	of	ADP
ejpam-5279	23	4	approach	approach	NOUN
ejpam-5279	23	5	group	group	NOUN
ejpam-5279	23	6	along	along	ADP
ejpam-5279	23	7	with	with	ADP
ejpam-5279	23	8	its	its	PRON
ejpam-5279	23	9	uniformization	uniformization	NOUN
ejpam-5279	23	10	first	first	ADV
ejpam-5279	23	11	appeared	appear	VERB
ejpam-5279	23	12	in	in	ADP
ejpam-5279	23	13	[	[	X
ejpam-5279	23	14	18	18	NUM
ejpam-5279	23	15	]	]	PUNCT
ejpam-5279	23	16	,	,	PUNCT
ejpam-5279	23	17	and	and	CCONJ
ejpam-5279	23	18	later	later	ADV
ejpam-5279	23	19	,	,	PUNCT
ejpam-5279	23	20	we	we	PRON
ejpam-5279	23	21	modified	modify	VERB
ejpam-5279	23	22	this	this	DET
ejpam-5279	23	23	concept	concept	NOUN
ejpam-5279	23	24	further	far	ADV
ejpam-5279	23	25	in	in	ADP
ejpam-5279	23	26	[	[	X
ejpam-5279	23	27	4	4	X
ejpam-5279	23	28	]	]	PUNCT
ejpam-5279	23	29	in	in	ADP
ejpam-5279	23	30	a	a	DET
ejpam-5279	23	31	wider	wide	ADJ
ejpam-5279	23	32	context	context	NOUN
ejpam-5279	23	33	.	.	PUNCT
ejpam-5279	24	1	furthermore	furthermore	ADV
ejpam-5279	24	2	,	,	PUNCT
ejpam-5279	24	3	we	we	PRON
ejpam-5279	24	4	identified	identify	VERB
ejpam-5279	24	5	this	this	DET
ejpam-5279	24	6	approach	approach	NOUN
ejpam-5279	24	7	group	group	NOUN
ejpam-5279	24	8	with	with	ADP
ejpam-5279	24	9	some	some	DET
ejpam-5279	24	10	other	other	ADJ
ejpam-5279	24	11	structures	structure	NOUN
ejpam-5279	24	12	cf	cf	VERB
ejpam-5279	24	13	.	.	PUNCT
ejpam-5279	25	1	[	[	X
ejpam-5279	25	2	2	2	NUM
ejpam-5279	25	3	,	,	PUNCT
ejpam-5279	25	4	3	3	NUM
ejpam-5279	25	5	]	]	PUNCT
ejpam-5279	25	6	.	.	PUNCT
ejpam-5279	26	1	in	in	ADP
ejpam-5279	26	2	this	this	DET
ejpam-5279	26	3	paper	paper	NOUN
ejpam-5279	26	4	,	,	PUNCT
ejpam-5279	26	5	we	we	PRON
ejpam-5279	26	6	characterize	characterize	VERB
ejpam-5279	26	7	approach	approach	NOUN
ejpam-5279	26	8	groups	group	NOUN
ejpam-5279	26	9	vis	vis	X
ejpam-5279	26	10	-	-	PUNCT
ejpam-5279	26	11	à-vis	à-vis	VERB
ejpam-5279	26	12	ultra	ultra	ADJ
ejpam-5279	26	13	approach	approach	NOUN
ejpam-5279	26	14	groups	group	NOUN
ejpam-5279	26	15	,	,	PUNCT
ejpam-5279	26	16	and	and	CCONJ
ejpam-5279	26	17	study	study	VERB
ejpam-5279	26	18	some	some	PRON
ejpam-5279	26	19	of	of	ADP
ejpam-5279	26	20	their	their	PRON
ejpam-5279	26	21	related	relate	VERB
ejpam-5279	26	22	results	result	NOUN
ejpam-5279	26	23	.	.	PUNCT
ejpam-5279	27	1	thus	thus	ADV
ejpam-5279	27	2	,	,	PUNCT
ejpam-5279	27	3	we	we	PRON
ejpam-5279	27	4	concentrate	concentrate	VERB
ejpam-5279	27	5	on	on	ADP
ejpam-5279	27	6	three	three	NUM
ejpam-5279	27	7	main	main	ADJ
ejpam-5279	27	8	issues	issue	NOUN
ejpam-5279	27	9	in	in	ADP
ejpam-5279	27	10	relation	relation	NOUN
ejpam-5279	27	11	with	with	ADP
ejpam-5279	27	12	approach	approach	NOUN
ejpam-5279	27	13	groups	group	NOUN
ejpam-5279	27	14	,	,	PUNCT
ejpam-5279	27	15	such	such	ADJ
ejpam-5279	27	16	as	as	ADP
ejpam-5279	27	17	,	,	PUNCT
ejpam-5279	27	18	(	(	PUNCT
ejpam-5279	27	19	a	a	X
ejpam-5279	27	20	)	)	PUNCT
ejpam-5279	27	21	ultra	ultra	ADJ
ejpam-5279	27	22	approach	approach	NOUN
ejpam-5279	27	23	groups	group	NOUN
ejpam-5279	27	24	and	and	CCONJ
ejpam-5279	27	25	some	some	PRON
ejpam-5279	27	26	of	of	ADP
ejpam-5279	27	27	their	their	PRON
ejpam-5279	27	28	characterizations	characterization	NOUN
ejpam-5279	27	29	including	include	VERB
ejpam-5279	27	30	ultra	ultra	ADJ
ejpam-5279	27	31	uniformization	uniformization	NOUN
ejpam-5279	27	32	of	of	ADP
ejpam-5279	27	33	ultra	ultra	ADJ
ejpam-5279	27	34	approach	approach	NOUN
ejpam-5279	27	35	groups	group	NOUN
ejpam-5279	27	36	which	which	PRON
ejpam-5279	27	37	however	however	ADV
ejpam-5279	27	38	have	have	AUX
ejpam-5279	27	39	not	not	PART
ejpam-5279	27	40	been	be	AUX
ejpam-5279	27	41	considered	consider	VERB
ejpam-5279	27	42	in	in	ADP
ejpam-5279	27	43	[	[	X
ejpam-5279	27	44	18	18	NUM
ejpam-5279	27	45	]	]	PUNCT
ejpam-5279	27	46	although	although	SCONJ
ejpam-5279	27	47	the	the	DET
ejpam-5279	27	48	idea	idea	NOUN
ejpam-5279	27	49	of	of	ADP
ejpam-5279	27	50	ultra	ultra	ADJ
ejpam-5279	27	51	approach	approach	NOUN
ejpam-5279	27	52	spaces	space	NOUN
ejpam-5279	27	53	,	,	PUNCT
ejpam-5279	27	54	and	and	CCONJ
ejpam-5279	27	55	ultra	ultra	ADJ
ejpam-5279	27	56	uniform	uniform	ADJ
ejpam-5279	27	57	spaces	space	NOUN
ejpam-5279	27	58	are	be	AUX
ejpam-5279	27	59	crept	creep	VERB
ejpam-5279	27	60	inside	inside	ADV
ejpam-5279	27	61	in	in	ADP
ejpam-5279	27	62	[	[	X
ejpam-5279	27	63	14	14	NUM
ejpam-5279	27	64	]	]	PUNCT
ejpam-5279	27	65	in	in	ADP
ejpam-5279	27	66	addition	addition	NOUN
ejpam-5279	27	67	to	to	ADP
ejpam-5279	27	68	some	some	DET
ejpam-5279	27	69	other	other	ADJ
ejpam-5279	27	70	papers	paper	NOUN
ejpam-5279	27	71	;	;	PUNCT
ejpam-5279	27	72	(	(	PUNCT
ejpam-5279	27	73	b	b	X
ejpam-5279	27	74	)	)	PUNCT
ejpam-5279	27	75	considering	consider	VERB
ejpam-5279	27	76	natural	natural	ADJ
ejpam-5279	27	77	connection	connection	NOUN
ejpam-5279	27	78	of	of	ADP
ejpam-5279	27	79	neighborhood	neighborhood	NOUN
ejpam-5279	27	80	system	system	NOUN
ejpam-5279	27	81	with	with	ADP
ejpam-5279	27	82	approach	approach	NOUN
ejpam-5279	27	83	spaces	space	NOUN
ejpam-5279	27	84	,	,	PUNCT
ejpam-5279	27	85	we	we	PRON
ejpam-5279	27	86	characterize	characterize	VERB
ejpam-5279	27	87	approach	approach	NOUN
ejpam-5279	27	88	group	group	NOUN
ejpam-5279	27	89	by	by	ADP
ejpam-5279	27	90	compatible	compatible	ADJ
ejpam-5279	27	91	neighborhood	neighborhood	NOUN
ejpam-5279	27	92	system	system	NOUN
ejpam-5279	27	93	on	on	ADP
ejpam-5279	27	94	group	group	NOUN
ejpam-5279	27	95	structure	structure	NOUN
ejpam-5279	27	96	;	;	PUNCT
ejpam-5279	27	97	(	(	PUNCT
ejpam-5279	27	98	c	c	X
ejpam-5279	27	99	)	)	PUNCT
ejpam-5279	27	100	considering	consider	VERB
ejpam-5279	27	101	approach	approach	NOUN
ejpam-5279	27	102	-	-	PUNCT
ejpam-5279	27	103	cauchy	cauchy	NOUN
ejpam-5279	27	104	structures	structure	NOUN
ejpam-5279	27	105	,	,	PUNCT
ejpam-5279	27	106	we	we	PRON
ejpam-5279	27	107	show	show	VERB
ejpam-5279	27	108	that	that	SCONJ
ejpam-5279	27	109	for	for	ADP
ejpam-5279	27	110	a	a	DET
ejpam-5279	27	111	group	group	NOUN
ejpam-5279	27	112	,	,	PUNCT
ejpam-5279	27	113	there	there	PRON
ejpam-5279	27	114	is	be	VERB
ejpam-5279	27	115	a	a	DET
ejpam-5279	27	116	one	one	NUM
ejpam-5279	27	117	-	-	PUNCT
ejpam-5279	27	118	to	to	ADP
ejpam-5279	27	119	-	-	PUNCT
ejpam-5279	27	120	one	one	NUM
ejpam-5279	27	121	correspondence	correspondence	NOUN
ejpam-5279	27	122	between	between	ADP
ejpam-5279	27	123	ultra	ultra	ADJ
ejpam-5279	27	124	approach	approach	NOUN
ejpam-5279	27	125	-	-	PUNCT
ejpam-5279	27	126	cauchy	cauchy	NOUN
ejpam-5279	27	127	group	group	NOUN
ejpam-5279	27	128	structures	structure	NOUN
ejpam-5279	27	129	and	and	CCONJ
ejpam-5279	27	130	strongly	strongly	ADV
ejpam-5279	27	131	normal	normal	ADJ
ejpam-5279	27	132	ultra	ultra	ADJ
ejpam-5279	27	133	approach	approach	NOUN
ejpam-5279	27	134	limit	limit	NOUN
ejpam-5279	27	135	group	group	NOUN
ejpam-5279	27	136	structures	structure	NOUN
ejpam-5279	27	137	,	,	PUNCT
ejpam-5279	27	138	the	the	DET
ejpam-5279	27	139	idea	idea	NOUN
ejpam-5279	27	140	of	of	ADP
ejpam-5279	27	141	strong	strong	ADJ
ejpam-5279	27	142	normality	normality	NOUN
ejpam-5279	27	143	first	first	ADV
ejpam-5279	27	144	appeared	appear	VERB
ejpam-5279	27	145	in	in	ADP
ejpam-5279	27	146	[	[	X
ejpam-5279	27	147	5	5	NUM
ejpam-5279	27	148	]	]	PUNCT
ejpam-5279	27	149	;	;	PUNCT
ejpam-5279	27	150	in	in	ADP
ejpam-5279	27	151	fact	fact	NOUN
ejpam-5279	27	152	,	,	PUNCT
ejpam-5279	27	153	we	we	PRON
ejpam-5279	27	154	prove	prove	VERB
ejpam-5279	27	155	here	here	ADV
ejpam-5279	27	156	that	that	SCONJ
ejpam-5279	27	157	the	the	DET
ejpam-5279	27	158	category	category	NOUN
ejpam-5279	27	159	of	of	ADP
ejpam-5279	27	160	ultra	ultra	ADJ
ejpam-5279	27	161	approach	approach	NOUN
ejpam-5279	27	162	-	-	PUNCT
ejpam-5279	27	163	cauchy	cauchy	NOUN
ejpam-5279	27	164	groups	group	NOUN
ejpam-5279	27	165	and	and	CCONJ
ejpam-5279	27	166	the	the	DET
ejpam-5279	27	167	category	category	NOUN
ejpam-5279	27	168	of	of	ADP
ejpam-5279	27	169	strongly	strongly	ADV
ejpam-5279	27	170	normal	normal	ADJ
ejpam-5279	27	171	ultra	ultra	ADJ
ejpam-5279	27	172	approach	approach	NOUN
ejpam-5279	27	173	limit	limit	NOUN
ejpam-5279	27	174	groups	group	NOUN
ejpam-5279	27	175	are	be	AUX
ejpam-5279	27	176	isomorphic	isomorphic	ADJ
ejpam-5279	27	177	.	.	PUNCT
ejpam-5279	28	1	we	we	PRON
ejpam-5279	28	2	arrange	arrange	VERB
ejpam-5279	28	3	these	these	DET
ejpam-5279	28	4	findings	finding	NOUN
ejpam-5279	28	5	as	as	SCONJ
ejpam-5279	28	6	follows	follow	VERB
ejpam-5279	28	7	.	.	PUNCT
ejpam-5279	29	1	in	in	ADP
ejpam-5279	29	2	section	section	NOUN
ejpam-5279	29	3	2	2	NUM
ejpam-5279	29	4	,	,	PUNCT
ejpam-5279	29	5	we	we	PRON
ejpam-5279	29	6	consider	consider	VERB
ejpam-5279	29	7	some	some	DET
ejpam-5279	29	8	basic	basic	ADJ
ejpam-5279	29	9	facts	fact	NOUN
ejpam-5279	29	10	that	that	PRON
ejpam-5279	29	11	are	be	AUX
ejpam-5279	29	12	used	use	VERB
ejpam-5279	29	13	in	in	ADP
ejpam-5279	29	14	the	the	DET
ejpam-5279	29	15	sequel	sequel	NOUN
ejpam-5279	29	16	.	.	PUNCT
ejpam-5279	30	1	we	we	PRON
ejpam-5279	30	2	present	present	VERB
ejpam-5279	30	3	the	the	DET
ejpam-5279	30	4	notion	notion	NOUN
ejpam-5279	30	5	of	of	ADP
ejpam-5279	30	6	ultra	ultra	ADJ
ejpam-5279	30	7	approach	approach	NOUN
ejpam-5279	30	8	groups	group	NOUN
ejpam-5279	30	9	in	in	ADP
ejpam-5279	30	10	section	section	NOUN
ejpam-5279	30	11	3	3	NUM
ejpam-5279	30	12	,	,	PUNCT
ejpam-5279	30	13	provide	provide	VERB
ejpam-5279	30	14	characterization	characterization	NOUN
ejpam-5279	30	15	theorems	theorem	NOUN
ejpam-5279	30	16	,	,	PUNCT
ejpam-5279	30	17	and	and	CCONJ
ejpam-5279	30	18	ultra	ultra	ADJ
ejpam-5279	30	19	uniformization	uniformization	NOUN
ejpam-5279	30	20	of	of	ADP
ejpam-5279	30	21	ultra	ultra	ADJ
ejpam-5279	30	22	approach	approach	NOUN
ejpam-5279	30	23	groups	group	NOUN
ejpam-5279	30	24	.	.	PUNCT
ejpam-5279	31	1	using	use	VERB
ejpam-5279	31	2	the	the	DET
ejpam-5279	31	3	notion	notion	NOUN
ejpam-5279	31	4	of	of	ADP
ejpam-5279	31	5	neighborhood	neighborhood	NOUN
ejpam-5279	31	6	system	system	NOUN
ejpam-5279	31	7	as	as	SCONJ
ejpam-5279	31	8	defined	define	VERB
ejpam-5279	31	9	in	in	ADP
ejpam-5279	31	10	approach	approach	NOUN
ejpam-5279	31	11	space	space	NOUN
ejpam-5279	31	12	,	,	PUNCT
ejpam-5279	31	13	we	we	PRON
ejpam-5279	31	14	give	give	VERB
ejpam-5279	31	15	characterization	characterization	NOUN
ejpam-5279	31	16	theorem	theorem	VERB
ejpam-5279	31	17	on	on	ADP
ejpam-5279	31	18	approach	approach	NOUN
ejpam-5279	31	19	group	group	NOUN
ejpam-5279	31	20	in	in	ADP
ejpam-5279	31	21	section4	section4	PROPN
ejpam-5279	31	22	.	.	PUNCT
ejpam-5279	32	1	relation	relation	NOUN
ejpam-5279	32	2	between	between	ADP
ejpam-5279	32	3	approach	approach	NOUN
ejpam-5279	32	4	groups	group	NOUN
ejpam-5279	32	5	and	and	CCONJ
ejpam-5279	32	6	approach	approach	NOUN
ejpam-5279	32	7	limit	limit	NOUN
ejpam-5279	32	8	groups	group	NOUN
ejpam-5279	32	9	are	be	AUX
ejpam-5279	32	10	discussed	discuss	VERB
ejpam-5279	32	11	in	in	ADP
ejpam-5279	32	12	section	section	NOUN
ejpam-5279	32	13	5	5	NUM
ejpam-5279	32	14	.	.	PUNCT
ejpam-5279	33	1	finally	finally	ADV
ejpam-5279	33	2	,	,	PUNCT
ejpam-5279	33	3	we	we	PRON
ejpam-5279	33	4	describe	describe	VERB
ejpam-5279	33	5	the	the	DET
ejpam-5279	33	6	connection	connection	NOUN
ejpam-5279	33	7	between	between	ADP
ejpam-5279	33	8	ultra	ultra	ADJ
ejpam-5279	33	9	approach	approach	NOUN
ejpam-5279	33	10	-	-	PUNCT
ejpam-5279	33	11	cauchy	cauchy	NOUN
ejpam-5279	33	12	groups	group	NOUN
ejpam-5279	33	13	and	and	CCONJ
ejpam-5279	33	14	strongly	strongly	ADV
ejpam-5279	33	15	normal	normal	ADJ
ejpam-5279	33	16	approach	approach	NOUN
ejpam-5279	33	17	-	-	PUNCT
ejpam-5279	33	18	cauchy	cauchy	NOUN
ejpam-5279	33	19	groups	group	NOUN
ejpam-5279	33	20	in	in	ADP
ejpam-5279	33	21	section	section	NOUN
ejpam-5279	33	22	6	6	NUM
ejpam-5279	33	23	.	.	NOUN
ejpam-5279	33	24	2	2	NUM
ejpam-5279	33	25	.	.	X
ejpam-5279	33	26	preliminaries	preliminary	NOUN
ejpam-5279	33	27	we	we	PRON
ejpam-5279	33	28	denote	denote	VERB
ejpam-5279	33	29	the	the	DET
ejpam-5279	33	30	set	set	NOUN
ejpam-5279	33	31	of	of	ADP
ejpam-5279	33	32	all	all	DET
ejpam-5279	33	33	filters	filter	NOUN
ejpam-5279	33	34	f	f	NOUN
ejpam-5279	33	35	,	,	PUNCT
ejpam-5279	33	36	g	g	PROPN
ejpam-5279	33	37	,	,	PUNCT
ejpam-5279	33	38	...	...	PUNCT
ejpam-5279	33	39	on	on	ADP
ejpam-5279	33	40	a	a	DET
ejpam-5279	33	41	set	set	NOUN
ejpam-5279	33	42	x	x	PUNCT
ejpam-5279	33	43	by	by	ADP
ejpam-5279	33	44	f(x	f(x	PROPN
ejpam-5279	33	45	)	)	PUNCT
ejpam-5279	33	46	.	.	PUNCT
ejpam-5279	34	1	the	the	DET
ejpam-5279	34	2	point	point	NOUN
ejpam-5279	34	3	filter	filter	NOUN
ejpam-5279	34	4	of	of	ADP
ejpam-5279	34	5	a	a	DET
ejpam-5279	34	6	point	point	NOUN
ejpam-5279	34	7	x	x	X
ejpam-5279	34	8	∈	∈	NOUN
ejpam-5279	34	9	x	x	PUNCT
ejpam-5279	34	10	is	be	AUX
ejpam-5279	34	11	defined	define	VERB
ejpam-5279	34	12	by	by	ADP
ejpam-5279	34	13	ẋ	ẋ	PROPN
ejpam-5279	35	1	=	=	PRON
ejpam-5279	35	2	{	{	PUNCT
ejpam-5279	35	3	a	a	PRON
ejpam-5279	35	4	⊆	⊆	NUM
ejpam-5279	35	5	x	x	SYM
ejpam-5279	35	6	:	:	PUNCT
ejpam-5279	35	7	x	x	SYM
ejpam-5279	35	8	∈	∈	PROPN
ejpam-5279	35	9	a	a	PRON
ejpam-5279	35	10	}	}	PUNCT
ejpam-5279	35	11	,	,	PUNCT
ejpam-5279	35	12	or	or	CCONJ
ejpam-5279	35	13	by	by	ADP
ejpam-5279	35	14	[	[	X
ejpam-5279	35	15	x	x	X
ejpam-5279	35	16	]	]	X
ejpam-5279	35	17	.	.	PUNCT
ejpam-5279	36	1	the	the	DET
ejpam-5279	36	2	set	set	ADJ
ejpam-5279	36	3	f(x	f(x	PROPN
ejpam-5279	36	4	)	)	PUNCT
ejpam-5279	36	5	is	be	AUX
ejpam-5279	36	6	ordered	order	VERB
ejpam-5279	36	7	by	by	ADP
ejpam-5279	36	8	set	set	VERB
ejpam-5279	36	9	inclusion	inclusion	NOUN
ejpam-5279	36	10	,	,	PUNCT
ejpam-5279	36	11	i.e.	i.e.	X
ejpam-5279	36	12	,	,	PUNCT
ejpam-5279	36	13	we	we	PRON
ejpam-5279	36	14	write	write	VERB
ejpam-5279	36	15	f	f	PROPN
ejpam-5279	36	16	≤	≤	ADV
ejpam-5279	36	17	g	g	NOUN
ejpam-5279	37	1	if	if	SCONJ
ejpam-5279	37	2	f	f	PROPN
ejpam-5279	37	3	⊆	⊆	NUM
ejpam-5279	37	4	g.	g.	NOUN
ejpam-5279	37	5	if	if	SCONJ
ejpam-5279	37	6	(	(	PUNCT
ejpam-5279	37	7	fj)j∈j	fj)j∈j	NUM
ejpam-5279	37	8	is	be	AUX
ejpam-5279	37	9	a	a	DET
ejpam-5279	37	10	family	family	NOUN
ejpam-5279	37	11	of	of	ADP
ejpam-5279	37	12	filters	filter	NOUN
ejpam-5279	37	13	on	on	ADP
ejpam-5279	37	14	a	a	DET
ejpam-5279	37	15	set	set	NOUN
ejpam-5279	37	16	x	x	NOUN
ejpam-5279	37	17	,	,	PUNCT
ejpam-5279	37	18	then	then	ADV
ejpam-5279	37	19	for	for	ADP
ejpam-5279	37	20	a	a	DET
ejpam-5279	37	21	filter	filter	NOUN
ejpam-5279	37	22	u	u	NOUN
ejpam-5279	37	23	on	on	ADP
ejpam-5279	37	24	j	j	PROPN
ejpam-5279	37	25	,	,	PUNCT
ejpam-5279	37	26	the	the	DET
ejpam-5279	37	27	compressed	compress	VERB
ejpam-5279	37	28	operator	operator	NOUN
ejpam-5279	37	29	κ	κ	NOUN
ejpam-5279	37	30	(	(	PUNCT
ejpam-5279	37	31	u	u	NOUN
ejpam-5279	37	32	,	,	PUNCT
ejpam-5279	37	33	(	(	PUNCT
ejpam-5279	37	34	fj)j∈j	fj)j∈j	NUM
ejpam-5279	37	35	)	)	PUNCT
ejpam-5279	37	36	is	be	AUX
ejpam-5279	37	37	defined	define	VERB
ejpam-5279	37	38	by	by	ADP
ejpam-5279	37	39	[	[	X
ejpam-5279	37	40	15	15	NUM
ejpam-5279	37	41	]	]	X
ejpam-5279	37	42	κ	κ	X
ejpam-5279	37	43	(	(	PUNCT
ejpam-5279	37	44	u	u	NOUN
ejpam-5279	37	45	,	,	PUNCT
ejpam-5279	37	46	(	(	PUNCT
ejpam-5279	37	47	fj)j∈j	fj)j∈j	NUM
ejpam-5279	37	48	)	)	PUNCT
ejpam-5279	37	49	=	=	PUNCT
ejpam-5279	38	1	∨	∨	NUM
ejpam-5279	38	2	v	v	NUM
ejpam-5279	38	3	∈u	∈u	VERB
ejpam-5279	38	4	∧	∧	PROPN
ejpam-5279	38	5	j∈v	j∈v	NOUN
ejpam-5279	38	6	fj	fj	INTJ
ejpam-5279	38	7	.	.	PUNCT
ejpam-5279	39	1	if	if	SCONJ
ejpam-5279	39	2	f	f	X
ejpam-5279	39	3	,	,	PUNCT
ejpam-5279	39	4	g	g	PROPN
ejpam-5279	39	5	∈	∈	PROPN
ejpam-5279	39	6	f(x	f(x	PROPN
ejpam-5279	39	7	)	)	PUNCT
ejpam-5279	39	8	,	,	PUNCT
ejpam-5279	39	9	then	then	ADV
ejpam-5279	39	10	the	the	DET
ejpam-5279	39	11	product	product	NOUN
ejpam-5279	39	12	filter	filter	NOUN
ejpam-5279	39	13	f	f	NOUN
ejpam-5279	39	14	×	×	NOUN
ejpam-5279	39	15	g	g	PROPN
ejpam-5279	39	16	=	=	PRON
ejpam-5279	39	17	<	<	X
ejpam-5279	39	18	{	{	PUNCT
ejpam-5279	39	19	f	f	PROPN
ejpam-5279	39	20	×	×	PROPN
ejpam-5279	39	21	g|f	g|f	PROPN
ejpam-5279	39	22	∈	∈	PROPN
ejpam-5279	39	23	f	f	AUX
ejpam-5279	39	24	,	,	PUNCT
ejpam-5279	39	25	g	g	PROPN
ejpam-5279	39	26	∈	∈	PROPN
ejpam-5279	39	27	g	g	PROPN
ejpam-5279	39	28	}	}	PUNCT
ejpam-5279	39	29	>	>	X
ejpam-5279	39	30	,	,	PUNCT
ejpam-5279	39	31	i.e.	i.e.	X
ejpam-5279	39	32	,	,	PUNCT
ejpam-5279	39	33	we	we	PRON
ejpam-5279	39	34	have	have	VERB
ejpam-5279	39	35	{	{	PUNCT
ejpam-5279	39	36	f	f	PROPN
ejpam-5279	39	37	×g|f	×g|f	NOUN
ejpam-5279	39	38	∈	∈	PROPN
ejpam-5279	39	39	f	f	X
ejpam-5279	39	40	,	,	PUNCT
ejpam-5279	39	41	g	g	PROPN
ejpam-5279	39	42	∈	∈	PROPN
ejpam-5279	39	43	g	g	PROPN
ejpam-5279	39	44	}	}	PUNCT
ejpam-5279	39	45	as	as	ADP
ejpam-5279	39	46	a	a	DET
ejpam-5279	39	47	basis	basis	NOUN
ejpam-5279	39	48	for	for	ADP
ejpam-5279	39	49	f×g	f×g	PROPN
ejpam-5279	39	50	.	.	PUNCT
ejpam-5279	40	1	for	for	ADP
ejpam-5279	40	2	f	f	PROPN
ejpam-5279	40	3	,	,	PUNCT
ejpam-5279	40	4	g	g	PROPN
ejpam-5279	40	5	∈	∈	PROPN
ejpam-5279	40	6	f(x	f(x	PROPN
ejpam-5279	40	7	)	)	PUNCT
ejpam-5279	40	8	we	we	PRON
ejpam-5279	40	9	define	define	VERB
ejpam-5279	40	10	t.m.g	t.m.g	ADJ
ejpam-5279	40	11	.	.	PUNCT
ejpam-5279	41	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	41	2	,	,	PUNCT
ejpam-5279	41	3	fawzi	fawzi	PROPN
ejpam-5279	41	4	al	al	PROPN
ejpam-5279	41	5	-	-	PUNCT
ejpam-5279	41	6	thukair	thukair	NOUN
ejpam-5279	41	7	/	/	SYM
ejpam-5279	41	8	eur	eur	NOUN
ejpam-5279	41	9	.	.	PUNCT
ejpam-5279	42	1	j.	j.	PROPN
ejpam-5279	42	2	pure	pure	PROPN
ejpam-5279	42	3	appl	appl	PROPN
ejpam-5279	42	4	.	.	PROPN
ejpam-5279	42	5	math	math	PROPN
ejpam-5279	42	6	,	,	PUNCT
ejpam-5279	42	7	17	17	NUM
ejpam-5279	42	8	(	(	PUNCT
ejpam-5279	42	9	3	3	NUM
ejpam-5279	42	10	)	)	PUNCT
ejpam-5279	42	11	(	(	PUNCT
ejpam-5279	42	12	2024	2024	NUM
ejpam-5279	42	13	)	)	PUNCT
ejpam-5279	42	14	,	,	PUNCT
ejpam-5279	42	15	1762	1762	NUM
ejpam-5279	42	16	-	-	SYM
ejpam-5279	42	17	1778	1778	NUM
ejpam-5279	42	18	1764	1764	NUM
ejpam-5279	42	19	f⊙g	f⊙g	NOUN
ejpam-5279	43	1	=	=	SYM
ejpam-5279	43	2	m	m	PROPN
ejpam-5279	43	3	(	(	PUNCT
ejpam-5279	43	4	f×g	f×g	NUM
ejpam-5279	43	5	)	)	PUNCT
ejpam-5279	43	6	and	and	CCONJ
ejpam-5279	43	7	f−1	f−1	PROPN
ejpam-5279	43	8	=	=	SYM
ejpam-5279	43	9	i(f	i(f	PROPN
ejpam-5279	43	10	)	)	PUNCT
ejpam-5279	43	11	.	.	PUNCT
ejpam-5279	44	1	noting	note	VERB
ejpam-5279	44	2	that	that	SCONJ
ejpam-5279	44	3	m	m	PROPN
ejpam-5279	44	4	(	(	PUNCT
ejpam-5279	44	5	f	f	NOUN
ejpam-5279	44	6	×g	×g	NOUN
ejpam-5279	44	7	)	)	PUNCT
ejpam-5279	44	8	=	=	PRON
ejpam-5279	44	9	{	{	PUNCT
ejpam-5279	44	10	xy|x	xy|x	PROPN
ejpam-5279	44	11	∈	∈	PROPN
ejpam-5279	45	1	f	f	X
ejpam-5279	45	2	,	,	PUNCT
ejpam-5279	45	3	y	y	PROPN
ejpam-5279	45	4	∈	∈	PROPN
ejpam-5279	45	5	g	g	PROPN
ejpam-5279	45	6	}	}	PUNCT
ejpam-5279	45	7	=	=	SYM
ejpam-5279	45	8	f	f	PROPN
ejpam-5279	45	9	⊙g	⊙g	PROPN
ejpam-5279	45	10	,	,	PUNCT
ejpam-5279	45	11	we	we	PRON
ejpam-5279	45	12	have	have	VERB
ejpam-5279	45	13	{	{	PUNCT
ejpam-5279	45	14	f	f	NOUN
ejpam-5279	45	15	⊙g|f	⊙g|f	NOUN
ejpam-5279	45	16	∈	∈	PROPN
ejpam-5279	46	1	f	f	X
ejpam-5279	46	2	,	,	PUNCT
ejpam-5279	46	3	g	g	PROPN
ejpam-5279	46	4	∈	∈	PROPN
ejpam-5279	46	5	g	g	PROPN
ejpam-5279	46	6	}	}	PUNCT
ejpam-5279	46	7	as	as	ADP
ejpam-5279	46	8	a	a	DET
ejpam-5279	46	9	basis	basis	NOUN
ejpam-5279	46	10	for	for	ADP
ejpam-5279	46	11	f⊙g	f⊙g	NOUN
ejpam-5279	46	12	.	.	PUNCT
ejpam-5279	47	1	similarly	similarly	ADV
ejpam-5279	47	2	,	,	PUNCT
ejpam-5279	47	3	we	we	PRON
ejpam-5279	47	4	find	find	VERB
ejpam-5279	47	5	{	{	PUNCT
ejpam-5279	47	6	f−1|f	f−1|f	PROPN
ejpam-5279	47	7	∈	∈	PROPN
ejpam-5279	47	8	f	f	X
ejpam-5279	47	9	}	}	PUNCT
ejpam-5279	47	10	as	as	ADP
ejpam-5279	47	11	a	a	DET
ejpam-5279	47	12	basis	basis	NOUN
ejpam-5279	47	13	for	for	ADP
ejpam-5279	47	14	f−1	f−1	PROPN
ejpam-5279	47	15	,	,	PUNCT
ejpam-5279	47	16	where	where	SCONJ
ejpam-5279	47	17	f−1	f−1	PROPN
ejpam-5279	47	18	=	=	SYM
ejpam-5279	47	19	{	{	PUNCT
ejpam-5279	47	20	x−1|x	x−1|x	PROPN
ejpam-5279	47	21	∈	∈	PROPN
ejpam-5279	47	22	f	f	X
ejpam-5279	47	23	}	}	PUNCT
ejpam-5279	47	24	.	.	PUNCT
ejpam-5279	48	1	throughout	throughout	ADP
ejpam-5279	48	2	the	the	DET
ejpam-5279	48	3	text	text	NOUN
ejpam-5279	48	4	for	for	ADP
ejpam-5279	48	5	a	a	DET
ejpam-5279	48	6	group	group	NOUN
ejpam-5279	48	7	(	(	PUNCT
ejpam-5279	48	8	x	x	X
ejpam-5279	48	9	,	,	PUNCT
ejpam-5279	48	10	·	·	PUNCT
ejpam-5279	48	11	)	)	PUNCT
ejpam-5279	48	12	,	,	PUNCT
ejpam-5279	48	13	we	we	PRON
ejpam-5279	48	14	consider	consider	VERB
ejpam-5279	48	15	e	e	NOUN
ejpam-5279	48	16	as	as	ADP
ejpam-5279	48	17	the	the	DET
ejpam-5279	48	18	identity	identity	NOUN
ejpam-5279	48	19	element	element	NOUN
ejpam-5279	48	20	.	.	PUNCT
ejpam-5279	49	1	lemma	lemma	PROPN
ejpam-5279	49	2	1	1	X
ejpam-5279	49	3	.	.	PUNCT
ejpam-5279	50	1	let	let	VERB
ejpam-5279	50	2	x	x	PRON
ejpam-5279	50	3	and	and	CCONJ
ejpam-5279	50	4	y	y	PROPN
ejpam-5279	50	5	be	be	AUX
ejpam-5279	50	6	groups	group	NOUN
ejpam-5279	50	7	,	,	PUNCT
ejpam-5279	50	8	f	f	X
ejpam-5279	50	9	,	,	PUNCT
ejpam-5279	50	10	g	g	PROPN
ejpam-5279	50	11	,	,	PUNCT
ejpam-5279	50	12	h	h	NOUN
ejpam-5279	50	13	∈	∈	PROPN
ejpam-5279	50	14	f(x	f(x	PROPN
ejpam-5279	50	15	)	)	PUNCT
ejpam-5279	50	16	and	and	CCONJ
ejpam-5279	50	17	f	f	X
ejpam-5279	50	18	:	:	PUNCT
ejpam-5279	50	19	x	x	PUNCT
ejpam-5279	50	20	−→	−→	NOUN
ejpam-5279	50	21	y	y	PROPN
ejpam-5279	50	22	a	a	DET
ejpam-5279	50	23	group	group	NOUN
ejpam-5279	50	24	homomorphism	homomorphism	NOUN
ejpam-5279	50	25	,	,	PUNCT
ejpam-5279	50	26	then	then	ADV
ejpam-5279	50	27	we	we	PRON
ejpam-5279	50	28	have	have	AUX
ejpam-5279	50	29	(	(	PUNCT
ejpam-5279	50	30	i	i	NOUN
ejpam-5279	50	31	)	)	PUNCT
ejpam-5279	50	32	f⊙	f⊙	VERB
ejpam-5279	50	33	f−1	f−1	PROPN
ejpam-5279	50	34	≤	≤	ADV
ejpam-5279	50	35	ė	ė	PROPN
ejpam-5279	50	36	and	and	CCONJ
ejpam-5279	50	37	f−1	f−1	PROPN
ejpam-5279	50	38	⊙	⊙	NOUN
ejpam-5279	50	39	f	f	PROPN
ejpam-5279	50	40	≤	≤	PROPN
ejpam-5279	50	41	ė	ė	PROPN
ejpam-5279	50	42	;	;	PUNCT
ejpam-5279	50	43	(	(	PUNCT
ejpam-5279	50	44	ii	ii	NOUN
ejpam-5279	50	45	)	)	PUNCT
ejpam-5279	50	46	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	51	1	(	(	PUNCT
ejpam-5279	51	2	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	51	3	=	=	PRON
ejpam-5279	51	4	(	(	PUNCT
ejpam-5279	51	5	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	51	6	⊙	⊙	PROPN
ejpam-5279	51	7	ẋ	ẋ	PROPN
ejpam-5279	52	1	=	=	SYM
ejpam-5279	52	2	ė	ė	PROPN
ejpam-5279	52	3	;	;	PUNCT
ejpam-5279	52	4	(	(	PUNCT
ejpam-5279	52	5	iii	iii	X
ejpam-5279	52	6	)	)	PUNCT
ejpam-5279	52	7	˙̂xy	˙̂xy	PROPN
ejpam-5279	52	8	=	=	SYM
ejpam-5279	53	1	ẋ⊙	ẋ⊙	NUM
ejpam-5279	53	2	ẏ	ẏ	PROPN
ejpam-5279	53	3	;	;	PUNCT
ejpam-5279	53	4	(	(	PUNCT
ejpam-5279	53	5	iv	iv	X
ejpam-5279	53	6	)	)	PUNCT
ejpam-5279	53	7	˙̂	˙̂	NOUN
ejpam-5279	53	8	x−1	x−1	PROPN
ejpam-5279	54	1	=	=	PUNCT
ejpam-5279	54	2	(	(	PUNCT
ejpam-5279	54	3	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	54	4	;	;	PUNCT
ejpam-5279	54	5	(	(	PUNCT
ejpam-5279	54	6	v	v	NOUN
ejpam-5279	54	7	)	)	PUNCT
ejpam-5279	54	8	(	(	PUNCT
ejpam-5279	54	9	f⊙g)⊙h	f⊙g)⊙h	NOUN
ejpam-5279	54	10	=	=	SYM
ejpam-5279	54	11	f⊙	f⊙	NOUN
ejpam-5279	54	12	(	(	PUNCT
ejpam-5279	54	13	g⊙h	g⊙h	ADJ
ejpam-5279	54	14	)	)	PUNCT
ejpam-5279	54	15	;	;	PUNCT
ejpam-5279	54	16	(	(	PUNCT
ejpam-5279	54	17	vi	vi	NOUN
ejpam-5279	54	18	)	)	PUNCT
ejpam-5279	54	19	(	(	PUNCT
ejpam-5279	54	20	f−1)−1	f−1)−1	X
ejpam-5279	54	21	=	=	SYM
ejpam-5279	54	22	f	f	X
ejpam-5279	54	23	;	;	PUNCT
ejpam-5279	54	24	(	(	PUNCT
ejpam-5279	54	25	vii	vii	PROPN
ejpam-5279	54	26	)	)	PUNCT
ejpam-5279	54	27	(	(	PUNCT
ejpam-5279	54	28	f⊙g)−1	f⊙g)−1	NOUN
ejpam-5279	54	29	=	=	SYM
ejpam-5279	54	30	g−1	g−1	PROPN
ejpam-5279	54	31	⊙	⊙	PROPN
ejpam-5279	55	1	f−1	f−1	PROPN
ejpam-5279	55	2	;	;	PUNCT
ejpam-5279	55	3	(	(	PUNCT
ejpam-5279	55	4	viii	viii	NOUN
ejpam-5279	55	5	)	)	PUNCT
ejpam-5279	55	6	ė⊙	ė⊙	X
ejpam-5279	56	1	f	f	NOUN
ejpam-5279	56	2	=	=	PRON
ejpam-5279	56	3	f⊙	f⊙	VERB
ejpam-5279	56	4	ė	ė	ADJ
ejpam-5279	56	5	=	=	SYM
ejpam-5279	56	6	f	f	X
ejpam-5279	56	7	;	;	PUNCT
ejpam-5279	56	8	(	(	PUNCT
ejpam-5279	56	9	ix	ix	X
ejpam-5279	56	10	)	)	PUNCT
ejpam-5279	56	11	(	(	PUNCT
ejpam-5279	56	12	f	f	NOUN
ejpam-5279	56	13	∧g)−1	∧g)−1	NOUN
ejpam-5279	56	14	=	=	SYM
ejpam-5279	56	15	f−1	f−1	PROPN
ejpam-5279	56	16	∧g−1	∧g−1	NUM
ejpam-5279	56	17	;	;	PUNCT
ejpam-5279	56	18	(	(	PUNCT
ejpam-5279	56	19	x	x	X
ejpam-5279	56	20	)	)	PUNCT
ejpam-5279	56	21	(	(	PUNCT
ejpam-5279	56	22	f	f	X
ejpam-5279	56	23	∧g)⊙h	∧g)⊙h	NUM
ejpam-5279	56	24	=	=	SYM
ejpam-5279	56	25	(	(	PUNCT
ejpam-5279	56	26	f⊙h	f⊙h	ADJ
ejpam-5279	56	27	)	)	PUNCT
ejpam-5279	56	28	∧	∧	NOUN
ejpam-5279	56	29	(	(	PUNCT
ejpam-5279	56	30	g⊙h	g⊙h	ADJ
ejpam-5279	56	31	)	)	PUNCT
ejpam-5279	56	32	;	;	PUNCT
ejpam-5279	56	33	(	(	PUNCT
ejpam-5279	56	34	xi	xi	X
ejpam-5279	56	35	)	)	PUNCT
ejpam-5279	56	36	f	f	PROPN
ejpam-5279	56	37	≤	≤	PROPN
ejpam-5279	56	38	ẋ⊙g	ẋ⊙g	PUNCT
ejpam-5279	57	1	⇔	⇔	X
ejpam-5279	57	2	(	(	PUNCT
ejpam-5279	57	3	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	57	4	⊙	⊙	PROPN
ejpam-5279	57	5	f	f	PROPN
ejpam-5279	57	6	≤	≤	PROPN
ejpam-5279	57	7	g	g	PROPN
ejpam-5279	57	8	(	(	PUNCT
ejpam-5279	57	9	resp	resp	NOUN
ejpam-5279	57	10	.	.	PUNCT
ejpam-5279	58	1	f	f	PROPN
ejpam-5279	58	2	≤	≤	PROPN
ejpam-5279	58	3	g⊙	g⊙	PROPN
ejpam-5279	58	4	ẋ	ẋ	PROPN
ejpam-5279	59	1	⇔	⇔	PROPN
ejpam-5279	59	2	f⊙	f⊙	PROPN
ejpam-5279	59	3	(	(	PUNCT
ejpam-5279	59	4	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	59	5	≤	≤	PROPN
ejpam-5279	59	6	g	g	NOUN
ejpam-5279	59	7	)	)	PUNCT
ejpam-5279	59	8	;	;	PUNCT
ejpam-5279	59	9	(	(	PUNCT
ejpam-5279	59	10	xii	xii	NOUN
ejpam-5279	59	11	)	)	PUNCT
ejpam-5279	59	12	f(f⊙g	f(f⊙g	NUM
ejpam-5279	59	13	)	)	PUNCT
ejpam-5279	59	14	=	=	PRON
ejpam-5279	59	15	f(f)⊙	f(f)⊙	PROPN
ejpam-5279	59	16	f(g	f(g	PROPN
ejpam-5279	59	17	)	)	PUNCT
ejpam-5279	59	18	;	;	PUNCT
ejpam-5279	59	19	(	(	PUNCT
ejpam-5279	59	20	xiii	xiii	X
ejpam-5279	59	21	)	)	PUNCT
ejpam-5279	59	22	f(f−1	f(f−1	PROPN
ejpam-5279	59	23	)	)	PUNCT
ejpam-5279	59	24	=	=	SYM
ejpam-5279	59	25	(	(	PUNCT
ejpam-5279	59	26	f(f))−1	f(f))−1	NOUN
ejpam-5279	59	27	.	.	PUNCT
ejpam-5279	60	1	a	a	DET
ejpam-5279	60	2	subset	subset	NOUN
ejpam-5279	60	3	ω	ω	X
ejpam-5279	60	4	⊂	⊂	PROPN
ejpam-5279	61	1	[	[	X
ejpam-5279	61	2	0,∞]x	0,∞]x	X
ejpam-5279	61	3	is	be	AUX
ejpam-5279	61	4	called	call	VERB
ejpam-5279	61	5	an	an	DET
ejpam-5279	61	6	ideal	ideal	NOUN
ejpam-5279	61	7	in	in	ADP
ejpam-5279	61	8	[	[	X
ejpam-5279	61	9	0,∞]x	0,∞]x	X
ejpam-5279	61	10	if	if	SCONJ
ejpam-5279	61	11	for	for	ADP
ejpam-5279	61	12	any	any	DET
ejpam-5279	61	13	ξ1	ξ1	NOUN
ejpam-5279	61	14	,	,	PUNCT
ejpam-5279	61	15	ξ2	ξ2	PROPN
ejpam-5279	61	16	∈	∈	PROPN
ejpam-5279	61	17	ω	ω	PROPN
ejpam-5279	61	18	,	,	PUNCT
ejpam-5279	61	19	ξ1	ξ1	PROPN
ejpam-5279	61	20	∨	∨	NUM
ejpam-5279	61	21	ξ2	ξ2	PROPN
ejpam-5279	61	22	∈	∈	PROPN
ejpam-5279	61	23	ω	ω	PROPN
ejpam-5279	61	24	and	and	CCONJ
ejpam-5279	61	25	that	that	SCONJ
ejpam-5279	61	26	for	for	ADP
ejpam-5279	61	27	any	any	DET
ejpam-5279	61	28	ξ	ξ	PROPN
ejpam-5279	61	29	∈	∈	PROPN
ejpam-5279	61	30	ω	ω	NOUN
ejpam-5279	61	31	with	with	ADP
ejpam-5279	61	32	ν	ν	PROPN
ejpam-5279	61	33	≤	≤	NUM
ejpam-5279	61	34	ξ	ξ	PROPN
ejpam-5279	61	35	implies	imply	VERB
ejpam-5279	61	36	ν	ν	PROPN
ejpam-5279	61	37	∈	∈	PROPN
ejpam-5279	61	38	ω	ω	PROPN
ejpam-5279	61	39	,	,	PUNCT
ejpam-5279	61	40	where	where	SCONJ
ejpam-5279	61	41	the	the	DET
ejpam-5279	61	42	lattice	lattice	NOUN
ejpam-5279	61	43	[	[	X
ejpam-5279	61	44	0,∞]x	0,∞]x	X
ejpam-5279	61	45	is	be	AUX
ejpam-5279	61	46	equipped	equip	VERB
ejpam-5279	61	47	with	with	ADP
ejpam-5279	61	48	the	the	DET
ejpam-5279	61	49	point	point	NOUN
ejpam-5279	61	50	-	-	PUNCT
ejpam-5279	61	51	wise	wise	ADJ
ejpam-5279	61	52	order	order	NOUN
ejpam-5279	61	53	.	.	PUNCT
ejpam-5279	62	1	definition	definition	NOUN
ejpam-5279	62	2	1	1	NUM
ejpam-5279	62	3	.	.	PUNCT
ejpam-5279	63	1	[	[	X
ejpam-5279	63	2	15	15	NUM
ejpam-5279	63	3	]	]	X
ejpam-5279	63	4	a	a	DET
ejpam-5279	63	5	collection	collection	NOUN
ejpam-5279	63	6	of	of	ADP
ejpam-5279	63	7	ideals	ideal	NOUN
ejpam-5279	63	8	ω	ω	NUM
ejpam-5279	63	9	=	=	SYM
ejpam-5279	63	10	(	(	PUNCT
ejpam-5279	63	11	ω(x))x∈x	ω(x))x∈x	VERB
ejpam-5279	63	12	in	in	ADP
ejpam-5279	63	13	[	[	X
ejpam-5279	63	14	0,∞]x	0,∞]x	NOUN
ejpam-5279	63	15	indexed	index	VERB
ejpam-5279	63	16	by	by	ADP
ejpam-5279	63	17	the	the	DET
ejpam-5279	63	18	points	point	NOUN
ejpam-5279	63	19	of	of	ADP
ejpam-5279	63	20	x	x	SYM
ejpam-5279	63	21	is	be	AUX
ejpam-5279	63	22	called	call	VERB
ejpam-5279	63	23	an	an	DET
ejpam-5279	63	24	approach	approach	NOUN
ejpam-5279	63	25	system	system	NOUN
ejpam-5279	63	26	on	on	ADP
ejpam-5279	63	27	x	x	PUNCT
ejpam-5279	63	28	if	if	SCONJ
ejpam-5279	63	29	and	and	CCONJ
ejpam-5279	63	30	only	only	ADV
ejpam-5279	63	31	if	if	SCONJ
ejpam-5279	63	32	the	the	DET
ejpam-5279	63	33	following	follow	VERB
ejpam-5279	63	34	conditions	condition	NOUN
ejpam-5279	63	35	are	be	AUX
ejpam-5279	63	36	fulfilled	fulfil	VERB
ejpam-5279	63	37	:	:	PUNCT
ejpam-5279	63	38	(	(	PUNCT
ejpam-5279	63	39	as1	as1	NOUN
ejpam-5279	63	40	)	)	PUNCT
ejpam-5279	63	41	∀x	∀x	VERB
ejpam-5279	63	42	∈	∈	PROPN
ejpam-5279	63	43	x	x	SYM
ejpam-5279	63	44	,	,	PUNCT
ejpam-5279	63	45	∀ν	∀ν	PROPN
ejpam-5279	63	46	∈	∈	PROPN
ejpam-5279	63	47	ω(x	ω(x	NOUN
ejpam-5279	63	48	):	):	PUNCT
ejpam-5279	63	49	ν(x	ν(x	PROPN
ejpam-5279	63	50	)	)	PUNCT
ejpam-5279	64	1	=	=	SYM
ejpam-5279	64	2	0	0	X
ejpam-5279	64	3	.	.	PUNCT
ejpam-5279	64	4	(	(	PUNCT
ejpam-5279	64	5	as2	as2	NOUN
ejpam-5279	64	6	)	)	PUNCT
ejpam-5279	64	7	∀x	∀x	VERB
ejpam-5279	64	8	∈	∈	PROPN
ejpam-5279	64	9	x,∀ν	x,∀ν	X
ejpam-5279	64	10	∈	∈	PROPN
ejpam-5279	65	1	[	[	X
ejpam-5279	65	2	0,∞]x	0,∞]x	X
ejpam-5279	65	3	,	,	PUNCT
ejpam-5279	65	4	∀ϵ	∀ϵ	NOUN
ejpam-5279	65	5	>	>	X
ejpam-5279	65	6	0	0	NUM
ejpam-5279	65	7	,	,	PUNCT
ejpam-5279	65	8	∀n	∀n	PUNCT
ejpam-5279	65	9	<	<	X
ejpam-5279	65	10	∞	∞	PROPN
ejpam-5279	65	11	,	,	PUNCT
ejpam-5279	65	12	there	there	PRON
ejpam-5279	65	13	exists	exist	VERB
ejpam-5279	65	14	νnϵ	νnϵ	NOUN
ejpam-5279	65	15	∈	∈	PROPN
ejpam-5279	65	16	ω(x	ω(x	NOUN
ejpam-5279	65	17	)	)	PUNCT
ejpam-5279	65	18	such	such	ADJ
ejpam-5279	65	19	that	that	SCONJ
ejpam-5279	65	20	ν	ν	PROPN
ejpam-5279	65	21	∧n	∧n	PROPN
ejpam-5279	65	22	≤	≤	NUM
ejpam-5279	65	23	νnϵ	νnϵ	NOUN
ejpam-5279	65	24	+	+	CCONJ
ejpam-5279	65	25	ϵ	ϵ	PROPN
ejpam-5279	65	26	implies	imply	VERB
ejpam-5279	65	27	ν	ν	PROPN
ejpam-5279	65	28	∈	∈	PROPN
ejpam-5279	65	29	ω(x	ω(x	NOUN
ejpam-5279	65	30	)	)	PUNCT
ejpam-5279	65	31	.	.	PUNCT
ejpam-5279	66	1	(	(	PUNCT
ejpam-5279	66	2	as3	as3	PROPN
ejpam-5279	66	3	)	)	PUNCT
ejpam-5279	66	4	∀x	∀x	VERB
ejpam-5279	66	5	∈	∈	PROPN
ejpam-5279	66	6	x,∀ν	x,∀ν	PUNCT
ejpam-5279	66	7	∈	∈	PROPN
ejpam-5279	66	8	ω(x	ω(x	NOUN
ejpam-5279	66	9	)	)	PUNCT
ejpam-5279	66	10	,	,	PUNCT
ejpam-5279	66	11	∀ϵ	∀ϵ	NOUN
ejpam-5279	66	12	>	>	X
ejpam-5279	66	13	0	0	PROPN
ejpam-5279	66	14	,	,	PUNCT
ejpam-5279	66	15	n	n	CCONJ
ejpam-5279	66	16	<	<	X
ejpam-5279	66	17	∞	∞	NOUN
ejpam-5279	66	18	there	there	ADV
ejpam-5279	66	19	exists	exist	VERB
ejpam-5279	66	20	(	(	PUNCT
ejpam-5279	66	21	νz	νz	NOUN
ejpam-5279	66	22	)	)	PUNCT
ejpam-5279	66	23	∈	∈	PROPN
ejpam-5279	66	24	∏	∏	PROPN
ejpam-5279	66	25	z∈x	z∈x	NOUN
ejpam-5279	66	26	ω(z	ω(z	PUNCT
ejpam-5279	66	27	)	)	PUNCT
ejpam-5279	66	28	such	such	ADJ
ejpam-5279	66	29	that	that	PRON
ejpam-5279	66	30	for	for	ADP
ejpam-5279	66	31	any	any	DET
ejpam-5279	66	32	y	y	NOUN
ejpam-5279	66	33	,	,	PUNCT
ejpam-5279	66	34	z	z	PROPN
ejpam-5279	66	35	∈	∈	PROPN
ejpam-5279	67	1	x	x	NOUN
ejpam-5279	67	2	:	:	PUNCT
ejpam-5279	67	3	ν(y	ν(y	PROPN
ejpam-5279	67	4	)	)	PUNCT
ejpam-5279	67	5	∧n	∧n	VERB
ejpam-5279	67	6	≤	≤	NOUN
ejpam-5279	67	7	νx(z	νx(z	ADV
ejpam-5279	67	8	)	)	PUNCT
ejpam-5279	67	9	+	+	CCONJ
ejpam-5279	67	10	νz(y	νz(y	NOUN
ejpam-5279	67	11	)	)	PUNCT
ejpam-5279	68	1	+	+	CCONJ
ejpam-5279	68	2	ϵ.	ϵ.	NOUN
ejpam-5279	68	3	for	for	ADP
ejpam-5279	68	4	any	any	DET
ejpam-5279	68	5	x	x	SYM
ejpam-5279	68	6	∈	∈	PROPN
ejpam-5279	68	7	x	x	NOUN
ejpam-5279	68	8	,	,	PUNCT
ejpam-5279	68	9	ν	ν	PROPN
ejpam-5279	68	10	∈	∈	PROPN
ejpam-5279	68	11	ω(x	ω(x	NOUN
ejpam-5279	68	12	)	)	PUNCT
ejpam-5279	68	13	is	be	AUX
ejpam-5279	68	14	called	call	VERB
ejpam-5279	68	15	a	a	DET
ejpam-5279	68	16	local	local	ADJ
ejpam-5279	68	17	distance	distance	NOUN
ejpam-5279	68	18	in	in	ADP
ejpam-5279	68	19	x	x	NOUN
ejpam-5279	68	20	,	,	PUNCT
ejpam-5279	68	21	and	and	CCONJ
ejpam-5279	68	22	the	the	DET
ejpam-5279	68	23	value	value	NOUN
ejpam-5279	68	24	ν(t	ν(t	NOUN
ejpam-5279	68	25	)	)	PUNCT
ejpam-5279	68	26	of	of	ADP
ejpam-5279	68	27	a	a	DET
ejpam-5279	68	28	local	local	ADJ
ejpam-5279	68	29	distance	distance	NOUN
ejpam-5279	68	30	ν	ν	X
ejpam-5279	68	31	∈	∈	PROPN
ejpam-5279	68	32	ω(x	ω(x	NOUN
ejpam-5279	68	33	)	)	PUNCT
ejpam-5279	68	34	at	at	ADP
ejpam-5279	68	35	a	a	DET
ejpam-5279	68	36	point	point	NOUN
ejpam-5279	68	37	t	t	X
ejpam-5279	68	38	∈	∈	PROPN
ejpam-5279	68	39	x	x	PUNCT
ejpam-5279	68	40	is	be	AUX
ejpam-5279	68	41	interpreted	interpret	VERB
ejpam-5279	68	42	as	as	ADP
ejpam-5279	68	43	the	the	DET
ejpam-5279	68	44	distance	distance	NOUN
ejpam-5279	68	45	from	from	ADP
ejpam-5279	68	46	x	x	PUNCT
ejpam-5279	68	47	to	to	ADP
ejpam-5279	68	48	t	t	NOUN
ejpam-5279	68	49	according	accord	VERB
ejpam-5279	68	50	to	to	ADP
ejpam-5279	68	51	ν	ν	PROPN
ejpam-5279	68	52	.	.	PUNCT
ejpam-5279	69	1	each	each	DET
ejpam-5279	69	2	local	local	ADJ
ejpam-5279	69	3	distance	distance	NOUN
ejpam-5279	69	4	makes	make	VERB
ejpam-5279	69	5	its	its	PRON
ejpam-5279	69	6	own	own	ADJ
ejpam-5279	69	7	measurement	measurement	NOUN
ejpam-5279	69	8	of	of	ADP
ejpam-5279	69	9	the	the	DET
ejpam-5279	69	10	distance	distance	NOUN
ejpam-5279	69	11	other	other	ADJ
ejpam-5279	69	12	points	point	NOUN
ejpam-5279	69	13	in	in	ADP
ejpam-5279	69	14	the	the	DET
ejpam-5279	69	15	space	space	NOUN
ejpam-5279	69	16	are	be	AUX
ejpam-5279	69	17	away	away	ADV
ejpam-5279	69	18	from	from	ADP
ejpam-5279	69	19	the	the	DET
ejpam-5279	69	20	given	give	VERB
ejpam-5279	69	21	point	point	NOUN
ejpam-5279	69	22	.	.	PUNCT
ejpam-5279	70	1	a	a	DET
ejpam-5279	70	2	subset	subset	NOUN
ejpam-5279	70	3	b	b	X
ejpam-5279	71	1	⊂	⊂	PROPN
ejpam-5279	71	2	[	[	X
ejpam-5279	71	3	0,∞]x	0,∞]x	X
ejpam-5279	71	4	is	be	AUX
ejpam-5279	71	5	called	call	VERB
ejpam-5279	71	6	a	a	DET
ejpam-5279	71	7	ideal	ideal	ADJ
ejpam-5279	71	8	basis	basis	NOUN
ejpam-5279	71	9	in	in	ADP
ejpam-5279	71	10	[	[	X
ejpam-5279	71	11	0,∞]x	0,∞]x	X
ejpam-5279	71	12	if	if	SCONJ
ejpam-5279	71	13	for	for	ADP
ejpam-5279	71	14	any	any	DET
ejpam-5279	71	15	β1	β1	NOUN
ejpam-5279	71	16	,	,	PUNCT
ejpam-5279	71	17	β2	β2	NOUN
ejpam-5279	71	18	∈	∈	PROPN
ejpam-5279	71	19	b	b	ADP
ejpam-5279	71	20	there	there	PRON
ejpam-5279	71	21	is	be	VERB
ejpam-5279	71	22	a	a	DET
ejpam-5279	71	23	β	β	X
ejpam-5279	71	24	∈	∈	PROPN
ejpam-5279	71	25	b	b	NOUN
ejpam-5279	71	26	such	such	ADJ
ejpam-5279	71	27	that	that	DET
ejpam-5279	71	28	β1	β1	PROPN
ejpam-5279	71	29	∨	∨	NUM
ejpam-5279	71	30	β2	β2	PROPN
ejpam-5279	71	31	≤	≤	NOUN
ejpam-5279	71	32	β	β	X
ejpam-5279	71	33	.	.	PUNCT
ejpam-5279	72	1	definition	definition	NOUN
ejpam-5279	72	2	2	2	NUM
ejpam-5279	72	3	.	.	PUNCT
ejpam-5279	73	1	[	[	X
ejpam-5279	73	2	15	15	NUM
ejpam-5279	73	3	]	]	X
ejpam-5279	73	4	a	a	DET
ejpam-5279	73	5	collection	collection	NOUN
ejpam-5279	73	6	of	of	ADP
ejpam-5279	73	7	ideal	ideal	ADJ
ejpam-5279	73	8	bases	basis	NOUN
ejpam-5279	73	9	b	b	X
ejpam-5279	73	10	=	=	PUNCT
ejpam-5279	73	11	(	(	PUNCT
ejpam-5279	73	12	b(x))x∈x	b(x))x∈x	PROPN
ejpam-5279	73	13	in	in	ADP
ejpam-5279	73	14	[	[	X
ejpam-5279	73	15	0,∞]x	0,∞]x	X
ejpam-5279	73	16	is	be	AUX
ejpam-5279	73	17	called	call	VERB
ejpam-5279	73	18	an	an	DET
ejpam-5279	73	19	approach	approach	NOUN
ejpam-5279	73	20	basis	basis	NOUN
ejpam-5279	73	21	if	if	SCONJ
ejpam-5279	73	22	and	and	CCONJ
ejpam-5279	73	23	only	only	ADV
ejpam-5279	73	24	if	if	SCONJ
ejpam-5279	73	25	the	the	DET
ejpam-5279	73	26	following	follow	VERB
ejpam-5279	73	27	statements	statement	NOUN
ejpam-5279	73	28	are	be	AUX
ejpam-5279	73	29	true	true	ADJ
ejpam-5279	73	30	.	.	PUNCT
ejpam-5279	74	1	(	(	PUNCT
ejpam-5279	74	2	ab1	ab1	X
ejpam-5279	74	3	)	)	PUNCT
ejpam-5279	74	4	∀x	∀x	VERB
ejpam-5279	74	5	∈	∈	PROPN
ejpam-5279	74	6	x	x	PRON
ejpam-5279	74	7	,	,	PUNCT
ejpam-5279	74	8	∀β	∀β	PROPN
ejpam-5279	74	9	∈	∈	NOUN
ejpam-5279	74	10	b(x	b(x	NOUN
ejpam-5279	74	11	):	):	PUNCT
ejpam-5279	74	12	β(x	β(x	NOUN
ejpam-5279	74	13	)	)	PUNCT
ejpam-5279	74	14	=	=	SYM
ejpam-5279	74	15	0	0	X
ejpam-5279	74	16	.	.	PUNCT
ejpam-5279	74	17	t.m.g	t.m.g	ADJ
ejpam-5279	74	18	.	.	PUNCT
ejpam-5279	75	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	75	2	,	,	PUNCT
ejpam-5279	75	3	fawzi	fawzi	PROPN
ejpam-5279	75	4	al	al	PROPN
ejpam-5279	75	5	-	-	PUNCT
ejpam-5279	75	6	thukair	thukair	NOUN
ejpam-5279	75	7	/	/	SYM
ejpam-5279	75	8	eur	eur	NOUN
ejpam-5279	75	9	.	.	PUNCT
ejpam-5279	76	1	j.	j.	PROPN
ejpam-5279	76	2	pure	pure	PROPN
ejpam-5279	76	3	appl	appl	PROPN
ejpam-5279	76	4	.	.	PROPN
ejpam-5279	76	5	math	math	PROPN
ejpam-5279	76	6	,	,	PUNCT
ejpam-5279	76	7	17	17	NUM
ejpam-5279	76	8	(	(	PUNCT
ejpam-5279	76	9	3	3	NUM
ejpam-5279	76	10	)	)	PUNCT
ejpam-5279	76	11	(	(	PUNCT
ejpam-5279	76	12	2024	2024	NUM
ejpam-5279	76	13	)	)	PUNCT
ejpam-5279	76	14	,	,	PUNCT
ejpam-5279	76	15	1762	1762	NUM
ejpam-5279	76	16	-	-	SYM
ejpam-5279	76	17	1778	1778	NUM
ejpam-5279	76	18	1765	1765	NUM
ejpam-5279	76	19	(	(	PUNCT
ejpam-5279	76	20	ab2	ab2	PROPN
ejpam-5279	76	21	)	)	PUNCT
ejpam-5279	76	22	∀x	∀x	VERB
ejpam-5279	76	23	∈	∈	PROPN
ejpam-5279	76	24	x	x	PRON
ejpam-5279	76	25	,	,	PUNCT
ejpam-5279	76	26	∀β	∀β	PROPN
ejpam-5279	76	27	∈	∈	NOUN
ejpam-5279	76	28	b(x	b(x	NOUN
ejpam-5279	76	29	)	)	PUNCT
ejpam-5279	76	30	,	,	PUNCT
ejpam-5279	76	31	∀ϵ	∀ϵ	PROPN
ejpam-5279	76	32	>	>	X
ejpam-5279	76	33	0,∀n	0,∀n	PROPN
ejpam-5279	76	34	<	<	X
ejpam-5279	76	35	∞	∞	PROPN
ejpam-5279	76	36	,	,	PUNCT
ejpam-5279	76	37	there	there	PRON
ejpam-5279	76	38	exists	exist	VERB
ejpam-5279	76	39	(	(	PUNCT
ejpam-5279	76	40	βz)z∈x	βz)z∈x	NUM
ejpam-5279	76	41	∈	∈	PROPN
ejpam-5279	76	42	∏	∏	PROPN
ejpam-5279	76	43	z∈x	z∈x	NOUN
ejpam-5279	76	44	b(z	b(z	NOUN
ejpam-5279	76	45	)	)	PUNCT
ejpam-5279	76	46	such	such	ADJ
ejpam-5279	76	47	that	that	DET
ejpam-5279	76	48	∀y	∀y	NUM
ejpam-5279	76	49	,	,	PUNCT
ejpam-5279	76	50	z	z	PROPN
ejpam-5279	76	51	∈	∈	PROPN
ejpam-5279	76	52	x	x	X
ejpam-5279	76	53	:	:	PUNCT
ejpam-5279	76	54	β(y	β(y	NOUN
ejpam-5279	76	55	)	)	PUNCT
ejpam-5279	76	56	∧n	∧n	X
ejpam-5279	76	57	≤	≤	NOUN
ejpam-5279	76	58	βx(z	βx(z	PUNCT
ejpam-5279	76	59	)	)	PUNCT
ejpam-5279	77	1	+	+	CCONJ
ejpam-5279	77	2	βz(y	βz(y	PUNCT
ejpam-5279	77	3	)	)	PUNCT
ejpam-5279	78	1	+	+	CCONJ
ejpam-5279	78	2	ϵ.	ϵ.	NOUN
ejpam-5279	78	3	definition	definition	NOUN
ejpam-5279	78	4	3	3	NUM
ejpam-5279	78	5	.	.	PUNCT
ejpam-5279	79	1	[	[	X
ejpam-5279	79	2	15	15	NUM
ejpam-5279	79	3	]	]	X
ejpam-5279	79	4	if	if	SCONJ
ejpam-5279	79	5	ω	ω	PROPN
ejpam-5279	79	6	is	be	AUX
ejpam-5279	79	7	an	an	DET
ejpam-5279	79	8	approach	approach	NOUN
ejpam-5279	79	9	system	system	NOUN
ejpam-5279	79	10	,	,	PUNCT
ejpam-5279	79	11	then	then	ADV
ejpam-5279	79	12	b	b	X
ejpam-5279	79	13	=	=	PUNCT
ejpam-5279	79	14	(	(	PUNCT
ejpam-5279	79	15	b(x))x∈x	b(x))x∈x	PROPN
ejpam-5279	79	16	is	be	AUX
ejpam-5279	79	17	called	call	VERB
ejpam-5279	79	18	a	a	DET
ejpam-5279	79	19	basis	basis	NOUN
ejpam-5279	79	20	for	for	ADP
ejpam-5279	79	21	ω	ω	NUM
ejpam-5279	79	22	if	if	SCONJ
ejpam-5279	80	1	and	and	CCONJ
ejpam-5279	80	2	only	only	ADV
ejpam-5279	80	3	if	if	SCONJ
ejpam-5279	80	4	the	the	DET
ejpam-5279	80	5	following	following	NOUN
ejpam-5279	80	6	are	be	AUX
ejpam-5279	80	7	satisfied	satisfied	ADJ
ejpam-5279	80	8	.	.	PUNCT
ejpam-5279	81	1	(	(	PUNCT
ejpam-5279	81	2	ab1	ab1	X
ejpam-5279	81	3	)	)	PUNCT
ejpam-5279	81	4	∀x	∀x	VERB
ejpam-5279	81	5	∈	∈	PROPN
ejpam-5279	81	6	x	x	NOUN
ejpam-5279	81	7	,	,	PUNCT
ejpam-5279	81	8	b(x	b(x	NOUN
ejpam-5279	81	9	)	)	PUNCT
ejpam-5279	81	10	is	be	AUX
ejpam-5279	81	11	a	a	DET
ejpam-5279	81	12	basis	basis	NOUN
ejpam-5279	81	13	for	for	ADP
ejpam-5279	81	14	an	an	DET
ejpam-5279	81	15	ideal	ideal	NOUN
ejpam-5279	81	16	.	.	PUNCT
ejpam-5279	82	1	(	(	PUNCT
ejpam-5279	82	2	ab2	ab2	ADJ
ejpam-5279	82	3	)	)	PUNCT
ejpam-5279	82	4	∀x	∀x	VERB
ejpam-5279	82	5	∈	∈	PROPN
ejpam-5279	82	6	x	x	PRON
ejpam-5279	82	7	:	:	PUNCT
ejpam-5279	82	8	ω(x	ω(x	X
ejpam-5279	82	9	)	)	PUNCT
ejpam-5279	82	10	=	=	SYM
ejpam-5279	83	1	b̂(x	b̂(x	NOUN
ejpam-5279	83	2	)	)	PUNCT
ejpam-5279	83	3	,	,	PUNCT
ejpam-5279	83	4	where	where	SCONJ
ejpam-5279	83	5	b̂(x	b̂(x	ADP
ejpam-5279	83	6	)	)	PUNCT
ejpam-5279	83	7	=	=	PRON
ejpam-5279	83	8	{	{	PUNCT
ejpam-5279	83	9	ν	ν	X
ejpam-5279	83	10	∈	∈	PROPN
ejpam-5279	84	1	[	[	X
ejpam-5279	84	2	0,∞]x	0,∞]x	X
