id	sid	tid	token	lemma	pos
ejpam-3842	1	1	european	european	PROPN
ejpam-3842	1	2	journal	journal	PROPN
ejpam-3842	1	3	of	of	ADP
ejpam-3842	1	4	pure	pure	ADJ
ejpam-3842	1	5	and	and	CCONJ
ejpam-3842	1	6	applied	apply	VERB
ejpam-3842	1	7	mathematics	mathematic	NOUN
ejpam-3842	1	8	vol	vol	NOUN
ejpam-3842	1	9	.	.	PROPN
ejpam-3842	2	1	13	13	NUM
ejpam-3842	2	2	,	,	PUNCT
ejpam-3842	2	3	no	no	INTJ
ejpam-3842	2	4	.	.	NOUN
ejpam-3842	2	5	4	4	NUM
ejpam-3842	2	6	,	,	PUNCT
ejpam-3842	2	7	2020	2020	NUM
ejpam-3842	2	8	,	,	PUNCT
ejpam-3842	2	9	730	730	NUM
ejpam-3842	2	10	-	-	SYM
ejpam-3842	2	11	738	738	NUM
ejpam-3842	2	12	issn	issn	PROPN
ejpam-3842	2	13	1307	1307	NUM
ejpam-3842	2	14	-	-	SYM
ejpam-3842	2	15	5543	5543	NUM
ejpam-3842	2	16	–	–	PUNCT
ejpam-3842	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3842	2	18	published	publish	VERB
ejpam-3842	2	19	by	by	ADP
ejpam-3842	2	20	new	new	PROPN
ejpam-3842	2	21	york	york	PROPN
ejpam-3842	2	22	business	business	PROPN
ejpam-3842	2	23	global	global	PROPN
ejpam-3842	2	24	another	another	DET
ejpam-3842	2	25	look	look	NOUN
ejpam-3842	2	26	at	at	ADP
ejpam-3842	2	27	topological	topological	ADJ
ejpam-3842	2	28	bch	bch	PROPN
ejpam-3842	2	29	-	-	PUNCT
ejpam-3842	2	30	algebras	algebras	PROPN
ejpam-3842	2	31	jemil	jemil	PROPN
ejpam-3842	2	32	d.	d.	PROPN
ejpam-3842	2	33	mancao1,∗	mancao1,∗	PROPN
ejpam-3842	2	34	,	,	PUNCT
ejpam-3842	2	35	sergio	sergio	PROPN
ejpam-3842	2	36	r.	r.	PROPN
ejpam-3842	2	37	canoy	canoy	PROPN
ejpam-3842	2	38	,	,	PUNCT
ejpam-3842	2	39	jr.1	jr.1	PROPN
ejpam-3842	2	40	1	1	NUM
ejpam-3842	2	41	department	department	NOUN
ejpam-3842	2	42	of	of	ADP
ejpam-3842	2	43	mathematics	mathematic	NOUN
ejpam-3842	2	44	and	and	CCONJ
ejpam-3842	2	45	statistics	statistic	NOUN
ejpam-3842	2	46	,	,	PUNCT
ejpam-3842	2	47	college	college	NOUN
ejpam-3842	2	48	of	of	ADP
ejpam-3842	2	49	science	science	NOUN
ejpam-3842	2	50	and	and	CCONJ
ejpam-3842	2	51	mathematics	mathematic	NOUN
ejpam-3842	2	52	,	,	PUNCT
ejpam-3842	2	53	center	center	NOUN
ejpam-3842	2	54	of	of	ADP
ejpam-3842	2	55	graph	graph	NOUN
ejpam-3842	2	56	theory	theory	NOUN
ejpam-3842	2	57	,	,	PUNCT
ejpam-3842	2	58	algebra	algebra	NOUN
ejpam-3842	2	59	and	and	CCONJ
ejpam-3842	2	60	analysis	analysis	NOUN
ejpam-3842	2	61	,	,	PUNCT
ejpam-3842	2	62	premier	premier	PROPN
ejpam-3842	2	63	research	research	PROPN
ejpam-3842	2	64	institute	institute	PROPN
ejpam-3842	2	65	of	of	ADP
ejpam-3842	2	66	science	science	NOUN
ejpam-3842	2	67	and	and	CCONJ
ejpam-3842	2	68	mathematics	mathematic	NOUN
ejpam-3842	2	69	,	,	PUNCT
ejpam-3842	2	70	mindanao	mindanao	PROPN
ejpam-3842	2	71	state	state	PROPN
ejpam-3842	2	72	university	university	PROPN
ejpam-3842	2	73	-	-	PUNCT
ejpam-3842	2	74	iligan	iligan	PROPN
ejpam-3842	2	75	institute	institute	PROPN
ejpam-3842	2	76	of	of	ADP
ejpam-3842	2	77	technology	technology	PROPN
ejpam-3842	2	78	,	,	PUNCT
ejpam-3842	2	79	9200	9200	NUM
ejpam-3842	2	80	iligan	iligan	ADJ
ejpam-3842	2	81	city	city	NOUN
ejpam-3842	2	82	,	,	PUNCT
ejpam-3842	2	83	philippines	philippine	NOUN
ejpam-3842	2	84	abstract	abstract	ADJ
ejpam-3842	2	85	.	.	PUNCT
ejpam-3842	3	1	a	a	DET
ejpam-3842	3	2	bch	bch	NOUN
ejpam-3842	3	3	-	-	PUNCT
ejpam-3842	3	4	algebra	algebra	NOUN
ejpam-3842	3	5	(	(	PUNCT
ejpam-3842	3	6	h	h	NOUN
ejpam-3842	3	7	,	,	PUNCT
ejpam-3842	3	8	∗	∗	NOUN
ejpam-3842	3	9	,	,	PUNCT
ejpam-3842	3	10	0	0	NUM
ejpam-3842	3	11	)	)	PUNCT
ejpam-3842	3	12	furnished	furnish	VERB
ejpam-3842	3	13	with	with	ADP
ejpam-3842	3	14	a	a	DET
ejpam-3842	3	15	topology	topology	NOUN
ejpam-3842	3	16	τ	τ	PROPN
ejpam-3842	3	17	on	on	ADP
ejpam-3842	3	18	h	h	PROPN
ejpam-3842	3	19	(	(	PUNCT
ejpam-3842	3	20	also	also	ADV
ejpam-3842	3	21	called	call	VERB
ejpam-3842	3	22	a	a	DET
ejpam-3842	3	23	bch	bch	NOUN
ejpam-3842	3	24	-	-	PUNCT
ejpam-3842	3	25	topology	topology	NOUN
ejpam-3842	3	26	on	on	ADP
ejpam-3842	3	27	h	h	NOUN
ejpam-3842	3	28	)	)	PUNCT
ejpam-3842	3	29	is	be	AUX
ejpam-3842	3	30	called	call	VERB
ejpam-3842	3	31	a	a	DET
ejpam-3842	3	32	topological	topological	ADJ
ejpam-3842	3	33	bch	bch	NOUN
ejpam-3842	3	34	-	-	PUNCT
ejpam-3842	3	35	algebra	algebra	PROPN
ejpam-3842	3	36	(	(	PUNCT
ejpam-3842	3	37	or	or	CCONJ
ejpam-3842	3	38	tbch	tbch	NOUN
ejpam-3842	3	39	-	-	PUNCT
ejpam-3842	3	40	algebra	algebra	NOUN
ejpam-3842	3	41	)	)	PUNCT
ejpam-3842	3	42	if	if	SCONJ
ejpam-3842	3	43	the	the	DET
ejpam-3842	3	44	function	function	NOUN
ejpam-3842	3	45	∗	∗	VERB
ejpam-3842	3	46	:	:	PUNCT
ejpam-3842	4	1	h	h	PROPN
ejpam-3842	4	2	×	×	NOUN
ejpam-3842	4	3	h	h	NOUN
ejpam-3842	4	4	→	→	SYM
ejpam-3842	4	5	h	h	PROPN
ejpam-3842	4	6	,	,	PUNCT
ejpam-3842	4	7	defined	define	VERB
ejpam-3842	4	8	by	by	ADP
ejpam-3842	4	9	∗((x	∗((x	NOUN
ejpam-3842	4	10	,	,	PUNCT
ejpam-3842	4	11	y	y	NOUN
ejpam-3842	4	12	)	)	PUNCT
ejpam-3842	4	13	)	)	PUNCT
ejpam-3842	5	1	=	=	PUNCT
ejpam-3842	5	2	x	x	X
ejpam-3842	5	3	∗	∗	X
ejpam-3842	5	4	y	y	PROPN
ejpam-3842	5	5	for	for	ADP
ejpam-3842	5	6	any	any	DET
ejpam-3842	5	7	x	x	NOUN
ejpam-3842	5	8	,	,	PUNCT
ejpam-3842	5	9	y	y	PROPN
ejpam-3842	5	10	∈	∈	PROPN
ejpam-3842	5	11	h	h	NOUN
ejpam-3842	5	12	,	,	PUNCT
ejpam-3842	5	13	is	be	AUX
ejpam-3842	5	14	continuous	continuous	ADJ
ejpam-3842	5	15	,	,	PUNCT
ejpam-3842	5	16	where	where	SCONJ
ejpam-3842	5	17	the	the	DET
ejpam-3842	5	18	cartesian	cartesian	ADJ
ejpam-3842	5	19	product	product	NOUN
ejpam-3842	5	20	topology	topology	NOUN
ejpam-3842	5	21	on	on	ADP
ejpam-3842	5	22	h	h	PROPN
ejpam-3842	5	23	×	×	PROPN
ejpam-3842	5	24	h	h	NOUN
ejpam-3842	5	25	is	be	AUX
ejpam-3842	5	26	furnished	furnish	VERB
ejpam-3842	5	27	by	by	ADP
ejpam-3842	5	28	τ	τ	PROPN
ejpam-3842	5	29	.	.	PUNCT
ejpam-3842	6	1	in	in	ADP
ejpam-3842	6	2	this	this	DET
ejpam-3842	6	3	paper	paper	NOUN
ejpam-3842	6	4	,	,	PUNCT
ejpam-3842	6	5	we	we	PRON
ejpam-3842	6	6	give	give	VERB
ejpam-3842	6	7	other	other	ADJ
ejpam-3842	6	8	structural	structural	ADJ
ejpam-3842	6	9	properties	property	NOUN
ejpam-3842	6	10	of	of	ADP
ejpam-3842	6	11	topological	topological	ADJ
ejpam-3842	6	12	bch	bch	PROPN
ejpam-3842	6	13	-	-	PUNCT
ejpam-3842	6	14	algebras	algebras	PROPN
ejpam-3842	6	15	.	.	PUNCT
ejpam-3842	7	1	2020	2020	NUM
ejpam-3842	7	2	mathematics	mathematics	PROPN
ejpam-3842	7	3	subject	subject	NOUN
ejpam-3842	7	4	classifications	classification	NOUN
ejpam-3842	7	5	:	:	PUNCT
ejpam-3842	7	6	06f35	06f35	NUM
ejpam-3842	7	7	,	,	PUNCT
ejpam-3842	7	8	03g25	03g25	NOUN
ejpam-3842	7	9	key	key	ADJ
ejpam-3842	7	10	words	word	NOUN
ejpam-3842	7	11	and	and	CCONJ
ejpam-3842	7	12	phrases	phrase	NOUN
ejpam-3842	7	13	:	:	PUNCT
ejpam-3842	7	14	bch	bch	NOUN
ejpam-3842	7	15	-	-	PUNCT
ejpam-3842	7	16	algebra	algebra	NOUN
ejpam-3842	7	17	,	,	PUNCT
ejpam-3842	7	18	topology	topology	NOUN
ejpam-3842	7	19	,	,	PUNCT
ejpam-3842	7	20	tbch	tbch	NOUN
ejpam-3842	7	21	-	-	PUNCT
ejpam-3842	7	22	algebra	algebra	NOUN
ejpam-3842	7	23	,	,	PUNCT
ejpam-3842	7	24	separation	separation	NOUN
ejpam-3842	7	25	axioms	axiom	VERB
ejpam-3842	7	26	1	1	NUM
ejpam-3842	7	27	.	.	PUNCT
ejpam-3842	8	1	introduction	introduction	NOUN
ejpam-3842	8	2	in	in	ADP
ejpam-3842	8	3	1983	1983	NUM
ejpam-3842	8	4	,	,	PUNCT
ejpam-3842	8	5	hu	hu	PROPN
ejpam-3842	8	6	and	and	CCONJ
ejpam-3842	8	7	li	li	PROPN
ejpam-3842	9	1	[	[	X
ejpam-3842	9	2	5	5	NUM
ejpam-3842	9	3	,	,	PUNCT
ejpam-3842	9	4	6	6	NUM
ejpam-3842	9	5	]	]	PUNCT
ejpam-3842	9	6	introduced	introduce	VERB
ejpam-3842	9	7	the	the	DET
ejpam-3842	9	8	notion	notion	NOUN
ejpam-3842	9	9	of	of	ADP
ejpam-3842	9	10	a	a	DET
ejpam-3842	9	11	bch	bch	NOUN
ejpam-3842	9	12	-	-	PUNCT
ejpam-3842	9	13	algebra	algebra	NOUN
ejpam-3842	9	14	which	which	PRON
ejpam-3842	9	15	is	be	AUX
ejpam-3842	9	16	a	a	DET
ejpam-3842	9	17	generalization	generalization	NOUN
ejpam-3842	9	18	of	of	ADP
ejpam-3842	9	19	bck	bck	PROPN
ejpam-3842	9	20	and	and	CCONJ
ejpam-3842	9	21	bci	bci	NOUN
ejpam-3842	9	22	-	-	PUNCT
ejpam-3842	9	23	algebras	algebras	X
ejpam-3842	9	24	.	.	PUNCT
ejpam-3842	10	1	in	in	ADP
ejpam-3842	10	2	the	the	DET
ejpam-3842	10	3	same	same	ADJ
ejpam-3842	10	4	paper	paper	NOUN
ejpam-3842	10	5	,	,	PUNCT
ejpam-3842	10	6	the	the	DET
ejpam-3842	10	7	concept	concept	NOUN
ejpam-3842	10	8	of	of	ADP
ejpam-3842	10	9	associative	associative	ADJ
ejpam-3842	10	10	bch	bch	PROPN
ejpam-3842	10	11	-	-	PUNCT
ejpam-3842	10	12	algebra	algebra	PROPN
ejpam-3842	10	13	was	be	AUX
ejpam-3842	10	14	also	also	ADV
ejpam-3842	10	15	introduced	introduce	VERB
ejpam-3842	10	16	.	.	PUNCT
ejpam-3842	11	1	dar	dar	PROPN
ejpam-3842	11	2	,	,	PUNCT
ejpam-3842	11	3	k.	k.	PROPN
ejpam-3842	11	4	h.	h.	PROPN
ejpam-3842	11	5	,	,	PUNCT
ejpam-3842	11	6	and	and	CCONJ
ejpam-3842	11	7	akram	akram	PROPN
ejpam-3842	11	8	,	,	PUNCT
ejpam-3842	11	9	m.	m.	NOUN
ejpam-3842	12	1	[	[	X
ejpam-3842	12	2	2	2	X
ejpam-3842	12	3	]	]	PUNCT
ejpam-3842	12	4	defined	define	VERB
ejpam-3842	12	5	the	the	DET
ejpam-3842	12	6	concepts	concept	NOUN
ejpam-3842	12	7	of	of	ADP
ejpam-3842	12	8	bch	bch	PROPN
ejpam-3842	12	9	-	-	PUNCT
ejpam-3842	12	10	ideal	ideal	ADJ
ejpam-3842	12	11	,	,	PUNCT
ejpam-3842	12	12	bch	bch	NOUN
ejpam-3842	12	13	-	-	PUNCT
ejpam-3842	12	14	subalgebra	subalgebra	NOUN
ejpam-3842	12	15	,	,	PUNCT
ejpam-3842	12	16	∗-commutative	∗-commutative	ADJ
ejpam-3842	12	17	,	,	PUNCT
ejpam-3842	12	18	left	left	ADJ
ejpam-3842	12	19	and	and	CCONJ
ejpam-3842	12	20	right	right	ADJ
ejpam-3842	12	21	mappings	mapping	NOUN
ejpam-3842	12	22	on	on	ADP
ejpam-3842	12	23	a	a	DET
ejpam-3842	12	24	bch	bch	NOUN
ejpam-3842	12	25	-	-	PUNCT
ejpam-3842	12	26	algebra	algebra	NOUN
ejpam-3842	12	27	and	and	CCONJ
ejpam-3842	12	28	some	some	DET
ejpam-3842	12	29	properties	property	NOUN
ejpam-3842	12	30	structures	structure	NOUN
ejpam-3842	12	31	were	be	AUX
ejpam-3842	12	32	investigated	investigate	VERB
ejpam-3842	12	33	.	.	PUNCT
ejpam-3842	13	1	in	in	ADP
ejpam-3842	13	2	[	[	X
ejpam-3842	13	3	8	8	NUM
ejpam-3842	13	4	]	]	PUNCT
ejpam-3842	13	5	and	and	CCONJ
ejpam-3842	13	6	[	[	X
ejpam-3842	13	7	4	4	NUM
ejpam-3842	13	8	]	]	PUNCT
ejpam-3842	13	9	,	,	PUNCT
ejpam-3842	13	10	the	the	DET
ejpam-3842	13	11	concepts	concept	NOUN
ejpam-3842	13	12	of	of	ADP
ejpam-3842	13	13	topological	topological	ADJ
ejpam-3842	13	14	bck	bck	NOUN
ejpam-3842	13	15	-	-	PUNCT
ejpam-3842	13	16	algebra	algebra	PROPN
ejpam-3842	13	17	and	and	CCONJ
ejpam-3842	13	18	topological	topological	ADJ
ejpam-3842	13	19	bci	bci	NOUN
ejpam-3842	13	20	-	-	PUNCT
ejpam-3842	13	21	algebra	algebra	NOUN
ejpam-3842	13	22	were	be	AUX
ejpam-3842	13	23	defined	define	VERB
ejpam-3842	13	24	and	and	CCONJ
ejpam-3842	13	25	some	some	DET
ejpam-3842	13	26	properties	property	NOUN
ejpam-3842	13	27	of	of	ADP
ejpam-3842	13	28	each	each	DET
ejpam-3842	13	29	newly	newly	ADV
ejpam-3842	13	30	defined	define	VERB
ejpam-3842	13	31	concepts	concept	NOUN
ejpam-3842	13	32	were	be	AUX
ejpam-3842	13	33	investigated	investigate	VERB
ejpam-3842	13	34	.	.	PUNCT
ejpam-3842	14	1	in	in	ADP
ejpam-3842	14	2	2017	2017	NUM
ejpam-3842	14	3	,	,	PUNCT
ejpam-3842	14	4	m.	m.	NOUN
ejpam-3842	14	5	jansi	jansi	PROPN
ejpam-3842	14	6	and	and	CCONJ
ejpam-3842	14	7	v.	v.	ADP
ejpam-3842	14	8	thiruveni	thiruveni	NOUN
ejpam-3842	14	9	[	[	X
ejpam-3842	14	10	7	7	NUM
ejpam-3842	14	11	]	]	PUNCT
ejpam-3842	14	12	introduced	introduce	VERB
ejpam-3842	14	13	the	the	DET
ejpam-3842	14	14	concept	concept	NOUN
ejpam-3842	14	15	of	of	ADP
ejpam-3842	14	16	topological	topological	ADJ
ejpam-3842	14	17	bch	bch	PROPN
ejpam-3842	14	18	-	-	PUNCT
ejpam-3842	14	19	algebra	algebra	PROPN
ejpam-3842	14	20	(	(	PUNCT
ejpam-3842	14	21	or	or	CCONJ
ejpam-3842	14	22	tbch	tbch	NOUN
ejpam-3842	14	23	-	-	PUNCT
ejpam-3842	14	24	algebra	algebra	NOUN
ejpam-3842	14	25	)	)	PUNCT
ejpam-3842	14	26	and	and	CCONJ
ejpam-3842	14	27	investigated	investigate	VERB
ejpam-3842	14	28	some	some	PRON
ejpam-3842	14	29	of	of	ADP
ejpam-3842	14	30	its	its	PRON
ejpam-3842	14	31	algebraic	algebraic	ADJ
ejpam-3842	14	32	and	and	CCONJ
ejpam-3842	14	33	topological	topological	ADJ
ejpam-3842	14	34	properties	property	NOUN
ejpam-3842	14	35	.	.	PUNCT
ejpam-3842	15	1	the	the	DET
ejpam-3842	15	2	aim	aim	NOUN
ejpam-3842	15	3	of	of	ADP
ejpam-3842	15	4	this	this	DET
ejpam-3842	15	5	paper	paper	NOUN
ejpam-3842	15	6	is	be	AUX
ejpam-3842	15	7	to	to	PART
ejpam-3842	15	8	give	give	VERB
ejpam-3842	15	9	other	other	ADJ
ejpam-3842	15	10	structural	structural	ADJ
ejpam-3842	15	11	properties	property	NOUN
ejpam-3842	15	12	of	of	ADP
ejpam-3842	15	13	topological	topological	ADJ
ejpam-3842	15	14	bch	bch	PROPN
ejpam-3842	15	15	-	-	PUNCT
ejpam-3842	15	16	algebras	algebras	X
ejpam-3842	15	17	.	.	PUNCT
ejpam-3842	16	1	∗corresponding	∗corresponde	VERB
ejpam-3842	16	2	author	author	NOUN
ejpam-3842	16	3	.	.	PUNCT
ejpam-3842	17	1	doi	doi	NOUN
ejpam-3842	17	2	:	:	PUNCT
ejpam-3842	17	3	https://doi.org/10.29020/nybg.ejpam.v13i4.3842	https://doi.org/10.29020/nybg.ejpam.v13i4.3842	ADJ
ejpam-3842	17	4	email	email	NOUN
ejpam-3842	17	5	addresses	address	NOUN
ejpam-3842	17	6	:	:	PUNCT
ejpam-3842	17	7	jemil.mancao@g.msuiit.edu.ph	jemil.mancao@g.msuiit.edu.ph	PROPN
ejpam-3842	17	8	(	(	PUNCT
ejpam-3842	17	9	j.	j.	PROPN
ejpam-3842	17	10	mancao	mancao	PROPN
ejpam-3842	17	11	)	)	PUNCT
ejpam-3842	17	12	,	,	PUNCT
ejpam-3842	17	13	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-3842	17	14	(	(	PUNCT
ejpam-3842	17	15	s.	s.	PROPN
ejpam-3842	17	16	canoy	canoy	PROPN
ejpam-3842	17	17	)	)	PUNCT
ejpam-3842	17	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-3842	18	1	730	730	NUM
ejpam-3842	18	2	c	c	NOUN
ejpam-3842	18	3	©	©	NOUN
ejpam-3842	18	4	2020	2020	NUM
ejpam-3842	18	5	ejpam	ejpam	VERB
ejpam-3842	18	6	all	all	DET
ejpam-3842	18	7	rights	right	NOUN
ejpam-3842	18	8	reserved	reserve	VERB
ejpam-3842	18	9	.	.	PUNCT
ejpam-3842	19	1	j.	j.	PROPN
ejpam-3842	19	2	mancao	mancao	PROPN
ejpam-3842	19	3	,	,	PUNCT
ejpam-3842	19	4	s.	s.	PROPN
ejpam-3842	19	5	canoy	canoy	PROPN
ejpam-3842	19	6	/	/	SYM
ejpam-3842	19	7	eur	eur	PROPN
ejpam-3842	19	8	.	.	PUNCT
ejpam-3842	20	1	j.	j.	PROPN
ejpam-3842	20	2	pure	pure	PROPN
ejpam-3842	20	3	appl	appl	PROPN
ejpam-3842	20	4	.	.	PROPN
ejpam-3842	20	5	math	math	PROPN
ejpam-3842	20	6	,	,	PUNCT
ejpam-3842	20	7	13	13	NUM
ejpam-3842	20	8	(	(	PUNCT
ejpam-3842	20	9	4	4	NUM
ejpam-3842	20	10	)	)	PUNCT
ejpam-3842	20	11	(	(	PUNCT
ejpam-3842	20	12	2020	2020	NUM
ejpam-3842	20	13	)	)	PUNCT
ejpam-3842	20	14	,	,	PUNCT
ejpam-3842	20	15	730	730	NUM
ejpam-3842	20	16	-	-	SYM
ejpam-3842	20	17	738	738	NUM
ejpam-3842	20	18	731	731	NUM
ejpam-3842	20	19	2	2	NUM
ejpam-3842	20	20	.	.	PUNCT
ejpam-3842	20	21	preliminaries	preliminary	NOUN
ejpam-3842	20	22	and	and	CCONJ
ejpam-3842	20	23	known	know	VERB
ejpam-3842	20	24	results	result	NOUN
ejpam-3842	20	25	definition	definition	NOUN
ejpam-3842	20	26	1	1	NUM
ejpam-3842	20	27	.	.	PUNCT
ejpam-3842	21	1	[	[	X
ejpam-3842	21	2	3	3	X
ejpam-3842	21	3	]	]	X
ejpam-3842	21	4	let	let	VERB
ejpam-3842	21	5	(	(	PUNCT
ejpam-3842	21	6	x	x	NOUN
ejpam-3842	21	7	,	,	PUNCT
ejpam-3842	21	8	τ	τ	X
ejpam-3842	21	9	)	)	PUNCT
ejpam-3842	21	10	be	be	VERB
ejpam-3842	21	11	a	a	DET
ejpam-3842	21	12	topological	topological	ADJ
ejpam-3842	21	13	space	space	NOUN
ejpam-3842	21	14	and	and	CCONJ
ejpam-3842	21	15	let	let	VERB
ejpam-3842	21	16	x	x	X
ejpam-3842	21	17	∈	∈	VERB
ejpam-3842	21	18	x.	x.	NOUN
ejpam-3842	22	1	any	any	DET
ejpam-3842	22	2	set	set	NOUN
ejpam-3842	22	3	u	u	NOUN
ejpam-3842	22	4	∈	∈	PROPN
ejpam-3842	22	5	τ	τ	X
ejpam-3842	22	6	containing	contain	VERB
ejpam-3842	22	7	x	x	PROPN
ejpam-3842	22	8	is	be	AUX
ejpam-3842	22	9	called	call	VERB
ejpam-3842	22	10	a	a	DET
ejpam-3842	22	11	neighborhood	neighborhood	NOUN
ejpam-3842	22	12	(	(	PUNCT
ejpam-3842	22	13	sometimes	sometimes	ADV
ejpam-3842	22	14	written	write	VERB
ejpam-3842	22	15	as	as	ADP
ejpam-3842	22	16	nbhd	nbhd	NOUN
ejpam-3842	22	17	or	or	CCONJ
ejpam-3842	22	18	τ	τ	NUM
ejpam-3842	22	19	-nbhd	-nbhd	NOUN
ejpam-3842	22	20	)	)	PUNCT
ejpam-3842	22	21	of	of	ADP
ejpam-3842	22	22	x.	x.	NOUN
ejpam-3842	22	23	definition	definition	NOUN
ejpam-3842	22	24	2	2	NUM
ejpam-3842	22	25	.	.	PUNCT
ejpam-3842	23	1	[	[	X
ejpam-3842	23	2	3	3	X
ejpam-3842	23	3	]	]	X
ejpam-3842	23	4	let	let	VERB
ejpam-3842	23	5	(	(	PUNCT
ejpam-3842	23	6	x	x	NOUN
ejpam-3842	23	7	,	,	PUNCT
ejpam-3842	23	8	τ	τ	X
ejpam-3842	23	9	)	)	PUNCT
ejpam-3842	23	10	be	be	VERB
ejpam-3842	23	11	a	a	DET
ejpam-3842	23	12	topological	topological	ADJ
ejpam-3842	23	13	space	space	NOUN
ejpam-3842	23	14	.	.	PUNCT
ejpam-3842	24	1	then	then	ADV
ejpam-3842	24	2	(	(	PUNCT
ejpam-3842	24	3	i	i	NOUN
ejpam-3842	24	4	)	)	PUNCT
ejpam-3842	24	5	(	(	PUNCT
ejpam-3842	24	6	x	x	X
ejpam-3842	24	7	,	,	PUNCT
ejpam-3842	24	8	τ	τ	X
ejpam-3842	24	9	)	)	PUNCT
ejpam-3842	24	10	is	be	AUX
ejpam-3842	24	11	a	a	DET
ejpam-3842	24	12	t0	t0	NOUN
ejpam-3842	24	13	-	-	NOUN
ejpam-3842	24	14	space	space	NOUN
ejpam-3842	24	15	if	if	SCONJ
ejpam-3842	24	16	for	for	ADP
ejpam-3842	24	17	any	any	DET
ejpam-3842	24	18	x	x	NOUN
ejpam-3842	24	19	,	,	PUNCT
ejpam-3842	24	20	y	y	PROPN
ejpam-3842	24	21	∈	∈	PROPN
ejpam-3842	24	22	x	x	PUNCT
ejpam-3842	24	23	with	with	ADP
ejpam-3842	24	24	x	x	SYM
ejpam-3842	24	25	6=	6=	PROPN
ejpam-3842	24	26	y	y	PROPN
ejpam-3842	24	27	,	,	PUNCT
ejpam-3842	24	28	there	there	PRON
ejpam-3842	24	29	exists	exist	VERB
ejpam-3842	24	30	an	an	DET
ejpam-3842	24	31	open	open	ADJ
ejpam-3842	24	32	set	set	NOUN
ejpam-3842	24	33	u	u	NOUN
ejpam-3842	24	34	containing	contain	VERB
ejpam-3842	24	35	one	one	NUM
ejpam-3842	24	36	but	but	CCONJ
ejpam-3842	24	37	not	not	PART
ejpam-3842	24	38	the	the	DET
ejpam-3842	24	39	other	other	ADJ
ejpam-3842	24	40	;	;	PUNCT
ejpam-3842	24	41	(	(	PUNCT
ejpam-3842	24	42	ii	ii	NOUN
ejpam-3842	24	43	)	)	PUNCT
ejpam-3842	24	44	(	(	PUNCT
ejpam-3842	24	45	x	x	X
ejpam-3842	24	46	,	,	PUNCT
ejpam-3842	24	47	τ	τ	X
ejpam-3842	24	48	)	)	PUNCT
ejpam-3842	24	49	is	be	AUX
ejpam-3842	24	50	a	a	DET
ejpam-3842	24	51	t1	t1	NOUN
ejpam-3842	24	52	-	-	PUNCT
ejpam-3842	24	53	space	space	NOUN
ejpam-3842	24	54	if	if	SCONJ
ejpam-3842	24	55	for	for	ADP
ejpam-3842	24	56	any	any	DET
ejpam-3842	24	57	x	x	NOUN
ejpam-3842	24	58	,	,	PUNCT
ejpam-3842	24	59	y	y	PROPN
ejpam-3842	24	60	∈	∈	PROPN
ejpam-3842	24	61	x	x	PUNCT
ejpam-3842	24	62	with	with	ADP
ejpam-3842	24	63	x	x	SYM
ejpam-3842	24	64	6=	6=	PROPN
ejpam-3842	24	65	y	y	PROPN
ejpam-3842	24	66	,	,	PUNCT
ejpam-3842	24	67	there	there	PRON
ejpam-3842	24	68	exist	exist	VERB
ejpam-3842	24	69	nbhds	nbhds	ADJ
ejpam-3842	24	70	u	u	NOUN
ejpam-3842	24	71	and	and	CCONJ
ejpam-3842	24	72	v	v	NOUN
ejpam-3842	24	73	of	of	ADP
ejpam-3842	24	74	x	x	PROPN
ejpam-3842	24	75	and	and	CCONJ
ejpam-3842	24	76	y	y	PROPN
ejpam-3842	24	77	,	,	PUNCT
ejpam-3842	24	78	respectively	respectively	ADV
ejpam-3842	24	79	,	,	PUNCT
ejpam-3842	24	80	such	such	ADJ
ejpam-3842	24	81	that	that	SCONJ
ejpam-3842	24	82	x	x	SYM
ejpam-3842	24	83	/∈	/∈	PUNCT
ejpam-3842	24	84	v	v	NOUN
ejpam-3842	24	85	and	and	CCONJ
ejpam-3842	24	86	y	y	PROPN
ejpam-3842	24	87	/∈	/∈	PUNCT
ejpam-3842	24	88	u	u	PROPN
ejpam-3842	24	89	;	;	PUNCT
ejpam-3842	24	90	(	(	PUNCT
ejpam-3842	24	91	iii	iii	X
ejpam-3842	24	92	)	)	PUNCT
ejpam-3842	24	93	(	(	PUNCT
ejpam-3842	24	94	x	x	X
ejpam-3842	24	95	,	,	PUNCT
ejpam-3842	24	96	τ	τ	X
ejpam-3842	24	97	)	)	PUNCT
ejpam-3842	24	98	is	be	AUX
ejpam-3842	24	99	a	a	DET
ejpam-3842	24	100	t2	t2	NOUN
ejpam-3842	24	101	-	-	PUNCT
ejpam-3842	24	102	space	space	NOUN
ejpam-3842	24	103	(	(	PUNCT
ejpam-3842	24	104	or	or	CCONJ
ejpam-3842	24	105	hausdorff	hausdorff	NOUN
ejpam-3842	24	106	space	space	NOUN
ejpam-3842	24	107	)	)	PUNCT
ejpam-3842	24	108	if	if	SCONJ
ejpam-3842	24	109	for	for	ADP
ejpam-3842	24	110	any	any	DET
ejpam-3842	24	111	x	x	NOUN
ejpam-3842	24	112	,	,	PUNCT
ejpam-3842	24	113	y	y	PROPN
ejpam-3842	24	114	∈	∈	PROPN
ejpam-3842	24	115	x	x	PUNCT
ejpam-3842	24	116	with	with	ADP
ejpam-3842	24	117	x	x	SYM
ejpam-3842	24	118	6=	6=	PROPN
ejpam-3842	24	119	y	y	PROPN
ejpam-3842	24	120	,	,	PUNCT
ejpam-3842	24	121	there	there	PRON
ejpam-3842	24	122	exist	exist	VERB
ejpam-3842	24	123	disjoint	disjoint	NOUN
ejpam-3842	24	124	nbhds	nbhds	PROPN
ejpam-3842	24	125	u	u	PROPN
ejpam-3842	24	126	and	and	CCONJ
ejpam-3842	24	127	v	v	NOUN
ejpam-3842	24	128	of	of	ADP
ejpam-3842	24	129	x	x	PROPN
ejpam-3842	24	130	and	and	CCONJ
ejpam-3842	24	131	y	y	PROPN
ejpam-3842	24	132	,	,	PUNCT
ejpam-3842	24	133	respectively	respectively	ADV
ejpam-3842	24	134	.	.	PUNCT
ejpam-3842	25	1	remark	remark	PROPN
ejpam-3842	25	2	1	1	NUM
ejpam-3842	25	3	.	.	PUNCT
ejpam-3842	26	1	[	[	X
ejpam-3842	26	2	3	3	X
