id	sid	tid	token	lemma	pos
ma-156	1	1	2023	2023	NUM
ma-156	1	2	ada	ada	PROPN
ma-156	1	3	academica	academica	PROPN
ma-156	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-156	1	5	.	.	PUNCT
ma-156	2	1	j.	j.	PROPN
ma-156	2	2	math	math	PROPN
ma-156	2	3	.	.	PUNCT
ma-156	3	1	anal	anal	ADJ
ma-156	3	2	.	.	PUNCT
ma-156	4	1	3	3	NUM
ma-156	4	2	(	(	PUNCT
ma-156	4	3	2023	2023	NUM
ma-156	4	4	)	)	PUNCT
ma-156	4	5	16doi	16doi	NUM
ma-156	4	6	:	:	PUNCT
ma-156	4	7	10.28924	10.28924	NUM
ma-156	4	8	/	/	SYM
ma-156	4	9	ada	ada	PROPN
ma-156	4	10	/	/	SYM
ma-156	4	11	ma.3.16	ma.3.16	PROPN
ma-156	4	12	best	good	ADJ
ma-156	4	13	proximity	proximity	NOUN
ma-156	4	14	points	point	NOUN
ma-156	4	15	for	for	ADP
ma-156	4	16	generalized	generalized	ADJ
ma-156	4	17	geraghty	geraghty	PROPN
ma-156	4	18	quasi	quasi	PROPN
ma-156	4	19	-	-	NOUN
ma-156	4	20	contraction	contraction	NOUN
ma-156	4	21	type	type	NOUN
ma-156	4	22	mappings	mapping	NOUN
ma-156	4	23	in	in	ADP
ma-156	4	24	metric	metric	ADJ
ma-156	4	25	spaces	space	NOUN
ma-156	4	26	j.	j.	PROPN
ma-156	4	27	c.	c.	PROPN
ma-156	4	28	umudu1,∗	umudu1,∗	PROPN
ma-156	4	29	,	,	PUNCT
ma-156	4	30	j.	j.	PROPN
ma-156	4	31	o.	o.	PROPN
ma-156	4	32	olaleru2	olaleru2	PROPN
ma-156	4	33	,	,	PUNCT
ma-156	4	34	h.	h.	PROPN
ma-156	4	35	olaoluwa2	olaoluwa2	PROPN
ma-156	4	36	,	,	PUNCT
ma-156	4	37	a.	a.	NOUN
ma-156	4	38	a.	a.	NOUN
ma-156	4	39	mogbademu2	mogbademu2	PROPN
ma-156	4	40	1department	1department	NUM
ma-156	4	41	of	of	ADP
ma-156	4	42	mathematics	mathematic	NOUN
ma-156	4	43	,	,	PUNCT
ma-156	4	44	faculty	faculty	NOUN
ma-156	4	45	of	of	ADP
ma-156	4	46	natural	natural	ADJ
ma-156	4	47	sciences	science	NOUN
ma-156	4	48	,	,	PUNCT
ma-156	4	49	university	university	NOUN
ma-156	4	50	of	of	ADP
ma-156	4	51	jos	jos	PROPN
ma-156	4	52	,	,	PUNCT
ma-156	4	53	nigeria	nigeria	PROPN
ma-156	4	54	umuduj@unijos.edu.ng	umuduj@unijos.edu.ng	NOUN
ma-156	4	55	2department	2department	NUM
ma-156	4	56	of	of	ADP
ma-156	4	57	mathematics	mathematic	NOUN
ma-156	4	58	,	,	PUNCT
ma-156	4	59	faculty	faculty	NOUN
ma-156	4	60	of	of	ADP
ma-156	4	61	science	science	NOUN
ma-156	4	62	,	,	PUNCT
ma-156	4	63	university	university	NOUN
ma-156	4	64	of	of	ADP
ma-156	4	65	lagos	lagos	PROPN
ma-156	4	66	,	,	PUNCT
ma-156	4	67	nigeria	nigeria	PROPN
ma-156	4	68	jolaleru@unilag.edu.ng	jolaleru@unilag.edu.ng	NOUN
ma-156	4	69	,	,	PUNCT
ma-156	4	70	holaoluwa@unilag.edu.ng	holaoluwa@unilag.edu.ng	ADV
ma-156	4	71	,	,	PUNCT
ma-156	4	72	amogbademu@unilag.edu.ng	amogbademu@unilag.edu.ng	ADJ
ma-156	4	73	∗correspondence	∗correspondence	NOUN
ma-156	4	74	:	:	PUNCT
ma-156	4	75	umuduj@unijos.edu.ng	umuduj@unijos.edu.ng	NOUN
ma-156	4	76	abstract	abstract	NOUN
ma-156	4	77	.	.	PUNCT
ma-156	5	1	in	in	ADP
ma-156	5	2	this	this	DET
ma-156	5	3	paper	paper	NOUN
ma-156	5	4	,	,	PUNCT
ma-156	5	5	we	we	PRON
ma-156	5	6	introduce	introduce	VERB
ma-156	5	7	a	a	DET
ma-156	5	8	new	new	ADJ
ma-156	5	9	concept	concept	NOUN
ma-156	5	10	of	of	ADP
ma-156	5	11	α	α	PROPN
ma-156	5	12	-	-	PUNCT
ma-156	5	13	φ	φ	VERB
ma-156	5	14	-	-	PUNCT
ma-156	5	15	geraghty	geraghty	VERB
ma-156	5	16	proximal	proximal	ADJ
ma-156	5	17	quasi	quasi	ADJ
ma-156	5	18	-	-	ADJ
ma-156	5	19	contractiontype	contractiontype	ADJ
ma-156	5	20	mappings	mapping	NOUN
ma-156	5	21	and	and	CCONJ
ma-156	5	22	establish	establish	VERB
ma-156	5	23	best	good	ADJ
ma-156	5	24	proximity	proximity	NOUN
ma-156	5	25	point	point	NOUN
ma-156	5	26	theorems	theorem	NOUN
ma-156	5	27	for	for	ADP
ma-156	5	28	those	those	DET
ma-156	5	29	mappings	mapping	NOUN
ma-156	5	30	in	in	ADP
ma-156	5	31	proximal	proximal	ADJ
ma-156	5	32	t	t	NOUN
ma-156	5	33	-orbitallycomplete	-orbitallycomplete	ADJ
ma-156	5	34	metric	metric	ADJ
ma-156	5	35	spaces	space	NOUN
ma-156	5	36	.	.	PUNCT
ma-156	6	1	this	this	PRON
ma-156	6	2	generalizes	generalize	VERB
ma-156	6	3	and	and	CCONJ
ma-156	6	4	complements	complement	VERB
ma-156	6	5	the	the	DET
ma-156	6	6	proofs	proof	NOUN
ma-156	6	7	of	of	ADP
ma-156	6	8	some	some	DET
ma-156	6	9	known	know	VERB
ma-156	6	10	fixed	fix	VERB
ma-156	6	11	and	and	CCONJ
ma-156	6	12	bestproximity	bestproximity	NOUN
ma-156	6	13	point	point	NOUN
ma-156	6	14	results	result	NOUN
ma-156	6	15	.	.	PUNCT
ma-156	7	1	1	1	X
ma-156	7	2	.	.	X
ma-156	7	3	introduction	introduction	NOUN
ma-156	7	4	let	let	VERB
ma-156	7	5	a	a	PRON
ma-156	7	6	and	and	CCONJ
ma-156	7	7	b	b	NOUN
ma-156	7	8	be	be	AUX
ma-156	7	9	two	two	NUM
ma-156	7	10	nonempty	nonempty	ADJ
ma-156	7	11	subsets	subset	NOUN
ma-156	7	12	of	of	ADP
ma-156	7	13	a	a	DET
ma-156	7	14	metric	metric	ADJ
ma-156	7	15	space	space	NOUN
ma-156	7	16	(	(	PUNCT
ma-156	7	17	x	x	X
ma-156	7	18	,	,	PUNCT
ma-156	7	19	d	d	NOUN
ma-156	7	20	)	)	PUNCT
ma-156	7	21	.	.	PUNCT
ma-156	8	1	a	a	DET
ma-156	8	2	best	good	ADJ
ma-156	8	3	proximity	proximity	NOUN
ma-156	8	4	point	point	NOUN
ma-156	8	5	of	of	ADP
ma-156	8	6	anon	anon	ADJ
ma-156	8	7	-	-	ADJ
ma-156	8	8	self	self	NOUN
ma-156	8	9	mapping	mapping	NOUN
ma-156	8	10	t	t	NOUN
ma-156	8	11	:	:	PUNCT
ma-156	8	12	a	a	DET
ma-156	8	13	→	→	SYM
ma-156	8	14	b	b	NOUN
ma-156	8	15	,	,	PUNCT
ma-156	8	16	is	be	AUX
ma-156	8	17	the	the	DET
ma-156	8	18	point	point	NOUN
ma-156	8	19	x	x	X
ma-156	8	20	∈	∈	PROPN
ma-156	8	21	a	a	DET
ma-156	8	22	,	,	PUNCT
ma-156	8	23	satisfying	satisfying	ADJ
ma-156	8	24	d(x	d(x	PROPN
ma-156	8	25	,	,	PUNCT
ma-156	8	26	t	t	NOUN
ma-156	8	27	x	x	X
ma-156	8	28	)	)	PUNCT
ma-156	8	29	=	=	SYM
ma-156	9	1	d(a	d(a	PROPN
ma-156	9	2	,	,	PUNCT
ma-156	9	3	b	b	NOUN
ma-156	9	4	)	)	PUNCT
ma-156	9	5	.	.	PUNCT
ma-156	10	1	numerousresults	numerousresult	NOUN
ma-156	10	2	on	on	ADP
ma-156	10	3	best	good	ADJ
ma-156	10	4	proximity	proximity	NOUN
ma-156	10	5	point	point	NOUN
ma-156	10	6	theory	theory	NOUN
ma-156	10	7	were	be	AUX
ma-156	10	8	studied	study	VERB
ma-156	10	9	by	by	ADP
ma-156	10	10	several	several	ADJ
ma-156	10	11	authors	author	NOUN
ma-156	10	12	(	(	PUNCT
ma-156	10	13	[	[	X
ma-156	10	14	1	1	NUM
ma-156	10	15	]	]	PUNCT
ma-156	10	16	,	,	PUNCT
ma-156	11	1	[	[	X
ma-156	11	2	3	3	NUM
ma-156	11	3	]	]	PUNCT
ma-156	11	4	,	,	PUNCT
ma-156	11	5	[	[	X
ma-156	11	6	4	4	NUM
ma-156	11	7	]	]	PUNCT
ma-156	11	8	,	,	PUNCT
ma-156	11	9	[	[	X
ma-156	11	10	5	5	NUM
ma-156	11	11	]	]	PUNCT
ma-156	11	12	)	)	PUNCT
ma-156	11	13	imposingsufficient	imposingsufficient	ADJ
ma-156	11	14	conditions	condition	NOUN
ma-156	11	15	that	that	PRON
ma-156	11	16	would	would	AUX
ma-156	11	17	assure	assure	VERB
ma-156	11	18	the	the	DET
ma-156	11	19	existence	existence	NOUN
ma-156	11	20	and	and	CCONJ
ma-156	11	21	uniqueness	uniqueness	NOUN
ma-156	11	22	of	of	ADP
ma-156	11	23	such	such	ADJ
ma-156	11	24	points	point	NOUN
ma-156	11	25	.	.	PUNCT
ma-156	12	1	these	these	DET
ma-156	12	2	resultsare	resultsare	NOUN
ma-156	12	3	generalizations	generalization	NOUN
ma-156	12	4	of	of	ADP
ma-156	12	5	the	the	DET
ma-156	12	6	contraction	contraction	NOUN
ma-156	12	7	principle	principle	NOUN
ma-156	12	8	and	and	CCONJ
ma-156	12	9	other	other	ADJ
ma-156	12	10	contractive	contractive	ADJ
ma-156	12	11	mappings	mapping	NOUN
ma-156	12	12	(	(	PUNCT
ma-156	12	13	[	[	X
ma-156	12	14	2	2	NUM
ma-156	12	15	]	]	PUNCT
ma-156	12	16	,	,	PUNCT
ma-156	12	17	[	[	X
ma-156	12	18	6	6	NUM
ma-156	12	19	]	]	PUNCT
ma-156	12	20	,	,	PUNCT
ma-156	12	21	[	[	X
ma-156	12	22	8	8	NUM
ma-156	12	23	]	]	PUNCT
ma-156	12	24	,	,	PUNCT
ma-156	12	25	[	[	X
ma-156	12	26	16],[21	16],[21	X
ma-156	12	27	]	]	PUNCT
ma-156	12	28	,	,	PUNCT
ma-156	12	29	[	[	X
ma-156	12	30	22	22	NUM
ma-156	12	31	]	]	PUNCT
ma-156	12	32	,	,	PUNCT
ma-156	12	33	[	[	X
ma-156	12	34	24	24	NUM
ma-156	12	35	]	]	PUNCT
ma-156	12	36	)	)	PUNCT
ma-156	12	37	in	in	ADP
ma-156	12	38	the	the	DET
ma-156	12	39	case	case	NOUN
ma-156	12	40	of	of	ADP
ma-156	12	41	self	self	NOUN
ma-156	12	42	-	-	PUNCT
ma-156	12	43	mappings	mapping	NOUN
ma-156	12	44	,	,	PUNCT
ma-156	12	45	which	which	PRON
ma-156	12	46	reduces	reduce	VERB
ma-156	12	47	to	to	ADP
ma-156	12	48	a	a	DET
ma-156	12	49	fixed	fix	VERB
ma-156	12	50	point	point	NOUN
ma-156	12	51	if	if	SCONJ
ma-156	12	52	the	the	DET
ma-156	12	53	mapping	mapping	NOUN
ma-156	12	54	underconsideration	underconsideration	NOUN
ma-156	12	55	is	be	AUX
ma-156	12	56	a	a	DET
ma-156	12	57	self	self	NOUN
ma-156	12	58	-	-	PUNCT
ma-156	12	59	mapping	mapping	NOUN
ma-156	12	60	.	.	PUNCT
ma-156	13	1	the	the	DET
ma-156	13	2	notion	notion	NOUN
ma-156	13	3	of	of	ADP
ma-156	13	4	best	good	ADJ
ma-156	13	5	proximity	proximity	NOUN
ma-156	13	6	point	point	NOUN
ma-156	13	7	was	be	AUX
ma-156	13	8	introduced	introduce	VERB
ma-156	13	9	in	in	ADP
ma-156	13	10	[	[	X
ma-156	13	11	14	14	NUM
ma-156	13	12	]	]	PUNCT
ma-156	13	13	,	,	PUNCT
ma-156	13	14	the	the	DET
ma-156	13	15	classof	classof	ADJ
ma-156	13	16	proximal	proximal	ADJ
ma-156	13	17	quasi	quasi	NOUN
ma-156	13	18	contraction	contraction	NOUN
ma-156	13	19	mappings	mapping	NOUN
ma-156	13	20	was	be	AUX
ma-156	13	21	introduced	introduce	VERB
ma-156	13	22	in	in	ADP
ma-156	13	23	[	[	X
ma-156	13	24	11	11	NUM
ma-156	13	25	]	]	PUNCT
ma-156	13	26	and	and	CCONJ
ma-156	13	27	thereafter	thereafter	ADV
ma-156	13	28	,	,	PUNCT
ma-156	13	29	several	several	ADJ
ma-156	13	30	known	know	VERB
ma-156	13	31	resultswere	resultswere	NOUN
ma-156	13	32	derived	derive	VERB
ma-156	13	33	(	(	PUNCT
ma-156	13	34	[	[	X
ma-156	13	35	10	10	NUM
ma-156	13	36	]	]	PUNCT
ma-156	13	37	,	,	PUNCT
ma-156	14	1	[	[	X
ma-156	14	2	12	12	NUM
ma-156	14	3	]	]	PUNCT
ma-156	14	4	,	,	PUNCT
ma-156	14	5	[	[	X
ma-156	14	6	13	13	NUM
ma-156	14	7	]	]	NUM
ma-156	14	8	)	)	PUNCT
ma-156	14	9	.	.	PUNCT
ma-156	15	1	best	good	ADJ
ma-156	15	2	proximity	proximity	NOUN
ma-156	15	3	pair	pair	NOUN
ma-156	15	4	theorems	theorem	NOUN
ma-156	15	5	analyse	analyse	VERB
ma-156	15	6	the	the	DET
ma-156	15	7	conditions	condition	NOUN
ma-156	15	8	under	under	ADP
ma-156	15	9	which	which	PRON
ma-156	15	10	theoptimization	theoptimization	NOUN
ma-156	15	11	problem	problem	NOUN
ma-156	15	12	,	,	PUNCT
ma-156	15	13	namely	namely	ADV
ma-156	15	14	minx∈a	minx∈a	VERB
ma-156	15	15	d(x	d(x	PROPN
ma-156	15	16	,	,	PUNCT
ma-156	15	17	t	t	PROPN
ma-156	15	18	x	x	VERB
ma-156	15	19	)	)	PUNCT
ma-156	15	20	has	have	VERB
ma-156	15	21	a	a	DET
ma-156	15	22	solution	solution	NOUN
ma-156	15	23	and	and	CCONJ
ma-156	15	24	is	be	AUX
ma-156	15	25	known	know	VERB
ma-156	15	26	to	to	PART
ma-156	15	27	have	have	VERB
ma-156	15	28	applicationsin	applicationsin	NOUN
ma-156	15	29	game	game	NOUN
ma-156	15	30	theory	theory	NOUN
ma-156	15	31	.	.	PUNCT
ma-156	16	1	for	for	ADP
ma-156	16	2	additional	additional	ADJ
ma-156	16	3	information	information	NOUN
ma-156	16	4	on	on	ADP
ma-156	16	5	best	good	ADJ
ma-156	16	6	proximity	proximity	NOUN
ma-156	16	7	point	point	NOUN
ma-156	16	8	,	,	PUNCT
ma-156	16	9	see	see	VERB
ma-156	16	10	[	[	X
ma-156	16	11	7	7	NUM
ma-156	16	12	]	]	PUNCT
ma-156	16	13	,	,	PUNCT
ma-156	17	1	[	[	X
ma-156	17	2	9	9	NUM
ma-156	17	3	]	]	PUNCT
ma-156	17	4	,	,	PUNCT
ma-156	17	5	[	[	X
ma-156	17	6	10	10	NUM
ma-156	17	7	]	]	PUNCT
ma-156	17	8	,	,	PUNCT
ma-156	18	1	[	[	X
ma-156	18	2	11	11	NUM
ma-156	18	3	]	]	PUNCT
ma-156	18	4	,	,	PUNCT
ma-156	18	5	[	[	X
ma-156	18	6	12	12	NUM
ma-156	18	7	]	]	PUNCT
ma-156	18	8	,	,	PUNCT
ma-156	19	1	[	[	X
ma-156	19	2	13],[14	13],[14	PROPN
ma-156	19	3	]	]	X
ma-156	19	4	,	,	PUNCT
ma-156	20	1	[	[	X
ma-156	20	2	15	15	NUM
ma-156	20	3	]	]	PUNCT
ma-156	20	4	,	,	PUNCT
ma-156	21	1	[	[	X
ma-156	21	2	17	17	NUM
ma-156	21	3	]	]	PUNCT
ma-156	21	4	,	,	PUNCT
ma-156	21	5	[	[	X
ma-156	21	6	18	18	NUM
ma-156	21	7	]	]	PUNCT
ma-156	21	8	,	,	PUNCT
ma-156	21	9	[	[	X
ma-156	21	10	20	20	NUM
ma-156	21	11	]	]	PUNCT
ma-156	21	12	,	,	PUNCT
ma-156	21	13	[	[	X
ma-156	21	14	23	23	NUM
ma-156	21	15	]	]	PUNCT
ma-156	21	16	.	.	PUNCT
ma-156	22	1	definition	definition	NOUN
ma-156	22	2	1.1	1.1	NUM
ma-156	22	3	[	[	X
ma-156	22	4	4	4	NUM
ma-156	22	5	]	]	PUNCT
ma-156	22	6	.	.	PUNCT
ma-156	23	1	let	let	VERB
ma-156	23	2	t	t	NOUN
ma-156	23	3	:	:	PUNCT
ma-156	23	4	x	x	X
ma-156	23	5	→	→	PUNCT
ma-156	23	6	x	x	PUNCT
ma-156	23	7	be	be	AUX
ma-156	23	8	a	a	DET
ma-156	23	9	map	map	NOUN
ma-156	23	10	on	on	ADP
ma-156	23	11	metric	metric	ADJ
ma-156	23	12	space	space	NOUN
ma-156	23	13	.	.	PUNCT
ma-156	24	1	for	for	ADP
ma-156	24	2	each	each	DET
ma-156	24	3	x	x	SYM
ma-156	24	4	∈	∈	PROPN
ma-156	24	5	x	x	X
ma-156	24	6	and	and	CCONJ
ma-156	24	7	for	for	ADP
ma-156	24	8	any	any	DET
ma-156	24	9	positiveinteger	positiveinteger	NOUN
ma-156	24	10	n	n	NOUN
ma-156	24	11	,	,	PUNCT
ma-156	24	12	ot	ot	INTJ
ma-156	24	13	(	(	PUNCT
ma-156	24	14	x	x	NOUN
ma-156	24	15	,	,	PUNCT
ma-156	24	16	n	n	CCONJ
ma-156	24	17	)	)	PUNCT
ma-156	24	18	=	=	PRON
ma-156	24	19	{	{	PUNCT
ma-156	24	20	x	x	PROPN
ma-156	24	21	,	,	PUNCT
ma-156	24	22	t	t	PROPN
ma-156	24	23	x	x	X
ma-156	24	24	,	,	PUNCT
ma-156	24	25	...	...	PUNCT
ma-156	24	26	,	,	PUNCT
ma-156	24	27	t	t	PROPN
ma-156	24	28	nx	nx	X
ma-156	24	29	}	}	PUNCT
ma-156	24	30	and	and	CCONJ
ma-156	24	31	ot	ot	INTJ
ma-156	24	32	(	(	PUNCT
ma-156	24	33	x,∞	x,∞	PROPN
ma-156	24	34	)	)	PUNCT
ma-156	24	35	=	=	SYM
ma-156	24	36	{	{	PUNCT
ma-156	24	37	x	x	PROPN
ma-156	24	38	,	,	PUNCT
ma-156	24	39	t	t	PROPN
ma-156	24	40	x	x	X
ma-156	24	41	,	,	PUNCT
ma-156	24	42	...	...	PUNCT
ma-156	24	43	,	,	PUNCT
ma-156	24	44	t	t	PROPN
ma-156	24	45	nx	nx	PROPN
ma-156	24	46	,	,	PUNCT
ma-156	24	47	...	...	PUNCT
ma-156	24	48	}	}	PUNCT
ma-156	24	49	.	.	PUNCT
ma-156	25	1	received	receive	VERB
ma-156	25	2	:	:	PUNCT
ma-156	25	3	8	8	NUM
ma-156	25	4	feb	feb	NOUN
ma-156	25	5	2023	2023	NUM
ma-156	25	6	.	.	PUNCT
ma-156	26	1	key	key	ADJ
ma-156	26	2	words	word	NOUN
ma-156	26	3	and	and	CCONJ
ma-156	26	4	phrases	phrase	NOUN
ma-156	26	5	.	.	PUNCT
ma-156	27	1	best	good	ADJ
ma-156	27	2	proximity	proximity	NOUN
ma-156	27	3	;	;	PUNCT
ma-156	27	4	quasi	quasi	NOUN
ma-156	27	5	-	-	NOUN
ma-156	27	6	contraction	contraction	NOUN
ma-156	27	7	;	;	PUNCT
ma-156	27	8	metric	metric	ADJ
ma-156	27	9	space.1	space.1	PROPN
ma-156	27	10	https://adac.ee	https://adac.ee	PROPN
ma-156	27	11	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	27	12	eur	eur	PROPN
ma-156	27	13	.	.	PUNCT
ma-156	28	1	j.	j.	PROPN
ma-156	28	2	math	math	PROPN
ma-156	28	3	.	.	PUNCT
ma-156	29	1	anal	anal	PROPN
ma-156	29	2	.	.	PUNCT
ma-156	30	1	10.28924	10.28924	NUM
ma-156	30	2	/	/	SYM
ma-156	30	3	ada	ada	PROPN
ma-156	30	4	/	/	SYM
ma-156	30	5	ma.3.16	ma.3.16	PROPN
ma-156	31	1	2the	2the	PROPN
ma-156	31	2	set	set	NOUN
ma-156	31	3	ot	ot	INTJ
ma-156	31	4	(	(	PUNCT
ma-156	31	5	x,∞	x,∞	PROPN
ma-156	31	6	)	)	PUNCT
ma-156	31	7	is	be	AUX
ma-156	31	8	called	call	VERB
ma-156	31	9	the	the	DET
ma-156	31	10	orbit	orbit	NOUN
ma-156	31	11	of	of	ADP
ma-156	31	12	t	t	PROPN
ma-156	31	13	at	at	ADP
ma-156	31	14	x	x	X
ma-156	31	15	and	and	CCONJ
ma-156	31	16	the	the	DET
ma-156	31	17	metric	metric	ADJ
ma-156	31	18	space	space	NOUN
ma-156	31	19	x	x	PUNCT
ma-156	31	20	is	be	AUX
ma-156	31	21	called	call	VERB
ma-156	31	22	t	t	PROPN
ma-156	31	23	-orbitally	-orbitally	NOUN
ma-156	31	24	completeif	completeif	VERB
ma-156	31	25	every	every	DET
ma-156	31	26	cauchy	cauchy	ADJ
ma-156	31	27	sequence	sequence	NOUN
ma-156	31	28	in	in	ADP
ma-156	31	29	ot	ot	INTJ
ma-156	31	30	(	(	PUNCT
ma-156	31	31	x,∞	x,∞	PROPN
ma-156	31	32	)	)	PUNCT
ma-156	31	33	is	be	AUX
ma-156	31	34	convergent	convergent	ADJ
ma-156	31	35	in	in	ADP
ma-156	31	36	x.	x.	PROPN
ma-156	31	37	quasi	quasi	PROPN
ma-156	31	38	contraction	contraction	PROPN
ma-156	31	39	mapping	mapping	NOUN
ma-156	31	40	is	be	AUX
ma-156	31	41	known	know	VERB
ma-156	31	42	in	in	ADP
ma-156	31	43	literature	literature	NOUN
ma-156	31	44	as	as	ADP
ma-156	31	45	one	one	NUM
ma-156	31	46	of	of	ADP
ma-156	31	47	the	the	DET
ma-156	31	48	most	most	ADV
ma-156	31	49	generalized	generalized	ADJ
ma-156	31	50	contractive	contractive	ADJ
ma-156	31	51	map	map	NOUN
ma-156	31	52	-	-	PUNCT
ma-156	31	53	pings	ping	NOUN
ma-156	31	54	and	and	CCONJ
ma-156	31	55	is	be	AUX
ma-156	31	56	defined	define	VERB
ma-156	31	57	as	as	ADP
ma-156	31	58	follows	follow	VERB
ma-156	31	59	.	.	PUNCT
ma-156	32	1	definition	definition	NOUN
ma-156	32	2	1.2	1.2	NUM
ma-156	32	3	[	[	X
ma-156	32	4	6	6	NUM
ma-156	32	5	]	]	PUNCT
ma-156	32	6	.	.	PUNCT
ma-156	33	1	a	a	DET
ma-156	33	2	mapping	mapping	NOUN
ma-156	33	3	t	t	NOUN
ma-156	33	4	:	:	PUNCT
ma-156	33	5	x	x	X
ma-156	33	6	→	→	SYM
ma-156	33	7	x	x	X
ma-156	33	8	of	of	ADP
ma-156	33	9	a	a	DET
ma-156	33	10	metric	metric	ADJ
ma-156	33	11	space	space	NOUN
ma-156	33	12	x	x	PUNCT
ma-156	33	13	into	into	ADP
ma-156	33	14	itself	itself	PRON
ma-156	33	15	is	be	AUX
ma-156	33	16	said	say	VERB
ma-156	33	17	to	to	PART
ma-156	33	18	be	be	AUX
ma-156	33	19	a	a	DET
ma-156	33	20	quasi	quasi	NOUN
ma-156	33	21	-	-	NOUN
ma-156	33	22	contraction	contraction	NOUN
ma-156	33	23	if	if	SCONJ
ma-156	33	24	and	and	CCONJ
ma-156	33	25	only	only	ADV
ma-156	33	26	if	if	SCONJ
ma-156	33	27	there	there	PRON
ma-156	33	28	exists	exist	VERB
ma-156	33	29	a	a	DET
ma-156	33	30	number	number	NOUN
ma-156	33	31	k	k	NOUN
ma-156	33	32	,	,	PUNCT
ma-156	33	33	0	0	NUM
ma-156	33	34	≤	≤	PUNCT
ma-156	34	1	k	k	X
ma-156	34	2	<	<	X
ma-156	34	3	1	1	NUM
ma-156	34	4	,	,	PUNCT
ma-156	34	5	such	such	ADJ
ma-156	34	6	that	that	SCONJ
ma-156	34	7	d(tx	d(tx	PROPN
ma-156	34	8	,	,	PUNCT
ma-156	34	9	t	t	PROPN
ma-156	34	10	y	y	NOUN
ma-156	34	11	)	)	PUNCT
ma-156	34	12	≤	≤	NOUN
ma-156	35	1	k	k	X
ma-156	35	2	max{d(x	max{d(x	PROPN
ma-156	35	3	,	,	PUNCT
ma-156	35	4	y	y	PROPN
ma-156	35	5	)	)	PUNCT
ma-156	35	6	;	;	PUNCT
ma-156	35	7	d(x	d(x	PROPN
ma-156	35	8	,	,	PUNCT
ma-156	35	9	t	t	PROPN
ma-156	35	10	x	x	PROPN
ma-156	35	11	)	)	PUNCT
ma-156	35	12	;	;	PUNCT
ma-156	35	13	d(y	d(y	NOUN
ma-156	35	14	,	,	PUNCT
ma-156	35	15	t	t	PROPN
ma-156	35	16	y	y	PROPN
ma-156	35	17	)	)	PUNCT
ma-156	35	18	;	;	PUNCT
ma-156	35	19	d(x	d(x	PROPN
ma-156	35	20	,	,	PUNCT
ma-156	35	21	t	t	PROPN
ma-156	35	22	y	y	PROPN
ma-156	35	23	)	)	PUNCT
ma-156	35	24	;	;	PUNCT
ma-156	35	25	d(y	d(y	PROPN
ma-156	35	26	,	,	PUNCT
ma-156	35	27	t	t	PROPN
ma-156	35	28	x	x	PROPN
ma-156	35	29	)	)	PUNCT
ma-156	35	30	}	}	PUNCT