ejpam-5279	84	3	|	|	NOUN
ejpam-5279	84	4	∀ϵ	∀ϵ	NOUN
ejpam-5279	84	5	>	>	X
ejpam-5279	84	6	0,∀n	0,∀n	PROPN
ejpam-5279	84	7	≤	≤	NOUN
ejpam-5279	84	8	∞∃	∞∃	NOUN
ejpam-5279	84	9	ξ	ξ	PRON
ejpam-5279	84	10	∈	∈	PROPN
ejpam-5279	84	11	b(x	b(x	NOUN
ejpam-5279	84	12	)	)	PUNCT
ejpam-5279	84	13	:	:	PUNCT
ejpam-5279	85	1	ν	ν	X
ejpam-5279	85	2	∧n	∧n	VERB
ejpam-5279	85	3	≤	≤	NOUN
ejpam-5279	85	4	ξ	ξ	PROPN
ejpam-5279	86	1	+	+	CCONJ
ejpam-5279	86	2	ϵ	ϵ	X
ejpam-5279	86	3	}	}	PUNCT
ejpam-5279	86	4	.	.	PUNCT
ejpam-5279	87	1	definition	definition	NOUN
ejpam-5279	87	2	4	4	NUM
ejpam-5279	87	3	.	.	PUNCT
ejpam-5279	88	1	[	[	X
ejpam-5279	88	2	15	15	NUM
ejpam-5279	88	3	]	]	X
ejpam-5279	88	4	a	a	DET
ejpam-5279	88	5	function	function	NOUN
ejpam-5279	88	6	λ	λ	NOUN
ejpam-5279	88	7	:	:	PUNCT
ejpam-5279	88	8	f(x	f(x	PROPN
ejpam-5279	88	9	)	)	PUNCT
ejpam-5279	89	1	−→	−→	NOUN
ejpam-5279	90	1	[	[	X
ejpam-5279	90	2	0,∞]x	0,∞]x	X
ejpam-5279	90	3	is	be	AUX
ejpam-5279	90	4	called	call	VERB
ejpam-5279	90	5	a	a	DET
ejpam-5279	90	6	limit	limit	NOUN
ejpam-5279	90	7	operator	operator	NOUN
ejpam-5279	90	8	if	if	SCONJ
ejpam-5279	90	9	and	and	CCONJ
ejpam-5279	90	10	only	only	ADV
ejpam-5279	90	11	if	if	SCONJ
ejpam-5279	90	12	the	the	DET
ejpam-5279	90	13	following	follow	VERB
ejpam-5279	90	14	conditions	condition	NOUN
ejpam-5279	90	15	are	be	AUX
ejpam-5279	90	16	fulfilled	fulfil	VERB
ejpam-5279	90	17	:	:	PUNCT
ejpam-5279	90	18	(	(	PUNCT
ejpam-5279	90	19	al1	al1	PROPN
ejpam-5279	90	20	)	)	PUNCT
ejpam-5279	90	21	∀x	∀x	VERB
ejpam-5279	90	22	∈	∈	PROPN
ejpam-5279	90	23	x	x	X
ejpam-5279	90	24	:	:	PUNCT
ejpam-5279	90	25	λ(ẋ)(x	λ(ẋ)(x	NOUN
ejpam-5279	90	26	)	)	PUNCT
ejpam-5279	90	27	=	=	SYM
ejpam-5279	90	28	0	0	NUM
ejpam-5279	90	29	;	;	PUNCT
ejpam-5279	90	30	(	(	PUNCT
ejpam-5279	90	31	al2	al2	PROPN
ejpam-5279	90	32	)	)	PUNCT
ejpam-5279	90	33	∀f	∀f	PROPN
ejpam-5279	90	34	,	,	PUNCT
ejpam-5279	90	35	g	g	PROPN
ejpam-5279	90	36	∈	∈	PROPN
ejpam-5279	90	37	f(x	f(x	PROPN
ejpam-5279	90	38	)	)	PUNCT
ejpam-5279	90	39	,	,	PUNCT
ejpam-5279	90	40	x	x	PUNCT
ejpam-5279	90	41	∈	∈	NOUN
ejpam-5279	90	42	x	x	X
ejpam-5279	90	43	:	:	PUNCT
ejpam-5279	90	44	f	f	X
ejpam-5279	90	45	≤	≤	NUM
ejpam-5279	90	46	g	g	PROPN
ejpam-5279	90	47	implies	imply	VERB
ejpam-5279	90	48	λ(g)(x	λ(g)(x	PROPN
ejpam-5279	90	49	)	)	PUNCT
ejpam-5279	90	50	≤	≤	NOUN
ejpam-5279	90	51	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	90	52	)	)	PUNCT
ejpam-5279	90	53	;	;	PUNCT
ejpam-5279	90	54	(	(	PUNCT
ejpam-5279	90	55	al3	al3	PROPN
ejpam-5279	90	56	)	)	PUNCT
ejpam-5279	90	57	∀	∀	X
ejpam-5279	90	58	(	(	PUNCT
ejpam-5279	90	59	fj)j∈j	fj)j∈j	NUM
ejpam-5279	90	60	∈	∈	PROPN
ejpam-5279	90	61	f(x)j	f(x)j	PROPN
ejpam-5279	90	62	,	,	PUNCT
ejpam-5279	90	63	x	x	PUNCT
ejpam-5279	90	64	∈	∈	NOUN
ejpam-5279	90	65	x	x	X
ejpam-5279	90	66	:	:	PUNCT
ejpam-5279	90	67	λ	λ	X
ejpam-5279	90	68	(	(	PUNCT
ejpam-5279	90	69	∧	∧	PROPN
ejpam-5279	90	70	j	j	PROPN
ejpam-5279	90	71	fj	fj	PROPN
ejpam-5279	90	72	)	)	PUNCT
ejpam-5279	90	73	(	(	PUNCT
ejpam-5279	90	74	x	x	X
ejpam-5279	90	75	)	)	PUNCT
ejpam-5279	90	76	=	=	SYM
ejpam-5279	90	77	∨	∨	NUM
ejpam-5279	90	78	j	j	PROPN
ejpam-5279	90	79	λ	λ	PROPN
ejpam-5279	90	80	(	(	PUNCT
ejpam-5279	90	81	fj	fj	PROPN
ejpam-5279	90	82	)	)	PUNCT
ejpam-5279	90	83	(	(	PUNCT
ejpam-5279	90	84	x	x	NOUN
ejpam-5279	90	85	)	)	PUNCT
ejpam-5279	90	86	;	;	PUNCT
ejpam-5279	90	87	(	(	PUNCT
ejpam-5279	90	88	al4	al4	NOUN
ejpam-5279	90	89	)	)	PUNCT
ejpam-5279	90	90	∀g	∀g	NOUN
ejpam-5279	90	91	∈	∈	PROPN
ejpam-5279	90	92	f(x	f(x	PROPN
ejpam-5279	90	93	)	)	PUNCT
ejpam-5279	90	94	,	,	PUNCT
ejpam-5279	90	95	(	(	PUNCT
ejpam-5279	90	96	fy)y∈x	fy)y∈x	NUM
ejpam-5279	90	97	∈	∈	NOUN
ejpam-5279	90	98	f(x)x	f(x)x	NOUN
ejpam-5279	90	99	,	,	PUNCT
ejpam-5279	90	100	x	x	PUNCT
ejpam-5279	90	101	∈	∈	NOUN
ejpam-5279	90	102	x	x	X
ejpam-5279	90	103	:	:	PUNCT
ejpam-5279	90	104	λ	λ	X
ejpam-5279	90	105	(	(	PUNCT
ejpam-5279	90	106	κ	κ	X
ejpam-5279	90	107	(	(	PUNCT
ejpam-5279	90	108	g	g	PROPN
ejpam-5279	90	109	,	,	PUNCT
ejpam-5279	90	110	(	(	PUNCT
ejpam-5279	90	111	fy)y∈x	fy)y∈x	NUM
ejpam-5279	90	112	)	)	PUNCT
ejpam-5279	90	113	)	)	PUNCT
ejpam-5279	90	114	(	(	PUNCT
ejpam-5279	90	115	x	x	X
ejpam-5279	90	116	)	)	PUNCT
ejpam-5279	90	117	≤	≤	NOUN
ejpam-5279	90	118	λ(g)(x	λ(g)(x	PROPN
ejpam-5279	90	119	)	)	PUNCT
ejpam-5279	91	1	+	+	NOUN
ejpam-5279	91	2	∨	∨	NOUN
ejpam-5279	91	3	y∈x	y∈x	NOUN
ejpam-5279	91	4	λ	λ	PROPN
ejpam-5279	91	5	(	(	PUNCT
ejpam-5279	91	6	fy	fy	PROPN
ejpam-5279	91	7	)	)	PUNCT
ejpam-5279	91	8	(	(	PUNCT
ejpam-5279	91	9	y	y	NOUN
ejpam-5279	91	10	)	)	PUNCT
ejpam-5279	91	11	.	.	PUNCT
ejpam-5279	92	1	then	then	ADV
ejpam-5279	92	2	the	the	DET
ejpam-5279	92	3	pair	pair	NOUN
ejpam-5279	92	4	(	(	PUNCT
ejpam-5279	92	5	x	x	NOUN
ejpam-5279	92	6	,	,	PUNCT
ejpam-5279	92	7	λ	λ	X
ejpam-5279	92	8	)	)	PUNCT
ejpam-5279	92	9	is	be	AUX
ejpam-5279	92	10	called	call	VERB
ejpam-5279	92	11	an	an	DET
ejpam-5279	92	12	approach	approach	NOUN
ejpam-5279	92	13	space	space	NOUN
ejpam-5279	92	14	.	.	PUNCT
ejpam-5279	93	1	a	a	DET
ejpam-5279	93	2	map	map	NOUN
ejpam-5279	93	3	f	f	X
ejpam-5279	93	4	:	:	PUNCT
ejpam-5279	93	5	(	(	PUNCT
ejpam-5279	93	6	x	x	X
ejpam-5279	93	7	,	,	PUNCT
ejpam-5279	93	8	λ	λ	NOUN
ejpam-5279	93	9	)	)	PUNCT
ejpam-5279	93	10	−→	−→	NOUN
ejpam-5279	93	11	(	(	PUNCT
ejpam-5279	93	12	y	y	NOUN
ejpam-5279	93	13	,	,	PUNCT
ejpam-5279	93	14	λ′	λ′	PROPN
ejpam-5279	93	15	)	)	PUNCT
ejpam-5279	93	16	between	between	ADP
ejpam-5279	93	17	two	two	NUM
ejpam-5279	93	18	approach	approach	NOUN
ejpam-5279	93	19	spaces	space	NOUN
ejpam-5279	93	20	is	be	AUX
ejpam-5279	93	21	called	call	VERB
ejpam-5279	93	22	a	a	DET
ejpam-5279	93	23	contraction	contraction	NOUN
ejpam-5279	93	24	if	if	SCONJ
ejpam-5279	93	25	λ′	λ′	X
ejpam-5279	93	26	(	(	PUNCT
ejpam-5279	93	27	f(f	f(f	PROPN
ejpam-5279	93	28	)	)	PUNCT
ejpam-5279	93	29	)	)	PUNCT
ejpam-5279	94	1	(	(	PUNCT
ejpam-5279	94	2	f(x	f(x	PROPN
ejpam-5279	94	3	)	)	PUNCT
ejpam-5279	94	4	)	)	PUNCT
ejpam-5279	95	1	≤	≤	NUM
ejpam-5279	95	2	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	95	3	)	)	PUNCT
ejpam-5279	95	4	,	,	PUNCT
ejpam-5279	95	5	∀x	∀x	X
ejpam-5279	95	6	∈	∈	PROPN
ejpam-5279	95	7	x.	x.	NOUN
ejpam-5279	95	8	the	the	DET
ejpam-5279	95	9	category	category	NOUN
ejpam-5279	95	10	of	of	ADP
ejpam-5279	95	11	approach	approach	NOUN
ejpam-5279	95	12	spaces	space	NOUN
ejpam-5279	95	13	and	and	CCONJ
ejpam-5279	95	14	contraction	contraction	NOUN
ejpam-5279	95	15	mappings	mapping	NOUN
ejpam-5279	95	16	is	be	AUX
ejpam-5279	95	17	denoted	denote	VERB
ejpam-5279	95	18	by	by	ADP
ejpam-5279	95	19	ap	ap	PROPN
ejpam-5279	95	20	.	.	PUNCT
ejpam-5279	96	1	theorem	theorem	PROPN
ejpam-5279	96	2	1	1	NUM
ejpam-5279	96	3	.	.	PUNCT
ejpam-5279	97	1	[	[	X
ejpam-5279	97	2	15	15	NUM
ejpam-5279	97	3	]	]	X
ejpam-5279	97	4	let	let	VERB
ejpam-5279	97	5	(	(	PUNCT
ejpam-5279	97	6	fj	fj	INTJ
ejpam-5279	97	7	:	:	PUNCT
ejpam-5279	97	8	x	x	PUNCT
ejpam-5279	97	9	−→	−→	NOUN
ejpam-5279	97	10	(	(	PUNCT
ejpam-5279	97	11	xj	xj	PROPN
ejpam-5279	97	12	,	,	PUNCT
ejpam-5279	97	13	ω	ω	PROPN
ejpam-5279	97	14	j	j	PROPN
ejpam-5279	97	15	)	)	PUNCT
ejpam-5279	97	16	j∈j	j∈j	NOUN
ejpam-5279	97	17	)	)	PUNCT
ejpam-5279	97	18	be	be	AUX
ejpam-5279	97	19	a	a	DET
ejpam-5279	97	20	structured	structured	ADJ
ejpam-5279	97	21	source	source	NOUN
ejpam-5279	97	22	in	in	ADP
ejpam-5279	97	23	ap	ap	PROPN
ejpam-5279	97	24	.	.	PUNCT
ejpam-5279	98	1	then	then	ADV
ejpam-5279	98	2	an	an	DET
ejpam-5279	98	3	approach	approach	NOUN
ejpam-5279	98	4	basis	basis	NOUN
ejpam-5279	98	5	on	on	ADP
ejpam-5279	98	6	x	x	PUNCT
ejpam-5279	98	7	for	for	ADP
ejpam-5279	98	8	the	the	DET
ejpam-5279	98	9	unique	unique	ADJ
ejpam-5279	98	10	initial	initial	ADJ
ejpam-5279	98	11	lift	lift	NOUN
ejpam-5279	98	12	of	of	ADP
ejpam-5279	98	13	this	this	DET
ejpam-5279	98	14	source	source	NOUN
ejpam-5279	98	15	in	in	ADP
ejpam-5279	98	16	ap	ap	PROPN
ejpam-5279	98	17	is	be	AUX
ejpam-5279	98	18	given	give	VERB
ejpam-5279	98	19	by	by	ADP
ejpam-5279	98	20	b(x	b(x	NOUN
ejpam-5279	98	21	)	)	PUNCT
ejpam-5279	98	22	=	=	SYM
ejpam-5279	98	23	{	{	PUNCT
ejpam-5279	98	24	∨	∨	NUM
ejpam-5279	98	25	j∈k	j∈k	PROPN
ejpam-5279	98	26	νj	νj	PROPN
ejpam-5279	98	27	◦	◦	PROPN
ejpam-5279	98	28	fj	fj	ADP
ejpam-5279	98	29	|k	|k	NOUN
ejpam-5279	98	30	∈	∈	PROPN
ejpam-5279	98	31	2(j),∀j	2(j),∀j	NUM
ejpam-5279	99	1	∈	∈	NOUN
ejpam-5279	99	2	k	k	NOUN
ejpam-5279	99	3	:	:	PUNCT
ejpam-5279	99	4	νj	νj	X
ejpam-5279	99	5	∈	∈	PROPN
ejpam-5279	99	6	ωj	ωj	ADP
ejpam-5279	99	7	(	(	PUNCT
ejpam-5279	99	8	fj(x))},∀x	fj(x))},∀x	PROPN
ejpam-5279	99	9	∈	∈	PROPN
ejpam-5279	99	10	x.	x.	NOUN
ejpam-5279	99	11	given	give	VERB
ejpam-5279	99	12	an	an	DET
ejpam-5279	99	13	approach	approach	NOUN
ejpam-5279	99	14	space	space	NOUN
ejpam-5279	99	15	(	(	PUNCT
ejpam-5279	99	16	x	x	X
ejpam-5279	99	17	,	,	PUNCT
ejpam-5279	99	18	λ	λ	NOUN
ejpam-5279	99	19	)	)	PUNCT
ejpam-5279	99	20	,	,	PUNCT
ejpam-5279	99	21	one	one	NUM
ejpam-5279	99	22	defines	define	NOUN
ejpam-5279	99	23	for	for	ADP
ejpam-5279	99	24	α	α	PRON
ejpam-5279	99	25	∈	∈	PROPN
ejpam-5279	100	1	[	[	X
ejpam-5279	100	2	0,∞	0,∞	X
ejpam-5279	100	3	]	]	PUNCT
ejpam-5279	100	4	and	and	CCONJ
ejpam-5279	100	5	x	x	PUNCT
ejpam-5279	100	6	∈	∈	PROPN
ejpam-5279	100	7	x	x	NOUN
ejpam-5279	100	8	,	,	PUNCT
ejpam-5279	100	9	the	the	DET
ejpam-5279	100	10	αneighborhood	αneighborhood	ADJ
ejpam-5279	100	11	filter	filter	NOUN
ejpam-5279	100	12	at	at	ADP
ejpam-5279	100	13	x	x	X
ejpam-5279	100	14	∈	∈	PROPN
ejpam-5279	100	15	x	x	PUNCT
ejpam-5279	100	16	given	give	VERB
ejpam-5279	100	17	in	in	ADP
ejpam-5279	100	18	[	[	NOUN
ejpam-5279	100	19	13	13	NUM
ejpam-5279	100	20	]	]	PUNCT
ejpam-5279	100	21	by	by	ADP
ejpam-5279	100	22	ux	ux	PROPN
ejpam-5279	100	23	α	α	PROPN
ejpam-5279	100	24	=	=	SYM
ejpam-5279	100	25	∧	∧	PROPN
ejpam-5279	100	26	{	{	PUNCT
ejpam-5279	100	27	f	f	PROPN
ejpam-5279	100	28	∈	∈	PROPN
ejpam-5279	100	29	f(x)|λ(f)(x	f(x)|λ(f)(x	PROPN
ejpam-5279	100	30	)	)	PUNCT
ejpam-5279	100	31	≤	≤	NOUN
ejpam-5279	100	32	α	α	X
ejpam-5279	100	33	}	}	PUNCT
ejpam-5279	100	34	.	.	PUNCT
ejpam-5279	101	1	remark	remark	NOUN
ejpam-5279	101	2	1	1	NUM
ejpam-5279	101	3	.	.	PUNCT
ejpam-5279	102	1	in	in	ADP
ejpam-5279	102	2	an	an	DET
ejpam-5279	102	3	approach	approach	NOUN
ejpam-5279	102	4	space	space	NOUN
ejpam-5279	102	5	(	(	PUNCT
ejpam-5279	102	6	x	x	X
ejpam-5279	102	7	,	,	PUNCT
ejpam-5279	102	8	λ	λ	NOUN
ejpam-5279	102	9	)	)	PUNCT
ejpam-5279	102	10	,	,	PUNCT
ejpam-5279	102	11	∀f	∀f	PROPN
ejpam-5279	102	12	∈	∈	PROPN
ejpam-5279	102	13	f(x	f(x	PROPN
ejpam-5279	102	14	)	)	PUNCT
ejpam-5279	102	15	,	,	PUNCT
ejpam-5279	102	16	∀x	∀x	VERB
ejpam-5279	102	17	∈	∈	PROPN
ejpam-5279	102	18	x	x	X
ejpam-5279	102	19	and	and	CCONJ
ejpam-5279	102	20	∀α	∀α	NOUN
ejpam-5279	102	21	∈	∈	PROPN
ejpam-5279	103	1	[	[	X
ejpam-5279	103	2	0,∞	0,∞	X
ejpam-5279	103	3	]	]	X
ejpam-5279	103	4	:	:	PUNCT
ejpam-5279	103	5	λ(f)(x	λ(f)(x	X
ejpam-5279	103	6	)	)	PUNCT
ejpam-5279	103	7	≤	≤	NOUN
ejpam-5279	103	8	α	α	PROPN
ejpam-5279	103	9	⇔	⇔	PROPN
ejpam-5279	103	10	f	f	PROPN
ejpam-5279	103	11	≥	≥	PROPN
ejpam-5279	103	12	ux	ux	PROPN
ejpam-5279	103	13	α	α	X
ejpam-5279	103	14	.	.	PUNCT
ejpam-5279	104	1	[	[	X
ejpam-5279	104	2	13	13	NUM
ejpam-5279	104	3	]	]	PUNCT
ejpam-5279	104	4	theorem	theorem	NOUN
ejpam-5279	104	5	2	2	NUM
ejpam-5279	104	6	.	.	PUNCT
ejpam-5279	105	1	[	[	X
ejpam-5279	105	2	13	13	NUM
ejpam-5279	105	3	]	]	PUNCT
ejpam-5279	105	4	let	let	VERB
ejpam-5279	105	5	(	(	PUNCT
ejpam-5279	105	6	x	x	NOUN
ejpam-5279	105	7	,	,	PUNCT
ejpam-5279	105	8	λ	λ	NOUN
ejpam-5279	105	9	)	)	PUNCT
ejpam-5279	105	10	∈	∈	PROPN
ejpam-5279	105	11	|	|	NOUN
ejpam-5279	105	12	ap|	ap|	PROPN
ejpam-5279	105	13	.	.	PUNCT
ejpam-5279	106	1	the	the	DET
ejpam-5279	106	2	system	system	NOUN
ejpam-5279	106	3	u	u	NOUN
ejpam-5279	106	4	=	=	X
ejpam-5279	106	5	(	(	PUNCT
ejpam-5279	106	6	ux	ux	PROPN
ejpam-5279	106	7	α)x∈x	α)x∈x	PROPN
ejpam-5279	106	8	,	,	PUNCT
ejpam-5279	106	9	α∈[0,∞	α∈[0,∞	NUM
ejpam-5279	106	10	]	]	PUNCT
ejpam-5279	106	11	has	have	VERB
ejpam-5279	106	12	the	the	DET
ejpam-5279	106	13	following	follow	VERB
ejpam-5279	106	14	properties	property	NOUN
ejpam-5279	106	15	:	:	PUNCT
ejpam-5279	106	16	(	(	PUNCT
ejpam-5279	106	17	u0	u0	ADJ
ejpam-5279	106	18	)	)	PUNCT
ejpam-5279	106	19	ux	ux	NOUN
ejpam-5279	107	1	α	α	PROPN
ejpam-5279	107	2	∈	∈	PROPN
ejpam-5279	107	3	f(x	f(x	PROPN
ejpam-5279	107	4	)	)	PUNCT
ejpam-5279	107	5	for	for	ADP
ejpam-5279	107	6	all	all	DET
ejpam-5279	107	7	x	x	SYM
ejpam-5279	107	8	∈	∈	PROPN
ejpam-5279	107	9	x	x	X
ejpam-5279	107	10	and	and	CCONJ
ejpam-5279	107	11	α	α	NOUN
ejpam-5279	107	12	∈	∈	PROPN
ejpam-5279	108	1	[	[	X
ejpam-5279	108	2	0,∞	0,∞	X
ejpam-5279	108	3	]	]	X
ejpam-5279	108	4	;	;	PUNCT
ejpam-5279	108	5	(	(	PUNCT
ejpam-5279	108	6	u1	u1	NOUN
ejpam-5279	108	7	)	)	PUNCT
ejpam-5279	108	8	ux	ux	NOUN
ejpam-5279	108	9	α	α	PROPN
ejpam-5279	108	10	≤	≤	PROPN
ejpam-5279	108	11	ẋ	ẋ	PROPN
ejpam-5279	108	12	,	,	PUNCT
ejpam-5279	108	13	for	for	ADP
ejpam-5279	108	14	all	all	DET
ejpam-5279	108	15	x	x	SYM
ejpam-5279	108	16	∈	∈	PROPN
ejpam-5279	108	17	x	x	X
ejpam-5279	108	18	and	and	CCONJ
ejpam-5279	108	19	α	α	NOUN
ejpam-5279	108	20	∈	∈	PROPN
ejpam-5279	109	1	[	[	X
ejpam-5279	109	2	0,∞	0,∞	X
ejpam-5279	109	3	]	]	X
ejpam-5279	109	4	;	;	PUNCT
ejpam-5279	109	5	(	(	PUNCT
ejpam-5279	109	6	u2	u2	NOUN
ejpam-5279	109	7	)	)	PUNCT
ejpam-5279	109	8	ux	ux	NOUN
ejpam-5279	109	9	α+β	α+β	PROPN
ejpam-5279	109	10	≤	≤	NUM
ejpam-5279	109	11	κ	κ	NOUN
ejpam-5279	109	12	(	(	PUNCT
ejpam-5279	109	13	ux	ux	PROPN
ejpam-5279	109	14	β	β	X
ejpam-5279	109	15	,	,	PUNCT
ejpam-5279	109	16	(	(	PUNCT
ejpam-5279	109	17	(	(	PUNCT
ejpam-5279	109	18	u	u	NOUN
ejpam-5279	109	19	y	y	PROPN
ejpam-5279	109	20	α)y∈x	α)y∈x	PROPN
ejpam-5279	109	21	)	)	PUNCT
ejpam-5279	109	22	)	)	PUNCT
ejpam-5279	109	23	,	,	PUNCT
ejpam-5279	109	24	for	for	ADP
ejpam-5279	109	25	all	all	DET
ejpam-5279	109	26	x	x	SYM
ejpam-5279	109	27	∈	∈	PROPN
ejpam-5279	109	28	x	x	X
ejpam-5279	109	29	and	and	CCONJ
ejpam-5279	109	30	α	α	NOUN
ejpam-5279	109	31	,	,	PUNCT
ejpam-5279	109	32	β	β	X
ejpam-5279	109	33	∈	∈	PROPN
ejpam-5279	110	1	[	[	X
ejpam-5279	110	2	0,∞	0,∞	X
ejpam-5279	110	3	]	]	X
ejpam-5279	110	4	;	;	PUNCT
ejpam-5279	110	5	(	(	PUNCT
ejpam-5279	110	6	u3	u3	PROPN
ejpam-5279	110	7	)	)	PUNCT
ejpam-5279	110	8	0	0	NUM
ejpam-5279	110	9	≤	≤	NUM
ejpam-5279	110	10	α	α	NOUN
ejpam-5279	110	11	≤	≤	NOUN
ejpam-5279	110	12	β	β	NOUN
ejpam-5279	110	13	implies	imply	VERB
ejpam-5279	110	14	ux	ux	PROPN
ejpam-5279	110	15	β	β	X
ejpam-5279	110	16	≤	≤	NUM
ejpam-5279	110	17	ux	ux	PROPN
ejpam-5279	111	1	α	α	PROPN
ejpam-5279	111	2	;	;	PUNCT
ejpam-5279	111	3	(	(	PUNCT
ejpam-5279	111	4	u4	u4	NOUN
ejpam-5279	111	5	)	)	PUNCT
ejpam-5279	111	6	for	for	ADP
ejpam-5279	111	7	all	all	DET
ejpam-5279	111	8	∅	∅	NOUN
ejpam-5279	111	9	=	=	NOUN
ejpam-5279	111	10	̸	̸	VERB
ejpam-5279	111	11	a	a	DET
ejpam-5279	111	12	⊂	⊂	PROPN
ejpam-5279	112	1	[	[	X
ejpam-5279	112	2	0,∞	0,∞	X
ejpam-5279	112	3	]	]	X
ejpam-5279	112	4	:	:	PUNCT
ejpam-5279	112	5	∨	∨	NUM
ejpam-5279	112	6	α∈aux	α∈aux	PROPN
ejpam-5279	112	7	α	α	PROPN
ejpam-5279	112	8	=	=	SYM
ejpam-5279	112	9	ux	ux	PROPN
ejpam-5279	112	10	∧a	∧a	PROPN
ejpam-5279	112	11	.	.	PUNCT
ejpam-5279	113	1	we	we	PRON
ejpam-5279	113	2	call	call	VERB
ejpam-5279	113	3	the	the	DET
ejpam-5279	113	4	system	system	NOUN
ejpam-5279	113	5	u	u	NOUN
ejpam-5279	113	6	as	as	ADP
ejpam-5279	113	7	the	the	DET
ejpam-5279	113	8	corresponding	correspond	VERB
ejpam-5279	113	9	neighborhood	neighborhood	NOUN
ejpam-5279	113	10	system	system	NOUN
ejpam-5279	113	11	of	of	ADP
ejpam-5279	113	12	the	the	DET
ejpam-5279	113	13	approach	approach	NOUN
ejpam-5279	113	14	space	space	NOUN
ejpam-5279	113	15	(	(	PUNCT
ejpam-5279	113	16	x	x	X
ejpam-5279	113	17	,	,	PUNCT
ejpam-5279	113	18	λ	λ	NOUN
ejpam-5279	113	19	)	)	PUNCT
ejpam-5279	113	20	.	.	PUNCT
ejpam-5279	114	1	t.m.g	t.m.g	ADJ
ejpam-5279	114	2	.	.	PUNCT
ejpam-5279	115	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	115	2	,	,	PUNCT
ejpam-5279	115	3	fawzi	fawzi	PROPN
ejpam-5279	115	4	al	al	PROPN
ejpam-5279	115	5	-	-	PUNCT
ejpam-5279	115	6	thukair	thukair	NOUN
ejpam-5279	115	7	/	/	SYM
ejpam-5279	115	8	eur	eur	NOUN
ejpam-5279	115	9	.	.	PUNCT
ejpam-5279	116	1	j.	j.	PROPN
ejpam-5279	116	2	pure	pure	PROPN
ejpam-5279	116	3	appl	appl	PROPN
ejpam-5279	116	4	.	.	PROPN
ejpam-5279	116	5	math	math	PROPN
ejpam-5279	116	6	,	,	PUNCT
ejpam-5279	116	7	17	17	NUM
ejpam-5279	116	8	(	(	PUNCT
ejpam-5279	116	9	3	3	NUM
ejpam-5279	116	10	)	)	PUNCT
ejpam-5279	116	11	(	(	PUNCT
ejpam-5279	116	12	2024	2024	NUM
ejpam-5279	116	13	)	)	PUNCT
ejpam-5279	116	14	,	,	PUNCT
ejpam-5279	116	15	1762	1762	NUM
ejpam-5279	116	16	-	-	SYM
ejpam-5279	116	17	1778	1778	NUM
ejpam-5279	116	18	1766	1766	NUM
ejpam-5279	116	19	if	if	SCONJ
ejpam-5279	116	20	(	(	PUNCT
ejpam-5279	116	21	x	x	NOUN
ejpam-5279	116	22	,	,	PUNCT
ejpam-5279	116	23	·	·	PUNCT
ejpam-5279	116	24	)	)	PUNCT
ejpam-5279	116	25	is	be	AUX
ejpam-5279	116	26	a	a	DET
ejpam-5279	116	27	group	group	NOUN
ejpam-5279	116	28	and	and	CCONJ
ejpam-5279	116	29	ν	ν	NOUN
ejpam-5279	116	30	∈	∈	PROPN
ejpam-5279	117	1	[	[	X
ejpam-5279	117	2	0,∞]x	0,∞]x	X
ejpam-5279	117	3	,	,	PUNCT
ejpam-5279	117	4	then	then	ADV
ejpam-5279	117	5	for	for	ADP
ejpam-5279	117	6	any	any	DET
ejpam-5279	117	7	x	x	SYM
ejpam-5279	117	8	∈	∈	PROPN
ejpam-5279	117	9	x	x	X
ejpam-5279	117	10	,	,	PUNCT
ejpam-5279	117	11	we	we	PRON
ejpam-5279	117	12	write	write	VERB
ejpam-5279	117	13	x	x	PROPN
ejpam-5279	117	14	⊙	⊙	PROPN
ejpam-5279	117	15	ν	ν	PROPN
ejpam-5279	117	16	:	:	PUNCT
ejpam-5279	117	17	x	x	PUNCT
ejpam-5279	117	18	−→	−→	NOUN
ejpam-5279	117	19	[	[	X
ejpam-5279	117	20	0,∞	0,∞	X
ejpam-5279	117	21	]	]	X
ejpam-5279	117	22	,	,	PUNCT
ejpam-5279	117	23	y	y	PROPN
ejpam-5279	117	24	7→	7→	PROPN
ejpam-5279	117	25	x⊙ν(y	x⊙ν(y	PUNCT
ejpam-5279	117	26	)	)	PUNCT
ejpam-5279	118	1	=	=	SYM
ejpam-5279	118	2	ν(xy	ν(xy	PROPN
ejpam-5279	118	3	)	)	PUNCT
ejpam-5279	118	4	.	.	PUNCT
ejpam-5279	119	1	similarly	similarly	ADV
ejpam-5279	119	2	,	,	PUNCT
ejpam-5279	119	3	we	we	PRON
ejpam-5279	119	4	also	also	ADV
ejpam-5279	119	5	write	write	VERB
ejpam-5279	119	6	ν⊙x	ν⊙x	NOUN
ejpam-5279	119	7	:	:	PUNCT
ejpam-5279	119	8	x	x	X
ejpam-5279	119	9	−→	−→	NOUN
ejpam-5279	119	10	[	[	X
ejpam-5279	119	11	0,∞	0,∞	X
ejpam-5279	119	12	]	]	PUNCT
ejpam-5279	119	13	,	,	PUNCT
ejpam-5279	119	14	y	y	PROPN
ejpam-5279	119	15	7→	7→	NUM
ejpam-5279	119	16	ν⊙x(y	ν⊙x(y	NOUN
ejpam-5279	119	17	)	)	PUNCT
ejpam-5279	119	18	=	=	SYM
ejpam-5279	120	1	ν(yx	ν(yx	NOUN
ejpam-5279	120	2	)	)	PUNCT
ejpam-5279	120	3	.	.	PUNCT
ejpam-5279	121	1	if	if	SCONJ
ejpam-5279	121	2	ν	ν	X
ejpam-5279	121	3	∈	∈	PROPN
ejpam-5279	121	4	[	[	X
ejpam-5279	121	5	0,∞]x	0,∞]x	X
ejpam-5279	121	6	,	,	PUNCT
ejpam-5279	121	7	then	then	ADV
ejpam-5279	121	8	ν−1	ν−1	ADV
ejpam-5279	121	9	:	:	PUNCT
ejpam-5279	121	10	x	x	X
ejpam-5279	121	11	→	→	PUNCT
ejpam-5279	121	12	[	[	X
ejpam-5279	121	13	0,∞	0,∞	X
ejpam-5279	121	14	]	]	PUNCT
ejpam-5279	121	15	is	be	AUX
ejpam-5279	121	16	defined	define	VERB
ejpam-5279	121	17	by	by	ADP
ejpam-5279	121	18	ν−1(x	ν−1(x	NOUN
ejpam-5279	121	19	)	)	PUNCT
ejpam-5279	121	20	=	=	PUNCT
ejpam-5279	122	1	ν(x−1	ν(x−1	NOUN
ejpam-5279	122	2	)	)	PUNCT
ejpam-5279	122	3	.	.	PUNCT
ejpam-5279	123	1	if	if	SCONJ
ejpam-5279	123	2	b	b	X
ejpam-5279	123	3	⊂	⊂	PROPN
ejpam-5279	123	4	[	[	X
ejpam-5279	123	5	0,∞]x	0,∞]x	X
ejpam-5279	123	6	,	,	PUNCT
ejpam-5279	123	7	put	put	VERB
ejpam-5279	123	8	x⊙	x⊙	PROPN
ejpam-5279	123	9	b	b	NOUN
ejpam-5279	123	10	=	=	PRON
ejpam-5279	123	11	{	{	PUNCT
ejpam-5279	123	12	x⊙	x⊙	PROPN
ejpam-5279	123	13	ν|	ν|	PROPN
ejpam-5279	123	14	ν	ν	X
ejpam-5279	123	15	∈	∈	PROPN
ejpam-5279	123	16	b	b	NOUN
ejpam-5279	123	17	}	}	PUNCT
ejpam-5279	123	18	.	.	PUNCT
ejpam-5279	124	1	note	note	VERB
ejpam-5279	124	2	that	that	SCONJ
ejpam-5279	125	1	⟨x⊙	⟨x⊙	INTJ
ejpam-5279	125	2	b⟩	b⟩	PRON
ejpam-5279	126	1	=	=	PUNCT
ejpam-5279	126	2	x⊙	x⊙	PROPN
ejpam-5279	127	1	⟨b⟩.	⟨b⟩.	PROPN
ejpam-5279	127	2	now	now	ADV
ejpam-5279	127	3	for	for	ADP
ejpam-5279	127	4	the	the	DET
ejpam-5279	127	5	convenience	convenience	NOUN
ejpam-5279	127	6	of	of	ADP
ejpam-5279	127	7	the	the	DET
ejpam-5279	127	8	reader	reader	NOUN
ejpam-5279	127	9	,	,	PUNCT
ejpam-5279	127	10	we	we	PRON
ejpam-5279	127	11	recall	recall	VERB
ejpam-5279	127	12	some	some	DET
ejpam-5279	127	13	essential	essential	ADJ
ejpam-5279	127	14	categorical	categorical	ADJ
ejpam-5279	127	15	terms	term	NOUN
ejpam-5279	127	16	that	that	PRON
ejpam-5279	127	17	are	be	AUX
ejpam-5279	127	18	needed	need	VERB
ejpam-5279	127	19	in	in	ADP
ejpam-5279	127	20	the	the	DET
ejpam-5279	127	21	sequel	sequel	NOUN
ejpam-5279	127	22	,	,	PUNCT
ejpam-5279	127	23	for	for	ADP
ejpam-5279	127	24	the	the	DET
ejpam-5279	127	25	details	detail	NOUN
ejpam-5279	127	26	,	,	PUNCT
ejpam-5279	127	27	we	we	PRON
ejpam-5279	127	28	refer	refer	VERB
ejpam-5279	127	29	to	to	ADP
ejpam-5279	127	30	[	[	X
ejpam-5279	127	31	1	1	NUM
ejpam-5279	127	32	]	]	PUNCT
ejpam-5279	127	33	.	.	PUNCT
ejpam-5279	128	1	a	a	DET
ejpam-5279	128	2	functor	functor	PROPN
ejpam-5279	128	3	f	f	PROPN
ejpam-5279	128	4	:	:	PUNCT
ejpam-5279	128	5	c	c	AUX
ejpam-5279	129	1	−→	−→	NOUN
ejpam-5279	129	2	d	d	NOUN
ejpam-5279	129	3	is	be	AUX
ejpam-5279	129	4	a	a	DET
ejpam-5279	129	5	morphism	morphism	NOUN
ejpam-5279	129	6	between	between	ADP
ejpam-5279	129	7	categories	category	NOUN
ejpam-5279	129	8	,	,	PUNCT
ejpam-5279	129	9	consists	consist	VERB
ejpam-5279	129	10	of	of	ADP
ejpam-5279	129	11	mappings	mapping	NOUN
ejpam-5279	129	12	between	between	ADP
ejpam-5279	129	13	objects	object	NOUN
ejpam-5279	129	14	of	of	ADP
ejpam-5279	129	15	c	c	NOUN
ejpam-5279	129	16	and	and	CCONJ
ejpam-5279	129	17	objects	object	NOUN
ejpam-5279	129	18	of	of	ADP
ejpam-5279	129	19	d(sometimes	d(sometime	NOUN
ejpam-5279	129	20	we	we	PRON
ejpam-5279	129	21	write	write	VERB
ejpam-5279	129	22	as	as	ADP
ejpam-5279	129	23	|c|	|c|	PROPN
ejpam-5279	129	24	to	to	PART
ejpam-5279	129	25	denote	denote	VERB
ejpam-5279	129	26	the	the	DET
ejpam-5279	129	27	objects	object	NOUN
ejpam-5279	129	28	of	of	ADP
ejpam-5279	129	29	c	c	NOUN
ejpam-5279	129	30	)	)	PUNCT
ejpam-5279	129	31	and	and	CCONJ
ejpam-5279	129	32	the	the	DET
ejpam-5279	129	33	mapping	mapping	NOUN
ejpam-5279	129	34	between	between	ADP
ejpam-5279	129	35	morphisms	morphism	NOUN
ejpam-5279	129	36	of	of	ADP
ejpam-5279	129	37	c	c	NOUN
ejpam-5279	129	38	and	and	CCONJ
ejpam-5279	129	39	morphisms	morphism	NOUN
ejpam-5279	129	40	of	of	ADP
ejpam-5279	129	41	d	d	PROPN
ejpam-5279	129	42	such	such	ADJ
ejpam-5279	129	43	that	that	SCONJ
ejpam-5279	129	44	(	(	PUNCT
ejpam-5279	129	45	i	i	NOUN
ejpam-5279	129	46	)	)	PUNCT
ejpam-5279	129	47	if	if	SCONJ
ejpam-5279	129	48	f	f	PROPN
ejpam-5279	129	49	:	:	PUNCT
ejpam-5279	129	50	s	s	VERB
ejpam-5279	129	51	−→	−→	NOUN
ejpam-5279	129	52	t	t	NOUN
ejpam-5279	129	53	,	,	PUNCT
ejpam-5279	129	54	then	then	ADV
ejpam-5279	129	55	f(f	f(f	PROPN
ejpam-5279	129	56	)	)	PUNCT
ejpam-5279	129	57	:	:	PUNCT
ejpam-5279	129	58	c(s	c(	VERB
ejpam-5279	129	59	)	)	PUNCT
ejpam-5279	129	60	−→	−→	ADJ
ejpam-5279	129	61	d(t	d(t	PROPN
ejpam-5279	129	62	)	)	PUNCT
ejpam-5279	129	63	;	;	PUNCT
ejpam-5279	129	64	(	(	PUNCT
ejpam-5279	129	65	ii	ii	X
ejpam-5279	129	66	)	)	PUNCT
ejpam-5279	129	67	f(f	f(f	PROPN
ejpam-5279	129	68	◦	◦	VERB
ejpam-5279	129	69	g	g	NOUN
ejpam-5279	129	70	)	)	PUNCT
ejpam-5279	129	71	=	=	SYM
ejpam-5279	129	72	f(f	f(f	X
ejpam-5279	129	73	)	)	PUNCT
ejpam-5279	129	74	◦	◦	NOUN
ejpam-5279	129	75	f(g	f(g	NOUN
ejpam-5279	129	76	)	)	PUNCT
ejpam-5279	129	77	,	,	PUNCT
ejpam-5279	129	78	whenever	whenever	SCONJ
ejpam-5279	129	79	f	f	AUX
ejpam-5279	129	80	◦	◦	NOUN
ejpam-5279	129	81	g	g	NOUN
ejpam-5279	129	82	is	be	AUX
ejpam-5279	129	83	defined	define	VERB
ejpam-5279	129	84	;	;	PUNCT
ejpam-5279	129	85	(	(	PUNCT
ejpam-5279	129	86	iii	iii	X
ejpam-5279	129	87	)	)	PUNCT
ejpam-5279	129	88	f(ids	f(id	NOUN
ejpam-5279	129	89	)	)	PUNCT
ejpam-5279	129	90	=	=	SYM
ejpam-5279	129	91	idf(s	idf(s	PROPN
ejpam-5279	129	92	)	)	PUNCT
ejpam-5279	129	93	.	.	PUNCT
ejpam-5279	130	1	the	the	DET
ejpam-5279	130	2	functor	functor	PROPN
ejpam-5279	130	3	f	f	PROPN
ejpam-5279	130	4	is	be	AUX
ejpam-5279	130	5	called	call	VERB
ejpam-5279	130	6	an	an	DET
ejpam-5279	130	7	embedding	embedding	NOUN
ejpam-5279	130	8	if	if	SCONJ
ejpam-5279	130	9	it	it	PRON
ejpam-5279	130	10	is	be	AUX
ejpam-5279	130	11	injective	injective	ADJ
ejpam-5279	130	12	on	on	ADP
ejpam-5279	130	13	objects	object	NOUN
ejpam-5279	130	14	.	.	PUNCT
ejpam-5279	131	1	if	if	SCONJ
ejpam-5279	131	2	e	e	PROPN
ejpam-5279	131	3	is	be	AUX
ejpam-5279	131	4	a	a	DET
ejpam-5279	131	5	category	category	NOUN
ejpam-5279	131	6	,	,	PUNCT
ejpam-5279	131	7	then	then	ADV
ejpam-5279	131	8	by	by	ADP
ejpam-5279	131	9	a	a	DET
ejpam-5279	131	10	concrete	concrete	ADJ
ejpam-5279	131	11	category	category	NOUN
ejpam-5279	131	12	over	over	ADP
ejpam-5279	131	13	e	e	NOUN
ejpam-5279	131	14	,	,	PUNCT
ejpam-5279	131	15	we	we	PRON
ejpam-5279	131	16	understand	understand	VERB
ejpam-5279	131	17	a	a	DET
ejpam-5279	131	18	pair	pair	NOUN
ejpam-5279	131	19	(	(	PUNCT
ejpam-5279	131	20	g	g	NOUN
ejpam-5279	131	21	,	,	PUNCT
ejpam-5279	131	22	f	f	PROPN
ejpam-5279	131	23	)	)	PUNCT
ejpam-5279	131	24	,	,	PUNCT
ejpam-5279	131	25	where	where	SCONJ
ejpam-5279	131	26	c	c	NOUN
ejpam-5279	131	27	is	be	AUX
ejpam-5279	131	28	a	a	DET
ejpam-5279	131	29	category	category	NOUN
ejpam-5279	131	30	and	and	CCONJ
ejpam-5279	131	31	f	f	NOUN
ejpam-5279	131	32	:	:	PUNCT
ejpam-5279	131	33	g	g	ADP
ejpam-5279	131	34	−→	−→	NOUN
ejpam-5279	131	35	e	e	NOUN
ejpam-5279	131	36	is	be	AUX
ejpam-5279	131	37	a	a	DET
ejpam-5279	131	38	faithful	faithful	ADJ
ejpam-5279	131	39	functor	functor	NOUN
ejpam-5279	131	40	.	.	PUNCT
ejpam-5279	132	1	a	a	DET
ejpam-5279	132	2	construct	construct	NOUN
ejpam-5279	132	3	is	be	AUX
ejpam-5279	132	4	a	a	DET
ejpam-5279	132	5	concrete	concrete	ADJ
ejpam-5279	132	6	category	category	NOUN
ejpam-5279	132	7	over	over	ADP
ejpam-5279	132	8	set	set	NOUN
ejpam-5279	132	9	,	,	PUNCT
ejpam-5279	132	10	the	the	DET
ejpam-5279	132	11	category	category	NOUN
ejpam-5279	132	12	of	of	ADP
ejpam-5279	132	13	sets	set	NOUN
ejpam-5279	132	14	,	,	PUNCT
ejpam-5279	132	15	and	and	CCONJ
ejpam-5279	132	16	we	we	PRON
ejpam-5279	132	17	consider	consider	VERB
ejpam-5279	132	18	the	the	DET
ejpam-5279	132	19	objects	object	NOUN
ejpam-5279	132	20	of	of	ADP
ejpam-5279	132	21	a	a	DET
ejpam-5279	132	22	construct	construct	NOUN
ejpam-5279	132	23	as	as	ADP
ejpam-5279	132	24	structured	structure	VERB
ejpam-5279	132	25	set	set	NOUN
ejpam-5279	132	26	(	(	PUNCT
ejpam-5279	132	27	s	s	PROPN
ejpam-5279	132	28	,	,	PUNCT
ejpam-5279	132	29	ξ	ξ	NOUN
ejpam-5279	132	30	)	)	PUNCT
ejpam-5279	132	31	,	,	PUNCT
ejpam-5279	132	32	and	and	CCONJ
ejpam-5279	132	33	morphisms	morphism	NOUN
ejpam-5279	132	34	are	be	AUX
ejpam-5279	132	35	suitable	suitable	ADJ
ejpam-5279	132	36	mappings	mapping	NOUN
ejpam-5279	132	37	between	between	ADP
ejpam-5279	132	38	the	the	DET
ejpam-5279	132	39	underlying	underlie	VERB
ejpam-5279	132	40	sets	set	NOUN
ejpam-5279	132	41	.	.	PUNCT
ejpam-5279	133	1	a	a	DET
ejpam-5279	133	2	construct	construct	NOUN
ejpam-5279	133	3	is	be	AUX
ejpam-5279	133	4	called	call	VERB
ejpam-5279	133	5	topological	topological	ADJ
ejpam-5279	133	6	if	if	SCONJ
ejpam-5279	133	7	it	it	PRON
ejpam-5279	133	8	allows	allow	VERB
ejpam-5279	133	9	initial	initial	ADJ
ejpam-5279	133	10	constructions	construction	NOUN
ejpam-5279	133	11	,	,	PUNCT
ejpam-5279	133	12	that	that	ADV
ejpam-5279	133	13	is	is	ADV
ejpam-5279	133	14	,	,	PUNCT
ejpam-5279	133	15	for	for	ADP
ejpam-5279	133	16	any	any	DET
ejpam-5279	133	17	source	source	NOUN
ejpam-5279	133	18	(	(	PUNCT
ejpam-5279	133	19	fj	fj	INTJ
ejpam-5279	133	20	:	:	PUNCT
ejpam-5279	133	21	s	s	AUX
ejpam-5279	133	22	−→	−→	NOUN
ejpam-5279	133	23	(	(	PUNCT
ejpam-5279	133	24	sj	sj	INTJ
ejpam-5279	133	25	,	,	PUNCT
ejpam-5279	133	26	ςj))j∈j	ςj))j∈j	NUM
ejpam-5279	133	27	,	,	PUNCT
ejpam-5279	133	28	there	there	PRON
ejpam-5279	133	29	is	be	VERB
ejpam-5279	133	30	a	a	DET
ejpam-5279	133	31	unique	unique	ADJ
ejpam-5279	133	32	structure	structure	NOUN
ejpam-5279	133	33	ς	ς	NOUN
ejpam-5279	133	34	on	on	ADP
ejpam-5279	133	35	s	s	PRON
ejpam-5279	133	36	such	such	ADJ
ejpam-5279	133	37	that	that	SCONJ
ejpam-5279	133	38	a	a	DET
ejpam-5279	133	39	mapping	mapping	NOUN
ejpam-5279	133	40	g	g	NOUN
ejpam-5279	133	41	:	:	PUNCT
ejpam-5279	133	42	(	(	PUNCT
ejpam-5279	133	43	t	t	PROPN
ejpam-5279	133	44	,	,	PUNCT
ejpam-5279	133	45	β	β	NOUN
ejpam-5279	133	46	)	)	PUNCT
ejpam-5279	133	47	−→	−→	NOUN
ejpam-5279	133	48	(	(	PUNCT
ejpam-5279	133	49	s	s	PROPN
ejpam-5279	133	50	,	,	PUNCT
ejpam-5279	133	51	ς	ς	NOUN
ejpam-5279	133	52	)	)	PUNCT
ejpam-5279	133	53	is	be	AUX
ejpam-5279	133	54	a	a	DET
ejpam-5279	133	55	morphism	morphism	NOUN
ejpam-5279	133	56	if	if	SCONJ
ejpam-5279	133	57	and	and	CCONJ
ejpam-5279	133	58	only	only	ADV
ejpam-5279	133	59	if	if	SCONJ
ejpam-5279	133	60	for	for	ADP
ejpam-5279	133	61	each	each	DET
ejpam-5279	133	62	j	j	PROPN
ejpam-5279	133	63	∈	∈	PROPN
ejpam-5279	133	64	j	j	PROPN
ejpam-5279	133	65	the	the	DET
ejpam-5279	133	66	composition	composition	NOUN
ejpam-5279	133	67	fj	fj	INTJ
ejpam-5279	133	68	◦	◦	NOUN
ejpam-5279	133	69	g	g	NOUN
ejpam-5279	133	70	:	:	PUNCT
ejpam-5279	133	71	(	(	PUNCT
ejpam-5279	133	72	t	t	NOUN
ejpam-5279	133	73	,	,	PUNCT
ejpam-5279	133	74	β	β	NOUN
ejpam-5279	133	75	)	)	PUNCT
ejpam-5279	133	76	−→	−→	NOUN
ejpam-5279	133	77	(	(	PUNCT
ejpam-5279	133	78	sj	sj	INTJ
ejpam-5279	133	79	,	,	PUNCT
ejpam-5279	133	80	ςj	ςj	PROPN
ejpam-5279	133	81	)	)	PUNCT
ejpam-5279	133	82	is	be	AUX
ejpam-5279	133	83	a	a	DET
ejpam-5279	133	84	morphism	morphism	NOUN
ejpam-5279	133	85	,	,	PUNCT
ejpam-5279	133	86	where	where	SCONJ
ejpam-5279	133	87	(	(	PUNCT
ejpam-5279	133	88	t	t	PROPN
ejpam-5279	133	89	,	,	PUNCT
ejpam-5279	133	90	β	β	NOUN
ejpam-5279	133	91	)	)	PUNCT
ejpam-5279	133	92	is	be	AUX
ejpam-5279	133	93	a	a	DET
ejpam-5279	133	94	structured	structured	ADJ
ejpam-5279	133	95	set	set	NOUN
ejpam-5279	133	96	.	.	PUNCT
ejpam-5279	134	1	a	a	DET
ejpam-5279	134	2	functor	functor	PROPN
ejpam-5279	134	3	f	f	PROPN
ejpam-5279	134	4	:	:	PUNCT
ejpam-5279	134	5	c	c	AUX
ejpam-5279	134	6	−→	−→	NOUN
ejpam-5279	134	7	d	d	X
ejpam-5279	134	8	between	between	ADP
ejpam-5279	134	9	categories	category	NOUN
ejpam-5279	134	10	c	c	PROPN
ejpam-5279	134	11	and	and	CCONJ
ejpam-5279	134	12	d	d	PROPN
ejpam-5279	134	13	is	be	AUX
ejpam-5279	134	14	called	call	VERB
ejpam-5279	134	15	an	an	DET
ejpam-5279	134	16	isomorphism	isomorphism	NOUN
ejpam-5279	134	17	if	if	SCONJ
ejpam-5279	134	18	there	there	PRON
ejpam-5279	134	19	is	be	VERB
ejpam-5279	134	20	a	a	DET
ejpam-5279	134	21	functor	functor	PROPN
ejpam-5279	134	22	h	h	NOUN
ejpam-5279	134	23	:	:	PUNCT
ejpam-5279	135	1	d	d	X
ejpam-5279	135	2	−→	−→	NOUN
ejpam-5279	135	3	c	c	ADP
ejpam-5279	135	4	such	such	ADJ
ejpam-5279	135	5	that	that	DET
ejpam-5279	135	6	h	h	PROPN
ejpam-5279	135	7	◦	◦	NOUN
ejpam-5279	135	8	f	f	NOUN
ejpam-5279	135	9	=	=	SYM
ejpam-5279	135	10	idc	idc	PROPN
ejpam-5279	135	11	and	and	CCONJ
ejpam-5279	135	12	f	f	PROPN
ejpam-5279	135	13	◦	◦	NOUN
ejpam-5279	135	14	h	h	NOUN
ejpam-5279	135	15	=	=	SYM
ejpam-5279	135	16	idd	idd	X
ejpam-5279	135	17	.	.	PUNCT
ejpam-5279	136	1	two	two	NUM
ejpam-5279	136	2	categories	category	NOUN
ejpam-5279	136	3	c	c	NOUN
ejpam-5279	136	4	and	and	CCONJ
ejpam-5279	136	5	d	d	PROPN
ejpam-5279	136	6	are	be	AUX
ejpam-5279	136	7	said	say	VERB
ejpam-5279	136	8	to	to	PART
ejpam-5279	136	9	be	be	AUX
ejpam-5279	136	10	isomorphic	isomorphic	ADJ
ejpam-5279	136	11	if	if	SCONJ
ejpam-5279	136	12	there	there	PRON
ejpam-5279	136	13	is	be	VERB
ejpam-5279	136	14	an	an	DET
ejpam-5279	136	15	isomorphism	isomorphism	NOUN
ejpam-5279	136	16	.	.	PUNCT
ejpam-5279	137	1	3	3	X
ejpam-5279	137	2	.	.	X
ejpam-5279	137	3	approach	approach	NOUN
ejpam-5279	137	4	groups	group	NOUN
ejpam-5279	137	5	,	,	PUNCT
ejpam-5279	137	6	characterizations	characterization	NOUN
ejpam-5279	137	7	and	and	CCONJ
ejpam-5279	137	8	ultra	ultra	ADJ
ejpam-5279	137	9	uniformization	uniformization	NOUN
ejpam-5279	137	10	definition	definition	NOUN
ejpam-5279	137	11	5	5	NUM
ejpam-5279	137	12	.	.	PUNCT
ejpam-5279	138	1	[	[	X
ejpam-5279	138	2	15	15	NUM
ejpam-5279	138	3	]	]	PUNCT
ejpam-5279	138	4	let	let	VERB
ejpam-5279	138	5	x	x	PRON
ejpam-5279	138	6	be	be	AUX
ejpam-5279	138	7	a	a	DET
ejpam-5279	138	8	set	set	NOUN
ejpam-5279	138	9	.	.	PUNCT
ejpam-5279	139	1	a	a	DET
ejpam-5279	139	2	family	family	NOUN
ejpam-5279	139	3	of	of	ADP
ejpam-5279	139	4	ideals	ideal	NOUN
ejpam-5279	139	5	(	(	PUNCT
ejpam-5279	139	6	ω(x))x∈x	ω(x))x∈x	VERB
ejpam-5279	139	7	in	in	ADP
ejpam-5279	139	8	[	[	X
ejpam-5279	139	9	0,∞]x	0,∞]x	PRON
ejpam-5279	139	10	is	be	AUX
ejpam-5279	139	11	called	call	VERB
ejpam-5279	139	12	an	an	DET
ejpam-5279	139	13	ultra	ultra	ADJ
ejpam-5279	139	14	approach	approach	NOUN
ejpam-5279	139	15	system	system	NOUN
ejpam-5279	139	16	on	on	ADP
ejpam-5279	139	17	x	x	PUNCT
ejpam-5279	139	18	if	if	SCONJ
ejpam-5279	139	19	and	and	CCONJ
ejpam-5279	139	20	only	only	ADV
ejpam-5279	139	21	if	if	SCONJ
ejpam-5279	139	22	for	for	ADP
ejpam-5279	139	23	all	all	DET
ejpam-5279	139	24	x	x	SYM
ejpam-5279	139	25	∈	∈	PROPN
ejpam-5279	139	26	x	x	NOUN
ejpam-5279	139	27	,	,	PUNCT
ejpam-5279	139	28	the	the	DET
ejpam-5279	139	29	following	follow	VERB
ejpam-5279	139	30	properties	property	NOUN
ejpam-5279	139	31	are	be	AUX
ejpam-5279	139	32	satisfied	satisfied	ADJ
ejpam-5279	139	33	:	:	PUNCT
ejpam-5279	139	34	(	(	PUNCT
ejpam-5279	139	35	as1	as1	NOUN
ejpam-5279	139	36	)	)	PUNCT
ejpam-5279	139	37	∀ν	∀ν	PROPN
ejpam-5279	139	38	∈	∈	PROPN
ejpam-5279	139	39	ω(x	ω(x	NOUN
ejpam-5279	139	40	):	):	PUNCT
ejpam-5279	139	41	ν(x	ν(x	PROPN
ejpam-5279	139	42	)	)	PUNCT
ejpam-5279	139	43	=	=	SYM
ejpam-5279	140	1	0	0	X
ejpam-5279	140	2	.	.	PUNCT
ejpam-5279	140	3	(	(	PUNCT
ejpam-5279	140	4	as2	as2	PROPN
ejpam-5279	140	5	)	)	PUNCT
ejpam-5279	140	6	∀ν	∀ν	PROPN
ejpam-5279	140	7	∈	∈	PROPN
ejpam-5279	141	1	[	[	X
ejpam-5279	141	2	0,∞]x	0,∞]x	X
ejpam-5279	141	3	:	:	PUNCT
ejpam-5279	141	4	(	(	PUNCT
ejpam-5279	141	5	∀ϵ	∀ϵ	NOUN
ejpam-5279	141	6	>	>	X
ejpam-5279	141	7	0,∀n	0,∀n	PROPN
ejpam-5279	141	8	<	<	X
ejpam-5279	141	9	∞,∃νnϵ	∞,∃νnϵ	PROPN
ejpam-5279	141	10	∈	∈	PROPN
ejpam-5279	141	11	ω(x	ω(x	NOUN
ejpam-5279	141	12	)	)	PUNCT
ejpam-5279	141	13	s.t	s.t	PROPN
ejpam-5279	141	14	.	.	PUNCT
ejpam-5279	141	15	ν∧n	ν∧n	PROPN
ejpam-5279	141	16	≤	≤	NUM
ejpam-5279	141	17	νnϵ	νnϵ	NOUN
ejpam-5279	141	18	+	+	NOUN
ejpam-5279	141	19	ϵ	ϵ	NOUN
ejpam-5279	141	20	)	)	PUNCT
ejpam-5279	141	21	implies	imply	VERB
ejpam-5279	141	22	ν	ν	PROPN
ejpam-5279	141	23	∈	∈	PROPN
ejpam-5279	141	24	ω(x	ω(x	NOUN
ejpam-5279	141	25	)	)	PUNCT
ejpam-5279	141	26	.	.	PUNCT
ejpam-5279	142	1	(	(	PUNCT
ejpam-5279	142	2	as3	as3	PROPN
ejpam-5279	142	3	)	)	PUNCT
ejpam-5279	142	4	∀ν	∀ν	PROPN
ejpam-5279	142	5	∈	∈	PROPN
ejpam-5279	142	6	ω(x	ω(x	NOUN
ejpam-5279	142	7	)	)	PUNCT
ejpam-5279	142	8	,	,	PUNCT
ejpam-5279	142	9	∀ϵ	∀ϵ	NOUN
ejpam-5279	142	10	>	>	X
ejpam-5279	142	11	0	0	NUM
ejpam-5279	142	12	,	,	PUNCT
ejpam-5279	142	13	∀n	∀n	NUM
ejpam-5279	142	14	<	<	X
ejpam-5279	142	15	∞,∃	∞,∃	PROPN
ejpam-5279	142	16	a	a	DET
ejpam-5279	142	17	family	family	NOUN
ejpam-5279	142	18	(	(	PUNCT
ejpam-5279	142	19	ω(z))z∈x	ω(z))z∈x	X
ejpam-5279	142	20	such	such	ADJ
ejpam-5279	142	21	that	that	PRON
ejpam-5279	142	22	νz	νz	PROPN
ejpam-5279	142	23	∈	∈	PROPN
ejpam-5279	142	24	ωz	ωz	PROPN
ejpam-5279	142	25	,	,	PUNCT
ejpam-5279	142	26	∀z	∀z	PROPN
ejpam-5279	142	27	∈	∈	PROPN
ejpam-5279	142	28	x	x	X
ejpam-5279	142	29	and	and	CCONJ
ejpam-5279	142	30	that	that	SCONJ
ejpam-5279	142	31	∀y	∀y	NUM
ejpam-5279	142	32	∈	∈	PROPN
ejpam-5279	142	33	x	x	SYM
ejpam-5279	142	34	,	,	PUNCT
ejpam-5279	142	35	ν(y	ν(y	PROPN
ejpam-5279	142	36	)	)	PUNCT
ejpam-5279	142	37	∧n	∧n	VERB
ejpam-5279	142	38	≤	≤	NOUN
ejpam-5279	142	39	νx(z	νx(z	ADV
ejpam-5279	142	40	)	)	PUNCT
ejpam-5279	142	41	∨	∨	NUM
ejpam-5279	142	42	νz(y	νz(y	NOUN
ejpam-5279	142	43	)	)	PUNCT
ejpam-5279	142	44	+	+	CCONJ
ejpam-5279	142	45	ϵ.	ϵ.	NOUN
ejpam-5279	142	46	definition	definition	NOUN
ejpam-5279	142	47	6	6	NUM
ejpam-5279	142	48	.	.	PUNCT
ejpam-5279	143	1	[	[	X
ejpam-5279	143	2	15	15	NUM
ejpam-5279	143	3	]	]	X
ejpam-5279	143	4	if	if	SCONJ
ejpam-5279	143	5	(	(	PUNCT
ejpam-5279	143	6	x	x	X
ejpam-5279	143	7	,	,	PUNCT
ejpam-5279	143	8	ω	ω	NUM
ejpam-5279	143	9	)	)	PUNCT
ejpam-5279	143	10	and	and	CCONJ
ejpam-5279	143	11	(	(	PUNCT
ejpam-5279	143	12	x	x	SYM
ejpam-5279	143	13	′,ω′	′,ω′	NOUN
ejpam-5279	143	14	)	)	PUNCT
ejpam-5279	143	15	are	be	AUX
ejpam-5279	143	16	approach	approach	NOUN
ejpam-5279	143	17	spaces	space	NOUN
ejpam-5279	143	18	(	(	PUNCT
ejpam-5279	143	19	resp.ultra	resp.ultra	ADJ
ejpam-5279	143	20	approach	approach	NOUN
ejpam-5279	143	21	spaces	space	VERB
ejpam-5279	143	22	)	)	PUNCT
ejpam-5279	143	23	,	,	PUNCT
ejpam-5279	143	24	then	then	ADV
ejpam-5279	143	25	a	a	DET
ejpam-5279	143	26	map	map	NOUN
ejpam-5279	144	1	f	f	X
ejpam-5279	144	2	:	:	PUNCT
ejpam-5279	144	3	(	(	PUNCT
ejpam-5279	144	4	x	x	X
ejpam-5279	144	5	,	,	PUNCT
ejpam-5279	144	6	ω	ω	NUM
ejpam-5279	144	7	)	)	PUNCT
ejpam-5279	144	8	→	→	SYM
ejpam-5279	144	9	(	(	PUNCT
ejpam-5279	144	10	x	x	SYM
ejpam-5279	144	11	′,ω′	′,ω′	NOUN
ejpam-5279	144	12	)	)	PUNCT
ejpam-5279	144	13	is	be	AUX
ejpam-5279	144	14	called	call	VERB
ejpam-5279	144	15	contracting	contract	VERB
ejpam-5279	144	16	at	at	ADP
ejpam-5279	144	17	x	x	X
ejpam-5279	144	18	∈	∈	PROPN
ejpam-5279	144	19	x	x	SYM
ejpam-5279	144	20	if	if	SCONJ
ejpam-5279	144	21	and	and	CCONJ
ejpam-5279	144	22	only	only	ADV
ejpam-5279	144	23	if	if	SCONJ
ejpam-5279	144	24	for	for	ADP
ejpam-5279	144	25	all	all	DET
ejpam-5279	144	26	ν	ν	NOUN
ejpam-5279	144	27	′	′	NUM
ejpam-5279	144	28	∈	∈	PROPN
ejpam-5279	144	29	ω′(f(x	ω′(f(x	NOUN
ejpam-5279	144	30	)	)	PUNCT
ejpam-5279	144	31	)	)	PUNCT
ejpam-5279	144	32	,	,	PUNCT
ejpam-5279	144	33	for	for	ADP
ejpam-5279	144	34	all	all	DET
ejpam-5279	144	35	ϵ	ϵ	X
ejpam-5279	144	36	>	>	X
ejpam-5279	144	37	0	0	PUNCT
ejpam-5279	144	38	and	and	CCONJ
ejpam-5279	144	39	for	for	ADP
ejpam-5279	144	40	all	all	PRON
ejpam-5279	144	41	n	n	DET
ejpam-5279	144	42	<	<	X
ejpam-5279	144	43	∞	∞	NOUN
ejpam-5279	144	44	there	there	ADV
ejpam-5279	144	45	exists	exist	VERB
ejpam-5279	144	46	a	a	DET
ejpam-5279	144	47	ν	ν	NOUN
ejpam-5279	144	48	∈	∈	PROPN
ejpam-5279	144	49	ω(x	ω(x	NOUN
ejpam-5279	144	50	)	)	PUNCT
ejpam-5279	145	1	such	such	ADJ
ejpam-5279	145	2	that	that	PRON
ejpam-5279	145	3	(	(	PUNCT
ejpam-5279	145	4	ν	ν	NOUN
ejpam-5279	145	5	′	′	NUM
ejpam-5279	145	6	◦	◦	NOUN
ejpam-5279	145	7	f	f	X
ejpam-5279	145	8	)	)	PUNCT
ejpam-5279	145	9	∧n	∧n	VERB
ejpam-5279	145	10	≤	≤	NOUN
ejpam-5279	145	11	ν	ν	X
ejpam-5279	145	12	+	+	CCONJ
ejpam-5279	145	13	ϵ.	ϵ.	NOUN
ejpam-5279	145	14	the	the	DET
ejpam-5279	145	15	map	map	NOUN
ejpam-5279	146	1	f	f	PROPN
ejpam-5279	146	2	is	be	AUX
ejpam-5279	146	3	called	call	VERB
ejpam-5279	146	4	a	a	DET
ejpam-5279	146	5	contraction	contraction	NOUN
ejpam-5279	146	6	if	if	SCONJ
ejpam-5279	146	7	and	and	CCONJ
ejpam-5279	146	8	only	only	ADV
ejpam-5279	146	9	if	if	SCONJ
ejpam-5279	146	10	it	it	PRON
ejpam-5279	146	11	is	be	AUX
ejpam-5279	146	12	contracting	contract	VERB
ejpam-5279	146	13	in	in	ADP
ejpam-5279	146	14	each	each	DET
ejpam-5279	146	15	x	x	SYM
ejpam-5279	146	16	∈	∈	PROPN
ejpam-5279	146	17	x.	x.	NOUN
ejpam-5279	146	18	definition	definition	NOUN
ejpam-5279	146	19	7	7	NUM
ejpam-5279	146	20	.	.	PUNCT
ejpam-5279	147	1	[	[	X
ejpam-5279	147	2	14	14	NUM
ejpam-5279	147	3	]	]	PUNCT
ejpam-5279	147	4	a	a	DET
ejpam-5279	147	5	family	family	NOUN
ejpam-5279	147	6	of	of	ADP
ejpam-5279	147	7	ideals	ideal	NOUN
ejpam-5279	147	8	ξ	ξ	X
ejpam-5279	147	9	in	in	ADP
ejpam-5279	147	10	[	[	NOUN
ejpam-5279	147	11	0,∞]x×x	0,∞]x×x	PROPN
ejpam-5279	147	12	is	be	AUX
ejpam-5279	147	13	called	call	VERB
ejpam-5279	147	14	an	an	DET
ejpam-5279	147	15	ultra	ultra	ADJ
ejpam-5279	147	16	approach	approach	NOUN
ejpam-5279	147	17	uniformity	uniformity	NOUN
ejpam-5279	147	18	on	on	ADP
ejpam-5279	147	19	x	x	PUNCT
ejpam-5279	147	20	if	if	SCONJ
ejpam-5279	147	21	and	and	CCONJ
ejpam-5279	147	22	only	only	ADV
ejpam-5279	147	23	if	if	SCONJ
ejpam-5279	147	24	the	the	DET
ejpam-5279	147	25	following	follow	VERB
ejpam-5279	147	26	properties	property	NOUN
ejpam-5279	147	27	are	be	AUX
ejpam-5279	147	28	satisfied	satisfied	ADJ
ejpam-5279	147	29	:	:	PUNCT
ejpam-5279	147	30	(	(	PUNCT
ejpam-5279	147	31	uau1	uau1	PROPN
ejpam-5279	147	32	)	)	PUNCT
ejpam-5279	147	33	∀ξ	∀ξ	X
ejpam-5279	147	34	∈	∈	NOUN
ejpam-5279	147	35	ξ,∀x	ξ,∀x	NOUN
ejpam-5279	147	36	∈	∈	NOUN
ejpam-5279	147	37	x	x	NOUN
ejpam-5279	147	38	:	:	PUNCT
ejpam-5279	147	39	ξ(x	ξ(x	NOUN
