id	sid	tid	token	lemma	pos
ejpam-3464	1	1	european	european	PROPN
ejpam-3464	1	2	journal	journal	PROPN
ejpam-3464	1	3	of	of	ADP
ejpam-3464	1	4	pure	pure	ADJ
ejpam-3464	1	5	and	and	CCONJ
ejpam-3464	1	6	applied	apply	VERB
ejpam-3464	1	7	mathematics	mathematic	NOUN
ejpam-3464	1	8	vol	vol	NOUN
ejpam-3464	1	9	.	.	PROPN
ejpam-3464	2	1	12	12	NUM
ejpam-3464	2	2	,	,	PUNCT
ejpam-3464	2	3	no	no	INTJ
ejpam-3464	2	4	.	.	NOUN
ejpam-3464	2	5	3	3	NUM
ejpam-3464	2	6	,	,	PUNCT
ejpam-3464	2	7	2019	2019	NUM
ejpam-3464	2	8	,	,	PUNCT
ejpam-3464	2	9	749	749	NUM
ejpam-3464	2	10	-	-	SYM
ejpam-3464	2	11	755	755	NUM
ejpam-3464	2	12	issn	issn	PROPN
ejpam-3464	2	13	1307	1307	NUM
ejpam-3464	2	14	-	-	SYM
ejpam-3464	2	15	5543	5543	NUM
ejpam-3464	2	16	–	–	PUNCT
ejpam-3464	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3464	2	18	published	publish	VERB
ejpam-3464	2	19	by	by	ADP
ejpam-3464	2	20	new	new	PROPN
ejpam-3464	2	21	york	york	PROPN
ejpam-3464	2	22	business	business	PROPN
ejpam-3464	2	23	global	global	ADJ
ejpam-3464	2	24	topologies	topology	NOUN
ejpam-3464	2	25	induced	induce	VERB
ejpam-3464	2	26	by	by	ADP
ejpam-3464	2	27	neighborhoods	neighborhood	NOUN
ejpam-3464	2	28	of	of	ADP
ejpam-3464	2	29	a	a	DET
ejpam-3464	2	30	graph	graph	NOUN
ejpam-3464	2	31	under	under	ADP
ejpam-3464	2	32	some	some	DET
ejpam-3464	2	33	binary	binary	ADJ
ejpam-3464	2	34	operation	operation	NOUN
ejpam-3464	2	35	anabel	anabel	PROPN
ejpam-3464	2	36	e.	e.	PROPN
ejpam-3464	2	37	gamorez1,∗	gamorez1,∗	PROPN
ejpam-3464	2	38	,	,	PUNCT
ejpam-3464	2	39	caen	caen	PROPN
ejpam-3464	2	40	grace	grace	PROPN
ejpam-3464	2	41	s.	s.	PROPN
ejpam-3464	2	42	nianga2,3	nianga2,3	PROPN
ejpam-3464	2	43	,	,	PUNCT
ejpam-3464	3	1	sergio	sergio	PROPN
ejpam-3464	3	2	r.	r.	PROPN
ejpam-3464	3	3	canoy	canoy	PROPN
ejpam-3464	3	4	jr.2,3	jr.2,3	PROPN
ejpam-3464	3	5	1	1	NUM
ejpam-3464	3	6	department	department	NOUN
ejpam-3464	3	7	of	of	ADP
ejpam-3464	3	8	mathematics	mathematic	NOUN
ejpam-3464	3	9	and	and	CCONJ
ejpam-3464	3	10	statistics	statistic	NOUN
ejpam-3464	3	11	,	,	PUNCT
ejpam-3464	3	12	college	college	NOUN
ejpam-3464	3	13	of	of	ADP
ejpam-3464	3	14	science	science	NOUN
ejpam-3464	3	15	and	and	CCONJ
ejpam-3464	3	16	mathematics	mathematic	NOUN
ejpam-3464	3	17	,	,	PUNCT
ejpam-3464	3	18	western	western	ADJ
ejpam-3464	3	19	mindanao	mindanao	PROPN
ejpam-3464	3	20	state	state	PROPN
ejpam-3464	3	21	university	university	PROPN
ejpam-3464	3	22	,	,	PUNCT
ejpam-3464	3	23	7000	7000	NUM
ejpam-3464	3	24	zamboanga	zamboanga	PROPN
ejpam-3464	3	25	city	city	PROPN
ejpam-3464	3	26	,	,	PUNCT
ejpam-3464	3	27	philippines	philippines	PROPN
ejpam-3464	3	28	2	2	NUM
ejpam-3464	3	29	department	department	NOUN
ejpam-3464	3	30	of	of	ADP
ejpam-3464	3	31	mathematics	mathematic	NOUN
ejpam-3464	3	32	and	and	CCONJ
ejpam-3464	3	33	statistics	statistic	NOUN
ejpam-3464	3	34	,	,	PUNCT
ejpam-3464	3	35	college	college	NOUN
ejpam-3464	3	36	of	of	ADP
ejpam-3464	3	37	science	science	NOUN
ejpam-3464	3	38	and	and	CCONJ
ejpam-3464	3	39	mathematics	mathematic	NOUN
ejpam-3464	3	40	,	,	PUNCT
ejpam-3464	3	41	mindanao	mindanao	PROPN
ejpam-3464	3	42	state	state	PROPN
ejpam-3464	3	43	university	university	PROPN
ejpam-3464	3	44	-	-	PUNCT
ejpam-3464	3	45	iligan	iligan	PROPN
ejpam-3464	3	46	institute	institute	PROPN
ejpam-3464	3	47	of	of	ADP
ejpam-3464	3	48	technology	technology	PROPN
ejpam-3464	3	49	,	,	PUNCT
ejpam-3464	3	50	9200	9200	NUM
ejpam-3464	3	51	iligan	iligan	ADJ
ejpam-3464	3	52	city	city	NOUN
ejpam-3464	3	53	,	,	PUNCT
ejpam-3464	3	54	philippines	philippine	NOUN
ejpam-3464	3	55	3	3	NUM
ejpam-3464	3	56	center	center	NOUN
ejpam-3464	3	57	for	for	ADP
ejpam-3464	3	58	graph	graph	NOUN
ejpam-3464	3	59	theory	theory	NOUN
ejpam-3464	3	60	,	,	PUNCT
ejpam-3464	3	61	algebra	algebra	NOUN
ejpam-3464	3	62	and	and	CCONJ
ejpam-3464	3	63	analysis	analysis	NOUN
ejpam-3464	3	64	,	,	PUNCT
ejpam-3464	3	65	premier	premier	PROPN
ejpam-3464	3	66	research	research	PROPN
ejpam-3464	3	67	institute	institute	PROPN
ejpam-3464	3	68	of	of	ADP
ejpam-3464	3	69	science	science	NOUN
ejpam-3464	3	70	and	and	CCONJ
ejpam-3464	3	71	mathematics	mathematic	NOUN
ejpam-3464	3	72	,	,	PUNCT
ejpam-3464	3	73	mindanao	mindanao	PROPN
ejpam-3464	3	74	state	state	PROPN
ejpam-3464	3	75	university	university	PROPN
ejpam-3464	3	76	-	-	PUNCT
ejpam-3464	3	77	iligan	iligan	PROPN
ejpam-3464	3	78	institute	institute	PROPN
ejpam-3464	3	79	of	of	ADP
ejpam-3464	3	80	technology	technology	PROPN
ejpam-3464	3	81	,	,	PUNCT
ejpam-3464	3	82	9200	9200	NUM
ejpam-3464	3	83	iligan	iligan	ADJ
ejpam-3464	3	84	city	city	NOUN
ejpam-3464	3	85	,	,	PUNCT
ejpam-3464	3	86	philippines	philippine	NOUN
ejpam-3464	3	87	abstract	abstract	ADJ
ejpam-3464	3	88	.	.	PUNCT
ejpam-3464	4	1	let	let	VERB
ejpam-3464	4	2	g	g	PROPN
ejpam-3464	4	3	=	=	SYM
ejpam-3464	4	4	(	(	PUNCT
ejpam-3464	4	5	v	v	NOUN
ejpam-3464	4	6	(	(	PUNCT
ejpam-3464	4	7	g	g	NOUN
ejpam-3464	4	8	)	)	PUNCT
ejpam-3464	4	9	,	,	PUNCT
ejpam-3464	4	10	e(g	e(g	PROPN
ejpam-3464	4	11	)	)	PUNCT
ejpam-3464	4	12	)	)	PUNCT
ejpam-3464	4	13	be	be	AUX
ejpam-3464	4	14	any	any	DET
ejpam-3464	4	15	undirected	undirected	ADJ
ejpam-3464	4	16	graph	graph	NOUN
ejpam-3464	4	17	.	.	PUNCT
ejpam-3464	5	1	then	then	ADV
ejpam-3464	5	2	g	g	PROPN
ejpam-3464	5	3	induces	induce	VERB
ejpam-3464	5	4	a	a	DET
ejpam-3464	5	5	topology	topology	NOUN
ejpam-3464	5	6	τg	τg	NOUN
ejpam-3464	5	7	on	on	ADP
ejpam-3464	5	8	v	v	ADP
ejpam-3464	5	9	(	(	PUNCT
ejpam-3464	5	10	g	g	NOUN
ejpam-3464	5	11	)	)	PUNCT
ejpam-3464	5	12	with	with	ADP
ejpam-3464	5	13	base	base	NOUN
ejpam-3464	5	14	consisting	consist	VERB
ejpam-3464	5	15	of	of	ADP
ejpam-3464	5	16	sets	set	NOUN
ejpam-3464	5	17	of	of	ADP
ejpam-3464	5	18	the	the	DET
ejpam-3464	5	19	form	form	NOUN
ejpam-3464	5	20	fg[a	fg[a	PROPN
ejpam-3464	5	21	]	]	X
ejpam-3464	5	22	=	=	SYM
ejpam-3464	5	23	v	v	X
ejpam-3464	5	24	(	(	PUNCT
ejpam-3464	5	25	g)\ng[a	g)\ng[a	PROPN
ejpam-3464	5	26	]	]	X
ejpam-3464	5	27	,	,	PUNCT
ejpam-3464	5	28	where	where	SCONJ
ejpam-3464	5	29	ng[a	ng[a	NOUN
ejpam-3464	5	30	]	]	X
ejpam-3464	5	31	=	=	PUNCT
ejpam-3464	5	32	a	a	DET
ejpam-3464	5	33	∪	∪	X
ejpam-3464	5	34	{	{	PUNCT
ejpam-3464	5	35	x	x	NOUN
ejpam-3464	5	36	:	:	PUNCT
ejpam-3464	5	37	xa	xa	PROPN
ejpam-3464	5	38	∈	∈	PROPN
ejpam-3464	5	39	e(g	e(g	PROPN
ejpam-3464	5	40	)	)	PUNCT
ejpam-3464	5	41	for	for	ADP
ejpam-3464	5	42	some	some	DET
ejpam-3464	5	43	a	a	DET
ejpam-3464	5	44	∈	∈	PROPN
ejpam-3464	5	45	a	a	PRON
ejpam-3464	5	46	}	}	PUNCT
ejpam-3464	5	47	and	and	CCONJ
ejpam-3464	5	48	a	a	DET
ejpam-3464	5	49	ranges	range	NOUN
ejpam-3464	5	50	over	over	ADP
ejpam-3464	5	51	all	all	DET
ejpam-3464	5	52	subsets	subset	NOUN
ejpam-3464	5	53	of	of	ADP
ejpam-3464	5	54	v	v	NOUN
ejpam-3464	5	55	(	(	PUNCT
ejpam-3464	5	56	g	g	NOUN
ejpam-3464	5	57	)	)	PUNCT
ejpam-3464	5	58	.	.	PUNCT
ejpam-3464	6	1	in	in	ADP
ejpam-3464	6	2	this	this	DET
ejpam-3464	6	3	paper	paper	NOUN
ejpam-3464	6	4	,	,	PUNCT
ejpam-3464	6	5	we	we	PRON
ejpam-3464	6	6	describe	describe	VERB
ejpam-3464	6	7	the	the	DET
ejpam-3464	6	8	topologies	topology	NOUN
ejpam-3464	6	9	induced	induce	VERB
ejpam-3464	6	10	by	by	ADP
ejpam-3464	6	11	the	the	DET
ejpam-3464	6	12	corona	corona	PROPN
ejpam-3464	6	13	,	,	PUNCT
ejpam-3464	6	14	edge	edge	NOUN
ejpam-3464	6	15	corona	corona	NOUN
ejpam-3464	6	16	,	,	PUNCT
ejpam-3464	6	17	disjunction	disjunction	NOUN
ejpam-3464	6	18	,	,	PUNCT
ejpam-3464	6	19	symmetric	symmetric	ADJ
ejpam-3464	6	20	difference	difference	NOUN
ejpam-3464	6	21	,	,	PUNCT
ejpam-3464	6	22	tensor	tensor	NOUN
ejpam-3464	6	23	product	product	NOUN
ejpam-3464	6	24	,	,	PUNCT
ejpam-3464	6	25	and	and	CCONJ
ejpam-3464	6	26	the	the	DET
ejpam-3464	6	27	strong	strong	ADJ
ejpam-3464	6	28	product	product	NOUN
ejpam-3464	6	29	of	of	ADP
ejpam-3464	6	30	two	two	NUM
ejpam-3464	6	31	graphs	graph	NOUN
ejpam-3464	6	32	by	by	ADP
ejpam-3464	6	33	determining	determine	VERB
ejpam-3464	6	34	the	the	DET
ejpam-3464	6	35	subbasic	subbasic	ADJ
ejpam-3464	6	36	open	open	ADJ
ejpam-3464	6	37	sets	set	NOUN
ejpam-3464	6	38	.	.	PUNCT
ejpam-3464	7	1	2010	2010	NUM
ejpam-3464	7	2	mathematics	mathematic	NOUN
ejpam-3464	7	3	subject	subject	NOUN
ejpam-3464	7	4	classifications	classification	NOUN
ejpam-3464	7	5	:	:	PUNCT
ejpam-3464	7	6	05c76	05c76	NUM
ejpam-3464	7	7	key	key	ADJ
ejpam-3464	7	8	words	word	NOUN
ejpam-3464	7	9	and	and	CCONJ
ejpam-3464	7	10	phrases	phrase	NOUN
ejpam-3464	7	11	:	:	PUNCT
ejpam-3464	7	12	topology	topology	NOUN
ejpam-3464	7	13	,	,	PUNCT
ejpam-3464	7	14	graph	graph	NOUN
ejpam-3464	7	15	,	,	PUNCT
ejpam-3464	7	16	edge	edge	NOUN
ejpam-3464	7	17	corona	corona	NOUN
ejpam-3464	7	18	,	,	PUNCT
ejpam-3464	7	19	disjunction	disjunction	NOUN
ejpam-3464	7	20	,	,	PUNCT
ejpam-3464	7	21	symmetric	symmetric	ADJ
ejpam-3464	7	22	difference	difference	NOUN
ejpam-3464	7	23	1	1	NUM
ejpam-3464	7	24	.	.	PUNCT
ejpam-3464	8	1	introduction	introduction	NOUN
ejpam-3464	8	2	let	let	VERB
ejpam-3464	8	3	g	g	NOUN
ejpam-3464	8	4	=	=	SYM
ejpam-3464	8	5	(	(	PUNCT
ejpam-3464	8	6	v	v	NOUN
ejpam-3464	8	7	(	(	PUNCT
ejpam-3464	8	8	g	g	NOUN
ejpam-3464	8	9	)	)	PUNCT
ejpam-3464	8	10	,	,	PUNCT
ejpam-3464	8	11	e(g	e(g	PROPN
ejpam-3464	8	12	)	)	PUNCT
ejpam-3464	8	13	)	)	PUNCT
ejpam-3464	8	14	be	be	AUX
ejpam-3464	8	15	any	any	DET
ejpam-3464	8	16	undirected	undirected	ADJ
ejpam-3464	8	17	(	(	PUNCT
ejpam-3464	8	18	simple	simple	ADJ
ejpam-3464	8	19	)	)	PUNCT
ejpam-3464	8	20	graph	graph	NOUN
ejpam-3464	8	21	and	and	CCONJ
ejpam-3464	8	22	let	let	VERB
ejpam-3464	8	23	v	v	NUM
ejpam-3464	8	24	∈	∈	PROPN
ejpam-3464	8	25	v	v	NOUN
ejpam-3464	8	26	(	(	PUNCT
ejpam-3464	8	27	g	g	NOUN
ejpam-3464	8	28	)	)	PUNCT
ejpam-3464	8	29	.	.	PUNCT
ejpam-3464	9	1	the	the	DET
ejpam-3464	9	2	open	open	ADJ
ejpam-3464	9	3	neighborhood	neighborhood	NOUN
ejpam-3464	9	4	of	of	ADP
ejpam-3464	9	5	v	v	NOUN
ejpam-3464	9	6	is	be	AUX
ejpam-3464	9	7	the	the	DET
ejpam-3464	9	8	set	set	NOUN
ejpam-3464	9	9	ng(v	ng(v	PUNCT
ejpam-3464	9	10	)	)	PUNCT
ejpam-3464	9	11	=	=	SYM
ejpam-3464	10	1	{	{	PUNCT
ejpam-3464	10	2	u	u	NOUN
ejpam-3464	10	3	∈	∈	PROPN
ejpam-3464	10	4	v	v	NOUN
ejpam-3464	10	5	(	(	PUNCT
ejpam-3464	10	6	g	g	NOUN
ejpam-3464	10	7	)	)	PUNCT
ejpam-3464	10	8	:	:	PUNCT
ejpam-3464	10	9	uv	uv	PROPN
ejpam-3464	10	10	∈	∈	PROPN
ejpam-3464	10	11	e(g	e(g	PROPN
ejpam-3464	10	12	)	)	PUNCT
ejpam-3464	10	13	}	}	PUNCT
ejpam-3464	10	14	and	and	CCONJ
ejpam-3464	10	15	its	its	PRON
ejpam-3464	10	16	closed	closed	ADJ
ejpam-3464	10	17	neighborhood	neighborhood	NOUN
ejpam-3464	10	18	is	be	AUX
ejpam-3464	10	19	ng[v	ng[v	ADJ
ejpam-3464	10	20	]	]	X
ejpam-3464	11	1	=	=	SYM
ejpam-3464	11	2	{	{	PUNCT
ejpam-3464	11	3	v	v	NOUN
ejpam-3464	11	4	}	}	PUNCT
ejpam-3464	11	5	∪	∪	ADJ
ejpam-3464	11	6	ng(v	ng(v	NOUN
ejpam-3464	11	7	)	)	PUNCT
ejpam-3464	11	8	.	.	PUNCT
ejpam-3464	12	1	if	if	SCONJ
ejpam-3464	12	2	a	a	DET
ejpam-3464	12	3	⊆	⊆	NUM
ejpam-3464	12	4	v	v	NOUN
ejpam-3464	12	5	(	(	PUNCT
ejpam-3464	12	6	g	g	NOUN
ejpam-3464	12	7	)	)	PUNCT
ejpam-3464	12	8	,	,	PUNCT
ejpam-3464	12	9	then	then	ADV
ejpam-3464	12	10	the	the	DET
ejpam-3464	12	11	open	open	ADJ
ejpam-3464	12	12	neighborhood	neighborhood	NOUN
ejpam-3464	12	13	of	of	ADP
ejpam-3464	12	14	a	a	PRON
ejpam-3464	12	15	is	be	AUX
ejpam-3464	12	16	the	the	DET
ejpam-3464	12	17	set	set	VERB
ejpam-3464	12	18	ng(a	ng(a	NOUN
ejpam-3464	12	19	)	)	PUNCT
ejpam-3464	12	20	=	=	SYM
ejpam-3464	12	21	∪v∈ang(v	∪v∈ang(v	PROPN
ejpam-3464	12	22	)	)	PUNCT
ejpam-3464	12	23	.	.	PUNCT
ejpam-3464	13	1	the	the	DET
ejpam-3464	13	2	closed	closed	ADJ
ejpam-3464	13	3	neighborhood	neighborhood	NOUN
ejpam-3464	13	4	of	of	ADP
ejpam-3464	13	5	a	a	PRON
ejpam-3464	13	6	is	be	AUX
ejpam-3464	13	7	ng[a	ng[a	ADJ
ejpam-3464	13	8	]	]	X
ejpam-3464	13	9	=	=	PUNCT
ejpam-3464	13	10	a	a	DET
ejpam-3464	13	11	∪	∪	ADJ
ejpam-3464	13	12	ng(a	ng(a	NOUN
ejpam-3464	13	13	)	)	PUNCT
ejpam-3464	13	14	.	.	PUNCT
ejpam-3464	14	1	clearly	clearly	ADV
ejpam-3464	14	2	,	,	PUNCT
ejpam-3464	14	3	if	if	SCONJ
ejpam-3464	14	4	a	a	PRON
ejpam-3464	14	5	=	=	X
ejpam-3464	14	6	{	{	PUNCT
ejpam-3464	14	7	v	v	NOUN
ejpam-3464	14	8	}	}	PUNCT
ejpam-3464	14	9	,	,	PUNCT
ejpam-3464	14	10	then	then	ADV
ejpam-3464	14	11	ng(a	ng(a	X
ejpam-3464	14	12	)	)	PUNCT
ejpam-3464	14	13	=	=	PUNCT
ejpam-3464	14	14	ng(v	ng(v	X
ejpam-3464	14	15	)	)	PUNCT
ejpam-3464	14	16	and	and	CCONJ
ejpam-3464	14	17	ng[a	ng[a	NOUN
ejpam-3464	14	18	]	]	X
ejpam-3464	14	19	=	=	PUNCT
ejpam-3464	15	1	ng[v	ng[v	X
ejpam-3464	15	2	]	]	PUNCT
ejpam-3464	15	3	.	.	PUNCT
ejpam-3464	16	1	the	the	DET
ejpam-3464	16	2	degree	degree	NOUN
ejpam-3464	16	3	of	of	ADP
ejpam-3464	16	4	v	v	NOUN
ejpam-3464	16	5	,	,	PUNCT
ejpam-3464	16	6	denoted	denote	VERB
ejpam-3464	16	7	by	by	ADP
ejpam-3464	16	8	degg(v	degg(v	PROPN
ejpam-3464	16	9	)	)	PUNCT
ejpam-3464	16	10	,	,	PUNCT
ejpam-3464	16	11	is	be	AUX
ejpam-3464	16	12	equal	equal	ADJ
ejpam-3464	16	13	to	to	ADP
ejpam-3464	16	14	|ng(v)|	|ng(v)|	NOUN
ejpam-3464	16	15	.	.	PUNCT
ejpam-3464	17	1	the	the	DET
ejpam-3464	17	2	distance	distance	NOUN
ejpam-3464	17	3	between	between	ADP
ejpam-3464	17	4	vertices	vertex	NOUN
ejpam-3464	17	5	u	u	NOUN
ejpam-3464	17	6	and	and	CCONJ
ejpam-3464	17	7	v	v	NOUN
ejpam-3464	17	8	of	of	ADP
ejpam-3464	17	9	g	g	NOUN
ejpam-3464	17	10	,	,	PUNCT
ejpam-3464	17	11	denoted	denote	VERB
ejpam-3464	17	12	by	by	ADP
ejpam-3464	17	13	dg(u	dg(u	NOUN
ejpam-3464	17	14	,	,	PUNCT
ejpam-3464	17	15	v	v	NOUN
ejpam-3464	17	16	)	)	PUNCT
ejpam-3464	17	17	,	,	PUNCT
ejpam-3464	17	18	is	be	AUX
ejpam-3464	17	19	the	the	DET
ejpam-3464	17	20	length	length	NOUN
ejpam-3464	17	21	of	of	ADP
ejpam-3464	17	22	a	a	DET
ejpam-3464	17	23	shortest	short	ADJ
ejpam-3464	17	24	path	path	NOUN
ejpam-3464	17	25	connecting	connect	VERB
ejpam-3464	17	26	u	u	NOUN
ejpam-3464	17	27	and	and	CCONJ
ejpam-3464	17	28	v	v	NOUN
ejpam-3464	17	29	(	(	PUNCT
ejpam-3464	17	30	or	or	CCONJ
ejpam-3464	17	31	length	length	NOUN
ejpam-3464	17	32	of	of	ADP
ejpam-3464	17	33	a	a	DET
ejpam-3464	17	34	shortest	short	ADJ
ejpam-3464	17	35	u	u	NOUN
ejpam-3464	17	36	-	-	NOUN
ejpam-3464	17	37	v	v	ADJ
ejpam-3464	17	38	path	path	NOUN
ejpam-3464	17	39	)	)	PUNCT
ejpam-3464	17	40	.	.	PUNCT
ejpam-3464	18	1	a	a	DET
ejpam-3464	18	2	way	way	NOUN
ejpam-3464	18	3	to	to	PART
ejpam-3464	18	4	relate	relate	VERB
ejpam-3464	18	5	graph	graph	NOUN
ejpam-3464	18	6	theory	theory	NOUN
ejpam-3464	18	7	to	to	ADP
ejpam-3464	18	8	topology	topology	NOUN
ejpam-3464	18	9	is	be	AUX
ejpam-3464	18	10	to	to	PART
ejpam-3464	18	11	find	find	VERB
ejpam-3464	18	12	a	a	DET
ejpam-3464	18	13	way	way	NOUN
ejpam-3464	18	14	of	of	ADP
ejpam-3464	18	15	constructing	construct	VERB
ejpam-3464	18	16	a	a	DET
ejpam-3464	18	17	topological	topological	ADJ
ejpam-3464	18	18	space	space	NOUN
ejpam-3464	18	19	from	from	ADP
ejpam-3464	18	20	a	a	DET
ejpam-3464	18	21	given	give	VERB
ejpam-3464	18	22	graph	graph	NOUN
ejpam-3464	18	23	or	or	CCONJ
ejpam-3464	18	24	devise	devise	VERB
ejpam-3464	18	25	a	a	DET
ejpam-3464	18	26	method	method	NOUN
ejpam-3464	18	27	of	of	ADP
ejpam-3464	18	28	generating	generate	VERB
ejpam-3464	18	29	a	a	DET
ejpam-3464	18	30	graph	graph	NOUN
ejpam-3464	18	31	from	from	ADP
ejpam-3464	18	32	a	a	DET
ejpam-3464	18	33	given	give	VERB
ejpam-3464	18	34	(	(	PUNCT
ejpam-3464	18	35	finite	finite	ADJ
ejpam-3464	18	36	)	)	PUNCT
ejpam-3464	18	37	topological	topological	ADJ
ejpam-3464	18	38	space	space	NOUN
ejpam-3464	18	39	.	.	PUNCT
ejpam-3464	19	1	in	in	ADP
ejpam-3464	19	2	1983	1983	NUM
ejpam-3464	19	3	,	,	PUNCT
ejpam-3464	19	4	gervacio	gervacio	NOUN
ejpam-3464	19	5	and	and	CCONJ
ejpam-3464	19	6	diesto	diesto	ADJ
ejpam-3464	19	7	in	in	ADP
ejpam-3464	19	8	[	[	X
ejpam-3464	19	9	1	1	NUM
ejpam-3464	19	10	]	]	PUNCT
ejpam-3464	19	11	introduced	introduce	VERB
ejpam-3464	19	12	a	a	DET
ejpam-3464	19	13	way	way	NOUN
ejpam-3464	19	14	of	of	ADP
ejpam-3464	19	15	constructing	construct	VERB
ejpam-3464	19	16	∗corresponding	∗corresponde	VERB
ejpam-3464	19	17	author	author	NOUN
ejpam-3464	19	18	.	.	PUNCT
ejpam-3464	20	1	doi	doi	NOUN
ejpam-3464	20	2	:	:	PUNCT
ejpam-3464	20	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3464	https://doi.org/10.29020/nybg.ejpam.v12i3.3464	PROPN
ejpam-3464	20	4	email	email	NOUN
ejpam-3464	20	5	addresses	address	NOUN
ejpam-3464	20	6	:	:	PUNCT
ejpam-3464	20	7	anabel.gamorez@gmail.com	anabel.gamorez@gmail.com	X
ejpam-3464	20	8	(	(	PUNCT
ejpam-3464	20	9	a.	a.	NOUN
ejpam-3464	20	10	gamorez	gamorez	PROPN
ejpam-3464	20	11	)	)	PUNCT
ejpam-3464	20	12	,	,	PUNCT
ejpam-3464	20	13	caengrace1997@gmail.com	caengrace1997@gmail.com	X
ejpam-3464	21	1	(	(	PUNCT
ejpam-3464	21	2	c.	c.	PROPN
ejpam-3464	21	3	nianga	nianga	PROPN
ejpam-3464	21	4	)	)	PUNCT
ejpam-3464	21	5	,	,	PUNCT
ejpam-3464	21	6	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-3464	21	7	(	(	PUNCT
ejpam-3464	21	8	s.	s.	PROPN
ejpam-3464	21	9	canoy	canoy	PROPN
ejpam-3464	21	10	jr	jr	PROPN
ejpam-3464	21	11	.	.	PUNCT
ejpam-3464	21	12	)	)	PUNCT
ejpam-3464	21	13	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3464	22	1	749	749	NUM
ejpam-3464	22	2	c	c	X
ejpam-3464	22	3	©	©	PROPN
ejpam-3464	22	4	2019	2019	NUM
ejpam-3464	22	5	ejpam	ejpam	NOUN
ejpam-3464	22	6	all	all	DET
ejpam-3464	22	7	rights	right	NOUN
ejpam-3464	22	8	reserved	reserve	VERB
ejpam-3464	22	9	.	.	PUNCT
ejpam-3464	23	1	a.	a.	NOUN
ejpam-3464	23	2	gamorez	gamorez	PROPN
ejpam-3464	23	3	,	,	PUNCT
ejpam-3464	23	4	c.	c.	PROPN
ejpam-3464	23	5	nianga	nianga	PROPN
