id	sid	tid	token	lemma	pos
cana-3848	1	1	communications	communication	NOUN
cana-3848	1	2	on	on	ADP
cana-3848	1	3	applied	apply	VERB
cana-3848	1	4	nonlinear	nonlinear	ADJ
cana-3848	1	5	analysis	analysis	NOUN
cana-3848	1	6	issn	issn	NOUN
cana-3848	1	7	:	:	PUNCT
cana-3848	1	8	1074	1074	NUM
cana-3848	1	9	-	-	PUNCT
cana-3848	1	10	133x	133x	NUM
cana-3848	1	11	vol	vol	NOUN
cana-3848	1	12	32	32	NUM
cana-3848	1	13	no	no	NOUN
cana-3848	1	14	.	.	PUNCT
cana-3848	2	1	9s	9s	NUM
cana-3848	2	2	(	(	PUNCT
cana-3848	2	3	2025	2025	NUM
cana-3848	2	4	)	)	PUNCT
cana-3848	2	5	199	199	NUM
cana-3848	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	2	7	quasinormed	quasinorme	VERB
cana-3848	2	8	cones	cone	NOUN
cana-3848	2	9	and	and	CCONJ
cana-3848	2	10	bicompletion	bicompletion	NOUN
cana-3848	2	11	isometries	isometry	NOUN
cana-3848	2	12	manal	manal	PROPN
cana-3848	2	13	yagoub	yagoub	PROPN
cana-3848	2	14	ahmed	ahmed	PROPN
cana-3848	2	15	juma	juma	PROPN
cana-3848	2	16	department	department	PROPN
cana-3848	2	17	of	of	ADP
cana-3848	2	18	mathematic	mathematic	PROPN
cana-3848	2	19	,	,	PUNCT
cana-3848	2	20	college	college	NOUN
cana-3848	2	21	of	of	ADP
cana-3848	2	22	science	science	NOUN
cana-3848	2	23	,	,	PUNCT
cana-3848	2	24	qassim	qassim	PROPN
cana-3848	2	25	university	university	PROPN
cana-3848	2	26	,	,	PUNCT
cana-3848	2	27	buraidah	buraidah	PROPN
cana-3848	2	28	,	,	PUNCT
cana-3848	2	29	saudi	saudi	PROPN
cana-3848	2	30	arabia	arabia	PROPN
cana-3848	2	31	.	.	PUNCT
cana-3848	3	1	m.juma@qu.edu.sa	m.juma@qu.edu.sa	PROPN
cana-3848	3	2	article	article	NOUN
cana-3848	3	3	history	history	NOUN
cana-3848	3	4	:	:	PUNCT
cana-3848	3	5	received	receive	VERB
cana-3848	3	6	:	:	PUNCT
cana-3848	3	7	13	13	NUM
cana-3848	3	8	-	-	SYM
cana-3848	3	9	11	11	NUM
cana-3848	3	10	-	-	PUNCT
cana-3848	3	11	2024	2024	NUM
cana-3848	3	12	revised:25	revised:25	NOUN
cana-3848	3	13	-	-	PUNCT
cana-3848	3	14	12	12	NUM
cana-3848	3	15	-	-	PUNCT
cana-3848	3	16	2024	2024	NUM
cana-3848	3	17	accepted:09	accepted:09	NOUN
cana-3848	3	18	-	-	PUNCT
cana-3848	3	19	01	01	NUM
cana-3848	3	20	-	-	PUNCT
cana-3848	3	21	2025	2025	NUM
cana-3848	3	22	abstract	abstract	NOUN
cana-3848	3	23	:	:	PUNCT
cana-3848	3	24	since	since	SCONJ
cana-3848	3	25	(	(	PUNCT
cana-3848	3	26	x	x	X
cana-3848	3	27	,	,	PUNCT
cana-3848	3	28	e_(p_j	e_(p_j	PROPN
cana-3848	3	29	)	)	PUNCT
cana-3848	3	30	)	)	PUNCT
cana-3848	3	31	is	be	AUX
cana-3848	3	32	an	an	DET
cana-3848	3	33	extended	extended	ADJ
cana-3848	3	34	quasi	quasi	ADJ
cana-3848	3	35	-	-	ADJ
cana-3848	3	36	metric	metric	ADJ
cana-3848	3	37	cone	cone	NOUN
cana-3848	3	38	,	,	PUNCT
cana-3848	3	39	we	we	PRON
cana-3848	3	40	demonstrate	demonstrate	VERB
cana-3848	3	41	how	how	SCONJ
cana-3848	3	42	any	any	DET
cana-3848	3	43	quasinorm	quasinorm	NOUN
cana-3848	3	44	p_jon	p_jon	NOUN
cana-3848	3	45	an	an	DET
cana-3848	3	46	actual	actual	ADJ
cana-3848	3	47	cancellative	cancellative	ADJ
cana-3848	3	48	cone	cone	NOUN
cana-3848	3	49	x	x	PUNCT
cana-3848	3	50	naturally	naturally	ADV
cana-3848	3	51	implies	imply	VERB
cana-3848	3	52	an	an	DET
cana-3848	3	53	extended	extended	ADJ
cana-3848	3	54	quasi	quasi	ADJ
cana-3848	3	55	-	-	ADJ
cana-3848	3	56	metric	metric	ADJ
cana-3848	3	57	e_(p_j	e_(p_j	PROPN
cana-3848	3	58	)	)	PUNCT
cana-3848	3	59	on	on	ADP
cana-3848	3	60	that	that	DET
cana-3848	3	61	cone	cone	NOUN
cana-3848	3	62	.	.	PUNCT
cana-3848	4	1	we	we	PRON
cana-3848	4	2	demonstrate	demonstrate	VERB
cana-3848	4	3	that	that	SCONJ
cana-3848	4	4	bicompletion	bicompletion	NOUN
cana-3848	4	5	respects	respect	VERB
cana-3848	4	6	the	the	DET
cana-3848	4	7	structure	structure	NOUN
cana-3848	4	8	of	of	ADP
cana-3848	4	9	a	a	DET
cana-3848	4	10	quasi	quasi	NOUN
cana-3848	4	11	-	-	VERB
cana-3848	4	12	normalized	normalize	VERB
cana-3848	4	13	cone	cone	NOUN
cana-3848	4	14	under	under	ADP
cana-3848	4	15	bijective	bijective	ADJ
cana-3848	4	16	isometries	isometry	NOUN
cana-3848	4	17	.	.	PUNCT
cana-3848	5	1	in	in	ADP
cana-3848	5	2	fact	fact	NOUN
cana-3848	5	3	,	,	PUNCT
cana-3848	5	4	we	we	PRON
cana-3848	5	5	find	find	VERB
cana-3848	5	6	that	that	SCONJ
cana-3848	5	7	isometries	isometry	NOUN
cana-3848	5	8	are	be	AUX
cana-3848	5	9	not	not	PART
cana-3848	5	10	generally	generally	ADV
cana-3848	5	11	injective	injective	ADJ
cana-3848	5	12	in	in	ADP
cana-3848	5	13	this	this	DET
cana-3848	5	14	case	case	NOUN
cana-3848	5	15	.	.	PUNCT
cana-3848	6	1	in	in	ADP
cana-3848	6	2	addition	addition	NOUN
cana-3848	6	3	,	,	PUNCT
cana-3848	6	4	a	a	DET
cana-3848	6	5	few	few	ADJ
cana-3848	6	6	situations	situation	NOUN
cana-3848	6	7	are	be	AUX
cana-3848	6	8	shown	show	VERB
cana-3848	6	9	.	.	PUNCT
cana-3848	7	1	key	key	ADJ
cana-3848	7	2	words	word	NOUN
cana-3848	7	3	:	:	PUNCT
cana-3848	7	4	bicompletion	bicompletion	NOUN
cana-3848	7	5	,	,	PUNCT
cana-3848	7	6	quasi	quasi	ADJ
cana-3848	7	7	-	-	NOUN
cana-3848	7	8	cone	cone	ADJ
cana-3848	7	9	,	,	PUNCT
cana-3848	7	10	calculative	calculative	ADJ
cana-3848	7	11	,	,	PUNCT
cana-3848	7	12	bijective	bijective	ADJ
cana-3848	7	13	isometry	isometry	NOUN
cana-3848	7	14	,	,	PUNCT
cana-3848	7	15	injective	injective	ADJ
cana-3848	7	16	.	.	PUNCT
cana-3848	8	1	1	1	X
cana-3848	8	2	.	.	X
cana-3848	8	3	introduction	introduction	NOUN
cana-3848	8	4	and	and	CCONJ
cana-3848	8	5	preliminary	preliminary	ADJ
cana-3848	8	6	information	information	NOUN
cana-3848	8	7	the	the	DET
cana-3848	8	8	letters	letter	NOUN
cana-3848	8	9	ℝ+	ℝ+	ADP
cana-3848	8	10	,	,	PUNCT
cana-3848	8	11	𝜔	𝜔	ADP
cana-3848	8	12	,	,	PUNCT
cana-3848	8	13	and	and	CCONJ
cana-3848	8	14	n	n	PRON
cana-3848	8	15	will	will	AUX
cana-3848	8	16	be	be	AUX
cana-3848	8	17	used	use	VERB
cana-3848	8	18	for	for	ADP
cana-3848	8	19	the	the	DET
cana-3848	8	20	sets	set	NOUN
cana-3848	8	21	of	of	ADP
cana-3848	8	22	nonnegative	nonnegative	ADJ
cana-3848	8	23	real	real	ADJ
cana-3848	8	24	numbers	number	NOUN
cana-3848	8	25	,	,	PUNCT
cana-3848	8	26	nonnegative	nonnegative	ADJ
cana-3848	8	27	integer	integer	NOUN
cana-3848	8	28	numbers	number	NOUN
cana-3848	8	29	,	,	PUNCT
cana-3848	8	30	and	and	CCONJ
cana-3848	8	31	positive	positive	ADJ
cana-3848	8	32	integer	integer	NOUN
cana-3848	8	33	numbers	number	NOUN
cana-3848	8	34	,	,	PUNCT
cana-3848	8	35	respectively	respectively	ADV
cana-3848	8	36	,	,	PUNCT
cana-3848	8	37	throughout	throughout	ADP
cana-3848	8	38	this	this	DET
cana-3848	8	39	essay	essay	NOUN
cana-3848	8	40	.	.	PUNCT
cana-3848	9	1	the	the	DET
cana-3848	9	2	letters	letter	NOUN
cana-3848	9	3	ℝ+	ℝ+	ADP
cana-3848	9	4	,	,	PUNCT
cana-3848	9	5	𝜔	𝜔	ADP
cana-3848	9	6	,	,	PUNCT
cana-3848	9	7	and	and	CCONJ
cana-3848	9	8	n	n	PRON
cana-3848	9	9	will	will	AUX
cana-3848	9	10	be	be	AUX
cana-3848	9	11	used	use	VERB
cana-3848	9	12	for	for	ADP
cana-3848	9	13	the	the	DET
cana-3848	9	14	sets	set	NOUN
cana-3848	9	15	of	of	ADP
cana-3848	9	16	nonnegative	nonnegative	ADJ
cana-3848	9	17	real	real	ADJ
cana-3848	9	18	numbers	number	NOUN
cana-3848	9	19	,	,	PUNCT
cana-3848	9	20	nonnegative	nonnegative	ADJ
cana-3848	9	21	integer	integer	NOUN
cana-3848	9	22	numbers	number	NOUN
cana-3848	9	23	,	,	PUNCT
cana-3848	9	24	and	and	CCONJ
cana-3848	9	25	positive	positive	ADJ
cana-3848	9	26	integer	integer	NOUN
cana-3848	9	27	numbers	number	NOUN
cana-3848	9	28	,	,	PUNCT
cana-3848	9	29	respectively	respectively	ADV
cana-3848	9	30	,	,	PUNCT
cana-3848	9	31	throughout	throughout	ADP
cana-3848	9	32	this	this	DET
cana-3848	9	33	essay	essay	NOUN
cana-3848	9	34	.	.	PUNCT
cana-3848	10	1	remind	remind	VERB
cana-3848	10	2	that	that	SCONJ
cana-3848	10	3	a	a	DET
cana-3848	10	4	semigroup	semigroup	NOUN
cana-3848	10	5	(	(	PUNCT
cana-3848	10	6	𝑋	𝑋	NOUN
cana-3848	10	7	,	,	PUNCT
cana-3848	10	8	ℝ+	ℝ+	PUNCT
cana-3848	10	9	)	)	PUNCT
cana-3848	10	10	is	be	AUX
cana-3848	10	11	a	a	DET
cana-3848	10	12	monoid	monoid	NOUN
cana-3848	10	13	if	if	SCONJ
cana-3848	10	14	its	its	PRON
cana-3848	10	15	neutral	neutral	ADJ
cana-3848	10	16	element	element	NOUN
cana-3848	10	17	is	be	AUX
cana-3848	10	18	0	0	NUM
cana-3848	10	19	[	[	X
cana-3848	10	20	2	2	NUM
cana-3848	10	21	]	]	PUNCT
cana-3848	10	22	.	.	PUNCT
cana-3848	11	1	for	for	ADP
cana-3848	11	2	each	each	DET
cana-3848	11	3	𝑥	𝑥	PROPN
cana-3848	11	4	,	,	PUNCT
cana-3848	11	5	𝑦	𝑦	PRON
cana-3848	11	6	𝑖𝑛	𝑖𝑛	PRON
cana-3848	11	7	𝑋	𝑋	NOUN
cana-3848	11	8	and	and	CCONJ
cana-3848	11	9	𝑟	𝑟	NOUN
cana-3848	11	10	,	,	PUNCT
cana-3848	11	11	𝑠	𝑠	INTJ
cana-3848	11	12	𝑖𝑛	𝑖𝑛	NOUN
cana-3848	11	13	ℝ+	ℝ+	PUNCT
cana-3848	11	14	,	,	PUNCT
cana-3848	11	15	the	the	DET
cana-3848	11	16	cone	cone	NOUN
cana-3848	11	17	(	(	PUNCT
cana-3848	11	18	on	on	ADP
cana-3848	11	19	ℝ+	ℝ+	ADJ
cana-3848	11	20	)	)	PUNCT
cana-3848	11	21	is	be	AUX
cana-3848	11	22	defined	define	VERB
cana-3848	11	23	as	as	ADP
cana-3848	11	24	a	a	DET
cana-3848	11	25	triple	triple	ADJ
cana-3848	11	26	(	(	PUNCT
cana-3848	11	27	(	(	PUNCT
cana-3848	11	28	𝑋	𝑋	PROPN
cana-3848	11	29	,	,	PUNCT
cana-3848	11	30	ℝ+,⋅	ℝ+,⋅	PROPN
cana-3848	11	31	)	)	PUNCT
cana-3848	11	32	where	where	SCONJ
cana-3848	11	33	(	(	PUNCT
cana-3848	11	34	𝑋	𝑋	NOUN
cana-3848	11	35	,	,	PUNCT
cana-3848	11	36	+	+	PROPN
cana-3848	11	37	)	)	PUNCT
cana-3848	11	38	is	be	AUX
cana-3848	11	39	an	an	DET
cana-3848	11	40	abelian	abelian	ADJ
cana-3848	11	41	monoid	monoid	NOUN
cana-3848	11	42	and	and	CCONJ
cana-3848	11	43	⋅	⋅	PROPN
cana-3848	11	44	is	be	AUX
cana-3848	11	45	a	a	DET
cana-3848	11	46	function	function	NOUN
cana-3848	11	47	from	from	ADP
cana-3848	11	48	ℝ+	ℝ+	PUNCT
cana-3848	11	49	×	×	PROPN
cana-3848	11	50	𝑋	𝑋	NOUN
cana-3848	11	51	to	to	ADP
cana-3848	11	52	𝑋.	𝑋.	PROPN
cana-3848	11	53	(	(	PUNCT
cana-3848	11	54	𝑎	𝑎	NOUN
cana-3848	11	55	)	)	PUNCT
cana-3848	11	56	𝑟	𝑟	NOUN
cana-3848	12	1	⋅	⋅	X
cana-3848	12	2	(	(	PUNCT
cana-3848	12	3	𝑠	𝑠	PROPN
cana-3848	12	4	⋅	⋅	PROPN
cana-3848	12	5	𝑥	𝑥	PROPN
cana-3848	12	6	)	)	PUNCT
cana-3848	12	7	=	=	SYM
cana-3848	12	8	(	(	PUNCT
cana-3848	12	9	𝑟𝑠	𝑟𝑠	NOUN
cana-3848	12	10	)	)	PUNCT
cana-3848	12	11	⋅	⋅	PROPN
cana-3848	12	12	𝑥	𝑥	NOUN
cana-3848	12	13	;	;	PUNCT
cana-3848	12	14	(	(	PUNCT
cana-3848	12	15	b	b	X
cana-3848	12	16	)	)	PUNCT
cana-3848	12	17	𝑟	𝑟	NOUN
cana-3848	12	18	⋅	⋅	X
cana-3848	12	19	(	(	PUNCT
cana-3848	12	20	𝑥	𝑥	PROPN
cana-3848	12	21	+	+	NUM
cana-3848	12	22	𝑦	𝑦	X
cana-3848	12	23	)	)	PUNCT
cana-3848	12	24	=	=	SYM
cana-3848	12	25	(	(	PUNCT
cana-3848	12	26	𝑟	𝑟	NOUN
cana-3848	12	27	⋅	⋅	PROPN
cana-3848	12	28	𝑥	𝑥	NOUN
cana-3848	12	29	)	)	PUNCT
cana-3848	12	30	+	+	CCONJ
cana-3848	12	31	(	(	PUNCT
cana-3848	12	32	𝑟	𝑟	NOUN
cana-3848	12	33	⋅	⋅	PROPN
cana-3848	12	34	𝑦	𝑦	NUM
cana-3848	12	35	)	)	PUNCT
cana-3848	12	36	;	;	PUNCT
cana-3848	12	37	(	(	PUNCT
cana-3848	12	38	c	c	X
cana-3848	12	39	)	)	PUNCT
cana-3848	12	40	(	(	PUNCT
cana-3848	12	41	𝑟	𝑟	X
cana-3848	12	42	+	+	SYM
cana-3848	12	43	𝑠	𝑠	X
cana-3848	12	44	)	)	PUNCT
cana-3848	12	45	⋅	⋅	PROPN
cana-3848	12	46	𝑥	𝑥	NOUN
cana-3848	12	47	=	=	SYM
cana-3848	12	48	(	(	PUNCT
cana-3848	12	49	𝑟	𝑟	NOUN
cana-3848	12	50	⋅	⋅	PROPN
cana-3848	12	51	𝑥	𝑥	NOUN
cana-3848	12	52	)	)	PUNCT
cana-3848	12	53	+	+	CCONJ
cana-3848	12	54	(	(	PUNCT
cana-3848	12	55	𝑠	𝑠	PROPN
cana-3848	12	56	⋅	⋅	PROPN
cana-3848	12	57	𝑥	𝑥	PROPN
cana-3848	12	58	)	)	PUNCT
cana-3848	12	59	;	;	PUNCT
cana-3848	12	60	(	(	PUNCT
cana-3848	12	61	𝑑)1	𝑑)1	NOUN
cana-3848	12	62	⋅	⋅	PROPN
cana-3848	12	63	𝑥	𝑥	NOUN
cana-3848	12	64	=	=	PUNCT
cana-3848	12	65	𝑥.	𝑥.	DET
cana-3848	12	66	every	every	DET
cana-3848	12	67	element	element	NOUN
cana-3848	12	68	𝑥	𝑥	DET
cana-3848	12	69	∈	∈	PROPN
cana-3848	12	70	𝑋	𝑋	NOUN
cana-3848	12	71	that	that	PRON
cana-3848	12	72	allows	allow	VERB
cana-3848	12	73	an	an	DET
cana-3848	12	74	inverse	inverse	NOUN
cana-3848	12	75	is	be	AUX
cana-3848	12	76	distinct	distinct	ADJ
cana-3848	12	77	and	and	CCONJ
cana-3848	12	78	is	be	AUX
cana-3848	12	79	denoted	denote	VERB
cana-3848	12	80	by	by	ADP
cana-3848	12	81	-x	-x	PROPN
cana-3848	12	82	,	,	PUNCT
cana-3848	12	83	as	as	SCONJ
cana-3848	12	84	is	be	AUX
cana-3848	12	85	normal	normal	ADJ
cana-3848	12	86	.	.	PUNCT
cana-3848	13	1	when	when	SCONJ
cana-3848	13	2	y	y	PROPN
cana-3848	13	3	is	be	AUX
cana-3848	13	4	a	a	DET
cana-3848	13	5	part	part	NOUN
cana-3848	13	6	of	of	ADP
cana-3848	13	7	x	x	X
cana-3848	13	8	and	and	CCONJ
cana-3848	13	9	+	+	NOUN
cana-3848	13	10	|𝑌	|𝑌	PROPN
cana-3848	13	11	and	and	CCONJ
cana-3848	13	12	⋅|𝑌are	⋅|𝑌are	PROPN
cana-3848	13	13	the	the	DET
cana-3848	13	14	limits	limit	NOUN
cana-3848	13	15	of	of	ADP
cana-3848	13	16	+	+	CCONJ
cana-3848	13	17	and	and	CCONJ
cana-3848	13	18	to	to	ADP
cana-3848	13	19	𝑌	𝑌	PROPN
cana-3848	13	20	,	,	PUNCT
cana-3848	13	21	respectively	respectively	ADV
cana-3848	13	22	,	,	PUNCT
cana-3848	13	23	the	the	DET
cana-3848	13	24	cone	cone	NOUN
cana-3848	13	25	(	(	PUNCT
cana-3848	13	26	𝑌	𝑌	PROPN
cana-3848	13	27	,	,	PUNCT
cana-3848	13	28	+	+	NOUN
cana-3848	13	29	|𝑌	|𝑌	NOUN
cana-3848	13	30	,	,	PUNCT
cana-3848	13	31	⋅|𝑌	⋅|𝑌	PROPN
cana-3848	13	32	)	)	PUNCT
cana-3848	13	33	)	)	PUNCT
cana-3848	13	34	is	be	AUX
cana-3848	13	35	said	say	VERB
cana-3848	13	36	to	to	PART
cana-3848	13	37	be	be	AUX
cana-3848	13	38	a	a	DET
cana-3848	13	39	subcone	subcone	NOUN
cana-3848	13	40	of	of	ADP
cana-3848	13	41	a	a	DET
cana-3848	13	42	cone	cone	NOUN
cana-3848	13	43	(	(	PUNCT
cana-3848	13	44	𝑋	𝑋	PROPN
cana-3848	13	45	,	,	PUNCT
cana-3848	13	46	+	+	PROPN
cana-3848	13	47	,	,	PUNCT
cana-3848	13	48	⋅).assume	⋅).assume	PROPN
cana-3848	13	49	that	that	SCONJ
cana-3848	13	50	(	(	PUNCT
cana-3848	13	51	𝑋	𝑋	PROPN
cana-3848	13	52	,	,	PUNCT
cana-3848	13	53	+	+	PROPN
cana-3848	13	54	,	,	PUNCT
cana-3848	13	55	⋅	⋅	PROPN
cana-3848	13	56	)	)	PUNCT
cana-3848	13	57	is	be	AUX
cana-3848	13	58	a	a	DET
cana-3848	13	59	cone	cone	NOUN
cana-3848	13	60	.	.	PUNCT
cana-3848	14	1	as	as	ADV
cana-3848	14	2	long	long	ADV
cana-3848	14	3	as	as	ADP
cana-3848	14	4	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	14	5	+	+	CCONJ
cana-3848	14	6	𝑦	𝑦	X
cana-3848	14	7	)	)	PUNCT
cana-3848	14	8	≤	≤	NOUN
cana-3848	14	9	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	14	10	)	)	PUNCT
cana-3848	15	1	+	+	CCONJ
cana-3848	15	2	𝑓𝑗(𝑦	𝑓𝑗(𝑦	X
cana-3848	15	3	)	)	PUNCT
cana-3848	15	4	definition	definition	NOUN
cana-3848	15	5	(	(	PUNCT
cana-3848	15	6	1.1	1.1	NUM
cana-3848	15	7	):	):	PUNCT
cana-3848	15	8	the	the	DET
cana-3848	15	9	function	function	NOUN
cana-3848	15	10	𝑓𝑗	𝑓𝑗	NOUN
cana-3848	15	11	∶	∶	NOUN
cana-3848	15	12	𝑋	𝑋	PROPN
cana-3848	15	13	→	→	PUNCT
cana-3848	15	14	ℝ	ℝ	PROPN
cana-3848	15	15	is	be	AUX
cana-3848	15	16	considered	consider	VERB
cana-3848	15	17	subadditive	subadditive	ADJ
cana-3848	15	18	for	for	ADP
cana-3848	15	19	any	any	DET
cana-3848	15	20	𝑥	𝑥	PROPN
cana-3848	15	21	,	,	PUNCT
cana-3848	15	22	𝑦	𝑦	NOUN
cana-3848	15	23	∈	∈	PROPN
cana-3848	15	24	𝑋.	𝑋.	PROPN
cana-3848	15	25	for	for	ADP
cana-3848	15	26	any	any	DET
cana-3848	15	27	x	x	NOUN
cana-3848	15	28	in	in	ADP
cana-3848	15	29	𝑋	𝑋	NOUN
cana-3848	15	30	and	and	CCONJ
cana-3848	15	31	𝑟	𝑟	PRON
cana-3848	15	32	𝑖𝑛	𝑖𝑛	INTJ
cana-3848	15	33	,	,	PUNCT
cana-3848	15	34	ℝ+	ℝ+	ADP
cana-3848	15	35	,	,	PUNCT
cana-3848	15	36	a	a	DET
cana-3848	15	37	quasi	quasi	NOUN
cana-3848	15	38	-	-	NOUN
cana-3848	15	39	norm	norm	NOUN
cana-3848	15	40	on	on	ADP
cana-3848	15	41	a	a	DET
cana-3848	15	42	cone	cone	NOUN
cana-3848	15	43	(	(	PUNCT
cana-3848	15	44	x,+,⋅	x,+,⋅	PROPN
cana-3848	15	45	)	)	PUNCT
cana-3848	15	46	is	be	AUX
cana-3848	15	47	a	a	DET
cana-3848	15	48	subadditive	subadditive	ADJ
cana-3848	15	49	function	function	NOUN
cana-3848	15	50	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	15	51	∶	∶	NOUN
cana-3848	15	52	𝑋	𝑋	PROPN
cana-3848	15	53	→	→	PUNCT
cana-3848	16	1	ℝ+	ℝ+	PUNCT
cana-3848	16	2	such	such	ADJ
cana-3848	16	3	that	that	PRON
cana-3848	16	4	:	:	PUNCT
cana-3848	16	5	(	(	PUNCT
cana-3848	16	6	a	a	X
cana-3848	16	7	)	)	PUNCT
cana-3848	16	8	𝑥	𝑥	NOUN
cana-3848	17	1	=	=	SYM
cana-3848	17	2	0	0	PUNCT
cana-3848	18	1	if	if	SCONJ
cana-3848	18	2	and	and	CCONJ
cana-3848	18	3	only	only	ADV
cana-3848	18	4	if	if	SCONJ
cana-3848	18	5	−𝑥	−𝑥	NOUN
cana-3848	18	6	∈	∈	PROPN
cana-3848	18	7	𝑋	𝑋	NOUN
cana-3848	18	8	and	and	CCONJ
cana-3848	18	9	𝑝𝑗(𝑥	𝑝𝑗(𝑥	NOUN
cana-3848	18	10	)	)	PUNCT
cana-3848	18	11	=	=	SYM
cana-3848	18	12	𝑝𝑗(−𝑥	𝑝𝑗(−𝑥	NOUN
cana-3848	18	13	)	)	PUNCT
cana-3848	18	14	=	=	SYM
cana-3848	19	1	0,and	0,and	NUM
cana-3848	19	2	(	(	PUNCT
cana-3848	19	3	b	b	NOUN
cana-3848	19	4	)	)	PUNCT
cana-3848	19	5	𝑝𝑗(𝑟	𝑝𝑗(𝑟	NOUN
cana-3848	19	6	⋅	⋅	PROPN
cana-3848	19	7	𝑥	𝑥	NOUN
cana-3848	19	8	)	)	PUNCT
cana-3848	19	9	=	=	SYM
cana-3848	19	10	𝑟𝑝𝑗(𝑥	𝑟𝑝𝑗(𝑥	PROPN
cana-3848	19	11	)	)	PUNCT
cana-3848	19	12	.	.	PUNCT
cana-3848	20	1	definition	definition	NOUN
cana-3848	20	2	(	(	PUNCT
cana-3848	20	3	2.1	2.1	NUM
cana-3848	20	4	):	):	PUNCT
cana-3848	20	5	a	a	DET
cana-3848	20	6	quasi	quasi	NOUN
cana-3848	20	7	-	-	NOUN
cana-3848	20	8	norm	norm	ADJ
cana-3848	20	9	p	p	NOUN
cana-3848	20	10	on	on	ADP
cana-3848	20	11	𝑋	𝑋	PROPN
cana-3848	20	12	is	be	AUX
cana-3848	20	13	a	a	DET
cana-3848	20	14	norm	norm	NOUN
cana-3848	20	15	on	on	ADP
cana-3848	20	16	a	a	DET
cana-3848	20	17	cone	cone	NOUN
cana-3848	20	18	(	(	PUNCT
cana-3848	20	19	𝑋	𝑋	PROPN
cana-3848	20	20	,	,	PUNCT
cana-3848	20	21	+	+	PROPN
cana-3848	20	22	,	,	PUNCT
cana-3848	20	23	⋅	⋅	PROPN
cana-3848	20	24	)	)	PUNCT
cana-3848	20	25	if	if	SCONJ
cana-3848	20	26	it	it	PRON
cana-3848	20	27	satisfies	satisfy	VERB
cana-3848	20	28	the	the	DET
cana-3848	20	29	following	follow	VERB
cana-3848	20	30	needs	need	NOUN
cana-3848	20	31	:	:	PUNCT
cana-3848	20	32	communications	communication	NOUN
cana-3848	20	33	on	on	ADP
cana-3848	20	34	applied	apply	VERB
cana-3848	20	35	nonlinear	nonlinear	ADJ
cana-3848	20	36	analysis	analysis	NOUN
cana-3848	20	37	issn	issn	NOUN
cana-3848	20	38	:	:	PUNCT
cana-3848	20	39	1074	1074	NUM
cana-3848	20	40	-	-	PUNCT
cana-3848	20	41	133x	133x	NUM
cana-3848	20	42	vol	vol	NOUN
cana-3848	20	43	32	32	NUM
cana-3848	21	1	no	no	NOUN
cana-3848	21	2	.	.	PUNCT
cana-3848	22	1	9s	9s	NUM
cana-3848	22	2	(	(	PUNCT
cana-3848	22	3	2025	2025	NUM
cana-3848	22	4	)	)	PUNCT
cana-3848	22	5	200	200	NUM
cana-3848	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	22	7	𝑝𝑗(𝑥	𝑝𝑗(𝑥	X
cana-3848	22	8	)	)	PUNCT
cana-3848	22	9	=	=	SYM
cana-3848	22	10	0	0	PUNCT
cana-3848	22	11	if	if	SCONJ
cana-3848	22	12	only	only	ADV
cana-3848	22	13	if	if	SCONJ
cana-3848	22	14	𝑥	𝑥	PROPN
cana-3848	22	15	=	=	SYM
cana-3848	22	16	0	0	X
cana-3848	22	17	.	.	PUNCT
cana-3848	23	1	definition	definition	NOUN
cana-3848	23	2	(	(	PUNCT
cana-3848	23	3	3.1	3.1	NUM
cana-3848	23	4	):	):	PUNCT
cana-3848	23	5	the	the	DET
cana-3848	23	6	cancellative	cancellative	ADJ
cana-3848	23	7	cone	cone	NOUN
cana-3848	23	8	(	(	PUNCT
cana-3848	23	9	𝑋	𝑋	PROPN
cana-3848	23	10	,	,	PUNCT
cana-3848	23	11	ℝ+,⋅)is	ℝ+,⋅)is	PROPN
cana-3848	23	12	defined	define	VERB
cana-3848	23	13	as	as	SCONJ
cana-3848	23	14	follows	follow	VERB
cana-3848	23	15	:	:	PUNCT
cana-3848	23	16	𝑥	𝑥	NOUN
cana-3848	23	17	,	,	PUNCT
cana-3848	23	18	𝑦	𝑦	NOUN
cana-3848	23	19	,	,	PUNCT
cana-3848	23	20	𝑧	𝑧	DET
cana-3848	23	21	∈	∈	PROPN
cana-3848	23	22	𝑋	𝑋	PROPN
cana-3848	23	23	,	,	PUNCT
cana-3848	23	24	𝑧	𝑧	PROPN
cana-3848	23	25	+	+	X
cana-3848	23	26	𝑥	𝑥	NOUN
cana-3848	23	27	=	=	X
cana-3848	23	28	𝑧	𝑧	NOUN
cana-3848	23	29	+	+	NOUN
cana-3848	23	30	𝑦	𝑦	NOUN
cana-3848	23	31	implies	imply	VERB
cana-3848	23	32	𝑥	𝑥	X
cana-3848	23	33	=	=	SYM
cana-3848	23	34	𝑦	𝑦	NOUN
cana-3848	23	35	for	for	ADP
cana-3848	23	36	any	any	DET
cana-3848	23	37	𝑥	𝑥	PROPN
cana-3848	23	38	,	,	PUNCT
cana-3848	23	39	𝑦	𝑦	NOUN
cana-3848	23	40	,	,	PUNCT
cana-3848	23	41	𝑧	𝑧	DET
cana-3848	23	42	∈	∈	PROPN
cana-3848	23	43	𝑋.	𝑋.	PROPN
cana-3848	23	44	one	one	NUM
cana-3848	23	45	relatively	relatively	ADV
cana-3848	23	46	direct	direct	ADJ
cana-3848	23	47	way	way	NOUN
cana-3848	23	48	of	of	ADP
cana-3848	23	49	interpreting	interpret	VERB
cana-3848	23	50	any	any	DET
cana-3848	23	51	linear	linear	ADJ
cana-3848	23	52	space	space	NOUN
cana-3848	23	53	x	x	NOUN
cana-3848	23	54	,	,	PUNCT
cana-3848	23	55	r+	r+	X
cana-3848	23	56	,	,	PUNCT
cana-3848	23	57	·	·	PUNCT
cana-3848	23	58	as	as	SCONJ
cana-3848	23	59	cone	cone	NOUN
cana-3848	23	60	consists	consist	VERB
cana-3848	23	61	in	in	ADP
cana-3848	23	62	the	the	DET
cana-3848	23	63	fact	fact	NOUN
cana-3848	23	64	that	that	SCONJ
cana-3848	23	65	the	the	DET
cana-3848	23	66	operation	operation	NOUN
cana-3848	23	67	·	·	PUNCT
cana-3848	23	68	has	have	AUX
cana-3848	23	69	to	to	PART
cana-3848	23	70	be	be	AUX
cana-3848	23	71	confined	confine	VERB
cana-3848	23	72	to	to	ADP
cana-3848	23	73	r+	r+	NOUN
cana-3848	23	74	×	×	PROPN
cana-3848	23	75	x.	x.	NOUN
cana-3848	23	76	as	as	SCONJ
cana-3848	23	77	is	be	AUX
cana-3848	23	78	well	well	ADV
cana-3848	23	79	documented	document	VERB
cana-3848	23	80	,	,	PUNCT
cana-3848	23	81	every	every	DET
cana-3848	23	82	norm	norm	NOUN
cana-3848	23	83	on	on	ADP
cana-3848	23	84	a	a	DET
cana-3848	23	85	space	space	NOUN
cana-3848	23	86	x	x	PRON
cana-3848	23	87	generates	generate	VERB
cana-3848	23	88	a	a	DET
cana-3848	23	89	metric	metric	NOUN
cana-3848	23	90	on	on	ADP
cana-3848	23	91	x.	x.	NOUN
cana-3848	24	1	we	we	PRON
cana-3848	24	2	continue	continue	VERB
cana-3848	24	3	this	this	DET
cana-3848	24	4	useful	useful	ADJ
cana-3848	24	5	result	result	NOUN
cana-3848	24	6	by	by	ADP
cana-3848	24	7	showing	show	VERB
cana-3848	24	8	how	how	SCONJ
cana-3848	24	9	quasi	quasi	ADJ
cana-3848	24	10	-	-	NOUN
cana-3848	24	11	metrics	metric	NOUN
cana-3848	24	12	can	can	AUX
cana-3848	24	13	naturally	naturally	ADV
cana-3848	24	14	arise	arise	VERB
cana-3848	24	15	from	from	ADP
cana-3848	24	16	quasi	quasi	NOUN
cana-3848	24	17	-	-	NOUN
cana-3848	24	18	norms	norm	NOUN
cana-3848	24	19	on	on	ADP
cana-3848	24	20	cancellative	cancellative	ADJ
cana-3848	24	21	cones	cone	NOUN
cana-3848	24	22	.	.	PUNCT
cana-3848	25	1	in	in	ADP
cana-3848	25	2	this	this	DET
cana-3848	25	3	work	work	NOUN
cana-3848	25	4	,	,	PUNCT
cana-3848	25	5	we	we	PRON
cana-3848	25	6	analyze	analyze	VERB
cana-3848	25	7	the	the	DET
cana-3848	25	8	bicompletion	bicompletion	NOUN
cana-3848	25	9	of	of	ADP
cana-3848	25	10	these	these	DET
cana-3848	25	11	structures	structure	NOUN
cana-3848	25	12	for	for	ADP
cana-3848	25	13	bijective	bijective	ADJ
cana-3848	25	14	isometries	isometry	NOUN
cana-3848	25	15	.	.	PUNCT
cana-3848	26	1	this	this	PRON
cana-3848	26	2	we	we	PRON
cana-3848	26	3	prove	prove	VERB
cana-3848	26	4	to	to	PART
cana-3848	26	5	be	be	AUX
cana-3848	26	6	a	a	DET
cana-3848	26	7	noninjective	noninjective	ADJ
cana-3848	26	8	isometry	isometry	NOUN
cana-3848	26	9	between	between	ADP
cana-3848	26	10	quasi	quasi	ADJ
cana-3848	26	11	-	-	ADJ
cana-3848	26	12	normed	normed	ADJ
cana-3848	26	13	cones	cone	NOUN
cana-3848	26	14	.	.	PUNCT
cana-3848	27	1	we	we	PRON
cana-3848	27	2	note	note	VERB
cana-3848	27	3	that	that	SCONJ
cana-3848	27	4	the	the	DET
cana-3848	27	5	standard	standard	ADJ
cana-3848	27	6	sorgenfrey	sorgenfrey	ADJ
cana-3848	27	7	line	line	NOUN
cana-3848	27	8	on	on	ADP
cana-3848	27	9	r+	r+	NOUN
cana-3848	27	10	can	can	AUX
cana-3848	27	11	be	be	AUX
cana-3848	27	12	obtained	obtain	VERB
cana-3848	27	13	through	through	ADP
cana-3848	27	14	the	the	DET
cana-3848	27	15	extended	extended	ADJ
cana-3848	27	16	quasi	quasi	ADJ
cana-3848	27	17	-	-	ADJ
cana-3848	27	18	metric	metric	ADJ
cana-3848	27	19	created	create	VERB
cana-3848	27	20	by	by	ADP
cana-3848	27	21	a	a	DET
cana-3848	27	22	norm	norm	NOUN
cana-3848	27	23	on	on	ADP
cana-3848	27	24	r+	r+	NOUN
cana-3848	27	25	itself	itself	PRON
cana-3848	27	26	.	.	PUNCT
cana-3848	28	1	moreover	moreover	ADV
cana-3848	28	2	,	,	PUNCT
cana-3848	28	3	we	we	PRON
cana-3848	28	4	extend	extend	VERB
cana-3848	28	5	the	the	DET
cana-3848	28	6	approach	approach	NOUN
cana-3848	28	7	to	to	ADP
cana-3848	28	8	complexity	complexity	NOUN
cana-3848	28	9	functions	function	NOUN
cana-3848	28	10	,	,	PUNCT
cana-3848	28	11	giving	give	VERB
cana-3848	28	12	a	a	DET
cana-3848	28	13	rather	rather	ADV
cana-3848	28	14	remarkable	remarkable	ADJ
cana-3848	28	15	example	example	NOUN
cana-3848	28	16	of	of	ADP
cana-3848	28	17	a	a	DET
cana-3848	28	18	space	space	NOUN
cana-3848	28	19	documented	document	VERB
cana-3848	28	20	in	in	ADP
cana-3848	28	21	different	different	ADJ
cana-3848	28	22	parts	part	NOUN
cana-3848	28	23	of	of	ADP