ejpam-5279	147	40	,	,	PUNCT
ejpam-5279	147	41	x	x	NOUN
ejpam-5279	147	42	)	)	PUNCT
ejpam-5279	147	43	=	=	SYM
ejpam-5279	148	1	0	0	X
ejpam-5279	148	2	.	.	PUNCT
ejpam-5279	149	1	(	(	PUNCT
ejpam-5279	149	2	uau2	uau2	NOUN
ejpam-5279	149	3	)	)	PUNCT
ejpam-5279	149	4	∀ξ	∀ξ	X
ejpam-5279	149	5	∈	∈	NOUN
ejpam-5279	150	1	[	[	X
ejpam-5279	150	2	0,∞]x×x	0,∞]x×x	NUM
ejpam-5279	150	3	:	:	PUNCT
ejpam-5279	150	4	(	(	PUNCT
ejpam-5279	150	5	∀ϵ	∀ϵ	NOUN
ejpam-5279	150	6	>	>	X
ejpam-5279	150	7	0,∀n	0,∀n	PROPN
ejpam-5279	150	8	<	<	X
ejpam-5279	150	9	∞,∃ξnϵ	∞,∃ξnϵ	ADP
ejpam-5279	150	10	∈	∈	PROPN
ejpam-5279	150	11	ξ	ξ	PROPN
ejpam-5279	150	12	s.t	s.t	PROPN
ejpam-5279	150	13	.	.	PROPN
ejpam-5279	151	1	ξ	ξ	PROPN
ejpam-5279	151	2	∧n	∧n	X
ejpam-5279	151	3	≤	≤	NUM
ejpam-5279	151	4	ξnϵ	ξnϵ	NOUN
ejpam-5279	151	5	+	+	CCONJ
ejpam-5279	151	6	ϵ	ϵ	X
ejpam-5279	151	7	)	)	PUNCT
ejpam-5279	151	8	implies	imply	VERB
ejpam-5279	151	9	ξ	ξ	PROPN
ejpam-5279	151	10	∈	∈	PROPN
ejpam-5279	151	11	ξ	ξ	PROPN
ejpam-5279	151	12	.	.	PUNCT
ejpam-5279	152	1	(	(	PUNCT
ejpam-5279	152	2	uau3	uau3	NOUN
ejpam-5279	152	3	)	)	PUNCT
ejpam-5279	152	4	∀ξ	∀ξ	X
ejpam-5279	152	5	∈	∈	NOUN
ejpam-5279	152	6	ξ	ξ	NOUN
ejpam-5279	152	7	:	:	PUNCT
ejpam-5279	152	8	ξs	ξs	PROPN
ejpam-5279	152	9	∈	∈	PROPN
ejpam-5279	152	10	ξ	ξ	PROPN
ejpam-5279	152	11	,	,	PUNCT
ejpam-5279	152	12	where	where	SCONJ
ejpam-5279	152	13	ξs(x	ξs(x	NUM
ejpam-5279	152	14	,	,	PUNCT
ejpam-5279	152	15	y	y	NOUN
ejpam-5279	152	16	)	)	PUNCT
ejpam-5279	152	17	=	=	PUNCT
ejpam-5279	152	18	ξ(y	ξ(y	PROPN
ejpam-5279	152	19	,	,	PUNCT
ejpam-5279	152	20	x	x	NOUN
ejpam-5279	152	21	)	)	PUNCT
ejpam-5279	152	22	,	,	PUNCT
ejpam-5279	152	23	∀(x	∀(x	X
ejpam-5279	152	24	,	,	PUNCT
ejpam-5279	152	25	y	y	NOUN
ejpam-5279	152	26	)	)	PUNCT
ejpam-5279	152	27	∈	∈	PROPN
ejpam-5279	152	28	x	x	SYM
ejpam-5279	152	29	×x	×x	PROPN
ejpam-5279	152	30	.	.	PUNCT
ejpam-5279	153	1	t.m.g	t.m.g	PROPN
ejpam-5279	153	2	.	.	PUNCT
ejpam-5279	154	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	154	2	,	,	PUNCT
ejpam-5279	154	3	fawzi	fawzi	PROPN
ejpam-5279	154	4	al	al	PROPN
ejpam-5279	154	5	-	-	PUNCT
ejpam-5279	154	6	thukair	thukair	NOUN
ejpam-5279	154	7	/	/	SYM
ejpam-5279	154	8	eur	eur	NOUN
ejpam-5279	154	9	.	.	PUNCT
ejpam-5279	155	1	j.	j.	PROPN
ejpam-5279	155	2	pure	pure	PROPN
ejpam-5279	155	3	appl	appl	PROPN
ejpam-5279	155	4	.	.	PROPN
ejpam-5279	155	5	math	math	PROPN
ejpam-5279	155	6	,	,	PUNCT
ejpam-5279	155	7	17	17	NUM
ejpam-5279	155	8	(	(	PUNCT
ejpam-5279	155	9	3	3	NUM
ejpam-5279	155	10	)	)	PUNCT
ejpam-5279	155	11	(	(	PUNCT
ejpam-5279	155	12	2024	2024	NUM
ejpam-5279	155	13	)	)	PUNCT
ejpam-5279	155	14	,	,	PUNCT
ejpam-5279	155	15	1762	1762	NUM
ejpam-5279	155	16	-	-	SYM
ejpam-5279	155	17	1778	1778	NUM
ejpam-5279	155	18	1767	1767	NUM
ejpam-5279	155	19	(	(	PUNCT
ejpam-5279	155	20	uau4	uau4	PROPN
ejpam-5279	155	21	)	)	PUNCT
ejpam-5279	155	22	∀ξ	∀ξ	X
ejpam-5279	155	23	∈	∈	PROPN
ejpam-5279	155	24	ξ	ξ	PROPN
ejpam-5279	155	25	,	,	PUNCT
ejpam-5279	155	26	∀ϵ	∀ϵ	NOUN
ejpam-5279	155	27	>	>	X
ejpam-5279	155	28	0,∀n	0,∀n	PROPN
ejpam-5279	156	1	≤	≤	NUM
ejpam-5279	156	2	∞,∃ξn	∞,∃ξn	NOUN
ejpam-5279	156	3	∈	∈	PROPN
ejpam-5279	156	4	ξ	ξ	PROPN
ejpam-5279	156	5	s.t	s.t	PROPN
ejpam-5279	156	6	.	.	PROPN
ejpam-5279	156	7	∀x	∀x	PROPN
ejpam-5279	156	8	,	,	PUNCT
ejpam-5279	156	9	y	y	PROPN
ejpam-5279	156	10	,	,	PUNCT
ejpam-5279	156	11	z	z	NOUN
ejpam-5279	156	12	∈	∈	PROPN
ejpam-5279	156	13	x	x	X
ejpam-5279	156	14	:	:	PUNCT
ejpam-5279	156	15	ξ(x	ξ(x	NOUN
ejpam-5279	156	16	,	,	PUNCT
ejpam-5279	156	17	z	z	NOUN
ejpam-5279	156	18	)	)	PUNCT
ejpam-5279	156	19	∧	∧	PROPN
ejpam-5279	156	20	n	n	CCONJ
ejpam-5279	156	21	≤	≤	NOUN
ejpam-5279	156	22	ξn	ξn	PROPN
ejpam-5279	156	23	(	(	PUNCT
ejpam-5279	156	24	x	x	PROPN
ejpam-5279	156	25	,	,	PUNCT
ejpam-5279	156	26	y	y	PROPN
ejpam-5279	156	27	)	)	PUNCT
ejpam-5279	156	28	∨	∨	PROPN
ejpam-5279	156	29	ξn	ξn	PROPN
ejpam-5279	156	30	(	(	PUNCT
ejpam-5279	156	31	y	y	PROPN
ejpam-5279	156	32	,	,	PUNCT
ejpam-5279	156	33	z	z	NOUN
ejpam-5279	156	34	)	)	PUNCT
ejpam-5279	157	1	+	+	CCONJ
ejpam-5279	157	2	ϵ.	ϵ.	NOUN
ejpam-5279	157	3	if	if	SCONJ
ejpam-5279	157	4	(	(	PUNCT
ejpam-5279	157	5	x	x	X
ejpam-5279	157	6	,	,	PUNCT
ejpam-5279	157	7	ξ	ξ	X
ejpam-5279	157	8	)	)	PUNCT
ejpam-5279	157	9	is	be	AUX
ejpam-5279	157	10	an	an	DET
ejpam-5279	157	11	ultra	ultra	ADJ
ejpam-5279	157	12	approach	approach	NOUN
ejpam-5279	157	13	uniform	uniform	ADJ
ejpam-5279	157	14	space	space	NOUN
ejpam-5279	157	15	,	,	PUNCT
ejpam-5279	157	16	then	then	ADV
ejpam-5279	157	17	the	the	DET
ejpam-5279	157	18	underlying	underlie	VERB
ejpam-5279	157	19	approach	approach	NOUN
ejpam-5279	157	20	system	system	NOUN
ejpam-5279	157	21	of	of	ADP
ejpam-5279	157	22	ξ	ξ	PROPN
ejpam-5279	157	23	is	be	AUX
ejpam-5279	157	24	given	give	VERB
ejpam-5279	157	25	by	by	ADP
ejpam-5279	157	26	ω(x	ω(x	NOUN
ejpam-5279	157	27	)	)	PUNCT
ejpam-5279	157	28	=	=	SYM
ejpam-5279	157	29	{	{	PUNCT
ejpam-5279	157	30	ξ(x	ξ(x	NOUN
ejpam-5279	157	31	,	,	PUNCT
ejpam-5279	157	32	.)|ξ	.)|ξ	NOUN
ejpam-5279	157	33	∈	∈	PROPN
ejpam-5279	157	34	ξ	ξ	X
ejpam-5279	157	35	}	}	PUNCT
ejpam-5279	157	36	,	,	PUNCT
ejpam-5279	157	37	or	or	CCONJ
ejpam-5279	157	38	by	by	ADP
ejpam-5279	157	39	ωx	ωx	PROPN
ejpam-5279	157	40	.	.	PUNCT
ejpam-5279	157	41	definition	definition	NOUN
ejpam-5279	157	42	8	8	NUM
ejpam-5279	157	43	.	.	PUNCT
ejpam-5279	158	1	let	let	VERB
ejpam-5279	158	2	(	(	PUNCT
ejpam-5279	158	3	x	x	NOUN
ejpam-5279	158	4	,	,	PUNCT
ejpam-5279	158	5	·	·	PUNCT
ejpam-5279	158	6	)	)	PUNCT
ejpam-5279	158	7	be	be	AUX
ejpam-5279	158	8	a	a	DET
ejpam-5279	158	9	group	group	NOUN
ejpam-5279	158	10	,	,	PUNCT
ejpam-5279	158	11	and	and	CCONJ
ejpam-5279	158	12	ω	ω	X
ejpam-5279	158	13	=	=	PUNCT
ejpam-5279	158	14	(	(	PUNCT
ejpam-5279	158	15	ω(x))x∈x	ω(x))x∈x	NOUN
ejpam-5279	158	16	be	be	AUX
ejpam-5279	158	17	a	a	DET
ejpam-5279	158	18	family	family	NOUN
ejpam-5279	158	19	of	of	ADP
ejpam-5279	158	20	ultra	ultra	ADJ
ejpam-5279	158	21	approach	approach	NOUN
ejpam-5279	158	22	system	system	NOUN
ejpam-5279	158	23	on	on	ADP
ejpam-5279	158	24	x.	x.	NOUN
ejpam-5279	158	25	then	then	ADV
ejpam-5279	158	26	the	the	DET
ejpam-5279	158	27	triple	triple	ADJ
ejpam-5279	158	28	(	(	PUNCT
ejpam-5279	158	29	x	x	NOUN
ejpam-5279	158	30	,	,	PUNCT
ejpam-5279	158	31	·	·	PUNCT
ejpam-5279	158	32	,	,	PUNCT
ejpam-5279	158	33	ω	ω	X
ejpam-5279	158	34	=	=	SYM
ejpam-5279	158	35	(	(	PUNCT
ejpam-5279	158	36	ω(x))x∈x	ω(x))x∈x	PROPN
ejpam-5279	158	37	)	)	PUNCT
ejpam-5279	158	38	is	be	AUX
ejpam-5279	158	39	called	call	VERB
ejpam-5279	158	40	an	an	DET
ejpam-5279	158	41	ultra	ultra	ADJ
ejpam-5279	158	42	approach	approach	NOUN
ejpam-5279	158	43	group	group	NOUN
ejpam-5279	158	44	if	if	SCONJ
ejpam-5279	158	45	and	and	CCONJ
ejpam-5279	158	46	only	only	ADV
ejpam-5279	158	47	if	if	SCONJ
ejpam-5279	158	48	the	the	DET
ejpam-5279	158	49	following	follow	VERB
ejpam-5279	158	50	conditions	condition	NOUN
ejpam-5279	158	51	are	be	AUX
ejpam-5279	158	52	fulfilled	fulfil	VERB
ejpam-5279	158	53	:	:	PUNCT
ejpam-5279	158	54	(	(	PUNCT
ejpam-5279	158	55	uag1	uag1	NOUN
ejpam-5279	158	56	)	)	PUNCT
ejpam-5279	158	57	the	the	DET
ejpam-5279	158	58	mapping	mapping	NOUN
ejpam-5279	158	59	m	m	VERB
ejpam-5279	158	60	:	:	PUNCT
ejpam-5279	158	61	x	x	X
ejpam-5279	158	62	×x	×x	X
ejpam-5279	158	63	→	→	SYM
ejpam-5279	158	64	x	x	SYM
ejpam-5279	158	65	,	,	PUNCT
ejpam-5279	158	66	(	(	PUNCT
ejpam-5279	158	67	x	x	NOUN
ejpam-5279	158	68	,	,	PUNCT
ejpam-5279	158	69	y	y	PROPN
ejpam-5279	158	70	)	)	PUNCT
ejpam-5279	159	1	7→	7→	NUM
ejpam-5279	159	2	xy	xy	NOUN
ejpam-5279	159	3	is	be	AUX
ejpam-5279	159	4	a	a	DET
ejpam-5279	159	5	contraction	contraction	NOUN
ejpam-5279	159	6	;	;	PUNCT
ejpam-5279	159	7	(	(	PUNCT
ejpam-5279	159	8	uag2	uag2	PROPN
ejpam-5279	159	9	)	)	PUNCT
ejpam-5279	159	10	the	the	DET
ejpam-5279	159	11	inversion	inversion	NOUN
ejpam-5279	159	12	ȷ	ȷ	NOUN
ejpam-5279	159	13	:	:	PUNCT
ejpam-5279	159	14	x	x	X
ejpam-5279	159	15	→	→	SYM
ejpam-5279	159	16	x	x	SYM
ejpam-5279	159	17	,	,	PUNCT
ejpam-5279	159	18	x	x	PROPN
ejpam-5279	159	19	7→	7→	NUM
ejpam-5279	159	20	x−1	x−1	PROPN
ejpam-5279	159	21	is	be	AUX
ejpam-5279	159	22	a	a	DET
ejpam-5279	159	23	contraction	contraction	NOUN
ejpam-5279	159	24	.	.	PUNCT
ejpam-5279	160	1	lemma	lemma	PROPN
ejpam-5279	160	2	2	2	X
ejpam-5279	160	3	.	.	PUNCT
ejpam-5279	161	1	if	if	SCONJ
ejpam-5279	161	2	(	(	PUNCT
ejpam-5279	161	3	x	x	X
ejpam-5279	161	4	,	,	PUNCT
ejpam-5279	161	5	·	·	PUNCT
ejpam-5279	161	6	,	,	PUNCT
ejpam-5279	161	7	ω	ω	X
ejpam-5279	161	8	=	=	SYM
ejpam-5279	161	9	(	(	PUNCT
ejpam-5279	161	10	ω(x))x∈x	ω(x))x∈x	PROPN
ejpam-5279	161	11	)	)	PUNCT
ejpam-5279	161	12	is	be	AUX
ejpam-5279	161	13	an	an	DET
ejpam-5279	161	14	approach	approach	NOUN
ejpam-5279	161	15	group	group	NOUN
ejpam-5279	161	16	,	,	PUNCT
ejpam-5279	161	17	then	then	ADV
ejpam-5279	161	18	for	for	ADP
ejpam-5279	161	19	any	any	DET
ejpam-5279	161	20	a	a	DET
ejpam-5279	161	21	∈	∈	PROPN
ejpam-5279	161	22	x	x	NOUN
ejpam-5279	161	23	,	,	PUNCT
ejpam-5279	161	24	la	la	ADP
ejpam-5279	161	25	the	the	DET
ejpam-5279	161	26	left	left	ADJ
ejpam-5279	161	27	(	(	PUNCT
ejpam-5279	161	28	resp	resp	NOUN
ejpam-5279	161	29	.	.	PUNCT
ejpam-5279	162	1	ra	ra	PROPN
ejpam-5279	162	2	the	the	DET
ejpam-5279	162	3	right	right	NOUN
ejpam-5279	162	4	)	)	PUNCT
ejpam-5279	162	5	translation	translation	NOUN
ejpam-5279	162	6	is	be	AUX
ejpam-5279	162	7	a	a	DET
ejpam-5279	162	8	homeomorphism	homeomorphism	NOUN
ejpam-5279	162	9	.	.	PUNCT
ejpam-5279	163	1	that	that	PRON
ejpam-5279	163	2	is	be	AUX
ejpam-5279	163	3	,	,	PUNCT
ejpam-5279	163	4	bijective	bijective	ADJ
ejpam-5279	163	5	and	and	CCONJ
ejpam-5279	163	6	bi	bi	NOUN
ejpam-5279	163	7	-	-	NOUN
ejpam-5279	163	8	contraction	contraction	NOUN
ejpam-5279	163	9	.	.	PUNCT
ejpam-5279	164	1	proof	proof	NOUN
ejpam-5279	164	2	.	.	PUNCT
ejpam-5279	165	1	this	this	PRON
ejpam-5279	165	2	goes	go	VERB
ejpam-5279	165	3	almost	almost	ADV
ejpam-5279	165	4	in	in	ADP
ejpam-5279	165	5	the	the	DET
ejpam-5279	165	6	same	same	ADJ
ejpam-5279	165	7	way	way	NOUN
ejpam-5279	165	8	as	as	ADP
ejpam-5279	165	9	in	in	ADP
ejpam-5279	165	10	the	the	DET
ejpam-5279	165	11	proof	proof	NOUN
ejpam-5279	165	12	of	of	ADP
ejpam-5279	165	13	proposition	proposition	NOUN
ejpam-5279	165	14	2.3	2.3	NUM
ejpam-5279	165	15	[	[	X
ejpam-5279	165	16	18	18	NUM
ejpam-5279	165	17	]	]	PUNCT
ejpam-5279	165	18	with	with	ADP
ejpam-5279	165	19	la	la	ADV
ejpam-5279	165	20	:	:	PUNCT
ejpam-5279	165	21	x	x	X
ejpam-5279	165	22	→	→	SYM
ejpam-5279	165	23	x	x	SYM
ejpam-5279	165	24	,	,	PUNCT
ejpam-5279	165	25	x	x	SYM
ejpam-5279	165	26	7→	7→	NUM
ejpam-5279	165	27	ax	ax	NOUN
ejpam-5279	165	28	.	.	PUNCT
ejpam-5279	166	1	proposition	proposition	NOUN
ejpam-5279	166	2	1	1	NUM
ejpam-5279	166	3	.	.	PUNCT
ejpam-5279	167	1	let	let	VERB
ejpam-5279	167	2	(	(	PUNCT
ejpam-5279	167	3	x	x	X
ejpam-5279	167	4	,	,	PUNCT
ejpam-5279	167	5	·	·	PUNCT
ejpam-5279	167	6	,	,	PUNCT
ejpam-5279	167	7	ω	ω	X
ejpam-5279	167	8	=	=	SYM
ejpam-5279	167	9	(	(	PUNCT
ejpam-5279	167	10	ω(x))x∈x	ω(x))x∈x	PROPN
ejpam-5279	167	11	)	)	PUNCT
ejpam-5279	167	12	be	be	AUX
ejpam-5279	167	13	an	an	DET
ejpam-5279	167	14	approach	approach	NOUN
ejpam-5279	167	15	group	group	NOUN
ejpam-5279	167	16	.	.	PUNCT
ejpam-5279	168	1	then	then	ADV
ejpam-5279	168	2	for	for	ADP
ejpam-5279	168	3	any	any	DET
ejpam-5279	168	4	x	x	SYM
ejpam-5279	168	5	∈	∈	PROPN
ejpam-5279	168	6	x	x	X
ejpam-5279	168	7	,	,	PUNCT
ejpam-5279	168	8	ω(x	ω(x	X
ejpam-5279	168	9	)	)	PUNCT
ejpam-5279	168	10	=	=	PRON
ejpam-5279	168	11	{	{	PUNCT
ejpam-5279	168	12	ν	ν	PART
ejpam-5279	168	13	◦	◦	NOUN
ejpam-5279	168	14	lx−1	lx−1	NOUN
ejpam-5279	168	15	|ν	|ν	NOUN
ejpam-5279	168	16	∈	∈	PROPN
ejpam-5279	168	17	ω(e	ω(e	NOUN
ejpam-5279	168	18	)	)	PUNCT
ejpam-5279	168	19	}	}	PUNCT
ejpam-5279	168	20	(	(	PUNCT
ejpam-5279	168	21	respectively	respectively	ADV
ejpam-5279	168	22	,	,	PUNCT
ejpam-5279	168	23	ω(x	ω(x	X
ejpam-5279	168	24	)	)	PUNCT
ejpam-5279	168	25	=	=	PRON
ejpam-5279	168	26	{	{	PUNCT
ejpam-5279	168	27	ν	ν	PART
ejpam-5279	168	28	◦	◦	NOUN
ejpam-5279	168	29	rx−1	rx−1	NOUN
ejpam-5279	168	30	|ν	|ν	NOUN
ejpam-5279	168	31	∈	∈	PROPN
ejpam-5279	168	32	ω(e	ω(e	NOUN
ejpam-5279	168	33	)	)	PUNCT
ejpam-5279	168	34	}	}	PUNCT
ejpam-5279	168	35	)	)	PUNCT
ejpam-5279	168	36	.	.	PUNCT
ejpam-5279	169	1	proof	proof	NOUN
ejpam-5279	169	2	.	.	PUNCT
ejpam-5279	170	1	let	let	VERB
ejpam-5279	170	2	x	x	PUNCT
ejpam-5279	170	3	∈	∈	PROPN
ejpam-5279	170	4	x.	x.	NOUN
ejpam-5279	170	5	if	if	SCONJ
ejpam-5279	170	6	ν	ν	PROPN
ejpam-5279	170	7	∈	∈	PROPN
ejpam-5279	170	8	ω(e	ω(e	PROPN
ejpam-5279	170	9	)	)	PUNCT
ejpam-5279	170	10	,	,	PUNCT
ejpam-5279	170	11	then	then	ADV
ejpam-5279	170	12	ν	ν	PROPN
ejpam-5279	170	13	∈	∈	PROPN
ejpam-5279	170	14	ω	ω	PROPN
ejpam-5279	170	15	(	(	PUNCT
ejpam-5279	170	16	lx−1(x	lx−1(x	NOUN
ejpam-5279	170	17	)	)	PUNCT
ejpam-5279	170	18	)	)	PUNCT
ejpam-5279	170	19	.	.	PUNCT
ejpam-5279	171	1	since	since	SCONJ
ejpam-5279	171	2	lx−1	lx−1	PROPN
ejpam-5279	171	3	is	be	AUX
ejpam-5279	171	4	a	a	DET
ejpam-5279	171	5	contraction	contraction	NOUN
ejpam-5279	171	6	,	,	PUNCT
ejpam-5279	171	7	by	by	ADP
ejpam-5279	171	8	definition	definition	NOUN
ejpam-5279	171	9	,	,	PUNCT
ejpam-5279	171	10	we	we	PRON
ejpam-5279	171	11	have	have	VERB
ejpam-5279	171	12	ν	ν	NOUN
ejpam-5279	171	13	◦	◦	VERB
ejpam-5279	171	14	lx−1	lx−1	NOUN
ejpam-5279	171	15	∈	∈	PROPN
ejpam-5279	171	16	ω(x	ω(x	NOUN
ejpam-5279	171	17	)	)	PUNCT
ejpam-5279	171	18	.	.	PUNCT
ejpam-5279	172	1	conversely	conversely	ADV
ejpam-5279	172	2	,	,	PUNCT
ejpam-5279	172	3	if	if	SCONJ
ejpam-5279	172	4	ν	ν	ADP
ejpam-5279	172	5	◦	◦	NOUN
ejpam-5279	172	6	lx−1	lx−1	NOUN
ejpam-5279	172	7	∈	∈	PROPN
ejpam-5279	172	8	ω(x	ω(x	NOUN
ejpam-5279	172	9	)	)	PUNCT
ejpam-5279	172	10	,	,	PUNCT
ejpam-5279	172	11	then	then	ADV
ejpam-5279	172	12	ν	ν	X
ejpam-5279	172	13	◦	◦	NOUN
ejpam-5279	172	14	lx−1	lx−1	NOUN
ejpam-5279	172	15	∈	∈	PROPN
ejpam-5279	172	16	ω	ω	PROPN
ejpam-5279	172	17	(	(	PUNCT
ejpam-5279	172	18	lx(e	lx(e	PROPN
ejpam-5279	172	19	)	)	PUNCT
ejpam-5279	172	20	)	)	PUNCT
ejpam-5279	172	21	,	,	PUNCT
ejpam-5279	172	22	which	which	PRON
ejpam-5279	172	23	by	by	ADP
ejpam-5279	172	24	definition	definition	NOUN
ejpam-5279	172	25	of	of	ADP
ejpam-5279	172	26	contraction	contraction	NOUN
ejpam-5279	172	27	yields	yield	NOUN
ejpam-5279	172	28	that	that	DET
ejpam-5279	172	29	(	(	PUNCT
ejpam-5279	172	30	ν	ν	X
ejpam-5279	172	31	◦	◦	NOUN
ejpam-5279	172	32	lx−1	lx−1	NOUN
ejpam-5279	172	33	)	)	PUNCT
ejpam-5279	172	34	◦	◦	VERB
ejpam-5279	172	35	lx	lx	ADP
ejpam-5279	172	36	∈	∈	PROPN
ejpam-5279	172	37	ω(e	ω(e	PROPN
ejpam-5279	172	38	)	)	PUNCT
ejpam-5279	172	39	.	.	PUNCT
ejpam-5279	173	1	but	but	CCONJ
ejpam-5279	173	2	(	(	PUNCT
ejpam-5279	173	3	ν	ν	X
ejpam-5279	173	4	◦	◦	NOUN
ejpam-5279	173	5	lx−1	lx−1	NOUN
ejpam-5279	173	6	)	)	PUNCT
ejpam-5279	173	7	◦	◦	NOUN
ejpam-5279	173	8	lx	lx	NOUN
ejpam-5279	173	9	=	=	SYM
ejpam-5279	173	10	ν	ν	NOUN
ejpam-5279	173	11	,	,	PUNCT
ejpam-5279	173	12	and	and	CCONJ
ejpam-5279	173	13	hence	hence	ADV
ejpam-5279	173	14	ν	ν	X
ejpam-5279	173	15	∈	∈	PROPN
ejpam-5279	173	16	ω(e	ω(e	PROPN
ejpam-5279	173	17	)	)	PUNCT
ejpam-5279	173	18	.	.	PUNCT
ejpam-5279	174	1	this	this	PRON
ejpam-5279	174	2	shows	show	VERB
ejpam-5279	174	3	that	that	SCONJ
ejpam-5279	174	4	ν	ν	PROPN
ejpam-5279	174	5	∈	∈	PROPN
ejpam-5279	174	6	ω(e	ω(e	PROPN
ejpam-5279	174	7	)	)	PUNCT
ejpam-5279	174	8	⇔	⇔	PROPN
ejpam-5279	174	9	ν	ν	ADP
ejpam-5279	174	10	◦	◦	NOUN
ejpam-5279	174	11	lx−1	lx−1	NOUN
ejpam-5279	174	12	∈	∈	PROPN
ejpam-5279	174	13	ω(x	ω(x	NOUN
ejpam-5279	174	14	)	)	PUNCT
ejpam-5279	174	15	.	.	PUNCT
ejpam-5279	175	1	the	the	DET
ejpam-5279	175	2	other	other	ADJ
ejpam-5279	175	3	part	part	NOUN
ejpam-5279	175	4	follows	follow	VERB
ejpam-5279	175	5	exactly	exactly	ADV
ejpam-5279	175	6	the	the	DET
ejpam-5279	175	7	same	same	ADJ
ejpam-5279	175	8	way	way	NOUN
ejpam-5279	175	9	.	.	PUNCT
ejpam-5279	176	1	theorem	theorem	NOUN
ejpam-5279	176	2	3	3	X
ejpam-5279	176	3	.	.	PUNCT
ejpam-5279	177	1	let	let	VERB
ejpam-5279	177	2	(	(	PUNCT
ejpam-5279	177	3	x	x	NOUN
ejpam-5279	177	4	,	,	PUNCT
ejpam-5279	177	5	·	·	PUNCT
ejpam-5279	177	6	)	)	PUNCT
ejpam-5279	177	7	be	be	AUX
ejpam-5279	177	8	a	a	DET
ejpam-5279	177	9	group	group	NOUN
ejpam-5279	177	10	and	and	CCONJ
ejpam-5279	177	11	ω	ω	NUM
ejpam-5279	177	12	=	=	PUNCT
ejpam-5279	177	13	(	(	PUNCT
ejpam-5279	177	14	ω(x))x∈x	ω(x))x∈x	NOUN
ejpam-5279	177	15	be	be	AUX
ejpam-5279	177	16	an	an	DET
ejpam-5279	177	17	approach	approach	NOUN
ejpam-5279	177	18	system	system	NOUN
ejpam-5279	177	19	on	on	ADP
ejpam-5279	177	20	x.	x.	NOUN
ejpam-5279	177	21	then	then	ADV
ejpam-5279	177	22	the	the	DET
ejpam-5279	177	23	triple	triple	ADJ
ejpam-5279	177	24	(	(	PUNCT
ejpam-5279	177	25	x	x	NOUN
ejpam-5279	177	26	,	,	PUNCT
ejpam-5279	177	27	·	·	PUNCT
ejpam-5279	177	28	,	,	PUNCT
ejpam-5279	177	29	ω	ω	X
ejpam-5279	177	30	=	=	SYM
ejpam-5279	177	31	(	(	PUNCT
ejpam-5279	177	32	ω(x))x∈x	ω(x))x∈x	PROPN
ejpam-5279	177	33	)	)	PUNCT
ejpam-5279	177	34	is	be	AUX
ejpam-5279	177	35	an	an	DET
ejpam-5279	177	36	approach	approach	NOUN
ejpam-5279	177	37	group	group	NOUN
ejpam-5279	178	1	if	if	SCONJ
ejpam-5279	178	2	and	and	CCONJ
ejpam-5279	178	3	only	only	ADV
ejpam-5279	178	4	if	if	SCONJ
ejpam-5279	178	5	the	the	DET
ejpam-5279	178	6	following	follow	VERB
ejpam-5279	178	7	are	be	AUX
ejpam-5279	178	8	fulfilled	fulfil	VERB
ejpam-5279	178	9	:	:	PUNCT
ejpam-5279	178	10	(	(	PUNCT
ejpam-5279	178	11	a	a	X
ejpam-5279	178	12	)	)	PUNCT
ejpam-5279	178	13	∀x	∀x	VERB
ejpam-5279	178	14	∈	∈	PROPN
ejpam-5279	178	15	x	x	NOUN
ejpam-5279	178	16	:	:	PUNCT
ejpam-5279	178	17	ω(x	ω(x	X
ejpam-5279	178	18	)	)	PUNCT
ejpam-5279	178	19	=	=	PRON
ejpam-5279	178	20	{	{	PUNCT
ejpam-5279	178	21	x−1	x−1	PROPN
ejpam-5279	178	22	⊙	⊙	PROPN
ejpam-5279	178	23	ν|ν	ν|ν	PUNCT
ejpam-5279	179	1	∈	∈	PROPN
ejpam-5279	179	2	ω(e	ω(e	PROPN
ejpam-5279	179	3	)	)	PUNCT
ejpam-5279	179	4	}	}	PUNCT
ejpam-5279	179	5	,	,	PUNCT
ejpam-5279	179	6	where	where	SCONJ
ejpam-5279	179	7	x−1	x−1	PROPN
ejpam-5279	179	8	⊙	⊙	PROPN
ejpam-5279	179	9	ν	ν	X
ejpam-5279	179	10	=	=	PUNCT
ejpam-5279	179	11	ν	ν	PART
ejpam-5279	179	12	◦	◦	NOUN
ejpam-5279	179	13	lx−1	lx−1	NOUN
ejpam-5279	179	14	(	(	PUNCT
ejpam-5279	179	15	alternatively	alternatively	ADV
ejpam-5279	179	16	,	,	PUNCT
ejpam-5279	179	17	ω(x	ω(x	NOUN
ejpam-5279	179	18	)	)	PUNCT
ejpam-5279	179	19	=	=	SYM
ejpam-5279	179	20	{	{	PUNCT
ejpam-5279	179	21	ν	ν	X
ejpam-5279	179	22	⊙	⊙	X
ejpam-5279	179	23	x−1|	x−1|	PROPN
ejpam-5279	179	24	ν	ν	PROPN
ejpam-5279	179	25	∈	∈	PROPN
ejpam-5279	179	26	ω(e	ω(e	PROPN
ejpam-5279	179	27	)	)	PUNCT
ejpam-5279	179	28	}	}	PUNCT
ejpam-5279	179	29	,	,	PUNCT
ejpam-5279	179	30	where	where	SCONJ
ejpam-5279	179	31	ν	ν	PROPN
ejpam-5279	179	32	⊙	⊙	X
ejpam-5279	179	33	x−1	x−1	PUNCT
ejpam-5279	180	1	=	=	PUNCT
ejpam-5279	180	2	ν	ν	PRON
ejpam-5279	180	3	◦	◦	NOUN
ejpam-5279	180	4	rx−1	rx−1	NOUN
ejpam-5279	180	5	)	)	PUNCT
ejpam-5279	180	6	;	;	PUNCT
ejpam-5279	180	7	(	(	PUNCT
ejpam-5279	180	8	b	b	X
ejpam-5279	180	9	)	)	PUNCT
ejpam-5279	180	10	∀ν	∀ν	PROPN
ejpam-5279	180	11	∈	∈	PROPN
ejpam-5279	180	12	ω(e	ω(e	PROPN
ejpam-5279	180	13	)	)	PUNCT
ejpam-5279	180	14	,	,	PUNCT
ejpam-5279	180	15	∀ϵ	∀ϵ	NOUN
ejpam-5279	180	16	>	>	X
ejpam-5279	180	17	0	0	NUM
ejpam-5279	180	18	,	,	PUNCT
ejpam-5279	180	19	∀n	∀n	PUNCT
ejpam-5279	180	20	<	<	X
ejpam-5279	180	21	∞	∞	PROPN
ejpam-5279	180	22	,	,	PUNCT
ejpam-5279	180	23	there	there	PRON
ejpam-5279	180	24	exists	exist	VERB
ejpam-5279	180	25	µ	µ	PRON
ejpam-5279	180	26	∈	∈	PROPN
ejpam-5279	180	27	ω(e	ω(e	PROPN
ejpam-5279	180	28	)	)	PUNCT
ejpam-5279	180	29	,	,	PUNCT
ejpam-5279	180	30	ν−1	ν−1	PROPN
ejpam-5279	180	31	∧	∧	PROPN
ejpam-5279	180	32	n	n	CCONJ
ejpam-5279	180	33	≤	≤	X
ejpam-5279	180	34	µ	µ	X
ejpam-5279	180	35	+	+	X
ejpam-5279	180	36	ϵ	ϵ	ADP
ejpam-5279	180	37	,	,	PUNCT
ejpam-5279	180	38	i.e.	i.e.	X
ejpam-5279	180	39	ȷ	ȷ	NOUN
ejpam-5279	180	40	:	:	PUNCT
ejpam-5279	180	41	x	x	X
ejpam-5279	180	42	→	→	SYM
ejpam-5279	180	43	x	x	SYM
ejpam-5279	180	44	,	,	PUNCT
ejpam-5279	180	45	x	x	PROPN
ejpam-5279	180	46	7→	7→	NUM
ejpam-5279	180	47	x−1	x−1	PROPN
ejpam-5279	180	48	is	be	AUX
ejpam-5279	180	49	contracting	contract	VERB
ejpam-5279	180	50	at	at	ADP
ejpam-5279	180	51	e	e	NOUN
ejpam-5279	180	52	;	;	PUNCT
ejpam-5279	180	53	(	(	PUNCT
ejpam-5279	180	54	c	c	X
ejpam-5279	180	55	)	)	PUNCT
ejpam-5279	180	56	∀ν	∀ν	PROPN
ejpam-5279	180	57	∈	∈	X
ejpam-5279	180	58	ω(e),∀ϵ	ω(e),∀ϵ	PUNCT
ejpam-5279	180	59	>	>	X
ejpam-5279	180	60	0,∀n	0,∀n	PROPN
ejpam-5279	180	61	<	<	X
ejpam-5279	180	62	∞	∞	PROPN
ejpam-5279	180	63	,	,	PUNCT
ejpam-5279	180	64	there	there	PRON
ejpam-5279	180	65	exists	exist	VERB
ejpam-5279	180	66	µ	µ	PRON
ejpam-5279	180	67	∈	∈	PROPN
ejpam-5279	180	68	ω(e	ω(e	NOUN
ejpam-5279	180	69	)	)	PUNCT
ejpam-5279	180	70	such	such	ADJ
ejpam-5279	180	71	that	that	SCONJ
ejpam-5279	180	72	∀x	∀x	NUM
ejpam-5279	180	73	,	,	PUNCT
ejpam-5279	180	74	y	y	PROPN
ejpam-5279	180	75	∈	∈	PROPN
ejpam-5279	181	1	x	x	X
ejpam-5279	181	2	:	:	PUNCT
ejpam-5279	181	3	ν(xy)∧n	ν(xy)∧n	X
ejpam-5279	181	4	≤	≤	ADJ
ejpam-5279	181	5	µ(x)∨µ(y)+ϵ	µ(x)∨µ(y)+ϵ	ADJ
ejpam-5279	181	6	,	,	PUNCT
ejpam-5279	181	7	i.e.	i.e.	X
ejpam-5279	181	8	m	m	VERB
ejpam-5279	181	9	:	:	PUNCT
ejpam-5279	181	10	(	(	PUNCT
ejpam-5279	181	11	x	x	X
ejpam-5279	181	12	,	,	PUNCT
ejpam-5279	181	13	y	y	PROPN
ejpam-5279	181	14	)	)	PUNCT
ejpam-5279	181	15	7→	7→	NUM
ejpam-5279	181	16	xy	xy	NOUN
ejpam-5279	181	17	is	be	AUX
ejpam-5279	181	18	contracting	contract	VERB
ejpam-5279	181	19	at	at	ADP
ejpam-5279	181	20	(	(	PUNCT
ejpam-5279	181	21	e	e	NOUN
ejpam-5279	181	22	,	,	PUNCT
ejpam-5279	181	23	e	e	NOUN
ejpam-5279	181	24	)	)	PUNCT
ejpam-5279	181	25	∈	∈	PROPN
ejpam-5279	181	26	x×x	x×x	PROPN
ejpam-5279	181	27	;	;	PUNCT
ejpam-5279	181	28	(	(	PUNCT
ejpam-5279	181	29	d	d	X
ejpam-5279	181	30	)	)	PUNCT
ejpam-5279	181	31	∀ν	∀ν	PROPN
ejpam-5279	181	32	∈	∈	PROPN
ejpam-5279	181	33	ω(e	ω(e	PROPN
ejpam-5279	181	34	)	)	PUNCT
ejpam-5279	181	35	,	,	PUNCT
ejpam-5279	181	36	∀ϵ	∀ϵ	NOUN
ejpam-5279	181	37	>	>	X
ejpam-5279	181	38	0	0	NUM
ejpam-5279	181	39	,	,	PUNCT
ejpam-5279	181	40	∀n	∀n	PUNCT
ejpam-5279	181	41	<	<	X
ejpam-5279	181	42	∞	∞	PROPN
ejpam-5279	181	43	,	,	PUNCT
ejpam-5279	181	44	∀x	∀x	VERB
ejpam-5279	181	45	∈	∈	PROPN
ejpam-5279	181	46	x	x	PRON
ejpam-5279	181	47	,	,	PUNCT
ejpam-5279	181	48	there	there	PRON
ejpam-5279	181	49	exists	exist	VERB
ejpam-5279	181	50	µ	µ	PRON
ejpam-5279	181	51	∈	∈	PROPN
ejpam-5279	181	52	ω(e	ω(e	NOUN
ejpam-5279	181	53	)	)	PUNCT
ejpam-5279	181	54	such	such	ADJ
ejpam-5279	181	55	that	that	SCONJ
ejpam-5279	181	56	(	(	PUNCT
ejpam-5279	181	57	x⊙	x⊙	PROPN
ejpam-5279	181	58	ν	ν	PROPN
ejpam-5279	181	59	⊙	⊙	PROPN
ejpam-5279	181	60	x−1	x−1	PROPN
ejpam-5279	181	61	)	)	PUNCT
ejpam-5279	181	62	∧	∧	PROPN
ejpam-5279	181	63	n	n	CCONJ
ejpam-5279	181	64	≤	≤	NOUN
ejpam-5279	181	65	µ+	µ+	PRON
ejpam-5279	181	66	ϵ	ϵ	NOUN
ejpam-5279	181	67	,	,	PUNCT
ejpam-5279	181	68	i.e.	i.e.	X
ejpam-5279	181	69	intx	intx	X
ejpam-5279	181	70	:	:	PUNCT
ejpam-5279	181	71	z	z	X
ejpam-5279	181	72	7→	7→	NUM
ejpam-5279	181	73	xzx−1	xzx−1	PROPN
ejpam-5279	181	74	is	be	AUX
ejpam-5279	181	75	contracting	contract	VERB
ejpam-5279	181	76	at	at	ADP
ejpam-5279	181	77	e.	e.	PROPN
ejpam-5279	181	78	proof	proof	PROPN
ejpam-5279	181	79	.	.	PUNCT
ejpam-5279	182	1	if	if	SCONJ
ejpam-5279	182	2	(	(	PUNCT
ejpam-5279	182	3	x	x	X
ejpam-5279	182	4	,	,	PUNCT
ejpam-5279	182	5	·	·	PUNCT
ejpam-5279	182	6	,	,	PUNCT
ejpam-5279	182	7	ω	ω	X
ejpam-5279	182	8	=	=	SYM
ejpam-5279	182	9	(	(	PUNCT
ejpam-5279	182	10	ω(x))x∈x	ω(x))x∈x	PROPN
ejpam-5279	182	11	)	)	PUNCT
ejpam-5279	182	12	is	be	AUX
ejpam-5279	182	13	an	an	DET
ejpam-5279	182	14	approach	approach	NOUN
ejpam-5279	182	15	group	group	NOUN
ejpam-5279	182	16	,	,	PUNCT
ejpam-5279	182	17	then	then	ADV
ejpam-5279	182	18	(	(	PUNCT
ejpam-5279	182	19	a	a	X
ejpam-5279	182	20	)	)	PUNCT
ejpam-5279	182	21	follows	follow	VERB
ejpam-5279	182	22	from	from	ADP
ejpam-5279	182	23	the	the	DET
ejpam-5279	182	24	proposition	proposition	NOUN
ejpam-5279	182	25	1	1	NUM
ejpam-5279	182	26	,	,	PUNCT
ejpam-5279	182	27	and	and	CCONJ
ejpam-5279	182	28	(	(	PUNCT
ejpam-5279	182	29	b	b	X
ejpam-5279	182	30	)	)	PUNCT
ejpam-5279	182	31	follows	follow	VERB
ejpam-5279	182	32	from	from	ADP
ejpam-5279	182	33	the	the	DET
ejpam-5279	182	34	definition	definition	NOUN
ejpam-5279	182	35	3.1	3.1	NUM
ejpam-5279	183	1	[	[	X
ejpam-5279	183	2	15	15	NUM
ejpam-5279	183	3	]	]	PUNCT
ejpam-5279	183	4	,	,	PUNCT
ejpam-5279	183	5	while	while	SCONJ
ejpam-5279	183	6	(	(	PUNCT
ejpam-5279	183	7	c	c	X
ejpam-5279	183	8	)	)	PUNCT
ejpam-5279	183	9	follows	follow	VERB
ejpam-5279	183	10	the	the	DET
ejpam-5279	183	11	definition	definition	NOUN
ejpam-5279	183	12	.	.	PUNCT
ejpam-5279	184	1	as	as	ADP
ejpam-5279	184	2	for	for	ADP
ejpam-5279	184	3	(	(	PUNCT
ejpam-5279	184	4	d	d	NOUN
ejpam-5279	184	5	)	)	PUNCT
ejpam-5279	184	6	,	,	PUNCT
ejpam-5279	184	7	we	we	PRON
ejpam-5279	184	8	employ	employ	VERB
ejpam-5279	184	9	the	the	DET
ejpam-5279	184	10	theorem	theorem	NOUN
ejpam-5279	184	11	3.4(b	3.4(b	NUM
ejpam-5279	184	12	)	)	PUNCT
ejpam-5279	185	1	[	[	X
ejpam-5279	185	2	15	15	NUM
ejpam-5279	185	3	]	]	PUNCT
ejpam-5279	185	4	since	since	SCONJ
ejpam-5279	185	5	intx	intx	PROPN
ejpam-5279	185	6	=	=	PUNCT
ejpam-5279	185	7	lx	lx	ADP
ejpam-5279	185	8	◦	◦	NOUN
ejpam-5279	185	9	rx−1	rx−1	NOUN
ejpam-5279	185	10	,	,	PUNCT
ejpam-5279	185	11	and	and	CCONJ
ejpam-5279	185	12	both	both	CCONJ
ejpam-5279	185	13	the	the	DET
ejpam-5279	185	14	translations	translation	NOUN
ejpam-5279	185	15	are	be	AUX
ejpam-5279	185	16	contraction	contraction	NOUN
ejpam-5279	185	17	maps	map	NOUN
ejpam-5279	185	18	.	.	PUNCT
ejpam-5279	186	1	to	to	PART
ejpam-5279	186	2	show	show	VERB
ejpam-5279	186	3	the	the	DET
ejpam-5279	186	4	converse	converse	NOUN
ejpam-5279	186	5	,	,	PUNCT
ejpam-5279	186	6	assume	assume	VERB
ejpam-5279	186	7	that	that	SCONJ
ejpam-5279	186	8	(	(	PUNCT
ejpam-5279	186	9	a)-(d	a)-(d	NOUN
ejpam-5279	186	10	)	)	PUNCT
ejpam-5279	186	11	are	be	AUX
ejpam-5279	186	12	true	true	ADJ
ejpam-5279	186	13	.	.	PUNCT
ejpam-5279	187	1	first	first	ADV
ejpam-5279	187	2	,	,	PUNCT
ejpam-5279	187	3	we	we	PRON
ejpam-5279	187	4	show	show	VERB
ejpam-5279	187	5	that	that	SCONJ
ejpam-5279	187	6	the	the	DET
ejpam-5279	187	7	inversion	inversion	NOUN
ejpam-5279	187	8	map	map	NOUN
ejpam-5279	187	9	ȷ	ȷ	NOUN
ejpam-5279	187	10	:	:	PUNCT
ejpam-5279	187	11	x	x	X
ejpam-5279	187	12	→	→	SYM
ejpam-5279	187	13	x	x	SYM
ejpam-5279	187	14	,	,	PUNCT
ejpam-5279	187	15	x	x	PROPN
ejpam-5279	187	16	7→	7→	NUM
ejpam-5279	187	17	x−1	x−1	PROPN
ejpam-5279	187	18	is	be	AUX
ejpam-5279	187	19	a	a	DET
ejpam-5279	187	20	contraction	contraction	NOUN
ejpam-5279	187	21	.	.	PUNCT
ejpam-5279	188	1	let	let	VERB
ejpam-5279	188	2	x	x	SYM
ejpam-5279	188	3	∈	∈	PROPN
ejpam-5279	188	4	x	x	NOUN
ejpam-5279	188	5	,	,	PUNCT
ejpam-5279	188	6	ν	ν	PROPN
ejpam-5279	188	7	∈	∈	PROPN
ejpam-5279	188	8	ω	ω	X
ejpam-5279	188	9	(	(	PUNCT
ejpam-5279	188	10	ȷ(x	ȷ(x	NOUN
ejpam-5279	188	11	)	)	PUNCT
ejpam-5279	188	12	)	)	PUNCT
ejpam-5279	188	13	,	,	PUNCT
ejpam-5279	188	14	t.m.g	t.m.g	X
ejpam-5279	188	15	.	.	PUNCT
ejpam-5279	189	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	189	2	,	,	PUNCT
ejpam-5279	189	3	fawzi	fawzi	PROPN
ejpam-5279	189	4	al	al	PROPN
ejpam-5279	189	5	-	-	PUNCT
ejpam-5279	189	6	thukair	thukair	NOUN
ejpam-5279	189	7	/	/	SYM
ejpam-5279	189	8	eur	eur	NOUN
ejpam-5279	189	9	.	.	PUNCT
ejpam-5279	190	1	j.	j.	PROPN
ejpam-5279	190	2	pure	pure	PROPN
ejpam-5279	190	3	appl	appl	PROPN
ejpam-5279	190	4	.	.	PROPN
ejpam-5279	190	5	math	math	PROPN
ejpam-5279	190	6	,	,	PUNCT
ejpam-5279	190	7	17	17	NUM
ejpam-5279	190	8	(	(	PUNCT
ejpam-5279	190	9	3	3	NUM
ejpam-5279	190	10	)	)	PUNCT
ejpam-5279	190	11	(	(	PUNCT
ejpam-5279	190	12	2024	2024	NUM
ejpam-5279	190	13	)	)	PUNCT
ejpam-5279	190	14	,	,	PUNCT
ejpam-5279	190	15	1762	1762	NUM
ejpam-5279	190	16	-	-	SYM
ejpam-5279	190	17	1778	1778	NUM
ejpam-5279	190	18	1768	1768	NUM
ejpam-5279	190	19	ϵ	ϵ	X
ejpam-5279	190	20	>	>	X
ejpam-5279	190	21	0	0	NUM
ejpam-5279	190	22	,	,	PUNCT
ejpam-5279	190	23	and	and	CCONJ
ejpam-5279	190	24	n	n	CCONJ
ejpam-5279	190	25	<	<	X
ejpam-5279	190	26	∞	∞	PROPN
ejpam-5279	190	27	,	,	PUNCT
ejpam-5279	190	28	we	we	PRON
ejpam-5279	190	29	want	want	VERB
ejpam-5279	190	30	to	to	PART
ejpam-5279	190	31	find	find	VERB
ejpam-5279	190	32	a	a	DET
ejpam-5279	190	33	θ	θ	PROPN
ejpam-5279	190	34	∈	∈	PROPN
ejpam-5279	190	35	ω(x	ω(x	NOUN
ejpam-5279	190	36	)	)	PUNCT
ejpam-5279	190	37	(	(	PUNCT
ejpam-5279	190	38	say	say	INTJ
ejpam-5279	190	39	)	)	PUNCT
ejpam-5279	190	40	such	such	ADJ
ejpam-5279	190	41	that	that	SCONJ
ejpam-5279	190	42	(	(	PUNCT
ejpam-5279	190	43	ν	ν	X
ejpam-5279	190	44	◦	◦	NOUN
ejpam-5279	190	45	ȷ	ȷ	NOUN
ejpam-5279	190	46	)	)	PUNCT
ejpam-5279	190	47	∧n	∧n	VERB
ejpam-5279	190	48	≤	≤	NUM
ejpam-5279	190	49	θ	θ	PROPN
ejpam-5279	191	1	+	+	CCONJ
ejpam-5279	191	2	ϵ.	ϵ.	NOUN
ejpam-5279	191	3	since	since	SCONJ
ejpam-5279	191	4	ν	ν	PROPN
ejpam-5279	191	5	∈	∈	PROPN
ejpam-5279	191	6	ω	ω	PROPN
ejpam-5279	191	7	(	(	PUNCT
ejpam-5279	191	8	ȷ(x	ȷ(x	NOUN
ejpam-5279	191	9	)	)	PUNCT
ejpam-5279	191	10	)	)	PUNCT
ejpam-5279	191	11	,	,	PUNCT
ejpam-5279	191	12	there	there	PRON
ejpam-5279	191	13	exists	exist	VERB
ejpam-5279	191	14	a	a	DET
ejpam-5279	191	15	µ	µ	PROPN
ejpam-5279	191	16	∈	∈	PROPN
ejpam-5279	191	17	ω(e	ω(e	NOUN
ejpam-5279	191	18	)	)	PUNCT
ejpam-5279	191	19	such	such	ADJ
ejpam-5279	191	20	that	that	SCONJ
ejpam-5279	191	21	ν	ν	X
ejpam-5279	191	22	=	=	X
ejpam-5279	191	23	x⊙µ.	x⊙µ.	PROPN
ejpam-5279	191	24	consequently	consequently	ADV
ejpam-5279	191	25	,	,	PUNCT
ejpam-5279	191	26	due	due	ADP
ejpam-5279	191	27	to	to	ADP
ejpam-5279	191	28	axiom	axiom	NOUN
ejpam-5279	191	29	(	(	PUNCT
ejpam-5279	191	30	b	b	NOUN
ejpam-5279	191	31	)	)	PUNCT
ejpam-5279	191	32	,	,	PUNCT
ejpam-5279	191	33	there	there	PRON
ejpam-5279	191	34	exists	exist	VERB
ejpam-5279	191	35	a	a	DET
ejpam-5279	191	36	µ1	µ1	PROPN
ejpam-5279	191	37	∈	∈	PROPN
ejpam-5279	191	38	ω(e	ω(e	NOUN
ejpam-5279	191	39	)	)	PUNCT
ejpam-5279	192	1	such	such	ADJ
ejpam-5279	192	2	that	that	SCONJ
ejpam-5279	192	3	µ−1∧n	µ−1∧n	PROPN
ejpam-5279	192	4	≤	≤	PUNCT
ejpam-5279	192	5	µ1	µ1	PROPN
ejpam-5279	192	6	+	+	X
ejpam-5279	192	7	ϵ.	ϵ.	NOUN
ejpam-5279	193	1	then	then	ADV
ejpam-5279	193	2	for	for	ADP
ejpam-5279	193	3	any	any	DET
ejpam-5279	193	4	z	z	NOUN
ejpam-5279	193	5	∈	∈	PROPN
ejpam-5279	193	6	x	x	X
ejpam-5279	193	7	,	,	PUNCT
ejpam-5279	193	8	(	(	PUNCT
ejpam-5279	193	9	ν	ν	X
ejpam-5279	193	10	◦	◦	NOUN
ejpam-5279	193	11	ȷ	ȷ	NOUN
ejpam-5279	193	12	)	)	PUNCT
ejpam-5279	193	13	(	(	PUNCT
ejpam-5279	193	14	z)∧n	z)∧n	NOUN
ejpam-5279	193	15	=	=	PUNCT
ejpam-5279	194	1	[	[	X
ejpam-5279	194	2	(	(	PUNCT
ejpam-5279	194	3	x⊙	x⊙	PROPN
ejpam-5279	194	4	µ	µ	NOUN
ejpam-5279	194	5	)	)	PUNCT
ejpam-5279	194	6	◦	◦	NOUN
ejpam-5279	194	7	ȷ](z)∧n	ȷ](z)∧n	X
ejpam-5279	195	1	=	=	X
ejpam-5279	195	2	µ(xz−1)∧n	µ(xz−1)∧n	X
ejpam-5279	195	3	=	=	X
ejpam-5279	195	4	µ−1	µ−1	PROPN
ejpam-5279	195	5	(	(	PUNCT
ejpam-5279	195	6	zx−1	zx−1	NOUN
ejpam-5279	195	7	)	)	PUNCT
ejpam-5279	195	8	∧n	∧n	VERB
ejpam-5279	195	9	≤	≤	NOUN
ejpam-5279	195	10	µ1(zx	µ1(zx	NUM
ejpam-5279	195	11	−1	−1	NOUN
ejpam-5279	195	12	)	)	PUNCT
ejpam-5279	196	1	+	+	CCONJ
ejpam-5279	196	2	ϵ	ϵ	X
ejpam-5279	196	3	=	=	SYM
ejpam-5279	196	4	(	(	PUNCT
ejpam-5279	196	5	µ1	µ1	PROPN
ejpam-5279	196	6	⊙	⊙	PROPN
ejpam-5279	196	7	x−1	x−1	PROPN
ejpam-5279	196	8	)	)	PUNCT
ejpam-5279	197	1	(	(	PUNCT
ejpam-5279	197	2	z	z	NOUN
ejpam-5279	197	3	)	)	PUNCT
ejpam-5279	197	4	+	+	CCONJ
ejpam-5279	197	5	ϵ	ϵ	X
ejpam-5279	197	6	⇒	⇒	NOUN
ejpam-5279	197	7	(	(	PUNCT
ejpam-5279	197	8	ν	ν	X
ejpam-5279	197	9	◦	◦	NOUN
ejpam-5279	197	10	ȷ	ȷ	NOUN
ejpam-5279	197	11	)	)	PUNCT
ejpam-5279	197	12	∧n	∧n	VERB
ejpam-5279	197	13	≤	≤	NUM
ejpam-5279	197	14	θ	θ	PROPN
ejpam-5279	198	1	+	+	CCONJ
ejpam-5279	198	2	ϵ	ϵ	X
ejpam-5279	198	3	,	,	PUNCT
ejpam-5279	198	4	with	with	ADP
ejpam-5279	198	5	θ	θ	PROPN
ejpam-5279	198	6	:	:	PUNCT
ejpam-5279	198	7	=	=	PUNCT
ejpam-5279	198	8	µ1	µ1	PROPN
ejpam-5279	198	9	⊙	⊙	PROPN
ejpam-5279	198	10	x−1	x−1	PROPN
ejpam-5279	198	11	,	,	PUNCT
ejpam-5279	198	12	θ	θ	PROPN
ejpam-5279	198	13	∈	∈	PROPN
ejpam-5279	198	14	ω(x	ω(x	NOUN
ejpam-5279	198	15	)	)	PUNCT
ejpam-5279	198	16	.	.	PUNCT
ejpam-5279	199	1	thus	thus	ADV
ejpam-5279	199	2	we	we	PRON
ejpam-5279	199	3	have	have	AUX
ejpam-5279	199	4	proved	prove	VERB
ejpam-5279	199	5	that	that	SCONJ
ejpam-5279	199	6	for	for	ADP
ejpam-5279	199	7	any	any	DET
ejpam-5279	199	8	ν	ν	NOUN
ejpam-5279	199	9	∈	∈	PROPN
ejpam-5279	199	10	ω	ω	X
ejpam-5279	199	11	(	(	PUNCT
ejpam-5279	199	12	ȷ(x	ȷ(x	NOUN
ejpam-5279	199	13	)	)	PUNCT
ejpam-5279	199	14	)	)	PUNCT
ejpam-5279	199	15	,	,	PUNCT
ejpam-5279	199	16	∀ϵ	∀ϵ	NOUN
ejpam-5279	199	17	>	>	X
ejpam-5279	199	18	0	0	NUM
ejpam-5279	199	19	,	,	PUNCT
ejpam-5279	199	20	∀n	∀n	PUNCT
ejpam-5279	199	21	<	<	X
ejpam-5279	199	22	∞	∞	PROPN
ejpam-5279	199	23	,	,	PUNCT
ejpam-5279	199	24	there	there	PRON
ejpam-5279	199	25	exists	exist	VERB
ejpam-5279	199	26	a	a	DET
ejpam-5279	199	27	θ	θ	PROPN
ejpam-5279	199	28	∈	∈	PROPN
ejpam-5279	199	29	ω(x	ω(x	NOUN
ejpam-5279	199	30	)	)	PUNCT
ejpam-5279	199	31	such	such	ADJ
ejpam-5279	199	32	that	that	SCONJ
ejpam-5279	199	33	(	(	PUNCT
ejpam-5279	199	34	ν	ν	AUX
ejpam-5279	199	35	◦	◦	NOUN
ejpam-5279	199	36	ȷ)∧n	ȷ)∧n	X
ejpam-5279	199	37	≤	≤	NOUN
ejpam-5279	199	38	θ+ϵ	θ+ϵ	NUM
ejpam-5279	199	39	,	,	PUNCT
ejpam-5279	199	40	that	that	ADV
ejpam-5279	199	41	is	is	ADV
ejpam-5279	199	42	,	,	PUNCT
ejpam-5279	199	43	ȷ	ȷ	X
ejpam-5279	199	44	:	:	PUNCT
ejpam-5279	199	45	x	x	X
ejpam-5279	199	46	→	→	SYM
ejpam-5279	199	47	x	x	SYM
ejpam-5279	199	48	,	,	PUNCT
ejpam-5279	199	49	x	x	PROPN
ejpam-5279	199	50	7→	7→	NUM
ejpam-5279	199	51	x−1	x−1	PROPN
ejpam-5279	199	52	is	be	AUX
ejpam-5279	199	53	contracting	contract	VERB
ejpam-5279	199	54	at	at	ADP
ejpam-5279	199	55	x	x	NOUN
ejpam-5279	199	56	,	,	PUNCT
ejpam-5279	199	57	and	and	CCONJ
ejpam-5279	199	58	hence	hence	ADV
ejpam-5279	199	59	it	it	PRON
ejpam-5279	199	60	is	be	AUX
ejpam-5279	199	61	contracting	contract	VERB
ejpam-5279	199	62	in	in	ADP
ejpam-5279	199	63	each	each	DET
ejpam-5279	199	64	x	x	NOUN
ejpam-5279	199	65	,	,	PUNCT
ejpam-5279	199	66	and	and	CCONJ
ejpam-5279	199	67	so	so	ADV
ejpam-5279	199	68	,	,	PUNCT
ejpam-5279	199	69	the	the	DET
ejpam-5279	199	70	inversion	inversion	NOUN
ejpam-5279	199	71	ȷ	ȷ	NOUN
ejpam-5279	199	72	:	:	PUNCT
ejpam-5279	199	73	x	x	X
ejpam-5279	199	74	→	→	PUNCT
ejpam-5279	199	75	x	x	X
ejpam-5279	199	76	is	be	AUX
ejpam-5279	199	77	a	a	DET
ejpam-5279	199	78	contraction	contraction	NOUN
ejpam-5279	199	79	.	.	PUNCT
ejpam-5279	200	1	to	to	PART
ejpam-5279	200	2	prove	prove	VERB
ejpam-5279	200	3	that	that	SCONJ
ejpam-5279	200	4	the	the	DET
ejpam-5279	200	5	map	map	NOUN
ejpam-5279	200	6	m	m	VERB
ejpam-5279	200	7	:	:	PUNCT
ejpam-5279	200	8	(	(	PUNCT
ejpam-5279	200	9	x	x	X
ejpam-5279	200	10	,	,	PUNCT
ejpam-5279	200	11	y	y	PROPN
ejpam-5279	200	12	)	)	PUNCT
ejpam-5279	200	13	7→	7→	NUM
ejpam-5279	200	14	xy	xy	NOUN
ejpam-5279	200	15	is	be	AUX
ejpam-5279	200	16	contacting	contact	VERB
ejpam-5279	200	17	in	in	ADP
ejpam-5279	200	18	(	(	PUNCT
ejpam-5279	200	19	a	a	PRON
ejpam-5279	200	20	,	,	PUNCT
ejpam-5279	200	21	b	b	NOUN
ejpam-5279	200	22	)	)	PUNCT
ejpam-5279	200	23	∈	∈	PROPN
ejpam-5279	200	24	x	x	SYM
ejpam-5279	200	25	×x	×x	PROPN
ejpam-5279	200	26	,	,	PUNCT
ejpam-5279	200	27	we	we	PRON
ejpam-5279	200	28	employ	employ	VERB
ejpam-5279	200	29	axioms	axiom	NOUN
ejpam-5279	200	30	(	(	PUNCT
ejpam-5279	200	31	a)-(d	a)-(d	NOUN
ejpam-5279	200	32	)	)	PUNCT
ejpam-5279	200	33	in	in	ADP
ejpam-5279	200	34	conjunction	conjunction	NOUN
ejpam-5279	200	35	with	with	ADP
ejpam-5279	200	36	theorem	theorem	ADJ
ejpam-5279	200	37	3.4	3.4	NUM
ejpam-5279	200	38	[	[	SYM
ejpam-5279	200	39	15	15	NUM
ejpam-5279	200	40	]	]	PUNCT
ejpam-5279	200	41	to	to	ADP
ejpam-5279	200	42	the	the	DET
ejpam-5279	200	43	contracting	contracting	NOUN
ejpam-5279	200	44	maps	maps	PROPN
ejpam-5279	200	45	m	m	PROPN
ejpam-5279	200	46	,	,	PUNCT
ejpam-5279	200	47	la−1	la−1	NOUN
ejpam-5279	200	48	,	,	PUNCT
ejpam-5279	200	49	lb−1	lb−1	PROPN
ejpam-5279	200	50	,	,	PUNCT
ejpam-5279	200	51	la	la	PROPN
ejpam-5279	200	52	and	and	CCONJ
ejpam-5279	200	53	intb	intb	VERB
ejpam-5279	200	54	which	which	PRON
ejpam-5279	200	55	are	be	AUX
ejpam-5279	200	56	respectively	respectively	ADV
ejpam-5279	200	57	contracting	contract	VERB
ejpam-5279	200	58	at	at	ADP
ejpam-5279	200	59	(	(	PUNCT
ejpam-5279	200	60	e	e	NOUN
ejpam-5279	200	61	,	,	PUNCT
ejpam-5279	200	62	e	e	NOUN
ejpam-5279	200	63	)	)	PUNCT
ejpam-5279	200	64	,	,	PUNCT
ejpam-5279	200	65	a	a	DET
ejpam-5279	200	66	,	,	PUNCT
ejpam-5279	200	67	b	b	NOUN
ejpam-5279	200	68	,	,	PUNCT
ejpam-5279	200	69	and	and	CCONJ
ejpam-5279	200	70	e	e	NOUN
ejpam-5279	200	71	,	,	PUNCT
ejpam-5279	200	72	to	to	PART
ejpam-5279	200	73	get	get	VERB
ejpam-5279	200	74	the	the	DET
ejpam-5279	200	75	compositions	composition	NOUN
ejpam-5279	200	76	:	:	PUNCT
ejpam-5279	200	77	m(a	m(a	NOUN
ejpam-5279	200	78	,	,	PUNCT
ejpam-5279	200	79	b	b	NOUN
ejpam-5279	200	80	)	)	PUNCT
ejpam-5279	200	81	=	=	NOUN
ejpam-5279	201	1	[	[	X
ejpam-5279	201	2	la	la	ADP
ejpam-5279	201	3	◦	◦	NOUN
ejpam-5279	201	4	intb	intb	VERB
ejpam-5279	201	5	◦	◦	NOUN
ejpam-5279	201	6	m	m	NOUN
ejpam-5279	201	7	◦	◦	NOUN
ejpam-5279	201	8	(	(	PUNCT
ejpam-5279	201	9	la−1	la−1	NOUN
ejpam-5279	201	10	×	×	PROPN
ejpam-5279	201	11	lb−1)](a	lb−1)](a	NOUN
ejpam-5279	201	12	,	,	PUNCT
ejpam-5279	201	13	b	b	NOUN
ejpam-5279	201	14	)	)	PUNCT
ejpam-5279	202	1	=	=	SYM
ejpam-5279	202	2	ab	ab	PROPN
ejpam-5279	202	3	.	.	PUNCT
ejpam-5279	203	1	thus	thus	ADV
ejpam-5279	203	2	,	,	PUNCT
ejpam-5279	203	3	we	we	PRON
ejpam-5279	203	4	have	have	VERB
ejpam-5279	203	5	the	the	DET
ejpam-5279	203	6	m	m	NOUN
ejpam-5279	203	7	:	:	PUNCT
ejpam-5279	203	8	x	x	X
ejpam-5279	203	9	×x	×x	X
ejpam-5279	203	10	→	→	SYM
ejpam-5279	203	11	x	x	SYM
ejpam-5279	203	12	,	,	PUNCT
ejpam-5279	203	13	(	(	PUNCT
ejpam-5279	203	14	x	x	NOUN
ejpam-5279	203	15	,	,	PUNCT
ejpam-5279	203	16	y	y	PROPN
ejpam-5279	203	17	)	)	PUNCT
ejpam-5279	203	18	7→	7→	NUM
ejpam-5279	203	19	xy	xy	NOUN
ejpam-5279	203	20	is	be	AUX
ejpam-5279	203	21	a	a	DET
ejpam-5279	203	22	contraction	contraction	NOUN
ejpam-5279	203	23	map	map	NOUN
ejpam-5279	203	24	.	.	PUNCT
ejpam-5279	204	1	theorem	theorem	ADJ
ejpam-5279	204	2	4	4	NUM
ejpam-5279	204	3	.	.	PUNCT
ejpam-5279	205	1	let	let	VERB
ejpam-5279	205	2	(	(	PUNCT
ejpam-5279	205	3	x	x	NOUN
ejpam-5279	205	4	,	,	PUNCT
ejpam-5279	205	5	·	·	PUNCT
ejpam-5279	205	6	)	)	PUNCT
ejpam-5279	205	7	be	be	AUX
ejpam-5279	205	8	a	a	DET
ejpam-5279	205	9	group	group	NOUN
ejpam-5279	205	10	,	,	PUNCT
ejpam-5279	205	11	and	and	CCONJ
ejpam-5279	205	12	b	b	X
ejpam-5279	205	13	be	be	AUX
ejpam-5279	205	14	a	a	DET
ejpam-5279	205	15	family	family	NOUN
ejpam-5279	205	16	of	of	ADP
ejpam-5279	205	17	ideals	ideal	NOUN
ejpam-5279	205	18	in	in	ADP
ejpam-5279	205	19	[	[	X
ejpam-5279	205	20	0,∞]x	0,∞]x	NOUN
ejpam-5279	205	21	such	such	ADJ
ejpam-5279	205	22	that	that	SCONJ
ejpam-5279	205	23	the	the	DET
ejpam-5279	205	24	following	follow	VERB
ejpam-5279	205	25	are	be	AUX
ejpam-5279	205	26	fulfilled	fulfil	VERB
ejpam-5279	205	27	:	:	PUNCT
ejpam-5279	205	28	(	(	PUNCT
ejpam-5279	205	29	1	1	X
ejpam-5279	205	30	)	)	PUNCT
ejpam-5279	205	31	b	b	NOUN
ejpam-5279	205	32	is	be	AUX
ejpam-5279	205	33	an	an	DET
ejpam-5279	205	34	ideal	ideal	ADJ
ejpam-5279	205	35	basis	basis	NOUN
ejpam-5279	205	36	,	,	PUNCT
ejpam-5279	205	37	such	such	ADJ
ejpam-5279	205	38	that	that	SCONJ
ejpam-5279	205	39	∀	∀	NOUN
ejpam-5279	205	40	ν	ν	X
ejpam-5279	205	41	∈	∈	PROPN
ejpam-5279	205	42	b	b	NOUN
ejpam-5279	205	43	:	:	PUNCT
ejpam-5279	205	44	ν(e	ν(e	PROPN
ejpam-5279	205	45	)	)	PUNCT
ejpam-5279	205	46	=	=	SYM
ejpam-5279	205	47	0	0	NUM
ejpam-5279	205	48	;	;	PUNCT
ejpam-5279	205	49	(	(	PUNCT
ejpam-5279	205	50	2	2	X
ejpam-5279	205	51	)	)	PUNCT
ejpam-5279	205	52	∀ν	∀ν	PROPN
ejpam-5279	205	53	∈	∈	PROPN
ejpam-5279	205	54	b	b	PROPN
ejpam-5279	205	55	,	,	PUNCT
ejpam-5279	205	56	∀ϵ	∀ϵ	NOUN
ejpam-5279	205	57	>	>	X
ejpam-5279	205	58	0	0	NUM
ejpam-5279	205	59	,	,	PUNCT
ejpam-5279	205	60	∀n	∀n	PUNCT
ejpam-5279	205	61	<	<	X
ejpam-5279	205	62	∞	∞	PROPN
ejpam-5279	205	63	,	,	PUNCT
ejpam-5279	205	64	there	there	PRON
ejpam-5279	205	65	exists	exist	VERB
ejpam-5279	205	66	µ	µ	PRON
ejpam-5279	205	67	∈	∈	PROPN
ejpam-5279	205	68	b	b	NOUN
ejpam-5279	205	69	such	such	ADJ
ejpam-5279	205	70	that	that	PRON
ejpam-5279	205	71	ν−1	ν−1	PROPN
ejpam-5279	205	72	∧n	∧n	VERB
ejpam-5279	205	73	≤	≤	NOUN
ejpam-5279	205	74	µ+	µ+	PRON
ejpam-5279	205	75	ϵ	ϵ	X
ejpam-5279	205	76	;	;	PUNCT
