id	sid	tid	token	lemma	pos
ejpam-4744	1	1	european	european	PROPN
ejpam-4744	1	2	journal	journal	PROPN
ejpam-4744	1	3	of	of	ADP
ejpam-4744	1	4	pure	pure	ADJ
ejpam-4744	1	5	and	and	CCONJ
ejpam-4744	1	6	applied	apply	VERB
ejpam-4744	1	7	mathematics	mathematic	NOUN
ejpam-4744	1	8	vol	vol	NOUN
ejpam-4744	1	9	.	.	PUNCT
ejpam-4744	2	1	16	16	NUM
ejpam-4744	2	2	,	,	PUNCT
ejpam-4744	2	3	no	no	INTJ
ejpam-4744	2	4	.	.	NOUN
ejpam-4744	2	5	2	2	NUM
ejpam-4744	2	6	,	,	PUNCT
ejpam-4744	2	7	2023	2023	NUM
ejpam-4744	2	8	,	,	PUNCT
ejpam-4744	2	9	1318	1318	NUM
ejpam-4744	2	10	-	-	SYM
ejpam-4744	2	11	1325	1325	NUM
ejpam-4744	2	12	issn	issn	PROPN
ejpam-4744	2	13	1307	1307	NUM
ejpam-4744	2	14	-	-	SYM
ejpam-4744	2	15	5543	5543	NUM
ejpam-4744	2	16	–	–	PUNCT
ejpam-4744	2	17	ejpam.com	ejpam.com	X
ejpam-4744	2	18	published	publish	VERB
ejpam-4744	2	19	by	by	ADP
ejpam-4744	2	20	new	new	PROPN
ejpam-4744	2	21	york	york	PROPN
ejpam-4744	2	22	business	business	PROPN
ejpam-4744	2	23	global	global	ADJ
ejpam-4744	2	24	bounds	bound	NOUN
ejpam-4744	2	25	on	on	ADP
ejpam-4744	2	26	intersection	intersection	NOUN
ejpam-4744	2	27	number	number	NOUN
ejpam-4744	2	28	in	in	ADP
ejpam-4744	2	29	the	the	DET
ejpam-4744	2	30	join	join	NOUN
ejpam-4744	2	31	and	and	CCONJ
ejpam-4744	2	32	corona	corona	NOUN
ejpam-4744	2	33	of	of	ADP
ejpam-4744	2	34	graphs	graph	NOUN
ejpam-4744	2	35	jesrael	jesrael	PROPN
ejpam-4744	2	36	b.	b.	PROPN
ejpam-4744	2	37	palco1,∗	palco1,∗	PROPN
ejpam-4744	2	38	,	,	PUNCT
ejpam-4744	2	39	rolando	rolando	PROPN
ejpam-4744	2	40	n.	n.	PROPN
ejpam-4744	2	41	paluga2	paluga2	PROPN
ejpam-4744	3	1	1	1	NUM
ejpam-4744	3	2	department	department	NOUN
ejpam-4744	3	3	of	of	ADP
ejpam-4744	3	4	physical	physical	ADJ
ejpam-4744	3	5	sciences	sciences	PROPN
ejpam-4744	3	6	and	and	CCONJ
ejpam-4744	3	7	mathematics	mathematic	NOUN
ejpam-4744	3	8	,	,	PUNCT
ejpam-4744	3	9	college	college	NOUN
ejpam-4744	3	10	of	of	ADP
ejpam-4744	3	11	marine	marine	PROPN
ejpam-4744	3	12	and	and	CCONJ
ejpam-4744	3	13	allied	ally	VERB
ejpam-4744	3	14	sciences	science	NOUN
ejpam-4744	3	15	,	,	PUNCT
ejpam-4744	3	16	mindanao	mindanao	PROPN
ejpam-4744	3	17	state	state	PROPN
ejpam-4744	3	18	university	university	PROPN
ejpam-4744	3	19	at	at	ADP
ejpam-4744	3	20	naawan	naawan	PROPN
ejpam-4744	3	21	,	,	PUNCT
ejpam-4744	3	22	9023	9023	NUM
ejpam-4744	3	23	,	,	PUNCT
ejpam-4744	3	24	philippines	philippine	NOUN
ejpam-4744	3	25	2	2	NUM
ejpam-4744	3	26	department	department	NOUN
ejpam-4744	3	27	of	of	ADP
ejpam-4744	3	28	mathematics	mathematic	NOUN
ejpam-4744	3	29	,	,	PUNCT
ejpam-4744	3	30	college	college	NOUN
ejpam-4744	3	31	of	of	ADP
ejpam-4744	3	32	mathematics	mathematic	NOUN
ejpam-4744	3	33	and	and	CCONJ
ejpam-4744	3	34	natural	natural	ADJ
ejpam-4744	3	35	sciences	science	NOUN
ejpam-4744	3	36	,	,	PUNCT
ejpam-4744	3	37	caraga	caraga	PROPN
ejpam-4744	3	38	state	state	PROPN
ejpam-4744	3	39	university	university	PROPN
ejpam-4744	3	40	,	,	PUNCT
ejpam-4744	3	41	8600	8600	NUM
ejpam-4744	3	42	,	,	PUNCT
ejpam-4744	3	43	philippines	philippine	NOUN
ejpam-4744	3	44	abstract	abstract	ADJ
ejpam-4744	3	45	.	.	PUNCT
ejpam-4744	4	1	in	in	ADP
ejpam-4744	4	2	this	this	DET
ejpam-4744	4	3	paper	paper	NOUN
ejpam-4744	4	4	,	,	PUNCT
ejpam-4744	4	5	we	we	PRON
ejpam-4744	4	6	provide	provide	VERB
ejpam-4744	4	7	an	an	DET
ejpam-4744	4	8	upper	upper	ADJ
ejpam-4744	4	9	bound	bind	VERB
ejpam-4744	4	10	for	for	ADP
ejpam-4744	4	11	the	the	DET
ejpam-4744	4	12	intersection	intersection	NOUN
ejpam-4744	4	13	number	number	NOUN
ejpam-4744	4	14	in	in	ADP
ejpam-4744	4	15	the	the	DET
ejpam-4744	4	16	join	join	NOUN
ejpam-4744	4	17	and	and	CCONJ
ejpam-4744	4	18	corona	corona	NOUN
ejpam-4744	4	19	of	of	ADP
ejpam-4744	4	20	graphs	graph	NOUN
ejpam-4744	4	21	.	.	PUNCT
ejpam-4744	5	1	moreover	moreover	ADV
ejpam-4744	5	2	,	,	PUNCT
ejpam-4744	5	3	we	we	PRON
ejpam-4744	5	4	give	give	VERB
ejpam-4744	5	5	formulas	formula	NOUN
ejpam-4744	5	6	for	for	ADP
ejpam-4744	5	7	the	the	DET
ejpam-4744	5	8	intersection	intersection	NOUN
ejpam-4744	5	9	number	number	NOUN
ejpam-4744	5	10	of	of	ADP
ejpam-4744	5	11	kn	kn	PROPN
ejpam-4744	5	12	◦	◦	PROPN
ejpam-4744	5	13	g	g	PROPN
ejpam-4744	5	14	,	,	PUNCT
ejpam-4744	5	15	pn	pn	PROPN
ejpam-4744	5	16	◦	◦	NOUN
ejpam-4744	5	17	g	g	PROPN
ejpam-4744	5	18	,	,	PUNCT
ejpam-4744	5	19	cn	cn	PROPN
ejpam-4744	5	20	◦	◦	NOUN
ejpam-4744	5	21	g	g	PROPN
ejpam-4744	5	22	and	and	CCONJ
ejpam-4744	5	23	crn	crn	PROPN
ejpam-4744	5	24	.	.	PUNCT
ejpam-4744	6	1	2020	2020	NUM
ejpam-4744	6	2	mathematics	mathematics	PROPN
ejpam-4744	6	3	subject	subject	NOUN
ejpam-4744	6	4	classifications	classification	NOUN
ejpam-4744	6	5	:	:	PUNCT
ejpam-4744	6	6	05c69	05c69	X
ejpam-4744	6	7	key	key	ADJ
ejpam-4744	6	8	words	word	NOUN
ejpam-4744	6	9	and	and	CCONJ
ejpam-4744	6	10	phrases	phrase	NOUN
ejpam-4744	6	11	:	:	PUNCT
ejpam-4744	6	12	intersection	intersection	NOUN
ejpam-4744	6	13	number	number	NOUN
ejpam-4744	6	14	,	,	PUNCT
ejpam-4744	6	15	extreme	extreme	ADJ
ejpam-4744	6	16	intersection	intersection	NOUN
ejpam-4744	6	17	graph	graph	NOUN
ejpam-4744	6	18	,	,	PUNCT
ejpam-4744	6	19	join	join	VERB
ejpam-4744	6	20	and	and	CCONJ
ejpam-4744	6	21	corona	corona	PROPN
ejpam-4744	6	22	1	1	NUM
ejpam-4744	6	23	.	.	PUNCT
ejpam-4744	7	1	introduction	introduction	NOUN
ejpam-4744	7	2	let	let	VERB
ejpam-4744	7	3	s	s	PRON
ejpam-4744	7	4	be	be	AUX
ejpam-4744	7	5	a	a	DET
ejpam-4744	7	6	set	set	NOUN
ejpam-4744	7	7	and	and	CCONJ
ejpam-4744	7	8	f	f	NOUN
ejpam-4744	7	9	=	=	SYM
ejpam-4744	7	10	{	{	PUNCT
ejpam-4744	7	11	s1	s1	NOUN
ejpam-4744	7	12	,	,	PUNCT
ejpam-4744	7	13	s2	s2	PROPN
ejpam-4744	7	14	,	,	PUNCT
ejpam-4744	7	15	·	·	PUNCT
ejpam-4744	7	16	·	·	PUNCT
ejpam-4744	7	17	·	·	PUNCT
ejpam-4744	7	18	,	,	PUNCT
ejpam-4744	7	19	sp	sp	ADP
ejpam-4744	7	20	}	}	PUNCT
ejpam-4744	7	21	,	,	PUNCT
ejpam-4744	7	22	for	for	ADP
ejpam-4744	7	23	some	some	DET
ejpam-4744	7	24	integer	integer	NOUN
ejpam-4744	7	25	p	p	NOUN
ejpam-4744	7	26	,	,	PUNCT
ejpam-4744	7	27	a	a	DET
ejpam-4744	7	28	nonempty	nonempty	ADJ
ejpam-4744	7	29	family	family	NOUN
ejpam-4744	7	30	of	of	ADP
ejpam-4744	7	31	distinct	distinct	ADJ
ejpam-4744	7	32	nonempty	nonempty	ADJ
ejpam-4744	7	33	subsets	subset	NOUN
ejpam-4744	7	34	of	of	ADP
ejpam-4744	7	35	s	s	NOUN
ejpam-4744	7	36	whose	whose	DET
ejpam-4744	7	37	union	union	NOUN
ejpam-4744	7	38	is	be	AUX
ejpam-4744	7	39	s.	s.	PROPN
ejpam-4744	7	40	the	the	DET
ejpam-4744	7	41	intersection	intersection	NOUN
ejpam-4744	7	42	graph	graph	NOUN
ejpam-4744	7	43	of	of	ADP
ejpam-4744	7	44	f	f	PROPN
ejpam-4744	7	45	is	be	AUX
ejpam-4744	7	46	denoted	denote	VERB
ejpam-4744	7	47	by	by	ADP
ejpam-4744	7	48	ω(f	ω(f	PROPN
ejpam-4744	7	49	)	)	PUNCT
ejpam-4744	7	50	and	and	CCONJ
ejpam-4744	7	51	defined	define	VERB
ejpam-4744	7	52	by	by	ADP
ejpam-4744	7	53	v	v	NUM
ejpam-4744	7	54	(	(	PUNCT
ejpam-4744	7	55	ω(f	ω(f	ADJ
ejpam-4744	7	56	)	)	PUNCT
ejpam-4744	7	57	)	)	PUNCT
ejpam-4744	8	1	=	=	SYM
ejpam-4744	8	2	f	f	PROPN
ejpam-4744	8	3	,	,	PUNCT
ejpam-4744	8	4	with	with	ADP
ejpam-4744	8	5	si	si	PROPN
ejpam-4744	8	6	and	and	CCONJ
ejpam-4744	8	7	sj	sj	INTJ
ejpam-4744	8	8	adjacent	adjacent	ADJ
ejpam-4744	8	9	whenever	whenever	SCONJ
ejpam-4744	8	10	i	i	PRON
ejpam-4744	8	11	̸=	̸=	PROPN
ejpam-4744	8	12	j	j	PROPN
ejpam-4744	8	13	and	and	CCONJ
ejpam-4744	8	14	si	si	PROPN
ejpam-4744	8	15	∩	∩	NOUN
ejpam-4744	8	16	sj	sj	PROPN
ejpam-4744	8	17	̸=	̸=	PROPN
ejpam-4744	8	18	∅.	∅.	ADP
ejpam-4744	8	19	a	a	DET
ejpam-4744	8	20	graph	graph	NOUN
ejpam-4744	8	21	g	g	NOUN
ejpam-4744	8	22	is	be	AUX
ejpam-4744	8	23	an	an	DET
ejpam-4744	8	24	intersection	intersection	NOUN
ejpam-4744	8	25	graph	graph	NOUN
ejpam-4744	8	26	on	on	ADP
ejpam-4744	8	27	s	s	PRON
ejpam-4744	8	28	if	if	SCONJ
ejpam-4744	8	29	there	there	PRON
ejpam-4744	8	30	exists	exist	VERB
ejpam-4744	8	31	a	a	DET
ejpam-4744	8	32	family	family	NOUN
ejpam-4744	8	33	f	f	NOUN
ejpam-4744	8	34	of	of	ADP
ejpam-4744	8	35	subsets	subset	NOUN
ejpam-4744	8	36	of	of	ADP
ejpam-4744	8	37	s	s	PRON
ejpam-4744	8	38	for	for	ADP
ejpam-4744	8	39	which	which	PRON
ejpam-4744	8	40	g	g	NOUN
ejpam-4744	8	41	∼=	∼=	PROPN
ejpam-4744	8	42	ω(f	ω(f	PUNCT
ejpam-4744	8	43	)	)	PUNCT
ejpam-4744	8	44	.	.	PUNCT
ejpam-4744	9	1	the	the	DET
ejpam-4744	9	2	intersection	intersection	NOUN
ejpam-4744	9	3	number	number	NOUN
ejpam-4744	9	4	ω(g	ω(g	NOUN
ejpam-4744	9	5	)	)	PUNCT
ejpam-4744	9	6	of	of	ADP
ejpam-4744	9	7	a	a	DET
ejpam-4744	9	8	given	give	VERB
ejpam-4744	9	9	graph	graph	NOUN
ejpam-4744	9	10	g	g	PROPN
ejpam-4744	9	11	is	be	AUX
ejpam-4744	9	12	the	the	DET
ejpam-4744	9	13	minimum	minimum	ADJ
ejpam-4744	9	14	number	number	NOUN
ejpam-4744	9	15	of	of	ADP
ejpam-4744	9	16	elements	element	NOUN
ejpam-4744	9	17	in	in	ADP
ejpam-4744	9	18	a	a	DET
ejpam-4744	9	19	set	set	NOUN
ejpam-4744	9	20	s	s	PRON
ejpam-4744	9	21	such	such	ADJ
ejpam-4744	9	22	that	that	SCONJ
ejpam-4744	9	23	g	g	PROPN
ejpam-4744	9	24	is	be	AUX
ejpam-4744	9	25	an	an	DET
ejpam-4744	9	26	intersection	intersection	NOUN
ejpam-4744	9	27	graph	graph	NOUN
ejpam-4744	9	28	on	on	ADP
ejpam-4744	9	29	s.	s.	PROPN
ejpam-4744	9	30	the	the	DET
ejpam-4744	9	31	intersection	intersection	NOUN
ejpam-4744	9	32	number	number	NOUN
ejpam-4744	9	33	has	have	AUX
ejpam-4744	9	34	been	be	AUX
ejpam-4744	9	35	studied	study	VERB
ejpam-4744	9	36	by	by	ADP
ejpam-4744	9	37	[	[	X
ejpam-4744	9	38	1	1	NUM
ejpam-4744	9	39	]	]	PUNCT
ejpam-4744	9	40	.	.	PUNCT
ejpam-4744	10	1	they	they	PRON
ejpam-4744	10	2	obtained	obtain	VERB
ejpam-4744	10	3	the	the	DET
ejpam-4744	10	4	best	good	ADJ
ejpam-4744	10	5	possible	possible	ADJ
ejpam-4744	10	6	upper	upper	ADJ
ejpam-4744	10	7	bound	bind	VERB
ejpam-4744	10	8	for	for	ADP
ejpam-4744	10	9	the	the	DET
ejpam-4744	10	10	intersection	intersection	NOUN
ejpam-4744	10	11	number	number	NOUN
ejpam-4744	10	12	of	of	ADP
ejpam-4744	10	13	a	a	DET
ejpam-4744	10	14	graph	graph	NOUN
ejpam-4744	10	15	with	with	ADP
ejpam-4744	10	16	a	a	DET
ejpam-4744	10	17	given	give	VERB
ejpam-4744	10	18	number	number	NOUN
ejpam-4744	10	19	of	of	ADP
ejpam-4744	10	20	points	point	NOUN
ejpam-4744	10	21	.	.	PUNCT
ejpam-4744	11	1	in	in	ADP
ejpam-4744	11	2	[	[	X
ejpam-4744	11	3	2	2	NUM
ejpam-4744	11	4	]	]	PUNCT
ejpam-4744	11	5	,	,	PUNCT
ejpam-4744	11	6	frank	frank	PROPN
ejpam-4744	11	7	harary	harary	PROPN
ejpam-4744	11	8	provided	provide	VERB
ejpam-4744	11	9	an	an	DET
ejpam-4744	11	10	upper	upper	ADJ
ejpam-4744	11	11	bound	bind	VERB
ejpam-4744	11	12	for	for	ADP
ejpam-4744	11	13	the	the	DET
ejpam-4744	11	14	intersection	intersection	NOUN
ejpam-4744	11	15	number	number	NOUN
ejpam-4744	11	16	of	of	ADP
ejpam-4744	11	17	a	a	DET
ejpam-4744	11	18	graph	graph	NOUN
ejpam-4744	11	19	g.	g.	NOUN
ejpam-4744	12	1	he	he	PRON
ejpam-4744	12	2	showed	show	VERB
ejpam-4744	12	3	that	that	SCONJ
ejpam-4744	12	4	ω(g	ω(g	NOUN
ejpam-4744	12	5	)	)	PUNCT
ejpam-4744	12	6	≤	≤	PUNCT
ejpam-4744	13	1	|e(g)|	|e(g)|	PROPN
ejpam-4744	13	2	.	.	PUNCT
ejpam-4744	14	1	in	in	ADP
ejpam-4744	14	2	[	[	X
ejpam-4744	14	3	3	3	NUM
ejpam-4744	14	4	]	]	PUNCT
ejpam-4744	14	5	,	,	PUNCT
ejpam-4744	14	6	the	the	DET
ejpam-4744	14	7	authors	author	NOUN
ejpam-4744	14	8	provided	provide	VERB
ejpam-4744	14	9	a	a	DET
ejpam-4744	14	10	lower	lower	ADV
ejpam-4744	14	11	bound	bind	VERB
ejpam-4744	14	12	for	for	ADP
ejpam-4744	14	13	the	the	DET
ejpam-4744	14	14	intersection	intersection	NOUN
ejpam-4744	14	15	number	number	NOUN
ejpam-4744	14	16	of	of	ADP
ejpam-4744	14	17	a	a	DET
ejpam-4744	14	18	graph	graph	NOUN
ejpam-4744	14	19	g.	g.	NOUN
ejpam-4744	15	1	they	they	PRON
ejpam-4744	15	2	showed	show	VERB
ejpam-4744	15	3	that	that	SCONJ
ejpam-4744	15	4	log2(|v	log2(|v	PROPN
ejpam-4744	15	5	(	(	PUNCT
ejpam-4744	15	6	g)|+1	g)|+1	NOUN
ejpam-4744	15	7	)	)	PUNCT
ejpam-4744	15	8	≤	≤	NOUN
ejpam-4744	15	9	ω(g	ω(g	NOUN
ejpam-4744	15	10	)	)	PUNCT
ejpam-4744	15	11	.	.	PUNCT
ejpam-4744	16	1	moreover	moreover	ADV
ejpam-4744	16	2	,	,	PUNCT
ejpam-4744	16	3	the	the	DET
ejpam-4744	16	4	authors	author	NOUN
ejpam-4744	16	5	provided	provide	VERB
ejpam-4744	16	6	formulas	formula	NOUN
ejpam-4744	16	7	for	for	ADP
ejpam-4744	16	8	the	the	DET
ejpam-4744	16	9	intersection	intersection	NOUN
ejpam-4744	16	10	numbers	number	NOUN
ejpam-4744	16	11	of	of	ADP
ejpam-4744	16	12	pn	pn	PROPN
ejpam-4744	16	13	,	,	PUNCT
ejpam-4744	16	14	cn	cn	PROPN
ejpam-4744	16	15	,	,	PUNCT
ejpam-4744	16	16	wn	wn	PROPN
ejpam-4744	16	17	,	,	PUNCT
ejpam-4744	16	18	fn	fn	PROPN
ejpam-4744	16	19	,	,	PUNCT
ejpam-4744	16	20	kn	kn	PROPN
ejpam-4744	16	21	,	,	PUNCT
ejpam-4744	16	22	and	and	CCONJ
ejpam-4744	16	23	g+k1	g+k1	NOUN
ejpam-4744	16	24	for	for	ADP
ejpam-4744	16	25	any	any	DET
ejpam-4744	16	26	connected	connected	ADJ
ejpam-4744	16	27	graph	graph	NOUN
ejpam-4744	16	28	g.	g.	NOUN
ejpam-4744	16	29	they	they	PRON
ejpam-4744	16	30	also	also	ADV
ejpam-4744	16	31	defined	define	VERB
ejpam-4744	16	32	the	the	DET
ejpam-4744	16	33	concept	concept	NOUN
ejpam-4744	16	34	of	of	ADP
ejpam-4744	16	35	an	an	DET
ejpam-4744	16	36	extreme	extreme	ADJ
ejpam-4744	16	37	intersection	intersection	NOUN
ejpam-4744	16	38	graph	graph	NOUN
ejpam-4744	16	39	.	.	PUNCT
ejpam-4744	17	1	a	a	DET
ejpam-4744	17	2	graph	graph	NOUN
ejpam-4744	17	3	g	g	NOUN
ejpam-4744	17	4	is	be	AUX
ejpam-4744	17	5	an	an	DET
ejpam-4744	17	6	extreme	extreme	ADJ
ejpam-4744	17	7	intersection	intersection	NOUN
ejpam-4744	17	8	graph	graph	NOUN
ejpam-4744	17	9	if	if	SCONJ
ejpam-4744	17	10	for	for	ADP
ejpam-4744	17	11	any	any	DET
ejpam-4744	17	12	family	family	NOUN
ejpam-4744	17	13	f	f	NOUN
ejpam-4744	17	14	of	of	ADP
ejpam-4744	17	15	subsets	subset	NOUN
ejpam-4744	17	16	of	of	ADP
ejpam-4744	17	17	s	s	NOUN
ejpam-4744	17	18	=	=	PUNCT
ejpam-4744	17	19	{	{	PUNCT
ejpam-4744	17	20	1	1	NUM
ejpam-4744	17	21	,	,	PUNCT
ejpam-4744	17	22	2	2	NUM
ejpam-4744	17	23	,	,	PUNCT
ejpam-4744	17	24	3	3	NUM
ejpam-4744	17	25	,	,	PUNCT
ejpam-4744	17	26	...	...	PUNCT
ejpam-4744	17	27	,	,	PUNCT
ejpam-4744	17	28	ω(g	ω(g	NOUN
ejpam-4744	17	29	)	)	PUNCT
ejpam-4744	17	30	}	}	PUNCT
ejpam-4744	17	31	such	such	ADJ
ejpam-4744	17	32	that	that	SCONJ
ejpam-4744	17	33	ω(f	ω(f	PROPN
ejpam-4744	17	34	)	)	PUNCT
ejpam-4744	17	35	∼=	∼=	ADP
ejpam-4744	17	36	g	g	NOUN
ejpam-4744	17	37	,	,	PUNCT
ejpam-4744	17	38	then	then	ADV
ejpam-4744	17	39	s	s	VERB
ejpam-4744	17	40	∈	∈	PROPN
ejpam-4744	17	41	f	f	X
ejpam-4744	17	42	.	.	PUNCT
ejpam-4744	18	1	∗corresponding	∗corresponde	VERB
ejpam-4744	18	2	author	author	NOUN
ejpam-4744	18	3	.	.	PUNCT
ejpam-4744	19	1	doi	doi	NOUN
ejpam-4744	19	2	:	:	PUNCT
ejpam-4744	19	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4744	https://doi.org/10.29020/nybg.ejpam.v16i2.4744	ADJ
ejpam-4744	19	4	email	email	NOUN
ejpam-4744	19	5	addresses	address	VERB
ejpam-4744	19	6	:	:	PUNCT
ejpam-4744	20	1	jesrael.palco@msunaawan.edu.ph	jesrael.palco@msunaawan.edu.ph	PROPN
ejpam-4744	20	2	(	(	PUNCT
ejpam-4744	20	3	jesrael	jesrael	PROPN
ejpam-4744	20	4	b.	b.	PROPN
ejpam-4744	20	5	palco	palco	PROPN
ejpam-4744	20	6	)	)	PUNCT
ejpam-4744	20	7	,	,	PUNCT
ejpam-4744	20	8	rnpaluga@carsu.edu.ph	rnpaluga@carsu.edu.ph	NOUN
ejpam-4744	20	9	(	(	PUNCT
ejpam-4744	20	10	rolando	rolando	PROPN
ejpam-4744	20	11	n.	n.	PROPN
ejpam-4744	20	12	paluga	paluga	PROPN
ejpam-4744	20	13	)	)	PUNCT
ejpam-4744	20	14	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4744	20	15	1318	1318	NUM
ejpam-4744	21	1	©	©	PROPN
ejpam-4744	21	2	2023	2023	NUM
ejpam-4744	21	3	ejpam	ejpam	NOUN
ejpam-4744	21	4	all	all	DET
ejpam-4744	21	5	rights	right	NOUN
ejpam-4744	21	6	reserved	reserve	VERB
ejpam-4744	21	7	.	.	PUNCT
ejpam-4744	22	1	j.	j.	PROPN
ejpam-4744	22	2	b.	b.	PROPN
ejpam-4744	22	3	palco	palco	PROPN
ejpam-4744	22	4	,	,	PUNCT
ejpam-4744	22	5	r.	r.	PROPN
ejpam-4744	22	6	n.	n.	PROPN
ejpam-4744	22	7	paluga	paluga	PROPN
ejpam-4744	22	8	/	/	SYM
ejpam-4744	22	9	eur	eur	PROPN
ejpam-4744	22	10	.	.	PUNCT
ejpam-4744	23	1	j.	j.	PROPN
ejpam-4744	23	2	pure	pure	PROPN
ejpam-4744	23	3	appl	appl	PROPN
ejpam-4744	23	4	.	.	PROPN
ejpam-4744	23	5	math	math	PROPN
ejpam-4744	23	6	,	,	PUNCT
ejpam-4744	23	7	16	16	NUM
ejpam-4744	23	8	(	(	PUNCT
ejpam-4744	23	9	2	2	NUM
ejpam-4744	23	10	)	)	PUNCT
ejpam-4744	23	11	(	(	PUNCT
ejpam-4744	23	12	2023	2023	NUM
ejpam-4744	23	13	)	)	PUNCT
ejpam-4744	23	14	,	,	PUNCT
ejpam-4744	23	15	1318	1318	NUM
ejpam-4744	23	16	-	-	SYM
ejpam-4744	23	17	1325	1325	NUM
ejpam-4744	23	18	1319	1319	NUM
ejpam-4744	23	19	2	2	NUM
ejpam-4744	23	20	.	.	PUNCT
ejpam-4744	23	21	results	result	VERB
ejpam-4744	23	22	the	the	DET
ejpam-4744	23	23	join	join	NOUN
ejpam-4744	23	24	of	of	ADP
ejpam-4744	23	25	two	two	NUM
ejpam-4744	23	26	graphs	graph	NOUN
ejpam-4744	23	27	g	g	NOUN
ejpam-4744	23	28	and	and	CCONJ
ejpam-4744	23	29	h	h	NOUN
ejpam-4744	23	30	,	,	PUNCT
ejpam-4744	23	31	denoted	denote	VERB
ejpam-4744	23	32	by	by	ADP
ejpam-4744	23	33	g	g	PROPN
ejpam-4744	23	34	+	+	PROPN
ejpam-4744	23	35	h	h	NOUN
ejpam-4744	23	36	,	,	PUNCT
ejpam-4744	23	37	is	be	AUX
ejpam-4744	23	38	the	the	DET
ejpam-4744	23	39	graph	graph	NOUN
ejpam-4744	23	40	with	with	ADP
ejpam-4744	23	41	v	v	NOUN
ejpam-4744	23	42	(	(	PUNCT
ejpam-4744	23	43	g+h	g+h	NOUN
ejpam-4744	23	44	)	)	PUNCT
ejpam-4744	24	1	=	=	SYM
ejpam-4744	24	2	v	v	X
ejpam-4744	24	3	(	(	PUNCT
ejpam-4744	24	4	g)∪v	g)∪v	NOUN
ejpam-4744	24	5	(	(	PUNCT
ejpam-4744	24	6	h	h	NOUN
ejpam-4744	24	7	)	)	PUNCT
ejpam-4744	24	8	and	and	CCONJ
ejpam-4744	24	9	e(g+h	e(g+h	NUM
ejpam-4744	24	10	)	)	PUNCT
ejpam-4744	24	11	=	=	PRON
ejpam-4744	25	1	e(g)∪e(h)∪{aibj	e(g)∪e(h)∪{aibj	VERB
ejpam-4744	25	2	:	:	PUNCT
ejpam-4744	25	3	ai	ai	VERB
ejpam-4744	25	4	∈	∈	PROPN
ejpam-4744	25	5	v	v	ADP
ejpam-4744	25	6	(	(	PUNCT
ejpam-4744	25	7	g	g	NOUN
ejpam-4744	25	8	)	)	PUNCT
ejpam-4744	25	9	and	and	CCONJ
ejpam-4744	25	10	bj	bj	VERB
ejpam-4744	25	11	∈	∈	PROPN
ejpam-4744	25	12	v	v	NOUN
ejpam-4744	25	13	(	(	PUNCT
ejpam-4744	25	14	h	h	NOUN
ejpam-4744	25	15	)	)	PUNCT
ejpam-4744	25	16	}	}	PUNCT
ejpam-4744	25	17	.	.	PUNCT
ejpam-4744	26	1	theorem	theorem	NOUN
ejpam-4744	26	2	1	1	X
ejpam-4744	26	3	.	.	PUNCT
ejpam-4744	26	4	suppose	suppose	VERB
ejpam-4744	26	5	g	g	PROPN
ejpam-4744	26	6	is	be	AUX
ejpam-4744	26	7	not	not	PART
ejpam-4744	26	8	an	an	DET
ejpam-4744	26	9	extreme	extreme	ADJ
ejpam-4744	26	10	intersection	intersection	NOUN
ejpam-4744	26	11	graph	graph	NOUN
ejpam-4744	26	12	.	.	PUNCT
ejpam-4744	27	1	then	then	ADV
ejpam-4744	27	2	for	for	ADP
ejpam-4744	27	3	any	any	DET
ejpam-4744	27	4	graph	graph	NOUN
ejpam-4744	27	5	h	h	NOUN
ejpam-4744	27	6	,	,	PUNCT
ejpam-4744	27	7	ω(g+h	ω(g+h	NUM
ejpam-4744	27	8	)	)	PUNCT
ejpam-4744	27	9	≤	≤	NUM
ejpam-4744	27	10	ω(g)ω(h	ω(g)ω(h	NOUN
ejpam-4744	27	11	)	)	PUNCT
ejpam-4744	27	12	.	.	PUNCT
ejpam-4744	28	1	proof	proof	NOUN
ejpam-4744	28	2	.	.	PUNCT
ejpam-4744	29	1	let	let	VERB
ejpam-4744	29	2	g	g	NOUN