ejpam-3464	23	6	,	,	PUNCT
ejpam-3464	23	7	s.	s.	PROPN
ejpam-3464	23	8	canoy	canoy	PROPN
ejpam-3464	23	9	/	/	SYM
ejpam-3464	23	10	eur	eur	PROPN
ejpam-3464	23	11	.	.	PUNCT
ejpam-3464	24	1	j.	j.	PROPN
ejpam-3464	24	2	pure	pure	PROPN
ejpam-3464	24	3	appl	appl	PROPN
ejpam-3464	24	4	.	.	PROPN
ejpam-3464	24	5	math	math	PROPN
ejpam-3464	24	6	,	,	PUNCT
ejpam-3464	24	7	12	12	NUM
ejpam-3464	24	8	(	(	PUNCT
ejpam-3464	24	9	3	3	NUM
ejpam-3464	24	10	)	)	PUNCT
ejpam-3464	24	11	(	(	PUNCT
ejpam-3464	24	12	2019	2019	NUM
ejpam-3464	24	13	)	)	PUNCT
ejpam-3464	24	14	,	,	PUNCT
ejpam-3464	24	15	749	749	NUM
ejpam-3464	24	16	-	-	SYM
ejpam-3464	24	17	755	755	NUM
ejpam-3464	24	18	750	750	NUM
ejpam-3464	24	19	a	a	DET
ejpam-3464	24	20	topological	topological	ADJ
ejpam-3464	24	21	space	space	NOUN
ejpam-3464	24	22	from	from	ADP
ejpam-3464	24	23	a	a	DET
ejpam-3464	24	24	given	give	VERB
ejpam-3464	24	25	undirected	undirected	ADJ
ejpam-3464	24	26	graph	graph	NOUN
ejpam-3464	24	27	.	.	PUNCT
ejpam-3464	25	1	specifically	specifically	ADV
ejpam-3464	25	2	,	,	PUNCT
ejpam-3464	25	3	they	they	PRON
ejpam-3464	25	4	used	use	VERB
ejpam-3464	25	5	the	the	DET
ejpam-3464	25	6	closed	closed	ADJ
ejpam-3464	25	7	neighborhood	neighborhood	NOUN
ejpam-3464	25	8	subsets	subset	NOUN
ejpam-3464	25	9	of	of	ADP
ejpam-3464	25	10	the	the	DET
ejpam-3464	25	11	vertex	vertex	NOUN
ejpam-3464	25	12	set	set	NOUN
ejpam-3464	25	13	of	of	ADP
ejpam-3464	25	14	a	a	DET
ejpam-3464	25	15	graph	graph	NOUN
ejpam-3464	25	16	to	to	PART
ejpam-3464	25	17	obtain	obtain	VERB
ejpam-3464	25	18	a	a	DET
ejpam-3464	25	19	base	base	NOUN
ejpam-3464	25	20	for	for	ADP
ejpam-3464	25	21	some	some	DET
ejpam-3464	25	22	topology	topology	NOUN
ejpam-3464	25	23	on	on	ADP
ejpam-3464	25	24	its	its	PRON
ejpam-3464	25	25	vertex	vertex	NOUN
ejpam-3464	25	26	set	set	NOUN
ejpam-3464	25	27	.	.	PUNCT
ejpam-3464	26	1	this	this	DET
ejpam-3464	26	2	type	type	NOUN
ejpam-3464	26	3	of	of	ADP
ejpam-3464	26	4	construction	construction	NOUN
ejpam-3464	26	5	of	of	ADP
ejpam-3464	26	6	a	a	DET
ejpam-3464	26	7	topological	topological	ADJ
ejpam-3464	26	8	space	space	NOUN
ejpam-3464	26	9	was	be	AUX
ejpam-3464	26	10	also	also	ADV
ejpam-3464	26	11	studied	study	VERB
ejpam-3464	26	12	by	by	ADP
ejpam-3464	26	13	guerrero	guerrero	PROPN
ejpam-3464	26	14	and	and	CCONJ
ejpam-3464	26	15	gervacio	gervacio	NOUN
ejpam-3464	26	16	in	in	ADP
ejpam-3464	26	17	[	[	X
ejpam-3464	26	18	2	2	NUM
ejpam-3464	26	19	]	]	PUNCT
ejpam-3464	26	20	where	where	SCONJ
ejpam-3464	26	21	they	they	PRON
ejpam-3464	26	22	characterized	characterize	VERB
ejpam-3464	26	23	those	those	DET
ejpam-3464	26	24	graphs	graph	NOUN
ejpam-3464	26	25	which	which	PRON
ejpam-3464	26	26	induce	induce	VERB
ejpam-3464	26	27	the	the	DET
ejpam-3464	26	28	indiscrete	indiscrete	ADJ
ejpam-3464	26	29	topology	topology	NOUN
ejpam-3464	26	30	and	and	CCONJ
ejpam-3464	26	31	the	the	DET
ejpam-3464	26	32	discrete	discrete	ADJ
ejpam-3464	26	33	topology	topology	NOUN
ejpam-3464	26	34	.	.	PUNCT
ejpam-3464	27	1	canoy	canoy	NOUN
ejpam-3464	27	2	and	and	CCONJ
ejpam-3464	27	3	lemence	lemence	ADV
ejpam-3464	27	4	in	in	ADP
ejpam-3464	27	5	[	[	X
ejpam-3464	27	6	4	4	NUM
ejpam-3464	27	7	]	]	PUNCT
ejpam-3464	27	8	investigated	investigate	VERB
ejpam-3464	27	9	further	far	ADV
ejpam-3464	27	10	this	this	DET
ejpam-3464	27	11	construction	construction	NOUN
ejpam-3464	27	12	and	and	CCONJ
ejpam-3464	27	13	obtained	obtain	VERB
ejpam-3464	27	14	a	a	DET
ejpam-3464	27	15	subbase	subbase	NOUN
ejpam-3464	27	16	for	for	ADP
ejpam-3464	27	17	the	the	DET
ejpam-3464	27	18	generated	generate	VERB
ejpam-3464	27	19	topology	topology	NOUN
ejpam-3464	27	20	.	.	PUNCT
ejpam-3464	28	1	using	use	VERB
ejpam-3464	28	2	this	this	DET
ejpam-3464	28	3	particular	particular	ADJ
ejpam-3464	28	4	result	result	NOUN
ejpam-3464	28	5	,	,	PUNCT
ejpam-3464	28	6	they	they	PRON
ejpam-3464	28	7	described	describe	VERB
ejpam-3464	28	8	the	the	DET
ejpam-3464	28	9	subbases	subbase	NOUN
ejpam-3464	28	10	of	of	ADP
ejpam-3464	28	11	the	the	DET
ejpam-3464	28	12	topologies	topology	NOUN
ejpam-3464	28	13	induced	induce	VERB
ejpam-3464	28	14	by	by	ADP
ejpam-3464	28	15	a	a	DET
ejpam-3464	28	16	path	path	NOUN
ejpam-3464	28	17	,	,	PUNCT
ejpam-3464	28	18	fan	fan	NOUN
ejpam-3464	28	19	,	,	PUNCT
ejpam-3464	28	20	complement	complement	NOUN
ejpam-3464	28	21	of	of	ADP
ejpam-3464	28	22	a	a	DET
ejpam-3464	28	23	graph	graph	NOUN
ejpam-3464	28	24	,	,	PUNCT
ejpam-3464	28	25	and	and	CCONJ
ejpam-3464	28	26	graphs	graph	NOUN
ejpam-3464	28	27	resulting	result	VERB
ejpam-3464	28	28	from	from	ADP
ejpam-3464	28	29	the	the	DET
ejpam-3464	28	30	join	join	NOUN
ejpam-3464	28	31	,	,	PUNCT
ejpam-3464	28	32	cartesian	cartesian	ADJ
ejpam-3464	28	33	product	product	NOUN
ejpam-3464	28	34	,	,	PUNCT
ejpam-3464	28	35	and	and	CCONJ
ejpam-3464	28	36	composition	composition	NOUN
ejpam-3464	28	37	of	of	ADP
ejpam-3464	28	38	two	two	NUM
ejpam-3464	28	39	graphs	graph	NOUN
ejpam-3464	28	40	.	.	PUNCT
ejpam-3464	29	1	recently	recently	ADV
ejpam-3464	29	2	,	,	PUNCT
ejpam-3464	29	3	nianga	nianga	ADV
ejpam-3464	29	4	and	and	CCONJ
ejpam-3464	29	5	canoy	canoy	ADJ
ejpam-3464	29	6	in	in	ADP
ejpam-3464	29	7	[	[	X
ejpam-3464	29	8	7	7	NUM
ejpam-3464	29	9	]	]	PUNCT
ejpam-3464	29	10	used	use	VERB
ejpam-3464	29	11	the	the	DET
ejpam-3464	29	12	hop	hop	NOUN
ejpam-3464	29	13	neighborhoods	neighborhood	NOUN
ejpam-3464	29	14	of	of	ADP
ejpam-3464	29	15	a	a	DET
ejpam-3464	29	16	graph	graph	NOUN
ejpam-3464	29	17	to	to	PART
ejpam-3464	29	18	define	define	VERB
ejpam-3464	29	19	a	a	DET
ejpam-3464	29	20	topology	topology	NOUN
ejpam-3464	29	21	on	on	ADP
ejpam-3464	29	22	its	its	PRON
ejpam-3464	29	23	vertex	vertex	NOUN
ejpam-3464	29	24	set	set	NOUN
ejpam-3464	29	25	.	.	PUNCT
ejpam-3464	30	1	in	in	ADP
ejpam-3464	30	2	[	[	X
ejpam-3464	30	3	8	8	NUM
ejpam-3464	30	4	]	]	PUNCT
ejpam-3464	30	5	they	they	PRON
ejpam-3464	30	6	describe	describe	VERB
ejpam-3464	30	7	the	the	DET
ejpam-3464	30	8	subbasic	subbasic	ADJ
ejpam-3464	30	9	open	open	ADJ
ejpam-3464	30	10	sets	set	NOUN
ejpam-3464	30	11	in	in	ADP
ejpam-3464	30	12	graphs	graph	NOUN
ejpam-3464	30	13	under	under	ADP
ejpam-3464	30	14	some	some	DET
ejpam-3464	30	15	unary	unary	ADJ
ejpam-3464	30	16	and	and	CCONJ
ejpam-3464	30	17	binary	binary	ADJ
ejpam-3464	30	18	operations	operation	NOUN
ejpam-3464	30	19	.	.	PUNCT
ejpam-3464	31	1	in	in	ADP
ejpam-3464	31	2	this	this	DET
ejpam-3464	31	3	paper	paper	NOUN
ejpam-3464	31	4	,	,	PUNCT
ejpam-3464	31	5	we	we	PRON
ejpam-3464	31	6	revisit	revisit	VERB
ejpam-3464	31	7	the	the	DET
ejpam-3464	31	8	construction	construction	NOUN
ejpam-3464	31	9	of	of	ADP
ejpam-3464	31	10	a	a	DET
ejpam-3464	31	11	topological	topological	ADJ
ejpam-3464	31	12	space	space	NOUN
ejpam-3464	31	13	given	give	VERB
ejpam-3464	31	14	in	in	ADP
ejpam-3464	31	15	[	[	X
ejpam-3464	31	16	1	1	NUM
ejpam-3464	31	17	]	]	PUNCT
ejpam-3464	31	18	and	and	CCONJ
ejpam-3464	31	19	describe	describe	VERB
ejpam-3464	31	20	the	the	DET
ejpam-3464	31	21	topologies	topology	NOUN
ejpam-3464	31	22	induced	induce	VERB
ejpam-3464	31	23	by	by	ADP
ejpam-3464	31	24	the	the	DET
ejpam-3464	31	25	corona	corona	PROPN
ejpam-3464	31	26	,	,	PUNCT
ejpam-3464	31	27	edge	edge	NOUN
ejpam-3464	31	28	corona	corona	NOUN
ejpam-3464	31	29	,	,	PUNCT
ejpam-3464	31	30	disjunction	disjunction	NOUN
ejpam-3464	31	31	,	,	PUNCT
ejpam-3464	31	32	symmetric	symmetric	ADJ
ejpam-3464	31	33	difference	difference	NOUN
ejpam-3464	31	34	,	,	PUNCT
ejpam-3464	31	35	tensor	tensor	NOUN
ejpam-3464	31	36	product	product	NOUN
ejpam-3464	31	37	,	,	PUNCT
ejpam-3464	31	38	and	and	CCONJ
ejpam-3464	31	39	the	the	DET
ejpam-3464	31	40	strong	strong	ADJ
ejpam-3464	31	41	product	product	NOUN
ejpam-3464	31	42	of	of	ADP
ejpam-3464	31	43	two	two	NUM
ejpam-3464	31	44	graphs	graph	NOUN
ejpam-3464	31	45	.	.	PUNCT
ejpam-3464	32	1	it	it	PRON
ejpam-3464	32	2	can	can	AUX
ejpam-3464	32	3	be	be	AUX
ejpam-3464	32	4	observed	observe	VERB
ejpam-3464	32	5	from	from	ADP
ejpam-3464	32	6	the	the	DET
ejpam-3464	32	7	subbasic	subbasic	ADJ
ejpam-3464	32	8	open	open	ADJ
ejpam-3464	32	9	sets	set	NOUN
ejpam-3464	32	10	that	that	PRON
ejpam-3464	32	11	,	,	PUNCT
ejpam-3464	32	12	generally	generally	ADV
ejpam-3464	32	13	,	,	PUNCT
ejpam-3464	32	14	it	it	PRON
ejpam-3464	32	15	’s	’	VERB
ejpam-3464	32	16	not	not	PART
ejpam-3464	32	17	easy	easy	ADJ
ejpam-3464	32	18	to	to	PART
ejpam-3464	32	19	determine	determine	VERB
ejpam-3464	32	20	the	the	DET
ejpam-3464	32	21	basic	basic	ADJ
ejpam-3464	32	22	open	open	ADJ
ejpam-3464	32	23	sets	set	NOUN
ejpam-3464	32	24	and	and	CCONJ
ejpam-3464	32	25	exact	exact	ADJ
ejpam-3464	32	26	topologies	topology	NOUN
ejpam-3464	32	27	these	these	DET
ejpam-3464	32	28	graphs	graph	NOUN
ejpam-3464	32	29	induced	induce	VERB
ejpam-3464	32	30	.	.	PUNCT
ejpam-3464	33	1	for	for	ADP
ejpam-3464	33	2	some	some	DET
ejpam-3464	33	3	basic	basic	ADJ
ejpam-3464	33	4	concepts	concept	NOUN
ejpam-3464	33	5	in	in	ADP
ejpam-3464	33	6	graph	graph	NOUN
ejpam-3464	33	7	theory	theory	NOUN
ejpam-3464	33	8	and	and	CCONJ
ejpam-3464	33	9	topology	topology	NOUN
ejpam-3464	33	10	,	,	PUNCT
ejpam-3464	33	11	we	we	PRON
ejpam-3464	33	12	refer	refer	VERB
ejpam-3464	33	13	readers	reader	NOUN
ejpam-3464	33	14	to	to	ADP
ejpam-3464	33	15	[	[	X
ejpam-3464	33	16	3	3	NUM
ejpam-3464	33	17	]	]	PUNCT
ejpam-3464	33	18	and	and	CCONJ
ejpam-3464	33	19	[	[	X
ejpam-3464	33	20	6	6	NUM
ejpam-3464	33	21	]	]	PUNCT
ejpam-3464	33	22	.	.	PUNCT
ejpam-3464	34	1	2	2	X
ejpam-3464	34	2	.	.	X
ejpam-3464	34	3	results	result	NOUN
ejpam-3464	34	4	if	if	SCONJ
ejpam-3464	34	5	a	a	DET
ejpam-3464	34	6	⊆	⊆	NUM
ejpam-3464	34	7	v	v	NOUN
ejpam-3464	34	8	(	(	PUNCT
ejpam-3464	34	9	g	g	NOUN
ejpam-3464	34	10	)	)	PUNCT
ejpam-3464	34	11	,	,	PUNCT
ejpam-3464	34	12	we	we	PRON
ejpam-3464	34	13	denote	denote	VERB
ejpam-3464	34	14	by	by	ADP
ejpam-3464	34	15	fg(a	fg(a	NOUN
ejpam-3464	34	16	)	)	PUNCT
ejpam-3464	34	17	and	and	CCONJ
ejpam-3464	34	18	fg[a	fg[a	PROPN
ejpam-3464	34	19	]	]	X
ejpam-3464	35	1	the	the	DET
ejpam-3464	35	2	complements	complement	NOUN
ejpam-3464	35	3	of	of	ADP
ejpam-3464	35	4	ng(a	ng(a	NOUN
ejpam-3464	35	5	)	)	PUNCT
ejpam-3464	35	6	and	and	CCONJ
ejpam-3464	35	7	ng[a	ng[a	NOUN
ejpam-3464	35	8	]	]	X
ejpam-3464	35	9	,	,	PUNCT
ejpam-3464	35	10	respectively	respectively	ADV
ejpam-3464	35	11	,	,	PUNCT
ejpam-3464	35	12	that	that	ADV
ejpam-3464	35	13	is	is	ADV
ejpam-3464	35	14	,	,	PUNCT
ejpam-3464	35	15	fg(a	fg(a	PUNCT
ejpam-3464	35	16	)	)	PUNCT
ejpam-3464	35	17	=	=	SYM
ejpam-3464	35	18	v	v	X
ejpam-3464	35	19	(	(	PUNCT
ejpam-3464	35	20	g	g	NOUN
ejpam-3464	35	21	)	)	PUNCT
ejpam-3464	35	22	\	\	NOUN
ejpam-3464	35	23	ng(a	ng(a	NOUN
ejpam-3464	35	24	)	)	PUNCT
ejpam-3464	35	25	and	and	CCONJ
ejpam-3464	35	26	fg[a	fg[a	PROPN
ejpam-3464	35	27	]	]	X
ejpam-3464	36	1	=	=	SYM
ejpam-3464	36	2	v	v	X
ejpam-3464	36	3	(	(	PUNCT
ejpam-3464	36	4	g	g	NOUN
ejpam-3464	36	5	)	)	PUNCT
ejpam-3464	36	6	\	\	PUNCT
ejpam-3464	36	7	ng[a	ng[a	NOUN
ejpam-3464	36	8	]	]	X
ejpam-3464	36	9	.	.	PUNCT
ejpam-3464	37	1	if	if	SCONJ
ejpam-3464	37	2	a	a	DET
ejpam-3464	37	3	=	=	X
ejpam-3464	37	4	{	{	PUNCT
ejpam-3464	37	5	v	v	NOUN
ejpam-3464	37	6	}	}	PUNCT
ejpam-3464	37	7	,	,	PUNCT
ejpam-3464	37	8	then	then	ADV
ejpam-3464	37	9	we	we	PRON
ejpam-3464	37	10	write	write	VERB
ejpam-3464	37	11	fg(a	fg(a	PUNCT
ejpam-3464	37	12	)	)	PUNCT
ejpam-3464	37	13	=	=	PUNCT
ejpam-3464	37	14	fg(v	fg(v	X
ejpam-3464	37	15	)	)	PUNCT
ejpam-3464	37	16	and	and	CCONJ
ejpam-3464	37	17	fg[a	fg[a	PROPN
ejpam-3464	37	18	]	]	X
ejpam-3464	38	1	=	=	PUNCT
ejpam-3464	38	2	fg[v	fg[v	PROPN
ejpam-3464	38	3	]	]	PUNCT
ejpam-3464	38	4	.	.	PUNCT
ejpam-3464	39	1	clearly	clearly	ADV
ejpam-3464	39	2	,	,	PUNCT
ejpam-3464	39	3	fg(v	fg(v	X
ejpam-3464	39	4	)	)	PUNCT
ejpam-3464	39	5	=	=	SYM
ejpam-3464	39	6	fg[v	fg[v	PROPN
ejpam-3464	39	7	]	]	PUNCT
ejpam-3464	39	8	∪	∪	X
ejpam-3464	39	9	{	{	PUNCT
ejpam-3464	39	10	v	v	NOUN
ejpam-3464	39	11	}	}	PUNCT
ejpam-3464	39	12	.	.	PUNCT
ejpam-3464	40	1	the	the	DET
ejpam-3464	40	2	first	first	ADJ
ejpam-3464	40	3	two	two	NUM
ejpam-3464	40	4	results	result	NOUN
ejpam-3464	40	5	are	be	AUX
ejpam-3464	40	6	found	find	VERB
ejpam-3464	40	7	in	in	ADP
ejpam-3464	40	8	[	[	X
ejpam-3464	40	9	1	1	NUM
ejpam-3464	40	10	]	]	PUNCT
ejpam-3464	40	11	and	and	CCONJ
ejpam-3464	40	12	[	[	X
ejpam-3464	40	13	5	5	NUM
ejpam-3464	40	14	]	]	PUNCT
ejpam-3464	40	15	,	,	PUNCT
ejpam-3464	40	16	respectively	respectively	ADV
ejpam-3464	40	17	,	,	PUNCT
ejpam-3464	40	18	and	and	CCONJ
ejpam-3464	40	19	play	play	VERB
ejpam-3464	40	20	vital	vital	ADJ
ejpam-3464	40	21	roles	role	NOUN
ejpam-3464	40	22	in	in	ADP
ejpam-3464	40	23	the	the	DET
ejpam-3464	40	24	next	next	ADJ
ejpam-3464	40	25	results	result	NOUN
ejpam-3464	40	26	.	.	PUNCT
ejpam-3464	41	1	theorem	theorem	NOUN
ejpam-3464	41	2	1	1	NUM
ejpam-3464	41	3	.	.	PUNCT
ejpam-3464	42	1	let	let	VERB
ejpam-3464	42	2	g	g	PRON
ejpam-3464	42	3	be	be	AUX
ejpam-3464	42	4	a	a	DET
ejpam-3464	42	5	graph	graph	NOUN
ejpam-3464	42	6	.	.	PUNCT
ejpam-3464	43	1	then	then	ADV
ejpam-3464	43	2	bg	bg	PROPN
ejpam-3464	43	3	=	=	PUNCT
ejpam-3464	43	4	{	{	PUNCT
ejpam-3464	43	5	fg[a	fg[a	PROPN
ejpam-3464	43	6	]	]	X
ejpam-3464	43	7	:	:	PUNCT
ejpam-3464	43	8	a	a	DET
ejpam-3464	43	9	⊆	⊆	NUM
ejpam-3464	43	10	v	v	NOUN
ejpam-3464	43	11	(	(	PUNCT
ejpam-3464	43	12	g	g	NOUN
ejpam-3464	43	13	)	)	PUNCT
ejpam-3464	43	14	}	}	PUNCT
ejpam-3464	43	15	is	be	AUX
ejpam-3464	43	16	a	a	DET
ejpam-3464	43	17	base	base	NOUN
ejpam-3464	43	18	for	for	ADP
ejpam-3464	43	19	some	some	DET
ejpam-3464	43	20	topology	topology	NOUN
ejpam-3464	43	21	on	on	ADP
ejpam-3464	43	22	v	v	ADP
ejpam-3464	43	23	(	(	PUNCT
ejpam-3464	43	24	g	g	NOUN
ejpam-3464	43	25	)	)	PUNCT
ejpam-3464	43	26	.	.	PUNCT
ejpam-3464	44	1	throughout	throughout	ADP
ejpam-3464	44	2	this	this	DET
ejpam-3464	44	3	paper	paper	NOUN
ejpam-3464	44	4	,	,	PUNCT
ejpam-3464	44	5	we	we	PRON
ejpam-3464	44	6	denote	denote	VERB
ejpam-3464	44	7	by	by	ADP
ejpam-3464	44	8	τg	τg	NUM
ejpam-3464	44	9	the	the	DET
ejpam-3464	44	10	topology	topology	NOUN
ejpam-3464	44	11	on	on	ADP
ejpam-3464	44	12	v	v	ADP
ejpam-3464	44	13	(	(	PUNCT
ejpam-3464	44	14	g	g	NOUN
ejpam-3464	44	15	)	)	PUNCT
ejpam-3464	44	16	generated	generate	VERB
ejpam-3464	44	17	by	by	ADP
ejpam-3464	44	18	the	the	DET
ejpam-3464	44	19	family	family	NOUN
ejpam-3464	44	20	bg	bg	PROPN
ejpam-3464	44	21	in	in	ADP
ejpam-3464	44	22	theorem	theorem	NOUN
ejpam-3464	44	23	1	1	NUM
ejpam-3464	44	24	.	.	PUNCT
ejpam-3464	45	1	this	this	DET
ejpam-3464	45	2	topology	topology	NOUN
ejpam-3464	45	3	is	be	AUX
ejpam-3464	45	4	also	also	ADV
ejpam-3464	45	5	called	call	VERB
ejpam-3464	45	6	the	the	DET
ejpam-3464	45	7	topology	topology	NOUN
ejpam-3464	45	8	induced	induce	VERB
ejpam-3464	45	9	by	by	ADP
ejpam-3464	45	10	g.	g.	PROPN
ejpam-3464	45	11	theorem	theorem	PROPN
ejpam-3464	45	12	2	2	X
ejpam-3464	45	13	.	.	PUNCT
ejpam-3464	46	1	let	let	VERB
ejpam-3464	46	2	g	g	PRON
ejpam-3464	46	3	be	be	AUX
ejpam-3464	46	4	a	a	DET
ejpam-3464	46	5	graph	graph	NOUN
ejpam-3464	46	6	.	.	PUNCT
ejpam-3464	47	1	then	then	ADV
ejpam-3464	47	2	sg	sg	ADV
ejpam-3464	47	3	=	=	SYM
ejpam-3464	47	4	{	{	PUNCT
ejpam-3464	47	5	fg[v	fg[v	PROPN
ejpam-3464	47	6	]	]	X
ejpam-3464	47	7	:	:	PUNCT
ejpam-3464	47	8	v	v	X
ejpam-3464	47	9	∈	∈	PROPN
ejpam-3464	47	10	v	v	NOUN
ejpam-3464	47	11	(	(	PUNCT
ejpam-3464	47	12	g	g	NOUN
ejpam-3464	47	13	)	)	PUNCT
ejpam-3464	47	14	}	}	PUNCT
ejpam-3464	47	15	is	be	AUX
ejpam-3464	47	16	a	a	DET
ejpam-3464	47	17	subbase	subbase	NOUN
ejpam-3464	47	18	for	for	ADP
ejpam-3464	47	19	τg	τg	PROPN
ejpam-3464	47	20	.	.	PUNCT
ejpam-3464	48	1	definition	definition	NOUN
ejpam-3464	48	2	1	1	NUM
ejpam-3464	48	3	.	.	PUNCT
ejpam-3464	49	1	[	[	X
ejpam-3464	49	2	3	3	X
ejpam-3464	49	3	]	]	PUNCT
ejpam-3464	49	4	the	the	DET
ejpam-3464	49	5	corona	corona	NOUN
ejpam-3464	49	6	g	g	PROPN
ejpam-3464	49	7	◦	◦	NOUN
ejpam-3464	49	8	h	h	NOUN
ejpam-3464	49	9	of	of	ADP
ejpam-3464	49	10	graphs	graph	NOUN
ejpam-3464	49	11	g	g	NOUN
ejpam-3464	49	12	and	and	CCONJ
ejpam-3464	49	13	h	h	NOUN
ejpam-3464	49	14	is	be	AUX
ejpam-3464	49	15	the	the	DET
ejpam-3464	49	16	graph	graph	NOUN
ejpam-3464	49	17	obtained	obtain	VERB
ejpam-3464	49	18	by	by	ADP
ejpam-3464	49	19	taking	take	VERB
ejpam-3464	49	20	one	one	NUM
ejpam-3464	49	21	copy	copy	NOUN
ejpam-3464	49	22	of	of	ADP
ejpam-3464	49	23	g	g	NOUN
ejpam-3464	49	24	and	and	CCONJ
ejpam-3464	49	25	|	|	ADV
ejpam-3464	49	26	v	v	NOUN
ejpam-3464	49	27	(	(	PUNCT
ejpam-3464	49	28	g	g	NOUN
ejpam-3464	49	29	)	)	PUNCT
ejpam-3464	49	30	|	|	ADV
ejpam-3464	49	31	copies	copy	VERB
ejpam-3464	49	32	h	h	NOUN
ejpam-3464	49	33	and	and	CCONJ
ejpam-3464	49	34	then	then	ADV
ejpam-3464	49	35	forming	form	VERB
ejpam-3464	49	36	the	the	DET
ejpam-3464	49	37	join	join	NOUN
ejpam-3464	49	38	<	<	X
ejpam-3464	49	39	v	v	X
ejpam-3464	49	40	>	>	X
ejpam-3464	49	41	+	+	PROPN
ejpam-3464	49	42	hv	hv	NOUN
ejpam-3464	49	43	=	=	SYM
ejpam-3464	49	44	v+hv	v+hv	PROPN
ejpam-3464	49	45	for	for	ADP
ejpam-3464	49	46	each	each	DET
ejpam-3464	49	47	v	v	NUM
ejpam-3464	49	48	∈	∈	PROPN
ejpam-3464	49	49	v	v	NOUN
ejpam-3464	49	50	(	(	PUNCT
ejpam-3464	49	51	g	g	NOUN
ejpam-3464	49	52	)	)	PUNCT
ejpam-3464	49	53	,	,	PUNCT
ejpam-3464	49	54	where	where	SCONJ
ejpam-3464	49	55	hv	hv	PROPN
ejpam-3464	49	56	is	be	AUX
ejpam-3464	49	57	a	a	DET
ejpam-3464	49	58	copy	copy	NOUN