ejpam-5279	205	77	(	(	PUNCT
ejpam-5279	205	78	3	3	X
ejpam-5279	205	79	)	)	PUNCT
ejpam-5279	205	80	∀ν	∀ν	PROPN
ejpam-5279	205	81	∈	∈	PROPN
ejpam-5279	205	82	b	b	PROPN
ejpam-5279	205	83	,	,	PUNCT
ejpam-5279	205	84	∀ϵ	∀ϵ	NOUN
ejpam-5279	205	85	>	>	X
ejpam-5279	205	86	0	0	NUM
ejpam-5279	205	87	,	,	PUNCT
ejpam-5279	205	88	∀n	∀n	PUNCT
ejpam-5279	205	89	<	<	X
ejpam-5279	205	90	∞	∞	PROPN
ejpam-5279	205	91	,	,	PUNCT
ejpam-5279	205	92	there	there	PRON
ejpam-5279	205	93	exists	exist	VERB
ejpam-5279	205	94	µ	µ	PRON
ejpam-5279	205	95	∈	∈	PROPN
ejpam-5279	205	96	b	b	NOUN
ejpam-5279	205	97	such	such	ADJ
ejpam-5279	205	98	that	that	SCONJ
ejpam-5279	205	99	∀x	∀x	NUM
ejpam-5279	205	100	,	,	PUNCT
ejpam-5279	205	101	y	y	PROPN
ejpam-5279	205	102	∈	∈	PROPN
ejpam-5279	206	1	x	x	X
ejpam-5279	206	2	:	:	PUNCT
ejpam-5279	206	3	ν(xy	ν(xy	NUM
ejpam-5279	206	4	)	)	PUNCT
ejpam-5279	206	5	∧n	∧n	VERB
ejpam-5279	206	6	≤	≤	NOUN
ejpam-5279	206	7	µ(x	µ(x	X
ejpam-5279	206	8	)	)	PUNCT
ejpam-5279	206	9	∨	∨	NUM
ejpam-5279	206	10	µ(y	µ(y	PROPN
ejpam-5279	206	11	)	)	PUNCT
ejpam-5279	207	1	+	+	CCONJ
ejpam-5279	207	2	ϵ	ϵ	X
ejpam-5279	207	3	;	;	PUNCT
ejpam-5279	207	4	(	(	PUNCT
ejpam-5279	207	5	4	4	X
ejpam-5279	207	6	)	)	PUNCT
ejpam-5279	207	7	∀ν	∀ν	PROPN
ejpam-5279	207	8	∈	∈	PROPN
ejpam-5279	207	9	b	b	PROPN
ejpam-5279	207	10	,	,	PUNCT
ejpam-5279	207	11	∀ϵ	∀ϵ	NOUN
ejpam-5279	207	12	>	>	X
ejpam-5279	207	13	0	0	NUM
ejpam-5279	207	14	,	,	PUNCT
ejpam-5279	207	15	∀n	∀n	PUNCT
ejpam-5279	207	16	<	<	X
ejpam-5279	207	17	∞	∞	PROPN
ejpam-5279	207	18	,	,	PUNCT
ejpam-5279	207	19	∀x	∀x	VERB
ejpam-5279	207	20	∈	∈	PROPN
ejpam-5279	207	21	x	x	PRON
ejpam-5279	207	22	,	,	PUNCT
ejpam-5279	207	23	there	there	PRON
ejpam-5279	207	24	exists	exist	VERB
ejpam-5279	207	25	µ	µ	PRON
ejpam-5279	207	26	∈	∈	PROPN
ejpam-5279	207	27	b	b	NOUN
ejpam-5279	207	28	such	such	ADJ
ejpam-5279	207	29	that	that	PRON
ejpam-5279	207	30	(	(	PUNCT
ejpam-5279	207	31	x⊙	x⊙	PROPN
ejpam-5279	207	32	ν	ν	PROPN
ejpam-5279	207	33	⊙	⊙	PROPN
ejpam-5279	207	34	x−1	x−1	PROPN
ejpam-5279	207	35	)	)	PUNCT
ejpam-5279	207	36	∧n	∧n	VERB
ejpam-5279	207	37	≤	≤	NOUN
ejpam-5279	207	38	µ+	µ+	PUNCT
ejpam-5279	207	39	ϵ.	ϵ.	NOUN
ejpam-5279	207	40	then	then	ADV
ejpam-5279	207	41	there	there	PRON
ejpam-5279	207	42	exists	exist	VERB
ejpam-5279	207	43	a	a	DET
ejpam-5279	207	44	unique	unique	ADJ
ejpam-5279	207	45	approach	approach	NOUN
ejpam-5279	207	46	system	system	NOUN
ejpam-5279	207	47	such	such	ADJ
ejpam-5279	207	48	that	that	SCONJ
ejpam-5279	207	49	b	b	NOUN
ejpam-5279	207	50	is	be	AUX
ejpam-5279	207	51	a	a	DET
ejpam-5279	207	52	basis	basis	NOUN
ejpam-5279	207	53	for	for	ADP
ejpam-5279	207	54	the	the	DET
ejpam-5279	207	55	approach	approach	NOUN
ejpam-5279	207	56	system	system	NOUN
ejpam-5279	207	57	at	at	ADP
ejpam-5279	207	58	e	e	NOUN
ejpam-5279	207	59	and	and	CCONJ
ejpam-5279	207	60	compatible	compatible	ADJ
ejpam-5279	207	61	with	with	ADP
ejpam-5279	207	62	group	group	NOUN
ejpam-5279	207	63	structure	structure	NOUN
ejpam-5279	207	64	of	of	ADP
ejpam-5279	207	65	x.	x.	NOUN
ejpam-5279	207	66	this	this	DET
ejpam-5279	207	67	approach	approach	NOUN
ejpam-5279	207	68	system	system	NOUN
ejpam-5279	207	69	is	be	AUX
ejpam-5279	207	70	given	give	VERB
ejpam-5279	207	71	by	by	ADP
ejpam-5279	207	72	:	:	PUNCT
ejpam-5279	207	73	a(x	a(x	PROPN
ejpam-5279	207	74	)	)	PUNCT
ejpam-5279	207	75	=	=	PUNCT
ejpam-5279	207	76	⟨{x−1	⟨{x−1	NOUN
ejpam-5279	207	77	⊙	⊙	NOUN
ejpam-5279	208	1	ν|	ν|	ADP
ejpam-5279	208	2	ν	ν	X
ejpam-5279	208	3	∈	∈	NOUN
ejpam-5279	208	4	b}⟩	b}⟩	NOUN
ejpam-5279	208	5	=	=	PUNCT
ejpam-5279	209	1	⟨{ν	⟨{ν	NOUN
ejpam-5279	209	2	⊙	⊙	NOUN
ejpam-5279	209	3	x−1|	x−1|	PROPN
ejpam-5279	210	1	ν	ν	PROPN
ejpam-5279	210	2	∈	∈	PROPN
ejpam-5279	210	3	b}⟩.	b}⟩.	NOUN
ejpam-5279	210	4	proof	proof	NOUN
ejpam-5279	210	5	.	.	PUNCT
ejpam-5279	211	1	in	in	ADP
ejpam-5279	211	2	view	view	NOUN
ejpam-5279	211	3	of	of	ADP
ejpam-5279	211	4	the	the	DET
ejpam-5279	211	5	preceding	precede	VERB
ejpam-5279	211	6	theorem	theorem	NOUN
ejpam-5279	211	7	,	,	PUNCT
ejpam-5279	211	8	we	we	PRON
ejpam-5279	211	9	only	only	ADV
ejpam-5279	211	10	prove	prove	VERB
ejpam-5279	211	11	(	(	PUNCT
ejpam-5279	211	12	as3	as3	PROPN
ejpam-5279	211	13	)	)	PUNCT
ejpam-5279	211	14	,	,	PUNCT
ejpam-5279	211	15	for	for	ADP
ejpam-5279	211	16	this	this	PRON
ejpam-5279	211	17	we	we	PRON
ejpam-5279	211	18	proceed	proceed	VERB
ejpam-5279	211	19	as	as	SCONJ
ejpam-5279	211	20	follows	follow	VERB
ejpam-5279	211	21	.	.	PUNCT
ejpam-5279	212	1	let	let	VERB
ejpam-5279	212	2	ξ	ξ	X
ejpam-5279	212	3	=	=	SYM
ejpam-5279	212	4	x−1	x−1	PROPN
ejpam-5279	212	5	⊙	⊙	PROPN
ejpam-5279	212	6	ν	ν	PROPN
ejpam-5279	212	7	∈	∈	PROPN
ejpam-5279	212	8	a(x	a(x	PROPN
ejpam-5279	212	9	)	)	PUNCT
ejpam-5279	212	10	with	with	ADP
ejpam-5279	212	11	ν	ν	PROPN
ejpam-5279	212	12	∈	∈	PROPN
ejpam-5279	212	13	b	b	PROPN
ejpam-5279	212	14	,	,	PUNCT
ejpam-5279	212	15	let	let	VERB
ejpam-5279	212	16	ϵ	ϵ	PRON
ejpam-5279	212	17	>	>	X
ejpam-5279	212	18	0	0	NUM
ejpam-5279	212	19	,	,	PUNCT
ejpam-5279	212	20	and	and	CCONJ
ejpam-5279	212	21	n	n	CCONJ
ejpam-5279	212	22	<	<	X
ejpam-5279	212	23	∞.	∞.	PROPN
ejpam-5279	212	24	choose	choose	VERB
ejpam-5279	212	25	η	η	PROPN
ejpam-5279	212	26	∈	∈	PROPN
ejpam-5279	212	27	b	b	PROPN
ejpam-5279	212	28	such	such	ADJ
ejpam-5279	212	29	that	that	PRON
ejpam-5279	212	30	ν(xy	ν(xy	NOUN
ejpam-5279	212	31	)	)	PUNCT
ejpam-5279	212	32	∧	∧	PROPN
ejpam-5279	212	33	n	n	CCONJ
ejpam-5279	212	34	≤	≤	NOUN
ejpam-5279	212	35	η(x	η(x	NOUN
ejpam-5279	212	36	)	)	PUNCT
ejpam-5279	212	37	∨	∨	NUM
ejpam-5279	212	38	η(y	η(y	NOUN
ejpam-5279	212	39	)	)	PUNCT
ejpam-5279	213	1	+	+	CCONJ
ejpam-5279	213	2	ϵ.	ϵ.	NOUN
ejpam-5279	213	3	if	if	SCONJ
ejpam-5279	213	4	ξz	ξz	PRON
ejpam-5279	213	5	=	=	SYM
ejpam-5279	213	6	z−1	z−1	PROPN
ejpam-5279	213	7	⊙	⊙	PROPN
ejpam-5279	213	8	η	η	PROPN
ejpam-5279	213	9	,	,	PUNCT
ejpam-5279	213	10	then	then	ADV
ejpam-5279	213	11	ξ(y	ξ(y	PROPN
ejpam-5279	213	12	)	)	PUNCT
ejpam-5279	213	13	∧	∧	PROPN
ejpam-5279	213	14	n	n	NOUN
ejpam-5279	213	15	=	=	SYM
ejpam-5279	213	16	(	(	PUNCT
ejpam-5279	213	17	x−1	x−1	PROPN
ejpam-5279	213	18	⊙	⊙	PROPN
ejpam-5279	213	19	ν	ν	PROPN
ejpam-5279	213	20	)	)	PUNCT
ejpam-5279	213	21	(	(	PUNCT
ejpam-5279	213	22	y	y	X
ejpam-5279	213	23	)	)	PUNCT
ejpam-5279	213	24	∧	∧	PROPN
ejpam-5279	213	25	n	n	PROPN
ejpam-5279	213	26	=	=	NOUN
ejpam-5279	213	27	ν(x−1y	ν(x−1y	ADJ
ejpam-5279	213	28	)	)	PUNCT
ejpam-5279	213	29	∧n	∧n	X
ejpam-5279	213	30	=	=	SYM
ejpam-5279	213	31	ν	ν	X
ejpam-5279	213	32	(	(	PUNCT
ejpam-5279	213	33	x−1zz−1y	x−1zz−1y	PROPN
ejpam-5279	213	34	)	)	PUNCT
ejpam-5279	213	35	∧n	∧n	VERB
ejpam-5279	213	36	≤	≤	NUM
ejpam-5279	213	37	η(x−1z	η(x−1z	NOUN
ejpam-5279	213	38	)	)	PUNCT
ejpam-5279	213	39	∨	∨	NUM
ejpam-5279	213	40	η(z−1y	η(z−1y	NOUN
ejpam-5279	213	41	)	)	PUNCT
ejpam-5279	214	1	+	+	CCONJ
ejpam-5279	215	1	ϵ	ϵ	X
ejpam-5279	215	2	=	=	PUNCT
ejpam-5279	215	3	ξx(z	ξx(z	NUM
ejpam-5279	215	4	)	)	PUNCT
ejpam-5279	215	5	∨	∨	NUM
ejpam-5279	215	6	ξz(y	ξz(y	NUM
ejpam-5279	215	7	)	)	PUNCT
ejpam-5279	215	8	+	+	CCONJ
ejpam-5279	215	9	ϵ.	ϵ.	NOUN
ejpam-5279	215	10	proposition	proposition	NOUN
ejpam-5279	215	11	2	2	NUM
ejpam-5279	215	12	.	.	PUNCT
ejpam-5279	216	1	every	every	DET
ejpam-5279	216	2	ultra	ultra	ADJ
ejpam-5279	216	3	approach	approach	NOUN
ejpam-5279	216	4	group	group	NOUN
ejpam-5279	216	5	is	be	AUX
ejpam-5279	216	6	ultra	ultra	ADJ
ejpam-5279	216	7	approach	approach	NOUN
ejpam-5279	216	8	uniformizable	uniformizable	ADJ
ejpam-5279	216	9	.	.	PUNCT
ejpam-5279	217	1	proof	proof	NOUN
ejpam-5279	217	2	.	.	PUNCT
ejpam-5279	218	1	let	let	VERB
ejpam-5279	218	2	(	(	PUNCT
ejpam-5279	218	3	x	x	NOUN
ejpam-5279	218	4	,	,	PUNCT
ejpam-5279	218	5	·	·	PUNCT
ejpam-5279	218	6	,	,	PUNCT
ejpam-5279	218	7	ω	ω	NUM
ejpam-5279	218	8	)	)	PUNCT
ejpam-5279	218	9	be	be	VERB
ejpam-5279	218	10	an	an	DET
ejpam-5279	218	11	ultra	ultra	ADJ
ejpam-5279	218	12	approach	approach	NOUN
ejpam-5279	218	13	group	group	NOUN
ejpam-5279	218	14	.	.	PUNCT
ejpam-5279	219	1	define	define	VERB
ejpam-5279	219	2	νl	νl	ADP
ejpam-5279	219	3	:	:	PUNCT
ejpam-5279	219	4	x	x	PROPN
ejpam-5279	219	5	×x	×x	X
ejpam-5279	219	6	→	→	X
ejpam-5279	219	7	[	[	X
ejpam-5279	219	8	0,∞	0,∞	X
ejpam-5279	219	9	]	]	PUNCT
ejpam-5279	219	10	,	,	PUNCT
ejpam-5279	219	11	(	(	PUNCT
ejpam-5279	219	12	x	x	NOUN
ejpam-5279	219	13	,	,	PUNCT
ejpam-5279	219	14	y	y	PROPN
ejpam-5279	219	15	)	)	PUNCT
ejpam-5279	219	16	7→	7→	NUM
ejpam-5279	219	17	νl(x	νl(x	NOUN
ejpam-5279	219	18	,	,	PUNCT
ejpam-5279	219	19	y	y	NOUN
ejpam-5279	219	20	)	)	PUNCT
ejpam-5279	219	21	=	=	SYM
ejpam-5279	219	22	ν(x−1y	ν(x−1y	ADJ
ejpam-5279	219	23	)	)	PUNCT
ejpam-5279	219	24	and	and	CCONJ
ejpam-5279	219	25	γ	γ	X
ejpam-5279	219	26	=	=	SYM
ejpam-5279	219	27	<	<	X
ejpam-5279	219	28	{	{	PUNCT
ejpam-5279	219	29	νl	νl	NOUN
ejpam-5279	219	30	∈	∈	PROPN
ejpam-5279	220	1	[	[	X
ejpam-5279	220	2	0,∞]x×x	0,∞]x×x	NUM
ejpam-5279	220	3	|	|	ADV
ejpam-5279	220	4	ν	ν	X
ejpam-5279	220	5	∈	∈	PROPN
ejpam-5279	220	6	ωe	ωe	ADV
ejpam-5279	220	7	}	}	PUNCT
ejpam-5279	220	8	>	>	PUNCT
ejpam-5279	220	9	.	.	PUNCT
ejpam-5279	221	1	(	(	PUNCT
ejpam-5279	221	2	uau1	uau1	PROPN
ejpam-5279	221	3	)	)	PUNCT
ejpam-5279	221	4	let	let	VERB
ejpam-5279	221	5	x	x	PUNCT
ejpam-5279	221	6	∈	∈	PROPN
ejpam-5279	221	7	x	x	X
ejpam-5279	221	8	and	and	CCONJ
ejpam-5279	221	9	γ	γ	PROPN
ejpam-5279	221	10	∈	∈	PROPN
ejpam-5279	221	11	γ	γ	X
ejpam-5279	221	12	.	.	PUNCT
ejpam-5279	222	1	then	then	ADV
ejpam-5279	222	2	there	there	PRON
ejpam-5279	222	3	is	be	VERB
ejpam-5279	222	4	a	a	DET
ejpam-5279	222	5	ν	ν	X
ejpam-5279	222	6	∈	∈	NOUN
ejpam-5279	222	7	ωe	ωe	ADV
ejpam-5279	222	8	such	such	ADJ
ejpam-5279	222	9	that	that	PRON
ejpam-5279	222	10	γ(x	γ(x	NOUN
ejpam-5279	222	11	,	,	PUNCT
ejpam-5279	222	12	x	x	NOUN
ejpam-5279	222	13	)	)	PUNCT
ejpam-5279	222	14	=	=	NOUN
ejpam-5279	222	15	νl(x	νl(x	NOUN
ejpam-5279	222	16	,	,	PUNCT
ejpam-5279	222	17	x	x	X
ejpam-5279	222	18	)	)	PUNCT
ejpam-5279	222	19	=	=	SYM
ejpam-5279	222	20	ν(e	ν(e	PROPN
ejpam-5279	222	21	)	)	PUNCT
ejpam-5279	222	22	=	=	SYM
ejpam-5279	222	23	0	0	X
ejpam-5279	222	24	.	.	PUNCT
ejpam-5279	223	1	(	(	PUNCT
ejpam-5279	223	2	uau3	uau3	NOUN
ejpam-5279	223	3	)	)	PUNCT
ejpam-5279	223	4	let	let	VERB
ejpam-5279	223	5	ϵ	ϵ	PRON
ejpam-5279	223	6	>	>	X
ejpam-5279	223	7	0	0	PUNCT
ejpam-5279	223	8	and	and	CCONJ
ejpam-5279	223	9	γ	γ	PROPN
ejpam-5279	223	10	∈	∈	PROPN
ejpam-5279	223	11	γ	γ	X
ejpam-5279	223	12	.	.	PUNCT
ejpam-5279	224	1	then	then	ADV
ejpam-5279	224	2	there	there	PRON
ejpam-5279	224	3	exists	exist	VERB
ejpam-5279	224	4	ν	ν	X
ejpam-5279	224	5	∈	∈	PROPN
ejpam-5279	224	6	ωe	ωe	ADV
ejpam-5279	224	7	such	such	ADJ
ejpam-5279	224	8	that	that	PRON
ejpam-5279	224	9	γ(x	γ(x	PROPN
ejpam-5279	224	10	,	,	PUNCT
ejpam-5279	224	11	y	y	NOUN
ejpam-5279	224	12	)	)	PUNCT
ejpam-5279	224	13	=	=	NOUN
ejpam-5279	224	14	νl(x	νl(x	PROPN
ejpam-5279	224	15	,	,	PUNCT
ejpam-5279	224	16	y	y	NOUN
ejpam-5279	224	17	)	)	PUNCT
ejpam-5279	224	18	=	=	SYM
ejpam-5279	224	19	ν(x−1y	ν(x−1y	ADJ
ejpam-5279	224	20	)	)	PUNCT
ejpam-5279	224	21	.	.	PUNCT
ejpam-5279	225	1	since	since	SCONJ
ejpam-5279	225	2	by	by	ADP
ejpam-5279	225	3	contraction	contraction	NOUN
ejpam-5279	225	4	of	of	ADP
ejpam-5279	225	5	r	r	NOUN
ejpam-5279	225	6	,	,	PUNCT
ejpam-5279	225	7	one	one	NOUN
ejpam-5279	225	8	obtains	obtain	VERB
ejpam-5279	225	9	νl	νl	ADP
ejpam-5279	225	10	◦	◦	NOUN
ejpam-5279	225	11	r	r	NOUN
ejpam-5279	225	12	∈	∈	PROPN
ejpam-5279	225	13	ωe	ωe	ADP
ejpam-5279	225	14	,	,	PUNCT
ejpam-5279	225	15	yields	yield	VERB
ejpam-5279	225	16	that	that	SCONJ
ejpam-5279	225	17	ν−1	ν−1	PROPN
ejpam-5279	225	18	∈	∈	NOUN
ejpam-5279	225	19	ωe	ωe	PRON
ejpam-5279	225	20	.	.	PUNCT
ejpam-5279	226	1	thus	thus	ADV
ejpam-5279	226	2	,	,	PUNCT
ejpam-5279	226	3	we	we	PRON
ejpam-5279	226	4	have	have	VERB
ejpam-5279	226	5	γs(x	γs(x	PROPN
ejpam-5279	226	6	,	,	PUNCT
ejpam-5279	226	7	y	y	NOUN
ejpam-5279	226	8	)	)	PUNCT
ejpam-5279	226	9	=	=	SYM
ejpam-5279	227	1	γ(y	γ(y	PROPN
ejpam-5279	227	2	,	,	PUNCT
ejpam-5279	227	3	x	x	NOUN
ejpam-5279	227	4	)	)	PUNCT
ejpam-5279	227	5	=	=	SYM
ejpam-5279	227	6	νl(y	νl(y	X
ejpam-5279	227	7	,	,	PUNCT
ejpam-5279	227	8	x	x	X
ejpam-5279	227	9	)	)	PUNCT
ejpam-5279	227	10	=	=	SYM
ejpam-5279	227	11	ν−1(x−1y	ν−1(x−1y	NOUN
ejpam-5279	227	12	)	)	PUNCT
ejpam-5279	227	13	=	=	PUNCT
ejpam-5279	227	14	(	(	PUNCT
ejpam-5279	227	15	ν−1)l(x	ν−1)l(x	PROPN
ejpam-5279	227	16	,	,	PUNCT
ejpam-5279	227	17	y	y	PROPN
ejpam-5279	227	18	)	)	PUNCT
ejpam-5279	227	19	.	.	PUNCT
ejpam-5279	228	1	so	so	ADV
ejpam-5279	228	2	,	,	PUNCT
ejpam-5279	228	3	γ	γ	PROPN
ejpam-5279	228	4	s	s	PROPN
ejpam-5279	228	5	∈	∈	PROPN
ejpam-5279	228	6	γ	γ	X
ejpam-5279	228	7	.	.	PUNCT
ejpam-5279	228	8	(	(	PUNCT
ejpam-5279	228	9	uau4	uau4	PROPN
ejpam-5279	228	10	)	)	PUNCT
ejpam-5279	228	11	let	let	VERB
ejpam-5279	228	12	γ	γ	X
ejpam-5279	228	13	∈	∈	PROPN
ejpam-5279	228	14	γ	γ	NOUN
ejpam-5279	228	15	be	be	AUX
ejpam-5279	228	16	such	such	ADJ
ejpam-5279	228	17	that	that	SCONJ
ejpam-5279	228	18	γ(x	γ(x	PROPN
ejpam-5279	228	19	,	,	PUNCT
ejpam-5279	228	20	y	y	NOUN
ejpam-5279	228	21	)	)	PUNCT
ejpam-5279	228	22	=	=	NOUN
ejpam-5279	228	23	νl(x	νl(x	PROPN
ejpam-5279	228	24	,	,	PUNCT
ejpam-5279	228	25	y	y	NOUN
ejpam-5279	228	26	)	)	PUNCT
ejpam-5279	228	27	=	=	SYM
ejpam-5279	228	28	ν(x−1y	ν(x−1y	ADJ
ejpam-5279	228	29	)	)	PUNCT
ejpam-5279	228	30	.	.	PUNCT
ejpam-5279	229	1	then	then	ADV
ejpam-5279	229	2	for	for	ADP
ejpam-5279	229	3	each	each	DET
ejpam-5279	229	4	ϵ	ϵ	X
ejpam-5279	229	5	>	>	X
ejpam-5279	229	6	0	0	PUNCT
ejpam-5279	229	7	and	and	CCONJ
ejpam-5279	229	8	n	n	CCONJ
ejpam-5279	229	9	<	<	X
ejpam-5279	229	10	∞	∞	PROPN
ejpam-5279	229	11	,	,	PUNCT
ejpam-5279	229	12	there	there	PRON
ejpam-5279	229	13	exists	exist	VERB
ejpam-5279	229	14	νnϵ	νnϵ	VERB
ejpam-5279	229	15	∈	∈	PROPN
ejpam-5279	229	16	ωe	ωe	ADV
ejpam-5279	229	17	such	such	ADJ
ejpam-5279	229	18	that	that	SCONJ
ejpam-5279	229	19	ν(xy	ν(xy	NOUN
ejpam-5279	229	20	)	)	PUNCT
ejpam-5279	229	21	∧	∧	PROPN
ejpam-5279	229	22	n	n	CCONJ
ejpam-5279	229	23	≤	≤	NUM
ejpam-5279	229	24	νnϵ	νnϵ	NOUN
ejpam-5279	229	25	(	(	PUNCT
ejpam-5279	229	26	x	x	NOUN
ejpam-5279	229	27	)	)	PUNCT
ejpam-5279	229	28	∨	∨	NUM
ejpam-5279	229	29	νnϵ	νnϵ	NOUN
ejpam-5279	229	30	(	(	PUNCT
ejpam-5279	229	31	y	y	NOUN
ejpam-5279	229	32	)	)	PUNCT
ejpam-5279	229	33	.	.	PUNCT
ejpam-5279	230	1	if	if	SCONJ
ejpam-5279	230	2	we	we	PRON
ejpam-5279	230	3	put	put	VERB
ejpam-5279	230	4	t.m.g	t.m.g	ADJ
ejpam-5279	230	5	.	.	PUNCT
ejpam-5279	231	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	231	2	,	,	PUNCT
ejpam-5279	231	3	fawzi	fawzi	PROPN
ejpam-5279	231	4	al	al	PROPN
ejpam-5279	231	5	-	-	PUNCT
ejpam-5279	231	6	thukair	thukair	NOUN
ejpam-5279	231	7	/	/	SYM
ejpam-5279	231	8	eur	eur	NOUN
ejpam-5279	231	9	.	.	PUNCT
ejpam-5279	232	1	j.	j.	PROPN
ejpam-5279	232	2	pure	pure	PROPN
ejpam-5279	232	3	appl	appl	PROPN
ejpam-5279	232	4	.	.	PROPN
ejpam-5279	232	5	math	math	PROPN
ejpam-5279	232	6	,	,	PUNCT
ejpam-5279	232	7	17	17	NUM
ejpam-5279	232	8	(	(	PUNCT
ejpam-5279	232	9	3	3	NUM
ejpam-5279	232	10	)	)	PUNCT
ejpam-5279	232	11	(	(	PUNCT
ejpam-5279	232	12	2024	2024	NUM
ejpam-5279	232	13	)	)	PUNCT
ejpam-5279	232	14	,	,	PUNCT
ejpam-5279	232	15	1762	1762	NUM
ejpam-5279	232	16	-	-	SYM
ejpam-5279	232	17	1778	1778	NUM
ejpam-5279	232	18	1769	1769	NUM
ejpam-5279	232	19	γnϵ	γnϵ	NOUN
ejpam-5279	232	20	(	(	PUNCT
ejpam-5279	232	21	x	x	NOUN
ejpam-5279	232	22	,	,	PUNCT
ejpam-5279	232	23	y	y	NOUN
ejpam-5279	232	24	)	)	PUNCT
ejpam-5279	232	25	=	=	NOUN
ejpam-5279	233	1	νnϵ	νnϵ	NOUN
ejpam-5279	233	2	(	(	PUNCT
ejpam-5279	233	3	x−1y	x−1y	PROPN
ejpam-5279	233	4	)	)	PUNCT
ejpam-5279	233	5	,	,	PUNCT
ejpam-5279	233	6	then	then	ADV
ejpam-5279	233	7	γ(x	γ(x	PROPN
ejpam-5279	233	8	,	,	PUNCT
ejpam-5279	233	9	y	y	NOUN
ejpam-5279	233	10	)	)	PUNCT
ejpam-5279	233	11	∧n	∧n	NOUN
ejpam-5279	233	12	=	=	SYM
ejpam-5279	233	13	ν(x−1y	ν(x−1y	ADJ
ejpam-5279	233	14	)	)	PUNCT
ejpam-5279	233	15	∧n	∧n	X
ejpam-5279	233	16	=	=	SYM
ejpam-5279	233	17	ν	ν	X
ejpam-5279	233	18	(	(	PUNCT
ejpam-5279	233	19	x−1zz−1y	x−1zz−1y	PROPN
ejpam-5279	233	20	)	)	PUNCT
ejpam-5279	233	21	∧n	∧n	PROPN
ejpam-5279	233	22	≤	≤	NUM
ejpam-5279	233	23	νnϵ	νnϵ	NOUN
ejpam-5279	233	24	(	(	PUNCT
ejpam-5279	233	25	x−1z	x−1z	PROPN
ejpam-5279	233	26	)	)	PUNCT
ejpam-5279	233	27	∨	∨	NUM
ejpam-5279	233	28	νnϵ	νnϵ	NOUN
ejpam-5279	233	29	(	(	PUNCT
ejpam-5279	233	30	z−1y	z−1y	NOUN
ejpam-5279	233	31	)	)	PUNCT
ejpam-5279	233	32	≤	≤	NOUN
ejpam-5279	233	33	γnϵ	γnϵ	NOUN
ejpam-5279	233	34	(	(	PUNCT
ejpam-5279	233	35	x	x	X
ejpam-5279	233	36	,	,	PUNCT
ejpam-5279	233	37	z	z	NOUN
ejpam-5279	233	38	)	)	PUNCT
ejpam-5279	233	39	∨	∨	NUM
ejpam-5279	233	40	γnϵ	γnϵ	NOUN
ejpam-5279	233	41	(	(	PUNCT
ejpam-5279	233	42	z	z	NOUN
ejpam-5279	233	43	,	,	PUNCT
ejpam-5279	233	44	y	y	PROPN
ejpam-5279	233	45	)	)	PUNCT
ejpam-5279	234	1	+	+	CCONJ
ejpam-5279	234	2	ϵ.	ϵ.	NOUN
ejpam-5279	234	3	the	the	DET
ejpam-5279	234	4	underlying	underlie	VERB
ejpam-5279	234	5	ultra	ultra	ADJ
ejpam-5279	234	6	-	-	ADJ
ejpam-5279	234	7	approach	approach	ADJ
ejpam-5279	234	8	structure	structure	NOUN
ejpam-5279	234	9	of	of	ADP
ejpam-5279	234	10	γ	γ	PROPN
ejpam-5279	234	11	is	be	AUX
ejpam-5279	234	12	given	give	VERB
ejpam-5279	234	13	by	by	ADP
ejpam-5279	234	14	ω′	ω′	X
ejpam-5279	234	15	x	x	X
ejpam-5279	234	16	=	=	PUNCT
ejpam-5279	234	17	{	{	PUNCT
ejpam-5279	234	18	γ(x	γ(x	PROPN
ejpam-5279	234	19	,	,	PUNCT
ejpam-5279	234	20	.)|γ	.)|γ	PROPN
ejpam-5279	234	21	∈	∈	PROPN
ejpam-5279	234	22	γ	γ	X
ejpam-5279	234	23	}	}	PUNCT
ejpam-5279	234	24	=	=	SYM
ejpam-5279	234	25	{	{	PUNCT
ejpam-5279	234	26	γ	γ	X
ejpam-5279	234	27	(	(	PUNCT
ejpam-5279	234	28	.	.	PUNCT
ejpam-5279	234	29	,	,	PUNCT
ejpam-5279	234	30	x)|γ	x)|γ	PROPN
ejpam-5279	234	31	∈	∈	PROPN
ejpam-5279	234	32	γ	γ	X
ejpam-5279	234	33	}	}	PUNCT
ejpam-5279	234	34	=	=	NOUN
ejpam-5279	234	35	{	{	PUNCT
ejpam-5279	234	36	ν	ν	PART
ejpam-5279	234	37	◦	◦	NOUN
ejpam-5279	234	38	lx−1	lx−1	NOUN
ejpam-5279	234	39	|ν	|ν	NOUN
ejpam-5279	234	40	∈	∈	NOUN
ejpam-5279	234	41	ωe	ωe	ADP
ejpam-5279	234	42	}	}	PUNCT
ejpam-5279	234	43	.	.	PUNCT
ejpam-5279	235	1	in	in	ADP
ejpam-5279	235	2	fact	fact	NOUN
ejpam-5279	235	3	,	,	PUNCT
ejpam-5279	235	4	γ(x	γ(x	ADP
ejpam-5279	235	5	,	,	PUNCT
ejpam-5279	235	6	.)(y	.)(y	PUNCT
ejpam-5279	235	7	)	)	PUNCT
ejpam-5279	236	1	=	=	SYM
ejpam-5279	236	2	γ(x	γ(x	PROPN
ejpam-5279	236	3	,	,	PUNCT
ejpam-5279	236	4	y	y	NOUN
ejpam-5279	236	5	)	)	PUNCT
ejpam-5279	236	6	=	=	SYM
ejpam-5279	236	7	ν(x−1y	ν(x−1y	ADJ
ejpam-5279	236	8	)	)	PUNCT
ejpam-5279	236	9	=	=	PUNCT
ejpam-5279	237	1	ν	ν	NOUN
ejpam-5279	237	2	◦	◦	NOUN
ejpam-5279	237	3	lx−1(y	lx−1(y	PROPN
ejpam-5279	237	4	)	)	PUNCT
ejpam-5279	237	5	,	,	PUNCT
ejpam-5279	237	6	for	for	ADP
ejpam-5279	237	7	any	any	DET
ejpam-5279	237	8	y	y	PROPN
ejpam-5279	237	9	∈	∈	PROPN
ejpam-5279	237	10	x.	x.	NOUN
ejpam-5279	237	11	thus	thus	ADV
ejpam-5279	237	12	by	by	ADP
ejpam-5279	237	13	proposition	proposition	NOUN
ejpam-5279	237	14	1	1	NUM
ejpam-5279	237	15	,	,	PUNCT
ejpam-5279	237	16	we	we	PRON
ejpam-5279	237	17	have	have	VERB
ejpam-5279	237	18	ω′	ω′	NUM
ejpam-5279	237	19	x	x	NOUN
ejpam-5279	238	1	=	=	SYM
ejpam-5279	238	2	ωx	ωx	PROPN
ejpam-5279	238	3	.	.	NOUN
ejpam-5279	238	4	4	4	NUM
ejpam-5279	238	5	.	.	X
ejpam-5279	238	6	characterization	characterization	NOUN
ejpam-5279	238	7	of	of	ADP
ejpam-5279	238	8	approach	approach	NOUN
ejpam-5279	238	9	groups	group	NOUN
ejpam-5279	238	10	by	by	ADP
ejpam-5279	238	11	neighborhood	neighborhood	NOUN
ejpam-5279	238	12	systems	system	NOUN
ejpam-5279	238	13	definition	definition	NOUN
ejpam-5279	238	14	9	9	NUM
ejpam-5279	238	15	.	.	PUNCT
ejpam-5279	239	1	let	let	VERB
ejpam-5279	239	2	(	(	PUNCT
ejpam-5279	239	3	x	x	NOUN
ejpam-5279	239	4	,	,	PUNCT
ejpam-5279	239	5	·	·	PUNCT
ejpam-5279	239	6	)	)	PUNCT
ejpam-5279	239	7	be	be	AUX
ejpam-5279	239	8	a	a	DET
ejpam-5279	239	9	group	group	NOUN
ejpam-5279	239	10	,	,	PUNCT
ejpam-5279	239	11	(	(	PUNCT
ejpam-5279	239	12	x	x	NOUN
ejpam-5279	239	13	,	,	PUNCT
ejpam-5279	239	14	λ	λ	NOUN
ejpam-5279	239	15	)	)	PUNCT
ejpam-5279	239	16	be	be	VERB
ejpam-5279	239	17	an	an	DET
ejpam-5279	239	18	approach	approach	NOUN
ejpam-5279	239	19	space	space	NOUN
ejpam-5279	239	20	,	,	PUNCT
ejpam-5279	239	21	and	and	CCONJ
ejpam-5279	239	22	u	u	X
ejpam-5279	239	23	=	=	PUNCT
ejpam-5279	239	24	(	(	PUNCT
ejpam-5279	239	25	ux	ux	PROPN
ejpam-5279	239	26	α)x∈x	α)x∈x	PROPN
ejpam-5279	239	27	,	,	PUNCT
ejpam-5279	239	28	α∈[0,∞	α∈[0,∞	NUM
ejpam-5279	239	29	]	]	PUNCT
ejpam-5279	239	30	be	be	AUX
ejpam-5279	239	31	a	a	DET
ejpam-5279	239	32	corresponding	correspond	VERB
ejpam-5279	239	33	neighborhood	neighborhood	NOUN
ejpam-5279	239	34	system	system	NOUN
ejpam-5279	239	35	of	of	ADP
ejpam-5279	239	36	the	the	DET
ejpam-5279	239	37	approach	approach	NOUN
ejpam-5279	239	38	space	space	NOUN
ejpam-5279	239	39	(	(	PUNCT
ejpam-5279	239	40	x	x	X
ejpam-5279	239	41	,	,	PUNCT
ejpam-5279	239	42	λ	λ	NOUN
ejpam-5279	239	43	)	)	PUNCT
ejpam-5279	239	44	.	.	PUNCT
ejpam-5279	240	1	then	then	ADV
ejpam-5279	240	2	the	the	DET
ejpam-5279	240	3	triple	triple	ADJ
ejpam-5279	240	4	(	(	PUNCT
ejpam-5279	240	5	x	x	NOUN
ejpam-5279	240	6	,	,	PUNCT
ejpam-5279	240	7	·	·	PUNCT
ejpam-5279	240	8	,	,	PUNCT
ejpam-5279	240	9	u	u	NOUN
ejpam-5279	240	10	=	=	PUNCT
ejpam-5279	240	11	(	(	PUNCT
ejpam-5279	240	12	ux	ux	PROPN
ejpam-5279	240	13	α)x∈x	α)x∈x	PROPN
ejpam-5279	240	14	,	,	PUNCT
ejpam-5279	240	15	α∈[0,∞	α∈[0,∞	NUM
ejpam-5279	240	16	]	]	PUNCT
ejpam-5279	240	17	)	)	PUNCT
ejpam-5279	240	18	is	be	AUX
ejpam-5279	240	19	called	call	VERB
ejpam-5279	240	20	a	a	DET
ejpam-5279	240	21	neighborhood	neighborhood	NOUN
ejpam-5279	240	22	approach	approach	NOUN
ejpam-5279	240	23	group	group	NOUN
ejpam-5279	240	24	if	if	SCONJ
ejpam-5279	240	25	and	and	CCONJ
ejpam-5279	240	26	only	only	ADV
ejpam-5279	240	27	if	if	SCONJ
ejpam-5279	240	28	the	the	DET
ejpam-5279	240	29	following	follow	VERB
ejpam-5279	240	30	are	be	AUX
ejpam-5279	240	31	fulfilled	fulfil	VERB
ejpam-5279	240	32	:	:	PUNCT
ejpam-5279	240	33	(	(	PUNCT
ejpam-5279	240	34	nagm	nagm	PROPN
ejpam-5279	240	35	)	)	PUNCT
ejpam-5279	240	36	uxy	uxy	VERB
ejpam-5279	240	37	α∨β	α∨β	VERB
ejpam-5279	240	38	≤	≤	NUM
ejpam-5279	241	1	ux	ux	ADP
ejpam-5279	241	2	α	α	PROPN
ejpam-5279	241	3	⊙	⊙	VERB
ejpam-5279	241	4	uy	uy	PROPN
ejpam-5279	241	5	β	β	PROPN
ejpam-5279	241	6	,	,	PUNCT
ejpam-5279	241	7	∀x	∀x	NUM
ejpam-5279	241	8	,	,	PUNCT
ejpam-5279	241	9	y	y	PROPN
ejpam-5279	241	10	∈	∈	PROPN
ejpam-5279	241	11	x	x	X
ejpam-5279	241	12	and	and	CCONJ
ejpam-5279	241	13	∀α	∀α	NOUN
ejpam-5279	241	14	,	,	PUNCT
ejpam-5279	241	15	β	β	X
ejpam-5279	241	16	∈	∈	PROPN
ejpam-5279	242	1	[	[	X
ejpam-5279	242	2	0,∞	0,∞	X
ejpam-5279	242	3	]	]	PUNCT
ejpam-5279	242	4	.	.	PUNCT
ejpam-5279	243	1	(	(	PUNCT
ejpam-5279	243	2	nagi	nagi	PROPN
ejpam-5279	243	3	)	)	PUNCT
ejpam-5279	243	4	ux−1	ux−1	NOUN
ejpam-5279	243	5	α	α	NOUN
ejpam-5279	243	6	≤	≤	NOUN
ejpam-5279	243	7	(	(	PUNCT
ejpam-5279	243	8	ux	ux	NOUN
ejpam-5279	243	9	α	α	NOUN
ejpam-5279	243	10	)	)	PUNCT
ejpam-5279	243	11	−1	−1	NOUN
ejpam-5279	243	12	.	.	PUNCT
ejpam-5279	244	1	theorem	theorem	ADJ
ejpam-5279	244	2	5	5	NUM
ejpam-5279	244	3	.	.	PUNCT
ejpam-5279	245	1	let	let	VERB
ejpam-5279	245	2	(	(	PUNCT
ejpam-5279	245	3	x	x	NOUN
ejpam-5279	245	4	,	,	PUNCT
ejpam-5279	245	5	·	·	PUNCT
ejpam-5279	245	6	)	)	PUNCT
ejpam-5279	245	7	be	be	AUX
ejpam-5279	245	8	a	a	DET
ejpam-5279	245	9	group	group	NOUN
ejpam-5279	245	10	and	and	CCONJ
ejpam-5279	245	11	u	u	NOUN
ejpam-5279	246	1	=	=	PUNCT
ejpam-5279	246	2	(	(	PUNCT
ejpam-5279	246	3	ux	ux	INTJ
ejpam-5279	246	4	α)α∈[0,∞],x∈x	α)α∈[0,∞],x∈x	PROPN
ejpam-5279	246	5	be	be	VERB
ejpam-5279	246	6	the	the	DET
ejpam-5279	246	7	neighborhood	neighborhood	NOUN
ejpam-5279	246	8	approach	approach	NOUN
ejpam-5279	246	9	system	system	NOUN
ejpam-5279	246	10	corresponding	correspond	VERB
ejpam-5279	246	11	to	to	PART
ejpam-5279	246	12	approach	approach	VERB
ejpam-5279	246	13	space	space	NOUN
ejpam-5279	246	14	(	(	PUNCT
ejpam-5279	246	15	x	x	X
ejpam-5279	246	16	,	,	PUNCT
ejpam-5279	246	17	λ	λ	NOUN
ejpam-5279	246	18	)	)	PUNCT
ejpam-5279	246	19	.	.	PUNCT
ejpam-5279	247	1	then	then	ADV
ejpam-5279	247	2	(	(	PUNCT
ejpam-5279	247	3	x	x	X
ejpam-5279	247	4	,	,	PUNCT
ejpam-5279	247	5	·	·	PUNCT
ejpam-5279	247	6	,	,	PUNCT
ejpam-5279	247	7	u	u	NOUN
ejpam-5279	247	8	=	=	PUNCT
ejpam-5279	247	9	(	(	PUNCT
ejpam-5279	247	10	ux	ux	INTJ
ejpam-5279	247	11	α)α∈[0,∞],x∈x	α)α∈[0,∞],x∈x	PROPN
ejpam-5279	247	12	)	)	PUNCT
ejpam-5279	247	13	is	be	AUX
ejpam-5279	247	14	a	a	DET
ejpam-5279	247	15	neighborhood	neighborhood	NOUN
ejpam-5279	247	16	approach	approach	NOUN
ejpam-5279	247	17	group	group	NOUN
ejpam-5279	247	18	if	if	SCONJ
ejpam-5279	247	19	and	and	CCONJ
ejpam-5279	247	20	only	only	ADV
ejpam-5279	247	21	if	if	SCONJ
ejpam-5279	247	22	the	the	DET
ejpam-5279	247	23	following	follow	VERB
ejpam-5279	247	24	axioms	axiom	NOUN
ejpam-5279	247	25	are	be	AUX
ejpam-5279	247	26	fulfilled	fulfil	VERB
ejpam-5279	247	27	.	.	PUNCT
ejpam-5279	248	1	(	(	PUNCT
ejpam-5279	248	2	1	1	X
ejpam-5279	248	3	)	)	PUNCT
ejpam-5279	248	4	ue	ue	PROPN
ejpam-5279	248	5	α	α	PROPN
ejpam-5279	248	6	∈	∈	PROPN
ejpam-5279	248	7	f(x	f(x	PROPN
ejpam-5279	248	8	)	)	PUNCT
ejpam-5279	248	9	,	,	PUNCT
ejpam-5279	248	10	∀α	∀α	VERB
ejpam-5279	248	11	∈	∈	PROPN
ejpam-5279	249	1	[	[	X
ejpam-5279	249	2	0	0	NUM
ejpam-5279	249	3	,	,	PUNCT
ejpam-5279	249	4	1	1	NUM
ejpam-5279	249	5	]	]	PUNCT
ejpam-5279	249	6	;	;	PUNCT
ejpam-5279	249	7	(	(	PUNCT
ejpam-5279	249	8	2	2	X
ejpam-5279	249	9	)	)	PUNCT
ejpam-5279	249	10	ue	ue	PROPN
ejpam-5279	249	11	α	α	PROPN
ejpam-5279	249	12	≤	≤	PROPN
ejpam-5279	249	13	ė	ė	PROPN
ejpam-5279	249	14	,	,	PUNCT
ejpam-5279	249	15	∀α	∀α	VERB
ejpam-5279	249	16	∈	∈	PROPN
ejpam-5279	250	1	[	[	X
ejpam-5279	250	2	0,∞	0,∞	X
ejpam-5279	250	3	]	]	X
ejpam-5279	250	4	;	;	PUNCT
ejpam-5279	250	5	(	(	PUNCT
ejpam-5279	250	6	3	3	X
ejpam-5279	250	7	)	)	PUNCT
ejpam-5279	250	8	ue	ue	PROPN
ejpam-5279	250	9	α+β	α+β	PROPN
ejpam-5279	250	10	≤	≤	NUM
ejpam-5279	250	11	κ	κ	PROPN
ejpam-5279	250	12	(	(	PUNCT
ejpam-5279	250	13	ue	ue	INTJ
ejpam-5279	250	14	β	β	X
ejpam-5279	250	15	,	,	PUNCT
ejpam-5279	250	16	(	(	PUNCT
ejpam-5279	250	17	u	u	NOUN
ejpam-5279	250	18	y	y	PROPN
ejpam-5279	250	19	α)y∈x	α)y∈x	PROPN
ejpam-5279	250	20	)	)	PUNCT
ejpam-5279	250	21	,	,	PUNCT
ejpam-5279	250	22	∀α	∀α	NOUN
ejpam-5279	250	23	,	,	PUNCT
ejpam-5279	250	24	β	β	X
ejpam-5279	250	25	∈	∈	NOUN
ejpam-5279	251	1	[	[	X
ejpam-5279	251	2	0,∞	0,∞	X
ejpam-5279	251	3	]	]	X
ejpam-5279	251	4	;	;	PUNCT
ejpam-5279	251	5	(	(	PUNCT
ejpam-5279	251	6	4	4	X
ejpam-5279	251	7	)	)	PUNCT
ejpam-5279	251	8	if	if	SCONJ
ejpam-5279	251	9	0	0	NUM
ejpam-5279	251	10	≤	≤	NUM
ejpam-5279	251	11	α	α	NOUN
ejpam-5279	251	12	≤	≤	NOUN
ejpam-5279	251	13	β	β	NOUN
ejpam-5279	251	14	,	,	PUNCT
ejpam-5279	251	15	then	then	ADV
ejpam-5279	251	16	ue	ue	INTJ
ejpam-5279	251	17	β	β	X
ejpam-5279	251	18	≤	≤	NUM
ejpam-5279	251	19	ue	ue	PROPN
ejpam-5279	251	20	α	α	X
ejpam-5279	251	21	;	;	PUNCT
ejpam-5279	251	22	(	(	PUNCT
ejpam-5279	251	23	5	5	X
ejpam-5279	251	24	)	)	PUNCT
ejpam-5279	251	25	ue	ue	PROPN
ejpam-5279	251	26	α	α	PROPN
ejpam-5279	251	27	=	=	PUNCT
ejpam-5279	251	28	∨	∨	NUM
ejpam-5279	251	29	α	α	X
ejpam-5279	251	30	<	<	X
ejpam-5279	251	31	β	β	X
ejpam-5279	251	32	ue	ue	PROPN
ejpam-5279	251	33	β	β	PROPN
ejpam-5279	251	34	;	;	PUNCT
ejpam-5279	251	35	(	(	PUNCT
ejpam-5279	251	36	6	6	X
ejpam-5279	251	37	)	)	PUNCT
ejpam-5279	251	38	ue	ue	PROPN
ejpam-5279	251	39	α∨β	α∨β	PROPN
ejpam-5279	251	40	≤	≤	PUNCT
ejpam-5279	251	41	ue	ue	INTJ
ejpam-5279	251	42	α	α	PROPN
ejpam-5279	251	43	⊙	⊙	PROPN
ejpam-5279	251	44	ue	ue	PROPN
ejpam-5279	252	1	β	β	PROPN
ejpam-5279	252	2	,	,	PUNCT
ejpam-5279	252	3	∀α	∀α	NOUN
ejpam-5279	252	4	,	,	PUNCT
ejpam-5279	252	5	β	β	X
ejpam-5279	252	6	∈	∈	PROPN
ejpam-5279	253	1	[	[	X
ejpam-5279	253	2	0,∞	0,∞	X
ejpam-5279	253	3	]	]	X
ejpam-5279	253	4	;	;	PUNCT
ejpam-5279	253	5	(	(	PUNCT
ejpam-5279	253	6	7	7	X
ejpam-5279	253	7	)	)	PUNCT
ejpam-5279	253	8	ue	ue	PROPN
ejpam-5279	253	9	α	α	PROPN
ejpam-5279	253	10	≤	≤	PROPN
ejpam-5279	253	11	(	(	PUNCT
ejpam-5279	253	12	ue	ue	PROPN
ejpam-5279	253	13	α	α	NOUN
ejpam-5279	253	14	)	)	PUNCT
ejpam-5279	253	15	−1	−1	NOUN
ejpam-5279	253	16	;	;	PUNCT
ejpam-5279	253	17	(	(	PUNCT
ejpam-5279	253	18	8)	8)	NUM
ejpam-5279	253	19	∀α	∀α	NOUN
ejpam-5279	253	20	∈	∈	NOUN
ejpam-5279	254	1	[	[	X
ejpam-5279	254	2	0	0	NUM
ejpam-5279	254	3	,	,	PUNCT
ejpam-5279	254	4	1	1	NUM
ejpam-5279	254	5	]	]	PUNCT
ejpam-5279	254	6	,	,	PUNCT
ejpam-5279	254	7	∀x	∀x	VERB
ejpam-5279	254	8	∈	∈	PROPN
ejpam-5279	254	9	x	x	X
ejpam-5279	254	10	:	:	PUNCT
ejpam-5279	254	11	ux	ux	PROPN
ejpam-5279	254	12	α	α	PROPN
ejpam-5279	254	13	=	=	PUNCT
ejpam-5279	255	1	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	255	2	ue	ue	PROPN
ejpam-5279	255	3	α	α	PROPN
ejpam-5279	255	4	=	=	PUNCT
ejpam-5279	255	5	ue	ue	PROPN
ejpam-5279	255	6	α	α	PROPN
ejpam-5279	255	7	⊙	⊙	PROPN
ejpam-5279	256	1	ẋ.	ẋ.	PROPN
ejpam-5279	256	2	proof	proof	NOUN
ejpam-5279	256	3	.	.	PUNCT
ejpam-5279	257	1	let	let	VERB
ejpam-5279	257	2	(	(	PUNCT
ejpam-5279	257	3	x	x	X
ejpam-5279	257	4	,	,	PUNCT
ejpam-5279	257	5	·	·	PUNCT
ejpam-5279	257	6	,	,	PUNCT
ejpam-5279	257	7	u	u	NOUN
ejpam-5279	257	8	=	=	PUNCT
ejpam-5279	257	9	(	(	PUNCT
ejpam-5279	257	10	ux	ux	INTJ
ejpam-5279	257	11	α)α∈[0,∞],x∈x	α)α∈[0,∞],x∈x	PROPN
ejpam-5279	257	12	)	)	PUNCT
ejpam-5279	257	13	be	be	VERB
ejpam-5279	257	14	a	a	DET
ejpam-5279	257	15	neighborhood	neighborhood	NOUN
ejpam-5279	257	16	approach	approach	NOUN
ejpam-5279	257	17	group	group	NOUN
ejpam-5279	257	18	.	.	PUNCT
ejpam-5279	258	1	then	then	ADV
ejpam-5279	258	2	conditions	condition	NOUN
ejpam-5279	258	3	(	(	PUNCT
ejpam-5279	258	4	1)-(7	1)-(7	NUM
ejpam-5279	258	5	)	)	PUNCT
ejpam-5279	258	6	follow	follow	VERB
ejpam-5279	258	7	immediately	immediately	ADV
ejpam-5279	258	8	.	.	PUNCT
ejpam-5279	259	1	we	we	PRON
ejpam-5279	259	2	prove	prove	VERB
ejpam-5279	259	3	only	only	ADV
ejpam-5279	259	4	(	(	PUNCT
ejpam-5279	259	5	8)	8)	NUM
ejpam-5279	259	6	.	.	PUNCT
ejpam-5279	260	1	since	since	SCONJ
ejpam-5279	260	2	ẋ	ẋ	PROPN
ejpam-5279	260	3	≥	≥	PROPN
ejpam-5279	260	4	ux	ux	PROPN
ejpam-5279	260	5	α	α	X
ejpam-5279	260	6	,	,	PUNCT
ejpam-5279	260	7	we	we	PRON
ejpam-5279	260	8	have	have	VERB
ejpam-5279	260	9	ẋ	ẋ	PROPN
ejpam-5279	260	10	⊙	⊙	PROPN
ejpam-5279	260	11	ue	ue	PROPN
ejpam-5279	261	1	α	α	PROPN
ejpam-5279	261	2	≥	≥	NUM
ejpam-5279	261	3	ux	ux	INTJ
ejpam-5279	261	4	α	α	PROPN
ejpam-5279	261	5	⊙	⊙	PROPN
ejpam-5279	262	1	ue	ue	PROPN
ejpam-5279	262	2	α	α	PROPN
ejpam-5279	262	3	≥	≥	NOUN
ejpam-5279	262	4	uxe	uxe	PROPN
ejpam-5279	262	5	α∨α	α∨α	NOUN
ejpam-5279	262	6	=	=	SYM
ejpam-5279	262	7	ux	ux	PROPN
ejpam-5279	262	8	α	α	X
ejpam-5279	262	9	.	.	PUNCT
ejpam-5279	263	1	next	next	ADV
ejpam-5279	263	2	,	,	PUNCT
ejpam-5279	263	3	we	we	PRON
ejpam-5279	263	4	have	have	VERB
ejpam-5279	263	5	:	:	PUNCT
ejpam-5279	264	1	ux	ux	ADV
ejpam-5279	264	2	α	α	PROPN
ejpam-5279	264	3	=	=	SYM
ejpam-5279	264	4	ė	ė	PROPN
ejpam-5279	264	5	⊙	⊙	PROPN
ejpam-5279	264	6	ux	ux	PROPN
ejpam-5279	265	1	α	α	PROPN
ejpam-5279	265	2	=	=	SYM
ejpam-5279	266	1	(	(	PUNCT
ejpam-5279	266	2	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	266	3	(	(	PUNCT
ejpam-5279	266	4	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	266	5	)	)	PUNCT
ejpam-5279	266	6	⊙	⊙	PROPN
ejpam-5279	267	1	ux	ux	PROPN
ejpam-5279	267	2	α	α	PROPN
ejpam-5279	267	3	=	=	SYM
ejpam-5279	267	4	(	(	PUNCT
ejpam-5279	267	5	ẋ)⊙	ẋ)⊙	PROPN
ejpam-5279	267	6	(	(	PUNCT
ejpam-5279	267	7	(	(	PUNCT
ejpam-5279	267	8	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	267	9	⊙	⊙	PROPN
ejpam-5279	267	10	ux	ux	PROPN
ejpam-5279	267	11	α	α	PROPN
ejpam-5279	267	12	)	)	PUNCT
ejpam-5279	267	13	≥	≥	NOUN
ejpam-5279	267	14	ẋ⊙	ẋ⊙	INTJ
ejpam-5279	267	15	(	(	PUNCT
ejpam-5279	267	16	(	(	PUNCT
ejpam-5279	267	17	ux	ux	NOUN
ejpam-5279	267	18	α	α	NOUN
ejpam-5279	267	19	)	)	PUNCT
ejpam-5279	267	20	−1	−1	NOUN
ejpam-5279	268	1	⊙	⊙	PROPN
ejpam-5279	268	2	ux	ux	PROPN
ejpam-5279	268	3	α	α	PROPN
ejpam-5279	268	4	)	)	PUNCT
ejpam-5279	268	5	≥	≥	NOUN
ejpam-5279	268	6	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	269	1	(	(	PUNCT
ejpam-5279	269	2	ux−1	ux−1	PROPN
ejpam-5279	269	3	α	α	PROPN
ejpam-5279	269	4	⊙	⊙	PROPN
ejpam-5279	269	5	ux	ux	PROPN
ejpam-5279	269	6	α	α	PROPN
ejpam-5279	269	7	)	)	PUNCT
ejpam-5279	269	8	≥	≥	PROPN
ejpam-5279	269	9	ẋ⊙ux−1x	ẋ⊙ux−1x	NOUN
ejpam-5279	269	10	α∨α	α∨α	PROPN
ejpam-5279	269	11	=	=	PUNCT
ejpam-5279	269	12	ẋ⊙ue	ẋ⊙ue	PROPN
ejpam-5279	270	1	α	α	NOUN
ejpam-5279	270	2	.	.	PUNCT
ejpam-5279	271	1	this	this	PRON
ejpam-5279	271	2	ends	end	VERB
ejpam-5279	271	3	the	the	DET
ejpam-5279	271	4	proof	proof	NOUN
ejpam-5279	271	5	that	that	SCONJ
ejpam-5279	271	6	ux	ux	PROPN
ejpam-5279	271	7	α	α	PROPN
ejpam-5279	271	8	=	=	SYM
ejpam-5279	271	9	ẋ	ẋ	PROPN
ejpam-5279	271	10	⊙	⊙	PROPN
ejpam-5279	271	11	ue	ue	PROPN
ejpam-5279	272	1	α	α	PROPN
ejpam-5279	272	2	.	.	PUNCT
ejpam-5279	273	1	similarly	similarly	ADV
ejpam-5279	273	2	,	,	PUNCT
ejpam-5279	273	3	one	one	PRON
ejpam-5279	273	4	can	can	AUX
ejpam-5279	273	5	obtain	obtain	VERB
ejpam-5279	273	6	the	the	DET
ejpam-5279	273	7	right	right	ADJ
ejpam-5279	273	8	part	part	NOUN
ejpam-5279	273	9	.	.	PUNCT
ejpam-5279	274	1	hence	hence	ADV
ejpam-5279	274	2	the	the	DET
ejpam-5279	274	3	results	result	NOUN
ejpam-5279	274	4	follows	follow	VERB
ejpam-5279	274	5	.	.	PUNCT
ejpam-5279	275	1	conversely	conversely	ADV
ejpam-5279	275	2	,	,	PUNCT
ejpam-5279	275	3	assume	assume	VERB
ejpam-5279	275	4	that	that	SCONJ
ejpam-5279	275	5	all	all	DET
ejpam-5279	275	6	the	the	DET
ejpam-5279	275	7	conditions	condition	NOUN
ejpam-5279	275	8	(	(	PUNCT
ejpam-5279	275	9	1)-(8	1)-(8	NUM
ejpam-5279	275	10	)	)	PUNCT
ejpam-5279	275	11	are	be	AUX
ejpam-5279	275	12	true	true	ADJ
ejpam-5279	275	13	.	.	PUNCT
ejpam-5279	276	1	we	we	PRON
ejpam-5279	276	2	need	need	VERB
ejpam-5279	276	3	to	to	PART
ejpam-5279	276	4	show	show	VERB
ejpam-5279	276	5	that	that	SCONJ
ejpam-5279	276	6	(	(	PUNCT
ejpam-5279	276	7	x	x	X
ejpam-5279	276	8	,	,	PUNCT
ejpam-5279	276	9	·	·	PUNCT
ejpam-5279	276	10	,	,	PUNCT
ejpam-5279	276	11	u	u	NOUN
ejpam-5279	276	12	=	=	PUNCT
ejpam-5279	276	13	(	(	PUNCT
ejpam-5279	276	14	ux	ux	INTJ
ejpam-5279	276	15	α)α∈[0,∞],x∈x	α)α∈[0,∞],x∈x	PROPN
ejpam-5279	276	16	)	)	PUNCT
ejpam-5279	276	17	is	be	AUX
ejpam-5279	276	18	a	a	DET
ejpam-5279	276	19	neighborhood	neighborhood	NOUN
ejpam-5279	276	20	approach	approach	NOUN
ejpam-5279	276	21	group	group	NOUN
ejpam-5279	276	22	.	.	PUNCT
ejpam-5279	277	1	as	as	ADP
ejpam-5279	277	2	t.m.g	t.m.g	ADJ
ejpam-5279	277	3	.	.	PUNCT
ejpam-5279	278	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	278	2	,	,	PUNCT
ejpam-5279	278	3	fawzi	fawzi	PROPN
ejpam-5279	278	4	al	al	PROPN
ejpam-5279	278	5	-	-	PUNCT
ejpam-5279	278	6	thukair	thukair	NOUN
ejpam-5279	278	7	/	/	SYM
ejpam-5279	278	8	eur	eur	NOUN
ejpam-5279	278	9	.	.	PUNCT
ejpam-5279	279	1	j.	j.	PROPN
ejpam-5279	279	2	pure	pure	PROPN
ejpam-5279	279	3	appl	appl	PROPN
ejpam-5279	279	4	.	.	PROPN
ejpam-5279	279	5	math	math	PROPN
ejpam-5279	279	6	,	,	PUNCT
ejpam-5279	279	7	17	17	NUM
ejpam-5279	279	8	(	(	PUNCT
ejpam-5279	279	9	3	3	NUM
ejpam-5279	279	10	)	)	PUNCT
ejpam-5279	279	11	(	(	PUNCT
ejpam-5279	279	12	2024	2024	NUM
ejpam-5279	279	13	)	)	PUNCT
ejpam-5279	279	14	,	,	PUNCT
ejpam-5279	279	15	1762	1762	NUM
ejpam-5279	279	16	-	-	SYM
ejpam-5279	279	17	1778	1778	NUM
ejpam-5279	279	18	1770	1770	NUM
ejpam-5279	279	19	it	it	PRON
ejpam-5279	279	20	is	be	AUX
ejpam-5279	279	21	already	already	ADV
ejpam-5279	279	22	a	a	DET
ejpam-5279	279	23	neighborhood	neighborhood	NOUN
ejpam-5279	279	24	approach	approach	NOUN
ejpam-5279	279	25	space	space	NOUN
ejpam-5279	280	1	,	,	PUNCT
ejpam-5279	280	2	we	we	PRON
ejpam-5279	280	3	first	first	ADV
ejpam-5279	280	4	prove	prove	VERB
ejpam-5279	280	5	the	the	DET
ejpam-5279	280	6	condition	condition	NOUN
ejpam-5279	280	7	(	(	PUNCT
ejpam-5279	280	8	nagm	nagm	PROPN
ejpam-5279	280	9	)	)	PUNCT
ejpam-5279	280	10	.	.	PUNCT
ejpam-5279	281	1	let	let	VERB
ejpam-5279	281	2	α	α	PRON
ejpam-5279	281	3	,	,	PUNCT
ejpam-5279	281	4	β	β	X
ejpam-5279	281	5	∈	∈	PROPN
ejpam-5279	282	1	[	[	X
ejpam-5279	282	2	0,∞	0,∞	X
ejpam-5279	282	3	]	]	PUNCT
ejpam-5279	282	4	and	and	CCONJ
ejpam-5279	282	5	x	x	X
ejpam-5279	282	6	,	,	PUNCT
ejpam-5279	282	7	y	y	PROPN
ejpam-5279	282	8	∈	∈	PROPN
ejpam-5279	282	9	x.	x.	NOUN
ejpam-5279	282	10	then	then	ADV
ejpam-5279	282	11	by	by	ADP
ejpam-5279	282	12	using	use	VERB
ejpam-5279	282	13	lemma	lemma	PROPN
ejpam-5279	282	14	1	1	NUM
ejpam-5279	282	15	repeatedly	repeatedly	ADV
ejpam-5279	282	16	we	we	PRON
ejpam-5279	282	17	get	get	VERB
ejpam-5279	282	18	:	:	PUNCT
ejpam-5279	282	19	uxy	uxy	PROPN
ejpam-5279	283	1	α∨β	α∨β	PROPN
ejpam-5279	283	2	=	=	SYM
ejpam-5279	283	3	˙̂xy	˙̂xy	PROPN
ejpam-5279	283	4	⊙	⊙	PROPN
ejpam-5279	283	5	ue	ue	PROPN
ejpam-5279	283	6	α∨β	α∨β	PROPN
ejpam-5279	283	7	=	=	SYM
ejpam-5279	283	8	(	(	PUNCT
ejpam-5279	283	9	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	283	10	ẏ)⊙	ẏ)⊙	NUM
ejpam-5279	283	11	ue	ue	PROPN
ejpam-5279	283	12	α∨β	α∨β	PROPN
ejpam-5279	283	13	=	=	SYM
ejpam-5279	283	14	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	283	15	(	(	PUNCT
ejpam-5279	283	16	ẏ	ẏ	PROPN
ejpam-5279	283	17	⊙	⊙	PROPN
ejpam-5279	283	18	ue	ue	PROPN
ejpam-5279	283	19	α∨β	α∨β	PROPN
ejpam-5279	283	20	)	)	PUNCT
ejpam-5279	283	21	≤	≤	NOUN
ejpam-5279	284	1	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	284	2	(	(	PUNCT
ejpam-5279	284	3	ẏ	ẏ	PROPN
ejpam-5279	284	4	⊙	⊙	PROPN
ejpam-5279	284	5	(	(	PUNCT
ejpam-5279	284	6	ue	ue	INTJ
ejpam-5279	284	7	α	α	PROPN
ejpam-5279	284	8	⊙	⊙	PROPN
ejpam-5279	284	9	ue	ue	PROPN
ejpam-5279	284	10	β	β	PROPN
ejpam-5279	284	11	)	)	PUNCT
ejpam-5279	284	12	)	)	PUNCT
ejpam-5279	284	13	(	(	PUNCT
ejpam-5279	284	14	by	by	ADP
ejpam-5279	284	15	(	(	PUNCT
ejpam-5279	284	16	6	6	NUM
ejpam-5279	284	17	)	)	PUNCT
ejpam-5279	284	18	)	)	PUNCT
ejpam-5279	285	1	=	=	SYM
ejpam-5279	286	1	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	286	2	(	(	PUNCT
ejpam-5279	286	3	(	(	PUNCT
ejpam-5279	286	4	ẏ	ẏ	PROPN
ejpam-5279	286	5	⊙	⊙	VERB
ejpam-5279	286	6	ue	ue	PROPN
ejpam-5279	286	7	α)⊙	α)⊙	PROPN
ejpam-5279	286	8	ue	ue	PROPN
ejpam-5279	286	9	β	β	NOUN
ejpam-5279	286	10	)	)	PUNCT
ejpam-5279	287	1	=	=	SYM
ejpam-5279	288	1	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	288	2	(	(	PUNCT
ejpam-5279	288	3	(	(	PUNCT
ejpam-5279	288	4	ue	ue	INTJ
ejpam-5279	288	5	α	α	PROPN
ejpam-5279	288	6	⊙	⊙	PROPN
ejpam-5279	288	7	ẏ)⊙	ẏ)⊙	PROPN
ejpam-5279	288	8	ue	ue	PROPN
ejpam-5279	288	9	β	β	PROPN
ejpam-5279	288	10	)	)	PUNCT
ejpam-5279	288	11	(	(	PUNCT
ejpam-5279	288	12	applying	apply	VERB
ejpam-5279	288	13	(	(	PUNCT
ejpam-5279	288	14	8)	8)	NUM
ejpam-5279	288	15	)	)	PUNCT
ejpam-5279	288	16	=	=	SYM
ejpam-5279	289	1	(	(	PUNCT
ejpam-5279	289	2	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	289	3	ue	ue	ADJ
ejpam-5279	289	4	α)⊙	α)⊙	PROPN
ejpam-5279	289	5	(	(	PUNCT
ejpam-5279	289	6	ẏ	ẏ	PROPN
ejpam-5279	289	7	⊙	⊙	VERB
ejpam-5279	289	8	ue	ue	PROPN
ejpam-5279	289	9	β	β	PROPN
ejpam-5279	289	10	)	)	PUNCT
ejpam-5279	290	1	=	=	PUNCT
ejpam-5279	290	2	ux	ux	PROPN
ejpam-5279	291	1	α	α	PROPN
ejpam-5279	291	2	⊙	⊙	VERB
ejpam-5279	291	3	uy	uy	PROPN
ejpam-5279	292	1	β	β	PROPN
ejpam-5279	292	2	(	(	PUNCT
ejpam-5279	292	3	again	again	ADV
ejpam-5279	292	4	by	by	ADP
ejpam-5279	292	5	applying	apply	VERB
ejpam-5279	292	6	(	(	PUNCT
ejpam-5279	292	7	8)	8)	NUM
ejpam-5279	292	8	)	)	PUNCT
ejpam-5279	292	9	.	.	PUNCT
ejpam-5279	293	1	to	to	PART
ejpam-5279	293	2	prove	prove	VERB
ejpam-5279	293	3	(	(	PUNCT
ejpam-5279	293	4	nagi	nagi	PROPN
ejpam-5279	293	5	)	)	PUNCT
ejpam-5279	293	6	,	,	PUNCT
ejpam-5279	293	7	note	note	VERB
ejpam-5279	293	8	that	that	SCONJ
ejpam-5279	293	9	by	by	ADP
ejpam-5279	293	10	applying	apply	VERB
ejpam-5279	293	11	(	(	PUNCT
ejpam-5279	293	12	7	7	NUM
ejpam-5279	293	13	)	)	PUNCT
ejpam-5279	293	14	,	,	PUNCT
ejpam-5279	293	15	(	(	PUNCT
ejpam-5279	293	16	8)	8)	NUM
ejpam-5279	293	17	and	and	CCONJ
ejpam-5279	293	18	lemma	lemma	PROPN
ejpam-5279	293	19	1	1	NUM
ejpam-5279	293	20	,	,	PUNCT
ejpam-5279	293	21	we	we	PRON
ejpam-5279	293	22	have	have	VERB
ejpam-5279	293	23	:	:	PUNCT
ejpam-5279	293	24	(	(	PUNCT
ejpam-5279	293	25	ux	ux	PROPN
ejpam-5279	293	26	α	α	X
ejpam-5279	293	27	)	)	PUNCT
ejpam-5279	293	28	−1	−1	NOUN
ejpam-5279	293	29	=	=	SYM
ejpam-5279	294	1	(	(	PUNCT
ejpam-5279	294	2	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	294	3	ue	ue	PROPN
ejpam-5279	294	4	α	α	NOUN
ejpam-5279	294	5	)	)	PUNCT
ejpam-5279	294	6	−1	−1	NOUN
ejpam-5279	294	7	=	=	SYM
ejpam-5279	294	8	(	(	PUNCT
ejpam-5279	294	9	ue	ue	PROPN