ejpam-4744	29	3	be	be	AUX
ejpam-4744	29	4	not	not	PART
ejpam-4744	29	5	an	an	DET
ejpam-4744	29	6	extreme	extreme	ADJ
ejpam-4744	29	7	intersection	intersection	NOUN
ejpam-4744	29	8	graph	graph	NOUN
ejpam-4744	29	9	.	.	PUNCT
ejpam-4744	30	1	then	then	ADV
ejpam-4744	30	2	there	there	PRON
ejpam-4744	30	3	exists	exist	VERB
ejpam-4744	30	4	a	a	DET
ejpam-4744	30	5	family	family	NOUN
ejpam-4744	30	6	f1	f1	NOUN
ejpam-4744	30	7	of	of	ADP
ejpam-4744	30	8	nonempty	nonempty	ADJ
ejpam-4744	30	9	subsets	subset	NOUN
ejpam-4744	30	10	of	of	ADP
ejpam-4744	30	11	a	a	DET
ejpam-4744	30	12	set	set	ADJ
ejpam-4744	30	13	s1	s1	NOUN
ejpam-4744	30	14	such	such	ADJ
ejpam-4744	30	15	that	that	DET
ejpam-4744	30	16	s1	s1	PROPN
ejpam-4744	30	17	/∈	/∈	PUNCT
ejpam-4744	30	18	f1	f1	PROPN
ejpam-4744	30	19	and	and	CCONJ
ejpam-4744	30	20	ω(f1	ω(f1	ADJ
ejpam-4744	30	21	)	)	PUNCT
ejpam-4744	30	22	∼=	∼=	NOUN
ejpam-4744	30	23	g.	g.	NOUN
ejpam-4744	30	24	that	that	PRON
ejpam-4744	30	25	is	be	AUX
ejpam-4744	30	26	,	,	PUNCT
ejpam-4744	30	27	there	there	PRON
ejpam-4744	30	28	is	be	VERB
ejpam-4744	30	29	an	an	DET
ejpam-4744	30	30	isomorphism	isomorphism	NOUN
ejpam-4744	30	31	ϕ1	ϕ1	NOUN
ejpam-4744	30	32	:	:	PUNCT
ejpam-4744	30	33	v	v	X
ejpam-4744	30	34	(	(	PUNCT
ejpam-4744	30	35	g	g	NOUN
ejpam-4744	30	36	)	)	PUNCT
ejpam-4744	30	37	→	→	SYM
ejpam-4744	30	38	f1	f1	NOUN
ejpam-4744	30	39	such	such	ADJ
ejpam-4744	30	40	that	that	DET
ejpam-4744	30	41	ϕ1(x	ϕ1(x	NOUN
ejpam-4744	30	42	)	)	PUNCT
ejpam-4744	30	43	̸=	̸=	PROPN
ejpam-4744	30	44	s1	s1	NOUN
ejpam-4744	30	45	,	,	PUNCT
ejpam-4744	30	46	for	for	ADP
ejpam-4744	30	47	all	all	PRON
ejpam-4744	30	48	x	x	SYM
ejpam-4744	30	49	∈	∈	NOUN
ejpam-4744	30	50	v	v	NOUN
ejpam-4744	30	51	(	(	PUNCT
ejpam-4744	30	52	g	g	NOUN
ejpam-4744	30	53	)	)	PUNCT
ejpam-4744	30	54	.	.	PUNCT
ejpam-4744	31	1	let	let	VERB
ejpam-4744	31	2	h	h	PRON
ejpam-4744	31	3	be	be	AUX
ejpam-4744	31	4	any	any	DET
ejpam-4744	31	5	graph	graph	NOUN
ejpam-4744	31	6	and	and	CCONJ
ejpam-4744	31	7	suppose	suppose	VERB
ejpam-4744	31	8	ω(h	ω(h	NUM
ejpam-4744	31	9	)	)	PUNCT
ejpam-4744	32	1	=	=	SYM
ejpam-4744	32	2	m.	m.	NOUN
ejpam-4744	32	3	let	let	VERB
ejpam-4744	32	4	s2	s2	VERB
ejpam-4744	32	5	=	=	PUNCT
ejpam-4744	32	6	{	{	PUNCT
ejpam-4744	32	7	1	1	NUM
ejpam-4744	32	8	,	,	PUNCT
ejpam-4744	32	9	2	2	NUM
ejpam-4744	32	10	,	,	PUNCT
ejpam-4744	32	11	..	..	PUNCT
ejpam-4744	32	12	,	,	PUNCT
ejpam-4744	32	13	m	m	VERB
ejpam-4744	32	14	}	}	PUNCT
ejpam-4744	32	15	and	and	CCONJ
ejpam-4744	32	16	f2	f2	PROPN
ejpam-4744	32	17	be	be	VERB
ejpam-4744	32	18	a	a	DET
ejpam-4744	32	19	nonempty	nonempty	ADJ
ejpam-4744	32	20	subset	subset	NOUN
ejpam-4744	32	21	of	of	ADP
ejpam-4744	32	22	a	a	DET
ejpam-4744	32	23	set	set	VERB
ejpam-4744	32	24	s2	s2	NOUN
ejpam-4744	32	25	for	for	ADP
ejpam-4744	32	26	which	which	PRON
ejpam-4744	32	27	ω(f2	ω(f2	NOUN
ejpam-4744	32	28	)	)	PUNCT
ejpam-4744	32	29	∼=	∼=	PROPN
ejpam-4744	32	30	h.	h.	NOUN
ejpam-4744	32	31	that	that	PRON
ejpam-4744	32	32	is	be	AUX
ejpam-4744	32	33	,	,	PUNCT
ejpam-4744	32	34	there	there	PRON
ejpam-4744	32	35	is	be	VERB
ejpam-4744	32	36	an	an	DET
ejpam-4744	32	37	isomorphism	isomorphism	NOUN
ejpam-4744	32	38	ϕ2	ϕ2	ADV
ejpam-4744	32	39	:	:	PUNCT
ejpam-4744	32	40	v	v	X
ejpam-4744	32	41	(	(	PUNCT
ejpam-4744	32	42	h	h	NOUN
ejpam-4744	32	43	)	)	PUNCT
ejpam-4744	32	44	→	→	SYM
ejpam-4744	32	45	f2	f2	PROPN
ejpam-4744	32	46	.	.	PUNCT
ejpam-4744	33	1	let	let	VERB
ejpam-4744	33	2	s	s	PRON
ejpam-4744	33	3	=	=	SYM
ejpam-4744	33	4	s1×s2	s1×s2	PROPN
ejpam-4744	33	5	,	,	PUNCT
ejpam-4744	33	6	and	and	CCONJ
ejpam-4744	33	7	f	f	X
ejpam-4744	33	8	=	=	PRON
ejpam-4744	33	9	(	(	PUNCT
ejpam-4744	33	10	∪{a	∪{a	X
ejpam-4744	33	11	×	×	PROPN
ejpam-4744	33	12	s2	s2	NOUN
ejpam-4744	33	13	:	:	PUNCT
ejpam-4744	33	14	a	a	DET
ejpam-4744	33	15	∈	∈	PROPN
ejpam-4744	33	16	f1	f1	NOUN
ejpam-4744	33	17	}	}	PUNCT
ejpam-4744	33	18	)	)	PUNCT
ejpam-4744	33	19	∪	∪	NOUN
ejpam-4744	33	20	(	(	PUNCT
ejpam-4744	33	21	∪{s1	∪{s1	NOUN
ejpam-4744	33	22	×	×	NOUN
ejpam-4744	33	23	b	b	NOUN
ejpam-4744	33	24	:	:	PUNCT
ejpam-4744	33	25	b	b	PROPN
ejpam-4744	33	26	∈	∈	PROPN
ejpam-4744	33	27	f2	f2	PROPN
ejpam-4744	33	28	}	}	PUNCT
ejpam-4744	33	29	)	)	PUNCT
ejpam-4744	33	30	.	.	PUNCT
ejpam-4744	34	1	let	let	VERB
ejpam-4744	34	2	ϕ	ϕ	NOUN
ejpam-4744	34	3	:	:	PUNCT
ejpam-4744	34	4	v	v	X
ejpam-4744	34	5	(	(	PUNCT
ejpam-4744	34	6	g	g	PROPN
ejpam-4744	34	7	+	+	NOUN
ejpam-4744	34	8	h	h	NOUN
ejpam-4744	34	9	)	)	PUNCT
ejpam-4744	34	10	→	→	SYM
ejpam-4744	34	11	f	f	X
ejpam-4744	34	12	be	be	AUX
ejpam-4744	34	13	a	a	DET
ejpam-4744	34	14	mapping	mapping	NOUN
ejpam-4744	34	15	defined	define	VERB
ejpam-4744	34	16	by	by	ADP
ejpam-4744	34	17	ϕ(x	ϕ(x	NOUN
ejpam-4744	34	18	)	)	PUNCT
ejpam-4744	35	1	=	=	PRON
ejpam-4744	35	2	{	{	PUNCT
ejpam-4744	35	3	ϕ1(x)×	ϕ1(x)×	PROPN
ejpam-4744	35	4	s2	s2	PROPN
ejpam-4744	35	5	,	,	PUNCT
ejpam-4744	35	6	if	if	SCONJ
ejpam-4744	35	7	x	x	PROPN
ejpam-4744	35	8	∈	∈	PROPN
ejpam-4744	35	9	v	v	ADP
ejpam-4744	35	10	(	(	PUNCT
ejpam-4744	35	11	g	g	NOUN
ejpam-4744	35	12	)	)	PUNCT
ejpam-4744	35	13	s1	s1	PROPN
ejpam-4744	35	14	×	×	PROPN
ejpam-4744	35	15	ϕ2(x	ϕ2(x	PROPN
ejpam-4744	35	16	)	)	PUNCT
ejpam-4744	35	17	,	,	PUNCT
ejpam-4744	35	18	if	if	SCONJ
ejpam-4744	35	19	x	x	SYM
ejpam-4744	35	20	∈	∈	PROPN
ejpam-4744	35	21	v	v	X
ejpam-4744	35	22	(	(	PUNCT
ejpam-4744	35	23	h	h	NOUN
ejpam-4744	35	24	)	)	PUNCT
ejpam-4744	35	25	.	.	PUNCT
ejpam-4744	36	1	let	let	VERB
ejpam-4744	36	2	x1	x1	NUM
ejpam-4744	36	3	,	,	PUNCT
ejpam-4744	36	4	x2	x2	PROPN
ejpam-4744	36	5	∈	∈	PROPN
ejpam-4744	36	6	v	v	NOUN
ejpam-4744	36	7	(	(	PUNCT
ejpam-4744	36	8	g+h	g+h	NOUN
ejpam-4744	36	9	)	)	PUNCT
ejpam-4744	36	10	such	such	ADJ
ejpam-4744	36	11	that	that	SCONJ
ejpam-4744	36	12	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	36	13	)	)	PUNCT
ejpam-4744	36	14	=	=	SYM
ejpam-4744	36	15	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	36	16	)	)	PUNCT
ejpam-4744	36	17	.	.	PUNCT
ejpam-4744	37	1	the	the	DET
ejpam-4744	37	2	case	case	NOUN
ejpam-4744	37	3	x1	x1	PROPN
ejpam-4744	37	4	∈	∈	PROPN
ejpam-4744	37	5	v	v	ADP
ejpam-4744	37	6	(	(	PUNCT
ejpam-4744	37	7	g	g	NOUN
ejpam-4744	37	8	)	)	PUNCT
ejpam-4744	37	9	and	and	CCONJ
ejpam-4744	37	10	x2	x2	PROPN
ejpam-4744	37	11	∈	∈	PROPN
ejpam-4744	37	12	v	v	ADP
ejpam-4744	37	13	(	(	PUNCT
ejpam-4744	37	14	h	h	NOUN
ejpam-4744	37	15	)	)	PUNCT
ejpam-4744	37	16	is	be	AUX
ejpam-4744	37	17	not	not	PART
ejpam-4744	37	18	possible	possible	ADJ
ejpam-4744	37	19	.	.	PUNCT
ejpam-4744	38	1	since	since	SCONJ
ejpam-4744	38	2	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	38	3	)	)	PUNCT
ejpam-4744	38	4	=	=	SYM
ejpam-4744	38	5	ϕ1(x1)×s2	ϕ1(x1)×s2	PROPN
ejpam-4744	38	6	and	and	CCONJ
ejpam-4744	38	7	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	38	8	)	)	PUNCT
ejpam-4744	38	9	=	=	SYM
ejpam-4744	38	10	s1×ϕ2(x2	s1×ϕ2(x2	NOUN
ejpam-4744	38	11	)	)	PUNCT
ejpam-4744	38	12	.	.	PUNCT
ejpam-4744	39	1	consider	consider	VERB
ejpam-4744	39	2	the	the	DET
ejpam-4744	39	3	following	follow	VERB
ejpam-4744	39	4	cases	case	NOUN
ejpam-4744	39	5	:	:	PUNCT
ejpam-4744	39	6	case	case	NOUN
ejpam-4744	39	7	1	1	NUM
ejpam-4744	39	8	.	.	PUNCT
ejpam-4744	39	9	suppose	suppose	VERB
ejpam-4744	39	10	x1	x1	NUM
ejpam-4744	39	11	,	,	PUNCT
ejpam-4744	39	12	x2	x2	PROPN
ejpam-4744	39	13	∈	∈	PROPN
ejpam-4744	39	14	v	v	ADP
ejpam-4744	39	15	(	(	PUNCT
ejpam-4744	39	16	g	g	NOUN
ejpam-4744	39	17	)	)	PUNCT
ejpam-4744	39	18	.	.	PUNCT
ejpam-4744	40	1	then	then	ADV
ejpam-4744	40	2	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	40	3	)	)	PUNCT
ejpam-4744	41	1	=	=	SYM
ejpam-4744	41	2	ϕ1(x1	ϕ1(x1	ADJ
ejpam-4744	41	3	)	)	PUNCT
ejpam-4744	41	4	×	×	NOUN
ejpam-4744	41	5	s2	s2	NOUN
ejpam-4744	41	6	and	and	CCONJ
ejpam-4744	41	7	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	41	8	)	)	PUNCT
ejpam-4744	42	1	=	=	SYM
ejpam-4744	42	2	ϕ1(x2	ϕ1(x2	PROPN
ejpam-4744	42	3	)	)	PUNCT
ejpam-4744	42	4	×	×	PROPN
ejpam-4744	42	5	s2	s2	PROPN
ejpam-4744	42	6	.	.	PUNCT
ejpam-4744	43	1	note	note	VERB
ejpam-4744	43	2	that	that	SCONJ
ejpam-4744	43	3	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	43	4	)	)	PUNCT
ejpam-4744	43	5	=	=	SYM
ejpam-4744	43	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	43	7	)	)	PUNCT
ejpam-4744	43	8	,	,	PUNCT
ejpam-4744	43	9	so	so	ADV
ejpam-4744	43	10	we	we	PRON
ejpam-4744	43	11	have	have	VERB
ejpam-4744	43	12	ϕ1(x1	ϕ1(x1	NOUN
ejpam-4744	43	13	)	)	PUNCT
ejpam-4744	43	14	=	=	SYM
ejpam-4744	43	15	ϕ1(x2	ϕ1(x2	PROPN
ejpam-4744	43	16	)	)	PUNCT
ejpam-4744	43	17	.	.	PUNCT
ejpam-4744	44	1	since	since	SCONJ
ejpam-4744	44	2	ϕ1	ϕ1	PROPN
ejpam-4744	44	3	is	be	AUX
ejpam-4744	44	4	one	one	NUM
ejpam-4744	44	5	to	to	ADP
ejpam-4744	44	6	one	one	NUM
ejpam-4744	44	7	,	,	PUNCT
ejpam-4744	44	8	x1	x1	PROPN
ejpam-4744	44	9	=	=	SYM
ejpam-4744	44	10	x2	x2	PROPN
ejpam-4744	44	11	.	.	PUNCT
ejpam-4744	45	1	case	case	NOUN
ejpam-4744	45	2	2	2	X
ejpam-4744	45	3	.	.	PUNCT
ejpam-4744	45	4	suppose	suppose	VERB
ejpam-4744	46	1	x1	x1	PROPN
ejpam-4744	46	2	,	,	PUNCT
ejpam-4744	46	3	x2	x2	PROPN
ejpam-4744	46	4	∈	∈	PROPN
ejpam-4744	46	5	v	v	ADP
ejpam-4744	46	6	(	(	PUNCT
ejpam-4744	46	7	h	h	NOUN
ejpam-4744	46	8	)	)	PUNCT
ejpam-4744	46	9	.	.	PUNCT
ejpam-4744	47	1	then	then	ADV
ejpam-4744	47	2	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	47	3	)	)	PUNCT
ejpam-4744	48	1	=	=	PUNCT
ejpam-4744	48	2	s1	s1	PROPN
ejpam-4744	48	3	×	×	PROPN
ejpam-4744	48	4	ϕ2(x1	ϕ2(x1	PROPN
ejpam-4744	48	5	)	)	PUNCT
ejpam-4744	48	6	and	and	CCONJ
ejpam-4744	48	7	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	48	8	)	)	PUNCT
ejpam-4744	49	1	=	=	SYM
ejpam-4744	49	2	s1	s1	PROPN
ejpam-4744	49	3	×	×	PROPN
ejpam-4744	49	4	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	49	5	)	)	PUNCT
ejpam-4744	49	6	.	.	PUNCT
ejpam-4744	50	1	note	note	VERB
ejpam-4744	50	2	that	that	SCONJ
ejpam-4744	50	3	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	50	4	)	)	PUNCT
ejpam-4744	50	5	=	=	SYM
ejpam-4744	50	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	50	7	)	)	PUNCT
ejpam-4744	50	8	,	,	PUNCT
ejpam-4744	50	9	so	so	ADV
ejpam-4744	50	10	we	we	PRON
ejpam-4744	50	11	have	have	VERB
ejpam-4744	50	12	ϕ2(x1	ϕ2(x1	ADP
ejpam-4744	50	13	)	)	PUNCT
ejpam-4744	50	14	=	=	SYM
ejpam-4744	50	15	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	50	16	)	)	PUNCT
ejpam-4744	50	17	.	.	PUNCT
ejpam-4744	51	1	since	since	SCONJ
ejpam-4744	51	2	ϕ2	ϕ2	ADV
ejpam-4744	51	3	is	be	AUX
ejpam-4744	51	4	one	one	NUM
ejpam-4744	51	5	to	to	ADP
ejpam-4744	51	6	one	one	NUM
ejpam-4744	51	7	,	,	PUNCT
ejpam-4744	51	8	x1	x1	PROPN
ejpam-4744	51	9	=	=	SYM
ejpam-4744	51	10	x2	x2	PROPN
ejpam-4744	51	11	.	.	PUNCT
ejpam-4744	52	1	therefore	therefore	ADV
ejpam-4744	52	2	,	,	PUNCT
ejpam-4744	52	3	ϕ	ϕ	PROPN
ejpam-4744	52	4	is	be	AUX
ejpam-4744	52	5	one	one	NUM
ejpam-4744	52	6	to	to	ADP
ejpam-4744	52	7	one	one	NUM
ejpam-4744	52	8	.	.	PUNCT
ejpam-4744	53	1	let	let	VERB
ejpam-4744	53	2	u	u	PRON
ejpam-4744	53	3	∈	∈	PROPN
ejpam-4744	53	4	f	f	X
ejpam-4744	53	5	.	.	PUNCT
ejpam-4744	54	1	if	if	SCONJ
ejpam-4744	54	2	u	u	PROPN
ejpam-4744	54	3	=	=	SYM
ejpam-4744	54	4	s1×b	s1×b	NOUN
ejpam-4744	54	5	,	,	PUNCT
ejpam-4744	54	6	b	b	PROPN
ejpam-4744	54	7	∈	∈	PROPN
ejpam-4744	54	8	f2	f2	PROPN
ejpam-4744	54	9	.	.	PUNCT
ejpam-4744	55	1	since	since	SCONJ
ejpam-4744	55	2	ϕ2	ϕ2	ADV
ejpam-4744	55	3	is	be	AUX
ejpam-4744	55	4	onto	onto	ADP
ejpam-4744	55	5	,	,	PUNCT
ejpam-4744	55	6	there	there	PRON
ejpam-4744	55	7	exists	exist	VERB
ejpam-4744	55	8	x	x	X
ejpam-4744	55	9	∈	∈	PROPN
ejpam-4744	55	10	v	v	ADP
ejpam-4744	55	11	(	(	PUNCT
ejpam-4744	55	12	h	h	NOUN
ejpam-4744	55	13	)	)	PUNCT
ejpam-4744	55	14	⊆	⊆	NUM
ejpam-4744	55	15	v	v	NOUN
ejpam-4744	55	16	(	(	PUNCT
ejpam-4744	55	17	g+h	g+h	NOUN
ejpam-4744	55	18	)	)	PUNCT
ejpam-4744	55	19	such	such	ADJ
ejpam-4744	55	20	that	that	PRON
ejpam-4744	55	21	ϕ2(x	ϕ2(x	NOUN
ejpam-4744	55	22	)	)	PUNCT
ejpam-4744	55	23	=	=	SYM
ejpam-4744	55	24	b.	b.	PROPN
ejpam-4744	55	25	thus	thus	ADV
ejpam-4744	55	26	,	,	PUNCT
ejpam-4744	55	27	ϕ(x	ϕ(x	X
ejpam-4744	55	28	)	)	PUNCT
ejpam-4744	56	1	=	=	SYM
ejpam-4744	56	2	s1	s1	PROPN
ejpam-4744	56	3	×	×	PROPN
ejpam-4744	56	4	ϕ2(x	ϕ2(x	PROPN
ejpam-4744	56	5	)	)	PUNCT
ejpam-4744	57	1	=	=	SYM
ejpam-4744	57	2	s1	s1	NOUN
ejpam-4744	57	3	×b	×b	NOUN
ejpam-4744	57	4	=	=	PUNCT
ejpam-4744	57	5	u.	u.	PROPN
ejpam-4744	57	6	therefore	therefore	ADV
ejpam-4744	57	7	,	,	PUNCT
ejpam-4744	57	8	ϕ	ϕ	PROPN
ejpam-4744	57	9	is	be	AUX
ejpam-4744	57	10	onto	onto	ADP
ejpam-4744	57	11	.	.	PUNCT
ejpam-4744	58	1	if	if	SCONJ
ejpam-4744	58	2	u	u	PROPN
ejpam-4744	58	3	=	=	SYM
ejpam-4744	58	4	a×	a×	PROPN
ejpam-4744	58	5	s2	s2	PROPN
ejpam-4744	58	6	,	,	PUNCT
ejpam-4744	58	7	a	a	DET
ejpam-4744	58	8	∈	∈	PROPN
ejpam-4744	58	9	f1	f1	NOUN
ejpam-4744	58	10	.	.	PUNCT
ejpam-4744	59	1	since	since	SCONJ
ejpam-4744	59	2	ϕ1	ϕ1	PROPN
ejpam-4744	59	3	is	be	AUX
ejpam-4744	59	4	onto	onto	ADP
ejpam-4744	59	5	,	,	PUNCT
ejpam-4744	59	6	there	there	PRON
ejpam-4744	59	7	exists	exist	VERB
ejpam-4744	59	8	x	x	X
ejpam-4744	59	9	∈	∈	PROPN
ejpam-4744	59	10	v	v	X
ejpam-4744	59	11	(	(	PUNCT
ejpam-4744	59	12	g	g	NOUN
ejpam-4744	59	13	)	)	PUNCT
ejpam-4744	59	14	⊆	⊆	NUM
ejpam-4744	59	15	v	v	NOUN
ejpam-4744	59	16	(	(	PUNCT
ejpam-4744	59	17	g+h	g+h	NOUN
ejpam-4744	59	18	)	)	PUNCT
ejpam-4744	59	19	such	such	ADJ
ejpam-4744	59	20	that	that	PRON
ejpam-4744	59	21	ϕ1(x	ϕ1(x	NOUN
ejpam-4744	59	22	)	)	PUNCT
ejpam-4744	59	23	=	=	SYM
ejpam-4744	59	24	a.	a.	NOUN
ejpam-4744	59	25	thus	thus	ADV
ejpam-4744	59	26	,	,	PUNCT
ejpam-4744	59	27	ϕ(x	ϕ(x	X
ejpam-4744	59	28	)	)	PUNCT
ejpam-4744	59	29	=	=	SYM
ejpam-4744	60	1	ϕ1(x)×	ϕ1(x)×	PROPN
ejpam-4744	60	2	s2	s2	NOUN
ejpam-4744	60	3	=	=	PUNCT
ejpam-4744	60	4	a×	a×	PROPN
ejpam-4744	60	5	s2	s2	NOUN
ejpam-4744	60	6	=	=	PUNCT
ejpam-4744	60	7	u.	u.	PROPN
ejpam-4744	60	8	therefore	therefore	ADV
ejpam-4744	60	9	,	,	PUNCT
ejpam-4744	60	10	ϕ	ϕ	PROPN
ejpam-4744	60	11	is	be	AUX
ejpam-4744	60	12	onto	onto	ADP
ejpam-4744	60	13	.	.	PUNCT
ejpam-4744	61	1	let	let	VERB
ejpam-4744	61	2	x1	x1	PROPN
ejpam-4744	61	3	and	and	CCONJ
ejpam-4744	61	4	x2	x2	PROPN
ejpam-4744	61	5	be	be	VERB
ejpam-4744	61	6	adjacent	adjacent	ADJ
ejpam-4744	61	7	in	in	ADP
ejpam-4744	61	8	g+h	g+h	PROPN
ejpam-4744	61	9	.	.	PUNCT
ejpam-4744	62	1	consider	consider	VERB
ejpam-4744	62	2	the	the	DET
ejpam-4744	62	3	following	follow	VERB
ejpam-4744	62	4	cases	case	NOUN
ejpam-4744	62	5	:	:	PUNCT
ejpam-4744	62	6	case	case	NOUN
ejpam-4744	62	7	1	1	NUM
ejpam-4744	62	8	.	.	PUNCT
ejpam-4744	62	9	suppose	suppose	VERB
ejpam-4744	62	10	x1	x1	PROPN
ejpam-4744	62	11	and	and	CCONJ
ejpam-4744	62	12	x2	x2	PROPN
ejpam-4744	62	13	are	be	AUX
ejpam-4744	62	14	adjacent	adjacent	ADJ
ejpam-4744	62	15	in	in	ADP
ejpam-4744	62	16	g.	g.	PROPN
ejpam-4744	62	17	then	then	ADV
ejpam-4744	62	18	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	62	19	)	)	PUNCT
ejpam-4744	63	1	=	=	SYM
ejpam-4744	63	2	ϕ1(x1	ϕ1(x1	ADJ
ejpam-4744	63	3	)	)	PUNCT
ejpam-4744	63	4	×	×	NOUN
ejpam-4744	63	5	s2	s2	NOUN
ejpam-4744	63	6	and	and	CCONJ
ejpam-4744	63	7	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	63	8	)	)	PUNCT
ejpam-4744	63	9	=	=	SYM
ejpam-4744	63	10	ϕ1(x2)×	ϕ1(x2)×	NUM
ejpam-4744	63	11	s2	s2	PROPN
ejpam-4744	63	12	.	.	PUNCT
ejpam-4744	64	1	now	now	ADV
ejpam-4744	64	2	,	,	PUNCT
ejpam-4744	64	3	ϕ(x1	ϕ(x1	ADJ
ejpam-4744	64	4	)	)	PUNCT
ejpam-4744	64	5	∩	∩	ADJ
ejpam-4744	64	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	64	7	)	)	PUNCT
ejpam-4744	64	8	=	=	SYM
ejpam-4744	64	9	(	(	PUNCT
ejpam-4744	64	10	ϕ1(x1)×	ϕ1(x1)×	PROPN
ejpam-4744	64	11	s2	s2	PROPN
ejpam-4744	64	12	)	)	PUNCT
ejpam-4744	65	1	∩	∩	NOUN
ejpam-4744	65	2	(	(	PUNCT
ejpam-4744	65	3	ϕ1(x2)×	ϕ1(x2)×	X
ejpam-4744	65	4	s2	s2	PROPN
ejpam-4744	65	5	)	)	PUNCT
ejpam-4744	65	6	=	=	PUNCT
ejpam-4744	65	7	(	(	PUNCT
ejpam-4744	65	8	ϕ1(x1	ϕ1(x1	NOUN
ejpam-4744	65	9	)	)	PUNCT
ejpam-4744	65	10	∩	∩	NOUN
ejpam-4744	65	11	ϕ1(x2))×	ϕ1(x2))×	PROPN
ejpam-4744	65	12	s2	s2	VERB
ejpam-4744	65	13	̸=	̸=	PROPN
ejpam-4744	65	14	∅	∅	NOUN
ejpam-4744	65	15	,	,	PUNCT
ejpam-4744	65	16	since	since	SCONJ
ejpam-4744	65	17	ϕ1	ϕ1	PROPN
ejpam-4744	65	18	preserves	preserve	VERB
ejpam-4744	65	19	adjacency	adjacency	NOUN
ejpam-4744	65	20	.	.	PUNCT
ejpam-4744	66	1	therefore	therefore	ADV
ejpam-4744	66	2	,	,	PUNCT
ejpam-4744	66	3	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	66	4	)	)	PUNCT
ejpam-4744	66	5	and	and	CCONJ
ejpam-4744	66	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	66	7	)	)	PUNCT
ejpam-4744	66	8	are	be	AUX
ejpam-4744	66	9	adjacent	adjacent	ADJ
ejpam-4744	66	10	in	in	ADP
ejpam-4744	66	11	ω(f	ω(f	PROPN
ejpam-4744	66	12	)	)	PUNCT
ejpam-4744	66	13	.	.	PUNCT
ejpam-4744	67	1	case	case	NOUN
ejpam-4744	67	2	2	2	X
ejpam-4744	67	3	.	.	PUNCT
ejpam-4744	67	4	suppose	suppose	VERB
ejpam-4744	67	5	x1	x1	PROPN
ejpam-4744	67	6	and	and	CCONJ
ejpam-4744	67	7	x2	x2	PROPN
ejpam-4744	67	8	are	be	AUX
ejpam-4744	67	9	adjacent	adjacent	ADJ
ejpam-4744	67	10	in	in	ADP
ejpam-4744	67	11	h.	h.	PROPN
ejpam-4744	67	12	then	then	ADV
ejpam-4744	67	13	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	67	14	)	)	PUNCT
ejpam-4744	68	1	=	=	SYM
ejpam-4744	68	2	s1	s1	PROPN
ejpam-4744	68	3	×	×	PROPN
ejpam-4744	68	4	ϕ2(x1	ϕ2(x1	PROPN
ejpam-4744	68	5	)	)	PUNCT
ejpam-4744	68	6	and	and	CCONJ
ejpam-4744	68	7	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	68	8	)	)	PUNCT
ejpam-4744	69	1	=	=	SYM
ejpam-4744	69	2	s1	s1	PROPN
ejpam-4744	69	3	×	×	PROPN
ejpam-4744	69	4	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	69	5	)	)	PUNCT
ejpam-4744	69	6	.	.	PUNCT
ejpam-4744	70	1	now	now	ADV
ejpam-4744	70	2	,	,	PUNCT
ejpam-4744	70	3	ϕ(x1	ϕ(x1	ADJ
ejpam-4744	70	4	)	)	PUNCT
ejpam-4744	70	5	∩	∩	ADJ
ejpam-4744	70	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	70	7	)	)	PUNCT
ejpam-4744	70	8	=	=	SYM
ejpam-4744	71	1	(	(	PUNCT
ejpam-4744	71	2	s1	s1	PROPN
ejpam-4744	71	3	×	×	PROPN
ejpam-4744	71	4	ϕ2(x1	ϕ2(x1	PROPN
ejpam-4744	71	5	)	)	PUNCT
ejpam-4744	71	6	)	)	PUNCT
ejpam-4744	71	7	∩	∩	NOUN
ejpam-4744	71	8	(	(	PUNCT
ejpam-4744	71	9	s1	s1	PROPN
ejpam-4744	71	10	×	×	PROPN
ejpam-4744	71	11	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	71	12	)	)	PUNCT
ejpam-4744	71	13	)	)	PUNCT
ejpam-4744	72	1	=	=	PUNCT
ejpam-4744	72	2	s1	s1	PROPN
ejpam-4744	72	3	×	×	NOUN
ejpam-4744	72	4	(	(	PUNCT
ejpam-4744	72	5	ϕ2(x1	ϕ2(x1	PROPN
ejpam-4744	72	6	)	)	PUNCT
ejpam-4744	72	7	∩	∩	ADJ
ejpam-4744	72	8	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	72	9	)	)	PUNCT
ejpam-4744	72	10	)	)	PUNCT
ejpam-4744	73	1	j.	j.	PROPN
ejpam-4744	73	2	b.	b.	PROPN
ejpam-4744	73	3	palco	palco	PROPN
ejpam-4744	73	4	,	,	PUNCT
ejpam-4744	73	5	r.	r.	PROPN
ejpam-4744	73	6	n.	n.	PROPN
ejpam-4744	73	7	paluga	paluga	PROPN
