id	sid	tid	token	lemma	pos
ijassa-1958	1	1	adv	adv	PROPN
ijassa-1958	1	2	syst	syst	PROPN
ijassa-1958	1	3	sci	sci	PROPN
ijassa-1958	1	4	appl	appl	PROPN
ijassa-1958	1	5	2024	2024	NUM
ijassa-1958	1	6	;	;	PUNCT
ijassa-1958	1	7	03:1–10	03:1–10	NUM
ijassa-1958	1	8	published	publish	VERB
ijassa-1958	1	9	online	online	ADV
ijassa-1958	1	10	at	at	ADP
ijassa-1958	1	11	https://ijassa.ipu.ru	https://ijassa.ipu.ru	ADV
ijassa-1958	1	12	.	.	PUNCT
ijassa-1958	2	1	on	on	ADP
ijassa-1958	2	2	the	the	DET
ijassa-1958	2	3	compatibility	compatibility	NOUN
ijassa-1958	2	4	of	of	ADP
ijassa-1958	2	5	metric	metric	ADJ
ijassa-1958	2	6	and	and	CCONJ
ijassa-1958	2	7	linear	linear	ADJ
ijassa-1958	2	8	order	order	NOUN
ijassa-1958	2	9	petr	petr	PROPN
ijassa-1958	2	10	klimov1	klimov1	PROPN
ijassa-1958	2	11	,	,	PUNCT
ijassa-1958	2	12	richik	richik	ADJ
ijassa-1958	2	13	sengupta2,3	sengupta2,3	PROPN
ijassa-1958	2	14	*	*	SYM
ijassa-1958	2	15	1moscow	1moscow	NUM
ijassa-1958	2	16	institute	institute	PROPN
ijassa-1958	2	17	of	of	ADP
ijassa-1958	2	18	physics	physics	PROPN
ijassa-1958	2	19	and	and	CCONJ
ijassa-1958	2	20	technology	technology	NOUN
ijassa-1958	2	21	,	,	PUNCT
ijassa-1958	2	22	moscow	moscow	PROPN
ijassa-1958	2	23	,	,	PUNCT
ijassa-1958	2	24	russia	russia	PROPN
ijassa-1958	2	25	2skolkovo	2skolkovo	PROPN
ijassa-1958	2	26	institute	institute	NOUN
ijassa-1958	2	27	of	of	ADP
ijassa-1958	2	28	science	science	NOUN
ijassa-1958	2	29	and	and	CCONJ
ijassa-1958	2	30	technology	technology	NOUN
ijassa-1958	2	31	,	,	PUNCT
ijassa-1958	2	32	moscow	moscow	PROPN
ijassa-1958	2	33	,	,	PUNCT
ijassa-1958	2	34	russia	russia	PROPN
ijassa-1958	2	35	3derzhavin	3derzhavin	NUM
ijassa-1958	2	36	tambov	tambov	PROPN
ijassa-1958	2	37	state	state	PROPN
ijassa-1958	2	38	university	university	PROPN
ijassa-1958	2	39	,	,	PUNCT
ijassa-1958	2	40	tambov	tambov	PROPN
ijassa-1958	2	41	,	,	PUNCT
ijassa-1958	2	42	russia	russia	PROPN
ijassa-1958	2	43	abstract	abstract	NOUN
ijassa-1958	2	44	:	:	PUNCT
ijassa-1958	2	45	there	there	PRON
ijassa-1958	2	46	is	be	VERB
ijassa-1958	2	47	a	a	DET
ijassa-1958	2	48	classic	classic	ADJ
ijassa-1958	2	49	question	question	NOUN
ijassa-1958	2	50	of	of	ADP
ijassa-1958	2	51	introducing	introduce	VERB
ijassa-1958	2	52	some	some	DET
ijassa-1958	2	53	reasonable	reasonable	ADJ
ijassa-1958	2	54	order	order	NOUN
ijassa-1958	2	55	on	on	ADP
ijassa-1958	2	56	the	the	DET
ijassa-1958	2	57	real	real	ADJ
ijassa-1958	2	58	plane	plane	NOUN
ijassa-1958	2	59	r2	r2	NOUN
ijassa-1958	2	60	.	.	PUNCT
ijassa-1958	3	1	we	we	PRON
ijassa-1958	3	2	consider	consider	VERB
ijassa-1958	3	3	this	this	DET
ijassa-1958	3	4	problem	problem	NOUN
ijassa-1958	3	5	through	through	ADP
ijassa-1958	3	6	the	the	DET
ijassa-1958	3	7	prism	prism	NOUN
ijassa-1958	3	8	of	of	ADP
ijassa-1958	3	9	the	the	DET
ijassa-1958	3	10	relation	relation	NOUN
ijassa-1958	3	11	between	between	ADP
ijassa-1958	3	12	metric	metric	ADJ
ijassa-1958	3	13	proximity	proximity	NOUN
ijassa-1958	3	14	and	and	CCONJ
ijassa-1958	3	15	linear	linear	ADJ
ijassa-1958	3	16	order	order	NOUN
ijassa-1958	3	17	proximity	proximity	NOUN
ijassa-1958	3	18	,	,	PUNCT
ijassa-1958	3	19	introducing	introduce	VERB
ijassa-1958	3	20	a	a	DET
ijassa-1958	3	21	way	way	NOUN
ijassa-1958	3	22	to	to	PART
ijassa-1958	3	23	determine	determine	VERB
ijassa-1958	3	24	the	the	DET
ijassa-1958	3	25	compatibility	compatibility	NOUN
ijassa-1958	3	26	relation	relation	NOUN
ijassa-1958	3	27	between	between	ADP
ijassa-1958	3	28	metric	metric	ADJ
ijassa-1958	3	29	and	and	CCONJ
ijassa-1958	3	30	linearly	linearly	ADV
ijassa-1958	3	31	ordered	order	VERB
ijassa-1958	3	32	spaces	space	NOUN
ijassa-1958	3	33	,	,	PUNCT
ijassa-1958	3	34	and	and	CCONJ
ijassa-1958	3	35	study	study	VERB
ijassa-1958	3	36	the	the	DET
ijassa-1958	3	37	basic	basic	ADJ
ijassa-1958	3	38	properties	property	NOUN
ijassa-1958	3	39	of	of	ADP
ijassa-1958	3	40	this	this	DET
ijassa-1958	3	41	connection	connection	NOUN
ijassa-1958	3	42	.	.	PUNCT
ijassa-1958	4	1	for	for	ADP
ijassa-1958	4	2	r	r	NOUN
ijassa-1958	4	3	and	and	CCONJ
ijassa-1958	4	4	its	its	PRON
ijassa-1958	4	5	metric	metric	ADJ
ijassa-1958	4	6	subspaces	subspace	NOUN
ijassa-1958	4	7	,	,	PUNCT
ijassa-1958	4	8	the	the	DET
ijassa-1958	4	9	order	order	NOUN
ijassa-1958	4	10	on	on	ADP
ijassa-1958	4	11	the	the	DET
ijassa-1958	4	12	line	line	NOUN
ijassa-1958	4	13	is	be	AUX
ijassa-1958	4	14	compatible	compatible	ADJ
ijassa-1958	4	15	with	with	ADP
ijassa-1958	4	16	the	the	DET
ijassa-1958	4	17	metric	metric	NOUN
ijassa-1958	4	18	according	accord	VERB
ijassa-1958	4	19	to	to	ADP
ijassa-1958	4	20	the	the	DET
ijassa-1958	4	21	definition	definition	NOUN
ijassa-1958	4	22	we	we	PRON
ijassa-1958	4	23	introduce	introduce	VERB
ijassa-1958	4	24	.	.	PUNCT
ijassa-1958	5	1	it	it	PRON
ijassa-1958	5	2	turns	turn	VERB
ijassa-1958	5	3	out	out	ADP
ijassa-1958	5	4	that	that	SCONJ
ijassa-1958	5	5	for	for	ADP
ijassa-1958	5	6	rn	rn	PROPN
ijassa-1958	5	7	for	for	ADP
ijassa-1958	5	8	n	n	PROPN
ijassa-1958	5	9	>	>	X
ijassa-1958	5	10	1	1	NUM
ijassa-1958	5	11	,	,	PUNCT
ijassa-1958	5	12	as	as	ADV
ijassa-1958	5	13	well	well	ADV
ijassa-1958	5	14	as	as	ADP
ijassa-1958	5	15	for	for	ADP
ijassa-1958	5	16	other	other	ADJ
ijassa-1958	5	17	similar	similar	ADJ
ijassa-1958	5	18	metric	metric	ADJ
ijassa-1958	5	19	spaces	space	NOUN
ijassa-1958	5	20	in	in	ADP
ijassa-1958	5	21	which	which	PRON
ijassa-1958	5	22	there	there	PRON
ijassa-1958	5	23	is	be	VERB
ijassa-1958	5	24	a	a	DET
ijassa-1958	5	25	subspace	subspace	NOUN
ijassa-1958	5	26	isometric	isometric	ADJ
ijassa-1958	5	27	to	to	ADP
ijassa-1958	5	28	an	an	DET
ijassa-1958	5	29	open	open	ADJ
ijassa-1958	5	30	ball	ball	NOUN
ijassa-1958	5	31	in	in	ADP
ijassa-1958	5	32	rn	rn	PROPN
ijassa-1958	5	33	,	,	PUNCT
ijassa-1958	5	34	it	it	PRON
ijassa-1958	5	35	is	be	AUX
ijassa-1958	5	36	impossible	impossible	ADJ
ijassa-1958	5	37	to	to	PART
ijassa-1958	5	38	introduce	introduce	VERB
ijassa-1958	5	39	an	an	DET
ijassa-1958	5	40	order	order	NOUN
ijassa-1958	5	41	that	that	PRON
ijassa-1958	5	42	is	be	AUX
ijassa-1958	5	43	compatible	compatible	ADJ
ijassa-1958	5	44	with	with	ADP
ijassa-1958	5	45	the	the	DET
ijassa-1958	5	46	metric	metric	NOUN
ijassa-1958	5	47	.	.	PUNCT
ijassa-1958	6	1	at	at	ADP
ijassa-1958	6	2	the	the	DET
ijassa-1958	6	3	same	same	ADJ
ijassa-1958	6	4	time	time	NOUN
ijassa-1958	6	5	,	,	PUNCT
ijassa-1958	6	6	we	we	PRON
ijassa-1958	6	7	also	also	ADV
ijassa-1958	6	8	introduce	introduce	VERB
ijassa-1958	6	9	a	a	DET
ijassa-1958	6	10	family	family	NOUN
ijassa-1958	6	11	of	of	ADP
ijassa-1958	6	12	spaces	space	NOUN
ijassa-1958	6	13	called	call	VERB
ijassa-1958	6	14	discrete	discrete	ADJ
ijassa-1958	6	15	curved	curved	ADJ
ijassa-1958	6	16	lines	line	NOUN
ijassa-1958	6	17	,	,	PUNCT
ijassa-1958	6	18	which	which	PRON
ijassa-1958	6	19	are	be	AUX
ijassa-1958	6	20	not	not	PART
ijassa-1958	6	21	generally	generally	ADV
ijassa-1958	6	22	isometric	isometric	ADJ
ijassa-1958	6	23	to	to	ADP
ijassa-1958	6	24	the	the	DET
ijassa-1958	6	25	subspaces	subspace	NOUN
ijassa-1958	6	26	of	of	ADP
ijassa-1958	6	27	r	r	NOUN
ijassa-1958	6	28	,	,	PUNCT
ijassa-1958	6	29	but	but	CCONJ
ijassa-1958	6	30	for	for	ADP
ijassa-1958	6	31	which	which	PRON
ijassa-1958	6	32	it	it	PRON
ijassa-1958	6	33	is	be	AUX
ijassa-1958	6	34	possible	possible	ADJ
ijassa-1958	6	35	to	to	PART
ijassa-1958	6	36	introduce	introduce	VERB
ijassa-1958	6	37	a	a	DET
ijassa-1958	6	38	linear	linear	ADJ
ijassa-1958	6	39	order	order	NOUN
ijassa-1958	6	40	compatible	compatible	ADJ
ijassa-1958	6	41	with	with	ADP
ijassa-1958	6	42	the	the	DET
ijassa-1958	6	43	natural	natural	ADJ
ijassa-1958	6	44	metric	metric	NOUN
ijassa-1958	6	45	on	on	ADP
ijassa-1958	6	46	them	they	PRON
ijassa-1958	6	47	.	.	PUNCT
ijassa-1958	7	1	for	for	ADP
ijassa-1958	7	2	them	they	PRON
ijassa-1958	7	3	we	we	PRON
ijassa-1958	7	4	prove	prove	VERB
ijassa-1958	7	5	sufficient	sufficient	ADJ
ijassa-1958	7	6	conditions	condition	NOUN
ijassa-1958	7	7	for	for	ADP
ijassa-1958	7	8	the	the	DET
ijassa-1958	7	9	compatibility	compatibility	NOUN
ijassa-1958	7	10	of	of	ADP
ijassa-1958	7	11	the	the	DET
ijassa-1958	7	12	metric	metric	NOUN
ijassa-1958	7	13	with	with	ADP
ijassa-1958	7	14	the	the	DET
ijassa-1958	7	15	order	order	NOUN
ijassa-1958	7	16	.	.	PUNCT
ijassa-1958	8	1	using	use	VERB
ijassa-1958	8	2	them	they	PRON
ijassa-1958	8	3	,	,	PUNCT
ijassa-1958	8	4	we	we	PRON
ijassa-1958	8	5	construct	construct	VERB
ijassa-1958	8	6	spaces	space	NOUN
ijassa-1958	8	7	with	with	ADP
ijassa-1958	8	8	non	non	ADJ
ijassa-1958	8	9	-	-	ADJ
ijassa-1958	8	10	trivial	trivial	ADJ
ijassa-1958	8	11	properties	property	NOUN
ijassa-1958	8	12	in	in	ADP
ijassa-1958	8	13	which	which	PRON
ijassa-1958	8	14	the	the	DET
ijassa-1958	8	15	order	order	NOUN
ijassa-1958	8	16	is	be	AUX
ijassa-1958	8	17	compatible	compatible	ADJ
ijassa-1958	8	18	with	with	ADP
ijassa-1958	8	19	the	the	DET
ijassa-1958	8	20	metric	metric	NOUN
ijassa-1958	8	21	.	.	PUNCT
ijassa-1958	9	1	in	in	ADP
ijassa-1958	9	2	particular	particular	ADJ
ijassa-1958	9	3	,	,	PUNCT
ijassa-1958	9	4	we	we	PRON
ijassa-1958	9	5	show	show	VERB
ijassa-1958	9	6	that	that	SCONJ
ijassa-1958	9	7	although	although	SCONJ
ijassa-1958	9	8	on	on	ADP
ijassa-1958	9	9	all	all	PRON
ijassa-1958	9	10	of	of	ADP
ijassa-1958	9	11	rn	rn	PROPN
ijassa-1958	9	12	no	no	DET
ijassa-1958	9	13	order	order	NOUN
ijassa-1958	9	14	is	be	AUX
ijassa-1958	9	15	compatible	compatible	ADJ
ijassa-1958	9	16	with	with	ADP
ijassa-1958	9	17	the	the	DET
ijassa-1958	9	18	metric	metric	NOUN
ijassa-1958	9	19	,	,	PUNCT
ijassa-1958	9	20	in	in	ADP
ijassa-1958	9	21	any	any	DET
ijassa-1958	9	22	normed	normed	ADJ
ijassa-1958	9	23	space	space	NOUN
ijassa-1958	9	24	one	one	PRON
ijassa-1958	9	25	can	can	AUX
ijassa-1958	9	26	introduce	introduce	VERB
ijassa-1958	9	27	non	non	ADJ
ijassa-1958	9	28	-	-	ADJ
ijassa-1958	9	29	trivial	trivial	ADJ
ijassa-1958	9	30	metric	metric	ADJ
ijassa-1958	9	31	subspaces	subspace	NOUN
ijassa-1958	9	32	compatible	compatible	ADJ
ijassa-1958	9	33	with	with	ADP
ijassa-1958	9	34	some	some	DET
ijassa-1958	9	35	order	order	NOUN
ijassa-1958	9	36	(	(	PUNCT
ijassa-1958	9	37	in	in	ADP
ijassa-1958	9	38	the	the	DET
ijassa-1958	9	39	case	case	NOUN
ijassa-1958	9	40	of	of	ADP
ijassa-1958	9	41	infinite	infinite	ADJ
ijassa-1958	9	42	-	-	PUNCT
ijassa-1958	9	43	dimensional	dimensional	ADJ
ijassa-1958	9	44	normed	normed	ADJ
ijassa-1958	9	45	spaces	space	NOUN
ijassa-1958	9	46	,	,	PUNCT
ijassa-1958	9	47	this	this	DET
ijassa-1958	9	48	subspace	subspace	NOUN
ijassa-1958	9	49	can	can	AUX
ijassa-1958	9	50	be	be	AUX
ijassa-1958	9	51	constructed	construct	VERB
ijassa-1958	9	52	to	to	PART
ijassa-1958	9	53	be	be	AUX
ijassa-1958	9	54	non	non	ADJ
ijassa-1958	9	55	-	-	ADJ
ijassa-1958	9	56	embeddable	embeddable	ADJ
ijassa-1958	9	57	in	in	ADP
ijassa-1958	9	58	any	any	DET
ijassa-1958	9	59	finite	finite	ADJ
ijassa-1958	9	60	-	-	ADJ
ijassa-1958	9	61	dimensional	dimensional	ADJ
ijassa-1958	9	62	subspace	subspace	NOUN
ijassa-1958	9	63	)	)	PUNCT
ijassa-1958	9	64	keywords	keyword	NOUN
ijassa-1958	9	65	:	:	PUNCT
ijassa-1958	9	66	ordered	order	VERB
ijassa-1958	9	67	metric	metric	ADJ
ijassa-1958	9	68	spaces	space	NOUN
ijassa-1958	9	69	,	,	PUNCT
ijassa-1958	9	70	metric	metric	ADJ
ijassa-1958	9	71	spaces	space	NOUN
ijassa-1958	9	72	,	,	PUNCT
ijassa-1958	9	73	linear	linear	ADJ
ijassa-1958	9	74	order	order	NOUN
ijassa-1958	9	75	.	.	PUNCT
ijassa-1958	10	1	1	1	X
ijassa-1958	10	2	.	.	X
ijassa-1958	10	3	introduction	introduction	NOUN
ijassa-1958	10	4	one	one	NUM
ijassa-1958	10	5	of	of	ADP
ijassa-1958	10	6	the	the	DET
ijassa-1958	10	7	natural	natural	ADJ
ijassa-1958	10	8	questions	question	NOUN
ijassa-1958	10	9	one	one	PRON
ijassa-1958	10	10	might	might	AUX
ijassa-1958	10	11	have	have	VERB
ijassa-1958	10	12	is	be	AUX
ijassa-1958	10	13	how	how	SCONJ
ijassa-1958	10	14	one	one	PRON
ijassa-1958	10	15	can	can	AUX
ijassa-1958	10	16	reasonably	reasonably	ADV
ijassa-1958	10	17	introduce	introduce	VERB
ijassa-1958	10	18	an	an	DET
ijassa-1958	10	19	order	order	NOUN
ijassa-1958	10	20	on	on	ADP
ijassa-1958	10	21	a	a	DET
ijassa-1958	10	22	real	real	ADJ
ijassa-1958	10	23	plane	plane	NOUN
ijassa-1958	10	24	r2	r2	NOUN
ijassa-1958	10	25	in	in	ADP
ijassa-1958	10	26	general	general	ADJ
ijassa-1958	10	27	given	give	VERB
ijassa-1958	10	28	that	that	SCONJ
ijassa-1958	10	29	there	there	PRON
ijassa-1958	10	30	is	be	VERB
ijassa-1958	10	31	a	a	DET
ijassa-1958	10	32	standard	standard	ADJ
ijassa-1958	10	33	order	order	NOUN
ijassa-1958	10	34	on	on	ADP
ijassa-1958	10	35	the	the	DET
ijassa-1958	10	36	real	real	ADJ
ijassa-1958	10	37	line	line	NOUN
ijassa-1958	10	38	r.	r.	PROPN
ijassa-1958	10	39	a	a	DET
ijassa-1958	10	40	common	common	ADJ
ijassa-1958	10	41	way	way	NOUN
ijassa-1958	10	42	is	be	AUX
ijassa-1958	10	43	to	to	PART
ijassa-1958	10	44	introduce	introduce	VERB
ijassa-1958	10	45	lexicographic	lexicographic	ADJ
ijassa-1958	10	46	order	order	NOUN
ijassa-1958	10	47	,	,	PUNCT
ijassa-1958	10	48	and	and	CCONJ
ijassa-1958	10	49	there	there	PRON
ijassa-1958	10	50	are	be	VERB
ijassa-1958	10	51	other	other	ADJ
ijassa-1958	10	52	ways	way	NOUN
ijassa-1958	10	53	to	to	PART
ijassa-1958	10	54	introduce	introduce	VERB
ijassa-1958	10	55	an	an	DET
ijassa-1958	10	56	order	order	NOUN
ijassa-1958	10	57	on	on	ADP
ijassa-1958	10	58	a	a	DET
ijassa-1958	10	59	plane	plane	NOUN
ijassa-1958	10	60	,	,	PUNCT
ijassa-1958	10	61	but	but	CCONJ
ijassa-1958	10	62	all	all	PRON
ijassa-1958	10	63	of	of	ADP
ijassa-1958	10	64	these	these	DET
ijassa-1958	10	65	methods	method	NOUN
ijassa-1958	10	66	lose	lose	VERB
ijassa-1958	10	67	some	some	PRON
ijassa-1958	10	68	of	of	ADP
ijassa-1958	10	69	the	the	DET
ijassa-1958	10	70	natural	natural	ADJ
ijassa-1958	10	71	properties	property	NOUN
ijassa-1958	10	72	that	that	PRON
ijassa-1958	10	73	order	order	VERB
ijassa-1958	10	74	on	on	ADP
ijassa-1958	10	75	a	a	DET
ijassa-1958	10	76	straight	straight	ADJ
ijassa-1958	10	77	line	line	NOUN
ijassa-1958	10	78	possessed	possess	VERB
ijassa-1958	10	79	.	.	PUNCT
ijassa-1958	11	1	lexicographic	lexicographic	ADJ
ijassa-1958	11	2	order	order	NOUN
ijassa-1958	11	3	is	be	AUX
ijassa-1958	11	4	very	very	ADV
ijassa-1958	11	5	inconsistent	inconsistent	ADJ
ijassa-1958	11	6	with	with	ADP
ijassa-1958	11	7	our	our	PRON
ijassa-1958	11	8	sense	sense	NOUN
ijassa-1958	11	9	of	of	ADP
ijassa-1958	11	10	nearness	nearness	NOUN
ijassa-1958	11	11	.	.	PUNCT
ijassa-1958	12	1	according	accord	VERB
ijassa-1958	12	2	to	to	ADP
ijassa-1958	12	3	the	the	DET
ijassa-1958	12	4	lexicographic	lexicographic	ADJ
ijassa-1958	12	5	order	order	NOUN
ijassa-1958	12	6	,	,	PUNCT
ijassa-1958	12	7	it	it	PRON
ijassa-1958	12	8	turns	turn	VERB
ijassa-1958	12	9	out	out	ADP
ijassa-1958	12	10	that	that	SCONJ
ijassa-1958	12	11	(	(	PUNCT
ijassa-1958	12	12	a	a	DET
ijassa-1958	12	13	,	,	PUNCT
ijassa-1958	12	14	b	b	NOUN
ijassa-1958	12	15	)	)	PUNCT
ijassa-1958	12	16	≤	≤	NOUN
ijassa-1958	12	17	(	(	PUNCT
ijassa-1958	12	18	a	a	DET
ijassa-1958	12	19	,	,	PUNCT
ijassa-1958	12	20	b+	b+	X
ijassa-1958	12	21	ϵ	ϵ	X
ijassa-1958	12	22	)	)	PUNCT
ijassa-1958	12	23	≤	≤	NOUN
ijassa-1958	12	24	(	(	PUNCT
ijassa-1958	12	25	a+	a+	PUNCT
ijassa-1958	12	26	δ	δ	PROPN
ijassa-1958	12	27	,	,	PUNCT
ijassa-1958	12	28	b	b	NOUN
ijassa-1958	12	29	)	)	PUNCT
ijassa-1958	12	30	,	,	PUNCT
ijassa-1958	12	31	as	as	ADP
ijassa-1958	12	32	arbitrarily	arbitrarily	ADV
ijassa-1958	12	33	small	small	ADJ
ijassa-1958	12	34	we	we	PRON
ijassa-1958	12	35	do	do	AUX
ijassa-1958	12	36	not	not	PART
ijassa-1958	12	37	take	take	VERB
ijassa-1958	12	38	δ	δ	PROPN
ijassa-1958	12	39	>	>	X
ijassa-1958	12	40	0	0	PUNCT
ijassa-1958	13	1	and	and	CCONJ
ijassa-1958	13	2	as	as	ADV
ijassa-1958	13	3	arbitrarily	arbitrarily	ADV
ijassa-1958	13	4	large	large	ADJ
ijassa-1958	13	5	we	we	PRON
ijassa-1958	13	6	do	do	AUX
ijassa-1958	13	7	not	not	PART
ijassa-1958	13	8	take	take	VERB
ijassa-1958	13	9	ϵ	ϵ	X
ijassa-1958	13	10	>	>	X
ijassa-1958	13	11	0	0	NUM
ijassa-1958	13	12	.	.	PUNCT
ijassa-1958	14	1	this	this	PRON
ijassa-1958	14	2	is	be	AUX
ijassa-1958	14	3	very	very	ADV
ijassa-1958	14	4	inconsistent	inconsistent	ADJ
ijassa-1958	14	5	with	with	ADP
ijassa-1958	14	6	our	our	PRON
ijassa-1958	14	7	understanding	understanding	NOUN
ijassa-1958	14	8	that	that	SCONJ
ijassa-1958	14	9	in	in	ADP
ijassa-1958	14	10	some	some	DET
ijassa-1958	14	11	ways	way	NOUN
ijassa-1958	14	12	the	the	DET
ijassa-1958	14	13	point	point	NOUN
ijassa-1958	14	14	(	(	PUNCT
ijassa-1958	14	15	a	a	DET
ijassa-1958	14	16	,	,	PUNCT
ijassa-1958	14	17	b	b	NOUN
ijassa-1958	14	18	)	)	PUNCT
ijassa-1958	14	19	is	be	AUX
ijassa-1958	14	20	much	much	ADV
ijassa-1958	14	21	closer	close	ADJ
ijassa-1958	14	22	to	to	ADP
ijassa-1958	14	23	(	(	PUNCT
ijassa-1958	14	24	a+	a+	X
ijassa-1958	14	25	δ	δ	PROPN
ijassa-1958	14	26	,	,	PUNCT
ijassa-1958	14	27	b	b	NOUN
ijassa-1958	14	28	)	)	PUNCT
ijassa-1958	14	29	than	than	ADP
ijassa-1958	14	30	to	to	ADP
ijassa-1958	14	31	(	(	PUNCT
ijassa-1958	14	32	a	a	PRON
ijassa-1958	14	33	,	,	PUNCT
ijassa-1958	14	34	b+	b+	X
ijassa-1958	14	35	ϵ	ϵ	X
ijassa-1958	14	36	)	)	PUNCT
ijassa-1958	14	37	.	.	PUNCT
ijassa-1958	15	1	what	what	PRON
ijassa-1958	15	2	gives	give	VERB
ijassa-1958	15	3	rise	rise	NOUN
ijassa-1958	15	4	to	to	ADP
ijassa-1958	15	5	this	this	DET
ijassa-1958	15	6	feeling	feeling	NOUN
ijassa-1958	15	7	?	?	PUNCT
ijassa-1958	16	1	this	this	DET
ijassa-1958	16	2	understanding	understanding	NOUN
ijassa-1958	16	3	is	be	AUX
ijassa-1958	16	4	based	base	VERB
ijassa-1958	16	5	on	on	ADP
ijassa-1958	16	6	the	the	DET
ijassa-1958	16	7	notion	notion	NOUN
ijassa-1958	16	8	of	of	ADP
ijassa-1958	16	9	metric	metric	ADJ
ijassa-1958	16	10	proximity	proximity	NOUN
ijassa-1958	16	11	by	by	ADP
ijassa-1958	16	12	the	the	DET
ijassa-1958	16	13	natural	natural	ADJ
ijassa-1958	16	14	classical	classical	ADJ
ijassa-1958	16	15	euclidean	euclidean	ADJ
ijassa-1958	16	16	metric	metric	NOUN
ijassa-1958	16	17	on	on	ADP
ijassa-1958	16	18	r2	r2	PROPN
ijassa-1958	16	19	.	.	PUNCT
ijassa-1958	17	1	∗corresponding	∗corresponde	VERB
ijassa-1958	17	2	author	author	NOUN
ijassa-1958	17	3	:	:	PUNCT
ijassa-1958	17	4	r.sengupta@skoltech.ru	r.sengupta@skoltech.ru	X
ijassa-1958	17	5	2	2	NUM
ijassa-1958	17	6	p.	p.	NOUN
ijassa-1958	17	7	klimov	klimov	PROPN
ijassa-1958	17	8	,	,	PUNCT
ijassa-1958	17	9	r.	r.	PROPN
ijassa-1958	17	10	sengupta	sengupta	PROPN
ijassa-1958	17	11	this	this	PRON
ijassa-1958	17	12	gives	give	VERB
ijassa-1958	17	13	rise	rise	NOUN
ijassa-1958	17	14	to	to	ADP
ijassa-1958	17	15	a	a	DET
ijassa-1958	17	16	natural	natural	ADJ
ijassa-1958	17	17	way	way	NOUN
ijassa-1958	17	18	to	to	PART
ijassa-1958	17	19	impose	impose	VERB
ijassa-1958	17	20	the	the	DET
ijassa-1958	17	21	requirement	requirement	NOUN
ijassa-1958	17	22	to	to	PART
ijassa-1958	17	23	impose	impose	VERB
ijassa-1958	17	24	compatibility	compatibility	NOUN
ijassa-1958	17	25	in	in	ADP
ijassa-1958	17	26	the	the	DET
ijassa-1958	17	27	definitions	definition	NOUN
ijassa-1958	17	28	of	of	ADP
ijassa-1958	17	29	metric	metric	ADJ
ijassa-1958	17	30	proximity	proximity	NOUN
ijassa-1958	18	1	[	[	X
ijassa-1958	18	2	2–4	2–4	NUM
ijassa-1958	18	3	,	,	PUNCT
ijassa-1958	18	4	6	6	NUM
ijassa-1958	18	5	,	,	PUNCT
ijassa-1958	18	6	8	8	NUM
ijassa-1958	18	7	]	]	PUNCT
ijassa-1958	18	8	and	and	CCONJ
ijassa-1958	18	9	linear	linear	ADJ
ijassa-1958	18	10	proximity	proximity	NOUN
ijassa-1958	19	1	[	[	X
ijassa-1958	19	2	1	1	NUM
ijassa-1958	19	3	,	,	PUNCT
ijassa-1958	19	4	5	5	NUM
ijassa-1958	19	5	,	,	PUNCT
ijassa-1958	19	6	10	10	NUM
ijassa-1958	19	7	,	,	PUNCT
ijassa-1958	19	8	11	11	NUM
ijassa-1958	19	9	]	]	PUNCT
ijassa-1958	19	10	,	,	PUNCT
ijassa-1958	19	11	which	which	PRON
ijassa-1958	19	12	we	we	PRON
ijassa-1958	19	13	do	do	VERB
ijassa-1958	19	14	in	in	ADP
ijassa-1958	19	15	our	our	PRON
ijassa-1958	19	16	article	article	NOUN
ijassa-1958	19	17	.	.	PUNCT
ijassa-1958	20	1	it	it	PRON
ijassa-1958	20	2	turns	turn	VERB
ijassa-1958	20	3	out	out	ADP
ijassa-1958	20	4	that	that	SCONJ
ijassa-1958	20	5	there	there	PRON
ijassa-1958	20	6	are	be	VERB
ijassa-1958	20	7	no	no	DET
ijassa-1958	20	8	linear	linear	ADJ
ijassa-1958	20	9	spaces	space	NOUN
ijassa-1958	20	10	compatible	compatible	ADJ
ijassa-1958	20	11	with	with	ADP
ijassa-1958	20	12	the	the	DET
ijassa-1958	20	13	classical	classical	ADJ
ijassa-1958	20	14	metric	metric	NOUN
ijassa-1958	20	15	on	on	ADP
ijassa-1958	20	16	rn	rn	PROPN
ijassa-1958	20	17	for	for	ADP
ijassa-1958	20	18	n	n	PROPN
ijassa-1958	20	19	>	>	SYM
ijassa-1958	20	20	1	1	NUM
ijassa-1958	20	21	and	and	CCONJ
ijassa-1958	20	22	similar	similar	ADJ
ijassa-1958	20	23	metric	metric	ADJ
ijassa-1958	20	24	spaces	space	NOUN
ijassa-1958	20	25	with	with	ADP
ijassa-1958	20	26	linear	linear	ADJ
ijassa-1958	20	27	orders	order	NOUN
ijassa-1958	20	28	according	accord	VERB
ijassa-1958	20	29	to	to	ADP
ijassa-1958	20	30	our	our	PRON
ijassa-1958	20	31	definition	definition	NOUN
ijassa-1958	20	32	.	.	PUNCT
ijassa-1958	21	1	further	far	ADV
ijassa-1958	21	2	we	we	PRON
ijassa-1958	21	3	introduce	introduce	VERB
ijassa-1958	21	4	a	a	DET
ijassa-1958	21	5	series	series	NOUN
ijassa-1958	21	6	of	of	ADP
ijassa-1958	21	7	non	non	ADJ
ijassa-1958	21	8	-	-	ADJ
ijassa-1958	21	9	trivial	trivial	ADJ
ijassa-1958	21	10	metric	metric	ADJ
ijassa-1958	21	11	spaces	space	NOUN
ijassa-1958	21	12	that	that	PRON
ijassa-1958	21	13	are	be	AUX
ijassa-1958	21	14	compatible	compatible	ADJ
ijassa-1958	21	15	with	with	ADP
ijassa-1958	21	16	respect	respect	NOUN
ijassa-1958	21	17	to	to	ADP
ijassa-1958	21	18	the	the	DET
ijassa-1958	21	19	order	order	NOUN
ijassa-1958	21	20	but	but	CCONJ
ijassa-1958	21	21	are	be	AUX
ijassa-1958	21	22	not	not	PART
ijassa-1958	21	23	isometric	isometric	ADJ
ijassa-1958	21	24	to	to	ADP
ijassa-1958	21	25	subsets	subset	NOUN
ijassa-1958	21	26	of	of	ADP
ijassa-1958	21	27	the	the	DET
ijassa-1958	21	28	real	real	ADJ
ijassa-1958	21	29	line	line	NOUN
ijassa-1958	21	30	.	.	PUNCT
ijassa-1958	22	1	2	2	X
ijassa-1958	22	2	.	.	X
ijassa-1958	22	3	basic	basic	ADJ
ijassa-1958	22	4	definitions	definition	NOUN
ijassa-1958	22	5	let	let	VERB
ijassa-1958	22	6	(	(	PUNCT
ijassa-1958	22	7	x	x	NOUN
ijassa-1958	22	8	,	,	PUNCT
ijassa-1958	22	9	d,≤	d,≤	PUNCT
ijassa-1958	22	10	)	)	PUNCT
ijassa-1958	22	11	be	be	VERB
ijassa-1958	22	12	a	a	DET
ijassa-1958	22	13	set	set	NOUN
ijassa-1958	22	14	equipped	equip	VERB
ijassa-1958	22	15	with	with	ADP
ijassa-1958	22	16	a	a	DET
ijassa-1958	22	17	metric	metric	ADJ
ijassa-1958	22	18	d	d	NOUN
ijassa-1958	22	19	and	and	CCONJ
ijassa-1958	22	20	a	a	DET
ijassa-1958	22	21	non	non	ADJ
ijassa-1958	22	22	-	-	ADJ
ijassa-1958	22	23	strict	strict	ADJ
ijassa-1958	22	24	linear	linear	ADJ
ijassa-1958	22	25	order	order	NOUN
ijassa-1958	22	26	denoted	denote	VERB
ijassa-1958	22	27	by	by	ADP
ijassa-1958	22	28	≤.	≤.	NOUN
ijassa-1958	22	29	let	let	VERB
ijassa-1958	22	30	us	we	PRON
ijassa-1958	22	31	introduce	introduce	VERB
ijassa-1958	22	32	what	what	PRON
ijassa-1958	22	33	it	it	PRON
ijassa-1958	22	34	means	mean	VERB
ijassa-1958	22	35	for	for	ADP
ijassa-1958	22	36	a	a	DET
ijassa-1958	22	37	metric	metric	NOUN
ijassa-1958	22	38	to	to	PART
ijassa-1958	22	39	be	be	AUX
ijassa-1958	22	40	compatible	compatible	ADJ
ijassa-1958	22	41	with	with	ADP
ijassa-1958	22	42	an	an	DET
ijassa-1958	22	43	order	order	NOUN
ijassa-1958	22	44	.	.	PUNCT
ijassa-1958	23	1	definition	definition	NOUN
ijassa-1958	23	2	2.1	2.1	NUM
ijassa-1958	23	3	:	:	PUNCT
ijassa-1958	23	4	we	we	PRON
ijassa-1958	23	5	will	will	AUX
ijassa-1958	23	6	call	call	VERB
ijassa-1958	23	7	the	the	DET
ijassa-1958	23	8	order	order	NOUN
ijassa-1958	23	9	≤	≤	NOUN
ijassa-1958	23	10	on	on	ADP
ijassa-1958	23	11	x	x	SYM
ijassa-1958	23	12	compatible	compatible	ADJ
ijassa-1958	23	13	with	with	ADP
ijassa-1958	23	14	the	the	DET
ijassa-1958	23	15	metric	metric	ADJ
ijassa-1958	23	16	d	d	PROPN
ijassa-1958	23	17	on	on	ADP
ijassa-1958	23	18	the	the	DET
ijassa-1958	23	19	left	left	NOUN
ijassa-1958	23	20	if	if	SCONJ
ijassa-1958	23	21	d(x	d(x	PROPN
ijassa-1958	23	22	,	,	PUNCT
ijassa-1958	23	23	y	y	NOUN
ijassa-1958	23	24	)	)	PUNCT
ijassa-1958	23	25	≤	≤	NOUN
ijassa-1958	23	26	d(x	d(x	NOUN
ijassa-1958	23	27	,	,	PUNCT
ijassa-1958	23	28	z	z	NOUN
ijassa-1958	23	29	)	)	PUNCT
ijassa-1958	23	30	for	for	ADP
ijassa-1958	23	31	all	all	DET
ijassa-1958	23	32	x	x	NOUN
ijassa-1958	23	33	,	,	PUNCT
ijassa-1958	23	34	y	y	PROPN
ijassa-1958	23	35	,	,	PUNCT
ijassa-1958	23	36	z	z	NOUN
ijassa-1958	23	37	∈	∈	PROPN
ijassa-1958	23	38	x	x	X
ijassa-1958	23	39	:	:	PUNCT