ejpam-5279	294	10	α	α	NOUN
ejpam-5279	294	11	)	)	PUNCT
ejpam-5279	294	12	−1	−1	NOUN
ejpam-5279	294	13	⊙	⊙	NOUN
ejpam-5279	294	14	(	(	PUNCT
ejpam-5279	294	15	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	294	16	≥	≥	PROPN
ejpam-5279	294	17	ue	ue	PROPN
ejpam-5279	295	1	α	α	PROPN
ejpam-5279	295	2	⊙	⊙	PROPN
ejpam-5279	295	3	(	(	PUNCT
ejpam-5279	295	4	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	295	5	=	=	SYM
ejpam-5279	295	6	ux−1	ux−1	PROPN
ejpam-5279	295	7	α	α	NOUN
ejpam-5279	295	8	.	.	PUNCT
ejpam-5279	296	1	5	5	X
ejpam-5279	296	2	.	.	X
ejpam-5279	296	3	ultra	ultra	ADJ
ejpam-5279	296	4	approach	approach	NOUN
ejpam-5279	296	5	limit	limit	NOUN
ejpam-5279	296	6	group	group	NOUN
ejpam-5279	296	7	and	and	CCONJ
ejpam-5279	296	8	its	its	PRON
ejpam-5279	296	9	relationship	relationship	NOUN
ejpam-5279	296	10	with	with	ADP
ejpam-5279	296	11	neighborhood	neighborhood	NOUN
ejpam-5279	296	12	approach	approach	NOUN
ejpam-5279	296	13	group	group	NOUN
ejpam-5279	296	14	definition	definition	NOUN
ejpam-5279	296	15	10	10	NUM
ejpam-5279	296	16	.	.	PUNCT
ejpam-5279	297	1	[	[	X
ejpam-5279	297	2	4	4	X
ejpam-5279	297	3	]	]	X
ejpam-5279	297	4	let	let	VERB
ejpam-5279	297	5	(	(	PUNCT
ejpam-5279	297	6	x	x	NOUN
ejpam-5279	297	7	,	,	PUNCT
ejpam-5279	297	8	·	·	PUNCT
ejpam-5279	297	9	)	)	PUNCT
ejpam-5279	297	10	be	be	AUX
ejpam-5279	297	11	a	a	DET
ejpam-5279	297	12	group	group	NOUN
ejpam-5279	297	13	and	and	CCONJ
ejpam-5279	297	14	(	(	PUNCT
ejpam-5279	297	15	x	x	NOUN
ejpam-5279	297	16	,	,	PUNCT
ejpam-5279	297	17	λ	λ	NOUN
ejpam-5279	297	18	)	)	PUNCT
ejpam-5279	297	19	be	be	VERB
ejpam-5279	297	20	an	an	DET
ejpam-5279	297	21	ultra	ultra	ADJ
ejpam-5279	297	22	approach	approach	NOUN
ejpam-5279	297	23	limit	limit	NOUN
ejpam-5279	297	24	space	space	NOUN
ejpam-5279	297	25	.	.	PUNCT
ejpam-5279	298	1	we	we	PRON
ejpam-5279	298	2	call	call	VERB
ejpam-5279	298	3	the	the	DET
ejpam-5279	298	4	triple	triple	ADJ
ejpam-5279	298	5	(	(	PUNCT
ejpam-5279	298	6	x	x	NOUN
ejpam-5279	298	7	,	,	PUNCT
ejpam-5279	298	8	·	·	PUNCT
ejpam-5279	298	9	,	,	PUNCT
ejpam-5279	298	10	λ	λ	PROPN
ejpam-5279	298	11	)	)	PUNCT
ejpam-5279	298	12	an	an	DET
ejpam-5279	298	13	ultra	ultra	ADJ
ejpam-5279	298	14	-	-	ADJ
ejpam-5279	298	15	approach	approach	ADJ
ejpam-5279	298	16	limit	limit	NOUN
ejpam-5279	298	17	group	group	NOUN
ejpam-5279	298	18	if	if	SCONJ
ejpam-5279	298	19	the	the	DET
ejpam-5279	298	20	following	follow	VERB
ejpam-5279	298	21	axioms	axiom	NOUN
ejpam-5279	298	22	are	be	AUX
ejpam-5279	298	23	satisfied	satisfied	ADJ
ejpam-5279	298	24	:	:	PUNCT
ejpam-5279	298	25	(	(	PUNCT
ejpam-5279	298	26	ualm	ualm	PROPN
ejpam-5279	298	27	)	)	PUNCT
ejpam-5279	298	28	∀f	∀f	PROPN
ejpam-5279	298	29	,	,	PUNCT
ejpam-5279	298	30	g	g	PROPN
ejpam-5279	298	31	∈	∈	PROPN
ejpam-5279	298	32	f(x	f(x	PROPN
ejpam-5279	298	33	)	)	PUNCT
ejpam-5279	298	34	,	,	PUNCT
ejpam-5279	298	35	x	x	X
ejpam-5279	298	36	,	,	PUNCT
ejpam-5279	298	37	y	y	PROPN
ejpam-5279	298	38	∈	∈	PROPN
ejpam-5279	298	39	x	x	X
ejpam-5279	298	40	:	:	PUNCT
ejpam-5279	298	41	λ(f⊙g)(xy	λ(f⊙g)(xy	NOUN
ejpam-5279	298	42	)	)	PUNCT
ejpam-5279	298	43	≤	≤	NOUN
ejpam-5279	298	44	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	298	45	)	)	PUNCT
ejpam-5279	298	46	∨	∨	NOUN
ejpam-5279	298	47	λ(g)(y	λ(g)(y	NUM
ejpam-5279	298	48	)	)	PUNCT
ejpam-5279	298	49	.	.	PUNCT
ejpam-5279	299	1	(	(	PUNCT
ejpam-5279	299	2	uali	uali	PROPN
ejpam-5279	299	3	)	)	PUNCT
ejpam-5279	299	4	∀f	∀f	PROPN
ejpam-5279	299	5	∈	∈	PROPN
ejpam-5279	299	6	f(x	f(x	PROPN
ejpam-5279	299	7	)	)	PUNCT
ejpam-5279	299	8	,	,	PUNCT
ejpam-5279	299	9	x	x	PUNCT
ejpam-5279	299	10	∈	∈	NOUN
ejpam-5279	299	11	x	x	X
ejpam-5279	299	12	:	:	PUNCT
ejpam-5279	299	13	λ(f−1)(x−1	λ(f−1)(x−1	X
ejpam-5279	299	14	)	)	PUNCT
ejpam-5279	299	15	≤	≤	NUM
ejpam-5279	299	16	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	299	17	)	)	PUNCT
ejpam-5279	299	18	.	.	PUNCT
ejpam-5279	300	1	one	one	PRON
ejpam-5279	300	2	can	can	AUX
ejpam-5279	300	3	notice	notice	VERB
ejpam-5279	300	4	from	from	ADP
ejpam-5279	300	5	[	[	X
ejpam-5279	300	6	4	4	X
ejpam-5279	300	7	]	]	PUNCT
ejpam-5279	300	8	that	that	SCONJ
ejpam-5279	300	9	the	the	DET
ejpam-5279	300	10	conditions	condition	NOUN
ejpam-5279	300	11	(	(	PUNCT
ejpam-5279	300	12	ualm	ualm	PROPN
ejpam-5279	300	13	)	)	PUNCT
ejpam-5279	300	14	and	and	CCONJ
ejpam-5279	300	15	(	(	PUNCT
ejpam-5279	300	16	uali	uali	PROPN
ejpam-5279	300	17	)	)	PUNCT
ejpam-5279	300	18	can	can	AUX
ejpam-5279	300	19	be	be	AUX
ejpam-5279	300	20	replaced	replace	VERB
ejpam-5279	300	21	by	by	ADP
ejpam-5279	300	22	a	a	DET
ejpam-5279	300	23	single	single	ADJ
ejpam-5279	300	24	condition	condition	NOUN
ejpam-5279	300	25	,	,	PUNCT
ejpam-5279	300	26	i.e.	i.e.	X
ejpam-5279	300	27	,	,	PUNCT
ejpam-5279	300	28	for	for	ADP
ejpam-5279	300	29	all	all	DET
ejpam-5279	300	30	f	f	NOUN
ejpam-5279	300	31	,	,	PUNCT
ejpam-5279	300	32	g	g	PROPN
ejpam-5279	300	33	∈	∈	PROPN
ejpam-5279	300	34	f(x	f(x	PROPN
ejpam-5279	300	35	)	)	PUNCT
ejpam-5279	300	36	,	,	PUNCT
ejpam-5279	300	37	x	x	X
ejpam-5279	300	38	,	,	PUNCT
ejpam-5279	300	39	y	y	PROPN
ejpam-5279	300	40	∈	∈	PROPN
ejpam-5279	300	41	x	x	X
ejpam-5279	300	42	:	:	PUNCT
ejpam-5279	300	43	λ(f⊙g)(xy−1	λ(f⊙g)(xy−1	ADJ
ejpam-5279	300	44	)	)	PUNCT
ejpam-5279	300	45	≤	≤	NUM
ejpam-5279	300	46	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	300	47	)	)	PUNCT
ejpam-5279	300	48	∨	∨	NOUN
ejpam-5279	300	49	λ(g)(y	λ(g)(y	NUM
ejpam-5279	300	50	)	)	PUNCT
ejpam-5279	300	51	.	.	PUNCT
ejpam-5279	301	1	theorem	theorem	VERB
ejpam-5279	301	2	6	6	NUM
ejpam-5279	301	3	.	.	PUNCT
ejpam-5279	302	1	if	if	SCONJ
ejpam-5279	302	2	(	(	PUNCT
ejpam-5279	302	3	x	x	X
ejpam-5279	302	4	,	,	PUNCT
ejpam-5279	302	5	·	·	PUNCT
ejpam-5279	302	6	,	,	PUNCT
ejpam-5279	302	7	λ	λ	X
ejpam-5279	302	8	)	)	PUNCT
ejpam-5279	302	9	is	be	AUX
ejpam-5279	302	10	an	an	DET
ejpam-5279	302	11	ultra	ultra	ADJ
ejpam-5279	302	12	approach	approach	NOUN
ejpam-5279	302	13	limit	limit	NOUN
ejpam-5279	302	14	group	group	NOUN
ejpam-5279	302	15	,	,	PUNCT
ejpam-5279	302	16	then	then	ADV
ejpam-5279	302	17	(	(	PUNCT
ejpam-5279	302	18	x	x	NOUN
ejpam-5279	302	19	,	,	PUNCT
ejpam-5279	302	20	·	·	PUNCT
ejpam-5279	302	21	,	,	PUNCT
ejpam-5279	302	22	uλ	uλ	ADP
ejpam-5279	302	23	=	=	SYM
ejpam-5279	302	24	(	(	PUNCT
ejpam-5279	302	25	ux	ux	INTJ
ejpam-5279	302	26	α)α∈[0,∞],x∈x	α)α∈[0,∞],x∈x	PROPN
ejpam-5279	302	27	)	)	PUNCT
ejpam-5279	302	28	is	be	AUX
ejpam-5279	302	29	a	a	DET
ejpam-5279	302	30	neighborhood	neighborhood	NOUN
ejpam-5279	302	31	approach	approach	NOUN
ejpam-5279	302	32	group	group	NOUN
ejpam-5279	302	33	,	,	PUNCT
ejpam-5279	302	34	where	where	SCONJ
ejpam-5279	302	35	ux	ux	ADV
ejpam-5279	302	36	α	α	NOUN
ejpam-5279	302	37	=	=	SYM
ejpam-5279	302	38	∧	∧	PROPN
ejpam-5279	302	39	{	{	PUNCT
ejpam-5279	302	40	f	f	PROPN
ejpam-5279	302	41	∈	∈	PROPN
ejpam-5279	302	42	f(x	f(x	PROPN
ejpam-5279	302	43	)	)	PUNCT
ejpam-5279	302	44	|λ(f)(x	|λ(f)(x	NOUN
ejpam-5279	302	45	)	)	PUNCT
ejpam-5279	302	46	≤	≤	NOUN
ejpam-5279	302	47	α	α	NUM
ejpam-5279	302	48	}	}	PUNCT
ejpam-5279	302	49	,	,	PUNCT
ejpam-5279	302	50	for	for	ADP
ejpam-5279	302	51	any	any	DET
ejpam-5279	302	52	α	α	NOUN
ejpam-5279	302	53	∈	∈	PROPN
ejpam-5279	303	1	[	[	X
ejpam-5279	303	2	0,∞	0,∞	X
ejpam-5279	303	3	]	]	PUNCT
ejpam-5279	303	4	and	and	CCONJ
ejpam-5279	303	5	x	x	PUNCT
ejpam-5279	303	6	∈	∈	NOUN
ejpam-5279	303	7	x.	x.	NOUN
ejpam-5279	303	8	conversely	conversely	ADV
ejpam-5279	303	9	,	,	PUNCT
ejpam-5279	303	10	if	if	SCONJ
ejpam-5279	303	11	(	(	PUNCT
ejpam-5279	303	12	x	x	X
ejpam-5279	303	13	,	,	PUNCT
ejpam-5279	303	14	·	·	PUNCT
ejpam-5279	303	15	,	,	PUNCT
ejpam-5279	303	16	u	u	NOUN
ejpam-5279	303	17	=	=	PUNCT
ejpam-5279	303	18	(	(	PUNCT
ejpam-5279	303	19	ux	ux	INTJ
ejpam-5279	303	20	α)α∈[0,∞],x∈x	α)α∈[0,∞],x∈x	PROPN
ejpam-5279	303	21	)	)	PUNCT
ejpam-5279	303	22	is	be	AUX
ejpam-5279	303	23	a	a	DET
ejpam-5279	303	24	neighborhood	neighborhood	NOUN
ejpam-5279	303	25	approach	approach	NOUN
ejpam-5279	303	26	group	group	NOUN
ejpam-5279	303	27	,	,	PUNCT
ejpam-5279	303	28	then	then	ADV
ejpam-5279	303	29	(	(	PUNCT
ejpam-5279	303	30	x	x	X
ejpam-5279	303	31	,	,	PUNCT
ejpam-5279	303	32	·	·	PUNCT
ejpam-5279	303	33	,	,	PUNCT
ejpam-5279	303	34	λu	λu	X
ejpam-5279	303	35	)	)	PUNCT
ejpam-5279	303	36	is	be	AUX
ejpam-5279	303	37	an	an	DET
ejpam-5279	303	38	ultra	ultra	ADJ
ejpam-5279	303	39	-	-	ADJ
ejpam-5279	303	40	approach	approach	ADJ
ejpam-5279	303	41	limit	limit	NOUN
ejpam-5279	303	42	group	group	NOUN
ejpam-5279	303	43	,	,	PUNCT
ejpam-5279	303	44	where	where	SCONJ
ejpam-5279	303	45	λu	λu	PROPN
ejpam-5279	303	46	(	(	PUNCT
ejpam-5279	303	47	f)(x	f)(x	PROPN
ejpam-5279	303	48	)	)	PUNCT
ejpam-5279	304	1	=	=	SYM
ejpam-5279	305	1	∧	∧	NOUN
ejpam-5279	305	2	{	{	PUNCT
ejpam-5279	305	3	α	α	NOUN
ejpam-5279	305	4	∈	∈	PROPN
ejpam-5279	306	1	[	[	X
ejpam-5279	306	2	0,∞	0,∞	X
ejpam-5279	306	3	]	]	X
ejpam-5279	306	4	|ux	|ux	NUM
ejpam-5279	306	5	α	α	NOUN
ejpam-5279	306	6	≤	≤	NOUN
ejpam-5279	307	1	f	f	X
ejpam-5279	307	2	}	}	PUNCT
ejpam-5279	307	3	,	,	PUNCT
ejpam-5279	307	4	for	for	ADP
ejpam-5279	307	5	any	any	DET
ejpam-5279	307	6	f	f	PROPN
ejpam-5279	307	7	∈	∈	PROPN
ejpam-5279	307	8	f(x	f(x	PROPN
ejpam-5279	307	9	)	)	PUNCT
ejpam-5279	307	10	,	,	PUNCT
ejpam-5279	307	11	and	and	CCONJ
ejpam-5279	307	12	x	x	PUNCT
ejpam-5279	307	13	∈	∈	NOUN
ejpam-5279	307	14	x.	x.	NOUN
ejpam-5279	307	15	proof	proof	NOUN
ejpam-5279	307	16	.	.	PUNCT
ejpam-5279	308	1	let	let	AUX
ejpam-5279	308	2	(	(	PUNCT
ejpam-5279	308	3	x	x	NOUN
ejpam-5279	308	4	,	,	PUNCT
ejpam-5279	308	5	·	·	PUNCT
ejpam-5279	308	6	,	,	PUNCT
ejpam-5279	308	7	λ	λ	X
ejpam-5279	308	8	)	)	PUNCT
ejpam-5279	308	9	be	be	VERB
ejpam-5279	308	10	an	an	DET
ejpam-5279	308	11	ultra	ultra	ADJ
ejpam-5279	308	12	approach	approach	NOUN
ejpam-5279	308	13	limit	limit	NOUN
ejpam-5279	308	14	group	group	NOUN
ejpam-5279	308	15	.	.	PUNCT
ejpam-5279	309	1	for	for	ADP
ejpam-5279	309	2	α	α	NOUN
ejpam-5279	309	3	,	,	PUNCT
ejpam-5279	309	4	β	β	X
ejpam-5279	309	5	∈	∈	PROPN
ejpam-5279	310	1	[	[	X
ejpam-5279	310	2	0,∞	0,∞	X
ejpam-5279	310	3	]	]	PUNCT
ejpam-5279	310	4	,	,	PUNCT
ejpam-5279	310	5	and	and	CCONJ
ejpam-5279	310	6	x	x	X
ejpam-5279	310	7	,	,	PUNCT
ejpam-5279	310	8	y	y	PROPN
ejpam-5279	310	9	∈	∈	PROPN
ejpam-5279	310	10	x	x	X
ejpam-5279	310	11	,	,	PUNCT
ejpam-5279	310	12	we	we	PRON
ejpam-5279	310	13	put	put	VERB
ejpam-5279	310	14	f	f	NOUN
ejpam-5279	310	15	=	=	SYM
ejpam-5279	310	16	ux	ux	PROPN
ejpam-5279	310	17	α	α	NOUN
ejpam-5279	310	18	and	and	CCONJ
ejpam-5279	310	19	g	g	NOUN
ejpam-5279	310	20	=	=	X
ejpam-5279	310	21	uy	uy	PROPN
ejpam-5279	310	22	β	β	NOUN
ejpam-5279	310	23	.	.	PUNCT
ejpam-5279	311	1	then	then	ADV
ejpam-5279	311	2	λ	λ	INTJ
ejpam-5279	311	3	(	(	PUNCT
ejpam-5279	311	4	ux	ux	INTJ
ejpam-5279	311	5	α	α	PROPN
ejpam-5279	311	6	⊙	⊙	VERB
ejpam-5279	311	7	uy	uy	PROPN
ejpam-5279	311	8	β	β	PROPN
ejpam-5279	311	9	)	)	PUNCT
ejpam-5279	312	1	(	(	PUNCT
ejpam-5279	312	2	xy	xy	NOUN
ejpam-5279	312	3	)	)	PUNCT
ejpam-5279	312	4	=	=	SYM
ejpam-5279	312	5	λ	λ	X
ejpam-5279	312	6	(	(	PUNCT
ejpam-5279	312	7	f⊙g	f⊙g	NOUN
ejpam-5279	312	8	)	)	PUNCT
ejpam-5279	312	9	(	(	PUNCT
ejpam-5279	312	10	xy	xy	NOUN
ejpam-5279	312	11	)	)	PUNCT
ejpam-5279	312	12	≤	≤	NOUN
ejpam-5279	312	13	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	312	14	)	)	PUNCT
ejpam-5279	312	15	∨	∨	NOUN
ejpam-5279	312	16	λ(g)(y	λ(g)(y	X
ejpam-5279	312	17	)	)	PUNCT
ejpam-5279	312	18	(	(	PUNCT
ejpam-5279	312	19	by	by	ADP
ejpam-5279	312	20	(	(	PUNCT
ejpam-5279	312	21	ualm	ualm	PROPN
ejpam-5279	312	22	)	)	PUNCT
ejpam-5279	312	23	)	)	PUNCT
ejpam-5279	312	24	≤	≤	NUM
ejpam-5279	313	1	α	α	PRON
ejpam-5279	313	2	∨	∨	X
ejpam-5279	313	3	β	β	X
ejpam-5279	313	4	.	.	PUNCT
ejpam-5279	314	1	this	this	PRON
ejpam-5279	314	2	implies	imply	VERB
ejpam-5279	314	3	that	that	SCONJ
ejpam-5279	314	4	λ	λ	PROPN
ejpam-5279	314	5	(	(	PUNCT
ejpam-5279	314	6	ux	ux	INTJ
ejpam-5279	314	7	α	α	PROPN
ejpam-5279	314	8	⊙	⊙	VERB
ejpam-5279	314	9	uy	uy	PROPN
ejpam-5279	314	10	β	β	PROPN
ejpam-5279	314	11	)	)	PUNCT
ejpam-5279	314	12	(	(	PUNCT
ejpam-5279	314	13	xy	xy	NOUN
ejpam-5279	314	14	)	)	PUNCT
ejpam-5279	314	15	≤	≤	NOUN
ejpam-5279	314	16	α	α	PROPN
ejpam-5279	314	17	∨	∨	PROPN
ejpam-5279	314	18	β	β	X
ejpam-5279	314	19	which	which	PRON
ejpam-5279	314	20	in	in	ADP
ejpam-5279	314	21	view	view	NOUN
ejpam-5279	314	22	of	of	ADP
ejpam-5279	314	23	remark	remark	NOUN
ejpam-5279	314	24	1	1	NUM
ejpam-5279	314	25	yields	yield	NOUN
ejpam-5279	314	26	that	that	PRON
ejpam-5279	314	27	ux	ux	PROPN
ejpam-5279	314	28	α	α	PROPN
ejpam-5279	314	29	⊙	⊙	PROPN
ejpam-5279	314	30	t.m.g	t.m.g	PROPN
ejpam-5279	314	31	.	.	PUNCT
ejpam-5279	315	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	315	2	,	,	PUNCT
ejpam-5279	315	3	fawzi	fawzi	PROPN
ejpam-5279	315	4	al	al	PROPN
ejpam-5279	315	5	-	-	PUNCT
ejpam-5279	315	6	thukair	thukair	NOUN
ejpam-5279	315	7	/	/	SYM
ejpam-5279	315	8	eur	eur	NOUN
ejpam-5279	315	9	.	.	PUNCT
ejpam-5279	316	1	j.	j.	PROPN
ejpam-5279	316	2	pure	pure	PROPN
ejpam-5279	316	3	appl	appl	PROPN
ejpam-5279	316	4	.	.	PROPN
ejpam-5279	316	5	math	math	PROPN
ejpam-5279	316	6	,	,	PUNCT
ejpam-5279	316	7	17	17	NUM
ejpam-5279	316	8	(	(	PUNCT
ejpam-5279	316	9	3	3	NUM
ejpam-5279	316	10	)	)	PUNCT
ejpam-5279	316	11	(	(	PUNCT
ejpam-5279	316	12	2024	2024	NUM
ejpam-5279	316	13	)	)	PUNCT
ejpam-5279	316	14	,	,	PUNCT
ejpam-5279	316	15	1762	1762	NUM
ejpam-5279	316	16	-	-	SYM
ejpam-5279	316	17	1778	1778	NUM
ejpam-5279	316	18	1771	1771	NUM
ejpam-5279	316	19	uy	uy	NOUN
ejpam-5279	316	20	β	β	PROPN
ejpam-5279	316	21	≥	≥	PROPN
ejpam-5279	316	22	uxy	uxy	PROPN
ejpam-5279	316	23	α∨β	α∨β	VERB
ejpam-5279	316	24	,	,	PUNCT
ejpam-5279	316	25	i.e.	i.e.	X
ejpam-5279	316	26	,	,	PUNCT
ejpam-5279	316	27	the	the	DET
ejpam-5279	316	28	condition	condition	NOUN
ejpam-5279	316	29	(	(	PUNCT
ejpam-5279	316	30	nagm	nagm	PROPN
ejpam-5279	316	31	)	)	PUNCT
ejpam-5279	316	32	is	be	AUX
ejpam-5279	316	33	proved	prove	VERB
ejpam-5279	316	34	.	.	PUNCT
ejpam-5279	317	1	the	the	DET
ejpam-5279	317	2	condition	condition	NOUN
ejpam-5279	317	3	(	(	PUNCT
ejpam-5279	317	4	nagi	nagi	PROPN
ejpam-5279	317	5	)	)	PUNCT
ejpam-5279	317	6	is	be	AUX
ejpam-5279	317	7	an	an	DET
ejpam-5279	317	8	immediate	immediate	ADJ
ejpam-5279	317	9	consequence	consequence	NOUN
ejpam-5279	317	10	of	of	ADP
ejpam-5279	317	11	lemma	lemma	PROPN
ejpam-5279	317	12	3.7	3.7	NUM
ejpam-5279	318	1	[	[	X
ejpam-5279	318	2	13	13	NUM
ejpam-5279	318	3	]	]	PUNCT
ejpam-5279	318	4	.	.	PUNCT
ejpam-5279	319	1	conversely	conversely	ADV
ejpam-5279	319	2	,	,	PUNCT
ejpam-5279	319	3	in	in	ADP
ejpam-5279	319	4	view	view	NOUN
ejpam-5279	319	5	of	of	ADP
ejpam-5279	319	6	lemma	lemma	PROPN
ejpam-5279	319	7	3.4[13	3.4[13	PROPN
ejpam-5279	319	8	]	]	PUNCT
ejpam-5279	319	9	,	,	PUNCT
ejpam-5279	319	10	we	we	PRON
ejpam-5279	319	11	only	only	ADV
ejpam-5279	319	12	show	show	VERB
ejpam-5279	319	13	condition	condition	NOUN
ejpam-5279	319	14	(	(	PUNCT
ejpam-5279	319	15	ualm	ualm	PROPN
ejpam-5279	319	16	)	)	PUNCT
ejpam-5279	319	17	.	.	PUNCT
ejpam-5279	320	1	assume	assume	VERB
ejpam-5279	320	2	that	that	SCONJ
ejpam-5279	320	3	(	(	PUNCT
ejpam-5279	320	4	nagm	nagm	PROPN
ejpam-5279	320	5	)	)	PUNCT
ejpam-5279	320	6	is	be	AUX
ejpam-5279	320	7	true	true	ADJ
ejpam-5279	320	8	.	.	PUNCT
ejpam-5279	321	1	let	let	VERB
ejpam-5279	321	2	f	f	X
ejpam-5279	321	3	,	,	PUNCT
ejpam-5279	321	4	g	g	PROPN
ejpam-5279	321	5	∈	∈	PROPN
ejpam-5279	321	6	f(x	f(x	PROPN
ejpam-5279	321	7	)	)	PUNCT
ejpam-5279	321	8	,	,	PUNCT
ejpam-5279	321	9	and	and	CCONJ
ejpam-5279	321	10	x	x	X
ejpam-5279	321	11	,	,	PUNCT
ejpam-5279	321	12	y	y	PROPN
ejpam-5279	321	13	∈	∈	PROPN
ejpam-5279	321	14	x.	x.	NOUN
ejpam-5279	322	1	then	then	ADV
ejpam-5279	322	2	λu	λu	X
ejpam-5279	322	3	(	(	PUNCT
ejpam-5279	322	4	f)(x	f)(x	PROPN
ejpam-5279	322	5	)	)	PUNCT
ejpam-5279	322	6	∨	∨	NUM
ejpam-5279	322	7	λu	λu	X
ejpam-5279	322	8	(	(	PUNCT
ejpam-5279	322	9	g)(y	g)(y	PROPN
ejpam-5279	322	10	)	)	PUNCT
ejpam-5279	323	1	=	=	PRON
ejpam-5279	323	2	(	(	PUNCT
ejpam-5279	323	3	∧	∧	PROPN
ejpam-5279	323	4	{	{	PUNCT
ejpam-5279	323	5	α	α	NOUN
ejpam-5279	323	6	∈	∈	PROPN
ejpam-5279	324	1	[	[	X
ejpam-5279	324	2	0,∞	0,∞	X
ejpam-5279	324	3	]	]	X
ejpam-5279	324	4	|ux	|ux	NUM
ejpam-5279	324	5	α	α	NOUN
ejpam-5279	324	6	≤	≤	NUM
ejpam-5279	325	1	f	f	X
ejpam-5279	325	2	}	}	PUNCT
ejpam-5279	325	3	)	)	PUNCT
ejpam-5279	325	4	∨	∨	PROPN
ejpam-5279	325	5	(	(	PUNCT
ejpam-5279	325	6	∧	∧	PROPN
ejpam-5279	325	7	{	{	PUNCT
ejpam-5279	325	8	β	β	X
ejpam-5279	325	9	∈	∈	PROPN
ejpam-5279	326	1	[	[	X
ejpam-5279	326	2	0,∞]|uy	0,∞]|uy	VERB
ejpam-5279	326	3	β	β	NOUN
ejpam-5279	326	4	≤	≤	ADV
ejpam-5279	326	5	g	g	NOUN
ejpam-5279	326	6	}	}	PUNCT
ejpam-5279	326	7	)	)	PUNCT
ejpam-5279	327	1	=	=	SYM
ejpam-5279	327	2	∧	∧	PROPN
ejpam-5279	327	3	{	{	PUNCT
ejpam-5279	327	4	α	α	PROPN
ejpam-5279	327	5	∨	∨	X
ejpam-5279	327	6	β	β	X
ejpam-5279	327	7	∈	∈	PROPN
ejpam-5279	328	1	[	[	X
ejpam-5279	328	2	0,∞	0,∞	X
ejpam-5279	328	3	]	]	PUNCT
ejpam-5279	328	4	|ux	|ux	NUM
ejpam-5279	328	5	α	α	NOUN
ejpam-5279	328	6	≤	≤	NUM
ejpam-5279	328	7	f	f	X
ejpam-5279	328	8	,	,	PUNCT
ejpam-5279	328	9	uy	uy	ADV
ejpam-5279	328	10	β	β	NOUN
ejpam-5279	328	11	≤	≤	ADJ
ejpam-5279	328	12	g	g	NOUN
ejpam-5279	328	13	}	}	PUNCT
ejpam-5279	328	14	(	(	PUNCT
ejpam-5279	328	15	by	by	ADP
ejpam-5279	328	16	using	use	VERB
ejpam-5279	328	17	lemma	lemma	PROPN
ejpam-5279	328	18	2.8[4	2.8[4	X
ejpam-5279	328	19	]	]	SYM
ejpam-5279	328	20	)	)	PUNCT
ejpam-5279	328	21	≥	≥	NOUN
ejpam-5279	329	1	∧	∧	PROPN
ejpam-5279	329	2	{	{	PUNCT
ejpam-5279	329	3	α	α	PROPN
ejpam-5279	329	4	∨	∨	X
ejpam-5279	329	5	β	β	X
ejpam-5279	329	6	∈	∈	PROPN
ejpam-5279	330	1	[	[	X
ejpam-5279	330	2	0,∞	0,∞	X
ejpam-5279	330	3	]	]	PUNCT
ejpam-5279	330	4	|uxy	|uxy	PUNCT
ejpam-5279	330	5	α∨β	α∨β	VERB
ejpam-5279	330	6	≤	≤	NOUN
ejpam-5279	330	7	f⊙g	f⊙g	NOUN
ejpam-5279	330	8	}	}	PUNCT
ejpam-5279	330	9	(	(	PUNCT
ejpam-5279	330	10	as	as	SCONJ
ejpam-5279	330	11	because	because	SCONJ
ejpam-5279	330	12	uxy	uxy	PROPN
ejpam-5279	330	13	α∨β	α∨β	VERB
ejpam-5279	330	14	≤	≤	NUM
ejpam-5279	330	15	ux	ux	ADP
ejpam-5279	331	1	α	α	PROPN
ejpam-5279	331	2	⊙	⊙	VERB
ejpam-5279	331	3	uy	uy	PROPN
ejpam-5279	331	4	β	β	PROPN
ejpam-5279	331	5	≤	≤	PROPN
ejpam-5279	331	6	f⊙g	f⊙g	CCONJ
ejpam-5279	331	7	,	,	PUNCT
ejpam-5279	331	8	and	and	CCONJ
ejpam-5279	331	9	with	with	ADP
ejpam-5279	331	10	the	the	DET
ejpam-5279	331	11	assumption	assumption	NOUN
ejpam-5279	331	12	that	that	SCONJ
ejpam-5279	331	13	f⊙g	f⊙g	NOUN
ejpam-5279	331	14	exists	exist	VERB
ejpam-5279	331	15	,	,	PUNCT
ejpam-5279	331	16	so	so	ADV
ejpam-5279	331	17	is	be	AUX
ejpam-5279	331	18	ux	ux	PROPN
ejpam-5279	331	19	α	α	PROPN
ejpam-5279	331	20	⊙	⊙	VERB
ejpam-5279	331	21	uy	uy	PROPN
ejpam-5279	331	22	β	β	X
ejpam-5279	331	23	by	by	ADP
ejpam-5279	331	24	using	use	VERB
ejpam-5279	331	25	lemma	lemma	PROPN
ejpam-5279	331	26	4.2[4	4.2[4	NOUN
ejpam-5279	331	27	]	]	X
ejpam-5279	331	28	)	)	PUNCT
ejpam-5279	332	1	=	=	SYM
ejpam-5279	332	2	λu	λu	X
ejpam-5279	332	3	(	(	PUNCT
ejpam-5279	332	4	f⊙g	f⊙g	NOUN
ejpam-5279	332	5	)	)	PUNCT
ejpam-5279	332	6	(	(	PUNCT
ejpam-5279	332	7	xy	xy	NOUN
ejpam-5279	332	8	)	)	PUNCT
ejpam-5279	332	9	,	,	PUNCT
ejpam-5279	332	10	showing	show	VERB
ejpam-5279	332	11	the	the	DET
ejpam-5279	332	12	condition	condition	NOUN
ejpam-5279	332	13	(	(	PUNCT
ejpam-5279	332	14	ualm	ualm	PROPN
ejpam-5279	332	15	)	)	PUNCT
ejpam-5279	332	16	is	be	AUX
ejpam-5279	332	17	proved	prove	VERB
ejpam-5279	332	18	.	.	PUNCT
ejpam-5279	333	1	the	the	DET
ejpam-5279	333	2	last	last	ADJ
ejpam-5279	333	3	condition	condition	NOUN
ejpam-5279	333	4	follows	follow	VERB
ejpam-5279	333	5	immediately	immediately	ADV
ejpam-5279	333	6	by	by	ADP
ejpam-5279	333	7	using	use	VERB
ejpam-5279	333	8	(	(	PUNCT
ejpam-5279	333	9	nagi	nagi	PROPN
ejpam-5279	333	10	)	)	PUNCT
ejpam-5279	333	11	coupled	couple	VERB
ejpam-5279	333	12	with	with	ADP
ejpam-5279	333	13	lemma	lemma	PROPN
ejpam-5279	333	14	3.7[13	3.7[13	NUM
ejpam-5279	333	15	]	]	PUNCT
ejpam-5279	333	16	.	.	PUNCT
ejpam-5279	334	1	corollary	corollary	ADJ
ejpam-5279	334	2	1	1	NUM
ejpam-5279	334	3	.	.	PUNCT
ejpam-5279	335	1	let	let	VERB
ejpam-5279	335	2	(	(	PUNCT
ejpam-5279	335	3	x	x	X
ejpam-5279	335	4	,	,	PUNCT
ejpam-5279	335	5	·	·	PUNCT
ejpam-5279	335	6	,	,	PUNCT
ejpam-5279	335	7	u	u	NOUN
ejpam-5279	335	8	=	=	PUNCT
ejpam-5279	335	9	(	(	PUNCT
ejpam-5279	335	10	ux	ux	INTJ
ejpam-5279	335	11	α)α∈[0,∞],x∈x	α)α∈[0,∞],x∈x	PROPN
ejpam-5279	335	12	)	)	PUNCT
ejpam-5279	335	13	be	be	VERB
ejpam-5279	335	14	a	a	DET
ejpam-5279	335	15	neighborhood	neighborhood	NOUN
ejpam-5279	335	16	approach	approach	NOUN
ejpam-5279	335	17	group	group	NOUN
ejpam-5279	335	18	.	.	PUNCT
ejpam-5279	336	1	then	then	ADV
ejpam-5279	336	2	for	for	ADP
ejpam-5279	336	3	any	any	DET
ejpam-5279	336	4	α	α	NOUN
ejpam-5279	336	5	∈	∈	PROPN
ejpam-5279	337	1	[	[	X
ejpam-5279	337	2	0,∞	0,∞	X
ejpam-5279	337	3	]	]	PUNCT
ejpam-5279	337	4	and	and	CCONJ
ejpam-5279	337	5	x	x	PUNCT
ejpam-5279	337	6	∈	∈	PROPN
ejpam-5279	337	7	x	x	NOUN
ejpam-5279	337	8	:	:	PUNCT
ejpam-5279	337	9	ux	ux	PROPN
ejpam-5279	337	10	α	α	PROPN
ejpam-5279	337	11	=	=	PUNCT
ejpam-5279	338	1	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	338	2	ue	ue	PROPN
ejpam-5279	338	3	α	α	PROPN
ejpam-5279	338	4	=	=	PUNCT
ejpam-5279	338	5	ue	ue	PROPN
ejpam-5279	338	6	α	α	PROPN
ejpam-5279	338	7	⊙	⊙	PROPN
ejpam-5279	339	1	ẋ.	ẋ.	PROPN
ejpam-5279	339	2	proof	proof	NOUN
ejpam-5279	339	3	.	.	PUNCT
ejpam-5279	340	1	let	let	VERB
ejpam-5279	340	2	x	x	PUNCT
ejpam-5279	340	3	∈	∈	PROPN
ejpam-5279	340	4	x	x	X
ejpam-5279	340	5	and	and	CCONJ
ejpam-5279	340	6	α	α	NOUN
ejpam-5279	340	7	∈	∈	PROPN
ejpam-5279	341	1	[	[	X
ejpam-5279	341	2	0,∞	0,∞	X
ejpam-5279	341	3	]	]	PUNCT
ejpam-5279	341	4	.	.	PUNCT
ejpam-5279	342	1	then	then	ADV
ejpam-5279	342	2	in	in	ADP
ejpam-5279	342	3	view	view	NOUN
ejpam-5279	342	4	of	of	ADP
ejpam-5279	342	5	the	the	DET
ejpam-5279	342	6	preceding	precede	VERB
ejpam-5279	342	7	theorem	theorem	NOUN
ejpam-5279	342	8	,	,	PUNCT
ejpam-5279	342	9	lemma	lemma	PROPN
ejpam-5279	342	10	1	1	NUM
ejpam-5279	342	11	and	and	CCONJ
ejpam-5279	342	12	lemma	lemma	PROPN
ejpam-5279	342	13	3.8[4	3.8[4	NUM
ejpam-5279	342	14	]	]	X
ejpam-5279	342	15	,	,	PUNCT
ejpam-5279	342	16	we	we	PRON
ejpam-5279	342	17	have	have	VERB
ejpam-5279	342	18	ux	ux	PROPN
ejpam-5279	342	19	α	α	NOUN
ejpam-5279	342	20	=	=	SYM
ejpam-5279	342	21	∧	∧	PROPN
ejpam-5279	342	22	{	{	PUNCT
ejpam-5279	342	23	f	f	PROPN
ejpam-5279	342	24	∈	∈	PROPN
ejpam-5279	342	25	f(x	f(x	PROPN
ejpam-5279	342	26	)	)	PUNCT
ejpam-5279	342	27	|λ(f)(x	|λ(f)(x	NOUN
ejpam-5279	342	28	)	)	PUNCT
ejpam-5279	342	29	≤	≤	NOUN
ejpam-5279	342	30	α	α	X
ejpam-5279	342	31	}	}	PUNCT
ejpam-5279	342	32	=	=	SYM
ejpam-5279	342	33	∧	∧	PROPN
ejpam-5279	342	34	{	{	PUNCT
ejpam-5279	342	35	f	f	PROPN
ejpam-5279	342	36	∈	∈	PROPN
ejpam-5279	342	37	f(x	f(x	PROPN
ejpam-5279	342	38	)	)	PUNCT
ejpam-5279	343	1	|λ	|λ	CCONJ
ejpam-5279	343	2	(	(	PUNCT
ejpam-5279	343	3	(	(	PUNCT
ejpam-5279	343	4	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	343	5	⊙	⊙	PROPN
ejpam-5279	343	6	f	f	PROPN
ejpam-5279	343	7	)	)	PUNCT
ejpam-5279	344	1	(	(	PUNCT
ejpam-5279	344	2	e	e	NOUN
ejpam-5279	344	3	)	)	PUNCT
ejpam-5279	344	4	≤	≤	NOUN
ejpam-5279	344	5	α	α	X
ejpam-5279	344	6	}	}	PUNCT
ejpam-5279	344	7	=	=	SYM
ejpam-5279	344	8	∧	∧	PROPN
ejpam-5279	344	9	{	{	PUNCT
ejpam-5279	344	10	f	f	PROPN
ejpam-5279	344	11	∈	∈	PROPN
ejpam-5279	344	12	f(x	f(x	PROPN
ejpam-5279	344	13	)	)	PUNCT
ejpam-5279	345	1	|	|	ADV
ejpam-5279	345	2	(	(	PUNCT
ejpam-5279	345	3	ẋ)−1	ẋ)−1	PROPN
ejpam-5279	345	4	⊙	⊙	PROPN
ejpam-5279	345	5	f	f	PROPN
ejpam-5279	345	6	≥	≥	PROPN
ejpam-5279	345	7	ue	ue	PROPN
ejpam-5279	345	8	α	α	X
ejpam-5279	345	9	}	}	PUNCT
ejpam-5279	345	10	(	(	PUNCT
ejpam-5279	345	11	by	by	ADP
ejpam-5279	345	12	remark	remark	NOUN
ejpam-5279	345	13	1	1	NUM
ejpam-5279	345	14	)	)	PUNCT
ejpam-5279	346	1	=	=	SYM
ejpam-5279	346	2	∧	∧	PROPN
ejpam-5279	346	3	{	{	PUNCT
ejpam-5279	346	4	f	f	PROPN
ejpam-5279	346	5	∈	∈	PROPN
ejpam-5279	346	6	f(x	f(x	PROPN
ejpam-5279	346	7	)	)	PUNCT
ejpam-5279	346	8	|f	|f	PROPN
ejpam-5279	346	9	≥	≥	PROPN
ejpam-5279	346	10	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	346	11	ue	ue	PROPN
ejpam-5279	346	12	α	α	NOUN
ejpam-5279	346	13	}	}	PUNCT
ejpam-5279	346	14	=	=	PUNCT
ejpam-5279	346	15	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	346	16	ue	ue	PROPN
ejpam-5279	347	1	α	α	PROPN
ejpam-5279	347	2	.	.	PUNCT
ejpam-5279	348	1	similarly	similarly	ADV
ejpam-5279	348	2	,	,	PUNCT
ejpam-5279	348	3	one	one	PRON
ejpam-5279	348	4	can	can	AUX
ejpam-5279	348	5	show	show	VERB
ejpam-5279	348	6	that	that	SCONJ
ejpam-5279	348	7	ux	ux	PROPN
ejpam-5279	348	8	α	α	PROPN
ejpam-5279	348	9	=	=	NOUN
ejpam-5279	348	10	ue	ue	PROPN
ejpam-5279	348	11	α	α	PROPN
ejpam-5279	348	12	⊙	⊙	PROPN
ejpam-5279	348	13	ẋ.	ẋ.	PROPN
ejpam-5279	349	1	if	if	SCONJ
ejpam-5279	349	2	we	we	PRON
ejpam-5279	349	3	denote	denote	VERB
ejpam-5279	349	4	uaplimgrp	uaplimgrp	ADJ
ejpam-5279	349	5	as	as	SCONJ
ejpam-5279	349	6	the	the	DET
ejpam-5279	349	7	category	category	NOUN
ejpam-5279	349	8	of	of	ADP
ejpam-5279	349	9	all	all	DET
ejpam-5279	349	10	ultra	ultra	ADJ
ejpam-5279	349	11	-	-	ADJ
ejpam-5279	349	12	approach	approach	ADJ
ejpam-5279	349	13	limit	limit	NOUN
ejpam-5279	349	14	groups	group	NOUN
ejpam-5279	349	15	andnapgrp	andnapgrp	VERB
ejpam-5279	349	16	,	,	PUNCT
ejpam-5279	349	17	the	the	DET
ejpam-5279	349	18	category	category	NOUN
ejpam-5279	349	19	of	of	ADP
ejpam-5279	349	20	neighborhood	neighborhood	NOUN
ejpam-5279	349	21	approach	approach	NOUN
ejpam-5279	349	22	groups	group	NOUN
ejpam-5279	349	23	associated	associate	VERB
ejpam-5279	349	24	with	with	ADP
ejpam-5279	349	25	approach	approach	NOUN
ejpam-5279	349	26	spaces	space	NOUN
ejpam-5279	349	27	,	,	PUNCT
ejpam-5279	349	28	then	then	ADV
ejpam-5279	349	29	if	if	SCONJ
ejpam-5279	349	30	follows	follow	VERB
ejpam-5279	349	31	from	from	ADP
ejpam-5279	349	32	[	[	X
ejpam-5279	349	33	13	13	NUM
ejpam-5279	349	34	]	]	PUNCT
ejpam-5279	349	35	in	in	ADP
ejpam-5279	349	36	conjunction	conjunction	NOUN
ejpam-5279	349	37	with	with	ADP
ejpam-5279	349	38	the	the	DET
ejpam-5279	349	39	lemma	lemma	PROPN
ejpam-5279	349	40	3.7[13	3.7[13	NUM
ejpam-5279	349	41	]	]	PUNCT
ejpam-5279	349	42	and	and	CCONJ
ejpam-5279	349	43	the	the	DET
ejpam-5279	349	44	theorem	theorem	NOUN
ejpam-5279	349	45	6	6	NUM
ejpam-5279	349	46	above	above	ADV
ejpam-5279	349	47	,	,	PUNCT
ejpam-5279	349	48	these	these	DET
ejpam-5279	349	49	two	two	NUM
ejpam-5279	349	50	categories	category	NOUN
ejpam-5279	349	51	are	be	AUX
ejpam-5279	349	52	isomorphic	isomorphic	ADJ
ejpam-5279	349	53	,	,	PUNCT
ejpam-5279	349	54	we	we	PRON
ejpam-5279	349	55	leave	leave	VERB
ejpam-5279	349	56	details	detail	NOUN
ejpam-5279	349	57	for	for	ADP
ejpam-5279	349	58	the	the	DET
ejpam-5279	349	59	interested	interested	ADJ
ejpam-5279	349	60	reader	reader	NOUN
ejpam-5279	349	61	.	.	PUNCT
ejpam-5279	350	1	however	however	ADV
ejpam-5279	350	2	,	,	PUNCT
ejpam-5279	350	3	the	the	DET
ejpam-5279	350	4	functors	functor	NOUN
ejpam-5279	350	5	in	in	ADP
ejpam-5279	350	6	question	question	NOUN
ejpam-5279	350	7	,	,	PUNCT
ejpam-5279	350	8	say	say	VERB
ejpam-5279	350	9	for	for	ADP
ejpam-5279	350	10	instance	instance	NOUN
ejpam-5279	350	11	,	,	PUNCT
ejpam-5279	350	12	f	f	PROPN
ejpam-5279	350	13	and	and	CCONJ
ejpam-5279	350	14	g	g	PROPN
ejpam-5279	350	15	are	be	AUX
ejpam-5279	350	16	connected	connect	VERB
ejpam-5279	350	17	as	as	SCONJ
ejpam-5279	350	18	described	describe	VERB
ejpam-5279	350	19	below	below	ADV
ejpam-5279	350	20	:	:	PUNCT
ejpam-5279	350	21	f	f	X
ejpam-5279	350	22	:	:	PUNCT
ejpam-5279	350	23			PUNCT
ejpam-5279	350	24	uaplimgrp	uaplimgrp	ADJ
ejpam-5279	350	25	−→	−→	ADJ
ejpam-5279	350	26	napgrp	napgrp	NOUN
ejpam-5279	350	27	(	(	PUNCT
ejpam-5279	350	28	x	x	X
ejpam-5279	350	29	,	,	PUNCT
ejpam-5279	350	30	·	·	PUNCT
ejpam-5279	350	31	,	,	PUNCT
ejpam-5279	350	32	λ	λ	NOUN
ejpam-5279	350	33	)	)	PUNCT
ejpam-5279	350	34	7−→	7−→	NOUN
ejpam-5279	350	35	(	(	PUNCT
ejpam-5279	350	36	x	x	NOUN
ejpam-5279	350	37	,	,	PUNCT
ejpam-5279	350	38	·	·	PUNCT
ejpam-5279	350	39	,	,	PUNCT
ejpam-5279	350	40	uλ	uλ	NOUN
ejpam-5279	350	41	)	)	PUNCT
ejpam-5279	351	1	f	f	NOUN
ejpam-5279	351	2	7−→	7−→	PROPN
ejpam-5279	351	3	f	f	PROPN
ejpam-5279	351	4	and	and	CCONJ
ejpam-5279	351	5	g	g	NOUN
ejpam-5279	351	6	:	:	PUNCT
ejpam-5279	351	7			PUNCT
ejpam-5279	351	8	napgrp	napgrp	NOUN
ejpam-5279	351	9	−→	−→	NOUN
ejpam-5279	351	10	uaplimgrp	uaplimgrp	NOUN
ejpam-5279	351	11	(	(	PUNCT
ejpam-5279	351	12	x	x	X
ejpam-5279	351	13	,	,	PUNCT
ejpam-5279	351	14	·	·	PUNCT
ejpam-5279	351	15	,	,	PUNCT
ejpam-5279	351	16	u	u	NOUN
ejpam-5279	351	17	)	)	PUNCT
ejpam-5279	351	18	7−→	7−→	NOUN
ejpam-5279	351	19	(	(	PUNCT
ejpam-5279	351	20	x	x	X
ejpam-5279	351	21	,	,	PUNCT
ejpam-5279	351	22	·	·	PUNCT
ejpam-5279	351	23	,	,	PUNCT
ejpam-5279	351	24	λu	λu	X
ejpam-5279	351	25	)	)	PUNCT
ejpam-5279	352	1	f	f	PROPN
ejpam-5279	353	1	7−→	7−→	NOUN
ejpam-5279	353	2	f	f	NOUN
ejpam-5279	353	3	6	6	NUM
ejpam-5279	353	4	.	.	PUNCT
ejpam-5279	353	5	ultra	ultra	ADJ
ejpam-5279	353	6	approach	approach	NOUN
ejpam-5279	353	7	-	-	PUNCT
ejpam-5279	353	8	cauchy	cauchy	NOUN
ejpam-5279	353	9	groups	group	NOUN
ejpam-5279	353	10	,	,	PUNCT
ejpam-5279	353	11	ultra	ultra	ADJ
ejpam-5279	353	12	approach	approach	NOUN
ejpam-5279	353	13	limit	limit	NOUN
ejpam-5279	353	14	groups	group	NOUN
ejpam-5279	353	15	and	and	CCONJ
ejpam-5279	353	16	strongly	strongly	ADV
ejpam-5279	353	17	normal	normal	ADJ
ejpam-5279	353	18	ultra	ultra	ADJ
ejpam-5279	353	19	approach	approach	NOUN
ejpam-5279	353	20	limit	limit	NOUN
ejpam-5279	353	21	in	in	ADP
ejpam-5279	353	22	the	the	DET
ejpam-5279	353	23	light	light	NOUN
ejpam-5279	353	24	of	of	ADP
ejpam-5279	353	25	the	the	DET
ejpam-5279	353	26	section	section	NOUN
ejpam-5279	353	27	5[4	5[4	NOUN
ejpam-5279	353	28	]	]	PUNCT
ejpam-5279	353	29	,	,	PUNCT
ejpam-5279	353	30	we	we	PRON
ejpam-5279	353	31	add	add	VERB
ejpam-5279	353	32	some	some	DET
ejpam-5279	353	33	results	result	NOUN
ejpam-5279	353	34	in	in	ADP
ejpam-5279	353	35	this	this	DET
ejpam-5279	353	36	section	section	NOUN
ejpam-5279	353	37	;	;	PUNCT
ejpam-5279	353	38	specifically	specifically	ADV
ejpam-5279	353	39	,	,	PUNCT
ejpam-5279	353	40	our	our	PRON
ejpam-5279	353	41	main	main	ADJ
ejpam-5279	353	42	aim	aim	NOUN
ejpam-5279	353	43	here	here	ADV
ejpam-5279	353	44	is	be	AUX
ejpam-5279	353	45	to	to	PART
ejpam-5279	353	46	show	show	VERB
ejpam-5279	353	47	that	that	SCONJ
ejpam-5279	353	48	the	the	DET
ejpam-5279	353	49	categories	category	NOUN
ejpam-5279	353	50	uapchygrp	uapchygrp	VERB
ejpam-5279	353	51	(	(	PUNCT
ejpam-5279	353	52	the	the	DET
ejpam-5279	353	53	category	category	NOUN
ejpam-5279	353	54	of	of	ADP
ejpam-5279	353	55	ultra	ultra	ADJ
ejpam-5279	353	56	approacht.m.g	approacht.m.g	NOUN
ejpam-5279	353	57	.	.	PUNCT
ejpam-5279	354	1	ahsanullah	ahsanullah	VERB
ejpam-5279	354	2	,	,	PUNCT
ejpam-5279	354	3	fawzi	fawzi	PROPN
ejpam-5279	354	4	al	al	PROPN
ejpam-5279	354	5	-	-	PUNCT
ejpam-5279	354	6	thukair	thukair	NOUN
ejpam-5279	354	7	/	/	SYM
ejpam-5279	354	8	eur	eur	NOUN
ejpam-5279	354	9	.	.	PUNCT
ejpam-5279	355	1	j.	j.	PROPN
ejpam-5279	355	2	pure	pure	PROPN
ejpam-5279	355	3	appl	appl	PROPN
ejpam-5279	355	4	.	.	PROPN
ejpam-5279	355	5	math	math	PROPN
ejpam-5279	355	6	,	,	PUNCT
ejpam-5279	355	7	17	17	NUM
ejpam-5279	355	8	(	(	PUNCT
ejpam-5279	355	9	3	3	NUM
ejpam-5279	355	10	)	)	PUNCT
ejpam-5279	355	11	(	(	PUNCT
ejpam-5279	355	12	2024	2024	NUM
ejpam-5279	355	13	)	)	PUNCT
ejpam-5279	355	14	,	,	PUNCT
ejpam-5279	355	15	1762	1762	NUM
ejpam-5279	355	16	-	-	SYM
ejpam-5279	355	17	1778	1778	NUM
ejpam-5279	355	18	1772	1772	NUM
ejpam-5279	355	19	cauchy	cauchy	NOUN
ejpam-5279	355	20	groups	group	NOUN
ejpam-5279	355	21	)	)	PUNCT
ejpam-5279	355	22	and	and	CCONJ
ejpam-5279	355	23	snuaplimgrp	snuaplimgrp	NOUN
ejpam-5279	355	24	(	(	PUNCT
ejpam-5279	355	25	the	the	DET
ejpam-5279	355	26	category	category	NOUN
ejpam-5279	355	27	of	of	ADP
ejpam-5279	355	28	strongly	strongly	ADV
ejpam-5279	355	29	normal	normal	ADJ
ejpam-5279	355	30	ultra	ultra	ADJ
ejpam-5279	355	31	approach	approach	NOUN
ejpam-5279	355	32	limit	limit	NOUN
ejpam-5279	355	33	groups	group	NOUN
ejpam-5279	355	34	)	)	PUNCT
ejpam-5279	355	35	are	be	AUX
ejpam-5279	355	36	isomorphic	isomorphic	ADJ
ejpam-5279	355	37	.	.	PUNCT
ejpam-5279	356	1	definition	definition	NOUN
ejpam-5279	356	2	11	11	NUM
ejpam-5279	356	3	.	.	PUNCT
ejpam-5279	357	1	[	[	X
ejpam-5279	357	2	17	17	NUM
ejpam-5279	357	3	]	]	PUNCT
ejpam-5279	357	4	let	let	VERB
ejpam-5279	357	5	x	x	PRON
ejpam-5279	357	6	be	be	AUX
ejpam-5279	357	7	a	a	DET
ejpam-5279	357	8	set	set	NOUN
ejpam-5279	357	9	.	.	PUNCT
ejpam-5279	358	1	a	a	DET
ejpam-5279	358	2	mapping	mapping	NOUN
ejpam-5279	358	3	υ	υ	NOUN
ejpam-5279	358	4	:	:	PUNCT
ejpam-5279	358	5	f(x	f(x	PROPN
ejpam-5279	358	6	)	)	PUNCT
ejpam-5279	359	1	−→	−→	NOUN
ejpam-5279	360	1	[	[	X
ejpam-5279	360	2	0,∞	0,∞	X
ejpam-5279	360	3	]	]	PUNCT
ejpam-5279	360	4	is	be	AUX
ejpam-5279	360	5	called	call	VERB
ejpam-5279	360	6	an	an	DET
ejpam-5279	360	7	approachcauchy	approachcauchy	ADJ
ejpam-5279	360	8	structure	structure	NOUN
ejpam-5279	360	9	if	if	SCONJ
ejpam-5279	360	10	and	and	CCONJ
ejpam-5279	360	11	only	only	ADV
ejpam-5279	360	12	if	if	SCONJ
ejpam-5279	360	13	the	the	DET
ejpam-5279	360	14	following	follow	VERB
ejpam-5279	360	15	conditions	condition	NOUN
ejpam-5279	360	16	are	be	AUX
ejpam-5279	360	17	fulfilled	fulfil	VERB
ejpam-5279	360	18	:	:	PUNCT
ejpam-5279	360	19	(	(	PUNCT
ejpam-5279	360	20	achy1	achy1	X
ejpam-5279	360	21	)	)	PUNCT
ejpam-5279	360	22	υ(ẋ	υ(ẋ	NOUN
ejpam-5279	360	23	)	)	PUNCT
ejpam-5279	360	24	=	=	SYM
ejpam-5279	360	25	0	0	NUM
ejpam-5279	360	26	;	;	PUNCT
ejpam-5279	360	27	(	(	PUNCT
ejpam-5279	360	28	achy2	achy2	NOUN
ejpam-5279	360	29	)	)	PUNCT
ejpam-5279	360	30	f	f	NOUN
ejpam-5279	360	31	≤	≤	NOUN
ejpam-5279	360	32	g	g	PROPN
ejpam-5279	360	33	implies	imply	VERB
ejpam-5279	360	34	υ(g	υ(g	NOUN
ejpam-5279	360	35	)	)	PUNCT
ejpam-5279	360	36	≤	≤	NUM
ejpam-5279	360	37	υ(f	υ(f	PROPN
ejpam-5279	360	38	)	)	PUNCT
ejpam-5279	360	39	for	for	ADP
ejpam-5279	360	40	all	all	DET
ejpam-5279	360	41	f	f	NOUN
ejpam-5279	360	42	,	,	PUNCT
ejpam-5279	360	43	g	g	PROPN
ejpam-5279	360	44	∈	∈	PROPN
ejpam-5279	360	45	f(x	f(x	PROPN
ejpam-5279	360	46	)	)	PUNCT
ejpam-5279	360	47	;	;	PUNCT
ejpam-5279	360	48	(	(	PUNCT
ejpam-5279	360	49	achy3	achy3	NOUN
ejpam-5279	360	50	)	)	PUNCT
ejpam-5279	360	51	υ(f	υ(f	PROPN
ejpam-5279	360	52	∧g	∧g	PROPN
ejpam-5279	360	53	)	)	PUNCT
ejpam-5279	360	54	≤	≤	NUM
ejpam-5279	360	55	υ(f	υ(f	PROPN
ejpam-5279	360	56	)	)	PUNCT
ejpam-5279	361	1	+	+	CCONJ
ejpam-5279	361	2	υ(g	υ(g	NOUN
ejpam-5279	361	3	)	)	PUNCT
ejpam-5279	361	4	.	.	PUNCT
ejpam-5279	362	1	then	then	ADV
ejpam-5279	362	2	the	the	DET
ejpam-5279	362	3	pair	pair	NOUN
ejpam-5279	362	4	(	(	PUNCT
ejpam-5279	362	5	x	x	NOUN
ejpam-5279	362	6	,	,	PUNCT
ejpam-5279	362	7	υ	υ	NOUN
ejpam-5279	362	8	)	)	PUNCT
ejpam-5279	362	9	is	be	AUX
ejpam-5279	362	10	called	call	VERB
ejpam-5279	362	11	an	an	DET
ejpam-5279	362	12	approach	approach	NOUN
ejpam-5279	362	13	-	-	PUNCT
ejpam-5279	362	14	cauchy	cauchy	NOUN
ejpam-5279	362	15	space	space	NOUN
ejpam-5279	362	16	.	.	PUNCT
ejpam-5279	363	1	a	a	DET
ejpam-5279	363	2	mapping	mapping	NOUN
ejpam-5279	363	3	f	f	NOUN
ejpam-5279	363	4	:	:	PUNCT
ejpam-5279	363	5	(	(	PUNCT
ejpam-5279	363	6	x	x	NOUN
ejpam-5279	363	7	,	,	PUNCT
ejpam-5279	363	8	υ	υ	NOUN
ejpam-5279	363	9	)	)	PUNCT
ejpam-5279	363	10	−→	−→	NOUN
ejpam-5279	363	11	(	(	PUNCT
ejpam-5279	363	12	x	x	SYM
ejpam-5279	363	13	′,υ′	′,υ′	PROPN
ejpam-5279	363	14	)	)	PUNCT
ejpam-5279	363	15	between	between	ADP
ejpam-5279	363	16	approach	approach	NOUN
ejpam-5279	363	17	-	-	PUNCT
ejpam-5279	363	18	cauchy	cauchy	NOUN
ejpam-5279	363	19	spaces	space	NOUN
ejpam-5279	363	20	is	be	AUX
ejpam-5279	363	21	called	call	VERB
ejpam-5279	363	22	approachcauchy	approachcauchy	ADJ
ejpam-5279	363	23	contraction	contraction	NOUN
ejpam-5279	363	24	if	if	SCONJ
ejpam-5279	363	25	and	and	CCONJ
ejpam-5279	363	26	only	only	ADV
ejpam-5279	363	27	if	if	SCONJ
ejpam-5279	363	28	for	for	ADP
ejpam-5279	363	29	all	all	DET
ejpam-5279	363	30	f	f	PROPN
ejpam-5279	363	31	∈	∈	PROPN
ejpam-5279	363	32	f(x	f(x	PROPN
ejpam-5279	363	33	)	)	PUNCT
ejpam-5279	363	34	,	,	PUNCT
ejpam-5279	363	35	υ′(f(f	υ′(f(f	NOUN
ejpam-5279	363	36	)	)	PUNCT
ejpam-5279	363	37	)	)	PUNCT
ejpam-5279	363	38	≤	≤	PUNCT
ejpam-5279	363	39	υ(f	υ(f	PROPN
ejpam-5279	363	40	)	)	PUNCT
ejpam-5279	363	41	.	.	PUNCT
ejpam-5279	364	1	the	the	DET
ejpam-5279	364	2	category	category	NOUN
ejpam-5279	364	3	of	of	ADP
ejpam-5279	364	4	all	all	DET
ejpam-5279	364	5	approach	approach	NOUN
ejpam-5279	364	6	-	-	PUNCT
ejpam-5279	364	7	cauchy	cauchy	NOUN
ejpam-5279	364	8	spaces	space	NOUN
ejpam-5279	364	9	and	and	CCONJ
ejpam-5279	364	10	approach	approach	NOUN
ejpam-5279	364	11	-	-	PUNCT
ejpam-5279	364	12	cauchy	cauchy	NOUN
ejpam-5279	364	13	contractions	contraction	NOUN
ejpam-5279	364	14	is	be	AUX
ejpam-5279	364	15	denoted	denote	VERB
ejpam-5279	364	16	by	by	ADP
ejpam-5279	364	17	apchy	apchy	ADJ
ejpam-5279	364	18	.	.	PUNCT
ejpam-5279	365	1	definition	definition	NOUN
ejpam-5279	365	2	12	12	NUM
ejpam-5279	365	3	.	.	PUNCT
ejpam-5279	366	1	let	let	VERB
ejpam-5279	366	2	(	(	PUNCT
ejpam-5279	366	3	x	x	NOUN
ejpam-5279	366	4	,	,	PUNCT
ejpam-5279	366	5	·	·	PUNCT
ejpam-5279	366	6	)	)	PUNCT
ejpam-5279	366	7	be	be	AUX
ejpam-5279	366	8	a	a	DET
ejpam-5279	366	9	group	group	NOUN
ejpam-5279	366	10	and	and	CCONJ
ejpam-5279	366	11	(	(	PUNCT
ejpam-5279	366	12	x	x	NOUN
ejpam-5279	366	13	,	,	PUNCT
ejpam-5279	366	14	υ	υ	NOUN
ejpam-5279	366	15	)	)	PUNCT
ejpam-5279	366	16	be	be	AUX
ejpam-5279	366	17	an	an	DET
ejpam-5279	366	18	approach	approach	NOUN
ejpam-5279	366	19	-	-	PUNCT
ejpam-5279	366	20	cauchy	cauchy	NOUN
ejpam-5279	366	21	space	space	NOUN
ejpam-5279	366	22	.	.	PUNCT
ejpam-5279	367	1	then	then	ADV
ejpam-5279	367	2	the	the	DET
ejpam-5279	367	3	triple	triple	ADJ
ejpam-5279	367	4	(	(	PUNCT
ejpam-5279	367	5	x	x	NOUN
ejpam-5279	367	6	,	,	PUNCT
ejpam-5279	367	7	·	·	PUNCT
ejpam-5279	367	8	,	,	PUNCT
ejpam-5279	367	9	υ	υ	NOUN
ejpam-5279	367	10	)	)	PUNCT
ejpam-5279	367	11	is	be	AUX
ejpam-5279	367	12	called	call	VERB
ejpam-5279	367	13	an	an	DET
ejpam-5279	367	14	approach	approach	NOUN
ejpam-5279	367	15	-	-	PUNCT
ejpam-5279	367	16	cauchy	cauchy	NOUN
ejpam-5279	367	17	group	group	NOUN
ejpam-5279	367	18	if	if	SCONJ
ejpam-5279	367	19	and	and	CCONJ
ejpam-5279	367	20	only	only	ADV
ejpam-5279	367	21	if	if	SCONJ
ejpam-5279	367	22	for	for	ADP
ejpam-5279	367	23	all	all	DET
ejpam-5279	367	24	f	f	NOUN
ejpam-5279	367	25	,	,	PUNCT
ejpam-5279	367	26	g	g	PROPN
ejpam-5279	367	27	∈	∈	PROPN
ejpam-5279	367	28	f(x	f(x	PROPN
ejpam-5279	367	29	):	):	PUNCT
ejpam-5279	367	30	υ(f⊙g−1	υ(f⊙g−1	X
ejpam-5279	367	31	)	)	PUNCT
ejpam-5279	367	32	≤	≤	NUM
ejpam-5279	367	33	υ(f	υ(f	PROPN
ejpam-5279	367	34	)	)	PUNCT
ejpam-5279	368	1	+	+	CCONJ
ejpam-5279	368	2	υ(g	υ(g	NOUN
ejpam-5279	368	3	)	)	PUNCT
ejpam-5279	368	4	.	.	PUNCT
ejpam-5279	369	1	the	the	DET
ejpam-5279	369	2	category	category	NOUN
ejpam-5279	369	3	of	of	ADP
ejpam-5279	369	4	all	all	DET
ejpam-5279	369	5	approach	approach	NOUN
ejpam-5279	369	6	-	-	PUNCT
ejpam-5279	369	7	cauchy	cauchy	NOUN
ejpam-5279	369	8	groups	group	NOUN
ejpam-5279	369	9	and	and	CCONJ
ejpam-5279	369	10	approach	approach	NOUN
ejpam-5279	369	11	-	-	PUNCT
ejpam-5279	369	12	cauchy	cauchy	NOUN
ejpam-5279	369	13	contractions	contraction	NOUN
ejpam-5279	369	14	which	which	PRON
ejpam-5279	369	15	are	be	AUX
ejpam-5279	369	16	homomorphisms	homomorphism	NOUN
ejpam-5279	369	17	denoted	denote	VERB
ejpam-5279	369	18	by	by	ADP
ejpam-5279	369	19	apchygrp	apchygrp	PROPN
ejpam-5279	369	20	.	.	PUNCT
ejpam-5279	370	1	definition	definition	NOUN
ejpam-5279	370	2	13	13	NUM
ejpam-5279	370	3	.	.	PUNCT
ejpam-5279	371	1	[	[	X
ejpam-5279	371	2	4	4	NUM
ejpam-5279	371	3	,	,	PUNCT
ejpam-5279	371	4	17	17	NUM
ejpam-5279	371	5	]	]	PUNCT
ejpam-5279	371	6	a	a	DET
ejpam-5279	371	7	map	map	NOUN
ejpam-5279	371	8	υ	υ	X
ejpam-5279	371	9	:	:	PUNCT
ejpam-5279	371	10	f(x	f(x	PROPN
ejpam-5279	371	11	)	)	PUNCT
ejpam-5279	371	12	→	→	PUNCT
ejpam-5279	372	1	[	[	X
ejpam-5279	372	2	0,∞	0,∞	X
ejpam-5279	372	3	]	]	PUNCT
ejpam-5279	372	4	is	be	AUX
ejpam-5279	372	5	called	call	VERB
ejpam-5279	372	6	an	an	DET
ejpam-5279	372	7	ultra	ultra	ADJ
ejpam-5279	372	8	approach	approach	NOUN
ejpam-5279	372	9	-	-	PUNCT
ejpam-5279	372	10	cauchy	cauchy	NOUN
ejpam-5279	372	11	structure	structure	NOUN
ejpam-5279	372	12	on	on	ADP
ejpam-5279	372	13	x	x	SYM
ejpam-5279	372	14	if	if	SCONJ
ejpam-5279	372	15	and	and	CCONJ