ejpam-4744	73	8	/	/	SYM
ejpam-4744	73	9	eur	eur	PROPN
ejpam-4744	73	10	.	.	PUNCT
ejpam-4744	74	1	j.	j.	PROPN
ejpam-4744	74	2	pure	pure	PROPN
ejpam-4744	74	3	appl	appl	PROPN
ejpam-4744	74	4	.	.	PROPN
ejpam-4744	74	5	math	math	PROPN
ejpam-4744	74	6	,	,	PUNCT
ejpam-4744	74	7	16	16	NUM
ejpam-4744	74	8	(	(	PUNCT
ejpam-4744	74	9	2	2	NUM
ejpam-4744	74	10	)	)	PUNCT
ejpam-4744	74	11	(	(	PUNCT
ejpam-4744	74	12	2023	2023	NUM
ejpam-4744	74	13	)	)	PUNCT
ejpam-4744	74	14	,	,	PUNCT
ejpam-4744	74	15	1318	1318	NUM
ejpam-4744	74	16	-	-	SYM
ejpam-4744	74	17	1325	1325	NUM
ejpam-4744	74	18	1320	1320	NUM
ejpam-4744	74	19	̸=	̸=	PROPN
ejpam-4744	74	20	∅	∅	NOUN
ejpam-4744	74	21	,	,	PUNCT
ejpam-4744	74	22	since	since	SCONJ
ejpam-4744	74	23	ϕ2	ϕ2	ADV
ejpam-4744	74	24	preserves	preserve	VERB
ejpam-4744	74	25	adjacency	adjacency	NOUN
ejpam-4744	74	26	.	.	PUNCT
ejpam-4744	75	1	therefore	therefore	ADV
ejpam-4744	75	2	,	,	PUNCT
ejpam-4744	75	3	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	75	4	)	)	PUNCT
ejpam-4744	75	5	and	and	CCONJ
ejpam-4744	75	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	75	7	)	)	PUNCT
ejpam-4744	75	8	are	be	AUX
ejpam-4744	75	9	adjacent	adjacent	ADJ
ejpam-4744	75	10	ω(f	ω(f	PROPN
ejpam-4744	75	11	)	)	PUNCT
ejpam-4744	75	12	.	.	PUNCT
ejpam-4744	76	1	case	case	NOUN
ejpam-4744	76	2	3	3	X
ejpam-4744	76	3	.	.	PUNCT
ejpam-4744	76	4	suppose	suppose	VERB
ejpam-4744	76	5	x1	x1	PROPN
ejpam-4744	76	6	∈	∈	PROPN
ejpam-4744	76	7	v	v	ADP
ejpam-4744	76	8	(	(	PUNCT
ejpam-4744	76	9	g	g	NOUN
ejpam-4744	76	10	)	)	PUNCT
ejpam-4744	76	11	and	and	CCONJ
ejpam-4744	76	12	x2	x2	PROPN
ejpam-4744	76	13	∈	∈	PROPN
ejpam-4744	76	14	v	v	ADP
ejpam-4744	76	15	(	(	PUNCT
ejpam-4744	76	16	h	h	NOUN
ejpam-4744	76	17	)	)	PUNCT
ejpam-4744	76	18	.	.	PUNCT
ejpam-4744	77	1	then	then	ADV
ejpam-4744	77	2	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	77	3	)	)	PUNCT
ejpam-4744	78	1	=	=	SYM
ejpam-4744	78	2	ϕ1(x1	ϕ1(x1	ADJ
ejpam-4744	78	3	)	)	PUNCT
ejpam-4744	78	4	×	×	NOUN
ejpam-4744	78	5	s2	s2	NOUN
ejpam-4744	78	6	and	and	CCONJ
ejpam-4744	78	7	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	78	8	)	)	PUNCT
ejpam-4744	79	1	=	=	SYM
ejpam-4744	79	2	s1	s1	PROPN
ejpam-4744	79	3	×	×	PROPN
ejpam-4744	79	4	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	79	5	)	)	PUNCT
ejpam-4744	79	6	.	.	PUNCT
ejpam-4744	80	1	now	now	ADV
ejpam-4744	80	2	,	,	PUNCT
ejpam-4744	80	3	ϕ(x1	ϕ(x1	ADJ
ejpam-4744	80	4	)	)	PUNCT
ejpam-4744	80	5	∩	∩	ADJ
ejpam-4744	80	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	80	7	)	)	PUNCT
ejpam-4744	80	8	=	=	SYM
ejpam-4744	80	9	(	(	PUNCT
ejpam-4744	80	10	ϕ1(x1)×	ϕ1(x1)×	PROPN
ejpam-4744	80	11	s2	s2	PROPN
ejpam-4744	80	12	)	)	PUNCT
ejpam-4744	81	1	∩	∩	NOUN
ejpam-4744	81	2	(	(	PUNCT
ejpam-4744	81	3	s1	s1	PROPN
ejpam-4744	81	4	×	×	PROPN
ejpam-4744	81	5	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	81	6	)	)	PUNCT
ejpam-4744	81	7	)	)	PUNCT
ejpam-4744	82	1	=	=	PRON
ejpam-4744	82	2	(	(	PUNCT
ejpam-4744	82	3	ϕ1(x1	ϕ1(x1	ADJ
ejpam-4744	82	4	)	)	PUNCT
ejpam-4744	82	5	∩	∩	NOUN
ejpam-4744	82	6	s1)×	s1)×	PROPN
ejpam-4744	82	7	(	(	PUNCT
ejpam-4744	82	8	s2	s2	VERB
ejpam-4744	82	9	∩	∩	ADJ
ejpam-4744	82	10	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	82	11	)	)	PUNCT
ejpam-4744	82	12	)	)	PUNCT
ejpam-4744	83	1	=	=	SYM
ejpam-4744	84	1	ϕ1(x1)×	ϕ1(x1)×	PROPN
ejpam-4744	84	2	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	84	3	)	)	PUNCT
ejpam-4744	84	4	,	,	PUNCT
ejpam-4744	84	5	since	since	SCONJ
ejpam-4744	84	6	ϕ1(x1	ϕ1(x1	ADJ
ejpam-4744	84	7	)	)	PUNCT
ejpam-4744	84	8	⊆	⊆	NUM
ejpam-4744	84	9	s1	s1	NOUN
ejpam-4744	84	10	and	and	CCONJ
ejpam-4744	84	11	ϕ2(x2	ϕ2(x2	NOUN
ejpam-4744	84	12	)	)	PUNCT
ejpam-4744	85	1	⊆	⊆	NUM
ejpam-4744	85	2	s2	s2	NOUN
ejpam-4744	85	3	̸=	̸=	PROPN
ejpam-4744	85	4	∅.	∅.	PRON
ejpam-4744	85	5	therefore	therefore	ADV
ejpam-4744	85	6	,	,	PUNCT
ejpam-4744	85	7	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	85	8	)	)	PUNCT
ejpam-4744	85	9	and	and	CCONJ
ejpam-4744	85	10	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	85	11	)	)	PUNCT
ejpam-4744	85	12	are	be	AUX
ejpam-4744	85	13	adjacent	adjacent	ADJ
ejpam-4744	85	14	ω(f	ω(f	PROPN
ejpam-4744	85	15	)	)	PUNCT
ejpam-4744	85	16	.	.	PUNCT
ejpam-4744	86	1	let	let	VERB
ejpam-4744	86	2	u	u	NOUN
ejpam-4744	86	3	,	,	PUNCT
ejpam-4744	86	4	v	v	PROPN
ejpam-4744	86	5	∈	∈	NOUN
ejpam-4744	86	6	f	f	NOUN
ejpam-4744	86	7	.	.	PUNCT
ejpam-4744	87	1	if	if	SCONJ
ejpam-4744	87	2	u	u	PRON
ejpam-4744	87	3	=	=	PUNCT
ejpam-4744	87	4	a	a	DET
ejpam-4744	87	5	×	×	NOUN
ejpam-4744	87	6	s2	s2	NOUN
ejpam-4744	87	7	and	and	CCONJ
ejpam-4744	87	8	v	v	NOUN
ejpam-4744	87	9	=	=	SYM
ejpam-4744	87	10	s1	s1	PROPN
ejpam-4744	87	11	×	×	PROPN
ejpam-4744	87	12	b	b	PROPN
ejpam-4744	87	13	for	for	ADP
ejpam-4744	87	14	some	some	DET
ejpam-4744	87	15	a	a	DET
ejpam-4744	87	16	∈	∈	PROPN
ejpam-4744	87	17	f1	f1	NOUN
ejpam-4744	87	18	and	and	CCONJ
ejpam-4744	87	19	b	b	PROPN
ejpam-4744	87	20	∈	∈	PROPN
ejpam-4744	87	21	f2	f2	PROPN
ejpam-4744	87	22	,	,	PUNCT
ejpam-4744	87	23	then	then	ADV
ejpam-4744	87	24	u	u	X
ejpam-4744	87	25	=	=	PROPN
ejpam-4744	87	26	ϕ1(x	ϕ1(x	PROPN
ejpam-4744	87	27	)	)	PUNCT
ejpam-4744	87	28	×	×	NOUN
ejpam-4744	87	29	s2	s2	NOUN
ejpam-4744	87	30	and	and	CCONJ
ejpam-4744	87	31	v	v	NOUN
ejpam-4744	87	32	=	=	SYM
ejpam-4744	87	33	s1	s1	PROPN
ejpam-4744	87	34	×	×	NOUN
ejpam-4744	87	35	ϕ2(y	ϕ2(y	NOUN
ejpam-4744	87	36	)	)	PUNCT
ejpam-4744	87	37	for	for	ADP
ejpam-4744	87	38	some	some	PRON
ejpam-4744	87	39	x	x	SYM
ejpam-4744	87	40	∈	∈	PROPN
ejpam-4744	87	41	v	v	ADP
ejpam-4744	87	42	(	(	PUNCT
ejpam-4744	87	43	g	g	NOUN
ejpam-4744	87	44	)	)	PUNCT
ejpam-4744	87	45	and	and	CCONJ
ejpam-4744	87	46	y	y	PROPN
ejpam-4744	87	47	∈	∈	PROPN
ejpam-4744	87	48	v	v	ADP
ejpam-4744	87	49	(	(	PUNCT
ejpam-4744	87	50	h	h	NOUN
ejpam-4744	87	51	)	)	PUNCT
ejpam-4744	87	52	.	.	PUNCT
ejpam-4744	88	1	consequently	consequently	ADV
ejpam-4744	88	2	,	,	PUNCT
ejpam-4744	88	3	ϕ−1(u	ϕ−1(u	PROPN
ejpam-4744	88	4	)	)	PUNCT
ejpam-4744	89	1	=	=	PUNCT
ejpam-4744	89	2	x	x	PUNCT
ejpam-4744	89	3	∈	∈	PROPN
ejpam-4744	89	4	v	v	ADP
ejpam-4744	89	5	(	(	PUNCT
ejpam-4744	89	6	g	g	NOUN
ejpam-4744	89	7	)	)	PUNCT
ejpam-4744	89	8	and	and	CCONJ
ejpam-4744	89	9	ϕ−1(v	ϕ−1(v	PROPN
ejpam-4744	89	10	)	)	PUNCT
ejpam-4744	90	1	=	=	PUNCT
ejpam-4744	90	2	y	y	PROPN
ejpam-4744	90	3	∈	∈	PROPN
ejpam-4744	90	4	v	v	ADP
ejpam-4744	90	5	(	(	PUNCT
ejpam-4744	90	6	h	h	NOUN
ejpam-4744	90	7	)	)	PUNCT
ejpam-4744	90	8	.	.	PUNCT
ejpam-4744	91	1	it	it	PRON
ejpam-4744	91	2	follows	follow	VERB
ejpam-4744	91	3	that	that	SCONJ
ejpam-4744	91	4	x	x	PROPN
ejpam-4744	91	5	and	and	CCONJ
ejpam-4744	91	6	y	y	PROPN
ejpam-4744	91	7	are	be	AUX
ejpam-4744	91	8	adjacent	adjacent	ADJ
ejpam-4744	91	9	in	in	ADP
ejpam-4744	91	10	g+h	g+h	PROPN
ejpam-4744	91	11	.	.	PUNCT
ejpam-4744	92	1	if	if	SCONJ
ejpam-4744	92	2	u	u	PRON
ejpam-4744	92	3	=	=	NOUN
ejpam-4744	92	4	a1	a1	NOUN
ejpam-4744	92	5	×	×	NOUN
ejpam-4744	92	6	s2	s2	NOUN
ejpam-4744	92	7	and	and	CCONJ
ejpam-4744	92	8	v	v	NOUN
ejpam-4744	92	9	=	=	SYM
ejpam-4744	92	10	a2	a2	PROPN
ejpam-4744	92	11	×	×	PROPN
ejpam-4744	92	12	s2	s2	PROPN
ejpam-4744	92	13	,	,	PUNCT
ejpam-4744	92	14	for	for	ADP
ejpam-4744	92	15	some	some	DET
ejpam-4744	92	16	a1	a1	NOUN
ejpam-4744	92	17	,	,	PUNCT
ejpam-4744	92	18	a2	a2	PROPN
ejpam-4744	92	19	∈	∈	PROPN
ejpam-4744	92	20	f1	f1	NOUN
ejpam-4744	92	21	then	then	ADV
ejpam-4744	92	22	u	u	NOUN
ejpam-4744	92	23	=	=	PROPN
ejpam-4744	92	24	ϕ1(x1)×	ϕ1(x1)×	PROPN
ejpam-4744	92	25	s2	s2	PROPN
ejpam-4744	92	26	=	=	SYM
ejpam-4744	92	27	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	92	28	)	)	PUNCT
ejpam-4744	92	29	and	and	CCONJ
ejpam-4744	92	30	v	v	NOUN
ejpam-4744	92	31	=	=	SYM
ejpam-4744	92	32	ϕ1(x2)×s2	ϕ1(x2)×s2	PUNCT
ejpam-4744	92	33	=	=	NOUN
ejpam-4744	92	34	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	92	35	)	)	PUNCT
ejpam-4744	92	36	,	,	PUNCT
ejpam-4744	92	37	for	for	ADP
ejpam-4744	92	38	some	some	DET
ejpam-4744	92	39	x1	x1	PROPN
ejpam-4744	92	40	,	,	PUNCT
ejpam-4744	92	41	x2	x2	PROPN
ejpam-4744	92	42	∈	∈	PROPN
ejpam-4744	92	43	v	v	ADP
ejpam-4744	92	44	(	(	PUNCT
ejpam-4744	92	45	g	g	NOUN
ejpam-4744	92	46	)	)	PUNCT
ejpam-4744	92	47	.	.	PUNCT
ejpam-4744	93	1	consequently	consequently	ADV
ejpam-4744	93	2	,	,	PUNCT
ejpam-4744	93	3	ϕ−1(u	ϕ−1(u	PROPN
ejpam-4744	93	4	)	)	PUNCT
ejpam-4744	94	1	=	=	PUNCT
ejpam-4744	95	1	x1	x1	PROPN
ejpam-4744	95	2	∈	∈	PROPN
ejpam-4744	95	3	v	v	ADP
ejpam-4744	95	4	(	(	PUNCT
ejpam-4744	95	5	g	g	NOUN
ejpam-4744	95	6	)	)	PUNCT
ejpam-4744	95	7	and	and	CCONJ
ejpam-4744	95	8	ϕ−1(v	ϕ−1(v	PROPN
ejpam-4744	95	9	)	)	PUNCT
ejpam-4744	96	1	=	=	PUNCT
ejpam-4744	96	2	x2	x2	PROPN
ejpam-4744	96	3	∈	∈	PROPN
ejpam-4744	96	4	v	v	ADP
ejpam-4744	96	5	(	(	PUNCT
ejpam-4744	96	6	g	g	NOUN
ejpam-4744	96	7	)	)	PUNCT
ejpam-4744	96	8	.	.	PUNCT
ejpam-4744	97	1	thus	thus	ADV
ejpam-4744	97	2	,	,	PUNCT
ejpam-4744	97	3	x1	x1	PROPN
ejpam-4744	97	4	and	and	CCONJ
ejpam-4744	97	5	x2	x2	PROPN
ejpam-4744	97	6	are	be	AUX
ejpam-4744	97	7	adjacent	adjacent	ADJ
ejpam-4744	97	8	in	in	ADP
ejpam-4744	97	9	g.	g.	PROPN
ejpam-4744	97	10	if	if	SCONJ
ejpam-4744	97	11	u	u	PROPN
ejpam-4744	97	12	=	=	PROPN
ejpam-4744	97	13	s1	s1	PROPN
ejpam-4744	97	14	×	×	NOUN
ejpam-4744	97	15	b1	b1	NOUN
ejpam-4744	97	16	and	and	CCONJ
ejpam-4744	97	17	v	v	NOUN
ejpam-4744	97	18	=	=	SYM
ejpam-4744	97	19	s1	s1	PROPN
ejpam-4744	97	20	×	×	PROPN
ejpam-4744	97	21	b2	b2	NOUN
ejpam-4744	97	22	,	,	PUNCT
ejpam-4744	97	23	for	for	ADP
ejpam-4744	97	24	some	some	DET
ejpam-4744	97	25	b1	b1	NOUN
ejpam-4744	97	26	,	,	PUNCT
ejpam-4744	97	27	b2	b2	NOUN
ejpam-4744	97	28	∈	∈	PROPN
ejpam-4744	97	29	f2	f2	PROPN
ejpam-4744	97	30	then	then	ADV
ejpam-4744	97	31	u	u	NOUN
ejpam-4744	97	32	=	=	PROPN
ejpam-4744	97	33	s1	s1	PROPN
ejpam-4744	97	34	×	×	PROPN
ejpam-4744	97	35	ϕ1(y1	ϕ1(y1	NOUN
ejpam-4744	97	36	)	)	PUNCT
ejpam-4744	97	37	=	=	SYM
ejpam-4744	97	38	ϕ(y1	ϕ(y1	NOUN
ejpam-4744	97	39	)	)	PUNCT
ejpam-4744	97	40	and	and	CCONJ
ejpam-4744	97	41	v	v	NOUN
ejpam-4744	97	42	=	=	SYM
ejpam-4744	97	43	s1	s1	PROPN
ejpam-4744	97	44	×	×	PROPN
ejpam-4744	97	45	ϕ1(y2	ϕ1(y2	PROPN
ejpam-4744	97	46	)	)	PUNCT
ejpam-4744	97	47	=	=	SYM
ejpam-4744	97	48	ϕ(y2	ϕ(y2	PROPN
ejpam-4744	97	49	)	)	PUNCT
ejpam-4744	97	50	for	for	ADP
ejpam-4744	97	51	some	some	DET
ejpam-4744	97	52	y1	y1	NOUN
ejpam-4744	97	53	,	,	PUNCT
ejpam-4744	97	54	y2	y2	PROPN
ejpam-4744	97	55	∈	∈	PROPN
ejpam-4744	97	56	v	v	NOUN
ejpam-4744	97	57	(	(	PUNCT
ejpam-4744	97	58	h	h	NOUN
ejpam-4744	97	59	)	)	PUNCT
ejpam-4744	97	60	.	.	PUNCT
ejpam-4744	98	1	consequently	consequently	ADV
ejpam-4744	98	2	,	,	PUNCT
ejpam-4744	98	3	ϕ−1(u	ϕ−1(u	PROPN
ejpam-4744	98	4	)	)	PUNCT
ejpam-4744	98	5	=	=	PUNCT
ejpam-4744	98	6	y1	y1	NOUN
ejpam-4744	98	7	∈	∈	PROPN
ejpam-4744	98	8	v	v	NOUN
ejpam-4744	98	9	(	(	PUNCT
ejpam-4744	98	10	h	h	NOUN
ejpam-4744	98	11	)	)	PUNCT
ejpam-4744	98	12	and	and	CCONJ
ejpam-4744	98	13	ϕ−1(v	ϕ−1(v	PROPN
ejpam-4744	98	14	)	)	PUNCT
ejpam-4744	99	1	=	=	PUNCT
ejpam-4744	100	1	y2	y2	NOUN
ejpam-4744	100	2	∈	∈	NOUN
ejpam-4744	100	3	v	v	NOUN
ejpam-4744	100	4	(	(	PUNCT
ejpam-4744	100	5	h	h	NOUN
ejpam-4744	100	6	)	)	PUNCT
ejpam-4744	100	7	.	.	PUNCT
ejpam-4744	101	1	thus	thus	ADV
ejpam-4744	101	2	,	,	PUNCT
ejpam-4744	101	3	y1	y1	INTJ
ejpam-4744	101	4	and	and	CCONJ
ejpam-4744	101	5	y2	y2	NOUN
ejpam-4744	101	6	are	be	AUX
ejpam-4744	101	7	adjacent	adjacent	ADJ
ejpam-4744	101	8	in	in	ADP
ejpam-4744	101	9	h.	h.	PROPN
ejpam-4744	101	10	therefore	therefore	ADV
ejpam-4744	101	11	,	,	PUNCT
ejpam-4744	101	12	ϕ	ϕ	PROPN
ejpam-4744	101	13	preserves	preserve	VERB
ejpam-4744	101	14	adjacency	adjacency	NOUN
ejpam-4744	101	15	.	.	PUNCT
ejpam-4744	102	1	hence	hence	ADV
ejpam-4744	102	2	,	,	PUNCT
ejpam-4744	102	3	ω(f	ω(f	ADJ
ejpam-4744	102	4	)	)	PUNCT
ejpam-4744	102	5	∼=	∼=	PROPN
ejpam-4744	102	6	g+h	g+h	NOUN
ejpam-4744	102	7	accordingly	accordingly	ADV
ejpam-4744	102	8	,	,	PUNCT
ejpam-4744	102	9	ω(g+h	ω(g+h	NUM
ejpam-4744	102	10	)	)	PUNCT
ejpam-4744	102	11	≤	≤	NOUN
ejpam-4744	102	12	|s|	|s|	PROPN
ejpam-4744	102	13	,	,	PUNCT
ejpam-4744	102	14	since	since	SCONJ
ejpam-4744	102	15	s	s	PART
ejpam-4744	102	16	=	=	SYM
ejpam-4744	102	17	s1	s1	PROPN
ejpam-4744	102	18	×	×	PROPN
ejpam-4744	102	19	s2	s2	PROPN
ejpam-4744	102	20	.	.	PUNCT
ejpam-4744	103	1	then	then	ADV
ejpam-4744	103	2	|s|	|s|	PROPN
ejpam-4744	103	3	=	=	SYM
ejpam-4744	103	4	|s1||s2|	|s1||s2|	PROPN
ejpam-4744	103	5	=	=	SYM
ejpam-4744	103	6	ω(g)ω(h	ω(g)ω(h	PROPN
ejpam-4744	103	7	)	)	PUNCT
ejpam-4744	103	8	.	.	PUNCT
ejpam-4744	104	1	hence	hence	ADV
ejpam-4744	104	2	,	,	PUNCT
ejpam-4744	104	3	ω(g+h	ω(g+h	NUM
ejpam-4744	104	4	)	)	PUNCT
ejpam-4744	104	5	≤	≤	NUM
ejpam-4744	104	6	ω(g)ω(h	ω(g)ω(h	NOUN
ejpam-4744	104	7	)	)	PUNCT
ejpam-4744	104	8	.	.	PUNCT
ejpam-4744	105	1	let	let	VERB
ejpam-4744	105	2	g	g	PRON
ejpam-4744	105	3	be	be	AUX
ejpam-4744	105	4	a	a	DET
ejpam-4744	105	5	connected	connected	ADJ
ejpam-4744	105	6	graph	graph	NOUN
ejpam-4744	105	7	.	.	PUNCT
ejpam-4744	106	1	a	a	DET
ejpam-4744	106	2	subset	subset	NOUN
ejpam-4744	106	3	s	s	NOUN
ejpam-4744	106	4	of	of	ADP
ejpam-4744	106	5	v	v	NOUN
ejpam-4744	106	6	(	(	PUNCT
ejpam-4744	106	7	g	g	NOUN
ejpam-4744	106	8	)	)	PUNCT
ejpam-4744	106	9	is	be	AUX
ejpam-4744	106	10	a	a	DET
ejpam-4744	106	11	clique	clique	NOUN
ejpam-4744	106	12	if	if	SCONJ
ejpam-4744	106	13	⟨s⟩	⟨s⟩	PROPN
ejpam-4744	106	14	is	be	AUX
ejpam-4744	106	15	a	a	DET
ejpam-4744	106	16	complete	complete	ADJ
ejpam-4744	106	17	graph	graph	NOUN
ejpam-4744	106	18	.	.	PUNCT
ejpam-4744	107	1	a	a	DET
ejpam-4744	107	2	clique	clique	NOUN
ejpam-4744	107	3	m	m	VERB
ejpam-4744	107	4	is	be	AUX
ejpam-4744	107	5	maximal	maximal	ADJ
ejpam-4744	107	6	if	if	SCONJ
ejpam-4744	107	7	a	a	DET
ejpam-4744	107	8	∈	∈	PROPN
ejpam-4744	107	9	v	v	NOUN
ejpam-4744	107	10	(	(	PUNCT
ejpam-4744	107	11	g	g	NOUN
ejpam-4744	107	12	)	)	PUNCT
ejpam-4744	107	13	−m	−m	NOUN
ejpam-4744	107	14	,	,	PUNCT
ejpam-4744	107	15	then	then	ADV
ejpam-4744	107	16	m	m	VERB
ejpam-4744	107	17	∪	∪	ADJ
ejpam-4744	107	18	{	{	PUNCT
ejpam-4744	107	19	a	a	PRON
ejpam-4744	107	20	}	}	PUNCT
ejpam-4744	107	21	is	be	AUX
ejpam-4744	107	22	no	no	PRON
ejpam-4744	107	23	longer	long	ADV
ejpam-4744	107	24	a	a	DET
ejpam-4744	107	25	clique	clique	NOUN
ejpam-4744	107	26	in	in	ADP
ejpam-4744	107	27	g.	g.	PROPN
ejpam-4744	107	28	the	the	DET
ejpam-4744	107	29	clique	clique	NOUN
ejpam-4744	107	30	graph	graph	NOUN
ejpam-4744	107	31	of	of	ADP
ejpam-4744	107	32	g	g	NOUN
ejpam-4744	107	33	,	,	PUNCT
ejpam-4744	107	34	denoted	denote	VERB
ejpam-4744	107	35	by	by	ADP
ejpam-4744	107	36	ζ(g	ζ(g	PROPN
ejpam-4744	107	37	)	)	PUNCT
ejpam-4744	107	38	,	,	PUNCT
ejpam-4744	107	39	is	be	AUX
ejpam-4744	107	40	the	the	DET
ejpam-4744	107	41	intersection	intersection	NOUN
ejpam-4744	107	42	graph	graph	NOUN
ejpam-4744	107	43	of	of	ADP
ejpam-4744	107	44	the	the	DET
ejpam-4744	107	45	set	set	NOUN
ejpam-4744	107	46	of	of	ADP
ejpam-4744	107	47	all	all	DET
ejpam-4744	107	48	maximal	maximal	ADJ
ejpam-4744	107	49	cliques	clique	NOUN
ejpam-4744	107	50	of	of	ADP
ejpam-4744	107	51	g.	g.	PROPN
ejpam-4744	107	52	the	the	DET
ejpam-4744	107	53	clique	clique	ADJ
ejpam-4744	107	54	order	order	NOUN
ejpam-4744	107	55	of	of	ADP
ejpam-4744	107	56	g	g	NOUN
ejpam-4744	107	57	,	,	PUNCT
ejpam-4744	107	58	denoted	denote	VERB
ejpam-4744	107	59	by	by	ADP
ejpam-4744	107	60	co(g	co(g	NOUN
ejpam-4744	107	61	)	)	PUNCT
ejpam-4744	107	62	,	,	PUNCT
ejpam-4744	107	63	is	be	AUX
ejpam-4744	107	64	|v	|v	PROPN
ejpam-4744	107	65	(	(	PUNCT
ejpam-4744	107	66	ζ(g))|	ζ(g))|	PROPN
ejpam-4744	107	67	.	.	PUNCT
ejpam-4744	108	1	that	that	PRON
ejpam-4744	108	2	is	be	AUX
ejpam-4744	108	3	,	,	PUNCT
ejpam-4744	108	4	co(g	co(g	NOUN
ejpam-4744	108	5	)	)	PUNCT
ejpam-4744	108	6	is	be	AUX
ejpam-4744	108	7	the	the	DET
ejpam-4744	108	8	number	number	NOUN
ejpam-4744	108	9	of	of	ADP
ejpam-4744	108	10	maximal	maximal	ADJ
ejpam-4744	108	11	cliques	clique	NOUN
ejpam-4744	108	12	in	in	ADP
ejpam-4744	108	13	g.	g.	PROPN
ejpam-4744	108	14	theorem	theorem	PROPN
ejpam-4744	108	15	2	2	X
ejpam-4744	108	16	.	.	PUNCT
ejpam-4744	109	1	let	let	VERB
ejpam-4744	109	2	kn	kn	PROPN
ejpam-4744	109	3	,	,	PUNCT
ejpam-4744	109	4	pn	pn	PROPN
ejpam-4744	109	5	and	and	CCONJ
ejpam-4744	109	6	cn	cn	PROPN
ejpam-4744	109	7	be	be	AUX
ejpam-4744	109	8	a	a	DET
ejpam-4744	109	9	complete	complete	ADJ
ejpam-4744	109	10	graph	graph	NOUN
ejpam-4744	109	11	,	,	PUNCT
ejpam-4744	109	12	path	path	NOUN
ejpam-4744	109	13	and	and	CCONJ
ejpam-4744	109	14	cycle	cycle	NOUN
ejpam-4744	109	15	,	,	PUNCT
ejpam-4744	109	16	respectively	respectively	ADV
ejpam-4744	109	17	.	.	PUNCT
ejpam-4744	110	1	then	then	ADV
ejpam-4744	110	2	(	(	PUNCT
ejpam-4744	110	3	i	i	NOUN
ejpam-4744	110	4	)	)	PUNCT
ejpam-4744	110	5	co(kn	co(kn	NOUN
ejpam-4744	110	6	)	)	PUNCT
ejpam-4744	111	1	=	=	SYM
ejpam-4744	111	2	1	1	NUM
ejpam-4744	111	3	,	,	PUNCT
ejpam-4744	111	4	n	n	PRON
ejpam-4744	111	5	≥	≥	NOUN
ejpam-4744	111	6	1	1	NUM
ejpam-4744	111	7	(	(	PUNCT
ejpam-4744	111	8	ii	ii	NOUN
ejpam-4744	111	9	)	)	PUNCT
ejpam-4744	111	10	co(pn	co(pn	PROPN
ejpam-4744	111	11	)	)	PUNCT
ejpam-4744	112	1	=	=	PUNCT
ejpam-4744	112	2	n−	n−	NOUN
ejpam-4744	112	3	1	1	NUM
ejpam-4744	112	4	,	,	PUNCT
ejpam-4744	112	5	n	n	PRON
ejpam-4744	112	6	≥	≥	NOUN
ejpam-4744	112	7	2	2	NUM
ejpam-4744	112	8	(	(	PUNCT
ejpam-4744	112	9	iii	iii	NOUN
ejpam-4744	112	10	)	)	PUNCT
ejpam-4744	112	11	co(cn	co(cn	PROPN
ejpam-4744	112	12	)	)	PUNCT
ejpam-4744	113	1	=	=	PRON
ejpam-4744	113	2	{	{	PUNCT
ejpam-4744	113	3	1	1	NUM
ejpam-4744	113	4	,	,	PUNCT
ejpam-4744	113	5	if	if	SCONJ
ejpam-4744	113	6	n	n	CCONJ
ejpam-4744	113	7	=	=	SYM
ejpam-4744	113	8	3	3	NUM
ejpam-4744	113	9	n	n	CCONJ
ejpam-4744	113	10	,	,	PUNCT
ejpam-4744	113	11	if	if	SCONJ
ejpam-4744	113	12	n	n	PRON
ejpam-4744	113	13	≥	≥	VERB
ejpam-4744	113	14	4	4	NUM
ejpam-4744	113	15	the	the	DET
ejpam-4744	113	16	corona	corona	NOUN
ejpam-4744	113	17	g	g	PROPN
ejpam-4744	113	18	◦	◦	NOUN
ejpam-4744	113	19	h	h	NOUN
ejpam-4744	113	20	of	of	ADP
ejpam-4744	113	21	two	two	NUM
ejpam-4744	113	22	graphs	graph	NOUN
ejpam-4744	113	23	g	g	NOUN
ejpam-4744	113	24	and	and	CCONJ
ejpam-4744	113	25	h	h	NOUN
ejpam-4744	113	26	,	,	PUNCT
ejpam-4744	113	27	is	be	AUX
ejpam-4744	113	28	the	the	DET
ejpam-4744	113	29	graph	graph	NOUN
ejpam-4744	113	30	obtained	obtain	VERB
ejpam-4744	113	31	by	by	ADP
ejpam-4744	113	32	making	make	VERB
ejpam-4744	113	33	n	n	PRON
ejpam-4744	113	34	copies	copy	NOUN
ejpam-4744	113	35	(	(	PUNCT
ejpam-4744	113	36	n	n	X
ejpam-4744	113	37	is	be	AUX
ejpam-4744	113	38	the	the	DET
ejpam-4744	113	39	ordered	ordered	NOUN
ejpam-4744	113	40	of	of	ADP
ejpam-4744	113	41	g	g	NOUN
ejpam-4744	113	42	)	)	PUNCT