cana-3848	28	24	theoretical	theoretical	ADJ
cana-3848	28	25	computer	computer	NOUN
cana-3848	28	26	science	science	NOUN
cana-3848	28	27	(	(	PUNCT
cana-3848	28	28	see	see	VERB
cana-3848	28	29	example	example	NOUN
cana-3848	28	30	3.2	3.2	NUM
cana-3848	28	31	below	below	ADP
cana-3848	28	32	)	)	PUNCT
cana-3848	28	33	.	.	PUNCT
cana-3848	29	1	the	the	DET
cana-3848	29	2	notion	notion	NOUN
cana-3848	29	3	of	of	ADP
cana-3848	29	4	quasi	quasi	ADJ
cana-3848	29	5	-	-	ADJ
cana-3848	29	6	metric	metric	ADJ
cana-3848	29	7	space	space	NOUN
cana-3848	29	8	is	be	AUX
cana-3848	29	9	specially	specially	ADV
cana-3848	29	10	introduced	introduce	VERB
cana-3848	29	11	and	and	CCONJ
cana-3848	29	12	developed	develop	VERB
cana-3848	29	13	in	in	ADP
cana-3848	29	14	[	[	X
cana-3848	29	15	1	1	NUM
cana-3848	29	16	]	]	PUNCT
cana-3848	29	17	.	.	PUNCT
cana-3848	30	1	a	a	DET
cana-3848	30	2	quasi	quasi	ADJ
cana-3848	30	3	-	-	ADJ
cana-3848	30	4	metric	metric	ADJ
cana-3848	30	5	x	x	X
cana-3848	30	6	→	→	SYM
cana-3848	30	7	ℝ	ℝ	NOUN
cana-3848	30	8	+	+	X
cana-3848	30	9	explains	explain	VERB
cana-3848	30	10	if	if	SCONJ
cana-3848	30	11	the	the	DET
cana-3848	30	12	set	set	NOUN
cana-3848	30	13	x	x	PART
cana-3848	30	14	describes	describe	VERB
cana-3848	30	15	a	a	DET
cana-3848	30	16	nonnegative	nonnegative	ADJ
cana-3848	30	17	real	real	ADJ
cana-3848	30	18	number	number	NOUN
cana-3848	30	19	element	element	NOUN
cana-3848	30	20	in	in	ADP
cana-3848	30	21	the	the	DET
cana-3848	30	22	set	set	NOUN
cana-3848	30	23	x.	x.	NOUN
cana-3848	30	24	however	however	ADV
cana-3848	30	25	,	,	PUNCT
cana-3848	30	26	this	this	DET
cana-3848	30	27	definition	definition	NOUN
cana-3848	30	28	must	must	AUX
cana-3848	30	29	be	be	AUX
cana-3848	30	30	true	true	ADJ
cana-3848	30	31	for	for	ADP
cana-3848	30	32	all	all	DET
cana-3848	30	33	sets	set	NOUN
cana-3848	30	34	of	of	ADP
cana-3848	30	35	elements	element	NOUN
cana-3848	30	36	taken	take	VERB
cana-3848	30	37	from	from	ADP
cana-3848	30	38	elements	element	NOUN
cana-3848	30	39	set	set	VERB
cana-3848	30	40	x	x	SYM
cana-3848	30	41	,	,	PUNCT
cana-3848	30	42	y	y	PROPN
cana-3848	30	43	,	,	PUNCT
cana-3848	30	44	and	and	CCONJ
cana-3848	30	45	z	z	X
cana-3848	30	46	,	,	PUNCT
cana-3848	30	47	elements	element	NOUN
cana-3848	30	48	set	set	VERB
cana-3848	30	49	x	x	NOUN
cana-3848	30	50	:	:	PUNCT
cana-3848	30	51	(	(	PUNCT
cana-3848	30	52	a	a	X
cana-3848	30	53	)	)	PUNCT
cana-3848	30	54	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3848	30	55	,	,	PUNCT
cana-3848	30	56	𝑦	𝑦	X
cana-3848	30	57	)	)	PUNCT
cana-3848	30	58	=	=	SYM
cana-3848	30	59	𝑑(𝑦	𝑑(𝑦	NOUN
cana-3848	30	60	,	,	PUNCT
cana-3848	30	61	𝑥	𝑥	NOUN
cana-3848	30	62	)	)	PUNCT
cana-3848	30	63	=	=	SYM
cana-3848	30	64	0	0	PUNCT
cana-3848	31	1	if	if	SCONJ
cana-3848	31	2	and	and	CCONJ
cana-3848	31	3	only	only	ADV
cana-3848	31	4	if	if	SCONJ
cana-3848	31	5	𝑥	𝑥	PRON
cana-3848	31	6	=	=	SYM
cana-3848	31	7	𝑦	𝑦	PROPN
cana-3848	31	8	and	and	CCONJ
cana-3848	31	9	(	(	PUNCT
cana-3848	31	10	b	b	NOUN
cana-3848	31	11	)	)	PUNCT
cana-3848	31	12	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3848	31	13	,	,	PUNCT
cana-3848	31	14	𝑧	𝑧	NOUN
cana-3848	31	15	)	)	PUNCT
cana-3848	31	16	≤	≤	NOUN
cana-3848	31	17	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3848	31	18	,	,	PUNCT
cana-3848	31	19	𝑦	𝑦	NOUN
cana-3848	31	20	)	)	PUNCT
cana-3848	31	21	+	+	CCONJ
cana-3848	31	22	𝑑(𝑦	𝑑(𝑦	NOUN
cana-3848	31	23	,	,	PUNCT
cana-3848	31	24	𝑧	𝑧	PART
cana-3848	31	25	):	):	PUNCT
cana-3848	31	26	we	we	PRON
cana-3848	31	27	'll	will	AUX
cana-3848	31	28	also	also	ADV
cana-3848	31	29	talk	talk	VERB
cana-3848	31	30	about	about	ADP
cana-3848	31	31	extended	extended	ADJ
cana-3848	31	32	quasi	quasi	NOUN
cana-3848	31	33	-	-	ADJ
cana-3848	31	34	metric	metric	ADJ
cana-3848	31	35	.	.	PUNCT
cana-3848	32	1	except	except	SCONJ
cana-3848	32	2	for	for	ADP
cana-3848	32	3	the	the	DET
cana-3848	32	4	fact	fact	NOUN
cana-3848	32	5	that	that	SCONJ
cana-3848	32	6	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3848	32	7	,	,	PUNCT
cana-3848	32	8	𝑦	𝑦	NOUN
cana-3848	32	9	)	)	PUNCT
cana-3848	32	10	=	=	PUNCT
cana-3848	33	1	+	+	ADV
cana-3848	33	2	∞.is	∞.is	PROPN
cana-3848	33	3	permitted	permit	VERB
cana-3848	33	4	,	,	PUNCT
cana-3848	33	5	they	they	PRON
cana-3848	33	6	adhere	adhere	VERB
cana-3848	33	7	to	to	ADP
cana-3848	33	8	the	the	DET
cana-3848	33	9	first	first	ADJ
cana-3848	33	10	three	three	NUM
cana-3848	33	11	axioms.a	axioms.a	NUM
cana-3848	33	12	(	(	PUNCT
cana-3848	33	13	n	n	CCONJ
cana-3848	33	14	extended	extended	ADJ
cana-3848	33	15	)	)	PUNCT
cana-3848	33	16	quasi	quasi	ADJ
cana-3848	33	17	-	-	ADJ
cana-3848	33	18	metric	metric	ADJ
cana-3848	33	19	space	space	NOUN
cana-3848	33	20	is	be	AUX
cana-3848	33	21	a	a	DET
cana-3848	33	22	pair	pair	NOUN
cana-3848	33	23	(	(	PUNCT
cana-3848	33	24	𝑋	𝑋	PROPN
cana-3848	33	25	,	,	PUNCT
cana-3848	33	26	𝑑	𝑑	NOUN
cana-3848	33	27	)	)	PUNCT
cana-3848	33	28	in	in	ADP
cana-3848	33	29	which	which	PRON
cana-3848	33	30	𝑋	𝑋	NOUN
cana-3848	33	31	is	be	AUX
cana-3848	33	32	a	a	DET
cana-3848	33	33	(	(	PUNCT
cana-3848	33	34	nonempty	nonempty	NOUN
cana-3848	33	35	)	)	PUNCT
cana-3848	33	36	set	set	NOUN
cana-3848	33	37	and	and	CCONJ
cana-3848	33	38	d	d	NOUN
cana-3848	33	39	is	be	AUX
cana-3848	33	40	a	a	DET
cana-3848	33	41	(	(	PUNCT
cana-3848	33	42	n	n	ADP
cana-3848	33	43	extended	extended	ADJ
cana-3848	33	44	)	)	PUNCT
cana-3848	33	45	quasi	quasi	NOUN
cana-3848	33	46	-	-	ADJ
cana-3848	33	47	metric	metric	ADJ
cana-3848	33	48	on	on	ADP
cana-3848	33	49	𝑋.	𝑋.	PROPN
cana-3848	33	50	with	with	ADP
cana-3848	33	51	the	the	DET
cana-3848	33	52	family	family	NOUN
cana-3848	33	53	of	of	ADP
cana-3848	33	54	open	open	ADJ
cana-3848	33	55	dballs	dball	NOUN
cana-3848	33	56	{	{	PUNCT
cana-3848	33	57	𝐵𝑑(𝑥	𝐵𝑑(𝑥	PROPN
cana-3848	33	58	,	,	PUNCT
cana-3848	33	59	𝜌	𝜌	X
cana-3848	33	60	)	)	PUNCT
cana-3848	33	61	∶	∶	NOUN
cana-3848	33	62	𝑥	𝑥	X
cana-3848	33	63	∈	∈	PROPN
cana-3848	33	64	𝑋	𝑋	PROPN
cana-3848	33	65	,	,	PUNCT
cana-3848	33	66	𝜌	𝜌	X
cana-3848	33	67	>	>	X
cana-3848	33	68	0	0	NUM
cana-3848	33	69	}	}	PUNCT
cana-3848	33	70	,	,	PUNCT
cana-3848	33	71	as	as	ADP
cana-3848	33	72	its	its	PRON
cana-3848	33	73	basis	basis	NOUN
cana-3848	33	74	,	,	PUNCT
cana-3848	33	75	every	every	DET
cana-3848	33	76	extended	extended	ADJ
cana-3848	33	77	quasi	quasi	ADJ
cana-3848	33	78	-	-	ADJ
cana-3848	33	79	metric	metric	ADJ
cana-3848	33	80	d	d	NOUN
cana-3848	33	81	on	on	ADP
cana-3848	33	82	a	a	DET
cana-3848	33	83	set	set	NOUN
cana-3848	33	84	𝑋	𝑋	NOUN
cana-3848	33	85	yields	yield	VERB
cana-3848	33	86	a	a	DET
cana-3848	33	87	𝑇0	𝑇0	NOUN
cana-3848	33	88	topology	topology	NOUN
cana-3848	33	89	𝒯(𝑑)on	𝒯(𝑑)on	PROPN
cana-3848	33	90	𝑋	𝑋	PROPN
cana-3848	33	91	;	;	PUNCT
cana-3848	34	1	for	for	ADP
cana-3848	34	2	all	all	PRON
cana-3848	34	3	𝑥	𝑥	DET
cana-3848	34	4	∈	∈	PROPN
cana-3848	34	5	𝑋	𝑋	NOUN
cana-3848	34	6	and	and	CCONJ
cana-3848	34	7	𝜌	𝜌	X
cana-3848	34	8	>	>	X
cana-3848	34	9	0	0	NUM
cana-3848	34	10	,	,	PUNCT
cana-3848	34	11	𝐵𝑑(𝑥	𝐵𝑑(𝑥	NOUN
cana-3848	34	12	,	,	PUNCT
cana-3848	34	13	𝜌	𝜌	X
cana-3848	34	14	)	)	PUNCT
cana-3848	34	15	=	=	SYM
cana-3848	34	16	{	{	PUNCT
cana-3848	34	17	𝑦	𝑦	NOUN
cana-3848	34	18	∈	∈	NOUN
cana-3848	34	19	𝑋	𝑋	PROPN
cana-3848	34	20	∶	∶	PROPN
cana-3848	34	21	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3848	34	22	,	,	PUNCT
cana-3848	34	23	𝑦	𝑦	NOUN
cana-3848	34	24	)	)	PUNCT
cana-3848	34	25	<	<	X
cana-3848	34	26	𝜌	𝜌	ADP
cana-3848	34	27	}	}	PUNCT
cana-3848	34	28	𝑑𝑠(𝑥	𝑑𝑠(𝑥	NUM
cana-3848	34	29	,	,	PUNCT
cana-3848	34	30	𝑦	𝑦	X
cana-3848	34	31	)	)	PUNCT
cana-3848	34	32	=	=	SYM
cana-3848	34	33	max{𝑑(𝑥	max{𝑑(𝑥	PROPN
cana-3848	34	34	,	,	PUNCT
cana-3848	34	35	𝑦	𝑦	NOUN
cana-3848	34	36	)	)	PUNCT
cana-3848	34	37	,	,	PUNCT
cana-3848	34	38	𝑑(𝑦	𝑑(𝑦	PROPN
cana-3848	34	39	,	,	PUNCT
cana-3848	34	40	𝑥	𝑥	NOUN
cana-3848	34	41	)	)	PUNCT
cana-3848	34	42	}	}	PUNCT
cana-3848	34	43	,	,	PUNCT
cana-3848	34	44	defined	define	VERB
cana-3848	34	45	on	on	ADP
cana-3848	34	46	𝑋	𝑋	PROPN
cana-3848	34	47	×	×	PROPN
cana-3848	34	48	𝑋	𝑋	PROPN
cana-3848	34	49	,	,	PUNCT
cana-3848	34	50	is	be	AUX
cana-3848	34	51	a	a	DET
cana-3848	34	52	(	(	PUNCT
cana-3848	34	53	n	n	ADP
cana-3848	34	54	extended	extended	ADJ
cana-3848	34	55	)	)	PUNCT
cana-3848	34	56	metric	metric	NOUN
cana-3848	34	57	on	on	ADP
cana-3848	34	58	𝑋	𝑋	PROPN
cana-3848	34	59	if	if	SCONJ
cana-3848	34	60	d	d	PROPN
cana-3848	34	61	is	be	AUX
cana-3848	34	62	a	a	DET
cana-3848	34	63	(	(	PUNCT
cana-3848	34	64	n	n	ADP
cana-3848	34	65	extended	extended	ADJ
cana-3848	34	66	)	)	PUNCT
cana-3848	34	67	quasi	quasi	NOUN
cana-3848	34	68	-	-	ADJ
cana-3848	34	69	metric	metric	ADJ
cana-3848	34	70	on	on	ADP
cana-3848	34	71	a	a	DET
cana-3848	34	72	set	set	NOUN
cana-3848	34	73	x.	x.	NOUN
cana-3848	34	74	when	when	SCONJ
cana-3848	34	75	𝑑𝑠	𝑑𝑠	PRON
cana-3848	34	76	is	be	AUX
cana-3848	34	77	a	a	DET
cana-3848	34	78	complete	complete	ADJ
cana-3848	34	79	extended	extend	VERB
cana-3848	34	80	metric	metric	NOUN
cana-3848	34	81	on	on	ADP
cana-3848	34	82	a	a	DET
cana-3848	34	83	set	set	ADJ
cana-3848	34	84	𝑋	𝑋	NOUN
cana-3848	34	85	,	,	PUNCT
cana-3848	34	86	then	then	ADV
cana-3848	34	87	d	d	X
cana-3848	34	88	on	on	ADP
cana-3848	34	89	𝑋	𝑋	PROPN
cana-3848	34	90	is	be	AUX
cana-3848	34	91	a	a	DET
cana-3848	34	92	bicomplete	bicomplete	ADJ
cana-3848	34	93	extended	extend	VERB
cana-3848	34	94	quasimetric	quasimetric	ADJ
cana-3848	34	95	.	.	PUNCT
cana-3848	35	1	2	2	X
cana-3848	35	2	.	.	X
cana-3848	35	3	producing	produce	VERB
cana-3848	35	4	extended	extend	VERB
cana-3848	35	5	quasi	quasi	NOUN
cana-3848	35	6	metrics	metric	NOUN
cana-3848	35	7	an	an	DET
cana-3848	35	8	extended	extended	ADJ
cana-3848	35	9	quasi	quasi	ADJ
cana-3848	35	10	-	-	ADJ
cana-3848	35	11	metric	metric	ADJ
cana-3848	35	12	d	d	NOUN
cana-3848	35	13	on	on	ADP
cana-3848	35	14	a	a	DET
cana-3848	35	15	cone	cone	NOUN
cana-3848	35	16	(	(	PUNCT
cana-3848	35	17	𝑋	𝑋	PROPN
cana-3848	35	18	,	,	PUNCT
cana-3848	35	19	+	+	PROPN
cana-3848	35	20	,	,	PUNCT
cana-3848	35	21	⋅)is	⋅)is	ADV
cana-3848	35	22	considered	consider	VERB
cana-3848	35	23	constant	constant	ADJ
cana-3848	35	24	if	if	SCONJ
cana-3848	35	25	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3848	35	26	+	+	CCONJ
cana-3848	35	27	𝑧	𝑧	PROPN
cana-3848	35	28	,	,	PUNCT
cana-3848	35	29	𝑦	𝑦	NOUN
cana-3848	35	30	+	+	X
cana-3848	35	31	𝑧	𝑧	X
cana-3848	35	32	)	)	PUNCT
cana-3848	35	33	=	=	SYM
cana-3848	35	34	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3848	35	35	,	,	PUNCT
cana-3848	35	36	𝑦	𝑦	NOUN
cana-3848	35	37	)	)	PUNCT
cana-3848	35	38	and	and	CCONJ
cana-3848	35	39	𝑑(𝜌𝑥	𝑑(𝜌𝑥	NOUN
cana-3848	35	40	,	,	PUNCT
cana-3848	35	41	𝜌𝑦	𝜌𝑦	X
cana-3848	35	42	)	)	PUNCT
cana-3848	35	43	=	=	SYM
cana-3848	35	44	𝜌𝑑(𝑥	𝜌𝑑(𝑥	PROPN
cana-3848	35	45	,	,	PUNCT
cana-3848	35	46	𝑦	𝑦	NOUN
cana-3848	35	47	)	)	PUNCT
cana-3848	35	48	,	,	PUNCT
cana-3848	35	49	as	as	SCONJ
cana-3848	35	50	in	in	ADP
cana-3848	35	51	[	[	PUNCT
cana-3848	35	52	3].assuming	3].assuming	NUM
cana-3848	35	53	𝜌	𝜌	PART
cana-3848	35	54	∈ℝ+	∈ℝ+	NUM
cana-3848	35	55	and	and	CCONJ
cana-3848	35	56	𝑥	𝑥	NOUN
cana-3848	35	57	,	,	PUNCT
cana-3848	35	58	𝑦	𝑦	PRON
cana-3848	35	59	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3848	35	60	𝑧	𝑧	PROPN
cana-3848	35	61	∈	∈	PROPN
cana-3848	35	62	𝑋.	𝑋.	PROPN
cana-3848	35	63	definition	definition	NOUN
cana-3848	35	64	(	(	PUNCT
cana-3848	35	65	2.1	2.1	NUM
cana-3848	35	66	):	):	PUNCT
cana-3848	35	67	a	a	DET
cana-3848	35	68	pair	pair	NOUN
cana-3848	35	69	(	(	PUNCT
cana-3848	35	70	𝑋	𝑋	PROPN
cana-3848	35	71	,	,	PUNCT
cana-3848	35	72	𝑑	𝑑	NOUN
cana-3848	35	73	)	)	PUNCT
cana-3848	35	74	is	be	AUX
cana-3848	35	75	considered	consider	VERB
cana-3848	35	76	to	to	PART
cana-3848	35	77	be	be	AUX
cana-3848	35	78	an	an	DET
cana-3848	35	79	extended	extended	ADJ
cana-3848	35	80	quasi	quasi	ADJ
cana-3848	35	81	-	-	ADJ
cana-3848	35	82	metric	metric	ADJ
cana-3848	35	83	cone	cone	NOUN
cana-3848	35	84	if	if	SCONJ
cana-3848	35	85	𝑋	𝑋	NOUN
cana-3848	35	86	is	be	AUX
cana-3848	35	87	a	a	DET
cana-3848	35	88	cone	cone	NOUN
cana-3848	35	89	and	and	CCONJ
cana-3848	35	90	d	d	NOUN
cana-3848	35	91	is	be	AUX
cana-3848	35	92	an	an	DET
cana-3848	35	93	invariant	invariant	ADJ
cana-3848	35	94	extended	extend	VERB
cana-3848	35	95	quasi	quasi	NOUN
cana-3848	35	96	-	-	ADJ
cana-3848	35	97	metric	metric	ADJ
cana-3848	35	98	on	on	ADP
cana-3848	35	99	𝑋.	𝑋.	PROPN
cana-3848	35	100	assume	assume	PROPN
cana-3848	35	101	(	(	PUNCT
cana-3848	35	102	𝑋	𝑋	PROPN
cana-3848	35	103	,	,	PUNCT
cana-3848	35	104	+	+	PROPN
cana-3848	35	105	,	,	PUNCT
cana-3848	35	106	⋅	⋅	PROPN
cana-3848	35	107	)	)	PUNCT
cana-3848	35	108	is	be	AUX
cana-3848	35	109	a	a	DET
cana-3848	35	110	cone	cone	NOUN
cana-3848	35	111	.	.	PUNCT
cana-3848	36	1	for	for	ADP
cana-3848	36	2	each	each	DET
cana-3848	36	3	𝑥	𝑥	PRON
cana-3848	36	4	∈	∈	PROPN
cana-3848	36	5	𝑋	𝑋	PROPN
cana-3848	36	6	,	,	PUNCT
cana-3848	36	7	the	the	DET
cana-3848	36	8	formula	formula	NOUN
cana-3848	36	9	𝑥	𝑥	X
cana-3848	36	10	+	+	CCONJ
cana-3848	36	11	𝑋	𝑋	NOUN
cana-3848	36	12	=	=	SYM
cana-3848	36	13	{	{	PUNCT
cana-3848	36	14	𝑥	𝑥	PROPN
cana-3848	36	15	+	+	NUM
cana-3848	36	16	𝑦	𝑦	X
cana-3848	36	17	:	:	PUNCT
cana-3848	36	18	𝑦	𝑦	PROPN
cana-3848	36	19	∈	∈	PROPN
cana-3848	36	20	𝑋	𝑋	PROPN
cana-3848	36	21	}	}	PUNCT
cana-3848	36	22	is	be	AUX
cana-3848	36	23	defined	define	VERB
cana-3848	36	24	.	.	PUNCT
cana-3848	37	1	proposition	proposition	NOUN
cana-3848	37	2	(	(	PUNCT
cana-3848	37	3	2.2	2.2	NUM
cana-3848	37	4	):	):	PUNCT
cana-3848	37	5	consider	consider	VERB
cana-3848	37	6	that	that	SCONJ
cana-3848	37	7	on	on	ADP
cana-3848	37	8	the	the	DET
cana-3848	37	9	cancellative	cancellative	ADJ
cana-3848	37	10	cone	cone	NOUN
cana-3848	37	11	(	(	PUNCT
cana-3848	37	12	𝑋	𝑋	PROPN
cana-3848	37	13	,	,	PUNCT
cana-3848	37	14	+	+	PROPN
cana-3848	37	15	,	,	PUNCT
cana-3848	37	16	⋅	⋅	PROPN
cana-3848	37	17	)	)	PUNCT
cana-3848	37	18	.	.	PUNCT
cana-3848	38	1	p	p	NOUN
cana-3848	38	2	is	be	AUX
cana-3848	38	3	a	a	DET
cana-3848	38	4	quasi	quasi	NOUN
cana-3848	38	5	-	-	NOUN
cana-3848	38	6	norm	norm	NOUN
cana-3848	38	7	.	.	PUNCT
cana-3848	39	1	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	39	2	defined	define	VERB
cana-3848	39	3	on	on	ADP
cana-3848	39	4	𝑋	𝑋	PROPN
cana-3848	39	5	×	×	NOUN
cana-3848	39	6	𝑋	𝑋	PROPN
cana-3848	39	7	is	be	AUX
cana-3848	39	8	an	an	DET
cana-3848	39	9	invariant	invariant	ADJ
cana-3848	39	10	extended	extend	VERB
cana-3848	39	11	quasi	quasi	NOUN
cana-3848	39	12	-	-	ADJ
cana-3848	39	13	metric	metric	ADJ
cana-3848	39	14	on	on	ADP
cana-3848	39	15	𝑋	𝑋	PROPN
cana-3848	39	16	if	if	SCONJ
cana-3848	39	17	𝑥	𝑥	PRON
cana-3848	39	18	∈	∈	PROPN
cana-3848	39	19	𝑋	𝑋	NOUN
cana-3848	39	20	and	and	CCONJ
cana-3848	39	21	𝑦	𝑦	NOUN
cana-3848	39	22	∈	∈	NOUN
cana-3848	40	1	𝑥	𝑥	NOUN
cana-3848	40	2	+	+	CCONJ
cana-3848	40	3	𝑋	𝑋	NOUN
cana-3848	40	4	with	with	ADP
cana-3848	40	5	𝑦	𝑦	NOUN
cana-3848	40	6	=	=	SYM
cana-3848	40	7	𝑥	𝑥	PROPN
cana-3848	40	8	+	+	NUM
cana-3848	40	9	𝑎	𝑎	ADP
cana-3848	40	10	,	,	PUNCT
cana-3848	40	11	communications	communication	NOUN
cana-3848	40	12	on	on	ADP
cana-3848	40	13	applied	apply	VERB
cana-3848	40	14	nonlinear	nonlinear	ADJ
cana-3848	40	15	analysis	analysis	NOUN
cana-3848	40	16	issn	issn	NOUN
cana-3848	40	17	:	:	PUNCT
cana-3848	40	18	1074	1074	NUM
cana-3848	40	19	-	-	PUNCT
cana-3848	40	20	133x	133x	NUM
cana-3848	40	21	vol	vol	NOUN
cana-3848	40	22	32	32	NUM
cana-3848	40	23	no	no	NOUN
cana-3848	40	24	.	.	PUNCT
cana-3848	41	1	9s	9s	NUM
cana-3848	41	2	(	(	PUNCT
cana-3848	41	3	2025	2025	NUM
cana-3848	41	4	)	)	PUNCT
cana-3848	41	5	201	201	NUM
cana-3848	41	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	41	7	and	and	CCONJ
cana-3848	41	8	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	41	9	(	(	PUNCT
cana-3848	41	10	𝑥	𝑥	NOUN
cana-3848	41	11	,	,	PUNCT
cana-3848	41	12	𝑦	𝑦	NOUN
cana-3848	41	13	)	)	PUNCT
cana-3848	41	14	=	=	PUNCT
cana-3848	42	1	+	+	NUM
cana-3848	42	2	∞	∞	NUM
cana-3848	42	3	if	if	SCONJ
cana-3848	42	4	𝑥	𝑥	PROPN
cana-3848	42	5	∈	∈	PROPN
cana-3848	42	6	𝑋	𝑋	PROPN
cana-3848	42	7	and	and	CCONJ
cana-3848	42	8	𝑦	𝑦	NOUN
cana-3848	42	9	∉	∉	PROPN
cana-3848	42	10	𝑥	𝑥	X
cana-3848	42	11	+	+	PROPN
cana-3848	42	12	𝑋	𝑋	PROPN
cana-3848	42	13	,	,	PUNCT
cana-3848	42	14	respectively	respectively	ADV
cana-3848	42	15	.	.	PUNCT
cana-3848	43	1	consequently	consequently	ADV
cana-3848	43	2	,	,	PUNCT
cana-3848	43	3	(	(	PUNCT
cana-3848	43	4	𝑋	𝑋	PROPN
cana-3848	43	5	,	,	PUNCT
cana-3848	43	6	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	43	7	)	)	PUNCT
cana-3848	43	8	is	be	AUX
cana-3848	43	9	a	a	DET
cana-3848	43	10	stretched	stretch	VERB
cana-3848	43	11	quasi	quasi	ADJ
cana-3848	43	12	-	-	ADJ
cana-3848	43	13	metric	metric	ADJ
cana-3848	43	14	cone	cone	NOUN
cana-3848	43	15	.	.	PUNCT
cana-3848	44	1	also	also	ADV
cana-3848	44	2	,	,	PUNCT
cana-3848	44	3	for	for	ADP
cana-3848	44	4	every	every	DET
cana-3848	44	5	x	x	NOUN
cana-3848	44	6	in	in	ADP
cana-3848	44	7	x	x	PROPN
cana-3848	44	8	,	,	PUNCT
cana-3848	44	9	ρ	ρ	PROPN
cana-3848	44	10	in	in	ADP
cana-3848	44	11	ℝ+	ℝ+	NOUN
cana-3848	44	12	,	,	PUNCT
cana-3848	44	13	and	and	CCONJ
cana-3848	44	14	every	every	PRON
cana-3848	44	15	𝜇	𝜇	X
cana-3848	44	16	>	>	X
cana-3848	44	17	−1	−1	NOUN
cana-3848	44	18	,	,	PUNCT
cana-3848	44	19	the	the	DET
cana-3848	44	20	translations	translation	NOUN
cana-3848	44	21	are	be	AUX
cana-3848	44	22	𝑇	𝑇	PROPN
cana-3848	44	23	(	(	PUNCT
cana-3848	44	24	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	44	25	)	)	PUNCT
cana-3848	44	26	--open	--open	PROPN
cana-3848	44	27	,	,	PUNCT
cana-3848	44	28	and	and	CCONJ
cana-3848	44	29	𝐵𝑒𝑝𝑗	𝐵𝑒𝑝𝑗	PROPN
cana-3848	44	30	(	(	PUNCT
cana-3848	44	31	𝑥	𝑥	NOUN
cana-3848	44	32	,	,	PUNCT
cana-3848	44	33	𝜇	𝜇	ADP
cana-3848	44	34	+	+	NOUN
cana-3848	44	35	1	1	NUM
cana-3848	44	36	)	)	PUNCT
cana-3848	44	37	=	=	PUNCT
cana-3848	44	38	𝛿𝑥	𝛿𝑥	NOUN
cana-3848	44	39	+	+	X
cana-3848	44	40	{	{	PUNCT
cana-3848	44	41	𝑦	𝑦	NOUN
cana-3848	44	42	∈	∈	NOUN
cana-3848	44	43	𝑋	𝑋	NOUN
cana-3848	44	44	∶	∶	NOUN
cana-3848	44	45	𝑝𝑗(𝑦	𝑝𝑗(𝑦	NOUN
cana-3848	44	46	)	)	PUNCT
cana-3848	44	47	<	<	X
cana-3848	45	1	𝜌(𝜇	𝜌(𝜇	PUNCT
cana-3848	45	2	+	+	NOUN
cana-3848	45	3	1	1	NUM
cana-3848	45	4	)	)	PUNCT
cana-3848	45	5	}	}	PUNCT
cana-3848	45	6	proof	proof	NOUN
cana-3848	45	7	:	:	PUNCT
cana-3848	45	8	if	if	SCONJ
cana-3848	45	9	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	45	10	(	(	PUNCT
cana-3848	45	11	𝑥	𝑥	NOUN
cana-3848	45	12	,	,	PUNCT
cana-3848	45	13	𝑥	𝑥	NOUN
cana-3848	45	14	)	)	PUNCT
cana-3848	45	15	=	=	SYM
cana-3848	45	16	𝑝𝑗(0	𝑝𝑗(0	NOUN
cana-3848	45	17	)	)	PUNCT
cana-3848	45	18	=	=	SYM
cana-3848	45	19	0	0	NUM
cana-3848	45	20	,	,	PUNCT
cana-3848	45	21	then	then	ADV
cana-3848	45	22	for	for	ADP
cana-3848	45	23	any	any	DET
cana-3848	45	24	𝑥	𝑥	NOUN
cana-3848	45	25	𝑖𝑛	𝑖𝑛	PRON
cana-3848	45	26	𝑋.	𝑋.	PROPN
cana-3848	45	27	allow	allow	VERB
cana-3848	45	28	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	45	29	(	(	PUNCT
cana-3848	45	30	𝑥	𝑥	PROPN
cana-3848	45	31	,	,	PUNCT
cana-3848	45	32	𝑦	𝑦	NOUN
cana-3848	45	33	)	)	PUNCT
cana-3848	45	34	=	=	SYM
cana-3848	45	35	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	45	36	(	(	PUNCT
cana-3848	45	37	𝑦	𝑦	NOUN
cana-3848	45	38	,	,	PUNCT
cana-3848	45	39	𝑥	𝑥	NOUN
cana-3848	45	40	)	)	PUNCT
cana-3848	45	41	=	=	SYM
cana-3848	46	1	0at	0at	NOUN
cana-3848	46	2	this	this	DET
cana-3848	46	3	point.hence	point.hence	NOUN
cana-3848	46	4	,	,	PUNCT
cana-3848	46	5	𝑦	𝑦	NOUN
cana-3848	46	6	=	=	SYM
cana-3848	46	7	𝑥	𝑥	PROPN
cana-3848	47	1	+	+	NUM
cana-3848	47	2	𝑎	𝑎	ADP
cana-3848	47	3	,	,	PUNCT
cana-3848	47	4	𝑥	𝑥	NOUN
cana-3848	47	5	=	=	SYM
cana-3848	47	6	𝑦	𝑦	PROPN
cana-3848	47	7	+	+	CCONJ
cana-3848	47	8	𝑏	𝑏	NOUN
cana-3848	47	9	,	,	PUNCT
cana-3848	47	10	occur	occur	VERB
cana-3848	47	11	when	when	SCONJ
cana-3848	47	12	𝑎	𝑎	X
cana-3848	47	13	,	,	PUNCT
cana-3848	47	14	𝑏	𝑏	PROPN
cana-3848	47	15	∈	∈	PROPN
cana-3848	47	16	𝑋.	𝑋.	PROPN
cana-3848	47	17	given	give	VERB
cana-3848	47	18	that	that	SCONJ
cana-3848	47	19	𝑎	𝑎	PROPN
cana-3848	47	20	+	+	NOUN
cana-3848	47	21	𝑏	𝑏	NOUN
cana-3848	47	22	=	=	SYM
cana-3848	47	23	0	0	NUM
cana-3848	47	24	and	and	CCONJ
cana-3848	47	25	𝑋	𝑋	PROPN
cana-3848	47	26	is	be	AUX
cana-3848	47	27	cancellative	cancellative	ADJ
cana-3848	47	28	,	,	PUNCT
cana-3848	47	29	we	we	PRON
cana-3848	47	30	can	can	AUX
cana-3848	47	31	deduce	deduce	VERB
cana-3848	47	32	that	that	DET
cana-3848	47	33	b=-a.thus	b=-a.thus	NOUN
cana-3848	47	34	,	,	PUNCT
cana-3848	47	35	𝑎	𝑎	PROPN
cana-3848	47	36	=	=	SYM
cana-3848	47	37	0	0	NUM
cana-3848	47	38	since	since	SCONJ
cana-3848	47	39	𝑝𝑗(𝑎	𝑝𝑗(𝑎	NOUN
cana-3848	47	40	)	)	PUNCT
cana-3848	47	41	=	=	SYM
cana-3848	47	42	𝑝𝑗(−𝑎	𝑝𝑗(−𝑎	NOUN
cana-3848	47	43	)	)	PUNCT
cana-3848	48	1	=	=	PUNCT
cana-3848	48	2	0	0	X
cana-3848	48	3	.	.	PUNCT
cana-3848	49	1	therefore	therefore	ADV
cana-3848	49	2	,	,	PUNCT
cana-3848	49	3	𝑥	𝑥	PROPN
cana-3848	49	4	=	=	PUNCT
cana-3848	49	5	𝑦.	𝑦.	PROPN
cana-3848	49	6	furthermore	furthermore	ADV
cana-3848	49	7	,	,	PUNCT
cana-3848	49	8	we	we	PRON
cana-3848	49	9	show	show	VERB
cana-3848	49	10	that	that	SCONJ
cana-3848	49	11	for	for	ADP
cana-3848	49	12	each	each	DET
cana-3848	49	13	𝑥	𝑥	PROPN
cana-3848	49	14	,	,	PUNCT
cana-3848	49	15	𝑦	𝑦	NOUN
cana-3848	49	16	,	,	PUNCT
cana-3848	49	17	𝑧	𝑧	PRON
cana-3848	49	18	𝑖𝑛	𝑖𝑛	ADJ
cana-3848	49	19	𝑋	𝑋	PROPN
cana-3848	49	20	,	,	PUNCT
cana-3848	49	21	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	49	22	(	(	PUNCT
cana-3848	49	23	𝑥	𝑥	NOUN
cana-3848	49	24	,	,	PUNCT
cana-3848	49	25	𝑧)𝑒𝑝𝑗	𝑧)𝑒𝑝𝑗	PROPN
cana-3848	49	26	(	(	PUNCT
cana-3848	49	27	𝑥	𝑥	PROPN
cana-3848	49	28	,	,	PUNCT
cana-3848	49	29	𝑦	𝑦	NOUN
cana-3848	49	30	)	)	PUNCT
cana-3848	49	31	+	+	CCONJ
cana-3848	49	32	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	49	33	(	(	PUNCT
cana-3848	49	34	𝑦	𝑦	NOUN
cana-3848	49	35	,	,	PUNCT
cana-3848	49	36	𝑧),think	𝑧),think	VERB
cana-3848	49	37	about	about	ADV
cana-3848	49	38	just	just	ADV
cana-3848	49	39	the	the	DET
cana-3848	49	40	case	case	NOUN
cana-3848	49	41	when	when	SCONJ
cana-3848	49	42	𝑦	𝑦	X
cana-3848	49	43	∈	∈	NOUN
cana-3848	49	44	𝑥	𝑥	X
cana-3848	49	45	+	+	CCONJ
cana-3848	49	46	𝑋	𝑋	NOUN
cana-3848	49	47	and	and	CCONJ
cana-3848	49	48	𝑧	𝑧	PRON
cana-3848	49	49	∈	∈	PROPN
cana-3848	49	50	𝑦	𝑦	NOUN
cana-3848	49	51	+	+	X
cana-3848	49	52	𝑋.	𝑋.	PROPN
cana-3848	49	53	at	at	ADP
cana-3848	49	54	that	that	DET
cana-3848	49	55	point	point	NOUN
cana-3848	49	56	,	,	PUNCT
cana-3848	49	57	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	49	58	(	(	PUNCT
cana-3848	49	59	𝑥	𝑥	PROPN
cana-3848	49	60	,	,	PUNCT
cana-3848	49	61	𝑦	𝑦	NOUN
cana-3848	49	62	)	)	PUNCT
cana-3848	49	63	=	=	SYM
cana-3848	49	64	𝑝𝑗(𝑎	𝑝𝑗(𝑎	X
cana-3848	49	65	)	)	PUNCT
cana-3848	49	66	and	and	CCONJ
cana-3848	49	67	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	49	68	(	(	PUNCT
cana-3848	49	69	𝑦	𝑦	NOUN
cana-3848	49	70	,	,	PUNCT
cana-3848	49	71	𝑧	𝑧	NOUN
cana-3848	49	72	)	)	PUNCT
cana-3848	49	73	=	=	SYM
cana-3848	49	74	𝑝𝑗(𝑏	𝑝𝑗(𝑏	NOUN
cana-3848	49	75	)	)	PUNCT
cana-3848	49	76	.	.	PUNCT
cana-3848	50	1	have	have	VERB
cana-3848	50	2	the	the	DET
cana-3848	50	3	properties	property	NOUN
cana-3848	50	4	𝑦	𝑦	NOUN
cana-3848	50	5	=	=	SYM
cana-3848	50	6	𝑥	𝑥	PROPN
cana-3848	50	7	+	+	NUM
cana-3848	50	8	𝑎	𝑎	ADP
cana-3848	50	9	,	,	PUNCT
cana-3848	50	10	𝑧	𝑧	NOUN
cana-3848	50	11	=	=	PUNCT
cana-3848	50	12	𝑦	𝑦	NOUN
cana-3848	50	13	+	+	X
cana-3848	50	14	𝑏	𝑏	NOUN
cana-3848	50	15	for	for	ADP
cana-3848	50	16	each	each	DET
cana-3848	50	17	𝑎	𝑎	NOUN
cana-3848	50	18	,	,	PUNCT
cana-3848	50	19	𝑏	𝑏	PROPN
cana-3848	50	20	∈	∈	PROPN
cana-3848	50	21	𝑋.	𝑋.	PROPN
cana-3848	50	22	because	because	SCONJ
cana-3848	50	23	of	of	ADP
cana-3848	50	24	this	this	PRON
cana-3848	50	25	,	,	PUNCT
cana-3848	50	26	𝑧	𝑧	PROPN
cana-3848	50	27	=	=	X
cana-3848	50	28	𝑥	𝑥	PROPN
cana-3848	51	1	+	+	CCONJ
cana-3848	51	2	𝑎	𝑎	X
cana-3848	51	3	+	+	NOUN
cana-3848	51	4	𝑏	𝑏	NOUN
cana-3848	51	5	,	,	PUNCT
cana-3848	51	6	and	and	CCONJ
cana-3848	51	7	so	so	ADV
cana-3848	51	8	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	51	9	(	(	PUNCT
cana-3848	51	10	𝑥	𝑥	NOUN
cana-3848	51	11	,	,	PUNCT
cana-3848	51	12	𝑧	𝑧	NOUN
cana-3848	51	13	)	)	PUNCT
