id	sid	tid	token	lemma	pos
ejpam-3548	1	1	european	european	PROPN
ejpam-3548	1	2	journal	journal	PROPN
ejpam-3548	1	3	of	of	ADP
ejpam-3548	1	4	pure	pure	ADJ
ejpam-3548	1	5	and	and	CCONJ
ejpam-3548	1	6	applied	apply	VERB
ejpam-3548	1	7	mathematics	mathematic	NOUN
ejpam-3548	1	8	vol	vol	NOUN
ejpam-3548	1	9	.	.	PROPN
ejpam-3548	2	1	12	12	NUM
ejpam-3548	2	2	,	,	PUNCT
ejpam-3548	2	3	no	no	INTJ
ejpam-3548	2	4	.	.	NOUN
ejpam-3548	2	5	4	4	NUM
ejpam-3548	2	6	,	,	PUNCT
ejpam-3548	2	7	2019	2019	NUM
ejpam-3548	2	8	,	,	PUNCT
ejpam-3548	2	9	1584	1584	NUM
ejpam-3548	2	10	-	-	SYM
ejpam-3548	2	11	1594	1594	NUM
ejpam-3548	2	12	issn	issn	PROPN
ejpam-3548	2	13	1307	1307	NUM
ejpam-3548	2	14	-	-	SYM
ejpam-3548	2	15	5543	5543	NUM
ejpam-3548	2	16	–	–	PUNCT
ejpam-3548	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3548	2	18	published	publish	VERB
ejpam-3548	2	19	by	by	ADP
ejpam-3548	2	20	new	new	PROPN
ejpam-3548	2	21	york	york	PROPN
ejpam-3548	2	22	business	business	PROPN
ejpam-3548	2	23	global	global	ADJ
ejpam-3548	2	24	topology	topology	NOUN
ejpam-3548	2	25	on	on	ADP
ejpam-3548	2	26	a	a	DET
ejpam-3548	2	27	be	be	NOUN
ejpam-3548	2	28	-	-	PUNCT
ejpam-3548	2	29	algebra	algebra	NOUN
ejpam-3548	2	30	induced	induce	VERB
ejpam-3548	2	31	by	by	ADP
ejpam-3548	2	32	right	right	ADJ
ejpam-3548	2	33	application	application	NOUN
ejpam-3548	2	34	of	of	ADP
ejpam-3548	2	35	be	be	NOUN
ejpam-3548	2	36	-	-	PUNCT
ejpam-3548	2	37	ordering	order	VERB
ejpam-3548	2	38	jimboy	jimboy	PROPN
ejpam-3548	2	39	r.	r.	PROPN
ejpam-3548	2	40	albaracin1,∗	albaracin1,∗	PROPN
ejpam-3548	2	41	,	,	PUNCT
ejpam-3548	2	42	jocelyn	jocelyn	PROPN
ejpam-3548	2	43	p.	p.	PROPN
ejpam-3548	2	44	vilela2	vilela2	PROPN
ejpam-3548	2	45	1	1	NUM
ejpam-3548	2	46	department	department	NOUN
ejpam-3548	2	47	of	of	ADP
ejpam-3548	2	48	mathematics	mathematic	NOUN
ejpam-3548	2	49	and	and	CCONJ
ejpam-3548	2	50	statistics	statistic	NOUN
ejpam-3548	2	51	,	,	PUNCT
ejpam-3548	2	52	college	college	NOUN
ejpam-3548	2	53	of	of	ADP
ejpam-3548	2	54	science	science	NOUN
ejpam-3548	2	55	and	and	CCONJ
ejpam-3548	2	56	mathematics	mathematic	NOUN
ejpam-3548	2	57	,	,	PUNCT
ejpam-3548	2	58	mindanao	mindanao	PROPN
ejpam-3548	2	59	state	state	PROPN
ejpam-3548	2	60	university	university	PROPN
ejpam-3548	2	61	-	-	PUNCT
ejpam-3548	2	62	iligan	iligan	PROPN
ejpam-3548	2	63	institute	institute	PROPN
ejpam-3548	2	64	of	of	ADP
ejpam-3548	2	65	technology	technology	PROPN
ejpam-3548	2	66	,	,	PUNCT
ejpam-3548	2	67	9200	9200	NUM
ejpam-3548	2	68	iligan	iligan	ADJ
ejpam-3548	2	69	city	city	NOUN
ejpam-3548	2	70	,	,	PUNCT
ejpam-3548	2	71	philippines	philippine	NOUN
ejpam-3548	2	72	2	2	NUM
ejpam-3548	2	73	center	center	NOUN
ejpam-3548	2	74	for	for	ADP
ejpam-3548	2	75	graph	graph	NOUN
ejpam-3548	2	76	theory	theory	NOUN
ejpam-3548	2	77	,	,	PUNCT
ejpam-3548	2	78	algebra	algebra	NOUN
ejpam-3548	2	79	and	and	CCONJ
ejpam-3548	2	80	analysis	analysis	NOUN
ejpam-3548	2	81	,	,	PUNCT
ejpam-3548	2	82	premier	premier	ADJ
ejpam-3548	2	83	research	research	NOUN
ejpam-3548	2	84	of	of	ADP
ejpam-3548	2	85	institute	institute	PROPN
ejpam-3548	2	86	of	of	ADP
ejpam-3548	2	87	science	science	NOUN
ejpam-3548	2	88	and	and	CCONJ
ejpam-3548	2	89	mathematics	mathematic	NOUN
ejpam-3548	2	90	,	,	PUNCT
ejpam-3548	2	91	mindanao	mindanao	PROPN
ejpam-3548	2	92	state	state	PROPN
ejpam-3548	2	93	university	university	PROPN
ejpam-3548	2	94	-	-	PUNCT
ejpam-3548	2	95	iligan	iligan	PROPN
ejpam-3548	2	96	institute	institute	PROPN
ejpam-3548	2	97	of	of	ADP
ejpam-3548	2	98	technology	technology	PROPN
ejpam-3548	2	99	,	,	PUNCT
ejpam-3548	2	100	9200	9200	NUM
ejpam-3548	2	101	iligan	iligan	ADJ
ejpam-3548	2	102	city	city	NOUN
ejpam-3548	2	103	,	,	PUNCT
ejpam-3548	2	104	philippines	philippine	NOUN
ejpam-3548	2	105	abstract	abstract	ADJ
ejpam-3548	2	106	.	.	PUNCT
ejpam-3548	3	1	this	this	DET
ejpam-3548	3	2	study	study	NOUN
ejpam-3548	3	3	deals	deal	VERB
ejpam-3548	3	4	with	with	ADP
ejpam-3548	3	5	the	the	DET
ejpam-3548	3	6	topology	topology	NOUN
ejpam-3548	3	7	generated	generate	VERB
ejpam-3548	3	8	by	by	ADP
ejpam-3548	3	9	the	the	DET
ejpam-3548	3	10	family	family	NOUN
ejpam-3548	3	11	of	of	ADP
ejpam-3548	3	12	subsets	subset	NOUN
ejpam-3548	3	13	determined	determine	VERB
ejpam-3548	3	14	by	by	ADP
ejpam-3548	3	15	the	the	DET
ejpam-3548	3	16	right	right	ADJ
ejpam-3548	3	17	application	application	NOUN
ejpam-3548	3	18	of	of	ADP
ejpam-3548	3	19	be	be	NOUN
ejpam-3548	3	20	-	-	PUNCT
ejpam-3548	3	21	ordering	ordering	NOUN
ejpam-3548	3	22	of	of	ADP
ejpam-3548	3	23	a	a	DET
ejpam-3548	3	24	be	be	NOUN
ejpam-3548	3	25	-	-	PUNCT
ejpam-3548	3	26	algebra	algebra	NOUN
ejpam-3548	3	27	and	and	CCONJ
ejpam-3548	3	28	investigates	investigate	VERB
ejpam-3548	3	29	some	some	PRON
ejpam-3548	3	30	of	of	ADP
ejpam-3548	3	31	its	its	PRON
ejpam-3548	3	32	properties	property	NOUN
ejpam-3548	3	33	.	.	PUNCT
ejpam-3548	4	1	characterizations	characterization	NOUN
ejpam-3548	4	2	of	of	ADP
ejpam-3548	4	3	some	some	DET
ejpam-3548	4	4	elementary	elementary	ADJ
ejpam-3548	4	5	topological	topological	ADJ
ejpam-3548	4	6	concepts	concept	NOUN
ejpam-3548	4	7	as	as	ADV
ejpam-3548	4	8	well	well	ADV
ejpam-3548	4	9	as	as	ADP
ejpam-3548	4	10	the	the	DET
ejpam-3548	4	11	concepts	concept	NOUN
ejpam-3548	4	12	of	of	ADP
ejpam-3548	4	13	continuous	continuous	ADJ
ejpam-3548	4	14	,	,	PUNCT
ejpam-3548	4	15	open	open	ADJ
ejpam-3548	4	16	,	,	PUNCT
ejpam-3548	4	17	and	and	CCONJ
ejpam-3548	4	18	closed	closed	ADJ
ejpam-3548	4	19	maps	map	NOUN
ejpam-3548	4	20	associated	associate	VERB
ejpam-3548	4	21	with	with	ADP
ejpam-3548	4	22	this	this	DET
ejpam-3548	4	23	topological	topological	ADJ
ejpam-3548	4	24	space	space	NOUN
ejpam-3548	4	25	are	be	AUX
ejpam-3548	4	26	obtained	obtain	VERB
ejpam-3548	4	27	.	.	PUNCT
ejpam-3548	5	1	2010	2010	NUM
ejpam-3548	5	2	mathematics	mathematic	NOUN
ejpam-3548	5	3	subject	subject	NOUN
ejpam-3548	5	4	classifications	classification	NOUN
ejpam-3548	5	5	:	:	PUNCT
ejpam-3548	5	6	06f35	06f35	NUM
ejpam-3548	5	7	,	,	PUNCT
ejpam-3548	5	8	03g25	03g25	NOUN
ejpam-3548	5	9	key	key	ADJ
ejpam-3548	5	10	words	word	NOUN
ejpam-3548	5	11	and	and	CCONJ
ejpam-3548	5	12	phrases	phrase	NOUN
ejpam-3548	5	13	:	:	PUNCT
ejpam-3548	5	14	topology	topology	NOUN
ejpam-3548	5	15	,	,	PUNCT
ejpam-3548	5	16	be	be	AUX
ejpam-3548	5	17	-	-	PUNCT
ejpam-3548	5	18	algebra	algebra	NOUN
ejpam-3548	5	19	,	,	PUNCT
ejpam-3548	5	20	be	be	AUX
ejpam-3548	5	21	-	-	PUNCT
ejpam-3548	5	22	ordering	order	VERB
ejpam-3548	5	23	1	1	NUM
ejpam-3548	5	24	.	.	PUNCT
ejpam-3548	5	25	introduction	introduction	NOUN
ejpam-3548	5	26	the	the	DET
ejpam-3548	5	27	study	study	NOUN
ejpam-3548	5	28	of	of	ADP
ejpam-3548	5	29	bck	bck	PROPN
ejpam-3548	5	30	-	-	PUNCT
ejpam-3548	5	31	algebras	algebras	PROPN
ejpam-3548	5	32	was	be	AUX
ejpam-3548	5	33	initiated	initiate	VERB
ejpam-3548	5	34	by	by	ADP
ejpam-3548	5	35	y.	y.	PROPN
ejpam-3548	5	36	imai	imai	PROPN
ejpam-3548	5	37	and	and	CCONJ
ejpam-3548	5	38	k.	k.	PROPN
ejpam-3548	5	39	iséki	iséki	PROPN
ejpam-3548	6	1	[	[	X
ejpam-3548	6	2	3	3	X
ejpam-3548	6	3	]	]	PUNCT
ejpam-3548	6	4	in	in	ADP
ejpam-3548	6	5	1966	1966	NUM
ejpam-3548	6	6	as	as	ADP
ejpam-3548	6	7	a	a	DET
ejpam-3548	6	8	generalization	generalization	NOUN
ejpam-3548	6	9	of	of	ADP
ejpam-3548	6	10	the	the	DET
ejpam-3548	6	11	concept	concept	NOUN
ejpam-3548	6	12	of	of	ADP
ejpam-3548	6	13	set	set	VERB
ejpam-3548	6	14	theoretic	theoretic	ADJ
ejpam-3548	6	15	difference	difference	NOUN
ejpam-3548	6	16	and	and	CCONJ
ejpam-3548	6	17	propositional	propositional	ADJ
ejpam-3548	6	18	calculi	calculi	NOUN
ejpam-3548	6	19	.	.	PUNCT
ejpam-3548	7	1	in	in	ADP
ejpam-3548	7	2	[	[	X
ejpam-3548	7	3	5	5	NUM
ejpam-3548	7	4	]	]	PUNCT
ejpam-3548	7	5	,	,	PUNCT
ejpam-3548	7	6	k.	k.	PROPN
ejpam-3548	7	7	h.	h.	PROPN
ejpam-3548	7	8	kim	kim	PROPN
ejpam-3548	7	9	and	and	CCONJ
ejpam-3548	7	10	y.	y.	PROPN
ejpam-3548	7	11	h.	h.	PROPN
ejpam-3548	7	12	yon	yon	PROPN
ejpam-3548	7	13	introduced	introduce	VERB
ejpam-3548	7	14	the	the	DET
ejpam-3548	7	15	dual	dual	ADJ
ejpam-3548	7	16	bck	bck	NOUN
ejpam-3548	7	17	-	-	PUNCT
ejpam-3548	7	18	algebra	algebra	NOUN
ejpam-3548	7	19	and	and	CCONJ
ejpam-3548	7	20	study	study	VERB
ejpam-3548	7	21	its	its	PRON
ejpam-3548	7	22	relation	relation	NOUN
ejpam-3548	7	23	to	to	ADP
ejpam-3548	7	24	mv	mv	NOUN
ejpam-3548	7	25	-	-	NOUN
ejpam-3548	7	26	algebra	algebra	NOUN
ejpam-3548	7	27	.	.	PUNCT
ejpam-3548	8	1	as	as	ADP
ejpam-3548	8	2	a	a	DET
ejpam-3548	8	3	generalization	generalization	NOUN
ejpam-3548	8	4	of	of	ADP
ejpam-3548	8	5	dual	dual	ADJ
ejpam-3548	8	6	bck	bck	NOUN
ejpam-3548	8	7	-	-	PUNCT
ejpam-3548	8	8	algebra	algebra	NOUN
ejpam-3548	8	9	,	,	PUNCT
ejpam-3548	8	10	h.	h.	PROPN
ejpam-3548	8	11	s.	s.	PROPN
ejpam-3548	8	12	kim	kim	PROPN
ejpam-3548	8	13	and	and	CCONJ
ejpam-3548	8	14	y.	y.	PROPN
ejpam-3548	8	15	h.	h.	PROPN
ejpam-3548	8	16	kim	kim	PROPN
ejpam-3548	9	1	[	[	X
ejpam-3548	9	2	4	4	X
ejpam-3548	9	3	]	]	PUNCT
ejpam-3548	9	4	introduced	introduce	VERB
ejpam-3548	9	5	the	the	DET
ejpam-3548	9	6	be	be	NOUN
ejpam-3548	9	7	-	-	PUNCT
ejpam-3548	9	8	algebra	algebra	NOUN
ejpam-3548	9	9	.	.	PUNCT
ejpam-3548	10	1	today	today	NOUN
ejpam-3548	10	2	,	,	PUNCT
ejpam-3548	10	3	be	be	AUX
ejpam-3548	10	4	-	-	PUNCT
ejpam-3548	10	5	algebras	algebra	NOUN
ejpam-3548	10	6	have	have	AUX
ejpam-3548	10	7	been	be	AUX
ejpam-3548	10	8	studied	study	VERB
ejpam-3548	10	9	by	by	ADP
ejpam-3548	10	10	many	many	ADJ
ejpam-3548	10	11	authors	author	NOUN
ejpam-3548	10	12	and	and	CCONJ
ejpam-3548	10	13	many	many	ADJ
ejpam-3548	10	14	branches	branch	NOUN
ejpam-3548	10	15	of	of	ADP
ejpam-3548	10	16	mathematics	mathematic	NOUN
ejpam-3548	10	17	have	have	AUX
ejpam-3548	10	18	been	be	AUX
ejpam-3548	10	19	applied	apply	VERB
ejpam-3548	10	20	to	to	PART
ejpam-3548	10	21	be	be	AUX
ejpam-3548	10	22	-	-	PUNCT
ejpam-3548	10	23	algebras	algebra	NOUN
ejpam-3548	10	24	,	,	PUNCT
ejpam-3548	10	25	such	such	ADJ
ejpam-3548	10	26	as	as	ADP
ejpam-3548	10	27	probability	probability	NOUN
ejpam-3548	10	28	theory	theory	NOUN
ejpam-3548	10	29	,	,	PUNCT
ejpam-3548	10	30	topology	topology	NOUN
ejpam-3548	10	31	,	,	PUNCT
ejpam-3548	10	32	fuzzy	fuzzy	ADJ
ejpam-3548	10	33	set	set	NOUN
ejpam-3548	10	34	theory	theory	NOUN
ejpam-3548	10	35	and	and	CCONJ
ejpam-3548	10	36	so	so	ADV
ejpam-3548	10	37	on	on	ADV
ejpam-3548	10	38	.	.	PUNCT
ejpam-3548	11	1	various	various	ADJ
ejpam-3548	11	2	authors	author	NOUN
ejpam-3548	11	3	studied	study	VERB
ejpam-3548	11	4	the	the	DET
ejpam-3548	11	5	topological	topological	ADJ
ejpam-3548	11	6	aspects	aspect	NOUN
ejpam-3548	11	7	of	of	ADP
ejpam-3548	11	8	be	be	NOUN
ejpam-3548	11	9	-	-	PUNCT
ejpam-3548	11	10	algebras	algebra	NOUN
ejpam-3548	11	11	.	.	PUNCT
ejpam-3548	12	1	in	in	ADP
ejpam-3548	12	2	[	[	X
ejpam-3548	12	3	7	7	NUM
ejpam-3548	12	4	]	]	PUNCT
ejpam-3548	12	5	,	,	PUNCT
ejpam-3548	12	6	s.	s.	PROPN
ejpam-3548	12	7	mehrshad	mehrshad	VERB
ejpam-3548	12	8	and	and	CCONJ
ejpam-3548	12	9	j.	j.	PROPN
ejpam-3548	12	10	golzarpoor	golzarpoor	PROPN
ejpam-3548	12	11	studied	study	VERB
ejpam-3548	12	12	some	some	DET
ejpam-3548	12	13	properties	property	NOUN
ejpam-3548	12	14	of	of	ADP
ejpam-3548	12	15	uniform	uniform	ADJ
ejpam-3548	12	16	topology	topology	NOUN
ejpam-3548	12	17	and	and	CCONJ
ejpam-3548	12	18	topological	topological	ADJ
ejpam-3548	12	19	be	be	VERB
ejpam-3548	12	20	-	-	PUNCT
ejpam-3548	12	21	algebras	algebra	NOUN
ejpam-3548	12	22	and	and	CCONJ
ejpam-3548	12	23	compare	compare	VERB
ejpam-3548	12	24	these	these	DET
ejpam-3548	12	25	topologies	topology	NOUN
ejpam-3548	12	26	.	.	PUNCT
ejpam-3548	13	1	in	in	ADP
ejpam-3548	13	2	[	[	X
ejpam-3548	13	3	8	8	NUM
ejpam-3548	13	4	]	]	PUNCT
ejpam-3548	13	5	,	,	PUNCT
ejpam-3548	13	6	the	the	DET
ejpam-3548	13	7	author	author	NOUN
ejpam-3548	13	8	produced	produce	VERB
ejpam-3548	13	9	a	a	DET
ejpam-3548	13	10	basis	basis	NOUN
ejpam-3548	13	11	for	for	ADP
ejpam-3548	13	12	a	a	DET
ejpam-3548	13	13	topology	topology	NOUN
ejpam-3548	13	14	using	use	VERB
ejpam-3548	13	15	left	left	ADJ
ejpam-3548	13	16	and	and	CCONJ
ejpam-3548	13	17	right	right	ADJ
ejpam-3548	13	18	stabilizers	stabilizer	NOUN
ejpam-3548	13	19	of	of	ADP
ejpam-3548	13	20	a	a	DET
ejpam-3548	13	21	be	be	NOUN
ejpam-3548	13	22	-	-	PUNCT
ejpam-3548	13	23	algebra	algebra	NOUN
ejpam-3548	13	24	.	.	PUNCT
ejpam-3548	14	1	it	it	PRON
ejpam-3548	14	2	is	be	AUX
ejpam-3548	14	3	proved	prove	VERB
ejpam-3548	14	4	that	that	SCONJ
ejpam-3548	14	5	the	the	DET
ejpam-3548	14	6	generated	generate	VERB
ejpam-3548	14	7	topological	topological	ADJ
ejpam-3548	14	8	space	space	NOUN
ejpam-3548	14	9	is	be	AUX
ejpam-3548	14	10	a	a	DET
ejpam-3548	14	11	bair	bair	NOUN
ejpam-3548	14	12	,	,	PUNCT
ejpam-3548	14	13	locally	locally	ADV
ejpam-3548	14	14	connected	connected	ADJ
ejpam-3548	14	15	and	and	CCONJ
ejpam-3548	14	16	separable	separable	ADJ
ejpam-3548	14	17	space	space	NOUN
ejpam-3548	14	18	.	.	PUNCT
ejpam-3548	15	1	some	some	DET
ejpam-3548	15	2	other	other	ADJ
ejpam-3548	15	3	topological	topological	ADJ
ejpam-3548	15	4	properties	property	NOUN
ejpam-3548	15	5	are	be	AUX
ejpam-3548	15	6	studied	study	VERB
ejpam-3548	15	7	using	use	VERB
ejpam-3548	15	8	left	left	ADJ
ejpam-3548	15	9	and	and	CCONJ
ejpam-3548	15	10	right	right	ADJ
ejpam-3548	15	11	stabilizers	stabilizer	NOUN
ejpam-3548	15	12	.	.	PUNCT
ejpam-3548	16	1	motivated	motivate	VERB
ejpam-3548	16	2	by	by	ADP
ejpam-3548	16	3	these	these	DET
ejpam-3548	16	4	works	work	NOUN
ejpam-3548	16	5	,	,	PUNCT
ejpam-3548	16	6	this	this	DET
ejpam-3548	16	7	paper	paper	NOUN
ejpam-3548	16	8	introduces	introduce	VERB
ejpam-3548	16	9	the	the	DET
ejpam-3548	16	10	topology	topology	NOUN
ejpam-3548	16	11	induced	induce	VERB
ejpam-3548	16	12	by	by	ADP
ejpam-3548	16	13	a	a	DET
ejpam-3548	16	14	be	be	NOUN
ejpam-3548	16	15	-	-	PUNCT
ejpam-3548	16	16	algebra	algebra	NOUN
ejpam-3548	16	17	using	use	VERB
ejpam-3548	16	18	the	the	DET
ejpam-3548	16	19	right	right	ADJ
ejpam-3548	16	20	application	application	NOUN
ejpam-3548	16	21	of	of	ADP
ejpam-3548	16	22	be	be	NOUN
ejpam-3548	16	23	-	-	PUNCT
ejpam-3548	16	24	ordering	ordering	NOUN
ejpam-3548	16	25	and	and	CCONJ
ejpam-3548	16	26	investigates	investigate	VERB
ejpam-3548	16	27	some	some	PRON
ejpam-3548	16	28	of	of	ADP
ejpam-3548	16	29	its	its	PRON
ejpam-3548	16	30	properties	property	NOUN
ejpam-3548	16	31	.	.	PUNCT
ejpam-3548	17	1	∗corresponding	∗corresponde	VERB
ejpam-3548	17	2	author	author	NOUN
ejpam-3548	17	3	.	.	PUNCT
ejpam-3548	18	1	doi	doi	NOUN
ejpam-3548	18	2	:	:	PUNCT
ejpam-3548	18	3	https://doi.org/10.29020/nybg.ejpam.v12i4.3548	https://doi.org/10.29020/nybg.ejpam.v12i4.3548	PROPN
ejpam-3548	18	4	email	email	NOUN
ejpam-3548	18	5	addresses	address	VERB
ejpam-3548	18	6	:	:	PUNCT
ejpam-3548	18	7	jimboy.albaracin@g.msuiit.edu.ph	jimboy.albaracin@g.msuiit.edu.ph	PROPN
ejpam-3548	18	8	(	(	PUNCT
ejpam-3548	18	9	j.	j.	PROPN
ejpam-3548	18	10	albaracin	albaracin	PROPN
ejpam-3548	18	11	)	)	PUNCT
ejpam-3548	18	12	,	,	PUNCT
ejpam-3548	18	13	jocelyn.vilela@g.msuiit.edu.ph	jocelyn.vilela@g.msuiit.edu.ph	PROPN
ejpam-3548	18	14	(	(	PUNCT
ejpam-3548	18	15	j.	j.	PROPN
ejpam-3548	18	16	vilela	vilela	PROPN
ejpam-3548	18	17	)	)	PUNCT
ejpam-3548	18	18	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3548	19	1	1584	1584	NUM
ejpam-3548	19	2	c	c	X
ejpam-3548	19	3	©	©	PROPN
ejpam-3548	19	4	2019	2019	NUM
ejpam-3548	19	5	ejpam	ejpam	NOUN
ejpam-3548	19	6	all	all	DET
ejpam-3548	19	7	rights	right	NOUN
ejpam-3548	19	8	reserved	reserve	VERB
ejpam-3548	19	9	.	.	PUNCT
ejpam-3548	20	1	j.	j.	PROPN
ejpam-3548	20	2	albaracin	albaracin	PROPN
ejpam-3548	20	3	,	,	PUNCT
ejpam-3548	20	4	j.	j.	PROPN
ejpam-3548	20	5	vilela	vilela	PROPN
ejpam-3548	20	6	/	/	SYM
ejpam-3548	20	7	eur	eur	PROPN
ejpam-3548	20	8	.	.	PUNCT
ejpam-3548	21	1	j.	j.	PROPN
ejpam-3548	21	2	pure	pure	PROPN
ejpam-3548	21	3	appl	appl	PROPN
ejpam-3548	21	4	.	.	PROPN
ejpam-3548	21	5	math	math	PROPN
ejpam-3548	21	6	,	,	PUNCT
ejpam-3548	21	7	12	12	NUM
ejpam-3548	21	8	(	(	PUNCT
ejpam-3548	21	9	4	4	NUM
ejpam-3548	21	10	)	)	PUNCT
ejpam-3548	21	11	(	(	PUNCT
ejpam-3548	21	12	2019	2019	NUM
ejpam-3548	21	13	)	)	PUNCT
ejpam-3548	21	14	,	,	PUNCT
ejpam-3548	21	15	1584	1584	NUM
ejpam-3548	21	16	-	-	SYM
ejpam-3548	21	17	1594	1594	NUM
ejpam-3548	21	18	1585	1585	NUM
ejpam-3548	21	19	an	an	DET
ejpam-3548	21	20	algebra	algebra	NOUN
ejpam-3548	21	21	(	(	PUNCT
ejpam-3548	21	22	x	x	NOUN
ejpam-3548	21	23	;	;	PUNCT
ejpam-3548	21	24	∗	∗	NOUN
ejpam-3548	21	25	,	,	PUNCT
ejpam-3548	21	26	1x	1x	NUM
ejpam-3548	21	27	)	)	PUNCT
ejpam-3548	21	28	is	be	AUX
ejpam-3548	21	29	called	call	VERB
ejpam-3548	21	30	a	a	DET
ejpam-3548	21	31	be	be	NOUN
ejpam-3548	21	32	-	-	PUNCT
ejpam-3548	21	33	algebra	algebra	NOUN
ejpam-3548	21	34	if	if	SCONJ
ejpam-3548	21	35	the	the	DET
ejpam-3548	21	36	following	follow	VERB
ejpam-3548	21	37	hold	hold	NOUN
ejpam-3548	21	38	:	:	PUNCT
ejpam-3548	21	39	for	for	ADP
ejpam-3548	21	40	all	all	DET
ejpam-3548	21	41	x	x	NOUN
ejpam-3548	21	42	,	,	PUNCT
ejpam-3548	21	43	y	y	PROPN
ejpam-3548	21	44	,	,	PUNCT
ejpam-3548	21	45	z	z	PROPN
ejpam-3548	21	46	∈	∈	PROPN
ejpam-3548	21	47	x	x	X
ejpam-3548	21	48	,	,	PUNCT
ejpam-3548	21	49	(	(	PUNCT
ejpam-3548	21	50	be1	be1	NOUN
ejpam-3548	21	51	)	)	PUNCT
ejpam-3548	21	52	x∗x	x∗x	PUNCT
ejpam-3548	22	1	=	=	X
ejpam-3548	22	2	1x	1x	NUM
ejpam-3548	22	3	;	;	PUNCT
ejpam-3548	22	4	(	(	PUNCT
ejpam-3548	22	5	be2	be2	NOUN
ejpam-3548	22	6	)	)	PUNCT
ejpam-3548	22	7	x∗1x	x∗1x	X
ejpam-3548	23	1	=	=	PUNCT
ejpam-3548	23	2	1x	1x	NUM
ejpam-3548	23	3	;	;	PUNCT
ejpam-3548	23	4	(	(	PUNCT
ejpam-3548	23	5	be3	be3	PROPN
ejpam-3548	23	6	)	)	PUNCT
ejpam-3548	23	7	1x	1x	NUM
ejpam-3548	24	1	∗x	∗x	NOUN
ejpam-3548	24	2	=	=	SYM
ejpam-3548	24	3	x	x	X
ejpam-3548	24	4	;	;	PUNCT
ejpam-3548	24	5	and	and	CCONJ
ejpam-3548	24	6	(	(	PUNCT
ejpam-3548	24	7	be4	be4	NOUN
ejpam-3548	24	8	)	)	PUNCT
ejpam-3548	24	9	x∗	x∗	PROPN
ejpam-3548	24	10	(	(	PUNCT
ejpam-3548	24	11	y	y	PROPN
ejpam-3548	24	12	∗z	∗z	PROPN
ejpam-3548	24	13	)	)	PUNCT
ejpam-3548	25	1	=	=	SYM
ejpam-3548	25	2	y	y	PROPN
ejpam-3548	25	3	∗	∗	NOUN
ejpam-3548	25	4	(	(	PUNCT
ejpam-3548	25	5	x∗z	x∗z	NOUN
ejpam-3548	25	6	)	)	PUNCT
ejpam-3548	25	7	.	.	PUNCT
ejpam-3548	26	1	a	a	DET
ejpam-3548	26	2	relation	relation	NOUN
ejpam-3548	26	3	“	"	PUNCT
ejpam-3548	26	4	≤	≤	NOUN
ejpam-3548	26	5	”	"	PUNCT
ejpam-3548	26	6	on	on	ADP
ejpam-3548	26	7	x	x	PRON
ejpam-3548	26	8	,	,	PUNCT
ejpam-3548	26	9	called	call	VERB
ejpam-3548	26	10	be	be	NOUN
ejpam-3548	26	11	-	-	PUNCT
ejpam-3548	26	12	ordering	ordering	NOUN
ejpam-3548	26	13	,	,	PUNCT
ejpam-3548	26	14	is	be	AUX
ejpam-3548	26	15	defined	define	VERB
ejpam-3548	26	16	by	by	ADP
ejpam-3548	26	17	x	x	PROPN
ejpam-3548	26	18	≤	≤	NUM
ejpam-3548	26	19	y	y	NOUN
ejpam-3548	26	20	if	if	SCONJ
ejpam-3548	27	1	and	and	CCONJ
ejpam-3548	27	2	only	only	ADV
ejpam-3548	27	3	if	if	SCONJ
ejpam-3548	27	4	x	x	X
ejpam-3548	27	5	∗	∗	VERB
ejpam-3548	27	6	y	y	NOUN
ejpam-3548	27	7	=	=	SYM
ejpam-3548	27	8	1x	1x	PROPN
ejpam-3548	27	9	.	.	PUNCT
ejpam-3548	28	1	throughout	throughout	ADP
ejpam-3548	28	2	this	this	DET
ejpam-3548	28	3	paper	paper	NOUN
ejpam-3548	28	4	,	,	PUNCT
ejpam-3548	28	5	we	we	PRON
ejpam-3548	28	6	denote	denote	VERB
ejpam-3548	28	7	a	a	DET
ejpam-3548	28	8	be	be	NOUN
ejpam-3548	28	9	-	-	PUNCT
ejpam-3548	28	10	algebra	algebra	NOUN
ejpam-3548	28	11	(	(	PUNCT
ejpam-3548	28	12	x	x	X
ejpam-3548	28	13	,	,	PUNCT
ejpam-3548	28	14	∗	∗	NOUN
ejpam-3548	28	15	,	,	PUNCT
ejpam-3548	28	16	1x	1x	NUM
ejpam-3548	28	17	)	)	PUNCT
ejpam-3548	28	18	simply	simply	ADV
ejpam-3548	28	19	by	by	ADP
ejpam-3548	28	20	x	x	PRON
ejpam-3548	28	21	if	if	SCONJ
ejpam-3548	28	22	no	no	DET
ejpam-3548	28	23	confusion	confusion	NOUN
ejpam-3548	28	24	arises	arise	VERB
ejpam-3548	28	25	.	.	PUNCT
ejpam-3548	29	1	a	a	DET
ejpam-3548	29	2	non	non	ADJ
ejpam-3548	29	3	-	-	ADJ
ejpam-3548	29	4	empty	empty	ADJ
ejpam-3548	29	5	subset	subset	NOUN
ejpam-3548	29	6	s	s	NOUN
ejpam-3548	29	7	of	of	ADP
ejpam-3548	29	8	x	x	VERB
ejpam-3548	29	9	is	be	AUX
ejpam-3548	29	10	said	say	VERB
ejpam-3548	29	11	to	to	PART
ejpam-3548	29	12	be	be	AUX
ejpam-3548	29	13	a	a	DET
ejpam-3548	29	14	subalgebra	subalgebra	NOUN
ejpam-3548	29	15	of	of	ADP
ejpam-3548	29	16	x	x	PRON
ejpam-3548	29	17	if	if	SCONJ
ejpam-3548	29	18	x	x	PROPN
ejpam-3548	29	19	∗	∗	VERB
ejpam-3548	29	20	y	y	PROPN
ejpam-3548	29	21	∈	∈	PROPN
ejpam-3548	29	22	s	s	PROPN
ejpam-3548	29	23	for	for	ADP
ejpam-3548	29	24	all	all	DET
ejpam-3548	29	25	x	x	NOUN
ejpam-3548	29	26	,	,	PUNCT
ejpam-3548	29	27	y	y	PROPN
ejpam-3548	29	28	∈	∈	PROPN
ejpam-3548	29	29	s.	s.	PROPN
ejpam-3548	29	30	a	a	DET
ejpam-3548	29	31	be	be	NOUN
ejpam-3548	29	32	-	-	PUNCT
ejpam-3548	29	33	algebra	algebra	NOUN
ejpam-3548	29	34	x	x	PUNCT
ejpam-3548	29	35	is	be	AUX
ejpam-3548	29	36	said	say	VERB
ejpam-3548	29	37	to	to	PART
ejpam-3548	29	38	be	be	AUX
ejpam-3548	29	39	self	self	NOUN
ejpam-3548	29	40	distributive	distributive	ADJ
ejpam-3548	29	41	if	if	SCONJ
ejpam-3548	29	42	x	x	SYM
ejpam-3548	29	43	∗	∗	NOUN
ejpam-3548	29	44	(	(	PUNCT
ejpam-3548	29	45	y	y	PROPN
ejpam-3548	29	46	∗	∗	PROPN
ejpam-3548	29	47	z	z	NOUN
ejpam-3548	29	48	)	)	PUNCT
ejpam-3548	29	49	=	=	SYM
ejpam-3548	29	50	(	(	PUNCT
ejpam-3548	29	51	x	x	X
ejpam-3548	29	52	∗	∗	PROPN
ejpam-3548	29	53	y	y	NOUN
ejpam-3548	29	54	)	)	PUNCT
ejpam-3548	29	55	∗	∗	NOUN
ejpam-3548	29	56	(	(	PUNCT
ejpam-3548	29	57	x	x	X
ejpam-3548	29	58	∗	∗	PROPN
ejpam-3548	29	59	z	z	NOUN
ejpam-3548	29	60	)	)	PUNCT
ejpam-3548	29	61	for	for	ADP
ejpam-3548	29	62	all	all	DET
ejpam-3548	29	63	x	x	NOUN
ejpam-3548	29	64	,	,	PUNCT
ejpam-3548	29	65	y	y	PROPN
ejpam-3548	29	66	,	,	PUNCT
ejpam-3548	29	67	z	z	NOUN
ejpam-3548	29	68	∈	∈	PROPN
ejpam-3548	29	69	x.	x.	NOUN
ejpam-3548	30	1	it	it	PRON
ejpam-3548	30	2	is	be	AUX
ejpam-3548	30	3	called	call	VERB
ejpam-3548	30	4	commutative	commutative	ADJ
ejpam-3548	30	5	if	if	SCONJ
ejpam-3548	30	6	satisfies	satisfie	NOUN
ejpam-3548	30	7	(	(	PUNCT
ejpam-3548	30	8	x	x	X
ejpam-3548	30	9	∗	∗	PROPN
ejpam-3548	30	10	y	y	NOUN
ejpam-3548	30	11	)	)	PUNCT
ejpam-3548	30	12	∗	∗	NOUN
ejpam-3548	30	13	y	y	NOUN
ejpam-3548	30	14	=	=	PUNCT
ejpam-3548	30	15	(	(	PUNCT
ejpam-3548	30	16	y	y	PROPN
ejpam-3548	30	17	∗	∗	X
ejpam-3548	30	18	x	x	NOUN
ejpam-3548	30	19	)	)	PUNCT
ejpam-3548	30	20	∗	∗	NOUN
ejpam-3548	30	21	x	x	PUNCT
ejpam-3548	30	22	for	for	ADP
ejpam-3548	30	23	all	all	DET
ejpam-3548	30	24	x	x	NOUN
ejpam-3548	30	25	,	,	PUNCT
ejpam-3548	30	26	y	y	PROPN
ejpam-3548	30	27	∈	∈	PROPN
ejpam-3548	30	28	x.	x.	NOUN
ejpam-3548	31	1	it	it	PRON
ejpam-3548	31	2	is	be	AUX
ejpam-3548	31	3	said	say	VERB
ejpam-3548	31	4	to	to	PART
ejpam-3548	31	5	be	be	AUX
ejpam-3548	31	6	a	a	DET
ejpam-3548	31	7	transitive	transitive	ADJ
ejpam-3548	31	8	be	be	NOUN
ejpam-3548	31	9	-	-	PUNCT
ejpam-3548	31	10	algebra	algebra	NOUN
ejpam-3548	31	11	if	if	SCONJ
ejpam-3548	31	12	it	it	PRON
ejpam-3548	31	13	satisfies	satisfy	VERB
ejpam-3548	31	14	the	the	DET
ejpam-3548	31	15	condition	condition	NOUN
ejpam-3548	31	16	:	:	PUNCT
ejpam-3548	31	17	y	y	PROPN
ejpam-3548	31	18	∗	∗	NOUN
ejpam-3548	31	19	z	z	NOUN
ejpam-3548	31	20	≤	≤	NOUN
ejpam-3548	31	21	(	(	PUNCT
ejpam-3548	31	22	x	x	X
ejpam-3548	31	23	∗	∗	PROPN
ejpam-3548	31	24	y	y	NOUN
ejpam-3548	31	25	)	)	PUNCT
ejpam-3548	31	26	∗	∗	NOUN
ejpam-3548	31	27	(	(	PUNCT
ejpam-3548	31	28	x	x	X
ejpam-3548	31	29	∗	∗	PROPN
ejpam-3548	31	30	z	z	NOUN
ejpam-3548	31	31	)	)	PUNCT
ejpam-3548	31	32	for	for	ADP
ejpam-3548	31	33	all	all	DET
ejpam-3548	31	34	x	x	NOUN
ejpam-3548	31	35	,	,	PUNCT
ejpam-3548	31	36	y	y	PROPN
ejpam-3548	31	37	,	,	PUNCT
ejpam-3548	31	38	z	z	NOUN
ejpam-3548	31	39	∈	∈	PROPN
ejpam-3548	31	40	x.	x.	NOUN
ejpam-3548	32	1	if	if	SCONJ
ejpam-3548	32	2	x	x	PRON
ejpam-3548	32	3	is	be	AUX
ejpam-3548	32	4	a	a	DET
ejpam-3548	32	5	transitive	transitive	ADJ
ejpam-3548	32	6	be	be	NOUN
ejpam-3548	32	7	-	-	PUNCT
ejpam-3548	32	8	algebra	algebra	NOUN
ejpam-3548	32	9	,	,	PUNCT
ejpam-3548	32	10	then	then	ADV
ejpam-3548	32	11	the	the	DET
ejpam-3548	32	12	relation	relation	NOUN
ejpam-3548	32	13	“	"	PUNCT
ejpam-3548	32	14	≤	≤	NUM
ejpam-3548	32	15	”	"	PUNCT
ejpam-3548	32	16	is	be	AUX
ejpam-3548	32	17	transitive	transitive	ADJ
ejpam-3548	32	18	.	.	PUNCT
ejpam-3548	33	1	let	let	VERB
ejpam-3548	33	2	f	f	PRON
ejpam-3548	33	3	be	be	AUX
ejpam-3548	33	4	a	a	DET
ejpam-3548	33	5	non	non	ADJ
ejpam-3548	33	6	-	-	ADJ
ejpam-3548	33	7	empty	empty	ADJ
ejpam-3548	33	8	subset	subset	NOUN
ejpam-3548	33	9	of	of	ADP
ejpam-3548	33	10	x.	x.	PROPN
ejpam-3548	33	11	then	then	ADV
ejpam-3548	33	12	f	f	PROPN
ejpam-3548	33	13	is	be	AUX
ejpam-3548	33	14	said	say	VERB
ejpam-3548	33	15	to	to	PART
ejpam-3548	33	16	be	be	AUX
ejpam-3548	33	17	a	a	DET
ejpam-3548	33	18	filter	filter	NOUN
ejpam-3548	33	19	of	of	ADP
ejpam-3548	33	20	x	x	SYM
ejpam-3548	33	21	if	if	SCONJ
ejpam-3548	33	22	:	:	PUNCT
ejpam-3548	33	23	(	(	PUNCT
ejpam-3548	33	24	f1	f1	NOUN
ejpam-3548	33	25	)	)	PUNCT
ejpam-3548	33	26	1x	1x	NOUN
ejpam-3548	34	1	∈	∈	PROPN
ejpam-3548	34	2	f	f	X
ejpam-3548	34	3	;	;	PUNCT
ejpam-3548	34	4	and	and	CCONJ
ejpam-3548	34	5	(	(	PUNCT
ejpam-3548	34	6	f2	f2	PROPN
ejpam-3548	34	7	)	)	PUNCT
ejpam-3548	34	8	x	x	SYM
ejpam-3548	34	9	∗	∗	NOUN
ejpam-3548	34	10	y	y	PROPN
ejpam-3548	34	11	∈	∈	PROPN
ejpam-3548	34	12	f	f	PROPN
ejpam-3548	34	13	and	and	CCONJ
ejpam-3548	34	14	x	x	PROPN
ejpam-3548	34	15	∈	∈	NOUN
ejpam-3548	35	1	f	f	X
ejpam-3548	35	2	imply	imply	VERB
ejpam-3548	35	3	y	y	PROPN
ejpam-3548	35	4	∈	∈	PROPN
ejpam-3548	36	1	f	f	PROPN
ejpam-3548	36	2	.	.	PUNCT
ejpam-3548	37	1	a	a	DET
ejpam-3548	37	2	non	non	ADJ
ejpam-3548	37	3	-	-	ADJ
ejpam-3548	37	4	empty	empty	ADJ
ejpam-3548	37	5	subset	subset	NOUN
ejpam-3548	37	6	i	i	PRON
ejpam-3548	37	7	of	of	ADP
ejpam-3548	37	8	x	x	PRON
ejpam-3548	37	9	is	be	AUX
ejpam-3548	37	10	called	call	VERB
ejpam-3548	37	11	an	an	DET
ejpam-3548	37	12	ideal	ideal	NOUN
ejpam-3548	37	13	of	of	ADP
ejpam-3548	37	14	x	x	PRON
ejpam-3548	37	15	if	if	SCONJ
ejpam-3548	37	16	it	it	PRON
ejpam-3548	37	17	satisfies	satisfy	VERB
ejpam-3548	37	18	:	:	PUNCT
ejpam-3548	37	19	for	for	ADP
ejpam-3548	37	20	all	all	PRON
ejpam-3548	37	21	x	x	SYM
ejpam-3548	37	22	∈	∈	ADJ
ejpam-3548	37	23	x	x	X
ejpam-3548	37	24	and	and	CCONJ
ejpam-3548	37	25	for	for	ADP
ejpam-3548	37	26	all	all	DET
ejpam-3548	37	27	a	a	PRON
ejpam-3548	37	28	,	,	PUNCT
ejpam-3548	37	29	b	b	X
ejpam-3548	37	30	∈	∈	PROPN
ejpam-3548	37	31	i	i	PRON
ejpam-3548	37	32	,	,	PUNCT
ejpam-3548	37	33	(	(	PUNCT
ejpam-3548	37	34	i1	i1	PROPN
ejpam-3548	37	35	)	)	PUNCT
ejpam-3548	38	1	x	x	PROPN
ejpam-3548	38	2	∗	∗	VERB
ejpam-3548	38	3	a	a	DET
ejpam-3548	38	4	∈	∈	NOUN
ejpam-3548	39	1	i	i	PRON
ejpam-3548	39	2	,	,	PUNCT
ejpam-3548	39	3	that	that	ADV
ejpam-3548	39	4	is	is	ADV
ejpam-3548	39	5	,	,	PUNCT
ejpam-3548	39	6	x	x	PUNCT
ejpam-3548	39	7	∗	∗	NOUN
ejpam-3548	39	8	i	i	PRON
ejpam-3548	39	9	⊆	⊆	NUM
ejpam-3548	39	10	i	i	PRON
ejpam-3548	39	11	;	;	PUNCT
ejpam-3548	39	12	and	and	CCONJ
ejpam-3548	39	13	(	(	PUNCT
ejpam-3548	39	14	i2	i2	PROPN
ejpam-3548	39	15	)	)	PUNCT
ejpam-3548	39	16	(	(	PUNCT
ejpam-3548	39	17	a	a	DET
ejpam-3548	39	18	∗	∗	NOUN
ejpam-3548	39	19	(	(	PUNCT
ejpam-3548	39	20	b	b	NOUN
ejpam-3548	39	21	∗	∗	NOUN
ejpam-3548	39	22	x	x	NOUN
ejpam-3548	39	23	)	)	PUNCT
ejpam-3548	39	24	)	)	PUNCT
ejpam-3548	39	25	∗	∗	NOUN
ejpam-3548	39	26	x	x	PUNCT
ejpam-3548	39	27	∈	∈	PROPN
ejpam-3548	39	28	i.	i.	NOUN
ejpam-3548	39	29	the	the	DET
ejpam-3548	39	30	set	set	NOUN
ejpam-3548	40	1	[	[	X
ejpam-3548	40	2	a	a	X