ejpam-5279	372	16	only	only	ADV
ejpam-5279	372	17	if	if	SCONJ
ejpam-5279	372	18	the	the	DET
ejpam-5279	372	19	following	follow	VERB
ejpam-5279	372	20	axioms	axiom	NOUN
ejpam-5279	372	21	are	be	AUX
ejpam-5279	372	22	fulfilled	fulfil	VERB
ejpam-5279	372	23	:	:	PUNCT
ejpam-5279	372	24	(	(	PUNCT
ejpam-5279	372	25	uachy1	uachy1	X
ejpam-5279	372	26	)	)	PUNCT
ejpam-5279	373	1	∀x	∀x	VERB
ejpam-5279	373	2	∈	∈	PROPN
ejpam-5279	373	3	x	x	NOUN
ejpam-5279	373	4	:	:	PUNCT
ejpam-5279	373	5	υ(ẋ	υ(ẋ	ADJ
ejpam-5279	373	6	)	)	PUNCT
ejpam-5279	373	7	=	=	SYM
ejpam-5279	373	8	0	0	NUM
ejpam-5279	373	9	;	;	PUNCT
ejpam-5279	373	10	(	(	PUNCT
ejpam-5279	373	11	uachy2	uachy2	NOUN
ejpam-5279	373	12	)	)	PUNCT
ejpam-5279	373	13	∀f	∀f	PROPN
ejpam-5279	373	14	,	,	PUNCT
ejpam-5279	373	15	g	g	PROPN
ejpam-5279	373	16	∈	∈	PROPN
ejpam-5279	373	17	f(x	f(x	PROPN
ejpam-5279	373	18	)	)	PUNCT
ejpam-5279	373	19	with	with	ADP
ejpam-5279	373	20	f	f	PROPN
ejpam-5279	373	21	≤	≤	PROPN
ejpam-5279	373	22	g	g	PROPN
ejpam-5279	373	23	,	,	PUNCT
ejpam-5279	373	24	υ(g	υ(g	NOUN
ejpam-5279	373	25	)	)	PUNCT
ejpam-5279	373	26	≤	≤	NUM
ejpam-5279	373	27	υ(f	υ(f	PROPN
ejpam-5279	373	28	)	)	PUNCT
ejpam-5279	373	29	,	,	PUNCT
ejpam-5279	373	30	(	(	PUNCT
ejpam-5279	373	31	uachy3	uachy3	NOUN
ejpam-5279	373	32	)	)	PUNCT
ejpam-5279	373	33	∀f	∀f	PROPN
ejpam-5279	373	34	,	,	PUNCT
ejpam-5279	373	35	g	g	PROPN
ejpam-5279	373	36	∈	∈	PROPN
ejpam-5279	373	37	f(x	f(x	PROPN
ejpam-5279	373	38	)	)	PUNCT
ejpam-5279	373	39	,	,	PUNCT
ejpam-5279	373	40	if	if	SCONJ
ejpam-5279	373	41	f	f	PROPN
ejpam-5279	373	42	∨g	∨g	PROPN
ejpam-5279	373	43	exists	exist	VERB
ejpam-5279	373	44	,	,	PUNCT
ejpam-5279	373	45	then	then	ADV
ejpam-5279	373	46	υ(f	υ(f	PROPN
ejpam-5279	373	47	∩g	∩g	PROPN
ejpam-5279	373	48	)	)	PUNCT
ejpam-5279	373	49	≤	≤	PUNCT
ejpam-5279	374	1	υ(f	υ(f	PROPN
ejpam-5279	374	2	)	)	PUNCT
ejpam-5279	374	3	∨υ(g	∨υ(g	PROPN
ejpam-5279	374	4	)	)	PUNCT
ejpam-5279	374	5	.	.	PUNCT
ejpam-5279	375	1	a	a	DET
ejpam-5279	375	2	mapping	mapping	NOUN
ejpam-5279	375	3	f	f	NOUN
ejpam-5279	375	4	:	:	PUNCT
ejpam-5279	375	5	(	(	PUNCT
ejpam-5279	375	6	x	x	NOUN
ejpam-5279	375	7	,	,	PUNCT
ejpam-5279	375	8	υ	υ	NOUN
ejpam-5279	375	9	)	)	PUNCT
ejpam-5279	375	10	−→	−→	NOUN
ejpam-5279	375	11	(	(	PUNCT
ejpam-5279	375	12	y	y	NOUN
ejpam-5279	375	13	,	,	PUNCT
ejpam-5279	375	14	υ′	υ′	NOUN
ejpam-5279	375	15	)	)	PUNCT
ejpam-5279	375	16	between	between	ADP
ejpam-5279	375	17	ultra	ultra	ADJ
ejpam-5279	375	18	approach	approach	NOUN
ejpam-5279	375	19	-	-	PUNCT
ejpam-5279	375	20	cauchy	cauchy	NOUN
ejpam-5279	375	21	spaces	space	NOUN
ejpam-5279	375	22	is	be	AUX
ejpam-5279	375	23	called	call	VERB
ejpam-5279	375	24	ultra	ultra	ADJ
ejpam-5279	375	25	approach	approach	NOUN
ejpam-5279	375	26	-	-	PUNCT
ejpam-5279	375	27	cauchy	cauchy	NOUN
ejpam-5279	375	28	contraction	contraction	NOUN
ejpam-5279	375	29	or	or	CCONJ
ejpam-5279	375	30	cauchy	cauchy	ADJ
ejpam-5279	375	31	contraction	contraction	NOUN
ejpam-5279	375	32	if	if	SCONJ
ejpam-5279	375	33	and	and	CCONJ
ejpam-5279	375	34	only	only	ADV
ejpam-5279	375	35	if	if	SCONJ
ejpam-5279	375	36	for	for	ADP
ejpam-5279	375	37	all	all	DET
ejpam-5279	375	38	f	f	PROPN
ejpam-5279	375	39	∈	∈	PROPN
ejpam-5279	375	40	f(x	f(x	PROPN
ejpam-5279	375	41	)	)	PUNCT
ejpam-5279	375	42	,	,	PUNCT
ejpam-5279	375	43	υ′(f(f	υ′(f(f	NOUN
ejpam-5279	375	44	)	)	PUNCT
ejpam-5279	375	45	)	)	PUNCT
ejpam-5279	375	46	≤	≤	PUNCT
ejpam-5279	375	47	υ(f	υ(f	PROPN
ejpam-5279	375	48	)	)	PUNCT
ejpam-5279	375	49	.	.	PUNCT
ejpam-5279	376	1	the	the	DET
ejpam-5279	376	2	category	category	NOUN
ejpam-5279	376	3	of	of	ADP
ejpam-5279	376	4	all	all	DET
ejpam-5279	376	5	ultra	ultra	ADJ
ejpam-5279	376	6	approach	approach	NOUN
ejpam-5279	376	7	-	-	PUNCT
ejpam-5279	376	8	cauchy	cauchy	NOUN
ejpam-5279	376	9	spaces	space	NOUN
ejpam-5279	376	10	and	and	CCONJ
ejpam-5279	376	11	contractions	contraction	NOUN
ejpam-5279	376	12	is	be	AUX
ejpam-5279	376	13	denoted	denote	VERB
ejpam-5279	376	14	by	by	ADP
ejpam-5279	376	15	uapchy	uapchy	NOUN
ejpam-5279	376	16	.	.	PUNCT
ejpam-5279	377	1	definition	definition	NOUN
ejpam-5279	377	2	14	14	NUM
ejpam-5279	377	3	.	.	PUNCT
ejpam-5279	378	1	let	let	VERB
ejpam-5279	378	2	υ	υ	NOUN
ejpam-5279	378	3	:	:	PUNCT
ejpam-5279	378	4	f(x	f(x	PROPN
ejpam-5279	378	5	)	)	PUNCT
ejpam-5279	378	6	→	→	PUNCT
ejpam-5279	379	1	[	[	X
ejpam-5279	379	2	0,∞	0,∞	X
ejpam-5279	379	3	]	]	PUNCT
ejpam-5279	379	4	be	be	VERB
ejpam-5279	379	5	an	an	DET
ejpam-5279	379	6	ultra	ultra	ADJ
ejpam-5279	379	7	approach	approach	NOUN
ejpam-5279	379	8	-	-	PUNCT
ejpam-5279	379	9	cauchy	cauchy	NOUN
ejpam-5279	379	10	structure	structure	NOUN
ejpam-5279	379	11	on	on	ADP
ejpam-5279	379	12	a	a	DET
ejpam-5279	379	13	group	group	NOUN
ejpam-5279	379	14	(	(	PUNCT
ejpam-5279	379	15	x	x	X
ejpam-5279	379	16	,	,	PUNCT
ejpam-5279	379	17	·	·	PUNCT
ejpam-5279	379	18	)	)	PUNCT
ejpam-5279	379	19	,	,	PUNCT
ejpam-5279	379	20	then	then	ADV
ejpam-5279	379	21	the	the	DET
ejpam-5279	379	22	triple	triple	ADJ
ejpam-5279	379	23	(	(	PUNCT
ejpam-5279	379	24	x	x	NOUN
ejpam-5279	379	25	,	,	PUNCT
ejpam-5279	379	26	·	·	PUNCT
ejpam-5279	379	27	,	,	PUNCT
ejpam-5279	379	28	υ	υ	NOUN
ejpam-5279	379	29	)	)	PUNCT
ejpam-5279	379	30	is	be	AUX
ejpam-5279	379	31	called	call	VERB
ejpam-5279	379	32	an	an	DET
ejpam-5279	379	33	ultra	ultra	ADJ
ejpam-5279	379	34	approach	approach	NOUN
ejpam-5279	379	35	-	-	PUNCT
ejpam-5279	379	36	cauchy	cauchy	NOUN
ejpam-5279	379	37	group	group	NOUN
ejpam-5279	379	38	if	if	SCONJ
ejpam-5279	379	39	and	and	CCONJ
ejpam-5279	379	40	only	only	ADV
ejpam-5279	379	41	if	if	SCONJ
ejpam-5279	379	42	the	the	DET
ejpam-5279	379	43	mapping	mapping	NOUN
ejpam-5279	379	44	h	h	NOUN
ejpam-5279	379	45	:	:	PUNCT
ejpam-5279	379	46	(	(	PUNCT
ejpam-5279	379	47	x	x	X
ejpam-5279	379	48	×x	×x	X
ejpam-5279	379	49	,	,	PUNCT
ejpam-5279	379	50	υ×υ	υ×υ	PROPN
ejpam-5279	379	51	)	)	PUNCT
ejpam-5279	379	52	→	→	SYM
ejpam-5279	379	53	(	(	PUNCT
ejpam-5279	379	54	x	x	NOUN
ejpam-5279	379	55	,	,	PUNCT
ejpam-5279	379	56	υ	υ	NOUN
ejpam-5279	379	57	)	)	PUNCT
ejpam-5279	379	58	,	,	PUNCT
ejpam-5279	379	59	(	(	PUNCT
ejpam-5279	379	60	x	x	X
ejpam-5279	379	61	,	,	PUNCT
ejpam-5279	379	62	y	y	NOUN
ejpam-5279	379	63	)	)	PUNCT
ejpam-5279	379	64	7→	7→	NUM
ejpam-5279	379	65	x−1y	x−1y	NOUN
ejpam-5279	379	66	is	be	AUX
ejpam-5279	379	67	a	a	DET
ejpam-5279	379	68	contraction	contraction	NOUN
ejpam-5279	379	69	,	,	PUNCT
ejpam-5279	379	70	(	(	PUNCT
ejpam-5279	379	71	or	or	CCONJ
ejpam-5279	379	72	,	,	PUNCT
ejpam-5279	379	73	equivalently	equivalently	ADV
ejpam-5279	379	74	,	,	PUNCT
ejpam-5279	379	75	∀f	∀f	PROPN
ejpam-5279	379	76	,	,	PUNCT
ejpam-5279	379	77	g	g	PROPN
ejpam-5279	379	78	∈	∈	PROPN
ejpam-5279	379	79	f(x	f(x	PROPN
ejpam-5279	379	80	)	)	PUNCT
ejpam-5279	379	81	,	,	PUNCT
ejpam-5279	380	1	υ	υ	PROPN
ejpam-5279	380	2	(	(	PUNCT
ejpam-5279	380	3	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	380	4	)	)	PUNCT
ejpam-5279	380	5	≤	≤	PUNCT
ejpam-5279	381	1	υ(f	υ(f	PROPN
ejpam-5279	381	2	)	)	PUNCT
ejpam-5279	381	3	∨υ(g	∨υ(g	PROPN
ejpam-5279	381	4	)	)	PUNCT
ejpam-5279	381	5	.	.	PUNCT
ejpam-5279	381	6	)	)	PUNCT
ejpam-5279	381	7	note	note	VERB
ejpam-5279	381	8	that	that	SCONJ
ejpam-5279	381	9	∀f	∀f	PROPN
ejpam-5279	381	10	∈	∈	PROPN
ejpam-5279	381	11	f(x	f(x	PROPN
ejpam-5279	381	12	)	)	PUNCT
ejpam-5279	381	13	,	,	PUNCT
ejpam-5279	381	14	υ(f−1	υ(f−1	NOUN
ejpam-5279	381	15	)	)	PUNCT
ejpam-5279	381	16	≤	≤	NUM
ejpam-5279	381	17	υ(f	υ(f	PROPN
ejpam-5279	381	18	)	)	PUNCT
ejpam-5279	381	19	,	,	PUNCT
ejpam-5279	381	20	and	and	CCONJ
ejpam-5279	381	21	∀f	∀f	PROPN
ejpam-5279	381	22	,	,	PUNCT
ejpam-5279	381	23	g	g	PROPN
ejpam-5279	381	24	∈	∈	PROPN
ejpam-5279	381	25	f(x	f(x	PROPN
ejpam-5279	381	26	)	)	PUNCT
ejpam-5279	381	27	,	,	PUNCT
ejpam-5279	381	28	υ(f⊙g	υ(f⊙g	NOUN
ejpam-5279	381	29	)	)	PUNCT
ejpam-5279	381	30	≤	≤	NOUN
ejpam-5279	381	31	υ(f)∨υ(g	υ(f)∨υ(g	NOUN
ejpam-5279	381	32	)	)	PUNCT
ejpam-5279	381	33	,	,	PUNCT
ejpam-5279	381	34	are	be	AUX
ejpam-5279	381	35	also	also	ADV
ejpam-5279	381	36	hold	hold	VERB
ejpam-5279	381	37	good	good	ADJ
ejpam-5279	381	38	.	.	PUNCT
ejpam-5279	382	1	the	the	DET
ejpam-5279	382	2	category	category	NOUN
ejpam-5279	382	3	of	of	ADP
ejpam-5279	382	4	ultra	ultra	ADJ
ejpam-5279	382	5	approach	approach	NOUN
ejpam-5279	382	6	-	-	PUNCT
ejpam-5279	382	7	cauchy	cauchy	NOUN
ejpam-5279	382	8	groups	group	NOUN
ejpam-5279	382	9	and	and	CCONJ
ejpam-5279	382	10	cauchy	cauchy	PROPN
ejpam-5279	382	11	contractive	contractive	ADJ
ejpam-5279	382	12	homomorphisms	homomorphism	NOUN
ejpam-5279	382	13	is	be	AUX
ejpam-5279	382	14	denoted	denote	VERB
ejpam-5279	382	15	by	by	ADP
ejpam-5279	382	16	uapchygrp	uapchygrp	NOUN
ejpam-5279	382	17	.	.	PUNCT
ejpam-5279	383	1	proposition	proposition	NOUN
ejpam-5279	383	2	3	3	NUM
ejpam-5279	383	3	.	.	PUNCT
ejpam-5279	383	4	uapchygrp	uapchygrp	PROPN
ejpam-5279	383	5	is	be	AUX
ejpam-5279	383	6	a	a	DET
ejpam-5279	383	7	topological	topological	ADJ
ejpam-5279	383	8	category	category	NOUN
ejpam-5279	383	9	.	.	PUNCT
ejpam-5279	384	1	proof	proof	NOUN
ejpam-5279	384	2	.	.	PUNCT
ejpam-5279	385	1	let	let	VERB
ejpam-5279	385	2	(	(	PUNCT
ejpam-5279	385	3	x	x	NOUN
ejpam-5279	385	4	,	,	PUNCT
ejpam-5279	385	5	·	·	PUNCT
ejpam-5279	385	6	)	)	PUNCT
ejpam-5279	385	7	be	be	AUX
ejpam-5279	385	8	a	a	DET
ejpam-5279	385	9	group	group	NOUN
ejpam-5279	385	10	,	,	PUNCT
ejpam-5279	385	11	fj	fj	PROPN
ejpam-5279	385	12	:	:	PUNCT
ejpam-5279	385	13	x	x	X
ejpam-5279	385	14	−→	−→	NOUN
ejpam-5279	385	15	xj	xj	PROPN
ejpam-5279	385	16	a	a	DET
ejpam-5279	385	17	group	group	NOUN
ejpam-5279	385	18	homomorphism	homomorphism	NOUN
ejpam-5279	385	19	,	,	PUNCT
ejpam-5279	385	20	and	and	CCONJ
ejpam-5279	385	21	(	(	PUNCT
ejpam-5279	385	22	xj	xj	PROPN
ejpam-5279	385	23	,	,	PUNCT
ejpam-5279	385	24	·	·	PUNCT
ejpam-5279	385	25	,	,	PUNCT
ejpam-5279	385	26	(	(	PUNCT
ejpam-5279	385	27	υj)j∈j	υj)j∈j	NUM
ejpam-5279	385	28	)	)	PUNCT
ejpam-5279	385	29	be	be	VERB
ejpam-5279	385	30	a	a	DET
ejpam-5279	385	31	family	family	NOUN
ejpam-5279	385	32	of	of	ADP
ejpam-5279	385	33	ultra	ultra	ADJ
ejpam-5279	385	34	approach	approach	NOUN
ejpam-5279	385	35	-	-	PUNCT
ejpam-5279	385	36	cauchy	cauchy	NOUN
ejpam-5279	385	37	groups	group	NOUN
ejpam-5279	385	38	.	.	PUNCT
ejpam-5279	386	1	consider	consider	VERB
ejpam-5279	386	2	a	a	DET
ejpam-5279	386	3	source	source	NOUN
ejpam-5279	386	4	s	s	PART
ejpam-5279	386	5	=	=	X
ejpam-5279	386	6	(	(	PUNCT
ejpam-5279	386	7	fj	fj	INTJ
ejpam-5279	386	8	:	:	PUNCT
ejpam-5279	386	9	x	x	PUNCT
ejpam-5279	386	10	−→	−→	NOUN
ejpam-5279	386	11	(	(	PUNCT
ejpam-5279	386	12	xj	xj	PROPN
ejpam-5279	386	13	,	,	PUNCT
ejpam-5279	386	14	·	·	PUNCT
ejpam-5279	386	15	,	,	PUNCT
ejpam-5279	386	16	υj))j∈j	υj))j∈j	PROPN
ejpam-5279	386	17	.	.	PUNCT
ejpam-5279	387	1	then	then	ADV
ejpam-5279	387	2	in	in	ADP
ejpam-5279	387	3	view	view	NOUN
ejpam-5279	387	4	of	of	ADP
ejpam-5279	387	5	the	the	DET
ejpam-5279	387	6	section	section	NOUN
ejpam-5279	387	7	5[17	5[17	NUM
ejpam-5279	387	8	]	]	PUNCT
ejpam-5279	387	9	the	the	DET
ejpam-5279	387	10	ultra	ultra	ADJ
ejpam-5279	387	11	approach	approach	NOUN
ejpam-5279	387	12	-	-	PUNCT
ejpam-5279	387	13	cauchy	cauchy	NOUN
ejpam-5279	387	14	structure	structure	NOUN
ejpam-5279	387	15	on	on	ADP
ejpam-5279	387	16	x	x	PROPN
ejpam-5279	387	17	is	be	AUX
ejpam-5279	387	18	given	give	VERB
ejpam-5279	387	19	for	for	ADP
ejpam-5279	387	20	any	any	DET
ejpam-5279	387	21	f	f	PROPN
ejpam-5279	387	22	∈	∈	PROPN
ejpam-5279	387	23	f(x	f(x	PROPN
ejpam-5279	387	24	)	)	PUNCT
ejpam-5279	387	25	,	,	PUNCT
ejpam-5279	387	26	by	by	ADP
ejpam-5279	387	27	υ(f	υ(f	PROPN
ejpam-5279	387	28	)	)	PUNCT
ejpam-5279	388	1	=	=	PUNCT
ejpam-5279	388	2	∨	∨	NUM
ejpam-5279	388	3	j∈j	j∈j	PROPN
ejpam-5279	388	4	υj(f(j(f	υj(f(j(f	NOUN
ejpam-5279	388	5	)	)	PUNCT
ejpam-5279	388	6	)	)	PUNCT
ejpam-5279	388	7	)	)	PUNCT
ejpam-5279	388	8	.	.	PUNCT
ejpam-5279	389	1	it	it	PRON
ejpam-5279	389	2	is	be	AUX
ejpam-5279	389	3	proved	prove	VERB
ejpam-5279	389	4	in	in	ADP
ejpam-5279	389	5	[	[	X
ejpam-5279	389	6	17	17	NUM
ejpam-5279	389	7	]	]	PUNCT
ejpam-5279	389	8	that	that	SCONJ
ejpam-5279	389	9	the	the	DET
ejpam-5279	389	10	source	source	NOUN
ejpam-5279	389	11	s	s	NOUN
ejpam-5279	389	12	has	have	AUX
ejpam-5279	389	13	t.m.g	t.m.g	ADJ
ejpam-5279	389	14	.	.	PUNCT
ejpam-5279	390	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	390	2	,	,	PUNCT
ejpam-5279	390	3	fawzi	fawzi	PROPN
ejpam-5279	390	4	al	al	PROPN
ejpam-5279	390	5	-	-	PUNCT
ejpam-5279	390	6	thukair	thukair	NOUN
ejpam-5279	390	7	/	/	SYM
ejpam-5279	390	8	eur	eur	NOUN
ejpam-5279	390	9	.	.	PUNCT
ejpam-5279	391	1	j.	j.	PROPN
ejpam-5279	391	2	pure	pure	PROPN
ejpam-5279	391	3	appl	appl	PROPN
ejpam-5279	391	4	.	.	PROPN
ejpam-5279	391	5	math	math	PROPN
ejpam-5279	391	6	,	,	PUNCT
ejpam-5279	391	7	17	17	NUM
ejpam-5279	391	8	(	(	PUNCT
ejpam-5279	391	9	3	3	NUM
ejpam-5279	391	10	)	)	PUNCT
ejpam-5279	391	11	(	(	PUNCT
ejpam-5279	391	12	2024	2024	NUM
ejpam-5279	391	13	)	)	PUNCT
ejpam-5279	391	14	,	,	PUNCT
ejpam-5279	391	15	1762	1762	NUM
ejpam-5279	391	16	-	-	SYM
ejpam-5279	391	17	1778	1778	NUM
ejpam-5279	391	18	1773	1773	NUM
ejpam-5279	391	19	initial	initial	ADJ
ejpam-5279	391	20	structure	structure	NOUN
ejpam-5279	391	21	and	and	CCONJ
ejpam-5279	391	22	that	that	SCONJ
ejpam-5279	391	23	the	the	DET
ejpam-5279	391	24	pair	pair	NOUN
ejpam-5279	391	25	(	(	PUNCT
ejpam-5279	391	26	x	x	NOUN
ejpam-5279	391	27	,	,	PUNCT
ejpam-5279	391	28	υ	υ	NOUN
ejpam-5279	391	29	)	)	PUNCT
ejpam-5279	391	30	is	be	AUX
ejpam-5279	391	31	an	an	DET
ejpam-5279	391	32	ultra	ultra	ADJ
ejpam-5279	391	33	approach	approach	NOUN
ejpam-5279	391	34	-	-	PUNCT
ejpam-5279	391	35	cauchy	cauchy	NOUN
ejpam-5279	391	36	space	space	NOUN
ejpam-5279	391	37	.	.	PUNCT
ejpam-5279	392	1	it	it	PRON
ejpam-5279	392	2	remains	remain	VERB
ejpam-5279	392	3	to	to	PART
ejpam-5279	392	4	be	be	AUX
ejpam-5279	392	5	checked	check	VERB
ejpam-5279	392	6	the	the	DET
ejpam-5279	392	7	following	follow	VERB
ejpam-5279	392	8	single	single	ADJ
ejpam-5279	392	9	condition	condition	NOUN
ejpam-5279	392	10	:	:	PUNCT
ejpam-5279	392	11	thus	thus	ADV
ejpam-5279	392	12	,	,	PUNCT
ejpam-5279	392	13	for	for	ADP
ejpam-5279	392	14	any	any	DET
ejpam-5279	392	15	f	f	NOUN
ejpam-5279	392	16	,	,	PUNCT
ejpam-5279	392	17	g	g	PROPN
ejpam-5279	392	18	∈	∈	PROPN
ejpam-5279	392	19	f(x	f(x	PROPN
ejpam-5279	392	20	)	)	PUNCT
ejpam-5279	392	21	,	,	PUNCT
ejpam-5279	392	22	we	we	PRON
ejpam-5279	392	23	have	have	VERB
ejpam-5279	392	24	υ	υ	PRON
ejpam-5279	392	25	(	(	PUNCT
ejpam-5279	392	26	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	392	27	)	)	PUNCT
ejpam-5279	393	1	=	=	PUNCT
ejpam-5279	393	2	∨	∨	NUM
ejpam-5279	393	3	j∈j	j∈j	NOUN
ejpam-5279	393	4	υj∈j	υj∈j	ADP
ejpam-5279	393	5	(	(	PUNCT
ejpam-5279	393	6	fj(f⊙g−1	fj(f⊙g−1	PROPN
ejpam-5279	393	7	)	)	PUNCT
ejpam-5279	393	8	)	)	PUNCT
ejpam-5279	394	1	=	=	PUNCT
ejpam-5279	394	2	∨	∨	NUM
ejpam-5279	394	3	j∈j	j∈j	NOUN
ejpam-5279	394	4	υj	υj	X
ejpam-5279	394	5	(	(	PUNCT
ejpam-5279	394	6	fj(f)⊙	fj(f)⊙	NOUN
ejpam-5279	394	7	fj(g)−1	fj(g)−1	PUNCT
ejpam-5279	394	8	)	)	PUNCT
ejpam-5279	394	9	≤	≤	PROPN
ejpam-5279	394	10	∨	∨	NUM
ejpam-5279	394	11	υj∈j(fj(f	υj∈j(fj(f	PROPN
ejpam-5279	394	12	)	)	PUNCT
ejpam-5279	394	13	)	)	PUNCT
ejpam-5279	394	14	∨	∨	NUM
ejpam-5279	394	15	∨	∨	NUM
ejpam-5279	394	16	j∈j	j∈j	NOUN
ejpam-5279	394	17	υj(fj(g	υj(fj(g	NOUN
ejpam-5279	394	18	)	)	PUNCT
ejpam-5279	394	19	)	)	PUNCT
ejpam-5279	395	1	=	=	SYM
ejpam-5279	395	2	υ(f	υ(f	PROPN
ejpam-5279	395	3	)	)	PUNCT
ejpam-5279	395	4	∨υ(g	∨υ(g	PROPN
ejpam-5279	395	5	)	)	PUNCT
ejpam-5279	395	6	,	,	PUNCT
ejpam-5279	395	7	for	for	ADP
ejpam-5279	395	8	all	all	DET
ejpam-5279	395	9	j	j	PROPN
ejpam-5279	395	10	∈	∈	PROPN
ejpam-5279	395	11	j	j	PROPN
ejpam-5279	395	12	.	.	PUNCT
ejpam-5279	396	1	finally	finally	ADV
ejpam-5279	396	2	,	,	PUNCT
ejpam-5279	396	3	we	we	PRON
ejpam-5279	396	4	show	show	VERB
ejpam-5279	396	5	that	that	SCONJ
ejpam-5279	396	6	for	for	ADP
ejpam-5279	396	7	any	any	DET
ejpam-5279	396	8	ultra	ultra	ADJ
ejpam-5279	396	9	approach	approach	NOUN
ejpam-5279	396	10	-	-	PUNCT
ejpam-5279	396	11	cauchy	cauchy	NOUN
ejpam-5279	396	12	group	group	NOUN
ejpam-5279	396	13	(	(	PUNCT
ejpam-5279	396	14	y	y	PROPN
ejpam-5279	396	15	,	,	PUNCT
ejpam-5279	396	16	·	·	PUNCT
ejpam-5279	396	17	,	,	PUNCT
ejpam-5279	396	18	υ′	υ′	NUM
ejpam-5279	396	19	)	)	PUNCT
ejpam-5279	396	20	,	,	PUNCT
ejpam-5279	396	21	a	a	DET
ejpam-5279	396	22	group	group	NOUN
ejpam-5279	396	23	homomorphism	homomorphism	NOUN
ejpam-5279	396	24	g	g	NOUN
ejpam-5279	396	25	:	:	PUNCT
ejpam-5279	396	26	(	(	PUNCT
ejpam-5279	396	27	y	y	NOUN
ejpam-5279	396	28	,	,	PUNCT
ejpam-5279	396	29	·	·	PUNCT
ejpam-5279	396	30	,	,	PUNCT
ejpam-5279	396	31	υ′	υ′	NUM
ejpam-5279	396	32	)	)	PUNCT
ejpam-5279	396	33	−→	−→	NOUN
ejpam-5279	396	34	(	(	PUNCT
ejpam-5279	396	35	x	x	NOUN
ejpam-5279	396	36	,	,	PUNCT
ejpam-5279	396	37	·	·	PUNCT
ejpam-5279	396	38	,	,	PUNCT
ejpam-5279	396	39	υ	υ	NOUN
ejpam-5279	396	40	)	)	PUNCT
ejpam-5279	396	41	is	be	AUX
ejpam-5279	396	42	a	a	DET
ejpam-5279	396	43	contraction	contraction	NOUN
ejpam-5279	396	44	if	if	SCONJ
ejpam-5279	396	45	and	and	CCONJ
ejpam-5279	396	46	only	only	ADV
ejpam-5279	396	47	if	if	SCONJ
ejpam-5279	396	48	for	for	ADP
ejpam-5279	396	49	all	all	DET
ejpam-5279	396	50	j	j	PROPN
ejpam-5279	396	51	∈	∈	PROPN
ejpam-5279	396	52	j	j	PROPN
ejpam-5279	396	53	,	,	PUNCT
ejpam-5279	396	54	the	the	DET
ejpam-5279	396	55	map	map	NOUN
ejpam-5279	396	56	fj	fj	INTJ
ejpam-5279	396	57	◦	◦	NOUN
ejpam-5279	396	58	g	g	NOUN
ejpam-5279	396	59	:	:	PUNCT
ejpam-5279	396	60	(	(	PUNCT
ejpam-5279	396	61	y	y	NOUN
ejpam-5279	396	62	,	,	PUNCT
ejpam-5279	396	63	·	·	PUNCT
ejpam-5279	396	64	,	,	PUNCT
ejpam-5279	396	65	υ′	υ′	NUM
ejpam-5279	396	66	)	)	PUNCT
ejpam-5279	396	67	−→	−→	NOUN
ejpam-5279	396	68	(	(	PUNCT
ejpam-5279	396	69	xj	xj	PROPN
ejpam-5279	396	70	,	,	PUNCT
ejpam-5279	396	71	·	·	PUNCT
ejpam-5279	396	72	,	,	PUNCT
ejpam-5279	396	73	υj	υj	CCONJ
ejpam-5279	396	74	)	)	PUNCT
ejpam-5279	396	75	is	be	AUX
ejpam-5279	396	76	a	a	DET
ejpam-5279	396	77	contraction	contraction	NOUN
ejpam-5279	396	78	.	.	PUNCT
ejpam-5279	397	1	but	but	CCONJ
ejpam-5279	397	2	the	the	DET
ejpam-5279	397	3	composition	composition	NOUN
ejpam-5279	397	4	fj	fj	INTJ
ejpam-5279	397	5	◦	◦	NOUN
ejpam-5279	397	6	g	g	NOUN
ejpam-5279	397	7	is	be	AUX
ejpam-5279	397	8	clearly	clearly	ADV
ejpam-5279	397	9	contraction	contraction	NOUN
ejpam-5279	397	10	,	,	PUNCT
ejpam-5279	397	11	while	while	SCONJ
ejpam-5279	397	12	the	the	DET
ejpam-5279	397	13	contraction	contraction	NOUN
ejpam-5279	397	14	of	of	ADP
ejpam-5279	397	15	g	g	PROPN
ejpam-5279	397	16	follows	follow	VERB
ejpam-5279	397	17	at	at	ADP
ejpam-5279	397	18	once	once	ADV
ejpam-5279	397	19	from	from	ADP
ejpam-5279	397	20	[	[	X
ejpam-5279	397	21	17	17	NUM
ejpam-5279	397	22	]	]	PUNCT
ejpam-5279	397	23	.	.	PUNCT
ejpam-5279	398	1	recall	recall	VERB
ejpam-5279	398	2	that	that	SCONJ
ejpam-5279	398	3	an	an	DET
ejpam-5279	398	4	ultra	ultra	ADJ
ejpam-5279	398	5	approach	approach	NOUN
ejpam-5279	398	6	limit	limit	NOUN
ejpam-5279	398	7	group	group	NOUN
ejpam-5279	398	8	,	,	PUNCT
ejpam-5279	398	9	[	[	X
ejpam-5279	398	10	3	3	NUM
ejpam-5279	398	11	]	]	PUNCT
ejpam-5279	398	12	,	,	PUNCT
ejpam-5279	398	13	is	be	AUX
ejpam-5279	398	14	a	a	DET
ejpam-5279	398	15	triple	triple	ADJ
ejpam-5279	398	16	(	(	PUNCT
ejpam-5279	398	17	x	x	NOUN
ejpam-5279	398	18	,	,	PUNCT
ejpam-5279	398	19	·	·	PUNCT
ejpam-5279	398	20	,	,	PUNCT
ejpam-5279	398	21	λ	λ	NOUN
ejpam-5279	398	22	)	)	PUNCT
ejpam-5279	398	23	consisting	consist	VERB
ejpam-5279	398	24	of	of	ADP
ejpam-5279	398	25	a	a	DET
ejpam-5279	398	26	group	group	NOUN
ejpam-5279	398	27	(	(	PUNCT
ejpam-5279	398	28	x	x	X
ejpam-5279	398	29	,	,	PUNCT
ejpam-5279	398	30	·	·	PUNCT
ejpam-5279	398	31	)	)	PUNCT
ejpam-5279	398	32	and	and	CCONJ
ejpam-5279	398	33	an	an	DET
ejpam-5279	398	34	ultra	ultra	ADJ
ejpam-5279	398	35	approach	approach	NOUN
ejpam-5279	398	36	limit	limit	NOUN
ejpam-5279	398	37	structure	structure	NOUN
ejpam-5279	398	38	λ	λ	NOUN
ejpam-5279	398	39	:	:	PUNCT
ejpam-5279	398	40	f(x	f(x	PROPN
ejpam-5279	398	41	)	)	PUNCT
ejpam-5279	398	42	→	→	PUNCT
ejpam-5279	399	1	[	[	X
ejpam-5279	399	2	0,∞]x	0,∞]x	VERB
ejpam-5279	399	3	meaning	meaning	NOUN
ejpam-5279	399	4	for	for	ADP
ejpam-5279	399	5	all	all	DET
ejpam-5279	399	6	x	x	SYM
ejpam-5279	399	7	∈	∈	PROPN
ejpam-5279	399	8	x	x	X
ejpam-5279	399	9	,	,	PUNCT
ejpam-5279	399	10	λ(ẋ	λ(ẋ	PROPN
ejpam-5279	399	11	)	)	PUNCT
ejpam-5279	399	12	=	=	SYM
ejpam-5279	399	13	0	0	NUM
ejpam-5279	399	14	,	,	PUNCT
ejpam-5279	399	15	for	for	ADP
ejpam-5279	399	16	all	all	DET
ejpam-5279	399	17	f	f	NOUN
ejpam-5279	399	18	,	,	PUNCT
ejpam-5279	399	19	g	g	PROPN
ejpam-5279	399	20	∈	∈	PROPN
ejpam-5279	399	21	f(x	f(x	PROPN
ejpam-5279	399	22	)	)	PUNCT
ejpam-5279	399	23	with	with	ADP
ejpam-5279	399	24	f	f	PROPN
ejpam-5279	399	25	≤	≤	PROPN
ejpam-5279	399	26	g	g	PROPN
ejpam-5279	399	27	implies	imply	VERB
ejpam-5279	399	28	λ(g	λ(g	NOUN
ejpam-5279	399	29	)	)	PUNCT
ejpam-5279	399	30	≤	≤	NOUN
ejpam-5279	399	31	λ(f	λ(f	NOUN
ejpam-5279	399	32	)	)	PUNCT
ejpam-5279	399	33	and	and	CCONJ
ejpam-5279	399	34	λ(f∧g	λ(f∧g	ADJ
ejpam-5279	399	35	)	)	PUNCT
ejpam-5279	399	36	=	=	SYM
ejpam-5279	399	37	λ(f)∨λ(g	λ(f)∨λ(g	NOUN
ejpam-5279	399	38	)	)	PUNCT
ejpam-5279	399	39	,	,	PUNCT
ejpam-5279	399	40	such	such	ADJ
ejpam-5279	399	41	that	that	SCONJ
ejpam-5279	399	42	the	the	DET
ejpam-5279	399	43	group	group	NOUN
ejpam-5279	399	44	operation	operation	NOUN
ejpam-5279	399	45	h	h	NOUN
ejpam-5279	399	46	:	:	PUNCT
ejpam-5279	399	47	x	x	PUNCT
ejpam-5279	399	48	×	×	NOUN
ejpam-5279	399	49	x	x	PUNCT
ejpam-5279	399	50	→	→	SYM
ejpam-5279	399	51	x	x	SYM
ejpam-5279	399	52	,	,	PUNCT
ejpam-5279	399	53	(	(	PUNCT
ejpam-5279	399	54	x	x	NOUN
ejpam-5279	399	55	,	,	PUNCT
ejpam-5279	399	56	y	y	NOUN
ejpam-5279	399	57	)	)	PUNCT
ejpam-5279	399	58	7→	7→	NUM
ejpam-5279	400	1	xy−1	xy−1	PROPN
ejpam-5279	400	2	is	be	AUX
ejpam-5279	400	3	a	a	DET
ejpam-5279	400	4	contraction	contraction	NOUN
ejpam-5279	400	5	,	,	PUNCT
ejpam-5279	400	6	i.e.	i.e.	X
ejpam-5279	400	7	,	,	PUNCT
ejpam-5279	400	8	∀f	∀f	PROPN
ejpam-5279	400	9	,	,	PUNCT
ejpam-5279	400	10	g	g	PROPN
ejpam-5279	400	11	∈	∈	PROPN
ejpam-5279	400	12	f(x	f(x	PROPN
ejpam-5279	400	13	)	)	PUNCT
ejpam-5279	400	14	,	,	PUNCT
ejpam-5279	401	1	λ	λ	X
ejpam-5279	401	2	(	(	PUNCT
ejpam-5279	401	3	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	401	4	)	)	PUNCT
ejpam-5279	401	5	(	(	PUNCT
ejpam-5279	401	6	xy−1	xy−1	NOUN
ejpam-5279	401	7	)	)	PUNCT
ejpam-5279	401	8	≤	≤	NUM
ejpam-5279	401	9	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	401	10	)	)	PUNCT
ejpam-5279	401	11	∨	∨	NOUN
ejpam-5279	401	12	λ(g)(y	λ(g)(y	NUM
ejpam-5279	401	13	)	)	PUNCT
ejpam-5279	401	14	.	.	PUNCT
ejpam-5279	402	1	with	with	ADP
ejpam-5279	402	2	each	each	DET
ejpam-5279	402	3	ultra	ultra	ADJ
ejpam-5279	402	4	approach	approach	NOUN
ejpam-5279	402	5	limit	limit	NOUN
ejpam-5279	402	6	group	group	NOUN
ejpam-5279	402	7	(	(	PUNCT
ejpam-5279	402	8	x	x	X
ejpam-5279	402	9	,	,	PUNCT
ejpam-5279	402	10	·	·	PUNCT
ejpam-5279	402	11	,	,	PUNCT
ejpam-5279	402	12	λ	λ	PROPN
ejpam-5279	402	13	)	)	PUNCT
ejpam-5279	402	14	,	,	PUNCT
ejpam-5279	402	15	there	there	PRON
ejpam-5279	402	16	is	be	VERB
ejpam-5279	402	17	associated	associate	VERB
ejpam-5279	402	18	a	a	DET
ejpam-5279	402	19	natural	natural	ADJ
ejpam-5279	402	20	ultra	ultra	ADJ
ejpam-5279	402	21	approach	approach	NOUN
ejpam-5279	402	22	-	-	PUNCT
ejpam-5279	402	23	cauchy	cauchy	NOUN
ejpam-5279	402	24	structure	structure	NOUN
ejpam-5279	402	25	defined	define	VERB
ejpam-5279	402	26	as	as	SCONJ
ejpam-5279	402	27	follows	follow	VERB
ejpam-5279	402	28	:	:	PUNCT
ejpam-5279	402	29	υλ	υλ	NOUN
ejpam-5279	402	30	:	:	PUNCT
ejpam-5279	402	31	f(x	f(x	PROPN
ejpam-5279	402	32	)	)	PUNCT
ejpam-5279	402	33	→	→	PUNCT
ejpam-5279	403	1	[	[	X
ejpam-5279	403	2	0,∞],f	0,∞],f	NUM
ejpam-5279	403	3	7→	7→	NUM
ejpam-5279	403	4	υλ(f	υλ(f	NOUN
ejpam-5279	403	5	)	)	PUNCT
ejpam-5279	403	6	=	=	SYM
ejpam-5279	403	7	λ	λ	NOUN
ejpam-5279	403	8	(	(	PUNCT
ejpam-5279	403	9	f−1	f−1	PROPN
ejpam-5279	403	10	⊙	⊙	NOUN
ejpam-5279	403	11	f	f	PROPN
ejpam-5279	403	12	)	)	PUNCT
ejpam-5279	403	13	(	(	PUNCT
ejpam-5279	403	14	e	e	NOUN
ejpam-5279	403	15	)	)	PUNCT
ejpam-5279	403	16	∨	∨	NUM
ejpam-5279	403	17	λ	λ	PROPN
ejpam-5279	403	18	(	(	PUNCT
ejpam-5279	403	19	f⊙	f⊙	VERB
ejpam-5279	403	20	f−1	f−1	PROPN
ejpam-5279	403	21	)	)	PUNCT
ejpam-5279	403	22	(	(	PUNCT
ejpam-5279	403	23	e	e	NOUN
ejpam-5279	403	24	)	)	PUNCT
ejpam-5279	403	25	.	.	PUNCT
ejpam-5279	404	1	on	on	ADP
ejpam-5279	404	2	the	the	DET
ejpam-5279	404	3	other	other	ADJ
ejpam-5279	404	4	hand	hand	NOUN
ejpam-5279	404	5	,	,	PUNCT
ejpam-5279	404	6	every	every	DET
ejpam-5279	404	7	ultra	ultra	ADJ
ejpam-5279	404	8	approach	approach	NOUN
ejpam-5279	404	9	cauchy	cauchy	PROPN
ejpam-5279	404	10	group	group	NOUN
ejpam-5279	404	11	(	(	PUNCT
ejpam-5279	404	12	x	x	X
ejpam-5279	404	13	,	,	PUNCT
ejpam-5279	404	14	·	·	PUNCT
ejpam-5279	404	15	,	,	PUNCT
ejpam-5279	404	16	υ	υ	NOUN
ejpam-5279	404	17	)	)	PUNCT
ejpam-5279	404	18	gives	give	VERB
ejpam-5279	404	19	rise	rise	NOUN
ejpam-5279	404	20	to	to	ADP
ejpam-5279	404	21	an	an	DET
ejpam-5279	404	22	ultra	ultra	ADJ
ejpam-5279	404	23	approach	approach	NOUN
ejpam-5279	404	24	limit	limit	NOUN
ejpam-5279	404	25	structure	structure	NOUN
ejpam-5279	404	26	given	give	VERB
ejpam-5279	404	27	by	by	ADP
ejpam-5279	404	28	:	:	PUNCT
ejpam-5279	404	29	λυ	λυ	ADP
ejpam-5279	404	30	:	:	PUNCT
ejpam-5279	404	31	f(x	f(x	PROPN
ejpam-5279	404	32	)	)	PUNCT
ejpam-5279	404	33	→	→	PUNCT
ejpam-5279	405	1	[	[	X
ejpam-5279	405	2	0,∞]x	0,∞]x	X
ejpam-5279	405	3	,	,	PUNCT
ejpam-5279	405	4	f	f	PROPN
ejpam-5279	405	5	7→	7→	NUM
ejpam-5279	405	6	λυ(f)(x	λυ(f)(x	NOUN
ejpam-5279	405	7	)	)	PUNCT
ejpam-5279	406	1	=	=	SYM
ejpam-5279	406	2	υ	υ	PROPN
ejpam-5279	406	3	(	(	PUNCT
ejpam-5279	406	4	f	f	PROPN
ejpam-5279	406	5	∩	∩	PROPN
ejpam-5279	406	6	ẋ	ẋ	PROPN
ejpam-5279	406	7	)	)	PUNCT
ejpam-5279	406	8	.	.	PUNCT
ejpam-5279	407	1	lemma	lemma	PROPN
ejpam-5279	407	2	3	3	X
ejpam-5279	407	3	.	.	PUNCT
ejpam-5279	408	1	[	[	X
ejpam-5279	408	2	4	4	X
ejpam-5279	408	3	]	]	X
ejpam-5279	408	4	let	let	VERB
ejpam-5279	408	5	(	(	PUNCT
ejpam-5279	408	6	x	x	NOUN
ejpam-5279	408	7	,	,	PUNCT
ejpam-5279	408	8	·	·	PUNCT
ejpam-5279	408	9	,	,	PUNCT
ejpam-5279	408	10	λ	λ	X
ejpam-5279	408	11	)	)	PUNCT
ejpam-5279	408	12	be	be	VERB
ejpam-5279	408	13	an	an	DET
ejpam-5279	408	14	ultra	ultra	ADJ
ejpam-5279	408	15	approach	approach	NOUN
ejpam-5279	408	16	limit	limit	NOUN
ejpam-5279	408	17	group	group	NOUN
ejpam-5279	408	18	,	,	PUNCT
ejpam-5279	408	19	f	f	PROPN
ejpam-5279	408	20	∈	∈	PROPN
ejpam-5279	408	21	f(x	f(x	PROPN
ejpam-5279	408	22	)	)	PUNCT
ejpam-5279	408	23	and	and	CCONJ
ejpam-5279	408	24	x	x	PUNCT
ejpam-5279	408	25	∈	∈	PROPN
ejpam-5279	408	26	x.	x.	NOUN
ejpam-5279	408	27	then	then	ADV
ejpam-5279	408	28	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	408	29	)	)	PUNCT
ejpam-5279	408	30	=	=	PUNCT
ejpam-5279	409	1	λ([x]−1	λ([x]−1	PUNCT
ejpam-5279	409	2	⊙	⊙	NOUN
ejpam-5279	409	3	f)(e	f)(e	PRON
ejpam-5279	409	4	)	)	PUNCT
ejpam-5279	409	5	=	=	SYM
ejpam-5279	410	1	λ(f⊙	λ(f⊙	NOUN
ejpam-5279	410	2	[	[	X
ejpam-5279	410	3	x]−1)(e	x]−1)(e	NUM
ejpam-5279	410	4	)	)	PUNCT
ejpam-5279	410	5	.	.	PUNCT
ejpam-5279	411	1	proposition	proposition	NOUN
ejpam-5279	411	2	4	4	NUM
ejpam-5279	411	3	.	.	PUNCT
ejpam-5279	412	1	if	if	SCONJ
ejpam-5279	412	2	(	(	PUNCT
ejpam-5279	412	3	x	x	X
ejpam-5279	412	4	,	,	PUNCT
ejpam-5279	412	5	·	·	PUNCT
ejpam-5279	412	6	,	,	PUNCT
ejpam-5279	412	7	λ	λ	X
ejpam-5279	412	8	)	)	PUNCT
ejpam-5279	412	9	is	be	AUX
ejpam-5279	412	10	an	an	DET
ejpam-5279	412	11	ultra	ultra	ADJ
ejpam-5279	412	12	approach	approach	NOUN
ejpam-5279	412	13	limit	limit	NOUN
ejpam-5279	412	14	group	group	NOUN
ejpam-5279	412	15	,	,	PUNCT
ejpam-5279	412	16	then	then	ADV
ejpam-5279	412	17	υλ	υλ	VERB
ejpam-5279	412	18	:	:	PUNCT
ejpam-5279	412	19	f(x	f(x	PROPN
ejpam-5279	412	20	)	)	PUNCT
ejpam-5279	412	21	→	→	PUNCT
ejpam-5279	413	1	[	[	X
ejpam-5279	413	2	0,∞	0,∞	X
ejpam-5279	413	3	]	]	PUNCT
ejpam-5279	413	4	defined	define	VERB
ejpam-5279	413	5	by	by	ADP
ejpam-5279	413	6	υλ(f	υλ(f	NOUN
ejpam-5279	413	7	)	)	PUNCT
ejpam-5279	413	8	=	=	SYM
ejpam-5279	413	9	λ	λ	NOUN
ejpam-5279	413	10	(	(	PUNCT
ejpam-5279	413	11	f−1	f−1	PROPN
ejpam-5279	413	12	⊙	⊙	NOUN
ejpam-5279	413	13	f	f	PROPN
ejpam-5279	413	14	)	)	PUNCT
ejpam-5279	413	15	(	(	PUNCT
ejpam-5279	413	16	e	e	NOUN
ejpam-5279	413	17	)	)	PUNCT
ejpam-5279	413	18	∨	∨	NUM
ejpam-5279	413	19	λ	λ	PROPN
ejpam-5279	413	20	(	(	PUNCT
ejpam-5279	413	21	f⊙	f⊙	VERB
ejpam-5279	413	22	f−1	f−1	PROPN
ejpam-5279	413	23	)	)	PUNCT
ejpam-5279	413	24	(	(	PUNCT
ejpam-5279	413	25	e	e	NOUN
ejpam-5279	413	26	)	)	PUNCT
ejpam-5279	413	27	,	,	PUNCT
ejpam-5279	413	28	∀f	∀f	PROPN
ejpam-5279	413	29	∈	∈	PROPN
ejpam-5279	413	30	f(x	f(x	PROPN
ejpam-5279	413	31	)	)	PUNCT
ejpam-5279	413	32	gives	give	VERB
ejpam-5279	413	33	rise	rise	NOUN
ejpam-5279	413	34	to	to	ADP
ejpam-5279	413	35	an	an	DET
ejpam-5279	413	36	ultra	ultra	ADJ
ejpam-5279	413	37	approach	approach	NOUN
ejpam-5279	413	38	-	-	PUNCT
ejpam-5279	413	39	cauchy	cauchy	NOUN
ejpam-5279	413	40	structure	structure	NOUN
ejpam-5279	413	41	.	.	PUNCT
ejpam-5279	414	1	proof	proof	NOUN
ejpam-5279	414	2	.	.	PUNCT
ejpam-5279	415	1	(	(	PUNCT
ejpam-5279	415	2	uachy1	uachy1	NOUN
ejpam-5279	415	3	)	)	PUNCT
ejpam-5279	415	4	for	for	ADP
ejpam-5279	415	5	any	any	DET
ejpam-5279	415	6	x	x	SYM
ejpam-5279	415	7	∈	∈	PROPN
ejpam-5279	415	8	x	x	NOUN
ejpam-5279	415	9	,	,	PUNCT
ejpam-5279	415	10	υλ(ẋ	υλ(ẋ	ADJ
ejpam-5279	415	11	)	)	PUNCT
ejpam-5279	415	12	=	=	SYM
ejpam-5279	415	13	λ	λ	NOUN
ejpam-5279	415	14	(	(	PUNCT
ejpam-5279	415	15	ẋ−1	ẋ−1	PROPN
ejpam-5279	415	16	⊙	⊙	PROPN
ejpam-5279	415	17	ẋ	ẋ	PROPN
ejpam-5279	415	18	)	)	PUNCT
ejpam-5279	416	1	(	(	PUNCT
ejpam-5279	416	2	e)∨λ	e)∨λ	ADV
ejpam-5279	416	3	(	(	PUNCT
ejpam-5279	416	4	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	416	5	ẋ−1	ẋ−1	PROPN
ejpam-5279	416	6	)	)	PUNCT
ejpam-5279	416	7	(	(	PUNCT
ejpam-5279	416	8	e	e	NOUN
ejpam-5279	416	9	)	)	PUNCT
ejpam-5279	416	10	=	=	NOUN
ejpam-5279	416	11	λ(ė)(e)∨	λ(ė)(e)∨	NOUN
ejpam-5279	416	12	λ(ė)(e	λ(ė)(e	NOUN
ejpam-5279	416	13	)	)	PUNCT
ejpam-5279	416	14	=	=	SYM
ejpam-5279	416	15	0	0	X
ejpam-5279	416	16	.	.	PUNCT
ejpam-5279	417	1	(	(	PUNCT
ejpam-5279	417	2	uachy2	uachy2	PROPN
ejpam-5279	417	3	)	)	PUNCT
ejpam-5279	417	4	if	if	SCONJ
ejpam-5279	417	5	f	f	PROPN
ejpam-5279	417	6	≤	≤	X
ejpam-5279	417	7	g	g	PROPN
ejpam-5279	417	8	,	,	PUNCT
ejpam-5279	417	9	then	then	ADV
ejpam-5279	417	10	since	since	SCONJ
ejpam-5279	417	11	f−1	f−1	PROPN
ejpam-5279	417	12	⊙	⊙	VERB
ejpam-5279	417	13	f	f	PROPN
ejpam-5279	417	14	≤	≤	PROPN
ejpam-5279	417	15	g−1	g−1	PROPN
ejpam-5279	417	16	⊙g	⊙g	PROPN
ejpam-5279	417	17	,	,	PUNCT
ejpam-5279	417	18	we	we	PRON
ejpam-5279	417	19	have	have	VERB
ejpam-5279	417	20	υλ(g	υλ(g	NOUN
ejpam-5279	417	21	)	)	PUNCT
ejpam-5279	417	22	≤	≤	NOUN
ejpam-5279	417	23	υλ(f	υλ(f	NOUN
ejpam-5279	417	24	)	)	PUNCT
ejpam-5279	417	25	.	.	PUNCT
ejpam-5279	418	1	(	(	PUNCT
ejpam-5279	418	2	uachy3	uachy3	NOUN
ejpam-5279	418	3	)	)	PUNCT
ejpam-5279	418	4	let	let	VERB
ejpam-5279	418	5	f	f	X
ejpam-5279	418	6	,	,	PUNCT
ejpam-5279	418	7	g	g	PROPN
ejpam-5279	418	8	∈	∈	PROPN
ejpam-5279	418	9	f(x	f(x	PROPN
ejpam-5279	418	10	)	)	PUNCT
ejpam-5279	418	11	such	such	ADJ
ejpam-5279	418	12	that	that	PRON
ejpam-5279	418	13	f∨g	f∨g	PROPN
ejpam-5279	418	14	exists	exist	VERB
ejpam-5279	418	15	,	,	PUNCT
ejpam-5279	418	16	then	then	ADV
ejpam-5279	418	17	f−1⊙g	f−1⊙g	PROPN
ejpam-5279	418	18	≤	≤	PROPN
ejpam-5279	418	19	ė	ė	ADJ
ejpam-5279	418	20	and	and	CCONJ
ejpam-5279	418	21	also	also	ADV
ejpam-5279	418	22	,	,	PUNCT
ejpam-5279	418	23	g−1⊙f	g−1⊙f	PROPN
ejpam-5279	418	24	≤	≤	PROPN
ejpam-5279	418	25	ė	ė	ADJ
ejpam-5279	418	26	,	,	PUNCT
ejpam-5279	418	27	hence	hence	ADV
ejpam-5279	418	28	upon	upon	SCONJ
ejpam-5279	418	29	using	use	VERB
ejpam-5279	418	30	these	these	PRON
ejpam-5279	418	31	we	we	PRON
ejpam-5279	418	32	have	have	VERB
ejpam-5279	418	33	υλ(f	υλ(f	NOUN
ejpam-5279	418	34	)	)	PUNCT
ejpam-5279	418	35	∨υλ(g	∨υλ(g	PROPN
ejpam-5279	418	36	)	)	PUNCT
ejpam-5279	418	37	=	=	SYM
ejpam-5279	419	1	λ	λ	NOUN
ejpam-5279	419	2	(	(	PUNCT
ejpam-5279	419	3	f−1	f−1	PROPN
ejpam-5279	419	4	⊙	⊙	NOUN
ejpam-5279	419	5	f	f	PROPN
ejpam-5279	419	6	)	)	PUNCT
ejpam-5279	419	7	(	(	PUNCT
ejpam-5279	419	8	e	e	NOUN
ejpam-5279	419	9	)	)	PUNCT
ejpam-5279	419	10	∨	∨	NUM
ejpam-5279	419	11	λ	λ	PROPN
ejpam-5279	419	12	(	(	PUNCT
ejpam-5279	419	13	f⊙	f⊙	VERB
ejpam-5279	419	14	f−1	f−1	PROPN
ejpam-5279	419	15	)	)	PUNCT
ejpam-5279	419	16	(	(	PUNCT
ejpam-5279	419	17	e	e	NOUN
ejpam-5279	419	18	)	)	PUNCT
ejpam-5279	419	19	∨	∨	NOUN
ejpam-5279	419	20	(	(	PUNCT
ejpam-5279	419	21	g−1	g−1	PROPN
ejpam-5279	419	22	⊙g	⊙g	NOUN
ejpam-5279	419	23	)	)	PUNCT
ejpam-5279	419	24	(	(	PUNCT
ejpam-5279	419	25	e	e	NOUN
ejpam-5279	419	26	)	)	PUNCT
ejpam-5279	419	27	∨	∨	NUM
ejpam-5279	419	28	λ	λ	PROPN
ejpam-5279	419	29	(	(	PUNCT
ejpam-5279	419	30	g⊙g−1	g⊙g−1	PROPN
ejpam-5279	419	31	)	)	PUNCT
ejpam-5279	419	32	(	(	PUNCT
ejpam-5279	419	33	e	e	NOUN
ejpam-5279	419	34	)	)	PUNCT
ejpam-5279	419	35	=	=	NOUN
ejpam-5279	420	1	[	[	X
ejpam-5279	420	2	λ	λ	X
ejpam-5279	420	3	(	(	PUNCT
ejpam-5279	420	4	f−1	f−1	PROPN
ejpam-5279	420	5	⊙	⊙	NOUN
ejpam-5279	420	6	f	f	PROPN
ejpam-5279	420	7	)	)	PUNCT
ejpam-5279	420	8	(	(	PUNCT
ejpam-5279	420	9	e	e	NOUN
ejpam-5279	420	10	)	)	PUNCT
ejpam-5279	420	11	∨	∨	NOUN
ejpam-5279	420	12	(	(	PUNCT
ejpam-5279	420	13	g−1	g−1	PROPN
ejpam-5279	420	14	⊙g	⊙g	NOUN
ejpam-5279	420	15	)	)	PUNCT
ejpam-5279	420	16	(	(	PUNCT
ejpam-5279	420	17	e	e	NOUN
ejpam-5279	420	18	)	)	PUNCT
ejpam-5279	420	19	]	]	PUNCT
ejpam-5279	421	1	∨	∨	NUM
ejpam-5279	421	2	[	[	X
ejpam-5279	421	3	λ	λ	X
ejpam-5279	421	4	(	(	PUNCT
ejpam-5279	421	5	f⊙	f⊙	NOUN
ejpam-5279	421	6	f−1	f−1	PROPN
ejpam-5279	421	7	)	)	PUNCT
ejpam-5279	421	8	(	(	PUNCT
ejpam-5279	421	9	e	e	NOUN
ejpam-5279	421	10	)	)	PUNCT
ejpam-5279	421	11	∨	∨	NUM
ejpam-5279	421	12	λ	λ	PROPN
ejpam-5279	421	13	(	(	PUNCT
ejpam-5279	421	14	g⊙g−1	g⊙g−1	PROPN
ejpam-5279	421	15	)	)	PUNCT
ejpam-5279	421	16	(	(	PUNCT
ejpam-5279	421	17	e	e	NOUN
ejpam-5279	421	18	)	)	PUNCT
ejpam-5279	421	19	]	]	PUNCT
ejpam-5279	421	20	≥	≥	X
ejpam-5279	421	21	λ	λ	X
ejpam-5279	421	22	(	(	PUNCT
ejpam-5279	421	23	f−1	f−1	PROPN
ejpam-5279	421	24	⊙g	⊙g	PROPN
ejpam-5279	421	25	)	)	PUNCT
ejpam-5279	421	26	(	(	PUNCT
ejpam-5279	421	27	e	e	NOUN
ejpam-5279	421	28	)	)	PUNCT
ejpam-5279	421	29	∨	∨	NUM
ejpam-5279	421	30	λ	λ	PROPN
ejpam-5279	421	31	(	(	PUNCT
ejpam-5279	421	32	g−1	g−1	PROPN
ejpam-5279	421	33	⊙	⊙	PROPN
ejpam-5279	421	34	f	f	PROPN
ejpam-5279	421	35	)	)	PUNCT
ejpam-5279	421	36	(	(	PUNCT
ejpam-5279	421	37	e	e	NOUN
ejpam-5279	421	38	)	)	PUNCT
ejpam-5279	421	39	thus	thus	ADV
ejpam-5279	421	40	,	,	PUNCT
ejpam-5279	421	41	we	we	PRON
ejpam-5279	421	42	have	have	VERB
ejpam-5279	421	43	υλ	υλ	NUM
ejpam-5279	421	44	(	(	PUNCT
ejpam-5279	421	45	f	f	NOUN
ejpam-5279	421	46	∩g	∩g	PROPN
ejpam-5279	421	47	)	)	PUNCT
ejpam-5279	422	1	=	=	SYM
ejpam-5279	422	2	λ	λ	INTJ
ejpam-5279	422	3	(	(	PUNCT
ejpam-5279	422	4	(	(	PUNCT
ejpam-5279	422	5	f	f	PROPN
ejpam-5279	422	6	∩g)−1	∩g)−1	NUM
ejpam-5279	422	7	⊙	⊙	NOUN
ejpam-5279	422	8	(	(	PUNCT
ejpam-5279	422	9	f	f	PROPN
ejpam-5279	422	10	∩g	∩g	PROPN
ejpam-5279	422	11	)	)	PUNCT
ejpam-5279	422	12	)	)	PUNCT
ejpam-5279	422	13	(	(	PUNCT
ejpam-5279	422	14	e	e	NOUN
ejpam-5279	422	15	)	)	PUNCT
ejpam-5279	422	16	∨	∨	NUM
ejpam-5279	422	17	λ	λ	PROPN
ejpam-5279	422	18	(	(	PUNCT
ejpam-5279	422	19	(	(	PUNCT
ejpam-5279	422	20	f	f	X
ejpam-5279	422	21	∩g)⊙	∩g)⊙	PROPN
ejpam-5279	422	22	(	(	PUNCT
ejpam-5279	422	23	f	f	PROPN
ejpam-5279	422	24	∩g)−1	∩g)−1	NUM
ejpam-5279	422	25	)	)	PUNCT
ejpam-5279	422	26	(	(	PUNCT
ejpam-5279	422	27	e	e	NOUN
ejpam-5279	422	28	)	)	PUNCT
ejpam-5279	422	29	=	=	SYM
ejpam-5279	422	30	λ	λ	NOUN
ejpam-5279	422	31	(	(	PUNCT
ejpam-5279	422	32	(	(	PUNCT
ejpam-5279	422	33	f−1	f−1	PROPN
ejpam-5279	422	34	⊙	⊙	NOUN
ejpam-5279	422	35	f	f	PROPN
ejpam-5279	422	36	)	)	PUNCT
ejpam-5279	422	37	∩	∩	NOUN
ejpam-5279	422	38	(	(	PUNCT
ejpam-5279	422	39	f−1	f−1	PROPN
ejpam-5279	422	40	⊙g	⊙g	NOUN
ejpam-5279	422	41	)	)	PUNCT
ejpam-5279	422	42	∩	∩	NOUN
ejpam-5279	422	43	(	(	PUNCT
ejpam-5279	422	44	g−1	g−1	PROPN
ejpam-5279	422	45	⊙	⊙	PROPN
ejpam-5279	422	46	f	f	PROPN
ejpam-5279	422	47	)	)	PUNCT
ejpam-5279	422	48	∩	∩	NOUN
ejpam-5279	422	49	(	(	PUNCT
ejpam-5279	422	50	g−1	g−1	PROPN
ejpam-5279	422	51	⊙g	⊙g	PROPN
ejpam-5279	422	52	)	)	PUNCT
ejpam-5279	422	53	)	)	PUNCT
ejpam-5279	422	54	(	(	PUNCT
ejpam-5279	422	55	e	e	NOUN
ejpam-5279	422	56	)	)	PUNCT
ejpam-5279	422	57	∨	∨	NUM
ejpam-5279	422	58	λ	λ	PROPN
ejpam-5279	422	59	(	(	PUNCT
ejpam-5279	422	60	(	(	PUNCT
ejpam-5279	422	61	f⊙	f⊙	VERB
ejpam-5279	422	62	f−1	f−1	PROPN
ejpam-5279	422	63	)	)	PUNCT
ejpam-5279	422	64	∩	∩	NOUN
ejpam-5279	422	65	(	(	PUNCT
ejpam-5279	422	66	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	422	67	)	)	PUNCT
ejpam-5279	422	68	∩	∩	NOUN
ejpam-5279	422	69	(	(	PUNCT
ejpam-5279	422	70	g⊙	g⊙	PROPN
ejpam-5279	422	71	f−1	f−1	PROPN
ejpam-5279	422	72	)	)	PUNCT
ejpam-5279	422	73	∩	∩	NOUN
ejpam-5279	422	74	(	(	PUNCT
ejpam-5279	422	75	g⊙g−1	g⊙g−1	PROPN
ejpam-5279	422	76	)	)	PUNCT
ejpam-5279	422	77	)	)	PUNCT
ejpam-5279	422	78	(	(	PUNCT
ejpam-5279	422	79	e	e	NOUN
ejpam-5279	422	80	)	)	PUNCT
ejpam-5279	422	81	t.m.g	t.m.g	ADJ
ejpam-5279	422	82	.	.	PUNCT
ejpam-5279	423	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	423	2	,	,	PUNCT
ejpam-5279	423	3	fawzi	fawzi	PROPN
ejpam-5279	423	4	al	al	PROPN
ejpam-5279	423	5	-	-	PUNCT
ejpam-5279	423	6	thukair	thukair	NOUN
ejpam-5279	423	7	/	/	SYM
ejpam-5279	423	8	eur	eur	NOUN
ejpam-5279	423	9	.	.	PUNCT
ejpam-5279	424	1	j.	j.	PROPN
ejpam-5279	424	2	pure	pure	PROPN
ejpam-5279	424	3	appl	appl	PROPN
ejpam-5279	424	4	.	.	PROPN
ejpam-5279	424	5	math	math	PROPN
ejpam-5279	424	6	,	,	PUNCT
ejpam-5279	424	7	17	17	NUM
ejpam-5279	424	8	(	(	PUNCT
ejpam-5279	424	9	3	3	NUM
ejpam-5279	424	10	)	)	PUNCT
ejpam-5279	424	11	(	(	PUNCT
ejpam-5279	424	12	2024	2024	NUM
ejpam-5279	424	13	)	)	PUNCT
ejpam-5279	424	14	,	,	PUNCT
ejpam-5279	424	15	1762	1762	NUM
ejpam-5279	424	16	-	-	SYM
ejpam-5279	424	17	1778	1778	NUM
ejpam-5279	424	18	1774	1774	NUM
ejpam-5279	424	19	=	=	PUNCT
ejpam-5279	425	1	[	[	X
ejpam-5279	425	2	λ	λ	X
ejpam-5279	425	3	(	(	PUNCT
ejpam-5279	425	4	f−1	f−1	PROPN
ejpam-5279	425	5	⊙	⊙	NOUN
ejpam-5279	425	6	f	f	PROPN
ejpam-5279	425	7	)	)	PUNCT
ejpam-5279	425	8	(	(	PUNCT
ejpam-5279	425	9	e	e	NOUN
ejpam-5279	425	10	)	)	PUNCT
ejpam-5279	425	11	∨	∨	NUM
ejpam-5279	425	12	λ	λ	PROPN
ejpam-5279	425	13	(	(	PUNCT
ejpam-5279	425	14	f⊙	f⊙	VERB
ejpam-5279	425	15	f−1	f−1	PROPN
ejpam-5279	425	16	)	)	PUNCT
ejpam-5279	425	17	(	(	PUNCT
ejpam-5279	425	18	e	e	NOUN
ejpam-5279	425	19	)	)	PUNCT
ejpam-5279	425	20	]	]	PUNCT
ejpam-5279	426	1	∨	∨	PUNCT
ejpam-5279	427	1	[	[	X
ejpam-5279	427	2	λ	λ	X
ejpam-5279	427	3	(	(	PUNCT
ejpam-5279	427	4	g−1	g−1	PROPN
ejpam-5279	427	5	⊙g	⊙g	PROPN
ejpam-5279	427	6	)	)	PUNCT
ejpam-5279	427	7	(	(	PUNCT
ejpam-5279	427	8	e	e	NOUN
ejpam-5279	427	9	)	)	PUNCT
ejpam-5279	427	10	∨	∨	NUM
ejpam-5279	427	11	λ	λ	PROPN
ejpam-5279	427	12	(	(	PUNCT
ejpam-5279	427	13	g⊙g−1	g⊙g−1	PROPN
ejpam-5279	427	14	)	)	PUNCT
ejpam-5279	427	15	(	(	PUNCT
ejpam-5279	427	16	e)]∨	e)]∨	PROPN
ejpam-5279	427	17	[	[	X
ejpam-5279	427	18	λ	λ	X
ejpam-5279	427	19	(	(	PUNCT
ejpam-5279	427	20	f−1	f−1	PROPN
ejpam-5279	427	21	⊙g	⊙g	PROPN
ejpam-5279	427	22	)	)	PUNCT
ejpam-5279	427	23	(	(	PUNCT
ejpam-5279	427	24	e	e	NOUN
ejpam-5279	427	25	)	)	PUNCT
ejpam-5279	427	26	∨	∨	NUM
ejpam-5279	427	27	λ	λ	PROPN
ejpam-5279	427	28	(	(	PUNCT
ejpam-5279	427	29	g⊙	g⊙	PROPN
ejpam-5279	427	30	f−1	f−1	PROPN
ejpam-5279	427	31	)	)	PUNCT
ejpam-5279	427	32	(	(	PUNCT
ejpam-5279	427	33	e	e	NOUN
ejpam-5279	427	34	)	)	PUNCT
ejpam-5279	427	35	]	]	PUNCT
ejpam-5279	428	1	∨	∨	PUNCT
ejpam-5279	429	1	[	[	X
ejpam-5279	429	2	λ	λ	X
ejpam-5279	429	3	(	(	PUNCT
ejpam-5279	429	4	g−1	g−1	PROPN
ejpam-5279	429	5	⊙	⊙	PROPN
ejpam-5279	429	6	f	f	PROPN
ejpam-5279	429	7	)	)	PUNCT
ejpam-5279	429	8	(	(	PUNCT
ejpam-5279	429	9	e	e	NOUN
ejpam-5279	429	10	)	)	PUNCT
ejpam-5279	429	11	∨	∨	NUM
ejpam-5279	429	12	λ	λ	PROPN
ejpam-5279	429	13	(	(	PUNCT
ejpam-5279	429	14	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	429	15	)	)	PUNCT
ejpam-5279	429	16	(	(	PUNCT
ejpam-5279	429	17	e	e	NOUN
ejpam-5279	429	18	)	)	PUNCT
ejpam-5279	429	19	]	]	PUNCT
ejpam-5279	429	20	≤	≤	NUM
ejpam-5279	429	21	υλ(f	υλ(f	NOUN
ejpam-5279	429	22	)	)	PUNCT
ejpam-5279	429	23	∨υλ(g	∨υλ(g	PROPN
ejpam-5279	429	24	)	)	PUNCT