ejpam-3842	26	3	]	]	X
ejpam-3842	26	4	t2	t2	PROPN
ejpam-3842	26	5	⇒	⇒	PROPN
ejpam-3842	26	6	t1	t1	PROPN
ejpam-3842	26	7	⇒	⇒	PROPN
ejpam-3842	26	8	t0	t0	PROPN
ejpam-3842	26	9	but	but	CCONJ
ejpam-3842	26	10	not	not	PART
ejpam-3842	26	11	conversely	conversely	ADV
ejpam-3842	26	12	.	.	PUNCT
ejpam-3842	27	1	theorem	theorem	NOUN
ejpam-3842	27	2	1	1	NUM
ejpam-3842	27	3	.	.	PUNCT
ejpam-3842	28	1	[	[	X
ejpam-3842	28	2	3	3	X
ejpam-3842	28	3	]	]	X
ejpam-3842	28	4	let	let	VERB
ejpam-3842	28	5	(	(	PUNCT
ejpam-3842	28	6	x	x	NOUN
ejpam-3842	28	7	,	,	PUNCT
ejpam-3842	28	8	τ	τ	X
ejpam-3842	28	9	)	)	PUNCT
ejpam-3842	28	10	be	be	VERB
ejpam-3842	28	11	a	a	DET
ejpam-3842	28	12	topological	topological	ADJ
ejpam-3842	28	13	space	space	NOUN
ejpam-3842	28	14	.	.	PUNCT
ejpam-3842	29	1	x	x	PUNCT
ejpam-3842	29	2	is	be	AUX
ejpam-3842	29	3	a	a	DET
ejpam-3842	29	4	t1	t1	NOUN
ejpam-3842	29	5	-	-	PUNCT
ejpam-3842	29	6	space	space	NOUN
ejpam-3842	29	7	if	if	SCONJ
ejpam-3842	29	8	and	and	CCONJ
ejpam-3842	29	9	only	only	ADV
ejpam-3842	29	10	if	if	SCONJ
ejpam-3842	29	11	for	for	ADP
ejpam-3842	29	12	each	each	DET
ejpam-3842	29	13	x	x	SYM
ejpam-3842	29	14	∈	∈	PROPN
ejpam-3842	29	15	x	x	X
ejpam-3842	29	16	,	,	PUNCT
ejpam-3842	29	17	{	{	PUNCT
ejpam-3842	29	18	x	x	X
ejpam-3842	29	19	}	}	PUNCT
ejpam-3842	29	20	is	be	AUX
ejpam-3842	29	21	a	a	DET
ejpam-3842	29	22	closed	closed	ADJ
ejpam-3842	29	23	set	set	NOUN
ejpam-3842	29	24	in	in	ADP
ejpam-3842	29	25	x.	x.	NOUN
ejpam-3842	29	26	definition	definition	NOUN
ejpam-3842	29	27	3	3	NUM
ejpam-3842	29	28	.	.	PUNCT
ejpam-3842	30	1	[	[	X
ejpam-3842	30	2	5	5	NUM
ejpam-3842	30	3	]	]	PUNCT
ejpam-3842	30	4	a	a	DET
ejpam-3842	30	5	bch	bch	NOUN
ejpam-3842	30	6	-	-	PUNCT
ejpam-3842	30	7	algebra	algebra	NOUN
ejpam-3842	30	8	is	be	AUX
ejpam-3842	30	9	a	a	DET
ejpam-3842	30	10	nonempty	nonempty	ADV
ejpam-3842	30	11	set	set	VERB
ejpam-3842	30	12	h	h	NOUN
ejpam-3842	30	13	endowed	endow	VERB
ejpam-3842	30	14	with	with	ADP
ejpam-3842	30	15	a	a	DET
ejpam-3842	30	16	operation	operation	NOUN
ejpam-3842	30	17	“	"	PUNCT
ejpam-3842	30	18	∗	∗	NOUN
ejpam-3842	30	19	”	"	PUNCT
ejpam-3842	30	20	and	and	CCONJ
ejpam-3842	30	21	constant	constant	ADJ
ejpam-3842	30	22	0	0	NUM
ejpam-3842	30	23	satisfying	satisfy	VERB
ejpam-3842	30	24	the	the	DET
ejpam-3842	30	25	following	follow	VERB
ejpam-3842	30	26	axioms	axiom	NOUN
ejpam-3842	30	27	:	:	PUNCT
ejpam-3842	30	28	for	for	ADP
ejpam-3842	30	29	all	all	DET
ejpam-3842	30	30	x	x	NOUN
ejpam-3842	30	31	,	,	PUNCT
ejpam-3842	30	32	y	y	PROPN
ejpam-3842	30	33	,	,	PUNCT
ejpam-3842	30	34	z	z	PROPN
ejpam-3842	30	35	∈	∈	PROPN
ejpam-3842	30	36	h	h	NOUN
ejpam-3842	30	37	,	,	PUNCT
ejpam-3842	30	38	(	(	PUNCT
ejpam-3842	30	39	b1	b1	NOUN
ejpam-3842	30	40	)	)	PUNCT
ejpam-3842	30	41	x	x	SYM
ejpam-3842	30	42	∗	∗	NOUN
ejpam-3842	30	43	x	x	SYM
ejpam-3842	30	44	=	=	SYM
ejpam-3842	30	45	0	0	NUM
ejpam-3842	30	46	,	,	PUNCT
ejpam-3842	30	47	(	(	PUNCT
ejpam-3842	30	48	b2	b2	NOUN
ejpam-3842	30	49	)	)	PUNCT
ejpam-3842	30	50	x	x	SYM
ejpam-3842	30	51	∗	∗	NOUN
ejpam-3842	30	52	y	y	NOUN
ejpam-3842	30	53	=	=	SYM
ejpam-3842	30	54	0	0	PROPN
ejpam-3842	31	1	and	and	CCONJ
ejpam-3842	31	2	y	y	PROPN
ejpam-3842	31	3	∗	∗	NOUN
ejpam-3842	31	4	x	x	PUNCT
ejpam-3842	31	5	=	=	SYM
ejpam-3842	31	6	0	0	NUM
ejpam-3842	31	7	implies	imply	VERB
ejpam-3842	31	8	x	x	PUNCT
ejpam-3842	31	9	=	=	SYM
ejpam-3842	31	10	y.	y.	NOUN
ejpam-3842	31	11	(	(	PUNCT
ejpam-3842	31	12	b3	b3	PROPN
ejpam-3842	31	13	)	)	PUNCT
ejpam-3842	31	14	(	(	PUNCT
ejpam-3842	31	15	x	x	SYM
ejpam-3842	31	16	∗	∗	PROPN
ejpam-3842	31	17	y	y	NOUN
ejpam-3842	31	18	)	)	PUNCT
ejpam-3842	31	19	∗	∗	NOUN
ejpam-3842	31	20	z	z	NOUN
ejpam-3842	31	21	=	=	SYM
ejpam-3842	31	22	(	(	PUNCT
ejpam-3842	31	23	x	x	X
ejpam-3842	31	24	∗	∗	PROPN
ejpam-3842	31	25	z	z	NOUN
ejpam-3842	31	26	)	)	PUNCT
ejpam-3842	31	27	∗	∗	PROPN
ejpam-3842	31	28	y	y	PROPN
ejpam-3842	31	29	,	,	PUNCT
ejpam-3842	31	30	remark	remark	NOUN
ejpam-3842	31	31	2	2	NUM
ejpam-3842	31	32	.	.	PUNCT
ejpam-3842	32	1	[	[	X
ejpam-3842	32	2	5	5	NUM
ejpam-3842	32	3	,	,	PUNCT
ejpam-3842	32	4	6	6	NUM
ejpam-3842	32	5	]	]	PUNCT
ejpam-3842	32	6	in	in	ADP
ejpam-3842	32	7	any	any	DET
ejpam-3842	32	8	bch	bch	NOUN
ejpam-3842	32	9	-	-	PUNCT
ejpam-3842	32	10	algebra	algebra	NOUN
ejpam-3842	32	11	(	(	PUNCT
ejpam-3842	32	12	x	x	X
ejpam-3842	32	13	,	,	PUNCT
ejpam-3842	32	14	∗	∗	NOUN
ejpam-3842	32	15	,	,	PUNCT
ejpam-3842	32	16	0	0	NUM
ejpam-3842	32	17	)	)	PUNCT
ejpam-3842	32	18	,	,	PUNCT
ejpam-3842	32	19	the	the	DET
ejpam-3842	32	20	following	follow	VERB
ejpam-3842	32	21	hold	hold	NOUN
ejpam-3842	32	22	:	:	PUNCT
ejpam-3842	32	23	(	(	PUNCT
ejpam-3842	32	24	i	i	NOUN
ejpam-3842	32	25	)	)	PUNCT
ejpam-3842	32	26	x	x	SYM
ejpam-3842	33	1	∗	∗	NOUN
ejpam-3842	33	2	0	0	NUM
ejpam-3842	34	1	=	=	SYM
ejpam-3842	34	2	x	x	X
ejpam-3842	34	3	;	;	PUNCT
ejpam-3842	34	4	(	(	PUNCT
ejpam-3842	34	5	ii	ii	NOUN
ejpam-3842	34	6	)	)	PUNCT
ejpam-3842	34	7	x	x	SYM
ejpam-3842	35	1	∗	∗	NOUN
ejpam-3842	35	2	0	0	NUM
ejpam-3842	36	1	=	=	SYM
ejpam-3842	36	2	0	0	NUM
ejpam-3842	36	3	implies	imply	VERB
ejpam-3842	36	4	x	x	PUNCT
ejpam-3842	36	5	=	=	SYM
ejpam-3842	36	6	0	0	NUM
ejpam-3842	36	7	;	;	PUNCT
ejpam-3842	36	8	(	(	PUNCT
ejpam-3842	36	9	iii	iii	NOUN
ejpam-3842	36	10	)	)	PUNCT
ejpam-3842	36	11	0	0	NUM
ejpam-3842	36	12	∗	∗	NOUN
ejpam-3842	36	13	(	(	PUNCT
ejpam-3842	36	14	x	x	X
ejpam-3842	36	15	∗	∗	PROPN
ejpam-3842	36	16	y	y	NOUN
ejpam-3842	36	17	)	)	PUNCT
ejpam-3842	36	18	=	=	SYM
ejpam-3842	36	19	(	(	PUNCT
ejpam-3842	36	20	0	0	NUM
ejpam-3842	36	21	∗	∗	NOUN
ejpam-3842	36	22	x	x	NOUN
ejpam-3842	36	23	)	)	PUNCT
ejpam-3842	36	24	∗	∗	NOUN
ejpam-3842	36	25	(	(	PUNCT
ejpam-3842	36	26	0	0	NUM
ejpam-3842	36	27	∗	∗	NOUN
ejpam-3842	36	28	y	y	PROPN
ejpam-3842	36	29	)	)	PUNCT
ejpam-3842	36	30	;	;	PUNCT
ejpam-3842	36	31	(	(	PUNCT
ejpam-3842	36	32	iv	iv	X
ejpam-3842	36	33	)	)	PUNCT
ejpam-3842	36	34	(	(	PUNCT
ejpam-3842	36	35	x	x	SYM
ejpam-3842	36	36	∗	∗	NOUN
ejpam-3842	36	37	(	(	PUNCT
ejpam-3842	36	38	x	x	X
ejpam-3842	36	39	∗	∗	PROPN
ejpam-3842	36	40	y	y	NOUN
ejpam-3842	36	41	)	)	PUNCT
ejpam-3842	36	42	)	)	PUNCT
ejpam-3842	37	1	∗	∗	NOUN
ejpam-3842	37	2	y	y	NOUN
ejpam-3842	37	3	=	=	SYM
ejpam-3842	37	4	0	0	PROPN
ejpam-3842	37	5	.	.	PUNCT
ejpam-3842	38	1	definition	definition	NOUN
ejpam-3842	38	2	4	4	NUM
ejpam-3842	38	3	.	.	PUNCT
ejpam-3842	39	1	[	[	X
ejpam-3842	39	2	7	7	X
ejpam-3842	39	3	]	]	X
ejpam-3842	39	4	let	let	VERB
ejpam-3842	39	5	(	(	PUNCT
ejpam-3842	39	6	x	x	X
ejpam-3842	39	7	,	,	PUNCT
ejpam-3842	39	8	∗	∗	NOUN
ejpam-3842	39	9	,	,	PUNCT
ejpam-3842	39	10	0	0	NUM
ejpam-3842	39	11	)	)	PUNCT
ejpam-3842	39	12	be	be	AUX
ejpam-3842	39	13	a	a	DET
ejpam-3842	39	14	bch	bch	NOUN
ejpam-3842	39	15	-	-	PUNCT
ejpam-3842	39	16	algebra	algebra	NOUN
ejpam-3842	39	17	and	and	CCONJ
ejpam-3842	39	18	u	u	NOUN
ejpam-3842	39	19	,	,	PUNCT
ejpam-3842	39	20	v	v	X
ejpam-3842	39	21	be	be	AUX
ejpam-3842	39	22	any	any	DET
ejpam-3842	39	23	nonempty	nonempty	ADJ
ejpam-3842	39	24	subsets	subset	NOUN
ejpam-3842	39	25	of	of	ADP
ejpam-3842	39	26	x.	x.	NOUN
ejpam-3842	39	27	we	we	PRON
ejpam-3842	39	28	define	define	VERB
ejpam-3842	39	29	a	a	DET
ejpam-3842	39	30	subset	subset	NOUN
ejpam-3842	39	31	u	u	NOUN
ejpam-3842	39	32	∗	∗	NOUN
ejpam-3842	39	33	v	v	NOUN
ejpam-3842	39	34	of	of	ADP
ejpam-3842	39	35	x	x	PUNCT
ejpam-3842	39	36	by	by	ADP
ejpam-3842	39	37	u	u	NOUN
ejpam-3842	39	38	∗	∗	NOUN
ejpam-3842	39	39	v	v	NOUN
ejpam-3842	39	40	=	=	SYM
ejpam-3842	39	41	{	{	PUNCT
ejpam-3842	39	42	x	x	X
ejpam-3842	39	43	∗	∗	NOUN
ejpam-3842	39	44	y	y	NOUN
ejpam-3842	39	45	:	:	PUNCT
ejpam-3842	39	46	x	x	X
ejpam-3842	39	47	∈	∈	PROPN
ejpam-3842	39	48	u	u	NOUN
ejpam-3842	39	49	,	,	PUNCT
ejpam-3842	39	50	y	y	PROPN
ejpam-3842	39	51	∈	∈	PROPN
ejpam-3842	39	52	v	v	ADP
ejpam-3842	39	53	}	}	PUNCT
ejpam-3842	39	54	.	.	PUNCT
ejpam-3842	40	1	remark	remark	PROPN
ejpam-3842	40	2	3	3	NUM
ejpam-3842	40	3	.	.	PUNCT
ejpam-3842	41	1	let	let	VERB
ejpam-3842	41	2	(	(	PUNCT
ejpam-3842	41	3	x	x	X
ejpam-3842	41	4	,	,	PUNCT
ejpam-3842	41	5	∗	∗	NOUN
ejpam-3842	41	6	,	,	PUNCT
ejpam-3842	41	7	0	0	NUM
ejpam-3842	41	8	)	)	PUNCT
ejpam-3842	41	9	be	be	AUX
ejpam-3842	41	10	a	a	DET
ejpam-3842	41	11	bch	bch	NOUN
ejpam-3842	41	12	-	-	PUNCT
ejpam-3842	41	13	algebra	algebra	NOUN
ejpam-3842	41	14	.	.	PUNCT
ejpam-3842	42	1	then	then	ADV
ejpam-3842	42	2	∗(a	∗(a	VERB
ejpam-3842	42	3	×	×	PROPN
ejpam-3842	42	4	b	b	NOUN
ejpam-3842	42	5	)	)	PUNCT
ejpam-3842	42	6	=	=	PUNCT
ejpam-3842	42	7	a	a	DET
ejpam-3842	42	8	∗	∗	NOUN
ejpam-3842	42	9	b	b	NOUN
ejpam-3842	42	10	for	for	ADP
ejpam-3842	42	11	any	any	DET
ejpam-3842	42	12	nonempty	nonempty	NOUN
ejpam-3842	42	13	subsets	subset	NOUN
ejpam-3842	42	14	a	a	DET
ejpam-3842	42	15	and	and	CCONJ
ejpam-3842	42	16	b	b	PROPN
ejpam-3842	42	17	of	of	ADP
ejpam-3842	42	18	x.	x.	NOUN
ejpam-3842	42	19	remark	remark	PROPN
ejpam-3842	42	20	4	4	NUM
ejpam-3842	42	21	.	.	PUNCT
ejpam-3842	43	1	let	let	VERB
ejpam-3842	43	2	(	(	PUNCT
ejpam-3842	43	3	x	x	X
ejpam-3842	43	4	,	,	PUNCT
ejpam-3842	43	5	∗	∗	NOUN
ejpam-3842	43	6	,	,	PUNCT
ejpam-3842	43	7	0	0	NUM
ejpam-3842	43	8	)	)	PUNCT
ejpam-3842	43	9	be	be	AUX
ejpam-3842	43	10	a	a	DET
ejpam-3842	43	11	bch	bch	NOUN
ejpam-3842	43	12	-	-	PUNCT
ejpam-3842	43	13	algebra	algebra	NOUN
ejpam-3842	43	14	and	and	CCONJ
ejpam-3842	43	15	a	a	PRON
ejpam-3842	43	16	,	,	PUNCT
ejpam-3842	43	17	b	b	PROPN
ejpam-3842	43	18	⊆	⊆	NUM
ejpam-3842	43	19	x.	x.	NOUN
ejpam-3842	43	20	if	if	SCONJ
ejpam-3842	43	21	a∩b	a∩b	PROPN
ejpam-3842	43	22	6=	6=	ADP
ejpam-3842	43	23	∅	∅	NOUN
ejpam-3842	43	24	,	,	PUNCT
ejpam-3842	43	25	then	then	ADV
ejpam-3842	43	26	0	0	NUM
ejpam-3842	43	27	∈	∈	PROPN
ejpam-3842	43	28	a	a	DET
ejpam-3842	43	29	∗b	∗b	PROPN
ejpam-3842	43	30	.	.	PUNCT
ejpam-3842	44	1	definition	definition	NOUN
ejpam-3842	44	2	5	5	NUM
ejpam-3842	44	3	.	.	PUNCT
ejpam-3842	45	1	[	[	X
ejpam-3842	45	2	2	2	NUM
ejpam-3842	45	3	]	]	X
ejpam-3842	45	4	let	let	VERB
ejpam-3842	45	5	(	(	PUNCT
ejpam-3842	45	6	x	x	NOUN
ejpam-3842	45	7	,	,	PUNCT
ejpam-3842	45	8	∗	∗	NOUN
ejpam-3842	45	9	,	,	PUNCT
ejpam-3842	45	10	0	0	NUM
ejpam-3842	45	11	)	)	PUNCT
ejpam-3842	45	12	be	be	AUX
ejpam-3842	45	13	a	a	DET
ejpam-3842	45	14	bch	bch	NOUN
ejpam-3842	45	15	-	-	PUNCT
ejpam-3842	45	16	algebra	algebra	NOUN
ejpam-3842	45	17	.	.	PUNCT
ejpam-3842	46	1	a	a	DET
ejpam-3842	46	2	nonempty	nonempty	ADJ
ejpam-3842	46	3	subset	subset	VERB
ejpam-3842	46	4	s	s	NOUN
ejpam-3842	46	5	of	of	ADP
ejpam-3842	46	6	x	x	PRON
ejpam-3842	46	7	is	be	AUX
ejpam-3842	46	8	a	a	DET
ejpam-3842	46	9	bchsubalgebra	bchsubalgebra	NOUN
ejpam-3842	46	10	if	if	SCONJ
ejpam-3842	46	11	for	for	ADP
ejpam-3842	46	12	each	each	DET
ejpam-3842	46	13	x	x	NOUN
ejpam-3842	46	14	,	,	PUNCT
ejpam-3842	46	15	y	y	PROPN
ejpam-3842	46	16	∈	∈	PROPN
ejpam-3842	46	17	s	s	PART
ejpam-3842	46	18	,	,	PUNCT
ejpam-3842	46	19	x	x	PUNCT
ejpam-3842	46	20	∗	∗	NOUN
ejpam-3842	46	21	y	y	PROPN
ejpam-3842	46	22	∈	∈	PROPN
ejpam-3842	46	23	s.	s.	PROPN
ejpam-3842	46	24	j.	j.	PROPN
ejpam-3842	46	25	mancao	mancao	PROPN
ejpam-3842	46	26	,	,	PUNCT
ejpam-3842	46	27	s.	s.	PROPN
ejpam-3842	46	28	canoy	canoy	PROPN
ejpam-3842	46	29	/	/	SYM
ejpam-3842	46	30	eur	eur	PROPN
ejpam-3842	46	31	.	.	PUNCT
ejpam-3842	47	1	j.	j.	PROPN
ejpam-3842	47	2	pure	pure	PROPN
ejpam-3842	47	3	appl	appl	PROPN
ejpam-3842	47	4	.	.	PROPN
ejpam-3842	47	5	math	math	PROPN
ejpam-3842	47	6	,	,	PUNCT
ejpam-3842	47	7	13	13	NUM
ejpam-3842	47	8	(	(	PUNCT
ejpam-3842	47	9	4	4	NUM
ejpam-3842	47	10	)	)	PUNCT
ejpam-3842	47	11	(	(	PUNCT
ejpam-3842	47	12	2020	2020	NUM
ejpam-3842	47	13	)	)	PUNCT
ejpam-3842	47	14	,	,	PUNCT
ejpam-3842	47	15	730	730	NUM
ejpam-3842	47	16	-	-	SYM
ejpam-3842	47	17	738	738	NUM
ejpam-3842	47	18	732	732	NUM
ejpam-3842	47	19	definition	definition	NOUN
ejpam-3842	47	20	6	6	NUM
ejpam-3842	47	21	.	.	PUNCT
ejpam-3842	48	1	[	[	X
ejpam-3842	48	2	7	7	X
ejpam-3842	48	3	]	]	X
ejpam-3842	48	4	let	let	NOUN
ejpam-3842	48	5	(	(	PUNCT
ejpam-3842	48	6	h	h	NOUN
ejpam-3842	48	7	,	,	PUNCT
ejpam-3842	48	8	∗	∗	NOUN
ejpam-3842	48	9	,	,	PUNCT
ejpam-3842	48	10	0	0	NUM
ejpam-3842	48	11	)	)	PUNCT
ejpam-3842	48	12	be	be	AUX
ejpam-3842	48	13	a	a	DET
ejpam-3842	48	14	bch	bch	NOUN
ejpam-3842	48	15	-	-	PUNCT
ejpam-3842	48	16	algebra	algebra	NOUN
ejpam-3842	48	17	.	.	PUNCT
ejpam-3842	49	1	a	a	DET
ejpam-3842	49	2	topology	topology	NOUN
ejpam-3842	49	3	τ	τ	PROPN
ejpam-3842	49	4	furnished	furnish	VERB
ejpam-3842	49	5	on	on	ADP
ejpam-3842	49	6	h	h	NOUN
ejpam-3842	49	7	is	be	AUX
ejpam-3842	49	8	called	call	VERB
ejpam-3842	49	9	a	a	DET
ejpam-3842	49	10	bch	bch	NOUN
ejpam-3842	49	11	-	-	PUNCT
ejpam-3842	49	12	topology	topology	NOUN
ejpam-3842	49	13	on	on	ADP
ejpam-3842	49	14	h.	h.	PROPN
ejpam-3842	49	15	in	in	ADP
ejpam-3842	49	16	addition	addition	NOUN
ejpam-3842	49	17	,	,	PUNCT
ejpam-3842	49	18	(	(	PUNCT
ejpam-3842	49	19	h	h	NOUN
ejpam-3842	49	20	,	,	PUNCT
ejpam-3842	49	21	τ	τ	X
ejpam-3842	49	22	)	)	PUNCT
ejpam-3842	49	23	is	be	AUX
ejpam-3842	49	24	called	call	VERB
ejpam-3842	49	25	a	a	DET
ejpam-3842	49	26	topological	topological	ADJ
ejpam-3842	49	27	bch	bch	NOUN
ejpam-3842	49	28	-	-	PUNCT
ejpam-3842	49	29	algebra	algebra	PROPN
ejpam-3842	49	30	(	(	PUNCT
ejpam-3842	49	31	or	or	CCONJ
ejpam-3842	49	32	tbchalgebra	tbchalgebra	ADJ
ejpam-3842	49	33	)	)	PUNCT
ejpam-3842	49	34	if	if	SCONJ
ejpam-3842	49	35	τ	τ	PROPN
ejpam-3842	49	36	is	be	AUX
ejpam-3842	49	37	a	a	DET
ejpam-3842	49	38	bch	bch	NOUN
ejpam-3842	49	39	-	-	PUNCT
ejpam-3842	49	40	topology	topology	NOUN
ejpam-3842	49	41	on	on	ADP
ejpam-3842	49	42	h	h	NOUN
ejpam-3842	49	43	and	and	CCONJ
ejpam-3842	49	44	the	the	DET
ejpam-3842	49	45	function	function	NOUN
ejpam-3842	49	46	∗	∗	NOUN
ejpam-3842	49	47	:	:	PUNCT
ejpam-3842	49	48	h	h	PROPN
ejpam-3842	49	49	×	×	NOUN
ejpam-3842	49	50	h	h	NOUN
ejpam-3842	49	51	→	→	SYM
ejpam-3842	49	52	h	h	PRON
ejpam-3842	49	53	defined	define	VERB
ejpam-3842	49	54	as	as	ADP
ejpam-3842	49	55	∗((x	∗((x	NOUN
ejpam-3842	49	56	,	,	PUNCT
ejpam-3842	49	57	y	y	NOUN
ejpam-3842	49	58	)	)	PUNCT
ejpam-3842	49	59	)	)	PUNCT
ejpam-3842	50	1	=	=	PRON
ejpam-3842	50	2	x∗y	x∗y	X
ejpam-3842	50	3	is	be	AUX
ejpam-3842	50	4	continuous	continuous	ADJ
ejpam-3842	50	5	,	,	PUNCT
ejpam-3842	50	6	where	where	SCONJ
ejpam-3842	50	7	the	the	DET
ejpam-3842	50	8	cartesian	cartesian	ADJ
ejpam-3842	50	9	product	product	NOUN
ejpam-3842	50	10	topology	topology	NOUN
ejpam-3842	50	11	on	on	ADP
ejpam-3842	50	12	h×h	h×h	PROPN
ejpam-3842	50	13	is	be	AUX
ejpam-3842	50	14	furnished	furnish	VERB
ejpam-3842	50	15	by	by	ADP
ejpam-3842	50	16	τ	τ	PROPN
ejpam-3842	50	17	.	.	PUNCT
ejpam-3842	50	18	example	example	NOUN
ejpam-3842	51	1	1	1	NUM
ejpam-3842	51	2	.	.	PUNCT
ejpam-3842	51	3	let	let	VERB
ejpam-3842	51	4	x	x	PUNCT
ejpam-3842	51	5	=	=	PUNCT
ejpam-3842	51	6	{	{	PUNCT
ejpam-3842	51	7	0	0	NUM
ejpam-3842	51	8	,	,	PUNCT
ejpam-3842	51	9	1	1	NUM
ejpam-3842	51	10	,	,	PUNCT
ejpam-3842	51	11	2	2	NUM
ejpam-3842	51	12	,	,	PUNCT
ejpam-3842	51	13	3	3	NUM
ejpam-3842	51	14	,	,	PUNCT
ejpam-3842	51	15	4	4	NUM
ejpam-3842	51	16	}	}	PUNCT
ejpam-3842	51	17	and	and	CCONJ
ejpam-3842	51	18	define	define	VERB
ejpam-3842	51	19	∗	∗	NOUN
ejpam-3842	51	20	as	as	SCONJ
ejpam-3842	51	21	follows	follow	VERB
ejpam-3842	51	22	:	:	PUNCT
ejpam-3842	51	23	∗	∗	NOUN
ejpam-3842	51	24	0	0	NUM
ejpam-3842	52	1	1	1	NUM
ejpam-3842	52	2	2	2	NUM
ejpam-3842	52	3	3	3	NUM
ejpam-3842	52	4	4	4	NUM
ejpam-3842	52	5	0	0	NUM
ejpam-3842	52	6	0	0	NUM
ejpam-3842	52	7	0	0	NUM
ejpam-3842	52	8	0	0	NUM
ejpam-3842	52	9	0	0	NUM
ejpam-3842	52	10	4	4	NUM
ejpam-3842	52	11	1	1	NUM
ejpam-3842	52	12	1	1	NUM
ejpam-3842	52	13	0	0	NUM
ejpam-3842	52	14	0	0	NUM
ejpam-3842	52	15	1	1	NUM
ejpam-3842	52	16	4	4	NUM
ejpam-3842	52	17	2	2	NUM
ejpam-3842	52	18	2	2	NUM
ejpam-3842	52	19	2	2	NUM
ejpam-3842	52	20	0	0	NUM
ejpam-3842	52	21	0	0	NUM
ejpam-3842	52	22	4	4	NUM
ejpam-3842	52	23	3	3	NUM
ejpam-3842	52	24	3	3	NUM
ejpam-3842	52	25	3	3	NUM
ejpam-3842	52	26	3	3	NUM
ejpam-3842	52	27	0	0	NUM
ejpam-3842	52	28	4	4	NUM
ejpam-3842	52	29	4	4	NUM
ejpam-3842	52	30	4	4	NUM
ejpam-3842	52	31	4	4	NUM
ejpam-3842	52	32	4	4	NUM
ejpam-3842	52	33	4	4	NUM
ejpam-3842	52	34	0	0	NUM
ejpam-3842	52	35	then	then	ADV
ejpam-3842	52	36	,	,	PUNCT
ejpam-3842	52	37	(	(	PUNCT
ejpam-3842	52	38	x	x	X
ejpam-3842	52	39	,	,	PUNCT
ejpam-3842	52	40	∗	∗	NOUN
ejpam-3842	52	41	,	,	PUNCT
ejpam-3842	52	42	0	0	NUM
ejpam-3842	52	43	)	)	PUNCT
ejpam-3842	52	44	is	be	AUX
ejpam-3842	52	45	a	a	DET
ejpam-3842	52	46	bch	bch	NOUN
ejpam-3842	52	47	-	-	PUNCT
ejpam-3842	52	48	algebra	algebra	NOUN
ejpam-3842	52	49	[	[	X
ejpam-3842	52	50	1	1	NUM
ejpam-3842	52	51	]	]	PUNCT
ejpam-3842	52	52	.	.	PUNCT
ejpam-3842	53	1	let	let	VERB
ejpam-3842	53	2	τ	τ	PROPN
ejpam-3842	53	3	=	=	PRON
ejpam-3842	53	4	{	{	PUNCT
ejpam-3842	53	5	x,∅	x,∅	PROPN
ejpam-3842	53	6	,	,	PUNCT
ejpam-3842	53	7	{	{	PUNCT
ejpam-3842	53	8	4	4	NUM
ejpam-3842	53	9	}	}	PUNCT
ejpam-3842	53	10	,	,	PUNCT
ejpam-3842	53	11	{	{	PUNCT
ejpam-3842	53	12	0	0	NUM
ejpam-3842	53	13	,	,	PUNCT
ejpam-3842	53	14	1	1	NUM
ejpam-3842	53	15	,	,	PUNCT
ejpam-3842	53	16	2	2	NUM
ejpam-3842	53	17	,	,	PUNCT
ejpam-3842	53	18	3	3	NUM
ejpam-3842	53	19	}	}	PUNCT
ejpam-3842	53	20	}	}	PUNCT
ejpam-3842	53	21	.	.	PUNCT
ejpam-3842	54	1	then	then	ADV
ejpam-3842	54	2	τ	τ	PROPN
ejpam-3842	54	3	is	be	AUX
ejpam-3842	54	4	a	a	DET
ejpam-3842	54	5	bch	bch	NOUN
ejpam-3842	54	6	-	-	PUNCT
ejpam-3842	54	7	topology	topology	NOUN
ejpam-3842	54	8	on	on	ADP
ejpam-3842	54	9	x.	x.	NOUN
ejpam-3842	54	10	moreover	moreover	ADV
ejpam-3842	54	11	,	,	PUNCT
ejpam-3842	54	12	∗−1(x	∗−1(x	PROPN
ejpam-3842	54	13	)	)	PUNCT
ejpam-3842	54	14	=	=	SYM
ejpam-3842	54	15	x	x	X
ejpam-3842	54	16	×x	×x	X
ejpam-3842	54	17	∗−1(∅	∗−1(∅	NOUN
ejpam-3842	54	18	)	)	PUNCT
ejpam-3842	54	19	=	=	SYM
ejpam-3842	54	20	∅	∅	NOUN
ejpam-3842	54	21	∗−1({4	∗−1({4	NOUN
ejpam-3842	54	22	}	}	PUNCT
ejpam-3842	54	23	)	)	PUNCT
ejpam-3842	54	24	=	=	SYM
ejpam-3842	54	25	(	(	PUNCT
ejpam-3842	54	26	{	{	PUNCT
ejpam-3842	54	27	0	0	NUM
ejpam-3842	54	28	,	,	PUNCT
ejpam-3842	54	29	1	1	NUM
ejpam-3842	54	30	,	,	PUNCT
ejpam-3842	54	31	2	2	NUM
ejpam-3842	54	32	,	,	PUNCT
ejpam-3842	54	33	3	3	NUM
ejpam-3842	54	34	}	}	SYM
ejpam-3842	54	35	×	×	NOUN
ejpam-3842	54	36	{	{	PUNCT
ejpam-3842	54	37	4	4	NUM
ejpam-3842	54	38	}	}	PUNCT
ejpam-3842	54	39	)	)	PUNCT
ejpam-3842	54	40	∪	∪	X
ejpam-3842	54	41	(	(	PUNCT
ejpam-3842	54	42	{	{	PUNCT
ejpam-3842	54	43	4	4	NUM
ejpam-3842	54	44	}	}	PUNCT
ejpam-3842	54	45	×	×	NOUN
ejpam-3842	54	46	{	{	PUNCT
ejpam-3842	54	47	0	0	NUM
ejpam-3842	54	48	,	,	PUNCT
ejpam-3842	54	49	1	1	NUM
ejpam-3842	54	50	,	,	PUNCT
ejpam-3842	54	51	2	2	NUM
ejpam-3842	54	52	,	,	PUNCT
ejpam-3842	54	53	3	3	NUM
ejpam-3842	54	54	}	}	PUNCT
ejpam-3842	54	55	)	)	PUNCT
ejpam-3842	55	1	∗−1({0	∗−1({0	NOUN
ejpam-3842	55	2	,	,	PUNCT
ejpam-3842	55	3	1	1	NUM
ejpam-3842	55	4	,	,	PUNCT
ejpam-3842	55	5	2	2	NUM
ejpam-3842	55	6	,	,	PUNCT
ejpam-3842	55	7	3	3	NUM
ejpam-3842	55	8	}	}	PUNCT
ejpam-3842	55	9	)	)	PUNCT
ejpam-3842	56	1	=	=	SYM
ejpam-3842	56	2	(	(	PUNCT
ejpam-3842	56	3	{	{	PUNCT
ejpam-3842	56	4	0	0	NUM
ejpam-3842	56	5	,	,	PUNCT
ejpam-3842	56	6	1	1	NUM
ejpam-3842	56	7	,	,	PUNCT
ejpam-3842	56	8	2	2	NUM
ejpam-3842	56	9	,	,	PUNCT
ejpam-3842	56	10	3	3	NUM
ejpam-3842	56	11	}	}	SYM
ejpam-3842	56	12	×	×	NOUN
ejpam-3842	56	13	{	{	PUNCT
ejpam-3842	56	14	0	0	NUM
ejpam-3842	56	15	,	,	PUNCT
ejpam-3842	56	16	1	1	NUM
ejpam-3842	56	17	,	,	PUNCT
ejpam-3842	56	18	2	2	NUM
ejpam-3842	56	19	,	,	PUNCT
ejpam-3842	56	20	3	3	NUM
ejpam-3842	56	21	}	}	PUNCT
ejpam-3842	56	22	)	)	PUNCT
ejpam-3842	56	23	∪	∪	X
ejpam-3842	56	24	(	(	PUNCT
ejpam-3842	56	25	{	{	PUNCT
ejpam-3842	56	26	4	4	NUM
ejpam-3842	56	27	}	}	PUNCT
ejpam-3842	56	28	×	×	NOUN
ejpam-3842	56	29	{	{	PUNCT
ejpam-3842	56	30	4	4	NUM
ejpam-3842	56	31	}	}	PUNCT
ejpam-3842	56	32	)	)	PUNCT
ejpam-3842	56	33	.	.	PUNCT
ejpam-3842	57	1	this	this	PRON
ejpam-3842	57	2	implies	imply	VERB
ejpam-3842	57	3	that	that	SCONJ
ejpam-3842	57	4	∗	∗	NOUN
ejpam-3842	57	5	is	be	AUX
ejpam-3842	57	6	continuous	continuous	ADJ
ejpam-3842	57	7	.	.	PUNCT
ejpam-3842	58	1	thus	thus	ADV
ejpam-3842	58	2	,	,	PUNCT
ejpam-3842	58	3	(	(	PUNCT
ejpam-3842	58	4	x	x	X
ejpam-3842	58	5	,	,	PUNCT
ejpam-3842	58	6	τ	τ	X
ejpam-3842	58	7	)	)	PUNCT
ejpam-3842	58	8	is	be	AUX
ejpam-3842	58	9	a	a	DET
ejpam-3842	58	10	tbch	tbch	NOUN
ejpam-3842	58	11	-	-	PUNCT
ejpam-3842	58	12	algebra	algebra	NOUN
ejpam-3842	58	13	.	.	PUNCT
ejpam-3842	59	1	3	3	X
ejpam-3842	59	2	.	.	NOUN