cana-3848	51	14	=	=	NOUN
cana-3848	51	15	𝑝𝑗(𝑎	𝑝𝑗(𝑎	X
cana-3848	51	16	+	+	CCONJ
cana-3848	51	17	𝑏	𝑏	NOUN
cana-3848	51	18	)	)	PUNCT
cana-3848	51	19	≤	≤	NOUN
cana-3848	51	20	𝑝𝑗(𝑎	𝑝𝑗(𝑎	NOUN
cana-3848	51	21	)	)	PUNCT
cana-3848	51	22	+	+	NUM
cana-3848	51	23	𝑝𝑗(𝑏	𝑝𝑗(𝑏	X
cana-3848	51	24	)	)	PUNCT
cana-3848	52	1	=	=	VERB
cana-3848	52	2	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	52	3	(	(	PUNCT
cana-3848	52	4	𝑥	𝑥	PROPN
cana-3848	52	5	,	,	PUNCT
cana-3848	52	6	𝑦	𝑦	NOUN
cana-3848	52	7	)	)	PUNCT
cana-3848	52	8	+	+	CCONJ
cana-3848	52	9	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	52	10	(	(	PUNCT
cana-3848	52	11	𝑦	𝑦	NOUN
cana-3848	52	12	,	,	PUNCT
cana-3848	52	13	𝑧	𝑧	NOUN
cana-3848	52	14	)	)	PUNCT
cana-3848	52	15	.	.	PUNCT
cana-3848	53	1	from	from	ADP
cana-3848	53	2	this	this	PRON
cana-3848	53	3	,	,	PUNCT
cana-3848	53	4	we	we	PRON
cana-3848	53	5	conclude	conclude	VERB
cana-3848	53	6	that	that	SCONJ
cana-3848	53	7	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	53	8	is	be	AUX
cana-3848	53	9	an	an	DET
cana-3848	53	10	extended	extended	ADJ
cana-3848	53	11	quasi	quasi	NOUN
cana-3848	53	12	-	-	ADJ
cana-3848	53	13	metric	metric	ADJ
cana-3848	53	14	on	on	ADP
cana-3848	53	15	𝑋.	𝑋.	PROPN
cana-3848	53	16	we	we	PRON
cana-3848	53	17	then	then	ADV
cana-3848	53	18	verified	verify	VERB
cana-3848	53	19	that	that	SCONJ
cana-3848	53	20	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	53	21	is	be	AUX
cana-3848	53	22	invariant	invariant	ADJ
cana-3848	53	23	.	.	PUNCT
cana-3848	54	1	let	let	VERB
cana-3848	54	2	𝑥	𝑥	PRON
cana-3848	54	3	,	,	PUNCT
cana-3848	54	4	𝑦	𝑦	NOUN
cana-3848	54	5	,	,	PUNCT
cana-3848	54	6	and	and	CCONJ
cana-3848	54	7	𝑧	𝑧	PRON
cana-3848	54	8	be	be	AUX
cana-3848	54	9	part	part	NOUN
cana-3848	54	10	of	of	ADP
cana-3848	54	11	𝑋.	𝑋.	PROPN
cana-3848	54	12	if	if	SCONJ
cana-3848	54	13	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	54	14	(	(	PUNCT
cana-3848	54	15	𝑥	𝑥	PROPN
cana-3848	54	16	+	+	CCONJ
cana-3848	54	17	𝑧	𝑧	PROPN
cana-3848	54	18	,	,	PUNCT
cana-3848	54	19	𝑦	𝑦	NOUN
cana-3848	54	20	+	+	X
cana-3848	54	21	𝑧	𝑧	X
cana-3848	54	22	)	)	PUNCT
cana-3848	54	23	=	=	PUNCT
cana-3848	55	1	+	+	PROPN
cana-3848	55	2	∞.	∞.	PROPN
cana-3848	55	3	,	,	PUNCT
cana-3848	55	4	then	then	ADV
cana-3848	55	5	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	55	6	(	(	PUNCT
cana-3848	55	7	𝑥	𝑥	PROPN
cana-3848	55	8	,	,	PUNCT
cana-3848	55	9	𝑦	𝑦	NOUN
cana-3848	55	10	)	)	PUNCT
cana-3848	55	11	=	=	PUNCT
cana-3848	56	1	+	+	NUM
cana-3848	56	2	∞	∞	NOUN
cana-3848	56	3	as	as	SCONJ
cana-3848	56	4	𝑋	𝑋	PROPN
cana-3848	56	5	is	be	AUX
cana-3848	56	6	cancellative	cancellative	ADJ
cana-3848	56	7	.	.	PUNCT
cana-3848	57	1	otherwise	otherwise	ADV
cana-3848	57	2	,	,	PUNCT
cana-3848	57	3	assume	assume	VERB
cana-3848	57	4	that	that	SCONJ
cana-3848	57	5	an	an	PRON
cana-3848	57	6	is	be	AUX
cana-3848	57	7	such	such	ADJ
cana-3848	57	8	that	that	SCONJ
cana-3848	57	9	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	57	10	(	(	PUNCT
cana-3848	57	11	𝑥	𝑥	PROPN
cana-3848	58	1	+	+	CCONJ
cana-3848	58	2	𝑧	𝑧	PROPN
cana-3848	58	3	,	,	PUNCT
cana-3848	58	4	𝑦	𝑦	NOUN
cana-3848	58	5	+	+	X
cana-3848	58	6	𝑧	𝑧	X
cana-3848	58	7	)	)	PUNCT
cana-3848	58	8	=	=	NOUN
cana-3848	58	9	𝑝𝑗(𝑎	𝑝𝑗(𝑎	NOUN
cana-3848	58	10	)	)	PUNCT
cana-3848	58	11	.	.	PUNCT
cana-3848	59	1	if	if	SCONJ
cana-3848	59	2	+	+	ADJ
cana-3848	59	3	𝑧	𝑧	NOUN
cana-3848	59	4	=	=	X
cana-3848	59	5	𝑥	𝑥	PROPN
cana-3848	60	1	+	+	NOUN
cana-3848	60	2	𝑧	𝑧	VERB
cana-3848	60	3	+	+	NUM
cana-3848	61	1	𝑎	𝑎	NOUN
cana-3848	61	2	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
cana-3848	61	3	𝑦	𝑦	NOUN
cana-3848	61	4	=	=	SYM
cana-3848	61	5	𝑥	𝑥	PROPN
cana-3848	61	6	+	+	NUM
cana-3848	61	7	𝑎	𝑎	NOUN
cana-3848	61	8	,	,	PUNCT
cana-3848	61	9	and	and	CCONJ
cana-3848	61	10	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	61	11	(	(	PUNCT
cana-3848	61	12	𝑥	𝑥	NOUN
cana-3848	61	13	,	,	PUNCT
cana-3848	61	14	𝑦	𝑦	NOUN
cana-3848	61	15	)	)	PUNCT
cana-3848	61	16	=	=	SYM
cana-3848	61	17	𝑝𝑗(𝑎	𝑝𝑗(𝑎	NOUN
cana-3848	61	18	)	)	PUNCT
cana-3848	61	19	.	.	PUNCT
cana-3848	62	1	the	the	DET
cana-3848	62	2	same	same	ADJ
cana-3848	62	3	as	as	SCONJ
cana-3848	62	4	we	we	PRON
cana-3848	62	5	derive	derive	VERB
cana-3848	62	6	that	that	SCONJ
cana-3848	62	7	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	62	8	(	(	PUNCT
cana-3848	62	9	𝜌𝑥	𝜌𝑥	NOUN
cana-3848	62	10	,	,	PUNCT
cana-3848	62	11	𝜌𝑦	𝜌𝑦	NOUN
cana-3848	62	12	)	)	PUNCT
cana-3848	62	13	=	=	PUNCT
cana-3848	62	14	𝜌𝑒𝑝𝑗	𝜌𝑒𝑝𝑗	VERB
cana-3848	62	15	(	(	PUNCT
cana-3848	62	16	𝑥	𝑥	NOUN
cana-3848	62	17	,	,	PUNCT
cana-3848	62	18	𝑦	𝑦	NOUN
cana-3848	62	19	)	)	PUNCT
cana-3848	62	20	.	.	PUNCT
cana-3848	62	21	)	)	PUNCT
cana-3848	62	22	for	for	ADP
cana-3848	62	23	all	all	PRON
cana-3848	62	24	𝑥	𝑥	PROPN
cana-3848	62	25	,	,	PUNCT
cana-3848	62	26	𝑦	𝑦	NOUN
cana-3848	62	27	∈	∈	NOUN
cana-3848	62	28	𝑋	𝑋	NOUN
cana-3848	62	29	and	and	CCONJ
cana-3848	62	30	𝜌	𝜌	ADP
cana-3848	62	31	∈	∈	PROPN
cana-3848	62	32	ℝ+	ℝ+	PROPN
cana-3848	62	33	.	.	PUNCT
cana-3848	62	34	similarly	similarly	ADV
cana-3848	62	35	,	,	PUNCT
cana-3848	62	36	for	for	ADP
cana-3848	62	37	any	any	DET
cana-3848	62	38	𝑥	𝑥	NOUN
cana-3848	62	39	,	,	PUNCT
cana-3848	62	40	𝑦	𝑦	NOUN
cana-3848	62	41	∈	∈	NOUN
cana-3848	62	42	𝑋	𝑋	NOUN
cana-3848	62	43	and	and	CCONJ
cana-3848	62	44	𝜌	𝜌	ADP
cana-3848	62	45	∈	∈	PROPN
cana-3848	62	46	ℝ+	ℝ+	PUNCT
cana-3848	62	47	,	,	PUNCT
cana-3848	62	48	we	we	PRON
cana-3848	62	49	find	find	VERB
cana-3848	62	50	that	that	SCONJ
cana-3848	62	51	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	62	52	(	(	PUNCT
cana-3848	62	53	𝜌𝑥	𝜌𝑥	NOUN
cana-3848	62	54	,	,	PUNCT
cana-3848	62	55	𝜌𝑦	𝜌𝑦	NOUN
cana-3848	62	56	)	)	PUNCT
cana-3848	62	57	=	=	PUNCT
cana-3848	62	58	𝜌𝑒𝑝𝑗	𝜌𝑒𝑝𝑗	VERB
cana-3848	62	59	(	(	PUNCT
cana-3848	62	60	𝑥	𝑥	NOUN
cana-3848	62	61	,	,	PUNCT
cana-3848	62	62	𝑦	𝑦	NOUN
cana-3848	62	63	)	)	PUNCT
cana-3848	62	64	.	.	PUNCT
cana-3848	63	1	finally	finally	ADV
cana-3848	63	2	,	,	PUNCT
cana-3848	63	3	recall	recall	VERB
cana-3848	63	4	that	that	PRON
cana-3848	63	5	for	for	ADP
cana-3848	63	6	each	each	DET
cana-3848	63	7	x	x	PUNCT
cana-3848	63	8	in	in	ADP
cana-3848	63	9	x	x	X
cana-3848	63	10	and	and	CCONJ
cana-3848	63	11	𝜌	𝜌	X
cana-3848	63	12	𝑖𝑛	𝑖𝑛	PRON
cana-3848	63	13	ℝ+	ℝ+	PROPN
cana-3848	63	14	,	,	PUNCT
cana-3848	63	15	𝑒𝑝𝑗	𝑒𝑝𝑗	X
cana-3848	63	16	(	(	PUNCT
cana-3848	63	17	0	0	NUM
cana-3848	63	18	,	,	PUNCT
cana-3848	63	19	𝑥	𝑥	NOUN
cana-3848	63	20	)	)	PUNCT
cana-3848	63	21	=	=	SYM
cana-3848	63	22	𝑝𝑗(𝑥	𝑝𝑗(𝑥	NOUN
cana-3848	63	23	)	)	PUNCT
cana-3848	63	24	.thus	.thus	ADV
cana-3848	63	25	,	,	PUNCT
cana-3848	63	26	for	for	ADP
cana-3848	63	27	any	any	DET
cana-3848	63	28	𝜇	𝜇	ADP
cana-3848	63	29	>	>	X
cana-3848	63	30	−1	−1	NOUN
cana-3848	63	31	,	,	PUNCT
cana-3848	63	32	we	we	PRON
cana-3848	63	33	get	get	VERB
cana-3848	63	34	rb_(e_(p_j	rb_(e_(p_j	PROPN
cana-3848	63	35	)	)	PUNCT
cana-3848	63	36	)	)	PUNCT
cana-3848	64	1	(	(	PUNCT
cana-3848	64	2	0,δ)=b_(e_(p_j	0,δ)=b_(e_(p_j	PROPN
cana-3848	64	3	)	)	PUNCT
cana-3848	64	4	)	)	PUNCT
cana-3848	65	1	(	(	PUNCT
cana-3848	65	2	0,ρδ	0,ρδ	NUM
cana-3848	65	3	)	)	PUNCT
cana-3848	65	4	and	and	CCONJ
cana-3848	65	5	𝐵𝑒𝑝𝑗	𝐵𝑒𝑝𝑗	PROPN
cana-3848	65	6	(	(	PUNCT
cana-3848	65	7	0	0	NUM
cana-3848	65	8	,	,	PUNCT
cana-3848	65	9	𝜇	𝜇	ADP
cana-3848	65	10	+	+	NOUN
cana-3848	65	11	1	1	NUM
cana-3848	65	12	)	)	PUNCT
cana-3848	65	13	=	=	PRON
cana-3848	65	14	{	{	PUNCT
cana-3848	65	15	𝑥	𝑥	PUNCT
cana-3848	65	16	∈	∈	PROPN
cana-3848	65	17	𝑋	𝑋	PROPN
cana-3848	65	18	∶	∶	NOUN
cana-3848	65	19	𝑝𝑗(𝑥	𝑝𝑗(𝑥	PUNCT
cana-3848	65	20	)	)	PUNCT
cana-3848	65	21	<	<	X
cana-3848	65	22	𝛿	𝛿	PROPN
cana-3848	65	23	}	}	PUNCT
cana-3848	65	24	.it	.it	PUNCT
cana-3848	65	25	is	be	AUX
cana-3848	65	26	obvious	obvious	ADJ
cana-3848	65	27	that	that	SCONJ
cana-3848	65	28	for	for	ADP
cana-3848	65	29	each	each	DET
cana-3848	65	30	𝑥	𝑥	ADP
cana-3848	65	31	𝑖𝑛	𝑖𝑛	DET
cana-3848	65	32	𝑋	𝑋	NOUN
cana-3848	65	33	and	and	CCONJ
cana-3848	65	34	each	each	DET
cana-3848	65	35	𝛿	𝛿	X
cana-3848	65	36	>	>	X
cana-3848	65	37	0	0	NUM
cana-3848	65	38	,	,	PUNCT
cana-3848	65	39	𝜌𝐵𝑒𝑝𝑗	𝜌𝐵𝑒𝑝𝑗	PROPN
cana-3848	65	40	(	(	PUNCT
cana-3848	65	41	𝑥	𝑥	PROPN
cana-3848	65	42	,	,	PUNCT
cana-3848	65	43	𝛿	𝛿	ADJ
cana-3848	65	44	)	)	PUNCT
cana-3848	65	45	=	=	SYM
cana-3848	65	46	𝜌𝑥	𝜌𝑥	PROPN
cana-3848	65	47	+	+	X
cana-3848	65	48	𝐵𝑒𝑝𝑗	𝐵𝑒𝑝𝑗	PROPN
cana-3848	65	49	(	(	PUNCT
cana-3848	65	50	0	0	NUM
cana-3848	65	51	,	,	PUNCT
cana-3848	65	52	𝜌𝛿	𝜌𝛿	ADJ
cana-3848	65	53	)	)	PUNCT
cana-3848	65	54	,	,	PUNCT
cana-3848	65	55	𝒯	𝒯	PROPN
cana-3848	65	56	(	(	PUNCT
cana-3848	65	57	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	65	58	)	)	PUNCT
cana-3848	65	59	−open	−open	PROPN
cana-3848	65	60	is	be	AUX
cana-3848	65	61	the	the	DET
cana-3848	65	62	translation	translation	NOUN
cana-3848	65	63	for	for	ADP
cana-3848	65	64	+	+	SYM
cana-3848	65	65	and	and	CCONJ
cana-3848	65	66	⋅	⋅	PROPN
cana-3848	65	67	as	as	ADP
cana-3848	65	68	a	a	DET
cana-3848	65	69	result	result	NOUN
cana-3848	65	70	.	.	PUNCT
cana-3848	66	1	example	example	NOUN
cana-3848	66	2	(	(	PUNCT
cana-3848	66	3	2.3	2.3	NUM
cana-3848	66	4	):	):	PUNCT
cana-3848	66	5	for	for	ADP
cana-3848	66	6	every	every	DET
cana-3848	66	7	x	x	NOUN
cana-3848	66	8	in	in	ADP
cana-3848	66	9	ℝ+	ℝ+	ADV
cana-3848	66	10	,	,	PUNCT
cana-3848	66	11	find	find	VERB
cana-3848	66	12	a	a	DET
cana-3848	66	13	quasi	quasi	NOUN
cana-3848	66	14	-	-	NOUN
cana-3848	66	15	norm	norm	ADJ
cana-3848	66	16	p	p	NOUN
cana-3848	66	17	such	such	ADJ
cana-3848	66	18	that	that	PRON
cana-3848	66	19	𝑝𝑗(𝑥	𝑝𝑗(𝑥	NOUN
cana-3848	66	20	)	)	PUNCT
cana-3848	66	21	=	=	SYM
cana-3848	66	22	0	0	NUM
cana-3848	66	23	,	,	PUNCT
cana-3848	66	24	given	give	VERB
cana-3848	66	25	the	the	DET
cana-3848	66	26	standard	standard	ADJ
cana-3848	66	27	addition	addition	NOUN
cana-3848	66	28	and	and	CCONJ
cana-3848	66	29	product	product	NOUN
cana-3848	66	30	on	on	ADP
cana-3848	66	31	ℝ+.the	ℝ+.the	DET
cana-3848	66	32	alexandr	alexandr	NOUN
cana-3848	66	33	off	off	ADP
cana-3848	66	34	extended	extended	ADJ
cana-3848	66	35	quasi	quasi	NOUN
cana-3848	66	36	-	-	ADJ
cana-3848	66	37	metric	metric	ADJ
cana-3848	66	38	on	on	ADV
cana-3848	66	39	ℝ+	ℝ+	PUNCT
cana-3848	66	40	is	be	AUX
cana-3848	66	41	the	the	DET
cana-3848	66	42	extended	extended	ADJ
cana-3848	66	43	quasi	quasi	NOUN
cana-3848	66	44	-	-	ADJ
cana-3848	66	45	metric	metric	ADJ
cana-3848	66	46	in	in	ADP
cana-3848	66	47	this	this	DET
cana-3848	66	48	instance	instance	NOUN
cana-3848	66	49	,	,	PUNCT
cana-3848	66	50	or	or	CCONJ
cana-3848	66	51	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	66	52	(	(	PUNCT
cana-3848	66	53	𝑥	𝑥	NOUN
cana-3848	66	54	,	,	PUNCT
cana-3848	66	55	𝑦	𝑦	NOUN
cana-3848	66	56	)	)	PUNCT
cana-3848	66	57	=	=	SYM
cana-3848	66	58	0	0	PUNCT
cana-3848	67	1	if	if	SCONJ
cana-3848	67	2	𝑥	𝑥	PRON
cana-3848	67	3	≤	≤	NUM
cana-3848	67	4	𝑦	𝑦	NOUN
cana-3848	67	5	and	and	CCONJ
cana-3848	67	6	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	67	7	(	(	PUNCT
cana-3848	67	8	𝑥	𝑥	NOUN
cana-3848	67	9	,	,	PUNCT
cana-3848	67	10	𝑦	𝑦	NOUN
cana-3848	67	11	)	)	PUNCT
cana-3848	67	12	=	=	PUNCT
cana-3848	68	1	+	+	PUNCT
cana-3848	68	2	∞	∞	NOUN
cana-3848	68	3	otherwise	otherwise	ADV
cana-3848	68	4	.	.	PUNCT
cana-3848	69	1	example	example	NOUN
cana-3848	69	2	(	(	PUNCT
cana-3848	69	3	2.4	2.4	NUM
cana-3848	69	4	):	):	PUNCT
cana-3848	69	5	for	for	ADP
cana-3848	69	6	each	each	DET
cana-3848	69	7	x	x	PUNCT
cana-3848	69	8	in	in	ADP
cana-3848	69	9	ℝ+	ℝ+	NOUN
cana-3848	69	10	,	,	PUNCT
cana-3848	69	11	𝑝𝑗(𝑥	𝑝𝑗(𝑥	NOUN
cana-3848	69	12	)	)	PUNCT
cana-3848	70	1	=	=	SYM
cana-3848	70	2	𝑥	𝑥	NOUN
cana-3848	70	3	,	,	PUNCT
cana-3848	70	4	therefore	therefore	ADV
cana-3848	70	5	let	let	VERB
cana-3848	70	6	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	70	7	∶	∶	NOUN
cana-3848	70	8	ℝ+	ℝ+	PUNCT
cana-3848	70	9	→	→	SYM
cana-3848	70	10	ℝ+	ℝ+	PUNCT
cana-3848	70	11	.	.	PUNCT
cana-3848	71	1	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	71	2	is	be	AUX
cana-3848	71	3	clearly	clearly	ADV
cana-3848	71	4	a	a	DET
cana-3848	71	5	norm	norm	NOUN
cana-3848	71	6	on	on	ADP
cana-3848	71	7	ℝ+	ℝ+	NOUN
cana-3848	71	8	,	,	PUNCT
cana-3848	71	9	and	and	CCONJ
cana-3848	71	10	if	if	SCONJ
cana-3848	71	11	𝑥	𝑥	PRON
cana-3848	71	12	≤	≤	NUM
cana-3848	71	13	𝑦	𝑦	NUM
cana-3848	71	14	,	,	PUNCT
cana-3848	71	15	then	then	ADV
cana-3848	71	16	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	71	17	(	(	PUNCT
cana-3848	71	18	𝑥	𝑥	PROPN
cana-3848	71	19	,	,	PUNCT
cana-3848	71	20	𝑦	𝑦	NOUN
cana-3848	71	21	)	)	PUNCT
cana-3848	71	22	=	=	PUNCT
cana-3848	72	1	+	+	PUNCT
cana-3848	72	2	∞	∞	NUM
cana-3848	72	3	and	and	CCONJ
cana-3848	72	4	𝑦	𝑦	NOUN
cana-3848	72	5	−	−	X
cana-3848	72	6	𝑥.	𝑥.	VERB
cana-3848	72	7	the	the	DET
cana-3848	72	8	sorgenfrey	sorgenfrey	PROPN
cana-3848	72	9	topology	topology	NOUN
cana-3848	72	10	on	on	ADV
cana-3848	72	11	ℝ+	ℝ+	PUNCT
cana-3848	72	12	is	be	AUX
cana-3848	72	13	thus	thus	ADV
cana-3848	72	14	the	the	DET
cana-3848	72	15	topology	topology	NOUN
cana-3848	72	16	generated	generate	VERB
cana-3848	72	17	by	by	ADP
cana-3848	72	18	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	72	19	,	,	PUNCT
cana-3848	72	20	since	since	SCONJ
cana-3848	72	21	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	72	22	is	be	AUX
cana-3848	72	23	the	the	DET
cana-3848	72	24	sorgenfrey	sorgenfrey	PROPN
cana-3848	72	25	extended	extended	ADJ
cana-3848	72	26	quasi	quasi	NOUN
cana-3848	72	27	-	-	ADJ
cana-3848	72	28	metric	metric	ADJ
cana-3848	72	29	on	on	ADP
cana-3848	72	30	ℝ+	ℝ+	NOUN
cana-3848	72	31	.	.	PUNCT
cana-3848	72	32	communications	communication	NOUN
cana-3848	72	33	on	on	ADP
cana-3848	72	34	applied	apply	VERB
cana-3848	72	35	nonlinear	nonlinear	ADJ
cana-3848	72	36	analysis	analysis	NOUN
cana-3848	72	37	issn	issn	NOUN
cana-3848	72	38	:	:	PUNCT
cana-3848	72	39	1074	1074	NUM
cana-3848	72	40	-	-	PUNCT
cana-3848	72	41	133x	133x	NUM
cana-3848	72	42	vol	vol	NOUN
cana-3848	72	43	32	32	NUM
cana-3848	72	44	no	no	NOUN
cana-3848	72	45	.	.	PUNCT
cana-3848	73	1	9s	9s	NUM
cana-3848	73	2	(	(	PUNCT
cana-3848	73	3	2025	2025	NUM
cana-3848	73	4	)	)	PUNCT
cana-3848	73	5	202	202	NUM
cana-3848	73	6	https://internationalpubls.com	https://internationalpubls.com	NOUN
cana-3848	73	7	remark	remark	NOUN
cana-3848	73	8	(	(	PUNCT
cana-3848	73	9	2.5):because	2.5):because	NUM
cana-3848	73	10	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	73	11	is	be	AUX
cana-3848	73	12	a	a	DET
cana-3848	73	13	(	(	PUNCT
cana-3848	73	14	quasi	quasi	ADJ
cana-3848	73	15	)	)	PUNCT
cana-3848	73	16	norm	norm	NOUN
cana-3848	73	17	on	on	ADP
cana-3848	73	18	a	a	DET
cana-3848	73	19	linear	linear	ADJ
cana-3848	73	20	space	space	NOUN
cana-3848	73	21	(	(	PUNCT
cana-3848	73	22	𝑋	𝑋	PROPN
cana-3848	73	23	,	,	PUNCT
cana-3848	73	24	+	+	PROPN
cana-3848	73	25	,	,	PUNCT
cana-3848	73	26	⋅	⋅	PROPN
cana-3848	73	27	)	)	PUNCT
cana-3848	73	28	,	,	PUNCT
cana-3848	73	29	the	the	DET
cana-3848	73	30	(	(	PUNCT
cana-3848	73	31	extended	extended	ADJ
cana-3848	73	32	)	)	PUNCT
cana-3848	73	33	quasimetric	quasimetric	ADJ
cana-3848	73	34	𝑝𝑗of	𝑝𝑗of	NOUN
cana-3848	73	35	proposition	proposition	NOUN
cana-3848	73	36	2.2	2.2	NUM
cana-3848	73	37	is	be	AUX
cana-3848	73	38	the	the	DET
cana-3848	73	39	classical	classical	ADJ
cana-3848	73	40	(	(	PUNCT
cana-3848	73	41	quasi)metric	quasi)metric	NOUN
cana-3848	73	42	on	on	ADP
cana-3848	73	43	𝑋	𝑋	PROPN
cana-3848	73	44	that	that	SCONJ
cana-3848	73	45	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	73	46	generates	generate	NOUN
cana-3848	73	47	.	.	PUNCT
cana-3848	74	1	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	74	2	(	(	PUNCT
cana-3848	74	3	𝑥	𝑥	PROPN
cana-3848	74	4	,	,	PUNCT
cana-3848	74	5	𝑦	𝑦	NOUN
cana-3848	74	6	)	)	PUNCT
cana-3848	74	7	=	=	SYM
cana-3848	74	8	𝑝𝑗(𝑦	𝑝𝑗(𝑦	NOUN
cana-3848	74	9	−	−	PROPN
cana-3848	74	10	𝑥	𝑥	NOUN
cana-3848	74	11	)	)	PUNCT
cana-3848	74	12	,	,	PUNCT
cana-3848	74	13	for	for	ADP
cana-3848	74	14	all	all	DET
cana-3848	74	15	𝑥	𝑥	PROPN
cana-3848	74	16	,	,	PUNCT
cana-3848	74	17	𝑦	𝑦	NOUN
cana-3848	74	18	belonging	belong	VERB
cana-3848	74	19	to	to	ADP
cana-3848	74	20	𝑋.since	𝑋.since	NOUN
cana-3848	74	21	there	there	PRON
cana-3848	74	22	is	be	VERB
cana-3848	74	23	a	a	DET
cana-3848	74	24	good	good	ADJ
cana-3848	74	25	solution	solution	NOUN
cana-3848	74	26	to	to	ADP
cana-3848	74	27	the	the	DET
cana-3848	74	28	bicompletion	bicompletion	NOUN
cana-3848	74	29	problem	problem	NOUN
cana-3848	74	30	in	in	ADP
cana-3848	74	31	the	the	DET
cana-3848	74	32	context	context	NOUN
cana-3848	74	33	of	of	ADP
cana-3848	74	34	quasi	quasi	ADJ
cana-3848	74	35	-	-	ADJ
cana-3848	74	36	metric	metric	ADJ
cana-3848	74	37	spaces	space	NOUN
cana-3848	74	38	(	(	PUNCT
cana-3848	74	39	[	[	X
cana-3848	74	40	5	5	NUM
cana-3848	74	41	]	]	NUM
cana-3848	74	42	)	)	PUNCT
cana-3848	74	43	,	,	PUNCT
cana-3848	74	44	we	we	PRON
cana-3848	74	45	will	will	AUX
cana-3848	74	46	focus	focus	VERB
cana-3848	74	47	on	on	ADP
cana-3848	74	48	quasi	quasi	ADJ
cana-3848	74	49	-	-	NOUN
cana-3848	74	50	norms	norm	NOUN
cana-3848	74	51	defined	define	VERB
cana-3848	74	52	on	on	ADP
cana-3848	74	53	cancellative	cancellative	ADJ
cana-3848	74	54	cones	cone	NOUN
cana-3848	74	55	.	.	PUNCT
cana-3848	75	1	however	however	ADV
cana-3848	75	2	,	,	PUNCT
cana-3848	75	3	some	some	DET
cana-3848	75	4	instances	instance	NOUN
cana-3848	75	5	of	of	ADP
cana-3848	75	6	spaces	space	NOUN
cana-3848	75	7	that	that	PRON
cana-3848	75	8	naturally	naturally	ADV
cana-3848	75	9	emerge	emerge	VERB
cana-3848	75	10	from	from	ADP
cana-3848	75	11	modeling	model	VERB
cana-3848	75	12	particular	particular	ADJ
cana-3848	75	13	processes	process	NOUN
cana-3848	75	14	in	in	ADP
cana-3848	75	15	theoretical	theoretical	ADJ
cana-3848	75	16	computer	computer	NOUN
cana-3848	75	17	science	science	NOUN
cana-3848	75	18	can	can	AUX
cana-3848	75	19	be	be	AUX
cana-3848	75	20	viewed	view	VERB
cana-3848	75	21	as	as	ADP
cana-3848	75	22	extended	extend	VERB
cana-3848	75	23	cancellative	cancellative	ADJ
cana-3848	75	24	quasimetric	quasimetric	ADJ
cana-3848	75	25	cones	cone	NOUN
cana-3848	75	26	(	(	PUNCT
cana-3848	75	27	see	see	VERB
cana-3848	75	28	example	example	NOUN
cana-3848	75	29	3.2	3.2	NUM
cana-3848	75	30	)	)	PUNCT
cana-3848	75	31	below	below	ADP
cana-3848	75	32	)	)	PUNCT
cana-3848	75	33	.	.	PUNCT
cana-3848	76	1	consequently	consequently	ADV
cana-3848	76	2	,	,	PUNCT
cana-3848	76	3	we	we	PRON
cana-3848	76	4	propose	propose	VERB
cana-3848	76	5	the	the	DET
cana-3848	76	6	following	following	ADJ
cana-3848	76	7	notion	notion	NOUN
cana-3848	76	8	.	.	PUNCT
cana-3848	77	1	definition	definition	NOUN
cana-3848	77	2	(	(	PUNCT
cana-3848	77	3	2.6	2.6	NUM
cana-3848	77	4	):	):	PUNCT
cana-3848	77	5	quasi	quasi	ADJ
cana-3848	77	6	-	-	ADJ
cana-3848	77	7	normed	norme	VERB
cana-3848	77	8	cones	cone	NOUN
cana-3848	77	9	are	be	AUX
cana-3848	77	10	pairs	pair	NOUN
cana-3848	77	11	(	(	PUNCT
cana-3848	77	12	𝑋	𝑋	NOUN
cana-3848	77	13	,	,	PUNCT
cana-3848	77	14	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	77	15	)	)	PUNCT
cana-3848	77	16	in	in	ADP
cana-3848	77	17	which	which	PRON
cana-3848	77	18	𝑋	𝑋	NOUN
cana-3848	77	19	is	be	AUX
cana-3848	77	20	a	a	DET
cana-3848	77	21	cancellative	cancellative	ADJ
cana-3848	77	22	cone	cone	NOUN
cana-3848	77	23	and	and	CCONJ
cana-3848	77	24	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	77	25	is	be	AUX
cana-3848	77	26	a	a	DET
cana-3848	77	27	quasi	quasi	NOUN
cana-3848	77	28	-	-	NOUN
cana-3848	77	29	norm	norm	NOUN
cana-3848	77	30	on	on	ADP
cana-3848	77	31	𝑋.	𝑋.	PROPN
cana-3848	77	32	3	3	NUM
cana-3848	77	33	.	.	PUNCT
cana-3848	78	1	the	the	DET
cana-3848	78	2	semi	semi	ADJ
cana-3848	78	3	-	-	ADJ
cana-3848	78	4	normal	normal	ADJ
cana-3848	78	5	cone	cone	NOUN
cana-3848	78	6	's	's	PART
cana-3848	78	7	bicompletion	bicompletion	NOUN
cana-3848	78	8	remember	remember	VERB
cana-3848	78	9	that	that	SCONJ
cana-3848	78	10	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	78	11	∶	∶	NOUN
cana-3848	78	12	𝑋	𝑋	PROPN
cana-3848	78	13	→	→	SYM
cana-3848	78	14	𝑌	𝑌	PROPN
cana-3848	78	15	is	be	AUX
cana-3848	78	16	a	a	DET
cana-3848	78	17	linear	linear	ADJ
cana-3848	78	18	function	function	NOUN
cana-3848	78	19	from	from	ADP
cana-3848	78	20	a	a	DET
cana-3848	78	21	cone	cone	NOUN
cana-3848	78	22	(	(	PUNCT
cana-3848	78	23	𝑋	𝑋	PROPN
cana-3848	78	24	,	,	PUNCT
cana-3848	78	25	+	+	PROPN
cana-3848	78	26	,	,	PUNCT
cana-3848	78	27	⋅	⋅	PROPN
cana-3848	78	28	)	)	PUNCT
cana-3848	78	29	to	to	ADP
cana-3848	78	30	a	a	DET
cana-3848	78	31	cone	cone	NOUN
cana-3848	78	32	(	(	PUNCT
cana-3848	78	33	𝑌,⊕,⊗	𝑌,⊕,⊗	NOUN
cana-3848	78	34	)	)	PUNCT
cana-3848	78	35	.	.	PUNCT
cana-3848	79	1	such	such	ADJ
cana-3848	79	2	that	that	PRON
cana-3848	79	3	𝑓𝑗(𝛼	𝑓𝑗(𝛼	PUNCT
cana-3848	79	4	⋅	⋅	PROPN
cana-3848	79	5	𝑥	𝑥	PROPN
cana-3848	79	6	+	+	PUNCT
cana-3848	79	7	𝛽	𝛽	PROPN
cana-3848	79	8	⋅	⋅	PROPN
cana-3848	79	9	𝑦	𝑦	NUM
cana-3848	79	10	)	)	PUNCT
cana-3848	79	11	=	=	SYM
cana-3848	79	12	𝛼	𝛼	NOUN
cana-3848	79	13	⊗	⊗	PROPN
cana-3848	79	14	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	79	15	)	)	PUNCT
cana-3848	79	16	⊕	⊕	PROPN
cana-3848	79	17	𝛽	𝛽	NOUN
cana-3848	79	18	⊗	⊗	PROPN
cana-3848	79	19	𝑓𝑗(𝑦	𝑓𝑗(𝑦	NUM
cana-3848	79	20	)	)	PUNCT
cana-3848	79	21	.	.	PUNCT
cana-3848	80	1	definition	definition	NOUN
cana-3848	80	2	(	(	PUNCT
cana-3848	80	3	3.1):the	3.1):the	DET
cana-3848	80	4	quasi	quasi	ADJ
cana-3848	80	5	-	-	ADJ
cana-3848	80	6	normed	normed	ADJ
cana-3848	80	7	cones	cone	NOUN
cana-3848	80	8	(	(	PUNCT
cana-3848	80	9	𝑋	𝑋	NOUN
cana-3848	80	10	,	,	PUNCT
cana-3848	80	11	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	80	12	)	)	PUNCT
cana-3848	80	13	and	and	CCONJ
cana-3848	80	14	(	(	PUNCT
cana-3848	80	15	𝑌	𝑌	PROPN
cana-3848	80	16	,	,	PUNCT
cana-3848	80	17	𝑞𝑗	𝑞𝑗	PROPN
cana-3848	80	18	)	)	PUNCT
cana-3848	80	19	.	.	PUNCT
cana-3848	81	1	are	be	AUX
cana-3848	81	2	isometric	isometric	ADJ
cana-3848	81	3	to	to	ADP
cana-3848	81	4	a	a	DET
cana-3848	81	5	linear	linear	ADJ
cana-3848	81	6	function	function	NOUN
cana-3848	81	7	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	81	8	∶	∶	NOUN
cana-3848	81	9	𝑋	𝑋	PROPN
cana-3848	81	10	→	→	SYM
cana-3848	81	11	𝑌.	𝑌.	PROPN
cana-3848	81	12	it	it	PRON
cana-3848	81	13	guarantees	guarantee	VERB
cana-3848	81	14	that	that	SCONJ
cana-3848	81	15	for	for	ADP
cana-3848	81	16	every	every	DET
cana-3848	81	17	𝑥	𝑥	PROPN
cana-3848	81	18	𝑖𝑛	𝑖𝑛	NOUN
cana-3848	81	19	𝑋	𝑋	PROPN
cana-3848	81	20	,	,	PUNCT
cana-3848	81	21	𝑞𝑗	𝑞𝑗	VERB
cana-3848	81	22	(	(	PUNCT
cana-3848	81	23	𝑓𝑗(𝑥	𝑓𝑗(𝑥	NOUN
cana-3848	81	24	)	)	PUNCT
cana-3848	81	25	)	)	PUNCT
cana-3848	81	26	=	=	SYM
cana-3848	81	27	𝑝𝑗(𝑥	𝑝𝑗(𝑥	NOUN
cana-3848	81	28	)	)	PUNCT
cana-3848	81	29	.	.	PUNCT
cana-3848	82	1	the	the	DET
cana-3848	82	2	following	follow	VERB
cana-3848	82	3	illustration	illustration	NOUN
cana-3848	82	4	shows	show	VERB
cana-3848	82	5	that	that	SCONJ
cana-3848	82	6	,	,	PUNCT
cana-3848	82	7	unlike	unlike	ADP
cana-3848	82	8	the	the	DET
cana-3848	82	9	quasi	quasi	ADJ
cana-3848	82	10	-	-	ADJ
cana-3848	82	11	metric	metric	ADJ
cana-3848	82	12	case	case	NOUN
cana-3848	82	13	,	,	PUNCT
cana-3848	82	14	there	there	PRON
cana-3848	82	15	are	be	VERB
cana-3848	82	16	non	non	ADJ
cana-3848	82	17	-	-	ADJ
cana-3848	82	18	injective	injective	ADJ
cana-3848	82	19	isometries	isometry	NOUN
cana-3848	82	20	between	between	ADP
cana-3848	82	21	quasinormed	quasinorme	VERB
cana-3848	82	22	cones	cone	NOUN
cana-3848	82	23	.	.	PUNCT
cana-3848	83	1	example	example	NOUN
cana-3848	83	2	(	(	PUNCT
cana-3848	83	3	3.2	3.2	NUM
cana-3848	83	4	):	):	PUNCT
cana-3848	83	5	taking	take	VERB
cana-3848	83	6	cue	cue	NOUN
cana-3848	83	7	from	from	ADP
cana-3848	83	8	[	[	X
cana-3848	83	9	6	6	NUM
cana-3848	83	10	]	]	SYM
cana-3848	83	11	's	's	PART
cana-3848	83	12	applications	application	NOUN
cana-3848	83	13	for	for	ADP
cana-3848	83	14	program	program	NOUN
cana-3848	83	15	and	and	CCONJ
cana-3848	83	16	algorithmic	algorithmic	ADJ
cana-3848	83	17	complexity	complexity	NOUN
cana-3848	83	18	analysis	analysis	NOUN
cana-3848	83	19	,	,	PUNCT
cana-3848	83	20	[	[	X
cana-3848	83	21	4	4	NUM
cana-3848	83	22	]	]	ADJ
cana-3848	83	23	introduces	introduce	NOUN
cana-3848	83	24	and	and	CCONJ
