id	sid	tid	token	lemma	pos
ejpam-5571	1	1	european	european	PROPN
ejpam-5571	1	2	journal	journal	PROPN
ejpam-5571	1	3	of	of	ADP
ejpam-5571	1	4	pure	pure	ADJ
ejpam-5571	1	5	and	and	CCONJ
ejpam-5571	1	6	applied	apply	VERB
ejpam-5571	1	7	mathematics	mathematic	NOUN
ejpam-5571	1	8	vol	vol	NOUN
ejpam-5571	1	9	.	.	PROPN
ejpam-5571	2	1	17	17	NUM
ejpam-5571	2	2	,	,	PUNCT
ejpam-5571	2	3	no	no	INTJ
ejpam-5571	2	4	.	.	NOUN
ejpam-5571	2	5	4	4	NUM
ejpam-5571	2	6	,	,	PUNCT
ejpam-5571	2	7	2024	2024	NUM
ejpam-5571	2	8	,	,	PUNCT
ejpam-5571	2	9	3932	3932	NUM
ejpam-5571	2	10	-	-	SYM
ejpam-5571	2	11	3944	3944	NUM
ejpam-5571	2	12	issn	issn	PROPN
ejpam-5571	2	13	1307	1307	NUM
ejpam-5571	2	14	-	-	SYM
ejpam-5571	2	15	5543	5543	NUM
ejpam-5571	2	16	–	–	PUNCT
ejpam-5571	2	17	ejpam.com	ejpam.com	X
ejpam-5571	2	18	published	publish	VERB
ejpam-5571	2	19	by	by	ADP
ejpam-5571	2	20	new	new	PROPN
ejpam-5571	2	21	york	york	PROPN
ejpam-5571	2	22	business	business	NOUN
ejpam-5571	2	23	global	global	ADJ
ejpam-5571	2	24	some	some	DET
ejpam-5571	2	25	rigidity	rigidity	NOUN
ejpam-5571	2	26	theorems	theorem	NOUN
ejpam-5571	2	27	of	of	ADP
ejpam-5571	2	28	closed	closed	ADJ
ejpam-5571	2	29	geodesic	geodesic	ADJ
ejpam-5571	2	30	polygons	polygon	NOUN
ejpam-5571	2	31	and	and	CCONJ
ejpam-5571	2	32	spherical	spherical	ADJ
ejpam-5571	2	33	curves	curve	NOUN
ejpam-5571	2	34	in	in	ADP
ejpam-5571	2	35	metric	metric	ADJ
ejpam-5571	2	36	spaces	space	NOUN
ejpam-5571	2	37	with	with	ADP
ejpam-5571	2	38	curvature	curvature	NOUN
ejpam-5571	2	39	bounded	bound	VERB
ejpam-5571	2	40	below	below	ADP
ejpam-5571	2	41	chanpen	chanpen	ADJ
ejpam-5571	2	42	phokaew1	phokaew1	PROPN
ejpam-5571	2	43	,	,	PUNCT
ejpam-5571	2	44	areeyuth	areeyuth	NOUN
ejpam-5571	2	45	sama	sama	NOUN
ejpam-5571	2	46	-	-	PUNCT
ejpam-5571	2	47	ae1,∗	ae1,∗	PROPN
ejpam-5571	2	48	1	1	NUM
ejpam-5571	2	49	department	department	NOUN
ejpam-5571	2	50	of	of	ADP
ejpam-5571	2	51	mathematics	mathematic	NOUN
ejpam-5571	2	52	and	and	CCONJ
ejpam-5571	2	53	computer	computer	NOUN
ejpam-5571	2	54	science	science	NOUN
ejpam-5571	2	55	,	,	PUNCT
ejpam-5571	2	56	faculty	faculty	NOUN
ejpam-5571	2	57	of	of	ADP
ejpam-5571	2	58	science	science	NOUN
ejpam-5571	2	59	and	and	CCONJ
ejpam-5571	2	60	technology	technology	NOUN
ejpam-5571	2	61	,	,	PUNCT
ejpam-5571	2	62	prince	prince	NOUN
ejpam-5571	2	63	of	of	ADP
ejpam-5571	2	64	songkla	songkla	PROPN
ejpam-5571	2	65	university	university	PROPN
ejpam-5571	2	66	,	,	PUNCT
ejpam-5571	2	67	pattani	pattani	NOUN
ejpam-5571	2	68	campus	campus	NOUN
ejpam-5571	2	69	,	,	PUNCT
ejpam-5571	2	70	pattani	pattani	NOUN
ejpam-5571	2	71	,	,	PUNCT
ejpam-5571	2	72	94000	94000	NUM
ejpam-5571	2	73	,	,	PUNCT
ejpam-5571	2	74	thailand	thailand	PROPN
ejpam-5571	2	75	abstract	abstract	PROPN
ejpam-5571	2	76	.	.	PUNCT
ejpam-5571	3	1	this	this	DET
ejpam-5571	3	2	paper	paper	NOUN
ejpam-5571	3	3	examines	examine	VERB
ejpam-5571	3	4	characterizations	characterization	NOUN
ejpam-5571	3	5	of	of	ADP
ejpam-5571	3	6	closed	closed	ADJ
ejpam-5571	3	7	curves	curve	NOUN
ejpam-5571	3	8	in	in	ADP
ejpam-5571	3	9	a	a	DET
ejpam-5571	3	10	geodesic	geodesic	ADJ
ejpam-5571	3	11	metric	metric	ADJ
ejpam-5571	3	12	space	space	NOUN
ejpam-5571	3	13	with	with	ADP
ejpam-5571	3	14	curvature	curvature	NOUN
ejpam-5571	3	15	bounded	bound	VERB
ejpam-5571	3	16	below	below	ADV
ejpam-5571	3	17	,	,	PUNCT
ejpam-5571	3	18	including	include	VERB
ejpam-5571	3	19	closed	close	VERB
ejpam-5571	3	20	geodesic	geodesic	ADJ
ejpam-5571	3	21	polygons	polygon	NOUN
ejpam-5571	3	22	and	and	CCONJ
ejpam-5571	3	23	closed	close	VERB
ejpam-5571	3	24	spherical	spherical	ADJ
ejpam-5571	3	25	curves	curve	NOUN
ejpam-5571	3	26	bounding	bounding	NOUN
ejpam-5571	3	27	surfaces	surface	NOUN
ejpam-5571	3	28	isometric	isometric	ADJ
ejpam-5571	3	29	to	to	PART
ejpam-5571	3	30	convex	convex	VERB
ejpam-5571	3	31	polygons	polygon	NOUN
ejpam-5571	3	32	and	and	CCONJ
ejpam-5571	3	33	circles	circle	NOUN
ejpam-5571	3	34	in	in	ADP
ejpam-5571	3	35	the	the	DET
ejpam-5571	3	36	model	model	NOUN
ejpam-5571	3	37	space	space	NOUN
ejpam-5571	3	38	.	.	PUNCT
ejpam-5571	4	1	2020	2020	NUM
ejpam-5571	4	2	mathematics	mathematic	NOUN
ejpam-5571	4	3	subject	subject	NOUN
ejpam-5571	4	4	classifications	classification	NOUN
ejpam-5571	4	5	:	:	PUNCT
ejpam-5571	4	6	51k05	51k05	NUM
ejpam-5571	4	7	,	,	PUNCT
ejpam-5571	4	8	54e35	54e35	NUM
ejpam-5571	4	9	,	,	PUNCT
ejpam-5571	4	10	54e40	54e40	NUM
ejpam-5571	4	11	key	key	ADJ
ejpam-5571	4	12	words	word	NOUN
ejpam-5571	4	13	and	and	CCONJ
ejpam-5571	4	14	phrases	phrase	NOUN
ejpam-5571	4	15	:	:	PUNCT
ejpam-5571	4	16	spaces	space	NOUN
ejpam-5571	4	17	with	with	ADP
ejpam-5571	4	18	curvature	curvature	NOUN
ejpam-5571	4	19	bounded	bound	VERB
ejpam-5571	4	20	below	below	ADV
ejpam-5571	4	21	,	,	PUNCT
ejpam-5571	4	22	convex	convex	NOUN
ejpam-5571	4	23	hull	hull	NOUN
ejpam-5571	4	24	,	,	PUNCT
ejpam-5571	4	25	closed	close	VERB
ejpam-5571	4	26	geodesic	geodesic	ADJ
ejpam-5571	4	27	polygon	polygon	NOUN
ejpam-5571	4	28	,	,	PUNCT
ejpam-5571	4	29	spherical	spherical	ADJ
ejpam-5571	4	30	curve	curve	NOUN
ejpam-5571	4	31	,	,	PUNCT
ejpam-5571	4	32	isometry	isometry	PROPN
ejpam-5571	4	33	1	1	NUM
ejpam-5571	4	34	.	.	PUNCT
ejpam-5571	4	35	introduction	introduction	NOUN
ejpam-5571	4	36	and	and	CCONJ
ejpam-5571	4	37	preliminaries	preliminary	NOUN
ejpam-5571	4	38	in	in	ADP
ejpam-5571	4	39	this	this	DET
ejpam-5571	4	40	paper	paper	NOUN
ejpam-5571	4	41	,	,	PUNCT
ejpam-5571	4	42	we	we	PRON
ejpam-5571	4	43	investigate	investigate	VERB
ejpam-5571	4	44	the	the	DET
ejpam-5571	4	45	characterizations	characterization	NOUN
ejpam-5571	4	46	of	of	ADP
ejpam-5571	4	47	closed	closed	ADJ
ejpam-5571	4	48	curves	curve	NOUN
ejpam-5571	4	49	in	in	ADP
ejpam-5571	4	50	a	a	DET
ejpam-5571	4	51	geodesic	geodesic	ADJ
ejpam-5571	4	52	metric	metric	ADJ
ejpam-5571	4	53	space	space	NOUN
ejpam-5571	4	54	with	with	ADP
ejpam-5571	4	55	curvature	curvature	NOUN
ejpam-5571	4	56	bound	bind	VERB
ejpam-5571	4	57	below	below	ADV
ejpam-5571	4	58	in	in	ADP
ejpam-5571	4	59	the	the	DET
ejpam-5571	4	60	sense	sense	NOUN
ejpam-5571	4	61	of	of	ADP
ejpam-5571	4	62	alexandrov	alexandrov	PROPN
ejpam-5571	4	63	.	.	PUNCT
ejpam-5571	5	1	we	we	PRON
ejpam-5571	5	2	discuss	discuss	VERB
ejpam-5571	5	3	the	the	DET
ejpam-5571	5	4	characterizations	characterization	NOUN
ejpam-5571	5	5	for	for	ADP
ejpam-5571	5	6	closed	closed	ADJ
ejpam-5571	5	7	geodesic	geodesic	ADJ
ejpam-5571	5	8	polygons	polygon	NOUN
ejpam-5571	5	9	in	in	ADP
ejpam-5571	5	10	the	the	DET
ejpam-5571	5	11	space	space	NOUN
ejpam-5571	5	12	that	that	PRON
ejpam-5571	5	13	bound	bind	VERB
ejpam-5571	5	14	surfaces	surface	NOUN
ejpam-5571	5	15	isometric	isometric	ADJ
ejpam-5571	5	16	to	to	ADP
ejpam-5571	5	17	regions	region	NOUN
ejpam-5571	5	18	bounded	bound	VERB
ejpam-5571	5	19	by	by	ADP
ejpam-5571	5	20	closed	close	VERB
ejpam-5571	5	21	convex	convex	NOUN
ejpam-5571	5	22	polygons	polygon	NOUN
ejpam-5571	5	23	in	in	ADP
ejpam-5571	5	24	the	the	DET
ejpam-5571	5	25	model	model	NOUN
ejpam-5571	5	26	space	space	NOUN
ejpam-5571	5	27	rk	rk	PROPN
ejpam-5571	5	28	,	,	PUNCT
ejpam-5571	5	29	and	and	CCONJ
ejpam-5571	5	30	we	we	PRON
ejpam-5571	5	31	also	also	ADV
ejpam-5571	5	32	look	look	VERB
ejpam-5571	5	33	at	at	ADP
ejpam-5571	5	34	the	the	DET
ejpam-5571	5	35	characterizations	characterization	NOUN
ejpam-5571	5	36	for	for	ADP
ejpam-5571	5	37	closed	closed	ADJ
ejpam-5571	5	38	spherical	spherical	ADJ
ejpam-5571	5	39	curves	curve	NOUN
ejpam-5571	5	40	in	in	ADP
ejpam-5571	5	41	the	the	DET
ejpam-5571	5	42	space	space	NOUN
ejpam-5571	5	43	that	that	PRON
ejpam-5571	5	44	bound	bind	VERB
ejpam-5571	5	45	surfaces	surface	NOUN
ejpam-5571	5	46	isometric	isometric	ADJ
ejpam-5571	5	47	to	to	ADP
ejpam-5571	5	48	regions	region	NOUN
ejpam-5571	5	49	bounded	bound	VERB
ejpam-5571	5	50	by	by	ADP
ejpam-5571	5	51	circles	circle	NOUN
ejpam-5571	5	52	in	in	ADP
ejpam-5571	5	53	the	the	DET
ejpam-5571	5	54	model	model	NOUN
ejpam-5571	5	55	space	space	NOUN
ejpam-5571	5	56	rk	rk	NOUN
ejpam-5571	5	57	with	with	ADP
ejpam-5571	5	58	the	the	DET
ejpam-5571	5	59	same	same	ADJ
ejpam-5571	5	60	perimeter	perimeter	NOUN
ejpam-5571	5	61	.	.	PUNCT
ejpam-5571	6	1	alexandrov	alexandrov	PROPN
ejpam-5571	7	1	[	[	X
ejpam-5571	7	2	1–5	1–5	NUM
ejpam-5571	7	3	,	,	PUNCT
ejpam-5571	7	4	13	13	NUM
ejpam-5571	7	5	]	]	PUNCT
ejpam-5571	7	6	introduced	introduce	VERB
ejpam-5571	7	7	lower	low	ADJ
ejpam-5571	7	8	and	and	CCONJ
ejpam-5571	7	9	upper	upper	ADJ
ejpam-5571	7	10	curvature	curvature	NOUN
ejpam-5571	7	11	bounds	bound	NOUN
ejpam-5571	7	12	on	on	ADP
ejpam-5571	7	13	metric	metric	ADJ
ejpam-5571	7	14	spaces	space	NOUN
ejpam-5571	7	15	without	without	ADP
ejpam-5571	7	16	riemannian	riemannian	ADJ
ejpam-5571	7	17	structure	structure	NOUN
ejpam-5571	7	18	,	,	PUNCT
ejpam-5571	7	19	which	which	PRON
ejpam-5571	7	20	extended	extend	VERB
ejpam-5571	7	21	concepts	concept	NOUN
ejpam-5571	7	22	to	to	ADP
ejpam-5571	7	23	arbitrary	arbitrary	ADJ
ejpam-5571	7	24	spaces	space	NOUN
ejpam-5571	7	25	.	.	PUNCT
ejpam-5571	8	1	this	this	PRON
ejpam-5571	8	2	led	lead	VERB
ejpam-5571	8	3	to	to	ADP
ejpam-5571	8	4	the	the	DET
ejpam-5571	8	5	theorems	theorem	NOUN
ejpam-5571	8	6	of	of	ADP
ejpam-5571	8	7	riemannian	riemannian	ADJ
ejpam-5571	8	8	geometry	geometry	NOUN
ejpam-5571	8	9	,	,	PUNCT
ejpam-5571	8	10	which	which	PRON
ejpam-5571	8	11	defined	define	VERB
ejpam-5571	8	12	bounded	bounded	ADJ
ejpam-5571	8	13	curvature	curvature	NOUN
ejpam-5571	8	14	as	as	ADP
ejpam-5571	8	15	bounded	bounded	ADJ
ejpam-5571	8	16	sectional	sectional	ADJ
ejpam-5571	8	17	curvature	curvature	NOUN
ejpam-5571	8	18	.	.	PUNCT
ejpam-5571	9	1	examples	example	NOUN
ejpam-5571	9	2	include	include	VERB
ejpam-5571	9	3	the	the	DET
ejpam-5571	9	4	remannian	remannian	NOUN
ejpam-5571	9	5	manifolds	manifold	NOUN
ejpam-5571	9	6	with	with	ADP
ejpam-5571	9	7	sectional	sectional	ADJ
ejpam-5571	9	8	curvature	curvature	NOUN
ejpam-5571	9	9	are	be	AUX
ejpam-5571	9	10	not	not	PART
ejpam-5571	9	11	less	less	ADJ
ejpam-5571	9	12	than	than	SCONJ
ejpam-5571	9	13	k	k	PROPN
ejpam-5571	9	14	and	and	CCONJ
ejpam-5571	9	15	its	its	PRON
ejpam-5571	9	16	convex	convex	NOUN
ejpam-5571	9	17	subset	subset	NOUN
ejpam-5571	9	18	,	,	PUNCT
ejpam-5571	9	19	and	and	CCONJ
ejpam-5571	9	20	hilbert	hilbert	NOUN
ejpam-5571	9	21	spaces	space	NOUN
ejpam-5571	9	22	.	.	PUNCT
ejpam-5571	10	1	let	let	VERB
ejpam-5571	10	2	(	(	PUNCT
ejpam-5571	10	3	x	x	NOUN
ejpam-5571	10	4	,	,	PUNCT
ejpam-5571	10	5	d	d	NOUN
ejpam-5571	10	6	)	)	PUNCT
ejpam-5571	10	7	be	be	AUX
ejpam-5571	10	8	a	a	DET
ejpam-5571	10	9	metric	metric	ADJ
ejpam-5571	10	10	space	space	NOUN
ejpam-5571	10	11	and	and	CCONJ
ejpam-5571	10	12	γ	γ	X
ejpam-5571	10	13	:	:	PUNCT
ejpam-5571	11	1	[	[	X
ejpam-5571	11	2	a	a	X
ejpam-5571	11	3	,	,	PUNCT
ejpam-5571	11	4	b	b	NOUN
ejpam-5571	11	5	]	]	X
ejpam-5571	11	6	→	→	PUNCT
ejpam-5571	11	7	x	x	X
ejpam-5571	11	8	a	a	DET
ejpam-5571	11	9	curve	curve	NOUN
ejpam-5571	11	10	in	in	ADP
ejpam-5571	11	11	x.	x.	NOUN
ejpam-5571	11	12	the	the	DET
ejpam-5571	11	13	length	length	NOUN
ejpam-5571	11	14	ℓ(γ	ℓ(γ	PROPN
ejpam-5571	11	15	)	)	PUNCT
ejpam-5571	11	16	of	of	ADP
ejpam-5571	11	17	γ	γ	PROPN
ejpam-5571	11	18	is	be	AUX
ejpam-5571	11	19	defined	define	VERB
ejpam-5571	11	20	by	by	ADP
ejpam-5571	11	21	ℓ(γ	ℓ(γ	NOUN
ejpam-5571	11	22	)	)	PUNCT
ejpam-5571	12	1	=	=	SYM
ejpam-5571	12	2	sup	sup	NUM
ejpam-5571	12	3	k∑	k∑	PROPN
ejpam-5571	12	4	i=1	i=1	PROPN
ejpam-5571	12	5	d(γ(ti−1	d(γ(ti−1	PROPN
ejpam-5571	12	6	)	)	PUNCT
ejpam-5571	12	7	,	,	PUNCT
ejpam-5571	12	8	γ(ti	γ(ti	NOUN
ejpam-5571	12	9	)	)	PUNCT
ejpam-5571	12	10	)	)	PUNCT
ejpam-5571	12	11	,	,	PUNCT
ejpam-5571	12	12	∗corresponding	∗corresponde	VERB
ejpam-5571	12	13	author	author	NOUN
ejpam-5571	12	14	.	.	PUNCT
ejpam-5571	13	1	doi	doi	NOUN
ejpam-5571	13	2	:	:	PUNCT
ejpam-5571	13	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5571	https://doi.org/10.29020/nybg.ejpam.v17i4.5571	PROPN
ejpam-5571	13	4	email	email	NOUN
ejpam-5571	13	5	addresses	address	VERB
ejpam-5571	13	6	:	:	PUNCT
ejpam-5571	13	7	chunpen.t@psu.ac.th	chunpen.t@psu.ac.th	INTJ
ejpam-5571	13	8	(	(	PUNCT
ejpam-5571	13	9	c.	c.	PROPN
ejpam-5571	13	10	phokaew	phokaew	PROPN
ejpam-5571	13	11	)	)	PUNCT
ejpam-5571	13	12	,	,	PUNCT
ejpam-5571	13	13	areeyuth.s@psu.ac.th	areeyuth.s@psu.ac.th	X
ejpam-5571	13	14	(	(	PUNCT
ejpam-5571	13	15	a.	a.	PROPN
ejpam-5571	13	16	sama	sama	PROPN
ejpam-5571	13	17	-	-	PUNCT
ejpam-5571	13	18	ae	ae	PROPN
ejpam-5571	13	19	)	)	PUNCT
ejpam-5571	13	20	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5571	13	21	3932	3932	NUM
ejpam-5571	14	1	copyright	copyright	NOUN
ejpam-5571	14	2	:	:	PUNCT
ejpam-5571	14	3	©	©	PROPN
ejpam-5571	14	4	2024	2024	NUM
ejpam-5571	14	5	the	the	DET
ejpam-5571	14	6	author(s	author(s	NOUN
ejpam-5571	14	7	)	)	PUNCT
ejpam-5571	14	8	.	.	PUNCT
ejpam-5571	15	1	(	(	PUNCT
ejpam-5571	15	2	cc	cc	NOUN
ejpam-5571	15	3	by	by	ADP
ejpam-5571	15	4	-	-	PUNCT
ejpam-5571	15	5	nc	nc	PROPN
ejpam-5571	15	6	4.0	4.0	NUM
ejpam-5571	15	7	)	)	PUNCT
ejpam-5571	15	8	c.	c.	NOUN
ejpam-5571	15	9	phokaew	phokaew	PROPN
ejpam-5571	15	10	,	,	PUNCT
ejpam-5571	15	11	a.	a.	PROPN
ejpam-5571	15	12	sama	sama	PROPN
ejpam-5571	15	13	-	-	PUNCT
ejpam-5571	15	14	ae	ae	PROPN
ejpam-5571	15	15	,	,	PUNCT
ejpam-5571	15	16	/	/	SYM
ejpam-5571	15	17	eur	eur	NOUN
ejpam-5571	15	18	.	.	PUNCT
ejpam-5571	16	1	j.	j.	PROPN
ejpam-5571	16	2	pure	pure	PROPN
ejpam-5571	16	3	appl	appl	PROPN
ejpam-5571	16	4	.	.	PROPN
ejpam-5571	16	5	math	math	PROPN
ejpam-5571	16	6	,	,	PUNCT
ejpam-5571	16	7	17	17	NUM
ejpam-5571	16	8	(	(	PUNCT
ejpam-5571	16	9	4	4	NUM
ejpam-5571	16	10	)	)	PUNCT
ejpam-5571	16	11	(	(	PUNCT
ejpam-5571	16	12	2024	2024	NUM
ejpam-5571	16	13	)	)	PUNCT
ejpam-5571	16	14	,	,	PUNCT
ejpam-5571	16	15	3932	3932	NUM
ejpam-5571	16	16	-	-	SYM
ejpam-5571	16	17	3944	3944	NUM
ejpam-5571	16	18	3933	3933	NUM
ejpam-5571	16	19	where	where	SCONJ
ejpam-5571	16	20	the	the	DET
ejpam-5571	16	21	supremum	supremum	NOUN
ejpam-5571	16	22	is	be	AUX
ejpam-5571	16	23	taken	take	VERB
ejpam-5571	16	24	over	over	ADP
ejpam-5571	16	25	all	all	DET
ejpam-5571	16	26	partitions	partition	NOUN
ejpam-5571	17	1	a	a	DET
ejpam-5571	17	2	=	=	SYM
ejpam-5571	17	3	t0	t0	PROPN
ejpam-5571	17	4	<	<	X
ejpam-5571	17	5	t1	t1	NOUN
ejpam-5571	17	6	<	<	X
ejpam-5571	17	7	·	·	PUNCT
ejpam-5571	17	8	·	·	PUNCT
ejpam-5571	17	9	·	·	PUNCT
ejpam-5571	17	10	<	<	X
ejpam-5571	17	11	tk	tk	PROPN
ejpam-5571	17	12	=	=	SYM
ejpam-5571	17	13	b	b	PROPN
ejpam-5571	17	14	of	of	ADP
ejpam-5571	17	15	[	[	X
ejpam-5571	17	16	a	a	X
ejpam-5571	17	17	,	,	PUNCT
ejpam-5571	17	18	b	b	NOUN
ejpam-5571	17	19	]	]	X
ejpam-5571	17	20	.	.	PUNCT
ejpam-5571	18	1	hence	hence	ADV
ejpam-5571	18	2	,	,	PUNCT
ejpam-5571	18	3	d∗(x	d∗(x	PROPN
ejpam-5571	18	4	,	,	PUNCT
ejpam-5571	18	5	y	y	PROPN
ejpam-5571	18	6	)	)	PUNCT
ejpam-5571	18	7	:	:	PUNCT
ejpam-5571	19	1	=	=	PUNCT
ejpam-5571	19	2	inf{ℓ(γ)|	inf{ℓ(γ)|	PROPN
ejpam-5571	19	3	γ	γ	PROPN
ejpam-5571	19	4	is	be	AUX
ejpam-5571	19	5	a	a	DET
ejpam-5571	19	6	curve	curve	NOUN
ejpam-5571	19	7	from	from	ADP
ejpam-5571	19	8	x	x	PUNCT
ejpam-5571	19	9	to	to	ADP
ejpam-5571	19	10	y	y	PROPN
ejpam-5571	19	11	}	}	PUNCT
ejpam-5571	19	12	,	,	PUNCT
ejpam-5571	19	13	for	for	ADP
ejpam-5571	19	14	all	all	DET
ejpam-5571	19	15	x	x	SYM
ejpam-5571	19	16	and	and	CCONJ
ejpam-5571	19	17	y	y	PROPN
ejpam-5571	19	18	∈	∈	PROPN
ejpam-5571	19	19	x	x	X
ejpam-5571	19	20	,	,	PUNCT
ejpam-5571	19	21	defines	define	VERB
ejpam-5571	19	22	a	a	DET
ejpam-5571	19	23	metric	metric	NOUN
ejpam-5571	19	24	on	on	ADP
ejpam-5571	19	25	x	x	PUNCT
ejpam-5571	19	26	with	with	ADP
ejpam-5571	19	27	distance	distance	NOUN
ejpam-5571	19	28	values	value	NOUN
ejpam-5571	19	29	in	in	ADP
ejpam-5571	19	30	[	[	X
ejpam-5571	19	31	0,∞	0,∞	NOUN
ejpam-5571	19	32	]	]	PUNCT
ejpam-5571	19	33	.	.	PUNCT
ejpam-5571	20	1	if	if	SCONJ
ejpam-5571	20	2	d	d	PROPN
ejpam-5571	20	3	=	=	SYM
ejpam-5571	20	4	d∗	d∗	PROPN
ejpam-5571	20	5	,	,	PUNCT
ejpam-5571	20	6	then	then	ADV
ejpam-5571	20	7	(	(	PUNCT
ejpam-5571	20	8	x	x	X
ejpam-5571	20	9	,	,	PUNCT
ejpam-5571	20	10	d	d	NOUN
ejpam-5571	20	11	)	)	PUNCT
ejpam-5571	20	12	is	be	AUX
ejpam-5571	20	13	called	call	VERB
ejpam-5571	20	14	a	a	DET
ejpam-5571	20	15	length	length	NOUN
ejpam-5571	20	16	space	space	NOUN
ejpam-5571	20	17	.	.	PUNCT
ejpam-5571	21	1	a	a	DET
ejpam-5571	21	2	geodesic	geodesic	NOUN
ejpam-5571	21	3	in	in	ADP
ejpam-5571	21	4	x	x	PROPN
ejpam-5571	21	5	is	be	AUX
ejpam-5571	21	6	an	an	DET
ejpam-5571	21	7	isometry	isometry	NOUN
ejpam-5571	21	8	from	from	ADP
ejpam-5571	21	9	r	r	NOUN
ejpam-5571	21	10	=	=	PUNCT
ejpam-5571	21	11	(	(	PUNCT
ejpam-5571	21	12	−∞,∞	−∞,∞	NOUN
ejpam-5571	21	13	)	)	PUNCT
ejpam-5571	21	14	into	into	ADP
ejpam-5571	21	15	x.	x.	NOUN
ejpam-5571	21	16	we	we	PRON
ejpam-5571	21	17	may	may	AUX
ejpam-5571	21	18	also	also	ADV
ejpam-5571	21	19	refer	refer	VERB
ejpam-5571	21	20	to	to	ADP
ejpam-5571	21	21	the	the	DET
ejpam-5571	21	22	image	image	NOUN
ejpam-5571	21	23	of	of	ADP
ejpam-5571	21	24	this	this	DET
ejpam-5571	21	25	isometry	isometry	NOUN
ejpam-5571	21	26	as	as	ADP
ejpam-5571	21	27	a	a	DET
ejpam-5571	21	28	geodesic	geodesic	NOUN
ejpam-5571	21	29	.	.	PUNCT
ejpam-5571	22	1	a	a	DET
ejpam-5571	22	2	geodesic	geodesic	ADJ
ejpam-5571	22	3	path	path	NOUN
ejpam-5571	22	4	joining	join	VERB
ejpam-5571	22	5	two	two	NUM
ejpam-5571	22	6	points	point	NOUN
ejpam-5571	22	7	x	x	PUNCT
ejpam-5571	22	8	and	and	CCONJ
ejpam-5571	22	9	y	y	PROPN
ejpam-5571	22	10	is	be	AUX
ejpam-5571	22	11	a	a	DET
ejpam-5571	22	12	map	map	NOUN
ejpam-5571	22	13	c	c	NOUN
ejpam-5571	22	14	:	:	PUNCT
ejpam-5571	23	1	[	[	X
ejpam-5571	23	2	0	0	NUM
ejpam-5571	23	3	,	,	PUNCT
ejpam-5571	23	4	a	a	DET
ejpam-5571	23	5	]	]	X
ejpam-5571	23	6	⊂	⊂	X
ejpam-5571	23	7	r	r	NOUN
ejpam-5571	23	8	→	→	PUNCT
ejpam-5571	23	9	x	x	X
ejpam-5571	23	10	such	such	ADJ
ejpam-5571	23	11	that	that	DET
ejpam-5571	23	12	c(0	c(0	NOUN
ejpam-5571	23	13	)	)	PUNCT
ejpam-5571	23	14	=	=	SYM
ejpam-5571	24	1	x	x	X
ejpam-5571	24	2	and	and	CCONJ
ejpam-5571	24	3	c(a	c(a	ADV
ejpam-5571	24	4	)	)	PUNCT
ejpam-5571	24	5	=	=	SYM
ejpam-5571	24	6	y	y	PROPN
ejpam-5571	24	7	,	,	PUNCT
ejpam-5571	24	8	and	and	CCONJ
ejpam-5571	24	9	d(c(t	d(c(t	PROPN
ejpam-5571	24	10	)	)	PUNCT
ejpam-5571	24	11	,	,	PUNCT
ejpam-5571	24	12	c(t′	c(t′	NOUN
ejpam-5571	24	13	)	)	PUNCT
ejpam-5571	24	14	)	)	PUNCT
ejpam-5571	25	1	=	=	PRON
ejpam-5571	25	2	|t	|t	VERB
ejpam-5571	25	3	−	−	PROPN
ejpam-5571	25	4	t′|	t′|	ADV
ejpam-5571	25	5	for	for	ADP
ejpam-5571	25	6	all	all	DET
ejpam-5571	25	7	t	t	NOUN
ejpam-5571	25	8	,	,	PUNCT
ejpam-5571	25	9	t′	t′	X
ejpam-5571	25	10	∈	∈	PROPN
ejpam-5571	26	1	[	[	X
ejpam-5571	26	2	0	0	NUM
ejpam-5571	26	3	,	,	PUNCT
ejpam-5571	26	4	a	a	PRON
ejpam-5571	26	5	]	]	X
ejpam-5571	26	6	.	.	PUNCT
ejpam-5571	27	1	usually	usually	ADV
ejpam-5571	27	2	,	,	PUNCT
ejpam-5571	27	3	the	the	DET
ejpam-5571	27	4	image	image	NOUN
ejpam-5571	27	5	c([0	c([0	VERB
ejpam-5571	27	6	,	,	PUNCT
ejpam-5571	27	7	a	a	DET
ejpam-5571	27	8	]	]	X
ejpam-5571	27	9	)	)	PUNCT
ejpam-5571	27	10	is	be	AUX
ejpam-5571	27	11	called	call	VERB
ejpam-5571	27	12	a	a	DET
ejpam-5571	27	13	geodesic	geodesic	ADJ
ejpam-5571	27	14	segment	segment	NOUN
ejpam-5571	27	15	joining	join	VERB
ejpam-5571	27	16	x	x	PUNCT
ejpam-5571	27	17	and	and	CCONJ
ejpam-5571	27	18	y	y	PROPN
ejpam-5571	27	19	,	,	PUNCT
ejpam-5571	27	20	and	and	CCONJ
ejpam-5571	27	21	if	if	SCONJ
ejpam-5571	27	22	there	there	PRON
ejpam-5571	27	23	is	be	VERB
ejpam-5571	27	24	a	a	DET
ejpam-5571	27	25	unique	unique	ADJ
ejpam-5571	27	26	geodesic	geodesic	ADJ
ejpam-5571	27	27	segment	segment	NOUN
ejpam-5571	27	28	joining	join	VERB
ejpam-5571	27	29	two	two	NUM
ejpam-5571	27	30	points	point	NOUN
ejpam-5571	27	31	x	x	PUNCT
ejpam-5571	27	32	and	and	CCONJ
ejpam-5571	27	33	y	y	PROPN
ejpam-5571	27	34	,	,	PUNCT
ejpam-5571	27	35	then	then	ADV
ejpam-5571	27	36	[	[	X
ejpam-5571	27	37	x	x	X
ejpam-5571	27	38	,	,	PUNCT
ejpam-5571	27	39	y	y	PROPN
ejpam-5571	27	40	]	]	PUNCT
ejpam-5571	27	41	is	be	AUX
ejpam-5571	27	42	denoted	denote	VERB
ejpam-5571	27	43	the	the	DET
ejpam-5571	27	44	geodesic	geodesic	ADJ
ejpam-5571	27	45	segment	segment	NOUN
ejpam-5571	27	46	joining	join	VERB
ejpam-5571	27	47	them	they	PRON
ejpam-5571	27	48	.	.	PUNCT
ejpam-5571	28	1	the	the	DET
ejpam-5571	28	2	metric	metric	ADJ
ejpam-5571	28	3	space	space	NOUN
ejpam-5571	28	4	(	(	PUNCT
ejpam-5571	28	5	x	x	X
ejpam-5571	28	6	,	,	PUNCT
ejpam-5571	28	7	d	d	NOUN
ejpam-5571	28	8	)	)	PUNCT
ejpam-5571	28	9	is	be	AUX
ejpam-5571	28	10	called	call	VERB
ejpam-5571	28	11	a	a	DET
ejpam-5571	28	12	geodesic	geodesic	ADJ
ejpam-5571	28	13	space	space	NOUN
ejpam-5571	28	14	if	if	SCONJ
ejpam-5571	28	15	each	each	DET
ejpam-5571	28	16	pair	pair	NOUN
ejpam-5571	28	17	of	of	ADP
ejpam-5571	28	18	two	two	NUM
ejpam-5571	28	19	points	point	NOUN
ejpam-5571	28	20	of	of	ADP
ejpam-5571	28	21	x	x	PUNCT
ejpam-5571	28	22	is	be	AUX
ejpam-5571	28	23	joined	join	VERB
ejpam-5571	28	24	by	by	ADP
ejpam-5571	28	25	a	a	DET
ejpam-5571	28	26	geodesic	geodesic	ADJ
ejpam-5571	28	27	segment	segment	NOUN
ejpam-5571	28	28	.	.	PUNCT
ejpam-5571	29	1	definition	definition	NOUN
ejpam-5571	29	2	1	1	NUM
ejpam-5571	29	3	.	.	PUNCT
ejpam-5571	30	1	[	[	X
ejpam-5571	30	2	5	5	X
ejpam-5571	30	3	]	]	PUNCT
ejpam-5571	30	4	let	let	VERB
ejpam-5571	30	5	k	k	PRON
ejpam-5571	30	6	be	be	AUX
ejpam-5571	30	7	a	a	DET
ejpam-5571	30	8	real	real	ADJ
ejpam-5571	30	9	number	number	NOUN
ejpam-5571	30	10	.	.	PUNCT
ejpam-5571	31	1	the	the	DET
ejpam-5571	31	2	rk	rk	NOUN
ejpam-5571	31	3	is	be	AUX
ejpam-5571	31	4	one	one	NUM
ejpam-5571	31	5	of	of	ADP
ejpam-5571	31	6	the	the	DET
ejpam-5571	31	7	following	following	ADJ
ejpam-5571	31	8	spaces	space	NOUN
ejpam-5571	31	9	,	,	PUNCT
ejpam-5571	31	10	depending	depend	VERB
ejpam-5571	31	11	on	on	ADP
ejpam-5571	31	12	the	the	DET
ejpam-5571	31	13	sign	sign	NOUN
ejpam-5571	31	14	of	of	ADP
ejpam-5571	31	15	k	k	NOUN
ejpam-5571	31	16	:	:	PUNCT
ejpam-5571	31	17	r2	r2	NOUN
ejpam-5571	31	18	,	,	PUNCT
ejpam-5571	31	19	if	if	SCONJ
ejpam-5571	31	20	k	k	PROPN
ejpam-5571	31	21	=	=	SYM
ejpam-5571	31	22	0	0	PROPN
ejpam-5571	31	23	,	,	PUNCT
ejpam-5571	31	24	the	the	DET
ejpam-5571	31	25	euclidean	euclidean	ADJ
ejpam-5571	31	26	sphere	sphere	NOUN
ejpam-5571	31	27	of	of	ADP
ejpam-5571	31	28	radius	radius	NOUN
ejpam-5571	31	29	1/	1/	NUM
ejpam-5571	31	30	√	√	NUM
ejpam-5571	32	1	k	k	NOUN
ejpam-5571	32	2	,	,	PUNCT
ejpam-5571	32	3	if	if	SCONJ
ejpam-5571	32	4	k	k	PROPN
ejpam-5571	32	5	>	>	X
ejpam-5571	32	6	0	0	PROPN
ejpam-5571	32	7	,	,	PUNCT
ejpam-5571	32	8	and	and	CCONJ
ejpam-5571	32	9	the	the	DET
ejpam-5571	32	10	hyperbolic	hyperbolic	ADJ
ejpam-5571	32	11	plane	plane	NOUN
ejpam-5571	32	12	with	with	ADP
ejpam-5571	32	13	curvature	curvature	NOUN
ejpam-5571	32	14	k	k	PROPN
ejpam-5571	32	15	,	,	PUNCT
ejpam-5571	32	16	if	if	SCONJ
ejpam-5571	32	17	k	k	PROPN
ejpam-5571	32	18	<	<	X
ejpam-5571	32	19	0	0	X
ejpam-5571	32	20	.	.	PUNCT
ejpam-5571	33	1	we	we	PRON
ejpam-5571	33	2	can	can	AUX
ejpam-5571	33	3	learn	learn	VERB
ejpam-5571	33	4	more	more	ADJ
ejpam-5571	33	5	about	about	ADP
ejpam-5571	33	6	the	the	DET
ejpam-5571	33	7	rk	rk	NOUN
ejpam-5571	33	8	spaces	space	VERB
ejpam-5571	33	9	in	in	ADP
ejpam-5571	33	10	[	[	X
ejpam-5571	33	11	4	4	NUM
ejpam-5571	33	12	,	,	PUNCT
ejpam-5571	33	13	7–11	7–11	NOUN
ejpam-5571	33	14	]	]	PUNCT
ejpam-5571	33	15	.	.	PUNCT
ejpam-5571	34	1	a	a	DET
ejpam-5571	34	2	geodesic	geodesic	ADJ
ejpam-5571	34	3	triangle	triangle	NOUN
ejpam-5571	34	4	△	△	X
ejpam-5571	34	5	(p	(p	NOUN
ejpam-5571	34	6	,	,	PUNCT
ejpam-5571	34	7	q	q	NOUN
ejpam-5571	34	8	,	,	PUNCT
ejpam-5571	34	9	r	r	NOUN
ejpam-5571	34	10	)	)	PUNCT
ejpam-5571	34	11	in	in	ADP
ejpam-5571	34	12	x	x	PRON
ejpam-5571	34	13	is	be	AUX
ejpam-5571	34	14	a	a	DET
ejpam-5571	34	15	triangle	triangle	NOUN
ejpam-5571	34	16	with	with	ADP
ejpam-5571	34	17	points	point	NOUN
ejpam-5571	34	18	p	p	X
ejpam-5571	34	19	,	,	PUNCT
ejpam-5571	34	20	q	q	ADJ
ejpam-5571	34	21	,	,	PUNCT
ejpam-5571	34	22	r	r	NOUN
ejpam-5571	34	23	as	as	ADP
ejpam-5571	34	24	its	its	PRON
ejpam-5571	34	25	vertices	vertex	NOUN
ejpam-5571	34	26	and	and	CCONJ
ejpam-5571	34	27	three	three	NUM
ejpam-5571	34	28	chosen	choose	VERB
ejpam-5571	34	29	geodesics	geodesic	NOUN
ejpam-5571	35	1	[	[	X
ejpam-5571	35	2	p	p	X
ejpam-5571	35	3	,	,	PUNCT
ejpam-5571	35	4	q	q	X
ejpam-5571	35	5	]	]	X
ejpam-5571	35	6	,	,	PUNCT
ejpam-5571	35	7	[	[	X
ejpam-5571	35	8	q	q	X
ejpam-5571	35	9	,	,	PUNCT
ejpam-5571	35	10	r	r	NOUN
ejpam-5571	35	11	]	]	X
ejpam-5571	35	12	,	,	PUNCT
ejpam-5571	35	13	[	[	X
ejpam-5571	35	14	p	p	X
ejpam-5571	35	15	,	,	PUNCT
ejpam-5571	35	16	r	r	NOUN
ejpam-5571	35	17	]	]	X
ejpam-5571	35	18	as	as	ADP
ejpam-5571	35	19	its	its	PRON
ejpam-5571	35	20	sides	side	NOUN
ejpam-5571	35	21	.	.	PUNCT
ejpam-5571	36	1	a	a	DET
ejpam-5571	36	2	comparison	comparison	NOUN
ejpam-5571	36	3	triangle	triangle	NOUN
ejpam-5571	36	4	in	in	ADP
ejpam-5571	36	5	rk	rk	NOUN
ejpam-5571	36	6	for	for	ADP
ejpam-5571	36	7	the	the	DET
ejpam-5571	36	8	geodesic	geodesic	ADJ
ejpam-5571	36	9	triangle	triangle	NOUN
ejpam-5571	36	10	△	△	X
ejpam-5571	36	11	(	(	PUNCT
ejpam-5571	36	12	p	p	X
ejpam-5571	36	13	,	,	PUNCT
ejpam-5571	36	14	q	q	ADJ
ejpam-5571	36	15	,	,	PUNCT
ejpam-5571	36	16	r	r	NOUN
ejpam-5571	36	17	)	)	PUNCT
ejpam-5571	36	18	in	in	ADP
ejpam-5571	36	19	x	x	PRON
ejpam-5571	36	20	is	be	AUX
ejpam-5571	36	21	a	a	DET
ejpam-5571	36	22	triangle	triangle	NOUN
ejpam-5571	36	23	△	△	X
ejpam-5571	36	24	(	(	PUNCT
ejpam-5571	36	25	p̃	p̃	PROPN
ejpam-5571	36	26	,	,	PUNCT
ejpam-5571	36	27	q̃	q̃	PROPN
ejpam-5571	36	28	,	,	PUNCT
ejpam-5571	36	29	r̃	r̃	PROPN
ejpam-5571	36	30	)	)	PUNCT
ejpam-5571	36	31	in	in	ADP
ejpam-5571	36	32	rk	rk	NOUN
ejpam-5571	36	33	such	such	ADJ
ejpam-5571	36	34	that	that	SCONJ
ejpam-5571	36	35	d(p	d(p	PROPN
ejpam-5571	36	36	,	,	PUNCT
ejpam-5571	36	37	q	q	NOUN
ejpam-5571	36	38	)	)	PUNCT
ejpam-5571	36	39	=	=	SYM
ejpam-5571	36	40	dk(p̃	dk(p̃	PROPN
ejpam-5571	36	41	,	,	PUNCT
ejpam-5571	36	42	q̃	q̃	PROPN
ejpam-5571	36	43	)	)	PUNCT
ejpam-5571	36	44	,	,	PUNCT
ejpam-5571	36	45	d(q	d(q	PROPN
ejpam-5571	36	46	,	,	PUNCT
ejpam-5571	36	47	r	r	NOUN
ejpam-5571	36	48	)	)	PUNCT
ejpam-5571	36	49	=	=	SYM
ejpam-5571	36	50	d(q̃	d(q̃	PROPN
ejpam-5571	36	51	,	,	PUNCT
ejpam-5571	36	52	r̃	r̃	PROPN
ejpam-5571	36	53	)	)	PUNCT
ejpam-5571	36	54	,	,	PUNCT
ejpam-5571	36	55	and	and	CCONJ
ejpam-5571	36	56	d(p	d(p	PROPN
ejpam-5571	36	57	,	,	PUNCT
ejpam-5571	36	58	r	r	NOUN
ejpam-5571	36	59	)	)	PUNCT
ejpam-5571	36	60	=	=	SYM
ejpam-5571	36	61	d(p̃	d(p̃	PROPN
ejpam-5571	36	62	,	,	PUNCT
ejpam-5571	36	63	r̃	r̃	PROPN
ejpam-5571	36	64	)	)	PUNCT
ejpam-5571	36	65	.	.	PUNCT
ejpam-5571	37	1	such	such	DET
ejpam-5571	37	2	a	a	DET
ejpam-5571	37	3	triangle	triangle	NOUN
ejpam-5571	37	4	△	△	X
ejpam-5571	37	5	(	(	PUNCT
ejpam-5571	37	6	p̃	p̃	PROPN
ejpam-5571	37	7	,	,	PUNCT
ejpam-5571	37	8	q̃	q̃	PROPN
ejpam-5571	37	9	,	,	PUNCT
ejpam-5571	37	10	r̃	r̃	PROPN
ejpam-5571	37	11	)	)	PUNCT
ejpam-5571	37	12	always	always	ADV
ejpam-5571	37	13	exists	exist	VERB
ejpam-5571	37	14	if	if	SCONJ
ejpam-5571	37	15	d(p	d(p	PROPN
ejpam-5571	37	16	,	,	PUNCT
ejpam-5571	37	17	q	q	X
ejpam-5571	37	18	)	)	PUNCT
ejpam-5571	38	1	+	+	CCONJ
ejpam-5571	38	2	d(q	d(q	PROPN
ejpam-5571	38	3	,	,	PUNCT
ejpam-5571	38	4	r	r	NOUN
ejpam-5571	38	5	)	)	PUNCT
ejpam-5571	38	6	+	+	CCONJ
ejpam-5571	38	7	d(p	d(p	PROPN
ejpam-5571	38	8	,	,	PUNCT
ejpam-5571	38	9	r	r	NOUN
ejpam-5571	38	10	)	)	PUNCT
ejpam-5571	38	11	<	<	X
ejpam-5571	39	1	2π√	2π√	PROPN
ejpam-5571	39	2	k	k	NOUN
ejpam-5571	40	1	and	and	CCONJ
ejpam-5571	40	2	it	it	PRON
ejpam-5571	40	3	is	be	AUX
ejpam-5571	40	4	unique	unique	ADJ
ejpam-5571	40	5	up	up	ADP
ejpam-5571	40	6	to	to	ADP
ejpam-5571	40	7	isometries	isometry	NOUN
ejpam-5571	40	8	.	.	PUNCT
ejpam-5571	41	1	given	give	VERB
ejpam-5571	41	2	a	a	DET
ejpam-5571	41	3	pair	pair	NOUN
ejpam-5571	41	4	of	of	ADP
ejpam-5571	41	5	a	a	DET
ejpam-5571	41	6	triangle	triangle	NOUN
ejpam-5571	41	7	△	△	X
ejpam-5571	41	8	(	(	PUNCT
ejpam-5571	41	9	p	p	X
ejpam-5571	41	10	,	,	PUNCT
ejpam-5571	41	11	q	q	ADJ
ejpam-5571	41	12	,	,	PUNCT
ejpam-5571	41	13	r	r	NOUN
ejpam-5571	41	14	)	)	PUNCT
ejpam-5571	41	15	in	in	ADP
ejpam-5571	41	16	x	x	PUNCT
ejpam-5571	41	17	and	and	CCONJ
ejpam-5571	41	18	its	its	PRON
ejpam-5571	41	19	comparison	comparison	NOUN
ejpam-5571	41	20	triangle	triangle	NOUN
ejpam-5571	41	21	△	△	PROPN
ejpam-5571	41	22	(	(	PUNCT
ejpam-5571	41	23	p̃	p̃	PROPN
ejpam-5571	41	24	,	,	PUNCT
ejpam-5571	41	25	q̃	q̃	PROPN
ejpam-5571	41	26	,	,	PUNCT
ejpam-5571	41	27	r̃	r̃	PROPN
ejpam-5571	41	28	)	)	PUNCT
ejpam-5571	41	29	in	in	ADP
ejpam-5571	41	30	rk	rk	PROPN
ejpam-5571	41	31	,	,	PUNCT
ejpam-5571	41	32	the	the	DET
ejpam-5571	41	33	comparison	comparison	NOUN
ejpam-5571	41	34	point	point	NOUN
ejpam-5571	41	35	for	for	ADP
ejpam-5571	41	36	a	a	DET
ejpam-5571	41	37	point	point	NOUN
ejpam-5571	41	38	x	x	X
ejpam-5571	41	39	∈	∈	PROPN
ejpam-5571	42	1	[	[	X
ejpam-5571	42	2	q	q	X
ejpam-5571	42	3	,	,	PUNCT
ejpam-5571	42	4	r	r	X
ejpam-5571	42	5	]	]	X
ejpam-5571	42	6	is	be	AUX
ejpam-5571	42	7	the	the	DET
ejpam-5571	42	8	point	point	NOUN
ejpam-5571	42	9	denoted	denote	VERB
ejpam-5571	42	10	by	by	ADP
ejpam-5571	42	11	x̃	x̃	PROPN
ejpam-5571	42	12	in	in	ADP
ejpam-5571	42	13	[	[	X
ejpam-5571	42	14	q̃	q̃	PROPN
ejpam-5571	42	15	,	,	PUNCT
ejpam-5571	42	16	r̃	r̃	PROPN
ejpam-5571	42	17	]	]	PUNCT
ejpam-5571	42	18	such	such	ADJ
ejpam-5571	42	19	that	that	SCONJ
ejpam-5571	42	20	d(q	d(q	PROPN
ejpam-5571	42	21	,	,	PUNCT
ejpam-5571	42	22	x	x	NOUN
ejpam-5571	42	23	)	)	PUNCT
ejpam-5571	42	24	=	=	SYM
ejpam-5571	42	25	d(q̃	d(q̃	PROPN
ejpam-5571	42	26	,	,	PUNCT
ejpam-5571	42	27	x̃	x̃	PROPN
ejpam-5571	42	28	)	)	PUNCT
ejpam-5571	42	29	,	,	PUNCT
ejpam-5571	42	30	and	and	CCONJ
ejpam-5571	42	31	the	the	DET
ejpam-5571	42	32	comparison	comparison	NOUN
ejpam-5571	42	33	angle	angle	NOUN
ejpam-5571	42	34	at	at	ADP
ejpam-5571	42	35	q	q	NOUN
ejpam-5571	42	36	of	of	ADP
ejpam-5571	42	37	the	the	DET
ejpam-5571	42	38	triangle	triangle	NOUN
ejpam-5571	42	39	△	△	PROPN
ejpam-5571	42	40	(	(	PUNCT
ejpam-5571	42	41	p	p	X
ejpam-5571	42	42	,	,	PUNCT
ejpam-5571	42	43	q	q	ADJ
ejpam-5571	42	44	,	,	PUNCT
ejpam-5571	42	45	r	r	NOUN
ejpam-5571	42	46	)	)	PUNCT
ejpam-5571	42	47	is	be	AUX
ejpam-5571	42	48	the	the	DET
ejpam-5571	42	49	angle	angle	NOUN
ejpam-5571	42	50	at	at	ADP
ejpam-5571	42	51	q̃	q̃	PROPN
ejpam-5571	42	52	of	of	ADP
ejpam-5571	42	53	triangle	triangle	NOUN
ejpam-5571	42	54	△	△	X
ejpam-5571	42	55	(	(	PUNCT
ejpam-5571	42	56	p̃	p̃	PROPN
ejpam-5571	42	57	,	,	PUNCT
ejpam-5571	42	58	q̃	q̃	PROPN
ejpam-5571	42	59	,	,	PUNCT
ejpam-5571	42	60	r̃	r̃	PROPN
ejpam-5571	42	61	)	)	PUNCT
ejpam-5571	42	62	.	.	PUNCT
ejpam-5571	43	1	∠p(q	∠p(q	ADV
ejpam-5571	43	2	,	,	PUNCT
ejpam-5571	43	3	r	r	NOUN
ejpam-5571	43	4	)	)	PUNCT
ejpam-5571	43	5	denotes	denote	VERB
ejpam-5571	43	6	the	the	DET
ejpam-5571	43	7	angle	angle	NOUN
ejpam-5571	43	8	at	at	ADP
ejpam-5571	43	9	p	p	NOUN
ejpam-5571	43	10	of	of	ADP
ejpam-5571	43	11	△	△	X
ejpam-5571	43	12	(	(	PUNCT
ejpam-5571	43	13	p	p	X
ejpam-5571	43	14	,	,	PUNCT
ejpam-5571	43	15	q	q	ADJ
ejpam-5571	43	16	,	,	PUNCT
ejpam-5571	43	17	r	r	NOUN
ejpam-5571	43	18	)	)	PUNCT
ejpam-5571	43	19	in	in	ADP
ejpam-5571	43	20	x.	x.	NOUN
ejpam-5571	44	1	we	we	PRON
ejpam-5571	44	2	let	let	VERB
ejpam-5571	44	3	∠̃p(q	∠̃p(q	NOUN
ejpam-5571	44	4	,	,	PUNCT
ejpam-5571	44	5	r	r	NOUN
ejpam-5571	44	6	)	)	PUNCT
ejpam-5571	44	7	or	or	CCONJ
ejpam-5571	44	8	∠p̃(q̃	∠p̃(q̃	PROPN
ejpam-5571	44	9	,	,	PUNCT
ejpam-5571	44	10	r̃	r̃	PROPN
ejpam-5571	44	11	)	)	PUNCT
ejpam-5571	44	12	denote	denote	VERB
ejpam-5571	44	13	the	the	DET
ejpam-5571	44	14	angle	angle	NOUN
ejpam-5571	44	15	at	at	ADP
ejpam-5571	44	16	p̃	p̃	PROPN
ejpam-5571	44	17	of	of	ADP
ejpam-5571	44	18	a	a	DET
ejpam-5571	44	19	triangle	triangle	NOUN
ejpam-5571	44	20	△	△	X
ejpam-5571	44	21	(	(	PUNCT
ejpam-5571	44	22	q̃	q̃	PROPN
ejpam-5571	44	23	,	,	PUNCT
ejpam-5571	44	24	p̃	p̃	PROPN
ejpam-5571	44	25	,	,	PUNCT
ejpam-5571	44	26	r̃	r̃	PROPN
ejpam-5571	44	27	)	)	PUNCT
ejpam-5571	44	28	in	in	ADP
ejpam-5571	44	29	rk	rk	PROPN
ejpam-5571	44	30	.	.	PUNCT
ejpam-5571	45	1	sometimes	sometimes	ADV
ejpam-5571	45	2	,	,	PUNCT
ejpam-5571	45	3	for	for	ADP
ejpam-5571	45	4	convenience	convenience	NOUN
ejpam-5571	45	5	we	we	PRON
ejpam-5571	45	6	let	let	VERB
ejpam-5571	45	7	a	a	DET
ejpam-5571	45	8	triangle	triangle	NOUN
ejpam-5571	45	9	△	△	NOUN
ejpam-5571	45	10	̃(p	̃(p	NOUN
ejpam-5571	45	11	,	,	PUNCT
ejpam-5571	45	12	q	q	NOUN
ejpam-5571	45	13	,	,	PUNCT
ejpam-5571	45	14	r	r	NOUN
ejpam-5571	45	15	)	)	PUNCT
ejpam-5571	45	16	in	in	ADP
ejpam-5571	45	17	rk	rk	NOUN
ejpam-5571	45	18	be	be	AUX
ejpam-5571	45	19	a	a	DET
ejpam-5571	45	20	comparison	comparison	NOUN
ejpam-5571	45	21	triangle	triangle	NOUN
ejpam-5571	45	22	of	of	ADP
ejpam-5571	45	23	△	△	X
ejpam-5571	45	24	(	(	PUNCT
ejpam-5571	45	25	p	p	X
ejpam-5571	45	26	,	,	PUNCT
ejpam-5571	45	27	q	q	ADJ
ejpam-5571	45	28	,	,	PUNCT
ejpam-5571	45	29	r	r	NOUN
ejpam-5571	45	30	)	)	PUNCT
ejpam-5571	45	31	in	in	ADP
ejpam-5571	45	32	x.	x.	NOUN
ejpam-5571	45	33	definition	definition	NOUN
ejpam-5571	45	34	2	2	NUM
ejpam-5571	45	35	.	.	PUNCT
ejpam-5571	46	1	[	[	X
ejpam-5571	46	2	6	6	NUM
ejpam-5571	46	3	]	]	PUNCT
ejpam-5571	46	4	let	let	VERB
ejpam-5571	46	5	x	x	PRON
ejpam-5571	46	6	be	be	AUX
ejpam-5571	46	7	a	a	DET
ejpam-5571	46	8	length	length	NOUN
ejpam-5571	46	9	space	space	NOUN
ejpam-5571	46	10	.	.	PUNCT
ejpam-5571	47	1	a	a	DET
ejpam-5571	47	2	locally	locally	ADV
ejpam-5571	47	3	complete	complete	ADJ
ejpam-5571	47	4	space	space	NOUN
ejpam-5571	47	5	x	x	PUNCT
ejpam-5571	47	6	is	be	AUX
ejpam-5571	47	7	a	a	DET
ejpam-5571	47	8	space	space	NOUN
ejpam-5571	47	9	with	with	ADP
ejpam-5571	47	10	curvature	curvature	NOUN
ejpam-5571	47	11	bounded	bound	VERB
ejpam-5571	47	12	below	below	ADV
ejpam-5571	47	13	by	by	ADP
ejpam-5571	47	14	a	a	DET
ejpam-5571	47	15	real	real	ADJ
ejpam-5571	47	16	number	number	NOUN
ejpam-5571	47	17	k	k	NOUN
ejpam-5571	47	18	if	if	SCONJ
ejpam-5571	47	19	every	every	DET
ejpam-5571	47	20	point	point	NOUN
ejpam-5571	47	21	x	x	X
ejpam-5571	47	22	∈	∈	NOUN
ejpam-5571	47	23	x	x	PUNCT
ejpam-5571	47	24	has	have	VERB
ejpam-5571	47	25	a	a	DET
ejpam-5571	47	26	neighborhood	neighborhood	NOUN
ejpam-5571	47	27	u(x	u(x	NOUN
ejpam-5571	47	28	)	)	PUNCT
ejpam-5571	47	29	the	the	DET
ejpam-5571	47	30	following	follow	VERB
ejpam-5571	47	31	condition	condition	NOUN
ejpam-5571	47	32	is	be	AUX
ejpam-5571	47	33	satisfied	satisfied	ADJ
ejpam-5571	47	34	:	:	PUNCT
ejpam-5571	47	35	(	(	PUNCT
ejpam-5571	47	36	a	a	X
ejpam-5571	47	37	)	)	PUNCT
ejpam-5571	47	38	for	for	ADP
ejpam-5571	47	39	any	any	DET
ejpam-5571	47	40	four	four	NUM
ejpam-5571	47	41	distinct	distinct	ADJ
ejpam-5571	47	42	points	point	NOUN
ejpam-5571	47	43	p	p	X
ejpam-5571	47	44	,	,	PUNCT
ejpam-5571	47	45	q	q	ADJ
ejpam-5571	47	46	,	,	PUNCT
ejpam-5571	47	47	r	r	NOUN
ejpam-5571	47	48	,	,	PUNCT
ejpam-5571	47	49	s	s	PART
ejpam-5571	47	50	∈	∈	NOUN
ejpam-5571	47	51	u(x	u(x	NOUN
ejpam-5571	47	52	)	)	PUNCT
ejpam-5571	47	53	,	,	PUNCT
ejpam-5571	47	54	∠̃s(q	∠̃s(q	PROPN
ejpam-5571	47	55	,	,	PUNCT
ejpam-5571	47	56	p	p	NOUN
ejpam-5571	47	57	)	)	PUNCT
ejpam-5571	48	1	+	+	CCONJ
ejpam-5571	48	2	∠̃s(q	∠̃s(q	NOUN
ejpam-5571	48	3	,	,	PUNCT
ejpam-5571	48	4	r	r	NOUN
ejpam-5571	48	5	)	)	PUNCT
ejpam-5571	48	6	+	+	CCONJ
ejpam-5571	49	1	∠̃s(p	∠̃s(p	PROPN
ejpam-5571	49	2	,	,	PUNCT
ejpam-5571	49	3	r	r	NOUN
ejpam-5571	49	4	)	)	PUNCT
ejpam-5571	49	5	≤	≤	NOUN
ejpam-5571	49	6	2π	2π	NOUN
ejpam-5571	49	7	.	.	PUNCT
ejpam-5571	50	1	for	for	ADP
ejpam-5571	50	2	spaces	space	NOUN
ejpam-5571	50	3	in	in	ADP
ejpam-5571	50	4	which	which	PRON
ejpam-5571	50	5	,	,	PUNCT
ejpam-5571	50	6	locally	locally	ADV
ejpam-5571	50	7	,	,	PUNCT
ejpam-5571	50	8	any	any	DET
ejpam-5571	50	9	two	two	NUM
ejpam-5571	50	10	points	point	NOUN
ejpam-5571	50	11	are	be	AUX
ejpam-5571	50	12	joined	join	VERB
ejpam-5571	50	13	by	by	ADP
ejpam-5571	50	14	a	a	DET
ejpam-5571	50	15	geodesic	geodesic	NOUN
ejpam-5571	50	16	,	,	PUNCT
ejpam-5571	50	17	in	in	ADP
ejpam-5571	50	18	particular	particular	ADJ
ejpam-5571	50	19	for	for	ADP
ejpam-5571	50	20	locally	locally	ADV
ejpam-5571	50	21	compact	compact	ADJ
ejpam-5571	50	22	spaces	space	NOUN
ejpam-5571	50	23	,	,	PUNCT
ejpam-5571	50	24	the	the	DET
ejpam-5571	50	25	condition	condition	NOUN
ejpam-5571	50	26	(	(	PUNCT
ejpam-5571	50	27	a	a	X
ejpam-5571	50	28	)	)	PUNCT
ejpam-5571	50	29	in	in	ADP
ejpam-5571	50	30	definition	definition	NOUN
ejpam-5571	50	31	2	2	NUM
ejpam-5571	50	32	can	can	AUX
ejpam-5571	50	33	be	be	AUX
ejpam-5571	50	34	replaced	replace	VERB
ejpam-5571	50	35	by	by	ADP
ejpam-5571	50	36	the	the	DET
ejpam-5571	50	37	condition	condition	NOUN
ejpam-5571	50	38	:	:	PUNCT
ejpam-5571	50	39	(	(	PUNCT
ejpam-5571	50	40	b	b	X
ejpam-5571	50	41	)	)	PUNCT
ejpam-5571	50	42	for	for	ADP
ejpam-5571	50	43	any	any	DET
ejpam-5571	50	44	triangle	triangle	NOUN
ejpam-5571	50	45	△	△	PUNCT
ejpam-5571	50	46	(	(	PUNCT
ejpam-5571	50	47	p	p	X
ejpam-5571	50	48	,	,	PUNCT
ejpam-5571	50	49	q	q	ADJ
ejpam-5571	50	50	,	,	PUNCT
ejpam-5571	50	51	r	r	NOUN
ejpam-5571	50	52	)	)	PUNCT
ejpam-5571	50	53	in	in	ADP
ejpam-5571	50	54	u(x	u(x	NOUN
ejpam-5571	50	55	)	)	PUNCT
ejpam-5571	50	56	and	and	CCONJ
ejpam-5571	50	57	any	any	DET
ejpam-5571	50	58	point	point	NOUN
ejpam-5571	50	59	s	s	VERB
ejpam-5571	50	60	on	on	ADP
ejpam-5571	50	61	the	the	DET
ejpam-5571	50	62	side	side	NOUN
ejpam-5571	51	1	[	[	X
ejpam-5571	51	2	q	q	X
ejpam-5571	51	3	,	,	PUNCT
ejpam-5571	51	4	r	r	X
ejpam-5571	51	5	]	]	X
ejpam-5571	51	6	the	the	DET
ejpam-5571	51	7	inequality	inequality	NOUN
ejpam-5571	51	8	d(p	d(p	PROPN
ejpam-5571	51	9	,	,	PUNCT
ejpam-5571	51	10	s	s	NOUN
ejpam-5571	51	11	)	)	PUNCT
ejpam-5571	51	12	≥	≥	NOUN
ejpam-5571	51	13	d(p̃	d(p̃	PROPN
ejpam-5571	51	14	,	,	PUNCT
ejpam-5571	51	15	s̃	s̃	PROPN
ejpam-5571	51	16	)	)	PUNCT
ejpam-5571	51	17	is	be	AUX
ejpam-5571	51	18	satisfied	satisfied	ADJ
ejpam-5571	51	19	,	,	PUNCT
ejpam-5571	51	20	where	where	SCONJ
ejpam-5571	51	21	s̃	s̃	PROPN
ejpam-5571	51	22	is	be	AUX
ejpam-5571	51	23	the	the	DET
ejpam-5571	51	24	corresponding	corresponding	ADJ
ejpam-5571	51	25	point	point	NOUN
ejpam-5571	51	26	of	of	ADP
ejpam-5571	51	27	s	s	PRON
ejpam-5571	51	28	on	on	ADP
ejpam-5571	51	29	the	the	DET
ejpam-5571	51	30	side	side	NOUN
ejpam-5571	52	1	[	[	X
ejpam-5571	52	2	q̃	q̃	PROPN
ejpam-5571	52	3	,	,	PUNCT
ejpam-5571	52	4	r̃	r̃	PROPN
ejpam-5571	52	5	]	]	PUNCT
ejpam-5571	52	6	of	of	ADP
ejpam-5571	52	7	comparison	comparison	NOUN
ejpam-5571	52	8	triangle	triangle	NOUN
ejpam-5571	52	9	△	△	NOUN
ejpam-5571	52	10	̃(p	̃(p	NOUN
ejpam-5571	52	11	,	,	PUNCT
ejpam-5571	52	12	q	q	NOUN
ejpam-5571	52	13	,	,	PUNCT