ma-156	35	31	holds	hold	VERB
ma-156	35	32	for	for	ADP
ma-156	35	33	every	every	DET
ma-156	35	34	x	x	NOUN
ma-156	35	35	,	,	PUNCT
ma-156	35	36	y	y	PROPN
ma-156	35	37	∈	∈	PROPN
ma-156	35	38	x.	x.	NOUN
ma-156	35	39	consider	consider	VERB
ma-156	35	40	the	the	DET
ma-156	35	41	class	class	NOUN
ma-156	35	42	f	f	PROPN
ma-156	35	43	of	of	ADP
ma-156	35	44	functions	function	NOUN
ma-156	35	45	β	β	NOUN
ma-156	35	46	:	:	PUNCT
ma-156	36	1	[	[	X
ma-156	36	2	0,∞)→	0,∞)→	NOUN
ma-156	36	3	[	[	X
ma-156	36	4	0	0	NUM
ma-156	36	5	,	,	PUNCT
ma-156	36	6	1	1	NUM
ma-156	36	7	)	)	PUNCT
ma-156	36	8	satisfying	satisfy	VERB
ma-156	36	9	the	the	DET
ma-156	36	10	condition	condition	NOUN
ma-156	36	11	:	:	PUNCT
ma-156	36	12	lim	lim	PROPN
ma-156	36	13	n→∞	n→∞	X
ma-156	36	14	β(tn	β(tn	NUM
ma-156	36	15	)	)	PUNCT
ma-156	36	16	=	=	SYM
ma-156	36	17	1	1	NUM
ma-156	36	18	implies	imply	VERB
ma-156	36	19	lim	lim	PROPN
ma-156	36	20	n→∞	n→∞	NUM
ma-156	36	21	tn	tn	PROPN
ma-156	36	22	=	=	SYM
ma-156	36	23	0	0	X
ma-156	36	24	.	.	PUNCT
ma-156	36	25	recently	recently	ADV
ma-156	36	26	,	,	PUNCT
ma-156	36	27	using	use	VERB
ma-156	36	28	these	these	DET
ma-156	36	29	class	class	NOUN
ma-156	36	30	of	of	ADP
ma-156	36	31	functions	function	NOUN
ma-156	36	32	,	,	PUNCT
ma-156	36	33	umudu	umudu	PROPN
ma-156	36	34	et	et	PROPN
ma-156	36	35	al	al	PROPN
ma-156	36	36	.	.	PUNCT
ma-156	37	1	[	[	X
ma-156	37	2	22	22	NUM
ma-156	37	3	]	]	PUNCT
ma-156	37	4	introduced	introduce	VERB
ma-156	37	5	a	a	DET
ma-156	37	6	new	new	ADJ
ma-156	37	7	class	class	NOUN
ma-156	37	8	of	of	ADP
ma-156	37	9	quasi	quasi	ADJ
ma-156	37	10	-	-	NOUN
ma-156	37	11	contraction	contraction	NOUN
ma-156	37	12	type	type	NOUN
ma-156	37	13	mappings	mapping	NOUN
ma-156	37	14	called	call	VERB
ma-156	37	15	generalized	generalized	ADJ
ma-156	37	16	α	α	PROPN
ma-156	37	17	-	-	PUNCT
ma-156	37	18	φ	φ	VERB
ma-156	37	19	-	-	PUNCT
ma-156	37	20	geraghty	geraghty	VERB
ma-156	37	21	quasi	quasi	PROPN
ma-156	37	22	-	-	NOUN
ma-156	37	23	contraction	contraction	NOUN
ma-156	37	24	type	type	NOUN
ma-156	37	25	mappings	mapping	NOUN
ma-156	37	26	andproved	andprove	VERB
ma-156	37	27	the	the	DET
ma-156	37	28	existence	existence	NOUN
ma-156	37	29	of	of	ADP
ma-156	37	30	its	its	PRON
ma-156	37	31	unique	unique	ADJ
ma-156	37	32	fixed	fix	VERB
ma-156	37	33	point	point	NOUN
ma-156	37	34	as	as	SCONJ
ma-156	37	35	follows	follow	VERB
ma-156	37	36	.	.	PUNCT
ma-156	38	1	definition	definition	NOUN
ma-156	38	2	1.3	1.3	NUM
ma-156	39	1	[	[	X
ma-156	39	2	22	22	NUM
ma-156	39	3	]	]	PUNCT
ma-156	39	4	.	.	PUNCT
ma-156	40	1	let	let	VERB
ma-156	40	2	(	(	PUNCT
ma-156	40	3	x	x	X
ma-156	40	4	,	,	PUNCT
ma-156	40	5	d	d	NOUN
ma-156	40	6	)	)	PUNCT
ma-156	40	7	be	be	AUX
ma-156	40	8	a	a	DET
ma-156	40	9	metric	metric	ADJ
ma-156	40	10	space	space	NOUN
ma-156	40	11	and	and	CCONJ
ma-156	40	12	α	α	NOUN
ma-156	40	13	:	:	PUNCT
ma-156	40	14	x	x	SYM
ma-156	40	15	×	×	NOUN
ma-156	40	16	x	x	PUNCT
ma-156	40	17	→	→	X
ma-156	40	18	r+	r+	X
ma-156	40	19	.	.	PUNCT
ma-156	41	1	a	a	DET
ma-156	41	2	mapping	mapping	NOUN
ma-156	41	3	t	t	NOUN
ma-156	41	4	:	:	PUNCT
ma-156	41	5	x	x	X
ma-156	41	6	→	→	PUNCT
ma-156	41	7	x	x	PUNCT
ma-156	41	8	iscalled	iscalle	VERB
ma-156	41	9	a	a	DET
ma-156	41	10	generalized	generalized	ADJ
ma-156	41	11	α	α	NOUN
ma-156	41	12	-	-	PUNCT
ma-156	41	13	geraghty	geraghty	VERB
ma-156	41	14	quasi	quasi	PROPN
ma-156	41	15	-	-	NOUN
ma-156	41	16	contraction	contraction	NOUN
ma-156	41	17	type	type	NOUN
ma-156	41	18	mapping	mapping	NOUN
ma-156	41	19	if	if	SCONJ
ma-156	41	20	there	there	PRON
ma-156	41	21	exists	exist	VERB
ma-156	41	22	β	β	X
ma-156	41	23	∈	∈	PROPN
ma-156	41	24	f	f	PROPN
ma-156	41	25	such	such	ADJ
ma-156	41	26	thatfor	thatfor	VERB
ma-156	41	27	all	all	DET
ma-156	41	28	x	x	NOUN
ma-156	41	29	,	,	PUNCT
ma-156	41	30	y	y	PROPN
ma-156	41	31	∈	∈	PROPN
ma-156	41	32	x	x	PROPN
ma-156	41	33	,	,	PUNCT
ma-156	41	34	α(x	α(x	PROPN
ma-156	41	35	,	,	PUNCT
ma-156	41	36	y)(d(tx	y)(d(tx	NUM
ma-156	41	37	,	,	PUNCT
ma-156	41	38	t	t	PROPN
ma-156	41	39	y	y	PROPN
ma-156	41	40	)	)	PUNCT
ma-156	41	41	)	)	PUNCT
ma-156	42	1	≤	≤	NUM
ma-156	42	2	β(mt	β(mt	ADJ
ma-156	42	3	(	(	PUNCT
ma-156	42	4	x	x	X
ma-156	42	5	,	,	PUNCT
ma-156	42	6	y))(mt	y))(mt	PUNCT
ma-156	42	7	(	(	PUNCT
ma-156	42	8	x	x	X
ma-156	42	9	,	,	PUNCT
ma-156	42	10	y	y	PROPN
ma-156	42	11	)	)	PUNCT
ma-156	42	12	)	)	PUNCT
ma-156	42	13	,	,	PUNCT
ma-156	42	14	(	(	PUNCT
ma-156	42	15	1	1	X
ma-156	42	16	)	)	PUNCT
ma-156	42	17	where	where	SCONJ
ma-156	42	18	mt	mt	PROPN
ma-156	42	19	(	(	PUNCT
ma-156	42	20	x	x	PROPN
ma-156	42	21	,	,	PUNCT
ma-156	42	22	y	y	NOUN
ma-156	42	23	)	)	PUNCT
ma-156	42	24	=	=	PUNCT
ma-156	43	1	max{d(x	max{d(x	PROPN
ma-156	43	2	,	,	PUNCT
ma-156	43	3	y	y	NOUN
ma-156	43	4	)	)	PUNCT
ma-156	43	5	,	,	PUNCT
ma-156	43	6	d(x	d(x	PROPN
ma-156	43	7	,	,	PUNCT
ma-156	43	8	t	t	PROPN
ma-156	43	9	x	x	PROPN
ma-156	43	10	)	)	PUNCT
ma-156	43	11	,	,	PUNCT
ma-156	43	12	d(y	d(y	PROPN
ma-156	43	13	,	,	PUNCT
ma-156	43	14	t	t	PROPN
ma-156	43	15	y	y	PROPN
ma-156	43	16	)	)	PUNCT
ma-156	43	17	,	,	PUNCT
ma-156	43	18	d(x	d(x	PROPN
ma-156	43	19	,	,	PUNCT
ma-156	43	20	t	t	PROPN
ma-156	43	21	y	y	PROPN
ma-156	43	22	)	)	PUNCT
ma-156	43	23	,	,	PUNCT
ma-156	43	24	d(y	d(y	PROPN
ma-156	43	25	,	,	PUNCT
ma-156	43	26	t	t	PROPN
ma-156	43	27	x	x	PROPN
ma-156	43	28	)	)	PUNCT
ma-156	43	29	}	}	PUNCT
ma-156	43	30	.	.	PUNCT
ma-156	44	1	let	let	VERB
ma-156	44	2	φ	φ	PROPN
ma-156	44	3	denote	denote	VERB
ma-156	44	4	the	the	DET
ma-156	44	5	class	class	NOUN
ma-156	44	6	of	of	ADP
ma-156	44	7	the	the	DET
ma-156	44	8	functions	function	NOUN
ma-156	44	9	φ	φ	X
ma-156	44	10	:	:	PUNCT
ma-156	45	1	[	[	X
ma-156	45	2	0,∞)→	0,∞)→	NOUN
ma-156	45	3	[	[	X
ma-156	45	4	0,∞	0,∞	NOUN
ma-156	45	5	)	)	PUNCT
ma-156	45	6	which	which	PRON
ma-156	45	7	satisfies	satisfy	VERB
ma-156	45	8	the	the	DET
ma-156	45	9	following	follow	VERB
ma-156	45	10	conditions:(i	conditions:(i	NOUN
ma-156	45	11	)	)	PUNCT
ma-156	45	12	φ	φ	PROPN
ma-156	45	13	is	be	AUX
ma-156	45	14	nondecreasing;(ii	nondecreasing;(ii	PROPN
ma-156	45	15	)	)	PUNCT
ma-156	45	16	φ	φ	PROPN
ma-156	45	17	is	be	AUX
ma-156	45	18	continuous;(iii	continuous;(iii	X
ma-156	45	19	)	)	PUNCT
ma-156	45	20	φ(t	φ(t	PROPN
ma-156	45	21	)	)	PUNCT
ma-156	46	1	=	=	SYM
ma-156	46	2	0	0	NUM
ma-156	47	1	⇐	⇐	ADJ
ma-156	47	2	⇒	⇒	NOUN
ma-156	47	3	t	t	NOUN
ma-156	47	4	=	=	SYM
ma-156	47	5	0	0	X
ma-156	47	6	.	.	PUNCT
ma-156	48	1	definition	definition	NOUN
ma-156	48	2	1.4	1.4	NUM
ma-156	49	1	[	[	X
ma-156	49	2	22	22	NUM
ma-156	49	3	]	]	PUNCT
ma-156	49	4	.	.	PUNCT
ma-156	50	1	let	let	VERB
ma-156	50	2	(	(	PUNCT
ma-156	50	3	x	x	X
ma-156	50	4	,	,	PUNCT
ma-156	50	5	d	d	NOUN
ma-156	50	6	)	)	PUNCT
ma-156	50	7	be	be	AUX
ma-156	50	8	a	a	DET
ma-156	50	9	metric	metric	ADJ
ma-156	50	10	space	space	NOUN
ma-156	50	11	and	and	CCONJ
ma-156	50	12	α	α	NOUN
ma-156	50	13	:	:	PUNCT
ma-156	51	1	x×x	x×x	PROPN
ma-156	51	2	→	→	SYM
ma-156	51	3	r+	r+	X
ma-156	51	4	.	.	PUNCT
ma-156	52	1	a	a	DET
ma-156	52	2	self	self	NOUN
ma-156	52	3	mapping	mapping	NOUN
ma-156	52	4	t	t	NOUN
ma-156	52	5	:	:	PUNCT
ma-156	52	6	x	x	X
ma-156	52	7	→	→	SYM
ma-156	52	8	xis	xis	PROPN
ma-156	52	9	called	call	VERB
ma-156	52	10	a	a	DET
ma-156	52	11	generalized	generalized	ADJ
ma-156	52	12	α	α	PROPN
ma-156	52	13	-	-	PUNCT
ma-156	52	14	φ	φ	VERB
ma-156	52	15	-	-	PUNCT
ma-156	52	16	geraghty	geraghty	VERB
ma-156	52	17	quasi	quasi	PROPN
ma-156	52	18	-	-	NOUN
ma-156	52	19	contraction	contraction	NOUN
ma-156	52	20	type	type	NOUN
ma-156	52	21	mapping	mapping	NOUN
ma-156	52	22	if	if	SCONJ
ma-156	52	23	there	there	PRON
ma-156	52	24	exists	exist	VERB
ma-156	52	25	β	β	X
ma-156	52	26	∈	∈	PROPN
ma-156	52	27	f	f	PROPN
ma-156	52	28	suchthat	suchthat	PROPN
ma-156	52	29	for	for	ADP
ma-156	52	30	all	all	DET
ma-156	52	31	x	x	NOUN
ma-156	52	32	,	,	PUNCT
ma-156	52	33	y	y	PROPN
ma-156	52	34	∈	∈	PROPN
ma-156	52	35	x	x	PROPN
ma-156	52	36	,	,	PUNCT
ma-156	52	37	α(x	α(x	PROPN
ma-156	52	38	,	,	PUNCT
ma-156	52	39	y)φ(d(tx	y)φ(d(tx	PROPN
ma-156	52	40	,	,	PUNCT
ma-156	52	41	t	t	PROPN
ma-156	52	42	y	y	PROPN
ma-156	52	43	)	)	PUNCT
ma-156	52	44	)	)	PUNCT
ma-156	52	45	≤	≤	NUM
ma-156	52	46	β(φ(mt	β(φ(mt	PUNCT
ma-156	52	47	(	(	PUNCT
ma-156	52	48	x	x	NOUN
ma-156	52	49	,	,	PUNCT
ma-156	52	50	y)))φ(mt	y)))φ(mt	ADJ
ma-156	52	51	(	(	PUNCT
ma-156	52	52	x	x	X
ma-156	52	53	,	,	PUNCT
ma-156	52	54	y	y	PROPN
ma-156	52	55	)	)	PUNCT
ma-156	52	56	)	)	PUNCT
ma-156	52	57	,	,	PUNCT
ma-156	52	58	(	(	PUNCT
ma-156	52	59	2	2	X
ma-156	52	60	)	)	PUNCT
ma-156	53	1	where	where	SCONJ
ma-156	53	2	mt	mt	PROPN
ma-156	53	3	(	(	PUNCT
ma-156	53	4	x	x	PROPN
ma-156	53	5	,	,	PUNCT
ma-156	53	6	y	y	NOUN
ma-156	53	7	)	)	PUNCT
ma-156	53	8	=	=	PUNCT
ma-156	53	9	max{d(x	max{d(x	PROPN
ma-156	53	10	,	,	PUNCT
ma-156	53	11	y	y	NOUN
ma-156	53	12	)	)	PUNCT
ma-156	53	13	,	,	PUNCT
ma-156	53	14	d(x	d(x	PROPN
ma-156	53	15	,	,	PUNCT
ma-156	53	16	t	t	PROPN
ma-156	53	17	x	x	PROPN
ma-156	53	18	)	)	PUNCT
ma-156	53	19	,	,	PUNCT
ma-156	53	20	d(y	d(y	PROPN
ma-156	53	21	,	,	PUNCT
ma-156	53	22	t	t	PROPN
ma-156	53	23	y	y	PROPN
ma-156	53	24	)	)	PUNCT
ma-156	53	25	,	,	PUNCT
ma-156	53	26	d(x	d(x	PROPN
ma-156	53	27	,	,	PUNCT
ma-156	53	28	t	t	PROPN
ma-156	53	29	y	y	PROPN
ma-156	53	30	)	)	PUNCT
ma-156	53	31	,	,	PUNCT
ma-156	53	32	d(y	d(y	PROPN
ma-156	53	33	,	,	PUNCT
ma-156	53	34	t	t	PROPN
ma-156	53	35	x	x	PROPN
ma-156	53	36	)	)	PUNCT
ma-156	53	37	}	}	PUNCT
ma-156	53	38	,	,	PUNCT
ma-156	53	39	and	and	CCONJ
ma-156	53	40	φ	φ	PROPN
ma-156	53	41	∈	∈	PROPN
ma-156	53	42	φ	φ	NOUN
ma-156	53	43	.	.	PUNCT
ma-156	54	1	if	if	SCONJ
ma-156	54	2	φ(t	φ(t	PROPN
ma-156	54	3	)	)	PUNCT
ma-156	54	4	=	=	SYM
ma-156	54	5	t	t	PROPN
ma-156	54	6	,	,	PUNCT
ma-156	54	7	inequality	inequality	NOUN
ma-156	54	8	(	(	PUNCT
ma-156	54	9	2	2	NUM
ma-156	54	10	)	)	PUNCT
ma-156	54	11	reduces	reduce	VERB
ma-156	54	12	to	to	ADP
ma-156	54	13	inequality	inequality	NOUN
ma-156	54	14	(	(	PUNCT
ma-156	54	15	1	1	NUM
ma-156	54	16	)	)	PUNCT
ma-156	54	17	.	.	PUNCT
ma-156	55	1	the	the	DET
ma-156	55	2	generalized	generalized	ADJ
ma-156	55	3	α	α	PROPN
ma-156	55	4	-	-	PUNCT
ma-156	55	5	φ	φ	VERB
ma-156	55	6	-	-	PUNCT
ma-156	55	7	geraghty	geraghty	VERB
ma-156	55	8	quasi	quasi	PROPN
ma-156	55	9	-	-	NOUN
ma-156	55	10	contraction	contraction	NOUN
ma-156	55	11	type	type	NOUN
ma-156	55	12	self	self	NOUN
ma-156	55	13	mapping	mapping	NOUN
ma-156	55	14	is	be	AUX
ma-156	55	15	a	a	DET
ma-156	55	16	generalization	generalization	NOUN
ma-156	55	17	of	of	ADP
ma-156	55	18	other	other	ADJ
ma-156	55	19	quasi	quasi	ADJ
ma-156	55	20	-	-	NOUN
ma-156	55	21	contraction	contraction	NOUN
ma-156	55	22	type	type	NOUN
ma-156	55	23	self	self	NOUN
ma-156	55	24	mappingsin	mappingsin	NOUN
ma-156	55	25	literature	literature	NOUN
ma-156	55	26	.	.	PUNCT
ma-156	56	1	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	56	2	eur	eur	PROPN
ma-156	56	3	.	.	PUNCT
ma-156	57	1	j.	j.	PROPN
ma-156	57	2	math	math	PROPN
ma-156	57	3	.	.	PUNCT
ma-156	58	1	anal	anal	PROPN
ma-156	58	2	.	.	PUNCT
ma-156	59	1	10.28924	10.28924	NUM
ma-156	59	2	/	/	SYM
ma-156	59	3	ada	ada	PROPN
ma-156	59	4	/	/	SYM
ma-156	59	5	ma.3.16	ma.3.16	PROPN
ma-156	59	6	3the	3the	DET
ma-156	59	7	following	follow	VERB
ma-156	59	8	mappings	mapping	NOUN
ma-156	59	9	introduced	introduce	VERB
ma-156	59	10	by	by	ADP
ma-156	59	11	popescu	popescu	PROPN
ma-156	59	12	[	[	X
ma-156	59	13	19	19	NUM
ma-156	59	14	]	]	PUNCT
ma-156	59	15	and	and	CCONJ
ma-156	59	16	used	use	VERB
ma-156	59	17	by	by	ADP
ma-156	59	18	umudu	umudu	NOUN
ma-156	59	19	et	et	PROPN
ma-156	59	20	al	al	PROPN
ma-156	59	21	.	.	PUNCT
ma-156	60	1	[	[	X
ma-156	60	2	22	22	NUM
ma-156	60	3	]	]	PUNCT
ma-156	60	4	to	to	PART
ma-156	60	5	establish	establish	VERB
ma-156	60	6	theexistence	theexistence	NOUN
ma-156	60	7	of	of	ADP
ma-156	60	8	a	a	DET
ma-156	60	9	fixed	fix	VERB
ma-156	60	10	point	point	NOUN
ma-156	60	11	will	will	AUX
ma-156	60	12	also	also	ADV
ma-156	60	13	be	be	AUX
ma-156	60	14	needed	need	VERB
ma-156	60	15	in	in	ADP
ma-156	60	16	this	this	DET
ma-156	60	17	paper	paper	NOUN
ma-156	60	18	.	.	PUNCT
ma-156	61	1	definition	definition	NOUN
ma-156	61	2	1.5	1.5	NUM
ma-156	61	3	[	[	X
ma-156	61	4	19	19	NUM
ma-156	61	5	]	]	PUNCT
ma-156	61	6	.	.	PUNCT
ma-156	62	1	let	let	VERB
ma-156	62	2	t	t	NOUN
ma-156	62	3	:	:	PUNCT
ma-156	62	4	x	x	X
ma-156	62	5	→	→	PUNCT
ma-156	62	6	x	x	PUNCT
ma-156	62	7	be	be	AUX
ma-156	62	8	a	a	DET
ma-156	62	9	self	self	NOUN
ma-156	62	10	-	-	PUNCT
ma-156	62	11	mapping	mapping	NOUN
ma-156	62	12	and	and	CCONJ
ma-156	62	13	α	α	NOUN
ma-156	62	14	:	:	PUNCT
ma-156	62	15	x	x	SYM
ma-156	62	16	×	×	NOUN
ma-156	62	17	x	x	PUNCT
ma-156	62	18	→	→	X
ma-156	62	19	r+	r+	NOUN
ma-156	62	20	be	be	AUX
ma-156	62	21	a	a	DET
ma-156	62	22	function	function	NOUN
ma-156	62	23	.	.	PUNCT
ma-156	63	1	then	then	ADV
ma-156	63	2	t	t	PROPN
ma-156	63	3	is	be	AUX
ma-156	63	4	said	say	VERB
ma-156	63	5	to	to	PART
ma-156	63	6	be	be	AUX
ma-156	63	7	α	α	PRON
ma-156	63	8	-	-	ADJ
ma-156	63	9	orbital	orbital	ADJ
ma-156	63	10	admissible	admissible	NOUN
ma-156	63	11	if	if	SCONJ
ma-156	63	12	α(x	α(x	PROPN
ma-156	63	13	,	,	PUNCT
ma-156	63	14	t	t	PROPN
ma-156	63	15	x	x	PROPN
ma-156	63	16	)	)	PUNCT
ma-156	63	17	≥	≥	NOUN
ma-156	63	18	1	1	NUM
ma-156	63	19	implies	imply	VERB
ma-156	63	20	α(tx	α(tx	PROPN
ma-156	63	21	,	,	PUNCT
ma-156	63	22	t	t	PROPN
ma-156	63	23	2x	2x	NUM
ma-156	63	24	)	)	PUNCT
ma-156	63	25	≥	≥	NOUN
ma-156	64	1	1	1	NUM
ma-156	64	2	.	.	PUNCT
ma-156	64	3	definition	definition	NOUN
ma-156	64	4	1.6	1.6	NUM
ma-156	64	5	[	[	X
ma-156	64	6	19	19	NUM
ma-156	64	7	]	]	PUNCT
ma-156	64	8	.	.	PUNCT
ma-156	65	1	let	let	VERB
ma-156	65	2	t	t	NOUN
ma-156	65	3	:	:	PUNCT
ma-156	65	4	x	x	X
ma-156	65	5	→	→	PUNCT
ma-156	65	6	x	x	PUNCT
ma-156	65	7	be	be	AUX
ma-156	65	8	a	a	DET
ma-156	65	9	self	self	NOUN
ma-156	65	10	-	-	PUNCT
ma-156	65	11	mapping	mapping	NOUN
ma-156	65	12	and	and	CCONJ
ma-156	65	13	α	α	NOUN
ma-156	65	14	:	:	PUNCT
ma-156	65	15	x	x	SYM
ma-156	65	16	×	×	NOUN
ma-156	65	17	x	x	PUNCT
ma-156	65	18	→	→	X
ma-156	65	19	r+	r+	NOUN
ma-156	65	20	be	be	AUX
ma-156	65	21	a	a	DET
ma-156	65	22	function.then	function.then	PROPN
ma-156	65	23	t	t	PROPN
ma-156	65	24	is	be	AUX
ma-156	65	25	said	say	VERB
ma-156	65	26	to	to	PART
ma-156	65	27	be	be	AUX
ma-156	65	28	triangular	triangular	VERB
ma-156	65	29	α	α	PRON
ma-156	65	30	-	-	ADJ
ma-156	65	31	orbital	orbital	ADJ
ma-156	65	32	admissible	admissible	NOUN
ma-156	65	33	if	if	SCONJ
ma-156	65	34	t	t	PROPN
ma-156	65	35	is	be	AUX
ma-156	65	36	α	α	DET
ma-156	65	37	-	-	ADJ
ma-156	65	38	orbital	orbital	ADJ
ma-156	65	39	admissible	admissible	ADJ
ma-156	65	40	,	,	PUNCT
ma-156	65	41	α(x	α(x	NOUN
ma-156	65	42	,	,	PUNCT
ma-156	65	43	y	y	PROPN
ma-156	65	44	)	)	PUNCT
ma-156	65	45	≥	≥	NOUN
ma-156	65	46	1	1	NUM
ma-156	65	47	and	and	CCONJ
ma-156	65	48	α(y	α(y	NOUN
ma-156	65	49	,	,	PUNCT
ma-156	65	50	t	t	PROPN
ma-156	65	51	y	y	PROPN
ma-156	65	52	)	)	PUNCT
ma-156	65	53	≥	≥	NOUN
ma-156	65	54	1	1	NUM
ma-156	65	55	imply	imply	VERB
ma-156	65	56	α(x	α(x	PROPN
ma-156	65	57	,	,	PUNCT
ma-156	65	58	t	t	PROPN
ma-156	65	59	y	y	PROPN
ma-156	65	60	)	)	PUNCT
ma-156	65	61	≥	≥	NOUN
ma-156	65	62	1	1	NUM
ma-156	65	63	.	.	PUNCT
ma-156	66	1	the	the	DET
ma-156	66	2	main	main	ADJ
ma-156	66	3	result	result	NOUN
ma-156	66	4	obtained	obtain	VERB
ma-156	66	5	in	in	ADP
ma-156	66	6	[	[	X
ma-156	66	7	22	22	NUM
ma-156	66	8	]	]	PUNCT
ma-156	66	9	is	be	AUX
ma-156	66	10	the	the	DET
ma-156	66	11	following	following	NOUN
ma-156	66	12	.	.	PUNCT
ma-156	67	1	theorem	theorem	VERB
ma-156	67	2	1.7	1.7	NUM
ma-156	67	3	.	.	PUNCT
ma-156	68	1	let	let	VERB
ma-156	68	2	(	(	PUNCT
ma-156	68	3	x	x	NOUN
ma-156	68	4	,	,	PUNCT
ma-156	68	5	d	d	X
ma-156	68	6	)	)	PUNCT
ma-156	68	7	be	be	AUX
ma-156	68	8	a	a	DET
ma-156	68	9	t	t	NOUN
ma-156	68	10	orbitally	orbitally	ADV
ma-156	68	11	complete	complete	ADJ
ma-156	68	12	metric	metric	ADJ
ma-156	68	13	space	space	NOUN
ma-156	68	14	,	,	PUNCT
ma-156	68	15	α	α	NOUN
ma-156	68	16	:	:	PUNCT
ma-156	68	17	x	x	SYM
ma-156	68	18	×	×	NOUN
ma-156	68	19	x	x	PUNCT
ma-156	68	20	→	→	X
ma-156	68	21	r+	r+	NOUN
ma-156	68	22	be	be	AUX
ma-156	68	23	a	a	DET
ma-156	68	24	function	function	NOUN
ma-156	68	25	,	,	PUNCT
ma-156	68	26	and	and	CCONJ
ma-156	68	27	let	let	VERB
ma-156	68	28	t	t	NOUN
ma-156	68	29	:	:	PUNCT
ma-156	68	30	x	x	X
ma-156	68	31	→	→	PUNCT
ma-156	68	32	x	x	PUNCT
ma-156	68	33	be	be	AUX
ma-156	68	34	a	a	DET
ma-156	68	35	self	self	NOUN
ma-156	68	36	-	-	PUNCT
ma-156	68	37	mapping	mapping	NOUN
ma-156	68	38	.	.	PUNCT
ma-156	69	1	suppose	suppose	VERB
ma-156	69	2	that	that	SCONJ
ma-156	69	3	the	the	DET
ma-156	69	4	following	follow	VERB
ma-156	69	5	conditions	condition	NOUN
ma-156	69	6	are	be	AUX
ma-156	69	7	satisfied	satisfied	ADJ
ma-156	69	8	:	:	PUNCT
ma-156	69	9	(	(	PUNCT
ma-156	69	10	i	i	NOUN
ma-156	69	11	)	)	PUNCT
ma-156	69	12	t	t	PROPN
ma-156	69	13	is	be	AUX
ma-156	69	14	a	a	DET
ma-156	69	15	generalized	generalized	ADJ
ma-156	69	16	α	α	NOUN
ma-156	69	17	-	-	PUNCT
ma-156	69	18	φ	φ	VERB
ma-156	69	19	-	-	PUNCT
ma-156	69	20	geraghty	geraghty	VERB
ma-156	69	21	quasi	quasi	PROPN
ma-156	69	22	-	-	NOUN
ma-156	69	23	contraction	contraction	NOUN
ma-156	69	24	type	type	NOUN
ma-156	69	25	mapping;(ii	mapping;(ii	PROPN
ma-156	69	26	)	)	PUNCT
ma-156	69	27	t	t	PROPN
ma-156	69	28	is	be	AUX
ma-156	69	29	triangular	triangular	ADJ
ma-156	69	30	α	α	DET
ma-156	69	31	-	-	ADJ
ma-156	69	32	orbital	orbital	ADJ
ma-156	69	33	admissible	admissible	ADJ
ma-156	69	34	mapping;(iii	mapping;(iii	NOUN
ma-156	69	35	)	)	PUNCT
ma-156	69	36	there	there	PRON
ma-156	69	37	exists	exist	VERB
ma-156	69	38	x1	x1	PROPN
ma-156	69	39	∈	∈	PROPN
ma-156	69	40	x	x	PUNCT
ma-156	69	41	such	such	ADJ
ma-156	69	42	that	that	DET
ma-156	69	43	α(x1	α(x1	NOUN
ma-156	69	44	,	,	PUNCT
ma-156	69	45	t	t	PROPN
ma-156	69	46	x1	x1	NUM
ma-156	69	47	)	)	PUNCT
ma-156	69	48	≥	≥	NOUN
ma-156	69	49	1	1	NUM
ma-156	69	50	;	;	PUNCT
ma-156	69	51	then	then	ADV
ma-156	69	52	t	t	PROPN
ma-156	69	53	has	have	VERB
ma-156	69	54	a	a	DET