ejpam-4744	113	43	of	of	ADP
ejpam-4744	113	44	h	h	NOUN
ejpam-4744	113	45	and	and	CCONJ
ejpam-4744	113	46	joining	join	VERB
ejpam-4744	113	47	every	every	DET
ejpam-4744	113	48	vertex	vertex	NOUN
ejpam-4744	113	49	of	of	ADP
ejpam-4744	113	50	the	the	DET
ejpam-4744	113	51	ith	ith	PROPN
ejpam-4744	113	52	copy	copy	NOUN
ejpam-4744	113	53	of	of	ADP
ejpam-4744	113	54	h	h	PROPN
ejpam-4744	113	55	with	with	ADP
ejpam-4744	113	56	the	the	DET
ejpam-4744	113	57	vertex	vertex	NOUN
ejpam-4744	113	58	vi	vi	PROPN
ejpam-4744	113	59	of	of	ADP
ejpam-4744	113	60	g.	g.	PROPN
ejpam-4744	113	61	for	for	ADP
ejpam-4744	113	62	each	each	PRON
ejpam-4744	113	63	a	a	DET
ejpam-4744	113	64	∈	∈	PROPN
ejpam-4744	113	65	v	v	NOUN
ejpam-4744	113	66	(	(	PUNCT
ejpam-4744	113	67	g	g	NOUN
ejpam-4744	113	68	)	)	PUNCT
ejpam-4744	113	69	,	,	PUNCT
ejpam-4744	113	70	we	we	PRON
ejpam-4744	113	71	denote	denote	VERB
ejpam-4744	113	72	by	by	ADP
ejpam-4744	113	73	ha	ha	INTJ
ejpam-4744	113	74	the	the	DET
ejpam-4744	113	75	copy	copy	NOUN
ejpam-4744	113	76	of	of	ADP
ejpam-4744	113	77	h	h	NOUN
ejpam-4744	113	78	corresponding	correspond	VERB
ejpam-4744	113	79	to	to	ADP
ejpam-4744	113	80	the	the	DET
ejpam-4744	113	81	vertex	vertex	NOUN
ejpam-4744	113	82	a.	a.	PROPN
ejpam-4744	113	83	j.	j.	PROPN
ejpam-4744	113	84	b.	b.	PROPN
ejpam-4744	113	85	palco	palco	PROPN
ejpam-4744	113	86	,	,	PUNCT
ejpam-4744	113	87	r.	r.	PROPN
ejpam-4744	113	88	n.	n.	PROPN
ejpam-4744	113	89	paluga	paluga	PROPN
ejpam-4744	113	90	/	/	SYM
ejpam-4744	113	91	eur	eur	PROPN
ejpam-4744	113	92	.	.	PUNCT
ejpam-4744	114	1	j.	j.	PROPN
ejpam-4744	114	2	pure	pure	PROPN
ejpam-4744	114	3	appl	appl	PROPN
ejpam-4744	114	4	.	.	PROPN
ejpam-4744	114	5	math	math	PROPN
ejpam-4744	114	6	,	,	PUNCT
ejpam-4744	114	7	16	16	NUM
ejpam-4744	114	8	(	(	PUNCT
ejpam-4744	114	9	2	2	NUM
ejpam-4744	114	10	)	)	PUNCT
ejpam-4744	114	11	(	(	PUNCT
ejpam-4744	114	12	2023	2023	NUM
ejpam-4744	114	13	)	)	PUNCT
ejpam-4744	114	14	,	,	PUNCT
ejpam-4744	114	15	1318	1318	NUM
ejpam-4744	114	16	-	-	SYM
ejpam-4744	114	17	1325	1325	NUM
ejpam-4744	114	18	1321	1321	NUM
ejpam-4744	114	19	theorem	theorem	NOUN
ejpam-4744	114	20	3	3	X
ejpam-4744	114	21	.	.	PUNCT
ejpam-4744	115	1	let	let	VERB
ejpam-4744	115	2	g	g	PRON
ejpam-4744	115	3	be	be	AUX
ejpam-4744	115	4	a	a	DET
ejpam-4744	115	5	connected	connected	ADJ
ejpam-4744	115	6	graph	graph	NOUN
ejpam-4744	115	7	and	and	CCONJ
ejpam-4744	115	8	h	h	NOUN
ejpam-4744	115	9	be	be	AUX
ejpam-4744	115	10	any	any	DET
ejpam-4744	115	11	graph	graph	NOUN
ejpam-4744	115	12	.	.	PUNCT
ejpam-4744	116	1	then	then	ADV
ejpam-4744	116	2	ω(g	ω(g	VERB
ejpam-4744	116	3	◦	◦	NOUN
ejpam-4744	116	4	h	h	NOUN
ejpam-4744	116	5	)	)	PUNCT
ejpam-4744	116	6	≤	≤	NUM
ejpam-4744	116	7	co(g	co(g	NOUN
ejpam-4744	116	8	)	)	PUNCT
ejpam-4744	117	1	+	+	CCONJ
ejpam-4744	117	2	|v	|v	X
ejpam-4744	117	3	(	(	PUNCT
ejpam-4744	117	4	g)|	g)|	NOUN
ejpam-4744	117	5	·	·	PUNCT
ejpam-4744	117	6	ω(h	ω(h	NUM
ejpam-4744	117	7	)	)	PUNCT
ejpam-4744	117	8	.	.	PUNCT
ejpam-4744	118	1	proof	proof	NOUN
ejpam-4744	118	2	.	.	PUNCT
ejpam-4744	119	1	let	let	VERB
ejpam-4744	119	2	v	v	X
ejpam-4744	119	3	(	(	PUNCT
ejpam-4744	119	4	g	g	NOUN
ejpam-4744	119	5	)	)	PUNCT
ejpam-4744	119	6	=	=	SYM
ejpam-4744	119	7	{	{	PUNCT
ejpam-4744	119	8	a1	a1	PROPN
ejpam-4744	119	9	,	,	PUNCT
ejpam-4744	119	10	a2	a2	PROPN
ejpam-4744	119	11	,	,	PUNCT
ejpam-4744	119	12	a3	a3	NOUN
ejpam-4744	119	13	,	,	PUNCT
ejpam-4744	119	14	...	...	PUNCT
ejpam-4744	119	15	,	,	PUNCT
ejpam-4744	119	16	an	an	PRON
ejpam-4744	119	17	}	}	PUNCT
ejpam-4744	119	18	and	and	CCONJ
ejpam-4744	119	19	v	v	NOUN
ejpam-4744	119	20	(	(	PUNCT
ejpam-4744	119	21	ζ(g	ζ(g	NOUN
ejpam-4744	119	22	)	)	PUNCT
ejpam-4744	119	23	)	)	PUNCT
ejpam-4744	120	1	=	=	PRON
ejpam-4744	120	2	{	{	PUNCT
ejpam-4744	120	3	b1	b1	NOUN
ejpam-4744	120	4	,	,	PUNCT
ejpam-4744	120	5	b2	b2	NOUN
ejpam-4744	120	6	,	,	PUNCT
ejpam-4744	120	7	...	...	PUNCT
ejpam-4744	120	8	,	,	PUNCT
ejpam-4744	120	9	bco(g	bco(g	PROPN
ejpam-4744	120	10	)	)	PUNCT
ejpam-4744	120	11	}	}	PUNCT
ejpam-4744	120	12	.	.	PUNCT
ejpam-4744	121	1	for	for	ADP
ejpam-4744	121	2	each	each	DET
ejpam-4744	121	3	i	i	NOUN
ejpam-4744	121	4	=	=	NOUN
ejpam-4744	121	5	1	1	NUM
ejpam-4744	121	6	,	,	PUNCT
ejpam-4744	121	7	2	2	NUM
ejpam-4744	121	8	,	,	PUNCT
ejpam-4744	121	9	...	...	PUNCT
ejpam-4744	121	10	,	,	PUNCT
ejpam-4744	121	11	n	n	CCONJ
ejpam-4744	121	12	,	,	PUNCT
ejpam-4744	121	13	let	let	VERB
ejpam-4744	121	14	fi	fi	NOUN
ejpam-4744	121	15	be	be	AUX
ejpam-4744	121	16	a	a	DET
ejpam-4744	121	17	collection	collection	NOUN
ejpam-4744	121	18	of	of	ADP
ejpam-4744	121	19	nonempty	nonempty	ADJ
ejpam-4744	121	20	subsets	subset	NOUN
ejpam-4744	121	21	of	of	ADP
ejpam-4744	121	22	si	si	X
ejpam-4744	121	23	=	=	PUNCT
ejpam-4744	121	24	{	{	PUNCT
ejpam-4744	121	25	(	(	PUNCT
ejpam-4744	121	26	i	i	PROPN
ejpam-4744	121	27	,	,	PUNCT
ejpam-4744	121	28	j	j	PROPN
ejpam-4744	121	29	)	)	PUNCT
ejpam-4744	121	30	:	:	PUNCT
ejpam-4744	121	31	1	1	NUM
ejpam-4744	121	32	≤	≤	NUM
ejpam-4744	121	33	j	j	PROPN
ejpam-4744	121	34	≤	≤	PROPN
ejpam-4744	121	35	ω(h	ω(h	NUM
ejpam-4744	121	36	)	)	PUNCT
ejpam-4744	121	37	}	}	PUNCT
ejpam-4744	121	38	such	such	ADJ
ejpam-4744	121	39	that	that	SCONJ
ejpam-4744	121	40	ω(fi	ω(fi	ADJ
ejpam-4744	121	41	)	)	PUNCT
ejpam-4744	121	42	∼=	∼=	PROPN
ejpam-4744	121	43	hai	hai	NOUN
ejpam-4744	121	44	.	.	PUNCT
ejpam-4744	122	1	for	for	ADP
ejpam-4744	122	2	each	each	DET
ejpam-4744	122	3	i	i	NOUN
ejpam-4744	122	4	=	=	NOUN
ejpam-4744	122	5	1	1	NUM
ejpam-4744	122	6	,	,	PUNCT
ejpam-4744	122	7	2	2	NUM
ejpam-4744	122	8	,	,	PUNCT
ejpam-4744	122	9	...	...	PUNCT
ejpam-4744	122	10	,	,	PUNCT
ejpam-4744	122	11	n	n	CCONJ
ejpam-4744	122	12	,	,	PUNCT
ejpam-4744	122	13	let	let	VERB
ejpam-4744	122	14	ϕi	ϕi	ADP
ejpam-4744	122	15	:	:	PUNCT
ejpam-4744	122	16	v	v	X
ejpam-4744	122	17	(	(	PUNCT
ejpam-4744	122	18	hai	hai	NOUN
ejpam-4744	122	19	)	)	PUNCT
ejpam-4744	122	20	→	→	SYM
ejpam-4744	122	21	fi	fi	NOUN
ejpam-4744	122	22	be	be	AUX
ejpam-4744	122	23	an	an	DET
ejpam-4744	122	24	isomorphism	isomorphism	NOUN
ejpam-4744	122	25	.	.	PUNCT
ejpam-4744	123	1	let	let	VERB
ejpam-4744	123	2	so	so	ADV
ejpam-4744	123	3	=	=	PRON
ejpam-4744	123	4	{	{	PUNCT
ejpam-4744	123	5	(	(	PUNCT
ejpam-4744	123	6	0	0	NUM
ejpam-4744	123	7	,	,	PUNCT
ejpam-4744	123	8	j	j	NOUN
ejpam-4744	123	9	)	)	PUNCT
ejpam-4744	123	10	:	:	PUNCT
ejpam-4744	123	11	1	1	NUM
ejpam-4744	123	12	≤	≤	NUM
ejpam-4744	123	13	j	j	PROPN
ejpam-4744	123	14	≤	≤	NUM
ejpam-4744	123	15	co(g	co(g	NOUN
ejpam-4744	123	16	)	)	PUNCT
ejpam-4744	123	17	}	}	PUNCT
ejpam-4744	123	18	and	and	CCONJ
ejpam-4744	123	19	s	s	PROPN
ejpam-4744	123	20	=	=	PROPN
ejpam-4744	123	21	⋃n	⋃n	PROPN
ejpam-4744	123	22	i=0	i=0	PROPN
ejpam-4744	123	23	si	si	X
ejpam-4744	123	24	.	.	PROPN
ejpam-4744	124	1	for	for	ADP
ejpam-4744	124	2	each	each	DET
ejpam-4744	124	3	i	i	NOUN
ejpam-4744	124	4	=	=	NOUN
ejpam-4744	124	5	1	1	NUM
ejpam-4744	124	6	,	,	PUNCT
ejpam-4744	124	7	2	2	NUM
ejpam-4744	124	8	,	,	PUNCT
ejpam-4744	124	9	...	...	PUNCT
ejpam-4744	124	10	,	,	PUNCT
ejpam-4744	124	11	n	n	CCONJ
ejpam-4744	124	12	,	,	PUNCT
ejpam-4744	124	13	let	let	VERB
ejpam-4744	124	14	ti	ti	NOUN
ejpam-4744	124	15	=	=	PRON
ejpam-4744	124	16	{	{	PUNCT
ejpam-4744	124	17	(	(	PUNCT
ejpam-4744	124	18	0	0	NUM
ejpam-4744	124	19	,	,	PUNCT
ejpam-4744	124	20	j	j	NOUN
ejpam-4744	124	21	)	)	PUNCT
ejpam-4744	124	22	:	:	PUNCT
ejpam-4744	124	23	ai	ai	VERB
ejpam-4744	124	24	∈	∈	NOUN
ejpam-4744	124	25	bj	bj	NOUN
ejpam-4744	124	26	,	,	PUNCT
ejpam-4744	124	27	for	for	ADP
ejpam-4744	124	28	some	some	DET
ejpam-4744	124	29	j	j	NOUN
ejpam-4744	124	30	}	}	PUNCT
ejpam-4744	124	31	.	.	PUNCT
ejpam-4744	125	1	let	let	VERB
ejpam-4744	125	2	f	f	PROPN
ejpam-4744	125	3	=	=	PRON
ejpam-4744	125	4	(	(	PUNCT
ejpam-4744	125	5	⋃n	⋃n	PROPN
ejpam-4744	125	6	i=1	i=1	X
ejpam-4744	125	7	fi	fi	NOUN
ejpam-4744	125	8	)	)	PUNCT
ejpam-4744	125	9	⋃	⋃	NOUN
ejpam-4744	125	10	{	{	PUNCT
ejpam-4744	125	11	si	si	X
ejpam-4744	125	12	⋃	⋃	NOUN
ejpam-4744	125	13	ti	ti	NOUN
ejpam-4744	125	14	:	:	PUNCT
ejpam-4744	125	15	1	1	NUM
ejpam-4744	125	16	≤	≤	NUM
ejpam-4744	125	17	i	i	PRON
ejpam-4744	125	18	≤	≤	NOUN
ejpam-4744	125	19	n	n	CCONJ
ejpam-4744	125	20	}	}	PUNCT
ejpam-4744	125	21	.	.	PUNCT
ejpam-4744	126	1	define	define	VERB
ejpam-4744	126	2	a	a	DET
ejpam-4744	126	3	mapping	mapping	NOUN
ejpam-4744	126	4	ϕ	ϕ	NOUN
ejpam-4744	126	5	:	:	PUNCT
ejpam-4744	126	6	v	v	NOUN
ejpam-4744	126	7	(	(	PUNCT
ejpam-4744	126	8	g	g	PROPN
ejpam-4744	126	9	◦	◦	NOUN
ejpam-4744	126	10	h	h	NOUN
ejpam-4744	126	11	)	)	PUNCT
ejpam-4744	126	12	→	→	SYM
ejpam-4744	126	13	f	f	PROPN
ejpam-4744	126	14	as	as	SCONJ
ejpam-4744	126	15	follows	follow	VERB
ejpam-4744	126	16	ϕ(x	ϕ(x	X
ejpam-4744	126	17	)	)	PUNCT
ejpam-4744	126	18	=	=	PRON
ejpam-4744	126	19	{	{	PUNCT
ejpam-4744	126	20	ϕi(x	ϕi(x	NOUN
ejpam-4744	126	21	)	)	PUNCT
ejpam-4744	126	22	,	,	PUNCT
ejpam-4744	126	23	if	if	SCONJ
ejpam-4744	126	24	x	x	SYM
ejpam-4744	126	25	∈	∈	PROPN
ejpam-4744	126	26	v	v	NOUN
ejpam-4744	126	27	(	(	PUNCT
ejpam-4744	126	28	hai	hai	NOUN
ejpam-4744	126	29	)	)	PUNCT
ejpam-4744	126	30	,	,	PUNCT
ejpam-4744	126	31	for	for	ADP
ejpam-4744	126	32	some	some	PRON
ejpam-4744	127	1	i	i	PRON
ejpam-4744	127	2	si	si	NOUN
ejpam-4744	127	3	∪	∪	ADP
ejpam-4744	127	4	ti	ti	NOUN
ejpam-4744	127	5	,	,	PUNCT
ejpam-4744	127	6	for	for	SCONJ
ejpam-4744	127	7	some	some	DET
ejpam-4744	127	8	i.	i.	NOUN
ejpam-4744	127	9	let	let	VERB
ejpam-4744	127	10	x1	x1	PROPN
ejpam-4744	127	11	,	,	PUNCT
ejpam-4744	127	12	x2	x2	PROPN
ejpam-4744	127	13	∈	∈	PROPN
ejpam-4744	127	14	v	v	NOUN
ejpam-4744	127	15	(	(	PUNCT
ejpam-4744	127	16	g	g	PROPN
ejpam-4744	127	17	◦	◦	NOUN
ejpam-4744	127	18	h	h	NOUN
ejpam-4744	127	19	)	)	PUNCT
ejpam-4744	127	20	such	such	ADJ
ejpam-4744	127	21	that	that	SCONJ
ejpam-4744	127	22	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	127	23	)	)	PUNCT
ejpam-4744	127	24	=	=	SYM
ejpam-4744	127	25	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	127	26	)	)	PUNCT
ejpam-4744	127	27	.	.	PUNCT
ejpam-4744	128	1	suppose	suppose	VERB
ejpam-4744	128	2	x1	x1	PROPN
ejpam-4744	128	3	∈	∈	PROPN
ejpam-4744	128	4	v	v	ADP
ejpam-4744	128	5	(	(	PUNCT
ejpam-4744	128	6	g	g	NOUN
ejpam-4744	128	7	)	)	PUNCT
ejpam-4744	128	8	and	and	CCONJ
ejpam-4744	128	9	x2	x2	PROPN
ejpam-4744	128	10	∈	∈	PROPN
ejpam-4744	128	11	v	v	X
ejpam-4744	128	12	(	(	PUNCT
ejpam-4744	128	13	hai	hai	NOUN
ejpam-4744	128	14	)	)	PUNCT
ejpam-4744	128	15	for	for	ADP
ejpam-4744	128	16	some	some	DET
ejpam-4744	128	17	i.	i.	NOUN
ejpam-4744	128	18	then	then	ADV
ejpam-4744	128	19	x1	x1	PROPN
ejpam-4744	128	20	∈	∈	PROPN
ejpam-4744	128	21	bj	bj	VERB
ejpam-4744	128	22	for	for	ADP
ejpam-4744	128	23	some	some	DET
ejpam-4744	128	24	j.	j.	PROPN
ejpam-4744	128	25	thus	thus	ADV
ejpam-4744	128	26	,	,	PUNCT
ejpam-4744	128	27	(	(	PUNCT
ejpam-4744	128	28	0	0	NUM
ejpam-4744	128	29	,	,	PUNCT
ejpam-4744	128	30	j	j	NOUN
ejpam-4744	128	31	)	)	PUNCT
ejpam-4744	128	32	∈	∈	PROPN
ejpam-4744	128	33	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	128	34	)	)	PUNCT
ejpam-4744	128	35	.	.	PUNCT
ejpam-4744	129	1	now	now	ADV
ejpam-4744	129	2	,	,	PUNCT
ejpam-4744	129	3	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	129	4	)	)	PUNCT
ejpam-4744	129	5	=	=	SYM
ejpam-4744	129	6	ϕi(x2	ϕi(x2	NOUN
ejpam-4744	129	7	)	)	PUNCT
ejpam-4744	129	8	⊆	⊆	NUM
ejpam-4744	129	9	si	si	NOUN
ejpam-4744	129	10	,	,	PUNCT
ejpam-4744	129	11	so	so	CCONJ
ejpam-4744	129	12	(	(	PUNCT
ejpam-4744	129	13	0	0	NUM
ejpam-4744	129	14	,	,	PUNCT
ejpam-4744	129	15	j	j	NOUN
ejpam-4744	129	16	)	)	PUNCT
ejpam-4744	129	17	/∈	/∈	PUNCT
ejpam-4744	130	1	sj	sj	INTJ
ejpam-4744	130	2	.	.	PUNCT
ejpam-4744	131	1	this	this	PRON
ejpam-4744	131	2	is	be	AUX
ejpam-4744	131	3	a	a	DET
ejpam-4744	131	4	contradiction	contradiction	NOUN
ejpam-4744	131	5	.	.	PUNCT
ejpam-4744	132	1	therefore	therefore	ADV
ejpam-4744	132	2	,	,	PUNCT
ejpam-4744	132	3	the	the	DET
ejpam-4744	132	4	case	case	NOUN
ejpam-4744	132	5	x1	x1	PROPN
ejpam-4744	132	6	∈	∈	PROPN
ejpam-4744	132	7	v	v	ADP
ejpam-4744	132	8	(	(	PUNCT
ejpam-4744	132	9	g	g	NOUN
ejpam-4744	132	10	)	)	PUNCT
ejpam-4744	132	11	and	and	CCONJ
ejpam-4744	132	12	x2	x2	PROPN
ejpam-4744	132	13	∈	∈	PROPN
ejpam-4744	132	14	v	v	X
ejpam-4744	132	15	(	(	PUNCT
ejpam-4744	132	16	hai	hai	NOUN
ejpam-4744	132	17	)	)	PUNCT
ejpam-4744	132	18	is	be	AUX
ejpam-4744	132	19	not	not	PART
ejpam-4744	132	20	possible	possible	ADJ
ejpam-4744	132	21	.	.	PUNCT
ejpam-4744	133	1	consider	consider	VERB
ejpam-4744	133	2	the	the	DET
ejpam-4744	133	3	following	follow	VERB
ejpam-4744	133	4	cases	case	NOUN
ejpam-4744	133	5	:	:	PUNCT
ejpam-4744	133	6	case	case	NOUN
ejpam-4744	133	7	1	1	NUM
ejpam-4744	133	8	.	.	PUNCT
ejpam-4744	133	9	suppose	suppose	VERB
ejpam-4744	133	10	x1	x1	NUM
ejpam-4744	133	11	,	,	PUNCT
ejpam-4744	133	12	x2	x2	PROPN
ejpam-4744	133	13	∈	∈	PROPN
ejpam-4744	133	14	v	v	ADP
ejpam-4744	133	15	(	(	PUNCT
ejpam-4744	133	16	g	g	NOUN
ejpam-4744	133	17	)	)	PUNCT
ejpam-4744	133	18	.	.	PUNCT
ejpam-4744	134	1	then	then	ADV
ejpam-4744	134	2	x1	x1	PROPN
ejpam-4744	134	3	=	=	PUNCT
ejpam-4744	134	4	ai	ai	VERB
ejpam-4744	134	5	and	and	CCONJ
ejpam-4744	134	6	x2	x2	PROPN
ejpam-4744	134	7	=	=	PROPN
ejpam-4744	134	8	aj	aj	PROPN
ejpam-4744	134	9	.	.	PUNCT
ejpam-4744	135	1	thus	thus	ADV
ejpam-4744	135	2	,	,	PUNCT
ejpam-4744	135	3	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	135	4	)	)	PUNCT
ejpam-4744	136	1	=	=	SYM
ejpam-4744	136	2	si	si	X
ejpam-4744	136	3	∪	∪	X
ejpam-4744	136	4	ti	ti	NOUN
ejpam-4744	136	5	and	and	CCONJ
ejpam-4744	136	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	136	7	)	)	PUNCT
ejpam-4744	137	1	=	=	VERB
ejpam-4744	137	2	sj	sj	VERB
ejpam-4744	137	3	∪	∪	PROPN
ejpam-4744	137	4	tj	tj	PROPN
ejpam-4744	137	5	.	.	PUNCT
ejpam-4744	138	1	note	note	VERB
ejpam-4744	138	2	that	that	SCONJ
ejpam-4744	138	3	(	(	PUNCT
ejpam-4744	138	4	i	i	PRON
ejpam-4744	138	5	,	,	PUNCT
ejpam-4744	138	6	1	1	X
ejpam-4744	138	7	)	)	PUNCT
ejpam-4744	138	8	∈	∈	NOUN
ejpam-4744	138	9	si	si	PROPN
ejpam-4744	138	10	⊆	⊆	NUM
ejpam-4744	138	11	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	138	12	)	)	PUNCT
ejpam-4744	138	13	=	=	SYM
ejpam-4744	138	14	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	138	15	)	)	PUNCT
ejpam-4744	138	16	.	.	PUNCT
ejpam-4744	139	1	it	it	PRON
ejpam-4744	139	2	follows	follow	VERB
ejpam-4744	139	3	that	that	SCONJ
ejpam-4744	139	4	(	(	PUNCT
ejpam-4744	139	5	i	i	PRON
ejpam-4744	139	6	,	,	PUNCT
ejpam-4744	139	7	1	1	X
ejpam-4744	139	8	)	)	PUNCT
ejpam-4744	139	9	∈	∈	NOUN
ejpam-4744	139	10	sj	sj	NOUN
ejpam-4744	139	11	=	=	SYM
ejpam-4744	139	12	{	{	PUNCT
ejpam-4744	139	13	(	(	PUNCT
ejpam-4744	139	14	j	j	NOUN
ejpam-4744	139	15	,	,	PUNCT
ejpam-4744	139	16	1	1	NUM
ejpam-4744	139	17	)	)	PUNCT
ejpam-4744	139	18	,	,	PUNCT
ejpam-4744	139	19	(	(	PUNCT
ejpam-4744	139	20	j	j	NOUN
ejpam-4744	139	21	,	,	PUNCT
ejpam-4744	139	22	2	2	NUM
ejpam-4744	139	23	)	)	PUNCT
ejpam-4744	139	24	,	,	PUNCT
ejpam-4744	139	25	...	...	PUNCT
ejpam-4744	139	26	,	,	PUNCT
ejpam-4744	139	27	(	(	PUNCT
ejpam-4744	139	28	j	j	NOUN
ejpam-4744	139	29	,	,	PUNCT
ejpam-4744	139	30	ω(h	ω(h	NUM
ejpam-4744	139	31	)	)	PUNCT
ejpam-4744	139	32	)	)	PUNCT
ejpam-4744	139	33	}	}	PUNCT
ejpam-4744	139	34	.	.	PUNCT
ejpam-4744	140	1	consequently	consequently	ADV
ejpam-4744	140	2	,	,	PUNCT
ejpam-4744	140	3	i	i	PRON
ejpam-4744	140	4	=	=	PUNCT
ejpam-4744	140	5	j.	j.	PROPN
ejpam-4744	140	6	in	in	ADP
ejpam-4744	140	7	effect	effect	NOUN
ejpam-4744	140	8	x1	x1	X
ejpam-4744	140	9	=	=	SYM
ejpam-4744	140	10	x2	x2	PROPN
ejpam-4744	140	11	.	.	PUNCT
ejpam-4744	141	1	case	case	NOUN
ejpam-4744	141	2	2	2	X
ejpam-4744	141	3	.	.	PUNCT
ejpam-4744	141	4	suppose	suppose	VERB
ejpam-4744	141	5	x1	x1	PROPN
ejpam-4744	141	6	∈	∈	PROPN
ejpam-4744	141	7	v	v	PROPN
ejpam-4744	141	8	(	(	PUNCT
ejpam-4744	141	9	hai	hai	NOUN
ejpam-4744	141	10	)	)	PUNCT
ejpam-4744	141	11	and	and	CCONJ
ejpam-4744	141	12	x2	x2	PROPN
ejpam-4744	141	13	∈	∈	PROPN
ejpam-4744	141	14	v	v	X
ejpam-4744	141	15	(	(	PUNCT
ejpam-4744	141	16	haj	haj	PROPN
ejpam-4744	141	17	)	)	PUNCT
ejpam-4744	141	18	.	.	PUNCT
ejpam-4744	142	1	suppose	suppose	VERB
ejpam-4744	143	1	i	i	PRON
ejpam-4744	143	2	̸=	̸=	PROPN
ejpam-4744	143	3	j.	j.	PROPN
ejpam-4744	143	4	then	then	ADV
ejpam-4744	143	5	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	143	6	)	)	PUNCT
ejpam-4744	143	7	∩	∩	ADJ
ejpam-4744	143	8	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	143	9	)	)	PUNCT
ejpam-4744	143	10	=	=	SYM
ejpam-4744	143	11	ϕi(x1	ϕi(x1	NOUN
ejpam-4744	143	12	)	)	PUNCT
ejpam-4744	143	13	∩	∩	ADJ
ejpam-4744	143	14	ϕj(x2	ϕj(x2	NOUN
ejpam-4744	143	15	)	)	PUNCT
ejpam-4744	143	16	⊆	⊆	NUM
ejpam-4744	143	17	si	si	NOUN
ejpam-4744	143	18	∩	∩	NOUN
ejpam-4744	143	19	sj	sj	PROPN
ejpam-4744	143	20	̸=	̸=	PROPN
ejpam-4744	143	21	∅.	∅.	ADP
ejpam-4744	143	22	this	this	PRON
ejpam-4744	143	23	is	be	AUX
ejpam-4744	143	24	a	a	DET
ejpam-4744	143	25	contradiction	contradiction	NOUN
ejpam-4744	143	26	.	.	PUNCT
ejpam-4744	144	1	hence	hence	ADV
ejpam-4744	144	2	,	,	PUNCT
ejpam-4744	144	3	i	i	PRON
ejpam-4744	144	4	=	=	PUNCT
ejpam-4744	144	5	j.	j.	PROPN
ejpam-4744	144	6	consequently	consequently	ADV
ejpam-4744	144	7	,	,	PUNCT
ejpam-4744	144	8	ϕi(x1	ϕi(x1	PROPN
ejpam-4744	144	9	)	)	PUNCT
ejpam-4744	144	10	=	=	SYM
ejpam-4744	144	11	ϕ(x1	ϕ(x1	ADJ
ejpam-4744	144	12	)	)	PUNCT
ejpam-4744	144	13	=	=	SYM
ejpam-4744	144	14	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	144	15	)	)	PUNCT
ejpam-4744	144	16	=	=	SYM
ejpam-4744	144	17	ϕj(x2	ϕj(x2	NOUN
ejpam-4744	144	18	)	)	PUNCT
ejpam-4744	144	19	=	=	SYM
ejpam-4744	144	20	ϕi(x2	ϕi(x2	NOUN
ejpam-4744	144	21	)	)	PUNCT
ejpam-4744	144	22	.	.	PUNCT
ejpam-4744	145	1	since	since	SCONJ
ejpam-4744	145	2	ϕi	ϕi	PRON
ejpam-4744	145	3	is	be	AUX
ejpam-4744	145	4	one	one	NUM
ejpam-4744	145	5	to	to	ADP
ejpam-4744	145	6	one	one	NUM
ejpam-4744	145	7	,	,	PUNCT
ejpam-4744	145	8	x1	x1	PROPN
ejpam-4744	145	9	=	=	SYM
ejpam-4744	145	10	x2	x2	PROPN
ejpam-4744	145	11	.	.	PUNCT
ejpam-4744	146	1	therefore	therefore	ADV
ejpam-4744	146	2	,	,	PUNCT
ejpam-4744	146	3	ϕ	ϕ	PROPN
ejpam-4744	146	4	is	be	AUX
ejpam-4744	146	5	one	one	NUM
ejpam-4744	146	6	to	to	ADP
ejpam-4744	146	7	one	one	NUM
ejpam-4744	146	8	.	.	PUNCT
ejpam-4744	147	1	suppose	suppose	VERB
ejpam-4744	147	2	b	b	X
ejpam-4744	147	3	∈	∈	PROPN
ejpam-4744	147	4	fi	fi	NOUN
ejpam-4744	147	5	for	for	ADP
ejpam-4744	147	6	some	some	DET
ejpam-4744	147	7	i.	i.	NOUN
ejpam-4744	147	8	since	since	SCONJ
ejpam-4744	147	9	ϕi	ϕi	ADP
ejpam-4744	147	10	:	:	PUNCT
ejpam-4744	147	11	v	v	NOUN
ejpam-4744	147	12	(	(	PUNCT
ejpam-4744	147	13	hai	hai	NOUN
ejpam-4744	147	14	)	)	PUNCT
ejpam-4744	147	15	→	→	SYM
ejpam-4744	147	16	fi	fi	NOUN