ejpam-3842	59	3	results	result	NOUN
ejpam-3842	59	4	throughout	throughout	ADP
ejpam-3842	59	5	this	this	DET
ejpam-3842	59	6	study	study	NOUN
ejpam-3842	59	7	,	,	PUNCT
ejpam-3842	59	8	we	we	PRON
ejpam-3842	59	9	denote	denote	VERB
ejpam-3842	59	10	a	a	DET
ejpam-3842	59	11	bch	bch	NOUN
ejpam-3842	59	12	-	-	PUNCT
ejpam-3842	59	13	algebra	algebra	NOUN
ejpam-3842	59	14	(	(	PUNCT
ejpam-3842	59	15	x	x	X
ejpam-3842	59	16	,	,	PUNCT
ejpam-3842	59	17	∗	∗	NOUN
ejpam-3842	59	18	,	,	PUNCT
ejpam-3842	59	19	0	0	NUM
ejpam-3842	59	20	)	)	PUNCT
ejpam-3842	59	21	by	by	ADP
ejpam-3842	59	22	x	x	X
ejpam-3842	59	23	,	,	PUNCT
ejpam-3842	59	24	unless	unless	SCONJ
ejpam-3842	59	25	otherwise	otherwise	ADV
ejpam-3842	59	26	specified	specify	VERB
ejpam-3842	59	27	.	.	PUNCT
ejpam-3842	60	1	theorem	theorem	NOUN
ejpam-3842	60	2	2	2	NUM
ejpam-3842	60	3	.	.	PUNCT
ejpam-3842	61	1	let	let	VERB
ejpam-3842	61	2	τ	τ	PROPN
ejpam-3842	61	3	be	be	AUX
ejpam-3842	61	4	a	a	DET
ejpam-3842	61	5	bch	bch	NOUN
ejpam-3842	61	6	-	-	PUNCT
ejpam-3842	61	7	topology	topology	NOUN
ejpam-3842	61	8	on	on	ADP
ejpam-3842	61	9	x.	x.	NOUN
ejpam-3842	61	10	then	then	ADV
ejpam-3842	61	11	,	,	PUNCT
ejpam-3842	61	12	(	(	PUNCT
ejpam-3842	61	13	x	x	X
ejpam-3842	61	14	,	,	PUNCT
ejpam-3842	61	15	τ	τ	X
ejpam-3842	61	16	)	)	PUNCT
ejpam-3842	61	17	is	be	AUX
ejpam-3842	61	18	a	a	DET
ejpam-3842	61	19	tbch	tbch	NOUN
ejpam-3842	61	20	-	-	PUNCT
ejpam-3842	61	21	algebra	algebra	NOUN
ejpam-3842	61	22	if	if	SCONJ
ejpam-3842	61	23	and	and	CCONJ
ejpam-3842	61	24	only	only	ADV
ejpam-3842	61	25	if	if	SCONJ
ejpam-3842	61	26	for	for	ADP
ejpam-3842	61	27	each	each	DET
ejpam-3842	61	28	x	x	NOUN
ejpam-3842	61	29	,	,	PUNCT
ejpam-3842	62	1	y	y	PROPN
ejpam-3842	62	2	∈	∈	PROPN
ejpam-3842	62	3	x	x	X
ejpam-3842	62	4	and	and	CCONJ
ejpam-3842	62	5	each	each	DET
ejpam-3842	62	6	nbhd	nbhd	NOUN
ejpam-3842	62	7	w	w	ADP
ejpam-3842	62	8	of	of	ADP
ejpam-3842	62	9	x	x	PROPN
ejpam-3842	62	10	∗	∗	PROPN
ejpam-3842	62	11	y	y	PROPN
ejpam-3842	62	12	,	,	PUNCT
ejpam-3842	62	13	there	there	PRON
ejpam-3842	62	14	exist	exist	VERB
ejpam-3842	62	15	nbhds	nbhds	ADJ
ejpam-3842	62	16	u	u	NOUN
ejpam-3842	62	17	and	and	CCONJ
ejpam-3842	62	18	v	v	NOUN
ejpam-3842	62	19	of	of	ADP
ejpam-3842	62	20	x	x	PROPN
ejpam-3842	62	21	and	and	CCONJ
ejpam-3842	62	22	y	y	PROPN
ejpam-3842	62	23	,	,	PUNCT
ejpam-3842	62	24	respectively	respectively	ADV
ejpam-3842	62	25	,	,	PUNCT
ejpam-3842	62	26	such	such	ADJ
ejpam-3842	62	27	that	that	SCONJ
ejpam-3842	62	28	u	u	PROPN
ejpam-3842	62	29	∗	∗	NOUN
ejpam-3842	62	30	v	v	NOUN
ejpam-3842	62	31	⊆w	⊆w	NOUN
ejpam-3842	62	32	.	.	PUNCT
ejpam-3842	63	1	proof	proof	NOUN
ejpam-3842	63	2	.	.	PUNCT
ejpam-3842	64	1	let	let	VERB
ejpam-3842	64	2	x	x	PRON
ejpam-3842	64	3	be	be	AUX
ejpam-3842	64	4	a	a	DET
ejpam-3842	64	5	tbch	tbch	NOUN
ejpam-3842	64	6	-	-	PUNCT
ejpam-3842	64	7	algebra	algebra	NOUN
ejpam-3842	64	8	.	.	PUNCT
ejpam-3842	65	1	let	let	VERB
ejpam-3842	65	2	x	x	PRON
ejpam-3842	65	3	,	,	PUNCT
ejpam-3842	65	4	y	y	PROPN
ejpam-3842	65	5	∈	∈	PROPN
ejpam-3842	65	6	x	x	X
ejpam-3842	65	7	and	and	CCONJ
ejpam-3842	65	8	a	a	DET
ejpam-3842	65	9	nbhd	nbhd	NOUN
ejpam-3842	65	10	w	w	NOUN
ejpam-3842	65	11	of	of	ADP
ejpam-3842	65	12	x	x	PROPN
ejpam-3842	65	13	∗	∗	NOUN
ejpam-3842	65	14	y.	y.	NOUN
ejpam-3842	65	15	since	since	SCONJ
ejpam-3842	65	16	∗	∗	NOUN
ejpam-3842	65	17	is	be	AUX
ejpam-3842	65	18	continuous	continuous	ADJ
ejpam-3842	65	19	,	,	PUNCT
ejpam-3842	65	20	∗−1(w	∗−1(w	PROPN
ejpam-3842	65	21	)	)	PUNCT
ejpam-3842	65	22	is	be	AUX
ejpam-3842	65	23	a	a	DET
ejpam-3842	65	24	nbhd	nbhd	NOUN
ejpam-3842	65	25	of	of	ADP
ejpam-3842	65	26	(	(	PUNCT
ejpam-3842	65	27	x	x	NOUN
ejpam-3842	65	28	,	,	PUNCT
ejpam-3842	65	29	y	y	NOUN
ejpam-3842	65	30	)	)	PUNCT
ejpam-3842	65	31	in	in	ADP
ejpam-3842	65	32	x	x	X
ejpam-3842	65	33	×	×	NOUN
ejpam-3842	65	34	x.	x.	NOUN
ejpam-3842	65	35	by	by	ADP
ejpam-3842	65	36	definition	definition	NOUN
ejpam-3842	65	37	of	of	ADP
ejpam-3842	65	38	cartesian	cartesian	ADJ
ejpam-3842	65	39	product	product	NOUN
ejpam-3842	65	40	topology	topology	NOUN
ejpam-3842	65	41	,	,	PUNCT
ejpam-3842	65	42	there	there	PRON
ejpam-3842	65	43	exist	exist	VERB
ejpam-3842	65	44	nbhds	nbhds	ADJ
ejpam-3842	65	45	u	u	NOUN
ejpam-3842	65	46	and	and	CCONJ
ejpam-3842	65	47	v	v	NOUN
ejpam-3842	65	48	of	of	ADP
ejpam-3842	65	49	x	x	PROPN
ejpam-3842	65	50	and	and	CCONJ
ejpam-3842	65	51	y	y	PROPN
ejpam-3842	65	52	,	,	PUNCT
ejpam-3842	65	53	respectively	respectively	ADV
ejpam-3842	65	54	,	,	PUNCT
ejpam-3842	65	55	such	such	ADJ
ejpam-3842	65	56	that	that	SCONJ
ejpam-3842	65	57	u	u	PROPN
ejpam-3842	65	58	×	×	NOUN
ejpam-3842	65	59	v	v	ADP
ejpam-3842	65	60	⊆	⊆	NUM
ejpam-3842	65	61	∗−1(w	∗−1(w	NOUN
ejpam-3842	65	62	)	)	PUNCT
ejpam-3842	65	63	.	.	PUNCT
ejpam-3842	66	1	by	by	ADP
ejpam-3842	66	2	remark	remark	NOUN
ejpam-3842	66	3	3	3	NUM
ejpam-3842	66	4	,	,	PUNCT
ejpam-3842	66	5	u	u	NOUN
ejpam-3842	66	6	∗	∗	NOUN
ejpam-3842	66	7	v	v	NOUN
ejpam-3842	66	8	=	=	NOUN
ejpam-3842	67	1	∗(u	∗(u	NOUN
ejpam-3842	67	2	×	×	NOUN
ejpam-3842	67	3	v	v	NOUN
ejpam-3842	67	4	)	)	PUNCT
ejpam-3842	67	5	.	.	PUNCT
ejpam-3842	68	1	it	it	PRON
ejpam-3842	68	2	follows	follow	VERB
ejpam-3842	68	3	that	that	SCONJ
ejpam-3842	68	4	u	u	PRON
ejpam-3842	68	5	∗	∗	NOUN
ejpam-3842	68	6	v	v	NOUN
ejpam-3842	68	7	⊆	⊆	NUM
ejpam-3842	68	8	∗(∗−1(w	∗(∗−1(w	PROPN
ejpam-3842	68	9	)	)	PUNCT
ejpam-3842	68	10	)	)	PUNCT
ejpam-3842	68	11	⊆w	⊆w	NOUN
ejpam-3842	68	12	.	.	PUNCT
ejpam-3842	69	1	conversely	conversely	ADV
ejpam-3842	69	2	,	,	PUNCT
ejpam-3842	69	3	suppose	suppose	VERB
ejpam-3842	69	4	that	that	SCONJ
ejpam-3842	69	5	for	for	ADP
ejpam-3842	69	6	each	each	DET
ejpam-3842	69	7	x	x	NOUN
ejpam-3842	69	8	,	,	PUNCT
ejpam-3842	69	9	y	y	PROPN
ejpam-3842	69	10	∈	∈	PROPN
ejpam-3842	69	11	x	x	X
ejpam-3842	69	12	and	and	CCONJ
ejpam-3842	69	13	each	each	DET
ejpam-3842	69	14	nbhd	nbhd	NOUN
ejpam-3842	69	15	w	w	ADP
ejpam-3842	69	16	of	of	ADP
ejpam-3842	69	17	x	x	PROPN
ejpam-3842	69	18	∗	∗	PROPN
ejpam-3842	69	19	y	y	PROPN
ejpam-3842	69	20	,	,	PUNCT
ejpam-3842	69	21	there	there	PRON
ejpam-3842	69	22	are	be	VERB
ejpam-3842	69	23	nbhds	nbhds	ADJ
ejpam-3842	69	24	u	u	NOUN
ejpam-3842	69	25	and	and	CCONJ
ejpam-3842	69	26	v	v	NOUN
ejpam-3842	69	27	of	of	ADP
ejpam-3842	69	28	x	x	PROPN
ejpam-3842	69	29	and	and	CCONJ
ejpam-3842	69	30	y	y	PROPN
ejpam-3842	69	31	,	,	PUNCT
ejpam-3842	69	32	respectively	respectively	ADV
ejpam-3842	69	33	,	,	PUNCT
ejpam-3842	69	34	such	such	ADJ
ejpam-3842	69	35	that	that	SCONJ
ejpam-3842	69	36	u	u	NOUN
ejpam-3842	69	37	∗v	∗v	ADJ
ejpam-3842	69	38	⊆w	⊆w	NOUN
ejpam-3842	69	39	.	.	PUNCT
ejpam-3842	70	1	by	by	ADP
ejpam-3842	70	2	definition	definition	NOUN
ejpam-3842	70	3	of	of	ADP
ejpam-3842	70	4	cartesian	cartesian	ADJ
ejpam-3842	70	5	product	product	NOUN
ejpam-3842	70	6	topology	topology	NOUN
ejpam-3842	70	7	,	,	PUNCT
ejpam-3842	70	8	u	u	PRON
ejpam-3842	70	9	×	×	NOUN
ejpam-3842	70	10	v	v	NOUN
ejpam-3842	70	11	is	be	AUX
ejpam-3842	70	12	a	a	DET
ejpam-3842	70	13	nbhd	nbhd	NOUN
ejpam-3842	70	14	of	of	ADP
ejpam-3842	70	15	(	(	PUNCT
ejpam-3842	70	16	x	x	NOUN
ejpam-3842	70	17	,	,	PUNCT
ejpam-3842	70	18	y	y	NOUN
ejpam-3842	70	19	)	)	PUNCT
ejpam-3842	70	20	in	in	ADP
ejpam-3842	70	21	x	x	X
ejpam-3842	70	22	×	×	NOUN
ejpam-3842	70	23	x.	x.	NOUN
ejpam-3842	70	24	by	by	ADP
ejpam-3842	70	25	remark	remark	NOUN
ejpam-3842	70	26	3	3	NUM
ejpam-3842	70	27	,	,	PUNCT
ejpam-3842	70	28	∗(u	∗(u	PROPN
ejpam-3842	70	29	×	×	NOUN
ejpam-3842	70	30	v	v	NOUN
ejpam-3842	70	31	)	)	PUNCT
ejpam-3842	70	32	=	=	SYM
ejpam-3842	70	33	u	u	NOUN
ejpam-3842	70	34	∗	∗	NOUN
ejpam-3842	70	35	v	v	NOUN
ejpam-3842	70	36	⊆	⊆	NUM
ejpam-3842	70	37	w	w	NOUN
ejpam-3842	70	38	.	.	PUNCT
ejpam-3842	71	1	therefore	therefore	ADV
ejpam-3842	71	2	,	,	PUNCT
ejpam-3842	71	3	∗	∗	NOUN
ejpam-3842	71	4	is	be	AUX
ejpam-3842	71	5	continuous	continuous	ADJ
ejpam-3842	71	6	.	.	PUNCT
ejpam-3842	72	1	corollary	corollary	ADJ
ejpam-3842	72	2	1	1	NUM
ejpam-3842	72	3	.	.	PUNCT
ejpam-3842	73	1	let	let	VERB
ejpam-3842	73	2	x	x	PRON
ejpam-3842	73	3	be	be	AUX
ejpam-3842	73	4	a	a	DET
ejpam-3842	73	5	tbch	tbch	NOUN
ejpam-3842	73	6	-	-	PUNCT
ejpam-3842	73	7	algebra	algebra	NOUN
ejpam-3842	73	8	and	and	CCONJ
ejpam-3842	73	9	a	a	DET
ejpam-3842	73	10	⊆	⊆	NUM
ejpam-3842	73	11	x.	x.	NOUN
ejpam-3842	73	12	if	if	SCONJ
ejpam-3842	73	13	z	z	NOUN
ejpam-3842	73	14	is	be	AUX
ejpam-3842	73	15	an	an	DET
ejpam-3842	73	16	interior	interior	ADJ
ejpam-3842	73	17	point	point	NOUN
ejpam-3842	73	18	of	of	ADP
ejpam-3842	73	19	a	a	PRON
ejpam-3842	73	20	,	,	PUNCT
ejpam-3842	73	21	then	then	ADV
ejpam-3842	73	22	there	there	PRON
ejpam-3842	73	23	exist	exist	VERB
ejpam-3842	73	24	elements	element	NOUN
ejpam-3842	73	25	x	x	X
ejpam-3842	73	26	,	,	PUNCT
ejpam-3842	73	27	y	y	PROPN
ejpam-3842	73	28	∈	∈	PROPN
ejpam-3842	73	29	x	x	X
ejpam-3842	73	30	and	and	CCONJ
ejpam-3842	73	31	nbhds	nbhds	PROPN
ejpam-3842	73	32	nx	nx	PROPN
ejpam-3842	73	33	,	,	PUNCT
ejpam-3842	73	34	ny	ny	PROPN
ejpam-3842	73	35	and	and	CCONJ
ejpam-3842	73	36	nz	nz	PROPN
ejpam-3842	73	37	of	of	ADP
ejpam-3842	73	38	x	x	PROPN
ejpam-3842	73	39	,	,	PUNCT
ejpam-3842	73	40	y	y	PROPN
ejpam-3842	73	41	and	and	CCONJ
ejpam-3842	73	42	z	z	PROPN
ejpam-3842	73	43	,	,	PUNCT
ejpam-3842	73	44	respectively	respectively	ADV
ejpam-3842	73	45	,	,	PUNCT
ejpam-3842	73	46	such	such	ADJ
ejpam-3842	73	47	that	that	SCONJ
ejpam-3842	73	48	z	z	NOUN
ejpam-3842	73	49	=	=	PUNCT
ejpam-3842	73	50	x	x	SYM
ejpam-3842	73	51	∗	∗	NOUN
ejpam-3842	73	52	y	y	PROPN
ejpam-3842	73	53	and	and	CCONJ
ejpam-3842	73	54	nx	nx	NUM
ejpam-3842	73	55	∗ny	∗ny	NUM
ejpam-3842	73	56	⊆	⊆	NUM
ejpam-3842	73	57	nz	nz	X
ejpam-3842	73	58	=	=	SYM
ejpam-3842	73	59	nx∗y	nx∗y	PROPN
ejpam-3842	74	1	.	.	PUNCT
ejpam-3842	74	2	j.	j.	PROPN
ejpam-3842	74	3	mancao	mancao	PROPN
ejpam-3842	74	4	,	,	PUNCT
ejpam-3842	74	5	s.	s.	PROPN
ejpam-3842	74	6	canoy	canoy	PROPN
ejpam-3842	74	7	/	/	SYM
ejpam-3842	74	8	eur	eur	PROPN
ejpam-3842	74	9	.	.	PUNCT
ejpam-3842	75	1	j.	j.	PROPN
ejpam-3842	75	2	pure	pure	PROPN
ejpam-3842	75	3	appl	appl	PROPN
ejpam-3842	75	4	.	.	PROPN
ejpam-3842	75	5	math	math	PROPN
ejpam-3842	75	6	,	,	PUNCT
ejpam-3842	75	7	13	13	NUM
ejpam-3842	75	8	(	(	PUNCT
ejpam-3842	75	9	4	4	NUM
ejpam-3842	75	10	)	)	PUNCT
ejpam-3842	75	11	(	(	PUNCT
ejpam-3842	75	12	2020	2020	NUM
ejpam-3842	75	13	)	)	PUNCT
ejpam-3842	75	14	,	,	PUNCT
ejpam-3842	75	15	730	730	NUM
ejpam-3842	75	16	-	-	SYM
ejpam-3842	75	17	738	738	NUM
ejpam-3842	75	18	733	733	NUM
ejpam-3842	75	19	proof	proof	NOUN
ejpam-3842	75	20	.	.	PUNCT
ejpam-3842	76	1	suppose	suppose	VERB
ejpam-3842	76	2	z	z	NOUN
ejpam-3842	76	3	is	be	AUX
ejpam-3842	76	4	an	an	DET
ejpam-3842	76	5	interior	interior	ADJ
ejpam-3842	76	6	point	point	NOUN
ejpam-3842	76	7	of	of	ADP
ejpam-3842	76	8	a.	a.	NOUN
ejpam-3842	76	9	then	then	ADV
ejpam-3842	76	10	there	there	PRON
ejpam-3842	76	11	exists	exist	VERB
ejpam-3842	76	12	a	a	DET
ejpam-3842	76	13	nbhd	nbhd	NOUN
ejpam-3842	76	14	nz	nz	NOUN
ejpam-3842	76	15	of	of	ADP
ejpam-3842	76	16	z	z	NOUN
ejpam-3842	76	17	such	such	ADJ
ejpam-3842	76	18	that	that	SCONJ
ejpam-3842	76	19	nz	nz	PROPN
ejpam-3842	76	20	⊆	⊆	NUM
ejpam-3842	76	21	a.	a.	NOUN
ejpam-3842	76	22	since	since	SCONJ
ejpam-3842	76	23	z	z	PROPN
ejpam-3842	76	24	∈	∈	PROPN
ejpam-3842	76	25	x	x	X
ejpam-3842	76	26	,	,	PUNCT
ejpam-3842	76	27	z	z	NOUN
ejpam-3842	76	28	=	=	SYM
ejpam-3842	77	1	x	x	SYM
ejpam-3842	77	2	∗	∗	X
ejpam-3842	77	3	y	y	PROPN
ejpam-3842	77	4	for	for	ADP
ejpam-3842	77	5	some	some	DET
ejpam-3842	77	6	x	x	NOUN
ejpam-3842	77	7	,	,	PUNCT
ejpam-3842	77	8	y	y	PROPN
ejpam-3842	77	9	∈	∈	PROPN
ejpam-3842	77	10	x	x	INTJ
ejpam-3842	77	11	(	(	PUNCT
ejpam-3842	77	12	say	say	INTJ
ejpam-3842	77	13	,	,	PUNCT
ejpam-3842	77	14	x	x	PUNCT
ejpam-3842	77	15	=	=	PUNCT
ejpam-3842	77	16	z	z	PROPN
ejpam-3842	77	17	and	and	CCONJ
ejpam-3842	77	18	y	y	PROPN
ejpam-3842	77	19	=	=	PROPN
ejpam-3842	77	20	0	0	NUM
ejpam-3842	77	21	)	)	PUNCT
ejpam-3842	77	22	.	.	PUNCT
ejpam-3842	78	1	by	by	ADP
ejpam-3842	78	2	theorem	theorem	NOUN
ejpam-3842	78	3	2	2	NUM
ejpam-3842	78	4	,	,	PUNCT
ejpam-3842	78	5	there	there	PRON
ejpam-3842	78	6	exist	exist	VERB
ejpam-3842	78	7	nbhds	nbhds	ADV
ejpam-3842	78	8	nx	nx	PROPN
ejpam-3842	78	9	and	and	CCONJ
ejpam-3842	78	10	ny	ny	PROPN
ejpam-3842	78	11	of	of	ADP
ejpam-3842	78	12	x	x	PROPN
ejpam-3842	78	13	and	and	CCONJ
ejpam-3842	78	14	y	y	PROPN
ejpam-3842	78	15	,	,	PUNCT
ejpam-3842	78	16	respectively	respectively	ADV
ejpam-3842	78	17	,	,	PUNCT
ejpam-3842	78	18	such	such	ADJ
ejpam-3842	78	19	that	that	SCONJ
ejpam-3842	78	20	nx	nx	PROPN
ejpam-3842	78	21	∗ny	∗ny	PROPN
ejpam-3842	78	22	⊆	⊆	NUM
ejpam-3842	78	23	nz	nz	X
ejpam-3842	78	24	=	=	SYM
ejpam-3842	78	25	nx∗y	nx∗y	PROPN
ejpam-3842	78	26	.	.	PUNCT
ejpam-3842	79	1	the	the	DET
ejpam-3842	79	2	next	next	ADJ
ejpam-3842	79	3	theorem	theorem	NOUN
ejpam-3842	79	4	asserts	assert	VERB
ejpam-3842	79	5	that	that	SCONJ
ejpam-3842	79	6	the	the	DET
ejpam-3842	79	7	topology	topology	NOUN
ejpam-3842	79	8	associated	associate	VERB
ejpam-3842	79	9	in	in	ADP
ejpam-3842	79	10	a	a	DET
ejpam-3842	79	11	tbch	tbch	NOUN
ejpam-3842	79	12	-	-	PUNCT
ejpam-3842	79	13	algebra	algebra	NOUN
ejpam-3842	79	14	having	have	VERB
ejpam-3842	79	15	{	{	PUNCT
ejpam-3842	79	16	0	0	NUM
ejpam-3842	79	17	}	}	PUNCT
ejpam-3842	79	18	as	as	ADP
ejpam-3842	79	19	an	an	DET
ejpam-3842	79	20	open	open	ADJ
ejpam-3842	79	21	set	set	NOUN
ejpam-3842	79	22	is	be	AUX
ejpam-3842	79	23	the	the	DET
ejpam-3842	79	24	discrete	discrete	ADJ
ejpam-3842	79	25	topology	topology	NOUN
ejpam-3842	79	26	.	.	PUNCT
ejpam-3842	80	1	theorem	theorem	NOUN
ejpam-3842	80	2	3	3	X
ejpam-3842	80	3	.	.	PUNCT
ejpam-3842	81	1	let	let	VERB
ejpam-3842	81	2	x	x	PRON
ejpam-3842	81	3	be	be	AUX
ejpam-3842	81	4	a	a	DET
ejpam-3842	81	5	tbch	tbch	NOUN
ejpam-3842	81	6	-	-	PUNCT
ejpam-3842	81	7	algebra	algebra	NOUN
ejpam-3842	81	8	.	.	PUNCT
ejpam-3842	82	1	then	then	ADV
ejpam-3842	82	2	{	{	PUNCT
ejpam-3842	82	3	0	0	X
ejpam-3842	82	4	}	}	PUNCT
ejpam-3842	82	5	is	be	AUX
ejpam-3842	82	6	an	an	DET
ejpam-3842	82	7	open	open	ADJ
ejpam-3842	82	8	set	set	NOUN
ejpam-3842	82	9	in	in	ADP
ejpam-3842	82	10	x	x	PUNCT
ejpam-3842	82	11	if	if	SCONJ
ejpam-3842	83	1	and	and	CCONJ
ejpam-3842	83	2	only	only	ADV
ejpam-3842	83	3	if	if	SCONJ
ejpam-3842	83	4	x	x	PRON
ejpam-3842	83	5	is	be	AUX
ejpam-3842	83	6	a	a	DET
ejpam-3842	83	7	discrete	discrete	ADJ
ejpam-3842	83	8	space	space	NOUN
ejpam-3842	83	9	.	.	PUNCT
ejpam-3842	84	1	proof	proof	NOUN
ejpam-3842	84	2	.	.	PUNCT
ejpam-3842	85	1	suppose	suppose	VERB
ejpam-3842	85	2	that	that	SCONJ
ejpam-3842	85	3	{	{	PUNCT
ejpam-3842	85	4	0	0	X
ejpam-3842	85	5	}	}	PUNCT
ejpam-3842	85	6	is	be	AUX
ejpam-3842	85	7	an	an	DET
ejpam-3842	85	8	open	open	ADJ
ejpam-3842	85	9	set	set	NOUN
ejpam-3842	85	10	in	in	ADP
ejpam-3842	85	11	x	x	PUNCT
ejpam-3842	85	12	and	and	CCONJ
ejpam-3842	85	13	let	let	VERB
ejpam-3842	85	14	x	x	X
ejpam-3842	85	15	∈	∈	PROPN
ejpam-3842	85	16	x.	x.	NOUN
ejpam-3842	85	17	then	then	ADV
ejpam-3842	85	18	,	,	PUNCT
ejpam-3842	85	19	x	x	X
ejpam-3842	85	20	∗	∗	NOUN
ejpam-3842	85	21	x	x	X
ejpam-3842	85	22	=	=	SYM
ejpam-3842	85	23	0	0	SYM
ejpam-3842	85	24	∈	∈	PROPN
ejpam-3842	85	25	{	{	PUNCT
ejpam-3842	85	26	0	0	NUM
ejpam-3842	85	27	}	}	PUNCT
ejpam-3842	85	28	by	by	ADP
ejpam-3842	85	29	(	(	PUNCT
ejpam-3842	85	30	b1	b1	NOUN
ejpam-3842	85	31	)	)	PUNCT
ejpam-3842	85	32	.	.	PUNCT
ejpam-3842	86	1	since	since	SCONJ
ejpam-3842	86	2	{	{	PUNCT
ejpam-3842	86	3	0	0	NUM
ejpam-3842	86	4	}	}	PUNCT
ejpam-3842	86	5	is	be	AUX
ejpam-3842	86	6	an	an	DET
ejpam-3842	86	7	open	open	ADJ
ejpam-3842	86	8	set	set	NOUN
ejpam-3842	86	9	in	in	ADP
ejpam-3842	86	10	x	x	NOUN
ejpam-3842	86	11	,	,	PUNCT
ejpam-3842	86	12	there	there	PRON
ejpam-3842	86	13	exist	exist	VERB
ejpam-3842	86	14	nbhds	nbhds	ADJ
ejpam-3842	86	15	u	u	NOUN
ejpam-3842	86	16	and	and	CCONJ
ejpam-3842	86	17	v	v	NOUN
ejpam-3842	86	18	of	of	ADP
ejpam-3842	86	19	x	x	PUNCT
ejpam-3842	86	20	such	such	ADJ
ejpam-3842	86	21	that	that	SCONJ
ejpam-3842	86	22	u	u	NOUN
ejpam-3842	86	23	∗v	∗v	NOUN
ejpam-3842	86	24	=	=	SYM
ejpam-3842	86	25	{	{	PUNCT
ejpam-3842	86	26	0	0	NUM
ejpam-3842	86	27	}	}	PUNCT
ejpam-3842	86	28	by	by	ADP
ejpam-3842	86	29	theorem	theorem	NOUN
ejpam-3842	86	30	2	2	NUM
ejpam-3842	86	31	.	.	PUNCT
ejpam-3842	87	1	let	let	VERB
ejpam-3842	87	2	w	w	NOUN
ejpam-3842	87	3	=	=	PUNCT
ejpam-3842	87	4	u	u	NOUN
ejpam-3842	87	5	∩	∩	NOUN
ejpam-3842	87	6	v	v	NOUN
ejpam-3842	87	7	.	.	PUNCT
ejpam-3842	88	1	then	then	ADV
ejpam-3842	88	2	,	,	PUNCT
ejpam-3842	88	3	w	w	PROPN
ejpam-3842	88	4	is	be	AUX
ejpam-3842	88	5	a	a	DET
ejpam-3842	88	6	nbhd	nbhd	NOUN
ejpam-3842	88	7	of	of	ADP
ejpam-3842	88	8	x	x	X
ejpam-3842	88	9	and	and	CCONJ
ejpam-3842	88	10	w	w	PROPN
ejpam-3842	88	11	∗w	∗w	PROPN
ejpam-3842	89	1	⊆	⊆	NUM
ejpam-3842	89	2	u	u	NOUN
ejpam-3842	89	3	∗	∗	NOUN
ejpam-3842	89	4	v	v	NOUN
ejpam-3842	89	5	.	.	PUNCT
ejpam-3842	90	1	hence	hence	ADV
ejpam-3842	90	2	,	,	PUNCT
ejpam-3842	90	3	w	w	PROPN
ejpam-3842	90	4	∗w	∗w	PROPN
ejpam-3842	90	5	=	=	PUNCT
ejpam-3842	90	6	{	{	PUNCT
ejpam-3842	90	7	0	0	NUM
ejpam-3842	90	8	}	}	PUNCT
ejpam-3842	90	9	.	.	PUNCT
ejpam-3842	91	1	let	let	VERB
ejpam-3842	91	2	y	y	PROPN
ejpam-3842	91	3	∈	∈	PROPN
ejpam-3842	91	4	w	w	PROPN
ejpam-3842	91	5	.	.	PUNCT
ejpam-3842	92	1	then	then	ADV
ejpam-3842	92	2	x	x	X
ejpam-3842	92	3	∗	∗	NOUN
ejpam-3842	92	4	y	y	NOUN
ejpam-3842	92	5	=	=	SYM
ejpam-3842	92	6	0	0	PUNCT
ejpam-3842	93	1	=	=	SYM
ejpam-3842	93	2	y	y	PROPN
ejpam-3842	93	3	∗	∗	NOUN
ejpam-3842	93	4	x.	x.	NOUN
ejpam-3842	93	5	by	by	ADP
ejpam-3842	93	6	(	(	PUNCT
ejpam-3842	93	7	b2	b2	NOUN
ejpam-3842	93	8	)	)	PUNCT
ejpam-3842	93	9	,	,	PUNCT
ejpam-3842	93	10	y	y	PROPN
ejpam-3842	93	11	=	=	PUNCT
ejpam-3842	93	12	x.	x.	PUNCT
ejpam-3842	93	13	thus	thus	ADV
ejpam-3842	93	14	,	,	PUNCT
ejpam-3842	93	15	w	w	PROPN
ejpam-3842	93	16	=	=	SYM
ejpam-3842	93	17	{	{	PUNCT
ejpam-3842	93	18	x	x	NOUN
ejpam-3842	93	19	}	}	PUNCT
ejpam-3842	93	20	,	,	PUNCT
ejpam-3842	93	21	showing	show	VERB
ejpam-3842	93	22	that	that	SCONJ
ejpam-3842	93	23	x	x	PRON
ejpam-3842	93	24	is	be	AUX
ejpam-3842	93	25	a	a	DET
ejpam-3842	93	26	discrete	discrete	ADJ
ejpam-3842	93	27	space	space	NOUN
ejpam-3842	93	28	.	.	PUNCT
ejpam-3842	94	1	conversely	conversely	ADV
ejpam-3842	94	2	,	,	PUNCT
ejpam-3842	94	3	suppose	suppose	VERB
ejpam-3842	94	4	x	x	PRON
ejpam-3842	94	5	is	be	AUX
ejpam-3842	94	6	the	the	DET
ejpam-3842	94	7	discrete	discrete	ADJ
ejpam-3842	94	8	space	space	NOUN
ejpam-3842	94	9	.	.	PUNCT
ejpam-3842	95	1	then	then	ADV
ejpam-3842	95	2	,	,	PUNCT
ejpam-3842	95	3	{	{	PUNCT
ejpam-3842	95	4	0	0	X
ejpam-3842	95	5	}	}	PUNCT
ejpam-3842	95	6	is	be	AUX
ejpam-3842	95	7	an	an	DET
ejpam-3842	95	8	open	open	ADJ
ejpam-3842	95	9	set	set	NOUN
ejpam-3842	95	10	in	in	ADP
ejpam-3842	95	11	x.	x.	NOUN
ejpam-3842	95	12	corollary	corollary	NOUN
ejpam-3842	95	13	2	2	NUM
ejpam-3842	95	14	.	.	PUNCT
ejpam-3842	96	1	if	if	SCONJ
ejpam-3842	96	2	{	{	PUNCT
ejpam-3842	96	3	0	0	X
ejpam-3842	96	4	}	}	PUNCT
ejpam-3842	96	5	is	be	AUX
ejpam-3842	96	6	an	an	DET
ejpam-3842	96	7	open	open	ADJ
ejpam-3842	96	8	set	set	NOUN
ejpam-3842	96	9	in	in	ADP
ejpam-3842	96	10	a	a	DET
ejpam-3842	96	11	tbch	tbch	NOUN
ejpam-3842	96	12	-	-	PUNCT
ejpam-3842	96	13	algebra	algebra	NOUN
ejpam-3842	96	14	x	x	NOUN
ejpam-3842	96	15	,	,	PUNCT
ejpam-3842	96	16	then	then	ADV
ejpam-3842	96	17	every	every	DET
ejpam-3842	96	18	subset	subset	NOUN
ejpam-3842	96	19	of	of	ADP
ejpam-3842	96	20	x	x	SYM
ejpam-3842	96	21	is	be	AUX
ejpam-3842	96	22	both	both	CCONJ
ejpam-3842	96	23	open	open	ADJ
ejpam-3842	96	24	and	and	CCONJ
ejpam-3842	96	25	closed	close	VERB
ejpam-3842	96	26	set	set	VERB
ejpam-3842	96	27	in	in	ADP
ejpam-3842	96	28	x.	x.	NOUN
ejpam-3842	96	29	in	in	ADP
ejpam-3842	96	30	particular	particular	ADJ
ejpam-3842	96	31	,	,	PUNCT
ejpam-3842	96	32	if	if	SCONJ
ejpam-3842	96	33	|x|	|x|	PROPN
ejpam-3842	96	34	≥	≥	NUM
ejpam-3842	96	35	2	2	NUM
ejpam-3842	96	36	,	,	PUNCT
ejpam-3842	96	37	then	then	ADV
ejpam-3842	96	38	x	x	PUNCT
ejpam-3842	96	39	is	be	AUX
ejpam-3842	96	40	a	a	DET
ejpam-3842	96	41	disconnected	disconnected	ADJ
ejpam-3842	96	42	space	space	NOUN
ejpam-3842	96	43	.	.	PUNCT
ejpam-3842	97	1	remark	remark	NOUN
ejpam-3842	97	2	5	5	NUM
ejpam-3842	97	3	.	.	PUNCT
ejpam-3842	98	1	if	if	SCONJ
ejpam-3842	98	2	a	a	DET
ejpam-3842	98	3	bch	bch	ADJ
ejpam-3842	98	4	-	-	PUNCT
ejpam-3842	98	5	topological	topological	ADJ