ma-156	69	55	fixed	fix	VERB
ma-156	69	56	point	point	NOUN
ma-156	69	57	x∗	x∗	PROPN
ma-156	69	58	∈	∈	PROPN
ma-156	69	59	x	x	X
ma-156	69	60	and	and	CCONJ
ma-156	69	61	{	{	PUNCT
ma-156	69	62	t	t	PROPN
ma-156	69	63	nx1	nx1	PROPN
ma-156	69	64	}	}	PUNCT
ma-156	69	65	converges	converge	NOUN
ma-156	69	66	to	to	ADP
ma-156	69	67	x∗.	x∗.	NOUN
ma-156	69	68	in	in	ADP
ma-156	69	69	this	this	DET
ma-156	69	70	paper	paper	NOUN
ma-156	69	71	,	,	PUNCT
ma-156	69	72	we	we	PRON
ma-156	69	73	extend	extend	VERB
ma-156	69	74	the	the	DET
ma-156	69	75	concept	concept	NOUN
ma-156	69	76	of	of	ADP
ma-156	69	77	generalized	generalized	ADJ
ma-156	69	78	α	α	PROPN
ma-156	69	79	-	-	PUNCT
ma-156	69	80	φ	φ	VERB
ma-156	69	81	-	-	PUNCT
ma-156	69	82	geraghty	geraghty	VERB
ma-156	69	83	quasi	quasi	NOUN
ma-156	69	84	-	-	NOUN
ma-156	69	85	contraction	contraction	NOUN
ma-156	69	86	typemapping	typemappe	VERB
ma-156	69	87	to	to	ADP
ma-156	69	88	generalized	generalized	ADJ
ma-156	69	89	α	α	PROPN
ma-156	69	90	-	-	PUNCT
ma-156	69	91	φ	φ	VERB
ma-156	69	92	-	-	PUNCT
ma-156	69	93	geraghty	geraghty	VERB
ma-156	69	94	proximal	proximal	ADJ
ma-156	69	95	quasi	quasi	PROPN
ma-156	69	96	-	-	NOUN
ma-156	69	97	contraction	contraction	NOUN
ma-156	69	98	type	type	NOUN
ma-156	69	99	mapping	mapping	NOUN
ma-156	69	100	in	in	ADP
ma-156	69	101	the	the	DET
ma-156	69	102	case	case	NOUN
ma-156	69	103	ofnon	ofnon	ADJ
ma-156	69	104	-	-	PUNCT
ma-156	69	105	self	self	NOUN
ma-156	69	106	mappings	mapping	NOUN
ma-156	69	107	.	.	PUNCT
ma-156	70	1	more	more	ADV
ma-156	70	2	precisely	precisely	ADV
ma-156	70	3	,	,	PUNCT
ma-156	70	4	we	we	PRON
ma-156	70	5	study	study	VERB
ma-156	70	6	the	the	DET
ma-156	70	7	existence	existence	NOUN
ma-156	70	8	and	and	CCONJ
ma-156	70	9	uniqueness	uniqueness	NOUN
ma-156	70	10	of	of	ADP
ma-156	70	11	best	good	ADJ
ma-156	70	12	proximitypoints	proximitypoint	NOUN
ma-156	70	13	for	for	ADP
ma-156	70	14	generalized	generalized	ADJ
ma-156	70	15	α	α	PROPN
ma-156	70	16	-	-	PUNCT
ma-156	70	17	φ	φ	VERB
ma-156	70	18	-	-	PUNCT
ma-156	70	19	geraghty	geraghty	VERB
ma-156	70	20	proximal	proximal	ADJ
ma-156	70	21	quasi	quasi	NOUN
ma-156	70	22	-	-	NOUN
ma-156	70	23	contraction	contraction	NOUN
ma-156	70	24	for	for	ADP
ma-156	70	25	non	non	ADJ
ma-156	70	26	-	-	ADJ
ma-156	70	27	self	self	NOUN
ma-156	70	28	mappings	mapping	NOUN
ma-156	70	29	.	.	PUNCT
ma-156	71	1	2	2	X
ma-156	71	2	.	.	X
ma-156	71	3	preliminaries	preliminary	NOUN
ma-156	71	4	we	we	PRON
ma-156	71	5	start	start	VERB
ma-156	71	6	this	this	DET
ma-156	71	7	section	section	NOUN
ma-156	71	8	with	with	ADP
ma-156	71	9	the	the	DET
ma-156	71	10	following	follow	VERB
ma-156	71	11	definitions.let	definitions.let	X
ma-156	71	12	a	a	NOUN
ma-156	71	13	and	and	CCONJ
ma-156	71	14	b	b	NOUN
ma-156	71	15	be	be	AUX
ma-156	71	16	non	non	ADJ
ma-156	71	17	-	-	ADJ
ma-156	71	18	empty	empty	ADJ
ma-156	71	19	subsets	subset	NOUN
ma-156	71	20	of	of	ADP
ma-156	71	21	a	a	DET
ma-156	71	22	metric	metric	ADJ
ma-156	71	23	space	space	NOUN
ma-156	71	24	(	(	PUNCT
ma-156	71	25	x	x	X
ma-156	71	26	,	,	PUNCT
ma-156	71	27	d	d	NOUN
ma-156	71	28	)	)	PUNCT
ma-156	71	29	.	.	PUNCT
ma-156	72	1	we	we	PRON
ma-156	72	2	denote	denote	VERB
ma-156	72	3	by	by	ADP
ma-156	72	4	a0	a0	PROPN
ma-156	72	5	and	and	CCONJ
ma-156	72	6	b0	b0	VERB
ma-156	72	7	the	the	DET
ma-156	72	8	followingsets	followingset	NOUN
ma-156	72	9	:	:	PUNCT
ma-156	73	1	d(a	d(a	PROPN
ma-156	73	2	,	,	PUNCT
ma-156	73	3	b	b	NOUN
ma-156	73	4	)	)	PUNCT
ma-156	73	5	=	=	SYM
ma-156	73	6	inf{d(a	inf{d(a	PROPN
ma-156	73	7	,	,	PUNCT
ma-156	73	8	b	b	NOUN
ma-156	73	9	)	)	PUNCT
ma-156	73	10	:	:	PUNCT
ma-156	73	11	a	a	DET
ma-156	73	12	∈	∈	PROPN
ma-156	73	13	a	a	PRON
ma-156	73	14	,	,	PUNCT
ma-156	73	15	b	b	PROPN
ma-156	73	16	∈	∈	PROPN
ma-156	73	17	b	b	NOUN
ma-156	73	18	}	}	PUNCT
ma-156	73	19	.	.	PUNCT
ma-156	74	1	a0	a0	PROPN
ma-156	74	2	=	=	PUNCT
ma-156	74	3	{	{	PUNCT
ma-156	74	4	x	x	PROPN
ma-156	74	5	∈	∈	PROPN
ma-156	74	6	a	a	DET
ma-156	74	7	:	:	PUNCT
ma-156	74	8	d(x	d(x	PROPN
ma-156	74	9	,	,	PUNCT
ma-156	74	10	y	y	NOUN
ma-156	74	11	)	)	PUNCT
ma-156	74	12	=	=	SYM
ma-156	75	1	d(a	d(a	PROPN
ma-156	75	2	,	,	PUNCT
ma-156	75	3	b	b	NOUN
ma-156	75	4	)	)	PUNCT
ma-156	75	5	for	for	ADP
ma-156	75	6	some	some	DET
ma-156	75	7	y	y	PROPN
ma-156	75	8	∈	∈	PROPN
ma-156	75	9	b	b	PROPN
ma-156	75	10	}	}	PUNCT
ma-156	75	11	.	.	PUNCT
ma-156	76	1	b0	b0	NOUN
ma-156	76	2	=	=	SYM
ma-156	76	3	{	{	PUNCT
ma-156	76	4	y	y	PROPN
ma-156	76	5	∈	∈	PROPN
ma-156	76	6	b	b	PROPN
ma-156	76	7	:	:	PUNCT
ma-156	76	8	d(x	d(x	PROPN
ma-156	76	9	,	,	PUNCT
ma-156	76	10	y	y	NOUN
ma-156	76	11	)	)	PUNCT
ma-156	76	12	=	=	SYM
ma-156	77	1	d(a	d(a	PROPN
ma-156	77	2	,	,	PUNCT
ma-156	77	3	b	b	NOUN
ma-156	77	4	)	)	PUNCT
ma-156	77	5	for	for	ADP
ma-156	77	6	some	some	DET
ma-156	77	7	x	x	SYM
ma-156	77	8	∈	∈	PROPN
ma-156	77	9	a	a	PRON
ma-156	77	10	}	}	PUNCT
ma-156	77	11	.	.	PUNCT
ma-156	78	1	definition	definition	NOUN
ma-156	78	2	2.1	2.1	NUM
ma-156	78	3	[	[	X
ma-156	78	4	14	14	NUM
ma-156	78	5	]	]	PUNCT
ma-156	78	6	.	.	PUNCT
ma-156	79	1	an	an	DET
ma-156	79	2	element	element	NOUN
ma-156	79	3	x	x	SYM
ma-156	79	4	∈	∈	PROPN
ma-156	79	5	a	a	PRON
ma-156	79	6	is	be	AUX
ma-156	79	7	said	say	VERB
ma-156	79	8	to	to	PART
ma-156	79	9	be	be	AUX
ma-156	79	10	a	a	DET
ma-156	79	11	best	good	ADJ
ma-156	79	12	proximity	proximity	NOUN
ma-156	79	13	point	point	NOUN
ma-156	79	14	of	of	ADP
ma-156	79	15	the	the	DET
ma-156	79	16	non	non	ADJ
ma-156	79	17	-	-	ADJ
ma-156	79	18	self	self	NOUN
ma-156	79	19	-	-	PUNCT
ma-156	79	20	mapping	mapping	NOUN
ma-156	79	21	t	t	NOUN
ma-156	79	22	:	:	PUNCT
ma-156	79	23	a→	a→	PROPN
ma-156	79	24	b	b	X
ma-156	79	25	if	if	SCONJ
ma-156	79	26	it	it	PRON
ma-156	79	27	satisfies	satisfy	VERB
ma-156	79	28	the	the	DET
ma-156	79	29	condition	condition	NOUN
ma-156	79	30	that	that	SCONJ
ma-156	79	31	d(x	d(x	PROPN
ma-156	79	32	,	,	PUNCT
ma-156	79	33	t	t	NOUN
ma-156	79	34	x	x	X
ma-156	79	35	)	)	PUNCT
ma-156	80	1	=	=	SYM
ma-156	80	2	d(a	d(a	PROPN
ma-156	80	3	,	,	PUNCT
ma-156	80	4	b).we	b).we	NOUN
ma-156	80	5	denote	denote	VERB
ma-156	80	6	the	the	DET
ma-156	80	7	set	set	NOUN
ma-156	80	8	of	of	ADP
ma-156	80	9	all	all	DET
ma-156	80	10	best	good	ADJ
ma-156	80	11	proximity	proximity	NOUN
ma-156	80	12	points	point	NOUN
ma-156	80	13	of	of	ADP
ma-156	80	14	t	t	NOUN
ma-156	80	15	by	by	ADP
ma-156	80	16	pt	pt	X
ma-156	80	17	(	(	PUNCT
ma-156	80	18	a	a	NOUN
ma-156	80	19	)	)	PUNCT
ma-156	80	20	,	,	PUNCT
ma-156	80	21	that	that	ADV
ma-156	80	22	is	is	ADV
ma-156	80	23	,	,	PUNCT
ma-156	80	24	pt	pt	X
ma-156	80	25	(	(	PUNCT
ma-156	80	26	a	a	NOUN
ma-156	80	27	)	)	PUNCT
ma-156	80	28	:	:	PUNCT
ma-156	80	29	=	=	SYM
ma-156	80	30	{	{	PUNCT
ma-156	80	31	x	x	PUNCT
ma-156	80	32	∈	∈	PROPN
ma-156	80	33	a	a	DET
ma-156	80	34	:	:	PUNCT
ma-156	80	35	d(x	d(x	PROPN
ma-156	80	36	,	,	PUNCT
ma-156	80	37	t	t	NOUN
ma-156	80	38	x	x	X
ma-156	80	39	)	)	PUNCT
ma-156	80	40	=	=	SYM
ma-156	81	1	d(a	d(a	PROPN
ma-156	81	2	,	,	PUNCT
ma-156	81	3	b	b	NOUN
ma-156	81	4	)	)	PUNCT
ma-156	81	5	}	}	PUNCT
ma-156	81	6	.	.	PUNCT
ma-156	82	1	the	the	DET
ma-156	82	2	following	follow	VERB
ma-156	82	3	were	be	AUX
ma-156	82	4	introduced	introduce	VERB
ma-156	82	5	by	by	ADP
ma-156	82	6	[	[	X
ma-156	82	7	11	11	NUM
ma-156	82	8	]	]	PUNCT
ma-156	82	9	.	.	PUNCT
ma-156	83	1	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	83	2	eur	eur	PROPN
ma-156	83	3	.	.	PUNCT
ma-156	84	1	j.	j.	PROPN
ma-156	84	2	math	math	PROPN
ma-156	84	3	.	.	PUNCT
ma-156	85	1	anal	anal	PROPN
ma-156	85	2	.	.	PUNCT
ma-156	86	1	10.28924	10.28924	NUM
ma-156	86	2	/	/	SYM
ma-156	86	3	ada	ada	PROPN
ma-156	86	4	/	/	SYM
ma-156	86	5	ma.3.16	ma.3.16	PROPN
ma-156	86	6	4	4	NUM
ma-156	86	7	definition	definition	NOUN
ma-156	86	8	2.2	2.2	NUM
ma-156	86	9	[	[	X
ma-156	86	10	11	11	NUM
ma-156	86	11	]	]	PUNCT
ma-156	86	12	.	.	PUNCT
ma-156	87	1	a	a	DET
ma-156	87	2	non	non	ADJ
ma-156	87	3	-	-	ADJ
ma-156	87	4	self	self	ADJ
ma-156	87	5	mapping	mapping	NOUN
ma-156	87	6	t	t	NOUN
ma-156	87	7	:	:	PUNCT
ma-156	87	8	a	a	DET
ma-156	87	9	→	→	SYM
ma-156	87	10	b	b	PROPN
ma-156	87	11	is	be	AUX
ma-156	87	12	said	say	VERB
ma-156	87	13	to	to	PART
ma-156	87	14	be	be	AUX
ma-156	87	15	a	a	DET
ma-156	87	16	proximal	proximal	ADJ
ma-156	87	17	quasi	quasi	NOUN
ma-156	87	18	-	-	NOUN
ma-156	87	19	contraction	contraction	NOUN
ma-156	87	20	ifand	ifand	NOUN
ma-156	87	21	only	only	ADV
ma-156	87	22	if	if	SCONJ
ma-156	87	23	there	there	PRON
ma-156	87	24	exists	exist	VERB
ma-156	87	25	a	a	DET
ma-156	87	26	number	number	NOUN
ma-156	87	27	q	q	NOUN
ma-156	87	28	,	,	PUNCT
ma-156	87	29	0	0	NUM
ma-156	87	30	≤	≤	NUM
ma-156	87	31	q	q	X
ma-156	87	32	<	<	X
ma-156	87	33	1	1	NUM
ma-156	87	34	,	,	PUNCT
ma-156	87	35	such	such	ADJ
ma-156	87	36	that	that	SCONJ
ma-156	87	37	{	{	PUNCT
ma-156	87	38	d(u	d(u	PROPN
ma-156	87	39	,	,	PUNCT
ma-156	87	40	t	t	NOUN
ma-156	87	41	x	x	NOUN
ma-156	87	42	)	)	PUNCT
ma-156	87	43	=	=	SYM
ma-156	88	1	d(a	d(a	PROPN
ma-156	88	2	,	,	PUNCT
ma-156	88	3	b	b	NOUN
ma-156	88	4	)	)	PUNCT
ma-156	88	5	d(v	d(v	PROPN
ma-156	88	6	,	,	PUNCT
ma-156	88	7	t	t	PROPN
ma-156	88	8	y	y	PROPN
ma-156	88	9	)	)	PUNCT
ma-156	89	1	=	=	SYM
ma-156	90	1	d(a	d(a	PROPN
ma-156	90	2	,	,	PUNCT
ma-156	90	3	b	b	NOUN
ma-156	90	4	)	)	PUNCT
ma-156	90	5	=	=	NOUN
ma-156	90	6	⇒	⇒	VERB
ma-156	90	7	d(u	d(u	PROPN
ma-156	90	8	,	,	PUNCT
ma-156	90	9	v	v	NOUN
ma-156	90	10	)	)	PUNCT
ma-156	90	11	≤	≤	NOUN
ma-156	90	12	qmax{d(x	qmax{d(x	NOUN
ma-156	90	13	,	,	PUNCT
ma-156	90	14	y	y	NOUN
ma-156	90	15	)	)	PUNCT
ma-156	90	16	;	;	PUNCT
ma-156	90	17	d(x	d(x	PROPN
ma-156	90	18	,	,	PUNCT
ma-156	90	19	u	u	NOUN
ma-156	90	20	)	)	PUNCT
ma-156	90	21	;	;	PUNCT
ma-156	90	22	d(y	d(y	NOUN
ma-156	90	23	,	,	PUNCT
ma-156	90	24	v	v	NOUN
ma-156	90	25	)	)	PUNCT
ma-156	90	26	;	;	PUNCT
ma-156	90	27	d(x	d(x	PROPN
ma-156	90	28	,	,	PUNCT
ma-156	90	29	v	v	NOUN
ma-156	90	30	)	)	PUNCT
ma-156	90	31	;	;	PUNCT
ma-156	90	32	d(y	d(y	NOUN
ma-156	90	33	,	,	PUNCT
ma-156	90	34	u	u	NOUN
ma-156	90	35	)	)	PUNCT
ma-156	90	36	}	}	PUNCT
ma-156	90	37	,	,	PUNCT
ma-156	90	38	where	where	SCONJ
ma-156	90	39	x	x	X
ma-156	90	40	,	,	PUNCT
ma-156	90	41	y	y	PROPN
ma-156	90	42	,	,	PUNCT
ma-156	90	43	u	u	PROPN
ma-156	90	44	,	,	PUNCT
ma-156	90	45	v	v	NOUN
ma-156	90	46	∈	∈	NOUN
ma-156	90	47	a.	a.	NOUN
ma-156	90	48	if	if	SCONJ
ma-156	90	49	t	t	PROPN
ma-156	90	50	is	be	AUX
ma-156	90	51	a	a	DET
ma-156	90	52	self	self	NOUN
ma-156	90	53	mapping	mapping	NOUN
ma-156	90	54	on	on	ADP
ma-156	90	55	a	a	PRON
ma-156	90	56	,	,	PUNCT
ma-156	90	57	then	then	ADV
ma-156	90	58	definition	definition	NOUN
ma-156	90	59	2.2	2.2	NUM
ma-156	90	60	reduces	reduce	VERB
ma-156	90	61	to	to	ADP
ma-156	90	62	definition	definition	NOUN
ma-156	90	63	1.2	1.2	NUM
ma-156	90	64	.	.	PUNCT
ma-156	91	1	lemma	lemma	PROPN
ma-156	91	2	2.3	2.3	NUM
ma-156	92	1	[	[	X
ma-156	92	2	11	11	NUM
ma-156	92	3	]	]	PUNCT
ma-156	92	4	.	.	PUNCT
ma-156	93	1	let	let	VERB
ma-156	93	2	t	t	NOUN
ma-156	93	3	:	:	PUNCT
ma-156	93	4	a	a	DET
ma-156	93	5	→	→	SYM
ma-156	93	6	b	b	X
ma-156	93	7	be	be	AUX
ma-156	93	8	a	a	DET
ma-156	93	9	non	non	ADJ
ma-156	93	10	-	-	ADJ
ma-156	93	11	self	self	ADJ
ma-156	93	12	mapping	mapping	NOUN
ma-156	93	13	.	.	PUNCT
ma-156	94	1	suppose	suppose	VERB
ma-156	94	2	that	that	SCONJ
ma-156	94	3	the	the	DET
ma-156	94	4	following	follow	VERB
ma-156	94	5	conditionshold:(i	conditionshold:(i	PROPN
ma-156	94	6	)	)	PUNCT
ma-156	94	7	a0	a0	PROPN
ma-156	94	8	6=	6=	PROPN
ma-156	94	9	∅;(ii	∅;(ii	PROPN
ma-156	94	10	)	)	PUNCT
ma-156	94	11	t	t	PROPN
ma-156	94	12	(	(	PUNCT
ma-156	94	13	a0	a0	PROPN
ma-156	94	14	)	)	PUNCT
ma-156	94	15	⊆	⊆	NUM
ma-156	94	16	b0.then	b0.then	NOUN
ma-156	94	17	,	,	PUNCT
ma-156	94	18	for	for	ADP
ma-156	94	19	all	all	DET
ma-156	94	20	a	a	DET
ma-156	94	21	∈	∈	PROPN
ma-156	94	22	a0	a0	NOUN
ma-156	94	23	,	,	PUNCT
ma-156	94	24	there	there	PRON
ma-156	94	25	exists	exist	VERB
ma-156	94	26	a	a	DET
ma-156	94	27	sequence	sequence	NOUN
ma-156	94	28	{	{	PUNCT
ma-156	94	29	xn	xn	NOUN
ma-156	94	30	}	}	PUNCT
ma-156	94	31	⊂	⊂	PROPN
ma-156	94	32	a0	a0	PROPN
ma-156	94	33	such	such	ADJ
ma-156	94	34	that	that	SCONJ
ma-156	94	35	{	{	PUNCT
ma-156	95	1	x0	x0	PROPN
ma-156	95	2	=	=	PUNCT
ma-156	95	3	a	a	PROPN
ma-156	95	4	,	,	PUNCT
ma-156	95	5	d(xn+1	d(xn+1	PROPN
ma-156	95	6	,	,	PUNCT
ma-156	95	7	t	t	PROPN
ma-156	95	8	xn	xn	PROPN
ma-156	95	9	)	)	PUNCT
ma-156	96	1	=	=	SYM
ma-156	96	2	d(a	d(a	PROPN
ma-156	96	3	,	,	PUNCT
ma-156	96	4	b	b	NOUN
ma-156	96	5	)	)	PUNCT
ma-156	96	6	,	,	PUNCT
ma-156	96	7	∀n	∀n	NUM
ma-156	96	8	∈	∈	VERB
ma-156	96	9	n.any	n.any	ADJ
ma-156	96	10	sequence	sequence	NOUN
ma-156	96	11	{	{	PUNCT
ma-156	96	12	xn	xn	NOUN
ma-156	96	13	}	}	PUNCT
ma-156	96	14	⊂	⊂	PROPN
ma-156	96	15	a0	a0	PROPN
ma-156	96	16	satisfying	satisfy	VERB
ma-156	96	17	the	the	DET
ma-156	96	18	equation	equation	NOUN
ma-156	96	19	in	in	ADP
ma-156	96	20	lemma	lemma	PROPN
ma-156	96	21	2.3	2.3	NUM
ma-156	96	22	is	be	AUX
ma-156	96	23	called	call	VERB
ma-156	96	24	a	a	DET
ma-156	96	25	proximal	proximal	ADJ
ma-156	96	26	picardsequence	picardsequence	NOUN
ma-156	96	27	associated	associate	VERB
ma-156	96	28	to	to	ADP
ma-156	96	29	a	a	DET
ma-156	96	30	∈	∈	PROPN
ma-156	96	31	a0	a0	NOUN
ma-156	96	32	and	and	CCONJ
ma-156	96	33	we	we	PRON
ma-156	96	34	denote	denote	VERB
ma-156	96	35	by	by	ADP
ma-156	96	36	pp(a	pp(a	NOUN
ma-156	96	37	)	)	PUNCT
ma-156	96	38	the	the	DET
ma-156	96	39	set	set	NOUN
ma-156	96	40	of	of	ADP
ma-156	96	41	all	all	DET
ma-156	96	42	proximal	proximal	ADJ
ma-156	96	43	picard	picard	NOUN
ma-156	96	44	sequencesassociated	sequencesassociate	VERB
ma-156	96	45	to	to	PART
ma-156	96	46	a.	a.	NOUN
ma-156	96	47	suppose	suppose	VERB
ma-156	96	48	a	a	DET
ma-156	96	49	∈	∈	PROPN
ma-156	96	50	a0	a0	NOUN
ma-156	96	51	and	and	CCONJ
ma-156	96	52	{	{	PUNCT
ma-156	96	53	xn	xn	NOUN
ma-156	96	54	}	}	PUNCT
ma-156	96	55	∈	∈	PROPN
ma-156	96	56	pp(a	pp(a	NOUN
ma-156	96	57	)	)	PUNCT
ma-156	96	58	.	.	PUNCT
ma-156	97	1	for	for	ADP
ma-156	97	2	all	all	PRON
ma-156	97	3	(	(	PUNCT
ma-156	97	4	i	i	PROPN
ma-156	97	5	,	,	PUNCT
ma-156	97	6	j	j	PROPN
ma-156	97	7	)	)	PUNCT
ma-156	97	8	∈	∈	PROPN
ma-156	97	9	n2	n2	NOUN
ma-156	97	10	,	,	PUNCT
ma-156	97	11	the	the	DET
ma-156	97	12	following	follow	VERB
ma-156	97	13	sets	set	NOUN
ma-156	97	14	are	be	AUX
ma-156	97	15	definedby	definedby	ADJ
ma-156	97	16	:	:	PUNCT
ma-156	97	17	ot	ot	INTJ
ma-156	97	18	(	(	PUNCT
ma-156	97	19	xi	xi	PROPN
ma-156	97	20	,	,	PUNCT
ma-156	97	21	j	j	PROPN
ma-156	97	22	)	)	PUNCT
ma-156	97	23	:	:	PUNCT
ma-156	98	1	=	=	SYM
ma-156	98	2	{	{	PUNCT
ma-156	98	3	xl	xl	PROPN
ma-156	98	4	:	:	PUNCT
ma-156	99	1	i	i	PRON
ma-156	99	2	≤	≤	PUNCT
ma-156	99	3	l	l	X
ma-156	99	4	≤	≤	NUM
ma-156	99	5	j	j	PROPN
ma-156	100	1	+	+	CCONJ
ma-156	100	2	i	i	PROPN
ma-156	100	3	}	}	PUNCT
ma-156	100	4	and	and	CCONJ
ma-156	100	5	ot	ot	INTJ
ma-156	100	6	(	(	PUNCT
ma-156	100	7	xi	xi	INTJ
ma-156	100	8	,	,	PUNCT
ma-156	100	9	∞	∞	PROPN
ma-156	100	10	)	)	PUNCT
ma-156	100	11	:	:	PUNCT
ma-156	101	1	=	=	SYM
ma-156	101	2	{	{	PUNCT
ma-156	101	3	xl	xl	PROPN
ma-156	101	4	:	:	PUNCT
ma-156	101	5	l	l	NOUN
ma-156	101	6	≥	≥	NOUN
ma-156	101	7	i	i	NOUN
ma-156	101	8	}	}	PUNCT
ma-156	101	9	.	.	PUNCT
ma-156	102	1	definition	definition	NOUN
ma-156	102	2	2.4	2.4	NUM
ma-156	102	3	[	[	SYM
ma-156	102	4	11	11	NUM
ma-156	102	5	]	]	X
ma-156	102	6	a0	a0	NOUN
ma-156	102	7	is	be	AUX
ma-156	102	8	said	say	VERB
ma-156	102	9	to	to	PART
ma-156	102	10	be	be	AUX
ma-156	102	11	proximal	proximal	ADJ
ma-156	102	12	t	t	NOUN
ma-156	102	13	-orbitally	-orbitally	ADV
ma-156	102	14	complete	complete	ADJ
ma-156	102	15	if	if	SCONJ
ma-156	102	16	and	and	CCONJ
ma-156	102	17	only	only	ADV
ma-156	102	18	if	if	SCONJ
ma-156	102	19	every	every	DET
ma-156	102	20	cauchysequence	cauchysequence	NOUN
ma-156	102	21	{	{	PUNCT
ma-156	102	22	xn	xn	NOUN
ma-156	102	23	}	}	PUNCT
ma-156	102	24	∈	∈	PROPN
ma-156	102	25	pp(a	pp(a	NOUN
ma-156	102	26	)	)	PUNCT
ma-156	102	27	for	for	ADP
ma-156	102	28	some	some	PRON
ma-156	102	29	a	a	DET
ma-156	102	30	∈	∈	PROPN
ma-156	102	31	a0	a0	NOUN
ma-156	102	32	,	,	PUNCT
ma-156	102	33	converges	converge	VERB
ma-156	102	34	to	to	ADP
ma-156	102	35	an	an	DET
ma-156	102	36	element	element	NOUN
ma-156	102	37	in	in	ADP
ma-156	102	38	a0.if	a0.if	PROPN
ma-156	102	39	t	t	NOUN
ma-156	102	40	is	be	AUX
ma-156	102	41	a	a	DET
ma-156	102	42	self	self	NOUN
ma-156	102	43	mapping	mapping	NOUN
ma-156	102	44	on	on	ADP
ma-156	102	45	a	a	PRON
ma-156	102	46	,	,	PUNCT
ma-156	102	47	then	then	ADV
ma-156	102	48	the	the	DET
ma-156	102	49	preceding	precede	VERB
ma-156	102	50	definition	definition	NOUN
ma-156	102	51	reduces	reduce	VERB
ma-156	102	52	to	to	ADP
ma-156	102	53	the	the	DET
ma-156	102	54	condition	condition	NOUN
ma-156	102	55	that	that	SCONJ
ma-156	102	56	a	a	PRON
ma-156	102	57	is	be	AUX
ma-156	102	58	t	t	NOUN
ma-156	102	59	-orbitally	-orbitally	ADV
ma-156	102	60	complete	complete	ADJ
ma-156	102	61	.	.	PUNCT
ma-156	103	1	the	the	DET
ma-156	103	2	concepts	concept	NOUN
ma-156	103	3	of	of	ADP
ma-156	103	4	α	α	NOUN
ma-156	103	5	-	-	ADJ
ma-156	103	6	orbital	orbital	ADJ
ma-156	103	7	proximal	proximal	ADJ
ma-156	103	8	admissible	admissible	ADJ
ma-156	103	9	mapping	mapping	NOUN
ma-156	103	10	and	and	CCONJ
ma-156	103	11	triangular	triangular	NOUN
ma-156	103	12	α	α	ADJ
ma-156	103	13	-	-	ADJ
ma-156	103	14	orbital	orbital	ADJ
ma-156	103	15	proximaladmissible	proximaladmissible	ADJ
ma-156	103	16	mapping	mapping	NOUN
ma-156	103	17	are	be	AUX
ma-156	103	18	hereby	hereby	ADV
ma-156	103	19	introduced	introduce	VERB
ma-156	103	20	as	as	SCONJ
ma-156	103	21	follows	follow	VERB
ma-156	103	22	.	.	PUNCT
ma-156	104	1	definition	definition	NOUN
ma-156	104	2	2.5	2.5	NUM
ma-156	104	3	let	let	VERB
ma-156	104	4	t	t	NOUN
ma-156	104	5	:	:	PUNCT
ma-156	104	6	a	a	DET
ma-156	104	7	→	→	SYM
ma-156	104	8	b	b	X
ma-156	104	9	be	be	AUX
ma-156	104	10	a	a	DET
ma-156	104	11	non	non	ADJ
ma-156	104	12	-	-	ADJ
ma-156	104	13	self	self	ADJ
ma-156	104	14	mapping	mapping	NOUN
ma-156	104	15	and	and	CCONJ