ejpam-3548	40	3	,	,	PUNCT
ejpam-3548	40	4	1x	1x	NUM
ejpam-3548	40	5	]	]	PUNCT
ejpam-3548	41	1	=	=	PUNCT
ejpam-3548	41	2	{	{	PUNCT
ejpam-3548	41	3	x	x	SYM
ejpam-3548	41	4	∈	∈	PROPN
ejpam-3548	41	5	x	x	PUNCT
ejpam-3548	41	6	|	|	ADV
ejpam-3548	41	7	a	a	DET
ejpam-3548	41	8	∗	∗	NOUN
ejpam-3548	41	9	x	x	PUNCT
ejpam-3548	41	10	=	=	SYM
ejpam-3548	41	11	1x	1x	NUM
ejpam-3548	41	12	}	}	PUNCT
ejpam-3548	41	13	for	for	ADP
ejpam-3548	41	14	all	all	DET
ejpam-3548	41	15	a	a	DET
ejpam-3548	41	16	∈	∈	NOUN
ejpam-3548	41	17	x	x	PUNCT
ejpam-3548	41	18	is	be	AUX
ejpam-3548	41	19	called	call	VERB
ejpam-3548	41	20	the	the	DET
ejpam-3548	41	21	final	final	ADJ
ejpam-3548	41	22	segment	segment	NOUN
ejpam-3548	41	23	of	of	ADP
ejpam-3548	41	24	x.	x.	NOUN
ejpam-3548	41	25	an	an	DET
ejpam-3548	41	26	element	element	NOUN
ejpam-3548	41	27	a	a	DET
ejpam-3548	41	28	6=	6=	NUM
ejpam-3548	41	29	1x	1x	NUM
ejpam-3548	41	30	of	of	ADP
ejpam-3548	41	31	a	a	DET
ejpam-3548	41	32	be	be	NOUN
ejpam-3548	41	33	-	-	PUNCT
ejpam-3548	41	34	algebra	algebra	NOUN
ejpam-3548	41	35	x	x	PUNCT
ejpam-3548	41	36	is	be	AUX
ejpam-3548	41	37	said	say	VERB
ejpam-3548	41	38	to	to	PART
ejpam-3548	41	39	be	be	AUX
ejpam-3548	41	40	a	a	DET
ejpam-3548	41	41	dual	dual	ADJ
ejpam-3548	41	42	atom	atom	NOUN
ejpam-3548	41	43	of	of	ADP
ejpam-3548	41	44	x	x	PRON
ejpam-3548	41	45	if	if	SCONJ
ejpam-3548	41	46	a	a	DET
ejpam-3548	41	47	≤	≤	NOUN
ejpam-3548	41	48	x	x	AUX
ejpam-3548	41	49	implies	imply	VERB
ejpam-3548	41	50	either	either	CCONJ
ejpam-3548	41	51	a	a	DET
ejpam-3548	41	52	=	=	SYM
ejpam-3548	41	53	x	x	X
ejpam-3548	41	54	or	or	CCONJ
ejpam-3548	41	55	x	x	SYM
ejpam-3548	41	56	=	=	SYM
ejpam-3548	41	57	1x	1x	NUM
ejpam-3548	41	58	for	for	ADP
ejpam-3548	41	59	all	all	PRON
ejpam-3548	41	60	x	x	SYM
ejpam-3548	41	61	∈	∈	ADJ
ejpam-3548	41	62	x.	x.	NOUN
ejpam-3548	41	63	we	we	PRON
ejpam-3548	41	64	will	will	AUX
ejpam-3548	41	65	denote	denote	VERB
ejpam-3548	41	66	by	by	ADP
ejpam-3548	41	67	a(x	a(x	NOUN
ejpam-3548	41	68	)	)	PUNCT
ejpam-3548	41	69	the	the	DET
ejpam-3548	41	70	set	set	NOUN
ejpam-3548	41	71	of	of	ADP
ejpam-3548	41	72	all	all	DET
ejpam-3548	41	73	dual	dual	ADJ
ejpam-3548	41	74	atoms	atom	NOUN
ejpam-3548	41	75	of	of	ADP
ejpam-3548	41	76	x	x	PRON
ejpam-3548	41	77	unless	unless	SCONJ
ejpam-3548	41	78	otherwise	otherwise	ADV
ejpam-3548	41	79	mentioned	mention	VERB
ejpam-3548	41	80	.	.	PUNCT
ejpam-3548	42	1	hence	hence	ADV
ejpam-3548	42	2	,	,	PUNCT
ejpam-3548	42	3	a(x	a(x	PROPN
ejpam-3548	42	4	)	)	PUNCT
ejpam-3548	42	5	=	=	PRON
ejpam-3548	42	6	{	{	PUNCT
ejpam-3548	42	7	x	x	PUNCT
ejpam-3548	42	8	∈	∈	NOUN
ejpam-3548	42	9	x	x	INTJ
ejpam-3548	42	10	|	|	ADV
ejpam-3548	42	11	x	x	INTJ
ejpam-3548	42	12	is	be	AUX
ejpam-3548	42	13	a	a	DET
ejpam-3548	42	14	dual	dual	ADJ
ejpam-3548	42	15	atom	atom	NOUN
ejpam-3548	42	16	}	}	PUNCT
ejpam-3548	42	17	.	.	PUNCT
ejpam-3548	43	1	we	we	PRON
ejpam-3548	43	2	will	will	AUX
ejpam-3548	43	3	consider	consider	VERB
ejpam-3548	43	4	a1(x	a1(x	NOUN
ejpam-3548	43	5	)	)	PUNCT
ejpam-3548	43	6	=	=	SYM
ejpam-3548	43	7	a(x)∪{1x	a(x)∪{1x	NOUN
ejpam-3548	43	8	}	}	PUNCT
ejpam-3548	43	9	.	.	PUNCT
ejpam-3548	44	1	a	a	DET
ejpam-3548	44	2	be	be	NOUN
ejpam-3548	44	3	-	-	PUNCT
ejpam-3548	44	4	algebra	algebra	NOUN
ejpam-3548	44	5	x	x	PUNCT
ejpam-3548	44	6	is	be	AUX
ejpam-3548	44	7	called	call	VERB
ejpam-3548	44	8	dual	dual	ADJ
ejpam-3548	44	9	atomistic	atomistic	ADJ
ejpam-3548	44	10	if	if	SCONJ
ejpam-3548	44	11	every	every	DET
ejpam-3548	44	12	non	non	ADJ
ejpam-3548	44	13	-	-	ADJ
ejpam-3548	44	14	unit	unit	ADJ
ejpam-3548	44	15	element	element	NOUN
ejpam-3548	44	16	of	of	ADP
ejpam-3548	44	17	x	x	PUNCT
ejpam-3548	44	18	is	be	AUX
ejpam-3548	44	19	a	a	DET
ejpam-3548	44	20	dual	dual	ADJ
ejpam-3548	44	21	atom	atom	NOUN
ejpam-3548	44	22	in	in	ADP
ejpam-3548	44	23	x	x	NOUN
ejpam-3548	44	24	,	,	PUNCT
ejpam-3548	44	25	that	that	ADV
ejpam-3548	44	26	is	is	ADV
ejpam-3548	44	27	,	,	PUNCT
ejpam-3548	44	28	x	x	PUNCT
ejpam-3548	44	29	=	=	SYM
ejpam-3548	44	30	a1(x	a1(x	NOUN
ejpam-3548	44	31	)	)	PUNCT
ejpam-3548	44	32	,	,	PUNCT
ejpam-3548	44	33	see	see	VERB
ejpam-3548	44	34	[	[	X
ejpam-3548	44	35	1	1	NUM
ejpam-3548	44	36	,	,	PUNCT
ejpam-3548	44	37	8	8	NUM
ejpam-3548	44	38	]	]	PUNCT
ejpam-3548	44	39	.	.	PUNCT
ejpam-3548	45	1	example	example	NOUN
ejpam-3548	46	1	1	1	NUM
ejpam-3548	46	2	.	.	PUNCT
ejpam-3548	47	1	[	[	X
ejpam-3548	47	2	8	8	NUM
ejpam-3548	47	3	]	]	PUNCT
ejpam-3548	47	4	let	let	VERB
ejpam-3548	47	5	n0	n0	X
ejpam-3548	47	6	=	=	PUNCT
ejpam-3548	47	7	n	n	CCONJ
ejpam-3548	47	8	∪	∪	X
ejpam-3548	47	9	{	{	PUNCT
ejpam-3548	47	10	0	0	NUM
ejpam-3548	47	11	}	}	PUNCT
ejpam-3548	47	12	and	and	CCONJ
ejpam-3548	47	13	let	let	VERB
ejpam-3548	47	14	∗	∗	NOUN
ejpam-3548	47	15	be	be	AUX
ejpam-3548	47	16	the	the	DET
ejpam-3548	47	17	binary	binary	ADJ
ejpam-3548	47	18	operation	operation	NOUN
ejpam-3548	47	19	on	on	ADP
ejpam-3548	47	20	n0	n0	NUM
ejpam-3548	47	21	defined	define	VERB
ejpam-3548	47	22	by	by	ADP
ejpam-3548	47	23	x	x	PROPN
ejpam-3548	47	24	∗	∗	NOUN
ejpam-3548	47	25	y	y	NOUN
ejpam-3548	47	26	=	=	PUNCT
ejpam-3548	47	27	{	{	PUNCT
ejpam-3548	47	28	0	0	NUM
ejpam-3548	47	29	if	if	SCONJ
ejpam-3548	47	30	y	y	PROPN
ejpam-3548	47	31	≤	≤	X
ejpam-3548	47	32	x	x	PUNCT
ejpam-3548	48	1	y	y	NOUN
ejpam-3548	48	2	−	−	NOUN
ejpam-3548	48	3	x	x	SYM
ejpam-3548	48	4	if	if	SCONJ
ejpam-3548	48	5	x	x	X
ejpam-3548	48	6	<	<	X
ejpam-3548	48	7	y.	y.	NOUN
ejpam-3548	48	8	then	then	ADV
ejpam-3548	48	9	(	(	PUNCT
ejpam-3548	48	10	n0	n0	X
ejpam-3548	48	11	;	;	PUNCT
ejpam-3548	48	12	∗	∗	NOUN
ejpam-3548	48	13	,	,	PUNCT
ejpam-3548	48	14	0	0	NUM
ejpam-3548	48	15	)	)	PUNCT
ejpam-3548	48	16	is	be	AUX
ejpam-3548	48	17	a	a	DET
ejpam-3548	48	18	commutative	commutative	ADJ
ejpam-3548	48	19	be	be	NOUN
ejpam-3548	48	20	-	-	PUNCT
ejpam-3548	48	21	algebra	algebra	NOUN
ejpam-3548	48	22	where	where	SCONJ
ejpam-3548	48	23	1n0	1n0	NUM
ejpam-3548	48	24	=	=	SYM
ejpam-3548	48	25	0	0	X
ejpam-3548	48	26	.	.	PUNCT
ejpam-3548	49	1	it	it	PRON
ejpam-3548	49	2	can	can	AUX
ejpam-3548	49	3	be	be	AUX
ejpam-3548	49	4	seen	see	VERB
ejpam-3548	49	5	that	that	SCONJ
ejpam-3548	49	6	a(n0	a(n0	VERB
ejpam-3548	49	7	)	)	PUNCT
ejpam-3548	49	8	=	=	SYM
ejpam-3548	49	9	{	{	PUNCT
ejpam-3548	49	10	1	1	NUM
ejpam-3548	49	11	}	}	PUNCT
ejpam-3548	49	12	.	.	PUNCT
ejpam-3548	50	1	lemma	lemma	PROPN
ejpam-3548	50	2	1	1	NUM
ejpam-3548	50	3	.	.	PUNCT
ejpam-3548	51	1	[	[	X
ejpam-3548	51	2	9	9	NUM
ejpam-3548	51	3	]	]	X
ejpam-3548	51	4	let	let	VERB
ejpam-3548	51	5	(	(	PUNCT
ejpam-3548	51	6	x	x	X
ejpam-3548	51	7	;	;	PUNCT
ejpam-3548	51	8	∗	∗	NOUN
ejpam-3548	51	9	,	,	PUNCT
ejpam-3548	51	10	1x	1x	NUM
ejpam-3548	51	11	)	)	PUNCT
ejpam-3548	51	12	be	be	AUX
ejpam-3548	51	13	a	a	DET
ejpam-3548	51	14	be	be	NOUN
ejpam-3548	51	15	-	-	PUNCT
ejpam-3548	51	16	algebra	algebra	NOUN
ejpam-3548	51	17	and	and	CCONJ
ejpam-3548	51	18	let	let	VERB
ejpam-3548	51	19	i	i	PRON
ejpam-3548	51	20	be	be	AUX
ejpam-3548	51	21	a	a	DET
ejpam-3548	51	22	non	non	ADJ
ejpam-3548	51	23	-	-	ADJ
ejpam-3548	51	24	empty	empty	ADJ
ejpam-3548	51	25	subset	subset	NOUN
ejpam-3548	51	26	of	of	ADP
ejpam-3548	51	27	x.	x.	NOUN
ejpam-3548	52	1	then	then	ADV
ejpam-3548	52	2	i	i	PRON
ejpam-3548	52	3	is	be	AUX
ejpam-3548	52	4	an	an	DET
ejpam-3548	52	5	ideal	ideal	NOUN
ejpam-3548	52	6	of	of	ADP
ejpam-3548	52	7	x	x	PUNCT
ejpam-3548	52	8	if	if	SCONJ
ejpam-3548	52	9	and	and	CCONJ
ejpam-3548	52	10	only	only	ADV
ejpam-3548	52	11	if	if	SCONJ
ejpam-3548	52	12	it	it	PRON
ejpam-3548	52	13	satisfies	satisfy	VERB
ejpam-3548	52	14	(	(	PUNCT
ejpam-3548	52	15	i	i	NOUN
ejpam-3548	52	16	)	)	PUNCT
ejpam-3548	52	17	1x	1x	PROPN
ejpam-3548	53	1	∈	∈	PROPN
ejpam-3548	54	1	i	i	PRON
ejpam-3548	54	2	;	;	PUNCT
ejpam-3548	54	3	and	and	CCONJ
ejpam-3548	54	4	(	(	PUNCT
ejpam-3548	54	5	ii	ii	NOUN
ejpam-3548	54	6	)	)	PUNCT
ejpam-3548	54	7	for	for	ADP
ejpam-3548	54	8	all	all	DET
ejpam-3548	54	9	x	x	NOUN
ejpam-3548	54	10	,	,	PUNCT
ejpam-3548	54	11	z	z	NOUN
ejpam-3548	54	12	∈	∈	PROPN
ejpam-3548	54	13	x	x	X
ejpam-3548	54	14	and	and	CCONJ
ejpam-3548	54	15	for	for	ADP
ejpam-3548	54	16	all	all	DET
ejpam-3548	54	17	y	y	PROPN
ejpam-3548	54	18	∈	∈	PROPN
ejpam-3548	54	19	i	i	PRON
ejpam-3548	54	20	,	,	PUNCT
ejpam-3548	54	21	(	(	PUNCT
ejpam-3548	54	22	x	x	X
ejpam-3548	54	23	∗	∗	NOUN
ejpam-3548	54	24	(	(	PUNCT
ejpam-3548	54	25	y	y	PROPN
ejpam-3548	54	26	∗	∗	PROPN
ejpam-3548	54	27	z	z	NOUN
ejpam-3548	54	28	)	)	PUNCT
ejpam-3548	54	29	)	)	PUNCT
ejpam-3548	55	1	∈	∈	PROPN
ejpam-3548	55	2	i	i	PRON
ejpam-3548	55	3	implies	imply	VERB
ejpam-3548	55	4	x	x	X
ejpam-3548	55	5	∗	∗	NOUN
ejpam-3548	55	6	z	z	PROPN
ejpam-3548	55	7	∈	∈	PROPN
ejpam-3548	55	8	i.	i.	NOUN
ejpam-3548	55	9	let	let	VERB
ejpam-3548	55	10	y	y	PRON
ejpam-3548	55	11	be	be	AUX
ejpam-3548	55	12	a	a	DET
ejpam-3548	55	13	non	non	ADJ
ejpam-3548	55	14	-	-	ADJ
ejpam-3548	55	15	empty	empty	ADJ
ejpam-3548	55	16	set	set	NOUN
ejpam-3548	55	17	.	.	PUNCT
ejpam-3548	56	1	a	a	DET
ejpam-3548	56	2	collection	collection	NOUN
ejpam-3548	56	3	τ	τ	PROPN
ejpam-3548	56	4	of	of	ADP
ejpam-3548	56	5	subsets	subset	NOUN
ejpam-3548	56	6	of	of	ADP
ejpam-3548	56	7	y	y	PROPN
ejpam-3548	56	8	is	be	AUX
ejpam-3548	56	9	a	a	DET
ejpam-3548	56	10	topology	topology	NOUN
ejpam-3548	56	11	on	on	ADP
ejpam-3548	56	12	y	y	PROPN
ejpam-3548	56	13	if	if	SCONJ
ejpam-3548	56	14	it	it	PRON
ejpam-3548	56	15	satisfies	satisfy	VERB
ejpam-3548	56	16	the	the	DET
ejpam-3548	56	17	following	follow	VERB
ejpam-3548	56	18	axioms	axiom	NOUN
ejpam-3548	56	19	:	:	PUNCT
ejpam-3548	56	20	(	(	PUNCT
ejpam-3548	56	21	g1	g1	X
ejpam-3548	56	22	)	)	PUNCT
ejpam-3548	56	23	∅	∅	NOUN
ejpam-3548	56	24	and	and	CCONJ
ejpam-3548	56	25	y	y	PROPN
ejpam-3548	56	26	belong	belong	VERB
ejpam-3548	56	27	to	to	ADP
ejpam-3548	56	28	τ	τ	PROPN
ejpam-3548	56	29	;	;	PUNCT
ejpam-3548	56	30	(	(	PUNCT
ejpam-3548	56	31	g2	g2	PROPN
ejpam-3548	56	32	)	)	PUNCT
ejpam-3548	56	33	if	if	SCONJ
ejpam-3548	56	34	g1	g1	PROPN
ejpam-3548	56	35	and	and	CCONJ
ejpam-3548	56	36	g2	g2	PROPN
ejpam-3548	56	37	are	be	AUX
ejpam-3548	56	38	elements	element	NOUN
ejpam-3548	56	39	of	of	ADP
ejpam-3548	56	40	τ	τ	PROPN
ejpam-3548	56	41	,	,	PUNCT
ejpam-3548	56	42	then	then	ADV
ejpam-3548	56	43	g1	g1	VERB
ejpam-3548	56	44	∩g2	∩g2	PROPN
ejpam-3548	56	45	∈	∈	PROPN
ejpam-3548	56	46	τ	τ	X
ejpam-3548	56	47	;	;	PUNCT
ejpam-3548	56	48	and	and	CCONJ
ejpam-3548	56	49	(	(	PUNCT
ejpam-3548	56	50	g3	g3	NOUN
ejpam-3548	56	51	)	)	PUNCT
ejpam-3548	56	52	if	if	SCONJ
ejpam-3548	56	53	{	{	PUNCT
ejpam-3548	56	54	gi	gi	X
ejpam-3548	56	55	:	:	PUNCT
ejpam-3548	56	56	i	i	PRON
ejpam-3548	56	57	∈	∈	VERB
ejpam-3548	56	58	i	i	PRON
ejpam-3548	56	59	}	}	PUNCT
ejpam-3548	56	60	⊆	⊆	NUM
ejpam-3548	56	61	τ	τ	X
ejpam-3548	56	62	,	,	PUNCT
ejpam-3548	56	63	then	then	ADV
ejpam-3548	56	64	⋃	⋃	PUNCT
ejpam-3548	56	65	i∈i	i∈i	ADJ
ejpam-3548	56	66	gi	gi	NOUN
ejpam-3548	56	67	∈	∈	PROPN
ejpam-3548	56	68	τ	τ	X
ejpam-3548	56	69	.	.	PUNCT
ejpam-3548	57	1	if	if	SCONJ
ejpam-3548	57	2	τ	τ	PROPN
ejpam-3548	57	3	is	be	AUX
ejpam-3548	57	4	a	a	DET
ejpam-3548	57	5	topology	topology	NOUN
ejpam-3548	57	6	on	on	ADP
ejpam-3548	57	7	y	y	PROPN
ejpam-3548	57	8	,	,	PUNCT
ejpam-3548	57	9	then	then	ADV
ejpam-3548	57	10	the	the	DET
ejpam-3548	57	11	ordered	ordered	ADJ
ejpam-3548	57	12	pair	pair	NOUN
ejpam-3548	57	13	(	(	PUNCT
ejpam-3548	57	14	y	y	PROPN
ejpam-3548	57	15	,	,	PUNCT
ejpam-3548	57	16	τ	τ	X
ejpam-3548	57	17	)	)	PUNCT
ejpam-3548	57	18	is	be	AUX
ejpam-3548	57	19	called	call	VERB
ejpam-3548	57	20	a	a	DET
ejpam-3548	57	21	topological	topological	ADJ
ejpam-3548	57	22	space	space	NOUN
ejpam-3548	57	23	.	.	PUNCT
ejpam-3548	58	1	an	an	DET
ejpam-3548	58	2	element	element	NOUN
ejpam-3548	58	3	o	o	NOUN
ejpam-3548	58	4	of	of	ADP
ejpam-3548	58	5	τ	τ	PROPN
ejpam-3548	58	6	is	be	AUX
ejpam-3548	58	7	called	call	VERB
ejpam-3548	58	8	a	a	DET
ejpam-3548	58	9	τ	τ	X
ejpam-3548	58	10	-open	-open	NOUN
ejpam-3548	58	11	set	set	NOUN
ejpam-3548	58	12	(	(	PUNCT
ejpam-3548	58	13	or	or	CCONJ
ejpam-3548	58	14	simply	simply	ADV
ejpam-3548	58	15	open	open	VERB
ejpam-3548	58	16	set	set	NOUN
ejpam-3548	58	17	)	)	PUNCT
ejpam-3548	58	18	and	and	CCONJ
ejpam-3548	58	19	the	the	DET
ejpam-3548	58	20	complement	complement	NOUN
ejpam-3548	58	21	of	of	ADP
ejpam-3548	58	22	o	o	PROPN
ejpam-3548	58	23	is	be	AUX
ejpam-3548	58	24	called	call	VERB
ejpam-3548	58	25	a	a	DET
ejpam-3548	58	26	τ	τ	X
ejpam-3548	58	27	-closed	-close	VERB
ejpam-3548	58	28	set	set	NOUN
ejpam-3548	58	29	(	(	PUNCT
ejpam-3548	58	30	or	or	CCONJ
ejpam-3548	58	31	simply	simply	ADV
ejpam-3548	58	32	closed	close	VERB
ejpam-3548	58	33	set	set	NOUN
ejpam-3548	58	34	)	)	PUNCT
ejpam-3548	58	35	.	.	PUNCT
ejpam-3548	59	1	the	the	DET
ejpam-3548	59	2	discrete	discrete	ADJ
ejpam-3548	59	3	topology	topology	NOUN
ejpam-3548	59	4	on	on	ADP
ejpam-3548	59	5	y	y	PROPN
ejpam-3548	59	6	is	be	AUX
ejpam-3548	59	7	dy	dy	X
ejpam-3548	59	8	=	=	SYM
ejpam-3548	59	9	p(y	p(y	PROPN
ejpam-3548	59	10	)	)	PUNCT
ejpam-3548	59	11	.	.	PUNCT
ejpam-3548	60	1	a	a	DET
ejpam-3548	60	2	class	class	NOUN
ejpam-3548	60	3	b	b	NOUN
ejpam-3548	60	4	⊆	⊆	NUM
ejpam-3548	60	5	τ	τ	X
ejpam-3548	60	6	is	be	AUX
ejpam-3548	60	7	a	a	DET
ejpam-3548	60	8	basis	basis	NOUN
ejpam-3548	60	9	for	for	ADP
ejpam-3548	60	10	τ	τ	PROPN
ejpam-3548	60	11	if	if	SCONJ
ejpam-3548	60	12	each	each	DET
ejpam-3548	60	13	open	open	ADJ
ejpam-3548	60	14	set	set	NOUN
ejpam-3548	60	15	is	be	AUX
ejpam-3548	60	16	the	the	DET
ejpam-3548	60	17	union	union	NOUN
ejpam-3548	60	18	of	of	ADP
ejpam-3548	60	19	members	member	NOUN
ejpam-3548	60	20	of	of	ADP
ejpam-3548	60	21	b.	b.	PROPN
ejpam-3548	60	22	the	the	DET
ejpam-3548	60	23	elements	element	NOUN
ejpam-3548	60	24	of	of	ADP
ejpam-3548	60	25	a	a	DET
ejpam-3548	60	26	basis	basis	NOUN
ejpam-3548	60	27	are	be	AUX
ejpam-3548	60	28	called	call	VERB
ejpam-3548	60	29	basic	basic	ADJ
ejpam-3548	60	30	open	open	ADJ
ejpam-3548	60	31	sets	set	NOUN
ejpam-3548	60	32	.	.	PUNCT
ejpam-3548	61	1	the	the	DET
ejpam-3548	61	2	topology	topology	NOUN
ejpam-3548	61	3	τ	τ	PROPN
ejpam-3548	61	4	is	be	AUX
ejpam-3548	61	5	said	say	VERB
ejpam-3548	61	6	to	to	PART
ejpam-3548	61	7	be	be	AUX
ejpam-3548	61	8	generated	generate	VERB
ejpam-3548	61	9	by	by	ADP
ejpam-3548	61	10	a	a	DET
ejpam-3548	61	11	basis	basis	NOUN
ejpam-3548	61	12	b	b	NOUN
ejpam-3548	61	13	if	if	SCONJ
ejpam-3548	61	14	the	the	DET
ejpam-3548	61	15	family	family	NOUN
ejpam-3548	61	16	τ	τ	PROPN
ejpam-3548	61	17	consists	consist	VERB
ejpam-3548	61	18	∅	∅	NOUN
ejpam-3548	61	19	,	,	PUNCT
ejpam-3548	61	20	y	y	PROPN
ejpam-3548	61	21	,	,	PUNCT
ejpam-3548	61	22	and	and	CCONJ
ejpam-3548	61	23	all	all	DET
ejpam-3548	61	24	unions	union	NOUN
ejpam-3548	61	25	of	of	ADP
ejpam-3548	61	26	members	member	NOUN
ejpam-3548	61	27	of	of	ADP
ejpam-3548	61	28	b.	b.	PROPN
ejpam-3548	61	29	a	a	DET
ejpam-3548	61	30	class	class	NOUN
ejpam-3548	61	31	s	s	X
ejpam-3548	61	32	of	of	ADP
ejpam-3548	61	33	open	open	ADJ
ejpam-3548	61	34	subsets	subset	NOUN
ejpam-3548	61	35	of	of	ADP
ejpam-3548	61	36	y	y	PROPN
ejpam-3548	61	37	,	,	PUNCT
ejpam-3548	61	38	that	that	ADV
ejpam-3548	61	39	is	is	ADV
ejpam-3548	61	40	,	,	PUNCT
ejpam-3548	61	41	s	s	VERB
ejpam-3548	61	42	⊆	⊆	NUM
ejpam-3548	61	43	τ	τ	X
ejpam-3548	61	44	,	,	PUNCT
ejpam-3548	61	45	is	be	AUX
ejpam-3548	61	46	a	a	DET
ejpam-3548	61	47	subbase	subbase	NOUN
ejpam-3548	61	48	or	or	CCONJ
ejpam-3548	61	49	subbasis	subbasis	VERB
ejpam-3548	61	50	for	for	ADP
ejpam-3548	61	51	the	the	DET
ejpam-3548	61	52	topology	topology	NOUN
ejpam-3548	61	53	τ	τ	PROPN
ejpam-3548	61	54	on	on	ADP
ejpam-3548	61	55	y	y	PROPN
ejpam-3548	61	56	if	if	SCONJ
ejpam-3548	62	1	and	and	CCONJ
ejpam-3548	62	2	only	only	ADV
ejpam-3548	62	3	if	if	SCONJ
ejpam-3548	62	4	finite	finite	ADJ
ejpam-3548	62	5	intersections	intersection	NOUN
ejpam-3548	62	6	of	of	ADP
ejpam-3548	62	7	members	member	NOUN
ejpam-3548	62	8	of	of	ADP
ejpam-3548	62	9	s	s	NOUN
ejpam-3548	62	10	form	form	NOUN
ejpam-3548	62	11	a	a	DET
ejpam-3548	62	12	basis	basis	NOUN
ejpam-3548	62	13	for	for	ADP
ejpam-3548	62	14	τ	τ	PROPN
ejpam-3548	62	15	.	.	PUNCT
ejpam-3548	62	16	suppose	suppose	VERB
ejpam-3548	62	17	that	that	SCONJ
ejpam-3548	62	18	x	x	SYM
ejpam-3548	62	19	∈	∈	PROPN
ejpam-3548	62	20	y	y	NOUN
ejpam-3548	62	21	and	and	CCONJ
ejpam-3548	62	22	u	u	NOUN
ejpam-3548	62	23	⊆	⊆	NUM
ejpam-3548	62	24	y	y	NOUN
ejpam-3548	62	25	.	.	PUNCT
ejpam-3548	63	1	u	u	NOUN
ejpam-3548	63	2	is	be	AUX
ejpam-3548	63	3	a	a	DET
ejpam-3548	63	4	neighborhood	neighborhood	NOUN
ejpam-3548	63	5	of	of	ADP
ejpam-3548	63	6	x	x	PART
ejpam-3548	63	7	(	(	PUNCT
ejpam-3548	63	8	briefly	briefly	NOUN
ejpam-3548	63	9	nbd	nbd	PROPN
ejpam-3548	63	10	u(x	u(x	PROPN
ejpam-3548	63	11	)	)	PUNCT
ejpam-3548	63	12	)	)	PUNCT
ejpam-3548	64	1	if	if	SCONJ
ejpam-3548	64	2	x	x	SYM
ejpam-3548	64	3	∈	∈	PROPN
ejpam-3548	64	4	u	u	NOUN
ejpam-3548	64	5	and	and	CCONJ
ejpam-3548	64	6	u	u	NOUN
ejpam-3548	64	7	∈	∈	PROPN
ejpam-3548	64	8	τ	τ	X
ejpam-3548	64	9	.	.	PUNCT
ejpam-3548	65	1	throughout	throughout	ADP
ejpam-3548	65	2	this	this	DET
ejpam-3548	65	3	paper	paper	NOUN
ejpam-3548	65	4	,	,	PUNCT
ejpam-3548	65	5	we	we	PRON
ejpam-3548	65	6	denote	denote	VERB
ejpam-3548	65	7	a	a	DET
ejpam-3548	65	8	topological	topological	ADJ
ejpam-3548	65	9	space	space	NOUN
ejpam-3548	65	10	(	(	PUNCT
ejpam-3548	65	11	y	y	PROPN
ejpam-3548	65	12	,	,	PUNCT
ejpam-3548	65	13	τ	τ	PROPN
ejpam-3548	65	14	)	)	PUNCT
ejpam-3548	65	15	by	by	ADP
ejpam-3548	65	16	y	y	PROPN
ejpam-3548	65	17	,	,	PUNCT
ejpam-3548	65	18	unless	unless	SCONJ
ejpam-3548	65	19	otherwise	otherwise	ADV
ejpam-3548	65	20	specified	specify	VERB
ejpam-3548	65	21	.	.	PUNCT
ejpam-3548	66	1	let	let	VERB
ejpam-3548	66	2	a	a	DET
ejpam-3548	66	3	be	be	AUX
ejpam-3548	66	4	a	a	DET
ejpam-3548	66	5	subset	subset	NOUN
ejpam-3548	66	6	of	of	ADP
ejpam-3548	66	7	a	a	DET
ejpam-3548	66	8	topological	topological	ADJ
ejpam-3548	66	9	space	space	NOUN
ejpam-3548	66	10	y	y	PROPN
ejpam-3548	66	11	.	.	PUNCT
ejpam-3548	67	1	a	a	DET
ejpam-3548	67	2	j.	j.	PROPN
ejpam-3548	67	3	albaracin	albaracin	PROPN
ejpam-3548	67	4	,	,	PUNCT
ejpam-3548	67	5	j.	j.	PROPN
ejpam-3548	67	6	vilela	vilela	PROPN
ejpam-3548	67	7	/	/	SYM
ejpam-3548	67	8	eur	eur	PROPN
ejpam-3548	67	9	.	.	PUNCT
ejpam-3548	68	1	j.	j.	PROPN
ejpam-3548	68	2	pure	pure	PROPN
ejpam-3548	68	3	appl	appl	PROPN
ejpam-3548	68	4	.	.	PROPN
ejpam-3548	68	5	math	math	PROPN
ejpam-3548	68	6	,	,	PUNCT
ejpam-3548	68	7	12	12	NUM
ejpam-3548	68	8	(	(	PUNCT
ejpam-3548	68	9	4	4	NUM
ejpam-3548	68	10	)	)	PUNCT
ejpam-3548	68	11	(	(	PUNCT
ejpam-3548	68	12	2019	2019	NUM
ejpam-3548	68	13	)	)	PUNCT
ejpam-3548	68	14	,	,	PUNCT
ejpam-3548	68	15	1584	1584	NUM
ejpam-3548	68	16	-	-	SYM
ejpam-3548	68	17	1594	1594	NUM
ejpam-3548	68	18	1586	1586	NUM
ejpam-3548	68	19	point	point	NOUN
ejpam-3548	68	20	x	x	X
ejpam-3548	68	21	∈	∈	NOUN
ejpam-3548	68	22	y	y	NOUN
ejpam-3548	68	23	is	be	AUX
ejpam-3548	68	24	adherent	adherent	ADJ
ejpam-3548	68	25	to	to	ADP
ejpam-3548	68	26	a	a	DET
ejpam-3548	68	27	or	or	CCONJ
ejpam-3548	68	28	closure	closure	NOUN
ejpam-3548	68	29	point	point	NOUN
ejpam-3548	68	30	of	of	ADP
ejpam-3548	68	31	a	a	PRON
ejpam-3548	68	32	if	if	SCONJ
ejpam-3548	68	33	each	each	DET
ejpam-3548	68	34	neighborhood	neighborhood	NOUN
ejpam-3548	68	35	of	of	ADP
ejpam-3548	68	36	x	x	PUNCT
ejpam-3548	68	37	contains	contain	VERB
ejpam-3548	68	38	at	at	ADV
ejpam-3548	68	39	least	least	ADV
ejpam-3548	68	40	one	one	NUM
ejpam-3548	68	41	point	point	NOUN
ejpam-3548	68	42	of	of	ADP
ejpam-3548	68	43	a	a	PRON
ejpam-3548	68	44	(	(	PUNCT
ejpam-3548	68	45	which	which	PRON
ejpam-3548	68	46	maybe	maybe	ADV
ejpam-3548	68	47	x	x	X
ejpam-3548	68	48	itself	itself	PRON
ejpam-3548	68	49	)	)	PUNCT
ejpam-3548	68	50	.	.	PUNCT
ejpam-3548	69	1	the	the	DET
ejpam-3548	69	2	set	set	NOUN
ejpam-3548	69	3	of	of	ADP
ejpam-3548	69	4	all	all	DET
ejpam-3548	69	5	points	point	NOUN
ejpam-3548	69	6	in	in	ADP
ejpam-3548	69	7	y	y	PROPN
ejpam-3548	69	8	adherent	adherent	NOUN
ejpam-3548	69	9	to	to	ADP
ejpam-3548	69	10	a	a	PRON
ejpam-3548	69	11	,	,	PUNCT
ejpam-3548	69	12	denoted	denote	VERB
ejpam-3548	69	13	by	by	ADP
ejpam-3548	69	14	a	a	PRON
ejpam-3548	69	15	,	,	PUNCT
ejpam-3548	69	16	is	be	AUX
ejpam-3548	69	17	called	call	VERB
ejpam-3548	69	18	the	the	DET
ejpam-3548	69	19	closure	closure	NOUN
ejpam-3548	69	20	of	of	ADP
ejpam-3548	69	21	a	a	PRON
ejpam-3548	69	22	,	,	PUNCT
ejpam-3548	69	23	that	that	ADV
ejpam-3548	69	24	is	is	ADV
ejpam-3548	69	25	,	,	PUNCT
ejpam-3548	69	26	a	a	DET
ejpam-3548	69	27	=	=	X
ejpam-3548	69	28	{	{	PUNCT
ejpam-3548	69	29	x	x	SYM
ejpam-3548	69	30	∈	∈	PROPN
ejpam-3548	69	31	y	y	NOUN
ejpam-3548	69	32	|	|	ADV
ejpam-3548	69	33	∀	∀	NOUN
ejpam-3548	69	34	u(x	u(x	NOUN
ejpam-3548	69	35	)	)	PUNCT
ejpam-3548	69	36	:	:	PUNCT
ejpam-3548	69	37	u(x	u(x	PROPN
ejpam-3548	69	38	)	)	PUNCT
ejpam-3548	69	39	∩a	∩a	NOUN
ejpam-3548	69	40	6=	6=	NUM
ejpam-3548	69	41	∅	∅	NOUN
ejpam-3548	69	42	}	}	PUNCT
ejpam-3548	69	43	.	.	PUNCT
ejpam-3548	70	1	a	a	DET
ejpam-3548	70	2	point	point	NOUN
ejpam-3548	70	3	p	p	X
ejpam-3548	70	4	∈	∈	PROPN
ejpam-3548	70	5	a	a	PRON
ejpam-3548	70	6	is	be	AUX
ejpam-3548	70	7	called	call	VERB
ejpam-3548	70	8	an	an	DET
ejpam-3548	70	9	interior	interior	ADJ
ejpam-3548	70	10	point	point	NOUN
ejpam-3548	70	11	of	of	ADP
ejpam-3548	70	12	a	a	PRON
ejpam-3548	70	13	if	if	SCONJ
ejpam-3548	70	14	p	p	NOUN
ejpam-3548	70	15	belongs	belong	VERB
ejpam-3548	70	16	to	to	ADP
ejpam-3548	70	17	an	an	DET
ejpam-3548	70	18	open	open	ADJ
ejpam-3548	70	19	set	set	NOUN
ejpam-3548	70	20	g	g	NOUN
ejpam-3548	70	21	in	in	ADP
ejpam-3548	70	22	y	y	PROPN
ejpam-3548	70	23	contained	contain	VERB
ejpam-3548	70	24	in	in	ADP
ejpam-3548	70	25	a	a	PRON
ejpam-3548	70	26	,	,	PUNCT
ejpam-3548	70	27	that	that	ADV
ejpam-3548	70	28	is	is	ADV
ejpam-3548	70	29	,	,	PUNCT
ejpam-3548	70	30	p	p	PROPN
ejpam-3548	70	31	∈	∈	PROPN
ejpam-3548	70	32	g	g	ADP
ejpam-3548	70	33	⊆	⊆	NUM
ejpam-3548	70	34	a.	a.	NOUN
ejpam-3548	70	35	the	the	DET
ejpam-3548	70	36	set	set	NOUN
ejpam-3548	70	37	of	of	ADP
ejpam-3548	70	38	all	all	DET
ejpam-3548	70	39	interior	interior	ADJ
ejpam-3548	70	40	points	point	NOUN
ejpam-3548	70	41	of	of	ADP
ejpam-3548	70	42	a	a	PRON
ejpam-3548	70	43	,	,	PUNCT
ejpam-3548	70	44	denoted	denote	VERB
ejpam-3548	70	45	by	by	ADP
ejpam-3548	70	46	int(a	int(a	PROPN
ejpam-3548	70	47	)	)	PUNCT
ejpam-3548	70	48	,	,	PUNCT
ejpam-3548	70	49	is	be	AUX
ejpam-3548	70	50	called	call	VERB
ejpam-3548	70	51	the	the	DET
ejpam-3548	70	52	interior	interior	NOUN
ejpam-3548	70	53	of	of	ADP
ejpam-3548	70	54	a	a	PRON
ejpam-3548	70	55	,	,	PUNCT
ejpam-3548	70	56	that	that	ADV
ejpam-3548	70	57	is	is	ADV
ejpam-3548	70	58	,	,	PUNCT
ejpam-3548	70	59	the	the	DET
ejpam-3548	70	60	interior	interior	NOUN
ejpam-3548	70	61	of	of	ADP
ejpam-3548	70	62	a	a	PRON
ejpam-3548	70	63	is	be	AUX
ejpam-3548	70	64	the	the	DET
ejpam-3548	70	65	largest	large	ADJ
ejpam-3548	70	66	open	open	ADJ
ejpam-3548	70	67	set	set	NOUN
ejpam-3548	70	68	contained	contain	VERB
ejpam-3548	70	69	in	in	ADP
ejpam-3548	70	70	a	a	PRON
ejpam-3548	70	71	,	,	PUNCT
ejpam-3548	70	72	or	or	CCONJ
ejpam-3548	70	73	,	,	PUNCT
ejpam-3548	70	74	int(a	int(a	PROPN
ejpam-3548	70	75	)	)	PUNCT
ejpam-3548	70	76	=	=	SYM
ejpam-3548	70	77	⋃	⋃	NOUN
ejpam-3548	70	78	{	{	PUNCT
ejpam-3548	70	79	u	u	NOUN
ejpam-3548	70	80	|	|	ADV
ejpam-3548	70	81	u	u	NOUN
ejpam-3548	70	82	is	be	AUX
ejpam-3548	70	83	open	open	ADJ
ejpam-3548	70	84	and	and	CCONJ
ejpam-3548	70	85	u	u	PRON
ejpam-3548	70	86	⊆	⊆	NUM
ejpam-3548	70	87	a	a	PRON
ejpam-3548	70	88	}	}	PUNCT
ejpam-3548	70	89	.	.	PUNCT
ejpam-3548	71	1	d	d	NOUN
ejpam-3548	71	2	⊆	⊆	NUM
ejpam-3548	71	3	y	y	PROPN
ejpam-3548	71	4	is	be	AUX
ejpam-3548	71	5	dense	dense	ADJ
ejpam-3548	71	6	in	in	ADP
ejpam-3548	71	7	y	y	PROPN
ejpam-3548	71	8	if	if	SCONJ
ejpam-3548	71	9	d	d	PROPN
ejpam-3548	71	10	=	=	SYM
ejpam-3548	71	11	y	y	PROPN
ejpam-3548	71	12	.	.	PUNCT
ejpam-3548	72	1	let	let	VERB
ejpam-3548	72	2	z	z	PROPN
ejpam-3548	72	3	⊆	⊆	NUM
ejpam-3548	72	4	y	y	PROPN
ejpam-3548	72	5	.	.	PUNCT
ejpam-3548	73	1	the	the	DET
ejpam-3548	73	2	topology	topology	NOUN
ejpam-3548	73	3	τz	τz	ADP
ejpam-3548	73	4	on	on	ADP
ejpam-3548	73	5	y	y	PROPN
ejpam-3548	73	6	defined	define	VERB
ejpam-3548	73	7	as	as	ADP
ejpam-3548	73	8	τz	τz	ADP
ejpam-3548	73	9	=	=	VERB
ejpam-3548	73	10	{	{	PUNCT
ejpam-3548	73	11	z	z	NOUN
ejpam-3548	73	12	∩	∩	ADJ
ejpam-3548	73	13	o	o	NOUN
ejpam-3548	73	14	:	:	PUNCT
ejpam-3548	73	15	o	o	X
ejpam-3548	73	16	∈	∈	PROPN
ejpam-3548	73	17	τ	τ	PROPN
ejpam-3548	73	18	}	}	PUNCT
ejpam-3548	73	19	is	be	AUX
ejpam-3548	73	20	called	call	VERB
ejpam-3548	73	21	the	the	DET
ejpam-3548	73	22	relative	relative	ADJ
ejpam-3548	73	23	topology	topology	NOUN
ejpam-3548	73	24	on	on	ADP
ejpam-3548	73	25	y	y	PROPN
ejpam-3548	73	26	.	.	PUNCT
ejpam-3548	74	1	in	in	ADP
ejpam-3548	74	2	this	this	DET
ejpam-3548	74	3	case	case	NOUN
ejpam-3548	74	4	,	,	PUNCT
ejpam-3548	74	5	(	(	PUNCT
ejpam-3548	74	6	z	z	NOUN
ejpam-3548	74	7	,	,	PUNCT
ejpam-3548	74	8	τz	τz	ADP
ejpam-3548	74	9	)	)	PUNCT
ejpam-3548	74	10	is	be	AUX
ejpam-3548	74	11	called	call	VERB
ejpam-3548	74	12	a	a	DET
ejpam-3548	74	13	subspace	subspace	NOUN
ejpam-3548	74	14	of	of	ADP
ejpam-3548	74	15	(	(	PUNCT
ejpam-3548	74	16	y	y	PROPN
ejpam-3548	74	17	,	,	PUNCT
ejpam-3548	74	18	τ	τ	PROPN
ejpam-3548	74	19	)	)	PUNCT
ejpam-3548	74	20	.	.	PUNCT
ejpam-3548	75	1	let	let	VERB
ejpam-3548	75	2	y	y	PRON
ejpam-3548	75	3	and	and	CCONJ
ejpam-3548	75	4	z	z	PROPN
ejpam-3548	75	5	be	be	AUX
ejpam-3548	75	6	topological	topological	ADJ
ejpam-3548	75	7	spaces	space	NOUN
ejpam-3548	75	8	.	.	PUNCT
ejpam-3548	76	1	a	a	DET
ejpam-3548	76	2	function	function	NOUN
ejpam-3548	76	3	f	f	NOUN
ejpam-3548	76	4	:	:	PUNCT
ejpam-3548	76	5	y	y	PROPN
ejpam-3548	76	6	→	→	SYM
ejpam-3548	76	7	z	z	PROPN
ejpam-3548	76	8	is	be	AUX
ejpam-3548	76	9	said	say	VERB
ejpam-3548	76	10	to	to	PART
ejpam-3548	76	11	be	be	AUX
ejpam-3548	76	12	continuous	continuous	ADJ
ejpam-3548	76	13	if	if	SCONJ
ejpam-3548	76	14	the	the	DET
ejpam-3548	76	15	inverse	inverse	ADJ
ejpam-3548	76	16	image	image	NOUN
ejpam-3548	76	17	of	of	ADP
ejpam-3548	76	18	each	each	DET
ejpam-3548	76	19	open	open	ADJ
ejpam-3548	76	20	set	set	NOUN
ejpam-3548	76	21	in	in	ADP
ejpam-3548	76	22	z	z	PROPN
ejpam-3548	76	23	is	be	AUX
ejpam-3548	76	24	open	open	ADJ
ejpam-3548	76	25	in	in	ADP
ejpam-3548	76	26	y	y	PROPN
ejpam-3548	76	27	;	;	PUNCT
ejpam-3548	76	28	open	open	VERB
ejpam-3548	76	29	if	if	SCONJ
ejpam-3548	76	30	the	the	DET
ejpam-3548	76	31	image	image	NOUN
ejpam-3548	76	32	of	of	ADP
ejpam-3548	76	33	each	each	DET
ejpam-3548	76	34	open	open	ADJ
ejpam-3548	76	35	set	set	NOUN
ejpam-3548	76	36	in	in	ADP
ejpam-3548	76	37	y	y	PROPN
ejpam-3548	76	38	is	be	AUX
ejpam-3548	76	39	open	open	ADJ
ejpam-3548	76	40	in	in	ADP
ejpam-3548	76	41	z	z	NOUN
ejpam-3548	76	42	;	;	PUNCT
ejpam-3548	76	43	and	and	CCONJ
ejpam-3548	76	44	closed	close	VERB
ejpam-3548	76	45	if	if	SCONJ
ejpam-3548	76	46	the	the	DET
ejpam-3548	76	47	image	image	NOUN
ejpam-3548	76	48	of	of	ADP
ejpam-3548	76	49	each	each	DET
ejpam-3548	76	50	closed	close	VERB
ejpam-3548	76	51	set	set	VERB
ejpam-3548	76	52	in	in	ADP
ejpam-3548	76	53	y	y	PROPN
ejpam-3548	76	54	is	be	AUX
ejpam-3548	76	55	closed	close	VERB
ejpam-3548	76	56	in	in	ADP
ejpam-3548	76	57	y	y	PROPN
ejpam-3548	76	58	.	.	PUNCT
ejpam-3548	77	1	a	a	DET
ejpam-3548	77	2	space	space	NOUN
ejpam-3548	77	3	y	y	NOUN
ejpam-3548	77	4	is	be	AUX
ejpam-3548	77	5	connected	connect	VERB
ejpam-3548	77	6	if	if	SCONJ