ijassa-1958	23	40	x	x	SYM
ijassa-1958	23	41	≤	≤	NUM
ijassa-1958	23	42	y	y	PROPN
ijassa-1958	23	43	≤	≤	PROPN
ijassa-1958	23	44	z.	z.	PROPN
ijassa-1958	23	45	definition	definition	NOUN
ijassa-1958	23	46	2.2	2.2	NUM
ijassa-1958	23	47	:	:	PUNCT
ijassa-1958	23	48	we	we	PRON
ijassa-1958	23	49	will	will	AUX
ijassa-1958	23	50	call	call	VERB
ijassa-1958	23	51	the	the	DET
ijassa-1958	23	52	order	order	NOUN
ijassa-1958	23	53	≤	≤	NOUN
ijassa-1958	23	54	on	on	ADP
ijassa-1958	23	55	x	x	SYM
ijassa-1958	23	56	compatible	compatible	ADJ
ijassa-1958	23	57	with	with	ADP
ijassa-1958	23	58	the	the	DET
ijassa-1958	23	59	metric	metric	ADJ
ijassa-1958	23	60	d	d	PROPN
ijassa-1958	23	61	on	on	ADP
ijassa-1958	23	62	the	the	DET
ijassa-1958	23	63	right	right	NOUN
ijassa-1958	23	64	if	if	SCONJ
ijassa-1958	23	65	d(y	d(y	PROPN
ijassa-1958	23	66	,	,	PUNCT
ijassa-1958	23	67	z	z	NOUN
ijassa-1958	23	68	)	)	PUNCT
ijassa-1958	23	69	≤	≤	NOUN
ijassa-1958	23	70	d(x	d(x	NOUN
ijassa-1958	23	71	,	,	PUNCT
ijassa-1958	23	72	z	z	NOUN
ijassa-1958	23	73	)	)	PUNCT
ijassa-1958	23	74	for	for	ADP
ijassa-1958	23	75	all	all	DET
ijassa-1958	23	76	x	x	NOUN
ijassa-1958	23	77	,	,	PUNCT
ijassa-1958	23	78	y	y	PROPN
ijassa-1958	23	79	,	,	PUNCT
ijassa-1958	23	80	z	z	NOUN
ijassa-1958	23	81	∈	∈	PROPN
ijassa-1958	23	82	x	x	X
ijassa-1958	23	83	:	:	PUNCT
ijassa-1958	23	84	x	x	SYM
ijassa-1958	23	85	≤	≤	NUM
ijassa-1958	23	86	y	y	PROPN
ijassa-1958	23	87	≤	≤	PROPN
ijassa-1958	23	88	z.	z.	PROPN
ijassa-1958	23	89	definition	definition	NOUN
ijassa-1958	23	90	2.3	2.3	NUM
ijassa-1958	23	91	:	:	PUNCT
ijassa-1958	23	92	we	we	PRON
ijassa-1958	23	93	will	will	AUX
ijassa-1958	23	94	say	say	VERB
ijassa-1958	23	95	that	that	SCONJ
ijassa-1958	23	96	the	the	DET
ijassa-1958	23	97	order	order	NOUN
ijassa-1958	23	98	of	of	ADP
ijassa-1958	23	99	x	x	PUNCT
ijassa-1958	23	100	is	be	AUX
ijassa-1958	23	101	compatible	compatible	ADJ
ijassa-1958	23	102	with	with	ADP
ijassa-1958	23	103	the	the	DET
ijassa-1958	23	104	metric	metric	ADJ
ijassa-1958	23	105	d	d	NOUN
ijassa-1958	23	106	if	if	SCONJ
ijassa-1958	23	107	it	it	PRON
ijassa-1958	23	108	is	be	AUX
ijassa-1958	23	109	compatible	compatible	ADJ
ijassa-1958	23	110	on	on	ADP
ijassa-1958	23	111	the	the	DET
ijassa-1958	23	112	right	right	NOUN
ijassa-1958	23	113	and	and	CCONJ
ijassa-1958	23	114	left	leave	VERB
ijassa-1958	23	115	sides	side	NOUN
ijassa-1958	23	116	.	.	PUNCT
ijassa-1958	24	1	remark	remark	VERB
ijassa-1958	24	2	2.1	2.1	NUM
ijassa-1958	24	3	:	:	PUNCT
ijassa-1958	24	4	further	far	ADV
ijassa-1958	24	5	,	,	PUNCT
ijassa-1958	24	6	we	we	PRON
ijassa-1958	24	7	will	will	AUX
ijassa-1958	24	8	say	say	VERB
ijassa-1958	24	9	both	both	PRON
ijassa-1958	24	10	”	"	PUNCT
ijassa-1958	24	11	the	the	DET
ijassa-1958	24	12	order	order	NOUN
ijassa-1958	24	13	is	be	AUX
ijassa-1958	24	14	compatible	compatible	ADJ
ijassa-1958	24	15	with	with	ADP
ijassa-1958	24	16	the	the	DET
ijassa-1958	24	17	metric	metric	NOUN
ijassa-1958	24	18	”	"	PUNCT
ijassa-1958	24	19	and	and	CCONJ
ijassa-1958	24	20	”	"	PUNCT
ijassa-1958	24	21	the	the	DET
ijassa-1958	24	22	metric	metric	NOUN
ijassa-1958	24	23	is	be	AUX
ijassa-1958	24	24	compatible	compatible	ADJ
ijassa-1958	24	25	with	with	ADP
ijassa-1958	24	26	the	the	DET
ijassa-1958	24	27	order	order	NOUN
ijassa-1958	24	28	”	"	PUNCT
ijassa-1958	24	29	.	.	PUNCT
ijassa-1958	25	1	in	in	ADP
ijassa-1958	25	2	other	other	ADJ
ijassa-1958	25	3	words	word	NOUN
ijassa-1958	25	4	,	,	PUNCT
ijassa-1958	25	5	this	this	DET
ijassa-1958	25	6	property	property	NOUN
ijassa-1958	25	7	connects	connect	VERB
ijassa-1958	25	8	both	both	PRON
ijassa-1958	25	9	of	of	ADP
ijassa-1958	25	10	these	these	DET
ijassa-1958	25	11	structures	structure	NOUN
ijassa-1958	25	12	symmetrically	symmetrically	ADV
ijassa-1958	25	13	.	.	PUNCT
ijassa-1958	26	1	the	the	DET
ijassa-1958	26	2	spaces	space	NOUN
ijassa-1958	26	3	where	where	SCONJ
ijassa-1958	26	4	this	this	DET
ijassa-1958	26	5	property	property	NOUN
ijassa-1958	26	6	holds	hold	VERB
ijassa-1958	26	7	will	will	AUX
ijassa-1958	26	8	be	be	AUX
ijassa-1958	26	9	called	call	VERB
ijassa-1958	26	10	”	"	PUNCT
ijassa-1958	26	11	order	order	NOUN
ijassa-1958	26	12	-	-	PUNCT
ijassa-1958	26	13	compatible	compatible	ADJ
ijassa-1958	26	14	metric	metric	ADJ
ijassa-1958	26	15	spaces	space	NOUN
ijassa-1958	26	16	.	.	PUNCT
ijassa-1958	26	17	”	"	PUNCT
ijassa-1958	27	1	in	in	ADP
ijassa-1958	27	2	this	this	DET
ijassa-1958	27	3	article	article	NOUN
ijassa-1958	27	4	,	,	PUNCT
ijassa-1958	27	5	we	we	PRON
ijassa-1958	27	6	will	will	AUX
ijassa-1958	27	7	focus	focus	VERB
ijassa-1958	27	8	on	on	ADP
ijassa-1958	27	9	studying	study	VERB
ijassa-1958	27	10	the	the	DET
ijassa-1958	27	11	compatibility	compatibility	NOUN
ijassa-1958	27	12	of	of	ADP
ijassa-1958	27	13	certain	certain	ADJ
ijassa-1958	27	14	metrics	metric	NOUN
ijassa-1958	27	15	with	with	ADP
ijassa-1958	27	16	order	order	NOUN
ijassa-1958	27	17	in	in	ADP
ijassa-1958	27	18	the	the	DET
ijassa-1958	27	19	sense	sense	NOUN
ijassa-1958	27	20	of	of	ADP
ijassa-1958	27	21	definition	definition	NOUN
ijassa-1958	27	22	2.3	2.3	NUM
ijassa-1958	27	23	.	.	PUNCT
ijassa-1958	28	1	definition	definition	NOUN
ijassa-1958	28	2	2.4	2.4	NUM
ijassa-1958	28	3	:	:	PUNCT
ijassa-1958	28	4	if	if	SCONJ
ijassa-1958	28	5	x	x	PRON
ijassa-1958	28	6	is	be	AUX
ijassa-1958	28	7	a	a	DET
ijassa-1958	28	8	vector	vector	NOUN
ijassa-1958	28	9	space	space	NOUN
ijassa-1958	28	10	,	,	PUNCT
ijassa-1958	28	11	we	we	PRON
ijassa-1958	28	12	will	will	AUX
ijassa-1958	28	13	call	call	VERB
ijassa-1958	28	14	the	the	DET
ijassa-1958	28	15	order	order	NOUN
ijassa-1958	28	16	≤	≤	NOUN
ijassa-1958	28	17	on	on	ADP
ijassa-1958	28	18	x	x	SYM
ijassa-1958	28	19	compatible	compatible	ADJ
ijassa-1958	28	20	with	with	ADP
ijassa-1958	28	21	the	the	DET
ijassa-1958	28	22	norm	norm	NOUN
ijassa-1958	28	23	||	||	PROPN
ijassa-1958	28	24	·	·	PUNCT
ijassa-1958	29	1	||	||	NUM
ijassa-1958	30	1	:	:	PUNCT
ijassa-1958	30	2	x	x	X
ijassa-1958	30	3	→	→	SYM
ijassa-1958	30	4	r	r	NOUN
ijassa-1958	30	5	if	if	SCONJ
ijassa-1958	30	6	it	it	PRON
ijassa-1958	30	7	is	be	AUX
ijassa-1958	30	8	compatible	compatible	ADJ
ijassa-1958	30	9	with	with	ADP
ijassa-1958	30	10	the	the	DET
ijassa-1958	30	11	metric	metric	ADJ
ijassa-1958	30	12	d(x	d(x	PROPN
ijassa-1958	30	13	,	,	PUNCT
ijassa-1958	30	14	y	y	NOUN
ijassa-1958	30	15	)	)	PUNCT
ijassa-1958	30	16	=	=	PUNCT
ijassa-1958	31	1	||x−	||x−	NOUN
ijassa-1958	31	2	y||	y||	NOUN
ijassa-1958	31	3	induced	induce	VERB
ijassa-1958	31	4	by	by	ADP
ijassa-1958	31	5	it	it	PRON
ijassa-1958	31	6	.	.	PUNCT
ijassa-1958	32	1	remark	remark	VERB
ijassa-1958	32	2	2.2	2.2	NUM
ijassa-1958	32	3	:	:	PUNCT
ijassa-1958	32	4	it	it	PRON
ijassa-1958	32	5	is	be	AUX
ijassa-1958	32	6	easy	easy	ADJ
ijassa-1958	32	7	to	to	PART
ijassa-1958	32	8	see	see	VERB
ijassa-1958	32	9	that	that	SCONJ
ijassa-1958	32	10	the	the	DET
ijassa-1958	32	11	metric	metric	ADJ
ijassa-1958	32	12	d	d	PROPN
ijassa-1958	32	13	in	in	ADP
ijassa-1958	32	14	the	the	DET
ijassa-1958	32	15	space	space	NOUN
ijassa-1958	32	16	x	x	PUNCT
ijassa-1958	32	17	is	be	AUX
ijassa-1958	32	18	compatible	compatible	ADJ
ijassa-1958	32	19	with	with	ADP
ijassa-1958	32	20	the	the	DET
ijassa-1958	32	21	order	order	NOUN
ijassa-1958	32	22	≤	≤	NOUN
ijassa-1958	33	1	if	if	SCONJ
ijassa-1958	33	2	and	and	CCONJ
ijassa-1958	33	3	only	only	ADV
ijassa-1958	33	4	if	if	SCONJ
ijassa-1958	33	5	d(x	d(x	PROPN
ijassa-1958	33	6	,	,	PUNCT
ijassa-1958	33	7	z	z	NOUN
ijassa-1958	33	8	)	)	PUNCT
ijassa-1958	33	9	≥	≥	NOUN
ijassa-1958	33	10	max{d(y	max{d(y	PROPN
ijassa-1958	33	11	,	,	PUNCT
ijassa-1958	33	12	z	z	NOUN
ijassa-1958	33	13	)	)	PUNCT
ijassa-1958	33	14	,	,	PUNCT
ijassa-1958	33	15	d(x	d(x	PROPN
ijassa-1958	33	16	,	,	PUNCT
ijassa-1958	33	17	y	y	NOUN
ijassa-1958	33	18	)	)	PUNCT
ijassa-1958	33	19	}	}	PUNCT
ijassa-1958	33	20	∀x	∀x	NUM
ijassa-1958	33	21	,	,	PUNCT
ijassa-1958	33	22	y	y	PROPN
ijassa-1958	33	23	,	,	PUNCT
ijassa-1958	33	24	z	z	NOUN
ijassa-1958	33	25	∈	∈	PROPN
ijassa-1958	33	26	x	x	X
ijassa-1958	33	27	:	:	PUNCT
ijassa-1958	33	28	x	x	SYM
ijassa-1958	33	29	≤	≤	NUM
ijassa-1958	33	30	y	y	PROPN
ijassa-1958	33	31	≤	≤	PROPN
ijassa-1958	33	32	z.	z.	PROPN
ijassa-1958	34	1	obviously	obviously	ADV
ijassa-1958	34	2	,	,	PUNCT
ijassa-1958	34	3	r	r	NOUN
ijassa-1958	34	4	is	be	AUX
ijassa-1958	34	5	an	an	DET
ijassa-1958	34	6	order	order	NOUN
ijassa-1958	34	7	-	-	PUNCT
ijassa-1958	34	8	compatible	compatible	ADJ
ijassa-1958	34	9	metric	metric	ADJ
ijassa-1958	34	10	space	space	NOUN
ijassa-1958	34	11	.	.	PUNCT
ijassa-1958	35	1	in	in	ADP
ijassa-1958	35	2	addition	addition	NOUN
ijassa-1958	35	3	,	,	PUNCT
ijassa-1958	35	4	there	there	PRON
ijassa-1958	35	5	are	be	VERB
ijassa-1958	35	6	other	other	ADJ
ijassa-1958	35	7	examples	example	NOUN
ijassa-1958	35	8	of	of	ADP
ijassa-1958	35	9	order	order	NOUN
ijassa-1958	35	10	-	-	PUNCT
ijassa-1958	35	11	compatible	compatible	ADJ
ijassa-1958	35	12	metric	metric	ADJ
ijassa-1958	35	13	spaces	space	NOUN
ijassa-1958	35	14	.	.	PUNCT
ijassa-1958	36	1	a	a	DET
ijassa-1958	36	2	trivial	trivial	ADJ
ijassa-1958	36	3	example	example	NOUN
ijassa-1958	36	4	is	be	AUX
ijassa-1958	36	5	a	a	DET
ijassa-1958	36	6	discrete	discrete	ADJ
ijassa-1958	36	7	metric	metric	ADJ
ijassa-1958	36	8	space	space	NOUN
ijassa-1958	36	9	with	with	ADP
ijassa-1958	36	10	an	an	DET
ijassa-1958	36	11	arbitrary	arbitrary	ADJ
ijassa-1958	36	12	linear	linear	ADJ
ijassa-1958	36	13	order	order	NOUN
ijassa-1958	36	14	.	.	PUNCT
ijassa-1958	37	1	the	the	DET
ijassa-1958	37	2	following	follow	VERB
ijassa-1958	37	3	lemma	lemma	PROPN
ijassa-1958	37	4	follows	follow	VERB
ijassa-1958	37	5	from	from	ADP
ijassa-1958	37	6	the	the	DET
ijassa-1958	37	7	definition	definition	NOUN
ijassa-1958	37	8	of	of	ADP
ijassa-1958	37	9	compatibility	compatibility	NOUN
ijassa-1958	37	10	.	.	PUNCT
ijassa-1958	38	1	lemma	lemma	PROPN
ijassa-1958	38	2	2.1	2.1	NUM
ijassa-1958	38	3	:	:	PUNCT
ijassa-1958	38	4	in	in	ADP
ijassa-1958	38	5	the	the	DET
ijassa-1958	38	6	metric	metric	ADJ
ijassa-1958	38	7	space	space	NOUN
ijassa-1958	38	8	(	(	PUNCT
ijassa-1958	38	9	x	x	X
ijassa-1958	38	10	,	,	PUNCT
ijassa-1958	38	11	d	d	PROPN
ijassa-1958	38	12	)	)	PUNCT
ijassa-1958	38	13	,	,	PUNCT
ijassa-1958	38	14	let	let	VERB
ijassa-1958	38	15	the	the	DET
ijassa-1958	38	16	metric	metric	NOUN
ijassa-1958	38	17	d	d	AUX
ijassa-1958	38	18	be	be	AUX
ijassa-1958	38	19	compatible	compatible	ADJ
ijassa-1958	38	20	with	with	ADP
ijassa-1958	38	21	the	the	DET
ijassa-1958	38	22	order	order	NOUN
ijassa-1958	38	23	≤.	≤.	NOUN
ijassa-1958	38	24	then	then	ADV
ijassa-1958	38	25	the	the	DET
ijassa-1958	38	26	induced	induced	ADJ
ijassa-1958	38	27	metric	metric	NOUN
ijassa-1958	38	28	on	on	ADP
ijassa-1958	38	29	any	any	DET
ijassa-1958	38	30	subset	subset	NOUN
ijassa-1958	38	31	of	of	ADP
ijassa-1958	38	32	x	x	PRON
ijassa-1958	38	33	will	will	AUX
ijassa-1958	38	34	be	be	AUX
ijassa-1958	38	35	compatible	compatible	ADJ
ijassa-1958	38	36	with	with	ADP
ijassa-1958	38	37	the	the	DET
ijassa-1958	38	38	induced	induced	ADJ
ijassa-1958	38	39	order	order	NOUN
ijassa-1958	38	40	.	.	PUNCT
ijassa-1958	39	1	recall	recall	VERB
ijassa-1958	39	2	that	that	PRON
ijassa-1958	39	3	on	on	ADP
ijassa-1958	39	4	a	a	DET
ijassa-1958	39	5	linearly	linearly	ADV
ijassa-1958	39	6	ordered	order	VERB
ijassa-1958	39	7	set	set	NOUN
ijassa-1958	39	8	(	(	PUNCT
ijassa-1958	39	9	x	x	X
ijassa-1958	39	10	,	,	PUNCT
ijassa-1958	39	11	<	<	X
ijassa-1958	39	12	)	)	PUNCT
ijassa-1958	39	13	with	with	ADP
ijassa-1958	39	14	strict	strict	ADJ
ijassa-1958	39	15	order	order	NOUN
ijassa-1958	39	16	<	<	X
ijassa-1958	39	17	,	,	PUNCT
ijassa-1958	39	18	we	we	PRON
ijassa-1958	39	19	can	can	AUX
ijassa-1958	39	20	define	define	VERB
ijassa-1958	39	21	an	an	DET
ijassa-1958	39	22	order	order	NOUN
ijassa-1958	39	23	topology	topology	NOUN
ijassa-1958	40	1	[	[	X
ijassa-1958	40	2	7	7	X
ijassa-1958	40	3	]	]	PUNCT
ijassa-1958	40	4	as	as	ADP
ijassa-1958	40	5	one	one	NUM
ijassa-1958	40	6	with	with	ADP
ijassa-1958	40	7	a	a	DET
ijassa-1958	40	8	pre	pre	NOUN
ijassa-1958	40	9	-	-	NOUN
ijassa-1958	40	10	base	base	NOUN
ijassa-1958	40	11	of	of	ADP
ijassa-1958	40	12	”	"	PUNCT
ijassa-1958	40	13	open	open	ADJ
ijassa-1958	40	14	rays	ray	NOUN
ijassa-1958	40	15	”	"	PUNCT
ijassa-1958	40	16	{	{	PUNCT
ijassa-1958	40	17	x	x	X
ijassa-1958	40	18	|	|	ADV
ijassa-1958	40	19	a	a	DET
ijassa-1958	40	20	<	<	X
ijassa-1958	40	21	x	x	X
ijassa-1958	40	22	}	}	PUNCT
ijassa-1958	40	23	,	,	PUNCT
ijassa-1958	40	24	{	{	PUNCT
ijassa-1958	40	25	x	x	X
ijassa-1958	41	1	|	|	ADV
ijassa-1958	41	2	x	x	X
ijassa-1958	41	3	<	<	X
ijassa-1958	41	4	b	b	X
ijassa-1958	41	5	}	}	PUNCT
ijassa-1958	41	6	,	,	PUNCT
ijassa-1958	41	7	a	a	DET
ijassa-1958	41	8	,	,	PUNCT
ijassa-1958	41	9	b	b	X
ijassa-1958	41	10	∈	∈	PROPN
ijassa-1958	41	11	x.	x.	NOUN
ijassa-1958	41	12	we	we	PRON
ijassa-1958	41	13	might	might	AUX
ijassa-1958	41	14	assume	assume	VERB
ijassa-1958	41	15	that	that	SCONJ
ijassa-1958	41	16	the	the	DET
ijassa-1958	41	17	definition	definition	NOUN
ijassa-1958	41	18	of	of	ADP
ijassa-1958	41	19	compatibility	compatibility	NOUN
ijassa-1958	41	20	between	between	ADP
ijassa-1958	41	21	metric	metric	ADJ
ijassa-1958	41	22	spaces	space	NOUN
ijassa-1958	41	23	and	and	CCONJ
ijassa-1958	41	24	linear	linear	ADJ
ijassa-1958	41	25	orders	order	NOUN
ijassa-1958	41	26	reduces	reduce	VERB
ijassa-1958	41	27	to	to	ADP
ijassa-1958	41	28	an	an	DET
ijassa-1958	41	29	order	order	NOUN
ijassa-1958	41	30	topology	topology	NOUN
ijassa-1958	41	31	.	.	PUNCT
ijassa-1958	42	1	we	we	PRON
ijassa-1958	42	2	will	will	AUX
ijassa-1958	42	3	show	show	VERB
ijassa-1958	42	4	that	that	SCONJ
ijassa-1958	42	5	this	this	PRON
ijassa-1958	42	6	is	be	AUX
ijassa-1958	42	7	not	not	PART
ijassa-1958	42	8	the	the	DET
ijassa-1958	42	9	case	case	NOUN
ijassa-1958	42	10	.	.	PUNCT
ijassa-1958	43	1	remark	remark	VERB
ijassa-1958	43	2	2.3	2.3	NUM
ijassa-1958	43	3	:	:	PUNCT
ijassa-1958	43	4	metrizable	metrizable	ADJ
ijassa-1958	43	5	order	order	NOUN
ijassa-1958	43	6	topologies	topology	NOUN
ijassa-1958	43	7	and	and	CCONJ
ijassa-1958	43	8	metric	metric	ADJ
ijassa-1958	43	9	spaces	space	NOUN
ijassa-1958	43	10	in	in	ADP
ijassa-1958	43	11	which	which	PRON
ijassa-1958	43	12	the	the	DET
ijassa-1958	43	13	metric	metric	NOUN
ijassa-1958	43	14	is	be	AUX
ijassa-1958	43	15	compatible	compatible	ADJ
ijassa-1958	43	16	with	with	ADP
ijassa-1958	43	17	copyright	copyright	NOUN
ijassa-1958	43	18	©	©	PROPN
ijassa-1958	43	19	2024	2024	NUM
ijassa-1958	43	20	assa	assa	NOUN
ijassa-1958	43	21	.	.	PUNCT
ijassa-1958	44	1	adv	adv	PROPN
ijassa-1958	44	2	syst	syst	PROPN
ijassa-1958	44	3	sci	sci	PROPN
ijassa-1958	44	4	appl	appl	PROPN
ijassa-1958	44	5	(	(	PUNCT
ijassa-1958	44	6	2024	2024	NUM
ijassa-1958	44	7	)	)	PUNCT
ijassa-1958	44	8	on	on	ADP
ijassa-1958	44	9	the	the	DET
ijassa-1958	44	10	compatibility	compatibility	NOUN
ijassa-1958	44	11	of	of	ADP
ijassa-1958	44	12	metric	metric	ADJ
ijassa-1958	44	13	and	and	CCONJ
ijassa-1958	44	14	linear	linear	ADJ
ijassa-1958	44	15	order	order	NOUN
ijassa-1958	44	16	3	3	NUM
ijassa-1958	44	17	the	the	DET
ijassa-1958	44	18	order	order	NOUN
ijassa-1958	44	19	do	do	AUX
ijassa-1958	44	20	not	not	PART
ijassa-1958	44	21	”	"	PUNCT
ijassa-1958	44	22	naturally	naturally	ADV
ijassa-1958	44	23	”	"	PUNCT
ijassa-1958	44	24	reduce	reduce	VERB
ijassa-1958	44	25	to	to	ADP
ijassa-1958	44	26	each	each	DET
ijassa-1958	44	27	other	other	ADJ
ijassa-1958	44	28	.	.	PUNCT
ijassa-1958	45	1	it	it	PRON
ijassa-1958	45	2	is	be	AUX
ijassa-1958	45	3	not	not	PART
ijassa-1958	45	4	difficult	difficult	ADJ
ijassa-1958	45	5	to	to	PART
ijassa-1958	45	6	see	see	VERB
ijassa-1958	45	7	that	that	SCONJ
ijassa-1958	45	8	we	we	PRON
ijassa-1958	45	9	can	can	AUX
ijassa-1958	45	10	unambiguously	unambiguously	ADV
ijassa-1958	45	11	obtain	obtain	VERB
ijassa-1958	45	12	an	an	DET
ijassa-1958	45	13	order	order	NOUN
ijassa-1958	45	14	topology	topology	NOUN
ijassa-1958	45	15	from	from	ADP
ijassa-1958	45	16	an	an	DET
ijassa-1958	45	17	order	order	NOUN
ijassa-1958	45	18	and	and	CCONJ
ijassa-1958	45	19	an	an	DET
ijassa-1958	45	20	order	order	NOUN
ijassa-1958	45	21	from	from	ADP
ijassa-1958	45	22	an	an	DET
ijassa-1958	45	23	order	order	NOUN
ijassa-1958	45	24	topology	topology	NOUN
ijassa-1958	45	25	.	.	PUNCT
ijassa-1958	46	1	unlike	unlike	ADP
ijassa-1958	46	2	defining	define	VERB
ijassa-1958	46	3	an	an	DET
ijassa-1958	46	4	order	order	NOUN
ijassa-1958	46	5	topology	topology	NOUN
ijassa-1958	46	6	,	,	PUNCT
ijassa-1958	46	7	an	an	DET
ijassa-1958	46	8	order	order	NOUN
ijassa-1958	46	9	can	can	AUX
ijassa-1958	46	10	be	be	AUX
ijassa-1958	46	11	compatible	compatible	ADJ
ijassa-1958	46	12	with	with	ADP
ijassa-1958	46	13	different	different	ADJ
ijassa-1958	46	14	metrics	metric	NOUN
ijassa-1958	46	15	,	,	PUNCT
ijassa-1958	46	16	including	include	VERB
ijassa-1958	46	17	from	from	ADP
ijassa-1958	46	18	the	the	DET
ijassa-1958	46	19	point	point	NOUN
ijassa-1958	46	20	of	of	ADP
ijassa-1958	46	21	view	view	NOUN
ijassa-1958	46	22	of	of	ADP
ijassa-1958	46	23	the	the	DET
ijassa-1958	46	24	generated	generate	VERB
ijassa-1958	46	25	topology	topology	NOUN
ijassa-1958	46	26	,	,	PUNCT
ijassa-1958	46	27	and	and	CCONJ
ijassa-1958	46	28	this	this	DET
ijassa-1958	46	29	metric	metric	NOUN
ijassa-1958	46	30	can	can	AUX
ijassa-1958	46	31	be	be	AUX
ijassa-1958	46	32	compatible	compatible	ADJ
ijassa-1958	46	33	with	with	ADP
ijassa-1958	46	34	different	different	ADJ
ijassa-1958	46	35	orders	order	NOUN
ijassa-1958	46	36	.	.	PUNCT
ijassa-1958	47	1	for	for	ADP
ijassa-1958	47	2	example	example	NOUN
ijassa-1958	47	3	,	,	PUNCT
ijassa-1958	47	4	the	the	DET
ijassa-1958	47	5	discrete	discrete	ADJ
ijassa-1958	47	6	metric	metric	NOUN
ijassa-1958	47	7	and	and	CCONJ
ijassa-1958	47	8	the	the	DET
ijassa-1958	47	9	euclidean	euclidean	ADJ
ijassa-1958	47	10	metric	metric	NOUN
ijassa-1958	47	11	on	on	ADP
ijassa-1958	47	12	a	a	DET
ijassa-1958	47	13	straight	straight	ADJ
ijassa-1958	47	14	line	line	NOUN
ijassa-1958	47	15	are	be	AUX
ijassa-1958	47	16	compatible	compatible	ADJ
ijassa-1958	47	17	with	with	ADP
ijassa-1958	47	18	the	the	DET
ijassa-1958	47	19	standard	standard	ADJ
ijassa-1958	47	20	linear	linear	ADJ
ijassa-1958	47	21	order	order	NOUN
ijassa-1958	47	22	on	on	ADP
ijassa-1958	47	23	a	a	DET
ijassa-1958	47	24	straight	straight	ADJ
ijassa-1958	47	25	line	line	NOUN
ijassa-1958	47	26	,	,	PUNCT
ijassa-1958	47	27	but	but	CCONJ
ijassa-1958	47	28	they	they	PRON
ijassa-1958	47	29	generate	generate	VERB
ijassa-1958	47	30	different	different	ADJ
ijassa-1958	47	31	topologies	topology	NOUN
ijassa-1958	47	32	.	.	PUNCT
ijassa-1958	48	1	further	far	ADV
ijassa-1958	48	2	,	,	PUNCT
ijassa-1958	48	3	a	a	DET
ijassa-1958	48	4	discrete	discrete	ADJ
ijassa-1958	48	5	metric	metric	NOUN
ijassa-1958	48	6	on	on	ADP
ijassa-1958	48	7	any	any	DET
ijassa-1958	48	8	set	set	NOUN
ijassa-1958	48	9	is	be	AUX
ijassa-1958	48	10	compatible	compatible	ADJ
ijassa-1958	48	11	with	with	ADP
ijassa-1958	48	12	any	any	DET
ijassa-1958	48	13	order	order	NOUN
ijassa-1958	48	14	on	on	ADP
ijassa-1958	48	15	that	that	DET
ijassa-1958	48	16	set	set	NOUN
ijassa-1958	48	17	.	.	PUNCT
ijassa-1958	49	1	3	3	X
ijassa-1958	49	2	.	.	X
ijassa-1958	49	3	incompatibility	incompatibility	NOUN
ijassa-1958	49	4	with	with	ADP
ijassa-1958	49	5	order	order	NOUN
ijassa-1958	49	6	of	of	ADP
ijassa-1958	49	7	classic	classic	ADJ
ijassa-1958	49	8	multidimensional	multidimensional	ADJ
ijassa-1958	49	9	spaces	space	NOUN
ijassa-1958	49	10	in	in	ADP
ijassa-1958	49	11	this	this	DET
ijassa-1958	49	12	section	section	NOUN
ijassa-1958	49	13	,	,	PUNCT
ijassa-1958	49	14	we	we	PRON
ijassa-1958	49	15	will	will	AUX
ijassa-1958	49	16	construct	construct	VERB
ijassa-1958	49	17	examples	example	NOUN
ijassa-1958	49	18	of	of	ADP
ijassa-1958	49	19	metric	metric	ADJ
ijassa-1958	49	20	spaces	space	NOUN
ijassa-1958	49	21	where	where	SCONJ
ijassa-1958	49	22	it	it	PRON
ijassa-1958	49	23	is	be	AUX
ijassa-1958	49	24	impossible	impossible	ADJ
ijassa-1958	49	25	to	to	PART
ijassa-1958	49	26	define	define	VERB
ijassa-1958	49	27	any	any	DET
ijassa-1958	49	28	linear	linear	ADJ
ijassa-1958	49	29	order	order	NOUN
ijassa-1958	49	30	that	that	PRON
ijassa-1958	49	31	is	be	AUX
ijassa-1958	49	32	compatible	compatible	ADJ
ijassa-1958	49	33	with	with	ADP
ijassa-1958	49	34	the	the	DET
ijassa-1958	49	35	metric	metric	NOUN
ijassa-1958	49	36	.	.	PUNCT
ijassa-1958	50	1	we	we	PRON
ijassa-1958	50	2	will	will	AUX
ijassa-1958	50	3	demonstrate	demonstrate	VERB
ijassa-1958	50	4	for	for	ADP
ijassa-1958	50	5	a	a	DET
ijassa-1958	50	6	broad	broad	ADJ
ijassa-1958	50	7	range	range	NOUN
ijassa-1958	50	8	of	of	ADP
ijassa-1958	50	9	multidimensional	multidimensional	ADJ
ijassa-1958	50	10	normed	normed	ADJ
ijassa-1958	50	11	spaces	space	NOUN
ijassa-1958	50	12	that	that	SCONJ
ijassa-1958	50	13	it	it	PRON
ijassa-1958	50	14	is	be	AUX
ijassa-1958	50	15	not	not	PART
ijassa-1958	50	16	possible	possible	ADJ
ijassa-1958	50	17	to	to	PART
ijassa-1958	50	18	establish	establish	VERB
ijassa-1958	50	19	any	any	DET
ijassa-1958	50	20	linear	linear	ADJ
ijassa-1958	50	21	order	order	NOUN
ijassa-1958	50	22	that	that	PRON
ijassa-1958	50	23	is	be	AUX
ijassa-1958	50	24	compatible	compatible	ADJ
ijassa-1958	50	25	with	with	ADP
ijassa-1958	50	26	their	their	PRON
ijassa-1958	50	27	norm	norm	NOUN
ijassa-1958	50	28	.	.	PUNCT
ijassa-1958	51	1	proposition	proposition	NOUN
ijassa-1958	51	2	3.1	3.1	NUM
ijassa-1958	51	3	:	:	PUNCT
ijassa-1958	51	4	for	for	ADP
ijassa-1958	51	5	any	any	DET
ijassa-1958	51	6	metric	metric	ADJ
ijassa-1958	51	7	space	space	NOUN
ijassa-1958	51	8	with	with	ADP
ijassa-1958	51	9	at	at	ADP
ijassa-1958	51	10	most	most	ADV
ijassa-1958	51	11	three	three	NUM
ijassa-1958	51	12	points	point	NOUN
ijassa-1958	51	13	,	,	PUNCT
ijassa-1958	51	14	you	you	PRON
ijassa-1958	51	15	can	can	AUX
ijassa-1958	51	16	always	always	ADV
ijassa-1958	51	17	create	create	VERB
ijassa-1958	51	18	an	an	DET
ijassa-1958	51	19	order	order	NOUN
ijassa-1958	51	20	that	that	PRON
ijassa-1958	51	21	is	be	AUX
ijassa-1958	51	22	compatible	compatible	ADJ
ijassa-1958	51	23	with	with	ADP
ijassa-1958	51	24	the	the	DET
ijassa-1958	51	25	metric	metric	NOUN
ijassa-1958	51	26	.	.	PUNCT
ijassa-1958	52	1	it	it	PRON
ijassa-1958	52	2	turns	turn	VERB
ijassa-1958	52	3	out	out	ADP
ijassa-1958	52	4	that	that	SCONJ
ijassa-1958	52	5	this	this	PRON
ijassa-1958	52	6	is	be	AUX
ijassa-1958	52	7	no	no	ADV
ijassa-1958	52	8	longer	long	ADV
ijassa-1958	52	9	always	always	ADV
ijassa-1958	52	10	true	true	ADJ
ijassa-1958	52	11	for	for	ADP
ijassa-1958	52	12	four	four	NUM
ijassa-1958	52	13	points	point	NOUN
ijassa-1958	52	14	.	.	PUNCT
ijassa-1958	53	1	theorem	theorem	VERB
ijassa-1958	53	2	3.1	3.1	NUM
ijassa-1958	53	3	:	:	PUNCT
ijassa-1958	53	4	let	let	VERB
ijassa-1958	53	5	there	there	PRON
ijassa-1958	53	6	be	be	AUX
ijassa-1958	53	7	four	four	NUM
ijassa-1958	53	8	distinct	distinct	ADJ
ijassa-1958	53	9	points	point	NOUN
ijassa-1958	53	10	x	x	NOUN
ijassa-1958	53	11	,	,	PUNCT
ijassa-1958	53	12	y	y	PROPN
ijassa-1958	53	13	,	,	PUNCT
ijassa-1958	53	14	z	z	PROPN
ijassa-1958	53	15	,	,	PUNCT
ijassa-1958	53	16	t	t	PROPN
ijassa-1958	53	17	in	in	ADP
ijassa-1958	53	18	the	the	DET
ijassa-1958	53	19	metric	metric	ADJ
ijassa-1958	53	20	space	space	NOUN
ijassa-1958	53	21	m	m	VERB
ijassa-1958	53	22	such	such	ADJ
ijassa-1958	53	23	that	that	SCONJ
ijassa-1958	53	24	min{d(x	min{d(x	PROPN
ijassa-1958	53	25	,	,	PUNCT
ijassa-1958	53	26	z	z	NOUN
ijassa-1958	53	27	)	)	PUNCT
ijassa-1958	53	28	,	,	PUNCT
ijassa-1958	53	29	d(y	d(y	PROPN
ijassa-1958	53	30	,	,	PUNCT
ijassa-1958	53	31	t	t	NOUN
ijassa-1958	53	32	)	)	PUNCT
ijassa-1958	53	33	}	}	PUNCT
ijassa-1958	53	34	≥	≥	NUM
ijassa-1958	53	35	max{d(y	max{d(y	PROPN
ijassa-1958	53	36	,	,	PUNCT
ijassa-1958	53	37	z	z	NOUN
ijassa-1958	53	38	)	)	PUNCT
ijassa-1958	53	39	,	,	PUNCT
ijassa-1958	53	40	d(x	d(x	PROPN
ijassa-1958	53	41	,	,	PUNCT
ijassa-1958	53	42	y	y	PROPN
ijassa-1958	53	43	)	)	PUNCT
ijassa-1958	53	44	,	,	PUNCT
ijassa-1958	53	45	d(x	d(x	PROPN
ijassa-1958	53	46	,	,	PUNCT
ijassa-1958	53	47	t	t	PROPN
ijassa-1958	53	48	)	)	PUNCT
ijassa-1958	53	49	,	,	PUNCT
ijassa-1958	53	50	d(t	d(t	PROPN
ijassa-1958	53	51	,	,	PUNCT
ijassa-1958	53	52	z	z	NOUN
ijassa-1958	53	53	)	)	PUNCT
ijassa-1958	53	54	}	}	PUNCT
ijassa-1958	53	55	.	.	PUNCT
ijassa-1958	54	1	then	then	ADV
ijassa-1958	54	2	there	there	PRON
ijassa-1958	54	3	does	do	AUX
ijassa-1958	54	4	not	not	PART
ijassa-1958	54	5	exist	exist	VERB