ma-156	104	16	α	α	NOUN
ma-156	104	17	:	:	PUNCT
ma-156	105	1	a×	a×	VERB
ma-156	105	2	a	a	X
ma-156	105	3	→	→	X
ma-156	105	4	[	[	X
ma-156	105	5	0,∞	0,∞	NOUN
ma-156	105	6	)	)	PUNCT
ma-156	105	7	be	be	AUX
ma-156	105	8	a	a	DET
ma-156	105	9	function	function	NOUN
ma-156	105	10	.	.	PUNCT
ma-156	106	1	themapping	themappe	VERB
ma-156	106	2	t	t	PROPN
ma-156	106	3	is	be	AUX
ma-156	106	4	said	say	VERB
ma-156	106	5	to	to	PART
ma-156	106	6	be	be	AUX
ma-156	106	7	α	α	PRON
ma-156	106	8	-	-	ADJ
ma-156	106	9	orbital	orbital	ADJ
ma-156	106	10	proximal	proximal	ADJ
ma-156	106	11	admissible	admissible	ADJ
ma-156	106	12	if	if	SCONJ
ma-156	106	13			PROPN
ma-156	106	14	α(x	α(x	PROPN
ma-156	106	15	,	,	PUNCT
ma-156	106	16	u	u	NOUN
ma-156	106	17	)	)	PUNCT
ma-156	106	18	≥	≥	PROPN
ma-156	106	19	1	1	NUM
ma-156	106	20	d(u	d(u	PROPN
ma-156	106	21	,	,	PUNCT
ma-156	106	22	t	t	NOUN
ma-156	106	23	x	x	NOUN
ma-156	106	24	)	)	PUNCT
ma-156	106	25	=	=	SYM
ma-156	107	1	d(a	d(a	PROPN
ma-156	107	2	,	,	PUNCT
ma-156	107	3	b	b	NOUN
ma-156	107	4	)	)	PUNCT
ma-156	107	5	d(v	d(v	PROPN
ma-156	107	6	,	,	PUNCT
ma-156	107	7	tu	tu	PROPN
ma-156	107	8	)	)	PUNCT
ma-156	107	9	=	=	SYM
ma-156	108	1	d(a	d(a	PROPN
ma-156	108	2	,	,	PUNCT
ma-156	108	3	b	b	NOUN
ma-156	108	4	)	)	PUNCT
ma-156	108	5	=	=	NOUN
ma-156	108	6	⇒	⇒	X
ma-156	108	7	α(u	α(u	PROPN
ma-156	108	8	,	,	PUNCT
ma-156	108	9	v	v	NOUN
ma-156	108	10	)	)	PUNCT
ma-156	108	11	≥	≥	NOUN
ma-156	108	12	1	1	NUM
ma-156	108	13	,	,	PUNCT
ma-156	108	14	for	for	ADP
ma-156	108	15	all	all	DET
ma-156	108	16	x	x	NOUN
ma-156	108	17	,	,	PUNCT
ma-156	108	18	u	u	NOUN
ma-156	108	19	,	,	PUNCT
ma-156	108	20	v	v	PROPN
ma-156	108	21	∈	∈	PROPN
ma-156	108	22	a.	a.	NOUN
ma-156	108	23	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	108	24	eur	eur	PROPN
ma-156	108	25	.	.	PUNCT
ma-156	109	1	j.	j.	PROPN
ma-156	109	2	math	math	PROPN
ma-156	109	3	.	.	PUNCT
ma-156	110	1	anal	anal	PROPN
ma-156	110	2	.	.	PUNCT
ma-156	111	1	10.28924	10.28924	NUM
ma-156	111	2	/	/	SYM
ma-156	111	3	ada	ada	PROPN
ma-156	111	4	/	/	SYM
ma-156	111	5	ma.3.16	ma.3.16	PROPN
ma-156	111	6	5	5	NUM
ma-156	111	7	definition	definition	NOUN
ma-156	111	8	2.6	2.6	NUM
ma-156	111	9	let	let	VERB
ma-156	111	10	t	t	NOUN
ma-156	111	11	:	:	PUNCT
ma-156	111	12	a	a	DET
ma-156	111	13	→	→	SYM
ma-156	111	14	b	b	X
ma-156	111	15	be	be	AUX
ma-156	111	16	a	a	DET
ma-156	111	17	non	non	ADJ
ma-156	111	18	-	-	ADJ
ma-156	111	19	self	self	ADJ
ma-156	111	20	mapping	mapping	NOUN
ma-156	111	21	and	and	CCONJ
ma-156	111	22	α	α	NOUN
ma-156	111	23	:	:	PUNCT
ma-156	111	24	a	a	DET
ma-156	111	25	×	×	NOUN
ma-156	111	26	a	a	PRON
ma-156	111	27	→	→	X
ma-156	111	28	[	[	X
ma-156	111	29	0,∞	0,∞	NOUN
ma-156	111	30	)	)	PUNCT
ma-156	111	31	be	be	AUX
ma-156	111	32	a	a	DET
ma-156	111	33	function.the	function.the	DET
ma-156	111	34	mapping	mapping	NOUN
ma-156	111	35	t	t	NOUN
ma-156	111	36	is	be	AUX
ma-156	111	37	said	say	VERB
ma-156	111	38	to	to	PART
ma-156	111	39	be	be	AUX
ma-156	111	40	triangular	triangular	VERB
ma-156	111	41	α	α	DET
ma-156	111	42	-	-	ADJ
ma-156	111	43	orbital	orbital	ADJ
ma-156	111	44	proximal	proximal	ADJ
ma-156	111	45	admissible	admissible	NOUN
ma-156	111	46	if	if	SCONJ
ma-156	111	47	it	it	PRON
ma-156	111	48	is	be	AUX
ma-156	111	49	α	α	DET
ma-156	111	50	-	-	ADJ
ma-156	111	51	orbital	orbital	ADJ
ma-156	111	52	proximaladmissible	proximaladmissible	NOUN
ma-156	111	53	and	and	CCONJ
ma-156	111	54			PROPN
ma-156	111	55	α(x	α(x	PROPN
ma-156	111	56	,	,	PUNCT
ma-156	111	57	y	y	PROPN
ma-156	111	58	)	)	PUNCT
ma-156	111	59	≥	≥	NOUN
ma-156	111	60	1	1	NUM
ma-156	111	61	α(y	α(y	NOUN
ma-156	111	62	,	,	PUNCT
ma-156	111	63	u	u	NOUN
ma-156	111	64	)	)	PUNCT
ma-156	111	65	≥	≥	PROPN
ma-156	111	66	1	1	NUM
ma-156	111	67	d(u	d(u	PROPN
ma-156	111	68	,	,	PUNCT
ma-156	111	69	t	t	PROPN
ma-156	111	70	y	y	PROPN
ma-156	111	71	)	)	PUNCT
ma-156	112	1	=	=	SYM
ma-156	113	1	d(a	d(a	PROPN
ma-156	113	2	,	,	PUNCT
ma-156	113	3	b	b	NOUN
ma-156	113	4	)	)	PUNCT
ma-156	113	5	=	=	NOUN
ma-156	113	6	⇒	⇒	NOUN
ma-156	113	7	α(x	α(x	PROPN
ma-156	113	8	,	,	PUNCT
ma-156	113	9	u	u	NOUN
ma-156	113	10	)	)	PUNCT
ma-156	113	11	≥	≥	NOUN
ma-156	113	12	1	1	NUM
ma-156	113	13	,	,	PUNCT
ma-156	113	14	for	for	ADP
ma-156	113	15	all	all	DET
ma-156	113	16	x	x	NOUN
ma-156	113	17	,	,	PUNCT
ma-156	113	18	y	y	PROPN
ma-156	113	19	,	,	PUNCT
ma-156	113	20	u	u	PROPN
ma-156	113	21	∈	∈	PROPN
ma-156	113	22	a.	a.	NOUN
ma-156	113	23	remark	remark	NOUN
ma-156	113	24	2.7	2.7	NUM
ma-156	113	25	.	.	PUNCT
ma-156	114	1	if	if	SCONJ
ma-156	114	2	t	t	PROPN
ma-156	114	3	is	be	AUX
ma-156	114	4	a	a	DET
ma-156	114	5	self	self	NOUN
ma-156	114	6	mapping	mapping	NOUN
ma-156	114	7	,	,	PUNCT
ma-156	114	8	that	that	ADV
ma-156	114	9	is	is	ADV
ma-156	114	10	,	,	PUNCT
ma-156	114	11	if	if	SCONJ
ma-156	114	12	a	a	DET
ma-156	114	13	=	=	SYM
ma-156	114	14	b	b	PROPN
ma-156	114	15	,	,	PUNCT
ma-156	114	16	α	α	NOUN
ma-156	114	17	-	-	ADJ
ma-156	114	18	orbital	orbital	ADJ
ma-156	114	19	proximal	proximal	ADJ
ma-156	114	20	admissible	admissible	ADJ
ma-156	114	21	mappingreduces	mappingreduce	NOUN
ma-156	114	22	to	to	ADP
ma-156	114	23	α	α	NOUN
ma-156	114	24	-	-	PUNCT
ma-156	114	25	orbital	orbital	ADJ
ma-156	114	26	admissible	admissible	ADJ
ma-156	114	27	mapping	mapping	NOUN
ma-156	114	28	while	while	SCONJ
ma-156	114	29	triangular	triangular	NOUN
ma-156	114	30	α	α	NUM
ma-156	114	31	-	-	ADJ
ma-156	114	32	orbital	orbital	ADJ
ma-156	114	33	proximal	proximal	ADJ
ma-156	114	34	admissible	admissible	ADJ
ma-156	114	35	mappingreduces	mappingreduce	NOUN
ma-156	114	36	to	to	PART
ma-156	114	37	triangular	triangular	VERB
ma-156	114	38	α	α	PRON
ma-156	114	39	-	-	ADJ
ma-156	114	40	orbital	orbital	ADJ
ma-156	114	41	admissible	admissible	ADJ
ma-156	114	42	mapping	mapping	NOUN
ma-156	114	43	defined	define	VERB
ma-156	114	44	in	in	ADP
ma-156	114	45	[	[	X
ma-156	114	46	19	19	NUM
ma-156	114	47	]	]	PUNCT
ma-156	114	48	.	.	PUNCT
ma-156	115	1	example	example	NOUN
ma-156	116	1	2.8	2.8	NUM
ma-156	116	2	.	.	PUNCT
ma-156	117	1	let	let	VERB
ma-156	117	2	x	x	PRON
ma-156	117	3	be	be	AUX
ma-156	117	4	the	the	DET
ma-156	117	5	euclidean	euclidean	ADJ
ma-156	117	6	plane	plane	NOUN
ma-156	117	7	r2	r2	NOUN
ma-156	117	8	and	and	CCONJ
ma-156	117	9	consider	consider	VERB
ma-156	117	10	the	the	DET
ma-156	117	11	two	two	NUM
ma-156	117	12	subsets	subset	NOUN
ma-156	117	13	:	:	PUNCT
ma-156	117	14	a	a	DET
ma-156	117	15	=	=	X
ma-156	117	16	{	{	PUNCT
ma-156	117	17	(	(	PUNCT
ma-156	117	18	0	0	NUM
ma-156	117	19	,	,	PUNCT
ma-156	117	20	0	0	NUM
ma-156	117	21	)	)	PUNCT
ma-156	117	22	,	,	PUNCT
ma-156	117	23	(	(	PUNCT
ma-156	117	24	0	0	NUM
ma-156	117	25	,	,	PUNCT
ma-156	117	26	1	1	NUM
ma-156	117	27	)	)	PUNCT
ma-156	117	28	,	,	PUNCT
ma-156	117	29	(	(	PUNCT
ma-156	117	30	0	0	NUM
ma-156	117	31	,	,	PUNCT
ma-156	117	32	2	2	NUM
ma-156	117	33	)	)	PUNCT
ma-156	117	34	,	,	PUNCT
ma-156	117	35	(	(	PUNCT
ma-156	117	36	0	0	NUM
ma-156	117	37	,	,	PUNCT
ma-156	117	38	3	3	NUM
ma-156	117	39	)	)	PUNCT
ma-156	117	40	}	}	PUNCT
ma-156	117	41	b	b	X
ma-156	117	42	=	=	PRON
ma-156	117	43	{	{	PUNCT
ma-156	117	44	(	(	PUNCT
ma-156	117	45	1	1	NUM
ma-156	117	46	,	,	PUNCT
ma-156	117	47	0	0	NUM
ma-156	117	48	)	)	PUNCT
ma-156	117	49	,	,	PUNCT
ma-156	117	50	(	(	PUNCT
ma-156	117	51	2	2	NUM
ma-156	117	52	,	,	PUNCT
ma-156	117	53	1	1	NUM
ma-156	117	54	)	)	PUNCT
ma-156	117	55	,	,	PUNCT
ma-156	117	56	(	(	PUNCT
ma-156	117	57	2	2	NUM
ma-156	117	58	,	,	PUNCT
ma-156	117	59	2	2	NUM
ma-156	117	60	)	)	PUNCT
ma-156	117	61	,	,	PUNCT
ma-156	117	62	(	(	PUNCT
ma-156	117	63	1	1	X
ma-156	117	64	,	,	PUNCT
ma-156	117	65	3)}define	3)}define	NUM
ma-156	117	66	a	a	DET
ma-156	117	67	mapping	mapping	NOUN
ma-156	117	68	t	t	NOUN
ma-156	117	69	:	:	PUNCT
ma-156	117	70	a	a	DET
ma-156	117	71	→	→	SYM
ma-156	117	72	b	b	X
ma-156	117	73	such	such	ADJ
ma-156	117	74	that	that	DET
ma-156	117	75	t	t	PROPN
ma-156	117	76	(	(	PUNCT
ma-156	117	77	0	0	NUM
ma-156	117	78	,	,	PUNCT
ma-156	117	79	0	0	NUM
ma-156	117	80	)	)	PUNCT
ma-156	117	81	=	=	NOUN
ma-156	117	82	(	(	PUNCT
ma-156	117	83	1	1	NUM
ma-156	117	84	,	,	PUNCT
ma-156	117	85	0	0	NUM
ma-156	117	86	)	)	PUNCT
ma-156	117	87	,	,	PUNCT
ma-156	117	88	t	t	PROPN
ma-156	117	89	(	(	PUNCT
ma-156	117	90	0	0	NUM
ma-156	117	91	,	,	PUNCT
ma-156	117	92	1	1	NUM
ma-156	117	93	)	)	PUNCT
ma-156	117	94	=	=	NOUN
ma-156	117	95	(	(	PUNCT
ma-156	117	96	2	2	NUM
ma-156	117	97	,	,	PUNCT
ma-156	117	98	2	2	NUM
ma-156	117	99	)	)	PUNCT
ma-156	117	100	,	,	PUNCT
ma-156	117	101	t	t	PROPN
ma-156	117	102	(	(	PUNCT
ma-156	117	103	0	0	NUM
ma-156	117	104	,	,	PUNCT
ma-156	117	105	2	2	NUM
ma-156	117	106	)	)	PUNCT
ma-156	117	107	=	=	NOUN
ma-156	117	108	(	(	PUNCT
ma-156	117	109	2	2	NUM
ma-156	117	110	,	,	PUNCT
ma-156	117	111	1	1	NUM
ma-156	117	112	)	)	PUNCT
ma-156	117	113	and	and	CCONJ
ma-156	117	114	t	t	PROPN
ma-156	117	115	(	(	PUNCT
ma-156	117	116	0	0	NUM
ma-156	117	117	,	,	PUNCT
ma-156	117	118	3	3	NUM
ma-156	117	119	)	)	PUNCT
ma-156	117	120	=	=	SYM
ma-156	117	121	(	(	PUNCT
ma-156	117	122	1	1	NUM
ma-156	117	123	,	,	PUNCT
ma-156	117	124	3).also	3).also	PRON
ma-156	117	125	define	define	VERB
ma-156	117	126	a	a	DET
ma-156	117	127	mapping	mapping	NOUN
ma-156	117	128	α	α	NOUN
ma-156	117	129	:	:	PUNCT
ma-156	118	1	a×	a×	X
ma-156	118	2	a→	a→	VERB
ma-156	118	3	[	[	X
ma-156	118	4	0,∞	0,∞	NOUN
ma-156	118	5	)	)	PUNCT
ma-156	118	6	such	such	ADJ
ma-156	118	7	that	that	DET
ma-156	118	8	α(x	α(x	NOUN
ma-156	118	9	,	,	PUNCT
ma-156	118	10	y	y	PROPN
ma-156	118	11	)	)	PUNCT
ma-156	118	12	=	=	SYM
ma-156	118	13			NUM
ma-156	118	14	1	1	NUM
ma-156	118	15	,	,	PUNCT
ma-156	118	16	if	if	SCONJ
ma-156	118	17	x	x	ADP
ma-156	118	18	=	=	VERB
ma-156	118	19	y	y	PROPN
ma-156	118	20	∈	∈	PROPN
ma-156	118	21	{	{	PUNCT
ma-156	118	22	(	(	PUNCT
ma-156	118	23	0	0	NUM
ma-156	118	24	,	,	PUNCT
ma-156	118	25	0	0	NUM
ma-156	118	26	)	)	PUNCT
ma-156	118	27	,	,	PUNCT
ma-156	118	28	(	(	PUNCT
ma-156	118	29	0	0	NUM
ma-156	118	30	,	,	PUNCT
ma-156	118	31	3	3	NUM
ma-156	118	32	)	)	PUNCT
ma-156	118	33	}	}	PUNCT
ma-156	118	34	0	0	NUM
ma-156	118	35	elsewhere.for	elsewhere.for	ADP
ma-156	118	36	all	all	DET
ma-156	118	37	x	x	NOUN
ma-156	118	38	,	,	PUNCT
ma-156	118	39	y	y	PROPN
ma-156	118	40	∈	∈	PROPN
ma-156	118	41	a.	a.	NOUN
ma-156	118	42	one	one	NOUN
ma-156	118	43	can	can	AUX
ma-156	118	44	see	see	VERB
ma-156	118	45	that	that	SCONJ
ma-156	118	46	d(a	d(a	PROPN
ma-156	118	47	,	,	PUNCT
ma-156	118	48	b	b	NOUN
ma-156	118	49	)	)	PUNCT
ma-156	118	50	=	=	SYM
ma-156	118	51	1	1	X
ma-156	118	52	.	.	PUNCT
ma-156	118	53	let	let	VERB
ma-156	118	54	u	u	NOUN
ma-156	118	55	,	,	PUNCT
ma-156	118	56	v	v	INTJ
ma-156	118	57	,	,	PUNCT
ma-156	118	58	x	x	SYM
ma-156	118	59	∈	∈	NOUN
ma-156	118	60	a.	a.	NOUN
ma-156	118	61	one	one	NOUN
ma-156	118	62	can	can	AUX
ma-156	118	63	check	check	VERB
ma-156	118	64	that	that	NUM
ma-156	118	65	α(x	α(x	NOUN
ma-156	118	66	,	,	PUNCT
ma-156	118	67	u	u	NOUN
ma-156	118	68	)	)	PUNCT
ma-156	118	69	≥	≥	PROPN
ma-156	118	70	1	1	NUM
ma-156	118	71	d(u	d(u	PROPN
ma-156	118	72	,	,	PUNCT
ma-156	118	73	t	t	NOUN
ma-156	118	74	x	x	NOUN
ma-156	118	75	)	)	PUNCT
ma-156	118	76	=	=	SYM
ma-156	118	77	1	1	NUM
ma-156	118	78	d(v	d(v	PROPN
ma-156	118	79	,	,	PUNCT
ma-156	118	80	tu	tu	PROPN
ma-156	118	81	)	)	PUNCT
ma-156	118	82	=	=	SYM
ma-156	119	1	1	1	NUM
ma-156	119	2	=	=	NOUN
ma-156	119	3	⇒	⇒	NOUN
ma-156	119	4	x	x	PUNCT
ma-156	120	1	=	=	SYM
ma-156	120	2	u	u	NOUN
ma-156	120	3	=	=	SYM
ma-156	120	4	v	v	PROPN
ma-156	120	5	∈	∈	PROPN
ma-156	120	6	{	{	PUNCT
ma-156	120	7	(	(	PUNCT
ma-156	120	8	0	0	NUM
ma-156	120	9	,	,	PUNCT
ma-156	120	10	0	0	NUM
ma-156	120	11	)	)	PUNCT
ma-156	120	12	,	,	PUNCT
ma-156	120	13	(	(	PUNCT
ma-156	120	14	0	0	NUM
ma-156	120	15	,	,	PUNCT
ma-156	120	16	3	3	NUM
ma-156	120	17	)	)	PUNCT
ma-156	120	18	}	}	PUNCT
ma-156	121	1	=	=	VERB
ma-156	121	2	⇒	⇒	NOUN
ma-156	121	3	α(u	α(u	NOUN
ma-156	121	4	,	,	PUNCT
ma-156	121	5	v	v	NOUN
ma-156	121	6	)	)	PUNCT
ma-156	121	7	=	=	SYM
ma-156	121	8	1	1	X
ma-156	121	9	.	.	PUNCT
ma-156	122	1	hence	hence	ADV
ma-156	122	2	,	,	PUNCT
ma-156	122	3	t	t	PROPN
ma-156	122	4	is	be	AUX
ma-156	122	5	α	α	DET
ma-156	122	6	-	-	ADJ
ma-156	122	7	orbital	orbital	ADJ
ma-156	122	8	proximal	proximal	ADJ
ma-156	122	9	admissible	admissible	NOUN
ma-156	122	10	.	.	PUNCT
ma-156	123	1	let	let	VERB
ma-156	123	2	u	u	NOUN
ma-156	123	3	,	,	PUNCT
ma-156	123	4	x	x	PRON
ma-156	123	5	,	,	PUNCT
ma-156	123	6	y	y	PROPN
ma-156	123	7	∈	∈	PROPN
ma-156	123	8	a.	a.	NOUN
ma-156	123	9	one	one	NOUN
ma-156	123	10	can	can	AUX
ma-156	123	11	check	check	VERB
ma-156	123	12	that	that	DET
ma-156	123	13			PROPN
ma-156	123	14	α(x	α(x	PROPN
ma-156	123	15	,	,	PUNCT
ma-156	123	16	u	u	NOUN
ma-156	123	17	)	)	PUNCT
ma-156	123	18	≥	≥	NOUN
ma-156	123	19	1	1	NUM
ma-156	123	20	α(y	α(y	NOUN
ma-156	123	21	,	,	PUNCT
ma-156	123	22	u	u	NOUN
ma-156	123	23	)	)	PUNCT
ma-156	123	24	≥	≥	PROPN
ma-156	123	25	1	1	NUM
ma-156	123	26	d(u	d(u	PROPN
ma-156	123	27	,	,	PUNCT
ma-156	123	28	t	t	PROPN
ma-156	123	29	y	y	PROPN
ma-156	123	30	)	)	PUNCT
ma-156	123	31	=	=	SYM
ma-156	124	1	1	1	NUM
ma-156	124	2	=	=	NOUN
ma-156	124	3	⇒	⇒	NOUN
ma-156	124	4	x	x	PUNCT
ma-156	124	5	=	=	SYM
ma-156	124	6	y	y	PROPN
ma-156	124	7	=	=	PUNCT
ma-156	124	8	u	u	PROPN
ma-156	124	9	∈	∈	PROPN
ma-156	124	10	{	{	PUNCT
ma-156	124	11	(	(	PUNCT
ma-156	124	12	0	0	NUM
ma-156	124	13	,	,	PUNCT
ma-156	124	14	0	0	NUM
ma-156	124	15	)	)	PUNCT
ma-156	124	16	,	,	PUNCT
ma-156	124	17	(	(	PUNCT
ma-156	124	18	0	0	NUM
ma-156	124	19	,	,	PUNCT
ma-156	124	20	3	3	NUM
ma-156	124	21	)	)	PUNCT
ma-156	124	22	}	}	PUNCT
ma-156	125	1	=	=	VERB
ma-156	125	2	⇒	⇒	NOUN
ma-156	125	3	α(x	α(x	PROPN
ma-156	125	4	,	,	PUNCT
ma-156	125	5	u	u	NOUN
ma-156	125	6	)	)	PUNCT
ma-156	125	7	=	=	SYM
ma-156	125	8	1	1	X
ma-156	125	9	.	.	PUNCT
ma-156	125	10	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	125	11	eur	eur	PROPN
ma-156	125	12	.	.	PUNCT
ma-156	126	1	j.	j.	PROPN
ma-156	126	2	math	math	PROPN
ma-156	126	3	.	.	PUNCT
ma-156	127	1	anal	anal	PROPN
ma-156	127	2	.	.	PUNCT
ma-156	128	1	10.28924	10.28924	NUM
ma-156	128	2	/	/	SYM
ma-156	128	3	ada	ada	PROPN
ma-156	128	4	/	/	SYM
ma-156	128	5	ma.3.16	ma.3.16	PROPN
ma-156	128	6	6thus	6thus	NUM
ma-156	128	7	,	,	PUNCT
ma-156	128	8	t	t	PROPN
ma-156	128	9	is	be	AUX
ma-156	128	10	also	also	ADV
ma-156	128	11	triangular	triangular	ADJ
ma-156	128	12	α	α	DET
ma-156	128	13	-	-	ADJ
ma-156	128	14	orbital	orbital	ADJ
ma-156	128	15	proximal	proximal	ADJ
ma-156	128	16	admissible	admissible	NOUN
ma-156	128	17	.	.	PUNCT
ma-156	129	1	we	we	PRON
ma-156	129	2	introduce	introduce	VERB
ma-156	129	3	the	the	DET
ma-156	129	4	following	follow	VERB
ma-156	129	5	new	new	ADJ
ma-156	129	6	classes	class	NOUN
ma-156	129	7	of	of	ADP
ma-156	129	8	non	non	ADJ
ma-156	129	9	-	-	ADJ
ma-156	129	10	self	self	NOUN
ma-156	129	11	mappings	mapping	NOUN
ma-156	129	12	.	.	PUNCT
ma-156	130	1	definition	definition	NOUN
ma-156	130	2	2.9	2.9	NUM
ma-156	130	3	let	let	VERB
ma-156	130	4	a	a	PRON
ma-156	130	5	and	and	CCONJ
ma-156	130	6	b	b	NOUN
ma-156	130	7	be	be	AUX
ma-156	130	8	two	two	NUM
ma-156	130	9	nonempty	nonempty	ADJ
ma-156	130	10	subsets	subset	NOUN
ma-156	130	11	of	of	ADP
ma-156	130	12	a	a	DET
ma-156	130	13	metric	metric	ADJ
ma-156	130	14	space	space	NOUN
ma-156	130	15	(	(	PUNCT
ma-156	130	16	x	x	X
ma-156	130	17	,	,	PUNCT
ma-156	130	18	d	d	NOUN
ma-156	130	19	)	)	PUNCT
ma-156	130	20	and	and	CCONJ
ma-156	130	21	α	α	PRON
ma-156	130	22	:	:	PUNCT
ma-156	130	23	a×a→	a×a→	PROPN
ma-156	130	24	r+be	r+be	PROPN
ma-156	130	25	a	a	DET
ma-156	130	26	function	function	NOUN
ma-156	130	27	.	.	PUNCT
ma-156	131	1	a	a	DET
ma-156	131	2	non	non	ADJ
ma-156	131	3	-	-	ADJ
ma-156	131	4	self	self	ADJ
ma-156	131	5	mapping	mapping	NOUN
ma-156	131	6	t	t	NOUN
ma-156	131	7	:	:	PUNCT
ma-156	131	8	a	a	DET
ma-156	131	9	→	→	SYM
ma-156	131	10	b	b	PROPN
ma-156	131	11	is	be	AUX
ma-156	131	12	called	call	VERB
ma-156	131	13	a	a	DET
ma-156	131	14	generalized	generalized	ADJ
ma-156	131	15	α	α	NOUN
ma-156	131	16	-	-	PUNCT
ma-156	131	17	φ	φ	VERB
ma-156	131	18	-	-	PUNCT
ma-156	131	19	geraghty	geraghty	VERB
ma-156	131	20	proximalquasi	proximalquasi	NOUN
ma-156	131	21	-	-	PUNCT
ma-156	131	22	contraction	contraction	NOUN
ma-156	131	23	type	type	NOUN
ma-156	131	24	mapping	mapping	NOUN
ma-156	131	25	if	if	SCONJ
ma-156	131	26	there	there	PRON
ma-156	131	27	exists	exist	VERB
ma-156	131	28	β	β	X
ma-156	131	29	∈	∈	PROPN
ma-156	131	30	f	f	PROPN
ma-156	131	31	such	such	ADJ
ma-156	131	32	that	that	PRON
ma-156	131	33	for	for	ADP
ma-156	131	34	all	all	DET
ma-156	131	35	x	x	NOUN
ma-156	131	36	,	,	PUNCT
ma-156	131	37	y	y	PROPN
ma-156	131	38	,	,	PUNCT
ma-156	131	39	u	u	PROPN
ma-156	131	40	,	,	PUNCT
ma-156	131	41	v	v	ADP
ma-156	131	42	∈	∈	PROPN
ma-156	131	43	a	a	PRON
ma-156	131	44	,	,	PUNCT
ma-156	131	45	{	{	PUNCT
ma-156	131	46	d(u	d(u	PROPN
ma-156	131	47	,	,	PUNCT
ma-156	131	48	t	t	NOUN
ma-156	131	49	x	x	NOUN
ma-156	131	50	)	)	PUNCT
ma-156	131	51	=	=	SYM
ma-156	132	1	d(a	d(a	PROPN
ma-156	132	2	,	,	PUNCT
ma-156	132	3	b	b	NOUN
ma-156	132	4	)	)	PUNCT
ma-156	132	5	d(v	d(v	PROPN
ma-156	132	6	,	,	PUNCT
ma-156	132	7	t	t	PROPN
ma-156	132	8	y	y	PROPN
ma-156	132	9	)	)	PUNCT
ma-156	133	1	=	=	SYM
ma-156	134	1	d(a	d(a	PROPN
ma-156	134	2	,	,	PUNCT
ma-156	134	3	b	b	NOUN
ma-156	134	4	)	)	PUNCT
ma-156	134	5	=	=	NOUN
ma-156	134	6	⇒	⇒	NOUN
ma-156	134	7	α(x	α(x	PROPN
ma-156	134	8	,	,	PUNCT
ma-156	134	9	y)φ(d(u	y)φ(d(u	SYM
ma-156	134	10	,	,	PUNCT
ma-156	134	11	v	v	NOUN
ma-156	134	12	)	)	PUNCT
ma-156	134	13	)	)	PUNCT
ma-156	134	14	≤	≤	NUM
ma-156	134	15	β(φ(mt	β(φ(mt	PUNCT
ma-156	134	16	(	(	PUNCT
ma-156	134	17	x	x	NOUN
ma-156	134	18	,	,	PUNCT
ma-156	134	19	y)))φ(mt	y)))φ(mt	ADJ
ma-156	134	20	(	(	PUNCT
ma-156	134	21	x	x	X
ma-156	134	22	,	,	PUNCT
ma-156	134	23	y	y	PROPN
ma-156	134	24	)	)	PUNCT
ma-156	134	25	)	)	PUNCT
ma-156	134	26	,	,	PUNCT
ma-156	134	27	(	(	PUNCT
ma-156	134	28	3	3	X
ma-156	134	29	)	)	PUNCT
ma-156	134	30	where	where	SCONJ
ma-156	134	31	mt	mt	PROPN
ma-156	134	32	(	(	PUNCT