cana-3848	83	25	investigates	investigate	VERB
cana-3848	83	26	the	the	DET
cana-3848	83	27	idea	idea	NOUN
cana-3848	83	28	of	of	ADP
cana-3848	83	29	the	the	DET
cana-3848	83	30	so	so	ADV
cana-3848	83	31	-	-	PUNCT
cana-3848	83	32	called	call	VERB
cana-3848	83	33	dual	dual	ADJ
cana-3848	83	34	complexity	complexity	NOUN
cana-3848	83	35	space	space	NOUN
cana-3848	83	36	,	,	PUNCT
cana-3848	83	37	which	which	PRON
cana-3848	83	38	consists	consist	VERB
cana-3848	83	39	of	of	ADP
cana-3848	83	40	the	the	DET
cana-3848	83	41	pair(𝒞∗	pair(𝒞∗	NOUN
cana-3848	83	42	,	,	PUNCT
cana-3848	83	43	𝑑𝒞∗	𝑑𝒞∗	NOUN
cana-3848	83	44	)	)	PUNCT
cana-3848	83	45	,	,	PUNCT
cana-3848	83	46	where	where	SCONJ
cana-3848	83	47	𝒞∗	𝒞∗	PROPN
cana-3848	83	48	=	=	PUNCT
cana-3848	83	49	{	{	PUNCT
cana-3848	83	50	𝑓𝑗	𝑓𝑗	NOUN
cana-3848	83	51	∈	∈	NOUN
cana-3848	83	52	(	(	PUNCT
cana-3848	83	53	ℝ+)𝜔	ℝ+)𝜔	NOUN
cana-3848	83	54	∶	∶	NOUN
cana-3848	83	55	∑	∑	PUNCT
cana-3848	83	56	(	(	PUNCT
cana-3848	83	57	∑	∑	NOUN
cana-3848	83	58	2−𝑛𝑓𝑗(𝑛	2−𝑛𝑓𝑗(𝑛	NUM
cana-3848	83	59	)	)	PUNCT
cana-3848	83	60	∞	∞	NUM
cana-3848	83	61	𝑛=0	𝑛=0	X
cana-3848	83	62	<	<	X
cana-3848	84	1	+	+	NOUN
cana-3848	84	2	∞	∞	NOUN
cana-3848	84	3	)	)	PUNCT
cana-3848	84	4	𝑗	𝑗	NOUN
cana-3848	84	5	}	}	PUNCT
cana-3848	84	6	,	,	PUNCT
cana-3848	84	7	and	and	CCONJ
cana-3848	84	8	𝑑𝒞∗	𝑑𝒞∗	NOUN
cana-3848	84	9	is	be	AUX
cana-3848	84	10	the	the	DET
cana-3848	84	11	quasi	quasi	NOUN
cana-3848	84	12	-	-	ADJ
cana-3848	84	13	metric	metric	ADJ
cana-3848	84	14	on	on	ADP
cana-3848	84	15	𝒞∗	𝒞∗	PROPN
cana-3848	84	16	given	give	VERB
cana-3848	84	17	by	by	ADP
cana-3848	84	18	𝑑𝒞∗(𝑓𝑗	𝑑𝒞∗(𝑓𝑗	PROPN
cana-3848	84	19	,	,	PUNCT
cana-3848	84	20	𝑔𝑗	𝑔𝑗	PROPN
cana-3848	84	21	)	)	PUNCT
cana-3848	84	22	=	=	SYM
cana-3848	85	1	∑	∑	PUNCT
cana-3848	85	2	∑	∑	PROPN
cana-3848	85	3	2−𝑛	2−𝑛	NUM
cana-3848	85	4	∞	∞	NUM
cana-3848	85	5	𝑛=0	𝑛=0	PROPN
cana-3848	86	1	[	[	X
cana-3848	86	2	(	(	PUNCT
cana-3848	86	3	𝑔𝑗(𝑛	𝑔𝑗(𝑛	NOUN
cana-3848	86	4	)	)	PUNCT
cana-3848	86	5	−	−	NOUN
cana-3848	86	6	𝑓𝑗(𝑛	𝑓𝑗(𝑛	NOUN
cana-3848	86	7	)	)	PUNCT
cana-3848	86	8	)	)	PUNCT
cana-3848	86	9	⋁0	⋁0	PROPN
cana-3848	86	10	]	]	X
cana-3848	86	11	𝑗	𝑗	X
cana-3848	86	12	.	.	PUNCT
cana-3848	87	1	there	there	PRON
cana-3848	87	2	are	be	VERB
cana-3848	87	3	several	several	ADJ
cana-3848	87	4	𝑑𝒞∗properties	𝑑𝒞∗propertie	NOUN
cana-3848	87	5	discussed	discuss	VERB
cana-3848	87	6	in	in	ADP
cana-3848	87	7	[	[	X
cana-3848	87	8	4	4	NUM
cana-3848	87	9	]	]	PUNCT
cana-3848	87	10	.	.	PUNCT
cana-3848	88	1	note	note	VERB
cana-3848	88	2	specifically	specifically	ADV
cana-3848	88	3	that	that	SCONJ
cana-3848	88	4	𝑇1	𝑇1	NOUN
cana-3848	88	5	topology	topology	NOUN
cana-3848	88	6	is	be	AUX
cana-3848	88	7	not	not	PART
cana-3848	88	8	caused	cause	VERB
cana-3848	88	9	by	by	ADP
cana-3848	88	10	𝑑𝒞∗	𝑑𝒞∗	NOUN
cana-3848	88	11	.	.	PUNCT
cana-3848	89	1	with	with	ADP
cana-3848	89	2	the	the	DET
cana-3848	89	3	neutral	neutral	ADJ
cana-3848	89	4	element	element	NOUN
cana-3848	89	5	𝑓0𝑗	𝑓0𝑗	PROPN
cana-3848	89	6	∈	∈	PROPN
cana-3848	89	7	𝒞∗	𝒞∗	PROPN
cana-3848	89	8	given	give	VERB
cana-3848	89	9	by	by	ADP
cana-3848	89	10	𝑓0𝑗	𝑓0𝑗	PROPN
cana-3848	89	11	(	(	PUNCT
cana-3848	89	12	𝑛	𝑛	NOUN
cana-3848	89	13	)	)	PUNCT
cana-3848	89	14	=	=	SYM
cana-3848	89	15	0	0	NUM
cana-3848	89	16	for	for	ADP
cana-3848	89	17	all	all	DET
cana-3848	89	18	𝑛	𝑛	DET
cana-3848	89	19	∈	∈	NOUN
cana-3848	89	20	𝜔	𝜔	NOUN
cana-3848	89	21	,	,	PUNCT
cana-3848	89	22	and	and	CCONJ
cana-3848	89	23	⋅	⋅	PROPN
cana-3848	89	24	being	be	AUX
cana-3848	89	25	the	the	DET
cana-3848	89	26	operation	operation	NOUN
cana-3848	89	27	specified	specify	VERB
cana-3848	89	28	by	by	ADP
cana-3848	89	29	(	(	PUNCT
cana-3848	89	30	𝜆	𝜆	DET
cana-3848	89	31	⋅	⋅	PROPN
cana-3848	89	32	𝑓𝑗)(𝑛	𝑓𝑗)(𝑛	PROPN
cana-3848	89	33	)	)	PUNCT
cana-3848	89	34	=	=	SYM
cana-3848	89	35	𝜆𝑓𝑗(𝑛	𝜆𝑓𝑗(𝑛	X
cana-3848	89	36	)	)	PUNCT
cana-3848	89	37	for	for	ADP
cana-3848	89	38	all	all	DET
cana-3848	89	39	𝑛	𝑛	DET
cana-3848	89	40	∈	∈	NOUN
cana-3848	89	41	𝜔	𝜔	NOUN
cana-3848	89	42	,	,	PUNCT
cana-3848	89	43	(	(	PUNCT
cana-3848	89	44	𝒞∗	𝒞∗	PROPN
cana-3848	89	45	,	,	PUNCT
cana-3848	89	46	+	+	NOUN
cana-3848	89	47	,	,	PUNCT
cana-3848	89	48	⋅	⋅	PROPN
cana-3848	89	49	)	)	PUNCT
cana-3848	89	50	is	be	AUX
cana-3848	89	51	clearly	clearly	ADV
cana-3848	89	52	a	a	DET
cana-3848	89	53	cancellative	cancellative	ADJ
cana-3848	89	54	cone	cone	NOUN
cana-3848	89	55	.	.	PUNCT
cana-3848	90	1	let	let	VERB
cana-3848	90	2	's	us	PRON
cana-3848	90	3	say	say	VERB
cana-3848	90	4	that	that	SCONJ
cana-3848	90	5	∑	∑	PUNCT
cana-3848	90	6	𝑝𝑗(𝑓𝑗	𝑝𝑗(𝑓𝑗	ADJ
cana-3848	90	7	)	)	PUNCT
cana-3848	90	8	𝑗	𝑗	NOUN
cana-3848	90	9	=	=	PUNCT
cana-3848	90	10	∑	∑	PUNCT
cana-3848	90	11	(	(	PUNCT
cana-3848	90	12	∑	∑	PROPN
cana-3848	90	13	2−𝑛	2−𝑛	NUM
cana-3848	90	14	∞	∞	NUM
cana-3848	90	15	𝑛=0	𝑛=0	NOUN
cana-3848	90	16	𝑓𝑗(𝑛	𝑓𝑗(𝑛	NOUN
cana-3848	90	17	)	)	PUNCT
cana-3848	90	18	)	)	PUNCT
cana-3848	91	1	𝑗	𝑗	PROPN
cana-3848	91	2	communications	communication	NOUN
cana-3848	91	3	on	on	ADP
cana-3848	91	4	applied	apply	VERB
cana-3848	91	5	nonlinear	nonlinear	ADJ
cana-3848	91	6	analysis	analysis	NOUN
cana-3848	91	7	issn	issn	NOUN
cana-3848	91	8	:	:	PUNCT
cana-3848	91	9	1074	1074	NUM
cana-3848	91	10	-	-	PUNCT
cana-3848	91	11	133x	133x	NUM
cana-3848	91	12	vol	vol	NOUN
cana-3848	91	13	32	32	NUM
cana-3848	91	14	no	no	NOUN
cana-3848	91	15	.	.	PUNCT
cana-3848	92	1	9s	9s	NUM
cana-3848	92	2	(	(	PUNCT
cana-3848	92	3	2025	2025	NUM
cana-3848	92	4	)	)	PUNCT
cana-3848	92	5	203	203	NUM
cana-3848	92	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	92	7	suppose	suppose	VERB
cana-3848	92	8	that	that	SCONJ
cana-3848	92	9	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	92	10	∶	∶	NOUN
cana-3848	92	11	𝒞∗	𝒞∗	NUM
cana-3848	92	12	→	→	PUNCT
cana-3848	92	13	ℝ+.the	ℝ+.the	DET
cana-3848	92	14	fact	fact	NOUN
cana-3848	92	15	that	that	SCONJ
cana-3848	92	16	p_j	p_j	PROPN
cana-3848	92	17	is	be	AUX
cana-3848	92	18	a	a	DET
cana-3848	92	19	quasi	quasi	NOUN
cana-3848	92	20	-	-	NOUN
cana-3848	92	21	norm	norm	NOUN
cana-3848	92	22	on	on	ADP
cana-3848	92	23	𝒞∗	𝒞∗	PROPN
cana-3848	92	24	is	be	AUX
cana-3848	92	25	well	well	ADV
cana-3848	92	26	accepted	accept	VERB
cana-3848	92	27	.	.	PUNCT
cana-3848	93	1	next	next	ADV
cana-3848	93	2	,	,	PUNCT
cana-3848	93	3	the	the	DET
cana-3848	93	4	induced	induce	VERB
cana-3848	93	5	extended	extended	ADJ
cana-3848	93	6	quasi	quasi	ADJ
cana-3848	93	7	-	-	ADJ
cana-3848	93	8	metric	metric	ADJ
cana-3848	93	9	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	93	10	on	on	ADP
cana-3848	93	11	𝒞∗	𝒞∗	PROPN
cana-3848	93	12	is	be	AUX
cana-3848	93	13	given	give	VERB
cana-3848	93	14	by	by	ADP
cana-3848	93	15	∑	∑	ADV
cana-3848	93	16	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	93	17	(	(	PUNCT
cana-3848	93	18	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	93	19	,	,	PUNCT
cana-3848	93	20	𝑔𝑗	𝑔𝑗	NOUN
cana-3848	93	21	)	)	PUNCT
cana-3848	93	22	∗	∗	NOUN
cana-3848	93	23	𝑗	𝑗	NOUN
cana-3848	93	24	=	=	X
cana-3848	93	25	∑	∑	PUNCT
cana-3848	93	26	(	(	PUNCT
cana-3848	93	27	∑	∑	PROPN
cana-3848	93	28	2−𝑛	2−𝑛	NUM
cana-3848	93	29	(	(	PUNCT
cana-3848	93	30	𝑔𝑗(𝑛	𝑔𝑗(𝑛	NOUN
cana-3848	93	31	)	)	PUNCT
cana-3848	93	32	−	−	NOUN
cana-3848	93	33	𝑓𝑗(𝑛	𝑓𝑗(𝑛	NOUN
cana-3848	93	34	)	)	PUNCT
cana-3848	93	35	)	)	PUNCT
cana-3848	94	1	∞	∞	NUM
cana-3848	94	2	𝑛=0	𝑛=0	NOUN
cana-3848	94	3	)	)	PUNCT
cana-3848	95	1	𝑗	𝑗	INTJ
cana-3848	95	2	if	if	SCONJ
cana-3848	95	3	𝑓𝑗	𝑓𝑗	PRON
cana-3848	95	4	≤	≤	NUM
cana-3848	95	5	𝑔𝑗	𝑔𝑗	PROPN
cana-3848	95	6	,	,	PUNCT
cana-3848	95	7	and	and	CCONJ
cana-3848	95	8	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	95	9	(	(	PUNCT
cana-3848	95	10	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	95	11	,	,	PUNCT
cana-3848	95	12	𝑔𝑗	𝑔𝑗	NOUN
cana-3848	95	13	)	)	PUNCT
cana-3848	95	14	=	=	PUNCT
cana-3848	96	1	+	+	PUNCT
cana-3848	96	2	∞	∞	NOUN
cana-3848	96	3	otherwise	otherwise	ADV
cana-3848	96	4	.	.	PUNCT
cana-3848	97	1	let	let	VERB
cana-3848	97	2	𝑋	𝑋	PROPN
cana-3848	97	3	=	=	SYM
cana-3848	97	4	{	{	PUNCT
cana-3848	97	5	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	97	6	∈	∈	PROPN
cana-3848	97	7	𝒞∗	𝒞∗	PROPN
cana-3848	97	8	∶	∶	PROPN
cana-3848	97	9	𝑓𝑗(0	𝑓𝑗(0	PROPN
cana-3848	97	10	)	)	PUNCT
cana-3848	97	11	>	>	X
cana-3848	97	12	0	0	X
cana-3848	97	13	}	}	PUNCT
cana-3848	97	14	∪	∪	X
cana-3848	97	15	{	{	PUNCT
cana-3848	97	16	𝑓𝑗0	𝑓𝑗0	NOUN
cana-3848	97	17	}	}	PUNCT
cana-3848	97	18	.	.	PUNCT
cana-3848	98	1	it	it	PRON
cana-3848	98	2	is	be	AUX
cana-3848	98	3	frequently	frequently	ADV
cana-3848	98	4	observed	observe	VERB
cana-3848	98	5	that	that	SCONJ
cana-3848	98	6	𝑋	𝑋	PROPN
cana-3848	98	7	is	be	AUX
cana-3848	98	8	a	a	DET
cana-3848	98	9	subclone	subclone	NOUN
cana-3848	98	10	of	of	ADP
cana-3848	98	11	𝒞∗	𝒞∗	PROPN
cana-3848	98	12	𝑞(𝑓𝑗	𝑞(𝑓𝑗	PROPN
cana-3848	98	13	)	)	PUNCT
cana-3848	98	14	=	=	SYM
cana-3848	98	15	𝑓𝑗(0	𝑓𝑗(0	PROPN
cana-3848	98	16	)	)	PUNCT
cana-3848	98	17	is	be	AUX
cana-3848	98	18	the	the	DET
cana-3848	98	19	definition	definition	NOUN
cana-3848	98	20	of	of	ADP
cana-3848	98	21	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	98	22	∶	∶	NOUN
cana-3848	98	23	𝑋	𝑋	NOUN
cana-3848	98	24	→	→	SYM
cana-3848	98	25	ℝ+	ℝ+	PROPN
cana-3848	98	26	.	.	PUNCT
cana-3848	99	1	there	there	PRON
cana-3848	99	2	is	be	VERB
cana-3848	99	3	no	no	DET
cana-3848	99	4	doubt	doubt	NOUN
cana-3848	99	5	that	that	SCONJ
cana-3848	99	6	𝑞𝑗	𝑞𝑗	VERB
cana-3848	99	7	is	be	AUX
cana-3848	99	8	a	a	DET
cana-3848	99	9	quasi	quasi	NOUN
cana-3848	99	10	-	-	NOUN
cana-3848	99	11	norm	norm	NOUN
cana-3848	99	12	on	on	ADP
cana-3848	99	13	x.	x.	NOUN
cana-3848	99	14	let	let	VERB
cana-3848	99	15	𝐹(𝑓𝑗)(0	𝐹(𝑓𝑗)(0	NUM
cana-3848	99	16	)	)	PUNCT
cana-3848	100	1	=	=	SYM
cana-3848	100	2	𝑓𝑗(0	𝑓𝑗(0	NOUN
cana-3848	100	3	)	)	PUNCT
cana-3848	100	4	and	and	CCONJ
cana-3848	100	5	𝐹(𝑓𝑗)(𝑛	𝐹(𝑓𝑗)(𝑛	NUM
cana-3848	100	6	)	)	PUNCT
cana-3848	101	1	=	=	SYM
cana-3848	101	2	0	0	NUM
cana-3848	101	3	define	define	VERB
cana-3848	101	4	𝐹	𝐹	PROPN
cana-3848	101	5	∶	∶	NOUN
cana-3848	101	6	𝑋	𝑋	PROPN
cana-3848	101	7	→	→	SYM
cana-3848	101	8	𝒞∗	𝒞∗	X
cana-3848	101	9	for	for	ADP
cana-3848	101	10	each	each	DET
cana-3848	101	11	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	101	12	∈	∈	PROPN
cana-3848	101	13	𝑋and	𝑋and	PROPN
cana-3848	101	14	𝑛	𝑛	PRON
cana-3848	101	15	∈	∈	PROPN
cana-3848	101	16	ℕ.	ℕ.	PROPN
cana-3848	101	17	f	f	PROPN
cana-3848	101	18	is	be	AUX
cana-3848	101	19	obviously	obviously	ADV
cana-3848	101	20	linear	linear	ADJ
cana-3848	101	21	from	from	ADP
cana-3848	101	22	(	(	PUNCT
cana-3848	101	23	𝑋	𝑋	PROPN
cana-3848	101	24	,	,	PUNCT
cana-3848	101	25	+	+	PROPN
cana-3848	101	26	,	,	PUNCT
cana-3848	101	27	⋅	⋅	PROPN
cana-3848	101	28	)	)	PUNCT
cana-3848	101	29	to	to	ADP
cana-3848	101	30	(	(	PUNCT
cana-3848	101	31	𝒞∗	𝒞∗	PROPN
cana-3848	101	32	,	,	PUNCT
cana-3848	101	33	+	+	NOUN
cana-3848	101	34	,	,	PUNCT
cana-3848	101	35	⋅	⋅	PROPN
cana-3848	101	36	)	)	PUNCT
cana-3848	101	37	.	.	PUNCT
cana-3848	102	1	moreover	moreover	ADV
cana-3848	102	2	,	,	PUNCT
cana-3848	102	3	for	for	ADP
cana-3848	102	4	any	any	DET
cana-3848	102	5	𝑓𝑗𝑖𝑛	𝑓𝑗𝑖𝑛	PROPN
cana-3848	102	6	𝑋	𝑋	PROPN
cana-3848	102	7	,	,	PUNCT
cana-3848	102	8	∑	∑	ADV
cana-3848	102	9	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	102	10	(	(	PUNCT
cana-3848	102	11	(	(	PUNCT
cana-3848	102	12	𝐹(𝑓𝑗	𝐹(𝑓𝑗	NOUN
cana-3848	102	13	)	)	PUNCT
cana-3848	102	14	)	)	PUNCT
cana-3848	103	1	𝑗	𝑗	X
cana-3848	103	2	=	=	PUNCT
cana-3848	103	3	∑	∑	PUNCT
cana-3848	103	4	(	(	PUNCT
cana-3848	103	5	∑	∑	PROPN
cana-3848	103	6	2−𝑛𝐹	2−𝑛𝐹	PROPN
cana-3848	103	7	(	(	PUNCT
cana-3848	103	8	𝑓𝑗(𝑛	𝑓𝑗(𝑛	NOUN
cana-3848	103	9	)	)	PUNCT
cana-3848	103	10	)	)	PUNCT
cana-3848	104	1	∞	∞	NUM
cana-3848	104	2	𝑛=0	𝑛=0	PROPN
cana-3848	104	3	)	)	PUNCT
cana-3848	105	1	𝑗	𝑗	NOUN
cana-3848	105	2	=	=	PUNCT
cana-3848	105	3	∑	∑	PUNCT
cana-3848	105	4	𝑓𝑗(0	𝑓𝑗(0	PROPN
cana-3848	105	5	)	)	PUNCT
cana-3848	105	6	𝑗	𝑗	PROPN
cana-3848	105	7	𝑗	𝑗	NOUN
cana-3848	105	8	=	=	SYM
cana-3848	105	9	∑	∑	PUNCT
cana-3848	105	10	𝑞𝑗(𝑓𝑗	𝑞𝑗(𝑓𝑗	PROPN
cana-3848	105	11	)	)	PUNCT
cana-3848	106	1	𝑗	𝑗	PROPN
cana-3848	106	2	f	f	NOUN
cana-3848	106	3	,	,	PUNCT
cana-3848	106	4	thus	thus	ADV
cana-3848	106	5	,	,	PUNCT
cana-3848	106	6	is	be	AUX
cana-3848	106	7	an	an	DET
cana-3848	106	8	isometry	isometry	NOUN
cana-3848	106	9	between	between	ADP
cana-3848	106	10	(	(	PUNCT
cana-3848	106	11	𝑋	𝑋	PROPN
cana-3848	106	12	,	,	PUNCT
cana-3848	106	13	𝑞𝑗	𝑞𝑗	PROPN
cana-3848	106	14	)	)	PUNCT
cana-3848	106	15	and	and	CCONJ
cana-3848	106	16	(	(	PUNCT
cana-3848	106	17	𝒞∗	𝒞∗	PROPN
cana-3848	106	18	,	,	PUNCT
cana-3848	106	19	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	106	20	)	)	PUNCT
cana-3848	106	21	.	.	PUNCT
cana-3848	107	1	but	but	CCONJ
cana-3848	107	2	𝐹	𝐹	PROPN
cana-3848	107	3	is	be	AUX
cana-3848	107	4	not	not	PART
cana-3848	107	5	injective	injective	ADJ
cana-3848	107	6	if	if	SCONJ
cana-3848	107	7	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	107	8	,	,	PUNCT
cana-3848	107	9	𝑔𝑗	𝑔𝑗	NOUN
cana-3848	107	10	∈	∈	NOUN
cana-3848	107	11	𝑋	𝑋	PROPN
cana-3848	107	12	satisfy	satisfy	NOUN
cana-3848	107	13	𝑓𝑗(0	𝑓𝑗(0	PROPN
cana-3848	107	14	)	)	PUNCT
cana-3848	107	15	=	=	SYM
cana-3848	107	16	𝑔𝑗(0	𝑔𝑗(0	NOUN
cana-3848	107	17	)	)	PUNCT
cana-3848	107	18	and	and	CCONJ
cana-3848	107	19	𝑓𝑗(1	𝑓𝑗(1	NOUN
cana-3848	107	20	)	)	PUNCT
cana-3848	107	21	≠	≠	PROPN
cana-3848	107	22	𝑔𝑗(1).this	𝑔𝑗(1).this	PRON
cana-3848	107	23	is	be	AUX
cana-3848	107	24	because	because	SCONJ
cana-3848	107	25	we	we	PRON
cana-3848	107	26	get	get	VERB
cana-3848	107	27	𝐹(𝑓𝑗	𝐹(𝑓𝑗	PRON
cana-3848	107	28	)	)	PUNCT
cana-3848	108	1	=	=	SYM
cana-3848	108	2	𝐹(𝑔𝑗).f	𝐹(𝑔𝑗).f	NOUN
cana-3848	108	3	,	,	PUNCT
cana-3848	108	4	thus	thus	ADV
cana-3848	108	5	,	,	PUNCT
cana-3848	108	6	is	be	AUX
cana-3848	108	7	an	an	DET
cana-3848	108	8	isometry	isometry	NOUN
cana-3848	108	9	between	between	ADP
cana-3848	108	10	(	(	PUNCT
cana-3848	108	11	𝑋	𝑋	PROPN
cana-3848	108	12	,	,	PUNCT
cana-3848	108	13	𝑞𝑗	𝑞𝑗	PROPN
cana-3848	108	14	)	)	PUNCT
cana-3848	108	15	and	and	CCONJ
cana-3848	108	16	(	(	PUNCT
cana-3848	108	17	𝒞∗	𝒞∗	PROPN
cana-3848	108	18	,	,	PUNCT
cana-3848	108	19	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	108	20	)	)	PUNCT
cana-3848	108	21	.	.	PUNCT
cana-3848	109	1	however	however	ADV
cana-3848	109	2	,	,	PUNCT
cana-3848	109	3	if	if	SCONJ
cana-3848	109	4	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	109	5	,	,	PUNCT
cana-3848	109	6	𝑔𝑗	𝑔𝑗	NOUN
cana-3848	109	7	∈	∈	NOUN
cana-3848	109	8	𝑋	𝑋	PROPN
cana-3848	109	9	complete	complete	ADJ
cana-3848	109	10	𝑓𝑗(0	𝑓𝑗(0	PROPN
cana-3848	109	11	)	)	PUNCT
cana-3848	109	12	=	=	SYM
cana-3848	109	13	𝑔𝑗(0	𝑔𝑗(0	NOUN
cana-3848	109	14	)	)	PUNCT
cana-3848	109	15	and	and	CCONJ
cana-3848	109	16	𝑓𝑗(1	𝑓𝑗(1	NOUN
cana-3848	109	17	)	)	PUNCT
cana-3848	109	18	≠	≠	PROPN
cana-3848	109	19	𝑔𝑗(1	𝑔𝑗(1	PROPN
cana-3848	109	20	)	)	PUNCT
cana-3848	109	21	,	,	PUNCT
cana-3848	109	22	then	then	ADV
cana-3848	109	23	f	f	PROPN
cana-3848	109	24	is	be	AUX
cana-3848	109	25	not	not	PART
cana-3848	109	26	injective	injective	ADJ
cana-3848	109	27	.	.	PUNCT
cana-3848	110	1	the	the	DET
cana-3848	110	2	reason	reason	NOUN
cana-3848	110	3	for	for	ADP
cana-3848	110	4	this	this	PRON
cana-3848	110	5	is	be	AUX
cana-3848	110	6	that	that	SCONJ
cana-3848	110	7	we	we	PRON
cana-3848	110	8	obtain	obtain	VERB
cana-3848	110	9	𝐹(𝑓𝑗	𝐹(𝑓𝑗	PRON
cana-3848	110	10	)	)	PUNCT
cana-3848	111	1	=	=	SYM
cana-3848	111	2	𝐹(𝑔𝑗	𝐹(𝑔𝑗	NUM
cana-3848	111	3	)	)	PUNCT
cana-3848	111	4	.	.	PUNCT
cana-3848	112	1	definition	definition	NOUN
cana-3848	112	2	(	(	PUNCT
cana-3848	112	3	3.3	3.3	NUM
cana-3848	112	4	):	):	PUNCT
cana-3848	112	5	it	it	PRON
cana-3848	112	6	is	be	AUX
cana-3848	112	7	contended	contend	VERB
cana-3848	112	8	that	that	SCONJ
cana-3848	112	9	two	two	NUM
cana-3848	112	10	quasi	quasi	ADJ
cana-3848	112	11	-	-	ADJ
cana-3848	112	12	normed	normed	ADJ
cana-3848	112	13	cones	cone	NOUN
cana-3848	112	14	(	(	PUNCT
cana-3848	112	15	𝑋	𝑋	NOUN
cana-3848	112	16	,	,	PUNCT
cana-3848	112	17	𝑓𝑗	𝑓𝑗	NOUN
cana-3848	112	18	)	)	PUNCT
cana-3848	112	19	and	and	CCONJ
cana-3848	112	20	(	(	PUNCT
cana-3848	112	21	𝑌	𝑌	PROPN
cana-3848	112	22	,	,	PUNCT
cana-3848	112	23	𝑞𝑗	𝑞𝑗	PROPN
cana-3848	112	24	)	)	PUNCT
cana-3848	112	25	,	,	PUNCT
cana-3848	112	26	are	be	AUX
cana-3848	112	27	isometric	isometric	ADJ
cana-3848	112	28	if	if	SCONJ
cana-3848	112	29	there	there	PRON
cana-3848	112	30	is	be	VERB
cana-3848	112	31	a	a	DET
cana-3848	112	32	bijective	bijective	ADJ
cana-3848	112	33	isometry	isometry	NOUN
cana-3848	112	34	𝑓𝑗	𝑓𝑗	PROPN
cana-3848	112	35	∶	∶	NOUN
cana-3848	112	36	𝑋	𝑋	PROPN
cana-3848	112	37	→	→	SYM
cana-3848	112	38	𝑌	𝑌	PROPN
cana-3848	112	39	between	between	ADP
cana-3848	112	40	them	they	PRON
cana-3848	112	41	.	.	PUNCT
cana-3848	113	1	proposition	proposition	NOUN
cana-3848	113	2	(	(	PUNCT
cana-3848	113	3	3.4	3.4	NUM
cana-3848	113	4	):	):	PUNCT
cana-3848	113	5	the	the	DET
cana-3848	113	6	quasi	quasi	ADJ
cana-3848	113	7	-	-	ADJ
cana-3848	113	8	metric	metric	ADJ
cana-3848	113	9	spaces	space	NOUN
cana-3848	113	10	(	(	PUNCT
cana-3848	113	11	𝑋	𝑋	PROPN
cana-3848	113	12	,	,	PUNCT
cana-3848	113	13	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	113	14	)	)	PUNCT
cana-3848	113	15	and	and	CCONJ
cana-3848	113	16	(	(	PUNCT
cana-3848	113	17	𝑌	𝑌	PROPN
cana-3848	113	18	,	,	PUNCT
cana-3848	113	19	𝑞𝑗	𝑞𝑗	VERB
cana-3848	113	20	)	)	PUNCT
cana-3848	113	21	are	be	AUX
cana-3848	113	22	also	also	ADV
cana-3848	113	23	isometric	isometric	ADJ
cana-3848	113	24	to	to	ADP
cana-3848	113	25	𝑓𝑗	𝑓𝑗	VERB
cana-3848	113	26	if	if	SCONJ
cana-3848	113	27	a	a	DET
cana-3848	113	28	(	(	PUNCT
cana-3848	113	29	bijective	bijective	ADJ
cana-3848	113	30	)	)	PUNCT
cana-3848	113	31	isometry	isometry	PROPN
cana-3848	113	32	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	113	33	isometric	isometric	ADJ
cana-3848	113	34	to(𝑋	to(𝑋	PROPN
cana-3848	113	35	,	,	PUNCT
cana-3848	113	36	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	113	37	)	)	PUNCT
cana-3848	113	38	and	and	CCONJ
cana-3848	113	39	(	(	PUNCT
cana-3848	113	40	𝑌	𝑌	PROPN
cana-3848	113	41	,	,	PUNCT
cana-3848	113	42	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	113	43	)	)	PUNCT
cana-3848	113	44	quasi	quasi	ADJ
cana-3848	113	45	-	-	ADJ
cana-3848	113	46	normed	normed	ADJ
cana-3848	113	47	cones	cone	NOUN
cana-3848	113	48	.	.	PUNCT
cana-3848	114	1	proof	proof	NOUN
cana-3848	114	2	:	:	PUNCT
cana-3848	114	3	let	let	VERB
cana-3848	114	4	𝑥	𝑥	PRON
cana-3848	114	5	,	,	PUNCT
cana-3848	114	6	𝑦	𝑦	PRON
cana-3848	114	7	are	be	AUX
cana-3848	114	8	in	in	ADP
cana-3848	114	9	𝑋.	𝑋.	PROPN
cana-3848	114	10	𝑒𝑞𝑗	𝑒𝑞𝑗	PROPN
cana-3848	114	11	(	(	PUNCT
cana-3848	114	12	𝑓𝑗(𝑥	𝑓𝑗(𝑥	NOUN
cana-3848	114	13	)	)	PUNCT
cana-3848	114	14	,	,	PUNCT
cana-3848	114	15	𝑓𝑗(𝑦	𝑓𝑗(𝑦	NOUN
cana-3848	114	16	)	)	PUNCT
cana-3848	114	17	)	)	PUNCT
cana-3848	115	1	=	=	SYM
cana-3848	115	2	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	115	3	(	(	PUNCT
cana-3848	115	4	𝑥	𝑥	PROPN
cana-3848	115	5	,	,	PUNCT
cana-3848	115	6	𝑦	𝑦	NOUN
cana-3848	115	7	)	)	PUNCT
cana-3848	115	8	=	=	PUNCT
cana-3848	116	1	+	+	NUM
cana-3848	116	2	∞.	∞.	PROPN
cana-3848	116	3	is	be	AUX
cana-3848	116	4	equal	equal	ADJ
cana-3848	116	5	to	to	ADP
cana-3848	116	6	e_(q_j	e_(q_j	PROPN
cana-3848	116	7	)	)	PUNCT
cana-3848	116	8	.	.	PUNCT
cana-3848	117	1	whether	whether	SCONJ
cana-3848	117	2	𝑓𝑗(𝑦	𝑓𝑗(𝑦	NOUN
cana-3848	117	3	)	)	PUNCT
cana-3848	117	4	∈	∈	NOUN
cana-3848	117	5	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	117	6	)	)	PUNCT
cana-3848	118	1	+	+	CCONJ
cana-3848	118	2	𝑌.	𝑌.	PROPN
cana-3848	118	3	for	for	ADP
cana-3848	118	4	some	some	DET
cana-3848	118	5	𝑧	𝑧	PRON
cana-3848	118	6	𝑖𝑛	𝑖𝑛	NOUN
cana-3848	118	7	𝑌	𝑌	PROPN
cana-3848	118	8	,	,	PUNCT
cana-3848	118	9	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	118	10	(	(	PUNCT
cana-3848	118	11	𝑓𝑗(𝑥	𝑓𝑗(𝑥	NOUN
cana-3848	118	12	)	)	PUNCT
cana-3848	118	13	,	,	PUNCT
cana-3848	118	14	𝑓𝑗(𝑦	𝑓𝑗(𝑦	NOUN
cana-3848	118	15	)	)	PUNCT
cana-3848	118	16	)	)	PUNCT
cana-3848	119	1	=	=	SYM
cana-3848	119	2	𝑞𝑗(𝑧)if	𝑞𝑗(𝑧)if	NOUN
cana-3848	119	3	𝑓𝑗(𝑦	𝑓𝑗(𝑦	NOUN
cana-3848	119	4	)	)	PUNCT
cana-3848	119	5	=	=	SYM
cana-3848	119	6	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	119	7	)	)	PUNCT
cana-3848	120	1	+	+	CCONJ
cana-3848	120	2	𝑧	𝑧	X
cana-3848	120	3	.	.	PUNCT
cana-3848	121	1	only	only	ADV
cana-3848	121	2	one	one	NUM
cana-3848	121	3	𝑎	𝑎	NOUN
cana-3848	121	4	∈	∈	NOUN
cana-3848	121	5	𝑋	𝑋	NOUN
cana-3848	121	6	can	can	AUX
cana-3848	121	7	be	be	AUX
cana-3848	121	8	covered	cover	VERB
cana-3848	121	9	by	by	ADP
cana-3848	121	10	𝑓𝑗(𝑎	𝑓𝑗(𝑎	NOUN
cana-3848	121	11	)	)	PUNCT
cana-3848	122	1	=	=	SYM
cana-3848	122	2	𝑧	𝑧	PROPN
cana-3848	122	3	since	since	SCONJ
cana-3848	122	4	𝑓	𝑓	PRON
cana-3848	122	5	is	be	AUX
cana-3848	122	6	bijective	bijective	ADJ
cana-3848	122	7	.	.	PUNCT
cana-3848	123	1	because	because	SCONJ
cana-3848	123	2	of	of	ADP
cana-3848	123	3	this	this	PRON
cana-3848	123	4	𝑓𝑗(𝑦	𝑓𝑗(𝑦	NOUN
cana-3848	123	5	)	)	PUNCT
cana-3848	123	6	=	=	SYM
cana-3848	123	7	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	123	8	)	)	PUNCT
cana-3848	123	9	+	+	CCONJ
cana-3848	123	10	𝑓𝑗(𝑎	𝑓𝑗(𝑎	NOUN
cana-3848	123	11	)	)	PUNCT
cana-3848	123	12	=	=	PUNCT
cana-3848	123	13	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	123	14	+	+	X
cana-3848	123	15	𝑎	𝑎	X
cana-3848	123	16	)	)	PUNCT
cana-3848	123	17	.	.	PUNCT
cana-3848	124	1	since	since	SCONJ
cana-3848	124	2	𝑦	𝑦	NOUN
cana-3848	124	3	=	=	SYM
cana-3848	124	4	𝑥	𝑥	PROPN
cana-3848	124	5	+	+	NUM
cana-3848	124	6	𝑎	𝑎	ADP
cana-3848	124	7	,	,	PUNCT
cana-3848	124	8	𝑒𝑝𝑗	𝑒𝑝𝑗	X
cana-3848	124	9	(	(	PUNCT
cana-3848	124	10	𝑥	𝑥	PROPN
cana-3848	124	11	,	,	PUNCT
cana-3848	124	12	𝑦	𝑦	NOUN
cana-3848	124	13	)	)	PUNCT
cana-3848	124	14	=	=	SYM
cana-3848	124	15	𝑝𝑗(𝑎	𝑝𝑗(𝑎	NOUN
cana-3848	124	16	)	)	PUNCT
cana-3848	124	17	.	.	PUNCT
cana-3848	124	18	follows	follow	VERB
cana-3848	124	19	.	.	PUNCT
cana-3848	125	1	(	(	PUNCT
cana-3848	125	2	𝑋	𝑋	NOUN
cana-3848	125	3	,	,	PUNCT
cana-3848	125	4	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	125	5	)	)	PUNCT
cana-3848	125	6	and	and	CCONJ
cana-3848	125	7	(	(	PUNCT
cana-3848	125	8	𝑌	𝑌	PROPN
cana-3848	125	9	,	,	PUNCT
cana-3848	125	10	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	125	11	)	)	PUNCT
cana-3848	125	12	are	be	AUX
cana-3848	125	13	isometrically	isometrically	PROPN
cana-3848	125	14	represented	represent	VERB
cana-3848	125	15	by	by	ADP
cana-3848	125	16	𝑓	𝑓	PRON
cana-3848	125	17	,	,	PUNCT
cana-3848	125	18	we	we	PRON
cana-3848	125	19	find	find	VERB
cana-3848	125	20	.	.	PUNCT
cana-3848	126	1	definition	definition	NOUN
cana-3848	126	2	(	(	PUNCT
cana-3848	126	3	3.5	3.5	NUM
cana-3848	126	4	):	):	PUNCT
cana-3848	126	5	an	an	DET
cana-3848	126	6	extended	extended	ADJ
cana-3848	126	7	quasi	quasi	ADJ
cana-3848	126	8	-	-	ADJ
cana-3848	126	9	metric	metric	ADJ
cana-3848	126	10	𝑒𝑝𝑗	𝑒𝑝𝑗	NOUN
cana-3848	126	11	is	be	AUX
cana-3848	126	12	bicomplete	bicomplete	ADJ
cana-3848	126	13	on	on	ADP
cana-3848	126	14	𝑋	𝑋	PROPN
cana-3848	126	15	,	,	PUNCT
cana-3848	126	16	and	and	CCONJ
cana-3848	126	17	a	a	DET
cana-3848	126	18	quasi	quasi	ADJ
cana-3848	126	19	-	-	ADJ
cana-3848	126	20	normed	normed	ADJ
cana-3848	126	21	cone	cone	NOUN
cana-3848	126	22	(	(	PUNCT
cana-3848	126	23	𝑋	𝑋	PROPN
cana-3848	126	24	,	,	PUNCT
cana-3848	126	25	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	126	26	)	)	PUNCT
cana-3848	126	27	is	be	AUX
cana-3848	126	28	bicomplete	bicomplete	ADJ
cana-3848	126	29	if	if	SCONJ
cana-3848	126	30	it	it	PRON
cana-3848	126	31	is	be	AUX
cana-3848	126	32	.	.	PUNCT
cana-3848	127	1	communications	communication	NOUN
cana-3848	127	2	on	on	ADP
cana-3848	127	3	applied	apply	VERB
cana-3848	127	4	nonlinear	nonlinear	ADJ
cana-3848	127	5	analysis	analysis	NOUN
cana-3848	127	6	issn	issn	NOUN
cana-3848	127	7	:	:	PUNCT
cana-3848	127	8	1074	1074	NUM
cana-3848	127	9	-	-	PUNCT
cana-3848	127	10	133x	133x	NUM
cana-3848	127	11	vol	vol	NOUN
cana-3848	127	12	32	32	NUM
cana-3848	127	13	no	no	NOUN
cana-3848	127	14	.	.	PUNCT
cana-3848	128	1	9s	9s	NUM
cana-3848	128	2	(	(	PUNCT
cana-3848	128	3	2025	2025	NUM
cana-3848	128	4	)	)	PUNCT
cana-3848	128	5	204	204	NUM