ejpam-3548	77	7	it	it	PRON
ejpam-3548	77	8	is	be	AUX
ejpam-3548	77	9	not	not	PART
ejpam-3548	77	10	the	the	DET
ejpam-3548	77	11	union	union	NOUN
ejpam-3548	77	12	of	of	ADP
ejpam-3548	77	13	two	two	NUM
ejpam-3548	77	14	non	non	ADJ
ejpam-3548	77	15	-	-	ADJ
ejpam-3548	77	16	empty	empty	ADJ
ejpam-3548	77	17	disjoint	disjoint	ADJ
ejpam-3548	77	18	open	open	ADJ
ejpam-3548	77	19	sets	set	NOUN
ejpam-3548	77	20	.	.	PUNCT
ejpam-3548	78	1	a	a	DET
ejpam-3548	78	2	subset	subset	NOUN
ejpam-3548	78	3	b	b	NOUN
ejpam-3548	78	4	of	of	ADP
ejpam-3548	78	5	y	y	PROPN
ejpam-3548	78	6	is	be	AUX
ejpam-3548	78	7	connected	connect	VERB
ejpam-3548	78	8	if	if	SCONJ
ejpam-3548	78	9	it	it	PRON
ejpam-3548	78	10	is	be	AUX
ejpam-3548	78	11	connected	connect	VERB
ejpam-3548	78	12	as	as	ADP
ejpam-3548	78	13	a	a	DET
ejpam-3548	78	14	subspace	subspace	NOUN
ejpam-3548	78	15	of	of	ADP
ejpam-3548	78	16	y	y	PROPN
ejpam-3548	78	17	.	.	PUNCT
ejpam-3548	79	1	a	a	DET
ejpam-3548	79	2	space	space	NOUN
ejpam-3548	79	3	y	y	NOUN
ejpam-3548	79	4	is	be	AUX
ejpam-3548	79	5	disconnected	disconnected	ADJ
ejpam-3548	79	6	if	if	SCONJ
ejpam-3548	79	7	y	y	PROPN
ejpam-3548	79	8	=	=	PUNCT
ejpam-3548	79	9	a	a	PRON
ejpam-3548	79	10	∪	∪	X
ejpam-3548	79	11	b	b	NOUN
ejpam-3548	79	12	where	where	SCONJ
ejpam-3548	79	13	∅	∅	NOUN
ejpam-3548	79	14	6=	6=	ADP
ejpam-3548	79	15	a	a	PRON
ejpam-3548	79	16	,	,	PUNCT
ejpam-3548	79	17	b	b	X
ejpam-3548	79	18	∈	∈	PROPN
ejpam-3548	79	19	τ	τ	X
ejpam-3548	79	20	such	such	ADJ
ejpam-3548	79	21	that	that	SCONJ
ejpam-3548	79	22	a	a	DET
ejpam-3548	79	23	∩	∩	ADJ
ejpam-3548	79	24	b	b	NOUN
ejpam-3548	79	25	=	=	PUNCT
ejpam-3548	79	26	∅.	∅.	NOUN
ejpam-3548	79	27	then	then	ADV
ejpam-3548	79	28	a	a	DET
ejpam-3548	79	29	∪	∪	X
ejpam-3548	79	30	b	b	NOUN
ejpam-3548	79	31	is	be	AUX
ejpam-3548	79	32	a	a	DET
ejpam-3548	79	33	decomposition	decomposition	NOUN
ejpam-3548	79	34	of	of	ADP
ejpam-3548	79	35	y	y	PROPN
ejpam-3548	79	36	.	.	PUNCT
ejpam-3548	80	1	let	let	VERB
ejpam-3548	80	2	p	p	PRON
ejpam-3548	80	3	∈	∈	PROPN
ejpam-3548	80	4	y	y	PROPN
ejpam-3548	80	5	.	.	PUNCT
ejpam-3548	81	1	the	the	DET
ejpam-3548	81	2	topology	topology	NOUN
ejpam-3548	81	3	τp	τp	NOUN
ejpam-3548	81	4	given	give	VERB
ejpam-3548	81	5	by	by	ADP
ejpam-3548	81	6	τp	τp	PRON
ejpam-3548	81	7	=	=	SYM
ejpam-3548	81	8	{	{	PUNCT
ejpam-3548	81	9	∅	∅	NOUN
ejpam-3548	81	10	}	}	PUNCT
ejpam-3548	81	11	∪	∪	X
ejpam-3548	81	12	{	{	PUNCT
ejpam-3548	81	13	a	a	DET
ejpam-3548	81	14	⊆	⊆	NUM
ejpam-3548	81	15	y	y	NOUN
ejpam-3548	81	16	:	:	PUNCT
ejpam-3548	81	17	p	p	X
ejpam-3548	81	18	∈	∈	PROPN
ejpam-3548	81	19	a	a	PRON
ejpam-3548	81	20	}	}	PUNCT
ejpam-3548	81	21	is	be	AUX
ejpam-3548	81	22	called	call	VERB
ejpam-3548	81	23	a	a	DET
ejpam-3548	81	24	particular	particular	ADJ
ejpam-3548	81	25	point	point	NOUN
ejpam-3548	81	26	topology	topology	NOUN
ejpam-3548	81	27	on	on	ADP
ejpam-3548	81	28	y	y	PROPN
ejpam-3548	81	29	.	.	PUNCT
ejpam-3548	82	1	all	all	DET
ejpam-3548	82	2	topological	topological	ADJ
ejpam-3548	82	3	concepts	concept	NOUN
ejpam-3548	82	4	above	above	ADV
ejpam-3548	82	5	are	be	AUX
ejpam-3548	82	6	found	find	VERB
ejpam-3548	82	7	in	in	ADP
ejpam-3548	82	8	[	[	X
ejpam-3548	82	9	2	2	NUM
ejpam-3548	82	10	,	,	PUNCT
ejpam-3548	82	11	6	6	NUM
ejpam-3548	82	12	,	,	PUNCT
ejpam-3548	82	13	10	10	NUM
ejpam-3548	82	14	]	]	PUNCT
ejpam-3548	82	15	.	.	PUNCT
ejpam-3548	83	1	theorem	theorem	NOUN
ejpam-3548	83	2	1	1	NUM
ejpam-3548	83	3	.	.	PUNCT
ejpam-3548	84	1	[	[	X
ejpam-3548	84	2	6	6	NUM
ejpam-3548	84	3	]	]	PUNCT
ejpam-3548	84	4	let	let	VERB
ejpam-3548	84	5	b	b	PRON
ejpam-3548	84	6	be	be	AUX
ejpam-3548	84	7	a	a	DET
ejpam-3548	84	8	class	class	NOUN
ejpam-3548	84	9	of	of	ADP
ejpam-3548	84	10	subsets	subset	NOUN
ejpam-3548	84	11	of	of	ADP
ejpam-3548	84	12	a	a	DET
ejpam-3548	84	13	nonempty	nonempty	ADV
ejpam-3548	84	14	set	set	VERB
ejpam-3548	84	15	y	y	PROPN
ejpam-3548	84	16	.	.	PUNCT
ejpam-3548	85	1	then	then	ADV
ejpam-3548	85	2	b	b	X
ejpam-3548	85	3	is	be	AUX
ejpam-3548	85	4	a	a	DET
ejpam-3548	85	5	base	base	NOUN
ejpam-3548	85	6	for	for	ADP
ejpam-3548	85	7	some	some	DET
ejpam-3548	85	8	topology	topology	NOUN
ejpam-3548	85	9	on	on	ADP
ejpam-3548	85	10	y	y	PROPN
ejpam-3548	85	11	if	if	SCONJ
ejpam-3548	86	1	and	and	CCONJ
ejpam-3548	86	2	only	only	ADV
ejpam-3548	86	3	if	if	SCONJ
ejpam-3548	86	4	it	it	PRON
ejpam-3548	86	5	possesses	possess	VERB
ejpam-3548	86	6	the	the	DET
ejpam-3548	86	7	following	follow	VERB
ejpam-3548	86	8	two	two	NUM
ejpam-3548	86	9	properties	property	NOUN
ejpam-3548	86	10	:	:	PUNCT
ejpam-3548	86	11	(	(	PUNCT
ejpam-3548	86	12	i	i	NOUN
ejpam-3548	86	13	)	)	PUNCT
ejpam-3548	86	14	y	y	PROPN
ejpam-3548	87	1	=	=	PUNCT
ejpam-3548	87	2	⋃	⋃	NOUN
ejpam-3548	87	3	{	{	PUNCT
ejpam-3548	87	4	b	b	NOUN
ejpam-3548	87	5	:	:	PUNCT
ejpam-3548	87	6	b	b	X
ejpam-3548	87	7	∈	∈	PROPN
ejpam-3548	87	8	b	b	NOUN
ejpam-3548	87	9	}	}	PUNCT
ejpam-3548	87	10	.	.	PUNCT
ejpam-3548	88	1	(	(	PUNCT
ejpam-3548	88	2	ii	ii	NOUN
ejpam-3548	88	3	)	)	PUNCT
ejpam-3548	88	4	for	for	ADP
ejpam-3548	88	5	any	any	DET
ejpam-3548	88	6	b	b	NOUN
ejpam-3548	88	7	,	,	PUNCT
ejpam-3548	88	8	b∗	b∗	ADJ
ejpam-3548	88	9	∈	∈	PROPN
ejpam-3548	88	10	b	b	PROPN
ejpam-3548	88	11	,	,	PUNCT
ejpam-3548	88	12	b	b	PROPN
ejpam-3548	88	13	∩	∩	NOUN
ejpam-3548	88	14	b∗	b∗	ADJ
ejpam-3548	88	15	is	be	AUX
ejpam-3548	88	16	the	the	DET
ejpam-3548	88	17	union	union	NOUN
ejpam-3548	88	18	of	of	ADP
ejpam-3548	88	19	members	member	NOUN
ejpam-3548	88	20	of	of	ADP
ejpam-3548	88	21	b	b	PROPN
ejpam-3548	88	22	,	,	PUNCT
ejpam-3548	88	23	or	or	CCONJ
ejpam-3548	88	24	,	,	PUNCT
ejpam-3548	88	25	equivalently	equivalently	ADV
ejpam-3548	88	26	,	,	PUNCT
ejpam-3548	88	27	if	if	SCONJ
ejpam-3548	88	28	p	p	PROPN
ejpam-3548	88	29	∈	∈	PROPN
ejpam-3548	88	30	b	b	PROPN
ejpam-3548	88	31	∩b∗	∩b∗	PUNCT
ejpam-3548	88	32	then	then	ADV
ejpam-3548	88	33	there	there	PRON
ejpam-3548	88	34	exists	exist	VERB
ejpam-3548	88	35	bp	bp	PROPN
ejpam-3548	88	36	such	such	ADJ
ejpam-3548	88	37	that	that	SCONJ
ejpam-3548	88	38	p	p	PROPN
ejpam-3548	88	39	∈	∈	PROPN
ejpam-3548	88	40	bp	bp	PROPN
ejpam-3548	88	41	⊆	⊆	NUM
ejpam-3548	88	42	b	b	PROPN
ejpam-3548	88	43	∩b∗.	∩b∗.	PROPN
ejpam-3548	88	44	theorem	theorem	NOUN
ejpam-3548	88	45	2	2	NUM
ejpam-3548	88	46	.	.	PUNCT
ejpam-3548	89	1	[	[	X
ejpam-3548	89	2	2	2	X
ejpam-3548	89	3	]	]	PUNCT
ejpam-3548	89	4	let	let	VERB
ejpam-3548	89	5	y	y	PRON
ejpam-3548	89	6	be	be	AUX
ejpam-3548	89	7	a	a	DET
ejpam-3548	89	8	topological	topological	ADJ
ejpam-3548	89	9	space	space	NOUN
ejpam-3548	89	10	,	,	PUNCT
ejpam-3548	89	11	and	and	CCONJ
ejpam-3548	89	12	b	b	X
ejpam-3548	89	13	⊆	⊆	NUM
ejpam-3548	89	14	τ	τ	X
ejpam-3548	89	15	.	.	PUNCT
ejpam-3548	90	1	then	then	ADV
ejpam-3548	90	2	b	b	X
ejpam-3548	90	3	is	be	AUX
ejpam-3548	90	4	a	a	DET
ejpam-3548	90	5	basis	basis	NOUN
ejpam-3548	90	6	for	for	ADP
ejpam-3548	90	7	τ	τ	PROPN
ejpam-3548	90	8	if	if	SCONJ
ejpam-3548	90	9	and	and	CCONJ
ejpam-3548	90	10	only	only	ADV
ejpam-3548	90	11	if	if	SCONJ
ejpam-3548	90	12	for	for	ADP
ejpam-3548	90	13	each	each	DET
ejpam-3548	90	14	g	g	PROPN
ejpam-3548	90	15	∈	∈	PROPN
ejpam-3548	90	16	τ	τ	X
ejpam-3548	90	17	and	and	CCONJ
ejpam-3548	90	18	for	for	ADP
ejpam-3548	90	19	each	each	DET
ejpam-3548	90	20	x	x	SYM
ejpam-3548	90	21	∈	∈	PROPN
ejpam-3548	90	22	g	g	NOUN
ejpam-3548	90	23	,	,	PUNCT
ejpam-3548	90	24	there	there	PRON
ejpam-3548	90	25	exists	exist	VERB
ejpam-3548	90	26	u	u	PROPN
ejpam-3548	90	27	∈	∈	PROPN
ejpam-3548	90	28	b	b	PROPN
ejpam-3548	90	29	such	such	ADJ
ejpam-3548	90	30	that	that	SCONJ
ejpam-3548	90	31	x	x	SYM
ejpam-3548	90	32	∈	∈	PROPN
ejpam-3548	90	33	u	u	NOUN
ejpam-3548	90	34	⊆	⊆	NUM
ejpam-3548	90	35	g.	g.	NOUN
ejpam-3548	90	36	theorem	theorem	VERB
ejpam-3548	90	37	3	3	NUM
ejpam-3548	90	38	.	.	PUNCT
ejpam-3548	91	1	[	[	X
ejpam-3548	91	2	2	2	NUM
ejpam-3548	91	3	]	]	X
ejpam-3548	91	4	let	let	VERB
ejpam-3548	91	5	(	(	PUNCT
ejpam-3548	91	6	y	y	PROPN
ejpam-3548	91	7	,	,	PUNCT
ejpam-3548	91	8	τ	τ	PROPN
ejpam-3548	91	9	)	)	PUNCT
ejpam-3548	91	10	be	be	VERB
ejpam-3548	91	11	a	a	DET
ejpam-3548	91	12	topological	topological	ADJ
ejpam-3548	91	13	space	space	NOUN
ejpam-3548	91	14	and	and	CCONJ
ejpam-3548	91	15	(	(	PUNCT
ejpam-3548	91	16	z	z	NOUN
ejpam-3548	91	17	,	,	PUNCT
ejpam-3548	91	18	τz	τz	AUX
ejpam-3548	91	19	)	)	PUNCT
ejpam-3548	91	20	be	be	AUX
ejpam-3548	91	21	a	a	DET
ejpam-3548	91	22	subspace	subspace	NOUN
ejpam-3548	91	23	.	.	PUNCT
ejpam-3548	92	1	if	if	SCONJ
ejpam-3548	92	2	{	{	PUNCT
ejpam-3548	92	3	uα	uα	X
ejpam-3548	92	4	:	:	PUNCT
ejpam-3548	92	5	α	α	PROPN
ejpam-3548	92	6	∈	∈	PROPN
ejpam-3548	92	7	a	a	PRON
ejpam-3548	92	8	}	}	PUNCT
ejpam-3548	92	9	is	be	AUX
ejpam-3548	92	10	a	a	DET
ejpam-3548	92	11	basis	basis	NOUN
ejpam-3548	92	12	(	(	PUNCT
ejpam-3548	92	13	subbasis	subbasis	NOUN
ejpam-3548	92	14	)	)	PUNCT
ejpam-3548	92	15	for	for	ADP
ejpam-3548	92	16	τ	τ	PROPN
ejpam-3548	92	17	,	,	PUNCT
ejpam-3548	92	18	then	then	ADV
ejpam-3548	92	19	{	{	PUNCT
ejpam-3548	92	20	z	z	NOUN
ejpam-3548	92	21	∩	∩	X
ejpam-3548	92	22	uα	uα	NOUN
ejpam-3548	92	23	:	:	PUNCT
ejpam-3548	92	24	α	α	PROPN
ejpam-3548	92	25	∈	∈	PROPN
ejpam-3548	92	26	a	a	PRON
ejpam-3548	92	27	}	}	PUNCT
ejpam-3548	92	28	is	be	AUX
ejpam-3548	92	29	a	a	DET
ejpam-3548	92	30	basis	basis	NOUN
ejpam-3548	92	31	(	(	PUNCT
ejpam-3548	92	32	subbasis	subbasis	NOUN
ejpam-3548	92	33	)	)	PUNCT
ejpam-3548	92	34	for	for	ADP
ejpam-3548	92	35	τz	τz	ADV
ejpam-3548	92	36	.	.	PUNCT
ejpam-3548	93	1	theorem	theorem	ADJ
ejpam-3548	93	2	4	4	NUM
ejpam-3548	93	3	.	.	PUNCT
ejpam-3548	94	1	[	[	X
ejpam-3548	94	2	2	2	X
ejpam-3548	94	3	]	]	PUNCT
ejpam-3548	94	4	let	let	VERB
ejpam-3548	94	5	y	y	PRON
ejpam-3548	94	6	,	,	PUNCT
ejpam-3548	94	7	z	z	PROPN
ejpam-3548	94	8	be	be	VERB
ejpam-3548	94	9	topological	topological	ADJ
ejpam-3548	94	10	spaces	space	NOUN
ejpam-3548	94	11	and	and	CCONJ
ejpam-3548	94	12	f	f	NOUN
ejpam-3548	94	13	:	:	PUNCT
ejpam-3548	94	14	y	y	PROPN
ejpam-3548	94	15	→	→	PROPN
ejpam-3548	94	16	z	z	NOUN
ejpam-3548	94	17	a	a	DET
ejpam-3548	94	18	map	map	NOUN
ejpam-3548	94	19	.	.	PUNCT
ejpam-3548	95	1	the	the	DET
ejpam-3548	95	2	following	follow	VERB
ejpam-3548	95	3	statements	statement	NOUN
ejpam-3548	95	4	are	be	AUX
ejpam-3548	95	5	equivalent	equivalent	ADJ
ejpam-3548	95	6	:	:	PUNCT
ejpam-3548	95	7	(	(	PUNCT
ejpam-3548	95	8	i	i	NOUN
ejpam-3548	95	9	)	)	PUNCT
ejpam-3548	95	10	f	f	PROPN
ejpam-3548	95	11	is	be	AUX
ejpam-3548	95	12	continuous	continuous	ADJ
ejpam-3548	95	13	.	.	PUNCT
ejpam-3548	96	1	(	(	PUNCT
ejpam-3548	96	2	ii	ii	NOUN
ejpam-3548	96	3	)	)	PUNCT
ejpam-3548	96	4	the	the	DET
ejpam-3548	96	5	inverse	inverse	ADJ
ejpam-3548	96	6	image	image	NOUN
ejpam-3548	96	7	of	of	ADP
ejpam-3548	96	8	each	each	DET
ejpam-3548	96	9	closed	close	VERB
ejpam-3548	96	10	set	set	VERB
ejpam-3548	96	11	in	in	ADP
ejpam-3548	96	12	z	z	PROPN
ejpam-3548	96	13	is	be	AUX
ejpam-3548	96	14	closed	close	VERB
ejpam-3548	96	15	in	in	ADP
ejpam-3548	96	16	y	y	PROPN
ejpam-3548	96	17	.	.	PUNCT
ejpam-3548	97	1	(	(	PUNCT
ejpam-3548	97	2	iii	iii	X
ejpam-3548	97	3	)	)	PUNCT
ejpam-3548	97	4	the	the	DET
ejpam-3548	97	5	inverse	inverse	ADJ
ejpam-3548	97	6	image	image	NOUN
ejpam-3548	97	7	of	of	ADP
ejpam-3548	97	8	each	each	DET
ejpam-3548	97	9	member	member	NOUN
ejpam-3548	97	10	of	of	ADP
ejpam-3548	97	11	a	a	DET
ejpam-3548	97	12	subbasis	subbasis	NOUN
ejpam-3548	97	13	(	(	PUNCT
ejpam-3548	97	14	basis	basis	NOUN
ejpam-3548	97	15	)	)	PUNCT
ejpam-3548	97	16	for	for	ADP
ejpam-3548	97	17	z	z	PROPN
ejpam-3548	97	18	is	be	AUX
ejpam-3548	97	19	open	open	ADJ
ejpam-3548	97	20	in	in	ADP
ejpam-3548	97	21	y	y	PROPN
ejpam-3548	97	22	(	(	PUNCT
ejpam-3548	97	23	not	not	PART
ejpam-3548	97	24	necessarily	necessarily	ADV
ejpam-3548	97	25	a	a	DET
ejpam-3548	97	26	member	member	NOUN
ejpam-3548	97	27	of	of	ADP
ejpam-3548	97	28	subbasis	subbasis	NOUN
ejpam-3548	97	29	,	,	PUNCT
ejpam-3548	97	30	or	or	CCONJ
ejpam-3548	97	31	basis	basis	NOUN
ejpam-3548	97	32	for	for	ADP
ejpam-3548	97	33	y	y	PROPN
ejpam-3548	97	34	)	)	PUNCT
ejpam-3548	97	35	.	.	PUNCT
ejpam-3548	98	1	2	2	X
ejpam-3548	98	2	.	.	X
ejpam-3548	98	3	some	some	DET
ejpam-3548	98	4	properties	property	NOUN
ejpam-3548	98	5	of	of	ADP
ejpam-3548	98	6	rx(a	rx(a	NOUN
ejpam-3548	98	7	)	)	PUNCT
ejpam-3548	98	8	definition	definition	NOUN
ejpam-3548	98	9	1	1	NUM
ejpam-3548	98	10	.	.	PUNCT
ejpam-3548	98	11	let	let	VERB
ejpam-3548	98	12	x	x	PRON
ejpam-3548	98	13	be	be	AUX
ejpam-3548	98	14	a	a	DET
ejpam-3548	98	15	be	be	NOUN
ejpam-3548	98	16	-	-	PUNCT
ejpam-3548	98	17	algebra	algebra	NOUN
ejpam-3548	98	18	.	.	PUNCT
ejpam-3548	99	1	for	for	ADP
ejpam-3548	99	2	any	any	DET
ejpam-3548	99	3	a	a	DET
ejpam-3548	99	4	⊆	⊆	NUM
ejpam-3548	99	5	x	x	NOUN
ejpam-3548	99	6	,	,	PUNCT
ejpam-3548	99	7	the	the	DET
ejpam-3548	99	8	set	set	NOUN
ejpam-3548	99	9	rx(a	rx(a	NOUN
ejpam-3548	99	10	)	)	PUNCT
ejpam-3548	99	11	=	=	PRON
ejpam-3548	99	12	{	{	PUNCT
ejpam-3548	99	13	x	x	PUNCT
ejpam-3548	99	14	∈	∈	NOUN
ejpam-3548	99	15	x	x	PUNCT
ejpam-3548	99	16	|	|	ADV
ejpam-3548	99	17	a	a	DET
ejpam-3548	99	18	∗	∗	NOUN
ejpam-3548	99	19	x	x	PUNCT
ejpam-3548	99	20	=	=	SYM
ejpam-3548	99	21	1x	1x	NUM
ejpam-3548	99	22	,	,	PUNCT
ejpam-3548	99	23	∀a	∀a	NOUN
ejpam-3548	99	24	∈	∈	PROPN
ejpam-3548	99	25	a	a	PRON
ejpam-3548	99	26	}	}	PUNCT
ejpam-3548	99	27	is	be	AUX
ejpam-3548	99	28	called	call	VERB
ejpam-3548	99	29	the	the	DET
ejpam-3548	99	30	subset	subset	NOUN
ejpam-3548	99	31	of	of	ADP
ejpam-3548	99	32	x	x	SYM
ejpam-3548	99	33	determined	determine	VERB
ejpam-3548	99	34	by	by	ADP
ejpam-3548	99	35	right	right	ADJ
ejpam-3548	99	36	application	application	NOUN
ejpam-3548	99	37	of	of	ADP
ejpam-3548	99	38	be	be	NOUN
ejpam-3548	99	39	-	-	PUNCT
ejpam-3548	99	40	ordering	ordering	NOUN
ejpam-3548	99	41	on	on	ADP
ejpam-3548	99	42	a.	a.	NOUN
ejpam-3548	99	43	note	note	NOUN
ejpam-3548	99	44	that	that	PRON
ejpam-3548	99	45	rx({a	rx({a	VERB
ejpam-3548	99	46	}	}	PUNCT
ejpam-3548	99	47	)	)	PUNCT
ejpam-3548	100	1	=	=	PUNCT
ejpam-3548	101	1	[	[	X
ejpam-3548	101	2	a	a	X
ejpam-3548	101	3	,	,	PUNCT
ejpam-3548	101	4	1x	1x	NUM
ejpam-3548	101	5	]	]	PUNCT
ejpam-3548	101	6	for	for	ADP
ejpam-3548	101	7	all	all	DET
ejpam-3548	101	8	a	a	DET
ejpam-3548	101	9	∈	∈	PROPN
ejpam-3548	101	10	x.	x.	NOUN
ejpam-3548	101	11	j.	j.	PROPN
ejpam-3548	101	12	albaracin	albaracin	PROPN
ejpam-3548	101	13	,	,	PUNCT
ejpam-3548	101	14	j.	j.	PROPN
ejpam-3548	101	15	vilela	vilela	PROPN
ejpam-3548	101	16	/	/	SYM
ejpam-3548	101	17	eur	eur	PROPN
ejpam-3548	101	18	.	.	PUNCT
ejpam-3548	102	1	j.	j.	PROPN
ejpam-3548	102	2	pure	pure	PROPN
ejpam-3548	102	3	appl	appl	PROPN
ejpam-3548	102	4	.	.	PROPN
ejpam-3548	102	5	math	math	PROPN
ejpam-3548	102	6	,	,	PUNCT
ejpam-3548	102	7	12	12	NUM
ejpam-3548	102	8	(	(	PUNCT
ejpam-3548	102	9	4	4	NUM
ejpam-3548	102	10	)	)	PUNCT
ejpam-3548	102	11	(	(	PUNCT
ejpam-3548	102	12	2019	2019	NUM
ejpam-3548	102	13	)	)	PUNCT
ejpam-3548	102	14	,	,	PUNCT
ejpam-3548	102	15	1584	1584	NUM
ejpam-3548	102	16	-	-	SYM
ejpam-3548	102	17	1594	1594	NUM
ejpam-3548	102	18	1587	1587	NUM
ejpam-3548	102	19	theorem	theorem	NOUN
ejpam-3548	102	20	5	5	NUM
ejpam-3548	102	21	.	.	PUNCT
ejpam-3548	103	1	let	let	VERB
ejpam-3548	103	2	a	a	PRON
ejpam-3548	103	3	and	and	CCONJ
ejpam-3548	103	4	b	b	NOUN
ejpam-3548	103	5	be	be	AUX
ejpam-3548	103	6	subsets	subset	NOUN
ejpam-3548	103	7	of	of	ADP
ejpam-3548	103	8	x.	x.	NOUN
ejpam-3548	103	9	then	then	ADV
ejpam-3548	103	10	the	the	DET
ejpam-3548	103	11	following	follow	VERB
ejpam-3548	103	12	hold	hold	NOUN
ejpam-3548	103	13	:	:	PUNCT
ejpam-3548	103	14	(	(	PUNCT
ejpam-3548	103	15	i	i	NOUN
ejpam-3548	103	16	)	)	PUNCT
ejpam-3548	103	17	rx(∅	rx(∅	PUNCT
ejpam-3548	103	18	)	)	PUNCT
ejpam-3548	104	1	=	=	PUNCT
ejpam-3548	104	2	x.	x.	NOUN
ejpam-3548	104	3	(	(	PUNCT
ejpam-3548	104	4	ii	ii	PROPN
ejpam-3548	104	5	)	)	PUNCT
ejpam-3548	104	6	if	if	SCONJ
ejpam-3548	104	7	a	a	DET
ejpam-3548	104	8	⊆	⊆	NUM
ejpam-3548	104	9	b	b	NOUN
ejpam-3548	104	10	,	,	PUNCT
ejpam-3548	104	11	then	then	ADV
ejpam-3548	104	12	rx(b	rx(b	NOUN
ejpam-3548	104	13	)	)	PUNCT
ejpam-3548	104	14	⊆	⊆	NUM
ejpam-3548	104	15	rx(a	rx(a	NOUN
ejpam-3548	104	16	)	)	PUNCT
ejpam-3548	104	17	.	.	PUNCT
ejpam-3548	105	1	(	(	PUNCT
ejpam-3548	105	2	iii	iii	X
ejpam-3548	105	3	)	)	PUNCT
ejpam-3548	105	4	if	if	SCONJ
ejpam-3548	105	5	x	x	PRON
ejpam-3548	105	6	is	be	AUX
ejpam-3548	105	7	a	a	DET
ejpam-3548	105	8	transitive	transitive	ADJ
ejpam-3548	105	9	be	be	NOUN
ejpam-3548	105	10	-	-	PUNCT
ejpam-3548	105	11	algebra	algebra	NOUN
ejpam-3548	105	12	,	,	PUNCT
ejpam-3548	105	13	then	then	ADV
ejpam-3548	105	14	rx(rx(a	rx(rx(a	NOUN
ejpam-3548	105	15	)	)	PUNCT
ejpam-3548	105	16	)	)	PUNCT
ejpam-3548	106	1	⊆	⊆	NUM
ejpam-3548	106	2	rx(a	rx(a	NOUN
ejpam-3548	106	3	)	)	PUNCT
ejpam-3548	106	4	.	.	PUNCT
ejpam-3548	107	1	proof	proof	NOUN
ejpam-3548	107	2	.	.	PUNCT
ejpam-3548	108	1	to	to	PART
ejpam-3548	108	2	prove	prove	VERB
ejpam-3548	108	3	(	(	PUNCT
ejpam-3548	108	4	i	i	NOUN
ejpam-3548	108	5	)	)	PUNCT
ejpam-3548	108	6	,	,	PUNCT
ejpam-3548	108	7	suppose	suppose	VERB
ejpam-3548	108	8	rx(∅	rx(∅	PROPN
ejpam-3548	108	9	)	)	PUNCT
ejpam-3548	108	10	6=	6=	PUNCT
ejpam-3548	108	11	x.	x.	NOUN
ejpam-3548	108	12	then	then	ADV
ejpam-3548	108	13	there	there	PRON
ejpam-3548	108	14	exists	exist	VERB
ejpam-3548	108	15	x	x	X
ejpam-3548	108	16	∈	∈	PROPN
ejpam-3548	108	17	x	x	X
ejpam-3548	108	18	such	such	ADJ
ejpam-3548	108	19	that	that	PRON
ejpam-3548	108	20	x	x	X
ejpam-3548	108	21	/∈	/∈	PUNCT
ejpam-3548	108	22	rx(∅	rx(∅	ADJ
ejpam-3548	108	23	)	)	PUNCT
ejpam-3548	108	24	.	.	PUNCT
ejpam-3548	109	1	thus	thus	ADV
ejpam-3548	109	2	,	,	PUNCT
ejpam-3548	109	3	there	there	PRON
ejpam-3548	109	4	exists	exist	VERB
ejpam-3548	109	5	a	a	DET
ejpam-3548	109	6	∈	∈	NOUN
ejpam-3548	109	7	∅	∅	NOUN
ejpam-3548	109	8	such	such	ADJ
ejpam-3548	109	9	that	that	SCONJ
ejpam-3548	109	10	a	a	DET
ejpam-3548	109	11	∗	∗	NOUN
ejpam-3548	109	12	x	x	PUNCT
ejpam-3548	109	13	6=	6=	ADP
ejpam-3548	109	14	1x	1x	NUM
ejpam-3548	109	15	,	,	PUNCT
ejpam-3548	109	16	a	a	DET
ejpam-3548	109	17	contradiction	contradiction	NOUN
ejpam-3548	109	18	.	.	PUNCT
ejpam-3548	110	1	therefore	therefore	ADV
ejpam-3548	110	2	,	,	PUNCT
ejpam-3548	110	3	rx(∅	rx(∅	PROPN
ejpam-3548	110	4	)	)	PUNCT
ejpam-3548	110	5	=	=	PUNCT
ejpam-3548	110	6	x.	x.	NOUN
ejpam-3548	110	7	to	to	PART
ejpam-3548	110	8	prove	prove	VERB
ejpam-3548	110	9	(	(	PUNCT
ejpam-3548	110	10	ii	ii	NOUN
ejpam-3548	110	11	)	)	PUNCT
ejpam-3548	110	12	,	,	PUNCT
ejpam-3548	110	13	let	let	VERB
ejpam-3548	110	14	x	x	X
ejpam-3548	110	15	∈	∈	PROPN
ejpam-3548	110	16	rx(b	rx(b	VERB
ejpam-3548	110	17	)	)	PUNCT
ejpam-3548	110	18	.	.	PUNCT
ejpam-3548	111	1	then	then	ADV
ejpam-3548	111	2	b	b	X
ejpam-3548	111	3	∗	∗	NOUN
ejpam-3548	111	4	x	x	PUNCT
ejpam-3548	112	1	=	=	SYM
ejpam-3548	112	2	1x	1x	NUM
ejpam-3548	112	3	for	for	ADP
ejpam-3548	112	4	all	all	DET
ejpam-3548	112	5	b	b	PROPN
ejpam-3548	112	6	∈	∈	PROPN
ejpam-3548	112	7	b.	b.	PROPN
ejpam-3548	112	8	since	since	SCONJ
ejpam-3548	112	9	a	a	DET
ejpam-3548	112	10	⊆	⊆	NUM
ejpam-3548	112	11	b	b	NOUN
ejpam-3548	112	12	,	,	PUNCT
ejpam-3548	112	13	a	a	DET
ejpam-3548	112	14	∗	∗	NOUN
ejpam-3548	112	15	x	x	X
ejpam-3548	113	1	=	=	SYM
ejpam-3548	113	2	1x	1x	NUM
ejpam-3548	113	3	for	for	ADP
ejpam-3548	113	4	all	all	DET
ejpam-3548	113	5	a	a	DET
ejpam-3548	113	6	∈	∈	PROPN
ejpam-3548	113	7	a.	a.	NOUN
ejpam-3548	113	8	thus	thus	ADV
ejpam-3548	113	9	,	,	PUNCT
ejpam-3548	113	10	x	x	PROPN
ejpam-3548	113	11	∈	∈	NOUN
ejpam-3548	113	12	rx(a	rx(a	NOUN
ejpam-3548	113	13	)	)	PUNCT
ejpam-3548	113	14	.	.	PUNCT
ejpam-3548	114	1	hence	hence	ADV
ejpam-3548	114	2	,	,	PUNCT
ejpam-3548	114	3	rx(b	rx(b	ADJ
ejpam-3548	114	4	)	)	PUNCT
ejpam-3548	114	5	⊆	⊆	NUM
ejpam-3548	114	6	rx(a	rx(a	NOUN
ejpam-3548	114	7	)	)	PUNCT
ejpam-3548	114	8	.	.	PUNCT
ejpam-3548	115	1	to	to	PART
ejpam-3548	115	2	prove	prove	VERB
ejpam-3548	115	3	(	(	PUNCT
ejpam-3548	115	4	iii	iii	NOUN
ejpam-3548	115	5	)	)	PUNCT
ejpam-3548	115	6	,	,	PUNCT
ejpam-3548	115	7	let	let	VERB
ejpam-3548	115	8	x	x	X
ejpam-3548	115	9	∈	∈	NOUN
ejpam-3548	115	10	rx(rx(a	rx(rx(a	NOUN
ejpam-3548	115	11	)	)	PUNCT
ejpam-3548	115	12	)	)	PUNCT
ejpam-3548	115	13	.	.	PUNCT
ejpam-3548	116	1	then	then	ADV
ejpam-3548	116	2	b	b	X
ejpam-3548	116	3	∗	∗	NOUN
ejpam-3548	116	4	x	x	PUNCT
ejpam-3548	117	1	=	=	SYM
ejpam-3548	117	2	1x	1x	NUM
ejpam-3548	117	3	for	for	ADP
ejpam-3548	117	4	all	all	DET
ejpam-3548	117	5	b	b	PROPN
ejpam-3548	117	6	∈	∈	NOUN
ejpam-3548	117	7	rx(a	rx(a	NOUN
ejpam-3548	117	8	)	)	PUNCT
ejpam-3548	117	9	.	.	PUNCT
ejpam-3548	118	1	since	since	SCONJ
ejpam-3548	118	2	a	a	DET
ejpam-3548	118	3	∗	∗	NOUN
ejpam-3548	118	4	b	b	NOUN
ejpam-3548	118	5	=	=	X
ejpam-3548	118	6	1x	1x	NUM
ejpam-3548	118	7	for	for	ADP
ejpam-3548	118	8	all	all	DET
ejpam-3548	118	9	a	a	DET
ejpam-3548	118	10	∈	∈	PROPN
ejpam-3548	118	11	a	a	PRON
ejpam-3548	118	12	and	and	CCONJ
ejpam-3548	118	13	x	x	NOUN
ejpam-3548	118	14	is	be	AUX
ejpam-3548	118	15	transitive	transitive	ADJ
ejpam-3548	118	16	,	,	PUNCT
ejpam-3548	118	17	it	it	PRON
ejpam-3548	118	18	follows	follow	VERB
ejpam-3548	118	19	that	that	SCONJ
ejpam-3548	118	20	a∗x	a∗x	NUM
ejpam-3548	118	21	=	=	SYM
ejpam-3548	118	22	1x	1x	NUM
ejpam-3548	118	23	for	for	ADP
ejpam-3548	118	24	all	all	DET
ejpam-3548	118	25	a	a	DET
ejpam-3548	118	26	∈	∈	PROPN
ejpam-3548	118	27	a.	a.	NOUN
ejpam-3548	118	28	thus	thus	ADV
ejpam-3548	118	29	,	,	PUNCT
ejpam-3548	118	30	x	x	PROPN
ejpam-3548	118	31	∈	∈	NOUN
ejpam-3548	118	32	rx(a	rx(a	NOUN
ejpam-3548	118	33	)	)	PUNCT
ejpam-3548	118	34	.	.	PUNCT
ejpam-3548	119	1	hence	hence	ADV
ejpam-3548	119	2	,	,	PUNCT
ejpam-3548	119	3	rx(rx(a	rx(rx(a	NOUN
ejpam-3548	119	4	)	)	PUNCT
ejpam-3548	119	5	)	)	PUNCT
ejpam-3548	119	6	⊆	⊆	NUM
ejpam-3548	119	7	rx(a	rx(a	NOUN
ejpam-3548	119	8	)	)	PUNCT
ejpam-3548	119	9	.	.	PUNCT
ejpam-3548	120	1	whitehsdgkjgaskdj	whitehsdgkjgaskdj	PROPN
ejpam-3548	120	2	theorem	theorem	VERB
ejpam-3548	120	3	6	6	NUM
ejpam-3548	120	4	.	.	PUNCT
ejpam-3548	121	1	let	let	VERB
ejpam-3548	121	2	x	x	PRON
ejpam-3548	121	3	be	be	AUX
ejpam-3548	121	4	a	a	DET
ejpam-3548	121	5	be	be	NOUN
ejpam-3548	121	6	-	-	PUNCT
ejpam-3548	121	7	algebra	algebra	NOUN
ejpam-3548	121	8	and	and	CCONJ
ejpam-3548	121	9	a	a	DET
ejpam-3548	121	10	⊆	⊆	NUM
ejpam-3548	121	11	x.	x.	NOUN
ejpam-3548	121	12	then	then	ADV
ejpam-3548	121	13	rx(a	rx(a	NOUN
ejpam-3548	121	14	)	)	PUNCT
ejpam-3548	122	1	=	=	SYM
ejpam-3548	122	2	⋂	⋂	PROPN
ejpam-3548	122	3	a∈a	a∈a	VERB
ejpam-3548	122	4	[	[	X
ejpam-3548	122	5	a	a	X
ejpam-3548	122	6	,	,	PUNCT
ejpam-3548	122	7	1x	1x	NUM
ejpam-3548	122	8	]	]	PUNCT
ejpam-3548	122	9	and	and	CCONJ
ejpam-3548	122	10	1x	1x	NUM
ejpam-3548	122	11	∈	∈	PROPN
ejpam-3548	122	12	rx(a	rx(a	PROPN
ejpam-3548	122	13	)	)	PUNCT
ejpam-3548	122	14	.	.	PUNCT
ejpam-3548	123	1	furthermore	furthermore	ADV
ejpam-3548	123	2	,	,	PUNCT
ejpam-3548	123	3	if	if	SCONJ
ejpam-3548	123	4	1x	1x	PROPN
ejpam-3548	123	5	∈	∈	PROPN
ejpam-3548	123	6	a	a	PRON
ejpam-3548	123	7	,	,	PUNCT
ejpam-3548	123	8	then	then	ADV
ejpam-3548	123	9	rx(a	rx(a	NOUN
ejpam-3548	123	10	)	)	PUNCT
ejpam-3548	123	11	=	=	SYM
ejpam-3548	123	12	{	{	PUNCT
ejpam-3548	123	13	1x	1x	NUM
ejpam-3548	123	14	}	}	PUNCT
ejpam-3548	123	15	.	.	PUNCT
ejpam-3548	124	1	proof	proof	NOUN
ejpam-3548	124	2	.	.	PUNCT
ejpam-3548	125	1	note	note	VERB
ejpam-3548	125	2	that	that	SCONJ
ejpam-3548	125	3	rx(a	rx(a	NOUN
ejpam-3548	125	4	)	)	PUNCT
ejpam-3548	125	5	=	=	SYM
ejpam-3548	126	1	{	{	PUNCT
ejpam-3548	126	2	x	x	PUNCT
ejpam-3548	126	3	∈	∈	NOUN
ejpam-3548	126	4	x	x	PUNCT
ejpam-3548	126	5	|	|	ADV
ejpam-3548	126	6	a	a	DET
ejpam-3548	126	7	∗	∗	NOUN
ejpam-3548	126	8	x	x	PUNCT
ejpam-3548	127	1	=	=	SYM
ejpam-3548	127	2	1x	1x	NUM
ejpam-3548	127	3	,	,	PUNCT
ejpam-3548	127	4	∀a	∀a	NOUN
ejpam-3548	127	5	∈	∈	NOUN
ejpam-3548	127	6	a	a	PRON
ejpam-3548	127	7	}	}	PUNCT
ejpam-3548	127	8	=	=	SYM
ejpam-3548	127	9	{	{	PUNCT
ejpam-3548	127	10	x	x	PUNCT
ejpam-3548	127	11	∈	∈	NOUN
ejpam-3548	127	12	x	x	PUNCT
ejpam-3548	128	1	|	|	ADV
ejpam-3548	128	2	x	x	SYM
ejpam-3548	128	3	∈	∈	NOUN
ejpam-3548	128	4	rx({a	rx({a	NOUN
ejpam-3548	128	5	}	}	PUNCT
ejpam-3548	128	6	)	)	PUNCT
ejpam-3548	128	7	,	,	PUNCT
ejpam-3548	128	8	∀a	∀a	VERB
ejpam-3548	128	9	∈	∈	PROPN
ejpam-3548	128	10	a	a	PRON
ejpam-3548	128	11	}	}	PUNCT
ejpam-3548	128	12	=	=	SYM
ejpam-3548	128	13	⋂	⋂	PROPN
ejpam-3548	128	14	a∈a	a∈a	ADJ
ejpam-3548	128	15	rx({a	rx({a	NOUN
ejpam-3548	128	16	}	}	PUNCT
ejpam-3548	128	17	)	)	PUNCT
ejpam-3548	129	1	=	=	SYM
ejpam-3548	129	2	⋂	⋂	PROPN
ejpam-3548	129	3	a∈a	a∈a	VERB
ejpam-3548	129	4	[	[	X
ejpam-3548	129	5	a	a	X
ejpam-3548	129	6	,	,	PUNCT
ejpam-3548	129	7	1x	1x	NUM
ejpam-3548	129	8	]	]	PUNCT
ejpam-3548	129	9	.	.	PUNCT
ejpam-3548	130	1	let	let	VERB
ejpam-3548	130	2	a	a	DET
ejpam-3548	130	3	∈	∈	NOUN
ejpam-3548	130	4	a.	a.	NOUN
ejpam-3548	130	5	then	then	ADV
ejpam-3548	130	6	a	a	DET
ejpam-3548	130	7	∗	∗	NOUN
ejpam-3548	130	8	1x	1x	NUM
ejpam-3548	131	1	=	=	SYM
ejpam-3548	131	2	1x	1x	NUM
ejpam-3548	131	3	for	for	ADP
ejpam-3548	131	4	all	all	DET
ejpam-3548	131	5	a	a	DET
ejpam-3548	131	6	∈	∈	PROPN
ejpam-3548	131	7	a.	a.	NOUN
ejpam-3548	131	8	thus	thus	ADV
ejpam-3548	131	9	,	,	PUNCT
ejpam-3548	131	10	1x	1x	PROPN
ejpam-3548	131	11	∈	∈	PROPN
ejpam-3548	131	12	rx(a	rx(a	PROPN
ejpam-3548	131	13	)	)	PUNCT
ejpam-3548	131	14	.	.	PUNCT
ejpam-3548	132	1	now	now	ADV
ejpam-3548	132	2	,	,	PUNCT
ejpam-3548	132	3	rx({1x	rx({1x	NOUN
ejpam-3548	132	4	}	}	PUNCT
ejpam-3548	132	5	)	)	PUNCT
ejpam-3548	133	1	=	=	PRON
ejpam-3548	133	2	{	{	PUNCT
ejpam-3548	133	3	y	y	PROPN
ejpam-3548	133	4	∈	∈	PROPN
ejpam-3548	133	5	x	x	PUNCT
ejpam-3548	133	6	|	|	ADV
ejpam-3548	133	7	1x	1x	NUM
ejpam-3548	134	1	∗	∗	NOUN
ejpam-3548	134	2	y	y	PROPN
ejpam-3548	134	3	=	=	SYM
ejpam-3548	134	4	1x	1x	NUM
ejpam-3548	134	5	}	}	PUNCT
ejpam-3548	134	6	=	=	SYM
ejpam-3548	134	7	{	{	PUNCT
ejpam-3548	134	8	1x	1x	NUM
ejpam-3548	134	9	}	}	PUNCT
ejpam-3548	134	10	.	.	PUNCT
ejpam-3548	135	1	thus	thus	ADV
ejpam-3548	135	2	,	,	PUNCT
ejpam-3548	135	3	if	if	SCONJ
ejpam-3548	135	4	1x	1x	PROPN
ejpam-3548	135	5	∈	∈	PROPN
ejpam-3548	135	6	a	a	PRON
ejpam-3548	135	7	,	,	PUNCT
ejpam-3548	135	8	then	then	ADV
ejpam-3548	135	9	rx(a	rx(a	NOUN
ejpam-3548	135	10	)	)	PUNCT
ejpam-3548	135	11	⊆	⊆	NUM
ejpam-3548	135	12	rx({1x	rx({1x	NOUN
ejpam-3548	135	13	}	}	PUNCT
ejpam-3548	135	14	)	)	PUNCT
ejpam-3548	135	15	=	=	SYM
ejpam-3548	135	16	{	{	PUNCT
ejpam-3548	135	17	1x	1x	NUM
ejpam-3548	135	18	}	}	PUNCT
ejpam-3548	135	19	,	,	PUNCT
ejpam-3548	135	20	that	that	ADV
ejpam-3548	135	21	is	is	ADV
ejpam-3548	135	22	,	,	PUNCT
ejpam-3548	135	23	rx(a	rx(a	NOUN
ejpam-3548	135	24	)	)	PUNCT
ejpam-3548	135	25	=	=	SYM
ejpam-3548	135	26	{	{	PUNCT
ejpam-3548	135	27	1x	1x	NUM
ejpam-3548	135	28	}	}	PUNCT
ejpam-3548	135	29	theorem	theorem	VERB
ejpam-3548	135	30	7	7	NUM
ejpam-3548	135	31	.	.	PUNCT
ejpam-3548	136	1	let	let	VERB
ejpam-3548	136	2	x	x	PRON
ejpam-3548	136	3	be	be	AUX
ejpam-3548	136	4	a	a	DET
ejpam-3548	136	5	self	self	NOUN
ejpam-3548	136	6	distributive	distributive	ADJ
ejpam-3548	136	7	be	be	NOUN
ejpam-3548	136	8	-	-	PUNCT
ejpam-3548	136	9	algebra	algebra	NOUN
ejpam-3548	136	10	and	and	CCONJ
ejpam-3548	136	11	a	a	DET
ejpam-3548	136	12	be	be	NOUN
ejpam-3548	136	13	a	a	DET
ejpam-3548	136	14	nonempty	nonempty	ADJ
ejpam-3548	136	15	subset	subset	NOUN
ejpam-3548	136	16	of	of	ADP
ejpam-3548	136	17	x.	x.	NOUN
ejpam-3548	136	18	then	then	ADV
ejpam-3548	136	19	rx(a	rx(a	NOUN
ejpam-3548	136	20	)	)	PUNCT