ijassa-1958	54	6	an	an	DET
ijassa-1958	54	7	order	order	NOUN
ijassa-1958	54	8	that	that	PRON
ijassa-1958	54	9	is	be	AUX
ijassa-1958	54	10	compatible	compatible	ADJ
ijassa-1958	54	11	with	with	ADP
ijassa-1958	54	12	the	the	DET
ijassa-1958	54	13	metric	metric	NOUN
ijassa-1958	54	14	.	.	PUNCT
ijassa-1958	55	1	proof	proof	NOUN
ijassa-1958	55	2	let	let	VERB
ijassa-1958	55	3	us	we	PRON
ijassa-1958	55	4	assume	assume	VERB
ijassa-1958	55	5	that	that	SCONJ
ijassa-1958	55	6	in	in	ADP
ijassa-1958	55	7	m	m	DET
ijassa-1958	55	8	an	an	DET
ijassa-1958	55	9	order	order	NOUN
ijassa-1958	55	10	-	-	PUNCT
ijassa-1958	55	11	compatible	compatible	ADJ
ijassa-1958	55	12	metric	metric	NOUN
ijassa-1958	55	13	can	can	AUX
ijassa-1958	55	14	be	be	AUX
ijassa-1958	55	15	introduced	introduce	VERB
ijassa-1958	55	16	.	.	PUNCT
ijassa-1958	56	1	since	since	SCONJ
ijassa-1958	56	2	d(x	d(x	PROPN
ijassa-1958	56	3	,	,	PUNCT
ijassa-1958	56	4	z	z	NOUN
ijassa-1958	56	5	)	)	PUNCT
ijassa-1958	56	6	≥	≥	NOUN
ijassa-1958	56	7	max{d(y	max{d(y	PROPN
ijassa-1958	56	8	,	,	PUNCT
ijassa-1958	56	9	z	z	NOUN
ijassa-1958	56	10	)	)	PUNCT
ijassa-1958	56	11	,	,	PUNCT
ijassa-1958	56	12	d(x	d(x	PROPN
ijassa-1958	56	13	,	,	PUNCT
ijassa-1958	56	14	y	y	PROPN
ijassa-1958	56	15	)	)	PUNCT
ijassa-1958	56	16	}	}	PUNCT
ijassa-1958	56	17	by	by	ADP
ijassa-1958	56	18	the	the	DET
ijassa-1958	56	19	definition	definition	NOUN
ijassa-1958	56	20	of	of	ADP
ijassa-1958	56	21	compatibility	compatibility	NOUN
ijassa-1958	56	22	,	,	PUNCT
ijassa-1958	56	23	we	we	PRON
ijassa-1958	56	24	have	have	VERB
ijassa-1958	56	25	x	x	NOUN
ijassa-1958	56	26	≤	≤	NUM
ijassa-1958	56	27	y	y	PROPN
ijassa-1958	56	28	≤	≤	PROPN
ijassa-1958	56	29	z	z	NOUN
ijassa-1958	56	30	or	or	CCONJ
ijassa-1958	56	31	z	z	NOUN
ijassa-1958	56	32	≤	≤	NOUN
ijassa-1958	56	33	y	y	PROPN
ijassa-1958	56	34	≤	≤	PROPN
ijassa-1958	56	35	x.	x.	NOUN
ijassa-1958	56	36	without	without	ADP
ijassa-1958	56	37	loss	loss	NOUN
ijassa-1958	56	38	of	of	ADP
ijassa-1958	56	39	generality	generality	NOUN
ijassa-1958	56	40	,	,	PUNCT
ijassa-1958	56	41	we	we	PRON
ijassa-1958	56	42	will	will	AUX
ijassa-1958	56	43	assume	assume	VERB
ijassa-1958	56	44	that	that	SCONJ
ijassa-1958	56	45	x	x	X
ijassa-1958	56	46	≤	≤	NUM
ijassa-1958	56	47	y	y	PROPN
ijassa-1958	56	48	≤	≤	PROPN
ijassa-1958	56	49	z.	z.	PROPN
ijassa-1958	56	50	therefore	therefore	ADV
ijassa-1958	56	51	,	,	PUNCT
ijassa-1958	56	52	since	since	SCONJ
ijassa-1958	56	53	d(x	d(x	PROPN
ijassa-1958	56	54	,	,	PUNCT
ijassa-1958	56	55	z	z	NOUN
ijassa-1958	56	56	)	)	PUNCT
ijassa-1958	56	57	≥	≥	NOUN
ijassa-1958	56	58	max{d(x	max{d(x	PROPN
ijassa-1958	56	59	,	,	PUNCT
ijassa-1958	56	60	t	t	PROPN
ijassa-1958	56	61	)	)	PUNCT
ijassa-1958	56	62	,	,	PUNCT
ijassa-1958	56	63	d(t	d(t	PROPN
ijassa-1958	56	64	,	,	PUNCT
ijassa-1958	56	65	z	z	NOUN
ijassa-1958	56	66	)	)	PUNCT
ijassa-1958	56	67	}	}	PUNCT
ijassa-1958	56	68	,	,	PUNCT
ijassa-1958	56	69	we	we	PRON
ijassa-1958	56	70	have	have	VERB
ijassa-1958	56	71	that	that	PRON
ijassa-1958	56	72	x	x	SYM
ijassa-1958	56	73	≤	≤	PROPN
ijassa-1958	56	74	t	t	PROPN
ijassa-1958	56	75	≤	≤	PROPN
ijassa-1958	56	76	z.	z.	PROPN
ijassa-1958	56	77	since	since	SCONJ
ijassa-1958	56	78	d(y	d(y	PROPN
ijassa-1958	56	79	,	,	PUNCT
ijassa-1958	56	80	t	t	PROPN
ijassa-1958	56	81	)	)	PUNCT
ijassa-1958	56	82	≥	≥	PROPN
ijassa-1958	56	83	max{d(y	max{d(y	PROPN
ijassa-1958	56	84	,	,	PUNCT
ijassa-1958	56	85	x	x	NOUN
ijassa-1958	56	86	)	)	PUNCT
ijassa-1958	56	87	,	,	PUNCT
ijassa-1958	56	88	d(t	d(t	PROPN
ijassa-1958	56	89	,	,	PUNCT
ijassa-1958	56	90	x	x	NOUN
ijassa-1958	56	91	)	)	PUNCT
ijassa-1958	56	92	}	}	PUNCT
ijassa-1958	56	93	,	,	PUNCT
ijassa-1958	56	94	we	we	PRON
ijassa-1958	56	95	have	have	VERB
ijassa-1958	56	96	t	t	NOUN
ijassa-1958	56	97	≤	≤	NUM
ijassa-1958	56	98	x	x	PUNCT
ijassa-1958	56	99	≤	≤	NUM
ijassa-1958	56	100	y	y	PROPN
ijassa-1958	56	101	or	or	CCONJ
ijassa-1958	56	102	y	y	PROPN
ijassa-1958	56	103	≤	≤	NUM
ijassa-1958	56	104	x	x	PUNCT
ijassa-1958	56	105	≤	≤	NUM
ijassa-1958	56	106	t.	t.	NOUN
ijassa-1958	56	107	it	it	PRON
ijassa-1958	56	108	follows	follow	VERB
ijassa-1958	56	109	that	that	SCONJ
ijassa-1958	56	110	x	x	X
ijassa-1958	56	111	=	=	SYM
ijassa-1958	56	112	t	t	PROPN
ijassa-1958	56	113	or	or	CCONJ
ijassa-1958	56	114	x	x	X
ijassa-1958	56	115	=	=	SYM
ijassa-1958	56	116	y.	y.	PROPN
ijassa-1958	56	117	hence	hence	ADV
ijassa-1958	56	118	,	,	PUNCT
ijassa-1958	56	119	we	we	PRON
ijassa-1958	56	120	have	have	AUX
ijassa-1958	56	121	obtained	obtain	VERB
ijassa-1958	56	122	a	a	DET
ijassa-1958	56	123	contradiction	contradiction	NOUN
ijassa-1958	56	124	.	.	PUNCT
ijassa-1958	57	1	this	this	DET
ijassa-1958	57	2	construction	construction	NOUN
ijassa-1958	57	3	can	can	AUX
ijassa-1958	57	4	be	be	AUX
ijassa-1958	57	5	used	use	VERB
ijassa-1958	57	6	further	far	ADV
ijassa-1958	57	7	.	.	PUNCT
ijassa-1958	58	1	to	to	PART
ijassa-1958	58	2	do	do	VERB
ijassa-1958	58	3	so	so	ADV
ijassa-1958	58	4	,	,	PUNCT
ijassa-1958	58	5	we	we	PRON
ijassa-1958	58	6	need	need	VERB
ijassa-1958	58	7	the	the	DET
ijassa-1958	58	8	following	follow	VERB
ijassa-1958	58	9	definitions	definition	NOUN
ijassa-1958	58	10	:	:	PUNCT
ijassa-1958	58	11	definition	definition	NOUN
ijassa-1958	58	12	3.1	3.1	NUM
ijassa-1958	58	13	:	:	PUNCT
ijassa-1958	58	14	let	let	VERB
ijassa-1958	58	15	the	the	DET
ijassa-1958	58	16	real	real	ADJ
ijassa-1958	58	17	numbers	number	NOUN
ijassa-1958	58	18	r	r	NOUN
ijassa-1958	58	19	,	,	PUNCT
ijassa-1958	58	20	x	x	X
ijassa-1958	58	21	and	and	CCONJ
ijassa-1958	58	22	y	y	PROPN
ijassa-1958	58	23	with	with	ADP
ijassa-1958	58	24	r	r	NOUN
ijassa-1958	58	25	>	>	X
ijassa-1958	58	26	0	0	NUM
ijassa-1958	58	27	be	be	AUX
ijassa-1958	58	28	given	give	VERB
ijassa-1958	58	29	.	.	PUNCT
ijassa-1958	59	1	we	we	PRON
ijassa-1958	59	2	will	will	AUX
ijassa-1958	59	3	call	call	VERB
ijassa-1958	59	4	(	(	PUNCT
ijassa-1958	59	5	r	r	NOUN
ijassa-1958	59	6	,	,	PUNCT
ijassa-1958	59	7	x	x	NOUN
ijassa-1958	59	8	,	,	PUNCT
ijassa-1958	59	9	y)1	y)1	NOUN
ijassa-1958	59	10	system	system	NOUN
ijassa-1958	59	11	a	a	DET
ijassa-1958	59	12	set	set	NOUN
ijassa-1958	59	13	of	of	ADP
ijassa-1958	59	14	points	point	NOUN
ijassa-1958	59	15	on	on	ADP
ijassa-1958	59	16	the	the	DET
ijassa-1958	59	17	plane	plane	NOUN
ijassa-1958	59	18	{	{	PUNCT
ijassa-1958	59	19	x	x	NOUN
ijassa-1958	59	20	,	,	PUNCT
ijassa-1958	59	21	y	y	PROPN
ijassa-1958	59	22	,	,	PUNCT
ijassa-1958	59	23	z	z	PROPN
ijassa-1958	59	24	,	,	PUNCT
ijassa-1958	59	25	t	t	PROPN
ijassa-1958	59	26	}	}	PUNCT
ijassa-1958	59	27	,	,	PUNCT
ijassa-1958	59	28	x	x	PUNCT
ijassa-1958	59	29	=	=	PUNCT
ijassa-1958	59	30	(	(	PUNCT
ijassa-1958	59	31	r	r	NOUN
ijassa-1958	59	32	+	+	NUM
ijassa-1958	59	33	x	x	NOUN
ijassa-1958	59	34	,	,	PUNCT
ijassa-1958	59	35	r	r	NOUN
ijassa-1958	59	36	+	+	NOUN
ijassa-1958	59	37	y	y	NOUN
ijassa-1958	59	38	)	)	PUNCT
ijassa-1958	59	39	,	,	PUNCT
ijassa-1958	59	40	y	y	PROPN
ijassa-1958	59	41	=	=	PRON
ijassa-1958	59	42	(	(	PUNCT
ijassa-1958	59	43	−r	−r	PROPN
ijassa-1958	60	1	+	+	X
ijassa-1958	60	2	x	x	NOUN
ijassa-1958	60	3	,	,	PUNCT
ijassa-1958	60	4	r	r	NOUN
ijassa-1958	60	5	+	+	NOUN
ijassa-1958	60	6	y	y	NOUN
ijassa-1958	60	7	)	)	PUNCT
ijassa-1958	60	8	,	,	PUNCT
ijassa-1958	60	9	z	z	NOUN
ijassa-1958	60	10	=	=	SYM
ijassa-1958	60	11	(	(	PUNCT
ijassa-1958	60	12	−r	−r	PROPN
ijassa-1958	60	13	+	+	X
ijassa-1958	60	14	x,−r	x,−r	PROPN
ijassa-1958	61	1	+	+	NUM
ijassa-1958	61	2	y	y	NOUN
ijassa-1958	61	3	)	)	PUNCT
ijassa-1958	61	4	,	,	PUNCT
ijassa-1958	62	1	x	x	PUNCT
ijassa-1958	62	2	=	=	PUNCT
ijassa-1958	62	3	(	(	PUNCT
ijassa-1958	62	4	r	r	NOUN
ijassa-1958	62	5	+	+	NOUN
ijassa-1958	62	6	x,−r	x,−r	PROPN
ijassa-1958	62	7	+	+	NUM
ijassa-1958	62	8	y	y	NOUN
ijassa-1958	62	9	)	)	PUNCT
ijassa-1958	62	10	.	.	PUNCT
ijassa-1958	63	1	we	we	PRON
ijassa-1958	63	2	will	will	AUX
ijassa-1958	63	3	call	call	VERB
ijassa-1958	63	4	(	(	PUNCT
ijassa-1958	63	5	r	r	NOUN
ijassa-1958	63	6	,	,	PUNCT
ijassa-1958	63	7	x	x	PRON
ijassa-1958	63	8	,	,	PUNCT
ijassa-1958	63	9	y)∞	y)∞	NOUN
ijassa-1958	63	10	system	system	VERB
ijassa-1958	63	11	a	a	DET
ijassa-1958	63	12	set	set	NOUN
ijassa-1958	63	13	of	of	ADP
ijassa-1958	63	14	points	point	NOUN
ijassa-1958	63	15	on	on	ADP
ijassa-1958	63	16	the	the	DET
ijassa-1958	63	17	plane	plane	NOUN
ijassa-1958	63	18	{	{	PUNCT
ijassa-1958	63	19	x	x	NOUN
ijassa-1958	63	20	,	,	PUNCT
ijassa-1958	63	21	y	y	PROPN
ijassa-1958	63	22	,	,	PUNCT
ijassa-1958	63	23	z	z	PROPN
ijassa-1958	63	24	,	,	PUNCT
ijassa-1958	63	25	t	t	PROPN
ijassa-1958	63	26	}	}	PUNCT
ijassa-1958	63	27	,	,	PUNCT
ijassa-1958	63	28	x	x	PUNCT
ijassa-1958	63	29	=	=	PUNCT
ijassa-1958	63	30	(	(	PUNCT
ijassa-1958	63	31	r	r	NOUN
ijassa-1958	63	32	+	+	NUM
ijassa-1958	63	33	x	x	PROPN
ijassa-1958	63	34	,	,	PUNCT
ijassa-1958	63	35	y	y	PROPN
ijassa-1958	63	36	)	)	PUNCT
ijassa-1958	63	37	,	,	PUNCT
ijassa-1958	63	38	y	y	PROPN
ijassa-1958	63	39	=	=	SYM
ijassa-1958	63	40	(	(	PUNCT
ijassa-1958	63	41	x	x	X
ijassa-1958	63	42	,	,	PUNCT
ijassa-1958	63	43	r	r	NOUN
ijassa-1958	63	44	+	+	NOUN
ijassa-1958	63	45	y	y	NOUN
ijassa-1958	63	46	)	)	PUNCT
ijassa-1958	63	47	,	,	PUNCT
ijassa-1958	63	48	z	z	NOUN
ijassa-1958	64	1	=	=	SYM
ijassa-1958	64	2	(	(	PUNCT
ijassa-1958	64	3	−r	−r	PROPN
ijassa-1958	64	4	+	+	NUM
ijassa-1958	64	5	x	x	NOUN
ijassa-1958	64	6	,	,	PUNCT
ijassa-1958	64	7	y	y	PROPN
ijassa-1958	64	8	)	)	PUNCT
ijassa-1958	64	9	,	,	PUNCT
ijassa-1958	64	10	t	t	PROPN
ijassa-1958	64	11	=	=	SYM
ijassa-1958	64	12	(	(	PUNCT
ijassa-1958	64	13	x,−r	x,−r	PROPN
ijassa-1958	64	14	+	+	NUM
ijassa-1958	64	15	y	y	NOUN
ijassa-1958	64	16	)	)	PUNCT
ijassa-1958	64	17	.	.	PUNCT
ijassa-1958	65	1	lemma	lemma	PROPN
ijassa-1958	65	2	3.1	3.1	NUM
ijassa-1958	65	3	:	:	PUNCT
ijassa-1958	65	4	on	on	ADP
ijassa-1958	65	5	the	the	DET
ijassa-1958	65	6	(	(	PUNCT
ijassa-1958	65	7	r	r	NOUN
ijassa-1958	65	8	,	,	PUNCT
ijassa-1958	65	9	x	x	NOUN
ijassa-1958	65	10	,	,	PUNCT
ijassa-1958	65	11	y)k	y)k	ADJ
ijassa-1958	65	12	-	-	NOUN
ijassa-1958	65	13	system	system	NOUN
ijassa-1958	65	14	,	,	PUNCT
ijassa-1958	65	15	it	it	PRON
ijassa-1958	65	16	is	be	AUX
ijassa-1958	65	17	impossible	impossible	ADJ
ijassa-1958	65	18	to	to	PART
ijassa-1958	65	19	introduce	introduce	VERB
ijassa-1958	65	20	an	an	DET
ijassa-1958	65	21	order	order	NOUN
ijassa-1958	65	22	compatible	compatible	ADJ
ijassa-1958	65	23	with	with	ADP
ijassa-1958	65	24	the	the	DET
ijassa-1958	65	25	p	p	NOUN
ijassa-1958	65	26	-	-	PUNCT
ijassa-1958	65	27	norm	norm	NOUN
ijassa-1958	65	28	induced	induce	VERB
ijassa-1958	65	29	on	on	ADP
ijassa-1958	65	30	the	the	DET
ijassa-1958	65	31	metric	metric	ADJ
ijassa-1958	65	32	plane	plane	NOUN
ijassa-1958	65	33	,	,	PUNCT
ijassa-1958	65	34	for	for	ADP
ijassa-1958	65	35	1	1	NUM
ijassa-1958	65	36	)	)	PUNCT
ijassa-1958	65	37	k	k	NOUN
ijassa-1958	66	1	=	=	SYM
ijassa-1958	66	2	1	1	NUM
ijassa-1958	66	3	and	and	CCONJ
ijassa-1958	66	4	1	1	NUM
ijassa-1958	66	5	≤	≤	NOUN
ijassa-1958	67	1	p	p	X
ijassa-1958	67	2	<	<	X
ijassa-1958	67	3	∞	∞	PROPN
ijassa-1958	67	4	;	;	PUNCT
ijassa-1958	67	5	2	2	X
ijassa-1958	67	6	)	)	PUNCT
ijassa-1958	67	7	k	k	NOUN
ijassa-1958	68	1	=	=	SYM
ijassa-1958	68	2	∞	∞	PROPN
ijassa-1958	68	3	and	and	CCONJ
ijassa-1958	68	4	1	1	NUM
ijassa-1958	68	5	<	<	X
ijassa-1958	68	6	p	p	X
ijassa-1958	68	7	≤	≤	PUNCT
ijassa-1958	68	8	∞.	∞.	PROPN
ijassa-1958	68	9	proof	proof	NOUN
ijassa-1958	68	10	the	the	DET
ijassa-1958	68	11	proof	proof	NOUN
ijassa-1958	68	12	follows	follow	VERB
ijassa-1958	68	13	directly	directly	ADV
ijassa-1958	68	14	from	from	ADP
ijassa-1958	68	15	checking	check	VERB
ijassa-1958	68	16	the	the	DET
ijassa-1958	68	17	condition	condition	NOUN
ijassa-1958	68	18	of	of	ADP
ijassa-1958	68	19	theorem	theorem	ADJ
ijassa-1958	68	20	3.1	3.1	NUM
ijassa-1958	68	21	.	.	PUNCT
ijassa-1958	69	1	copyright	copyright	NOUN
ijassa-1958	69	2	©	©	PROPN
ijassa-1958	69	3	2024	2024	NUM
ijassa-1958	69	4	assa	assa	NOUN
ijassa-1958	69	5	.	.	PUNCT
ijassa-1958	70	1	adv	adv	PROPN
ijassa-1958	70	2	syst	syst	PROPN
ijassa-1958	70	3	sci	sci	PROPN
ijassa-1958	70	4	appl	appl	PROPN
ijassa-1958	70	5	(	(	PUNCT
ijassa-1958	70	6	2024	2024	NUM
ijassa-1958	70	7	)	)	PUNCT
ijassa-1958	70	8	4	4	NUM
ijassa-1958	70	9	p.	p.	NOUN
ijassa-1958	70	10	klimov	klimov	PROPN
ijassa-1958	70	11	,	,	PUNCT
ijassa-1958	70	12	r.	r.	PROPN
ijassa-1958	70	13	sengupta	sengupta	PROPN
ijassa-1958	70	14	x	x	PUNCT
ijassa-1958	71	1	y	y	PROPN
ijassa-1958	71	2	z	z	PROPN
ijassa-1958	71	3	t	t	PROPN
ijassa-1958	71	4	fig	fig	NOUN
ijassa-1958	71	5	.	.	PUNCT
ijassa-1958	72	1	3.1	3.1	NUM
ijassa-1958	72	2	.	.	PUNCT
ijassa-1958	72	3	example	example	NOUN
ijassa-1958	72	4	of	of	ADP
ijassa-1958	72	5	(	(	PUNCT
ijassa-1958	72	6	r	r	NOUN
ijassa-1958	72	7	,	,	PUNCT
ijassa-1958	72	8	0	0	NUM
ijassa-1958	72	9	,	,	PUNCT
ijassa-1958	72	10	0)∞	0)∞	NUM
ijassa-1958	72	11	systems	system	NOUN
ijassa-1958	72	12	s	s	PART
ijassa-1958	72	13	=	=	SYM
ijassa-1958	72	14	{	{	PUNCT
ijassa-1958	72	15	x	x	PROPN
ijassa-1958	72	16	,	,	PUNCT
ijassa-1958	72	17	y	y	PROPN
ijassa-1958	72	18	,	,	PUNCT
ijassa-1958	72	19	z	z	PROPN
ijassa-1958	72	20	,	,	PUNCT
ijassa-1958	72	21	t	t	PROPN
ijassa-1958	72	22	}	}	PUNCT
ijassa-1958	72	23	on	on	ADP
ijassa-1958	72	24	the	the	DET
ijassa-1958	72	25	plane	plane	NOUN
ijassa-1958	72	26	around	around	ADP
ijassa-1958	72	27	the	the	DET
ijassa-1958	72	28	origin	origin	NOUN
ijassa-1958	72	29	.	.	PUNCT
ijassa-1958	73	1	it	it	PRON
ijassa-1958	73	2	is	be	AUX
ijassa-1958	73	3	easy	easy	ADJ
ijassa-1958	73	4	to	to	PART
ijassa-1958	73	5	see	see	VERB
ijassa-1958	73	6	that	that	SCONJ
ijassa-1958	73	7	the	the	DET
ijassa-1958	73	8	diameters	diameter	NOUN
ijassa-1958	73	9	of	of	ADP
ijassa-1958	73	10	the	the	DET
ijassa-1958	73	11	resulting	result	VERB
ijassa-1958	73	12	rhombus	rhombus	NOUN
ijassa-1958	73	13	with	with	ADP
ijassa-1958	73	14	vertices	vertex	NOUN
ijassa-1958	73	15	from	from	ADP
ijassa-1958	73	16	s	s	PRON
ijassa-1958	73	17	in	in	ADP
ijassa-1958	73	18	the	the	DET
ijassa-1958	73	19	2	2	NUM
ijassa-1958	73	20	norm	norm	NOUN
ijassa-1958	73	21	are	be	AUX
ijassa-1958	73	22	larger	large	ADJ
ijassa-1958	73	23	than	than	ADP
ijassa-1958	73	24	the	the	DET
ijassa-1958	73	25	sides	side	NOUN
ijassa-1958	73	26	,	,	PUNCT
ijassa-1958	73	27	which	which	PRON
ijassa-1958	73	28	shows	show	VERB
ijassa-1958	73	29	that	that	SCONJ
ijassa-1958	73	30	the	the	DET
ijassa-1958	73	31	condition	condition	NOUN
ijassa-1958	73	32	of	of	ADP
ijassa-1958	73	33	the	the	DET
ijassa-1958	73	34	theorem	theorem	ADJ
ijassa-1958	73	35	3.1	3.1	NUM
ijassa-1958	73	36	is	be	AUX
ijassa-1958	73	37	true	true	ADJ
ijassa-1958	73	38	.	.	PUNCT
ijassa-1958	74	1	the	the	DET
ijassa-1958	74	2	definition	definition	NOUN
ijassa-1958	74	3	and	and	CCONJ
ijassa-1958	74	4	lemma	lemma	PROPN
ijassa-1958	74	5	above	above	ADV
ijassa-1958	74	6	allow	allow	VERB
ijassa-1958	74	7	us	we	PRON
ijassa-1958	74	8	to	to	PART
ijassa-1958	74	9	obtain	obtain	VERB
ijassa-1958	74	10	the	the	DET
ijassa-1958	74	11	following	follow	VERB
ijassa-1958	74	12	useful	useful	ADJ
ijassa-1958	74	13	lemma	lemma	PROPN
ijassa-1958	74	14	,	,	PUNCT
ijassa-1958	74	15	which	which	PRON
ijassa-1958	74	16	can	can	AUX
ijassa-1958	74	17	be	be	AUX
ijassa-1958	74	18	used	use	VERB
ijassa-1958	74	19	to	to	PART
ijassa-1958	74	20	prove	prove	VERB
ijassa-1958	74	21	that	that	SCONJ
ijassa-1958	74	22	the	the	DET
ijassa-1958	74	23	metric	metric	NOUN
ijassa-1958	74	24	is	be	AUX
ijassa-1958	74	25	incompatible	incompatible	ADJ
ijassa-1958	74	26	with	with	ADP
ijassa-1958	74	27	the	the	DET
ijassa-1958	74	28	order	order	NOUN
ijassa-1958	74	29	for	for	ADP
ijassa-1958	74	30	a	a	DET
ijassa-1958	74	31	number	number	NOUN
ijassa-1958	74	32	of	of	ADP
ijassa-1958	74	33	spaces	space	NOUN
ijassa-1958	74	34	:	:	PUNCT
ijassa-1958	74	35	lemma	lemma	PROPN
ijassa-1958	74	36	3.2	3.2	NUM
ijassa-1958	74	37	:	:	PUNCT
ijassa-1958	74	38	on	on	ADP
ijassa-1958	74	39	an	an	DET
ijassa-1958	74	40	open	open	ADJ
ijassa-1958	74	41	ball	ball	NOUN
ijassa-1958	74	42	br(x	br(x	PUNCT
ijassa-1958	74	43	,	,	PUNCT
ijassa-1958	74	44	y	y	NOUN
ijassa-1958	74	45	)	)	PUNCT
ijassa-1958	74	46	in	in	ADP
ijassa-1958	74	47	r2	r2	PROPN
ijassa-1958	74	48	,	,	PUNCT
ijassa-1958	74	49	for	for	ADP
ijassa-1958	74	50	all	all	PRON
ijassa-1958	74	51	1	1	NUM
ijassa-1958	74	52	<	<	X
ijassa-1958	74	53	p	p	X
ijassa-1958	74	54	≤	≤	NUM
ijassa-1958	74	55	∞	∞	PROPN
ijassa-1958	74	56	,	,	PUNCT
ijassa-1958	74	57	it	it	PRON
ijassa-1958	74	58	is	be	AUX
ijassa-1958	74	59	impossible	impossible	ADJ
ijassa-1958	74	60	to	to	PART
ijassa-1958	74	61	introduce	introduce	VERB
ijassa-1958	74	62	an	an	DET
ijassa-1958	74	63	order	order	NOUN
ijassa-1958	74	64	that	that	PRON
ijassa-1958	74	65	is	be	AUX
ijassa-1958	74	66	compatible	compatible	ADJ
ijassa-1958	74	67	with	with	ADP
ijassa-1958	74	68	the	the	DET
ijassa-1958	74	69	metric	metric	NOUN
ijassa-1958	74	70	induced	induce	VERB
ijassa-1958	74	71	from	from	ADP
ijassa-1958	74	72	the	the	DET
ijassa-1958	74	73	p	p	NOUN
ijassa-1958	74	74	-	-	PUNCT
ijassa-1958	74	75	norm	norm	NOUN
ijassa-1958	74	76	.	.	PUNCT
ijassa-1958	75	1	proof	proof	NOUN
ijassa-1958	75	2	it	it	PRON
ijassa-1958	75	3	is	be	AUX
ijassa-1958	75	4	easy	easy	ADJ
ijassa-1958	75	5	to	to	PART
ijassa-1958	75	6	see	see	VERB
ijassa-1958	75	7	that	that	PRON
ijassa-1958	75	8	for	for	ADP
ijassa-1958	75	9	any	any	DET
ijassa-1958	75	10	x	x	NOUN
ijassa-1958	75	11	,	,	PUNCT
ijassa-1958	75	12	y	y	PROPN
ijassa-1958	75	13	∈	∈	PROPN
ijassa-1958	75	14	r	r	NOUN
ijassa-1958	75	15	,	,	PUNCT
ijassa-1958	75	16	r	r	NOUN
ijassa-1958	75	17	>	>	X
ijassa-1958	75	18	0	0	NUM
ijassa-1958	75	19	,	,	PUNCT
ijassa-1958	75	20	1	1	NUM
ijassa-1958	75	21	≤	≤	NOUN
ijassa-1958	75	22	p	p	NOUN
ijassa-1958	75	23	≤	≤	NOUN
ijassa-1958	75	24	∞	∞	NUM
ijassa-1958	75	25	we	we	PRON
ijassa-1958	75	26	have	have	VERB
ijassa-1958	75	27	(	(	PUNCT
ijassa-1958	75	28	r	r	NOUN
ijassa-1958	75	29	3	3	NUM
ijassa-1958	75	30	,	,	PUNCT
ijassa-1958	75	31	x	x	PROPN
ijassa-1958	75	32	,	,	PUNCT
ijassa-1958	75	33	y)1	y)1	PROPN
ijassa-1958	75	34	⊂	⊂	PROPN
ijassa-1958	75	35	br(x	br(x	PROPN
ijassa-1958	75	36	,	,	PUNCT
ijassa-1958	75	37	y	y	PROPN
ijassa-1958	75	38	)	)	PUNCT
ijassa-1958	75	39	and	and	CCONJ
ijassa-1958	75	40	(	(	PUNCT
ijassa-1958	75	41	r	r	NOUN
ijassa-1958	75	42	2	2	NUM
ijassa-1958	75	43	,	,	PUNCT
ijassa-1958	75	44	x	x	PROPN
ijassa-1958	75	45	,	,	PUNCT
ijassa-1958	75	46	y)∞	y)∞	PROPN
ijassa-1958	75	47	⊂	⊂	PROPN
ijassa-1958	75	48	br(x	br(x	PROPN
ijassa-1958	75	49	,	,	PUNCT
ijassa-1958	75	50	y	y	NOUN
ijassa-1958	75	51	)	)	PUNCT
ijassa-1958	75	52	.	.	PUNCT
ijassa-1958	76	1	theorem	theorem	ADJ
ijassa-1958	76	2	3.2	3.2	NUM
ijassa-1958	76	3	:	:	PUNCT
ijassa-1958	76	4	there	there	PRON
ijassa-1958	76	5	is	be	VERB
ijassa-1958	76	6	no	no	DET
ijassa-1958	76	7	metric	metric	ADJ
ijassa-1958	76	8	-	-	PUNCT
ijassa-1958	76	9	compatible	compatible	ADJ
ijassa-1958	76	10	order	order	NOUN
ijassa-1958	76	11	for	for	ADP
ijassa-1958	76	12	any	any	DET
ijassa-1958	76	13	metric	metric	ADJ
ijassa-1958	76	14	space	space	NOUN
ijassa-1958	76	15	that	that	PRON
ijassa-1958	76	16	contains	contain	VERB
ijassa-1958	76	17	a	a	DET
ijassa-1958	76	18	subset	subset	NOUN
ijassa-1958	76	19	that	that	PRON
ijassa-1958	76	20	is	be	AUX
ijassa-1958	76	21	isometric	isometric	ADJ
ijassa-1958	76	22	to	to	ADP
ijassa-1958	76	23	an	an	DET
ijassa-1958	76	24	open	open	ADJ
ijassa-1958	76	25	ball	ball	NOUN
ijassa-1958	76	26	in	in	ADP
ijassa-1958	76	27	the	the	DET
ijassa-1958	76	28	plane	plane	NOUN
ijassa-1958	76	29	with	with	ADP
ijassa-1958	76	30	p	p	NOUN
ijassa-1958	76	31	-	-	PUNCT
ijassa-1958	76	32	norm	norm	NOUN
ijassa-1958	76	33	,	,	PUNCT
ijassa-1958	76	34	1	1	NUM
ijassa-1958	76	35	≤	≤	NOUN
ijassa-1958	76	36	p	p	NOUN
ijassa-1958	76	37	≤	≤	NOUN
ijassa-1958	76	38	∞	∞	PROPN
ijassa-1958	76	39	.	.	PUNCT
ijassa-1958	77	1	in	in	ADP
ijassa-1958	77	2	particular	particular	ADJ
ijassa-1958	77	3	,	,	PUNCT
ijassa-1958	77	4	there	there	PRON
ijassa-1958	77	5	is	be	VERB
ijassa-1958	77	6	no	no	DET
ijassa-1958	77	7	metric	metric	ADJ
ijassa-1958	77	8	-	-	PUNCT
ijassa-1958	77	9	compatible	compatible	ADJ
ijassa-1958	77	10	order	order	NOUN
ijassa-1958	77	11	in	in	ADP
ijassa-1958	77	12	the	the	DET
ijassa-1958	77	13	following	follow	VERB
ijassa-1958	77	14	spaces	space	NOUN
ijassa-1958	77	15	:	:	PUNCT
ijassa-1958	77	16	1	1	X
ijassa-1958	77	17	.	.	X
ijassa-1958	77	18	rn	rn	PROPN
ijassa-1958	77	19	with	with	ADP
ijassa-1958	77	20	the	the	DET
ijassa-1958	77	21	p	p	PROPN
ijassa-1958	77	22	norm	norm	NOUN
ijassa-1958	77	23	for	for	ADP
ijassa-1958	77	24	n	n	X
ijassa-1958	77	25	>	>	X
ijassa-1958	77	26	1	1	NUM
ijassa-1958	77	27	,	,	PUNCT
ijassa-1958	77	28	1	1	NUM
ijassa-1958	77	29	≤	≤	NOUN
ijassa-1958	77	30	p	p	NOUN
ijassa-1958	77	31	≤	≤	NUM
ijassa-1958	77	32	∞	∞	NUM
ijassa-1958	77	33	;	;	PUNCT
ijassa-1958	77	34	2	2	X
ijassa-1958	77	35	.	.	X
ijassa-1958	77	36	ℓp	ℓp	NOUN
ijassa-1958	77	37	for	for	ADP
ijassa-1958	77	38	1	1	NUM
ijassa-1958	77	39	≤	≤	NOUN
ijassa-1958	77	40	p	p	NOUN
ijassa-1958	77	41	≤	≤	NUM
ijassa-1958	77	42	∞	∞	NUM
ijassa-1958	77	43	;	;	PUNCT
ijassa-1958	77	44	3	3	X
ijassa-1958	77	45	.	.	X
ijassa-1958	77	46	lp[a	lp[a	PROPN
ijassa-1958	77	47	,	,	PUNCT
ijassa-1958	77	48	b	b	NOUN
ijassa-1958	77	49	]	]	X
ijassa-1958	77	50	for	for	ADP
ijassa-1958	77	51	[	[	X
ijassa-1958	77	52	a	a	X
ijassa-1958	77	53	,	,	PUNCT
ijassa-1958	77	54	b	b	NOUN
ijassa-1958	77	55	]	]	X
ijassa-1958	77	56	⊂	⊂	PROPN
ijassa-1958	77	57	r	r	X
ijassa-1958	77	58	,	,	PUNCT
ijassa-1958	77	59	1	1	NUM
ijassa-1958	77	60	≤	≤	NOUN
ijassa-1958	77	61	p	p	NOUN
ijassa-1958	77	62	≤	≤	NUM
ijassa-1958	77	63	∞	∞	NUM
ijassa-1958	77	64	;	;	PUNCT
ijassa-1958	77	65	4	4	NUM
ijassa-1958	77	66	.	.	PUNCT
ijassa-1958	77	67	lp(a	lp(a	NUM
ijassa-1958	77	68	)	)	PUNCT
ijassa-1958	77	69	,	,	PUNCT
ijassa-1958	77	70	where	where	SCONJ
ijassa-1958	77	71	a	a	PRON
ijassa-1958	77	72	is	be	AUX
ijassa-1958	77	73	a	a	DET
ijassa-1958	77	74	measurable	measurable	ADJ
ijassa-1958	77	75	space	space	NOUN
ijassa-1958	77	76	that	that	PRON
ijassa-1958	77	77	contains	contain	VERB
ijassa-1958	77	78	two	two	NUM
ijassa-1958	77	79	disjoint	disjoint	NOUN
ijassa-1958	77	80	subsets	subset	NOUN
ijassa-1958	77	81	a1	a1	NOUN
ijassa-1958	77	82	and	and	CCONJ
ijassa-1958	77	83	a2	a2	PROPN
ijassa-1958	77	84	with	with	ADP
ijassa-1958	77	85	a	a	DET
ijassa-1958	77	86	finite	finite	ADJ
ijassa-1958	77	87	nonzero	nonzero	PROPN
ijassa-1958	77	88	measure	measure	NOUN
ijassa-1958	77	89	,	,	PUNCT
ijassa-1958	77	90	1	1	NUM
ijassa-1958	77	91	≤	≤	NOUN
ijassa-1958	77	92	p	p	NOUN
ijassa-1958	77	93	≤	≤	NUM
ijassa-1958	77	94	∞	∞	NUM
ijassa-1958	77	95	;	;	PUNCT
ijassa-1958	77	96	5	5	NUM
ijassa-1958	77	97	.	.	X
ijassa-1958	77	98	an	an	DET
ijassa-1958	77	99	open	open	ADJ
ijassa-1958	77	100	ball	ball	NOUN
ijassa-1958	77	101	in	in	ADP
ijassa-1958	77	102	any	any	PRON
ijassa-1958	77	103	of	of	ADP
ijassa-1958	77	104	the	the	DET
ijassa-1958	77	105	spaces	space	NOUN
ijassa-1958	77	106	listed	list	VERB
ijassa-1958	77	107	above	above	ADV
ijassa-1958	77	108	.	.	PUNCT
ijassa-1958	78	1	proof	proof	NOUN
ijassa-1958	78	2	from	from	ADP
ijassa-1958	78	3	the	the	DET
ijassa-1958	78	4	statement	statement	NOUN
ijassa-1958	78	5	of	of	ADP
ijassa-1958	78	6	point	point	NOUN
ijassa-1958	78	7	(	(	PUNCT
ijassa-1958	78	8	5	5	NUM
ijassa-1958	78	9	)	)	PUNCT
ijassa-1958	78	10	,	,	PUNCT
ijassa-1958	78	11	the	the	DET
ijassa-1958	78	12	statements	statement	NOUN
ijassa-1958	78	13	of	of	ADP
ijassa-1958	78	14	points	point	NOUN
ijassa-1958	78	15	(	(	PUNCT
ijassa-1958	78	16	1)-(4	1)-(4	NUM
ijassa-1958	78	17	)	)	PUNCT
ijassa-1958	78	18	follow	follow	NOUN
ijassa-1958	78	19	by	by	ADP
ijassa-1958	78	20	lemma	lemma	PROPN
ijassa-1958	78	21	2.1	2.1	NUM
ijassa-1958	78	22	.	.	PUNCT
ijassa-1958	79	1	we	we	PRON
ijassa-1958	79	2	prove	prove	VERB
ijassa-1958	79	3	point	point	NOUN
ijassa-1958	79	4	(	(	PUNCT
ijassa-1958	79	5	5	5	NUM
ijassa-1958	79	6	)	)	PUNCT
ijassa-1958	79	7	for	for	ADP
ijassa-1958	79	8	each	each	PRON
ijassa-1958	79	9	of	of	ADP
ijassa-1958	79	10	these	these	DET
ijassa-1958	79	11	spaces	space	NOUN
ijassa-1958	79	12	.	.	PUNCT
ijassa-1958	80	1	to	to	PART
ijassa-1958	80	2	do	do	VERB
ijassa-1958	80	3	this	this	PRON
ijassa-1958	80	4	,	,	PUNCT
ijassa-1958	80	5	we	we	PRON
ijassa-1958	80	6	isometrically	isometrically	PROPN
ijassa-1958	80	7	embed	embe	VERB
ijassa-1958	80	8	an	an	DET
ijassa-1958	80	9	open	open	ADJ
ijassa-1958	80	10	ball	ball	NOUN
ijassa-1958	80	11	from	from	ADP
ijassa-1958	80	12	r2	r2	PROPN
ijassa-1958	80	13	in	in	ADP