cana-3848	128	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	128	7	definition	definition	NOUN
cana-3848	128	8	(	(	PUNCT
cana-3848	128	9	3.6	3.6	NUM
cana-3848	128	10	):	):	PUNCT
cana-3848	128	11	consider	consider	VERB
cana-3848	128	12	the	the	DET
cana-3848	128	13	quasi	quasi	ADJ
cana-3848	128	14	-	-	ADJ
cana-3848	128	15	normed	normed	ADJ
cana-3848	128	16	cone	cone	NOUN
cana-3848	128	17	(	(	PUNCT
cana-3848	128	18	𝑋	𝑋	PROPN
cana-3848	128	19	,	,	PUNCT
cana-3848	128	20	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	128	21	)	)	PUNCT
cana-3848	128	22	.if	.if	PUNCT
cana-3848	129	1	(	(	PUNCT
cana-3848	129	2	𝑋	𝑋	NOUN
cana-3848	129	3	,	,	PUNCT
cana-3848	129	4	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	129	5	)	)	PUNCT
cana-3848	129	6	.is	.is	PUNCT
cana-3848	129	7	isometric	isometric	ADJ
cana-3848	129	8	to	to	ADP
cana-3848	129	9	a	a	DET
cana-3848	129	10	dense	dense	ADJ
cana-3848	129	11	subspace	subspace	NOUN
cana-3848	129	12	of	of	ADP
cana-3848	129	13	(	(	PUNCT
cana-3848	129	14	𝑌	𝑌	PROPN
cana-3848	129	15	,	,	PUNCT
cana-3848	129	16	𝑞𝑗	𝑞𝑗	VERB
cana-3848	129	17	)	)	PUNCT
cana-3848	129	18	in	in	ADP
cana-3848	129	19	the	the	DET
cana-3848	129	20	extended	extended	ADJ
cana-3848	129	21	metric	metric	ADJ
cana-3848	129	22	space	space	NOUN
cana-3848	129	23	(	(	PUNCT
cana-3848	129	24	𝑦	𝑦	NOUN
cana-3848	129	25	,	,	PUNCT
cana-3848	129	26	(	(	PUNCT
cana-3848	129	27	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	129	28	)	)	PUNCT
cana-3848	129	29	𝑠	𝑠	PROPN
cana-3848	129	30	)	)	PUNCT
cana-3848	129	31	,	,	PUNCT
cana-3848	129	32	then	then	ADV
cana-3848	129	33	(	(	PUNCT
cana-3848	129	34	𝑋	𝑋	PROPN
cana-3848	129	35	,	,	PUNCT
cana-3848	129	36	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	129	37	)	)	PUNCT
cana-3848	129	38	is	be	AUX
cana-3848	129	39	a	a	DET
cana-3848	129	40	bicompletion	bicompletion	NOUN
cana-3848	129	41	in	in	ADP
cana-3848	129	42	terms	term	NOUN
cana-3848	129	43	of	of	ADP
cana-3848	129	44	a	a	DET
cana-3848	129	45	bicomplete	bicomplete	ADJ
cana-3848	129	46	quasi	quasi	NOUN
cana-3848	129	47	-	-	ADJ
cana-3848	129	48	normed	normed	ADJ
cana-3848	129	49	cone	cone	NOUN
cana-3848	129	50	(	(	PUNCT
cana-3848	129	51	𝑌	𝑌	PROPN
cana-3848	129	52	,	,	PUNCT
cana-3848	129	53	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	129	54	)	)	PUNCT
cana-3848	129	55	.	.	PUNCT
cana-3848	130	1	one	one	NUM
cana-3848	130	2	bicompletion	bicompletion	NOUN
cana-3848	130	3	of	of	ADP
cana-3848	130	4	each	each	DET
cana-3848	130	5	quasi	quasi	ADJ
cana-3848	130	6	-	-	ADJ
cana-3848	130	7	normed	normed	ADJ
cana-3848	130	8	cone	cone	NOUN
cana-3848	130	9	(	(	PUNCT
cana-3848	130	10	𝑋	𝑋	PROPN
cana-3848	130	11	,	,	PUNCT
cana-3848	130	12	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	130	13	)	)	PUNCT
cana-3848	130	14	is	be	AUX
cana-3848	130	15	(	(	PUNCT
cana-3848	130	16	�	�	PROPN
cana-3848	130	17	̃	̃	PROPN
cana-3848	130	18	�	�	PROPN
cana-3848	130	19	,	,	PUNCT
cana-3848	130	20	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	130	21	)	)	PUNCT
cana-3848	130	22	which	which	PRON
cana-3848	130	23	is	be	AUX
cana-3848	130	24	isometric	isometric	ADJ
cana-3848	130	25	to	to	ADP
cana-3848	130	26	any	any	DET
cana-3848	130	27	bicompletion	bicompletion	NOUN
cana-3848	130	28	of	of	ADP
cana-3848	130	29	(	(	PUNCT
cana-3848	130	30	𝑋	𝑋	PROPN
cana-3848	130	31	,	,	PUNCT
cana-3848	130	32	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	130	33	)	)	PUNCT
cana-3848	130	34	.this	.this	PRON
cana-3848	130	35	is	be	AUX
cana-3848	130	36	what	what	PRON
cana-3848	130	37	we	we	PRON
cana-3848	130	38	shall	shall	AUX
cana-3848	130	39	demonstrate	demonstrate	VERB
cana-3848	130	40	.	.	PUNCT
cana-3848	131	1	(	(	PUNCT
cana-3848	131	2	𝑋	𝑋	PROPN
cana-3848	131	3	,	,	PUNCT
cana-3848	131	4	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	131	5	)	)	PUNCT
cana-3848	131	6	is	be	AUX
cana-3848	131	7	the	the	DET
cana-3848	131	8	symbol	symbol	NOUN
cana-3848	131	9	for	for	ADP
cana-3848	131	10	the	the	DET
cana-3848	131	11	extended	extended	ADJ
cana-3848	131	12	quasi	quasi	ADJ
cana-3848	131	13	-	-	ADJ
cana-3848	131	14	metric	metric	ADJ
cana-3848	131	15	space	space	NOUN
cana-3848	131	16	created	create	VERB
cana-3848	131	17	by	by	ADP
cana-3848	131	18	(	(	PUNCT
cana-3848	131	19	𝑋	𝑋	PROPN
cana-3848	131	20	,	,	PUNCT
cana-3848	131	21	𝑝𝑗).in	𝑝𝑗).in	ADJ
cana-3848	131	22	the	the	DET
cana-3848	131	23	extended	extended	ADJ
cana-3848	131	24	metric	metric	ADJ
cana-3848	131	25	space	space	NOUN
cana-3848	131	26	(	(	PUNCT
cana-3848	131	27	𝑋	𝑋	NOUN
cana-3848	131	28	,	,	PUNCT
cana-3848	131	29	(	(	PUNCT
cana-3848	131	30	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	131	31	)	)	PUNCT
cana-3848	131	32	𝑠	𝑠	PROPN
cana-3848	131	33	)	)	PUNCT
cana-3848	131	34	,	,	PUNCT
cana-3848	131	35	�	�	PROPN
cana-3848	131	36	̃	̃	PROPN
cana-3848	131	37	�	�	PROPN
cana-3848	131	38	̃	̃	PROPN
cana-3848	131	39	is	be	AUX
cana-3848	131	40	the	the	DET
cana-3848	131	41	sum	sum	NOUN
cana-3848	131	42	of	of	ADP
cana-3848	131	43	all	all	DET
cana-3848	131	44	cauchy	cauchy	ADJ
cana-3848	131	45	sequences	sequence	NOUN
cana-3848	131	46	.	.	PUNCT
cana-3848	132	1	consider	consider	VERB
cana-3848	132	2	that	that	PRON
cana-3848	132	3	for	for	ADP
cana-3848	132	4	each	each	PRON
cana-3848	132	5	𝜇	𝜇	X
cana-3848	132	6	>	>	X
cana-3848	132	7	−1	−1	NOUN
cana-3848	132	8	,	,	PUNCT
cana-3848	132	9	there	there	PRON
cana-3848	132	10	exists	exist	VERB
cana-3848	132	11	𝑛0	𝑛0	VERB
cana-3848	132	12	∈n	∈n	NOUN
cana-3848	132	13	such	such	ADJ
cana-3848	132	14	that	that	SCONJ
cana-3848	132	15	(	(	PUNCT
cana-3848	132	16	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	132	17	)	)	PUNCT
cana-3848	132	18	𝑠	𝑠	PROPN
cana-3848	133	1	(	(	PUNCT
cana-3848	133	2	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	133	3	,	,	PUNCT
cana-3848	133	4	𝑥𝑚	𝑥𝑚	NOUN
cana-3848	133	5	)	)	PUNCT
cana-3848	133	6	<	<	X
cana-3848	133	7	𝜇	𝜇	ADP
cana-3848	133	8	+	+	X
cana-3848	133	9	1,and	1,and	NUM
cana-3848	133	10	for	for	ADP
cana-3848	133	11	any	any	DET
cana-3848	133	12	𝑚	𝑚	NOUN
cana-3848	133	13	,	,	PUNCT
cana-3848	133	14	𝑛	𝑛	DET
cana-3848	133	15	≥	≥	NOUN
cana-3848	133	16	𝑛0	𝑛0	VERB
cana-3848	133	17	,	,	PUNCT
cana-3848	133	18	𝑥	𝑥	PROPN
cana-3848	133	19	𝑚	𝑚	X
cana-3848	133	20	∈	∈	PROPN
cana-3848	133	21	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	134	1	+	+	CCONJ
cana-3848	134	2	𝑋	𝑋	NOUN
cana-3848	134	3	if	if	SCONJ
cana-3848	134	4	̃	̃	PROPN
cana-3848	134	5	𝑥	𝑥	PRON
cana-3848	134	6	∶=	∶=	NUM
cana-3848	134	7	(	(	PUNCT
cana-3848	134	8	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	134	9	∈	∈	PROPN
cana-3848	134	10	�	�	PROPN
cana-3848	134	11	̃	̃	PROPN
cana-3848	134	12	�	�	PROPN
cana-3848	134	13	r	r	NOUN
cana-3848	134	14	is	be	AUX
cana-3848	134	15	a	a	DET
cana-3848	134	16	relation	relation	NOUN
cana-3848	134	17	of	of	ADP
cana-3848	134	18	�	�	PROPN
cana-3848	134	19	̃	̃	PROPN
cana-3848	134	20	�	�	PROPN
cana-3848	134	21	that	that	PRON
cana-3848	134	22	has	have	VERB
cana-3848	134	23	the	the	DET
cana-3848	134	24	following	follow	VERB
cana-3848	134	25	definition	definition	NOUN
cana-3848	134	26	:	:	PUNCT
cana-3848	134	27	for	for	ADP
cana-3848	134	28	each	each	DET
cana-3848	134	29	𝑥	𝑥	NOUN
cana-3848	134	30	∶=	∶=	NUM
cana-3848	134	31	(	(	PUNCT
cana-3848	134	32	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	134	33	and	and	CCONJ
cana-3848	134	34	𝑦	𝑦	NOUN
cana-3848	134	35	∶=	∶=	NUM
cana-3848	134	36	(	(	PUNCT
cana-3848	134	37	𝑦𝑛)𝑛∈𝑁	𝑦𝑛)𝑛∈𝑁	NOUN
cana-3848	134	38	in	in	ADP
cana-3848	134	39	�	�	PROPN
cana-3848	134	40	̂	̂	SYM
cana-3848	134	41	�	�	PROPN
cana-3848	134	42	,	,	PUNCT
cana-3848	134	43	set	set	VERB
cana-3848	134	44	𝑅𝑦	𝑅𝑦	PROPN
cana-3848	134	45	⟺	⟺	PROPN
cana-3848	134	46	lim	lim	PROPN
cana-3848	134	47	𝑛→∞	𝑛→∞	NUM
cana-3848	134	48	(	(	PUNCT
cana-3848	134	49	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	134	50	)	)	PUNCT
cana-3848	135	1	𝑠	𝑠	PROPN
cana-3848	135	2	(	(	PUNCT
cana-3848	135	3	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	135	4	,	,	PUNCT
cana-3848	135	5	𝑦𝑛	𝑦𝑛	NOUN
cana-3848	135	6	)	)	PUNCT
cana-3848	135	7	=	=	SYM
cana-3848	135	8	0	0	NUM
cana-3848	135	9	.	.	PUNCT
cana-3848	136	1	r	r	NOUN
cana-3848	136	2	is	be	AUX
cana-3848	136	3	an	an	DET
cana-3848	136	4	equivalency	equivalency	NOUN
cana-3848	136	5	relation	relation	NOUN
cana-3848	136	6	on	on	ADP
cana-3848	136	7	x	x	PUNCT
cana-3848	136	8	̂	̂	PUNCT
cana-3848	136	9	in	in	ADP
cana-3848	136	10	this	this	DET
cana-3848	136	11	situation	situation	NOUN
cana-3848	136	12	.	.	PUNCT
cana-3848	137	1	show	show	VERB
cana-3848	137	2	the	the	DET
cana-3848	137	3	quotient	quotient	NOUN
cana-3848	137	4	of	of	ADP
cana-3848	137	5	�	�	PROPN
cana-3848	137	6	̂	̂	PROPN
cana-3848	137	7	�	�	PROPN
cana-3848	137	8	𝑅⁄	𝑅⁄	PROPN
cana-3848	137	9	by	by	ADP
cana-3848	137	10	�	�	PROPN
cana-3848	137	11	̂�.for	̂�.for	ADP
cana-3848	137	12	any	any	DET
cana-3848	137	13	𝑥	𝑥	NOUN
cana-3848	137	14	in	in	ADP
cana-3848	137	15	�	�	PROPN
cana-3848	137	16	̂	̂	SYM
cana-3848	137	17	�	�	PROPN
cana-3848	137	18	therefore	therefore	ADV
cana-3848	137	19	,	,	PUNCT
cana-3848	137	20	�	�	PROPN
cana-3848	137	21	̂	̂	NOUN
cana-3848	137	22	�	�	NOUN
cana-3848	137	23	=	=	SYM
cana-3848	137	24	{	{	PUNCT
cana-3848	137	25	[	[	X
cana-3848	137	26	𝑥	𝑥	X
cana-3848	137	27	]	]	X
cana-3848	137	28	∶	∶	NOUN
cana-3848	137	29	𝑥	𝑥	X
cana-3848	137	30	∈	∈	PROPN
cana-3848	137	31	�	�	PROPN
cana-3848	137	32	̂	̂	NOUN
cana-3848	137	33	�	�	NOUN
cana-3848	137	34	},where[𝑥	},where[𝑥	NOUN
cana-3848	137	35	]	]	PUNCT
cana-3848	138	1	=	=	SYM
cana-3848	138	2	{	{	PUNCT
cana-3848	138	3	𝑦	𝑦	PROPN
cana-3848	138	4	∈	∈	PROPN
cana-3848	138	5	�	�	PROPN
cana-3848	138	6	̂	̂	PROPN
cana-3848	138	7	�	�	NOUN
cana-3848	138	8	∶	∶	NOUN
cana-3848	138	9	𝑥𝑅𝑦	𝑥𝑅𝑦	NOUN
cana-3848	138	10	}	}	PUNCT
cana-3848	138	11	.	.	PUNCT
cana-3848	139	1	insert	insert	VERB
cana-3848	140	1	[	[	X
cana-3848	140	2	𝑥	𝑥	X
cana-3848	140	3	]	]	X
cana-3848	141	1	+	+	CCONJ
cana-3848	142	1	[	[	X
cana-3848	142	2	𝑦	𝑦	X
cana-3848	142	3	]	]	X
cana-3848	142	4	=	=	PUNCT
cana-3848	143	1	[	[	X
cana-3848	143	2	𝑥	𝑥	X
cana-3848	143	3	+	+	NUM
cana-3848	143	4	𝑦	𝑦	NOUN
cana-3848	143	5	]	]	X
cana-3848	143	6	and	and	CCONJ
cana-3848	143	7	𝑎	𝑎	DET
cana-3848	143	8	⋅	⋅	PROPN
cana-3848	143	9	[	[	X
cana-3848	143	10	𝑥	𝑥	X
cana-3848	143	11	]	]	X
cana-3848	143	12	=	=	PUNCT
cana-3848	144	1	[	[	X
cana-3848	144	2	𝑎𝑥	𝑎𝑥	X
cana-3848	144	3	]	]	X
cana-3848	144	4	for	for	ADP
cana-3848	144	5	each	each	DET
cana-3848	144	6	𝑥	𝑥	NOUN
cana-3848	144	7	∶=	∶=	NUM
cana-3848	144	8	(	(	PUNCT
cana-3848	144	9	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	144	10	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
cana-3848	144	11	𝑦	𝑦	PROPN
cana-3848	144	12	∶=	∶=	NUM
cana-3848	144	13	(	(	PUNCT
cana-3848	144	14	𝑦𝑛)𝑛∈ℕ	𝑦𝑛)𝑛∈ℕ	NOUN
cana-3848	144	15	in	in	ADP
cana-3848	144	16	�	�	PROPN
cana-3848	144	17	̂	̂	SYM
cana-3848	144	18	�	�	PROPN
cana-3848	144	19	and	and	CCONJ
cana-3848	144	20	every	every	DET
cana-3848	144	21	𝑎	𝑎	PRON
cana-3848	144	22	∈	∈	NOUN
cana-3848	144	23	ℝ+	ℝ+	NOUN
cana-3848	144	24	,	,	PUNCT
cana-3848	144	25	where	where	SCONJ
cana-3848	144	26	+	+	NOUN
cana-3848	144	27	𝑦	𝑦	NOUN
cana-3848	144	28	=	=	PUNCT
cana-3848	144	29	(	(	PUNCT
cana-3848	144	30	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	144	31	+	+	NUM
cana-3848	144	32	𝑦𝑛)𝑛∈ℕ	𝑦𝑛)𝑛∈ℕ	PROPN
cana-3848	144	33	,	,	PUNCT
cana-3848	144	34	and	and	CCONJ
cana-3848	144	35	𝑎𝑥	𝑎𝑥	X
cana-3848	144	36	=	=	SYM
cana-3848	144	37	(	(	PUNCT
cana-3848	144	38	𝑎𝑥𝑛)𝑛∈ℕ.	𝑎𝑥𝑛)𝑛∈ℕ.	PROPN
cana-3848	144	39	these	these	DET
cana-3848	144	40	processes	process	NOUN
cana-3848	144	41	are	be	AUX
cana-3848	144	42	simple	simple	ADJ
cana-3848	144	43	to	to	PART
cana-3848	144	44	comprehend	comprehend	VERB
cana-3848	144	45	due	due	ADP
cana-3848	144	46	to	to	ADP
cana-3848	144	47	their	their	PRON
cana-3848	144	48	precise	precise	ADJ
cana-3848	144	49	specification	specification	NOUN
cana-3848	144	50	.	.	PUNCT
cana-3848	145	1	this	this	PRON
cana-3848	145	2	is	be	AUX
cana-3848	145	3	the	the	DET
cana-3848	145	4	result	result	NOUN
cana-3848	145	5	that	that	PRON
cana-3848	145	6	comes	come	VERB
cana-3848	145	7	next	next	ADV
cana-3848	145	8	.	.	PUNCT
cana-3848	146	1	lemma	lemma	PROPN
cana-3848	146	2	(	(	PUNCT
cana-3848	146	3	3.7	3.7	NUM
cana-3848	146	4	):	):	PUNCT
cana-3848	146	5	take	take	VERB
cana-3848	146	6	the	the	DET
cana-3848	146	7	case	case	NOUN
cana-3848	146	8	of	of	ADP
cana-3848	146	9	a	a	DET
cana-3848	146	10	quasi	quasi	ADJ
cana-3848	146	11	-	-	ADJ
cana-3848	146	12	normed	normed	ADJ
cana-3848	146	13	cone	cone	NOUN
cana-3848	146	14	(	(	PUNCT
cana-3848	146	15	𝑋	𝑋	PROPN
cana-3848	146	16	,	,	PUNCT
cana-3848	146	17	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	146	18	)	)	PUNCT
cana-3848	146	19	.	.	PUNCT
cana-3848	147	1	a	a	DET
cana-3848	147	2	cancellative	cancellative	ADJ
cana-3848	147	3	cone	cone	NOUN
cana-3848	147	4	is	be	AUX
cana-3848	147	5	therefore	therefore	ADV
cana-3848	147	6	(	(	PUNCT
cana-3848	147	7	�	�	PROPN
cana-3848	147	8	̃	̃	NOUN
cana-3848	147	9	�	�	PROPN
cana-3848	147	10	,	,	PUNCT
cana-3848	147	11	+	+	NOUN
cana-3848	147	12	,	,	PUNCT
cana-3848	147	13	⋅	⋅	ADJ
cana-3848	147	14	)	)	PUNCT
cana-3848	147	15	proof	proof	NOUN
cana-3848	147	16	:	:	PUNCT
cana-3848	147	17	(	(	PUNCT
cana-3848	147	18	�	�	PROPN
cana-3848	147	19	̃	̃	NOUN
cana-3848	147	20	�	�	PROPN
cana-3848	147	21	,	,	PUNCT
cana-3848	147	22	+	+	PUNCT
cana-3848	147	23	)	)	PUNCT
cana-3848	147	24	is	be	AUX
cana-3848	147	25	a	a	DET
cana-3848	147	26	cancellative	cancellative	ADJ
cana-3848	147	27	abelian	abelian	NOUN
cana-3848	147	28	semigroup	semigroup	NOUN
cana-3848	147	29	with	with	ADP
cana-3848	147	30	neutral	neutral	ADJ
cana-3848	147	31	element	element	NOUN
cana-3848	147	32	[	[	X
cana-3848	147	33	0	0	X
cana-3848	147	34	]	]	X
cana-3848	147	35	∈	∈	PROPN
cana-3848	147	36	�	�	PROPN
cana-3848	147	37	̃	̃	PROPN
cana-3848	147	38	�	�	PROPN
cana-3848	147	39	̃	̃	PROPN
cana-3848	147	40	,	,	PUNCT
cana-3848	147	41	as	as	SCONJ
cana-3848	147	42	we	we	PRON
cana-3848	147	43	can	can	AUX
cana-3848	147	44	easily	easily	ADV
cana-3848	147	45	conclude	conclude	VERB
cana-3848	147	46	from	from	ADP
cana-3848	147	47	(	(	PUNCT
cana-3848	147	48	𝑋	𝑋	PROPN
cana-3848	147	49	,	,	PUNCT
cana-3848	147	50	+	+	PROPN
cana-3848	147	51	,	,	PUNCT
cana-3848	147	52	⋅	⋅	PROPN
cana-3848	147	53	)	)	PUNCT
cana-3848	147	54	being	be	AUX
cana-3848	147	55	a	a	DET
cana-3848	147	56	cancellative	cancellative	ADJ
cana-3848	147	57	cone	cone	NOUN
cana-3848	147	58	.	.	PUNCT
cana-3848	148	1	furthermore	furthermore	ADV
cana-3848	148	2	,	,	PUNCT
cana-3848	148	3	the	the	DET
cana-3848	148	4	previously	previously	ADV
cana-3848	148	5	defined	define	VERB
cana-3848	148	6	operation	operation	NOUN
cana-3848	148	7	⋅	⋅	PROPN
cana-3848	148	8	,	,	PUNCT
cana-3848	148	9	which	which	PRON
cana-3848	148	10	was	be	AUX
cana-3848	148	11	defined	define	VERB
cana-3848	148	12	as	as	ADP
cana-3848	148	13	ℝ+	ℝ+	X
cana-3848	148	14	×	×	PROPN
cana-3848	148	15	�	�	PROPN
cana-3848	148	16	̃	̃	PROPN
cana-3848	148	17	�	�	PROPN
cana-3848	148	18	to	to	ADP
cana-3848	148	19	�	�	PROPN
cana-3848	148	20	̃	̃	PROPN
cana-3848	148	21	�	�	PROPN
cana-3848	148	22	,	,	PUNCT
cana-3848	148	23	satisfies	satisfie	NOUN
cana-3848	148	24	for	for	ADP
cana-3848	148	25	any	any	DET
cana-3848	148	26	[	[	X
cana-3848	148	27	𝑥	𝑥	X
cana-3848	148	28	]	]	X
cana-3848	148	29	,	,	PUNCT
cana-3848	148	30	[	[	X
cana-3848	148	31	𝑦	𝑦	X
cana-3848	148	32	]	]	X
cana-3848	148	33	∈	∈	PROPN
cana-3848	148	34	�	�	PROPN
cana-3848	148	35	̃	̃	PROPN
cana-3848	148	36	�	�	PROPN
cana-3848	148	37	and	and	CCONJ
cana-3848	148	38	𝑟	𝑟	NOUN
cana-3848	148	39	,	,	PUNCT
cana-3848	148	40	𝑠	𝑠	PROPN
cana-3848	148	41	∈	∈	PROPN
cana-3848	148	42	ℝ+	ℝ+	PROPN
cana-3848	148	43	:	:	PUNCT
cana-3848	148	44	(	(	PUNCT
cana-3848	148	45	𝑎	𝑎	NOUN
cana-3848	148	46	)	)	PUNCT
cana-3848	148	47	𝑟	𝑟	NOUN
cana-3848	148	48	⋅	⋅	X
cana-3848	148	49	(	(	PUNCT
cana-3848	148	50	𝑠[𝑥	𝑠[𝑥	PROPN
cana-3848	148	51	]	]	X
cana-3848	148	52	)	)	PUNCT
cana-3848	148	53	=	=	SYM
cana-3848	148	54	(	(	PUNCT
cana-3848	148	55	𝑟𝑠	𝑟𝑠	NOUN
cana-3848	148	56	)	)	PUNCT
cana-3848	148	57	⋅	⋅	PROPN
cana-3848	149	1	[	[	X
cana-3848	149	2	𝑥	𝑥	X
cana-3848	149	3	]	]	X
cana-3848	149	4	;	;	PUNCT
cana-3848	149	5	(	(	PUNCT
cana-3848	149	6	𝑏	𝑏	NOUN
cana-3848	149	7	)	)	PUNCT
cana-3848	149	8	𝑟	𝑟	NOUN
cana-3848	149	9	⋅	⋅	X
cana-3848	149	10	(	(	PUNCT
cana-3848	149	11	[	[	X
cana-3848	149	12	𝑥	𝑥	X
cana-3848	149	13	]	]	X
cana-3848	150	1	+	+	CCONJ
cana-3848	150	2	[	[	X
cana-3848	150	3	𝑦	𝑦	X
cana-3848	150	4	]	]	X
cana-3848	150	5	)	)	PUNCT
cana-3848	150	6	=	=	SYM
cana-3848	151	1	(	(	PUNCT
cana-3848	151	2	𝑟	𝑟	X
cana-3848	151	3	⋅	⋅	X
cana-3848	151	4	[	[	X
cana-3848	151	5	𝑥	𝑥	X
cana-3848	151	6	]	]	X
cana-3848	151	7	)	)	PUNCT
cana-3848	152	1	+	+	CCONJ
cana-3848	152	2	(	(	PUNCT
cana-3848	152	3	𝑟	𝑟	NOUN
cana-3848	152	4	⋅	⋅	X
cana-3848	152	5	[	[	X
cana-3848	152	6	𝑦	𝑦	X
cana-3848	152	7	]	]	X
cana-3848	152	8	)	)	PUNCT
cana-3848	152	9	;	;	PUNCT
cana-3848	152	10	(	(	PUNCT
cana-3848	152	11	𝑐	𝑐	X
cana-3848	152	12	)	)	PUNCT
cana-3848	152	13	(	(	PUNCT
cana-3848	152	14	𝑟	𝑟	X
cana-3848	152	15	+	+	NUM
cana-3848	152	16	𝑠	𝑠	NOUN
cana-3848	152	17	)	)	PUNCT
cana-3848	152	18	⋅	⋅	PROPN
cana-3848	153	1	[	[	X
cana-3848	153	2	𝑥	𝑥	X
cana-3848	153	3	]	]	X
cana-3848	153	4	=	=	SYM
cana-3848	153	5	(	(	PUNCT
cana-3848	153	6	𝑟	𝑟	X
cana-3848	153	7	⋅	⋅	X
cana-3848	153	8	[	[	X
cana-3848	153	9	𝑥	𝑥	X
cana-3848	153	10	]	]	X
cana-3848	153	11	)	)	PUNCT
cana-3848	154	1	+	+	CCONJ
cana-3848	154	2	(	(	PUNCT
cana-3848	154	3	𝑠	𝑠	PROPN
cana-3848	154	4	⋅	⋅	PROPN
cana-3848	155	1	[	[	X
cana-3848	155	2	𝑥	𝑥	X
cana-3848	155	3	]	]	X
cana-3848	155	4	)	)	PUNCT
cana-3848	155	5	;	;	PUNCT
cana-3848	155	6	(	(	PUNCT
cana-3848	155	7	𝑑)1	𝑑)1	NOUN
cana-3848	155	8	⋅	⋅	PROPN
cana-3848	156	1	[	[	X
cana-3848	156	2	𝑥	𝑥	X
cana-3848	156	3	]	]	X
cana-3848	156	4	=	=	PUNCT
cana-3848	157	1	[	[	X
cana-3848	157	2	𝑥	𝑥	X
cana-3848	157	3	]	]	X
cana-3848	157	4	.	.	PUNCT
cana-3848	158	1	the	the	DET
cana-3848	158	2	proof	proof	NOUN
cana-3848	158	3	of	of	ADP
cana-3848	158	4	the	the	DET
cana-3848	158	5	following	following	ADJ
cana-3848	158	6	result	result	NOUN
cana-3848	158	7	on	on	ADP
cana-3848	158	8	a	a	DET
cana-3848	158	9	general	general	ADJ
cana-3848	158	10	monoid	monoid	NOUN
cana-3848	158	11	context	context	NOUN
cana-3848	158	12	can	can	AUX
cana-3848	158	13	be	be	AUX
cana-3848	158	14	found	find	VERB
cana-3848	158	15	in	in	ADP
cana-3848	158	16	[	[	X
cana-3848	158	17	99	99	NUM
cana-3848	158	18	]	]	PUNCT
cana-3848	158	19	.	.	PUNCT
cana-3848	159	1	lemma	lemma	PROPN
cana-3848	159	2	(	(	PUNCT
cana-3848	159	3	3	3	NUM
cana-3848	159	4	..	..	SYM
cana-3848	159	5	8)	8)	NUM
cana-3848	159	6	:	:	PUNCT
cana-3848	159	7	assume	assume	VERB
cana-3848	159	8	a	a	DET
cana-3848	159	9	quasi	quasi	ADJ
cana-3848	159	10	-	-	ADJ
cana-3848	159	11	normed	normed	ADJ
cana-3848	159	12	cone	cone	NOUN
cana-3848	159	13	(	(	PUNCT
cana-3848	159	14	𝑋	𝑋	PROPN
cana-3848	159	15	,	,	PUNCT
cana-3848	159	16	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	159	17	)	)	PUNCT
cana-3848	159	18	and	and	CCONJ
cana-3848	159	19	that	that	SCONJ
cana-3848	159	20	𝑥	𝑥	PRON
cana-3848	159	21	∶=	∶=	NUM
cana-3848	159	22	(	(	PUNCT
cana-3848	159	23	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	159	24	∈	∈	PROPN
cana-3848	159	25	�	�	PROPN
cana-3848	159	26	̂	̂	PROPN
cana-3848	159	27	�	�	PROPN
cana-3848	159	28	then	then	ADV
cana-3848	159	29	:	:	PUNCT
cana-3848	159	30	(	(	PUNCT
cana-3848	159	31	𝑎	𝑎	NOUN
cana-3848	159	32	)	)	PUNCT
cana-3848	159	33	lim	lim	NOUN
cana-3848	159	34	𝑛→∞	𝑛→∞	NUM
cana-3848	159	35	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	159	36	)	)	PUNCT
cana-3848	159	37	exists	exist	VERB
cana-3848	159	38	and	and	CCONJ
cana-3848	159	39	is	be	AUX
cana-3848	159	40	finite	finite	ADJ
cana-3848	159	41	.	.	PUNCT
cana-3848	160	1	(	(	PUNCT
cana-3848	160	2	𝑏	𝑏	NOUN
cana-3848	160	3	)	)	PUNCT
cana-3848	160	4	lim	lim	NOUN
cana-3848	160	5	𝑛→∞	𝑛→∞	NUM
cana-3848	160	6	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	160	7	)	)	PUNCT
cana-3848	161	1	=	=	SYM
cana-3848	161	2	lim	lim	NOUN
cana-3848	161	3	𝑛→∞	𝑛→∞	NUM
cana-3848	161	4	𝑝𝑗(𝑦𝑛	𝑝𝑗(𝑦𝑛	NOUN
cana-3848	161	5	)	)	PUNCT
cana-3848	161	6	for	for	ADP
cana-3848	161	7	all	all	DET
cana-3848	161	8	𝑦	𝑦	NOUN
cana-3848	161	9	∈	∈	NOUN
cana-3848	162	1	[	[	X
cana-3848	162	2	𝑥	𝑥	X
cana-3848	162	3	]	]	X
cana-3848	162	4	.	.	PUNCT
cana-3848	163	1	we	we	PRON
cana-3848	163	2	may	may	AUX
cana-3848	163	3	define	define	VERB
cana-3848	163	4	a	a	DET
cana-3848	163	5	function	function	NOUN
cana-3848	163	6	𝑝	𝑝	PROPN
cana-3848	163	7	�	�	PROPN
cana-3848	163	8	̃	̃	PROPN
cana-3848	163	9	�	�	PROPN
cana-3848	163	10	∶	∶	PROPN
cana-3848	163	11	�	�	PROPN
cana-3848	163	12	̃	̃	PROPN
cana-3848	163	13	�	�	PROPN
cana-3848	163	14	→	→	SYM
cana-3848	163	15	ℝ+	ℝ+	PUNCT
cana-3848	163	16	given	give	VERB
cana-3848	163	17	the	the	DET
cana-3848	163	18	earlier	early	ADJ
cana-3848	163	19	lemma	lemma	PROPN
cana-3848	163	20	,	,	PUNCT
cana-3848	163	21	which	which	PRON
cana-3848	163	22	is	be	AUX
cana-3848	163	23	defined	define	VERB
cana-3848	163	24	as	as	ADP
cana-3848	163	25	communications	communication	NOUN
cana-3848	163	26	on	on	ADP
cana-3848	163	27	applied	apply	VERB
cana-3848	163	28	nonlinear	nonlinear	ADJ
cana-3848	163	29	analysis	analysis	NOUN
cana-3848	163	30	issn	issn	NOUN
cana-3848	163	31	:	:	PUNCT
cana-3848	163	32	1074	1074	NUM
cana-3848	163	33	-	-	PUNCT
cana-3848	163	34	133x	133x	NUM
cana-3848	163	35	vol	vol	NOUN
cana-3848	163	36	32	32	NUM
cana-3848	163	37	no	no	NOUN
cana-3848	163	38	.	.	PUNCT
cana-3848	164	1	9s	9s	NUM
cana-3848	164	2	(	(	PUNCT
cana-3848	164	3	2025	2025	NUM
cana-3848	164	4	)	)	PUNCT
cana-3848	164	5	205	205	NUM
cana-3848	164	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-3848	164	7	𝑝	𝑝	PROPN
cana-3848	164	8	�	�	PROPN
cana-3848	164	9	̃	̃	PROPN
cana-3848	164	10	�	�	PROPN
cana-3848	164	11	([𝑥	([𝑥	PROPN
cana-3848	164	12	]	]	NOUN
cana-3848	164	13	)	)	PUNCT
cana-3848	165	1	=	=	SYM
cana-3848	165	2	lim	lim	PROPN
cana-3848	165	3	𝑛→∞	𝑛→∞	NUM
cana-3848	165	4	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	165	5	)	)	PUNCT
cana-3848	165	6	for	for	ADP
cana-3848	165	7	all	all	DET
cana-3848	165	8	𝑥	𝑥	PRON
cana-3848	165	9	∈	∈	PROPN
cana-3848	165	10	�	�	PROPN
cana-3848	165	11	̂	̂	NOUN
cana-3848	165	12	�	�	NOUN
cana-3848	165	13	.	.	PUNCT
cana-3848	166	1	we	we	PRON
cana-3848	166	2	will	will	AUX
cana-3848	166	3	show	show	VERB
cana-3848	166	4	how	how	SCONJ
cana-3848	166	5	(	(	PUNCT
cana-3848	166	6	�	�	PROPN
cana-3848	166	7	̃	̃	NOUN
cana-3848	166	8	�	�	PROPN
cana-3848	166	9	,	,	PUNCT
cana-3848	166	10	𝑝	𝑝	PROPN
cana-3848	166	11	�	�	PROPN
cana-3848	166	12	̃	̃	PROPN
cana-3848	166	13	�	�	PROPN
cana-3848	166	14	)	)	PUNCT
cana-3848	166	15	is	be	AUX
cana-3848	166	16	a	a	DET
cana-3848	166	17	bicomplete	bicomplete	ADJ
cana-3848	166	18	quasi	quasi	NOUN
cana-3848	166	19	-	-	ADJ
cana-3848	166	20	normed	normed	ADJ
cana-3848	166	21	cone	cone	NOUN
cana-3848	166	22	in	in	ADP
cana-3848	166	23	the	the	DET
cana-3848	166	24	following	follow	VERB
cana-3848	166	25	lemma	lemma	PROPN
cana-3848	166	26	3:9	3:9	NUM
cana-3848	166	27	.	.	PUNCT
cana-3848	167	1	lemma	lemma	PROPN
cana-3848	167	2	(	(	PUNCT
cana-3848	167	3	3.9	3.9	NUM
cana-3848	167	4	):	):	PUNCT
cana-3848	167	5	consider	consider	VERB
cana-3848	167	6	the	the	PRON
cana-3848	167	7	following	follow	VERB
cana-3848	167	8	two	two	NUM
cana-3848	167	9	cones	cone	NOUN
cana-3848	167	10	(	(	PUNCT
cana-3848	167	11	𝑋	𝑋	PROPN
cana-3848	167	12	,	,	PUNCT
cana-3848	167	13	+	+	PROPN
cana-3848	167	14	,	,	PUNCT
cana-3848	167	15	⋅	⋅	PROPN
cana-3848	167	16	)	)	PUNCT
cana-3848	167	17	and	and	CCONJ
cana-3848	167	18	(	(	PUNCT
cana-3848	167	19	𝑌,⊕,⊗	𝑌,⊕,⊗	NOUN
cana-3848	167	20	)	)	PUNCT
cana-3848	167	21	if	if	SCONJ
cana-3848	167	22	𝑓𝑗	𝑓𝑗	PRON
cana-3848	167	23	∶	∶	NOUN
cana-3848	167	24	𝐴	𝐴	PROPN
cana-3848	167	25	→	→	SYM
cana-3848	167	26	𝑌	𝑌	PROPN
cana-3848	167	27	is	be	AUX
cana-3848	167	28	a	a	DET
cana-3848	167	29	linear	linear	ADJ
cana-3848	167	30	function	function	NOUN
cana-3848	167	31	and	and	CCONJ
cana-3848	167	32	a	a	PRON
cana-3848	167	33	is	be	AUX