ejpam-5279	429	25	∨υλ(g	∨υλ(g	PROPN
ejpam-5279	429	26	)	)	PUNCT
ejpam-5279	429	27	∨υλ(f	∨υλ(f	PROPN
ejpam-5279	429	28	)	)	PUNCT
ejpam-5279	429	29	=	=	SYM
ejpam-5279	429	30	υλ(f	υλ(f	X
ejpam-5279	429	31	)	)	PUNCT
ejpam-5279	429	32	∨υλ(g	∨υλ(g	PROPN
ejpam-5279	429	33	)	)	PUNCT
ejpam-5279	429	34	.	.	PUNCT
ejpam-5279	430	1	definition	definition	NOUN
ejpam-5279	430	2	15	15	NUM
ejpam-5279	430	3	.	.	PUNCT
ejpam-5279	431	1	an	an	DET
ejpam-5279	431	2	ultra	ultra	ADJ
ejpam-5279	431	3	approach	approach	NOUN
ejpam-5279	431	4	limit	limit	NOUN
ejpam-5279	431	5	group	group	NOUN
ejpam-5279	431	6	(	(	PUNCT
ejpam-5279	431	7	x	x	X
ejpam-5279	431	8	,	,	PUNCT
ejpam-5279	431	9	·	·	PUNCT
ejpam-5279	431	10	,	,	PUNCT
ejpam-5279	431	11	λ	λ	X
ejpam-5279	431	12	)	)	PUNCT
ejpam-5279	431	13	is	be	AUX
ejpam-5279	431	14	called	call	VERB
ejpam-5279	431	15	strongly	strongly	ADV
ejpam-5279	431	16	normal	normal	ADJ
ejpam-5279	431	17	if	if	SCONJ
ejpam-5279	431	18	and	and	CCONJ
ejpam-5279	431	19	only	only	ADV
ejpam-5279	431	20	if	if	SCONJ
ejpam-5279	431	21	for	for	ADP
ejpam-5279	431	22	all	all	DET
ejpam-5279	431	23	f	f	NOUN
ejpam-5279	431	24	,	,	PUNCT
ejpam-5279	431	25	g	g	PROPN
ejpam-5279	431	26	∈	∈	PROPN
ejpam-5279	431	27	f(x	f(x	PROPN
ejpam-5279	431	28	)	)	PUNCT
ejpam-5279	431	29	,	,	PUNCT
ejpam-5279	431	30	λ	λ	PROPN
ejpam-5279	431	31	(	(	PUNCT
ejpam-5279	431	32	f⊙g⊙	f⊙g⊙	NOUN
ejpam-5279	431	33	f−1	f−1	PROPN
ejpam-5279	431	34	)	)	PUNCT
ejpam-5279	431	35	(	(	PUNCT
ejpam-5279	431	36	e	e	NOUN
ejpam-5279	431	37	)	)	PUNCT
ejpam-5279	431	38	≤	≤	NUM
ejpam-5279	431	39	λ	λ	PROPN
ejpam-5279	431	40	(	(	PUNCT
ejpam-5279	431	41	f⊙	f⊙	VERB
ejpam-5279	431	42	f−1	f−1	PROPN
ejpam-5279	431	43	)	)	PUNCT
ejpam-5279	431	44	(	(	PUNCT
ejpam-5279	431	45	e	e	NOUN
ejpam-5279	431	46	)	)	PUNCT
ejpam-5279	431	47	∨	∨	NUM
ejpam-5279	431	48	λ	λ	PROPN
ejpam-5279	431	49	(	(	PUNCT
ejpam-5279	431	50	f−1	f−1	PROPN
ejpam-5279	431	51	⊙	⊙	NOUN
ejpam-5279	431	52	f	f	PROPN
ejpam-5279	431	53	)	)	PUNCT
ejpam-5279	431	54	(	(	PUNCT
ejpam-5279	431	55	e	e	NOUN
ejpam-5279	431	56	)	)	PUNCT
ejpam-5279	431	57	∨	∨	NUM
ejpam-5279	431	58	λ(g)(e	λ(g)(e	NOUN
ejpam-5279	431	59	)	)	PUNCT
ejpam-5279	431	60	.	.	PUNCT
ejpam-5279	432	1	proposition	proposition	NOUN
ejpam-5279	432	2	5	5	NUM
ejpam-5279	432	3	.	.	PUNCT
ejpam-5279	433	1	let	let	VERB
ejpam-5279	433	2	(	(	PUNCT
ejpam-5279	433	3	x	x	NOUN
ejpam-5279	433	4	,	,	PUNCT
ejpam-5279	433	5	·	·	PUNCT
ejpam-5279	433	6	,	,	PUNCT
ejpam-5279	433	7	λ	λ	NOUN
ejpam-5279	433	8	)	)	PUNCT
ejpam-5279	433	9	,	,	PUNCT
ejpam-5279	433	10	and	and	CCONJ
ejpam-5279	433	11	(	(	PUNCT
ejpam-5279	433	12	x	x	NOUN
ejpam-5279	433	13	′	′	NUM
ejpam-5279	433	14	,	,	PUNCT
ejpam-5279	433	15	·	·	PUNCT
ejpam-5279	433	16	,	,	PUNCT
ejpam-5279	433	17	λ′	λ′	NUM
ejpam-5279	433	18	)	)	PUNCT
ejpam-5279	433	19	be	be	VERB
ejpam-5279	433	20	strongly	strongly	ADV
ejpam-5279	433	21	normal	normal	ADJ
ejpam-5279	433	22	ultra	ultra	ADJ
ejpam-5279	433	23	approach	approach	NOUN
ejpam-5279	433	24	limit	limit	NOUN
ejpam-5279	433	25	groups	group	NOUN
ejpam-5279	433	26	.	.	PUNCT
ejpam-5279	434	1	if	if	SCONJ
ejpam-5279	434	2	f	f	PROPN
ejpam-5279	434	3	:	:	PUNCT
ejpam-5279	434	4	x	x	PUNCT
ejpam-5279	434	5	−→	−→	NOUN
ejpam-5279	434	6	x	x	INTJ
ejpam-5279	434	7	′	′	NOUN
ejpam-5279	434	8	is	be	AUX
ejpam-5279	434	9	a	a	DET
ejpam-5279	434	10	group	group	NOUN
ejpam-5279	434	11	homomorphism	homomorphism	NOUN
ejpam-5279	434	12	,	,	PUNCT
ejpam-5279	434	13	then	then	ADV
ejpam-5279	434	14	the	the	DET
ejpam-5279	434	15	following	follow	VERB
ejpam-5279	434	16	assertions	assertion	NOUN
ejpam-5279	434	17	are	be	AUX
ejpam-5279	434	18	equivalent	equivalent	ADJ
ejpam-5279	434	19	:	:	PUNCT
ejpam-5279	434	20	(	(	PUNCT
ejpam-5279	434	21	i	i	NOUN
ejpam-5279	434	22	)	)	PUNCT
ejpam-5279	434	23	f	f	NOUN
ejpam-5279	434	24	:	:	PUNCT
ejpam-5279	434	25	(	(	PUNCT
ejpam-5279	434	26	x	x	X
ejpam-5279	434	27	,	,	PUNCT
ejpam-5279	434	28	λ	λ	NOUN
ejpam-5279	434	29	)	)	PUNCT
ejpam-5279	434	30	−→	−→	NOUN
ejpam-5279	434	31	(	(	PUNCT
ejpam-5279	434	32	x	x	NOUN
ejpam-5279	434	33	′	′	NUM
ejpam-5279	434	34	,	,	PUNCT
ejpam-5279	434	35	λ′	λ′	NUM
ejpam-5279	434	36	)	)	PUNCT
ejpam-5279	434	37	is	be	AUX
ejpam-5279	434	38	a	a	DET
ejpam-5279	434	39	contraction	contraction	NOUN
ejpam-5279	434	40	;	;	PUNCT
ejpam-5279	434	41	(	(	PUNCT
ejpam-5279	434	42	ii	ii	NOUN
ejpam-5279	434	43	)	)	PUNCT
ejpam-5279	434	44	f	f	NOUN
ejpam-5279	434	45	:	:	PUNCT
ejpam-5279	434	46	(	(	PUNCT
ejpam-5279	434	47	x	x	NOUN
ejpam-5279	434	48	,	,	PUNCT
ejpam-5279	434	49	υλ	υλ	NOUN
ejpam-5279	434	50	)	)	PUNCT
ejpam-5279	434	51	−→	−→	NOUN
ejpam-5279	434	52	(	(	PUNCT
ejpam-5279	434	53	x	x	NOUN
ejpam-5279	434	54	′,υλ′	′,υλ′	NOUN
ejpam-5279	434	55	)	)	PUNCT
ejpam-5279	434	56	is	be	AUX
ejpam-5279	434	57	cauchy	cauchy	ADJ
ejpam-5279	434	58	contraction	contraction	NOUN
ejpam-5279	434	59	.	.	PUNCT
ejpam-5279	435	1	proof	proof	NOUN
ejpam-5279	435	2	.	.	PUNCT
ejpam-5279	436	1	assume	assume	VERB
ejpam-5279	436	2	(	(	PUNCT
ejpam-5279	436	3	i	i	NOUN
ejpam-5279	436	4	)	)	PUNCT
ejpam-5279	436	5	holds	hold	VERB
ejpam-5279	436	6	.	.	PUNCT
ejpam-5279	437	1	then	then	ADV
ejpam-5279	437	2	using	use	VERB
ejpam-5279	437	3	lemma	lemma	PROPN
ejpam-5279	437	4	1	1	NUM
ejpam-5279	437	5	,	,	PUNCT
ejpam-5279	437	6	for	for	ADP
ejpam-5279	437	7	any	any	DET
ejpam-5279	437	8	f	f	PROPN
ejpam-5279	437	9	∈	∈	PROPN
ejpam-5279	437	10	f(x	f(x	PROPN
ejpam-5279	437	11	)	)	PUNCT
ejpam-5279	437	12	,	,	PUNCT
ejpam-5279	437	13	υλ′(f(f	υλ′(f(f	NOUN
ejpam-5279	437	14	)	)	PUNCT
ejpam-5279	437	15	)	)	PUNCT
ejpam-5279	438	1	=	=	SYM
ejpam-5279	438	2	λ′	λ′	X
ejpam-5279	438	3	(	(	PUNCT
ejpam-5279	438	4	f(f)−1	f(f)−1	PROPN
ejpam-5279	438	5	⊙	⊙	PROPN
ejpam-5279	438	6	f(f	f(f	PROPN
ejpam-5279	438	7	)	)	PUNCT
ejpam-5279	438	8	)	)	PUNCT
ejpam-5279	439	1	(	(	PUNCT
ejpam-5279	439	2	f(e))∨λ′	f(e))∨λ′	PROPN
ejpam-5279	439	3	(	(	PUNCT
ejpam-5279	439	4	f(f)⊙	f(f)⊙	PROPN
ejpam-5279	439	5	f(f)−1	f(f)−1	PROPN
ejpam-5279	439	6	)	)	PUNCT
ejpam-5279	439	7	)	)	PUNCT
ejpam-5279	439	8	(	(	PUNCT
ejpam-5279	439	9	f(e	f(e	NOUN
ejpam-5279	439	10	)	)	PUNCT
ejpam-5279	439	11	)	)	PUNCT
ejpam-5279	440	1	=	=	SYM
ejpam-5279	440	2	λ′	λ′	X
ejpam-5279	440	3	(	(	PUNCT
ejpam-5279	440	4	f(f−1	f(f−1	PROPN
ejpam-5279	440	5	⊙	⊙	PROPN
ejpam-5279	440	6	f	f	NOUN
ejpam-5279	440	7	)	)	PUNCT
ejpam-5279	440	8	)	)	PUNCT
ejpam-5279	441	1	(	(	PUNCT
ejpam-5279	441	2	f(e))∨	f(e))∨	VERB
ejpam-5279	441	3	λ′	λ′	X
ejpam-5279	441	4	(	(	PUNCT
ejpam-5279	441	5	f(f⊙	f(f⊙	ADV
ejpam-5279	441	6	f−1	f−1	PROPN
ejpam-5279	441	7	)	)	PUNCT
ejpam-5279	441	8	)	)	PUNCT
ejpam-5279	441	9	(	(	PUNCT
ejpam-5279	441	10	f(e	f(e	NOUN
ejpam-5279	441	11	)	)	PUNCT
ejpam-5279	441	12	)	)	PUNCT
ejpam-5279	441	13	≤	≤	NUM
ejpam-5279	442	1	λ(f−1	λ(f−1	PROPN
ejpam-5279	442	2	⊙	⊙	NOUN
ejpam-5279	442	3	f)(e)∨	f)(e)∨	PROPN
ejpam-5279	442	4	λ(f⊙	λ(f⊙	PROPN
ejpam-5279	442	5	f−1)(e	f−1)(e	NOUN
ejpam-5279	442	6	)	)	PUNCT
ejpam-5279	442	7	=	=	SYM
ejpam-5279	442	8	υλ(f	υλ(f	NOUN
ejpam-5279	442	9	)	)	PUNCT
ejpam-5279	442	10	,	,	PUNCT
ejpam-5279	442	11	i.e.	i.e.	X
ejpam-5279	442	12	,	,	PUNCT
ejpam-5279	442	13	υλ′(f(f	υλ′(f(f	NOUN
ejpam-5279	442	14	)	)	PUNCT
ejpam-5279	442	15	)	)	PUNCT
ejpam-5279	442	16	≤	≤	NOUN
ejpam-5279	442	17	υλ(f	υλ(f	NOUN
ejpam-5279	442	18	)	)	PUNCT
ejpam-5279	442	19	.	.	PUNCT
ejpam-5279	443	1	now	now	ADV
ejpam-5279	443	2	assume	assume	VERB
ejpam-5279	443	3	(	(	PUNCT
ejpam-5279	443	4	ii	ii	NOUN
ejpam-5279	443	5	)	)	PUNCT
ejpam-5279	443	6	,	,	PUNCT
ejpam-5279	443	7	and	and	CCONJ
ejpam-5279	443	8	let	let	VERB
ejpam-5279	443	9	f	f	PROPN
ejpam-5279	443	10	∈	∈	PROPN
ejpam-5279	443	11	f(x	f(x	PROPN
ejpam-5279	443	12	)	)	PUNCT
ejpam-5279	443	13	,	,	PUNCT
ejpam-5279	443	14	and	and	CCONJ
ejpam-5279	443	15	x	x	PUNCT
ejpam-5279	443	16	∈	∈	PROPN
ejpam-5279	443	17	x.	x.	NOUN
ejpam-5279	443	18	then	then	ADV
ejpam-5279	443	19	upon	upon	SCONJ
ejpam-5279	443	20	using	use	VERB
ejpam-5279	443	21	lemma	lemma	PROPN
ejpam-5279	443	22	3	3	NUM
ejpam-5279	443	23	,	,	PUNCT
ejpam-5279	443	24	we	we	PRON
ejpam-5279	443	25	get	get	VERB
ejpam-5279	443	26	λ′(f(f))(f(x	λ′(f(f))(f(x	NOUN
ejpam-5279	443	27	)	)	PUNCT
ejpam-5279	443	28	)	)	PUNCT
ejpam-5279	444	1	=	=	SYM
ejpam-5279	444	2	λ′	λ′	X
ejpam-5279	444	3	(	(	PUNCT
ejpam-5279	444	4	[	[	X
ejpam-5279	444	5	f(x)]−1	f(x)]−1	PROPN
ejpam-5279	444	6	⊙	⊙	PROPN
ejpam-5279	444	7	f(f	f(f	PROPN
ejpam-5279	444	8	)	)	PUNCT
ejpam-5279	444	9	)	)	PUNCT
ejpam-5279	444	10	(	(	PUNCT
ejpam-5279	444	11	f(e	f(e	NOUN
ejpam-5279	444	12	)	)	PUNCT
ejpam-5279	444	13	)	)	PUNCT
ejpam-5279	444	14	∨	∨	NUM
ejpam-5279	444	15	λ′	λ′	X
ejpam-5279	444	16	(	(	PUNCT
ejpam-5279	444	17	f(f)⊙	f(f)⊙	PROPN
ejpam-5279	444	18	[	[	X
ejpam-5279	444	19	f(x)]−1	f(x)]−1	NOUN
ejpam-5279	444	20	)	)	PUNCT
ejpam-5279	444	21	(	(	PUNCT
ejpam-5279	444	22	f(e	f(e	NOUN
ejpam-5279	444	23	)	)	PUNCT
ejpam-5279	444	24	)	)	PUNCT
ejpam-5279	444	25	≤	≤	NUM
ejpam-5279	444	26	λ′	λ′	X
ejpam-5279	444	27	(	(	PUNCT
ejpam-5279	444	28	(	(	PUNCT
ejpam-5279	444	29	[	[	X
ejpam-5279	444	30	f(x	f(x	PROPN
ejpam-5279	444	31	)	)	PUNCT
ejpam-5279	444	32	]	]	PUNCT
ejpam-5279	445	1	∧	∧	PROPN
ejpam-5279	445	2	f(f	f(f	PROPN
ejpam-5279	445	3	)	)	PUNCT
ejpam-5279	445	4	)	)	PUNCT
ejpam-5279	445	5	−1	−1	NOUN
ejpam-5279	445	6	⊙	⊙	NOUN
ejpam-5279	445	7	(	(	PUNCT
ejpam-5279	445	8	[	[	X
ejpam-5279	445	9	f(x	f(x	PROPN
ejpam-5279	445	10	)	)	PUNCT
ejpam-5279	445	11	]	]	PUNCT
ejpam-5279	445	12	∧	∧	PROPN
ejpam-5279	445	13	f(f	f(f	PROPN
ejpam-5279	445	14	)	)	PUNCT
ejpam-5279	445	15	)	)	PUNCT
ejpam-5279	446	1	(	(	PUNCT
ejpam-5279	446	2	f(e))∨λ′	f(e))∨λ′	PROPN
ejpam-5279	446	3	(	(	PUNCT
ejpam-5279	446	4	f(f	f(f	PROPN
ejpam-5279	446	5	)	)	PUNCT
ejpam-5279	446	6	∧	∧	PROPN
ejpam-5279	446	7	[	[	X
ejpam-5279	446	8	f(x)]⊙	f(x)]⊙	NOUN
ejpam-5279	446	9	(	(	PUNCT
ejpam-5279	446	10	f(f	f(f	PROPN
ejpam-5279	446	11	)	)	PUNCT
ejpam-5279	446	12	∧	∧	PROPN
ejpam-5279	447	1	[	[	X
ejpam-5279	447	2	f(x)])−1	f(x)])−1	PROPN
ejpam-5279	447	3	)	)	PUNCT
ejpam-5279	447	4	(	(	PUNCT
ejpam-5279	447	5	f(e	f(e	NOUN
ejpam-5279	447	6	)	)	PUNCT
ejpam-5279	447	7	)	)	PUNCT
ejpam-5279	448	1	=	=	SYM
ejpam-5279	448	2	υλ′(f(f)∧[f(x	υλ′(f(f)∧[f(x	PROPN
ejpam-5279	448	3	)	)	PUNCT
ejpam-5279	448	4	]	]	PUNCT
ejpam-5279	448	5	)	)	PUNCT
ejpam-5279	448	6	≤	≤	ADV
ejpam-5279	448	7	υλ(f∧[x	υλ(f∧[x	NOUN
ejpam-5279	448	8	]	]	PUNCT
ejpam-5279	448	9	)	)	PUNCT
ejpam-5279	449	1	=	=	PUNCT
ejpam-5279	450	1	λ((f∧[x])−1⊙(f∧[x])(e)∨λ((f∧[x])⊙(f∧[x])−1)(e	λ((f∧[x])−1⊙(f∧[x])(e)∨λ((f∧[x])⊙(f∧[x])−1)(e	ADJ
ejpam-5279	450	2	)	)	PUNCT
ejpam-5279	450	3	≤	≤	NUM
ejpam-5279	450	4	λ(f−1⊙f)(e	λ(f−1⊙f)(e	NOUN
ejpam-5279	451	1	=	=	PUNCT
ejpam-5279	451	2	x−1x)∨λ(f⊙f)(e	x−1x)∨λ(f⊙f)(e	PROPN
ejpam-5279	451	3	=	=	SYM
ejpam-5279	451	4	xx−1	xx−1	PROPN
ejpam-5279	451	5	)	)	PUNCT
ejpam-5279	451	6	≤	≤	NOUN
ejpam-5279	451	7	λ(f)(x)∨λ(f−1)(x−1	λ(f)(x)∨λ(f−1)(x−1	NOUN
ejpam-5279	451	8	)	)	PUNCT
ejpam-5279	451	9	≤	≤	NUM
ejpam-5279	451	10	λ(f)(x)∨λ(f)(x	λ(f)(x)∨λ(f)(x	PUNCT
ejpam-5279	451	11	)	)	PUNCT
ejpam-5279	451	12	=	=	PUNCT
ejpam-5279	451	13	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	451	14	)	)	PUNCT
ejpam-5279	451	15	,	,	PUNCT
ejpam-5279	451	16	which	which	PRON
ejpam-5279	451	17	proves	prove	VERB
ejpam-5279	451	18	that	that	SCONJ
ejpam-5279	451	19	λ′(f(f))(f(x	λ′(f(f))(f(x	NOUN
ejpam-5279	451	20	)	)	PUNCT
ejpam-5279	451	21	)	)	PUNCT
ejpam-5279	451	22	≤	≤	NUM
ejpam-5279	451	23	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	451	24	)	)	PUNCT
ejpam-5279	451	25	,	,	PUNCT
ejpam-5279	451	26	i.e.	i.e.	X
ejpam-5279	451	27	,	,	PUNCT
ejpam-5279	451	28	f	f	X
ejpam-5279	451	29	:	:	PUNCT
ejpam-5279	451	30	x	x	PUNCT
ejpam-5279	451	31	−→	−→	NOUN
ejpam-5279	451	32	x	x	INTJ
ejpam-5279	451	33	′	′	NOUN
ejpam-5279	451	34	is	be	AUX
ejpam-5279	451	35	a	a	DET
ejpam-5279	451	36	contraction	contraction	NOUN
ejpam-5279	451	37	.	.	PUNCT
ejpam-5279	452	1	lemma	lemma	PROPN
ejpam-5279	452	2	4	4	X
ejpam-5279	452	3	.	.	PUNCT
ejpam-5279	453	1	if	if	SCONJ
ejpam-5279	453	2	(	(	PUNCT
ejpam-5279	453	3	x	x	X
ejpam-5279	453	4	,	,	PUNCT
ejpam-5279	453	5	·	·	PUNCT
ejpam-5279	453	6	,	,	PUNCT
ejpam-5279	453	7	υ	υ	NOUN
ejpam-5279	453	8	)	)	PUNCT
ejpam-5279	453	9	is	be	AUX
ejpam-5279	453	10	a	a	DET
ejpam-5279	453	11	ultra	ultra	ADJ
ejpam-5279	453	12	approach	approach	NOUN
ejpam-5279	453	13	-	-	PUNCT
ejpam-5279	453	14	cauchy	cauchy	NOUN
ejpam-5279	453	15	group	group	NOUN
ejpam-5279	453	16	,	,	PUNCT
ejpam-5279	453	17	then	then	ADV
ejpam-5279	453	18	(	(	PUNCT
ejpam-5279	453	19	x	x	X
ejpam-5279	453	20	,	,	PUNCT
ejpam-5279	453	21	·	·	PUNCT
ejpam-5279	453	22	,	,	PUNCT
ejpam-5279	453	23	λυ	λυ	ADP
ejpam-5279	453	24	)	)	PUNCT
ejpam-5279	453	25	is	be	AUX
ejpam-5279	453	26	an	an	DET
ejpam-5279	453	27	ultra	ultra	ADJ
ejpam-5279	453	28	approach	approach	NOUN
ejpam-5279	453	29	-	-	PUNCT
ejpam-5279	453	30	limit	limit	NOUN
ejpam-5279	453	31	group	group	NOUN
ejpam-5279	453	32	.	.	PUNCT
ejpam-5279	454	1	proof	proof	NOUN
ejpam-5279	454	2	.	.	PUNCT
ejpam-5279	455	1	in	in	ADP
ejpam-5279	455	2	view	view	NOUN
ejpam-5279	455	3	of	of	ADP
ejpam-5279	455	4	the	the	DET
ejpam-5279	455	5	proposition	proposition	NOUN
ejpam-5279	455	6	5.16	5.16	NUM
ejpam-5279	455	7	[	[	X
ejpam-5279	455	8	17	17	NUM
ejpam-5279	455	9	]	]	PUNCT
ejpam-5279	455	10	,	,	PUNCT
ejpam-5279	455	11	we	we	PRON
ejpam-5279	455	12	only	only	ADV
ejpam-5279	455	13	need	need	VERB
ejpam-5279	455	14	to	to	PART
ejpam-5279	455	15	show	show	VERB
ejpam-5279	455	16	that	that	SCONJ
ejpam-5279	455	17	the	the	DET
ejpam-5279	455	18	group	group	NOUN
ejpam-5279	455	19	operation	operation	NOUN
ejpam-5279	455	20	h	h	NOUN
ejpam-5279	455	21	:	:	PUNCT
ejpam-5279	455	22	x	x	X
ejpam-5279	455	23	×x	×x	X
ejpam-5279	455	24	→	→	SYM
ejpam-5279	455	25	x	x	SYM
ejpam-5279	455	26	,	,	PUNCT
ejpam-5279	455	27	(	(	PUNCT
ejpam-5279	455	28	x	x	NOUN
ejpam-5279	455	29	,	,	PUNCT
ejpam-5279	455	30	y	y	NOUN
ejpam-5279	455	31	)	)	PUNCT
ejpam-5279	455	32	7→	7→	NUM
ejpam-5279	455	33	xy−1	xy−1	PROPN
ejpam-5279	455	34	is	be	AUX
ejpam-5279	455	35	a	a	DET
ejpam-5279	455	36	contraction	contraction	NOUN
ejpam-5279	455	37	.	.	PUNCT
ejpam-5279	456	1	if	if	SCONJ
ejpam-5279	456	2	f	f	X
ejpam-5279	456	3	,	,	PUNCT
ejpam-5279	456	4	g	g	PROPN
ejpam-5279	456	5	∈	∈	PROPN
ejpam-5279	456	6	f(x	f(x	PROPN
ejpam-5279	456	7	)	)	PUNCT
ejpam-5279	456	8	and	and	CCONJ
ejpam-5279	456	9	x	x	X
ejpam-5279	456	10	,	,	PUNCT
ejpam-5279	456	11	y	y	PROPN
ejpam-5279	456	12	∈	∈	PROPN
ejpam-5279	456	13	x	x	X
ejpam-5279	456	14	,	,	PUNCT
ejpam-5279	456	15	then	then	ADV
ejpam-5279	456	16	by	by	ADP
ejpam-5279	456	17	the	the	DET
ejpam-5279	456	18	lemma	lemma	PROPN
ejpam-5279	456	19	1(iii	1(iii	NUM
ejpam-5279	456	20	)	)	PUNCT
ejpam-5279	456	21	,	,	PUNCT
ejpam-5279	456	22	we	we	PRON
ejpam-5279	456	23	have	have	VERB
ejpam-5279	456	24	(	(	PUNCT
ejpam-5279	456	25	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	456	26	)	)	PUNCT
ejpam-5279	456	27	∩	∩	PROPN
ejpam-5279	456	28	˙̂	˙̂	VERB
ejpam-5279	456	29	xy−1	xy−1	PROPN
ejpam-5279	456	30	=	=	PRON
ejpam-5279	456	31	(	(	PUNCT
ejpam-5279	456	32	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	456	33	)	)	PUNCT
ejpam-5279	456	34	∩	∩	NOUN
ejpam-5279	456	35	(	(	PUNCT
ejpam-5279	456	36	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	456	37	ẏ−1	ẏ−1	PROPN
ejpam-5279	456	38	)	)	PUNCT
ejpam-5279	456	39	≥	≥	NOUN
ejpam-5279	456	40	(	(	PUNCT
ejpam-5279	456	41	f	f	PROPN
ejpam-5279	456	42	∩	∩	PROPN
ejpam-5279	456	43	ẋ)⊙	ẋ)⊙	PROPN
ejpam-5279	456	44	(	(	PUNCT
ejpam-5279	456	45	g−1	g−1	X
ejpam-5279	456	46	∩	∩	X
ejpam-5279	456	47	ẏ−1	ẏ−1	PROPN
ejpam-5279	456	48	)	)	PUNCT
ejpam-5279	456	49	.	.	PUNCT
ejpam-5279	457	1	upon	upon	SCONJ
ejpam-5279	457	2	using	use	VERB
ejpam-5279	457	3	(	(	PUNCT
ejpam-5279	457	4	uachy2	uachy2	PROPN
ejpam-5279	457	5	)	)	PUNCT
ejpam-5279	457	6	,	,	PUNCT
ejpam-5279	457	7	we	we	PRON
ejpam-5279	457	8	get	get	VERB
ejpam-5279	457	9	υ	υ	PRON
ejpam-5279	457	10	(	(	PUNCT
ejpam-5279	457	11	(	(	PUNCT
ejpam-5279	457	12	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	457	13	)	)	PUNCT
ejpam-5279	457	14	∩	∩	NOUN
ejpam-5279	457	15	(	(	PUNCT
ejpam-5279	457	16	ẋ⊙	ẋ⊙	PROPN
ejpam-5279	457	17	ẏ−1	ẏ−1	PROPN
ejpam-5279	457	18	)	)	PUNCT
ejpam-5279	457	19	)	)	PUNCT
ejpam-5279	457	20	≤	≤	NUM
ejpam-5279	458	1	υ	υ	NOUN
ejpam-5279	458	2	(	(	PUNCT
ejpam-5279	458	3	(	(	PUNCT
ejpam-5279	458	4	f	f	PROPN
ejpam-5279	458	5	∩	∩	PROPN
ejpam-5279	458	6	ẋ)⊙	ẋ)⊙	PROPN
ejpam-5279	458	7	(	(	PUNCT
ejpam-5279	458	8	g−1	g−1	X
ejpam-5279	458	9	∩	∩	X
ejpam-5279	458	10	ẏ−1	ẏ−1	PROPN
ejpam-5279	458	11	)	)	PUNCT
ejpam-5279	458	12	)	)	PUNCT
ejpam-5279	459	1	=	=	SYM
ejpam-5279	459	2	υ	υ	PROPN
ejpam-5279	459	3	(	(	PUNCT
ejpam-5279	459	4	(	(	PUNCT
ejpam-5279	459	5	f	f	PROPN
ejpam-5279	459	6	∩	∩	PROPN
ejpam-5279	459	7	ẋ)⊙	ẋ)⊙	PROPN
ejpam-5279	459	8	(	(	PUNCT
ejpam-5279	459	9	g	g	PROPN
ejpam-5279	459	10	∩	∩	X
ejpam-5279	459	11	ẏ)−1	ẏ)−1	NOUN
ejpam-5279	459	12	)	)	PUNCT
ejpam-5279	459	13	≤	≤	PUNCT
ejpam-5279	460	1	υ(f	υ(f	PROPN
ejpam-5279	460	2	∩	∩	PROPN
ejpam-5279	460	3	ẋ	ẋ	PROPN
ejpam-5279	460	4	)	)	PUNCT
ejpam-5279	460	5	∨υ(g	∨υ(g	NOUN
ejpam-5279	460	6	∩	∩	ADJ
ejpam-5279	460	7	ẏ	ẏ	NOUN
ejpam-5279	460	8	)	)	PUNCT
ejpam-5279	460	9	=	=	SYM
ejpam-5279	460	10	λυ(f)(x	λυ(f)(x	NOUN
ejpam-5279	460	11	)	)	PUNCT
ejpam-5279	460	12	∨	∨	NOUN
ejpam-5279	460	13	λ(g)(y	λ(g)(y	NUM
ejpam-5279	460	14	)	)	PUNCT
ejpam-5279	460	15	.	.	PUNCT
ejpam-5279	461	1	consequently	consequently	ADV
ejpam-5279	461	2	,	,	PUNCT
ejpam-5279	461	3	we	we	PRON
ejpam-5279	461	4	have	have	VERB
ejpam-5279	461	5	λυ	λυ	ADP
ejpam-5279	461	6	(	(	PUNCT
ejpam-5279	461	7	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	461	8	)	)	PUNCT
ejpam-5279	462	1	(	(	PUNCT
ejpam-5279	462	2	xy−1	xy−1	PROPN
ejpam-5279	462	3	)	)	PUNCT
ejpam-5279	463	1	=	=	SYM
ejpam-5279	463	2	υ	υ	PROPN
ejpam-5279	463	3	(	(	PUNCT
ejpam-5279	463	4	(	(	PUNCT
ejpam-5279	463	5	f⊙g−1	f⊙g−1	PROPN
ejpam-5279	463	6	)	)	PUNCT
ejpam-5279	463	7	∩	∩	PROPN
ejpam-5279	463	8	˙̂	˙̂	PROPN
ejpam-5279	463	9	xy−1	xy−1	PROPN
ejpam-5279	463	10	)	)	PUNCT
ejpam-5279	464	1	≤	≤	NUM
ejpam-5279	464	2	λυ(f)(x	λυ(f)(x	NOUN
ejpam-5279	464	3	)	)	PUNCT
ejpam-5279	464	4	∨	∨	NUM
ejpam-5279	464	5	λυ(g)(y	λυ(g)(y	NUM
ejpam-5279	464	6	)	)	PUNCT
ejpam-5279	464	7	.	.	PUNCT
ejpam-5279	465	1	lemma	lemma	PROPN
ejpam-5279	465	2	5	5	NUM
ejpam-5279	465	3	.	.	PUNCT
ejpam-5279	466	1	if	if	SCONJ
ejpam-5279	466	2	(	(	PUNCT
ejpam-5279	466	3	x	x	X
ejpam-5279	466	4	,	,	PUNCT
ejpam-5279	466	5	·	·	PUNCT
ejpam-5279	466	6	,	,	PUNCT
ejpam-5279	466	7	λ	λ	X
ejpam-5279	466	8	)	)	PUNCT
ejpam-5279	466	9	is	be	AUX
ejpam-5279	466	10	an	an	DET
ejpam-5279	466	11	ultra	ultra	ADJ
ejpam-5279	466	12	approach	approach	NOUN
ejpam-5279	466	13	limit	limit	NOUN
ejpam-5279	466	14	group	group	NOUN
ejpam-5279	466	15	,	,	PUNCT
ejpam-5279	466	16	then	then	ADV
ejpam-5279	466	17	(	(	PUNCT
ejpam-5279	466	18	x	x	X
ejpam-5279	466	19	,	,	PUNCT
ejpam-5279	466	20	·	·	PUNCT
ejpam-5279	466	21	,	,	PUNCT
ejpam-5279	466	22	υλ	υλ	NOUN
ejpam-5279	466	23	)	)	PUNCT
ejpam-5279	466	24	is	be	AUX
ejpam-5279	466	25	an	an	DET
ejpam-5279	466	26	ultra	ultra	ADJ
ejpam-5279	466	27	approachcauchy	approachcauchy	NOUN
ejpam-5279	466	28	group	group	NOUN
ejpam-5279	466	29	if	if	SCONJ
ejpam-5279	467	1	and	and	CCONJ
ejpam-5279	467	2	only	only	ADV
ejpam-5279	467	3	if	if	SCONJ
ejpam-5279	467	4	it	it	PRON
ejpam-5279	467	5	is	be	AUX
ejpam-5279	467	6	strongly	strongly	ADV
ejpam-5279	467	7	normal	normal	ADJ
ejpam-5279	467	8	approach	approach	NOUN
ejpam-5279	467	9	limit	limit	NOUN
ejpam-5279	467	10	groups	group	NOUN
ejpam-5279	467	11	.	.	PUNCT
ejpam-5279	468	1	proof	proof	NOUN
ejpam-5279	468	2	.	.	PUNCT
ejpam-5279	469	1	define	define	VERB
ejpam-5279	469	2	υ(f	υ(f	PROPN
ejpam-5279	469	3	)	)	PUNCT
ejpam-5279	470	1	=	=	PRON
ejpam-5279	471	1	λ(f−1	λ(f−1	PROPN
ejpam-5279	471	2	⊙	⊙	NOUN
ejpam-5279	471	3	f)(e	f)(e	NOUN
ejpam-5279	471	4	)	)	PUNCT
ejpam-5279	471	5	∨	∨	NUM
ejpam-5279	471	6	λ(f⊙	λ(f⊙	SYM
ejpam-5279	471	7	f−1)(e	f−1)(e	NOUN
ejpam-5279	471	8	)	)	PUNCT
ejpam-5279	471	9	∨	∨	NUM
ejpam-5279	471	10	λ(g)(e	λ(g)(e	NOUN
ejpam-5279	471	11	)	)	PUNCT
ejpam-5279	471	12	,	,	PUNCT
ejpam-5279	471	13	for	for	ADP
ejpam-5279	471	14	all	all	DET
ejpam-5279	471	15	f	f	NOUN
ejpam-5279	471	16	,	,	PUNCT
ejpam-5279	471	17	g	g	PROPN
ejpam-5279	471	18	∈	∈	PROPN
ejpam-5279	471	19	f(x	f(x	PROPN
ejpam-5279	471	20	)	)	PUNCT
ejpam-5279	471	21	assume	assume	VERB
ejpam-5279	471	22	that	that	SCONJ
ejpam-5279	471	23	(	(	PUNCT
ejpam-5279	471	24	x	x	X
ejpam-5279	471	25	,	,	PUNCT
ejpam-5279	471	26	·	·	PUNCT
ejpam-5279	471	27	,	,	PUNCT
ejpam-5279	471	28	λ	λ	X
ejpam-5279	471	29	)	)	PUNCT
ejpam-5279	471	30	be	be	VERB
ejpam-5279	471	31	a	a	DET
ejpam-5279	471	32	strongly	strongly	ADV
ejpam-5279	471	33	normal	normal	ADJ
ejpam-5279	471	34	approach	approach	NOUN
ejpam-5279	471	35	limit	limit	NOUN
ejpam-5279	471	36	group	group	NOUN
ejpam-5279	471	37	,	,	PUNCT
ejpam-5279	471	38	we	we	PRON
ejpam-5279	471	39	prove	prove	VERB
ejpam-5279	471	40	that	that	SCONJ
ejpam-5279	471	41	(	(	PUNCT
ejpam-5279	471	42	x	x	X
ejpam-5279	471	43	,	,	PUNCT
ejpam-5279	471	44	·	·	PUNCT
ejpam-5279	471	45	,	,	PUNCT
ejpam-5279	471	46	υλ	υλ	NOUN
ejpam-5279	471	47	)	)	PUNCT
ejpam-5279	471	48	t.m.g	t.m.g	ADJ
ejpam-5279	471	49	.	.	PUNCT
ejpam-5279	472	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	472	2	,	,	PUNCT
ejpam-5279	472	3	fawzi	fawzi	PROPN
ejpam-5279	472	4	al	al	PROPN
ejpam-5279	472	5	-	-	PUNCT
ejpam-5279	472	6	thukair	thukair	NOUN
ejpam-5279	472	7	/	/	SYM
ejpam-5279	472	8	eur	eur	NOUN
ejpam-5279	472	9	.	.	PUNCT
ejpam-5279	473	1	j.	j.	PROPN
ejpam-5279	473	2	pure	pure	PROPN
ejpam-5279	473	3	appl	appl	PROPN
ejpam-5279	473	4	.	.	PROPN
ejpam-5279	473	5	math	math	PROPN
ejpam-5279	473	6	,	,	PUNCT
ejpam-5279	473	7	17	17	NUM
ejpam-5279	473	8	(	(	PUNCT
ejpam-5279	473	9	3	3	NUM
ejpam-5279	473	10	)	)	PUNCT
ejpam-5279	473	11	(	(	PUNCT
ejpam-5279	473	12	2024	2024	NUM
ejpam-5279	473	13	)	)	PUNCT
ejpam-5279	473	14	,	,	PUNCT
ejpam-5279	473	15	1762	1762	NUM
ejpam-5279	473	16	-	-	SYM
ejpam-5279	473	17	1778	1778	NUM
ejpam-5279	473	18	1775	1775	NUM
ejpam-5279	473	19	is	be	AUX
ejpam-5279	473	20	an	an	DET
ejpam-5279	473	21	ultra	ultra	ADJ
ejpam-5279	473	22	approach	approach	NOUN
ejpam-5279	473	23	-	-	PUNCT
ejpam-5279	473	24	cauchy	cauchy	NOUN
ejpam-5279	473	25	group	group	NOUN
ejpam-5279	473	26	.	.	PUNCT
ejpam-5279	474	1	in	in	ADP
ejpam-5279	474	2	view	view	NOUN
ejpam-5279	474	3	of	of	ADP
ejpam-5279	474	4	the	the	DET
ejpam-5279	474	5	proposition	proposition	NOUN
ejpam-5279	474	6	4	4	NUM
ejpam-5279	474	7	,	,	PUNCT
ejpam-5279	474	8	only	only	ADV
ejpam-5279	474	9	we	we	PRON
ejpam-5279	474	10	need	need	VERB
ejpam-5279	474	11	to	to	PART
ejpam-5279	474	12	prove	prove	VERB
ejpam-5279	474	13	that	that	SCONJ
ejpam-5279	474	14	the	the	DET
ejpam-5279	474	15	group	group	NOUN
ejpam-5279	474	16	operations	operation	NOUN
ejpam-5279	474	17	are	be	AUX
ejpam-5279	474	18	contractive	contractive	ADJ
ejpam-5279	474	19	.	.	PUNCT
ejpam-5279	475	1	since	since	SCONJ
ejpam-5279	475	2	(	(	PUNCT
ejpam-5279	475	3	x	x	X
ejpam-5279	475	4	,	,	PUNCT
ejpam-5279	475	5	·	·	PUNCT
ejpam-5279	475	6	,	,	PUNCT
ejpam-5279	475	7	λ	λ	X
ejpam-5279	475	8	)	)	PUNCT
ejpam-5279	475	9	is	be	AUX
ejpam-5279	475	10	strongly	strongly	ADV
ejpam-5279	475	11	normal	normal	ADJ
ejpam-5279	475	12	,	,	PUNCT
ejpam-5279	475	13	we	we	PRON
ejpam-5279	475	14	have	have	VERB
ejpam-5279	475	15	for	for	ADP
ejpam-5279	475	16	any	any	DET
ejpam-5279	475	17	f	f	NOUN
ejpam-5279	475	18	,	,	PUNCT
ejpam-5279	475	19	g	g	PROPN
ejpam-5279	475	20	∈	∈	PROPN
ejpam-5279	475	21	f(x	f(x	PROPN
ejpam-5279	475	22	)	)	PUNCT
ejpam-5279	475	23	,	,	PUNCT
ejpam-5279	475	24	υ(f⊙g	υ(f⊙g	NOUN
ejpam-5279	475	25	)	)	PUNCT
ejpam-5279	476	1	=	=	SYM
ejpam-5279	476	2	λ((f⊙g)⊙	λ((f⊙g)⊙	NOUN
ejpam-5279	476	3	(	(	PUNCT
ejpam-5279	476	4	f⊙g)−1)(e	f⊙g)−1)(e	ADJ
ejpam-5279	476	5	)	)	PUNCT
ejpam-5279	476	6	∨	∨	PROPN
ejpam-5279	476	7	λ((f⊙g)−1	λ((f⊙g)−1	NOUN
ejpam-5279	476	8	⊙	⊙	NOUN
ejpam-5279	476	9	(	(	PUNCT
ejpam-5279	476	10	f⊙g))(e	f⊙g))(e	PROPN
ejpam-5279	476	11	)	)	PUNCT
ejpam-5279	476	12	=	=	SYM
ejpam-5279	476	13	λ	λ	PROPN
ejpam-5279	476	14	(	(	PUNCT
ejpam-5279	476	15	f⊙g⊙g−1	f⊙g⊙g−1	NOUN
ejpam-5279	476	16	⊙	⊙	PROPN
ejpam-5279	476	17	f−1	f−1	PROPN
ejpam-5279	476	18	)	)	PUNCT
ejpam-5279	476	19	(	(	PUNCT
ejpam-5279	476	20	e)∨λ(g−1⊙f−1⊙f⊙g)(e	e)∨λ(g−1⊙f−1⊙f⊙g)(e	PROPN
ejpam-5279	476	21	)	)	PUNCT
ejpam-5279	476	22	≤	≤	NOUN
ejpam-5279	476	23	λ(f−1⊙f)(e)∨λ(f⊙f−1)(e)∨	λ(f−1⊙f)(e)∨λ(f⊙f−1)(e)∨	X
ejpam-5279	476	24	λ(g	λ(g	PROPN
ejpam-5279	476	25	⊙	⊙	PROPN
ejpam-5279	476	26	g−1)(e	g−1)(e	PROPN
ejpam-5279	476	27	)	)	PUNCT
ejpam-5279	476	28	∨	∨	PROPN
ejpam-5279	476	29	λ((g	λ((g	PROPN
ejpam-5279	476	30	⊙	⊙	PROPN
ejpam-5279	476	31	g))(e	g))(e	PROPN
ejpam-5279	476	32	)	)	PUNCT
ejpam-5279	476	33	∨	∨	PROPN
ejpam-5279	476	34	λ(g−1	λ(g−1	PROPN
ejpam-5279	476	35	⊙	⊙	PROPN
ejpam-5279	476	36	g−1))(e	g−1))(e	PROPN
ejpam-5279	476	37	)	)	PUNCT
ejpam-5279	476	38	∨	∨	NUM
ejpam-5279	476	39	λ(f−1	λ(f−1	PROPN
ejpam-5279	476	40	⊙	⊙	NOUN
ejpam-5279	476	41	f)(e	f)(e	NOUN
ejpam-5279	476	42	)	)	PUNCT
ejpam-5279	476	43	(	(	PUNCT
ejpam-5279	476	44	by	by	ADP
ejpam-5279	476	45	applying	apply	VERB
ejpam-5279	476	46	λ	λ	NOUN
ejpam-5279	476	47	to	to	ADP
ejpam-5279	476	48	[	[	X
ejpam-5279	476	49	e	e	X
ejpam-5279	476	50	]	]	X
ejpam-5279	476	51	=	=	PUNCT
ejpam-5279	477	1	[	[	X
ejpam-5279	477	2	e]⊙	e]⊙	X
ejpam-5279	477	3	[	[	X
ejpam-5279	477	4	e	e	X
ejpam-5279	477	5	]	]	X
ejpam-5279	477	6	≤	≤	NUM
ejpam-5279	477	7	g⊙g	g⊙g	NOUN
ejpam-5279	477	8	)	)	PUNCT
ejpam-5279	477	9	≤	≤	NUM
ejpam-5279	477	10	λ(f−1⊙f)(e)∨λ(f⊙f−1)(e)∨λ(g⊙g−1)(e)∨λ(g−1⊙g)(e	λ(f−1⊙f)(e)∨λ(f⊙f−1)(e)∨λ(g⊙g−1)(e)∨λ(g−1⊙g)(e	PROPN
ejpam-5279	477	11	)	)	PUNCT
ejpam-5279	477	12	=	=	SYM
ejpam-5279	477	13	υ(f)∨υ(g	υ(f)∨υ(g	NOUN
ejpam-5279	477	14	)	)	PUNCT
ejpam-5279	477	15	,	,	PUNCT
ejpam-5279	477	16	proving	prove	VERB
ejpam-5279	477	17	that	that	PRON
ejpam-5279	477	18	υ(f⊙g	υ(f⊙g	NOUN
ejpam-5279	477	19	)	)	PUNCT
ejpam-5279	477	20	≤	≤	PUNCT
ejpam-5279	477	21	υ(f	υ(f	PROPN
ejpam-5279	477	22	)	)	PUNCT
ejpam-5279	477	23	∨υ(g	∨υ(g	PROPN
ejpam-5279	477	24	)	)	PUNCT
ejpam-5279	477	25	.	.	PUNCT
ejpam-5279	478	1	finaly	finaly	VERB
ejpam-5279	478	2	,	,	PUNCT
ejpam-5279	478	3	for	for	ADP
ejpam-5279	478	4	any	any	DET
ejpam-5279	478	5	f	f	PROPN
ejpam-5279	478	6	∈	∈	PROPN
ejpam-5279	478	7	f(x	f(x	PROPN
ejpam-5279	478	8	)	)	PUNCT
ejpam-5279	478	9	,	,	PUNCT
ejpam-5279	478	10	we	we	PRON
ejpam-5279	478	11	have	have	VERB
ejpam-5279	478	12	υ(f−1	υ(f−1	NOUN
ejpam-5279	478	13	)	)	PUNCT
ejpam-5279	478	14	=	=	PUNCT
ejpam-5279	479	1	λ((f−1)−1	λ((f−1)−1	PROPN
ejpam-5279	479	2	⊙	⊙	X
ejpam-5279	479	3	f−1)(e	f−1)(e	PROPN
ejpam-5279	479	4	)	)	PUNCT
ejpam-5279	479	5	∨	∨	NUM
ejpam-5279	479	6	λ(f−1	λ(f−1	PROPN
ejpam-5279	479	7	⊙	⊙	X
ejpam-5279	479	8	(	(	PUNCT
ejpam-5279	479	9	f−1)−1)(e	f−1)−1)(e	NOUN
ejpam-5279	479	10	)	)	PUNCT
ejpam-5279	479	11	λ((f	λ((f	PUNCT
ejpam-5279	479	12	⊙	⊙	PROPN
ejpam-5279	479	13	f−1)(e	f−1)(e	PROPN
ejpam-5279	479	14	)	)	PUNCT
ejpam-5279	479	15	∨	∨	NUM
ejpam-5279	479	16	λ(f−1	λ(f−1	PROPN
ejpam-5279	479	17	⊙	⊙	X
ejpam-5279	479	18	(	(	PUNCT
ejpam-5279	479	19	f)(e	f)(e	X
ejpam-5279	479	20	)	)	PUNCT
ejpam-5279	479	21	=	=	SYM
ejpam-5279	479	22	υ(f	υ(f	PROPN
ejpam-5279	479	23	)	)	PUNCT
ejpam-5279	479	24	.	.	PUNCT
ejpam-5279	480	1	this	this	PRON
ejpam-5279	480	2	ends	end	VERB
ejpam-5279	480	3	the	the	DET
ejpam-5279	480	4	prove	prove	VERB
ejpam-5279	480	5	that	that	SCONJ
ejpam-5279	480	6	(	(	PUNCT
ejpam-5279	480	7	x	x	X
ejpam-5279	480	8	,	,	PUNCT
ejpam-5279	480	9	·	·	PUNCT
ejpam-5279	480	10	,	,	PUNCT
ejpam-5279	480	11	υ	υ	NOUN
ejpam-5279	480	12	)	)	PUNCT
ejpam-5279	480	13	is	be	AUX
ejpam-5279	480	14	an	an	DET
ejpam-5279	480	15	ultra	ultra	ADJ
ejpam-5279	480	16	approach	approach	NOUN
ejpam-5279	480	17	-	-	PUNCT
ejpam-5279	480	18	cauchy	cauchy	NOUN
ejpam-5279	480	19	group	group	NOUN
ejpam-5279	480	20	.	.	PUNCT
ejpam-5279	481	1	thus	thus	ADV
ejpam-5279	481	2	,	,	PUNCT
ejpam-5279	481	3	there	there	PRON
ejpam-5279	481	4	are	be	VERB
ejpam-5279	481	5	two	two	NUM
ejpam-5279	481	6	functors	functor	NOUN
ejpam-5279	481	7	a	a	PRON
ejpam-5279	481	8	and	and	CCONJ
ejpam-5279	481	9	b	b	NOUN
ejpam-5279	481	10	which	which	PRON
ejpam-5279	481	11	in	in	ADP
ejpam-5279	481	12	conjunction	conjunction	NOUN
ejpam-5279	481	13	with	with	ADP
ejpam-5279	481	14	the	the	DET
ejpam-5279	481	15	proposition	proposition	NOUN
ejpam-5279	481	16	5	5	NUM
ejpam-5279	481	17	yields	yield	VERB
ejpam-5279	481	18	the	the	DET
ejpam-5279	481	19	following	following	NOUN
ejpam-5279	481	20	:	:	PUNCT
ejpam-5279	481	21	a	a	DET
ejpam-5279	481	22	:	:	PUNCT
ejpam-5279	481	23			PUNCT
ejpam-5279	481	24	uapchygrp	uapchygrp	ADJ
ejpam-5279	481	25	−→	−→	NOUN
ejpam-5279	481	26	snaplimgrp	snaplimgrp	NOUN
ejpam-5279	481	27	(	(	PUNCT
ejpam-5279	481	28	x	x	NOUN
ejpam-5279	481	29	,	,	PUNCT
ejpam-5279	481	30	·	·	PUNCT
ejpam-5279	481	31	,	,	PUNCT
ejpam-5279	481	32	υ	υ	NOUN
ejpam-5279	481	33	)	)	PUNCT
ejpam-5279	481	34	7−→	7−→	NOUN
ejpam-5279	481	35	(	(	PUNCT
ejpam-5279	481	36	x	x	NOUN
ejpam-5279	481	37	,	,	PUNCT
ejpam-5279	481	38	·	·	PUNCT
ejpam-5279	481	39	,	,	PUNCT
ejpam-5279	481	40	λυ	λυ	ADP
ejpam-5279	481	41	)	)	PUNCT
ejpam-5279	481	42	f	f	NOUN
ejpam-5279	482	1	7−→	7−→	PROPN
ejpam-5279	482	2	f	f	PROPN
ejpam-5279	482	3	and	and	CCONJ
ejpam-5279	482	4	b	b	NOUN
ejpam-5279	482	5	:	:	PUNCT
ejpam-5279	482	6			PUNCT
ejpam-5279	482	7	snaplimgrp	snaplimgrp	VERB
ejpam-5279	482	8	−→	−→	ADJ
ejpam-5279	482	9	uapchygrp	uapchygrp	NOUN
ejpam-5279	482	10	(	(	PUNCT
ejpam-5279	482	11	x	x	X
ejpam-5279	482	12	,	,	PUNCT
ejpam-5279	482	13	·	·	PUNCT
ejpam-5279	482	14	,	,	PUNCT
ejpam-5279	482	15	λ	λ	NOUN
ejpam-5279	482	16	)	)	PUNCT
ejpam-5279	482	17	7−→	7−→	NOUN
ejpam-5279	482	18	(	(	PUNCT
ejpam-5279	482	19	x	x	NOUN
ejpam-5279	482	20	,	,	PUNCT
ejpam-5279	482	21	·	·	PUNCT
ejpam-5279	482	22	,	,	PUNCT
ejpam-5279	482	23	υλ	υλ	NOUN
ejpam-5279	482	24	)	)	PUNCT
ejpam-5279	483	1	f	f	NOUN
ejpam-5279	483	2	7−→	7−→	NOUN
ejpam-5279	483	3	f	f	PROPN
ejpam-5279	483	4	theorem	theorem	VERB
ejpam-5279	483	5	7	7	NUM
ejpam-5279	483	6	.	.	PUNCT
ejpam-5279	483	7	uapchygrp	uapchygrp	PROPN
ejpam-5279	483	8	is	be	AUX
ejpam-5279	483	9	isomorphic	isomorphic	ADJ
ejpam-5279	483	10	to	to	ADP
ejpam-5279	483	11	snaplimgrp	snaplimgrp	VERB
ejpam-5279	483	12	.	.	PUNCT
ejpam-5279	484	1	proof	proof	NOUN
ejpam-5279	484	2	.	.	PUNCT
ejpam-5279	485	1	observed	observe	VERB
ejpam-5279	485	2	that	that	SCONJ
ejpam-5279	485	3	snaplimgrp	snaplimgrp	VERB
ejpam-5279	485	4	b−→uapchygrp	b−→uapchygrp	NOUN
ejpam-5279	485	5	a−→snaplimgrp	a−→snaplimgrp	PROPN
ejpam-5279	485	6	:	:	PUNCT
ejpam-5279	485	7	(	(	PUNCT
ejpam-5279	485	8	x	x	X
ejpam-5279	485	9	,	,	PUNCT
ejpam-5279	485	10	·	·	PUNCT
ejpam-5279	485	11	,	,	PUNCT
ejpam-5279	485	12	λ	λ	NOUN
ejpam-5279	485	13	)	)	PUNCT
ejpam-5279	485	14	7−→	7−→	NOUN
ejpam-5279	485	15	(	(	PUNCT
ejpam-5279	485	16	x	x	NOUN
ejpam-5279	485	17	,	,	PUNCT
ejpam-5279	485	18	·	·	PUNCT
ejpam-5279	485	19	,	,	PUNCT
ejpam-5279	485	20	υλ	υλ	NOUN
ejpam-5279	485	21	)	)	PUNCT
ejpam-5279	485	22	7−→	7−→	NOUN
ejpam-5279	485	23	(	(	PUNCT
ejpam-5279	485	24	x	x	NOUN
ejpam-5279	485	25	,	,	PUNCT
ejpam-5279	485	26	·	·	PUNCT
ejpam-5279	485	27	,	,	PUNCT
ejpam-5279	485	28	λ	λ	PROPN
ejpam-5279	485	29	)	)	PUNCT
ejpam-5279	485	30	;	;	PUNCT
ejpam-5279	485	31	then	then	ADV
ejpam-5279	485	32	one	one	PRON
ejpam-5279	485	33	can	can	AUX
ejpam-5279	485	34	check	check	VERB
ejpam-5279	485	35	that	that	PRON
ejpam-5279	485	36	a	a	DET
ejpam-5279	485	37	◦	◦	NOUN
ejpam-5279	485	38	b	b	NOUN
ejpam-5279	485	39	=	=	SYM
ejpam-5279	485	40	idsnaplimgrp	idsnaplimgrp	ADJ
ejpam-5279	485	41	,	,	PUNCT
ejpam-5279	485	42	i.e.	i.e.	X
ejpam-5279	485	43	,	,	PUNCT
ejpam-5279	485	44	λυλ	λυλ	NOUN
ejpam-5279	486	1	=	=	SYM
ejpam-5279	486	2	λ	λ	PROPN
ejpam-5279	486	3	.	.	PUNCT
ejpam-5279	487	1	in	in	ADP
ejpam-5279	487	2	fact	fact	NOUN
ejpam-5279	487	3	,	,	PUNCT
ejpam-5279	487	4	for	for	ADP
ejpam-5279	487	5	any	any	DET
ejpam-5279	487	6	f	f	PROPN
ejpam-5279	487	7	∈	∈	PROPN
ejpam-5279	487	8	f(x	f(x	PROPN
ejpam-5279	487	9	)	)	PUNCT
ejpam-5279	487	10	,	,	PUNCT
ejpam-5279	487	11	and	and	CCONJ
ejpam-5279	487	12	x	x	X
ejpam-5279	487	13	∈	∈	NOUN
ejpam-5279	487	14	x	x	X
ejpam-5279	487	15	,	,	PUNCT
ejpam-5279	487	16	we	we	PRON
ejpam-5279	487	17	have	have	VERB
ejpam-5279	487	18	λυλ	λυλ	PROPN
ejpam-5279	487	19	(	(	PUNCT
ejpam-5279	487	20	f)(x	f)(x	NOUN
ejpam-5279	487	21	)	)	PUNCT
ejpam-5279	487	22	=	=	SYM
ejpam-5279	487	23	υλ(f	υλ(f	X
ejpam-5279	488	1	∧	∧	PROPN
ejpam-5279	488	2	[	[	X
ejpam-5279	488	3	x	x	X
ejpam-5279	488	4	]	]	X
ejpam-5279	488	5	)	)	PUNCT
ejpam-5279	488	6	=	=	SYM
ejpam-5279	488	7	λ	λ	INTJ
ejpam-5279	488	8	(	(	PUNCT
ejpam-5279	488	9	(	(	PUNCT
ejpam-5279	488	10	f	f	X
ejpam-5279	488	11	∧	∧	PROPN
ejpam-5279	489	1	[	[	X
ejpam-5279	489	2	x])−1	x])−1	PROPN
ejpam-5279	489	3	⊙	⊙	PROPN
ejpam-5279	489	4	(	(	PUNCT
ejpam-5279	489	5	f	f	PROPN
ejpam-5279	489	6	∧	∧	PROPN
ejpam-5279	490	1	[	[	X
ejpam-5279	490	2	x	x	X
ejpam-5279	490	3	]	]	X
ejpam-5279	490	4	)	)	PUNCT
ejpam-5279	490	5	)	)	PUNCT
ejpam-5279	491	1	(	(	PUNCT
ejpam-5279	491	2	e	e	NOUN
ejpam-5279	491	3	)	)	PUNCT
ejpam-5279	491	4	∨	∨	NUM
ejpam-5279	491	5	λ	λ	X
ejpam-5279	491	6	(	(	PUNCT
ejpam-5279	491	7	(	(	PUNCT
ejpam-5279	491	8	f	f	X
ejpam-5279	491	9	∧	∧	PROPN
ejpam-5279	492	1	[	[	X
ejpam-5279	492	2	x])⊙	x])⊙	X
ejpam-5279	492	3	(	(	PUNCT
ejpam-5279	492	4	f	f	PROPN
ejpam-5279	492	5	∧	∧	PROPN
ejpam-5279	492	6	[	[	X
ejpam-5279	492	7	x])−1	x])−1	PROPN
ejpam-5279	492	8	)	)	PUNCT
ejpam-5279	492	9	(	(	PUNCT
ejpam-5279	492	10	e	e	X
ejpam-5279	492	11	)	)	PUNCT
ejpam-5279	492	12	=	=	SYM
ejpam-5279	492	13	λ	λ	INTJ
ejpam-5279	492	14	(	(	PUNCT
ejpam-5279	492	15	(	(	PUNCT
ejpam-5279	492	16	[	[	X
ejpam-5279	492	17	x]−1	x]−1	X
ejpam-5279	492	18	∧	∧	PROPN
ejpam-5279	492	19	f−1)⊙	f−1)⊙	PROPN
ejpam-5279	492	20	(	(	PUNCT
ejpam-5279	492	21	f	f	PROPN
ejpam-5279	492	22	∧	∧	PROPN
ejpam-5279	493	1	[	[	X
ejpam-5279	493	2	x	x	X
ejpam-5279	493	3	]	]	X
ejpam-5279	493	4	)	)	PUNCT
ejpam-5279	493	5	)	)	PUNCT
ejpam-5279	494	1	(	(	PUNCT
ejpam-5279	494	2	e	e	NOUN
ejpam-5279	494	3	)	)	PUNCT
ejpam-5279	494	4	∨	∨	NUM
ejpam-5279	494	5	λ	λ	X
ejpam-5279	494	6	(	(	PUNCT
ejpam-5279	494	7	(	(	PUNCT
ejpam-5279	494	8	f	f	X
ejpam-5279	494	9	∧	∧	PROPN
ejpam-5279	495	1	[	[	X
ejpam-5279	495	2	x])⊙	x])⊙	X
ejpam-5279	495	3	(	(	PUNCT
ejpam-5279	495	4	f−1	f−1	PROPN
ejpam-5279	495	5	∧	∧	PROPN
ejpam-5279	495	6	[	[	X
ejpam-5279	495	7	x]−1	x]−1	NOUN
ejpam-5279	495	8	)	)	PUNCT
ejpam-5279	495	9	)	)	PUNCT
ejpam-5279	496	1	(	(	PUNCT
ejpam-5279	496	2	e	e	NOUN
ejpam-5279	496	3	)	)	PUNCT
ejpam-5279	496	4	≥	≥	NOUN
ejpam-5279	496	5	λ	λ	INTJ
ejpam-5279	496	6	(	(	PUNCT
ejpam-5279	496	7	(	(	PUNCT
ejpam-5279	496	8	[	[	X
ejpam-5279	496	9	x]−1	x]−1	X
ejpam-5279	496	10	∧	∧	PROPN
ejpam-5279	496	11	f−1)⊙	f−1)⊙	PROPN
ejpam-5279	496	12	(	(	PUNCT
ejpam-5279	496	13	f	f	PROPN
ejpam-5279	496	14	∧	∧	PROPN
ejpam-5279	496	15	[	[	X
ejpam-5279	496	16	x	x	X
ejpam-5279	496	17	]	]	X
ejpam-5279	496	18	)	)	PUNCT
ejpam-5279	496	19	)	)	PUNCT
ejpam-5279	496	20	(	(	PUNCT
ejpam-5279	496	21	e	e	NOUN
ejpam-5279	496	22	)	)	PUNCT
ejpam-5279	496	23	≥	≥	NOUN
ejpam-5279	496	24	λ(f⊙	λ(f⊙	NOUN
ejpam-5279	497	1	[	[	X
ejpam-5279	497	2	x]−1	x]−1	X
ejpam-5279	497	3	)	)	PUNCT
ejpam-5279	497	4	=	=	PUNCT
ejpam-5279	497	5	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	497	6	)	)	PUNCT
ejpam-5279	497	7	,	,	PUNCT
ejpam-5279	497	8	this	this	PRON
ejpam-5279	497	9	is	be	AUX
ejpam-5279	497	10	so	so	ADV
ejpam-5279	497	11	,	,	PUNCT
ejpam-5279	497	12	because	because	SCONJ
ejpam-5279	497	13	of	of	ADP
ejpam-5279	497	14	the	the	DET
ejpam-5279	497	15	fact	fact	NOUN
ejpam-5279	497	16	that	that	SCONJ
ejpam-5279	497	17	[	[	X
ejpam-5279	497	18	x]−1	x]−1	X
ejpam-5279	497	19	∧	∧	PROPN
ejpam-5279	497	20	f−1)⊙	f−1)⊙	PROPN
ejpam-5279	497	21	(	(	PUNCT
ejpam-5279	497	22	f	f	PROPN
ejpam-5279	497	23	∧	∧	PROPN
ejpam-5279	497	24	[	[	X
ejpam-5279	497	25	x	x	X
ejpam-5279	497	26	]	]	X
ejpam-5279	497	27	)	)	PUNCT
ejpam-5279	497	28	≤	≤	NOUN
ejpam-5279	498	1	[	[	X
ejpam-5279	498	2	x]−1	x]−1	X
ejpam-5279	498	3	⊙	⊙	PROPN
ejpam-5279	498	4	f	f	PROPN
ejpam-5279	498	5	,	,	PUNCT
ejpam-5279	498	6	the	the	DET
ejpam-5279	498	7	applying	apply	VERB
ejpam-5279	498	8	λ	λ	PROPN
ejpam-5279	498	9	to	to	PART
ejpam-5279	498	10	get	get	VERB
ejpam-5279	498	11	λυλ	λυλ	PROPN
ejpam-5279	498	12	≥	≥	PROPN
ejpam-5279	498	13	λ	λ	NOUN
ejpam-5279	498	14	.	.	PUNCT
ejpam-5279	498	15	conversely	conversely	ADV
ejpam-5279	498	16	,	,	PUNCT
ejpam-5279	498	17	for	for	ADP
ejpam-5279	498	18	any	any	DET
ejpam-5279	498	19	f	f	PROPN
ejpam-5279	498	20	∈	∈	PROPN
ejpam-5279	498	21	f(x	f(x	PROPN
ejpam-5279	498	22	)	)	PUNCT
ejpam-5279	498	23	and	and	CCONJ
ejpam-5279	498	24	x	x	PUNCT
ejpam-5279	498	25	∈	∈	NOUN
ejpam-5279	498	26	x	x	X
ejpam-5279	498	27	,	,	PUNCT
ejpam-5279	498	28	we	we	PRON
ejpam-5279	498	29	have	have	VERB
ejpam-5279	498	30	λυλ	λυλ	PROPN
ejpam-5279	498	31	(	(	PUNCT
ejpam-5279	498	32	f)(x	f)(x	NOUN
ejpam-5279	498	33	)	)	PUNCT
ejpam-5279	498	34	=	=	SYM
ejpam-5279	498	35	υλ(f	υλ(f	X
ejpam-5279	499	1	∧	∧	PROPN
ejpam-5279	499	2	[	[	X
ejpam-5279	499	3	x	x	X
ejpam-5279	499	4	]	]	X
ejpam-5279	499	5	)	)	PUNCT
ejpam-5279	499	6	=	=	SYM
ejpam-5279	499	7	λ	λ	INTJ
ejpam-5279	499	8	(	(	PUNCT
ejpam-5279	499	9	(	(	PUNCT
ejpam-5279	499	10	f	f	X
ejpam-5279	499	11	∧	∧	PROPN
ejpam-5279	500	1	[	[	X
ejpam-5279	500	2	x])−1	x])−1	PROPN
ejpam-5279	500	3	⊙	⊙	PROPN
ejpam-5279	500	4	(	(	PUNCT
ejpam-5279	500	5	f	f	PROPN
ejpam-5279	500	6	∧	∧	PROPN
ejpam-5279	501	1	[	[	X
ejpam-5279	501	2	x	x	X
ejpam-5279	501	3	]	]	X
ejpam-5279	501	4	)	)	PUNCT
ejpam-5279	501	5	)	)	PUNCT
ejpam-5279	502	1	∨	∨	NUM
ejpam-5279	502	2	λ	λ	PROPN
ejpam-5279	502	3	(	(	PUNCT
ejpam-5279	502	4	(	(	PUNCT
ejpam-5279	502	5	f	f	X
ejpam-5279	502	6	∧	∧	PROPN
ejpam-5279	503	1	[	[	X
ejpam-5279	503	2	x])⊙	x])⊙	X
ejpam-5279	503	3	(	(	PUNCT
ejpam-5279	503	4	f	f	PROPN
ejpam-5279	503	5	∧	∧	PROPN
ejpam-5279	503	6	[	[	X
ejpam-5279	503	7	x])−1	x])−1	PROPN
ejpam-5279	503	8	)	)	PUNCT
ejpam-5279	503	9	)	)	PUNCT
ejpam-5279	504	1	=	=	PUNCT
ejpam-5279	504	2	λ	λ	INTJ
ejpam-5279	504	3	(	(	PUNCT
ejpam-5279	504	4	(	(	PUNCT
ejpam-5279	504	5	f	f	X
ejpam-5279	504	6	∧	∧	PROPN
ejpam-5279	505	1	[	[	X
ejpam-5279	505	2	x])−1	x])−1	PROPN
ejpam-5279	505	3	⊙	⊙	PROPN
ejpam-5279	505	4	(	(	PUNCT
ejpam-5279	505	5	f	f	PROPN
ejpam-5279	505	6	∧	∧	PROPN
ejpam-5279	506	1	[	[	X
ejpam-5279	506	2	x	x	X
ejpam-5279	506	3	]	]	X
ejpam-5279	506	4	)	)	PUNCT
ejpam-5279	506	5	)	)	PUNCT
ejpam-5279	507	1	∨	∨	NUM
ejpam-5279	507	2	λ	λ	PROPN
ejpam-5279	507	3	(	(	PUNCT
ejpam-5279	507	4	(	(	PUNCT
ejpam-5279	507	5	f	f	X
ejpam-5279	507	6	∧	∧	PROPN
ejpam-5279	508	1	[	[	X
ejpam-5279	508	2	x])⊙	x])⊙	X
ejpam-5279	508	3	(	(	PUNCT
ejpam-5279	508	4	f−1	f−1	PROPN
ejpam-5279	508	5	∧	∧	PROPN
ejpam-5279	508	6	[	[	X
ejpam-5279	508	7	x]−1	x]−1	NOUN
ejpam-5279	508	8	)	)	PUNCT
ejpam-5279	508	9	)	)	PUNCT
ejpam-5279	508	10	.	.	PUNCT
ejpam-5279	509	1	since	since	SCONJ
ejpam-5279	509	2	in	in	ADP
ejpam-5279	509	3	one	one	NUM
ejpam-5279	509	4	hand	hand	NOUN
ejpam-5279	509	5	,	,	PUNCT
ejpam-5279	509	6	(	(	PUNCT
ejpam-5279	509	7	f∧	f∧	PROPN
ejpam-5279	510	1	[	[	X
ejpam-5279	510	2	x])⊙(f−1∧	x])⊙(f−1∧	X
ejpam-5279	510	3	[	[	X
ejpam-5279	510	4	x]−1	x]−1	X
ejpam-5279	510	5	)	)	PUNCT
ejpam-5279	510	6	=	=	SYM
ejpam-5279	510	7	(	(	PUNCT
ejpam-5279	510	8	f⊙f−1)∧(f⊙	f⊙f−1)∧(f⊙	X
ejpam-5279	511	1	[	[	X
ejpam-5279	511	2	x]−1)∧([x]⊙f−1)∧	x]−1)∧([x]⊙f−1)∧	X
ejpam-5279	512	1	[	[	X
ejpam-5279	512	2	x]⊙	x]⊙	X
ejpam-5279	513	1	[	[	X
ejpam-5279	513	2	x]−1	x]−1	X
ejpam-5279	513	3	and	and	CCONJ
ejpam-5279	513	4	the	the	DET
ejpam-5279	513	5	other	other	ADJ
ejpam-5279	513	6	(	(	PUNCT
ejpam-5279	513	7	f∧	f∧	PROPN
ejpam-5279	513	8	[	[	X
ejpam-5279	513	9	x])−1	x])−1	X
ejpam-5279	513	10	⊙	⊙	PROPN
ejpam-5279	513	11	(	(	PUNCT
ejpam-5279	513	12	f∧	f∧	PROPN
ejpam-5279	514	1	[	[	X
ejpam-5279	514	2	x	x	X
ejpam-5279	514	3	]	]	X
ejpam-5279	514	4	)	)	PUNCT
ejpam-5279	514	5	=	=	SYM
ejpam-5279	514	6	(	(	PUNCT
ejpam-5279	514	7	f−1	f−1	PROPN
ejpam-5279	514	8	⊙f)∧	⊙f)∧	NOUN
ejpam-5279	514	9	(	(	PUNCT
ejpam-5279	514	10	f−1	f−1	PROPN
ejpam-5279	514	11	⊙	⊙	VERB
ejpam-5279	515	1	[	[	X
ejpam-5279	515	2	x])∧	x])∧	X
ejpam-5279	515	3	(	(	PUNCT
ejpam-5279	515	4	[	[	X
ejpam-5279	515	5	x]−1	x]−1	X
ejpam-5279	515	6	⊙f)∧	⊙f)∧	NOUN
ejpam-5279	515	7	(	(	PUNCT
ejpam-5279	515	8	[	[	X
ejpam-5279	515	9	x]−1	x]−1	X
ejpam-5279	515	10	⊙	⊙	NOUN
ejpam-5279	516	1	[	[	X
ejpam-5279	516	2	x	x	X
ejpam-5279	516	3	]	]	X
ejpam-5279	516	4	)	)	PUNCT
ejpam-5279	516	5	,	,	PUNCT