ejpam-5571	52	14	r	r	NOUN
ejpam-5571	52	15	)	)	PUNCT
ejpam-5571	52	16	.	.	PUNCT
ejpam-5571	53	1	let	let	VERB
ejpam-5571	53	2	x	x	PRON
ejpam-5571	53	3	be	be	AUX
ejpam-5571	53	4	a	a	DET
ejpam-5571	53	5	space	space	NOUN
ejpam-5571	53	6	with	with	ADP
ejpam-5571	53	7	curvature	curvature	NOUN
ejpam-5571	53	8	bounded	bound	VERB
ejpam-5571	53	9	below	below	ADV
ejpam-5571	53	10	by	by	ADP
ejpam-5571	53	11	k	k	PROPN
ejpam-5571	53	12	and	and	CCONJ
ejpam-5571	53	13	α	α	PROPN
ejpam-5571	53	14	and	and	CCONJ
ejpam-5571	53	15	β	β	X
ejpam-5571	53	16	be	be	AUX
ejpam-5571	53	17	two	two	NUM
ejpam-5571	53	18	geodesics	geodesic	NOUN
ejpam-5571	53	19	starting	start	VERB
ejpam-5571	53	20	at	at	ADP
ejpam-5571	53	21	a	a	DET
ejpam-5571	53	22	point	point	NOUN
ejpam-5571	53	23	p	p	NOUN
ejpam-5571	53	24	in	in	ADP
ejpam-5571	53	25	x.	x.	NOUN
ejpam-5571	53	26	the	the	DET
ejpam-5571	53	27	angle	angle	NOUN
ejpam-5571	53	28	between	between	ADP
ejpam-5571	53	29	α	α	PROPN
ejpam-5571	53	30	and	and	CCONJ
ejpam-5571	53	31	β	β	X
ejpam-5571	53	32	is	be	AUX
ejpam-5571	53	33	defined	define	VERB
ejpam-5571	53	34	by	by	ADP
ejpam-5571	53	35	lim	lim	PROPN
ejpam-5571	53	36	s→0	s→0	PROPN
ejpam-5571	53	37	cos−1	cos−1	PROPN
ejpam-5571	53	38	(	(	PUNCT
ejpam-5571	53	39	d2(p	d2(p	PROPN
ejpam-5571	53	40	,	,	PUNCT
ejpam-5571	53	41	α(s	α(s	PROPN
ejpam-5571	53	42	)	)	PUNCT
ejpam-5571	53	43	)	)	PUNCT
ejpam-5571	54	1	+	+	CCONJ
ejpam-5571	54	2	d2(p	d2(p	PROPN
ejpam-5571	54	3	,	,	PUNCT
ejpam-5571	54	4	β(s))−	β(s))−	ADJ
ejpam-5571	54	5	d2((α(s	d2((α(s	PROPN
ejpam-5571	54	6	)	)	PUNCT
ejpam-5571	54	7	,	,	PUNCT
ejpam-5571	54	8	β(s	β(	NOUN
ejpam-5571	54	9	)	)	PUNCT
ejpam-5571	54	10	)	)	PUNCT
ejpam-5571	55	1	2d(p	2d(p	NUM
ejpam-5571	55	2	,	,	PUNCT
ejpam-5571	55	3	α(s))d(p	α(s))d(p	NOUN
ejpam-5571	55	4	,	,	PUNCT
ejpam-5571	55	5	β(s	β(	NOUN
ejpam-5571	55	6	)	)	PUNCT
ejpam-5571	55	7	)	)	PUNCT
ejpam-5571	55	8	)	)	PUNCT
ejpam-5571	55	9	.	.	PUNCT
ejpam-5571	56	1	c.	c.	PROPN
ejpam-5571	56	2	phokaew	phokaew	PROPN
ejpam-5571	56	3	,	,	PUNCT
ejpam-5571	56	4	a.	a.	PROPN
ejpam-5571	56	5	sama	sama	PROPN
ejpam-5571	56	6	-	-	PUNCT
ejpam-5571	56	7	ae	ae	PROPN
ejpam-5571	56	8	,	,	PUNCT
ejpam-5571	56	9	/	/	SYM
ejpam-5571	56	10	eur	eur	NOUN
ejpam-5571	56	11	.	.	PUNCT
ejpam-5571	57	1	j.	j.	PROPN
ejpam-5571	57	2	pure	pure	PROPN
ejpam-5571	57	3	appl	appl	PROPN
ejpam-5571	57	4	.	.	PROPN
ejpam-5571	57	5	math	math	PROPN
ejpam-5571	57	6	,	,	PUNCT
ejpam-5571	57	7	17	17	NUM
ejpam-5571	57	8	(	(	PUNCT
ejpam-5571	57	9	4	4	NUM
ejpam-5571	57	10	)	)	PUNCT
ejpam-5571	57	11	(	(	PUNCT
ejpam-5571	57	12	2024	2024	NUM
ejpam-5571	57	13	)	)	PUNCT
ejpam-5571	57	14	,	,	PUNCT
ejpam-5571	57	15	3932	3932	NUM
ejpam-5571	57	16	-	-	SYM
ejpam-5571	57	17	3944	3944	NUM
ejpam-5571	57	18	3934	3934	NUM
ejpam-5571	57	19	the	the	DET
ejpam-5571	57	20	angle	angle	NOUN
ejpam-5571	57	21	at	at	ADP
ejpam-5571	57	22	p	p	NOUN
ejpam-5571	57	23	of	of	ADP
ejpam-5571	57	24	a	a	DET
ejpam-5571	57	25	triangle	triangle	NOUN
ejpam-5571	57	26	△	△	X
ejpam-5571	57	27	(	(	PUNCT
ejpam-5571	57	28	q	q	ADJ
ejpam-5571	57	29	,	,	PUNCT
ejpam-5571	57	30	p	p	X
ejpam-5571	57	31	,	,	PUNCT
ejpam-5571	57	32	r	r	NOUN
ejpam-5571	57	33	)	)	PUNCT
ejpam-5571	57	34	is	be	AUX
ejpam-5571	57	35	the	the	DET
ejpam-5571	57	36	angle	angle	NOUN
ejpam-5571	57	37	between	between	ADP
ejpam-5571	57	38	[	[	X
ejpam-5571	57	39	p	p	X
ejpam-5571	57	40	,	,	PUNCT
ejpam-5571	57	41	q	q	X
ejpam-5571	57	42	]	]	X
ejpam-5571	57	43	and	and	CCONJ
ejpam-5571	57	44	[	[	X
ejpam-5571	57	45	p	p	X
ejpam-5571	57	46	,	,	PUNCT
ejpam-5571	57	47	r	r	NOUN
ejpam-5571	57	48	]	]	PUNCT
ejpam-5571	57	49	.	.	PUNCT
ejpam-5571	58	1	the	the	DET
ejpam-5571	58	2	condition	condition	NOUN
ejpam-5571	58	3	(	(	PUNCT
ejpam-5571	58	4	b	b	NOUN
ejpam-5571	58	5	)	)	PUNCT
ejpam-5571	58	6	is	be	AUX
ejpam-5571	58	7	equivalent	equivalent	ADJ
ejpam-5571	58	8	to	to	ADP
ejpam-5571	58	9	the	the	DET
ejpam-5571	58	10	following	follow	VERB
ejpam-5571	58	11	condition	condition	NOUN
ejpam-5571	58	12	:	:	PUNCT
ejpam-5571	58	13	(	(	PUNCT
ejpam-5571	58	14	b̃	b̃	PROPN
ejpam-5571	58	15	)	)	PUNCT
ejpam-5571	58	16	for	for	ADP
ejpam-5571	58	17	any	any	DET
ejpam-5571	58	18	triangle	triangle	NOUN
ejpam-5571	58	19	△	△	PUNCT
ejpam-5571	58	20	(	(	PUNCT
ejpam-5571	58	21	p	p	X
ejpam-5571	58	22	,	,	PUNCT
ejpam-5571	58	23	q	q	ADJ
ejpam-5571	58	24	,	,	PUNCT
ejpam-5571	58	25	r	r	NOUN
ejpam-5571	58	26	)	)	PUNCT
ejpam-5571	58	27	in	in	ADP
ejpam-5571	58	28	u(x	u(x	NOUN
ejpam-5571	58	29	)	)	PUNCT
ejpam-5571	58	30	,	,	PUNCT
ejpam-5571	58	31	∠p(q	∠p(q	ADV
ejpam-5571	58	32	,	,	PUNCT
ejpam-5571	58	33	r	r	NOUN
ejpam-5571	58	34	)	)	PUNCT
ejpam-5571	58	35	≥	≥	NOUN
ejpam-5571	58	36	∠̃p(q	∠̃p(q	NOUN
ejpam-5571	58	37	,	,	PUNCT
ejpam-5571	58	38	r	r	NOUN
ejpam-5571	58	39	)	)	PUNCT
ejpam-5571	58	40	,	,	PUNCT
ejpam-5571	58	41	∠q(p	∠q(p	NOUN
ejpam-5571	58	42	,	,	PUNCT
ejpam-5571	58	43	r	r	NOUN
ejpam-5571	58	44	)	)	PUNCT
ejpam-5571	58	45	≥	≥	NOUN
ejpam-5571	59	1	∠̃q(p	∠̃q(p	NOUN
ejpam-5571	59	2	,	,	PUNCT
ejpam-5571	59	3	r	r	NOUN
ejpam-5571	59	4	)	)	PUNCT
ejpam-5571	59	5	,	,	PUNCT
ejpam-5571	59	6	and	and	CCONJ
ejpam-5571	59	7	∠r(p	∠r(p	ADJ
ejpam-5571	59	8	,	,	PUNCT
ejpam-5571	59	9	q	q	ADJ
ejpam-5571	59	10	)	)	PUNCT
ejpam-5571	59	11	≥	≥	NOUN
ejpam-5571	60	1	∠̃r(p	∠̃r(p	NOUN
ejpam-5571	60	2	,	,	PUNCT
ejpam-5571	60	3	q	q	NOUN
ejpam-5571	60	4	)	)	PUNCT
ejpam-5571	60	5	,	,	PUNCT
ejpam-5571	60	6	where	where	SCONJ
ejpam-5571	60	7	△	△	NOUN
ejpam-5571	60	8	̃(p	̃(p	NOUN
ejpam-5571	60	9	,	,	PUNCT
ejpam-5571	60	10	q	q	NOUN
ejpam-5571	60	11	,	,	PUNCT
ejpam-5571	60	12	r	r	NOUN
ejpam-5571	60	13	)	)	PUNCT
ejpam-5571	60	14	is	be	AUX
ejpam-5571	60	15	a	a	DET
ejpam-5571	60	16	comparision	comparision	NOUN
ejpam-5571	60	17	triangle	triangle	NOUN
ejpam-5571	60	18	in	in	ADP
ejpam-5571	60	19	rk	rk	NOUN
ejpam-5571	60	20	of	of	ADP
ejpam-5571	60	21	the	the	DET
ejpam-5571	60	22	triangle	triangle	NOUN
ejpam-5571	60	23	△	△	PROPN
ejpam-5571	60	24	(	(	PUNCT
ejpam-5571	60	25	p	p	X
ejpam-5571	60	26	,	,	PUNCT
ejpam-5571	60	27	q	q	ADJ
ejpam-5571	60	28	,	,	PUNCT
ejpam-5571	60	29	r	r	NOUN
ejpam-5571	60	30	)	)	PUNCT
ejpam-5571	60	31	.	.	PUNCT
ejpam-5571	61	1	spaces	space	NOUN
ejpam-5571	61	2	with	with	ADP
ejpam-5571	61	3	curvature	curvature	NOUN
ejpam-5571	61	4	bounded	bound	VERB
ejpam-5571	61	5	below	below	ADV
ejpam-5571	61	6	were	be	AUX
ejpam-5571	61	7	defined	define	VERB
ejpam-5571	61	8	above	above	ADP
ejpam-5571	61	9	using	use	VERB
ejpam-5571	61	10	local	local	ADJ
ejpam-5571	61	11	conditions	condition	NOUN
ejpam-5571	61	12	.	.	PUNCT
ejpam-5571	62	1	however	however	ADV
ejpam-5571	62	2	,	,	PUNCT
ejpam-5571	62	3	for	for	ADP
ejpam-5571	62	4	complete	complete	ADJ
ejpam-5571	62	5	spaces	space	NOUN
ejpam-5571	62	6	,	,	PUNCT
ejpam-5571	62	7	the	the	DET
ejpam-5571	62	8	global	global	ADJ
ejpam-5571	62	9	conditions	condition	NOUN
ejpam-5571	62	10	may	may	AUX
ejpam-5571	62	11	be	be	AUX
ejpam-5571	62	12	deduced	deduce	VERB
ejpam-5571	62	13	from	from	ADP
ejpam-5571	62	14	the	the	DET
ejpam-5571	62	15	corresponding	corresponding	ADJ
ejpam-5571	62	16	local	local	ADJ
ejpam-5571	62	17	ones	one	NOUN
ejpam-5571	62	18	.	.	PUNCT
ejpam-5571	63	1	the	the	DET
ejpam-5571	63	2	metric	metric	ADJ
ejpam-5571	63	3	space	space	NOUN
ejpam-5571	63	4	x	x	PRON
ejpam-5571	63	5	considered	consider	VERB
ejpam-5571	63	6	in	in	ADP
ejpam-5571	63	7	this	this	DET
ejpam-5571	63	8	work	work	NOUN
ejpam-5571	63	9	is	be	AUX
ejpam-5571	63	10	complete	complete	ADJ
ejpam-5571	63	11	.	.	PUNCT
ejpam-5571	64	1	we	we	PRON
ejpam-5571	64	2	then	then	ADV
ejpam-5571	64	3	call	call	VERB
ejpam-5571	64	4	x	x	PUNCT
ejpam-5571	64	5	a	a	DET
ejpam-5571	64	6	metric	metric	ADJ
ejpam-5571	64	7	space	space	NOUN
ejpam-5571	64	8	with	with	ADP
ejpam-5571	64	9	curvature	curvature	NOUN
ejpam-5571	64	10	bounded	bound	VERB
ejpam-5571	64	11	below	below	ADV
ejpam-5571	64	12	in	in	ADP
ejpam-5571	64	13	the	the	DET
ejpam-5571	64	14	large	large	ADJ
ejpam-5571	64	15	.	.	PUNCT
ejpam-5571	65	1	theorem	theorem	NOUN
ejpam-5571	65	2	1	1	NUM
ejpam-5571	65	3	.	.	PUNCT
ejpam-5571	66	1	[	[	X
ejpam-5571	66	2	5	5	X
ejpam-5571	66	3	]	]	PUNCT
ejpam-5571	66	4	if	if	SCONJ
ejpam-5571	66	5	x	x	PRON
ejpam-5571	66	6	is	be	AUX
ejpam-5571	66	7	a	a	DET
ejpam-5571	66	8	metric	metric	ADJ
ejpam-5571	66	9	space	space	NOUN
ejpam-5571	66	10	with	with	ADP
ejpam-5571	66	11	curvature	curvature	NOUN
ejpam-5571	66	12	bounded	bound	VERB
ejpam-5571	66	13	below	below	ADV
ejpam-5571	66	14	by	by	ADP
ejpam-5571	66	15	k	k	PROPN
ejpam-5571	66	16	in	in	ADP
ejpam-5571	66	17	the	the	DET
ejpam-5571	66	18	large	large	ADJ
ejpam-5571	66	19	,	,	PUNCT
ejpam-5571	66	20	where	where	SCONJ
ejpam-5571	66	21	k	k	PROPN
ejpam-5571	66	22	>	>	X
ejpam-5571	66	23	0	0	PROPN
ejpam-5571	66	24	,	,	PUNCT
ejpam-5571	66	25	then	then	ADV
ejpam-5571	66	26	dim(x	dim(x	PROPN
ejpam-5571	66	27	)	)	PUNCT
ejpam-5571	66	28	≤	≤	NOUN
ejpam-5571	66	29	π/	π/	ADP
ejpam-5571	66	30	√	√	PROPN
ejpam-5571	66	31	k	k	NOUN
ejpam-5571	66	32	and	and	CCONJ
ejpam-5571	66	33	any	any	DET
ejpam-5571	66	34	triangle	triangle	NOUN
ejpam-5571	66	35	in	in	ADP
ejpam-5571	66	36	x	x	PROPN
ejpam-5571	66	37	has	have	VERB
ejpam-5571	66	38	perimeter	perimeter	NOUN
ejpam-5571	66	39	no	no	ADV
ejpam-5571	66	40	greater	great	ADJ
ejpam-5571	66	41	than	than	ADP
ejpam-5571	66	42	2π/	2π/	NUM
ejpam-5571	66	43	√	√	PROPN
ejpam-5571	66	44	k.	k.	PROPN
ejpam-5571	66	45	theorem	theorem	PROPN
ejpam-5571	66	46	2	2	NUM
ejpam-5571	66	47	.	.	PUNCT
ejpam-5571	67	1	[	[	X
ejpam-5571	67	2	12	12	NUM
ejpam-5571	67	3	]	]	PUNCT
ejpam-5571	67	4	let	let	VERB
ejpam-5571	67	5	x	x	PRON
ejpam-5571	67	6	be	be	AUX
ejpam-5571	67	7	a	a	DET
ejpam-5571	67	8	metric	metric	ADJ
ejpam-5571	67	9	space	space	NOUN
ejpam-5571	67	10	with	with	ADP
ejpam-5571	67	11	curvature	curvature	NOUN
ejpam-5571	67	12	bounded	bound	VERB
ejpam-5571	67	13	below	below	ADV
ejpam-5571	67	14	by	by	ADP
ejpam-5571	67	15	k	k	PROPN
ejpam-5571	67	16	in	in	ADP
ejpam-5571	67	17	the	the	DET
ejpam-5571	67	18	large	large	ADJ
ejpam-5571	67	19	,	,	PUNCT
ejpam-5571	67	20	△	△	X
ejpam-5571	67	21	(	(	PUNCT
ejpam-5571	67	22	p	p	X
ejpam-5571	67	23	,	,	PUNCT
ejpam-5571	67	24	q	q	ADJ
ejpam-5571	67	25	,	,	PUNCT
ejpam-5571	67	26	r	r	NOUN
ejpam-5571	67	27	)	)	PUNCT
ejpam-5571	67	28	a	a	DET
ejpam-5571	67	29	triangle	triangle	NOUN
ejpam-5571	67	30	in	in	ADP
ejpam-5571	67	31	x	x	X
ejpam-5571	67	32	and	and	CCONJ
ejpam-5571	67	33	△	△	PROPN
ejpam-5571	67	34	(	(	PUNCT
ejpam-5571	67	35	p̃	p̃	PROPN
ejpam-5571	67	36	,	,	PUNCT
ejpam-5571	67	37	q̃	q̃	PROPN
ejpam-5571	67	38	,	,	PUNCT
ejpam-5571	67	39	r̃	r̃	PROPN
ejpam-5571	67	40	)	)	PUNCT
ejpam-5571	67	41	a	a	DET
ejpam-5571	67	42	triangle	triangle	NOUN
ejpam-5571	67	43	in	in	ADP
ejpam-5571	67	44	rk	rk	NOUN
ejpam-5571	67	45	.	.	PUNCT
ejpam-5571	68	1	if	if	SCONJ
ejpam-5571	68	2	d(p	d(p	PROPN
ejpam-5571	68	3	,	,	PUNCT
ejpam-5571	68	4	q	q	NOUN
ejpam-5571	68	5	)	)	PUNCT
ejpam-5571	68	6	=	=	SYM
ejpam-5571	68	7	d(p̃	d(p̃	PROPN
ejpam-5571	68	8	,	,	PUNCT
ejpam-5571	68	9	q̃	q̃	PROPN
ejpam-5571	68	10	)	)	PUNCT
ejpam-5571	68	11	,	,	PUNCT
ejpam-5571	68	12	d(p	d(p	PROPN
ejpam-5571	68	13	,	,	PUNCT
ejpam-5571	68	14	r	r	NOUN
ejpam-5571	68	15	)	)	PUNCT
ejpam-5571	68	16	=	=	SYM
ejpam-5571	68	17	d(p̃	d(p̃	PROPN
ejpam-5571	68	18	,	,	PUNCT
ejpam-5571	68	19	r̃	r̃	PROPN
ejpam-5571	68	20	)	)	PUNCT
ejpam-5571	68	21	,	,	PUNCT
ejpam-5571	68	22	and	and	CCONJ
ejpam-5571	68	23	∠p(q	∠p(q	ADV
ejpam-5571	68	24	,	,	PUNCT
ejpam-5571	68	25	r	r	NOUN
ejpam-5571	68	26	)	)	PUNCT
ejpam-5571	68	27	=	=	SYM
ejpam-5571	68	28	∠p̃(q̃	∠p̃(q̃	NOUN
ejpam-5571	68	29	,	,	PUNCT
ejpam-5571	68	30	r̃	r̃	PROPN
ejpam-5571	68	31	)	)	PUNCT
ejpam-5571	68	32	,	,	PUNCT
ejpam-5571	68	33	then	then	ADV
ejpam-5571	68	34	d(q	d(q	PROPN
ejpam-5571	68	35	,	,	PUNCT
ejpam-5571	68	36	r	r	NOUN
ejpam-5571	68	37	)	)	PUNCT
ejpam-5571	68	38	≤	≤	NUM
ejpam-5571	68	39	d(q̃	d(q̃	NOUN
ejpam-5571	68	40	,	,	PUNCT
ejpam-5571	68	41	r̃	r̃	PROPN
ejpam-5571	68	42	)	)	PUNCT
ejpam-5571	68	43	.	.	PUNCT
ejpam-5571	69	1	2	2	X
ejpam-5571	69	2	.	.	NUM
ejpam-5571	69	3	closed	close	VERB
ejpam-5571	69	4	geodesic	geodesic	ADJ
ejpam-5571	69	5	polygons	polygon	NOUN
ejpam-5571	69	6	a	a	DET
ejpam-5571	69	7	closed	closed	ADJ
ejpam-5571	69	8	curve	curve	NOUN
ejpam-5571	69	9	in	in	ADP
ejpam-5571	69	10	a	a	DET
ejpam-5571	69	11	metric	metric	ADJ
ejpam-5571	69	12	space	space	NOUN
ejpam-5571	69	13	x	x	PUNCT
ejpam-5571	69	14	is	be	AUX
ejpam-5571	69	15	a	a	DET
ejpam-5571	69	16	continuous	continuous	ADJ
ejpam-5571	69	17	map	map	NOUN
ejpam-5571	69	18	of	of	ADP
ejpam-5571	69	19	an	an	DET
ejpam-5571	69	20	oriented	orient	VERB
ejpam-5571	69	21	circle	circle	NOUN
ejpam-5571	69	22	in	in	ADP
ejpam-5571	69	23	the	the	DET
ejpam-5571	69	24	2dimensional	2dimensional	ADJ
ejpam-5571	69	25	euclidean	euclidean	ADJ
ejpam-5571	69	26	space	space	NOUN
ejpam-5571	69	27	.	.	PUNCT
ejpam-5571	70	1	a	a	DET
ejpam-5571	70	2	chain	chain	NOUN
ejpam-5571	70	3	v	v	NOUN
ejpam-5571	70	4	on	on	ADP
ejpam-5571	70	5	a	a	DET
ejpam-5571	70	6	closed	closed	ADJ
ejpam-5571	70	7	curve	curve	NOUN
ejpam-5571	70	8	γ	γ	NOUN
ejpam-5571	70	9	is	be	AUX
ejpam-5571	70	10	a	a	DET
ejpam-5571	70	11	set	set	NOUN
ejpam-5571	70	12	of	of	ADP
ejpam-5571	70	13	points	point	NOUN
ejpam-5571	70	14	corresponding	correspond	VERB
ejpam-5571	70	15	to	to	ADP
ejpam-5571	70	16	finitely	finitely	ADV
ejpam-5571	70	17	many	many	ADJ
ejpam-5571	70	18	parameter	parameter	NOUN
ejpam-5571	70	19	values	value	NOUN
ejpam-5571	70	20	in	in	ADP
ejpam-5571	70	21	order	order	NOUN
ejpam-5571	70	22	.	.	PUNCT
ejpam-5571	71	1	the	the	DET
ejpam-5571	71	2	points	point	NOUN
ejpam-5571	71	3	in	in	ADP
ejpam-5571	71	4	v	v	NOUN
ejpam-5571	71	5	are	be	AUX
ejpam-5571	71	6	called	call	VERB
ejpam-5571	71	7	the	the	DET
ejpam-5571	71	8	vertices	vertex	NOUN
ejpam-5571	71	9	of	of	ADP
ejpam-5571	71	10	the	the	DET
ejpam-5571	71	11	chain	chain	NOUN
ejpam-5571	71	12	.	.	PUNCT
ejpam-5571	72	1	if	if	SCONJ
ejpam-5571	72	2	γ	γ	PROPN
ejpam-5571	72	3	consists	consist	VERB
ejpam-5571	72	4	of	of	ADP
ejpam-5571	72	5	geodesic	geodesic	ADJ
ejpam-5571	72	6	segments	segment	NOUN
ejpam-5571	72	7	joining	join	VERB
ejpam-5571	72	8	adjacent	adjacent	ADJ
ejpam-5571	72	9	pairs	pair	NOUN
ejpam-5571	72	10	in	in	ADP
ejpam-5571	72	11	v	v	NUM
ejpam-5571	72	12	,	,	PUNCT
ejpam-5571	72	13	then	then	ADV
ejpam-5571	72	14	γ	γ	PROPN
ejpam-5571	72	15	and	and	CCONJ
ejpam-5571	72	16	v	v	ADP
ejpam-5571	72	17	form	form	NOUN
ejpam-5571	72	18	a	a	DET
ejpam-5571	72	19	closed	closed	ADJ
ejpam-5571	72	20	geodesic	geodesic	ADJ
ejpam-5571	72	21	polygon	polygon	NOUN
ejpam-5571	72	22	with	with	ADP
ejpam-5571	72	23	a	a	DET
ejpam-5571	72	24	vertex	vertex	NOUN
ejpam-5571	72	25	chain	chain	NOUN
ejpam-5571	72	26	in	in	ADP
ejpam-5571	72	27	v	v	NUM
ejpam-5571	72	28	.	.	PUNCT
ejpam-5571	73	1	a	a	DET
ejpam-5571	73	2	subset	subset	NOUN
ejpam-5571	73	3	a	a	PRON
ejpam-5571	73	4	of	of	ADP
ejpam-5571	73	5	a	a	DET
ejpam-5571	73	6	metric	metric	ADJ
ejpam-5571	73	7	space	space	NOUN
ejpam-5571	73	8	(	(	PUNCT
ejpam-5571	73	9	x	x	X
ejpam-5571	73	10	,	,	PUNCT
ejpam-5571	73	11	d	d	NOUN
ejpam-5571	73	12	)	)	PUNCT
ejpam-5571	73	13	is	be	AUX
ejpam-5571	73	14	defined	define	VERB
ejpam-5571	73	15	as	as	ADP
ejpam-5571	73	16	convex	convex	NOUN
ejpam-5571	73	17	if	if	SCONJ
ejpam-5571	73	18	,	,	PUNCT
ejpam-5571	73	19	for	for	ADP
ejpam-5571	73	20	any	any	DET
ejpam-5571	73	21	two	two	NUM
ejpam-5571	73	22	points	point	NOUN
ejpam-5571	73	23	x	x	X
ejpam-5571	73	24	,	,	PUNCT
ejpam-5571	73	25	y	y	PROPN
ejpam-5571	73	26	∈	∈	PROPN
ejpam-5571	74	1	a	a	PRON
ejpam-5571	74	2	,	,	PUNCT
ejpam-5571	74	3	the	the	DET
ejpam-5571	74	4	segment	segment	NOUN
ejpam-5571	74	5	joining	join	VERB
ejpam-5571	74	6	x	x	PUNCT
ejpam-5571	74	7	and	and	CCONJ
ejpam-5571	74	8	y	y	PROPN
ejpam-5571	74	9	is	be	AUX
ejpam-5571	74	10	also	also	ADV
ejpam-5571	74	11	included	include	VERB
ejpam-5571	74	12	in	in	ADP
ejpam-5571	74	13	a.	a.	NOUN
ejpam-5571	74	14	c(a	c(a	PROPN
ejpam-5571	74	15	)	)	PUNCT
ejpam-5571	74	16	represents	represent	VERB
ejpam-5571	74	17	the	the	DET
ejpam-5571	74	18	convex	convex	PROPN
ejpam-5571	74	19	hull	hull	NOUN
ejpam-5571	74	20	of	of	ADP
ejpam-5571	74	21	a	a	DET
ejpam-5571	74	22	subset	subset	NOUN
ejpam-5571	74	23	a	a	X
ejpam-5571	74	24	,	,	PUNCT
ejpam-5571	74	25	defined	define	VERB
ejpam-5571	74	26	as	as	ADP
ejpam-5571	74	27	the	the	DET
ejpam-5571	74	28	smallest	small	ADJ
ejpam-5571	74	29	convex	convex	NOUN
ejpam-5571	74	30	set	set	NOUN
ejpam-5571	74	31	that	that	PRON
ejpam-5571	74	32	contains	contain	VERB
ejpam-5571	74	33	a.	a.	NOUN
ejpam-5571	74	34	an	an	DET
ejpam-5571	74	35	isometry	isometry	NOUN
ejpam-5571	74	36	between	between	ADP
ejpam-5571	74	37	two	two	NUM
ejpam-5571	74	38	metric	metric	ADJ
ejpam-5571	74	39	spaces	space	NOUN
ejpam-5571	74	40	(	(	PUNCT
ejpam-5571	74	41	x	x	X
ejpam-5571	74	42	,	,	PUNCT
ejpam-5571	74	43	d	d	NOUN
ejpam-5571	74	44	)	)	PUNCT
ejpam-5571	74	45	and	and	CCONJ
ejpam-5571	74	46	(	(	PUNCT
ejpam-5571	74	47	y	y	PROPN
ejpam-5571	74	48	,	,	PUNCT
ejpam-5571	74	49	d∗	d∗	PROPN
ejpam-5571	74	50	)	)	PUNCT
ejpam-5571	74	51	is	be	AUX
ejpam-5571	74	52	a	a	DET
ejpam-5571	74	53	function	function	NOUN
ejpam-5571	75	1	i	i	PRON
ejpam-5571	75	2	:	:	PUNCT
ejpam-5571	75	3	x	x	X
ejpam-5571	75	4	→	→	PUNCT
ejpam-5571	75	5	y	y	PROPN
ejpam-5571	75	6	such	such	ADJ
ejpam-5571	75	7	that	that	PRON
ejpam-5571	75	8	d(x	d(x	PROPN
ejpam-5571	75	9	,	,	PUNCT
ejpam-5571	75	10	y	y	NOUN
ejpam-5571	75	11	)	)	PUNCT
ejpam-5571	75	12	=	=	SYM
ejpam-5571	75	13	d∗(i(x	d∗(i(x	PROPN
ejpam-5571	75	14	)	)	PUNCT
ejpam-5571	75	15	,	,	PUNCT
ejpam-5571	75	16	i(y	i(y	NOUN
ejpam-5571	75	17	)	)	PUNCT
ejpam-5571	75	18	)	)	PUNCT
ejpam-5571	75	19	,	,	PUNCT
ejpam-5571	75	20	for	for	ADP
ejpam-5571	75	21	all	all	DET
ejpam-5571	75	22	points	point	NOUN
ejpam-5571	75	23	x	x	PRON
ejpam-5571	75	24	,	,	PUNCT
ejpam-5571	75	25	y	y	PROPN
ejpam-5571	75	26	∈	∈	PROPN
ejpam-5571	75	27	x.	x.	NOUN
ejpam-5571	76	1	we	we	PRON
ejpam-5571	76	2	will	will	AUX
ejpam-5571	76	3	start	start	VERB
ejpam-5571	76	4	by	by	ADP
ejpam-5571	76	5	describing	describe	VERB
ejpam-5571	76	6	a	a	DET
ejpam-5571	76	7	triangle	triangle	NOUN
ejpam-5571	76	8	in	in	ADP
ejpam-5571	76	9	metric	metric	ADJ
ejpam-5571	76	10	space	space	NOUN
ejpam-5571	76	11	with	with	ADP
ejpam-5571	76	12	curvature	curvature	NOUN
ejpam-5571	76	13	bounded	bound	VERB
ejpam-5571	76	14	below	below	ADP
ejpam-5571	76	15	whose	whose	DET
ejpam-5571	76	16	convex	convex	NOUN
ejpam-5571	76	17	hull	hull	NOUN
ejpam-5571	76	18	is	be	AUX
ejpam-5571	76	19	isometric	isometric	ADJ
ejpam-5571	76	20	to	to	ADP
ejpam-5571	76	21	a	a	DET
ejpam-5571	76	22	comparison	comparison	NOUN
ejpam-5571	76	23	triangle	triangle	NOUN
ejpam-5571	76	24	’s	’s	PART
ejpam-5571	76	25	convex	convex	PROPN
ejpam-5571	76	26	hull	hull	NOUN
ejpam-5571	76	27	in	in	ADP
ejpam-5571	76	28	model	model	NOUN
ejpam-5571	76	29	space	space	NOUN
ejpam-5571	76	30	rk	rk	NOUN
ejpam-5571	76	31	.	.	PUNCT
ejpam-5571	77	1	we	we	PRON
ejpam-5571	77	2	note	note	VERB
ejpam-5571	77	3	that	that	SCONJ
ejpam-5571	77	4	if	if	SCONJ
ejpam-5571	77	5	p1	p1	PROPN
ejpam-5571	77	6	,	,	PUNCT
ejpam-5571	77	7	p2	p2	NOUN
ejpam-5571	77	8	,	,	PUNCT
ejpam-5571	77	9	p3	p3	NOUN
ejpam-5571	77	10	,	,	PUNCT
ejpam-5571	77	11	p4	p4	ADJ
ejpam-5571	77	12	,	,	PUNCT
ejpam-5571	77	13	p5	p5	ADJ
ejpam-5571	77	14	=	=	SYM
ejpam-5571	77	15	p1	p1	NOUN
ejpam-5571	77	16	is	be	AUX
ejpam-5571	77	17	a	a	DET
ejpam-5571	77	18	closed	closed	ADJ
ejpam-5571	77	19	polygon	polygon	NOUN
ejpam-5571	77	20	in	in	ADP
ejpam-5571	77	21	x	x	PROPN
ejpam-5571	77	22	,	,	PUNCT
ejpam-5571	77	23	a	a	DET
ejpam-5571	77	24	metric	metric	ADJ
ejpam-5571	77	25	space	space	NOUN
ejpam-5571	77	26	with	with	ADP
ejpam-5571	77	27	curvature	curvature	NOUN
ejpam-5571	77	28	bounded	bound	VERB
ejpam-5571	77	29	below	below	ADV
ejpam-5571	77	30	by	by	ADP
ejpam-5571	77	31	k	k	PROPN
ejpam-5571	77	32	,	,	PUNCT
ejpam-5571	77	33	and	and	CCONJ
ejpam-5571	77	34	p′1	p′1	NOUN
ejpam-5571	77	35	,	,	PUNCT
ejpam-5571	77	36	p	p	NOUN
ejpam-5571	77	37	′	′	NOUN
ejpam-5571	77	38	2	2	NUM
ejpam-5571	77	39	,	,	PUNCT
ejpam-5571	77	40	p	p	NOUN
ejpam-5571	77	41	′	′	NOUN
ejpam-5571	77	42	3	3	NUM
ejpam-5571	77	43	,	,	PUNCT
ejpam-5571	77	44	p	p	NOUN
ejpam-5571	77	45	′	′	NOUN
ejpam-5571	77	46	4	4	NUM
ejpam-5571	77	47	,	,	PUNCT
ejpam-5571	77	48	p	p	NOUN
ejpam-5571	77	49	′	′	NOUN
ejpam-5571	77	50	5	5	NUM
ejpam-5571	77	51	=	=	NOUN
ejpam-5571	77	52	p′1	p′1	NOUN
ejpam-5571	77	53	is	be	AUX
ejpam-5571	77	54	a	a	DET
ejpam-5571	77	55	closed	closed	ADJ
ejpam-5571	77	56	polygon	polygon	NOUN
ejpam-5571	77	57	in	in	ADP
ejpam-5571	77	58	rk	rk	NOUN
ejpam-5571	77	59	with	with	ADP
ejpam-5571	77	60	d(p1	d(p1	NOUN
ejpam-5571	77	61	,	,	PUNCT
ejpam-5571	77	62	p2	p2	X
ejpam-5571	77	63	)	)	PUNCT
ejpam-5571	77	64	=	=	SYM
ejpam-5571	77	65	d(p′1	d(p′1	PROPN
ejpam-5571	77	66	,	,	PUNCT
ejpam-5571	77	67	p	p	NOUN
ejpam-5571	77	68	′	′	NOUN
ejpam-5571	77	69	2	2	NUM
ejpam-5571	77	70	)	)	PUNCT
ejpam-5571	77	71	,	,	PUNCT
ejpam-5571	77	72	d(p2	d(p2	ADJ
ejpam-5571	77	73	,	,	PUNCT
ejpam-5571	77	74	p3	p3	PROPN
ejpam-5571	77	75	)	)	PUNCT
ejpam-5571	77	76	=	=	SYM
ejpam-5571	77	77	d(p′2	d(p′2	PROPN
ejpam-5571	77	78	,	,	PUNCT
ejpam-5571	77	79	p	p	NOUN
ejpam-5571	77	80	′	′	NOUN
ejpam-5571	77	81	3	3	NUM
ejpam-5571	77	82	)	)	PUNCT
ejpam-5571	77	83	,	,	PUNCT
ejpam-5571	77	84	d(p3	d(p3	PROPN
ejpam-5571	77	85	,	,	PUNCT
ejpam-5571	77	86	p4	p4	ADJ
ejpam-5571	77	87	)	)	PUNCT
ejpam-5571	77	88	=	=	SYM
ejpam-5571	77	89	d(p′3	d(p′3	PROPN
ejpam-5571	77	90	,	,	PUNCT
ejpam-5571	77	91	p	p	NOUN
ejpam-5571	77	92	′	′	NOUN
ejpam-5571	77	93	4	4	NUM
ejpam-5571	77	94	)	)	PUNCT
ejpam-5571	77	95	,	,	PUNCT
ejpam-5571	77	96	d(p4	d(p4	NOUN
ejpam-5571	77	97	,	,	PUNCT
ejpam-5571	77	98	p5	p5	ADJ
ejpam-5571	77	99	)	)	PUNCT
ejpam-5571	77	100	=	=	SYM
ejpam-5571	78	1	d(p′4	d(p′4	PROPN
ejpam-5571	78	2	,	,	PUNCT
ejpam-5571	78	3	p	p	NOUN
ejpam-5571	78	4	′	′	NOUN
ejpam-5571	78	5	5	5	NUM
ejpam-5571	78	6	)	)	PUNCT
ejpam-5571	78	7	and	and	CCONJ
ejpam-5571	78	8	d(p1	d(p1	NOUN
ejpam-5571	78	9	,	,	PUNCT
ejpam-5571	78	10	p4	p4	ADJ
ejpam-5571	78	11	)	)	PUNCT
ejpam-5571	78	12	=	=	SYM
ejpam-5571	78	13	d(p′1	d(p′1	PROPN
ejpam-5571	78	14	,	,	PUNCT
ejpam-5571	78	15	p	p	NOUN
ejpam-5571	78	16	′	′	NUM
ejpam-5571	78	17	4	4	NUM
ejpam-5571	78	18	then	then	ADV
ejpam-5571	78	19	∠p1(p2	∠p1(p2	ADJ
ejpam-5571	78	20	,	,	PUNCT
ejpam-5571	78	21	p4	p4	ADJ
ejpam-5571	78	22	)	)	PUNCT
ejpam-5571	78	23	≤	≤	NUM
ejpam-5571	78	24	∠p1(p2	∠p1(p2	NOUN
ejpam-5571	78	25	,	,	PUNCT
ejpam-5571	78	26	p3	p3	PROPN
ejpam-5571	78	27	)	)	PUNCT
ejpam-5571	79	1	+	+	CCONJ
ejpam-5571	79	2	∠p1(p4	∠p1(p4	ADJ
ejpam-5571	79	3	,	,	PUNCT
ejpam-5571	79	4	p3	p3	PROPN
ejpam-5571	79	5	)	)	PUNCT
ejpam-5571	79	6	≥	≥	X
ejpam-5571	79	7	∠p′1	∠p′1	X
ejpam-5571	79	8	(	(	PUNCT
ejpam-5571	79	9	p′2	p′2	INTJ
ejpam-5571	79	10	,	,	PUNCT
ejpam-5571	79	11	p	p	NOUN
ejpam-5571	79	12	′	′	NOUN
ejpam-5571	79	13	3	3	NUM
ejpam-5571	79	14	)	)	PUNCT
ejpam-5571	80	1	+	+	CCONJ
ejpam-5571	80	2	∠p′1	∠p′1	X
ejpam-5571	80	3	(	(	PUNCT
ejpam-5571	80	4	p′4	p′4	NOUN
ejpam-5571	80	5	,	,	PUNCT
ejpam-5571	80	6	p	p	NOUN
ejpam-5571	80	7	′	′	NOUN
ejpam-5571	80	8	3	3	NUM
ejpam-5571	80	9	)	)	PUNCT
ejpam-5571	80	10	≥	≥	NOUN
ejpam-5571	80	11	∠p′1	∠p′1	X
ejpam-5571	80	12	(	(	PUNCT
ejpam-5571	80	13	p′2	p′2	INTJ
ejpam-5571	80	14	,	,	PUNCT
ejpam-5571	80	15	p	p	NOUN
ejpam-5571	80	16	′	′	NOUN
ejpam-5571	80	17	4	4	NUM
ejpam-5571	80	18	)	)	PUNCT
ejpam-5571	80	19	.	.	PUNCT
ejpam-5571	81	1	hence	hence	ADV
ejpam-5571	81	2	,	,	PUNCT
ejpam-5571	81	3	it	it	PRON
ejpam-5571	81	4	is	be	AUX
ejpam-5571	81	5	not	not	PART
ejpam-5571	81	6	possible	possible	ADJ
ejpam-5571	81	7	to	to	PART
ejpam-5571	81	8	compare	compare	VERB
ejpam-5571	81	9	the	the	DET
ejpam-5571	81	10	values	value	NOUN
ejpam-5571	81	11	of	of	ADP
ejpam-5571	81	12	∠p1(p2	∠p1(p2	ADJ
ejpam-5571	81	13	,	,	PUNCT
ejpam-5571	81	14	p4	p4	ADJ
ejpam-5571	81	15	)	)	PUNCT
ejpam-5571	81	16	and	and	CCONJ
ejpam-5571	81	17	∠p′1	∠p′1	X
ejpam-5571	81	18	(	(	PUNCT
ejpam-5571	81	19	p′2	p′2	NOUN
ejpam-5571	81	20	,	,	PUNCT
ejpam-5571	81	21	p	p	NOUN
ejpam-5571	81	22	′	′	NOUN
ejpam-5571	81	23	4	4	NUM
ejpam-5571	81	24	)	)	PUNCT
ejpam-5571	81	25	.	.	PUNCT
ejpam-5571	82	1	therefore	therefore	ADV
ejpam-5571	82	2	,	,	PUNCT
ejpam-5571	82	3	we	we	PRON
ejpam-5571	82	4	must	must	AUX
ejpam-5571	82	5	suppose	suppose	VERB
ejpam-5571	82	6	that	that	SCONJ
ejpam-5571	82	7	the	the	DET
ejpam-5571	82	8	equation	equation	NOUN
ejpam-5571	82	9	∠p1(p2	∠p1(p2	NOUN
ejpam-5571	82	10	,	,	PUNCT
ejpam-5571	82	11	p4	p4	ADJ
ejpam-5571	82	12	)	)	PUNCT
ejpam-5571	82	13	=	=	SYM
ejpam-5571	82	14	∠p1(p2	∠p1(p2	ADJ
ejpam-5571	82	15	,	,	PUNCT
ejpam-5571	82	16	p3	p3	PROPN
ejpam-5571	82	17	)	)	PUNCT
ejpam-5571	83	1	+	+	VERB
ejpam-5571	83	2	∠p1(p4	∠p1(p4	ADJ
ejpam-5571	83	3	,	,	PUNCT
ejpam-5571	83	4	p3	p3	NOUN
ejpam-5571	83	5	)	)	PUNCT
ejpam-5571	83	6	is	be	AUX
ejpam-5571	83	7	true	true	ADJ
ejpam-5571	83	8	in	in	ADP
ejpam-5571	83	9	order	order	NOUN
ejpam-5571	83	10	to	to	PART
ejpam-5571	83	11	obtain	obtain	VERB
ejpam-5571	83	12	∠p1(p2	∠p1(p2	ADJ
ejpam-5571	83	13	,	,	PUNCT
ejpam-5571	83	14	p4	p4	ADJ
ejpam-5571	83	15	)	)	PUNCT
ejpam-5571	83	16	≥	≥	NOUN
ejpam-5571	83	17	∠p′1	∠p′1	X
ejpam-5571	83	18	(	(	PUNCT
ejpam-5571	83	19	p′2	p′2	INTJ
ejpam-5571	83	20	,	,	PUNCT
ejpam-5571	83	21	p	p	NOUN
ejpam-5571	83	22	′	′	NOUN
ejpam-5571	83	23	4	4	NUM
ejpam-5571	83	24	)	)	PUNCT
ejpam-5571	83	25	.	.	PUNCT
ejpam-5571	84	1	lemma	lemma	PROPN
ejpam-5571	84	2	1	1	X
ejpam-5571	84	3	.	.	PUNCT
ejpam-5571	85	1	let	let	VERB
ejpam-5571	85	2	x	x	PRON
ejpam-5571	85	3	be	be	AUX
ejpam-5571	85	4	a	a	DET
ejpam-5571	85	5	metric	metric	ADJ
ejpam-5571	85	6	space	space	NOUN
ejpam-5571	85	7	with	with	ADP
ejpam-5571	85	8	curvature	curvature	NOUN
ejpam-5571	85	9	bounded	bound	VERB
ejpam-5571	85	10	below	below	ADV
ejpam-5571	85	11	by	by	ADP
ejpam-5571	85	12	k	k	PROPN
ejpam-5571	85	13	in	in	ADP
ejpam-5571	85	14	the	the	DET
ejpam-5571	85	15	large	large	NOUN
ejpam-5571	85	16	.	.	PUNCT
ejpam-5571	86	1	let	let	VERB
ejpam-5571	86	2	△	△	X
ejpam-5571	86	3	(	(	PUNCT
ejpam-5571	86	4	p	p	X
ejpam-5571	86	5	,	,	PUNCT
ejpam-5571	86	6	q	q	ADJ
ejpam-5571	86	7	,	,	PUNCT
ejpam-5571	86	8	r	r	NOUN
ejpam-5571	86	9	)	)	PUNCT
ejpam-5571	86	10	be	be	AUX
ejpam-5571	86	11	a	a	DET
ejpam-5571	86	12	triangle	triangle	NOUN
ejpam-5571	86	13	in	in	ADP
ejpam-5571	86	14	x	x	X
ejpam-5571	86	15	and	and	CCONJ
ejpam-5571	86	16	△	△	PROPN
ejpam-5571	86	17	(	(	PUNCT
ejpam-5571	86	18	p′	p′	NOUN
ejpam-5571	86	19	,	,	PUNCT
ejpam-5571	86	20	q′	q′	NOUN
ejpam-5571	86	21	,	,	PUNCT
ejpam-5571	86	22	r′	r′	PROPN
ejpam-5571	86	23	)	)	PUNCT
ejpam-5571	86	24	be	be	VERB
ejpam-5571	86	25	its	its	PRON
ejpam-5571	86	26	comparison	comparison	NOUN
ejpam-5571	86	27	triangle	triangle	NOUN
ejpam-5571	86	28	in	in	ADP
ejpam-5571	86	29	rk	rk	PROPN
ejpam-5571	86	30	,	,	PUNCT
ejpam-5571	86	31	x	x	ADJ
ejpam-5571	86	32	be	be	AUX
ejpam-5571	86	33	a	a	DET
ejpam-5571	86	34	point	point	NOUN
ejpam-5571	86	35	on	on	ADP
ejpam-5571	86	36	[	[	X
ejpam-5571	86	37	p	p	X
ejpam-5571	86	38	,	,	PUNCT
ejpam-5571	86	39	q	q	X
ejpam-5571	86	40	]	]	X
ejpam-5571	86	41	,	,	PUNCT
ejpam-5571	86	42	and	and	CCONJ
ejpam-5571	86	43	x′	x′	PROPN
ejpam-5571	86	44	∈	∈	PROPN
ejpam-5571	86	45	[	[	X
ejpam-5571	86	46	p′	p′	NOUN
ejpam-5571	86	47	,	,	PUNCT
ejpam-5571	86	48	q′	q′	NOUN
ejpam-5571	86	49	]	]	PUNCT
ejpam-5571	86	50	be	be	VERB
ejpam-5571	86	51	a	a	DET
ejpam-5571	86	52	comparison	comparison	NOUN
ejpam-5571	86	53	point	point	NOUN
ejpam-5571	86	54	of	of	ADP
ejpam-5571	86	55	x.	x.	NOUN
ejpam-5571	87	1	if	if	SCONJ
ejpam-5571	87	2	∠p(x	∠p(x	NOUN
ejpam-5571	87	3	,	,	PUNCT
ejpam-5571	87	4	r	r	NOUN
ejpam-5571	87	5	)	)	PUNCT
ejpam-5571	87	6	=	=	SYM
ejpam-5571	87	7	∠p′(x	∠p′(x	NOUN
ejpam-5571	87	8	′	′	NUM
ejpam-5571	87	9	,	,	PUNCT
ejpam-5571	87	10	r′	r′	PROPN
ejpam-5571	87	11	)	)	PUNCT
ejpam-5571	87	12	,	,	PUNCT
ejpam-5571	87	13	then	then	ADV
ejpam-5571	87	14	d(x	d(x	PROPN
ejpam-5571	87	15	,	,	PUNCT
ejpam-5571	87	16	r	r	NOUN
ejpam-5571	87	17	)	)	PUNCT
ejpam-5571	87	18	=	=	PUNCT
ejpam-5571	87	19	d(x′	d(x′	PROPN
ejpam-5571	87	20	,	,	PUNCT
ejpam-5571	87	21	r′	r′	PROPN
ejpam-5571	87	22	)	)	PUNCT
ejpam-5571	87	23	.	.	PUNCT
ejpam-5571	88	1	c.	c.	PROPN
ejpam-5571	88	2	phokaew	phokaew	PROPN
ejpam-5571	88	3	,	,	PUNCT
ejpam-5571	88	4	a.	a.	PROPN
ejpam-5571	88	5	sama	sama	PROPN
ejpam-5571	88	6	-	-	PUNCT
ejpam-5571	88	7	ae	ae	PROPN
ejpam-5571	88	8	,	,	PUNCT
ejpam-5571	88	9	/	/	SYM
ejpam-5571	88	10	eur	eur	NOUN
ejpam-5571	88	11	.	.	PUNCT
ejpam-5571	89	1	j.	j.	PROPN
ejpam-5571	89	2	pure	pure	PROPN
ejpam-5571	89	3	appl	appl	PROPN
ejpam-5571	89	4	.	.	PROPN
ejpam-5571	89	5	math	math	PROPN
ejpam-5571	89	6	,	,	PUNCT
ejpam-5571	89	7	17	17	NUM
ejpam-5571	89	8	(	(	PUNCT
ejpam-5571	89	9	4	4	NUM
ejpam-5571	89	10	)	)	PUNCT
ejpam-5571	89	11	(	(	PUNCT
ejpam-5571	89	12	2024	2024	NUM
ejpam-5571	89	13	)	)	PUNCT
ejpam-5571	89	14	,	,	PUNCT
ejpam-5571	89	15	3932	3932	NUM
ejpam-5571	89	16	-	-	SYM
ejpam-5571	89	17	3944	3944	NUM
ejpam-5571	89	18	3935	3935	NUM
ejpam-5571	89	19	proof	proof	NOUN
ejpam-5571	89	20	.	.	PUNCT
ejpam-5571	90	1	since	since	SCONJ
ejpam-5571	90	2	x	x	PRON
ejpam-5571	90	3	is	be	AUX
ejpam-5571	90	4	a	a	DET
ejpam-5571	90	5	metric	metric	ADJ
ejpam-5571	90	6	space	space	NOUN
ejpam-5571	90	7	with	with	ADP
ejpam-5571	90	8	curvature	curvature	NOUN
ejpam-5571	90	9	bounded	bound	VERB
ejpam-5571	90	10	below	below	ADV
ejpam-5571	90	11	by	by	ADP
ejpam-5571	90	12	k	k	PROPN
ejpam-5571	90	13	in	in	ADP
ejpam-5571	90	14	the	the	DET
ejpam-5571	90	15	large	large	ADJ
ejpam-5571	90	16	,	,	PUNCT
ejpam-5571	90	17	we	we	PRON
ejpam-5571	90	18	have	have	VERB
ejpam-5571	90	19	that	that	DET
ejpam-5571	90	20	d(x	d(x	NOUN
ejpam-5571	90	21	,	,	PUNCT
ejpam-5571	90	22	r	r	NOUN
ejpam-5571	90	23	)	)	PUNCT
ejpam-5571	90	24	≥	≥	NOUN
ejpam-5571	90	25	d(x′	d(x′	PROPN
ejpam-5571	90	26	,	,	PUNCT
ejpam-5571	90	27	r′	r′	NUM
ejpam-5571	90	28	)	)	PUNCT
ejpam-5571	90	29	.	.	PUNCT
ejpam-5571	91	1	using	use	VERB
ejpam-5571	91	2	theorem	theorem	NOUN
ejpam-5571	91	3	2	2	NUM
ejpam-5571	91	4	,	,	PUNCT
ejpam-5571	91	5	we	we	PRON
ejpam-5571	91	6	have	have	VERB
ejpam-5571	91	7	d(x	d(x	NOUN
ejpam-5571	91	8	,	,	PUNCT
ejpam-5571	91	9	r	r	NOUN
ejpam-5571	91	10	)	)	PUNCT
ejpam-5571	91	11	≤	≤	NOUN
ejpam-5571	91	12	d(x′	d(x′	PROPN
ejpam-5571	91	13	,	,	PUNCT
ejpam-5571	91	14	r′	r′	PROPN
ejpam-5571	91	15	)	)	PUNCT
ejpam-5571	91	16	.	.	PUNCT
ejpam-5571	92	1	hence	hence	ADV
ejpam-5571	92	2	,	,	PUNCT
ejpam-5571	92	3	we	we	PRON
ejpam-5571	92	4	have	have	VERB
ejpam-5571	92	5	the	the	DET
ejpam-5571	92	6	result	result	NOUN
ejpam-5571	92	7	.	.	PUNCT
ejpam-5571	93	1	theorem	theorem	NOUN
ejpam-5571	93	2	3	3	X
ejpam-5571	93	3	.	.	PUNCT
ejpam-5571	94	1	let	let	VERB
ejpam-5571	94	2	x	x	PRON
ejpam-5571	94	3	be	be	AUX
ejpam-5571	94	4	a	a	DET
ejpam-5571	94	5	metric	metric	ADJ
ejpam-5571	94	6	space	space	NOUN
ejpam-5571	94	7	with	with	ADP
ejpam-5571	94	8	curvature	curvature	NOUN
ejpam-5571	94	9	bounded	bound	VERB
ejpam-5571	94	10	below	below	ADV
ejpam-5571	94	11	by	by	ADP
ejpam-5571	94	12	k	k	PROPN
ejpam-5571	94	13	in	in	ADP
ejpam-5571	94	14	the	the	DET
ejpam-5571	94	15	large	large	NOUN
ejpam-5571	94	16	.	.	PUNCT
ejpam-5571	95	1	let	let	VERB
ejpam-5571	95	2	△	△	X
ejpam-5571	95	3	(	(	PUNCT
ejpam-5571	95	4	p	p	X
ejpam-5571	95	5	,	,	PUNCT
ejpam-5571	95	6	q	q	ADJ
ejpam-5571	95	7	,	,	PUNCT
ejpam-5571	95	8	r	r	NOUN
ejpam-5571	95	9	)	)	PUNCT
ejpam-5571	95	10	be	be	AUX
ejpam-5571	95	11	a	a	DET
ejpam-5571	95	12	triangle	triangle	NOUN
ejpam-5571	95	13	in	in	ADP
ejpam-5571	95	14	x	x	X
ejpam-5571	95	15	and	and	CCONJ
ejpam-5571	95	16	△	△	PROPN
ejpam-5571	95	17	(	(	PUNCT
ejpam-5571	95	18	p′	p′	NOUN
ejpam-5571	95	19	,	,	PUNCT
ejpam-5571	95	20	q′	q′	NOUN
ejpam-5571	95	21	,	,	PUNCT
ejpam-5571	95	22	r′	r′	PROPN
ejpam-5571	95	23	)	)	PUNCT
ejpam-5571	95	24	be	be	VERB
ejpam-5571	95	25	its	its	PRON
ejpam-5571	95	26	comparison	comparison	NOUN
ejpam-5571	95	27	triangle	triangle	NOUN
ejpam-5571	95	28	in	in	ADP
ejpam-5571	95	29	rk	rk	PROPN
ejpam-5571	95	30	.	.	PUNCT
ejpam-5571	96	1	suppose	suppose	VERB
ejpam-5571	96	2	that	that	SCONJ
ejpam-5571	96	3	the	the	DET
ejpam-5571	96	4	following	follow	VERB
ejpam-5571	96	5	statements	statement	NOUN
ejpam-5571	96	6	hold	hold	VERB
ejpam-5571	96	7	:	:	PUNCT
ejpam-5571	96	8	(	(	PUNCT
ejpam-5571	96	9	i	i	NOUN
ejpam-5571	96	10	)	)	PUNCT
ejpam-5571	96	11	∠r(p	∠r(p	ADJ
ejpam-5571	96	12	,	,	PUNCT
ejpam-5571	96	13	q	q	NOUN
ejpam-5571	96	14	)	)	PUNCT
ejpam-5571	96	15	=	=	SYM
ejpam-5571	96	16	∠r′(p	∠r′(p	NOUN
ejpam-5571	96	17	′	′	NOUN
ejpam-5571	96	18	,	,	PUNCT
ejpam-5571	96	19	q′	q′	NOUN
ejpam-5571	96	20	)	)	PUNCT
ejpam-5571	96	21	;	;	PUNCT
ejpam-5571	96	22	(	(	PUNCT
ejpam-5571	96	23	ii	ii	NOUN
ejpam-5571	96	24	)	)	PUNCT
ejpam-5571	96	25	∠p(x	∠p(x	NOUN
ejpam-5571	96	26	,	,	PUNCT
ejpam-5571	96	27	r	r	NOUN
ejpam-5571	96	28	)	)	PUNCT
ejpam-5571	96	29	=	=	PUNCT
ejpam-5571	96	30	∠p(q	∠p(q	ADV
ejpam-5571	96	31	,	,	PUNCT
ejpam-5571	96	32	r	r	NOUN
ejpam-5571	96	33	)	)	PUNCT
ejpam-5571	96	34	,	,	PUNCT
ejpam-5571	96	35	∠q(x	∠q(x	NOUN
ejpam-5571	96	36	,	,	PUNCT
ejpam-5571	96	37	r	r	NOUN
ejpam-5571	96	38	)	)	PUNCT
ejpam-5571	96	39	=	=	SYM
ejpam-5571	97	1	∠q(p	∠q(p	NOUN
ejpam-5571	97	2	,	,	PUNCT
ejpam-5571	97	3	r	r	NOUN
ejpam-5571	97	4	)	)	PUNCT
ejpam-5571	97	5	and	and	CCONJ
ejpam-5571	97	6	∠r(p	∠r(p	NOUN
ejpam-5571	97	7	,	,	PUNCT
ejpam-5571	97	8	q	q	NOUN
ejpam-5571	97	9	)	)	PUNCT
ejpam-5571	97	10	=	=	SYM
ejpam-5571	97	11	∠r(p	∠r(p	NOUN
ejpam-5571	97	12	,	,	PUNCT
ejpam-5571	97	13	x	x	X
ejpam-5571	97	14	)	)	PUNCT
ejpam-5571	98	1	+	+	CCONJ
ejpam-5571	98	2	∠r(q	∠r(q	NOUN
ejpam-5571	98	3	,	,	PUNCT
ejpam-5571	98	4	x	x	X
ejpam-5571	98	5	)	)	PUNCT
ejpam-5571	98	6	for	for	ADP
ejpam-5571	98	7	any	any	DET
ejpam-5571	98	8	x	x	SYM
ejpam-5571	98	9	∈	∈	PROPN
ejpam-5571	98	10	[	[	X
ejpam-5571	98	11	p	p	X
ejpam-5571	98	12	,	,	PUNCT
ejpam-5571	98	13	q	q	X
ejpam-5571	98	14	]	]	X
ejpam-5571	98	15	.	.	PUNCT
ejpam-5571	99	1	then	then	ADV
ejpam-5571	99	2	the	the	DET
ejpam-5571	99	3	convex	convex	PROPN
ejpam-5571	99	4	hull	hull	NOUN
ejpam-5571	99	5	of	of	ADP
ejpam-5571	99	6	△	△	X
ejpam-5571	99	7	(	(	PUNCT
ejpam-5571	99	8	p	p	X
ejpam-5571	99	9	,	,	PUNCT
ejpam-5571	99	10	q	q	ADJ
ejpam-5571	99	11	,	,	PUNCT
ejpam-5571	99	12	r	r	NOUN
ejpam-5571	99	13	)	)	PUNCT
ejpam-5571	99	14	is	be	AUX
ejpam-5571	99	15	isometric	isometric	ADJ
ejpam-5571	99	16	to	to	ADP
ejpam-5571	99	17	the	the	DET
ejpam-5571	99	18	convex	convex	PROPN
ejpam-5571	99	19	hull	hull	NOUN
ejpam-5571	99	20	of	of	ADP
ejpam-5571	99	21	△	△	X
ejpam-5571	99	22	(	(	PUNCT
ejpam-5571	99	23	p′	p′	NOUN
ejpam-5571	99	24	,	,	PUNCT
ejpam-5571	99	25	q′	q′	NOUN
ejpam-5571	99	26	,	,	PUNCT
ejpam-5571	99	27	r′	r′	PROPN
ejpam-5571	99	28	)	)	PUNCT
ejpam-5571	99	29	.	.	PUNCT
ejpam-5571	100	1	proof	proof	NOUN
ejpam-5571	100	2	.	.	PUNCT
ejpam-5571	101	1	by	by	ADP
ejpam-5571	101	2	lemma	lemma	PROPN
ejpam-5571	101	3	1	1	NUM
ejpam-5571	101	4	,	,	PUNCT
ejpam-5571	101	5	we	we	PRON
ejpam-5571	101	6	have	have	VERB
ejpam-5571	101	7	that	that	DET
ejpam-5571	101	8	d(x	d(x	NOUN
ejpam-5571	101	9	,	,	PUNCT
ejpam-5571	101	10	r	r	NOUN
ejpam-5571	101	11	)	)	PUNCT
ejpam-5571	101	12	=	=	PUNCT
ejpam-5571	101	13	d(x′	d(x′	PROPN
ejpam-5571	101	14	,	,	PUNCT
ejpam-5571	101	15	r′	r′	PROPN
ejpam-5571	101	16	)	)	PUNCT
ejpam-5571	101	17	if	if	SCONJ
ejpam-5571	101	18	x	x	PUNCT
ejpam-5571	101	19	∈	∈	PROPN
ejpam-5571	101	20	[	[	X
ejpam-5571	101	21	p	p	X
ejpam-5571	101	22	,	,	PUNCT
ejpam-5571	101	23	q	q	X
ejpam-5571	101	24	]	]	PUNCT
ejpam-5571	101	25	and	and	CCONJ
ejpam-5571	101	26	x′	x′	PROPN
ejpam-5571	101	27	∈	∈	PROPN
ejpam-5571	101	28	[	[	X
ejpam-5571	101	29	p′	p′	NOUN
ejpam-5571	101	30	,	,	PUNCT
ejpam-5571	101	31	q′	q′	NOUN
ejpam-5571	101	32	]	]	PUNCT
ejpam-5571	101	33	such	such	ADJ
ejpam-5571	101	34	that	that	SCONJ
ejpam-5571	101	35	d(x	d(x	PROPN
ejpam-5571	101	36	,	,	PUNCT
ejpam-5571	101	37	p	p	X
ejpam-5571	101	38	)	)	PUNCT
ejpam-5571	101	39	=	=	SYM
ejpam-5571	101	40	d(x′	d(x′	PROPN
ejpam-5571	101	41	,	,	PUNCT
ejpam-5571	101	42	p′	p′	NOUN
ejpam-5571	101	43	)	)	PUNCT
ejpam-5571	101	44	.	.	PUNCT
ejpam-5571	102	1	let	let	VERB
ejpam-5571	102	2	j	j	PROPN
ejpam-5571	102	3	be	be	AUX
ejpam-5571	102	4	the	the	DET
ejpam-5571	102	5	map	map	NOUN
ejpam-5571	102	6	from	from	ADP
ejpam-5571	102	7	the	the	DET
ejpam-5571	102	8	convex	convex	PROPN
ejpam-5571	102	9	hull	hull	NOUN
ejpam-5571	102	10	c(	c(	X
ejpam-5571	102	11	△	△	X
ejpam-5571	102	12	(p′	(p′	X
ejpam-5571	102	13	,	,	PUNCT
ejpam-5571	102	14	q′	q′	NOUN
ejpam-5571	102	15	,	,	PUNCT
ejpam-5571	102	16	r′	r′	PROPN
ejpam-5571	102	17	)	)	PUNCT
ejpam-5571	102	18	)	)	PUNCT
ejpam-5571	102	19	in	in	ADP
ejpam-5571	102	20	rk	rk	NOUN
ejpam-5571	102	21	to	to	ADP
ejpam-5571	102	22	the	the	DET
ejpam-5571	102	23	convex	convex	PROPN
ejpam-5571	102	24	hull	hull	NOUN
ejpam-5571	102	25	c(	c(	X
ejpam-5571	102	26	△	△	X
ejpam-5571	102	27	(p	(p	NOUN
ejpam-5571	102	28	,	,	PUNCT
ejpam-5571	102	29	q	q	NOUN
ejpam-5571	102	30	,	,	PUNCT
ejpam-5571	102	31	r	r	NOUN
ejpam-5571	102	32	)	)	PUNCT
ejpam-5571	102	33	)	)	PUNCT
ejpam-5571	102	34	in	in	ADP
ejpam-5571	102	35	x	x	PUNCT
ejpam-5571	102	36	which	which	PRON
ejpam-5571	102	37	,	,	PUNCT
ejpam-5571	102	38	for	for	ADP
ejpam-5571	102	39	every	every	DET
ejpam-5571	102	40	x′	x′	PROPN
ejpam-5571	102	41	∈	∈	PROPN
ejpam-5571	102	42	[	[	X
ejpam-5571	102	43	p′	p′	NOUN
ejpam-5571	102	44	,	,	PUNCT
ejpam-5571	102	45	q′	q′	NOUN
ejpam-5571	102	46	]	]	PUNCT
ejpam-5571	102	47	,	,	PUNCT
ejpam-5571	102	48	sends	send	VERB
ejpam-5571	102	49	the	the	DET
ejpam-5571	102	50	geodesic	geodesic	ADJ
ejpam-5571	102	51	segment	segment	NOUN
ejpam-5571	102	52	[	[	X
ejpam-5571	102	53	r′	r′	X
ejpam-5571	102	54	,	,	PUNCT
ejpam-5571	102	55	x′	x′	PROPN
ejpam-5571	102	56	]	]	X
ejpam-5571	103	1	isometrically	isometrically	PROPN
ejpam-5571	103	2	onto	onto	ADP
ejpam-5571	103	3	the	the	DET
ejpam-5571	103	4	geodesic	geodesic	ADJ
ejpam-5571	103	5	segment	segment	NOUN
ejpam-5571	104	1	[	[	X
ejpam-5571	104	2	r	r	X
ejpam-5571	104	3	,	,	PUNCT
ejpam-5571	104	4	x	x	NOUN
ejpam-5571	104	5	]	]	X
ejpam-5571	104	6	.	.	PUNCT
ejpam-5571	105	1	we	we	PRON
ejpam-5571	105	2	claim	claim	VERB
ejpam-5571	105	3	that	that	SCONJ
ejpam-5571	105	4	j	j	PROPN
ejpam-5571	105	5	is	be	AUX
ejpam-5571	105	6	an	an	DET
ejpam-5571	105	7	isometry	isometry	NOUN
ejpam-5571	105	8	onto	onto	ADP
ejpam-5571	105	9	its	its	PRON
ejpam-5571	105	10	image	image	NOUN
ejpam-5571	105	11	;	;	PUNCT
ejpam-5571	105	12	it	it	PRON
ejpam-5571	105	13	then	then	ADV
ejpam-5571	105	14	follows	follow	VERB
ejpam-5571	105	15	that	that	SCONJ
ejpam-5571	105	16	the	the	DET