ejpam-3464	49	59	of	of	ADP
ejpam-3464	49	60	h	h	NOUN
ejpam-3464	49	61	corresponding	correspond	VERB
ejpam-3464	49	62	to	to	ADP
ejpam-3464	49	63	the	the	DET
ejpam-3464	49	64	vertex	vertex	NOUN
ejpam-3464	50	1	v.	v.	ADP
ejpam-3464	50	2	we	we	PRON
ejpam-3464	50	3	now	now	ADV
ejpam-3464	50	4	describe	describe	VERB
ejpam-3464	50	5	the	the	DET
ejpam-3464	50	6	subbasic	subbasic	ADJ
ejpam-3464	50	7	open	open	ADJ
ejpam-3464	50	8	sets	set	NOUN
ejpam-3464	50	9	in	in	ADP
ejpam-3464	50	10	the	the	DET
ejpam-3464	50	11	space	space	NOUN
ejpam-3464	50	12	(	(	PUNCT
ejpam-3464	50	13	v	v	NOUN
ejpam-3464	50	14	(	(	PUNCT
ejpam-3464	50	15	g	g	PROPN
ejpam-3464	50	16	◦	◦	NOUN
ejpam-3464	50	17	h	h	NOUN
ejpam-3464	50	18	)	)	PUNCT
ejpam-3464	50	19	,	,	PUNCT
ejpam-3464	50	20	τg	τg	NUM
ejpam-3464	50	21	◦	◦	NOUN
ejpam-3464	50	22	h	h	NOUN
ejpam-3464	50	23	)	)	PUNCT
ejpam-3464	50	24	.	.	PUNCT
ejpam-3464	51	1	theorem	theorem	NOUN
ejpam-3464	51	2	3	3	X
ejpam-3464	51	3	.	.	PUNCT
ejpam-3464	52	1	let	let	VERB
ejpam-3464	52	2	k	k	NOUN
ejpam-3464	52	3	=	=	PUNCT
ejpam-3464	52	4	g	g	PROPN
ejpam-3464	52	5	◦	◦	NOUN
ejpam-3464	52	6	h	h	NOUN
ejpam-3464	52	7	=	=	SYM
ejpam-3464	52	8	(	(	PUNCT
ejpam-3464	52	9	v	v	NOUN
ejpam-3464	52	10	(	(	PUNCT
ejpam-3464	52	11	k	k	NOUN
ejpam-3464	52	12	)	)	PUNCT
ejpam-3464	52	13	,	,	PUNCT
ejpam-3464	52	14	e(k	e(k	NOUN
ejpam-3464	52	15	)	)	PUNCT
ejpam-3464	52	16	)	)	PUNCT
ejpam-3464	52	17	,	,	PUNCT
ejpam-3464	52	18	and	and	CCONJ
ejpam-3464	52	19	let	let	VERB
ejpam-3464	52	20	a	a	DET
ejpam-3464	52	21	∈	∈	PROPN
ejpam-3464	52	22	v	v	NOUN
ejpam-3464	52	23	(	(	PUNCT
ejpam-3464	52	24	k	k	NOUN
ejpam-3464	52	25	)	)	PUNCT
ejpam-3464	52	26	.	.	PUNCT
ejpam-3464	53	1	(	(	PUNCT
ejpam-3464	53	2	i	i	NOUN
ejpam-3464	53	3	)	)	PUNCT
ejpam-3464	53	4	if	if	SCONJ
ejpam-3464	53	5	a	a	DET
ejpam-3464	53	6	∈	∈	PROPN
ejpam-3464	53	7	v	v	NOUN
ejpam-3464	53	8	(	(	PUNCT
ejpam-3464	53	9	g	g	NOUN
ejpam-3464	53	10	)	)	PUNCT
ejpam-3464	53	11	,	,	PUNCT
ejpam-3464	53	12	then	then	ADV
ejpam-3464	53	13	fk	fk	INTJ
ejpam-3464	54	1	[	[	X
ejpam-3464	54	2	a	a	X
ejpam-3464	54	3	]	]	X
ejpam-3464	54	4	=	=	PUNCT
ejpam-3464	54	5	fg[a	fg[a	PROPN
ejpam-3464	54	6	]	]	PUNCT
ejpam-3464	54	7	∪	∪	ADP
ejpam-3464	54	8			NOUN
ejpam-3464	54	9	⋃	⋃	NOUN
ejpam-3464	54	10	u∈v	u∈v	NOUN
ejpam-3464	54	11	(	(	PUNCT
ejpam-3464	54	12	g)\{a	g)\{a	PROPN
ejpam-3464	54	13	}	}	PUNCT
ejpam-3464	54	14	v	v	NOUN
ejpam-3464	54	15	(	(	PUNCT
ejpam-3464	54	16	hu	hu	NOUN
ejpam-3464	54	17	)	)	PUNCT
ejpam-3464	54	18			NOUN
ejpam-3464	54	19	.	.	PUNCT
ejpam-3464	55	1	a.	a.	NOUN
ejpam-3464	55	2	gamorez	gamorez	PROPN
ejpam-3464	55	3	,	,	PUNCT
ejpam-3464	55	4	c.	c.	PROPN
ejpam-3464	55	5	nianga	nianga	PROPN
ejpam-3464	55	6	,	,	PUNCT
ejpam-3464	55	7	s.	s.	PROPN
ejpam-3464	55	8	canoy	canoy	PROPN
ejpam-3464	55	9	/	/	SYM
ejpam-3464	55	10	eur	eur	PROPN
ejpam-3464	55	11	.	.	PUNCT
ejpam-3464	56	1	j.	j.	PROPN
ejpam-3464	56	2	pure	pure	PROPN
ejpam-3464	56	3	appl	appl	PROPN
ejpam-3464	56	4	.	.	PROPN
ejpam-3464	56	5	math	math	PROPN
ejpam-3464	56	6	,	,	PUNCT
ejpam-3464	56	7	12	12	NUM
ejpam-3464	56	8	(	(	PUNCT
ejpam-3464	56	9	3	3	NUM
ejpam-3464	56	10	)	)	PUNCT
ejpam-3464	56	11	(	(	PUNCT
ejpam-3464	56	12	2019	2019	NUM
ejpam-3464	56	13	)	)	PUNCT
ejpam-3464	56	14	,	,	PUNCT
ejpam-3464	56	15	749	749	NUM
ejpam-3464	56	16	-	-	SYM
ejpam-3464	56	17	755	755	NUM
ejpam-3464	56	18	751	751	NUM
ejpam-3464	56	19	(	(	PUNCT
ejpam-3464	56	20	ii	ii	NOUN
ejpam-3464	56	21	)	)	PUNCT
ejpam-3464	56	22	if	if	SCONJ
ejpam-3464	56	23	a	a	DET
ejpam-3464	56	24	∈	∈	PROPN
ejpam-3464	56	25	v	v	NOUN
ejpam-3464	56	26	(	(	PUNCT
ejpam-3464	56	27	hw	hw	NOUN
ejpam-3464	56	28	)	)	PUNCT
ejpam-3464	56	29	for	for	ADP
ejpam-3464	56	30	some	some	DET
ejpam-3464	56	31	w	w	PROPN
ejpam-3464	56	32	∈	∈	PROPN
ejpam-3464	56	33	v	v	ADP
ejpam-3464	56	34	(	(	PUNCT
ejpam-3464	56	35	g	g	NOUN
ejpam-3464	56	36	)	)	PUNCT
ejpam-3464	56	37	,	,	PUNCT
ejpam-3464	56	38	then	then	ADV
ejpam-3464	56	39	fk	fk	INTJ
ejpam-3464	57	1	[	[	X
ejpam-3464	57	2	a	a	X
ejpam-3464	57	3	]	]	X
ejpam-3464	57	4	=	=	PUNCT
ejpam-3464	58	1	[	[	X
ejpam-3464	58	2	v	v	X
ejpam-3464	58	3	(	(	PUNCT
ejpam-3464	58	4	g)\{w	g)\{w	NOUN
ejpam-3464	58	5	}	}	PUNCT
ejpam-3464	58	6	]	]	PUNCT
ejpam-3464	58	7	∪	∪	ADP
ejpam-3464	58	8	fhw	fhw	PROPN
ejpam-3464	58	9	[	[	X
ejpam-3464	58	10	a	a	X
ejpam-3464	58	11	]	]	X
ejpam-3464	58	12	∪	∪	ADP
ejpam-3464	58	13			NOUN
ejpam-3464	58	14	⋃	⋃	NOUN
ejpam-3464	58	15	z∈v	z∈v	NOUN
ejpam-3464	58	16	(	(	PUNCT
ejpam-3464	58	17	g)\{w	g)\{w	NOUN
ejpam-3464	58	18	}	}	PUNCT
ejpam-3464	58	19	v	v	NOUN
ejpam-3464	58	20	(	(	PUNCT
ejpam-3464	58	21	hz	hz	NOUN
ejpam-3464	58	22	)	)	PUNCT
ejpam-3464	58	23			NOUN
ejpam-3464	58	24	proof	proof	NOUN
ejpam-3464	58	25	.	.	PUNCT
ejpam-3464	59	1	(	(	PUNCT
ejpam-3464	59	2	i	i	NOUN
ejpam-3464	59	3	)	)	PUNCT
ejpam-3464	59	4	suppose	suppose	VERB
ejpam-3464	59	5	a	a	DET
ejpam-3464	59	6	∈	∈	PROPN
ejpam-3464	59	7	v	v	NOUN
ejpam-3464	59	8	(	(	PUNCT
ejpam-3464	59	9	g	g	NOUN
ejpam-3464	59	10	)	)	PUNCT
ejpam-3464	59	11	.	.	PUNCT
ejpam-3464	60	1	then	then	ADV
ejpam-3464	60	2	nk	nk	PROPN
ejpam-3464	61	1	[	[	X
ejpam-3464	61	2	a	a	X
ejpam-3464	61	3	]	]	X
ejpam-3464	61	4	=	=	SYM
ejpam-3464	61	5	ng[a	ng[a	NOUN
ejpam-3464	61	6	]	]	X
ejpam-3464	61	7	∪	∪	X
ejpam-3464	61	8	v	v	NOUN
ejpam-3464	61	9	(	(	PUNCT
ejpam-3464	61	10	ha	ha	INTJ
ejpam-3464	61	11	)	)	PUNCT
ejpam-3464	61	12	by	by	ADP
ejpam-3464	61	13	definition	definition	NOUN
ejpam-3464	61	14	1	1	NUM
ejpam-3464	61	15	.	.	PUNCT
ejpam-3464	62	1	hence	hence	ADV
ejpam-3464	62	2	,	,	PUNCT
ejpam-3464	62	3	fk	fk	INTJ
ejpam-3464	63	1	[	[	X
ejpam-3464	63	2	a	a	X
ejpam-3464	63	3	]	]	X
ejpam-3464	63	4	=	=	SYM
ejpam-3464	63	5	v	v	X
ejpam-3464	63	6	(	(	PUNCT
ejpam-3464	63	7	k)\(ng[a	k)\(ng[a	NOUN
ejpam-3464	63	8	]	]	X
ejpam-3464	63	9	∪	∪	X
ejpam-3464	63	10	v	v	NOUN
ejpam-3464	63	11	(	(	PUNCT
ejpam-3464	63	12	ha	ha	INTJ
ejpam-3464	63	13	)	)	PUNCT
ejpam-3464	63	14	)	)	PUNCT
ejpam-3464	64	1	=	=	SYM
ejpam-3464	64	2	fg[a	fg[a	PROPN
ejpam-3464	64	3	]	]	PUNCT
ejpam-3464	64	4	∪	∪	ADP
ejpam-3464	64	5			NOUN
ejpam-3464	64	6	⋃	⋃	NOUN
ejpam-3464	64	7	u∈v	u∈v	NOUN
ejpam-3464	64	8	(	(	PUNCT
ejpam-3464	64	9	g)\{a	g)\{a	PROPN
ejpam-3464	64	10	}	}	PUNCT
ejpam-3464	64	11	v	v	NOUN
ejpam-3464	64	12	(	(	PUNCT
ejpam-3464	64	13	hu	hu	NOUN
ejpam-3464	64	14	)	)	PUNCT
ejpam-3464	64	15			NOUN
ejpam-3464	64	16	.	.	PUNCT
ejpam-3464	65	1	(	(	PUNCT
ejpam-3464	65	2	ii	ii	NOUN
ejpam-3464	65	3	)	)	PUNCT
ejpam-3464	65	4	suppose	suppose	VERB
ejpam-3464	65	5	a	a	DET
ejpam-3464	65	6	∈	∈	PROPN
ejpam-3464	65	7	v	v	NOUN
ejpam-3464	65	8	(	(	PUNCT
ejpam-3464	65	9	hw	hw	NOUN
ejpam-3464	65	10	)	)	PUNCT
ejpam-3464	65	11	for	for	ADP
ejpam-3464	65	12	some	some	DET
ejpam-3464	65	13	w	w	PROPN
ejpam-3464	65	14	∈	∈	PROPN
ejpam-3464	65	15	v	v	ADP
ejpam-3464	65	16	(	(	PUNCT
ejpam-3464	65	17	g	g	NOUN
ejpam-3464	65	18	)	)	PUNCT
ejpam-3464	65	19	.	.	PUNCT
ejpam-3464	66	1	then	then	ADV
ejpam-3464	66	2	nk	nk	PROPN
ejpam-3464	67	1	[	[	X
ejpam-3464	67	2	a	a	X
ejpam-3464	67	3	]	]	X
ejpam-3464	67	4	=	=	X
ejpam-3464	67	5	{	{	PUNCT
ejpam-3464	67	6	w}∪nhw	w}∪nhw	VERB
ejpam-3464	67	7	[	[	X
ejpam-3464	67	8	a	a	X
ejpam-3464	67	9	]	]	X
ejpam-3464	67	10	by	by	ADP
ejpam-3464	67	11	definition	definition	NOUN
ejpam-3464	67	12	1	1	NUM
ejpam-3464	67	13	.	.	PUNCT
ejpam-3464	68	1	thus	thus	ADV
ejpam-3464	68	2	,	,	PUNCT
ejpam-3464	68	3	fk	fk	INTJ
ejpam-3464	68	4	[	[	X
ejpam-3464	68	5	a	a	X
ejpam-3464	68	6	]	]	X
ejpam-3464	68	7	=	=	SYM
ejpam-3464	68	8	v	v	ADJ
ejpam-3464	68	9	(	(	PUNCT
ejpam-3464	68	10	k)\({w	k)\({w	NOUN
ejpam-3464	68	11	}	}	PUNCT
ejpam-3464	68	12	∪nhw	∪nhw	VERB
ejpam-3464	69	1	[	[	X
ejpam-3464	69	2	a	a	X
ejpam-3464	69	3	]	]	X
ejpam-3464	69	4	)	)	PUNCT
ejpam-3464	70	1	=	=	PUNCT
ejpam-3464	71	1	[	[	X
ejpam-3464	71	2	v	v	X
ejpam-3464	71	3	(	(	PUNCT
ejpam-3464	71	4	g)\{w	g)\{w	NOUN
ejpam-3464	71	5	}	}	PUNCT
ejpam-3464	71	6	]	]	PUNCT
ejpam-3464	71	7	∪	∪	ADP
ejpam-3464	71	8	fhw	fhw	PROPN
ejpam-3464	71	9	[	[	X
ejpam-3464	71	10	a	a	X
ejpam-3464	71	11	]	]	X
ejpam-3464	71	12	∪	∪	ADP
ejpam-3464	71	13			NOUN
ejpam-3464	71	14	⋃	⋃	NOUN
ejpam-3464	71	15	z∈v	z∈v	NOUN
ejpam-3464	71	16	(	(	PUNCT
ejpam-3464	71	17	g)\{w	g)\{w	NOUN
ejpam-3464	71	18	}	}	PUNCT
ejpam-3464	71	19	v	v	NOUN
ejpam-3464	71	20	(	(	PUNCT
ejpam-3464	71	21	hz	hz	NOUN
ejpam-3464	71	22	)	)	PUNCT
ejpam-3464	71	23			NOUN
ejpam-3464	71	24	this	this	PRON
ejpam-3464	71	25	proves	prove	VERB
ejpam-3464	71	26	the	the	DET
ejpam-3464	71	27	assertion	assertion	NOUN
ejpam-3464	71	28	.	.	PUNCT
ejpam-3464	72	1	definition	definition	NOUN
ejpam-3464	72	2	2	2	NUM
ejpam-3464	72	3	.	.	PUNCT
ejpam-3464	73	1	[	[	X
ejpam-3464	73	2	3	3	X
ejpam-3464	73	3	]	]	X
ejpam-3464	73	4	the	the	DET
ejpam-3464	73	5	edge	edge	NOUN
ejpam-3464	73	6	corona	corona	NOUN
ejpam-3464	73	7	g	g	PROPN
ejpam-3464	73	8	�	�	PROPN
ejpam-3464	73	9	h	h	PROPN
ejpam-3464	73	10	of	of	ADP
ejpam-3464	73	11	graphs	graph	NOUN
ejpam-3464	73	12	g	g	NOUN
ejpam-3464	73	13	and	and	CCONJ
ejpam-3464	73	14	h	h	NOUN
ejpam-3464	73	15	is	be	AUX
ejpam-3464	73	16	the	the	DET
ejpam-3464	73	17	graph	graph	NOUN
ejpam-3464	73	18	obtained	obtain	VERB
ejpam-3464	73	19	by	by	ADP
ejpam-3464	73	20	taking	take	VERB
ejpam-3464	73	21	one	one	NUM
ejpam-3464	73	22	copy	copy	NOUN
ejpam-3464	73	23	of	of	ADP
ejpam-3464	73	24	g	g	PROPN
ejpam-3464	73	25	and	and	CCONJ
ejpam-3464	73	26	|	|	ADV
ejpam-3464	73	27	e(g	e(g	PROPN
ejpam-3464	73	28	)	)	PUNCT
ejpam-3464	74	1	|	|	ADV
ejpam-3464	74	2	copies	copy	VERB
ejpam-3464	74	3	h	h	NOUN
ejpam-3464	74	4	and	and	CCONJ
ejpam-3464	74	5	joining	join	VERB
ejpam-3464	74	6	each	each	PRON
ejpam-3464	74	7	of	of	ADP
ejpam-3464	74	8	the	the	DET
ejpam-3464	74	9	end	end	NOUN
ejpam-3464	74	10	vertices	vertice	VERB
ejpam-3464	74	11	u	u	NOUN
ejpam-3464	74	12	and	and	CCONJ
ejpam-3464	74	13	v	v	NOUN
ejpam-3464	74	14	of	of	ADP
ejpam-3464	74	15	every	every	DET
ejpam-3464	74	16	edge	edge	NOUN
ejpam-3464	74	17	uv	uv	NOUN
ejpam-3464	74	18	to	to	ADP
ejpam-3464	74	19	every	every	DET
ejpam-3464	74	20	vertex	vertex	NOUN
ejpam-3464	74	21	of	of	ADP
ejpam-3464	74	22	the	the	DET
ejpam-3464	74	23	copy	copy	NOUN
ejpam-3464	74	24	huv	huv	PROPN
ejpam-3464	74	25	of	of	ADP
ejpam-3464	74	26	h	h	PROPN
ejpam-3464	74	27	(	(	PUNCT
ejpam-3464	74	28	that	that	ADV
ejpam-3464	74	29	is	is	ADV
ejpam-3464	74	30	,	,	PUNCT
ejpam-3464	74	31	forming	form	VERB
ejpam-3464	74	32	the	the	DET
ejpam-3464	74	33	join	join	NOUN
ejpam-3464	74	34	〈	〈	PROPN
ejpam-3464	74	35	{	{	PUNCT
ejpam-3464	74	36	u	u	NOUN
ejpam-3464	74	37	,	,	PUNCT
ejpam-3464	74	38	v}〉+huv	v}〉+huv	VERB
ejpam-3464	74	39	for	for	ADP
ejpam-3464	74	40	each	each	DET
ejpam-3464	74	41	uv	uv	PROPN
ejpam-3464	74	42	∈	∈	PROPN
ejpam-3464	74	43	e(g	e(g	PROPN
ejpam-3464	74	44	)	)	PUNCT
ejpam-3464	74	45	)	)	PUNCT
ejpam-3464	74	46	.	.	PUNCT
ejpam-3464	75	1	theorem	theorem	ADJ
ejpam-3464	75	2	4	4	NUM
ejpam-3464	75	3	.	.	PUNCT
ejpam-3464	76	1	let	let	VERB
ejpam-3464	76	2	k	k	NOUN
ejpam-3464	76	3	=	=	PUNCT
ejpam-3464	76	4	g	g	PROPN
ejpam-3464	76	5	�	�	PROPN
ejpam-3464	76	6	h	h	NOUN
ejpam-3464	76	7	=	=	SYM
ejpam-3464	76	8	(	(	PUNCT
ejpam-3464	76	9	v	v	NOUN
ejpam-3464	76	10	(	(	PUNCT
ejpam-3464	76	11	k	k	NOUN
ejpam-3464	76	12	)	)	PUNCT
ejpam-3464	76	13	,	,	PUNCT
ejpam-3464	76	14	e(k	e(k	NOUN
ejpam-3464	76	15	)	)	PUNCT
ejpam-3464	76	16	)	)	PUNCT
ejpam-3464	76	17	and	and	CCONJ
ejpam-3464	76	18	let	let	VERB
ejpam-3464	76	19	a	a	DET
ejpam-3464	76	20	∈	∈	PROPN
ejpam-3464	76	21	v	v	NOUN
ejpam-3464	76	22	(	(	PUNCT
ejpam-3464	76	23	g	g	NOUN
ejpam-3464	76	24	)	)	PUNCT
ejpam-3464	76	25	.	.	PUNCT
ejpam-3464	77	1	(	(	PUNCT
ejpam-3464	77	2	i	i	NOUN
ejpam-3464	77	3	)	)	PUNCT
ejpam-3464	77	4	if	if	SCONJ
ejpam-3464	77	5	a	a	DET
ejpam-3464	77	6	∈	∈	PROPN
ejpam-3464	77	7	v	v	NOUN
ejpam-3464	77	8	(	(	PUNCT
ejpam-3464	77	9	g	g	NOUN
ejpam-3464	77	10	)	)	PUNCT
ejpam-3464	77	11	,	,	PUNCT
ejpam-3464	77	12	then	then	ADV
ejpam-3464	77	13	fk	fk	INTJ
ejpam-3464	78	1	[	[	X
ejpam-3464	78	2	a	a	X
ejpam-3464	78	3	]	]	X
ejpam-3464	78	4	=	=	PUNCT
ejpam-3464	78	5	fg[a	fg[a	PROPN
ejpam-3464	78	6	]	]	PUNCT
ejpam-3464	78	7	∪	∪	ADP
ejpam-3464	78	8			PROPN
ejpam-3464	78	9	⋃	⋃	PROPN
ejpam-3464	78	10	u	u	NOUN
ejpam-3464	78	11	,	,	PUNCT
ejpam-3464	78	12	v	v	ADP
ejpam-3464	78	13	6	6	NUM
ejpam-3464	78	14	=	=	NOUN
ejpam-3464	78	15	a	a	DET
ejpam-3464	78	16	v	v	NOUN
ejpam-3464	78	17	(	(	PUNCT
ejpam-3464	78	18	huv	huv	PROPN
ejpam-3464	78	19	)	)	PUNCT
ejpam-3464	78	20			NOUN
ejpam-3464	78	21	.	.	PUNCT
ejpam-3464	79	1	(	(	PUNCT
ejpam-3464	79	2	ii	ii	NOUN
ejpam-3464	79	3	)	)	PUNCT
ejpam-3464	79	4	if	if	SCONJ
ejpam-3464	79	5	a	a	DET
ejpam-3464	79	6	∈	∈	PROPN
ejpam-3464	79	7	v	v	NOUN
ejpam-3464	79	8	(	(	PUNCT
ejpam-3464	79	9	hwz	hwz	PROPN
ejpam-3464	79	10	)	)	PUNCT
ejpam-3464	79	11	for	for	ADP
ejpam-3464	79	12	some	some	DET
ejpam-3464	79	13	wz	wz	PROPN
ejpam-3464	79	14	∈	∈	PROPN
ejpam-3464	79	15	e(g	e(g	PROPN
ejpam-3464	79	16	)	)	PUNCT
ejpam-3464	79	17	,	,	PUNCT
ejpam-3464	79	18	then	then	ADV
ejpam-3464	79	19	fk	fk	INTJ
ejpam-3464	80	1	[	[	X
ejpam-3464	80	2	a	a	X
ejpam-3464	80	3	]	]	X
ejpam-3464	80	4	=	=	SYM
ejpam-3464	80	5	v	v	NOUN
ejpam-3464	80	6	(	(	PUNCT
ejpam-3464	80	7	g)\{w	g)\{w	NOUN
ejpam-3464	80	8	,	,	PUNCT
ejpam-3464	80	9	z	z	NOUN
ejpam-3464	80	10	}	}	PUNCT
ejpam-3464	80	11	∪	∪	X
ejpam-3464	80	12	fhwz	fhwz	VERB
ejpam-3464	80	13	[	[	X
ejpam-3464	80	14	a	a	X
ejpam-3464	80	15	]	]	PUNCT
ejpam-3464	80	16	∪	∪	ADP
ejpam-3464	80	17			NOUN
ejpam-3464	80	18	⋃	⋃	NOUN
ejpam-3464	80	19	pq∈e(g)\{wz	pq∈e(g)\{wz	NOUN
ejpam-3464	80	20	}	}	SYM
ejpam-3464	80	21	v	v	PROPN
ejpam-3464	80	22	(	(	PUNCT
ejpam-3464	80	23	hpq	hpq	PROPN
ejpam-3464	80	24	)	)	PUNCT
ejpam-3464	80	25			NOUN
ejpam-3464	80	26	a.	a.	NOUN
ejpam-3464	80	27	gamorez	gamorez	PROPN
ejpam-3464	80	28	,	,	PUNCT
ejpam-3464	80	29	c.	c.	PROPN
ejpam-3464	80	30	nianga	nianga	PROPN
ejpam-3464	80	31	,	,	PUNCT
ejpam-3464	80	32	s.	s.	PROPN
ejpam-3464	80	33	canoy	canoy	PROPN
ejpam-3464	80	34	/	/	SYM
ejpam-3464	80	35	eur	eur	PROPN
ejpam-3464	80	36	.	.	PUNCT
ejpam-3464	81	1	j.	j.	PROPN
ejpam-3464	81	2	pure	pure	PROPN
ejpam-3464	81	3	appl	appl	PROPN
ejpam-3464	81	4	.	.	PROPN
ejpam-3464	81	5	math	math	PROPN
ejpam-3464	81	6	,	,	PUNCT
ejpam-3464	81	7	12	12	NUM
ejpam-3464	81	8	(	(	PUNCT
ejpam-3464	81	9	3	3	NUM
ejpam-3464	81	10	)	)	PUNCT
ejpam-3464	81	11	(	(	PUNCT
ejpam-3464	81	12	2019	2019	NUM
ejpam-3464	81	13	)	)	PUNCT
ejpam-3464	81	14	,	,	PUNCT
ejpam-3464	81	15	749	749	NUM
ejpam-3464	81	16	-	-	SYM
ejpam-3464	81	17	755	755	NUM
ejpam-3464	81	18	752	752	NUM
ejpam-3464	81	19	proof	proof	NOUN
ejpam-3464	81	20	.	.	PUNCT
ejpam-3464	82	1	(	(	PUNCT
ejpam-3464	82	2	i	i	NOUN
ejpam-3464	82	3	)	)	PUNCT
ejpam-3464	82	4	suppose	suppose	VERB
ejpam-3464	82	5	a	a	DET
ejpam-3464	82	6	∈	∈	PROPN
ejpam-3464	82	7	v	v	NOUN
ejpam-3464	82	8	(	(	PUNCT
ejpam-3464	82	9	g	g	NOUN
ejpam-3464	82	10	)	)	PUNCT
ejpam-3464	82	11	.	.	PUNCT
ejpam-3464	83	1	then	then	ADV
ejpam-3464	83	2	,	,	PUNCT
ejpam-3464	83	3	by	by	ADP
ejpam-3464	83	4	definition	definition	NOUN
ejpam-3464	83	5	2	2	NUM
ejpam-3464	83	6	,	,	PUNCT
ejpam-3464	83	7	nk	nk	PROPN
ejpam-3464	84	1	[	[	X
ejpam-3464	84	2	a	a	X
ejpam-3464	84	3	]	]	X
ejpam-3464	84	4	=	=	SYM
ejpam-3464	84	5	ng[a	ng[a	NOUN
ejpam-3464	84	6	]	]	PUNCT
ejpam-3464	84	7	∪	∪	ADP
ejpam-3464	84	8			PROPN
ejpam-3464	84	9	⋃	⋃	PROPN
ejpam-3464	84	10	z∈ng(a	z∈ng(a	PROPN
ejpam-3464	84	11	)	)	PUNCT
ejpam-3464	84	12	v	v	NOUN
ejpam-3464	84	13	(	(	PUNCT
ejpam-3464	84	14	haz	haz	PROPN
ejpam-3464	84	15	)	)	PUNCT
ejpam-3464	84	16			NOUN
ejpam-3464	84	17	.	.	PUNCT
ejpam-3464	85	1	consequently	consequently	ADV
ejpam-3464	85	2	,	,	PUNCT
ejpam-3464	85	3	fk	fk	INTJ
ejpam-3464	86	1	[	[	X
ejpam-3464	86	2	a	a	X
ejpam-3464	86	3	]	]	X
ejpam-3464	86	4	=	=	PUNCT
ejpam-3464	86	5	fg[a	fg[a	PROPN
ejpam-3464	86	6	]	]	PUNCT
ejpam-3464	86	7	∪	∪	ADP
ejpam-3464	86	8			PROPN
ejpam-3464	86	9	⋃	⋃	PROPN
ejpam-3464	86	10	u	u	NOUN
ejpam-3464	86	11	,	,	PUNCT
ejpam-3464	86	12	v	v	ADP
ejpam-3464	86	13	6	6	NUM
ejpam-3464	86	14	=	=	NOUN
ejpam-3464	86	15	a	a	DET
ejpam-3464	86	16	v	v	NOUN
ejpam-3464	86	17	(	(	PUNCT
ejpam-3464	86	18	huv	huv	PROPN
ejpam-3464	86	19	)	)	PUNCT
ejpam-3464	86	20			NOUN
ejpam-3464	86	21	.	.	PUNCT
ejpam-3464	87	1	(	(	PUNCT
ejpam-3464	87	2	ii	ii	NOUN
ejpam-3464	87	3	)	)	PUNCT
ejpam-3464	87	4	suppose	suppose	VERB
ejpam-3464	87	5	that	that	SCONJ
ejpam-3464	87	6	a	a	DET
ejpam-3464	87	7	∈	∈	PROPN
ejpam-3464	87	8	v	v	NOUN
ejpam-3464	87	9	(	(	PUNCT
ejpam-3464	87	10	hwz	hwz	PROPN
ejpam-3464	87	11	)	)	PUNCT
ejpam-3464	87	12	for	for	ADP
ejpam-3464	87	13	some	some	DET
ejpam-3464	87	14	wz	wz	PROPN
ejpam-3464	87	15	∈	∈	PROPN
ejpam-3464	87	16	e(g	e(g	PROPN
ejpam-3464	87	17	)	)	PUNCT
ejpam-3464	87	18	.	.	PUNCT
ejpam-3464	88	1	then	then	ADV
ejpam-3464	88	2	,	,	PUNCT
ejpam-3464	88	3	by	by	ADP
ejpam-3464	88	4	definition	definition	NOUN
ejpam-3464	88	5	2	2	NUM
ejpam-3464	88	6	,	,	PUNCT
ejpam-3464	88	7	nk	nk	PROPN
ejpam-3464	89	1	[	[	X
ejpam-3464	89	2	a	a	X
ejpam-3464	89	3	]	]	X
ejpam-3464	89	4	=	=	SYM
ejpam-3464	89	5	{	{	PUNCT
ejpam-3464	89	6	w	w	PROPN
ejpam-3464	89	7	,	,	PUNCT
ejpam-3464	89	8	z	z	NOUN
ejpam-3464	89	9	}	}	PUNCT
ejpam-3464	89	10	∪nhwz	∪nhwz	NOUN
ejpam-3464	90	1	[	[	X
ejpam-3464	90	2	a	a	X
ejpam-3464	90	3	]	]	X
ejpam-3464	90	4	.	.	PUNCT
ejpam-3464	91	1	therefore	therefore	ADV
ejpam-3464	91	2	,	,	PUNCT
ejpam-3464	91	3	fk	fk	INTJ