ejpam-3548	136	21	is	be	AUX
ejpam-3548	136	22	an	an	DET
ejpam-3548	136	23	ideal	ideal	NOUN
ejpam-3548	136	24	and	and	CCONJ
ejpam-3548	136	25	a	a	DET
ejpam-3548	136	26	filter	filter	NOUN
ejpam-3548	136	27	.	.	PUNCT
ejpam-3548	137	1	proof	proof	NOUN
ejpam-3548	137	2	.	.	PUNCT
ejpam-3548	138	1	by	by	ADP
ejpam-3548	138	2	theorem	theorem	NOUN
ejpam-3548	138	3	6	6	NUM
ejpam-3548	138	4	,	,	PUNCT
ejpam-3548	138	5	1x	1x	NUM
ejpam-3548	138	6	∈	∈	PROPN
ejpam-3548	138	7	rx(a	rx(a	PROPN
ejpam-3548	138	8	)	)	PUNCT
ejpam-3548	138	9	.	.	PUNCT
ejpam-3548	139	1	let	let	VERB
ejpam-3548	139	2	x	x	PRON
ejpam-3548	139	3	,	,	PUNCT
ejpam-3548	139	4	y	y	PROPN
ejpam-3548	139	5	,	,	PUNCT
ejpam-3548	139	6	z	z	PROPN
ejpam-3548	139	7	∈	∈	PROPN
ejpam-3548	139	8	x.	x.	NOUN
ejpam-3548	139	9	suppose	suppose	VERB
ejpam-3548	139	10	that	that	SCONJ
ejpam-3548	139	11	y	y	PROPN
ejpam-3548	139	12	∈	∈	PROPN
ejpam-3548	139	13	rx(a	rx(a	PROPN
ejpam-3548	139	14	)	)	PUNCT
ejpam-3548	139	15	.	.	PUNCT
ejpam-3548	140	1	then	then	ADV
ejpam-3548	140	2	a	a	DET
ejpam-3548	140	3	∗	∗	NOUN
ejpam-3548	140	4	y	y	NOUN
ejpam-3548	140	5	=	=	SYM
ejpam-3548	140	6	1x	1x	NUM
ejpam-3548	140	7	for	for	ADP
ejpam-3548	140	8	all	all	DET
ejpam-3548	140	9	a	a	DET
ejpam-3548	140	10	∈	∈	NOUN
ejpam-3548	140	11	a.	a.	NOUN
ejpam-3548	140	12	let	let	NOUN
ejpam-3548	140	13	x	x	PRON
ejpam-3548	140	14	∗	∗	X
ejpam-3548	140	15	(	(	PUNCT
ejpam-3548	140	16	y	y	PROPN
ejpam-3548	140	17	∗	∗	PROPN
ejpam-3548	140	18	z	z	PROPN
ejpam-3548	140	19	)	)	PUNCT
ejpam-3548	140	20	∈	∈	PROPN
ejpam-3548	140	21	rx(a	rx(a	NOUN
ejpam-3548	140	22	)	)	PUNCT
ejpam-3548	140	23	.	.	PUNCT
ejpam-3548	141	1	then	then	ADV
ejpam-3548	141	2	a	a	DET
ejpam-3548	141	3	∗	∗	NOUN
ejpam-3548	141	4	(	(	PUNCT
ejpam-3548	141	5	x	x	SYM
ejpam-3548	141	6	∗	∗	NOUN
ejpam-3548	141	7	(	(	PUNCT
ejpam-3548	141	8	y	y	PROPN
ejpam-3548	141	9	∗	∗	PROPN
ejpam-3548	141	10	z	z	NOUN
ejpam-3548	141	11	)	)	PUNCT
ejpam-3548	141	12	)	)	PUNCT
ejpam-3548	142	1	=	=	SYM
ejpam-3548	142	2	1x	1x	NUM
ejpam-3548	142	3	for	for	ADP
ejpam-3548	142	4	all	all	DET
ejpam-3548	142	5	a	a	DET
ejpam-3548	142	6	∈	∈	NOUN
ejpam-3548	142	7	a.	a.	NOUN
ejpam-3548	142	8	since	since	SCONJ
ejpam-3548	142	9	x	x	PROPN
ejpam-3548	142	10	∗	∗	NOUN
ejpam-3548	142	11	(	(	PUNCT
ejpam-3548	142	12	y	y	PROPN
ejpam-3548	142	13	∗	∗	PROPN
ejpam-3548	142	14	z	z	NOUN
ejpam-3548	142	15	)	)	PUNCT
ejpam-3548	143	1	=	=	SYM
ejpam-3548	143	2	y	y	PROPN
ejpam-3548	143	3	∗	∗	NOUN
ejpam-3548	143	4	(	(	PUNCT
ejpam-3548	143	5	x	x	X
ejpam-3548	143	6	∗	∗	PROPN
ejpam-3548	143	7	z	z	PROPN
ejpam-3548	143	8	)	)	PUNCT
ejpam-3548	143	9	,	,	PUNCT
ejpam-3548	143	10	a	a	DET
ejpam-3548	143	11	∗	∗	NOUN
ejpam-3548	143	12	(	(	PUNCT
ejpam-3548	143	13	y	y	PROPN
ejpam-3548	143	14	∗	∗	NOUN
ejpam-3548	143	15	(	(	PUNCT
ejpam-3548	143	16	x	x	X
ejpam-3548	143	17	∗	∗	PROPN
ejpam-3548	143	18	z	z	NOUN
ejpam-3548	143	19	)	)	PUNCT
ejpam-3548	143	20	)	)	PUNCT
ejpam-3548	144	1	=	=	SYM
ejpam-3548	144	2	1x	1x	NUM
ejpam-3548	144	3	for	for	ADP
ejpam-3548	144	4	all	all	DET
ejpam-3548	144	5	a	a	DET
ejpam-3548	144	6	∈	∈	NOUN
ejpam-3548	144	7	a.	a.	NOUN
ejpam-3548	144	8	since	since	SCONJ
ejpam-3548	144	9	x	x	PRON
ejpam-3548	144	10	is	be	AUX
ejpam-3548	144	11	self	self	NOUN
ejpam-3548	144	12	distributive	distributive	ADJ
ejpam-3548	144	13	,	,	PUNCT
ejpam-3548	144	14	(	(	PUNCT
ejpam-3548	144	15	a	a	DET
ejpam-3548	144	16	∗	∗	NOUN
ejpam-3548	144	17	y	y	NOUN
ejpam-3548	144	18	)	)	PUNCT
ejpam-3548	144	19	∗	∗	NOUN
ejpam-3548	144	20	(	(	PUNCT
ejpam-3548	144	21	a	a	DET
ejpam-3548	144	22	∗	∗	NOUN
ejpam-3548	144	23	(	(	PUNCT
ejpam-3548	144	24	x	x	X
ejpam-3548	144	25	∗	∗	PROPN
ejpam-3548	144	26	z	z	NOUN
ejpam-3548	144	27	)	)	PUNCT
ejpam-3548	144	28	)	)	PUNCT
ejpam-3548	145	1	=	=	SYM
ejpam-3548	145	2	1x	1x	NUM
ejpam-3548	145	3	for	for	ADP
ejpam-3548	145	4	all	all	DET
ejpam-3548	145	5	a	a	DET
ejpam-3548	145	6	∈	∈	NOUN
ejpam-3548	145	7	a.	a.	NOUN
ejpam-3548	145	8	since	since	SCONJ
ejpam-3548	145	9	a	a	DET
ejpam-3548	145	10	∗	∗	NOUN
ejpam-3548	145	11	y	y	NOUN
ejpam-3548	145	12	=	=	SYM
ejpam-3548	145	13	1x	1x	NUM
ejpam-3548	145	14	for	for	ADP
ejpam-3548	145	15	all	all	DET
ejpam-3548	145	16	a	a	DET
ejpam-3548	145	17	∈	∈	PROPN
ejpam-3548	145	18	a	a	PRON
ejpam-3548	145	19	,	,	PUNCT
ejpam-3548	145	20	1x	1x	NUM
ejpam-3548	145	21	∗	∗	NOUN
ejpam-3548	145	22	(	(	PUNCT
ejpam-3548	145	23	a	a	DET
ejpam-3548	145	24	∗	∗	NOUN
ejpam-3548	145	25	(	(	PUNCT
ejpam-3548	145	26	x	x	X
ejpam-3548	145	27	∗	∗	PROPN
ejpam-3548	145	28	z	z	NOUN
ejpam-3548	145	29	)	)	PUNCT
ejpam-3548	145	30	)	)	PUNCT
ejpam-3548	146	1	=	=	SYM
ejpam-3548	146	2	1x	1x	NUM
ejpam-3548	146	3	for	for	ADP
ejpam-3548	146	4	all	all	DET
ejpam-3548	146	5	a	a	DET
ejpam-3548	146	6	∈	∈	NOUN
ejpam-3548	146	7	a.	a.	NOUN
ejpam-3548	146	8	this	this	PRON
ejpam-3548	146	9	implies	imply	VERB
ejpam-3548	146	10	that	that	SCONJ
ejpam-3548	146	11	a	a	DET
ejpam-3548	146	12	∗	∗	NOUN
ejpam-3548	146	13	(	(	PUNCT
ejpam-3548	146	14	x	x	X
ejpam-3548	146	15	∗	∗	NOUN
ejpam-3548	146	16	z	z	NOUN
ejpam-3548	146	17	)	)	PUNCT
ejpam-3548	146	18	=	=	SYM
ejpam-3548	146	19	1x	1x	NUM
ejpam-3548	146	20	.	.	PUNCT
ejpam-3548	147	1	hence	hence	ADV
ejpam-3548	147	2	,	,	PUNCT
ejpam-3548	147	3	x	x	X
ejpam-3548	147	4	∗	∗	NOUN
ejpam-3548	147	5	z	z	NOUN
ejpam-3548	147	6	∈	∈	PROPN
ejpam-3548	147	7	rx(a	rx(a	NOUN
ejpam-3548	147	8	)	)	PUNCT
ejpam-3548	147	9	.	.	PUNCT
ejpam-3548	148	1	by	by	ADP
ejpam-3548	148	2	lemma	lemma	PROPN
ejpam-3548	148	3	1	1	NUM
ejpam-3548	148	4	,	,	PUNCT
ejpam-3548	148	5	rx(a	rx(a	NOUN
ejpam-3548	148	6	)	)	PUNCT
ejpam-3548	148	7	is	be	AUX
ejpam-3548	148	8	an	an	DET
ejpam-3548	148	9	ideal	ideal	NOUN
ejpam-3548	148	10	.	.	PUNCT
ejpam-3548	149	1	suppose	suppose	VERB
ejpam-3548	149	2	that	that	SCONJ
ejpam-3548	149	3	x	x	PROPN
ejpam-3548	149	4	∈	∈	PROPN
ejpam-3548	149	5	rx(a	rx(a	PROPN
ejpam-3548	149	6	)	)	PUNCT
ejpam-3548	149	7	and	and	CCONJ
ejpam-3548	149	8	x∗y	x∗y	PUNCT
ejpam-3548	149	9	∈	∈	PROPN
ejpam-3548	149	10	rx(a	rx(a	PROPN
ejpam-3548	149	11	)	)	PUNCT
ejpam-3548	149	12	.	.	PUNCT
ejpam-3548	150	1	then	then	ADV
ejpam-3548	150	2	a∗x	a∗x	NUM
ejpam-3548	150	3	=	=	SYM
ejpam-3548	150	4	1x	1x	NUM
ejpam-3548	150	5	and	and	CCONJ
ejpam-3548	150	6	a∗(x∗y	a∗(x∗y	PROPN
ejpam-3548	150	7	)	)	PUNCT
ejpam-3548	150	8	=	=	SYM
ejpam-3548	150	9	1x	1x	NUM
ejpam-3548	150	10	for	for	ADP
ejpam-3548	150	11	all	all	DET
ejpam-3548	150	12	a	a	DET
ejpam-3548	150	13	∈	∈	NOUN
ejpam-3548	150	14	a.	a.	NOUN
ejpam-3548	150	15	since	since	SCONJ
ejpam-3548	150	16	x	x	PRON
ejpam-3548	150	17	is	be	AUX
ejpam-3548	150	18	self	self	NOUN
ejpam-3548	150	19	distributive	distributive	ADJ
ejpam-3548	150	20	,	,	PUNCT
ejpam-3548	150	21	a	a	DET
ejpam-3548	150	22	∗	∗	NOUN
ejpam-3548	150	23	y	y	NOUN
ejpam-3548	150	24	=	=	SYM
ejpam-3548	150	25	1x	1x	PROPN
ejpam-3548	150	26	∗	∗	NOUN
ejpam-3548	150	27	(	(	PUNCT
ejpam-3548	150	28	a	a	DET
ejpam-3548	150	29	∗	∗	NOUN
ejpam-3548	150	30	y	y	NOUN
ejpam-3548	150	31	)	)	PUNCT
ejpam-3548	150	32	=	=	PUNCT
ejpam-3548	150	33	(	(	PUNCT
ejpam-3548	150	34	a	a	DET
ejpam-3548	150	35	∗	∗	NOUN
ejpam-3548	150	36	x	x	NOUN
ejpam-3548	150	37	)	)	PUNCT
ejpam-3548	150	38	∗	∗	NOUN
ejpam-3548	150	39	(	(	PUNCT
ejpam-3548	150	40	a	a	DET
ejpam-3548	150	41	∗	∗	NOUN
ejpam-3548	150	42	y	y	NOUN
ejpam-3548	150	43	)	)	PUNCT
ejpam-3548	150	44	=	=	PUNCT
ejpam-3548	150	45	a	a	DET
ejpam-3548	150	46	∗	∗	NOUN
ejpam-3548	150	47	(	(	PUNCT
ejpam-3548	150	48	x	x	X
ejpam-3548	150	49	∗	∗	NOUN
ejpam-3548	150	50	y	y	NOUN
ejpam-3548	150	51	)	)	PUNCT
ejpam-3548	150	52	=	=	SYM
ejpam-3548	150	53	1x	1x	NUM
ejpam-3548	150	54	for	for	ADP
ejpam-3548	150	55	all	all	DET
ejpam-3548	150	56	a	a	DET
ejpam-3548	150	57	∈	∈	PROPN
ejpam-3548	150	58	a.	a.	NOUN
ejpam-3548	150	59	thus	thus	ADV
ejpam-3548	150	60	,	,	PUNCT
ejpam-3548	150	61	y	y	PROPN
ejpam-3548	150	62	∈	∈	PROPN
ejpam-3548	150	63	rx(a	rx(a	NOUN
ejpam-3548	150	64	)	)	PUNCT
ejpam-3548	150	65	.	.	PUNCT
ejpam-3548	151	1	therefore	therefore	ADV
ejpam-3548	151	2	,	,	PUNCT
ejpam-3548	151	3	rx(a	rx(a	NOUN
ejpam-3548	151	4	)	)	PUNCT
ejpam-3548	151	5	is	be	AUX
ejpam-3548	151	6	a	a	DET
ejpam-3548	151	7	filter	filter	NOUN
ejpam-3548	151	8	.	.	PUNCT
ejpam-3548	152	1	3	3	X
ejpam-3548	152	2	.	.	X
ejpam-3548	152	3	a	a	DET
ejpam-3548	152	4	basis	basis	NOUN
ejpam-3548	152	5	br(x	br(x	NOUN
ejpam-3548	152	6	)	)	PUNCT
ejpam-3548	152	7	for	for	ADP
ejpam-3548	152	8	a	a	DET
ejpam-3548	152	9	topology	topology	NOUN
ejpam-3548	152	10	on	on	ADP
ejpam-3548	152	11	x	x	X
ejpam-3548	152	12	lemma	lemma	PROPN
ejpam-3548	152	13	2	2	X
ejpam-3548	152	14	.	.	PUNCT
ejpam-3548	152	15	let	let	VERB
ejpam-3548	152	16	x	x	PRON
ejpam-3548	152	17	be	be	AUX
ejpam-3548	152	18	a	a	DET
ejpam-3548	152	19	be	be	NOUN
ejpam-3548	152	20	-	-	PUNCT
ejpam-3548	152	21	algebra	algebra	NOUN
ejpam-3548	152	22	and	and	CCONJ
ejpam-3548	152	23	let	let	VERB
ejpam-3548	152	24	{	{	PUNCT
ejpam-3548	152	25	aα	aα	NOUN
ejpam-3548	152	26	:	:	PUNCT
ejpam-3548	152	27	α	α	PROPN
ejpam-3548	152	28	∈	∈	PROPN
ejpam-3548	153	1	i	i	PRON
ejpam-3548	153	2	}	}	PUNCT
ejpam-3548	153	3	be	be	VERB
ejpam-3548	153	4	a	a	DET
ejpam-3548	153	5	collection	collection	NOUN
ejpam-3548	153	6	of	of	ADP
ejpam-3548	153	7	subsets	subset	NOUN
ejpam-3548	153	8	of	of	ADP
ejpam-3548	153	9	x.	x.	NOUN
ejpam-3548	153	10	then	then	ADV
ejpam-3548	153	11	⋂	⋂	PROPN
ejpam-3548	153	12	α∈i	α∈i	NUM
ejpam-3548	153	13	rx(aα	rx(aα	X
ejpam-3548	153	14	)	)	PUNCT
ejpam-3548	153	15	=	=	SYM
ejpam-3548	153	16	rx	rx	X
ejpam-3548	153	17	(	(	PUNCT
ejpam-3548	153	18	⋃	⋃	PROPN
ejpam-3548	153	19	α∈i	α∈i	NUM
ejpam-3548	153	20	aα	aα	NOUN
ejpam-3548	153	21	)	)	PUNCT
ejpam-3548	153	22	.	.	PUNCT
ejpam-3548	154	1	j.	j.	PROPN
ejpam-3548	154	2	albaracin	albaracin	PROPN
ejpam-3548	154	3	,	,	PUNCT
ejpam-3548	154	4	j.	j.	PROPN
ejpam-3548	154	5	vilela	vilela	PROPN
ejpam-3548	154	6	/	/	SYM
ejpam-3548	154	7	eur	eur	PROPN
ejpam-3548	154	8	.	.	PUNCT
ejpam-3548	155	1	j.	j.	PROPN
ejpam-3548	155	2	pure	pure	PROPN
ejpam-3548	155	3	appl	appl	PROPN
ejpam-3548	155	4	.	.	PROPN
ejpam-3548	155	5	math	math	PROPN
ejpam-3548	155	6	,	,	PUNCT
ejpam-3548	155	7	12	12	NUM
ejpam-3548	155	8	(	(	PUNCT
ejpam-3548	155	9	4	4	NUM
ejpam-3548	155	10	)	)	PUNCT
ejpam-3548	155	11	(	(	PUNCT
ejpam-3548	155	12	2019	2019	NUM
ejpam-3548	155	13	)	)	PUNCT
ejpam-3548	155	14	,	,	PUNCT
ejpam-3548	155	15	1584	1584	NUM
ejpam-3548	155	16	-	-	SYM
ejpam-3548	155	17	1594	1594	NUM
ejpam-3548	155	18	1588	1588	NUM
ejpam-3548	155	19	proof	proof	NOUN
ejpam-3548	155	20	.	.	PUNCT
ejpam-3548	156	1	let	let	VERB
ejpam-3548	156	2	x	x	SYM
ejpam-3548	156	3	∈	∈	PROPN
ejpam-3548	156	4	⋂	⋂	PROPN
ejpam-3548	156	5	α∈i	α∈i	NOUN
ejpam-3548	156	6	rx(aα	rx(aα	NOUN
ejpam-3548	156	7	)	)	PUNCT
ejpam-3548	156	8	.	.	PUNCT
ejpam-3548	157	1	then	then	ADV
ejpam-3548	157	2	x	x	X
ejpam-3548	157	3	∈	∈	PROPN
ejpam-3548	157	4	rx(aα	rx(aα	PROPN
ejpam-3548	157	5	)	)	PUNCT
ejpam-3548	157	6	for	for	ADP
ejpam-3548	157	7	all	all	DET
ejpam-3548	157	8	α	α	PROPN
ejpam-3548	157	9	∈	∈	PROPN
ejpam-3548	157	10	i.	i.	NOUN
ejpam-3548	157	11	thus	thus	ADV
ejpam-3548	157	12	,	,	PUNCT
ejpam-3548	157	13	a	a	DET
ejpam-3548	157	14	∗	∗	NOUN
ejpam-3548	157	15	x	x	X
ejpam-3548	157	16	=	=	SYM
ejpam-3548	157	17	1x	1x	NUM
ejpam-3548	157	18	for	for	ADP
ejpam-3548	157	19	all	all	DET
ejpam-3548	157	20	a	a	DET
ejpam-3548	157	21	∈	∈	ADJ
ejpam-3548	157	22	aα	aα	NOUN
ejpam-3548	157	23	and	and	CCONJ
ejpam-3548	157	24	for	for	ADP
ejpam-3548	157	25	all	all	DET
ejpam-3548	157	26	α	α	DET
ejpam-3548	157	27	∈	∈	PROPN
ejpam-3548	157	28	i.	i.	NOUN
ejpam-3548	157	29	hence	hence	ADV
ejpam-3548	157	30	,	,	PUNCT
ejpam-3548	157	31	a	a	DET
ejpam-3548	157	32	∗	∗	NOUN
ejpam-3548	157	33	x	x	X
ejpam-3548	157	34	=	=	SYM
ejpam-3548	157	35	1x	1x	NUM
ejpam-3548	157	36	for	for	ADP
ejpam-3548	157	37	all	all	DET
ejpam-3548	157	38	a	a	DET
ejpam-3548	157	39	∈	∈	NOUN
ejpam-3548	157	40	⋃	⋃	NOUN
ejpam-3548	157	41	α∈i	α∈i	NUM
ejpam-3548	157	42	aα	aα	NOUN
ejpam-3548	157	43	.	.	PUNCT
ejpam-3548	158	1	so	so	ADV
ejpam-3548	158	2	,	,	PUNCT
ejpam-3548	158	3	x	x	PUNCT
ejpam-3548	158	4	∈	∈	NOUN
ejpam-3548	158	5	rx	rx	X
ejpam-3548	158	6	(	(	PUNCT
ejpam-3548	158	7	⋃	⋃	PROPN
ejpam-3548	158	8	α∈i	α∈i	NUM
ejpam-3548	158	9	aα	aα	NOUN
ejpam-3548	158	10	)	)	PUNCT
ejpam-3548	158	11	and	and	CCONJ
ejpam-3548	158	12	⋂	⋂	PROPN
ejpam-3548	158	13	α∈i	α∈i	NUM
ejpam-3548	158	14	rx(aα	rx(aα	NOUN
ejpam-3548	158	15	)	)	PUNCT
ejpam-3548	158	16	⊆	⊆	NUM
ejpam-3548	158	17	rx	rx	X
ejpam-3548	158	18	(	(	PUNCT
ejpam-3548	158	19	⋃	⋃	PROPN
ejpam-3548	158	20	α∈i	α∈i	NUM
ejpam-3548	158	21	aα	aα	NOUN
ejpam-3548	158	22	)	)	PUNCT
ejpam-3548	158	23	.	.	PUNCT
ejpam-3548	159	1	the	the	DET
ejpam-3548	159	2	other	other	ADJ
ejpam-3548	159	3	inclusion	inclusion	NOUN
ejpam-3548	159	4	is	be	AUX
ejpam-3548	159	5	proved	prove	VERB
ejpam-3548	159	6	similarly	similarly	ADV
ejpam-3548	159	7	.	.	PUNCT
ejpam-3548	160	1	therefore	therefore	ADV
ejpam-3548	160	2	,	,	PUNCT
ejpam-3548	160	3	the	the	DET
ejpam-3548	160	4	equality	equality	NOUN
ejpam-3548	160	5	is	be	AUX
ejpam-3548	160	6	true	true	ADJ
ejpam-3548	160	7	.	.	PUNCT
ejpam-3548	161	1	theorem	theorem	ADJ
ejpam-3548	161	2	8	8	NUM
ejpam-3548	161	3	.	.	PUNCT
ejpam-3548	162	1	let	let	VERB
ejpam-3548	162	2	x	x	PRON
ejpam-3548	162	3	be	be	AUX
ejpam-3548	162	4	a	a	DET
ejpam-3548	162	5	be	be	NOUN
ejpam-3548	162	6	-	-	PUNCT
ejpam-3548	162	7	algebra	algebra	NOUN
ejpam-3548	162	8	.	.	PUNCT
ejpam-3548	163	1	then	then	ADV
ejpam-3548	163	2	br(x	br(x	PUNCT
ejpam-3548	163	3	)	)	PUNCT
ejpam-3548	163	4	=	=	SYM
ejpam-3548	163	5	{	{	PUNCT
ejpam-3548	163	6	rx(a	rx(a	NOUN
ejpam-3548	163	7	)	)	PUNCT
ejpam-3548	163	8	:	:	PUNCT
ejpam-3548	163	9	∅	∅	NOUN
ejpam-3548	163	10	6=	6=	ADP
ejpam-3548	163	11	a	a	DET
ejpam-3548	163	12	⊆	⊆	NUM
ejpam-3548	163	13	x	x	SYM
ejpam-3548	163	14	}	}	PUNCT
ejpam-3548	163	15	is	be	AUX
ejpam-3548	163	16	a	a	DET
ejpam-3548	163	17	basis	basis	NOUN
ejpam-3548	163	18	for	for	ADP
ejpam-3548	163	19	some	some	DET
ejpam-3548	163	20	topology	topology	NOUN
ejpam-3548	163	21	on	on	ADP
ejpam-3548	163	22	x.	x.	NOUN
ejpam-3548	163	23	proof	proof	NOUN
ejpam-3548	163	24	.	.	PUNCT
ejpam-3548	164	1	clearly	clearly	ADV
ejpam-3548	164	2	,	,	PUNCT
ejpam-3548	164	3	x	x	X
ejpam-3548	164	4	=	=	PUNCT
ejpam-3548	164	5	⋃	⋃	ADP
ejpam-3548	164	6	a∈x	a∈x	NOUN
ejpam-3548	164	7	rx({a	rx({a	VERB
ejpam-3548	164	8	}	}	PUNCT
ejpam-3548	164	9	)	)	PUNCT
ejpam-3548	164	10	.	.	PUNCT
ejpam-3548	165	1	suppose	suppose	VERB
ejpam-3548	165	2	that	that	SCONJ
ejpam-3548	165	3	∅	∅	NOUN
ejpam-3548	165	4	6=	6=	ADP
ejpam-3548	165	5	a	a	PRON
ejpam-3548	165	6	,	,	PUNCT
ejpam-3548	165	7	b	b	NOUN
ejpam-3548	165	8	⊆	⊆	NUM
ejpam-3548	165	9	x.	x.	NOUN
ejpam-3548	165	10	by	by	ADP
ejpam-3548	165	11	lemma	lemma	PROPN
ejpam-3548	165	12	2	2	NUM
ejpam-3548	165	13	,	,	PUNCT
ejpam-3548	165	14	rx(a)∩	rx(a)∩	X
ejpam-3548	165	15	rx(b	rx(b	ADJ
ejpam-3548	165	16	)	)	PUNCT
ejpam-3548	165	17	=	=	SYM
ejpam-3548	165	18	rx(a∪b	rx(a∪b	X
ejpam-3548	165	19	)	)	PUNCT
ejpam-3548	165	20	∈	∈	PROPN
ejpam-3548	165	21	br(x	br(x	NUM
ejpam-3548	165	22	)	)	PUNCT
ejpam-3548	165	23	.	.	PUNCT
ejpam-3548	166	1	by	by	ADP
ejpam-3548	166	2	theorem	theorem	NOUN
ejpam-3548	166	3	1	1	NUM
ejpam-3548	166	4	,	,	PUNCT
ejpam-3548	166	5	br(x	br(x	NUM
ejpam-3548	166	6	)	)	PUNCT
ejpam-3548	166	7	is	be	AUX
ejpam-3548	166	8	a	a	DET
ejpam-3548	166	9	basis	basis	NOUN
ejpam-3548	166	10	for	for	ADP
ejpam-3548	166	11	some	some	DET
ejpam-3548	166	12	topology	topology	NOUN
ejpam-3548	166	13	on	on	ADP
ejpam-3548	166	14	x.	x.	NOUN
ejpam-3548	166	15	we	we	PRON
ejpam-3548	166	16	denote	denote	VERB
ejpam-3548	166	17	by	by	ADP
ejpam-3548	166	18	τr(x	τr(x	NOUN
ejpam-3548	166	19	)	)	PUNCT
ejpam-3548	166	20	the	the	DET
ejpam-3548	166	21	topology	topology	NOUN
ejpam-3548	166	22	generated	generate	VERB
ejpam-3548	166	23	by	by	ADP
ejpam-3548	166	24	br(x	br(x	NOUN
ejpam-3548	166	25	)	)	PUNCT
ejpam-3548	166	26	.	.	PUNCT
ejpam-3548	167	1	example	example	NOUN
ejpam-3548	168	1	2	2	NUM
ejpam-3548	168	2	.	.	X
ejpam-3548	168	3	consider	consider	VERB
ejpam-3548	168	4	the	the	DET
ejpam-3548	168	5	be	be	AUX
ejpam-3548	168	6	-	-	PUNCT
ejpam-3548	168	7	algebra	algebra	NOUN
ejpam-3548	168	8	n0	n0	NUM
ejpam-3548	168	9	in	in	ADP
ejpam-3548	168	10	example	example	NOUN
ejpam-3548	168	11	1	1	X
ejpam-3548	168	12	.	.	PUNCT
ejpam-3548	169	1	let	let	VERB
ejpam-3548	169	2	z	z	NOUN
ejpam-3548	169	3	∈	∈	PROPN
ejpam-3548	169	4	n0	n0	PROPN
ejpam-3548	169	5	.	.	PUNCT
ejpam-3548	170	1	then	then	ADV
ejpam-3548	170	2	rn0(z	rn0(z	X
ejpam-3548	170	3	)	)	PUNCT
ejpam-3548	171	1	=	=	PUNCT
ejpam-3548	171	2	{	{	PUNCT
ejpam-3548	171	3	0	0	NUM
ejpam-3548	171	4	,	,	PUNCT
ejpam-3548	171	5	1	1	NUM
ejpam-3548	171	6	,	,	PUNCT
ejpam-3548	171	7	2	2	NUM
ejpam-3548	171	8	,	,	PUNCT
ejpam-3548	171	9	.	.	PUNCT
ejpam-3548	171	10	.	.	PUNCT
ejpam-3548	172	1	.	.	PUNCT
ejpam-3548	173	1	,	,	PUNCT
ejpam-3548	173	2	z	z	X
ejpam-3548	173	3	}	}	PUNCT
ejpam-3548	173	4	.	.	PUNCT
ejpam-3548	174	1	it	it	PRON
ejpam-3548	174	2	is	be	AUX
ejpam-3548	174	3	easy	easy	ADJ
ejpam-3548	174	4	to	to	PART
ejpam-3548	174	5	see	see	VERB
ejpam-3548	174	6	that	that	DET
ejpam-3548	174	7	b	b	NOUN
ejpam-3548	174	8	=	=	PRON
ejpam-3548	174	9	{	{	PUNCT
ejpam-3548	174	10	rn0(z	rn0(z	NOUN
ejpam-3548	174	11	)	)	PUNCT
ejpam-3548	174	12	:	:	PUNCT
ejpam-3548	174	13	z	z	PROPN
ejpam-3548	174	14	∈	∈	PROPN
ejpam-3548	174	15	n0	n0	PROPN
ejpam-3548	174	16	}	}	PUNCT
ejpam-3548	174	17	⊆	⊆	NUM
ejpam-3548	174	18	br(n0	br(n0	NOUN
ejpam-3548	174	19	)	)	PUNCT
ejpam-3548	174	20	.	.	PUNCT
ejpam-3548	175	1	suppose	suppose	VERB
ejpam-3548	175	2	that	that	SCONJ
ejpam-3548	175	3	∅	∅	NOUN
ejpam-3548	175	4	6=	6=	ADP
ejpam-3548	175	5	a	a	DET
ejpam-3548	175	6	⊆	⊆	NUM
ejpam-3548	175	7	n0	n0	NOUN
ejpam-3548	175	8	and	and	CCONJ
ejpam-3548	175	9	w	w	PROPN
ejpam-3548	175	10	=	=	PUNCT
ejpam-3548	175	11	mina	mina	PROPN
ejpam-3548	175	12	.	.	PUNCT
ejpam-3548	176	1	by	by	ADP
ejpam-3548	176	2	theorem	theorem	NOUN
ejpam-3548	176	3	6	6	NUM
ejpam-3548	176	4	,	,	PUNCT
ejpam-3548	176	5	rn0(a	rn0(a	PROPN
ejpam-3548	176	6	)	)	PUNCT
ejpam-3548	176	7	=	=	PUNCT
ejpam-3548	177	1	⋂	⋂	PROPN
ejpam-3548	177	2	a∈a	a∈a	ADJ
ejpam-3548	177	3	rn0({a	rn0({a	NOUN
ejpam-3548	177	4	}	}	PUNCT
ejpam-3548	177	5	)	)	PUNCT
ejpam-3548	177	6	.	.	PUNCT
ejpam-3548	178	1	it	it	PRON
ejpam-3548	178	2	follows	follow	VERB
ejpam-3548	178	3	that	that	PRON
ejpam-3548	178	4	rn0(a	rn0(a	PROPN
ejpam-3548	178	5	)	)	PUNCT
ejpam-3548	178	6	=	=	PRON
ejpam-3548	178	7	{	{	PUNCT
ejpam-3548	178	8	0	0	NUM
ejpam-3548	178	9	,	,	PUNCT
ejpam-3548	178	10	1	1	NUM
ejpam-3548	178	11	,	,	PUNCT
ejpam-3548	178	12	2	2	NUM
ejpam-3548	178	13	,	,	PUNCT
ejpam-3548	178	14	.	.	PUNCT
ejpam-3548	178	15	.	.	PUNCT
ejpam-3548	179	1	.	.	PUNCT
ejpam-3548	180	1	,	,	PUNCT
ejpam-3548	180	2	w	w	X
ejpam-3548	180	3	}	}	PUNCT
ejpam-3548	180	4	=	=	SYM
ejpam-3548	180	5	rn0(w	rn0(w	ADJ
ejpam-3548	180	6	)	)	PUNCT
ejpam-3548	180	7	∈	∈	PROPN
ejpam-3548	180	8	b.	b.	PROPN
ejpam-3548	180	9	hence	hence	ADV
ejpam-3548	180	10	,	,	PUNCT
ejpam-3548	180	11	br(n0	br(n0	PROPN
ejpam-3548	180	12	)	)	PUNCT
ejpam-3548	181	1	=	=	SYM
ejpam-3548	181	2	b	b	X
ejpam-3548	181	3	=	=	PRON
ejpam-3548	181	4	{	{	PUNCT
ejpam-3548	181	5	rn0(z	rn0(z	NOUN
ejpam-3548	181	6	)	)	PUNCT
ejpam-3548	181	7	:	:	PUNCT
ejpam-3548	181	8	z	z	PROPN
ejpam-3548	181	9	∈	∈	PROPN
ejpam-3548	181	10	n0	n0	NUM
ejpam-3548	181	11	}	}	PUNCT
ejpam-3548	181	12	.	.	PUNCT
ejpam-3548	182	1	let	let	VERB
ejpam-3548	182	2	∅	∅	NOUN
ejpam-3548	182	3	6=	6=	ADP
ejpam-3548	182	4	g	g	PROPN
ejpam-3548	182	5	∈	∈	PROPN
ejpam-3548	182	6	τr(n0	τr(n0	NUM
ejpam-3548	182	7	)	)	PUNCT
ejpam-3548	182	8	.	.	PUNCT
ejpam-3548	183	1	then	then	ADV
ejpam-3548	183	2	g	g	PROPN
ejpam-3548	183	3	=	=	PUNCT
ejpam-3548	183	4	⋃	⋃	PROPN
ejpam-3548	183	5	x∈k	x∈k	PROPN
ejpam-3548	183	6	rn0(x	rn0(x	NOUN
ejpam-3548	183	7	)	)	PUNCT
ejpam-3548	183	8	for	for	ADP
ejpam-3548	183	9	some	some	DET
ejpam-3548	183	10	∅	∅	NOUN
ejpam-3548	183	11	6=	6=	ADP
ejpam-3548	183	12	k	k	PROPN
ejpam-3548	183	13	⊆	⊆	NUM
ejpam-3548	183	14	n0	n0	NUM
ejpam-3548	183	15	.	.	PUNCT
ejpam-3548	184	1	clearly	clearly	ADV
ejpam-3548	184	2	,	,	PUNCT
ejpam-3548	184	3	k	k	PROPN
ejpam-3548	184	4	⊆	⊆	NUM
ejpam-3548	184	5	g.	g.	NOUN
ejpam-3548	184	6	suppose	suppose	VERB
ejpam-3548	184	7	first	first	ADV
ejpam-3548	184	8	that	that	SCONJ
ejpam-3548	184	9	|g|	|g|	PROPN
ejpam-3548	184	10	<	<	X
ejpam-3548	184	11	∞	∞	PROPN
ejpam-3548	184	12	and	and	CCONJ
ejpam-3548	184	13	let	let	VERB
ejpam-3548	184	14	v	v	NOUN
ejpam-3548	184	15	=	=	PUNCT
ejpam-3548	184	16	maxk	maxk	NOUN
ejpam-3548	184	17	.	.	PUNCT
ejpam-3548	185	1	then	then	ADV
ejpam-3548	185	2	g	g	PROPN
ejpam-3548	185	3	=	=	PROPN
ejpam-3548	185	4	rn0(v	rn0(v	PROPN
ejpam-3548	185	5	)	)	PUNCT
ejpam-3548	185	6	.	.	PUNCT
ejpam-3548	186	1	next	next	ADV
ejpam-3548	186	2	,	,	PUNCT
ejpam-3548	186	3	suppose	suppose	VERB
ejpam-3548	186	4	that	that	SCONJ
ejpam-3548	186	5	g	g	PROPN
ejpam-3548	186	6	is	be	AUX
ejpam-3548	186	7	an	an	DET
ejpam-3548	186	8	infinite	infinite	ADJ
ejpam-3548	186	9	set	set	NOUN
ejpam-3548	186	10	.	.	PUNCT
ejpam-3548	187	1	suppose	suppose	VERB
ejpam-3548	187	2	further	far	ADV
ejpam-3548	187	3	that	that	PRON
ejpam-3548	187	4	g	g	PROPN
ejpam-3548	187	5	6=	6=	PROPN
ejpam-3548	187	6	n0	n0	PROPN
ejpam-3548	187	7	,	,	PUNCT
ejpam-3548	187	8	say	say	VERB
ejpam-3548	188	1	m	m	VERB
ejpam-3548	188	2	∈	∈	PROPN
ejpam-3548	188	3	n0	n0	PROPN
ejpam-3548	188	4	\	\	PROPN
ejpam-3548	188	5	g.	g.	PROPN
ejpam-3548	188	6	then	then	ADV
ejpam-3548	188	7	m	m	PROPN
ejpam-3548	188	8	/∈	/∈	PUNCT
ejpam-3548	188	9	rn0(x	rn0(x	NOUN
ejpam-3548	188	10	)	)	PUNCT
ejpam-3548	188	11	for	for	ADP
ejpam-3548	188	12	all	all	DET
ejpam-3548	188	13	x	x	SYM
ejpam-3548	188	14	∈	∈	PROPN
ejpam-3548	188	15	k.	k.	NOUN
ejpam-3548	189	1	this	this	PRON
ejpam-3548	189	2	implies	imply	VERB
ejpam-3548	189	3	that	that	SCONJ
ejpam-3548	189	4	x	x	PUNCT
ejpam-3548	189	5	<	<	X
ejpam-3548	189	6	m	m	VERB
ejpam-3548	189	7	for	for	ADP
ejpam-3548	189	8	all	all	DET
ejpam-3548	189	9	x	x	SYM
ejpam-3548	189	10	∈	∈	PROPN
ejpam-3548	189	11	k.	k.	NOUN
ejpam-3548	189	12	hence	hence	ADV
ejpam-3548	189	13	,	,	PUNCT
ejpam-3548	189	14	g	g	PROPN
ejpam-3548	189	15	⊆	⊆	NUM
ejpam-3548	189	16	rn0(m	rn0(m	PROPN
ejpam-3548	189	17	)	)	PUNCT
ejpam-3548	189	18	,	,	PUNCT
ejpam-3548	189	19	contrary	contrary	ADV
ejpam-3548	189	20	to	to	ADP
ejpam-3548	189	21	the	the	DET
ejpam-3548	189	22	assumption	assumption	NOUN
ejpam-3548	189	23	that	that	SCONJ
ejpam-3548	189	24	g	g	PROPN
ejpam-3548	189	25	is	be	AUX
ejpam-3548	189	26	an	an	DET
ejpam-3548	189	27	infinite	infinite	ADJ
ejpam-3548	189	28	set	set	NOUN
ejpam-3548	189	29	.	.	PUNCT
ejpam-3548	190	1	therefore	therefore	ADV
ejpam-3548	190	2	,	,	PUNCT
ejpam-3548	190	3	g	g	PROPN
ejpam-3548	190	4	=	=	SYM
ejpam-3548	190	5	n0	n0	PROPN
ejpam-3548	190	6	.	.	PUNCT
ejpam-3548	190	7	accordingly	accordingly	ADV
ejpam-3548	190	8	,	,	PUNCT
ejpam-3548	190	9	τr(n0	τr(n0	X
ejpam-3548	190	10	)	)	PUNCT
ejpam-3548	191	1	=	=	SYM
ejpam-3548	191	2	{	{	PUNCT
ejpam-3548	191	3	∅	∅	NOUN
ejpam-3548	191	4	,	,	PUNCT
ejpam-3548	191	5	n0	n0	ADJ
ejpam-3548	191	6	}	}	PUNCT
ejpam-3548	191	7	∪	∪	X
ejpam-3548	191	8	{	{	PUNCT
ejpam-3548	191	9	rn0(z	rn0(z	NOUN
ejpam-3548	191	10	)	)	PUNCT
ejpam-3548	191	11	:	:	PUNCT
ejpam-3548	191	12	z	z	PROPN
ejpam-3548	191	13	∈	∈	PROPN
ejpam-3548	191	14	n0	n0	PROPN
ejpam-3548	191	15	}	}	PUNCT
ejpam-3548	191	16	=	=	SYM
ejpam-3548	191	17	{	{	PUNCT
ejpam-3548	191	18	∅	∅	NOUN
ejpam-3548	191	19	,	,	PUNCT
ejpam-3548	191	20	n0	n0	ADJ
ejpam-3548	191	21	}	}	PUNCT
ejpam-3548	191	22	∪	∪	VERB
ejpam-3548	191	23	br(n0	br(n0	PROPN
ejpam-3548	191	24	)	)	PUNCT
ejpam-3548	191	25	.	.	PUNCT
ejpam-3548	192	1	theorem	theorem	NOUN
ejpam-3548	192	2	9	9	NUM
ejpam-3548	192	3	.	.	PUNCT
ejpam-3548	193	1	let	let	VERB
ejpam-3548	193	2	x	x	PRON
ejpam-3548	193	3	be	be	AUX
ejpam-3548	193	4	a	a	DET
ejpam-3548	193	5	be	be	NOUN
ejpam-3548	193	6	-	-	PUNCT
ejpam-3548	193	7	algebra	algebra	NOUN
ejpam-3548	193	8	.	.	PUNCT
ejpam-3548	194	1	then	then	ADV
ejpam-3548	194	2	(	(	PUNCT
ejpam-3548	194	3	x	x	NOUN
ejpam-3548	194	4	,	,	PUNCT
ejpam-3548	194	5	τr(x	τr(x	NUM
ejpam-3548	194	6	)	)	PUNCT
ejpam-3548	194	7	)	)	PUNCT
ejpam-3548	194	8	is	be	AUX
ejpam-3548	194	9	connected	connect	VERB
ejpam-3548	194	10	.	.	PUNCT
ejpam-3548	195	1	proof	proof	NOUN
ejpam-3548	195	2	.	.	PUNCT
ejpam-3548	196	1	let	let	VERB
ejpam-3548	196	2	∅	∅	NOUN
ejpam-3548	196	3	6=	6=	ADP
ejpam-3548	196	4	g	g	PROPN
ejpam-3548	196	5	∈	∈	PROPN
ejpam-3548	196	6	τr(x	τr(x	NUM
ejpam-3548	196	7	)	)	PUNCT
ejpam-3548	196	8	.	.	PUNCT
ejpam-3548	197	1	by	by	ADP
ejpam-3548	197	2	theorem	theorem	NOUN
ejpam-3548	197	3	2	2	NUM
ejpam-3548	197	4	,	,	PUNCT
ejpam-3548	197	5	there	there	PRON
ejpam-3548	197	6	exists	exist	VERB
ejpam-3548	197	7	a	a	DET
ejpam-3548	197	8	⊆	⊆	NUM
ejpam-3548	197	9	x	x	PUNCT
ejpam-3548	197	10	such	such	ADJ
ejpam-3548	197	11	that	that	SCONJ
ejpam-3548	197	12	rx(a	rx(a	NOUN
ejpam-3548	197	13	)	)	PUNCT
ejpam-3548	197	14	⊆	⊆	NUM
ejpam-3548	197	15	g.	g.	NOUN
ejpam-3548	197	16	by	by	ADP
ejpam-3548	197	17	theorem	theorem	NOUN
ejpam-3548	197	18	6	6	NUM
ejpam-3548	197	19	,	,	PUNCT
ejpam-3548	197	20	1x	1x	PROPN
ejpam-3548	197	21	∈	∈	PROPN
ejpam-3548	197	22	g.	g.	PROPN
ejpam-3548	198	1	thus	thus	ADV
ejpam-3548	198	2	,	,	PUNCT
ejpam-3548	198	3	if	if	SCONJ
ejpam-3548	198	4	u	u	NOUN
ejpam-3548	198	5	is	be	AUX
ejpam-3548	198	6	a	a	DET
ejpam-3548	198	7	nonempty	nonempty	ADJ
ejpam-3548	198	8	open	open	ADJ
ejpam-3548	198	9	set	set	NOUN
ejpam-3548	198	10	such	such	ADJ
ejpam-3548	198	11	that	that	DET
ejpam-3548	198	12	u	u	PROPN
ejpam-3548	198	13	6=	6=	ADP
ejpam-3548	198	14	g	g	PROPN
ejpam-3548	198	15	,	,	PUNCT
ejpam-3548	198	16	then	then	ADV
ejpam-3548	198	17	g	g	PROPN
ejpam-3548	198	18	∩	∩	ADJ
ejpam-3548	198	19	u	u	PROPN
ejpam-3548	198	20	6=	6=	NOUN
ejpam-3548	198	21	∅	∅	NOUN
ejpam-3548	198	22	since	since	SCONJ
ejpam-3548	198	23	1x	1x	PROPN
ejpam-3548	198	24	∈	∈	PROPN
ejpam-3548	198	25	u	u	NOUN
ejpam-3548	198	26	.	.	PUNCT
ejpam-3548	199	1	hence	hence	ADV
ejpam-3548	199	2	,	,	PUNCT
ejpam-3548	199	3	x	x	PRON
ejpam-3548	199	4	can	can	AUX
ejpam-3548	199	5	not	not	PART
ejpam-3548	199	6	have	have	VERB
ejpam-3548	199	7	a	a	DET
ejpam-3548	199	8	decomposition	decomposition	NOUN
ejpam-3548	199	9	,	,	PUNCT
ejpam-3548	199	10	that	that	ADV
ejpam-3548	199	11	is	is	ADV
ejpam-3548	199	12	,	,	PUNCT
ejpam-3548	199	13	(	(	PUNCT
ejpam-3548	199	14	x	x	X
ejpam-3548	199	15	,	,	PUNCT
ejpam-3548	199	16	τr(x	τr(x	NUM
ejpam-3548	199	17	)	)	PUNCT
ejpam-3548	199	18	)	)	PUNCT
ejpam-3548	199	19	is	be	AUX
ejpam-3548	199	20	connected	connect	VERB
ejpam-3548	199	21	.	.	PUNCT
ejpam-3548	200	1	lemma	lemma	PROPN
ejpam-3548	200	2	3	3	X
ejpam-3548	200	3	.	.	PUNCT
ejpam-3548	201	1	let	let	VERB
ejpam-3548	201	2	x	x	PRON
ejpam-3548	201	3	be	be	AUX
ejpam-3548	201	4	a	a	DET
ejpam-3548	201	5	be	be	NOUN
ejpam-3548	201	6	-	-	PUNCT
ejpam-3548	201	7	algebra	algebra	NOUN
ejpam-3548	201	8	and	and	CCONJ
ejpam-3548	201	9	x	x	SYM
ejpam-3548	201	10	∈	∈	PROPN
ejpam-3548	201	11	x.	x.	NOUN
ejpam-3548	201	12	then	then	ADV
ejpam-3548	201	13	{	{	PUNCT
ejpam-3548	201	14	x	x	NOUN
ejpam-3548	201	15	}	}	PUNCT
ejpam-3548	201	16	∈	∈	PROPN
ejpam-3548	201	17	τr(x	τr(x	NUM
ejpam-3548	201	18	)	)	PUNCT
ejpam-3548	201	19	if	if	SCONJ
ejpam-3548	201	20	and	and	CCONJ
ejpam-3548	201	21	only	only	ADV
ejpam-3548	201	22	if	if	SCONJ
ejpam-3548	201	23	x	x	X