ejpam-5571	105	17	unique	unique	ADJ
ejpam-5571	105	18	geodesic	geodesic	NOUN
ejpam-5571	105	19	joining	join	VERB
ejpam-5571	105	20	any	any	DET
ejpam-5571	105	21	two	two	NUM
ejpam-5571	105	22	points	point	NOUN
ejpam-5571	105	23	of	of	ADP
ejpam-5571	105	24	the	the	DET
ejpam-5571	105	25	image	image	NOUN
ejpam-5571	105	26	of	of	ADP
ejpam-5571	105	27	j	j	PROPN
ejpam-5571	105	28	will	will	AUX
ejpam-5571	105	29	be	be	AUX
ejpam-5571	105	30	contained	contain	VERB
ejpam-5571	105	31	in	in	ADP
ejpam-5571	105	32	the	the	DET
ejpam-5571	105	33	image	image	NOUN
ejpam-5571	105	34	,	,	PUNCT
ejpam-5571	105	35	so	so	ADV
ejpam-5571	105	36	j	j	PROPN
ejpam-5571	105	37	maps	map	VERB
ejpam-5571	105	38	c(	c(	ADP
ejpam-5571	105	39	△	△	NOUN
ejpam-5571	105	40	(p′	(p′	X
ejpam-5571	105	41	,	,	PUNCT
ejpam-5571	105	42	q′	q′	NOUN
ejpam-5571	105	43	,	,	PUNCT
ejpam-5571	105	44	r′	r′	PROPN
ejpam-5571	105	45	)	)	PUNCT
ejpam-5571	105	46	)	)	PUNCT
ejpam-5571	105	47	onto	onto	ADP
ejpam-5571	105	48	c(	c(	ADV
ejpam-5571	105	49	△	△	X
ejpam-5571	105	50	(p	(p	NOUN
ejpam-5571	105	51	,	,	PUNCT
ejpam-5571	105	52	q	q	NOUN
ejpam-5571	105	53	,	,	PUNCT
ejpam-5571	105	54	r	r	NOUN
ejpam-5571	105	55	)	)	PUNCT
ejpam-5571	105	56	)	)	PUNCT
ejpam-5571	105	57	.	.	PUNCT
ejpam-5571	106	1	consider	consider	VERB
ejpam-5571	106	2	two	two	NUM
ejpam-5571	106	3	points	point	NOUN
ejpam-5571	106	4	a′	a′	PROPN
ejpam-5571	106	5	∈	∈	PROPN
ejpam-5571	107	1	[	[	X
ejpam-5571	107	2	r′	r′	X
ejpam-5571	107	3	,	,	PUNCT
ejpam-5571	107	4	x′	x′	NUM
ejpam-5571	107	5	]	]	PUNCT
ejpam-5571	107	6	and	and	CCONJ
ejpam-5571	107	7	b′	b′	NUM
ejpam-5571	107	8	∈	∈	PROPN
ejpam-5571	107	9	[	[	X
ejpam-5571	107	10	r′	r′	X
ejpam-5571	107	11	,	,	PUNCT
ejpam-5571	107	12	y′	y′	NUM
ejpam-5571	107	13	]	]	PUNCT
ejpam-5571	107	14	in	in	ADP
ejpam-5571	107	15	c(	c(	ADJ
ejpam-5571	107	16	△	△	NOUN
ejpam-5571	107	17	(p′	(p′	X
ejpam-5571	107	18	,	,	PUNCT
ejpam-5571	107	19	q′	q′	NOUN
ejpam-5571	107	20	,	,	PUNCT
ejpam-5571	107	21	r′	r′	PROPN
ejpam-5571	107	22	)	)	PUNCT
ejpam-5571	107	23	)	)	PUNCT
ejpam-5571	107	24	,	,	PUNCT
ejpam-5571	107	25	where	where	SCONJ
ejpam-5571	107	26	x′	x′	PROPN
ejpam-5571	107	27	,	,	PUNCT
ejpam-5571	107	28	y′	y′	NOUN
ejpam-5571	107	29	∈	∈	PROPN
ejpam-5571	107	30	[	[	X
ejpam-5571	107	31	p′	p′	NOUN
ejpam-5571	107	32	,	,	PUNCT
ejpam-5571	107	33	q′	q′	NOUN
ejpam-5571	107	34	]	]	PUNCT
ejpam-5571	107	35	and	and	CCONJ
ejpam-5571	107	36	x′	x′	PROPN
ejpam-5571	107	37	is	be	AUX
ejpam-5571	107	38	between	between	ADP
ejpam-5571	107	39	q′	q′	NOUN
ejpam-5571	107	40	and	and	CCONJ
ejpam-5571	107	41	y′.	y′.	PROPN
ejpam-5571	107	42	let	let	VERB
ejpam-5571	107	43	x	x	SYM
ejpam-5571	107	44	=	=	PROPN
ejpam-5571	107	45	j(x′	j(x′	PROPN
ejpam-5571	107	46	)	)	PUNCT
ejpam-5571	107	47	,	,	PUNCT
ejpam-5571	107	48	y	y	PROPN
ejpam-5571	107	49	=	=	PUNCT
ejpam-5571	107	50	j(y′	j(y′	PROPN
ejpam-5571	107	51	)	)	PUNCT
ejpam-5571	107	52	,	,	PUNCT
ejpam-5571	107	53	a	a	DET
ejpam-5571	107	54	=	=	NOUN
ejpam-5571	107	55	j(a′	j(a′	NOUN
ejpam-5571	107	56	)	)	PUNCT
ejpam-5571	107	57	and	and	CCONJ
ejpam-5571	107	58	b	b	X
ejpam-5571	107	59	=	=	PROPN
ejpam-5571	107	60	j(b′	j(b′	PROPN
ejpam-5571	107	61	)	)	PUNCT
ejpam-5571	107	62	.	.	PUNCT
ejpam-5571	108	1	since	since	SCONJ
ejpam-5571	108	2	d(r	d(r	PROPN
ejpam-5571	108	3	,	,	PUNCT
ejpam-5571	108	4	x	x	NOUN
ejpam-5571	108	5	)	)	PUNCT
ejpam-5571	108	6	=	=	SYM
ejpam-5571	108	7	d(r′	d(r′	PROPN
ejpam-5571	108	8	,	,	PUNCT
ejpam-5571	108	9	x′	x′	NUM
ejpam-5571	108	10	)	)	PUNCT
ejpam-5571	108	11	,	,	PUNCT
ejpam-5571	108	12	d(r	d(r	PROPN
ejpam-5571	108	13	,	,	PUNCT
ejpam-5571	108	14	y	y	NOUN
ejpam-5571	108	15	)	)	PUNCT
ejpam-5571	108	16	=	=	PUNCT
ejpam-5571	108	17	d(r′	d(r′	PROPN
ejpam-5571	108	18	,	,	PUNCT
ejpam-5571	108	19	y′	y′	NUM
ejpam-5571	108	20	)	)	PUNCT
ejpam-5571	108	21	and	and	CCONJ
ejpam-5571	108	22	d(x	d(x	PROPN
ejpam-5571	108	23	,	,	PUNCT
ejpam-5571	108	24	y	y	NOUN
ejpam-5571	108	25	)	)	PUNCT
ejpam-5571	108	26	=	=	SYM
ejpam-5571	108	27	d(x′	d(x′	PROPN
ejpam-5571	108	28	,	,	PUNCT
ejpam-5571	108	29	y′	y′	NUM
ejpam-5571	108	30	)	)	PUNCT
ejpam-5571	108	31	,	,	PUNCT
ejpam-5571	108	32	we	we	PRON
ejpam-5571	108	33	have	have	VERB
ejpam-5571	108	34	that	that	PRON
ejpam-5571	108	35	△	△	X
ejpam-5571	108	36	(	(	PUNCT
ejpam-5571	108	37	r′	r′	NUM
ejpam-5571	108	38	,	,	PUNCT
ejpam-5571	108	39	x′	x′	NUM
ejpam-5571	108	40	,	,	PUNCT
ejpam-5571	108	41	y′	y′	NUM
ejpam-5571	108	42	)	)	PUNCT
ejpam-5571	108	43	is	be	AUX
ejpam-5571	108	44	a	a	DET
ejpam-5571	108	45	comparison	comparison	NOUN
ejpam-5571	108	46	triangle	triangle	NOUN
ejpam-5571	108	47	of	of	ADP
ejpam-5571	108	48	△	△	X
ejpam-5571	108	49	(	(	PUNCT
ejpam-5571	108	50	r	r	NOUN
ejpam-5571	108	51	,	,	PUNCT
ejpam-5571	108	52	x	x	NOUN
ejpam-5571	108	53	,	,	PUNCT
ejpam-5571	108	54	y	y	PROPN
ejpam-5571	108	55	)	)	PUNCT
ejpam-5571	108	56	.	.	PUNCT
ejpam-5571	109	1	since	since	SCONJ
ejpam-5571	109	2	x	x	PRON
ejpam-5571	109	3	is	be	AUX
ejpam-5571	109	4	a	a	DET
ejpam-5571	109	5	metric	metric	ADJ
ejpam-5571	109	6	space	space	NOUN
ejpam-5571	109	7	with	with	ADP
ejpam-5571	109	8	curvature	curvature	NOUN
ejpam-5571	109	9	bounded	bound	VERB
ejpam-5571	109	10	below	below	ADV
ejpam-5571	109	11	by	by	ADP
ejpam-5571	109	12	k	k	PROPN
ejpam-5571	109	13	in	in	ADP
ejpam-5571	109	14	the	the	DET
ejpam-5571	109	15	large	large	ADJ
ejpam-5571	109	16	,	,	PUNCT
ejpam-5571	109	17	we	we	PRON
ejpam-5571	109	18	have	have	VERB
ejpam-5571	109	19	d(a	d(a	PROPN
ejpam-5571	109	20	,	,	PUNCT
ejpam-5571	109	21	b	b	NOUN
ejpam-5571	109	22	)	)	PUNCT
ejpam-5571	109	23	≥	≥	NOUN
ejpam-5571	109	24	d(a′	d(a′	NOUN
ejpam-5571	109	25	,	,	PUNCT
ejpam-5571	109	26	b′	b′	NUM
ejpam-5571	109	27	)	)	PUNCT
ejpam-5571	109	28	.	.	PUNCT
ejpam-5571	110	1	as	as	ADP
ejpam-5571	110	2	∠r(a	∠r(a	ADJ
ejpam-5571	110	3	,	,	PUNCT
ejpam-5571	110	4	b	b	NOUN
ejpam-5571	110	5	)	)	PUNCT
ejpam-5571	110	6	≤	≤	NOUN
ejpam-5571	110	7	∠r(x	∠r(x	X
ejpam-5571	110	8	,	,	PUNCT
ejpam-5571	110	9	y	y	NOUN
ejpam-5571	110	10	)	)	PUNCT
ejpam-5571	111	1	=	=	SYM
ejpam-5571	111	2	∠r′(x	∠r′(x	PROPN
ejpam-5571	111	3	′	′	NUM
ejpam-5571	111	4	,	,	PUNCT
ejpam-5571	111	5	y′	y′	NUM
ejpam-5571	111	6	)	)	PUNCT
ejpam-5571	111	7	=	=	SYM
ejpam-5571	111	8	∠r′(a	∠r′(a	NOUN
ejpam-5571	111	9	′	′	NOUN
ejpam-5571	111	10	,	,	PUNCT
ejpam-5571	111	11	b′	b′	NUM
ejpam-5571	111	12	)	)	PUNCT
ejpam-5571	111	13	,	,	PUNCT
ejpam-5571	111	14	applying	apply	VERB
ejpam-5571	111	15	theorem	theorem	NOUN
ejpam-5571	111	16	2	2	NUM
ejpam-5571	111	17	,	,	PUNCT
ejpam-5571	111	18	we	we	PRON
ejpam-5571	111	19	get	get	VERB
ejpam-5571	111	20	that	that	SCONJ
ejpam-5571	111	21	d(a	d(a	PROPN
ejpam-5571	111	22	,	,	PUNCT
ejpam-5571	111	23	b	b	NOUN
ejpam-5571	111	24	)	)	PUNCT
ejpam-5571	111	25	≤	≤	NOUN
ejpam-5571	111	26	d(a′	d(a′	NOUN
ejpam-5571	111	27	,	,	PUNCT
ejpam-5571	111	28	b′	b′	NUM
ejpam-5571	111	29	)	)	PUNCT
ejpam-5571	111	30	.	.	PUNCT
ejpam-5571	112	1	therefore	therefore	ADV
ejpam-5571	112	2	,	,	PUNCT
ejpam-5571	112	3	d(a	d(a	PROPN
ejpam-5571	112	4	,	,	PUNCT
ejpam-5571	112	5	b	b	NOUN
ejpam-5571	112	6	)	)	PUNCT
ejpam-5571	112	7	=	=	VERB
ejpam-5571	112	8	d(a′	d(a′	NOUN
ejpam-5571	112	9	,	,	PUNCT
ejpam-5571	112	10	b′	b′	NUM
ejpam-5571	112	11	)	)	PUNCT
ejpam-5571	112	12	,	,	PUNCT
ejpam-5571	112	13	as	as	SCONJ
ejpam-5571	112	14	required	require	VERB
ejpam-5571	112	15	.	.	PUNCT
ejpam-5571	113	1	we	we	PRON
ejpam-5571	113	2	then	then	ADV
ejpam-5571	113	3	describe	describe	VERB
ejpam-5571	113	4	that	that	SCONJ
ejpam-5571	113	5	the	the	DET
ejpam-5571	113	6	convex	convex	PROPN
ejpam-5571	113	7	hull	hull	NOUN
ejpam-5571	113	8	of	of	ADP
ejpam-5571	113	9	a	a	DET
ejpam-5571	113	10	closed	closed	ADJ
ejpam-5571	113	11	geodesic	geodesic	ADJ
ejpam-5571	113	12	polygon	polygon	NOUN
ejpam-5571	113	13	in	in	ADP
ejpam-5571	113	14	a	a	DET
ejpam-5571	113	15	metric	metric	ADJ
ejpam-5571	113	16	space	space	NOUN
ejpam-5571	113	17	with	with	ADP
ejpam-5571	113	18	curvature	curvature	NOUN
ejpam-5571	113	19	bounded	bound	VERB
ejpam-5571	113	20	below	below	ADV
ejpam-5571	113	21	is	be	AUX
ejpam-5571	113	22	isometric	isometric	ADJ
ejpam-5571	113	23	to	to	ADP
ejpam-5571	113	24	that	that	PRON
ejpam-5571	113	25	of	of	ADP
ejpam-5571	113	26	a	a	DET
ejpam-5571	113	27	polygon	polygon	NOUN
ejpam-5571	113	28	in	in	ADP
ejpam-5571	113	29	the	the	DET
ejpam-5571	113	30	model	model	NOUN
ejpam-5571	113	31	space	space	NOUN
ejpam-5571	113	32	rk	rk	PROPN
ejpam-5571	113	33	.	.	PUNCT
ejpam-5571	114	1	theorem	theorem	VERB
ejpam-5571	114	2	4	4	NUM
ejpam-5571	114	3	.	.	PUNCT
ejpam-5571	115	1	let	let	VERB
ejpam-5571	115	2	x	x	PRON
ejpam-5571	115	3	be	be	AUX
ejpam-5571	115	4	a	a	DET
ejpam-5571	115	5	metric	metric	ADJ
ejpam-5571	115	6	space	space	NOUN
ejpam-5571	115	7	with	with	ADP
ejpam-5571	115	8	curvature	curvature	NOUN
ejpam-5571	115	9	bounded	bound	VERB
ejpam-5571	115	10	below	below	ADV
ejpam-5571	115	11	by	by	ADP
ejpam-5571	115	12	k	k	PROPN
ejpam-5571	115	13	in	in	ADP
ejpam-5571	115	14	the	the	DET
ejpam-5571	115	15	large	large	NOUN
ejpam-5571	115	16	.	.	PUNCT
ejpam-5571	116	1	let	let	VERB
ejpam-5571	116	2	σ	σ	NOUN
ejpam-5571	116	3	be	be	AUX
ejpam-5571	116	4	a	a	DET
ejpam-5571	116	5	closed	closed	ADJ
ejpam-5571	116	6	geodesic	geodesic	ADJ
ejpam-5571	116	7	polygon	polygon	NOUN
ejpam-5571	116	8	with	with	ADP
ejpam-5571	116	9	ordered	order	VERB
ejpam-5571	116	10	vertices	vertex	NOUN
ejpam-5571	116	11	p1	p1	NOUN
ejpam-5571	116	12	,	,	PUNCT
ejpam-5571	116	13	p2	p2	NOUN
ejpam-5571	116	14	,	,	PUNCT
ejpam-5571	116	15	p3	p3	NOUN
ejpam-5571	116	16	,	,	PUNCT
ejpam-5571	116	17	p4	p4	ADJ
ejpam-5571	116	18	,	,	PUNCT
ejpam-5571	116	19	p1	p1	NOUN
ejpam-5571	116	20	with	with	ADP
ejpam-5571	116	21	perimeter	perimeter	NOUN
ejpam-5571	117	1	less	less	ADJ
ejpam-5571	117	2	than	than	ADP
ejpam-5571	117	3	π/	π/	ADV
ejpam-5571	117	4	√	√	VERB
ejpam-5571	117	5	k	k	NOUN
ejpam-5571	118	1	in	in	ADP
ejpam-5571	118	2	x	x	PUNCT
ejpam-5571	118	3	and	and	CCONJ
ejpam-5571	118	4	let	let	VERB
ejpam-5571	118	5	σ′	σ′	PROPN
ejpam-5571	118	6	be	be	AUX
ejpam-5571	118	7	a	a	DET
ejpam-5571	118	8	convex	convex	NOUN
ejpam-5571	118	9	polygon	polygon	NOUN
ejpam-5571	118	10	with	with	ADP
ejpam-5571	118	11	ordered	order	VERB
ejpam-5571	118	12	vertices	vertex	NOUN
ejpam-5571	118	13	p′1	p′1	NOUN
ejpam-5571	118	14	,	,	PUNCT
ejpam-5571	118	15	p	p	NOUN
ejpam-5571	118	16	′	′	NOUN
ejpam-5571	118	17	2	2	NUM
ejpam-5571	118	18	,	,	PUNCT
ejpam-5571	118	19	p	p	NOUN
ejpam-5571	118	20	′	′	NOUN
ejpam-5571	118	21	3	3	NUM
ejpam-5571	118	22	,	,	PUNCT
ejpam-5571	118	23	p	p	NOUN
ejpam-5571	118	24	′	′	NOUN
ejpam-5571	118	25	4	4	NUM
ejpam-5571	118	26	,	,	PUNCT
ejpam-5571	118	27	p	p	NOUN
ejpam-5571	118	28	′	′	NOUN
ejpam-5571	118	29	1	1	NUM
ejpam-5571	118	30	in	in	ADP
ejpam-5571	118	31	the	the	DET
ejpam-5571	118	32	model	model	NOUN
ejpam-5571	118	33	space	space	NOUN
ejpam-5571	118	34	rk	rk	PROPN
ejpam-5571	118	35	.	.	PUNCT
ejpam-5571	119	1	suppose	suppose	VERB
ejpam-5571	119	2	the	the	DET
ejpam-5571	119	3	following	follow	VERB
ejpam-5571	119	4	statements	statement	NOUN
ejpam-5571	119	5	hold	hold	VERB
ejpam-5571	119	6	:	:	PUNCT
ejpam-5571	119	7	(	(	PUNCT
ejpam-5571	119	8	i	i	NOUN
ejpam-5571	119	9	)	)	PUNCT
ejpam-5571	119	10	c({p1	c({p1	PROPN
ejpam-5571	119	11	,	,	PUNCT
ejpam-5571	119	12	p2	p2	NOUN
ejpam-5571	119	13	,	,	PUNCT
ejpam-5571	119	14	p3	p3	NOUN
ejpam-5571	119	15	}	}	PUNCT
ejpam-5571	119	16	)	)	PUNCT
ejpam-5571	119	17	and	and	CCONJ
ejpam-5571	119	18	c({p1	c({p1	PROPN
ejpam-5571	119	19	,	,	PUNCT
ejpam-5571	119	20	p3	p3	PROPN
ejpam-5571	119	21	,	,	PUNCT
ejpam-5571	119	22	p4	p4	ADJ
ejpam-5571	119	23	}	}	PUNCT
ejpam-5571	119	24	)	)	PUNCT
ejpam-5571	119	25	are	be	AUX
ejpam-5571	119	26	isometric	isometric	ADJ
ejpam-5571	119	27	to	to	ADP
ejpam-5571	119	28	c({p′1	c({p′1	PROPN
ejpam-5571	119	29	,	,	PUNCT
ejpam-5571	119	30	p′2	p′2	NOUN
ejpam-5571	119	31	,	,	PUNCT
ejpam-5571	119	32	p′3	p′3	NOUN
ejpam-5571	119	33	}	}	PUNCT
ejpam-5571	119	34	)	)	PUNCT
ejpam-5571	119	35	and	and	CCONJ
ejpam-5571	119	36	c({p′1	c({p′1	PROPN
ejpam-5571	119	37	,	,	PUNCT
ejpam-5571	119	38	p′3	p′3	NOUN
ejpam-5571	119	39	,	,	PUNCT
ejpam-5571	119	40	p′4	p′4	PROPN
ejpam-5571	119	41	}	}	PUNCT
ejpam-5571	119	42	)	)	PUNCT
ejpam-5571	119	43	,	,	PUNCT
ejpam-5571	119	44	respectively	respectively	ADV
ejpam-5571	119	45	;	;	PUNCT
ejpam-5571	119	46	(	(	PUNCT
ejpam-5571	119	47	ii	ii	X
ejpam-5571	119	48	)	)	PUNCT
ejpam-5571	119	49	the	the	DET
ejpam-5571	119	50	geodesic	geodesic	NOUN
ejpam-5571	120	1	[	[	X
ejpam-5571	120	2	p1	p1	NOUN
ejpam-5571	120	3	,	,	PUNCT
ejpam-5571	120	4	p3	p3	PROPN
ejpam-5571	120	5	]	]	PUNCT
ejpam-5571	120	6	intersects	intersect	VERB
ejpam-5571	120	7	the	the	DET
ejpam-5571	120	8	geodesic	geodesic	NOUN
ejpam-5571	120	9	[	[	X
ejpam-5571	120	10	p2	p2	NOUN
ejpam-5571	120	11	,	,	PUNCT
ejpam-5571	120	12	p4	p4	ADJ
ejpam-5571	120	13	]	]	PUNCT
ejpam-5571	120	14	at	at	ADP
ejpam-5571	120	15	a	a	DET
ejpam-5571	120	16	point	point	NOUN
ejpam-5571	120	17	p	p	X
ejpam-5571	120	18	;	;	PUNCT
ejpam-5571	120	19	(	(	PUNCT
ejpam-5571	120	20	iii	iii	X
ejpam-5571	120	21	)	)	PUNCT
ejpam-5571	120	22	∠p2(p̃	∠p2(p̃	PROPN
ejpam-5571	120	23	,	,	PUNCT
ejpam-5571	120	24	p3	p3	PROPN
ejpam-5571	120	25	)	)	PUNCT
ejpam-5571	120	26	=	=	PUNCT
ejpam-5571	120	27	∠p′2	∠p′2	X
ejpam-5571	120	28	(	(	PUNCT
ejpam-5571	120	29	p′	p′	NOUN
ejpam-5571	120	30	,	,	PUNCT
ejpam-5571	120	31	p′3	p′3	NOUN
ejpam-5571	120	32	)	)	PUNCT
ejpam-5571	120	33	,	,	PUNCT
ejpam-5571	120	34	∠p2(p̃	∠p2(p̃	PROPN
ejpam-5571	120	35	,	,	PUNCT
ejpam-5571	120	36	p1	p1	PROPN
ejpam-5571	120	37	)	)	PUNCT
ejpam-5571	120	38	=	=	PUNCT
ejpam-5571	120	39	∠p′2	∠p′2	X
ejpam-5571	120	40	(	(	PUNCT
ejpam-5571	120	41	p′	p′	NOUN
ejpam-5571	120	42	,	,	PUNCT
ejpam-5571	120	43	p′1	p′1	NOUN
ejpam-5571	120	44	)	)	PUNCT
ejpam-5571	120	45	,	,	PUNCT
ejpam-5571	120	46	∠p4(p̃	∠p4(p̃	PROPN
ejpam-5571	120	47	,	,	PUNCT
ejpam-5571	120	48	p1	p1	PROPN
ejpam-5571	120	49	)	)	PUNCT
ejpam-5571	120	50	=	=	SYM
ejpam-5571	120	51	∠p′4	∠p′4	X
ejpam-5571	120	52	(	(	PUNCT
ejpam-5571	120	53	p′	p′	NOUN
ejpam-5571	120	54	,	,	PUNCT
ejpam-5571	120	55	p′1	p′1	NOUN
ejpam-5571	120	56	)	)	PUNCT
ejpam-5571	120	57	and	and	CCONJ
ejpam-5571	120	58	∠p4(p̃	∠p4(p̃	PROPN
ejpam-5571	120	59	,	,	PUNCT
ejpam-5571	120	60	p3	p3	PROPN
ejpam-5571	120	61	)	)	PUNCT
ejpam-5571	121	1	=	=	SYM
ejpam-5571	121	2	∠p′4	∠p′4	X
ejpam-5571	121	3	(	(	PUNCT
ejpam-5571	121	4	p′	p′	NOUN
ejpam-5571	121	5	,	,	PUNCT
ejpam-5571	121	6	p′3	p′3	NOUN
ejpam-5571	121	7	)	)	PUNCT
ejpam-5571	121	8	for	for	ADP
ejpam-5571	121	9	all	all	DET
ejpam-5571	121	10	p̃	p̃	PROPN
ejpam-5571	121	11	∈	∈	PROPN
ejpam-5571	121	12	[	[	X
ejpam-5571	121	13	p2	p2	NOUN
ejpam-5571	121	14	,	,	PUNCT
ejpam-5571	121	15	p4	p4	ADJ
ejpam-5571	121	16	]	]	PUNCT
ejpam-5571	121	17	and	and	CCONJ
ejpam-5571	121	18	p′	p′	NOUN
ejpam-5571	121	19	is	be	AUX
ejpam-5571	121	20	the	the	DET
ejpam-5571	121	21	intersection	intersection	NOUN
ejpam-5571	121	22	of	of	ADP
ejpam-5571	121	23	[	[	X
ejpam-5571	121	24	p′1	p′1	NOUN
ejpam-5571	121	25	,	,	PUNCT
ejpam-5571	121	26	p	p	NOUN
ejpam-5571	121	27	′	′	NOUN
ejpam-5571	121	28	3	3	NUM
ejpam-5571	121	29	]	]	PUNCT
ejpam-5571	121	30	and	and	CCONJ
ejpam-5571	121	31	[	[	X
ejpam-5571	121	32	p′2	p′2	X
ejpam-5571	121	33	,	,	PUNCT
ejpam-5571	121	34	p	p	NOUN
ejpam-5571	121	35	′	′	NUM
ejpam-5571	121	36	4	4	NUM
ejpam-5571	121	37	]	]	PUNCT
ejpam-5571	121	38	;	;	PUNCT
ejpam-5571	121	39	(	(	PUNCT
ejpam-5571	121	40	iv	iv	X
ejpam-5571	121	41	)	)	PUNCT
ejpam-5571	121	42	∠p1(p2	∠p1(p2	ADJ
ejpam-5571	121	43	,	,	PUNCT
ejpam-5571	121	44	p4	p4	ADJ
ejpam-5571	121	45	)	)	PUNCT
ejpam-5571	121	46	=	=	SYM
ejpam-5571	121	47	∠p1(p2	∠p1(p2	ADJ
ejpam-5571	121	48	,	,	PUNCT
ejpam-5571	121	49	p̃	p̃	PROPN
ejpam-5571	121	50	)	)	PUNCT
ejpam-5571	121	51	+	+	NUM
ejpam-5571	121	52	∠p1(p̃	∠p1(p̃	PROPN
ejpam-5571	121	53	,	,	PUNCT
ejpam-5571	121	54	p4	p4	ADJ
ejpam-5571	121	55	)	)	PUNCT
ejpam-5571	121	56	and	and	CCONJ
ejpam-5571	121	57	∠p3(p2	∠p3(p2	ADJ
ejpam-5571	121	58	,	,	PUNCT
ejpam-5571	121	59	p4	p4	ADJ
ejpam-5571	121	60	)	)	PUNCT
ejpam-5571	121	61	=	=	SYM
ejpam-5571	121	62	∠p3(p2	∠p3(p2	ADJ
ejpam-5571	121	63	,	,	PUNCT
ejpam-5571	121	64	p̃	p̃	PROPN
ejpam-5571	121	65	)	)	PUNCT
ejpam-5571	121	66	+	+	NUM
ejpam-5571	121	67	∠p3(p̃	∠p3(p̃	PROPN
ejpam-5571	121	68	,	,	PUNCT
ejpam-5571	121	69	p4	p4	ADJ
ejpam-5571	121	70	)	)	PUNCT
ejpam-5571	121	71	for	for	ADP
ejpam-5571	121	72	all	all	DET
ejpam-5571	121	73	p̃	p̃	PROPN
ejpam-5571	121	74	∈	∈	PROPN
ejpam-5571	121	75	[	[	X
ejpam-5571	121	76	p2	p2	NOUN
ejpam-5571	121	77	,	,	PUNCT
ejpam-5571	121	78	p4	p4	ADJ
ejpam-5571	121	79	]	]	PUNCT
ejpam-5571	121	80	.	.	PUNCT
ejpam-5571	122	1	c.	c.	PROPN
ejpam-5571	122	2	phokaew	phokaew	PROPN
ejpam-5571	122	3	,	,	PUNCT
ejpam-5571	122	4	a.	a.	PROPN
ejpam-5571	122	5	sama	sama	PROPN
ejpam-5571	122	6	-	-	PUNCT
ejpam-5571	122	7	ae	ae	PROPN
ejpam-5571	122	8	,	,	PUNCT
ejpam-5571	122	9	/	/	SYM
ejpam-5571	122	10	eur	eur	NOUN
ejpam-5571	122	11	.	.	PUNCT
ejpam-5571	123	1	j.	j.	PROPN
ejpam-5571	123	2	pure	pure	PROPN
ejpam-5571	123	3	appl	appl	PROPN
ejpam-5571	123	4	.	.	PROPN
ejpam-5571	123	5	math	math	PROPN
ejpam-5571	123	6	,	,	PUNCT
ejpam-5571	123	7	17	17	NUM
ejpam-5571	123	8	(	(	PUNCT
ejpam-5571	123	9	4	4	NUM
ejpam-5571	123	10	)	)	PUNCT
ejpam-5571	123	11	(	(	PUNCT
ejpam-5571	123	12	2024	2024	NUM
ejpam-5571	123	13	)	)	PUNCT
ejpam-5571	123	14	,	,	PUNCT
ejpam-5571	123	15	3932	3932	NUM
ejpam-5571	123	16	-	-	SYM
ejpam-5571	123	17	3944	3944	NUM
ejpam-5571	123	18	3936	3936	NUM
ejpam-5571	123	19	then	then	ADV
ejpam-5571	123	20	the	the	DET
ejpam-5571	123	21	convex	convex	PROPN
ejpam-5571	123	22	hull	hull	NOUN
ejpam-5571	123	23	of	of	ADP
ejpam-5571	123	24	σ	σ	PROPN
ejpam-5571	123	25	is	be	AUX
ejpam-5571	123	26	isometric	isometric	ADJ
ejpam-5571	123	27	the	the	DET
ejpam-5571	123	28	convex	convex	NOUN
ejpam-5571	123	29	hull	hull	NOUN
ejpam-5571	123	30	of	of	ADP
ejpam-5571	123	31	σ′.	σ′.	NOUN
ejpam-5571	123	32	proof	proof	NOUN
ejpam-5571	123	33	.	.	PUNCT
ejpam-5571	124	1	first	first	ADV
ejpam-5571	124	2	,	,	PUNCT
ejpam-5571	124	3	we	we	PRON
ejpam-5571	124	4	shall	shall	AUX
ejpam-5571	124	5	prove	prove	VERB
ejpam-5571	124	6	that	that	SCONJ
ejpam-5571	124	7	the	the	DET
ejpam-5571	124	8	point	point	NOUN
ejpam-5571	124	9	p	p	NOUN
ejpam-5571	124	10	is	be	AUX
ejpam-5571	124	11	a	a	DET
ejpam-5571	124	12	corresponding	corresponding	ADJ
ejpam-5571	124	13	point	point	NOUN
ejpam-5571	124	14	under	under	ADP
ejpam-5571	124	15	both	both	DET
ejpam-5571	124	16	isometries	isometry	NOUN
ejpam-5571	124	17	to	to	ADP
ejpam-5571	124	18	the	the	DET
ejpam-5571	124	19	point	point	NOUN
ejpam-5571	124	20	p′.	p′.	NOUN
ejpam-5571	124	21	let	let	VERB
ejpam-5571	124	22	p∗	p∗	NOUN
ejpam-5571	124	23	be	be	AUX
ejpam-5571	124	24	a	a	DET
ejpam-5571	124	25	point	point	NOUN
ejpam-5571	124	26	on	on	ADP
ejpam-5571	124	27	the	the	DET
ejpam-5571	124	28	geodesic	geodesic	ADJ
ejpam-5571	124	29	segment	segment	NOUN
ejpam-5571	124	30	[	[	X
ejpam-5571	124	31	p1	p1	PROPN
ejpam-5571	124	32	,	,	PUNCT
ejpam-5571	124	33	p3	p3	PROPN
ejpam-5571	124	34	]	]	PUNCT
ejpam-5571	124	35	such	such	ADJ
ejpam-5571	124	36	that	that	SCONJ
ejpam-5571	124	37	d(p∗	d(p∗	ADV
ejpam-5571	124	38	,	,	PUNCT
ejpam-5571	124	39	p1	p1	PROPN
ejpam-5571	124	40	)	)	PUNCT
ejpam-5571	124	41	=	=	SYM
ejpam-5571	124	42	d(p′	d(p′	PROPN
ejpam-5571	124	43	,	,	PUNCT
ejpam-5571	124	44	p′1	p′1	NOUN
ejpam-5571	124	45	)	)	PUNCT
ejpam-5571	124	46	and	and	CCONJ
ejpam-5571	124	47	let	let	VERB
ejpam-5571	124	48	p′′	p′′	PROPN
ejpam-5571	124	49	∈	∈	PROPN
ejpam-5571	124	50	[	[	X
ejpam-5571	124	51	p′1	p′1	X
ejpam-5571	124	52	,	,	PUNCT
ejpam-5571	124	53	p	p	NOUN
ejpam-5571	124	54	′	′	NOUN
ejpam-5571	124	55	3	3	NUM
ejpam-5571	124	56	]	]	PUNCT
ejpam-5571	124	57	be	be	AUX
ejpam-5571	124	58	a	a	DET
ejpam-5571	124	59	corresponding	corresponding	ADJ
ejpam-5571	124	60	point	point	NOUN
ejpam-5571	124	61	of	of	ADP
ejpam-5571	124	62	p	p	NOUN
ejpam-5571	124	63	under	under	ADP
ejpam-5571	124	64	both	both	DET
ejpam-5571	124	65	isometries	isometry	NOUN
ejpam-5571	124	66	.	.	PUNCT
ejpam-5571	125	1	thus	thus	ADV
ejpam-5571	125	2	,	,	PUNCT
ejpam-5571	125	3	d(p1	d(p1	NOUN
ejpam-5571	125	4	,	,	PUNCT
ejpam-5571	125	5	p3	p3	PROPN
ejpam-5571	125	6	)	)	PUNCT
ejpam-5571	125	7	=	=	PUNCT
ejpam-5571	126	1	d(p1	d(p1	NOUN
ejpam-5571	126	2	,	,	PUNCT
ejpam-5571	126	3	p	p	NOUN
ejpam-5571	126	4	)	)	PUNCT
ejpam-5571	127	1	+	+	CCONJ
ejpam-5571	127	2	d(p	d(p	PROPN
ejpam-5571	127	3	,	,	PUNCT
ejpam-5571	127	4	p3	p3	PROPN
ejpam-5571	127	5	)	)	PUNCT
ejpam-5571	127	6	≤	≤	NUM
ejpam-5571	127	7	d(p1	d(p1	NOUN
ejpam-5571	127	8	,	,	PUNCT
ejpam-5571	127	9	p	p	NOUN
ejpam-5571	127	10	∗	∗	NOUN
ejpam-5571	127	11	)	)	PUNCT
ejpam-5571	127	12	+	+	CCONJ
ejpam-5571	127	13	d(p∗	d(p∗	ADV
ejpam-5571	127	14	,	,	PUNCT
ejpam-5571	127	15	p3	p3	PROPN
ejpam-5571	127	16	)	)	PUNCT
ejpam-5571	128	1	=	=	SYM
ejpam-5571	128	2	d(p′1	d(p′1	PROPN
ejpam-5571	128	3	,	,	PUNCT
ejpam-5571	128	4	p	p	NOUN
ejpam-5571	128	5	′	′	NOUN
ejpam-5571	128	6	)	)	PUNCT
ejpam-5571	128	7	+	+	CCONJ
ejpam-5571	128	8	d(p′	d(p′	PROPN
ejpam-5571	128	9	,	,	PUNCT
ejpam-5571	128	10	p′3	p′3	NOUN
ejpam-5571	128	11	)	)	PUNCT
ejpam-5571	128	12	=	=	SYM
ejpam-5571	129	1	d(p′1	d(p′1	PROPN
ejpam-5571	129	2	,	,	PUNCT
ejpam-5571	129	3	p	p	NOUN
ejpam-5571	129	4	′	′	NOUN
ejpam-5571	129	5	3	3	NUM
ejpam-5571	129	6	)	)	PUNCT
ejpam-5571	129	7	,	,	PUNCT
ejpam-5571	129	8	and	and	CCONJ
ejpam-5571	129	9	d(p′1	d(p′1	ADP
ejpam-5571	129	10	,	,	PUNCT
ejpam-5571	129	11	p	p	NOUN
ejpam-5571	129	12	′	′	NOUN
ejpam-5571	129	13	3	3	NUM
ejpam-5571	129	14	)	)	PUNCT
ejpam-5571	129	15	=	=	SYM
ejpam-5571	130	1	d(p′1	d(p′1	PROPN
ejpam-5571	130	2	,	,	PUNCT
ejpam-5571	130	3	p	p	NOUN
ejpam-5571	130	4	′	′	NOUN
ejpam-5571	130	5	)	)	PUNCT
ejpam-5571	130	6	+	+	CCONJ
ejpam-5571	130	7	d(p′	d(p′	PROPN
ejpam-5571	130	8	,	,	PUNCT
ejpam-5571	130	9	p′3	p′3	NOUN
ejpam-5571	130	10	)	)	PUNCT
ejpam-5571	130	11	≤	≤	NOUN
ejpam-5571	130	12	d(p′1	d(p′1	PROPN
ejpam-5571	130	13	,	,	PUNCT
ejpam-5571	130	14	p	p	PROPN
ejpam-5571	130	15	′′	′′	PROPN
ejpam-5571	130	16	)	)	PUNCT
ejpam-5571	130	17	+	+	CCONJ
ejpam-5571	130	18	d(p′′	d(p′′	NOUN
ejpam-5571	130	19	,	,	PUNCT
ejpam-5571	130	20	p′3	p′3	NOUN
ejpam-5571	130	21	)	)	PUNCT
ejpam-5571	130	22	=	=	SYM
ejpam-5571	130	23	d(p1	d(p1	NOUN
ejpam-5571	130	24	,	,	PUNCT
ejpam-5571	130	25	p	p	NOUN
ejpam-5571	130	26	)	)	PUNCT
ejpam-5571	130	27	+	+	CCONJ
ejpam-5571	130	28	d(p	d(p	PROPN
ejpam-5571	130	29	,	,	PUNCT
ejpam-5571	130	30	p3	p3	PROPN
ejpam-5571	130	31	)	)	PUNCT
ejpam-5571	130	32	=	=	SYM
ejpam-5571	130	33	d(p1	d(p1	NOUN
ejpam-5571	130	34	,	,	PUNCT
ejpam-5571	130	35	p3	p3	PROPN
ejpam-5571	130	36	)	)	PUNCT
ejpam-5571	130	37	,	,	PUNCT
ejpam-5571	130	38	and	and	CCONJ
ejpam-5571	130	39	hence	hence	ADV
ejpam-5571	130	40	,	,	PUNCT
ejpam-5571	130	41	d(p1	d(p1	NOUN
ejpam-5571	130	42	,	,	PUNCT
ejpam-5571	130	43	p3	p3	PROPN
ejpam-5571	130	44	)	)	PUNCT
ejpam-5571	130	45	=	=	SYM
ejpam-5571	131	1	d(p′1	d(p′1	PROPN
ejpam-5571	131	2	,	,	PUNCT
ejpam-5571	131	3	p	p	NOUN
ejpam-5571	131	4	′	′	NOUN
ejpam-5571	131	5	3	3	NUM
ejpam-5571	131	6	)	)	PUNCT
ejpam-5571	131	7	.	.	PUNCT
ejpam-5571	132	1	so	so	ADV
ejpam-5571	132	2	we	we	PRON
ejpam-5571	132	3	have	have	VERB
ejpam-5571	132	4	p	p	NOUN
ejpam-5571	132	5	=	=	SYM
ejpam-5571	132	6	p∗	p∗	NOUN
ejpam-5571	132	7	and	and	CCONJ
ejpam-5571	132	8	p′	p′	NOUN
ejpam-5571	132	9	=	=	SYM
ejpam-5571	132	10	p′′	p′′	NOUN
ejpam-5571	132	11	,	,	PUNCT
ejpam-5571	132	12	as	as	SCONJ
ejpam-5571	132	13	required	require	VERB
ejpam-5571	132	14	.	.	PUNCT
ejpam-5571	133	1	by	by	ADP
ejpam-5571	133	2	(	(	PUNCT
ejpam-5571	133	3	iii	iii	NOUN
ejpam-5571	133	4	)	)	PUNCT
ejpam-5571	133	5	and	and	CCONJ
ejpam-5571	133	6	(	(	PUNCT
ejpam-5571	133	7	iv	iv	X
ejpam-5571	133	8	)	)	PUNCT
ejpam-5571	133	9	,	,	PUNCT
ejpam-5571	133	10	we	we	PRON
ejpam-5571	133	11	employ	employ	VERB
ejpam-5571	133	12	theorem	theorem	ADJ
ejpam-5571	133	13	3	3	NUM
ejpam-5571	133	14	,	,	PUNCT
ejpam-5571	133	15	c({p2	c({p2	PROPN
ejpam-5571	133	16	,	,	PUNCT
ejpam-5571	133	17	p3	p3	PROPN
ejpam-5571	133	18	,	,	PUNCT
ejpam-5571	133	19	p4	p4	ADJ
ejpam-5571	133	20	}	}	PUNCT
ejpam-5571	133	21	)	)	PUNCT
ejpam-5571	133	22	is	be	AUX
ejpam-5571	133	23	isometric	isometric	ADJ
ejpam-5571	133	24	to	to	ADP
ejpam-5571	133	25	c({p′2	c({p′2	PROPN
ejpam-5571	133	26	,	,	PUNCT
ejpam-5571	133	27	p′3	p′3	NOUN
ejpam-5571	133	28	,	,	PUNCT
ejpam-5571	133	29	p′4	p′4	PROPN
ejpam-5571	133	30	}	}	PUNCT
ejpam-5571	133	31	)	)	PUNCT
ejpam-5571	133	32	and	and	CCONJ
ejpam-5571	133	33	c({p1	c({p1	PROPN
ejpam-5571	133	34	,	,	PUNCT
ejpam-5571	133	35	p2	p2	NOUN
ejpam-5571	133	36	,	,	PUNCT
ejpam-5571	133	37	p4	p4	ADJ
ejpam-5571	133	38	}	}	PUNCT
ejpam-5571	133	39	)	)	PUNCT
ejpam-5571	133	40	is	be	AUX
ejpam-5571	133	41	isometric	isometric	ADJ
ejpam-5571	133	42	to	to	ADP
ejpam-5571	133	43	c({p′1	c({p′1	PROPN
ejpam-5571	133	44	,	,	PUNCT
ejpam-5571	133	45	p′2	p′2	NOUN
ejpam-5571	133	46	,	,	PUNCT
ejpam-5571	133	47	p′4	p′4	PROPN
ejpam-5571	133	48	}	}	PUNCT
ejpam-5571	133	49	)	)	PUNCT
ejpam-5571	133	50	.	.	PUNCT
ejpam-5571	134	1	the	the	DET
ejpam-5571	134	2	next	next	ADJ
ejpam-5571	134	3	step	step	NOUN
ejpam-5571	134	4	is	be	AUX
ejpam-5571	134	5	to	to	PART
ejpam-5571	134	6	confirm	confirm	VERB
ejpam-5571	134	7	that	that	SCONJ
ejpam-5571	134	8	c(σ	c(σ	PROPN
ejpam-5571	134	9	)	)	PUNCT
ejpam-5571	134	10	and	and	CCONJ
ejpam-5571	134	11	c(σ′	c(σ′	PROPN
ejpam-5571	134	12	)	)	PUNCT
ejpam-5571	134	13	are	be	AUX
ejpam-5571	134	14	isometric	isometric	ADJ
ejpam-5571	134	15	to	to	ADP
ejpam-5571	134	16	each	each	DET
ejpam-5571	134	17	other	other	ADJ
ejpam-5571	134	18	.	.	PUNCT
ejpam-5571	135	1	by	by	ADP
ejpam-5571	135	2	the	the	DET
ejpam-5571	135	3	definition	definition	NOUN
ejpam-5571	135	4	of	of	ADP
ejpam-5571	135	5	convex	convex	PROPN
ejpam-5571	135	6	hull	hull	NOUN
ejpam-5571	135	7	,	,	PUNCT
ejpam-5571	135	8	c(σ	c(σ	PROPN
ejpam-5571	135	9	)	)	PUNCT
ejpam-5571	135	10	exists	exist	VERB
ejpam-5571	135	11	and	and	CCONJ
ejpam-5571	135	12	is	be	AUX
ejpam-5571	135	13	distinct	distinct	ADJ
ejpam-5571	135	14	,	,	PUNCT
ejpam-5571	135	15	as	as	SCONJ
ejpam-5571	135	16	we	we	PRON
ejpam-5571	135	17	have	have	AUX
ejpam-5571	135	18	noted	note	VERB
ejpam-5571	135	19	.	.	PUNCT
ejpam-5571	136	1	let	let	VERB
ejpam-5571	136	2	i1	i1	PROPN
ejpam-5571	136	3	:	:	PUNCT
ejpam-5571	136	4	c({p1	c({p1	PROPN
ejpam-5571	136	5	,	,	PUNCT
ejpam-5571	136	6	p2	p2	NOUN
ejpam-5571	136	7	,	,	PUNCT
ejpam-5571	136	8	p3	p3	NOUN
ejpam-5571	136	9	}	}	PUNCT
ejpam-5571	136	10	)	)	PUNCT
ejpam-5571	137	1	→	→	SYM
ejpam-5571	137	2	c({p′1	c({p′1	PROPN
ejpam-5571	137	3	,	,	PUNCT
ejpam-5571	137	4	p′2	p′2	NOUN
ejpam-5571	137	5	,	,	PUNCT
ejpam-5571	137	6	p′3	p′3	NOUN
ejpam-5571	137	7	}	}	PUNCT
ejpam-5571	137	8	)	)	PUNCT
ejpam-5571	137	9	and	and	CCONJ
ejpam-5571	137	10	i2	i2	PROPN
ejpam-5571	137	11	:	:	PUNCT
ejpam-5571	138	1	c({p1	c({p1	PROPN
ejpam-5571	138	2	,	,	PUNCT
ejpam-5571	138	3	p3	p3	PROPN
ejpam-5571	138	4	,	,	PUNCT
ejpam-5571	138	5	p4	p4	ADJ
ejpam-5571	138	6	}	}	PUNCT
ejpam-5571	138	7	)	)	PUNCT
ejpam-5571	138	8	→	→	SYM
ejpam-5571	138	9	c({p′1	c({p′1	PROPN
ejpam-5571	138	10	,	,	PUNCT
ejpam-5571	138	11	p′3	p′3	NOUN
ejpam-5571	138	12	,	,	PUNCT
ejpam-5571	138	13	p′4	p′4	PROPN
ejpam-5571	138	14	}	}	PUNCT
ejpam-5571	138	15	)	)	PUNCT
ejpam-5571	138	16	be	be	AUX
ejpam-5571	138	17	such	such	ADJ
ejpam-5571	139	1	that	that	SCONJ
ejpam-5571	139	2	i1(pj	i1(pj	NOUN
ejpam-5571	139	3	)	)	PUNCT
ejpam-5571	139	4	=	=	SYM
ejpam-5571	140	1	p′j	p′j	PROPN
ejpam-5571	140	2	,	,	PUNCT
ejpam-5571	140	3	j	j	PROPN
ejpam-5571	140	4	=	=	SYM
ejpam-5571	140	5	1	1	NUM
ejpam-5571	140	6	,	,	PUNCT
ejpam-5571	140	7	2	2	NUM
ejpam-5571	140	8	,	,	PUNCT
ejpam-5571	140	9	3	3	NUM
ejpam-5571	140	10	and	and	CCONJ
ejpam-5571	140	11	i2(pk	i2(pk	NUM
ejpam-5571	140	12	)	)	PUNCT
ejpam-5571	141	1	=	=	SYM
ejpam-5571	141	2	p′k	p′k	PROPN
ejpam-5571	141	3	,	,	PUNCT
ejpam-5571	141	4	k	k	NOUN
ejpam-5571	141	5	=	=	SYM
ejpam-5571	141	6	1	1	NUM
ejpam-5571	141	7	,	,	PUNCT
ejpam-5571	141	8	3	3	NUM
ejpam-5571	141	9	,	,	PUNCT
ejpam-5571	141	10	4	4	NUM
ejpam-5571	141	11	.	.	PUNCT
ejpam-5571	142	1	let	let	VERB
ejpam-5571	142	2	i	i	PRON
ejpam-5571	142	3	be	be	AUX
ejpam-5571	142	4	a	a	DET
ejpam-5571	142	5	map	map	NOUN
ejpam-5571	142	6	from	from	ADP
ejpam-5571	142	7	c(σ	c(σ	PROPN
ejpam-5571	142	8	)	)	PUNCT
ejpam-5571	142	9	to	to	PART
ejpam-5571	142	10	c(σ′	c(σ′	VERB
ejpam-5571	142	11	)	)	PUNCT
ejpam-5571	143	1	=	=	SYM
ejpam-5571	143	2	c({p′1	c({p′1	PROPN
ejpam-5571	143	3	,	,	PUNCT
ejpam-5571	143	4	p′2	p′2	NOUN
ejpam-5571	143	5	,	,	PUNCT
ejpam-5571	143	6	p′3	p′3	NOUN
ejpam-5571	143	7	}	}	PUNCT
ejpam-5571	143	8	)	)	PUNCT
ejpam-5571	143	9	∪	∪	ADP
ejpam-5571	143	10	c({p′1	c({p′1	PROPN
ejpam-5571	143	11	,	,	PUNCT
ejpam-5571	143	12	p′3	p′3	NOUN
ejpam-5571	143	13	,	,	PUNCT
ejpam-5571	143	14	p′4	p′4	PROPN
ejpam-5571	143	15	}	}	PUNCT
ejpam-5571	143	16	)	)	PUNCT
ejpam-5571	143	17	such	such	ADJ
ejpam-5571	143	18	that	that	SCONJ
ejpam-5571	143	19	i|c({p1,p2,p3	i|c({p1,p2,p3	NOUN
ejpam-5571	143	20	}	}	PUNCT
ejpam-5571	143	21	)	)	PUNCT
ejpam-5571	143	22	=	=	SYM
ejpam-5571	143	23	i1	i1	PROPN
ejpam-5571	143	24	and	and	CCONJ
ejpam-5571	143	25	i|c({p1,p3,p4	i|c({p1,p3,p4	NOUN
ejpam-5571	143	26	}	}	PUNCT
ejpam-5571	143	27	)	)	PUNCT
ejpam-5571	143	28	=	=	SYM
ejpam-5571	143	29	i2	i2	PROPN
ejpam-5571	143	30	.	.	PUNCT
ejpam-5571	144	1	we	we	PRON
ejpam-5571	144	2	must	must	AUX
ejpam-5571	144	3	demonstrate	demonstrate	VERB
ejpam-5571	144	4	that	that	SCONJ
ejpam-5571	144	5	i	i	PRON
ejpam-5571	144	6	is	be	AUX
ejpam-5571	144	7	an	an	DET
ejpam-5571	144	8	isometry	isometry	NOUN
ejpam-5571	144	9	from	from	ADP
ejpam-5571	144	10	c(σ	c(σ	PROPN
ejpam-5571	144	11	)	)	PUNCT
ejpam-5571	144	12	to	to	ADP
ejpam-5571	144	13	c(σ′	c(σ′	PROPN
ejpam-5571	144	14	)	)	PUNCT
ejpam-5571	144	15	by	by	ADP
ejpam-5571	144	16	verifying	verify	VERB
ejpam-5571	144	17	that	that	PRON
ejpam-5571	144	18	(	(	PUNCT
ejpam-5571	144	19	∗	∗	NOUN
ejpam-5571	144	20	)	)	PUNCT
ejpam-5571	145	1	i	i	PRON
ejpam-5571	145	2	is	be	AUX
ejpam-5571	145	3	an	an	DET
ejpam-5571	145	4	isometry	isometry	NOUN
ejpam-5571	145	5	onto	onto	ADP
ejpam-5571	145	6	its	its	PRON
ejpam-5571	145	7	image	image	NOUN
ejpam-5571	145	8	,	,	PUNCT
ejpam-5571	145	9	and	and	CCONJ
ejpam-5571	145	10	(	(	PUNCT
ejpam-5571	145	11	∗∗	∗∗	NOUN
ejpam-5571	145	12	)	)	PUNCT
ejpam-5571	145	13	c(σ	c(σ	PROPN
ejpam-5571	145	14	)	)	PUNCT
ejpam-5571	145	15	=	=	SYM
ejpam-5571	145	16	c({p1	c({p1	PROPN
ejpam-5571	145	17	,	,	PUNCT
ejpam-5571	145	18	p2	p2	NOUN
ejpam-5571	145	19	,	,	PUNCT
ejpam-5571	145	20	p3	p3	PROPN
ejpam-5571	145	21	}	}	PUNCT
ejpam-5571	145	22	)	)	PUNCT
ejpam-5571	145	23	∪	∪	ADP
ejpam-5571	145	24	c({p1	c({p1	PROPN
ejpam-5571	145	25	,	,	PUNCT
ejpam-5571	145	26	p3	p3	PROPN
ejpam-5571	145	27	,	,	PUNCT
ejpam-5571	145	28	p4	p4	ADJ
ejpam-5571	145	29	}	}	PUNCT
ejpam-5571	145	30	)	)	PUNCT
ejpam-5571	145	31	.	.	PUNCT
ejpam-5571	146	1	that	that	SCONJ
ejpam-5571	146	2	i	i	PRON
ejpam-5571	146	3	is	be	AUX
ejpam-5571	146	4	surjective	surjective	ADJ
ejpam-5571	146	5	is	be	AUX
ejpam-5571	146	6	obvious	obvious	ADJ
ejpam-5571	146	7	.	.	PUNCT
ejpam-5571	147	1	additionally	additionally	ADV
ejpam-5571	147	2	,	,	PUNCT
ejpam-5571	147	3	i	i	PRON
ejpam-5571	147	4	is	be	AUX
ejpam-5571	147	5	injective	injective	ADJ
ejpam-5571	147	6	due	due	ADP
ejpam-5571	147	7	to	to	ADP
ejpam-5571	147	8	the	the	DET
ejpam-5571	147	9	circumstances	circumstance	NOUN
ejpam-5571	147	10	of	of	ADP
ejpam-5571	147	11	intersecting	intersect	VERB
ejpam-5571	147	12	geodesic	geodesic	ADJ
ejpam-5571	147	13	segments	segment	NOUN
ejpam-5571	147	14	and	and	CCONJ
ejpam-5571	147	15	isometric	isometric	ADJ
ejpam-5571	147	16	convex	convex	NOUN
ejpam-5571	147	17	hulls	hull	NOUN
ejpam-5571	147	18	.	.	PUNCT
ejpam-5571	148	1	to	to	PART
ejpam-5571	148	2	prove	prove	VERB
ejpam-5571	148	3	(	(	PUNCT
ejpam-5571	148	4	∗	∗	NOUN
ejpam-5571	148	5	)	)	PUNCT
ejpam-5571	148	6	,	,	PUNCT
ejpam-5571	148	7	let	let	VERB
ejpam-5571	148	8	x1	x1	NUM
ejpam-5571	148	9	,	,	PUNCT
ejpam-5571	148	10	x2	x2	PROPN
ejpam-5571	148	11	∈	∈	PROPN
ejpam-5571	148	12	c(σ	c(σ	PROPN
ejpam-5571	148	13	)	)	PUNCT
ejpam-5571	148	14	,	,	PUNCT
ejpam-5571	148	15	x′1	x′1	NOUN
ejpam-5571	149	1	=	=	SYM
ejpam-5571	149	2	i(x1	i(x1	X
ejpam-5571	149	3	)	)	PUNCT
ejpam-5571	149	4	and	and	CCONJ
ejpam-5571	149	5	x′2	x′2	NOUN
ejpam-5571	149	6	=	=	NOUN
ejpam-5571	149	7	i(x2	i(x2	NOUN
ejpam-5571	149	8	)	)	PUNCT
ejpam-5571	149	9	.	.	PUNCT
ejpam-5571	150	1	we	we	PRON
ejpam-5571	150	2	shall	shall	AUX
ejpam-5571	150	3	verify	verify	VERB
ejpam-5571	150	4	that	that	DET
ejpam-5571	150	5	d(x1	d(x1	NOUN
ejpam-5571	150	6	,	,	PUNCT
ejpam-5571	150	7	x2	x2	PROPN
ejpam-5571	150	8	)	)	PUNCT
ejpam-5571	150	9	=	=	SYM
ejpam-5571	150	10	d(x′1	d(x′1	PROPN
ejpam-5571	150	11	,	,	PUNCT
ejpam-5571	150	12	x	x	SYM
ejpam-5571	150	13	′	′	NOUN
ejpam-5571	150	14	2	2	NUM
ejpam-5571	150	15	)	)	PUNCT
ejpam-5571	150	16	.	.	PUNCT
ejpam-5571	151	1	there	there	PRON
ejpam-5571	151	2	is	be	VERB
ejpam-5571	151	3	nothing	nothing	PRON
ejpam-5571	151	4	to	to	PART
ejpam-5571	151	5	prove	prove	VERB
ejpam-5571	151	6	if	if	SCONJ
ejpam-5571	151	7	x1	x1	PROPN
ejpam-5571	151	8	,	,	PUNCT
ejpam-5571	151	9	x2	x2	PROPN
ejpam-5571	151	10	∈	∈	PROPN
ejpam-5571	151	11	c({p1	c({p1	PROPN
ejpam-5571	151	12	,	,	PUNCT
ejpam-5571	151	13	p2	p2	NOUN
ejpam-5571	151	14	,	,	PUNCT
ejpam-5571	151	15	p3	p3	NOUN
ejpam-5571	151	16	}	}	PUNCT
ejpam-5571	151	17	)	)	PUNCT
ejpam-5571	151	18	or	or	CCONJ
ejpam-5571	151	19	x1	x1	NUM
ejpam-5571	151	20	,	,	PUNCT
ejpam-5571	151	21	x2	x2	PROPN
ejpam-5571	151	22	∈	∈	PROPN
ejpam-5571	151	23	c({p1	c({p1	PROPN
ejpam-5571	151	24	,	,	PUNCT
ejpam-5571	151	25	p2	p2	NOUN
ejpam-5571	151	26	,	,	PUNCT
ejpam-5571	151	27	p4	p4	ADJ
ejpam-5571	151	28	}	}	PUNCT
ejpam-5571	151	29	)	)	PUNCT
ejpam-5571	151	30	or	or	CCONJ
ejpam-5571	151	31	x1	x1	NUM
ejpam-5571	151	32	,	,	PUNCT
ejpam-5571	151	33	x2	x2	PROPN
ejpam-5571	151	34	∈	∈	PROPN
ejpam-5571	151	35	c({p1	c({p1	PROPN
ejpam-5571	151	36	,	,	PUNCT
ejpam-5571	151	37	p3	p3	PROPN
ejpam-5571	151	38	,	,	PUNCT
ejpam-5571	151	39	p4	p4	ADJ
ejpam-5571	151	40	}	}	PUNCT
ejpam-5571	151	41	)	)	PUNCT
ejpam-5571	151	42	or	or	CCONJ
ejpam-5571	151	43	x1	x1	NUM
ejpam-5571	151	44	,	,	PUNCT
ejpam-5571	151	45	x2	x2	PROPN
ejpam-5571	151	46	∈	∈	PROPN
ejpam-5571	151	47	c({p2	c({p2	PROPN
ejpam-5571	151	48	,	,	PUNCT
ejpam-5571	151	49	p3	p3	PROPN
ejpam-5571	151	50	,	,	PUNCT
ejpam-5571	151	51	p4	p4	ADJ
ejpam-5571	151	52	}	}	PUNCT
ejpam-5571	151	53	)	)	PUNCT
ejpam-5571	151	54	.	.	PUNCT
ejpam-5571	152	1	without	without	ADP
ejpam-5571	152	2	loss	loss	NOUN
ejpam-5571	152	3	of	of	ADP
ejpam-5571	152	4	generality	generality	NOUN
ejpam-5571	152	5	,	,	PUNCT
ejpam-5571	152	6	we	we	PRON
ejpam-5571	152	7	assume	assume	VERB
ejpam-5571	152	8	that	that	SCONJ
ejpam-5571	152	9	x1	x1	PROPN
ejpam-5571	152	10	is	be	AUX
ejpam-5571	152	11	in	in	ADP
ejpam-5571	152	12	c(p1	c(p1	NOUN
ejpam-5571	152	13	,	,	PUNCT
ejpam-5571	152	14	p2	p2	NOUN
ejpam-5571	152	15	,	,	PUNCT
ejpam-5571	152	16	p	p	NOUN
ejpam-5571	152	17	)	)	PUNCT
ejpam-5571	152	18	and	and	CCONJ
ejpam-5571	152	19	x2	x2	PROPN
ejpam-5571	152	20	is	be	AUX
ejpam-5571	152	21	in	in	ADP
ejpam-5571	152	22	c(p3	c(p3	NOUN
ejpam-5571	152	23	,	,	PUNCT
ejpam-5571	152	24	p4	p4	ADJ
ejpam-5571	152	25	,	,	PUNCT
ejpam-5571	152	26	p	p	NOUN
ejpam-5571	152	27	)	)	PUNCT
ejpam-5571	152	28	,	,	PUNCT
ejpam-5571	152	29	where	where	SCONJ
ejpam-5571	152	30	p	p	NOUN
ejpam-5571	152	31	is	be	AUX
ejpam-5571	152	32	the	the	DET
ejpam-5571	152	33	point	point	NOUN
ejpam-5571	152	34	at	at	ADP
ejpam-5571	152	35	which	which	PRON
ejpam-5571	152	36	the	the	DET
ejpam-5571	152	37	geodesic	geodesic	ADJ
ejpam-5571	152	38	segments	segment	NOUN
ejpam-5571	152	39	[	[	X
ejpam-5571	152	40	p1	p1	NOUN
ejpam-5571	152	41	,	,	PUNCT
ejpam-5571	152	42	p3	p3	PROPN
ejpam-5571	152	43	]	]	PUNCT
ejpam-5571	152	44	and	and	CCONJ
ejpam-5571	152	45	[	[	X
ejpam-5571	152	46	p2	p2	NOUN
ejpam-5571	152	47	,	,	PUNCT
ejpam-5571	152	48	p4	p4	ADJ
ejpam-5571	152	49	]	]	PUNCT
ejpam-5571	152	50	cross	cross	NOUN
ejpam-5571	152	51	.	.	PUNCT
ejpam-5571	153	1	let	let	VERB
ejpam-5571	153	2	x′1	x′1	PROPN
ejpam-5571	154	1	and	and	CCONJ
ejpam-5571	154	2	x′2	x′2	NOUN
ejpam-5571	154	3	be	be	VERB
ejpam-5571	154	4	corresponding	correspond	VERB
ejpam-5571	154	5	points	point	NOUN
ejpam-5571	154	6	of	of	ADP
ejpam-5571	154	7	x1	x1	PROPN
ejpam-5571	154	8	and	and	CCONJ
ejpam-5571	154	9	x2	x2	PROPN
ejpam-5571	154	10	,	,	PUNCT
ejpam-5571	154	11	respectively	respectively	ADV
ejpam-5571	154	12	.	.	PUNCT
ejpam-5571	155	1	we	we	PRON
ejpam-5571	155	2	suppose	suppose	VERB
ejpam-5571	155	3	that	that	SCONJ
ejpam-5571	155	4	the	the	DET
ejpam-5571	155	5	segment	segment	NOUN
ejpam-5571	155	6	[	[	X
ejpam-5571	155	7	x′1	x′1	X
ejpam-5571	155	8	,	,	PUNCT
ejpam-5571	155	9	x	x	X
ejpam-5571	155	10	′	′	NOUN
ejpam-5571	155	11	2	2	NUM
ejpam-5571	155	12	]	]	PUNCT
ejpam-5571	155	13	meets	meet	VERB
ejpam-5571	155	14	the	the	DET
ejpam-5571	155	15	segment	segment	NOUN
ejpam-5571	155	16	[	[	X
ejpam-5571	155	17	p′1	p′1	NOUN
ejpam-5571	155	18	,	,	PUNCT
ejpam-5571	155	19	p	p	NOUN
ejpam-5571	155	20	′	′	NOUN
ejpam-5571	155	21	3	3	NUM
ejpam-5571	155	22	]	]	PUNCT
ejpam-5571	155	23	at	at	ADP
ejpam-5571	155	24	a	a	DET
ejpam-5571	155	25	point	point	NOUN
ejpam-5571	155	26	x′3	x′3	PROPN
ejpam-5571	155	27	and	and	CCONJ
ejpam-5571	155	28	meets	meet	VERB
ejpam-5571	155	29	the	the	DET
ejpam-5571	155	30	segment	segment	NOUN
ejpam-5571	156	1	[	[	X
ejpam-5571	156	2	p′2	p′2	X
ejpam-5571	156	3	,	,	PUNCT
ejpam-5571	156	4	p	p	NOUN
ejpam-5571	156	5	′	′	NOUN
ejpam-5571	156	6	4	4	NUM
ejpam-5571	156	7	]	]	PUNCT
ejpam-5571	156	8	at	at	ADP
ejpam-5571	156	9	a	a	DET
ejpam-5571	156	10	point	point	NOUN
ejpam-5571	156	11	x′4	x′4	PUNCT
ejpam-5571	157	1	such	such	ADJ
ejpam-5571	157	2	that	that	SCONJ
ejpam-5571	157	3	x′3	x′3	PROPN
ejpam-5571	157	4	∈	∈	PROPN
ejpam-5571	158	1	[	[	X
ejpam-5571	158	2	x′1	x′1	X
ejpam-5571	158	3	,	,	PUNCT
ejpam-5571	158	4	x	x	X
ejpam-5571	158	5	′	′	NOUN
ejpam-5571	158	6	4	4	NUM
ejpam-5571	158	7	]	]	PUNCT
ejpam-5571	158	8	(	(	PUNCT
ejpam-5571	158	9	if	if	SCONJ
ejpam-5571	158	10	x	x	SYM
ejpam-5571	158	11	′	′	NOUN
ejpam-5571	158	12	4	4	NUM
ejpam-5571	158	13	∈	∈	NOUN
ejpam-5571	159	1	[	[	X
ejpam-5571	159	2	x′1	x′1	X
ejpam-5571	159	3	,	,	PUNCT
ejpam-5571	159	4	x	x	X
ejpam-5571	159	5	′	′	NOUN
ejpam-5571	159	6	3	3	NUM
ejpam-5571	159	7	]	]	PUNCT
ejpam-5571	159	8	we	we	PRON
ejpam-5571	159	9	can	can	AUX
ejpam-5571	159	10	prove	prove	VERB
ejpam-5571	159	11	in	in	ADP
ejpam-5571	159	12	the	the	DET
ejpam-5571	159	13	same	same	ADJ