ejpam-3464	92	1	[	[	X
ejpam-3464	92	2	a	a	X
ejpam-3464	92	3	]	]	X
ejpam-3464	92	4	=	=	SYM
ejpam-3464	92	5	v	v	NOUN
ejpam-3464	92	6	(	(	PUNCT
ejpam-3464	92	7	g)\{w	g)\{w	NOUN
ejpam-3464	92	8	,	,	PUNCT
ejpam-3464	92	9	z	z	NOUN
ejpam-3464	92	10	}	}	PUNCT
ejpam-3464	92	11	∪	∪	X
ejpam-3464	92	12	fhwz	fhwz	VERB
ejpam-3464	92	13	[	[	X
ejpam-3464	92	14	a	a	X
ejpam-3464	92	15	]	]	PUNCT
ejpam-3464	92	16	∪	∪	ADP
ejpam-3464	92	17			NOUN
ejpam-3464	92	18	⋃	⋃	NOUN
ejpam-3464	92	19	pq∈e(g)\{wz	pq∈e(g)\{wz	NOUN
ejpam-3464	92	20	}	}	SYM
ejpam-3464	92	21	v	v	PROPN
ejpam-3464	92	22	(	(	PUNCT
ejpam-3464	92	23	hpq	hpq	PROPN
ejpam-3464	92	24	)	)	PUNCT
ejpam-3464	92	25			NOUN
ejpam-3464	92	26	,	,	PUNCT
ejpam-3464	92	27	showing	show	VERB
ejpam-3464	92	28	the	the	DET
ejpam-3464	92	29	desired	desire	VERB
ejpam-3464	92	30	equality	equality	NOUN
ejpam-3464	92	31	.	.	PUNCT
ejpam-3464	93	1	definition	definition	NOUN
ejpam-3464	93	2	3	3	NUM
ejpam-3464	93	3	.	.	PUNCT
ejpam-3464	94	1	[	[	X
ejpam-3464	94	2	3	3	X
ejpam-3464	94	3	]	]	PUNCT
ejpam-3464	94	4	the	the	DET
ejpam-3464	94	5	tensor	tensor	NOUN
ejpam-3464	94	6	product	product	NOUN
ejpam-3464	94	7	g	g	PROPN
ejpam-3464	94	8	�	�	PROPN
ejpam-3464	94	9	h	h	NOUN
ejpam-3464	94	10	of	of	ADP
ejpam-3464	94	11	graphs	graph	NOUN
ejpam-3464	94	12	g	g	NOUN
ejpam-3464	94	13	and	and	CCONJ
ejpam-3464	94	14	h	h	NOUN
ejpam-3464	94	15	is	be	AUX
ejpam-3464	94	16	the	the	DET
ejpam-3464	94	17	graph	graph	NOUN
ejpam-3464	94	18	with	with	ADP
ejpam-3464	94	19	vertex	vertex	NOUN
ejpam-3464	94	20	set	set	VERB
ejpam-3464	94	21	v	v	NOUN
ejpam-3464	94	22	(	(	PUNCT
ejpam-3464	94	23	g)×v	g)×v	PROPN
ejpam-3464	94	24	(	(	PUNCT
ejpam-3464	94	25	h	h	NOUN
ejpam-3464	94	26	)	)	PUNCT
ejpam-3464	94	27	and	and	CCONJ
ejpam-3464	94	28	(	(	PUNCT
ejpam-3464	94	29	u	u	NOUN
ejpam-3464	94	30	,	,	PUNCT
ejpam-3464	94	31	v	v	NOUN
ejpam-3464	94	32	)	)	PUNCT
ejpam-3464	94	33	is	be	AUX
ejpam-3464	94	34	adjacent	adjacent	ADJ
ejpam-3464	94	35	with	with	ADP
ejpam-3464	94	36	(	(	PUNCT
ejpam-3464	94	37	u	u	NOUN
ejpam-3464	94	38	′	′	NOUN
ejpam-3464	94	39	,	,	PUNCT
ejpam-3464	94	40	v	v	NOUN
ejpam-3464	94	41	′	′	NOUN
ejpam-3464	94	42	)	)	PUNCT
ejpam-3464	95	1	whenever	whenever	SCONJ
ejpam-3464	95	2	uu	uu	ADP
ejpam-3464	95	3	′	′	NUM
ejpam-3464	95	4	∈	∈	PROPN
ejpam-3464	95	5	e(g	e(g	PROPN
ejpam-3464	95	6	)	)	PUNCT
ejpam-3464	95	7	and	and	CCONJ
ejpam-3464	95	8	vv	vv	INTJ
ejpam-3464	95	9	′	′	NUM
ejpam-3464	95	10	∈	∈	PROPN
ejpam-3464	95	11	e(h	e(h	PROPN
ejpam-3464	95	12	)	)	PUNCT
ejpam-3464	95	13	.	.	PUNCT
ejpam-3464	96	1	theorem	theorem	NOUN
ejpam-3464	96	2	5	5	NUM
ejpam-3464	96	3	.	.	PUNCT
ejpam-3464	97	1	let	let	VERB
ejpam-3464	97	2	k	k	NOUN
ejpam-3464	97	3	=	=	PUNCT
ejpam-3464	97	4	g	g	PROPN
ejpam-3464	97	5	�	�	PROPN
ejpam-3464	97	6	h	h	NOUN
ejpam-3464	97	7	=	=	PUNCT
ejpam-3464	97	8	(	(	PUNCT
ejpam-3464	97	9	v	v	NOUN
ejpam-3464	97	10	(	(	PUNCT
ejpam-3464	97	11	k	k	NOUN
ejpam-3464	97	12	)	)	PUNCT
ejpam-3464	97	13	,	,	PUNCT
ejpam-3464	97	14	e(k	e(k	NOUN
ejpam-3464	97	15	)	)	PUNCT
ejpam-3464	97	16	)	)	PUNCT
ejpam-3464	97	17	,	,	PUNCT
ejpam-3464	97	18	where	where	SCONJ
ejpam-3464	97	19	g	g	NOUN
ejpam-3464	97	20	=	=	SYM
ejpam-3464	97	21	(	(	PUNCT
ejpam-3464	97	22	v	v	NOUN
ejpam-3464	97	23	(	(	PUNCT
ejpam-3464	97	24	g	g	NOUN
ejpam-3464	97	25	)	)	PUNCT
ejpam-3464	97	26	,	,	PUNCT
ejpam-3464	97	27	e(g	e(g	PROPN
ejpam-3464	97	28	)	)	PUNCT
ejpam-3464	97	29	)	)	PUNCT
ejpam-3464	97	30	and	and	CCONJ
ejpam-3464	97	31	h	h	NOUN
ejpam-3464	97	32	=	=	SYM
ejpam-3464	97	33	(	(	PUNCT
ejpam-3464	97	34	v	v	NOUN
ejpam-3464	97	35	(	(	PUNCT
ejpam-3464	97	36	h	h	NOUN
ejpam-3464	97	37	)	)	PUNCT
ejpam-3464	97	38	,	,	PUNCT
ejpam-3464	97	39	e(h	e(h	PROPN
ejpam-3464	97	40	)	)	PUNCT
ejpam-3464	97	41	)	)	PUNCT
ejpam-3464	97	42	are	be	AUX
ejpam-3464	97	43	non	non	ADJ
ejpam-3464	97	44	trivial	trivial	ADJ
ejpam-3464	97	45	graphs	graph	NOUN
ejpam-3464	97	46	.	.	PUNCT
ejpam-3464	98	1	then	then	ADV
ejpam-3464	98	2	,	,	PUNCT
ejpam-3464	98	3	for	for	ADP
ejpam-3464	98	4	each	each	DET
ejpam-3464	98	5	(	(	PUNCT
ejpam-3464	98	6	v	v	NOUN
ejpam-3464	98	7	,	,	PUNCT
ejpam-3464	98	8	a	a	PRON
ejpam-3464	98	9	)	)	PUNCT
ejpam-3464	98	10	∈	∈	NOUN
ejpam-3464	98	11	v	v	NOUN
ejpam-3464	98	12	(	(	PUNCT
ejpam-3464	98	13	k	k	NOUN
ejpam-3464	98	14	)	)	PUNCT
ejpam-3464	98	15	,	,	PUNCT
ejpam-3464	98	16	fk	fk	INTJ
ejpam-3464	98	17	[	[	X
ejpam-3464	98	18	(	(	PUNCT
ejpam-3464	98	19	v	v	NOUN
ejpam-3464	98	20	,	,	PUNCT
ejpam-3464	98	21	a	a	NOUN
ejpam-3464	98	22	)	)	PUNCT
ejpam-3464	98	23	]	]	PUNCT
ejpam-3464	99	1	=	=	PUNCT
ejpam-3464	100	1	[	[	X
ejpam-3464	100	2	fg(v)×	fg(v)×	PROPN
ejpam-3464	100	3	(	(	PUNCT
ejpam-3464	100	4	v	v	NOUN
ejpam-3464	100	5	(	(	PUNCT
ejpam-3464	100	6	h)\{a	h)\{a	NOUN
ejpam-3464	100	7	}	}	PUNCT
ejpam-3464	100	8	)	)	PUNCT
ejpam-3464	100	9	]	]	PUNCT
ejpam-3464	100	10	∪	∪	ADP
ejpam-3464	100	11	[	[	X
ejpam-3464	100	12	(	(	PUNCT
ejpam-3464	100	13	v	v	NOUN
ejpam-3464	100	14	(	(	PUNCT
ejpam-3464	100	15	g)\{v})×	g)\{v})×	NOUN
ejpam-3464	100	16	fh(a	fh(a	PROPN
ejpam-3464	100	17	)	)	PUNCT
ejpam-3464	100	18	]	]	PUNCT
ejpam-3464	100	19	.	.	PUNCT
ejpam-3464	100	20	proof	proof	NOUN
ejpam-3464	100	21	.	.	PUNCT
ejpam-3464	101	1	suppose	suppose	VERB
ejpam-3464	101	2	(	(	PUNCT
ejpam-3464	101	3	v	v	NOUN
ejpam-3464	101	4	,	,	PUNCT
ejpam-3464	101	5	a	a	PRON
ejpam-3464	101	6	)	)	PUNCT
ejpam-3464	101	7	∈	∈	NOUN
ejpam-3464	101	8	v	v	NOUN
ejpam-3464	101	9	(	(	PUNCT
ejpam-3464	101	10	g	g	PROPN
ejpam-3464	101	11	�	�	NOUN
ejpam-3464	101	12	h	h	NOUN
ejpam-3464	101	13	)	)	PUNCT
ejpam-3464	101	14	=	=	NOUN
ejpam-3464	101	15	v	v	X
ejpam-3464	101	16	(	(	PUNCT
ejpam-3464	101	17	k	k	NOUN
ejpam-3464	101	18	)	)	PUNCT
ejpam-3464	101	19	.	.	PUNCT
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ejpam-3464	102	2	definition	definition	NOUN
ejpam-3464	102	3	3	3	NUM
ejpam-3464	102	4	,	,	PUNCT
ejpam-3464	102	5	nk	nk	PROPN
ejpam-3464	102	6	[	[	X
ejpam-3464	102	7	(	(	PUNCT
ejpam-3464	102	8	v	v	NOUN
ejpam-3464	102	9	,	,	PUNCT
ejpam-3464	102	10	a	a	NOUN
ejpam-3464	102	11	)	)	PUNCT
ejpam-3464	102	12	]	]	PUNCT
ejpam-3464	102	13	=	=	PUNCT
ejpam-3464	102	14	{	{	PUNCT
ejpam-3464	102	15	(	(	PUNCT
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ejpam-3464	102	17	,	,	PUNCT
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ejpam-3464	102	20	:	:	PUNCT
ejpam-3464	102	21	x	x	PUNCT
ejpam-3464	102	22	∈	∈	NOUN
ejpam-3464	102	23	ng(v	ng(v	PUNCT
ejpam-3464	102	24	)	)	PUNCT
ejpam-3464	102	25	and	and	CCONJ
ejpam-3464	102	26	y	y	PROPN
ejpam-3464	102	27	∈	∈	PROPN
ejpam-3464	102	28	nh(a	nh(a	NUM
ejpam-3464	102	29	)	)	PUNCT
ejpam-3464	102	30	}	}	PUNCT
ejpam-3464	102	31	∪	∪	VERB
ejpam-3464	102	32	{	{	PUNCT
ejpam-3464	102	33	(	(	PUNCT
ejpam-3464	102	34	v	v	NOUN
ejpam-3464	102	35	,	,	PUNCT
ejpam-3464	102	36	a	a	NOUN
ejpam-3464	102	37	)	)	PUNCT
ejpam-3464	102	38	}	}	PUNCT
ejpam-3464	102	39	=	=	SYM
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ejpam-3464	103	2	(	(	PUNCT
ejpam-3464	103	3	ng(v)×	ng(v)×	PROPN
ejpam-3464	103	4	v	v	INTJ
ejpam-3464	103	5	(	(	PUNCT
ejpam-3464	103	6	h	h	NOUN
ejpam-3464	103	7	)	)	PUNCT
ejpam-3464	103	8	)	)	PUNCT
ejpam-3464	103	9	∩	∩	NOUN
ejpam-3464	103	10	(	(	PUNCT
ejpam-3464	103	11	v	v	NOUN
ejpam-3464	103	12	(	(	PUNCT
ejpam-3464	103	13	g)×nh(a	g)×nh(a	NOUN
ejpam-3464	103	14	)	)	PUNCT
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ejpam-3464	103	16	]	]	PUNCT
ejpam-3464	103	17	∪	∪	X
ejpam-3464	103	18	{	{	PUNCT
ejpam-3464	103	19	(	(	PUNCT
ejpam-3464	103	20	v	v	NOUN
ejpam-3464	103	21	,	,	PUNCT
ejpam-3464	103	22	a	a	NOUN
ejpam-3464	103	23	)	)	PUNCT
ejpam-3464	103	24	}	}	PUNCT
ejpam-3464	103	25	=	=	SYM
ejpam-3464	103	26	(	(	PUNCT
ejpam-3464	103	27	ng(v)×nh(a	ng(v)×nh(a	NOUN
ejpam-3464	103	28	)	)	PUNCT
ejpam-3464	103	29	)	)	PUNCT
ejpam-3464	103	30	∪	∪	ADP
ejpam-3464	103	31	{	{	PUNCT
ejpam-3464	103	32	(	(	PUNCT
ejpam-3464	103	33	v	v	NOUN
ejpam-3464	103	34	,	,	PUNCT
ejpam-3464	103	35	a	a	PRON
ejpam-3464	103	36	)	)	PUNCT
ejpam-3464	103	37	}	}	PUNCT
ejpam-3464	103	38	.	.	PUNCT
ejpam-3464	104	1	hence	hence	ADV
ejpam-3464	104	2	,	,	PUNCT
ejpam-3464	104	3	fk	fk	INTJ
ejpam-3464	104	4	[	[	X
ejpam-3464	104	5	(	(	PUNCT
ejpam-3464	104	6	v	v	NOUN
ejpam-3464	104	7	,	,	PUNCT
ejpam-3464	104	8	a	a	NOUN
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ejpam-3464	105	1	=	=	PUNCT
ejpam-3464	106	1	[	[	X
ejpam-3464	106	2	(	(	PUNCT
ejpam-3464	106	3	fg(v)×	fg(v)×	PROPN
ejpam-3464	106	4	v	v	PROPN
ejpam-3464	106	5	(	(	PUNCT
ejpam-3464	106	6	h	h	NOUN
ejpam-3464	106	7	)	)	PUNCT
ejpam-3464	106	8	)	)	PUNCT
ejpam-3464	106	9	∪	∪	X
ejpam-3464	106	10	(	(	PUNCT
ejpam-3464	106	11	(	(	PUNCT
ejpam-3464	106	12	v	v	NOUN
ejpam-3464	106	13	(	(	PUNCT
ejpam-3464	106	14	g)×	g)×	NOUN
ejpam-3464	106	15	fh(a	fh(a	NOUN
ejpam-3464	106	16	)	)	PUNCT
ejpam-3464	106	17	)	)	PUNCT
ejpam-3464	106	18	]	]	PUNCT
ejpam-3464	107	1	\{(v	\{(v	PROPN
ejpam-3464	107	2	,	,	PUNCT
ejpam-3464	107	3	a	a	PRON
ejpam-3464	107	4	)	)	PUNCT
ejpam-3464	107	5	}	}	PUNCT
ejpam-3464	107	6	=	=	PUNCT
ejpam-3464	108	1	[	[	X
ejpam-3464	108	2	fg(v)×	fg(v)×	PROPN
ejpam-3464	108	3	(	(	PUNCT
ejpam-3464	108	4	v	v	NOUN
ejpam-3464	108	5	(	(	PUNCT
ejpam-3464	108	6	h)\{a	h)\{a	NOUN
ejpam-3464	108	7	}	}	PUNCT
ejpam-3464	108	8	)	)	PUNCT
ejpam-3464	108	9	]	]	PUNCT
ejpam-3464	108	10	∪	∪	ADP
ejpam-3464	108	11	[	[	X
ejpam-3464	108	12	(	(	PUNCT
ejpam-3464	108	13	v	v	NOUN
ejpam-3464	108	14	(	(	PUNCT
ejpam-3464	108	15	g)\{v})×	g)\{v})×	NOUN
ejpam-3464	108	16	fh(a	fh(a	PROPN
ejpam-3464	108	17	)	)	PUNCT
ejpam-3464	108	18	]	]	PUNCT
ejpam-3464	108	19	,	,	PUNCT
ejpam-3464	108	20	showing	show	VERB
ejpam-3464	108	21	the	the	DET
ejpam-3464	108	22	desired	desire	VERB
ejpam-3464	108	23	result	result	NOUN
ejpam-3464	108	24	.	.	PUNCT
ejpam-3464	109	1	corollary	corollary	ADJ
ejpam-3464	109	2	1	1	NUM
ejpam-3464	109	3	.	.	PUNCT
ejpam-3464	110	1	let	let	VERB
ejpam-3464	110	2	g	g	NOUN
ejpam-3464	110	3	be	be	AUX
ejpam-3464	110	4	any	any	DET
ejpam-3464	110	5	graph	graph	NOUN
ejpam-3464	110	6	and	and	CCONJ
ejpam-3464	110	7	let	let	VERB
ejpam-3464	110	8	(	(	PUNCT
ejpam-3464	110	9	v	v	NOUN
ejpam-3464	110	10	,	,	PUNCT
ejpam-3464	110	11	a	a	PRON
ejpam-3464	110	12	)	)	PUNCT
ejpam-3464	110	13	∈	∈	NOUN
ejpam-3464	110	14	v	v	NOUN
ejpam-3464	110	15	(	(	PUNCT
ejpam-3464	110	16	g	g	PROPN
ejpam-3464	110	17	�	�	PROPN
ejpam-3464	110	18	kn	kn	PROPN
ejpam-3464	110	19	)	)	PUNCT
ejpam-3464	110	20	.	.	PUNCT
ejpam-3464	111	1	then	then	ADV
ejpam-3464	111	2	fg	fg	PROPN
ejpam-3464	111	3	�	�	PROPN
ejpam-3464	111	4	kn	kn	PROPN
ejpam-3464	111	5	[	[	X
ejpam-3464	111	6	(	(	PUNCT
ejpam-3464	111	7	v	v	NOUN
ejpam-3464	111	8	,	,	PUNCT
ejpam-3464	111	9	a	a	NOUN
ejpam-3464	111	10	)	)	PUNCT
ejpam-3464	111	11	]	]	PUNCT
ejpam-3464	112	1	=	=	PUNCT
ejpam-3464	113	1	[	[	X
ejpam-3464	113	2	fg(v)×	fg(v)×	PROPN
ejpam-3464	113	3	(	(	PUNCT
ejpam-3464	113	4	v	v	PROPN
ejpam-3464	113	5	(	(	PUNCT
ejpam-3464	113	6	kn	kn	PROPN
ejpam-3464	113	7	)	)	PUNCT
ejpam-3464	113	8	\	\	PROPN
ejpam-3464	113	9	{	{	PUNCT
ejpam-3464	113	10	a	a	NOUN
ejpam-3464	113	11	}	}	PUNCT
ejpam-3464	113	12	)	)	PUNCT
ejpam-3464	113	13	]	]	PUNCT
ejpam-3464	113	14	∪	∪	ADP
ejpam-3464	113	15	[	[	X
ejpam-3464	113	16	(	(	PUNCT
ejpam-3464	113	17	v	v	NOUN
ejpam-3464	113	18	(	(	PUNCT
ejpam-3464	113	19	g	g	NOUN
ejpam-3464	113	20	)	)	PUNCT
ejpam-3464	113	21	\	\	NOUN
ejpam-3464	113	22	{	{	PUNCT
ejpam-3464	113	23	v})×	v})×	PROPN
ejpam-3464	113	24	{	{	PUNCT
ejpam-3464	113	25	a	a	NOUN
ejpam-3464	113	26	}	}	PUNCT
ejpam-3464	113	27	]	]	PUNCT
ejpam-3464	113	28	.	.	PUNCT
ejpam-3464	114	1	a.	a.	PROPN
ejpam-3464	114	2	gamorez	gamorez	PROPN
ejpam-3464	114	3	,	,	PUNCT
ejpam-3464	114	4	c.	c.	PROPN
ejpam-3464	114	5	nianga	nianga	PROPN
ejpam-3464	114	6	,	,	PUNCT
ejpam-3464	114	7	s.	s.	PROPN
ejpam-3464	114	8	canoy	canoy	PROPN
ejpam-3464	114	9	/	/	SYM
ejpam-3464	114	10	eur	eur	PROPN
ejpam-3464	114	11	.	.	PUNCT
ejpam-3464	115	1	j.	j.	PROPN
ejpam-3464	115	2	pure	pure	PROPN
ejpam-3464	115	3	appl	appl	PROPN
ejpam-3464	115	4	.	.	PROPN
ejpam-3464	115	5	math	math	PROPN
ejpam-3464	115	6	,	,	PUNCT
ejpam-3464	115	7	12	12	NUM
ejpam-3464	115	8	(	(	PUNCT
ejpam-3464	115	9	3	3	NUM
ejpam-3464	115	10	)	)	PUNCT
ejpam-3464	115	11	(	(	PUNCT
ejpam-3464	115	12	2019	2019	NUM
ejpam-3464	115	13	)	)	PUNCT
ejpam-3464	115	14	,	,	PUNCT
ejpam-3464	115	15	749	749	NUM
ejpam-3464	115	16	-	-	SYM
ejpam-3464	115	17	755	755	NUM
ejpam-3464	115	18	753	753	NUM
ejpam-3464	115	19	proof	proof	NOUN
ejpam-3464	115	20	.	.	PUNCT
ejpam-3464	116	1	since	since	SCONJ
ejpam-3464	116	2	a	a	DET
ejpam-3464	116	3	∈	∈	PROPN
ejpam-3464	116	4	v	v	NOUN
ejpam-3464	116	5	(	(	PUNCT
ejpam-3464	116	6	kn	kn	PROPN
ejpam-3464	116	7	)	)	PUNCT
ejpam-3464	116	8	,	,	PUNCT
ejpam-3464	116	9	fkn	fkn	PROPN
ejpam-3464	117	1	[	[	X
ejpam-3464	117	2	a	a	X
ejpam-3464	117	3	]	]	X
ejpam-3464	117	4	=	=	PUNCT
ejpam-3464	117	5	∅.	∅.	VERB
ejpam-3464	117	6	hence	hence	ADV
ejpam-3464	117	7	,	,	PUNCT
ejpam-3464	117	8	fkn(a	fkn(a	PROPN
ejpam-3464	117	9	)	)	PUNCT
ejpam-3464	117	10	=	=	PRON
ejpam-3464	117	11	{	{	PUNCT
ejpam-3464	117	12	a	a	NOUN
ejpam-3464	117	13	}	}	PUNCT
ejpam-3464	117	14	.	.	PUNCT
ejpam-3464	118	1	therefore	therefore	ADV
ejpam-3464	118	2	,	,	PUNCT
ejpam-3464	118	3	by	by	ADP
ejpam-3464	118	4	theorem	theorem	NOUN
ejpam-3464	118	5	5	5	NUM
ejpam-3464	118	6	,	,	PUNCT
ejpam-3464	118	7	fg	fg	PROPN
ejpam-3464	118	8	�	�	PROPN
ejpam-3464	118	9	kn	kn	PROPN
ejpam-3464	118	10	[	[	X
ejpam-3464	118	11	(	(	PUNCT
ejpam-3464	118	12	v	v	NOUN
ejpam-3464	118	13	,	,	PUNCT
ejpam-3464	118	14	a	a	NOUN
ejpam-3464	118	15	)	)	PUNCT
ejpam-3464	118	16	]	]	PUNCT
ejpam-3464	119	1	=	=	PUNCT
ejpam-3464	120	1	[	[	X
ejpam-3464	120	2	fg(v)×	fg(v)×	PROPN
ejpam-3464	120	3	(	(	PUNCT
ejpam-3464	120	4	v	v	PROPN
ejpam-3464	120	5	(	(	PUNCT
ejpam-3464	120	6	kn	kn	PROPN
ejpam-3464	120	7	)	)	PUNCT
ejpam-3464	120	8	\	\	PROPN
ejpam-3464	120	9	{	{	PUNCT
ejpam-3464	120	10	a	a	NOUN
ejpam-3464	120	11	}	}	PUNCT
ejpam-3464	120	12	)	)	PUNCT
ejpam-3464	120	13	]	]	PUNCT
ejpam-3464	120	14	∪	∪	ADP
ejpam-3464	120	15	[	[	X
ejpam-3464	120	16	(	(	PUNCT
ejpam-3464	120	17	v	v	NOUN
ejpam-3464	120	18	(	(	PUNCT
ejpam-3464	120	19	g	g	NOUN
ejpam-3464	120	20	)	)	PUNCT
ejpam-3464	120	21	\	\	NOUN
ejpam-3464	120	22	{	{	PUNCT
ejpam-3464	120	23	v})×	v})×	PROPN
ejpam-3464	120	24	{	{	PUNCT
ejpam-3464	120	25	a	a	NOUN
ejpam-3464	120	26	}	}	PUNCT
ejpam-3464	120	27	]	]	PUNCT
ejpam-3464	120	28	.	.	PUNCT
ejpam-3464	121	1	this	this	PRON
ejpam-3464	121	2	proves	prove	VERB
ejpam-3464	121	3	the	the	DET
ejpam-3464	121	4	assertion	assertion	NOUN
ejpam-3464	121	5	.	.	PUNCT
ejpam-3464	122	1	definition	definition	NOUN
ejpam-3464	122	2	4	4	NUM
ejpam-3464	122	3	.	.	PUNCT
ejpam-3464	123	1	[	[	X
ejpam-3464	123	2	3	3	X
ejpam-3464	123	3	]	]	X
ejpam-3464	123	4	the	the	DET
ejpam-3464	123	5	disjunction	disjunction	NOUN
ejpam-3464	123	6	g	g	PROPN
ejpam-3464	123	7	∨h	∨h	NOUN
ejpam-3464	123	8	of	of	ADP
ejpam-3464	123	9	graphs	graph	NOUN
ejpam-3464	123	10	g	g	PROPN
ejpam-3464	123	11	and	and	CCONJ
ejpam-3464	123	12	h	h	NOUN
ejpam-3464	123	13	is	be	AUX
ejpam-3464	123	14	the	the	DET
ejpam-3464	123	15	graph	graph	NOUN
ejpam-3464	123	16	with	with	ADP
ejpam-3464	123	17	vertex	vertex	NOUN
ejpam-3464	123	18	set	set	VERB
ejpam-3464	123	19	v	v	NOUN
ejpam-3464	123	20	(	(	PUNCT
ejpam-3464	123	21	g)×	g)×	NOUN
ejpam-3464	123	22	v	v	NOUN
ejpam-3464	123	23	(	(	PUNCT
ejpam-3464	123	24	h	h	NOUN
ejpam-3464	123	25	)	)	PUNCT
ejpam-3464	123	26	and	and	CCONJ
ejpam-3464	123	27	(	(	PUNCT
ejpam-3464	123	28	u	u	NOUN
ejpam-3464	123	29	,	,	PUNCT
ejpam-3464	123	30	v	v	NOUN
ejpam-3464	123	31	)	)	PUNCT
ejpam-3464	123	32	is	be	AUX
ejpam-3464	123	33	adjacent	adjacent	ADJ
ejpam-3464	123	34	with	with	ADP
ejpam-3464	123	35	(	(	PUNCT
ejpam-3464	123	36	u	u	NOUN
ejpam-3464	123	37	′	′	NOUN
ejpam-3464	123	38	,	,	PUNCT
ejpam-3464	123	39	v	v	NOUN
ejpam-3464	123	40	′	′	NOUN
ejpam-3464	123	41	)	)	PUNCT
ejpam-3464	124	1	whenever	whenever	SCONJ
ejpam-3464	124	2	uu	uu	ADP
ejpam-3464	124	3	′	′	NUM
ejpam-3464	124	4	∈	∈	PROPN
ejpam-3464	124	5	e(g	e(g	PROPN
ejpam-3464	124	6	)	)	PUNCT
ejpam-3464	124	7	or	or	CCONJ
ejpam-3464	124	8	vv	vv	INTJ
ejpam-3464	124	9	′	′	NUM
ejpam-3464	124	10	∈	∈	PROPN
ejpam-3464	124	11	e(h	e(h	PROPN
ejpam-3464	124	12	)	)	PUNCT
ejpam-3464	124	13	.	.	PUNCT
ejpam-3464	125	1	theorem	theorem	VERB
ejpam-3464	125	2	6	6	NUM
ejpam-3464	125	3	.	.	PUNCT
ejpam-3464	126	1	let	let	VERB
ejpam-3464	126	2	k	k	NOUN
ejpam-3464	126	3	=	=	PUNCT
ejpam-3464	126	4	g	g	PROPN
ejpam-3464	126	5	∨	∨	NUM
ejpam-3464	126	6	h	h	NOUN
ejpam-3464	126	7	=	=	SYM
ejpam-3464	126	8	(	(	PUNCT
ejpam-3464	126	9	v	v	NOUN
ejpam-3464	126	10	(	(	PUNCT
ejpam-3464	126	11	k	k	NOUN
ejpam-3464	126	12	)	)	PUNCT
ejpam-3464	126	13	,	,	PUNCT
ejpam-3464	126	14	e(k	e(k	NOUN
ejpam-3464	126	15	)	)	PUNCT
ejpam-3464	126	16	)	)	PUNCT
ejpam-3464	126	17	,	,	PUNCT
ejpam-3464	126	18	where	where	SCONJ
ejpam-3464	126	19	g	g	NOUN
ejpam-3464	126	20	=	=	SYM
ejpam-3464	126	21	(	(	PUNCT
ejpam-3464	126	22	v	v	NOUN
ejpam-3464	126	23	(	(	PUNCT
ejpam-3464	126	24	g	g	NOUN
ejpam-3464	126	25	)	)	PUNCT
ejpam-3464	126	26	,	,	PUNCT
ejpam-3464	126	27	e(g	e(g	PROPN
ejpam-3464	126	28	)	)	PUNCT
ejpam-3464	126	29	)	)	PUNCT
ejpam-3464	126	30	and	and	CCONJ
ejpam-3464	126	31	h	h	NOUN
ejpam-3464	126	32	=	=	SYM
ejpam-3464	126	33	(	(	PUNCT
ejpam-3464	126	34	v	v	NOUN