ma-156	134	33	x	x	PROPN
ma-156	134	34	,	,	PUNCT
ma-156	134	35	y	y	NOUN
ma-156	134	36	)	)	PUNCT
ma-156	134	37	=	=	PUNCT
ma-156	135	1	max{d(x	max{d(x	PROPN
ma-156	135	2	,	,	PUNCT
ma-156	135	3	y	y	NOUN
ma-156	135	4	)	)	PUNCT
ma-156	135	5	,	,	PUNCT
ma-156	135	6	d(x	d(x	PROPN
ma-156	135	7	,	,	PUNCT
ma-156	135	8	u	u	NOUN
ma-156	135	9	)	)	PUNCT
ma-156	135	10	,	,	PUNCT
ma-156	135	11	d(y	d(y	NOUN
ma-156	135	12	,	,	PUNCT
ma-156	135	13	v	v	NOUN
ma-156	135	14	)	)	PUNCT
ma-156	135	15	,	,	PUNCT
ma-156	135	16	d(x	d(x	PROPN
ma-156	135	17	,	,	PUNCT
ma-156	135	18	v	v	NOUN
ma-156	135	19	)	)	PUNCT
ma-156	135	20	,	,	PUNCT
ma-156	135	21	d(y	d(y	NOUN
ma-156	135	22	,	,	PUNCT
ma-156	135	23	u	u	NOUN
ma-156	135	24	)	)	PUNCT
ma-156	135	25	}	}	PUNCT
ma-156	135	26	,	,	PUNCT
ma-156	135	27	for	for	ADP
ma-156	135	28	all	all	DET
ma-156	135	29	x	x	NOUN
ma-156	135	30	,	,	PUNCT
ma-156	135	31	y	y	PROPN
ma-156	135	32	,	,	PUNCT
ma-156	135	33	u	u	PROPN
ma-156	135	34	,	,	PUNCT
ma-156	135	35	v	v	ADP
ma-156	135	36	∈	∈	PROPN
ma-156	135	37	a	a	PRON
ma-156	135	38	and	and	CCONJ
ma-156	135	39	φ	φ	PROPN
ma-156	135	40	∈	∈	PROPN
ma-156	135	41	φ	φ	NOUN
ma-156	135	42	.	.	PUNCT
ma-156	136	1	if	if	SCONJ
ma-156	136	2	φ(t	φ(t	PROPN
ma-156	136	3	)	)	PUNCT
ma-156	137	1	=	=	SYM
ma-156	137	2	t	t	PROPN
ma-156	137	3	,	,	PUNCT
ma-156	137	4	then	then	ADV
ma-156	137	5	definition	definition	NOUN
ma-156	137	6	2.9	2.9	NUM
ma-156	137	7	reduces	reduce	VERB
ma-156	137	8	to	to	ADP
ma-156	137	9	the	the	DET
ma-156	137	10	following	following	NOUN
ma-156	137	11	.	.	PUNCT
ma-156	138	1	definition	definition	NOUN
ma-156	138	2	2.10	2.10	NUM
ma-156	138	3	let	let	VERB
ma-156	138	4	a	a	PRON
ma-156	138	5	and	and	CCONJ
ma-156	138	6	b	b	NOUN
ma-156	138	7	be	be	AUX
ma-156	138	8	two	two	NUM
ma-156	138	9	nonempty	nonempty	ADJ
ma-156	138	10	subsets	subset	NOUN
ma-156	138	11	of	of	ADP
ma-156	138	12	a	a	DET
ma-156	138	13	metric	metric	ADJ
ma-156	138	14	space	space	NOUN
ma-156	138	15	(	(	PUNCT
ma-156	138	16	x	x	X
ma-156	138	17	,	,	PUNCT
ma-156	138	18	d	d	NOUN
ma-156	138	19	)	)	PUNCT
ma-156	138	20	and	and	CCONJ
ma-156	138	21	α	α	PRON
ma-156	138	22	:	:	PUNCT
ma-156	138	23	a×a→	a×a→	PROPN
ma-156	138	24	r+be	r+be	PROPN
ma-156	138	25	a	a	DET
ma-156	138	26	function	function	NOUN
ma-156	138	27	.	.	PUNCT
ma-156	139	1	a	a	DET
ma-156	139	2	non	non	ADJ
ma-156	139	3	-	-	ADJ
ma-156	139	4	self	self	ADJ
ma-156	139	5	mapping	mapping	NOUN
ma-156	139	6	t	t	NOUN
ma-156	139	7	:	:	PUNCT
ma-156	139	8	a→	a→	PROPN
ma-156	139	9	b	b	NOUN
ma-156	139	10	is	be	AUX
ma-156	139	11	called	call	VERB
ma-156	139	12	an	an	DET
ma-156	139	13	α	α	NOUN
ma-156	139	14	-	-	PUNCT
ma-156	139	15	geraghty	geraghty	VERB
ma-156	139	16	proximal	proximal	ADJ
ma-156	139	17	quasi	quasi	ADJ
ma-156	139	18	-	-	ADJ
ma-156	139	19	contractiontype	contractiontype	ADJ
ma-156	139	20	mapping	mapping	NOUN
ma-156	139	21	if	if	SCONJ
ma-156	139	22	there	there	PRON
ma-156	139	23	exists	exist	VERB
ma-156	139	24	β	β	X
ma-156	139	25	∈	∈	PROPN
ma-156	139	26	f	f	PROPN
ma-156	139	27	such	such	ADJ
ma-156	139	28	that	that	PRON
ma-156	139	29	for	for	ADP
ma-156	139	30	all	all	DET
ma-156	139	31	x	x	NOUN
ma-156	139	32	,	,	PUNCT
ma-156	139	33	y	y	PROPN
ma-156	139	34	,	,	PUNCT
ma-156	139	35	u	u	PROPN
ma-156	139	36	,	,	PUNCT
ma-156	139	37	v	v	ADP
ma-156	139	38	∈	∈	PROPN
ma-156	139	39	a	a	PRON
ma-156	139	40	,	,	PUNCT
ma-156	139	41	{	{	PUNCT
ma-156	139	42	d(u	d(u	PROPN
ma-156	139	43	,	,	PUNCT
ma-156	139	44	t	t	NOUN
ma-156	139	45	x	x	NOUN
ma-156	139	46	)	)	PUNCT
ma-156	140	1	=	=	SYM
ma-156	141	1	d(a	d(a	PROPN
ma-156	141	2	,	,	PUNCT
ma-156	141	3	b	b	NOUN
ma-156	141	4	)	)	PUNCT
ma-156	141	5	d(v	d(v	PROPN
ma-156	141	6	,	,	PUNCT
ma-156	141	7	t	t	PROPN
ma-156	141	8	y	y	PROPN
ma-156	141	9	)	)	PUNCT
ma-156	142	1	=	=	SYM
ma-156	143	1	d(a	d(a	PROPN
ma-156	143	2	,	,	PUNCT
ma-156	143	3	b	b	NOUN
ma-156	143	4	)	)	PUNCT
ma-156	143	5	=	=	NOUN
ma-156	143	6	⇒	⇒	NOUN
ma-156	143	7	α(x	α(x	PROPN
ma-156	143	8	,	,	PUNCT
ma-156	143	9	y)d(u	y)d(u	PROPN
ma-156	143	10	,	,	PUNCT
ma-156	143	11	v	v	NOUN
ma-156	143	12	)	)	PUNCT
ma-156	143	13	≤	≤	NUM
ma-156	143	14	β(mt	β(mt	ADJ
ma-156	143	15	(	(	PUNCT
ma-156	143	16	x	x	X
ma-156	143	17	,	,	PUNCT
ma-156	143	18	y))(mt	y))(mt	PUNCT
ma-156	143	19	(	(	PUNCT
ma-156	143	20	x	x	X
ma-156	143	21	,	,	PUNCT
ma-156	143	22	y	y	PROPN
ma-156	143	23	)	)	PUNCT
ma-156	143	24	)	)	PUNCT
ma-156	143	25	,	,	PUNCT
ma-156	143	26	(	(	PUNCT
ma-156	143	27	4	4	X
ma-156	143	28	)	)	PUNCT
ma-156	143	29	for	for	ADP
ma-156	143	30	all	all	DET
ma-156	143	31	x	x	PROPN
ma-156	143	32	,	,	PUNCT
ma-156	143	33	y	y	PROPN
ma-156	143	34	,	,	PUNCT
ma-156	143	35	u	u	PROPN
ma-156	143	36	,	,	PUNCT
ma-156	143	37	v	v	NOUN
ma-156	143	38	∈	∈	NOUN
ma-156	143	39	a.	a.	NOUN
ma-156	143	40	where	where	SCONJ
ma-156	143	41	mt	mt	PROPN
ma-156	143	42	(	(	PUNCT
ma-156	143	43	x	x	PROPN
ma-156	143	44	,	,	PUNCT
ma-156	143	45	y	y	NOUN
ma-156	143	46	)	)	PUNCT
ma-156	143	47	=	=	PUNCT
ma-156	144	1	max{d(x	max{d(x	PROPN
ma-156	144	2	,	,	PUNCT
ma-156	144	3	y	y	NOUN
ma-156	144	4	)	)	PUNCT
ma-156	144	5	,	,	PUNCT
ma-156	144	6	d(x	d(x	PROPN
ma-156	144	7	,	,	PUNCT
ma-156	144	8	u	u	NOUN
ma-156	144	9	)	)	PUNCT
ma-156	144	10	,	,	PUNCT
ma-156	144	11	d(y	d(y	NOUN
ma-156	144	12	,	,	PUNCT
ma-156	144	13	v	v	NOUN
ma-156	144	14	)	)	PUNCT
ma-156	144	15	,	,	PUNCT
ma-156	144	16	d(x	d(x	PROPN
ma-156	144	17	,	,	PUNCT
ma-156	144	18	v	v	NOUN
ma-156	144	19	)	)	PUNCT
ma-156	144	20	,	,	PUNCT
ma-156	144	21	d(y	d(y	PROPN
ma-156	144	22	,	,	PUNCT
ma-156	144	23	u	u	NOUN
ma-156	144	24	)	)	PUNCT
ma-156	144	25	}	}	PUNCT
ma-156	144	26	for	for	ADP
ma-156	144	27	all	all	DET
ma-156	144	28	x	x	NOUN
ma-156	144	29	,	,	PUNCT
ma-156	144	30	y	y	PROPN
ma-156	144	31	,	,	PUNCT
ma-156	144	32	u	u	PROPN
ma-156	144	33	,	,	PUNCT
ma-156	144	34	v	v	NOUN
ma-156	144	35	∈	∈	PROPN
ma-156	144	36	a.	a.	NOUN
ma-156	144	37	3	3	NUM
ma-156	144	38	.	.	X
ma-156	144	39	main	main	ADJ
ma-156	144	40	results	result	NOUN
ma-156	144	41	now	now	ADV
ma-156	144	42	we	we	PRON
ma-156	144	43	state	state	VERB
ma-156	144	44	and	and	CCONJ
ma-156	144	45	prove	prove	VERB
ma-156	144	46	our	our	PRON
ma-156	144	47	main	main	ADJ
ma-156	144	48	results	result	NOUN
ma-156	144	49	.	.	PUNCT
ma-156	145	1	theorem	theorem	VERB
ma-156	145	2	3.1	3.1	NUM
ma-156	145	3	.	.	PUNCT
ma-156	146	1	let	let	VERB
ma-156	146	2	a	a	PRON
ma-156	146	3	and	and	CCONJ
ma-156	146	4	b	b	NOUN
ma-156	146	5	be	be	AUX
ma-156	146	6	two	two	NUM
ma-156	146	7	nonempty	nonempty	ADJ
ma-156	146	8	subsets	subset	NOUN
ma-156	146	9	of	of	ADP
ma-156	146	10	a	a	DET
ma-156	146	11	metric	metric	ADJ
ma-156	146	12	space	space	NOUN
ma-156	146	13	such	such	ADJ
ma-156	146	14	that	that	SCONJ
ma-156	146	15	a0	a0	PROPN
ma-156	146	16	isproximal	isproximal	PROPN
ma-156	146	17	t	t	PROPN
ma-156	146	18	-orbitally	-orbitally	PROPN
ma-156	146	19	complete	complete	ADJ
ma-156	146	20	,	,	PUNCT
ma-156	146	21	where	where	SCONJ
ma-156	146	22	t	t	NOUN
ma-156	146	23	:	:	PUNCT
ma-156	146	24	a	a	DET
ma-156	146	25	→	→	SYM
ma-156	146	26	b	b	PROPN
ma-156	146	27	is	be	AUX
ma-156	146	28	a	a	DET
ma-156	146	29	non	non	ADJ
ma-156	146	30	-	-	ADJ
ma-156	146	31	self	self	ADJ
ma-156	146	32	mapping	mapping	NOUN
ma-156	146	33	,	,	PUNCT
ma-156	146	34	α	α	X
ma-156	146	35	:	:	PUNCT
ma-156	146	36	a	a	DET
ma-156	146	37	×	×	NOUN
ma-156	146	38	a	a	PRON
ma-156	146	39	→	→	PUNCT
ma-156	146	40	r+	r+	NOUN
ma-156	146	41	is	be	AUX
ma-156	146	42	afunction	afunction	NOUN
ma-156	146	43	and	and	CCONJ
ma-156	146	44	the	the	DET
ma-156	146	45	following	follow	VERB
ma-156	146	46	conditions	condition	NOUN
ma-156	146	47	are	be	AUX
ma-156	146	48	satisfied:(i	satisfied:(i	NOUN
ma-156	146	49	)	)	PUNCT
ma-156	147	1	t	t	PROPN
ma-156	147	2	is	be	AUX
ma-156	147	3	a	a	DET
ma-156	147	4	generalized	generalized	ADJ
ma-156	147	5	α	α	NOUN
ma-156	147	6	-	-	PUNCT
ma-156	147	7	φ	φ	VERB
ma-156	147	8	-	-	PUNCT
ma-156	147	9	geraghty	geraghty	VERB
ma-156	147	10	proximal	proximal	ADJ
ma-156	147	11	quasi	quasi	NOUN
ma-156	147	12	-	-	NOUN
ma-156	147	13	contraction	contraction	NOUN
ma-156	147	14	type	type	NOUN
ma-156	147	15	mapping;(ii	mapping;(ii	PROPN
ma-156	147	16	)	)	PUNCT
ma-156	147	17	t	t	PROPN
ma-156	147	18	(	(	PUNCT
ma-156	147	19	a0	a0	PROPN
ma-156	147	20	)	)	PUNCT
ma-156	147	21	⊆	⊆	NUM
ma-156	147	22	b0	b0	NOUN
ma-156	147	23	and	and	CCONJ
ma-156	147	24	t	t	NOUN
ma-156	147	25	is	be	AUX
ma-156	147	26	a	a	DET
ma-156	147	27	triangular	triangular	NOUN
ma-156	147	28	α	α	DET
ma-156	147	29	-	-	ADJ
ma-156	147	30	orbital	orbital	ADJ
ma-156	147	31	proximal	proximal	ADJ
ma-156	147	32	admissible	admissible	ADJ
ma-156	147	33	mapping;(iii	mapping;(iii	NOUN
ma-156	147	34	)	)	PUNCT
ma-156	147	35	there	there	PRON
ma-156	147	36	exists	exist	VERB
ma-156	147	37	x0	x0	PROPN
ma-156	147	38	,	,	PUNCT
ma-156	147	39	x1	x1	PROPN
ma-156	147	40	∈	∈	PROPN
ma-156	147	41	a0	a0	NOUN
ma-156	147	42	such	such	ADJ
ma-156	147	43	that	that	DET
ma-156	147	44	d(x1	d(x1	NOUN
ma-156	147	45	,	,	PUNCT
ma-156	147	46	t	t	NOUN
ma-156	147	47	x0	x0	NUM
ma-156	147	48	)	)	PUNCT
ma-156	148	1	=	=	SYM
ma-156	148	2	d(a	d(a	PROPN
ma-156	148	3	,	,	PUNCT
ma-156	148	4	b	b	NOUN
ma-156	148	5	)	)	PUNCT
ma-156	148	6	and	and	CCONJ
ma-156	148	7	α(x0	α(x0	ADJ
ma-156	148	8	,	,	PUNCT
ma-156	148	9	x1	x1	PROPN
ma-156	148	10	)	)	PUNCT
ma-156	148	11	≥	≥	NOUN
ma-156	148	12	1.then	1.then	NUM
ma-156	148	13	there	there	PRON
ma-156	148	14	exists	exist	VERB
ma-156	148	15	an	an	DET
ma-156	148	16	element	element	NOUN
ma-156	148	17	x∗	x∗	PROPN
ma-156	148	18	∈	∈	PROPN
ma-156	148	19	a0	a0	NOUN
ma-156	148	20	such	such	ADJ
ma-156	148	21	that	that	DET
ma-156	148	22	d(x∗	d(x∗	NOUN
ma-156	148	23	,	,	PUNCT
ma-156	148	24	t	t	PROPN
ma-156	148	25	x∗	x∗	X
ma-156	148	26	)	)	PUNCT
ma-156	149	1	=	=	SYM
ma-156	149	2	d(a	d(a	PROPN
ma-156	149	3	,	,	PUNCT
ma-156	149	4	b	b	NOUN
ma-156	149	5	)	)	PUNCT
ma-156	149	6	.	.	PUNCT
ma-156	150	1	moreover	moreover	ADV
ma-156	150	2	,	,	PUNCT
ma-156	150	3	if	if	SCONJ
ma-156	150	4	α(x	α(x	PROPN
ma-156	150	5	,	,	PUNCT
ma-156	150	6	y	y	PROPN
ma-156	150	7	)	)	PUNCT
ma-156	150	8	≥	≥	NOUN
ma-156	150	9	1	1	NUM
ma-156	150	10	for	for	ADP
ma-156	150	11	all	all	DET
ma-156	150	12	x	x	NOUN
ma-156	150	13	,	,	PUNCT
ma-156	150	14	y	y	PROPN
ma-156	150	15	∈	∈	PROPN
ma-156	150	16	pt	pt	X
ma-156	150	17	(	(	PUNCT
ma-156	150	18	a	a	NOUN
ma-156	150	19	)	)	PUNCT
ma-156	150	20	,	,	PUNCT
ma-156	150	21	then	then	ADV
ma-156	150	22	x∗	x∗	PROPN
ma-156	150	23	is	be	AUX
ma-156	150	24	the	the	DET
ma-156	150	25	unique	unique	ADJ
ma-156	150	26	best	good	ADJ
ma-156	150	27	proximity	proximity	NOUN
ma-156	150	28	point	point	NOUN
ma-156	150	29	of	of	ADP
ma-156	150	30	t	t	PROPN
ma-156	150	31	.	.	PUNCT
ma-156	151	1	proof.let	proof.let	X
ma-156	151	2	x0	x0	PROPN
ma-156	151	3	,	,	PUNCT
ma-156	151	4	x1	x1	PROPN
ma-156	151	5	∈	∈	PROPN
ma-156	151	6	a0	a0	NOUN
ma-156	151	7	be	be	VERB
ma-156	151	8	such	such	ADJ
ma-156	151	9	that	that	SCONJ
ma-156	151	10	d(x1	d(x1	NOUN
ma-156	151	11	,	,	PUNCT
ma-156	151	12	t	t	NOUN
ma-156	151	13	x0	x0	NUM
ma-156	151	14	)	)	PUNCT
ma-156	152	1	=	=	SYM
ma-156	152	2	d(a	d(a	PROPN
ma-156	152	3	,	,	PUNCT
ma-156	152	4	b	b	NOUN
ma-156	152	5	)	)	PUNCT
ma-156	152	6	and	and	CCONJ
ma-156	152	7	α(x0	α(x0	ADJ
ma-156	152	8	,	,	PUNCT
ma-156	152	9	x1	x1	PROPN
ma-156	152	10	)	)	PUNCT
ma-156	152	11	≥	≥	NOUN
ma-156	152	12	1	1	NUM
ma-156	152	13	.	.	PUNCT
ma-156	153	1	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	153	2	eur	eur	PROPN
ma-156	153	3	.	.	PUNCT
ma-156	154	1	j.	j.	PROPN
ma-156	154	2	math	math	PROPN
ma-156	154	3	.	.	PUNCT
ma-156	155	1	anal	anal	PROPN
ma-156	155	2	.	.	PUNCT
ma-156	156	1	10.28924	10.28924	NUM
ma-156	156	2	/	/	SYM
ma-156	156	3	ada	ada	PROPN
ma-156	156	4	/	/	SYM
ma-156	156	5	ma.3.16	ma.3.16	PROPN
ma-156	156	6	7	7	NUM
ma-156	156	7	t	t	PROPN
ma-156	156	8	(	(	PUNCT
ma-156	156	9	a0	a0	PROPN
ma-156	156	10	)	)	PUNCT
ma-156	156	11	⊆	⊆	NUM
ma-156	156	12	b0	b0	NOUN
ma-156	156	13	and	and	CCONJ
ma-156	156	14	there	there	PRON
ma-156	156	15	exists	exist	VERB
ma-156	156	16	x2	x2	PROPN
ma-156	156	17	∈	∈	PROPN
ma-156	156	18	a0	a0	NOUN
ma-156	156	19	such	such	ADJ
ma-156	156	20	that	that	DET
ma-156	156	21	d(x2	d(x2	NOUN
ma-156	156	22	,	,	PUNCT
ma-156	156	23	t	t	PROPN
ma-156	156	24	x1	x1	NUM
ma-156	156	25	)	)	PUNCT
ma-156	157	1	=	=	SYM
ma-156	157	2	d(a	d(a	PROPN
ma-156	157	3	,	,	PUNCT
ma-156	157	4	b	b	NOUN
ma-156	157	5	)	)	PUNCT
ma-156	157	6	.	.	PUNCT
ma-156	158	1	now	now	ADV
ma-156	158	2	,	,	PUNCT
ma-156	158	3	we	we	PRON
ma-156	158	4	have	have	NUM
ma-156	158	5	α(x0	α(x0	NOUN
ma-156	158	6	,	,	PUNCT
ma-156	158	7	x1	x1	PROPN
ma-156	158	8	)	)	PUNCT
ma-156	158	9	≥	≥	NOUN
ma-156	158	10	1	1	NUM
ma-156	158	11	d(x1	d(x1	NOUN
ma-156	158	12	,	,	PUNCT
ma-156	158	13	t	t	NOUN
ma-156	158	14	x0	x0	NUM
ma-156	158	15	)	)	PUNCT
ma-156	159	1	=	=	SYM
ma-156	159	2	d(a	d(a	PROPN
ma-156	159	3	,	,	PUNCT
ma-156	159	4	b	b	NOUN
ma-156	159	5	)	)	PUNCT
ma-156	159	6	,	,	PUNCT
ma-156	159	7	d(x2	d(x2	NOUN
ma-156	159	8	,	,	PUNCT
ma-156	159	9	t	t	PROPN
ma-156	159	10	x1	x1	NUM
ma-156	159	11	)	)	PUNCT
ma-156	160	1	=	=	SYM
ma-156	160	2	d(a	d(a	PROPN
ma-156	160	3	,	,	PUNCT
ma-156	160	4	b	b	NOUN
ma-156	160	5	)	)	PUNCT
ma-156	160	6	.	.	PUNCT
ma-156	161	1	since	since	SCONJ
ma-156	161	2	t	t	PROPN
ma-156	161	3	is	be	AUX
ma-156	161	4	α	α	PRON
ma-156	161	5	-	-	ADJ
ma-156	161	6	orbital	orbital	ADJ
ma-156	161	7	proximal	proximal	ADJ
ma-156	161	8	admissible	admissible	NOUN
ma-156	161	9	,	,	PUNCT
ma-156	161	10	α(x1	α(x1	ADJ
ma-156	161	11	,	,	PUNCT
ma-156	161	12	x2	x2	NUM
ma-156	161	13	)	)	PUNCT
ma-156	161	14	≥	≥	NOUN
ma-156	161	15	1	1	NUM
ma-156	161	16	.	.	PUNCT
ma-156	162	1	thus	thus	ADV
ma-156	162	2	,	,	PUNCT
ma-156	162	3	we	we	PRON
ma-156	162	4	have	have	VERB
ma-156	162	5	d(x2	d(x2	NOUN
ma-156	162	6	,	,	PUNCT
ma-156	162	7	t	t	PROPN
ma-156	162	8	x1	x1	NUM
ma-156	162	9	)	)	PUNCT
ma-156	163	1	=	=	SYM
ma-156	163	2	d(a	d(a	PROPN
ma-156	163	3	,	,	PUNCT
ma-156	163	4	b	b	NOUN
ma-156	163	5	)	)	PUNCT
ma-156	163	6	and	and	CCONJ
ma-156	163	7	α(x1	α(x1	ADJ
ma-156	163	8	,	,	PUNCT
ma-156	163	9	x2	x2	PROPN
ma-156	163	10	)	)	PUNCT
ma-156	163	11	≥	≥	NOUN
ma-156	163	12	1	1	NUM
ma-156	163	13	.	.	PUNCT
ma-156	164	1	by	by	ADP
ma-156	164	2	induction	induction	NOUN
ma-156	164	3	,	,	PUNCT
ma-156	164	4	we	we	PRON
ma-156	164	5	can	can	AUX
ma-156	164	6	construct	construct	VERB
ma-156	164	7	a	a	DET
ma-156	164	8	sequence	sequence	NOUN
ma-156	164	9	{	{	PUNCT
ma-156	164	10	xi	xi	ADP
ma-156	164	11	}	}	PUNCT
ma-156	164	12	⊆	⊆	NUM
ma-156	164	13	a0	a0	NOUN
ma-156	164	14	such	such	ADJ
ma-156	164	15	that	that	DET
ma-156	164	16	d(xi+1	d(xi+1	NOUN
ma-156	164	17	,	,	PUNCT
ma-156	164	18	t	t	PROPN
ma-156	164	19	xi	xi	PROPN
ma-156	164	20	)	)	PUNCT
ma-156	165	1	=	=	SYM
ma-156	166	1	d(a	d(a	PROPN
ma-156	166	2	,	,	PUNCT
ma-156	166	3	b	b	NOUN
ma-156	166	4	)	)	PUNCT
ma-156	166	5	and	and	CCONJ
ma-156	166	6	α(xi	α(xi	PROPN
ma-156	166	7	,	,	PUNCT
ma-156	166	8	xi+1	xi+1	PROPN
ma-156	166	9	)	)	PUNCT
ma-156	166	10	≥	≥	NOUN
ma-156	166	11	1	1	NUM
ma-156	166	12	,	,	PUNCT
ma-156	166	13	f	f	PROPN
ma-156	166	14	or	or	CCONJ
ma-156	166	15	al	al	PROPN
ma-156	166	16	l	l	NOUN
ma-156	167	1	i	i	PRON
ma-156	167	2	∈	∈	PROPN
ma-156	167	3	n.	n.	NOUN
ma-156	167	4	(	(	PUNCT
ma-156	167	5	5	5	NUM
ma-156	167	6	)	)	PUNCT
ma-156	167	7	for	for	ADP
ma-156	167	8	all	all	PRON
ma-156	167	9	i	i	PRON
ma-156	167	10	≥	≥	VERB
ma-156	167	11	0	0	NUM
ma-156	167	12			NUM
ma-156	167	13	α(xi	α(xi	PROPN
ma-156	167	14	,	,	PUNCT
ma-156	167	15	xi+1	xi+1	PROPN
ma-156	167	16	)	)	PUNCT
ma-156	167	17	≥	≥	NOUN
ma-156	167	18	1	1	NUM
ma-156	167	19	α(xi+1	α(xi+1	NUM
ma-156	167	20	,	,	PUNCT
ma-156	167	21	xi+2	xi+2	NUM
ma-156	167	22	)	)	PUNCT
ma-156	167	23	≥	≥	NOUN
ma-156	167	24	1	1	NUM
ma-156	167	25	d(xi+2	d(xi+2	NOUN
ma-156	167	26	,	,	PUNCT
ma-156	167	27	t	t	PROPN
ma-156	167	28	xi−1	xi−1	PROPN
ma-156	167	29	)	)	PUNCT
ma-156	168	1	=	=	SYM
ma-156	169	1	d(a	d(a	PROPN
ma-156	169	2	,	,	PUNCT
ma-156	169	3	b	b	NOUN
ma-156	169	4	)	)	PUNCT
ma-156	169	5	,	,	PUNCT
ma-156	169	6	=	=	SYM
ma-156	169	7	⇒	⇒	VERB
ma-156	169	8	α(xi	α(xi	NUM
ma-156	169	9	,	,	PUNCT
ma-156	169	10	xi+2	xi+2	NUM
ma-156	169	11	)	)	PUNCT
ma-156	169	12	≥	≥	NOUN
ma-156	169	13	1	1	NUM
ma-156	169	14	,	,	PUNCT
ma-156	169	15	since	since	SCONJ
ma-156	169	16	t	t	PROPN
ma-156	169	17	is	be	AUX
ma-156	169	18	triangular	triangular	NOUN
ma-156	169	19	α	α	DET
ma-156	169	20	-	-	ADJ
ma-156	169	21	orbital	orbital	ADJ
ma-156	169	22	proximal	proximal	ADJ
ma-156	169	23	admissible	admissible	NOUN
ma-156	169	24	.	.	PUNCT
ma-156	170	1	thus	thus	ADV
ma-156	170	2	by	by	ADP
ma-156	170	3	induction	induction	NOUN
ma-156	170	4	,	,	PUNCT
ma-156	170	5	α(xi	α(xi	NUM
ma-156	170	6	,	,	PUNCT
ma-156	170	7	xj	xj	PROPN
ma-156	170	8	)	)	PUNCT
ma-156	170	9	≥	≥	NOUN
ma-156	170	10	1	1	NUM
ma-156	170	11	for	for	ADP
ma-156	170	12	all	all	DET
ma-156	170	13	i	i	PRON
ma-156	170	14	,	,	PUNCT
ma-156	170	15	jsuch	jsuch	VERB
ma-156	170	16	that	that	SCONJ
ma-156	170	17	0	0	NUM
ma-156	170	18	≤	≤	PUNCT
ma-156	171	1	i	i	PRON
ma-156	171	2	<	<	X
ma-156	171	3	j	j	PROPN
ma-156	171	4	.therefore	.therefore	ADV
ma-156	171	5	for	for	ADP
ma-156	171	6	any	any	DET
ma-156	171	7	i	i	PROPN
ma-156	171	8	∈	∈	PROPN
ma-156	171	9	n	n	CCONJ
ma-156	171	10	,	,	PUNCT
ma-156	171	11	we	we	PROPN
ma-156	171	12	have	have	NUM
ma-156	171	13	α(xi−1	α(xi−1	PROPN
ma-156	171	14	,	,	PUNCT
ma-156	171	15	xj−1	xj−1	PROPN
ma-156	171	16	)	)	PUNCT
ma-156	171	17	≥	≥	NOUN
ma-156	171	18	1	1	NUM
ma-156	171	19	d(xi	d(xi	PROPN
ma-156	171	20	,	,	PUNCT
ma-156	171	21	t	t	PROPN
ma-156	171	22	xi−1	xi−1	PROPN
ma-156	171	23	)	)	PUNCT
ma-156	172	1	=	=	SYM
ma-156	173	1	d(a	d(a	PROPN
ma-156	173	2	,	,	PUNCT
ma-156	173	3	b	b	NOUN
ma-156	173	4	)	)	PUNCT
ma-156	173	5	,	,	PUNCT
ma-156	173	6	d(xj	d(xj	PROPN
ma-156	173	7	,	,	PUNCT
ma-156	173	8	t	t	PROPN
ma-156	173	9	xj−1	xj−1	PROPN
ma-156	173	10	)	)	PUNCT
ma-156	173	11	=	=	SYM
ma-156	174	1	d(a	d(a	PROPN
ma-156	174	2	,	,	PUNCT
ma-156	174	3	b	b	NOUN
ma-156	174	4	)	)	PUNCT