ejpam-5571	159	14	manner	manner	NOUN
ejpam-5571	159	15	)	)	PUNCT
ejpam-5571	159	16	.	.	PUNCT
ejpam-5571	160	1	let	let	VERB
ejpam-5571	160	2	x3	x3	VERB
ejpam-5571	160	3	and	and	CCONJ
ejpam-5571	160	4	x4	x4	PROPN
ejpam-5571	160	5	be	be	VERB
ejpam-5571	160	6	two	two	NUM
ejpam-5571	160	7	points	point	NOUN
ejpam-5571	160	8	such	such	ADJ
ejpam-5571	160	9	that	that	SCONJ
ejpam-5571	160	10	x′3	x′3	PROPN
ejpam-5571	161	1	=	=	SYM
ejpam-5571	162	1	i1(x3	i1(x3	NUM
ejpam-5571	162	2	)	)	PUNCT
ejpam-5571	162	3	and	and	CCONJ
ejpam-5571	162	4	x′4	x′4	NOUN
ejpam-5571	162	5	=	=	SYM
ejpam-5571	162	6	i2(x4	i2(x4	NOUN
ejpam-5571	162	7	)	)	PUNCT
ejpam-5571	162	8	.	.	PUNCT
ejpam-5571	163	1	in	in	ADP
ejpam-5571	163	2	rk	rk	PROPN
ejpam-5571	163	3	,	,	PUNCT
ejpam-5571	163	4	we	we	PRON
ejpam-5571	163	5	have	have	VERB
ejpam-5571	163	6	[	[	X
ejpam-5571	163	7	x′1	x′1	X
ejpam-5571	163	8	,	,	PUNCT
ejpam-5571	163	9	x	x	NOUN
ejpam-5571	163	10	′	′	NOUN
ejpam-5571	163	11	4	4	NUM
ejpam-5571	163	12	]	]	PUNCT
ejpam-5571	163	13	=	=	PUNCT
ejpam-5571	164	1	[	[	X
ejpam-5571	164	2	x′1	x′1	X
ejpam-5571	164	3	,	,	PUNCT
ejpam-5571	164	4	x	x	X
ejpam-5571	164	5	′	′	NOUN
ejpam-5571	164	6	3	3	NUM
ejpam-5571	164	7	]	]	PUNCT
ejpam-5571	164	8	∪	∪	NOUN
ejpam-5571	164	9	[	[	X
ejpam-5571	164	10	x′3	x′3	X
ejpam-5571	164	11	,	,	PUNCT
ejpam-5571	164	12	x	x	NOUN
ejpam-5571	164	13	′	′	NOUN
ejpam-5571	164	14	4	4	NUM
ejpam-5571	164	15	]	]	PUNCT
ejpam-5571	164	16	and	and	CCONJ
ejpam-5571	165	1	[	[	X
ejpam-5571	165	2	x′3	x′3	X
ejpam-5571	165	3	,	,	PUNCT
ejpam-5571	165	4	x	x	X
ejpam-5571	165	5	′	′	NOUN
ejpam-5571	165	6	2	2	NUM
ejpam-5571	165	7	]	]	PUNCT
ejpam-5571	165	8	=	=	PUNCT
ejpam-5571	166	1	[	[	X
ejpam-5571	166	2	x′3	x′3	X
ejpam-5571	166	3	,	,	PUNCT
ejpam-5571	166	4	x	x	NOUN
ejpam-5571	166	5	′	′	NOUN
ejpam-5571	166	6	4	4	NUM
ejpam-5571	166	7	]	]	PUNCT
ejpam-5571	166	8	∪	∪	ADP
ejpam-5571	166	9	[	[	X
ejpam-5571	166	10	x′4	x′4	X
ejpam-5571	166	11	,	,	PUNCT
ejpam-5571	166	12	x	x	NOUN
ejpam-5571	166	13	′	′	NOUN
ejpam-5571	166	14	2	2	NUM
ejpam-5571	166	15	]	]	PUNCT
ejpam-5571	166	16	.	.	PUNCT
ejpam-5571	167	1	due	due	ADP
ejpam-5571	167	2	to	to	ADP
ejpam-5571	167	3	the	the	DET
ejpam-5571	167	4	fact	fact	NOUN
ejpam-5571	167	5	that	that	SCONJ
ejpam-5571	167	6	c({p1	c({p1	PROPN
ejpam-5571	167	7	,	,	PUNCT
ejpam-5571	167	8	p2	p2	NOUN
ejpam-5571	167	9	,	,	PUNCT
ejpam-5571	167	10	p4	p4	ADJ
ejpam-5571	167	11	}	}	PUNCT
ejpam-5571	167	12	)	)	PUNCT
ejpam-5571	167	13	is	be	AUX
ejpam-5571	167	14	isometric	isometric	ADJ
ejpam-5571	167	15	to	to	ADP
ejpam-5571	167	16	c({p′1	c({p′1	PROPN
ejpam-5571	167	17	,	,	PUNCT
ejpam-5571	167	18	p′2	p′2	NOUN
ejpam-5571	167	19	,	,	PUNCT
ejpam-5571	167	20	p′4	p′4	PROPN
ejpam-5571	167	21	}	}	PUNCT
ejpam-5571	167	22	)	)	PUNCT
ejpam-5571	167	23	and	and	CCONJ
ejpam-5571	167	24	[	[	X
ejpam-5571	167	25	x′1	x′1	X
ejpam-5571	167	26	,	,	PUNCT
ejpam-5571	167	27	x	x	X
ejpam-5571	167	28	′	′	NOUN
ejpam-5571	167	29	4	4	NUM
ejpam-5571	167	30	]	]	PUNCT
ejpam-5571	167	31	is	be	AUX
ejpam-5571	167	32	in	in	ADP
ejpam-5571	167	33	c({p′1	c({p′1	PROPN
ejpam-5571	167	34	,	,	PUNCT
ejpam-5571	167	35	p′2	p′2	NOUN
ejpam-5571	167	36	,	,	PUNCT
ejpam-5571	167	37	p′4	p′4	PROPN
ejpam-5571	167	38	}	}	PUNCT
ejpam-5571	167	39	)	)	PUNCT
ejpam-5571	167	40	,	,	PUNCT
ejpam-5571	167	41	we	we	PRON
ejpam-5571	167	42	have	have	VERB
ejpam-5571	167	43	c.	c.	PROPN
ejpam-5571	167	44	phokaew	phokaew	PROPN
ejpam-5571	167	45	,	,	PUNCT
ejpam-5571	167	46	a.	a.	PROPN
ejpam-5571	167	47	sama	sama	PROPN
ejpam-5571	167	48	-	-	PUNCT
ejpam-5571	167	49	ae	ae	PROPN
ejpam-5571	167	50	,	,	PUNCT
ejpam-5571	167	51	/	/	SYM
ejpam-5571	167	52	eur	eur	NOUN
ejpam-5571	167	53	.	.	PUNCT
ejpam-5571	168	1	j.	j.	PROPN
ejpam-5571	168	2	pure	pure	PROPN
ejpam-5571	168	3	appl	appl	PROPN
ejpam-5571	168	4	.	.	PROPN
ejpam-5571	168	5	math	math	PROPN
ejpam-5571	168	6	,	,	PUNCT
ejpam-5571	168	7	17	17	NUM
ejpam-5571	168	8	(	(	PUNCT
ejpam-5571	168	9	4	4	NUM
ejpam-5571	168	10	)	)	PUNCT
ejpam-5571	168	11	(	(	PUNCT
ejpam-5571	168	12	2024	2024	NUM
ejpam-5571	168	13	)	)	PUNCT
ejpam-5571	168	14	,	,	PUNCT
ejpam-5571	168	15	3932	3932	NUM
ejpam-5571	168	16	-	-	SYM
ejpam-5571	168	17	3944	3944	NUM
ejpam-5571	168	18	3937	3937	NUM
ejpam-5571	169	1	that	that	SCONJ
ejpam-5571	169	2	[	[	X
ejpam-5571	169	3	x1	x1	X
ejpam-5571	169	4	,	,	PUNCT
ejpam-5571	169	5	x4	x4	PROPN
ejpam-5571	169	6	]	]	PUNCT
ejpam-5571	169	7	=	=	PUNCT
ejpam-5571	170	1	[	[	X
ejpam-5571	170	2	x1	x1	PROPN
ejpam-5571	170	3	,	,	PUNCT
ejpam-5571	170	4	x3	x3	ADJ
ejpam-5571	170	5	]	]	PUNCT
ejpam-5571	170	6	∪	∪	ADP
ejpam-5571	170	7	[	[	X
ejpam-5571	170	8	x3	x3	ADJ
ejpam-5571	170	9	,	,	PUNCT
ejpam-5571	170	10	x4	x4	PROPN
ejpam-5571	170	11	]	]	PUNCT
ejpam-5571	170	12	is	be	AUX
ejpam-5571	170	13	in	in	ADP
ejpam-5571	170	14	c({p1	c({p1	PROPN
ejpam-5571	170	15	,	,	PUNCT
ejpam-5571	170	16	p3	p3	PROPN
ejpam-5571	170	17	,	,	PUNCT
ejpam-5571	170	18	p4	p4	ADJ
ejpam-5571	170	19	}	}	PUNCT
ejpam-5571	170	20	)	)	PUNCT
ejpam-5571	171	1	which	which	PRON
ejpam-5571	171	2	such	such	ADJ
ejpam-5571	171	3	that	that	DET
ejpam-5571	171	4	d(x1	d(x1	NOUN
ejpam-5571	171	5	,	,	PUNCT
ejpam-5571	171	6	x3	x3	ADJ
ejpam-5571	171	7	)	)	PUNCT
ejpam-5571	171	8	=	=	SYM
ejpam-5571	171	9	d(x′1	d(x′1	PROPN
ejpam-5571	171	10	,	,	PUNCT
ejpam-5571	171	11	x	x	X
ejpam-5571	171	12	′	′	NOUN
ejpam-5571	171	13	3	3	NUM
ejpam-5571	171	14	)	)	PUNCT
ejpam-5571	171	15	and	and	CCONJ
ejpam-5571	171	16	d(x3	d(x3	X
ejpam-5571	171	17	,	,	PUNCT
ejpam-5571	171	18	x4	x4	PROPN
ejpam-5571	171	19	)	)	PUNCT
ejpam-5571	171	20	=	=	SYM
ejpam-5571	171	21	d(x′3	d(x′3	PROPN
ejpam-5571	171	22	,	,	PUNCT
ejpam-5571	171	23	x	x	SYM
ejpam-5571	171	24	′	′	NOUN
ejpam-5571	171	25	4	4	NUM
ejpam-5571	171	26	)	)	PUNCT
ejpam-5571	171	27	,	,	PUNCT
ejpam-5571	171	28	and	and	CCONJ
ejpam-5571	171	29	then	then	ADV
ejpam-5571	171	30	,	,	PUNCT
ejpam-5571	171	31	d(x1	d(x1	NOUN
ejpam-5571	171	32	,	,	PUNCT
ejpam-5571	171	33	x4	x4	PROPN
ejpam-5571	171	34	)	)	PUNCT
ejpam-5571	171	35	=	=	SYM
ejpam-5571	171	36	d(x1	d(x1	NOUN
ejpam-5571	171	37	,	,	PUNCT
ejpam-5571	171	38	x3	x3	ADJ
ejpam-5571	171	39	)	)	PUNCT
ejpam-5571	172	1	+	+	CCONJ
ejpam-5571	172	2	d(x3	d(x3	X
ejpam-5571	172	3	,	,	PUNCT
ejpam-5571	172	4	x4	x4	PROPN
ejpam-5571	172	5	)	)	PUNCT
ejpam-5571	172	6	=	=	SYM
ejpam-5571	172	7	d(x′1	d(x′1	PROPN
ejpam-5571	172	8	,	,	PUNCT
ejpam-5571	172	9	x	x	X
ejpam-5571	172	10	′	′	NOUN
ejpam-5571	172	11	3	3	NUM
ejpam-5571	172	12	)	)	PUNCT
ejpam-5571	172	13	+	+	NUM
ejpam-5571	172	14	d(x′3	d(x′3	NOUN
ejpam-5571	172	15	,	,	PUNCT
ejpam-5571	172	16	x	x	SYM
ejpam-5571	172	17	′	′	NOUN
ejpam-5571	172	18	4	4	NUM
ejpam-5571	172	19	)	)	PUNCT
ejpam-5571	172	20	=	=	SYM
ejpam-5571	172	21	d(x′1	d(x′1	PROPN
ejpam-5571	172	22	,	,	PUNCT
ejpam-5571	172	23	x	x	SYM
ejpam-5571	172	24	′	′	NOUN
ejpam-5571	172	25	4	4	NUM
ejpam-5571	172	26	)	)	PUNCT
ejpam-5571	172	27	.	.	PUNCT
ejpam-5571	173	1	because	because	SCONJ
ejpam-5571	173	2	c({p1	c({p1	PROPN
ejpam-5571	173	3	,	,	PUNCT
ejpam-5571	173	4	p3	p3	PROPN
ejpam-5571	173	5	,	,	PUNCT
ejpam-5571	173	6	p4	p4	ADJ
ejpam-5571	173	7	}	}	PUNCT
ejpam-5571	173	8	)	)	PUNCT
ejpam-5571	173	9	is	be	AUX
ejpam-5571	173	10	isometric	isometric	ADJ
ejpam-5571	173	11	to	to	ADP
ejpam-5571	173	12	c({p′1	c({p′1	PROPN
ejpam-5571	173	13	,	,	PUNCT
ejpam-5571	173	14	p′3	p′3	NOUN
ejpam-5571	173	15	,	,	PUNCT
ejpam-5571	173	16	p′4	p′4	NOUN
ejpam-5571	173	17	}	}	PUNCT
ejpam-5571	173	18	)	)	PUNCT
ejpam-5571	174	1	and	and	CCONJ
ejpam-5571	175	1	[	[	X
ejpam-5571	175	2	x′3	x′3	X
ejpam-5571	175	3	,	,	PUNCT
ejpam-5571	175	4	x	x	X
ejpam-5571	175	5	′	′	NOUN
ejpam-5571	175	6	2	2	NUM
ejpam-5571	175	7	]	]	PUNCT
ejpam-5571	175	8	is	be	AUX
ejpam-5571	175	9	in	in	ADP
ejpam-5571	175	10	c({p′1	c({p′1	PROPN
ejpam-5571	175	11	,	,	PUNCT
ejpam-5571	175	12	p′3	p′3	NOUN
ejpam-5571	175	13	,	,	PUNCT
ejpam-5571	175	14	p′4	p′4	PROPN
ejpam-5571	175	15	}	}	PUNCT
ejpam-5571	175	16	)	)	PUNCT
ejpam-5571	175	17	,	,	PUNCT
ejpam-5571	175	18	we	we	PRON
ejpam-5571	175	19	have	have	VERB
ejpam-5571	175	20	[	[	X
ejpam-5571	175	21	x3	x3	ADJ
ejpam-5571	175	22	,	,	PUNCT
ejpam-5571	175	23	x2	x2	PROPN
ejpam-5571	175	24	]	]	X
ejpam-5571	175	25	=	=	PUNCT
ejpam-5571	176	1	[	[	X
ejpam-5571	176	2	x3	x3	ADJ
ejpam-5571	176	3	,	,	PUNCT
ejpam-5571	176	4	x4]∪	x4]∪	PUNCT
ejpam-5571	177	1	[	[	X
ejpam-5571	177	2	x4	x4	PROPN
ejpam-5571	177	3	,	,	PUNCT
ejpam-5571	177	4	x2	x2	PROPN
ejpam-5571	177	5	]	]	PUNCT
ejpam-5571	177	6	is	be	AUX
ejpam-5571	177	7	in	in	ADP
ejpam-5571	177	8	c({p1	c({p1	PROPN
ejpam-5571	177	9	,	,	PUNCT
ejpam-5571	177	10	p3	p3	PROPN
ejpam-5571	177	11	,	,	PUNCT
ejpam-5571	177	12	p4	p4	ADJ
ejpam-5571	177	13	}	}	PUNCT
ejpam-5571	177	14	)	)	PUNCT
ejpam-5571	177	15	,	,	PUNCT
ejpam-5571	177	16	which	which	PRON
ejpam-5571	177	17	such	such	ADJ
ejpam-5571	177	18	that	that	PRON
ejpam-5571	177	19	d(x3	d(x3	NOUN
ejpam-5571	177	20	,	,	PUNCT
ejpam-5571	177	21	x4	x4	PROPN
ejpam-5571	177	22	)	)	PUNCT
ejpam-5571	177	23	=	=	SYM
ejpam-5571	177	24	d(x′3	d(x′3	PROPN
ejpam-5571	177	25	,	,	PUNCT
ejpam-5571	177	26	x	x	SYM
ejpam-5571	177	27	′	′	NOUN
ejpam-5571	177	28	4	4	NUM
ejpam-5571	177	29	)	)	PUNCT
ejpam-5571	177	30	and	and	CCONJ
ejpam-5571	177	31	d(x4	d(x4	NOUN
ejpam-5571	177	32	,	,	PUNCT
ejpam-5571	177	33	x2	x2	PROPN
ejpam-5571	177	34	)	)	PUNCT
ejpam-5571	177	35	=	=	SYM
ejpam-5571	178	1	d(x′4	d(x′4	PROPN
ejpam-5571	178	2	,	,	PUNCT
ejpam-5571	178	3	x	x	NOUN
ejpam-5571	178	4	′	′	NOUN
ejpam-5571	178	5	2	2	NUM
ejpam-5571	178	6	)	)	PUNCT
ejpam-5571	178	7	,	,	PUNCT
ejpam-5571	178	8	and	and	CCONJ
ejpam-5571	178	9	thus	thus	ADV
ejpam-5571	178	10	,	,	PUNCT
ejpam-5571	178	11	d(x3	d(x3	ADJ
ejpam-5571	178	12	,	,	PUNCT
ejpam-5571	178	13	x2	x2	PROPN
ejpam-5571	178	14	)	)	PUNCT
ejpam-5571	178	15	=	=	PUNCT
ejpam-5571	178	16	d(x3	d(x3	X
ejpam-5571	178	17	,	,	PUNCT
ejpam-5571	178	18	x4	x4	PROPN
ejpam-5571	178	19	)	)	PUNCT
ejpam-5571	178	20	+	+	CCONJ
ejpam-5571	178	21	d(x4	d(x4	ADJ
ejpam-5571	178	22	,	,	PUNCT
ejpam-5571	178	23	x2	x2	PROPN
ejpam-5571	178	24	)	)	PUNCT
ejpam-5571	178	25	=	=	SYM
ejpam-5571	178	26	d(x′3	d(x′3	PROPN
ejpam-5571	178	27	,	,	PUNCT
ejpam-5571	178	28	x	x	SYM
ejpam-5571	178	29	′	′	NOUN
ejpam-5571	178	30	4	4	NUM
ejpam-5571	178	31	)	)	PUNCT
ejpam-5571	178	32	+	+	CCONJ
ejpam-5571	178	33	d(x′4	d(x′4	NOUN
ejpam-5571	178	34	,	,	PUNCT
ejpam-5571	178	35	x	x	NOUN
ejpam-5571	178	36	′	′	NOUN
ejpam-5571	178	37	2	2	NUM
ejpam-5571	178	38	)	)	PUNCT
ejpam-5571	178	39	=	=	SYM
ejpam-5571	178	40	d(x′3	d(x′3	NOUN
ejpam-5571	178	41	,	,	PUNCT
ejpam-5571	178	42	x	x	NOUN
ejpam-5571	178	43	′	′	NOUN
ejpam-5571	178	44	2	2	NUM
ejpam-5571	178	45	)	)	PUNCT
ejpam-5571	178	46	.	.	PUNCT
ejpam-5571	179	1	hence	hence	ADV
ejpam-5571	179	2	,	,	PUNCT
ejpam-5571	179	3	[	[	X
ejpam-5571	179	4	x1	x1	X
ejpam-5571	179	5	,	,	PUNCT
ejpam-5571	179	6	x2	x2	PROPN
ejpam-5571	179	7	]	]	X
ejpam-5571	179	8	=	=	PUNCT
ejpam-5571	180	1	[	[	X
ejpam-5571	180	2	x1	x1	PROPN
ejpam-5571	180	3	,	,	PUNCT
ejpam-5571	180	4	x3	x3	ADJ
ejpam-5571	180	5	]	]	PUNCT
ejpam-5571	180	6	∪	∪	ADP
ejpam-5571	180	7	[	[	X
ejpam-5571	180	8	x3	x3	ADJ
ejpam-5571	180	9	,	,	PUNCT
ejpam-5571	180	10	x4	x4	PROPN
ejpam-5571	180	11	]	]	PUNCT
ejpam-5571	180	12	∪	∪	X
ejpam-5571	180	13	[	[	X
ejpam-5571	180	14	x4	x4	PROPN
ejpam-5571	180	15	,	,	PUNCT
ejpam-5571	180	16	x2	x2	PROPN
ejpam-5571	180	17	]	]	PUNCT
ejpam-5571	180	18	forms	form	VERB
ejpam-5571	180	19	a	a	DET
ejpam-5571	180	20	geodesic	geodesic	NOUN
ejpam-5571	180	21	seqment	seqment	ADJ
ejpam-5571	180	22	,	,	PUNCT
ejpam-5571	180	23	and	and	CCONJ
ejpam-5571	180	24	therefore	therefore	ADV
ejpam-5571	180	25	,	,	PUNCT
ejpam-5571	180	26	d(x1	d(x1	NOUN
ejpam-5571	180	27	,	,	PUNCT
ejpam-5571	180	28	x2	x2	PROPN
ejpam-5571	180	29	)	)	PUNCT
ejpam-5571	180	30	=	=	SYM
ejpam-5571	180	31	d(x1	d(x1	NOUN
ejpam-5571	180	32	,	,	PUNCT
ejpam-5571	180	33	x3	x3	ADJ
ejpam-5571	180	34	)	)	PUNCT
ejpam-5571	180	35	+	+	CCONJ
ejpam-5571	180	36	d(x3	d(x3	X
ejpam-5571	180	37	,	,	PUNCT
ejpam-5571	180	38	x4	x4	PROPN
ejpam-5571	180	39	)	)	PUNCT
ejpam-5571	180	40	+	+	CCONJ
ejpam-5571	180	41	d(x4	d(x4	ADJ
ejpam-5571	180	42	,	,	PUNCT
ejpam-5571	180	43	x2	x2	PROPN
ejpam-5571	180	44	)	)	PUNCT
ejpam-5571	180	45	=	=	SYM
ejpam-5571	181	1	d(x′1	d(x′1	PROPN
ejpam-5571	181	2	,	,	PUNCT
ejpam-5571	181	3	x	x	X
ejpam-5571	181	4	′	′	NOUN
ejpam-5571	181	5	3	3	NUM
ejpam-5571	181	6	)	)	PUNCT
ejpam-5571	181	7	+	+	NUM
ejpam-5571	181	8	d(x′3	d(x′3	NOUN
ejpam-5571	181	9	,	,	PUNCT
ejpam-5571	181	10	x	x	SYM
ejpam-5571	181	11	′	′	NOUN
ejpam-5571	181	12	4	4	NUM
ejpam-5571	181	13	)	)	PUNCT
ejpam-5571	182	1	+	+	CCONJ
ejpam-5571	182	2	d(x′4	d(x′4	NOUN
ejpam-5571	182	3	,	,	PUNCT
ejpam-5571	182	4	x	x	NOUN
ejpam-5571	182	5	′	′	NOUN
ejpam-5571	182	6	2	2	NUM
ejpam-5571	182	7	)	)	PUNCT
ejpam-5571	182	8	=	=	SYM
ejpam-5571	182	9	d(x′1	d(x′1	PROPN
ejpam-5571	182	10	,	,	PUNCT
ejpam-5571	182	11	x	x	SYM
ejpam-5571	182	12	′	′	NOUN
ejpam-5571	182	13	2	2	NUM
ejpam-5571	182	14	)	)	PUNCT
ejpam-5571	182	15	.	.	PUNCT
ejpam-5571	183	1	we	we	PRON
ejpam-5571	183	2	now	now	ADV
ejpam-5571	183	3	demonstrate	demonstrate	VERB
ejpam-5571	183	4	(	(	PUNCT
ejpam-5571	183	5	∗∗	∗∗	NOUN
ejpam-5571	183	6	)	)	PUNCT
ejpam-5571	183	7	.	.	PUNCT
ejpam-5571	184	1	for	for	ADP
ejpam-5571	184	2	convenience	convenience	NOUN
ejpam-5571	184	3	,	,	PUNCT
ejpam-5571	184	4	we	we	PRON
ejpam-5571	184	5	set	set	VERB
ejpam-5571	184	6	a	a	PRON
ejpam-5571	184	7	=	=	X
ejpam-5571	184	8	{	{	PUNCT
ejpam-5571	184	9	p1	p1	NOUN
ejpam-5571	184	10	,	,	PUNCT
ejpam-5571	184	11	p2	p2	NOUN
ejpam-5571	184	12	,	,	PUNCT
ejpam-5571	184	13	p3	p3	NOUN
ejpam-5571	184	14	}	}	PUNCT
ejpam-5571	184	15	and	and	CCONJ
ejpam-5571	184	16	b	b	X
ejpam-5571	184	17	=	=	SYM
ejpam-5571	184	18	{	{	PUNCT
ejpam-5571	184	19	p1	p1	PROPN
ejpam-5571	184	20	,	,	PUNCT
ejpam-5571	184	21	p3	p3	PROPN
ejpam-5571	184	22	,	,	PUNCT
ejpam-5571	184	23	p4	p4	ADJ
ejpam-5571	184	24	}	}	PUNCT
ejpam-5571	184	25	.	.	PUNCT
ejpam-5571	185	1	first	first	ADV
ejpam-5571	185	2	,	,	PUNCT
ejpam-5571	185	3	we	we	PRON
ejpam-5571	185	4	must	must	AUX
ejpam-5571	185	5	establish	establish	VERB
ejpam-5571	185	6	the	the	DET
ejpam-5571	185	7	convexity	convexity	NOUN
ejpam-5571	185	8	of	of	ADP
ejpam-5571	185	9	c(a	c(a	NOUN
ejpam-5571	185	10	)	)	PUNCT
ejpam-5571	185	11	∪	∪	ADP
ejpam-5571	185	12	c(b	c(b	PROPN
ejpam-5571	185	13	)	)	PUNCT
ejpam-5571	185	14	.	.	PUNCT
ejpam-5571	186	1	let	let	VERB
ejpam-5571	186	2	x1	x1	PRON
ejpam-5571	186	3	,	,	PUNCT
ejpam-5571	186	4	x2	x2	PRON
ejpam-5571	186	5	be	be	VERB
ejpam-5571	186	6	two	two	NUM
ejpam-5571	186	7	points	point	NOUN
ejpam-5571	186	8	in	in	ADP
ejpam-5571	186	9	c(a	c(a	NOUN
ejpam-5571	186	10	)	)	PUNCT
ejpam-5571	186	11	∪	∪	ADP
ejpam-5571	186	12	c(b	c(b	PROPN
ejpam-5571	186	13	)	)	PUNCT
ejpam-5571	186	14	.	.	PUNCT
ejpam-5571	187	1	the	the	DET
ejpam-5571	187	2	geodesic	geodesic	ADJ
ejpam-5571	187	3	segment	segment	NOUN
ejpam-5571	187	4	[	[	X
ejpam-5571	187	5	x1	x1	PROPN
ejpam-5571	187	6	,	,	PUNCT
ejpam-5571	187	7	x2	x2	PROPN
ejpam-5571	187	8	]	]	PUNCT
ejpam-5571	187	9	must	must	AUX
ejpam-5571	187	10	be	be	AUX
ejpam-5571	187	11	declared	declare	VERB
ejpam-5571	187	12	to	to	PART
ejpam-5571	187	13	be	be	AUX
ejpam-5571	187	14	in	in	ADP
ejpam-5571	187	15	c(a	c(a	ADV
ejpam-5571	187	16	)	)	PUNCT
ejpam-5571	187	17	∪	∪	ADP
ejpam-5571	187	18	c(b	c(b	PROPN
ejpam-5571	187	19	)	)	PUNCT
ejpam-5571	187	20	.	.	PUNCT
ejpam-5571	188	1	it	it	PRON
ejpam-5571	188	2	makes	make	VERB
ejpam-5571	188	3	no	no	DET
ejpam-5571	188	4	difference	difference	NOUN
ejpam-5571	188	5	if	if	SCONJ
ejpam-5571	188	6	x1	x1	PROPN
ejpam-5571	188	7	,	,	PUNCT
ejpam-5571	188	8	x2	x2	PROPN
ejpam-5571	188	9	are	be	AUX
ejpam-5571	188	10	both	both	PRON
ejpam-5571	188	11	in	in	ADP
ejpam-5571	188	12	c(a	c(a	PROPN
ejpam-5571	188	13	)	)	PUNCT
ejpam-5571	188	14	or	or	CCONJ
ejpam-5571	188	15	c(b	c(b	PROPN
ejpam-5571	188	16	)	)	PUNCT
ejpam-5571	188	17	.	.	PUNCT
ejpam-5571	189	1	we	we	PRON
ejpam-5571	189	2	assume	assume	VERB
ejpam-5571	189	3	,	,	PUNCT
ejpam-5571	189	4	without	without	ADP
ejpam-5571	189	5	loss	loss	NOUN
ejpam-5571	189	6	of	of	ADP
ejpam-5571	189	7	generality	generality	NOUN
ejpam-5571	189	8	,	,	PUNCT
ejpam-5571	189	9	that	that	SCONJ
ejpam-5571	189	10	x1	x1	PROPN
ejpam-5571	189	11	∈	∈	PROPN
ejpam-5571	189	12	c(a	c(a	PROPN
ejpam-5571	189	13	)	)	PUNCT
ejpam-5571	189	14	and	and	CCONJ
ejpam-5571	189	15	x2	x2	PROPN
ejpam-5571	189	16	∈	∈	PROPN
ejpam-5571	189	17	c(b	c(b	PROPN
ejpam-5571	189	18	)	)	PUNCT
ejpam-5571	189	19	.	.	PUNCT
ejpam-5571	190	1	let	let	VERB
ejpam-5571	190	2	x′1	x′1	PROPN
ejpam-5571	191	1	and	and	CCONJ
ejpam-5571	191	2	x′2	x′2	NOUN
ejpam-5571	191	3	be	be	AUX
ejpam-5571	191	4	corresponding	correspond	VERB
ejpam-5571	191	5	points	point	NOUN
ejpam-5571	191	6	to	to	ADP
ejpam-5571	191	7	x1	x1	PROPN
ejpam-5571	191	8	and	and	CCONJ
ejpam-5571	191	9	x2	x2	PROPN
ejpam-5571	191	10	,	,	PUNCT
ejpam-5571	191	11	respectively	respectively	ADV
ejpam-5571	191	12	,	,	PUNCT
ejpam-5571	191	13	and	and	CCONJ
ejpam-5571	191	14	t′	t′	NUM
ejpam-5571	191	15	the	the	DET
ejpam-5571	191	16	point	point	NOUN
ejpam-5571	191	17	of	of	ADP
ejpam-5571	191	18	intersection	intersection	NOUN
ejpam-5571	191	19	of	of	ADP
ejpam-5571	191	20	the	the	DET
ejpam-5571	191	21	two	two	NUM
ejpam-5571	191	22	segments	segment	NOUN
ejpam-5571	191	23	[	[	X
ejpam-5571	191	24	x′1	x′1	X
ejpam-5571	191	25	,	,	PUNCT
ejpam-5571	191	26	x	x	X
ejpam-5571	191	27	′	′	NOUN
ejpam-5571	191	28	2	2	NUM
ejpam-5571	191	29	]	]	PUNCT
ejpam-5571	191	30	and	and	CCONJ
ejpam-5571	191	31	[	[	X
ejpam-5571	191	32	p′1	p′1	X
ejpam-5571	191	33	,	,	PUNCT
ejpam-5571	191	34	p	p	NOUN
ejpam-5571	191	35	′	′	NOUN
ejpam-5571	191	36	3	3	NUM
ejpam-5571	191	37	]	]	PUNCT
ejpam-5571	191	38	and	and	CCONJ
ejpam-5571	191	39	let	let	VERB
ejpam-5571	191	40	t′	t′	NUM
ejpam-5571	191	41	=	=	SYM
ejpam-5571	191	42	i(t	i(t	PROPN
ejpam-5571	191	43	)	)	PUNCT
ejpam-5571	191	44	.	.	PUNCT
ejpam-5571	192	1	thus	thus	ADV
ejpam-5571	192	2	d(x1	d(x1	VERB
ejpam-5571	192	3	,	,	PUNCT
ejpam-5571	192	4	x2	x2	PROPN
ejpam-5571	192	5	)	)	PUNCT
ejpam-5571	192	6	=	=	SYM
ejpam-5571	192	7	d(x′1	d(x′1	PROPN
ejpam-5571	192	8	,	,	PUNCT
ejpam-5571	192	9	x	x	SYM
ejpam-5571	192	10	′	′	NOUN
ejpam-5571	192	11	2	2	NUM
ejpam-5571	192	12	)	)	PUNCT
ejpam-5571	192	13	=	=	SYM
ejpam-5571	192	14	d(x′1	d(x′1	NUM
ejpam-5571	192	15	,	,	PUNCT
ejpam-5571	192	16	t	t	NOUN
ejpam-5571	192	17	′	′	NUM
ejpam-5571	192	18	)	)	PUNCT
ejpam-5571	193	1	+	+	CCONJ
ejpam-5571	193	2	d(t′	d(t′	NOUN
ejpam-5571	193	3	,	,	PUNCT
ejpam-5571	193	4	x′2	x′2	NOUN
ejpam-5571	193	5	)	)	PUNCT
ejpam-5571	194	1	=	=	SYM
ejpam-5571	194	2	d(x1	d(x1	NOUN
ejpam-5571	194	3	,	,	PUNCT
ejpam-5571	194	4	t	t	PROPN
ejpam-5571	194	5	)	)	PUNCT
ejpam-5571	194	6	+	+	CCONJ
ejpam-5571	194	7	d(t	d(t	PROPN
ejpam-5571	194	8	,	,	PUNCT
ejpam-5571	194	9	x2	x2	PROPN
ejpam-5571	194	10	)	)	PUNCT
ejpam-5571	194	11	.	.	PUNCT
ejpam-5571	195	1	this	this	PRON
ejpam-5571	195	2	suggests	suggest	VERB
ejpam-5571	195	3	that	that	SCONJ
ejpam-5571	195	4	geodesic	geodesic	ADJ
ejpam-5571	195	5	segments	segment	NOUN
ejpam-5571	195	6	[	[	X
ejpam-5571	195	7	x1	x1	PROPN
ejpam-5571	195	8	,	,	PUNCT
ejpam-5571	195	9	t	t	PROPN
ejpam-5571	195	10	]	]	PUNCT
ejpam-5571	195	11	and	and	CCONJ
ejpam-5571	195	12	[	[	X
ejpam-5571	195	13	t	t	X
ejpam-5571	195	14	,	,	PUNCT
ejpam-5571	195	15	x2	x2	PROPN
ejpam-5571	195	16	]	]	PUNCT
ejpam-5571	195	17	form	form	VERB
ejpam-5571	195	18	a	a	DET
ejpam-5571	195	19	geodesic	geodesic	ADJ
ejpam-5571	195	20	segment	segment	NOUN
ejpam-5571	195	21	connecting	connect	VERB
ejpam-5571	195	22	points	point	NOUN
ejpam-5571	195	23	x1	x1	PROPN
ejpam-5571	195	24	and	and	CCONJ
ejpam-5571	195	25	x2	x2	NOUN
ejpam-5571	195	26	.	.	PUNCT
ejpam-5571	196	1	that	that	PRON
ejpam-5571	196	2	is	be	AUX
ejpam-5571	196	3	[	[	X
ejpam-5571	196	4	x1	x1	PROPN
ejpam-5571	196	5	,	,	PUNCT
ejpam-5571	196	6	x2	x2	PROPN
ejpam-5571	196	7	]	]	X
ejpam-5571	196	8	=	=	PUNCT
ejpam-5571	197	1	[	[	X
ejpam-5571	197	2	x1	x1	PROPN
ejpam-5571	197	3	,	,	PUNCT
ejpam-5571	197	4	t	t	PROPN
ejpam-5571	197	5	]	]	PUNCT
ejpam-5571	197	6	∪	∪	ADP
ejpam-5571	197	7	[	[	X
ejpam-5571	197	8	t	t	PROPN
ejpam-5571	197	9	,	,	PUNCT
ejpam-5571	197	10	x2	x2	PROPN
ejpam-5571	197	11	]	]	X
ejpam-5571	197	12	⊂	⊂	PROPN
ejpam-5571	197	13	c(a	c(a	PROPN
ejpam-5571	197	14	)	)	PUNCT
ejpam-5571	197	15	∪	∪	ADP
ejpam-5571	197	16	c(b	c(b	PROPN
ejpam-5571	197	17	)	)	PUNCT
ejpam-5571	197	18	.	.	PUNCT
ejpam-5571	198	1	accordingly	accordingly	ADV
ejpam-5571	198	2	,	,	PUNCT
ejpam-5571	198	3	c(a	c(a	ADV
ejpam-5571	198	4	)	)	PUNCT
ejpam-5571	198	5	∪	∪	ADP
ejpam-5571	198	6	c(b	c(b	PROPN
ejpam-5571	198	7	)	)	PUNCT
ejpam-5571	198	8	is	be	AUX
ejpam-5571	198	9	convex	convex	ADJ
ejpam-5571	198	10	.	.	PUNCT
ejpam-5571	199	1	we	we	PRON
ejpam-5571	199	2	obtain	obtain	VERB
ejpam-5571	199	3	that	that	SCONJ
ejpam-5571	199	4	c(a	c(a	PROPN
ejpam-5571	199	5	∪	∪	ADP
ejpam-5571	199	6	b	b	NOUN
ejpam-5571	199	7	)	)	PUNCT
ejpam-5571	199	8	⊂	⊂	PROPN
ejpam-5571	199	9	c(a	c(a	PROPN
ejpam-5571	199	10	)	)	PUNCT
ejpam-5571	199	11	∪	∪	ADP
ejpam-5571	199	12	c(b	c(b	PROPN
ejpam-5571	199	13	)	)	PUNCT
ejpam-5571	199	14	,	,	PUNCT
ejpam-5571	199	15	because	because	SCONJ
ejpam-5571	199	16	c(a	c(a	NOUN
ejpam-5571	199	17	∪	∪	X
ejpam-5571	199	18	b	b	NOUN
ejpam-5571	199	19	)	)	PUNCT
ejpam-5571	199	20	is	be	AUX
ejpam-5571	199	21	the	the	DET
ejpam-5571	199	22	smallest	small	ADJ
ejpam-5571	199	23	convex	convex	NOUN
ejpam-5571	199	24	set	set	NOUN
ejpam-5571	199	25	containing	contain	VERB
ejpam-5571	199	26	a	a	DET
ejpam-5571	199	27	∪b	∪b	NOUN
ejpam-5571	199	28	.	.	PUNCT
ejpam-5571	200	1	we	we	PRON
ejpam-5571	200	2	also	also	ADV
ejpam-5571	200	3	obtain	obtain	VERB
ejpam-5571	200	4	c(a	c(a	ADV
ejpam-5571	200	5	)	)	PUNCT
ejpam-5571	200	6	∪	∪	ADP
ejpam-5571	200	7	c(b	c(b	PROPN
ejpam-5571	200	8	)	)	PUNCT
ejpam-5571	201	1	⊂	⊂	PROPN
ejpam-5571	201	2	c(a	c(a	ADV
ejpam-5571	201	3	∪b	∪b	VERB
ejpam-5571	201	4	)	)	PUNCT
ejpam-5571	201	5	because	because	SCONJ
ejpam-5571	201	6	both	both	DET
ejpam-5571	201	7	c(a	c(a	NOUN
ejpam-5571	201	8	)	)	PUNCT
ejpam-5571	201	9	and	and	CCONJ
ejpam-5571	201	10	c(b	c(b	PROPN
ejpam-5571	201	11	)	)	PUNCT
ejpam-5571	201	12	are	be	AUX
ejpam-5571	201	13	subsets	subset	NOUN
ejpam-5571	201	14	of	of	ADP
ejpam-5571	201	15	c(a	c(a	PROPN
ejpam-5571	201	16	∪b	∪b	NOUN
ejpam-5571	201	17	)	)	PUNCT
ejpam-5571	201	18	.	.	PUNCT
ejpam-5571	202	1	therefore	therefore	ADV
ejpam-5571	202	2	,	,	PUNCT
ejpam-5571	202	3	c(a	c(a	ADV
ejpam-5571	202	4	∪b	∪b	X
ejpam-5571	202	5	)	)	PUNCT
ejpam-5571	202	6	=	=	SYM
ejpam-5571	202	7	c(a	c(a	PROPN
ejpam-5571	202	8	)	)	PUNCT
ejpam-5571	202	9	∪	∪	ADP
ejpam-5571	202	10	c(b	c(b	PROPN
ejpam-5571	202	11	)	)	PUNCT
ejpam-5571	202	12	.	.	PUNCT
ejpam-5571	203	1	it	it	PRON
ejpam-5571	203	2	is	be	AUX
ejpam-5571	203	3	important	important	ADJ
ejpam-5571	203	4	to	to	PART
ejpam-5571	203	5	note	note	VERB
ejpam-5571	203	6	that	that	SCONJ
ejpam-5571	203	7	theorem	theorem	NOUN
ejpam-5571	203	8	4	4	NUM
ejpam-5571	203	9	depends	depend	VERB
ejpam-5571	203	10	on	on	ADP
ejpam-5571	203	11	the	the	DET
ejpam-5571	203	12	intersection	intersection	NOUN
ejpam-5571	203	13	of	of	ADP
ejpam-5571	203	14	two	two	NUM
ejpam-5571	203	15	geodesics	geodesic	NOUN
ejpam-5571	203	16	.	.	PUNCT
ejpam-5571	204	1	we	we	PRON
ejpam-5571	204	2	consider	consider	VERB
ejpam-5571	204	3	the	the	DET
ejpam-5571	204	4	space	space	NOUN
ejpam-5571	204	5	r3	r3	PROPN
ejpam-5571	204	6	with	with	ADP
ejpam-5571	204	7	usual	usual	ADJ
ejpam-5571	204	8	metric	metric	NOUN
ejpam-5571	204	9	as	as	ADP
ejpam-5571	204	10	a	a	DET
ejpam-5571	204	11	metric	metric	ADJ
ejpam-5571	204	12	space	space	NOUN
ejpam-5571	204	13	with	with	ADP
ejpam-5571	204	14	curvature	curvature	NOUN
ejpam-5571	204	15	bounded	bound	VERB
ejpam-5571	204	16	below	below	ADV
ejpam-5571	204	17	by	by	ADP
ejpam-5571	204	18	0	0	NUM
ejpam-5571	204	19	in	in	ADP
ejpam-5571	204	20	the	the	DET
ejpam-5571	204	21	large	large	NOUN
ejpam-5571	204	22	.	.	PUNCT
ejpam-5571	205	1	let	let	VERB
ejpam-5571	205	2	x	x	PRON
ejpam-5571	205	3	be	be	AUX
ejpam-5571	205	4	a	a	DET
ejpam-5571	205	5	triangle	triangle	NOUN
ejpam-5571	205	6	with	with	ADP
ejpam-5571	205	7	points	point	NOUN
ejpam-5571	205	8	(	(	PUNCT
ejpam-5571	205	9	−1	−1	NOUN
ejpam-5571	205	10	,	,	PUNCT
ejpam-5571	205	11	0	0	NUM
ejpam-5571	205	12	,	,	PUNCT
ejpam-5571	205	13	0	0	NUM
ejpam-5571	205	14	)	)	PUNCT
ejpam-5571	205	15	,	,	PUNCT
ejpam-5571	205	16	(	(	PUNCT
ejpam-5571	205	17	1	1	NUM
ejpam-5571	205	18	,	,	PUNCT
ejpam-5571	205	19	0	0	NUM
ejpam-5571	205	20	,	,	PUNCT
ejpam-5571	205	21	0	0	NUM
ejpam-5571	205	22	)	)	PUNCT
ejpam-5571	205	23	and	and	CCONJ
ejpam-5571	205	24	(	(	PUNCT
ejpam-5571	205	25	0	0	NUM
ejpam-5571	205	26	,	,	PUNCT
ejpam-5571	205	27	1	1	NUM
ejpam-5571	205	28	,	,	PUNCT
ejpam-5571	205	29	0	0	NUM
ejpam-5571	205	30	)	)	PUNCT
ejpam-5571	205	31	and	and	CCONJ
ejpam-5571	205	32	y	y	PROPN
ejpam-5571	205	33	be	be	AUX
ejpam-5571	205	34	a	a	DET
ejpam-5571	205	35	triangle	triangle	NOUN
ejpam-5571	205	36	with	with	ADP
ejpam-5571	205	37	points	point	NOUN
ejpam-5571	205	38	(	(	PUNCT
ejpam-5571	205	39	−1	−1	NOUN
ejpam-5571	205	40	,	,	PUNCT
ejpam-5571	205	41	0	0	NUM
ejpam-5571	205	42	,	,	PUNCT
ejpam-5571	205	43	0	0	NUM
ejpam-5571	205	44	)	)	PUNCT
ejpam-5571	205	45	,	,	PUNCT
ejpam-5571	205	46	(	(	PUNCT
ejpam-5571	205	47	1	1	NUM
ejpam-5571	205	48	,	,	PUNCT
ejpam-5571	205	49	0	0	NUM
ejpam-5571	205	50	,	,	PUNCT
ejpam-5571	205	51	0	0	NUM
ejpam-5571	205	52	)	)	PUNCT
ejpam-5571	205	53	and	and	CCONJ
ejpam-5571	205	54	(	(	PUNCT
ejpam-5571	205	55	0	0	NUM
ejpam-5571	205	56	,	,	PUNCT
ejpam-5571	205	57	0	0	NUM
ejpam-5571	205	58	,	,	PUNCT
ejpam-5571	205	59	1	1	NUM
ejpam-5571	205	60	)	)	PUNCT
ejpam-5571	205	61	.	.	PUNCT
ejpam-5571	206	1	in	in	ADP
ejpam-5571	206	2	r0	r0	NOUN
ejpam-5571	206	3	space	space	NOUN
ejpam-5571	206	4	,	,	PUNCT
ejpam-5571	206	5	we	we	PRON
ejpam-5571	206	6	let	let	VERB
ejpam-5571	206	7	x	x	PART
ejpam-5571	206	8	′	′	PRON
ejpam-5571	206	9	be	be	AUX
ejpam-5571	206	10	a	a	DET
ejpam-5571	206	11	triangle	triangle	NOUN
ejpam-5571	206	12	with	with	ADP
ejpam-5571	206	13	points	point	NOUN
ejpam-5571	206	14	(	(	PUNCT
ejpam-5571	206	15	0,−1	0,−1	NUM
ejpam-5571	206	16	)	)	PUNCT
ejpam-5571	206	17	,	,	PUNCT
ejpam-5571	206	18	(	(	PUNCT
ejpam-5571	206	19	0	0	NUM
ejpam-5571	206	20	,	,	PUNCT
ejpam-5571	206	21	1	1	NUM
ejpam-5571	206	22	)	)	PUNCT
ejpam-5571	206	23	and	and	CCONJ
ejpam-5571	206	24	(	(	PUNCT
ejpam-5571	206	25	1	1	NUM
ejpam-5571	206	26	,	,	PUNCT
ejpam-5571	206	27	0	0	NUM
ejpam-5571	206	28	)	)	PUNCT
ejpam-5571	206	29	and	and	CCONJ
ejpam-5571	206	30	b′	b′	NUM
ejpam-5571	206	31	be	be	AUX
ejpam-5571	206	32	a	a	DET
ejpam-5571	206	33	triangle	triangle	NOUN
ejpam-5571	206	34	with	with	ADP
ejpam-5571	206	35	points	point	NOUN
ejpam-5571	206	36	(	(	PUNCT
ejpam-5571	206	37	0,−1	0,−1	NUM
ejpam-5571	206	38	)	)	PUNCT
ejpam-5571	206	39	,	,	PUNCT
ejpam-5571	206	40	(	(	PUNCT
ejpam-5571	206	41	0	0	NUM
ejpam-5571	206	42	,	,	PUNCT
ejpam-5571	206	43	1	1	NUM
ejpam-5571	206	44	)	)	PUNCT
ejpam-5571	206	45	and	and	CCONJ
ejpam-5571	206	46	(	(	PUNCT
ejpam-5571	206	47	−1	−1	NOUN
ejpam-5571	206	48	,	,	PUNCT
ejpam-5571	206	49	0	0	NUM
ejpam-5571	206	50	)	)	PUNCT
ejpam-5571	206	51	.	.	PUNCT
ejpam-5571	207	1	thus	thus	ADV
ejpam-5571	207	2	,	,	PUNCT
ejpam-5571	207	3	we	we	PRON
ejpam-5571	207	4	have	have	VERB
ejpam-5571	207	5	that	that	PRON
ejpam-5571	207	6	x	x	SYM
ejpam-5571	208	1	′	′	NOUN
ejpam-5571	209	1	and	and	CCONJ
ejpam-5571	209	2	y	y	PROPN
ejpam-5571	209	3	′	′	NOUN
ejpam-5571	209	4	are	be	AUX
ejpam-5571	209	5	corresponding	correspond	VERB
ejpam-5571	209	6	triangles	triangle	NOUN
ejpam-5571	209	7	of	of	ADP
ejpam-5571	209	8	x	x	X
ejpam-5571	209	9	and	and	CCONJ
ejpam-5571	209	10	y	y	PROPN
ejpam-5571	209	11	,	,	PUNCT
ejpam-5571	209	12	respectively	respectively	ADV
ejpam-5571	209	13	.	.	PUNCT
ejpam-5571	210	1	it	it	PRON
ejpam-5571	210	2	is	be	AUX
ejpam-5571	210	3	evident	evident	ADJ
ejpam-5571	210	4	that	that	SCONJ
ejpam-5571	210	5	c(x	c(x	NOUN
ejpam-5571	210	6	′	′	NOUN
ejpam-5571	210	7	)	)	PUNCT
ejpam-5571	210	8	is	be	AUX
ejpam-5571	210	9	isometric	isometric	ADJ
ejpam-5571	210	10	to	to	ADP
ejpam-5571	210	11	c(x	c(x	NOUN
ejpam-5571	210	12	)	)	PUNCT
ejpam-5571	210	13	and	and	CCONJ
ejpam-5571	210	14	c(y	c(y	PROPN
ejpam-5571	210	15	′	′	NUM
ejpam-5571	210	16	)	)	PUNCT
ejpam-5571	210	17	is	be	AUX
ejpam-5571	210	18	isometric	isometric	ADJ
ejpam-5571	210	19	to	to	ADP
ejpam-5571	210	20	c(y	c(y	PROPN
ejpam-5571	210	21	)	)	PUNCT
ejpam-5571	210	22	but	but	CCONJ
ejpam-5571	210	23	that	that	PRON
ejpam-5571	210	24	c(x	c(x	NOUN
ejpam-5571	210	25	′∪y	′∪y	NOUN
ejpam-5571	210	26	′	′	NUM
ejpam-5571	210	27	)	)	PUNCT
ejpam-5571	210	28	is	be	AUX
ejpam-5571	210	29	not	not	PART
ejpam-5571	210	30	isometric	isometric	ADJ
ejpam-5571	210	31	to	to	PART
ejpam-5571	210	32	c(x∪y	c(x∪y	VERB
ejpam-5571	210	33	)	)	PUNCT
ejpam-5571	210	34	since	since	SCONJ
ejpam-5571	210	35	the	the	DET
ejpam-5571	210	36	segment	segment	NOUN
ejpam-5571	210	37	connecting	connect	VERB
ejpam-5571	210	38	points	point	NOUN
ejpam-5571	210	39	(	(	PUNCT
ejpam-5571	210	40	−1	−1	NOUN
ejpam-5571	210	41	,	,	PUNCT
ejpam-5571	210	42	0	0	NUM
ejpam-5571	210	43	,	,	PUNCT
ejpam-5571	210	44	0	0	NUM
ejpam-5571	210	45	)	)	PUNCT
ejpam-5571	210	46	and	and	CCONJ
ejpam-5571	210	47	(	(	PUNCT
ejpam-5571	210	48	1	1	NUM
ejpam-5571	210	49	,	,	PUNCT
ejpam-5571	210	50	0	0	NUM
ejpam-5571	210	51	,	,	PUNCT
ejpam-5571	210	52	0	0	NUM
ejpam-5571	210	53	)	)	PUNCT
ejpam-5571	210	54	do	do	AUX
ejpam-5571	210	55	not	not	PART
ejpam-5571	210	56	meet	meet	VERB
ejpam-5571	210	57	the	the	DET
ejpam-5571	210	58	segment	segment	NOUN
ejpam-5571	210	59	connecting	connect	VERB
ejpam-5571	210	60	points	point	NOUN
ejpam-5571	210	61	(	(	PUNCT
ejpam-5571	210	62	0	0	NUM
ejpam-5571	210	63	,	,	PUNCT
ejpam-5571	210	64	1	1	NUM
ejpam-5571	210	65	,	,	PUNCT
ejpam-5571	210	66	0	0	NUM
ejpam-5571	210	67	)	)	PUNCT
ejpam-5571	210	68	and	and	CCONJ
ejpam-5571	210	69	(	(	PUNCT
ejpam-5571	210	70	0	0	NUM
ejpam-5571	210	71	,	,	PUNCT
ejpam-5571	210	72	0	0	NUM
ejpam-5571	210	73	,	,	PUNCT
ejpam-5571	210	74	1	1	NUM
ejpam-5571	210	75	)	)	PUNCT
ejpam-5571	210	76	.	.	PUNCT
ejpam-5571	211	1	the	the	DET
ejpam-5571	211	2	following	follow	VERB
ejpam-5571	211	3	theorem	theorem	NOUN
ejpam-5571	211	4	can	can	AUX
ejpam-5571	211	5	be	be	AUX
ejpam-5571	211	6	proven	prove	VERB
ejpam-5571	211	7	using	use	VERB
ejpam-5571	211	8	the	the	DET
ejpam-5571	211	9	concept	concept	NOUN
ejpam-5571	211	10	of	of	ADP
ejpam-5571	211	11	the	the	DET
ejpam-5571	211	12	proof	proof	NOUN
ejpam-5571	211	13	of	of	ADP
ejpam-5571	211	14	theorem	theorem	ADJ
ejpam-5571	211	15	4	4	NUM
ejpam-5571	211	16	theorem	theorem	NOUN
ejpam-5571	211	17	5	5	NUM
ejpam-5571	211	18	.	.	PUNCT
ejpam-5571	212	1	let	let	VERB
ejpam-5571	212	2	x	x	PRON
ejpam-5571	212	3	be	be	AUX
ejpam-5571	212	4	a	a	DET
ejpam-5571	212	5	metric	metric	ADJ
ejpam-5571	212	6	space	space	NOUN
ejpam-5571	212	7	with	with	ADP
ejpam-5571	212	8	curvature	curvature	NOUN
ejpam-5571	212	9	bounded	bound	VERB
ejpam-5571	212	10	below	below	ADV
ejpam-5571	212	11	by	by	ADP
ejpam-5571	212	12	k	k	PROPN
ejpam-5571	212	13	in	in	ADP
ejpam-5571	212	14	the	the	DET
ejpam-5571	212	15	large	large	NOUN
ejpam-5571	212	16	.	.	PUNCT
ejpam-5571	213	1	let	let	VERB
ejpam-5571	213	2	σ	σ	NOUN
ejpam-5571	213	3	be	be	AUX
ejpam-5571	213	4	a	a	DET
ejpam-5571	213	5	closed	closed	ADJ
ejpam-5571	213	6	geodesic	geodesic	ADJ
ejpam-5571	213	7	polygon	polygon	NOUN
ejpam-5571	213	8	with	with	ADP
ejpam-5571	213	9	ordered	order	VERB
ejpam-5571	213	10	vertices	vertex	NOUN
ejpam-5571	213	11	p1	p1	NOUN
ejpam-5571	213	12	,	,	PUNCT
ejpam-5571	213	13	p2	p2	NOUN
ejpam-5571	213	14	,	,	PUNCT
ejpam-5571	213	15	p3	p3	PROPN
ejpam-5571	213	16	,	,	PUNCT
ejpam-5571	213	17	...	...	PUNCT
ejpam-5571	213	18	,	,	PUNCT
ejpam-5571	213	19	pn	pn	PROPN
ejpam-5571	213	20	,	,	PUNCT
ejpam-5571	213	21	p1	p1	PROPN
ejpam-5571	213	22	with	with	ADP
ejpam-5571	213	23	perimeter	perimeter	NOUN
ejpam-5571	214	1	less	less	ADJ
ejpam-5571	214	2	than	than	ADP
ejpam-5571	214	3	π/	π/	ADV
ejpam-5571	214	4	√	√	VERB
ejpam-5571	214	5	k	k	NOUN
ejpam-5571	215	1	in	in	ADP
ejpam-5571	215	2	x	x	X
ejpam-5571	215	3	and	and	CCONJ
ejpam-5571	215	4	σ′	σ′	PROPN
ejpam-5571	215	5	be	be	AUX
ejpam-5571	215	6	a	a	DET
ejpam-5571	215	7	convex	convex	NOUN
ejpam-5571	215	8	polygon	polygon	NOUN
ejpam-5571	215	9	with	with	ADP
ejpam-5571	215	10	ordered	order	VERB
ejpam-5571	215	11	vertices	vertex	NOUN
ejpam-5571	215	12	p′1	p′1	NOUN
ejpam-5571	215	13	,	,	PUNCT
ejpam-5571	215	14	p	p	NOUN
ejpam-5571	215	15	′	′	NOUN
ejpam-5571	215	16	2	2	NUM
ejpam-5571	215	17	,	,	PUNCT
ejpam-5571	215	18	p	p	NOUN
ejpam-5571	215	19	′	′	NOUN
ejpam-5571	215	20	3	3	NUM
ejpam-5571	215	21	,	,	PUNCT
ejpam-5571	215	22	...	...	PUNCT
ejpam-5571	215	23	,	,	PUNCT
ejpam-5571	215	24	p	p	NOUN
ejpam-5571	215	25	′	′	NUM
ejpam-5571	215	26	n	n	CCONJ
ejpam-5571	215	27	,	,	PUNCT
ejpam-5571	215	28	p	p	NOUN
ejpam-5571	215	29	′	′	NOUN
ejpam-5571	215	30	1	1	NUM
ejpam-5571	215	31	in	in	ADP
ejpam-5571	215	32	the	the	DET
ejpam-5571	215	33	model	model	NOUN
ejpam-5571	215	34	space	space	NOUN
ejpam-5571	215	35	rk	rk	NOUN
ejpam-5571	215	36	,	,	PUNCT
ejpam-5571	215	37	for	for	ADP
ejpam-5571	215	38	n	n	X
ejpam-5571	215	39	>	>	X
ejpam-5571	215	40	4	4	X
ejpam-5571	215	41	.	.	PUNCT
ejpam-5571	215	42	suppose	suppose	VERB
ejpam-5571	215	43	that	that	SCONJ
ejpam-5571	215	44	the	the	DET
ejpam-5571	215	45	following	follow	VERB
ejpam-5571	215	46	statements	statement	NOUN
ejpam-5571	215	47	hold	hold	VERB
ejpam-5571	215	48	:	:	PUNCT
ejpam-5571	215	49	c.	c.	PROPN
ejpam-5571	215	50	phokaew	phokaew	PROPN
ejpam-5571	215	51	,	,	PUNCT
ejpam-5571	215	52	a.	a.	PROPN
ejpam-5571	215	53	sama	sama	PROPN
ejpam-5571	215	54	-	-	PUNCT
ejpam-5571	215	55	ae	ae	PROPN
ejpam-5571	215	56	,	,	PUNCT
ejpam-5571	215	57	/	/	SYM
ejpam-5571	215	58	eur	eur	NOUN
ejpam-5571	215	59	.	.	PUNCT
ejpam-5571	216	1	j.	j.	PROPN
ejpam-5571	216	2	pure	pure	PROPN
ejpam-5571	216	3	appl	appl	PROPN
ejpam-5571	216	4	.	.	PROPN
ejpam-5571	216	5	math	math	PROPN
ejpam-5571	216	6	,	,	PUNCT
ejpam-5571	216	7	17	17	NUM
ejpam-5571	216	8	(	(	PUNCT
ejpam-5571	216	9	4	4	NUM
ejpam-5571	216	10	)	)	PUNCT
ejpam-5571	216	11	(	(	PUNCT
ejpam-5571	216	12	2024	2024	NUM
ejpam-5571	216	13	)	)	PUNCT
ejpam-5571	216	14	,	,	PUNCT
ejpam-5571	216	15	3932	3932	NUM
ejpam-5571	216	16	-	-	SYM
ejpam-5571	216	17	3944	3944	NUM
ejpam-5571	216	18	3938	3938	NUM
ejpam-5571	216	19	(	(	PUNCT
ejpam-5571	216	20	i	i	NOUN
ejpam-5571	216	21	)	)	PUNCT
ejpam-5571	216	22	c({p1	c({p1	PROPN
ejpam-5571	216	23	,	,	PUNCT
ejpam-5571	216	24	p2	p2	NOUN
ejpam-5571	216	25	,	,	PUNCT
ejpam-5571	216	26	...	...	PUNCT
ejpam-5571	216	27	,	,	PUNCT
ejpam-5571	216	28	pt	pt	PROPN
ejpam-5571	216	29	}	}	PUNCT
ejpam-5571	216	30	)	)	PUNCT
ejpam-5571	216	31	is	be	AUX
ejpam-5571	216	32	isometric	isometric	ADJ
ejpam-5571	216	33	to	to	ADP
ejpam-5571	216	34	c({p′1	c({p′1	PROPN
ejpam-5571	216	35	,	,	PUNCT
ejpam-5571	216	36	p′2	p′2	NOUN
ejpam-5571	216	37	,	,	PUNCT
ejpam-5571	216	38	...	...	PUNCT
ejpam-5571	216	39	,	,	PUNCT
ejpam-5571	216	40	p′t	p′t	NOUN
ejpam-5571	216	41	}	}	PUNCT
ejpam-5571	216	42	)	)	PUNCT
ejpam-5571	216	43	and	and	CCONJ
ejpam-5571	216	44	c({pt	c({pt	PROPN
ejpam-5571	216	45	,	,	PUNCT
ejpam-5571	216	46	pt+1	pt+1	PROPN
ejpam-5571	216	47	,	,	PUNCT
ejpam-5571	216	48	...	...	PUNCT
ejpam-5571	216	49	,	,	PUNCT
ejpam-5571	216	50	pn	pn	NOUN
ejpam-5571	216	51	}	}	PUNCT
ejpam-5571	216	52	)	)	PUNCT
ejpam-5571	216	53	is	be	AUX
ejpam-5571	216	54	isometric	isometric	ADJ
ejpam-5571	216	55	to	to	ADP
ejpam-5571	216	56	c({p′t	c({p′t	PROPN
ejpam-5571	216	57	,	,	PUNCT
ejpam-5571	216	58	p′t+1	p′t+1	PROPN
ejpam-5571	216	59	,	,	PUNCT
ejpam-5571	216	60	...	...	PUNCT
ejpam-5571	216	61	,	,	PUNCT
ejpam-5571	216	62	p	p	NOUN
ejpam-5571	216	63	′	′	NOUN
ejpam-5571	216	64	n	n	CCONJ
ejpam-5571	216	65	}	}	PUNCT
ejpam-5571	216	66	)	)	PUNCT
ejpam-5571	216	67	;	;	PUNCT
ejpam-5571	216	68	(	(	PUNCT
ejpam-5571	216	69	ii	ii	X
ejpam-5571	216	70	)	)	PUNCT
ejpam-5571	216	71	the	the	DET
ejpam-5571	216	72	geodesic	geodesic	ADJ
ejpam-5571	216	73	segment	segment	NOUN
ejpam-5571	216	74	[	[	X
ejpam-5571	216	75	p1	p1	NOUN
ejpam-5571	216	76	,	,	PUNCT
ejpam-5571	216	77	pt	pt	X
ejpam-5571	216	78	]	]	X
ejpam-5571	216	79	intersects	intersect	VERB
ejpam-5571	216	80	the	the	DET
ejpam-5571	216	81	geodesic	geodesic	ADJ
ejpam-5571	216	82	segment	segment	NOUN
ejpam-5571	217	1	[	[	X
ejpam-5571	217	2	pi	pi	NOUN
ejpam-5571	217	3	,	,	PUNCT
ejpam-5571	217	4	pj	pj	PROPN
ejpam-5571	217	5	]	]	PUNCT
ejpam-5571	217	6	at	at	ADP
ejpam-5571	217	7	a	a	DET
ejpam-5571	217	8	point	point	NOUN
ejpam-5571	217	9	for	for	ADP
ejpam-5571	217	10	some	some	DET
ejpam-5571	217	11	i	i	PRON
ejpam-5571	217	12	∈	∈	PROPN
ejpam-5571	217	13	{	{	PUNCT
ejpam-5571	217	14	1	1	NUM
ejpam-5571	217	15	,	,	PUNCT
ejpam-5571	217	16	2	2	NUM
ejpam-5571	217	17	,	,	PUNCT
ejpam-5571	217	18	...	...	PUNCT
ejpam-5571	217	19	,	,	PUNCT
ejpam-5571	217	20	t−	t−	PROPN
ejpam-5571	217	21	1	1	NUM
ejpam-5571	217	22	}	}	PUNCT
ejpam-5571	217	23	and	and	CCONJ
ejpam-5571	217	24	j	j	PROPN
ejpam-5571	217	25	∈	∈	PROPN
ejpam-5571	217	26	{	{	PUNCT
ejpam-5571	217	27	t+	t+	NOUN
ejpam-5571	217	28	1	1	NUM
ejpam-5571	217	29	,	,	PUNCT
ejpam-5571	217	30	t+	t+	VERB
ejpam-5571	217	31	2	2	NUM
ejpam-5571	217	32	,	,	PUNCT
ejpam-5571	217	33	...	...	PUNCT
ejpam-5571	217	34	,	,	PUNCT
ejpam-5571	217	35	n−	n−	NOUN
ejpam-5571	217	36	1	1	NUM
ejpam-5571	217	37	}	}	PUNCT
ejpam-5571	217	38	.	.	PUNCT
ejpam-5571	218	1	(	(	PUNCT
ejpam-5571	218	2	iii	iii	NOUN
ejpam-5571	218	3	)	)	PUNCT
ejpam-5571	218	4	∠pi(p	∠pi(p	NOUN
ejpam-5571	218	5	∗	∗	NOUN
ejpam-5571	218	6	,	,	PUNCT
ejpam-5571	218	7	pt	pt	NOUN
ejpam-5571	218	8	)	)	PUNCT
ejpam-5571	218	9	=	=	NOUN
ejpam-5571	218	10	∠p′i	∠p′i	NOUN
ejpam-5571	218	11	(	(	PUNCT
ejpam-5571	218	12	p′	p′	NOUN
ejpam-5571	218	13	,	,	PUNCT
ejpam-5571	218	14	p′t	p′t	NOUN
ejpam-5571	218	15	)	)	PUNCT
ejpam-5571	218	16	,	,	PUNCT
ejpam-5571	218	17	∠pi(p	∠pi(p	NOUN
ejpam-5571	218	18	∗	∗	NOUN
ejpam-5571	218	19	,	,	PUNCT
ejpam-5571	218	20	p1	p1	NOUN
ejpam-5571	218	21	)	)	PUNCT
ejpam-5571	218	22	=	=	SYM
ejpam-5571	218	23	∠p′i	∠p′i	NOUN
ejpam-5571	218	24	(	(	PUNCT
ejpam-5571	218	25	p′	p′	NOUN
ejpam-5571	218	26	,	,	PUNCT
ejpam-5571	218	27	p′1	p′1	NOUN
ejpam-5571	218	28	)	)	PUNCT
ejpam-5571	218	29	,	,	PUNCT
ejpam-5571	218	30	∠pj	∠pj	X
ejpam-5571	218	31	(	(	PUNCT
ejpam-5571	218	32	p	p	NOUN
ejpam-5571	218	33	∗	∗	NOUN
ejpam-5571	218	34	,	,	PUNCT
ejpam-5571	218	35	p1	p1	NOUN
ejpam-5571	218	36	)	)	PUNCT
ejpam-5571	218	37	=	=	VERB