ejpam-3548	201	24	=	=	PUNCT
ejpam-3548	201	25	1x	1x	NUM
ejpam-3548	201	26	.	.	PUNCT
ejpam-3548	202	1	proof	proof	NOUN
ejpam-3548	202	2	.	.	PUNCT
ejpam-3548	203	1	suppose	suppose	VERB
ejpam-3548	203	2	that	that	SCONJ
ejpam-3548	203	3	x	x	PROPN
ejpam-3548	203	4	=	=	PUNCT
ejpam-3548	203	5	1x	1x	NUM
ejpam-3548	203	6	.	.	PUNCT
ejpam-3548	204	1	then	then	ADV
ejpam-3548	204	2	{	{	PUNCT
ejpam-3548	204	3	1x	1x	NUM
ejpam-3548	204	4	}	}	PUNCT
ejpam-3548	204	5	=	=	PUNCT
ejpam-3548	204	6	rx	rx	X
ejpam-3548	204	7	(	(	PUNCT
ejpam-3548	204	8	1x	1x	NUM
ejpam-3548	204	9	)	)	PUNCT
ejpam-3548	204	10	∈	∈	PROPN
ejpam-3548	204	11	br(x	br(x	NUM
ejpam-3548	204	12	)	)	PUNCT
ejpam-3548	204	13	.	.	PUNCT
ejpam-3548	205	1	hence	hence	ADV
ejpam-3548	205	2	,	,	PUNCT
ejpam-3548	205	3	{	{	PUNCT
ejpam-3548	205	4	1x	1x	NUM
ejpam-3548	205	5	}	}	PUNCT
ejpam-3548	205	6	∈	∈	PROPN
ejpam-3548	205	7	τr(x	τr(x	NUM
ejpam-3548	205	8	)	)	PUNCT
ejpam-3548	205	9	.	.	PUNCT
ejpam-3548	205	10	suppose	suppose	VERB
ejpam-3548	205	11	that	that	SCONJ
ejpam-3548	205	12	{	{	PUNCT
ejpam-3548	205	13	x	x	X
ejpam-3548	205	14	}	}	PUNCT
ejpam-3548	205	15	∈	∈	PROPN
ejpam-3548	205	16	τr(x	τr(x	NUM
ejpam-3548	205	17	)	)	PUNCT
ejpam-3548	205	18	.	.	PUNCT
ejpam-3548	206	1	then	then	ADV
ejpam-3548	206	2	there	there	PRON
ejpam-3548	206	3	exists	exist	VERB
ejpam-3548	206	4	∅	∅	NOUN
ejpam-3548	206	5	6=	6=	ADP
ejpam-3548	206	6	a	a	DET
ejpam-3548	206	7	⊆	⊆	NUM
ejpam-3548	206	8	x	x	PUNCT
ejpam-3548	206	9	such	such	ADJ
ejpam-3548	206	10	that	that	SCONJ
ejpam-3548	206	11	rx(a	rx(a	NOUN
ejpam-3548	206	12	)	)	PUNCT
ejpam-3548	206	13	=	=	SYM
ejpam-3548	206	14	{	{	PUNCT
ejpam-3548	206	15	x	x	NOUN
ejpam-3548	206	16	}	}	PUNCT
ejpam-3548	206	17	.	.	PUNCT
ejpam-3548	207	1	since	since	SCONJ
ejpam-3548	207	2	1x	1x	PROPN
ejpam-3548	207	3	∈	∈	PROPN
ejpam-3548	207	4	rx(a	rx(a	NOUN
ejpam-3548	207	5	)	)	PUNCT
ejpam-3548	207	6	,	,	PUNCT
ejpam-3548	207	7	it	it	PRON
ejpam-3548	207	8	follows	follow	VERB
ejpam-3548	207	9	that	that	SCONJ
ejpam-3548	207	10	x	x	PUNCT
ejpam-3548	207	11	=	=	SYM
ejpam-3548	207	12	1x	1x	NUM
ejpam-3548	207	13	.	.	PUNCT
ejpam-3548	208	1	corollary	corollary	ADJ
ejpam-3548	208	2	1	1	NUM
ejpam-3548	208	3	.	.	PUNCT
ejpam-3548	209	1	let	let	VERB
ejpam-3548	209	2	x	x	PRON
ejpam-3548	209	3	be	be	AUX
ejpam-3548	209	4	a	a	DET
ejpam-3548	209	5	be	be	NOUN
ejpam-3548	209	6	-	-	PUNCT
ejpam-3548	209	7	algebra	algebra	NOUN
ejpam-3548	209	8	.	.	PUNCT
ejpam-3548	210	1	then	then	ADV
ejpam-3548	210	2	τr(x	τr(x	NUM
ejpam-3548	210	3	)	)	PUNCT
ejpam-3548	210	4	is	be	AUX
ejpam-3548	210	5	the	the	DET
ejpam-3548	210	6	discrete	discrete	ADJ
ejpam-3548	210	7	topology	topology	NOUN
ejpam-3548	210	8	on	on	ADP
ejpam-3548	210	9	x	x	SYM
ejpam-3548	210	10	if	if	SCONJ
ejpam-3548	210	11	and	and	CCONJ
ejpam-3548	210	12	only	only	ADV
ejpam-3548	210	13	if	if	SCONJ
ejpam-3548	210	14	x	x	X
ejpam-3548	210	15	=	=	PRON
ejpam-3548	210	16	{	{	PUNCT
ejpam-3548	210	17	1x	1x	NUM
ejpam-3548	210	18	}	}	PUNCT
ejpam-3548	210	19	.	.	PUNCT
ejpam-3548	211	1	j.	j.	PROPN
ejpam-3548	211	2	albaracin	albaracin	PROPN
ejpam-3548	211	3	,	,	PUNCT
ejpam-3548	211	4	j.	j.	PROPN
ejpam-3548	211	5	vilela	vilela	PROPN
ejpam-3548	211	6	/	/	SYM
ejpam-3548	211	7	eur	eur	PROPN
ejpam-3548	211	8	.	.	PUNCT
ejpam-3548	212	1	j.	j.	PROPN
ejpam-3548	212	2	pure	pure	PROPN
ejpam-3548	212	3	appl	appl	PROPN
ejpam-3548	212	4	.	.	PROPN
ejpam-3548	212	5	math	math	PROPN
ejpam-3548	212	6	,	,	PUNCT
ejpam-3548	212	7	12	12	NUM
ejpam-3548	212	8	(	(	PUNCT
ejpam-3548	212	9	4	4	NUM
ejpam-3548	212	10	)	)	PUNCT
ejpam-3548	212	11	(	(	PUNCT
ejpam-3548	212	12	2019	2019	NUM
ejpam-3548	212	13	)	)	PUNCT
ejpam-3548	212	14	,	,	PUNCT
ejpam-3548	212	15	1584	1584	NUM
ejpam-3548	212	16	-	-	SYM
ejpam-3548	212	17	1594	1594	NUM
ejpam-3548	212	18	1589	1589	NUM
ejpam-3548	212	19	proof	proof	NOUN
ejpam-3548	212	20	.	.	PUNCT
ejpam-3548	212	21	suppose	suppose	VERB
ejpam-3548	212	22	that	that	SCONJ
ejpam-3548	212	23	x	x	X
ejpam-3548	212	24	=	=	PRON
ejpam-3548	212	25	{	{	PUNCT
ejpam-3548	212	26	1x	1x	NUM
ejpam-3548	212	27	}	}	PUNCT
ejpam-3548	212	28	.	.	PUNCT
ejpam-3548	213	1	then	then	ADV
ejpam-3548	213	2	br(x	br(x	PUNCT
ejpam-3548	213	3	)	)	PUNCT
ejpam-3548	213	4	=	=	SYM
ejpam-3548	213	5	{	{	PUNCT
ejpam-3548	213	6	rx(1x	rx(1x	NOUN
ejpam-3548	213	7	)	)	PUNCT
ejpam-3548	213	8	}	}	PUNCT
ejpam-3548	213	9	=	=	SYM
ejpam-3548	213	10	{	{	PUNCT
ejpam-3548	213	11	{	{	PUNCT
ejpam-3548	213	12	1x	1x	NUM
ejpam-3548	213	13	}	}	PUNCT
ejpam-3548	213	14	}	}	PUNCT
ejpam-3548	213	15	.	.	PUNCT
ejpam-3548	214	1	hence	hence	ADV
ejpam-3548	214	2	,	,	PUNCT
ejpam-3548	214	3	τr(x	τr(x	NUM
ejpam-3548	214	4	)	)	PUNCT
ejpam-3548	214	5	=	=	PRON
ejpam-3548	214	6	{	{	PUNCT
ejpam-3548	214	7	∅	∅	NOUN
ejpam-3548	214	8	,	,	PUNCT
ejpam-3548	214	9	x	x	NOUN
ejpam-3548	214	10	}	}	PUNCT
ejpam-3548	214	11	,	,	PUNCT
ejpam-3548	214	12	the	the	DET
ejpam-3548	214	13	discrete	discrete	ADJ
ejpam-3548	214	14	topology	topology	NOUN
ejpam-3548	214	15	on	on	ADP
ejpam-3548	214	16	x.	x.	NOUN
ejpam-3548	214	17	conversely	conversely	ADV
ejpam-3548	214	18	,	,	PUNCT
ejpam-3548	214	19	suppose	suppose	VERB
ejpam-3548	214	20	that	that	SCONJ
ejpam-3548	214	21	τr(x	τr(x	NUM
ejpam-3548	214	22	)	)	PUNCT
ejpam-3548	214	23	is	be	AUX
ejpam-3548	214	24	the	the	DET
ejpam-3548	214	25	discrete	discrete	ADJ
ejpam-3548	214	26	topology	topology	NOUN
ejpam-3548	214	27	on	on	ADP
ejpam-3548	214	28	x.	x.	NOUN
ejpam-3548	214	29	then	then	ADV
ejpam-3548	214	30	{	{	PUNCT
ejpam-3548	214	31	x	x	NOUN
ejpam-3548	214	32	}	}	PUNCT
ejpam-3548	214	33	∈	∈	PROPN
ejpam-3548	214	34	τr(x	τr(x	NUM
ejpam-3548	214	35	)	)	PUNCT
ejpam-3548	214	36	for	for	ADP
ejpam-3548	214	37	all	all	PRON
ejpam-3548	214	38	x	x	SYM
ejpam-3548	214	39	∈	∈	PROPN
ejpam-3548	214	40	x.	x.	NOUN
ejpam-3548	214	41	by	by	ADP
ejpam-3548	214	42	lemma	lemma	PROPN
ejpam-3548	214	43	3	3	NUM
ejpam-3548	214	44	,	,	PUNCT
ejpam-3548	214	45	x	x	PUNCT
ejpam-3548	214	46	=	=	PRON
ejpam-3548	214	47	{	{	PUNCT
ejpam-3548	214	48	1x	1x	NUM
ejpam-3548	214	49	}	}	PUNCT
ejpam-3548	214	50	.	.	PUNCT
ejpam-3548	215	1	theorem	theorem	ADJ
ejpam-3548	215	2	10	10	NUM
ejpam-3548	215	3	.	.	PUNCT
ejpam-3548	216	1	if	if	SCONJ
ejpam-3548	216	2	x	x	PRON
ejpam-3548	216	3	is	be	AUX
ejpam-3548	216	4	a	a	DET
ejpam-3548	216	5	finite	finite	NOUN
ejpam-3548	216	6	be	be	NOUN
ejpam-3548	216	7	-	-	PUNCT
ejpam-3548	216	8	algebra	algebra	NOUN
ejpam-3548	216	9	,	,	PUNCT
ejpam-3548	216	10	then	then	ADV
ejpam-3548	216	11	sr(x	sr(x	NOUN
ejpam-3548	216	12	)	)	PUNCT
ejpam-3548	217	1	=	=	SYM
ejpam-3548	217	2	{	{	PUNCT
ejpam-3548	217	3	rx({a	rx({a	NOUN
ejpam-3548	217	4	}	}	PUNCT
ejpam-3548	217	5	)	)	PUNCT
ejpam-3548	217	6	:	:	PUNCT
ejpam-3548	217	7	a	a	DET
ejpam-3548	217	8	∈	∈	PROPN
ejpam-3548	217	9	x	x	PRON
ejpam-3548	217	10	}	}	PUNCT
ejpam-3548	217	11	is	be	AUX
ejpam-3548	217	12	a	a	DET
ejpam-3548	217	13	subbase	subbase	NOUN
ejpam-3548	217	14	of	of	ADP
ejpam-3548	217	15	τr(x	τr(x	NUM
ejpam-3548	217	16	)	)	PUNCT
ejpam-3548	217	17	.	.	PUNCT
ejpam-3548	218	1	proof	proof	NOUN
ejpam-3548	218	2	.	.	PUNCT
ejpam-3548	219	1	clearly	clearly	ADV
ejpam-3548	219	2	,	,	PUNCT
ejpam-3548	219	3	sr(x	sr(x	NOUN
ejpam-3548	219	4	)	)	PUNCT
ejpam-3548	219	5	⊆	⊆	NUM
ejpam-3548	219	6	τr(x	τr(x	NUM
ejpam-3548	219	7	)	)	PUNCT
ejpam-3548	219	8	.	.	PUNCT
ejpam-3548	220	1	by	by	ADP
ejpam-3548	220	2	theorem	theorem	NOUN
ejpam-3548	220	3	6	6	NUM
ejpam-3548	220	4	,	,	PUNCT
ejpam-3548	220	5	rx(a	rx(a	NOUN
ejpam-3548	220	6	)	)	PUNCT
ejpam-3548	220	7	=	=	SYM
ejpam-3548	220	8	⋂	⋂	PROPN
ejpam-3548	220	9	a∈a	a∈a	ADJ
ejpam-3548	220	10	rx({a	rx({a	NOUN
ejpam-3548	220	11	}	}	PUNCT
ejpam-3548	220	12	)	)	PUNCT
ejpam-3548	220	13	for	for	ADP
ejpam-3548	220	14	each	each	DET
ejpam-3548	220	15	∅	∅	NOUN
ejpam-3548	220	16	6=	6=	ADP
ejpam-3548	220	17	a	a	DET
ejpam-3548	220	18	⊆	⊆	NUM
ejpam-3548	220	19	x.	x.	NOUN
ejpam-3548	220	20	since	since	SCONJ
ejpam-3548	220	21	x	x	PROPN
ejpam-3548	220	22	is	be	AUX
ejpam-3548	220	23	finite	finite	ADJ
ejpam-3548	220	24	,	,	PUNCT
ejpam-3548	220	25	it	it	PRON
ejpam-3548	220	26	follows	follow	VERB
ejpam-3548	220	27	that	that	SCONJ
ejpam-3548	220	28	every	every	DET
ejpam-3548	220	29	element	element	NOUN
ejpam-3548	220	30	of	of	ADP
ejpam-3548	220	31	br(x	br(x	PROPN
ejpam-3548	220	32	)	)	PUNCT
ejpam-3548	220	33	is	be	AUX
ejpam-3548	220	34	a	a	DET
ejpam-3548	220	35	finite	finite	ADJ
ejpam-3548	220	36	intersection	intersection	NOUN
ejpam-3548	220	37	of	of	ADP
ejpam-3548	220	38	members	member	NOUN
ejpam-3548	220	39	of	of	ADP
ejpam-3548	220	40	sr(x	sr(x	NOUN
ejpam-3548	220	41	)	)	PUNCT
ejpam-3548	220	42	.	.	PUNCT
ejpam-3548	221	1	thus	thus	ADV
ejpam-3548	221	2	,	,	PUNCT
ejpam-3548	221	3	sr(x	sr(x	NOUN
ejpam-3548	221	4	)	)	PUNCT
ejpam-3548	221	5	is	be	AUX
ejpam-3548	221	6	a	a	DET
ejpam-3548	221	7	subbase	subbase	NOUN
ejpam-3548	221	8	of	of	ADP
ejpam-3548	221	9	τr(x	τr(x	NUM
ejpam-3548	221	10	)	)	PUNCT
ejpam-3548	221	11	.	.	PUNCT
ejpam-3548	222	1	whitejgvgvkhg	whitejgvgvkhg	NOUN
ejpam-3548	222	2	lemma	lemma	PROPN
ejpam-3548	222	3	4	4	X
ejpam-3548	222	4	.	.	PUNCT
ejpam-3548	223	1	let	let	VERB
ejpam-3548	223	2	x	x	PRON
ejpam-3548	223	3	be	be	AUX
ejpam-3548	223	4	a	a	DET
ejpam-3548	223	5	be	be	NOUN
ejpam-3548	223	6	-	-	PUNCT
ejpam-3548	223	7	algebra	algebra	NOUN
ejpam-3548	223	8	and	and	CCONJ
ejpam-3548	223	9	a	a	DET
ejpam-3548	223	10	∈	∈	NOUN
ejpam-3548	223	11	x	x	X
ejpam-3548	223	12	\	\	X
ejpam-3548	223	13	{	{	PUNCT
ejpam-3548	223	14	1x	1x	NUM
ejpam-3548	223	15	}	}	PUNCT
ejpam-3548	223	16	.	.	PUNCT
ejpam-3548	224	1	then	then	ADV
ejpam-3548	224	2	a	a	DET
ejpam-3548	224	3	∈	∈	PROPN
ejpam-3548	224	4	a(x	a(x	PROPN
ejpam-3548	224	5	)	)	PUNCT
ejpam-3548	224	6	if	if	SCONJ
ejpam-3548	224	7	and	and	CCONJ
ejpam-3548	224	8	only	only	ADV
ejpam-3548	224	9	if	if	SCONJ
ejpam-3548	224	10	rx	rx	VERB
ejpam-3548	224	11	(	(	PUNCT
ejpam-3548	224	12	{	{	PUNCT
ejpam-3548	224	13	a	a	NOUN
ejpam-3548	224	14	}	}	PUNCT
ejpam-3548	224	15	)	)	PUNCT
ejpam-3548	224	16	=	=	SYM
ejpam-3548	224	17	{	{	PUNCT
ejpam-3548	224	18	1x	1x	NOUN
ejpam-3548	224	19	,	,	PUNCT
ejpam-3548	224	20	a	a	PRON
ejpam-3548	224	21	}	}	PUNCT
ejpam-3548	224	22	.	.	PUNCT
ejpam-3548	225	1	proof	proof	NOUN
ejpam-3548	225	2	.	.	PUNCT
ejpam-3548	226	1	suppose	suppose	VERB
ejpam-3548	226	2	that	that	SCONJ
ejpam-3548	226	3	a	a	DET
ejpam-3548	226	4	∈	∈	PROPN
ejpam-3548	226	5	a(x	a(x	NOUN
ejpam-3548	226	6	)	)	PUNCT
ejpam-3548	226	7	and	and	CCONJ
ejpam-3548	226	8	let	let	VERB
ejpam-3548	226	9	x	x	X
ejpam-3548	226	10	∈	∈	PROPN
ejpam-3548	226	11	rx({a	rx({a	NOUN
ejpam-3548	226	12	}	}	PUNCT
ejpam-3548	226	13	)	)	PUNCT
ejpam-3548	226	14	.	.	PUNCT
ejpam-3548	227	1	then	then	ADV
ejpam-3548	227	2	a	a	DET
ejpam-3548	227	3	≤	≤	NUM
ejpam-3548	227	4	x.	x.	NOUN
ejpam-3548	227	5	since	since	SCONJ
ejpam-3548	227	6	a	a	DET
ejpam-3548	227	7	∈	∈	PROPN
ejpam-3548	227	8	a(x	a(x	NOUN
ejpam-3548	227	9	)	)	PUNCT
ejpam-3548	227	10	,	,	PUNCT
ejpam-3548	227	11	x	x	PUNCT
ejpam-3548	228	1	=	=	PUNCT
ejpam-3548	228	2	1x	1x	NUM
ejpam-3548	228	3	or	or	CCONJ
ejpam-3548	228	4	x	x	X
ejpam-3548	228	5	=	=	PUNCT
ejpam-3548	228	6	a.	a.	NOUN
ejpam-3548	228	7	thus	thus	ADV
ejpam-3548	228	8	,	,	PUNCT
ejpam-3548	228	9	rx({a	rx({a	ADJ
ejpam-3548	228	10	}	}	PUNCT
ejpam-3548	228	11	)	)	PUNCT
ejpam-3548	229	1	=	=	PRON
ejpam-3548	229	2	{	{	PUNCT
ejpam-3548	229	3	1x	1x	NOUN
ejpam-3548	229	4	,	,	PUNCT
ejpam-3548	229	5	a	a	PRON
ejpam-3548	229	6	}	}	PUNCT
ejpam-3548	229	7	.	.	PUNCT
ejpam-3548	230	1	conversely	conversely	ADV
ejpam-3548	230	2	,	,	PUNCT
ejpam-3548	230	3	suppose	suppose	VERB
ejpam-3548	230	4	that	that	SCONJ
ejpam-3548	230	5	rx({a	rx({a	NOUN
ejpam-3548	230	6	}	}	PUNCT
ejpam-3548	230	7	)	)	PUNCT
ejpam-3548	231	1	=	=	PRON
ejpam-3548	231	2	{	{	PUNCT
ejpam-3548	231	3	1x	1x	NOUN
ejpam-3548	231	4	,	,	PUNCT
ejpam-3548	231	5	a	a	PRON
ejpam-3548	231	6	}	}	PUNCT
ejpam-3548	231	7	.	.	PUNCT
ejpam-3548	232	1	then	then	ADV
ejpam-3548	232	2	a	a	DET
ejpam-3548	232	3	≤	≤	NOUN
ejpam-3548	232	4	x	x	AUX
ejpam-3548	232	5	implies	imply	VERB
ejpam-3548	232	6	that	that	SCONJ
ejpam-3548	232	7	x	x	X
ejpam-3548	232	8	=	=	PUNCT
ejpam-3548	232	9	1x	1x	NUM
ejpam-3548	232	10	or	or	CCONJ
ejpam-3548	232	11	x	x	X
ejpam-3548	232	12	=	=	PUNCT
ejpam-3548	232	13	a.	a.	NOUN
ejpam-3548	232	14	therefore	therefore	ADV
ejpam-3548	232	15	,	,	PUNCT
ejpam-3548	232	16	a	a	DET
ejpam-3548	232	17	∈	∈	PROPN
ejpam-3548	232	18	a(x	a(x	NOUN
ejpam-3548	232	19	)	)	PUNCT
ejpam-3548	232	20	.	.	PUNCT
ejpam-3548	233	1	white	white	PROPN
ejpam-3548	233	2	hgzgjykuiahlu	hgzgjykuiahlu	PROPN
ejpam-3548	233	3	theorem	theorem	NOUN
ejpam-3548	233	4	11	11	NUM
ejpam-3548	233	5	.	.	PUNCT
ejpam-3548	234	1	let	let	VERB
ejpam-3548	234	2	x	x	PRON
ejpam-3548	234	3	be	be	AUX
ejpam-3548	234	4	a	a	DET
ejpam-3548	234	5	be	be	NOUN
ejpam-3548	234	6	-	-	PUNCT
ejpam-3548	234	7	algebra	algebra	NOUN
ejpam-3548	234	8	with	with	ADP
ejpam-3548	234	9	|x|	|x|	PROPN
ejpam-3548	234	10	≥	≥	NUM
ejpam-3548	234	11	2	2	NUM
ejpam-3548	234	12	.	.	PUNCT
ejpam-3548	234	13	then	then	ADV
ejpam-3548	234	14	br(x	br(x	PUNCT
ejpam-3548	234	15	)	)	PUNCT
ejpam-3548	235	1	=	=	PRON
ejpam-3548	235	2	{	{	PUNCT
ejpam-3548	235	3	{	{	PUNCT
ejpam-3548	235	4	1x	1x	NUM
ejpam-3548	235	5	,	,	PUNCT
ejpam-3548	235	6	a	a	PRON
ejpam-3548	235	7	}	}	PUNCT
ejpam-3548	235	8	:	:	PUNCT
ejpam-3548	235	9	a	a	DET
ejpam-3548	235	10	∈	∈	PROPN
ejpam-3548	235	11	a(x	a(x	NOUN
ejpam-3548	235	12	)	)	PUNCT
ejpam-3548	235	13	}	}	PUNCT
ejpam-3548	235	14	⋃	⋃	NOUN
ejpam-3548	235	15	{	{	PUNCT
ejpam-3548	235	16	rx(a	rx(a	NOUN
ejpam-3548	235	17	)	)	PUNCT
ejpam-3548	235	18	:	:	PUNCT
ejpam-3548	235	19	a	a	DET
ejpam-3548	235	20	∩	∩	ADJ
ejpam-3548	235	21	a(x	a(x	NOUN
ejpam-3548	235	22	)	)	PUNCT
ejpam-3548	235	23	=	=	NOUN
ejpam-3548	235	24	∅	∅	NOUN
ejpam-3548	235	25	}	}	PUNCT
ejpam-3548	235	26	.	.	PUNCT
ejpam-3548	236	1	proof	proof	NOUN
ejpam-3548	236	2	.	.	PUNCT
ejpam-3548	237	1	by	by	ADP
ejpam-3548	237	2	lemma	lemma	PROPN
ejpam-3548	237	3	4	4	NUM
ejpam-3548	237	4	,	,	PUNCT
ejpam-3548	237	5	rx	rx	VERB
ejpam-3548	237	6	(	(	PUNCT
ejpam-3548	237	7	a	a	X
ejpam-3548	237	8	)	)	PUNCT
ejpam-3548	237	9	=	=	SYM
ejpam-3548	237	10	{	{	PUNCT
ejpam-3548	237	11	1x	1x	NOUN
ejpam-3548	237	12	,	,	PUNCT
ejpam-3548	237	13	a	a	DET
ejpam-3548	237	14	}	}	PUNCT
ejpam-3548	237	15	∈	∈	PROPN
ejpam-3548	237	16	br(x	br(x	NOUN
ejpam-3548	237	17	)	)	PUNCT
ejpam-3548	237	18	for	for	ADP
ejpam-3548	237	19	each	each	PRON
ejpam-3548	237	20	a	a	DET
ejpam-3548	237	21	∈	∈	PROPN
ejpam-3548	237	22	a(x	a(x	NOUN
ejpam-3548	237	23	)	)	PUNCT
ejpam-3548	237	24	.	.	PUNCT
ejpam-3548	238	1	let	let	VERB
ejpam-3548	238	2	∅	∅	NOUN
ejpam-3548	238	3	6=	6=	ADP
ejpam-3548	238	4	a	a	DET
ejpam-3548	238	5	⊆	⊆	NUM
ejpam-3548	238	6	x	x	PUNCT
ejpam-3548	238	7	such	such	ADJ
ejpam-3548	238	8	that	that	DET
ejpam-3548	238	9	a∩a(x	a∩a(x	NOUN
ejpam-3548	238	10	)	)	PUNCT
ejpam-3548	238	11	6=	6=	ADP
ejpam-3548	238	12	∅	∅	NOUN
ejpam-3548	238	13	,	,	PUNCT
ejpam-3548	238	14	say	say	VERB
ejpam-3548	238	15	z	z	PROPN
ejpam-3548	238	16	∈	∈	PROPN
ejpam-3548	238	17	a∩a(x	a∩a(x	NOUN
ejpam-3548	238	18	)	)	PUNCT
ejpam-3548	238	19	.	.	PUNCT
ejpam-3548	239	1	if	if	SCONJ
ejpam-3548	239	2	1x	1x	PROPN
ejpam-3548	239	3	∈	∈	PROPN
ejpam-3548	239	4	a	a	PRON
ejpam-3548	239	5	,	,	PUNCT
ejpam-3548	239	6	then	then	ADV
ejpam-3548	239	7	by	by	ADP
ejpam-3548	239	8	theorem	theorem	NOUN
ejpam-3548	239	9	6	6	NUM
ejpam-3548	239	10	,	,	PUNCT
ejpam-3548	239	11	rx(a	rx(a	NOUN
ejpam-3548	239	12	)	)	PUNCT
ejpam-3548	239	13	=	=	SYM
ejpam-3548	239	14	{	{	PUNCT
ejpam-3548	239	15	1x	1x	NUM
ejpam-3548	239	16	}	}	PUNCT
ejpam-3548	239	17	.	.	PUNCT
ejpam-3548	240	1	suppose	suppose	VERB
ejpam-3548	241	1	that	that	SCONJ
ejpam-3548	241	2	1x	1x	PROPN
ejpam-3548	241	3	/∈	/∈	PUNCT
ejpam-3548	241	4	a.	a.	NOUN
ejpam-3548	241	5	since	since	SCONJ
ejpam-3548	241	6	z	z	PROPN
ejpam-3548	241	7	∈	∈	PROPN
ejpam-3548	241	8	a(x	a(x	PROPN
ejpam-3548	241	9	)	)	PUNCT
ejpam-3548	241	10	and	and	CCONJ
ejpam-3548	241	11	by	by	ADP
ejpam-3548	241	12	theorem	theorem	NOUN
ejpam-3548	241	13	5(ii	5(ii	NUM
ejpam-3548	241	14	)	)	PUNCT
ejpam-3548	241	15	,	,	PUNCT
ejpam-3548	241	16	rx(a	rx(a	PROPN
ejpam-3548	241	17	)	)	PUNCT
ejpam-3548	241	18	⊆	⊆	NUM
ejpam-3548	241	19	rx	rx	X
ejpam-3548	241	20	(	(	PUNCT
ejpam-3548	241	21	z	z	NOUN
ejpam-3548	241	22	)	)	PUNCT
ejpam-3548	241	23	=	=	PRON
ejpam-3548	242	1	{	{	PUNCT
ejpam-3548	242	2	1x	1x	X
ejpam-3548	242	3	,	,	PUNCT
ejpam-3548	242	4	z	z	NOUN
ejpam-3548	242	5	}	}	PUNCT
ejpam-3548	242	6	and	and	CCONJ
ejpam-3548	242	7	1x	1x	NUM
ejpam-3548	242	8	∈	∈	PROPN
ejpam-3548	242	9	rx(a	rx(a	PROPN
ejpam-3548	242	10	)	)	PUNCT
ejpam-3548	242	11	,	,	PUNCT
ejpam-3548	242	12	it	it	PRON
ejpam-3548	242	13	follows	follow	VERB
ejpam-3548	242	14	that	that	SCONJ
ejpam-3548	242	15	rx(a	rx(a	NOUN
ejpam-3548	242	16	)	)	PUNCT
ejpam-3548	242	17	=	=	SYM
ejpam-3548	242	18	{	{	PUNCT
ejpam-3548	242	19	1x	1x	NUM
ejpam-3548	242	20	}	}	PUNCT
ejpam-3548	242	21	or	or	CCONJ
ejpam-3548	242	22	rx(a	rx(a	NOUN
ejpam-3548	242	23	)	)	PUNCT
ejpam-3548	242	24	=	=	PRON
ejpam-3548	242	25	{	{	PUNCT
ejpam-3548	242	26	1x	1x	X
ejpam-3548	242	27	,	,	PUNCT
ejpam-3548	242	28	z	z	NOUN
ejpam-3548	242	29	}	}	PUNCT
ejpam-3548	242	30	.	.	PUNCT
ejpam-3548	243	1	this	this	PRON
ejpam-3548	243	2	proves	prove	VERB
ejpam-3548	243	3	the	the	DET
ejpam-3548	243	4	assertion	assertion	NOUN
ejpam-3548	243	5	.	.	PUNCT
ejpam-3548	244	1	corollary	corollary	ADJ
ejpam-3548	244	2	2	2	NUM
ejpam-3548	244	3	.	.	PUNCT
ejpam-3548	245	1	let	let	VERB
ejpam-3548	245	2	x	x	PRON
ejpam-3548	245	3	be	be	AUX
ejpam-3548	245	4	a	a	DET
ejpam-3548	245	5	be	be	NOUN
ejpam-3548	245	6	-	-	PUNCT
ejpam-3548	245	7	algebra	algebra	NOUN
ejpam-3548	245	8	with	with	ADP
ejpam-3548	245	9	|x|	|x|	PROPN
ejpam-3548	245	10	≥	≥	NUM
ejpam-3548	245	11	2	2	NUM
ejpam-3548	245	12	.	.	PUNCT
ejpam-3548	246	1	if	if	SCONJ
ejpam-3548	246	2	a(x	a(x	NOUN
ejpam-3548	246	3	)	)	PUNCT
ejpam-3548	246	4	=	=	NOUN
ejpam-3548	246	5	{	{	PUNCT
ejpam-3548	246	6	a	a	NOUN
ejpam-3548	246	7	}	}	PUNCT
ejpam-3548	246	8	,	,	PUNCT
ejpam-3548	246	9	then	then	ADV
ejpam-3548	246	10	br(x	br(x	PUNCT
ejpam-3548	246	11	)	)	PUNCT
ejpam-3548	247	1	=	=	PRON
ejpam-3548	247	2	{	{	PUNCT
ejpam-3548	247	3	{	{	PUNCT
ejpam-3548	247	4	1x	1x	NUM
ejpam-3548	247	5	,	,	PUNCT
ejpam-3548	247	6	a	a	PRON
ejpam-3548	247	7	}	}	PUNCT
ejpam-3548	247	8	}	}	PUNCT
ejpam-3548	247	9	∪	∪	ADJ
ejpam-3548	247	10	{	{	PUNCT
ejpam-3548	247	11	rx(a	rx(a	NOUN
ejpam-3548	247	12	)	)	PUNCT
ejpam-3548	247	13	:	:	PUNCT
ejpam-3548	247	14	a	a	DET
ejpam-3548	247	15	/∈	/∈	NOUN
ejpam-3548	247	16	a	a	PRON
ejpam-3548	247	17	}	}	PUNCT
ejpam-3548	247	18	.	.	PUNCT
ejpam-3548	248	1	example	example	NOUN
ejpam-3548	249	1	3	3	X
ejpam-3548	249	2	.	.	X
ejpam-3548	249	3	consider	consider	VERB
ejpam-3548	249	4	the	the	DET
ejpam-3548	249	5	be	be	AUX
ejpam-3548	249	6	-	-	PUNCT
ejpam-3548	249	7	algebra	algebra	NOUN
ejpam-3548	249	8	n0	n0	NUM
ejpam-3548	249	9	in	in	ADP
ejpam-3548	249	10	example	example	NOUN
ejpam-3548	249	11	2	2	NUM
ejpam-3548	249	12	.	.	X
ejpam-3548	250	1	for	for	ADP
ejpam-3548	250	2	any	any	DET
ejpam-3548	250	3	x	x	SYM
ejpam-3548	250	4	∈	∈	PROPN
ejpam-3548	250	5	n0	n0	PROPN
ejpam-3548	250	6	,	,	PUNCT
ejpam-3548	250	7	rn0(x	rn0(x	PROPN
ejpam-3548	250	8	)	)	PUNCT
ejpam-3548	250	9	=	=	PRON
ejpam-3548	250	10	{	{	PUNCT
ejpam-3548	250	11	0	0	NUM
ejpam-3548	250	12	,	,	PUNCT
ejpam-3548	250	13	1	1	NUM
ejpam-3548	250	14	,	,	PUNCT
ejpam-3548	250	15	.	.	PUNCT
ejpam-3548	250	16	.	.	PUNCT
ejpam-3548	250	17	.	.	PUNCT
ejpam-3548	251	1	,	,	PUNCT
ejpam-3548	251	2	x	x	X
ejpam-3548	251	3	}	}	PUNCT
ejpam-3548	251	4	.	.	PUNCT
ejpam-3548	252	1	hence	hence	ADV
ejpam-3548	252	2	,	,	PUNCT
ejpam-3548	252	3	a(n0	a(n0	ADJ
ejpam-3548	252	4	)	)	PUNCT
ejpam-3548	252	5	=	=	SYM
ejpam-3548	252	6	{	{	PUNCT
ejpam-3548	252	7	1	1	NUM
ejpam-3548	252	8	}	}	PUNCT
ejpam-3548	252	9	.	.	PUNCT
ejpam-3548	253	1	by	by	ADP
ejpam-3548	253	2	corollary	corollary	ADJ
ejpam-3548	253	3	2	2	NUM
ejpam-3548	253	4	,	,	PUNCT
ejpam-3548	253	5	br(n0	br(n0	NOUN
ejpam-3548	253	6	)	)	PUNCT
ejpam-3548	253	7	=	=	PRON
ejpam-3548	253	8	{	{	PUNCT
ejpam-3548	253	9	{	{	PUNCT
ejpam-3548	253	10	0	0	NUM
ejpam-3548	253	11	,	,	PUNCT
ejpam-3548	253	12	1	1	NUM
ejpam-3548	253	13	}	}	PUNCT
ejpam-3548	253	14	}	}	PUNCT
ejpam-3548	253	15	∪	∪	ADP
ejpam-3548	253	16	{	{	PUNCT
ejpam-3548	253	17	rn0(a	rn0(a	PROPN
ejpam-3548	253	18	)	)	PUNCT
ejpam-3548	253	19	:	:	PUNCT
ejpam-3548	253	20	1	1	NUM
ejpam-3548	253	21	/∈	/∈	SYM
ejpam-3548	253	22	a	a	X
ejpam-3548	253	23	}	}	PUNCT
ejpam-3548	253	24	=	=	SYM
ejpam-3548	253	25	{	{	PUNCT
ejpam-3548	253	26	rn0(y	rn0(y	PROPN
ejpam-3548	253	27	)	)	PUNCT
ejpam-3548	253	28	:	:	PUNCT
ejpam-3548	254	1	y	y	PROPN
ejpam-3548	254	2	∈	∈	PROPN
ejpam-3548	254	3	n0	n0	PROPN
ejpam-3548	254	4	}	}	PUNCT
ejpam-3548	254	5	.	.	PUNCT
ejpam-3548	255	1	therefore	therefore	ADV
ejpam-3548	255	2	,	,	PUNCT
ejpam-3548	255	3	τr(n0	τr(n0	X
ejpam-3548	255	4	)	)	PUNCT
ejpam-3548	255	5	=	=	SYM
ejpam-3548	255	6	{	{	PUNCT
ejpam-3548	255	7	∅	∅	NOUN
ejpam-3548	255	8	,	,	PUNCT
ejpam-3548	255	9	n0	n0	ADJ
ejpam-3548	255	10	}	}	PUNCT
ejpam-3548	255	11	∪	∪	X
ejpam-3548	255	12	{	{	PUNCT
ejpam-3548	255	13	rn0(y	rn0(y	PROPN
ejpam-3548	255	14	)	)	PUNCT
ejpam-3548	255	15	:	:	PUNCT
ejpam-3548	255	16	y	y	PROPN
ejpam-3548	255	17	∈	∈	PROPN
ejpam-3548	255	18	n0	n0	PROPN
ejpam-3548	255	19	}	}	PUNCT
ejpam-3548	255	20	.	.	PUNCT
ejpam-3548	256	1	theorem	theorem	NOUN
ejpam-3548	256	2	12	12	NUM
ejpam-3548	256	3	.	.	PUNCT
ejpam-3548	257	1	let	let	VERB
ejpam-3548	257	2	x	x	PRON
ejpam-3548	257	3	be	be	AUX
ejpam-3548	257	4	a	a	DET
ejpam-3548	257	5	be	be	NOUN
ejpam-3548	257	6	-	-	PUNCT
ejpam-3548	257	7	algebra	algebra	NOUN
ejpam-3548	257	8	with	with	ADP
ejpam-3548	257	9	|x|	|x|	PROPN
ejpam-3548	257	10	≥	≥	NUM
ejpam-3548	257	11	2	2	NUM
ejpam-3548	257	12	.	.	PUNCT
ejpam-3548	257	13	then	then	ADV
ejpam-3548	257	14	br(x	br(x	PUNCT
ejpam-3548	257	15	)	)	PUNCT
ejpam-3548	258	1	=	=	PRON
ejpam-3548	258	2	{	{	PUNCT
ejpam-3548	258	3	{	{	PUNCT
ejpam-3548	258	4	1x	1x	NUM
ejpam-3548	258	5	}	}	PUNCT
ejpam-3548	258	6	}	}	PUNCT
ejpam-3548	258	7	∪	∪	X
ejpam-3548	258	8	{	{	PUNCT
ejpam-3548	258	9	{	{	PUNCT
ejpam-3548	258	10	1x	1x	NUM
ejpam-3548	258	11	,	,	PUNCT
ejpam-3548	258	12	a	a	PRON
ejpam-3548	258	13	}	}	PUNCT
ejpam-3548	258	14	:	:	PUNCT
ejpam-3548	258	15	a	a	DET
ejpam-3548	258	16	∈	∈	PROPN
ejpam-3548	258	17	x	x	X
ejpam-3548	258	18	\	\	X
ejpam-3548	258	19	{	{	PUNCT
ejpam-3548	258	20	1x	1x	NUM
ejpam-3548	258	21	}	}	PUNCT
ejpam-3548	258	22	}	}	PUNCT
ejpam-3548	258	23	if	if	SCONJ
ejpam-3548	258	24	and	and	CCONJ
ejpam-3548	258	25	only	only	ADV
ejpam-3548	258	26	if	if	SCONJ
ejpam-3548	258	27	x	x	PRON
ejpam-3548	258	28	is	be	AUX
ejpam-3548	258	29	dual	dual	ADV
ejpam-3548	258	30	atomistic	atomistic	ADJ
ejpam-3548	258	31	.	.	PUNCT
ejpam-3548	259	1	proof	proof	NOUN
ejpam-3548	259	2	.	.	PUNCT
ejpam-3548	260	1	suppose	suppose	VERB
ejpam-3548	260	2	that	that	SCONJ
ejpam-3548	260	3	x	x	PRON
ejpam-3548	260	4	is	be	AUX
ejpam-3548	260	5	dual	dual	ADV
ejpam-3548	260	6	atomistic	atomistic	ADJ
ejpam-3548	260	7	.	.	PUNCT
ejpam-3548	261	1	by	by	ADP
ejpam-3548	261	2	lemma	lemma	PROPN
ejpam-3548	261	3	4	4	NUM
ejpam-3548	261	4	,	,	PUNCT
ejpam-3548	261	5	rx	rx	VERB
ejpam-3548	261	6	(	(	PUNCT
ejpam-3548	261	7	a	a	X
ejpam-3548	261	8	)	)	PUNCT
ejpam-3548	261	9	=	=	SYM
ejpam-3548	261	10	{	{	PUNCT
ejpam-3548	261	11	1x	1x	X
ejpam-3548	261	12	,	,	PUNCT
ejpam-3548	261	13	a	a	PRON
ejpam-3548	261	14	}	}	PUNCT
ejpam-3548	261	15	for	for	ADP
ejpam-3548	261	16	all	all	DET
ejpam-3548	261	17	a	a	DET
ejpam-3548	261	18	∈	∈	NOUN
ejpam-3548	261	19	x	x	SYM
ejpam-3548	261	20	\	\	X
ejpam-3548	261	21	{	{	PUNCT
ejpam-3548	261	22	1x	1x	NUM
ejpam-3548	261	23	}	}	PUNCT
ejpam-3548	261	24	.	.	PUNCT
ejpam-3548	262	1	the	the	DET
ejpam-3548	262	2	only	only	ADJ
ejpam-3548	262	3	∅	∅	NOUN
ejpam-3548	262	4	6=	6=	ADP
ejpam-3548	262	5	a	a	DET
ejpam-3548	262	6	⊆	⊆	NUM
ejpam-3548	262	7	x	x	SYM
ejpam-3548	262	8	such	such	ADJ
ejpam-3548	262	9	that	that	SCONJ
ejpam-3548	262	10	a	a	DET
ejpam-3548	262	11	∩	∩	NOUN
ejpam-3548	262	12	a(x	a(x	NOUN
ejpam-3548	262	13	)	)	PUNCT
ejpam-3548	262	14	=	=	NOUN
ejpam-3548	262	15	∅	∅	NOUN
ejpam-3548	262	16	is	be	AUX
ejpam-3548	262	17	a	a	DET
ejpam-3548	262	18	=	=	PUNCT
ejpam-3548	262	19	{	{	PUNCT
ejpam-3548	262	20	1x	1x	NUM
ejpam-3548	262	21	}	}	PUNCT
ejpam-3548	262	22	.	.	PUNCT
ejpam-3548	263	1	by	by	ADP
ejpam-3548	263	2	theorem	theorem	NOUN
ejpam-3548	263	3	6	6	NUM
ejpam-3548	263	4	,	,	PUNCT
ejpam-3548	263	5	rx(a	rx(a	NOUN
ejpam-3548	263	6	)	)	PUNCT
ejpam-3548	263	7	=	=	SYM
ejpam-3548	263	8	{	{	PUNCT
ejpam-3548	263	9	1x	1x	NUM
ejpam-3548	263	10	}	}	PUNCT
ejpam-3548	263	11	.	.	PUNCT
ejpam-3548	264	1	thus	thus	ADV
ejpam-3548	264	2	,	,	PUNCT
ejpam-3548	264	3	br(x	br(x	X
ejpam-3548	264	4	)	)	PUNCT
ejpam-3548	264	5	=	=	SYM
ejpam-3548	264	6	{	{	PUNCT
ejpam-3548	264	7	rx({a	rx({a	NOUN
ejpam-3548	264	8	}	}	PUNCT
ejpam-3548	264	9	)	)	PUNCT
ejpam-3548	265	1	:	:	PUNCT
ejpam-3548	265	2	a	a	DET
ejpam-3548	265	3	∈	∈	ADJ
ejpam-3548	265	4	x	x	NOUN
ejpam-3548	265	5	}	}	PUNCT
ejpam-3548	265	6	=	=	PUNCT
ejpam-3548	265	7	{	{	PUNCT
ejpam-3548	265	8	1x}∪{{1x	1x}∪{{1x	NUM
ejpam-3548	265	9	,	,	PUNCT
ejpam-3548	265	10	a	a	PROPN
ejpam-3548	265	11	}	}	PUNCT
ejpam-3548	265	12	:	:	PUNCT
ejpam-3548	265	13	a	a	DET
ejpam-3548	265	14	∈	∈	PROPN
ejpam-3548	265	15	x	x	X
ejpam-3548	265	16	\	\	X
ejpam-3548	265	17	{	{	PUNCT
ejpam-3548	265	18	1x	1x	NUM
ejpam-3548	265	19	}	}	PUNCT
ejpam-3548	265	20	}	}	PUNCT
ejpam-3548	265	21	.	.	PUNCT
ejpam-3548	266	1	conversely	conversely	ADV
ejpam-3548	266	2	,	,	PUNCT
ejpam-3548	266	3	suppose	suppose	VERB
ejpam-3548	266	4	that	that	SCONJ
ejpam-3548	266	5	br(x	br(x	NOUN
ejpam-3548	266	6	)	)	PUNCT
ejpam-3548	266	7	is	be	AUX
ejpam-3548	266	8	the	the	DET
ejpam-3548	266	9	given	give	VERB
ejpam-3548	266	10	family	family	NOUN
ejpam-3548	266	11	of	of	ADP
ejpam-3548	266	12	subsets	subset	NOUN
ejpam-3548	266	13	of	of	ADP
ejpam-3548	266	14	x.	x.	NOUN
ejpam-3548	266	15	let	let	VERB
ejpam-3548	266	16	a	a	DET
ejpam-3548	266	17	∈	∈	NOUN
ejpam-3548	266	18	x	x	SYM
ejpam-3548	266	19	\	\	X
ejpam-3548	266	20	{	{	PUNCT
ejpam-3548	266	21	1x	1x	NUM
ejpam-3548	266	22	}	}	PUNCT
ejpam-3548	266	23	.	.	PUNCT
ejpam-3548	267	1	then	then	ADV
ejpam-3548	267	2	rx({a	rx({a	VERB
ejpam-3548	267	3	}	}	PUNCT
ejpam-3548	267	4	)	)	PUNCT
ejpam-3548	268	1	=	=	PRON
ejpam-3548	268	2	{	{	PUNCT
ejpam-3548	268	3	1x	1x	NOUN
ejpam-3548	268	4	,	,	PUNCT
ejpam-3548	268	5	a	a	PRON
ejpam-3548	268	6	}	}	PUNCT
ejpam-3548	268	7	.	.	PUNCT
ejpam-3548	269	1	hence	hence	ADV
ejpam-3548	269	2	,	,	PUNCT
ejpam-3548	269	3	if	if	SCONJ
ejpam-3548	269	4	x	x	SYM
ejpam-3548	269	5	∈	∈	PROPN
ejpam-3548	269	6	x	x	X
ejpam-3548	269	7	and	and	CCONJ
ejpam-3548	269	8	a	a	DET
ejpam-3548	269	9	≤	≤	NOUN
ejpam-3548	269	10	x	x	PUNCT
ejpam-3548	269	11	,	,	PUNCT
ejpam-3548	269	12	then	then	ADV
ejpam-3548	269	13	x	x	X
ejpam-3548	269	14	=	=	PUNCT
ejpam-3548	269	15	a	a	PRON
ejpam-3548	269	16	or	or	CCONJ
ejpam-3548	269	17	x	x	SYM
ejpam-3548	269	18	=	=	SYM
ejpam-3548	269	19	1x	1x	NUM
ejpam-3548	269	20	.	.	PUNCT
ejpam-3548	270	1	thus	thus	ADV
ejpam-3548	270	2	,	,	PUNCT
ejpam-3548	270	3	a	a	DET
ejpam-3548	270	4	∈	∈	PROPN
ejpam-3548	270	5	a(x	a(x	NOUN
ejpam-3548	270	6	)	)	PUNCT
ejpam-3548	270	7	.	.	PUNCT
ejpam-3548	271	1	accordingly	accordingly	ADV
ejpam-3548	271	2	,	,	PUNCT
ejpam-3548	271	3	x	x	X