ijassa-1958	80	14	the	the	DET
ijassa-1958	80	15	corresponding	corresponding	ADJ
ijassa-1958	80	16	space	space	NOUN
ijassa-1958	80	17	and	and	CCONJ
ijassa-1958	80	18	use	use	VERB
ijassa-1958	80	19	the	the	DET
ijassa-1958	80	20	lemma	lemma	PROPN
ijassa-1958	80	21	3.2	3.2	NUM
ijassa-1958	80	22	.	.	NOUN
ijassa-1958	81	1	1	1	NUM
ijassa-1958	81	2	.	.	X
ijassa-1958	81	3	let	let	VERB
ijassa-1958	81	4	x	x	PROPN
ijassa-1958	81	5	∈	∈	PROPN
ijassa-1958	81	6	rn	rn	PROPN
ijassa-1958	81	7	.	.	PUNCT
ijassa-1958	82	1	we	we	PRON
ijassa-1958	82	2	can	can	AUX
ijassa-1958	82	3	embed	embed	VERB
ijassa-1958	82	4	isometrically	isometrically	PROPN
ijassa-1958	82	5	br(0	br(0	PROPN
ijassa-1958	82	6	,	,	PUNCT
ijassa-1958	82	7	0	0	NUM
ijassa-1958	82	8	)	)	PUNCT
ijassa-1958	82	9	into	into	ADP
ijassa-1958	82	10	br(x	br(x	NOUN
ijassa-1958	82	11	)	)	PUNCT
ijassa-1958	82	12	with	with	ADP
ijassa-1958	82	13	the	the	DET
ijassa-1958	82	14	p	p	PROPN
ijassa-1958	82	15	norm	norm	NOUN
ijassa-1958	82	16	:	:	PUNCT
ijassa-1958	82	17	f(y1	f(y1	NOUN
ijassa-1958	82	18	,	,	PUNCT
ijassa-1958	82	19	y2	y2	NOUN
ijassa-1958	82	20	)	)	PUNCT
ijassa-1958	82	21	=	=	SYM
ijassa-1958	83	1	(	(	PUNCT
ijassa-1958	83	2	x1	x1	PROPN
ijassa-1958	83	3	+	+	NUM
ijassa-1958	83	4	y1	y1	NOUN
ijassa-1958	83	5	,	,	PUNCT
ijassa-1958	83	6	x2	x2	PROPN
ijassa-1958	83	7	+	+	CCONJ
ijassa-1958	83	8	y2	y2	ADJ
ijassa-1958	83	9	,	,	PUNCT
ijassa-1958	83	10	x3	x3	ADJ
ijassa-1958	83	11	,	,	PUNCT
ijassa-1958	83	12	.	.	PUNCT
ijassa-1958	83	13	.	.	PUNCT
ijassa-1958	83	14	.	.	PUNCT
ijassa-1958	83	15	,	,	PUNCT
ijassa-1958	83	16	xn	xn	PROPN
ijassa-1958	83	17	)	)	PUNCT
ijassa-1958	83	18	.	.	PUNCT
ijassa-1958	84	1	2	2	X
ijassa-1958	84	2	.	.	X
ijassa-1958	84	3	let	let	VERB
ijassa-1958	84	4	x	x	PUNCT
ijassa-1958	84	5	∈	∈	PROPN
ijassa-1958	84	6	ℓp	ℓp	NOUN
ijassa-1958	84	7	.	.	PUNCT
ijassa-1958	85	1	we	we	PRON
ijassa-1958	85	2	can	can	AUX
ijassa-1958	85	3	isometrically	isometrically	PROPN
ijassa-1958	85	4	embed	embed	VERB
ijassa-1958	85	5	br(0	br(0	PROPN
ijassa-1958	85	6	,	,	PUNCT
ijassa-1958	85	7	0	0	NUM
ijassa-1958	85	8	)	)	PUNCT
ijassa-1958	85	9	with	with	ADP
ijassa-1958	85	10	the	the	DET
ijassa-1958	85	11	p	p	NOUN
ijassa-1958	85	12	-	-	PUNCT
ijassa-1958	85	13	norm	norm	NOUN
ijassa-1958	85	14	into	into	ADP
ijassa-1958	85	15	br(x	br(x	NOUN
ijassa-1958	85	16	)	)	PUNCT
ijassa-1958	85	17	with	with	ADP
ijassa-1958	85	18	the	the	DET
ijassa-1958	85	19	ℓp	ℓp	ADJ
ijassa-1958	85	20	norm	norm	NOUN
ijassa-1958	85	21	:	:	PUNCT
ijassa-1958	85	22	f(y1	f(y1	NOUN
ijassa-1958	85	23	,	,	PUNCT
ijassa-1958	85	24	y2	y2	NOUN
ijassa-1958	85	25	)	)	PUNCT
ijassa-1958	85	26	=	=	SYM
ijassa-1958	86	1	(	(	PUNCT
ijassa-1958	86	2	x1	x1	PROPN
ijassa-1958	86	3	+	+	NUM
ijassa-1958	86	4	y1	y1	NOUN
ijassa-1958	86	5	,	,	PUNCT
ijassa-1958	86	6	x2	x2	PROPN
ijassa-1958	86	7	+	+	CCONJ
ijassa-1958	86	8	y2	y2	ADJ
ijassa-1958	86	9	,	,	PUNCT
ijassa-1958	86	10	x3	x3	ADJ
ijassa-1958	86	11	,	,	PUNCT
ijassa-1958	86	12	.	.	PUNCT
ijassa-1958	86	13	.	.	PUNCT
ijassa-1958	86	14	.	.	PUNCT
ijassa-1958	86	15	)	)	PUNCT
ijassa-1958	86	16	.	.	PUNCT
ijassa-1958	87	1	3	3	X
ijassa-1958	87	2	.	.	X
ijassa-1958	87	3	we	we	PRON
ijassa-1958	87	4	can	can	AUX
ijassa-1958	87	5	embed	embed	VERB
ijassa-1958	87	6	r2	r2	PROPN
ijassa-1958	87	7	isometrically	isometrically	PROPN
ijassa-1958	87	8	with	with	ADP
ijassa-1958	87	9	the	the	DET
ijassa-1958	87	10	p	p	PROPN
ijassa-1958	87	11	norm	norm	NOUN
ijassa-1958	87	12	into	into	ADP
ijassa-1958	87	13	lp[a	lp[a	PROPN
ijassa-1958	87	14	,	,	PUNCT
ijassa-1958	87	15	b	b	NOUN
ijassa-1958	87	16	]	]	X
ijassa-1958	87	17	:	:	PUNCT
ijassa-1958	87	18	copyright	copyright	NOUN
ijassa-1958	87	19	©	©	PROPN
ijassa-1958	87	20	2024	2024	NUM
ijassa-1958	87	21	assa	assa	NOUN
ijassa-1958	87	22	.	.	PUNCT
ijassa-1958	88	1	adv	adv	PROPN
ijassa-1958	88	2	syst	syst	PROPN
ijassa-1958	88	3	sci	sci	PROPN
ijassa-1958	88	4	appl	appl	PROPN
ijassa-1958	88	5	(	(	PUNCT
ijassa-1958	88	6	2024	2024	NUM
ijassa-1958	88	7	)	)	PUNCT
ijassa-1958	88	8	on	on	ADP
ijassa-1958	88	9	the	the	DET
ijassa-1958	88	10	compatibility	compatibility	NOUN
ijassa-1958	88	11	of	of	ADP
ijassa-1958	88	12	metric	metric	ADJ
ijassa-1958	88	13	and	and	CCONJ
ijassa-1958	88	14	linear	linear	ADJ
ijassa-1958	88	15	order	order	NOUN
ijassa-1958	88	16	5	5	NUM
ijassa-1958	88	17	f(y1	f(y1	NOUN
ijassa-1958	88	18	,	,	PUNCT
ijassa-1958	88	19	y2	y2	NOUN
ijassa-1958	88	20	)	)	PUNCT
ijassa-1958	88	21	=	=	PUNCT
ijassa-1958	89	1	y1	y1	PROPN
ijassa-1958	89	2	p√	p√	X
ijassa-1958	89	3	b−a	b−a	AUX
ijassa-1958	89	4	2	2	NUM
ijassa-1958	89	5	i	i	PRON
ijassa-1958	90	1	[	[	X
ijassa-1958	90	2	a	a	X
ijassa-1958	90	3	,	,	PUNCT
ijassa-1958	90	4	a+	a+	PRON
ijassa-1958	90	5	(	(	PUNCT
ijassa-1958	90	6	b−a	b−a	NOUN
ijassa-1958	90	7	)	)	PUNCT
ijassa-1958	90	8	2	2	NUM
ijassa-1958	90	9	]	]	PUNCT
ijassa-1958	91	1	+	+	CCONJ
ijassa-1958	91	2	y2	y2	PROPN
ijassa-1958	91	3	p√	p√	X
ijassa-1958	91	4	b−a	b−a	ADP
ijassa-1958	91	5	2	2	NUM
ijassa-1958	91	6	i	i	PRON
ijassa-1958	91	7	[	[	X
ijassa-1958	91	8	a+	a+	PUNCT
ijassa-1958	91	9	(	(	PUNCT
ijassa-1958	91	10	b−a	b−a	NOUN
ijassa-1958	91	11	)	)	PUNCT
ijassa-1958	91	12	2	2	NUM
ijassa-1958	91	13	,	,	PUNCT
ijassa-1958	91	14	b	b	NOUN
ijassa-1958	91	15	]	]	PUNCT
ijassa-1958	91	16	.	.	PUNCT
ijassa-1958	92	1	we	we	PRON
ijassa-1958	92	2	show	show	VERB
ijassa-1958	92	3	the	the	DET
ijassa-1958	92	4	isometricity	isometricity	NOUN
ijassa-1958	92	5	of	of	ADP
ijassa-1958	92	6	f.	f.	PROPN
ijassa-1958	92	7	for	for	ADP
ijassa-1958	92	8	(	(	PUNCT
ijassa-1958	92	9	y1	y1	INTJ
ijassa-1958	92	10	,	,	PUNCT
ijassa-1958	92	11	y2	y2	PROPN
ijassa-1958	92	12	)	)	PUNCT
ijassa-1958	92	13	and	and	CCONJ
ijassa-1958	92	14	(	(	PUNCT
ijassa-1958	92	15	y′1	y′1	X
ijassa-1958	92	16	,	,	PUNCT
ijassa-1958	92	17	y	y	PROPN
ijassa-1958	92	18	′	′	NOUN
ijassa-1958	92	19	2	2	NUM
ijassa-1958	92	20	)	)	PUNCT
ijassa-1958	92	21	:	:	PUNCT
ijassa-1958	92	22	d((y1	d((y1	NOUN
ijassa-1958	92	23	,	,	PUNCT
ijassa-1958	92	24	y2	y2	PROPN
ijassa-1958	92	25	)	)	PUNCT
ijassa-1958	92	26	,	,	PUNCT
ijassa-1958	92	27	(	(	PUNCT
ijassa-1958	92	28	y	y	NOUN
ijassa-1958	92	29	′	′	NUM
ijassa-1958	92	30	1	1	NUM
ijassa-1958	92	31	,	,	PUNCT
ijassa-1958	92	32	y	y	PROPN
ijassa-1958	92	33	′	′	NOUN
ijassa-1958	92	34	2	2	NUM
ijassa-1958	92	35	)	)	PUNCT
ijassa-1958	92	36	)	)	PUNCT
ijassa-1958	93	1	=	=	PUNCT
ijassa-1958	94	1	p	p	NOUN
ijassa-1958	94	2	√	√	NUM
ijassa-1958	94	3	|y1	|y1	VERB
ijassa-1958	94	4	−	−	VERB
ijassa-1958	94	5	y′1|p	y′1|p	PROPN
ijassa-1958	95	1	+	+	CCONJ
ijassa-1958	95	2	|y2	|y2	VERB
ijassa-1958	95	3	−	−	NOUN
ijassa-1958	95	4	y′2|p	y′2|p	NOUN
ijassa-1958	96	1	=	=	PUNCT
ijassa-1958	96	2	=	=	PUNCT
ijassa-1958	97	1	p	p	X
ijassa-1958	97	2	√√√√√	√√√√√	ADP
ijassa-1958	97	3	b∫	b∫	PROPN
ijassa-1958	97	4	a	a	DET
ijassa-1958	97	5	2	2	NUM
ijassa-1958	97	6	b−	b−	NOUN
ijassa-1958	97	7	a	a	PRON
ijassa-1958	97	8	(	(	PUNCT
ijassa-1958	97	9	|y1	|y1	NUM
ijassa-1958	97	10	−	−	VERB
ijassa-1958	97	11	y′1|pi[a	y′1|pi[a	NOUN
ijassa-1958	97	12	,	,	PUNCT
ijassa-1958	97	13	a+	a+	PUNCT
ijassa-1958	97	14	(	(	PUNCT
ijassa-1958	97	15	b−a	b−a	NOUN
ijassa-1958	97	16	)	)	PUNCT
ijassa-1958	97	17	2	2	NUM
ijassa-1958	97	18	]	]	PUNCT
ijassa-1958	98	1	+	+	CCONJ
ijassa-1958	98	2	|y2	|y2	VERB
ijassa-1958	98	3	−	−	NOUN
ijassa-1958	98	4	y′2|pi[a+	y′2|pi[a+	SYM
ijassa-1958	98	5	(	(	PUNCT
ijassa-1958	98	6	b−a	b−a	NOUN
ijassa-1958	98	7	)	)	PUNCT
ijassa-1958	98	8	2	2	NUM
ijassa-1958	98	9	,	,	PUNCT
ijassa-1958	98	10	b	b	NOUN
ijassa-1958	98	11	]	]	PUNCT
ijassa-1958	98	12	)	)	PUNCT
ijassa-1958	98	13	(	(	PUNCT
ijassa-1958	98	14	z)dz	z)dz	PROPN
ijassa-1958	98	15	=	=	SYM
ijassa-1958	98	16	=	=	PUNCT
ijassa-1958	98	17	d(f(y1	d(f(y1	X
ijassa-1958	98	18	,	,	PUNCT
ijassa-1958	98	19	y2	y2	PROPN
ijassa-1958	98	20	)	)	PUNCT
ijassa-1958	98	21	,	,	PUNCT
ijassa-1958	98	22	f(y	f(y	NOUN
ijassa-1958	98	23	′	′	NUM
ijassa-1958	98	24	1	1	NUM
ijassa-1958	98	25	,	,	PUNCT
ijassa-1958	98	26	y	y	PROPN
ijassa-1958	98	27	′	′	NOUN
ijassa-1958	98	28	2	2	NUM
ijassa-1958	98	29	)	)	PUNCT
ijassa-1958	98	30	)	)	PUNCT
ijassa-1958	99	1	and	and	CCONJ
ijassa-1958	99	2	f	f	PROPN
ijassa-1958	99	3	is	be	AUX
ijassa-1958	99	4	injective	injective	ADJ
ijassa-1958	99	5	.	.	PUNCT
ijassa-1958	100	1	for	for	ADP
ijassa-1958	100	2	any	any	DET
ijassa-1958	100	3	br(x	br(x	NOUN
ijassa-1958	100	4	)	)	PUNCT
ijassa-1958	100	5	⊂	⊂	PROPN
ijassa-1958	100	6	lp[a	lp[a	PROPN
ijassa-1958	100	7	,	,	PUNCT
ijassa-1958	100	8	b	b	NOUN
ijassa-1958	100	9	]	]	PUNCT
ijassa-1958	100	10	,	,	PUNCT
ijassa-1958	100	11	we	we	PRON
ijassa-1958	100	12	can	can	AUX
ijassa-1958	100	13	restrict	restrict	VERB
ijassa-1958	100	14	f	f	NOUN
ijassa-1958	100	15	to	to	ADP
ijassa-1958	100	16	some	some	DET
ijassa-1958	100	17	open	open	ADJ
ijassa-1958	100	18	ball	ball	NOUN
ijassa-1958	100	19	lying	lie	VERB
ijassa-1958	100	20	in	in	ADP
ijassa-1958	100	21	f−1(br(x	f−1(br(x	NOUN
ijassa-1958	100	22	)	)	PUNCT
ijassa-1958	100	23	)	)	PUNCT
ijassa-1958	100	24	,	,	PUNCT
ijassa-1958	100	25	thus	thus	ADV
ijassa-1958	100	26	constructing	construct	VERB
ijassa-1958	100	27	an	an	DET
ijassa-1958	100	28	isometric	isometric	ADJ
ijassa-1958	100	29	embedding	embedding	NOUN
ijassa-1958	100	30	of	of	ADP
ijassa-1958	100	31	the	the	DET
ijassa-1958	100	32	open	open	ADJ
ijassa-1958	100	33	ball	ball	NOUN
ijassa-1958	100	34	from	from	ADP
ijassa-1958	100	35	r2	r2	PROPN
ijassa-1958	100	36	into	into	ADP
ijassa-1958	100	37	br(x	br(x	NOUN
ijassa-1958	100	38	)	)	PUNCT
ijassa-1958	100	39	.	.	PUNCT
ijassa-1958	101	1	4	4	X
ijassa-1958	101	2	.	.	X
ijassa-1958	101	3	it	it	PRON
ijassa-1958	101	4	is	be	AUX
ijassa-1958	101	5	enough	enough	ADJ
ijassa-1958	101	6	to	to	PART
ijassa-1958	101	7	repeat	repeat	VERB
ijassa-1958	101	8	the	the	DET
ijassa-1958	101	9	reasoning	reasoning	NOUN
ijassa-1958	101	10	of	of	ADP
ijassa-1958	101	11	point	point	NOUN
ijassa-1958	101	12	3	3	NUM
ijassa-1958	101	13	for	for	ADP
ijassa-1958	101	14	f(y1	f(y1	NOUN
ijassa-1958	101	15	,	,	PUNCT
ijassa-1958	101	16	y2	y2	NOUN
ijassa-1958	101	17	)	)	PUNCT
ijassa-1958	102	1	=	=	PUNCT
ijassa-1958	102	2	y1	y1	PROPN
ijassa-1958	102	3	p√	p√	PROPN
ijassa-1958	102	4	µ(a1	µ(a1	NOUN
ijassa-1958	102	5	)	)	PUNCT
ijassa-1958	102	6	ia1	ia1	NOUN
ijassa-1958	102	7	+	+	CCONJ
ijassa-1958	102	8	y2	y2	PROPN
ijassa-1958	102	9	p√	p√	PROPN
ijassa-1958	102	10	µ(a2	µ(a2	PROPN
ijassa-1958	102	11	)	)	PUNCT
ijassa-1958	102	12	ia2	ia2	NOUN
ijassa-1958	102	13	.	.	PUNCT
ijassa-1958	103	1	4	4	X
ijassa-1958	103	2	.	.	X
ijassa-1958	103	3	constructions	construction	NOUN
ijassa-1958	103	4	of	of	ADP
ijassa-1958	103	5	order	order	NOUN
ijassa-1958	103	6	-	-	PUNCT
ijassa-1958	103	7	compatible	compatible	ADJ
ijassa-1958	103	8	metric	metric	ADJ
ijassa-1958	103	9	spaces	space	NOUN
ijassa-1958	103	10	in	in	ADP
ijassa-1958	103	11	the	the	DET
ijassa-1958	103	12	previous	previous	ADJ
ijassa-1958	103	13	section	section	NOUN
ijassa-1958	103	14	,	,	PUNCT
ijassa-1958	103	15	we	we	PRON
ijassa-1958	103	16	provided	provide	VERB
ijassa-1958	103	17	numerous	numerous	ADJ
ijassa-1958	103	18	examples	example	NOUN
ijassa-1958	103	19	of	of	ADP
ijassa-1958	103	20	spaces	space	NOUN
ijassa-1958	103	21	that	that	PRON
ijassa-1958	103	22	are	be	AUX
ijassa-1958	103	23	not	not	PART
ijassa-1958	103	24	compatible	compatible	ADJ
ijassa-1958	103	25	with	with	ADP
ijassa-1958	103	26	any	any	DET
ijassa-1958	103	27	linear	linear	ADJ
ijassa-1958	103	28	order	order	NOUN
ijassa-1958	103	29	.	.	PUNCT
ijassa-1958	104	1	these	these	DET
ijassa-1958	104	2	spaces	space	NOUN
ijassa-1958	104	3	were	be	AUX
ijassa-1958	104	4	primarily	primarily	ADV
ijassa-1958	104	5	created	create	VERB
ijassa-1958	104	6	by	by	ADP
ijassa-1958	104	7	naturally	naturally	ADV
ijassa-1958	104	8	embedding	embed	VERB
ijassa-1958	104	9	certain	certain	ADJ
ijassa-1958	104	10	subdomains	subdomain	NOUN
ijassa-1958	104	11	of	of	ADP
ijassa-1958	104	12	rn	rn	PROPN
ijassa-1958	104	13	into	into	ADP
ijassa-1958	104	14	them	they	PRON
ijassa-1958	104	15	,	,	PUNCT
ijassa-1958	104	16	meaning	mean	VERB
ijassa-1958	104	17	they	they	PRON
ijassa-1958	104	18	were	be	AUX
ijassa-1958	104	19	”	"	PUNCT
ijassa-1958	104	20	locally	locally	ADV
ijassa-1958	104	21	multidimensional	multidimensional	ADJ
ijassa-1958	104	22	”	"	PUNCT
ijassa-1958	104	23	and	and	CCONJ
ijassa-1958	104	24	”	"	PUNCT
ijassa-1958	104	25	closely	closely	ADV
ijassa-1958	104	26	resembled	resemble	VERB
ijassa-1958	104	27	”	"	PUNCT
ijassa-1958	104	28	a	a	DET
ijassa-1958	104	29	multidimensional	multidimensional	ADJ
ijassa-1958	104	30	real	real	ADJ
ijassa-1958	104	31	space	space	NOUN
ijassa-1958	104	32	.	.	PUNCT
ijassa-1958	105	1	at	at	ADP
ijassa-1958	105	2	the	the	DET
ijassa-1958	105	3	same	same	ADJ
ijassa-1958	105	4	time	time	NOUN
ijassa-1958	105	5	,	,	PUNCT
ijassa-1958	105	6	only	only	ADV
ijassa-1958	105	7	degenerate	degenerate	ADJ
ijassa-1958	105	8	finite	finite	ADJ
ijassa-1958	105	9	metric	metric	ADJ
ijassa-1958	105	10	spaces	space	NOUN
ijassa-1958	105	11	(	(	PUNCT
ijassa-1958	105	12	consisting	consist	VERB
ijassa-1958	105	13	of	of	ADP
ijassa-1958	105	14	no	no	DET
ijassa-1958	105	15	more	more	ADJ
ijassa-1958	105	16	than	than	ADP
ijassa-1958	105	17	3	3	NUM
ijassa-1958	105	18	points	point	NOUN
ijassa-1958	105	19	)	)	PUNCT
ijassa-1958	105	20	,	,	PUNCT
ijassa-1958	105	21	discrete	discrete	ADJ
ijassa-1958	105	22	metric	metric	ADJ
ijassa-1958	105	23	spaces	space	NOUN
ijassa-1958	105	24	,	,	PUNCT
ijassa-1958	105	25	and	and	CCONJ
ijassa-1958	105	26	r	r	NOUN
ijassa-1958	105	27	fit	fit	VERB
ijassa-1958	105	28	the	the	DET
ijassa-1958	105	29	conditions	condition	NOUN
ijassa-1958	105	30	for	for	ADP
ijassa-1958	105	31	compatibility	compatibility	NOUN
ijassa-1958	105	32	with	with	ADP
ijassa-1958	105	33	some	some	DET
ijassa-1958	105	34	order	order	NOUN
ijassa-1958	105	35	.	.	PUNCT
ijassa-1958	106	1	let	let	VERB
ijassa-1958	106	2	us	we	PRON
ijassa-1958	106	3	construct	construct	VERB
ijassa-1958	106	4	an	an	DET
ijassa-1958	106	5	example	example	NOUN
ijassa-1958	106	6	of	of	ADP
ijassa-1958	106	7	a	a	DET
ijassa-1958	106	8	more	more	ADV
ijassa-1958	106	9	nontrivial	nontrivial	ADJ
ijassa-1958	106	10	space	space	NOUN
ijassa-1958	106	11	with	with	ADP
ijassa-1958	106	12	such	such	ADJ
ijassa-1958	106	13	properties	property	NOUN
ijassa-1958	106	14	obtained	obtain	VERB
ijassa-1958	106	15	by	by	ADP
ijassa-1958	106	16	a	a	DET
ijassa-1958	106	17	certain	certain	ADJ
ijassa-1958	106	18	deformation	deformation	NOUN
ijassa-1958	106	19	of	of	ADP
ijassa-1958	106	20	a	a	DET
ijassa-1958	106	21	countable	countable	ADJ
ijassa-1958	106	22	subset	subset	NOUN
ijassa-1958	106	23	of	of	ADP
ijassa-1958	106	24	points	point	NOUN
ijassa-1958	106	25	on	on	ADP
ijassa-1958	106	26	a	a	DET
ijassa-1958	106	27	line	line	NOUN
ijassa-1958	106	28	.	.	PUNCT
ijassa-1958	107	1	to	to	PART
ijassa-1958	107	2	begin	begin	VERB
ijassa-1958	107	3	with	with	ADP
ijassa-1958	107	4	,	,	PUNCT
ijassa-1958	107	5	we	we	PRON
ijassa-1958	107	6	recall	recall	VERB
ijassa-1958	107	7	the	the	DET
ijassa-1958	107	8	definition	definition	NOUN
ijassa-1958	107	9	of	of	ADP
ijassa-1958	107	10	a	a	DET
ijassa-1958	107	11	uniformly	uniformly	ADV
ijassa-1958	107	12	discrete	discrete	ADJ
ijassa-1958	107	13	set	set	NOUN
ijassa-1958	107	14	[	[	X
ijassa-1958	107	15	9	9	NUM
ijassa-1958	107	16	]	]	PUNCT
ijassa-1958	107	17	.	.	PUNCT
ijassa-1958	108	1	definition	definition	NOUN
ijassa-1958	108	2	4.1	4.1	NUM
ijassa-1958	108	3	:	:	PUNCT
ijassa-1958	108	4	if	if	SCONJ
ijassa-1958	108	5	(	(	PUNCT
ijassa-1958	108	6	m	m	NOUN
ijassa-1958	108	7	,	,	PUNCT
ijassa-1958	108	8	d	d	NOUN
ijassa-1958	108	9	)	)	PUNCT
ijassa-1958	108	10	is	be	AUX
ijassa-1958	108	11	a	a	DET
ijassa-1958	108	12	metric	metric	ADJ
ijassa-1958	108	13	space	space	NOUN
ijassa-1958	108	14	,	,	PUNCT
ijassa-1958	108	15	and	and	CCONJ
ijassa-1958	108	16	a	a	DET
ijassa-1958	108	17	⊂	⊂	PROPN
ijassa-1958	108	18	m	m	NOUN
ijassa-1958	108	19	,	,	PUNCT
ijassa-1958	108	20	then	then	ADV
ijassa-1958	108	21	the	the	DET
ijassa-1958	108	22	packing	packing	NOUN
ijassa-1958	108	23	radius	radius	NOUN
ijassa-1958	108	24	r	r	NOUN
ijassa-1958	108	25	of	of	ADP
ijassa-1958	108	26	the	the	DET
ijassa-1958	108	27	set	set	NOUN
ijassa-1958	108	28	a	a	PRON
ijassa-1958	108	29	is	be	AUX
ijassa-1958	108	30	defined	define	VERB
ijassa-1958	108	31	as	as	ADP
ijassa-1958	108	32	:	:	PUNCT
ijassa-1958	108	33	r	r	NOUN
ijassa-1958	108	34	=	=	SYM
ijassa-1958	108	35	1	1	NUM
ijassa-1958	108	36	2	2	NUM
ijassa-1958	108	37	inf	inf	NOUN
ijassa-1958	108	38	x	x	NOUN
ijassa-1958	108	39	,	,	PUNCT
ijassa-1958	108	40	y∈a	y∈a	ADJ
ijassa-1958	108	41	d(x	d(x	PROPN
ijassa-1958	108	42	,	,	PUNCT
ijassa-1958	108	43	y	y	NOUN
ijassa-1958	108	44	)	)	PUNCT
ijassa-1958	108	45	.	.	PUNCT
ijassa-1958	109	1	definition	definition	NOUN
ijassa-1958	109	2	4.2	4.2	NUM
ijassa-1958	109	3	:	:	PUNCT
ijassa-1958	109	4	a	a	DET
ijassa-1958	109	5	set	set	NOUN
ijassa-1958	109	6	is	be	AUX
ijassa-1958	109	7	called	call	VERB
ijassa-1958	109	8	uniformly	uniformly	ADV
ijassa-1958	109	9	discrete	discrete	ADJ
ijassa-1958	109	10	if	if	SCONJ
ijassa-1958	109	11	it	it	PRON
ijassa-1958	109	12	has	have	VERB
ijassa-1958	109	13	a	a	DET
ijassa-1958	109	14	nonzero	nonzero	NOUN
ijassa-1958	109	15	covering	covering	NOUN
ijassa-1958	109	16	radius	radius	NOUN
ijassa-1958	109	17	.	.	PUNCT
ijassa-1958	110	1	now	now	ADV
ijassa-1958	110	2	we	we	PRON
ijassa-1958	110	3	introduce	introduce	VERB
ijassa-1958	110	4	the	the	DET
ijassa-1958	110	5	definition	definition	NOUN
ijassa-1958	110	6	of	of	ADP
ijassa-1958	110	7	a	a	DET
ijassa-1958	110	8	discrete	discrete	ADJ
ijassa-1958	110	9	curved	curved	ADJ
ijassa-1958	110	10	line	line	NOUN
ijassa-1958	110	11	,	,	PUNCT
ijassa-1958	110	12	which	which	PRON
ijassa-1958	110	13	will	will	AUX
ijassa-1958	110	14	be	be	AUX
ijassa-1958	110	15	a	a	DET
ijassa-1958	110	16	model	model	NOUN
ijassa-1958	110	17	example	example	NOUN
ijassa-1958	110	18	of	of	ADP
ijassa-1958	110	19	a	a	DET
ijassa-1958	110	20	set	set	NOUN
ijassa-1958	110	21	for	for	ADP
ijassa-1958	110	22	many	many	ADJ
ijassa-1958	110	23	constructions	construction	NOUN
ijassa-1958	110	24	in	in	ADP
ijassa-1958	110	25	this	this	DET
ijassa-1958	110	26	section	section	NOUN
ijassa-1958	110	27	.	.	PUNCT
ijassa-1958	111	1	definition	definition	NOUN
ijassa-1958	111	2	4.3	4.3	NUM
ijassa-1958	111	3	:	:	PUNCT
ijassa-1958	111	4	let	let	VERB
ijassa-1958	111	5	(	(	PUNCT
ijassa-1958	111	6	m	m	NOUN
ijassa-1958	111	7	,	,	PUNCT
ijassa-1958	111	8	d	d	NOUN
ijassa-1958	111	9	)	)	PUNCT
ijassa-1958	111	10	be	be	AUX
ijassa-1958	111	11	a	a	DET
ijassa-1958	111	12	metric	metric	ADJ
ijassa-1958	111	13	space	space	NOUN
ijassa-1958	111	14	.	.	PUNCT
ijassa-1958	112	1	let	let	VERB
ijassa-1958	112	2	d	d	PRON
ijassa-1958	112	3	be	be	AUX
ijassa-1958	112	4	a	a	DET
ijassa-1958	112	5	countable	countable	ADJ
ijassa-1958	112	6	uniformly	uniformly	ADV
ijassa-1958	112	7	discrete	discrete	ADJ
ijassa-1958	112	8	subset	subset	NOUN
ijassa-1958	112	9	of	of	ADP
ijassa-1958	112	10	r	r	NOUN
ijassa-1958	112	11	with	with	ADP
ijassa-1958	112	12	covering	cover	VERB
ijassa-1958	112	13	radius	radius	PROPN
ijassa-1958	112	14	r.	r.	PROPN
ijassa-1958	112	15	denote	denote	VERB
ijassa-1958	112	16	by	by	ADP
ijassa-1958	112	17	x	x	PUNCT
ijassa-1958	112	18	the	the	DET
ijassa-1958	112	19	image	image	NOUN
ijassa-1958	112	20	of	of	ADP
ijassa-1958	112	21	an	an	DET
ijassa-1958	112	22	isometric	isometric	ADJ
ijassa-1958	112	23	embedding	embedding	NOUN
ijassa-1958	112	24	of	of	ADP
ijassa-1958	112	25	d	d	PROPN
ijassa-1958	112	26	into	into	ADP
ijassa-1958	112	27	m.	m.	NOUN
ijassa-1958	112	28	let	let	VERB
ijassa-1958	112	29	ux	ux	PROPN
ijassa-1958	112	30	∈	∈	PROPN
ijassa-1958	112	31	m	m	AUX
ijassa-1958	112	32	be	be	AUX
ijassa-1958	112	33	chosen	choose	VERB
ijassa-1958	112	34	for	for	ADP
ijassa-1958	112	35	any	any	DET
ijassa-1958	112	36	x	x	SYM
ijassa-1958	112	37	∈	∈	PROPN
ijassa-1958	112	38	x	x	PUNCT
ijassa-1958	112	39	such	such	ADJ
ijassa-1958	112	40	that	that	SCONJ
ijassa-1958	112	41	there	there	PRON
ijassa-1958	112	42	exists	exist	VERB
ijassa-1958	112	43	c	c	NOUN
ijassa-1958	112	44	>	>	X
ijassa-1958	112	45	0	0	NUM
ijassa-1958	113	1	such	such	ADJ
ijassa-1958	113	2	that	that	PRON
ijassa-1958	113	3	for	for	ADP
ijassa-1958	113	4	all	all	DET
ijassa-1958	113	5	x	x	SYM
ijassa-1958	113	6	∈	∈	NOUN
ijassa-1958	113	7	x	x	INTJ
ijassa-1958	113	8	it	it	PRON
ijassa-1958	113	9	is	be	AUX
ijassa-1958	113	10	true	true	ADJ
ijassa-1958	113	11	that	that	SCONJ
ijassa-1958	113	12	d(x	d(x	NOUN
ijassa-1958	113	13	,	,	PUNCT
ijassa-1958	113	14	ux	ux	NOUN
ijassa-1958	113	15	)	)	PUNCT
ijassa-1958	113	16	≤	≤	NOUN
ijassa-1958	113	17	c.	c.	NOUN
ijassa-1958	113	18	let	let	VERB
ijassa-1958	113	19	ε	ε	PROPN
ijassa-1958	113	20	=	=	PRON
ijassa-1958	113	21	inf{c|	inf{c|	PROPN
ijassa-1958	113	22	∀x	∀x	X
ijassa-1958	113	23	∈	∈	PROPN
ijassa-1958	113	24	x	x	PUNCT
ijassa-1958	113	25	d(x	d(x	PROPN
ijassa-1958	113	26	,	,	PUNCT
ijassa-1958	113	27	ux	ux	NOUN
ijassa-1958	113	28	)	)	PUNCT
ijassa-1958	113	29	≤	≤	NUM
ijassa-1958	113	30	δ	δ	PROPN
ijassa-1958	113	31	}	}	PUNCT
ijassa-1958	113	32	.	.	PUNCT
ijassa-1958	114	1	we	we	PRON
ijassa-1958	114	2	refer	refer	VERB
ijassa-1958	114	3	to	to	ADP
ijassa-1958	114	4	a	a	DET
ijassa-1958	114	5	set	set	NOUN
ijassa-1958	114	6	{	{	PUNCT
ijassa-1958	114	7	ux}x∈x	ux}x∈x	NOUN
ijassa-1958	114	8	with	with	ADP
ijassa-1958	114	9	a	a	DET
ijassa-1958	114	10	metric	metric	NOUN
ijassa-1958	114	11	induced	induce	VERB
ijassa-1958	114	12	from	from	ADP
ijassa-1958	114	13	m	m	PROPN
ijassa-1958	114	14	and	and	CCONJ
ijassa-1958	114	15	an	an	DET
ijassa-1958	114	16	order	order	NOUN
ijassa-1958	114	17	induced	induce	VERB
ijassa-1958	114	18	from	from	ADP
ijassa-1958	114	19	x	x	PUNCT
ijassa-1958	114	20	(	(	PUNCT
ijassa-1958	114	21	meaning	mean	VERB
ijassa-1958	114	22	that	that	SCONJ
ijassa-1958	114	23	ux1	ux1	PROPN
ijassa-1958	114	24	≤	≤	X
ijassa-1958	114	25	ux2	ux2	PROPN
ijassa-1958	114	26	if	if	SCONJ
ijassa-1958	114	27	x1	x1	PROPN
ijassa-1958	114	28	<	<	X
ijassa-1958	114	29	x2	x2	PROPN
ijassa-1958	114	30	)	)	PUNCT
ijassa-1958	114	31	as	as	ADP
ijassa-1958	114	32	a	a	DET
ijassa-1958	114	33	discrete	discrete	ADJ
ijassa-1958	114	34	curved	curved	ADJ
ijassa-1958	114	35	line	line	NOUN
ijassa-1958	114	36	.	.	PUNCT
ijassa-1958	115	1	alternatively	alternatively	ADV
ijassa-1958	115	2	,	,	PUNCT
ijassa-1958	115	3	if	if	SCONJ
ijassa-1958	115	4	we	we	PRON
ijassa-1958	115	5	want	want	VERB
ijassa-1958	115	6	to	to	PART
ijassa-1958	115	7	highlight	highlight	VERB
ijassa-1958	115	8	the	the	DET
ijassa-1958	115	9	parameters	parameter	NOUN
ijassa-1958	115	10	,	,	PUNCT
ijassa-1958	115	11	we	we	PRON
ijassa-1958	115	12	can	can	AUX
ijassa-1958	115	13	call	call	VERB
ijassa-1958	115	14	it	it	PRON
ijassa-1958	115	15	a	a	DET
ijassa-1958	115	16	(	(	PUNCT
ijassa-1958	115	17	r	r	NOUN
ijassa-1958	115	18	,	,	PUNCT
ijassa-1958	115	19	ε)-discrete	ε)-discrete	ADJ
ijassa-1958	115	20	curved	curved	ADJ
ijassa-1958	115	21	line	line	NOUN
ijassa-1958	115	22	.	.	PUNCT
ijassa-1958	116	1	let	let	VERB
ijassa-1958	116	2	’s	’s	NOUN
ijassa-1958	116	3	try	try	VERB
ijassa-1958	116	4	to	to	PART
ijassa-1958	116	5	consider	consider	VERB
ijassa-1958	116	6	the	the	DET
ijassa-1958	116	7	motivation	motivation	NOUN
ijassa-1958	116	8	for	for	ADP
ijassa-1958	116	9	such	such	DET
ijassa-1958	116	10	a	a	DET
ijassa-1958	116	11	definition	definition	NOUN
ijassa-1958	116	12	.	.	PUNCT
ijassa-1958	117	1	it	it	PRON
ijassa-1958	117	2	comes	come	VERB
ijassa-1958	117	3	down	down	ADP
ijassa-1958	117	4	to	to	ADP
ijassa-1958	117	5	two	two	NUM
ijassa-1958	117	6	simple	simple	ADJ
ijassa-1958	117	7	concepts	concept	NOUN
ijassa-1958	117	8	:	:	PUNCT
ijassa-1958	117	9	copyright	copyright	NOUN
ijassa-1958	117	10	©	©	PROPN
ijassa-1958	117	11	2024	2024	NUM
ijassa-1958	117	12	assa	assa	NOUN
ijassa-1958	117	13	.	.	PUNCT
ijassa-1958	118	1	adv	adv	PROPN
ijassa-1958	118	2	syst	syst	PROPN
ijassa-1958	118	3	sci	sci	PROPN
ijassa-1958	118	4	appl	appl	PROPN
ijassa-1958	118	5	(	(	PUNCT
ijassa-1958	118	6	2024	2024	NUM
ijassa-1958	118	7	)	)	PUNCT
ijassa-1958	118	8	6	6	NUM
ijassa-1958	118	9	p.	p.	NOUN
ijassa-1958	118	10	klimov	klimov	PROPN
ijassa-1958	118	11	,	,	PUNCT
ijassa-1958	118	12	r.	r.	PROPN
ijassa-1958	118	13	sengupta	sengupta	PROPN
ijassa-1958	118	14	≥	≥	NUM
ijassa-1958	118	15	2r	2r	NUM
ijassa-1958	118	16	ε	ε	PROPN
ijassa-1958	118	17	ε	ε	PROPN
ijassa-1958	118	18	ε	ε	PROPN
ijassa-1958	118	19	f	f	PROPN