ejpam-5571	218	38	∠p′j	∠p′j	NOUN
ejpam-5571	218	39	(	(	PUNCT
ejpam-5571	218	40	p′	p′	NOUN
ejpam-5571	218	41	,	,	PUNCT
ejpam-5571	218	42	p′1	p′1	NOUN
ejpam-5571	218	43	)	)	PUNCT
ejpam-5571	218	44	and	and	CCONJ
ejpam-5571	218	45	∠pj	∠pj	X
ejpam-5571	218	46	(	(	PUNCT
ejpam-5571	218	47	p	p	NOUN
ejpam-5571	218	48	∗	∗	NOUN
ejpam-5571	218	49	,	,	PUNCT
ejpam-5571	218	50	pt	pt	NOUN
ejpam-5571	218	51	)	)	PUNCT
ejpam-5571	218	52	=	=	NOUN
ejpam-5571	218	53	∠p′j	∠p′j	NOUN
ejpam-5571	218	54	(	(	PUNCT
ejpam-5571	218	55	p′	p′	NOUN
ejpam-5571	218	56	,	,	PUNCT
ejpam-5571	218	57	p′t	p′t	NOUN
ejpam-5571	218	58	)	)	PUNCT
ejpam-5571	218	59	for	for	ADP
ejpam-5571	218	60	all	all	DET
ejpam-5571	218	61	p∗	p∗	PROPN
ejpam-5571	218	62	∈	∈	PROPN
ejpam-5571	218	63	[	[	X
ejpam-5571	218	64	pi	pi	NOUN
ejpam-5571	218	65	,	,	PUNCT
ejpam-5571	218	66	pj	pj	PROPN
ejpam-5571	218	67	]	]	X
ejpam-5571	218	68	;	;	PUNCT
ejpam-5571	218	69	(	(	PUNCT
ejpam-5571	218	70	iv	iv	X
ejpam-5571	218	71	)	)	PUNCT
ejpam-5571	218	72	∠p1(pi	∠p1(pi	PROPN
ejpam-5571	218	73	,	,	PUNCT
ejpam-5571	218	74	pj	pj	PROPN
ejpam-5571	218	75	)	)	PUNCT
ejpam-5571	218	76	=	=	SYM
ejpam-5571	219	1	∠p1(pi	∠p1(pi	PROPN
ejpam-5571	219	2	,	,	PUNCT
ejpam-5571	219	3	p	p	NOUN
ejpam-5571	219	4	∗	∗	NOUN
ejpam-5571	219	5	)	)	PUNCT
ejpam-5571	220	1	+	+	NUM
ejpam-5571	220	2	∠p1(p	∠p1(p	NOUN
ejpam-5571	220	3	∗	∗	NOUN
ejpam-5571	220	4	,	,	PUNCT
ejpam-5571	220	5	pj	pj	PROPN
ejpam-5571	220	6	)	)	PUNCT
ejpam-5571	220	7	and	and	CCONJ
ejpam-5571	220	8	∠pt(pi	∠pt(pi	ADP
ejpam-5571	220	9	,	,	PUNCT
ejpam-5571	220	10	pj	pj	PROPN
ejpam-5571	220	11	)	)	PUNCT
ejpam-5571	220	12	=	=	PUNCT
ejpam-5571	220	13	∠pt(pi	∠pt(pi	X
ejpam-5571	220	14	,	,	PUNCT
ejpam-5571	220	15	p	p	NOUN
ejpam-5571	220	16	∗	∗	NOUN
ejpam-5571	220	17	)	)	PUNCT
ejpam-5571	220	18	+	+	NUM
ejpam-5571	220	19	∠pt(p	∠pt(p	NOUN
ejpam-5571	220	20	∗	∗	NOUN
ejpam-5571	220	21	,	,	PUNCT
ejpam-5571	220	22	pj	pj	PROPN
ejpam-5571	220	23	)	)	PUNCT
ejpam-5571	220	24	for	for	ADP
ejpam-5571	220	25	all	all	DET
ejpam-5571	220	26	p∗	p∗	PROPN
ejpam-5571	220	27	∈	∈	PROPN
ejpam-5571	220	28	[	[	X
ejpam-5571	220	29	pi	pi	NOUN
ejpam-5571	220	30	,	,	PUNCT
ejpam-5571	220	31	pj	pj	PROPN
ejpam-5571	220	32	]	]	PUNCT
ejpam-5571	220	33	then	then	ADV
ejpam-5571	220	34	the	the	DET
ejpam-5571	220	35	convex	convex	PROPN
ejpam-5571	220	36	hull	hull	NOUN
ejpam-5571	220	37	of	of	ADP
ejpam-5571	220	38	σ	σ	PROPN
ejpam-5571	220	39	is	be	AUX
ejpam-5571	220	40	isometric	isometric	ADJ
ejpam-5571	220	41	the	the	DET
ejpam-5571	220	42	convex	convex	PROPN
ejpam-5571	220	43	hull	hull	NOUN
ejpam-5571	220	44	of	of	ADP
ejpam-5571	220	45	σ′	σ′	PROPN
ejpam-5571	220	46	,	,	PUNCT
ejpam-5571	220	47	that	that	PRON
ejpam-5571	220	48	is	be	AUX
ejpam-5571	220	49	the	the	DET
ejpam-5571	220	50	totally	totally	ADV
ejpam-5571	220	51	geodesic	geodesic	ADJ
ejpam-5571	220	52	surface	surface	NOUN
ejpam-5571	220	53	bounded	bound	VERB
ejpam-5571	220	54	by	by	ADP
ejpam-5571	220	55	σ	σ	PROPN
ejpam-5571	220	56	and	and	CCONJ
ejpam-5571	220	57	the	the	DET
ejpam-5571	220	58	region	region	NOUN
ejpam-5571	220	59	bounded	bound	VERB
ejpam-5571	220	60	by	by	ADP
ejpam-5571	220	61	σ′	σ′	PROPN
ejpam-5571	220	62	are	be	AUX
ejpam-5571	220	63	isometric	isometric	ADJ
ejpam-5571	220	64	to	to	ADP
ejpam-5571	220	65	each	each	DET
ejpam-5571	220	66	other	other	ADJ
ejpam-5571	220	67	.	.	PUNCT
ejpam-5571	221	1	3	3	X
ejpam-5571	221	2	.	.	X
ejpam-5571	221	3	spherical	spherical	ADJ
ejpam-5571	221	4	curves	curve	NOUN
ejpam-5571	221	5	if	if	SCONJ
ejpam-5571	221	6	there	there	PRON
ejpam-5571	221	7	is	be	VERB
ejpam-5571	221	8	a	a	DET
ejpam-5571	221	9	point	point	NOUN
ejpam-5571	221	10	p	p	NOUN
ejpam-5571	221	11	in	in	ADP
ejpam-5571	221	12	the	the	DET
ejpam-5571	221	13	metric	metric	ADJ
ejpam-5571	221	14	space	space	NOUN
ejpam-5571	221	15	x	x	PUNCT
ejpam-5571	221	16	and	and	CCONJ
ejpam-5571	221	17	a	a	DET
ejpam-5571	221	18	positive	positive	ADJ
ejpam-5571	221	19	real	real	ADJ
ejpam-5571	221	20	number	number	NOUN
ejpam-5571	221	21	r	r	NOUN
ejpam-5571	221	22	such	such	ADJ
ejpam-5571	221	23	that	that	DET
ejpam-5571	221	24	d(x	d(x	PROPN
ejpam-5571	221	25	,	,	PUNCT
ejpam-5571	221	26	p	p	X
ejpam-5571	221	27	)	)	PUNCT
ejpam-5571	221	28	=	=	SYM
ejpam-5571	221	29	r	r	NOUN
ejpam-5571	221	30	for	for	ADP
ejpam-5571	221	31	all	all	DET
ejpam-5571	221	32	x	x	NOUN
ejpam-5571	221	33	in	in	ADP
ejpam-5571	221	34	γ	γ	PROPN
ejpam-5571	221	35	,	,	PUNCT
ejpam-5571	221	36	the	the	DET
ejpam-5571	221	37	curve	curve	NOUN
ejpam-5571	221	38	γ	γ	NOUN
ejpam-5571	221	39	is	be	AUX
ejpam-5571	221	40	said	say	VERB
ejpam-5571	221	41	to	to	PART
ejpam-5571	221	42	be	be	AUX
ejpam-5571	221	43	spherical	spherical	ADJ
ejpam-5571	221	44	.	.	PUNCT
ejpam-5571	222	1	the	the	DET
ejpam-5571	222	2	radius	radius	NOUN
ejpam-5571	222	3	of	of	ADP
ejpam-5571	222	4	γ	γ	PROPN
ejpam-5571	222	5	is	be	AUX
ejpam-5571	222	6	the	the	DET
ejpam-5571	222	7	actual	actual	ADJ
ejpam-5571	222	8	value	value	NOUN
ejpam-5571	222	9	r.	r.	PROPN
ejpam-5571	222	10	for	for	ADP
ejpam-5571	222	11	example	example	NOUN
ejpam-5571	222	12	,	,	PUNCT
ejpam-5571	222	13	a	a	DET
ejpam-5571	222	14	circle	circle	NOUN
ejpam-5571	222	15	of	of	ADP
ejpam-5571	222	16	radius	radius	NOUN
ejpam-5571	222	17	r	r	NOUN
ejpam-5571	222	18	>	>	X
ejpam-5571	222	19	0	0	PUNCT
ejpam-5571	222	20	in	in	ADP
ejpam-5571	222	21	the	the	DET
ejpam-5571	222	22	model	model	NOUN
ejpam-5571	222	23	space	space	NOUN
ejpam-5571	222	24	rk	rk	NOUN
ejpam-5571	222	25	is	be	AUX
ejpam-5571	222	26	a	a	DET
ejpam-5571	222	27	closed	closed	ADJ
ejpam-5571	222	28	spherical	spherical	ADJ
ejpam-5571	222	29	curve	curve	NOUN
ejpam-5571	222	30	at	at	ADP
ejpam-5571	222	31	a	a	DET
ejpam-5571	222	32	distance	distance	NOUN
ejpam-5571	222	33	r	r	NOUN
ejpam-5571	222	34	from	from	ADP
ejpam-5571	222	35	its	its	PRON
ejpam-5571	222	36	center	center	NOUN
ejpam-5571	222	37	.	.	PUNCT
ejpam-5571	223	1	in	in	ADP
ejpam-5571	223	2	the	the	DET
ejpam-5571	223	3	subsequent	subsequent	ADJ
ejpam-5571	223	4	discussion	discussion	NOUN
ejpam-5571	223	5	,	,	PUNCT
ejpam-5571	223	6	we	we	PRON
ejpam-5571	223	7	define	define	VERB
ejpam-5571	223	8	γab	γab	ADV
ejpam-5571	223	9	as	as	ADP
ejpam-5571	223	10	a	a	DET
ejpam-5571	223	11	spherical	spherical	ADJ
ejpam-5571	223	12	curve	curve	NOUN
ejpam-5571	223	13	in	in	ADP
ejpam-5571	223	14	a	a	DET
ejpam-5571	223	15	metric	metric	ADJ
ejpam-5571	223	16	space	space	NOUN
ejpam-5571	223	17	with	with	ADP
ejpam-5571	223	18	curvature	curvature	NOUN
ejpam-5571	223	19	bound	bind	VERB
ejpam-5571	223	20	below	below	ADP
ejpam-5571	223	21	with	with	ADP
ejpam-5571	223	22	endpoints	endpoint	NOUN
ejpam-5571	223	23	a	a	PRON
ejpam-5571	223	24	and	and	CCONJ
ejpam-5571	223	25	b	b	NOUN
ejpam-5571	223	26	,	,	PUNCT
ejpam-5571	223	27	and	and	CCONJ
ejpam-5571	223	28	γ′a′b′	γ′a′b′	NOUN
ejpam-5571	223	29	as	as	ADP
ejpam-5571	223	30	a	a	DET
ejpam-5571	223	31	subarc	subarc	NOUN
ejpam-5571	223	32	of	of	ADP
ejpam-5571	223	33	a	a	DET
ejpam-5571	223	34	circle	circle	NOUN
ejpam-5571	223	35	in	in	ADP
ejpam-5571	223	36	the	the	DET
ejpam-5571	223	37	model	model	NOUN
ejpam-5571	223	38	space	space	NOUN
ejpam-5571	223	39	rk	rk	NOUN
ejpam-5571	223	40	with	with	ADP
ejpam-5571	223	41	endpoints	endpoint	NOUN
ejpam-5571	223	42	a′	a′	PROPN
ejpam-5571	223	43	and	and	CCONJ
ejpam-5571	223	44	b′.	b′.	PRON
ejpam-5571	223	45	we	we	PRON
ejpam-5571	223	46	describe	describe	VERB
ejpam-5571	223	47	a	a	DET
ejpam-5571	223	48	closed	closed	ADJ
ejpam-5571	223	49	spherical	spherical	ADJ
ejpam-5571	223	50	curve	curve	NOUN
ejpam-5571	223	51	bounding	bound	VERB
ejpam-5571	223	52	a	a	DET
ejpam-5571	223	53	surface	surface	NOUN
ejpam-5571	223	54	isometric	isometric	ADJ
ejpam-5571	223	55	to	to	ADP
ejpam-5571	223	56	a	a	DET
ejpam-5571	223	57	region	region	NOUN
ejpam-5571	223	58	bounded	bound	VERB
ejpam-5571	223	59	by	by	ADP
ejpam-5571	223	60	a	a	DET
ejpam-5571	223	61	circle	circle	NOUN
ejpam-5571	223	62	in	in	ADP
ejpam-5571	223	63	the	the	DET
ejpam-5571	223	64	model	model	NOUN
ejpam-5571	223	65	space	space	NOUN
ejpam-5571	223	66	rk	rk	PROPN
ejpam-5571	223	67	.	.	PUNCT
ejpam-5571	224	1	lemma	lemma	PROPN
ejpam-5571	224	2	2	2	X
ejpam-5571	224	3	.	.	PUNCT
ejpam-5571	225	1	let	let	VERB
ejpam-5571	225	2	x	x	PRON
ejpam-5571	225	3	be	be	AUX
ejpam-5571	225	4	a	a	DET
ejpam-5571	225	5	metric	metric	ADJ
ejpam-5571	225	6	space	space	NOUN
ejpam-5571	225	7	with	with	ADP
ejpam-5571	225	8	curvature	curvature	NOUN
ejpam-5571	225	9	bounded	bound	VERB
ejpam-5571	225	10	below	below	ADV
ejpam-5571	225	11	by	by	ADP
ejpam-5571	225	12	k	k	PROPN
ejpam-5571	225	13	in	in	ADP
ejpam-5571	225	14	the	the	DET
ejpam-5571	225	15	large	large	NOUN
ejpam-5571	225	16	.	.	PUNCT
ejpam-5571	226	1	let	let	VERB
ejpam-5571	226	2	γ	γ	NOUN
ejpam-5571	226	3	be	be	AUX
ejpam-5571	226	4	a	a	DET
ejpam-5571	226	5	spherical	spherical	ADJ
ejpam-5571	226	6	curve	curve	NOUN
ejpam-5571	226	7	at	at	ADP
ejpam-5571	226	8	a	a	DET
ejpam-5571	226	9	distance	distance	NOUN
ejpam-5571	227	1	r	r	NOUN
ejpam-5571	227	2	<	<	X
ejpam-5571	227	3	π	π	PROPN
ejpam-5571	227	4	2	2	NUM
ejpam-5571	227	5	√	√	PROPN
ejpam-5571	227	6	k	k	NOUN
ejpam-5571	227	7	from	from	ADP
ejpam-5571	227	8	a	a	DET
ejpam-5571	227	9	point	point	NOUN
ejpam-5571	227	10	p	p	NOUN
ejpam-5571	227	11	with	with	ADP
ejpam-5571	227	12	endpoints	endpoint	NOUN
ejpam-5571	227	13	a	a	PRON
ejpam-5571	227	14	,	,	PUNCT
ejpam-5571	227	15	b	b	NOUN
ejpam-5571	227	16	in	in	ADP
ejpam-5571	227	17	x	x	X
ejpam-5571	227	18	and	and	CCONJ
ejpam-5571	227	19	γ′	γ′	NOUN
ejpam-5571	227	20	be	be	AUX
ejpam-5571	227	21	a	a	DET
ejpam-5571	227	22	subarc	subarc	NOUN
ejpam-5571	227	23	of	of	ADP
ejpam-5571	227	24	a	a	DET
ejpam-5571	227	25	circle	circle	NOUN
ejpam-5571	227	26	of	of	ADP
ejpam-5571	227	27	radius	radius	NOUN
ejpam-5571	227	28	r	r	NOUN
ejpam-5571	227	29	centered	center	VERB
ejpam-5571	227	30	at	at	ADP
ejpam-5571	227	31	a	a	DET
ejpam-5571	227	32	point	point	NOUN
ejpam-5571	227	33	p′	p′	NOUN
ejpam-5571	227	34	in	in	ADP
ejpam-5571	227	35	rk	rk	NOUN
ejpam-5571	227	36	with	with	ADP
ejpam-5571	227	37	endpoints	endpoint	NOUN
ejpam-5571	227	38	a′	a′	PROPN
ejpam-5571	227	39	,	,	PUNCT
ejpam-5571	227	40	b′	b′	NUM
ejpam-5571	227	41	such	such	ADJ
ejpam-5571	227	42	that	that	SCONJ
ejpam-5571	227	43	d(a	d(a	PROPN
ejpam-5571	227	44	,	,	PUNCT
ejpam-5571	227	45	b	b	NOUN
ejpam-5571	227	46	)	)	PUNCT
ejpam-5571	227	47	=	=	VERB
ejpam-5571	227	48	d(a′	d(a′	NOUN
ejpam-5571	227	49	,	,	PUNCT
ejpam-5571	227	50	b′	b′	NUM
ejpam-5571	227	51	)	)	PUNCT
ejpam-5571	227	52	.	.	PUNCT
ejpam-5571	228	1	if	if	SCONJ
ejpam-5571	228	2	that	that	DET
ejpam-5571	228	3	c	c	PROPN
ejpam-5571	228	4	∈	∈	PROPN
ejpam-5571	228	5	γ	γ	X
ejpam-5571	228	6	is	be	AUX
ejpam-5571	228	7	between	between	ADP
ejpam-5571	228	8	a	a	PRON
ejpam-5571	228	9	,	,	PUNCT
ejpam-5571	228	10	b	b	NOUN
ejpam-5571	228	11	and	and	CCONJ
ejpam-5571	228	12	c′	c′	NOUN
ejpam-5571	228	13	∈	∈	PROPN
ejpam-5571	228	14	γ	γ	X
ejpam-5571	228	15	is	be	AUX
ejpam-5571	228	16	between	between	ADP
ejpam-5571	228	17	a′	a′	PROPN
ejpam-5571	228	18	,	,	PUNCT
ejpam-5571	228	19	b′	b′	NUM
ejpam-5571	228	20	with	with	ADP
ejpam-5571	228	21	conditions	condition	NOUN
ejpam-5571	228	22	d(a	d(a	PROPN
ejpam-5571	228	23	,	,	PUNCT
ejpam-5571	228	24	c	c	NOUN
ejpam-5571	228	25	)	)	PUNCT
ejpam-5571	228	26	=	=	SYM
ejpam-5571	228	27	d(a′	d(a′	NOUN
ejpam-5571	228	28	,	,	PUNCT
ejpam-5571	228	29	c′	c′	NUM
ejpam-5571	228	30	)	)	PUNCT
ejpam-5571	228	31	and	and	CCONJ
ejpam-5571	228	32	d(b	d(b	PROPN
ejpam-5571	228	33	,	,	PUNCT
ejpam-5571	228	34	c	c	NOUN
ejpam-5571	228	35	)	)	PUNCT
ejpam-5571	228	36	=	=	SYM
ejpam-5571	228	37	d(b′	d(b′	PROPN
ejpam-5571	228	38	,	,	PUNCT
ejpam-5571	228	39	c′	c′	NUM
ejpam-5571	228	40	)	)	PUNCT
ejpam-5571	228	41	,	,	PUNCT
ejpam-5571	228	42	then	then	ADV
ejpam-5571	228	43	the	the	DET
ejpam-5571	228	44	geodesic	geodesic	ADJ
ejpam-5571	228	45	segment	segment	NOUN
ejpam-5571	229	1	[	[	X
ejpam-5571	229	2	p	p	X
ejpam-5571	229	3	,	,	PUNCT
ejpam-5571	229	4	c	c	NOUN
ejpam-5571	229	5	]	]	PUNCT
ejpam-5571	229	6	intersects	intersect	VERB
ejpam-5571	229	7	the	the	DET
ejpam-5571	229	8	geodesic	geodesic	ADJ
ejpam-5571	229	9	segment	segment	NOUN
ejpam-5571	230	1	[	[	X
ejpam-5571	230	2	a	a	X
ejpam-5571	230	3	,	,	PUNCT
ejpam-5571	230	4	b	b	X
ejpam-5571	230	5	]	]	X
ejpam-5571	230	6	at	at	ADP
ejpam-5571	230	7	a	a	DET
ejpam-5571	230	8	point	point	NOUN
ejpam-5571	230	9	q	q	NOUN
ejpam-5571	230	10	which	which	PRON
ejpam-5571	230	11	corresponds	correspond	VERB
ejpam-5571	230	12	to	to	ADP
ejpam-5571	230	13	the	the	DET
ejpam-5571	230	14	point	point	NOUN
ejpam-5571	230	15	q′	q′	NOUN
ejpam-5571	230	16	of	of	ADP
ejpam-5571	230	17	intersection	intersection	NOUN
ejpam-5571	230	18	of	of	ADP
ejpam-5571	230	19	[	[	X
ejpam-5571	230	20	p′	p′	NOUN
ejpam-5571	230	21	,	,	PUNCT
ejpam-5571	230	22	c′	c′	NUM
ejpam-5571	230	23	]	]	PUNCT
ejpam-5571	230	24	and	and	CCONJ
ejpam-5571	230	25	[	[	X
ejpam-5571	230	26	a′	a′	PROPN
ejpam-5571	230	27	,	,	PUNCT
ejpam-5571	230	28	b′	b′	NOUN
ejpam-5571	230	29	]	]	PUNCT
ejpam-5571	230	30	.	.	PUNCT
ejpam-5571	231	1	proof	proof	NOUN
ejpam-5571	231	2	.	.	PUNCT
ejpam-5571	232	1	we	we	PRON
ejpam-5571	232	2	shall	shall	AUX
ejpam-5571	232	3	prove	prove	VERB
ejpam-5571	232	4	that	that	SCONJ
ejpam-5571	232	5	the	the	DET
ejpam-5571	232	6	segment	segment	NOUN
ejpam-5571	232	7	[	[	X
ejpam-5571	232	8	a	a	X
ejpam-5571	232	9	,	,	PUNCT
ejpam-5571	232	10	b	b	NOUN
ejpam-5571	232	11	]	]	PUNCT
ejpam-5571	232	12	intersects	intersect	VERB
ejpam-5571	232	13	the	the	DET
ejpam-5571	232	14	segment	segment	NOUN
ejpam-5571	233	1	[	[	X
ejpam-5571	233	2	p	p	X
ejpam-5571	233	3	,	,	PUNCT
ejpam-5571	233	4	c	c	X
ejpam-5571	233	5	]	]	X
ejpam-5571	233	6	at	at	ADP
ejpam-5571	233	7	a	a	DET
ejpam-5571	233	8	point	point	NOUN
ejpam-5571	233	9	.	.	PUNCT
ejpam-5571	234	1	suppose	suppose	VERB
ejpam-5571	234	2	that	that	SCONJ
ejpam-5571	234	3	q′	q′	NOUN
ejpam-5571	234	4	is	be	AUX
ejpam-5571	234	5	the	the	DET
ejpam-5571	234	6	point	point	NOUN
ejpam-5571	234	7	of	of	ADP
ejpam-5571	234	8	intersection	intersection	NOUN
ejpam-5571	234	9	of	of	ADP
ejpam-5571	234	10	the	the	DET
ejpam-5571	234	11	segments	segment	NOUN
ejpam-5571	234	12	[	[	X
ejpam-5571	234	13	a′	a′	PROPN
ejpam-5571	234	14	,	,	PUNCT
ejpam-5571	234	15	b′	b′	NOUN
ejpam-5571	234	16	]	]	PUNCT
ejpam-5571	234	17	and	and	CCONJ
ejpam-5571	234	18	[	[	X
ejpam-5571	234	19	p′	p′	NOUN
ejpam-5571	234	20	,	,	PUNCT
ejpam-5571	234	21	c′	c′	PRON
ejpam-5571	234	22	]	]	PUNCT
ejpam-5571	234	23	.	.	PUNCT
ejpam-5571	235	1	let	let	VERB
ejpam-5571	235	2	q	q	PRON
ejpam-5571	235	3	be	be	AUX
ejpam-5571	235	4	a	a	DET
ejpam-5571	235	5	point	point	NOUN
ejpam-5571	235	6	on	on	ADP
ejpam-5571	235	7	the	the	DET
ejpam-5571	235	8	segment	segment	NOUN
ejpam-5571	235	9	[	[	X
ejpam-5571	235	10	p	p	X
ejpam-5571	235	11	,	,	PUNCT
ejpam-5571	235	12	c	c	X
ejpam-5571	235	13	]	]	X
ejpam-5571	235	14	such	such	ADJ
ejpam-5571	235	15	that	that	SCONJ
ejpam-5571	235	16	d(p	d(p	PROPN
ejpam-5571	235	17	,	,	PUNCT
ejpam-5571	235	18	q	q	NOUN
ejpam-5571	235	19	)	)	PUNCT
ejpam-5571	235	20	=	=	SYM
ejpam-5571	235	21	d(p′	d(p′	PROPN
ejpam-5571	235	22	,	,	PUNCT
ejpam-5571	235	23	q′	q′	NOUN
ejpam-5571	235	24	)	)	PUNCT
ejpam-5571	235	25	.	.	PUNCT
ejpam-5571	236	1	thus	thus	ADV
ejpam-5571	236	2	,	,	PUNCT
ejpam-5571	236	3	q′	q′	NOUN
ejpam-5571	236	4	is	be	AUX
ejpam-5571	236	5	a	a	DET
ejpam-5571	236	6	corresponding	corresponding	ADJ
ejpam-5571	236	7	point	point	NOUN
ejpam-5571	236	8	of	of	ADP
ejpam-5571	236	9	q	q	NOUN
ejpam-5571	236	10	in	in	ADP
ejpam-5571	236	11	two	two	NUM
ejpam-5571	236	12	triangles	triangle	NOUN
ejpam-5571	236	13	△	△	X
ejpam-5571	236	14	(	(	PUNCT
ejpam-5571	236	15	p	p	X
ejpam-5571	236	16	,	,	PUNCT
ejpam-5571	236	17	a	a	PRON
ejpam-5571	236	18	,	,	PUNCT
ejpam-5571	236	19	c	c	NOUN
ejpam-5571	236	20	)	)	PUNCT
ejpam-5571	236	21	and	and	CCONJ
ejpam-5571	236	22	△	△	X
ejpam-5571	236	23	(	(	PUNCT
ejpam-5571	236	24	p	p	X
ejpam-5571	236	25	,	,	PUNCT
ejpam-5571	236	26	b	b	PROPN
ejpam-5571	236	27	,	,	PUNCT
ejpam-5571	236	28	c	c	NOUN
ejpam-5571	236	29	)	)	PUNCT
ejpam-5571	236	30	,	,	PUNCT
ejpam-5571	236	31	and	and	CCONJ
ejpam-5571	236	32	hence	hence	ADV
ejpam-5571	236	33	,	,	PUNCT
ejpam-5571	236	34	d(a	d(a	PROPN
ejpam-5571	236	35	,	,	PUNCT
ejpam-5571	236	36	q	q	NOUN
ejpam-5571	236	37	)	)	PUNCT
ejpam-5571	236	38	=	=	SYM
ejpam-5571	236	39	d(a′	d(a′	NOUN
ejpam-5571	236	40	,	,	PUNCT
ejpam-5571	236	41	q′	q′	NOUN
ejpam-5571	236	42	)	)	PUNCT
ejpam-5571	236	43	and	and	CCONJ
ejpam-5571	236	44	d(b	d(b	PROPN
ejpam-5571	236	45	,	,	PUNCT
ejpam-5571	236	46	q	q	X
ejpam-5571	236	47	)	)	PUNCT
ejpam-5571	236	48	=	=	SYM
ejpam-5571	236	49	d(b′	d(b′	PROPN
ejpam-5571	236	50	,	,	PUNCT
ejpam-5571	236	51	q′	q′	NOUN
ejpam-5571	236	52	)	)	PUNCT
ejpam-5571	236	53	.	.	PUNCT
ejpam-5571	237	1	consequently	consequently	ADV
ejpam-5571	237	2	,	,	PUNCT
ejpam-5571	237	3	d(a	d(a	PROPN
ejpam-5571	237	4	,	,	PUNCT
ejpam-5571	237	5	b	b	NOUN
ejpam-5571	237	6	)	)	PUNCT
ejpam-5571	237	7	≤	≤	NOUN
ejpam-5571	238	1	d(a	d(a	PROPN
ejpam-5571	238	2	,	,	PUNCT
ejpam-5571	238	3	q	q	NOUN
ejpam-5571	238	4	)	)	PUNCT
ejpam-5571	238	5	+	+	CCONJ
ejpam-5571	238	6	d(q	d(q	PROPN
ejpam-5571	238	7	,	,	PUNCT
ejpam-5571	238	8	b	b	NOUN
ejpam-5571	238	9	)	)	PUNCT
ejpam-5571	238	10	=	=	VERB
ejpam-5571	238	11	d(a′	d(a′	NOUN
ejpam-5571	238	12	,	,	PUNCT
ejpam-5571	238	13	q′	q′	NOUN
ejpam-5571	238	14	)	)	PUNCT
ejpam-5571	239	1	+	+	CCONJ
ejpam-5571	239	2	d(q′	d(q′	PROPN
ejpam-5571	239	3	,	,	PUNCT
ejpam-5571	239	4	b′	b′	NUM
ejpam-5571	239	5	)	)	PUNCT
ejpam-5571	239	6	=	=	SYM
ejpam-5571	239	7	d(a′	d(a′	NOUN
ejpam-5571	239	8	,	,	PUNCT
ejpam-5571	239	9	b′	b′	NUM
ejpam-5571	239	10	)	)	PUNCT
ejpam-5571	240	1	=	=	SYM
ejpam-5571	241	1	d(a	d(a	PROPN
ejpam-5571	241	2	,	,	PUNCT
ejpam-5571	241	3	b	b	NOUN
ejpam-5571	241	4	)	)	PUNCT
ejpam-5571	241	5	,	,	PUNCT
ejpam-5571	241	6	that	that	ADV
ejpam-5571	241	7	is	is	ADV
ejpam-5571	241	8	,	,	PUNCT
ejpam-5571	241	9	the	the	DET
ejpam-5571	241	10	point	point	NOUN
ejpam-5571	241	11	q	q	NOUN
ejpam-5571	241	12	is	be	AUX
ejpam-5571	241	13	the	the	DET
ejpam-5571	241	14	intersection	intersection	NOUN
ejpam-5571	241	15	of	of	ADP
ejpam-5571	241	16	[	[	X
ejpam-5571	241	17	a	a	X
ejpam-5571	241	18	,	,	PUNCT
ejpam-5571	241	19	b	b	NOUN
ejpam-5571	241	20	]	]	PUNCT
ejpam-5571	241	21	and	and	CCONJ
ejpam-5571	241	22	[	[	X
ejpam-5571	241	23	p	p	X
ejpam-5571	241	24	,	,	PUNCT
ejpam-5571	241	25	c	c	NOUN
ejpam-5571	241	26	]	]	PUNCT
ejpam-5571	241	27	.	.	PUNCT
ejpam-5571	242	1	c.	c.	PROPN
ejpam-5571	242	2	phokaew	phokaew	PROPN
ejpam-5571	242	3	,	,	PUNCT
ejpam-5571	242	4	a.	a.	PROPN
ejpam-5571	242	5	sama	sama	PROPN
ejpam-5571	242	6	-	-	PUNCT
ejpam-5571	242	7	ae	ae	PROPN
ejpam-5571	242	8	,	,	PUNCT
ejpam-5571	242	9	/	/	SYM
ejpam-5571	242	10	eur	eur	NOUN
ejpam-5571	242	11	.	.	PUNCT
ejpam-5571	243	1	j.	j.	PROPN
ejpam-5571	243	2	pure	pure	PROPN
ejpam-5571	243	3	appl	appl	PROPN
ejpam-5571	243	4	.	.	PROPN
ejpam-5571	243	5	math	math	PROPN
ejpam-5571	243	6	,	,	PUNCT
ejpam-5571	243	7	17	17	NUM
ejpam-5571	243	8	(	(	PUNCT
ejpam-5571	243	9	4	4	NUM
ejpam-5571	243	10	)	)	PUNCT
ejpam-5571	243	11	(	(	PUNCT
ejpam-5571	243	12	2024	2024	NUM
ejpam-5571	243	13	)	)	PUNCT
ejpam-5571	243	14	,	,	PUNCT
ejpam-5571	243	15	3932	3932	NUM
ejpam-5571	243	16	-	-	SYM
ejpam-5571	243	17	3944	3944	NUM
ejpam-5571	243	18	3939	3939	NUM
ejpam-5571	243	19	lemma	lemma	PROPN
ejpam-5571	243	20	3	3	X
ejpam-5571	243	21	.	.	PUNCT
ejpam-5571	244	1	let	let	VERB
ejpam-5571	244	2	x	x	PRON
ejpam-5571	244	3	be	be	AUX
ejpam-5571	244	4	a	a	DET
ejpam-5571	244	5	metric	metric	ADJ
ejpam-5571	244	6	space	space	NOUN
ejpam-5571	244	7	with	with	ADP
ejpam-5571	244	8	curvature	curvature	NOUN
ejpam-5571	244	9	bounded	bound	VERB
ejpam-5571	244	10	below	below	ADV
ejpam-5571	244	11	by	by	ADP
ejpam-5571	244	12	k	k	PROPN
ejpam-5571	244	13	in	in	ADP
ejpam-5571	244	14	the	the	DET
ejpam-5571	244	15	large	large	NOUN
ejpam-5571	244	16	.	.	PUNCT
ejpam-5571	245	1	let	let	VERB
ejpam-5571	245	2	γ	γ	NOUN
ejpam-5571	245	3	be	be	AUX
ejpam-5571	245	4	a	a	DET
ejpam-5571	245	5	spherical	spherical	ADJ
ejpam-5571	245	6	curve	curve	NOUN
ejpam-5571	245	7	at	at	ADP
ejpam-5571	245	8	a	a	DET
ejpam-5571	245	9	distance	distance	NOUN
ejpam-5571	246	1	r	r	NOUN
ejpam-5571	246	2	<	<	X
ejpam-5571	246	3	π	π	PROPN
ejpam-5571	246	4	2	2	NUM
ejpam-5571	246	5	√	√	PROPN
ejpam-5571	246	6	k	k	NOUN
ejpam-5571	246	7	from	from	ADP
ejpam-5571	246	8	a	a	DET
ejpam-5571	246	9	point	point	NOUN
ejpam-5571	246	10	p	p	NOUN
ejpam-5571	246	11	with	with	ADP
ejpam-5571	246	12	endpoints	endpoint	NOUN
ejpam-5571	246	13	a	a	PRON
ejpam-5571	246	14	,	,	PUNCT
ejpam-5571	246	15	b	b	NOUN
ejpam-5571	246	16	in	in	ADP
ejpam-5571	246	17	x	x	X
ejpam-5571	246	18	and	and	CCONJ
ejpam-5571	246	19	γ′	γ′	NOUN
ejpam-5571	246	20	be	be	AUX
ejpam-5571	246	21	a	a	DET
ejpam-5571	246	22	subarc	subarc	NOUN
ejpam-5571	246	23	of	of	ADP
ejpam-5571	246	24	a	a	DET
ejpam-5571	246	25	circle	circle	NOUN
ejpam-5571	246	26	of	of	ADP
ejpam-5571	246	27	radius	radius	NOUN
ejpam-5571	246	28	r	r	NOUN
ejpam-5571	246	29	centered	center	VERB
ejpam-5571	246	30	at	at	ADP
ejpam-5571	246	31	a	a	DET
ejpam-5571	246	32	point	point	NOUN
ejpam-5571	246	33	p′	p′	NOUN
ejpam-5571	246	34	in	in	ADP
ejpam-5571	246	35	rk	rk	NOUN
ejpam-5571	246	36	with	with	ADP
ejpam-5571	246	37	endpoints	endpoint	NOUN
ejpam-5571	246	38	a′	a′	PROPN
ejpam-5571	246	39	,	,	PUNCT
ejpam-5571	246	40	b′.	b′.	PROPN
ejpam-5571	246	41	suppose	suppose	VERB
ejpam-5571	246	42	that	that	SCONJ
ejpam-5571	246	43	c	c	PROPN
ejpam-5571	246	44	∈	∈	PROPN
ejpam-5571	246	45	γ	γ	X
ejpam-5571	246	46	is	be	AUX
ejpam-5571	246	47	between	between	ADP
ejpam-5571	246	48	a	a	PRON
ejpam-5571	246	49	,	,	PUNCT
ejpam-5571	246	50	b	b	NOUN
ejpam-5571	246	51	and	and	CCONJ
ejpam-5571	246	52	c′	c′	NOUN
ejpam-5571	246	53	∈	∈	PROPN
ejpam-5571	246	54	γ	γ	X
ejpam-5571	246	55	is	be	AUX
ejpam-5571	246	56	between	between	ADP
ejpam-5571	246	57	a′	a′	PROPN
ejpam-5571	246	58	,	,	PUNCT
ejpam-5571	246	59	b′.	b′.	PROPN
ejpam-5571	246	60	assume	assume	VERB
ejpam-5571	246	61	that	that	SCONJ
ejpam-5571	246	62	the	the	DET
ejpam-5571	246	63	following	follow	VERB
ejpam-5571	246	64	statements	statement	NOUN
ejpam-5571	246	65	hold	hold	VERB
ejpam-5571	246	66	:	:	PUNCT
ejpam-5571	246	67	(	(	PUNCT
ejpam-5571	246	68	i	i	NOUN
ejpam-5571	246	69	)	)	PUNCT
ejpam-5571	246	70	ℓ(γ	ℓ(γ	PROPN
ejpam-5571	246	71	)	)	PUNCT
ejpam-5571	246	72	=	=	SYM
ejpam-5571	246	73	ℓ(γ′	ℓ(γ′	PROPN
ejpam-5571	246	74	)	)	PUNCT
ejpam-5571	246	75	≤	≤	NOUN
ejpam-5571	246	76	π√	π√	PROPN
ejpam-5571	246	77	k	k	NOUN
ejpam-5571	246	78	;	;	PUNCT
ejpam-5571	246	79	(	(	PUNCT
ejpam-5571	246	80	ii	ii	NOUN
ejpam-5571	246	81	)	)	PUNCT
ejpam-5571	246	82	d(a	d(a	PROPN
ejpam-5571	246	83	,	,	PUNCT
ejpam-5571	246	84	b	b	NOUN
ejpam-5571	246	85	)	)	PUNCT
ejpam-5571	246	86	=	=	VERB
ejpam-5571	246	87	d(a′	d(a′	NOUN
ejpam-5571	246	88	,	,	PUNCT
ejpam-5571	246	89	b′	b′	NUM
ejpam-5571	246	90	)	)	PUNCT
ejpam-5571	246	91	;	;	PUNCT
ejpam-5571	246	92	(	(	PUNCT
ejpam-5571	246	93	iii	iii	X
ejpam-5571	246	94	)	)	PUNCT
ejpam-5571	246	95	d(a	d(a	PROPN
ejpam-5571	246	96	,	,	PUNCT
ejpam-5571	246	97	c	c	NOUN
ejpam-5571	246	98	)	)	PUNCT
ejpam-5571	246	99	=	=	SYM
ejpam-5571	246	100	d(a′	d(a′	NOUN
ejpam-5571	246	101	,	,	PUNCT
ejpam-5571	246	102	c′	c′	NUM
ejpam-5571	246	103	)	)	PUNCT
ejpam-5571	246	104	and	and	CCONJ
ejpam-5571	246	105	d(b	d(b	PROPN
ejpam-5571	246	106	,	,	PUNCT
ejpam-5571	246	107	c	c	NOUN
ejpam-5571	246	108	)	)	PUNCT
ejpam-5571	246	109	=	=	SYM
ejpam-5571	246	110	d(b′	d(b′	PROPN
ejpam-5571	246	111	,	,	PUNCT
ejpam-5571	246	112	c′	c′	NUM
ejpam-5571	246	113	)	)	PUNCT
ejpam-5571	246	114	;	;	PUNCT
ejpam-5571	246	115	(	(	PUNCT
ejpam-5571	246	116	iv	iv	X
ejpam-5571	246	117	)	)	PUNCT
ejpam-5571	246	118	∠p(a	∠p(a	ADJ
ejpam-5571	246	119	,	,	PUNCT
ejpam-5571	246	120	b	b	NOUN
ejpam-5571	246	121	)	)	PUNCT
ejpam-5571	246	122	=	=	SYM
ejpam-5571	246	123	∠p′(a	∠p′(a	PROPN
ejpam-5571	246	124	′	′	NOUN
ejpam-5571	246	125	,	,	PUNCT
ejpam-5571	246	126	b′	b′	NUM
ejpam-5571	246	127	)	)	PUNCT
ejpam-5571	246	128	and	and	CCONJ
ejpam-5571	246	129	∠c(a	∠c(a	NOUN
ejpam-5571	246	130	,	,	PUNCT
ejpam-5571	246	131	b	b	NOUN
ejpam-5571	246	132	)	)	PUNCT
ejpam-5571	246	133	=	=	SYM
ejpam-5571	246	134	∠c′(a	∠c′(a	PROPN
ejpam-5571	246	135	′	′	NOUN
ejpam-5571	246	136	,	,	PUNCT
ejpam-5571	246	137	b′	b′	NUM
ejpam-5571	246	138	)	)	PUNCT
ejpam-5571	246	139	;	;	PUNCT
ejpam-5571	246	140	(	(	PUNCT
ejpam-5571	246	141	v	v	NOUN
ejpam-5571	246	142	)	)	PUNCT
ejpam-5571	246	143	for	for	ADP
ejpam-5571	246	144	any	any	DET
ejpam-5571	246	145	triangle	triangle	NOUN
ejpam-5571	246	146	△	△	X
ejpam-5571	246	147	(	(	PUNCT
ejpam-5571	246	148	u	u	NOUN
ejpam-5571	246	149	,	,	PUNCT
ejpam-5571	246	150	v	v	NOUN
ejpam-5571	246	151	,	,	PUNCT
ejpam-5571	246	152	w	w	NOUN
ejpam-5571	246	153	)	)	PUNCT
ejpam-5571	246	154	in	in	ADP
ejpam-5571	246	155	x	x	NOUN
ejpam-5571	246	156	,	,	PUNCT
ejpam-5571	246	157	∠u(v	∠u(v	NOUN
ejpam-5571	246	158	,	,	PUNCT
ejpam-5571	246	159	x	x	NOUN
ejpam-5571	246	160	)	)	PUNCT
ejpam-5571	246	161	=	=	SYM
ejpam-5571	246	162	∠u(v	∠u(v	NOUN
ejpam-5571	246	163	,	,	PUNCT
ejpam-5571	246	164	w	w	NOUN
ejpam-5571	246	165	)	)	PUNCT
ejpam-5571	246	166	and	and	CCONJ
ejpam-5571	246	167	∠u(w	∠u(w	NUM
ejpam-5571	246	168	,	,	PUNCT
ejpam-5571	246	169	x	x	NOUN
ejpam-5571	246	170	)	)	PUNCT
ejpam-5571	246	171	=	=	SYM
ejpam-5571	246	172	∠u(w	∠u(w	PROPN
ejpam-5571	246	173	,	,	PUNCT
ejpam-5571	246	174	v	v	NOUN
ejpam-5571	246	175	)	)	PUNCT
ejpam-5571	246	176	for	for	ADP
ejpam-5571	246	177	all	all	DET
ejpam-5571	246	178	x	x	SYM
ejpam-5571	246	179	∈	∈	PROPN
ejpam-5571	247	1	[	[	X
ejpam-5571	247	2	v	v	NOUN
ejpam-5571	247	3	,	,	PUNCT
ejpam-5571	247	4	w	w	NOUN
ejpam-5571	247	5	]	]	X
ejpam-5571	247	6	;	;	PUNCT
ejpam-5571	247	7	(	(	PUNCT
ejpam-5571	247	8	vi	vi	NOUN
ejpam-5571	247	9	)	)	PUNCT
ejpam-5571	247	10	for	for	ADP
ejpam-5571	247	11	any	any	DET
ejpam-5571	247	12	triangle	triangle	NOUN
ejpam-5571	247	13	△	△	X
ejpam-5571	247	14	(	(	PUNCT
ejpam-5571	247	15	u	u	NOUN
ejpam-5571	247	16	,	,	PUNCT
ejpam-5571	247	17	v	v	NOUN
ejpam-5571	247	18	,	,	PUNCT
ejpam-5571	247	19	w	w	NOUN
ejpam-5571	247	20	)	)	PUNCT
ejpam-5571	247	21	in	in	ADP
ejpam-5571	247	22	x	x	NOUN
ejpam-5571	247	23	,	,	PUNCT
ejpam-5571	247	24	∠u(v	∠u(v	NOUN
ejpam-5571	247	25	,	,	PUNCT
ejpam-5571	247	26	w	w	NOUN
ejpam-5571	247	27	)	)	PUNCT
ejpam-5571	247	28	=	=	NOUN
ejpam-5571	247	29	∠u(v	∠u(v	NOUN
ejpam-5571	247	30	,	,	PUNCT
ejpam-5571	247	31	x	x	NOUN
ejpam-5571	247	32	)	)	PUNCT
ejpam-5571	247	33	+	+	CCONJ
ejpam-5571	247	34	∠u(x	∠u(x	ADJ
ejpam-5571	247	35	,	,	PUNCT
ejpam-5571	247	36	w	w	NOUN
ejpam-5571	247	37	)	)	PUNCT
ejpam-5571	247	38	for	for	ADP
ejpam-5571	247	39	all	all	DET
ejpam-5571	247	40	x	x	SYM
ejpam-5571	247	41	∈	∈	PROPN
ejpam-5571	248	1	[	[	X
ejpam-5571	248	2	v	v	NOUN
ejpam-5571	248	3	,	,	PUNCT
ejpam-5571	248	4	w	w	PROPN
ejpam-5571	248	5	]	]	X
ejpam-5571	248	6	;	;	PUNCT
ejpam-5571	248	7	then	then	ADV
ejpam-5571	248	8	c({p	c({p	PROPN
ejpam-5571	248	9	,	,	PUNCT
ejpam-5571	248	10	a	a	DET
ejpam-5571	248	11	,	,	PUNCT
ejpam-5571	248	12	c	c	NOUN
ejpam-5571	248	13	,	,	PUNCT
ejpam-5571	248	14	b	b	NOUN
ejpam-5571	248	15	}	}	PUNCT
ejpam-5571	248	16	)	)	PUNCT
ejpam-5571	248	17	is	be	AUX
ejpam-5571	248	18	isometric	isometric	ADJ
ejpam-5571	248	19	to	to	ADP
ejpam-5571	248	20	c({p′	c({p′	NOUN
ejpam-5571	248	21	,	,	PUNCT
ejpam-5571	248	22	a′	a′	PROPN
ejpam-5571	248	23	,	,	PUNCT
ejpam-5571	248	24	c′	c′	PRON
ejpam-5571	248	25	,	,	PUNCT
ejpam-5571	248	26	b′	b′	NUM
ejpam-5571	248	27	}	}	PUNCT
ejpam-5571	248	28	)	)	PUNCT
ejpam-5571	248	29	.	.	PUNCT
ejpam-5571	249	1	proof	proof	NOUN
ejpam-5571	249	2	.	.	PUNCT
ejpam-5571	250	1	by	by	ADP
ejpam-5571	250	2	(	(	PUNCT
ejpam-5571	250	3	ii	ii	NOUN
ejpam-5571	250	4	)	)	PUNCT
ejpam-5571	250	5	and	and	CCONJ
ejpam-5571	250	6	(	(	PUNCT
ejpam-5571	250	7	iii	iii	NOUN
ejpam-5571	250	8	)	)	PUNCT
ejpam-5571	250	9	,	,	PUNCT
ejpam-5571	250	10	we	we	PRON
ejpam-5571	250	11	have	have	VERB
ejpam-5571	250	12	that	that	DET
ejpam-5571	250	13	triangles	triangle	NOUN
ejpam-5571	250	14	△	△	X
ejpam-5571	250	15	(	(	PUNCT
ejpam-5571	250	16	a′	a′	PROPN
ejpam-5571	250	17	,	,	PUNCT
ejpam-5571	250	18	b′	b′	NUM
ejpam-5571	250	19	,	,	PUNCT
ejpam-5571	250	20	c′	c′	NUM
ejpam-5571	250	21	)	)	PUNCT
ejpam-5571	250	22	and	and	CCONJ
ejpam-5571	250	23	△	△	X
ejpam-5571	250	24	(	(	PUNCT
ejpam-5571	250	25	a′	a′	PROPN
ejpam-5571	250	26	,	,	PUNCT
ejpam-5571	250	27	b′	b′	NUM
ejpam-5571	250	28	,	,	PUNCT
ejpam-5571	250	29	p′	p′	NOUN
ejpam-5571	250	30	)	)	PUNCT
ejpam-5571	250	31	are	be	AUX
ejpam-5571	250	32	corresponding	correspond	VERB
ejpam-5571	250	33	triangles	triangle	NOUN
ejpam-5571	250	34	of	of	ADP
ejpam-5571	250	35	△	△	X
ejpam-5571	250	36	(	(	PUNCT
ejpam-5571	250	37	a	a	DET
ejpam-5571	250	38	,	,	PUNCT
ejpam-5571	250	39	b	b	NOUN
ejpam-5571	250	40	,	,	PUNCT
ejpam-5571	250	41	c	c	NOUN
ejpam-5571	250	42	)	)	PUNCT
ejpam-5571	250	43	and	and	CCONJ
ejpam-5571	250	44	△	△	X
ejpam-5571	250	45	(	(	PUNCT
ejpam-5571	250	46	a	a	PRON
ejpam-5571	250	47	,	,	PUNCT
ejpam-5571	250	48	b	b	NOUN
ejpam-5571	250	49	,	,	PUNCT
ejpam-5571	250	50	p	p	NOUN
ejpam-5571	250	51	)	)	PUNCT
ejpam-5571	250	52	,	,	PUNCT
ejpam-5571	250	53	respectively	respectively	ADV
ejpam-5571	250	54	.	.	PUNCT
ejpam-5571	251	1	by	by	ADP
ejpam-5571	251	2	(	(	PUNCT
ejpam-5571	251	3	iv	iv	X
ejpam-5571	251	4	)	)	PUNCT
ejpam-5571	251	5	and	and	CCONJ
ejpam-5571	251	6	(	(	PUNCT
ejpam-5571	251	7	v	v	NOUN
ejpam-5571	251	8	)	)	PUNCT
ejpam-5571	251	9	,	,	PUNCT
ejpam-5571	251	10	and	and	CCONJ
ejpam-5571	251	11	applying	apply	VERB
ejpam-5571	251	12	theorem	theorem	NOUN
ejpam-5571	251	13	4	4	NUM
ejpam-5571	251	14	,	,	PUNCT
ejpam-5571	251	15	we	we	PRON
ejpam-5571	251	16	get	get	VERB
ejpam-5571	251	17	that	that	SCONJ
ejpam-5571	251	18	the	the	DET
ejpam-5571	251	19	convex	convex	PROPN
ejpam-5571	251	20	hulls	hull	NOUN
ejpam-5571	251	21	of	of	ADP
ejpam-5571	251	22	triangles	triangle	NOUN
ejpam-5571	251	23	△	△	X
ejpam-5571	251	24	(	(	PUNCT
ejpam-5571	251	25	a′	a′	PROPN
ejpam-5571	251	26	,	,	PUNCT
ejpam-5571	251	27	b′	b′	NUM
ejpam-5571	251	28	,	,	PUNCT
ejpam-5571	251	29	c′	c′	NUM
ejpam-5571	251	30	)	)	PUNCT
ejpam-5571	251	31	and	and	CCONJ
ejpam-5571	251	32	△	△	X
ejpam-5571	251	33	(	(	PUNCT
ejpam-5571	251	34	a′	a′	PROPN
ejpam-5571	251	35	,	,	PUNCT
ejpam-5571	251	36	b′	b′	NUM
ejpam-5571	251	37	,	,	PUNCT
ejpam-5571	251	38	p′	p′	NOUN
ejpam-5571	251	39	)	)	PUNCT
ejpam-5571	251	40	are	be	AUX
ejpam-5571	251	41	isometric	isometric	ADJ
ejpam-5571	251	42	to	to	ADP
ejpam-5571	251	43	the	the	DET
ejpam-5571	251	44	convex	convex	PROPN
ejpam-5571	251	45	hulls	hull	NOUN
ejpam-5571	251	46	of	of	ADP
ejpam-5571	251	47	triangles	triangle	NOUN
ejpam-5571	251	48	△	△	X
ejpam-5571	251	49	(	(	PUNCT
ejpam-5571	251	50	a	a	PRON
ejpam-5571	251	51	,	,	PUNCT
ejpam-5571	251	52	b	b	NOUN
ejpam-5571	251	53	,	,	PUNCT
ejpam-5571	251	54	c	c	NOUN
ejpam-5571	251	55	)	)	PUNCT
ejpam-5571	251	56	and	and	CCONJ
ejpam-5571	251	57	△	△	X
ejpam-5571	251	58	(	(	PUNCT
ejpam-5571	251	59	a	a	PRON
ejpam-5571	251	60	,	,	PUNCT
ejpam-5571	251	61	b	b	NOUN
ejpam-5571	251	62	,	,	PUNCT
ejpam-5571	251	63	p	p	NOUN
ejpam-5571	251	64	)	)	PUNCT
ejpam-5571	251	65	,	,	PUNCT
ejpam-5571	251	66	respectively	respectively	ADV
ejpam-5571	251	67	.	.	PUNCT
ejpam-5571	252	1	by	by	ADP
ejpam-5571	252	2	(	(	PUNCT
ejpam-5571	252	3	vi	vi	NOUN
ejpam-5571	252	4	)	)	PUNCT
ejpam-5571	252	5	,	,	PUNCT
ejpam-5571	252	6	we	we	PRON
ejpam-5571	252	7	obtain	obtain	VERB
ejpam-5571	252	8	∠a(p	∠a(p	ADJ
ejpam-5571	252	9	,	,	PUNCT
ejpam-5571	252	10	c	c	NOUN
ejpam-5571	252	11	)	)	PUNCT
ejpam-5571	252	12	=	=	PUNCT
ejpam-5571	252	13	∠a(p	∠a(p	PROPN
ejpam-5571	252	14	,	,	PUNCT
ejpam-5571	252	15	q	q	ADJ
ejpam-5571	252	16	)	)	PUNCT
ejpam-5571	252	17	+	+	CCONJ
ejpam-5571	252	18	∠a(q	∠a(q	ADJ
ejpam-5571	252	19	,	,	PUNCT
ejpam-5571	252	20	c	c	NOUN
ejpam-5571	252	21	)	)	PUNCT
ejpam-5571	253	1	=	=	SYM
ejpam-5571	254	1	∠a′(p	∠a′(p	PROPN
ejpam-5571	254	2	′	′	NOUN
ejpam-5571	254	3	,	,	PUNCT
ejpam-5571	254	4	q′	q′	NOUN
ejpam-5571	254	5	)	)	PUNCT
ejpam-5571	255	1	+	+	CCONJ
ejpam-5571	255	2	∠a′(q	∠a′(q	PROPN
ejpam-5571	255	3	′	′	NOUN
ejpam-5571	255	4	,	,	PUNCT
ejpam-5571	255	5	c′	c′	NUM
ejpam-5571	255	6	)	)	PUNCT
ejpam-5571	256	1	=	=	PUNCT
ejpam-5571	256	2	∠a′(p	∠a′(p	PROPN
ejpam-5571	256	3	′	′	NOUN
ejpam-5571	256	4	,	,	PUNCT
ejpam-5571	256	5	c′	c′	NUM
ejpam-5571	256	6	)	)	PUNCT
ejpam-5571	256	7	and	and	CCONJ
ejpam-5571	256	8	∠b(p	∠b(p	PRON
ejpam-5571	256	9	,	,	PUNCT
ejpam-5571	256	10	c	c	NOUN
ejpam-5571	256	11	)	)	PUNCT
ejpam-5571	256	12	=	=	SYM
ejpam-5571	257	1	∠b(p	∠b(p	ADJ
ejpam-5571	257	2	,	,	PUNCT
ejpam-5571	257	3	q	q	ADJ
ejpam-5571	257	4	)	)	PUNCT
ejpam-5571	257	5	+	+	CCONJ
ejpam-5571	257	6	∠b(q	∠b(q	NOUN
ejpam-5571	257	7	,	,	PUNCT
ejpam-5571	257	8	c	c	NOUN
ejpam-5571	257	9	)	)	PUNCT
ejpam-5571	257	10	=	=	SYM
ejpam-5571	258	1	∠b′(p	∠b′(p	PROPN
ejpam-5571	258	2	′	′	NOUN
ejpam-5571	258	3	,	,	PUNCT
ejpam-5571	258	4	q′	q′	NOUN
ejpam-5571	258	5	)	)	PUNCT
ejpam-5571	259	1	+	+	CCONJ
ejpam-5571	259	2	∠b′(q	∠b′(q	PROPN
ejpam-5571	259	3	′	′	NOUN
ejpam-5571	259	4	,	,	PUNCT
ejpam-5571	259	5	c′	c′	NUM
ejpam-5571	259	6	)	)	PUNCT
ejpam-5571	260	1	=	=	PUNCT
ejpam-5571	260	2	∠b′(p	∠b′(p	PROPN
ejpam-5571	260	3	′	′	NOUN
ejpam-5571	260	4	,	,	PUNCT
ejpam-5571	260	5	c′	c′	NUM
ejpam-5571	260	6	)	)	PUNCT
ejpam-5571	260	7	,	,	PUNCT
ejpam-5571	260	8	and	and	CCONJ
ejpam-5571	260	9	using	use	VERB
ejpam-5571	260	10	(	(	PUNCT
ejpam-5571	260	11	iv	iv	NUM
ejpam-5571	260	12	)	)	PUNCT
ejpam-5571	260	13	and	and	CCONJ
ejpam-5571	260	14	theorem	theorem	VERB
ejpam-5571	260	15	3	3	NUM
ejpam-5571	260	16	,	,	PUNCT
ejpam-5571	260	17	we	we	PRON
ejpam-5571	260	18	have	have	VERB
ejpam-5571	260	19	that	that	SCONJ
ejpam-5571	260	20	the	the	DET
ejpam-5571	260	21	convex	convex	PROPN
ejpam-5571	260	22	hulls	hull	NOUN
ejpam-5571	260	23	of	of	ADP
ejpam-5571	260	24	triangles	triangle	NOUN
ejpam-5571	260	25	△	△	X
ejpam-5571	260	26	(	(	PUNCT
ejpam-5571	260	27	a′	a′	PROPN
ejpam-5571	260	28	,	,	PUNCT
ejpam-5571	260	29	p′	p′	NOUN
ejpam-5571	260	30	,	,	PUNCT
ejpam-5571	260	31	c′	c′	NUM
ejpam-5571	260	32	)	)	PUNCT
ejpam-5571	260	33	and	and	CCONJ
ejpam-5571	260	34	△	△	X
ejpam-5571	260	35	(	(	PUNCT
ejpam-5571	260	36	b′	b′	NUM
ejpam-5571	260	37	,	,	PUNCT
ejpam-5571	260	38	p′	p′	NOUN
ejpam-5571	260	39	,	,	PUNCT
ejpam-5571	260	40	c′	c′	NUM
ejpam-5571	260	41	)	)	PUNCT
ejpam-5571	260	42	are	be	AUX
ejpam-5571	260	43	isometric	isometric	ADJ
ejpam-5571	260	44	to	to	ADP
ejpam-5571	260	45	the	the	DET
ejpam-5571	260	46	convex	convex	PROPN
ejpam-5571	260	47	hulls	hull	NOUN
ejpam-5571	260	48	of	of	ADP
ejpam-5571	260	49	triangles	triangle	NOUN
ejpam-5571	260	50	△	△	X
ejpam-5571	260	51	(	(	PUNCT
ejpam-5571	260	52	a	a	DET
ejpam-5571	260	53	,	,	PUNCT
ejpam-5571	260	54	p	p	X
ejpam-5571	260	55	,	,	PUNCT
ejpam-5571	260	56	c	c	NOUN
ejpam-5571	260	57	)	)	PUNCT
ejpam-5571	260	58	and	and	CCONJ
ejpam-5571	260	59	△	△	X
ejpam-5571	260	60	(	(	PUNCT
ejpam-5571	260	61	b	b	NOUN
ejpam-5571	260	62	,	,	PUNCT
ejpam-5571	260	63	p	p	X
ejpam-5571	260	64	,	,	PUNCT
ejpam-5571	260	65	c	c	NOUN
ejpam-5571	260	66	)	)	PUNCT
ejpam-5571	260	67	,	,	PUNCT
ejpam-5571	260	68	respectively	respectively	ADV
ejpam-5571	260	69	.	.	PUNCT
ejpam-5571	261	1	we	we	PRON
ejpam-5571	261	2	prove	prove	VERB
ejpam-5571	261	3	,	,	PUNCT
ejpam-5571	261	4	as	as	ADP
ejpam-5571	261	5	the	the	DET
ejpam-5571	261	6	same	same	ADJ
ejpam-5571	261	7	proof	proof	NOUN
ejpam-5571	261	8	of	of	ADP
ejpam-5571	261	9	theorem	theorem	ADJ
ejpam-5571	261	10	4	4	NUM
ejpam-5571	261	11	,	,	PUNCT
ejpam-5571	261	12	that	that	DET
ejpam-5571	261	13	c({p	c({p	NOUN
ejpam-5571	261	14	,	,	PUNCT
ejpam-5571	261	15	a	a	DET
ejpam-5571	261	16	,	,	PUNCT
ejpam-5571	261	17	c	c	NOUN
ejpam-5571	261	18	,	,	PUNCT
ejpam-5571	261	19	b	b	NOUN
ejpam-5571	261	20	}	}	PUNCT
ejpam-5571	261	21	)	)	PUNCT
ejpam-5571	261	22	is	be	AUX
ejpam-5571	261	23	isometric	isometric	ADJ
ejpam-5571	261	24	to	to	ADP
ejpam-5571	261	25	c({p′	c({p′	NOUN
ejpam-5571	261	26	,	,	PUNCT
ejpam-5571	261	27	a′	a′	PROPN
ejpam-5571	261	28	,	,	PUNCT
ejpam-5571	261	29	c′	c′	PRON
ejpam-5571	261	30	,	,	PUNCT
ejpam-5571	261	31	b′	b′	NUM
ejpam-5571	261	32	}	}	PUNCT
ejpam-5571	261	33	)	)	PUNCT
ejpam-5571	261	34	.	.	PUNCT
ejpam-5571	262	1	theorem	theorem	VERB
ejpam-5571	262	2	6	6	NUM
ejpam-5571	262	3	.	.	PUNCT
ejpam-5571	263	1	let	let	VERB
ejpam-5571	263	2	x	x	PRON
ejpam-5571	263	3	be	be	AUX
ejpam-5571	263	4	a	a	DET
ejpam-5571	263	5	metric	metric	ADJ
ejpam-5571	263	6	space	space	NOUN
ejpam-5571	263	7	with	with	ADP
ejpam-5571	263	8	curvature	curvature	NOUN
ejpam-5571	263	9	bounded	bound	VERB
ejpam-5571	263	10	below	below	ADV
ejpam-5571	263	11	by	by	ADP
ejpam-5571	263	12	k	k	PROPN
ejpam-5571	263	13	in	in	ADP
ejpam-5571	263	14	the	the	DET
ejpam-5571	263	15	large	large	NOUN
ejpam-5571	263	16	.	.	PUNCT
ejpam-5571	264	1	let	let	VERB
ejpam-5571	264	2	γ	γ	NOUN
ejpam-5571	264	3	be	be	AUX
ejpam-5571	264	4	a	a	DET
ejpam-5571	264	5	spherical	spherical	ADJ
ejpam-5571	264	6	curve	curve	NOUN
ejpam-5571	264	7	at	at	ADP
ejpam-5571	264	8	a	a	DET
ejpam-5571	264	9	distance	distance	NOUN
ejpam-5571	265	1	r	r	NOUN
ejpam-5571	265	2	<	<	X
ejpam-5571	265	3	π	π	PROPN
ejpam-5571	265	4	2	2	NUM
ejpam-5571	265	5	√	√	PROPN
ejpam-5571	265	6	k	k	NOUN
ejpam-5571	265	7	from	from	ADP
ejpam-5571	265	8	a	a	DET
ejpam-5571	265	9	point	point	NOUN
ejpam-5571	265	10	p	p	NOUN
ejpam-5571	265	11	with	with	ADP
ejpam-5571	265	12	endpoints	endpoint	NOUN
ejpam-5571	265	13	a	a	PRON
ejpam-5571	265	14	,	,	PUNCT
ejpam-5571	265	15	b	b	NOUN
ejpam-5571	265	16	in	in	ADP
ejpam-5571	265	17	x	x	X
ejpam-5571	265	18	and	and	CCONJ
ejpam-5571	265	19	γ′	γ′	NOUN
ejpam-5571	265	20	be	be	AUX
ejpam-5571	265	21	a	a	DET
ejpam-5571	265	22	subarc	subarc	NOUN
ejpam-5571	265	23	of	of	ADP
ejpam-5571	265	24	a	a	DET
ejpam-5571	265	25	circle	circle	NOUN
ejpam-5571	265	26	of	of	ADP
ejpam-5571	265	27	radius	radius	NOUN
ejpam-5571	265	28	r	r	NOUN
ejpam-5571	265	29	centered	center	VERB
ejpam-5571	265	30	at	at	ADP
ejpam-5571	265	31	a	a	DET
ejpam-5571	265	32	point	point	NOUN
ejpam-5571	265	33	p′	p′	NOUN
ejpam-5571	265	34	in	in	ADP
ejpam-5571	265	35	rk	rk	NOUN
ejpam-5571	265	36	with	with	ADP
ejpam-5571	265	37	endpoints	endpoint	NOUN
ejpam-5571	265	38	a′	a′	PROPN
ejpam-5571	265	39	,	,	PUNCT
ejpam-5571	265	40	b′.	b′.	PROPN
ejpam-5571	265	41	suppose	suppose	VERB
ejpam-5571	265	42	that	that	SCONJ
ejpam-5571	265	43	c	c	PROPN
ejpam-5571	265	44	∈	∈	PROPN
ejpam-5571	265	45	γ	γ	X
ejpam-5571	265	46	is	be	AUX
ejpam-5571	265	47	between	between	ADP