ejpam-3842	98	6	space	space	NOUN
ejpam-3842	98	7	x	x	PUNCT
ejpam-3842	98	8	is	be	AUX
ejpam-3842	98	9	a	a	DET
ejpam-3842	98	10	discrete	discrete	ADJ
ejpam-3842	98	11	space	space	NOUN
ejpam-3842	98	12	,	,	PUNCT
ejpam-3842	98	13	then	then	ADV
ejpam-3842	98	14	x	x	PUNCT
ejpam-3842	98	15	is	be	AUX
ejpam-3842	98	16	a	a	DET
ejpam-3842	98	17	tbch	tbch	NOUN
ejpam-3842	98	18	-	-	PUNCT
ejpam-3842	98	19	algebra	algebra	NOUN
ejpam-3842	98	20	.	.	PUNCT
ejpam-3842	99	1	we	we	PRON
ejpam-3842	99	2	now	now	ADV
ejpam-3842	99	3	show	show	VERB
ejpam-3842	99	4	that	that	SCONJ
ejpam-3842	99	5	a	a	DET
ejpam-3842	99	6	bch	bch	NOUN
ejpam-3842	99	7	-	-	PUNCT
ejpam-3842	99	8	subalgebra	subalgebra	NOUN
ejpam-3842	99	9	of	of	ADP
ejpam-3842	99	10	a	a	DET
ejpam-3842	99	11	tbch	tbch	NOUN
ejpam-3842	99	12	-	-	PUNCT
ejpam-3842	99	13	algebra	algebra	NOUN
ejpam-3842	99	14	is	be	AUX
ejpam-3842	99	15	also	also	ADV
ejpam-3842	99	16	a	a	DET
ejpam-3842	99	17	tbch	tbch	NOUN
ejpam-3842	99	18	-	-	PUNCT
ejpam-3842	99	19	algebra	algebra	NOUN
ejpam-3842	99	20	.	.	PUNCT
ejpam-3842	100	1	theorem	theorem	NOUN
ejpam-3842	100	2	4	4	NUM
ejpam-3842	100	3	.	.	PUNCT
ejpam-3842	101	1	let	let	VERB
ejpam-3842	101	2	x	x	PRON
ejpam-3842	101	3	be	be	AUX
ejpam-3842	101	4	a	a	DET
ejpam-3842	101	5	tbch	tbch	NOUN
ejpam-3842	101	6	-	-	PUNCT
ejpam-3842	101	7	algebra	algebra	NOUN
ejpam-3842	101	8	and	and	CCONJ
ejpam-3842	101	9	h	h	NOUN
ejpam-3842	101	10	a	a	DET
ejpam-3842	101	11	bch	bch	NOUN
ejpam-3842	101	12	-	-	PUNCT
ejpam-3842	101	13	subalgebra	subalgebra	NOUN
ejpam-3842	101	14	of	of	ADP
ejpam-3842	101	15	x.	x.	NOUN
ejpam-3842	101	16	then	then	ADV
ejpam-3842	101	17	(	(	PUNCT
ejpam-3842	101	18	h	h	NOUN
ejpam-3842	101	19	,	,	PUNCT
ejpam-3842	101	20	τh	τh	ADP
ejpam-3842	101	21	)	)	PUNCT
ejpam-3842	101	22	is	be	AUX
ejpam-3842	101	23	a	a	DET
ejpam-3842	101	24	tbch	tbch	NOUN
ejpam-3842	101	25	-	-	PUNCT
ejpam-3842	101	26	algebra	algebra	NOUN
ejpam-3842	101	27	,	,	PUNCT
ejpam-3842	101	28	where	where	SCONJ
ejpam-3842	101	29	τh	τh	ADP
ejpam-3842	101	30	is	be	AUX
ejpam-3842	101	31	the	the	DET
ejpam-3842	101	32	relative	relative	ADJ
ejpam-3842	101	33	topology	topology	NOUN
ejpam-3842	101	34	on	on	ADP
ejpam-3842	101	35	h.	h.	PROPN
ejpam-3842	101	36	proof	proof	NOUN
ejpam-3842	101	37	.	.	PUNCT
ejpam-3842	102	1	let	let	VERB
ejpam-3842	102	2	x	x	PRON
ejpam-3842	102	3	,	,	PUNCT
ejpam-3842	102	4	y	y	PROPN
ejpam-3842	102	5	∈	∈	PROPN
ejpam-3842	102	6	h	h	NOUN
ejpam-3842	102	7	and	and	CCONJ
ejpam-3842	102	8	a	a	DET
ejpam-3842	102	9	nbhd	nbhd	NOUN
ejpam-3842	102	10	wh	wh	NOUN
ejpam-3842	102	11	of	of	ADP
ejpam-3842	102	12	x∗y	x∗y	NUM
ejpam-3842	102	13	in	in	ADP
ejpam-3842	102	14	the	the	DET
ejpam-3842	102	15	subspace	subspace	PROPN
ejpam-3842	102	16	h.	h.	PROPN
ejpam-3842	102	17	note	note	VERB
ejpam-3842	102	18	that	that	SCONJ
ejpam-3842	102	19	wh	wh	NOUN
ejpam-3842	102	20	may	may	AUX
ejpam-3842	102	21	be	be	AUX
ejpam-3842	102	22	written	write	VERB
ejpam-3842	102	23	as	as	ADP
ejpam-3842	102	24	the	the	DET
ejpam-3842	102	25	intersection	intersection	NOUN
ejpam-3842	102	26	with	with	ADP
ejpam-3842	102	27	h	h	NOUN
ejpam-3842	102	28	of	of	ADP
ejpam-3842	102	29	some	some	DET
ejpam-3842	102	30	nbhd	nbhd	NOUN
ejpam-3842	102	31	w	w	ADP
ejpam-3842	102	32	of	of	ADP
ejpam-3842	102	33	x	x	PROPN
ejpam-3842	102	34	∗	∗	NOUN
ejpam-3842	102	35	y	y	PROPN
ejpam-3842	102	36	in	in	ADP
ejpam-3842	102	37	x	x	PRON
ejpam-3842	102	38	,	,	PUNCT
ejpam-3842	102	39	that	that	ADV
ejpam-3842	102	40	is	is	ADV
ejpam-3842	102	41	,	,	PUNCT
ejpam-3842	102	42	wh	wh	NOUN
ejpam-3842	102	43	=	=	SYM
ejpam-3842	102	44	h	h	NOUN
ejpam-3842	102	45	∩w	∩w	NOUN
ejpam-3842	102	46	.	.	PUNCT
ejpam-3842	103	1	since	since	SCONJ
ejpam-3842	103	2	x	x	PRON
ejpam-3842	103	3	is	be	AUX
ejpam-3842	103	4	a	a	DET
ejpam-3842	103	5	tbch	tbch	NOUN
ejpam-3842	103	6	-	-	PUNCT
ejpam-3842	103	7	algebra	algebra	NOUN
ejpam-3842	103	8	,	,	PUNCT
ejpam-3842	103	9	there	there	PRON
ejpam-3842	103	10	exist	exist	VERB
ejpam-3842	103	11	nbhds	nbhds	ADJ
ejpam-3842	103	12	u	u	NOUN
ejpam-3842	103	13	and	and	CCONJ
ejpam-3842	103	14	v	v	NOUN
ejpam-3842	103	15	of	of	ADP
ejpam-3842	103	16	x	x	PROPN
ejpam-3842	103	17	and	and	CCONJ
ejpam-3842	103	18	y	y	PROPN
ejpam-3842	103	19	,	,	PUNCT
ejpam-3842	103	20	respectively	respectively	ADV
ejpam-3842	103	21	,	,	PUNCT
ejpam-3842	103	22	such	such	ADJ
ejpam-3842	103	23	that	that	SCONJ
ejpam-3842	103	24	u	u	PROPN
ejpam-3842	103	25	∗	∗	NOUN
ejpam-3842	103	26	v	v	NOUN
ejpam-3842	103	27	⊆	⊆	NUM
ejpam-3842	103	28	w	w	NOUN
ejpam-3842	103	29	by	by	ADP
ejpam-3842	103	30	theorem	theorem	NOUN
ejpam-3842	103	31	2	2	NUM
ejpam-3842	103	32	.	.	X
ejpam-3842	103	33	observe	observe	VERB
ejpam-3842	103	34	that	that	SCONJ
ejpam-3842	103	35	uh	uh	INTJ
ejpam-3842	103	36	=	=	ADJ
ejpam-3842	103	37	h	h	NOUN
ejpam-3842	103	38	∩	∩	NOUN
ejpam-3842	103	39	u	u	NOUN
ejpam-3842	103	40	and	and	CCONJ
ejpam-3842	103	41	vh	vh	PROPN
ejpam-3842	103	42	=	=	SYM
ejpam-3842	103	43	h	h	PROPN
ejpam-3842	103	44	∩	∩	NOUN
ejpam-3842	103	45	v	v	NOUN
ejpam-3842	103	46	are	be	AUX
ejpam-3842	103	47	nbhds	nbhds	NOUN
ejpam-3842	103	48	of	of	ADP
ejpam-3842	103	49	x	x	PROPN
ejpam-3842	103	50	and	and	CCONJ
ejpam-3842	103	51	y	y	PROPN
ejpam-3842	103	52	,	,	PUNCT
ejpam-3842	103	53	respectively	respectively	ADV
ejpam-3842	103	54	,	,	PUNCT
ejpam-3842	103	55	in	in	ADP
ejpam-3842	103	56	the	the	DET
ejpam-3842	103	57	subspace	subspace	NOUN
ejpam-3842	103	58	h.	h.	PROPN
ejpam-3842	103	59	furthermore	furthermore	ADV
ejpam-3842	103	60	.	.	PUNCT
ejpam-3842	104	1	uh	uh	INTJ
ejpam-3842	104	2	∗	∗	NOUN
ejpam-3842	104	3	vh	vh	NOUN
ejpam-3842	104	4	=	=	SYM
ejpam-3842	104	5	(	(	PUNCT
ejpam-3842	104	6	h	h	PROPN
ejpam-3842	104	7	∩	∩	ADJ
ejpam-3842	104	8	u	u	NOUN
ejpam-3842	104	9	)	)	PUNCT
ejpam-3842	104	10	∗	∗	NOUN
ejpam-3842	104	11	(	(	PUNCT
ejpam-3842	104	12	h	h	PROPN
ejpam-3842	104	13	∩	∩	NOUN
ejpam-3842	104	14	v	v	X
ejpam-3842	104	15	)	)	PUNCT
ejpam-3842	104	16	⊆	⊆	NUM
ejpam-3842	104	17	u	u	NOUN
ejpam-3842	104	18	∗	∗	NOUN
ejpam-3842	104	19	v	v	ADP
ejpam-3842	104	20	⊆w	⊆w	NOUN
ejpam-3842	104	21	.	.	PUNCT
ejpam-3842	105	1	since	since	SCONJ
ejpam-3842	105	2	h	h	NOUN
ejpam-3842	105	3	is	be	AUX
ejpam-3842	105	4	a	a	DET
ejpam-3842	105	5	bch	bch	PROPN
ejpam-3842	105	6	-subalgebra	-subalgebra	NOUN
ejpam-3842	105	7	,	,	PUNCT
ejpam-3842	105	8	uh	uh	INTJ
ejpam-3842	105	9	∗	∗	NOUN
ejpam-3842	105	10	vh	vh	NOUN
ejpam-3842	105	11	⊆	⊆	NUM
ejpam-3842	105	12	h	h	NOUN
ejpam-3842	105	13	∗h	∗h	VERB
ejpam-3842	105	14	⊆	⊆	NUM
ejpam-3842	105	15	h	h	NOUN
ejpam-3842	105	16	so	so	SCONJ
ejpam-3842	105	17	that	that	SCONJ
ejpam-3842	105	18	uh	uh	INTJ
ejpam-3842	105	19	∗	∗	NOUN
ejpam-3842	105	20	vh	vh	NOUN
ejpam-3842	105	21	⊆	⊆	NUM
ejpam-3842	105	22	h	h	NOUN
ejpam-3842	105	23	∩w	∩w	NOUN
ejpam-3842	106	1	=	=	PUNCT
ejpam-3842	106	2	wh	wh	NOUN
ejpam-3842	106	3	.	.	PUNCT
ejpam-3842	107	1	by	by	ADP
ejpam-3842	107	2	theorem	theorem	NOUN
ejpam-3842	107	3	2	2	NUM
ejpam-3842	107	4	,	,	PUNCT
ejpam-3842	107	5	(	(	PUNCT
ejpam-3842	107	6	h	h	NOUN
ejpam-3842	107	7	,	,	PUNCT
ejpam-3842	107	8	τh	τh	ADP
ejpam-3842	107	9	)	)	PUNCT
ejpam-3842	107	10	is	be	AUX
ejpam-3842	107	11	a	a	DET
ejpam-3842	107	12	tbch	tbch	NOUN
ejpam-3842	107	13	-	-	PUNCT
ejpam-3842	107	14	algebra	algebra	NOUN
ejpam-3842	107	15	.	.	PUNCT
ejpam-3842	108	1	j.	j.	PROPN
ejpam-3842	108	2	mancao	mancao	PROPN
ejpam-3842	108	3	,	,	PUNCT
ejpam-3842	108	4	s.	s.	PROPN
ejpam-3842	108	5	canoy	canoy	PROPN
ejpam-3842	108	6	/	/	SYM
ejpam-3842	108	7	eur	eur	PROPN
ejpam-3842	108	8	.	.	PUNCT
ejpam-3842	109	1	j.	j.	PROPN
ejpam-3842	109	2	pure	pure	PROPN
ejpam-3842	109	3	appl	appl	PROPN
ejpam-3842	109	4	.	.	PROPN
ejpam-3842	109	5	math	math	PROPN
ejpam-3842	109	6	,	,	PUNCT
ejpam-3842	109	7	13	13	NUM
ejpam-3842	109	8	(	(	PUNCT
ejpam-3842	109	9	4	4	NUM
ejpam-3842	109	10	)	)	PUNCT
ejpam-3842	109	11	(	(	PUNCT
ejpam-3842	109	12	2020	2020	NUM
ejpam-3842	109	13	)	)	PUNCT
ejpam-3842	109	14	,	,	PUNCT
ejpam-3842	109	15	730	730	NUM
ejpam-3842	109	16	-	-	SYM
ejpam-3842	109	17	738	738	NUM
ejpam-3842	109	18	734	734	NUM
ejpam-3842	109	19	theorem	theorem	NOUN
ejpam-3842	109	20	5	5	NUM
ejpam-3842	109	21	.	.	PUNCT
ejpam-3842	110	1	let	let	VERB
ejpam-3842	110	2	(	(	PUNCT
ejpam-3842	110	3	h1	h1	PROPN
ejpam-3842	110	4	,	,	PUNCT
ejpam-3842	110	5	∗1	∗1	PROPN
ejpam-3842	110	6	,	,	PUNCT
ejpam-3842	110	7	0	0	NUM
ejpam-3842	110	8	)	)	PUNCT
ejpam-3842	110	9	and	and	CCONJ
ejpam-3842	110	10	(	(	PUNCT
ejpam-3842	110	11	h2	h2	NOUN
ejpam-3842	110	12	,	,	PUNCT
ejpam-3842	110	13	∗2	∗2	PROPN
ejpam-3842	110	14	,	,	PUNCT
ejpam-3842	110	15	0	0	NUM
ejpam-3842	110	16	)	)	PUNCT
ejpam-3842	110	17	be	be	VERB
ejpam-3842	110	18	bch	bch	NOUN
ejpam-3842	110	19	-	-	PUNCT
ejpam-3842	110	20	algebras	algebra	NOUN
ejpam-3842	110	21	such	such	ADJ
ejpam-3842	110	22	that	that	SCONJ
ejpam-3842	110	23	h1∩h2	h1∩h2	PROPN
ejpam-3842	110	24	=	=	PUNCT
ejpam-3842	110	25	{	{	PUNCT
ejpam-3842	110	26	0	0	NUM
ejpam-3842	110	27	}	}	PUNCT
ejpam-3842	110	28	and	and	CCONJ
ejpam-3842	110	29	h	h	NOUN
ejpam-3842	110	30	=	=	NOUN
ejpam-3842	110	31	h1	h1	PROPN
ejpam-3842	110	32	∪h2	∪h2	PROPN
ejpam-3842	110	33	.	.	PUNCT
ejpam-3842	111	1	then	then	ADV
ejpam-3842	111	2	(	(	PUNCT
ejpam-3842	111	3	h	h	NOUN
ejpam-3842	111	4	,	,	PUNCT
ejpam-3842	111	5	∗	∗	NOUN
ejpam-3842	111	6	,	,	PUNCT
ejpam-3842	111	7	0	0	NUM
ejpam-3842	111	8	)	)	PUNCT
ejpam-3842	111	9	is	be	AUX
ejpam-3842	111	10	a	a	DET
ejpam-3842	111	11	bch	bch	NOUN
ejpam-3842	111	12	-	-	PUNCT
ejpam-3842	111	13	algebra	algebra	NOUN
ejpam-3842	111	14	,	,	PUNCT
ejpam-3842	111	15	denoted	denote	VERB
ejpam-3842	111	16	by	by	ADP
ejpam-3842	111	17	h1	h1	PROPN
ejpam-3842	111	18	⊕h2	⊕h2	NOUN
ejpam-3842	111	19	,	,	PUNCT
ejpam-3842	111	20	where	where	SCONJ
ejpam-3842	111	21	the	the	DET
ejpam-3842	111	22	operation	operation	NOUN
ejpam-3842	111	23	“	"	PUNCT
ejpam-3842	111	24	∗	∗	NOUN
ejpam-3842	111	25	”	"	PUNCT
ejpam-3842	111	26	on	on	ADP
ejpam-3842	111	27	h	h	NOUN
ejpam-3842	111	28	is	be	AUX
ejpam-3842	111	29	defined	define	VERB
ejpam-3842	111	30	for	for	ADP
ejpam-3842	111	31	all	all	DET
ejpam-3842	111	32	x	x	NOUN
ejpam-3842	111	33	,	,	PUNCT
ejpam-3842	111	34	y	y	PROPN
ejpam-3842	111	35	∈	∈	PROPN
ejpam-3842	111	36	h	h	NOUN
ejpam-3842	111	37	,	,	PUNCT
ejpam-3842	111	38	by	by	ADP
ejpam-3842	111	39	x	x	X
ejpam-3842	111	40	∗	∗	NOUN
ejpam-3842	112	1	y	y	NOUN
ejpam-3842	112	2	=	=	SYM
ejpam-3842	112	3			NOUN
ejpam-3842	112	4	x	x	X
ejpam-3842	112	5	∗1	∗1	PROPN
ejpam-3842	112	6	y	y	PROPN
ejpam-3842	112	7	if	if	SCONJ
ejpam-3842	112	8	x	x	PRON
ejpam-3842	112	9	,	,	PUNCT
ejpam-3842	112	10	y	y	PROPN
ejpam-3842	112	11	∈	∈	PROPN
ejpam-3842	112	12	h1	h1	PROPN
ejpam-3842	112	13	x	x	PUNCT
ejpam-3842	112	14	∗2	∗2	VERB
ejpam-3842	112	15	y	y	PROPN
ejpam-3842	112	16	if	if	SCONJ
ejpam-3842	112	17	x	x	PROPN
ejpam-3842	112	18	,	,	PUNCT
ejpam-3842	112	19	y	y	PROPN
ejpam-3842	112	20	∈	∈	PROPN
ejpam-3842	112	21	h2	h2	PROPN
ejpam-3842	112	22	x	x	X
ejpam-3842	112	23	otherwise	otherwise	ADV
ejpam-3842	112	24	.	.	PUNCT
ejpam-3842	113	1	proof	proof	NOUN
ejpam-3842	113	2	.	.	PUNCT
ejpam-3842	114	1	let	let	VERB
ejpam-3842	114	2	x	x	SYM
ejpam-3842	114	3	∈	∈	PROPN
ejpam-3842	114	4	h.	h.	NOUN
ejpam-3842	114	5	then	then	ADV
ejpam-3842	115	1	x	x	X
ejpam-3842	115	2	∗	∗	NOUN
ejpam-3842	115	3	x	x	X
ejpam-3842	115	4	=	=	PRON
ejpam-3842	115	5	{	{	PUNCT
ejpam-3842	115	6	x	x	X
ejpam-3842	115	7	∗1	∗1	PROPN
ejpam-3842	115	8	x	x	PUNCT
ejpam-3842	115	9	if	if	SCONJ
ejpam-3842	115	10	x	x	X
ejpam-3842	115	11	∈	∈	PROPN
ejpam-3842	115	12	h1	h1	NOUN
ejpam-3842	115	13	x	x	PART
ejpam-3842	115	14	∗2	∗2	NOUN
ejpam-3842	115	15	x	x	SYM
ejpam-3842	115	16	if	if	SCONJ
ejpam-3842	115	17	x	x	SYM
ejpam-3842	115	18	∈	∈	PROPN
ejpam-3842	115	19	h2	h2	NOUN
ejpam-3842	115	20	.	.	PUNCT
ejpam-3842	116	1	since	since	SCONJ
ejpam-3842	116	2	(	(	PUNCT
ejpam-3842	116	3	h1	h1	PROPN
ejpam-3842	116	4	,	,	PUNCT
ejpam-3842	116	5	∗1	∗1	PROPN
ejpam-3842	116	6	,	,	PUNCT
ejpam-3842	116	7	0	0	NUM
ejpam-3842	116	8	)	)	PUNCT
ejpam-3842	116	9	and	and	CCONJ
ejpam-3842	116	10	(	(	PUNCT
ejpam-3842	116	11	h2	h2	NOUN
ejpam-3842	116	12	,	,	PUNCT
ejpam-3842	116	13	∗2	∗2	PROPN
ejpam-3842	116	14	,	,	PUNCT
ejpam-3842	116	15	0	0	NUM
ejpam-3842	116	16	)	)	PUNCT
ejpam-3842	116	17	are	be	AUX
ejpam-3842	116	18	bch	bch	NOUN
ejpam-3842	116	19	-	-	PUNCT
ejpam-3842	116	20	algebras	algebras	PROPN
ejpam-3842	116	21	,	,	PUNCT
ejpam-3842	116	22	x	x	X
ejpam-3842	116	23	∗	∗	NOUN
ejpam-3842	116	24	x	x	SYM
ejpam-3842	117	1	=	=	SYM
ejpam-3842	117	2	0	0	NUM
ejpam-3842	117	3	by	by	ADP
ejpam-3842	117	4	property	property	NOUN
ejpam-3842	117	5	(	(	PUNCT
ejpam-3842	117	6	b1	b1	NOUN
ejpam-3842	117	7	)	)	PUNCT
ejpam-3842	117	8	.	.	PUNCT
ejpam-3842	118	1	next	next	ADV
ejpam-3842	118	2	,	,	PUNCT
ejpam-3842	118	3	let	let	VERB
ejpam-3842	118	4	x	x	PRON
ejpam-3842	118	5	,	,	PUNCT
ejpam-3842	118	6	y	y	PROPN
ejpam-3842	118	7	∈	∈	PROPN
ejpam-3842	118	8	h	h	NOUN
ejpam-3842	118	9	and	and	CCONJ
ejpam-3842	118	10	suppose	suppose	VERB
ejpam-3842	118	11	that	that	SCONJ
ejpam-3842	118	12	x	x	PROPN
ejpam-3842	118	13	∗	∗	NOUN
ejpam-3842	118	14	y	y	NOUN
ejpam-3842	118	15	=	=	SYM
ejpam-3842	118	16	0	0	PROPN
ejpam-3842	118	17	and	and	CCONJ
ejpam-3842	118	18	y	y	PROPN
ejpam-3842	118	19	∗	∗	NOUN
ejpam-3842	118	20	x	x	PUNCT
ejpam-3842	119	1	=	=	NOUN
ejpam-3842	119	2	0	0	X
ejpam-3842	119	3	.	.	PUNCT
ejpam-3842	119	4	consider	consider	VERB
ejpam-3842	119	5	the	the	DET
ejpam-3842	119	6	following	follow	VERB
ejpam-3842	119	7	cases	case	NOUN
ejpam-3842	119	8	:	:	PUNCT
ejpam-3842	119	9	case	case	NOUN
ejpam-3842	119	10	1	1	NUM
ejpam-3842	119	11	:	:	PUNCT
ejpam-3842	119	12	x	x	X
ejpam-3842	119	13	,	,	PUNCT
ejpam-3842	119	14	y	y	PROPN
ejpam-3842	119	15	∈	∈	PROPN
ejpam-3842	119	16	h1	h1	NOUN
ejpam-3842	119	17	(	(	PUNCT
ejpam-3842	119	18	or	or	CCONJ
ejpam-3842	119	19	x	x	X
ejpam-3842	119	20	,	,	PUNCT
ejpam-3842	119	21	y	y	PROPN
ejpam-3842	119	22	∈	∈	PROPN
ejpam-3842	119	23	h2	h2	PROPN
ejpam-3842	119	24	)	)	PUNCT
ejpam-3842	119	25	.	.	PUNCT
ejpam-3842	120	1	then	then	ADV
ejpam-3842	120	2	x	x	X
ejpam-3842	120	3	∗	∗	NOUN
ejpam-3842	120	4	y	y	NOUN
ejpam-3842	120	5	=	=	PUNCT
ejpam-3842	121	1	x	x	PUNCT
ejpam-3842	121	2	∗1	∗1	PROPN
ejpam-3842	121	3	y	y	PROPN
ejpam-3842	121	4	=	=	SYM
ejpam-3842	121	5	0	0	PROPN
ejpam-3842	121	6	and	and	CCONJ
ejpam-3842	121	7	y	y	PROPN
ejpam-3842	121	8	∗	∗	NOUN
ejpam-3842	121	9	x	x	PUNCT
ejpam-3842	122	1	=	=	PUNCT
ejpam-3842	122	2	y	y	NOUN
ejpam-3842	122	3	∗1	∗1	NOUN
ejpam-3842	123	1	x	x	PUNCT
ejpam-3842	124	1	=	=	PUNCT
ejpam-3842	125	1	0	0	X
ejpam-3842	125	2	.	.	PUNCT
ejpam-3842	126	1	since	since	SCONJ
ejpam-3842	126	2	(	(	PUNCT
ejpam-3842	126	3	h1	h1	PROPN
ejpam-3842	126	4	,	,	PUNCT
ejpam-3842	126	5	∗1	∗1	PROPN
ejpam-3842	126	6	,	,	PUNCT
ejpam-3842	126	7	0	0	NUM
ejpam-3842	126	8	)	)	PUNCT
ejpam-3842	126	9	is	be	AUX
ejpam-3842	126	10	a	a	DET
ejpam-3842	126	11	bch	bch	NOUN
ejpam-3842	126	12	-	-	PUNCT
ejpam-3842	126	13	algebra	algebra	NOUN
ejpam-3842	126	14	,	,	PUNCT
ejpam-3842	126	15	property	property	NOUN
ejpam-3842	126	16	(	(	PUNCT
ejpam-3842	126	17	b2	b2	NOUN
ejpam-3842	126	18	)	)	PUNCT
ejpam-3842	126	19	yields	yield	NOUN
ejpam-3842	126	20	x	x	PUNCT
ejpam-3842	127	1	=	=	PUNCT
ejpam-3842	127	2	y.	y.	NOUN
ejpam-3842	127	3	similarly	similarly	ADV
ejpam-3842	127	4	,	,	PUNCT
ejpam-3842	127	5	x	x	PUNCT
ejpam-3842	127	6	=	=	PUNCT
ejpam-3842	127	7	y	y	PROPN
ejpam-3842	127	8	if	if	SCONJ
ejpam-3842	127	9	x	x	PROPN
ejpam-3842	127	10	,	,	PUNCT
ejpam-3842	127	11	y	y	PROPN
ejpam-3842	127	12	∈	∈	PROPN
ejpam-3842	127	13	h2	h2	PROPN
ejpam-3842	127	14	.	.	PUNCT
ejpam-3842	128	1	case	case	NOUN
ejpam-3842	128	2	2	2	NUM
ejpam-3842	128	3	:	:	PUNCT
ejpam-3842	128	4	x	x	SYM
ejpam-3842	128	5	∈	∈	PROPN
ejpam-3842	128	6	h1	h1	PROPN
ejpam-3842	128	7	and	and	CCONJ
ejpam-3842	128	8	y	y	PROPN
ejpam-3842	128	9	∈	∈	PROPN
ejpam-3842	128	10	h2	h2	PROPN
ejpam-3842	128	11	(	(	PUNCT
ejpam-3842	128	12	or	or	CCONJ
ejpam-3842	128	13	y	y	PROPN
ejpam-3842	128	14	∈	∈	PROPN
ejpam-3842	128	15	h1	h1	PROPN
ejpam-3842	128	16	and	and	CCONJ
ejpam-3842	128	17	x	x	PUNCT
ejpam-3842	128	18	∈	∈	PROPN
ejpam-3842	128	19	h2	h2	NOUN
ejpam-3842	128	20	)	)	PUNCT
ejpam-3842	128	21	.	.	PUNCT
ejpam-3842	129	1	then	then	ADV
ejpam-3842	129	2	0	0	NUM
ejpam-3842	130	1	=	=	SYM
ejpam-3842	130	2	x	x	SYM
ejpam-3842	130	3	∗	∗	NOUN
ejpam-3842	130	4	y	y	NOUN
ejpam-3842	130	5	=	=	PUNCT
ejpam-3842	130	6	x	x	PROPN
ejpam-3842	130	7	and	and	CCONJ
ejpam-3842	130	8	0	0	NUM
ejpam-3842	130	9	=	=	SYM
ejpam-3842	130	10	y	y	NOUN
ejpam-3842	130	11	∗	∗	NOUN
ejpam-3842	130	12	x	x	PUNCT
ejpam-3842	131	1	=	=	PUNCT
ejpam-3842	131	2	y.	y.	PROPN
ejpam-3842	131	3	hence	hence	ADV
ejpam-3842	131	4	,	,	PUNCT
ejpam-3842	131	5	x	x	PUNCT
ejpam-3842	131	6	=	=	SYM
ejpam-3842	131	7	0	0	PUNCT
ejpam-3842	132	1	=	=	PUNCT
ejpam-3842	132	2	y.	y.	PROPN
ejpam-3842	132	3	finally	finally	ADV
ejpam-3842	132	4	,	,	PUNCT
ejpam-3842	132	5	let	let	VERB
ejpam-3842	132	6	x	x	PRON
ejpam-3842	132	7	,	,	PUNCT
ejpam-3842	132	8	y	y	PROPN
ejpam-3842	132	9	,	,	PUNCT
ejpam-3842	132	10	z	z	PROPN
ejpam-3842	132	11	∈	∈	PROPN
ejpam-3842	132	12	h.	h.	NOUN
ejpam-3842	132	13	consider	consider	VERB
ejpam-3842	132	14	the	the	DET
ejpam-3842	132	15	following	follow	VERB
ejpam-3842	132	16	cases	case	NOUN
ejpam-3842	132	17	:	:	PUNCT
ejpam-3842	132	18	case	case	NOUN
ejpam-3842	132	19	1	1	NUM
ejpam-3842	132	20	:	:	PUNCT
ejpam-3842	132	21	x	x	X
ejpam-3842	132	22	,	,	PUNCT
ejpam-3842	132	23	y	y	PROPN
ejpam-3842	132	24	∈	∈	PROPN
ejpam-3842	132	25	h1	h1	NOUN
ejpam-3842	132	26	(	(	PUNCT
ejpam-3842	132	27	or	or	CCONJ
ejpam-3842	132	28	x	x	X
ejpam-3842	132	29	,	,	PUNCT
ejpam-3842	132	30	y	y	PROPN
ejpam-3842	132	31	∈	∈	PROPN
ejpam-3842	132	32	h2	h2	PROPN
ejpam-3842	132	33	)	)	PUNCT
ejpam-3842	132	34	then	then	ADV
ejpam-3842	132	35	,	,	PUNCT
ejpam-3842	132	36	by	by	ADP
ejpam-3842	132	37	the	the	DET
ejpam-3842	132	38	definition	definition	NOUN
ejpam-3842	132	39	of	of	ADP
ejpam-3842	132	40	∗	∗	NOUN
ejpam-3842	132	41	,	,	PUNCT
ejpam-3842	132	42	(	(	PUNCT
ejpam-3842	132	43	x	x	X
ejpam-3842	132	44	∗	∗	PROPN
ejpam-3842	132	45	y	y	NOUN
ejpam-3842	132	46	)	)	PUNCT
ejpam-3842	132	47	∗	∗	NOUN
ejpam-3842	132	48	z	z	NOUN
ejpam-3842	132	49	=	=	PRON
ejpam-3842	132	50	{	{	PUNCT
ejpam-3842	132	51	(	(	PUNCT
ejpam-3842	132	52	x	x	SYM
ejpam-3842	132	53	∗1	∗1	PROPN
ejpam-3842	132	54	y	y	NOUN
ejpam-3842	132	55	)	)	PUNCT
ejpam-3842	133	1	∗1	∗1	PROPN
ejpam-3842	134	1	z	z	NOUN
ejpam-3842	134	2	if	if	SCONJ
ejpam-3842	134	3	z	z	PROPN
ejpam-3842	134	4	∈	∈	PROPN
ejpam-3842	134	5	h1	h1	NOUN
ejpam-3842	134	6	x	x	PUNCT
ejpam-3842	134	7	∗1	∗1	PROPN
ejpam-3842	134	8	y	y	NOUN
ejpam-3842	134	9	if	if	SCONJ
ejpam-3842	134	10	z	z	PROPN
ejpam-3842	134	11	∈	∈	PROPN
ejpam-3842	134	12	h2	h2	NOUN
ejpam-3842	134	13	.	.	PUNCT
ejpam-3842	135	1	and	and	CCONJ
ejpam-3842	135	2	(	(	PUNCT
ejpam-3842	135	3	x	x	PROPN
ejpam-3842	135	4	∗	∗	PROPN
ejpam-3842	135	5	z	z	NOUN
ejpam-3842	135	6	)	)	PUNCT
ejpam-3842	135	7	∗	∗	NOUN
ejpam-3842	135	8	y	y	NOUN
ejpam-3842	136	1	=	=	PRON
ejpam-3842	136	2	{	{	PUNCT
ejpam-3842	136	3	(	(	PUNCT
ejpam-3842	136	4	x	x	SYM
ejpam-3842	136	5	∗1	∗1	PROPN
ejpam-3842	136	6	z	z	X
ejpam-3842	136	7	)	)	PUNCT
ejpam-3842	136	8	∗1	∗1	PROPN
ejpam-3842	137	1	y	y	PROPN
ejpam-3842	137	2	if	if	SCONJ
ejpam-3842	137	3	z	z	PROPN
ejpam-3842	137	4	∈	∈	PROPN
ejpam-3842	137	5	h1	h1	NOUN
ejpam-3842	137	6	x	x	PUNCT
ejpam-3842	137	7	∗1	∗1	PROPN
ejpam-3842	137	8	y	y	NOUN
ejpam-3842	138	1	if	if	SCONJ
ejpam-3842	138	2	z	z	PROPN
ejpam-3842	138	3	∈	∈	PROPN
ejpam-3842	138	4	h2	h2	NOUN
ejpam-3842	138	5	.	.	PUNCT
ejpam-3842	139	1	since	since	SCONJ
ejpam-3842	139	2	(	(	PUNCT
ejpam-3842	139	3	h1	h1	PROPN
ejpam-3842	139	4	,	,	PUNCT
ejpam-3842	139	5	∗1	∗1	PROPN
ejpam-3842	139	6	,	,	PUNCT
ejpam-3842	139	7	0	0	NUM
ejpam-3842	139	8	)	)	PUNCT
ejpam-3842	139	9	is	be	AUX
ejpam-3842	139	10	a	a	DET
ejpam-3842	139	11	bch	bch	NOUN
ejpam-3842	139	12	-	-	PUNCT
ejpam-3842	139	13	algebra	algebra	NOUN
ejpam-3842	139	14	,	,	PUNCT
ejpam-3842	139	15	(	(	PUNCT
ejpam-3842	139	16	x∗1	x∗1	VERB
ejpam-3842	139	17	y)∗1	y)∗1	NOUN
ejpam-3842	139	18	z	z	NOUN
ejpam-3842	139	19	=	=	PUNCT
ejpam-3842	139	20	(	(	PUNCT
ejpam-3842	140	1	x∗1	x∗1	PROPN
ejpam-3842	140	2	z)∗1	z)∗1	PROPN
ejpam-3842	140	3	y	y	PROPN
ejpam-3842	140	4	if	if	SCONJ
ejpam-3842	140	5	z	z	PROPN
ejpam-3842	140	6	∈	∈	PROPN
ejpam-3842	140	7	h1	h1	PROPN
ejpam-3842	140	8	.	.	PUNCT