ejpam-3548	271	4	is	be	AUX
ejpam-3548	271	5	dual	dual	ADV
ejpam-3548	271	6	atomistic	atomistic	ADJ
ejpam-3548	271	7	.	.	PUNCT
ejpam-3548	272	1	j.	j.	PROPN
ejpam-3548	272	2	albaracin	albaracin	PROPN
ejpam-3548	272	3	,	,	PUNCT
ejpam-3548	272	4	j.	j.	PROPN
ejpam-3548	272	5	vilela	vilela	PROPN
ejpam-3548	272	6	/	/	SYM
ejpam-3548	272	7	eur	eur	PROPN
ejpam-3548	272	8	.	.	PUNCT
ejpam-3548	273	1	j.	j.	PROPN
ejpam-3548	273	2	pure	pure	PROPN
ejpam-3548	273	3	appl	appl	PROPN
ejpam-3548	273	4	.	.	PROPN
ejpam-3548	273	5	math	math	PROPN
ejpam-3548	273	6	,	,	PUNCT
ejpam-3548	273	7	12	12	NUM
ejpam-3548	273	8	(	(	PUNCT
ejpam-3548	273	9	4	4	NUM
ejpam-3548	273	10	)	)	PUNCT
ejpam-3548	273	11	(	(	PUNCT
ejpam-3548	273	12	2019	2019	NUM
ejpam-3548	273	13	)	)	PUNCT
ejpam-3548	273	14	,	,	PUNCT
ejpam-3548	273	15	1584	1584	NUM
ejpam-3548	273	16	-	-	SYM
ejpam-3548	273	17	1594	1594	NUM
ejpam-3548	273	18	1590	1590	NUM
ejpam-3548	273	19	4	4	NUM
ejpam-3548	273	20	.	.	PUNCT
ejpam-3548	274	1	characterizations	characterization	NOUN
ejpam-3548	274	2	involving	involve	VERB
ejpam-3548	274	3	the	the	DET
ejpam-3548	274	4	topology	topology	NOUN
ejpam-3548	274	5	τr(x	τr(x	PUNCT
ejpam-3548	274	6	)	)	PUNCT
ejpam-3548	274	7	this	this	DET
ejpam-3548	274	8	section	section	NOUN
ejpam-3548	274	9	gives	give	VERB
ejpam-3548	274	10	some	some	DET
ejpam-3548	274	11	characterizations	characterization	NOUN
ejpam-3548	274	12	of	of	ADP
ejpam-3548	274	13	the	the	DET
ejpam-3548	274	14	elementary	elementary	ADJ
ejpam-3548	274	15	concepts	concept	NOUN
ejpam-3548	274	16	associated	associate	VERB
ejpam-3548	274	17	with	with	ADP
ejpam-3548	274	18	the	the	DET
ejpam-3548	274	19	topological	topological	ADJ
ejpam-3548	274	20	space	space	NOUN
ejpam-3548	274	21	(	(	PUNCT
ejpam-3548	274	22	x	x	NOUN
ejpam-3548	274	23	,	,	PUNCT
ejpam-3548	274	24	τr(x	τr(x	NUM
ejpam-3548	274	25	)	)	PUNCT
ejpam-3548	274	26	)	)	PUNCT
ejpam-3548	274	27	.	.	PUNCT
ejpam-3548	275	1	theorem	theorem	VERB
ejpam-3548	275	2	13	13	NUM
ejpam-3548	275	3	.	.	PUNCT
ejpam-3548	276	1	let	let	VERB
ejpam-3548	276	2	x	x	PRON
ejpam-3548	276	3	be	be	AUX
ejpam-3548	276	4	a	a	DET
ejpam-3548	276	5	be	be	NOUN
ejpam-3548	276	6	-	-	PUNCT
ejpam-3548	276	7	algebra	algebra	NOUN
ejpam-3548	276	8	with	with	ADP
ejpam-3548	276	9	|x|	|x|	PROPN
ejpam-3548	276	10	≥	≥	NUM
ejpam-3548	276	11	2	2	NUM
ejpam-3548	276	12	.	.	PUNCT
ejpam-3548	276	13	then	then	ADV
ejpam-3548	276	14	τr(x	τr(x	NUM
ejpam-3548	276	15	)	)	PUNCT
ejpam-3548	276	16	is	be	AUX
ejpam-3548	276	17	the	the	DET
ejpam-3548	276	18	particular	particular	ADJ
ejpam-3548	276	19	point	point	NOUN
ejpam-3548	276	20	1x	1x	NUM
ejpam-3548	276	21	topology	topology	NOUN
ejpam-3548	276	22	τ1x	τ1x	PUNCT
ejpam-3548	276	23	on	on	ADP
ejpam-3548	276	24	x	x	SYM
ejpam-3548	276	25	if	if	SCONJ
ejpam-3548	276	26	and	and	CCONJ
ejpam-3548	276	27	only	only	ADV
ejpam-3548	276	28	if	if	SCONJ
ejpam-3548	276	29	x	x	PRON
ejpam-3548	276	30	is	be	AUX
ejpam-3548	276	31	dual	dual	ADV
ejpam-3548	276	32	atomistic	atomistic	ADJ
ejpam-3548	276	33	.	.	PUNCT
ejpam-3548	277	1	proof	proof	NOUN
ejpam-3548	277	2	.	.	PUNCT
ejpam-3548	278	1	suppose	suppose	VERB
ejpam-3548	278	2	that	that	SCONJ
ejpam-3548	278	3	τr(x	τr(x	PUNCT
ejpam-3548	278	4	)	)	PUNCT
ejpam-3548	279	1	=	=	PUNCT
ejpam-3548	279	2	τ1x	τ1x	PUNCT
ejpam-3548	280	1	=	=	PRON
ejpam-3548	280	2	{	{	PUNCT
ejpam-3548	280	3	∅	∅	NOUN
ejpam-3548	280	4	}	}	PUNCT
ejpam-3548	280	5	∪	∪	X
ejpam-3548	280	6	{	{	PUNCT
ejpam-3548	280	7	a	a	DET
ejpam-3548	280	8	⊆	⊆	NUM
ejpam-3548	280	9	x	x	SYM
ejpam-3548	280	10	:	:	PUNCT
ejpam-3548	280	11	1x	1x	PROPN
ejpam-3548	280	12	∈	∈	PROPN
ejpam-3548	280	13	a	a	X
ejpam-3548	280	14	}	}	PUNCT
ejpam-3548	280	15	and	and	CCONJ
ejpam-3548	280	16	let	let	VERB
ejpam-3548	280	17	a	a	DET
ejpam-3548	280	18	∈	∈	NOUN
ejpam-3548	280	19	τ1x	τ1x	PUNCT
ejpam-3548	280	20	\	\	PUNCT
ejpam-3548	280	21	{	{	PUNCT
ejpam-3548	280	22	∅	∅	NOUN
ejpam-3548	280	23	}	}	PUNCT
ejpam-3548	280	24	such	such	ADJ
ejpam-3548	280	25	that	that	SCONJ
ejpam-3548	280	26	|a|	|a|	PROPN
ejpam-3548	280	27	≥	≥	NOUN
ejpam-3548	280	28	2	2	NUM
ejpam-3548	280	29	.	.	PUNCT
ejpam-3548	281	1	then	then	ADV
ejpam-3548	281	2	a	a	DET
ejpam-3548	281	3	=	=	X
ejpam-3548	281	4	⋃	⋃	PROPN
ejpam-3548	281	5	a∈a	a∈a	ADJ
ejpam-3548	281	6	{	{	PUNCT
ejpam-3548	281	7	1x	1x	NUM
ejpam-3548	281	8	,	,	PUNCT
ejpam-3548	281	9	a	a	PRON
ejpam-3548	281	10	}	}	PUNCT
ejpam-3548	281	11	,	,	PUNCT
ejpam-3548	281	12	a	a	DET
ejpam-3548	281	13	6=	6=	NUM
ejpam-3548	281	14	1x	1x	NUM
ejpam-3548	281	15	.	.	PUNCT
ejpam-3548	282	1	if	if	SCONJ
ejpam-3548	282	2	|a|	|a|	PROPN
ejpam-3548	282	3	=	=	PROPN
ejpam-3548	282	4	1x	1x	PROPN
ejpam-3548	282	5	,	,	PUNCT
ejpam-3548	282	6	then	then	ADV
ejpam-3548	282	7	a	a	PRON
ejpam-3548	282	8	=	=	X
ejpam-3548	282	9	{	{	PUNCT
ejpam-3548	282	10	1x	1x	NUM
ejpam-3548	282	11	}	}	PUNCT
ejpam-3548	282	12	.	.	PUNCT
ejpam-3548	283	1	this	this	PRON
ejpam-3548	283	2	implies	imply	VERB
ejpam-3548	283	3	that	that	PRON
ejpam-3548	283	4	br(x	br(x	PUNCT
ejpam-3548	283	5	)	)	PUNCT
ejpam-3548	283	6	=	=	PRON
ejpam-3548	283	7	{	{	PUNCT
ejpam-3548	283	8	{	{	PUNCT
ejpam-3548	283	9	1x	1x	NUM
ejpam-3548	283	10	}	}	PUNCT
ejpam-3548	283	11	}	}	PUNCT
ejpam-3548	283	12	∪	∪	X
ejpam-3548	283	13	{	{	PUNCT
ejpam-3548	283	14	{	{	PUNCT
ejpam-3548	283	15	1x	1x	NUM
ejpam-3548	283	16	,	,	PUNCT
ejpam-3548	283	17	a	a	PRON
ejpam-3548	283	18	}	}	PUNCT
ejpam-3548	283	19	:	:	PUNCT
ejpam-3548	283	20	a	a	DET
ejpam-3548	283	21	∈	∈	PROPN
ejpam-3548	283	22	x	x	X
ejpam-3548	283	23	\	\	X
ejpam-3548	283	24	{	{	PUNCT
ejpam-3548	283	25	1x	1x	NUM
ejpam-3548	283	26	}	}	PUNCT
ejpam-3548	283	27	}	}	PUNCT
ejpam-3548	283	28	is	be	AUX
ejpam-3548	283	29	a	a	DET
ejpam-3548	283	30	basis	basis	NOUN
ejpam-3548	283	31	for	for	ADP
ejpam-3548	283	32	τ1x	τ1x	PRON
ejpam-3548	283	33	=	=	SYM
ejpam-3548	283	34	τr(x	τr(x	NUM
ejpam-3548	283	35	)	)	PUNCT
ejpam-3548	283	36	.	.	PUNCT
ejpam-3548	284	1	hence	hence	ADV
ejpam-3548	284	2	,	,	PUNCT
ejpam-3548	284	3	by	by	ADP
ejpam-3548	284	4	theorem	theorem	NOUN
ejpam-3548	284	5	12	12	NUM
ejpam-3548	284	6	,	,	PUNCT
ejpam-3548	284	7	x	x	X
ejpam-3548	284	8	is	be	AUX
ejpam-3548	284	9	dual	dual	ADV
ejpam-3548	284	10	atomistic	atomistic	ADJ
ejpam-3548	284	11	.	.	PUNCT
ejpam-3548	285	1	conversely	conversely	ADV
ejpam-3548	285	2	,	,	PUNCT
ejpam-3548	285	3	suppose	suppose	VERB
ejpam-3548	285	4	that	that	SCONJ
ejpam-3548	285	5	x	x	PRON
ejpam-3548	285	6	is	be	AUX
ejpam-3548	285	7	a	a	DET
ejpam-3548	285	8	dual	dual	ADJ
ejpam-3548	285	9	atomistic	atomistic	NOUN
ejpam-3548	285	10	.	.	PUNCT
ejpam-3548	286	1	by	by	ADP
ejpam-3548	286	2	theorem	theorem	NOUN
ejpam-3548	286	3	12	12	NUM
ejpam-3548	286	4	,	,	PUNCT
ejpam-3548	286	5	br(x	br(x	NUM
ejpam-3548	286	6	)	)	PUNCT
ejpam-3548	286	7	=	=	SYM
ejpam-3548	286	8	{	{	PUNCT
ejpam-3548	286	9	1x	1x	NUM
ejpam-3548	286	10	}	}	PUNCT
ejpam-3548	286	11	∪	∪	X
ejpam-3548	286	12	{	{	PUNCT
ejpam-3548	286	13	{	{	PUNCT
ejpam-3548	286	14	1x	1x	NUM
ejpam-3548	286	15	,	,	PUNCT
ejpam-3548	286	16	a	a	PRON
ejpam-3548	286	17	}	}	PUNCT
ejpam-3548	286	18	:	:	PUNCT
ejpam-3548	286	19	a	a	DET
ejpam-3548	286	20	∈	∈	PROPN
ejpam-3548	286	21	x	x	X
ejpam-3548	286	22	\	\	X
ejpam-3548	286	23	{	{	PUNCT
ejpam-3548	286	24	1x	1x	NUM
ejpam-3548	286	25	}	}	PUNCT
ejpam-3548	286	26	}	}	PUNCT
ejpam-3548	286	27	.	.	PUNCT
ejpam-3548	287	1	let	let	VERB
ejpam-3548	287	2	a	a	DET
ejpam-3548	287	3	∈	∈	NOUN
ejpam-3548	287	4	τr(x	τr(x	NUM
ejpam-3548	287	5	)	)	PUNCT
ejpam-3548	287	6	.	.	PUNCT
ejpam-3548	288	1	since	since	SCONJ
ejpam-3548	288	2	br(x	br(x	NUM
ejpam-3548	288	3	)	)	PUNCT
ejpam-3548	288	4	is	be	AUX
ejpam-3548	288	5	a	a	DET
ejpam-3548	288	6	basis	basis	NOUN
ejpam-3548	288	7	for	for	ADP
ejpam-3548	288	8	τr(x	τr(x	NUM
ejpam-3548	288	9	)	)	PUNCT
ejpam-3548	288	10	,	,	PUNCT
ejpam-3548	288	11	a	a	PRON
ejpam-3548	288	12	=	=	X
ejpam-3548	288	13	{	{	PUNCT
ejpam-3548	288	14	1x	1x	NUM
ejpam-3548	288	15	}	}	PUNCT
ejpam-3548	288	16	or	or	CCONJ
ejpam-3548	288	17	a	a	DET
ejpam-3548	288	18	=	=	X
ejpam-3548	288	19	⋃	⋃	NOUN
ejpam-3548	288	20	a∈a	a∈a	ADJ
ejpam-3548	288	21	{	{	PUNCT
ejpam-3548	288	22	1x	1x	NOUN
ejpam-3548	288	23	,	,	PUNCT
ejpam-3548	288	24	a	a	PRON
ejpam-3548	288	25	}	}	PUNCT
ejpam-3548	288	26	.	.	PUNCT
ejpam-3548	289	1	thus	thus	ADV
ejpam-3548	289	2	,	,	PUNCT
ejpam-3548	289	3	1x	1x	PROPN
ejpam-3548	289	4	∈	∈	PROPN
ejpam-3548	289	5	a	a	DET
ejpam-3548	289	6	implying	implying	NOUN
ejpam-3548	289	7	that	that	SCONJ
ejpam-3548	289	8	a	a	DET
ejpam-3548	289	9	∈	∈	NOUN
ejpam-3548	289	10	τ1x	τ1x	PUNCT
ejpam-3548	289	11	.	.	PUNCT
ejpam-3548	290	1	hence	hence	ADV
ejpam-3548	290	2	,	,	PUNCT
ejpam-3548	290	3	τr(x	τr(x	NUM
ejpam-3548	290	4	)	)	PUNCT
ejpam-3548	290	5	⊆	⊆	NUM
ejpam-3548	290	6	τ1x	τ1x	PUNCT
ejpam-3548	290	7	.	.	PUNCT
ejpam-3548	291	1	now	now	ADV
ejpam-3548	291	2	,	,	PUNCT
ejpam-3548	291	3	let	let	VERB
ejpam-3548	291	4	a	a	DET
ejpam-3548	291	5	∈	∈	NOUN
ejpam-3548	291	6	τ1x	τ1x	PUNCT
ejpam-3548	291	7	.	.	PUNCT
ejpam-3548	292	1	then	then	ADV
ejpam-3548	292	2	1x	1x	PROPN
ejpam-3548	292	3	∈	∈	PROPN
ejpam-3548	292	4	a.	a.	NOUN
ejpam-3548	292	5	since	since	SCONJ
ejpam-3548	292	6	br(x	br(x	NUM
ejpam-3548	292	7	)	)	PUNCT
ejpam-3548	292	8	is	be	AUX
ejpam-3548	292	9	a	a	DET
ejpam-3548	292	10	basis	basis	NOUN
ejpam-3548	292	11	for	for	ADP
ejpam-3548	292	12	τr(x	τr(x	NUM
ejpam-3548	292	13	)	)	PUNCT
ejpam-3548	292	14	,	,	PUNCT
ejpam-3548	292	15	a	a	PRON
ejpam-3548	292	16	=	=	X
ejpam-3548	292	17	{	{	PUNCT
ejpam-3548	292	18	1x	1x	NUM
ejpam-3548	292	19	}	}	PUNCT
ejpam-3548	292	20	or	or	CCONJ
ejpam-3548	292	21	a	a	DET
ejpam-3548	292	22	=	=	X
ejpam-3548	292	23	⋃	⋃	NOUN
ejpam-3548	292	24	a∈a	a∈a	ADJ
ejpam-3548	292	25	{	{	PUNCT
ejpam-3548	292	26	1x	1x	NOUN
ejpam-3548	292	27	,	,	PUNCT
ejpam-3548	292	28	a	a	PRON
ejpam-3548	292	29	}	}	PUNCT
ejpam-3548	292	30	.	.	PUNCT
ejpam-3548	293	1	therefore	therefore	ADV
ejpam-3548	293	2	,	,	PUNCT
ejpam-3548	293	3	a	a	DET
ejpam-3548	293	4	∈	∈	PROPN
ejpam-3548	293	5	τr(x	τr(x	NUM
ejpam-3548	293	6	)	)	PUNCT
ejpam-3548	293	7	and	and	CCONJ
ejpam-3548	293	8	τ1x	τ1x	NUM
ejpam-3548	293	9	⊆	⊆	NUM
ejpam-3548	293	10	τr(x	τr(x	NUM
ejpam-3548	293	11	)	)	PUNCT
ejpam-3548	293	12	.	.	PUNCT
ejpam-3548	294	1	consequently	consequently	ADV
ejpam-3548	294	2	,	,	PUNCT
ejpam-3548	294	3	τr(x	τr(x	NUM
ejpam-3548	294	4	)	)	PUNCT
ejpam-3548	295	1	=	=	SYM
ejpam-3548	295	2	τ1x	τ1x	PRON
ejpam-3548	295	3	.	.	PUNCT
ejpam-3548	296	1	in	in	ADP
ejpam-3548	296	2	a	a	DET
ejpam-3548	296	3	dual	dual	ADJ
ejpam-3548	296	4	atomistic	atomistic	ADJ
ejpam-3548	296	5	be	be	NOUN
ejpam-3548	296	6	-	-	PUNCT
ejpam-3548	296	7	algebra	algebra	NOUN
ejpam-3548	296	8	x	x	PUNCT
ejpam-3548	296	9	with	with	ADP
ejpam-3548	296	10	respect	respect	NOUN
ejpam-3548	296	11	to	to	ADP
ejpam-3548	296	12	τr(x	τr(x	NUM
ejpam-3548	296	13	)	)	PUNCT
ejpam-3548	296	14	,	,	PUNCT
ejpam-3548	296	15	every	every	DET
ejpam-3548	296	16	set	set	NOUN
ejpam-3548	296	17	that	that	PRON
ejpam-3548	296	18	contains	contain	VERB
ejpam-3548	296	19	1x	1x	NUM
ejpam-3548	296	20	is	be	AUX
ejpam-3548	296	21	open	open	ADJ
ejpam-3548	296	22	and	and	CCONJ
ejpam-3548	296	23	every	every	DET
ejpam-3548	296	24	set	set	NOUN
ejpam-3548	296	25	that	that	PRON
ejpam-3548	296	26	does	do	AUX
ejpam-3548	296	27	not	not	PART
ejpam-3548	296	28	contain	contain	VERB
ejpam-3548	296	29	1x	1x	NUM
ejpam-3548	296	30	is	be	AUX
ejpam-3548	296	31	closed	closed	ADJ
ejpam-3548	296	32	.	.	PUNCT
ejpam-3548	297	1	hence	hence	ADV
ejpam-3548	297	2	,	,	PUNCT
ejpam-3548	297	3	the	the	DET
ejpam-3548	297	4	following	follow	VERB
ejpam-3548	297	5	corollary	corollary	NOUN
ejpam-3548	297	6	is	be	AUX
ejpam-3548	297	7	true	true	ADJ
ejpam-3548	297	8	.	.	PUNCT
ejpam-3548	298	1	corollary	corollary	ADJ
ejpam-3548	298	2	3	3	X
ejpam-3548	298	3	.	.	PUNCT
ejpam-3548	299	1	let	let	VERB
ejpam-3548	299	2	x	x	PRON
ejpam-3548	299	3	be	be	AUX
ejpam-3548	299	4	a	a	DET
ejpam-3548	299	5	dual	dual	ADJ
ejpam-3548	299	6	atomistic	atomistic	ADJ
ejpam-3548	299	7	be	be	NOUN
ejpam-3548	299	8	-	-	PUNCT
ejpam-3548	299	9	algebra	algebra	NOUN
ejpam-3548	299	10	with	with	ADP
ejpam-3548	299	11	|x|	|x|	PROPN
ejpam-3548	299	12	≥	≥	NUM
ejpam-3548	299	13	2	2	NUM
ejpam-3548	299	14	and	and	CCONJ
ejpam-3548	299	15	let	let	VERB
ejpam-3548	299	16	o	o	NOUN
ejpam-3548	299	17	,	,	PUNCT
ejpam-3548	299	18	c	c	PROPN
ejpam-3548	299	19	⊆	⊆	NUM
ejpam-3548	299	20	x.	x.	NOUN
ejpam-3548	299	21	then	then	ADV
ejpam-3548	299	22	with	with	ADP
ejpam-3548	299	23	respect	respect	NOUN
ejpam-3548	299	24	to	to	ADP
ejpam-3548	299	25	τr(x	τr(x	NUM
ejpam-3548	299	26	)	)	PUNCT
ejpam-3548	299	27	,	,	PUNCT
ejpam-3548	299	28	we	we	PRON
ejpam-3548	299	29	have	have	VERB
ejpam-3548	299	30	(	(	PUNCT
ejpam-3548	299	31	i	i	NOUN
ejpam-3548	299	32	)	)	PUNCT
ejpam-3548	299	33	int(o	int(o	PROPN
ejpam-3548	299	34	)	)	PUNCT
ejpam-3548	299	35	=	=	PRON
ejpam-3548	299	36	{	{	PUNCT
ejpam-3548	299	37	∅	∅	NOUN
ejpam-3548	299	38	if	if	SCONJ
ejpam-3548	299	39	1x	1x	NUM
ejpam-3548	299	40	/∈	/∈	PUNCT
ejpam-3548	300	1	o	o	X
ejpam-3548	300	2	o	o	NOUN
ejpam-3548	301	1	if	if	SCONJ
ejpam-3548	301	2	1x	1x	NUM
ejpam-3548	301	3	∈	∈	PROPN
ejpam-3548	301	4	o	o	NOUN
ejpam-3548	301	5	,	,	PUNCT
ejpam-3548	301	6	and	and	CCONJ
ejpam-3548	301	7	(	(	PUNCT
ejpam-3548	301	8	ii	ii	NOUN
ejpam-3548	301	9	)	)	PUNCT
ejpam-3548	301	10	c	c	NOUN
ejpam-3548	301	11	=	=	PRON
ejpam-3548	302	1	{	{	PUNCT
ejpam-3548	302	2	x	x	X
ejpam-3548	302	3	if	if	SCONJ
ejpam-3548	302	4	1x	1x	NUM
ejpam-3548	302	5	∈	∈	PROPN
ejpam-3548	303	1	c	c	NOUN
ejpam-3548	303	2	c	c	NOUN
ejpam-3548	304	1	if	if	SCONJ
ejpam-3548	304	2	1x	1x	PROPN
ejpam-3548	304	3	/∈	/∈	PUNCT
ejpam-3548	304	4	c.	c.	PROPN
ejpam-3548	304	5	theorem	theorem	VERB
ejpam-3548	304	6	14	14	NUM
ejpam-3548	304	7	.	.	PUNCT
ejpam-3548	305	1	let	let	VERB
ejpam-3548	305	2	x	x	PRON
ejpam-3548	305	3	be	be	AUX
ejpam-3548	305	4	a	a	DET
ejpam-3548	305	5	be	be	NOUN
ejpam-3548	305	6	-	-	PUNCT
ejpam-3548	305	7	algebra	algebra	NOUN
ejpam-3548	305	8	and	and	CCONJ
ejpam-3548	305	9	let	let	VERB
ejpam-3548	305	10	d	d	PROPN
ejpam-3548	305	11	⊆	⊆	X
ejpam-3548	305	12	x.	x.	NOUN
ejpam-3548	305	13	then	then	ADV
ejpam-3548	305	14	with	with	ADP
ejpam-3548	305	15	respect	respect	NOUN
ejpam-3548	305	16	to	to	ADP
ejpam-3548	305	17	τr(x	τr(x	NUM
ejpam-3548	305	18	)	)	PUNCT
ejpam-3548	305	19	,	,	PUNCT
ejpam-3548	305	20	we	we	PRON
ejpam-3548	305	21	have	have	VERB
ejpam-3548	305	22	(	(	PUNCT
ejpam-3548	305	23	i	i	NOUN
ejpam-3548	305	24	)	)	PUNCT
ejpam-3548	305	25	z	z	PROPN
ejpam-3548	305	26	∈	∈	PROPN
ejpam-3548	305	27	int(d	int(d	PROPN
ejpam-3548	305	28	)	)	PUNCT
ejpam-3548	306	1	if	if	SCONJ
ejpam-3548	306	2	and	and	CCONJ
ejpam-3548	306	3	only	only	ADV
ejpam-3548	306	4	if	if	SCONJ
ejpam-3548	306	5	there	there	PRON
ejpam-3548	306	6	exists	exist	VERB
ejpam-3548	306	7	∅	∅	NOUN
ejpam-3548	306	8	6=	6=	ADP
ejpam-3548	306	9	b	b	X
ejpam-3548	306	10	⊆	⊆	NUM
ejpam-3548	306	11	x	x	SYM
ejpam-3548	306	12	such	such	ADJ
ejpam-3548	306	13	that	that	DET
ejpam-3548	306	14	b	b	NOUN
ejpam-3548	306	15	∗	∗	NOUN
ejpam-3548	306	16	z	z	NOUN
ejpam-3548	306	17	=	=	SYM
ejpam-3548	306	18	1x	1x	NUM
ejpam-3548	306	19	for	for	ADP
ejpam-3548	306	20	all	all	DET
ejpam-3548	306	21	b	b	PROPN
ejpam-3548	306	22	∈	∈	PROPN
ejpam-3548	306	23	b	b	NOUN
ejpam-3548	306	24	and	and	CCONJ
ejpam-3548	306	25	for	for	ADP
ejpam-3548	306	26	all	all	DET
ejpam-3548	306	27	x	x	SYM
ejpam-3548	306	28	∈	∈	PROPN
ejpam-3548	306	29	x	x	NOUN
ejpam-3548	306	30	,	,	PUNCT
ejpam-3548	307	1	x	x	SYM
ejpam-3548	307	2	∈	∈	PROPN
ejpam-3548	307	3	d	d	X
ejpam-3548	307	4	whenever	whenever	SCONJ
ejpam-3548	307	5	b	b	NOUN
ejpam-3548	307	6	∗	∗	NOUN
ejpam-3548	307	7	x	x	PUNCT
ejpam-3548	308	1	=	=	SYM
ejpam-3548	308	2	1x	1x	NUM
ejpam-3548	308	3	for	for	ADP
ejpam-3548	308	4	all	all	DET
ejpam-3548	308	5	b	b	PROPN
ejpam-3548	308	6	∈	∈	PROPN
ejpam-3548	308	7	b.	b.	PROPN
ejpam-3548	308	8	(	(	PUNCT
ejpam-3548	308	9	ii	ii	PROPN
ejpam-3548	308	10	)	)	PUNCT
ejpam-3548	308	11	y	y	PROPN
ejpam-3548	308	12	∈	∈	PROPN
ejpam-3548	309	1	d	d	NOUN
ejpam-3548	309	2	if	if	SCONJ
ejpam-3548	309	3	and	and	CCONJ
ejpam-3548	309	4	only	only	ADV
ejpam-3548	309	5	if	if	SCONJ
ejpam-3548	309	6	for	for	ADP
ejpam-3548	309	7	each	each	DET
ejpam-3548	309	8	∅	∅	NOUN
ejpam-3548	309	9	6=	6=	ADP
ejpam-3548	309	10	a	a	DET
ejpam-3548	309	11	⊆	⊆	NUM
ejpam-3548	309	12	x	x	PUNCT
ejpam-3548	309	13	with	with	ADP
ejpam-3548	309	14	a	a	DET
ejpam-3548	309	15	∗	∗	NOUN
ejpam-3548	309	16	y	y	NOUN
ejpam-3548	309	17	=	=	SYM
ejpam-3548	309	18	1x	1x	NUM
ejpam-3548	309	19	for	for	ADP
ejpam-3548	309	20	all	all	DET
ejpam-3548	309	21	a	a	DET
ejpam-3548	309	22	∈	∈	PROPN
ejpam-3548	309	23	a	a	PRON
ejpam-3548	309	24	,	,	PUNCT
ejpam-3548	309	25	there	there	PRON
ejpam-3548	309	26	exists	exist	VERB
ejpam-3548	309	27	d	d	X
ejpam-3548	309	28	∈	∈	PROPN
ejpam-3548	310	1	d	d	ADP
ejpam-3548	310	2	such	such	ADJ
ejpam-3548	310	3	that	that	SCONJ
ejpam-3548	310	4	a	a	DET
ejpam-3548	310	5	∗	∗	NOUN
ejpam-3548	310	6	d	d	X
ejpam-3548	310	7	=	=	SYM
ejpam-3548	310	8	1x	1x	NUM
ejpam-3548	310	9	for	for	ADP
ejpam-3548	310	10	all	all	DET
ejpam-3548	310	11	a	a	DET
ejpam-3548	310	12	∈	∈	PROPN
ejpam-3548	310	13	a.	a.	NOUN
ejpam-3548	310	14	(	(	PUNCT
ejpam-3548	310	15	iii	iii	X
ejpam-3548	310	16	)	)	PUNCT
ejpam-3548	310	17	d	d	NOUN
ejpam-3548	310	18	is	be	AUX
ejpam-3548	310	19	dense	dense	ADJ
ejpam-3548	310	20	in	in	ADP
ejpam-3548	310	21	x	x	SYM
ejpam-3548	310	22	if	if	SCONJ
ejpam-3548	310	23	and	and	CCONJ
ejpam-3548	310	24	only	only	ADV
ejpam-3548	310	25	if	if	SCONJ
ejpam-3548	310	26	1x	1x	PROPN
ejpam-3548	310	27	∈	∈	PROPN
ejpam-3548	310	28	d.	d.	PROPN
ejpam-3548	310	29	in	in	ADP
ejpam-3548	310	30	particular	particular	ADJ
ejpam-3548	310	31	,	,	PUNCT
ejpam-3548	310	32	{	{	PUNCT
ejpam-3548	310	33	1x	1x	NUM
ejpam-3548	310	34	}	}	PUNCT
ejpam-3548	310	35	is	be	AUX
ejpam-3548	310	36	dense	dense	ADJ
ejpam-3548	310	37	in	in	ADP
ejpam-3548	310	38	x.	x.	NOUN
ejpam-3548	310	39	proof	proof	NOUN
ejpam-3548	310	40	.	.	PUNCT
ejpam-3548	311	1	j.	j.	PROPN
ejpam-3548	311	2	albaracin	albaracin	PROPN
ejpam-3548	311	3	,	,	PUNCT
ejpam-3548	311	4	j.	j.	PROPN
ejpam-3548	311	5	vilela	vilela	PROPN
ejpam-3548	311	6	/	/	SYM
ejpam-3548	311	7	eur	eur	PROPN
ejpam-3548	311	8	.	.	PUNCT
ejpam-3548	312	1	j.	j.	PROPN
ejpam-3548	312	2	pure	pure	PROPN
ejpam-3548	312	3	appl	appl	PROPN
ejpam-3548	312	4	.	.	PROPN
ejpam-3548	312	5	math	math	PROPN
ejpam-3548	312	6	,	,	PUNCT
ejpam-3548	312	7	12	12	NUM
ejpam-3548	312	8	(	(	PUNCT
ejpam-3548	312	9	4	4	NUM
ejpam-3548	312	10	)	)	PUNCT
ejpam-3548	312	11	(	(	PUNCT
ejpam-3548	312	12	2019	2019	NUM
ejpam-3548	312	13	)	)	PUNCT
ejpam-3548	312	14	,	,	PUNCT
ejpam-3548	312	15	1584	1584	NUM
ejpam-3548	312	16	-	-	SYM
ejpam-3548	312	17	1594	1594	NUM
ejpam-3548	312	18	1591	1591	NUM
ejpam-3548	312	19	(	(	PUNCT
ejpam-3548	312	20	i	i	NOUN
ejpam-3548	312	21	)	)	PUNCT
ejpam-3548	312	22	by	by	ADP
ejpam-3548	312	23	definition	definition	NOUN
ejpam-3548	312	24	,	,	PUNCT
ejpam-3548	312	25	z	z	PROPN
ejpam-3548	312	26	∈	∈	PROPN
ejpam-3548	312	27	int(d	int(d	PROPN
ejpam-3548	312	28	)	)	PUNCT
ejpam-3548	313	1	if	if	SCONJ
ejpam-3548	314	1	and	and	CCONJ
ejpam-3548	314	2	only	only	ADV
ejpam-3548	314	3	if	if	SCONJ
ejpam-3548	314	4	there	there	PRON
ejpam-3548	314	5	exists	exist	VERB
ejpam-3548	314	6	∅	∅	NOUN
ejpam-3548	314	7	6=	6=	ADP
ejpam-3548	314	8	b	b	X
ejpam-3548	314	9	⊆	⊆	NUM
ejpam-3548	314	10	x	x	SYM
ejpam-3548	314	11	such	such	ADJ
ejpam-3548	314	12	that	that	SCONJ
ejpam-3548	314	13	z	z	PROPN
ejpam-3548	314	14	∈	∈	PROPN
ejpam-3548	314	15	rx(b	rx(b	VERB
ejpam-3548	314	16	)	)	PUNCT
ejpam-3548	314	17	⊆	⊆	NUM
ejpam-3548	314	18	d	d	NOUN
ejpam-3548	314	19	,	,	PUNCT
ejpam-3548	314	20	that	that	ADV
ejpam-3548	314	21	is	is	ADV
ejpam-3548	314	22	,	,	PUNCT
ejpam-3548	314	23	b	b	NOUN
ejpam-3548	314	24	∗	∗	NOUN
ejpam-3548	314	25	z	z	NOUN
ejpam-3548	314	26	=	=	SYM
ejpam-3548	314	27	1x	1x	NUM
ejpam-3548	314	28	for	for	ADP
ejpam-3548	314	29	all	all	DET
ejpam-3548	314	30	b	b	PROPN
ejpam-3548	314	31	∈	∈	PROPN
ejpam-3548	314	32	b	b	NOUN
ejpam-3548	314	33	and	and	CCONJ
ejpam-3548	314	34	for	for	ADP
ejpam-3548	314	35	all	all	DET
ejpam-3548	314	36	x	x	NOUN
ejpam-3548	314	37	in	in	ADP
ejpam-3548	314	38	x	x	PRON
ejpam-3548	314	39	,	,	PUNCT
ejpam-3548	314	40	x	x	SYM
ejpam-3548	314	41	∈	∈	PROPN
ejpam-3548	315	1	d	d	X
ejpam-3548	315	2	whenever	whenever	SCONJ
ejpam-3548	315	3	b	b	NOUN
ejpam-3548	315	4	∗	∗	NOUN
ejpam-3548	315	5	x	x	PUNCT
ejpam-3548	315	6	=	=	SYM
ejpam-3548	315	7	1x	1x	NUM
ejpam-3548	315	8	for	for	ADP
ejpam-3548	315	9	all	all	DET
ejpam-3548	315	10	b	b	PROPN
ejpam-3548	315	11	∈	∈	PROPN
ejpam-3548	315	12	b.	b.	PROPN
ejpam-3548	315	13	(	(	PUNCT
ejpam-3548	315	14	ii	ii	PROPN
ejpam-3548	315	15	)	)	PUNCT
ejpam-3548	315	16	by	by	ADP
ejpam-3548	315	17	definition	definition	NOUN
ejpam-3548	315	18	,	,	PUNCT
ejpam-3548	315	19	y	y	PROPN
ejpam-3548	315	20	∈	∈	PROPN
ejpam-3548	316	1	d	d	NOUN
ejpam-3548	316	2	if	if	SCONJ
ejpam-3548	316	3	and	and	CCONJ
ejpam-3548	316	4	only	only	ADV
ejpam-3548	316	5	if	if	SCONJ
ejpam-3548	316	6	for	for	ADP
ejpam-3548	316	7	each	each	DET
ejpam-3548	316	8	∅	∅	NOUN
ejpam-3548	316	9	6=	6=	ADP
ejpam-3548	316	10	a	a	DET
ejpam-3548	316	11	⊆	⊆	NUM
ejpam-3548	316	12	x	x	PUNCT
ejpam-3548	316	13	with	with	ADP
ejpam-3548	316	14	y	y	PROPN
ejpam-3548	316	15	∈	∈	PROPN
ejpam-3548	316	16	rx(a	rx(a	NOUN
ejpam-3548	316	17	)	)	PUNCT
ejpam-3548	316	18	,	,	PUNCT
ejpam-3548	316	19	we	we	PRON
ejpam-3548	316	20	have	have	VERB
ejpam-3548	316	21	d	d	NOUN
ejpam-3548	316	22	∩	∩	NOUN
ejpam-3548	316	23	rx(a	rx(a	X
ejpam-3548	316	24	)	)	PUNCT
ejpam-3548	316	25	6=	6=	ADP
ejpam-3548	316	26	∅	∅	NOUN
ejpam-3548	316	27	,	,	PUNCT
ejpam-3548	316	28	that	that	ADV
ejpam-3548	316	29	is	is	ADV
ejpam-3548	316	30	,	,	PUNCT
ejpam-3548	316	31	there	there	PRON
ejpam-3548	316	32	exists	exist	VERB
ejpam-3548	316	33	d	d	PROPN
ejpam-3548	316	34	∈	∈	PROPN
ejpam-3548	316	35	d	d	NOUN
ejpam-3548	316	36	∩	∩	NOUN
ejpam-3548	316	37	rx(a	rx(a	NOUN
ejpam-3548	316	38	)	)	PUNCT
ejpam-3548	316	39	.	.	PUNCT
ejpam-3548	317	1	thus	thus	ADV
ejpam-3548	317	2	,	,	PUNCT
ejpam-3548	317	3	(	(	PUNCT
ejpam-3548	317	4	ii	ii	NOUN
ejpam-3548	317	5	)	)	PUNCT
ejpam-3548	317	6	holds	hold	VERB
ejpam-3548	317	7	.	.	PUNCT
ejpam-3548	318	1	(	(	PUNCT
ejpam-3548	318	2	iii	iii	X
ejpam-3548	318	3	)	)	PUNCT
ejpam-3548	318	4	let	let	VERB
ejpam-3548	318	5	d	d	PRON
ejpam-3548	318	6	be	be	AUX
ejpam-3548	318	7	dense	dense	ADJ
ejpam-3548	318	8	in	in	ADP
ejpam-3548	318	9	x.	x.	NOUN
ejpam-3548	318	10	then	then	ADV
ejpam-3548	318	11	1x	1x	NUM
ejpam-3548	318	12	∈	∈	PROPN
ejpam-3548	319	1	d	d	X
ejpam-3548	319	2	=	=	PUNCT
ejpam-3548	319	3	x.	x.	NOUN
ejpam-3548	319	4	since	since	SCONJ
ejpam-3548	319	5	{	{	PUNCT
ejpam-3548	319	6	1x	1x	NUM
ejpam-3548	319	7	}	}	PUNCT
ejpam-3548	319	8	=	=	PUNCT
ejpam-3548	319	9	rx	rx	X
ejpam-3548	319	10	(	(	PUNCT
ejpam-3548	319	11	1x	1x	NUM
ejpam-3548	319	12	)	)	PUNCT
ejpam-3548	319	13	,	,	PUNCT
ejpam-3548	319	14	d	d	PROPN
ejpam-3548	319	15	∩	∩	X
ejpam-3548	319	16	rx	rx	X
ejpam-3548	319	17	(	(	PUNCT
ejpam-3548	319	18	1x	1x	NUM
ejpam-3548	319	19	)	)	PUNCT
ejpam-3548	319	20	6=	6=	ADP
ejpam-3548	319	21	∅	∅	NOUN
ejpam-3548	319	22	,	,	PUNCT
ejpam-3548	319	23	it	it	PRON
ejpam-3548	319	24	follows	follow	VERB
ejpam-3548	319	25	that	that	SCONJ
ejpam-3548	319	26	1x	1x	PROPN
ejpam-3548	319	27	∈	∈	PROPN
ejpam-3548	319	28	d.	d.	PROPN
ejpam-3548	319	29	conversely	conversely	ADV
ejpam-3548	319	30	,	,	PUNCT
ejpam-3548	319	31	suppose	suppose	VERB
ejpam-3548	319	32	that	that	SCONJ
ejpam-3548	319	33	1x	1x	PROPN
ejpam-3548	319	34	∈	∈	PROPN
ejpam-3548	319	35	d.	d.	PROPN
ejpam-3548	319	36	let	let	VERB
ejpam-3548	319	37	x	x	X
ejpam-3548	319	38	∈	∈	PROPN
ejpam-3548	319	39	x	x	PUNCT
ejpam-3548	319	40	and	and	CCONJ
ejpam-3548	319	41	let	let	VERB
ejpam-3548	319	42	∅	∅	NOUN
ejpam-3548	319	43	6=	6=	ADP
ejpam-3548	319	44	a	a	DET
ejpam-3548	319	45	⊆	⊆	NUM
ejpam-3548	319	46	x	x	SYM
ejpam-3548	319	47	such	such	ADJ
ejpam-3548	319	48	that	that	SCONJ
ejpam-3548	319	49	x	x	SYM
ejpam-3548	319	50	∈	∈	NOUN
ejpam-3548	319	51	rx(a	rx(a	NOUN
ejpam-3548	319	52	)	)	PUNCT
ejpam-3548	319	53	.	.	PUNCT
ejpam-3548	320	1	since	since	SCONJ
ejpam-3548	320	2	1x	1x	PROPN
ejpam-3548	320	3	∈	∈	PROPN
ejpam-3548	320	4	rx(a	rx(a	PROPN
ejpam-3548	320	5	)	)	PUNCT
ejpam-3548	320	6	by	by	ADP
ejpam-3548	320	7	theorem	theorem	NOUN
ejpam-3548	320	8	6	6	NUM
ejpam-3548	320	9	,	,	PUNCT
ejpam-3548	320	10	it	it	PRON
ejpam-3548	320	11	follows	follow	VERB
ejpam-3548	320	12	that	that	SCONJ
ejpam-3548	320	13	rx(a	rx(a	NOUN
ejpam-3548	320	14	)	)	PUNCT
ejpam-3548	320	15	∩d	∩d	VERB
ejpam-3548	321	1	6=	6=	ADP
ejpam-3548	321	2	∅.	∅.	ADP
ejpam-3548	321	3	thus	thus	ADV
ejpam-3548	321	4	,	,	PUNCT
ejpam-3548	321	5	x	x	PUNCT
ejpam-3548	321	6	∈	∈	PROPN
ejpam-3548	321	7	d	d	NOUN
ejpam-3548	321	8	,	,	PUNCT
ejpam-3548	321	9	showing	show	VERB
ejpam-3548	321	10	that	that	PRON
ejpam-3548	321	11	d	d	NOUN
ejpam-3548	321	12	=	=	PUNCT
ejpam-3548	321	13	x.	x.	NOUN
ejpam-3548	321	14	whiteyughjgkh	whiteyughjgkh	PROPN
ejpam-3548	321	15	lemma	lemma	PROPN
ejpam-3548	321	16	5	5	X
ejpam-3548	321	17	.	.	PUNCT
ejpam-3548	322	1	let	let	VERB
ejpam-3548	322	2	s	s	PRON
ejpam-3548	322	3	be	be	AUX
ejpam-3548	322	4	a	a	DET
ejpam-3548	322	5	subalgebra	subalgebra	NOUN
ejpam-3548	322	6	of	of	ADP
ejpam-3548	322	7	a	a	DET
ejpam-3548	322	8	be	be	NOUN
ejpam-3548	322	9	-	-	PUNCT
ejpam-3548	322	10	algebra	algebra	NOUN
ejpam-3548	322	11	x.	x.	NOUN
ejpam-3548	322	12	then	then	ADV
ejpam-3548	322	13	(	(	PUNCT
ejpam-3548	322	14	i	i	NOUN
ejpam-3548	322	15	)	)	PUNCT
ejpam-3548	322	16	a(x	a(x	PROPN
ejpam-3548	322	17	)	)	PUNCT
ejpam-3548	322	18	∩	∩	NOUN
ejpam-3548	322	19	s	s	PART
ejpam-3548	322	20	⊆	⊆	NUM
ejpam-3548	322	21	a(s	a(	NOUN
ejpam-3548	322	22	)	)	PUNCT
ejpam-3548	322	23	;	;	PUNCT
ejpam-3548	322	24	and	and	CCONJ
ejpam-3548	322	25	(	(	PUNCT
ejpam-3548	322	26	ii	ii	NOUN
ejpam-3548	322	27	)	)	PUNCT
ejpam-3548	322	28	rs(t	rs(t	X
ejpam-3548	322	29	)	)	PUNCT
ejpam-3548	323	1	=	=	SYM
ejpam-3548	323	2	rx(t	rx(t	NOUN
ejpam-3548	323	3	)	)	PUNCT
ejpam-3548	323	4	∩	∩	PROPN
ejpam-3548	323	5	s	s	PART
ejpam-3548	323	6	for	for	ADP
ejpam-3548	323	7	every	every	DET
ejpam-3548	323	8	t	t	PROPN
ejpam-3548	323	9	⊆	⊆	NUM
ejpam-3548	323	10	s.	s.	PROPN
ejpam-3548	323	11	proof	proof	NOUN
ejpam-3548	323	12	.	.	PUNCT
ejpam-3548	324	1	(	(	PUNCT
ejpam-3548	324	2	i	i	NOUN
ejpam-3548	324	3	)	)	PUNCT
ejpam-3548	324	4	let	let	VERB
ejpam-3548	324	5	a	a	DET
ejpam-3548	324	6	∈	∈	PROPN
ejpam-3548	324	7	a(x	a(x	PROPN
ejpam-3548	324	8	)	)	PUNCT
ejpam-3548	324	9	∩	∩	NOUN
ejpam-3548	324	10	s.	s.	PROPN
ejpam-3548	324	11	then	then	ADV
ejpam-3548	324	12	a	a	DET
ejpam-3548	324	13	∈	∈	PROPN
ejpam-3548	324	14	s	s	X
ejpam-3548	324	15	and	and	CCONJ
ejpam-3548	324	16	for	for	ADP
ejpam-3548	324	17	all	all	DET
ejpam-3548	324	18	x	x	SYM
ejpam-3548	324	19	∈	∈	PROPN
ejpam-3548	324	20	x	x	NOUN
ejpam-3548	324	21	,	,	PUNCT
ejpam-3548	324	22	a	a	DET
ejpam-3548	324	23	≤	≤	NOUN
ejpam-3548	324	24	x	x	PUNCT
ejpam-3548	324	25	implies	imply	VERB
ejpam-3548	324	26	that	that	SCONJ
ejpam-3548	324	27	x	x	X
ejpam-3548	324	28	=	=	PUNCT
ejpam-3548	324	29	a	a	PRON
ejpam-3548	324	30	or	or	CCONJ
ejpam-3548	324	31	x	x	SYM
ejpam-3548	324	32	=	=	SYM
ejpam-3548	324	33	1x	1x	NUM
ejpam-3548	324	34	.	.	PUNCT
ejpam-3548	325	1	hence	hence	ADV
ejpam-3548	325	2	,	,	PUNCT
ejpam-3548	325	3	in	in	ADP