ejpam-5571	265	48	a	a	PRON
ejpam-5571	265	49	,	,	PUNCT
ejpam-5571	265	50	b	b	NOUN
ejpam-5571	265	51	and	and	CCONJ
ejpam-5571	265	52	c′	c′	NOUN
ejpam-5571	265	53	∈	∈	PROPN
ejpam-5571	265	54	γ	γ	X
ejpam-5571	265	55	is	be	AUX
ejpam-5571	265	56	between	between	ADP
ejpam-5571	265	57	a′	a′	PROPN
ejpam-5571	265	58	,	,	PUNCT
ejpam-5571	265	59	b′.	b′.	PROPN
ejpam-5571	265	60	assume	assume	VERB
ejpam-5571	265	61	that	that	SCONJ
ejpam-5571	265	62	the	the	DET
ejpam-5571	265	63	following	follow	VERB
ejpam-5571	265	64	statements	statement	NOUN
ejpam-5571	265	65	hold	hold	VERB
ejpam-5571	265	66	:	:	PUNCT
ejpam-5571	265	67	(	(	PUNCT
ejpam-5571	265	68	i	i	NOUN
ejpam-5571	265	69	)	)	PUNCT
ejpam-5571	265	70	ℓ(γ	ℓ(γ	PROPN
ejpam-5571	265	71	)	)	PUNCT
ejpam-5571	265	72	=	=	SYM
ejpam-5571	265	73	ℓ(γ′	ℓ(γ′	PROPN
ejpam-5571	265	74	)	)	PUNCT
ejpam-5571	265	75	≤	≤	NOUN
ejpam-5571	265	76	π√	π√	PROPN
ejpam-5571	265	77	k	k	NOUN
ejpam-5571	265	78	;	;	PUNCT
ejpam-5571	265	79	(	(	PUNCT
ejpam-5571	265	80	ii	ii	NOUN
ejpam-5571	265	81	)	)	PUNCT
ejpam-5571	265	82	d(a	d(a	PROPN
ejpam-5571	265	83	,	,	PUNCT
ejpam-5571	265	84	b	b	NOUN
ejpam-5571	265	85	)	)	PUNCT
ejpam-5571	265	86	=	=	VERB
ejpam-5571	265	87	d(a′	d(a′	NOUN
ejpam-5571	265	88	,	,	PUNCT
ejpam-5571	265	89	b′	b′	NUM
ejpam-5571	265	90	)	)	PUNCT
ejpam-5571	265	91	;	;	PUNCT
ejpam-5571	265	92	(	(	PUNCT
ejpam-5571	265	93	iii	iii	X
ejpam-5571	265	94	)	)	PUNCT
ejpam-5571	265	95	c({p	c({p	NOUN
ejpam-5571	265	96	,	,	PUNCT
ejpam-5571	265	97	a	a	DET
ejpam-5571	265	98	,	,	PUNCT
ejpam-5571	265	99	c	c	NOUN
ejpam-5571	265	100	}	}	PUNCT
ejpam-5571	265	101	)	)	PUNCT
ejpam-5571	265	102	is	be	AUX
ejpam-5571	265	103	isometric	isometric	ADJ
ejpam-5571	265	104	to	to	ADP
ejpam-5571	265	105	c({p′	c({p′	NOUN
ejpam-5571	265	106	,	,	PUNCT
ejpam-5571	265	107	a′	a′	PROPN
ejpam-5571	265	108	,	,	PUNCT
ejpam-5571	265	109	c′	c′	NUM
ejpam-5571	265	110	}	}	PUNCT
ejpam-5571	265	111	)	)	PUNCT
ejpam-5571	265	112	and	and	CCONJ
ejpam-5571	265	113	c({p	c({p	PROPN
ejpam-5571	265	114	,	,	PUNCT
ejpam-5571	265	115	b	b	NOUN
ejpam-5571	265	116	,	,	PUNCT
ejpam-5571	265	117	c	c	NOUN
ejpam-5571	265	118	}	}	PUNCT
ejpam-5571	265	119	)	)	PUNCT
ejpam-5571	265	120	is	be	AUX
ejpam-5571	265	121	isometric	isometric	ADJ
ejpam-5571	265	122	to	to	ADP
ejpam-5571	265	123	c({p′	c({p′	PROPN
ejpam-5571	265	124	,	,	PUNCT
ejpam-5571	265	125	b′	b′	NUM
ejpam-5571	265	126	,	,	PUNCT
ejpam-5571	265	127	c′	c′	NUM
ejpam-5571	265	128	}	}	PUNCT
ejpam-5571	265	129	)	)	PUNCT
ejpam-5571	265	130	;	;	PUNCT
ejpam-5571	266	1	c.	c.	PROPN
ejpam-5571	266	2	phokaew	phokaew	PROPN
ejpam-5571	266	3	,	,	PUNCT
ejpam-5571	266	4	a.	a.	PROPN
ejpam-5571	266	5	sama	sama	PROPN
ejpam-5571	266	6	-	-	PUNCT
ejpam-5571	266	7	ae	ae	PROPN
ejpam-5571	266	8	,	,	PUNCT
ejpam-5571	266	9	/	/	SYM
ejpam-5571	266	10	eur	eur	NOUN
ejpam-5571	266	11	.	.	PUNCT
ejpam-5571	267	1	j.	j.	PROPN
ejpam-5571	267	2	pure	pure	PROPN
ejpam-5571	267	3	appl	appl	PROPN
ejpam-5571	267	4	.	.	PROPN
ejpam-5571	267	5	math	math	PROPN
ejpam-5571	267	6	,	,	PUNCT
ejpam-5571	267	7	17	17	NUM
ejpam-5571	267	8	(	(	PUNCT
ejpam-5571	267	9	4	4	NUM
ejpam-5571	267	10	)	)	PUNCT
ejpam-5571	267	11	(	(	PUNCT
ejpam-5571	267	12	2024	2024	NUM
ejpam-5571	267	13	)	)	PUNCT
ejpam-5571	267	14	,	,	PUNCT
ejpam-5571	267	15	3932	3932	NUM
ejpam-5571	267	16	-	-	SYM
ejpam-5571	267	17	3944	3944	NUM
ejpam-5571	267	18	3940	3940	NUM
ejpam-5571	267	19	(	(	PUNCT
ejpam-5571	267	20	iv	iv	X
ejpam-5571	267	21	)	)	PUNCT
ejpam-5571	267	22	∠c(a	∠c(a	NOUN
ejpam-5571	267	23	,	,	PUNCT
ejpam-5571	267	24	b	b	NOUN
ejpam-5571	267	25	)	)	PUNCT
ejpam-5571	267	26	=	=	SYM
ejpam-5571	267	27	∠c′(a	∠c′(a	PROPN
ejpam-5571	267	28	′	′	NOUN
ejpam-5571	267	29	,	,	PUNCT
ejpam-5571	267	30	b′	b′	NUM
ejpam-5571	267	31	)	)	PUNCT
ejpam-5571	267	32	and	and	CCONJ
ejpam-5571	267	33	∠p(a	∠p(a	ADJ
ejpam-5571	267	34	,	,	PUNCT
ejpam-5571	267	35	b	b	NOUN
ejpam-5571	267	36	)	)	PUNCT
ejpam-5571	267	37	=	=	SYM
ejpam-5571	267	38	∠p′(a	∠p′(a	PROPN
ejpam-5571	267	39	′	′	PROPN
ejpam-5571	267	40	,	,	PUNCT
ejpam-5571	267	41	b′	b′	NUM
ejpam-5571	267	42	)	)	PUNCT
ejpam-5571	267	43	;	;	PUNCT
ejpam-5571	267	44	(	(	PUNCT
ejpam-5571	267	45	v	v	NOUN
ejpam-5571	267	46	)	)	PUNCT
ejpam-5571	267	47	∠a(x	∠a(x	NOUN
ejpam-5571	267	48	,	,	PUNCT
ejpam-5571	267	49	c	c	NOUN
ejpam-5571	267	50	)	)	PUNCT
ejpam-5571	267	51	=	=	SYM
ejpam-5571	268	1	∠a(b	∠a(b	VERB
ejpam-5571	268	2	,	,	PUNCT
ejpam-5571	268	3	c	c	NOUN
ejpam-5571	268	4	)	)	PUNCT
ejpam-5571	268	5	,	,	PUNCT
ejpam-5571	268	6	∠b(x	∠b(x	NOUN
ejpam-5571	268	7	,	,	PUNCT
ejpam-5571	268	8	c	c	NOUN
ejpam-5571	268	9	)	)	PUNCT
ejpam-5571	268	10	=	=	SYM
ejpam-5571	268	11	∠b(a	∠b(a	NOUN
ejpam-5571	268	12	,	,	PUNCT
ejpam-5571	268	13	c	c	NOUN
ejpam-5571	268	14	)	)	PUNCT
ejpam-5571	268	15	and	and	CCONJ
ejpam-5571	268	16	∠c(a	∠c(a	NOUN
ejpam-5571	268	17	,	,	PUNCT
ejpam-5571	268	18	b	b	NOUN
ejpam-5571	268	19	)	)	PUNCT
ejpam-5571	268	20	=	=	SYM
ejpam-5571	268	21	∠c(a	∠c(a	NOUN
ejpam-5571	268	22	,	,	PUNCT
ejpam-5571	268	23	x	x	X
ejpam-5571	268	24	)	)	PUNCT
ejpam-5571	268	25	+	+	PUNCT
ejpam-5571	268	26	∠c(x	∠c(x	ADJ
ejpam-5571	268	27	,	,	PUNCT
ejpam-5571	268	28	b	b	NOUN
ejpam-5571	268	29	)	)	PUNCT
ejpam-5571	268	30	for	for	ADP
ejpam-5571	268	31	any	any	DET
ejpam-5571	268	32	x	x	SYM
ejpam-5571	268	33	∈	∈	PROPN
ejpam-5571	268	34	[	[	X
ejpam-5571	268	35	a	a	X
ejpam-5571	268	36	,	,	PUNCT
ejpam-5571	268	37	b	b	NOUN
ejpam-5571	268	38	]	]	X
ejpam-5571	268	39	;	;	PUNCT
ejpam-5571	268	40	(	(	PUNCT
ejpam-5571	268	41	vi	vi	NOUN
ejpam-5571	268	42	)	)	PUNCT
ejpam-5571	268	43	∠a(x	∠a(x	NOUN
ejpam-5571	268	44	,	,	PUNCT
ejpam-5571	268	45	p	p	X
ejpam-5571	268	46	)	)	PUNCT
ejpam-5571	268	47	=	=	PUNCT
ejpam-5571	268	48	∠a(b	∠a(b	VERB
ejpam-5571	268	49	,	,	PUNCT
ejpam-5571	268	50	p	p	NOUN
ejpam-5571	268	51	)	)	PUNCT
ejpam-5571	268	52	,	,	PUNCT
ejpam-5571	268	53	∠b(x	∠b(x	NOUN
ejpam-5571	268	54	,	,	PUNCT
ejpam-5571	268	55	p	p	NOUN
ejpam-5571	268	56	)	)	PUNCT
ejpam-5571	268	57	=	=	PUNCT
ejpam-5571	268	58	∠b(a	∠b(a	NOUN
ejpam-5571	268	59	,	,	PUNCT
ejpam-5571	268	60	p	p	NOUN
ejpam-5571	268	61	)	)	PUNCT
ejpam-5571	268	62	and	and	CCONJ
ejpam-5571	268	63	∠p(a	∠p(a	ADJ
ejpam-5571	268	64	,	,	PUNCT
ejpam-5571	268	65	b	b	NOUN
ejpam-5571	268	66	)	)	PUNCT
ejpam-5571	268	67	=	=	PUNCT
ejpam-5571	269	1	∠p(a	∠p(a	ADJ
ejpam-5571	269	2	,	,	PUNCT
ejpam-5571	269	3	x	x	PRON
ejpam-5571	269	4	)	)	PUNCT
ejpam-5571	269	5	+	+	CCONJ
ejpam-5571	269	6	∠p(x	∠p(x	NOUN
ejpam-5571	269	7	,	,	PUNCT
ejpam-5571	269	8	b	b	NOUN
ejpam-5571	269	9	)	)	PUNCT
ejpam-5571	269	10	for	for	ADP
ejpam-5571	269	11	any	any	DET
ejpam-5571	269	12	x	x	SYM
ejpam-5571	269	13	∈	∈	PROPN
ejpam-5571	269	14	[	[	X
ejpam-5571	269	15	a	a	X
ejpam-5571	269	16	,	,	PUNCT
ejpam-5571	269	17	b	b	NOUN
ejpam-5571	269	18	]	]	X
ejpam-5571	269	19	.	.	PUNCT
ejpam-5571	270	1	then	then	ADV
ejpam-5571	270	2	c({p	c({p	PROPN
ejpam-5571	270	3	,	,	PUNCT
ejpam-5571	270	4	a	a	DET
ejpam-5571	270	5	,	,	PUNCT
ejpam-5571	270	6	c	c	NOUN
ejpam-5571	270	7	,	,	PUNCT
ejpam-5571	270	8	b	b	NOUN
ejpam-5571	270	9	}	}	PUNCT
ejpam-5571	270	10	)	)	PUNCT
ejpam-5571	270	11	is	be	AUX
ejpam-5571	270	12	isometric	isometric	ADJ
ejpam-5571	270	13	to	to	ADP
ejpam-5571	270	14	c({p′	c({p′	NOUN
ejpam-5571	270	15	,	,	PUNCT
ejpam-5571	270	16	a′	a′	PROPN
ejpam-5571	270	17	,	,	PUNCT
ejpam-5571	270	18	c′	c′	PRON
ejpam-5571	270	19	,	,	PUNCT
ejpam-5571	270	20	b′	b′	NUM
ejpam-5571	270	21	}	}	PUNCT
ejpam-5571	270	22	)	)	PUNCT
ejpam-5571	270	23	.	.	PUNCT
ejpam-5571	271	1	proof	proof	NOUN
ejpam-5571	271	2	.	.	PUNCT
ejpam-5571	272	1	by	by	ADP
ejpam-5571	272	2	lemma	lemma	PROPN
ejpam-5571	272	3	2	2	NUM
ejpam-5571	272	4	,	,	PUNCT
ejpam-5571	272	5	the	the	DET
ejpam-5571	272	6	segment	segment	NOUN
ejpam-5571	272	7	[	[	X
ejpam-5571	272	8	a	a	X
ejpam-5571	272	9	,	,	PUNCT
ejpam-5571	272	10	b	b	NOUN
ejpam-5571	272	11	]	]	PUNCT
ejpam-5571	272	12	intersects	intersect	VERB
ejpam-5571	272	13	the	the	DET
ejpam-5571	272	14	segment	segment	NOUN
ejpam-5571	273	1	[	[	X
ejpam-5571	273	2	p	p	X
ejpam-5571	273	3	,	,	PUNCT
ejpam-5571	273	4	c	c	X
ejpam-5571	273	5	]	]	X
ejpam-5571	273	6	at	at	ADP
ejpam-5571	273	7	a	a	DET
ejpam-5571	273	8	point	point	NOUN
ejpam-5571	273	9	q	q	NOUN
ejpam-5571	273	10	and	and	CCONJ
ejpam-5571	273	11	using	use	VERB
ejpam-5571	273	12	theorem	theorem	NOUN
ejpam-5571	273	13	4	4	NUM
ejpam-5571	273	14	,	,	PUNCT
ejpam-5571	273	15	we	we	PRON
ejpam-5571	273	16	get	get	VERB
ejpam-5571	273	17	that	that	DET
ejpam-5571	273	18	c({p	c({p	NOUN
ejpam-5571	273	19	,	,	PUNCT
ejpam-5571	273	20	a	a	DET
ejpam-5571	273	21	,	,	PUNCT
ejpam-5571	273	22	c	c	NOUN
ejpam-5571	273	23	,	,	PUNCT
ejpam-5571	273	24	b	b	NOUN
ejpam-5571	273	25	}	}	PUNCT
ejpam-5571	273	26	)	)	PUNCT
ejpam-5571	274	1	is	be	AUX
ejpam-5571	274	2	isometric	isometric	ADJ
ejpam-5571	274	3	to	to	ADP
ejpam-5571	274	4	c({p′	c({p′	NOUN
ejpam-5571	274	5	,	,	PUNCT
ejpam-5571	274	6	a′	a′	PROPN
ejpam-5571	274	7	,	,	PUNCT
ejpam-5571	274	8	c′	c′	PRON
ejpam-5571	274	9	,	,	PUNCT
ejpam-5571	274	10	b′	b′	NUM
ejpam-5571	274	11	}	}	PUNCT
ejpam-5571	274	12	)	)	PUNCT
ejpam-5571	274	13	.	.	PUNCT
ejpam-5571	275	1	by	by	ADP
ejpam-5571	275	2	theorem	theorem	NOUN
ejpam-5571	275	3	6	6	NUM
ejpam-5571	275	4	,	,	PUNCT
ejpam-5571	275	5	we	we	PRON
ejpam-5571	275	6	have	have	VERB
ejpam-5571	275	7	the	the	DET
ejpam-5571	275	8	following	follow	VERB
ejpam-5571	275	9	corollary	corollary	NOUN
ejpam-5571	275	10	.	.	PUNCT
ejpam-5571	276	1	corollary	corollary	ADJ
ejpam-5571	276	2	1	1	NUM
ejpam-5571	276	3	.	.	PUNCT
ejpam-5571	277	1	let	let	VERB
ejpam-5571	277	2	x	x	PRON
ejpam-5571	277	3	be	be	AUX
ejpam-5571	277	4	a	a	DET
ejpam-5571	277	5	metric	metric	ADJ
ejpam-5571	277	6	space	space	NOUN
ejpam-5571	277	7	with	with	ADP
ejpam-5571	277	8	curvature	curvature	NOUN
ejpam-5571	277	9	bounded	bound	VERB
ejpam-5571	277	10	below	below	ADV
ejpam-5571	277	11	by	by	ADP
ejpam-5571	277	12	k	k	PROPN
ejpam-5571	277	13	in	in	ADP
ejpam-5571	277	14	the	the	DET
ejpam-5571	277	15	large	large	NOUN
ejpam-5571	277	16	.	.	PUNCT
ejpam-5571	278	1	let	let	VERB
ejpam-5571	278	2	γ	γ	NOUN
ejpam-5571	278	3	be	be	AUX
ejpam-5571	278	4	a	a	DET
ejpam-5571	278	5	spherical	spherical	ADJ
ejpam-5571	278	6	curve	curve	NOUN
ejpam-5571	278	7	at	at	ADP
ejpam-5571	278	8	a	a	DET
ejpam-5571	278	9	distance	distance	NOUN
ejpam-5571	279	1	r	r	NOUN
ejpam-5571	279	2	<	<	X
ejpam-5571	279	3	π	π	PROPN
ejpam-5571	279	4	2	2	NUM
ejpam-5571	279	5	√	√	PROPN
ejpam-5571	279	6	k	k	NOUN
ejpam-5571	279	7	from	from	ADP
ejpam-5571	279	8	a	a	DET
ejpam-5571	279	9	point	point	NOUN
ejpam-5571	279	10	p	p	NOUN
ejpam-5571	279	11	with	with	ADP
ejpam-5571	279	12	endpoints	endpoint	NOUN
ejpam-5571	279	13	a	a	PRON
ejpam-5571	279	14	,	,	PUNCT
ejpam-5571	279	15	b	b	NOUN
ejpam-5571	279	16	in	in	ADP
ejpam-5571	279	17	x	x	X
ejpam-5571	279	18	and	and	CCONJ
ejpam-5571	279	19	γ′	γ′	NOUN
ejpam-5571	279	20	be	be	AUX
ejpam-5571	279	21	a	a	DET
ejpam-5571	279	22	subarc	subarc	NOUN
ejpam-5571	279	23	of	of	ADP
ejpam-5571	279	24	a	a	DET
ejpam-5571	279	25	circle	circle	NOUN
ejpam-5571	279	26	of	of	ADP
ejpam-5571	279	27	radius	radius	NOUN
ejpam-5571	279	28	r	r	NOUN
ejpam-5571	279	29	centered	center	VERB
ejpam-5571	279	30	at	at	ADP
ejpam-5571	279	31	a	a	DET
ejpam-5571	279	32	point	point	NOUN
ejpam-5571	279	33	p′	p′	NOUN
ejpam-5571	279	34	in	in	ADP
ejpam-5571	279	35	rk	rk	NOUN
ejpam-5571	279	36	with	with	ADP
ejpam-5571	279	37	endpoints	endpoint	NOUN
ejpam-5571	279	38	a′	a′	PROPN
ejpam-5571	279	39	,	,	PUNCT
ejpam-5571	279	40	b′.	b′.	PROPN
ejpam-5571	279	41	suppose	suppose	VERB
ejpam-5571	279	42	that	that	SCONJ
ejpam-5571	279	43	a	a	DET
ejpam-5571	279	44	=	=	X
ejpam-5571	279	45	c1	c1	PROPN
ejpam-5571	279	46	,	,	PUNCT
ejpam-5571	279	47	c2	c2	PROPN
ejpam-5571	279	48	,	,	PUNCT
ejpam-5571	279	49	...	...	PUNCT
ejpam-5571	279	50	,	,	PUNCT
ejpam-5571	279	51	cn	cn	PROPN
ejpam-5571	280	1	=	=	SYM
ejpam-5571	280	2	b	b	PROPN
ejpam-5571	280	3	∈	∈	PROPN
ejpam-5571	280	4	γ	γ	NOUN
ejpam-5571	280	5	are	be	AUX
ejpam-5571	280	6	consecutive	consecutive	ADJ
ejpam-5571	280	7	points	point	NOUN
ejpam-5571	280	8	on	on	ADP
ejpam-5571	280	9	γ	γ	NOUN
ejpam-5571	280	10	and	and	CCONJ
ejpam-5571	280	11	a′	a′	PROPN
ejpam-5571	280	12	=	=	PUNCT
ejpam-5571	280	13	c′1	c′1	PROPN
ejpam-5571	280	14	,	,	PUNCT
ejpam-5571	280	15	c	c	NOUN
ejpam-5571	280	16	′	′	NOUN
ejpam-5571	280	17	2	2	NUM
ejpam-5571	280	18	,	,	PUNCT
ejpam-5571	280	19	...	...	PUNCT
ejpam-5571	280	20	,	,	PUNCT
ejpam-5571	280	21	c	c	NOUN
ejpam-5571	280	22	′	′	NUM
ejpam-5571	281	1	n	n	CCONJ
ejpam-5571	281	2	=	=	SYM
ejpam-5571	281	3	b′	b′	NUM
ejpam-5571	281	4	∈	∈	PROPN
ejpam-5571	281	5	γ	γ	NOUN
ejpam-5571	281	6	are	be	AUX
ejpam-5571	281	7	consecutive	consecutive	ADJ
ejpam-5571	281	8	points	point	NOUN
ejpam-5571	281	9	on	on	ADP
ejpam-5571	281	10	γ′.	γ′.	PROPN
ejpam-5571	281	11	assume	assume	VERB
ejpam-5571	281	12	that	that	SCONJ
ejpam-5571	281	13	the	the	DET
ejpam-5571	281	14	following	follow	VERB
ejpam-5571	281	15	statements	statement	NOUN
ejpam-5571	281	16	hold	hold	VERB
ejpam-5571	281	17	:	:	PUNCT
ejpam-5571	281	18	(	(	PUNCT
ejpam-5571	281	19	i	i	NOUN
ejpam-5571	281	20	)	)	PUNCT
ejpam-5571	281	21	ℓ(γ	ℓ(γ	PROPN
ejpam-5571	281	22	)	)	PUNCT
ejpam-5571	282	1	=	=	SYM
ejpam-5571	282	2	ℓ(γ′	ℓ(γ′	PROPN
ejpam-5571	282	3	)	)	PUNCT
ejpam-5571	282	4	≤	≤	NOUN
ejpam-5571	282	5	π√	π√	PROPN
ejpam-5571	282	6	k	k	NOUN
ejpam-5571	282	7	;	;	PUNCT
ejpam-5571	282	8	(	(	PUNCT
ejpam-5571	282	9	ii	ii	NOUN
ejpam-5571	282	10	)	)	PUNCT
ejpam-5571	282	11	d(a	d(a	PROPN
ejpam-5571	282	12	,	,	PUNCT
ejpam-5571	282	13	b	b	NOUN
ejpam-5571	282	14	)	)	PUNCT
ejpam-5571	282	15	=	=	VERB
ejpam-5571	282	16	d(a′	d(a′	NOUN
ejpam-5571	282	17	,	,	PUNCT
ejpam-5571	282	18	b′	b′	NUM
ejpam-5571	282	19	)	)	PUNCT
ejpam-5571	282	20	;	;	PUNCT
ejpam-5571	282	21	(	(	PUNCT
ejpam-5571	282	22	iii	iii	X
ejpam-5571	282	23	)	)	PUNCT
ejpam-5571	282	24	c({p	c({p	NOUN
ejpam-5571	282	25	,	,	PUNCT
ejpam-5571	282	26	c1	c1	PROPN
ejpam-5571	282	27	,	,	PUNCT
ejpam-5571	282	28	c2	c2	PROPN
ejpam-5571	282	29	,	,	PUNCT
ejpam-5571	282	30	...	...	PUNCT
ejpam-5571	282	31	,	,	PUNCT
ejpam-5571	282	32	ct	ct	PROPN
ejpam-5571	282	33	}	}	PUNCT
ejpam-5571	282	34	)	)	PUNCT
ejpam-5571	282	35	and	and	CCONJ
ejpam-5571	282	36	c({p	c({p	PROPN
ejpam-5571	282	37	,	,	PUNCT
ejpam-5571	282	38	ct	ct	PROPN
ejpam-5571	282	39	,	,	PUNCT
ejpam-5571	282	40	ct+1	ct+1	PROPN
ejpam-5571	282	41	,	,	PUNCT
ejpam-5571	282	42	...	...	PUNCT
ejpam-5571	282	43	,	,	PUNCT
ejpam-5571	282	44	cn	cn	PROPN
ejpam-5571	282	45	}	}	PUNCT
ejpam-5571	282	46	)	)	PUNCT
ejpam-5571	282	47	are	be	AUX
ejpam-5571	282	48	isometric	isometric	ADJ
ejpam-5571	282	49	to	to	ADP
ejpam-5571	282	50	c({p′	c({p′	PROPN
ejpam-5571	282	51	,	,	PUNCT
ejpam-5571	282	52	c′1	c′1	NOUN
ejpam-5571	282	53	,	,	PUNCT
ejpam-5571	282	54	c′2	c′2	NOUN
ejpam-5571	282	55	,	,	PUNCT
ejpam-5571	282	56	...	...	PUNCT
ejpam-5571	282	57	,	,	PUNCT
ejpam-5571	282	58	c′t	c′t	X
ejpam-5571	282	59	}	}	PUNCT
ejpam-5571	282	60	)	)	PUNCT
ejpam-5571	282	61	and	and	CCONJ
ejpam-5571	282	62	c({p′	c({p′	PROPN
ejpam-5571	282	63	,	,	PUNCT
ejpam-5571	282	64	c′t	c′t	PROPN
ejpam-5571	282	65	,	,	PUNCT
ejpam-5571	282	66	c′t+1	c′t+1	PROPN
ejpam-5571	282	67	,	,	PUNCT
ejpam-5571	282	68	,	,	PUNCT
ejpam-5571	282	69	...	...	PUNCT
ejpam-5571	282	70	,	,	PUNCT
ejpam-5571	282	71	cn	cn	PROPN
ejpam-5571	282	72	}	}	PUNCT
ejpam-5571	282	73	)	)	PUNCT
ejpam-5571	282	74	,	,	PUNCT
ejpam-5571	282	75	respectively	respectively	ADV
ejpam-5571	282	76	,	,	PUNCT
ejpam-5571	282	77	for	for	ADP
ejpam-5571	282	78	some	some	DET
ejpam-5571	282	79	t	t	NOUN
ejpam-5571	282	80	∈	∈	PROPN
ejpam-5571	282	81	{	{	PUNCT
ejpam-5571	282	82	2	2	NUM
ejpam-5571	282	83	,	,	PUNCT
ejpam-5571	282	84	3	3	NUM
ejpam-5571	282	85	,	,	PUNCT
ejpam-5571	282	86	...	...	PUNCT
ejpam-5571	282	87	,	,	PUNCT
ejpam-5571	282	88	n−	n−	NOUN
ejpam-5571	282	89	1	1	NUM
ejpam-5571	282	90	}	}	PUNCT
ejpam-5571	282	91	;	;	PUNCT
ejpam-5571	282	92	(	(	PUNCT
ejpam-5571	282	93	iv	iv	X
ejpam-5571	282	94	)	)	PUNCT
ejpam-5571	282	95	∠ct(a	∠ct(a	PROPN
ejpam-5571	282	96	,	,	PUNCT
ejpam-5571	282	97	b	b	NOUN
ejpam-5571	282	98	)	)	PUNCT
ejpam-5571	282	99	=	=	PUNCT
ejpam-5571	282	100	∠c′t	∠c′t	NOUN
ejpam-5571	282	101	(	(	PUNCT
ejpam-5571	282	102	a′	a′	PROPN
ejpam-5571	282	103	,	,	PUNCT
ejpam-5571	282	104	b′	b′	NUM
ejpam-5571	282	105	)	)	PUNCT
ejpam-5571	282	106	and	and	CCONJ
ejpam-5571	282	107	∠p(a	∠p(a	ADJ
ejpam-5571	282	108	,	,	PUNCT
ejpam-5571	282	109	b	b	NOUN
ejpam-5571	282	110	)	)	PUNCT
ejpam-5571	282	111	=	=	SYM
ejpam-5571	282	112	∠p′(a	∠p′(a	PROPN
ejpam-5571	282	113	′	′	PROPN
ejpam-5571	282	114	,	,	PUNCT
ejpam-5571	282	115	b′	b′	NUM
ejpam-5571	282	116	)	)	PUNCT
ejpam-5571	282	117	;	;	PUNCT
ejpam-5571	282	118	(	(	PUNCT
ejpam-5571	282	119	v	v	NOUN
ejpam-5571	282	120	)	)	PUNCT
ejpam-5571	282	121	∠a(x	∠a(x	NOUN
ejpam-5571	282	122	,	,	PUNCT
ejpam-5571	282	123	ct	ct	PROPN
ejpam-5571	282	124	)	)	PUNCT
ejpam-5571	282	125	=	=	PUNCT
ejpam-5571	282	126	∠a(b	∠a(b	VERB
ejpam-5571	282	127	,	,	PUNCT
ejpam-5571	282	128	ct	ct	PROPN
ejpam-5571	282	129	)	)	PUNCT
ejpam-5571	282	130	,	,	PUNCT
ejpam-5571	282	131	∠b(x	∠b(x	NOUN
ejpam-5571	282	132	,	,	PUNCT
ejpam-5571	282	133	ct	ct	PROPN
ejpam-5571	282	134	)	)	PUNCT
ejpam-5571	282	135	=	=	SYM
ejpam-5571	282	136	∠b(a	∠b(a	NOUN
ejpam-5571	282	137	,	,	PUNCT
ejpam-5571	282	138	ct	ct	NUM
ejpam-5571	282	139	)	)	PUNCT
ejpam-5571	282	140	and	and	CCONJ
ejpam-5571	282	141	∠ct(a	∠ct(a	PROPN
ejpam-5571	282	142	,	,	PUNCT
ejpam-5571	282	143	b	b	NOUN
ejpam-5571	282	144	)	)	PUNCT
ejpam-5571	282	145	=	=	SYM
ejpam-5571	282	146	∠ct(a	∠ct(a	PROPN
ejpam-5571	282	147	,	,	PUNCT
ejpam-5571	282	148	x	x	PRON
ejpam-5571	282	149	)	)	PUNCT
ejpam-5571	283	1	+	+	CCONJ
ejpam-5571	283	2	∠ct(x	∠ct(x	PROPN
ejpam-5571	283	3	,	,	PUNCT
ejpam-5571	283	4	b	b	NOUN
ejpam-5571	283	5	)	)	PUNCT
ejpam-5571	283	6	for	for	ADP
ejpam-5571	283	7	any	any	DET
ejpam-5571	283	8	x	x	SYM
ejpam-5571	283	9	∈	∈	PROPN
ejpam-5571	283	10	[	[	X
ejpam-5571	283	11	a	a	X
ejpam-5571	283	12	,	,	PUNCT
ejpam-5571	283	13	b	b	NOUN
ejpam-5571	283	14	]	]	X
ejpam-5571	283	15	;	;	PUNCT
ejpam-5571	283	16	(	(	PUNCT
ejpam-5571	283	17	vi	vi	NOUN
ejpam-5571	283	18	)	)	PUNCT
ejpam-5571	283	19	∠a(x	∠a(x	NOUN
ejpam-5571	283	20	,	,	PUNCT
ejpam-5571	283	21	p	p	X
ejpam-5571	283	22	)	)	PUNCT
ejpam-5571	283	23	=	=	PUNCT
ejpam-5571	284	1	∠a(b	∠a(b	VERB
ejpam-5571	284	2	,	,	PUNCT
ejpam-5571	284	3	p	p	NOUN
ejpam-5571	284	4	)	)	PUNCT
ejpam-5571	284	5	,	,	PUNCT
ejpam-5571	284	6	∠b(x	∠b(x	NOUN
ejpam-5571	284	7	,	,	PUNCT
ejpam-5571	284	8	p	p	NOUN
ejpam-5571	284	9	)	)	PUNCT
ejpam-5571	284	10	=	=	PUNCT
ejpam-5571	284	11	∠b(a	∠b(a	NOUN
ejpam-5571	284	12	,	,	PUNCT
ejpam-5571	284	13	p	p	NOUN
ejpam-5571	284	14	)	)	PUNCT
ejpam-5571	284	15	and	and	CCONJ
ejpam-5571	284	16	∠p(a	∠p(a	ADJ
ejpam-5571	284	17	,	,	PUNCT
ejpam-5571	284	18	b	b	NOUN
ejpam-5571	284	19	)	)	PUNCT
ejpam-5571	284	20	=	=	PUNCT
ejpam-5571	285	1	∠p(a	∠p(a	ADJ
ejpam-5571	285	2	,	,	PUNCT
ejpam-5571	285	3	x	x	PRON
ejpam-5571	285	4	)	)	PUNCT
ejpam-5571	285	5	+	+	CCONJ
ejpam-5571	285	6	∠p(x	∠p(x	NOUN
ejpam-5571	285	7	,	,	PUNCT
ejpam-5571	285	8	b	b	NOUN
ejpam-5571	285	9	)	)	PUNCT
ejpam-5571	285	10	for	for	ADP
ejpam-5571	285	11	any	any	DET
ejpam-5571	285	12	x	x	SYM
ejpam-5571	285	13	∈	∈	PROPN
ejpam-5571	285	14	[	[	X
ejpam-5571	285	15	a	a	X
ejpam-5571	285	16	,	,	PUNCT
ejpam-5571	285	17	b	b	NOUN
ejpam-5571	285	18	]	]	X
ejpam-5571	285	19	.	.	PUNCT
ejpam-5571	286	1	then	then	ADV
ejpam-5571	286	2	c({p	c({p	PROPN
ejpam-5571	286	3	,	,	PUNCT
ejpam-5571	286	4	c1	c1	PROPN
ejpam-5571	286	5	,	,	PUNCT
ejpam-5571	286	6	c2	c2	PROPN
ejpam-5571	286	7	,	,	PUNCT
ejpam-5571	286	8	...	...	PUNCT
ejpam-5571	286	9	,	,	PUNCT
ejpam-5571	286	10	cn	cn	PROPN
ejpam-5571	286	11	}	}	PUNCT
ejpam-5571	286	12	)	)	PUNCT
ejpam-5571	286	13	is	be	AUX
ejpam-5571	286	14	isometric	isometric	ADJ
ejpam-5571	286	15	to	to	ADP
ejpam-5571	286	16	c({p′	c({p′	PROPN
ejpam-5571	286	17	,	,	PUNCT
ejpam-5571	286	18	c′1	c′1	NOUN
ejpam-5571	286	19	,	,	PUNCT
ejpam-5571	286	20	c′2	c′2	NOUN
ejpam-5571	286	21	,	,	PUNCT
ejpam-5571	286	22	...	...	PUNCT
ejpam-5571	286	23	,	,	PUNCT
ejpam-5571	286	24	c′n	c′n	NOUN
ejpam-5571	286	25	}	}	PUNCT
ejpam-5571	286	26	)	)	PUNCT
ejpam-5571	286	27	.	.	PUNCT
ejpam-5571	287	1	proof	proof	NOUN
ejpam-5571	287	2	.	.	PUNCT
ejpam-5571	288	1	first	first	ADV
ejpam-5571	288	2	,	,	PUNCT
ejpam-5571	288	3	we	we	PRON
ejpam-5571	288	4	will	will	AUX
ejpam-5571	288	5	demonstrate	demonstrate	VERB
ejpam-5571	288	6	that	that	SCONJ
ejpam-5571	288	7	the	the	DET
ejpam-5571	288	8	segments	segment	NOUN
ejpam-5571	288	9	[	[	X
ejpam-5571	288	10	a	a	X
ejpam-5571	288	11	,	,	PUNCT
ejpam-5571	288	12	b	b	NOUN
ejpam-5571	288	13	]	]	PUNCT
ejpam-5571	288	14	and	and	CCONJ
ejpam-5571	288	15	[	[	X
ejpam-5571	288	16	p	p	X
ejpam-5571	288	17	,	,	PUNCT
ejpam-5571	288	18	c	c	NOUN
ejpam-5571	288	19	]	]	X
ejpam-5571	288	20	cross	cross	NOUN
ejpam-5571	288	21	at	at	ADP
ejpam-5571	288	22	a	a	DET
ejpam-5571	288	23	specific	specific	ADJ
ejpam-5571	288	24	location	location	NOUN
ejpam-5571	288	25	.	.	PUNCT
ejpam-5571	289	1	assume	assume	VERB
ejpam-5571	289	2	that	that	SCONJ
ejpam-5571	289	3	the	the	DET
ejpam-5571	289	4	intersection	intersection	NOUN
ejpam-5571	289	5	of	of	ADP
ejpam-5571	289	6	the	the	DET
ejpam-5571	289	7	segments	segment	NOUN
ejpam-5571	289	8	[	[	X
ejpam-5571	289	9	a′	a′	PROPN
ejpam-5571	289	10	,	,	PUNCT
ejpam-5571	289	11	b′	b′	NOUN
ejpam-5571	289	12	]	]	PUNCT
ejpam-5571	289	13	and	and	CCONJ
ejpam-5571	289	14	[	[	X
ejpam-5571	289	15	p′	p′	NOUN
ejpam-5571	289	16	,	,	PUNCT
ejpam-5571	289	17	c′	c′	NUM
ejpam-5571	289	18	]	]	PUNCT
ejpam-5571	289	19	is	be	AUX
ejpam-5571	289	20	at	at	ADP
ejpam-5571	289	21	q′.	q′.	DET
ejpam-5571	289	22	a	a	DET
ejpam-5571	289	23	point	point	NOUN
ejpam-5571	289	24	on	on	ADP
ejpam-5571	289	25	the	the	DET
ejpam-5571	289	26	segment	segment	NOUN
ejpam-5571	289	27	[	[	X
ejpam-5571	289	28	p	p	X
ejpam-5571	289	29	,	,	PUNCT
ejpam-5571	289	30	c	c	X
ejpam-5571	289	31	]	]	X
ejpam-5571	289	32	with	with	ADP
ejpam-5571	289	33	the	the	DET
ejpam-5571	289	34	property	property	NOUN
ejpam-5571	289	35	d(p	d(p	PROPN
ejpam-5571	289	36	,	,	PUNCT
ejpam-5571	289	37	q	q	NOUN
ejpam-5571	289	38	)	)	PUNCT
ejpam-5571	289	39	=	=	SYM
ejpam-5571	289	40	d(p′	d(p′	PROPN
ejpam-5571	289	41	,	,	PUNCT
ejpam-5571	289	42	q′	q′	NOUN
ejpam-5571	289	43	)	)	PUNCT
ejpam-5571	289	44	is	be	AUX
ejpam-5571	289	45	called	call	VERB
ejpam-5571	289	46	q.	q.	PROPN
ejpam-5571	289	47	because	because	SCONJ
ejpam-5571	289	48	q′	q′	NOUN
ejpam-5571	289	49	is	be	AUX
ejpam-5571	289	50	a	a	DET
ejpam-5571	289	51	point	point	NOUN
ejpam-5571	289	52	that	that	PRON
ejpam-5571	289	53	corresponds	correspond	VERB
ejpam-5571	289	54	to	to	ADP
ejpam-5571	289	55	q	q	NOUN
ejpam-5571	289	56	,	,	PUNCT
ejpam-5571	289	57	d(a	d(a	PROPN
ejpam-5571	289	58	,	,	PUNCT
ejpam-5571	289	59	q	q	NOUN
ejpam-5571	289	60	)	)	PUNCT
ejpam-5571	289	61	=	=	SYM
ejpam-5571	289	62	d(a′	d(a′	NOUN
ejpam-5571	289	63	,	,	PUNCT
ejpam-5571	289	64	q′	q′	NOUN
ejpam-5571	289	65	)	)	PUNCT
ejpam-5571	289	66	and	and	CCONJ
ejpam-5571	289	67	d(b	d(b	PROPN
ejpam-5571	289	68	,	,	PUNCT
ejpam-5571	289	69	q	q	X
ejpam-5571	289	70	)	)	PUNCT
ejpam-5571	289	71	=	=	SYM
ejpam-5571	289	72	d(b′	d(b′	PROPN
ejpam-5571	289	73	,	,	PUNCT
ejpam-5571	289	74	q′	q′	NOUN
ejpam-5571	289	75	)	)	PUNCT
ejpam-5571	289	76	.	.	PUNCT
ejpam-5571	290	1	consequently	consequently	ADV
ejpam-5571	290	2	,	,	PUNCT
ejpam-5571	290	3	d(a	d(a	PROPN
ejpam-5571	290	4	,	,	PUNCT
ejpam-5571	290	5	b	b	NOUN
ejpam-5571	290	6	)	)	PUNCT
ejpam-5571	290	7	≤	≤	NOUN
ejpam-5571	291	1	d(a	d(a	PROPN
ejpam-5571	291	2	,	,	PUNCT
ejpam-5571	291	3	q	q	NOUN
ejpam-5571	291	4	)	)	PUNCT
ejpam-5571	291	5	+	+	CCONJ
ejpam-5571	291	6	d(q	d(q	PROPN
ejpam-5571	291	7	,	,	PUNCT
ejpam-5571	291	8	b	b	NOUN
ejpam-5571	291	9	)	)	PUNCT
ejpam-5571	291	10	=	=	VERB
ejpam-5571	291	11	d(a′	d(a′	NOUN
ejpam-5571	291	12	,	,	PUNCT
ejpam-5571	291	13	q′	q′	NOUN
ejpam-5571	291	14	)	)	PUNCT
ejpam-5571	292	1	+	+	CCONJ
ejpam-5571	292	2	d(q′	d(q′	PROPN
ejpam-5571	292	3	,	,	PUNCT
ejpam-5571	292	4	b′	b′	NUM
ejpam-5571	292	5	)	)	PUNCT
ejpam-5571	292	6	=	=	SYM
ejpam-5571	292	7	d(a′	d(a′	NOUN
ejpam-5571	292	8	,	,	PUNCT
ejpam-5571	292	9	b′	b′	NUM
ejpam-5571	292	10	)	)	PUNCT
ejpam-5571	293	1	=	=	SYM
ejpam-5571	294	1	d(a	d(a	PROPN
ejpam-5571	294	2	,	,	PUNCT
ejpam-5571	294	3	b	b	NOUN
ejpam-5571	294	4	)	)	PUNCT
ejpam-5571	294	5	,	,	PUNCT
ejpam-5571	294	6	that	that	PRON
ejpam-5571	294	7	is	be	AUX
ejpam-5571	294	8	the	the	DET
ejpam-5571	294	9	point	point	NOUN
ejpam-5571	294	10	q	q	NOUN
ejpam-5571	294	11	is	be	AUX
ejpam-5571	294	12	the	the	DET
ejpam-5571	294	13	intersection	intersection	NOUN
ejpam-5571	294	14	of	of	ADP
ejpam-5571	294	15	[	[	X
ejpam-5571	294	16	a	a	X
ejpam-5571	294	17	,	,	PUNCT
ejpam-5571	294	18	b	b	NOUN
ejpam-5571	294	19	]	]	PUNCT
ejpam-5571	294	20	and	and	CCONJ
ejpam-5571	294	21	[	[	X
ejpam-5571	294	22	p	p	X
ejpam-5571	294	23	,	,	PUNCT
ejpam-5571	294	24	c	c	NOUN
ejpam-5571	294	25	]	]	PUNCT
ejpam-5571	294	26	.	.	PUNCT
ejpam-5571	295	1	we	we	PRON
ejpam-5571	295	2	can	can	AUX
ejpam-5571	295	3	see	see	VERB
ejpam-5571	295	4	from	from	ADP
ejpam-5571	295	5	(	(	PUNCT
ejpam-5571	295	6	iv	iv	NOUN
ejpam-5571	295	7	)	)	PUNCT
ejpam-5571	295	8	,	,	PUNCT
ejpam-5571	295	9	(	(	PUNCT
ejpam-5571	295	10	v	v	NOUN
ejpam-5571	295	11	)	)	PUNCT
ejpam-5571	295	12	,	,	PUNCT
ejpam-5571	295	13	and	and	CCONJ
ejpam-5571	295	14	(	(	PUNCT
ejpam-5571	295	15	vi	vi	NOUN
ejpam-5571	295	16	)	)	PUNCT
ejpam-5571	295	17	that	that	PRON
ejpam-5571	295	18	△	△	X
ejpam-5571	295	19	(	(	PUNCT
ejpam-5571	295	20	a	a	PRON
ejpam-5571	295	21	,	,	PUNCT
ejpam-5571	295	22	b	b	NOUN
ejpam-5571	295	23	,	,	PUNCT
ejpam-5571	295	24	c	c	NOUN
ejpam-5571	295	25	)	)	PUNCT
ejpam-5571	295	26	and	and	CCONJ
ejpam-5571	295	27	△	△	X
ejpam-5571	295	28	(	(	PUNCT
ejpam-5571	295	29	a	a	DET
ejpam-5571	295	30	,	,	PUNCT
ejpam-5571	295	31	b	b	NOUN
ejpam-5571	295	32	,	,	PUNCT
ejpam-5571	295	33	p	p	NOUN
ejpam-5571	295	34	)	)	PUNCT
ejpam-5571	295	35	are	be	AUX
ejpam-5571	295	36	isometric	isometric	ADJ
ejpam-5571	295	37	to	to	ADP
ejpam-5571	295	38	△	△	PROPN
ejpam-5571	295	39	(	(	PUNCT
ejpam-5571	295	40	a′	a′	PROPN
ejpam-5571	295	41	,	,	PUNCT
ejpam-5571	295	42	b′	b′	NUM
ejpam-5571	295	43	,	,	PUNCT
ejpam-5571	295	44	c′	c′	NUM
ejpam-5571	295	45	)	)	PUNCT
ejpam-5571	295	46	and	and	CCONJ
ejpam-5571	295	47	△	△	X
ejpam-5571	295	48	(	(	PUNCT
ejpam-5571	295	49	a′	a′	PROPN
ejpam-5571	295	50	,	,	PUNCT
ejpam-5571	295	51	b′	b′	NUM
ejpam-5571	295	52	,	,	PUNCT
ejpam-5571	295	53	p′	p′	NOUN
ejpam-5571	295	54	)	)	PUNCT
ejpam-5571	295	55	,	,	PUNCT
ejpam-5571	295	56	respectively	respectively	ADV
ejpam-5571	295	57	.	.	PUNCT
ejpam-5571	296	1	we	we	PRON
ejpam-5571	296	2	proceed	proceed	VERB
ejpam-5571	296	3	in	in	ADP
ejpam-5571	296	4	the	the	DET
ejpam-5571	296	5	same	same	ADJ
ejpam-5571	296	6	way	way	NOUN
ejpam-5571	296	7	as	as	SCONJ
ejpam-5571	296	8	theorem	theorem	ADJ
ejpam-5571	296	9	4	4	NUM
ejpam-5571	296	10	,	,	PUNCT
ejpam-5571	296	11	having	having	AUX
ejpam-5571	296	12	established	establish	VERB
ejpam-5571	296	13	that	that	DET
ejpam-5571	296	14	c({p	c({p	NOUN
ejpam-5571	296	15	,	,	PUNCT
ejpam-5571	296	16	a	a	DET
ejpam-5571	296	17	,	,	PUNCT
ejpam-5571	296	18	c	c	NOUN
ejpam-5571	296	19	,	,	PUNCT
ejpam-5571	296	20	b	b	NOUN
ejpam-5571	296	21	}	}	PUNCT
ejpam-5571	296	22	)	)	PUNCT
ejpam-5571	296	23	is	be	AUX
ejpam-5571	296	24	isometric	isometric	ADJ
ejpam-5571	296	25	to	to	ADP
ejpam-5571	296	26	c({p′	c({p′	NOUN
ejpam-5571	296	27	,	,	PUNCT
ejpam-5571	296	28	a′	a′	PROPN
ejpam-5571	296	29	,	,	PUNCT
ejpam-5571	296	30	c′	c′	PRON
ejpam-5571	296	31	,	,	PUNCT
ejpam-5571	296	32	b′	b′	NUM
ejpam-5571	296	33	}	}	PUNCT
ejpam-5571	296	34	)	)	PUNCT
ejpam-5571	296	35	.	.	PUNCT
ejpam-5571	297	1	c.	c.	PROPN
ejpam-5571	297	2	phokaew	phokaew	PROPN
ejpam-5571	297	3	,	,	PUNCT
ejpam-5571	297	4	a.	a.	PROPN
ejpam-5571	297	5	sama	sama	PROPN
ejpam-5571	297	6	-	-	PUNCT
ejpam-5571	297	7	ae	ae	PROPN
ejpam-5571	297	8	,	,	PUNCT
ejpam-5571	297	9	/	/	SYM
ejpam-5571	297	10	eur	eur	NOUN
ejpam-5571	297	11	.	.	PUNCT
ejpam-5571	298	1	j.	j.	PROPN
ejpam-5571	298	2	pure	pure	PROPN
ejpam-5571	298	3	appl	appl	PROPN
ejpam-5571	298	4	.	.	PROPN
ejpam-5571	298	5	math	math	PROPN
ejpam-5571	298	6	,	,	PUNCT
ejpam-5571	298	7	17	17	NUM
ejpam-5571	298	8	(	(	PUNCT
ejpam-5571	298	9	4	4	NUM
ejpam-5571	298	10	)	)	PUNCT
ejpam-5571	298	11	(	(	PUNCT
ejpam-5571	298	12	2024	2024	NUM
ejpam-5571	298	13	)	)	PUNCT
ejpam-5571	298	14	,	,	PUNCT
ejpam-5571	298	15	3932	3932	NUM
ejpam-5571	298	16	-	-	SYM
ejpam-5571	298	17	3944	3944	NUM
ejpam-5571	298	18	3941	3941	NUM
ejpam-5571	298	19	theorem	theorem	VERB
ejpam-5571	298	20	7	7	NUM
ejpam-5571	298	21	.	.	PUNCT
ejpam-5571	299	1	let	let	VERB
ejpam-5571	299	2	x	x	PRON
ejpam-5571	299	3	be	be	AUX
ejpam-5571	299	4	a	a	DET
ejpam-5571	299	5	metric	metric	ADJ
ejpam-5571	299	6	space	space	NOUN
ejpam-5571	299	7	with	with	ADP
ejpam-5571	299	8	curvature	curvature	NOUN
ejpam-5571	299	9	bounded	bound	VERB
ejpam-5571	299	10	below	below	ADV
ejpam-5571	299	11	by	by	ADP
ejpam-5571	299	12	k	k	PROPN
ejpam-5571	299	13	in	in	ADP
ejpam-5571	299	14	the	the	DET
ejpam-5571	299	15	large	large	ADJ
ejpam-5571	299	16	,	,	PUNCT
ejpam-5571	299	17	and	and	CCONJ
ejpam-5571	299	18	γ	γ	X
ejpam-5571	299	19	be	be	AUX
ejpam-5571	299	20	a	a	DET
ejpam-5571	299	21	spherical	spherical	ADJ
ejpam-5571	299	22	curve	curve	NOUN
ejpam-5571	299	23	at	at	ADP
ejpam-5571	299	24	a	a	DET
ejpam-5571	299	25	distance	distance	NOUN
ejpam-5571	300	1	r	r	NOUN
ejpam-5571	300	2	<	<	X
ejpam-5571	300	3	π	π	PROPN
ejpam-5571	300	4	2	2	NUM
ejpam-5571	300	5	√	√	PROPN
ejpam-5571	300	6	k	k	NOUN
ejpam-5571	300	7	from	from	ADP
ejpam-5571	300	8	a	a	DET
ejpam-5571	300	9	point	point	NOUN
ejpam-5571	300	10	p	p	NOUN
ejpam-5571	300	11	with	with	ADP
ejpam-5571	300	12	endpoints	endpoint	NOUN
ejpam-5571	300	13	a	a	PRON
ejpam-5571	300	14	,	,	PUNCT
ejpam-5571	300	15	b	b	X
ejpam-5571	300	16	in	in	ADP
ejpam-5571	300	17	x.	x.	NOUN
ejpam-5571	300	18	let	let	VERB
ejpam-5571	300	19	γ′	γ′	PRON
ejpam-5571	300	20	be	be	AUX
ejpam-5571	300	21	a	a	DET
ejpam-5571	300	22	subarc	subarc	NOUN
ejpam-5571	300	23	of	of	ADP
ejpam-5571	300	24	a	a	DET
ejpam-5571	300	25	circle	circle	NOUN
ejpam-5571	300	26	of	of	ADP
ejpam-5571	300	27	radius	radius	NOUN
ejpam-5571	300	28	r	r	NOUN
ejpam-5571	300	29	centered	center	VERB
ejpam-5571	300	30	at	at	ADP
ejpam-5571	300	31	a	a	DET
ejpam-5571	300	32	point	point	NOUN
ejpam-5571	300	33	p′	p′	NOUN
ejpam-5571	300	34	in	in	ADP
ejpam-5571	300	35	rk	rk	NOUN
ejpam-5571	300	36	with	with	ADP
ejpam-5571	300	37	endpoints	endpoint	NOUN
ejpam-5571	300	38	a′	a′	PROPN
ejpam-5571	300	39	,	,	PUNCT
ejpam-5571	300	40	b′.	b′.	PROPN
ejpam-5571	300	41	suppose	suppose	VERB
ejpam-5571	300	42	that	that	SCONJ
ejpam-5571	300	43	the	the	DET
ejpam-5571	300	44	following	follow	VERB
ejpam-5571	300	45	statements	statement	NOUN
ejpam-5571	300	46	hold	hold	VERB
ejpam-5571	300	47	:	:	PUNCT
ejpam-5571	300	48	(	(	PUNCT
ejpam-5571	300	49	i	i	NOUN
ejpam-5571	300	50	)	)	PUNCT
ejpam-5571	300	51	ℓ(γ	ℓ(γ	PROPN
ejpam-5571	300	52	)	)	PUNCT
ejpam-5571	300	53	=	=	SYM
ejpam-5571	300	54	ℓ(γ′	ℓ(γ′	PROPN
ejpam-5571	300	55	)	)	PUNCT
ejpam-5571	300	56	≤	≤	NOUN
ejpam-5571	300	57	π√	π√	PROPN
ejpam-5571	300	58	k	k	NOUN
ejpam-5571	300	59	;	;	PUNCT
ejpam-5571	300	60	(	(	PUNCT
ejpam-5571	300	61	ii	ii	NOUN
ejpam-5571	300	62	)	)	PUNCT
ejpam-5571	300	63	d(a	d(a	PROPN
ejpam-5571	300	64	,	,	PUNCT
ejpam-5571	300	65	b	b	NOUN
ejpam-5571	300	66	)	)	PUNCT
ejpam-5571	300	67	=	=	VERB
ejpam-5571	300	68	d(a′	d(a′	NOUN
ejpam-5571	300	69	,	,	PUNCT
ejpam-5571	300	70	b′	b′	NUM
ejpam-5571	300	71	)	)	PUNCT
ejpam-5571	300	72	;	;	PUNCT
ejpam-5571	300	73	(	(	PUNCT
ejpam-5571	300	74	iii	iii	X
ejpam-5571	300	75	)	)	PUNCT
ejpam-5571	300	76	ℓ(γxy	ℓ(γxy	PROPN
ejpam-5571	300	77	)	)	PUNCT
ejpam-5571	300	78	=	=	SYM
ejpam-5571	300	79	ℓ(γ′x′y′	ℓ(γ′x′y′	PROPN
ejpam-5571	300	80	)	)	PUNCT
ejpam-5571	301	1	if	if	SCONJ
ejpam-5571	301	2	and	and	CCONJ
ejpam-5571	301	3	only	only	ADV
ejpam-5571	301	4	if	if	SCONJ
ejpam-5571	301	5	d(x	d(x	PROPN
ejpam-5571	301	6	,	,	PUNCT
ejpam-5571	301	7	y	y	NOUN
ejpam-5571	301	8	)	)	PUNCT
ejpam-5571	301	9	=	=	SYM
ejpam-5571	301	10	d(x′	d(x′	PROPN
ejpam-5571	301	11	,	,	PUNCT
ejpam-5571	301	12	y′	y′	NUM
ejpam-5571	301	13	)	)	PUNCT
ejpam-5571	301	14	for	for	ADP
ejpam-5571	301	15	all	all	DET
ejpam-5571	301	16	x	x	NOUN
ejpam-5571	301	17	,	,	PUNCT
ejpam-5571	301	18	y	y	PROPN
ejpam-5571	301	19	∈	∈	PROPN
ejpam-5571	301	20	γ	γ	NOUN
ejpam-5571	301	21	and	and	CCONJ
ejpam-5571	301	22	x′	x′	NUM
ejpam-5571	301	23	,	,	PUNCT
ejpam-5571	301	24	y′	y′	NOUN
ejpam-5571	301	25	∈	∈	PROPN
ejpam-5571	301	26	γ′	γ′	NOUN
ejpam-5571	301	27	;	;	PUNCT
ejpam-5571	301	28	(	(	PUNCT
ejpam-5571	301	29	iv	iv	X
ejpam-5571	301	30	)	)	PUNCT
ejpam-5571	301	31	∠p(a	∠p(a	ADJ
ejpam-5571	301	32	,	,	PUNCT
ejpam-5571	301	33	b	b	NOUN
ejpam-5571	301	34	)	)	PUNCT
ejpam-5571	301	35	=	=	SYM
ejpam-5571	301	36	∠p′(a	∠p′(a	PROPN
ejpam-5571	301	37	′	′	NOUN
ejpam-5571	301	38	,	,	PUNCT
ejpam-5571	301	39	b′	b′	NUM
ejpam-5571	301	40	)	)	PUNCT
ejpam-5571	301	41	and	and	CCONJ
ejpam-5571	301	42	∠c(a	∠c(a	NOUN
ejpam-5571	301	43	,	,	PUNCT
ejpam-5571	301	44	b	b	NOUN
ejpam-5571	301	45	)	)	PUNCT
ejpam-5571	301	46	=	=	SYM
ejpam-5571	301	47	∠c′(a	∠c′(a	PROPN
ejpam-5571	301	48	′	′	NOUN
ejpam-5571	301	49	,	,	PUNCT
ejpam-5571	301	50	b′	b′	NUM
ejpam-5571	301	51	)	)	PUNCT
ejpam-5571	301	52	;	;	PUNCT
ejpam-5571	301	53	(	(	PUNCT
ejpam-5571	301	54	v	v	NOUN
ejpam-5571	301	55	)	)	PUNCT
ejpam-5571	301	56	for	for	ADP
ejpam-5571	301	57	any	any	DET
ejpam-5571	301	58	triangle	triangle	NOUN
ejpam-5571	301	59	△	△	X
ejpam-5571	301	60	(	(	PUNCT
ejpam-5571	301	61	u	u	NOUN
ejpam-5571	301	62	,	,	PUNCT
ejpam-5571	301	63	v	v	NOUN
ejpam-5571	301	64	,	,	PUNCT
ejpam-5571	301	65	w	w	NOUN
ejpam-5571	301	66	)	)	PUNCT
ejpam-5571	301	67	in	in	ADP
ejpam-5571	301	68	x	x	NOUN
ejpam-5571	301	69	,	,	PUNCT
ejpam-5571	301	70	∠u(v	∠u(v	NOUN
ejpam-5571	301	71	,	,	PUNCT
ejpam-5571	301	72	x	x	NOUN
ejpam-5571	301	73	)	)	PUNCT
ejpam-5571	301	74	=	=	SYM
ejpam-5571	301	75	∠u(v	∠u(v	NOUN
ejpam-5571	301	76	,	,	PUNCT
ejpam-5571	301	77	w	w	NOUN
ejpam-5571	301	78	)	)	PUNCT
ejpam-5571	301	79	and	and	CCONJ
ejpam-5571	301	80	∠u(w	∠u(w	NUM
ejpam-5571	301	81	,	,	PUNCT
ejpam-5571	301	82	x	x	NOUN
ejpam-5571	301	83	)	)	PUNCT
ejpam-5571	301	84	=	=	SYM
ejpam-5571	301	85	∠u(w	∠u(w	PROPN
ejpam-5571	301	86	,	,	PUNCT
ejpam-5571	301	87	v	v	NOUN
ejpam-5571	301	88	)	)	PUNCT
ejpam-5571	301	89	for	for	ADP
ejpam-5571	301	90	all	all	PRON
ejpam-5571	301	91	x	x	SYM
ejpam-5571	301	92	∈	∈	PROPN
ejpam-5571	302	1	[	[	X
ejpam-5571	302	2	v	v	NOUN
ejpam-5571	302	3	,	,	PUNCT
ejpam-5571	302	4	w	w	NOUN
ejpam-5571	302	5	]	]	X
ejpam-5571	302	6	;	;	PUNCT
ejpam-5571	302	7	(	(	PUNCT
ejpam-5571	302	8	vi	vi	NOUN
ejpam-5571	302	9	)	)	PUNCT
ejpam-5571	302	10	for	for	ADP
ejpam-5571	302	11	any	any	DET
ejpam-5571	302	12	triangle	triangle	NOUN
ejpam-5571	302	13	△	△	X
ejpam-5571	302	14	(	(	PUNCT
ejpam-5571	302	15	u	u	NOUN
ejpam-5571	302	16	,	,	PUNCT
ejpam-5571	302	17	v	v	NOUN
ejpam-5571	302	18	,	,	PUNCT
ejpam-5571	302	19	w	w	NOUN
ejpam-5571	302	20	)	)	PUNCT
ejpam-5571	302	21	in	in	ADP
ejpam-5571	302	22	x	x	NOUN
ejpam-5571	302	23	,	,	PUNCT
ejpam-5571	302	24	∠u(v	∠u(v	NOUN
ejpam-5571	302	25	,	,	PUNCT
ejpam-5571	302	26	w	w	NOUN
ejpam-5571	302	27	)	)	PUNCT
ejpam-5571	302	28	=	=	NOUN
ejpam-5571	302	29	∠u(v	∠u(v	NOUN
ejpam-5571	302	30	,	,	PUNCT
ejpam-5571	302	31	x	x	NOUN
ejpam-5571	302	32	)	)	PUNCT
ejpam-5571	302	33	+	+	CCONJ
ejpam-5571	302	34	∠u(x	∠u(x	ADJ
ejpam-5571	302	35	,	,	PUNCT
ejpam-5571	302	36	w	w	NOUN
ejpam-5571	302	37	)	)	PUNCT
ejpam-5571	302	38	for	for	ADP
ejpam-5571	302	39	all	all	DET
ejpam-5571	302	40	x	x	SYM
ejpam-5571	302	41	∈	∈	PROPN
ejpam-5571	303	1	[	[	X
ejpam-5571	303	2	v	v	NOUN
ejpam-5571	303	3	,	,	PUNCT
ejpam-5571	303	4	w	w	PROPN
ejpam-5571	303	5	]	]	X
ejpam-5571	303	6	;	;	PUNCT
ejpam-5571	303	7	then	then	ADV
ejpam-5571	303	8	∪e∈γ	∪e∈γ	VERB
ejpam-5571	304	1	[	[	X
ejpam-5571	304	2	p	p	X
ejpam-5571	304	3	,	,	PUNCT
ejpam-5571	304	4	e	e	X
ejpam-5571	304	5	]	]	X
ejpam-5571	304	6	=	=	SYM
ejpam-5571	304	7	c({p}∪γ	c({p}∪γ	NOUN
ejpam-5571	304	8	)	)	PUNCT
ejpam-5571	304	9	and	and	CCONJ
ejpam-5571	304	10	c({p}∪γ	c({p}∪γ	NOUN
ejpam-5571	304	11	)	)	PUNCT
ejpam-5571	304	12	is	be	AUX
ejpam-5571	304	13	isometric	isometric	ADJ
ejpam-5571	304	14	to	to	ADP
ejpam-5571	304	15	c({p′}∪γ′	c({p′}∪γ′	NOUN
ejpam-5571	304	16	)	)	PUNCT
ejpam-5571	304	17	,	,	PUNCT
ejpam-5571	304	18	that	that	PRON
ejpam-5571	304	19	is	be	AUX
ejpam-5571	304	20	the	the	DET
ejpam-5571	304	21	totally	totally	ADV
ejpam-5571	304	22	geodesic	geodesic	ADJ
ejpam-5571	304	23	surface	surface	NOUN
ejpam-5571	304	24	bounded	bound	VERB
ejpam-5571	304	25	by	by	ADP
ejpam-5571	304	26	γ	γ	NOUN
ejpam-5571	304	27	and	and	CCONJ
ejpam-5571	304	28	the	the	DET
ejpam-5571	304	29	region	region	NOUN
ejpam-5571	304	30	bounded	bound	VERB
ejpam-5571	304	31	by	by	ADP
ejpam-5571	304	32	γ′	γ′	PROPN
ejpam-5571	304	33	are	be	AUX
ejpam-5571	304	34	isometric	isometric	ADJ
ejpam-5571	304	35	to	to	ADP
ejpam-5571	304	36	each	each	DET
ejpam-5571	304	37	other	other	ADJ
ejpam-5571	304	38	.	.	PUNCT
ejpam-5571	305	1	proof	proof	NOUN
ejpam-5571	305	2	.	.	PUNCT
ejpam-5571	306	1	we	we	PRON
ejpam-5571	306	2	will	will	AUX
ejpam-5571	306	3	first	first	ADV
ejpam-5571	306	4	demonstrate	demonstrate	VERB
ejpam-5571	306	5	that	that	SCONJ
ejpam-5571	306	6	∪e∈γ	∪e∈γ	VERB
ejpam-5571	307	1	[	[	X
ejpam-5571	307	2	p	p	X
ejpam-5571	307	3	,	,	PUNCT
ejpam-5571	307	4	e	e	X
ejpam-5571	307	5	]	]	X
ejpam-5571	307	6	=	=	SYM
ejpam-5571	307	7	c({p	c({p	PROPN
ejpam-5571	307	8	}	}	PUNCT
ejpam-5571	307	9	∪	∪	X
ejpam-5571	307	10	γ	γ	NOUN
ejpam-5571	307	11	)	)	PUNCT
ejpam-5571	307	12	.	.	PUNCT
ejpam-5571	308	1	by	by	ADP
ejpam-5571	308	2	the	the	DET
ejpam-5571	308	3	definition	definition	NOUN
ejpam-5571	308	4	of	of	ADP
ejpam-5571	308	5	c({p	c({p	PROPN
ejpam-5571	308	6	}	}	PUNCT
ejpam-5571	308	7	∪	∪	X
ejpam-5571	308	8	γ	γ	NOUN
ejpam-5571	308	9	)	)	PUNCT
ejpam-5571	308	10	,	,	PUNCT
ejpam-5571	308	11	we	we	PRON
ejpam-5571	308	12	have	have	VERB
ejpam-5571	308	13	that	that	PRON
ejpam-5571	308	14	∪e∈γ	∪e∈γ	VERB
ejpam-5571	309	1	[	[	X
ejpam-5571	309	2	p	p	X
ejpam-5571	309	3	,	,	PUNCT
ejpam-5571	309	4	e	e	X
ejpam-5571	309	5	]	]	X
ejpam-5571	309	6	⊂	⊂	PROPN
ejpam-5571	309	7	c({p	c({p	PROPN
ejpam-5571	309	8	}	}	PUNCT
ejpam-5571	309	9	∪	∪	X
ejpam-5571	309	10	γ	γ	NOUN
ejpam-5571	309	11	)	)	PUNCT
ejpam-5571	309	12	.	.	PUNCT
ejpam-5571	310	1	the	the	DET
ejpam-5571	310	2	next	next	ADJ