ejpam-3842	141	1	hence	hence	ADV
ejpam-3842	141	2	,	,	PUNCT
ejpam-3842	141	3	(	(	PUNCT
ejpam-3842	141	4	x∗y)∗z	x∗y)∗z	NOUN
ejpam-3842	141	5	=	=	PUNCT
ejpam-3842	141	6	(	(	PUNCT
ejpam-3842	141	7	x	x	X
ejpam-3842	141	8	∗	∗	PROPN
ejpam-3842	141	9	z	z	NOUN
ejpam-3842	141	10	)	)	PUNCT
ejpam-3842	141	11	∗	∗	NOUN
ejpam-3842	141	12	y.	y.	PROPN
ejpam-3842	141	13	similarly	similarly	ADV
ejpam-3842	141	14	,	,	PUNCT
ejpam-3842	141	15	(	(	PUNCT
ejpam-3842	141	16	x	x	X
ejpam-3842	141	17	∗	∗	PROPN
ejpam-3842	141	18	y	y	NOUN
ejpam-3842	141	19	)	)	PUNCT
ejpam-3842	141	20	∗	∗	NOUN
ejpam-3842	141	21	z	z	NOUN
ejpam-3842	141	22	=	=	SYM
ejpam-3842	141	23	(	(	PUNCT
ejpam-3842	141	24	x	x	X
ejpam-3842	141	25	∗	∗	PROPN
ejpam-3842	141	26	z	z	NOUN
ejpam-3842	141	27	)	)	PUNCT
ejpam-3842	141	28	∗	∗	NOUN
ejpam-3842	141	29	y	y	PROPN
ejpam-3842	141	30	whenever	whenever	SCONJ
ejpam-3842	141	31	x	x	X
ejpam-3842	141	32	,	,	PUNCT
ejpam-3842	141	33	y	y	PROPN
ejpam-3842	141	34	∈	∈	PROPN
ejpam-3842	141	35	h2	h2	PROPN
ejpam-3842	141	36	.	.	PUNCT
ejpam-3842	142	1	case	case	NOUN
ejpam-3842	142	2	2	2	NUM
ejpam-3842	142	3	:	:	PUNCT
ejpam-3842	142	4	x	x	SYM
ejpam-3842	142	5	∈	∈	PROPN
ejpam-3842	142	6	h1	h1	PROPN
ejpam-3842	142	7	and	and	CCONJ
ejpam-3842	142	8	y	y	PROPN
ejpam-3842	142	9	∈	∈	PROPN
ejpam-3842	142	10	h2	h2	PROPN
ejpam-3842	142	11	(	(	PUNCT
ejpam-3842	142	12	or	or	CCONJ
ejpam-3842	142	13	y	y	PROPN
ejpam-3842	142	14	∈	∈	PROPN
ejpam-3842	142	15	h1	h1	PROPN
ejpam-3842	142	16	and	and	CCONJ
ejpam-3842	142	17	x	x	PUNCT
ejpam-3842	142	18	∈	∈	PROPN
ejpam-3842	142	19	h2	h2	NOUN
ejpam-3842	142	20	)	)	PUNCT
ejpam-3842	142	21	then	then	ADV
ejpam-3842	142	22	,	,	PUNCT
ejpam-3842	142	23	by	by	ADP
ejpam-3842	142	24	the	the	DET
ejpam-3842	142	25	definition	definition	NOUN
ejpam-3842	142	26	of	of	ADP
ejpam-3842	142	27	∗	∗	NOUN
ejpam-3842	142	28	,	,	PUNCT
ejpam-3842	142	29	(	(	PUNCT
ejpam-3842	142	30	x	x	X
ejpam-3842	142	31	∗	∗	PROPN
ejpam-3842	142	32	y	y	NOUN
ejpam-3842	142	33	)	)	PUNCT
ejpam-3842	142	34	∗	∗	NOUN
ejpam-3842	142	35	z	z	NOUN
ejpam-3842	142	36	=	=	PRON
ejpam-3842	142	37	{	{	PUNCT
ejpam-3842	143	1	x	x	X
ejpam-3842	143	2	∗1	∗1	PROPN
ejpam-3842	143	3	z	z	NOUN
ejpam-3842	143	4	if	if	SCONJ
ejpam-3842	143	5	z	z	PROPN
ejpam-3842	143	6	∈	∈	PROPN
ejpam-3842	143	7	h1	h1	PROPN
ejpam-3842	143	8	x	x	PUNCT
ejpam-3842	143	9	if	if	SCONJ
ejpam-3842	143	10	z	z	PROPN
ejpam-3842	143	11	∈	∈	PROPN
ejpam-3842	143	12	h2	h2	NOUN
ejpam-3842	143	13	.	.	PUNCT
ejpam-3842	144	1	and	and	CCONJ
ejpam-3842	144	2	(	(	PUNCT
ejpam-3842	144	3	x	x	PROPN
ejpam-3842	144	4	∗	∗	PROPN
ejpam-3842	144	5	z	z	NOUN
ejpam-3842	144	6	)	)	PUNCT
ejpam-3842	144	7	∗	∗	NOUN
ejpam-3842	144	8	y	y	NOUN
ejpam-3842	144	9	=	=	PUNCT
ejpam-3842	144	10	{	{	PUNCT
ejpam-3842	145	1	x	x	X
ejpam-3842	145	2	∗1	∗1	PROPN
ejpam-3842	145	3	z	z	NOUN
ejpam-3842	145	4	if	if	SCONJ
ejpam-3842	145	5	z	z	PROPN
ejpam-3842	145	6	∈	∈	PROPN
ejpam-3842	145	7	h1	h1	PROPN
ejpam-3842	145	8	x	x	PUNCT
ejpam-3842	145	9	if	if	SCONJ
ejpam-3842	145	10	z	z	PROPN
ejpam-3842	145	11	∈	∈	PROPN
ejpam-3842	145	12	h2	h2	PROPN
ejpam-3842	145	13	.	.	PUNCT
ejpam-3842	146	1	j.	j.	PROPN
ejpam-3842	146	2	mancao	mancao	PROPN
ejpam-3842	146	3	,	,	PUNCT
ejpam-3842	146	4	s.	s.	PROPN
ejpam-3842	146	5	canoy	canoy	PROPN
ejpam-3842	146	6	/	/	SYM
ejpam-3842	146	7	eur	eur	PROPN
ejpam-3842	146	8	.	.	PUNCT
ejpam-3842	147	1	j.	j.	PROPN
ejpam-3842	147	2	pure	pure	PROPN
ejpam-3842	147	3	appl	appl	PROPN
ejpam-3842	147	4	.	.	PROPN
ejpam-3842	147	5	math	math	PROPN
ejpam-3842	147	6	,	,	PUNCT
ejpam-3842	147	7	13	13	NUM
ejpam-3842	147	8	(	(	PUNCT
ejpam-3842	147	9	4	4	NUM
ejpam-3842	147	10	)	)	PUNCT
ejpam-3842	147	11	(	(	PUNCT
ejpam-3842	147	12	2020	2020	NUM
ejpam-3842	147	13	)	)	PUNCT
ejpam-3842	147	14	,	,	PUNCT
ejpam-3842	147	15	730	730	NUM
ejpam-3842	147	16	-	-	SYM
ejpam-3842	147	17	738	738	NUM
ejpam-3842	147	18	735	735	NUM
ejpam-3842	147	19	therefore	therefore	ADV
ejpam-3842	147	20	,	,	PUNCT
ejpam-3842	147	21	(	(	PUNCT
ejpam-3842	147	22	x	x	X
ejpam-3842	147	23	∗	∗	PROPN
ejpam-3842	147	24	y	y	NOUN
ejpam-3842	147	25	)	)	PUNCT
ejpam-3842	147	26	∗	∗	NOUN
ejpam-3842	147	27	z	z	NOUN
ejpam-3842	147	28	=	=	SYM
ejpam-3842	147	29	(	(	PUNCT
ejpam-3842	147	30	x	x	X
ejpam-3842	147	31	∗	∗	PROPN
ejpam-3842	147	32	z	z	NOUN
ejpam-3842	147	33	)	)	PUNCT
ejpam-3842	147	34	∗	∗	NOUN
ejpam-3842	147	35	y.	y.	PROPN
ejpam-3842	147	36	equality	equality	NOUN
ejpam-3842	147	37	is	be	AUX
ejpam-3842	147	38	also	also	ADV
ejpam-3842	147	39	obtained	obtain	VERB
ejpam-3842	147	40	if	if	SCONJ
ejpam-3842	147	41	y	y	PROPN
ejpam-3842	147	42	∈	∈	PROPN
ejpam-3842	147	43	h1	h1	PROPN
ejpam-3842	147	44	and	and	CCONJ
ejpam-3842	147	45	x	x	PUNCT
ejpam-3842	147	46	∈	∈	PROPN
ejpam-3842	147	47	h2	h2	NOUN
ejpam-3842	147	48	.	.	PUNCT
ejpam-3842	148	1	accordingly	accordingly	ADV
ejpam-3842	148	2	,	,	PUNCT
ejpam-3842	148	3	(	(	PUNCT
ejpam-3842	148	4	h	h	NOUN
ejpam-3842	148	5	,	,	PUNCT
ejpam-3842	148	6	∗	∗	NOUN
ejpam-3842	148	7	,	,	PUNCT
ejpam-3842	148	8	0	0	NUM
ejpam-3842	148	9	)	)	PUNCT
ejpam-3842	148	10	is	be	AUX
ejpam-3842	148	11	a	a	DET
ejpam-3842	148	12	bch	bch	NOUN
ejpam-3842	148	13	-	-	PUNCT
ejpam-3842	148	14	algebra	algebra	NOUN
ejpam-3842	148	15	.	.	PUNCT
ejpam-3842	149	1	lemma	lemma	PROPN
ejpam-3842	149	2	1	1	X
ejpam-3842	149	3	.	.	PUNCT
ejpam-3842	150	1	let	let	AUX
ejpam-3842	150	2	(	(	PUNCT
ejpam-3842	150	3	h	h	NOUN
ejpam-3842	150	4	,	,	PUNCT
ejpam-3842	150	5	∗1	∗1	PROPN
ejpam-3842	150	6	)	)	PUNCT
ejpam-3842	150	7	and	and	CCONJ
ejpam-3842	150	8	(	(	PUNCT
ejpam-3842	150	9	h2	h2	NOUN
ejpam-3842	150	10	,	,	PUNCT
ejpam-3842	150	11	∗2	∗2	PROPN
ejpam-3842	150	12	)	)	PUNCT
ejpam-3842	150	13	be	be	AUX
ejpam-3842	150	14	bch	bch	NOUN
ejpam-3842	150	15	-	-	PUNCT
ejpam-3842	150	16	algebras	algebra	NOUN
ejpam-3842	150	17	such	such	ADJ
ejpam-3842	150	18	that	that	SCONJ
ejpam-3842	150	19	h1	h1	PROPN
ejpam-3842	150	20	∩	∩	ADJ
ejpam-3842	150	21	h2	h2	NOUN
ejpam-3842	150	22	=	=	PUNCT
ejpam-3842	150	23	{	{	PUNCT
ejpam-3842	150	24	0	0	NUM
ejpam-3842	150	25	}	}	PUNCT
ejpam-3842	150	26	and	and	CCONJ
ejpam-3842	150	27	let	let	VERB
ejpam-3842	150	28	(	(	PUNCT
ejpam-3842	150	29	h	h	NOUN
ejpam-3842	150	30	,	,	PUNCT
ejpam-3842	150	31	∗	∗	NOUN
ejpam-3842	150	32	)	)	PUNCT
ejpam-3842	150	33	be	be	VERB
ejpam-3842	150	34	the	the	DET
ejpam-3842	150	35	sum	sum	NOUN
ejpam-3842	150	36	of	of	ADP
ejpam-3842	150	37	h1	h1	PROPN
ejpam-3842	150	38	and	and	CCONJ
ejpam-3842	150	39	h2	h2	NOUN
ejpam-3842	150	40	defined	define	VERB
ejpam-3842	150	41	in	in	ADP
ejpam-3842	150	42	theorem	theorem	NOUN
ejpam-3842	150	43	5	5	NUM
ejpam-3842	150	44	.	.	PUNCT
ejpam-3842	151	1	then	then	ADV
ejpam-3842	151	2	each	each	PRON
ejpam-3842	151	3	of	of	ADP
ejpam-3842	151	4	the	the	DET
ejpam-3842	151	5	following	follow	VERB
ejpam-3842	151	6	holds	hold	VERB
ejpam-3842	151	7	:	:	PUNCT
ejpam-3842	151	8	(	(	PUNCT
ejpam-3842	151	9	i	i	NOUN
ejpam-3842	151	10	)	)	PUNCT
ejpam-3842	151	11	if	if	SCONJ
ejpam-3842	151	12	u	u	NOUN
ejpam-3842	151	13	and	and	CCONJ
ejpam-3842	151	14	v	v	NOUN
ejpam-3842	151	15	are	be	AUX
ejpam-3842	151	16	subsets	subset	NOUN
ejpam-3842	151	17	of	of	ADP
ejpam-3842	151	18	h1	h1	PROPN
ejpam-3842	151	19	(	(	PUNCT
ejpam-3842	151	20	u	u	NOUN
ejpam-3842	151	21	and	and	CCONJ
ejpam-3842	151	22	v	v	NOUN
ejpam-3842	151	23	are	be	AUX
ejpam-3842	151	24	subsets	subset	NOUN
ejpam-3842	151	25	of	of	ADP
ejpam-3842	151	26	h2	h2	NOUN
ejpam-3842	151	27	)	)	PUNCT
ejpam-3842	151	28	,	,	PUNCT
ejpam-3842	151	29	then	then	ADV
ejpam-3842	151	30	u	u	X
ejpam-3842	151	31	∗1	∗1	PROPN
ejpam-3842	151	32	v	v	NOUN
ejpam-3842	151	33	=	=	SYM
ejpam-3842	151	34	u	u	PROPN
ejpam-3842	151	35	∗	∗	X
ejpam-3842	151	36	v	v	NOUN
ejpam-3842	151	37	(	(	PUNCT
ejpam-3842	151	38	resp	resp	NOUN
ejpam-3842	151	39	.	.	PUNCT
ejpam-3842	152	1	u	u	PRON
ejpam-3842	152	2	∗2	∗2	PROPN
ejpam-3842	152	3	v	v	NOUN
ejpam-3842	152	4	=	=	SYM
ejpam-3842	152	5	u	u	NOUN
ejpam-3842	152	6	∗	∗	NOUN
ejpam-3842	152	7	v	v	NOUN
ejpam-3842	152	8	)	)	PUNCT
ejpam-3842	152	9	.	.	PUNCT
ejpam-3842	153	1	(	(	PUNCT
ejpam-3842	153	2	ii	ii	NOUN
ejpam-3842	153	3	)	)	PUNCT
ejpam-3842	153	4	if	if	SCONJ
ejpam-3842	153	5	a	a	PRON
ejpam-3842	153	6	,	,	PUNCT
ejpam-3842	153	7	b	b	NOUN
ejpam-3842	153	8	⊆	⊆	NUM
ejpam-3842	153	9	h1	h1	NOUN
ejpam-3842	153	10	,	,	PUNCT
ejpam-3842	153	11	c	c	PROPN
ejpam-3842	153	12	⊆	⊆	NUM
ejpam-3842	153	13	h2	h2	NOUN
ejpam-3842	153	14	,	,	PUNCT
ejpam-3842	153	15	and	and	CCONJ
ejpam-3842	153	16	0	0	NUM
ejpam-3842	153	17	∈	∈	PROPN
ejpam-3842	153	18	b	b	NOUN
ejpam-3842	153	19	,	,	PUNCT
ejpam-3842	153	20	then	then	ADV
ejpam-3842	153	21	a	a	DET
ejpam-3842	153	22	⊆	⊆	NUM
ejpam-3842	153	23	a∗b	a∗b	NUM
ejpam-3842	153	24	and	and	CCONJ
ejpam-3842	153	25	a∗(b∪c	a∗(b∪c	ADJ
ejpam-3842	153	26	)	)	PUNCT
ejpam-3842	153	27	=	=	SYM
ejpam-3842	153	28	a∗1b	a∗1b	PROPN
ejpam-3842	153	29	=	=	PUNCT
ejpam-3842	153	30	a∗b	a∗b	NUM
ejpam-3842	153	31	.	.	PUNCT
ejpam-3842	154	1	proof	proof	NOUN
ejpam-3842	154	2	.	.	PUNCT
ejpam-3842	155	1	(	(	PUNCT
ejpam-3842	155	2	i	i	NOUN
ejpam-3842	155	3	)	)	PUNCT
ejpam-3842	155	4	suppose	suppose	VERB
ejpam-3842	155	5	u	u	NOUN
ejpam-3842	155	6	and	and	CCONJ
ejpam-3842	155	7	v	v	NOUN
ejpam-3842	155	8	are	be	AUX
ejpam-3842	155	9	subsets	subset	NOUN
ejpam-3842	155	10	ofh1	ofh1	PROPN
ejpam-3842	155	11	.	.	PUNCT
ejpam-3842	156	1	let	let	VERB
ejpam-3842	156	2	x	x	PUNCT
ejpam-3842	156	3	∈	∈	PROPN
ejpam-3842	156	4	u	u	NOUN
ejpam-3842	156	5	and	and	CCONJ
ejpam-3842	156	6	y	y	PROPN
ejpam-3842	156	7	∈	∈	PROPN
ejpam-3842	156	8	v	v	NOUN
ejpam-3842	156	9	.	.	PUNCT
ejpam-3842	157	1	since	since	SCONJ
ejpam-3842	157	2	x∗y	x∗y	X
ejpam-3842	157	3	=	=	SYM
ejpam-3842	157	4	x∗1y	x∗1y	PROPN
ejpam-3842	157	5	,	,	PUNCT
ejpam-3842	157	6	x∗y	x∗y	PROPN
ejpam-3842	157	7	∈	∈	PROPN
ejpam-3842	157	8	u	u	NOUN
ejpam-3842	157	9	∗v	∗v	PROPN
ejpam-3842	157	10	if	if	SCONJ
ejpam-3842	157	11	and	and	CCONJ
ejpam-3842	157	12	only	only	ADV
ejpam-3842	157	13	if	if	SCONJ
ejpam-3842	157	14	x∗1y	x∗1y	PROPN
ejpam-3842	157	15	∈	∈	PROPN
ejpam-3842	157	16	u	u	NOUN
ejpam-3842	157	17	∗1v	∗1v	PROPN
ejpam-3842	157	18	.	.	PUNCT
ejpam-3842	158	1	hence	hence	ADV
ejpam-3842	158	2	,	,	PUNCT
ejpam-3842	158	3	u	u	NOUN
ejpam-3842	158	4	∗1v	∗1v	PROPN
ejpam-3842	158	5	=	=	SYM
ejpam-3842	158	6	u	u	PROPN
ejpam-3842	158	7	∗v	∗v	NOUN
ejpam-3842	158	8	.	.	PUNCT
ejpam-3842	159	1	similarly	similarly	ADV
ejpam-3842	159	2	,	,	PUNCT
ejpam-3842	159	3	u	u	NOUN
ejpam-3842	159	4	∗2v	∗2v	PROPN
ejpam-3842	159	5	=	=	PUNCT
ejpam-3842	159	6	u	u	NOUN
ejpam-3842	159	7	∗v	∗v	NOUN
ejpam-3842	159	8	if	if	SCONJ
ejpam-3842	159	9	u	u	PROPN
ejpam-3842	159	10	and	and	CCONJ
ejpam-3842	159	11	v	v	NOUN
ejpam-3842	159	12	are	be	AUX
ejpam-3842	159	13	subsets	subset	NOUN
ejpam-3842	159	14	of	of	ADP
ejpam-3842	159	15	h2	h2	NOUN
ejpam-3842	159	16	.	.	PUNCT
ejpam-3842	160	1	(	(	PUNCT
ejpam-3842	160	2	ii	ii	NOUN
ejpam-3842	160	3	)	)	PUNCT
ejpam-3842	160	4	let	let	VERB
ejpam-3842	160	5	x	x	PUNCT
ejpam-3842	160	6	∈	∈	VERB
ejpam-3842	160	7	a.	a.	NOUN
ejpam-3842	160	8	then	then	ADV
ejpam-3842	160	9	x	x	X
ejpam-3842	161	1	=	=	PUNCT
ejpam-3842	161	2	x	x	SYM
ejpam-3842	161	3	∗	∗	NOUN
ejpam-3842	161	4	0	0	NUM
ejpam-3842	161	5	∈	∈	PROPN
ejpam-3842	161	6	a	a	DET
ejpam-3842	161	7	∗b	∗b	NOUN
ejpam-3842	161	8	since	since	SCONJ
ejpam-3842	161	9	0	0	NUM
ejpam-3842	161	10	∈	∈	PROPN
ejpam-3842	161	11	b.	b.	PROPN
ejpam-3842	161	12	hence	hence	ADV
ejpam-3842	161	13	,	,	PUNCT
ejpam-3842	161	14	a	a	DET
ejpam-3842	161	15	⊆	⊆	NUM
ejpam-3842	161	16	a	a	DET
ejpam-3842	161	17	∗b	∗b	NOUN
ejpam-3842	161	18	=	=	PUNCT
ejpam-3842	161	19	a	a	DET
ejpam-3842	161	20	∗1	∗1	PROPN
ejpam-3842	161	21	b.	b.	NOUN
ejpam-3842	161	22	to	to	PART
ejpam-3842	161	23	establish	establish	VERB
ejpam-3842	161	24	the	the	DET
ejpam-3842	161	25	equality	equality	NOUN
ejpam-3842	161	26	,	,	PUNCT
ejpam-3842	161	27	first	first	ADV
ejpam-3842	161	28	note	note	VERB
ejpam-3842	161	29	that	that	SCONJ
ejpam-3842	161	30	a	a	DET
ejpam-3842	161	31	∗1b	∗1b	PROPN
ejpam-3842	161	32	=	=	PUNCT
ejpam-3842	161	33	a	a	DET
ejpam-3842	161	34	∗b	∗b	PROPN
ejpam-3842	161	35	⊆	⊆	NUM
ejpam-3842	161	36	a	a	DET
ejpam-3842	161	37	∗	∗	NOUN
ejpam-3842	161	38	(	(	PUNCT
ejpam-3842	161	39	b	b	NOUN
ejpam-3842	161	40	∪c	∪c	NOUN
ejpam-3842	161	41	)	)	PUNCT
ejpam-3842	161	42	.	.	PUNCT
ejpam-3842	162	1	let	let	VERB
ejpam-3842	162	2	a	a	DET
ejpam-3842	162	3	∈	∈	PROPN
ejpam-3842	162	4	a	a	PRON
ejpam-3842	163	1	and	and	CCONJ
ejpam-3842	163	2	x	x	SYM
ejpam-3842	163	3	∈	∈	PROPN
ejpam-3842	163	4	(	(	PUNCT
ejpam-3842	163	5	b∪c	b∪c	ADJ
ejpam-3842	163	6	)	)	PUNCT
ejpam-3842	163	7	.	.	PUNCT
ejpam-3842	164	1	if	if	SCONJ
ejpam-3842	164	2	x	x	SYM
ejpam-3842	164	3	∈	∈	PROPN
ejpam-3842	164	4	b	b	PROPN
ejpam-3842	164	5	,	,	PUNCT
ejpam-3842	164	6	then	then	ADV
ejpam-3842	164	7	a∗x	a∗x	NUM
ejpam-3842	164	8	=	=	SYM
ejpam-3842	164	9	a∗1	a∗1	NOUN
ejpam-3842	164	10	x	x	PUNCT
ejpam-3842	164	11	∈	∈	PROPN
ejpam-3842	164	12	a∗1b	a∗1b	NOUN
ejpam-3842	164	13	.	.	PUNCT
ejpam-3842	165	1	if	if	SCONJ
ejpam-3842	165	2	x	x	SYM
ejpam-3842	165	3	∈	∈	PROPN
ejpam-3842	165	4	c	c	NOUN
ejpam-3842	165	5	,	,	PUNCT
ejpam-3842	165	6	then	then	ADV
ejpam-3842	165	7	a∗x	a∗x	PRON
ejpam-3842	165	8	=	=	PUNCT
ejpam-3842	165	9	a	a	DET
ejpam-3842	165	10	∈	∈	PROPN
ejpam-3842	165	11	a	a	DET
ejpam-3842	165	12	⊆	⊆	NUM
ejpam-3842	165	13	a∗1b	a∗1b	NOUN
ejpam-3842	165	14	.	.	PUNCT
ejpam-3842	166	1	thus	thus	ADV
ejpam-3842	166	2	,	,	PUNCT
ejpam-3842	166	3	a	a	DET
ejpam-3842	166	4	∗	∗	NOUN
ejpam-3842	166	5	(	(	PUNCT
ejpam-3842	166	6	b	b	NOUN
ejpam-3842	166	7	∪	∪	X
ejpam-3842	166	8	c	c	NOUN
ejpam-3842	166	9	)	)	PUNCT
ejpam-3842	166	10	=	=	PUNCT
ejpam-3842	167	1	a	a	DET
ejpam-3842	167	2	∗1	∗1	PROPN
ejpam-3842	167	3	b	b	X
ejpam-3842	167	4	=	=	PUNCT
ejpam-3842	167	5	a	a	DET
ejpam-3842	167	6	∗b	∗b	PROPN
ejpam-3842	167	7	.	.	PUNCT
ejpam-3842	167	8	theorem	theorem	NOUN
ejpam-3842	167	9	6	6	NUM
ejpam-3842	167	10	.	.	PUNCT
ejpam-3842	168	1	let	let	VERB
ejpam-3842	168	2	(	(	PUNCT
ejpam-3842	168	3	h1	h1	PROPN
ejpam-3842	168	4	,	,	PUNCT
ejpam-3842	168	5	∗1	∗1	PROPN
ejpam-3842	168	6	,	,	PUNCT
ejpam-3842	168	7	0	0	NUM
ejpam-3842	168	8	)	)	PUNCT
ejpam-3842	168	9	and	and	CCONJ
ejpam-3842	168	10	(	(	PUNCT
ejpam-3842	168	11	h2	h2	NOUN
ejpam-3842	168	12	,	,	PUNCT
ejpam-3842	168	13	∗2	∗2	PROPN
ejpam-3842	168	14	,	,	PUNCT
ejpam-3842	168	15	0	0	NUM
ejpam-3842	168	16	)	)	PUNCT
ejpam-3842	168	17	be	be	VERB
ejpam-3842	168	18	bch	bch	NOUN
ejpam-3842	168	19	-	-	PUNCT
ejpam-3842	168	20	algebras	algebra	NOUN
ejpam-3842	168	21	such	such	ADJ
ejpam-3842	168	22	that	that	SCONJ
ejpam-3842	168	23	h1∩h2	h1∩h2	PROPN
ejpam-3842	168	24	=	=	PUNCT
ejpam-3842	168	25	{	{	PUNCT
ejpam-3842	168	26	0	0	NUM
ejpam-3842	168	27	}	}	PUNCT
ejpam-3842	168	28	and	and	CCONJ
ejpam-3842	168	29	let	let	VERB
ejpam-3842	168	30	(	(	PUNCT
ejpam-3842	168	31	h	h	NOUN
ejpam-3842	168	32	,	,	PUNCT
ejpam-3842	168	33	∗	∗	NOUN
ejpam-3842	168	34	,	,	PUNCT
ejpam-3842	168	35	0	0	NUM
ejpam-3842	168	36	)	)	PUNCT
ejpam-3842	168	37	be	be	AUX
ejpam-3842	168	38	the	the	DET
ejpam-3842	168	39	sum	sum	NOUN
ejpam-3842	168	40	of	of	ADP
ejpam-3842	168	41	h1	h1	PROPN
ejpam-3842	168	42	and	and	CCONJ
ejpam-3842	168	43	h2	h2	PROPN
ejpam-3842	168	44	(	(	PUNCT
ejpam-3842	168	45	defined	define	VERB
ejpam-3842	168	46	in	in	ADP
ejpam-3842	168	47	theorem	theorem	NOUN
ejpam-3842	168	48	5	5	NUM
ejpam-3842	168	49	)	)	PUNCT
ejpam-3842	168	50	.	.	PUNCT
ejpam-3842	169	1	then	then	ADV
ejpam-3842	169	2	each	each	PRON
ejpam-3842	169	3	of	of	ADP
ejpam-3842	169	4	the	the	DET
ejpam-3842	169	5	following	follow	VERB
ejpam-3842	169	6	holds	hold	VERB
ejpam-3842	169	7	:	:	PUNCT
ejpam-3842	169	8	(	(	PUNCT
ejpam-3842	169	9	i	i	NOUN
ejpam-3842	169	10	)	)	PUNCT
ejpam-3842	169	11	(	(	PUNCT
ejpam-3842	169	12	h1	h1	PROPN
ejpam-3842	169	13	,	,	PUNCT
ejpam-3842	169	14	∗1	∗1	PROPN
ejpam-3842	169	15	,	,	PUNCT
ejpam-3842	169	16	0	0	NUM
ejpam-3842	169	17	)	)	PUNCT
ejpam-3842	169	18	and	and	CCONJ
ejpam-3842	169	19	(	(	PUNCT
ejpam-3842	169	20	h2	h2	NOUN
ejpam-3842	169	21	,	,	PUNCT
ejpam-3842	169	22	∗2	∗2	PROPN
ejpam-3842	169	23	,	,	PUNCT
ejpam-3842	169	24	0	0	NUM
ejpam-3842	169	25	)	)	PUNCT
ejpam-3842	169	26	are	be	AUX
ejpam-3842	169	27	bch	bch	NOUN
ejpam-3842	169	28	-	-	PUNCT
ejpam-3842	169	29	subalgebras	subalgebras	PROPN
ejpam-3842	169	30	of	of	ADP
ejpam-3842	169	31	h.	h.	PROPN
ejpam-3842	169	32	(	(	PUNCT
ejpam-3842	169	33	ii	ii	PROPN
ejpam-3842	169	34	)	)	PUNCT
ejpam-3842	169	35	(	(	PUNCT
ejpam-3842	169	36	h	h	NOUN
ejpam-3842	169	37	,	,	PUNCT
ejpam-3842	169	38	τh1	τh1	PROPN
ejpam-3842	169	39	)	)	PUNCT
ejpam-3842	169	40	and	and	CCONJ
ejpam-3842	169	41	(	(	PUNCT
ejpam-3842	169	42	h	h	NOUN
ejpam-3842	169	43	,	,	PUNCT
ejpam-3842	169	44	τh2	τh2	PRON
ejpam-3842	169	45	)	)	PUNCT
ejpam-3842	169	46	are	be	AUX
ejpam-3842	169	47	tbch	tbch	NOUN
ejpam-3842	169	48	-	-	PUNCT
ejpam-3842	169	49	algebras	algebra	NOUN
ejpam-3842	169	50	,	,	PUNCT
ejpam-3842	169	51	where	where	SCONJ
ejpam-3842	169	52	τh1	τh1	NOUN
ejpam-3842	169	53	=	=	PRON
ejpam-3842	169	54	{	{	PUNCT
ejpam-3842	169	55	∅	∅	NOUN
ejpam-3842	169	56	,	,	PUNCT
ejpam-3842	169	57	h1∪h2	h1∪h2	NOUN
ejpam-3842	169	58	,	,	PUNCT
ejpam-3842	169	59	h1	h1	NOUN
ejpam-3842	169	60	}	}	PUNCT
ejpam-3842	169	61	and	and	CCONJ
ejpam-3842	169	62	τh2	τh2	ADP
ejpam-3842	169	63	=	=	PRON
ejpam-3842	169	64	{	{	PUNCT
ejpam-3842	169	65	∅	∅	NOUN
ejpam-3842	169	66	,	,	PUNCT
ejpam-3842	169	67	h1	h1	PROPN
ejpam-3842	169	68	∪h2	∪h2	NOUN
ejpam-3842	169	69	,	,	PUNCT
ejpam-3842	169	70	h2	h2	NOUN
ejpam-3842	169	71	}	}	PUNCT
ejpam-3842	169	72	.	.	PUNCT
ejpam-3842	170	1	(	(	PUNCT
ejpam-3842	170	2	iii	iii	X
ejpam-3842	170	3	)	)	PUNCT
ejpam-3842	170	4	if	if	SCONJ
ejpam-3842	170	5	(	(	PUNCT
ejpam-3842	170	6	h	h	NOUN
ejpam-3842	170	7	,	,	PUNCT
ejpam-3842	170	8	τ	τ	X
ejpam-3842	170	9	)	)	PUNCT
ejpam-3842	170	10	is	be	AUX
ejpam-3842	170	11	a	a	DET
ejpam-3842	170	12	tbch	tbch	NOUN
ejpam-3842	170	13	-	-	PUNCT
ejpam-3842	170	14	algebra	algebra	NOUN
ejpam-3842	170	15	and	and	CCONJ
ejpam-3842	170	16	a	a	DET
ejpam-3842	170	17	,	,	PUNCT
ejpam-3842	170	18	b	b	PROPN
ejpam-3842	170	19	∈	∈	X
ejpam-3842	170	20	τ	τ	X
ejpam-3842	170	21	for	for	ADP
ejpam-3842	170	22	some	some	PRON
ejpam-3842	170	23	set	set	VERB
ejpam-3842	170	24	a	a	DET
ejpam-3842	170	25	⊆	⊆	NUM
ejpam-3842	170	26	h1	h1	NOUN
ejpam-3842	170	27	and	and	CCONJ
ejpam-3842	170	28	b	b	NOUN
ejpam-3842	170	29	⊆	⊆	NUM
ejpam-3842	170	30	h2	h2	NOUN
ejpam-3842	170	31	with	with	ADP
ejpam-3842	170	32	0	0	NUM
ejpam-3842	170	33	∈	∈	PROPN
ejpam-3842	170	34	a	a	DET
ejpam-3842	170	35	∩	∩	ADJ
ejpam-3842	170	36	b	b	NOUN
ejpam-3842	170	37	,	,	PUNCT
ejpam-3842	170	38	then	then	ADV
ejpam-3842	170	39	τ	τ	PROPN
ejpam-3842	170	40	is	be	AUX
ejpam-3842	170	41	the	the	DET
ejpam-3842	170	42	discrete	discrete	ADJ
ejpam-3842	170	43	topology	topology	NOUN
ejpam-3842	170	44	on	on	ADP
ejpam-3842	170	45	h.	h.	PROPN
ejpam-3842	170	46	in	in	ADP
ejpam-3842	170	47	particular	particular	ADJ
ejpam-3842	170	48	,	,	PUNCT
ejpam-3842	170	49	if	if	SCONJ
ejpam-3842	170	50	h1	h1	PROPN
ejpam-3842	170	51	,	,	PUNCT
ejpam-3842	170	52	h2	h2	PROPN
ejpam-3842	170	53	∈	∈	PROPN
ejpam-3842	170	54	τ	τ	X
ejpam-3842	170	55	,	,	PUNCT
ejpam-3842	170	56	then	then	ADV
ejpam-3842	170	57	τ	τ	PROPN
ejpam-3842	170	58	is	be	AUX
ejpam-3842	170	59	the	the	DET
ejpam-3842	170	60	discrete	discrete	ADJ
ejpam-3842	170	61	topology	topology	NOUN
ejpam-3842	170	62	on	on	ADP
ejpam-3842	170	63	h.	h.	PROPN
ejpam-3842	170	64	(	(	PUNCT
ejpam-3842	170	65	iv	iv	X
ejpam-3842	170	66	)	)	PUNCT
ejpam-3842	170	67	if	if	SCONJ
ejpam-3842	170	68	(	(	PUNCT
ejpam-3842	170	69	h	h	NOUN
ejpam-3842	170	70	,	,	PUNCT
ejpam-3842	170	71	τ	τ	X
ejpam-3842	170	72	)	)	PUNCT
ejpam-3842	170	73	is	be	AUX
ejpam-3842	170	74	a	a	DET
ejpam-3842	170	75	tbch	tbch	NOUN
ejpam-3842	170	76	-	-	PUNCT
ejpam-3842	170	77	algebra	algebra	NOUN
ejpam-3842	170	78	and	and	CCONJ
ejpam-3842	170	79	τ	τ	PROPN
ejpam-3842	171	1	⊆	⊆	NUM
ejpam-3842	171	2	p	p	X
ejpam-3842	171	3	(	(	PUNCT
ejpam-3842	171	4	h1)∪{h1∪h2	h1)∪{h1∪h2	PROPN
ejpam-3842	171	5	}	}	PUNCT
ejpam-3842	171	6	(	(	PUNCT
ejpam-3842	171	7	or	or	CCONJ
ejpam-3842	171	8	τ	τ	PROPN
ejpam-3842	171	9	⊆	⊆	NUM
ejpam-3842	171	10	p	p	PROPN
ejpam-3842	171	11	(	(	PUNCT
ejpam-3842	171	12	h2)∪{h1∪h2	h2)∪{h1∪h2	PROPN
ejpam-3842	171	13	}	}	PUNCT
ejpam-3842	171	14	)	)	PUNCT
ejpam-3842	171	15	,	,	PUNCT
ejpam-3842	171	16	where	where	SCONJ