ijassa-1958	118	20	u	u	PROPN
ijassa-1958	118	21	d	d	X
ijassa-1958	118	22	x	x	SYM
ijassa-1958	118	23	u	u	PROPN
ijassa-1958	118	24	e	e	X
ijassa-1958	118	25	fig	fig	NOUN
ijassa-1958	118	26	.	.	PUNCT
ijassa-1958	119	1	4.2	4.2	NUM
ijassa-1958	119	2	.	.	PUNCT
ijassa-1958	120	1	the	the	DET
ijassa-1958	120	2	concept	concept	NOUN
ijassa-1958	120	3	of	of	ADP
ijassa-1958	120	4	a	a	DET
ijassa-1958	120	5	discrete	discrete	ADJ
ijassa-1958	120	6	curved	curved	ADJ
ijassa-1958	120	7	line	line	NOUN
ijassa-1958	120	8	:	:	PUNCT
ijassa-1958	120	9	we	we	PRON
ijassa-1958	120	10	start	start	VERB
ijassa-1958	120	11	by	by	ADP
ijassa-1958	120	12	selecting	select	VERB
ijassa-1958	120	13	a	a	DET
ijassa-1958	120	14	set	set	NOUN
ijassa-1958	120	15	of	of	ADP
ijassa-1958	120	16	points	point	NOUN
ijassa-1958	120	17	in	in	ADP
ijassa-1958	120	18	d	d	PROPN
ijassa-1958	120	19	⊂	⊂	PROPN
ijassa-1958	120	20	r.	r.	PROPN
ijassa-1958	120	21	these	these	DET
ijassa-1958	120	22	points	point	NOUN
ijassa-1958	120	23	are	be	AUX
ijassa-1958	120	24	uniformly	uniformly	ADV
ijassa-1958	120	25	distributed	distribute	VERB
ijassa-1958	120	26	and	and	CCONJ
ijassa-1958	120	27	have	have	VERB
ijassa-1958	120	28	a	a	DET
ijassa-1958	120	29	covering	covering	NOUN
ijassa-1958	120	30	radius	radius	NOUN
ijassa-1958	120	31	of	of	ADP
ijassa-1958	120	32	r	r	NOUN
ijassa-1958	120	33	,	,	PUNCT
ijassa-1958	120	34	meaning	mean	VERB
ijassa-1958	120	35	that	that	SCONJ
ijassa-1958	120	36	no	no	DET
ijassa-1958	120	37	two	two	NUM
ijassa-1958	120	38	points	point	NOUN
ijassa-1958	120	39	are	be	AUX
ijassa-1958	120	40	closer	close	ADJ
ijassa-1958	120	41	than	than	ADP
ijassa-1958	120	42	2r	2r	NUM
ijassa-1958	120	43	.	.	PUNCT
ijassa-1958	121	1	we	we	PRON
ijassa-1958	121	2	then	then	ADV
ijassa-1958	121	3	embed	embed	VERB
ijassa-1958	121	4	this	this	DET
ijassa-1958	121	5	set	set	NOUN
ijassa-1958	121	6	into	into	ADP
ijassa-1958	121	7	a	a	DET
ijassa-1958	121	8	new	new	ADJ
ijassa-1958	121	9	metric	metric	ADJ
ijassa-1958	121	10	space	space	NOUN
ijassa-1958	121	11	,	,	PUNCT
ijassa-1958	121	12	creating	create	VERB
ijassa-1958	121	13	a	a	DET
ijassa-1958	121	14	new	new	ADJ
ijassa-1958	121	15	space	space	NOUN
ijassa-1958	121	16	called	call	VERB
ijassa-1958	121	17	x	x	INTJ
ijassa-1958	121	18	.	.	PUNCT
ijassa-1958	122	1	next	next	ADV
ijassa-1958	122	2	,	,	PUNCT
ijassa-1958	122	3	we	we	PRON
ijassa-1958	122	4	define	define	VERB
ijassa-1958	122	5	a	a	DET
ijassa-1958	122	6	subset	subset	NOUN
ijassa-1958	122	7	of	of	ADP
ijassa-1958	122	8	u	u	NOUN
ijassa-1958	122	9	by	by	ADP
ijassa-1958	122	10	shifting	shift	VERB
ijassa-1958	122	11	the	the	DET
ijassa-1958	122	12	elements	element	NOUN
ijassa-1958	122	13	of	of	ADP
ijassa-1958	122	14	x	x	PUNCT
ijassa-1958	122	15	by	by	ADP
ijassa-1958	122	16	no	no	PRON
ijassa-1958	122	17	more	more	ADJ
ijassa-1958	122	18	than	than	ADP
ijassa-1958	122	19	ε	ε	PROPN
ijassa-1958	122	20	.	.	PUNCT
ijassa-1958	123	1	this	this	PRON
ijassa-1958	123	2	allows	allow	VERB
ijassa-1958	123	3	us	we	PRON
ijassa-1958	123	4	to	to	PART
ijassa-1958	123	5	create	create	VERB
ijassa-1958	123	6	a	a	DET
ijassa-1958	123	7	curved	curved	ADJ
ijassa-1958	123	8	line	line	NOUN
ijassa-1958	123	9	structure	structure	NOUN
ijassa-1958	123	10	within	within	ADP
ijassa-1958	123	11	the	the	DET
ijassa-1958	123	12	space	space	NOUN
ijassa-1958	123	13	u	u	NOUN
ijassa-1958	123	14	.	.	PUNCT
ijassa-1958	124	1	1	1	X
ijassa-1958	124	2	.	.	X
ijassa-1958	124	3	we	we	PRON
ijassa-1958	124	4	select	select	VERB
ijassa-1958	124	5	a	a	DET
ijassa-1958	124	6	specific	specific	ADJ
ijassa-1958	124	7	subset	subset	NOUN
ijassa-1958	124	8	of	of	ADP
ijassa-1958	124	9	points	point	NOUN
ijassa-1958	124	10	from	from	ADP
ijassa-1958	124	11	a	a	DET
ijassa-1958	124	12	straight	straight	ADJ
ijassa-1958	124	13	line	line	NOUN
ijassa-1958	124	14	(	(	PUNCT
ijassa-1958	124	15	embedded	embed	VERB
ijassa-1958	124	16	in	in	ADP
ijassa-1958	124	17	another	another	DET
ijassa-1958	124	18	metric	metric	ADJ
ijassa-1958	124	19	space	space	NOUN
ijassa-1958	124	20	)	)	PUNCT
ijassa-1958	124	21	.	.	PUNCT
ijassa-1958	125	1	these	these	DET
ijassa-1958	125	2	points	point	NOUN
ijassa-1958	125	3	are	be	AUX
ijassa-1958	125	4	chosen	choose	VERB
ijassa-1958	125	5	such	such	ADJ
ijassa-1958	125	6	that	that	SCONJ
ijassa-1958	125	7	they	they	PRON
ijassa-1958	125	8	are	be	AUX
ijassa-1958	125	9	a	a	DET
ijassa-1958	125	10	sufficient	sufficient	ADJ
ijassa-1958	125	11	distance	distance	NOUN
ijassa-1958	125	12	apart	apart	ADV
ijassa-1958	125	13	from	from	ADP
ijassa-1958	125	14	each	each	DET
ijassa-1958	125	15	other	other	ADJ
ijassa-1958	125	16	,	,	PUNCT
ijassa-1958	125	17	for	for	ADP
ijassa-1958	125	18	example	example	NOUN
ijassa-1958	125	19	,	,	PUNCT
ijassa-1958	125	20	using	use	VERB
ijassa-1958	125	21	sets	set	NOUN
ijassa-1958	125	22	like	like	ADP
ijassa-1958	125	23	n	n	NOUN
ijassa-1958	125	24	or	or	CCONJ
ijassa-1958	125	25	z	z	NOUN
ijassa-1958	125	26	,	,	PUNCT
ijassa-1958	125	27	where	where	SCONJ
ijassa-1958	125	28	all	all	DET
ijassa-1958	125	29	points	point	NOUN
ijassa-1958	125	30	are	be	AUX
ijassa-1958	125	31	at	at	ADV
ijassa-1958	125	32	least	least	ADJ
ijassa-1958	125	33	1	1	NUM
ijassa-1958	125	34	unit	unit	NOUN
ijassa-1958	125	35	apart	apart	ADV
ijassa-1958	125	36	.	.	PUNCT
ijassa-1958	126	1	2	2	X
ijassa-1958	126	2	.	.	X
ijassa-1958	126	3	we	we	PRON
ijassa-1958	126	4	perform	perform	VERB
ijassa-1958	126	5	a	a	DET
ijassa-1958	126	6	small	small	ADJ
ijassa-1958	126	7	shift	shift	NOUN
ijassa-1958	126	8	on	on	ADP
ijassa-1958	126	9	each	each	DET
ijassa-1958	126	10	point	point	NOUN
ijassa-1958	126	11	,	,	PUNCT
ijassa-1958	126	12	relative	relative	ADJ
ijassa-1958	126	13	to	to	ADP
ijassa-1958	126	14	the	the	DET
ijassa-1958	126	15	distance	distance	NOUN
ijassa-1958	126	16	between	between	ADP
ijassa-1958	126	17	them	they	PRON
ijassa-1958	126	18	.	.	PUNCT
ijassa-1958	127	1	since	since	SCONJ
ijassa-1958	127	2	the	the	DET
ijassa-1958	127	3	shift	shift	NOUN
ijassa-1958	127	4	is	be	AUX
ijassa-1958	127	5	minimal	minimal	ADJ
ijassa-1958	127	6	,	,	PUNCT
ijassa-1958	127	7	we	we	PRON
ijassa-1958	127	8	can	can	AUX
ijassa-1958	127	9	expect	expect	VERB
ijassa-1958	127	10	that	that	SCONJ
ijassa-1958	127	11	after	after	ADP
ijassa-1958	127	12	the	the	DET
ijassa-1958	127	13	shift	shift	NOUN
ijassa-1958	127	14	,	,	PUNCT
ijassa-1958	127	15	for	for	ADP
ijassa-1958	127	16	any	any	DET
ijassa-1958	127	17	three	three	NUM
ijassa-1958	127	18	ordered	order	VERB
ijassa-1958	127	19	points	point	NOUN
ijassa-1958	127	20	x	x	PUNCT
ijassa-1958	127	21	≤	≤	NUM
ijassa-1958	127	22	y	y	PROPN
ijassa-1958	127	23	≤	≤	PROPN
ijassa-1958	127	24	z	z	PROPN
ijassa-1958	127	25	,	,	PUNCT
ijassa-1958	127	26	the	the	DET
ijassa-1958	127	27	distances	distance	NOUN
ijassa-1958	127	28	d′(x	d′(x	NOUN
ijassa-1958	127	29	,	,	PUNCT
ijassa-1958	127	30	z	z	NOUN
ijassa-1958	127	31	)	)	PUNCT
ijassa-1958	127	32	,	,	PUNCT
ijassa-1958	127	33	d′(y	d′(y	PROPN
ijassa-1958	127	34	,	,	PUNCT
ijassa-1958	127	35	y′	y′	NUM
ijassa-1958	127	36	)	)	PUNCT
ijassa-1958	127	37	,	,	PUNCT
ijassa-1958	127	38	and	and	CCONJ
ijassa-1958	127	39	d′(y′	d′(y′	NOUN
ijassa-1958	127	40	,	,	PUNCT
ijassa-1958	127	41	z′	z′	NUM
ijassa-1958	127	42	)	)	PUNCT
ijassa-1958	127	43	will	will	AUX
ijassa-1958	127	44	remain	remain	VERB
ijassa-1958	127	45	larger	large	ADJ
ijassa-1958	127	46	than	than	ADP
ijassa-1958	127	47	the	the	DET
ijassa-1958	127	48	original	original	ADJ
ijassa-1958	127	49	distances	distance	NOUN
ijassa-1958	127	50	d(x	d(x	PROPN
ijassa-1958	127	51	,	,	PUNCT
ijassa-1958	127	52	y	y	PROPN
ijassa-1958	127	53	)	)	PUNCT
ijassa-1958	127	54	,	,	PUNCT
ijassa-1958	127	55	d(y	d(y	PROPN
ijassa-1958	127	56	,	,	PUNCT
ijassa-1958	127	57	z	z	NOUN
ijassa-1958	127	58	)	)	PUNCT
ijassa-1958	127	59	,	,	PUNCT
ijassa-1958	127	60	and	and	CCONJ
ijassa-1958	127	61	d(x	d(x	PROPN
ijassa-1958	127	62	,	,	PUNCT
ijassa-1958	127	63	z	z	NOUN
ijassa-1958	127	64	)	)	PUNCT
ijassa-1958	127	65	,	,	PUNCT
ijassa-1958	127	66	respectively	respectively	ADV
ijassa-1958	127	67	.	.	PUNCT
ijassa-1958	128	1	this	this	PRON
ijassa-1958	128	2	is	be	AUX
ijassa-1958	128	3	because	because	SCONJ
ijassa-1958	128	4	all	all	DET
ijassa-1958	128	5	pairwise	pairwise	NOUN
ijassa-1958	128	6	distances	distance	NOUN
ijassa-1958	128	7	change	change	VERB
ijassa-1958	128	8	slightly	slightly	ADV
ijassa-1958	128	9	compared	compare	VERB
ijassa-1958	128	10	to	to	ADP
ijassa-1958	128	11	the	the	DET
ijassa-1958	128	12	significant	significant	ADJ
ijassa-1958	128	13	difference	difference	NOUN
ijassa-1958	128	14	between	between	ADP
ijassa-1958	128	15	d(x	d(x	PROPN
ijassa-1958	128	16	,	,	PUNCT
ijassa-1958	128	17	z	z	NOUN
ijassa-1958	128	18	)	)	PUNCT
ijassa-1958	128	19	and	and	CCONJ
ijassa-1958	128	20	the	the	DET
ijassa-1958	128	21	distances	distance	NOUN
ijassa-1958	128	22	d(x	d(x	PROPN
ijassa-1958	128	23	,	,	PUNCT
ijassa-1958	128	24	y	y	NOUN
ijassa-1958	128	25	)	)	PUNCT
ijassa-1958	128	26	and	and	CCONJ
ijassa-1958	128	27	d(y	d(y	PROPN
ijassa-1958	128	28	,	,	PUNCT
ijassa-1958	128	29	z	z	NOUN
ijassa-1958	128	30	)	)	PUNCT
ijassa-1958	128	31	.	.	PUNCT
ijassa-1958	129	1	a	a	DET
ijassa-1958	129	2	discrete	discrete	ADJ
ijassa-1958	129	3	curved	curve	VERB
ijassa-1958	129	4	straight	straight	ADJ
ijassa-1958	129	5	line	line	NOUN
ijassa-1958	129	6	allows	allow	VERB
ijassa-1958	129	7	us	we	PRON
ijassa-1958	129	8	to	to	PART
ijassa-1958	129	9	make	make	VERB
ijassa-1958	129	10	any	any	DET
ijassa-1958	129	11	desired	desire	VERB
ijassa-1958	129	12	shifts	shift	NOUN
ijassa-1958	129	13	relative	relative	ADJ
ijassa-1958	129	14	to	to	ADP
ijassa-1958	129	15	the	the	DET
ijassa-1958	129	16	original	original	ADJ
ijassa-1958	129	17	location	location	NOUN
ijassa-1958	129	18	of	of	ADP
ijassa-1958	129	19	points	point	NOUN
ijassa-1958	129	20	from	from	ADP
ijassa-1958	129	21	the	the	DET
ijassa-1958	129	22	perspective	perspective	NOUN
ijassa-1958	129	23	of	of	ADP
ijassa-1958	129	24	the	the	DET
ijassa-1958	129	25	geometry	geometry	NOUN
ijassa-1958	129	26	of	of	ADP
ijassa-1958	129	27	the	the	DET
ijassa-1958	129	28	space	space	NOUN
ijassa-1958	129	29	where	where	SCONJ
ijassa-1958	129	30	the	the	DET
ijassa-1958	129	31	embedding	embedding	NOUN
ijassa-1958	129	32	occurs	occur	VERB
ijassa-1958	129	33	.	.	PUNCT
ijassa-1958	130	1	now	now	ADV
ijassa-1958	130	2	we	we	PRON
ijassa-1958	130	3	will	will	AUX
ijassa-1958	130	4	prove	prove	VERB
ijassa-1958	130	5	the	the	DET
ijassa-1958	130	6	general	general	ADJ
ijassa-1958	130	7	theorem	theorem	NOUN
ijassa-1958	130	8	on	on	ADP
ijassa-1958	130	9	the	the	DET
ijassa-1958	130	10	necessary	necessary	ADJ
ijassa-1958	130	11	and	and	CCONJ
ijassa-1958	130	12	sufficient	sufficient	ADJ
ijassa-1958	130	13	conditions	condition	NOUN
ijassa-1958	130	14	for	for	ADP
ijassa-1958	130	15	the	the	DET
ijassa-1958	130	16	compatibility	compatibility	NOUN
ijassa-1958	130	17	of	of	ADP
ijassa-1958	130	18	the	the	DET
ijassa-1958	130	19	metric	metric	ADJ
ijassa-1958	130	20	and	and	CCONJ
ijassa-1958	130	21	order	order	NOUN
ijassa-1958	130	22	for	for	ADP
ijassa-1958	130	23	discrete	discrete	ADJ
ijassa-1958	130	24	curved	curved	ADJ
ijassa-1958	130	25	lines	line	NOUN
ijassa-1958	130	26	.	.	PUNCT
ijassa-1958	131	1	theorem	theorem	VERB
ijassa-1958	131	2	4.1	4.1	NUM
ijassa-1958	131	3	:	:	PUNCT
ijassa-1958	131	4	(	(	PUNCT
ijassa-1958	131	5	sufficient	sufficient	ADJ
ijassa-1958	131	6	condition	condition	NOUN
ijassa-1958	131	7	for	for	ADP
ijassa-1958	131	8	the	the	DET
ijassa-1958	131	9	compatibility	compatibility	NOUN
ijassa-1958	131	10	of	of	ADP
ijassa-1958	131	11	the	the	DET
ijassa-1958	131	12	metric	metric	NOUN
ijassa-1958	131	13	with	with	ADP
ijassa-1958	131	14	the	the	DET
ijassa-1958	131	15	order	order	NOUN
ijassa-1958	131	16	for	for	ADP
ijassa-1958	131	17	a	a	DET
ijassa-1958	131	18	discrete	discrete	ADJ
ijassa-1958	131	19	curved	curved	ADJ
ijassa-1958	131	20	line	line	NOUN
ijassa-1958	131	21	)	)	PUNCT
ijassa-1958	131	22	let	let	VERB
ijassa-1958	131	23	u	u	PRON
ijassa-1958	131	24	be	be	AUX
ijassa-1958	131	25	an	an	DET
ijassa-1958	131	26	(	(	PUNCT
ijassa-1958	131	27	r	r	NOUN
ijassa-1958	131	28	,	,	PUNCT
ijassa-1958	131	29	ε)-discrete	ε)-discrete	ADJ
ijassa-1958	131	30	curved	curved	ADJ
ijassa-1958	131	31	line	line	NOUN
ijassa-1958	131	32	in	in	ADP
ijassa-1958	131	33	the	the	DET
ijassa-1958	131	34	metric	metric	ADJ
ijassa-1958	131	35	space	space	NOUN
ijassa-1958	131	36	(	(	PUNCT
ijassa-1958	131	37	m	m	PROPN
ijassa-1958	131	38	,	,	PUNCT
ijassa-1958	131	39	d	d	NOUN
ijassa-1958	131	40	)	)	PUNCT
ijassa-1958	131	41	.	.	PUNCT
ijassa-1958	132	1	if	if	SCONJ
ijassa-1958	132	2	ε	ε	PROPN
ijassa-1958	132	3	≤	≤	NUM
ijassa-1958	132	4	r	r	NOUN
ijassa-1958	132	5	2	2	NUM
ijassa-1958	132	6	,	,	PUNCT
ijassa-1958	132	7	then	then	ADV
ijassa-1958	132	8	the	the	DET
ijassa-1958	132	9	metric	metric	NOUN
ijassa-1958	132	10	on	on	ADP
ijassa-1958	132	11	u	u	PROPN
ijassa-1958	132	12	is	be	AUX
ijassa-1958	132	13	compatible	compatible	ADJ
ijassa-1958	132	14	with	with	ADP
ijassa-1958	132	15	the	the	DET
ijassa-1958	132	16	order	order	NOUN
ijassa-1958	132	17	on	on	ADP
ijassa-1958	132	18	u	u	PROPN
ijassa-1958	132	19	.	.	PUNCT
ijassa-1958	133	1	copyright	copyright	NOUN
ijassa-1958	133	2	©	©	PROPN
ijassa-1958	133	3	2024	2024	NUM
ijassa-1958	133	4	assa	assa	NOUN
ijassa-1958	133	5	.	.	PUNCT
ijassa-1958	134	1	adv	adv	PROPN
ijassa-1958	134	2	syst	syst	PROPN
ijassa-1958	134	3	sci	sci	PROPN
ijassa-1958	134	4	appl	appl	PROPN
ijassa-1958	134	5	(	(	PUNCT
ijassa-1958	134	6	2024	2024	NUM
ijassa-1958	134	7	)	)	PUNCT
ijassa-1958	134	8	on	on	ADP
ijassa-1958	134	9	the	the	DET
ijassa-1958	134	10	compatibility	compatibility	NOUN
ijassa-1958	134	11	of	of	ADP
ijassa-1958	134	12	metric	metric	ADJ
ijassa-1958	134	13	and	and	CCONJ
ijassa-1958	134	14	linear	linear	ADJ
ijassa-1958	134	15	order	order	NOUN
ijassa-1958	134	16	7	7	NUM
ijassa-1958	134	17	proof	proof	NOUN
ijassa-1958	134	18	according	accord	VERB
ijassa-1958	134	19	to	to	ADP
ijassa-1958	134	20	the	the	DET
ijassa-1958	134	21	definition	definition	NOUN
ijassa-1958	134	22	4.3	4.3	NUM
ijassa-1958	134	23	for	for	ADP
ijassa-1958	134	24	u	u	NOUN
ijassa-1958	134	25	,	,	PUNCT
ijassa-1958	134	26	there	there	PRON
ijassa-1958	134	27	exists	exist	VERB
ijassa-1958	134	28	x	x	PUNCT
ijassa-1958	134	29	,	,	PUNCT
ijassa-1958	134	30	which	which	PRON
ijassa-1958	134	31	is	be	AUX
ijassa-1958	134	32	a	a	DET
ijassa-1958	134	33	uniformly	uniformly	ADV
ijassa-1958	134	34	discrete	discrete	ADJ
ijassa-1958	134	35	subset	subset	NOUN
ijassa-1958	134	36	of	of	ADP
ijassa-1958	134	37	r	r	NOUN
ijassa-1958	134	38	with	with	ADP
ijassa-1958	134	39	a	a	DET
ijassa-1958	134	40	covering	covering	NOUN
ijassa-1958	134	41	radius	radius	NOUN
ijassa-1958	134	42	of	of	ADP
ijassa-1958	134	43	r.	r.	PROPN
ijassa-1958	134	44	the	the	DET
ijassa-1958	134	45	set	set	NOUN
ijassa-1958	134	46	u	u	NOUN
ijassa-1958	134	47	is	be	AUX
ijassa-1958	134	48	defined	define	VERB
ijassa-1958	134	49	as	as	ADP
ijassa-1958	134	50	{	{	PUNCT
ijassa-1958	134	51	ux}x∈x	ux}x∈x	INTJ
ijassa-1958	134	52	,	,	PUNCT
ijassa-1958	134	53	where	where	SCONJ
ijassa-1958	134	54	d(x	d(x	NOUN
ijassa-1958	134	55	,	,	PUNCT
ijassa-1958	134	56	ux	ux	NOUN
ijassa-1958	134	57	)	)	PUNCT
ijassa-1958	134	58	<	<	X
ijassa-1958	134	59	ε	ε	PROPN
ijassa-1958	134	60	.	.	PUNCT
ijassa-1958	134	61	consider	consider	VERB
ijassa-1958	134	62	three	three	NUM
ijassa-1958	134	63	points	point	NOUN
ijassa-1958	134	64	:	:	PUNCT
ijassa-1958	134	65	ux1	ux1	PROPN
ijassa-1958	134	66	≤	≤	PUNCT
ijassa-1958	134	67	ux2	ux2	PROPN
ijassa-1958	134	68	≤	≤	ADJ
ijassa-1958	134	69	ux3	ux3	NOUN
ijassa-1958	134	70	(	(	PUNCT
ijassa-1958	134	71	which	which	PRON
ijassa-1958	134	72	implies	imply	VERB
ijassa-1958	134	73	x1	x1	PROPN
ijassa-1958	134	74	≤	≤	NUM
ijassa-1958	134	75	x2	x2	PROPN
ijassa-1958	134	76	≤	≤	NUM
ijassa-1958	134	77	x3	x3	ADJ
ijassa-1958	134	78	)	)	PUNCT
ijassa-1958	134	79	then	then	ADV
ijassa-1958	134	80	,	,	PUNCT
ijassa-1958	134	81	d(ux1	d(ux1	PROPN
ijassa-1958	134	82	,	,	PUNCT
ijassa-1958	134	83	ux2	ux2	PROPN
ijassa-1958	134	84	)	)	PUNCT
ijassa-1958	134	85	is	be	AUX
ijassa-1958	134	86	less	less	ADJ
ijassa-1958	134	87	than	than	ADP
ijassa-1958	134	88	or	or	CCONJ
ijassa-1958	134	89	equal	equal	ADJ
ijassa-1958	134	90	to	to	ADP
ijassa-1958	134	91	the	the	DET
ijassa-1958	134	92	sum	sum	NOUN
ijassa-1958	134	93	of	of	ADP
ijassa-1958	134	94	three	three	NUM
ijassa-1958	134	95	terms	term	NOUN
ijassa-1958	134	96	:	:	PUNCT
ijassa-1958	134	97	d(x1	d(x1	NOUN
ijassa-1958	134	98	,	,	PUNCT
ijassa-1958	134	99	x2	x2	PROPN
ijassa-1958	134	100	)	)	PUNCT
ijassa-1958	134	101	,	,	PUNCT
ijassa-1958	134	102	d(ux1	d(ux1	PROPN
ijassa-1958	134	103	,	,	PUNCT
ijassa-1958	134	104	x1	x1	PROPN
ijassa-1958	134	105	)	)	PUNCT
ijassa-1958	134	106	,	,	PUNCT
ijassa-1958	134	107	and	and	CCONJ
ijassa-1958	134	108	d(ux2	d(ux2	PROPN
ijassa-1958	134	109	,	,	PUNCT
ijassa-1958	134	110	x2	x2	PROPN
ijassa-1958	134	111	)	)	PUNCT
ijassa-1958	134	112	.	.	PUNCT
ijassa-1958	135	1	this	this	PRON
ijassa-1958	135	2	can	can	AUX
ijassa-1958	135	3	be	be	AUX
ijassa-1958	135	4	shown	show	VERB
ijassa-1958	135	5	using	use	VERB
ijassa-1958	135	6	the	the	DET
ijassa-1958	135	7	triangle	triangle	NOUN
ijassa-1958	135	8	inequality	inequality	NOUN
ijassa-1958	135	9	and	and	CCONJ
ijassa-1958	135	10	the	the	DET
ijassa-1958	135	11	definition	definition	NOUN
ijassa-1958	135	12	of	of	ADP
ijassa-1958	135	13	a	a	DET
ijassa-1958	135	14	uniformly	uniformly	ADV
ijassa-1958	135	15	discrete	discrete	ADJ
ijassa-1958	135	16	line	line	NOUN
ijassa-1958	135	17	.	.	PUNCT
ijassa-1958	136	1	thus	thus	ADV
ijassa-1958	136	2	,	,	PUNCT
ijassa-1958	136	3	we	we	PRON
ijassa-1958	136	4	have	have	VERB
ijassa-1958	136	5	d(ux1	d(ux1	PROPN
ijassa-1958	136	6	,	,	PUNCT
ijassa-1958	136	7	ux2	ux2	PROPN
ijassa-1958	136	8	)	)	PUNCT
ijassa-1958	136	9	≤	≤	NOUN
ijassa-1958	136	10	d(x1	d(x1	NOUN
ijassa-1958	136	11	,	,	PUNCT
ijassa-1958	136	12	x2	x2	PROPN
ijassa-1958	136	13	)	)	PUNCT
ijassa-1958	137	1	+	+	CCONJ
ijassa-1958	137	2	d(ux1	d(ux1	PROPN
ijassa-1958	137	3	,	,	PUNCT
ijassa-1958	137	4	x1	x1	PROPN
ijassa-1958	137	5	)	)	PUNCT
ijassa-1958	138	1	+	+	CCONJ
ijassa-1958	138	2	d(ux2	d(ux2	PROPN
ijassa-1958	138	3	,	,	PUNCT
ijassa-1958	138	4	x2	x2	PROPN
ijassa-1958	138	5	)	)	PUNCT
ijassa-1958	138	6	≤	≤	NUM
ijassa-1958	138	7	2r	2r	NUM
ijassa-1958	138	8	+	+	CCONJ
ijassa-1958	138	9	2ε	2ε	NUM
ijassa-1958	138	10	.	.	PUNCT
ijassa-1958	139	1	it	it	PRON
ijassa-1958	139	2	is	be	AUX
ijassa-1958	139	3	not	not	PART
ijassa-1958	139	4	difficult	difficult	ADJ
ijassa-1958	139	5	to	to	PART
ijassa-1958	139	6	see	see	VERB
ijassa-1958	139	7	that	that	PRON
ijassa-1958	139	8	on	on	ADP
ijassa-1958	139	9	x	x	X
ijassa-1958	139	10	(	(	PUNCT
ijassa-1958	139	11	since	since	SCONJ
ijassa-1958	139	12	x	x	PRON
ijassa-1958	139	13	is	be	AUX
ijassa-1958	139	14	isometric	isometric	ADJ
ijassa-1958	139	15	to	to	ADP
ijassa-1958	139	16	a	a	DET
ijassa-1958	139	17	subset	subset	NOUN
ijassa-1958	139	18	of	of	ADP
ijassa-1958	139	19	the	the	DET
ijassa-1958	139	20	real	real	ADJ
ijassa-1958	139	21	line	line	NOUN
ijassa-1958	139	22	with	with	ADP
ijassa-1958	139	23	covering	cover	VERB
ijassa-1958	139	24	radius	radius	NOUN
ijassa-1958	139	25	r	r	NOUN
ijassa-1958	139	26	)	)	PUNCT
ijassa-1958	139	27	d(x1	d(x1	NOUN
ijassa-1958	139	28	,	,	PUNCT
ijassa-1958	139	29	x3	x3	ADJ
ijassa-1958	139	30	)	)	PUNCT
ijassa-1958	139	31	=	=	SYM
ijassa-1958	139	32	d(x1	d(x1	NOUN
ijassa-1958	139	33	,	,	PUNCT
ijassa-1958	139	34	x2	x2	PROPN
ijassa-1958	139	35	)	)	PUNCT
ijassa-1958	140	1	+	+	NUM
ijassa-1958	140	2	d(x2	d(x2	NOUN
ijassa-1958	140	3	,	,	PUNCT
ijassa-1958	140	4	x3	x3	ADJ
ijassa-1958	140	5	)	)	PUNCT
ijassa-1958	140	6	≥	≥	NOUN
ijassa-1958	140	7	4r	4r	NOUN
ijassa-1958	140	8	.	.	PUNCT
ijassa-1958	141	1	similar	similar	ADJ
ijassa-1958	141	2	to	to	ADP
ijassa-1958	141	3	the	the	DET
ijassa-1958	141	4	reasoning	reasoning	NOUN
ijassa-1958	141	5	above	above	ADV
ijassa-1958	141	6	,	,	PUNCT
ijassa-1958	141	7	we	we	PRON
ijassa-1958	141	8	have	have	VERB
ijassa-1958	141	9	:	:	PUNCT
ijassa-1958	141	10	d(x1	d(x1	ADJ
ijassa-1958	141	11	,	,	PUNCT
ijassa-1958	141	12	x3	x3	ADJ
ijassa-1958	141	13	)	)	PUNCT
ijassa-1958	141	14	≤	≤	NOUN
ijassa-1958	141	15	d(ux1	d(ux1	PROPN
ijassa-1958	141	16	,	,	PUNCT
ijassa-1958	141	17	ux3	ux3	PROPN
ijassa-1958	141	18	)	)	PUNCT
ijassa-1958	141	19	+	+	CCONJ
ijassa-1958	141	20	d(ux1	d(ux1	PROPN
ijassa-1958	141	21	,	,	PUNCT
ijassa-1958	141	22	x1	x1	PROPN
ijassa-1958	141	23	)	)	PUNCT
ijassa-1958	142	1	+	+	CCONJ
ijassa-1958	142	2	d(ux3	d(ux3	NOUN
ijassa-1958	142	3	,	,	PUNCT
ijassa-1958	142	4	x3	x3	ADJ
ijassa-1958	142	5	)	)	PUNCT
ijassa-1958	142	6	≤	≤	NOUN
ijassa-1958	142	7	d(ux1	d(ux1	PROPN
ijassa-1958	142	8	,	,	PUNCT
ijassa-1958	142	9	ux3	ux3	PROPN
ijassa-1958	142	10	)	)	PUNCT
ijassa-1958	142	11	+	+	NUM
ijassa-1958	142	12	2ε	2ε	NOUN
ijassa-1958	142	13	from	from	ADP
ijassa-1958	142	14	which	which	PRON
ijassa-1958	142	15	it	it	PRON
ijassa-1958	142	16	follows	follow	VERB
ijassa-1958	142	17	that	that	SCONJ
ijassa-1958	142	18	d(ux1	d(ux1	PROPN
ijassa-1958	142	19	,	,	PUNCT
ijassa-1958	142	20	ux3	ux3	PROPN
ijassa-1958	142	21	)	)	PUNCT
ijassa-1958	142	22	≥	≥	NOUN
ijassa-1958	142	23	d(x1	d(x1	NOUN
ijassa-1958	142	24	,	,	PUNCT
ijassa-1958	142	25	x3)−	x3)−	PROPN
ijassa-1958	142	26	2ε	2ε	PROPN
ijassa-1958	142	27	≥	≥	NUM
ijassa-1958	142	28	4r	4r	NUM
ijassa-1958	142	29	−	−	PROPN
ijassa-1958	142	30	2ε	2ε	NUM
ijassa-1958	142	31	.	.	PUNCT
ijassa-1958	143	1	it	it	PRON
ijassa-1958	143	2	is	be	AUX
ijassa-1958	143	3	easy	easy	ADJ
ijassa-1958	143	4	to	to	PART
ijassa-1958	143	5	see	see	VERB
ijassa-1958	143	6	that	that	SCONJ
ijassa-1958	143	7	if	if	SCONJ
ijassa-1958	143	8	ε	ε	PROPN
ijassa-1958	143	9	≤	≤	NOUN
ijassa-1958	143	10	r	r	NOUN
ijassa-1958	143	11	2	2	NUM
ijassa-1958	143	12	then	then	ADV
ijassa-1958	143	13	d(ux1	d(ux1	PROPN
ijassa-1958	143	14	,	,	PUNCT
ijassa-1958	143	15	ux2	ux2	PROPN
ijassa-1958	143	16	)	)	PUNCT
ijassa-1958	143	17	≤	≤	NOUN
ijassa-1958	143	18	2r	2r	NUM
ijassa-1958	143	19	+	+	CCONJ
ijassa-1958	143	20	2ε	2ε	ADJ
ijassa-1958	143	21	≤	≤	ADJ
ijassa-1958	143	22	4r	4r	NOUN
ijassa-1958	143	23	−	−	PROPN
ijassa-1958	143	24	2ε	2ε	PROPN
ijassa-1958	143	25	≤	≤	PROPN
ijassa-1958	143	26	d(ux1	d(ux1	PROPN
ijassa-1958	143	27	,	,	PUNCT
ijassa-1958	143	28	ux3	ux3	PROPN
ijassa-1958	143	29	)	)	PUNCT
ijassa-1958	143	30	.	.	PUNCT
ijassa-1958	144	1	similarly	similarly	ADV
ijassa-1958	144	2	,	,	PUNCT
ijassa-1958	144	3	for	for	ADP
ijassa-1958	144	4	the	the	DET
ijassa-1958	144	5	case	case	NOUN
ijassa-1958	144	6	where	where	SCONJ
ijassa-1958	144	7	d(ux2	d(ux2	PROPN
ijassa-1958	144	8	,	,	PUNCT
ijassa-1958	144	9	ux3	ux3	PROPN
ijassa-1958	144	10	)	)	PUNCT
ijassa-1958	144	11	≤	≤	NOUN
ijassa-1958	144	12	d(ux1	d(ux1	PROPN
ijassa-1958	144	13	,	,	PUNCT
ijassa-1958	144	14	ux3	ux3	PROPN
ijassa-1958	144	15	)	)	PUNCT
ijassa-1958	144	16	,	,	PUNCT
ijassa-1958	144	17	the	the	DET
ijassa-1958	144	18	metric	metric	NOUN
ijassa-1958	144	19	is	be	AUX
ijassa-1958	144	20	compatible	compatible	ADJ
ijassa-1958	144	21	with	with	ADP
ijassa-1958	144	22	the	the	DET
ijassa-1958	144	23	order	order	NOUN
ijassa-1958	144	24	.	.	PUNCT
ijassa-1958	145	1	as	as	SCONJ
ijassa-1958	145	2	we	we	PRON
ijassa-1958	145	3	can	can	AUX
ijassa-1958	145	4	see	see	VERB
ijassa-1958	145	5	,	,	PUNCT
ijassa-1958	145	6	for	for	ADP
ijassa-1958	145	7	a	a	DET
ijassa-1958	145	8	small	small	ADJ
ijassa-1958	145	9	enough	enough	ADJ
ijassa-1958	145	10	value	value	NOUN
ijassa-1958	145	11	of	of	ADP
ijassa-1958	145	12	ε	ε	PROPN
ijassa-1958	145	13	,	,	PUNCT
ijassa-1958	145	14	a	a	DET
ijassa-1958	145	15	discrete	discrete	ADJ
ijassa-1958	145	16	curved	curved	ADJ
ijassa-1958	145	17	line	line	NOUN
ijassa-1958	145	18	’s	’s	PART
ijassa-1958	145	19	metric	metric	NOUN
ijassa-1958	145	20	is	be	AUX
ijassa-1958	145	21	compatible	compatible	ADJ
ijassa-1958	145	22	with	with	ADP
ijassa-1958	145	23	its	its	PRON
ijassa-1958	145	24	order	order	NOUN
ijassa-1958	145	25	.	.	PUNCT
ijassa-1958	146	1	this	this	PRON
ijassa-1958	146	2	is	be	AUX
ijassa-1958	146	3	an	an	DET
ijassa-1958	146	4	example	example	NOUN
ijassa-1958	146	5	of	of	ADP
ijassa-1958	146	6	a	a	DET
ijassa-1958	146	7	space	space	NOUN
ijassa-1958	146	8	with	with	ADP
ijassa-1958	146	9	a	a	DET
ijassa-1958	146	10	non	non	ADJ
ijassa-1958	146	11	-	-	ADJ
ijassa-1958	146	12	trivial	trivial	ADJ
ijassa-1958	146	13	metric	metric	NOUN
ijassa-1958	146	14	that	that	PRON