ejpam-5571	310	3	step	step	NOUN
ejpam-5571	310	4	is	be	AUX
ejpam-5571	310	5	to	to	PART
ejpam-5571	310	6	confirm	confirm	VERB
ejpam-5571	310	7	that	that	PRON
ejpam-5571	310	8	∪e∈γ	∪e∈γ	VERB
ejpam-5571	311	1	[	[	X
ejpam-5571	311	2	p	p	X
ejpam-5571	311	3	,	,	PUNCT
ejpam-5571	311	4	e	e	X
ejpam-5571	311	5	]	]	X
ejpam-5571	311	6	is	be	AUX
ejpam-5571	311	7	convex	convex	ADJ
ejpam-5571	311	8	.	.	PUNCT
ejpam-5571	312	1	let	let	VERB
ejpam-5571	312	2	m1,m2	m1,m2	PROPN
ejpam-5571	312	3	∈	∈	PROPN
ejpam-5571	312	4	∪e∈γ	∪e∈γ	VERB
ejpam-5571	312	5	[	[	X
ejpam-5571	312	6	p	p	X
ejpam-5571	312	7	,	,	PUNCT
ejpam-5571	312	8	e	e	NOUN
ejpam-5571	312	9	]	]	PUNCT
ejpam-5571	312	10	.	.	PUNCT
ejpam-5571	313	1	therefore	therefore	ADV
ejpam-5571	313	2	,	,	PUNCT
ejpam-5571	313	3	m1	m1	PROPN
ejpam-5571	313	4	∈	∈	PROPN
ejpam-5571	313	5	[	[	X
ejpam-5571	313	6	p	p	X
ejpam-5571	313	7	,	,	PUNCT
ejpam-5571	313	8	e1	e1	NOUN
ejpam-5571	313	9	]	]	PUNCT
ejpam-5571	313	10	and	and	CCONJ
ejpam-5571	313	11	m2	m2	PROPN
ejpam-5571	313	12	∈	∈	PROPN
ejpam-5571	314	1	[	[	X
ejpam-5571	314	2	p	p	X
ejpam-5571	314	3	,	,	PUNCT
ejpam-5571	314	4	e2	e2	PROPN
ejpam-5571	314	5	]	]	PUNCT
ejpam-5571	314	6	for	for	ADP
ejpam-5571	314	7	some	some	DET
ejpam-5571	314	8	e1	e1	NOUN
ejpam-5571	314	9	,	,	PUNCT
ejpam-5571	314	10	e2	e2	PROPN
ejpam-5571	314	11	∈	∈	PROPN
ejpam-5571	314	12	γ	γ	X
ejpam-5571	314	13	.	.	PROPN
ejpam-5571	315	1	if	if	SCONJ
ejpam-5571	315	2	e1	e1	PROPN
ejpam-5571	315	3	=	=	SYM
ejpam-5571	315	4	e2	e2	PROPN
ejpam-5571	315	5	,	,	PUNCT
ejpam-5571	315	6	there	there	PRON
ejpam-5571	315	7	is	be	VERB
ejpam-5571	315	8	nothing	nothing	PRON
ejpam-5571	315	9	to	to	PART
ejpam-5571	315	10	prove	prove	VERB
ejpam-5571	315	11	;	;	PUNCT
ejpam-5571	315	12	hence	hence	ADV
ejpam-5571	315	13	,	,	PUNCT
ejpam-5571	315	14	we	we	PRON
ejpam-5571	315	15	can	can	AUX
ejpam-5571	315	16	assume	assume	VERB
ejpam-5571	315	17	that	that	SCONJ
ejpam-5571	315	18	e1	e1	VERB
ejpam-5571	315	19	̸=	̸=	PROPN
ejpam-5571	315	20	e2	e2	PROPN
ejpam-5571	315	21	without	without	ADP
ejpam-5571	315	22	losing	lose	VERB
ejpam-5571	315	23	generality	generality	NOUN
ejpam-5571	315	24	.	.	PUNCT
ejpam-5571	316	1	let	let	VERB
ejpam-5571	316	2	e′1	e′1	NOUN
ejpam-5571	316	3	,	,	PUNCT
ejpam-5571	316	4	e	e	NOUN
ejpam-5571	316	5	′	′	NOUN
ejpam-5571	316	6	2	2	NUM
ejpam-5571	316	7	be	be	AUX
ejpam-5571	316	8	two	two	NUM
ejpam-5571	316	9	points	point	NOUN
ejpam-5571	316	10	on	on	ADP
ejpam-5571	316	11	γ	γ	PRON
ejpam-5571	316	12	such	such	ADJ
ejpam-5571	316	13	that	that	DET
ejpam-5571	316	14	ℓ(γ′a′e′1	ℓ(γ′a′e′1	NOUN
ejpam-5571	316	15	)	)	PUNCT
ejpam-5571	316	16	=	=	PUNCT
ejpam-5571	316	17	ℓ(γae1	ℓ(γae1	X
ejpam-5571	316	18	)	)	PUNCT
ejpam-5571	316	19	and	and	CCONJ
ejpam-5571	316	20	ℓ(γ′a′e′2	ℓ(γ′a′e′2	PROPN
ejpam-5571	316	21	)	)	PUNCT
ejpam-5571	316	22	=	=	SYM
ejpam-5571	316	23	ℓ(γae2	ℓ(γae2	NUM
ejpam-5571	316	24	)	)	PUNCT
ejpam-5571	316	25	.	.	PUNCT
ejpam-5571	317	1	assuming	assume	VERB
ejpam-5571	317	2	without	without	ADP
ejpam-5571	317	3	loss	loss	NOUN
ejpam-5571	317	4	of	of	ADP
ejpam-5571	317	5	generality	generality	NOUN
ejpam-5571	317	6	,	,	PUNCT
ejpam-5571	317	7	that	that	DET
ejpam-5571	317	8	e′1	e′1	NOUN
ejpam-5571	317	9	lies	lie	VERB
ejpam-5571	317	10	between	between	ADP
ejpam-5571	317	11	a′	a′	PROPN
ejpam-5571	317	12	and	and	CCONJ
ejpam-5571	317	13	e′2	e′2	PROPN
ejpam-5571	317	14	.	.	PUNCT
ejpam-5571	318	1	since	since	SCONJ
ejpam-5571	318	2	ℓ(γa	ℓ(γa	PROPN
ejpam-5571	318	3	,	,	PUNCT
ejpam-5571	318	4	e1	e1	NOUN
ejpam-5571	318	5	)	)	PUNCT
ejpam-5571	318	6	=	=	SYM
ejpam-5571	318	7	ℓ(γa′,e′1	ℓ(γa′,e′1	NOUN
ejpam-5571	318	8	)	)	PUNCT
ejpam-5571	318	9	and	and	CCONJ
ejpam-5571	318	10	ℓ(γa	ℓ(γa	PROPN
ejpam-5571	318	11	,	,	PUNCT
ejpam-5571	318	12	e2	e2	PROPN
ejpam-5571	318	13	)	)	PUNCT
ejpam-5571	318	14	=	=	SYM
ejpam-5571	319	1	ℓ(γa′,e′2	ℓ(γa′,e′2	PROPN
ejpam-5571	319	2	)	)	PUNCT
ejpam-5571	319	3	,	,	PUNCT
ejpam-5571	319	4	we	we	PRON
ejpam-5571	319	5	have	have	VERB
ejpam-5571	319	6	that	that	DET
ejpam-5571	319	7	ℓ(γe1e2	ℓ(γe1e2	NOUN
ejpam-5571	319	8	)	)	PUNCT
ejpam-5571	319	9	=	=	SYM
ejpam-5571	319	10	ℓ(γe′1e′2	ℓ(γe′1e′2	PROPN
ejpam-5571	319	11	)	)	PUNCT
ejpam-5571	319	12	.	.	PUNCT
ejpam-5571	320	1	by	by	ADP
ejpam-5571	320	2	(	(	PUNCT
ejpam-5571	320	3	iii	iii	NOUN
ejpam-5571	320	4	)	)	PUNCT
ejpam-5571	320	5	,	,	PUNCT
ejpam-5571	320	6	d(a	d(a	PROPN
ejpam-5571	320	7	,	,	PUNCT
ejpam-5571	320	8	e1	e1	PROPN
ejpam-5571	320	9	)	)	PUNCT
ejpam-5571	320	10	=	=	SYM
ejpam-5571	320	11	d(a′	d(a′	NOUN
ejpam-5571	320	12	,	,	PUNCT
ejpam-5571	320	13	e′1	e′1	NOUN
ejpam-5571	320	14	)	)	PUNCT
ejpam-5571	320	15	,	,	PUNCT
ejpam-5571	320	16	d(e1	d(e1	NOUN
ejpam-5571	320	17	,	,	PUNCT
ejpam-5571	320	18	e1	e1	NOUN
ejpam-5571	320	19	)	)	PUNCT
ejpam-5571	320	20	=	=	SYM
ejpam-5571	320	21	d(e′1	d(e′1	X
ejpam-5571	320	22	,	,	PUNCT
ejpam-5571	320	23	e	e	NOUN
ejpam-5571	320	24	′	′	NOUN
ejpam-5571	320	25	2	2	NUM
ejpam-5571	320	26	)	)	PUNCT
ejpam-5571	320	27	and	and	CCONJ
ejpam-5571	320	28	d(e2	d(e2	PROPN
ejpam-5571	320	29	,	,	PUNCT
ejpam-5571	320	30	b	b	NOUN
ejpam-5571	320	31	)	)	PUNCT
ejpam-5571	320	32	=	=	SYM
ejpam-5571	320	33	d(e′2	d(e′2	PROPN
ejpam-5571	320	34	,	,	PUNCT
ejpam-5571	320	35	b	b	NOUN
ejpam-5571	320	36	)	)	PUNCT
ejpam-5571	320	37	,	,	PUNCT
ejpam-5571	320	38	and	and	CCONJ
ejpam-5571	320	39	hence	hence	ADV
ejpam-5571	320	40	we	we	PRON
ejpam-5571	320	41	have	have	VERB
ejpam-5571	320	42	those	those	DET
ejpam-5571	320	43	triangles	triangle	NOUN
ejpam-5571	320	44	△	△	NOUN
ejpam-5571	320	45	(p′	(p′	X
ejpam-5571	320	46	,	,	PUNCT
ejpam-5571	320	47	a′	a′	PROPN
ejpam-5571	320	48	,	,	PUNCT
ejpam-5571	320	49	e′1	e′1	NOUN
ejpam-5571	320	50	)	)	PUNCT
ejpam-5571	320	51	,	,	PUNCT
ejpam-5571	320	52	△	△	X
ejpam-5571	320	53	(	(	PUNCT
ejpam-5571	320	54	p′	p′	NOUN
ejpam-5571	320	55	,	,	PUNCT
ejpam-5571	320	56	e′1	e′1	NOUN
ejpam-5571	320	57	,	,	PUNCT
ejpam-5571	320	58	e	e	NOUN
ejpam-5571	320	59	′	′	NOUN
ejpam-5571	320	60	2	2	NUM
ejpam-5571	320	61	)	)	PUNCT
ejpam-5571	320	62	and	and	CCONJ
ejpam-5571	320	63	△	△	X
ejpam-5571	320	64	(	(	PUNCT
ejpam-5571	320	65	p′	p′	PROPN
ejpam-5571	320	66	,	,	PUNCT
ejpam-5571	320	67	e′2	e′2	PROPN
ejpam-5571	320	68	,	,	PUNCT
ejpam-5571	320	69	b	b	PROPN
ejpam-5571	320	70	′	′	NOUN
ejpam-5571	320	71	)	)	PUNCT
ejpam-5571	320	72	are	be	AUX
ejpam-5571	320	73	comparison	comparison	NOUN
ejpam-5571	320	74	triangles	triangle	NOUN
ejpam-5571	320	75	of	of	ADP
ejpam-5571	320	76	△	△	X
ejpam-5571	320	77	(p	(p	NOUN
ejpam-5571	320	78	,	,	PUNCT
ejpam-5571	320	79	a	a	DET
ejpam-5571	320	80	,	,	PUNCT
ejpam-5571	320	81	e1	e1	NOUN
ejpam-5571	320	82	)	)	PUNCT
ejpam-5571	320	83	,	,	PUNCT
ejpam-5571	320	84	△	△	X
ejpam-5571	320	85	(	(	PUNCT
ejpam-5571	320	86	p	p	X
ejpam-5571	320	87	,	,	PUNCT
ejpam-5571	320	88	e1	e1	PROPN
ejpam-5571	320	89	,	,	PUNCT
ejpam-5571	320	90	e2	e2	PROPN
ejpam-5571	320	91	)	)	PUNCT
ejpam-5571	320	92	and	and	CCONJ
ejpam-5571	320	93	△	△	X
ejpam-5571	320	94	(	(	PUNCT
ejpam-5571	320	95	p	p	PROPN
ejpam-5571	320	96	,	,	PUNCT
ejpam-5571	320	97	e2	e2	PROPN
ejpam-5571	320	98	,	,	PUNCT
ejpam-5571	320	99	b	b	NOUN
ejpam-5571	320	100	)	)	PUNCT
ejpam-5571	320	101	,	,	PUNCT
ejpam-5571	320	102	respectively	respectively	ADV
ejpam-5571	320	103	.	.	PUNCT
ejpam-5571	321	1	as	as	SCONJ
ejpam-5571	321	2	x	x	PRON
ejpam-5571	321	3	is	be	AUX
ejpam-5571	321	4	metric	metric	ADJ
ejpam-5571	321	5	space	space	NOUN
ejpam-5571	321	6	with	with	ADP
ejpam-5571	321	7	curvature	curvature	NOUN
ejpam-5571	321	8	bounded	bound	VERB
ejpam-5571	321	9	below	below	ADV
ejpam-5571	321	10	,	,	PUNCT
ejpam-5571	321	11	∠p(a	∠p(a	ADJ
ejpam-5571	321	12	,	,	PUNCT
ejpam-5571	321	13	e1	e1	PROPN
ejpam-5571	321	14	)	)	PUNCT
ejpam-5571	321	15	≥	≥	NOUN
ejpam-5571	321	16	∠p′(a	∠p′(a	PROPN
ejpam-5571	321	17	′	′	PROPN
ejpam-5571	321	18	,	,	PUNCT
ejpam-5571	321	19	e′1	e′1	NOUN
ejpam-5571	321	20	)	)	PUNCT
ejpam-5571	321	21	,	,	PUNCT
ejpam-5571	321	22	∠p(e1	∠p(e1	PROPN
ejpam-5571	321	23	,	,	PUNCT
ejpam-5571	321	24	e2	e2	PROPN
ejpam-5571	321	25	)	)	PUNCT
ejpam-5571	321	26	≥	≥	NOUN
ejpam-5571	322	1	∠p′(e	∠p′(e	NOUN
ejpam-5571	322	2	′	′	NUM
ejpam-5571	322	3	1	1	NUM
ejpam-5571	322	4	,	,	PUNCT
ejpam-5571	322	5	e	e	NOUN
ejpam-5571	322	6	′	′	NOUN
ejpam-5571	322	7	2	2	NUM
ejpam-5571	322	8	)	)	PUNCT
ejpam-5571	322	9	and	and	CCONJ
ejpam-5571	322	10	∠p(e2	∠p(e2	PROPN
ejpam-5571	322	11	,	,	PUNCT
ejpam-5571	322	12	b	b	NOUN
ejpam-5571	322	13	)	)	PUNCT
ejpam-5571	322	14	≥	≥	NOUN
ejpam-5571	323	1	∠p′(e	∠p′(e	NOUN
ejpam-5571	323	2	′	′	NUM
ejpam-5571	323	3	2	2	NUM
ejpam-5571	323	4	,	,	PUNCT
ejpam-5571	323	5	b	b	NOUN
ejpam-5571	323	6	′	′	NOUN
ejpam-5571	323	7	)	)	PUNCT
ejpam-5571	323	8	.	.	PUNCT
ejpam-5571	324	1	since	since	SCONJ
ejpam-5571	324	2	∠p′(a	∠p′(a	PROPN
ejpam-5571	324	3	′	′	PROPN
ejpam-5571	324	4	,	,	PUNCT
ejpam-5571	324	5	b′	b′	NUM
ejpam-5571	324	6	)	)	PUNCT
ejpam-5571	325	1	=	=	SYM
ejpam-5571	325	2	∠p′(a	∠p′(a	PROPN
ejpam-5571	325	3	′	′	PROPN
ejpam-5571	325	4	,	,	PUNCT
ejpam-5571	325	5	e′1	e′1	NOUN
ejpam-5571	325	6	)	)	PUNCT
ejpam-5571	325	7	+	+	CCONJ
ejpam-5571	325	8	∠p′(e	∠p′(e	NUM
ejpam-5571	325	9	′	′	NUM
ejpam-5571	325	10	1	1	NUM
ejpam-5571	325	11	,	,	PUNCT
ejpam-5571	325	12	e	e	NOUN
ejpam-5571	325	13	′	′	NOUN
ejpam-5571	325	14	2	2	NUM
ejpam-5571	325	15	)	)	PUNCT
ejpam-5571	325	16	+	+	CCONJ
ejpam-5571	326	1	∠p′(e	∠p′(e	PROPN
ejpam-5571	326	2	′	′	NUM
ejpam-5571	326	3	2	2	NUM
ejpam-5571	326	4	,	,	PUNCT
ejpam-5571	326	5	b	b	NOUN
ejpam-5571	326	6	′	′	NOUN
ejpam-5571	326	7	)	)	PUNCT
ejpam-5571	326	8	≤	≤	NOUN
ejpam-5571	326	9	∠p(a	∠p(a	ADJ
ejpam-5571	326	10	,	,	PUNCT
ejpam-5571	326	11	e1	e1	PROPN
ejpam-5571	326	12	)	)	PUNCT
ejpam-5571	327	1	+	+	CCONJ
ejpam-5571	327	2	∠p(e1	∠p(e1	PROPN
ejpam-5571	327	3	,	,	PUNCT
ejpam-5571	327	4	e2	e2	PROPN
ejpam-5571	327	5	)	)	PUNCT
ejpam-5571	327	6	+	+	CCONJ
ejpam-5571	327	7	∠p(e2	∠p(e2	PROPN
ejpam-5571	327	8	,	,	PUNCT
ejpam-5571	327	9	b	b	NOUN
ejpam-5571	327	10	)	)	PUNCT
ejpam-5571	327	11	=	=	SYM
ejpam-5571	327	12	∠p(a	∠p(a	ADJ
ejpam-5571	327	13	,	,	PUNCT
ejpam-5571	327	14	b	b	NOUN
ejpam-5571	327	15	)	)	PUNCT
ejpam-5571	327	16	=	=	SYM
ejpam-5571	327	17	∠p′(a	∠p′(a	PROPN
ejpam-5571	327	18	′	′	PROPN
ejpam-5571	327	19	,	,	PUNCT
ejpam-5571	327	20	b′	b′	NUM
ejpam-5571	327	21	)	)	PUNCT
ejpam-5571	327	22	,	,	PUNCT
ejpam-5571	327	23	we	we	PRON
ejpam-5571	327	24	get	get	VERB
ejpam-5571	327	25	∠p(e1	∠p(e1	PROPN
ejpam-5571	327	26	,	,	PUNCT
ejpam-5571	327	27	e2	e2	PROPN
ejpam-5571	327	28	)	)	PUNCT
ejpam-5571	327	29	=	=	PUNCT
ejpam-5571	328	1	∠p′(e	∠p′(e	NOUN
ejpam-5571	328	2	′	′	NOUN
ejpam-5571	328	3	1	1	NUM
ejpam-5571	328	4	,	,	PUNCT
ejpam-5571	328	5	e	e	NOUN
ejpam-5571	328	6	′	′	NOUN
ejpam-5571	328	7	2	2	NUM
ejpam-5571	328	8	)	)	PUNCT
ejpam-5571	328	9	.	.	PUNCT
ejpam-5571	329	1	because	because	SCONJ
ejpam-5571	329	2	d(p	d(p	PROPN
ejpam-5571	329	3	,	,	PUNCT
ejpam-5571	329	4	m1	m1	NOUN
ejpam-5571	329	5	)	)	PUNCT
ejpam-5571	329	6	=	=	PRON
ejpam-5571	329	7	d(p′,m′	d(p′,m′	VERB
ejpam-5571	329	8	1	1	NUM
ejpam-5571	329	9	)	)	PUNCT
ejpam-5571	329	10	,	,	PUNCT
ejpam-5571	329	11	d(p	d(p	PROPN
ejpam-5571	329	12	,	,	PUNCT
ejpam-5571	329	13	m2	m2	PROPN
ejpam-5571	329	14	)	)	PUNCT
ejpam-5571	329	15	=	=	PRON
ejpam-5571	329	16	d(p′,m′	d(p′,m′	VERB
ejpam-5571	329	17	2	2	NUM
ejpam-5571	329	18	)	)	PUNCT
ejpam-5571	329	19	and	and	CCONJ
ejpam-5571	329	20	∠p(e1	∠p(e1	PROPN
ejpam-5571	329	21	,	,	PUNCT
ejpam-5571	329	22	e2	e2	PROPN
ejpam-5571	329	23	)	)	PUNCT
ejpam-5571	329	24	=	=	SYM
ejpam-5571	329	25	∠p(m1,m2	∠p(m1,m2	NOUN
ejpam-5571	329	26	)	)	PUNCT
ejpam-5571	329	27	,	,	PUNCT
ejpam-5571	329	28	by	by	ADP
ejpam-5571	329	29	using	use	VERB
ejpam-5571	329	30	theorem	theorem	ADJ
ejpam-5571	329	31	2	2	NUM
ejpam-5571	329	32	,	,	PUNCT
ejpam-5571	329	33	d(m1,m2	d(m1,m2	NOUN
ejpam-5571	329	34	)	)	PUNCT
ejpam-5571	329	35	≤	≤	NUM
ejpam-5571	329	36	d(m′	d(m′	NOUN
ejpam-5571	329	37	1,m	1,m	NOUN
ejpam-5571	329	38	′	′	NUM
ejpam-5571	329	39	2	2	NUM
ejpam-5571	329	40	)	)	PUNCT
ejpam-5571	329	41	.	.	PUNCT
ejpam-5571	330	1	as	as	ADP
ejpam-5571	330	2	ℓ(γe1e2	ℓ(γe1e2	NOUN
ejpam-5571	330	3	)	)	PUNCT
ejpam-5571	330	4	=	=	SYM
ejpam-5571	330	5	ℓ(γe′1e′2	ℓ(γe′1e′2	PROPN
ejpam-5571	330	6	)	)	PUNCT
ejpam-5571	330	7	,	,	PUNCT
ejpam-5571	330	8	by	by	ADP
ejpam-5571	330	9	(	(	PUNCT
ejpam-5571	330	10	iii	iii	NOUN
ejpam-5571	330	11	)	)	PUNCT
ejpam-5571	330	12	,	,	PUNCT
ejpam-5571	330	13	we	we	PRON
ejpam-5571	330	14	have	have	VERB
ejpam-5571	330	15	d(e1	d(e1	NOUN
ejpam-5571	330	16	,	,	PUNCT
ejpam-5571	330	17	e2	e2	PROPN
ejpam-5571	330	18	)	)	PUNCT
ejpam-5571	330	19	=	=	SYM
ejpam-5571	330	20	d(e′1	d(e′1	X
ejpam-5571	330	21	,	,	PUNCT
ejpam-5571	330	22	e	e	NOUN
ejpam-5571	330	23	′	′	NOUN
ejpam-5571	330	24	2	2	NUM
ejpam-5571	330	25	)	)	PUNCT
ejpam-5571	330	26	.	.	PUNCT
ejpam-5571	331	1	now	now	ADV
ejpam-5571	331	2	we	we	PRON
ejpam-5571	331	3	have	have	VERB
ejpam-5571	331	4	that	that	PRON
ejpam-5571	331	5	a	a	DET
ejpam-5571	331	6	triangle	triangle	NOUN
ejpam-5571	331	7	△	△	X
ejpam-5571	331	8	(	(	PUNCT
ejpam-5571	331	9	p′	p′	NOUN
ejpam-5571	331	10	,	,	PUNCT
ejpam-5571	331	11	e′1	e′1	NOUN
ejpam-5571	331	12	,	,	PUNCT
ejpam-5571	331	13	e	e	NOUN
ejpam-5571	331	14	′	′	NOUN
ejpam-5571	331	15	2	2	NUM
ejpam-5571	331	16	)	)	PUNCT
ejpam-5571	331	17	is	be	AUX
ejpam-5571	331	18	a	a	DET
ejpam-5571	331	19	comparison	comparison	NOUN
ejpam-5571	331	20	triangle	triangle	NOUN
ejpam-5571	331	21	of	of	ADP
ejpam-5571	331	22	a	a	DET
ejpam-5571	331	23	triangle	triangle	NOUN
ejpam-5571	332	1	△	△	X
ejpam-5571	332	2	(	(	PUNCT
ejpam-5571	332	3	p	p	X
ejpam-5571	332	4	,	,	PUNCT
ejpam-5571	332	5	e1	e1	PROPN
ejpam-5571	332	6	,	,	PUNCT
ejpam-5571	332	7	e2	e2	PROPN
ejpam-5571	332	8	)	)	PUNCT
ejpam-5571	332	9	.	.	PUNCT
ejpam-5571	333	1	as	as	ADP
ejpam-5571	333	2	a	a	DET
ejpam-5571	333	3	result	result	NOUN
ejpam-5571	333	4	,	,	PUNCT
ejpam-5571	333	5	x	x	PRON
ejpam-5571	333	6	is	be	AUX
ejpam-5571	333	7	a	a	DET
ejpam-5571	333	8	metric	metric	ADJ
ejpam-5571	333	9	space	space	NOUN
ejpam-5571	333	10	with	with	ADP
ejpam-5571	333	11	curvature	curvature	NOUN
ejpam-5571	333	12	bounded	bound	VERB
ejpam-5571	333	13	below	below	ADV
ejpam-5571	333	14	,	,	PUNCT
ejpam-5571	333	15	d(m1,m2	d(m1,m2	NOUN
ejpam-5571	333	16	)	)	PUNCT
ejpam-5571	333	17	≥	≥	NOUN
ejpam-5571	333	18	d(m′	d(m′	NOUN
ejpam-5571	333	19	1,m	1,m	NOUN
ejpam-5571	333	20	′	′	NUM
ejpam-5571	333	21	2	2	NUM
ejpam-5571	333	22	)	)	PUNCT
ejpam-5571	333	23	.	.	PUNCT
ejpam-5571	334	1	consequently	consequently	ADV
ejpam-5571	334	2	,	,	PUNCT
ejpam-5571	334	3	d(m1,m2	d(m1,m2	NOUN
ejpam-5571	334	4	)	)	PUNCT
ejpam-5571	334	5	=	=	VERB
ejpam-5571	335	1	d(m′	d(m′	NOUN
ejpam-5571	335	2	1,m	1,m	NOUN
ejpam-5571	335	3	′	′	NUM
ejpam-5571	335	4	2	2	NUM
ejpam-5571	335	5	)	)	PUNCT
ejpam-5571	335	6	.	.	PUNCT
ejpam-5571	336	1	that	that	PRON
ejpam-5571	336	2	means	mean	VERB
ejpam-5571	336	3	that	that	SCONJ
ejpam-5571	336	4	for	for	ADP
ejpam-5571	336	5	any	any	DET
ejpam-5571	336	6	point	point	NOUN
ejpam-5571	336	7	on	on	ADP
ejpam-5571	336	8	[	[	X
ejpam-5571	336	9	m1,m2	m1,m2	PROPN
ejpam-5571	336	10	]	]	PUNCT
ejpam-5571	336	11	,	,	PUNCT
ejpam-5571	336	12	lies	lie	VERB
ejpam-5571	336	13	on	on	ADP
ejpam-5571	336	14	a	a	DET
ejpam-5571	336	15	segment	segment	NOUN
ejpam-5571	336	16	[	[	X
ejpam-5571	336	17	p	p	X
ejpam-5571	336	18	,	,	PUNCT
ejpam-5571	336	19	t	t	PROPN
ejpam-5571	336	20	]	]	PUNCT
ejpam-5571	336	21	,	,	PUNCT
ejpam-5571	336	22	for	for	ADP
ejpam-5571	336	23	some	some	DET
ejpam-5571	336	24	t	t	NOUN
ejpam-5571	336	25	∈	∈	PROPN
ejpam-5571	336	26	γ	γ	X
ejpam-5571	336	27	.	.	PUNCT
ejpam-5571	337	1	we	we	PRON
ejpam-5571	337	2	may	may	AUX
ejpam-5571	337	3	infer	infer	VERB
ejpam-5571	337	4	that	that	SCONJ
ejpam-5571	337	5	,	,	PUNCT
ejpam-5571	337	6	the	the	DET
ejpam-5571	337	7	geodesic	geodesic	ADJ
ejpam-5571	337	8	segment	segment	NOUN
ejpam-5571	337	9	[	[	X
ejpam-5571	337	10	m1,m2	m1,m2	PROPN
ejpam-5571	337	11	]	]	X
ejpam-5571	337	12	is	be	AUX
ejpam-5571	337	13	contained	contain	VERB
ejpam-5571	337	14	in	in	ADP
ejpam-5571	337	15	∪e∈γ	∪e∈γ	PROPN
ejpam-5571	338	1	[	[	X
ejpam-5571	338	2	p	p	X
ejpam-5571	338	3	,	,	PUNCT
ejpam-5571	338	4	e	e	NOUN
ejpam-5571	338	5	]	]	PUNCT
ejpam-5571	338	6	.	.	PUNCT
ejpam-5571	339	1	c.	c.	PROPN
ejpam-5571	339	2	phokaew	phokaew	PROPN
ejpam-5571	339	3	,	,	PUNCT
ejpam-5571	339	4	a.	a.	PROPN
ejpam-5571	339	5	sama	sama	PROPN
ejpam-5571	339	6	-	-	PUNCT
ejpam-5571	339	7	ae	ae	PROPN
ejpam-5571	339	8	,	,	PUNCT
ejpam-5571	339	9	/	/	SYM
ejpam-5571	339	10	eur	eur	NOUN
ejpam-5571	339	11	.	.	PUNCT
ejpam-5571	340	1	j.	j.	PROPN
ejpam-5571	340	2	pure	pure	PROPN
ejpam-5571	340	3	appl	appl	PROPN
ejpam-5571	340	4	.	.	PROPN
ejpam-5571	340	5	math	math	PROPN
ejpam-5571	340	6	,	,	PUNCT
ejpam-5571	340	7	17	17	NUM
ejpam-5571	340	8	(	(	PUNCT
ejpam-5571	340	9	4	4	NUM
ejpam-5571	340	10	)	)	PUNCT
ejpam-5571	340	11	(	(	PUNCT
ejpam-5571	340	12	2024	2024	NUM
ejpam-5571	340	13	)	)	PUNCT
ejpam-5571	340	14	,	,	PUNCT
ejpam-5571	340	15	3932	3932	NUM
ejpam-5571	340	16	-	-	SYM
ejpam-5571	340	17	3944	3944	NUM
ejpam-5571	340	18	3942	3942	NUM
ejpam-5571	340	19	we	we	PRON
ejpam-5571	340	20	will	will	AUX
ejpam-5571	340	21	then	then	ADV
ejpam-5571	340	22	prove	prove	VERB
ejpam-5571	340	23	that	that	SCONJ
ejpam-5571	340	24	c({p′	c({p′	VERB
ejpam-5571	340	25	}	}	PUNCT
ejpam-5571	340	26	∪	∪	X
ejpam-5571	340	27	γ′	γ′	NOUN
ejpam-5571	340	28	)	)	PUNCT
ejpam-5571	340	29	is	be	AUX
ejpam-5571	340	30	isometric	isometric	ADJ
ejpam-5571	340	31	to	to	ADP
ejpam-5571	340	32	c({p	c({p	VERB
ejpam-5571	340	33	}	}	PUNCT
ejpam-5571	340	34	∪	∪	X
ejpam-5571	340	35	γ	γ	NOUN
ejpam-5571	340	36	)	)	PUNCT
ejpam-5571	340	37	.	.	PUNCT
ejpam-5571	341	1	define	define	VERB
ejpam-5571	341	2	a	a	DET
ejpam-5571	341	3	map	map	NOUN
ejpam-5571	341	4	i	i	PRON
ejpam-5571	341	5	from	from	ADP
ejpam-5571	341	6	c({p′	c({p′	PROPN
ejpam-5571	341	7	}	}	PUNCT
ejpam-5571	341	8	∪	∪	X
ejpam-5571	341	9	γ′	γ′	NOUN
ejpam-5571	341	10	)	)	PUNCT
ejpam-5571	341	11	to	to	PART
ejpam-5571	341	12	c({p	c({p	VERB
ejpam-5571	341	13	}	}	PUNCT
ejpam-5571	341	14	∪	∪	X
ejpam-5571	341	15	γ	γ	NOUN
ejpam-5571	341	16	)	)	PUNCT
ejpam-5571	341	17	in	in	ADP
ejpam-5571	341	18	such	such	DET
ejpam-5571	341	19	a	a	DET
ejpam-5571	341	20	way	way	NOUN
ejpam-5571	341	21	that	that	PRON
ejpam-5571	341	22	every	every	DET
ejpam-5571	341	23	geodesic	geodesic	ADJ
ejpam-5571	341	24	segment	segment	NOUN
ejpam-5571	341	25	[	[	X
ejpam-5571	341	26	p′	p′	NOUN
ejpam-5571	341	27	,	,	PUNCT
ejpam-5571	341	28	w′	w′	PROPN
ejpam-5571	341	29	]	]	PUNCT
ejpam-5571	341	30	from	from	ADP
ejpam-5571	341	31	p′	p′	NOUN
ejpam-5571	341	32	to	to	ADP
ejpam-5571	341	33	a	a	DET
ejpam-5571	341	34	point	point	NOUN
ejpam-5571	341	35	w′	w′	NOUN
ejpam-5571	341	36	on	on	ADP
ejpam-5571	341	37	γ′	γ′	PROPN
ejpam-5571	341	38	is	be	AUX
ejpam-5571	341	39	sent	send	VERB
ejpam-5571	341	40	isometrically	isometrically	PROPN
ejpam-5571	341	41	onto	onto	ADP
ejpam-5571	341	42	the	the	DET
ejpam-5571	341	43	segment	segment	NOUN
ejpam-5571	341	44	[	[	X
ejpam-5571	341	45	p	p	X
ejpam-5571	341	46	,	,	PUNCT
ejpam-5571	341	47	w	w	NOUN
ejpam-5571	341	48	]	]	X
ejpam-5571	341	49	where	where	SCONJ
ejpam-5571	341	50	w	w	NOUN
ejpam-5571	341	51	is	be	AUX
ejpam-5571	341	52	a	a	DET
ejpam-5571	341	53	point	point	NOUN
ejpam-5571	341	54	on	on	ADP
ejpam-5571	341	55	γ	γ	NOUN
ejpam-5571	341	56	with	with	ADP
ejpam-5571	341	57	ℓ(γaw	ℓ(γaw	PROPN
ejpam-5571	341	58	)	)	PUNCT
ejpam-5571	341	59	=	=	PUNCT
ejpam-5571	341	60	ℓ(γa′w′	ℓ(γa′w′	NOUN
ejpam-5571	341	61	)	)	PUNCT
ejpam-5571	341	62	.	.	PUNCT
ejpam-5571	342	1	the	the	DET
ejpam-5571	342	2	fact	fact	NOUN
ejpam-5571	342	3	that	that	SCONJ
ejpam-5571	342	4	i	i	PRON
ejpam-5571	342	5	is	be	AUX
ejpam-5571	342	6	a	a	DET
ejpam-5571	342	7	bijection	bijection	NOUN
ejpam-5571	342	8	is	be	AUX
ejpam-5571	342	9	evident	evident	ADJ
ejpam-5571	342	10	.	.	PUNCT
ejpam-5571	343	1	simply	simply	ADV
ejpam-5571	343	2	confirming	confirm	VERB
ejpam-5571	343	3	that	that	SCONJ
ejpam-5571	343	4	i	i	PRON
ejpam-5571	343	5	maintains	maintain	VERB
ejpam-5571	343	6	distances	distance	NOUN
ejpam-5571	343	7	between	between	ADP
ejpam-5571	343	8	points	point	NOUN
ejpam-5571	343	9	will	will	AUX
ejpam-5571	343	10	demonstrate	demonstrate	VERB
ejpam-5571	343	11	that	that	SCONJ
ejpam-5571	343	12	i	i	PRON
ejpam-5571	343	13	is	be	AUX
ejpam-5571	343	14	an	an	DET
ejpam-5571	343	15	isometry	isometry	NOUN
ejpam-5571	343	16	from	from	ADP
ejpam-5571	343	17	c({p′}∪γ′	c({p′}∪γ′	NOUN
ejpam-5571	343	18	)	)	PUNCT
ejpam-5571	343	19	onto	onto	ADP
ejpam-5571	343	20	c({p}∪γ	c({p}∪γ	NOUN
ejpam-5571	343	21	)	)	PUNCT
ejpam-5571	343	22	.	.	PUNCT
ejpam-5571	344	1	let	let	VERB
ejpam-5571	344	2	x′1	x′1	PROPN
ejpam-5571	345	1	and	and	CCONJ
ejpam-5571	345	2	x′2	x′2	NOUN
ejpam-5571	345	3	be	be	AUX
ejpam-5571	345	4	two	two	NUM
ejpam-5571	345	5	points	point	NOUN
ejpam-5571	345	6	on	on	ADP
ejpam-5571	345	7	segments	segment	NOUN
ejpam-5571	345	8	[	[	X
ejpam-5571	345	9	p′	p′	NOUN
ejpam-5571	345	10	,	,	PUNCT
ejpam-5571	345	11	y′1	y′1	X
ejpam-5571	345	12	]	]	PUNCT
ejpam-5571	345	13	and	and	CCONJ
ejpam-5571	345	14	[	[	X
ejpam-5571	345	15	p′	p′	NOUN
ejpam-5571	345	16	,	,	PUNCT
ejpam-5571	345	17	y′2	y′2	X
ejpam-5571	345	18	]	]	X
ejpam-5571	345	19	,	,	PUNCT
ejpam-5571	345	20	respectively	respectively	ADV
ejpam-5571	345	21	,	,	PUNCT
ejpam-5571	345	22	for	for	ADP
ejpam-5571	345	23	some	some	DET
ejpam-5571	345	24	y′1	y′1	NOUN
ejpam-5571	345	25	,	,	PUNCT
ejpam-5571	345	26	y	y	PROPN
ejpam-5571	345	27	′	′	NUM
ejpam-5571	345	28	2	2	NUM
ejpam-5571	345	29	∈	∈	NOUN
ejpam-5571	345	30	γ′.	γ′.	VERB
ejpam-5571	345	31	on	on	ADP
ejpam-5571	345	32	corresponding	correspond	VERB
ejpam-5571	345	33	geodesic	geodesic	ADJ
ejpam-5571	345	34	segments	segment	NOUN
ejpam-5571	345	35	[	[	X
ejpam-5571	345	36	p	p	X
ejpam-5571	345	37	,	,	PUNCT
ejpam-5571	345	38	y1	y1	NOUN
ejpam-5571	345	39	]	]	PUNCT
ejpam-5571	345	40	of	of	ADP
ejpam-5571	345	41	[	[	X
ejpam-5571	345	42	p	p	X
ejpam-5571	345	43	′	′	NOUN
ejpam-5571	345	44	,	,	PUNCT
ejpam-5571	345	45	y′1	y′1	X
ejpam-5571	345	46	]	]	PUNCT
ejpam-5571	345	47	and	and	CCONJ
ejpam-5571	345	48	[	[	X
ejpam-5571	345	49	p	p	X
ejpam-5571	345	50	,	,	PUNCT
ejpam-5571	345	51	y2	y2	NOUN
ejpam-5571	345	52	]	]	PUNCT
ejpam-5571	345	53	of	of	ADP
ejpam-5571	345	54	[	[	X
ejpam-5571	345	55	p′	p′	NOUN
ejpam-5571	345	56	,	,	PUNCT
ejpam-5571	345	57	y′2	y′2	X
ejpam-5571	345	58	]	]	X
ejpam-5571	345	59	,	,	PUNCT
ejpam-5571	345	60	we	we	PRON
ejpam-5571	345	61	let	let	VERB
ejpam-5571	345	62	x1	x1	PROPN
ejpam-5571	345	63	and	and	CCONJ
ejpam-5571	345	64	x2	x2	PROPN
ejpam-5571	345	65	be	be	VERB
ejpam-5571	345	66	the	the	DET
ejpam-5571	345	67	points	point	NOUN
ejpam-5571	345	68	corresponding	correspond	VERB
ejpam-5571	345	69	to	to	ADP
ejpam-5571	345	70	x′1	x′1	PROPN
ejpam-5571	345	71	and	and	CCONJ
ejpam-5571	345	72	x′2	x′2	NOUN
ejpam-5571	345	73	,	,	PUNCT
ejpam-5571	345	74	respectively	respectively	ADV
ejpam-5571	345	75	.	.	PUNCT
ejpam-5571	346	1	we	we	PRON
ejpam-5571	346	2	can	can	AUX
ejpam-5571	346	3	verify	verify	VERB
ejpam-5571	346	4	d(x1	d(x1	NOUN
ejpam-5571	346	5	,	,	PUNCT
ejpam-5571	346	6	x2	x2	PROPN
ejpam-5571	346	7	)	)	PUNCT
ejpam-5571	346	8	=	=	SYM
ejpam-5571	346	9	d(x′1	d(x′1	PROPN
ejpam-5571	346	10	,	,	PUNCT
ejpam-5571	346	11	x	x	SYM
ejpam-5571	346	12	′	′	NOUN
ejpam-5571	346	13	2	2	NUM
ejpam-5571	346	14	)	)	PUNCT
ejpam-5571	346	15	similarly	similarly	ADV
ejpam-5571	346	16	as	as	ADP
ejpam-5571	346	17	above	above	ADV
ejpam-5571	346	18	,	,	PUNCT
ejpam-5571	346	19	the	the	DET
ejpam-5571	346	20	result	result	NOUN
ejpam-5571	346	21	is	be	AUX
ejpam-5571	346	22	completely	completely	ADV
ejpam-5571	346	23	proven	prove	VERB
ejpam-5571	346	24	.	.	PUNCT
ejpam-5571	347	1	we	we	PRON
ejpam-5571	347	2	describe	describe	VERB
ejpam-5571	347	3	characterizations	characterization	NOUN
ejpam-5571	347	4	of	of	ADP
ejpam-5571	347	5	a	a	DET
ejpam-5571	347	6	closed	closed	ADJ
ejpam-5571	347	7	spherical	spherical	ADJ
ejpam-5571	347	8	curve	curve	NOUN
ejpam-5571	347	9	in	in	ADP
ejpam-5571	347	10	a	a	DET
ejpam-5571	347	11	metric	metric	ADJ
ejpam-5571	347	12	space	space	NOUN
ejpam-5571	347	13	with	with	ADP
ejpam-5571	347	14	curvature	curvature	NOUN
ejpam-5571	347	15	bounded	bound	VERB
ejpam-5571	347	16	below	below	ADV
ejpam-5571	347	17	by	by	ADP
ejpam-5571	347	18	k	k	PROPN
ejpam-5571	347	19	in	in	ADP
ejpam-5571	347	20	the	the	DET
ejpam-5571	347	21	large	large	ADJ
ejpam-5571	347	22	and	and	CCONJ
ejpam-5571	347	23	having	have	VERB
ejpam-5571	347	24	the	the	DET
ejpam-5571	347	25	same	same	ADJ
ejpam-5571	347	26	length	length	NOUN
ejpam-5571	347	27	as	as	ADP
ejpam-5571	347	28	a	a	DET
ejpam-5571	347	29	circle	circle	NOUN
ejpam-5571	347	30	in	in	ADP
ejpam-5571	347	31	the	the	DET
ejpam-5571	347	32	model	model	NOUN
ejpam-5571	347	33	space	space	NOUN
ejpam-5571	347	34	rk	rk	NOUN
ejpam-5571	347	35	in	in	ADP
ejpam-5571	347	36	the	the	DET
ejpam-5571	347	37	last	last	ADJ
ejpam-5571	347	38	theorem	theorem	NOUN
ejpam-5571	347	39	.	.	PUNCT
ejpam-5571	347	40	theorem	theorem	NOUN
ejpam-5571	347	41	8	8	NUM
ejpam-5571	347	42	.	.	PUNCT
ejpam-5571	348	1	let	let	VERB
ejpam-5571	348	2	x	x	PRON
ejpam-5571	348	3	be	be	AUX
ejpam-5571	348	4	a	a	DET
ejpam-5571	348	5	metric	metric	ADJ
ejpam-5571	348	6	space	space	NOUN
ejpam-5571	348	7	with	with	ADP
ejpam-5571	348	8	curvature	curvature	NOUN
ejpam-5571	348	9	bounded	bound	VERB
ejpam-5571	348	10	below	below	ADV
ejpam-5571	348	11	by	by	ADP
ejpam-5571	348	12	k	k	PROPN
ejpam-5571	348	13	in	in	ADP
ejpam-5571	348	14	the	the	DET
ejpam-5571	348	15	large	large	ADJ
ejpam-5571	348	16	,	,	PUNCT
ejpam-5571	348	17	and	and	CCONJ
ejpam-5571	348	18	γ	γ	X
ejpam-5571	348	19	be	be	AUX
ejpam-5571	348	20	a	a	DET
ejpam-5571	348	21	closed	closed	ADJ
ejpam-5571	348	22	spherical	spherical	ADJ
ejpam-5571	348	23	curve	curve	NOUN
ejpam-5571	348	24	at	at	ADP
ejpam-5571	348	25	a	a	DET
ejpam-5571	348	26	distance	distance	NOUN
ejpam-5571	349	1	r	r	NOUN
ejpam-5571	349	2	<	<	X
ejpam-5571	349	3	π	π	PROPN
ejpam-5571	349	4	2	2	NUM
ejpam-5571	349	5	√	√	PROPN
ejpam-5571	349	6	k	k	NOUN
ejpam-5571	349	7	from	from	ADP
ejpam-5571	349	8	a	a	DET
ejpam-5571	349	9	point	point	NOUN
ejpam-5571	349	10	p.	p.	NOUN
ejpam-5571	349	11	let	let	VERB
ejpam-5571	349	12	γ′	γ′	PRON
ejpam-5571	349	13	be	be	AUX
ejpam-5571	349	14	a	a	DET
ejpam-5571	349	15	circle	circle	NOUN
ejpam-5571	349	16	of	of	ADP
ejpam-5571	349	17	radius	radius	NOUN
ejpam-5571	349	18	r	r	NOUN
ejpam-5571	349	19	centered	center	VERB
ejpam-5571	349	20	at	at	ADP
ejpam-5571	349	21	a	a	DET
ejpam-5571	349	22	point	point	NOUN
ejpam-5571	349	23	p′	p′	NOUN
ejpam-5571	349	24	in	in	ADP
ejpam-5571	349	25	rk	rk	PROPN
ejpam-5571	349	26	.	.	PUNCT
ejpam-5571	350	1	suppose	suppose	VERB
ejpam-5571	350	2	that	that	SCONJ
ejpam-5571	350	3	the	the	DET
ejpam-5571	350	4	following	follow	VERB
ejpam-5571	350	5	statements	statement	NOUN
ejpam-5571	350	6	hold	hold	VERB
ejpam-5571	350	7	:	:	PUNCT
ejpam-5571	350	8	(	(	PUNCT
ejpam-5571	350	9	i	i	NOUN
ejpam-5571	350	10	)	)	PUNCT
ejpam-5571	350	11	ℓ(γ	ℓ(γ	PROPN
ejpam-5571	350	12	)	)	PUNCT
ejpam-5571	350	13	=	=	SYM
ejpam-5571	350	14	ℓ(γ′	ℓ(γ′	PROPN
ejpam-5571	350	15	)	)	PUNCT
ejpam-5571	350	16	;	;	PUNCT
ejpam-5571	351	1	(	(	PUNCT
ejpam-5571	351	2	ii	ii	NOUN
ejpam-5571	351	3	)	)	PUNCT
ejpam-5571	351	4	ℓ(γab	ℓ(γab	NOUN
ejpam-5571	351	5	)	)	PUNCT
ejpam-5571	351	6	=	=	SYM
ejpam-5571	351	7	ℓ(γ′a′b′	ℓ(γ′a′b′	PROPN
ejpam-5571	351	8	)	)	PUNCT
ejpam-5571	351	9	iff	iff	PROPN
ejpam-5571	351	10	d(a	d(a	PROPN
ejpam-5571	351	11	,	,	PUNCT
ejpam-5571	351	12	b	b	NOUN
ejpam-5571	351	13	)	)	PUNCT
ejpam-5571	351	14	=	=	VERB
ejpam-5571	352	1	d(a′	d(a′	NOUN
ejpam-5571	352	2	,	,	PUNCT
ejpam-5571	352	3	b′	b′	NUM
ejpam-5571	352	4	)	)	PUNCT
ejpam-5571	353	1	iff	iff	PROPN
ejpam-5571	353	2	∠p(a	∠p(a	ADJ
ejpam-5571	353	3	,	,	PUNCT
ejpam-5571	353	4	b	b	X
ejpam-5571	353	5	)	)	PUNCT
ejpam-5571	353	6	=	=	SYM
ejpam-5571	353	7	∠p′(a	∠p′(a	PROPN
ejpam-5571	353	8	′	′	PROPN
ejpam-5571	353	9	,	,	PUNCT
ejpam-5571	353	10	b′	b′	NUM
ejpam-5571	353	11	)	)	PUNCT
ejpam-5571	353	12	,	,	PUNCT
ejpam-5571	353	13	for	for	ADP
ejpam-5571	353	14	all	all	DET
ejpam-5571	353	15	a	a	PRON
ejpam-5571	353	16	,	,	PUNCT
ejpam-5571	353	17	b	b	X
ejpam-5571	353	18	∈	∈	PROPN
ejpam-5571	353	19	γ	γ	NOUN
ejpam-5571	353	20	and	and	CCONJ
ejpam-5571	353	21	a′	a′	PROPN
ejpam-5571	353	22	,	,	PUNCT
ejpam-5571	353	23	b′	b′	NUM
ejpam-5571	353	24	∈	∈	PROPN
ejpam-5571	353	25	γ′	γ′	NOUN
ejpam-5571	353	26	;	;	PUNCT
ejpam-5571	353	27	(	(	PUNCT
ejpam-5571	353	28	iii	iii	X
ejpam-5571	353	29	)	)	PUNCT
ejpam-5571	353	30	for	for	ADP
ejpam-5571	353	31	any	any	DET
ejpam-5571	353	32	triangle	triangle	NOUN
ejpam-5571	353	33	△	△	X
ejpam-5571	353	34	(	(	PUNCT
ejpam-5571	353	35	u	u	NOUN
ejpam-5571	353	36	,	,	PUNCT
ejpam-5571	353	37	v	v	NOUN
ejpam-5571	353	38	,	,	PUNCT
ejpam-5571	353	39	w	w	NOUN
ejpam-5571	353	40	)	)	PUNCT
ejpam-5571	353	41	in	in	ADP
ejpam-5571	353	42	x	x	NOUN
ejpam-5571	353	43	,	,	PUNCT
ejpam-5571	353	44	∠u(v	∠u(v	NOUN
ejpam-5571	353	45	,	,	PUNCT
ejpam-5571	353	46	x	x	NOUN
ejpam-5571	353	47	)	)	PUNCT
ejpam-5571	353	48	=	=	SYM
ejpam-5571	353	49	∠u(v	∠u(v	NOUN
ejpam-5571	353	50	,	,	PUNCT
ejpam-5571	353	51	w	w	NOUN
ejpam-5571	353	52	)	)	PUNCT
ejpam-5571	353	53	and	and	CCONJ
ejpam-5571	353	54	∠u(w	∠u(w	NUM
ejpam-5571	353	55	,	,	PUNCT
ejpam-5571	353	56	x	x	NOUN
ejpam-5571	353	57	)	)	PUNCT
ejpam-5571	353	58	=	=	SYM
ejpam-5571	353	59	∠u(w	∠u(w	PROPN
ejpam-5571	353	60	,	,	PUNCT
ejpam-5571	353	61	v	v	NOUN
ejpam-5571	353	62	)	)	PUNCT
ejpam-5571	353	63	for	for	ADP
ejpam-5571	353	64	all	all	DET
ejpam-5571	353	65	x	x	SYM
ejpam-5571	353	66	∈	∈	PROPN
ejpam-5571	354	1	[	[	X
ejpam-5571	354	2	v	v	NOUN
ejpam-5571	354	3	,	,	PUNCT
ejpam-5571	354	4	w	w	NOUN
ejpam-5571	354	5	]	]	X
ejpam-5571	354	6	;	;	PUNCT
ejpam-5571	354	7	(	(	PUNCT
ejpam-5571	354	8	iv	iv	X
ejpam-5571	354	9	)	)	PUNCT
ejpam-5571	354	10	for	for	ADP
ejpam-5571	354	11	any	any	DET
ejpam-5571	354	12	triangle	triangle	NOUN
ejpam-5571	354	13	△	△	X
ejpam-5571	354	14	(	(	PUNCT
ejpam-5571	354	15	u	u	NOUN
ejpam-5571	354	16	,	,	PUNCT
ejpam-5571	354	17	v	v	NOUN
ejpam-5571	354	18	,	,	PUNCT
ejpam-5571	354	19	w	w	NOUN
ejpam-5571	354	20	)	)	PUNCT
ejpam-5571	354	21	in	in	ADP
ejpam-5571	354	22	x	x	NOUN
ejpam-5571	354	23	,	,	PUNCT
ejpam-5571	354	24	∠u(v	∠u(v	NOUN
ejpam-5571	354	25	,	,	PUNCT
ejpam-5571	354	26	w	w	NOUN
ejpam-5571	354	27	)	)	PUNCT
ejpam-5571	354	28	=	=	NOUN
ejpam-5571	354	29	∠u(v	∠u(v	NOUN
ejpam-5571	354	30	,	,	PUNCT
ejpam-5571	354	31	x	x	NOUN
ejpam-5571	354	32	)	)	PUNCT
ejpam-5571	354	33	+	+	CCONJ
ejpam-5571	354	34	∠u(x	∠u(x	ADJ
ejpam-5571	354	35	,	,	PUNCT
ejpam-5571	354	36	w	w	NOUN
ejpam-5571	354	37	)	)	PUNCT
ejpam-5571	354	38	for	for	ADP
ejpam-5571	354	39	all	all	DET
ejpam-5571	354	40	x	x	SYM
ejpam-5571	354	41	∈	∈	PROPN
ejpam-5571	355	1	[	[	X
ejpam-5571	355	2	v	v	NOUN
ejpam-5571	355	3	,	,	PUNCT
ejpam-5571	355	4	w	w	NOUN
ejpam-5571	355	5	]	]	X
ejpam-5571	355	6	.	.	PUNCT
ejpam-5571	356	1	then	then	ADV
ejpam-5571	356	2	c(γ	c(γ	PROPN
ejpam-5571	356	3	)	)	PUNCT
ejpam-5571	356	4	is	be	AUX
ejpam-5571	356	5	isometric	isometric	ADJ
ejpam-5571	356	6	to	to	ADP
ejpam-5571	356	7	c(γ′	c(γ′	NOUN
ejpam-5571	356	8	)	)	PUNCT
ejpam-5571	356	9	,	,	PUNCT
ejpam-5571	356	10	that	that	ADV
ejpam-5571	356	11	is	is	ADV
ejpam-5571	356	12	,	,	PUNCT
ejpam-5571	356	13	the	the	DET
ejpam-5571	356	14	totally	totally	ADV
ejpam-5571	356	15	geodesic	geodesic	ADJ
ejpam-5571	356	16	surface	surface	NOUN
ejpam-5571	356	17	bounded	bound	VERB
ejpam-5571	356	18	by	by	ADP
ejpam-5571	356	19	γ	γ	NOUN
ejpam-5571	356	20	and	and	CCONJ
ejpam-5571	356	21	the	the	DET
ejpam-5571	356	22	disk	disk	NOUN
ejpam-5571	356	23	bounded	bound	VERB
ejpam-5571	356	24	by	by	ADP
ejpam-5571	356	25	γ′	γ′	PROPN
ejpam-5571	356	26	are	be	AUX
ejpam-5571	356	27	isometric	isometric	ADJ
ejpam-5571	356	28	to	to	ADP
ejpam-5571	356	29	each	each	DET
ejpam-5571	356	30	other	other	ADJ
ejpam-5571	356	31	.	.	PUNCT
ejpam-5571	357	1	proof	proof	NOUN
ejpam-5571	357	2	.	.	PUNCT
ejpam-5571	358	1	let	let	VERB
ejpam-5571	358	2	x	x	PRON
ejpam-5571	358	3	,	,	PUNCT
ejpam-5571	358	4	y	y	PROPN
ejpam-5571	358	5	∈	∈	PROPN
ejpam-5571	358	6	γ	γ	NOUN
ejpam-5571	358	7	and	and	CCONJ
ejpam-5571	358	8	x′	x′	NUM
ejpam-5571	358	9	,	,	PUNCT
ejpam-5571	358	10	y′	y′	NOUN
ejpam-5571	358	11	∈	∈	PROPN
ejpam-5571	358	12	γ′	γ′	PUNCT
ejpam-5571	358	13	be	be	AUX
ejpam-5571	358	14	such	such	ADJ
ejpam-5571	358	15	that	that	PRON
ejpam-5571	358	16	ℓ(γxy	ℓ(γxy	NOUN
ejpam-5571	358	17	)	)	PUNCT
ejpam-5571	358	18	=	=	SYM
ejpam-5571	358	19	ℓ(γ′x′y′	ℓ(γ′x′y′	PROPN
ejpam-5571	358	20	)	)	PUNCT
ejpam-5571	358	21	≤	≤	NOUN
ejpam-5571	358	22	π√	π√	PROPN
ejpam-5571	358	23	k	k	PROPN
ejpam-5571	358	24	.	.	PUNCT
ejpam-5571	359	1	we	we	PRON
ejpam-5571	359	2	can	can	AUX
ejpam-5571	359	3	conclude	conclude	VERB
ejpam-5571	359	4	from	from	ADP
ejpam-5571	359	5	(	(	PUNCT
ejpam-5571	359	6	ii	ii	NOUN
ejpam-5571	359	7	)	)	PUNCT
ejpam-5571	359	8	that	that	SCONJ
ejpam-5571	359	9	d(x	d(x	PROPN
ejpam-5571	359	10	,	,	PUNCT
ejpam-5571	359	11	y	y	NOUN
ejpam-5571	359	12	)	)	PUNCT
ejpam-5571	359	13	=	=	SYM
ejpam-5571	359	14	d(x′	d(x′	PROPN
ejpam-5571	359	15	,	,	PUNCT
ejpam-5571	359	16	y′	y′	NUM
ejpam-5571	359	17	)	)	PUNCT
ejpam-5571	359	18	.	.	PUNCT
ejpam-5571	360	1	we	we	PRON
ejpam-5571	360	2	establish	establish	VERB
ejpam-5571	360	3	a	a	DET
ejpam-5571	360	4	map	map	NOUN
ejpam-5571	360	5	j1	j1	NOUN
ejpam-5571	360	6	from	from	ADP
ejpam-5571	360	7	c({p′	c({p′	PROPN
ejpam-5571	360	8	}	}	PUNCT
ejpam-5571	360	9	∪	∪	ADP
ejpam-5571	360	10	γ′x′y′	γ′x′y′	NOUN
ejpam-5571	360	11	)	)	PUNCT
ejpam-5571	360	12	to	to	PART
ejpam-5571	360	13	c({p	c({p	VERB
ejpam-5571	360	14	}	}	PUNCT
ejpam-5571	360	15	∪	∪	X
ejpam-5571	360	16	γxy	γxy	NOUN
ejpam-5571	360	17	)	)	PUNCT
ejpam-5571	360	18	such	such	ADJ
ejpam-5571	360	19	that	that	SCONJ
ejpam-5571	360	20	each	each	DET
ejpam-5571	360	21	segment	segment	NOUN
ejpam-5571	360	22	[	[	X
ejpam-5571	360	23	p′	p′	NOUN
ejpam-5571	360	24	,	,	PUNCT
ejpam-5571	360	25	z′	z′	PROPN
ejpam-5571	360	26	]	]	PUNCT
ejpam-5571	360	27	from	from	ADP
ejpam-5571	360	28	p′	p′	NOUN
ejpam-5571	360	29	to	to	ADP
ejpam-5571	360	30	z′	z′	PROPN
ejpam-5571	360	31	on	on	ADP
ejpam-5571	360	32	γ′x′y′	γ′x′y′	PROPN
ejpam-5571	360	33	is	be	AUX
ejpam-5571	360	34	transferred	transfer	VERB
ejpam-5571	360	35	on	on	ADP
ejpam-5571	360	36	to	to	ADP
ejpam-5571	360	37	the	the	DET
ejpam-5571	360	38	geodesic	geodesic	ADJ
ejpam-5571	360	39	segment	segment	NOUN
ejpam-5571	360	40	[	[	X
ejpam-5571	360	41	p	p	X
ejpam-5571	360	42	,	,	PUNCT
ejpam-5571	360	43	z	z	X
ejpam-5571	360	44	]	]	X
ejpam-5571	360	45	from	from	ADP
ejpam-5571	360	46	p	p	PRON
ejpam-5571	360	47	to	to	ADP
ejpam-5571	360	48	a	a	DET
ejpam-5571	360	49	point	point	NOUN
ejpam-5571	360	50	z	z	NOUN
ejpam-5571	360	51	on	on	ADP
ejpam-5571	360	52	γxy	γxy	PROPN
ejpam-5571	360	53	,	,	PUNCT
ejpam-5571	360	54	where	where	SCONJ
ejpam-5571	360	55	z	z	NOUN
ejpam-5571	360	56	is	be	AUX
ejpam-5571	360	57	the	the	DET
ejpam-5571	360	58	point	point	NOUN
ejpam-5571	360	59	such	such	ADJ
ejpam-5571	360	60	that	that	SCONJ
ejpam-5571	360	61	ℓ(γxz	ℓ(γxz	X
ejpam-5571	360	62	)	)	PUNCT
ejpam-5571	360	63	=	=	SYM
ejpam-5571	360	64	ℓ(γ′x′z′	ℓ(γ′x′z′	PROPN
ejpam-5571	360	65	)	)	PUNCT
ejpam-5571	360	66	and	and	CCONJ
ejpam-5571	360	67	a	a	DET
ejpam-5571	360	68	map	map	NOUN
ejpam-5571	360	69	j2	j2	NOUN
ejpam-5571	360	70	is	be	AUX
ejpam-5571	360	71	defined	define	VERB
ejpam-5571	360	72	from	from	ADP
ejpam-5571	360	73	c({p′	c({p′	NOUN
ejpam-5571	360	74	}	}	PUNCT
ejpam-5571	360	75	∪	∪	NOUN
ejpam-5571	360	76	γ′y′x′	γ′y′x′	ADV
ejpam-5571	360	77	)	)	PUNCT
ejpam-5571	360	78	to	to	PART
ejpam-5571	360	79	c({p	c({p	VERB
ejpam-5571	360	80	}	}	PUNCT
ejpam-5571	360	81	∪	∪	ADP
ejpam-5571	360	82	γyx	γyx	NOUN
ejpam-5571	360	83	)	)	PUNCT
ejpam-5571	360	84	similar	similar	ADJ
ejpam-5571	360	85	to	to	ADP
ejpam-5571	360	86	j1	j1	PROPN
ejpam-5571	360	87	.	.	PUNCT
ejpam-5571	361	1	lemma	lemma	PROPN
ejpam-5571	361	2	3	3	NUM
ejpam-5571	361	3	indicates	indicate	VERB
ejpam-5571	361	4	that	that	SCONJ
ejpam-5571	361	5	j1	j1	PROPN
ejpam-5571	361	6	and	and	CCONJ
ejpam-5571	361	7	j2	j2	PROPN
ejpam-5571	361	8	are	be	AUX
ejpam-5571	361	9	isometries	isometry	NOUN
ejpam-5571	361	10	.	.	PUNCT
ejpam-5571	362	1	we	we	PRON
ejpam-5571	362	2	will	will	AUX
ejpam-5571	362	3	now	now	ADV
ejpam-5571	362	4	demonstrate	demonstrate	VERB
ejpam-5571	362	5	that	that	DET
ejpam-5571	362	6	c(γ′	c(γ′	NOUN
ejpam-5571	362	7	)	)	PUNCT
ejpam-5571	362	8	and	and	CCONJ
ejpam-5571	362	9	c(γ	c(γ	NOUN
ejpam-5571	362	10	)	)	PUNCT
ejpam-5571	362	11	are	be	AUX
ejpam-5571	362	12	isometric	isometric	ADJ
ejpam-5571	362	13	to	to	ADP
ejpam-5571	362	14	each	each	DET
ejpam-5571	362	15	other	other	ADJ
ejpam-5571	362	16	.	.	PUNCT
ejpam-5571	363	1	by	by	ADP
ejpam-5571	363	2	the	the	DET
ejpam-5571	363	3	definition	definition	NOUN
ejpam-5571	363	4	of	of	ADP
ejpam-5571	363	5	convex	convex	PROPN
ejpam-5571	363	6	hull	hull	NOUN
ejpam-5571	363	7	,	,	PUNCT
ejpam-5571	363	8	we	we	PRON
ejpam-5571	363	9	observe	observe	VERB
ejpam-5571	363	10	that	that	SCONJ
ejpam-5571	363	11	c(γ	c(γ	PROPN
ejpam-5571	363	12	)	)	PUNCT
ejpam-5571	363	13	exists	exist	VERB
ejpam-5571	363	14	and	and	CCONJ
ejpam-5571	363	15	is	be	AUX
ejpam-5571	363	16	unique	unique	ADJ
ejpam-5571	363	17	.	.	PUNCT
ejpam-5571	364	1	let	let	VERB
ejpam-5571	364	2	i	i	PRON
ejpam-5571	364	3	be	be	AUX
ejpam-5571	364	4	a	a	DET
ejpam-5571	364	5	map	map	NOUN
ejpam-5571	364	6	from	from	ADP
ejpam-5571	364	7	c(γ′	c(γ′	NOUN
ejpam-5571	364	8	)	)	PUNCT
ejpam-5571	365	1	=	=	PUNCT
ejpam-5571	365	2	c(γ′x′y′	c(γ′x′y′	PROPN
ejpam-5571	365	3	)	)	PUNCT
ejpam-5571	365	4	∪	∪	ADP
ejpam-5571	365	5	c(γ′y′x′	c(γ′y′x′	PROPN
ejpam-5571	365	6	)	)	PUNCT
ejpam-5571	365	7	to	to	ADP
ejpam-5571	365	8	c(γ	c(γ	PROPN
ejpam-5571	365	9	)	)	PUNCT
ejpam-5571	365	10	in	in	ADP
ejpam-5571	365	11	such	such	DET
ejpam-5571	365	12	a	a	DET
ejpam-5571	365	13	way	way	NOUN
ejpam-5571	365	14	that	that	PRON
ejpam-5571	365	15	the	the	DET
ejpam-5571	365	16	function	function	NOUN
ejpam-5571	365	17	i	i	PRON
ejpam-5571	365	18	on	on	ADP
ejpam-5571	365	19	c(γ′x′y′	c(γ′x′y′	PROPN
ejpam-5571	365	20	)	)	PUNCT
ejpam-5571	365	21	is	be	AUX
ejpam-5571	365	22	j1	j1	PROPN
ejpam-5571	365	23	and	and	CCONJ
ejpam-5571	365	24	on	on	ADP
ejpam-5571	365	25	c(γ′y′x′	c(γ′y′x′	PROPN
ejpam-5571	365	26	)	)	PUNCT
ejpam-5571	365	27	is	be	AUX
ejpam-5571	365	28	j2	j2	PROPN
ejpam-5571	365	29	.	.	PUNCT
ejpam-5571	366	1	we	we	PRON
ejpam-5571	366	2	must	must	AUX
ejpam-5571	366	3	demonstrate	demonstrate	VERB
ejpam-5571	366	4	that	that	SCONJ
ejpam-5571	366	5	i	i	PRON