ejpam-3842	171	17	p	p	PROPN
ejpam-3842	171	18	(	(	PUNCT
ejpam-3842	171	19	h1	h1	PROPN
ejpam-3842	171	20	)	)	PUNCT
ejpam-3842	171	21	and	and	CCONJ
ejpam-3842	171	22	p	p	PROPN
ejpam-3842	171	23	(	(	PUNCT
ejpam-3842	171	24	h2	h2	NOUN
ejpam-3842	171	25	)	)	PUNCT
ejpam-3842	171	26	are	be	AUX
ejpam-3842	171	27	the	the	DET
ejpam-3842	171	28	power	power	NOUN
ejpam-3842	171	29	sets	set	NOUN
ejpam-3842	171	30	of	of	ADP
ejpam-3842	171	31	h1	h1	NOUN
ejpam-3842	171	32	and	and	CCONJ
ejpam-3842	171	33	h2	h2	NOUN
ejpam-3842	171	34	,	,	PUNCT
ejpam-3842	171	35	respectively	respectively	ADV
ejpam-3842	171	36	,	,	PUNCT
ejpam-3842	171	37	then	then	ADV
ejpam-3842	171	38	0	0	NUM
ejpam-3842	171	39	∈w	∈w	NOUN
ejpam-3842	171	40	for	for	ADP
ejpam-3842	171	41	every	every	DET
ejpam-3842	171	42	w	w	PROPN
ejpam-3842	171	43	∈	∈	PROPN
ejpam-3842	171	44	τ	τ	X
ejpam-3842	171	45	\	\	X
ejpam-3842	171	46	{	{	PUNCT
ejpam-3842	171	47	∅	∅	NOUN
ejpam-3842	171	48	}	}	PUNCT
ejpam-3842	171	49	.	.	PUNCT
ejpam-3842	172	1	proof	proof	NOUN
ejpam-3842	172	2	.	.	PUNCT
ejpam-3842	173	1	(	(	PUNCT
ejpam-3842	173	2	i	i	NOUN
ejpam-3842	173	3	)	)	PUNCT
ejpam-3842	173	4	let	let	VERB
ejpam-3842	173	5	x	x	PRON
ejpam-3842	173	6	,	,	PUNCT
ejpam-3842	173	7	y	y	PROPN
ejpam-3842	173	8	∈	∈	PROPN
ejpam-3842	173	9	h1	h1	PROPN
ejpam-3842	173	10	.	.	PUNCT
ejpam-3842	174	1	then	then	ADV
ejpam-3842	174	2	x	x	X
ejpam-3842	174	3	∗	∗	NOUN
ejpam-3842	174	4	y	y	NOUN
ejpam-3842	174	5	=	=	PUNCT
ejpam-3842	175	1	x	x	SYM
ejpam-3842	175	2	∗1	∗1	PROPN
ejpam-3842	175	3	y	y	PROPN
ejpam-3842	175	4	∈	∈	PROPN
ejpam-3842	175	5	h1	h1	NOUN
ejpam-3842	175	6	by	by	ADP
ejpam-3842	175	7	theorem	theorem	NOUN
ejpam-3842	175	8	5	5	NUM
ejpam-3842	175	9	and	and	CCONJ
ejpam-3842	175	10	the	the	DET
ejpam-3842	175	11	fact	fact	NOUN
ejpam-3842	175	12	that	that	SCONJ
ejpam-3842	175	13	(	(	PUNCT
ejpam-3842	175	14	h1	h1	PROPN
ejpam-3842	175	15	,	,	PUNCT
ejpam-3842	175	16	∗1	∗1	PROPN
ejpam-3842	175	17	,	,	PUNCT
ejpam-3842	175	18	0	0	NUM
ejpam-3842	175	19	)	)	PUNCT
ejpam-3842	175	20	is	be	AUX
ejpam-3842	175	21	a	a	DET
ejpam-3842	175	22	bch	bch	NOUN
ejpam-3842	175	23	-	-	PUNCT
ejpam-3842	175	24	algebra	algebra	NOUN
ejpam-3842	175	25	.	.	PUNCT
ejpam-3842	176	1	therefore	therefore	ADV
ejpam-3842	176	2	,	,	PUNCT
ejpam-3842	176	3	(	(	PUNCT
ejpam-3842	176	4	h1	h1	PROPN
ejpam-3842	176	5	,	,	PUNCT
ejpam-3842	176	6	∗1	∗1	PROPN
ejpam-3842	176	7	,	,	PUNCT
ejpam-3842	176	8	0	0	NUM
ejpam-3842	176	9	)	)	PUNCT
ejpam-3842	176	10	=	=	PRON
ejpam-3842	176	11	(	(	PUNCT
ejpam-3842	176	12	h1	h1	PROPN
ejpam-3842	176	13	,	,	PUNCT
ejpam-3842	176	14	∗	∗	NOUN
ejpam-3842	176	15	,	,	PUNCT
ejpam-3842	176	16	0	0	NUM
ejpam-3842	176	17	)	)	PUNCT
ejpam-3842	176	18	is	be	AUX
ejpam-3842	176	19	a	a	DET
ejpam-3842	176	20	bch	bch	NOUN
ejpam-3842	176	21	-	-	PUNCT
ejpam-3842	176	22	subalgebra	subalgebra	NOUN
ejpam-3842	176	23	of	of	ADP
ejpam-3842	176	24	h.	h.	NOUN
ejpam-3842	176	25	similarly	similarly	ADV
ejpam-3842	176	26	,	,	PUNCT
ejpam-3842	176	27	(	(	PUNCT
ejpam-3842	176	28	h2	h2	NOUN
ejpam-3842	176	29	,	,	PUNCT
ejpam-3842	176	30	∗2	∗2	PROPN
ejpam-3842	176	31	,	,	PUNCT
ejpam-3842	176	32	0	0	NUM
ejpam-3842	176	33	)	)	PUNCT
ejpam-3842	176	34	is	be	AUX
ejpam-3842	176	35	a	a	DET
ejpam-3842	176	36	bch	bch	NOUN
ejpam-3842	176	37	-	-	PUNCT
ejpam-3842	176	38	subalgebra	subalgebra	NOUN
ejpam-3842	176	39	of	of	ADP
ejpam-3842	176	40	h.	h.	PROPN
ejpam-3842	176	41	(	(	PUNCT
ejpam-3842	176	42	ii	ii	PROPN
ejpam-3842	176	43	)	)	PUNCT
ejpam-3842	176	44	clearly	clearly	ADV
ejpam-3842	176	45	,	,	PUNCT
ejpam-3842	176	46	τh1	τh1	PROPN
ejpam-3842	176	47	are	be	AUX
ejpam-3842	176	48	τh2	τh2	ADV
ejpam-3842	176	49	are	be	AUX
ejpam-3842	176	50	topologies	topology	NOUN
ejpam-3842	176	51	on	on	ADP
ejpam-3842	176	52	h.	h.	PROPN
ejpam-3842	176	53	first	first	ADV
ejpam-3842	176	54	,	,	PUNCT
ejpam-3842	176	55	consider	consider	VERB
ejpam-3842	176	56	the	the	DET
ejpam-3842	176	57	space	space	NOUN
ejpam-3842	176	58	(	(	PUNCT
ejpam-3842	176	59	x	x	NOUN
ejpam-3842	176	60	,	,	PUNCT
ejpam-3842	176	61	τh1	τh1	PROPN
ejpam-3842	176	62	)	)	PUNCT
ejpam-3842	176	63	.	.	PUNCT
ejpam-3842	177	1	let	let	VERB
ejpam-3842	177	2	x	x	PRON
ejpam-3842	177	3	,	,	PUNCT
ejpam-3842	177	4	y	y	PROPN
ejpam-3842	177	5	∈	∈	PROPN
ejpam-3842	177	6	h	h	NOUN
ejpam-3842	177	7	and	and	CCONJ
ejpam-3842	177	8	let	let	VERB
ejpam-3842	177	9	w	w	NOUN
ejpam-3842	177	10	be	be	AUX
ejpam-3842	177	11	a	a	DET
ejpam-3842	177	12	τh1	τh1	NOUN
ejpam-3842	177	13	-	-	PUNCT
ejpam-3842	177	14	nbhd	nbhd	NOUN
ejpam-3842	177	15	of	of	ADP
ejpam-3842	177	16	x	x	PROPN
ejpam-3842	177	17	∗	∗	NOUN
ejpam-3842	177	18	y.	y.	NOUN
ejpam-3842	177	19	consider	consider	VERB
ejpam-3842	177	20	the	the	DET
ejpam-3842	177	21	following	follow	VERB
ejpam-3842	177	22	cases	case	NOUN
ejpam-3842	177	23	:	:	PUNCT
ejpam-3842	177	24	case	case	NOUN
ejpam-3842	177	25	1	1	NUM
ejpam-3842	177	26	:	:	PUNCT
ejpam-3842	177	27	x	x	X
ejpam-3842	177	28	,	,	PUNCT
ejpam-3842	177	29	y	y	PROPN
ejpam-3842	177	30	∈	∈	PROPN
ejpam-3842	177	31	h1	h1	PROPN
ejpam-3842	177	32	then	then	ADV
ejpam-3842	177	33	x	x	X
ejpam-3842	177	34	∗	∗	NOUN
ejpam-3842	177	35	y	y	NOUN
ejpam-3842	178	1	=	=	PUNCT
ejpam-3842	178	2	x	x	SYM
ejpam-3842	178	3	∗1	∗1	PROPN
ejpam-3842	178	4	y	y	PROPN
ejpam-3842	178	5	∈	∈	PROPN
ejpam-3842	178	6	h1	h1	PROPN
ejpam-3842	178	7	.	.	PUNCT
ejpam-3842	179	1	hence	hence	ADV
ejpam-3842	179	2	,	,	PUNCT
ejpam-3842	179	3	w	w	NOUN
ejpam-3842	179	4	=	=	PUNCT
ejpam-3842	179	5	h1	h1	NOUN
ejpam-3842	179	6	or	or	CCONJ
ejpam-3842	179	7	w	w	NOUN
ejpam-3842	179	8	=	=	NOUN
ejpam-3842	179	9	h1	h1	PROPN
ejpam-3842	179	10	∪h2	∪h2	PROPN
ejpam-3842	179	11	.	.	PUNCT
ejpam-3842	180	1	then	then	ADV
ejpam-3842	180	2	h1	h1	PROPN
ejpam-3842	180	3	is	be	AUX
ejpam-3842	180	4	a	a	DET
ejpam-3842	180	5	τh1	τh1	NOUN
ejpam-3842	180	6	-	-	PUNCT
ejpam-3842	180	7	nbhd	nbhd	NOUN
ejpam-3842	180	8	of	of	ADP
ejpam-3842	180	9	both	both	PRON
ejpam-3842	180	10	x	x	SYM
ejpam-3842	180	11	and	and	CCONJ
ejpam-3842	180	12	y	y	PROPN
ejpam-3842	180	13	,	,	PUNCT
ejpam-3842	180	14	and	and	CCONJ
ejpam-3842	180	15	by	by	ADP
ejpam-3842	180	16	lemma	lemma	PROPN
ejpam-3842	180	17	1(i	1(i	NUM
ejpam-3842	180	18	)	)	PUNCT
ejpam-3842	180	19	,	,	PUNCT
ejpam-3842	180	20	h1	h1	PROPN
ejpam-3842	180	21	∗h1	∗h1	NOUN
ejpam-3842	180	22	=	=	PUNCT
ejpam-3842	180	23	h1	h1	VERB
ejpam-3842	180	24	∗1	∗1	PROPN
ejpam-3842	180	25	h1	h1	PROPN
ejpam-3842	180	26	=	=	PUNCT
ejpam-3842	180	27	h1	h1	PROPN
ejpam-3842	180	28	⊂	⊂	PROPN
ejpam-3842	180	29	h1	h1	VERB
ejpam-3842	180	30	∪h2	∪h2	PROPN
ejpam-3842	180	31	.	.	PUNCT
ejpam-3842	181	1	case	case	NOUN
ejpam-3842	181	2	2	2	NUM
ejpam-3842	181	3	:	:	PUNCT
ejpam-3842	181	4	x	x	X
ejpam-3842	181	5	,	,	PUNCT
ejpam-3842	181	6	y	y	PROPN
ejpam-3842	181	7	∈	∈	PROPN
ejpam-3842	181	8	h2	h2	PROPN
ejpam-3842	181	9	or	or	CCONJ
ejpam-3842	181	10	[	[	X
ejpam-3842	181	11	x	x	X
ejpam-3842	181	12	∈	∈	PROPN
ejpam-3842	181	13	h2	h2	NOUN
ejpam-3842	181	14	and	and	CCONJ
ejpam-3842	181	15	y	y	PROPN
ejpam-3842	181	16	∈	∈	PROPN
ejpam-3842	181	17	h1	h1	PROPN
ejpam-3842	181	18	]	]	PUNCT
ejpam-3842	181	19	j.	j.	PROPN
ejpam-3842	181	20	mancao	mancao	PROPN
ejpam-3842	181	21	,	,	PUNCT
ejpam-3842	181	22	s.	s.	PROPN
ejpam-3842	181	23	canoy	canoy	PROPN
ejpam-3842	181	24	/	/	SYM
ejpam-3842	181	25	eur	eur	PROPN
ejpam-3842	181	26	.	.	PUNCT
ejpam-3842	182	1	j.	j.	PROPN
ejpam-3842	182	2	pure	pure	PROPN
ejpam-3842	182	3	appl	appl	PROPN
ejpam-3842	182	4	.	.	PROPN
ejpam-3842	182	5	math	math	PROPN
ejpam-3842	182	6	,	,	PUNCT
ejpam-3842	182	7	13	13	NUM
ejpam-3842	182	8	(	(	PUNCT
ejpam-3842	182	9	4	4	NUM
ejpam-3842	182	10	)	)	PUNCT
ejpam-3842	182	11	(	(	PUNCT
ejpam-3842	182	12	2020	2020	NUM
ejpam-3842	182	13	)	)	PUNCT
ejpam-3842	182	14	,	,	PUNCT
ejpam-3842	182	15	730	730	NUM
ejpam-3842	182	16	-	-	SYM
ejpam-3842	182	17	738	738	NUM
ejpam-3842	182	18	736	736	NUM
ejpam-3842	182	19	if	if	SCONJ
ejpam-3842	182	20	x	x	NOUN
ejpam-3842	182	21	,	,	PUNCT
ejpam-3842	182	22	y	y	PROPN
ejpam-3842	182	23	∈	∈	PROPN
ejpam-3842	182	24	h2	h2	NOUN
ejpam-3842	182	25	,	,	PUNCT
ejpam-3842	182	26	then	then	ADV
ejpam-3842	182	27	x	x	X
ejpam-3842	182	28	∗	∗	NOUN
ejpam-3842	182	29	y	y	NOUN
ejpam-3842	182	30	=	=	PUNCT
ejpam-3842	182	31	x	x	SYM
ejpam-3842	182	32	∗2	∗2	VERB
ejpam-3842	182	33	y	y	PROPN
ejpam-3842	182	34	∈	∈	PROPN
ejpam-3842	182	35	h2	h2	PROPN
ejpam-3842	182	36	.	.	PUNCT
ejpam-3842	183	1	hence	hence	ADV
ejpam-3842	183	2	,	,	PUNCT
ejpam-3842	183	3	w	w	PROPN
ejpam-3842	183	4	=	=	PUNCT
ejpam-3842	183	5	h1	h1	PROPN
ejpam-3842	183	6	∪h2	∪h2	NOUN
ejpam-3842	183	7	.	.	PUNCT
ejpam-3842	184	1	the	the	DET
ejpam-3842	184	2	set	set	NOUN
ejpam-3842	184	3	v	v	NOUN
ejpam-3842	184	4	=	=	SYM
ejpam-3842	184	5	h1	h1	ADJ
ejpam-3842	184	6	∪h2	∪h2	PROPN
ejpam-3842	184	7	is	be	AUX
ejpam-3842	184	8	a	a	DET
ejpam-3842	184	9	τh1	τh1	NOUN
ejpam-3842	184	10	-	-	PUNCT
ejpam-3842	184	11	nbhd	nbhd	NOUN
ejpam-3842	184	12	of	of	ADP
ejpam-3842	184	13	both	both	PRON
ejpam-3842	184	14	x	x	SYM
ejpam-3842	184	15	and	and	CCONJ
ejpam-3842	184	16	y	y	PROPN
ejpam-3842	184	17	,	,	PUNCT
ejpam-3842	184	18	and	and	CCONJ
ejpam-3842	184	19	v	v	ADP
ejpam-3842	184	20	∗	∗	NOUN
ejpam-3842	184	21	v	v	NOUN
ejpam-3842	184	22	=	=	X
ejpam-3842	184	23	h1	h1	PROPN
ejpam-3842	184	24	∪	∪	PROPN
ejpam-3842	184	25	h2	h2	NOUN
ejpam-3842	184	26	.	.	PUNCT
ejpam-3842	185	1	if	if	SCONJ
ejpam-3842	185	2	x	x	SYM
ejpam-3842	185	3	∈	∈	PROPN
ejpam-3842	185	4	h2	h2	NOUN
ejpam-3842	185	5	and	and	CCONJ
ejpam-3842	185	6	y	y	PROPN
ejpam-3842	185	7	∈	∈	PROPN
ejpam-3842	185	8	h1	h1	PROPN
ejpam-3842	185	9	,	,	PUNCT
ejpam-3842	185	10	then	then	ADV
ejpam-3842	185	11	x	x	X
ejpam-3842	185	12	∗	∗	NOUN
ejpam-3842	185	13	y	y	NOUN
ejpam-3842	185	14	=	=	PUNCT
ejpam-3842	185	15	x	x	SYM
ejpam-3842	185	16	∈	∈	PROPN
ejpam-3842	185	17	h2	h2	NOUN
ejpam-3842	185	18	.	.	PUNCT
ejpam-3842	186	1	again	again	ADV
ejpam-3842	186	2	,	,	PUNCT
ejpam-3842	186	3	w	w	PROPN
ejpam-3842	186	4	=	=	PUNCT
ejpam-3842	186	5	h1	h1	PROPN
ejpam-3842	186	6	∪h2	∪h2	NOUN
ejpam-3842	186	7	,	,	PUNCT
ejpam-3842	186	8	v	v	NOUN
ejpam-3842	186	9	=	=	PUNCT
ejpam-3842	186	10	h1	h1	NOUN
ejpam-3842	186	11	∪h1	∪h1	NOUN
ejpam-3842	186	12	is	be	AUX
ejpam-3842	186	13	a	a	DET
ejpam-3842	186	14	τh1	τh1	NOUN
ejpam-3842	186	15	-	-	PUNCT
ejpam-3842	186	16	nbhd	nbhd	NOUN
ejpam-3842	186	17	of	of	ADP
ejpam-3842	186	18	both	both	PRON
ejpam-3842	186	19	x	x	SYM
ejpam-3842	186	20	and	and	CCONJ
ejpam-3842	186	21	y	y	PROPN
ejpam-3842	186	22	,	,	PUNCT
ejpam-3842	186	23	and	and	CCONJ
ejpam-3842	186	24	v	v	ADP
ejpam-3842	186	25	∗	∗	NOUN
ejpam-3842	186	26	v	v	NOUN
ejpam-3842	186	27	=	=	X
ejpam-3842	186	28	h1	h1	PROPN
ejpam-3842	186	29	∪h2	∪h2	PROPN
ejpam-3842	186	30	.	.	PUNCT
ejpam-3842	187	1	case	case	NOUN
ejpam-3842	187	2	3	3	NUM
ejpam-3842	187	3	:	:	PUNCT
ejpam-3842	187	4	x	x	SYM
ejpam-3842	187	5	∈	∈	PROPN
ejpam-3842	187	6	h1	h1	PROPN
ejpam-3842	187	7	and	and	CCONJ
ejpam-3842	187	8	y	y	PROPN
ejpam-3842	187	9	∈	∈	PROPN
ejpam-3842	187	10	h2	h2	NOUN
ejpam-3842	187	11	then	then	ADV
ejpam-3842	187	12	x	x	X
ejpam-3842	187	13	∗	∗	NOUN
ejpam-3842	187	14	y	y	NOUN
ejpam-3842	187	15	=	=	PUNCT
ejpam-3842	187	16	x	x	PUNCT
ejpam-3842	187	17	∈	∈	PROPN
ejpam-3842	187	18	h1	h1	PROPN
ejpam-3842	187	19	.	.	PUNCT
ejpam-3842	188	1	hence	hence	ADV
ejpam-3842	188	2	,	,	PUNCT
ejpam-3842	188	3	w	w	NOUN
ejpam-3842	188	4	=	=	PUNCT
ejpam-3842	188	5	h1	h1	NOUN
ejpam-3842	188	6	or	or	CCONJ
ejpam-3842	188	7	w	w	NOUN
ejpam-3842	188	8	=	=	PUNCT
ejpam-3842	188	9	h1	h1	PROPN
ejpam-3842	188	10	∪	∪	PROPN
ejpam-3842	188	11	h2	h2	NOUN
ejpam-3842	188	12	.	.	PUNCT
ejpam-3842	189	1	let	let	VERB
ejpam-3842	189	2	vx	vx	INTJ
ejpam-3842	189	3	=	=	PUNCT
ejpam-3842	189	4	h1	h1	PROPN
ejpam-3842	189	5	and	and	CCONJ
ejpam-3842	189	6	vy	vy	VERB
ejpam-3842	189	7	=	=	NOUN
ejpam-3842	189	8	h1	h1	PROPN
ejpam-3842	189	9	∪	∪	PROPN
ejpam-3842	189	10	h2	h2	PROPN
ejpam-3842	189	11	.	.	PUNCT
ejpam-3842	190	1	then	then	ADV
ejpam-3842	190	2	vx	vx	PROPN
ejpam-3842	190	3	and	and	CCONJ
ejpam-3842	190	4	vy	vy	NOUN
ejpam-3842	190	5	are	be	AUX
ejpam-3842	190	6	τh1	τh1	ADJ
ejpam-3842	190	7	-	-	PUNCT
ejpam-3842	190	8	nbhds	nbhds	NOUN
ejpam-3842	190	9	of	of	ADP
ejpam-3842	190	10	x	x	PROPN
ejpam-3842	190	11	and	and	CCONJ
ejpam-3842	190	12	y	y	PROPN
ejpam-3842	190	13	,	,	PUNCT
ejpam-3842	190	14	respectively	respectively	ADV
ejpam-3842	190	15	,	,	PUNCT
ejpam-3842	190	16	and	and	CCONJ
ejpam-3842	190	17	by	by	ADP
ejpam-3842	190	18	lemma	lemma	PROPN
ejpam-3842	190	19	1(ii	1(ii	NUM
ejpam-3842	190	20	)	)	PUNCT
ejpam-3842	190	21	,	,	PUNCT
ejpam-3842	190	22	vx	vx	PROPN
ejpam-3842	190	23	∗	∗	PROPN
ejpam-3842	190	24	vy	vy	X
ejpam-3842	190	25	=	=	X
ejpam-3842	190	26	h1	h1	PROPN
ejpam-3842	190	27	∗	∗	NOUN
ejpam-3842	190	28	(	(	PUNCT
ejpam-3842	190	29	h1	h1	NOUN
ejpam-3842	190	30	∪h2	∪h2	NOUN
ejpam-3842	190	31	)	)	PUNCT
ejpam-3842	191	1	=	=	PRON
ejpam-3842	191	2	h1	h1	VERB
ejpam-3842	191	3	∗1	∗1	PROPN
ejpam-3842	191	4	h1	h1	PROPN
ejpam-3842	191	5	=	=	PUNCT
ejpam-3842	191	6	h1	h1	PROPN
ejpam-3842	191	7	⊂	⊂	PROPN
ejpam-3842	191	8	h1	h1	PROPN
ejpam-3842	191	9	∪h2	∪h2	PROPN
ejpam-3842	191	10	.	.	PUNCT
ejpam-3842	192	1	therefore	therefore	ADV
ejpam-3842	192	2	,	,	PUNCT
ejpam-3842	192	3	(	(	PUNCT
ejpam-3842	192	4	h	h	NOUN
ejpam-3842	192	5	,	,	PUNCT
ejpam-3842	192	6	τh1	τh1	PROPN
ejpam-3842	192	7	)	)	PUNCT
ejpam-3842	192	8	is	be	AUX
ejpam-3842	192	9	a	a	DET
ejpam-3842	192	10	tbch	tbch	ADJ
ejpam-3842	192	11	algebra	algebra	NOUN
ejpam-3842	192	12	.	.	PUNCT
ejpam-3842	193	1	similarly	similarly	ADV
ejpam-3842	193	2	,	,	PUNCT
ejpam-3842	193	3	(	(	PUNCT
ejpam-3842	193	4	h	h	NOUN
ejpam-3842	193	5	,	,	PUNCT
ejpam-3842	193	6	τh2	τh2	PRON
ejpam-3842	193	7	)	)	PUNCT
ejpam-3842	193	8	is	be	AUX
ejpam-3842	193	9	a	a	DET
ejpam-3842	193	10	tbch	tbch	ADJ
ejpam-3842	193	11	algebra	algebra	NOUN
ejpam-3842	193	12	.	.	PUNCT
ejpam-3842	194	1	(	(	PUNCT
ejpam-3842	194	2	iii	iii	X
ejpam-3842	194	3	)	)	PUNCT
ejpam-3842	194	4	suppose	suppose	VERB
ejpam-3842	194	5	a	a	DET
ejpam-3842	194	6	⊆	⊆	NUM
ejpam-3842	194	7	h1	h1	NOUN
ejpam-3842	194	8	,	,	PUNCT
ejpam-3842	194	9	b	b	PROPN
ejpam-3842	194	10	⊆	⊆	NUM
ejpam-3842	194	11	h2	h2	NOUN
ejpam-3842	194	12	,	,	PUNCT
ejpam-3842	194	13	0	0	NUM
ejpam-3842	194	14	∈	∈	PROPN
ejpam-3842	194	15	a	a	DET
ejpam-3842	194	16	∩	∩	ADJ
ejpam-3842	194	17	b	b	NOUN
ejpam-3842	194	18	,	,	PUNCT
ejpam-3842	194	19	and	and	CCONJ
ejpam-3842	194	20	a	a	DET
ejpam-3842	194	21	,	,	PUNCT
ejpam-3842	194	22	b	b	X
ejpam-3842	194	23	∈	∈	PROPN
ejpam-3842	194	24	τ	τ	X
ejpam-3842	194	25	.	.	PUNCT
ejpam-3842	195	1	since	since	SCONJ
ejpam-3842	195	2	h1	h1	PROPN
ejpam-3842	195	3	∩	∩	ADJ
ejpam-3842	195	4	h2	h2	NOUN
ejpam-3842	195	5	=	=	PUNCT
ejpam-3842	195	6	{	{	PUNCT
ejpam-3842	195	7	0	0	NUM
ejpam-3842	195	8	}	}	PUNCT
ejpam-3842	196	1	,	,	PUNCT
ejpam-3842	196	2	it	it	PRON
ejpam-3842	196	3	follows	follow	VERB
ejpam-3842	196	4	that	that	PRON
ejpam-3842	196	5	a∩b	a∩b	VERB
ejpam-3842	196	6	=	=	PUNCT
ejpam-3842	196	7	{	{	PUNCT
ejpam-3842	196	8	0	0	NUM
ejpam-3842	196	9	}	}	PUNCT
ejpam-3842	196	10	.	.	PUNCT
ejpam-3842	197	1	since	since	SCONJ
ejpam-3842	197	2	a	a	DET
ejpam-3842	197	3	,	,	PUNCT
ejpam-3842	197	4	b	b	PROPN
ejpam-3842	197	5	∈	∈	PROPN
ejpam-3842	197	6	τ	τ	X
ejpam-3842	197	7	,	,	PUNCT
ejpam-3842	197	8	{	{	PUNCT
ejpam-3842	197	9	0	0	NUM
ejpam-3842	197	10	}	}	PUNCT
ejpam-3842	197	11	∈	∈	PROPN
ejpam-3842	197	12	τ	τ	X
ejpam-3842	197	13	.	.	PUNCT
ejpam-3842	198	1	thus	thus	ADV
ejpam-3842	198	2	,	,	PUNCT
ejpam-3842	198	3	by	by	ADP
ejpam-3842	198	4	theorem	theorem	NOUN
ejpam-3842	198	5	3	3	NUM
ejpam-3842	198	6	,	,	PUNCT
ejpam-3842	198	7	τ	τ	PROPN
ejpam-3842	198	8	is	be	AUX
ejpam-3842	198	9	the	the	DET
ejpam-3842	198	10	discrete	discrete	ADJ
ejpam-3842	198	11	topology	topology	NOUN
ejpam-3842	198	12	on	on	ADP
ejpam-3842	198	13	h.	h.	PROPN
ejpam-3842	198	14	(	(	PUNCT
ejpam-3842	198	15	iv	iv	X
ejpam-3842	198	16	)	)	PUNCT
ejpam-3842	198	17	suppose	suppose	VERB
ejpam-3842	198	18	that	that	SCONJ
ejpam-3842	198	19	(	(	PUNCT
ejpam-3842	198	20	h	h	NOUN
ejpam-3842	198	21	,	,	PUNCT
ejpam-3842	198	22	τ	τ	X
ejpam-3842	198	23	)	)	PUNCT
ejpam-3842	198	24	is	be	AUX
ejpam-3842	198	25	a	a	DET
ejpam-3842	198	26	tbch	tbch	NOUN
ejpam-3842	198	27	-	-	PUNCT
ejpam-3842	198	28	algebra	algebra	NOUN
ejpam-3842	198	29	and	and	CCONJ
ejpam-3842	198	30	that	that	SCONJ
ejpam-3842	198	31	τ	τ	PROPN
ejpam-3842	198	32	⊆	⊆	NUM
ejpam-3842	198	33	p	p	PROPN
ejpam-3842	198	34	(	(	PUNCT
ejpam-3842	198	35	h1	h1	PROPN
ejpam-3842	198	36	)	)	PUNCT
ejpam-3842	198	37	∪	∪	NOUN
ejpam-3842	198	38	{	{	PUNCT
ejpam-3842	198	39	h1	h1	PROPN
ejpam-3842	198	40	∪	∪	ADJ
ejpam-3842	198	41	h2	h2	NOUN
ejpam-3842	198	42	}	}	PUNCT
ejpam-3842	198	43	.	.	PUNCT
ejpam-3842	199	1	let	let	VERB
ejpam-3842	199	2	w	w	NOUN
ejpam-3842	199	3	∈	∈	PROPN
ejpam-3842	199	4	τ	τ	X
ejpam-3842	199	5	\	\	X
ejpam-3842	199	6	{	{	PUNCT
ejpam-3842	199	7	∅	∅	NOUN
ejpam-3842	199	8	}	}	PUNCT
ejpam-3842	199	9	.	.	PUNCT
ejpam-3842	200	1	pick	pick	VERB
ejpam-3842	200	2	any	any	DET
ejpam-3842	200	3	x	x	SYM
ejpam-3842	200	4	∈	∈	PROPN
ejpam-3842	200	5	w	w	NOUN
ejpam-3842	200	6	and	and	CCONJ
ejpam-3842	200	7	y	y	PROPN
ejpam-3842	200	8	∈	∈	PROPN
ejpam-3842	200	9	h2	h2	PROPN
ejpam-3842	200	10	.	.	PUNCT
ejpam-3842	201	1	since	since	SCONJ
ejpam-3842	201	2	x	x	PROPN
ejpam-3842	201	3	∗	∗	NOUN
ejpam-3842	201	4	y	y	NOUN
ejpam-3842	201	5	=	=	SYM
ejpam-3842	201	6	x	x	PROPN
ejpam-3842	201	7	,	,	PUNCT
ejpam-3842	201	8	w	w	PROPN
ejpam-3842	201	9	is	be	AUX
ejpam-3842	201	10	a	a	DET
ejpam-3842	201	11	nbhd	nbhd	NOUN
ejpam-3842	201	12	of	of	ADP
ejpam-3842	201	13	x	x	PROPN
ejpam-3842	201	14	∗	∗	NOUN
ejpam-3842	201	15	y.	y.	NOUN
ejpam-3842	201	16	by	by	ADP
ejpam-3842	201	17	continuity	continuity	NOUN
ejpam-3842	201	18	of	of	ADP
ejpam-3842	201	19	∗	∗	NOUN
ejpam-3842	201	20	,	,	PUNCT
ejpam-3842	201	21	there	there	PRON
ejpam-3842	201	22	exist	exist	VERB
ejpam-3842	201	23	nbhds	nbhds	NOUN
ejpam-3842	201	24	vx	vx	PROPN
ejpam-3842	201	25	and	and	CCONJ
ejpam-3842	201	26	vy	vy	X
ejpam-3842	201	27	of	of	ADP
ejpam-3842	201	28	x	x	PROPN
ejpam-3842	201	29	and	and	CCONJ
ejpam-3842	201	30	y	y	PROPN
ejpam-3842	201	31	,	,	PUNCT
ejpam-3842	201	32	respectively	respectively	ADV
ejpam-3842	201	33	,	,	PUNCT
ejpam-3842	201	34	such	such	ADJ
ejpam-3842	201	35	that	that	SCONJ
ejpam-3842	201	36	vx∗vy	vx∗vy	PROPN
ejpam-3842	201	37	⊆w	⊆w	NOUN
ejpam-3842	201	38	.	.	PUNCT
ejpam-3842	202	1	now	now	ADV
ejpam-3842	202	2	,	,	PUNCT
ejpam-3842	202	3	since	since	SCONJ
ejpam-3842	202	4	τ	τ	X
ejpam-3842	202	5	⊆	⊆	NUM
ejpam-3842	202	6	p	p	PROPN
ejpam-3842	202	7	(	(	PUNCT
ejpam-3842	202	8	h1	h1	PROPN
ejpam-3842	202	9	)	)	PUNCT
ejpam-3842	202	10	∪	∪	NOUN
ejpam-3842	202	11	{	{	PUNCT
ejpam-3842	202	12	h1	h1	NOUN
ejpam-3842	202	13	∪h2	∪h2	NOUN
ejpam-3842	202	14	}	}	PUNCT
ejpam-3842	202	15	,	,	PUNCT
ejpam-3842	202	16	the	the	DET
ejpam-3842	202	17	only	only	ADJ
ejpam-3842	202	18	nbhd	nbhd	NOUN
ejpam-3842	202	19	of	of	ADP
ejpam-3842	202	20	y	y	PROPN
ejpam-3842	202	21	is	be	AUX
ejpam-3842	202	22	h1	h1	PROPN
ejpam-3842	202	23	∪h2	∪h2	NOUN
ejpam-3842	202	24	.	.	PUNCT
ejpam-3842	203	1	hence	hence	ADV
ejpam-3842	203	2	,	,	PUNCT
ejpam-3842	203	3	vy	vy	NOUN
ejpam-3842	203	4	=	=	PUNCT
ejpam-3842	203	5	h1	h1	PROPN
ejpam-3842	203	6	∪h2	∪h2	NOUN
ejpam-3842	203	7	and	and	CCONJ
ejpam-3842	203	8	by	by	ADP
ejpam-3842	203	9	lemma	lemma	PROPN
ejpam-3842	203	10	1(ii	1(ii	NUM
ejpam-3842	203	11	)	)	PUNCT
ejpam-3842	203	12	,	,	PUNCT
ejpam-3842	203	13	vx	vx	PROPN
ejpam-3842	203	14	∗	∗	NOUN
ejpam-3842	203	15	vy	vy	NOUN
ejpam-3842	203	16	=	=	SYM
ejpam-3842	203	17	vx	vx	PROPN
ejpam-3842	203	18	∗	∗	NOUN
ejpam-3842	203	19	(	(	PUNCT
ejpam-3842	203	20	h1	h1	PROPN
ejpam-3842	203	21	∪h2	∪h2	NOUN
ejpam-3842	203	22	)	)	PUNCT
ejpam-3842	203	23	=	=	SYM
ejpam-3842	203	24	vx	vx	PROPN
ejpam-3842	203	25	∗1h1	∗1h1	PROPN
ejpam-3842	203	26	.	.	PUNCT
ejpam-3842	204	1	since	since	SCONJ
ejpam-3842	204	2	x	x	PROPN
ejpam-3842	204	3	∈	∈	PROPN
ejpam-3842	204	4	h1	h1	PROPN
ejpam-3842	204	5	,	,	PUNCT
ejpam-3842	204	6	x	x	PUNCT
ejpam-3842	204	7	∗1	∗1	PUNCT
ejpam-3842	205	1	x	x	PUNCT
ejpam-3842	205	2	=	=	PUNCT
ejpam-3842	205	3	x	x	SYM
ejpam-3842	205	4	∗	∗	NOUN
ejpam-3842	205	5	x	x	X
ejpam-3842	205	6	=	=	SYM
ejpam-3842	205	7	0	0	NUM
ejpam-3842	205	8	∈	∈	PROPN
ejpam-3842	205	9	vx	vx	ADP
ejpam-3842	205	10	∗1	∗1	PROPN
ejpam-3842	205	11	h1	h1	PROPN
ejpam-3842	205	12	.	.	PUNCT