cana-3848	167	34	a	a	DET
cana-3848	167	35	sub	sub	NOUN
cana-3848	167	36	cone	cone	NOUN
cana-3848	167	37	of	of	ADP
cana-3848	167	38	𝑋	𝑋	PROPN
cana-3848	167	39	,	,	PUNCT
cana-3848	167	40	then	then	ADV
cana-3848	167	41	𝑓𝑗(𝐴	𝑓𝑗(𝐴	NOUN
cana-3848	167	42	)	)	PUNCT
cana-3848	167	43	is	be	AUX
cana-3848	167	44	a	a	DET
cana-3848	167	45	sub	sub	NOUN
cana-3848	167	46	cone	cone	NOUN
cana-3848	167	47	of	of	ADP
cana-3848	167	48	𝑌.	𝑌.	PROPN
cana-3848	167	49	lemma	lemma	PROPN
cana-3848	167	50	(	(	PUNCT
cana-3848	167	51	3.10	3.10	NUM
cana-3848	167	52	):	):	PUNCT
cana-3848	167	53	a	a	PRON
cana-3848	167	54	quasi	quasi	ADJ
cana-3848	167	55	-	-	ADJ
cana-3848	167	56	normed	normed	ADJ
cana-3848	167	57	cone	cone	NOUN
cana-3848	167	58	(	(	PUNCT
cana-3848	167	59	𝑋	𝑋	PROPN
cana-3848	167	60	,	,	PUNCT
cana-3848	167	61	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	167	62	)	)	PUNCT
cana-3848	167	63	is	be	AUX
cana-3848	167	64	considered	consider	VERB
cana-3848	167	65	.	.	PUNCT
cana-3848	168	1	then	then	ADV
cana-3848	168	2	the	the	DET
cana-3848	168	3	following	following	ADJ
cana-3848	168	4	statements	statement	NOUN
cana-3848	168	5	are	be	AUX
cana-3848	168	6	exact	exact	ADJ
cana-3848	168	7	:	:	PUNCT
cana-3848	168	8	(	(	PUNCT
cana-3848	168	9	a	a	X
cana-3848	168	10	)	)	PUNCT
cana-3848	168	11	the	the	DET
cana-3848	168	12	quasi	quasi	ADJ
cana-3848	168	13	-	-	ADJ
cana-3848	168	14	normed	normed	ADJ
cana-3848	168	15	cone	cone	X
cana-3848	168	16	(	(	PUNCT
cana-3848	168	17	�	�	PROPN
cana-3848	168	18	̃	̃	NOUN
cana-3848	168	19	�	�	PROPN
cana-3848	168	20	,	,	PUNCT
cana-3848	168	21	𝑝𝑗)bicomplete	𝑝𝑗)bicomplete	NOUN
cana-3848	168	22	.	.	PUNCT
cana-3848	169	1	(	(	PUNCT
cana-3848	169	2	b	b	X
cana-3848	169	3	)	)	PUNCT
cana-3848	169	4	(	(	PUNCT
cana-3848	169	5	𝑋	𝑋	PROPN
cana-3848	169	6	,	,	PUNCT
cana-3848	169	7	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	169	8	)	)	PUNCT
cana-3848	169	9	is	be	AUX
cana-3848	169	10	isometric	isometric	ADJ
cana-3848	169	11	to	to	ADP
cana-3848	169	12	a	a	DET
cana-3848	169	13	dense	dense	ADJ
cana-3848	169	14	subspace	subspace	NOUN
cana-3848	169	15	of	of	ADP
cana-3848	169	16	(	(	PUNCT
cana-3848	169	17	�	�	PROPN
cana-3848	169	18	̃	̃	PROPN
cana-3848	169	19	�	�	PROPN
cana-3848	169	20	,	,	PUNCT
cana-3848	169	21	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	169	22	)	)	PUNCT
cana-3848	169	23	in	in	ADP
cana-3848	169	24	the	the	DET
cana-3848	169	25	metric	metric	ADJ
cana-3848	169	26	space	space	NOUN
cana-3848	169	27	(	(	PUNCT
cana-3848	169	28	�	�	PROPN
cana-3848	169	29	̃	̃	PROPN
cana-3848	169	30	�	�	PROPN
cana-3848	169	31	,	,	PUNCT
cana-3848	169	32	(	(	PUNCT
cana-3848	169	33	𝑒	𝑒	PROPN
cana-3848	169	34	�	�	PROPN
cana-3848	169	35	̃	̃	NOUN
cana-3848	169	36	�	�	NOUN
cana-3848	169	37	𝑗	𝑗	NOUN
cana-3848	169	38	)	)	PUNCT
cana-3848	169	39	𝑠	𝑠	PROPN
cana-3848	169	40	)	)	PUNCT
cana-3848	169	41	.	.	PUNCT
cana-3848	170	1	isometric	isometric	ADJ
cana-3848	170	2	to	to	ADP
cana-3848	170	3	a	a	DET
cana-3848	170	4	dense	dense	ADJ
cana-3848	170	5	subspace	subspace	NOUN
cana-3848	170	6	of	of	ADP
cana-3848	170	7	the	the	DET
cana-3848	170	8	metric	metric	ADJ
cana-3848	170	9	space	space	NOUN
cana-3848	170	10	proof	proof	NOUN
cana-3848	170	11	:	:	PUNCT
cana-3848	170	12	(	(	PUNCT
cana-3848	170	13	a	a	X
cana-3848	170	14	)	)	PUNCT
cana-3848	170	15	the	the	DET
cana-3848	170	16	cancellative	cancellative	ADJ
cana-3848	170	17	condition	condition	NOUN
cana-3848	170	18	of	of	ADP
cana-3848	170	19	(	(	PUNCT
cana-3848	170	20	x	x	PROPN
cana-3848	170	21	̃,p	̃,p	PROPN
cana-3848	170	22	̃_j	̃_j	PROPN
cana-3848	170	23	)	)	PUNCT
cana-3848	170	24	is	be	AUX
cana-3848	170	25	obtained	obtain	VERB
cana-3848	170	26	from	from	ADP
cana-3848	170	27	lemma	lemma	PROPN
cana-3848	170	28	(	(	PUNCT
cana-3848	170	29	3.7	3.7	NUM
cana-3848	170	30	)	)	PUNCT
cana-3848	170	31	.	.	PUNCT
cana-3848	171	1	let	let	VERB
cana-3848	171	2	𝑥	𝑥	PRON
cana-3848	171	3	∶=	∶=	NUM
cana-3848	171	4	(	(	PUNCT
cana-3848	171	5	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	171	6	be	be	VERB
cana-3848	171	7	an	an	DET
cana-3848	171	8	element	element	NOUN
cana-3848	171	9	of	of	ADP
cana-3848	171	10	x	x	PUNCT
cana-3848	171	11	̂	̂	PUNCT
cana-3848	171	12	such	such	ADJ
cana-3848	171	13	that	that	PRON
cana-3848	171	14	–	–	PUNCT
cana-3848	172	1	[	[	X
cana-3848	172	2	𝑥	𝑥	X
cana-3848	172	3	]	]	X
cana-3848	172	4	∈	∈	PROPN
cana-3848	172	5	�	�	PROPN
cana-3848	172	6	̃	̃	PROPN
cana-3848	172	7	�	�	PROPN
cana-3848	172	8	and	and	CCONJ
cana-3848	172	9	𝑝𝑗([𝑥	𝑝𝑗([𝑥	PROPN
cana-3848	172	10	]	]	X
cana-3848	172	11	)	)	PUNCT
cana-3848	172	12	=	=	PUNCT
cana-3848	172	13	𝑝𝑗(−[𝑥	𝑝𝑗(−[𝑥	NOUN
cana-3848	172	14	]	]	PUNCT
cana-3848	172	15	)	)	PUNCT
cana-3848	172	16	=	=	SYM
cana-3848	173	1	0	0	X
cana-3848	173	2	.	.	PUNCT
cana-3848	174	1	since	since	SCONJ
cana-3848	174	2	lim	lim	PROPN
cana-3848	174	3	𝑛→∞	𝑛→∞	NUM
cana-3848	174	4	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	174	5	)	)	PUNCT
cana-3848	174	6	=	=	SYM
cana-3848	174	7	0	0	PUNCT
cana-3848	174	8	=	=	SYM
cana-3848	174	9	lim	lim	PROPN
cana-3848	174	10	𝑛→∞	𝑛→∞	NUM
cana-3848	174	11	𝑝𝑗(−𝑥𝑛	𝑝𝑗(−𝑥𝑛	PROPN
cana-3848	174	12	)	)	PUNCT
cana-3848	174	13	follows	follow	VERB
cana-3848	174	14	.	.	PUNCT
cana-3848	175	1	lim	lim	PROPN
cana-3848	175	2	𝑛→∞	𝑛→∞	NUM
cana-3848	175	3	(	(	PUNCT
cana-3848	175	4	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	175	5	)	)	PUNCT
cana-3848	175	6	𝑠	𝑠	PROPN
cana-3848	175	7	(	(	PUNCT
cana-3848	175	8	0	0	NUM
cana-3848	175	9	,	,	PUNCT
cana-3848	175	10	𝑥𝑛	𝑥𝑛	NOUN
cana-3848	175	11	)	)	PUNCT
cana-3848	175	12	=	=	SYM
cana-3848	175	13	0	0	PUNCT
cana-3848	175	14	because	because	SCONJ
cana-3848	175	15	,	,	PUNCT
cana-3848	175	16	finally	finally	ADV
cana-3848	175	17	,	,	PUNCT
cana-3848	175	18	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	175	19	(	(	PUNCT
cana-3848	175	20	0	0	NUM
cana-3848	175	21	,	,	PUNCT
cana-3848	175	22	𝑥𝑛	𝑥𝑛	NOUN
cana-3848	175	23	)	)	PUNCT
cana-3848	175	24	=	=	SYM
cana-3848	175	25	𝑝𝑗(𝑥𝑛)and	𝑝𝑗(𝑥𝑛)and	PROPN
cana-3848	175	26	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	175	27	(	(	PUNCT
cana-3848	175	28	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	175	29	,	,	PUNCT
cana-3848	175	30	0	0	NUM
cana-3848	175	31	)	)	PUNCT
cana-3848	175	32	=	=	SYM
cana-3848	175	33	𝑝𝑗(−𝑥𝑛	𝑝𝑗(−𝑥𝑛	PROPN
cana-3848	175	34	)	)	PUNCT
cana-3848	175	35	consequently	consequently	ADV
cana-3848	175	36	,	,	PUNCT
cana-3848	175	37	[	[	X
cana-3848	175	38	𝑥	𝑥	X
cana-3848	175	39	]	]	X
cana-3848	175	40	=	=	PUNCT
cana-3848	176	1	[	[	X
cana-3848	176	2	0	0	NUM
cana-3848	176	3	]	]	PUNCT
cana-3848	176	4	.	.	PUNCT
cana-3848	177	1	lemma	lemma	PROPN
cana-3848	177	2	(	(	PUNCT
cana-3848	177	3	3.7	3.7	NUM
cana-3848	177	4	)	)	PUNCT
cana-3848	177	5	yields	yield	VERB
cana-3848	177	6	the	the	DET
cana-3848	177	7	cancellative	cancellative	ADJ
cana-3848	177	8	condition	condition	NOUN
cana-3848	177	9	of	of	ADP
cana-3848	177	10	(	(	PUNCT
cana-3848	177	11	�	�	PROPN
cana-3848	177	12	̃	̃	PROPN
cana-3848	177	13	�	�	PROPN
cana-3848	177	14	,	,	PUNCT
cana-3848	177	15	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	177	16	)	)	PUNCT
cana-3848	177	17	.	.	PUNCT
cana-3848	178	1	consider	consider	VERB
cana-3848	178	2	an	an	DET
cana-3848	178	3	element	element	NOUN
cana-3848	178	4	of	of	ADP
cana-3848	178	5	�	�	PROPN
cana-3848	178	6	̂	̂	PROPN
cana-3848	178	7	�	�	PROPN
cana-3848	178	8	,	,	PUNCT
cana-3848	178	9	𝑥	𝑥	PRON
cana-3848	178	10	∶=	∶=	NUM
cana-3848	178	11	(	(	PUNCT
cana-3848	178	12	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	NOUN
cana-3848	178	13	,	,	PUNCT
cana-3848	178	14	such	such	ADJ
cana-3848	178	15	that	that	PRON
cana-3848	178	16	–	–	PUNCT
cana-3848	179	1	[	[	X
cana-3848	179	2	𝑥	𝑥	X
cana-3848	179	3	]	]	X
cana-3848	179	4	∈	∈	PROPN
cana-3848	179	5	�	�	PROPN
cana-3848	179	6	̃	̃	PROPN
cana-3848	179	7	�	�	PROPN
cana-3848	179	8	and	and	CCONJ
cana-3848	179	9	𝑝𝑗([𝑥	𝑝𝑗([𝑥	PROPN
cana-3848	179	10	]	]	X
cana-3848	179	11	)	)	PUNCT
cana-3848	179	12	=	=	PUNCT
cana-3848	179	13	𝑝𝑗(−[𝑥	𝑝𝑗(−[𝑥	NOUN
cana-3848	179	14	]	]	PUNCT
cana-3848	179	15	)	)	PUNCT
cana-3848	179	16	=	=	SYM
cana-3848	180	1	0	0	X
cana-3848	180	2	.	.	PUNCT
cana-3848	181	1	as	as	ADP
cana-3848	181	2	a	a	DET
cana-3848	181	3	result	result	NOUN
cana-3848	181	4	,	,	PUNCT
cana-3848	181	5	lim	lim	PROPN
cana-3848	181	6	𝑛→∞	𝑛→∞	NUM
cana-3848	181	7	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	181	8	)	)	PUNCT
cana-3848	181	9	=	=	SYM
cana-3848	181	10	0	0	PUNCT
cana-3848	182	1	=	=	SYM
cana-3848	182	2	lim	lim	NOUN
cana-3848	182	3	𝑛→∞	𝑛→∞	NUM
cana-3848	182	4	𝑝𝑗(−𝑥𝑛	𝑝𝑗(−𝑥𝑛	PROPN
cana-3848	182	5	)	)	PUNCT
cana-3848	182	6	.	.	PUNCT
cana-3848	183	1	lim	lim	PROPN
cana-3848	183	2	𝑛→∞	𝑛→∞	NUM
cana-3848	183	3	(	(	PUNCT
cana-3848	183	4	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	183	5	)	)	PUNCT
cana-3848	183	6	𝑠	𝑠	PROPN
cana-3848	183	7	(	(	PUNCT
cana-3848	183	8	0	0	NUM
cana-3848	183	9	,	,	PUNCT
cana-3848	183	10	𝑥𝑛	𝑥𝑛	NOUN
cana-3848	183	11	)	)	PUNCT
cana-3848	183	12	=	=	SYM
cana-3848	183	13	0	0	NUM
cana-3848	183	14	since	since	SCONJ
cana-3848	183	15	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	183	16	(	(	PUNCT
cana-3848	183	17	0	0	NUM
cana-3848	183	18	,	,	PUNCT
cana-3848	183	19	𝑥𝑛	𝑥𝑛	NOUN
cana-3848	183	20	)	)	PUNCT
cana-3848	183	21	=	=	SYM
cana-3848	183	22	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	183	23	)	)	PUNCT
cana-3848	183	24	and	and	CCONJ
cana-3848	183	25	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	183	26	(	(	PUNCT
cana-3848	183	27	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	183	28	,	,	PUNCT
cana-3848	183	29	0	0	NUM
cana-3848	183	30	)	)	PUNCT
cana-3848	183	31	=	=	SYM
cana-3848	183	32	𝑝𝑗(−𝑥𝑛	𝑝𝑗(−𝑥𝑛	PROPN
cana-3848	183	33	)	)	PUNCT
cana-3848	183	34	at	at	ADP
cana-3848	183	35	last	last	ADJ
cana-3848	183	36	.	.	PUNCT
cana-3848	184	1	as	as	SCONJ
cana-3848	184	2	a	a	DET
cana-3848	184	3	result	result	NOUN
cana-3848	184	4	,	,	PUNCT
cana-3848	184	5	[	[	X
cana-3848	184	6	𝑥	𝑥	X
cana-3848	184	7	]	]	X
cana-3848	184	8	=	=	PUNCT
cana-3848	185	1	[	[	X
cana-3848	185	2	0	0	NUM
cana-3848	185	3	]	]	PUNCT
cana-3848	185	4	.	.	PUNCT
cana-3848	186	1	we	we	PRON
cana-3848	186	2	have	have	VERB
cana-3848	186	3	𝑝𝑗(𝑎	𝑝𝑗(𝑎	NOUN
cana-3848	186	4	⋅	⋅	PROPN
cana-3848	187	1	[	[	X
cana-3848	187	2	𝑥	𝑥	X
cana-3848	187	3	]	]	X
cana-3848	187	4	)	)	PUNCT
cana-3848	187	5	=	=	SYM
cana-3848	187	6	lim	lim	NOUN
cana-3848	187	7	𝑛→∞	𝑛→∞	NUM
cana-3848	187	8	𝑝𝑗(𝑎𝑥𝑛	𝑝𝑗(𝑎𝑥𝑛	PROPN
cana-3848	187	9	)	)	PUNCT
cana-3848	187	10	=	=	SYM
cana-3848	187	11	lim	lim	NOUN
cana-3848	187	12	𝑛→∞	𝑛→∞	NUM
cana-3848	187	13	𝑎𝑝𝑗(𝑥𝑛	𝑎𝑝𝑗(𝑥𝑛	NUM
cana-3848	187	14	)	)	PUNCT
cana-3848	187	15	=	=	NOUN
cana-3848	188	1	𝑎	𝑎	PRON
cana-3848	188	2	lim	lim	NOUN
cana-3848	188	3	𝑛→∞	𝑛→∞	NUM
cana-3848	188	4	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	188	5	)	)	PUNCT
cana-3848	188	6	=	=	SYM
cana-3848	188	7	𝑎𝑝𝑗([𝑥	𝑎𝑝𝑗([𝑥	X
cana-3848	188	8	]	]	PUNCT
cana-3848	188	9	)	)	PUNCT
cana-3848	188	10	.	.	PUNCT
cana-3848	189	1	given	give	VERB
cana-3848	189	2	𝑥	𝑥	PRON
cana-3848	189	3	∶=	∶=	NUM
cana-3848	189	4	(	(	PUNCT
cana-3848	189	5	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	189	6	∈	∈	PROPN
cana-3848	189	7	�	�	PROPN
cana-3848	189	8	̂	̂	VERB
cana-3848	189	9	�	�	PROPN
cana-3848	189	10	and	and	CCONJ
cana-3848	189	11	𝑎	𝑎	PROPN
cana-3848	189	12	∈	∈	NOUN
cana-3848	189	13	ℝ+	ℝ+	NOUN
cana-3848	189	14	.	.	PUNCT
cana-3848	190	1	let	let	VERB
cana-3848	190	2	𝑥	𝑥	PRON
cana-3848	190	3	∶=	∶=	NUM
cana-3848	190	4	(	(	PUNCT
cana-3848	190	5	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	190	6	and	and	CCONJ
cana-3848	190	7	𝑦	𝑦	NOUN
cana-3848	190	8	∶=	∶=	NUM
cana-3848	190	9	(	(	PUNCT
cana-3848	190	10	𝑦𝑛)𝑛∈ℕ	𝑦𝑛)𝑛∈ℕ	NOUN
cana-3848	190	11	be	be	VERB
cana-3848	190	12	two	two	NUM
cana-3848	190	13	of	of	ADP
cana-3848	190	14	�	�	NOUN
cana-3848	190	15	̂�.elements	̂�.element	NOUN
cana-3848	190	16	.	.	PUNCT
cana-3848	191	1	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	191	2	+	+	CCONJ
cana-3848	191	3	𝑦𝑛	𝑦𝑛	NOUN
cana-3848	191	4	)	)	PUNCT
cana-3848	191	5	≤	≤	NOUN
cana-3848	191	6	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	191	7	)	)	PUNCT
cana-3848	192	1	+	+	NUM
cana-3848	192	2	𝑝𝑗(𝑦𝑛	𝑝𝑗(𝑦𝑛	NOUN
cana-3848	192	3	)	)	PUNCT
cana-3848	192	4	is	be	AUX
cana-3848	192	5	taken	take	VERB
cana-3848	192	6	into	into	ADP
cana-3848	192	7	consideration	consideration	NOUN
cana-3848	192	8	in	in	ADP
cana-3848	192	9	order	order	NOUN
cana-3848	192	10	to	to	PART
cana-3848	192	11	explain	explain	VERB
cana-3848	192	12	the	the	DET
cana-3848	192	13	triangle	triangle	NOUN
cana-3848	192	14	inequality	inequality	NOUN
cana-3848	192	15	;	;	PUNCT
cana-3848	192	16	therefore	therefore	ADV
cana-3848	192	17	,	,	PUNCT
cana-3848	192	18	lim	lim	PROPN
cana-3848	192	19	𝑛→∞	𝑛→∞	NUM
cana-3848	192	20	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	192	21	+	+	CCONJ
cana-3848	192	22	𝑦𝑛	𝑦𝑛	NOUN
cana-3848	192	23	)	)	PUNCT
cana-3848	192	24	≤	≤	NOUN
cana-3848	192	25	lim	lim	NOUN
cana-3848	192	26	𝑛→∞	𝑛→∞	NUM
cana-3848	192	27	𝑝𝑗(𝑥𝑛	𝑝𝑗(𝑥𝑛	PROPN
cana-3848	192	28	)	)	PUNCT
cana-3848	193	1	+	+	CCONJ
cana-3848	193	2	lim	lim	NOUN
cana-3848	193	3	𝑛→∞	𝑛→∞	NUM
cana-3848	193	4	𝑝𝑗(𝑦𝑛	𝑝𝑗(𝑦𝑛	NOUN
cana-3848	193	5	)	)	PUNCT
cana-3848	193	6	.	.	PUNCT
cana-3848	194	1	𝑝𝑗([𝑥	𝑝𝑗([𝑥	PROPN
cana-3848	194	2	]	]	PUNCT
cana-3848	195	1	+	+	CCONJ
cana-3848	196	1	[	[	X
cana-3848	196	2	𝑦	𝑦	X
cana-3848	196	3	]	]	X
cana-3848	196	4	)	)	PUNCT
cana-3848	196	5	≤	≤	PUNCT
cana-3848	197	1	𝑝𝑗([𝑥	𝑝𝑗([𝑥	PROPN
cana-3848	197	2	]	]	PUNCT
cana-3848	197	3	)	)	PUNCT
cana-3848	198	1	+	+	CCONJ
cana-3848	198	2	𝑝𝑗([𝑦	𝑝𝑗([𝑦	PROPN
cana-3848	198	3	]	]	PUNCT
cana-3848	198	4	)	)	PUNCT
cana-3848	198	5	,	,	PUNCT
cana-3848	198	6	for	for	ADP
cana-3848	198	7	this	this	PRON
cana-3848	198	8	.	.	PUNCT
cana-3848	199	1	so	so	ADV
cana-3848	199	2	𝑝	𝑝	PROPN
cana-3848	199	3	�	�	PROPN
cana-3848	199	4	̃	̃	PROPN
cana-3848	199	5	�	�	PROPN
cana-3848	199	6	is	be	AUX
cana-3848	199	7	a	a	DET
cana-3848	199	8	quasi	quasi	NOUN
cana-3848	199	9	-	-	NOUN
cana-3848	199	10	norm	norm	NOUN
cana-3848	199	11	for	for	ADP
cana-3848	199	12	�	�	PROPN
cana-3848	199	13	̃	̃	PROPN
cana-3848	199	14	�	�	PROPN
cana-3848	199	15	communications	communication	NOUN
cana-3848	199	16	on	on	ADP
cana-3848	199	17	applied	apply	VERB
cana-3848	199	18	nonlinear	nonlinear	ADJ
cana-3848	199	19	analysis	analysis	NOUN
cana-3848	199	20	issn	issn	NOUN
cana-3848	199	21	:	:	PUNCT
cana-3848	199	22	1074	1074	NUM
cana-3848	199	23	-	-	PUNCT
cana-3848	199	24	133x	133x	NUM
cana-3848	199	25	vol	vol	NOUN
cana-3848	199	26	32	32	NUM
cana-3848	199	27	no	no	NOUN
cana-3848	199	28	.	.	PUNCT
cana-3848	200	1	9s	9s	NUM
cana-3848	200	2	(	(	PUNCT
cana-3848	200	3	2025	2025	NUM
cana-3848	200	4	)	)	PUNCT
cana-3848	200	5	206	206	NUM
cana-3848	200	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	201	1	it	it	PRON
cana-3848	201	2	well	well	INTJ
cana-3848	201	3	known	know	VERB
cana-3848	201	4	that	that	SCONJ
cana-3848	201	5	the	the	DET
cana-3848	201	6	bicompletion	bicompletion	NOUN
cana-3848	201	7	of	of	ADP
cana-3848	201	8	the	the	DET
cana-3848	201	9	quasi	quasi	ADJ
cana-3848	201	10	-	-	ADJ
cana-3848	201	11	metric	metric	ADJ
cana-3848	201	12	space	space	NOUN
cana-3848	201	13	(	(	PUNCT
cana-3848	201	14	𝑋	𝑋	PROPN
cana-3848	201	15	,	,	PUNCT
cana-3848	201	16	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	201	17	)	)	PUNCT
cana-3848	201	18	is	be	AUX
cana-3848	201	19	a	a	DET
cana-3848	201	20	quasi	quasi	ADJ
cana-3848	201	21	-	-	ADJ
cana-3848	201	22	metric	metric	ADJ
cana-3848	201	23	space	space	NOUN
cana-3848	201	24	(	(	PUNCT
cana-3848	201	25	𝑋𝑏	𝑋𝑏	PROPN
cana-3848	201	26	,	,	PUNCT
cana-3848	201	27	(	(	PUNCT
cana-3848	201	28	𝑒𝑝𝑗	𝑒𝑝𝑗	ADV
cana-3848	201	29	)	)	PUNCT
cana-3848	201	30	𝑏	𝑏	NOUN
cana-3848	201	31	)	)	PUNCT
cana-3848	201	32	,	,	PUNCT
cana-3848	201	33	where	where	SCONJ
cana-3848	201	34	𝑋𝑏	𝑋𝑏	PROPN
cana-3848	201	35	=	=	SYM
cana-3848	201	36	{	{	PUNCT
cana-3848	202	1	[	[	X
cana-3848	202	2	𝑥	𝑥	X
cana-3848	202	3	]	]	X
cana-3848	202	4	∶	∶	NOUN
cana-3848	202	5	𝑥	𝑥	PROPN
cana-3848	202	6	is	be	AUX
cana-3848	202	7	a	a	DET
cana-3848	202	8	cauchy	cauchy	ADJ
cana-3848	202	9	sequence	sequence	NOUN
cana-3848	202	10	in	in	ADP
cana-3848	202	11	the	the	DET
cana-3848	202	12	metric	metric	ADJ
cana-3848	202	13	space	space	NOUN
cana-3848	202	14	(	(	PUNCT
cana-3848	202	15	𝑋	𝑋	NOUN
cana-3848	202	16	,	,	PUNCT
cana-3848	202	17	(	(	PUNCT
cana-3848	202	18	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	202	19	)	)	PUNCT
cana-3848	202	20	𝑠	𝑠	PROPN
cana-3848	202	21	)	)	PUNCT
cana-3848	202	22	}	}	PUNCT
cana-3848	202	23	,	,	PUNCT
cana-3848	202	24	(	(	PUNCT
cana-3848	202	25	𝑒𝑝𝑗	𝑒𝑝𝑗	ADV
cana-3848	202	26	)	)	PUNCT
cana-3848	203	1	𝑏	𝑏	NOUN
cana-3848	203	2	(	(	PUNCT
cana-3848	204	1	[	[	X
cana-3848	204	2	𝑥	𝑥	X
cana-3848	204	3	]	]	X
cana-3848	204	4	,	,	PUNCT
cana-3848	205	1	[	[	X
cana-3848	205	2	𝑦	𝑦	X
cana-3848	205	3	]	]	X
cana-3848	205	4	)	)	PUNCT
cana-3848	206	1	=	=	SYM
cana-3848	206	2	lim	lim	PROPN
cana-3848	206	3	𝑛→∞	𝑛→∞	NUM
cana-3848	206	4	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	206	5	(	(	PUNCT
cana-3848	206	6	𝑥𝑛	𝑥𝑛	NOUN
cana-3848	206	7	,	,	PUNCT
cana-3848	206	8	𝑦𝑛	𝑦𝑛	NOUN
cana-3848	206	9	)	)	PUNCT
cana-3848	206	10	for	for	ADP
cana-3848	206	11	all	all	PRON
cana-3848	206	12	[	[	X
cana-3848	206	13	𝑥	𝑥	X
cana-3848	206	14	]	]	X
cana-3848	206	15	,	,	PUNCT
cana-3848	206	16	[	[	X
cana-3848	206	17	𝑦	𝑦	X
cana-3848	206	18	]	]	X
cana-3848	206	19	∈	∈	PROPN
cana-3848	206	20	𝑋𝑏	𝑋𝑏	PROPN
cana-3848	206	21	,	,	PUNCT
cana-3848	206	22	and	and	CCONJ
cana-3848	206	23	for	for	ADP
cana-3848	206	24	each	each	DET
cana-3848	206	25	cauchy	cauchy	ADJ
cana-3848	206	26	sequence	sequence	NOUN
cana-3848	206	27	𝑥	𝑥	PROPN
cana-3848	206	28	∶=	∶=	NUM
cana-3848	206	29	(	(	PUNCT
cana-3848	206	30	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	206	31	𝑖𝑛	𝑖𝑛	X
cana-3848	206	32	(	(	PUNCT
cana-3848	206	33	𝑋	𝑋	PROPN
cana-3848	206	34	,	,	PUNCT
cana-3848	206	35	(	(	PUNCT
cana-3848	206	36	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	206	37	)	)	PUNCT
cana-3848	206	38	𝑠	𝑠	PROPN
cana-3848	206	39	)	)	PUNCT
cana-3848	206	40	,	,	PUNCT
cana-3848	207	1	[	[	X
cana-3848	207	2	𝑥	𝑥	X
cana-3848	207	3	]	]	X
cana-3848	207	4	=	=	PUNCT
cana-3848	207	5	{	{	PUNCT
cana-3848	207	6	𝑦	𝑦	NOUN
cana-3848	207	7	∶=	∶=	NUM
cana-3848	207	8	(	(	PUNCT
cana-3848	207	9	𝑦𝑛	𝑦𝑛	NOUN
cana-3848	207	10	)	)	PUNCT
cana-3848	207	11	∶	∶	NOUN
cana-3848	207	12	𝑦	𝑦	NOUN
cana-3848	207	13	is	be	AUX
cana-3848	207	14	a	a	DET
cana-3848	207	15	cauchy	cauchy	ADJ
cana-3848	207	16	sequence	sequence	NOUN
cana-3848	207	17	in	in	ADP
cana-3848	207	18	(	(	PUNCT
cana-3848	207	19	𝑋	𝑋	NOUN
cana-3848	207	20	,	,	PUNCT
cana-3848	207	21	(	(	PUNCT
cana-3848	207	22	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	207	23	)	)	PUNCT
cana-3848	207	24	𝑠	𝑠	PROPN
cana-3848	207	25	)	)	PUNCT
cana-3848	207	26	and	and	CCONJ
cana-3848	207	27	lim	lim	PROPN
cana-3848	207	28	𝑛→∞	𝑛→∞	NUM
cana-3848	207	29	(	(	PUNCT
cana-3848	207	30	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	207	31	)	)	PUNCT
cana-3848	208	1	𝑠	𝑠	PROPN
cana-3848	208	2	(	(	PUNCT
cana-3848	208	3	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	208	4	,	,	PUNCT
cana-3848	208	5	𝑦𝑛	𝑦𝑛	NOUN
cana-3848	208	6	)	)	PUNCT
cana-3848	208	7	=	=	SYM
cana-3848	208	8	0	0	NUM
cana-3848	208	9	}	}	PUNCT
cana-3848	208	10	.	.	PUNCT
cana-3848	209	1	𝑋𝑏	𝑋𝑏	PROPN
cana-3848	209	2	=	=	SYM
cana-3848	209	3	�	�	PROPN
cana-3848	209	4	̃	̃	PROPN
cana-3848	209	5	�	�	PROPN
cana-3848	209	6	and	and	CCONJ
cana-3848	209	7	(	(	PUNCT
cana-3848	209	8	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	209	9	)	)	PUNCT
cana-3848	210	1	𝑏	𝑏	PROPN
cana-3848	210	2	=	=	SYM
cana-3848	210	3	𝑒	𝑒	PROPN
cana-3848	210	4	�	�	PROPN
cana-3848	210	5	̃	̃	PROPN
cana-3848	210	6	�	�	NOUN
cana-3848	210	7	𝑗	𝑗	VERB
cana-3848	210	8	,	,	PUNCT
cana-3848	210	9	are	be	AUX
cana-3848	210	10	the	the	DET
cana-3848	210	11	results	result	NOUN
cana-3848	210	12	.	.	PUNCT
cana-3848	211	1	(	(	PUNCT
cana-3848	211	2	b	b	X
cana-3848	211	3	)	)	PUNCT
cana-3848	211	4	the	the	DET
cana-3848	211	5	constant	constant	ADJ
cana-3848	211	6	sequence	sequence	NOUN
cana-3848	211	7	𝑥	𝑥	PROPN
cana-3848	211	8	,	,	PUNCT
cana-3848	211	9	𝑥	𝑥	NOUN
cana-3848	211	10	,	,	PUNCT
cana-3848	211	11	…	…	PUNCT
cana-3848	211	12	𝑥	𝑥	NOUN
cana-3848	211	13	,	,	PUNCT
cana-3848	211	14	…	…	PUNCT
cana-3848	211	15	.	.	PUNCT
cana-3848	212	1	for	for	ADP
cana-3848	212	2	every	every	DET
cana-3848	212	3	x	x	NOUN
cana-3848	212	4	in	in	ADP
cana-3848	212	5	𝑋	𝑋	NOUN
cana-3848	212	6	is	be	AUX
cana-3848	212	7	defined	define	VERB
cana-3848	212	8	by	by	ADP
cana-3848	212	9	�	�	PROPN
cana-3848	212	10	̂	̂	NOUN
cana-3848	212	11	�	�	PROPN
cana-3848	212	12	.	.	PUNCT
cana-3848	212	13	provided	provide	VERB
cana-3848	212	14	that	that	PRON
cana-3848	212	15	(	(	PUNCT
cana-3848	212	16	𝑋𝑏	𝑋𝑏	PROPN
cana-3848	212	17	,	,	PUNCT
cana-3848	212	18	(	(	PUNCT
cana-3848	212	19	𝑒𝑝𝑗	𝑒𝑝𝑗	ADV
cana-3848	212	20	)	)	PUNCT
cana-3848	212	21	𝑏	𝑏	NOUN
cana-3848	212	22	)	)	PUNCT
cana-3848	212	23	is	be	AUX
cana-3848	212	24	the	the	DET
cana-3848	212	25	bicompletion	bicompletion	NOUN
cana-3848	212	26	of	of	ADP
cana-3848	212	27	(	(	PUNCT
cana-3848	212	28	𝑋	𝑋	PROPN
cana-3848	212	29	,	,	PUNCT
cana-3848	212	30	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	212	31	)	)	PUNCT
cana-3848	212	32	,	,	PUNCT
cana-3848	212	33	𝑖(𝑋	𝑖(𝑋	NUM
cana-3848	212	34	)	)	PUNCT
cana-3848	212	35	is	be	AUX
cana-3848	212	36	dense	dense	ADJ
cana-3848	212	37	in	in	ADP
cana-3848	212	38	(	(	PUNCT
cana-3848	212	39	�	�	PROPN
cana-3848	212	40	̃	̃	PROPN
cana-3848	212	41	�	�	PROPN
cana-3848	212	42	,	,	PUNCT
cana-3848	212	43	(	(	PUNCT
cana-3848	212	44	𝑒𝑝	𝑒𝑝	PROPN
cana-3848	212	45	�	�	PROPN
cana-3848	212	46	̃	̃	PROPN
cana-3848	212	47	�	�	PROPN
cana-3848	212	48	)	)	PUNCT
cana-3848	212	49	𝑠	𝑠	PROPN
cana-3848	212	50	)	)	PUNCT
cana-3848	212	51	,	,	PUNCT
cana-3848	212	52	where	where	SCONJ
cana-3848	212	53	i	i	PRON
cana-3848	212	54	is	be	AUX
cana-3848	212	55	the	the	DET
cana-3848	212	56	one	one	NUM
cana-3848	212	57	-	-	PUNCT
cana-3848	212	58	to	to	ADP
cana-3848	212	59	-	-	PUNCT
cana-3848	212	60	one	one	NUM
cana-3848	212	61	function	function	NOUN
cana-3848	212	62	from	from	ADP
cana-3848	212	63	𝑋	𝑋	PROPN
cana-3848	212	64	𝑡𝑜	𝑡𝑜	PROPN
cana-3848	212	65	�	�	PROPN
cana-3848	212	66	̃	̃	PROPN
cana-3848	212	67	�	�	PROPN
cana-3848	212	68	provided	provide	VERB
cana-3848	212	69	by	by	ADP
cana-3848	212	70	𝑖(𝑥)=	𝑖(𝑥)=	NOUN
cana-3848	212	71	[	[	PUNCT
cana-3848	212	72	[	[	X
cana-3848	212	73	�	�	NOUN
cana-3848	212	74	̂	̂	NOUN
cana-3848	212	75	�	�	NOUN
cana-3848	212	76	]	]	PUNCT
cana-3848	212	77	]	]	PUNCT
cana-3848	212	78	for	for	ADP
cana-3848	212	79	every	every	DET
cana-3848	212	80	𝑥	𝑥	PROPN
cana-3848	212	81	∈	∈	PROPN
cana-3848	212	82	𝑋.	𝑋.	PROPN
cana-3848	212	83	for	for	ADP
cana-3848	212	84	each	each	DET
cana-3848	212	85	𝑥	𝑥	PRON
cana-3848	213	1	𝑖𝑛	𝑖𝑛	PRON
cana-3848	213	2	𝑋	𝑋	NOUN
cana-3848	213	3	,	,	PUNCT
cana-3848	213	4	note	note	VERB
cana-3848	213	5	that	that	SCONJ
cana-3848	213	6	[	[	X
cana-3848	213	7	�	�	NOUN
cana-3848	213	8	̂	̂	SYM
cana-3848	213	9	�	�	NOUN
cana-3848	213	10	]	]	PUNCT
cana-3848	213	11	is	be	AUX
cana-3848	213	12	the	the	DET
cana-3848	213	13	collection	collection	NOUN
cana-3848	213	14	of	of	ADP
cana-3848	213	15	all	all	DET
cana-3848	213	16	sequences	sequence	NOUN
cana-3848	213	17	in	in	ADP
cana-3848	213	18	𝑋	𝑋	PROPN
cana-3848	213	19	that	that	PRON
cana-3848	213	20	converge	converge	VERB
cana-3848	213	21	to	to	ADP
cana-3848	213	22	x	x	PUNCT
cana-3848	213	23	in	in	ADP
cana-3848	213	24	the	the	DET
cana-3848	213	25	metric	metric	ADJ
cana-3848	213	26	space	space	NOUN
cana-3848	213	27	(	(	PUNCT
cana-3848	213	28	𝑋	𝑋	NOUN
cana-3848	213	29	,	,	PUNCT
cana-3848	213	30	(	(	PUNCT
cana-3848	213	31	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	213	32	)	)	PUNCT
cana-3848	214	1	𝑠	𝑠	PROPN
cana-3848	214	2	)	)	PUNCT
cana-3848	214	3	.since	.since	NOUN
cana-3848	215	1	it	it	PRON
cana-3848	215	2	is	be	AUX
cana-3848	215	3	standard	standard	ADJ
cana-3848	215	4	procedure	procedure	NOUN
cana-3848	215	5	to	to	PART