ejpam-5571	366	6	is	be	AUX
ejpam-5571	366	7	an	an	DET
ejpam-5571	366	8	isometry	isometry	NOUN
ejpam-5571	366	9	from	from	ADP
ejpam-5571	366	10	c(γ′	c(γ′	NOUN
ejpam-5571	366	11	)	)	PUNCT
ejpam-5571	366	12	to	to	ADP
ejpam-5571	366	13	c(γ	c(γ	PROPN
ejpam-5571	366	14	)	)	PUNCT
ejpam-5571	366	15	,	,	PUNCT
ejpam-5571	366	16	we	we	PRON
ejpam-5571	366	17	must	must	AUX
ejpam-5571	366	18	show	show	VERB
ejpam-5571	366	19	that	that	SCONJ
ejpam-5571	366	20	i	i	PRON
ejpam-5571	366	21	is	be	AUX
ejpam-5571	366	22	an	an	DET
ejpam-5571	366	23	isometry	isometry	NOUN
ejpam-5571	366	24	onto	onto	ADP
ejpam-5571	366	25	its	its	PRON
ejpam-5571	366	26	image	image	NOUN
ejpam-5571	366	27	and	and	CCONJ
ejpam-5571	366	28	c(γ	c(γ	NOUN
ejpam-5571	366	29	)	)	PUNCT
ejpam-5571	366	30	=	=	SYM
ejpam-5571	366	31	c(γxy	c(γxy	PROPN
ejpam-5571	366	32	)	)	PUNCT
ejpam-5571	366	33	∪	∪	ADP
ejpam-5571	366	34	c(γyx	c(γyx	NOUN
ejpam-5571	366	35	)	)	PUNCT
ejpam-5571	366	36	=	=	PUNCT
ejpam-5571	366	37	c(γxy	c(γxy	NOUN
ejpam-5571	366	38	∪	∪	ADJ
ejpam-5571	366	39	γyx	γyx	NOUN
ejpam-5571	366	40	)	)	PUNCT
ejpam-5571	366	41	.	.	PUNCT
ejpam-5571	367	1	it	it	PRON
ejpam-5571	367	2	is	be	AUX
ejpam-5571	367	3	clear	clear	ADJ
ejpam-5571	367	4	that	that	SCONJ
ejpam-5571	367	5	i	i	PRON
ejpam-5571	367	6	is	be	AUX
ejpam-5571	367	7	surjective	surjective	ADJ
ejpam-5571	367	8	.	.	PUNCT
ejpam-5571	368	1	additionally	additionally	ADV
ejpam-5571	368	2	,	,	PUNCT
ejpam-5571	368	3	as	as	SCONJ
ejpam-5571	368	4	we	we	PRON
ejpam-5571	368	5	shown	show	VERB
ejpam-5571	368	6	in	in	ADP
ejpam-5571	368	7	lemma	lemma	PROPN
ejpam-5571	368	8	3	3	NUM
ejpam-5571	368	9	,	,	PUNCT
ejpam-5571	368	10	i	i	PRON
ejpam-5571	368	11	is	be	AUX
ejpam-5571	368	12	injective	injective	ADJ
ejpam-5571	368	13	as	as	ADP
ejpam-5571	368	14	a	a	DET
ejpam-5571	368	15	result	result	NOUN
ejpam-5571	368	16	of	of	ADP
ejpam-5571	368	17	the	the	DET
ejpam-5571	368	18	requirements	requirement	NOUN
ejpam-5571	368	19	of	of	ADP
ejpam-5571	368	20	intersecting	intersecting	ADJ
ejpam-5571	368	21	geodesics	geodesic	NOUN
ejpam-5571	368	22	and	and	CCONJ
ejpam-5571	368	23	isometric	isometric	ADJ
ejpam-5571	368	24	convex	convex	NOUN
ejpam-5571	368	25	hulls	hull	NOUN
ejpam-5571	368	26	.	.	PUNCT
ejpam-5571	369	1	let	let	VERB
ejpam-5571	369	2	u′1	u′1	NOUN
ejpam-5571	369	3	and	and	CCONJ
ejpam-5571	369	4	u′2	u′2	PROPN
ejpam-5571	369	5	be	be	AUX
ejpam-5571	369	6	in	in	ADP
ejpam-5571	369	7	c(γ′	c(γ′	NOUN
ejpam-5571	369	8	)	)	PUNCT
ejpam-5571	369	9	and	and	CCONJ
ejpam-5571	369	10	u1	u1	NOUN
ejpam-5571	369	11	=	=	SYM
ejpam-5571	369	12	i(u′1	i(u′1	PART
ejpam-5571	369	13	)	)	PUNCT
ejpam-5571	369	14	and	and	CCONJ
ejpam-5571	369	15	u2	u2	PROPN
ejpam-5571	369	16	=	=	PUNCT
ejpam-5571	369	17	i(u′2	i(u′2	PROPN
ejpam-5571	369	18	)	)	PUNCT
ejpam-5571	369	19	.	.	PUNCT
ejpam-5571	370	1	we	we	PRON
ejpam-5571	370	2	will	will	AUX
ejpam-5571	370	3	demonstrate	demonstrate	VERB
ejpam-5571	370	4	that	that	DET
ejpam-5571	370	5	d(u1	d(u1	NOUN
ejpam-5571	370	6	,	,	PUNCT
ejpam-5571	370	7	u2	u2	NOUN
ejpam-5571	370	8	)	)	PUNCT
ejpam-5571	370	9	=	=	SYM
ejpam-5571	370	10	d(u′1	d(u′1	NOUN
ejpam-5571	370	11	,	,	PUNCT
ejpam-5571	370	12	u	u	NOUN
ejpam-5571	370	13	′	′	NOUN
ejpam-5571	370	14	2	2	NUM
ejpam-5571	370	15	)	)	PUNCT
ejpam-5571	370	16	.	.	PUNCT
ejpam-5571	371	1	if	if	SCONJ
ejpam-5571	371	2	u′1	u′1	NOUN
ejpam-5571	371	3	,	,	PUNCT
ejpam-5571	371	4	u	u	NOUN
ejpam-5571	371	5	′	′	NOUN
ejpam-5571	371	6	2	2	NUM
ejpam-5571	371	7	∈	∈	PROPN
ejpam-5571	371	8	c(γ′x′y′	c(γ′x′y′	NOUN
ejpam-5571	371	9	)	)	PUNCT
ejpam-5571	371	10	or	or	CCONJ
ejpam-5571	371	11	u′1	u′1	NOUN
ejpam-5571	371	12	,	,	PUNCT
ejpam-5571	371	13	u	u	NOUN
ejpam-5571	371	14	′	′	NOUN
ejpam-5571	371	15	2	2	NUM
ejpam-5571	371	16	∈	∈	PROPN
ejpam-5571	371	17	c(γ′y′x′	c(γ′y′x′	PROPN
ejpam-5571	371	18	)	)	PUNCT
ejpam-5571	371	19	,	,	PUNCT
ejpam-5571	371	20	neither	neither	DET
ejpam-5571	371	21	case	case	NOUN
ejpam-5571	371	22	can	can	AUX
ejpam-5571	371	23	be	be	AUX
ejpam-5571	371	24	proven	prove	VERB
ejpam-5571	371	25	.	.	PUNCT
ejpam-5571	372	1	we	we	PRON
ejpam-5571	372	2	assume	assume	VERB
ejpam-5571	372	3	that	that	SCONJ
ejpam-5571	372	4	c.	c.	PROPN
ejpam-5571	372	5	phokaew	phokaew	PROPN
ejpam-5571	372	6	,	,	PUNCT
ejpam-5571	372	7	a.	a.	PROPN
ejpam-5571	372	8	sama	sama	PROPN
ejpam-5571	372	9	-	-	PUNCT
ejpam-5571	372	10	ae	ae	PROPN
ejpam-5571	372	11	,	,	PUNCT
ejpam-5571	372	12	/	/	SYM
ejpam-5571	372	13	eur	eur	NOUN
ejpam-5571	372	14	.	.	PUNCT
ejpam-5571	373	1	j.	j.	PROPN
ejpam-5571	373	2	pure	pure	PROPN
ejpam-5571	373	3	appl	appl	PROPN
ejpam-5571	373	4	.	.	PROPN
ejpam-5571	373	5	math	math	PROPN
ejpam-5571	373	6	,	,	PUNCT
ejpam-5571	373	7	17	17	NUM
ejpam-5571	373	8	(	(	PUNCT
ejpam-5571	373	9	4	4	NUM
ejpam-5571	373	10	)	)	PUNCT
ejpam-5571	373	11	(	(	PUNCT
ejpam-5571	373	12	2024	2024	NUM
ejpam-5571	373	13	)	)	PUNCT
ejpam-5571	373	14	,	,	PUNCT
ejpam-5571	373	15	3932	3932	NUM
ejpam-5571	373	16	-	-	SYM
ejpam-5571	373	17	3944	3944	NUM
ejpam-5571	373	18	3943	3943	NUM
ejpam-5571	373	19	u′1	u′1	NOUN
ejpam-5571	373	20	∈	∈	PROPN
ejpam-5571	373	21	c(γ′x′y′	c(γ′x′y′	PROPN
ejpam-5571	373	22	)	)	PUNCT
ejpam-5571	373	23	and	and	CCONJ
ejpam-5571	373	24	u′2	u′2	PRON
ejpam-5571	373	25	∈	∈	PROPN
ejpam-5571	373	26	c(γ′y′x′	c(γ′y′x′	PROPN
ejpam-5571	373	27	)	)	PUNCT
ejpam-5571	373	28	.	.	PUNCT
ejpam-5571	374	1	let	let	VERB
ejpam-5571	374	2	u′1	u′1	VERB
ejpam-5571	374	3	∈	∈	PROPN
ejpam-5571	374	4	[	[	X
ejpam-5571	374	5	p′	p′	NOUN
ejpam-5571	374	6	,	,	PUNCT
ejpam-5571	374	7	v′1	v′1	VERB
ejpam-5571	374	8	]	]	PUNCT
ejpam-5571	374	9	and	and	CCONJ
ejpam-5571	374	10	u′2	u′2	PRON
ejpam-5571	374	11	∈	∈	PROPN
ejpam-5571	374	12	[	[	X
ejpam-5571	374	13	p′	p′	NOUN
ejpam-5571	374	14	,	,	PUNCT
ejpam-5571	374	15	v′2	v′2	X
ejpam-5571	374	16	]	]	X
ejpam-5571	374	17	for	for	ADP
ejpam-5571	374	18	some	some	DET
ejpam-5571	374	19	v′1	v′1	NOUN
ejpam-5571	374	20	∈	∈	PROPN
ejpam-5571	374	21	c(γ′x′y′	c(γ′x′y′	PROPN
ejpam-5571	374	22	)	)	PUNCT
ejpam-5571	374	23	and	and	CCONJ
ejpam-5571	374	24	v′2	v′2	NOUN
ejpam-5571	374	25	∈	∈	PROPN
ejpam-5571	374	26	c(γ′y′x′	c(γ′y′x′	PROPN
ejpam-5571	374	27	)	)	PUNCT
ejpam-5571	374	28	.	.	PUNCT
ejpam-5571	375	1	on	on	ADP
ejpam-5571	375	2	x	x	SYM
ejpam-5571	375	3	,	,	PUNCT
ejpam-5571	375	4	we	we	PRON
ejpam-5571	375	5	let	let	VERB
ejpam-5571	375	6	[	[	X
ejpam-5571	375	7	q	q	X
ejpam-5571	375	8	,	,	PUNCT
ejpam-5571	375	9	v1	v1	NOUN
ejpam-5571	375	10	]	]	PUNCT
ejpam-5571	375	11	be	be	VERB
ejpam-5571	375	12	the	the	DET
ejpam-5571	375	13	geodesic	geodesic	ADJ
ejpam-5571	375	14	segment	segment	NOUN
ejpam-5571	375	15	containing	contain	VERB
ejpam-5571	375	16	u1	u1	NOUN
ejpam-5571	375	17	and	and	CCONJ
ejpam-5571	375	18	let	let	VERB
ejpam-5571	375	19	[	[	X
ejpam-5571	375	20	q	q	X
ejpam-5571	375	21	,	,	PUNCT
ejpam-5571	375	22	v2	v2	PROPN
ejpam-5571	375	23	]	]	PUNCT
ejpam-5571	375	24	be	be	VERB
ejpam-5571	375	25	the	the	DET
ejpam-5571	375	26	geodesic	geodesic	ADJ
ejpam-5571	375	27	segment	segment	NOUN
ejpam-5571	375	28	containing	contain	VERB
ejpam-5571	375	29	u2	u2	NOUN
ejpam-5571	375	30	where	where	SCONJ
ejpam-5571	375	31	v1	v1	PROPN
ejpam-5571	375	32	∈	∈	PROPN
ejpam-5571	375	33	c(γxy	c(γxy	NOUN
ejpam-5571	375	34	)	)	PUNCT
ejpam-5571	375	35	and	and	CCONJ
ejpam-5571	375	36	v2	v2	PROPN
ejpam-5571	375	37	∈	∈	PROPN
ejpam-5571	375	38	c(γyx	c(γyx	NOUN
ejpam-5571	375	39	)	)	PUNCT
ejpam-5571	375	40	.	.	PUNCT
ejpam-5571	376	1	if	if	SCONJ
ejpam-5571	376	2	ℓ(γ′v′1v′2	ℓ(γ′v′1v′2	PROPN
ejpam-5571	376	3	)	)	PUNCT
ejpam-5571	376	4	≤	≤	PUNCT
ejpam-5571	377	1	ℓ(γ′)/2	ℓ(γ′)/2	NOUN
ejpam-5571	377	2	,	,	PUNCT
ejpam-5571	377	3	we	we	PRON
ejpam-5571	377	4	then	then	ADV
ejpam-5571	377	5	have	have	VERB
ejpam-5571	377	6	γ′v′1v′2	γ′v′1v′2	NOUN
ejpam-5571	377	7	=	=	SYM
ejpam-5571	377	8	γ′v′1y′	γ′v′1y′	PROPN
ejpam-5571	377	9	∪	∪	VERB
ejpam-5571	377	10	γ′y′v′2	γ′y′v′2	NOUN
ejpam-5571	377	11	.	.	PUNCT
ejpam-5571	378	1	by	by	ADP
ejpam-5571	378	2	(	(	PUNCT
ejpam-5571	378	3	ii	ii	NOUN
ejpam-5571	378	4	)	)	PUNCT
ejpam-5571	378	5	,	,	PUNCT
ejpam-5571	378	6	c(γ′v′1y′	c(γ′v′1y′	PROPN
ejpam-5571	378	7	)	)	PUNCT
ejpam-5571	378	8	is	be	AUX
ejpam-5571	378	9	isometric	isometric	ADJ
ejpam-5571	378	10	to	to	ADP
ejpam-5571	378	11	c(γv1y	c(γv1y	NOUN
ejpam-5571	378	12	)	)	PUNCT
ejpam-5571	378	13	by	by	ADP
ejpam-5571	378	14	j1	j1	PROPN
ejpam-5571	378	15	and	and	CCONJ
ejpam-5571	378	16	c(γ′v′2y′	c(γ′v′2y′	NOUN
ejpam-5571	378	17	)	)	PUNCT
ejpam-5571	378	18	is	be	AUX
ejpam-5571	378	19	isometric	isometric	ADJ
ejpam-5571	378	20	to	to	ADP
ejpam-5571	378	21	c(γv2y	c(γv2y	NUM
ejpam-5571	378	22	)	)	PUNCT
ejpam-5571	378	23	by	by	ADP
ejpam-5571	378	24	j2	j2	PROPN
ejpam-5571	378	25	,	,	PUNCT
ejpam-5571	378	26	we	we	PRON
ejpam-5571	378	27	thus	thus	ADV
ejpam-5571	378	28	get	get	VERB
ejpam-5571	378	29	that	that	DET
ejpam-5571	378	30	c(γ′v′1v′2	c(γ′v′1v′2	PROPN
ejpam-5571	378	31	)	)	PUNCT
ejpam-5571	378	32	is	be	AUX
ejpam-5571	378	33	isometric	isometric	ADJ
ejpam-5571	378	34	to	to	ADP
ejpam-5571	378	35	c(γv1v2	c(γv1v2	NOUN
ejpam-5571	378	36	)	)	PUNCT
ejpam-5571	378	37	by	by	ADP
ejpam-5571	378	38	i.	i.	PROPN
ejpam-5571	378	39	consequently	consequently	ADV
ejpam-5571	378	40	,	,	PUNCT
ejpam-5571	378	41	we	we	PRON
ejpam-5571	378	42	get	get	VERB
ejpam-5571	378	43	d(u1	d(u1	NOUN
ejpam-5571	378	44	,	,	PUNCT
ejpam-5571	378	45	u2	u2	NOUN
ejpam-5571	378	46	)	)	PUNCT
ejpam-5571	378	47	=	=	SYM
ejpam-5571	379	1	d(u′1	d(u′1	NOUN
ejpam-5571	379	2	,	,	PUNCT
ejpam-5571	379	3	u	u	NOUN
ejpam-5571	379	4	′	′	NOUN
ejpam-5571	379	5	2	2	NUM
ejpam-5571	379	6	)	)	PUNCT
ejpam-5571	379	7	.	.	PUNCT
ejpam-5571	380	1	additionally	additionally	ADV
ejpam-5571	380	2	,	,	PUNCT
ejpam-5571	380	3	we	we	PRON
ejpam-5571	380	4	also	also	ADV
ejpam-5571	380	5	have	have	VERB
ejpam-5571	380	6	d(u1	d(u1	NOUN
ejpam-5571	380	7	,	,	PUNCT
ejpam-5571	380	8	u2	u2	NOUN
ejpam-5571	380	9	)	)	PUNCT
ejpam-5571	380	10	=	=	SYM
ejpam-5571	381	1	d(u′1	d(u′1	NOUN
ejpam-5571	381	2	,	,	PUNCT
ejpam-5571	381	3	u	u	NOUN
ejpam-5571	381	4	′	′	NOUN
ejpam-5571	381	5	2	2	NUM
ejpam-5571	381	6	)	)	PUNCT
ejpam-5571	381	7	if	if	SCONJ
ejpam-5571	381	8	ℓ(γ′v′2v′1	ℓ(γ′v′2v′1	NOUN
ejpam-5571	381	9	)	)	PUNCT
ejpam-5571	381	10	≤	≤	PROPN
ejpam-5571	381	11	ℓ(γ′)/2	ℓ(γ′)/2	NOUN
ejpam-5571	381	12	.	.	PUNCT
ejpam-5571	382	1	we	we	PRON
ejpam-5571	382	2	will	will	AUX
ejpam-5571	382	3	now	now	ADV
ejpam-5571	382	4	demonstrate	demonstrate	VERB
ejpam-5571	382	5	that	that	SCONJ
ejpam-5571	382	6	c(γ	c(γ	PROPN
ejpam-5571	382	7	)	)	PUNCT
ejpam-5571	382	8	=	=	SYM
ejpam-5571	382	9	c(γxy	c(γxy	PROPN
ejpam-5571	382	10	)	)	PUNCT
ejpam-5571	382	11	∪	∪	ADP
ejpam-5571	382	12	c(γyx	c(γyx	NOUN
ejpam-5571	382	13	)	)	PUNCT
ejpam-5571	382	14	=	=	PUNCT
ejpam-5571	382	15	c(γxy	c(γxy	NOUN
ejpam-5571	382	16	∪	∪	ADJ
ejpam-5571	382	17	γyx	γyx	NOUN
ejpam-5571	382	18	)	)	PUNCT
ejpam-5571	382	19	.	.	PUNCT
ejpam-5571	383	1	it	it	PRON
ejpam-5571	383	2	is	be	AUX
ejpam-5571	383	3	necessary	necessary	ADJ
ejpam-5571	383	4	to	to	PART
ejpam-5571	383	5	demonstrate	demonstrate	VERB
ejpam-5571	383	6	that	that	SCONJ
ejpam-5571	383	7	the	the	DET
ejpam-5571	383	8	set	set	NOUN
ejpam-5571	383	9	c(γxy	c(γxy	PROPN
ejpam-5571	383	10	∪	∪	NOUN
ejpam-5571	383	11	γyx	γyx	NOUN
ejpam-5571	383	12	)	)	PUNCT
ejpam-5571	383	13	is	be	AUX
ejpam-5571	383	14	convex	convex	ADJ
ejpam-5571	383	15	.	.	PUNCT
ejpam-5571	384	1	without	without	ADP
ejpam-5571	384	2	losing	lose	VERB
ejpam-5571	384	3	generality	generality	NOUN
ejpam-5571	384	4	,	,	PUNCT
ejpam-5571	384	5	we	we	PRON
ejpam-5571	384	6	suppose	suppose	VERB
ejpam-5571	384	7	that	that	SCONJ
ejpam-5571	384	8	x1	x1	PROPN
ejpam-5571	384	9	is	be	AUX
ejpam-5571	384	10	in	in	ADP
ejpam-5571	384	11	c(γxy	c(γxy	PROPN
ejpam-5571	384	12	)	)	PUNCT
ejpam-5571	384	13	and	and	CCONJ
ejpam-5571	384	14	x2	x2	PROPN
ejpam-5571	384	15	is	be	AUX
ejpam-5571	384	16	in	in	ADP
ejpam-5571	384	17	c(γyx	c(γyx	NOUN
ejpam-5571	384	18	)	)	PUNCT
ejpam-5571	384	19	.	.	PUNCT
ejpam-5571	385	1	let	let	VERB
ejpam-5571	385	2	[	[	X
ejpam-5571	385	3	q	q	X
ejpam-5571	385	4	,	,	PUNCT
ejpam-5571	385	5	w1	w1	NOUN
ejpam-5571	385	6	]	]	PUNCT
ejpam-5571	385	7	and	and	CCONJ
ejpam-5571	385	8	[	[	X
ejpam-5571	385	9	q	q	X
ejpam-5571	385	10	,	,	PUNCT
ejpam-5571	385	11	w2	w2	NOUN
ejpam-5571	385	12	]	]	PUNCT
ejpam-5571	385	13	be	be	VERB
ejpam-5571	385	14	the	the	DET
ejpam-5571	385	15	segments	segment	NOUN
ejpam-5571	385	16	containing	contain	VERB
ejpam-5571	385	17	x1	x1	PROPN
ejpam-5571	385	18	and	and	CCONJ
ejpam-5571	385	19	x2	x2	PROPN
ejpam-5571	385	20	,	,	PUNCT
ejpam-5571	385	21	respectively	respectively	ADV
ejpam-5571	385	22	,	,	PUNCT
ejpam-5571	385	23	where	where	SCONJ
ejpam-5571	385	24	[	[	X
ejpam-5571	385	25	q	q	X
ejpam-5571	385	26	,	,	PUNCT
ejpam-5571	385	27	w1	w1	NOUN
ejpam-5571	385	28	]	]	PUNCT
ejpam-5571	385	29	is	be	AUX
ejpam-5571	385	30	the	the	DET
ejpam-5571	385	31	segment	segment	NOUN
ejpam-5571	385	32	containing	contain	VERB
ejpam-5571	385	33	x1	x1	PROPN
ejpam-5571	385	34	and	and	CCONJ
ejpam-5571	385	35	[	[	X
ejpam-5571	385	36	q	q	X
ejpam-5571	385	37	,	,	PUNCT
ejpam-5571	385	38	w2	w2	NOUN
ejpam-5571	385	39	]	]	PUNCT
ejpam-5571	385	40	is	be	AUX
ejpam-5571	385	41	the	the	DET
ejpam-5571	385	42	segment	segment	NOUN
ejpam-5571	385	43	containing	contain	VERB
ejpam-5571	385	44	x2	x2	PROPN
ejpam-5571	385	45	.	.	PUNCT
ejpam-5571	386	1	since	since	SCONJ
ejpam-5571	386	2	j1	j1	PROPN
ejpam-5571	386	3	is	be	AUX
ejpam-5571	386	4	the	the	DET
ejpam-5571	386	5	isometry	isometry	NOUN
ejpam-5571	386	6	from	from	ADP
ejpam-5571	386	7	c(γ′x′y′	c(γ′x′y′	PROPN
ejpam-5571	386	8	)	)	PUNCT
ejpam-5571	386	9	to	to	PART
ejpam-5571	386	10	c(γxy	c(γxy	VERB
ejpam-5571	386	11	)	)	PUNCT
ejpam-5571	386	12	and	and	CCONJ
ejpam-5571	386	13	j2	j2	PROPN
ejpam-5571	386	14	is	be	AUX
ejpam-5571	386	15	the	the	DET
ejpam-5571	386	16	isometry	isometry	NOUN
ejpam-5571	386	17	from	from	ADP
ejpam-5571	386	18	c(γ′y′x′	c(γ′y′x′	PROPN
ejpam-5571	386	19	)	)	PUNCT
ejpam-5571	386	20	to	to	PART
ejpam-5571	386	21	c(γyx	c(γyx	VERB
ejpam-5571	386	22	)	)	PUNCT
ejpam-5571	386	23	,	,	PUNCT
ejpam-5571	386	24	we	we	PRON
ejpam-5571	386	25	let	let	VERB
ejpam-5571	386	26	two	two	NUM
ejpam-5571	386	27	points	point	NOUN
ejpam-5571	386	28	w′	w′	PROPN
ejpam-5571	386	29	1	1	NUM
ejpam-5571	386	30	and	and	CCONJ
ejpam-5571	386	31	w′	w′	PROPN
ejpam-5571	386	32	2	2	NUM
ejpam-5571	386	33	in	in	ADP
ejpam-5571	386	34	rk	rk	NOUN
ejpam-5571	386	35	be	be	AUX
ejpam-5571	386	36	the	the	DET
ejpam-5571	386	37	points	point	NOUN
ejpam-5571	386	38	corresponding	correspond	VERB
ejpam-5571	386	39	to	to	ADP
ejpam-5571	386	40	w1	w1	NOUN
ejpam-5571	386	41	and	and	CCONJ
ejpam-5571	386	42	w2	w2	NOUN
ejpam-5571	386	43	,	,	PUNCT
ejpam-5571	386	44	respectively	respectively	ADV
ejpam-5571	386	45	,	,	PUNCT
ejpam-5571	386	46	and	and	CCONJ
ejpam-5571	386	47	let	let	VERB
ejpam-5571	386	48	two	two	NUM
ejpam-5571	386	49	points	point	NOUN
ejpam-5571	386	50	x′1	x′1	PROPN
ejpam-5571	387	1	and	and	CCONJ
ejpam-5571	387	2	x′2	x′2	NOUN
ejpam-5571	387	3	in	in	ADP
ejpam-5571	387	4	rk	rk	NOUN
ejpam-5571	387	5	be	be	AUX
ejpam-5571	387	6	the	the	DET
ejpam-5571	387	7	points	point	NOUN
ejpam-5571	387	8	corresponding	correspond	VERB
ejpam-5571	387	9	to	to	ADP
ejpam-5571	387	10	x1	x1	PROPN
ejpam-5571	387	11	and	and	CCONJ
ejpam-5571	387	12	x2	x2	PROPN
ejpam-5571	387	13	,	,	PUNCT
ejpam-5571	387	14	respectively	respectively	ADV
ejpam-5571	387	15	.	.	PUNCT
ejpam-5571	388	1	if	if	SCONJ
ejpam-5571	388	2	ℓ(γ′w′	ℓ(γ′w′	NOUN
ejpam-5571	388	3	1w	1w	VERB
ejpam-5571	388	4	′	′	NUM
ejpam-5571	388	5	2	2	NUM
ejpam-5571	388	6	)	)	PUNCT
ejpam-5571	388	7	≤	≤	NOUN
ejpam-5571	388	8	ℓ(γ′)/2	ℓ(γ′)/2	NOUN
ejpam-5571	388	9	,	,	PUNCT
ejpam-5571	388	10	then	then	ADV
ejpam-5571	388	11	γ′w′	γ′w′	ADP
ejpam-5571	388	12	1w	1w	NUM
ejpam-5571	388	13	′	′	NUM
ejpam-5571	388	14	2	2	NUM
ejpam-5571	388	15	=	=	SYM
ejpam-5571	388	16	γ′w′	γ′w′	ADP
ejpam-5571	388	17	1y	1y	NUM
ejpam-5571	388	18	′	′	NOUN
ejpam-5571	388	19	∪	∪	ADJ
ejpam-5571	388	20	γ′y′w′	γ′y′w′	VERB
ejpam-5571	388	21	2	2	NUM
ejpam-5571	388	22	is	be	AUX
ejpam-5571	388	23	the	the	DET
ejpam-5571	388	24	result	result	NOUN
ejpam-5571	388	25	.	.	PUNCT
ejpam-5571	389	1	as	as	SCONJ
ejpam-5571	389	2	c(γ′w′	c(γ′w′	NOUN
ejpam-5571	389	3	1y	1y	PROPN
ejpam-5571	389	4	′	′	NOUN
ejpam-5571	389	5	)	)	PUNCT
ejpam-5571	389	6	is	be	AUX
ejpam-5571	389	7	isometric	isometric	ADJ
ejpam-5571	389	8	to	to	ADP
ejpam-5571	389	9	c(γw1y	c(γw1y	NUM
ejpam-5571	389	10	)	)	PUNCT
ejpam-5571	389	11	by	by	ADP
ejpam-5571	389	12	j1	j1	PROPN
ejpam-5571	389	13	and	and	CCONJ
ejpam-5571	389	14	c(γ′w′	c(γ′w′	VERB
ejpam-5571	389	15	2y	2y	PROPN
ejpam-5571	389	16	′	′	NOUN
ejpam-5571	389	17	)	)	PUNCT
ejpam-5571	389	18	is	be	AUX
ejpam-5571	389	19	isometric	isometric	ADJ
ejpam-5571	389	20	to	to	ADP
ejpam-5571	389	21	c(γw2y	c(γw2y	NOUN
ejpam-5571	389	22	)	)	PUNCT
ejpam-5571	389	23	by	by	ADP
ejpam-5571	389	24	j2	j2	PROPN
ejpam-5571	389	25	,	,	PUNCT
ejpam-5571	389	26	we	we	PRON
ejpam-5571	389	27	thus	thus	ADV
ejpam-5571	389	28	obtain	obtain	VERB
ejpam-5571	389	29	that	that	SCONJ
ejpam-5571	389	30	c(γ′w′	c(γ′w′	NOUN
ejpam-5571	389	31	1w	1w	NUM
ejpam-5571	389	32	′	′	NUM
ejpam-5571	389	33	2	2	NUM
ejpam-5571	389	34	)	)	PUNCT
ejpam-5571	389	35	is	be	AUX
ejpam-5571	389	36	isometric	isometric	ADJ
ejpam-5571	389	37	to	to	PART
ejpam-5571	389	38	c(γw1w2	c(γw1w2	VERB
ejpam-5571	389	39	)	)	PUNCT
ejpam-5571	389	40	by	by	ADP
ejpam-5571	389	41	i.	i.	PROPN
ejpam-5571	389	42	consequently	consequently	ADV
ejpam-5571	389	43	,	,	PUNCT
ejpam-5571	389	44	d(x1	d(x1	NOUN
ejpam-5571	389	45	,	,	PUNCT
ejpam-5571	389	46	x2	x2	PROPN
ejpam-5571	389	47	)	)	PUNCT
ejpam-5571	389	48	=	=	SYM
ejpam-5571	389	49	d(x′1	d(x′1	PROPN
ejpam-5571	389	50	,	,	PUNCT
ejpam-5571	389	51	x	x	SYM
ejpam-5571	389	52	′	′	NOUN
ejpam-5571	389	53	2	2	NUM
ejpam-5571	389	54	)	)	PUNCT
ejpam-5571	389	55	is	be	AUX
ejpam-5571	389	56	what	what	PRON
ejpam-5571	389	57	we	we	PRON
ejpam-5571	389	58	have	have	VERB
ejpam-5571	389	59	.	.	PUNCT
ejpam-5571	390	1	let	let	VERB
ejpam-5571	390	2	x′′	x′′	PROPN
ejpam-5571	390	3	be	be	AUX
ejpam-5571	390	4	the	the	DET
ejpam-5571	390	5	point	point	NOUN
ejpam-5571	390	6	where	where	SCONJ
ejpam-5571	390	7	[	[	X
ejpam-5571	390	8	x′1	x′1	X
ejpam-5571	390	9	,	,	PUNCT
ejpam-5571	390	10	x	x	X
ejpam-5571	390	11	′	′	NOUN
ejpam-5571	390	12	2	2	NUM
ejpam-5571	390	13	]	]	PUNCT
ejpam-5571	390	14	and	and	CCONJ
ejpam-5571	390	15	[	[	X
ejpam-5571	390	16	x′	x′	PROPN
ejpam-5571	390	17	,	,	PUNCT
ejpam-5571	390	18	y′	y′	NUM
ejpam-5571	390	19	]	]	PUNCT
ejpam-5571	390	20	intersect	intersect	ADJ
ejpam-5571	390	21	,	,	PUNCT
ejpam-5571	390	22	and	and	CCONJ
ejpam-5571	390	23	let	let	VERB
ejpam-5571	390	24	x̂	x̂	PUNCT
ejpam-5571	391	1	=	=	PUNCT
ejpam-5571	391	2	j1(x	j1(x	PROPN
ejpam-5571	391	3	′′	′′	PROPN
ejpam-5571	391	4	)	)	PUNCT
ejpam-5571	391	5	=	=	PROPN
ejpam-5571	391	6	j2(x	j2(x	PROPN
ejpam-5571	391	7	′′	′′	PROPN
ejpam-5571	391	8	)	)	PUNCT
ejpam-5571	391	9	=	=	SYM
ejpam-5571	391	10	i(x′′	i(x′′	NOUN
ejpam-5571	391	11	)	)	PUNCT
ejpam-5571	391	12	.	.	PUNCT
ejpam-5571	392	1	hence	hence	ADV
ejpam-5571	392	2	,	,	PUNCT
ejpam-5571	392	3	d(x1	d(x1	NOUN
ejpam-5571	392	4	,	,	PUNCT
ejpam-5571	392	5	x2	x2	PROPN
ejpam-5571	392	6	)	)	PUNCT
ejpam-5571	392	7	=	=	SYM
ejpam-5571	392	8	d(x′1	d(x′1	PROPN
ejpam-5571	392	9	,	,	PUNCT
ejpam-5571	392	10	x	x	SYM
ejpam-5571	392	11	′	′	NOUN
ejpam-5571	392	12	2	2	NUM
ejpam-5571	392	13	)	)	PUNCT
ejpam-5571	392	14	=	=	SYM
ejpam-5571	392	15	d(x′1	d(x′1	NUM
ejpam-5571	392	16	,	,	PUNCT
ejpam-5571	392	17	x	x	SYM
ejpam-5571	392	18	′′	′′	PROPN
ejpam-5571	392	19	)	)	PUNCT
ejpam-5571	393	1	+	+	PROPN
ejpam-5571	393	2	d(x′′	d(x′′	PROPN
ejpam-5571	393	3	,	,	PUNCT
ejpam-5571	393	4	x′2	x′2	NOUN
ejpam-5571	393	5	)	)	PUNCT
ejpam-5571	393	6	=	=	SYM
ejpam-5571	393	7	d(x′1	d(x′1	NUM
ejpam-5571	393	8	,	,	PUNCT
ejpam-5571	393	9	x̂	x̂	NUM
ejpam-5571	393	10	)	)	PUNCT
ejpam-5571	394	1	+	+	CCONJ
ejpam-5571	394	2	d(x̂	d(x̂	NOUN
ejpam-5571	394	3	,	,	PUNCT
ejpam-5571	394	4	x′2	x′2	NOUN
ejpam-5571	394	5	)	)	PUNCT
ejpam-5571	394	6	,	,	PUNCT
ejpam-5571	394	7	so	so	CCONJ
ejpam-5571	395	1	[	[	X
ejpam-5571	395	2	x1	x1	X
ejpam-5571	395	3	,	,	PUNCT
ejpam-5571	395	4	x2	x2	PROPN
ejpam-5571	395	5	]	]	X
ejpam-5571	395	6	=	=	PUNCT
ejpam-5571	396	1	[	[	X
ejpam-5571	396	2	x1	x1	PROPN
ejpam-5571	396	3	,	,	PUNCT
ejpam-5571	396	4	x̂	x̂	NUM
ejpam-5571	396	5	]	]	PUNCT
ejpam-5571	396	6	∪	∪	ADP
ejpam-5571	396	7	[	[	X
ejpam-5571	396	8	x̂	x̂	NUM
ejpam-5571	396	9	,	,	PUNCT
ejpam-5571	396	10	x2	x2	PROPN
ejpam-5571	396	11	]	]	X
ejpam-5571	396	12	⊂	⊂	PROPN
ejpam-5571	396	13	c(γxy	c(γxy	PROPN
ejpam-5571	396	14	)	)	PUNCT
ejpam-5571	396	15	∪	∪	ADP
ejpam-5571	396	16	c(γyx	c(γyx	NOUN
ejpam-5571	396	17	)	)	PUNCT
ejpam-5571	396	18	.	.	PUNCT
ejpam-5571	397	1	therefore	therefore	ADV
ejpam-5571	397	2	,	,	PUNCT
ejpam-5571	397	3	c(γxy	c(γxy	PROPN
ejpam-5571	397	4	)	)	PUNCT
ejpam-5571	397	5	∪	∪	ADP
ejpam-5571	397	6	c(γyx	c(γyx	NOUN
ejpam-5571	397	7	)	)	PUNCT
ejpam-5571	397	8	is	be	AUX
ejpam-5571	397	9	a	a	DET
ejpam-5571	397	10	convex	convex	NOUN
ejpam-5571	397	11	set	set	NOUN
ejpam-5571	397	12	.	.	PUNCT
ejpam-5571	398	1	if	if	SCONJ
ejpam-5571	398	2	ℓ(γ′w′	ℓ(γ′w′	PROPN
ejpam-5571	398	3	2w	2w	NUM
ejpam-5571	398	4	′	′	NUM
ejpam-5571	398	5	1	1	NUM
ejpam-5571	398	6	)	)	PUNCT
ejpam-5571	398	7	≤	≤	NOUN
ejpam-5571	398	8	ℓ(γ′)/2	ℓ(γ′)/2	NOUN
ejpam-5571	398	9	,	,	PUNCT
ejpam-5571	398	10	we	we	PRON
ejpam-5571	398	11	proceed	proceed	VERB
ejpam-5571	398	12	in	in	ADP
ejpam-5571	398	13	the	the	DET
ejpam-5571	398	14	same	same	ADJ
ejpam-5571	398	15	proof	proof	NOUN
ejpam-5571	398	16	to	to	PART
ejpam-5571	398	17	have	have	VERB
ejpam-5571	398	18	that	that	DET
ejpam-5571	398	19	c(γxy)∪c(γyx	c(γxy)∪c(γyx	PROPN
ejpam-5571	398	20	)	)	PUNCT
ejpam-5571	398	21	is	be	AUX
ejpam-5571	398	22	a	a	DET
ejpam-5571	398	23	convex	convex	NOUN
ejpam-5571	398	24	set	set	VERB
ejpam-5571	398	25	as	as	ADP
ejpam-5571	398	26	in	in	ADP
ejpam-5571	398	27	the	the	DET
ejpam-5571	398	28	case	case	NOUN
ejpam-5571	398	29	ℓ(γ′w′	ℓ(γ′w′	NOUN
ejpam-5571	398	30	1w	1w	NUM
ejpam-5571	398	31	′	′	NUM
ejpam-5571	398	32	2	2	NUM
ejpam-5571	398	33	)	)	PUNCT
ejpam-5571	398	34	≤	≤	NOUN
ejpam-5571	398	35	ℓ(γ′)/2	ℓ(γ′)/2	NOUN
ejpam-5571	398	36	.	.	PUNCT
ejpam-5571	399	1	accordingly	accordingly	ADV
ejpam-5571	399	2	,	,	PUNCT
ejpam-5571	399	3	we	we	PRON
ejpam-5571	399	4	can	can	AUX
ejpam-5571	399	5	conclude	conclude	VERB
ejpam-5571	399	6	that	that	DET
ejpam-5571	399	7	c(γ′	c(γ′	NOUN
ejpam-5571	399	8	)	)	PUNCT
ejpam-5571	399	9	is	be	AUX
ejpam-5571	399	10	isometric	isometric	ADJ
ejpam-5571	399	11	to	to	ADP
ejpam-5571	399	12	c(γ	c(γ	PROPN
ejpam-5571	399	13	)	)	PUNCT
ejpam-5571	399	14	.	.	PUNCT
ejpam-5571	400	1	the	the	DET
ejpam-5571	400	2	theorem	theorem	VERB
ejpam-5571	400	3	’s	’s	PART
ejpam-5571	400	4	proof	proof	NOUN
ejpam-5571	400	5	is	be	AUX
ejpam-5571	400	6	now	now	ADV
ejpam-5571	400	7	complete	complete	ADJ
ejpam-5571	400	8	.	.	PUNCT
ejpam-5571	401	1	4	4	X
ejpam-5571	401	2	.	.	X
ejpam-5571	401	3	conclusion	conclusion	NOUN
ejpam-5571	401	4	the	the	DET
ejpam-5571	401	5	totally	totally	ADV
ejpam-5571	401	6	geodesic	geodesic	ADJ
ejpam-5571	401	7	surface	surface	NOUN
ejpam-5571	401	8	enclosed	enclose	VERB
ejpam-5571	401	9	by	by	ADP
ejpam-5571	401	10	a	a	DET
ejpam-5571	401	11	closed	closed	ADJ
ejpam-5571	401	12	spherical	spherical	ADJ
ejpam-5571	401	13	curve	curve	NOUN
ejpam-5571	401	14	at	at	ADP
ejpam-5571	401	15	a	a	DET
ejpam-5571	401	16	distance	distance	NOUN
ejpam-5571	402	1	r	r	NOUN
ejpam-5571	402	2	<	<	X
ejpam-5571	402	3	π	π	PROPN
ejpam-5571	402	4	2	2	NUM
ejpam-5571	402	5	√	√	PROPN
ejpam-5571	402	6	k	k	NOUN
ejpam-5571	402	7	from	from	ADP
ejpam-5571	402	8	a	a	DET
ejpam-5571	402	9	point	point	NOUN
ejpam-5571	402	10	in	in	ADP
ejpam-5571	402	11	a	a	DET
ejpam-5571	402	12	metric	metric	ADJ
ejpam-5571	402	13	space	space	NOUN
ejpam-5571	402	14	with	with	ADP
ejpam-5571	402	15	curvature	curvature	NOUN
ejpam-5571	402	16	bounded	bound	VERB
ejpam-5571	402	17	below	below	ADV
ejpam-5571	402	18	by	by	ADP
ejpam-5571	402	19	k	k	PROPN
ejpam-5571	402	20	is	be	AUX
ejpam-5571	402	21	isometric	isometric	ADJ
ejpam-5571	402	22	to	to	ADP
ejpam-5571	402	23	the	the	DET
ejpam-5571	402	24	region	region	NOUN
ejpam-5571	402	25	bounded	bound	VERB
ejpam-5571	402	26	by	by	ADP
ejpam-5571	402	27	a	a	DET
ejpam-5571	402	28	circle	circle	NOUN
ejpam-5571	402	29	of	of	ADP
ejpam-5571	402	30	radius	radius	NOUN
ejpam-5571	402	31	r	r	NOUN
ejpam-5571	402	32	in	in	ADP
ejpam-5571	402	33	rk	rk	NOUN
ejpam-5571	402	34	,	,	PUNCT
ejpam-5571	402	35	provided	provide	VERB
ejpam-5571	402	36	that	that	SCONJ
ejpam-5571	402	37	the	the	DET
ejpam-5571	402	38	closed	closed	ADJ
ejpam-5571	402	39	spherical	spherical	ADJ
ejpam-5571	402	40	curve	curve	NOUN
ejpam-5571	402	41	and	and	CCONJ
ejpam-5571	402	42	the	the	DET
ejpam-5571	402	43	circle	circle	NOUN
ejpam-5571	402	44	possess	possess	VERB
ejpam-5571	402	45	identical	identical	ADJ
ejpam-5571	402	46	lengths	length	NOUN
ejpam-5571	402	47	,	,	PUNCT
ejpam-5571	402	48	and	and	CCONJ
ejpam-5571	402	49	the	the	DET
ejpam-5571	402	50	angle	angle	NOUN
ejpam-5571	402	51	properties	property	NOUN
ejpam-5571	402	52	in	in	ADP
ejpam-5571	402	53	this	this	DET
ejpam-5571	402	54	metric	metric	ADJ
ejpam-5571	402	55	space	space	NOUN
ejpam-5571	402	56	have	have	VERB
ejpam-5571	402	57	similarities	similarity	NOUN
ejpam-5571	402	58	to	to	ADP
ejpam-5571	402	59	those	those	PRON
ejpam-5571	402	60	in	in	ADP
ejpam-5571	402	61	rk	rk	NOUN
ejpam-5571	402	62	.	.	PUNCT
ejpam-5571	403	1	acknowledgements	acknowledgement	VERB
ejpam-5571	403	2	the	the	DET
ejpam-5571	403	3	research	research	NOUN
ejpam-5571	403	4	received	receive	VERB
ejpam-5571	403	5	funding	funding	NOUN
ejpam-5571	403	6	from	from	ADP
ejpam-5571	403	7	prince	prince	PROPN
ejpam-5571	403	8	of	of	ADP
ejpam-5571	403	9	songkla	songkla	PROPN
ejpam-5571	403	10	university	university	PROPN
ejpam-5571	403	11	under	under	ADP
ejpam-5571	403	12	grant	grant	NOUN
ejpam-5571	403	13	no	no	INTJ
ejpam-5571	403	14	.	.	PUNCT
ejpam-5571	404	1	sat6403009s	sat6403009s	PROPN
ejpam-5571	404	2	.	.	PUNCT
ejpam-5571	405	1	references	reference	NOUN
ejpam-5571	405	2	3944	3944	NUM
ejpam-5571	405	3	references	reference	NOUN
ejpam-5571	405	4	[	[	X
ejpam-5571	405	5	1	1	NUM
ejpam-5571	405	6	]	]	X
ejpam-5571	405	7	a.d	a.d	PROPN
ejpam-5571	405	8	.	.	PROPN
ejpam-5571	405	9	alexandrov	alexandrov	PROPN
ejpam-5571	405	10	.	.	PROPN
ejpam-5571	406	1	die	die	PROPN
ejpam-5571	406	2	innere	innere	PROPN
ejpam-5571	406	3	geometrie	geometrie	PROPN
ejpam-5571	406	4	der	der	PROPN
ejpam-5571	406	5	konvexen	konvexen	PROPN
ejpam-5571	406	6	flächen	flächen	PROPN
ejpam-5571	406	7	.	.	PUNCT
ejpam-5571	407	1	akademie	akademie	PROPN
ejpam-5571	407	2	verlag	verlag	PROPN
ejpam-5571	407	3	,	,	PUNCT
ejpam-5571	407	4	berlin	berlin	PROPN
ejpam-5571	407	5	,	,	PUNCT
ejpam-5571	407	6	1955	1955	NUM
ejpam-5571	407	7	.	.	PUNCT
ejpam-5571	408	1	[	[	X
ejpam-5571	408	2	2	2	NUM
ejpam-5571	408	3	]	]	X
ejpam-5571	408	4	a.d	a.d	PROPN
ejpam-5571	408	5	.	.	PROPN
ejpam-5571	408	6	alexandrov	alexandrov	PROPN
ejpam-5571	408	7	.	.	PUNCT
ejpam-5571	409	1	über	über	PROPN
ejpam-5571	409	2	eine	eine	PROPN
ejpam-5571	409	3	verallgemeinerung	verallgemeinerung	PROPN
ejpam-5571	409	4	der	der	PROPN
ejpam-5571	409	5	riemannschen	riemannschen	PROPN
ejpam-5571	409	6	geometrie	geometrie	PROPN
ejpam-5571	409	7	.	.	PUNCT
ejpam-5571	410	1	schriftenreihe	schriftenreihe	PROPN
ejpam-5571	410	2	für	für	PROPN
ejpam-5571	410	3	forschung	forschung	VERB
ejpam-5571	410	4	i	i	PROPN
ejpam-5571	410	5	m	m	PROPN
ejpam-5571	410	6	gebiet	gebiet	PROPN
ejpam-5571	410	7	der	der	PROPN
ejpam-5571	410	8	mathematik	mathematik	PROPN
ejpam-5571	410	9	,	,	PUNCT
ejpam-5571	410	10	1:33–84	1:33–84	NUM
ejpam-5571	410	11	,	,	PUNCT
ejpam-5571	410	12	1957	1957	NUM
ejpam-5571	410	13	.	.	PUNCT
ejpam-5571	411	1	[	[	X
ejpam-5571	411	2	3	3	X
ejpam-5571	411	3	]	]	X
ejpam-5571	411	4	w.	w.	NOUN
ejpam-5571	411	5	ballmann	ballmann	PROPN
ejpam-5571	411	6	.	.	PUNCT
ejpam-5571	412	1	lectures	lecture	NOUN
ejpam-5571	412	2	on	on	ADP
ejpam-5571	412	3	spaces	space	NOUN
ejpam-5571	412	4	of	of	ADP
ejpam-5571	412	5	nonpositive	nonpositive	ADJ
ejpam-5571	412	6	curvature	curvature	NOUN
ejpam-5571	412	7	.	.	PUNCT
ejpam-5571	413	1	birkhauser	birkhauser	PROPN
ejpam-5571	413	2	,	,	PUNCT
ejpam-5571	413	3	basel	basel	PROPN
ejpam-5571	413	4	,	,	PUNCT
ejpam-5571	413	5	1995	1995	NUM
ejpam-5571	413	6	.	.	PUNCT
ejpam-5571	414	1	[	[	X
ejpam-5571	414	2	4	4	NUM
ejpam-5571	414	3	]	]	X
ejpam-5571	414	4	m.r	m.r	PROPN
ejpam-5571	414	5	.	.	PROPN
ejpam-5571	414	6	bridson	bridson	PROPN
ejpam-5571	414	7	and	and	CCONJ
ejpam-5571	414	8	a.	a.	NOUN
ejpam-5571	414	9	haefliger	haefliger	NOUN
ejpam-5571	414	10	.	.	PUNCT
ejpam-5571	415	1	metric	metric	ADJ
ejpam-5571	415	2	spaces	space	NOUN
ejpam-5571	415	3	of	of	ADP
ejpam-5571	415	4	nonpositive	nonpositive	ADJ
ejpam-5571	415	5	curvature	curvature	NOUN
ejpam-5571	415	6	.	.	PUNCT
ejpam-5571	416	1	springer	springer	NOUN
ejpam-5571	416	2	,	,	PUNCT
ejpam-5571	416	3	heidelberg	heidelberg	PROPN
ejpam-5571	416	4	,	,	PUNCT
ejpam-5571	416	5	1999	1999	NUM
ejpam-5571	416	6	.	.	PUNCT
ejpam-5571	417	1	[	[	X
ejpam-5571	417	2	5	5	X
ejpam-5571	417	3	]	]	PUNCT
ejpam-5571	417	4	d.	d.	PROPN
ejpam-5571	417	5	burago	burago	PROPN
ejpam-5571	417	6	,	,	PUNCT
ejpam-5571	417	7	yu	yu	PROPN
ejpam-5571	417	8	.	.	PROPN
ejpam-5571	417	9	burago	burago	PROPN
ejpam-5571	417	10	,	,	PUNCT
ejpam-5571	417	11	and	and	CCONJ
ejpam-5571	417	12	s.	s.	PROPN
ejpam-5571	417	13	ivanov	ivanov	PROPN
ejpam-5571	417	14	.	.	PUNCT
ejpam-5571	418	1	a	a	DET
ejpam-5571	418	2	course	course	NOUN
ejpam-5571	418	3	in	in	ADP
ejpam-5571	418	4	metric	metric	ADJ
ejpam-5571	418	5	geometry	geometry	NOUN
ejpam-5571	418	6	,	,	PUNCT
ejpam-5571	418	7	graduate	graduate	NOUN
ejpam-5571	418	8	studies	study	NOUN
ejpam-5571	418	9	in	in	ADP
ejpam-5571	418	10	mathematics	mathematic	NOUN
ejpam-5571	418	11	,	,	PUNCT
ejpam-5571	418	12	vol	vol	NOUN
ejpam-5571	418	13	.	.	PUNCT
ejpam-5571	419	1	33	33	NUM
ejpam-5571	419	2	.	.	PUNCT
ejpam-5571	420	1	american	american	PROPN
ejpam-5571	420	2	mathematical	mathematical	PROPN
ejpam-5571	420	3	society	society	NOUN
ejpam-5571	420	4	,	,	PUNCT
ejpam-5571	420	5	providence	providence	NOUN
ejpam-5571	420	6	,	,	PUNCT
ejpam-5571	420	7	rhode	rhode	NOUN
ejpam-5571	420	8	island	island	NOUN
ejpam-5571	420	9	,	,	PUNCT
ejpam-5571	420	10	2001	2001	NUM
ejpam-5571	420	11	.	.	PUNCT
ejpam-5571	421	1	[	[	X
ejpam-5571	421	2	6	6	NUM
ejpam-5571	421	3	]	]	X
ejpam-5571	421	4	yu	yu	PROPN
ejpam-5571	421	5	.	.	PROPN
ejpam-5571	421	6	burago	burago	PROPN
ejpam-5571	421	7	,	,	PUNCT
ejpam-5571	421	8	m.	m.	NOUN
ejpam-5571	421	9	gromov	gromov	NOUN
ejpam-5571	421	10	,	,	PUNCT
ejpam-5571	421	11	and	and	CCONJ
ejpam-5571	421	12	g.	g.	PROPN
ejpam-5571	421	13	perel’man	perel’man	PROPN
ejpam-5571	421	14	.	.	PROPN
ejpam-5571	422	1	a.d	a.d	PROPN
ejpam-5571	422	2	.	.	PUNCT
ejpam-5571	423	1	alexandrov	alexandrov	PROPN
ejpam-5571	423	2	spaces	space	VERB
ejpam-5571	423	3	with	with	ADP
ejpam-5571	423	4	curvature	curvature	NOUN
ejpam-5571	423	5	bounded	bound	VERB
ejpam-5571	423	6	below	below	ADV
ejpam-5571	423	7	.	.	PUNCT
ejpam-5571	424	1	russian	russian	ADJ
ejpam-5571	424	2	mathematical	mathematical	ADJ
ejpam-5571	424	3	surveys	survey	NOUN
ejpam-5571	424	4	,	,	PUNCT
ejpam-5571	424	5	47(2):1–58	47(2):1–58	NUM
ejpam-5571	424	6	,	,	PUNCT
ejpam-5571	424	7	1992	1992	NUM
ejpam-5571	424	8	.	.	PUNCT
ejpam-5571	425	1	[	[	X
ejpam-5571	425	2	7	7	X
ejpam-5571	425	3	]	]	X
ejpam-5571	425	4	r.	r.	PROPN
ejpam-5571	425	5	esṕınola	esṕınola	PROPN
ejpam-5571	425	6	,	,	PUNCT
ejpam-5571	425	7	c.	c.	PROPN
ejpam-5571	425	8	li	li	PROPN
ejpam-5571	425	9	,	,	PUNCT
ejpam-5571	425	10	and	and	CCONJ
ejpam-5571	425	11	g.	g.	PROPN
ejpam-5571	425	12	lópez	lópez	PROPN
ejpam-5571	425	13	.	.	PUNCT
ejpam-5571	426	1	nearest	near	ADJ
ejpam-5571	426	2	and	and	CCONJ
ejpam-5571	426	3	farthest	farth	ADJ
ejpam-5571	426	4	points	point	NOUN
ejpam-5571	426	5	in	in	ADP
ejpam-5571	426	6	spaces	space	NOUN
ejpam-5571	426	7	of	of	ADP
ejpam-5571	426	8	curvature	curvature	NOUN
ejpam-5571	426	9	bounded	bound	VERB
ejpam-5571	426	10	below	below	ADV
ejpam-5571	426	11	.	.	PUNCT
ejpam-5571	427	1	journal	journal	PROPN
ejpam-5571	427	2	of	of	ADP
ejpam-5571	427	3	approximation	approximation	NOUN
ejpam-5571	427	4	theory	theory	NOUN
ejpam-5571	427	5	,	,	PUNCT
ejpam-5571	427	6	162:1364–1380	162:1364–1380	NUM
ejpam-5571	427	7	,	,	PUNCT
ejpam-5571	427	8	2010	2010	NUM
ejpam-5571	427	9	.	.	PUNCT
ejpam-5571	428	1	[	[	X
ejpam-5571	428	2	8	8	X
ejpam-5571	428	3	]	]	PUNCT
ejpam-5571	428	4	s.	s.	PROPN
ejpam-5571	428	5	halbeisen	halbeisen	PROPN
ejpam-5571	428	6	.	.	PUNCT
ejpam-5571	429	1	on	on	ADP
ejpam-5571	429	2	tangent	tangent	ADJ
ejpam-5571	429	3	cones	cone	NOUN
ejpam-5571	429	4	of	of	ADP
ejpam-5571	429	5	alexandrov	alexandrov	NOUN
ejpam-5571	429	6	spaces	space	NOUN
ejpam-5571	429	7	with	with	ADP
ejpam-5571	429	8	curvature	curvature	NOUN
ejpam-5571	429	9	bounded	bound	VERB
ejpam-5571	429	10	below	below	ADV
ejpam-5571	429	11	.	.	PUNCT
ejpam-5571	430	1	manuscripta	manuscripta	NOUN
ejpam-5571	430	2	mathematica	mathematica	PROPN
ejpam-5571	430	3	,	,	PUNCT
ejpam-5571	430	4	103:169–182	103:169–182	NUM
ejpam-5571	430	5	,	,	PUNCT
ejpam-5571	430	6	2000	2000	NUM
ejpam-5571	430	7	.	.	PUNCT
ejpam-5571	431	1	[	[	X
ejpam-5571	431	2	9	9	NUM
ejpam-5571	431	3	]	]	X
ejpam-5571	431	4	u.	u.	PROPN
ejpam-5571	431	5	lang	lang	PROPN
ejpam-5571	431	6	and	and	CCONJ
ejpam-5571	431	7	v.	v.	ADP
ejpam-5571	431	8	schröder	schröder	NOUN
ejpam-5571	431	9	.	.	PUNCT
ejpam-5571	432	1	jung	jung	PROPN
ejpam-5571	432	2	’s	’s	PART
ejpam-5571	432	3	theorem	theorem	NOUN
ejpam-5571	432	4	for	for	ADP
ejpam-5571	432	5	alexandrov	alexandrov	ADJ
ejpam-5571	432	6	spaces	space	NOUN
ejpam-5571	432	7	of	of	ADP
ejpam-5571	432	8	curvature	curvature	NOUN
ejpam-5571	432	9	bounded	bound	VERB
ejpam-5571	432	10	above	above	ADV
ejpam-5571	432	11	.	.	PUNCT
ejpam-5571	433	1	annals	annal	NOUN
ejpam-5571	433	2	of	of	ADP
ejpam-5571	433	3	global	global	ADJ
ejpam-5571	433	4	analysis	analysis	NOUN
ejpam-5571	433	5	and	and	CCONJ
ejpam-5571	433	6	geometry	geometry	NOUN
ejpam-5571	433	7	,	,	PUNCT
ejpam-5571	433	8	15:263–275	15:263–275	NUM
ejpam-5571	433	9	,	,	PUNCT
ejpam-5571	433	10	1997	1997	NUM
ejpam-5571	433	11	.	.	PUNCT
ejpam-5571	434	1	[	[	X
ejpam-5571	434	2	10	10	NUM
ejpam-5571	434	3	]	]	X
ejpam-5571	434	4	n.	n.	PROPN
ejpam-5571	434	5	lebedeva	lebedeva	PROPN
ejpam-5571	434	6	and	and	CCONJ
ejpam-5571	434	7	a.	a.	NOUN
ejpam-5571	434	8	petrunin	petrunin	PROPN
ejpam-5571	434	9	.	.	PUNCT
ejpam-5571	435	1	curvature	curvature	NOUN
ejpam-5571	435	2	bounded	bound	VERB
ejpam-5571	435	3	below	below	ADV
ejpam-5571	435	4	:	:	PUNCT
ejpam-5571	435	5	a	a	DET
ejpam-5571	435	6	definition	definition	NOUN
ejpam-5571	435	7	a	a	PRON
ejpam-5571	435	8	la	la	X
ejpam-5571	435	9	bergnikolaev	bergnikolaev	X
ejpam-5571	435	10	.	.	PUNCT
ejpam-5571	436	1	electronic	electronic	ADJ
ejpam-5571	436	2	research	research	NOUN
ejpam-5571	436	3	announcements	announcement	NOUN
ejpam-5571	436	4	in	in	ADP
ejpam-5571	436	5	mathematical	mathematical	ADJ
ejpam-5571	436	6	sciences	science	NOUN
ejpam-5571	436	7	,	,	PUNCT
ejpam-5571	436	8	17:122–124	17:122–124	PROPN
ejpam-5571	436	9	,	,	PUNCT
ejpam-5571	436	10	2010	2010	NUM
ejpam-5571	436	11	.	.	PUNCT
ejpam-5571	437	1	[	[	X
ejpam-5571	437	2	11	11	NUM
ejpam-5571	437	3	]	]	PUNCT
ejpam-5571	437	4	a.	a.	NOUN
ejpam-5571	437	5	petrunin	petrunin	PROPN
ejpam-5571	437	6	.	.	PUNCT
ejpam-5571	438	1	parallel	parallel	ADJ
ejpam-5571	438	2	transportation	transportation	NOUN
ejpam-5571	438	3	for	for	ADP
ejpam-5571	438	4	alexandrov	alexandrov	NOUN
ejpam-5571	438	5	spaces	space	NOUN
ejpam-5571	438	6	with	with	ADP
ejpam-5571	438	7	curvature	curvature	NOUN
ejpam-5571	438	8	bounded	bound	VERB
ejpam-5571	438	9	below	below	ADV
ejpam-5571	438	10	.	.	PUNCT
ejpam-5571	439	1	gafa	gafa	PROPN
ejpam-5571	439	2	:	:	PUNCT
ejpam-5571	439	3	geometric	geometric	ADJ
ejpam-5571	439	4	functional	functional	ADJ
ejpam-5571	439	5	analysis	analysis	NOUN
ejpam-5571	439	6	,	,	PUNCT
ejpam-5571	439	7	8:123–148	8:123–148	NUM
ejpam-5571	439	8	,	,	PUNCT
ejpam-5571	439	9	1998	1998	NUM
ejpam-5571	439	10	.	.	PUNCT
ejpam-5571	440	1	[	[	X
ejpam-5571	440	2	12	12	NUM
ejpam-5571	440	3	]	]	X
ejpam-5571	440	4	a.	a.	NOUN
ejpam-5571	440	5	sama	sama	PROPN
ejpam-5571	440	6	-	-	PUNCT
ejpam-5571	440	7	ae	ae	PROPN
ejpam-5571	440	8	,	,	PUNCT
ejpam-5571	440	9	a.	a.	PROPN
ejpam-5571	440	10	phon	phon	PROPN
ejpam-5571	440	11	-	-	PUNCT
ejpam-5571	440	12	on	on	ADP
ejpam-5571	440	13	,	,	PUNCT
ejpam-5571	440	14	n.	n.	NOUN
ejpam-5571	440	15	makaje	makaje	NOUN
ejpam-5571	440	16	,	,	PUNCT
ejpam-5571	440	17	and	and	CCONJ
ejpam-5571	440	18	a.	a.	NOUN
ejpam-5571	440	19	hazanee	hazanee	NOUN
ejpam-5571	440	20	.	.	PUNCT
ejpam-5571	441	1	a	a	DET
ejpam-5571	441	2	distance	distance	NOUN
ejpam-5571	441	3	between	between	ADP
ejpam-5571	441	4	two	two	NUM
ejpam-5571	441	5	points	point	NOUN
ejpam-5571	441	6	and	and	CCONJ
ejpam-5571	441	7	nearest	near	ADJ
ejpam-5571	441	8	points	point	NOUN
ejpam-5571	441	9	in	in	ADP
ejpam-5571	441	10	a	a	DET
ejpam-5571	441	11	metric	metric	ADJ
ejpam-5571	441	12	space	space	NOUN
ejpam-5571	441	13	of	of	ADP
ejpam-5571	441	14	curvature	curvature	NOUN
ejpam-5571	441	15	bounded	bound	VERB
ejpam-5571	441	16	below	below	ADV
ejpam-5571	441	17	.	.	PUNCT
ejpam-5571	442	1	thai	thai	PROPN
ejpam-5571	442	2	journal	journal	PROPN
ejpam-5571	442	3	of	of	ADP
ejpam-5571	442	4	mathematics	mathematic	NOUN
ejpam-5571	442	5	,	,	PUNCT
ejpam-5571	442	6	special	special	ADJ
ejpam-5571	442	7	issue:229–239	issue:229–239	NUM
ejpam-5571	442	8	,	,	PUNCT
ejpam-5571	442	9	2022	2022	NUM
ejpam-5571	442	10	.	.	PUNCT
ejpam-5571	443	1	[	[	X
ejpam-5571	443	2	13	13	NUM
ejpam-5571	443	3	]	]	PUNCT
ejpam-5571	443	4	t.	t.	PROPN
ejpam-5571	443	5	yokota	yokota	PROPN
ejpam-5571	443	6	.	.	PUNCT
ejpam-5571	444	1	a	a	DET
ejpam-5571	444	2	rigidity	rigidity	NOUN
ejpam-5571	444	3	theorem	theorem	VERB
ejpam-5571	444	4	in	in	ADP
ejpam-5571	444	5	alexandrov	alexandrov	PROPN
ejpam-5571	444	6	spaces	space	NOUN
ejpam-5571	444	7	with	with	ADP
ejpam-5571	444	8	lower	low	ADJ
ejpam-5571	444	9	curvature	curvature	NOUN
ejpam-5571	444	10	bound	bind	VERB
ejpam-5571	444	11	.	.	PUNCT
ejpam-5571	445	1	mathematische	mathematische	PROPN
ejpam-5571	445	2	annalen	annalen	PROPN
ejpam-5571	445	3	,	,	PUNCT
ejpam-5571	445	4	353:305–331	353:305–331	NUM
ejpam-5571	445	5	,	,	PUNCT
ejpam-5571	445	6	2012	2012	NUM
ejpam-5571	445	7	.	.	PUNCT