ejpam-3548	325	4	particular	particular	ADJ
ejpam-3548	325	5	,	,	PUNCT
ejpam-3548	325	6	for	for	ADP
ejpam-3548	325	7	all	all	DET
ejpam-3548	325	8	y	y	PROPN
ejpam-3548	325	9	∈	∈	PROPN
ejpam-3548	325	10	s	s	PROPN
ejpam-3548	325	11	,	,	PUNCT
ejpam-3548	325	12	a	a	DET
ejpam-3548	325	13	≤	≤	ADJ
ejpam-3548	325	14	y	y	PROPN
ejpam-3548	325	15	implies	imply	VERB
ejpam-3548	325	16	that	that	SCONJ
ejpam-3548	325	17	y	y	PROPN
ejpam-3548	325	18	=	=	SYM
ejpam-3548	325	19	1x	1x	PROPN
ejpam-3548	325	20	or	or	CCONJ
ejpam-3548	325	21	y	y	NOUN
ejpam-3548	325	22	=	=	PUNCT
ejpam-3548	325	23	a.	a.	NOUN
ejpam-3548	325	24	thus	thus	ADV
ejpam-3548	325	25	,	,	PUNCT
ejpam-3548	325	26	a	a	DET
ejpam-3548	325	27	∈	∈	PROPN
ejpam-3548	325	28	a(s	a(s	PROPN
ejpam-3548	325	29	)	)	PUNCT
ejpam-3548	325	30	.	.	PUNCT
ejpam-3548	326	1	(	(	PUNCT
ejpam-3548	326	2	ii	ii	NOUN
ejpam-3548	326	3	)	)	PUNCT
ejpam-3548	326	4	let	let	VERB
ejpam-3548	326	5	t	t	PROPN
ejpam-3548	326	6	⊆	⊆	NUM
ejpam-3548	326	7	s.	s.	PROPN
ejpam-3548	326	8	then	then	ADV
ejpam-3548	326	9	z	z	PROPN
ejpam-3548	326	10	∈	∈	PROPN
ejpam-3548	326	11	rs(t	rs(t	PUNCT
ejpam-3548	326	12	)	)	PUNCT
ejpam-3548	326	13	if	if	SCONJ
ejpam-3548	326	14	and	and	CCONJ
ejpam-3548	326	15	only	only	ADV
ejpam-3548	326	16	if	if	SCONJ
ejpam-3548	326	17	z	z	PROPN
ejpam-3548	326	18	∈	∈	PROPN
ejpam-3548	326	19	s	s	PART
ejpam-3548	326	20	and	and	CCONJ
ejpam-3548	326	21	t	t	X
ejpam-3548	326	22	≤	≤	ADJ
ejpam-3548	326	23	z	z	NOUN
ejpam-3548	326	24	for	for	ADP
ejpam-3548	326	25	all	all	DET
ejpam-3548	326	26	t	t	NOUN
ejpam-3548	326	27	∈	∈	PROPN
ejpam-3548	326	28	t	t	PROPN
ejpam-3548	326	29	.	.	PUNCT
ejpam-3548	327	1	thus	thus	ADV
ejpam-3548	327	2	,	,	PUNCT
ejpam-3548	327	3	z	z	PROPN
ejpam-3548	327	4	∈	∈	PROPN
ejpam-3548	327	5	rs(t	rs(t	PUNCT
ejpam-3548	327	6	)	)	PUNCT
ejpam-3548	328	1	if	if	SCONJ
ejpam-3548	328	2	and	and	CCONJ
ejpam-3548	328	3	only	only	ADV
ejpam-3548	328	4	if	if	SCONJ
ejpam-3548	328	5	z	z	PROPN
ejpam-3548	328	6	∈	∈	PROPN
ejpam-3548	328	7	s	s	PART
ejpam-3548	328	8	∩	∩	NOUN
ejpam-3548	328	9	rx(t	rx(t	NOUN
ejpam-3548	328	10	)	)	PUNCT
ejpam-3548	328	11	for	for	ADP
ejpam-3548	328	12	each	each	DET
ejpam-3548	328	13	t	t	PROPN
ejpam-3548	328	14	∈	∈	PROPN
ejpam-3548	328	15	t	t	PROPN
ejpam-3548	328	16	⊆	⊆	NUM
ejpam-3548	328	17	s	s	ADP
ejpam-3548	328	18	⊆	⊆	NUM
ejpam-3548	328	19	x.	x.	NOUN
ejpam-3548	328	20	accordingly	accordingly	ADV
ejpam-3548	328	21	,	,	PUNCT
ejpam-3548	328	22	rs(t	rs(t	X
ejpam-3548	328	23	)	)	PUNCT
ejpam-3548	328	24	=	=	SYM
ejpam-3548	328	25	s	s	NOUN
ejpam-3548	328	26	∩	∩	NOUN
ejpam-3548	328	27	rx(t	rx(t	X
ejpam-3548	328	28	)	)	PUNCT
ejpam-3548	328	29	.	.	PUNCT
ejpam-3548	329	1	lemma	lemma	PROPN
ejpam-3548	329	2	6	6	NUM
ejpam-3548	329	3	.	.	PUNCT
ejpam-3548	330	1	let	let	VERB
ejpam-3548	330	2	s	s	PRON
ejpam-3548	330	3	be	be	AUX
ejpam-3548	330	4	a	a	DET
ejpam-3548	330	5	subalgebra	subalgebra	NOUN
ejpam-3548	330	6	of	of	ADP
ejpam-3548	330	7	a	a	DET
ejpam-3548	330	8	transitive	transitive	ADJ
ejpam-3548	330	9	be	be	NOUN
ejpam-3548	330	10	-	-	PUNCT
ejpam-3548	330	11	algebra	algebra	NOUN
ejpam-3548	330	12	x.	x.	NOUN
ejpam-3548	330	13	then	then	ADV
ejpam-3548	330	14	for	for	ADP
ejpam-3548	330	15	any	any	DET
ejpam-3548	330	16	∅	∅	NOUN
ejpam-3548	330	17	6=	6=	ADP
ejpam-3548	330	18	a	a	DET
ejpam-3548	330	19	⊆	⊆	NUM
ejpam-3548	330	20	x	x	NOUN
ejpam-3548	330	21	,	,	PUNCT
ejpam-3548	330	22	rx(a	rx(a	NOUN
ejpam-3548	330	23	)	)	PUNCT
ejpam-3548	330	24	∩	∩	NOUN
ejpam-3548	330	25	s	s	PART
ejpam-3548	330	26	=	=	PUNCT
ejpam-3548	330	27	⋃	⋃	ADJ
ejpam-3548	330	28	x∈rx(a)∩s	x∈rx(a)∩s	NOUN
ejpam-3548	330	29	rs(x	rs(x	X
ejpam-3548	330	30	)	)	PUNCT
ejpam-3548	330	31	.	.	PUNCT
ejpam-3548	331	1	proof	proof	NOUN
ejpam-3548	331	2	.	.	PUNCT
ejpam-3548	332	1	suppose	suppose	VERB
ejpam-3548	332	2	that	that	SCONJ
ejpam-3548	332	3	∅	∅	NOUN
ejpam-3548	332	4	6=	6=	ADP
ejpam-3548	332	5	a	a	DET
ejpam-3548	332	6	⊆	⊆	NUM
ejpam-3548	332	7	x	x	PUNCT
ejpam-3548	332	8	and	and	CCONJ
ejpam-3548	332	9	let	let	VERB
ejpam-3548	332	10	x	x	X
ejpam-3548	332	11	∈	∈	PROPN
ejpam-3548	332	12	rx(a	rx(a	NOUN
ejpam-3548	332	13	)	)	PUNCT
ejpam-3548	332	14	∩	∩	PROPN
ejpam-3548	332	15	s.	s.	PROPN
ejpam-3548	332	16	then	then	ADV
ejpam-3548	332	17	a	a	DET
ejpam-3548	332	18	∗	∗	NOUN
ejpam-3548	332	19	x	x	X
ejpam-3548	332	20	=	=	SYM
ejpam-3548	332	21	1x	1x	NUM
ejpam-3548	332	22	for	for	ADP
ejpam-3548	332	23	all	all	DET
ejpam-3548	332	24	a	a	DET
ejpam-3548	332	25	∈	∈	PROPN
ejpam-3548	332	26	a	a	PRON
ejpam-3548	332	27	and	and	CCONJ
ejpam-3548	332	28	x	x	SYM
ejpam-3548	332	29	∈	∈	PROPN
ejpam-3548	332	30	s.	s.	PROPN
ejpam-3548	332	31	let	let	VERB
ejpam-3548	332	32	y	y	PROPN
ejpam-3548	332	33	∈	∈	PROPN
ejpam-3548	332	34	rx(x	rx(x	ADV
ejpam-3548	332	35	)	)	PUNCT
ejpam-3548	332	36	∩	∩	PROPN
ejpam-3548	332	37	s.	s.	PROPN
ejpam-3548	332	38	then	then	ADV
ejpam-3548	332	39	x	x	X
ejpam-3548	332	40	∗	∗	NOUN
ejpam-3548	332	41	y	y	NOUN
ejpam-3548	332	42	=	=	SYM
ejpam-3548	332	43	1x	1x	PROPN
ejpam-3548	332	44	and	and	CCONJ
ejpam-3548	332	45	y	y	PROPN
ejpam-3548	332	46	∈	∈	PROPN
ejpam-3548	332	47	s.	s.	PROPN
ejpam-3548	332	48	since	since	SCONJ
ejpam-3548	332	49	x	x	PRON
ejpam-3548	332	50	is	be	AUX
ejpam-3548	332	51	transitive	transitive	ADJ
ejpam-3548	332	52	,	,	PUNCT
ejpam-3548	332	53	a	a	DET
ejpam-3548	332	54	∗	∗	NOUN
ejpam-3548	332	55	y	y	NOUN
ejpam-3548	332	56	=	=	SYM
ejpam-3548	332	57	1x	1x	NUM
ejpam-3548	332	58	for	for	ADP
ejpam-3548	332	59	all	all	DET
ejpam-3548	332	60	a	a	DET
ejpam-3548	332	61	∈	∈	NOUN
ejpam-3548	332	62	a.	a.	NOUN
ejpam-3548	332	63	hence	hence	ADV
ejpam-3548	332	64	,	,	PUNCT
ejpam-3548	332	65	y	y	PROPN
ejpam-3548	332	66	∈	∈	PROPN
ejpam-3548	332	67	rx(a	rx(a	NOUN
ejpam-3548	332	68	)	)	PUNCT
ejpam-3548	332	69	∩	∩	NOUN
ejpam-3548	332	70	s	s	VERB
ejpam-3548	332	71	showing	show	VERB
ejpam-3548	332	72	that	that	SCONJ
ejpam-3548	332	73	rx(x	rx(x	ADV
ejpam-3548	332	74	)	)	PUNCT
ejpam-3548	332	75	∩	∩	NOUN
ejpam-3548	332	76	s	s	PART
ejpam-3548	332	77	⊆	⊆	NUM
ejpam-3548	332	78	rx(a	rx(a	NOUN
ejpam-3548	332	79	)	)	PUNCT
ejpam-3548	332	80	∩	∩	PROPN
ejpam-3548	332	81	s.	s.	PROPN
ejpam-3548	332	82	consequently	consequently	ADV
ejpam-3548	332	83	,	,	PUNCT
ejpam-3548	332	84	⋃	⋃	PROPN
ejpam-3548	332	85	x∈rx(a)∩s	x∈rx(a)∩s	NOUN
ejpam-3548	332	86	(	(	PUNCT
ejpam-3548	332	87	rx(x	rx(x	ADJ
ejpam-3548	332	88	)	)	PUNCT
ejpam-3548	332	89	∩	∩	NOUN
ejpam-3548	332	90	s	s	PART
ejpam-3548	332	91	)	)	PUNCT
ejpam-3548	332	92	⊆	⊆	NUM
ejpam-3548	332	93	rx(a	rx(a	NOUN
ejpam-3548	332	94	)	)	PUNCT
ejpam-3548	332	95	∩	∩	PROPN
ejpam-3548	332	96	s.	s.	PROPN
ejpam-3548	332	97	next	next	ADV
ejpam-3548	332	98	,	,	PUNCT
ejpam-3548	332	99	let	let	VERB
ejpam-3548	332	100	z	z	PROPN
ejpam-3548	332	101	∈	∈	PROPN
ejpam-3548	332	102	rx(a	rx(a	NOUN
ejpam-3548	332	103	)	)	PUNCT
ejpam-3548	332	104	∩	∩	PROPN
ejpam-3548	332	105	s.	s.	PROPN
ejpam-3548	332	106	clearly	clearly	ADV
ejpam-3548	332	107	,	,	PUNCT
ejpam-3548	332	108	z	z	PROPN
ejpam-3548	332	109	∈	∈	PROPN
ejpam-3548	332	110	rx(z	rx(z	NOUN
ejpam-3548	332	111	)	)	PUNCT
ejpam-3548	332	112	.	.	PUNCT
ejpam-3548	333	1	it	it	PRON
ejpam-3548	333	2	follows	follow	VERB
ejpam-3548	333	3	that	that	SCONJ
ejpam-3548	333	4	z	z	PROPN
ejpam-3548	333	5	∈	∈	PROPN
ejpam-3548	333	6	rx(z	rx(z	NOUN
ejpam-3548	333	7	)	)	PUNCT
ejpam-3548	333	8	∩	∩	NOUN
ejpam-3548	333	9	s	s	VERB
ejpam-3548	333	10	showing	show	VERB
ejpam-3548	333	11	that	that	SCONJ
ejpam-3548	333	12	rx(a	rx(a	NOUN
ejpam-3548	333	13	)	)	PUNCT
ejpam-3548	333	14	∩	∩	NOUN
ejpam-3548	333	15	s	s	PART
ejpam-3548	333	16	⊆	⊆	NUM
ejpam-3548	333	17	rx(z	rx(z	NUM
ejpam-3548	333	18	)	)	PUNCT
ejpam-3548	333	19	∩	∩	PROPN
ejpam-3548	333	20	s.	s.	PROPN
ejpam-3548	333	21	thus	thus	ADV
ejpam-3548	333	22	,	,	PUNCT
ejpam-3548	333	23	rx(a	rx(a	NOUN
ejpam-3548	333	24	)	)	PUNCT
ejpam-3548	333	25	∩	∩	NOUN
ejpam-3548	333	26	s	s	PART
ejpam-3548	333	27	⊆	⊆	NUM
ejpam-3548	333	28	⋃	⋃	PROPN
ejpam-3548	333	29	x∈rx(a)∩s	x∈rx(a)∩s	NOUN
ejpam-3548	333	30	(	(	PUNCT
ejpam-3548	333	31	rx(x	rx(x	ADJ
ejpam-3548	333	32	)	)	PUNCT
ejpam-3548	333	33	∩	∩	NOUN
ejpam-3548	333	34	s	s	PART
ejpam-3548	333	35	)	)	PUNCT
ejpam-3548	333	36	.	.	PUNCT
ejpam-3548	334	1	therefore	therefore	ADV
ejpam-3548	334	2	,	,	PUNCT
ejpam-3548	334	3	by	by	ADP
ejpam-3548	334	4	lemma	lemma	PROPN
ejpam-3548	334	5	5(ii	5(ii	NUM
ejpam-3548	334	6	)	)	PUNCT
ejpam-3548	334	7	,	,	PUNCT
ejpam-3548	334	8	rx(a	rx(a	NOUN
ejpam-3548	334	9	)	)	PUNCT
ejpam-3548	334	10	∩	∩	NOUN
ejpam-3548	334	11	s	s	PART
ejpam-3548	334	12	=	=	PUNCT
ejpam-3548	334	13	⋃	⋃	PROPN
ejpam-3548	334	14	x∈rx(a)∩s	x∈rx(a)∩s	NOUN
ejpam-3548	334	15	(	(	PUNCT
ejpam-3548	334	16	rx(x	rx(x	ADJ
ejpam-3548	334	17	)	)	PUNCT
ejpam-3548	334	18	∩	∩	NOUN
ejpam-3548	334	19	s	s	PART
ejpam-3548	334	20	)	)	PUNCT
ejpam-3548	334	21	=	=	SYM
ejpam-3548	334	22	⋃	⋃	NOUN
ejpam-3548	334	23	x∈rx(a)∩s	x∈rx(a)∩s	NOUN
ejpam-3548	334	24	rs(x	rs(x	X
ejpam-3548	334	25	)	)	PUNCT
ejpam-3548	334	26	.	.	PUNCT
ejpam-3548	335	1	theorem	theorem	NOUN
ejpam-3548	335	2	15	15	NUM
ejpam-3548	335	3	.	.	PUNCT
ejpam-3548	336	1	let	let	VERB
ejpam-3548	336	2	s	s	PRON
ejpam-3548	336	3	be	be	AUX
ejpam-3548	336	4	a	a	DET
ejpam-3548	336	5	subalgebra	subalgebra	NOUN
ejpam-3548	336	6	of	of	ADP
ejpam-3548	336	7	a	a	DET
ejpam-3548	336	8	transitive	transitive	ADJ
ejpam-3548	336	9	be	be	NOUN
ejpam-3548	336	10	-	-	PUNCT
ejpam-3548	336	11	algebra	algebra	NOUN
ejpam-3548	336	12	x	x	PUNCT
ejpam-3548	336	13	with	with	ADP
ejpam-3548	336	14	|s|	|s|	NOUN
ejpam-3548	336	15	≥	≥	PROPN
ejpam-3548	336	16	2	2	NUM
ejpam-3548	336	17	.	.	PUNCT
ejpam-3548	336	18	then	then	ADV
ejpam-3548	336	19	τr(s	τr(s	PUNCT
ejpam-3548	336	20	)	)	PUNCT
ejpam-3548	336	21	coincides	coincide	VERB
ejpam-3548	336	22	with	with	ADP
ejpam-3548	336	23	the	the	DET
ejpam-3548	336	24	relative	relative	ADJ
ejpam-3548	336	25	topology	topology	NOUN
ejpam-3548	336	26	τs	τs	ADV
ejpam-3548	336	27	on	on	ADP
ejpam-3548	336	28	s.	s.	PROPN
ejpam-3548	336	29	j.	j.	PROPN
ejpam-3548	336	30	albaracin	albaracin	PROPN
ejpam-3548	336	31	,	,	PUNCT
ejpam-3548	336	32	j.	j.	PROPN
ejpam-3548	336	33	vilela	vilela	PROPN
ejpam-3548	336	34	/	/	SYM
ejpam-3548	336	35	eur	eur	PROPN
ejpam-3548	336	36	.	.	PUNCT
ejpam-3548	337	1	j.	j.	PROPN
ejpam-3548	337	2	pure	pure	PROPN
ejpam-3548	337	3	appl	appl	PROPN
ejpam-3548	337	4	.	.	PROPN
ejpam-3548	337	5	math	math	PROPN
ejpam-3548	337	6	,	,	PUNCT
ejpam-3548	337	7	12	12	NUM
ejpam-3548	337	8	(	(	PUNCT
ejpam-3548	337	9	4	4	NUM
ejpam-3548	337	10	)	)	PUNCT
ejpam-3548	337	11	(	(	PUNCT
ejpam-3548	337	12	2019	2019	NUM
ejpam-3548	337	13	)	)	PUNCT
ejpam-3548	337	14	,	,	PUNCT
ejpam-3548	337	15	1584	1584	NUM
ejpam-3548	337	16	-	-	SYM
ejpam-3548	337	17	1594	1594	NUM
ejpam-3548	337	18	1592	1592	NUM
ejpam-3548	337	19	proof	proof	NOUN
ejpam-3548	337	20	.	.	PUNCT
ejpam-3548	338	1	by	by	ADP
ejpam-3548	338	2	theorem	theorem	NOUN
ejpam-3548	338	3	11	11	NUM
ejpam-3548	338	4	,	,	PUNCT
ejpam-3548	338	5	a	a	DET
ejpam-3548	338	6	basis	basis	NOUN
ejpam-3548	338	7	for	for	ADP
ejpam-3548	338	8	τr(s	τr(	NOUN
ejpam-3548	338	9	)	)	PUNCT
ejpam-3548	338	10	is	be	AUX
ejpam-3548	338	11	the	the	DET
ejpam-3548	338	12	family	family	NOUN
ejpam-3548	338	13	br(s	br(s	PUNCT
ejpam-3548	338	14	)	)	PUNCT
ejpam-3548	339	1	=	=	PRON
ejpam-3548	339	2	{	{	PUNCT
ejpam-3548	339	3	{	{	PUNCT
ejpam-3548	339	4	1x	1x	NUM
ejpam-3548	339	5	}	}	PUNCT
ejpam-3548	339	6	}	}	PUNCT
ejpam-3548	339	7	∪	∪	X
ejpam-3548	339	8	{	{	PUNCT
ejpam-3548	339	9	{	{	PUNCT
ejpam-3548	339	10	1x	1x	NUM
ejpam-3548	339	11	,	,	PUNCT
ejpam-3548	339	12	a	a	PRON
ejpam-3548	339	13	}	}	PUNCT
ejpam-3548	339	14	:	:	PUNCT
ejpam-3548	339	15	a	a	DET
ejpam-3548	339	16	∈	∈	PROPN
ejpam-3548	339	17	a(s	a(s	PROPN
ejpam-3548	339	18	)	)	PUNCT
ejpam-3548	339	19	}	}	PUNCT
ejpam-3548	339	20	∪	∪	ADJ
ejpam-3548	339	21	{	{	PUNCT
ejpam-3548	339	22	rs(a	rs(a	NOUN
ejpam-3548	339	23	)	)	PUNCT
ejpam-3548	339	24	:	:	PUNCT
ejpam-3548	339	25	a	a	DET
ejpam-3548	339	26	⊆	⊆	NUM
ejpam-3548	339	27	s	s	NOUN
ejpam-3548	339	28	and	and	CCONJ
ejpam-3548	339	29	a	a	DET
ejpam-3548	339	30	∩	∩	NOUN
ejpam-3548	339	31	a(s	a(	NOUN
ejpam-3548	339	32	)	)	PUNCT
ejpam-3548	339	33	=	=	NOUN
ejpam-3548	339	34	∅	∅	NOUN
ejpam-3548	339	35	}	}	PUNCT
ejpam-3548	339	36	.	.	PUNCT
ejpam-3548	340	1	by	by	ADP
ejpam-3548	340	2	theorem	theorem	NOUN
ejpam-3548	340	3	3	3	NUM
ejpam-3548	340	4	,	,	PUNCT
ejpam-3548	340	5	a	a	DET
ejpam-3548	340	6	basis	basis	NOUN
ejpam-3548	340	7	for	for	ADP
ejpam-3548	340	8	the	the	DET
ejpam-3548	340	9	relative	relative	ADJ
ejpam-3548	340	10	topology	topology	NOUN
ejpam-3548	340	11	τs	τs	ADP
ejpam-3548	340	12	on	on	ADP
ejpam-3548	340	13	s	s	PROPN
ejpam-3548	340	14	is	be	AUX
ejpam-3548	340	15	given	give	VERB
ejpam-3548	340	16	by	by	ADP
ejpam-3548	340	17	bs	bs	NOUN
ejpam-3548	340	18	=	=	PUNCT
ejpam-3548	340	19	{	{	PUNCT
ejpam-3548	340	20	{	{	PUNCT
ejpam-3548	340	21	1x	1x	NUM
ejpam-3548	340	22	}	}	PUNCT
ejpam-3548	340	23	}	}	PUNCT
ejpam-3548	340	24	∪	∪	X
ejpam-3548	340	25	{	{	PUNCT
ejpam-3548	340	26	{	{	PUNCT
ejpam-3548	340	27	1x	1x	NUM
ejpam-3548	340	28	,	,	PUNCT
ejpam-3548	340	29	a	a	PRON
ejpam-3548	340	30	}	}	PUNCT
ejpam-3548	340	31	:	:	PUNCT
ejpam-3548	340	32	a	a	DET
ejpam-3548	340	33	∈	∈	PROPN
ejpam-3548	340	34	s	s	PART
ejpam-3548	340	35	∩	∩	NOUN
ejpam-3548	340	36	a(x	a(x	NOUN
ejpam-3548	340	37	)	)	PUNCT
ejpam-3548	340	38	}	}	PUNCT
ejpam-3548	340	39	∪	∪	X
ejpam-3548	340	40	{	{	PUNCT
ejpam-3548	340	41	rx(a	rx(a	NOUN
ejpam-3548	340	42	)	)	PUNCT
ejpam-3548	340	43	∩	∩	NOUN
ejpam-3548	340	44	s	s	PART
ejpam-3548	340	45	:	:	PUNCT
ejpam-3548	340	46	a	a	DET
ejpam-3548	340	47	⊆	⊆	NUM
ejpam-3548	340	48	s	s	NOUN
ejpam-3548	340	49	and	and	CCONJ
ejpam-3548	340	50	a	a	DET
ejpam-3548	340	51	∩	∩	ADJ
ejpam-3548	340	52	a(x	a(x	NOUN
ejpam-3548	340	53	)	)	PUNCT
ejpam-3548	340	54	=	=	NOUN
ejpam-3548	340	55	∅	∅	NOUN
ejpam-3548	340	56	}	}	PUNCT
ejpam-3548	340	57	.	.	PUNCT
ejpam-3548	341	1	suppose	suppose	VERB
ejpam-3548	341	2	that	that	SCONJ
ejpam-3548	341	3	a(s	a(	NOUN
ejpam-3548	341	4	)	)	PUNCT
ejpam-3548	341	5	\	\	PROPN
ejpam-3548	341	6	a(x	a(x	PROPN
ejpam-3548	341	7	)	)	PUNCT
ejpam-3548	341	8	=	=	PUNCT
ejpam-3548	341	9	∅.	∅.	NOUN
ejpam-3548	341	10	then	then	ADV
ejpam-3548	341	11	a(s	a(	NOUN
ejpam-3548	341	12	)	)	PUNCT
ejpam-3548	341	13	⊆	⊆	NUM
ejpam-3548	341	14	a(x	a(x	NOUN
ejpam-3548	341	15	)	)	PUNCT
ejpam-3548	341	16	.	.	PUNCT
ejpam-3548	342	1	since	since	SCONJ
ejpam-3548	342	2	a(s	a(s	PROPN
ejpam-3548	342	3	)	)	PUNCT
ejpam-3548	342	4	⊆	⊆	NUM
ejpam-3548	342	5	s	s	NOUN
ejpam-3548	342	6	,	,	PUNCT
ejpam-3548	342	7	for	for	ADP
ejpam-3548	342	8	every	every	DET
ejpam-3548	342	9	a	a	DET
ejpam-3548	342	10	∈	∈	PROPN
ejpam-3548	342	11	a(s	a(s	PROPN
ejpam-3548	342	12	)	)	PUNCT
ejpam-3548	342	13	,	,	PUNCT
ejpam-3548	342	14	we	we	PRON
ejpam-3548	342	15	have	have	VERB
ejpam-3548	342	16	a	a	DET
ejpam-3548	342	17	∈	∈	NOUN
ejpam-3548	342	18	s	s	PART
ejpam-3548	342	19	∩	∩	NOUN
ejpam-3548	342	20	a(x	a(x	NOUN
ejpam-3548	342	21	)	)	PUNCT
ejpam-3548	342	22	.	.	PUNCT
ejpam-3548	343	1	thus	thus	ADV
ejpam-3548	343	2	,	,	PUNCT
ejpam-3548	343	3	{	{	PUNCT
ejpam-3548	343	4	1x	1x	NUM
ejpam-3548	343	5	,	,	PUNCT
ejpam-3548	343	6	a	a	DET
ejpam-3548	343	7	}	}	PUNCT
ejpam-3548	343	8	∈	∈	NOUN
ejpam-3548	343	9	bs	bs	NOUN
ejpam-3548	343	10	.	.	PUNCT
ejpam-3548	344	1	now	now	ADV
ejpam-3548	344	2	,	,	PUNCT
ejpam-3548	344	3	suppose	suppose	VERB
ejpam-3548	344	4	that	that	SCONJ
ejpam-3548	344	5	a(s	a(	NOUN
ejpam-3548	344	6	)	)	PUNCT
ejpam-3548	344	7	\	\	PROPN
ejpam-3548	344	8	a(x	a(x	PROPN
ejpam-3548	344	9	)	)	PUNCT
ejpam-3548	344	10	6=	6=	PUNCT
ejpam-3548	344	11	∅.	∅.	NOUN
ejpam-3548	344	12	let	let	VERB
ejpam-3548	344	13	a	a	DET
ejpam-3548	344	14	∈	∈	PROPN
ejpam-3548	344	15	a(s	a(s	PROPN
ejpam-3548	344	16	)	)	PUNCT
ejpam-3548	344	17	\	\	PROPN
ejpam-3548	344	18	a(x	a(x	NOUN
ejpam-3548	344	19	)	)	PUNCT
ejpam-3548	344	20	such	such	ADJ
ejpam-3548	344	21	that	that	SCONJ
ejpam-3548	344	22	{	{	PUNCT
ejpam-3548	344	23	1x	1x	NUM
ejpam-3548	344	24	,	,	PUNCT
ejpam-3548	344	25	a	a	DET
ejpam-3548	344	26	}	}	PUNCT
ejpam-3548	344	27	∈	∈	PROPN
ejpam-3548	344	28	br(s	br(s	NOUN
ejpam-3548	344	29	)	)	PUNCT
ejpam-3548	344	30	.	.	PUNCT
ejpam-3548	345	1	then	then	ADV
ejpam-3548	345	2	{	{	PUNCT
ejpam-3548	345	3	a	a	NOUN
ejpam-3548	345	4	}	}	PUNCT
ejpam-3548	345	5	⊆	⊆	NUM
ejpam-3548	345	6	s	s	NOUN
ejpam-3548	345	7	and	and	CCONJ
ejpam-3548	345	8	{	{	PUNCT
ejpam-3548	345	9	a	a	PRON
ejpam-3548	345	10	}	}	PUNCT
ejpam-3548	345	11	∩	∩	NOUN
ejpam-3548	345	12	a(x	a(x	NOUN
ejpam-3548	345	13	)	)	PUNCT
ejpam-3548	345	14	=	=	PUNCT
ejpam-3548	345	15	∅.	∅.	X
ejpam-3548	345	16	by	by	ADP
ejpam-3548	345	17	lemma	lemma	PROPN
ejpam-3548	345	18	5(ii	5(ii	NUM
ejpam-3548	345	19	)	)	PUNCT
ejpam-3548	345	20	,	,	PUNCT
ejpam-3548	345	21	{	{	PUNCT
ejpam-3548	345	22	1x	1x	NUM
ejpam-3548	345	23	,	,	PUNCT
ejpam-3548	345	24	a	a	PRON
ejpam-3548	345	25	}	}	PUNCT
ejpam-3548	345	26	=	=	PUNCT
ejpam-3548	345	27	rs({a	rs({a	ADJ
ejpam-3548	345	28	}	}	PUNCT
ejpam-3548	345	29	)	)	PUNCT
ejpam-3548	345	30	=	=	SYM
ejpam-3548	345	31	rx({a	rx({a	NOUN
ejpam-3548	345	32	}	}	PUNCT
ejpam-3548	345	33	)	)	PUNCT
ejpam-3548	345	34	∩	∩	PROPN
ejpam-3548	345	35	s	s	PART
ejpam-3548	345	36	∈	∈	NOUN
ejpam-3548	345	37	bs	bs	NOUN
ejpam-3548	345	38	.	.	PUNCT
ejpam-3548	346	1	next	next	ADV
ejpam-3548	346	2	,	,	PUNCT
ejpam-3548	346	3	let	let	VERB
ejpam-3548	346	4	∅	∅	NOUN
ejpam-3548	346	5	6=	6=	ADP
ejpam-3548	346	6	a	a	DET
ejpam-3548	346	7	⊆	⊆	NUM
ejpam-3548	346	8	s	s	NOUN
ejpam-3548	346	9	such	such	ADJ
ejpam-3548	346	10	that	that	SCONJ
ejpam-3548	346	11	a	a	DET
ejpam-3548	346	12	∩	∩	NOUN
ejpam-3548	346	13	a(s	a(	NOUN
ejpam-3548	346	14	)	)	PUNCT
ejpam-3548	346	15	=	=	NOUN
ejpam-3548	346	16	∅.	∅.	NOUN
ejpam-3548	346	17	then	then	ADV
ejpam-3548	346	18	rs(a	rs(a	NOUN
ejpam-3548	346	19	)	)	PUNCT
ejpam-3548	346	20	∈	∈	PROPN
ejpam-3548	346	21	br(s	br(s	PROPN
ejpam-3548	346	22	)	)	PUNCT
ejpam-3548	346	23	.	.	PUNCT
ejpam-3548	347	1	since	since	SCONJ
ejpam-3548	347	2	by	by	ADP
ejpam-3548	347	3	lemma	lemma	PROPN
ejpam-3548	347	4	5(i	5(i	NUM
ejpam-3548	347	5	)	)	PUNCT
ejpam-3548	347	6	,	,	PUNCT
ejpam-3548	347	7	a	a	DET
ejpam-3548	347	8	∩	∩	ADJ
ejpam-3548	347	9	a(x	a(x	NOUN
ejpam-3548	347	10	)	)	PUNCT
ejpam-3548	347	11	=	=	NOUN
ejpam-3548	347	12	(	(	PUNCT
ejpam-3548	347	13	a	a	DET
ejpam-3548	347	14	∩	∩	ADJ
ejpam-3548	347	15	s	s	NOUN
ejpam-3548	347	16	)	)	PUNCT
ejpam-3548	347	17	∩	∩	NOUN
ejpam-3548	347	18	a(x	a(x	NOUN
ejpam-3548	347	19	)	)	PUNCT
ejpam-3548	347	20	=	=	NOUN
ejpam-3548	347	21	a	a	DET
ejpam-3548	347	22	∩	∩	NOUN
ejpam-3548	347	23	(	(	PUNCT
ejpam-3548	347	24	s	s	X
ejpam-3548	347	25	∩	∩	NOUN
ejpam-3548	347	26	a(x	a(x	NOUN
ejpam-3548	347	27	)	)	PUNCT
ejpam-3548	347	28	)	)	PUNCT
ejpam-3548	348	1	⊆	⊆	NUM
ejpam-3548	348	2	a	a	DET
ejpam-3548	348	3	∩	∩	NOUN
ejpam-3548	348	4	a(s	a(	NOUN
ejpam-3548	348	5	)	)	PUNCT
ejpam-3548	348	6	=	=	PUNCT
ejpam-3548	348	7	∅.	∅.	ADP
ejpam-3548	348	8	this	this	PRON
ejpam-3548	348	9	implies	imply	VERB
ejpam-3548	348	10	that	that	SCONJ
ejpam-3548	348	11	by	by	ADP
ejpam-3548	348	12	lemma	lemma	PROPN
ejpam-3548	348	13	5(ii	5(ii	NUM
ejpam-3548	348	14	)	)	PUNCT
ejpam-3548	348	15	,	,	PUNCT
ejpam-3548	348	16	rs(a	rs(a	NOUN
ejpam-3548	348	17	)	)	PUNCT
ejpam-3548	348	18	=	=	SYM
ejpam-3548	348	19	rx(a	rx(a	X
ejpam-3548	348	20	)	)	PUNCT
ejpam-3548	348	21	∩	∩	NOUN
ejpam-3548	348	22	s	s	PART
ejpam-3548	348	23	∈	∈	PROPN
ejpam-3548	348	24	bs	bs	NOUN
ejpam-3548	348	25	.	.	PUNCT
ejpam-3548	349	1	thus	thus	ADV
ejpam-3548	349	2	,	,	PUNCT
ejpam-3548	349	3	br(s	br(s	PROPN
ejpam-3548	349	4	)	)	PUNCT
ejpam-3548	349	5	⊆	⊆	NUM
ejpam-3548	349	6	bs	bs	NOUN
ejpam-3548	349	7	.	.	PUNCT
ejpam-3548	350	1	by	by	ADP
ejpam-3548	350	2	lemma	lemma	PROPN
ejpam-3548	350	3	5(i	5(i	NUM
ejpam-3548	350	4	)	)	PUNCT
ejpam-3548	350	5	,	,	PUNCT
ejpam-3548	350	6	{	{	PUNCT
ejpam-3548	350	7	{	{	PUNCT
ejpam-3548	350	8	1x}}∪{{1x	1x}}∪{{1x	NUM
ejpam-3548	350	9	,	,	PUNCT
ejpam-3548	350	10	a	a	PROPN
ejpam-3548	350	11	}	}	PUNCT
ejpam-3548	350	12	:	:	PUNCT
ejpam-3548	350	13	a	a	DET
ejpam-3548	350	14	∈	∈	PROPN
ejpam-3548	350	15	s∩a(x	s∩a(x	NOUN
ejpam-3548	350	16	)	)	PUNCT
ejpam-3548	350	17	}	}	PUNCT
ejpam-3548	350	18	⊆	⊆	NUM
ejpam-3548	350	19	{	{	PUNCT
ejpam-3548	350	20	{	{	PUNCT
ejpam-3548	350	21	1x}}∪{{1x	1x}}∪{{1x	NUM
ejpam-3548	350	22	,	,	PUNCT
ejpam-3548	350	23	a	a	PROPN
ejpam-3548	350	24	}	}	PUNCT
ejpam-3548	350	25	:	:	PUNCT
ejpam-3548	350	26	a	a	DET
ejpam-3548	350	27	∈	∈	PROPN
ejpam-3548	350	28	a(s	a(s	PROPN
ejpam-3548	350	29	)	)	PUNCT
ejpam-3548	350	30	}	}	PUNCT
ejpam-3548	350	31	.	.	PUNCT
ejpam-3548	351	1	let	let	VERB
ejpam-3548	351	2	∅	∅	NOUN
ejpam-3548	351	3	6=	6=	ADP
ejpam-3548	351	4	a	a	DET
ejpam-3548	351	5	⊆	⊆	NUM
ejpam-3548	351	6	s	s	NOUN
ejpam-3548	351	7	such	such	ADJ
ejpam-3548	351	8	that	that	SCONJ
ejpam-3548	351	9	a	a	DET
ejpam-3548	351	10	∩a(x	∩a(x	PROPN
ejpam-3548	351	11	)	)	PUNCT
ejpam-3548	351	12	=	=	PUNCT
ejpam-3548	351	13	∅.	∅.	NOUN
ejpam-3548	351	14	if	if	SCONJ
ejpam-3548	351	15	a	a	DET
ejpam-3548	351	16	∩a(s	∩a(s	PROPN
ejpam-3548	351	17	)	)	PUNCT
ejpam-3548	351	18	=	=	VERB
ejpam-3548	351	19	∅	∅	NOUN
ejpam-3548	351	20	,	,	PUNCT
ejpam-3548	351	21	then	then	ADV
ejpam-3548	351	22	rs(a	rs(a	VERB
ejpam-3548	351	23	)	)	PUNCT
ejpam-3548	351	24	∈	∈	PROPN
ejpam-3548	351	25	br(s	br(s	PROPN
ejpam-3548	351	26	)	)	PUNCT
ejpam-3548	351	27	.	.	PUNCT
ejpam-3548	352	1	suppose	suppose	VERB
ejpam-3548	352	2	that	that	SCONJ
ejpam-3548	352	3	a	a	DET
ejpam-3548	352	4	∩a(s	∩a(s	PROPN
ejpam-3548	352	5	)	)	PUNCT
ejpam-3548	352	6	6=	6=	ADP
ejpam-3548	352	7	∅	∅	NOUN
ejpam-3548	352	8	,	,	PUNCT
ejpam-3548	352	9	say	say	VERB
ejpam-3548	352	10	w	w	PROPN
ejpam-3548	352	11	∈	∈	PROPN
ejpam-3548	352	12	a	a	DET
ejpam-3548	352	13	∩	∩	NOUN
ejpam-3548	352	14	a(s	a(	NOUN
ejpam-3548	352	15	)	)	PUNCT
ejpam-3548	352	16	.	.	PUNCT
ejpam-3548	353	1	by	by	ADP
ejpam-3548	353	2	theorem	theorem	NOUN
ejpam-3548	353	3	5(ii	5(ii	NUM
ejpam-3548	353	4	)	)	PUNCT
ejpam-3548	353	5	,	,	PUNCT
ejpam-3548	353	6	rs(a	rs(a	NOUN
ejpam-3548	353	7	)	)	PUNCT
ejpam-3548	353	8	⊆	⊆	NUM
ejpam-3548	353	9	rs(w	rs(w	NUM
ejpam-3548	353	10	)	)	PUNCT
ejpam-3548	353	11	=	=	SYM
ejpam-3548	353	12	{	{	PUNCT
ejpam-3548	353	13	1x	1x	NOUN
ejpam-3548	353	14	,	,	PUNCT
ejpam-3548	353	15	w	w	NOUN
ejpam-3548	353	16	}	}	PUNCT
ejpam-3548	353	17	.	.	PUNCT
ejpam-3548	354	1	hence	hence	ADV
ejpam-3548	354	2	,	,	PUNCT
ejpam-3548	354	3	rs(a	rs(a	NOUN
ejpam-3548	354	4	)	)	PUNCT
ejpam-3548	354	5	=	=	SYM
ejpam-3548	354	6	{	{	PUNCT
ejpam-3548	354	7	1x	1x	NUM
ejpam-3548	354	8	}	}	PUNCT
ejpam-3548	354	9	or	or	CCONJ
ejpam-3548	354	10	rs(a	rs(a	X
ejpam-3548	354	11	)	)	PUNCT
ejpam-3548	354	12	=	=	SYM
ejpam-3548	354	13	{	{	PUNCT
ejpam-3548	354	14	1x	1x	NOUN
ejpam-3548	354	15	,	,	PUNCT
ejpam-3548	354	16	w	w	NOUN
ejpam-3548	354	17	}	}	PUNCT
ejpam-3548	354	18	.	.	PUNCT
ejpam-3548	355	1	thus	thus	ADV
ejpam-3548	355	2	,	,	PUNCT
ejpam-3548	355	3	rs(a	rs(a	X
ejpam-3548	355	4	)	)	PUNCT
ejpam-3548	355	5	∈	∈	NOUN
ejpam-3548	355	6	{	{	PUNCT
ejpam-3548	355	7	{	{	PUNCT
ejpam-3548	355	8	1x	1x	NUM
ejpam-3548	355	9	}	}	PUNCT
ejpam-3548	355	10	}	}	PUNCT
ejpam-3548	355	11	∪	∪	X
ejpam-3548	355	12	{	{	PUNCT
ejpam-3548	355	13	{	{	PUNCT
ejpam-3548	355	14	1x	1x	NUM
ejpam-3548	355	15	,	,	PUNCT
ejpam-3548	355	16	a	a	PRON
ejpam-3548	355	17	}	}	PUNCT
ejpam-3548	355	18	:	:	PUNCT
ejpam-3548	355	19	a	a	DET
ejpam-3548	355	20	∈	∈	PROPN
ejpam-3548	355	21	a(s	a(s	PROPN
ejpam-3548	355	22	)	)	PUNCT
ejpam-3548	355	23	}	}	PUNCT
ejpam-3548	355	24	⊆	⊆	NUM
ejpam-3548	355	25	br(s	br(s	NUM
ejpam-3548	355	26	)	)	PUNCT
ejpam-3548	355	27	.	.	PUNCT
ejpam-3548	356	1	therefore	therefore	ADV
ejpam-3548	356	2	,	,	PUNCT
ejpam-3548	356	3	bs	bs	PROPN
ejpam-3548	356	4	⊆	⊆	NUM
ejpam-3548	356	5	br(s	br(s	NUM
ejpam-3548	356	6	)	)	PUNCT
ejpam-3548	356	7	.	.	PUNCT
ejpam-3548	357	1	consequently	consequently	ADV
ejpam-3548	357	2	,	,	PUNCT
ejpam-3548	357	3	br(s	br(s	PROPN
ejpam-3548	357	4	)	)	PUNCT
ejpam-3548	357	5	=	=	SYM
ejpam-3548	357	6	bs	b	NOUN
ejpam-3548	357	7	,	,	PUNCT
ejpam-3548	357	8	showing	show	VERB
ejpam-3548	357	9	that	that	PRON
ejpam-3548	357	10	τr(s	τr(s	PUNCT
ejpam-3548	357	11	)	)	PUNCT
ejpam-3548	357	12	=	=	SYM
ejpam-3548	357	13	τs	τ	NOUN
ejpam-3548	357	14	.	.	PUNCT
ejpam-3548	358	1	theorem	theorem	VERB
ejpam-3548	358	2	16	16	NUM
ejpam-3548	358	3	.	.	PUNCT
ejpam-3548	359	1	let	let	VERB
ejpam-3548	359	2	(	(	PUNCT
ejpam-3548	359	3	x1	x1	ADJ
ejpam-3548	359	4	,	,	PUNCT
ejpam-3548	359	5	∗x1	∗x1	X
ejpam-3548	359	6	,	,	PUNCT
ejpam-3548	359	7	1x1	1x1	NUM
ejpam-3548	359	8	)	)	PUNCT
ejpam-3548	359	9	and	and	CCONJ
ejpam-3548	359	10	(	(	PUNCT
ejpam-3548	359	11	x2	x2	PROPN
ejpam-3548	359	12	,	,	PUNCT
ejpam-3548	359	13	∗x2	∗x2	PROPN
ejpam-3548	359	14	,	,	PUNCT
ejpam-3548	359	15	1x2	1x2	NUM
ejpam-3548	359	16	)	)	PUNCT
ejpam-3548	359	17	be	be	AUX
ejpam-3548	359	18	be	be	NOUN
ejpam-3548	359	19	-	-	PUNCT
ejpam-3548	359	20	algebras	algebras	X
ejpam-3548	359	21	.	.	PUNCT
ejpam-3548	360	1	then	then	ADV
ejpam-3548	360	2	a	a	DET
ejpam-3548	360	3	function	function	NOUN
ejpam-3548	360	4	f	f	NOUN
ejpam-3548	360	5	:	:	PUNCT
ejpam-3548	360	6	(	(	PUNCT
ejpam-3548	360	7	x1	x1	ADJ
ejpam-3548	360	8	,	,	PUNCT
ejpam-3548	360	9	τr(x1	τr(x1	NOUN
ejpam-3548	360	10	)	)	PUNCT
ejpam-3548	360	11	)	)	PUNCT
ejpam-3548	360	12	→	→	PUNCT
ejpam-3548	360	13	(	(	PUNCT
ejpam-3548	360	14	x2	x2	PROPN
ejpam-3548	360	15	,	,	PUNCT
ejpam-3548	360	16	τr(x2	τr(x2	NOUN
ejpam-3548	360	17	)	)	PUNCT
ejpam-3548	360	18	)	)	PUNCT
ejpam-3548	360	19	is	be	AUX
ejpam-3548	360	20	continuous	continuous	ADJ
ejpam-3548	360	21	on	on	ADP
ejpam-3548	360	22	x1	x1	PROPN
ejpam-3548	360	23	if	if	SCONJ
ejpam-3548	360	24	and	and	CCONJ
ejpam-3548	360	25	only	only	ADV
ejpam-3548	360	26	if	if	SCONJ
ejpam-3548	360	27	for	for	ADP
ejpam-3548	360	28	each	each	DET
ejpam-3548	360	29	b	b	PROPN
ejpam-3548	360	30	⊆	⊆	NUM
ejpam-3548	360	31	x2	x2	PROPN
ejpam-3548	360	32	and	and	CCONJ
ejpam-3548	360	33	for	for	ADP
ejpam-3548	360	34	each	each	DET
ejpam-3548	360	35	x	x	SYM
ejpam-3548	360	36	∈	∈	PROPN
ejpam-3548	360	37	x1	x1	NUM
ejpam-3548	360	38	such	such	ADJ
ejpam-3548	360	39	that	that	DET
ejpam-3548	360	40	b	b	PROPN
ejpam-3548	360	41	≤	≤	PROPN
ejpam-3548	360	42	f(x	f(x	PROPN
ejpam-3548	360	43	)	)	PUNCT
ejpam-3548	360	44	for	for	ADP
ejpam-3548	360	45	all	all	DET
ejpam-3548	360	46	b	b	PROPN
ejpam-3548	360	47	∈	∈	PROPN
ejpam-3548	360	48	b	b	NOUN
ejpam-3548	360	49	,	,	PUNCT
ejpam-3548	360	50	there	there	PRON
ejpam-3548	360	51	exists	exist	VERB
ejpam-3548	360	52	a	a	DET
ejpam-3548	360	53	⊆	⊆	NUM
ejpam-3548	360	54	x1	x1	NUM
ejpam-3548	360	55	satisfying	satisfy	VERB
ejpam-3548	360	56	the	the	DET
ejpam-3548	360	57	following	follow	VERB
ejpam-3548	360	58	conditions	condition	NOUN
ejpam-3548	360	59	:	:	PUNCT
ejpam-3548	360	60	(	(	PUNCT
ejpam-3548	360	61	i	i	NOUN
ejpam-3548	360	62	)	)	PUNCT
ejpam-3548	360	63	a	a	DET
ejpam-3548	360	64	≤	≤	NOUN
ejpam-3548	360	65	x	x	PUNCT
ejpam-3548	360	66	for	for	ADP
ejpam-3548	360	67	all	all	DET
ejpam-3548	360	68	a	a	DET
ejpam-3548	360	69	∈	∈	PROPN
ejpam-3548	360	70	a	a	DET
ejpam-3548	360	71	(	(	PUNCT
ejpam-3548	360	72	ii	ii	NOUN
ejpam-3548	360	73	)	)	PUNCT
ejpam-3548	360	74	b	b	PROPN
ejpam-3548	360	75	≤	≤	NUM
ejpam-3548	360	76	f(z	f(z	PROPN
ejpam-3548	360	77	)	)	PUNCT
ejpam-3548	360	78	for	for	ADP
ejpam-3548	360	79	all	all	DET
ejpam-3548	360	80	b	b	PROPN
ejpam-3548	360	81	∈	∈	ADP
ejpam-3548	360	82	b	b	NOUN