cana-3848	215	6	confirm	confirm	VERB
cana-3848	215	7	that	that	SCONJ
cana-3848	215	8	𝑖	𝑖	NOUN
cana-3848	215	9	is	be	AUX
cana-3848	215	10	a	a	DET
cana-3848	215	11	linear	linear	ADJ
cana-3848	215	12	function	function	NOUN
cana-3848	215	13	,	,	PUNCT
cana-3848	215	14	the	the	DET
cana-3848	215	15	preceding	precede	VERB
cana-3848	215	16	lemma	lemma	PROPN
cana-3848	215	17	states	state	NOUN
cana-3848	215	18	that	that	SCONJ
cana-3848	215	19	𝑖(𝑋	𝑖(𝑋	PROPN
cana-3848	215	20	)	)	PUNCT
cana-3848	215	21	is	be	AUX
cana-3848	215	22	a	a	DET
cana-3848	215	23	semi	semi	ADJ
cana-3848	215	24	linear	linear	ADJ
cana-3848	215	25	subspace	subspace	NOUN
cana-3848	215	26	of	of	ADP
cana-3848	215	27	�	�	PROPN
cana-3848	215	28	̃	̃	PROPN
cana-3848	215	29	�	�	PROPN
cana-3848	215	30	.	.	PUNCT
cana-3848	216	1	we	we	PRON
cana-3848	216	2	may	may	AUX
cana-3848	216	3	deduce	deduce	VERB
cana-3848	216	4	that	that	SCONJ
cana-3848	216	5	(	(	PUNCT
cana-3848	216	6	𝑋	𝑋	PROPN
cana-3848	216	7	,	,	PUNCT
cana-3848	216	8	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	216	9	)	)	PUNCT
cana-3848	216	10	and	and	CCONJ
cana-3848	216	11	(	(	PUNCT
cana-3848	216	12	𝑖(𝑋	𝑖(𝑋	PROPN
cana-3848	216	13	)	)	PUNCT
cana-3848	216	14	,	,	PUNCT
cana-3848	216	15	𝑝𝑗|	𝑝𝑗|	PROPN
cana-3848	216	16	𝑖(𝑋	𝑖(𝑋	PROPN
cana-3848	216	17	)	)	PUNCT
cana-3848	216	18	)	)	PUNCT
cana-3848	216	19	are	be	AUX
cana-3848	216	20	isometric	isometric	ADJ
cana-3848	216	21	quasi	quasi	ADJ
cana-3848	216	22	-	-	ADJ
cana-3848	216	23	normed	normed	ADJ
cana-3848	216	24	cones	cone	NOUN
cana-3848	216	25	because	because	SCONJ
cana-3848	216	26	,	,	PUNCT
cana-3848	216	27	for	for	ADP
cana-3848	216	28	all	all	PRON
cana-3848	216	29	𝑥	𝑥	DET
cana-3848	216	30	∈	∈	PROPN
cana-3848	216	31	𝑋	𝑋	PROPN
cana-3848	216	32	,	,	PUNCT
cana-3848	216	33	𝑝𝑗(𝑖(𝑥	𝑝𝑗(𝑖(𝑥	NOUN
cana-3848	216	34	)	)	PUNCT
cana-3848	216	35	)	)	PUNCT
cana-3848	217	1	=	=	PRON
cana-3848	217	2	𝑝𝑗([	𝑝𝑗([	PROPN
cana-3848	217	3	�	�	PROPN
cana-3848	217	4	̂	̂	NUM
cana-3848	217	5	�	�	NOUN
cana-3848	217	6	]	]	PUNCT
cana-3848	217	7	)	)	PUNCT
cana-3848	217	8	=	=	SYM
cana-3848	217	9	𝑝𝑗(𝑥	𝑝𝑗(𝑥	NOUN
cana-3848	217	10	)	)	PUNCT
cana-3848	217	11	.	.	PUNCT
cana-3848	218	1	the	the	DET
cana-3848	218	2	proof	proof	NOUN
cana-3848	218	3	is	be	AUX
cana-3848	218	4	complete	complete	ADJ
cana-3848	218	5	.	.	PUNCT
cana-3848	219	1	lemma	lemma	PROPN
cana-3848	219	2	(	(	PUNCT
cana-3848	219	3	3.11	3.11	NUM
cana-3848	219	4	):	):	PUNCT
cana-3848	219	5	take	take	VERB
cana-3848	219	6	a	a	DET
cana-3848	219	7	quasi	quasi	ADJ
cana-3848	219	8	-	-	ADJ
cana-3848	219	9	normed	normed	ADJ
cana-3848	219	10	cone	cone	NOUN
cana-3848	219	11	(	(	PUNCT
cana-3848	219	12	𝑋	𝑋	PROPN
cana-3848	219	13	,	,	PUNCT
cana-3848	219	14	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	219	15	)	)	PUNCT
cana-3848	219	16	,	,	PUNCT
cana-3848	219	17	a	a	DET
cana-3848	219	18	bicomplete	bicomplete	ADJ
cana-3848	219	19	quasi	quasi	NOUN
cana-3848	219	20	-	-	ADJ
cana-3848	219	21	normed	normed	ADJ
cana-3848	219	22	cone	cone	NOUN
cana-3848	219	23	(	(	PUNCT
cana-3848	219	24	𝑌	𝑌	PROPN
cana-3848	219	25	,	,	PUNCT
cana-3848	219	26	𝑞𝑗),and	𝑞𝑗),and	PRON
cana-3848	219	27	a	a	DET
cana-3848	219	28	sub	sub	NOUN
cana-3848	219	29	cone	cone	NOUN
cana-3848	219	30	a	a	PRON
cana-3848	219	31	of	of	ADP
cana-3848	219	32	𝑋	𝑋	NOUN
cana-3848	219	33	to	to	ADP
cana-3848	219	34	𝑌.	𝑌.	PROPN
cana-3848	219	35	if	if	SCONJ
cana-3848	219	36	a	a	PRON
cana-3848	219	37	is	be	AUX
cana-3848	219	38	dense	dense	ADJ
cana-3848	219	39	in	in	ADP
cana-3848	219	40	(	(	PUNCT
cana-3848	219	41	𝑋	𝑋	NOUN
cana-3848	219	42	,	,	PUNCT
cana-3848	219	43	(	(	PUNCT
cana-3848	219	44	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	219	45	)	)	PUNCT
cana-3848	219	46	𝑠	𝑠	PROPN
cana-3848	219	47	)	)	PUNCT
cana-3848	219	48	,	,	PUNCT
cana-3848	219	49	then	then	ADV
cana-3848	219	50	𝑓	𝑓	PRON
cana-3848	219	51	is	be	AUX
cana-3848	219	52	a	a	DET
cana-3848	219	53	one	one	NUM
cana-3848	219	54	-	-	PUNCT
cana-3848	219	55	to	to	ADP
cana-3848	219	56	-	-	PUNCT
cana-3848	219	57	one	one	NUM
cana-3848	219	58	isometry	isometry	NOUN
cana-3848	219	59	.	.	PUNCT
cana-3848	220	1	from	from	ADP
cana-3848	220	2	(	(	PUNCT
cana-3848	220	3	𝑋	𝑋	PROPN
cana-3848	220	4	,	,	PUNCT
cana-3848	220	5	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	220	6	)	)	PUNCT
cana-3848	220	7	to	to	ADP
cana-3848	220	8	(	(	PUNCT
cana-3848	220	9	𝑌	𝑌	PROPN
cana-3848	220	10	,	,	PUNCT
cana-3848	220	11	𝑞𝑗	𝑞𝑗	PROPN
cana-3848	220	12	)	)	PUNCT
cana-3848	220	13	,	,	PUNCT
cana-3848	220	14	𝑓𝑗	𝑓𝑗	ADV
cana-3848	220	15	then	then	ADV
cana-3848	220	16	extends	extend	VERB
cana-3848	220	17	uniquely	uniquely	ADV
cana-3848	220	18	to	to	ADP
cana-3848	220	19	a	a	DET
cana-3848	220	20	one	one	NUM
cana-3848	220	21	-	-	PUNCT
cana-3848	220	22	to	to	ADP
cana-3848	220	23	-	-	PUNCT
cana-3848	220	24	one	one	NUM
cana-3848	220	25	isometry	isometry	NOUN
cana-3848	220	26	.	.	PUNCT
cana-3848	221	1	proof	proof	NOUN
cana-3848	221	2	:	:	PUNCT
cana-3848	221	3	a	a	PRON
cana-3848	221	4	contains	contain	VERB
cana-3848	221	5	a	a	DET
cana-3848	221	6	sequence	sequence	NOUN
cana-3848	221	7	(	(	PUNCT
cana-3848	221	8	𝑥𝑛)𝑛∈ℕsuch	𝑥𝑛)𝑛∈ℕsuch	ADP
cana-3848	221	9	that	that	PRON
cana-3848	221	10	,	,	PUNCT
cana-3848	221	11	for	for	ADP
cana-3848	221	12	all	all	DET
cana-3848	221	13	𝑥	𝑥	DET
cana-3848	221	14	∈	∈	NOUN
cana-3848	221	15	𝑋\𝐴	𝑋\𝐴	PROPN
cana-3848	221	16	,	,	PUNCT
cana-3848	221	17	lim	lim	PROPN
cana-3848	221	18	𝑛→∞	𝑛→∞	NUM
cana-3848	221	19	(	(	PUNCT
cana-3848	221	20	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	221	21	)	)	PUNCT
cana-3848	222	1	𝑠	𝑠	PROPN
cana-3848	222	2	(	(	PUNCT
cana-3848	222	3	𝑥	𝑥	NOUN
cana-3848	222	4	,	,	PUNCT
cana-3848	222	5	𝑥𝑛	𝑥𝑛	NOUN
cana-3848	222	6	)	)	PUNCT
cana-3848	222	7	=	=	SYM
cana-3848	223	1	0	0	X
cana-3848	223	2	.	.	PUNCT
cana-3848	224	1	in	in	ADP
cana-3848	224	2	the	the	DET
cana-3848	224	3	metric	metric	ADJ
cana-3848	224	4	space	space	NOUN
cana-3848	224	5	(	(	PUNCT
cana-3848	224	6	𝑋	𝑋	NOUN
cana-3848	224	7	,	,	PUNCT
cana-3848	224	8	(	(	PUNCT
cana-3848	224	9	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	224	10	)	)	PUNCT
cana-3848	224	11	𝑠	𝑠	PROPN
cana-3848	224	12	)	)	PUNCT
cana-3848	224	13	,	,	PUNCT
cana-3848	224	14	the	the	DET
cana-3848	224	15	sequence	sequence	NOUN
cana-3848	224	16	(	(	PUNCT
cana-3848	224	17	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	224	18	(	(	PUNCT
cana-3848	224	19	related	relate	VERB
cana-3848	224	20	to	to	ADP
cana-3848	224	21	𝑥	𝑥	DET
cana-3848	224	22	∈	∈	NOUN
cana-3848	224	23	𝑋\𝐴	𝑋\𝐴	ADJ
cana-3848	224	24	is	be	AUX
cana-3848	224	25	a	a	DET
cana-3848	224	26	cauchy	cauchy	ADJ
cana-3848	224	27	sequence	sequence	NOUN
cana-3848	224	28	,	,	PUNCT
cana-3848	224	29	and	and	CCONJ
cana-3848	224	30	for	for	ADP
cana-3848	224	31	any	any	DET
cana-3848	224	32	𝑚	𝑚	NOUN
cana-3848	224	33	,	,	PUNCT
cana-3848	224	34	𝑛	𝑛	DET
cana-3848	224	35	≥	≥	NOUN
cana-3848	224	36	𝑛0	𝑛0	VERB
cana-3848	224	37	,	,	PUNCT
cana-3848	224	38	there	there	PRON
cana-3848	224	39	exists	exist	VERB
cana-3848	224	40	𝑛0	𝑛0	VERB
cana-3848	224	41	∈	∈	PROPN
cana-3848	224	42	ℕ	ℕ	PROPN
cana-3848	224	43	such	such	ADJ
cana-3848	224	44	that	that	SCONJ
cana-3848	224	45	(	(	PUNCT
cana-3848	224	46	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	224	47	)	)	PUNCT
cana-3848	224	48	𝑠	𝑠	PROPN
cana-3848	225	1	(	(	PUNCT
cana-3848	225	2	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	225	3	,	,	PUNCT
cana-3848	225	4	𝑥𝑚	𝑥𝑚	NOUN
cana-3848	225	5	)	)	PUNCT
cana-3848	225	6	<	<	X
cana-3848	225	7	𝜇	𝜇	X
cana-3848	225	8	+	+	X
cana-3848	225	9	1.for	1.for	X
cana-3848	225	10	any	any	DET
cana-3848	225	11	value	value	NOUN
cana-3848	225	12	of	of	ADP
cana-3848	225	13	𝜇	𝜇	ADP
cana-3848	225	14	>	>	X
cana-3848	225	15	−1	−1	NOUN
cana-3848	225	16	.	.	PUNCT
cana-3848	226	1	proposition	proposition	NOUN
cana-3848	226	2	(	(	PUNCT
cana-3848	226	3	3.2.9	3.2.9	NUM
cana-3848	226	4	)	)	PUNCT
cana-3848	226	5	states	state	VERB
cana-3848	226	6	that	that	SCONJ
cana-3848	226	7	for	for	ADP
cana-3848	226	8	any	any	DET
cana-3848	226	9	𝑚	𝑚	NOUN
cana-3848	226	10	,	,	PUNCT
cana-3848	226	11	𝑛	𝑛	DET
cana-3848	226	12	≥	≥	NOUN
cana-3848	226	13	𝑛0	𝑛0	VERB
cana-3848	226	14	,	,	PUNCT
cana-3848	226	15	(	(	PUNCT
cana-3848	226	16	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	226	17	)	)	PUNCT
cana-3848	226	18	𝑠	𝑠	PROPN
cana-3848	226	19	(	(	PUNCT
cana-3848	226	20	𝑓(𝑥𝑛	𝑓(𝑥𝑛	PROPN
cana-3848	226	21	)	)	PUNCT
cana-3848	226	22	,	,	PUNCT
cana-3848	226	23	𝑓(𝑥𝑚	𝑓(𝑥𝑚	NUM
cana-3848	226	24	)	)	PUNCT
cana-3848	226	25	<	<	X
cana-3848	226	26	𝜇	𝜇	X
cana-3848	226	27	+	+	NOUN
cana-3848	226	28	1	1	NUM
cana-3848	226	29	.	.	PUNCT
cana-3848	227	1	the	the	DET
cana-3848	227	2	metric	metric	ADJ
cana-3848	227	3	(	(	PUNCT
cana-3848	227	4	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	227	5	)	)	PUNCT
cana-3848	227	6	𝑠	𝑠	PROPN
cana-3848	227	7	(	(	PUNCT
cana-3848	227	8	𝑓(𝑥𝑛	𝑓(𝑥𝑛	PROPN
cana-3848	227	9	)	)	PUNCT
cana-3848	227	10	,	,	PUNCT
cana-3848	227	11	is	be	AUX
cana-3848	227	12	therefore	therefore	ADV
cana-3848	227	13	a	a	DET
cana-3848	227	14	cauchy	cauchy	ADJ
cana-3848	227	15	sequence	sequence	NOUN
cana-3848	227	16	in	in	ADP
cana-3848	227	17	the	the	DET
cana-3848	227	18	metric	metric	ADJ
cana-3848	227	19	space	space	NOUN
cana-3848	227	20	(	(	PUNCT
cana-3848	227	21	𝑌	𝑌	PROPN
cana-3848	227	22	,	,	PUNCT
cana-3848	227	23	(	(	PUNCT
cana-3848	227	24	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	227	25	)	)	PUNCT
cana-3848	227	26	𝑠	𝑠	PROPN
cana-3848	227	27	)	)	PUNCT
cana-3848	227	28	,	,	PUNCT
cana-3848	227	29	and	and	CCONJ
cana-3848	227	30	converges	converge	VERB
cana-3848	227	31	to	to	ADP
cana-3848	227	32	a	a	DET
cana-3848	227	33	point	point	NOUN
cana-3848	227	34	𝑥∗	𝑥∗	PROPN
cana-3848	227	35	∈	∈	PROPN
cana-3848	228	1	𝑌.	𝑌.	PROPN
cana-3848	228	2	define	define	VERB
cana-3848	228	3	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	228	4	∗	∗	NOUN
cana-3848	228	5	∶	∶	NOUN
cana-3848	228	6	𝑋	𝑋	PROPN
cana-3848	228	7	→	→	SYM
cana-3848	228	8	𝑌	𝑌	PROPN
cana-3848	228	9	for	for	ADP
cana-3848	228	10	each	each	DET
cana-3848	228	11	x	x	PUNCT
cana-3848	228	12	in	in	ADP
cana-3848	228	13	a	a	DET
cana-3848	228	14	,	,	PUNCT
cana-3848	228	15	where	where	SCONJ
cana-3848	228	16	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	228	17	∗(𝑥	∗(𝑥	NOUN
cana-3848	228	18	)	)	PUNCT
cana-3848	228	19	=	=	PUNCT
cana-3848	228	20	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	228	21	)	)	PUNCT
cana-3848	228	22	,	,	PUNCT
cana-3848	228	23	and	and	CCONJ
cana-3848	228	24	for	for	ADP
cana-3848	228	25	each	each	DET
cana-3848	228	26	𝑥	𝑥	ADP
cana-3848	228	27	𝑖𝑛	𝑖𝑛	NOUN
cana-3848	228	28	𝑋\𝐴	𝑋\𝐴	ADJ
cana-3848	228	29	,	,	PUNCT
cana-3848	228	30	where	where	SCONJ
cana-3848	228	31	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	228	32	∗(𝑥	∗(𝑥	NOUN
cana-3848	228	33	)	)	PUNCT
cana-3848	229	1	=	=	PUNCT
cana-3848	230	1	𝑥∗	𝑥∗	PROPN
cana-3848	230	2	*	*	PUNCT
cana-3848	230	3	.	.	PUNCT
cana-3848	231	1	it	it	PRON
cana-3848	231	2	should	should	AUX
cana-3848	231	3	be	be	AUX
cana-3848	231	4	noted	note	VERB
cana-3848	231	5	that	that	SCONJ
cana-3848	231	6	the	the	DET
cana-3848	231	7	definition	definition	NOUN
cana-3848	231	8	of	of	ADP
cana-3848	231	9	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	231	10	∗	∗	NOUN
cana-3848	231	11	is	be	AUX
cana-3848	231	12	not	not	PART
cana-3848	231	13	changed	change	VERB
cana-3848	231	14	by	by	ADP
cana-3848	231	15	the	the	DET
cana-3848	231	16	sequences	sequence	NOUN
cana-3848	231	17	(	(	PUNCT
cana-3848	231	18	𝑥𝑛)𝑛∈ℕin	𝑥𝑛)𝑛∈ℕin	NOUN
cana-3848	231	19	fact	fact	NOUN
cana-3848	231	20	,	,	PUNCT
cana-3848	231	21	if(𝑥𝑛)𝑛∈ℕ	if(𝑥𝑛)𝑛∈ℕ	NOUN
cana-3848	231	22	and	and	CCONJ
cana-3848	231	23	(	(	PUNCT
cana-3848	231	24	𝑦𝑛)𝑛∈ℕ	𝑦𝑛)𝑛∈ℕ	PROPN
cana-3848	231	25	)	)	PUNCT
cana-3848	231	26	are	be	AUX
cana-3848	231	27	sequences	sequence	NOUN
cana-3848	231	28	in	in	ADP
cana-3848	231	29	a	a	DET
cana-3848	231	30	that	that	DET
cana-3848	231	31	converge	converge	NOUN
cana-3848	231	32	to	to	ADP
cana-3848	231	33	a	a	DET
cana-3848	231	34	point	point	NOUN
cana-3848	232	1	𝑥	𝑥	INTJ
cana-3848	232	2	𝑖𝑛	𝑖𝑛	NOUN
cana-3848	232	3	𝑋\𝐴	𝑋\𝐴	ADJ
cana-3848	232	4	with	with	ADP
cana-3848	232	5	respect	respect	NOUN
cana-3848	232	6	to	to	ADP
cana-3848	232	7	the	the	DET
cana-3848	232	8	metric	metric	ADJ
cana-3848	232	9	(	(	PUNCT
cana-3848	232	10	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	232	11	)	)	PUNCT
cana-3848	232	12	𝑠	𝑠	PROPN
cana-3848	232	13	;	;	PUNCT
cana-3848	232	14	additionally	additionally	ADV
cana-3848	232	15	,	,	PUNCT
cana-3848	232	16	if	if	SCONJ
cana-3848	232	17	we	we	PRON
cana-3848	232	18	designate	designate	VERB
cana-3848	232	19	by	by	ADP
cana-3848	232	20	𝑥∗	𝑥∗	PROPN
cana-3848	232	21	and	and	CCONJ
cana-3848	232	22	𝑦∗the	𝑦∗the	DET
cana-3848	232	23	limit	limit	NOUN
cana-3848	232	24	points	point	VERB
cana-3848	232	25	in	in	ADP
cana-3848	232	26	communications	communication	NOUN
cana-3848	232	27	on	on	ADP
cana-3848	232	28	applied	apply	VERB
cana-3848	232	29	nonlinear	nonlinear	ADJ
cana-3848	232	30	analysis	analysis	NOUN
cana-3848	232	31	issn	issn	NOUN
cana-3848	232	32	:	:	PUNCT
cana-3848	232	33	1074	1074	NUM
cana-3848	232	34	-	-	PUNCT
cana-3848	232	35	133x	133x	NUM
cana-3848	232	36	vol	vol	NOUN
cana-3848	232	37	32	32	NUM
cana-3848	232	38	no	no	NOUN
cana-3848	232	39	.	.	PUNCT
cana-3848	233	1	9s	9s	NUM
cana-3848	233	2	(	(	PUNCT
cana-3848	233	3	2025	2025	NUM
cana-3848	233	4	)	)	PUNCT
cana-3848	233	5	207	207	NUM
cana-3848	233	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	233	7	(	(	PUNCT
cana-3848	233	8	𝑌	𝑌	PROPN
cana-3848	233	9	,	,	PUNCT
cana-3848	233	10	(	(	PUNCT
cana-3848	233	11	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	233	12	)	)	PUNCT
cana-3848	233	13	𝑠	𝑠	PROPN
cana-3848	233	14	)	)	PUNCT
cana-3848	233	15	of	of	ADP
cana-3848	233	16	lim	lim	PROPN
cana-3848	233	17	𝑛→∞	𝑛→∞	NUM
cana-3848	233	18	(	(	PUNCT
cana-3848	233	19	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	233	20	)	)	PUNCT
cana-3848	233	21	𝑠	𝑠	PROPN
cana-3848	233	22	𝑓𝑗(𝑥𝑛	𝑓𝑗(𝑥𝑛	PROPN
cana-3848	233	23	)	)	PUNCT
cana-3848	233	24	,	,	PUNCT
cana-3848	233	25	𝑓𝑗(𝑦𝑛	𝑓𝑗(𝑦𝑛	PROPN
cana-3848	233	26	)	)	PUNCT
cana-3848	234	1	=	=	PUNCT
cana-3848	234	2	0,since	0,since	NUM
cana-3848	234	3	lim	lim	NOUN
cana-3848	234	4	𝑛→∞	𝑛→∞	NUM
cana-3848	234	5	(	(	PUNCT
cana-3848	234	6	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	234	7	)	)	PUNCT
cana-3848	234	8	𝑠	𝑠	PROPN
cana-3848	234	9	(	(	PUNCT
cana-3848	234	10	𝑥𝑛	𝑥𝑛	PROPN
cana-3848	234	11	,	,	PUNCT
cana-3848	234	12	𝑦𝑛	𝑦𝑛	NOUN
cana-3848	234	13	)	)	PUNCT
cana-3848	234	14	=	=	SYM
cana-3848	235	1	0.therefore	0.therefore	NUM
cana-3848	235	2	,	,	PUNCT
cana-3848	235	3	𝑥∗	𝑥∗	PROPN
cana-3848	235	4	=	=	NOUN
cana-3848	235	5	𝑦∗.	𝑦∗.	NOUN
cana-3848	235	6	we	we	PRON
cana-3848	235	7	conclude	conclude	VERB
cana-3848	235	8	that	that	SCONJ
cana-3848	235	9	f	f	PROPN
cana-3848	235	10	extends	extend	VERB
cana-3848	235	11	uniquely	uniquely	ADV
cana-3848	235	12	to	to	PART
cana-3848	235	13	𝑓∗based	𝑓∗based	VERB
cana-3848	235	14	on	on	ADP
cana-3848	235	15	lemma	lemma	PROPN
cana-3848	235	16	(	(	PUNCT
cana-3848	235	17	3.12	3.12	NUM
cana-3848	235	18	)	)	PUNCT
cana-3848	235	19	of	of	ADP
cana-3848	235	20	[	[	X
cana-3848	235	21	7	7	NUM
cana-3848	235	22	]	]	PUNCT
cana-3848	235	23	,	,	PUNCT
cana-3848	235	24	where	where	SCONJ
cana-3848	235	25	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	235	26	∗	∗	NOUN
cana-3848	235	27	is	be	AUX
cana-3848	235	28	a	a	DET
cana-3848	235	29	one	one	NUM
cana-3848	235	30	-	-	PUNCT
cana-3848	235	31	toone	toone	NOUN
cana-3848	235	32	function	function	NOUN
cana-3848	235	33	such	such	ADJ
cana-3848	235	34	that	that	SCONJ
cana-3848	235	35	𝑞𝑗	𝑞𝑗	PROPN
cana-3848	235	36	(	(	PUNCT
cana-3848	235	37	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	235	38	∗(𝑥	∗(𝑥	NOUN
cana-3848	235	39	)	)	PUNCT
cana-3848	235	40	)	)	PUNCT
cana-3848	236	1	=	=	SYM
cana-3848	236	2	𝑝𝑗(𝑥	𝑝𝑗(𝑥	NOUN
cana-3848	236	3	)	)	PUNCT
cana-3848	236	4	,	,	PUNCT
cana-3848	236	5	(	(	PUNCT
cana-3848	236	6	𝑋	𝑋	NOUN
cana-3848	236	7	,	,	PUNCT
cana-3848	236	8	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	236	9	)	)	PUNCT
cana-3848	236	10	and	and	CCONJ
cana-3848	236	11	(	(	PUNCT
cana-3848	236	12	𝑌	𝑌	PROPN
cana-3848	236	13	,	,	PUNCT
cana-3848	236	14	𝑞𝑗	𝑞𝑗	VERB
cana-3848	236	15	)	)	PUNCT
cana-3848	236	16	are	be	AUX
cana-3848	236	17	quasi	quasi	ADJ
cana-3848	236	18	-	-	ADJ
cana-3848	236	19	normed	normed	ADJ
cana-3848	236	20	cones	cone	NOUN
cana-3848	236	21	.	.	PUNCT
cana-3848	237	1	we	we	PRON
cana-3848	237	2	simply	simply	ADV
cana-3848	237	3	show	show	VERB
cana-3848	237	4	that	that	SCONJ
cana-3848	237	5	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	237	6	∗	∗	NOUN
cana-3848	237	7	is	be	AUX
cana-3848	237	8	linear	linear	ADJ
cana-3848	237	9	on	on	ADP
cana-3848	237	10	𝑋	𝑋	NOUN
cana-3848	237	11	after	after	ADP
cana-3848	237	12	that	that	PRON
cana-3848	237	13	.	.	PUNCT
cana-3848	238	1	consider	consider	VERB
cana-3848	238	2	x	x	PRON
cana-3848	238	3	and	and	CCONJ
cana-3848	238	4	y	y	PROPN
cana-3848	238	5	to	to	PART
cana-3848	238	6	be	be	AUX
cana-3848	238	7	in	in	ADP
cana-3848	238	8	𝑋.	𝑋.	PROPN
cana-3848	238	9	only	only	ADV
cana-3848	238	10	the	the	DET
cana-3848	238	11	case	case	NOUN
cana-3848	238	12	where	where	SCONJ
cana-3848	238	13	𝑥	𝑥	NOUN
cana-3848	238	14	,	,	PUNCT
cana-3848	238	15	𝑦	𝑦	NOUN
cana-3848	238	16	∈	∈	NOUN
cana-3848	238	17	𝑋\𝐴	𝑋\𝐴	NOUN
cana-3848	238	18	,	,	PUNCT
cana-3848	238	19	which	which	PRON
cana-3848	238	20	is	be	AUX
cana-3848	238	21	linear	linear	ADJ
cana-3848	238	22	on	on	ADP
cana-3848	238	23	a	a	PRON
cana-3848	238	24	,	,	PUNCT
cana-3848	238	25	is	be	AUX
cana-3848	238	26	considered	consider	VERB
cana-3848	238	27	.	.	PUNCT
cana-3848	239	1	assume	assume	VERB
cana-3848	239	2	that	that	SCONJ
cana-3848	239	3	the	the	DET
cana-3848	239	4	sequences	sequence	NOUN
cana-3848	239	5	(	(	PUNCT
cana-3848	239	6	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	239	7	and	and	CCONJ
cana-3848	239	8	(	(	PUNCT
cana-3848	239	9	𝑦𝑛)𝑛∈ℕ	𝑦𝑛)𝑛∈ℕ	NOUN
cana-3848	239	10	be	be	VERB
cana-3848	239	11	in	in	ADP
cana-3848	239	12	a	a	DET
cana-3848	239	13	converge	converge	NOUN
cana-3848	239	14	to	to	ADP
cana-3848	239	15	x	x	PROPN
cana-3848	239	16	and	and	CCONJ
cana-3848	239	17	y	y	PROPN
cana-3848	239	18	,	,	PUNCT
cana-3848	239	19	respectively	respectively	ADV
cana-3848	239	20	,	,	PUNCT
cana-3848	239	21	in	in	ADP
cana-3848	239	22	the	the	DET
cana-3848	239	23	metric	metric	ADJ
cana-3848	239	24	space	space	NOUN
cana-3848	239	25	(	(	PUNCT
cana-3848	239	26	𝑋	𝑋	NOUN
cana-3848	239	27	,	,	PUNCT
cana-3848	239	28	(	(	PUNCT
cana-3848	239	29	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	239	30	)	)	PUNCT
cana-3848	239	31	𝑠	𝑠	PROPN
cana-3848	239	32	)	)	PUNCT
cana-3848	239	33	.	.	PUNCT
cana-3848	240	1	subsequently	subsequently	ADV
cana-3848	240	2	,	,	PUNCT
cana-3848	240	3	with	with	ADP
cana-3848	240	4	respect	respect	NOUN
cana-3848	240	5	to	to	ADP
cana-3848	240	6	(	(	PUNCT
cana-3848	240	7	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	240	8	)	)	PUNCT
cana-3848	240	9	𝑠	𝑠	PROPN
cana-3848	240	10	,	,	PUNCT
cana-3848	240	11	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	240	12	∗	∗	NOUN
cana-3848	240	13	converges	converge	NOUN
cana-3848	240	14	to	to	ADP
cana-3848	240	15	(	(	PUNCT
cana-3848	240	16	𝑓𝑗(𝑎𝑥𝑛	𝑓𝑗(𝑎𝑥𝑛	PROPN
cana-3848	240	17	+	+	NUM
cana-3848	240	18	𝑏𝑦𝑛	𝑏𝑦𝑛	NOUN
cana-3848	240	19	)	)	PUNCT
cana-3848	240	20	)	)	PUNCT
cana-3848	240	21	𝑛∈ℕ	𝑛∈ℕ	PROPN
cana-3848	240	22	.	.	PUNCT
cana-3848	241	1	that	that	PRON
cana-3848	241	2	's	be	AUX
cana-3848	241	3	because	because	SCONJ
cana-3848	241	4	,	,	PUNCT
cana-3848	241	5	with	with	ADP
cana-3848	241	6	considering	consider	VERB
cana-3848	241	7	(	(	PUNCT
cana-3848	241	8	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	241	9	)	)	PUNCT
cana-3848	241	10	𝑠	𝑠	PROPN
cana-3848	241	11	,	,	PUNCT
cana-3848	241	12	(	(	PUNCT
cana-3848	241	13	𝑎𝑥𝑛	𝑎𝑥𝑛	VERB
cana-3848	241	14	+	+	CCONJ
cana-3848	241	15	𝑏𝑦𝑛)𝑛∈𝑁	𝑏𝑦𝑛)𝑛∈𝑁	PROPN
cana-3848	241	16	converges	converge	VERB
cana-3848	241	17	to	to	ADP
cana-3848	241	18	𝑎𝑥	𝑎𝑥	NOUN
cana-3848	241	19	+	+	CCONJ
cana-3848	241	20	𝑏𝑦.	𝑏𝑦.	VERB
cana-3848	241	21	with	with	ADP
cana-3848	241	22	respect	respect	NOUN
cana-3848	241	23	to	to	ADP
cana-3848	241	24	(	(	PUNCT
cana-3848	241	25	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	241	26	)	)	PUNCT
cana-3848	241	27	𝑠	𝑠	PROPN
cana-3848	241	28	,	,	PUNCT
cana-3848	241	29	the	the	DET
cana-3848	241	30	sequence	sequence	NOUN
cana-3848	241	31	(	(	PUNCT
cana-3848	241	32	𝑎𝑓𝑗(𝑥𝑛	𝑎𝑓𝑗(𝑥𝑛	ADJ
cana-3848	241	33	)	)	PUNCT
cana-3848	241	34	+	+	NUM
cana-3848	241	35	𝑏𝑓𝑗(𝑦𝑛	𝑏𝑓𝑗(𝑦𝑛	NOUN
cana-3848	241	36	)	)	PUNCT
cana-3848	241	37	)	)	PUNCT
cana-3848	241	38	𝑛∈ℕ	𝑛∈ℕ	PROPN
cana-3848	241	39	converges	converge	VERB
cana-3848	241	40	to	to	ADP
cana-3848	241	41	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	241	42	∗(𝑎𝑥	∗(𝑎𝑥	NOUN
cana-3848	241	43	+	+	CCONJ
cana-3848	241	44	𝑏𝑦)since	𝑏𝑦)since	NOUN
cana-3848	241	45	f	f	PROPN
cana-3848	241	46	is	be	AUX
cana-3848	241	47	linear	linear	ADJ
cana-3848	241	48	on	on	ADP
cana-3848	241	49	a.on	a.on	PROPN
cana-3848	241	50	the	the	DET
cana-3848	241	51	other	other	ADJ
cana-3848	241	52	hand	hand	NOUN
cana-3848	241	53	,	,	PUNCT
cana-3848	241	54	(	(	PUNCT
cana-3848	241	55	𝑓𝑗(𝑥𝑛	𝑓𝑗(𝑥𝑛	NOUN
cana-3848	241	56	)	)	PUNCT
cana-3848	241	57	)	)	PUNCT
cana-3848	242	1	𝑛∈ℕ	𝑛∈ℕ	PROPN
cana-3848	242	2	converges	converge	VERB
cana-3848	242	3	to	to	ADP
cana-3848	242	4	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	242	5	∗(𝑥	∗(𝑥	NOUN
cana-3848	242	6	)	)	PUNCT
cana-3848	242	7	and	and	CCONJ
cana-3848	242	8	(	(	PUNCT
cana-3848	242	9	𝑓𝑗(𝑦𝑛	𝑓𝑗(𝑦𝑛	PROPN
cana-3848	242	10	)	)	PUNCT
cana-3848	242	11	)	)	PUNCT
cana-3848	242	12	𝑛∈ℕ	𝑛∈ℕ	PROPN
cana-3848	242	13	converges	converge	VERB
cana-3848	242	14	to	to	ADP
cana-3848	242	15	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	242	16	∗(𝑦	∗(𝑦	NOUN
cana-3848	242	17	)	)	PUNCT
cana-3848	242	18	with	with	ADP
cana-3848	242	19	respect	respect	NOUN
cana-3848	242	20	to	to	ADP
cana-3848	242	21	(	(	PUNCT
cana-3848	242	22	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	242	23	)	)	PUNCT
cana-3848	242	24	𝑠	𝑠	PROPN
cana-3848	242	25	,	,	PUNCT
cana-3848	242	26	as	as	ADP
cana-3848	242	27	per	per	ADP
cana-3848	242	28	the	the	DET
cana-3848	242	29	definition	definition	NOUN
cana-3848	242	30	of	of	ADP
cana-3848	242	31	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	242	32	∗.	∗.	PROPN
cana-3848	242	33	for	for	ADP
cana-3848	242	34	the	the	DET
cana-3848	242	35	metric	metric	ADJ
cana-3848	242	36	(	(	PUNCT
cana-3848	242	37	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	242	38	)	)	PUNCT
cana-3848	242	39	𝑠	𝑠	PROPN
cana-3848	242	40	,	,	PUNCT
cana-3848	242	41	then((𝑎𝑓𝑗(𝑥𝑛	then((𝑎𝑓𝑗(𝑥𝑛	PROPN
cana-3848	242	42	)	)	PUNCT
cana-3848	243	1	+	+	NUM
cana-3848	243	2	𝑏𝑓𝑗(𝑦𝑛	𝑏𝑓𝑗(𝑦𝑛	NOUN
cana-3848	243	3	)	)	PUNCT
cana-3848	243	4	)	)	PUNCT
cana-3848	243	5	)	)	PUNCT
cana-3848	244	1	𝑛∈ℕ	𝑛∈ℕ	PROPN
cana-3848	244	2	converges	converge	VERB
cana-3848	244	3	to	to	ADP
cana-3848	244	4	a	a	DET
cana-3848	244	5	𝑎𝑓𝑗	𝑎𝑓𝑗	NOUN
cana-3848	244	6	∗(𝑥	∗(𝑥	NOUN
cana-3848	244	7	)	)	PUNCT
cana-3848	245	1	+	+	CCONJ
cana-3848	245	2	𝑏𝑓𝑗	𝑏𝑓𝑗	PROPN
cana-3848	245	3	∗(𝑦).consequently	∗(𝑦).consequently	ADV
cana-3848	245	4	,	,	PUNCT
cana-3848	245	5	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	245	6	∗(𝑎𝑥	∗(𝑎𝑥	NOUN
cana-3848	245	7	+	+	CCONJ
cana-3848	245	8	𝑏𝑦	𝑏𝑦	NOUN
cana-3848	245	9	)	)	PUNCT
cana-3848	245	10	=	=	PUNCT
cana-3848	245	11	𝑎𝑓𝑗	𝑎𝑓𝑗	NOUN
cana-3848	245	12	∗(𝑥	∗(𝑥	NOUN
cana-3848	245	13	)	)	PUNCT
cana-3848	246	1	+	+	CCONJ
cana-3848	246	2	𝑏𝑓𝑗	𝑏𝑓𝑗	NOUN
cana-3848	246	3	∗(𝑦	∗(𝑦	NOUN
cana-3848	246	4	)	)	PUNCT
cana-3848	246	5	.	.	PUNCT
cana-3848	247	1	lemma	lemma	PROPN
cana-3848	247	2	(	(	PUNCT
cana-3848	247	3	3.2.12	3.2.12	NUM
cana-3848	247	4	):	):	PUNCT
cana-3848	247	5	a	a	PRON
cana-3848	247	6	quasi	quasi	ADJ
cana-3848	247	7	-	-	ADJ
cana-3848	247	8	normed	normed	ADJ
cana-3848	247	9	cone	cone	NOUN
cana-3848	247	10	(	(	PUNCT
cana-3848	247	11	𝑋	𝑋	PROPN
cana-3848	247	12	,	,	PUNCT
cana-3848	247	13	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	247	14	)	)	PUNCT