ma-156	174	5	for	for	ADP
ma-156	174	6	all	all	DET
ma-156	174	7	i	i	PRON
ma-156	174	8	,	,	PUNCT
ma-156	174	9	j	j	PROPN
ma-156	174	10	such	such	ADJ
ma-156	174	11	that	that	SCONJ
ma-156	174	12	1	1	NUM
ma-156	174	13	≤	≤	PUNCT
ma-156	175	1	i	i	PRON
ma-156	175	2	<	<	X
ma-156	175	3	j	j	PROPN
ma-156	175	4	.clearly	.clearly	ADV
ma-156	175	5	,	,	PUNCT
ma-156	175	6	if	if	SCONJ
ma-156	175	7	xi+1	xi+1	NUM
ma-156	175	8	=	=	SYM
ma-156	175	9	xi	xi	PROPN
ma-156	175	10	for	for	ADP
ma-156	175	11	some	some	DET
ma-156	175	12	i	i	PRON
ma-156	175	13	∈	∈	PROPN
ma-156	175	14	n	n	ADV
ma-156	175	15	from	from	ADP
ma-156	175	16	inequality	inequality	NOUN
ma-156	175	17	(	(	PUNCT
ma-156	175	18	5	5	NUM
ma-156	175	19	)	)	PUNCT
ma-156	175	20	,	,	PUNCT
ma-156	175	21	xi	xi	PROPN
ma-156	175	22	will	will	AUX
ma-156	175	23	be	be	AUX
ma-156	175	24	a	a	DET
ma-156	175	25	best	good	ADJ
ma-156	175	26	proximity	proximity	NOUN
ma-156	175	27	point	point	NOUN
ma-156	175	28	,	,	PUNCT
ma-156	175	29	sohenceforth	sohenceforth	NOUN
ma-156	175	30	,	,	PUNCT
ma-156	175	31	in	in	ADP
ma-156	175	32	this	this	DET
ma-156	175	33	proof	proof	NOUN
ma-156	175	34	,	,	PUNCT
ma-156	175	35	we	we	PRON
ma-156	175	36	assume	assume	VERB
ma-156	175	37	d(xi	d(xi	PROPN
ma-156	175	38	,	,	PUNCT
ma-156	175	39	xi+1	xi+1	PROPN
ma-156	175	40	)	)	PUNCT
ma-156	175	41	>	>	X
ma-156	176	1	0	0	NUM
ma-156	176	2	,	,	PUNCT
ma-156	176	3	∀	∀	VERB
ma-156	177	1	i	i	NOUN
ma-156	177	2	∈	∈	PROPN
ma-156	177	3	n.	n.	NOUN
ma-156	177	4	from	from	ADP
ma-156	177	5	inequality	inequality	NOUN
ma-156	177	6	(	(	PUNCT
ma-156	177	7	3	3	NUM
ma-156	177	8	)	)	PUNCT
ma-156	177	9	,	,	PUNCT
ma-156	177	10	we	we	PRON
ma-156	177	11	have	have	VERB
ma-156	177	12	φ(d(xi	φ(d(xi	NOUN
ma-156	177	13	,	,	PUNCT
ma-156	177	14	xj	xj	PROPN
ma-156	177	15	)	)	PUNCT
ma-156	177	16	)	)	PUNCT
ma-156	177	17	≤	≤	NOUN
ma-156	177	18	α(xi−1	α(xi−1	NUM
ma-156	177	19	,	,	PUNCT
ma-156	177	20	xj−1)φ(d(xi	xj−1)φ(d(xi	INTJ
ma-156	177	21	,	,	PUNCT
ma-156	177	22	xj	xj	PROPN
ma-156	177	23	)	)	PUNCT
ma-156	177	24	)	)	PUNCT
ma-156	178	1	≤	≤	NUM
ma-156	178	2	β(φ(mt	β(φ(mt	PUNCT
ma-156	178	3	(	(	PUNCT
ma-156	178	4	xi−1	xi−1	PROPN
ma-156	178	5	,	,	PUNCT
ma-156	178	6	xj−1)))φ(mt	xj−1)))φ(mt	PUNCT
ma-156	178	7	(	(	PUNCT
ma-156	178	8	xi−1	xi−1	PROPN
ma-156	178	9	,	,	PUNCT
ma-156	178	10	xj−1	xj−1	PROPN
ma-156	178	11	)	)	PUNCT
ma-156	178	12	)	)	PUNCT
ma-156	178	13	(	(	PUNCT
ma-156	178	14	6	6	X
ma-156	178	15	)	)	SYM
ma-156	178	16	1	1	NUM
ma-156	178	17	≤	≤	NUM
ma-156	179	1	i	i	PRON
ma-156	179	2	<	<	X
ma-156	179	3	j	j	X
ma-156	179	4	where	where	SCONJ
ma-156	179	5	φ(mt	φ(mt	X
ma-156	179	6	(	(	PUNCT
ma-156	179	7	xi−1	xi−1	PROPN
ma-156	179	8	,	,	PUNCT
ma-156	179	9	xj−1	xj−1	PROPN
ma-156	179	10	)	)	PUNCT
ma-156	179	11	)	)	PUNCT
ma-156	179	12	≤	≤	NUM
ma-156	179	13	φ(max{d(xi−1	φ(max{d(xi−1	NOUN
ma-156	179	14	,	,	PUNCT
ma-156	179	15	xj−1	xj−1	NOUN
ma-156	179	16	)	)	PUNCT
ma-156	179	17	,	,	PUNCT
ma-156	179	18	d(xi−1	d(xi−1	PROPN
ma-156	179	19	,	,	PUNCT
ma-156	179	20	xi	xi	PROPN
ma-156	179	21	)	)	PUNCT
ma-156	179	22	,	,	PUNCT
ma-156	179	23	d(xj−1	d(xj−1	NOUN
ma-156	179	24	,	,	PUNCT
ma-156	179	25	xj	xj	PROPN
ma-156	179	26	)	)	PUNCT
ma-156	179	27	,	,	PUNCT
ma-156	179	28	d(xi−1	d(xi−1	PROPN
ma-156	179	29	,	,	PUNCT
ma-156	179	30	xj	xj	PROPN
ma-156	179	31	)	)	PUNCT
ma-156	179	32	,	,	PUNCT
ma-156	179	33	d(xj−1	d(xj−1	NOUN
ma-156	179	34	,	,	PUNCT
ma-156	179	35	xi	xi	NOUN
ma-156	179	36	)	)	PUNCT
ma-156	179	37	}	}	PUNCT
ma-156	179	38	)	)	PUNCT
ma-156	179	39	≤	≤	PROPN
ma-156	179	40	φ(δ[ot	φ(δ[ot	ADP
ma-156	179	41	(	(	PUNCT
ma-156	179	42	xi−1	xi−1	PROPN
ma-156	179	43	,	,	PUNCT
ma-156	179	44	n	n	CCONJ
ma-156	179	45	)	)	PUNCT
ma-156	179	46	]	]	PUNCT
ma-156	179	47	)	)	PUNCT
ma-156	179	48	,	,	PUNCT
ma-156	179	49	f	f	PROPN
ma-156	179	50	or	or	CCONJ
ma-156	179	51	i	i	PRON
ma-156	179	52	≤	≤	NUM
ma-156	179	53	j	j	PROPN
ma-156	179	54	≤	≤	PROPN
ma-156	179	55	n	n	PROPN
ma-156	180	1	+	+	CCONJ
ma-156	180	2	i	i	NOUN
ma-156	180	3	.	.	PUNCT
ma-156	181	1	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	181	2	eur	eur	PROPN
ma-156	181	3	.	.	PUNCT
ma-156	182	1	j.	j.	PROPN
ma-156	182	2	math	math	PROPN
ma-156	182	3	.	.	PUNCT
ma-156	183	1	anal	anal	PROPN
ma-156	183	2	.	.	PUNCT
ma-156	184	1	10.28924	10.28924	NUM
ma-156	184	2	/	/	SYM
ma-156	184	3	ada	ada	PROPN
ma-156	184	4	/	/	SYM
ma-156	184	5	ma.3.16	ma.3.16	PROPN
ma-156	184	6	8note	8note	NUM
ma-156	184	7	that	that	SCONJ
ma-156	184	8	the	the	DET
ma-156	184	9	case	case	NOUN
ma-156	184	10	φ(mt	φ(mt	VERB
ma-156	184	11	(	(	PUNCT
ma-156	184	12	xi−1	xi−1	PROPN
ma-156	184	13	,	,	PUNCT
ma-156	184	14	xj−1	xj−1	NOUN
ma-156	184	15	)	)	PUNCT
ma-156	184	16	)	)	PUNCT
ma-156	185	1	=	=	SYM
ma-156	185	2	φ(d(xi	φ(d(xi	PROPN
ma-156	185	3	,	,	PUNCT
ma-156	185	4	xj	xj	PROPN
ma-156	185	5	)	)	PUNCT
ma-156	185	6	)	)	PUNCT
ma-156	185	7	is	be	AUX
ma-156	185	8	impossible	impossible	ADJ
ma-156	185	9	.	.	PUNCT
ma-156	186	1	indeed	indeed	ADV
ma-156	186	2	,	,	PUNCT
ma-156	186	3	by	by	ADP
ma-156	186	4	inequality	inequality	NOUN
ma-156	186	5	(	(	PUNCT
ma-156	186	6	6	6	NUM
ma-156	186	7	)	)	PUNCT
ma-156	186	8	,	,	PUNCT
ma-156	186	9	φ(d(xi	φ(d(xi	INTJ
ma-156	186	10	,	,	PUNCT
ma-156	186	11	xj	xj	PROPN
ma-156	186	12	)	)	PUNCT
ma-156	186	13	)	)	PUNCT
ma-156	186	14	≤	≤	NUM
ma-156	187	1	β(φ(mt	β(φ(mt	PUNCT
ma-156	187	2	(	(	PUNCT
ma-156	187	3	xi−1	xi−1	PROPN
ma-156	187	4	,	,	PUNCT
ma-156	187	5	xj−1)))φ(mt	xj−1)))φ(mt	PUNCT
ma-156	187	6	(	(	PUNCT
ma-156	187	7	xi−1	xi−1	PROPN
ma-156	187	8	,	,	PUNCT
ma-156	187	9	xj−1	xj−1	PROPN
ma-156	187	10	)	)	PUNCT
ma-156	187	11	)	)	PUNCT
ma-156	187	12	≤	≤	NUM
ma-156	187	13	β(φ(d(xi	β(φ(d(xi	PROPN
ma-156	187	14	,	,	PUNCT
ma-156	187	15	xj)))φ(d(xi	xj)))φ(d(xi	PROPN
ma-156	187	16	,	,	PUNCT
ma-156	187	17	xj	xj	PROPN
ma-156	187	18	)	)	PUNCT
ma-156	187	19	)	)	PUNCT
ma-156	187	20	<	<	X
ma-156	187	21	φ(d(xi	φ(d(xi	X
ma-156	187	22	,	,	PUNCT
ma-156	187	23	xj	xj	PROPN
ma-156	187	24	)	)	PUNCT
ma-156	187	25	)	)	PUNCT
ma-156	187	26	,	,	PUNCT
ma-156	187	27	is	be	AUX
ma-156	187	28	a	a	DET
ma-156	187	29	contradiction	contradiction	NOUN
ma-156	187	30	.	.	PUNCT
ma-156	188	1	thus	thus	ADV
ma-156	188	2	,	,	PUNCT
ma-156	188	3	we	we	PRON
ma-156	188	4	conclude	conclude	VERB
ma-156	188	5	that	that	SCONJ
ma-156	188	6	φ(d(xi	φ(d(xi	PROPN
ma-156	188	7	,	,	PUNCT
ma-156	188	8	xj	xj	PROPN
ma-156	188	9	)	)	PUNCT
ma-156	188	10	)	)	PUNCT
ma-156	189	1	<	<	X
ma-156	189	2	φ(d(xi−1	φ(d(xi−1	PROPN
ma-156	189	3	,	,	PUNCT
ma-156	189	4	xj−1	xj−1	PROPN
ma-156	189	5	)	)	PUNCT
ma-156	189	6	)	)	PUNCT
ma-156	189	7	for	for	ADP
ma-156	189	8	all	all	PRON
ma-156	189	9	0	0	NUM
ma-156	189	10	<	<	X
ma-156	190	1	i	i	X
ma-156	190	2	<	<	X
ma-156	190	3	j	j	PROPN
ma-156	190	4	and	and	CCONJ
ma-156	190	5	sothe	sothe	PROPN
ma-156	190	6	sequence	sequence	NOUN
ma-156	190	7	{	{	PUNCT
ma-156	190	8	φ(d(xi	φ(d(xi	INTJ
ma-156	190	9	,	,	PUNCT
ma-156	190	10	xj	xj	PROPN
ma-156	190	11	)	)	PUNCT
ma-156	190	12	)	)	PUNCT
ma-156	190	13	}	}	PUNCT
ma-156	190	14	is	be	AUX
ma-156	190	15	positive	positive	ADJ
ma-156	190	16	and	and	CCONJ
ma-156	190	17	decreasing	decrease	VERB
ma-156	190	18	.	.	PUNCT
ma-156	191	1	consequently	consequently	ADV
ma-156	191	2	,	,	PUNCT
ma-156	191	3	there	there	PRON
ma-156	191	4	exists	exist	VERB
ma-156	191	5	r	r	NOUN
ma-156	191	6	≥	≥	NOUN
ma-156	191	7	0	0	NUM
ma-156	191	8	such	such	ADJ
ma-156	191	9	that	that	SCONJ
ma-156	191	10	lim	lim	PROPN
ma-156	191	11	i	i	PRON
ma-156	191	12	,	,	PUNCT
ma-156	191	13	j→∞	j→∞	PROPN
ma-156	191	14	φ(d(xi	φ(d(xi	PROPN
ma-156	191	15	,	,	PUNCT
ma-156	191	16	xj	xj	PROPN
ma-156	191	17	)	)	PUNCT
ma-156	191	18	)	)	PUNCT
ma-156	192	1	=	=	PUNCT
ma-156	192	2	r.	r.	NOUN
ma-156	192	3	we	we	PRON
ma-156	192	4	claim	claim	VERB
ma-156	192	5	that	that	SCONJ
ma-156	192	6	r	r	NOUN
ma-156	192	7	=	=	SYM
ma-156	192	8	0	0	X
ma-156	192	9	.	.	PUNCT
ma-156	192	10	suppose	suppose	VERB
ma-156	192	11	,	,	PUNCT
ma-156	192	12	on	on	ADP
ma-156	192	13	the	the	DET
ma-156	192	14	contrary	contrary	NOUN
ma-156	192	15	,	,	PUNCT
ma-156	193	1	that	that	SCONJ
ma-156	193	2	r	r	NOUN
ma-156	193	3	>	>	X
ma-156	193	4	0	0	NUM
ma-156	193	5	.	.	PUNCT
ma-156	194	1	then	then	ADV
ma-156	194	2	we	we	PRON
ma-156	194	3	have	have	VERB
ma-156	194	4	φ(d(xi	φ(d(xi	NOUN
ma-156	194	5	,	,	PUNCT
ma-156	194	6	xj	xj	PROPN
ma-156	194	7	)	)	PUNCT
ma-156	194	8	)	)	PUNCT
ma-156	194	9	φ(d(xi−1	φ(d(xi−1	PROPN
ma-156	194	10	,	,	PUNCT
ma-156	194	11	xj−1	xj−1	PROPN
ma-156	194	12	)	)	PUNCT
ma-156	194	13	)	)	PUNCT
ma-156	194	14	≤	≤	NUM
ma-156	194	15	β(φ(mt	β(φ(mt	PUNCT
ma-156	194	16	(	(	PUNCT
ma-156	194	17	xi−1	xi−1	PROPN
ma-156	194	18	,	,	PUNCT
ma-156	194	19	xj−1	xj−1	PROPN
ma-156	194	20	)	)	PUNCT
ma-156	194	21	)	)	PUNCT
ma-156	194	22	)	)	PUNCT
ma-156	195	1	≤	≤	ADV
ma-156	195	2	1	1	NUM
ma-156	195	3	f	f	NOUN
ma-156	195	4	or	or	CCONJ
ma-156	195	5	each	each	PRON
ma-156	195	6	i	i	PRON
ma-156	195	7	,	,	PUNCT
ma-156	195	8	j	j	PROPN
ma-156	195	9	∈	∈	PROPN
ma-156	195	10	n	n	PRON
ma-156	195	11	such	such	ADJ
ma-156	195	12	that	that	SCONJ
ma-156	195	13	i	i	PRON
ma-156	195	14	<	<	X
ma-156	195	15	j.	j.	PROPN
ma-156	195	16	then	then	ADV
ma-156	195	17	,	,	PUNCT
ma-156	195	18	since	since	SCONJ
ma-156	195	19	β	β	PROPN
ma-156	195	20	∈	∈	PROPN
ma-156	195	21	f	f	PROPN
ma-156	195	22	,	,	PUNCT
ma-156	195	23	lim	lim	PROPN
ma-156	195	24	i	i	PRON
ma-156	195	25	,	,	PUNCT
ma-156	195	26	j→∞	j→∞	PROPN
ma-156	195	27	β(φ(mt	β(φ(mt	PUNCT
ma-156	195	28	(	(	PUNCT
ma-156	195	29	xi−1	xi−1	PROPN
ma-156	195	30	,	,	PUNCT
ma-156	195	31	xj−1	xj−1	PROPN
ma-156	195	32	)	)	PUNCT
ma-156	195	33	)	)	PUNCT
ma-156	195	34	)	)	PUNCT
ma-156	196	1	=	=	SYM
ma-156	196	2	1	1	NUM
ma-156	196	3	,	,	PUNCT
ma-156	196	4	implying	imply	VERB
ma-156	196	5	that	that	SCONJ
ma-156	196	6	lim	lim	PROPN
ma-156	196	7	i	i	PRON
ma-156	196	8	,	,	PUNCT
ma-156	196	9	j→∞	j→∞	ADV
ma-156	196	10	φ(mt	φ(mt	X
ma-156	196	11	(	(	PUNCT
ma-156	196	12	xi−1	xi−1	PROPN
ma-156	196	13	,	,	PUNCT
ma-156	196	14	xj−1	xj−1	PROPN
ma-156	196	15	)	)	PUNCT
ma-156	196	16	)	)	PUNCT
ma-156	196	17	=	=	PUNCT
ma-156	197	1	0	0	NUM
ma-156	197	2	,	,	PUNCT
ma-156	197	3	(	(	PUNCT
ma-156	197	4	7	7	NUM
ma-156	197	5	)	)	PUNCT
ma-156	197	6	and	and	CCONJ
ma-156	197	7	so	so	ADV
ma-156	197	8	by	by	ADP
ma-156	197	9	inequality	inequality	NOUN
ma-156	197	10	(	(	PUNCT
ma-156	197	11	6	6	NUM
ma-156	197	12	)	)	PUNCT
ma-156	198	1	lim	lim	NOUN
ma-156	198	2	i	i	PRON
ma-156	198	3	,	,	PUNCT
ma-156	198	4	j→∞	j→∞	PROPN
ma-156	198	5	φ(d(xi	φ(d(xi	PROPN
ma-156	198	6	,	,	PUNCT
ma-156	198	7	xj	xj	PROPN
ma-156	198	8	)	)	PUNCT
ma-156	198	9	)	)	PUNCT
ma-156	199	1	=	=	PUNCT
ma-156	199	2	0	0	NUM
ma-156	199	3	,	,	PUNCT
ma-156	199	4	which	which	PRON
ma-156	199	5	is	be	AUX
ma-156	199	6	a	a	DET
ma-156	199	7	contradiction	contradiction	NOUN
ma-156	199	8	.	.	PUNCT
ma-156	200	1	now	now	ADV
ma-156	200	2	,	,	PUNCT
ma-156	200	3	by	by	ADP
ma-156	200	4	the	the	DET
ma-156	200	5	continuity	continuity	NOUN
ma-156	200	6	property	property	NOUN
ma-156	200	7	of	of	ADP
ma-156	200	8	φ	φ	PROPN
ma-156	200	9	,	,	PUNCT
ma-156	200	10	φ	φ	PROPN
ma-156	200	11	(	(	PUNCT
ma-156	200	12	lim	lim	PROPN
ma-156	200	13	i	i	PRON
ma-156	200	14	,	,	PUNCT
ma-156	200	15	j→∞	j→∞	PROPN
ma-156	200	16	(	(	PUNCT
ma-156	200	17	d(xi	d(xi	PROPN
ma-156	200	18	,	,	PUNCT
ma-156	200	19	xj	xj	PROPN
ma-156	200	20	)	)	PUNCT
ma-156	200	21	)	)	PUNCT
ma-156	200	22	)	)	PUNCT
ma-156	201	1	=	=	PUNCT
ma-156	201	2	φ(0	φ(0	ADJ
ma-156	201	3	)	)	PUNCT
ma-156	201	4	.	.	PUNCT
ma-156	202	1	(	(	PUNCT
ma-156	202	2	8)	8)	NUM
ma-156	202	3	but	but	CCONJ
ma-156	202	4	φ(t	φ(t	PROPN
ma-156	202	5	)	)	PUNCT
ma-156	203	1	=	=	SYM
ma-156	203	2	0	0	PUNCT
ma-156	204	1	if	if	SCONJ
ma-156	204	2	and	and	CCONJ
ma-156	204	3	only	only	ADV
ma-156	204	4	if	if	SCONJ
ma-156	204	5	t	t	NOUN
ma-156	204	6	=	=	SYM
ma-156	204	7	0	0	PUNCT
ma-156	204	8	and	and	CCONJ
ma-156	204	9	so	so	ADV
ma-156	204	10	(	(	PUNCT
ma-156	204	11	8)	8)	NUM
ma-156	204	12	gives	give	VERB
ma-156	204	13	lim	lim	PROPN
ma-156	204	14	i	i	PRON
ma-156	204	15	,	,	PUNCT
ma-156	204	16	j→∞	j→∞	PROPN
ma-156	204	17	(	(	PUNCT
ma-156	204	18	d(xi	d(xi	PROPN
ma-156	204	19	,	,	PUNCT
ma-156	204	20	xj	xj	PROPN
ma-156	204	21	)	)	PUNCT
ma-156	204	22	)	)	PUNCT
ma-156	205	1	=	=	PUNCT
ma-156	205	2	0	0	X
ma-156	205	3	.	.	PUNCT
ma-156	205	4	therefore	therefore	ADV
ma-156	205	5	,	,	PUNCT
ma-156	205	6	{	{	PUNCT
ma-156	205	7	xn	xn	X
ma-156	205	8	}	}	PUNCT
ma-156	205	9	is	be	AUX
ma-156	205	10	a	a	DET
ma-156	205	11	cauchy	cauchy	ADJ
ma-156	205	12	sequence	sequence	NOUN
ma-156	205	13	in	in	ADP
ma-156	205	14	a0	a0	PROPN
ma-156	205	15	and	and	CCONJ
ma-156	205	16	since	since	SCONJ
ma-156	205	17	a0	a0	PROPN
ma-156	205	18	is	be	AUX
ma-156	205	19	proximal	proximal	ADJ
ma-156	205	20	t	t	NOUN
ma-156	205	21	-orbitally	-orbitally	ADV
ma-156	205	22	complete	complete	ADJ
ma-156	205	23	,	,	PUNCT
ma-156	205	24	thereexists	thereexist	NOUN
ma-156	205	25	x∗	x∗	PROPN
ma-156	205	26	∈	∈	PROPN
ma-156	205	27	a0	a0	NOUN
ma-156	205	28	such	such	ADJ
ma-156	205	29	that	that	SCONJ
ma-156	205	30	lim	lim	PROPN
ma-156	205	31	i→∞	i→∞	VERB
ma-156	205	32	xi	xi	X
ma-156	205	33	=	=	PUNCT
ma-156	205	34	x∗.	x∗.	PROPN
ma-156	206	1	also	also	ADV
ma-156	206	2	,	,	PUNCT
ma-156	206	3	since	since	SCONJ
ma-156	206	4	t	t	PROPN
ma-156	206	5	(	(	PUNCT
ma-156	206	6	a0	a0	PROPN
ma-156	206	7	)	)	PUNCT
ma-156	206	8	⊆	⊆	NUM
ma-156	206	9	b0	b0	NOUN
ma-156	206	10	,	,	PUNCT
ma-156	206	11	then	then	ADV
ma-156	206	12	there	there	PRON
ma-156	206	13	exists	exist	VERB
ma-156	206	14	y	y	PROPN
ma-156	206	15	∈	∈	PROPN
ma-156	206	16	a0	a0	NOUN
ma-156	206	17	such	such	ADJ
ma-156	206	18	that	that	DET
ma-156	206	19	d(y	d(y	NOUN
ma-156	206	20	,	,	PUNCT
ma-156	206	21	t	t	PROPN
ma-156	206	22	x∗	x∗	X
ma-156	206	23	)	)	PUNCT
ma-156	207	1	=	=	PUNCT
ma-156	207	2	d(xi	d(xi	PROPN
ma-156	207	3	,	,	PUNCT
ma-156	207	4	t	t	PROPN
ma-156	207	5	xi−1	xi−1	PROPN
ma-156	207	6	)	)	PUNCT
ma-156	207	7	=	=	SYM
ma-156	208	1	d(a	d(a	PROPN
ma-156	208	2	,	,	PUNCT
ma-156	208	3	b	b	NOUN
ma-156	208	4	)	)	PUNCT
ma-156	208	5	∀n	∀n	NUM
ma-156	208	6	∈	∈	PROPN
ma-156	208	7	n	n	CCONJ
ma-156	208	8	,	,	PUNCT
ma-156	208	9	∀i	∀i	NOUN
ma-156	208	10	≥	≥	NOUN
ma-156	208	11	0	0	NUM
ma-156	208	12	.	.	PUNCT
ma-156	209	1	t	t	PROPN
ma-156	209	2	being	be	AUX
ma-156	209	3	a	a	DET
ma-156	209	4	generalized	generalized	ADJ
ma-156	209	5	α	α	NOUN
ma-156	209	6	-	-	PUNCT
ma-156	209	7	φ	φ	VERB
ma-156	209	8	-	-	PUNCT
ma-156	209	9	geraghty	geraghty	VERB
ma-156	209	10	proximal	proximal	ADJ
ma-156	209	11	quasi	quasi	PROPN
ma-156	209	12	-	-	NOUN
ma-156	209	13	contraction	contraction	NOUN
ma-156	209	14	type	type	NOUN
ma-156	209	15	mapping	mapping	NOUN
ma-156	209	16	gives	give	VERB
ma-156	209	17	φ(d(y	φ(d(y	NUM
ma-156	209	18	,	,	PUNCT
ma-156	209	19	xi	xi	ADJ
ma-156	209	20	)	)	PUNCT
ma-156	209	21	)	)	PUNCT
ma-156	209	22	≤	≤	NUM
ma-156	209	23	α(x∗	α(x∗	NOUN
ma-156	209	24	,	,	PUNCT
ma-156	209	25	xi−1)φ(d(y	xi−1)φ(d(y	PROPN
ma-156	209	26	,	,	PUNCT
ma-156	209	27	xi	xi	PROPN
ma-156	209	28	)	)	PUNCT
ma-156	209	29	)	)	PUNCT
ma-156	209	30	≤	≤	NOUN
ma-156	209	31	β(φ(mt	β(φ(mt	PUNCT
ma-156	209	32	(	(	PUNCT
ma-156	209	33	x∗	x∗	PROPN
ma-156	209	34	,	,	PUNCT
ma-156	209	35	xi−1)φ(mt	xi−1)φ(mt	X
ma-156	209	36	(	(	PUNCT
ma-156	209	37	x∗	x∗	PROPN
ma-156	209	38	,	,	PUNCT
ma-156	209	39	xi−1	xi−1	PROPN
ma-156	209	40	)	)	PUNCT
ma-156	209	41	)	)	PUNCT
ma-156	210	1	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	210	2	eur	eur	PROPN
ma-156	210	3	.	.	PUNCT
ma-156	211	1	j.	j.	PROPN
ma-156	211	2	math	math	PROPN
ma-156	211	3	.	.	PUNCT
ma-156	212	1	anal	anal	PROPN
ma-156	212	2	.	.	PUNCT
ma-156	213	1	10.28924	10.28924	NUM
ma-156	213	2	/	/	SYM
ma-156	213	3	ada	ada	PROPN
ma-156	213	4	/	/	SYM
ma-156	213	5	ma.3.16	ma.3.16	PROPN
ma-156	213	6	9provided	9provided	NUM
ma-156	213	7	that	that	DET
ma-156	213	8	α(x∗	α(x∗	NOUN
ma-156	213	9	,	,	PUNCT
ma-156	213	10	xi−1	xi−1	PROPN
ma-156	213	11	)	)	PUNCT
ma-156	213	12	≥	≥	NOUN
ma-156	213	13	1	1	NUM
ma-156	213	14	where	where	SCONJ
ma-156	213	15	φ(mt	φ(mt	X
ma-156	213	16	(	(	PUNCT
ma-156	213	17	x∗	x∗	PROPN
ma-156	213	18	,	,	PUNCT
ma-156	213	19	xi−1	xi−1	PROPN
ma-156	213	20	)	)	PUNCT
ma-156	213	21	)	)	PUNCT
ma-156	214	1	=	=	SYM
ma-156	214	2	φ(max{d(x∗	φ(max{d(x∗	PROPN
ma-156	214	3	,	,	PUNCT
ma-156	214	4	xi−1	xi−1	PROPN
ma-156	214	5	)	)	PUNCT
ma-156	214	6	,	,	PUNCT
ma-156	214	7	d(x∗	d(x∗	NOUN
ma-156	214	8	,	,	PUNCT
ma-156	214	9	xi	xi	PROPN
ma-156	214	10	)	)	PUNCT
ma-156	214	11	,	,	PUNCT
ma-156	214	12	d(xi−1	d(xi−1	PROPN
ma-156	214	13	,	,	PUNCT
ma-156	214	14	xi	xi	PROPN
ma-156	214	15	)	)	PUNCT
ma-156	214	16	,	,	PUNCT
ma-156	214	17	d(x∗	d(x∗	PROPN
ma-156	214	18	,	,	PUNCT
ma-156	214	19	y	y	PROPN
ma-156	214	20	)	)	PUNCT
ma-156	214	21	,	,	PUNCT
ma-156	214	22	d(xi−1	d(xi−1	PROPN
ma-156	214	23	,	,	PUNCT
ma-156	214	24	y	y	NOUN
ma-156	214	25	)	)	PUNCT
ma-156	214	26	}	}	PUNCT
ma-156	214	27	)	)	PUNCT
ma-156	214	28	.	.	PUNCT
ma-156	215	1	but	but	CCONJ
ma-156	215	2	taking	take	VERB
ma-156	215	3	the	the	DET
ma-156	215	4	limit	limit	NOUN
ma-156	215	5	,	,	PUNCT
ma-156	215	6	φ(d(y	φ(d(y	NUM
ma-156	215	7	,	,	PUNCT
ma-156	215	8	x∗	x∗	PROPN
ma-156	215	9	)	)	PUNCT
ma-156	215	10	)	)	PUNCT
ma-156	215	11	≤	≤	PROPN
ma-156	215	12	lim	lim	PROPN
ma-156	215	13	i→∞	i→∞	VERB
ma-156	215	14	β(φ(mt	β(φ(mt	PUNCT
ma-156	215	15	(	(	PUNCT
ma-156	215	16	x∗	x∗	PROPN
ma-156	215	17	,	,	PUNCT
ma-156	215	18	xi−1)))φ(d(x∗	xi−1)))φ(d(x∗	PROPN
ma-156	215	19	,	,	PUNCT
ma-156	215	20	y	y	PROPN
ma-156	215	21	)	)	PUNCT
ma-156	215	22	)	)	PUNCT
ma-156	215	23	,	,	PUNCT
ma-156	215	24	which	which	PRON
ma-156	215	25	gives	give	VERB
ma-156	215	26	,	,	PUNCT
ma-156	215	27	1	1	NUM
ma-156	215	28	≤	≤	NUM
ma-156	215	29	lim	lim	PROPN
ma-156	215	30	i→∞	i→∞	VERB
ma-156	215	31	β(φ(mt	β(φ(mt	PUNCT
ma-156	215	32	(	(	PUNCT
ma-156	215	33	x∗	x∗	PROPN
ma-156	215	34	,	,	PUNCT
ma-156	215	35	xi−1	xi−1	PROPN
ma-156	215	36	)	)	PUNCT
ma-156	215	37	)	)	PUNCT
ma-156	215	38	)	)	PUNCT
ma-156	216	1	=	=	SYM
ma-156	216	2	β(φ(d(y	β(φ(d(y	PROPN