ejpam-4744	147	17	is	be	AUX
ejpam-4744	147	18	onto	onto	ADP
ejpam-4744	147	19	,	,	PUNCT
ejpam-4744	147	20	there	there	PRON
ejpam-4744	147	21	exists	exist	VERB
ejpam-4744	147	22	x	x	X
ejpam-4744	147	23	∈	∈	PROPN
ejpam-4744	147	24	v	v	X
ejpam-4744	147	25	(	(	PUNCT
ejpam-4744	147	26	hai	hai	NOUN
ejpam-4744	147	27	)	)	PUNCT
ejpam-4744	147	28	such	such	ADJ
ejpam-4744	147	29	that	that	DET
ejpam-4744	147	30	ϕi(x	ϕi(x	NOUN
ejpam-4744	147	31	)	)	PUNCT
ejpam-4744	147	32	=	=	SYM
ejpam-4744	147	33	b.	b.	PROPN
ejpam-4744	147	34	consequently	consequently	ADV
ejpam-4744	147	35	,	,	PUNCT
ejpam-4744	147	36	ϕ(x	ϕ(x	X
ejpam-4744	147	37	)	)	PUNCT
ejpam-4744	147	38	=	=	SYM
ejpam-4744	147	39	ϕi(x	ϕi(x	PROPN
ejpam-4744	147	40	)	)	PUNCT
ejpam-4744	147	41	=	=	SYM
ejpam-4744	148	1	b.	b.	PROPN
ejpam-4744	148	2	suppose	suppose	VERB
ejpam-4744	148	3	b	b	X
ejpam-4744	148	4	=	=	SYM
ejpam-4744	148	5	si	si	X
ejpam-4744	148	6	∪	∪	ADP
ejpam-4744	148	7	ti	ti	NOUN
ejpam-4744	148	8	,	,	PUNCT
ejpam-4744	148	9	for	for	SCONJ
ejpam-4744	148	10	some	some	DET
ejpam-4744	148	11	i.	i.	NOUN
ejpam-4744	148	12	take	take	NOUN
ejpam-4744	148	13	x	x	PUNCT
ejpam-4744	148	14	=	=	PRON
ejpam-4744	148	15	ai	ai	VERB
ejpam-4744	148	16	.	.	PUNCT
ejpam-4744	148	17	then	then	ADV
ejpam-4744	148	18	ϕ(x	ϕ(x	X
ejpam-4744	148	19	)	)	PUNCT
ejpam-4744	148	20	=	=	SYM
ejpam-4744	148	21	ϕ(ai	ϕ(ai	PROPN
ejpam-4744	148	22	)	)	PUNCT
ejpam-4744	148	23	=	=	SYM
ejpam-4744	148	24	b.	b.	PROPN
ejpam-4744	148	25	hence	hence	ADV
ejpam-4744	148	26	,	,	PUNCT
ejpam-4744	148	27	ϕ	ϕ	PROPN
ejpam-4744	148	28	is	be	AUX
ejpam-4744	148	29	onto	onto	ADP
ejpam-4744	148	30	.	.	PUNCT
ejpam-4744	149	1	let	let	VERB
ejpam-4744	149	2	x1	x1	PROPN
ejpam-4744	149	3	and	and	CCONJ
ejpam-4744	149	4	x2	x2	PROPN
ejpam-4744	149	5	be	be	VERB
ejpam-4744	149	6	adjacent	adjacent	ADJ
ejpam-4744	149	7	in	in	ADP
ejpam-4744	149	8	g	g	PROPN
ejpam-4744	149	9	◦	◦	NOUN
ejpam-4744	149	10	h.	h.	NOUN
ejpam-4744	149	11	consider	consider	VERB
ejpam-4744	149	12	the	the	DET
ejpam-4744	149	13	following	follow	VERB
ejpam-4744	149	14	cases	case	NOUN
ejpam-4744	149	15	:	:	PUNCT
ejpam-4744	149	16	case	case	NOUN
ejpam-4744	149	17	1	1	NUM
ejpam-4744	149	18	.	.	PUNCT
ejpam-4744	149	19	suppose	suppose	VERB
ejpam-4744	149	20	x1	x1	PROPN
ejpam-4744	149	21	and	and	CCONJ
ejpam-4744	149	22	x2	x2	PROPN
ejpam-4744	149	23	are	be	AUX
ejpam-4744	149	24	adjacent	adjacent	ADJ
ejpam-4744	149	25	in	in	ADP
ejpam-4744	149	26	g.	g.	PROPN
ejpam-4744	150	1	then	then	ADV
ejpam-4744	150	2	x1	x1	PROPN
ejpam-4744	150	3	=	=	PUNCT
ejpam-4744	150	4	ai	ai	VERB
ejpam-4744	150	5	and	and	CCONJ
ejpam-4744	150	6	x2	x2	PROPN
ejpam-4744	150	7	=	=	SYM
ejpam-4744	150	8	aj	aj	PROPN
ejpam-4744	150	9	,	,	PUNCT
ejpam-4744	150	10	for	for	ADP
ejpam-4744	150	11	some	some	DET
ejpam-4744	150	12	i	i	PROPN
ejpam-4744	150	13	and	and	CCONJ
ejpam-4744	150	14	j.	j.	PROPN
ejpam-4744	150	15	in	in	ADP
ejpam-4744	150	16	effect	effect	NOUN
ejpam-4744	150	17	,	,	PUNCT
ejpam-4744	150	18	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	150	19	)	)	PUNCT
ejpam-4744	150	20	=	=	SYM
ejpam-4744	151	1	si	si	X
ejpam-4744	151	2	∪	∪	X
ejpam-4744	151	3	ti	ti	NOUN
ejpam-4744	151	4	and	and	CCONJ
ejpam-4744	151	5	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	151	6	)	)	PUNCT
ejpam-4744	152	1	=	=	VERB
ejpam-4744	152	2	sj	sj	VERB
ejpam-4744	152	3	∪	∪	PROPN
ejpam-4744	152	4	tj	tj	PROPN
ejpam-4744	152	5	.	.	PUNCT
ejpam-4744	153	1	since	since	SCONJ
ejpam-4744	153	2	ai	ai	PROPN
ejpam-4744	153	3	and	and	CCONJ
ejpam-4744	153	4	aj	aj	PROPN
ejpam-4744	153	5	are	be	AUX
ejpam-4744	153	6	adjacent	adjacent	ADJ
ejpam-4744	153	7	in	in	ADP
ejpam-4744	153	8	g	g	PROPN
ejpam-4744	153	9	,	,	PUNCT
ejpam-4744	153	10	there	there	PRON
ejpam-4744	153	11	exists	exist	VERB
ejpam-4744	153	12	k	k	PROPN
ejpam-4744	153	13	such	such	ADJ
ejpam-4744	153	14	that	that	PRON
ejpam-4744	153	15	ai	ai	VERB
ejpam-4744	153	16	,	,	PUNCT
ejpam-4744	153	17	aj	aj	PROPN
ejpam-4744	153	18	∈	∈	PROPN
ejpam-4744	153	19	bk	bk	PROPN
ejpam-4744	153	20	.	.	PUNCT
ejpam-4744	154	1	this	this	PRON
ejpam-4744	154	2	implies	imply	VERB
ejpam-4744	154	3	that	that	SCONJ
ejpam-4744	154	4	(	(	PUNCT
ejpam-4744	154	5	0	0	NUM
ejpam-4744	154	6	,	,	PUNCT
ejpam-4744	154	7	k	k	NOUN
ejpam-4744	154	8	)	)	PUNCT
ejpam-4744	154	9	∈	∈	PROPN
ejpam-4744	154	10	ti	ti	NOUN
ejpam-4744	154	11	and	and	CCONJ
ejpam-4744	154	12	(	(	PUNCT
ejpam-4744	154	13	0	0	NUM
ejpam-4744	154	14	,	,	PUNCT
ejpam-4744	154	15	k	k	NOUN
ejpam-4744	154	16	)	)	PUNCT
ejpam-4744	154	17	∈	∈	PROPN
ejpam-4744	154	18	tj	tj	NOUN
ejpam-4744	154	19	.	.	PUNCT
ejpam-4744	155	1	it	it	PRON
ejpam-4744	155	2	follows	follow	VERB
ejpam-4744	155	3	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	155	4	)	)	PUNCT
ejpam-4744	155	5	∩	∩	ADJ
ejpam-4744	155	6	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	155	7	)	)	PUNCT
ejpam-4744	156	1	̸=	̸=	PROPN
ejpam-4744	156	2	∅.	∅.	PRON
ejpam-4744	156	3	therefore	therefore	ADV
ejpam-4744	156	4	,	,	PUNCT
ejpam-4744	156	5	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	156	6	)	)	PUNCT
ejpam-4744	156	7	and	and	CCONJ
ejpam-4744	156	8	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	156	9	)	)	PUNCT
ejpam-4744	156	10	are	be	AUX
ejpam-4744	156	11	adjacent	adjacent	ADJ
ejpam-4744	156	12	in	in	ADP
ejpam-4744	156	13	ω(f	ω(f	PROPN
ejpam-4744	156	14	)	)	PUNCT
ejpam-4744	156	15	.	.	PUNCT
ejpam-4744	157	1	case	case	NOUN
ejpam-4744	157	2	2	2	X
ejpam-4744	157	3	.	.	PUNCT
ejpam-4744	157	4	suppose	suppose	VERB
ejpam-4744	157	5	x1	x1	PROPN
ejpam-4744	157	6	and	and	CCONJ
ejpam-4744	157	7	x2	x2	PROPN
ejpam-4744	157	8	are	be	AUX
ejpam-4744	157	9	adjacent	adjacent	ADJ
ejpam-4744	157	10	in	in	ADP
ejpam-4744	157	11	hai	hai	PROPN
ejpam-4744	157	12	for	for	ADP
ejpam-4744	157	13	some	some	DET
ejpam-4744	157	14	i.	i.	NOUN
ejpam-4744	157	15	then	then	ADV
ejpam-4744	157	16	x1	x1	NUM
ejpam-4744	157	17	,	,	PUNCT
ejpam-4744	157	18	x2	x2	PROPN
ejpam-4744	157	19	∈	∈	PROPN
ejpam-4744	157	20	v	v	X
ejpam-4744	157	21	(	(	PUNCT
ejpam-4744	157	22	hai	hai	NOUN
ejpam-4744	157	23	)	)	PUNCT
ejpam-4744	157	24	.	.	PUNCT
ejpam-4744	158	1	it	it	PRON
ejpam-4744	158	2	follows	follow	VERB
ejpam-4744	158	3	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	158	4	)	)	PUNCT
ejpam-4744	159	1	=	=	SYM
ejpam-4744	159	2	ϕi(x1	ϕi(x1	PROPN
ejpam-4744	159	3	)	)	PUNCT
ejpam-4744	159	4	and	and	CCONJ
ejpam-4744	159	5	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	159	6	)	)	PUNCT
ejpam-4744	159	7	=	=	SYM
ejpam-4744	159	8	ϕi(x2	ϕi(x2	NOUN
ejpam-4744	159	9	)	)	PUNCT
ejpam-4744	159	10	.	.	PUNCT
ejpam-4744	160	1	since	since	SCONJ
ejpam-4744	160	2	ϕi	ϕi	ADP
ejpam-4744	160	3	preserves	preserves	PROPN
ejpam-4744	160	4	adjacency	adjacency	PROPN
ejpam-4744	160	5	,	,	PUNCT
ejpam-4744	160	6	ϕ(x1	ϕ(x1	ADJ
ejpam-4744	160	7	)	)	PUNCT
ejpam-4744	160	8	∩	∩	ADJ
ejpam-4744	160	9	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	160	10	)	)	PUNCT
ejpam-4744	160	11	=	=	SYM
ejpam-4744	160	12	ϕi(x1	ϕi(x1	NOUN
ejpam-4744	160	13	)	)	PUNCT
ejpam-4744	160	14	∩	∩	ADJ
ejpam-4744	160	15	ϕi(x2	ϕi(x2	NOUN
ejpam-4744	160	16	)	)	PUNCT
ejpam-4744	160	17	̸=	̸=	PROPN
ejpam-4744	160	18	∅.	∅.	ADP
ejpam-4744	160	19	thus	thus	ADV
ejpam-4744	160	20	,	,	PUNCT
ejpam-4744	160	21	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	160	22	)	)	PUNCT
ejpam-4744	160	23	and	and	CCONJ
ejpam-4744	160	24	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	160	25	)	)	PUNCT
ejpam-4744	160	26	are	be	AUX
ejpam-4744	160	27	adjacent	adjacent	ADJ
ejpam-4744	160	28	in	in	ADP
ejpam-4744	160	29	ω(f	ω(f	PROPN
ejpam-4744	160	30	)	)	PUNCT
ejpam-4744	160	31	.	.	PUNCT
ejpam-4744	161	1	case	case	NOUN
ejpam-4744	161	2	3	3	X
ejpam-4744	161	3	.	.	PUNCT
ejpam-4744	161	4	suppose	suppose	VERB
ejpam-4744	161	5	x1	x1	PROPN
ejpam-4744	161	6	=	=	PUNCT
ejpam-4744	161	7	ai	ai	VERB
ejpam-4744	161	8	and	and	CCONJ
ejpam-4744	161	9	x2	x2	PROPN
ejpam-4744	161	10	∈	∈	PROPN
ejpam-4744	161	11	v	v	X
ejpam-4744	161	12	(	(	PUNCT
ejpam-4744	161	13	hai	hai	NOUN
ejpam-4744	161	14	)	)	PUNCT
ejpam-4744	161	15	.	.	PUNCT
ejpam-4744	162	1	then	then	ADV
ejpam-4744	162	2	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	162	3	)	)	PUNCT
ejpam-4744	162	4	=	=	SYM
ejpam-4744	162	5	ϕ(ai	ϕ(ai	PROPN
ejpam-4744	162	6	)	)	PUNCT
ejpam-4744	162	7	=	=	NOUN
ejpam-4744	162	8	si	si	X
ejpam-4744	162	9	∪	∪	X
ejpam-4744	162	10	ti	ti	NOUN
ejpam-4744	162	11	and	and	CCONJ
ejpam-4744	162	12	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	162	13	)	)	PUNCT
ejpam-4744	162	14	=	=	SYM
ejpam-4744	162	15	ϕi(x2	ϕi(x2	NOUN
ejpam-4744	162	16	)	)	PUNCT
ejpam-4744	162	17	.	.	PUNCT
ejpam-4744	163	1	since	since	SCONJ
ejpam-4744	163	2	ϕi(x2	ϕi(x2	NOUN
ejpam-4744	163	3	)	)	PUNCT
ejpam-4744	163	4	⊆	⊆	NUM
ejpam-4744	163	5	si	si	X
ejpam-4744	163	6	,	,	PUNCT
ejpam-4744	163	7	ϕ(x1	ϕ(x1	ADJ
ejpam-4744	163	8	)	)	PUNCT
ejpam-4744	163	9	∩	∩	ADJ
ejpam-4744	163	10	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	163	11	)	)	PUNCT
ejpam-4744	163	12	̸=	̸=	PROPN
ejpam-4744	163	13	∅.	∅.	ADP
ejpam-4744	163	14	thus	thus	ADV
ejpam-4744	163	15	,	,	PUNCT
ejpam-4744	163	16	ϕ(x1	ϕ(x1	PROPN
ejpam-4744	163	17	)	)	PUNCT
ejpam-4744	163	18	and	and	CCONJ
ejpam-4744	163	19	ϕ(x2	ϕ(x2	NOUN
ejpam-4744	163	20	)	)	PUNCT
ejpam-4744	163	21	are	be	AUX
ejpam-4744	163	22	adjacent	adjacent	ADJ
ejpam-4744	163	23	in	in	ADP
ejpam-4744	163	24	ω(f	ω(f	PROPN
ejpam-4744	163	25	)	)	PUNCT
ejpam-4744	163	26	.	.	PUNCT
ejpam-4744	164	1	suppose	suppose	VERB
ejpam-4744	164	2	a	a	PRON
ejpam-4744	164	3	and	and	CCONJ
ejpam-4744	164	4	b	b	NOUN
ejpam-4744	164	5	are	be	AUX
ejpam-4744	164	6	adjacent	adjacent	ADJ
ejpam-4744	164	7	in	in	ADP
ejpam-4744	164	8	ω(f	ω(f	PROPN
ejpam-4744	164	9	)	)	PUNCT
ejpam-4744	164	10	.	.	PUNCT
ejpam-4744	165	1	that	that	PRON
ejpam-4744	165	2	is	be	AUX
ejpam-4744	165	3	,	,	PUNCT
ejpam-4744	165	4	a	a	DET
ejpam-4744	165	5	∩	∩	ADJ
ejpam-4744	165	6	b	b	NOUN
ejpam-4744	165	7	̸=	̸=	PROPN
ejpam-4744	165	8	∅.	∅.	ADP
ejpam-4744	165	9	the	the	DET
ejpam-4744	165	10	case	case	NOUN
ejpam-4744	165	11	a	a	DET
ejpam-4744	165	12	∈	∈	PROPN
ejpam-4744	165	13	fi	fi	NOUN
ejpam-4744	165	14	and	and	CCONJ
ejpam-4744	165	15	b	b	NOUN
ejpam-4744	165	16	∈	∈	PROPN
ejpam-4744	165	17	fj	fj	PROPN
ejpam-4744	165	18	,	,	PUNCT
ejpam-4744	165	19	where	where	SCONJ
ejpam-4744	165	20	i	i	PRON
ejpam-4744	165	21	,	,	PUNCT
ejpam-4744	165	22	j	j	PROPN
ejpam-4744	165	23	̸=	̸=	PROPN
ejpam-4744	165	24	0	0	NUM
ejpam-4744	166	1	and	and	CCONJ
ejpam-4744	166	2	i	i	PRON
ejpam-4744	166	3	̸=	̸=	PROPN
ejpam-4744	166	4	j	j	PROPN
ejpam-4744	166	5	,	,	PUNCT
ejpam-4744	166	6	is	be	AUX
ejpam-4744	166	7	not	not	PART
ejpam-4744	166	8	possible	possible	ADJ
ejpam-4744	166	9	,	,	PUNCT
ejpam-4744	166	10	since	since	SCONJ
ejpam-4744	166	11	si	si	PROPN
ejpam-4744	166	12	∩	∩	NOUN
ejpam-4744	166	13	sj	sj	NOUN
ejpam-4744	166	14	=	=	NOUN
ejpam-4744	166	15	∅	∅	NOUN
ejpam-4744	166	16	in	in	ADP
ejpam-4744	166	17	this	this	DET
ejpam-4744	166	18	case	case	NOUN
ejpam-4744	166	19	.	.	PUNCT
ejpam-4744	167	1	consider	consider	VERB
ejpam-4744	167	2	the	the	DET
ejpam-4744	167	3	following	follow	VERB
ejpam-4744	167	4	cases	case	NOUN
ejpam-4744	167	5	:	:	PUNCT
ejpam-4744	167	6	case	case	NOUN
ejpam-4744	167	7	1	1	NUM
ejpam-4744	167	8	.	.	PUNCT
ejpam-4744	167	9	suppose	suppose	VERB
ejpam-4744	167	10	a	a	DET
ejpam-4744	167	11	,	,	PUNCT
ejpam-4744	167	12	b	b	PROPN
ejpam-4744	167	13	∈	∈	PROPN
ejpam-4744	167	14	fi	fi	NOUN
ejpam-4744	167	15	,	,	PUNCT
ejpam-4744	167	16	for	for	ADP
ejpam-4744	167	17	some	some	DET
ejpam-4744	167	18	i.	i.	NOUN
ejpam-4744	167	19	since	since	SCONJ
ejpam-4744	167	20	ϕi	ϕi	PRON
ejpam-4744	167	21	is	be	AUX
ejpam-4744	167	22	onto	onto	ADP
ejpam-4744	167	23	,	,	PUNCT
ejpam-4744	167	24	there	there	PRON
ejpam-4744	167	25	exists	exist	VERB
ejpam-4744	167	26	x1	x1	PROPN
ejpam-4744	167	27	,	,	PUNCT
ejpam-4744	168	1	x2	x2	PROPN
ejpam-4744	168	2	∈	∈	PROPN
ejpam-4744	168	3	v	v	X
ejpam-4744	168	4	(	(	PUNCT
ejpam-4744	168	5	hai	hai	NOUN
ejpam-4744	168	6	)	)	PUNCT
ejpam-4744	168	7	such	such	ADJ
ejpam-4744	168	8	that	that	SCONJ
ejpam-4744	168	9	ϕi(x1	ϕi(x1	X
ejpam-4744	168	10	)	)	PUNCT
ejpam-4744	168	11	=	=	NOUN
ejpam-4744	168	12	a	a	PRON
ejpam-4744	168	13	and	and	CCONJ
ejpam-4744	168	14	ϕi(x2	ϕi(x2	NOUN
ejpam-4744	168	15	)	)	PUNCT
ejpam-4744	168	16	=	=	SYM
ejpam-4744	169	1	b.	b.	PROPN
ejpam-4744	169	2	since	since	SCONJ
ejpam-4744	169	3	ϕi	ϕi	ADP
ejpam-4744	169	4	preserves	preserves	PROPN
ejpam-4744	169	5	adjacency	adjacency	PROPN
ejpam-4744	169	6	,	,	PUNCT
ejpam-4744	169	7	x1	x1	PROPN
ejpam-4744	169	8	j.	j.	PROPN
ejpam-4744	169	9	b.	b.	PROPN
ejpam-4744	169	10	palco	palco	PROPN
ejpam-4744	169	11	,	,	PUNCT
ejpam-4744	169	12	r.	r.	PROPN
ejpam-4744	169	13	n.	n.	PROPN
ejpam-4744	169	14	paluga	paluga	PROPN
ejpam-4744	169	15	/	/	SYM
ejpam-4744	169	16	eur	eur	PROPN
ejpam-4744	169	17	.	.	PUNCT
ejpam-4744	170	1	j.	j.	PROPN
ejpam-4744	170	2	pure	pure	PROPN
ejpam-4744	170	3	appl	appl	PROPN
ejpam-4744	170	4	.	.	PROPN
ejpam-4744	170	5	math	math	PROPN
ejpam-4744	170	6	,	,	PUNCT
ejpam-4744	170	7	16	16	NUM
ejpam-4744	170	8	(	(	PUNCT
ejpam-4744	170	9	2	2	NUM
ejpam-4744	170	10	)	)	PUNCT
ejpam-4744	170	11	(	(	PUNCT
ejpam-4744	170	12	2023	2023	NUM
ejpam-4744	170	13	)	)	PUNCT
ejpam-4744	170	14	,	,	PUNCT
ejpam-4744	170	15	1318	1318	NUM
ejpam-4744	170	16	-	-	SYM
ejpam-4744	170	17	1325	1325	NUM
ejpam-4744	170	18	1322	1322	NUM
ejpam-4744	170	19	and	and	CCONJ
ejpam-4744	170	20	x2	x2	PRON
ejpam-4744	170	21	are	be	AUX
ejpam-4744	170	22	adjacent	adjacent	ADJ
ejpam-4744	170	23	in	in	ADP
ejpam-4744	170	24	hai	hai	PROPN
ejpam-4744	170	25	.	.	PUNCT
ejpam-4744	171	1	it	it	PRON
ejpam-4744	171	2	follows	follow	VERB
ejpam-4744	171	3	that	that	SCONJ
ejpam-4744	171	4	x1	x1	PROPN
ejpam-4744	171	5	and	and	CCONJ
ejpam-4744	171	6	x2	x2	PROPN
ejpam-4744	171	7	are	be	AUX
ejpam-4744	171	8	adjacent	adjacent	ADJ
ejpam-4744	171	9	in	in	ADP
ejpam-4744	171	10	g	g	PROPN
ejpam-4744	171	11	◦	◦	NOUN
ejpam-4744	171	12	h.	h.	NOUN
ejpam-4744	171	13	case	case	NOUN
ejpam-4744	171	14	2	2	X
ejpam-4744	171	15	.	.	PUNCT
ejpam-4744	171	16	suppose	suppose	VERB
ejpam-4744	171	17	a	a	DET
ejpam-4744	171	18	=	=	X
ejpam-4744	171	19	si	si	X
ejpam-4744	171	20	∪	∪	X
ejpam-4744	171	21	ti	ti	NOUN
ejpam-4744	171	22	and	and	CCONJ
ejpam-4744	171	23	b	b	X
ejpam-4744	171	24	=	=	PRON
ejpam-4744	171	25	sj	sj	VERB
ejpam-4744	171	26	∪	∪	NOUN
ejpam-4744	171	27	tj	tj	NOUN
ejpam-4744	171	28	for	for	ADP
ejpam-4744	171	29	some	some	DET
ejpam-4744	171	30	i	i	PROPN
ejpam-4744	171	31	,	,	PUNCT
ejpam-4744	171	32	j	j	PROPN
ejpam-4744	171	33	=	=	SYM
ejpam-4744	171	34	1	1	NUM
ejpam-4744	171	35	,	,	PUNCT
ejpam-4744	171	36	2	2	NUM
ejpam-4744	171	37	,	,	PUNCT
ejpam-4744	171	38	3	3	NUM
ejpam-4744	171	39	,	,	PUNCT
ejpam-4744	171	40	...	...	PUNCT
ejpam-4744	171	41	,	,	PUNCT
ejpam-4744	171	42	n	n	CCONJ
ejpam-4744	171	43	,	,	PUNCT
ejpam-4744	171	44	i	i	PRON
ejpam-4744	171	45	̸=	̸=	PROPN
ejpam-4744	171	46	j.	j.	PROPN
ejpam-4744	171	47	since	since	SCONJ
ejpam-4744	171	48	a	a	DET
ejpam-4744	171	49	∩	∩	ADJ
ejpam-4744	171	50	b	b	NOUN
ejpam-4744	171	51	̸=	̸=	PROPN
ejpam-4744	171	52	∅	∅	NOUN
ejpam-4744	171	53	,	,	PUNCT
ejpam-4744	171	54	(	(	PUNCT
ejpam-4744	171	55	si	si	X
ejpam-4744	171	56	∩	∩	ADJ
ejpam-4744	171	57	sj	sj	NOUN
ejpam-4744	171	58	)	)	PUNCT
ejpam-4744	171	59	∪	∪	NOUN
ejpam-4744	171	60	(	(	PUNCT
ejpam-4744	171	61	si	si	X
ejpam-4744	171	62	∩	∩	ADJ
ejpam-4744	171	63	tj	tj	NOUN
ejpam-4744	171	64	)	)	PUNCT
ejpam-4744	171	65	∪	∪	NOUN
ejpam-4744	171	66	(	(	PUNCT
ejpam-4744	171	67	ti	ti	X
ejpam-4744	171	68	∩	∩	ADJ
ejpam-4744	171	69	sj	sj	NOUN
ejpam-4744	171	70	)	)	PUNCT
ejpam-4744	171	71	∪	∪	NOUN
ejpam-4744	171	72	(	(	PUNCT
ejpam-4744	171	73	ti	ti	X
ejpam-4744	171	74	∩	∩	ADJ
ejpam-4744	171	75	tj	tj	NOUN
ejpam-4744	171	76	)	)	PUNCT
ejpam-4744	171	77	̸=	̸=	PROPN
ejpam-4744	171	78	∅.	∅.	AUX
ejpam-4744	171	79	note	note	VERB
ejpam-4744	171	80	that	that	SCONJ
ejpam-4744	171	81	si	si	NOUN
ejpam-4744	171	82	∩	∩	ADJ
ejpam-4744	171	83	sj	sj	NOUN
ejpam-4744	171	84	=	=	NOUN
ejpam-4744	171	85	∅	∅	NOUN
ejpam-4744	171	86	,	,	PUNCT
ejpam-4744	171	87	si∩tj	si∩tj	NOUN
ejpam-4744	171	88	=	=	NOUN
ejpam-4744	171	89	∅	∅	NOUN
ejpam-4744	171	90	,	,	PUNCT
ejpam-4744	171	91	ti∩sj	ti∩sj	VERB
ejpam-4744	171	92	=	=	PRON
ejpam-4744	171	93	∅.	∅.	VERB
ejpam-4744	171	94	consequently	consequently	ADV
ejpam-4744	171	95	,	,	PUNCT
ejpam-4744	171	96	(	(	PUNCT
ejpam-4744	171	97	ti∩tj	ti∩tj	NOUN
ejpam-4744	171	98	)	)	PUNCT
ejpam-4744	171	99	̸=	̸=	PROPN
ejpam-4744	171	100	∅.	∅.	ADP
ejpam-4744	171	101	moreover	moreover	ADV
ejpam-4744	171	102	,	,	PUNCT
ejpam-4744	171	103	ϕ(ai	ϕ(ai	PROPN
ejpam-4744	171	104	)	)	PUNCT
ejpam-4744	171	105	=	=	SYM
ejpam-4744	171	106	a	a	PRON
ejpam-4744	171	107	and	and	CCONJ
ejpam-4744	171	108	ϕ(aj	ϕ(aj	NUM
ejpam-4744	171	109	)	)	PUNCT
ejpam-4744	172	1	=	=	SYM
ejpam-4744	172	2	b.	b.	PROPN
ejpam-4744	172	3	let	let	VERB
ejpam-4744	172	4	t	t	PROPN
ejpam-4744	172	5	∈	∈	PROPN
ejpam-4744	172	6	ti	ti	PROPN
ejpam-4744	172	7	∩	∩	X
ejpam-4744	172	8	tj	tj	PROPN
ejpam-4744	172	9	.	.	PUNCT
ejpam-4744	173	1	then	then	ADV
ejpam-4744	173	2	t	t	PROPN
ejpam-4744	173	3	∈	∈	PROPN
ejpam-4744	173	4	ti	ti	NOUN
ejpam-4744	173	5	and	and	CCONJ
ejpam-4744	173	6	t	t	NOUN
ejpam-4744	173	7	∈	∈	PROPN
ejpam-4744	173	8	tj	tj	NOUN
ejpam-4744	173	9	.	.	PUNCT
ejpam-4744	174	1	this	this	PRON
ejpam-4744	174	2	implies	imply	VERB
ejpam-4744	174	3	that	that	SCONJ
ejpam-4744	174	4	t	t	NOUN
ejpam-4744	174	5	=	=	SYM
ejpam-4744	174	6	(	(	PUNCT
ejpam-4744	174	7	0	0	NUM
ejpam-4744	174	8	,	,	PUNCT
ejpam-4744	174	9	r	r	NOUN
ejpam-4744	174	10	)	)	PUNCT
ejpam-4744	174	11	where	where	SCONJ
ejpam-4744	174	12	ai	ai	VERB
ejpam-4744	174	13	∈	∈	PROPN
ejpam-4744	174	14	br	br	NOUN
ejpam-4744	174	15	and	and	CCONJ
ejpam-4744	174	16	t	t	NOUN
ejpam-4744	174	17	=	=	SYM
ejpam-4744	174	18	(	(	PUNCT
ejpam-4744	174	19	0	0	NUM
ejpam-4744	174	20	,	,	PUNCT
ejpam-4744	174	21	s	s	NOUN
ejpam-4744	174	22	)	)	PUNCT
ejpam-4744	174	23	where	where	SCONJ
ejpam-4744	174	24	aj	aj	PROPN
ejpam-4744	174	25	∈	∈	PROPN
ejpam-4744	174	26	bs	b	NOUN
ejpam-4744	174	27	.	.	PUNCT
ejpam-4744	175	1	obviously	obviously	ADV
ejpam-4744	175	2	,	,	PUNCT
ejpam-4744	175	3	r	r	NOUN
ejpam-4744	175	4	=	=	SYM
ejpam-4744	175	5	s	s	PROPN
ejpam-4744	175	6	and	and	CCONJ
ejpam-4744	175	7	ai	ai	VERB
ejpam-4744	175	8	,	,	PUNCT
ejpam-4744	175	9	aj	aj	PROPN
ejpam-4744	175	10	∈	∈	PROPN
ejpam-4744	175	11	br	br	PROPN
ejpam-4744	175	12	.	.	PUNCT
ejpam-4744	176	1	it	it	PRON
ejpam-4744	176	2	follows	follow	VERB
ejpam-4744	176	3	that	that	PRON
ejpam-4744	176	4	ai	ai	VERB
ejpam-4744	176	5	and	and	CCONJ
ejpam-4744	176	6	aj	aj	PROPN
ejpam-4744	176	7	are	be	AUX