ejpam-3464	126	35	(	(	PUNCT
ejpam-3464	126	36	h	h	NOUN
ejpam-3464	126	37	)	)	PUNCT
ejpam-3464	126	38	,	,	PUNCT
ejpam-3464	126	39	e(h	e(h	PROPN
ejpam-3464	126	40	)	)	PUNCT
ejpam-3464	126	41	)	)	PUNCT
ejpam-3464	126	42	.	.	PUNCT
ejpam-3464	127	1	then	then	ADV
ejpam-3464	127	2	,	,	PUNCT
ejpam-3464	127	3	for	for	ADP
ejpam-3464	127	4	each	each	DET
ejpam-3464	127	5	(	(	PUNCT
ejpam-3464	127	6	v	v	NOUN
ejpam-3464	127	7	,	,	PUNCT
ejpam-3464	127	8	a	a	PRON
ejpam-3464	127	9	)	)	PUNCT
ejpam-3464	127	10	∈	∈	NOUN
ejpam-3464	127	11	v	v	NOUN
ejpam-3464	127	12	(	(	PUNCT
ejpam-3464	127	13	k	k	NOUN
ejpam-3464	127	14	)	)	PUNCT
ejpam-3464	127	15	,	,	PUNCT
ejpam-3464	127	16	fk	fk	INTJ
ejpam-3464	127	17	[	[	X
ejpam-3464	127	18	(	(	PUNCT
ejpam-3464	127	19	v	v	NOUN
ejpam-3464	127	20	,	,	PUNCT
ejpam-3464	127	21	a	a	NOUN
ejpam-3464	127	22	)	)	PUNCT
ejpam-3464	127	23	]	]	PUNCT
ejpam-3464	128	1	=	=	PUNCT
ejpam-3464	128	2	[	[	X
ejpam-3464	128	3	{	{	PUNCT
ejpam-3464	128	4	v	v	NOUN
ejpam-3464	128	5	}	}	PUNCT
ejpam-3464	128	6	×	×	NOUN
ejpam-3464	128	7	fh	fh	PROPN
ejpam-3464	128	8	[	[	X
ejpam-3464	128	9	a	a	X
ejpam-3464	128	10	]	]	X
ejpam-3464	128	11	]	]	X
ejpam-3464	128	12	∪	∪	ADP
ejpam-3464	128	13	[	[	X
ejpam-3464	128	14	fg[v]×	fg[v]×	ADJ
ejpam-3464	128	15	fh(a	fh(a	NUM
ejpam-3464	128	16	)	)	PUNCT
ejpam-3464	128	17	]	]	PUNCT
ejpam-3464	128	18	.	.	PUNCT
ejpam-3464	129	1	proof	proof	NOUN
ejpam-3464	129	2	.	.	PUNCT
ejpam-3464	130	1	let	let	VERB
ejpam-3464	130	2	a	a	PRON
ejpam-3464	130	3	=	=	SYM
ejpam-3464	131	1	[	[	X
ejpam-3464	131	2	{	{	PUNCT
ejpam-3464	131	3	v}×fh	v}×fh	NOUN
ejpam-3464	131	4	[	[	X
ejpam-3464	131	5	a]]∪[fg[v]×fh(a	a]]∪[fg[v]×fh(a	NOUN
ejpam-3464	131	6	)	)	PUNCT
ejpam-3464	131	7	]	]	PUNCT
ejpam-3464	131	8	.	.	PUNCT
ejpam-3464	132	1	suppose	suppose	VERB
ejpam-3464	132	2	(	(	PUNCT
ejpam-3464	132	3	v	v	NOUN
ejpam-3464	132	4	,	,	PUNCT
ejpam-3464	132	5	a	a	PRON
ejpam-3464	132	6	)	)	PUNCT
ejpam-3464	132	7	∈	∈	PROPN
ejpam-3464	132	8	v	v	NOUN
ejpam-3464	132	9	(	(	PUNCT
ejpam-3464	132	10	g∨h	g∨h	PROPN
ejpam-3464	132	11	)	)	PUNCT
ejpam-3464	132	12	=	=	SYM
ejpam-3464	132	13	v	v	X
ejpam-3464	132	14	(	(	PUNCT
ejpam-3464	132	15	k	k	NOUN
ejpam-3464	132	16	)	)	PUNCT
ejpam-3464	132	17	and	and	CCONJ
ejpam-3464	132	18	(	(	PUNCT
ejpam-3464	132	19	x	x	X
ejpam-3464	132	20	,	,	PUNCT
ejpam-3464	132	21	q	q	NOUN
ejpam-3464	132	22	)	)	PUNCT
ejpam-3464	132	23	∈	∈	NOUN
ejpam-3464	132	24	fk	fk	INTJ
ejpam-3464	133	1	[	[	X
ejpam-3464	133	2	(	(	PUNCT
ejpam-3464	133	3	v	v	NOUN
ejpam-3464	133	4	,	,	PUNCT
ejpam-3464	133	5	a	a	NOUN
ejpam-3464	133	6	)	)	PUNCT
ejpam-3464	133	7	]	]	PUNCT
ejpam-3464	133	8	.	.	PUNCT
ejpam-3464	134	1	then	then	ADV
ejpam-3464	134	2	(	(	PUNCT
ejpam-3464	134	3	v	v	NOUN
ejpam-3464	134	4	,	,	PUNCT
ejpam-3464	134	5	a	a	PRON
ejpam-3464	134	6	)	)	PUNCT
ejpam-3464	134	7	6=	6=	NUM
ejpam-3464	134	8	(	(	PUNCT
ejpam-3464	134	9	x	x	X
ejpam-3464	134	10	,	,	PUNCT
ejpam-3464	134	11	q	q	NOUN
ejpam-3464	134	12	)	)	PUNCT
ejpam-3464	134	13	and	and	CCONJ
ejpam-3464	134	14	dk((v	dk((v	PROPN
ejpam-3464	134	15	,	,	PUNCT
ejpam-3464	134	16	a	a	PRON
ejpam-3464	134	17	)	)	PUNCT
ejpam-3464	134	18	,	,	PUNCT
ejpam-3464	134	19	(	(	PUNCT
ejpam-3464	134	20	x	x	X
ejpam-3464	134	21	,	,	PUNCT
ejpam-3464	134	22	q	q	NOUN
ejpam-3464	134	23	)	)	PUNCT
ejpam-3464	134	24	)	)	PUNCT
ejpam-3464	135	1	6=	6=	PRON
ejpam-3464	135	2	1	1	X
ejpam-3464	135	3	.	.	X
ejpam-3464	135	4	consider	consider	VERB
ejpam-3464	135	5	the	the	DET
ejpam-3464	135	6	following	follow	VERB
ejpam-3464	135	7	cases	case	NOUN
ejpam-3464	135	8	:	:	PUNCT
ejpam-3464	135	9	case	case	NOUN
ejpam-3464	135	10	1	1	NUM
ejpam-3464	135	11	.	.	X
ejpam-3464	135	12	assume	assume	VERB
ejpam-3464	135	13	that	that	SCONJ
ejpam-3464	135	14	x	x	PRON
ejpam-3464	135	15	=	=	PUNCT
ejpam-3464	135	16	v.	v.	ADP
ejpam-3464	135	17	then	then	ADV
ejpam-3464	135	18	q	q	X
ejpam-3464	136	1	6=	6=	ADP
ejpam-3464	136	2	a	a	PRON
ejpam-3464	136	3	and	and	CCONJ
ejpam-3464	136	4	dh(q	dh(q	ADV
ejpam-3464	136	5	,	,	PUNCT
ejpam-3464	136	6	a	a	PRON
ejpam-3464	136	7	)	)	PUNCT
ejpam-3464	136	8	=	=	SYM
ejpam-3464	136	9	dk((x	dk((x	ADJ
ejpam-3464	136	10	,	,	PUNCT
ejpam-3464	136	11	q	q	NOUN
ejpam-3464	136	12	)	)	PUNCT
ejpam-3464	136	13	,	,	PUNCT
ejpam-3464	136	14	(	(	PUNCT
ejpam-3464	136	15	x	x	X
ejpam-3464	136	16	,	,	PUNCT
ejpam-3464	136	17	a	a	NOUN
ejpam-3464	136	18	)	)	PUNCT
ejpam-3464	136	19	)	)	PUNCT
ejpam-3464	137	1	6=	6=	ADP
ejpam-3464	137	2	1	1	X
ejpam-3464	137	3	.	.	PUNCT
ejpam-3464	138	1	thus	thus	ADV
ejpam-3464	138	2	,	,	PUNCT
ejpam-3464	138	3	q	q	PROPN
ejpam-3464	138	4	∈	∈	PROPN
ejpam-3464	138	5	fh	fh	PROPN
ejpam-3464	139	1	[	[	X
ejpam-3464	139	2	a	a	X
ejpam-3464	139	3	]	]	X
ejpam-3464	139	4	and	and	CCONJ
ejpam-3464	139	5	so	so	ADV
ejpam-3464	139	6	,	,	PUNCT
ejpam-3464	139	7	(	(	PUNCT
ejpam-3464	139	8	x	x	NOUN
ejpam-3464	139	9	,	,	PUNCT
ejpam-3464	139	10	q	q	ADJ
ejpam-3464	139	11	)	)	PUNCT
ejpam-3464	139	12	∈	∈	PROPN
ejpam-3464	139	13	{	{	PUNCT
ejpam-3464	139	14	v	v	NOUN
ejpam-3464	139	15	}	}	PUNCT
ejpam-3464	139	16	×	×	NOUN
ejpam-3464	139	17	fh	fh	PROPN
ejpam-3464	140	1	[	[	X
ejpam-3464	140	2	a	a	X
ejpam-3464	140	3	]	]	X
ejpam-3464	140	4	.	.	PUNCT
ejpam-3464	141	1	case	case	NOUN
ejpam-3464	141	2	2	2	NUM
ejpam-3464	141	3	.	.	X
ejpam-3464	141	4	assume	assume	VERB
ejpam-3464	141	5	that	that	SCONJ
ejpam-3464	141	6	x	x	PRON
ejpam-3464	141	7	6=	6=	PROPN
ejpam-3464	141	8	v.	v.	CCONJ
ejpam-3464	141	9	suppose	suppose	VERB
ejpam-3464	141	10	q	q	X
ejpam-3464	141	11	=	=	PUNCT
ejpam-3464	141	12	a.	a.	NOUN
ejpam-3464	141	13	then	then	ADV
ejpam-3464	141	14	dg(x	dg(x	NUM
ejpam-3464	141	15	,	,	PUNCT
ejpam-3464	141	16	v	v	NOUN
ejpam-3464	141	17	)	)	PUNCT
ejpam-3464	141	18	=	=	SYM
ejpam-3464	141	19	dk((v	dk((v	PROPN
ejpam-3464	141	20	,	,	PUNCT
ejpam-3464	141	21	a	a	PRON
ejpam-3464	141	22	)	)	PUNCT
ejpam-3464	141	23	,	,	PUNCT
ejpam-3464	141	24	(	(	PUNCT
ejpam-3464	141	25	x	x	X
ejpam-3464	141	26	,	,	PUNCT
ejpam-3464	141	27	a	a	NOUN
ejpam-3464	141	28	)	)	PUNCT
ejpam-3464	141	29	)	)	PUNCT
ejpam-3464	141	30	6=	6=	ADP
ejpam-3464	142	1	1	1	X
ejpam-3464	142	2	.	.	PUNCT
ejpam-3464	143	1	it	it	PRON
ejpam-3464	143	2	follows	follow	VERB
ejpam-3464	143	3	that	that	SCONJ
ejpam-3464	143	4	x	x	PUNCT
ejpam-3464	143	5	∈	∈	PROPN
ejpam-3464	143	6	fg[v	fg[v	PROPN
ejpam-3464	143	7	]	]	X
ejpam-3464	143	8	and	and	CCONJ
ejpam-3464	143	9	(	(	PUNCT
ejpam-3464	143	10	x	x	NOUN
ejpam-3464	143	11	,	,	PUNCT
ejpam-3464	143	12	q	q	ADJ
ejpam-3464	143	13	)	)	PUNCT
ejpam-3464	143	14	∈	∈	PROPN
ejpam-3464	143	15	fg[v	fg[v	PROPN
ejpam-3464	143	16	]	]	X
ejpam-3464	143	17	×	×	NOUN
ejpam-3464	143	18	{	{	PUNCT
ejpam-3464	143	19	a	a	NOUN
ejpam-3464	143	20	}	}	PUNCT
ejpam-3464	143	21	.	.	PUNCT
ejpam-3464	144	1	suppose	suppose	VERB
ejpam-3464	144	2	q	q	X
ejpam-3464	144	3	6=	6=	NUM
ejpam-3464	144	4	a.	a.	NOUN
ejpam-3464	144	5	observe	observe	VERB
ejpam-3464	144	6	that	that	PRON
ejpam-3464	144	7	q	q	PROPN
ejpam-3464	144	8	/∈	/∈	PUNCT
ejpam-3464	144	9	nh	nh	PROPN
ejpam-3464	145	1	[	[	X
ejpam-3464	145	2	a	a	X
ejpam-3464	145	3	]	]	X
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ejpam-3464	145	5	x	x	PUNCT
ejpam-3464	145	6	/∈	/∈	PUNCT
ejpam-3464	146	1	ng[v	ng[v	ADJ
ejpam-3464	146	2	]	]	PUNCT
ejpam-3464	146	3	.	.	PUNCT
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ejpam-3464	147	2	,	,	PUNCT
ejpam-3464	147	3	q	q	PROPN
ejpam-3464	147	4	∈	∈	PROPN
ejpam-3464	147	5	fh	fh	PROPN
ejpam-3464	148	1	[	[	X
ejpam-3464	148	2	a	a	X
ejpam-3464	148	3	]	]	X
ejpam-3464	148	4	and	and	CCONJ
ejpam-3464	148	5	x	x	PROPN
ejpam-3464	148	6	∈	∈	PROPN
ejpam-3464	148	7	fg[v	fg[v	PROPN
ejpam-3464	148	8	]	]	X
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ejpam-3464	148	11	that	that	SCONJ
ejpam-3464	148	12	(	(	PUNCT
ejpam-3464	148	13	x	x	X
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ejpam-3464	148	17	∈	∈	NOUN
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ejpam-3464	148	19	[	[	X
ejpam-3464	148	20	a	a	X
ejpam-3464	148	21	]	]	X
ejpam-3464	148	22	.	.	PUNCT
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ejpam-3464	149	2	,	,	PUNCT
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ejpam-3464	149	11	⊆	⊆	NUM
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ejpam-3464	149	13	{	{	PUNCT
ejpam-3464	149	14	v	v	NOUN
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ejpam-3464	149	16	×	×	NOUN
ejpam-3464	149	17	fh	fh	PROPN
ejpam-3464	149	18	[	[	X
ejpam-3464	149	19	a	a	X
ejpam-3464	149	20	]	]	X
ejpam-3464	149	21	]	]	X
ejpam-3464	149	22	∪	∪	ADP
ejpam-3464	149	23	[	[	X
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ejpam-3464	149	25	{	{	PUNCT
ejpam-3464	149	26	a	a	X
ejpam-3464	149	27	}	}	PUNCT
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ejpam-3464	149	29	∪	∪	ADP
ejpam-3464	149	30	[	[	X
ejpam-3464	149	31	fg[v]×	fg[v]×	X
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ejpam-3464	149	33	[	[	X
ejpam-3464	149	34	a	a	X
ejpam-3464	149	35	]	]	X
ejpam-3464	149	36	]	]	X
ejpam-3464	149	37	=	=	PUNCT
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ejpam-3464	150	2	{	{	PUNCT
ejpam-3464	150	3	v	v	NOUN
ejpam-3464	150	4	}	}	PUNCT
ejpam-3464	150	5	×	×	NOUN
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ejpam-3464	150	7	[	[	X
ejpam-3464	150	8	a	a	X
ejpam-3464	150	9	]	]	X
ejpam-3464	150	10	]	]	X
ejpam-3464	150	11	∪	∪	ADP
ejpam-3464	150	12	[	[	X
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ejpam-3464	150	15	)	)	PUNCT
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ejpam-3464	152	9	v	v	NOUN
ejpam-3464	152	10	}	}	PUNCT
ejpam-3464	152	11	×	×	NOUN
ejpam-3464	152	12	fh	fh	PROPN
ejpam-3464	153	1	[	[	X
ejpam-3464	153	2	a	a	X
ejpam-3464	153	3	]	]	X
ejpam-3464	153	4	.	.	PUNCT
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ejpam-3464	154	2	u	u	X
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ejpam-3464	154	5	,	,	PUNCT
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ejpam-3464	154	7	6=	6=	ADP
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ejpam-3464	154	11	6=	6=	PROPN
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ejpam-3464	154	13	)	)	PUNCT
ejpam-3464	154	14	.	.	PUNCT
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ejpam-3464	155	2	,	,	PUNCT
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ejpam-3464	155	4	u	u	NOUN
ejpam-3464	155	5	,	,	PUNCT
ejpam-3464	155	6	z	z	NOUN
ejpam-3464	155	7	)	)	PUNCT
ejpam-3464	155	8	6=	6=	ADP
ejpam-3464	155	9	(	(	PUNCT
ejpam-3464	155	10	v	v	NOUN
ejpam-3464	155	11	,	,	PUNCT
ejpam-3464	155	12	a	a	PRON
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ejpam-3464	155	14	and	and	CCONJ
ejpam-3464	155	15	by	by	ADP
ejpam-3464	155	16	definition	definition	NOUN
ejpam-3464	155	17	4	4	NUM
ejpam-3464	155	18	,	,	PUNCT
ejpam-3464	155	19	(	(	PUNCT
ejpam-3464	155	20	u	u	NOUN
ejpam-3464	155	21	,	,	PUNCT
ejpam-3464	155	22	z)(v	z)(v	ADJ
ejpam-3464	155	23	,	,	PUNCT
ejpam-3464	155	24	a	a	PRON
ejpam-3464	155	25	)	)	PUNCT
ejpam-3464	155	26	/∈	/∈	PUNCT
ejpam-3464	155	27	e(k	e(k	NOUN
ejpam-3464	155	28	)	)	PUNCT
ejpam-3464	155	29	.	.	PUNCT
ejpam-3464	156	1	thus	thus	ADV
ejpam-3464	156	2	,	,	PUNCT
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ejpam-3464	156	4	u	u	NOUN
ejpam-3464	156	5	,	,	PUNCT
ejpam-3464	156	6	z	z	NOUN
ejpam-3464	156	7	)	)	PUNCT
ejpam-3464	156	8	∈	∈	PROPN
ejpam-3464	157	1	fk	fk	INTJ
ejpam-3464	157	2	[	[	X
ejpam-3464	157	3	(	(	PUNCT
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ejpam-3464	157	5	,	,	PUNCT
ejpam-3464	157	6	a	a	NOUN
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ejpam-3464	157	8	]	]	PUNCT
ejpam-3464	157	9	.	.	PUNCT
ejpam-3464	158	1	also	also	ADV
ejpam-3464	158	2	if	if	SCONJ
ejpam-3464	158	3	(	(	PUNCT
ejpam-3464	158	4	u	u	NOUN
ejpam-3464	158	5	,	,	PUNCT
ejpam-3464	158	6	z	z	NOUN
ejpam-3464	158	7	)	)	PUNCT
ejpam-3464	158	8	∈	∈	PROPN
ejpam-3464	158	9	fg[v	fg[v	PROPN
ejpam-3464	158	10	]	]	X
ejpam-3464	158	11	×	×	PROPN
ejpam-3464	158	12	fh(a	fh(a	NUM
ejpam-3464	158	13	)	)	PUNCT
ejpam-3464	158	14	,	,	PUNCT
ejpam-3464	158	15	u	u	PROPN
ejpam-3464	158	16	6=	6=	PROPN
ejpam-3464	158	17	v	v	NOUN
ejpam-3464	158	18	,	,	PUNCT
ejpam-3464	158	19	uv	uv	NOUN
ejpam-3464	158	20	/∈	/∈	PUNCT
ejpam-3464	158	21	e(g	e(g	PROPN
ejpam-3464	158	22	)	)	PUNCT
ejpam-3464	158	23	,	,	PUNCT
ejpam-3464	158	24	and	and	CCONJ
ejpam-3464	158	25	az	az	PROPN
ejpam-3464	158	26	/∈	/∈	PUNCT
ejpam-3464	159	1	e(h	e(h	PROPN
ejpam-3464	159	2	)	)	PUNCT
ejpam-3464	159	3	.	.	PUNCT
ejpam-3464	160	1	this	this	PRON
ejpam-3464	160	2	implies	imply	VERB
ejpam-3464	160	3	that	that	SCONJ
ejpam-3464	160	4	(	(	PUNCT
ejpam-3464	160	5	u	u	NOUN
ejpam-3464	160	6	,	,	PUNCT
ejpam-3464	160	7	z	z	NOUN
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ejpam-3464	160	9	6=	6=	ADP
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ejpam-3464	160	11	v	v	NOUN
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ejpam-3464	160	18	4	4	NUM
ejpam-3464	160	19	,	,	PUNCT
ejpam-3464	160	20	(	(	PUNCT
ejpam-3464	160	21	u	u	NOUN
ejpam-3464	160	22	,	,	PUNCT
ejpam-3464	160	23	z)(v	z)(v	ADJ
ejpam-3464	160	24	,	,	PUNCT
ejpam-3464	160	25	a	a	PRON
ejpam-3464	160	26	)	)	PUNCT
ejpam-3464	160	27	6=	6=	ADP
ejpam-3464	160	28	e(k	e(k	NOUN
ejpam-3464	160	29	)	)	PUNCT
ejpam-3464	160	30	.	.	PUNCT
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ejpam-3464	161	7	∈	∈	PROPN
ejpam-3464	161	8	fk	fk	INTJ
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ejpam-3464	162	3	v	v	NOUN
ejpam-3464	162	4	,	,	PUNCT
ejpam-3464	162	5	a	a	NOUN
ejpam-3464	162	6	)	)	PUNCT
ejpam-3464	162	7	]	]	PUNCT
ejpam-3464	162	8	.	.	PUNCT
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ejpam-3464	163	2	,	,	PUNCT
ejpam-3464	163	3	a	a	DET
ejpam-3464	163	4	⊆	⊆	NUM
ejpam-3464	163	5	fk	fk	NOUN
ejpam-3464	163	6	[	[	X
ejpam-3464	163	7	(	(	PUNCT
ejpam-3464	163	8	v	v	NOUN
ejpam-3464	163	9	,	,	PUNCT
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ejpam-3464	163	12	]	]	PUNCT
ejpam-3464	163	13	.	.	PUNCT
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ejpam-3464	164	2	,	,	PUNCT
ejpam-3464	164	3	fk	fk	INTJ
ejpam-3464	164	4	[	[	X
ejpam-3464	164	5	(	(	PUNCT
ejpam-3464	164	6	v	v	NOUN
ejpam-3464	164	7	,	,	PUNCT
ejpam-3464	164	8	a	a	NOUN
ejpam-3464	164	9	)	)	PUNCT
ejpam-3464	164	10	]	]	PUNCT
ejpam-3464	164	11	=	=	PUNCT
ejpam-3464	164	12	a.	a.	NOUN
ejpam-3464	164	13	corollary	corollary	NOUN
ejpam-3464	164	14	2	2	X
ejpam-3464	164	15	.	.	PUNCT
ejpam-3464	165	1	let	let	VERB
ejpam-3464	165	2	g	g	NOUN
ejpam-3464	165	3	be	be	AUX
ejpam-3464	165	4	any	any	DET
ejpam-3464	165	5	graph	graph	NOUN
ejpam-3464	165	6	and	and	CCONJ
ejpam-3464	165	7	let	let	VERB
ejpam-3464	165	8	(	(	PUNCT
ejpam-3464	165	9	v	v	NOUN
ejpam-3464	165	10	,	,	PUNCT
ejpam-3464	165	11	a	a	PRON
ejpam-3464	165	12	)	)	PUNCT
ejpam-3464	165	13	∈	∈	NOUN
ejpam-3464	165	14	v	v	NOUN
ejpam-3464	165	15	(	(	PUNCT
ejpam-3464	165	16	g	g	PROPN
ejpam-3464	165	17	∨	∨	PROPN
ejpam-3464	165	18	kn	kn	PROPN
ejpam-3464	165	19	)	)	PUNCT
ejpam-3464	165	20	.	.	PUNCT
ejpam-3464	166	1	then	then	ADV
ejpam-3464	166	2	fg∨kn	fg∨kn	X
ejpam-3464	166	3	[	[	X
ejpam-3464	166	4	(	(	PUNCT
ejpam-3464	166	5	v	v	NOUN
ejpam-3464	166	6	,	,	PUNCT
ejpam-3464	166	7	a	a	NOUN
ejpam-3464	166	8	)	)	PUNCT
ejpam-3464	166	9	]	]	PUNCT
ejpam-3464	167	1	=	=	PUNCT
ejpam-3464	167	2	fg[v]×	fg[v]×	X
ejpam-3464	167	3	{	{	PUNCT
ejpam-3464	167	4	a	a	X
ejpam-3464	167	5	}	}	PUNCT
ejpam-3464	167	6	.	.	PUNCT
ejpam-3464	168	1	proof	proof	NOUN
ejpam-3464	168	2	.	.	PUNCT
ejpam-3464	169	1	again	again	ADV
ejpam-3464	169	2	,	,	PUNCT
ejpam-3464	169	3	since	since	SCONJ
ejpam-3464	169	4	a	a	DET
ejpam-3464	169	5	∈	∈	PROPN
ejpam-3464	169	6	v	v	NOUN
ejpam-3464	169	7	(	(	PUNCT
ejpam-3464	169	8	kn	kn	PROPN
ejpam-3464	169	9	)	)	PUNCT
ejpam-3464	169	10	,	,	PUNCT
ejpam-3464	169	11	fkn(a	fkn(a	PROPN
ejpam-3464	169	12	)	)	PUNCT
ejpam-3464	169	13	=	=	PRON
ejpam-3464	169	14	{	{	PUNCT
ejpam-3464	169	15	a	a	NOUN
ejpam-3464	169	16	}	}	PUNCT
ejpam-3464	169	17	and	and	CCONJ
ejpam-3464	169	18	{	{	PUNCT
ejpam-3464	169	19	v	v	NOUN
ejpam-3464	169	20	}	}	PUNCT
ejpam-3464	169	21	×	×	NOUN
ejpam-3464	170	1	fkn	fkn	INTJ
ejpam-3464	171	1	[	[	X
ejpam-3464	171	2	a	a	X
ejpam-3464	171	3	]	]	X
ejpam-3464	171	4	=	=	PUNCT
ejpam-3464	171	5	∅.	∅.	VERB
ejpam-3464	171	6	hence	hence	ADV
ejpam-3464	171	7	,	,	PUNCT
ejpam-3464	171	8	by	by	ADP
ejpam-3464	171	9	theorem	theorem	NOUN
ejpam-3464	171	10	6	6	NUM
ejpam-3464	171	11	,	,	PUNCT
ejpam-3464	171	12	fg∨kn	fg∨kn	PROPN
ejpam-3464	172	1	[	[	X
ejpam-3464	172	2	(	(	PUNCT
ejpam-3464	172	3	v	v	NOUN
ejpam-3464	172	4	,	,	PUNCT
ejpam-3464	172	5	a	a	NOUN
ejpam-3464	172	6	)	)	PUNCT
ejpam-3464	172	7	]	]	PUNCT
ejpam-3464	173	1	=	=	PUNCT
ejpam-3464	173	2	fg[v]×	fg[v]×	X
ejpam-3464	173	3	{	{	PUNCT
ejpam-3464	173	4	a	a	X
ejpam-3464	173	5	}	}	PUNCT
ejpam-3464	173	6	.	.	PUNCT
ejpam-3464	174	1	definition	definition	NOUN
ejpam-3464	174	2	5	5	NUM
ejpam-3464	174	3	.	.	PUNCT
ejpam-3464	175	1	[	[	X
ejpam-3464	175	2	3	3	X
ejpam-3464	175	3	]	]	X
ejpam-3464	175	4	the	the	DET
ejpam-3464	175	5	symmetric	symmetric	ADJ
ejpam-3464	175	6	difference	difference	NOUN
ejpam-3464	175	7	g⊕h	g⊕h	NOUN
ejpam-3464	175	8	of	of	ADP
ejpam-3464	175	9	graphs	graph	NOUN
ejpam-3464	175	10	g	g	PROPN
ejpam-3464	175	11	and	and	CCONJ
ejpam-3464	175	12	h	h	NOUN
ejpam-3464	175	13	is	be	AUX
ejpam-3464	175	14	the	the	DET
ejpam-3464	175	15	graph	graph	NOUN