ma-156	216	3	,	,	PUNCT
ma-156	216	4	x∗	x∗	PROPN
ma-156	216	5	)	)	PUNCT
ma-156	216	6	)	)	PUNCT
ma-156	216	7	)	)	PUNCT
ma-156	217	1	=	=	SYM
ma-156	217	2	1	1	NUM
ma-156	217	3	implying	imply	VERB
ma-156	217	4	φ(d(y	φ(d(y	NUM
ma-156	217	5	,	,	PUNCT
ma-156	217	6	x∗	x∗	PROPN
ma-156	217	7	)	)	PUNCT
ma-156	217	8	)	)	PUNCT
ma-156	218	1	=	=	SYM
ma-156	218	2	0	0	NUM
ma-156	218	3	and	and	CCONJ
ma-156	218	4	d(y	d(y	NOUN
ma-156	218	5	,	,	PUNCT
ma-156	218	6	x∗	x∗	PROPN
ma-156	218	7	)	)	PUNCT
ma-156	219	1	=	=	SYM
ma-156	219	2	0	0	NUM
ma-156	220	1	i.e	i.e	VERB
ma-156	220	2	y	y	NOUN
ma-156	220	3	=	=	PUNCT
ma-156	220	4	x∗.	x∗.	PROPN
ma-156	221	1	we	we	PRON
ma-156	221	2	have	have	VERB
ma-156	221	3	d(x∗	d(x∗	PROPN
ma-156	221	4	,	,	PUNCT
ma-156	221	5	t	t	PROPN
ma-156	221	6	x∗	x∗	X
ma-156	221	7	)	)	PUNCT
ma-156	222	1	=	=	PUNCT
ma-156	222	2	d(y	d(y	NOUN
ma-156	222	3	,	,	PUNCT
ma-156	222	4	t	t	PROPN
ma-156	222	5	x∗	x∗	X
ma-156	222	6	)	)	PUNCT
ma-156	223	1	=	=	SYM
ma-156	223	2	d(a	d(a	PROPN
ma-156	223	3	,	,	PUNCT
ma-156	223	4	b	b	NOUN
ma-156	223	5	)	)	PUNCT
ma-156	223	6	and	and	CCONJ
ma-156	223	7	x∗	x∗	PROPN
ma-156	223	8	∈	∈	PROPN
ma-156	223	9	a0	a0	PROPN
ma-156	223	10	is	be	AUX
ma-156	223	11	a	a	DET
ma-156	223	12	bestproximity	bestproximity	NOUN
ma-156	223	13	point	point	NOUN
ma-156	223	14	of	of	ADP
ma-156	223	15	t	t	PROPN
ma-156	223	16	.for	.for	PUNCT
ma-156	223	17	uniqueness	uniqueness	NOUN
ma-156	223	18	,	,	PUNCT
ma-156	223	19	suppose	suppose	VERB
ma-156	223	20	the	the	DET
ma-156	223	21	best	good	ADJ
ma-156	223	22	proximity	proximity	NOUN
ma-156	223	23	point	point	NOUN
ma-156	223	24	of	of	ADP
ma-156	223	25	t	t	PROPN
ma-156	223	26	is	be	AUX
ma-156	223	27	not	not	PART
ma-156	223	28	unique	unique	ADJ
ma-156	223	29	.	.	PUNCT
ma-156	224	1	let	let	VERB
ma-156	224	2	x∗	x∗	PROPN
ma-156	224	3	,	,	PUNCT
ma-156	224	4	y∗	y∗	ADV
ma-156	224	5	be	be	AUX
ma-156	224	6	two	two	NUM
ma-156	224	7	bestproximity	bestproximity	NOUN
ma-156	224	8	points	point	NOUN
ma-156	224	9	of	of	ADP
ma-156	224	10	t	t	PROPN
ma-156	224	11	with	with	ADP
ma-156	224	12	x∗	x∗	PROPN
ma-156	224	13	6=	6=	SYM
ma-156	224	14	y∗.	y∗.	PROPN
ma-156	224	15	then,	then,	PROPN
ma-156	224	16	α(x∗	α(x∗	NOUN
ma-156	224	17	,	,	PUNCT
ma-156	224	18	y∗	y∗	PROPN
ma-156	224	19	)	)	PUNCT
ma-156	224	20	≥	≥	NOUN
ma-156	224	21	1	1	NUM
ma-156	224	22	d(x∗	d(x∗	NOUN
ma-156	224	23	,	,	PUNCT
ma-156	224	24	t	t	PROPN
ma-156	224	25	x∗	x∗	X
ma-156	224	26	)	)	PUNCT
ma-156	225	1	=	=	SYM
ma-156	225	2	d(a	d(a	PROPN
ma-156	225	3	,	,	PUNCT
ma-156	225	4	b	b	NOUN
ma-156	225	5	)	)	PUNCT
ma-156	225	6	d(y∗	d(y∗	NOUN
ma-156	225	7	,	,	PUNCT
ma-156	225	8	t	t	PROPN
ma-156	225	9	y∗	y∗	ADV
ma-156	225	10	)	)	PUNCT
ma-156	226	1	=	=	SYM
ma-156	226	2	d(a	d(a	PROPN
ma-156	226	3	,	,	PUNCT
ma-156	226	4	b	b	NOUN
ma-156	226	5	)	)	PUNCT
ma-156	226	6			NOUN
ma-156	226	7	since	since	SCONJ
ma-156	226	8	t	t	PROPN
ma-156	226	9	is	be	AUX
ma-156	226	10	a	a	DET
ma-156	226	11	generalized	generalized	ADJ
ma-156	226	12	α	α	NOUN
ma-156	226	13	-	-	PUNCT
ma-156	226	14	φ	φ	VERB
ma-156	226	15	-	-	PUNCT
ma-156	226	16	geraghty	geraghty	VERB
ma-156	226	17	proximal	proximal	ADJ
ma-156	226	18	quasi	quasi	PROPN
ma-156	226	19	-	-	NOUN
ma-156	226	20	contraction	contraction	NOUN
ma-156	226	21	type	type	NOUN
ma-156	226	22	mapping	mapping	NOUN
ma-156	226	23	,	,	PUNCT
ma-156	226	24	φ(d(x∗	φ(d(x∗	PROPN
ma-156	226	25	,	,	PUNCT
ma-156	226	26	y∗	y∗	PROPN
ma-156	226	27	)	)	PUNCT
ma-156	226	28	)	)	PUNCT
ma-156	226	29	≤	≤	NUM
ma-156	226	30	α(x∗	α(x∗	NOUN
ma-156	226	31	,	,	PUNCT
ma-156	226	32	y∗)φ(d(x∗	y∗)φ(d(x∗	NOUN
ma-156	226	33	,	,	PUNCT
ma-156	226	34	y∗	y∗	PROPN
ma-156	226	35	)	)	PUNCT
ma-156	226	36	)	)	PUNCT
ma-156	226	37	≤	≤	NUM
ma-156	226	38	β(mt	β(mt	ADJ
ma-156	226	39	(	(	PUNCT
ma-156	226	40	x∗	x∗	NOUN
ma-156	226	41	,	,	PUNCT
ma-156	226	42	y∗))φ(mt	y∗))φ(mt	PROPN
ma-156	226	43	(	(	PUNCT
ma-156	226	44	x∗	x∗	PROPN
ma-156	226	45	,	,	PUNCT
ma-156	226	46	y∗	y∗	PROPN
ma-156	226	47	)	)	PUNCT
ma-156	226	48	)	)	PUNCT
ma-156	226	49	<	<	X
ma-156	226	50	φ(mt	φ(mt	X
ma-156	226	51	(	(	PUNCT
ma-156	226	52	x∗	x∗	PROPN
ma-156	226	53	,	,	PUNCT
ma-156	226	54	y∗	y∗	PROPN
ma-156	226	55	)	)	PUNCT
ma-156	226	56	)	)	PUNCT
ma-156	226	57	where	where	SCONJ
ma-156	226	58	mt	mt	PROPN
ma-156	226	59	(	(	PUNCT
ma-156	226	60	x∗	x∗	PROPN
ma-156	226	61	,	,	PUNCT
ma-156	226	62	y∗	y∗	PROPN
ma-156	226	63	)	)	PUNCT
ma-156	226	64	=	=	SYM
ma-156	226	65	max{d(x∗	max{d(x∗	NOUN
ma-156	226	66	,	,	PUNCT
ma-156	226	67	y∗	y∗	PROPN
ma-156	226	68	)	)	PUNCT
ma-156	226	69	,	,	PUNCT
ma-156	226	70	d(x∗	d(x∗	NOUN
ma-156	226	71	,	,	PUNCT
ma-156	226	72	x∗	x∗	PROPN
ma-156	226	73	)	)	PUNCT
ma-156	226	74	,	,	PUNCT
ma-156	226	75	d(y∗	d(y∗	NOUN
ma-156	226	76	,	,	PUNCT
ma-156	226	77	y∗	y∗	PROPN
ma-156	226	78	)	)	PUNCT
ma-156	226	79	,	,	PUNCT
ma-156	226	80	d(x∗	d(x∗	NOUN
ma-156	226	81	,	,	PUNCT
ma-156	226	82	y∗	y∗	PROPN
ma-156	226	83	)	)	PUNCT
ma-156	226	84	,	,	PUNCT
ma-156	226	85	d(y∗	d(y∗	NOUN
ma-156	226	86	,	,	PUNCT
ma-156	226	87	x∗	x∗	PROPN
ma-156	226	88	)	)	PUNCT
ma-156	226	89	}	}	PUNCT
ma-156	226	90	=	=	SYM
ma-156	226	91	d(x∗	d(x∗	NOUN
ma-156	226	92	,	,	PUNCT
ma-156	226	93	y∗	y∗	PROPN
ma-156	226	94	)	)	PUNCT
ma-156	226	95	.	.	PUNCT
ma-156	227	1	this	this	PRON
ma-156	227	2	gives	give	VERB
ma-156	227	3	d(x∗	d(x∗	NOUN
ma-156	227	4	,	,	PUNCT
ma-156	227	5	y∗	y∗	PROPN
ma-156	227	6	)	)	PUNCT
ma-156	228	1	<	<	X
ma-156	228	2	d(x∗	d(x∗	PROPN
ma-156	228	3	,	,	PUNCT
ma-156	228	4	y∗	y∗	PROPN
ma-156	228	5	)	)	PUNCT
ma-156	228	6	,	,	PUNCT
ma-156	228	7	which	which	PRON
ma-156	228	8	is	be	AUX
ma-156	228	9	a	a	DET
ma-156	228	10	contradiction	contradiction	NOUN
ma-156	228	11	.	.	PUNCT
ma-156	229	1	therefore	therefore	ADV
ma-156	229	2	x∗	x∗	PROPN
ma-156	229	3	=	=	SYM
ma-156	229	4	y∗	y∗	PROPN
ma-156	229	5	,	,	PUNCT
ma-156	229	6	and	and	CCONJ
ma-156	229	7	the	the	DET
ma-156	229	8	bestproximity	bestproximity	NOUN
ma-156	229	9	point	point	NOUN
ma-156	229	10	of	of	ADP
ma-156	229	11	t	t	PROPN
ma-156	229	12	is	be	AUX
ma-156	229	13	unique	unique	ADJ
ma-156	229	14	.	.	PUNCT
ma-156	230	1	corollary	corollary	ADJ
ma-156	230	2	3.2	3.2	NUM
ma-156	230	3	.	.	PUNCT
ma-156	231	1	let	let	VERB
ma-156	231	2	a	a	PRON
ma-156	231	3	and	and	CCONJ
ma-156	231	4	b	b	NOUN
ma-156	231	5	be	be	AUX
ma-156	231	6	two	two	NUM
ma-156	231	7	nonempty	nonempty	ADJ
ma-156	231	8	subsets	subset	NOUN
ma-156	231	9	of	of	ADP
ma-156	231	10	a	a	DET
ma-156	231	11	metric	metric	ADJ
ma-156	231	12	space	space	NOUN
ma-156	231	13	such	such	ADJ
ma-156	231	14	that	that	SCONJ
ma-156	231	15	a0	a0	PROPN
ma-156	231	16	isproximal	isproximal	PROPN
ma-156	231	17	t	t	PROPN
ma-156	231	18	-orbitally	-orbitally	PROPN
ma-156	231	19	complete	complete	ADJ
ma-156	231	20	,	,	PUNCT
ma-156	231	21	where	where	SCONJ
ma-156	231	22	t	t	NOUN
ma-156	231	23	:	:	PUNCT
ma-156	231	24	a	a	DET
ma-156	231	25	→	→	SYM
ma-156	231	26	b	b	PROPN
ma-156	231	27	is	be	AUX
ma-156	231	28	a	a	DET
ma-156	231	29	non	non	ADJ
ma-156	231	30	-	-	ADJ
ma-156	231	31	self	self	ADJ
ma-156	231	32	mapping	mapping	NOUN
ma-156	231	33	,	,	PUNCT
ma-156	231	34	α	α	X
ma-156	231	35	:	:	PUNCT
ma-156	231	36	a	a	DET
ma-156	231	37	×	×	NOUN
ma-156	231	38	a	a	PRON
ma-156	231	39	→	→	PUNCT
ma-156	231	40	r+	r+	NOUN
ma-156	231	41	is	be	AUX
ma-156	231	42	afunction	afunction	NOUN
ma-156	231	43	and	and	CCONJ
ma-156	231	44	the	the	DET
ma-156	231	45	following	follow	VERB
ma-156	231	46	conditions	condition	NOUN
ma-156	231	47	are	be	AUX
ma-156	231	48	satisfied:(i	satisfied:(i	NOUN
ma-156	231	49	)	)	PUNCT
ma-156	232	1	t	t	PROPN
ma-156	232	2	is	be	AUX
ma-156	232	3	a	a	DET
ma-156	232	4	generalized	generalized	ADJ
ma-156	232	5	α	α	NOUN
ma-156	232	6	-	-	PUNCT
ma-156	232	7	geraghty	geraghty	VERB
ma-156	232	8	proximal	proximal	ADJ
ma-156	232	9	quasi	quasi	NOUN
ma-156	232	10	-	-	NOUN
ma-156	232	11	contraction	contraction	NOUN
ma-156	232	12	type	type	NOUN
ma-156	232	13	mapping;(ii	mapping;(ii	PROPN
ma-156	232	14	)	)	PUNCT
ma-156	232	15	t	t	PROPN
ma-156	232	16	(	(	PUNCT
ma-156	232	17	a0	a0	PROPN
ma-156	232	18	)	)	PUNCT
ma-156	232	19	⊆	⊆	NUM
ma-156	232	20	b0	b0	NOUN
ma-156	232	21	and	and	CCONJ
ma-156	232	22	t	t	NOUN
ma-156	232	23	is	be	AUX
ma-156	232	24	a	a	DET
ma-156	232	25	triangular	triangular	NOUN
ma-156	232	26	α	α	DET
ma-156	232	27	-	-	ADJ
ma-156	232	28	orbital	orbital	ADJ
ma-156	232	29	proximal	proximal	ADJ
ma-156	232	30	admissible	admissible	ADJ
ma-156	232	31	mapping;(iii	mapping;(iii	NOUN
ma-156	232	32	)	)	PUNCT
ma-156	232	33	there	there	PRON
ma-156	232	34	exists	exist	VERB
ma-156	232	35	x0	x0	PROPN
ma-156	232	36	,	,	PUNCT
ma-156	232	37	x1	x1	PROPN
ma-156	232	38	∈	∈	PROPN
ma-156	232	39	a0	a0	NOUN
ma-156	232	40	such	such	ADJ
ma-156	232	41	that	that	DET
ma-156	232	42	d(x1	d(x1	NOUN
ma-156	232	43	,	,	PUNCT
ma-156	232	44	t	t	NOUN
ma-156	232	45	x0	x0	NUM
ma-156	232	46	)	)	PUNCT
ma-156	233	1	=	=	SYM
ma-156	233	2	d(a	d(a	PROPN
ma-156	233	3	,	,	PUNCT
ma-156	233	4	b	b	NOUN
ma-156	233	5	)	)	PUNCT
ma-156	233	6	and	and	CCONJ
ma-156	233	7	α(x0	α(x0	ADJ
ma-156	233	8	,	,	PUNCT
ma-156	233	9	x1	x1	PROPN
ma-156	233	10	)	)	PUNCT
ma-156	233	11	≥	≥	NOUN
ma-156	233	12	1.then	1.then	NUM
ma-156	233	13	there	there	PRON
ma-156	233	14	exists	exist	VERB
ma-156	233	15	an	an	DET
ma-156	233	16	element	element	NOUN
ma-156	233	17	x∗	x∗	PROPN
ma-156	233	18	∈	∈	PROPN
ma-156	233	19	a0	a0	NOUN
ma-156	233	20	such	such	ADJ
ma-156	233	21	that	that	DET
ma-156	233	22	d(x∗	d(x∗	NOUN
ma-156	233	23	,	,	PUNCT
ma-156	233	24	t	t	PROPN
ma-156	233	25	x∗	x∗	X
ma-156	233	26	)	)	PUNCT
ma-156	234	1	=	=	SYM
ma-156	234	2	d(a	d(a	PROPN
ma-156	234	3	,	,	PUNCT
ma-156	234	4	b	b	NOUN
ma-156	234	5	)	)	PUNCT
ma-156	234	6	.	.	PUNCT
ma-156	235	1	moreover	moreover	ADV
ma-156	235	2	,	,	PUNCT
ma-156	235	3	if	if	SCONJ
ma-156	235	4	α(x	α(x	PROPN
ma-156	235	5	,	,	PUNCT
ma-156	235	6	y	y	PROPN
ma-156	235	7	)	)	PUNCT
ma-156	235	8	≥	≥	NOUN
ma-156	235	9	1	1	NUM
ma-156	235	10	for	for	ADP
ma-156	235	11	all	all	DET
ma-156	235	12	x	x	NOUN
ma-156	235	13	,	,	PUNCT
ma-156	235	14	y	y	PROPN
ma-156	235	15	∈	∈	PROPN
ma-156	235	16	pt	pt	X
ma-156	235	17	(	(	PUNCT
ma-156	235	18	a	a	NOUN
ma-156	235	19	)	)	PUNCT
ma-156	235	20	,	,	PUNCT
ma-156	235	21	then	then	ADV
ma-156	235	22	x∗	x∗	PROPN
ma-156	235	23	is	be	AUX
ma-156	235	24	the	the	DET
ma-156	235	25	unique	unique	ADJ
ma-156	235	26	best	good	ADJ
ma-156	235	27	proximity	proximity	NOUN
ma-156	235	28	point	point	NOUN
ma-156	235	29	of	of	ADP
ma-156	235	30	t	t	PROPN
ma-156	235	31	.	.	PUNCT
ma-156	236	1	https://doi.org/10.28924/ada/ma.3.16	https://doi.org/10.28924/ada/ma.3.16	PROPN
ma-156	236	2	eur	eur	PROPN
ma-156	236	3	.	.	PUNCT
ma-156	237	1	j.	j.	PROPN
ma-156	237	2	math	math	PROPN
ma-156	237	3	.	.	PUNCT
ma-156	238	1	anal	anal	PROPN
ma-156	238	2	.	.	PUNCT
ma-156	239	1	10.28924	10.28924	NUM
ma-156	239	2	/	/	SYM
ma-156	239	3	ada	ada	PROPN
ma-156	239	4	/	/	SYM
ma-156	239	5	ma.3.16	ma.3.16	PROPN
ma-156	239	6	104	104	NUM
ma-156	239	7	.	.	PUNCT
ma-156	239	8	conclusion	conclusion	NOUN
ma-156	239	9	in	in	ADP
ma-156	239	10	this	this	DET
ma-156	239	11	paper	paper	NOUN
ma-156	239	12	,	,	PUNCT
ma-156	239	13	we	we	PRON
ma-156	239	14	introduced	introduce	VERB
ma-156	239	15	the	the	DET
ma-156	239	16	notion	notion	NOUN
ma-156	239	17	of	of	ADP
ma-156	239	18	generalized	generalized	ADJ
ma-156	239	19	α	α	PROPN
ma-156	239	20	-	-	PUNCT
ma-156	239	21	φ	φ	VERB
ma-156	239	22	-	-	PUNCT
ma-156	239	23	geraghty	geraghty	VERB
ma-156	239	24	proximal	proximal	ADJ
ma-156	239	25	quasi	quasi	ADJ
ma-156	239	26	-	-	ADJ
ma-156	239	27	contractiontype	contractiontype	ADJ
ma-156	239	28	mappings	mapping	NOUN
ma-156	239	29	which	which	PRON
ma-156	239	30	,	,	PUNCT
ma-156	239	31	for	for	ADP
ma-156	239	32	a	a	DET
ma-156	239	33	self	self	NOUN
ma-156	239	34	mapping	mapping	NOUN
ma-156	239	35	,	,	PUNCT
ma-156	239	36	reduces	reduce	VERB
ma-156	239	37	to	to	ADP
ma-156	239	38	that	that	PRON
ma-156	239	39	in	in	ADP
ma-156	239	40	umudu	umudu	NOUN
ma-156	239	41	et	et	PROPN
ma-156	239	42	al	al	PROPN
ma-156	239	43	.	.	PUNCT
ma-156	240	1	[	[	X
ma-156	240	2	22	22	NUM
ma-156	240	3	]	]	PUNCT
ma-156	240	4	.	.	PUNCT
ma-156	241	1	equipped	equip	VERB
ma-156	241	2	withan	withan	AUX
ma-156	241	3	example	example	NOUN
ma-156	241	4	,	,	PUNCT
ma-156	241	5	we	we	PRON
ma-156	241	6	also	also	ADV
ma-156	241	7	introduced	introduce	VERB
ma-156	241	8	α	α	PRON
ma-156	241	9	-	-	ADJ
ma-156	241	10	orbital	orbital	ADJ
ma-156	241	11	proximal	proximal	ADJ
ma-156	241	12	admissible	admissible	ADJ
ma-156	241	13	mappings	mapping	NOUN
ma-156	241	14	and	and	CCONJ
ma-156	241	15	triangular	triangular	NOUN
ma-156	241	16	α	α	ADJ
ma-156	241	17	-	-	ADJ
ma-156	241	18	orbitalproximal	orbitalproximal	ADJ
ma-156	241	19	admissible	admissible	ADJ
ma-156	241	20	mappings	mapping	NOUN
ma-156	241	21	which	which	PRON
ma-156	241	22	include	include	VERB
ma-156	241	23	the	the	DET
ma-156	241	24	admissible	admissible	ADJ
ma-156	241	25	mappings	mapping	NOUN
ma-156	241	26	defined	define	VERB
ma-156	241	27	by	by	ADP
ma-156	241	28	popescu	popescu	PROPN
ma-156	241	29	[	[	X
ma-156	241	30	19].the	19].the	DET
ma-156	241	31	existence	existence	NOUN
ma-156	241	32	of	of	ADP
ma-156	241	33	best	good	ADJ
ma-156	241	34	proximity	proximity	NOUN
ma-156	241	35	point	point	NOUN
ma-156	241	36	was	be	AUX
ma-156	241	37	investigated	investigate	VERB
ma-156	241	38	for	for	ADP
ma-156	241	39	the	the	DET
ma-156	241	40	class	class	NOUN
ma-156	241	41	of	of	ADP
ma-156	241	42	mappings	mapping	NOUN
ma-156	241	43	in	in	ADP
ma-156	241	44	a	a	DET
ma-156	241	45	proximal	proximal	ADJ
ma-156	241	46	t	t	NOUN
ma-156	241	47	-orbitally	-orbitally	NOUN
ma-156	241	48	complete	complete	ADJ
ma-156	241	49	metric	metric	ADJ
ma-156	241	50	space	space	NOUN
ma-156	241	51	.	.	PUNCT
ma-156	242	1	competing	compete	VERB
ma-156	242	2	interests	interest	NOUN
ma-156	242	3	:	:	PUNCT
ma-156	242	4	the	the	DET
ma-156	242	5	authors	author	NOUN
ma-156	242	6	declare	declare	VERB
ma-156	242	7	that	that	SCONJ
ma-156	242	8	they	they	PRON
ma-156	242	9	have	have	VERB
ma-156	242	10	no	no	DET
ma-156	242	11	competing	compete	VERB
ma-156	242	12	interests	interest	NOUN
ma-156	242	13	.	.	PUNCT
ma-156	243	1	authors	author	NOUN
ma-156	243	2	’	'	PUNCT
ma-156	243	3	contributions	contribution	NOUN
ma-156	243	4	:	:	PUNCT
ma-156	243	5	all	all	DET
ma-156	243	6	authors	author	NOUN
ma-156	243	7	contributed	contribute	VERB
ma-156	243	8	equally	equally	ADV
ma-156	243	9	in	in	ADP
ma-156	243	10	the	the	DET
ma-156	243	11	preparation	preparation	NOUN
ma-156	243	12	of	of	ADP
ma-156	243	13	the	the	DET
ma-156	243	14	paper	paper	NOUN
ma-156	243	15	.	.	PUNCT
ma-156	244	1	the	the	DET
ma-156	244	2	authors	author	NOUN
ma-156	244	3	read	read	VERB
ma-156	244	4	and	and	CCONJ
ma-156	244	5	approvedthe	approvedthe	DET
ma-156	244	6	final	final	ADJ
ma-156	244	7	manuscript	manuscript	NOUN
ma-156	244	8	.	.	PUNCT
ma-156	245	1	references	reference	NOUN
ma-156	245	2	[	[	X
ma-156	245	3	1	1	NUM
ma-156	245	4	]	]	PUNCT
ma-156	245	5	a.	a.	NOUN
ma-156	245	6	abkar	abkar	PROPN
ma-156	245	7	,	,	PUNCT
ma-156	245	8	m.	m.	NOUN
ma-156	245	9	gabeleh	gabeleh	NOUN
ma-156	245	10	,	,	PUNCT
ma-156	245	11	best	good	ADJ
ma-156	245	12	proximity	proximity	NOUN
ma-156	245	13	points	point	NOUN
ma-156	245	14	for	for	ADP
ma-156	245	15	cyclic	cyclic	ADJ
ma-156	245	16	mappings	mapping	NOUN
ma-156	245	17	in	in	ADP
ma-156	245	18	ordered	order	VERB
ma-156	245	19	metric	metric	ADJ
ma-156	245	20	spaces	space	NOUN
ma-156	245	21	,	,	PUNCT
ma-156	245	22	j.	j.	PROPN
ma-156	245	23	optim	optim	PROPN
ma-156	245	24	.	.	PUNCT
ma-156	246	1	theory	theory	NOUN
ma-156	246	2	appl.150	appl.150	PROPN
ma-156	246	3	(	(	PUNCT
ma-156	246	4	2011	2011	NUM
ma-156	246	5	)	)	PUNCT
ma-156	246	6	,	,	PUNCT
ma-156	246	7	188	188	NUM
ma-156	246	8	-	-	SYM
ma-156	246	9	193	193	NUM
ma-156	246	10	.	.	PUNCT
ma-156	247	1	https://doi.org/10.1007/s10957-011-9810-x.[2	https://doi.org/10.1007/s10957-011-9810-x.[2	ADV
ma-156	247	2	]	]	PUNCT
ma-156	247	3	s.	s.	PROPN
ma-156	247	4	banach	banach	PROPN
ma-156	247	5	,	,	PUNCT
ma-156	247	6	sur	sur	PROPN
ma-156	247	7	les	les	PROPN
ma-156	247	8	operations	operation	NOUN
ma-156	247	9	dans	dan	NOUN
ma-156	247	10	les	le	NOUN
ma-156	247	11	ensembles	ensemble	NOUN
ma-156	247	12	abstraits	abstrait	NOUN
ma-156	247	13	et	et	PROPN
ma-156	247	14	leur	leur	PROPN
ma-156	247	15	applications	applications	PROPN
ma-156	247	16	aux	aux	PROPN
ma-156	247	17	equations	equation	NOUN
ma-156	247	18	integrales	integrale	NOUN
ma-156	247	19	,	,	PUNCT
ma-156	247	20	fund.math	fund.math	NOUN
ma-156	247	21	.	.	NOUN
ma-156	247	22	3	3	NUM
ma-156	247	23	(	(	PUNCT
ma-156	247	24	1922	1922	NUM
ma-156	247	25	)	)	PUNCT
ma-156	247	26	,	,	PUNCT
ma-156	247	27	133	133	NUM
ma-156	247	28	-	-	SYM
ma-156	247	29	181.[3	181.[3	NUM
ma-156	247	30	]	]	PUNCT
ma-156	247	31	s.	s.	PROPN
ma-156	247	32	sadiq	sadiq	PROPN
ma-156	247	33	basha	basha	PROPN
ma-156	247	34	,	,	PUNCT
ma-156	247	35	best	good	ADJ
ma-156	247	36	proximity	proximity	NOUN
ma-156	247	37	points	point	NOUN
ma-156	247	38	:	:	PUNCT
ma-156	247	39	optimal	optimal	ADJ
ma-156	247	40	solutions	solution	NOUN
ma-156	247	41	,	,	PUNCT
ma-156	247	42	j.	j.	PROPN
ma-156	247	43	optim	optim	PROPN
ma-156	247	44	.	.	PUNCT
ma-156	247	45	theory	theory	NOUN
ma-156	247	46	appl	appl	PROPN
ma-156	247	47	.	.	PUNCT
ma-156	248	1	151	151	NUM
ma-156	248	2	(	(	PUNCT
ma-156	248	3	2011	2011	NUM
ma-156	248	4	)	)	PUNCT
ma-156	248	5	,	,	PUNCT
ma-156	248	6	210	210	NUM
ma-156	248	7	-	-	SYM
ma-156	248	8	216	216	NUM
ma-156	248	9	.	.	PUNCT
ma-156	248	10	https	https	NOUN
ma-156	248	11	:	:	PUNCT
ma-156	248	12	//doi.org/10.1007	//doi.org/10.1007	PROPN
ma-156	248	13	/	/	SYM
ma-156	248	14	s10957	s10957	NOUN
ma-156	248	15	-	-	PUNCT
ma-156	248	16	011	011	NUM
ma-156	248	17	-	-	PUNCT
ma-156	248	18	9869	9869	NUM
ma-156	248	19	-	-	PUNCT
ma-156	248	20	4.[4	4.[4	PROPN
ma-156	248	21	]	]	X
ma-156	248	22	n.	n.	NOUN
ma-156	248	23	bilgili	bilgili	NOUN
ma-156	248	24	,	,	PUNCT