ejpam-4744	176	8	adjacent	adjacent	ADJ
ejpam-4744	176	9	in	in	ADP
ejpam-4744	176	10	g.	g.	PROPN
ejpam-4744	176	11	accordingly	accordingly	ADV
ejpam-4744	176	12	,	,	PUNCT
ejpam-4744	176	13	ai	ai	VERB
ejpam-4744	176	14	and	and	CCONJ
ejpam-4744	176	15	aj	aj	PROPN
ejpam-4744	176	16	are	be	AUX
ejpam-4744	176	17	adjacent	adjacent	ADJ
ejpam-4744	176	18	in	in	ADP
ejpam-4744	176	19	g	g	PROPN
ejpam-4744	176	20	◦	◦	NOUN
ejpam-4744	176	21	h.	h.	NOUN
ejpam-4744	176	22	case	case	NOUN
ejpam-4744	176	23	3	3	X
ejpam-4744	176	24	.	.	PUNCT
ejpam-4744	176	25	suppose	suppose	VERB
ejpam-4744	176	26	a	a	DET
ejpam-4744	176	27	∈	∈	PROPN
ejpam-4744	176	28	fi	fi	NOUN
ejpam-4744	176	29	and	and	CCONJ
ejpam-4744	176	30	b	b	NOUN
ejpam-4744	176	31	=	=	X
ejpam-4744	176	32	sj	sj	VERB
ejpam-4744	176	33	∪	∪	NOUN
ejpam-4744	176	34	tj	tj	NOUN
ejpam-4744	176	35	for	for	ADP
ejpam-4744	176	36	some	some	DET
ejpam-4744	176	37	i	i	PRON
ejpam-4744	176	38	and	and	CCONJ
ejpam-4744	176	39	j.	j.	PROPN
ejpam-4744	176	40	suppose	suppose	VERB
ejpam-4744	176	41	i	i	PRON
ejpam-4744	176	42	̸=	̸=	PROPN
ejpam-4744	176	43	j.	j.	PROPN
ejpam-4744	176	44	then	then	ADV
ejpam-4744	176	45	ϕ(a	ϕ(a	PROPN
ejpam-4744	176	46	)	)	PUNCT
ejpam-4744	176	47	=	=	SYM
ejpam-4744	176	48	ϕi(a	ϕi(a	PROPN
ejpam-4744	176	49	)	)	PUNCT
ejpam-4744	176	50	=	=	SYM
ejpam-4744	177	1	a	a	PRON
ejpam-4744	177	2	for	for	ADP
ejpam-4744	177	3	some	some	PRON
ejpam-4744	177	4	a	a	DET
ejpam-4744	177	5	∈	∈	PROPN
ejpam-4744	177	6	v	v	NOUN
ejpam-4744	177	7	(	(	PUNCT
ejpam-4744	177	8	hai	hai	NOUN
ejpam-4744	177	9	)	)	PUNCT
ejpam-4744	177	10	and	and	CCONJ
ejpam-4744	177	11	ϕ(aj	ϕ(aj	NUM
ejpam-4744	177	12	)	)	PUNCT
ejpam-4744	178	1	=	=	SYM
ejpam-4744	178	2	b.	b.	PROPN
ejpam-4744	178	3	since	since	SCONJ
ejpam-4744	178	4	a∩b	a∩b	PROPN
ejpam-4744	178	5	̸=	̸=	PROPN
ejpam-4744	178	6	∅	∅	NOUN
ejpam-4744	178	7	,	,	PUNCT
ejpam-4744	178	8	(	(	PUNCT
ejpam-4744	178	9	a∩	a∩	PROPN
ejpam-4744	178	10	sj)∪	sj)∪	NOUN
ejpam-4744	178	11	(	(	PUNCT
ejpam-4744	178	12	a∩	a∩	PROPN
ejpam-4744	178	13	tj	tj	PROPN
ejpam-4744	178	14	)	)	PUNCT
ejpam-4744	178	15	̸=	̸=	PROPN
ejpam-4744	178	16	∅.	∅.	ADV
ejpam-4744	178	17	since	since	SCONJ
ejpam-4744	178	18	a	a	DET
ejpam-4744	178	19	⊆	⊆	NUM
ejpam-4744	178	20	si	si	NOUN
ejpam-4744	178	21	,	,	PUNCT
ejpam-4744	178	22	a	a	DET
ejpam-4744	178	23	∩	∩	ADJ
ejpam-4744	178	24	tj	tj	ADP
ejpam-4744	178	25	⊆	⊆	NUM
ejpam-4744	178	26	si	si	X
ejpam-4744	178	27	∩	∩	ADJ
ejpam-4744	178	28	tj	tj	NOUN
ejpam-4744	178	29	=	=	NOUN
ejpam-4744	178	30	∅	∅	NOUN
ejpam-4744	178	31	and	and	CCONJ
ejpam-4744	178	32	a	a	DET
ejpam-4744	178	33	∩	∩	ADJ
ejpam-4744	178	34	sj	sj	ADP
ejpam-4744	178	35	⊆	⊆	NUM
ejpam-4744	178	36	si	si	X
ejpam-4744	178	37	∩	∩	NOUN
ejpam-4744	178	38	sj	sj	NOUN
ejpam-4744	178	39	=	=	PUNCT
ejpam-4744	178	40	∅.	∅.	PRON
ejpam-4744	178	41	this	this	PRON
ejpam-4744	178	42	is	be	AUX
ejpam-4744	178	43	a	a	DET
ejpam-4744	178	44	contradiction	contradiction	NOUN
ejpam-4744	178	45	.	.	PUNCT
ejpam-4744	179	1	thus	thus	ADV
ejpam-4744	179	2	,	,	PUNCT
ejpam-4744	179	3	i	i	PRON
ejpam-4744	179	4	=	=	PUNCT
ejpam-4744	179	5	j.	j.	PROPN
ejpam-4744	179	6	consequently	consequently	ADV
ejpam-4744	179	7	,	,	PUNCT
ejpam-4744	179	8	a	a	DET
ejpam-4744	179	9	∈	∈	PROPN
ejpam-4744	179	10	v	v	NOUN
ejpam-4744	179	11	(	(	PUNCT
ejpam-4744	179	12	haj	haj	PROPN
ejpam-4744	179	13	)	)	PUNCT
ejpam-4744	179	14	.	.	PUNCT
ejpam-4744	180	1	it	it	PRON
ejpam-4744	180	2	follows	follow	VERB
ejpam-4744	180	3	that	that	SCONJ
ejpam-4744	180	4	a	a	PRON
ejpam-4744	180	5	and	and	CCONJ
ejpam-4744	180	6	aj	aj	PROPN
ejpam-4744	180	7	are	be	AUX
ejpam-4744	180	8	adjacent	adjacent	ADJ
ejpam-4744	180	9	in	in	ADP
ejpam-4744	180	10	g	g	PROPN
ejpam-4744	180	11	◦	◦	NOUN
ejpam-4744	180	12	h.	h.	NOUN
ejpam-4744	180	13	hence	hence	ADV
ejpam-4744	180	14	ϕ	ϕ	PROPN
ejpam-4744	180	15	preserves	preserve	VERB
ejpam-4744	180	16	adjacency	adjacency	NOUN
ejpam-4744	180	17	.	.	PUNCT
ejpam-4744	181	1	therefore	therefore	ADV
ejpam-4744	181	2	,	,	PUNCT
ejpam-4744	181	3	ω(f	ω(f	ADJ
ejpam-4744	181	4	)	)	PUNCT
ejpam-4744	181	5	∼=	∼=	PROPN
ejpam-4744	181	6	g	g	NOUN
ejpam-4744	181	7	◦	◦	NOUN
ejpam-4744	181	8	h.	h.	NOUN
ejpam-4744	181	9	accordingly	accordingly	ADV
ejpam-4744	181	10	,	,	PUNCT
ejpam-4744	181	11	ω(g	ω(g	PROPN
ejpam-4744	181	12	◦	◦	NOUN
ejpam-4744	181	13	h	h	NOUN
ejpam-4744	181	14	)	)	PUNCT
ejpam-4744	181	15	≤	≤	NUM
ejpam-4744	181	16	|s|	|s|	PROPN
ejpam-4744	181	17	=	=	SYM
ejpam-4744	181	18	n∑	n∑	PROPN
ejpam-4744	181	19	i=0	i=0	PROPN
ejpam-4744	181	20	|si|	|si|	PROPN
ejpam-4744	182	1	=	=	SYM
ejpam-4744	182	2	|so|+	|so|+	NUM
ejpam-4744	182	3	n∑	n∑	NOUN
ejpam-4744	182	4	i=1	i=1	PROPN
ejpam-4744	182	5	|si|	|si|	PROPN
ejpam-4744	182	6	=	=	PUNCT
ejpam-4744	182	7	co(g	co(g	X
ejpam-4744	182	8	)	)	PUNCT
ejpam-4744	183	1	+	+	NUM
ejpam-4744	183	2	n∑	n∑	ADJ
ejpam-4744	183	3	i=1	i=1	PROPN
ejpam-4744	183	4	ω(h	ω(h	NUM
ejpam-4744	183	5	)	)	PUNCT
ejpam-4744	183	6	=	=	PRON
ejpam-4744	183	7	co(g	co(g	X
ejpam-4744	183	8	)	)	PUNCT
ejpam-4744	184	1	+	+	CCONJ
ejpam-4744	184	2	n	n	PROPN
ejpam-4744	184	3	·	·	PUNCT
ejpam-4744	184	4	ω(h	ω(h	NUM
ejpam-4744	184	5	)	)	PUNCT
ejpam-4744	184	6	=	=	PRON
ejpam-4744	184	7	co(g	co(g	X
ejpam-4744	184	8	)	)	PUNCT
ejpam-4744	185	1	+	+	CCONJ
ejpam-4744	185	2	|v	|v	X
ejpam-4744	185	3	(	(	PUNCT
ejpam-4744	185	4	g)|	g)|	NOUN
ejpam-4744	185	5	·	·	PUNCT
ejpam-4744	185	6	ω(h	ω(h	NUM
ejpam-4744	185	7	)	)	PUNCT
ejpam-4744	185	8	.	.	PUNCT
ejpam-4744	186	1	therefore	therefore	ADV
ejpam-4744	186	2	,	,	PUNCT
ejpam-4744	186	3	ω(g	ω(g	PROPN
ejpam-4744	186	4	◦	◦	NOUN
ejpam-4744	186	5	h	h	NOUN
ejpam-4744	186	6	)	)	PUNCT
ejpam-4744	186	7	≤	≤	NUM
ejpam-4744	186	8	co(g	co(g	NOUN
ejpam-4744	186	9	)	)	PUNCT
ejpam-4744	187	1	+	+	CCONJ
ejpam-4744	187	2	|v	|v	X
ejpam-4744	187	3	(	(	PUNCT
ejpam-4744	187	4	g)|	g)|	NOUN
ejpam-4744	187	5	·	·	PUNCT
ejpam-4744	187	6	ω(h	ω(h	NUM
ejpam-4744	187	7	)	)	PUNCT
ejpam-4744	187	8	.	.	PUNCT
ejpam-4744	188	1	corollary	corollary	ADJ
ejpam-4744	188	2	1	1	NUM
ejpam-4744	188	3	.	.	PUNCT
ejpam-4744	189	1	let	let	VERB
ejpam-4744	189	2	g	g	PRON
ejpam-4744	189	3	be	be	AUX
ejpam-4744	189	4	a	a	DET
ejpam-4744	189	5	connected	connected	ADJ
ejpam-4744	189	6	graph	graph	NOUN
ejpam-4744	189	7	and	and	CCONJ
ejpam-4744	189	8	n	n	PRON
ejpam-4744	189	9	≥	≥	NOUN
ejpam-4744	189	10	2	2	NUM
ejpam-4744	189	11	.	.	PUNCT
ejpam-4744	190	1	then	then	ADV
ejpam-4744	190	2	ω(kn	ω(kn	NUM
ejpam-4744	190	3	◦	◦	NOUN
ejpam-4744	190	4	g	g	NOUN
ejpam-4744	190	5	)	)	PUNCT
ejpam-4744	190	6	=	=	SYM
ejpam-4744	190	7	1	1	NUM
ejpam-4744	190	8	+	+	CCONJ
ejpam-4744	190	9	n	n	CCONJ
ejpam-4744	190	10	·	·	PUNCT
ejpam-4744	190	11	ω(g	ω(g	NOUN
ejpam-4744	190	12	)	)	PUNCT
ejpam-4744	190	13	.	.	PUNCT
ejpam-4744	191	1	proof	proof	NOUN
ejpam-4744	191	2	.	.	PUNCT
ejpam-4744	192	1	by	by	ADP
ejpam-4744	192	2	theorem	theorem	NOUN
ejpam-4744	192	3	3	3	NUM
ejpam-4744	192	4	,	,	PUNCT
ejpam-4744	192	5	ω(kn	ω(kn	NOUN
ejpam-4744	192	6	◦	◦	NOUN
ejpam-4744	192	7	g	g	NOUN
ejpam-4744	192	8	)	)	PUNCT
ejpam-4744	192	9	≤	≤	NOUN
ejpam-4744	192	10	co(kn)+|v	co(kn)+|v	PROPN
ejpam-4744	192	11	(	(	PUNCT
ejpam-4744	192	12	kn)|·ω(g	kn)|·ω(g	NOUN
ejpam-4744	192	13	)	)	PUNCT
ejpam-4744	192	14	.	.	PUNCT
ejpam-4744	193	1	by	by	ADP
ejpam-4744	193	2	theorem	theorem	NOUN
ejpam-4744	193	3	2	2	NUM
ejpam-4744	193	4	,	,	PUNCT
ejpam-4744	193	5	co(kn	co(kn	NOUN
ejpam-4744	193	6	)	)	PUNCT
ejpam-4744	193	7	=	=	SYM
ejpam-4744	193	8	1	1	X
ejpam-4744	193	9	.	.	PUNCT
ejpam-4744	193	10	thus	thus	ADV
ejpam-4744	193	11	,	,	PUNCT
ejpam-4744	193	12	ω(kn	ω(kn	X
ejpam-4744	193	13	◦	◦	NOUN
ejpam-4744	193	14	g	g	NOUN
ejpam-4744	193	15	)	)	PUNCT
ejpam-4744	193	16	≤	≤	NUM
ejpam-4744	193	17	co(kn	co(kn	NOUN
ejpam-4744	193	18	)	)	PUNCT
ejpam-4744	194	1	+	+	CCONJ
ejpam-4744	194	2	|v	|v	PROPN
ejpam-4744	194	3	(	(	PUNCT
ejpam-4744	194	4	kn)|	kn)|	X
ejpam-4744	194	5	·	·	PUNCT
ejpam-4744	194	6	ω(g	ω(g	PROPN
ejpam-4744	194	7	)	)	PUNCT
ejpam-4744	194	8	=	=	SYM
ejpam-4744	194	9	1	1	NUM
ejpam-4744	194	10	+	+	CCONJ
ejpam-4744	194	11	n	n	CCONJ
ejpam-4744	194	12	·	·	PUNCT
ejpam-4744	194	13	ω(g	ω(g	NOUN
ejpam-4744	194	14	)	)	PUNCT
ejpam-4744	194	15	.	.	PUNCT
ejpam-4744	195	1	suppose	suppose	VERB
ejpam-4744	195	2	ω(kn	ω(kn	VERB
ejpam-4744	195	3	◦	◦	NOUN
ejpam-4744	195	4	g	g	NOUN
ejpam-4744	195	5	)	)	PUNCT
ejpam-4744	195	6	<	<	X
ejpam-4744	195	7	1+n·ω(g	1+n·ω(g	NUM
ejpam-4744	195	8	)	)	PUNCT
ejpam-4744	195	9	.	.	PUNCT
ejpam-4744	196	1	let	let	VERB
ejpam-4744	196	2	v	v	X
ejpam-4744	196	3	(	(	PUNCT
ejpam-4744	196	4	kn	kn	PROPN
ejpam-4744	196	5	)	)	PUNCT
ejpam-4744	196	6	=	=	PRON
ejpam-4744	196	7	{	{	PUNCT
ejpam-4744	196	8	a1	a1	PROPN
ejpam-4744	196	9	,	,	PUNCT
ejpam-4744	196	10	a2	a2	PROPN
ejpam-4744	196	11	,	,	PUNCT
ejpam-4744	196	12	...	...	PUNCT
ejpam-4744	196	13	,	,	PUNCT
ejpam-4744	196	14	an	an	PRON
ejpam-4744	196	15	}	}	PUNCT
ejpam-4744	196	16	and	and	CCONJ
ejpam-4744	196	17	for	for	ADP
ejpam-4744	196	18	each	each	DET
ejpam-4744	196	19	i	i	PRON
ejpam-4744	196	20	,	,	PUNCT
ejpam-4744	196	21	1	1	NUM
ejpam-4744	196	22	≤	≤	NUM
ejpam-4744	196	23	i	i	PRON
ejpam-4744	196	24	≤	≤	PROPN
ejpam-4744	196	25	n	n	CCONJ
ejpam-4744	196	26	,	,	PUNCT
ejpam-4744	196	27	let	let	VERB
ejpam-4744	196	28	gi	gi	PART
ejpam-4744	196	29	be	be	AUX
ejpam-4744	196	30	the	the	DET
ejpam-4744	196	31	ith	ith	PROPN
ejpam-4744	196	32	copy	copy	NOUN
ejpam-4744	196	33	of	of	ADP
ejpam-4744	196	34	g	g	PROPN
ejpam-4744	196	35	corresponding	correspond	VERB
ejpam-4744	196	36	to	to	ADP
ejpam-4744	196	37	the	the	DET
ejpam-4744	196	38	vertex	vertex	NOUN
ejpam-4744	196	39	ai	ai	VERB
ejpam-4744	196	40	.	.	PUNCT
ejpam-4744	197	1	let	let	VERB
ejpam-4744	197	2	f	f	PRON
ejpam-4744	197	3	be	be	AUX
ejpam-4744	197	4	a	a	DET
ejpam-4744	197	5	collection	collection	NOUN
ejpam-4744	197	6	of	of	ADP
ejpam-4744	197	7	subsets	subset	NOUN
ejpam-4744	197	8	of	of	ADP
ejpam-4744	197	9	s	s	NOUN
ejpam-4744	197	10	=	=	PUNCT
ejpam-4744	197	11	{	{	PUNCT
ejpam-4744	197	12	1	1	NUM
ejpam-4744	197	13	,	,	PUNCT
ejpam-4744	197	14	2	2	NUM
ejpam-4744	197	15	,	,	PUNCT
ejpam-4744	197	16	3	3	NUM
ejpam-4744	197	17	,	,	PUNCT
ejpam-4744	197	18	...	...	PUNCT
ejpam-4744	197	19	,	,	PUNCT
ejpam-4744	197	20	ω(kn	ω(kn	X
ejpam-4744	197	21	◦	◦	VERB
ejpam-4744	197	22	g	g	NOUN
ejpam-4744	197	23	)	)	PUNCT
ejpam-4744	197	24	}	}	PUNCT
ejpam-4744	197	25	such	such	ADJ
ejpam-4744	197	26	that	that	SCONJ
ejpam-4744	197	27	ω(f	ω(f	PROPN
ejpam-4744	197	28	)	)	PUNCT
ejpam-4744	197	29	∼=	∼=	PROPN
ejpam-4744	197	30	kn	kn	NOUN
ejpam-4744	197	31	◦	◦	PROPN
ejpam-4744	197	32	g.	g.	PROPN
ejpam-4744	197	33	let	let	VERB
ejpam-4744	197	34	ϕ	ϕ	NOUN
ejpam-4744	197	35	:	:	PUNCT
ejpam-4744	197	36	v	v	X
ejpam-4744	197	37	(	(	PUNCT
ejpam-4744	197	38	kn	kn	NOUN
ejpam-4744	197	39	◦	◦	NOUN
ejpam-4744	197	40	g	g	NOUN
ejpam-4744	197	41	)	)	PUNCT
ejpam-4744	197	42	→	→	SYM
ejpam-4744	197	43	f	f	X
ejpam-4744	197	44	be	be	AUX
ejpam-4744	197	45	an	an	DET
ejpam-4744	197	46	isomorphism	isomorphism	NOUN
ejpam-4744	197	47	.	.	PUNCT
ejpam-4744	198	1	for	for	ADP
ejpam-4744	198	2	each	each	DET
ejpam-4744	198	3	i	i	PRON
ejpam-4744	198	4	,	,	PUNCT
ejpam-4744	198	5	1	1	NUM
ejpam-4744	198	6	≤	≤	NUM
ejpam-4744	198	7	i	i	PRON
ejpam-4744	198	8	≤	≤	NOUN
ejpam-4744	198	9	n	n	CCONJ
ejpam-4744	198	10	,	,	PUNCT
ejpam-4744	198	11	{	{	PUNCT
ejpam-4744	198	12	ϕ(x	ϕ(x	X
ejpam-4744	198	13	)	)	PUNCT
ejpam-4744	198	14	:	:	PUNCT
ejpam-4744	199	1	x	x	X
ejpam-4744	199	2	∈	∈	NOUN
ejpam-4744	199	3	v	v	X
ejpam-4744	199	4	(	(	PUNCT
ejpam-4744	199	5	gi	gi	NOUN
ejpam-4744	199	6	)	)	PUNCT
ejpam-4744	199	7	}	}	PUNCT
ejpam-4744	199	8	is	be	AUX
ejpam-4744	199	9	a	a	DET
ejpam-4744	199	10	set	set	VERB
ejpam-4744	199	11	representation	representation	NOUN
ejpam-4744	199	12	for	for	ADP
ejpam-4744	199	13	gi	gi	NOUN
ejpam-4744	199	14	.	.	PUNCT
ejpam-4744	200	1	thus	thus	ADV
ejpam-4744	200	2	,	,	PUNCT
ejpam-4744	200	3	|	|	NOUN
ejpam-4744	200	4	∪x∈v	∪x∈v	X
ejpam-4744	200	5	(	(	PUNCT
ejpam-4744	200	6	gi	gi	INTJ
ejpam-4744	200	7	)	)	PUNCT
ejpam-4744	200	8	ϕ(x)|	ϕ(x)|	PROPN
ejpam-4744	200	9	≥	≥	NUM
ejpam-4744	200	10	ω(gi	ω(gi	NUM
ejpam-4744	200	11	)	)	PUNCT
ejpam-4744	200	12	=	=	SYM
ejpam-4744	200	13	ω(g	ω(g	NOUN
ejpam-4744	200	14	)	)	PUNCT
ejpam-4744	200	15	.	.	PUNCT
ejpam-4744	201	1	note	note	VERB
ejpam-4744	201	2	that	that	SCONJ
ejpam-4744	201	3	for	for	ADP
ejpam-4744	201	4	each	each	DET
ejpam-4744	201	5	i	i	PROPN
ejpam-4744	201	6	,	,	PUNCT
ejpam-4744	201	7	j	j	PROPN
ejpam-4744	201	8	,	,	PUNCT
ejpam-4744	201	9	i	i	PROPN
ejpam-4744	201	10	̸=	̸=	PROPN
ejpam-4744	201	11	j	j	PROPN
ejpam-4744	201	12	,	,	PUNCT
ejpam-4744	201	13	and	and	CCONJ
ejpam-4744	201	14	each	each	PRON
ejpam-4744	201	15	a	a	DET
ejpam-4744	201	16	∈	∈	PROPN
ejpam-4744	201	17	gi	gi	NOUN
ejpam-4744	201	18	j.	j.	PROPN
ejpam-4744	201	19	b.	b.	PROPN
ejpam-4744	201	20	palco	palco	PROPN
ejpam-4744	201	21	,	,	PUNCT
ejpam-4744	201	22	r.	r.	PROPN
ejpam-4744	201	23	n.	n.	PROPN
ejpam-4744	201	24	paluga	paluga	PROPN
ejpam-4744	201	25	/	/	SYM
ejpam-4744	201	26	eur	eur	PROPN
ejpam-4744	201	27	.	.	PUNCT
ejpam-4744	202	1	j.	j.	PROPN
ejpam-4744	202	2	pure	pure	PROPN
ejpam-4744	202	3	appl	appl	PROPN
ejpam-4744	202	4	.	.	PROPN
ejpam-4744	202	5	math	math	PROPN
ejpam-4744	202	6	,	,	PUNCT
ejpam-4744	202	7	16	16	NUM
ejpam-4744	202	8	(	(	PUNCT
ejpam-4744	202	9	2	2	NUM
ejpam-4744	202	10	)	)	PUNCT
ejpam-4744	202	11	(	(	PUNCT
ejpam-4744	202	12	2023	2023	NUM
ejpam-4744	202	13	)	)	PUNCT
ejpam-4744	202	14	,	,	PUNCT
ejpam-4744	202	15	1318	1318	NUM
ejpam-4744	202	16	-	-	SYM
ejpam-4744	202	17	1325	1325	NUM
ejpam-4744	202	18	1323	1323	NUM
ejpam-4744	202	19	and	and	CCONJ
ejpam-4744	202	20	b	b	X
ejpam-4744	202	21	∈	∈	PROPN
ejpam-4744	202	22	gj	gj	NOUN
ejpam-4744	202	23	,	,	PUNCT
ejpam-4744	202	24	ab	ab	PROPN
ejpam-4744	202	25	/∈	/∈	PUNCT
ejpam-4744	203	1	e(kn	e(kn	PUNCT
ejpam-4744	203	2	◦	◦	NOUN
ejpam-4744	203	3	g	g	NOUN
ejpam-4744	203	4	)	)	PUNCT
ejpam-4744	203	5	.	.	PUNCT
ejpam-4744	204	1	consequently	consequently	ADV
ejpam-4744	204	2	,	,	PUNCT
ejpam-4744	204	3	ei	ei	NOUN
ejpam-4744	204	4	=	=	SYM
ejpam-4744	204	5	∪x∈v	∪x∈v	PROPN
ejpam-4744	204	6	(	(	PUNCT
ejpam-4744	204	7	gi)ϕ(x	gi)ϕ(x	X
ejpam-4744	204	8	)	)	PUNCT
ejpam-4744	204	9	and	and	CCONJ
ejpam-4744	204	10	ej	ej	X
ejpam-4744	204	11	=	=	SYM
ejpam-4744	204	12	∪x∈v	∪x∈v	PROPN
ejpam-4744	204	13	(	(	PUNCT
ejpam-4744	204	14	gj)ϕ(x	gj)ϕ(x	X
ejpam-4744	204	15	)	)	PUNCT
ejpam-4744	204	16	are	be	AUX
ejpam-4744	204	17	disjoint	disjoint	ADJ
ejpam-4744	204	18	whenever	whenever	SCONJ
ejpam-4744	204	19	i	i	PRON
ejpam-4744	204	20	̸=	̸=	PROPN
ejpam-4744	204	21	j.	j.	PROPN
ejpam-4744	204	22	now	now	ADV
ejpam-4744	204	23	,	,	PUNCT
ejpam-4744	204	24	|	|	ADV
ejpam-4744	204	25	∪n	∪n	X
ejpam-4744	204	26	i=1	i=1	PROPN
ejpam-4744	205	1	ei|	ei|	PROPN
ejpam-4744	205	2	=	=	PUNCT
ejpam-4744	205	3	n∑	n∑	PROPN
ejpam-4744	205	4	i=1	i=1	PROPN
ejpam-4744	206	1	|ei|	|ei|	PROPN
ejpam-4744	206	2	≥	≥	NUM
ejpam-4744	206	3	n∑	n∑	NOUN
ejpam-4744	206	4	i=1	i=1	PROPN
ejpam-4744	206	5	ω(g	ω(g	PROPN
ejpam-4744	206	6	)	)	PUNCT
ejpam-4744	206	7	=	=	SYM
ejpam-4744	206	8	n	n	X
ejpam-4744	206	9	·	·	PUNCT
ejpam-4744	206	10	ω(g	ω(g	NOUN
ejpam-4744	206	11	)	)	PUNCT
ejpam-4744	206	12	.	.	PUNCT
ejpam-4744	207	1	it	it	PRON
ejpam-4744	207	2	follows	follow	VERB
ejpam-4744	207	3	that	that	SCONJ
ejpam-4744	207	4	the	the	DET
ejpam-4744	207	5	elements	element	NOUN
ejpam-4744	207	6	of	of	ADP
ejpam-4744	207	7	s−	s−	PROPN
ejpam-4744	207	8	(	(	PUNCT
ejpam-4744	207	9	∪n	∪n	X
ejpam-4744	207	10	i=1ei	i=1ei	X
ejpam-4744	207	11	)	)	PUNCT
ejpam-4744	207	12	are	be	AUX
ejpam-4744	207	13	used	use	VERB
ejpam-4744	207	14	for	for	ADP
ejpam-4744	207	15	the	the	DET
ejpam-4744	207	16	set	set	NOUN
ejpam-4744	207	17	representation	representation	NOUN
ejpam-4744	207	18	of	of	ADP
ejpam-4744	207	19	g.	g.	PROPN
ejpam-4744	207	20	note	note	VERB
ejpam-4744	207	21	that	that	SCONJ
ejpam-4744	207	22	|s	|s	PROPN
ejpam-4744	208	1	−	−	PROPN
ejpam-4744	209	1	(	(	PUNCT
ejpam-4744	209	2	∪n	∪n	NUM
ejpam-4744	209	3	i=1ei)|	i=1ei)|	X
ejpam-4744	209	4	=	=	PUNCT
ejpam-4744	209	5	|s|	|s|	NOUN
ejpam-4744	209	6	−	−	NOUN
ejpam-4744	209	7	|(∪n	|(∪n	NOUN
ejpam-4744	209	8	i=1ei)|	i=1ei)|	NOUN
ejpam-4744	209	9	≤	≤	X
ejpam-4744	209	10	ω(kn	ω(kn	NUM
ejpam-4744	209	11	◦	◦	NOUN
ejpam-4744	209	12	g)−	g)−	PROPN
ejpam-4744	209	13	n	n	PRON
ejpam-4744	209	14	·	·	PUNCT
ejpam-4744	209	15	ω(g	ω(g	NOUN
ejpam-4744	209	16	)	)	PUNCT
ejpam-4744	209	17	,	,	PUNCT
ejpam-4744	209	18	since	since	SCONJ
ejpam-4744	209	19	we	we	PRON
ejpam-4744	209	20	suppose	suppose	VERB
ejpam-4744	209	21	ω(kn	ω(kn	SYM
ejpam-4744	209	22	◦	◦	NOUN
ejpam-4744	209	23	g	g	NOUN
ejpam-4744	209	24	)	)	PUNCT
ejpam-4744	209	25	<	<	X
ejpam-4744	209	26	1	1	NUM
ejpam-4744	209	27	+	+	CCONJ
ejpam-4744	209	28	n	n	CCONJ
ejpam-4744	209	29	·	·	PUNCT
ejpam-4744	209	30	ω(g	ω(g	NOUN
ejpam-4744	209	31	)	)	PUNCT
ejpam-4744	209	32	.	.	PUNCT
ejpam-4744	210	1	<	<	X
ejpam-4744	211	1	1	1	X
ejpam-4744	211	2	.	.	X
ejpam-4744	211	3	that	that	PRON
ejpam-4744	211	4	is	be	AUX
ejpam-4744	211	5	,	,	PUNCT
ejpam-4744	211	6	|s	|s	PROPN
ejpam-4744	211	7	−	−	PROPN
ejpam-4744	211	8	(	(	PUNCT
ejpam-4744	211	9	∪n	∪n	NUM
ejpam-4744	211	10	i=1ei)|	i=1ei)|	X
ejpam-4744	211	11	=	=	NOUN
ejpam-4744	211	12	0	0	X
ejpam-4744	211	13	.	.	PUNCT
ejpam-4744	212	1	this	this	PRON
ejpam-4744	212	2	implies	imply	VERB
ejpam-4744	212	3	,	,	PUNCT
ejpam-4744	212	4	s	s	PART
ejpam-4744	212	5	=	=	PUNCT
ejpam-4744	212	6	∪n	∪n	X
ejpam-4744	212	7	i=1ei	i=1ei	X
ejpam-4744	212	8	.	.	PUNCT
ejpam-4744	213	1	since	since	SCONJ
ejpam-4744	213	2	a1	a1	NOUN
ejpam-4744	213	3	and	and	CCONJ
ejpam-4744	213	4	a2	a2	PROPN
ejpam-4744	213	5	are	be	AUX
ejpam-4744	213	6	adjacent	adjacent	ADJ
ejpam-4744	213	7	,	,	PUNCT
ejpam-4744	213	8	ϕ(a1)∩ϕ(a2	ϕ(a1)∩ϕ(a2	PROPN
ejpam-4744	213	9	)	)	PUNCT
ejpam-4744	214	1	̸=	̸=	PROPN
ejpam-4744	214	2	∅.	∅.	ADV
ejpam-4744	214	3	let	let	VERB
ejpam-4744	214	4	t	t	PROPN
ejpam-4744	214	5	∈	∈	PROPN
ejpam-4744	214	6	ϕ(a1)∩ϕ(a2	ϕ(a1)∩ϕ(a2	PROPN
ejpam-4744	214	7	)	)	PUNCT
ejpam-4744	214	8	.	.	PUNCT
ejpam-4744	215	1	then	then	ADV
ejpam-4744	215	2	t	t	PROPN
ejpam-4744	215	3	∈	∈	PROPN
ejpam-4744	215	4	ϕ(a1	ϕ(a1	NOUN
ejpam-4744	215	5	)	)	PUNCT
ejpam-4744	215	6	and	and	CCONJ
ejpam-4744	215	7	t	t	PROPN
ejpam-4744	215	8	∈	∈	PROPN
ejpam-4744	215	9	ϕ(a2	ϕ(a2	PROPN
ejpam-4744	215	10	)	)	PUNCT
ejpam-4744	215	11	.	.	PUNCT
ejpam-4744	216	1	since	since	SCONJ
ejpam-4744	216	2	s	s	PART
ejpam-4744	216	3	=	=	PUNCT
ejpam-4744	216	4	∪n	∪n	X
ejpam-4744	216	5	i=1ei	i=1ei	ADV
ejpam-4744	216	6	,	,	PUNCT
ejpam-4744	216	7	t	t	PROPN
ejpam-4744	216	8	∈	∈	PROPN
ejpam-4744	216	9	er	er	INTJ
ejpam-4744	216	10	for	for	ADP
ejpam-4744	216	11	some	some	DET