ejpam-3464	175	16	with	with	ADP
ejpam-3464	175	17	vertex	vertex	NOUN
ejpam-3464	175	18	set	set	VERB
ejpam-3464	175	19	v	v	NOUN
ejpam-3464	175	20	(	(	PUNCT
ejpam-3464	175	21	g	g	NOUN
ejpam-3464	175	22	)	)	PUNCT
ejpam-3464	175	23	×	×	NOUN
ejpam-3464	175	24	v	v	NOUN
ejpam-3464	175	25	(	(	PUNCT
ejpam-3464	175	26	h	h	NOUN
ejpam-3464	175	27	)	)	PUNCT
ejpam-3464	175	28	and	and	CCONJ
ejpam-3464	175	29	(	(	PUNCT
ejpam-3464	175	30	u	u	NOUN
ejpam-3464	175	31	,	,	PUNCT
ejpam-3464	175	32	v	v	NOUN
ejpam-3464	175	33	)	)	PUNCT
ejpam-3464	175	34	is	be	AUX
ejpam-3464	175	35	adjacent	adjacent	ADJ
ejpam-3464	175	36	with	with	ADP
ejpam-3464	175	37	(	(	PUNCT
ejpam-3464	175	38	u	u	NOUN
ejpam-3464	175	39	′	′	NOUN
ejpam-3464	175	40	,	,	PUNCT
ejpam-3464	175	41	v	v	NOUN
ejpam-3464	175	42	′	′	NOUN
ejpam-3464	175	43	)	)	PUNCT
ejpam-3464	176	1	whenever	whenever	SCONJ
ejpam-3464	176	2	[	[	X
ejpam-3464	176	3	uu	uu	INTJ
ejpam-3464	176	4	′	′	NUM
ejpam-3464	176	5	∈	∈	PROPN
ejpam-3464	176	6	e(g	e(g	PROPN
ejpam-3464	176	7	)	)	PUNCT
ejpam-3464	176	8	]	]	PUNCT
ejpam-3464	176	9	or	or	CCONJ
ejpam-3464	176	10	[	[	X
ejpam-3464	176	11	vv	vv	NOUN
ejpam-3464	176	12	′	′	NOUN
ejpam-3464	176	13	∈	∈	PROPN
ejpam-3464	176	14	e(h	e(h	PROPN
ejpam-3464	176	15	)	)	PUNCT
ejpam-3464	176	16	]	]	PUNCT
ejpam-3464	176	17	but	but	CCONJ
ejpam-3464	176	18	not	not	PART
ejpam-3464	176	19	both	both	PRON
ejpam-3464	176	20	.	.	PUNCT
ejpam-3464	177	1	theorem	theorem	ADJ
ejpam-3464	177	2	7	7	NUM
ejpam-3464	177	3	.	.	PUNCT
ejpam-3464	178	1	let	let	VERB
ejpam-3464	178	2	k	k	NOUN
ejpam-3464	178	3	=	=	PUNCT
ejpam-3464	178	4	g	g	PROPN
ejpam-3464	178	5	⊕	⊕	PROPN
ejpam-3464	178	6	h	h	NOUN
ejpam-3464	179	1	=	=	SYM
ejpam-3464	179	2	(	(	PUNCT
ejpam-3464	179	3	v	v	NOUN
ejpam-3464	179	4	(	(	PUNCT
ejpam-3464	179	5	k	k	NOUN
ejpam-3464	179	6	)	)	PUNCT
ejpam-3464	179	7	,	,	PUNCT
ejpam-3464	179	8	e(k	e(k	NOUN
ejpam-3464	179	9	)	)	PUNCT
ejpam-3464	179	10	)	)	PUNCT
ejpam-3464	179	11	,	,	PUNCT
ejpam-3464	179	12	where	where	SCONJ
ejpam-3464	179	13	g	g	NOUN
ejpam-3464	179	14	=	=	SYM
ejpam-3464	179	15	(	(	PUNCT
ejpam-3464	179	16	v	v	NOUN
ejpam-3464	179	17	(	(	PUNCT
ejpam-3464	179	18	g	g	NOUN
ejpam-3464	179	19	)	)	PUNCT
ejpam-3464	179	20	,	,	PUNCT
ejpam-3464	179	21	e(g	e(g	PROPN
ejpam-3464	179	22	)	)	PUNCT
ejpam-3464	179	23	)	)	PUNCT
ejpam-3464	179	24	and	and	CCONJ
ejpam-3464	179	25	h	h	NOUN
ejpam-3464	179	26	=	=	SYM
ejpam-3464	179	27	(	(	PUNCT
ejpam-3464	179	28	v	v	NOUN
ejpam-3464	179	29	(	(	PUNCT
ejpam-3464	179	30	h	h	NOUN
ejpam-3464	179	31	)	)	PUNCT
ejpam-3464	179	32	,	,	PUNCT
ejpam-3464	179	33	e(h	e(h	PROPN
ejpam-3464	179	34	)	)	PUNCT
ejpam-3464	179	35	)	)	PUNCT
ejpam-3464	179	36	.	.	PUNCT
ejpam-3464	180	1	then	then	ADV
ejpam-3464	180	2	,	,	PUNCT
ejpam-3464	180	3	for	for	SCONJ
ejpam-3464	180	4	each	each	DET
ejpam-3464	180	5	(	(	PUNCT
ejpam-3464	180	6	v	v	NOUN
ejpam-3464	180	7	,	,	PUNCT
ejpam-3464	180	8	a	a	PRON
ejpam-3464	180	9	)	)	PUNCT
ejpam-3464	180	10	∈	∈	NOUN
ejpam-3464	180	11	v	v	NOUN
ejpam-3464	180	12	(	(	PUNCT
ejpam-3464	180	13	k	k	NOUN
ejpam-3464	180	14	)	)	PUNCT
ejpam-3464	180	15	,	,	PUNCT
ejpam-3464	180	16	fk	fk	INTJ
ejpam-3464	180	17	[	[	X
ejpam-3464	180	18	(	(	PUNCT
ejpam-3464	180	19	v	v	NOUN
ejpam-3464	180	20	,	,	PUNCT
ejpam-3464	180	21	a	a	NOUN
ejpam-3464	180	22	)	)	PUNCT
ejpam-3464	180	23	]	]	PUNCT
ejpam-3464	180	24	=	=	PUNCT
ejpam-3464	181	1	[	[	X
ejpam-3464	181	2	fg(v)×	fg(v)×	X
ejpam-3464	181	3	fh	fh	PROPN
ejpam-3464	182	1	[	[	X
ejpam-3464	182	2	a	a	X
ejpam-3464	182	3	]	]	X
ejpam-3464	182	4	]	]	X
ejpam-3464	182	5	∪	∪	ADP
ejpam-3464	182	6	[	[	X
ejpam-3464	182	7	ng(v)×nh(a	ng(v)×nh(a	NOUN
ejpam-3464	182	8	)	)	PUNCT
ejpam-3464	182	9	]	]	PUNCT
ejpam-3464	182	10	∪	∪	ADP
ejpam-3464	182	11	[	[	X
ejpam-3464	182	12	fg[v]×	fg[v]×	X
ejpam-3464	182	13	{	{	PUNCT
ejpam-3464	182	14	a	a	NOUN
ejpam-3464	182	15	}	}	PUNCT
ejpam-3464	182	16	]	]	PUNCT
ejpam-3464	182	17	.	.	PUNCT
ejpam-3464	183	1	a.	a.	PROPN
ejpam-3464	183	2	gamorez	gamorez	PROPN
ejpam-3464	183	3	,	,	PUNCT
ejpam-3464	183	4	c.	c.	PROPN
ejpam-3464	183	5	nianga	nianga	PROPN
ejpam-3464	183	6	,	,	PUNCT
ejpam-3464	183	7	s.	s.	PROPN
ejpam-3464	183	8	canoy	canoy	PROPN
ejpam-3464	183	9	/	/	SYM
ejpam-3464	183	10	eur	eur	PROPN
ejpam-3464	183	11	.	.	PUNCT
ejpam-3464	184	1	j.	j.	PROPN
ejpam-3464	184	2	pure	pure	PROPN
ejpam-3464	184	3	appl	appl	PROPN
ejpam-3464	184	4	.	.	PROPN
ejpam-3464	184	5	math	math	PROPN
ejpam-3464	184	6	,	,	PUNCT
ejpam-3464	184	7	12	12	NUM
ejpam-3464	184	8	(	(	PUNCT
ejpam-3464	184	9	3	3	NUM
ejpam-3464	184	10	)	)	PUNCT
ejpam-3464	184	11	(	(	PUNCT
ejpam-3464	184	12	2019	2019	NUM
ejpam-3464	184	13	)	)	PUNCT
ejpam-3464	184	14	,	,	PUNCT
ejpam-3464	184	15	749	749	NUM
ejpam-3464	184	16	-	-	SYM
ejpam-3464	184	17	755	755	NUM
ejpam-3464	184	18	754	754	NUM
ejpam-3464	184	19	proof	proof	NOUN
ejpam-3464	184	20	.	.	PUNCT
ejpam-3464	185	1	let	let	VERB
ejpam-3464	185	2	w	w	NOUN
ejpam-3464	185	3	=	=	PUNCT
ejpam-3464	186	1	[	[	X
ejpam-3464	186	2	fg(v)×	fg(v)×	X
ejpam-3464	186	3	fh	fh	PROPN
ejpam-3464	187	1	[	[	X
ejpam-3464	187	2	a	a	X
ejpam-3464	187	3	]	]	X
ejpam-3464	187	4	]	]	X
ejpam-3464	187	5	∪	∪	ADP
ejpam-3464	187	6	[	[	X
ejpam-3464	187	7	ng(v)×nh(a	ng(v)×nh(a	NOUN
ejpam-3464	187	8	)	)	PUNCT
ejpam-3464	187	9	]	]	PUNCT
ejpam-3464	187	10	∪	∪	ADP
ejpam-3464	187	11	[	[	X
ejpam-3464	187	12	fg[v]×	fg[v]×	X
ejpam-3464	187	13	{	{	PUNCT
ejpam-3464	187	14	a	a	NOUN
ejpam-3464	187	15	}	}	PUNCT
ejpam-3464	187	16	]	]	PUNCT
ejpam-3464	187	17	.	.	PUNCT
ejpam-3464	188	1	suppose	suppose	VERB
ejpam-3464	188	2	(	(	PUNCT
ejpam-3464	188	3	v	v	NOUN
ejpam-3464	188	4	,	,	PUNCT
ejpam-3464	188	5	a	a	PRON
ejpam-3464	188	6	)	)	PUNCT
ejpam-3464	188	7	∈	∈	NOUN
ejpam-3464	188	8	v	v	NOUN
ejpam-3464	188	9	(	(	PUNCT
ejpam-3464	188	10	g⊕h	g⊕h	NOUN
ejpam-3464	188	11	)	)	PUNCT
ejpam-3464	188	12	=	=	SYM
ejpam-3464	188	13	v	v	X
ejpam-3464	188	14	(	(	PUNCT
ejpam-3464	188	15	k	k	NOUN
ejpam-3464	188	16	)	)	PUNCT
ejpam-3464	188	17	and	and	CCONJ
ejpam-3464	188	18	let	let	VERB
ejpam-3464	188	19	(	(	PUNCT
ejpam-3464	188	20	x	x	NOUN
ejpam-3464	188	21	,	,	PUNCT
ejpam-3464	188	22	q	q	NOUN
ejpam-3464	188	23	)	)	PUNCT
ejpam-3464	188	24	∈	∈	NOUN
ejpam-3464	188	25	fk	fk	INTJ
ejpam-3464	189	1	[	[	X
ejpam-3464	189	2	(	(	PUNCT
ejpam-3464	189	3	v	v	NOUN
ejpam-3464	189	4	,	,	PUNCT
ejpam-3464	189	5	a	a	NOUN
ejpam-3464	189	6	)	)	PUNCT
ejpam-3464	189	7	]	]	PUNCT
ejpam-3464	189	8	.	.	PUNCT
ejpam-3464	190	1	then	then	ADV
ejpam-3464	190	2	(	(	PUNCT
ejpam-3464	190	3	v	v	NOUN
ejpam-3464	190	4	,	,	PUNCT
ejpam-3464	190	5	a	a	PRON
ejpam-3464	190	6	)	)	PUNCT
ejpam-3464	190	7	6=	6=	NUM
ejpam-3464	190	8	(	(	PUNCT
ejpam-3464	190	9	x	x	X
ejpam-3464	190	10	,	,	PUNCT
ejpam-3464	190	11	q	q	NOUN
ejpam-3464	190	12	)	)	PUNCT
ejpam-3464	190	13	and	and	CCONJ
ejpam-3464	190	14	dk((v	dk((v	PROPN
ejpam-3464	190	15	,	,	PUNCT
ejpam-3464	190	16	a	a	PRON
ejpam-3464	190	17	)	)	PUNCT
ejpam-3464	190	18	,	,	PUNCT
ejpam-3464	190	19	(	(	PUNCT
ejpam-3464	190	20	x	x	X
ejpam-3464	190	21	,	,	PUNCT
ejpam-3464	190	22	q	q	NOUN
ejpam-3464	190	23	)	)	PUNCT
ejpam-3464	190	24	)	)	PUNCT
ejpam-3464	191	1	6=	6=	ADP
ejpam-3464	191	2	1	1	X
ejpam-3464	191	3	.	.	PUNCT
ejpam-3464	191	4	now	now	ADV
ejpam-3464	191	5	,	,	PUNCT
ejpam-3464	191	6	consider	consider	VERB
ejpam-3464	191	7	the	the	DET
ejpam-3464	191	8	following	follow	VERB
ejpam-3464	191	9	cases	case	NOUN
ejpam-3464	191	10	:	:	PUNCT
ejpam-3464	191	11	case	case	NOUN
ejpam-3464	191	12	1	1	X
ejpam-3464	191	13	.	.	X
ejpam-3464	191	14	assume	assume	VERB
ejpam-3464	191	15	x	x	X
ejpam-3464	192	1	=	=	PUNCT
ejpam-3464	192	2	v.	v.	ADP
ejpam-3464	192	3	then	then	ADV
ejpam-3464	192	4	q	q	X
ejpam-3464	193	1	6=	6=	ADP
ejpam-3464	193	2	a	a	PRON
ejpam-3464	193	3	and	and	CCONJ
ejpam-3464	193	4	dh(q	dh(q	ADV
ejpam-3464	193	5	,	,	PUNCT
ejpam-3464	193	6	a	a	PRON
ejpam-3464	193	7	)	)	PUNCT
ejpam-3464	193	8	=	=	SYM
ejpam-3464	193	9	dk((x	dk((x	ADJ
ejpam-3464	193	10	,	,	PUNCT
ejpam-3464	193	11	q	q	NOUN
ejpam-3464	193	12	)	)	PUNCT
ejpam-3464	193	13	,	,	PUNCT
ejpam-3464	193	14	(	(	PUNCT
ejpam-3464	193	15	x	x	X
ejpam-3464	193	16	,	,	PUNCT
ejpam-3464	193	17	a	a	NOUN
ejpam-3464	193	18	)	)	PUNCT
ejpam-3464	193	19	)	)	PUNCT
ejpam-3464	194	1	6=	6=	ADP
ejpam-3464	194	2	1	1	X
ejpam-3464	194	3	.	.	PUNCT
ejpam-3464	195	1	hence	hence	ADV
ejpam-3464	195	2	,	,	PUNCT
ejpam-3464	195	3	q	q	PROPN
ejpam-3464	195	4	∈	∈	PROPN
ejpam-3464	195	5	fh	fh	PROPN
ejpam-3464	196	1	[	[	X
ejpam-3464	196	2	a	a	X
ejpam-3464	196	3	]	]	X
ejpam-3464	196	4	and	and	CCONJ
ejpam-3464	196	5	so	so	ADV
ejpam-3464	196	6	,	,	PUNCT
ejpam-3464	196	7	(	(	PUNCT
ejpam-3464	196	8	x	x	NOUN
ejpam-3464	196	9	,	,	PUNCT
ejpam-3464	196	10	q	q	ADJ
ejpam-3464	196	11	)	)	PUNCT
ejpam-3464	196	12	∈	∈	PROPN
ejpam-3464	196	13	{	{	PUNCT
ejpam-3464	196	14	v	v	NOUN
ejpam-3464	196	15	}	}	PUNCT
ejpam-3464	196	16	×	×	NOUN
ejpam-3464	196	17	fh	fh	PROPN
ejpam-3464	197	1	[	[	X
ejpam-3464	197	2	a	a	X
ejpam-3464	197	3	]	]	X
ejpam-3464	197	4	.	.	PUNCT
ejpam-3464	198	1	case	case	NOUN
ejpam-3464	198	2	2	2	NUM
ejpam-3464	198	3	.	.	X
ejpam-3464	198	4	assume	assume	VERB
ejpam-3464	198	5	that	that	SCONJ
ejpam-3464	198	6	x	x	PRON
ejpam-3464	198	7	6=	6=	PROPN
ejpam-3464	198	8	v.	v.	CCONJ
ejpam-3464	198	9	suppose	suppose	VERB
ejpam-3464	198	10	q	q	X
ejpam-3464	198	11	=	=	PUNCT
ejpam-3464	198	12	a.	a.	NOUN
ejpam-3464	198	13	then	then	ADV
ejpam-3464	198	14	dg(x	dg(x	NUM
ejpam-3464	198	15	,	,	PUNCT
ejpam-3464	198	16	v	v	NOUN
ejpam-3464	198	17	)	)	PUNCT
ejpam-3464	198	18	=	=	SYM
ejpam-3464	198	19	dk((v	dk((v	PROPN
ejpam-3464	198	20	,	,	PUNCT
ejpam-3464	198	21	a	a	PRON
ejpam-3464	198	22	)	)	PUNCT
ejpam-3464	198	23	,	,	PUNCT
ejpam-3464	198	24	(	(	PUNCT
ejpam-3464	198	25	x	x	X
ejpam-3464	198	26	,	,	PUNCT
ejpam-3464	198	27	a	a	NOUN
ejpam-3464	198	28	)	)	PUNCT
ejpam-3464	198	29	)	)	PUNCT
ejpam-3464	198	30	6=	6=	ADP
ejpam-3464	199	1	1	1	X
ejpam-3464	199	2	.	.	PUNCT
ejpam-3464	200	1	this	this	PRON
ejpam-3464	200	2	implies	imply	VERB
ejpam-3464	200	3	that	that	SCONJ
ejpam-3464	200	4	x	x	PROPN
ejpam-3464	200	5	∈	∈	PROPN
ejpam-3464	200	6	fg[v	fg[v	PROPN
ejpam-3464	200	7	]	]	X
ejpam-3464	200	8	and	and	CCONJ
ejpam-3464	200	9	(	(	PUNCT
ejpam-3464	200	10	x	x	NOUN
ejpam-3464	200	11	,	,	PUNCT
ejpam-3464	200	12	q	q	ADJ
ejpam-3464	200	13	)	)	PUNCT
ejpam-3464	200	14	∈	∈	PROPN
ejpam-3464	200	15	fg[v	fg[v	PROPN
ejpam-3464	200	16	]	]	X
ejpam-3464	200	17	×	×	NOUN
ejpam-3464	200	18	{	{	PUNCT
ejpam-3464	200	19	a	a	NOUN
ejpam-3464	200	20	}	}	PUNCT
ejpam-3464	200	21	.	.	PUNCT
ejpam-3464	201	1	suppose	suppose	VERB
ejpam-3464	201	2	q	q	X
ejpam-3464	202	1	6=	6=	NUM
ejpam-3464	202	2	a.	a.	NOUN
ejpam-3464	202	3	if	if	SCONJ
ejpam-3464	202	4	x	x	PROPN
ejpam-3464	202	5	∈	∈	NOUN
ejpam-3464	202	6	ng(v	ng(v	NOUN
ejpam-3464	202	7	)	)	PUNCT
ejpam-3464	202	8	,	,	PUNCT
ejpam-3464	202	9	then	then	ADV
ejpam-3464	202	10	q	q	PROPN
ejpam-3464	202	11	∈	∈	PROPN
ejpam-3464	202	12	nh(a	nh(a	NUM
ejpam-3464	202	13	)	)	PUNCT
ejpam-3464	202	14	.	.	PUNCT
ejpam-3464	203	1	if	if	SCONJ
ejpam-3464	203	2	x	x	PROPN
ejpam-3464	203	3	∈	∈	PROPN
ejpam-3464	203	4	fg[v	fg[v	PROPN
ejpam-3464	203	5	]	]	X
ejpam-3464	203	6	,	,	PUNCT
ejpam-3464	203	7	then	then	ADV
ejpam-3464	203	8	q	q	PROPN
ejpam-3464	203	9	/∈	/∈	PUNCT
ejpam-3464	203	10	nh(a	nh(a	NUM
ejpam-3464	203	11	)	)	PUNCT
ejpam-3464	203	12	and	and	CCONJ
ejpam-3464	203	13	so	so	ADV
ejpam-3464	203	14	,	,	PUNCT
ejpam-3464	203	15	q	q	PROPN
ejpam-3464	203	16	∈	∈	PROPN
ejpam-3464	203	17	fh	fh	PROPN
ejpam-3464	204	1	[	[	X
ejpam-3464	204	2	a	a	X
ejpam-3464	204	3	]	]	X
ejpam-3464	204	4	.	.	PUNCT
ejpam-3464	205	1	hence	hence	ADV
ejpam-3464	205	2	,	,	PUNCT
ejpam-3464	205	3	fk	fk	INTJ
ejpam-3464	205	4	[	[	X
ejpam-3464	205	5	(	(	PUNCT
ejpam-3464	205	6	v	v	NOUN
ejpam-3464	205	7	,	,	PUNCT
ejpam-3464	205	8	a	a	NOUN
ejpam-3464	205	9	)	)	PUNCT
ejpam-3464	205	10	]	]	PUNCT
ejpam-3464	206	1	⊆	⊆	NUM
ejpam-3464	206	2	[	[	X
ejpam-3464	206	3	{	{	PUNCT
ejpam-3464	206	4	v	v	NOUN
ejpam-3464	206	5	}	}	PUNCT
ejpam-3464	206	6	×	×	NOUN
ejpam-3464	206	7	fh	fh	PROPN
ejpam-3464	207	1	[	[	X
ejpam-3464	207	2	a	a	X
ejpam-3464	207	3	]	]	X
ejpam-3464	207	4	]	]	X
ejpam-3464	207	5	∪	∪	ADP
ejpam-3464	207	6	[	[	X
ejpam-3464	207	7	fg[v]×	fg[v]×	X
ejpam-3464	207	8	{	{	PUNCT
ejpam-3464	207	9	a	a	X
ejpam-3464	207	10	}	}	PUNCT
ejpam-3464	207	11	]	]	PUNCT
ejpam-3464	207	12	∪	∪	ADP
ejpam-3464	207	13	[	[	X
ejpam-3464	207	14	ng(v)×nh(a	ng(v)×nh(a	NOUN
ejpam-3464	207	15	)	)	PUNCT
ejpam-3464	207	16	]	]	X
ejpam-3464	207	17	]	]	PUNCT
ejpam-3464	207	18	∪	∪	ADP
ejpam-3464	207	19	[	[	X
ejpam-3464	207	20	fg[v]×	fg[v]×	X
ejpam-3464	207	21	fh	fh	PROPN
ejpam-3464	207	22	[	[	X
ejpam-3464	207	23	a	a	X
ejpam-3464	207	24	]	]	X
ejpam-3464	207	25	]	]	X
ejpam-3464	207	26	=	=	PUNCT
ejpam-3464	208	1	[	[	X
ejpam-3464	208	2	fg(v)×	fg(v)×	X
ejpam-3464	208	3	fh	fh	PROPN
ejpam-3464	209	1	[	[	X
ejpam-3464	209	2	a	a	X
ejpam-3464	209	3	]	]	X
ejpam-3464	209	4	]	]	X
ejpam-3464	209	5	∪	∪	ADP
ejpam-3464	209	6	[	[	X
ejpam-3464	209	7	ng(v)×nh(a	ng(v)×nh(a	NOUN
ejpam-3464	209	8	)	)	PUNCT
ejpam-3464	209	9	]	]	PUNCT
ejpam-3464	209	10	∪	∪	ADP
ejpam-3464	209	11	[	[	X
ejpam-3464	209	12	fg[v]×	fg[v]×	X
ejpam-3464	209	13	{	{	PUNCT
ejpam-3464	209	14	a	a	NOUN
ejpam-3464	209	15	}	}	PUNCT
ejpam-3464	209	16	]	]	PUNCT
ejpam-3464	209	17	=	=	SYM
ejpam-3464	209	18	w.	w.	NOUN
ejpam-3464	209	19	conversely	conversely	ADV
ejpam-3464	209	20	,	,	PUNCT
ejpam-3464	209	21	let	let	VERB
ejpam-3464	209	22	(	(	PUNCT
ejpam-3464	209	23	b	b	X
ejpam-3464	209	24	,	,	PUNCT
ejpam-3464	209	25	c	c	NOUN
ejpam-3464	209	26	)	)	PUNCT
ejpam-3464	209	27	∈	∈	PROPN
ejpam-3464	209	28	w.	w.	NOUN
ejpam-3464	210	1	if	if	SCONJ
ejpam-3464	210	2	(	(	PUNCT
ejpam-3464	210	3	b	b	NOUN
ejpam-3464	210	4	,	,	PUNCT
ejpam-3464	210	5	c	c	NOUN
ejpam-3464	210	6	)	)	PUNCT
ejpam-3464	210	7	∈	∈	NOUN
ejpam-3464	210	8	fg(v	fg(v	X
ejpam-3464	210	9	)	)	PUNCT
ejpam-3464	210	10	×	×	NOUN
ejpam-3464	210	11	fh	fh	PROPN
ejpam-3464	211	1	[	[	X
ejpam-3464	211	2	a	a	X
ejpam-3464	211	3	]	]	X
ejpam-3464	211	4	,	,	PUNCT
ejpam-3464	211	5	then	then	ADV
ejpam-3464	211	6	c	c	PROPN
ejpam-3464	211	7	6=	6=	PROPN
ejpam-3464	211	8	a	a	PRON
ejpam-3464	211	9	,	,	PUNCT
ejpam-3464	211	10	bv	bv	PROPN
ejpam-3464	211	11	/∈	/∈	PROPN
ejpam-3464	211	12	e(g	e(g	PROPN
ejpam-3464	211	13	)	)	PUNCT
ejpam-3464	211	14	,	,	PUNCT
ejpam-3464	211	15	and	and	CCONJ
ejpam-3464	211	16	ac	ac	PROPN
ejpam-3464	211	17	/∈	/∈	PUNCT
ejpam-3464	212	1	e(h	e(h	PROPN
ejpam-3464	212	2	)	)	PUNCT
ejpam-3464	212	3	.	.	PUNCT
ejpam-3464	213	1	hence	hence	ADV
ejpam-3464	213	2	,	,	PUNCT
ejpam-3464	213	3	(	(	PUNCT
ejpam-3464	213	4	b	b	X
ejpam-3464	213	5	,	,	PUNCT
ejpam-3464	213	6	c	c	NOUN
ejpam-3464	213	7	)	)	PUNCT
ejpam-3464	213	8	6=	6=	ADP
ejpam-3464	213	9	(	(	PUNCT
ejpam-3464	213	10	v	v	NOUN
ejpam-3464	213	11	,	,	PUNCT
ejpam-3464	213	12	a	a	PRON
ejpam-3464	213	13	)	)	PUNCT
ejpam-3464	213	14	and	and	CCONJ
ejpam-3464	213	15	,	,	PUNCT
ejpam-3464	213	16	by	by	ADP
ejpam-3464	213	17	definition	definition	NOUN
ejpam-3464	213	18	5	5	NUM
ejpam-3464	213	19	,	,	PUNCT
ejpam-3464	213	20	(	(	PUNCT
ejpam-3464	213	21	b	b	NOUN
ejpam-3464	213	22	,	,	PUNCT
ejpam-3464	213	23	c)(v	c)(v	NOUN
ejpam-3464	213	24	,	,	PUNCT
ejpam-3464	213	25	a	a	PRON
ejpam-3464	213	26	)	)	PUNCT
ejpam-3464	213	27	/∈	/∈	PUNCT
ejpam-3464	213	28	e(k	e(k	NOUN
ejpam-3464	213	29	)	)	PUNCT
ejpam-3464	213	30	.	.	PUNCT
ejpam-3464	214	1	thus	thus	ADV
ejpam-3464	214	2	,	,	PUNCT
ejpam-3464	214	3	(	(	PUNCT
ejpam-3464	214	4	b	b	X
ejpam-3464	214	5	,	,	PUNCT
ejpam-3464	214	6	c	c	NOUN
ejpam-3464	214	7	)	)	PUNCT
ejpam-3464	214	8	∈	∈	NOUN
ejpam-3464	214	9	fk	fk	INTJ
ejpam-3464	215	1	[	[	X
ejpam-3464	215	2	(	(	PUNCT
ejpam-3464	215	3	v	v	NOUN
ejpam-3464	215	4	,	,	PUNCT
ejpam-3464	215	5	a	a	NOUN
ejpam-3464	215	6	)	)	PUNCT
ejpam-3464	215	7	]	]	PUNCT
ejpam-3464	215	8	.	.	PUNCT
ejpam-3464	216	1	also	also	ADV
ejpam-3464	216	2	,	,	PUNCT
ejpam-3464	216	3	if	if	SCONJ
ejpam-3464	216	4	(	(	PUNCT
ejpam-3464	216	5	b	b	NOUN
ejpam-3464	216	6	,	,	PUNCT
ejpam-3464	216	7	c	c	NOUN
ejpam-3464	216	8	)	)	PUNCT
ejpam-3464	216	9	∈	∈	NOUN
ejpam-3464	216	10	ng(v	ng(v	PUNCT
ejpam-3464	216	11	)	)	PUNCT
ejpam-3464	216	12	×	×	NOUN
ejpam-3464	216	13	nh(a	nh(a	NUM
ejpam-3464	216	14	)	)	PUNCT
ejpam-3464	216	15	,	,	PUNCT
ejpam-3464	216	16	then	then	ADV
ejpam-3464	216	17	(	(	PUNCT
ejpam-3464	216	18	b	b	X
ejpam-3464	216	19	,	,	PUNCT
ejpam-3464	216	20	c	c	NOUN
ejpam-3464	216	21	)	)	PUNCT
ejpam-3464	216	22	6=	6=	ADP
ejpam-3464	216	23	(	(	PUNCT
ejpam-3464	216	24	v	v	NOUN
ejpam-3464	216	25	,	,	PUNCT
ejpam-3464	216	26	a	a	PRON
ejpam-3464	216	27	)	)	PUNCT
ejpam-3464	216	28	and	and	CCONJ
ejpam-3464	216	29	by	by	ADP
ejpam-3464	216	30	definition	definition	NOUN
ejpam-3464	216	31	5	5	NUM
ejpam-3464	216	32	,	,	PUNCT
ejpam-3464	216	33	(	(	PUNCT
ejpam-3464	216	34	b	b	NOUN
ejpam-3464	216	35	,	,	PUNCT
ejpam-3464	216	36	c)(v	c)(v	NOUN
ejpam-3464	216	37	,	,	PUNCT
ejpam-3464	216	38	a	a	PRON
ejpam-3464	216	39	)	)	PUNCT
ejpam-3464	216	40	/∈	/∈	PUNCT
ejpam-3464	216	41	e(k	e(k	NOUN
ejpam-3464	216	42	)	)	PUNCT
ejpam-3464	216	43	.	.	PUNCT
ejpam-3464	217	1	hence	hence	ADV
ejpam-3464	217	2	,	,	PUNCT
ejpam-3464	217	3	(	(	PUNCT
ejpam-3464	217	4	b	b	X
ejpam-3464	217	5	,	,	PUNCT
ejpam-3464	217	6	c	c	NOUN
ejpam-3464	217	7	)	)	PUNCT
ejpam-3464	217	8	∈	∈	NOUN
ejpam-3464	217	9	fk	fk	INTJ
ejpam-3464	218	1	[	[	X