cana-3848	247	15	has	have	VERB
cana-3848	247	16	bicompletions	bicompletion	NOUN
cana-3848	247	17	of	of	ADP
cana-3848	247	18	all	all	DET
cana-3848	247	19	kinds	kind	NOUN
cana-3848	247	20	that	that	PRON
cana-3848	247	21	are	be	AUX
cana-3848	247	22	isometric	isometric	ADJ
cana-3848	247	23	to	to	ADP
cana-3848	247	24	(	(	PUNCT
cana-3848	247	25	�	�	PROPN
cana-3848	247	26	̃	̃	NOUN
cana-3848	247	27	�	�	PROPN
cana-3848	247	28	,	,	PUNCT
cana-3848	247	29	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	247	30	)	)	PUNCT
cana-3848	247	31	.	.	PUNCT
cana-3848	248	1	proof	proof	NOUN
cana-3848	248	2	:	:	PUNCT
cana-3848	248	3	a	a	DET
cana-3848	248	4	bicompletion	bicompletion	NOUN
cana-3848	248	5	(	(	PUNCT
cana-3848	248	6	𝑌	𝑌	PROPN
cana-3848	248	7	,	,	PUNCT
cana-3848	248	8	𝑞𝑗	𝑞𝑗	VERB
cana-3848	248	9	)	)	PUNCT
cana-3848	248	10	is	be	AUX
cana-3848	248	11	assumed	assume	VERB
cana-3848	248	12	to	to	PART
cana-3848	248	13	exist	exist	VERB
cana-3848	248	14	for	for	ADP
cana-3848	248	15	(	(	PUNCT
cana-3848	248	16	𝑋	𝑋	PROPN
cana-3848	248	17	,	,	PUNCT
cana-3848	248	18	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	248	19	)	)	PUNCT
cana-3848	248	20	.	.	PUNCT
cana-3848	249	1	suppose	suppose	VERB
cana-3848	249	2	that	that	SCONJ
cana-3848	249	3	i	i	PRON
cana-3848	249	4	is	be	AUX
cana-3848	249	5	the	the	DET
cana-3848	249	6	one	one	NUM
cana-3848	249	7	-	-	PUNCT
cana-3848	249	8	to	to	ADP
cana-3848	249	9	-	-	PUNCT
cana-3848	249	10	one	one	NUM
cana-3848	249	11	isometry	isometry	NOUN
cana-3848	249	12	from	from	ADP
cana-3848	249	13	(	(	PUNCT
cana-3848	249	14	𝑋	𝑋	PROPN
cana-3848	249	15	,	,	PUNCT
cana-3848	249	16	𝑝	𝑝	NOUN
cana-3848	249	17	)	)	PUNCT
cana-3848	249	18	to	to	ADP
cana-3848	249	19	(	(	PUNCT
cana-3848	249	20	�	�	PROPN
cana-3848	249	21	̃	̃	NOUN
cana-3848	249	22	�	�	PROPN
cana-3848	249	23	,	,	PUNCT
cana-3848	249	24	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	249	25	)	)	PUNCT
cana-3848	249	26	is	be	AUX
cana-3848	249	27	defined	define	VERB
cana-3848	249	28	in	in	ADP
cana-3848	249	29	lemma	lemma	PROPN
cana-3848	249	30	(	(	PUNCT
cana-3848	249	31	3.2.14	3.2.14	NUM
cana-3848	249	32	)	)	PUNCT
cana-3848	249	33	.	.	PUNCT
cana-3848	250	1	furthermore	furthermore	ADV
cana-3848	250	2	,	,	PUNCT
cana-3848	250	3	because	because	SCONJ
cana-3848	250	4	x	x	PRON
cana-3848	250	5	is	be	AUX
cana-3848	250	6	dense	dense	ADJ
cana-3848	250	7	in	in	ADP
cana-3848	250	8	the	the	DET
cana-3848	250	9	metric	metric	ADJ
cana-3848	250	10	space	space	NOUN
cana-3848	250	11	(	(	PUNCT
cana-3848	250	12	𝑌	𝑌	PROPN
cana-3848	250	13	,	,	PUNCT
cana-3848	250	14	(	(	PUNCT
cana-3848	250	15	𝑒q𝑗	𝑒q𝑗	X
cana-3848	250	16	)	)	PUNCT
cana-3848	250	17	𝑠	𝑠	PROPN
cana-3848	250	18	)	)	PUNCT
cana-3848	250	19	,	,	PUNCT
cana-3848	250	20	the	the	DET
cana-3848	250	21	prior	prior	ADJ
cana-3848	250	22	lemma	lemma	PROPN
cana-3848	250	23	leads	lead	VERB
cana-3848	250	24	to	to	ADP
cana-3848	250	25	the	the	DET
cana-3848	250	26	conclusion	conclusion	NOUN
cana-3848	250	27	that	that	SCONJ
cana-3848	250	28	f	f	PROPN
cana-3848	250	29	has	have	AUX
cana-3848	250	30	a	a	DET
cana-3848	250	31	unique	unique	ADJ
cana-3848	250	32	one	one	NUM
cana-3848	250	33	-	-	PUNCT
cana-3848	250	34	to	to	ADP
cana-3848	250	35	-	-	PUNCT
cana-3848	250	36	one	one	NUM
cana-3848	250	37	isometry	isometry	NOUN
cana-3848	250	38	extension	extension	NOUN
cana-3848	250	39	𝑓∗	𝑓∗	NOUN
cana-3848	250	40	to	to	ADP
cana-3848	250	41	(	(	PUNCT
cana-3848	250	42	𝑌	𝑌	PROPN
cana-3848	250	43	,	,	PUNCT
cana-3848	250	44	𝑞𝑗	𝑞𝑗	PROPN
cana-3848	250	45	)	)	PUNCT
cana-3848	250	46	.	.	PUNCT
cana-3848	251	1	it	it	PRON
cana-3848	251	2	remains	remain	VERB
cana-3848	251	3	to	to	PART
cana-3848	251	4	be	be	AUX
cana-3848	251	5	shown	show	VERB
cana-3848	251	6	that	that	SCONJ
cana-3848	251	7	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	251	8	∗	∗	NOUN
cana-3848	251	9	∶	∶	NOUN
cana-3848	251	10	𝑌	𝑌	PROPN
cana-3848	251	11	→	→	SYM
cana-3848	251	12	�	�	PROPN
cana-3848	251	13	̃	̃	PROPN
cana-3848	251	14	�	�	PROPN
cana-3848	251	15	is	be	AUX
cana-3848	251	16	an	an	PRON
cana-3848	251	17	onto	onto	ADP
cana-3848	251	18	mapping	mapping	NOUN
cana-3848	251	19	.	.	PUNCT
cana-3848	252	1	let	let	VERB
cana-3848	252	2	x	x	PRON
cana-3848	252	3	actually	actually	ADV
cana-3848	252	4	stand	stand	VERB
cana-3848	252	5	for	for	ADP
cana-3848	252	6	any	any	DET
cana-3848	252	7	random	random	ADJ
cana-3848	252	8	�	�	PROPN
cana-3848	252	9	̃	̃	PROPN
cana-3848	252	10	�	�	PROPN
cana-3848	252	11	point	point	NOUN
cana-3848	252	12	.	.	PUNCT
cana-3848	253	1	for	for	ADP
cana-3848	253	2	a	a	DET
cana-3848	253	3	sequence	sequence	NOUN
cana-3848	253	4	(	(	PUNCT
cana-3848	253	5	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	253	6	in	in	ADP
cana-3848	253	7	𝑋	𝑋	PROPN
cana-3848	253	8	,	,	PUNCT
cana-3848	253	9	lim	lim	PROPN
cana-3848	253	10	𝑛→∞	𝑛→∞	NUM
cana-3848	253	11	𝑒	𝑒	PROPN
cana-3848	253	12	�	�	PROPN
cana-3848	253	13	̃	̃	NOUN
cana-3848	253	14	�	�	NOUN
cana-3848	253	15	𝑗	𝑗	PROPN
cana-3848	253	16	𝑠	𝑠	PROPN
cana-3848	253	17	(	(	PUNCT
cana-3848	253	18	𝑥	𝑥	NOUN
cana-3848	253	19	,	,	PUNCT
cana-3848	253	20	𝑓𝑗(𝑥𝑛	𝑓𝑗(𝑥𝑛	NOUN
cana-3848	253	21	)	)	PUNCT
cana-3848	253	22	)	)	PUNCT
cana-3848	253	23	=	=	SYM
cana-3848	253	24	0	0	NUM
cana-3848	253	25	,	,	PUNCT
cana-3848	253	26	given	give	VERB
cana-3848	253	27	that	that	DET
cana-3848	253	28	𝑓𝑗(𝑋	𝑓𝑗(𝑋	NOUN
cana-3848	253	29	)	)	PUNCT
cana-3848	253	30	is	be	AUX
cana-3848	253	31	dense	dense	ADJ
cana-3848	253	32	in	in	ADP
cana-3848	253	33	(	(	PUNCT
cana-3848	253	34	�	�	PROPN
cana-3848	253	35	̃	̃	PROPN
cana-3848	253	36	�	�	PROPN
cana-3848	253	37	,	,	PUNCT
cana-3848	253	38	(	(	PUNCT
cana-3848	253	39	𝑒	𝑒	PROPN
cana-3848	253	40	�	�	PROPN
cana-3848	253	41	̃	̃	NOUN
cana-3848	253	42	�	�	NOUN
cana-3848	253	43	𝑗	𝑗	NOUN
cana-3848	253	44	)	)	PUNCT
cana-3848	253	45	𝑠	𝑠	PROPN
cana-3848	253	46	)	)	PUNCT
cana-3848	253	47	.	.	PUNCT
cana-3848	254	1	a	a	DET
cana-3848	254	2	cauchy	cauchy	ADJ
cana-3848	254	3	sequence	sequence	NOUN
cana-3848	254	4	is	be	AUX
cana-3848	254	5	thus	thus	ADV
cana-3848	254	6	found	find	VERB
cana-3848	254	7	in	in	ADP
cana-3848	254	8	(	(	PUNCT
cana-3848	254	9	�	�	PROPN
cana-3848	254	10	̃	̃	PROPN
cana-3848	254	11	�	�	PROPN
cana-3848	254	12	,	,	PUNCT
cana-3848	254	13	(	(	PUNCT
cana-3848	254	14	𝑒	𝑒	PROPN
cana-3848	254	15	�	�	PROPN
cana-3848	254	16	̃	̃	NOUN
cana-3848	254	17	�	�	NOUN
cana-3848	254	18	𝑗	𝑗	NOUN
cana-3848	254	19	)	)	PUNCT
cana-3848	254	20	𝑠	𝑠	PROPN
cana-3848	254	21	)	)	PUNCT
cana-3848	254	22	,	,	PUNCT
cana-3848	254	23	(	(	PUNCT
cana-3848	254	24	𝑓𝑗(𝑥𝑛	𝑓𝑗(𝑥𝑛	NOUN
cana-3848	254	25	)	)	PUNCT
cana-3848	254	26	)	)	PUNCT
cana-3848	254	27	𝑛∈ℕ	𝑛∈ℕ	PROPN
cana-3848	254	28	.	.	PUNCT
cana-3848	255	1	in	in	ADP
cana-3848	255	2	(	(	PUNCT
cana-3848	255	3	𝑌	𝑌	PROPN
cana-3848	255	4	,	,	PUNCT
cana-3848	255	5	(	(	PUNCT
cana-3848	255	6	𝑒q𝑗	𝑒q𝑗	X
cana-3848	255	7	)	)	PUNCT
cana-3848	255	8	𝑠	𝑠	PROPN
cana-3848	255	9	)	)	PUNCT
cana-3848	255	10	,	,	PUNCT
cana-3848	255	11	(	(	PUNCT
cana-3848	255	12	𝑥𝑛)𝑛∈ℕ	𝑥𝑛)𝑛∈ℕ	PROPN
cana-3848	255	13	is	be	AUX
cana-3848	255	14	a	a	DET
cana-3848	255	15	cauchy	cauchy	ADJ
cana-3848	255	16	sequence	sequence	NOUN
cana-3848	255	17	since	since	SCONJ
cana-3848	255	18	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	255	19	∗	∗	NOUN
cana-3848	255	20	is	be	AUX
cana-3848	255	21	an	an	DET
cana-3848	255	22	isometry	isometry	NOUN
cana-3848	255	23	.	.	PUNCT
cana-3848	256	1	for	for	ADP
cana-3848	256	2	every	every	DET
cana-3848	256	3	𝑦	𝑦	NOUN
cana-3848	256	4	𝑖𝑛	𝑖𝑛	PRON
cana-3848	256	5	𝑌,if	𝑌,if	X
cana-3848	256	6	lim	lim	NOUN
cana-3848	256	7	𝑛→∞	𝑛→∞	NUM
cana-3848	256	8	(	(	PUNCT
cana-3848	256	9	𝑒q𝑗	𝑒q𝑗	X
cana-3848	256	10	)	)	PUNCT
cana-3848	256	11	𝑠	𝑠	PROPN
cana-3848	256	12	(	(	PUNCT
cana-3848	256	13	𝑦	𝑦	NOUN
cana-3848	256	14	,	,	PUNCT
cana-3848	256	15	𝑥𝑛	𝑥𝑛	NOUN
cana-3848	256	16	)	)	PUNCT
cana-3848	256	17	=	=	SYM
cana-3848	256	18	0	0	PROPN
cana-3848	256	19	,	,	PUNCT
cana-3848	256	20	lim	lim	PROPN
cana-3848	256	21	𝑛→∞	𝑛→∞	NUM
cana-3848	256	22	(	(	PUNCT
cana-3848	256	23	𝑒	𝑒	PROPN
cana-3848	256	24	�	�	PROPN
cana-3848	256	25	̃	̃	NOUN
cana-3848	256	26	�	�	NOUN
cana-3848	256	27	𝑗	𝑗	NOUN
cana-3848	256	28	)	)	PUNCT
cana-3848	256	29	𝑠	𝑠	PROPN
cana-3848	256	30	(	(	PUNCT
cana-3848	256	31	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	256	32	∗(𝑦	∗(𝑦	NOUN
cana-3848	256	33	)	)	PUNCT
cana-3848	256	34	,	,	PUNCT
cana-3848	256	35	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	256	36	∗(𝑥𝑛	∗(𝑥𝑛	NUM
cana-3848	256	37	)	)	PUNCT
cana-3848	256	38	)	)	PUNCT
cana-3848	256	39	=	=	PUNCT
cana-3848	257	1	0	0	X
cana-3848	257	2	.	.	X
cana-3848	258	1	for	for	ADP
cana-3848	258	2	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	258	3	∗(𝑦	∗(𝑦	NOUN
cana-3848	258	4	)	)	PUNCT
cana-3848	259	1	=	=	PUNCT
cana-3848	259	2	𝑥.	𝑥.	NOUN
cana-3848	259	3	this	this	PRON
cana-3848	259	4	completes	complete	VERB
cana-3848	259	5	the	the	DET
cana-3848	259	6	proof	proof	NOUN
cana-3848	259	7	.	.	PUNCT
cana-3848	260	1	from	from	ADP
cana-3848	260	2	the	the	DET
cana-3848	260	3	above	above	ADJ
cana-3848	260	4	stated	state	VERB
cana-3848	260	5	lemmas	lemmas	PROPN
cana-3848	260	6	,	,	PUNCT
cana-3848	260	7	the	the	DET
cana-3848	260	8	following	follow	VERB
cana-3848	260	9	can	can	AUX
cana-3848	260	10	be	be	AUX
cana-3848	260	11	deduced	deduce	VERB
cana-3848	260	12	immediately	immediately	ADV
cana-3848	260	13	.	.	PUNCT
cana-3848	261	1	theorem	theorem	NOUN
cana-3848	261	2	(	(	PUNCT
cana-3848	261	3	3.13	3.13	NUM
cana-3848	261	4	):	):	PUNCT
cana-3848	261	5	for	for	ADP
cana-3848	261	6	any	any	DET
cana-3848	261	7	quasi	quasi	ADJ
cana-3848	261	8	-	-	ADJ
cana-3848	261	9	normed	normed	ADJ
cana-3848	261	10	cone	cone	NOUN
cana-3848	261	11	(	(	PUNCT
cana-3848	261	12	𝑋	𝑋	PROPN
cana-3848	261	13	,	,	PUNCT
cana-3848	261	14	𝑝𝑗	𝑝𝑗	PROPN
cana-3848	261	15	)	)	PUNCT
cana-3848	261	16	,	,	PUNCT
cana-3848	261	17	there	there	PRON
cana-3848	261	18	is	be	VERB
cana-3848	261	19	only	only	ADV
cana-3848	261	20	one	one	NUM
cana-3848	261	21	bicompletion	bicompletion	NOUN
cana-3848	261	22	(	(	PUNCT
cana-3848	261	23	up	up	ADP
cana-3848	261	24	to	to	PART
cana-3848	261	25	bijective	bijective	ADJ
cana-3848	261	26	isometry	isometry	NOUN
cana-3848	261	27	)	)	PUNCT
cana-3848	261	28	.	.	PUNCT
cana-3848	262	1	communications	communication	NOUN
cana-3848	262	2	on	on	ADP
cana-3848	262	3	applied	apply	VERB
cana-3848	262	4	nonlinear	nonlinear	ADJ
cana-3848	262	5	analysis	analysis	NOUN
cana-3848	262	6	issn	issn	NOUN
cana-3848	262	7	:	:	PUNCT
cana-3848	262	8	1074	1074	NUM
cana-3848	262	9	-	-	PUNCT
cana-3848	262	10	133x	133x	NUM
cana-3848	262	11	vol	vol	NOUN
cana-3848	262	12	32	32	NUM
cana-3848	262	13	no	no	NOUN
cana-3848	262	14	.	.	PUNCT
cana-3848	263	1	9s	9s	NUM
cana-3848	263	2	(	(	PUNCT
cana-3848	263	3	2025	2025	NUM
cana-3848	263	4	)	)	PUNCT
cana-3848	264	1	208	208	NUM
cana-3848	264	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3848	264	3	corollary	corollary	NOUN
cana-3848	264	4	(	(	PUNCT
cana-3848	264	5	3.2.19	3.2.19	NUM
cana-3848	264	6	):	):	PUNCT
cana-3848	264	7	(	(	PUNCT
cana-3848	264	8	𝑋	𝑋	PROPN
cana-3848	264	9	,	,	PUNCT
cana-3848	264	10	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	264	11	)	)	PUNCT
cana-3848	264	12	and	and	CCONJ
cana-3848	264	13	(	(	PUNCT
cana-3848	264	14	𝑌	𝑌	PROPN
cana-3848	264	15	,	,	PUNCT
cana-3848	264	16	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	264	17	)	)	PUNCT
cana-3848	264	18	quasi	quasi	ADJ
cana-3848	264	19	-	-	ADJ
cana-3848	264	20	metric	metric	ADJ
cana-3848	264	21	spaces	space	NOUN
cana-3848	264	22	that	that	PRON
cana-3848	264	23	are	be	AUX
cana-3848	264	24	isometric	isometric	ADJ
cana-3848	264	25	by	by	ADP
cana-3848	264	26	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	264	27	if	if	SCONJ
cana-3848	264	28	(	(	PUNCT
cana-3848	264	29	𝑋	𝑋	PROPN
cana-3848	264	30	,	,	PUNCT
cana-3848	264	31	𝑝𝑗	𝑝𝑗	NOUN
cana-3848	264	32	)	)	PUNCT
cana-3848	264	33	and	and	CCONJ
cana-3848	264	34	(	(	PUNCT
cana-3848	264	35	𝑌	𝑌	PROPN
cana-3848	264	36	,	,	PUNCT
cana-3848	264	37	𝑞𝑗	𝑞𝑗	VERB
cana-3848	264	38	)	)	PUNCT
cana-3848	264	39	are	be	AUX
cana-3848	264	40	isometric	isometric	ADJ
cana-3848	264	41	quasi	quasi	ADJ
cana-3848	264	42	-	-	ADJ
cana-3848	264	43	normed	normed	ADJ
cana-3848	264	44	cones	cone	NOUN
cana-3848	264	45	by	by	ADP
cana-3848	264	46	a	a	DET
cana-3848	264	47	(	(	PUNCT
cana-3848	264	48	bijective	bijective	ADJ
cana-3848	264	49	)	)	PUNCT
cana-3848	264	50	isometry	isometry	NOUN
cana-3848	264	51	𝑓𝑗.	𝑓𝑗.	CCONJ
cana-3848	264	52	proof	proof	NOUN
cana-3848	264	53	:	:	PUNCT
cana-3848	264	54	suppose	suppose	VERB
cana-3848	264	55	that	that	SCONJ
cana-3848	264	56	𝑥	𝑥	NOUN
cana-3848	264	57	,	,	PUNCT
cana-3848	264	58	𝑥	𝑥	PROPN
cana-3848	265	1	+	+	NUM
cana-3848	265	2	2(𝜇	2(𝜇	NUM
cana-3848	265	3	+	+	CCONJ
cana-3848	265	4	1	1	NUM
cana-3848	265	5	)	)	PUNCT
cana-3848	265	6	∈	∈	PROPN
cana-3848	265	7	𝑋.	𝑋.	PROPN
cana-3848	265	8	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	265	9	+	+	X
cana-3848	265	10	2(𝜇	2(𝜇	NUM
cana-3848	265	11	+	+	CCONJ
cana-3848	265	12	1	1	NUM
cana-3848	265	13	)	)	PUNCT
cana-3848	265	14	)	)	PUNCT
cana-3848	266	1	=	=	SYM
cana-3848	266	2	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	266	3	(	(	PUNCT
cana-3848	266	4	𝑥	𝑥	NOUN
cana-3848	266	5	,	,	PUNCT
cana-3848	266	6	𝑥	𝑥	PROPN
cana-3848	267	1	+	+	NUM
cana-3848	267	2	2(𝜇	2(𝜇	NUM
cana-3848	267	3	+	+	CCONJ
cana-3848	267	4	1	1	NUM
cana-3848	267	5	)	)	PUNCT
cana-3848	267	6	)	)	PUNCT
cana-3848	268	1	=	=	PUNCT
cana-3848	269	1	+	+	NUM
cana-3848	269	2	∞.	∞.	PROPN
cana-3848	269	3	if	if	SCONJ
cana-3848	269	4	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	269	5	+	+	X
cana-3848	269	6	2(𝜇	2(𝜇	NUM
cana-3848	269	7	+	+	CCONJ
cana-3848	269	8	1	1	NUM
cana-3848	269	9	)	)	PUNCT
cana-3848	269	10	)	)	PUNCT
cana-3848	269	11	∈	∈	PROPN
cana-3848	269	12	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	269	13	)	)	PUNCT
cana-3848	270	1	+	+	CCONJ
cana-3848	270	2	𝑌.	𝑌.	PROPN
cana-3848	270	3	for	for	ADP
cana-3848	270	4	𝑧	𝑧	DET
cana-3848	270	5	∈	∈	PROPN
cana-3848	270	6	𝑌	𝑌	PROPN
cana-3848	270	7	such	such	ADJ
cana-3848	270	8	that	that	SCONJ
cana-3848	270	9	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	270	10	+	+	CCONJ
cana-3848	270	11	2(𝜇	2(𝜇	NUM
cana-3848	270	12	+	+	CCONJ
cana-3848	270	13	1	1	NUM
cana-3848	270	14	)	)	PUNCT
cana-3848	270	15	=	=	NOUN
cana-3848	270	16	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	270	17	)	)	PUNCT
cana-3848	271	1	+	+	CCONJ
cana-3848	271	2	𝑥	𝑥	X
cana-3848	271	3	+	+	CCONJ
cana-3848	271	4	𝜇	𝜇	X
cana-3848	271	5	+	+	NOUN
cana-3848	271	6	1	1	NUM
cana-3848	271	7	,	,	PUNCT
cana-3848	271	8	otherwise	otherwise	ADV
cana-3848	271	9	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	271	10	(	(	PUNCT
cana-3848	271	11	𝑓𝑗(𝑥	𝑓𝑗(𝑥	NOUN
cana-3848	271	12	)	)	PUNCT
cana-3848	271	13	,	,	PUNCT
cana-3848	271	14	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	271	15	+	+	CCONJ
cana-3848	271	16	2(𝜇	2(𝜇	NUM
cana-3848	271	17	+	+	CCONJ
cana-3848	271	18	1	1	NUM
cana-3848	271	19	)	)	PUNCT
cana-3848	271	20	)	)	PUNCT
cana-3848	271	21	)	)	PUNCT
cana-3848	272	1	=	=	SYM
cana-3848	272	2	𝑞𝑗(𝑥	𝑞𝑗(𝑥	X
cana-3848	273	1	+	+	CCONJ
cana-3848	273	2	𝜇	𝜇	X
cana-3848	273	3	+	+	NOUN
cana-3848	273	4	1	1	NUM
cana-3848	273	5	)	)	PUNCT
cana-3848	273	6	.	.	PUNCT
cana-3848	274	1	there	there	PRON
cana-3848	274	2	is	be	VERB
cana-3848	274	3	a	a	DET
cana-3848	274	4	single	single	ADJ
cana-3848	274	5	𝑎	𝑎	PROPN
cana-3848	274	6	∈	∈	NOUN
cana-3848	274	7	𝑋	𝑋	NOUN
cana-3848	274	8	with	with	ADP
cana-3848	274	9	𝑓𝑗(𝑎	𝑓𝑗(𝑎	NOUN
cana-3848	274	10	)	)	PUNCT
cana-3848	274	11	=	=	SYM
cana-3848	275	1	𝑥	𝑥	PROPN
cana-3848	276	1	+	+	CCONJ
cana-3848	276	2	𝜇	𝜇	X
cana-3848	276	3	+	+	ADP
cana-3848	276	4	1	1	NUM
cana-3848	276	5	since	since	SCONJ
cana-3848	276	6	𝑓𝑗	𝑓𝑗	ADJ
cana-3848	276	7	is	be	AUX
cana-3848	276	8	bijective	bijective	ADJ
cana-3848	276	9	.	.	PUNCT
cana-3848	277	1	consequently	consequently	ADV
cana-3848	277	2	,	,	PUNCT
cana-3848	277	3	𝑓𝑗(𝑥	𝑓𝑗(𝑥	PUNCT
cana-3848	277	4	+	+	CCONJ
cana-3848	277	5	2(𝜇	2(𝜇	NUM
cana-3848	277	6	+	+	CCONJ
cana-3848	277	7	1	1	NUM
cana-3848	277	8	)	)	PUNCT
cana-3848	277	9	)	)	PUNCT
cana-3848	278	1	=	=	SYM
cana-3848	278	2	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	278	3	)	)	PUNCT
cana-3848	279	1	+	+	CCONJ
cana-3848	279	2	𝑓𝑗(𝑎	𝑓𝑗(𝑎	NOUN
cana-3848	279	3	)	)	PUNCT
cana-3848	279	4	=	=	PUNCT
cana-3848	279	5	𝑓𝑗(𝑥	𝑓𝑗(𝑥	X
cana-3848	279	6	+	+	X
cana-3848	279	7	𝑎	𝑎	X
cana-3848	279	8	)	)	PUNCT
cana-3848	279	9	.	.	PUNCT
cana-3848	280	1	because	because	SCONJ
cana-3848	280	2	𝜇	𝜇	ADP
cana-3848	280	3	=	=	X
cana-3848	280	4	𝑎	𝑎	PRON
cana-3848	280	5	2	2	NUM
cana-3848	280	6	−	−	NUM
cana-3848	280	7	1	1	NUM
cana-3848	280	8	,	,	PUNCT
cana-3848	280	9	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	280	10	(	(	PUNCT
cana-3848	280	11	𝑥	𝑥	NOUN
cana-3848	280	12	,	,	PUNCT
cana-3848	280	13	𝑥	𝑥	PROPN
cana-3848	280	14	+	+	NUM
cana-3848	280	15	2(𝜇	2(𝜇	NUM
cana-3848	280	16	+	+	CCONJ
cana-3848	280	17	1	1	NUM
cana-3848	280	18	)	)	PUNCT
cana-3848	280	19	=	=	NOUN
cana-3848	280	20	𝑝𝑗(𝑎	𝑝𝑗(𝑎	NOUN
cana-3848	280	21	)	)	PUNCT
cana-3848	280	22	.	.	PUNCT
cana-3848	281	1	the	the	DET
cana-3848	281	2	isometry	isometry	NOUN
cana-3848	281	3	of	of	ADP
cana-3848	281	4	(	(	PUNCT
cana-3848	281	5	𝑋	𝑋	PROPN
cana-3848	281	6	,	,	PUNCT
cana-3848	281	7	𝑒𝑝𝑗	𝑒𝑝𝑗	PROPN
cana-3848	281	8	)	)	PUNCT
cana-3848	281	9	and	and	CCONJ
cana-3848	281	10	(	(	PUNCT
cana-3848	281	11	𝑌	𝑌	PROPN
cana-3848	281	12	,	,	PUNCT
cana-3848	281	13	𝑒𝑞𝑗	𝑒𝑞𝑗	NOUN
cana-3848	281	14	)	)	PUNCT
cana-3848	281	15	by	by	ADP
cana-3848	281	16	𝑓𝑗,is	𝑓𝑗,is	NOUN
cana-3848	281	17	what	what	PRON
cana-3848	281	18	we	we	PRON
cana-3848	281	19	deduce	deduce	VERB
cana-3848	281	20	.	.	PUNCT
cana-3848	281	21	references	reference	NOUN
cana-3848	281	22	1	1	NUM
cana-3848	281	23	.	.	PUNCT
cana-3848	282	1	p.	p.	NOUN
cana-3848	282	2	fletcher	fletcher	PROPN
cana-3848	282	3	and	and	CCONJ
cana-3848	282	4	w.	w.	PROPN
cana-3848	282	5	f.	f.	PROPN
cana-3848	282	6	lindgren	lindgren	PROPN
cana-3848	282	7	,	,	PUNCT
cana-3848	282	8	1982	1982	NUM
cana-3848	282	9	,	,	PUNCT
cana-3848	282	10	quasi	quasi	ADJ
cana-3848	282	11	-	-	ADJ
cana-3848	282	12	uniform	uniform	ADJ
cana-3848	282	13	spaces	space	NOUN
cana-3848	282	14	,	,	PUNCT
cana-3848	282	15	lecture	lecture	NOUN
cana-3848	282	16	notes	note	NOUN
cana-3848	282	17	in	in	ADP
cana-3848	282	18	pure	pure	ADJ
cana-3848	282	19	appl	appl	NOUN
cana-3848	282	20	.	.	PUNCT
cana-3848	283	1	math	math	NOUN
cana-3848	283	2	.	.	PUNCT
cana-3848	284	1	,77	,77	PROPN
cana-3848	284	2	,	,	PUNCT
cana-3848	284	3	marcel	marcel	PROPN
cana-3848	284	4	dekker	dekker	PROPN
cana-3848	284	5	,	,	PUNCT
cana-3848	284	6	new	new	PROPN
cana-3848	284	7	york	york	PROPN
cana-3848	284	8	.	.	PUNCT
cana-3848	285	1	2	2	X
cana-3848	285	2	.	.	X
cana-3848	285	3	k.	k.	PROPN
cana-3848	285	4	keimel	keimel	PROPN
cana-3848	285	5	and	and	CCONJ
cana-3848	285	6	w.	w.	PROPN
cana-3848	285	7	roth	roth	PROPN
cana-3848	285	8	,	,	PUNCT
cana-3848	285	9	1992	1992	NUM
cana-3848	285	10	,	,	PUNCT
cana-3848	285	11	ordered	order	VERB
cana-3848	285	12	cones	cone	NOUN
cana-3848	285	13	and	and	CCONJ
cana-3848	285	14	approximation	approximation	NOUN
cana-3848	285	15	,	,	PUNCT
cana-3848	285	16	springer	springer	NOUN
cana-3848	285	17	–	–	PUNCT
cana-3848	285	18	verlag	verlag	PROPN
cana-3848	285	19	,	,	PUNCT
cana-3848	285	20	berlin	berlin	PROPN
cana-3848	285	21	,	,	PUNCT
cana-3848	285	22	3	3	NUM
cana-3848	285	23	.	.	PUNCT
cana-3848	285	24	r.	r.	PROPN
cana-3848	285	25	kopperman	kopperman	PROPN
cana-3848	285	26	,	,	PUNCT
cana-3848	285	27	lengths	length	NOUN
cana-3848	285	28	on	on	ADP
cana-3848	285	29	semigroups	semigroup	NOUN
cana-3848	285	30	and	and	CCONJ
cana-3848	285	31	groups	group	NOUN
cana-3848	285	32	,	,	PUNCT
cana-3848	285	33	semigroup	semigroup	PROPN
cana-3848	285	34	forum	forum	PROPN
cana-3848	285	35	25	25	NUM
cana-3848	285	36	(	(	PUNCT
cana-3848	285	37	1982	1982	NUM
cana-3848	285	38	)	)	PUNCT
cana-3848	285	39	,	,	PUNCT
cana-3848	285	40	345–360	345–360	NUM
cana-3848	285	41	.	.	PUNCT
cana-3848	286	1	4	4	NUM
cana-3848	286	2	.	.	X
cana-3848	286	3	s.	s.	PROPN
cana-3848	286	4	romaguera	romaguera	PROPN
cana-3848	286	5	,	,	PUNCT
cana-3848	286	6	m.	m.	NOUN
cana-3848	286	7	schellekens	schellekens	PROPN
cana-3848	286	8	,	,	PUNCT
cana-3848	286	9	quasi	quasi	ADJ
cana-3848	286	10	–	–	PUNCT
cana-3848	286	11	metric	metric	ADJ
cana-3848	286	12	properties	property	NOUN
cana-3848	286	13	of	of	ADP
cana-3848	286	14	complexity	complexity	NOUN
cana-3848	286	15	,	,	PUNCT
cana-3848	286	16	1999	1999	NUM
cana-3848	286	17	,	,	PUNCT
cana-3848	286	18	p	p	NOUN
cana-3848	286	19	311	311	NUM
cana-3848	286	20	-	-	SYM
cana-3848	286	21	322	322	NUM
cana-3848	286	22	,	,	PUNCT
cana-3848	286	23	the	the	DET
cana-3848	286	24	journal	journal	NOUN
cana-3848	286	25	of	of	ADP
cana-3848	286	26	mathematical	mathematical	ADJ
cana-3848	286	27	analysis	analysis	NOUN
cana-3848	286	28	and	and	CCONJ
cana-3848	286	29	applications	application	NOUN
cana-3848	286	30	elsevier	elsevier	NOUN
cana-3848	286	31	.	.	PUNCT
cana-3848	287	1	5	5	X
cana-3848	287	2	.	.	PUNCT
cana-3848	287	3	s.	s.	PROPN
cana-3848	287	4	salbany	salbany	PROPN
cana-3848	287	5	,	,	PUNCT
cana-3848	287	6	bitopological	bitopological	ADJ
cana-3848	287	7	spaces	space	NOUN
cana-3848	287	8	,	,	PUNCT
cana-3848	287	9	compacti	compacti	ADJ
cana-3848	287	10	cations	cation	NOUN
cana-3848	287	11	and	and	CCONJ
cana-3848	287	12	completions	completion	NOUN
cana-3848	287	13	,	,	PUNCT
cana-3848	287	14	math	math	NOUN
cana-3848	287	15	.	.	PUNCT
cana-3848	288	1	monographs	monographs	PROPN
cana-3848	288	2	univ	univ	PROPN
cana-3848	288	3	.	.	PUNCT
cana-3848	289	1	cape	cape	NOUN
cana-3848	289	2	town	town	NOUN
cana-3848	289	3	,	,	PUNCT
cana-3848	289	4	no	no	INTJ
cana-3848	289	5	.	.	NOUN
cana-3848	289	6	1	1	NUM
cana-3848	289	7	,	,	PUNCT
cana-3848	289	8	1974	1974	NUM
cana-3848	289	9	.	.	PUNCT
cana-3848	290	1	6	6	NUM
cana-3848	290	2	.	.	X
cana-3848	290	3	m.	m.	NOUN
cana-3848	290	4	schellekens	schellekens	PROPN
cana-3848	290	5	,	,	PUNCT
cana-3848	290	6	the	the	DET
cana-3848	290	7	smyth	smyth	NOUN
cana-3848	290	8	completion	completion	NOUN
cana-3848	290	9	:	:	PUNCT
cana-3848	290	10	a	a	DET
cana-3848	290	11	common	common	ADJ
cana-3848	290	12	foundation	foundation	NOUN
cana-3848	290	13	for	for	ADP
cana-3848	290	14	denotational	denotational	ADJ
cana-3848	290	15	semantics	semantic	NOUN
cana-3848	290	16	and	and	CCONJ
cana-3848	290	17	complexity	complexity	NOUN
cana-3848	290	18	analysis	analysis	NOUN
cana-3848	290	19	,	,	PUNCT
cana-3848	290	20	in	in	ADP
cana-3848	290	21	proc	proc	NOUN
cana-3848	290	22	.	.	PUNCT
cana-3848	291	1	mfps	mfps	NOUN
cana-3848	291	2	11	11	NUM
cana-3848	291	3	,	,	PUNCT
cana-3848	291	4	electronic	electronic	ADJ
cana-3848	291	5	notes	note	NOUN
cana-3848	291	6	in	in	ADP
cana-3848	291	7	theoretical	theoretical	ADJ
cana-3848	291	8	comp	comp	NOUN
cana-3848	291	9	.	.	PUNCT
cana-3848	292	1	sci	sci	PROPN
cana-3848	292	2	.	.	PROPN
cana-3848	292	3	1	1	NUM
cana-3848	292	4	(	(	PUNCT
cana-3848	292	5	1995	1995	NUM
cana-3848	292	6	)	)	PUNCT
cana-3848	292	7	,	,	PUNCT
cana-3848	292	8	211232	211232	NUM
cana-3848	292	9	.	.	PUNCT
cana-3848	293	1	7	7	X
cana-3848	293	2	.	.	X
cana-3848	293	3	s.romaguera	s.romaguera	PROPN
cana-3848	293	4	,	,	PUNCT
cana-3848	293	5	(	(	PUNCT
cana-3848	293	6	valencia	valencia	PROPN
cana-3848	293	7	)	)	PUNCT
cana-3848	293	8	,	,	PUNCT
cana-3848	293	9	e.a	e.a	PROPN
cana-3848	293	10	sanchez	sanchez	PROPN
cana-3848	293	11	-	-	PUNCT
cana-3848	293	12	p	p	PROPN
cana-3848	293	13	´	´	NOUN
cana-3848	293	14	quasi	quasi	ADJ
cana-3848	293	15	-	-	ADJ
cana-3848	293	16	normed	normed	ADJ
cana-3848	293	17	monoids	monoid	NOUN
cana-3848	293	18	and	and	CCONJ
cana-3848	293	19	quasi	quasi	ADJ
cana-3848	293	20	-	-	NOUN
cana-3848	293	21	metrics	metric	NOUN
cana-3848	293	22	(	(	PUNCT
cana-3848	293	23	valencia	valencia	PROPN
cana-3848	293	24	)	)	PUNCT
cana-3848	293	25	53–69	53–69	NUM
cana-3848	293	26	(	(	PUNCT
cana-3848	293	27	2003	2003	NUM
cana-3848	293	28	)	)	PUNCT
cana-3848	293	29	.	.	PUNCT