ejpam-4744	216	12	r.	r.	NOUN
ejpam-4744	216	13	thus	thus	ADV
ejpam-4744	216	14	t	t	PROPN
ejpam-4744	216	15	∈	∈	PROPN
ejpam-4744	216	16	ϕ(x	ϕ(x	PROPN
ejpam-4744	216	17	)	)	PUNCT
ejpam-4744	216	18	for	for	ADP
ejpam-4744	216	19	x	x	PROPN
ejpam-4744	216	20	∈	∈	PROPN
ejpam-4744	216	21	v	v	PROPN
ejpam-4744	216	22	(	(	PUNCT
ejpam-4744	216	23	gr	gr	NOUN
ejpam-4744	216	24	)	)	PUNCT
ejpam-4744	216	25	.	.	PUNCT
ejpam-4744	217	1	therefore	therefore	ADV
ejpam-4744	217	2	,	,	PUNCT
ejpam-4744	217	3	⟨{x	⟨{x	PROPN
ejpam-4744	217	4	,	,	PUNCT
ejpam-4744	217	5	a1	a1	PROPN
ejpam-4744	217	6	,	,	PUNCT
ejpam-4744	217	7	a2}⟩	a2}⟩	ADV
ejpam-4744	217	8	is	be	AUX
ejpam-4744	217	9	complete	complete	ADJ
ejpam-4744	217	10	.	.	PUNCT
ejpam-4744	218	1	this	this	PRON
ejpam-4744	218	2	is	be	AUX
ejpam-4744	218	3	a	a	DET
ejpam-4744	218	4	contradiction	contradiction	NOUN
ejpam-4744	218	5	.	.	PUNCT
ejpam-4744	219	1	therefore	therefore	ADV
ejpam-4744	219	2	,	,	PUNCT
ejpam-4744	219	3	ω(kn	ω(kn	X
ejpam-4744	219	4	◦	◦	NOUN
ejpam-4744	219	5	g	g	NOUN
ejpam-4744	219	6	)	)	PUNCT
ejpam-4744	219	7	=	=	SYM
ejpam-4744	219	8	1	1	NUM
ejpam-4744	219	9	+	+	CCONJ
ejpam-4744	219	10	n	n	CCONJ
ejpam-4744	219	11	·	·	PUNCT
ejpam-4744	219	12	ω(g	ω(g	NOUN
ejpam-4744	219	13	)	)	PUNCT
ejpam-4744	219	14	.	.	PUNCT
ejpam-4744	220	1	corollary	corollary	ADJ
ejpam-4744	220	2	2	2	NUM
ejpam-4744	220	3	.	.	PUNCT
ejpam-4744	221	1	let	let	VERB
ejpam-4744	221	2	g	g	PRON
ejpam-4744	221	3	be	be	AUX
ejpam-4744	221	4	a	a	DET
ejpam-4744	221	5	connected	connected	ADJ
ejpam-4744	221	6	graph	graph	NOUN
ejpam-4744	221	7	and	and	CCONJ
ejpam-4744	221	8	n	n	PRON
ejpam-4744	221	9	≥	≥	NOUN
ejpam-4744	221	10	2	2	NUM
ejpam-4744	221	11	.	.	PUNCT
ejpam-4744	222	1	then	then	ADV
ejpam-4744	222	2	ω(pn	ω(pn	NUM
ejpam-4744	222	3	◦	◦	NOUN
ejpam-4744	222	4	g	g	NOUN
ejpam-4744	222	5	)	)	PUNCT
ejpam-4744	223	1	=	=	PUNCT
ejpam-4744	223	2	(	(	PUNCT
ejpam-4744	223	3	n−1)+n	n−1)+n	PROPN
ejpam-4744	223	4	·	·	SYM
ejpam-4744	223	5	ω(g	ω(g	NOUN
ejpam-4744	223	6	)	)	PUNCT
ejpam-4744	223	7	.	.	PUNCT
ejpam-4744	224	1	proof	proof	NOUN
ejpam-4744	224	2	.	.	PUNCT
ejpam-4744	225	1	by	by	ADP
ejpam-4744	225	2	theorem	theorem	NOUN
ejpam-4744	225	3	3	3	NUM
ejpam-4744	225	4	,	,	PUNCT
ejpam-4744	225	5	ω(pn	ω(pn	NUM
ejpam-4744	225	6	◦	◦	NOUN
ejpam-4744	225	7	g	g	NOUN
ejpam-4744	225	8	)	)	PUNCT
ejpam-4744	225	9	≤	≤	PROPN
ejpam-4744	225	10	co(pn	co(pn	PROPN
ejpam-4744	225	11	)	)	PUNCT
ejpam-4744	226	1	+	+	CCONJ
ejpam-4744	226	2	|v	|v	X
ejpam-4744	226	3	(	(	PUNCT
ejpam-4744	226	4	pn)|	pn)|	PROPN
ejpam-4744	226	5	·	·	SYM
ejpam-4744	226	6	ω(g	ω(g	NOUN
ejpam-4744	226	7	)	)	PUNCT
ejpam-4744	226	8	.	.	PUNCT
ejpam-4744	227	1	by	by	ADP
ejpam-4744	227	2	theorem	theorem	ADJ
ejpam-4744	227	3	2	2	NUM
ejpam-4744	227	4	,	,	PUNCT
ejpam-4744	227	5	co(pn	co(pn	PROPN
ejpam-4744	227	6	)	)	PUNCT
ejpam-4744	227	7	=	=	PUNCT
ejpam-4744	227	8	n−	n−	NOUN
ejpam-4744	227	9	1	1	NUM
ejpam-4744	227	10	.	.	PUNCT
ejpam-4744	228	1	thus	thus	ADV
ejpam-4744	228	2	,	,	PUNCT
ejpam-4744	228	3	ω(pn	ω(pn	NUM
ejpam-4744	228	4	◦	◦	NOUN
ejpam-4744	228	5	g	g	NOUN
ejpam-4744	228	6	)	)	PUNCT
ejpam-4744	228	7	≤	≤	NUM
ejpam-4744	228	8	co(pn	co(pn	PROPN
ejpam-4744	228	9	)	)	PUNCT
ejpam-4744	229	1	+	+	CCONJ
ejpam-4744	229	2	|v	|v	X
ejpam-4744	229	3	(	(	PUNCT
ejpam-4744	229	4	pn)|	pn)|	PROPN
ejpam-4744	229	5	·	·	SYM
ejpam-4744	229	6	ω(g	ω(g	PROPN
ejpam-4744	229	7	)	)	PUNCT
ejpam-4744	229	8	=	=	PUNCT
ejpam-4744	229	9	(	(	PUNCT
ejpam-4744	229	10	n−	n−	NOUN
ejpam-4744	229	11	1	1	NUM
ejpam-4744	229	12	)	)	PUNCT
ejpam-4744	229	13	+	+	CCONJ
ejpam-4744	229	14	n	n	CCONJ
ejpam-4744	229	15	·	·	PUNCT
ejpam-4744	229	16	ω(g	ω(g	NOUN
ejpam-4744	229	17	)	)	PUNCT
ejpam-4744	229	18	.	.	PUNCT
ejpam-4744	230	1	suppose	suppose	VERB
ejpam-4744	230	2	ω(pn	ω(pn	NUM
ejpam-4744	230	3	◦	◦	NOUN
ejpam-4744	230	4	g	g	NOUN
ejpam-4744	230	5	)	)	PUNCT
ejpam-4744	230	6	<	<	X
ejpam-4744	230	7	(	(	PUNCT
ejpam-4744	230	8	n−	n−	NOUN
ejpam-4744	230	9	1	1	NUM
ejpam-4744	230	10	)	)	PUNCT
ejpam-4744	230	11	+	+	CCONJ
ejpam-4744	230	12	n	n	CCONJ
ejpam-4744	230	13	·	·	PUNCT
ejpam-4744	230	14	ω(g	ω(g	NOUN
ejpam-4744	230	15	)	)	PUNCT
ejpam-4744	230	16	.	.	PUNCT
ejpam-4744	231	1	let	let	VERB
ejpam-4744	231	2	v	v	X
ejpam-4744	231	3	(	(	PUNCT
ejpam-4744	231	4	pn	pn	NOUN
ejpam-4744	231	5	)	)	PUNCT
ejpam-4744	231	6	=	=	SYM
ejpam-4744	231	7	{	{	PUNCT
ejpam-4744	231	8	a1	a1	PROPN
ejpam-4744	231	9	,	,	PUNCT
ejpam-4744	231	10	a2	a2	PROPN
ejpam-4744	231	11	,	,	PUNCT
ejpam-4744	231	12	...	...	PUNCT
ejpam-4744	231	13	,	,	PUNCT
ejpam-4744	231	14	an	an	PRON
ejpam-4744	231	15	}	}	PUNCT
ejpam-4744	231	16	,	,	PUNCT
ejpam-4744	231	17	e(pn	e(pn	NOUN
ejpam-4744	231	18	)	)	PUNCT
ejpam-4744	231	19	=	=	PRON
ejpam-4744	231	20	{	{	PUNCT
ejpam-4744	231	21	aiai+1	aiai+1	NOUN
ejpam-4744	231	22	:	:	PUNCT
ejpam-4744	231	23	1	1	NUM
ejpam-4744	231	24	≤	≤	NUM
ejpam-4744	231	25	i	i	PRON
ejpam-4744	231	26	≤	≤	ADJ
ejpam-4744	231	27	n	n	CCONJ
ejpam-4744	231	28	−	−	PROPN
ejpam-4744	231	29	1	1	NUM
ejpam-4744	231	30	}	}	PUNCT
ejpam-4744	231	31	and	and	CCONJ
ejpam-4744	231	32	for	for	ADP
ejpam-4744	231	33	each	each	DET
ejpam-4744	231	34	i	i	PRON
ejpam-4744	231	35	,	,	PUNCT
ejpam-4744	231	36	1	1	NUM
ejpam-4744	231	37	≤	≤	NUM
ejpam-4744	231	38	i	i	PRON
ejpam-4744	231	39	≤	≤	PROPN
ejpam-4744	231	40	n	n	CCONJ
ejpam-4744	231	41	,	,	PUNCT
ejpam-4744	231	42	let	let	VERB
ejpam-4744	231	43	gi	gi	PART
ejpam-4744	231	44	be	be	AUX
ejpam-4744	231	45	the	the	DET
ejpam-4744	231	46	ith	ith	PROPN
ejpam-4744	231	47	copy	copy	NOUN
ejpam-4744	231	48	of	of	ADP
ejpam-4744	231	49	g	g	PROPN
ejpam-4744	231	50	corresponding	correspond	VERB
ejpam-4744	231	51	to	to	ADP
ejpam-4744	231	52	the	the	DET
ejpam-4744	231	53	vertex	vertex	NOUN
ejpam-4744	231	54	ai	ai	VERB
ejpam-4744	231	55	.	.	PUNCT
ejpam-4744	232	1	let	let	VERB
ejpam-4744	232	2	f	f	PRON
ejpam-4744	232	3	be	be	AUX
ejpam-4744	232	4	a	a	DET
ejpam-4744	232	5	collection	collection	NOUN
ejpam-4744	232	6	of	of	ADP
ejpam-4744	232	7	subsets	subset	NOUN
ejpam-4744	232	8	of	of	ADP
ejpam-4744	232	9	s	s	NOUN
ejpam-4744	232	10	=	=	PUNCT
ejpam-4744	232	11	{	{	PUNCT
ejpam-4744	232	12	1	1	NUM
ejpam-4744	232	13	,	,	PUNCT
ejpam-4744	232	14	2	2	NUM
ejpam-4744	232	15	,	,	PUNCT
ejpam-4744	232	16	3	3	NUM
ejpam-4744	232	17	,	,	PUNCT
ejpam-4744	232	18	...	...	PUNCT
ejpam-4744	232	19	,	,	PUNCT
ejpam-4744	232	20	ω(pn	ω(pn	PROPN
ejpam-4744	232	21	◦	◦	NOUN
ejpam-4744	232	22	g	g	NOUN
ejpam-4744	232	23	)	)	PUNCT
ejpam-4744	232	24	}	}	PUNCT
ejpam-4744	232	25	such	such	ADJ
ejpam-4744	232	26	that	that	SCONJ
ejpam-4744	232	27	ω(f	ω(f	PROPN
ejpam-4744	232	28	)	)	PUNCT
ejpam-4744	232	29	∼=	∼=	PROPN
ejpam-4744	232	30	pn	pn	PROPN
ejpam-4744	232	31	◦	◦	PROPN
ejpam-4744	232	32	g.	g.	PROPN
ejpam-4744	232	33	let	let	VERB
ejpam-4744	232	34	ϕ	ϕ	NOUN
ejpam-4744	232	35	:	:	PUNCT
ejpam-4744	232	36	v	v	X
ejpam-4744	232	37	(	(	PUNCT
ejpam-4744	232	38	pn	pn	INTJ
ejpam-4744	232	39	◦	◦	NOUN
ejpam-4744	232	40	g	g	NOUN
ejpam-4744	232	41	)	)	PUNCT
ejpam-4744	232	42	→	→	SYM
ejpam-4744	232	43	f	f	X
ejpam-4744	232	44	be	be	AUX
ejpam-4744	232	45	an	an	DET
ejpam-4744	232	46	isomorphism	isomorphism	NOUN
ejpam-4744	232	47	.	.	PUNCT
ejpam-4744	233	1	for	for	ADP
ejpam-4744	233	2	each	each	DET
ejpam-4744	233	3	i	i	PRON
ejpam-4744	233	4	,	,	PUNCT
ejpam-4744	233	5	1	1	NUM
ejpam-4744	233	6	≤	≤	NUM
ejpam-4744	233	7	i	i	PRON
ejpam-4744	233	8	≤	≤	NOUN
ejpam-4744	233	9	n	n	CCONJ
ejpam-4744	233	10	,	,	PUNCT
ejpam-4744	233	11	{	{	PUNCT
ejpam-4744	233	12	ϕ(x	ϕ(x	X
ejpam-4744	233	13	)	)	PUNCT
ejpam-4744	233	14	:	:	PUNCT
ejpam-4744	234	1	x	x	X
ejpam-4744	234	2	∈	∈	NOUN
ejpam-4744	234	3	v	v	X
ejpam-4744	234	4	(	(	PUNCT
ejpam-4744	234	5	gi	gi	NOUN
ejpam-4744	234	6	)	)	PUNCT
ejpam-4744	234	7	}	}	PUNCT
ejpam-4744	234	8	is	be	AUX
ejpam-4744	234	9	a	a	DET
ejpam-4744	234	10	set	set	VERB
ejpam-4744	234	11	representation	representation	NOUN
ejpam-4744	234	12	for	for	ADP
ejpam-4744	234	13	gi	gi	NOUN
ejpam-4744	234	14	.	.	PUNCT
ejpam-4744	235	1	thus	thus	ADV
ejpam-4744	235	2	,	,	PUNCT
ejpam-4744	235	3	|	|	NOUN
ejpam-4744	235	4	∪x∈v	∪x∈v	X
ejpam-4744	235	5	(	(	PUNCT
ejpam-4744	235	6	gi	gi	INTJ
ejpam-4744	235	7	)	)	PUNCT
ejpam-4744	235	8	ϕ(x)|	ϕ(x)|	PROPN
ejpam-4744	235	9	≥	≥	NUM
ejpam-4744	235	10	ω(gi	ω(gi	NUM
ejpam-4744	235	11	)	)	PUNCT
ejpam-4744	235	12	=	=	SYM
ejpam-4744	235	13	ω(g	ω(g	NOUN
ejpam-4744	235	14	)	)	PUNCT
ejpam-4744	235	15	.	.	PUNCT
ejpam-4744	236	1	note	note	VERB
ejpam-4744	236	2	that	that	SCONJ
ejpam-4744	236	3	for	for	ADP
ejpam-4744	236	4	each	each	DET
ejpam-4744	236	5	i	i	PROPN
ejpam-4744	236	6	,	,	PUNCT
ejpam-4744	236	7	j	j	PROPN
ejpam-4744	236	8	,	,	PUNCT
ejpam-4744	236	9	i	i	PROPN
ejpam-4744	236	10	̸=	̸=	PROPN
ejpam-4744	236	11	j	j	PROPN
ejpam-4744	236	12	,	,	PUNCT
ejpam-4744	236	13	and	and	CCONJ
ejpam-4744	236	14	each	each	DET
ejpam-4744	236	15	a	a	DET
ejpam-4744	236	16	∈	∈	PROPN
ejpam-4744	236	17	gi	gi	NOUN
ejpam-4744	236	18	and	and	CCONJ
ejpam-4744	236	19	b	b	X
ejpam-4744	236	20	∈	∈	PROPN
ejpam-4744	236	21	gj	gj	NOUN
ejpam-4744	236	22	,	,	PUNCT
ejpam-4744	236	23	ab	ab	PROPN
ejpam-4744	236	24	/∈	/∈	PUNCT
ejpam-4744	237	1	e(pn	e(pn	PROPN
ejpam-4744	237	2	◦	◦	NOUN
ejpam-4744	237	3	g	g	NOUN
ejpam-4744	237	4	)	)	PUNCT
ejpam-4744	237	5	.	.	PUNCT
ejpam-4744	238	1	consequently	consequently	ADV
ejpam-4744	238	2	,	,	PUNCT
ejpam-4744	238	3	ei	ei	NOUN
ejpam-4744	238	4	=	=	SYM
ejpam-4744	238	5	∪x∈v	∪x∈v	PROPN
ejpam-4744	238	6	(	(	PUNCT
ejpam-4744	238	7	gi)ϕ(x	gi)ϕ(x	X
ejpam-4744	238	8	)	)	PUNCT
ejpam-4744	238	9	and	and	CCONJ
ejpam-4744	238	10	ej	ej	X
ejpam-4744	238	11	=	=	SYM
ejpam-4744	238	12	∪x∈v	∪x∈v	PROPN
ejpam-4744	238	13	(	(	PUNCT
ejpam-4744	238	14	gj)ϕ(x	gj)ϕ(x	X
ejpam-4744	238	15	)	)	PUNCT
ejpam-4744	238	16	are	be	AUX
ejpam-4744	238	17	disjoint	disjoint	ADJ
ejpam-4744	238	18	whenever	whenever	SCONJ
ejpam-4744	238	19	i	i	PRON
ejpam-4744	238	20	̸=	̸=	PROPN
ejpam-4744	238	21	j.	j.	PROPN
ejpam-4744	238	22	now	now	ADV
ejpam-4744	238	23	,	,	PUNCT
ejpam-4744	238	24	|	|	ADV
ejpam-4744	238	25	∪n	∪n	X
ejpam-4744	238	26	i=1	i=1	PROPN
ejpam-4744	239	1	ei|	ei|	PROPN
ejpam-4744	239	2	=	=	PUNCT
ejpam-4744	239	3	n∑	n∑	PROPN
ejpam-4744	239	4	i=1	i=1	PROPN
ejpam-4744	240	1	|ei|	|ei|	PROPN
ejpam-4744	240	2	≥	≥	NUM
ejpam-4744	240	3	n∑	n∑	NOUN
ejpam-4744	240	4	i=1	i=1	PROPN
ejpam-4744	240	5	ω(g	ω(g	PROPN
ejpam-4744	240	6	)	)	PUNCT
ejpam-4744	241	1	j.	j.	PROPN
ejpam-4744	241	2	b.	b.	PROPN
ejpam-4744	241	3	palco	palco	PROPN
ejpam-4744	241	4	,	,	PUNCT
ejpam-4744	241	5	r.	r.	PROPN
ejpam-4744	241	6	n.	n.	PROPN
ejpam-4744	241	7	paluga	paluga	PROPN
ejpam-4744	241	8	/	/	SYM
ejpam-4744	241	9	eur	eur	PROPN
ejpam-4744	241	10	.	.	PUNCT
ejpam-4744	242	1	j.	j.	PROPN
ejpam-4744	242	2	pure	pure	PROPN
ejpam-4744	242	3	appl	appl	PROPN
ejpam-4744	242	4	.	.	PROPN
ejpam-4744	242	5	math	math	PROPN
ejpam-4744	242	6	,	,	PUNCT
ejpam-4744	242	7	16	16	NUM
ejpam-4744	242	8	(	(	PUNCT
ejpam-4744	242	9	2	2	NUM
ejpam-4744	242	10	)	)	PUNCT
ejpam-4744	242	11	(	(	PUNCT
ejpam-4744	242	12	2023	2023	NUM
ejpam-4744	242	13	)	)	PUNCT
ejpam-4744	242	14	,	,	PUNCT
ejpam-4744	242	15	1318	1318	NUM
ejpam-4744	242	16	-	-	SYM
ejpam-4744	242	17	1325	1325	NUM
ejpam-4744	242	18	1324	1324	NUM
ejpam-4744	242	19	=	=	SYM
ejpam-4744	242	20	n	n	CCONJ
ejpam-4744	242	21	·	·	PUNCT
ejpam-4744	242	22	ω(g	ω(g	NOUN
ejpam-4744	242	23	)	)	PUNCT
ejpam-4744	242	24	.	.	PUNCT
ejpam-4744	243	1	it	it	PRON
ejpam-4744	243	2	follows	follow	VERB
ejpam-4744	243	3	that	that	SCONJ
ejpam-4744	243	4	the	the	DET
ejpam-4744	243	5	elements	element	NOUN
ejpam-4744	243	6	of	of	ADP
ejpam-4744	243	7	s−	s−	PROPN
ejpam-4744	243	8	(	(	PUNCT
ejpam-4744	243	9	∪n	∪n	X
ejpam-4744	243	10	i=1ei	i=1ei	X
ejpam-4744	243	11	)	)	PUNCT
ejpam-4744	243	12	are	be	AUX
ejpam-4744	243	13	used	use	VERB
ejpam-4744	243	14	for	for	ADP
ejpam-4744	243	15	the	the	DET
ejpam-4744	243	16	set	set	NOUN
ejpam-4744	243	17	representation	representation	NOUN
ejpam-4744	243	18	of	of	ADP
ejpam-4744	243	19	g.	g.	PROPN
ejpam-4744	243	20	note	note	VERB
ejpam-4744	243	21	that	that	SCONJ
ejpam-4744	243	22	|s	|s	PROPN
ejpam-4744	244	1	−	−	PROPN
ejpam-4744	245	1	(	(	PUNCT
ejpam-4744	245	2	∪n	∪n	NUM
ejpam-4744	245	3	i=1ei)|	i=1ei)|	X
ejpam-4744	245	4	=	=	PUNCT
ejpam-4744	245	5	|s|	|s|	NOUN
ejpam-4744	245	6	−	−	NOUN
ejpam-4744	245	7	|(∪n	|(∪n	NOUN
ejpam-4744	245	8	i=1ei)|	i=1ei)|	NOUN
ejpam-4744	245	9	≤	≤	NUM
ejpam-4744	245	10	ω(pn	ω(pn	NUM
ejpam-4744	245	11	◦	◦	NOUN
ejpam-4744	245	12	g)−	g)−	PROPN
ejpam-4744	245	13	n	n	PRON
ejpam-4744	245	14	·	·	PUNCT
ejpam-4744	245	15	ω(g	ω(g	NOUN
ejpam-4744	245	16	)	)	PUNCT
ejpam-4744	245	17	,	,	PUNCT
ejpam-4744	245	18	since	since	SCONJ
ejpam-4744	245	19	we	we	PRON
ejpam-4744	245	20	suppose	suppose	VERB
ejpam-4744	245	21	ω(pn	ω(pn	NUM
ejpam-4744	245	22	◦	◦	NOUN
ejpam-4744	245	23	g	g	NOUN
ejpam-4744	245	24	)	)	PUNCT
ejpam-4744	245	25	<	<	X
ejpam-4744	245	26	(	(	PUNCT
ejpam-4744	245	27	n−	n−	NOUN
ejpam-4744	245	28	1	1	NUM
ejpam-4744	245	29	)	)	PUNCT
ejpam-4744	245	30	+	+	CCONJ
ejpam-4744	245	31	n	n	CCONJ
ejpam-4744	245	32	·	·	PUNCT
ejpam-4744	245	33	ω(g	ω(g	NOUN
ejpam-4744	245	34	)	)	PUNCT
ejpam-4744	245	35	.	.	PUNCT
ejpam-4744	246	1	<	<	X
ejpam-4744	246	2	n−	n−	NOUN
ejpam-4744	246	3	1	1	NUM
ejpam-4744	246	4	.	.	PUNCT
ejpam-4744	247	1	since	since	SCONJ
ejpam-4744	247	2	ai	ai	ADV
ejpam-4744	247	3	and	and	CCONJ
ejpam-4744	247	4	ai+1	ai+1	NUM
ejpam-4744	247	5	are	be	AUX
ejpam-4744	247	6	adjacent	adjacent	ADJ
ejpam-4744	247	7	,	,	PUNCT
ejpam-4744	247	8	ϕ(ai	ϕ(ai	NOUN
ejpam-4744	247	9	)	)	PUNCT
ejpam-4744	247	10	∩	∩	ADJ
ejpam-4744	247	11	ϕ(ai+1	ϕ(ai+1	NOUN
ejpam-4744	247	12	)	)	PUNCT
ejpam-4744	247	13	̸=	̸=	NOUN
ejpam-4744	247	14	∅	∅	NOUN
ejpam-4744	247	15	,	,	PUNCT
ejpam-4744	247	16	for	for	ADP
ejpam-4744	247	17	every	every	DET
ejpam-4744	247	18	i	i	NOUN
ejpam-4744	247	19	,	,	PUNCT
ejpam-4744	247	20	1	1	NUM
ejpam-4744	247	21	≤	≤	NUM
ejpam-4744	247	22	i	i	PRON
ejpam-4744	247	23	≤	≤	NOUN
ejpam-4744	247	24	n	n	CCONJ
ejpam-4744	247	25	−	−	PROPN
ejpam-4744	247	26	1	1	X
ejpam-4744	247	27	.	.	PUNCT
ejpam-4744	248	1	let	let	VERB
ejpam-4744	248	2	ai	ai	VERB
ejpam-4744	248	3	=	=	PUNCT
ejpam-4744	248	4	ϕ(ai	ϕ(ai	PROPN
ejpam-4744	248	5	)	)	PUNCT
ejpam-4744	248	6	∩	∩	ADJ
ejpam-4744	248	7	ϕ(ai+1	ϕ(ai+1	PROPN
ejpam-4744	248	8	)	)	PUNCT
ejpam-4744	248	9	,	,	PUNCT
ejpam-4744	248	10	1	1	NUM
ejpam-4744	248	11	≤	≤	NUM
ejpam-4744	248	12	i	i	PRON
ejpam-4744	248	13	≤	≤	NOUN
ejpam-4744	248	14	n	n	CCONJ
ejpam-4744	248	15	−	−	PROPN
ejpam-4744	248	16	1	1	NUM
ejpam-4744	248	17	.	.	PUNCT
ejpam-4744	249	1	since	since	SCONJ
ejpam-4744	249	2	|s	|s	PROPN
ejpam-4744	249	3	−	−	PROPN
ejpam-4744	249	4	(	(	PUNCT
ejpam-4744	249	5	∪n	∪n	PROPN
ejpam-4744	249	6	i=1ei)|	i=1ei)|	VERB
ejpam-4744	249	7	<	<	X
ejpam-4744	249	8	n	n	CCONJ
ejpam-4744	249	9	−	−	PROPN
ejpam-4744	249	10	1	1	NUM
ejpam-4744	249	11	,	,	PUNCT
ejpam-4744	249	12	there	there	PRON
ejpam-4744	249	13	exist	exist	VERB
ejpam-4744	249	14	i	i	PRON
ejpam-4744	249	15	,	,	PUNCT
ejpam-4744	249	16	j	j	PROPN
ejpam-4744	249	17	with	with	ADP
ejpam-4744	249	18	i	i	PRON
ejpam-4744	249	19	<	<	X
ejpam-4744	249	20	j	j	PROPN
ejpam-4744	249	21	,	,	PUNCT
ejpam-4744	249	22	such	such	ADJ
ejpam-4744	249	23	that	that	SCONJ
ejpam-4744	249	24	ai	ai	VERB
ejpam-4744	249	25	∩	∩	PROPN
ejpam-4744	249	26	aj	aj	PROPN
ejpam-4744	249	27	̸=	̸=	PROPN
ejpam-4744	249	28	∅.	∅.	ADV
ejpam-4744	249	29	let	let	VERB
ejpam-4744	249	30	t	t	PROPN
ejpam-4744	249	31	∈	∈	PROPN
ejpam-4744	249	32	ai	ai	VERB
ejpam-4744	249	33	∩	∩	PROPN
ejpam-4744	249	34	aj	aj	PROPN
ejpam-4744	249	35	.	.	PUNCT
ejpam-4744	250	1	then	then	ADV
ejpam-4744	250	2	t	t	PROPN
ejpam-4744	250	3	∈	∈	PROPN
ejpam-4744	250	4	ai	ai	VERB
ejpam-4744	250	5	and	and	CCONJ
ejpam-4744	250	6	t	t	PROPN
ejpam-4744	250	7	∈	∈	PROPN
ejpam-4744	250	8	aj	aj	PROPN
ejpam-4744	250	9	.	.	PUNCT
ejpam-4744	251	1	it	it	PRON
ejpam-4744	251	2	follows	follow	VERB
ejpam-4744	251	3	that	that	SCONJ
ejpam-4744	251	4	t	t	PROPN
ejpam-4744	251	5	∈	∈	PROPN
ejpam-4744	251	6	ϕ(ai	ϕ(ai	PROPN
ejpam-4744	251	7	)	)	PUNCT
ejpam-4744	251	8	and	and	CCONJ
ejpam-4744	251	9	t	t	NOUN
ejpam-4744	251	10	∈	∈	PROPN
ejpam-4744	251	11	ϕ(aj+1	ϕ(aj+1	PROPN
ejpam-4744	251	12	)	)	PUNCT
ejpam-4744	251	13	.	.	PUNCT
ejpam-4744	252	1	note	note	VERB
ejpam-4744	252	2	that	that	SCONJ
ejpam-4744	252	3	j	j	PROPN
ejpam-4744	252	4	≥	≥	VERB
ejpam-4744	252	5	i	i	PRON
ejpam-4744	252	6	+	+	NOUN
ejpam-4744	252	7	1	1	NUM
ejpam-4744	252	8	,	,	PUNCT
ejpam-4744	252	9	it	it	PRON
ejpam-4744	252	10	follows	follow	VERB
ejpam-4744	252	11	ai	ai	VERB
ejpam-4744	252	12	and	and	CCONJ
ejpam-4744	252	13	aj+1	aj+1	NOUN
ejpam-4744	252	14	are	be	AUX
ejpam-4744	252	15	adjacent	adjacent	ADJ
ejpam-4744	252	16	.	.	PUNCT
ejpam-4744	253	1	this	this	PRON
ejpam-4744	253	2	is	be	AUX
ejpam-4744	253	3	a	a	DET
ejpam-4744	253	4	contradiction	contradiction	NOUN
ejpam-4744	253	5	.	.	PUNCT
ejpam-4744	254	1	hence	hence	ADV
ejpam-4744	254	2	,	,	PUNCT
ejpam-4744	254	3	ω(pn	ω(pn	NUM
ejpam-4744	254	4	◦	◦	NOUN
ejpam-4744	254	5	g	g	NOUN
ejpam-4744	254	6	)	)	PUNCT
ejpam-4744	254	7	=	=	PUNCT
ejpam-4744	255	1	(	(	PUNCT
ejpam-4744	255	2	n−	n−	NOUN
ejpam-4744	255	3	1	1	NUM
ejpam-4744	255	4	)	)	PUNCT
ejpam-4744	255	5	+	+	CCONJ
ejpam-4744	255	6	n	n	CCONJ
ejpam-4744	255	7	·	·	PUNCT
ejpam-4744	255	8	ω(g	ω(g	NOUN
ejpam-4744	255	9	)	)	PUNCT
ejpam-4744	255	10	.	.	PUNCT
ejpam-4744	256	1	corollary	corollary	ADJ
ejpam-4744	256	2	3	3	X
ejpam-4744	256	3	.	.	PUNCT
ejpam-4744	257	1	let	let	VERB
ejpam-4744	257	2	g	g	PRON
ejpam-4744	257	3	be	be	AUX
ejpam-4744	257	4	a	a	DET
ejpam-4744	257	5	connected	connected	ADJ
ejpam-4744	257	6	graph	graph	NOUN
ejpam-4744	257	7	.	.	PUNCT
ejpam-4744	258	1	then	then	ADV
ejpam-4744	258	2	ω(cn	ω(cn	NUM
ejpam-4744	258	3	◦	◦	NOUN
ejpam-4744	258	4	g	g	NOUN
ejpam-4744	258	5	)	)	PUNCT
ejpam-4744	258	6	=	=	PRON
ejpam-4744	258	7	{	{	PUNCT
ejpam-4744	258	8	1	1	NUM
ejpam-4744	258	9	+	+	NUM
ejpam-4744	258	10	3ω(g	3ω(g	NUM
ejpam-4744	258	11	)	)	PUNCT
ejpam-4744	258	12	,	,	PUNCT
ejpam-4744	258	13	if	if	SCONJ
ejpam-4744	258	14	n	n	PROPN
ejpam-4744	258	15	=	=	SYM
ejpam-4744	258	16	3	3	NUM
ejpam-4744	258	17	n+	n+	NUM
ejpam-4744	258	18	n	n	PRON
ejpam-4744	258	19	·	·	PUNCT
ejpam-4744	258	20	ω(g	ω(g	NOUN
ejpam-4744	258	21	)	)	PUNCT
ejpam-4744	258	22	,	,	PUNCT
ejpam-4744	258	23	if	if	SCONJ
ejpam-4744	258	24	n	n	PRON
ejpam-4744	258	25	≥	≥	NOUN
ejpam-4744	258	26	4	4	NUM
ejpam-4744	258	27	.	.	PUNCT
ejpam-4744	259	1	proof	proof	NOUN
ejpam-4744	259	2	.	.	PUNCT
ejpam-4744	260	1	by	by	ADP
ejpam-4744	260	2	theorem	theorem	ADJ
ejpam-4744	260	3	3	3	NUM
ejpam-4744	260	4	,	,	PUNCT
ejpam-4744	260	5	ω(cn	ω(cn	NUM
ejpam-4744	260	6	◦	◦	NOUN
ejpam-4744	260	7	g	g	NOUN
ejpam-4744	260	8	)	)	PUNCT