ejpam-3464	218	2	(	(	PUNCT
ejpam-3464	218	3	v	v	NOUN
ejpam-3464	218	4	,	,	PUNCT
ejpam-3464	218	5	a	a	NOUN
ejpam-3464	218	6	)	)	PUNCT
ejpam-3464	218	7	]	]	PUNCT
ejpam-3464	218	8	.	.	PUNCT
ejpam-3464	219	1	finally	finally	ADV
ejpam-3464	219	2	,	,	PUNCT
ejpam-3464	219	3	if	if	SCONJ
ejpam-3464	219	4	(	(	PUNCT
ejpam-3464	219	5	b	b	NOUN
ejpam-3464	219	6	,	,	PUNCT
ejpam-3464	219	7	c	c	NOUN
ejpam-3464	219	8	)	)	PUNCT
ejpam-3464	219	9	∈	∈	PROPN
ejpam-3464	219	10	fg[v]×	fg[v]×	ADP
ejpam-3464	219	11	{	{	PUNCT
ejpam-3464	219	12	a	a	X
ejpam-3464	219	13	}	}	PUNCT
ejpam-3464	219	14	,	,	PUNCT
ejpam-3464	219	15	then	then	ADV
ejpam-3464	219	16	b	b	PROPN
ejpam-3464	219	17	6=	6=	PROPN
ejpam-3464	219	18	v	v	PROPN
ejpam-3464	219	19	,	,	PUNCT
ejpam-3464	219	20	bv	bv	PROPN
ejpam-3464	219	21	/∈	/∈	PROPN
ejpam-3464	219	22	e(g	e(g	PROPN
ejpam-3464	219	23	)	)	PUNCT
ejpam-3464	219	24	,	,	PUNCT
ejpam-3464	219	25	and	and	CCONJ
ejpam-3464	219	26	ac	ac	PROPN
ejpam-3464	219	27	/∈	/∈	PUNCT
ejpam-3464	220	1	e(h	e(h	PROPN
ejpam-3464	220	2	)	)	PUNCT
ejpam-3464	220	3	.	.	PUNCT
ejpam-3464	221	1	it	it	PRON
ejpam-3464	221	2	follows	follow	VERB
ejpam-3464	221	3	that	that	SCONJ
ejpam-3464	221	4	(	(	PUNCT
ejpam-3464	221	5	b	b	X
ejpam-3464	221	6	,	,	PUNCT
ejpam-3464	221	7	c	c	NOUN
ejpam-3464	221	8	)	)	PUNCT
ejpam-3464	221	9	6=	6=	ADP
ejpam-3464	221	10	(	(	PUNCT
ejpam-3464	221	11	v	v	NOUN
ejpam-3464	221	12	,	,	PUNCT
ejpam-3464	221	13	a	a	PRON
ejpam-3464	221	14	)	)	PUNCT
ejpam-3464	221	15	and	and	CCONJ
ejpam-3464	221	16	by	by	ADP
ejpam-3464	221	17	definition	definition	NOUN
ejpam-3464	221	18	5	5	NUM
ejpam-3464	221	19	,	,	PUNCT
ejpam-3464	221	20	(	(	PUNCT
ejpam-3464	221	21	b	b	NOUN
ejpam-3464	221	22	,	,	PUNCT
ejpam-3464	221	23	c)(v	c)(v	NOUN
ejpam-3464	221	24	,	,	PUNCT
ejpam-3464	221	25	a	a	PRON
ejpam-3464	221	26	)	)	PUNCT
ejpam-3464	221	27	/∈	/∈	PUNCT
ejpam-3464	222	1	e(k	e(k	NOUN
ejpam-3464	222	2	)	)	PUNCT
ejpam-3464	222	3	which	which	PRON
ejpam-3464	222	4	shows	show	VERB
ejpam-3464	222	5	that	that	SCONJ
ejpam-3464	222	6	(	(	PUNCT
ejpam-3464	222	7	b	b	X
ejpam-3464	222	8	,	,	PUNCT
ejpam-3464	222	9	c	c	NOUN
ejpam-3464	222	10	)	)	PUNCT
ejpam-3464	222	11	∈	∈	NOUN
ejpam-3464	222	12	fk	fk	INTJ
ejpam-3464	223	1	[	[	X
ejpam-3464	223	2	(	(	PUNCT
ejpam-3464	223	3	v	v	NOUN
ejpam-3464	223	4	,	,	PUNCT
ejpam-3464	223	5	a	a	NOUN
ejpam-3464	223	6	)	)	PUNCT
ejpam-3464	223	7	]	]	PUNCT
ejpam-3464	223	8	.	.	PUNCT
ejpam-3464	224	1	thus	thus	ADV
ejpam-3464	224	2	,	,	PUNCT
ejpam-3464	224	3	w	w	PROPN
ejpam-3464	224	4	⊆	⊆	NUM
ejpam-3464	224	5	fk	fk	NOUN
ejpam-3464	224	6	[	[	X
ejpam-3464	224	7	(	(	PUNCT
ejpam-3464	224	8	v	v	NOUN
ejpam-3464	224	9	,	,	PUNCT
ejpam-3464	224	10	a	a	NOUN
ejpam-3464	224	11	)	)	PUNCT
ejpam-3464	224	12	]	]	PUNCT
ejpam-3464	224	13	.	.	PUNCT
ejpam-3464	225	1	therefore	therefore	ADV
ejpam-3464	225	2	,	,	PUNCT
ejpam-3464	225	3	fk	fk	INTJ
ejpam-3464	225	4	[	[	X
ejpam-3464	225	5	(	(	PUNCT
ejpam-3464	225	6	v	v	NOUN
ejpam-3464	225	7	,	,	PUNCT
ejpam-3464	225	8	a	a	NOUN
ejpam-3464	225	9	)	)	PUNCT
ejpam-3464	225	10	]	]	PUNCT
ejpam-3464	225	11	=	=	PUNCT
ejpam-3464	225	12	w.	w.	PROPN
ejpam-3464	225	13	corollary	corollary	NOUN
ejpam-3464	225	14	3	3	X
ejpam-3464	225	15	.	.	PUNCT
ejpam-3464	226	1	let	let	VERB
ejpam-3464	226	2	g	g	NOUN
ejpam-3464	226	3	be	be	AUX
ejpam-3464	226	4	any	any	DET
ejpam-3464	226	5	graph	graph	NOUN
ejpam-3464	226	6	and	and	CCONJ
ejpam-3464	226	7	let	let	VERB
ejpam-3464	226	8	(	(	PUNCT
ejpam-3464	226	9	v	v	NOUN
ejpam-3464	226	10	,	,	PUNCT
ejpam-3464	226	11	a	a	PRON
ejpam-3464	226	12	)	)	PUNCT
ejpam-3464	226	13	∈	∈	NOUN
ejpam-3464	226	14	v	v	NOUN
ejpam-3464	226	15	(	(	PUNCT
ejpam-3464	226	16	g	g	PROPN
ejpam-3464	226	17	⊕	⊕	PROPN
ejpam-3464	226	18	kn	kn	PROPN
ejpam-3464	226	19	)	)	PUNCT
ejpam-3464	226	20	.	.	PUNCT
ejpam-3464	227	1	then	then	ADV
ejpam-3464	227	2	fg⊕kn	fg⊕kn	PROPN
ejpam-3464	228	1	[	[	X
ejpam-3464	228	2	(	(	PUNCT
ejpam-3464	228	3	v	v	NOUN
ejpam-3464	228	4	,	,	PUNCT
ejpam-3464	228	5	a	a	NOUN
ejpam-3464	228	6	)	)	PUNCT
ejpam-3464	228	7	]	]	PUNCT
ejpam-3464	229	1	=	=	PUNCT
ejpam-3464	229	2	(	(	PUNCT
ejpam-3464	229	3	ng(v)×	ng(v)×	PROPN
ejpam-3464	229	4	[	[	X
ejpam-3464	229	5	v	v	X
ejpam-3464	229	6	(	(	PUNCT
ejpam-3464	229	7	kn	kn	PROPN
ejpam-3464	229	8	)	)	PUNCT
ejpam-3464	229	9	\	\	PROPN
ejpam-3464	229	10	{	{	PUNCT
ejpam-3464	229	11	a	a	NOUN
ejpam-3464	229	12	}	}	PUNCT
ejpam-3464	229	13	]	]	X
ejpam-3464	229	14	)	)	PUNCT
ejpam-3464	229	15	∪	∪	ADP
ejpam-3464	229	16	(	(	PUNCT
ejpam-3464	229	17	fg[v]×	fg[v]×	X
ejpam-3464	229	18	{	{	PUNCT
ejpam-3464	229	19	a	a	NOUN
ejpam-3464	229	20	}	}	PUNCT
ejpam-3464	229	21	)	)	PUNCT
ejpam-3464	229	22	.	.	PUNCT
ejpam-3464	230	1	proof	proof	NOUN
ejpam-3464	230	2	.	.	PUNCT
ejpam-3464	231	1	since	since	SCONJ
ejpam-3464	231	2	fkn	fkn	PROPN
ejpam-3464	232	1	[	[	X
ejpam-3464	232	2	a	a	X
ejpam-3464	232	3	]	]	X
ejpam-3464	232	4	=	=	SYM
ejpam-3464	232	5	∅	∅	NOUN
ejpam-3464	232	6	and	and	CCONJ
ejpam-3464	232	7	nkn(a	nkn(a	NUM
ejpam-3464	232	8	)	)	PUNCT
ejpam-3464	233	1	=	=	SYM
ejpam-3464	233	2	v	v	X
ejpam-3464	233	3	(	(	PUNCT
ejpam-3464	233	4	kn	kn	PROPN
ejpam-3464	233	5	)	)	PUNCT
ejpam-3464	233	6	\	\	PROPN
ejpam-3464	233	7	{	{	PUNCT
ejpam-3464	233	8	a	a	X
ejpam-3464	233	9	}	}	PUNCT
ejpam-3464	233	10	,	,	PUNCT
ejpam-3464	233	11	it	it	PRON
ejpam-3464	233	12	follows	follow	VERB
ejpam-3464	233	13	from	from	ADP
ejpam-3464	233	14	theorem	theorem	ADJ
ejpam-3464	233	15	7	7	NUM
ejpam-3464	233	16	that	that	SCONJ
ejpam-3464	233	17	fg⊕kn	fg⊕kn	PROPN
ejpam-3464	234	1	[	[	X
ejpam-3464	234	2	(	(	PUNCT
ejpam-3464	234	3	v	v	NOUN
ejpam-3464	234	4	,	,	PUNCT
ejpam-3464	234	5	a	a	NOUN
ejpam-3464	234	6	)	)	PUNCT
ejpam-3464	234	7	]	]	PUNCT
ejpam-3464	235	1	=	=	PUNCT
ejpam-3464	235	2	(	(	PUNCT
ejpam-3464	235	3	ng(v)×	ng(v)×	PROPN
ejpam-3464	235	4	[	[	X
ejpam-3464	235	5	v	v	X
ejpam-3464	235	6	(	(	PUNCT
ejpam-3464	235	7	kn	kn	PROPN
ejpam-3464	235	8	)	)	PUNCT
ejpam-3464	235	9	\	\	PROPN
ejpam-3464	235	10	{	{	PUNCT
ejpam-3464	235	11	a	a	NOUN
ejpam-3464	235	12	}	}	PUNCT
ejpam-3464	235	13	]	]	X
ejpam-3464	235	14	)	)	PUNCT
ejpam-3464	235	15	∪	∪	ADP
ejpam-3464	235	16	(	(	PUNCT
ejpam-3464	235	17	fg[v]×	fg[v]×	X
ejpam-3464	235	18	{	{	PUNCT
ejpam-3464	235	19	a	a	NOUN
ejpam-3464	235	20	}	}	PUNCT
ejpam-3464	235	21	)	)	PUNCT
ejpam-3464	235	22	.	.	PUNCT
ejpam-3464	236	1	definition	definition	NOUN
ejpam-3464	236	2	6	6	NUM
ejpam-3464	236	3	.	.	PUNCT
ejpam-3464	237	1	[	[	X
ejpam-3464	237	2	3	3	X
ejpam-3464	237	3	]	]	PUNCT
ejpam-3464	237	4	the	the	DET
ejpam-3464	237	5	strong	strong	ADJ
ejpam-3464	237	6	product	product	NOUN
ejpam-3464	237	7	g⊗h	g⊗h	NOUN
ejpam-3464	237	8	of	of	ADP
ejpam-3464	237	9	graphs	graph	NOUN
ejpam-3464	237	10	g	g	PROPN
ejpam-3464	237	11	and	and	CCONJ
ejpam-3464	237	12	h	h	NOUN
ejpam-3464	237	13	is	be	AUX
ejpam-3464	237	14	the	the	DET
ejpam-3464	237	15	graph	graph	NOUN
ejpam-3464	237	16	with	with	ADP
ejpam-3464	237	17	vertex	vertex	NOUN
ejpam-3464	237	18	set	set	VERB
ejpam-3464	237	19	v	v	NOUN
ejpam-3464	237	20	(	(	PUNCT
ejpam-3464	237	21	g)×	g)×	NOUN
ejpam-3464	237	22	v	v	NOUN
ejpam-3464	237	23	(	(	PUNCT
ejpam-3464	237	24	h	h	NOUN
ejpam-3464	237	25	)	)	PUNCT
ejpam-3464	237	26	and	and	CCONJ
ejpam-3464	237	27	(	(	PUNCT
ejpam-3464	237	28	u	u	NOUN
ejpam-3464	237	29	,	,	PUNCT
ejpam-3464	237	30	v	v	NOUN
ejpam-3464	237	31	)	)	PUNCT
ejpam-3464	237	32	is	be	AUX
ejpam-3464	237	33	adjacent	adjacent	ADJ
ejpam-3464	237	34	with	with	ADP
ejpam-3464	237	35	(	(	PUNCT
ejpam-3464	237	36	u	u	NOUN
ejpam-3464	237	37	′	′	NOUN
ejpam-3464	237	38	,	,	PUNCT
ejpam-3464	237	39	v	v	NOUN
ejpam-3464	237	40	′	′	NOUN
ejpam-3464	237	41	)	)	PUNCT
ejpam-3464	238	1	whenever	whenever	SCONJ
ejpam-3464	238	2	[	[	X
ejpam-3464	238	3	uu	uu	INTJ
ejpam-3464	238	4	′	′	NUM
ejpam-3464	238	5	∈	∈	PROPN
ejpam-3464	238	6	e(g	e(g	PROPN
ejpam-3464	238	7	)	)	PUNCT
ejpam-3464	238	8	and	and	CCONJ
ejpam-3464	238	9	v	v	X
ejpam-3464	238	10	=	=	SYM
ejpam-3464	238	11	v	v	NOUN
ejpam-3464	238	12	′	′	NOUN
ejpam-3464	238	13	]	]	PUNCT
ejpam-3464	238	14	or	or	CCONJ
ejpam-3464	238	15	[	[	X
ejpam-3464	238	16	vv	vv	NOUN
ejpam-3464	238	17	′	′	NOUN
ejpam-3464	238	18	∈	∈	PROPN
ejpam-3464	238	19	e(h	e(h	PROPN
ejpam-3464	238	20	)	)	PUNCT
ejpam-3464	238	21	and	and	CCONJ
ejpam-3464	238	22	u	u	X
ejpam-3464	238	23	=	=	SYM
ejpam-3464	238	24	u	u	NOUN
ejpam-3464	238	25	′	′	NOUN
ejpam-3464	238	26	]	]	PUNCT
ejpam-3464	238	27	or	or	CCONJ
ejpam-3464	238	28	[	[	X
ejpam-3464	238	29	uu	uu	INTJ
ejpam-3464	238	30	′	′	NUM
ejpam-3464	238	31	∈	∈	PROPN
ejpam-3464	238	32	e(g	e(g	PROPN
ejpam-3464	238	33	)	)	PUNCT
ejpam-3464	238	34	and	and	CCONJ
ejpam-3464	238	35	vv	vv	INTJ
ejpam-3464	238	36	′	′	NUM
ejpam-3464	238	37	∈	∈	PROPN
ejpam-3464	238	38	e(h	e(h	PROPN
ejpam-3464	238	39	)	)	PUNCT
ejpam-3464	238	40	]	]	PUNCT
ejpam-3464	238	41	.	.	PUNCT
ejpam-3464	239	1	theorem	theorem	ADJ
ejpam-3464	239	2	8	8	NUM
ejpam-3464	239	3	.	.	PUNCT
ejpam-3464	240	1	let	let	VERB
ejpam-3464	240	2	k	k	NOUN
ejpam-3464	240	3	=	=	PUNCT
ejpam-3464	241	1	g	g	PROPN
ejpam-3464	241	2	⊗	⊗	PROPN
ejpam-3464	241	3	h	h	NOUN
ejpam-3464	241	4	=	=	PUNCT
ejpam-3464	242	1	(	(	PUNCT
ejpam-3464	242	2	v	v	NOUN
ejpam-3464	242	3	(	(	PUNCT
ejpam-3464	242	4	k	k	NOUN
ejpam-3464	242	5	)	)	PUNCT
ejpam-3464	242	6	,	,	PUNCT
ejpam-3464	242	7	e(k	e(k	NOUN
ejpam-3464	242	8	)	)	PUNCT
ejpam-3464	242	9	)	)	PUNCT
ejpam-3464	242	10	where	where	SCONJ
ejpam-3464	242	11	g	g	NOUN
ejpam-3464	242	12	=	=	SYM
ejpam-3464	242	13	(	(	PUNCT
ejpam-3464	242	14	v	v	NOUN
ejpam-3464	242	15	(	(	PUNCT
ejpam-3464	242	16	g	g	NOUN
ejpam-3464	242	17	)	)	PUNCT
ejpam-3464	242	18	,	,	PUNCT
ejpam-3464	242	19	e(g	e(g	PROPN
ejpam-3464	242	20	)	)	PUNCT
ejpam-3464	242	21	)	)	PUNCT
ejpam-3464	242	22	and	and	CCONJ
ejpam-3464	242	23	h	h	NOUN
ejpam-3464	242	24	=	=	SYM
ejpam-3464	242	25	(	(	PUNCT
ejpam-3464	242	26	v	v	NOUN
ejpam-3464	242	27	(	(	PUNCT
ejpam-3464	242	28	h	h	NOUN
ejpam-3464	242	29	)	)	PUNCT
ejpam-3464	242	30	,	,	PUNCT
ejpam-3464	242	31	e(h	e(h	PROPN
ejpam-3464	242	32	)	)	PUNCT
ejpam-3464	242	33	)	)	PUNCT
ejpam-3464	242	34	.	.	PUNCT
ejpam-3464	243	1	then	then	ADV
ejpam-3464	243	2	,	,	PUNCT
ejpam-3464	243	3	for	for	ADP
ejpam-3464	243	4	each	each	DET
ejpam-3464	243	5	(	(	PUNCT
ejpam-3464	243	6	v	v	NOUN
ejpam-3464	243	7	,	,	PUNCT
ejpam-3464	243	8	a	a	PRON
ejpam-3464	243	9	)	)	PUNCT
ejpam-3464	243	10	∈	∈	NOUN
ejpam-3464	243	11	v	v	NOUN
ejpam-3464	243	12	(	(	PUNCT
ejpam-3464	243	13	k	k	NOUN
ejpam-3464	243	14	)	)	PUNCT
ejpam-3464	243	15	,	,	PUNCT
ejpam-3464	243	16	fk	fk	INTJ
ejpam-3464	243	17	[	[	X
ejpam-3464	243	18	(	(	PUNCT
ejpam-3464	243	19	v	v	NOUN
ejpam-3464	243	20	,	,	PUNCT
ejpam-3464	243	21	a	a	NOUN
ejpam-3464	243	22	)	)	PUNCT
ejpam-3464	243	23	]	]	PUNCT
ejpam-3464	244	1	=	=	PUNCT
ejpam-3464	245	1	[	[	X
ejpam-3464	245	2	fg[v]×	fg[v]×	ADP
ejpam-3464	245	3	v	v	PRON
ejpam-3464	245	4	(	(	PUNCT
ejpam-3464	245	5	h	h	NOUN
ejpam-3464	245	6	)	)	PUNCT
ejpam-3464	245	7	]	]	PUNCT
ejpam-3464	245	8	∪	∪	ADP
ejpam-3464	245	9	[	[	X
ejpam-3464	245	10	ng[v]×	ng[v]×	ADV
ejpam-3464	245	11	fh	fh	PROPN
ejpam-3464	246	1	[	[	X
ejpam-3464	246	2	a	a	X
ejpam-3464	246	3	]	]	X
ejpam-3464	246	4	]	]	PUNCT
ejpam-3464	246	5	.	.	PUNCT
ejpam-3464	247	1	proof	proof	NOUN
ejpam-3464	247	2	.	.	PUNCT
ejpam-3464	248	1	let	let	VERB
ejpam-3464	248	2	z	z	NOUN
ejpam-3464	248	3	=	=	PUNCT
ejpam-3464	249	1	[	[	X
ejpam-3464	249	2	fg[v]×v	fg[v]×v	PROPN
ejpam-3464	249	3	(	(	PUNCT
ejpam-3464	249	4	h)]∪	h)]∪	PROPN
ejpam-3464	250	1	[	[	X
ejpam-3464	250	2	ng[v]×fh	ng[v]×fh	PRON
ejpam-3464	250	3	[	[	X
ejpam-3464	250	4	a	a	X
ejpam-3464	250	5	]	]	X
ejpam-3464	250	6	]	]	PUNCT
ejpam-3464	250	7	.	.	PUNCT
ejpam-3464	251	1	suppose	suppose	VERB
ejpam-3464	251	2	(	(	PUNCT
ejpam-3464	251	3	v	v	NOUN
ejpam-3464	251	4	,	,	PUNCT
ejpam-3464	251	5	a	a	DET
ejpam-3464	251	6	)	)	PUNCT
ejpam-3464	251	7	∈	∈	PROPN
ejpam-3464	251	8	v	v	NOUN
ejpam-3464	251	9	(	(	PUNCT
ejpam-3464	251	10	g⊗h	g⊗h	PROPN
ejpam-3464	251	11	)	)	PUNCT
ejpam-3464	251	12	=	=	SYM
ejpam-3464	251	13	v	v	X
ejpam-3464	251	14	(	(	PUNCT
ejpam-3464	251	15	k	k	NOUN
ejpam-3464	251	16	)	)	PUNCT
ejpam-3464	251	17	.	.	PUNCT
ejpam-3464	252	1	then	then	ADV
ejpam-3464	252	2	(	(	PUNCT
ejpam-3464	252	3	x	x	X
ejpam-3464	252	4	,	,	PUNCT
ejpam-3464	252	5	q	q	NOUN
ejpam-3464	252	6	)	)	PUNCT
ejpam-3464	252	7	∈	∈	NOUN
ejpam-3464	252	8	fk	fk	INTJ
ejpam-3464	253	1	[	[	X
ejpam-3464	253	2	(	(	PUNCT
ejpam-3464	253	3	v	v	NOUN
ejpam-3464	253	4	,	,	PUNCT
ejpam-3464	253	5	a	a	NOUN
ejpam-3464	253	6	)	)	PUNCT
ejpam-3464	253	7	]	]	PUNCT
ejpam-3464	254	1	if	if	SCONJ
ejpam-3464	254	2	and	and	CCONJ
ejpam-3464	254	3	only	only	ADV
ejpam-3464	254	4	if	if	SCONJ
ejpam-3464	254	5	(	(	PUNCT
ejpam-3464	254	6	v	v	NOUN
ejpam-3464	254	7	,	,	PUNCT
ejpam-3464	254	8	a	a	PRON
ejpam-3464	254	9	)	)	PUNCT
ejpam-3464	254	10	6=	6=	NUM
ejpam-3464	254	11	(	(	PUNCT
ejpam-3464	254	12	x	x	X
ejpam-3464	254	13	,	,	PUNCT
ejpam-3464	254	14	q	q	NOUN
ejpam-3464	254	15	)	)	PUNCT
ejpam-3464	254	16	and	and	CCONJ
ejpam-3464	254	17	dk((v	dk((v	PROPN
ejpam-3464	254	18	,	,	PUNCT
ejpam-3464	254	19	a	a	PRON
ejpam-3464	254	20	)	)	PUNCT
ejpam-3464	254	21	,	,	PUNCT
ejpam-3464	254	22	(	(	PUNCT
ejpam-3464	254	23	x	x	X
ejpam-3464	254	24	,	,	PUNCT
ejpam-3464	254	25	q	q	NOUN
ejpam-3464	254	26	)	)	PUNCT
ejpam-3464	254	27	)	)	PUNCT
ejpam-3464	254	28	6=	6=	PRON
ejpam-3464	254	29	1	1	X
ejpam-3464	254	30	.	.	X
ejpam-3464	254	31	consider	consider	VERB
ejpam-3464	254	32	the	the	DET
ejpam-3464	254	33	following	follow	VERB
ejpam-3464	254	34	cases	case	NOUN
ejpam-3464	254	35	:	:	PUNCT
ejpam-3464	254	36	case	case	NOUN
ejpam-3464	254	37	1	1	NUM
ejpam-3464	254	38	.	.	X
ejpam-3464	254	39	assume	assume	VERB
ejpam-3464	254	40	that	that	SCONJ
ejpam-3464	254	41	x	x	PRON
ejpam-3464	254	42	=	=	PUNCT
ejpam-3464	254	43	v.	v.	ADP
ejpam-3464	254	44	then	then	ADV
ejpam-3464	254	45	q	q	X
ejpam-3464	255	1	6=	6=	ADP
ejpam-3464	255	2	a	a	PRON
ejpam-3464	255	3	and	and	CCONJ
ejpam-3464	255	4	dh(q	dh(q	ADV
ejpam-3464	255	5	,	,	PUNCT
ejpam-3464	255	6	a	a	PRON
ejpam-3464	255	7	)	)	PUNCT
ejpam-3464	255	8	=	=	SYM
ejpam-3464	255	9	dk((x	dk((x	ADJ
ejpam-3464	255	10	,	,	PUNCT
ejpam-3464	255	11	q	q	NOUN
ejpam-3464	255	12	)	)	PUNCT
ejpam-3464	255	13	,	,	PUNCT
ejpam-3464	255	14	(	(	PUNCT
ejpam-3464	255	15	x	x	X
ejpam-3464	255	16	,	,	PUNCT
ejpam-3464	255	17	a	a	NOUN
ejpam-3464	255	18	)	)	PUNCT
ejpam-3464	255	19	)	)	PUNCT
ejpam-3464	256	1	6=	6=	ADP
ejpam-3464	256	2	1	1	X
ejpam-3464	256	3	.	.	PUNCT
ejpam-3464	257	1	thus	thus	ADV
ejpam-3464	257	2	q	q	X
ejpam-3464	257	3	∈	∈	PROPN
ejpam-3464	257	4	fh	fh	PROPN
ejpam-3464	257	5	[	[	X
ejpam-3464	257	6	a	a	X
ejpam-3464	257	7	]	]	X
ejpam-3464	257	8	and	and	CCONJ
ejpam-3464	257	9	so	so	ADV
ejpam-3464	257	10	,	,	PUNCT
ejpam-3464	257	11	(	(	PUNCT
ejpam-3464	257	12	x	x	NOUN
ejpam-3464	257	13	,	,	PUNCT
ejpam-3464	257	14	q	q	ADJ
ejpam-3464	257	15	)	)	PUNCT
ejpam-3464	257	16	∈	∈	PROPN
ejpam-3464	257	17	{	{	PUNCT
ejpam-3464	257	18	v	v	NOUN
ejpam-3464	257	19	}	}	PUNCT
ejpam-3464	257	20	×	×	NOUN
ejpam-3464	257	21	fh	fh	PROPN
ejpam-3464	258	1	[	[	X
ejpam-3464	258	2	a	a	X
ejpam-3464	258	3	]	]	X
ejpam-3464	258	4	.	.	PUNCT
ejpam-3464	259	1	case	case	NOUN
ejpam-3464	259	2	2	2	NUM
ejpam-3464	259	3	.	.	X
ejpam-3464	259	4	assume	assume	VERB
ejpam-3464	259	5	that	that	SCONJ
ejpam-3464	259	6	x	x	PRON
ejpam-3464	259	7	6=	6=	PROPN
ejpam-3464	259	8	v.	v.	CCONJ
ejpam-3464	259	9	suppose	suppose	VERB
ejpam-3464	259	10	q	q	X
ejpam-3464	259	11	=	=	PUNCT
ejpam-3464	259	12	a.	a.	NOUN
ejpam-3464	259	13	then	then	ADV
ejpam-3464	259	14	dg(x	dg(x	NUM
ejpam-3464	259	15	,	,	PUNCT
ejpam-3464	259	16	v	v	NOUN
ejpam-3464	259	17	)	)	PUNCT
ejpam-3464	259	18	=	=	SYM
ejpam-3464	259	19	dk((v	dk((v	PROPN
ejpam-3464	259	20	,	,	PUNCT
ejpam-3464	259	21	a	a	PRON
ejpam-3464	259	22	)	)	PUNCT
ejpam-3464	259	23	,	,	PUNCT
ejpam-3464	259	24	(	(	PUNCT
ejpam-3464	259	25	x	x	X
ejpam-3464	259	26	,	,	PUNCT
ejpam-3464	259	27	a	a	NOUN
ejpam-3464	259	28	)	)	PUNCT
ejpam-3464	259	29	)	)	PUNCT
ejpam-3464	259	30	6=	6=	ADP
ejpam-3464	260	1	1	1	X
ejpam-3464	260	2	.	.	PUNCT
ejpam-3464	261	1	it	it	PRON
ejpam-3464	261	2	follows	follow	VERB
ejpam-3464	261	3	that	that	SCONJ
ejpam-3464	261	4	x	x	PUNCT
ejpam-3464	261	5	∈	∈	PROPN
ejpam-3464	261	6	fg[v	fg[v	PROPN
ejpam-3464	261	7	]	]	X
ejpam-3464	261	8	and	and	CCONJ
ejpam-3464	261	9	(	(	PUNCT
ejpam-3464	261	10	x	x	NOUN
ejpam-3464	261	11	,	,	PUNCT
ejpam-3464	261	12	q	q	ADJ
ejpam-3464	261	13	)	)	PUNCT
ejpam-3464	261	14	∈	∈	PROPN
ejpam-3464	261	15	fg[v	fg[v	PROPN
ejpam-3464	261	16	]	]	X
ejpam-3464	261	17	×	×	NOUN
ejpam-3464	261	18	{	{	PUNCT
ejpam-3464	261	19	a	a	NOUN
ejpam-3464	261	20	}	}	PUNCT
ejpam-3464	261	21	.	.	PUNCT
ejpam-3464	262	1	suppose	suppose	VERB
ejpam-3464	262	2	q	q	X
ejpam-3464	263	1	6=	6=	NUM
ejpam-3464	263	2	a.	a.	NOUN
ejpam-3464	263	3	if	if	SCONJ
ejpam-3464	263	4	x	x	PROPN
ejpam-3464	263	5	∈	∈	NOUN
ejpam-3464	263	6	ng(v	ng(v	NOUN
ejpam-3464	263	7	)	)	PUNCT
ejpam-3464	263	8	,	,	PUNCT
ejpam-3464	263	9	then	then	ADV
ejpam-3464	263	10	q	q	X
ejpam-3464	263	11	/∈	/∈	PUNCT
ejpam-3464	263	12	ng(a	ng(a	NOUN
ejpam-3464	263	13	)	)	PUNCT
ejpam-3464	263	14	and	and	CCONJ
ejpam-3464	263	15	so	so	ADV
ejpam-3464	263	16	,	,	PUNCT
ejpam-3464	263	17	q	q	PROPN
ejpam-3464	263	18	∈	∈	PROPN
ejpam-3464	263	19	fg[a	fg[a	PROPN
ejpam-3464	263	20	]	]	PUNCT
ejpam-3464	263	21	.	.	PUNCT