ijassa-1958	146	15	is	be	AUX
ijassa-1958	146	16	compatible	compatible	ADJ
ijassa-1958	146	17	with	with	ADP
ijassa-1958	146	18	a	a	DET
ijassa-1958	146	19	certain	certain	ADJ
ijassa-1958	146	20	order	order	NOUN
ijassa-1958	146	21	.	.	PUNCT
ijassa-1958	147	1	on	on	ADP
ijassa-1958	147	2	the	the	DET
ijassa-1958	147	3	other	other	ADJ
ijassa-1958	147	4	hand	hand	NOUN
ijassa-1958	147	5	,	,	PUNCT
ijassa-1958	147	6	if	if	SCONJ
ijassa-1958	147	7	ε	ε	PROPN
ijassa-1958	147	8	is	be	AUX
ijassa-1958	147	9	increased	increase	VERB
ijassa-1958	147	10	relative	relative	ADJ
ijassa-1958	147	11	to	to	ADP
ijassa-1958	147	12	r	r	NOUN
ijassa-1958	147	13	,	,	PUNCT
ijassa-1958	147	14	the	the	DET
ijassa-1958	147	15	order	order	NOUN
ijassa-1958	147	16	compatibility	compatibility	NOUN
ijassa-1958	147	17	property	property	NOUN
ijassa-1958	147	18	may	may	AUX
ijassa-1958	147	19	be	be	AUX
ijassa-1958	147	20	violated	violate	VERB
ijassa-1958	147	21	.	.	PUNCT
ijassa-1958	148	1	example	example	NOUN
ijassa-1958	148	2	4.1	4.1	NUM
ijassa-1958	148	3	:	:	PUNCT
ijassa-1958	148	4	(	(	PUNCT
ijassa-1958	148	5	left	leave	VERB
ijassa-1958	148	6	shift	shift	NOUN
ijassa-1958	148	7	of	of	ADP
ijassa-1958	148	8	a	a	DET
ijassa-1958	148	9	point	point	NOUN
ijassa-1958	148	10	of	of	ADP
ijassa-1958	148	11	the	the	DET
ijassa-1958	148	12	stretched	stretch	VERB
ijassa-1958	148	13	natural	natural	ADJ
ijassa-1958	148	14	number	number	NOUN
ijassa-1958	148	15	line	line	NOUN
ijassa-1958	148	16	)	)	PUNCT
ijassa-1958	148	17	consider	consider	VERB
ijassa-1958	148	18	the	the	DET
ijassa-1958	148	19	set	set	NOUN
ijassa-1958	148	20	of	of	ADP
ijassa-1958	148	21	natural	natural	ADJ
ijassa-1958	148	22	numbers	number	NOUN
ijassa-1958	148	23	scaled	scale	VERB
ijassa-1958	148	24	by	by	ADP
ijassa-1958	148	25	2r	2r	NUM
ijassa-1958	148	26	as	as	ADP
ijassa-1958	148	27	a	a	DET
ijassa-1958	148	28	subset	subset	NOUN
ijassa-1958	148	29	of	of	ADP
ijassa-1958	148	30	r	r	NOUN
ijassa-1958	148	31	with	with	ADP
ijassa-1958	148	32	the	the	DET
ijassa-1958	148	33	standard	standard	ADJ
ijassa-1958	148	34	metric	metric	NOUN
ijassa-1958	148	35	.	.	PUNCT
ijassa-1958	149	1	let	let	VERB
ijassa-1958	149	2	x	x	PUNCT
ijassa-1958	149	3	=	=	SYM
ijassa-1958	149	4	2rn	2rn	NOUN
ijassa-1958	149	5	.	.	PUNCT
ijassa-1958	150	1	move	move	VERB
ijassa-1958	150	2	the	the	DET
ijassa-1958	150	3	point	point	NOUN
ijassa-1958	150	4	4r	4r	NOUN
ijassa-1958	150	5	to	to	ADP
ijassa-1958	150	6	the	the	DET
ijassa-1958	150	7	left	left	NOUN
ijassa-1958	150	8	by	by	ADP
ijassa-1958	150	9	an	an	DET
ijassa-1958	150	10	amount	amount	NOUN
ijassa-1958	150	11	ε	ε	PROPN
ijassa-1958	150	12	>	>	X
ijassa-1958	150	13	2r	2r	NUM
ijassa-1958	150	14	.	.	PUNCT
ijassa-1958	151	1	let	let	VERB
ijassa-1958	151	2	u	u	PRON
ijassa-1958	151	3	=	=	PUNCT
ijassa-1958	151	4	{	{	PUNCT
ijassa-1958	151	5	ux}x∈x	ux}x∈x	NOUN
ijassa-1958	151	6	,	,	PUNCT
ijassa-1958	151	7	where	where	SCONJ
ijassa-1958	151	8	ux	ux	ADV
ijassa-1958	151	9	=	=	SYM
ijassa-1958	151	10	{	{	PUNCT
ijassa-1958	151	11	4r	4r	NUM
ijassa-1958	151	12	−	−	NOUN
ijassa-1958	151	13	ε	ε	PROPN
ijassa-1958	151	14	,	,	PUNCT
ijassa-1958	151	15	if	if	SCONJ
ijassa-1958	151	16	x	x	ADP
ijassa-1958	151	17	=	=	NOUN
ijassa-1958	151	18	4r	4r	NOUN
ijassa-1958	151	19	;	;	PUNCT
ijassa-1958	151	20	x	x	X
ijassa-1958	151	21	,	,	PUNCT
ijassa-1958	151	22	otherwise	otherwise	ADV
ijassa-1958	151	23	.	.	PUNCT
ijassa-1958	152	1	thus	thus	ADV
ijassa-1958	152	2	,	,	PUNCT
ijassa-1958	152	3	we	we	PRON
ijassa-1958	152	4	obtain	obtain	VERB
ijassa-1958	152	5	a	a	DET
ijassa-1958	152	6	(	(	PUNCT
ijassa-1958	152	7	r	r	NOUN
ijassa-1958	152	8	,	,	PUNCT
ijassa-1958	152	9	ε)-discrete	ε)-discrete	ADJ
ijassa-1958	152	10	curved	curved	ADJ
ijassa-1958	152	11	line	line	NOUN
ijassa-1958	152	12	,	,	PUNCT
ijassa-1958	152	13	where	where	SCONJ
ijassa-1958	152	14	the	the	DET
ijassa-1958	152	15	order	order	NOUN
ijassa-1958	152	16	is	be	AUX
ijassa-1958	152	17	not	not	PART
ijassa-1958	152	18	compatible	compatible	ADJ
ijassa-1958	152	19	with	with	ADP
ijassa-1958	152	20	the	the	DET
ijassa-1958	152	21	metric	metric	NOUN
ijassa-1958	152	22	.	.	PUNCT
ijassa-1958	153	1	generally	generally	ADV
ijassa-1958	153	2	,	,	PUNCT
ijassa-1958	153	3	when	when	SCONJ
ijassa-1958	153	4	we	we	PRON
ijassa-1958	153	5	create	create	VERB
ijassa-1958	153	6	various	various	ADJ
ijassa-1958	153	7	examples	example	NOUN
ijassa-1958	153	8	of	of	ADP
ijassa-1958	153	9	discrete	discrete	ADJ
ijassa-1958	153	10	curved	curved	ADJ
ijassa-1958	153	11	lines	line	NOUN
ijassa-1958	153	12	,	,	PUNCT
ijassa-1958	153	13	we	we	PRON
ijassa-1958	153	14	observe	observe	VERB
ijassa-1958	153	15	that	that	SCONJ
ijassa-1958	153	16	if	if	SCONJ
ijassa-1958	153	17	we	we	PRON
ijassa-1958	153	18	increase	increase	VERB
ijassa-1958	153	19	the	the	DET
ijassa-1958	153	20	parameter	parameter	NOUN
ijassa-1958	153	21	ε	ε	PROPN
ijassa-1958	153	22	compared	compare	VERB
ijassa-1958	153	23	to	to	ADP
ijassa-1958	153	24	r	r	NOUN
ijassa-1958	153	25	,	,	PUNCT
ijassa-1958	153	26	there	there	PRON
ijassa-1958	153	27	will	will	AUX
ijassa-1958	153	28	be	be	AUX
ijassa-1958	153	29	more	more	ADJ
ijassa-1958	153	30	instances	instance	NOUN
ijassa-1958	153	31	where	where	SCONJ
ijassa-1958	153	32	the	the	DET
ijassa-1958	153	33	metric	metric	NOUN
ijassa-1958	153	34	is	be	AUX
ijassa-1958	153	35	incompatible	incompatible	ADJ
ijassa-1958	153	36	with	with	ADP
ijassa-1958	153	37	the	the	DET
ijassa-1958	153	38	order	order	NOUN
ijassa-1958	153	39	.	.	PUNCT
ijassa-1958	154	1	it	it	PRON
ijassa-1958	154	2	would	would	AUX
ijassa-1958	154	3	be	be	AUX
ijassa-1958	154	4	natural	natural	ADJ
ijassa-1958	154	5	to	to	PART
ijassa-1958	154	6	assume	assume	VERB
ijassa-1958	154	7	that	that	SCONJ
ijassa-1958	154	8	for	for	ADP
ijassa-1958	154	9	large	large	ADJ
ijassa-1958	154	10	ε	ε	PROPN
ijassa-1958	154	11	,	,	PUNCT
ijassa-1958	154	12	the	the	DET
ijassa-1958	154	13	metric	metric	NOUN
ijassa-1958	154	14	is	be	AUX
ijassa-1958	154	15	guaranteed	guarantee	VERB
ijassa-1958	154	16	to	to	PART
ijassa-1958	154	17	cease	cease	VERB
ijassa-1958	154	18	to	to	PART
ijassa-1958	154	19	be	be	AUX
ijassa-1958	154	20	compatible	compatible	ADJ
ijassa-1958	154	21	with	with	ADP
ijassa-1958	154	22	the	the	DET
ijassa-1958	154	23	order	order	NOUN
ijassa-1958	154	24	.	.	PUNCT
ijassa-1958	155	1	however	however	ADV
ijassa-1958	155	2	,	,	PUNCT
ijassa-1958	155	3	we	we	PRON
ijassa-1958	155	4	will	will	AUX
ijassa-1958	155	5	demonstrate	demonstrate	VERB
ijassa-1958	155	6	a	a	DET
ijassa-1958	155	7	counterexample	counterexample	NOUN
ijassa-1958	155	8	.	.	PUNCT
ijassa-1958	155	9	example	example	NOUN
ijassa-1958	155	10	4.2	4.2	NUM
ijassa-1958	155	11	:	:	PUNCT
ijassa-1958	155	12	(	(	PUNCT
ijassa-1958	155	13	the	the	DET
ijassa-1958	155	14	set	set	NOUN
ijassa-1958	155	15	of	of	ADP
ijassa-1958	155	16	natural	natural	ADJ
ijassa-1958	155	17	numbers	number	NOUN
ijassa-1958	155	18	scaled	scale	VERB
ijassa-1958	155	19	and	and	CCONJ
ijassa-1958	155	20	shifted	shift	VERB
ijassa-1958	155	21	upwards	upwards	ADV
ijassa-1958	155	22	.	.	PUNCT
ijassa-1958	155	23	)	)	PUNCT
ijassa-1958	156	1	let	let	VERB
ijassa-1958	156	2	u	u	PRON
ijassa-1958	156	3	=	=	PUNCT
ijassa-1958	156	4	2rne1	2rne1	PROPN
ijassa-1958	156	5	+	+	CCONJ
ijassa-1958	156	6	εe2	εe2	NOUN
ijassa-1958	156	7	=	=	SYM
ijassa-1958	156	8	{	{	PUNCT
ijassa-1958	156	9	(	(	PUNCT
ijassa-1958	156	10	2rn	2rn	NOUN
ijassa-1958	156	11	,	,	PUNCT
ijassa-1958	156	12	ε)}n∈n	ε)}n∈n	PROPN
ijassa-1958	156	13	⊂	⊂	PROPN
ijassa-1958	156	14	r2	r2	PROPN
ijassa-1958	156	15	.	.	PUNCT
ijassa-1958	157	1	it	it	PRON
ijassa-1958	157	2	is	be	AUX
ijassa-1958	157	3	not	not	PART
ijassa-1958	157	4	difficult	difficult	ADJ
ijassa-1958	157	5	to	to	PART
ijassa-1958	157	6	see	see	VERB
ijassa-1958	157	7	that	that	SCONJ
ijassa-1958	157	8	this	this	DET
ijassa-1958	157	9	set	set	NOUN
ijassa-1958	157	10	is	be	AUX
ijassa-1958	157	11	(	(	PUNCT
ijassa-1958	157	12	r	r	NOUN
ijassa-1958	157	13	,	,	PUNCT
ijassa-1958	157	14	ε)discrete	ε)discrete	NUM
ijassa-1958	157	15	curved	curved	ADJ
ijassa-1958	157	16	line	line	NOUN
ijassa-1958	157	17	.	.	PUNCT
ijassa-1958	158	1	however	however	ADV
ijassa-1958	158	2	,	,	PUNCT
ijassa-1958	158	3	the	the	DET
ijassa-1958	158	4	order	order	NOUN
ijassa-1958	158	5	is	be	AUX
ijassa-1958	158	6	compatible	compatible	ADJ
ijassa-1958	158	7	with	with	ADP
ijassa-1958	158	8	the	the	DET
ijassa-1958	158	9	metric	metric	NOUN
ijassa-1958	158	10	,	,	PUNCT
ijassa-1958	158	11	no	no	ADV
ijassa-1958	158	12	matter	matter	ADV
ijassa-1958	158	13	how	how	SCONJ
ijassa-1958	158	14	small	small	ADJ
ijassa-1958	158	15	or	or	CCONJ
ijassa-1958	158	16	large	large	ADJ
ijassa-1958	158	17	ε	ε	PROPN
ijassa-1958	158	18	we	we	PRON
ijassa-1958	158	19	choose	choose	VERB
ijassa-1958	158	20	.	.	PUNCT
ijassa-1958	159	1	this	this	PRON
ijassa-1958	159	2	is	be	AUX
ijassa-1958	159	3	because	because	SCONJ
ijassa-1958	159	4	all	all	DET
ijassa-1958	159	5	metric	metric	ADJ
ijassa-1958	159	6	properties	property	NOUN
ijassa-1958	159	7	and	and	CCONJ
ijassa-1958	159	8	order	order	NOUN
ijassa-1958	159	9	properties	property	NOUN
ijassa-1958	159	10	remain	remain	VERB
ijassa-1958	159	11	unchanged	unchanged	ADJ
ijassa-1958	159	12	under	under	ADP
ijassa-1958	159	13	this	this	DET
ijassa-1958	159	14	isometric	isometric	ADJ
ijassa-1958	159	15	shift	shift	NOUN
ijassa-1958	159	16	.	.	PUNCT
ijassa-1958	160	1	thus	thus	ADV
ijassa-1958	160	2	,	,	PUNCT
ijassa-1958	160	3	we	we	PRON
ijassa-1958	160	4	see	see	VERB
ijassa-1958	160	5	the	the	DET
ijassa-1958	160	6	following	follow	VERB
ijassa-1958	160	7	picture	picture	NOUN
ijassa-1958	160	8	for	for	ADP
ijassa-1958	160	9	the	the	DET
ijassa-1958	160	10	(	(	PUNCT
ijassa-1958	160	11	r	r	NOUN
ijassa-1958	160	12	,	,	PUNCT
ijassa-1958	160	13	ε)-discrete	ε)-discrete	ADJ
ijassa-1958	160	14	curved	curved	ADJ
ijassa-1958	160	15	line	line	NOUN
ijassa-1958	160	16	:	:	PUNCT
ijassa-1958	160	17	1	1	X
ijassa-1958	160	18	)	)	PUNCT
ijassa-1958	160	19	for	for	ADP
ijassa-1958	160	20	a	a	DET
ijassa-1958	160	21	small	small	ADJ
ijassa-1958	160	22	ε	ε	NOUN
ijassa-1958	160	23	relative	relative	ADJ
ijassa-1958	160	24	to	to	ADP
ijassa-1958	160	25	r	r	NOUN
ijassa-1958	160	26	(	(	PUNCT
ijassa-1958	160	27	it	it	PRON
ijassa-1958	160	28	is	be	AUX
ijassa-1958	160	29	enough	enough	ADJ
ijassa-1958	160	30	if	if	SCONJ
ijassa-1958	160	31	ε	ε	PROPN
ijassa-1958	160	32	≤	≤	NOUN
ijassa-1958	160	33	r	r	NOUN
ijassa-1958	160	34	2	2	NUM
ijassa-1958	160	35	according	accord	VERB
ijassa-1958	160	36	to	to	ADP
ijassa-1958	160	37	theorem	theorem	NOUN
ijassa-1958	160	38	4.1	4.1	NUM
ijassa-1958	160	39	)	)	PUNCT
ijassa-1958	160	40	,	,	PUNCT
ijassa-1958	160	41	the	the	DET
ijassa-1958	160	42	metric	metric	NOUN
ijassa-1958	160	43	is	be	AUX
ijassa-1958	160	44	compatible	compatible	ADJ
ijassa-1958	160	45	with	with	ADP
ijassa-1958	160	46	the	the	DET
ijassa-1958	160	47	order	order	NOUN
ijassa-1958	160	48	.	.	PUNCT
ijassa-1958	161	1	2	2	X
ijassa-1958	161	2	)	)	PUNCT
ijassa-1958	161	3	if	if	SCONJ
ijassa-1958	161	4	we	we	PRON
ijassa-1958	161	5	exceed	exceed	VERB
ijassa-1958	161	6	a	a	DET
ijassa-1958	161	7	certain	certain	ADJ
ijassa-1958	161	8	limit	limit	NOUN
ijassa-1958	161	9	(	(	PUNCT
ijassa-1958	161	10	following	follow	VERB
ijassa-1958	161	11	the	the	DET
ijassa-1958	161	12	example	example	NOUN
ijassa-1958	161	13	of	of	ADP
ijassa-1958	161	14	4.1	4.1	NUM
ijassa-1958	161	15	,	,	PUNCT
ijassa-1958	161	16	it	it	PRON
ijassa-1958	161	17	is	be	AUX
ijassa-1958	161	18	enough	enough	ADJ
ijassa-1958	161	19	to	to	PART
ijassa-1958	161	20	have	have	VERB
ijassa-1958	161	21	ε	ε	PROPN
ijassa-1958	161	22	>	>	X
ijassa-1958	161	23	2r	2r	NUM
ijassa-1958	161	24	)	)	PUNCT
ijassa-1958	161	25	,	,	PUNCT
ijassa-1958	161	26	then	then	ADV
ijassa-1958	161	27	we	we	PRON
ijassa-1958	161	28	observe	observe	VERB
ijassa-1958	161	29	discrete	discrete	ADJ
ijassa-1958	161	30	curved	curved	ADJ
ijassa-1958	161	31	lines	line	NOUN
ijassa-1958	161	32	where	where	SCONJ
ijassa-1958	161	33	the	the	DET
ijassa-1958	161	34	metric	metric	NOUN
ijassa-1958	161	35	is	be	AUX
ijassa-1958	161	36	not	not	PART
ijassa-1958	161	37	compatible	compatible	ADJ
ijassa-1958	161	38	with	with	ADP
ijassa-1958	161	39	the	the	DET
ijassa-1958	161	40	order	order	NOUN
ijassa-1958	161	41	.	.	PUNCT
ijassa-1958	162	1	however	however	ADV
ijassa-1958	162	2	,	,	PUNCT
ijassa-1958	162	3	for	for	ADP
ijassa-1958	162	4	arbitrarily	arbitrarily	ADV
ijassa-1958	162	5	large	large	ADJ
ijassa-1958	162	6	ε	ε	NOUN
ijassa-1958	162	7	there	there	PRON
ijassa-1958	162	8	are	be	VERB
ijassa-1958	162	9	also	also	ADV
ijassa-1958	162	10	cases	case	NOUN
ijassa-1958	162	11	where	where	SCONJ
ijassa-1958	162	12	the	the	DET
ijassa-1958	162	13	metric	metric	NOUN
ijassa-1958	162	14	is	be	AUX
ijassa-1958	162	15	compatible	compatible	ADJ
ijassa-1958	162	16	with	with	ADP
ijassa-1958	162	17	the	the	DET
ijassa-1958	162	18	order	order	NOUN
ijassa-1958	162	19	,	,	PUNCT
ijassa-1958	162	20	as	as	SCONJ
ijassa-1958	162	21	demonstrated	demonstrate	VERB
ijassa-1958	162	22	by	by	ADP
ijassa-1958	162	23	the	the	DET
ijassa-1958	162	24	example	example	NOUN
ijassa-1958	162	25	4.2	4.2	NUM
ijassa-1958	162	26	.	.	PUNCT
ijassa-1958	163	1	copyright	copyright	NOUN
ijassa-1958	163	2	©	©	PROPN
ijassa-1958	163	3	2024	2024	NUM
ijassa-1958	163	4	assa	assa	NOUN
ijassa-1958	163	5	.	.	PUNCT
ijassa-1958	164	1	adv	adv	PROPN
ijassa-1958	164	2	syst	syst	PROPN
ijassa-1958	164	3	sci	sci	PROPN
ijassa-1958	164	4	appl	appl	PROPN
ijassa-1958	164	5	(	(	PUNCT
ijassa-1958	164	6	2024	2024	NUM
ijassa-1958	164	7	)	)	PUNCT
ijassa-1958	164	8	8	8	NUM
ijassa-1958	164	9	p.	p.	NOUN
ijassa-1958	164	10	klimov	klimov	PROPN
ijassa-1958	164	11	,	,	PUNCT
ijassa-1958	164	12	r.	r.	PROPN
ijassa-1958	164	13	sengupta	sengupta	PROPN
ijassa-1958	164	14	this	this	PRON
ijassa-1958	164	15	leads	lead	VERB
ijassa-1958	164	16	us	we	PRON
ijassa-1958	164	17	to	to	ADP
ijassa-1958	164	18	the	the	DET
ijassa-1958	164	19	question	question	NOUN
ijassa-1958	164	20	:	:	PUNCT
ijassa-1958	164	21	what	what	PRON
ijassa-1958	164	22	is	be	AUX
ijassa-1958	164	23	the	the	DET
ijassa-1958	164	24	highest	high	ADJ
ijassa-1958	164	25	possible	possible	ADJ
ijassa-1958	164	26	value	value	NOUN
ijassa-1958	164	27	of	of	ADP
ijassa-1958	164	28	ε	ε	PROPN
ijassa-1958	164	29	that	that	PRON
ijassa-1958	164	30	ensures	ensure	VERB
ijassa-1958	164	31	that	that	SCONJ
ijassa-1958	164	32	the	the	DET
ijassa-1958	164	33	order	order	NOUN
ijassa-1958	164	34	is	be	AUX
ijassa-1958	164	35	compatible	compatible	ADJ
ijassa-1958	164	36	with	with	ADP
ijassa-1958	164	37	the	the	DET
ijassa-1958	164	38	metric	metric	NOUN
ijassa-1958	164	39	?	?	PUNCT
ijassa-1958	165	1	problem	problem	NOUN
ijassa-1958	165	2	4.1	4.1	NUM
ijassa-1958	165	3	:	:	PUNCT
ijassa-1958	165	4	what	what	PRON
ijassa-1958	165	5	is	be	AUX
ijassa-1958	165	6	the	the	DET
ijassa-1958	165	7	minimum	minimum	ADJ
ijassa-1958	165	8	m(r	m(r	NOUN
ijassa-1958	165	9	)	)	PUNCT
ijassa-1958	165	10	such	such	ADJ
ijassa-1958	165	11	that	that	SCONJ
ijassa-1958	165	12	any	any	DET
ijassa-1958	165	13	(	(	PUNCT
ijassa-1958	165	14	r	r	NOUN
ijassa-1958	165	15	,	,	PUNCT
ijassa-1958	165	16	ε)-discrete	ε)-discrete	ADJ
ijassa-1958	165	17	curved	curved	ADJ
ijassa-1958	165	18	line	line	NOUN
ijassa-1958	165	19	with	with	ADP
ijassa-1958	165	20	ε	ε	PROPN
ijassa-1958	165	21	≤	≤	PROPN
ijassa-1958	165	22	m(r	m(r	PROPN
ijassa-1958	165	23	)	)	PUNCT
ijassa-1958	165	24	is	be	AUX
ijassa-1958	165	25	guaranteed	guarantee	VERB
ijassa-1958	165	26	to	to	PART
ijassa-1958	165	27	have	have	VERB
ijassa-1958	165	28	a	a	DET
ijassa-1958	165	29	compatible	compatible	ADJ
ijassa-1958	165	30	metric	metric	NOUN
ijassa-1958	165	31	with	with	ADP
ijassa-1958	165	32	the	the	DET
ijassa-1958	165	33	order	order	NOUN
ijassa-1958	165	34	?	?	PUNCT
ijassa-1958	166	1	according	accord	VERB
ijassa-1958	166	2	to	to	ADP
ijassa-1958	166	3	theorem	theorem	NOUN
ijassa-1958	166	4	4.1	4.1	NUM
ijassa-1958	166	5	and	and	CCONJ
ijassa-1958	166	6	the	the	DET
ijassa-1958	166	7	4.1	4.1	NUM
ijassa-1958	166	8	example	example	NOUN
ijassa-1958	166	9	,	,	PUNCT
ijassa-1958	166	10	it	it	PRON
ijassa-1958	166	11	is	be	AUX
ijassa-1958	166	12	true	true	ADJ
ijassa-1958	166	13	that	that	SCONJ
ijassa-1958	166	14	m(r	m(r	NOUN
ijassa-1958	166	15	)	)	PUNCT
ijassa-1958	166	16	is	be	AUX
ijassa-1958	166	17	between	between	ADP
ijassa-1958	166	18	r	r	NOUN
ijassa-1958	166	19	2	2	NUM
ijassa-1958	166	20	and	and	CCONJ
ijassa-1958	166	21	2r	2r	NUM
ijassa-1958	166	22	.	.	PUNCT
ijassa-1958	167	1	of	of	ADP
ijassa-1958	167	2	course	course	NOUN
ijassa-1958	167	3	,	,	PUNCT
ijassa-1958	167	4	the	the	DET
ijassa-1958	167	5	exact	exact	ADJ
ijassa-1958	167	6	conditions	condition	NOUN
ijassa-1958	167	7	on	on	ADP
ijassa-1958	167	8	ε	ε	PROPN
ijassa-1958	167	9	in	in	ADP
ijassa-1958	167	10	relation	relation	NOUN
ijassa-1958	167	11	to	to	ADP
ijassa-1958	167	12	r	r	NOUN
ijassa-1958	167	13	for	for	ADP
ijassa-1958	167	14	making	make	VERB
ijassa-1958	167	15	the	the	DET
ijassa-1958	167	16	metrics	metric	NOUN
ijassa-1958	167	17	compatible	compatible	ADJ
ijassa-1958	167	18	with	with	ADP
ijassa-1958	167	19	the	the	DET
ijassa-1958	167	20	order	order	NOUN
ijassa-1958	167	21	may	may	AUX
ijassa-1958	167	22	change	change	VERB
ijassa-1958	167	23	depending	depend	VERB
ijassa-1958	167	24	on	on	ADP
ijassa-1958	167	25	the	the	DET
ijassa-1958	167	26	specific	specific	ADJ
ijassa-1958	167	27	metric	metric	ADJ
ijassa-1958	167	28	spaces	space	NOUN
ijassa-1958	167	29	we	we	PRON
ijassa-1958	167	30	consider	consider	VERB
ijassa-1958	167	31	,	,	PUNCT
ijassa-1958	167	32	their	their	PRON
ijassa-1958	167	33	classes	class	NOUN
ijassa-1958	167	34	,	,	PUNCT
ijassa-1958	167	35	or	or	CCONJ
ijassa-1958	167	36	the	the	DET
ijassa-1958	167	37	methods	method	NOUN
ijassa-1958	167	38	we	we	PRON
ijassa-1958	167	39	use	use	VERB
ijassa-1958	167	40	to	to	PART
ijassa-1958	167	41	implement	implement	VERB
ijassa-1958	167	42	the	the	DET
ijassa-1958	167	43	isometric	isometric	ADJ
ijassa-1958	167	44	embedding	embedding	NOUN
ijassa-1958	167	45	and	and	CCONJ
ijassa-1958	167	46	select	select	VERB
ijassa-1958	167	47	new	new	ADJ
ijassa-1958	167	48	points	point	NOUN
ijassa-1958	167	49	.	.	PUNCT
ijassa-1958	168	1	one	one	PRON
ijassa-1958	168	2	might	might	AUX
ijassa-1958	168	3	think	think	VERB
ijassa-1958	168	4	that	that	SCONJ
ijassa-1958	168	5	all	all	DET
ijassa-1958	168	6	types	type	NOUN
ijassa-1958	168	7	of	of	ADP
ijassa-1958	168	8	order	order	NOUN
ijassa-1958	168	9	-	-	PUNCT
ijassa-1958	168	10	compatible	compatible	ADJ
ijassa-1958	168	11	discrete	discrete	ADJ
ijassa-1958	168	12	curved	curved	ADJ
ijassa-1958	168	13	lines	line	NOUN
ijassa-1958	168	14	,	,	PUNCT
ijassa-1958	168	15	in	in	ADP
ijassa-1958	168	16	terms	term	NOUN
ijassa-1958	168	17	of	of	ADP
ijassa-1958	168	18	their	their	PRON
ijassa-1958	168	19	metric	metric	ADJ
ijassa-1958	168	20	and	and	CCONJ
ijassa-1958	168	21	general	general	ADJ
ijassa-1958	168	22	properties	property	NOUN
ijassa-1958	168	23	,	,	PUNCT
ijassa-1958	168	24	are	be	AUX
ijassa-1958	168	25	exactly	exactly	ADV
ijassa-1958	168	26	the	the	DET
ijassa-1958	168	27	same	same	ADJ
ijassa-1958	168	28	as	as	ADP
ijassa-1958	168	29	an	an	DET
ijassa-1958	168	30	ordinary	ordinary	ADJ
ijassa-1958	168	31	lines	line	NOUN
ijassa-1958	168	32	.	.	PUNCT
ijassa-1958	169	1	this	this	PRON
ijassa-1958	169	2	is	be	AUX
ijassa-1958	169	3	because	because	SCONJ
ijassa-1958	169	4	they	they	PRON
ijassa-1958	169	5	are	be	AUX
ijassa-1958	169	6	created	create	VERB
ijassa-1958	169	7	by	by	ADP
ijassa-1958	169	8	simply	simply	ADV
ijassa-1958	169	9	shifting	shift	VERB
ijassa-1958	169	10	certain	certain	ADJ
ijassa-1958	169	11	points	point	NOUN
ijassa-1958	169	12	on	on	ADP
ijassa-1958	169	13	a	a	DET
ijassa-1958	169	14	line	line	NOUN
ijassa-1958	169	15	.	.	PUNCT
ijassa-1958	170	1	at	at	ADP
ijassa-1958	170	2	the	the	DET
ijassa-1958	170	3	same	same	ADJ
ijassa-1958	170	4	time	time	NOUN
ijassa-1958	170	5	,	,	PUNCT
ijassa-1958	170	6	we	we	PRON
ijassa-1958	170	7	can	can	AUX
ijassa-1958	170	8	select	select	VERB
ijassa-1958	170	9	discrete	discrete	ADJ
ijassa-1958	170	10	curved	curved	ADJ
ijassa-1958	170	11	lines	line	NOUN
ijassa-1958	170	12	that	that	PRON
ijassa-1958	170	13	can	can	AUX
ijassa-1958	170	14	significantly	significantly	ADV
ijassa-1958	170	15	differ	differ	VERB
ijassa-1958	170	16	from	from	ADP
ijassa-1958	170	17	a	a	DET
ijassa-1958	170	18	straight	straight	ADJ
ijassa-1958	170	19	line	line	NOUN
ijassa-1958	170	20	in	in	ADP
ijassa-1958	170	21	terms	term	NOUN
ijassa-1958	170	22	of	of	ADP
ijassa-1958	170	23	certain	certain	ADJ
ijassa-1958	170	24	properties	property	NOUN
ijassa-1958	170	25	.	.	PUNCT
ijassa-1958	171	1	this	this	PRON
ijassa-1958	171	2	allows	allow	VERB
ijassa-1958	171	3	us	we	PRON
ijassa-1958	171	4	to	to	PART
ijassa-1958	171	5	explore	explore	VERB
ijassa-1958	171	6	non	non	ADJ
ijassa-1958	171	7	-	-	ADJ
ijassa-1958	171	8	trivial	trivial	ADJ
ijassa-1958	171	9	examples	example	NOUN
ijassa-1958	171	10	of	of	ADP
ijassa-1958	171	11	spaces	space	NOUN
ijassa-1958	171	12	with	with	ADP
ijassa-1958	171	13	a	a	DET
ijassa-1958	171	14	metric	metric	NOUN
ijassa-1958	171	15	that	that	PRON
ijassa-1958	171	16	is	be	AUX
ijassa-1958	171	17	compatible	compatible	ADJ
ijassa-1958	171	18	with	with	ADP
ijassa-1958	171	19	an	an	DET
ijassa-1958	171	20	order	order	NOUN
ijassa-1958	171	21	.	.	PUNCT
ijassa-1958	172	1	lemma	lemma	PROPN
ijassa-1958	172	2	4.1	4.1	NUM
ijassa-1958	172	3	:	:	PUNCT
ijassa-1958	172	4	for	for	ADP
ijassa-1958	172	5	any	any	DET
ijassa-1958	172	6	infinite	infinite	ADJ
ijassa-1958	172	7	-	-	PUNCT
ijassa-1958	172	8	dimensional	dimensional	ADJ
ijassa-1958	172	9	normed	normed	ADJ
ijassa-1958	172	10	space	space	NOUN
ijassa-1958	172	11	v	v	NOUN
ijassa-1958	172	12	over	over	ADP
ijassa-1958	172	13	r	r	NOUN
ijassa-1958	172	14	,	,	PUNCT
ijassa-1958	172	15	there	there	PRON
ijassa-1958	172	16	exists	exist	VERB
ijassa-1958	172	17	a	a	DET
ijassa-1958	172	18	countable	countable	ADJ
ijassa-1958	172	19	subset	subset	NOUN
ijassa-1958	172	20	of	of	ADP
ijassa-1958	172	21	u	u	NOUN
ijassa-1958	172	22	compatible	compatible	ADJ
ijassa-1958	172	23	with	with	ADP
ijassa-1958	172	24	some	some	DET
ijassa-1958	172	25	order	order	NOUN
ijassa-1958	172	26	that	that	PRON
ijassa-1958	172	27	does	do	AUX
ijassa-1958	172	28	not	not	PART
ijassa-1958	172	29	does	do	AUX
ijassa-1958	172	30	not	not	PART
ijassa-1958	172	31	belong	belong	VERB
ijassa-1958	172	32	to	to	ADP
ijassa-1958	172	33	any	any	DET
ijassa-1958	172	34	finite	finite	ADJ
ijassa-1958	172	35	-	-	ADJ
ijassa-1958	172	36	dimensional	dimensional	ADJ
ijassa-1958	172	37	subspace	subspace	NOUN
ijassa-1958	172	38	,	,	PUNCT
ijassa-1958	172	39	meaning	mean	VERB
ijassa-1958	172	40	that	that	SCONJ
ijassa-1958	172	41	dim	dim	ADJ
ijassa-1958	172	42	span(u	span(u	NOUN
ijassa-1958	172	43	)	)	PUNCT
ijassa-1958	173	1	=	=	PUNCT
ijassa-1958	173	2	∞.	∞.	PROPN
ijassa-1958	173	3	proof	proof	NOUN
ijassa-1958	173	4	let	let	VERB
ijassa-1958	173	5	us	we	PRON
ijassa-1958	173	6	consider	consider	VERB
ijassa-1958	173	7	a	a	DET
ijassa-1958	173	8	specific	specific	ADJ
ijassa-1958	173	9	subset	subset	NOUN
ijassa-1958	173	10	of	of	ADP
ijassa-1958	173	11	the	the	DET
ijassa-1958	173	12	basis	basis	NOUN
ijassa-1958	173	13	vectors	vector	NOUN
ijassa-1958	173	14	:	:	PUNCT
ijassa-1958	173	15	{	{	PUNCT
ijassa-1958	173	16	e1	e1	NOUN
ijassa-1958	173	17	,	,	PUNCT
ijassa-1958	173	18	.	.	PUNCT
ijassa-1958	173	19	.	.	PUNCT
ijassa-1958	173	20	.	.	PUNCT
ijassa-1958	174	1	}	}	PUNCT
ijassa-1958	174	2	.	.	PUNCT
ijassa-1958	175	1	consider	consider	VERB
ijassa-1958	175	2	an	an	DET
ijassa-1958	175	3	(	(	PUNCT
ijassa-1958	175	4	r	r	NOUN
ijassa-1958	175	5	,	,	PUNCT
ijassa-1958	175	6	r	r	NOUN
ijassa-1958	175	7	2	2	NUM
ijassa-1958	175	8	)	)	PUNCT
ijassa-1958	175	9	-discrete	-discrete	NOUN
ijassa-1958	175	10	curved	curved	ADJ
ijassa-1958	175	11	line	line	NOUN
ijassa-1958	175	12	u	u	NOUN
ijassa-1958	175	13	in	in	ADP
ijassa-1958	175	14	the	the	DET
ijassa-1958	175	15	vector	vector	NOUN
ijassa-1958	175	16	space	space	NOUN
ijassa-1958	175	17	v	v	NOUN
ijassa-1958	175	18	.	.	PUNCT
ijassa-1958	176	1	let	let	VERB
ijassa-1958	176	2	u	u	PRON
ijassa-1958	176	3	be	be	AUX
ijassa-1958	176	4	a	a	DET
ijassa-1958	176	5	set	set	NOUN
ijassa-1958	176	6	of	of	ADP
ijassa-1958	176	7	points	point	NOUN
ijassa-1958	176	8	that	that	PRON
ijassa-1958	176	9	can	can	AUX
ijassa-1958	176	10	be	be	AUX