ejpam-4744	260	9	≤	≤	PROPN
ejpam-4744	260	10	co(cn	co(cn	PROPN
ejpam-4744	260	11	)	)	PUNCT
ejpam-4744	261	1	+	+	CCONJ
ejpam-4744	261	2	|v	|v	X
ejpam-4744	261	3	(	(	PUNCT
ejpam-4744	261	4	cn)|	cn)|	X
ejpam-4744	261	5	·	·	SYM
ejpam-4744	261	6	ω(g	ω(g	NOUN
ejpam-4744	261	7	)	)	PUNCT
ejpam-4744	261	8	.	.	PUNCT
ejpam-4744	262	1	by	by	ADP
ejpam-4744	262	2	theorem	theorem	NOUN
ejpam-4744	262	3	2	2	NUM
ejpam-4744	262	4	,	,	PUNCT
ejpam-4744	262	5	co(cn	co(cn	PROPN
ejpam-4744	262	6	)	)	PUNCT
ejpam-4744	262	7	=	=	PRON
ejpam-4744	262	8	{	{	PUNCT
ejpam-4744	262	9	1	1	NUM
ejpam-4744	262	10	,	,	PUNCT
ejpam-4744	262	11	if	if	SCONJ
ejpam-4744	262	12	n	n	CCONJ
ejpam-4744	262	13	=	=	SYM
ejpam-4744	262	14	3	3	NUM
ejpam-4744	262	15	n	n	CCONJ
ejpam-4744	262	16	,	,	PUNCT
ejpam-4744	262	17	if	if	SCONJ
ejpam-4744	262	18	n	n	PRON
ejpam-4744	262	19	≥	≥	VERB
ejpam-4744	262	20	4	4	NUM
ejpam-4744	262	21	the	the	DET
ejpam-4744	262	22	case	case	NOUN
ejpam-4744	262	23	n	n	NOUN
ejpam-4744	262	24	=	=	SYM
ejpam-4744	262	25	3	3	NUM
ejpam-4744	262	26	,	,	PUNCT
ejpam-4744	262	27	follows	follow	VERB
ejpam-4744	262	28	from	from	ADP
ejpam-4744	262	29	corollary	corollary	ADJ
ejpam-4744	262	30	1	1	NUM
ejpam-4744	262	31	and	and	CCONJ
ejpam-4744	262	32	for	for	ADP
ejpam-4744	262	33	n	n	PRON
ejpam-4744	262	34	≥	≥	NOUN
ejpam-4744	262	35	4	4	NUM
ejpam-4744	262	36	,	,	PUNCT
ejpam-4744	262	37	ω(cn	ω(cn	NUM
ejpam-4744	262	38	◦	◦	NOUN
ejpam-4744	262	39	g	g	NOUN
ejpam-4744	262	40	)	)	PUNCT
ejpam-4744	262	41	≤	≤	PROPN
ejpam-4744	262	42	co(cn	co(cn	PROPN
ejpam-4744	262	43	)	)	PUNCT
ejpam-4744	263	1	+	+	CCONJ
ejpam-4744	263	2	|v	|v	X
ejpam-4744	263	3	(	(	PUNCT
ejpam-4744	263	4	cn)|	cn)|	X
ejpam-4744	263	5	·	·	SYM
ejpam-4744	263	6	ω(g	ω(g	NOUN
ejpam-4744	263	7	)	)	PUNCT
ejpam-4744	263	8	=	=	SYM
ejpam-4744	263	9	n+	n+	X
ejpam-4744	263	10	n	n	PRON
ejpam-4744	263	11	·	·	PUNCT
ejpam-4744	263	12	ω(g	ω(g	NOUN
ejpam-4744	263	13	)	)	PUNCT
ejpam-4744	263	14	.	.	PUNCT
ejpam-4744	264	1	suppose	suppose	VERB
ejpam-4744	264	2	ω(cn	ω(cn	NUM
ejpam-4744	264	3	◦	◦	NOUN
ejpam-4744	264	4	g	g	NOUN
ejpam-4744	264	5	)	)	PUNCT
ejpam-4744	264	6	<	<	X
ejpam-4744	264	7	n+	n+	X
ejpam-4744	264	8	n	n	PRON
ejpam-4744	264	9	·	·	PUNCT
ejpam-4744	264	10	ω(g	ω(g	NOUN
ejpam-4744	264	11	)	)	PUNCT
ejpam-4744	264	12	.	.	PUNCT
ejpam-4744	265	1	let	let	VERB
ejpam-4744	265	2	v	v	X
ejpam-4744	265	3	(	(	PUNCT
ejpam-4744	265	4	cn	cn	PROPN
ejpam-4744	265	5	)	)	PUNCT
ejpam-4744	265	6	=	=	SYM
ejpam-4744	265	7	{	{	PUNCT
ejpam-4744	265	8	a1	a1	PROPN
ejpam-4744	265	9	,	,	PUNCT
ejpam-4744	265	10	a2	a2	PROPN
ejpam-4744	265	11	,	,	PUNCT
ejpam-4744	265	12	...	...	PUNCT
ejpam-4744	265	13	,	,	PUNCT
ejpam-4744	265	14	an	an	DET
ejpam-4744	265	15	}	}	PUNCT
ejpam-4744	265	16	,	,	PUNCT
ejpam-4744	265	17	e(cn	e(cn	NOUN
ejpam-4744	265	18	)	)	PUNCT
ejpam-4744	265	19	=	=	PRON
ejpam-4744	265	20	{	{	PUNCT
ejpam-4744	265	21	aiai+1	aiai+1	NOUN
ejpam-4744	265	22	:	:	PUNCT
ejpam-4744	265	23	1	1	NUM
ejpam-4744	265	24	≤	≤	NUM
ejpam-4744	265	25	i	i	PRON
ejpam-4744	265	26	≤	≤	ADJ
ejpam-4744	265	27	n	n	CCONJ
ejpam-4744	265	28	−	−	PROPN
ejpam-4744	265	29	1	1	NUM
ejpam-4744	265	30	}	}	PUNCT
ejpam-4744	265	31	∪	∪	ADJ
ejpam-4744	265	32	{	{	PUNCT
ejpam-4744	265	33	a1an	a1an	ADP
ejpam-4744	265	34	}	}	PUNCT
ejpam-4744	265	35	and	and	CCONJ
ejpam-4744	265	36	for	for	ADP
ejpam-4744	265	37	each	each	DET
ejpam-4744	265	38	i	i	PRON
ejpam-4744	265	39	,	,	PUNCT
ejpam-4744	265	40	1	1	NUM
ejpam-4744	265	41	≤	≤	NUM
ejpam-4744	265	42	i	i	PRON
ejpam-4744	265	43	≤	≤	PROPN
ejpam-4744	265	44	n	n	CCONJ
ejpam-4744	265	45	,	,	PUNCT
ejpam-4744	265	46	let	let	VERB
ejpam-4744	265	47	gi	gi	PART
ejpam-4744	265	48	be	be	AUX
ejpam-4744	265	49	the	the	DET
ejpam-4744	265	50	ith	ith	PROPN
ejpam-4744	265	51	copy	copy	NOUN
ejpam-4744	265	52	of	of	ADP
ejpam-4744	265	53	g	g	PROPN
ejpam-4744	265	54	corresponding	correspond	VERB
ejpam-4744	265	55	to	to	ADP
ejpam-4744	265	56	the	the	DET
ejpam-4744	265	57	vertex	vertex	NOUN
ejpam-4744	265	58	ai	ai	VERB
ejpam-4744	265	59	.	.	PUNCT
ejpam-4744	266	1	let	let	VERB
ejpam-4744	266	2	f	f	PRON
ejpam-4744	266	3	be	be	AUX
ejpam-4744	266	4	a	a	DET
ejpam-4744	266	5	collection	collection	NOUN
ejpam-4744	266	6	of	of	ADP
ejpam-4744	266	7	subsets	subset	NOUN
ejpam-4744	266	8	of	of	ADP
ejpam-4744	266	9	s	s	NOUN
ejpam-4744	266	10	=	=	PUNCT
ejpam-4744	266	11	{	{	PUNCT
ejpam-4744	266	12	1	1	NUM
ejpam-4744	266	13	,	,	PUNCT
ejpam-4744	266	14	2	2	NUM
ejpam-4744	266	15	,	,	PUNCT
ejpam-4744	266	16	3	3	NUM
ejpam-4744	266	17	,	,	PUNCT
ejpam-4744	266	18	...	...	PUNCT
ejpam-4744	266	19	,	,	PUNCT
ejpam-4744	266	20	ω(cn	ω(cn	PROPN
ejpam-4744	266	21	◦	◦	NOUN
ejpam-4744	266	22	g	g	NOUN
ejpam-4744	266	23	)	)	PUNCT
ejpam-4744	266	24	}	}	PUNCT
ejpam-4744	266	25	such	such	ADJ
ejpam-4744	266	26	that	that	SCONJ
ejpam-4744	266	27	ω(f	ω(f	PROPN
ejpam-4744	266	28	)	)	PUNCT
ejpam-4744	267	1	∼=	∼=	PROPN
ejpam-4744	267	2	cn	cn	PROPN
ejpam-4744	267	3	◦	◦	PROPN
ejpam-4744	267	4	g.	g.	PROPN
ejpam-4744	267	5	let	let	VERB
ejpam-4744	267	6	ϕ	ϕ	NOUN
ejpam-4744	267	7	:	:	PUNCT
ejpam-4744	267	8	v	v	X
ejpam-4744	267	9	(	(	PUNCT
ejpam-4744	267	10	cn	cn	INTJ
ejpam-4744	267	11	◦	◦	NOUN
ejpam-4744	267	12	g	g	NOUN
ejpam-4744	267	13	)	)	PUNCT
ejpam-4744	267	14	→	→	SYM
ejpam-4744	267	15	f	f	X
ejpam-4744	267	16	be	be	AUX
ejpam-4744	267	17	an	an	DET
ejpam-4744	267	18	isomorphism	isomorphism	NOUN
ejpam-4744	267	19	.	.	PUNCT
ejpam-4744	268	1	for	for	ADP
ejpam-4744	268	2	each	each	DET
ejpam-4744	268	3	i	i	PRON
ejpam-4744	268	4	,	,	PUNCT
ejpam-4744	268	5	1	1	NUM
ejpam-4744	268	6	≤	≤	NUM
ejpam-4744	268	7	i	i	PRON
ejpam-4744	268	8	≤	≤	NOUN
ejpam-4744	268	9	n	n	CCONJ
ejpam-4744	268	10	,	,	PUNCT
ejpam-4744	268	11	{	{	PUNCT
ejpam-4744	268	12	ϕ(x	ϕ(x	X
ejpam-4744	268	13	)	)	PUNCT
ejpam-4744	268	14	:	:	PUNCT
ejpam-4744	269	1	x	x	X
ejpam-4744	269	2	∈	∈	NOUN
ejpam-4744	269	3	v	v	X
ejpam-4744	269	4	(	(	PUNCT
ejpam-4744	269	5	gi	gi	NOUN
ejpam-4744	269	6	)	)	PUNCT
ejpam-4744	269	7	}	}	PUNCT
ejpam-4744	269	8	is	be	AUX
ejpam-4744	269	9	a	a	DET
ejpam-4744	269	10	set	set	VERB
ejpam-4744	269	11	representation	representation	NOUN
ejpam-4744	269	12	for	for	ADP
ejpam-4744	269	13	gi	gi	NOUN
ejpam-4744	269	14	.	.	PUNCT
ejpam-4744	270	1	thus	thus	ADV
ejpam-4744	270	2	,	,	PUNCT
ejpam-4744	270	3	|	|	NOUN
ejpam-4744	270	4	∪x∈v	∪x∈v	X
ejpam-4744	270	5	(	(	PUNCT
ejpam-4744	270	6	gi	gi	INTJ
ejpam-4744	270	7	)	)	PUNCT
ejpam-4744	270	8	ϕ(x)|	ϕ(x)|	PROPN
ejpam-4744	270	9	≥	≥	NUM
ejpam-4744	270	10	ω(gi	ω(gi	NUM
ejpam-4744	270	11	)	)	PUNCT
ejpam-4744	270	12	=	=	SYM
ejpam-4744	270	13	ω(g	ω(g	NOUN
ejpam-4744	270	14	)	)	PUNCT
ejpam-4744	270	15	.	.	PUNCT
ejpam-4744	271	1	note	note	VERB
ejpam-4744	271	2	that	that	SCONJ
ejpam-4744	271	3	for	for	ADP
ejpam-4744	271	4	each	each	DET
ejpam-4744	271	5	i	i	PROPN
ejpam-4744	271	6	,	,	PUNCT
ejpam-4744	271	7	j	j	PROPN
ejpam-4744	271	8	,	,	PUNCT
ejpam-4744	271	9	i	i	PROPN
ejpam-4744	271	10	̸=	̸=	PROPN
ejpam-4744	271	11	j	j	PROPN
ejpam-4744	271	12	,	,	PUNCT
ejpam-4744	271	13	and	and	CCONJ
ejpam-4744	271	14	each	each	DET
ejpam-4744	271	15	a	a	DET
ejpam-4744	271	16	∈	∈	PROPN
ejpam-4744	271	17	gi	gi	NOUN
ejpam-4744	271	18	and	and	CCONJ
ejpam-4744	271	19	b	b	X
ejpam-4744	271	20	∈	∈	PROPN
ejpam-4744	271	21	gj	gj	NOUN
ejpam-4744	271	22	,	,	PUNCT
ejpam-4744	272	1	ab	ab	PROPN
ejpam-4744	272	2	/∈	/∈	PUNCT
ejpam-4744	272	3	e(cn	e(cn	PROPN
ejpam-4744	272	4	◦	◦	NOUN
ejpam-4744	272	5	g	g	NOUN
ejpam-4744	272	6	)	)	PUNCT
ejpam-4744	272	7	.	.	PUNCT
ejpam-4744	273	1	consequently	consequently	ADV
ejpam-4744	273	2	,	,	PUNCT
ejpam-4744	273	3	ei	ei	NOUN
ejpam-4744	273	4	=	=	SYM
ejpam-4744	273	5	∪x∈v	∪x∈v	PROPN
ejpam-4744	273	6	(	(	PUNCT
ejpam-4744	273	7	gi)ϕ(x	gi)ϕ(x	X
ejpam-4744	273	8	)	)	PUNCT
ejpam-4744	273	9	and	and	CCONJ
ejpam-4744	273	10	ej	ej	X
ejpam-4744	273	11	=	=	SYM
ejpam-4744	273	12	∪x∈v	∪x∈v	PROPN
ejpam-4744	273	13	(	(	PUNCT
ejpam-4744	273	14	gj)ϕ(x	gj)ϕ(x	X
ejpam-4744	273	15	)	)	PUNCT
ejpam-4744	273	16	are	be	AUX
ejpam-4744	273	17	disjoint	disjoint	ADJ
ejpam-4744	273	18	whenever	whenever	SCONJ
ejpam-4744	273	19	i	i	PRON
ejpam-4744	273	20	̸=	̸=	PROPN
ejpam-4744	273	21	j.	j.	PROPN
ejpam-4744	273	22	now	now	ADV
ejpam-4744	273	23	,	,	PUNCT
ejpam-4744	273	24	|	|	ADV
ejpam-4744	273	25	∪n	∪n	X
ejpam-4744	273	26	i=1	i=1	PROPN
ejpam-4744	274	1	ei|	ei|	PROPN
ejpam-4744	274	2	=	=	PUNCT
ejpam-4744	274	3	n∑	n∑	PROPN
ejpam-4744	274	4	i=1	i=1	PROPN
ejpam-4744	274	5	|ei|	|ei|	PROPN
ejpam-4744	274	6	references	reference	NOUN
ejpam-4744	274	7	1325	1325	NUM
ejpam-4744	274	8	≥	≥	NUM
ejpam-4744	274	9	n∑	n∑	NOUN
ejpam-4744	274	10	i=1	i=1	PROPN
ejpam-4744	274	11	ω(g	ω(g	PROPN
ejpam-4744	274	12	)	)	PUNCT
ejpam-4744	274	13	=	=	SYM
ejpam-4744	274	14	n	n	X
ejpam-4744	274	15	·	·	PUNCT
ejpam-4744	274	16	ω(g	ω(g	NOUN
ejpam-4744	274	17	)	)	PUNCT
ejpam-4744	274	18	.	.	PUNCT
ejpam-4744	275	1	it	it	PRON
ejpam-4744	275	2	follows	follow	VERB
ejpam-4744	275	3	that	that	SCONJ
ejpam-4744	275	4	the	the	DET
ejpam-4744	275	5	elements	element	NOUN
ejpam-4744	275	6	of	of	ADP
ejpam-4744	275	7	s−	s−	PROPN
ejpam-4744	275	8	(	(	PUNCT
ejpam-4744	275	9	∪n	∪n	X
ejpam-4744	275	10	i=1ei	i=1ei	X
ejpam-4744	275	11	)	)	PUNCT
ejpam-4744	275	12	are	be	AUX
ejpam-4744	275	13	used	use	VERB
ejpam-4744	275	14	for	for	ADP
ejpam-4744	275	15	the	the	DET
ejpam-4744	275	16	set	set	NOUN
ejpam-4744	275	17	representation	representation	NOUN
ejpam-4744	275	18	of	of	ADP
ejpam-4744	275	19	g.	g.	PROPN
ejpam-4744	275	20	note	note	VERB
ejpam-4744	275	21	that	that	SCONJ
ejpam-4744	275	22	|s	|s	PROPN
ejpam-4744	276	1	−	−	PROPN
ejpam-4744	277	1	(	(	PUNCT
ejpam-4744	277	2	∪n	∪n	NUM
ejpam-4744	277	3	i=1ei)|	i=1ei)|	X
ejpam-4744	277	4	=	=	PUNCT
ejpam-4744	277	5	|s|	|s|	NOUN
ejpam-4744	277	6	−	−	NOUN
ejpam-4744	277	7	|(∪n	|(∪n	NOUN
ejpam-4744	277	8	i=1ei)|	i=1ei)|	ADJ
ejpam-4744	277	9	≤	≤	NUM
ejpam-4744	277	10	ω(cn	ω(cn	NUM
ejpam-4744	277	11	◦	◦	NOUN
ejpam-4744	277	12	g)−	g)−	PROPN
ejpam-4744	277	13	n	n	PRON
ejpam-4744	277	14	·	·	PUNCT
ejpam-4744	277	15	ω(g	ω(g	NOUN
ejpam-4744	277	16	)	)	PUNCT
ejpam-4744	277	17	,	,	PUNCT
ejpam-4744	277	18	since	since	SCONJ
ejpam-4744	277	19	we	we	PRON
ejpam-4744	277	20	suppose	suppose	VERB
ejpam-4744	277	21	ω(cn	ω(cn	NUM
ejpam-4744	277	22	◦	◦	NOUN
ejpam-4744	277	23	g	g	NOUN
ejpam-4744	277	24	)	)	PUNCT
ejpam-4744	277	25	<	<	X
ejpam-4744	277	26	n+	n+	X
ejpam-4744	277	27	n	n	PRON
ejpam-4744	277	28	·	·	PUNCT
ejpam-4744	277	29	ω(g	ω(g	NOUN
ejpam-4744	277	30	)	)	PUNCT
ejpam-4744	277	31	.	.	PUNCT
ejpam-4744	278	1	<	<	X
ejpam-4744	278	2	n.	n.	PROPN
ejpam-4744	278	3	since	since	SCONJ
ejpam-4744	278	4	ai	ai	PROPN
ejpam-4744	278	5	and	and	CCONJ
ejpam-4744	278	6	ai+1	ai+1	NUM
ejpam-4744	278	7	are	be	AUX
ejpam-4744	278	8	adjacent	adjacent	ADJ
ejpam-4744	278	9	,	,	PUNCT
ejpam-4744	278	10	ϕ(ai	ϕ(ai	NOUN
ejpam-4744	278	11	)	)	PUNCT
ejpam-4744	278	12	∩	∩	ADJ
ejpam-4744	278	13	ϕ(ai+1	ϕ(ai+1	NOUN
ejpam-4744	278	14	)	)	PUNCT
ejpam-4744	278	15	̸=	̸=	NOUN
ejpam-4744	278	16	∅	∅	NOUN
ejpam-4744	278	17	,	,	PUNCT
ejpam-4744	278	18	for	for	ADP
ejpam-4744	278	19	every	every	DET
ejpam-4744	278	20	i	i	NOUN
ejpam-4744	278	21	,	,	PUNCT
ejpam-4744	278	22	1	1	NUM
ejpam-4744	278	23	≤	≤	NUM
ejpam-4744	278	24	i	i	PRON
ejpam-4744	278	25	≤	≤	ADJ
ejpam-4744	278	26	n.	n.	NOUN
ejpam-4744	278	27	let	let	VERB
ejpam-4744	278	28	ai	ai	VERB
ejpam-4744	278	29	=	=	PUNCT
ejpam-4744	278	30	ϕ(ai)∩	ϕ(ai)∩	PROPN
ejpam-4744	278	31	ϕ(ai+1	ϕ(ai+1	NOUN
ejpam-4744	278	32	)	)	PUNCT
ejpam-4744	278	33	,	,	PUNCT
ejpam-4744	278	34	1	1	NUM
ejpam-4744	278	35	≤	≤	NUM
ejpam-4744	278	36	i	i	PRON
ejpam-4744	278	37	≤	≤	PROPN
ejpam-4744	278	38	n.	n.	NOUN
ejpam-4744	278	39	since	since	SCONJ
ejpam-4744	278	40	|s	|s	PROPN
ejpam-4744	278	41	−	−	PROPN
ejpam-4744	278	42	(	(	PUNCT
ejpam-4744	278	43	∪n	∪n	PROPN
ejpam-4744	278	44	i=1ei)|	i=1ei)|	VERB
ejpam-4744	278	45	<	<	X
ejpam-4744	278	46	n	n	CCONJ
ejpam-4744	278	47	,	,	PUNCT
ejpam-4744	278	48	there	there	PRON
ejpam-4744	278	49	exist	exist	VERB
ejpam-4744	278	50	i	i	PRON
ejpam-4744	278	51	,	,	PUNCT
ejpam-4744	278	52	j	j	PROPN
ejpam-4744	278	53	with	with	ADP
ejpam-4744	278	54	i	i	PRON
ejpam-4744	278	55	<	<	X
ejpam-4744	278	56	j	j	PROPN
ejpam-4744	278	57	,	,	PUNCT
ejpam-4744	278	58	such	such	ADJ
ejpam-4744	278	59	that	that	SCONJ
ejpam-4744	278	60	ai	ai	VERB
ejpam-4744	278	61	∩	∩	PROPN
ejpam-4744	278	62	aj	aj	PROPN
ejpam-4744	278	63	̸=	̸=	PROPN
ejpam-4744	278	64	∅.	∅.	ADV
ejpam-4744	278	65	let	let	VERB
ejpam-4744	278	66	t	t	PROPN
ejpam-4744	278	67	∈	∈	PROPN
ejpam-4744	278	68	ai	ai	VERB
ejpam-4744	278	69	∩	∩	PROPN
ejpam-4744	278	70	aj	aj	PROPN
ejpam-4744	278	71	.	.	PUNCT
ejpam-4744	279	1	then	then	ADV
ejpam-4744	279	2	t	t	PROPN
ejpam-4744	279	3	∈	∈	PROPN
ejpam-4744	279	4	ai	ai	VERB
ejpam-4744	279	5	and	and	CCONJ
ejpam-4744	279	6	t	t	PROPN
ejpam-4744	279	7	∈	∈	PROPN
ejpam-4744	279	8	aj	aj	PROPN
ejpam-4744	279	9	.	.	PUNCT
ejpam-4744	280	1	it	it	PRON
ejpam-4744	280	2	follows	follow	VERB
ejpam-4744	280	3	that	that	SCONJ
ejpam-4744	280	4	t	t	PROPN
ejpam-4744	280	5	∈	∈	PROPN
ejpam-4744	280	6	ϕ(ai	ϕ(ai	PROPN
ejpam-4744	280	7	)	)	PUNCT
ejpam-4744	280	8	and	and	CCONJ
ejpam-4744	280	9	t	t	NOUN
ejpam-4744	280	10	∈	∈	PROPN
ejpam-4744	280	11	ϕ(aj+1	ϕ(aj+1	PROPN
ejpam-4744	280	12	)	)	PUNCT
ejpam-4744	280	13	.	.	PUNCT
ejpam-4744	281	1	note	note	VERB
ejpam-4744	281	2	that	that	SCONJ
ejpam-4744	281	3	j	j	PROPN
ejpam-4744	281	4	≥	≥	VERB
ejpam-4744	281	5	i	i	PRON
ejpam-4744	281	6	+	+	NOUN
ejpam-4744	281	7	1	1	NUM
ejpam-4744	281	8	,	,	PUNCT
ejpam-4744	281	9	it	it	PRON
ejpam-4744	281	10	follows	follow	VERB
ejpam-4744	281	11	ai	ai	VERB
ejpam-4744	281	12	and	and	CCONJ
ejpam-4744	281	13	aj+1	aj+1	NOUN
ejpam-4744	281	14	are	be	AUX
ejpam-4744	281	15	adjacent	adjacent	ADJ
ejpam-4744	281	16	.	.	PUNCT
ejpam-4744	282	1	this	this	PRON
ejpam-4744	282	2	is	be	AUX
ejpam-4744	282	3	a	a	DET
ejpam-4744	282	4	contradiction	contradiction	NOUN
ejpam-4744	282	5	.	.	PUNCT
ejpam-4744	283	1	hence	hence	ADV
ejpam-4744	283	2	,	,	PUNCT
ejpam-4744	283	3	ω(cn	ω(cn	PROPN
ejpam-4744	283	4	◦	◦	NOUN
ejpam-4744	283	5	g	g	NOUN
ejpam-4744	283	6	)	)	PUNCT
ejpam-4744	283	7	=	=	SYM
ejpam-4744	283	8	n+	n+	X
ejpam-4744	283	9	n	n	PRON
ejpam-4744	283	10	·	·	PUNCT
ejpam-4744	283	11	ω(g	ω(g	NOUN
ejpam-4744	283	12	)	)	PUNCT
ejpam-4744	283	13	.	.	PUNCT
ejpam-4744	284	1	corollary	corollary	ADJ
ejpam-4744	284	2	4	4	NUM
ejpam-4744	284	3	.	.	PUNCT
ejpam-4744	285	1	let	let	VERB
ejpam-4744	285	2	n	n	PRON
ejpam-4744	285	3	≥	≥	NOUN
ejpam-4744	285	4	3	3	NUM
ejpam-4744	285	5	.	.	PUNCT
ejpam-4744	286	1	then	then	ADV
ejpam-4744	286	2	ω(crn	ω(crn	NUM
ejpam-4744	286	3	)	)	PUNCT
ejpam-4744	286	4	=	=	PRON
ejpam-4744	286	5	{	{	PUNCT
ejpam-4744	286	6	4	4	NUM
ejpam-4744	286	7	,	,	PUNCT
ejpam-4744	286	8	if	if	SCONJ
ejpam-4744	286	9	n	n	NOUN
ejpam-4744	286	10	=	=	SYM
ejpam-4744	286	11	3	3	NUM
ejpam-4744	286	12	2n	2n	NUM
ejpam-4744	286	13	,	,	PUNCT
ejpam-4744	286	14	if	if	SCONJ
ejpam-4744	286	15	n	n	PRON
ejpam-4744	286	16	≥	≥	NOUN
ejpam-4744	286	17	4	4	NUM
ejpam-4744	286	18	.	.	PUNCT
ejpam-4744	287	1	proof	proof	NOUN
ejpam-4744	287	2	.	.	PUNCT
ejpam-4744	288	1	the	the	DET
ejpam-4744	288	2	proof	proof	NOUN
ejpam-4744	288	3	follows	follow	VERB
ejpam-4744	288	4	from	from	ADP
ejpam-4744	288	5	corollary	corollary	ADJ
ejpam-4744	288	6	3	3	NUM
ejpam-4744	288	7	.	.	PUNCT
ejpam-4744	289	1	acknowledgements	acknowledgement	NOUN
ejpam-4744	289	2	the	the	DET
ejpam-4744	289	3	author	author	NOUN
ejpam-4744	289	4	would	would	AUX
ejpam-4744	289	5	like	like	VERB
ejpam-4744	289	6	to	to	PART
ejpam-4744	289	7	thank	thank	VERB
ejpam-4744	289	8	the	the	DET
ejpam-4744	289	9	peer	peer	NOUN
ejpam-4744	289	10	reviewers	reviewer	NOUN
ejpam-4744	289	11	of	of	ADP
ejpam-4744	289	12	the	the	DET
ejpam-4744	289	13	paper	paper	NOUN
ejpam-4744	289	14	and	and	CCONJ
ejpam-4744	289	15	this	this	DET
ejpam-4744	289	16	research	research	NOUN
ejpam-4744	289	17	is	be	AUX
ejpam-4744	289	18	funded	fund	VERB
ejpam-4744	289	19	by	by	ADP
ejpam-4744	289	20	the	the	DET
ejpam-4744	289	21	mindanao	mindanao	PROPN
ejpam-4744	289	22	state	state	PROPN
ejpam-4744	289	23	university	university	PROPN
ejpam-4744	289	24	at	at	ADP
ejpam-4744	289	25	naawan	naawan	PROPN
ejpam-4744	289	26	.	.	PUNCT
ejpam-4744	290	1	references	reference	NOUN
ejpam-4744	290	2	[	[	X
ejpam-4744	290	3	1	1	NUM
ejpam-4744	290	4	]	]	PUNCT
ejpam-4744	290	5	paul	paul	PROPN
ejpam-4744	290	6	erdos	erdos	PROPN
ejpam-4744	290	7	,	,	PUNCT
ejpam-4744	290	8	a	a	DET
ejpam-4744	290	9	goodman	goodman	NOUN
ejpam-4744	290	10	,	,	PUNCT
ejpam-4744	290	11	and	and	CCONJ
ejpam-4744	290	12	louis	louis	PROPN
ejpam-4744	290	13	posa	posa	PROPN
ejpam-4744	290	14	.	.	PUNCT
ejpam-4744	291	1	the	the	DET
ejpam-4744	291	2	representation	representation	NOUN
ejpam-4744	291	3	of	of	ADP
ejpam-4744	291	4	a	a	DET
ejpam-4744	291	5	graphing	graphing	NOUN
ejpam-4744	291	6	by	by	ADP
ejpam-4744	291	7	set	set	VERB
ejpam-4744	291	8	intersections	intersection	NOUN
ejpam-4744	291	9	.	.	PUNCT
ejpam-4744	292	1	canadian	canadian	ADJ
ejpam-4744	292	2	journal	journal	PROPN
ejpam-4744	292	3	of	of	ADP
ejpam-4744	292	4	mathematics	mathematic	NOUN
ejpam-4744	292	5	,	,	PUNCT
ejpam-4744	292	6	18:106–112	18:106–112	NUM
ejpam-4744	292	7	,	,	PUNCT
ejpam-4744	292	8	1966	1966	NUM
ejpam-4744	292	9	.	.	PUNCT
ejpam-4744	293	1	[	[	X
ejpam-4744	293	2	2	2	NUM
ejpam-4744	293	3	]	]	X
ejpam-4744	293	4	frank	frank	PROPN
ejpam-4744	293	5	harary	harary	PROPN
ejpam-4744	293	6	.	.	PUNCT
ejpam-4744	294	1	graph	graph	NOUN
ejpam-4744	294	2	theory	theory	NOUN
ejpam-4744	294	3	.	.	PUNCT
ejpam-4744	295	1	addison	addison	PROPN
ejpam-4744	295	2	-	-	PUNCT
ejpam-4744	295	3	wesly	wesly	ADV
ejpam-4744	295	4	publishing	publish	VERB
ejpam-4744	295	5	company	company	NOUN
ejpam-4744	295	6	,	,	PUNCT
ejpam-4744	295	7	massachusetts	massachusetts	PROPN
ejpam-4744	295	8	,	,	PUNCT
ejpam-4744	295	9	1972	1972	NUM
ejpam-4744	295	10	.	.	PUNCT
ejpam-4744	296	1	[	[	X
ejpam-4744	296	2	3	3	NUM
ejpam-4744	296	3	]	]	PUNCT
ejpam-4744	296	4	palco	palco	X
ejpam-4744	296	5	j	j	PROPN
ejpam-4744	296	6	and	and	CCONJ
ejpam-4744	296	7	paluga	paluga	PROPN
ejpam-4744	296	8	r.	r.	PROPN
ejpam-4744	296	9	intersection	intersection	NOUN
ejpam-4744	296	10	number	number	NOUN
ejpam-4744	296	11	of	of	ADP
ejpam-4744	296	12	some	some	DET
ejpam-4744	296	13	graphs	graph	NOUN
ejpam-4744	296	14	.	.	PUNCT
ejpam-4744	297	1	the	the	DET
ejpam-4744	297	2	mindanawan	mindanawan	PROPN
ejpam-4744	297	3	journal	journal	PROPN
ejpam-4744	297	4	of	of	ADP
ejpam-4744	297	5	mathematics	mathematic	NOUN
ejpam-4744	297	6	,	,	PUNCT
ejpam-4744	297	7	3:63–75	3:63–75	NUM
ejpam-4744	297	8	,	,	PUNCT
ejpam-4744	297	9	2012	2012	NUM
ejpam-4744	297	10	.	.	PUNCT