ejpam-3464	264	1	suppose	suppose	VERB
ejpam-3464	264	2	x	x	X
ejpam-3464	264	3	∈	∈	PROPN
ejpam-3464	264	4	fg[v	fg[v	PROPN
ejpam-3464	264	5	]	]	X
ejpam-3464	264	6	.	.	PUNCT
ejpam-3464	265	1	since	since	SCONJ
ejpam-3464	265	2	q	q	PROPN
ejpam-3464	265	3	6=	6=	PROPN
ejpam-3464	265	4	a	a	PRON
ejpam-3464	265	5	,	,	PUNCT
ejpam-3464	265	6	q	q	PROPN
ejpam-3464	265	7	∈	∈	PROPN
ejpam-3464	265	8	v	v	NOUN
ejpam-3464	265	9	(	(	PUNCT
ejpam-3464	265	10	h)\{a	h)\{a	NOUN
ejpam-3464	265	11	}	}	PUNCT
ejpam-3464	265	12	.	.	PUNCT
ejpam-3464	266	1	therefore	therefore	ADV
ejpam-3464	266	2	,	,	PUNCT
ejpam-3464	266	3	fk	fk	INTJ
ejpam-3464	266	4	[	[	X
ejpam-3464	266	5	(	(	PUNCT
ejpam-3464	266	6	v	v	NOUN
ejpam-3464	266	7	,	,	PUNCT
ejpam-3464	266	8	a	a	NOUN
ejpam-3464	266	9	)	)	PUNCT
ejpam-3464	266	10	]	]	PUNCT
ejpam-3464	267	1	⊆	⊆	NUM
ejpam-3464	267	2	[	[	X
ejpam-3464	267	3	{	{	PUNCT
ejpam-3464	267	4	v	v	NOUN
ejpam-3464	267	5	}	}	PUNCT
ejpam-3464	267	6	×	×	NOUN
ejpam-3464	267	7	fh	fh	PROPN
ejpam-3464	268	1	[	[	X
ejpam-3464	268	2	a	a	X
ejpam-3464	268	3	]	]	X
ejpam-3464	268	4	]	]	X
ejpam-3464	268	5	∪	∪	ADP
ejpam-3464	268	6	[	[	X
ejpam-3464	268	7	fg[v]×	fg[v]×	X
ejpam-3464	268	8	{	{	PUNCT
ejpam-3464	268	9	a	a	X
ejpam-3464	268	10	}	}	PUNCT
ejpam-3464	268	11	]	]	PUNCT
ejpam-3464	268	12	∪	∪	ADP
ejpam-3464	268	13	[	[	X
ejpam-3464	268	14	ng(v)×	ng(v)×	PROPN
ejpam-3464	268	15	fg[a	fg[a	PROPN
ejpam-3464	268	16	]	]	X
ejpam-3464	268	17	]	]	PUNCT
ejpam-3464	268	18	∪	∪	ADP
ejpam-3464	268	19	[	[	X
ejpam-3464	268	20	fg[v]×	fg[v]×	ADJ
ejpam-3464	268	21	v	v	NOUN
ejpam-3464	268	22	(	(	PUNCT
ejpam-3464	268	23	h)\{a	h)\{a	NOUN
ejpam-3464	268	24	}	}	PUNCT
ejpam-3464	268	25	]	]	PUNCT
ejpam-3464	268	26	references	reference	VERB
ejpam-3464	268	27	755	755	NUM
ejpam-3464	268	28	=	=	SYM
ejpam-3464	269	1	[	[	X
ejpam-3464	269	2	ng[v]×	ng[v]×	ADV
ejpam-3464	269	3	fg[a	fg[a	PROPN
ejpam-3464	269	4	]	]	X
ejpam-3464	269	5	]	]	PUNCT
ejpam-3464	269	6	∪	∪	ADP
ejpam-3464	269	7	[	[	X
ejpam-3464	269	8	fg[v]×	fg[v]×	ADJ
ejpam-3464	269	9	v	v	NOUN
ejpam-3464	269	10	(	(	PUNCT
ejpam-3464	269	11	h	h	NOUN
ejpam-3464	269	12	)	)	PUNCT
ejpam-3464	269	13	]	]	PUNCT
ejpam-3464	270	1	=	=	PUNCT
ejpam-3464	270	2	z.	z.	PROPN
ejpam-3464	270	3	next	next	ADV
ejpam-3464	270	4	,	,	PUNCT
ejpam-3464	270	5	let	let	VERB
ejpam-3464	270	6	(	(	PUNCT
ejpam-3464	270	7	w	w	PROPN
ejpam-3464	270	8	,	,	PUNCT
ejpam-3464	270	9	z	z	NOUN
ejpam-3464	270	10	)	)	PUNCT
ejpam-3464	270	11	∈	∈	PROPN
ejpam-3464	270	12	z.	z.	PROPN
ejpam-3464	271	1	if	if	SCONJ
ejpam-3464	271	2	(	(	PUNCT
ejpam-3464	271	3	w	w	PROPN
ejpam-3464	271	4	,	,	PUNCT
ejpam-3464	271	5	z	z	NOUN
ejpam-3464	271	6	)	)	PUNCT
ejpam-3464	271	7	∈	∈	PROPN
ejpam-3464	271	8	fg[v	fg[v	PROPN
ejpam-3464	271	9	]	]	X
ejpam-3464	271	10	×	×	PROPN
ejpam-3464	271	11	v	v	NOUN
ejpam-3464	271	12	(	(	PUNCT
ejpam-3464	271	13	h	h	NOUN
ejpam-3464	271	14	)	)	PUNCT
ejpam-3464	271	15	then	then	ADV
ejpam-3464	271	16	w	w	PROPN
ejpam-3464	271	17	6=	6=	PROPN
ejpam-3464	271	18	v	v	PROPN
ejpam-3464	271	19	and	and	CCONJ
ejpam-3464	271	20	wv	wv	PROPN
ejpam-3464	271	21	/∈	/∈	PUNCT
ejpam-3464	271	22	e(g	e(g	PROPN
ejpam-3464	271	23	)	)	PUNCT
ejpam-3464	271	24	.	.	PUNCT
ejpam-3464	272	1	hence	hence	ADV
ejpam-3464	272	2	(	(	PUNCT
ejpam-3464	272	3	w	w	PROPN
ejpam-3464	272	4	,	,	PUNCT
ejpam-3464	272	5	z	z	NOUN
ejpam-3464	272	6	)	)	PUNCT
ejpam-3464	272	7	6=	6=	ADP
ejpam-3464	272	8	(	(	PUNCT
ejpam-3464	272	9	v	v	NOUN
ejpam-3464	272	10	,	,	PUNCT
ejpam-3464	272	11	a	a	PRON
ejpam-3464	272	12	)	)	PUNCT
ejpam-3464	272	13	and	and	CCONJ
ejpam-3464	272	14	by	by	ADP
ejpam-3464	272	15	definition	definition	NOUN
ejpam-3464	272	16	6	6	NUM
ejpam-3464	272	17	,	,	PUNCT
ejpam-3464	272	18	(	(	PUNCT
ejpam-3464	272	19	w	w	PROPN
ejpam-3464	272	20	,	,	PUNCT
ejpam-3464	272	21	z	z	NOUN
ejpam-3464	272	22	)	)	PUNCT
ejpam-3464	272	23	/∈	/∈	PUNCT
ejpam-3464	272	24	nk((v	nk((v	NOUN
ejpam-3464	272	25	,	,	PUNCT
ejpam-3464	272	26	a	a	PRON
ejpam-3464	272	27	)	)	PUNCT
ejpam-3464	272	28	)	)	PUNCT
ejpam-3464	272	29	.	.	PUNCT
ejpam-3464	273	1	thus	thus	ADV
ejpam-3464	273	2	,	,	PUNCT
ejpam-3464	273	3	(	(	PUNCT
ejpam-3464	273	4	w	w	PROPN
ejpam-3464	273	5	,	,	PUNCT
ejpam-3464	273	6	z	z	NOUN
ejpam-3464	273	7	)	)	PUNCT
ejpam-3464	273	8	∈	∈	PROPN
ejpam-3464	273	9	fg[(v	fg[(v	NOUN
ejpam-3464	273	10	,	,	PUNCT
ejpam-3464	273	11	a	a	NOUN
ejpam-3464	273	12	)	)	PUNCT
ejpam-3464	273	13	]	]	PUNCT
ejpam-3464	273	14	.	.	PUNCT
ejpam-3464	274	1	if	if	SCONJ
ejpam-3464	274	2	(	(	PUNCT
ejpam-3464	274	3	w	w	PROPN
ejpam-3464	274	4	,	,	PUNCT
ejpam-3464	274	5	z	z	NOUN
ejpam-3464	274	6	)	)	PUNCT
ejpam-3464	274	7	∈	∈	PROPN
ejpam-3464	275	1	ng[v	ng[v	X
ejpam-3464	275	2	]	]	X
ejpam-3464	275	3	×	×	PROPN
ejpam-3464	275	4	fh	fh	PROPN
ejpam-3464	276	1	[	[	X
ejpam-3464	276	2	a	a	X
ejpam-3464	276	3	]	]	X
ejpam-3464	276	4	,	,	PUNCT
ejpam-3464	276	5	then	then	ADV
ejpam-3464	276	6	z	z	PROPN
ejpam-3464	276	7	6=	6=	PROPN
ejpam-3464	276	8	a	a	PRON
ejpam-3464	276	9	,	,	PUNCT
ejpam-3464	276	10	az	az	PROPN
ejpam-3464	276	11	/∈	/∈	PUNCT
ejpam-3464	277	1	e(h	e(h	PROPN
ejpam-3464	277	2	)	)	PUNCT
ejpam-3464	277	3	and	and	CCONJ
ejpam-3464	277	4	either	either	PRON
ejpam-3464	277	5	w	w	PROPN
ejpam-3464	277	6	=	=	SYM
ejpam-3464	277	7	v	v	PROPN
ejpam-3464	277	8	or	or	CCONJ
ejpam-3464	277	9	wv	wv	PROPN
ejpam-3464	277	10	∈	∈	PROPN
ejpam-3464	277	11	e(g	e(g	PROPN
ejpam-3464	277	12	)	)	PUNCT
ejpam-3464	277	13	.	.	PUNCT
ejpam-3464	278	1	this	this	PRON
ejpam-3464	278	2	means	mean	VERB
ejpam-3464	278	3	that	that	SCONJ
ejpam-3464	278	4	(	(	PUNCT
ejpam-3464	278	5	w	w	PROPN
ejpam-3464	278	6	,	,	PUNCT
ejpam-3464	278	7	z	z	NOUN
ejpam-3464	278	8	)	)	PUNCT
ejpam-3464	278	9	6=	6=	ADP
ejpam-3464	278	10	(	(	PUNCT
ejpam-3464	278	11	v	v	NOUN
ejpam-3464	278	12	,	,	PUNCT
ejpam-3464	278	13	a	a	PRON
ejpam-3464	278	14	)	)	PUNCT
ejpam-3464	278	15	and	and	CCONJ
ejpam-3464	278	16	by	by	ADP
ejpam-3464	278	17	definition	definition	NOUN
ejpam-3464	278	18	6	6	NUM
ejpam-3464	278	19	,	,	PUNCT
ejpam-3464	278	20	(	(	PUNCT
ejpam-3464	278	21	w	w	PROPN
ejpam-3464	278	22	,	,	PUNCT
ejpam-3464	278	23	z	z	NOUN
ejpam-3464	278	24	)	)	PUNCT
ejpam-3464	278	25	/∈	/∈	PUNCT
ejpam-3464	278	26	nk((v	nk((v	NOUN
ejpam-3464	278	27	,	,	PUNCT
ejpam-3464	278	28	a	a	PRON
ejpam-3464	278	29	)	)	PUNCT
ejpam-3464	278	30	)	)	PUNCT
ejpam-3464	278	31	.	.	PUNCT
ejpam-3464	279	1	it	it	PRON
ejpam-3464	279	2	follows	follow	VERB
ejpam-3464	279	3	that	that	SCONJ
ejpam-3464	279	4	(	(	PUNCT
ejpam-3464	279	5	w	w	PROPN
ejpam-3464	279	6	,	,	PUNCT
ejpam-3464	279	7	z	z	NOUN
ejpam-3464	279	8	)	)	PUNCT
ejpam-3464	279	9	∈	∈	NOUN
ejpam-3464	280	1	fk	fk	INTJ
ejpam-3464	281	1	[	[	X
ejpam-3464	281	2	(	(	PUNCT
ejpam-3464	281	3	u	u	NOUN
ejpam-3464	281	4	,	,	PUNCT
ejpam-3464	281	5	a	a	NOUN
ejpam-3464	281	6	)	)	PUNCT
ejpam-3464	281	7	]	]	PUNCT
ejpam-3464	281	8	.	.	PUNCT
ejpam-3464	282	1	this	this	PRON
ejpam-3464	282	2	shows	show	VERB
ejpam-3464	282	3	that	that	SCONJ
ejpam-3464	282	4	z	z	NOUN
ejpam-3464	282	5	⊆	⊆	NUM
ejpam-3464	282	6	fk	fk	X
ejpam-3464	282	7	[	[	X
ejpam-3464	282	8	(	(	PUNCT
ejpam-3464	282	9	v	v	NOUN
ejpam-3464	282	10	,	,	PUNCT
ejpam-3464	282	11	a	a	NOUN
ejpam-3464	282	12	)	)	PUNCT
ejpam-3464	282	13	]	]	PUNCT
ejpam-3464	282	14	.	.	PUNCT
ejpam-3464	283	1	therefore	therefore	ADV
ejpam-3464	283	2	,	,	PUNCT
ejpam-3464	283	3	fk	fk	INTJ
ejpam-3464	283	4	[	[	X
ejpam-3464	283	5	(	(	PUNCT
ejpam-3464	283	6	v	v	NOUN
ejpam-3464	283	7	,	,	PUNCT
ejpam-3464	283	8	a	a	NOUN
ejpam-3464	283	9	)	)	PUNCT
ejpam-3464	283	10	]	]	PUNCT
ejpam-3464	284	1	=	=	PUNCT
ejpam-3464	284	2	z.	z.	PROPN
ejpam-3464	284	3	corollary	corollary	NOUN
ejpam-3464	284	4	4	4	X
ejpam-3464	284	5	.	.	PUNCT
ejpam-3464	285	1	let	let	VERB
ejpam-3464	285	2	g	g	NOUN
ejpam-3464	285	3	be	be	AUX
ejpam-3464	285	4	any	any	DET
ejpam-3464	285	5	graph	graph	NOUN
ejpam-3464	285	6	and	and	CCONJ
ejpam-3464	285	7	let	let	VERB
ejpam-3464	285	8	(	(	PUNCT
ejpam-3464	285	9	v	v	NOUN
ejpam-3464	285	10	,	,	PUNCT
ejpam-3464	285	11	a	a	PRON
ejpam-3464	285	12	)	)	PUNCT
ejpam-3464	285	13	∈	∈	NOUN
ejpam-3464	285	14	v	v	NOUN
ejpam-3464	285	15	(	(	PUNCT
ejpam-3464	285	16	g	g	PROPN
ejpam-3464	285	17	⊗	⊗	PROPN
ejpam-3464	285	18	kn	kn	PROPN
ejpam-3464	285	19	)	)	PUNCT
ejpam-3464	285	20	.	.	PUNCT
ejpam-3464	286	1	then	then	ADV
ejpam-3464	286	2	fg⊗kn	fg⊗kn	VERB
ejpam-3464	286	3	[	[	X
ejpam-3464	286	4	(	(	PUNCT
ejpam-3464	286	5	v	v	NOUN
ejpam-3464	286	6	,	,	PUNCT
ejpam-3464	286	7	a	a	NOUN
ejpam-3464	286	8	)	)	PUNCT
ejpam-3464	286	9	]	]	PUNCT
ejpam-3464	287	1	=	=	PUNCT
ejpam-3464	287	2	fg[v]×	fg[v]×	ADP
ejpam-3464	287	3	v	v	X
ejpam-3464	287	4	(	(	PUNCT
ejpam-3464	287	5	kn	kn	PROPN
ejpam-3464	287	6	)	)	PUNCT
ejpam-3464	287	7	.	.	PUNCT
ejpam-3464	288	1	proof	proof	NOUN
ejpam-3464	288	2	.	.	PUNCT
ejpam-3464	289	1	since	since	SCONJ
ejpam-3464	289	2	fkn	fkn	PROPN
ejpam-3464	290	1	[	[	X
ejpam-3464	290	2	a	a	X
ejpam-3464	290	3	]	]	X
ejpam-3464	290	4	=	=	SYM
ejpam-3464	290	5	∅	∅	NOUN
ejpam-3464	290	6	,	,	PUNCT
ejpam-3464	290	7	theorem	theorem	VERB
ejpam-3464	290	8	8	8	NUM
ejpam-3464	290	9	would	would	AUX
ejpam-3464	290	10	imply	imply	VERB
ejpam-3464	290	11	that	that	DET
ejpam-3464	290	12	fg⊗kn	fg⊗kn	NOUN
ejpam-3464	290	13	[	[	X
ejpam-3464	290	14	(	(	PUNCT
ejpam-3464	290	15	v	v	NOUN
ejpam-3464	290	16	,	,	PUNCT
ejpam-3464	290	17	a	a	NOUN
ejpam-3464	290	18	)	)	PUNCT
ejpam-3464	290	19	]	]	PUNCT
ejpam-3464	290	20	=	=	SYM
ejpam-3464	290	21	fg[v	fg[v	PROPN
ejpam-3464	290	22	]	]	X
ejpam-3464	290	23	×	×	NOUN
ejpam-3464	290	24	v	v	NOUN
ejpam-3464	290	25	(	(	PUNCT
ejpam-3464	290	26	kn	kn	PROPN
ejpam-3464	290	27	)	)	PUNCT
ejpam-3464	290	28	.	.	PUNCT
ejpam-3464	291	1	acknowledgements	acknowledgement	NOUN
ejpam-3464	291	2	this	this	DET
ejpam-3464	291	3	research	research	NOUN
ejpam-3464	291	4	is	be	AUX
ejpam-3464	291	5	funded	fund	VERB
ejpam-3464	291	6	by	by	ADP
ejpam-3464	291	7	the	the	DET
ejpam-3464	291	8	commission	commission	NOUN
ejpam-3464	291	9	on	on	ADP
ejpam-3464	291	10	higher	high	ADJ
ejpam-3464	291	11	education	education	NOUN
ejpam-3464	291	12	(	(	PUNCT
ejpam-3464	291	13	ched	che	VERB
ejpam-3464	291	14	)	)	PUNCT
ejpam-3464	291	15	and	and	CCONJ
ejpam-3464	291	16	mindanao	mindanao	PROPN
ejpam-3464	291	17	state	state	PROPN
ejpam-3464	291	18	university	university	PROPN
ejpam-3464	291	19	-	-	PUNCT
ejpam-3464	291	20	iligan	iligan	PROPN
ejpam-3464	291	21	institute	institute	PROPN
ejpam-3464	291	22	of	of	ADP
ejpam-3464	291	23	technology	technology	PROPN
ejpam-3464	291	24	.	.	PUNCT
ejpam-3464	292	1	references	reference	NOUN
ejpam-3464	292	2	[	[	X
ejpam-3464	292	3	1	1	X
ejpam-3464	292	4	]	]	PUNCT
ejpam-3464	292	5	s.	s.	PROPN
ejpam-3464	292	6	diesto	diesto	PROPN
ejpam-3464	292	7	and	and	CCONJ
ejpam-3464	292	8	s.	s.	PROPN
ejpam-3464	292	9	gervacio	gervacio	PROPN
ejpam-3464	292	10	.	.	PUNCT
ejpam-3464	293	1	finite	finite	PROPN
ejpam-3464	293	2	topological	topological	ADJ
ejpam-3464	293	3	graphs	graph	NOUN
ejpam-3464	293	4	.	.	PUNCT
ejpam-3464	294	1	journal	journal	NOUN
ejpam-3464	294	2	of	of	ADP
ejpam-3464	294	3	research	research	NOUN
ejpam-3464	294	4	and	and	CCONJ
ejpam-3464	294	5	development	development	NOUN
ejpam-3464	294	6	,	,	PUNCT
ejpam-3464	294	7	1(1):76–81	1(1):76–81	NUM
ejpam-3464	294	8	,	,	PUNCT
ejpam-3464	294	9	1983	1983	NUM
ejpam-3464	294	10	.	.	PUNCT
ejpam-3464	295	1	[	[	X
ejpam-3464	295	2	2	2	NUM
ejpam-3464	295	3	]	]	PUNCT
ejpam-3464	295	4	r.	r.	PROPN
ejpam-3464	295	5	guerrero	guerrero	PROPN
ejpam-3464	295	6	and	and	CCONJ
ejpam-3464	295	7	s.	s.	PROPN
ejpam-3464	295	8	gervacio	gervacio	PROPN
ejpam-3464	295	9	.	.	PUNCT
ejpam-3464	296	1	characterization	characterization	NOUN
ejpam-3464	296	2	of	of	ADP
ejpam-3464	296	3	graphs	graph	NOUN
ejpam-3464	296	4	which	which	PRON
ejpam-3464	296	5	induce	induce	VERB
ejpam-3464	296	6	the	the	DET
ejpam-3464	296	7	discrete	discrete	ADJ
ejpam-3464	296	8	and	and	CCONJ
ejpam-3464	296	9	indiscrete	indiscrete	ADJ
ejpam-3464	296	10	topological	topological	ADJ
ejpam-3464	296	11	spaces	space	NOUN
ejpam-3464	296	12	.	.	PUNCT
ejpam-3464	297	1	matimyas	matimyas	PROPN
ejpam-3464	297	2	matematika	matematika	PROPN
ejpam-3464	297	3	,	,	PUNCT
ejpam-3464	297	4	1986	1986	NUM
ejpam-3464	297	5	.	.	PUNCT
ejpam-3464	298	1	[	[	X
ejpam-3464	298	2	3	3	X
ejpam-3464	298	3	]	]	X
ejpam-3464	298	4	f.	f.	PROPN
ejpam-3464	298	5	harary	harary	PROPN
ejpam-3464	298	6	.	.	PUNCT
ejpam-3464	299	1	graph	graph	NOUN
ejpam-3464	299	2	theory	theory	NOUN
ejpam-3464	299	3	.	.	PUNCT
ejpam-3464	300	1	addison	addison	PROPN
ejpam-3464	300	2	-	-	PUNCT
ejpam-3464	300	3	wesley	wesley	PROPN
ejpam-3464	300	4	publishing	publishing	PROPN
ejpam-3464	300	5	company	company	NOUN
ejpam-3464	300	6	,	,	PUNCT
ejpam-3464	300	7	usa	usa	PROPN
ejpam-3464	300	8	,	,	PUNCT
ejpam-3464	300	9	1969	1969	NUM
ejpam-3464	300	10	.	.	PUNCT
ejpam-3464	301	1	[	[	X
ejpam-3464	301	2	4	4	NUM
ejpam-3464	301	3	]	]	PUNCT
ejpam-3464	301	4	r.	r.	PROPN
ejpam-3464	301	5	lemence	lemence	PROPN
ejpam-3464	301	6	and	and	CCONJ
ejpam-3464	301	7	s.	s.	PROPN
ejpam-3464	301	8	canoy	canoy	PROPN
ejpam-3464	301	9	.	.	PUNCT
ejpam-3464	302	1	another	another	DET
ejpam-3464	302	2	look	look	NOUN
ejpam-3464	302	3	at	at	ADP
ejpam-3464	302	4	the	the	DET
ejpam-3464	302	5	topologies	topology	NOUN
ejpam-3464	302	6	induced	induce	VERB
ejpam-3464	302	7	by	by	ADP
ejpam-3464	302	8	graph	graph	NOUN
ejpam-3464	302	9	.	.	PUNCT
ejpam-3464	303	1	matimyas	matimyas	PROPN
ejpam-3464	303	2	matematika	matematika	PROPN
ejpam-3464	303	3	,	,	PUNCT
ejpam-3464	303	4	21(2):1–7	21(2):1–7	NUM
ejpam-3464	303	5	,	,	PUNCT
ejpam-3464	303	6	1998	1998	NUM
ejpam-3464	303	7	.	.	PUNCT
ejpam-3464	304	1	[	[	X
ejpam-3464	304	2	5	5	NUM
ejpam-3464	304	3	]	]	PUNCT
ejpam-3464	304	4	r.	r.	PROPN
ejpam-3464	304	5	lemence	lemence	PROPN
ejpam-3464	304	6	and	and	CCONJ
ejpam-3464	304	7	s.	s.	PROPN
ejpam-3464	304	8	canoy	canoy	PROPN
ejpam-3464	304	9	.	.	PUNCT
ejpam-3464	305	1	topologies	topology	NOUN
ejpam-3464	305	2	induced	induce	VERB
ejpam-3464	305	3	by	by	ADP
ejpam-3464	305	4	some	some	DET
ejpam-3464	305	5	special	special	ADJ
ejpam-3464	305	6	graphs	graph	NOUN
ejpam-3464	305	7	.	.	PUNCT
ejpam-3464	306	1	journal	journal	NOUN
ejpam-3464	306	2	of	of	ADP
ejpam-3464	306	3	mathematics	mathematic	NOUN
ejpam-3464	306	4	,	,	PUNCT
ejpam-3464	306	5	2(2):45–50	2(2):45–50	NUM
ejpam-3464	306	6	,	,	PUNCT
ejpam-3464	306	7	1999	1999	NUM
ejpam-3464	306	8	.	.	PUNCT
ejpam-3464	307	1	[	[	X
ejpam-3464	307	2	6	6	NUM
ejpam-3464	307	3	]	]	PUNCT
ejpam-3464	307	4	s.	s.	PROPN
ejpam-3464	307	5	lipschutz	lipschutz	PROPN
ejpam-3464	307	6	.	.	PUNCT
ejpam-3464	308	1	general	general	ADJ
ejpam-3464	308	2	topology	topology	PROPN
ejpam-3464	308	3	,	,	PUNCT
ejpam-3464	308	4	schaum	schaum	PROPN
ejpam-3464	308	5	’s	’s	PART
ejpam-3464	308	6	outline	outline	PROPN
ejpam-3464	308	7	series	series	PROPN
ejpam-3464	308	8	.	.	PUNCT
ejpam-3464	309	1	mcgraw	mcgraw	PROPN
ejpam-3464	309	2	hill	hill	PROPN
ejpam-3464	309	3	international	international	PROPN
ejpam-3464	309	4	publishing	publishing	PROPN
ejpam-3464	309	5	co.	co.	PROPN
ejpam-3464	309	6	,	,	PUNCT
ejpam-3464	309	7	1987	1987	NUM
ejpam-3464	309	8	.	.	PUNCT
ejpam-3464	310	1	[	[	X
ejpam-3464	310	2	7	7	X
ejpam-3464	310	3	]	]	X
ejpam-3464	310	4	c.	c.	PROPN
ejpam-3464	310	5	nianga	nianga	PROPN
ejpam-3464	310	6	and	and	CCONJ
ejpam-3464	310	7	s.	s.	PROPN
ejpam-3464	310	8	canoy	canoy	PROPN
ejpam-3464	310	9	.	.	PUNCT
ejpam-3464	311	1	on	on	ADP
ejpam-3464	311	2	a	a	DET
ejpam-3464	311	3	finite	finite	ADJ
ejpam-3464	311	4	topological	topological	ADJ
ejpam-3464	311	5	space	space	NOUN
ejpam-3464	311	6	induced	induce	VERB
ejpam-3464	311	7	by	by	ADP
ejpam-3464	311	8	hop	hop	NOUN
ejpam-3464	311	9	neighborhoods	neighborhood	NOUN
ejpam-3464	311	10	of	of	ADP
ejpam-3464	311	11	a	a	DET
ejpam-3464	311	12	graphs	graph	NOUN
ejpam-3464	311	13	.	.	PUNCT
ejpam-3464	312	1	advances	advance	NOUN
ejpam-3464	312	2	and	and	CCONJ
ejpam-3464	312	3	applications	application	NOUN
ejpam-3464	312	4	in	in	ADP
ejpam-3464	312	5	discrete	discrete	ADJ
ejpam-3464	312	6	mathematics	mathematic	NOUN
ejpam-3464	312	7	,	,	PUNCT
ejpam-3464	312	8	21(1):79–89	21(1):79–89	NUM
ejpam-3464	312	9	,	,	PUNCT
ejpam-3464	312	10	2019	2019	NUM
ejpam-3464	312	11	.	.	PUNCT
ejpam-3464	313	1	[	[	X
ejpam-3464	313	2	8	8	NUM
ejpam-3464	313	3	]	]	X
ejpam-3464	313	4	c.	c.	PROPN
ejpam-3464	313	5	nianga	nianga	PROPN
ejpam-3464	313	6	and	and	CCONJ
ejpam-3464	313	7	s.	s.	PROPN
ejpam-3464	313	8	canoy	canoy	PROPN
ejpam-3464	313	9	.	.	PUNCT
ejpam-3464	314	1	on	on	ADP
ejpam-3464	314	2	topologies	topology	NOUN
ejpam-3464	314	3	induced	induce	VERB
ejpam-3464	314	4	by	by	ADP
ejpam-3464	314	5	graphs	graph	NOUN
ejpam-3464	314	6	under	under	ADP
ejpam-3464	314	7	some	some	DET
ejpam-3464	314	8	unary	unary	ADJ
ejpam-3464	314	9	and	and	CCONJ
ejpam-3464	314	10	binary	binary	ADJ
ejpam-3464	314	11	operations	operation	NOUN
ejpam-3464	314	12	.	.	PUNCT
ejpam-3464	315	1	european	european	ADJ
ejpam-3464	315	2	journal	journal	PROPN
ejpam-3464	315	3	of	of	ADP
ejpam-3464	315	4	pure	pure	ADJ
ejpam-3464	315	5	and	and	CCONJ
ejpam-3464	315	6	applied	applied	ADJ
ejpam-3464	315	7	mathematics	mathematic	NOUN
ejpam-3464	315	8	,	,	PUNCT
ejpam-3464	315	9	12(2):499	12(2):499	NUM
ejpam-3464	315	10	–	–	PUNCT
ejpam-3464	315	11	505	505	NUM
ejpam-3464	315	12	,	,	PUNCT
ejpam-3464	315	13	2019	2019	NUM
ejpam-3464	315	14	.	.	PUNCT