ejpam-3842	206	1	therefore	therefore	ADV
ejpam-3842	206	2	,	,	PUNCT
ejpam-3842	206	3	0	0	NUM
ejpam-3842	206	4	∈w	∈w	NOUN
ejpam-3842	206	5	.	.	PUNCT
ejpam-3842	207	1	theorem	theorem	NOUN
ejpam-3842	207	2	7	7	NUM
ejpam-3842	207	3	.	.	PUNCT
ejpam-3842	208	1	let	let	VERB
ejpam-3842	208	2	x	x	PRON
ejpam-3842	208	3	be	be	AUX
ejpam-3842	208	4	a	a	DET
ejpam-3842	208	5	tbch	tbch	NOUN
ejpam-3842	208	6	-	-	PUNCT
ejpam-3842	208	7	algebra	algebra	NOUN
ejpam-3842	208	8	.	.	PUNCT
ejpam-3842	209	1	then	then	ADV
ejpam-3842	209	2	{	{	PUNCT
ejpam-3842	209	3	0	0	X
ejpam-3842	209	4	}	}	PUNCT
ejpam-3842	209	5	is	be	AUX
ejpam-3842	209	6	a	a	DET
ejpam-3842	209	7	closed	closed	ADJ
ejpam-3842	209	8	set	set	NOUN
ejpam-3842	209	9	in	in	ADP
ejpam-3842	209	10	x	x	PUNCT
ejpam-3842	209	11	if	if	SCONJ
ejpam-3842	210	1	and	and	CCONJ
ejpam-3842	210	2	only	only	ADV
ejpam-3842	210	3	if	if	SCONJ
ejpam-3842	210	4	x	x	PRON
ejpam-3842	210	5	is	be	AUX
ejpam-3842	210	6	a	a	DET
ejpam-3842	210	7	t2	t2	NOUN
ejpam-3842	210	8	-	-	PUNCT
ejpam-3842	210	9	space	space	NOUN
ejpam-3842	210	10	.	.	PUNCT
ejpam-3842	211	1	proof	proof	NOUN
ejpam-3842	211	2	.	.	PUNCT
ejpam-3842	212	1	suppose	suppose	VERB
ejpam-3842	212	2	{	{	PUNCT
ejpam-3842	212	3	0	0	NUM
ejpam-3842	212	4	}	}	PUNCT
ejpam-3842	212	5	is	be	AUX
ejpam-3842	212	6	a	a	DET
ejpam-3842	212	7	closed	closed	ADJ
ejpam-3842	212	8	set	set	NOUN
ejpam-3842	212	9	in	in	ADP
ejpam-3842	212	10	x.	x.	NOUN
ejpam-3842	212	11	let	let	VERB
ejpam-3842	212	12	x	x	PRON
ejpam-3842	212	13	,	,	PUNCT
ejpam-3842	212	14	y	y	PROPN
ejpam-3842	212	15	∈	∈	PROPN
ejpam-3842	212	16	x	x	PUNCT
ejpam-3842	212	17	with	with	ADP
ejpam-3842	212	18	x	x	SYM
ejpam-3842	212	19	6=	6=	ADP
ejpam-3842	212	20	y.	y.	NOUN
ejpam-3842	212	21	then	then	ADV
ejpam-3842	212	22	,	,	PUNCT
ejpam-3842	212	23	x	x	X
ejpam-3842	212	24	∗	∗	NOUN
ejpam-3842	212	25	y	y	PROPN
ejpam-3842	212	26	6=	6=	ADP
ejpam-3842	212	27	0	0	NUM
ejpam-3842	212	28	or	or	CCONJ
ejpam-3842	212	29	y	y	PROPN
ejpam-3842	212	30	∗x	∗x	PROPN
ejpam-3842	212	31	6=	6=	ADP
ejpam-3842	212	32	0	0	NUM
ejpam-3842	212	33	.	.	PUNCT
ejpam-3842	212	34	without	without	ADP
ejpam-3842	212	35	loss	loss	NOUN
ejpam-3842	212	36	of	of	ADP
ejpam-3842	212	37	generality	generality	NOUN
ejpam-3842	212	38	,	,	PUNCT
ejpam-3842	212	39	assume	assume	VERB
ejpam-3842	212	40	that	that	SCONJ
ejpam-3842	212	41	x	x	PROPN
ejpam-3842	212	42	∗	∗	VERB
ejpam-3842	212	43	y	y	PROPN
ejpam-3842	212	44	6=	6=	PROPN
ejpam-3842	212	45	0	0	NUM
ejpam-3842	212	46	.	.	PUNCT
ejpam-3842	212	47	note	note	VERB
ejpam-3842	212	48	that	that	SCONJ
ejpam-3842	212	49	x	x	PROPN
ejpam-3842	212	50	∗	∗	VERB
ejpam-3842	212	51	y	y	NOUN
ejpam-3842	212	52	∈	∈	PROPN
ejpam-3842	212	53	x	x	SYM
ejpam-3842	212	54	\	\	X
ejpam-3842	212	55	{	{	PUNCT
ejpam-3842	212	56	0	0	NUM
ejpam-3842	212	57	}	}	PUNCT
ejpam-3842	212	58	.	.	PUNCT
ejpam-3842	213	1	by	by	ADP
ejpam-3842	213	2	theorem	theorem	NOUN
ejpam-3842	213	3	2	2	NUM
ejpam-3842	213	4	,	,	PUNCT
ejpam-3842	213	5	there	there	PRON
ejpam-3842	213	6	exist	exist	VERB
ejpam-3842	213	7	nbhds	nbhds	ADJ
ejpam-3842	213	8	u	u	NOUN
ejpam-3842	213	9	and	and	CCONJ
ejpam-3842	213	10	v	v	NOUN
ejpam-3842	213	11	of	of	ADP
ejpam-3842	213	12	x	x	PROPN
ejpam-3842	213	13	and	and	CCONJ
ejpam-3842	213	14	y	y	PROPN
ejpam-3842	213	15	,	,	PUNCT
ejpam-3842	213	16	respectively	respectively	ADV
ejpam-3842	213	17	,	,	PUNCT
ejpam-3842	213	18	such	such	ADJ
ejpam-3842	213	19	that	that	SCONJ
ejpam-3842	213	20	u	u	PRON
ejpam-3842	213	21	∗v	∗v	NOUN
ejpam-3842	213	22	⊆	⊆	NUM
ejpam-3842	213	23	x	x	SYM
ejpam-3842	213	24	\{0	\{0	NOUN
ejpam-3842	213	25	}	}	PUNCT
ejpam-3842	213	26	.	.	PUNCT
ejpam-3842	214	1	suppose	suppose	VERB
ejpam-3842	214	2	u	u	NOUN
ejpam-3842	214	3	∩	∩	NOUN
ejpam-3842	214	4	v	v	ADP
ejpam-3842	214	5	6=	6=	PROPN
ejpam-3842	214	6	∅.	∅.	AUX
ejpam-3842	214	7	let	let	VERB
ejpam-3842	214	8	z	z	NOUN
ejpam-3842	214	9	∈	∈	PROPN
ejpam-3842	214	10	u	u	NOUN
ejpam-3842	214	11	∩	∩	NOUN
ejpam-3842	214	12	v	v	NOUN
ejpam-3842	214	13	.	.	PUNCT
ejpam-3842	215	1	then	then	ADV
ejpam-3842	215	2	,	,	PUNCT
ejpam-3842	215	3	z	z	PROPN
ejpam-3842	215	4	∈	∈	PROPN
ejpam-3842	215	5	u	u	NOUN
ejpam-3842	215	6	and	and	CCONJ
ejpam-3842	215	7	z	z	NOUN
ejpam-3842	215	8	∈	∈	PROPN
ejpam-3842	215	9	v	v	NOUN
ejpam-3842	215	10	.	.	PUNCT
ejpam-3842	216	1	hence	hence	ADV
ejpam-3842	216	2	,	,	PUNCT
ejpam-3842	216	3	by	by	ADP
ejpam-3842	216	4	(	(	PUNCT
ejpam-3842	216	5	b1	b1	NOUN
ejpam-3842	216	6	)	)	PUNCT
ejpam-3842	216	7	0	0	NUM
ejpam-3842	217	1	=	=	SYM
ejpam-3842	217	2	z	z	NOUN
ejpam-3842	217	3	∗	∗	NOUN
ejpam-3842	217	4	z	z	X
ejpam-3842	217	5	∈	∈	PROPN
ejpam-3842	217	6	u	u	NOUN
ejpam-3842	217	7	∗	∗	NOUN
ejpam-3842	217	8	v	v	NOUN
ejpam-3842	217	9	⊆	⊆	NUM
ejpam-3842	217	10	x	x	X
ejpam-3842	217	11	\	\	X
ejpam-3842	217	12	{	{	PUNCT
ejpam-3842	217	13	0	0	NUM
ejpam-3842	217	14	}	}	PUNCT
ejpam-3842	217	15	a	a	DET
ejpam-3842	217	16	contradiction	contradiction	NOUN
ejpam-3842	217	17	.	.	PUNCT
ejpam-3842	218	1	thus	thus	ADV
ejpam-3842	218	2	,	,	PUNCT
ejpam-3842	218	3	u	u	PROPN
ejpam-3842	218	4	∩	∩	NOUN
ejpam-3842	218	5	v	v	NOUN
ejpam-3842	218	6	=	=	NOUN
ejpam-3842	218	7	∅	∅	NOUN
ejpam-3842	218	8	and	and	CCONJ
ejpam-3842	218	9	so	so	ADV
ejpam-3842	218	10	x	x	PRON
ejpam-3842	218	11	is	be	AUX
ejpam-3842	218	12	a	a	DET
ejpam-3842	218	13	t2	t2	NOUN
ejpam-3842	218	14	-	-	PUNCT
ejpam-3842	218	15	space	space	NOUN
ejpam-3842	218	16	.	.	PUNCT
ejpam-3842	219	1	conversely	conversely	ADV
ejpam-3842	219	2	,	,	PUNCT
ejpam-3842	219	3	assume	assume	VERB
ejpam-3842	219	4	that	that	SCONJ
ejpam-3842	219	5	x	x	PRON
ejpam-3842	219	6	is	be	AUX
ejpam-3842	219	7	a	a	DET
ejpam-3842	219	8	t2	t2	NOUN
ejpam-3842	219	9	-	-	PUNCT
ejpam-3842	219	10	space	space	NOUN
ejpam-3842	219	11	.	.	PUNCT
ejpam-3842	220	1	let	let	VERB
ejpam-3842	220	2	x	x	SYM
ejpam-3842	220	3	∈	∈	PROPN
ejpam-3842	220	4	x	x	SYM
ejpam-3842	220	5	\	\	X
ejpam-3842	220	6	{	{	PUNCT
ejpam-3842	220	7	0	0	NUM
ejpam-3842	220	8	}	}	PUNCT
ejpam-3842	220	9	.	.	PUNCT
ejpam-3842	221	1	then	then	ADV
ejpam-3842	221	2	,	,	PUNCT
ejpam-3842	221	3	there	there	PRON
ejpam-3842	221	4	exist	exist	VERB
ejpam-3842	221	5	nbhds	nbhds	ADJ
ejpam-3842	221	6	u	u	NOUN
ejpam-3842	221	7	and	and	CCONJ
ejpam-3842	221	8	v	v	NOUN
ejpam-3842	221	9	of	of	ADP
ejpam-3842	221	10	x	x	X
ejpam-3842	221	11	and	and	CCONJ
ejpam-3842	221	12	0	0	NUM
ejpam-3842	221	13	,	,	PUNCT
ejpam-3842	221	14	respectively	respectively	ADV
ejpam-3842	221	15	,	,	PUNCT
ejpam-3842	221	16	such	such	ADJ
ejpam-3842	221	17	that	that	SCONJ
ejpam-3842	221	18	u	u	PROPN
ejpam-3842	221	19	∩	∩	NOUN
ejpam-3842	221	20	v	v	NOUN
ejpam-3842	221	21	=	=	PUNCT
ejpam-3842	221	22	∅.	∅.	NOUN
ejpam-3842	221	23	since	since	SCONJ
ejpam-3842	221	24	0	0	NUM
ejpam-3842	221	25	/∈	/∈	NUM
ejpam-3842	221	26	u	u	NOUN
ejpam-3842	221	27	,	,	PUNCT
ejpam-3842	221	28	x	x	PROPN
ejpam-3842	221	29	∈	∈	PROPN
ejpam-3842	221	30	u	u	NOUN
ejpam-3842	221	31	⊆	⊆	NUM
ejpam-3842	221	32	x	x	SYM
ejpam-3842	221	33	\	\	X
ejpam-3842	221	34	{	{	PUNCT
ejpam-3842	221	35	0	0	NUM
ejpam-3842	221	36	}	}	PUNCT
ejpam-3842	221	37	.	.	PUNCT
ejpam-3842	222	1	this	this	PRON
ejpam-3842	222	2	shows	show	VERB
ejpam-3842	222	3	that	that	SCONJ
ejpam-3842	222	4	x	x	SYM
ejpam-3842	222	5	\	\	X
ejpam-3842	222	6	{	{	PUNCT
ejpam-3842	222	7	0	0	NUM
ejpam-3842	222	8	}	}	PUNCT
ejpam-3842	222	9	is	be	AUX
ejpam-3842	222	10	open	open	ADJ
ejpam-3842	222	11	in	in	ADP
ejpam-3842	222	12	x.	x.	NOUN
ejpam-3842	222	13	therefore	therefore	ADV
ejpam-3842	222	14	,	,	PUNCT
ejpam-3842	222	15	{	{	PUNCT
ejpam-3842	222	16	0	0	X
ejpam-3842	222	17	}	}	PUNCT
ejpam-3842	222	18	is	be	AUX
ejpam-3842	222	19	a	a	DET
ejpam-3842	222	20	closed	closed	ADJ
ejpam-3842	222	21	set	set	NOUN
ejpam-3842	222	22	in	in	ADP
ejpam-3842	222	23	x.	x.	NOUN
ejpam-3842	222	24	the	the	DET
ejpam-3842	222	25	next	next	ADJ
ejpam-3842	222	26	theorem	theorem	NOUN
ejpam-3842	222	27	asserts	assert	VERB
ejpam-3842	222	28	that	that	SCONJ
ejpam-3842	222	29	t0	t0	NOUN
ejpam-3842	222	30	,	,	PUNCT
ejpam-3842	222	31	t1	t1	NOUN
ejpam-3842	222	32	and	and	CCONJ
ejpam-3842	222	33	t2	t2	PROPN
ejpam-3842	222	34	topological	topological	ADJ
ejpam-3842	222	35	spaces	space	NOUN
ejpam-3842	222	36	are	be	AUX
ejpam-3842	222	37	equivalent	equivalent	ADJ
ejpam-3842	222	38	in	in	ADP
ejpam-3842	222	39	a	a	DET
ejpam-3842	222	40	tbch	tbch	NOUN
ejpam-3842	222	41	-	-	PUNCT
ejpam-3842	222	42	algebra	algebra	NOUN
ejpam-3842	222	43	.	.	PUNCT
ejpam-3842	223	1	theorem	theorem	ADJ
ejpam-3842	223	2	8	8	NUM
ejpam-3842	223	3	.	.	PUNCT
ejpam-3842	224	1	let	let	VERB
ejpam-3842	224	2	x	x	PRON
ejpam-3842	224	3	be	be	AUX
ejpam-3842	224	4	a	a	DET
ejpam-3842	224	5	tbch	tbch	NOUN
ejpam-3842	224	6	-	-	PUNCT
ejpam-3842	224	7	algebra	algebra	NOUN
ejpam-3842	224	8	.	.	PUNCT
ejpam-3842	225	1	then	then	ADV
ejpam-3842	225	2	the	the	DET
ejpam-3842	225	3	following	follow	VERB
ejpam-3842	225	4	statements	statement	NOUN
ejpam-3842	225	5	are	be	AUX
ejpam-3842	225	6	equivalent	equivalent	ADJ
ejpam-3842	225	7	:	:	PUNCT
ejpam-3842	225	8	(	(	PUNCT
ejpam-3842	225	9	i	i	NOUN
ejpam-3842	225	10	)	)	PUNCT
ejpam-3842	225	11	x	x	X
ejpam-3842	225	12	is	be	AUX
ejpam-3842	225	13	a	a	DET
ejpam-3842	225	14	t0	t0	NOUN
ejpam-3842	225	15	-	-	NOUN
ejpam-3842	225	16	space	space	NOUN
ejpam-3842	225	17	(	(	PUNCT
ejpam-3842	225	18	ii	ii	NOUN
ejpam-3842	225	19	)	)	PUNCT
ejpam-3842	225	20	x	x	X
ejpam-3842	225	21	is	be	AUX
ejpam-3842	225	22	a	a	DET
ejpam-3842	225	23	t1	t1	NOUN
ejpam-3842	225	24	-	-	PUNCT
ejpam-3842	225	25	space	space	NOUN
ejpam-3842	225	26	(	(	PUNCT
ejpam-3842	225	27	iii	iii	NOUN
ejpam-3842	225	28	)	)	PUNCT
ejpam-3842	225	29	x	x	X
ejpam-3842	225	30	is	be	AUX
ejpam-3842	225	31	a	a	DET
ejpam-3842	225	32	t2	t2	NOUN
ejpam-3842	225	33	-	-	PUNCT
ejpam-3842	225	34	space	space	NOUN
ejpam-3842	225	35	.	.	PUNCT
ejpam-3842	226	1	j.	j.	PROPN
ejpam-3842	226	2	mancao	mancao	PROPN
ejpam-3842	226	3	,	,	PUNCT
ejpam-3842	226	4	s.	s.	PROPN
ejpam-3842	226	5	canoy	canoy	PROPN
ejpam-3842	226	6	/	/	SYM
ejpam-3842	226	7	eur	eur	PROPN
ejpam-3842	226	8	.	.	PUNCT
ejpam-3842	227	1	j.	j.	PROPN
ejpam-3842	227	2	pure	pure	PROPN
ejpam-3842	227	3	appl	appl	PROPN
ejpam-3842	227	4	.	.	PROPN
ejpam-3842	227	5	math	math	PROPN
ejpam-3842	227	6	,	,	PUNCT
ejpam-3842	227	7	13	13	NUM
ejpam-3842	227	8	(	(	PUNCT
ejpam-3842	227	9	4	4	NUM
ejpam-3842	227	10	)	)	PUNCT
ejpam-3842	227	11	(	(	PUNCT
ejpam-3842	227	12	2020	2020	NUM
ejpam-3842	227	13	)	)	PUNCT
ejpam-3842	227	14	,	,	PUNCT
ejpam-3842	227	15	730	730	NUM
ejpam-3842	227	16	-	-	SYM
ejpam-3842	227	17	738	738	NUM
ejpam-3842	227	18	737	737	NUM
ejpam-3842	227	19	proof	proof	NOUN
ejpam-3842	227	20	.	.	PUNCT
ejpam-3842	228	1	(	(	PUNCT
ejpam-3842	228	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-3842	228	3	):	):	PUNCT
ejpam-3842	228	4	suppose	suppose	VERB
ejpam-3842	228	5	x	x	PRON
ejpam-3842	228	6	is	be	AUX
ejpam-3842	228	7	a	a	DET
ejpam-3842	228	8	t0	t0	NOUN
ejpam-3842	228	9	-	-	NOUN
ejpam-3842	228	10	space	space	NOUN
ejpam-3842	228	11	.	.	PUNCT
ejpam-3842	229	1	let	let	VERB
ejpam-3842	229	2	x	x	PRON
ejpam-3842	229	3	,	,	PUNCT
ejpam-3842	229	4	y	y	PROPN
ejpam-3842	229	5	∈	∈	PROPN
ejpam-3842	229	6	x	x	PUNCT
ejpam-3842	229	7	with	with	ADP
ejpam-3842	229	8	x	x	SYM
ejpam-3842	229	9	6=	6=	ADP
ejpam-3842	229	10	y.	y.	NOUN
ejpam-3842	229	11	then	then	ADV
ejpam-3842	229	12	x	x	X
ejpam-3842	229	13	∗	∗	PROPN
ejpam-3842	229	14	y	y	PROPN
ejpam-3842	229	15	6=	6=	ADP
ejpam-3842	229	16	0	0	NUM
ejpam-3842	229	17	or	or	CCONJ
ejpam-3842	229	18	y	y	PROPN
ejpam-3842	229	19	∗	∗	NOUN
ejpam-3842	229	20	x	x	PUNCT
ejpam-3842	229	21	6=	6=	ADP
ejpam-3842	229	22	0	0	NUM
ejpam-3842	229	23	by	by	ADP
ejpam-3842	229	24	(	(	PUNCT
ejpam-3842	229	25	b2	b2	NOUN
ejpam-3842	229	26	)	)	PUNCT
ejpam-3842	229	27	.	.	PUNCT
ejpam-3842	230	1	without	without	ADP
ejpam-3842	230	2	loss	loss	NOUN
ejpam-3842	230	3	of	of	ADP
ejpam-3842	230	4	generality	generality	NOUN
ejpam-3842	230	5	,	,	PUNCT
ejpam-3842	230	6	assume	assume	VERB
ejpam-3842	230	7	that	that	SCONJ
ejpam-3842	230	8	x	x	PROPN
ejpam-3842	230	9	∗	∗	VERB
ejpam-3842	230	10	y	y	PROPN
ejpam-3842	230	11	6=	6=	PROPN
ejpam-3842	230	12	0	0	NUM
ejpam-3842	230	13	.	.	PUNCT
ejpam-3842	231	1	since	since	SCONJ
ejpam-3842	231	2	x	x	PRON
ejpam-3842	231	3	is	be	AUX
ejpam-3842	231	4	a	a	DET
ejpam-3842	231	5	t0	t0	NOUN
ejpam-3842	231	6	-	-	NOUN
ejpam-3842	231	7	space	space	NOUN
ejpam-3842	231	8	,	,	PUNCT
ejpam-3842	231	9	there	there	PRON
ejpam-3842	231	10	exists	exist	VERB
ejpam-3842	231	11	an	an	DET
ejpam-3842	231	12	open	open	ADJ
ejpam-3842	231	13	set	set	NOUN
ejpam-3842	231	14	u	u	PRON
ejpam-3842	231	15	such	such	ADJ
ejpam-3842	231	16	that	that	SCONJ
ejpam-3842	231	17	x∗y	x∗y	PUNCT
ejpam-3842	231	18	∈	∈	PROPN
ejpam-3842	231	19	u	u	NOUN
ejpam-3842	231	20	but	but	CCONJ
ejpam-3842	231	21	0	0	NUM
ejpam-3842	231	22	/∈	/∈	NUM
ejpam-3842	231	23	u	u	NOUN
ejpam-3842	231	24	or	or	CCONJ
ejpam-3842	231	25	0	0	NUM
ejpam-3842	231	26	∈	∈	NOUN
ejpam-3842	231	27	u	u	NOUN
ejpam-3842	231	28	but	but	CCONJ
ejpam-3842	231	29	x∗y	x∗y	NUM
ejpam-3842	231	30	/∈	/∈	PUNCT
ejpam-3842	232	1	u	u	INTJ
ejpam-3842	232	2	.	.	PUNCT
ejpam-3842	233	1	consider	consider	VERB
ejpam-3842	233	2	the	the	DET
ejpam-3842	233	3	following	follow	VERB
ejpam-3842	233	4	cases	case	NOUN
ejpam-3842	233	5	:	:	PUNCT
ejpam-3842	233	6	case	case	NOUN
ejpam-3842	233	7	1	1	NUM
ejpam-3842	233	8	.	.	PUNCT
ejpam-3842	233	9	x	x	SYM
ejpam-3842	233	10	∗	∗	NOUN
ejpam-3842	233	11	y	y	PROPN
ejpam-3842	233	12	∈	∈	PROPN
ejpam-3842	233	13	u	u	NOUN
ejpam-3842	233	14	(	(	PUNCT
ejpam-3842	233	15	but	but	CCONJ
ejpam-3842	233	16	0	0	NUM
ejpam-3842	233	17	/∈	/∈	NUM
ejpam-3842	233	18	u	u	NOUN
ejpam-3842	233	19	)	)	PUNCT
ejpam-3842	233	20	by	by	ADP
ejpam-3842	233	21	theorem	theorem	NOUN
ejpam-3842	233	22	2	2	NUM
ejpam-3842	233	23	,	,	PUNCT
ejpam-3842	233	24	there	there	PRON
ejpam-3842	233	25	exist	exist	VERB
ejpam-3842	233	26	nbhdsgx	nbhdsgx	ADJ
ejpam-3842	233	27	andhy	andhy	NOUN
ejpam-3842	233	28	of	of	ADP
ejpam-3842	233	29	x	x	PROPN
ejpam-3842	233	30	and	and	CCONJ
ejpam-3842	233	31	y	y	PROPN
ejpam-3842	233	32	,	,	PUNCT
ejpam-3842	233	33	respectively	respectively	ADV
ejpam-3842	233	34	,	,	PUNCT
ejpam-3842	233	35	such	such	ADJ
ejpam-3842	233	36	thatgx∗hy	thatgx∗hy	PROPN
ejpam-3842	233	37	⊆	⊆	NUM
ejpam-3842	233	38	u	u	NOUN
ejpam-3842	233	39	.	.	PUNCT
ejpam-3842	234	1	since	since	SCONJ
ejpam-3842	234	2	0	0	NUM
ejpam-3842	234	3	/∈	/∈	NUM
ejpam-3842	234	4	u	u	PROPN
ejpam-3842	234	5	,	,	PUNCT
ejpam-3842	234	6	0	0	PROPN
ejpam-3842	234	7	/∈	/∈	PUNCT
ejpam-3842	234	8	gx	gx	PROPN
ejpam-3842	234	9	∗hy	∗hy	NOUN
ejpam-3842	234	10	.	.	PUNCT
ejpam-3842	235	1	by	by	ADP
ejpam-3842	235	2	remark	remark	NOUN
ejpam-3842	235	3	4	4	NUM
ejpam-3842	235	4	,	,	PUNCT
ejpam-3842	235	5	gx	gx	PROPN
ejpam-3842	235	6	∩hy	∩hy	NOUN
ejpam-3842	235	7	=	=	PROPN
ejpam-3842	235	8	∅.	∅.	VERB
ejpam-3842	235	9	thus	thus	ADV
ejpam-3842	235	10	,	,	PUNCT
ejpam-3842	235	11	y	y	PROPN
ejpam-3842	235	12	/∈	/∈	PUNCT
ejpam-3842	235	13	gx	gx	PROPN
ejpam-3842	235	14	and	and	CCONJ
ejpam-3842	235	15	x	x	ADJ
ejpam-3842	235	16	/∈	/∈	PUNCT
ejpam-3842	235	17	hy	hy	INTJ
ejpam-3842	235	18	.	.	PUNCT
ejpam-3842	235	19	case	case	NOUN
ejpam-3842	235	20	2	2	NUM
ejpam-3842	235	21	.	.	NOUN
ejpam-3842	235	22	0	0	NUM
ejpam-3842	236	1	∈	∈	PROPN
ejpam-3842	236	2	u	u	NOUN
ejpam-3842	236	3	(	(	PUNCT
ejpam-3842	236	4	but	but	CCONJ
ejpam-3842	236	5	x	x	X
ejpam-3842	236	6	∗	∗	NOUN
ejpam-3842	236	7	y	y	PROPN
ejpam-3842	236	8	/∈	/∈	PUNCT
ejpam-3842	236	9	u	u	NOUN
ejpam-3842	236	10	)	)	PUNCT
ejpam-3842	236	11	.	.	PUNCT
ejpam-3842	237	1	by	by	ADP
ejpam-3842	237	2	(	(	PUNCT
ejpam-3842	237	3	b1	b1	NOUN
ejpam-3842	237	4	)	)	PUNCT
ejpam-3842	237	5	,	,	PUNCT
ejpam-3842	237	6	x	x	X
ejpam-3842	237	7	∗	∗	NOUN
ejpam-3842	237	8	x	x	X
ejpam-3842	237	9	=	=	SYM
ejpam-3842	237	10	0	0	NUM
ejpam-3842	237	11	∈	∈	PROPN
ejpam-3842	237	12	u	u	NOUN
ejpam-3842	237	13	.	.	PUNCT
ejpam-3842	238	1	by	by	ADP
ejpam-3842	238	2	theorem	theorem	NOUN
ejpam-3842	238	3	2	2	NUM
ejpam-3842	238	4	,	,	PUNCT
ejpam-3842	238	5	there	there	PRON
ejpam-3842	238	6	exist	exist	VERB
ejpam-3842	238	7	nbhds	nbhds	ADV
ejpam-3842	238	8	nx	nx	PROPN
ejpam-3842	238	9	and	and	CCONJ
ejpam-3842	238	10	mx	mx	PROPN
ejpam-3842	238	11	of	of	ADP
ejpam-3842	238	12	x	x	INTJ
ejpam-3842	239	1	such	such	ADJ
ejpam-3842	239	2	that	that	SCONJ
ejpam-3842	239	3	nx	nx	PROPN
ejpam-3842	239	4	∗mx	∗mx	PROPN
ejpam-3842	239	5	⊆	⊆	NUM
ejpam-3842	239	6	u	u	NOUN
ejpam-3842	239	7	.	.	PUNCT
ejpam-3842	240	1	since	since	SCONJ
ejpam-3842	240	2	x	x	PROPN
ejpam-3842	240	3	∗	∗	PROPN
ejpam-3842	240	4	y	y	PROPN
ejpam-3842	240	5	/∈	/∈	PUNCT
ejpam-3842	240	6	u	u	PROPN
ejpam-3842	240	7	,	,	PUNCT
ejpam-3842	240	8	x	x	PROPN
ejpam-3842	240	9	∗	∗	NOUN
ejpam-3842	240	10	y	y	PROPN
ejpam-3842	240	11	/∈	/∈	PUNCT
ejpam-3842	240	12	nx	nx	PROPN
ejpam-3842	240	13	∗mx	∗mx	PROPN
ejpam-3842	240	14	.	.	PUNCT
ejpam-3842	241	1	it	it	PRON
ejpam-3842	241	2	follows	follow	VERB
ejpam-3842	241	3	that	that	SCONJ
ejpam-3842	242	1	y	y	PROPN
ejpam-3842	242	2	/∈	/∈	PUNCT
ejpam-3842	242	3	mx	mx	PROPN
ejpam-3842	242	4	.	.	PROPN
ejpam-3842	242	5	similarly	similarly	ADV
ejpam-3842	242	6	,	,	PUNCT
ejpam-3842	242	7	since	since	SCONJ
ejpam-3842	242	8	y	y	PROPN
ejpam-3842	242	9	∗	∗	NOUN
ejpam-3842	242	10	y	y	NOUN
ejpam-3842	242	11	=	=	SYM
ejpam-3842	242	12	0	0	NUM
ejpam-3842	242	13	∈	∈	PROPN
ejpam-3842	242	14	u	u	NOUN
ejpam-3842	242	15	,	,	PUNCT
ejpam-3842	242	16	there	there	PRON
ejpam-3842	242	17	exist	exist	VERB
ejpam-3842	242	18	nbhds	nbhds	PROPN
ejpam-3842	242	19	ny	ny	PROPN
ejpam-3842	242	20	and	and	CCONJ
ejpam-3842	242	21	my	my	PRON
ejpam-3842	242	22	of	of	ADP
ejpam-3842	242	23	y	y	PRON
ejpam-3842	242	24	such	such	ADJ
ejpam-3842	242	25	that	that	SCONJ
ejpam-3842	242	26	ny	ny	NOUN
ejpam-3842	242	27	∗my	∗my	PROPN
ejpam-3842	242	28	⊆	⊆	NUM
ejpam-3842	242	29	u	u	NOUN
ejpam-3842	242	30	.	.	PUNCT
ejpam-3842	243	1	since	since	SCONJ
ejpam-3842	243	2	x	x	PROPN
ejpam-3842	243	3	∗	∗	PROPN
ejpam-3842	243	4	y	y	PROPN
ejpam-3842	243	5	/∈	/∈	PUNCT
ejpam-3842	243	6	u	u	PROPN
ejpam-3842	243	7	,	,	PUNCT
ejpam-3842	243	8	x	x	PROPN
ejpam-3842	243	9	∗	∗	NOUN
ejpam-3842	243	10	y	y	PROPN
ejpam-3842	243	11	/∈	/∈	PUNCT
ejpam-3842	243	12	ny	ny	PROPN
ejpam-3842	243	13	∗my	∗my	PROPN
ejpam-3842	243	14	.	.	PUNCT
ejpam-3842	244	1	it	it	PRON
ejpam-3842	244	2	follows	follow	VERB
ejpam-3842	244	3	that	that	SCONJ
ejpam-3842	244	4	x	x	PROPN
ejpam-3842	244	5	/∈	/∈	PROPN
ejpam-3842	244	6	ny	ny	PROPN
ejpam-3842	244	7	.	.	PROPN
ejpam-3842	245	1	hence	hence	ADV
ejpam-3842	245	2	,	,	PUNCT
ejpam-3842	245	3	there	there	PRON
ejpam-3842	245	4	exist	exist	VERB
ejpam-3842	245	5	nbhds	nbhds	PROPN
ejpam-3842	245	6	mx	mx	PROPN
ejpam-3842	245	7	and	and	CCONJ
ejpam-3842	245	8	ny	ny	PROPN
ejpam-3842	245	9	of	of	ADP
ejpam-3842	245	10	x	x	PROPN
ejpam-3842	245	11	and	and	CCONJ
ejpam-3842	245	12	y	y	PROPN
ejpam-3842	245	13	,	,	PUNCT
ejpam-3842	245	14	respectively	respectively	ADV
ejpam-3842	245	15	,	,	PUNCT
ejpam-3842	245	16	such	such	ADJ
ejpam-3842	245	17	that	that	SCONJ
ejpam-3842	245	18	y	y	PROPN
ejpam-3842	245	19	/∈mx	/∈mx	PUNCT
ejpam-3842	245	20	and	and	CCONJ
ejpam-3842	245	21	x	x	SYM
ejpam-3842	245	22	/∈	/∈	PROPN
ejpam-3842	245	23	ny	ny	PROPN
ejpam-3842	245	24	.	.	PROPN
ejpam-3842	245	25	therefore	therefore	ADV
ejpam-3842	245	26	,	,	PUNCT
ejpam-3842	245	27	x	x	X
ejpam-3842	245	28	is	be	AUX
ejpam-3842	245	29	a	a	DET
ejpam-3842	245	30	t1	t1	NOUN
ejpam-3842	245	31	-	-	PUNCT
ejpam-3842	245	32	space	space	NOUN
ejpam-3842	245	33	.	.	PUNCT
ejpam-3842	246	1	(	(	PUNCT
ejpam-3842	246	2	ii)⇒(iii	ii)⇒(iii	X
ejpam-3842	246	3	):	):	PUNCT
ejpam-3842	246	4	suppose	suppose	VERB
ejpam-3842	246	5	x	x	PRON
ejpam-3842	246	6	is	be	AUX
ejpam-3842	246	7	a	a	DET
ejpam-3842	246	8	t1	t1	NOUN
ejpam-3842	246	9	-	-	PUNCT
ejpam-3842	246	10	space	space	NOUN
ejpam-3842	246	11	.	.	PUNCT
ejpam-3842	247	1	by	by	ADP
ejpam-3842	247	2	theorem	theorem	NOUN
ejpam-3842	247	3	1	1	NUM
ejpam-3842	247	4	,	,	PUNCT
ejpam-3842	247	5	{	{	PUNCT
ejpam-3842	247	6	0	0	NUM
ejpam-3842	247	7	}	}	PUNCT
ejpam-3842	247	8	is	be	AUX
ejpam-3842	247	9	a	a	DET
ejpam-3842	247	10	closed	closed	ADJ
ejpam-3842	247	11	set	set	NOUN
ejpam-3842	247	12	in	in	ADP
ejpam-3842	247	13	x.	x.	NOUN
ejpam-3842	247	14	by	by	ADP
ejpam-3842	247	15	theorem	theorem	NOUN
ejpam-3842	247	16	7	7	NUM
ejpam-3842	247	17	,	,	PUNCT
ejpam-3842	247	18	x	x	X