ijassa-1958	176	11	represented	represent	VERB
ijassa-1958	176	12	as	as	ADP
ijassa-1958	176	13	un	un	PROPN
ijassa-1958	176	14	=	=	PROPN
ijassa-1958	177	1	2nre1	2nre1	PROPN
ijassa-1958	178	1	+	+	CCONJ
ijassa-1958	178	2	r	r	NOUN
ijassa-1958	178	3	2	2	NUM
ijassa-1958	178	4	en	en	ADP
ijassa-1958	178	5	,	,	PUNCT
ijassa-1958	178	6	where	where	SCONJ
ijassa-1958	178	7	en	en	X
ijassa-1958	178	8	is	be	AUX
ijassa-1958	178	9	the	the	DET
ijassa-1958	178	10	n	n	CCONJ
ijassa-1958	178	11	-	-	PUNCT
ijassa-1958	178	12	th	th	NOUN
ijassa-1958	178	13	basis	basis	NOUN
ijassa-1958	178	14	vector	vector	NOUN
ijassa-1958	178	15	.	.	PUNCT
ijassa-1958	179	1	it	it	PRON
ijassa-1958	179	2	is	be	AUX
ijassa-1958	179	3	easy	easy	ADJ
ijassa-1958	179	4	to	to	PART
ijassa-1958	179	5	see	see	VERB
ijassa-1958	179	6	that	that	SCONJ
ijassa-1958	179	7	u	u	NOUN
ijassa-1958	179	8	is	be	AUX
ijassa-1958	179	9	an	an	DET
ijassa-1958	179	10	(	(	PUNCT
ijassa-1958	179	11	r	r	NOUN
ijassa-1958	179	12	,	,	PUNCT
ijassa-1958	179	13	r	r	NOUN
ijassa-1958	179	14	2	2	NUM
ijassa-1958	179	15	)	)	PUNCT
ijassa-1958	179	16	-discrete	-discrete	ADJ
ijassa-1958	179	17	curved	curved	ADJ
ijassa-1958	179	18	line	line	NOUN
ijassa-1958	179	19	with	with	ADP
ijassa-1958	179	20	a	a	DET
ijassa-1958	179	21	specific	specific	ADJ
ijassa-1958	179	22	structure	structure	NOUN
ijassa-1958	179	23	.	.	PUNCT
ijassa-1958	180	1	it	it	PRON
ijassa-1958	180	2	is	be	AUX
ijassa-1958	180	3	obtained	obtain	VERB
ijassa-1958	180	4	by	by	ADP
ijassa-1958	180	5	shifting	shift	VERB
ijassa-1958	180	6	up	up	ADP
ijassa-1958	180	7	along	along	ADP
ijassa-1958	180	8	the	the	DET
ijassa-1958	180	9	nth	nth	NOUN
ijassa-1958	180	10	coordinate	coordinate	NOUN
ijassa-1958	180	11	of	of	ADP
ijassa-1958	180	12	the	the	DET
ijassa-1958	180	13	embedding	embedding	NOUN
ijassa-1958	180	14	of	of	ADP
ijassa-1958	180	15	natural	natural	ADJ
ijassa-1958	180	16	numbers	number	NOUN
ijassa-1958	180	17	,	,	PUNCT
ijassa-1958	180	18	which	which	PRON
ijassa-1958	180	19	are	be	AUX
ijassa-1958	180	20	scaled	scale	VERB
ijassa-1958	180	21	by	by	ADP
ijassa-1958	180	22	2r	2r	NUM
ijassa-1958	180	23	.	.	PUNCT
ijassa-1958	181	1	this	this	DET
ijassa-1958	181	2	scaled	scale	VERB
ijassa-1958	181	3	set	set	NOUN
ijassa-1958	181	4	is	be	AUX
ijassa-1958	181	5	then	then	ADV
ijassa-1958	181	6	embedded	embed	VERB
ijassa-1958	181	7	on	on	ADP
ijassa-1958	181	8	the	the	DET
ijassa-1958	181	9	first	first	ADJ
ijassa-1958	181	10	coordinate	coordinate	NOUN
ijassa-1958	181	11	axis	axis	NOUN
ijassa-1958	181	12	.	.	PUNCT
ijassa-1958	182	1	by	by	ADP
ijassa-1958	182	2	the	the	DET
ijassa-1958	182	3	theorem	theorem	NOUN
ijassa-1958	182	4	4.1	4.1	NUM
ijassa-1958	182	5	,	,	PUNCT
ijassa-1958	182	6	we	we	PRON
ijassa-1958	182	7	know	know	VERB
ijassa-1958	182	8	that	that	SCONJ
ijassa-1958	182	9	the	the	DET
ijassa-1958	182	10	metric	metric	NOUN
ijassa-1958	182	11	in	in	ADP
ijassa-1958	182	12	this	this	DET
ijassa-1958	182	13	curved	curved	ADJ
ijassa-1958	182	14	line	line	NOUN
ijassa-1958	182	15	is	be	AUX
ijassa-1958	182	16	compatible	compatible	ADJ
ijassa-1958	182	17	with	with	ADP
ijassa-1958	182	18	the	the	DET
ijassa-1958	182	19	order	order	NOUN
ijassa-1958	182	20	and	and	CCONJ
ijassa-1958	182	21	vectors	vector	NOUN
ijassa-1958	182	22	in	in	ADP
ijassa-1958	182	23	u	u	NOUN
ijassa-1958	182	24	are	be	AUX
ijassa-1958	182	25	linearly	linearly	ADV
ijassa-1958	182	26	independent	independent	ADJ
ijassa-1958	182	27	,	,	PUNCT
ijassa-1958	182	28	since	since	SCONJ
ijassa-1958	182	29	for	for	ADP
ijassa-1958	182	30	each	each	DET
ijassa-1958	182	31	unique	unique	ADJ
ijassa-1958	182	32	ui	ui	NOUN
ijassa-1958	182	33	in	in	ADP
ijassa-1958	182	34	u	u	NOUN
ijassa-1958	182	35	,	,	PUNCT
ijassa-1958	182	36	there	there	PRON
ijassa-1958	182	37	is	be	VERB
ijassa-1958	182	38	a	a	DET
ijassa-1958	182	39	corresponding	corresponding	ADJ
ijassa-1958	182	40	unique	unique	ADJ
ijassa-1958	182	41	ei	ei	NOUN
ijassa-1958	182	42	in	in	ADP
ijassa-1958	182	43	the	the	DET
ijassa-1958	182	44	basis	basis	NOUN
ijassa-1958	182	45	expansion	expansion	NOUN
ijassa-1958	182	46	.	.	PUNCT
ijassa-1958	183	1	remark	remark	VERB
ijassa-1958	183	2	4.1	4.1	NUM
ijassa-1958	183	3	:	:	PUNCT
ijassa-1958	183	4	theorem	theorem	VERB
ijassa-1958	183	5	4.3	4.3	NUM
ijassa-1958	183	6	shows	show	VERB
ijassa-1958	183	7	that	that	SCONJ
ijassa-1958	183	8	there	there	PRON
ijassa-1958	183	9	are	be	VERB
ijassa-1958	183	10	nontrivial	nontrivial	ADJ
ijassa-1958	183	11	examples	example	NOUN
ijassa-1958	183	12	of	of	ADP
ijassa-1958	183	13	metric	metric	ADJ
ijassa-1958	183	14	spaces	space	NOUN
ijassa-1958	183	15	that	that	PRON
ijassa-1958	183	16	are	be	AUX
ijassa-1958	183	17	not	not	PART
ijassa-1958	183	18	isometric	isometric	ADJ
ijassa-1958	183	19	to	to	ADP
ijassa-1958	183	20	any	any	DET
ijassa-1958	183	21	subset	subset	NOUN
ijassa-1958	183	22	of	of	ADP
ijassa-1958	183	23	the	the	DET
ijassa-1958	183	24	real	real	ADJ
ijassa-1958	183	25	line	line	NOUN
ijassa-1958	183	26	,	,	PUNCT
ijassa-1958	183	27	but	but	CCONJ
ijassa-1958	183	28	in	in	ADP
ijassa-1958	183	29	which	which	PRON
ijassa-1958	183	30	the	the	DET
ijassa-1958	183	31	order	order	NOUN
ijassa-1958	183	32	is	be	AUX
ijassa-1958	183	33	compatible	compatible	ADJ
ijassa-1958	183	34	with	with	ADP
ijassa-1958	183	35	the	the	DET
ijassa-1958	183	36	metric	metric	NOUN
ijassa-1958	183	37	.	.	PUNCT
ijassa-1958	184	1	theorem	theorem	VERB
ijassa-1958	184	2	4.1	4.1	NUM
ijassa-1958	184	3	gives	give	VERB
ijassa-1958	184	4	sufficient	sufficient	ADJ
ijassa-1958	184	5	conditions	condition	NOUN
ijassa-1958	184	6	for	for	SCONJ
ijassa-1958	184	7	the	the	DET
ijassa-1958	184	8	discrete	discrete	ADJ
ijassa-1958	184	9	curved	curved	ADJ
ijassa-1958	184	10	line	line	NOUN
ijassa-1958	184	11	to	to	PART
ijassa-1958	184	12	be	be	AUX
ijassa-1958	184	13	compatible	compatible	ADJ
ijassa-1958	184	14	with	with	ADP
ijassa-1958	184	15	the	the	DET
ijassa-1958	184	16	order	order	NOUN
ijassa-1958	184	17	.	.	PUNCT
ijassa-1958	185	1	at	at	ADP
ijassa-1958	185	2	the	the	DET
ijassa-1958	185	3	same	same	ADJ
ijassa-1958	185	4	time	time	NOUN
ijassa-1958	185	5	,	,	PUNCT
ijassa-1958	185	6	it	it	PRON
ijassa-1958	185	7	is	be	AUX
ijassa-1958	185	8	possible	possible	ADJ
ijassa-1958	185	9	to	to	PART
ijassa-1958	185	10	find	find	VERB
ijassa-1958	185	11	examples	example	NOUN
ijassa-1958	185	12	of	of	ADP
ijassa-1958	185	13	spaces	space	NOUN
ijassa-1958	185	14	where	where	SCONJ
ijassa-1958	185	15	the	the	DET
ijassa-1958	185	16	metric	metric	NOUN
ijassa-1958	185	17	is	be	AUX
ijassa-1958	185	18	compatible	compatible	ADJ
ijassa-1958	185	19	with	with	ADP
ijassa-1958	185	20	the	the	DET
ijassa-1958	185	21	order	order	NOUN
ijassa-1958	185	22	through	through	ADP
ijassa-1958	185	23	specific	specific	ADJ
ijassa-1958	185	24	directed	direct	VERB
ijassa-1958	185	25	deformations	deformation	NOUN
ijassa-1958	185	26	of	of	ADP
ijassa-1958	185	27	individual	individual	ADJ
ijassa-1958	185	28	points	point	NOUN
ijassa-1958	185	29	.	.	PUNCT
ijassa-1958	186	1	sometimes	sometimes	ADV
ijassa-1958	186	2	we	we	PRON
ijassa-1958	186	3	can	can	AUX
ijassa-1958	186	4	use	use	VERB
ijassa-1958	186	5	general	general	ADJ
ijassa-1958	186	6	conditions	condition	NOUN
ijassa-1958	186	7	for	for	ADP
ijassa-1958	186	8	the	the	DET
ijassa-1958	186	9	entire	entire	ADJ
ijassa-1958	186	10	line	line	NOUN
ijassa-1958	186	11	,	,	PUNCT
ijassa-1958	186	12	but	but	CCONJ
ijassa-1958	186	13	in	in	ADP
ijassa-1958	186	14	general	general	ADJ
ijassa-1958	186	15	it	it	PRON
ijassa-1958	186	16	is	be	AUX
ijassa-1958	186	17	preferred	prefer	VERB
ijassa-1958	186	18	to	to	PART
ijassa-1958	186	19	have	have	VERB
ijassa-1958	186	20	a	a	DET
ijassa-1958	186	21	more	more	ADV
ijassa-1958	186	22	precise	precise	ADJ
ijassa-1958	186	23	method	method	NOUN
ijassa-1958	186	24	for	for	ADP
ijassa-1958	186	25	evaluating	evaluate	VERB
ijassa-1958	186	26	the	the	DET
ijassa-1958	186	27	allowed	allow	VERB
ijassa-1958	186	28	changes	change	NOUN
ijassa-1958	186	29	at	at	ADP
ijassa-1958	186	30	a	a	DET
ijassa-1958	186	31	specific	specific	ADJ
ijassa-1958	186	32	point	point	NOUN
ijassa-1958	186	33	.	.	PUNCT
ijassa-1958	187	1	theorem	theorem	VERB
ijassa-1958	187	2	4.2	4.2	NUM
ijassa-1958	187	3	:	:	PUNCT
ijassa-1958	187	4	(	(	PUNCT
ijassa-1958	187	5	pointwise	pointwise	VERB
ijassa-1958	187	6	sufficient	sufficient	ADJ
ijassa-1958	187	7	condition	condition	NOUN
ijassa-1958	187	8	for	for	ADP
ijassa-1958	187	9	compatibility	compatibility	NOUN
ijassa-1958	187	10	of	of	ADP
ijassa-1958	187	11	the	the	DET
ijassa-1958	187	12	metric	metric	NOUN
ijassa-1958	187	13	with	with	ADP
ijassa-1958	187	14	an	an	DET
ijassa-1958	187	15	order	order	NOUN
ijassa-1958	187	16	for	for	ADP
ijassa-1958	187	17	a	a	DET
ijassa-1958	187	18	discrete	discrete	ADJ
ijassa-1958	187	19	curved	curved	ADJ
ijassa-1958	187	20	line	line	NOUN
ijassa-1958	187	21	)	)	PUNCT
ijassa-1958	187	22	let	let	VERB
ijassa-1958	187	23	us	we	PRON
ijassa-1958	187	24	consider	consider	VERB
ijassa-1958	187	25	a	a	DET
ijassa-1958	187	26	discrete	discrete	ADJ
ijassa-1958	187	27	curved	curved	ADJ
ijassa-1958	187	28	line	line	NOUN
ijassa-1958	187	29	u	u	NOUN
ijassa-1958	187	30	that	that	PRON
ijassa-1958	187	31	is	be	AUX
ijassa-1958	187	32	obtained	obtain	VERB
ijassa-1958	187	33	from	from	ADP
ijassa-1958	187	34	a	a	DET
ijassa-1958	187	35	set	set	NOUN
ijassa-1958	188	1	x	x	X
ijassa-1958	188	2	.	.	PUNCT
ijassa-1958	189	1	we	we	PRON
ijassa-1958	189	2	can	can	AUX
ijassa-1958	189	3	assign	assign	VERB
ijassa-1958	189	4	numbers	number	NOUN
ijassa-1958	189	5	to	to	ADP
ijassa-1958	189	6	the	the	DET
ijassa-1958	189	7	elements	element	NOUN
ijassa-1958	189	8	of	of	ADP
ijassa-1958	189	9	x	x	PUNCT
ijassa-1958	189	10	according	accord	VERB
ijassa-1958	189	11	to	to	ADP
ijassa-1958	189	12	their	their	PRON
ijassa-1958	189	13	order	order	NOUN
ijassa-1958	189	14	on	on	ADP
ijassa-1958	189	15	the	the	DET
ijassa-1958	189	16	real	real	ADJ
ijassa-1958	189	17	number	number	NOUN
ijassa-1958	189	18	line	line	NOUN
ijassa-1958	189	19	using	use	VERB
ijassa-1958	189	20	integers	integer	NOUN
ijassa-1958	189	21	.	.	PUNCT
ijassa-1958	190	1	if	if	SCONJ
ijassa-1958	190	2	there	there	PRON
ijassa-1958	190	3	is	be	VERB
ijassa-1958	190	4	an	an	DET
ijassa-1958	190	5	upper	upper	ADJ
ijassa-1958	190	6	or	or	CCONJ
ijassa-1958	190	7	lower	lower	ADV
ijassa-1958	190	8	bound	bind	VERB
ijassa-1958	190	9	,	,	PUNCT
ijassa-1958	190	10	we	we	PRON
ijassa-1958	190	11	can	can	AUX
ijassa-1958	190	12	use	use	VERB
ijassa-1958	190	13	positive	positive	ADJ
ijassa-1958	190	14	or	or	CCONJ
ijassa-1958	190	15	negative	negative	ADJ
ijassa-1958	190	16	integers	integer	NOUN
ijassa-1958	190	17	accordingly	accordingly	ADV
ijassa-1958	190	18	.	.	PUNCT
ijassa-1958	191	1	let	let	VERB
ijassa-1958	191	2	d(uxi	d(uxi	PROPN
ijassa-1958	191	3	,	,	PUNCT
ijassa-1958	191	4	xi	xi	PROPN
ijassa-1958	191	5	)	)	PUNCT
ijassa-1958	191	6	=	=	PRON
ijassa-1958	192	1	εi	εi	VERB
ijassa-1958	192	2	.	.	PUNCT
ijassa-1958	193	1	then	then	ADV
ijassa-1958	193	2	,	,	PUNCT
ijassa-1958	193	3	u	u	PRON
ijassa-1958	193	4	can	can	AUX
ijassa-1958	193	5	be	be	AUX
ijassa-1958	193	6	considered	consider	VERB
ijassa-1958	193	7	a	a	DET
ijassa-1958	193	8	order	order	NOUN
ijassa-1958	193	9	-	-	PUNCT
ijassa-1958	193	10	compatible	compatible	ADJ
ijassa-1958	193	11	metric	metric	ADJ
ijassa-1958	193	12	space	space	NOUN
ijassa-1958	193	13	if	if	SCONJ
ijassa-1958	193	14	the	the	DET
ijassa-1958	193	15	following	follow	VERB
ijassa-1958	193	16	condition	condition	NOUN
ijassa-1958	193	17	holds	hold	VERB
ijassa-1958	193	18	for	for	SCONJ
ijassa-1958	193	19	all	all	DET
ijassa-1958	193	20	i	i	PRON
ijassa-1958	193	21	:	:	PUNCT
ijassa-1958	193	22	εi	εi	VERB
ijassa-1958	193	23	≤	≤	NUM
ijassa-1958	193	24	min	min	PROPN
ijassa-1958	193	25	j	j	PROPN
ijassa-1958	193	26	<	<	X
ijassa-1958	193	27	i	i	PROPN
ijassa-1958	193	28	,	,	PUNCT
ijassa-1958	193	29	k	k	PROPN
ijassa-1958	193	30	>	>	X
ijassa-1958	193	31	i	i	PRON
ijassa-1958	193	32	max	max	PROPN
ijassa-1958	193	33	(	(	PUNCT
ijassa-1958	193	34	2r(k	2r(k	NUM
ijassa-1958	193	35	−	−	PROPN
ijassa-1958	193	36	i)−	i)−	PROPN
ijassa-1958	193	37	2εj	2εj	PROPN
ijassa-1958	193	38	−	−	PROPN
ijassa-1958	193	39	εk	εk	NOUN
ijassa-1958	193	40	,	,	PUNCT
ijassa-1958	193	41	2r(i−	2r(i−	NUM
ijassa-1958	193	42	j)−	j)−	PROPN
ijassa-1958	193	43	2εk	2εk	ADJ
ijassa-1958	193	44	−	−	PROPN
ijassa-1958	193	45	εj	εj	NOUN
ijassa-1958	193	46	)	)	PUNCT
ijassa-1958	193	47	.	.	PUNCT
ijassa-1958	194	1	proof	proof	NOUN
ijassa-1958	194	2	let	let	VERB
ijassa-1958	194	3	us	we	PRON
ijassa-1958	194	4	take	take	VERB
ijassa-1958	194	5	an	an	DET
ijassa-1958	194	6	arbitrary	arbitrary	ADJ
ijassa-1958	194	7	point	point	NOUN
ijassa-1958	194	8	xi	xi	PRON
ijassa-1958	194	9	and	and	CCONJ
ijassa-1958	194	10	consider	consider	VERB
ijassa-1958	194	11	arbitrary	arbitrary	ADJ
ijassa-1958	194	12	points	point	NOUN
ijassa-1958	194	13	xj	xj	PROPN
ijassa-1958	194	14	,	,	PUNCT
ijassa-1958	194	15	xk	xk	PROPN
ijassa-1958	194	16	,	,	PUNCT
ijassa-1958	194	17	xj	xj	PROPN
ijassa-1958	194	18	≤	≤	PROPN
ijassa-1958	194	19	xi	xi	ADP
ijassa-1958	194	20	≤	≤	NUM
ijassa-1958	195	1	xk	xk	PROPN
ijassa-1958	195	2	.	.	PUNCT
ijassa-1958	196	1	copyright	copyright	NOUN
ijassa-1958	196	2	©	©	PROPN
ijassa-1958	196	3	2024	2024	NUM
ijassa-1958	196	4	assa	assa	NOUN
ijassa-1958	196	5	.	.	PUNCT
ijassa-1958	197	1	adv	adv	PROPN
ijassa-1958	197	2	syst	syst	PROPN
ijassa-1958	197	3	sci	sci	PROPN
ijassa-1958	197	4	appl	appl	PROPN
ijassa-1958	197	5	(	(	PUNCT
ijassa-1958	197	6	2024	2024	NUM
ijassa-1958	197	7	)	)	PUNCT
ijassa-1958	197	8	on	on	ADP
ijassa-1958	197	9	the	the	DET
ijassa-1958	197	10	compatibility	compatibility	NOUN
ijassa-1958	197	11	of	of	ADP
ijassa-1958	197	12	metric	metric	ADJ
ijassa-1958	197	13	and	and	CCONJ
ijassa-1958	197	14	linear	linear	ADJ
ijassa-1958	197	15	order	order	NOUN
ijassa-1958	197	16	9	9	NUM
ijassa-1958	197	17	then	then	ADV
ijassa-1958	197	18	d(uxi	d(uxi	PROPN
ijassa-1958	197	19	,	,	PUNCT
ijassa-1958	197	20	uxj	uxj	ADJ
ijassa-1958	197	21	)	)	PUNCT
ijassa-1958	197	22	≤	≤	PUNCT
ijassa-1958	197	23	d(xi	d(xi	PROPN
ijassa-1958	197	24	,	,	PUNCT
ijassa-1958	197	25	xj	xj	PROPN
ijassa-1958	197	26	)	)	PUNCT
ijassa-1958	198	1	+	+	CCONJ
ijassa-1958	198	2	d(uxi	d(uxi	PROPN
ijassa-1958	198	3	,	,	PUNCT
ijassa-1958	198	4	xi	xi	ADJ
ijassa-1958	198	5	)	)	PUNCT
ijassa-1958	199	1	+	+	CCONJ
ijassa-1958	199	2	d(uxj	d(uxj	PROPN
ijassa-1958	199	3	,	,	PUNCT
ijassa-1958	199	4	xj	xj	PROPN
ijassa-1958	199	5	)	)	PUNCT
ijassa-1958	199	6	≤	≤	NOUN
ijassa-1958	200	1	2(i−	2(i−	NUM
ijassa-1958	200	2	j)r	j)r	NOUN
ijassa-1958	201	1	+	+	CCONJ
ijassa-1958	201	2	εi	εi	VERB
ijassa-1958	201	3	+	+	ADJ
ijassa-1958	201	4	εj	εj	NOUN
ijassa-1958	201	5	(	(	PUNCT
ijassa-1958	201	6	since	since	SCONJ
ijassa-1958	201	7	on	on	ADP
ijassa-1958	201	8	the	the	DET
ijassa-1958	201	9	straight	straight	ADJ
ijassa-1958	201	10	line	line	NOUN
ijassa-1958	201	11	between	between	ADP
ijassa-1958	201	12	xi	xi	PROPN
ijassa-1958	201	13	and	and	CCONJ
ijassa-1958	201	14	xj	xj	PROPN
ijassa-1958	201	15	there	there	PRON
ijassa-1958	201	16	is	be	VERB
ijassa-1958	201	17	exactly	exactly	ADV
ijassa-1958	201	18	j	j	PROPN
ijassa-1958	201	19	−	−	PROPN
ijassa-1958	202	1	i	i	PRON
ijassa-1958	202	2	times	time	VERB
ijassa-1958	202	3	2r	2r	NUM
ijassa-1958	202	4	)	)	PUNCT
ijassa-1958	202	5	.	.	PUNCT
ijassa-1958	203	1	similarly	similarly	ADV
ijassa-1958	203	2	d(uxj	d(uxj	ADJ
ijassa-1958	203	3	,	,	PUNCT
ijassa-1958	203	4	uxk	uxk	NOUN
ijassa-1958	203	5	)	)	PUNCT
ijassa-1958	203	6	≤	≤	NUM
ijassa-1958	203	7	2(k	2(k	NUM
ijassa-1958	203	8	−	−	NOUN
ijassa-1958	203	9	i)r	i)r	NOUN
ijassa-1958	203	10	+	+	CCONJ
ijassa-1958	203	11	εi	εi	ADJ
ijassa-1958	203	12	+	+	NUM
ijassa-1958	203	13	εk	εk	NOUN
ijassa-1958	203	14	.	.	NOUN
ijassa-1958	203	15	similar	similar	ADJ
ijassa-1958	203	16	to	to	ADP
ijassa-1958	203	17	the	the	DET
ijassa-1958	203	18	above	above	ADJ
ijassa-1958	203	19	d(xj	d(xj	ADJ
ijassa-1958	203	20	,	,	PUNCT
ijassa-1958	203	21	xk	xk	ADJ
ijassa-1958	203	22	)	)	PUNCT
ijassa-1958	203	23	≤	≤	NOUN
ijassa-1958	203	24	d(uxj	d(uxj	NOUN
ijassa-1958	203	25	,	,	PUNCT
ijassa-1958	203	26	uxk	uxk	NOUN
ijassa-1958	203	27	)	)	PUNCT
ijassa-1958	203	28	+	+	CCONJ
ijassa-1958	204	1	d(uxj	d(uxj	PROPN
ijassa-1958	204	2	,	,	PUNCT
ijassa-1958	204	3	xj	xj	PROPN
ijassa-1958	204	4	)	)	PUNCT
ijassa-1958	205	1	+	+	CCONJ
ijassa-1958	205	2	d(uxk	d(uxk	PROPN
ijassa-1958	205	3	,	,	PUNCT
ijassa-1958	205	4	xk	xk	ADJ
ijassa-1958	205	5	)	)	PUNCT
ijassa-1958	205	6	≤	≤	NOUN
ijassa-1958	205	7	d(uxj	d(uxj	NOUN
ijassa-1958	205	8	,	,	PUNCT
ijassa-1958	205	9	uxk	uxk	NOUN
ijassa-1958	205	10	)	)	PUNCT
ijassa-1958	206	1	+	+	CCONJ
ijassa-1958	206	2	εj	εj	NOUN
ijassa-1958	206	3	+	+	CCONJ
ijassa-1958	206	4	εk	εk	NOUN
ijassa-1958	206	5	which	which	PRON
ijassa-1958	206	6	implies	imply	VERB
ijassa-1958	206	7	that	that	DET
ijassa-1958	206	8	d(uxj	d(uxj	NOUN
ijassa-1958	206	9	,	,	PUNCT
ijassa-1958	206	10	uxk	uxk	NOUN
ijassa-1958	206	11	)	)	PUNCT
ijassa-1958	206	12	≥	≥	NOUN
ijassa-1958	206	13	d(xj	d(xj	PROPN
ijassa-1958	206	14	,	,	PUNCT
ijassa-1958	206	15	xk)−	xk)−	PROPN
ijassa-1958	206	16	εj	εj	PROPN
ijassa-1958	206	17	−	−	PROPN
ijassa-1958	206	18	εk	εk	ADJ
ijassa-1958	206	19	≥	≥	NOUN
ijassa-1958	206	20	(	(	PUNCT
ijassa-1958	206	21	k	k	PROPN
ijassa-1958	206	22	−	−	PROPN
ijassa-1958	206	23	j)2r	j)2r	NOUN
ijassa-1958	206	24	−	−	PROPN
ijassa-1958	206	25	εj	εj	NOUN
ijassa-1958	206	26	−	−	PROPN
ijassa-1958	206	27	εk	εk	NOUN
ijassa-1958	206	28	.	.	PUNCT
ijassa-1958	207	1	then	then	ADV
ijassa-1958	207	2	if	if	SCONJ
ijassa-1958	207	3	(	(	PUNCT
ijassa-1958	207	4	k	k	X
ijassa-1958	207	5	−	−	PROPN
ijassa-1958	207	6	j)2r	j)2r	NOUN
ijassa-1958	207	7	−	−	PROPN
ijassa-1958	208	1	εj	εj	NOUN
ijassa-1958	208	2	−	−	PROPN
ijassa-1958	208	3	εk	εk	PROPN
ijassa-1958	208	4	≥	≥	NOUN
ijassa-1958	208	5	max	max	PROPN
ijassa-1958	208	6	(	(	PUNCT
ijassa-1958	208	7	2(i−	2(i−	NUM
ijassa-1958	208	8	j)r	j)r	NOUN
ijassa-1958	209	1	+	+	CCONJ
ijassa-1958	209	2	εi	εi	ADJ
ijassa-1958	209	3	+	+	NUM
ijassa-1958	209	4	εj	εj	NOUN
ijassa-1958	209	5	,	,	PUNCT
ijassa-1958	209	6	2(k	2(k	NUM
ijassa-1958	209	7	−	−	NOUN
ijassa-1958	209	8	i)r	i)r	NOUN
ijassa-1958	209	9	+	+	CCONJ
ijassa-1958	209	10	εi	εi	ADJ
ijassa-1958	209	11	+	+	NUM
ijassa-1958	209	12	εk	εk	NOUN
ijassa-1958	209	13	)	)	PUNCT
ijassa-1958	209	14	,	,	PUNCT
ijassa-1958	209	15	then	then	ADV
ijassa-1958	209	16	if	if	SCONJ
ijassa-1958	209	17	εi	εi	VERB
ijassa-1958	209	18	≤	≤	NUM
ijassa-1958	209	19	max	max	X
ijassa-1958	209	20	(	(	PUNCT
ijassa-1958	209	21	2r(k	2r(k	NUM
ijassa-1958	210	1	−	−	PROPN
ijassa-1958	210	2	i)−	i)−	PROPN
ijassa-1958	210	3	2εj	2εj	PROPN
ijassa-1958	210	4	−	−	PROPN
ijassa-1958	210	5	εk	εk	NOUN
ijassa-1958	210	6	,	,	PUNCT
ijassa-1958	210	7	2r(i−	2r(i−	NUM
ijassa-1958	210	8	j)−	j)−	PROPN
ijassa-1958	210	9	2εk	2εk	ADJ
ijassa-1958	210	10	−	−	PROPN
ijassa-1958	210	11	εj	εj	NOUN
ijassa-1958	210	12	)	)	PUNCT
ijassa-1958	210	13	our	our	PRON
ijassa-1958	210	14	metric	metric	NOUN
ijassa-1958	210	15	will	will	AUX
ijassa-1958	210	16	be	be	AUX
ijassa-1958	210	17	compatible	compatible	ADJ
ijassa-1958	210	18	with	with	ADP
ijassa-1958	210	19	the	the	DET
ijassa-1958	210	20	order	order	NOUN
ijassa-1958	210	21	for	for	ADP
ijassa-1958	210	22	the	the	DET
ijassa-1958	210	23	triple	triple	ADJ
ijassa-1958	210	24	uxj	uxj	NOUN
ijassa-1958	210	25	,	,	PUNCT
ijassa-1958	210	26	uxi	uxi	NOUN
ijassa-1958	210	27	,	,	PUNCT
ijassa-1958	210	28	uxk	uxk	VERB
ijassa-1958	210	29	.	.	PUNCT
ijassa-1958	211	1	if	if	SCONJ
ijassa-1958	211	2	we	we	PRON
ijassa-1958	211	3	select	select	VERB
ijassa-1958	211	4	an	an	DET
ijassa-1958	211	5	arbitrary	arbitrary	ADJ
ijassa-1958	211	6	value	value	NOUN
ijassa-1958	211	7	of	of	ADP
ijassa-1958	211	8	i	i	PRON
ijassa-1958	211	9	and	and	CCONJ
ijassa-1958	211	10	then	then	ADV
ijassa-1958	211	11	choose	choose	VERB
ijassa-1958	211	12	any	any	DET
ijassa-1958	211	13	two	two	NUM
ijassa-1958	211	14	other	other	ADJ
ijassa-1958	211	15	values	value	NOUN
ijassa-1958	211	16	,	,	PUNCT
ijassa-1958	211	17	j	j	PROPN
ijassa-1958	211	18	and	and	CCONJ
ijassa-1958	211	19	k	k	PROPN
ijassa-1958	211	20	,	,	PUNCT
ijassa-1958	211	21	where	where	SCONJ
ijassa-1958	211	22	j	j	PROPN
ijassa-1958	211	23	is	be	AUX
ijassa-1958	211	24	less	less	ADJ
ijassa-1958	211	25	than	than	SCONJ
ijassa-1958	211	26	i	i	PRON
ijassa-1958	211	27	and	and	CCONJ
ijassa-1958	211	28	k	k	PROPN
ijassa-1958	211	29	is	be	AUX
ijassa-1958	211	30	greater	great	ADJ
ijassa-1958	211	31	than	than	ADP
ijassa-1958	211	32	i	i	PRON
ijassa-1958	211	33	,	,	PUNCT
ijassa-1958	211	34	and	and	CCONJ
ijassa-1958	211	35	keep	keep	VERB
ijassa-1958	211	36	in	in	ADP
ijassa-1958	211	37	mind	mind	NOUN
ijassa-1958	211	38	that	that	SCONJ
ijassa-1958	211	39	εi	εi	NOUN
ijassa-1958	211	40	is	be	AUX
ijassa-1958	211	41	bounded	bound	VERB
ijassa-1958	211	42	from	from	ADP
ijassa-1958	211	43	above	above	ADV
ijassa-1958	211	44	,	,	PUNCT
ijassa-1958	211	45	we	we	PRON
ijassa-1958	211	46	obtain	obtain	VERB
ijassa-1958	211	47	the	the	DET
ijassa-1958	211	48	conditions	condition	NOUN
ijassa-1958	211	49	of	of	ADP
ijassa-1958	211	50	the	the	DET
ijassa-1958	211	51	lemma	lemma	PROPN
ijassa-1958	211	52	.	.	PUNCT
ijassa-1958	212	1	all	all	DET
ijassa-1958	212	2	the	the	DET
ijassa-1958	212	3	examples	example	NOUN
ijassa-1958	212	4	of	of	ADP
ijassa-1958	212	5	order	order	NOUN
ijassa-1958	212	6	-	-	PUNCT
ijassa-1958	212	7	compatible	compatible	ADJ
ijassa-1958	212	8	discrete	discrete	ADJ
ijassa-1958	212	9	curved	curve	VERB
ijassa-1958	212	10	lines	line	NOUN
ijassa-1958	212	11	in	in	ADP
ijassa-1958	212	12	normed	normed	ADJ
ijassa-1958	212	13	spaces	space	NOUN
ijassa-1958	212	14	had	have	VERB
ijassa-1958	212	15	the	the	DET
ijassa-1958	212	16	property	property	NOUN
ijassa-1958	212	17	that	that	PRON
ijassa-1958	212	18	when	when	SCONJ
ijassa-1958	212	19	we	we	PRON
ijassa-1958	212	20	projected	project	VERB
ijassa-1958	212	21	them	they	PRON
ijassa-1958	212	22	onto	onto	ADP
ijassa-1958	212	23	the	the	DET
ijassa-1958	212	24	coordinate	coordinate	NOUN
ijassa-1958	212	25	axis	axis	NOUN
ijassa-1958	212	26	passing	pass	VERB
ijassa-1958	212	27	through	through	ADP
ijassa-1958	212	28	their	their	PRON
ijassa-1958	212	29	starting	starting	NOUN
ijassa-1958	212	30	points	point	NOUN
ijassa-1958	212	31	x	x	PRON
ijassa-1958	212	32	,	,	PUNCT
ijassa-1958	212	33	the	the	DET
ijassa-1958	212	34	order	order	NOUN
ijassa-1958	212	35	of	of	ADP
ijassa-1958	212	36	the	the	DET
ijassa-1958	212	37	points	point	NOUN
ijassa-1958	212	38	was	be	AUX
ijassa-1958	212	39	the	the	DET
ijassa-1958	212	40	same	same	ADJ
ijassa-1958	212	41	as	as	SCONJ
ijassa-1958	212	42	it	it	PRON
ijassa-1958	212	43	was	be	AUX
ijassa-1958	212	44	on	on	ADP
ijassa-1958	212	45	the	the	DET
ijassa-1958	212	46	original	original	ADJ
ijassa-1958	212	47	curved	curved	ADJ
ijassa-1958	212	48	line	line	NOUN
ijassa-1958	212	49	itself	itself	PRON
ijassa-1958	212	50	.	.	PUNCT
ijassa-1958	213	1	in	in	ADP
ijassa-1958	213	2	other	other	ADJ
ijassa-1958	213	3	words	word	NOUN
ijassa-1958	213	4	,	,	PUNCT
ijassa-1958	213	5	the	the	DET
ijassa-1958	213	6	points	point	NOUN
ijassa-1958	213	7	were	be	AUX
ijassa-1958	213	8	simply	simply	ADV
ijassa-1958	213	9	ordered	order	VERB
ijassa-1958	213	10	based	base	VERB
ijassa-1958	213	11	on	on	ADP
ijassa-1958	213	12	their	their	PRON
ijassa-1958	213	13	coordinate	coordinate	NOUN
ijassa-1958	213	14	along	along	ADP
ijassa-1958	213	15	the	the	DET
ijassa-1958	213	16	original	original	ADJ
ijassa-1958	213	17	straight	straight	ADJ
ijassa-1958	213	18	line	line	NOUN
ijassa-1958	213	19	.	.	PUNCT
ijassa-1958	214	1	fig	fig	NOUN
ijassa-1958	214	2	.	.	PUNCT
ijassa-1958	215	1	4.3	4.3	NUM
ijassa-1958	215	2	.	.	PUNCT
ijassa-1958	216	1	an	an	DET
ijassa-1958	216	2	example	example	NOUN
ijassa-1958	216	3	of	of	ADP
ijassa-1958	216	4	a	a	DET
ijassa-1958	216	5	discrete	discrete	ADJ
ijassa-1958	216	6	curved	curved	ADJ
ijassa-1958	216	7	line	line	NOUN
ijassa-1958	216	8	where	where	SCONJ
ijassa-1958	216	9	the	the	DET
ijassa-1958	216	10	metric	metric	NOUN
ijassa-1958	216	11	is	be	AUX
ijassa-1958	216	12	compatible	compatible	ADJ
ijassa-1958	216	13	with	with	ADP
ijassa-1958	216	14	the	the	DET
ijassa-1958	216	15	order	order	NOUN
ijassa-1958	216	16	.	.	PUNCT
ijassa-1958	217	1	since	since	SCONJ
ijassa-1958	217	2	we	we	PRON
ijassa-1958	217	3	make	make	VERB
ijassa-1958	217	4	a	a	DET
ijassa-1958	217	5	small	small	ADJ