ma-156	248	25	e.	e.	PROPN
ma-156	248	26	karapınar	karapınar	PROPN
ma-156	248	27	,	,	PUNCT
ma-156	248	28	k.	k.	PROPN
ma-156	248	29	sadarangani	sadarangani	PROPN
ma-156	248	30	,	,	PUNCT
ma-156	248	31	a	a	DET
ma-156	248	32	generalization	generalization	NOUN
ma-156	248	33	for	for	ADP
ma-156	248	34	the	the	DET
ma-156	248	35	best	good	ADJ
ma-156	248	36	proximity	proximity	NOUN
ma-156	248	37	point	point	NOUN
ma-156	248	38	of	of	ADP
ma-156	248	39	geraghty	geraghty	NOUN
ma-156	248	40	-	-	PUNCT
ma-156	248	41	contractions	contraction	NOUN
ma-156	248	42	,	,	PUNCT
ma-156	248	43	j.inequal	j.inequal	ADJ
ma-156	248	44	.	.	PUNCT
ma-156	249	1	appl	appl	PROPN
ma-156	249	2	.	.	PUNCT
ma-156	250	1	2013	2013	NUM
ma-156	250	2	(	(	PUNCT
ma-156	250	3	2013	2013	NUM
ma-156	250	4	)	)	PUNCT
ma-156	250	5	,	,	PUNCT
ma-156	250	6	286	286	NUM
ma-156	251	1	.	.	PUNCT
ma-156	251	2	https://doi.org/10.1186/1029-242x-2013-286.[5	https://doi.org/10.1186/1029-242x-2013-286.[5	PROPN
ma-156	251	3	]	]	PUNCT
ma-156	251	4	j.	j.	PROPN
ma-156	251	5	caballero	caballero	PROPN
ma-156	251	6	,	,	PUNCT
ma-156	251	7	j.	j.	PROPN
ma-156	251	8	harjani	harjani	PROPN
ma-156	251	9	,	,	PUNCT
ma-156	251	10	k.	k.	PROPN
ma-156	251	11	sadarangani	sadarangani	PROPN
ma-156	251	12	,	,	PUNCT
ma-156	251	13	a	a	DET
ma-156	251	14	best	good	ADJ
ma-156	251	15	proximity	proximity	NOUN
ma-156	251	16	point	point	NOUN
ma-156	251	17	theorem	theorem	NOUN
ma-156	251	18	for	for	ADP
ma-156	251	19	geraghty	geraghty	NOUN
ma-156	251	20	-	-	PUNCT
ma-156	251	21	contractions	contraction	NOUN
ma-156	251	22	,	,	PUNCT
ma-156	251	23	fixed	fix	VERB
ma-156	251	24	pointtheory	pointtheory	NOUN
ma-156	251	25	appl	appl	NOUN
ma-156	251	26	.	.	PUNCT
ma-156	252	1	2012	2012	NUM
ma-156	252	2	(	(	PUNCT
ma-156	252	3	2012	2012	NUM
ma-156	252	4	)	)	PUNCT
ma-156	252	5	,	,	PUNCT
ma-156	252	6	231	231	NUM
ma-156	252	7	.	.	PUNCT
ma-156	253	1	https://doi.org/10.1186/1687-1812-2012-231.[6	https://doi.org/10.1186/1687-1812-2012-231.[6	NOUN
ma-156	253	2	]	]	PUNCT
ma-156	253	3	l.b	l.b	PROPN
ma-156	253	4	.	.	PROPN
ma-156	253	5	ciric	ciric	PROPN
ma-156	253	6	,	,	PUNCT
ma-156	253	7	a	a	DET
ma-156	253	8	generalization	generalization	NOUN
ma-156	253	9	of	of	ADP
ma-156	253	10	banach	banach	NOUN
ma-156	253	11	’s	’s	PART
ma-156	253	12	contraction	contraction	NOUN
ma-156	253	13	principle	principle	NOUN
ma-156	253	14	,	,	PUNCT
ma-156	253	15	proc	proc	PROPN
ma-156	253	16	.	.	PUNCT
ma-156	254	1	amer	amer	PROPN
ma-156	254	2	.	.	PUNCT
ma-156	254	3	math	math	PROPN
ma-156	254	4	.	.	PUNCT
ma-156	255	1	soc	soc	PROPN
ma-156	255	2	.	.	PUNCT
ma-156	256	1	45	45	NUM
ma-156	256	2	(	(	PUNCT
ma-156	256	3	1974	1974	NUM
ma-156	256	4	)	)	PUNCT
ma-156	256	5	,	,	PUNCT
ma-156	256	6	267	267	NUM
ma-156	256	7	-	-	SYM
ma-156	256	8	273.[7	273.[7	NUM
ma-156	256	9	]	]	X
ma-156	256	10	a.a	a.a	PROPN
ma-156	256	11	.	.	PROPN
ma-156	256	12	eldred	eldred	PROPN
ma-156	256	13	,	,	PUNCT
ma-156	256	14	p.	p.	NOUN
ma-156	256	15	veeramani	veeramani	PROPN
ma-156	256	16	,	,	PUNCT
ma-156	256	17	existence	existence	NOUN
ma-156	256	18	and	and	CCONJ
ma-156	256	19	convergence	convergence	NOUN
ma-156	256	20	of	of	ADP
ma-156	256	21	best	good	ADJ
ma-156	256	22	proximity	proximity	NOUN
ma-156	256	23	points	point	NOUN
ma-156	256	24	,	,	PUNCT
ma-156	256	25	j.	j.	PROPN
ma-156	256	26	math	math	PROPN
ma-156	256	27	.	.	PUNCT
ma-156	257	1	anal	anal	PROPN
ma-156	257	2	.	.	PUNCT
ma-156	257	3	appl	appl	PROPN
ma-156	257	4	.	.	PROPN
ma-156	258	1	323	323	NUM
ma-156	258	2	(	(	PUNCT
ma-156	258	3	2006),1001	2006),1001	NUM
ma-156	258	4	-	-	SYM
ma-156	258	5	1006	1006	NUM
ma-156	258	6	.	.	PUNCT
ma-156	259	1	https://doi.org/10.1016/j.jmaa.2005.10.081.[8	https://doi.org/10.1016/j.jmaa.2005.10.081.[8	PROPN
ma-156	259	2	]	]	PUNCT
ma-156	259	3	m.	m.	PROPN
ma-156	259	4	geraghty	geraghty	PROPN
ma-156	259	5	,	,	PUNCT
ma-156	259	6	on	on	ADP
ma-156	259	7	contractive	contractive	ADJ
ma-156	259	8	mappings	mapping	NOUN
ma-156	259	9	,	,	PUNCT
ma-156	259	10	proc	proc	NOUN
ma-156	259	11	.	.	PUNCT
ma-156	260	1	amer	amer	PROPN
ma-156	260	2	.	.	PUNCT
ma-156	260	3	math	math	PROPN
ma-156	260	4	.	.	PUNCT
ma-156	261	1	soc	soc	PROPN
ma-156	261	2	.	.	PUNCT
ma-156	262	1	40	40	NUM
ma-156	262	2	(	(	PUNCT
ma-156	262	3	1973	1973	NUM
ma-156	262	4	)	)	PUNCT
ma-156	262	5	,	,	PUNCT
ma-156	262	6	604	604	NUM
ma-156	262	7	-	-	SYM
ma-156	262	8	608.[9	608.[9	NUM
ma-156	262	9	]	]	PUNCT
ma-156	262	10	j.	j.	PROPN
ma-156	262	11	hamzehnejadi	hamzehnejadi	PROPN
ma-156	262	12	,	,	PUNCT
ma-156	262	13	r.	r.	PROPN
ma-156	262	14	lashkaripour	lashkaripour	PROPN
ma-156	262	15	,	,	PUNCT
ma-156	262	16	best	best	ADJ
ma-156	262	17	proximity	proximity	NOUN
ma-156	262	18	points	point	NOUN
ma-156	262	19	for	for	ADP
ma-156	262	20	generalized	generalized	ADJ
ma-156	262	21	α	α	PROPN
ma-156	262	22	-	-	PUNCT
ma-156	262	23	φ	φ	VERB
ma-156	262	24	-	-	PUNCT
ma-156	262	25	geraghty	geraghty	VERB
ma-156	262	26	proximal	proximal	ADJ
ma-156	262	27	contrac	contrac	ADJ
ma-156	262	28	-	-	PUNCT
ma-156	262	29	tion	tion	NOUN
ma-156	262	30	mappings	mapping	NOUN
ma-156	262	31	and	and	CCONJ
ma-156	262	32	its	its	PRON
ma-156	262	33	applications	application	NOUN
ma-156	262	34	,	,	PUNCT
ma-156	262	35	fixed	fix	VERB
ma-156	262	36	point	point	NOUN
ma-156	262	37	theory	theory	NOUN
ma-156	262	38	appl	appl	NOUN
ma-156	262	39	.	.	PUNCT
ma-156	263	1	2016	2016	NUM
ma-156	263	2	(	(	PUNCT
ma-156	263	3	2016	2016	NUM
ma-156	263	4	)	)	PUNCT
ma-156	263	5	,	,	PUNCT
ma-156	263	6	72	72	NUM
ma-156	263	7	.	.	PUNCT
ma-156	264	1	https://doi.org/10.1186/	https://doi.org/10.1186/	PROPN
ma-156	264	2	s13663	s13663	PROPN
ma-156	264	3	-	-	PUNCT
ma-156	264	4	016	016	NUM
ma-156	264	5	-	-	PUNCT
ma-156	264	6	0561	0561	NUM
ma-156	264	7	-	-	PUNCT
ma-156	264	8	0.[10	0.[10	NOUN
ma-156	264	9	]	]	PUNCT
ma-156	264	10	m.	m.	NOUN
ma-156	264	11	jleli	jleli	PROPN
ma-156	264	12	,	,	PUNCT
ma-156	264	13	e.	e.	PROPN
ma-156	264	14	karapinar	karapinar	PROPN
ma-156	264	15	,	,	PUNCT
ma-156	264	16	b.	b.	PROPN
ma-156	264	17	samet	samet	PROPN
ma-156	264	18	,	,	PUNCT
ma-156	264	19	best	good	ADJ
ma-156	264	20	proximity	proximity	NOUN
ma-156	264	21	point	point	NOUN
ma-156	264	22	for	for	ADP
ma-156	264	23	generalized	generalized	ADJ
ma-156	264	24	α	α	NOUN
ma-156	264	25	-	-	PUNCT
ma-156	264	26	ψ	ψ	ADJ
ma-156	264	27	-	-	ADJ
ma-156	264	28	proximal	proximal	ADJ
ma-156	264	29	contraction	contraction	NOUN
ma-156	264	30	type	type	NOUN
ma-156	264	31	mapping	mapping	NOUN
ma-156	264	32	,	,	PUNCT
ma-156	264	33	j.appl	j.appl	NOUN
ma-156	264	34	.	.	PUNCT
ma-156	265	1	math	math	NOUN
ma-156	265	2	.	.	PUNCT
ma-156	266	1	2013	2013	NUM
ma-156	266	2	(	(	PUNCT
ma-156	266	3	2013	2013	NUM
ma-156	266	4	)	)	PUNCT
ma-156	266	5	,	,	PUNCT
ma-156	266	6	534127	534127	NUM
ma-156	266	7	.	.	PUNCT
ma-156	267	1	https://doi.org/10.1155/2013/534127.[11	https://doi.org/10.1155/2013/534127.[11	NOUN
ma-156	267	2	]	]	PUNCT
ma-156	267	3	m.	m.	NOUN
ma-156	267	4	jleli	jleli	PROPN
ma-156	267	5	,	,	PUNCT
ma-156	267	6	b.	b.	PROPN
ma-156	267	7	samet	samet	PROPN
ma-156	267	8	,	,	PUNCT
ma-156	267	9	an	an	DET
ma-156	267	10	optimization	optimization	NOUN
ma-156	267	11	problem	problem	NOUN
ma-156	267	12	involving	involve	VERB
ma-156	267	13	proximal	proximal	ADJ
ma-156	267	14	quasi	quasi	ADJ
ma-156	267	15	-	-	NOUN
ma-156	267	16	contraction	contraction	NOUN
ma-156	267	17	mappings	mapping	NOUN
ma-156	267	18	,	,	PUNCT
ma-156	267	19	fixed	fix	VERB
ma-156	267	20	point	point	NOUN
ma-156	267	21	theoryappl	theoryappl	ADJ
ma-156	267	22	.	.	PUNCT
ma-156	268	1	2014	2014	NUM
ma-156	268	2	(	(	PUNCT
ma-156	268	3	2014	2014	NUM
ma-156	268	4	)	)	PUNCT
ma-156	268	5	,	,	PUNCT
ma-156	268	6	141	141	NUM
ma-156	268	7	.	.	PUNCT
ma-156	269	1	https://doi.org/10.1186/1687-1812-2014-141.[12	https://doi.org/10.1186/1687-1812-2014-141.[12	PROPN
ma-156	269	2	]	]	X
ma-156	269	3	e.	e.	PROPN
ma-156	269	4	karapinar	karapinar	PROPN
ma-156	269	5	,	,	PUNCT
ma-156	269	6	i.m	i.m	PROPN
ma-156	269	7	.	.	PROPN
ma-156	269	8	erhan	erhan	PROPN
ma-156	269	9	,	,	PUNCT
ma-156	269	10	best	good	ADJ
ma-156	269	11	proximity	proximity	NOUN
ma-156	269	12	point	point	NOUN
ma-156	269	13	on	on	ADP
ma-156	269	14	different	different	ADJ
ma-156	269	15	type	type	NOUN
ma-156	269	16	of	of	ADP
ma-156	269	17	contractions	contraction	NOUN
ma-156	269	18	,	,	PUNCT
ma-156	269	19	appl	appl	PROPN
ma-156	269	20	.	.	PROPN
ma-156	269	21	math	math	PROPN
ma-156	269	22	.	.	PUNCT
ma-156	270	1	inf	inf	PROPN
ma-156	270	2	.	.	PUNCT
ma-156	271	1	sci	sci	PROPN
ma-156	271	2	.	.	PROPN
ma-156	271	3	5	5	NUM
ma-156	271	4	(	(	PUNCT
ma-156	271	5	2011),558	2011),558	NUM
ma-156	271	6	-	-	SYM
ma-156	271	7	569.[13	569.[13	NUM
ma-156	271	8	]	]	PUNCT
ma-156	271	9	e.	e.	PROPN
ma-156	271	10	karapinar	karapinar	PROPN
ma-156	271	11	,	,	PUNCT
ma-156	271	12	on	on	ADP
ma-156	271	13	best	good	ADJ
ma-156	271	14	proximity	proximity	NOUN
ma-156	271	15	point	point	NOUN
ma-156	271	16	of	of	ADP
ma-156	271	17	ψ	ψ	NOUN
ma-156	271	18	-	-	ADJ
ma-156	271	19	geraghty	geraghty	VERB
ma-156	271	20	contractions	contraction	NOUN
ma-156	271	21	,	,	PUNCT
ma-156	271	22	fixed	fix	VERB
ma-156	271	23	point	point	NOUN
ma-156	271	24	theory	theory	NOUN
ma-156	271	25	appl	appl	NOUN
ma-156	271	26	.	.	PUNCT
ma-156	272	1	2013	2013	NUM
ma-156	272	2	(	(	PUNCT
ma-156	272	3	2013	2013	NUM
ma-156	272	4	)	)	PUNCT
ma-156	272	5	,	,	PUNCT
ma-156	272	6	200	200	NUM
ma-156	272	7	.	.	PUNCT
ma-156	273	1	https://doi.org/10.1186/1687-1812-2013-200	https://doi.org/10.1186/1687-1812-2013-200	PROPN
ma-156	273	2	.	.	PUNCT
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ma-156	274	4	https://doi.org/10.1007/s10957-011-9869-4	https://doi.org/10.1007/s10957-011-9869-4	NUM
ma-156	275	1	https://doi.org/10.1186/1029-242x-2013-286	https://doi.org/10.1186/1029-242x-2013-286	ADJ
ma-156	275	2	https://doi.org/10.1186/1687-1812-2012-231	https://doi.org/10.1186/1687-1812-2012-231	ADJ
ma-156	275	3	https://doi.org/10.1016/j.jmaa.2005.10.081	https://doi.org/10.1016/j.jmaa.2005.10.081	PROPN
ma-156	275	4	https://doi.org/10.1186/s13663-016-0561-0	https://doi.org/10.1186/s13663-016-0561-0	NUM
ma-156	275	5	https://doi.org/10.1186/s13663-016-0561-0	https://doi.org/10.1186/s13663-016-0561-0	NUM
ma-156	275	6	https://doi.org/10.1155/2013/534127	https://doi.org/10.1155/2013/534127	NOUN
ma-156	275	7	https://doi.org/10.1186/1687-1812-2014-141	https://doi.org/10.1186/1687-1812-2014-141	VERB
ma-156	275	8	https://doi.org/10.1186/1687-1812-2013-200	https://doi.org/10.1186/1687-1812-2013-200	PROPN
ma-156	275	9	eur	eur	PROPN
ma-156	275	10	.	.	PUNCT
ma-156	276	1	j.	j.	PROPN
ma-156	276	2	math	math	PROPN
ma-156	276	3	.	.	PUNCT
ma-156	277	1	anal	anal	PROPN
ma-156	277	2	.	.	PUNCT
ma-156	278	1	10.28924	10.28924	NUM
ma-156	278	2	/	/	SYM
ma-156	278	3	ada	ada	PROPN
ma-156	278	4	/	/	SYM
ma-156	278	5	ma.3.16	ma.3.16	PROPN
ma-156	278	6	11	11	NUM
ma-156	279	1	[	[	X
ma-156	279	2	14	14	NUM
ma-156	279	3	]	]	X
ma-156	279	4	w.a	w.a	PROPN
ma-156	279	5	.	.	PROPN
ma-156	279	6	kirk	kirk	PROPN
ma-156	279	7	,	,	PUNCT
ma-156	279	8	p.s	p.s	PROPN
ma-156	279	9	.	.	PUNCT
ma-156	279	10	srinavasan	srinavasan	PROPN
ma-156	279	11	,	,	PUNCT
ma-156	279	12	p.	p.	NOUN
ma-156	279	13	veeramani	veeramani	PROPN
ma-156	279	14	,	,	PUNCT
ma-156	279	15	fixed	fix	VERB
ma-156	279	16	points	point	NOUN
ma-156	279	17	for	for	ADP
ma-156	279	18	mapping	mapping	NOUN
ma-156	279	19	satisfying	satisfy	VERB
ma-156	279	20	cyclical	cyclical	ADJ
ma-156	279	21	contractive	contractive	ADJ
ma-156	279	22	conditions	condition	NOUN
ma-156	279	23	,	,	PUNCT
ma-156	279	24	fixedpoint	fixedpoint	NOUN
ma-156	279	25	theory	theory	NOUN
ma-156	279	26	.	.	PUNCT
ma-156	280	1	4	4	NUM
ma-156	280	2	(	(	PUNCT
ma-156	280	3	2003	2003	NUM
ma-156	280	4	)	)	PUNCT
ma-156	280	5	,	,	PUNCT
ma-156	281	1	79	79	NUM
ma-156	281	2	-	-	SYM
ma-156	281	3	89.[15	89.[15	PROPN
ma-156	281	4	]	]	X
ma-156	281	5	c.	c.	PROPN
ma-156	281	6	mongkolkeha	mongkolkeha	PROPN
ma-156	281	7	,	,	PUNCT
ma-156	281	8	y.j	y.j	PROPN
ma-156	281	9	.	.	PUNCT
ma-156	281	10	cho	cho	PROPN
ma-156	281	11	,	,	PUNCT
ma-156	281	12	p.	p.	PROPN
ma-156	281	13	kumam	kumam	PROPN
ma-156	281	14	,	,	PUNCT
ma-156	281	15	best	good	ADJ
ma-156	281	16	proximity	proximity	NOUN
ma-156	281	17	points	point	NOUN
ma-156	281	18	for	for	ADP
ma-156	281	19	geraghty	geraghty	PROPN
ma-156	281	20	’s	’s	PART
ma-156	281	21	proximal	proximal	ADJ
ma-156	281	22	contraction	contraction	NOUN
ma-156	281	23	mappings	mapping	NOUN
ma-156	281	24	,	,	PUNCT
ma-156	281	25	fixedpoint	fixedpoint	NOUN
ma-156	281	26	theory	theory	NOUN
ma-156	281	27	appl	appl	NOUN
ma-156	281	28	.	.	PUNCT
ma-156	282	1	2013	2013	NUM
ma-156	282	2	(	(	PUNCT
ma-156	282	3	2013	2013	NUM
ma-156	282	4	)	)	PUNCT
ma-156	282	5	,	,	PUNCT
ma-156	282	6	180	180	NUM
ma-156	282	7	.	.	PUNCT
ma-156	283	1	https://doi.org/10.1186/1687-1812-2013-180.[16	https://doi.org/10.1186/1687-1812-2013-180.[16	PROPN
ma-156	283	2	]	]	PUNCT
ma-156	284	1	j.	j.	PROPN
ma-156	284	2	olaleru	olaleru	PROPN
ma-156	284	3	,	,	PUNCT
ma-156	284	4	a	a	DET
ma-156	284	5	comparison	comparison	NOUN
ma-156	284	6	of	of	ADP
ma-156	284	7	picard	picard	PROPN
ma-156	284	8	and	and	CCONJ
ma-156	284	9	mann	mann	PROPN
ma-156	284	10	iterations	iteration	NOUN
ma-156	284	11	for	for	ADP
ma-156	284	12	quasi	quasi	ADJ
ma-156	284	13	-	-	NOUN
ma-156	284	14	contraction	contraction	NOUN
ma-156	284	15	maps	map	NOUN
ma-156	284	16	,	,	PUNCT
ma-156	284	17	fixed	fix	VERB
ma-156	284	18	point	point	NOUN
ma-156	284	19	theory	theory	NOUN
ma-156	284	20	.	.	PUNCT
ma-156	285	1	8	8	NUM
ma-156	285	2	(	(	PUNCT
ma-156	285	3	2007),87	2007),87	NUM
ma-156	285	4	-	-	PUNCT
ma-156	285	5	95.[17	95.[17	NUM
ma-156	285	6	]	]	X
ma-156	285	7	j.	j.	PROPN
ma-156	286	1	olaleru	olaleru	PROPN
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ma-156	290	1	:	:	PUNCT
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ma-156	291	10	-	-	SYM
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ma-156	292	2	(	(	PUNCT
ma-156	292	3	2017	2017	NUM
ma-156	292	4	)	)	PUNCT
ma-156	292	5	,	,	PUNCT
ma-156	292	6	6173468	6173468	NUM
ma-156	292	7	.	.	PUNCT
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ma-156	293	9	point	point	NOUN
ma-156	293	10	theorems	theorem	NOUN
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ma-156	293	13	-	-	PUNCT
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ma-156	293	16	type	type	NOUN
ma-156	293	17	maps	map	NOUN
ma-156	293	18	in	in	ADP
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ma-156	293	20	spaces	space	NOUN
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ma-156	293	25	.	.	PUNCT
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ma-156	294	2	(	(	PUNCT
ma-156	294	3	2014	2014	NUM
ma-156	294	4	)	)	PUNCT
ma-156	294	5	,	,	PUNCT
ma-156	294	6	190	190	NUM
ma-156	294	7	.	.	PUNCT
ma-156	295	1	https://doi.org/10.1186/1687-1812-2014-190.[20	https://doi.org/10.1186/1687-1812-2014-190.[20	PROPN
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ma-156	296	8	point	point	NOUN
ma-156	296	9	theorem	theorem	NOUN
ma-156	296	10	for	for	ADP
ma-156	296	11	weakly	weakly	ADJ
ma-156	296	12	contractive	contractive	ADJ
ma-156	296	13	non	non	ADJ
ma-156	296	14	-	-	ADJ
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ma-156	296	16	-	-	PUNCT
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ma-156	296	18	,	,	PUNCT
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ma-156	296	20	anal	anal	NOUN
ma-156	296	21	.	.	PUNCT
ma-156	297	1	:	:	PUNCT
ma-156	298	1	theorymeth	theorymeth	PROPN
ma-156	298	2	.	.	PUNCT
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ma-156	298	4	.	.	PUNCT
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ma-156	299	2	(	(	PUNCT
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ma-156	299	4	)	)	PUNCT
ma-156	299	5	,	,	PUNCT
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ma-156	299	7	-	-	SYM
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ma-156	299	9	.	.	PUNCT
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ma-156	300	11	definitions	definition	NOUN
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ma-156	300	15	,	,	PUNCT
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ma-156	300	17	.	.	PROPN
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ma-156	301	2	.	.	PUNCT
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ma-156	301	4	.	.	PUNCT
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ma-156	302	2	.	.	PUNCT
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ma-156	303	2	(	(	PUNCT
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ma-156	303	4	)	)	PUNCT
ma-156	303	5	,	,	PUNCT
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ma-156	303	7	-	-	SYM
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ma-156	303	11	.	.	PROPN
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ma-156	303	13	,	,	PUNCT
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ma-156	304	1	.	.	PROPN
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ma-156	304	3	,	,	PUNCT
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ma-156	305	2	(	(	PUNCT
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ma-156	305	4	)	)	PUNCT
ma-156	305	5	,	,	PUNCT
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ma-156	305	7	.	.	PUNCT
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ma-156	306	5	-	-	PUNCT
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ma-156	306	7	-	-	PUNCT
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ma-156	307	7	,	,	PUNCT
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ma-156	307	9	-	-	SYM
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ma-156	307	11	.	.	PUNCT
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ma-156	309	9	1	1	NUM
ma-156	309	10	.	.	PUNCT
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ma-156	309	13	.	.	PUNCT
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ma-156	310	8	:	:	PUNCT
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