ejpam-3842	247	19	is	be	AUX
ejpam-3842	247	20	a	a	DET
ejpam-3842	247	21	t2	t2	NOUN
ejpam-3842	247	22	-	-	PUNCT
ejpam-3842	247	23	space	space	NOUN
ejpam-3842	247	24	.	.	PUNCT
ejpam-3842	248	1	by	by	ADP
ejpam-3842	248	2	remark	remark	NOUN
ejpam-3842	248	3	1	1	NUM
ejpam-3842	248	4	,	,	PUNCT
ejpam-3842	248	5	t2	t2	NOUN
ejpam-3842	248	6	⇒	⇒	PROPN
ejpam-3842	248	7	t1	t1	PROPN
ejpam-3842	248	8	⇒	⇒	PROPN
ejpam-3842	248	9	t0	t0	PROPN
ejpam-3842	248	10	.	.	PUNCT
ejpam-3842	249	1	therefore	therefore	ADV
ejpam-3842	249	2	,	,	PUNCT
ejpam-3842	249	3	(	(	PUNCT
ejpam-3842	249	4	i	i	NOUN
ejpam-3842	249	5	)	)	PUNCT
ejpam-3842	249	6	,	,	PUNCT
ejpam-3842	249	7	(	(	PUNCT
ejpam-3842	249	8	ii	ii	NOUN
ejpam-3842	249	9	)	)	PUNCT
ejpam-3842	249	10	,	,	PUNCT
ejpam-3842	249	11	and	and	CCONJ
ejpam-3842	249	12	(	(	PUNCT
ejpam-3842	249	13	iii	iii	X
ejpam-3842	249	14	)	)	PUNCT
ejpam-3842	249	15	are	be	AUX
ejpam-3842	249	16	equivalent	equivalent	ADJ
ejpam-3842	249	17	.	.	PUNCT
ejpam-3842	250	1	the	the	DET
ejpam-3842	250	2	following	follow	VERB
ejpam-3842	250	3	corollary	corollary	NOUN
ejpam-3842	250	4	follows	follow	VERB
ejpam-3842	250	5	from	from	ADP
ejpam-3842	250	6	theorems	theorem	NOUN
ejpam-3842	250	7	7	7	NUM
ejpam-3842	250	8	and	and	CCONJ
ejpam-3842	250	9	8	8	NUM
ejpam-3842	250	10	.	.	PUNCT
ejpam-3842	250	11	corollary	corollary	ADJ
ejpam-3842	250	12	3	3	X
ejpam-3842	250	13	.	.	PUNCT
ejpam-3842	251	1	let	let	VERB
ejpam-3842	251	2	x	x	PRON
ejpam-3842	251	3	be	be	AUX
ejpam-3842	251	4	a	a	DET
ejpam-3842	251	5	tbch	tbch	NOUN
ejpam-3842	251	6	-	-	PUNCT
ejpam-3842	251	7	algebra	algebra	NOUN
ejpam-3842	251	8	.	.	PUNCT
ejpam-3842	252	1	then	then	ADV
ejpam-3842	252	2	the	the	DET
ejpam-3842	252	3	following	follow	VERB
ejpam-3842	252	4	statements	statement	NOUN
ejpam-3842	252	5	are	be	AUX
ejpam-3842	252	6	equivalent	equivalent	ADJ
ejpam-3842	252	7	:	:	PUNCT
ejpam-3842	252	8	(	(	PUNCT
ejpam-3842	252	9	i	i	NOUN
ejpam-3842	252	10	)	)	PUNCT
ejpam-3842	252	11	x	x	X
ejpam-3842	252	12	is	be	AUX
ejpam-3842	252	13	a	a	DET
ejpam-3842	252	14	t0	t0	NOUN
ejpam-3842	252	15	-	-	NOUN
ejpam-3842	252	16	space	space	NOUN
ejpam-3842	252	17	(	(	PUNCT
ejpam-3842	252	18	ii	ii	NOUN
ejpam-3842	252	19	)	)	PUNCT
ejpam-3842	252	20	x	x	X
ejpam-3842	252	21	is	be	AUX
ejpam-3842	252	22	a	a	DET
ejpam-3842	252	23	t1	t1	NOUN
ejpam-3842	252	24	-	-	PUNCT
ejpam-3842	252	25	space	space	NOUN
ejpam-3842	252	26	(	(	PUNCT
ejpam-3842	252	27	iii	iii	NOUN
ejpam-3842	252	28	)	)	PUNCT
ejpam-3842	252	29	x	x	X
ejpam-3842	252	30	is	be	AUX
ejpam-3842	252	31	a	a	DET
ejpam-3842	252	32	t2	t2	NOUN
ejpam-3842	252	33	-	-	PUNCT
ejpam-3842	252	34	space	space	NOUN
ejpam-3842	252	35	(	(	PUNCT
ejpam-3842	252	36	iv	iv	X
ejpam-3842	252	37	)	)	PUNCT
ejpam-3842	252	38	{	{	PUNCT
ejpam-3842	252	39	0	0	NUM
ejpam-3842	252	40	}	}	PUNCT
ejpam-3842	252	41	is	be	AUX
ejpam-3842	252	42	a	a	DET
ejpam-3842	252	43	closed	closed	ADJ
ejpam-3842	252	44	set	set	NOUN
ejpam-3842	252	45	in	in	ADP
ejpam-3842	252	46	x.	x.	NOUN
ejpam-3842	252	47	theorem	theorem	VERB
ejpam-3842	252	48	9	9	NUM
ejpam-3842	252	49	.	.	PUNCT
ejpam-3842	253	1	let	let	VERB
ejpam-3842	253	2	x	x	PRON
ejpam-3842	253	3	be	be	AUX
ejpam-3842	253	4	a	a	DET
ejpam-3842	253	5	tbch	tbch	NOUN
ejpam-3842	253	6	-	-	PUNCT
ejpam-3842	253	7	algebra	algebra	NOUN
ejpam-3842	253	8	.	.	PUNCT
ejpam-3842	254	1	then	then	ADV
ejpam-3842	254	2	x	x	X
ejpam-3842	254	3	is	be	AUX
ejpam-3842	254	4	a	a	DET
ejpam-3842	254	5	t2	t2	NOUN
ejpam-3842	254	6	-	-	PUNCT
ejpam-3842	254	7	space	space	NOUN
ejpam-3842	254	8	if	if	SCONJ
ejpam-3842	254	9	and	and	CCONJ
ejpam-3842	254	10	only	only	ADV
ejpam-3842	254	11	if	if	SCONJ
ejpam-3842	254	12	for	for	ADP
ejpam-3842	254	13	any	any	DET
ejpam-3842	254	14	x	x	SYM
ejpam-3842	254	15	∈	∈	PROPN
ejpam-3842	254	16	x	x	PUNCT
ejpam-3842	254	17	with	with	ADP
ejpam-3842	254	18	x	x	SYM
ejpam-3842	254	19	6=	6=	ADP
ejpam-3842	254	20	0	0	NUM
ejpam-3842	254	21	,	,	PUNCT
ejpam-3842	254	22	there	there	PRON
ejpam-3842	254	23	exists	exist	VERB
ejpam-3842	254	24	a	a	DET
ejpam-3842	254	25	nbhd	nbhd	NOUN
ejpam-3842	254	26	u	u	NOUN
ejpam-3842	254	27	of	of	ADP
ejpam-3842	254	28	x	x	INTJ
ejpam-3842	254	29	such	such	ADJ
ejpam-3842	254	30	that	that	DET
ejpam-3842	254	31	0	0	NUM
ejpam-3842	254	32	/∈	/∈	NUM
ejpam-3842	254	33	u	u	NOUN
ejpam-3842	254	34	.	.	PUNCT
ejpam-3842	255	1	proof	proof	NOUN
ejpam-3842	255	2	.	.	PUNCT
ejpam-3842	256	1	clearly	clearly	ADV
ejpam-3842	256	2	,	,	PUNCT
ejpam-3842	256	3	if	if	SCONJ
ejpam-3842	256	4	x	x	PRON
ejpam-3842	256	5	is	be	AUX
ejpam-3842	256	6	a	a	DET
ejpam-3842	256	7	t2	t2	NOUN
ejpam-3842	256	8	-	-	PUNCT
ejpam-3842	256	9	space	space	NOUN
ejpam-3842	256	10	,	,	PUNCT
ejpam-3842	256	11	then	then	ADV
ejpam-3842	256	12	for	for	ADP
ejpam-3842	256	13	any	any	DET
ejpam-3842	256	14	x	x	SYM
ejpam-3842	256	15	∈	∈	PROPN
ejpam-3842	256	16	x	x	PUNCT
ejpam-3842	256	17	with	with	ADP
ejpam-3842	256	18	x	x	SYM
ejpam-3842	256	19	6=	6=	ADP
ejpam-3842	256	20	0	0	NUM
ejpam-3842	256	21	,	,	PUNCT
ejpam-3842	256	22	there	there	PRON
ejpam-3842	256	23	exists	exist	VERB
ejpam-3842	256	24	a	a	DET
ejpam-3842	256	25	nbhd	nbhd	NOUN
ejpam-3842	256	26	u	u	NOUN
ejpam-3842	256	27	of	of	ADP
ejpam-3842	256	28	x	x	INTJ
ejpam-3842	256	29	such	such	ADJ
ejpam-3842	256	30	that	that	DET
ejpam-3842	256	31	0	0	NUM
ejpam-3842	256	32	/∈	/∈	NUM
ejpam-3842	256	33	u	u	PROPN
ejpam-3842	256	34	.	.	PUNCT
ejpam-3842	257	1	for	for	ADP
ejpam-3842	257	2	the	the	DET
ejpam-3842	257	3	converse	converse	NOUN
ejpam-3842	257	4	,	,	PUNCT
ejpam-3842	257	5	suppose	suppose	VERB
ejpam-3842	257	6	that	that	SCONJ
ejpam-3842	257	7	for	for	ADP
ejpam-3842	257	8	any	any	DET
ejpam-3842	257	9	x	x	SYM
ejpam-3842	257	10	∈	∈	PROPN
ejpam-3842	257	11	x	x	PUNCT
ejpam-3842	257	12	with	with	ADP
ejpam-3842	257	13	x	x	SYM
ejpam-3842	257	14	6=	6=	ADP
ejpam-3842	257	15	0	0	NUM
ejpam-3842	257	16	,	,	PUNCT
ejpam-3842	257	17	there	there	PRON
ejpam-3842	257	18	exists	exist	VERB
ejpam-3842	257	19	a	a	DET
ejpam-3842	257	20	nbhd	nbhd	NOUN
ejpam-3842	257	21	u	u	NOUN
ejpam-3842	257	22	of	of	ADP
ejpam-3842	257	23	x	x	INTJ
ejpam-3842	257	24	such	such	ADJ
ejpam-3842	257	25	that	that	DET
ejpam-3842	257	26	0	0	NUM
ejpam-3842	257	27	/∈	/∈	NUM
ejpam-3842	257	28	u	u	INTJ
ejpam-3842	257	29	.	.	PUNCT
ejpam-3842	258	1	let	let	VERB
ejpam-3842	258	2	a	a	DET
ejpam-3842	258	3	,	,	PUNCT
ejpam-3842	258	4	b	b	X
ejpam-3842	258	5	∈	∈	PROPN
ejpam-3842	258	6	x	x	PUNCT
ejpam-3842	258	7	with	with	ADP
ejpam-3842	258	8	a	a	DET
ejpam-3842	258	9	6=	6=	NUM
ejpam-3842	258	10	b.	b.	PROPN
ejpam-3842	258	11	then	then	ADV
ejpam-3842	258	12	a∗b	a∗b	PROPN
ejpam-3842	258	13	6=	6=	SYM
ejpam-3842	258	14	0	0	NUM
ejpam-3842	258	15	or	or	CCONJ
ejpam-3842	258	16	b∗a	b∗a	NUM
ejpam-3842	258	17	6=	6=	ADP
ejpam-3842	258	18	0	0	NUM
ejpam-3842	258	19	by	by	ADP
ejpam-3842	258	20	(	(	PUNCT
ejpam-3842	258	21	b2	b2	NOUN
ejpam-3842	258	22	)	)	PUNCT
ejpam-3842	258	23	.	.	PUNCT
ejpam-3842	259	1	without	without	ADP
ejpam-3842	259	2	loss	loss	NOUN
ejpam-3842	259	3	of	of	ADP
ejpam-3842	259	4	generality	generality	NOUN
ejpam-3842	259	5	,	,	PUNCT
ejpam-3842	259	6	assume	assume	VERB
ejpam-3842	259	7	that	that	SCONJ
ejpam-3842	259	8	a	a	DET
ejpam-3842	259	9	∗	∗	NOUN
ejpam-3842	259	10	b	b	NOUN
ejpam-3842	259	11	6=	6=	NUM
ejpam-3842	259	12	0	0	NUM
ejpam-3842	259	13	.	.	PUNCT
ejpam-3842	260	1	then	then	ADV
ejpam-3842	260	2	,	,	PUNCT
ejpam-3842	260	3	by	by	ADP
ejpam-3842	260	4	assumption	assumption	NOUN
ejpam-3842	260	5	,	,	PUNCT
ejpam-3842	260	6	there	there	PRON
ejpam-3842	260	7	exists	exist	VERB
ejpam-3842	260	8	a	a	DET
ejpam-3842	260	9	nbhd	nbhd	NOUN
ejpam-3842	260	10	w	w	NOUN
ejpam-3842	260	11	of	of	ADP
ejpam-3842	260	12	a	a	DET
ejpam-3842	260	13	∗	∗	NOUN
ejpam-3842	260	14	b	b	NOUN
ejpam-3842	260	15	such	such	ADJ
ejpam-3842	260	16	that	that	DET
ejpam-3842	260	17	0	0	NUM
ejpam-3842	261	1	/∈	/∈	PUNCT
ejpam-3842	262	1	w	w	INTJ
ejpam-3842	262	2	.	.	PUNCT
ejpam-3842	263	1	by	by	ADP
ejpam-3842	263	2	theorem	theorem	NOUN
ejpam-3842	263	3	2	2	NUM
ejpam-3842	263	4	,	,	PUNCT
ejpam-3842	263	5	there	there	PRON
ejpam-3842	263	6	exist	exist	VERB
ejpam-3842	263	7	nbhds	nbhds	ADJ
ejpam-3842	263	8	wa	wa	PROPN
ejpam-3842	263	9	and	and	CCONJ
ejpam-3842	263	10	wb	wb	PROPN
ejpam-3842	263	11	of	of	ADP
ejpam-3842	263	12	a	a	PRON
ejpam-3842	263	13	and	and	CCONJ
ejpam-3842	263	14	b	b	NOUN
ejpam-3842	263	15	,	,	PUNCT
ejpam-3842	263	16	respectively	respectively	ADV
ejpam-3842	263	17	,	,	PUNCT
ejpam-3842	263	18	such	such	ADJ
ejpam-3842	263	19	that	that	DET
ejpam-3842	263	20	wa	wa	NOUN
ejpam-3842	263	21	∗wb	∗wb	PUNCT
ejpam-3842	263	22	⊆w	⊆w	NOUN
ejpam-3842	263	23	.	.	PUNCT
ejpam-3842	264	1	since	since	SCONJ
ejpam-3842	264	2	0	0	NUM
ejpam-3842	264	3	/∈w	/∈w	PUNCT
ejpam-3842	264	4	,	,	PUNCT
ejpam-3842	264	5	0	0	NUM
ejpam-3842	264	6	/∈wa	/∈wa	PUNCT
ejpam-3842	264	7	∗wb	∗wb	VERB
ejpam-3842	264	8	.	.	PUNCT
ejpam-3842	265	1	by	by	ADP
ejpam-3842	265	2	remark	remark	NOUN
ejpam-3842	265	3	4	4	NUM
ejpam-3842	265	4	,	,	PUNCT
ejpam-3842	265	5	wa	wa	NOUN
ejpam-3842	265	6	∩wb	∩wb	PROPN
ejpam-3842	265	7	=	=	PUNCT
ejpam-3842	265	8	∅.	∅.	VERB
ejpam-3842	265	9	thus	thus	ADV
ejpam-3842	265	10	,	,	PUNCT
ejpam-3842	265	11	x	x	PRON
ejpam-3842	265	12	is	be	AUX
ejpam-3842	265	13	a	a	DET
ejpam-3842	265	14	t2	t2	NOUN
ejpam-3842	265	15	-	-	PUNCT
ejpam-3842	265	16	space	space	NOUN
ejpam-3842	265	17	.	.	PUNCT
ejpam-3842	266	1	conclusion	conclusion	NOUN
ejpam-3842	266	2	:	:	PUNCT
ejpam-3842	266	3	given	give	VERB
ejpam-3842	266	4	two	two	NUM
ejpam-3842	266	5	bch	bch	NOUN
ejpam-3842	266	6	-	-	PUNCT
ejpam-3842	266	7	algebras	algebras	PROPN
ejpam-3842	266	8	h1	h1	PROPN
ejpam-3842	266	9	and	and	CCONJ
ejpam-3842	266	10	h2	h2	NOUN
ejpam-3842	266	11	such	such	ADJ
ejpam-3842	266	12	that	that	SCONJ
ejpam-3842	266	13	h1∩h2	h1∩h2	PROPN
ejpam-3842	266	14	=	=	PUNCT
ejpam-3842	266	15	{	{	PUNCT
ejpam-3842	266	16	0	0	NUM
ejpam-3842	266	17	}	}	PUNCT
ejpam-3842	266	18	,	,	PUNCT
ejpam-3842	266	19	an	an	DET
ejpam-3842	266	20	operation	operation	NOUN
ejpam-3842	266	21	“	"	PUNCT
ejpam-3842	266	22	∗	∗	NOUN
ejpam-3842	266	23	”	"	PUNCT
ejpam-3842	266	24	can	can	AUX
ejpam-3842	266	25	be	be	AUX
ejpam-3842	266	26	defined	define	VERB
ejpam-3842	266	27	on	on	ADP
ejpam-3842	266	28	h	h	NOUN
ejpam-3842	266	29	=	=	NOUN
ejpam-3842	266	30	h1	h1	PROPN
ejpam-3842	266	31	∪h2	∪h2	NOUN
ejpam-3842	267	1	so	so	SCONJ
ejpam-3842	267	2	that	that	SCONJ
ejpam-3842	267	3	(	(	PUNCT
ejpam-3842	267	4	h	h	NOUN
ejpam-3842	267	5	,	,	PUNCT
ejpam-3842	267	6	∗	∗	NOUN
ejpam-3842	267	7	)	)	PUNCT
ejpam-3842	267	8	is	be	AUX
ejpam-3842	267	9	a	a	DET
ejpam-3842	267	10	bch	bch	NOUN
ejpam-3842	267	11	-	-	PUNCT
ejpam-3842	267	12	algebra	algebra	NOUN
ejpam-3842	267	13	and	and	CCONJ
ejpam-3842	267	14	h1	h1	PROPN
ejpam-3842	267	15	and	and	CCONJ
ejpam-3842	267	16	h2	h2	PROPN
ejpam-3842	267	17	are	be	AUX
ejpam-3842	267	18	bch	bch	NOUN
ejpam-3842	267	19	-	-	PUNCT
ejpam-3842	267	20	subalgebras	subalgebras	PROPN
ejpam-3842	267	21	.	.	PUNCT
ejpam-3842	268	1	further	far	ADV
ejpam-3842	268	2	,	,	PUNCT
ejpam-3842	268	3	it	it	PRON
ejpam-3842	268	4	is	be	AUX
ejpam-3842	268	5	shown	show	VERB
ejpam-3842	268	6	that	that	SCONJ
ejpam-3842	268	7	t0	t0	NOUN
ejpam-3842	268	8	,	,	PUNCT
ejpam-3842	268	9	t1	t1	NOUN
ejpam-3842	268	10	and	and	CCONJ
ejpam-3842	268	11	t2	t2	NOUN
ejpam-3842	268	12	axioms	axiom	NOUN
ejpam-3842	268	13	are	be	AUX
ejpam-3842	268	14	equivalent	equivalent	ADJ
ejpam-3842	268	15	in	in	ADP
ejpam-3842	268	16	any	any	DET
ejpam-3842	268	17	topological	topological	ADJ
ejpam-3842	268	18	bch	bch	NOUN
ejpam-3842	268	19	-	-	PUNCT
ejpam-3842	268	20	algebra	algebra	NOUN
ejpam-3842	268	21	.	.	PUNCT
ejpam-3842	269	1	references	reference	NOUN
ejpam-3842	269	2	738	738	NUM
ejpam-3842	269	3	acknowledgements	acknowledgement	NOUN
ejpam-3842	269	4	the	the	DET
ejpam-3842	269	5	authors	author	NOUN
ejpam-3842	269	6	would	would	AUX
ejpam-3842	269	7	like	like	VERB
ejpam-3842	269	8	to	to	PART
ejpam-3842	269	9	thank	thank	VERB
ejpam-3842	269	10	the	the	DET
ejpam-3842	269	11	referees	referee	NOUN
ejpam-3842	269	12	for	for	ADP
ejpam-3842	269	13	reviewing	review	VERB
ejpam-3842	269	14	the	the	DET
ejpam-3842	269	15	initial	initial	ADJ
ejpam-3842	269	16	paper	paper	NOUN
ejpam-3842	269	17	and	and	CCONJ
ejpam-3842	269	18	for	for	ADP
ejpam-3842	269	19	the	the	DET
ejpam-3842	269	20	invaluable	invaluable	ADJ
ejpam-3842	269	21	comments	comment	NOUN
ejpam-3842	269	22	and	and	CCONJ
ejpam-3842	269	23	suggestions	suggestion	NOUN
ejpam-3842	269	24	that	that	PRON
ejpam-3842	269	25	eventually	eventually	ADV
ejpam-3842	269	26	led	lead	VERB
ejpam-3842	269	27	to	to	ADP
ejpam-3842	269	28	this	this	DET
ejpam-3842	269	29	much	much	ADV
ejpam-3842	269	30	improved	improved	ADJ
ejpam-3842	269	31	version	version	NOUN
ejpam-3842	269	32	of	of	ADP
ejpam-3842	269	33	the	the	DET
ejpam-3842	269	34	work	work	NOUN
ejpam-3842	269	35	.	.	PUNCT
ejpam-3842	270	1	this	this	DET
ejpam-3842	270	2	research	research	NOUN
ejpam-3842	270	3	is	be	AUX
ejpam-3842	270	4	funded	fund	VERB
ejpam-3842	270	5	by	by	ADP
ejpam-3842	270	6	the	the	DET
ejpam-3842	270	7	philippine	philippine	PROPN
ejpam-3842	270	8	department	department	PROPN
ejpam-3842	270	9	of	of	ADP
ejpam-3842	270	10	science	science	NOUN
ejpam-3842	270	11	and	and	CCONJ
ejpam-3842	270	12	technology	technology	NOUN
ejpam-3842	270	13	accelerated	accelerate	VERB
ejpam-3842	270	14	science	science	NOUN
ejpam-3842	270	15	and	and	CCONJ
ejpam-3842	270	16	technology	technology	NOUN
ejpam-3842	270	17	human	human	ADJ
ejpam-3842	270	18	resource	resource	NOUN
ejpam-3842	270	19	development	development	NOUN
ejpam-3842	270	20	program	program	NOUN
ejpam-3842	270	21	(	(	PUNCT
ejpam-3842	270	22	dostasthrdp	dostasthrdp	PROPN
ejpam-3842	270	23	)	)	PUNCT
ejpam-3842	270	24	.	.	PUNCT
ejpam-3842	271	1	references	reference	NOUN
ejpam-3842	271	2	[	[	X
ejpam-3842	271	3	1	1	NUM
ejpam-3842	271	4	]	]	X
ejpam-3842	271	5	m.a	m.a	PROPN
ejpam-3842	271	6	.	.	PROPN
ejpam-3842	271	7	chaudhry	chaudhry	PROPN
ejpam-3842	271	8	and	and	CCONJ
ejpam-3842	271	9	h.	h.	PROPN
ejpam-3842	271	10	fakhar	fakhar	PROPN
ejpam-3842	271	11	-	-	PUNCT
ejpam-3842	271	12	ud	ud	NOUN
ejpam-3842	271	13	-	-	PUNCT
ejpam-3842	271	14	din	din	NOUN
ejpam-3842	271	15	.	.	PROPN
ejpam-3842	271	16	on	on	ADP
ejpam-3842	271	17	some	some	DET
ejpam-3842	271	18	classes	class	NOUN
ejpam-3842	271	19	of	of	ADP
ejpam-3842	271	20	bch	bch	PROPN
ejpam-3842	271	21	-	-	PUNCT
ejpam-3842	271	22	algebras	algebras	PROPN
ejpam-3842	271	23	.	.	PUNCT
ejpam-3842	272	1	international	international	ADJ
ejpam-3842	272	2	journal	journal	PROPN
ejpam-3842	272	3	of	of	ADP
ejpam-3842	272	4	mathematics	mathematics	PROPN
ejpam-3842	272	5	and	and	CCONJ
ejpam-3842	272	6	mathematical	mathematical	ADJ
ejpam-3842	272	7	sciences	science	NOUN
ejpam-3842	272	8	,	,	PUNCT
ejpam-3842	272	9	25(3):205–211	25(3):205–211	PROPN
ejpam-3842	272	10	,	,	PUNCT
ejpam-3842	272	11	2001	2001	NUM
ejpam-3842	272	12	.	.	PUNCT
ejpam-3842	273	1	[	[	X
ejpam-3842	273	2	2	2	X
ejpam-3842	273	3	]	]	X
ejpam-3842	273	4	k.h	k.h	PROPN
ejpam-3842	273	5	.	.	PROPN
ejpam-3842	273	6	dar	dar	PROPN
ejpam-3842	273	7	and	and	CCONJ
ejpam-3842	273	8	m.	m.	PROPN
ejpam-3842	273	9	akram	akram	PROPN
ejpam-3842	273	10	.	.	PUNCT
ejpam-3842	274	1	on	on	ADP
ejpam-3842	274	2	endomorphisms	endomorphism	NOUN
ejpam-3842	274	3	of	of	ADP
ejpam-3842	274	4	bch	bch	PROPN
ejpam-3842	274	5	-	-	PUNCT
ejpam-3842	274	6	algebras	algebras	PROPN
ejpam-3842	274	7	.	.	PUNCT
ejpam-3842	275	1	annals	annal	NOUN
ejpam-3842	275	2	of	of	ADP
ejpam-3842	275	3	the	the	DET
ejpam-3842	275	4	university	university	NOUN
ejpam-3842	275	5	of	of	ADP
ejpam-3842	275	6	craiova	craiova	PROPN
ejpam-3842	275	7	-	-	PUNCT
ejpam-3842	275	8	mathematics	mathematic	NOUN
ejpam-3842	275	9	and	and	CCONJ
ejpam-3842	275	10	computer	computer	NOUN
ejpam-3842	275	11	science	science	NOUN
ejpam-3842	275	12	series	series	NOUN
ejpam-3842	275	13	,	,	PUNCT
ejpam-3842	275	14	33:227–234	33:227–234	NUM
ejpam-3842	275	15	,	,	PUNCT
ejpam-3842	275	16	2006	2006	NUM
ejpam-3842	275	17	.	.	PUNCT
ejpam-3842	276	1	[	[	X
ejpam-3842	276	2	3	3	X
ejpam-3842	276	3	]	]	X
ejpam-3842	276	4	j.	j.	PROPN
ejpam-3842	276	5	dugundji	dugundji	PROPN
ejpam-3842	276	6	.	.	PUNCT
ejpam-3842	276	7	topology	topology	PROPN
ejpam-3842	276	8	.	.	PUNCT
ejpam-3842	277	1	allyn	allyn	PROPN
ejpam-3842	277	2	and	and	CCONJ
ejpam-3842	277	3	bacon	bacon	PROPN
ejpam-3842	277	4	,	,	PUNCT
ejpam-3842	277	5	inc	inc	PROPN
ejpam-3842	277	6	.	.	PROPN
ejpam-3842	277	7	,	,	PUNCT
ejpam-3842	277	8	boston	boston	PROPN
ejpam-3842	277	9	,	,	PUNCT
ejpam-3842	277	10	1966	1966	NUM
ejpam-3842	277	11	.	.	PUNCT
ejpam-3842	278	1	[	[	X
ejpam-3842	278	2	4	4	NUM
ejpam-3842	278	3	]	]	X
ejpam-3842	278	4	y.b	y.b	PROPN
ejpam-3842	278	5	.	.	PROPN
ejpam-3842	278	6	jun	jun	PROPN
ejpam-3842	278	7	et	et	PROPN
ejpam-3842	278	8	al	al	PROPN
ejpam-3842	278	9	.	.	PROPN
ejpam-3842	279	1	on	on	ADP
ejpam-3842	279	2	topological	topological	ADJ
ejpam-3842	279	3	bci	bci	NOUN
ejpam-3842	279	4	-	-	PUNCT
ejpam-3842	279	5	algebras	algebra	NOUN
ejpam-3842	279	6	.	.	PUNCT
ejpam-3842	280	1	information	information	NOUN
ejpam-3842	280	2	sciences	sciences	PROPN
ejpam-3842	280	3	,	,	PUNCT
ejpam-3842	280	4	116(2	116(2	NUM
ejpam-3842	280	5	-	-	PUNCT
ejpam-3842	280	6	4):253–261	4):253–261	NUM
ejpam-3842	280	7	,	,	PUNCT
ejpam-3842	280	8	1999	1999	NUM
ejpam-3842	280	9	.	.	PUNCT
ejpam-3842	281	1	[	[	X
ejpam-3842	281	2	5	5	NUM
ejpam-3842	281	3	]	]	X
ejpam-3842	281	4	q.p	q.p	PROPN
ejpam-3842	281	5	.	.	PROPN
ejpam-3842	281	6	hu	hu	PROPN
ejpam-3842	281	7	and	and	CCONJ
ejpam-3842	281	8	x.	x.	PROPN
ejpam-3842	281	9	li	li	PROPN
ejpam-3842	281	10	.	.	PROPN
ejpam-3842	282	1	on	on	ADP
ejpam-3842	282	2	bch	bch	PROPN
ejpam-3842	282	3	-	-	PUNCT
ejpam-3842	282	4	algebras	algebras	PROPN
ejpam-3842	282	5	.	.	PUNCT
ejpam-3842	282	6	math	math	NOUN
ejpam-3842	282	7	.	.	PUNCT
ejpam-3842	283	1	seminar	seminar	NOUN
ejpam-3842	283	2	notes	note	NOUN
ejpam-3842	283	3	,	,	PUNCT
ejpam-3842	283	4	11(2):313–320	11(2):313–320	PROPN
ejpam-3842	283	5	,	,	PUNCT
ejpam-3842	283	6	1983	1983	NUM
ejpam-3842	283	7	.	.	PUNCT
ejpam-3842	284	1	[	[	X
ejpam-3842	284	2	6	6	NUM
ejpam-3842	284	3	]	]	X
ejpam-3842	284	4	q.p	q.p	PROPN
ejpam-3842	284	5	.	.	PROPN
ejpam-3842	284	6	hu	hu	PROPN
ejpam-3842	284	7	and	and	CCONJ
ejpam-3842	284	8	x.	x.	PROPN
ejpam-3842	284	9	li	li	PROPN
ejpam-3842	284	10	.	.	PROPN
ejpam-3842	285	1	on	on	ADP
ejpam-3842	285	2	proper	proper	ADJ
ejpam-3842	285	3	bch	bch	NOUN
ejpam-3842	285	4	-	-	PUNCT
ejpam-3842	285	5	algebras	algebras	PROPN
ejpam-3842	285	6	.	.	PUNCT
ejpam-3842	286	1	mathematica	mathematica	PROPN
ejpam-3842	286	2	japonica	japonica	PROPN
ejpam-3842	286	3	,	,	PUNCT
ejpam-3842	286	4	30(4):659–661	30(4):659–661	PROPN
ejpam-3842	286	5	,	,	PUNCT
ejpam-3842	286	6	1985	1985	NUM
ejpam-3842	286	7	.	.	PUNCT
ejpam-3842	287	1	[	[	X
ejpam-3842	287	2	7	7	X
ejpam-3842	287	3	]	]	X
ejpam-3842	287	4	m.	m.	NOUN
ejpam-3842	287	5	jansi	jansi	PROPN
ejpam-3842	287	6	and	and	CCONJ
ejpam-3842	287	7	v.	v.	ADP
ejpam-3842	287	8	thiruveni	thiruveni	NOUN
ejpam-3842	287	9	.	.	PUNCT
ejpam-3842	288	1	topological	topological	ADJ
ejpam-3842	288	2	structures	structure	NOUN
ejpam-3842	288	3	on	on	ADP
ejpam-3842	288	4	bch	bch	PROPN
ejpam-3842	288	5	-	-	PUNCT
ejpam-3842	288	6	algebras	algebras	PROPN
ejpam-3842	288	7	.	.	PUNCT
ejpam-3842	289	1	mathematica	mathematica	PROPN
ejpam-3842	289	2	japonicae	japonicae	PROPN
ejpam-3842	289	3	,	,	PUNCT
ejpam-3842	289	4	6:22594–22600	6:22594–22600	NUM
ejpam-3842	289	5	,	,	PUNCT
ejpam-3842	289	6	2017	2017	NUM
ejpam-3842	289	7	.	.	PUNCT
ejpam-3842	290	1	[	[	X
ejpam-3842	290	2	8	8	NUM
ejpam-3842	290	3	]	]	X
ejpam-3842	290	4	d.	d.	PROPN
ejpam-3842	290	5	s.	s.	PROPN
ejpam-3842	290	6	lee	lee	PROPN
ejpam-3842	290	7	and	and	CCONJ
ejpam-3842	290	8	d.	d.	PROPN
ejpam-3842	290	9	n.	n.	PROPN
ejpam-3842	290	10	ryu	ryu	PROPN
ejpam-3842	290	11	.	.	PUNCT
ejpam-3842	291	1	notes	note	NOUN
ejpam-3842	291	2	on	on	ADP
ejpam-3842	291	3	topological	topological	ADJ
ejpam-3842	291	4	bck	bck	PROPN
ejpam-3842	291	5	-	-	PUNCT
ejpam-3842	291	6	algebras	algebras	PROPN
ejpam-3842	291	7	.	.	PUNCT
ejpam-3842	292	1	sci	sci	PROPN
ejpam-3842	292	2	.	.	PROPN
ejpam-3842	292	3	math	math	PROPN
ejpam-3842	292	4	,	,	PUNCT
ejpam-3842	292	5	1:231–235	1:231–235	NUM
ejpam-3842	292	6	,	,	PUNCT
ejpam-3842	292	7	1998	1998	NUM
ejpam-3842	292	8	.	.	PUNCT