ejpam-5279	516	6	one	one	PRON
ejpam-5279	516	7	obtains	obtain	VERB
ejpam-5279	516	8	=	=	SYM
ejpam-5279	516	9	λ	λ	X
ejpam-5279	516	10	(	(	PUNCT
ejpam-5279	516	11	(	(	PUNCT
ejpam-5279	516	12	f	f	X
ejpam-5279	516	13	∧	∧	PROPN
ejpam-5279	517	1	[	[	X
ejpam-5279	517	2	x])−1	x])−1	PROPN
ejpam-5279	517	3	⊙	⊙	PROPN
ejpam-5279	517	4	(	(	PUNCT
ejpam-5279	517	5	f	f	PROPN
ejpam-5279	517	6	∧	∧	PROPN
ejpam-5279	518	1	[	[	X
ejpam-5279	518	2	x	x	X
ejpam-5279	518	3	]	]	X
ejpam-5279	518	4	)	)	PUNCT
ejpam-5279	518	5	)	)	PUNCT
ejpam-5279	519	1	∨	∨	NUM
ejpam-5279	519	2	λ	λ	PROPN
ejpam-5279	519	3	(	(	PUNCT
ejpam-5279	519	4	(	(	PUNCT
ejpam-5279	519	5	f	f	X
ejpam-5279	519	6	∧	∧	PROPN
ejpam-5279	520	1	[	[	X
ejpam-5279	520	2	x])⊙	x])⊙	X
ejpam-5279	520	3	(	(	PUNCT
ejpam-5279	520	4	f−1	f−1	PROPN
ejpam-5279	520	5	∧	∧	PROPN
ejpam-5279	520	6	[	[	X
ejpam-5279	520	7	x]−1	x]−1	NOUN
ejpam-5279	520	8	)	)	PUNCT
ejpam-5279	520	9	)	)	PUNCT
ejpam-5279	521	1	≤	≤	NUM
ejpam-5279	521	2	λ	λ	INTJ
ejpam-5279	521	3	(	(	PUNCT
ejpam-5279	521	4	(	(	PUNCT
ejpam-5279	521	5	f−1	f−1	PROPN
ejpam-5279	521	6	⊙	⊙	NOUN
ejpam-5279	521	7	f)(e	f)(e	NUM
ejpam-5279	521	8	=	=	SYM
ejpam-5279	521	9	x−1x	x−1x	PROPN
ejpam-5279	521	10	)	)	PUNCT
ejpam-5279	521	11	∨	∨	NUM
ejpam-5279	521	12	λ(f−1	λ(f−1	X
ejpam-5279	521	13	⊙	⊙	NOUN
ejpam-5279	521	14	[	[	X
ejpam-5279	521	15	x])(e	x])(e	PROPN
ejpam-5279	521	16	)	)	PUNCT
ejpam-5279	521	17	∧	∧	PROPN
ejpam-5279	521	18	λ(f⊙	λ(f⊙	ADP
ejpam-5279	521	19	[	[	X
ejpam-5279	521	20	x]−1)(x	x]−1)(x	NUM
ejpam-5279	521	21	)	)	PUNCT
ejpam-5279	521	22	∨	∨	NUM
ejpam-5279	521	23	λ([e])(e	λ([e])(e	PROPN
ejpam-5279	521	24	)	)	PUNCT
ejpam-5279	521	25	)	)	PUNCT
ejpam-5279	521	26	t.m.g	t.m.g	X
ejpam-5279	521	27	.	.	PUNCT
ejpam-5279	522	1	ahsanullah	ahsanullah	PROPN
ejpam-5279	522	2	,	,	PUNCT
ejpam-5279	522	3	fawzi	fawzi	PROPN
ejpam-5279	522	4	al	al	PROPN
ejpam-5279	522	5	-	-	PUNCT
ejpam-5279	522	6	thukair	thukair	NOUN
ejpam-5279	522	7	/	/	SYM
ejpam-5279	522	8	eur	eur	NOUN
ejpam-5279	522	9	.	.	PUNCT
ejpam-5279	523	1	j.	j.	PROPN
ejpam-5279	523	2	pure	pure	PROPN
ejpam-5279	523	3	appl	appl	PROPN
ejpam-5279	523	4	.	.	PROPN
ejpam-5279	523	5	math	math	PROPN
ejpam-5279	523	6	,	,	PUNCT
ejpam-5279	523	7	17	17	NUM
ejpam-5279	523	8	(	(	PUNCT
ejpam-5279	523	9	3	3	NUM
ejpam-5279	523	10	)	)	PUNCT
ejpam-5279	523	11	(	(	PUNCT
ejpam-5279	523	12	2024	2024	NUM
ejpam-5279	523	13	)	)	PUNCT
ejpam-5279	523	14	,	,	PUNCT
ejpam-5279	523	15	1762	1762	NUM
ejpam-5279	523	16	-	-	SYM
ejpam-5279	523	17	1778	1778	NUM
ejpam-5279	523	18	1776	1776	NUM
ejpam-5279	523	19	∨	∨	NUM
ejpam-5279	523	20	(	(	PUNCT
ejpam-5279	523	21	λ(f⊙	λ(f⊙	ADV
ejpam-5279	523	22	f−1)(e	f−1)(e	NOUN
ejpam-5279	523	23	=	=	SYM
ejpam-5279	523	24	xx−1	xx−1	PROPN
ejpam-5279	523	25	)	)	PUNCT
ejpam-5279	523	26	∨	∨	NUM
ejpam-5279	523	27	λ(f⊙	λ(f⊙	ADP
ejpam-5279	523	28	[	[	X
ejpam-5279	523	29	x]−1)(e	x]−1)(e	NUM
ejpam-5279	523	30	)	)	PUNCT
ejpam-5279	523	31	∨	∨	NUM
ejpam-5279	523	32	λ(([x	λ(([x	X
ejpam-5279	523	33	]	]	X
ejpam-5279	523	34	∧	∧	PROPN
ejpam-5279	523	35	f−1)(e	f−1)(e	PROPN
ejpam-5279	523	36	)	)	PUNCT
ejpam-5279	523	37	∨	∨	NUM
ejpam-5279	523	38	λ[e](e	λ[e](e	VERB
ejpam-5279	523	39	)	)	PUNCT
ejpam-5279	523	40	)	)	PUNCT
ejpam-5279	523	41	≤	≤	NUM
ejpam-5279	523	42	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	523	43	)	)	PUNCT
ejpam-5279	523	44	.	.	PUNCT
ejpam-5279	524	1	this	this	PRON
ejpam-5279	524	2	is	be	AUX
ejpam-5279	524	3	so	so	ADV
ejpam-5279	524	4	,	,	PUNCT
ejpam-5279	524	5	because	because	SCONJ
ejpam-5279	524	6	of	of	ADP
ejpam-5279	524	7	the	the	DET
ejpam-5279	524	8	fact	fact	NOUN
ejpam-5279	524	9	that	that	SCONJ
ejpam-5279	524	10	λ(f−1⊙f)(x−1x	λ(f−1⊙f)(x−1x	PROPN
ejpam-5279	524	11	)	)	PUNCT
ejpam-5279	524	12	≤	≤	NUM
ejpam-5279	524	13	λ(f−1)(x−1)∨λ(f(x	λ(f−1)(x−1)∨λ(f(x	NOUN
ejpam-5279	524	14	)	)	PUNCT
ejpam-5279	524	15	≤	≤	PUNCT
ejpam-5279	525	1	λ(f)(x)∨	λ(f)(x)∨	X
ejpam-5279	525	2	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	525	3	)	)	PUNCT
ejpam-5279	525	4	=	=	PUNCT
ejpam-5279	525	5	λ(f)(x	λ(f)(x	NOUN
ejpam-5279	525	6	)	)	PUNCT
ejpam-5279	525	7	,	,	PUNCT
ejpam-5279	525	8	and	and	CCONJ
ejpam-5279	525	9	continuing	continue	VERB
ejpam-5279	525	10	in	in	ADP
ejpam-5279	525	11	this	this	DET
ejpam-5279	525	12	way	way	NOUN
ejpam-5279	525	13	,	,	PUNCT
ejpam-5279	525	14	we	we	PRON
ejpam-5279	525	15	can	can	AUX
ejpam-5279	525	16	do	do	VERB
ejpam-5279	525	17	other	other	ADJ
ejpam-5279	525	18	parts	part	NOUN
ejpam-5279	525	19	including	include	VERB
ejpam-5279	525	20	applying	apply	VERB
ejpam-5279	525	21	homogeneity	homogeneity	NOUN
ejpam-5279	525	22	.	.	PUNCT
ejpam-5279	526	1	thus	thus	ADV
ejpam-5279	526	2	,	,	PUNCT
ejpam-5279	526	3	we	we	PRON
ejpam-5279	526	4	can	can	AUX
ejpam-5279	526	5	prove	prove	VERB
ejpam-5279	526	6	that	that	SCONJ
ejpam-5279	526	7	λυλ	λυλ	ADV
ejpam-5279	526	8	≤	≤	PROPN
ejpam-5279	526	9	λ	λ	PROPN
ejpam-5279	526	10	,	,	PUNCT
ejpam-5279	526	11	and	and	CCONJ
ejpam-5279	526	12	hence	hence	ADV
ejpam-5279	526	13	λυλ	λυλ	NOUN
ejpam-5279	527	1	=	=	SYM
ejpam-5279	527	2	λ	λ	PROPN
ejpam-5279	527	3	.	.	PROPN
ejpam-5279	527	4	for	for	ADP
ejpam-5279	527	5	the	the	DET
ejpam-5279	527	6	other	other	ADJ
ejpam-5279	527	7	direction	direction	NOUN
ejpam-5279	527	8	,	,	PUNCT
ejpam-5279	527	9	we	we	PRON
ejpam-5279	527	10	look	look	VERB
ejpam-5279	527	11	at	at	ADP
ejpam-5279	527	12	the	the	DET
ejpam-5279	527	13	scheme	scheme	NOUN
ejpam-5279	527	14	below	below	ADV
ejpam-5279	527	15	:	:	PUNCT
ejpam-5279	527	16	uapchygrp	uapchygrp	PROPN
ejpam-5279	527	17	b−→snaplimgrp	b−→snaplimgrp	PROPN
ejpam-5279	527	18	a−→uapchygrp	a−→uapchygrp	PROPN
ejpam-5279	527	19	:	:	PUNCT
ejpam-5279	527	20	(	(	PUNCT
ejpam-5279	527	21	x	x	X
ejpam-5279	527	22	,	,	PUNCT
ejpam-5279	527	23	·	·	PUNCT
ejpam-5279	527	24	,	,	PUNCT
ejpam-5279	527	25	υ	υ	NOUN
ejpam-5279	527	26	)	)	PUNCT
ejpam-5279	527	27	7−→	7−→	NOUN
ejpam-5279	527	28	(	(	PUNCT
ejpam-5279	527	29	x	x	NOUN
ejpam-5279	527	30	,	,	PUNCT
ejpam-5279	527	31	·	·	PUNCT
ejpam-5279	527	32	,	,	PUNCT
ejpam-5279	527	33	λυ	λυ	NOUN
ejpam-5279	527	34	)	)	PUNCT
ejpam-5279	527	35	7−→	7−→	NOUN
ejpam-5279	527	36	(	(	PUNCT
ejpam-5279	527	37	x	x	NOUN
ejpam-5279	527	38	,	,	PUNCT
ejpam-5279	527	39	·	·	PUNCT
ejpam-5279	527	40	,	,	PUNCT
ejpam-5279	527	41	υ	υ	NOUN
ejpam-5279	527	42	)	)	PUNCT
ejpam-5279	527	43	;	;	PUNCT
ejpam-5279	527	44	then	then	ADV
ejpam-5279	527	45	one	one	PRON
ejpam-5279	527	46	can	can	AUX
ejpam-5279	527	47	check	check	VERB
ejpam-5279	527	48	that	that	PRON
ejpam-5279	527	49	b	b	NOUN
ejpam-5279	527	50	◦	◦	NOUN
ejpam-5279	527	51	a	a	DET
ejpam-5279	527	52	=	=	NOUN
ejpam-5279	527	53	iduapchygrp	iduapchygrp	NOUN
ejpam-5279	527	54	,	,	PUNCT
ejpam-5279	527	55	i.e.	i.e.	X
ejpam-5279	527	56	,	,	PUNCT
ejpam-5279	527	57	υλυ	υλυ	PROPN
ejpam-5279	527	58	=	=	SYM
ejpam-5279	527	59	υ	υ	PROPN
ejpam-5279	527	60	.	.	PUNCT
ejpam-5279	528	1	in	in	ADP
ejpam-5279	528	2	fact	fact	NOUN
ejpam-5279	528	3	,	,	PUNCT
ejpam-5279	528	4	for	for	ADP
ejpam-5279	528	5	any	any	DET
ejpam-5279	528	6	f	f	PROPN
ejpam-5279	528	7	∈	∈	PROPN
ejpam-5279	528	8	f(x	f(x	PROPN
ejpam-5279	528	9	)	)	PUNCT
ejpam-5279	528	10	,	,	PUNCT
ejpam-5279	528	11	υλυ	υλυ	PROPN
ejpam-5279	528	12	(	(	PUNCT
ejpam-5279	528	13	f	f	X
ejpam-5279	528	14	)	)	PUNCT
ejpam-5279	528	15	=	=	SYM
ejpam-5279	528	16	λυ(f−1	λυ(f−1	PROPN
ejpam-5279	528	17	⊙	⊙	PROPN
ejpam-5279	528	18	f)(e	f)(e	NOUN
ejpam-5279	528	19	)	)	PUNCT
ejpam-5279	528	20	∨	∨	NUM
ejpam-5279	528	21	λυ(f⊙	λυ(f⊙	NOUN
ejpam-5279	528	22	f−1)(e	f−1)(e	NOUN
ejpam-5279	528	23	)	)	PUNCT
ejpam-5279	528	24	=	=	SYM
ejpam-5279	528	25	υ(f	υ(f	PROPN
ejpam-5279	528	26	)	)	PUNCT
ejpam-5279	528	27	.	.	PUNCT
ejpam-5279	529	1	hence	hence	ADV
ejpam-5279	529	2	the	the	DET
ejpam-5279	529	3	result	result	NOUN
ejpam-5279	529	4	follows	follow	VERB
ejpam-5279	529	5	.	.	PUNCT
ejpam-5279	530	1	corollary	corollary	ADJ
ejpam-5279	530	2	2	2	NUM
ejpam-5279	530	3	.	.	PUNCT
ejpam-5279	531	1	if	if	SCONJ
ejpam-5279	531	2	the	the	DET
ejpam-5279	531	3	underlying	underlie	VERB
ejpam-5279	531	4	group	group	NOUN
ejpam-5279	531	5	is	be	AUX
ejpam-5279	531	6	abelian	abelian	ADJ
ejpam-5279	531	7	,	,	PUNCT
ejpam-5279	531	8	then	then	ADV
ejpam-5279	531	9	uapchygrp	uapchygrp	VERB
ejpam-5279	531	10	is	be	AUX
ejpam-5279	531	11	isomorphic	isomorphic	ADJ
ejpam-5279	531	12	to	to	ADP
ejpam-5279	531	13	uaplimgrp	uaplimgrp	VERB
ejpam-5279	531	14	.	.	PUNCT
ejpam-5279	532	1	7	7	X
ejpam-5279	532	2	.	.	X
ejpam-5279	532	3	conclusion	conclusion	NOUN
ejpam-5279	532	4	in	in	ADP
ejpam-5279	532	5	this	this	DET
ejpam-5279	532	6	paper	paper	NOUN
ejpam-5279	532	7	,	,	PUNCT
ejpam-5279	532	8	from	from	ADP
ejpam-5279	532	9	categorical	categorical	ADJ
ejpam-5279	532	10	perspective	perspective	NOUN
ejpam-5279	532	11	,	,	PUNCT
ejpam-5279	532	12	we	we	PRON
ejpam-5279	532	13	considered	consider	VERB
ejpam-5279	532	14	two	two	NUM
ejpam-5279	532	15	isomorphisms	isomorphism	NOUN
ejpam-5279	532	16	,	,	PUNCT
ejpam-5279	532	17	one	one	NUM
ejpam-5279	532	18	between	between	ADP
ejpam-5279	532	19	the	the	DET
ejpam-5279	532	20	categories	category	NOUN
ejpam-5279	532	21	uaplimgrp	uaplimgrp	NOUN
ejpam-5279	532	22	(	(	PUNCT
ejpam-5279	532	23	the	the	DET
ejpam-5279	532	24	category	category	NOUN
ejpam-5279	532	25	of	of	ADP
ejpam-5279	532	26	all	all	DET
ejpam-5279	532	27	ultra	ultra	ADJ
ejpam-5279	532	28	-	-	ADJ
ejpam-5279	532	29	approach	approach	ADJ
ejpam-5279	532	30	limit	limit	NOUN
ejpam-5279	532	31	groups	group	NOUN
ejpam-5279	532	32	)	)	PUNCT
ejpam-5279	532	33	and	and	CCONJ
ejpam-5279	532	34	napgrp	napgrp	NOUN
ejpam-5279	532	35	(	(	PUNCT
ejpam-5279	532	36	the	the	DET
ejpam-5279	532	37	category	category	NOUN
ejpam-5279	532	38	of	of	ADP
ejpam-5279	532	39	neighborhood	neighborhood	NOUN
ejpam-5279	532	40	approach	approach	NOUN
ejpam-5279	532	41	groups	group	NOUN
ejpam-5279	532	42	associated	associate	VERB
ejpam-5279	532	43	with	with	ADP
ejpam-5279	532	44	approach	approach	NOUN
ejpam-5279	532	45	spaces	space	NOUN
ejpam-5279	532	46	)	)	PUNCT
ejpam-5279	532	47	;	;	PUNCT
ejpam-5279	532	48	another	another	PRON
ejpam-5279	532	49	,	,	PUNCT
ejpam-5279	532	50	between	between	ADP
ejpam-5279	532	51	the	the	DET
ejpam-5279	532	52	categories	category	NOUN
ejpam-5279	532	53	uapchygrp	uapchygrp	VERB
ejpam-5279	532	54	(	(	PUNCT
ejpam-5279	532	55	the	the	DET
ejpam-5279	532	56	category	category	NOUN
ejpam-5279	532	57	of	of	ADP
ejpam-5279	532	58	ultra	ultra	ADJ
ejpam-5279	532	59	approachcauchy	approachcauchy	NOUN
ejpam-5279	532	60	groups	group	NOUN
ejpam-5279	532	61	)	)	PUNCT
ejpam-5279	532	62	and	and	CCONJ
ejpam-5279	532	63	snaplimgrp	snaplimgrp	VERB
ejpam-5279	532	64	(	(	PUNCT
ejpam-5279	532	65	the	the	DET
ejpam-5279	532	66	category	category	NOUN
ejpam-5279	532	67	of	of	ADP
ejpam-5279	532	68	strongly	strongly	ADV
ejpam-5279	532	69	normal	normal	ADJ
ejpam-5279	532	70	approach	approach	NOUN
ejpam-5279	532	71	limit	limit	NOUN
ejpam-5279	532	72	groups	group	NOUN
ejpam-5279	532	73	)	)	PUNCT
ejpam-5279	532	74	besides	besides	SCONJ
ejpam-5279	532	75	discussing	discuss	VERB
ejpam-5279	532	76	some	some	DET
ejpam-5279	532	77	characterizations	characterization	NOUN
ejpam-5279	532	78	of	of	ADP
ejpam-5279	532	79	(	(	PUNCT
ejpam-5279	532	80	ultra)-approach	ultra)-approach	PROPN
ejpam-5279	532	81	groups	group	NOUN
ejpam-5279	532	82	.	.	PUNCT
ejpam-5279	533	1	since	since	SCONJ
ejpam-5279	533	2	ultra	ultra	ADJ
ejpam-5279	533	3	approach	approach	NOUN
ejpam-5279	533	4	structures	structure	NOUN
ejpam-5279	533	5	originated	originate	VERB
ejpam-5279	533	6	from	from	ADP
ejpam-5279	533	7	the	the	DET
ejpam-5279	533	8	idea	idea	NOUN
ejpam-5279	533	9	of	of	ADP
ejpam-5279	533	10	non	non	ADJ
ejpam-5279	533	11	-	-	ADJ
ejpam-5279	533	12	archimedean	archimedean	ADJ
ejpam-5279	533	13	structure	structure	NOUN
ejpam-5279	533	14	or	or	CCONJ
ejpam-5279	533	15	ultra	ultra	ADJ
ejpam-5279	533	16	metrics	metric	NOUN
ejpam-5279	533	17	,	,	PUNCT
ejpam-5279	533	18	we	we	PRON
ejpam-5279	533	19	are	be	AUX
ejpam-5279	533	20	interested	interested	ADJ
ejpam-5279	533	21	to	to	PART
ejpam-5279	533	22	associate	associate	VERB
ejpam-5279	533	23	all	all	PRON
ejpam-5279	533	24	of	of	ADP
ejpam-5279	533	25	these	these	DET
ejpam-5279	533	26	structures	structure	NOUN
ejpam-5279	533	27	in	in	ADP
ejpam-5279	533	28	relation	relation	NOUN
ejpam-5279	533	29	to	to	ADP
ejpam-5279	533	30	ultra	ultra	ADJ
ejpam-5279	533	31	approach	approach	NOUN
ejpam-5279	533	32	metric	metric	ADJ
ejpam-5279	533	33	groups	group	NOUN
ejpam-5279	533	34	,	,	PUNCT
ejpam-5279	533	35	and	and	CCONJ
ejpam-5279	533	36	the	the	DET
ejpam-5279	533	37	∞p	∞p	NOUN
ejpam-5279	533	38	-	-	PUNCT
ejpam-5279	533	39	metrizability	metrizability	NOUN
ejpam-5279	533	40	of	of	ADP
ejpam-5279	533	41	ultra	ultra	ADJ
ejpam-5279	533	42	approach	approach	NOUN
ejpam-5279	533	43	groups	group	NOUN
ejpam-5279	533	44	.	.	PUNCT
ejpam-5279	534	1	these	these	DET
ejpam-5279	534	2	questions	question	NOUN
ejpam-5279	534	3	are	be	AUX
ejpam-5279	534	4	yet	yet	ADV
ejpam-5279	534	5	to	to	PART
ejpam-5279	534	6	be	be	AUX
ejpam-5279	534	7	established	establish	VERB
ejpam-5279	534	8	.	.	PUNCT
ejpam-5279	535	1	we	we	PRON
ejpam-5279	535	2	hope	hope	VERB
ejpam-5279	535	3	to	to	PART
ejpam-5279	535	4	settle	settle	VERB
ejpam-5279	535	5	these	these	DET
ejpam-5279	535	6	issues	issue	NOUN
ejpam-5279	535	7	in	in	ADP
ejpam-5279	535	8	one	one	NUM
ejpam-5279	535	9	of	of	ADP
ejpam-5279	535	10	our	our	PRON
ejpam-5279	535	11	forthcoming	forthcoming	ADJ
ejpam-5279	535	12	papers	paper	NOUN
ejpam-5279	535	13	.	.	PUNCT
ejpam-5279	536	1	it	it	PRON
ejpam-5279	536	2	would	would	AUX
ejpam-5279	536	3	be	be	AUX
ejpam-5279	536	4	interesting	interesting	ADJ
ejpam-5279	536	5	to	to	PART
ejpam-5279	536	6	add	add	VERB
ejpam-5279	536	7	some	some	DET
ejpam-5279	536	8	applications	application	NOUN
ejpam-5279	536	9	like	like	ADP
ejpam-5279	536	10	those	those	PRON
ejpam-5279	536	11	in	in	ADP
ejpam-5279	536	12	[	[	X
ejpam-5279	536	13	10–12	10–12	NUM
ejpam-5279	536	14	]	]	PUNCT
ejpam-5279	536	15	that	that	PRON
ejpam-5279	536	16	are	be	AUX
ejpam-5279	536	17	based	base	VERB
ejpam-5279	536	18	on	on	ADP
ejpam-5279	536	19	rough	rough	ADJ
ejpam-5279	536	20	sets	set	NOUN
ejpam-5279	536	21	and	and	CCONJ
ejpam-5279	536	22	their	their	PRON
ejpam-5279	536	23	generalizations	generalization	NOUN
ejpam-5279	536	24	.	.	PUNCT
ejpam-5279	537	1	since	since	SCONJ
ejpam-5279	537	2	as	as	ADP
ejpam-5279	537	3	this	this	DET
ejpam-5279	537	4	stage	stage	NOUN
ejpam-5279	537	5	we	we	PRON
ejpam-5279	537	6	do	do	AUX
ejpam-5279	537	7	not	not	PART
ejpam-5279	537	8	see	see	VERB
ejpam-5279	537	9	any	any	DET
ejpam-5279	537	10	direct	direct	ADJ
ejpam-5279	537	11	relationship	relationship	NOUN
ejpam-5279	537	12	of	of	ADP
ejpam-5279	537	13	our	our	PRON
ejpam-5279	537	14	work	work	NOUN
ejpam-5279	537	15	with	with	ADP
ejpam-5279	537	16	rough	rough	ADJ
ejpam-5279	537	17	sets	set	NOUN
ejpam-5279	537	18	,	,	PUNCT
ejpam-5279	537	19	one	one	NUM
ejpam-5279	537	20	of	of	ADP
ejpam-5279	537	21	the	the	DET
ejpam-5279	537	22	reasons	reason	NOUN
ejpam-5279	537	23	could	could	AUX
ejpam-5279	537	24	be	be	AUX
ejpam-5279	537	25	that	that	DET
ejpam-5279	537	26	rough	rough	ADJ
ejpam-5279	537	27	sets	set	NOUN
ejpam-5279	537	28	are	be	AUX
ejpam-5279	537	29	generalization	generalization	NOUN
ejpam-5279	537	30	of	of	ADP
ejpam-5279	537	31	zadeh	zadeh	PROPN
ejpam-5279	537	32	’s	’s	PART
ejpam-5279	537	33	concept	concept	NOUN
ejpam-5279	537	34	of	of	ADP
ejpam-5279	537	35	fuzzy	fuzzy	ADJ
ejpam-5279	537	36	sets	set	NOUN
ejpam-5279	537	37	whereas	whereas	SCONJ
ejpam-5279	537	38	our	our	PRON
ejpam-5279	537	39	findings	finding	NOUN
ejpam-5279	537	40	are	be	AUX
ejpam-5279	537	41	based	base	VERB
ejpam-5279	537	42	on	on	ADP
ejpam-5279	537	43	nonfuzzy	nonfuzzy	NOUN
ejpam-5279	537	44	,	,	PUNCT
ejpam-5279	537	45	rather	rather	ADV
ejpam-5279	537	46	it	it	PRON
ejpam-5279	537	47	is	be	AUX
ejpam-5279	537	48	based	base	VERB
ejpam-5279	537	49	on	on	ADP
ejpam-5279	537	50	non	non	ADJ
ejpam-5279	537	51	-	-	ADJ
ejpam-5279	537	52	archimedean	archimedean	ADJ
ejpam-5279	537	53	metric	metric	ADJ
ejpam-5279	537	54	spaces	space	NOUN
ejpam-5279	537	55	;	;	PUNCT
ejpam-5279	537	56	however	however	ADV
ejpam-5279	537	57	,	,	PUNCT
ejpam-5279	537	58	we	we	PRON
ejpam-5279	537	59	will	will	AUX
ejpam-5279	537	60	look	look	VERB
ejpam-5279	537	61	into	into	ADP
ejpam-5279	537	62	this	this	DET
ejpam-5279	537	63	extraordinary	extraordinary	ADJ
ejpam-5279	537	64	situations	situation	NOUN
ejpam-5279	537	65	in	in	ADP
ejpam-5279	537	66	our	our	PRON
ejpam-5279	537	67	future	future	ADJ
ejpam-5279	537	68	research	research	NOUN
ejpam-5279	537	69	.	.	PUNCT
ejpam-5279	538	1	however	however	ADV
ejpam-5279	538	2	,	,	PUNCT
ejpam-5279	538	3	for	for	SCONJ
ejpam-5279	538	4	better	well	ADV
ejpam-5279	538	5	understand	understand	VERB
ejpam-5279	538	6	the	the	DET
ejpam-5279	538	7	soft	soft	ADJ
ejpam-5279	538	8	sets	set	NOUN
ejpam-5279	538	9	and	and	CCONJ
ejpam-5279	538	10	their	their	PRON
ejpam-5279	538	11	applications	application	NOUN
ejpam-5279	538	12	in	in	ADP
ejpam-5279	538	13	medical	medical	ADJ
ejpam-5279	538	14	science	science	NOUN
ejpam-5279	538	15	,	,	PUNCT
ejpam-5279	538	16	we	we	PRON
ejpam-5279	538	17	refer	refer	VERB
ejpam-5279	538	18	to	to	ADP
ejpam-5279	538	19	some	some	PRON
ejpam-5279	538	20	of	of	ADP
ejpam-5279	538	21	the	the	DET
ejpam-5279	538	22	papers	paper	NOUN
ejpam-5279	538	23	which	which	PRON
ejpam-5279	538	24	are	be	AUX
ejpam-5279	538	25	definitely	definitely	ADV
ejpam-5279	538	26	interesting	interesting	ADJ
ejpam-5279	538	27	in	in	ADP
ejpam-5279	538	28	their	their	PRON
ejpam-5279	538	29	own	own	ADJ
ejpam-5279	538	30	right	right	NOUN
ejpam-5279	538	31	such	such	ADJ
ejpam-5279	538	32	as	as	ADP
ejpam-5279	538	33	[	[	X
ejpam-5279	538	34	10–12	10–12	NUM
ejpam-5279	538	35	]	]	PUNCT
ejpam-5279	538	36	.	.	PUNCT
ejpam-5279	539	1	however	however	ADV
ejpam-5279	539	2	,	,	PUNCT
ejpam-5279	539	3	approach	approach	NOUN
ejpam-5279	539	4	spaces	space	NOUN
ejpam-5279	539	5	have	have	VERB
ejpam-5279	539	6	significant	significant	ADJ
ejpam-5279	539	7	applications	application	NOUN
ejpam-5279	539	8	within	within	ADP
ejpam-5279	539	9	mathematics	mathematic	NOUN
ejpam-5279	539	10	,	,	PUNCT
ejpam-5279	539	11	such	such	ADJ
ejpam-5279	539	12	as	as	ADP
ejpam-5279	539	13	,	,	PUNCT
ejpam-5279	539	14	functional	functional	ADJ
ejpam-5279	539	15	analysis	analysis	NOUN
ejpam-5279	539	16	,	,	PUNCT
ejpam-5279	539	17	and	and	CCONJ
ejpam-5279	539	18	much	much	ADV
ejpam-5279	539	19	beyond	beyond	ADP
ejpam-5279	539	20	which	which	PRON
ejpam-5279	539	21	can	can	AUX
ejpam-5279	539	22	be	be	AUX
ejpam-5279	539	23	find	find	VERB
ejpam-5279	539	24	out	out	ADP
ejpam-5279	539	25	in	in	ADP
ejpam-5279	539	26	[	[	X
ejpam-5279	539	27	16	16	NUM
ejpam-5279	539	28	]	]	PUNCT
ejpam-5279	539	29	,	,	PUNCT
ejpam-5279	539	30	where	where	SCONJ
ejpam-5279	539	31	one	one	PRON
ejpam-5279	539	32	can	can	AUX
ejpam-5279	539	33	also	also	ADV
ejpam-5279	539	34	find	find	VERB
ejpam-5279	539	35	plenty	plenty	NOUN
ejpam-5279	539	36	of	of	ADP
ejpam-5279	539	37	examples	example	NOUN
ejpam-5279	539	38	on	on	ADP
ejpam-5279	539	39	approach	approach	NOUN
ejpam-5279	539	40	spaces	space	NOUN
ejpam-5279	539	41	,	,	PUNCT
ejpam-5279	539	42	and	and	CCONJ
ejpam-5279	539	43	their	their	PRON
ejpam-5279	539	44	connected	connected	ADJ
ejpam-5279	539	45	branches	branch	NOUN
ejpam-5279	539	46	.	.	PUNCT
ejpam-5279	540	1	for	for	ADP
ejpam-5279	540	2	further	further	ADJ
ejpam-5279	540	3	examples	example	NOUN
ejpam-5279	540	4	on	on	ADP
ejpam-5279	540	5	approach	approach	NOUN
ejpam-5279	540	6	groups	group	NOUN
ejpam-5279	540	7	,	,	PUNCT
ejpam-5279	540	8	we	we	PRON
ejpam-5279	540	9	refer	refer	VERB
ejpam-5279	540	10	to	to	ADP
ejpam-5279	540	11	[	[	X
ejpam-5279	540	12	4	4	NUM
ejpam-5279	540	13	]	]	PUNCT
ejpam-5279	540	14	.	.	PUNCT
ejpam-5279	541	1	acknowledgements	acknowledgement	NOUN
ejpam-5279	541	2	we	we	PRON
ejpam-5279	541	3	are	be	AUX
ejpam-5279	541	4	sincerely	sincerely	ADV
ejpam-5279	541	5	grateful	grateful	ADJ
ejpam-5279	541	6	to	to	ADP
ejpam-5279	541	7	the	the	DET
ejpam-5279	541	8	reviewers	reviewer	NOUN
ejpam-5279	541	9	for	for	ADP
ejpam-5279	541	10	their	their	PRON
ejpam-5279	541	11	useful	useful	ADJ
ejpam-5279	541	12	comments	comment	NOUN
ejpam-5279	541	13	and	and	CCONJ
ejpam-5279	541	14	suggestions	suggestion	NOUN
ejpam-5279	541	15	.	.	PUNCT
ejpam-5279	542	1	references	reference	NOUN
ejpam-5279	542	2	1777	1777	NUM
ejpam-5279	542	3	references	reference	NOUN
ejpam-5279	542	4	[	[	X
ejpam-5279	542	5	1	1	NUM
ejpam-5279	542	6	]	]	PUNCT
ejpam-5279	542	7	j.	j.	PROPN
ejpam-5279	542	8	adámek	adámek	PROPN
ejpam-5279	542	9	,	,	PUNCT
ejpam-5279	542	10	h.	h.	PROPN
ejpam-5279	542	11	herrlich	herrlich	PROPN
ejpam-5279	542	12	,	,	PUNCT
ejpam-5279	542	13	and	and	CCONJ
ejpam-5279	542	14	g.	g.	PROPN
ejpam-5279	542	15	e.	e.	PROPN
ejpam-5279	542	16	strecker	strecker	PROPN
ejpam-5279	542	17	.	.	PUNCT
ejpam-5279	543	1	abstract	abstract	ADJ
ejpam-5279	543	2	and	and	CCONJ
ejpam-5279	543	3	concrete	concrete	ADJ
ejpam-5279	543	4	categories	category	NOUN
ejpam-5279	543	5	.	.	PUNCT
ejpam-5279	544	1	wiley	wiley	PROPN
ejpam-5279	544	2	,	,	PUNCT
ejpam-5279	544	3	new	new	PROPN
ejpam-5279	544	4	york	york	PROPN
ejpam-5279	544	5	,	,	PUNCT
ejpam-5279	544	6	1989	1989	NUM
ejpam-5279	544	7	.	.	PUNCT
ejpam-5279	545	1	[	[	X
ejpam-5279	545	2	2	2	X
ejpam-5279	545	3	]	]	PUNCT
ejpam-5279	545	4	t.	t.	NOUN
ejpam-5279	545	5	m.	m.	NOUN
ejpam-5279	545	6	g.	g.	PROPN
ejpam-5279	545	7	ahsanullah	ahsanullah	PROPN
ejpam-5279	545	8	,	,	PUNCT
ejpam-5279	545	9	fawzi	fawzi	PROPN
ejpam-5279	545	10	al	al	PROPN
ejpam-5279	545	11	-	-	PUNCT
ejpam-5279	545	12	thukair	thukair	NOUN
ejpam-5279	545	13	,	,	PUNCT
ejpam-5279	545	14	and	and	CCONJ
ejpam-5279	545	15	jawaher	jawaher	PROPN
ejpam-5279	545	16	al	al	PROPN
ejpam-5279	545	17	-	-	PUNCT
ejpam-5279	545	18	mufarrif	mufarrif	PROPN
ejpam-5279	545	19	.	.	PUNCT
ejpam-5279	546	1	on	on	ADP
ejpam-5279	546	2	the	the	DET
ejpam-5279	546	3	continuous	continuous	ADJ
ejpam-5279	546	4	action	action	NOUN
ejpam-5279	546	5	of	of	ADP
ejpam-5279	546	6	enriched	enriched	ADJ
ejpam-5279	546	7	lattice	lattice	NOUN
ejpam-5279	546	8	-	-	PUNCT
ejpam-5279	546	9	valued	value	VERB
ejpam-5279	546	10	convergence	convergence	NOUN
ejpam-5279	546	11	groups	group	NOUN
ejpam-5279	546	12	:	:	PUNCT
ejpam-5279	546	13	some	some	DET
ejpam-5279	546	14	examples	example	NOUN
ejpam-5279	546	15	.	.	PUNCT
ejpam-5279	547	1	filomat	filomat	NOUN
ejpam-5279	547	2	,	,	PUNCT
ejpam-5279	547	3	34(9):3045–3064	34(9):3045–3064	NOUN
ejpam-5279	547	4	,	,	PUNCT
ejpam-5279	547	5	2020	2020	NUM
ejpam-5279	547	6	.	.	PUNCT
ejpam-5279	548	1	[	[	X
ejpam-5279	548	2	3	3	X
ejpam-5279	548	3	]	]	PUNCT
ejpam-5279	548	4	t.	t.	NOUN
ejpam-5279	548	5	m.	m.	NOUN
ejpam-5279	548	6	g.	g.	PROPN
ejpam-5279	548	7	ahsanullah	ahsanullah	PROPN
ejpam-5279	548	8	,	,	PUNCT
ejpam-5279	548	9	david	david	PROPN
ejpam-5279	548	10	gould	gould	PROPN
ejpam-5279	548	11	,	,	PUNCT
ejpam-5279	548	12	jawaher	jawaher	PROPN
ejpam-5279	548	13	al	al	PROPN
ejpam-5279	548	14	-	-	PUNCT
ejpam-5279	548	15	mufarrij	mufarrij	PROPN
ejpam-5279	548	16	,	,	PUNCT
ejpam-5279	548	17	and	and	CCONJ
ejpam-5279	548	18	fawzi	fawzi	VERB
ejpam-5279	548	19	al	al	PROPN
ejpam-5279	548	20	-	-	PUNCT
ejpam-5279	548	21	thukair	thukair	NOUN
ejpam-5279	548	22	.	.	PUNCT
ejpam-5279	549	1	enriched	enrich	VERB
ejpam-5279	549	2	lattice	lattice	NOUN
ejpam-5279	549	3	-	-	PUNCT
ejpam-5279	549	4	valued	value	VERB
ejpam-5279	549	5	convergence	convergence	NOUN
ejpam-5279	549	6	groups	group	NOUN
ejpam-5279	549	7	.	.	PUNCT
ejpam-5279	550	1	fuzzy	fuzzy	ADJ
ejpam-5279	550	2	sets	set	NOUN
ejpam-5279	550	3	and	and	CCONJ
ejpam-5279	550	4	systems	system	NOUN
ejpam-5279	550	5	,	,	PUNCT
ejpam-5279	550	6	238(1):71–88	238(1):71–88	NUM
ejpam-5279	550	7	,	,	PUNCT
ejpam-5279	550	8	2014	2014	NUM
ejpam-5279	550	9	.	.	PUNCT
ejpam-5279	551	1	[	[	X
ejpam-5279	551	2	4	4	X
ejpam-5279	551	3	]	]	PUNCT
ejpam-5279	551	4	t.	t.	NOUN
ejpam-5279	551	5	m.	m.	NOUN
ejpam-5279	551	6	g.	g.	PROPN
ejpam-5279	551	7	ahsanullah	ahsanullah	PROPN
ejpam-5279	551	8	and	and	CCONJ
ejpam-5279	551	9	g.	g.	PROPN
ejpam-5279	551	10	jäger	jäger	PROPN
ejpam-5279	551	11	.	.	PUNCT
ejpam-5279	552	1	on	on	ADP
ejpam-5279	552	2	approach	approach	NOUN
ejpam-5279	552	3	limit	limit	NOUN
ejpam-5279	552	4	groups	group	NOUN
ejpam-5279	552	5	and	and	CCONJ
ejpam-5279	552	6	their	their	PRON
ejpam-5279	552	7	uniformization	uniformization	NOUN
ejpam-5279	552	8	.	.	PUNCT
ejpam-5279	553	1	internat	internat	PROPN
ejpam-5279	553	2	.	.	PUNCT
ejpam-5279	554	1	j.	j.	PROPN
ejpam-5279	554	2	contemp	contemp	PROPN
ejpam-5279	554	3	.	.	PUNCT
ejpam-5279	555	1	math	math	NOUN
ejpam-5279	555	2	.	.	PUNCT
ejpam-5279	556	1	sci	sci	PROPN
ejpam-5279	556	2	.	.	PROPN
ejpam-5279	556	3	,	,	PUNCT
ejpam-5279	556	4	9(5):195–213	9(5):195–213	NUM
ejpam-5279	556	5	,	,	PUNCT
ejpam-5279	556	6	2014	2014	NUM
ejpam-5279	556	7	.	.	PUNCT
ejpam-5279	557	1	[	[	X
ejpam-5279	557	2	5	5	NUM
ejpam-5279	557	3	]	]	PUNCT
ejpam-5279	557	4	r.	r.	PROPN
ejpam-5279	557	5	n.	n.	PROPN
ejpam-5279	557	6	ball	ball	PROPN
ejpam-5279	557	7	.	.	PUNCT
ejpam-5279	558	1	convergence	convergence	NOUN
ejpam-5279	558	2	and	and	CCONJ
ejpam-5279	558	3	cauchy	cauchy	NOUN
ejpam-5279	558	4	structures	structure	NOUN
ejpam-5279	558	5	on	on	ADP
ejpam-5279	558	6	lattice	lattice	NOUN
ejpam-5279	558	7	ordered	order	VERB
ejpam-5279	558	8	groups	group	NOUN
ejpam-5279	558	9	.	.	PUNCT
ejpam-5279	559	1	trans	trans	PROPN
ejpam-5279	559	2	.	.	PUNCT
ejpam-5279	560	1	amer	amer	PROPN
ejpam-5279	560	2	.	.	PUNCT
ejpam-5279	560	3	math	math	PROPN
ejpam-5279	560	4	.	.	PUNCT
ejpam-5279	561	1	soc	soc	PROPN
ejpam-5279	561	2	.	.	PROPN
ejpam-5279	561	3	,	,	PUNCT
ejpam-5279	561	4	259:357–392	259:357–392	NUM
ejpam-5279	561	5	,	,	PUNCT
ejpam-5279	561	6	1980	1980	NUM
ejpam-5279	561	7	.	.	PUNCT
ejpam-5279	562	1	[	[	X
ejpam-5279	562	2	6	6	NUM
ejpam-5279	562	3	]	]	PUNCT
ejpam-5279	562	4	m.	m.	NOUN
ejpam-5279	562	5	baran	baran	NOUN
ejpam-5279	562	6	and	and	CCONJ
ejpam-5279	562	7	m.	m.	PROPN
ejpam-5279	562	8	qasim	qasim	PROPN
ejpam-5279	562	9	.	.	PUNCT
ejpam-5279	563	1	t1	t1	NOUN
ejpam-5279	563	2	approach	approach	NOUN
ejpam-5279	563	3	spaces	space	VERB
ejpam-5279	563	4	.	.	PUNCT
ejpam-5279	564	1	commnunicat	commnunicat	PROPN
ejpam-5279	564	2	.	.	PUNCT
ejpam-5279	565	1	fac	fac	PROPN
ejpam-5279	565	2	.	.	PROPN
ejpam-5279	565	3	,	,	PUNCT
ejpam-5279	565	4	university	university	PROPN
ejpam-5279	565	5	of	of	ADP
ejpam-5279	565	6	ankara	ankara	PROPN
ejpam-5279	565	7	,	,	PUNCT
ejpam-5279	565	8	68(1):784–800	68(1):784–800	PROPN
ejpam-5279	565	9	,	,	PUNCT
ejpam-5279	565	10	2019	2019	NUM
ejpam-5279	565	11	.	.	PUNCT
ejpam-5279	566	1	[	[	X
ejpam-5279	566	2	7	7	X
ejpam-5279	566	3	]	]	X
ejpam-5279	566	4	m.	m.	NOUN
ejpam-5279	566	5	baran	baran	NOUN
ejpam-5279	566	6	and	and	CCONJ
ejpam-5279	566	7	m.	m.	PROPN
ejpam-5279	566	8	qasim	qasim	PROPN
ejpam-5279	566	9	.	.	PUNCT
ejpam-5279	567	1	t0	t0	PROPN
ejpam-5279	567	2	convergence	convergence	NOUN
ejpam-5279	567	3	approach	approach	NOUN
ejpam-5279	567	4	spaces	space	VERB
ejpam-5279	567	5	.	.	PUNCT
ejpam-5279	568	1	commnunicat	commnunicat	PROPN
ejpam-5279	568	2	.	.	PUNCT
ejpam-5279	569	1	fac	fac	PROPN
ejpam-5279	569	2	.	.	PROPN
ejpam-5279	569	3	,	,	PUNCT
ejpam-5279	569	4	university	university	PROPN
ejpam-5279	569	5	of	of	ADP
ejpam-5279	569	6	ankara	ankara	PROPN
ejpam-5279	569	7	,	,	PUNCT
ejpam-5279	569	8	69(1):603–612	69(1):603–612	NOUN
ejpam-5279	569	9	,	,	PUNCT
ejpam-5279	569	10	2020	2020	NUM
ejpam-5279	569	11	.	.	PUNCT
ejpam-5279	570	1	[	[	X
ejpam-5279	570	2	8	8	NUM
ejpam-5279	570	3	]	]	PUNCT
ejpam-5279	570	4	p.	p.	NOUN
ejpam-5279	570	5	brock	brock	PROPN
ejpam-5279	570	6	and	and	CCONJ
ejpam-5279	570	7	d.	d.	PROPN
ejpam-5279	570	8	c.	c.	PROPN
ejpam-5279	570	9	kent	kent	PROPN
ejpam-5279	570	10	.	.	PUNCT
ejpam-5279	571	1	approach	approach	NOUN
ejpam-5279	571	2	spaces	space	NOUN
ejpam-5279	571	3	,	,	PUNCT
ejpam-5279	571	4	limit	limit	VERB
ejpam-5279	571	5	tower	tower	NOUN
ejpam-5279	571	6	spaces	space	NOUN
ejpam-5279	571	7	,	,	PUNCT
ejpam-5279	571	8	and	and	CCONJ
ejpam-5279	571	9	probabilistic	probabilistic	ADJ
ejpam-5279	571	10	convergence	convergence	NOUN
ejpam-5279	571	11	spaces	space	NOUN
ejpam-5279	571	12	.	.	PUNCT
ejpam-5279	572	1	appl	appl	PROPN
ejpam-5279	572	2	.	.	PROPN
ejpam-5279	572	3	categor	categor	PROPN
ejpam-5279	572	4	.	.	PUNCT
ejpam-5279	573	1	struct	struct	NOUN
ejpam-5279	573	2	.	.	PUNCT
ejpam-5279	573	3	,	,	PUNCT
ejpam-5279	573	4	5:99–110	5:99–110	NUM
ejpam-5279	573	5	,	,	PUNCT
ejpam-5279	573	6	1997	1997	NUM
ejpam-5279	573	7	.	.	PUNCT
ejpam-5279	574	1	[	[	X
ejpam-5279	574	2	9	9	NUM
ejpam-5279	574	3	]	]	X
ejpam-5279	574	4	e.	e.	PROPN
ejpam-5279	574	5	colebunders	colebunders	PROPN
ejpam-5279	574	6	and	and	CCONJ
ejpam-5279	574	7	k.	k.	PROPN
ejpam-5279	574	8	van	van	PROPN
ejpam-5279	574	9	opdenbosch	opdenbosch	PROPN
ejpam-5279	574	10	.	.	PUNCT
ejpam-5279	575	1	topological	topological	ADJ
ejpam-5279	575	2	properties	property	NOUN
ejpam-5279	575	3	of	of	ADP
ejpam-5279	575	4	non	non	ADJ
ejpam-5279	575	5	-	-	ADJ
ejpam-5279	575	6	archimedean	archimedean	ADJ
ejpam-5279	575	7	approach	approach	NOUN
ejpam-5279	575	8	spaces	space	NOUN
ejpam-5279	575	9	.	.	PUNCT
ejpam-5279	576	1	theory	theory	NOUN
ejpam-5279	576	2	and	and	CCONJ
ejpam-5279	576	3	appl	appl	PROPN
ejpam-5279	576	4	.	.	PUNCT
ejpam-5279	576	5	cat	cat	PROPN
ejpam-5279	576	6	.	.	PUNCT
ejpam-5279	576	7	,	,	PUNCT
ejpam-5279	576	8	32(41):1454–1484	32(41):1454–1484	NUM
ejpam-5279	576	9	,	,	PUNCT
ejpam-5279	576	10	2017	2017	NUM
ejpam-5279	576	11	.	.	PUNCT
ejpam-5279	577	1	[	[	X
ejpam-5279	577	2	10	10	NUM
ejpam-5279	577	3	]	]	PUNCT
ejpam-5279	577	4	m.	m.	NOUN
ejpam-5279	577	5	k.	k.	PROPN
ejpam-5279	578	1	el	el	PROPN
ejpam-5279	578	2	-	-	PROPN
ejpam-5279	578	3	bably	bably	ADV
ejpam-5279	578	4	,	,	PUNCT
ejpam-5279	578	5	m.	m.	NOUN
ejpam-5279	578	6	i.	i.	PROPN
ejpam-5279	578	7	ali	ali	PROPN
ejpam-5279	578	8	,	,	PUNCT
ejpam-5279	578	9	and	and	CCONJ
ejpam-5279	578	10	e.	e.	PROPN
ejpam-5279	578	11	a.	a.	PROPN
ejpam-5279	578	12	abo	abo	PROPN
ejpam-5279	578	13	tabl	tabl	PROPN
ejpam-5279	578	14	.	.	PUNCT
ejpam-5279	579	1	new	new	ADJ
ejpam-5279	579	2	topological	topological	ADJ
ejpam-5279	579	3	approaches	approach	NOUN
ejpam-5279	579	4	to	to	ADP
ejpam-5279	579	5	generalized	generalize	VERB
ejpam-5279	579	6	soft	soft	ADJ
ejpam-5279	579	7	rough	rough	ADJ
ejpam-5279	579	8	sets	set	NOUN
ejpam-5279	579	9	with	with	ADP
ejpam-5279	579	10	medical	medical	ADJ
ejpam-5279	579	11	applications	application	NOUN
ejpam-5279	579	12	.	.	PUNCT
ejpam-5279	580	1	j.	j.	PROPN
ejpam-5279	580	2	math	math	PROPN
ejpam-5279	580	3	.	.	PUNCT
ejpam-5279	580	4	,	,	PUNCT
ejpam-5279	580	5	2021	2021	NUM
ejpam-5279	580	6	.	.	PUNCT
ejpam-5279	581	1	[	[	X
ejpam-5279	581	2	11	11	NUM
ejpam-5279	581	3	]	]	PUNCT
ejpam-5279	581	4	m.	m.	NOUN
ejpam-5279	581	5	k.	k.	PROPN
ejpam-5279	582	1	el	el	PROPN
ejpam-5279	582	2	-	-	PROPN
ejpam-5279	582	3	bably	bably	PROPN
ejpam-5279	582	4	and	and	CCONJ
ejpam-5279	582	5	m.	m.	NOUN
ejpam-5279	582	6	el	el	PROPN
ejpam-5279	582	7	-	-	PUNCT
ejpam-5279	582	8	sayed	say	VERB
ejpam-5279	582	9	.	.	PUNCT
ejpam-5279	583	1	three	three	NUM
ejpam-5279	583	2	methods	method	NOUN
ejpam-5279	583	3	to	to	PART
ejpam-5279	583	4	generalized	generalized	ADJ
ejpam-5279	583	5	pawlak	pawlak	ADJ
ejpam-5279	583	6	approximations	approximation	NOUN
ejpam-5279	583	7	via	via	ADP
ejpam-5279	583	8	simply	simply	ADV
ejpam-5279	583	9	open	open	ADJ
ejpam-5279	583	10	concepts	concept	NOUN
ejpam-5279	583	11	with	with	ADP
ejpam-5279	583	12	economic	economic	ADJ
ejpam-5279	583	13	applications	application	NOUN
ejpam-5279	583	14	.	.	PUNCT
ejpam-5279	584	1	soft	soft	ADJ
ejpam-5279	584	2	comput	comput	NOUN
ejpam-5279	584	3	.	.	PUNCT
ejpam-5279	585	1	,	,	PUNCT
ejpam-5279	585	2	26:4685	26:4685	NUM
ejpam-5279	585	3	–	–	PUNCT
ejpam-5279	585	4	4700	4700	NUM
ejpam-5279	585	5	,	,	PUNCT
ejpam-5279	585	6	2022	2022	NUM
ejpam-5279	585	7	.	.	PUNCT
ejpam-5279	586	1	[	[	X
ejpam-5279	586	2	12	12	NUM
ejpam-5279	586	3	]	]	PUNCT
ejpam-5279	586	4	m.	m.	NOUN
ejpam-5279	586	5	k.	k.	PROPN
ejpam-5279	587	1	el	el	PROPN
ejpam-5279	587	2	-	-	PROPN
ejpam-5279	587	3	bably	bably	PROPN
ejpam-5279	587	4	and	and	CCONJ
ejpam-5279	587	5	e.	e.	PROPN
ejpam-5279	587	6	a.	a.	PROPN
ejpam-5279	587	7	abo	abo	PROPN
ejpam-5279	587	8	tabl	tabl	PROPN
ejpam-5279	587	9	.	.	PUNCT
ejpam-5279	588	1	a	a	DET
ejpam-5279	588	2	topological	topological	ADJ
ejpam-5279	588	3	reduction	reduction	NOUN
ejpam-5279	588	4	for	for	ADP
ejpam-5279	588	5	predicting	predict	VERB
ejpam-5279	588	6	a	a	DET
ejpam-5279	588	7	lung	lung	NOUN
ejpam-5279	588	8	cancer	cancer	NOUN
ejpam-5279	588	9	disease	disease	NOUN
ejpam-5279	588	10	based	base	VERB
ejpam-5279	588	11	on	on	ADP
ejpam-5279	588	12	generalized	generalized	ADJ
ejpam-5279	588	13	rough	rough	ADJ
ejpam-5279	588	14	sets	set	NOUN
ejpam-5279	588	15	.	.	PUNCT
ejpam-5279	589	1	j.	j.	PROPN
ejpam-5279	589	2	intelligent	intelligent	ADJ
ejpam-5279	589	3	and	and	CCONJ
ejpam-5279	589	4	fuzzy	fuzzy	ADJ
ejpam-5279	589	5	syst	syst	NOUN
ejpam-5279	589	6	.	.	PUNCT
ejpam-5279	589	7	,	,	PUNCT
ejpam-5279	589	8	41(2):3045	41(2):3045	PROPN
ejpam-5279	589	9	–	–	PUNCT
ejpam-5279	589	10	3060	3060	NUM
ejpam-5279	589	11	,	,	PUNCT
ejpam-5279	589	12	2021	2021	NUM
ejpam-5279	589	13	.	.	PUNCT
ejpam-5279	590	1	[	[	X
ejpam-5279	590	2	13	13	NUM
ejpam-5279	590	3	]	]	X
ejpam-5279	590	4	g.	g.	PROPN
ejpam-5279	590	5	jäger	jäger	PROPN
ejpam-5279	590	6	.	.	PUNCT
ejpam-5279	591	1	a	a	DET
ejpam-5279	591	2	note	note	NOUN
ejpam-5279	591	3	on	on	ADP
ejpam-5279	591	4	neighbourhoods	neighbourhood	NOUN
ejpam-5279	591	5	for	for	ADP
ejpam-5279	591	6	approach	approach	NOUN
ejpam-5279	591	7	spaces	space	NOUN
ejpam-5279	591	8	.	.	PUNCT
ejpam-5279	592	1	hacettepe	hacettepe	PROPN
ejpam-5279	592	2	j.	j.	PROPN
ejpam-5279	592	3	math	math	PROPN
ejpam-5279	592	4	.	.	PUNCT
ejpam-5279	593	1	and	and	CCONJ
ejpam-5279	593	2	stat	stat	PROPN
ejpam-5279	593	3	.	.	PUNCT
ejpam-5279	593	4	,	,	PUNCT
ejpam-5279	593	5	41(2):283–290	41(2):283–290	PROPN
ejpam-5279	593	6	,	,	PUNCT
ejpam-5279	593	7	2012	2012	NUM
ejpam-5279	593	8	.	.	PUNCT
ejpam-5279	594	1	[	[	X
ejpam-5279	594	2	14	14	NUM
ejpam-5279	594	3	]	]	X
ejpam-5279	594	4	y.	y.	PROPN
ejpam-5279	594	5	j.	j.	PROPN
ejpam-5279	594	6	lee	lee	PROPN
ejpam-5279	594	7	and	and	CCONJ
ejpam-5279	594	8	b.	b.	PROPN
ejpam-5279	594	9	windels	windels	PROPN
ejpam-5279	594	10	.	.	PUNCT
ejpam-5279	595	1	transitivity	transitivity	NOUN
ejpam-5279	595	2	in	in	ADP
ejpam-5279	595	3	approach	approach	NOUN
ejpam-5279	595	4	theory	theory	NOUN
ejpam-5279	595	5	.	.	PUNCT
ejpam-5279	596	1	int	int	NOUN
ejpam-5279	596	2	.	.	PUNCT
ejpam-5279	597	1	j.	j.	PROPN
ejpam-5279	597	2	math	math	PROPN
ejpam-5279	597	3	.	.	PUNCT
ejpam-5279	598	1	sci	sci	PROPN
ejpam-5279	598	2	.	.	PROPN
ejpam-5279	598	3	,	,	PUNCT
ejpam-5279	598	4	32:707	32:707	NUM
ejpam-5279	598	5	–	–	PUNCT
ejpam-5279	598	6	720	720	NUM
ejpam-5279	598	7	,	,	PUNCT
ejpam-5279	598	8	2002	2002	NUM
ejpam-5279	598	9	.	.	PUNCT
ejpam-5279	599	1	references	reference	NOUN
ejpam-5279	599	2	1778	1778	NUM
ejpam-5279	599	3	[	[	X
ejpam-5279	599	4	15	15	NUM
ejpam-5279	599	5	]	]	X
ejpam-5279	599	6	r.	r.	PROPN
ejpam-5279	599	7	lowen	lowen	PROPN
ejpam-5279	599	8	.	.	PUNCT
ejpam-5279	600	1	approach	approach	NOUN
ejpam-5279	600	2	spaces	space	VERB
ejpam-5279	600	3	:	:	PUNCT
ejpam-5279	600	4	a	a	DET
ejpam-5279	600	5	common	common	ADJ
ejpam-5279	600	6	supercategory	supercategory	NOUN
ejpam-5279	600	7	of	of	ADP
ejpam-5279	600	8	top	top	NOUN
ejpam-5279	600	9	and	and	CCONJ
ejpam-5279	600	10	met	meet	VERB
ejpam-5279	600	11	.	.	PUNCT
ejpam-5279	601	1	math	math	NOUN
ejpam-5279	601	2	.	.	PUNCT
ejpam-5279	602	1	nachr	nachr	PROPN
ejpam-5279	602	2	.	.	PUNCT
ejpam-5279	602	3	,	,	PUNCT
ejpam-5279	602	4	141:183–226	141:183–226	NUM
ejpam-5279	602	5	,	,	PUNCT
ejpam-5279	602	6	1989	1989	NUM
ejpam-5279	602	7	.	.	PUNCT
ejpam-5279	603	1	[	[	X
ejpam-5279	603	2	16	16	NUM
ejpam-5279	603	3	]	]	X
ejpam-5279	603	4	r.	r.	PROPN
ejpam-5279	603	5	lowen	lowen	PROPN
ejpam-5279	603	6	.	.	PUNCT
ejpam-5279	604	1	index	index	NOUN
ejpam-5279	604	2	analysis	analysis	NOUN
ejpam-5279	604	3	:	:	PUNCT
ejpam-5279	604	4	approach	approach	NOUN
ejpam-5279	604	5	theory	theory	NOUN
ejpam-5279	604	6	at	at	ADP
ejpam-5279	604	7	work	work	NOUN
ejpam-5279	604	8	.	.	PUNCT
ejpam-5279	605	1	springer	springer	NOUN
ejpam-5279	605	2	,	,	PUNCT
ejpam-5279	605	3	2015	2015	NUM
ejpam-5279	605	4	.	.	PUNCT
ejpam-5279	606	1	[	[	X
ejpam-5279	606	2	17	17	NUM
ejpam-5279	606	3	]	]	X
ejpam-5279	606	4	r.	r.	PROPN
ejpam-5279	606	5	lowen	lowen	PROPN
ejpam-5279	606	6	and	and	CCONJ
ejpam-5279	606	7	y.	y.	PROPN
ejpam-5279	606	8	j.	j.	PROPN
ejpam-5279	606	9	lee	lee	PROPN
ejpam-5279	606	10	.	.	PROPN
ejpam-5279	606	11	approach	approach	PROPN
ejpam-5279	606	12	theory	theory	NOUN
ejpam-5279	606	13	in	in	ADP
ejpam-5279	606	14	merotopic	merotopic	ADJ
ejpam-5279	606	15	,	,	PUNCT
ejpam-5279	606	16	cauchy	cauchy	NOUN
ejpam-5279	606	17	and	and	CCONJ
ejpam-5279	606	18	convergence	convergence	NOUN
ejpam-5279	606	19	spaces	space	NOUN
ejpam-5279	606	20	i.	i.	PROPN
ejpam-5279	606	21	acta	acta	PROPN
ejpam-5279	606	22	math	math	PROPN
ejpam-5279	606	23	.	.	PUNCT
ejpam-5279	607	1	hungar	hungar	PROPN
ejpam-5279	607	2	.	.	PUNCT
ejpam-5279	607	3	,	,	PUNCT
ejpam-5279	607	4	83(3):189–207	83(3):189–207	PROPN
ejpam-5279	607	5	,	,	PUNCT
ejpam-5279	607	6	1999	1999	NUM
ejpam-5279	607	7	.	.	PUNCT
ejpam-5279	608	1	[	[	X
ejpam-5279	608	2	18	18	NUM
ejpam-5279	608	3	]	]	X
ejpam-5279	608	4	r.	r.	PROPN
ejpam-5279	608	5	lowen	lowen	PROPN
ejpam-5279	608	6	and	and	CCONJ
ejpam-5279	608	7	b.	b.	PROPN
ejpam-5279	608	8	windels	windels	PROPN
ejpam-5279	608	9	.	.	PUNCT
ejpam-5279	609	1	approach	approach	NOUN
ejpam-5279	609	2	groups	group	NOUN
ejpam-5279	609	3	.	.	PUNCT
ejpam-5279	610	1	rocky	rocky	ADJ
ejpam-5279	610	2	mountain	mountain	PROPN
ejpam-5279	610	3	j.	j.	PROPN
ejpam-5279	610	4	math	math	PROPN
ejpam-5279	610	5	.	.	PUNCT
ejpam-5279	610	6	,	,	PUNCT
ejpam-5279	610	7	30:1057–1073	30:1057–1073	NUM
ejpam-5279	610	8	,	,	PUNCT
ejpam-5279	610	9	2000	2000	NUM
ejpam-5279	610	10	.	.	PUNCT
ejpam-5279	611	1	[	[	X
ejpam-5279	611	2	19	19	NUM
ejpam-5279	611	3	]	]	PUNCT
ejpam-5279	611	4	m.	m.	NOUN
ejpam-5279	611	5	megrelishvili	megrelishvili	NOUN
ejpam-5279	611	6	and	and	CCONJ
ejpam-5279	611	7	m.	m.	NOUN
ejpam-5279	611	8	shlossberg	shlossberg	PROPN
ejpam-5279	611	9	.	.	PUNCT
ejpam-5279	612	1	notes	note	NOUN
ejpam-5279	612	2	on	on	ADP
ejpam-5279	612	3	non	non	ADJ
ejpam-5279	612	4	-	-	ADJ
ejpam-5279	612	5	archimedean	archimedean	ADJ
ejpam-5279	612	6	topological	topological	ADJ
ejpam-5279	612	7	groups	group	NOUN
ejpam-5279	612	8	.	.	PUNCT
ejpam-5279	613	1	top	top	ADJ
ejpam-5279	613	2	.	.	PUNCT
ejpam-5279	613	3	appl	appl	PROPN
ejpam-5279	613	4	.	.	PROPN
ejpam-5279	613	5	,	,	PUNCT
ejpam-5279	613	6	159:2497–2505	159:2497–2505	NUM
ejpam-5279	613	7	,	,	PUNCT
ejpam-5279	613	8	2012	2012	NUM
ejpam-5279	613	9	.	.	PUNCT
ejpam-5279	614	1	[	[	X
ejpam-5279	614	2	20	20	NUM
ejpam-5279	614	3	]	]	PUNCT
ejpam-5279	614	4	m.	m.	NOUN
ejpam-5279	614	5	qasim	qasim	PROPN
ejpam-5279	614	6	,	,	PUNCT
ejpam-5279	614	7	m	m	NOUN
ejpam-5279	614	8	baran	baran	NOUN
ejpam-5279	614	9	,	,	PUNCT
ejpam-5279	614	10	and	and	CCONJ
ejpam-5279	614	11	h.	h.	PROPN
ejpam-5279	614	12	abughalwa	abughalwa	PROPN
ejpam-5279	614	13	.	.	PUNCT
ejpam-5279	615	1	closure	closure	NOUN
ejpam-5279	615	2	operators	operator	NOUN
ejpam-5279	615	3	in	in	ADP
ejpam-5279	615	4	convergence	convergence	NOUN
ejpam-5279	615	5	approach	approach	NOUN
ejpam-5279	615	6	spaces	space	VERB
ejpam-5279	615	7	.	.	PUNCT
ejpam-5279	616	1	turkish	turkish	ADJ
ejpam-5279	616	2	j.	j.	PROPN
ejpam-5279	616	3	math	math	PROPN
ejpam-5279	616	4	.	.	PUNCT
ejpam-5279	616	5	,	,	PUNCT
ejpam-5279	616	6	45:139–152	45:139–152	PROPN
ejpam-5279	616	7	,	,	PUNCT
ejpam-5279	616	8	2021	2021	NUM
ejpam-5279	616	9	.	.	PUNCT
ejpam-5279	617	1	[	[	X
ejpam-5279	617	2	21	21	NUM
ejpam-5279	617	3	]	]	X
ejpam-5279	617	4	p.	p.	NOUN
ejpam-5279	617	5	schneider	schneider	PROPN
ejpam-5279	617	6	.	.	PUNCT
ejpam-5279	618	1	nonarchimedean	nonarchimedean	ADJ
ejpam-5279	618	2	functional	functional	ADJ
ejpam-5279	618	3	analysis	analysis	NOUN
ejpam-5279	618	4	.	.	PUNCT
ejpam-5279	619	1	springer	springer	NOUN
ejpam-5279	619	2	-	-	PUNCT
ejpam-5279	619	3	verlag	verlag	PROPN
ejpam-5279	619	4	,	,	PUNCT
ejpam-5279	619	5	heidelberg	heidelberg	PROPN
ejpam-5279	619	6	,	,	PUNCT
ejpam-5279	619	7	2002	2002	NUM
ejpam-5279	619	8	.	.	PUNCT