ijassa-1958	217	6	shift	shift	NOUN
ijassa-1958	217	7	,	,	PUNCT
ijassa-1958	217	8	the	the	DET
ijassa-1958	217	9	projection	projection	NOUN
ijassa-1958	217	10	preserves	preserve	VERB
ijassa-1958	217	11	the	the	DET
ijassa-1958	217	12	order	order	NOUN
ijassa-1958	217	13	on	on	ADP
ijassa-1958	217	14	the	the	DET
ijassa-1958	217	15	original	original	ADJ
ijassa-1958	217	16	set	set	NOUN
ijassa-1958	217	17	.	.	PUNCT
ijassa-1958	218	1	let	let	VERB
ijassa-1958	218	2	us	we	PRON
ijassa-1958	218	3	show	show	VERB
ijassa-1958	218	4	that	that	SCONJ
ijassa-1958	218	5	this	this	PRON
ijassa-1958	218	6	is	be	AUX
ijassa-1958	218	7	not	not	PART
ijassa-1958	218	8	always	always	ADV
ijassa-1958	218	9	the	the	DET
ijassa-1958	218	10	case	case	NOUN
ijassa-1958	218	11	.	.	PUNCT
ijassa-1958	219	1	example	example	NOUN
ijassa-1958	219	2	4.3	4.3	NUM
ijassa-1958	219	3	:	:	PUNCT
ijassa-1958	219	4	consider	consider	VERB
ijassa-1958	219	5	the	the	DET
ijassa-1958	219	6	set	set	NOUN
ijassa-1958	219	7	of	of	ADP
ijassa-1958	219	8	natural	natural	ADJ
ijassa-1958	219	9	numbers	number	NOUN
ijassa-1958	220	1	x	x	PUNCT
ijassa-1958	220	2	=	=	SYM
ijassa-1958	220	3	ne1	ne1	PROPN
ijassa-1958	220	4	embedded	embed	VERB
ijassa-1958	220	5	in	in	ADP
ijassa-1958	220	6	the	the	DET
ijassa-1958	220	7	plane	plane	NOUN
ijassa-1958	220	8	with	with	ADP
ijassa-1958	220	9	the	the	DET
ijassa-1958	220	10	euclidean	euclidean	ADJ
ijassa-1958	220	11	metric	metric	NOUN
ijassa-1958	220	12	.	.	PUNCT
ijassa-1958	221	1	then	then	ADV
ijassa-1958	221	2	the	the	DET
ijassa-1958	221	3	set	set	NOUN
ijassa-1958	221	4	u	u	NOUN
ijassa-1958	221	5	=	=	PUNCT
ijassa-1958	221	6	{	{	PUNCT
ijassa-1958	221	7	ux}x∈x	ux}x∈x	NOUN
ijassa-1958	221	8	,	,	PUNCT
ijassa-1958	221	9	where	where	SCONJ
ijassa-1958	221	10	ux	ux	ADV
ijassa-1958	221	11	=	=	PRON
ijassa-1958	221	12	{	{	PUNCT
ijassa-1958	221	13	(	(	PUNCT
ijassa-1958	221	14	0.5	0.5	NUM
ijassa-1958	221	15	,	,	PUNCT
ijassa-1958	221	16	1.5	1.5	NUM
ijassa-1958	221	17	)	)	PUNCT
ijassa-1958	221	18	,	,	PUNCT
ijassa-1958	221	19	if	if	SCONJ
ijassa-1958	221	20	x	x	X
ijassa-1958	221	21	=	=	SYM
ijassa-1958	221	22	(	(	PUNCT
ijassa-1958	221	23	2	2	NUM
ijassa-1958	221	24	,	,	PUNCT
ijassa-1958	221	25	0	0	NUM
ijassa-1958	221	26	)	)	PUNCT
ijassa-1958	221	27	;	;	PUNCT
ijassa-1958	221	28	x	x	X
ijassa-1958	221	29	,	,	PUNCT
ijassa-1958	221	30	otherwise	otherwise	ADV
ijassa-1958	221	31	will	will	AUX
ijassa-1958	221	32	be	be	AUX
ijassa-1958	221	33	a	a	DET
ijassa-1958	221	34	discrete	discrete	ADJ
ijassa-1958	221	35	curved	curved	ADJ
ijassa-1958	221	36	line	line	NOUN
ijassa-1958	221	37	with	with	ADP
ijassa-1958	221	38	the	the	DET
ijassa-1958	221	39	metric	metric	ADJ
ijassa-1958	221	40	compatible	compatible	ADJ
ijassa-1958	221	41	with	with	ADP
ijassa-1958	221	42	the	the	DET
ijassa-1958	221	43	order	order	NOUN
ijassa-1958	221	44	,	,	PUNCT
ijassa-1958	221	45	but	but	CCONJ
ijassa-1958	221	46	the	the	DET
ijassa-1958	221	47	projection	projection	NOUN
ijassa-1958	221	48	of	of	ADP
ijassa-1958	221	49	u	u	NOUN
ijassa-1958	221	50	on	on	ADP
ijassa-1958	221	51	the	the	DET
ijassa-1958	221	52	x	x	NOUN
ijassa-1958	221	53	axis	axis	NOUN
ijassa-1958	221	54	(	(	PUNCT
ijassa-1958	221	55	on	on	ADP
ijassa-1958	221	56	which	which	PRON
ijassa-1958	221	57	x	x	PRON
ijassa-1958	221	58	lies	lie	NOUN
ijassa-1958	221	59	)	)	PUNCT
ijassa-1958	221	60	has	have	VERB
ijassa-1958	221	61	an	an	DET
ijassa-1958	221	62	order	order	NOUN
ijassa-1958	221	63	that	that	PRON
ijassa-1958	221	64	does	do	AUX
ijassa-1958	221	65	not	not	PART
ijassa-1958	221	66	coincide	coincide	VERB
ijassa-1958	221	67	with	with	ADP
ijassa-1958	221	68	the	the	DET
ijassa-1958	221	69	order	order	NOUN
ijassa-1958	221	70	on	on	ADP
ijassa-1958	221	71	this	this	DET
ijassa-1958	221	72	axis	axis	NOUN
ijassa-1958	221	73	as	as	ADP
ijassa-1958	221	74	a	a	DET
ijassa-1958	221	75	straight	straight	ADJ
ijassa-1958	221	76	line	line	NOUN
ijassa-1958	221	77	.	.	PUNCT
ijassa-1958	222	1	proof	proof	NOUN
ijassa-1958	222	2	to	to	PART
ijassa-1958	222	3	check	check	VERB
ijassa-1958	222	4	compatibility	compatibility	NOUN
ijassa-1958	222	5	with	with	ADP
ijassa-1958	222	6	the	the	DET
ijassa-1958	222	7	order	order	NOUN
ijassa-1958	222	8	,	,	PUNCT
ijassa-1958	222	9	note	note	VERB
ijassa-1958	222	10	that	that	SCONJ
ijassa-1958	222	11	the	the	DET
ijassa-1958	222	12	only	only	ADJ
ijassa-1958	222	13	modified	modify	VERB
ijassa-1958	222	14	point	point	NOUN
ijassa-1958	222	15	is	be	AUX
ijassa-1958	222	16	u(2,0	u(2,0	NOUN
ijassa-1958	222	17	)	)	PUNCT
ijassa-1958	222	18	.	.	PUNCT
ijassa-1958	223	1	it	it	PRON
ijassa-1958	223	2	is	be	AUX
ijassa-1958	223	3	easy	easy	ADJ
ijassa-1958	223	4	to	to	PART
ijassa-1958	223	5	see	see	VERB
ijassa-1958	223	6	that	that	SCONJ
ijassa-1958	223	7	if	if	SCONJ
ijassa-1958	223	8	we	we	PRON
ijassa-1958	223	9	take	take	VERB
ijassa-1958	223	10	the	the	DET
ijassa-1958	223	11	points	point	NOUN
ijassa-1958	223	12	ux1	ux1	NOUN
ijassa-1958	223	13	=	=	PUNCT
ijassa-1958	223	14	(	(	PUNCT
ijassa-1958	223	15	1	1	NUM
ijassa-1958	223	16	,	,	PUNCT
ijassa-1958	223	17	0	0	NUM
ijassa-1958	223	18	)	)	PUNCT
ijassa-1958	223	19	,	,	PUNCT
ijassa-1958	223	20	ux2	ux2	PROPN
ijassa-1958	223	21	=	=	SYM
ijassa-1958	223	22	(	(	PUNCT
ijassa-1958	223	23	0.5	0.5	NUM
ijassa-1958	223	24	,	,	PUNCT
ijassa-1958	223	25	1.5	1.5	NUM
ijassa-1958	223	26	)	)	PUNCT
ijassa-1958	223	27	,	,	PUNCT
ijassa-1958	223	28	ux3	ux3	X
ijassa-1958	223	29	=	=	SYM
ijassa-1958	223	30	(	(	PUNCT
ijassa-1958	223	31	3	3	NUM
ijassa-1958	223	32	,	,	PUNCT
ijassa-1958	223	33	0	0	NUM
ijassa-1958	223	34	)	)	PUNCT
ijassa-1958	223	35	,	,	PUNCT
ijassa-1958	223	36	the	the	DET
ijassa-1958	223	37	compatibility	compatibility	NOUN
ijassa-1958	223	38	is	be	AUX
ijassa-1958	223	39	satisfied	satisfied	ADJ
ijassa-1958	223	40	(	(	PUNCT
ijassa-1958	223	41	checked	check	VERB
ijassa-1958	223	42	by	by	ADP
ijassa-1958	223	43	calculating	calculate	VERB
ijassa-1958	223	44	distances	distance	NOUN
ijassa-1958	223	45	)	)	PUNCT
ijassa-1958	223	46	.	.	PUNCT
ijassa-1958	224	1	the	the	DET
ijassa-1958	224	2	remaining	remain	VERB
ijassa-1958	224	3	cases	case	NOUN
ijassa-1958	224	4	follow	follow	VERB
ijassa-1958	224	5	from	from	ADP
ijassa-1958	224	6	theorem	theorem	ADJ
ijassa-1958	224	7	4.1	4.1	NUM
ijassa-1958	224	8	and	and	CCONJ
ijassa-1958	224	9	the	the	DET
ijassa-1958	224	10	fact	fact	NOUN
ijassa-1958	224	11	that	that	SCONJ
ijassa-1958	224	12	εi	εi	VERB
ijassa-1958	224	13	=	=	SYM
ijassa-1958	224	14	0	0	NUM
ijassa-1958	224	15	for	for	ADP
ijassa-1958	224	16	all	all	PRON
ijassa-1958	224	17	i	i	PRON
ijassa-1958	224	18	̸=	̸=	PROPN
ijassa-1958	224	19	2	2	NUM
ijassa-1958	224	20	.	.	NOUN
ijassa-1958	224	21	5	5	NUM
ijassa-1958	224	22	.	.	X
ijassa-1958	225	1	acknowledgements	acknowledgement	VERB
ijassa-1958	225	2	the	the	DET
ijassa-1958	225	3	research	research	NOUN
ijassa-1958	225	4	of	of	ADP
ijassa-1958	225	5	the	the	DET
ijassa-1958	225	6	second	second	ADJ
ijassa-1958	225	7	author	author	NOUN
ijassa-1958	225	8	was	be	AUX
ijassa-1958	225	9	supported	support	VERB
ijassa-1958	225	10	by	by	ADP
ijassa-1958	225	11	the	the	DET
ijassa-1958	225	12	russian	russian	PROPN
ijassa-1958	225	13	science	science	PROPN
ijassa-1958	225	14	foundation	foundation	PROPN
ijassa-1958	225	15	(	(	PUNCT
ijassa-1958	225	16	project	project	VERB
ijassa-1958	225	17	no	no	NOUN
ijassa-1958	225	18	.	.	NOUN
ijassa-1958	226	1	23	23	NUM
ijassa-1958	226	2	-	-	SYM
ijassa-1958	226	3	11	11	NUM
ijassa-1958	226	4	-	-	SYM
ijassa-1958	226	5	20020	20020	NUM
ijassa-1958	226	6	,	,	PUNCT
ijassa-1958	226	7	https://rscf.ru/en/project/23-11-20020/	https://rscf.ru/en/project/23-11-20020/	NUM
ijassa-1958	226	8	)	)	PUNCT
ijassa-1958	226	9	.	.	PUNCT
ijassa-1958	227	1	copyright	copyright	NOUN
ijassa-1958	227	2	©	©	PROPN
ijassa-1958	227	3	2024	2024	NUM
ijassa-1958	227	4	assa	assa	NOUN
ijassa-1958	227	5	.	.	PUNCT
ijassa-1958	228	1	adv	adv	PROPN
ijassa-1958	228	2	syst	syst	PROPN
ijassa-1958	228	3	sci	sci	PROPN
ijassa-1958	228	4	appl	appl	PROPN
ijassa-1958	228	5	(	(	PUNCT
ijassa-1958	228	6	2024	2024	NUM
ijassa-1958	228	7	)	)	PUNCT
ijassa-1958	228	8	10	10	NUM
ijassa-1958	228	9	p.	p.	NOUN
ijassa-1958	228	10	klimov	klimov	PROPN
ijassa-1958	228	11	,	,	PUNCT
ijassa-1958	228	12	r.	r.	PROPN
ijassa-1958	228	13	sengupta	sengupta	PROPN
ijassa-1958	228	14	references	reference	NOUN
ijassa-1958	228	15	1	1	NUM
ijassa-1958	228	16	.	.	PUNCT
ijassa-1958	228	17	arutyunov	arutyunov	PROPN
ijassa-1958	228	18	,	,	PUNCT
ijassa-1958	228	19	a.	a.	NOUN
ijassa-1958	228	20	v.	v.	PROPN
ijassa-1958	228	21	,	,	PUNCT
ijassa-1958	228	22	zhukovskiy	zhukovskiy	PROPN
ijassa-1958	228	23	,	,	PUNCT
ijassa-1958	228	24	e.	e.	PROPN
ijassa-1958	228	25	s.	s.	PROPN
ijassa-1958	228	26	,	,	PUNCT
ijassa-1958	228	27	&	&	CCONJ
ijassa-1958	228	28	zhukovskiy	zhukovskiy	PROPN
ijassa-1958	228	29	,	,	PUNCT
ijassa-1958	228	30	s.	s.	PROPN
ijassa-1958	228	31	e.	e.	PROPN
ijassa-1958	228	32	(	(	PUNCT
ijassa-1958	228	33	2016	2016	NUM
ijassa-1958	228	34	)	)	PUNCT
ijassa-1958	228	35	.	.	PUNCT
ijassa-1958	229	1	coincidence	coincidence	NOUN
ijassa-1958	229	2	points	point	VERB
ijassa-1958	229	3	principle	principle	NOUN
ijassa-1958	229	4	for	for	ADP
ijassa-1958	229	5	set	set	NOUN
ijassa-1958	229	6	-	-	PUNCT
ijassa-1958	229	7	valued	value	VERB
ijassa-1958	229	8	mappings	mapping	NOUN
ijassa-1958	229	9	in	in	ADP
ijassa-1958	229	10	partially	partially	ADV
ijassa-1958	229	11	ordered	order	VERB
ijassa-1958	229	12	spaces	space	NOUN
ijassa-1958	229	13	,	,	PUNCT
ijassa-1958	229	14	topology	topology	NOUN
ijassa-1958	229	15	and	and	CCONJ
ijassa-1958	229	16	its	its	PRON
ijassa-1958	229	17	applications	application	NOUN
ijassa-1958	229	18	,	,	PUNCT
ijassa-1958	229	19	201	201	NUM
ijassa-1958	229	20	,	,	PUNCT
ijassa-1958	229	21	330–343	330–343	NUM
ijassa-1958	229	22	.	.	PUNCT
ijassa-1958	230	1	doi	doi	NOUN
ijassa-1958	230	2	:	:	PUNCT
ijassa-1958	230	3	10.1016	10.1016	NUM
ijassa-1958	230	4	/	/	SYM
ijassa-1958	230	5	j.topol.2015.12.044	j.topol.2015.12.044	PROPN
ijassa-1958	230	6	.	.	PROPN
ijassa-1958	230	7	2	2	NUM
ijassa-1958	230	8	.	.	X
ijassa-1958	230	9	bowers	bower	NOUN
ijassa-1958	230	10	,	,	PUNCT
ijassa-1958	230	11	j.	j.	PROPN
ijassa-1958	230	12	c.	c.	PROPN
ijassa-1958	230	13	,	,	PUNCT
ijassa-1958	230	14	&	&	CCONJ
ijassa-1958	230	15	bowers	bower	NOUN
ijassa-1958	230	16	,	,	PUNCT
ijassa-1958	230	17	p.	p.	NOUN
ijassa-1958	230	18	l.	l.	PROPN
ijassa-1958	230	19	(	(	PUNCT
ijassa-1958	230	20	2017	2017	NUM
ijassa-1958	230	21	)	)	PUNCT
ijassa-1958	230	22	.	.	PUNCT
ijassa-1958	231	1	a	a	DET
ijassa-1958	231	2	menger	menger	PROPN
ijassa-1958	231	3	redux	redux	PROPN
ijassa-1958	231	4	:	:	PUNCT
ijassa-1958	231	5	embedding	embed	VERB
ijassa-1958	231	6	metric	metric	ADJ
ijassa-1958	231	7	spaces	space	NOUN
ijassa-1958	231	8	isometrically	isometrically	ADV
ijassa-1958	231	9	in	in	ADP
ijassa-1958	231	10	euclidean	euclidean	ADJ
ijassa-1958	231	11	space	space	NOUN
ijassa-1958	231	12	,	,	PUNCT
ijassa-1958	231	13	the	the	DET
ijassa-1958	231	14	american	american	PROPN
ijassa-1958	231	15	mathematical	mathematical	PROPN
ijassa-1958	231	16	monthly	monthly	ADV
ijassa-1958	231	17	,	,	PUNCT
ijassa-1958	231	18	124(7	124(7	NUM
ijassa-1958	231	19	)	)	PUNCT
ijassa-1958	231	20	,	,	PUNCT
ijassa-1958	231	21	621	621	NUM
ijassa-1958	231	22	.	.	PUNCT
ijassa-1958	232	1	doi	doi	NOUN
ijassa-1958	232	2	:	:	PUNCT
ijassa-1958	232	3	10.4169	10.4169	NUM
ijassa-1958	232	4	/	/	SYM
ijassa-1958	232	5	amer.math.monthly.124.7.621	amer.math.monthly.124.7.621	PROPN
ijassa-1958	232	6	.	.	PUNCT
ijassa-1958	233	1	3	3	NUM
ijassa-1958	233	2	.	.	NUM
ijassa-1958	233	3	burago	burago	PROPN
ijassa-1958	233	4	,	,	PUNCT
ijassa-1958	233	5	d.	d.	PROPN
ijassa-1958	233	6	,	,	PUNCT
ijassa-1958	233	7	burago	burago	VERB
ijassa-1958	233	8	,	,	PUNCT
ijassa-1958	233	9	y.	y.	PROPN
ijassa-1958	233	10	,	,	PUNCT
ijassa-1958	233	11	&	&	CCONJ
ijassa-1958	233	12	ivanov	ivanov	PROPN
ijassa-1958	233	13	,	,	PUNCT
ijassa-1958	233	14	s.	s.	PROPN
ijassa-1958	233	15	(	(	PUNCT
ijassa-1958	233	16	2001	2001	NUM
ijassa-1958	233	17	)	)	PUNCT
ijassa-1958	233	18	.	.	PUNCT
ijassa-1958	234	1	a	a	DET
ijassa-1958	234	2	course	course	NOUN
ijassa-1958	234	3	in	in	ADP
ijassa-1958	234	4	metric	metric	ADJ
ijassa-1958	234	5	geometry	geometry	NOUN
ijassa-1958	234	6	.	.	PUNCT
ijassa-1958	235	1	graduate	graduate	NOUN
ijassa-1958	235	2	studies	study	NOUN
ijassa-1958	235	3	in	in	ADP
ijassa-1958	235	4	math	math	NOUN
ijassa-1958	235	5	.	.	PUNCT
ijassa-1958	236	1	,	,	PUNCT
ijassa-1958	236	2	33	33	NUM
ijassa-1958	236	3	.	.	NOUN
ijassa-1958	236	4	4	4	NUM
ijassa-1958	236	5	.	.	X
ijassa-1958	236	6	fréchet	fréchet	NOUN
ijassa-1958	236	7	,	,	PUNCT
ijassa-1958	236	8	m.	m.	NOUN
ijassa-1958	236	9	(	(	PUNCT
ijassa-1958	236	10	1906	1906	NUM
ijassa-1958	236	11	)	)	PUNCT
ijassa-1958	236	12	.	.	PUNCT
ijassa-1958	237	1	sur	sur	PROPN
ijassa-1958	237	2	quelques	quelques	PROPN
ijassa-1958	237	3	points	point	NOUN
ijassa-1958	237	4	du	du	PROPN
ijassa-1958	237	5	calcul	calcul	PROPN
ijassa-1958	237	6	fonctionnel	fonctionnel	X
ijassa-1958	238	1	[	[	X
ijassa-1958	238	2	on	on	ADP
ijassa-1958	238	3	some	some	DET
ijassa-1958	238	4	points	point	NOUN
ijassa-1958	238	5	of	of	ADP
ijassa-1958	238	6	functional	functional	ADJ
ijassa-1958	238	7	calculation	calculation	NOUN
ijassa-1958	238	8	]	]	PUNCT
ijassa-1958	238	9	,	,	PUNCT
ijassa-1958	238	10	rendiconti	rendiconti	ADJ
ijassa-1958	238	11	del	del	X
ijassa-1958	238	12	circolo	circolo	PROPN
ijassa-1958	238	13	matematico	matematico	NOUN
ijassa-1958	238	14	di	di	NOUN
ijassa-1958	238	15	palermo	palermo	NOUN
ijassa-1958	238	16	,	,	PUNCT
ijassa-1958	238	17	22(1	22(1	NUM
ijassa-1958	238	18	)	)	PUNCT
ijassa-1958	238	19	,	,	PUNCT
ijassa-1958	238	20	1–72	1–72	NOUN
ijassa-1958	238	21	.	.	PUNCT
ijassa-1958	238	22	doi	doi	NOUN
ijassa-1958	238	23	:	:	PUNCT
ijassa-1958	238	24	10.1007	10.1007	NUM
ijassa-1958	238	25	/	/	SYM
ijassa-1958	238	26	bf03018603	bf03018603	NOUN
ijassa-1958	239	1	[	[	X
ijassa-1958	239	2	in	in	ADP
ijassa-1958	239	3	french	french	NOUN
ijassa-1958	239	4	]	]	PUNCT
ijassa-1958	239	5	.	.	PUNCT
ijassa-1958	240	1	5	5	X
ijassa-1958	240	2	.	.	X
ijassa-1958	240	3	fuchs	fuchs	PROPN
ijassa-1958	240	4	,	,	PUNCT
ijassa-1958	240	5	l.	l.	PROPN
ijassa-1958	240	6	(	(	PUNCT
ijassa-1958	240	7	2011	2011	NUM
ijassa-1958	240	8	)	)	PUNCT
ijassa-1958	240	9	.	.	PUNCT
ijassa-1958	241	1	partially	partially	ADV
ijassa-1958	241	2	ordered	order	VERB
ijassa-1958	241	3	algebraic	algebraic	ADJ
ijassa-1958	241	4	systems	system	NOUN
ijassa-1958	241	5	.	.	PUNCT
ijassa-1958	242	1	dover	dover	PROPN
ijassa-1958	242	2	,	,	PUNCT
ijassa-1958	242	3	uk	uk	PROPN
ijassa-1958	242	4	:	:	PUNCT
ijassa-1958	242	5	dover	dover	PROPN
ijassa-1958	242	6	publications	publication	NOUN
ijassa-1958	242	7	.	.	PUNCT
ijassa-1958	243	1	6	6	X
ijassa-1958	243	2	.	.	X
ijassa-1958	243	3	gromov	gromov	NOUN
ijassa-1958	243	4	,	,	PUNCT
ijassa-1958	243	5	m.	m.	NOUN
ijassa-1958	243	6	(	(	PUNCT
ijassa-1958	243	7	2007	2007	NUM
ijassa-1958	243	8	)	)	PUNCT
ijassa-1958	243	9	.	.	PUNCT
ijassa-1958	244	1	metric	metric	ADJ
ijassa-1958	244	2	structures	structure	NOUN
ijassa-1958	244	3	for	for	ADP
ijassa-1958	244	4	riemannian	riemannian	ADJ
ijassa-1958	244	5	and	and	CCONJ
ijassa-1958	244	6	non	non	ADJ
ijassa-1958	244	7	-	-	ADJ
ijassa-1958	244	8	riemannian	riemannian	ADJ
ijassa-1958	244	9	spaces	space	NOUN
ijassa-1958	244	10	.	.	PUNCT
ijassa-1958	245	1	berlin	berlin	PROPN
ijassa-1958	245	2	,	,	PUNCT
ijassa-1958	245	3	germany	germany	PROPN
ijassa-1958	245	4	:	:	PUNCT
ijassa-1958	245	5	springer	springer	NOUN
ijassa-1958	245	6	.	.	PUNCT
ijassa-1958	246	1	7	7	X
ijassa-1958	246	2	.	.	X
ijassa-1958	246	3	kelley	kelley	PROPN
ijassa-1958	246	4	,	,	PUNCT
ijassa-1958	246	5	j.	j.	PROPN
ijassa-1958	246	6	l.	l.	PROPN
ijassa-1958	246	7	(	(	PUNCT
ijassa-1958	246	8	2008	2008	NUM
ijassa-1958	246	9	)	)	PUNCT
ijassa-1958	246	10	.	.	PUNCT
ijassa-1958	247	1	general	general	ADJ
ijassa-1958	247	2	topology	topology	PROPN
ijassa-1958	247	3	.	.	PUNCT
ijassa-1958	248	1	mountain	mountain	NOUN
ijassa-1958	248	2	view	view	NOUN
ijassa-1958	248	3	,	,	PUNCT
ijassa-1958	248	4	ca	ca	NOUN
ijassa-1958	248	5	:	:	PUNCT
ijassa-1958	248	6	ishi	ishi	PROPN
ijassa-1958	248	7	press	press	PROPN
ijassa-1958	248	8	international	international	PROPN
ijassa-1958	248	9	.	.	PUNCT
ijassa-1958	249	1	8	8	X
ijassa-1958	249	2	.	.	X
ijassa-1958	249	3	liberti	liberti	PROPN
ijassa-1958	249	4	,	,	PUNCT
ijassa-1958	249	5	l.	l.	PROPN
ijassa-1958	249	6	,	,	PUNCT
ijassa-1958	249	7	&	&	CCONJ
ijassa-1958	249	8	lavor	lavor	PROPN
ijassa-1958	249	9	,	,	PUNCT
ijassa-1958	249	10	c.	c.	PROPN
ijassa-1958	249	11	(	(	PUNCT
ijassa-1958	249	12	2015	2015	NUM
ijassa-1958	249	13	)	)	PUNCT
ijassa-1958	249	14	.	.	PUNCT
ijassa-1958	250	1	six	six	NUM
ijassa-1958	250	2	mathematical	mathematical	ADJ
ijassa-1958	250	3	gems	gem	NOUN
ijassa-1958	250	4	from	from	ADP
ijassa-1958	250	5	the	the	DET
ijassa-1958	250	6	history	history	NOUN
ijassa-1958	250	7	of	of	ADP
ijassa-1958	250	8	distance	distance	NOUN
ijassa-1958	250	9	geometry	geometry	NOUN
ijassa-1958	250	10	,	,	PUNCT
ijassa-1958	250	11	international	international	ADJ
ijassa-1958	250	12	transactions	transaction	NOUN
ijassa-1958	250	13	in	in	ADP
ijassa-1958	250	14	operational	operational	ADJ
ijassa-1958	250	15	research	research	NOUN
ijassa-1958	250	16	,	,	PUNCT
ijassa-1958	250	17	23(5	23(5	NUM
ijassa-1958	250	18	)	)	PUNCT
ijassa-1958	250	19	,	,	PUNCT
ijassa-1958	250	20	897–920	897–920	NUM
ijassa-1958	250	21	.	.	PUNCT
ijassa-1958	250	22	doi	doi	NOUN
ijassa-1958	250	23	:	:	PUNCT
ijassa-1958	250	24	10.1111	10.1111	NUM
ijassa-1958	250	25	/	/	SYM
ijassa-1958	250	26	itor.12170	itor.12170	NOUN
ijassa-1958	250	27	.	.	NOUN
ijassa-1958	250	28	9	9	NUM
ijassa-1958	250	29	.	.	X
ijassa-1958	250	30	pleasants	pleasants	PROPN
ijassa-1958	250	31	,	,	PUNCT
ijassa-1958	250	32	p.	p.	PROPN
ijassa-1958	250	33	a.	a.	PROPN
ijassa-1958	250	34	b.	b.	PROPN
ijassa-1958	250	35	(	(	PUNCT
ijassa-1958	250	36	2000	2000	NUM
ijassa-1958	250	37	)	)	PUNCT
ijassa-1958	250	38	.	.	PUNCT
ijassa-1958	251	1	designer	designer	NOUN
ijassa-1958	251	2	quasicrystals	quasicrystal	NOUN
ijassa-1958	251	3	:	:	PUNCT
ijassa-1958	251	4	cut	cut	VERB
ijassa-1958	251	5	-	-	PUNCT
ijassa-1958	251	6	and	and	CCONJ
ijassa-1958	251	7	-	-	PUNCT
ijassa-1958	251	8	project	project	NOUN
ijassa-1958	251	9	sets	set	NOUN
ijassa-1958	251	10	with	with	ADP
ijassa-1958	251	11	preassigned	preassigned	ADJ
ijassa-1958	251	12	properties	property	NOUN
ijassa-1958	251	13	,	,	PUNCT
ijassa-1958	251	14	directions	direction	NOUN
ijassa-1958	251	15	in	in	ADP
ijassa-1958	251	16	mathematical	mathematical	ADJ
ijassa-1958	251	17	quasicrystals	quasicrystal	NOUN
ijassa-1958	251	18	,	,	PUNCT
ijassa-1958	251	19	95–141	95–141	NUM
ijassa-1958	251	20	.	.	NOUN
ijassa-1958	251	21	10	10	NUM
ijassa-1958	251	22	.	.	PUNCT
ijassa-1958	252	1	rubin	rubin	PROPN
ijassa-1958	252	2	,	,	PUNCT
ijassa-1958	252	3	m.	m.	NOUN
ijassa-1958	252	4	(	(	PUNCT
ijassa-1958	252	5	1974	1974	NUM
ijassa-1958	252	6	)	)	PUNCT
ijassa-1958	252	7	.	.	PUNCT
ijassa-1958	253	1	theories	theory	NOUN
ijassa-1958	253	2	of	of	ADP
ijassa-1958	253	3	linear	linear	ADJ
ijassa-1958	253	4	order	order	NOUN
ijassa-1958	253	5	,	,	PUNCT
ijassa-1958	253	6	israel	israel	PROPN
ijassa-1958	253	7	journal	journal	PROPN
ijassa-1958	253	8	of	of	ADP
ijassa-1958	253	9	mathematics	mathematic	NOUN
ijassa-1958	253	10	,	,	PUNCT
ijassa-1958	253	11	17(4	17(4	NUM
ijassa-1958	253	12	)	)	PUNCT
ijassa-1958	253	13	,	,	PUNCT
ijassa-1958	253	14	392	392	NUM
ijassa-1958	253	15	–	–	PUNCT
ijassa-1958	253	16	443	443	NUM
ijassa-1958	253	17	.	.	PUNCT
ijassa-1958	253	18	doi	doi	NOUN
ijassa-1958	253	19	:	:	PUNCT
ijassa-1958	253	20	10.1007	10.1007	NUM
ijassa-1958	253	21	/	/	SYM
ijassa-1958	253	22	bf02757141	bf02757141	NOUN
ijassa-1958	253	23	.	.	PUNCT
ijassa-1958	254	1	11	11	NUM
ijassa-1958	254	2	.	.	PUNCT
ijassa-1958	255	1	zermelo	zermelo	PROPN
ijassa-1958	255	2	,	,	PUNCT
ijassa-1958	255	3	e.	e.	PROPN
ijassa-1958	255	4	(	(	PUNCT
ijassa-1958	255	5	1908	1908	NUM
ijassa-1958	255	6	)	)	PUNCT
ijassa-1958	255	7	.	.	PUNCT
ijassa-1958	256	1	untersuchungen	untersuchungen	PROPN
ijassa-1958	256	2	über	über	PROPN
ijassa-1958	256	3	die	die	VERB
ijassa-1958	256	4	grundlagen	grundlagen	PROPN
ijassa-1958	256	5	der	der	PROPN
ijassa-1958	256	6	mengenlehre	mengenlehre	PROPN
ijassa-1958	256	7	.	.	PUNCT
ijassa-1958	257	1	i.	i.	PROPN
ijassa-1958	258	1	[	[	X
ijassa-1958	258	2	investigations	investigation	NOUN
ijassa-1958	258	3	into	into	ADP
ijassa-1958	258	4	the	the	DET
ijassa-1958	258	5	foundations	foundation	NOUN
ijassa-1958	258	6	of	of	ADP
ijassa-1958	258	7	set	set	NOUN
ijassa-1958	258	8	theory	theory	NOUN
ijassa-1958	258	9	.	.	PUNCT
ijassa-1958	259	1	i	i	PRON
ijassa-1958	259	2	]	]	X
ijassa-1958	259	3	,	,	PUNCT
ijassa-1958	259	4	mathematische	mathematische	PROPN
ijassa-1958	259	5	annalen	annalen	PROPN
ijassa-1958	259	6	,	,	PUNCT
ijassa-1958	259	7	65(2	65(2	NOUN
ijassa-1958	259	8	)	)	PUNCT
ijassa-1958	259	9	,	,	PUNCT
ijassa-1958	259	10	261–281	261–281	NUM
ijassa-1958	259	11	.	.	PUNCT
ijassa-1958	260	1	doi	doi	NOUN
ijassa-1958	260	2	:	:	PUNCT
ijassa-1958	260	3	10.1007	10.1007	NUM
ijassa-1958	260	4	/	/	SYM
ijassa-1958	260	5	bf01449999	bf01449999	NOUN
ijassa-1958	261	1	[	[	X
ijassa-1958	261	2	in	in	ADP
ijassa-1958	261	3	german	german	NOUN
ijassa-1958	261	4	]	]	X
ijassa-1958	261	5	.	.	PUNCT
ijassa-1958	262	1	12	12	NUM
ijassa-1958	262	2	.	.	PUNCT
ijassa-1958	263	1	zhukovskaya	zhukovskaya	PROPN
ijassa-1958	263	2	t.	t.	PROPN
ijassa-1958	263	3	v.	v.	PROPN
ijassa-1958	263	4	,	,	PUNCT
ijassa-1958	263	5	zhukovskiy	zhukovskiy	PROPN
ijassa-1958	263	6	e.	e.	PROPN
ijassa-1958	263	7	s.	s.	PROPN
ijassa-1958	263	8	,	,	PUNCT
ijassa-1958	263	9	&	&	CCONJ
ijassa-1958	263	10	serova	serova	PROPN
ijassa-1958	263	11	i.	i.	PROPN
ijassa-1958	263	12	d.	d.	PROPN
ijassa-1958	263	13	(	(	PUNCT
ijassa-1958	263	14	2020	2020	NUM
ijassa-1958	263	15	)	)	PUNCT
ijassa-1958	263	16	some	some	DET
ijassa-1958	263	17	questions	question	NOUN
ijassa-1958	263	18	of	of	ADP
ijassa-1958	263	19	the	the	DET
ijassa-1958	263	20	analysis	analysis	NOUN
ijassa-1958	263	21	of	of	ADP
ijassa-1958	263	22	mappings	mapping	NOUN
ijassa-1958	263	23	of	of	ADP
ijassa-1958	263	24	metric	metric	ADJ
ijassa-1958	263	25	and	and	CCONJ
ijassa-1958	263	26	partially	partially	ADV
ijassa-1958	263	27	ordered	order	VERB
ijassa-1958	263	28	spaces	space	NOUN
ijassa-1958	263	29	,	,	PUNCT
ijassa-1958	263	30	russian	russian	ADJ
ijassa-1958	263	31	universities	university	NOUN
ijassa-1958	263	32	reports	report	VERB
ijassa-1958	263	33	.	.	PUNCT
ijassa-1958	264	1	mathematics	mathematic	NOUN
ijassa-1958	264	2	,	,	PUNCT
ijassa-1958	264	3	25(132	25(132	NOUN
ijassa-1958	264	4	)	)	PUNCT
ijassa-1958	264	5	,	,	PUNCT
ijassa-1958	264	6	345–358	345–358	NUM
ijassa-1958	264	7	.	.	PUNCT
ijassa-1958	264	8	doi	doi	NOUN
ijassa-1958	264	9	:	:	PUNCT
ijassa-1958	264	10	10.20310/2686	10.20310/2686	NUM
ijassa-1958	264	11	-	-	PUNCT
ijassa-1958	264	12	9667	9667	NUM
ijassa-1958	264	13	-	-	PUNCT
ijassa-1958	264	14	2020	2020	NUM
ijassa-1958	264	15	-	-	PUNCT
ijassa-1958	264	16	25	25	NUM
ijassa-1958	264	17	-	-	PUNCT
ijassa-1958	264	18	132	132	NUM
ijassa-1958	264	19	-	-	NUM
ijassa-1958	264	20	345358	345358	NUM
ijassa-1958	264	21	.	.	PUNCT
ijassa-1958	265	1	copyright	copyright	NOUN
ijassa-1958	265	2	©	©	PROPN
ijassa-1958	265	3	2024	2024	NUM
ijassa-1958	265	4	assa	assa	NOUN
ijassa-1958	265	5	.	.	PUNCT
ijassa-1958	266	1	adv	adv	PROPN
ijassa-1958	266	2	syst	syst	PROPN
ijassa-1958	266	3	sci	sci	PROPN
ijassa-1958	266	4	appl	appl	PROPN
ijassa-1958	266	5	(	(	PUNCT
ijassa-1958	266	6	2024	2024	NUM
ijassa-1958	266	7	)	)	PUNCT
ijassa-1958	266	8	introduction	introduction	NOUN
ijassa-1958	266	9	basic	basic	ADJ
ijassa-1958	266	10	definitions	definition	NOUN
ijassa-1958	266	11	incompatibility	incompatibility	NOUN
ijassa-1958	266	12	with	with	ADP
ijassa-1958	266	13	order	order	NOUN
ijassa-1958	266	14	of	of	ADP
ijassa-1958	266	15	classic	classic	ADJ
ijassa-1958	266	16	multidimensional	multidimensional	ADJ
ijassa-1958	266	17	spaces	space	NOUN
ijassa-1958	266	18	constructions	construction	NOUN
ijassa-1958	266	19	of	of	ADP
ijassa-1958	266	20	order	order	NOUN
ijassa-1958	266	21	-	-	PUNCT
ijassa-1958	266	22	compatible	compatible	ADJ
ijassa-1958	266	23	metric	metric	ADJ
ijassa-1958	266	24	spaces	space	NOUN
ijassa-1958	266	25	acknowledgements	acknowledgement	NOUN