ejpam-3548	360	83	whenever	whenever	SCONJ
ejpam-3548	360	84	a	a	DET
ejpam-3548	360	85	≤	≤	ADJ
ejpam-3548	360	86	z	z	NOUN
ejpam-3548	360	87	for	for	ADP
ejpam-3548	360	88	all	all	DET
ejpam-3548	360	89	a	a	DET
ejpam-3548	360	90	∈	∈	NOUN
ejpam-3548	360	91	a.	a.	NOUN
ejpam-3548	360	92	proof	proof	NOUN
ejpam-3548	360	93	.	.	PUNCT
ejpam-3548	361	1	by	by	ADP
ejpam-3548	361	2	theorem	theorem	NOUN
ejpam-3548	361	3	4	4	NUM
ejpam-3548	361	4	,	,	PUNCT
ejpam-3548	361	5	f	f	PROPN
ejpam-3548	361	6	is	be	AUX
ejpam-3548	361	7	continuous	continuous	ADJ
ejpam-3548	361	8	on	on	ADP
ejpam-3548	361	9	x1	x1	PROPN
ejpam-3548	361	10	if	if	SCONJ
ejpam-3548	361	11	and	and	CCONJ
ejpam-3548	361	12	only	only	ADV
ejpam-3548	361	13	if	if	SCONJ
ejpam-3548	361	14	f−1(g	f−1(g	NUM
ejpam-3548	361	15	)	)	PUNCT
ejpam-3548	361	16	∈	∈	PROPN
ejpam-3548	361	17	τr(x1	τr(x1	NOUN
ejpam-3548	361	18	)	)	PUNCT
ejpam-3548	361	19	for	for	ADP
ejpam-3548	361	20	each	each	DET
ejpam-3548	361	21	g	g	PROPN
ejpam-3548	361	22	∈	∈	PROPN
ejpam-3548	361	23	br(x2	br(x2	NOUN
ejpam-3548	361	24	)	)	PUNCT
ejpam-3548	361	25	.	.	PUNCT
ejpam-3548	362	1	by	by	ADP
ejpam-3548	362	2	theorem	theorem	NOUN
ejpam-3548	362	3	8	8	NUM
ejpam-3548	362	4	,	,	PUNCT
ejpam-3548	362	5	f	f	PROPN
ejpam-3548	362	6	is	be	AUX
ejpam-3548	362	7	continuous	continuous	ADJ
ejpam-3548	362	8	if	if	SCONJ
ejpam-3548	362	9	and	and	CCONJ
ejpam-3548	362	10	only	only	ADV
ejpam-3548	362	11	if	if	SCONJ
ejpam-3548	362	12	for	for	ADP
ejpam-3548	362	13	each	each	DET
ejpam-3548	362	14	b	b	NOUN
ejpam-3548	362	15	⊆	⊆	NUM
ejpam-3548	362	16	x2	x2	NOUN
ejpam-3548	362	17	,	,	PUNCT
ejpam-3548	362	18	f−1	f−1	PROPN
ejpam-3548	362	19	(	(	PUNCT
ejpam-3548	362	20	rx2(b	rx2(b	NOUN
ejpam-3548	362	21	)	)	PUNCT
ejpam-3548	362	22	)	)	PUNCT
ejpam-3548	362	23	∈	∈	PROPN
ejpam-3548	362	24	τr(x1	τr(x1	NOUN
ejpam-3548	362	25	)	)	PUNCT
ejpam-3548	362	26	,	,	PUNCT
ejpam-3548	362	27	that	that	ADV
ejpam-3548	362	28	is	is	ADV
ejpam-3548	362	29	,	,	PUNCT
ejpam-3548	362	30	b	b	PROPN
ejpam-3548	362	31	≤	≤	NUM
ejpam-3548	362	32	f(x	f(x	PROPN
ejpam-3548	362	33	)	)	PUNCT
ejpam-3548	362	34	for	for	ADP
ejpam-3548	362	35	all	all	DET
ejpam-3548	362	36	b	b	PROPN
ejpam-3548	362	37	∈	∈	PROPN
ejpam-3548	362	38	b.	b.	NOUN
ejpam-3548	362	39	now	now	ADV
ejpam-3548	362	40	,	,	PUNCT
ejpam-3548	362	41	f−1	f−1	PROPN
ejpam-3548	362	42	(	(	PUNCT
ejpam-3548	362	43	rx2(b	rx2(b	NOUN
ejpam-3548	362	44	)	)	PUNCT
ejpam-3548	362	45	)	)	PUNCT
ejpam-3548	362	46	∈	∈	PROPN
ejpam-3548	362	47	τr(x1	τr(x1	NOUN
ejpam-3548	362	48	)	)	PUNCT
ejpam-3548	362	49	if	if	SCONJ
ejpam-3548	362	50	and	and	CCONJ
ejpam-3548	362	51	only	only	ADV
ejpam-3548	362	52	if	if	SCONJ
ejpam-3548	362	53	for	for	ADP
ejpam-3548	362	54	each	each	DET
ejpam-3548	362	55	x	x	SYM
ejpam-3548	362	56	∈	∈	PROPN
ejpam-3548	362	57	f−1	f−1	PROPN
ejpam-3548	362	58	(	(	PUNCT
ejpam-3548	362	59	rx2(b	rx2(b	PROPN
ejpam-3548	362	60	)	)	PUNCT
ejpam-3548	362	61	)	)	PUNCT
ejpam-3548	362	62	there	there	PRON
ejpam-3548	362	63	exists	exist	VERB
ejpam-3548	362	64	a	a	DET
ejpam-3548	362	65	⊆	⊆	NUM
ejpam-3548	362	66	x1	x1	PROPN
ejpam-3548	362	67	(	(	PUNCT
ejpam-3548	362	68	hence	hence	ADV
ejpam-3548	362	69	rx1(a	rx1(a	PROPN
ejpam-3548	362	70	)	)	PUNCT
ejpam-3548	362	71	∈	∈	PROPN
ejpam-3548	362	72	br(x1	br(x1	NOUN
ejpam-3548	362	73	)	)	PUNCT
ejpam-3548	362	74	)	)	PUNCT
ejpam-3548	363	1	such	such	ADJ
ejpam-3548	363	2	that	that	SCONJ
ejpam-3548	363	3	x	x	SYM
ejpam-3548	363	4	∈	∈	PROPN
ejpam-3548	363	5	rx1(a	rx1(a	PROPN
ejpam-3548	363	6	)	)	PUNCT
ejpam-3548	363	7	⊆	⊆	NUM
ejpam-3548	363	8	f−1	f−1	PROPN
ejpam-3548	363	9	(	(	PUNCT
ejpam-3548	363	10	rx2(b	rx2(b	PROPN
ejpam-3548	363	11	)	)	PUNCT
ejpam-3548	363	12	)	)	PUNCT
ejpam-3548	363	13	.	.	PUNCT
ejpam-3548	364	1	since	since	SCONJ
ejpam-3548	364	2	x	x	PROPN
ejpam-3548	364	3	∈	∈	PROPN
ejpam-3548	364	4	rx1(a	rx1(a	PROPN
ejpam-3548	364	5	)	)	PUNCT
ejpam-3548	364	6	,	,	PUNCT
ejpam-3548	364	7	a	a	DET
ejpam-3548	364	8	≤	≤	NOUN
ejpam-3548	364	9	x	x	PUNCT
ejpam-3548	364	10	for	for	ADP
ejpam-3548	364	11	all	all	DET
ejpam-3548	364	12	a	a	DET
ejpam-3548	364	13	∈	∈	NOUN
ejpam-3548	364	14	a.	a.	NOUN
ejpam-3548	364	15	now	now	ADV
ejpam-3548	364	16	,	,	PUNCT
ejpam-3548	364	17	suppose	suppose	VERB
ejpam-3548	364	18	that	that	SCONJ
ejpam-3548	364	19	a	a	DET
ejpam-3548	364	20	≤	≤	ADJ
ejpam-3548	364	21	z	z	NOUN
ejpam-3548	364	22	for	for	ADP
ejpam-3548	364	23	all	all	DET
ejpam-3548	364	24	a	a	DET
ejpam-3548	364	25	∈	∈	NOUN
ejpam-3548	364	26	a.	a.	NOUN
ejpam-3548	364	27	then	then	ADV
ejpam-3548	364	28	z	z	PROPN
ejpam-3548	364	29	∈	∈	PROPN
ejpam-3548	365	1	rx1(a	rx1(a	PROPN
ejpam-3548	365	2	)	)	PUNCT
ejpam-3548	365	3	⊆	⊆	NUM
ejpam-3548	365	4	f−1(rx2(b	f−1(rx2(b	NUM
ejpam-3548	365	5	)	)	PUNCT
ejpam-3548	365	6	)	)	PUNCT
ejpam-3548	365	7	.	.	PUNCT
ejpam-3548	366	1	thus	thus	ADV
ejpam-3548	366	2	,	,	PUNCT
ejpam-3548	366	3	z	z	PROPN
ejpam-3548	366	4	∈	∈	PROPN
ejpam-3548	366	5	f−1(rx2(b	f−1(rx2(b	NOUN
ejpam-3548	366	6	)	)	PUNCT
ejpam-3548	366	7	)	)	PUNCT
ejpam-3548	366	8	.	.	PUNCT
ejpam-3548	367	1	hence	hence	ADV
ejpam-3548	367	2	,	,	PUNCT
ejpam-3548	367	3	f(z	f(z	PROPN
ejpam-3548	367	4	)	)	PUNCT
ejpam-3548	367	5	∈	∈	PROPN
ejpam-3548	367	6	rx2(b	rx2(b	PROPN
ejpam-3548	367	7	)	)	PUNCT
ejpam-3548	367	8	.	.	PUNCT
ejpam-3548	368	1	therefore	therefore	ADV
ejpam-3548	368	2	,	,	PUNCT
ejpam-3548	368	3	b	b	PROPN
ejpam-3548	368	4	≤	≤	NUM
ejpam-3548	368	5	f(z	f(z	PROPN
ejpam-3548	368	6	)	)	PUNCT
ejpam-3548	368	7	for	for	ADP
ejpam-3548	368	8	all	all	DET
ejpam-3548	368	9	b	b	PROPN
ejpam-3548	368	10	∈	∈	PROPN
ejpam-3548	368	11	b.	b.	PROPN
ejpam-3548	368	12	theorem	theorem	VERB
ejpam-3548	368	13	17	17	NUM
ejpam-3548	368	14	.	.	PUNCT
ejpam-3548	369	1	let	let	VERB
ejpam-3548	369	2	(	(	PUNCT
ejpam-3548	369	3	x1	x1	ADJ
ejpam-3548	369	4	,	,	PUNCT
ejpam-3548	369	5	∗x1	∗x1	X
ejpam-3548	369	6	,	,	PUNCT
ejpam-3548	369	7	1x1	1x1	NUM
ejpam-3548	369	8	)	)	PUNCT
ejpam-3548	369	9	and	and	CCONJ
ejpam-3548	369	10	(	(	PUNCT
ejpam-3548	369	11	x2	x2	PROPN
ejpam-3548	369	12	,	,	PUNCT
ejpam-3548	369	13	∗x2	∗x2	PROPN
ejpam-3548	369	14	,	,	PUNCT
ejpam-3548	369	15	1x2	1x2	NUM
ejpam-3548	369	16	)	)	PUNCT
ejpam-3548	369	17	be	be	AUX
ejpam-3548	369	18	be	be	AUX
ejpam-3548	369	19	-	-	PUNCT
ejpam-3548	369	20	algebras	algebra	VERB
ejpam-3548	369	21	and	and	CCONJ
ejpam-3548	369	22	let	let	VERB
ejpam-3548	369	23	f	f	X
ejpam-3548	369	24	:	:	PUNCT
ejpam-3548	369	25	(	(	PUNCT
ejpam-3548	369	26	x1	x1	ADJ
ejpam-3548	369	27	,	,	PUNCT
ejpam-3548	369	28	τr(x1))→	τr(x1))→	PUNCT
ejpam-3548	369	29	(	(	PUNCT
ejpam-3548	369	30	x2	x2	INTJ
ejpam-3548	369	31	,	,	PUNCT
ejpam-3548	369	32	τr(x2	τr(x2	NOUN
ejpam-3548	369	33	)	)	PUNCT
ejpam-3548	369	34	)	)	PUNCT
ejpam-3548	369	35	be	be	AUX
ejpam-3548	369	36	a	a	DET
ejpam-3548	369	37	function	function	NOUN
ejpam-3548	369	38	.	.	PUNCT
ejpam-3548	370	1	then	then	ADV
ejpam-3548	370	2	(	(	PUNCT
ejpam-3548	370	3	i	i	NOUN
ejpam-3548	370	4	)	)	PUNCT
ejpam-3548	370	5	f	f	PROPN
ejpam-3548	370	6	is	be	AUX
ejpam-3548	370	7	open	open	ADJ
ejpam-3548	370	8	if	if	SCONJ
ejpam-3548	370	9	and	and	CCONJ
ejpam-3548	370	10	only	only	ADV
ejpam-3548	370	11	if	if	SCONJ
ejpam-3548	370	12	for	for	ADP
ejpam-3548	370	13	each	each	PRON
ejpam-3548	370	14	a	a	DET
ejpam-3548	370	15	⊆	⊆	NUM
ejpam-3548	370	16	x1	x1	PROPN
ejpam-3548	370	17	and	and	CCONJ
ejpam-3548	370	18	for	for	ADP
ejpam-3548	370	19	each	each	DET
ejpam-3548	370	20	x	x	SYM
ejpam-3548	370	21	∈	∈	PROPN
ejpam-3548	370	22	x1	x1	PROPN
ejpam-3548	370	23	with	with	ADP
ejpam-3548	370	24	a	a	DET
ejpam-3548	370	25	≤	≤	NOUN
ejpam-3548	370	26	x	x	PUNCT
ejpam-3548	370	27	for	for	ADP
ejpam-3548	370	28	all	all	DET
ejpam-3548	370	29	a	a	DET
ejpam-3548	370	30	∈	∈	PROPN
ejpam-3548	370	31	a	a	PRON
ejpam-3548	370	32	,	,	PUNCT
ejpam-3548	370	33	there	there	PRON
ejpam-3548	370	34	exists	exist	VERB
ejpam-3548	370	35	b	b	PROPN
ejpam-3548	370	36	⊆	⊆	NUM
ejpam-3548	370	37	x2	x2	NOUN
ejpam-3548	370	38	satisfying	satisfy	VERB
ejpam-3548	370	39	the	the	DET
ejpam-3548	370	40	following	follow	VERB
ejpam-3548	370	41	properties	property	NOUN
ejpam-3548	370	42	:	:	PUNCT
ejpam-3548	370	43	(	(	PUNCT
ejpam-3548	370	44	a	a	X
ejpam-3548	370	45	)	)	PUNCT
ejpam-3548	370	46	b	b	PROPN
ejpam-3548	370	47	≤	≤	NUM
ejpam-3548	370	48	f(x	f(x	PROPN
ejpam-3548	370	49	)	)	PUNCT
ejpam-3548	370	50	for	for	ADP
ejpam-3548	370	51	all	all	DET
ejpam-3548	370	52	b	b	PROPN
ejpam-3548	370	53	∈	∈	ADP
ejpam-3548	370	54	b	b	PROPN
ejpam-3548	370	55	(	(	PUNCT
ejpam-3548	370	56	b	b	NOUN
ejpam-3548	370	57	)	)	PUNCT
ejpam-3548	370	58	there	there	PRON
ejpam-3548	370	59	exists	exist	VERB
ejpam-3548	370	60	z	z	PROPN
ejpam-3548	370	61	∈	∈	PROPN
ejpam-3548	370	62	x1	x1	NUM
ejpam-3548	370	63	with	with	ADP
ejpam-3548	370	64	a	a	DET
ejpam-3548	370	65	≤	≤	ADJ
ejpam-3548	370	66	z	z	NOUN
ejpam-3548	370	67	for	for	ADP
ejpam-3548	370	68	all	all	DET
ejpam-3548	370	69	a	a	DET
ejpam-3548	370	70	∈	∈	PROPN
ejpam-3548	370	71	a	a	DET
ejpam-3548	370	72	and	and	CCONJ
ejpam-3548	370	73	f(z	f(z	PROPN
ejpam-3548	370	74	)	)	PUNCT
ejpam-3548	371	1	=	=	PUNCT
ejpam-3548	371	2	y	y	NOUN
ejpam-3548	371	3	whenever	whenever	SCONJ
ejpam-3548	371	4	a	a	DET
ejpam-3548	371	5	≤	≤	ADJ
ejpam-3548	371	6	f−1	f−1	PROPN
ejpam-3548	371	7	(	(	PUNCT
ejpam-3548	371	8	y	y	NOUN
ejpam-3548	371	9	)	)	PUNCT
ejpam-3548	371	10	for	for	ADP
ejpam-3548	371	11	all	all	DET
ejpam-3548	371	12	a	a	DET
ejpam-3548	371	13	∈	∈	PROPN
ejpam-3548	371	14	a	a	PRON
ejpam-3548	371	15	and	and	CCONJ
ejpam-3548	371	16	b	b	NOUN
ejpam-3548	371	17	≤	≤	NOUN
ejpam-3548	371	18	y	y	NOUN
ejpam-3548	371	19	for	for	ADP
ejpam-3548	371	20	all	all	DET
ejpam-3548	371	21	b	b	PROPN
ejpam-3548	371	22	∈	∈	PROPN
ejpam-3548	371	23	b.	b.	PROPN
ejpam-3548	371	24	j.	j.	PROPN
ejpam-3548	371	25	albaracin	albaracin	PROPN
ejpam-3548	371	26	,	,	PUNCT
ejpam-3548	371	27	j.	j.	PROPN
ejpam-3548	371	28	vilela	vilela	PROPN
ejpam-3548	371	29	/	/	SYM
ejpam-3548	371	30	eur	eur	PROPN
ejpam-3548	371	31	.	.	PUNCT
ejpam-3548	372	1	j.	j.	PROPN
ejpam-3548	372	2	pure	pure	PROPN
ejpam-3548	372	3	appl	appl	PROPN
ejpam-3548	372	4	.	.	PROPN
ejpam-3548	372	5	math	math	PROPN
ejpam-3548	372	6	,	,	PUNCT
ejpam-3548	372	7	12	12	NUM
ejpam-3548	372	8	(	(	PUNCT
ejpam-3548	372	9	4	4	NUM
ejpam-3548	372	10	)	)	PUNCT
ejpam-3548	372	11	(	(	PUNCT
ejpam-3548	372	12	2019	2019	NUM
ejpam-3548	372	13	)	)	PUNCT
ejpam-3548	372	14	,	,	PUNCT
ejpam-3548	372	15	1584	1584	NUM
ejpam-3548	372	16	-	-	SYM
ejpam-3548	372	17	1594	1594	NUM
ejpam-3548	372	18	1593	1593	NUM
ejpam-3548	372	19	(	(	PUNCT
ejpam-3548	372	20	ii	ii	NOUN
ejpam-3548	372	21	)	)	PUNCT
ejpam-3548	372	22	f	f	PROPN
ejpam-3548	372	23	is	be	AUX
ejpam-3548	372	24	closed	close	VERB
ejpam-3548	372	25	if	if	SCONJ
ejpam-3548	372	26	and	and	CCONJ
ejpam-3548	372	27	only	only	ADV
ejpam-3548	372	28	if	if	SCONJ
ejpam-3548	372	29	for	for	ADP
ejpam-3548	372	30	each	each	DET
ejpam-3548	372	31	τr(x1)-closed	τr(x1)-close	VERB
ejpam-3548	372	32	set	set	VERB
ejpam-3548	372	33	f	f	PROPN
ejpam-3548	372	34	and	and	CCONJ
ejpam-3548	372	35	for	for	ADP
ejpam-3548	372	36	all	all	DET
ejpam-3548	372	37	y	y	PROPN
ejpam-3548	372	38	∈	∈	PROPN
ejpam-3548	372	39	x2	x2	PROPN
ejpam-3548	372	40	with	with	ADP
ejpam-3548	372	41	y	y	PROPN
ejpam-3548	372	42	6=	6=	PROPN
ejpam-3548	372	43	f(x	f(x	PROPN
ejpam-3548	372	44	)	)	PUNCT
ejpam-3548	372	45	for	for	ADP
ejpam-3548	372	46	all	all	DET
ejpam-3548	372	47	x	x	SYM
ejpam-3548	372	48	∈	∈	PROPN
ejpam-3548	372	49	f	f	NOUN
ejpam-3548	372	50	,	,	PUNCT
ejpam-3548	372	51	there	there	PRON
ejpam-3548	372	52	exists	exist	VERB
ejpam-3548	372	53	ay	ay	PROPN
ejpam-3548	372	54	⊆	⊆	NUM
ejpam-3548	372	55	x2	x2	PROPN
ejpam-3548	372	56	such	such	ADJ
ejpam-3548	372	57	that	that	DET
ejpam-3548	372	58	rx2(ay	rx2(ay	PROPN
ejpam-3548	372	59	)	)	PUNCT
ejpam-3548	372	60	∩	∩	NOUN
ejpam-3548	372	61	f(f	f(f	PROPN
ejpam-3548	372	62	)	)	PUNCT
ejpam-3548	373	1	=	=	NOUN
ejpam-3548	373	2	∅	∅	NOUN
ejpam-3548	373	3	and	and	CCONJ
ejpam-3548	373	4	a	a	DET
ejpam-3548	373	5	≤	≤	ADJ
ejpam-3548	373	6	y	y	NOUN
ejpam-3548	373	7	for	for	ADP
ejpam-3548	373	8	all	all	DET
ejpam-3548	373	9	a	a	DET
ejpam-3548	373	10	∈	∈	PROPN
ejpam-3548	373	11	ay	ay	NOUN
ejpam-3548	373	12	.	.	PUNCT
ejpam-3548	373	13	proof	proof	NOUN
ejpam-3548	373	14	.	.	PUNCT
ejpam-3548	374	1	(	(	PUNCT
ejpam-3548	374	2	i	i	NOUN
ejpam-3548	374	3	)	)	PUNCT
ejpam-3548	374	4	by	by	ADP
ejpam-3548	374	5	definition	definition	NOUN
ejpam-3548	374	6	,	,	PUNCT
ejpam-3548	374	7	f	f	PROPN
ejpam-3548	374	8	is	be	AUX
ejpam-3548	374	9	open	open	ADJ
ejpam-3548	374	10	if	if	SCONJ
ejpam-3548	374	11	and	and	CCONJ
ejpam-3548	374	12	only	only	ADV
ejpam-3548	375	1	if	if	SCONJ
ejpam-3548	375	2	f	f	PROPN
ejpam-3548	375	3	(	(	PUNCT
ejpam-3548	375	4	rx1(a	rx1(a	PROPN
ejpam-3548	375	5	)	)	PUNCT
ejpam-3548	375	6	)	)	PUNCT
ejpam-3548	375	7	∈	∈	PROPN
ejpam-3548	375	8	τr(x2	τr(x2	NOUN
ejpam-3548	375	9	)	)	PUNCT
ejpam-3548	375	10	for	for	ADP
ejpam-3548	375	11	each	each	PRON
ejpam-3548	375	12	a	a	DET
ejpam-3548	375	13	⊆	⊆	NUM
ejpam-3548	375	14	x1	x1	PROPN
ejpam-3548	375	15	.	.	PUNCT
ejpam-3548	376	1	now	now	ADV
ejpam-3548	376	2	,	,	PUNCT
ejpam-3548	376	3	f	f	PROPN
ejpam-3548	376	4	(	(	PUNCT
ejpam-3548	376	5	rx1(a	rx1(a	PROPN
ejpam-3548	376	6	)	)	PUNCT
ejpam-3548	376	7	)	)	PUNCT
ejpam-3548	376	8	∈	∈	PROPN
ejpam-3548	376	9	τr(x2	τr(x2	NOUN
ejpam-3548	376	10	)	)	PUNCT
ejpam-3548	377	1	if	if	SCONJ
ejpam-3548	377	2	and	and	CCONJ
ejpam-3548	377	3	only	only	ADV
ejpam-3548	377	4	if	if	SCONJ
ejpam-3548	377	5	for	for	ADP
ejpam-3548	377	6	each	each	DET
ejpam-3548	377	7	x	x	SYM
ejpam-3548	377	8	∈	∈	PROPN
ejpam-3548	377	9	rx1(a	rx1(a	PROPN
ejpam-3548	377	10	)	)	PUNCT
ejpam-3548	377	11	,	,	PUNCT
ejpam-3548	377	12	there	there	PRON
ejpam-3548	377	13	exists	exist	VERB
ejpam-3548	377	14	b	b	PROPN
ejpam-3548	377	15	⊆	⊆	NUM
ejpam-3548	377	16	x2	x2	PROPN
ejpam-3548	377	17	such	such	ADJ
ejpam-3548	377	18	that	that	SCONJ
ejpam-3548	377	19	f(x	f(x	PROPN
ejpam-3548	377	20	)	)	PUNCT
ejpam-3548	377	21	∈	∈	PROPN
ejpam-3548	377	22	rx2(b	rx2(b	PROPN
ejpam-3548	377	23	)	)	PUNCT
ejpam-3548	377	24	⊆	⊆	NUM
ejpam-3548	377	25	f	f	X
ejpam-3548	377	26	(	(	PUNCT
ejpam-3548	377	27	rx1(a	rx1(a	PROPN
ejpam-3548	377	28	)	)	PUNCT
ejpam-3548	377	29	)	)	PUNCT
ejpam-3548	377	30	.	.	PUNCT
ejpam-3548	378	1	since	since	SCONJ
ejpam-3548	378	2	f(x	f(x	PROPN
ejpam-3548	378	3	)	)	PUNCT
ejpam-3548	378	4	∈	∈	PROPN
ejpam-3548	378	5	rx2(b	rx2(b	PROPN
ejpam-3548	378	6	)	)	PUNCT
ejpam-3548	378	7	,	,	PUNCT
ejpam-3548	378	8	b	b	PROPN
ejpam-3548	378	9	≤	≤	NUM
ejpam-3548	378	10	f(x	f(x	PROPN
ejpam-3548	378	11	)	)	PUNCT
ejpam-3548	378	12	for	for	ADP
ejpam-3548	378	13	all	all	DET
ejpam-3548	378	14	b	b	PROPN
ejpam-3548	378	15	∈	∈	PROPN
ejpam-3548	378	16	b.	b.	NOUN
ejpam-3548	379	1	moreover	moreover	ADV
ejpam-3548	379	2	,	,	PUNCT
ejpam-3548	379	3	if	if	SCONJ
ejpam-3548	379	4	a	a	DET
ejpam-3548	379	5	≤	≤	ADV
ejpam-3548	379	6	f−1(y	f−1(y	PROPN
ejpam-3548	379	7	)	)	PUNCT
ejpam-3548	379	8	for	for	ADP
ejpam-3548	379	9	all	all	DET
ejpam-3548	379	10	a	a	DET
ejpam-3548	379	11	∈	∈	PROPN
ejpam-3548	379	12	a	a	DET
ejpam-3548	379	13	and	and	CCONJ
ejpam-3548	379	14	b	b	NOUN
ejpam-3548	379	15	≤	≤	NOUN
ejpam-3548	379	16	y	y	NOUN
ejpam-3548	379	17	for	for	ADP
ejpam-3548	379	18	all	all	DET
ejpam-3548	379	19	b	b	PROPN
ejpam-3548	379	20	∈	∈	PROPN
ejpam-3548	379	21	b	b	NOUN
ejpam-3548	379	22	,	,	PUNCT
ejpam-3548	379	23	then	then	ADV
ejpam-3548	379	24	f−1(y	f−1(y	PROPN
ejpam-3548	379	25	)	)	PUNCT
ejpam-3548	379	26	∈	∈	PROPN
ejpam-3548	379	27	rx1(a	rx1(a	PROPN
ejpam-3548	379	28	)	)	PUNCT
ejpam-3548	379	29	and	and	CCONJ
ejpam-3548	379	30	y	y	PROPN
ejpam-3548	379	31	∈	∈	PROPN
ejpam-3548	379	32	rx2(b	rx2(b	PROPN
ejpam-3548	379	33	)	)	PUNCT
ejpam-3548	379	34	.	.	PUNCT
ejpam-3548	380	1	this	this	PRON
ejpam-3548	380	2	implies	imply	VERB
ejpam-3548	380	3	that	that	SCONJ
ejpam-3548	380	4	y	y	PROPN
ejpam-3548	380	5	∈	∈	PROPN
ejpam-3548	380	6	rx2(b	rx2(b	PROPN
ejpam-3548	380	7	)	)	PUNCT
ejpam-3548	380	8	⊆	⊆	NUM
ejpam-3548	380	9	f(rx1(a	f(rx1(a	NOUN
ejpam-3548	380	10	)	)	PUNCT
ejpam-3548	380	11	)	)	PUNCT
ejpam-3548	380	12	.	.	PUNCT
ejpam-3548	381	1	consequently	consequently	ADV
ejpam-3548	381	2	,	,	PUNCT
ejpam-3548	381	3	there	there	PRON
ejpam-3548	381	4	exists	exist	VERB
ejpam-3548	381	5	z	z	NOUN
ejpam-3548	381	6	∈	∈	PROPN
ejpam-3548	382	1	x1	x1	NUM
ejpam-3548	382	2	such	such	ADJ
ejpam-3548	382	3	that	that	SCONJ
ejpam-3548	382	4	z	z	PROPN
ejpam-3548	382	5	∈	∈	PROPN
ejpam-3548	382	6	rx1(a	rx1(a	PROPN
ejpam-3548	382	7	)	)	PUNCT
ejpam-3548	382	8	and	and	CCONJ
ejpam-3548	382	9	y	y	PROPN
ejpam-3548	382	10	=	=	PUNCT
ejpam-3548	382	11	f(z	f(z	PROPN
ejpam-3548	382	12	)	)	PUNCT
ejpam-3548	382	13	.	.	PUNCT
ejpam-3548	383	1	hence	hence	ADV
ejpam-3548	383	2	,	,	PUNCT
ejpam-3548	383	3	a	a	DET
ejpam-3548	383	4	≤	≤	ADJ
ejpam-3548	383	5	z	z	NOUN
ejpam-3548	383	6	for	for	ADP
ejpam-3548	383	7	all	all	DET
ejpam-3548	383	8	a	a	DET
ejpam-3548	383	9	∈	∈	PROPN
ejpam-3548	383	10	a	a	PRON
ejpam-3548	383	11	and	and	CCONJ
ejpam-3548	383	12	y	y	PROPN
ejpam-3548	383	13	=	=	PUNCT
ejpam-3548	383	14	f(z	f(z	PROPN
ejpam-3548	383	15	)	)	PUNCT
ejpam-3548	383	16	.	.	PUNCT
ejpam-3548	384	1	(	(	PUNCT
ejpam-3548	384	2	ii	ii	NOUN
ejpam-3548	384	3	)	)	PUNCT
ejpam-3548	384	4	suppose	suppose	VERB
ejpam-3548	384	5	that	that	SCONJ
ejpam-3548	384	6	f	f	PROPN
ejpam-3548	384	7	is	be	AUX
ejpam-3548	384	8	closed	close	VERB
ejpam-3548	384	9	and	and	CCONJ
ejpam-3548	384	10	let	let	VERB
ejpam-3548	384	11	f	f	PRON
ejpam-3548	384	12	be	be	AUX
ejpam-3548	384	13	a	a	DET
ejpam-3548	384	14	closed	closed	ADJ
ejpam-3548	384	15	subset	subset	NOUN
ejpam-3548	384	16	of	of	ADP
ejpam-3548	384	17	x1	x1	PROPN
ejpam-3548	384	18	.	.	PUNCT
ejpam-3548	385	1	then	then	ADV
ejpam-3548	385	2	by	by	ADP
ejpam-3548	385	3	definition	definition	NOUN
ejpam-3548	385	4	of	of	ADP
ejpam-3548	385	5	a	a	DET
ejpam-3548	385	6	closed	closed	ADJ
ejpam-3548	385	7	map	map	NOUN
ejpam-3548	385	8	,	,	PUNCT
ejpam-3548	385	9	f(f	f(f	PROPN
ejpam-3548	385	10	)	)	PUNCT
ejpam-3548	385	11	=	=	PRON
ejpam-3548	385	12	{	{	PUNCT
ejpam-3548	385	13	f(x	f(x	PROPN
ejpam-3548	385	14	)	)	PUNCT
ejpam-3548	385	15	:	:	PUNCT
ejpam-3548	386	1	x	x	X
ejpam-3548	386	2	∈	∈	PROPN
ejpam-3548	386	3	f	f	X
ejpam-3548	386	4	}	}	PUNCT
ejpam-3548	386	5	is	be	AUX
ejpam-3548	386	6	closed	close	VERB
ejpam-3548	386	7	in	in	ADP
ejpam-3548	386	8	x2	x2	PROPN
ejpam-3548	386	9	,	,	PUNCT
ejpam-3548	386	10	that	that	ADV
ejpam-3548	386	11	is	is	ADV
ejpam-3548	386	12	,	,	PUNCT
ejpam-3548	386	13	[	[	X
ejpam-3548	386	14	f(f	f(f	PROPN
ejpam-3548	386	15	)	)	PUNCT
ejpam-3548	387	1	]	]	X
ejpam-3548	387	2	c	c	X
ejpam-3548	387	3	=	=	SYM
ejpam-3548	387	4	{	{	PUNCT
ejpam-3548	387	5	f(x	f(x	PROPN
ejpam-3548	387	6	)	)	PUNCT
ejpam-3548	387	7	:	:	PUNCT
ejpam-3548	388	1	x	x	PUNCT
ejpam-3548	388	2	∈	∈	X
ejpam-3548	388	3	f}c	f}c	NOUN
ejpam-3548	388	4	=	=	PUNCT
ejpam-3548	388	5	⋃	⋃	ADP
ejpam-3548	388	6	a∈p2	a∈p2	NOUN
ejpam-3548	388	7	rx2(a	rx2(a	NOUN
ejpam-3548	388	8	)	)	PUNCT
ejpam-3548	388	9	,	,	PUNCT
ejpam-3548	388	10	where	where	SCONJ
ejpam-3548	388	11	p2	p2	PROPN
ejpam-3548	388	12	⊆	⊆	NUM
ejpam-3548	388	13	p(x2	p(x2	NOUN
ejpam-3548	388	14	)	)	PUNCT
ejpam-3548	388	15	\	\	NOUN
ejpam-3548	388	16	{	{	PUNCT
ejpam-3548	388	17	∅	∅	NOUN
ejpam-3548	388	18	}	}	PUNCT
ejpam-3548	388	19	.	.	PUNCT
ejpam-3548	389	1	hence	hence	ADV
ejpam-3548	389	2	,	,	PUNCT
ejpam-3548	389	3	for	for	ADP
ejpam-3548	389	4	each	each	DET
ejpam-3548	389	5	y	y	PROPN
ejpam-3548	389	6	∈	∈	PROPN
ejpam-3548	389	7	x2	x2	INTJ
ejpam-3548	389	8	such	such	ADJ
ejpam-3548	389	9	that	that	SCONJ
ejpam-3548	389	10	y	y	PROPN
ejpam-3548	389	11	6=	6=	PROPN
ejpam-3548	389	12	f(x	f(x	PROPN
ejpam-3548	389	13	)	)	PUNCT
ejpam-3548	389	14	for	for	ADP
ejpam-3548	389	15	all	all	DET
ejpam-3548	389	16	x	x	SYM
ejpam-3548	389	17	∈	∈	PROPN
ejpam-3548	389	18	f	f	NOUN
ejpam-3548	389	19	,	,	PUNCT
ejpam-3548	389	20	there	there	PRON
ejpam-3548	389	21	exists	exist	VERB
ejpam-3548	389	22	ay	ay	PROPN
ejpam-3548	389	23	⊆	⊆	NUM
ejpam-3548	389	24	x2	x2	PROPN
ejpam-3548	389	25	such	such	ADJ
ejpam-3548	389	26	that	that	DET
ejpam-3548	389	27	rx2(ay	rx2(ay	PROPN
ejpam-3548	389	28	)	)	PUNCT
ejpam-3548	389	29	⊆	⊆	NUM
ejpam-3548	390	1	[	[	X
ejpam-3548	390	2	f(f	f(f	PROPN
ejpam-3548	390	3	)	)	PUNCT
ejpam-3548	390	4	]	]	X
ejpam-3548	390	5	c	c	NOUN
ejpam-3548	390	6	and	and	CCONJ
ejpam-3548	390	7	a	a	DET
ejpam-3548	390	8	≤	≤	ADJ
ejpam-3548	390	9	y	y	NOUN
ejpam-3548	390	10	for	for	ADP
ejpam-3548	390	11	all	all	DET
ejpam-3548	390	12	a	a	DET
ejpam-3548	390	13	∈	∈	PROPN
ejpam-3548	390	14	ay	ay	NOUN
ejpam-3548	390	15	.	.	PUNCT
ejpam-3548	390	16	conversely	conversely	ADV
ejpam-3548	390	17	,	,	PUNCT
ejpam-3548	390	18	suppose	suppose	VERB
ejpam-3548	390	19	that	that	SCONJ
ejpam-3548	390	20	for	for	ADP
ejpam-3548	390	21	each	each	DET
ejpam-3548	390	22	closed	close	VERB
ejpam-3548	390	23	subset	subset	VERB
ejpam-3548	390	24	f	f	PROPN
ejpam-3548	390	25	of	of	ADP
ejpam-3548	390	26	x1	x1	PROPN
ejpam-3548	390	27	and	and	CCONJ
ejpam-3548	390	28	for	for	ADP
ejpam-3548	390	29	all	all	DET
ejpam-3548	390	30	y	y	PROPN
ejpam-3548	390	31	∈	∈	PROPN
ejpam-3548	390	32	x2	x2	PROPN
ejpam-3548	390	33	with	with	ADP
ejpam-3548	390	34	y	y	PROPN
ejpam-3548	390	35	6=	6=	PROPN
ejpam-3548	390	36	f(x	f(x	PROPN
ejpam-3548	390	37	)	)	PUNCT
ejpam-3548	390	38	for	for	ADP
ejpam-3548	390	39	all	all	DET
ejpam-3548	390	40	x	x	SYM
ejpam-3548	390	41	∈	∈	PROPN
ejpam-3548	390	42	f	f	NOUN
ejpam-3548	390	43	,	,	PUNCT
ejpam-3548	390	44	there	there	PRON
ejpam-3548	390	45	exists	exist	VERB
ejpam-3548	390	46	ay	ay	PROPN
ejpam-3548	390	47	⊆	⊆	NUM
ejpam-3548	390	48	x2	x2	PROPN
ejpam-3548	390	49	such	such	ADJ
ejpam-3548	390	50	that	that	DET
ejpam-3548	390	51	rx2(ay	rx2(ay	PROPN
ejpam-3548	390	52	)	)	PUNCT
ejpam-3548	390	53	∩	∩	NOUN
ejpam-3548	390	54	f(f	f(f	PROPN
ejpam-3548	390	55	)	)	PUNCT
ejpam-3548	391	1	=	=	NOUN
ejpam-3548	391	2	∅	∅	NOUN
ejpam-3548	391	3	and	and	CCONJ
ejpam-3548	391	4	a	a	DET
ejpam-3548	391	5	≤	≤	ADJ
ejpam-3548	391	6	y	y	NOUN
ejpam-3548	391	7	for	for	ADP
ejpam-3548	391	8	all	all	DET
ejpam-3548	391	9	a	a	DET
ejpam-3548	391	10	∈	∈	PROPN
ejpam-3548	391	11	ay	ay	NOUN
ejpam-3548	391	12	.	.	PUNCT
ejpam-3548	392	1	let	let	VERB
ejpam-3548	392	2	f	f	PROPN
ejpam-3548	392	3	∗	∗	NOUN
ejpam-3548	392	4	be	be	AUX
ejpam-3548	392	5	a	a	DET
ejpam-3548	392	6	closed	closed	ADJ
ejpam-3548	392	7	set	set	NOUN
ejpam-3548	392	8	in	in	ADP
ejpam-3548	392	9	x1	x1	PROPN
ejpam-3548	392	10	and	and	CCONJ
ejpam-3548	392	11	let	let	VERB
ejpam-3548	392	12	y	y	PROPN
ejpam-3548	392	13	∈	∈	PROPN
ejpam-3548	392	14	[	[	X
ejpam-3548	392	15	f(f	f(f	PROPN
ejpam-3548	392	16	∗)]c	∗)]c	PROPN
ejpam-3548	392	17	.	.	PUNCT
ejpam-3548	393	1	then	then	ADV
ejpam-3548	393	2	y	y	PROPN
ejpam-3548	393	3	∈	∈	PROPN
ejpam-3548	393	4	x2	x2	PROPN
ejpam-3548	393	5	and	and	CCONJ
ejpam-3548	393	6	y	y	PROPN
ejpam-3548	393	7	6=	6=	PROPN
ejpam-3548	393	8	f(x	f(x	PROPN
ejpam-3548	393	9	)	)	PUNCT
ejpam-3548	393	10	for	for	ADP
ejpam-3548	393	11	all	all	PRON
ejpam-3548	393	12	x	x	SYM
ejpam-3548	393	13	∈	∈	PROPN
ejpam-3548	393	14	f	f	X
ejpam-3548	393	15	∗.	∗.	PROPN
ejpam-3548	393	16	by	by	ADP
ejpam-3548	393	17	assumption	assumption	NOUN
ejpam-3548	393	18	,	,	PUNCT
ejpam-3548	393	19	there	there	PRON
ejpam-3548	393	20	exists	exist	VERB
ejpam-3548	393	21	ay	ay	PROPN
ejpam-3548	393	22	⊆	⊆	NUM
ejpam-3548	393	23	x2	x2	PROPN
ejpam-3548	393	24	such	such	ADJ
ejpam-3548	393	25	that	that	DET
ejpam-3548	393	26	rx2(ay	rx2(ay	PROPN
ejpam-3548	393	27	)	)	PUNCT
ejpam-3548	393	28	∩	∩	NOUN
ejpam-3548	393	29	f(f	f(f	PROPN
ejpam-3548	393	30	∗	∗	NOUN
ejpam-3548	393	31	)	)	PUNCT
ejpam-3548	394	1	=	=	SYM
ejpam-3548	394	2	∅	∅	NOUN
ejpam-3548	394	3	and	and	CCONJ
ejpam-3548	394	4	a	a	DET
ejpam-3548	394	5	≤	≤	ADJ
ejpam-3548	394	6	y	y	NOUN
ejpam-3548	394	7	for	for	ADP
ejpam-3548	394	8	all	all	DET
ejpam-3548	394	9	a	a	DET
ejpam-3548	394	10	∈	∈	PROPN
ejpam-3548	394	11	ay	ay	NOUN
ejpam-3548	394	12	,	,	PUNCT
ejpam-3548	394	13	that	that	ADV
ejpam-3548	394	14	is	is	ADV
ejpam-3548	394	15	,	,	PUNCT
ejpam-3548	394	16	y	y	PROPN
ejpam-3548	394	17	∈	∈	PROPN
ejpam-3548	394	18	rx2(ay	rx2(ay	PROPN
ejpam-3548	394	19	)	)	PUNCT
ejpam-3548	394	20	.	.	PUNCT
ejpam-3548	395	1	thus	thus	ADV
ejpam-3548	395	2	,	,	PUNCT
ejpam-3548	395	3	[	[	X
ejpam-3548	395	4	f(f	f(f	PROPN
ejpam-3548	395	5	∗)]c	∗)]c	PROPN
ejpam-3548	395	6	=	=	PUNCT
ejpam-3548	395	7	⋃	⋃	SCONJ
ejpam-3548	395	8	y∈[f(f	y∈[f(f	PROPN
ejpam-3548	395	9	∗)]c	∗)]c	PROPN
ejpam-3548	395	10	rx2(ay	rx2(ay	PROPN
ejpam-3548	395	11	)	)	PUNCT
ejpam-3548	395	12	.	.	PUNCT
ejpam-3548	396	1	hence	hence	ADV
ejpam-3548	396	2	,	,	PUNCT
ejpam-3548	396	3	[	[	X
ejpam-3548	396	4	f(f	f(f	PROPN
ejpam-3548	396	5	∗)]c	∗)]c	PROPN
ejpam-3548	396	6	is	be	AUX
ejpam-3548	396	7	τr(x2)-open	τr(x2)-open	ADJ
ejpam-3548	396	8	showing	show	VERB
ejpam-3548	396	9	that	that	SCONJ
ejpam-3548	396	10	f(f	f(f	PROPN
ejpam-3548	396	11	∗	∗	NOUN
ejpam-3548	396	12	)	)	PUNCT
ejpam-3548	396	13	is	be	AUX
ejpam-3548	396	14	a	a	DET
ejpam-3548	396	15	closed	closed	ADJ
ejpam-3548	396	16	subset	subset	NOUN
ejpam-3548	396	17	of	of	ADP
ejpam-3548	396	18	x2	x2	PROPN
ejpam-3548	396	19	.	.	PUNCT
ejpam-3548	397	1	therefore	therefore	ADV
ejpam-3548	397	2	,	,	PUNCT
ejpam-3548	397	3	f	f	PROPN
ejpam-3548	397	4	is	be	AUX
ejpam-3548	397	5	a	a	DET
ejpam-3548	397	6	closed	closed	ADJ
ejpam-3548	397	7	map	map	NOUN
ejpam-3548	397	8	.	.	PUNCT
ejpam-3548	398	1	5	5	X
ejpam-3548	398	2	.	.	X
ejpam-3548	398	3	conclusion	conclusion	NOUN
ejpam-3548	398	4	the	the	DET
ejpam-3548	398	5	topology	topology	NOUN
ejpam-3548	398	6	generated	generate	VERB
ejpam-3548	398	7	by	by	ADP
ejpam-3548	398	8	the	the	DET
ejpam-3548	398	9	family	family	NOUN
ejpam-3548	398	10	of	of	ADP
ejpam-3548	398	11	subsets	subset	NOUN
ejpam-3548	398	12	determined	determine	VERB
ejpam-3548	398	13	by	by	ADP
ejpam-3548	398	14	the	the	DET
ejpam-3548	398	15	right	right	ADJ
ejpam-3548	398	16	application	application	NOUN
ejpam-3548	398	17	of	of	ADP
ejpam-3548	398	18	be	be	NOUN
ejpam-3548	398	19	-	-	PUNCT
ejpam-3548	398	20	ordering	ordering	NOUN
ejpam-3548	398	21	of	of	ADP
ejpam-3548	398	22	a	a	DET
ejpam-3548	398	23	be	be	NOUN
ejpam-3548	398	24	-	-	PUNCT
ejpam-3548	398	25	algebra	algebra	NOUN
ejpam-3548	398	26	is	be	AUX
ejpam-3548	398	27	always	always	ADV
ejpam-3548	398	28	connected	connect	VERB
ejpam-3548	398	29	.	.	PUNCT
ejpam-3548	399	1	investigations	investigation	NOUN
ejpam-3548	399	2	for	for	ADP
ejpam-3548	399	3	some	some	DET
ejpam-3548	399	4	elementary	elementary	ADJ
ejpam-3548	399	5	topological	topological	ADJ
ejpam-3548	399	6	concepts	concept	NOUN
ejpam-3548	399	7	as	as	ADV
ejpam-3548	399	8	well	well	ADV
ejpam-3548	399	9	as	as	ADP
ejpam-3548	399	10	the	the	DET
ejpam-3548	399	11	concepts	concept	NOUN
ejpam-3548	399	12	of	of	ADP
ejpam-3548	399	13	continuous	continuous	ADJ
ejpam-3548	399	14	,	,	PUNCT
ejpam-3548	399	15	open	open	ADJ
ejpam-3548	399	16	,	,	PUNCT
ejpam-3548	399	17	and	and	CCONJ
ejpam-3548	399	18	closed	closed	ADJ
ejpam-3548	399	19	maps	map	NOUN
ejpam-3548	399	20	associated	associate	VERB
ejpam-3548	399	21	with	with	ADP
ejpam-3548	399	22	this	this	DET
ejpam-3548	399	23	topological	topological	ADJ
ejpam-3548	399	24	space	space	NOUN
ejpam-3548	399	25	are	be	AUX
ejpam-3548	399	26	obtained	obtain	VERB
ejpam-3548	399	27	.	.	PUNCT
ejpam-3548	400	1	this	this	DET
ejpam-3548	400	2	paper	paper	NOUN
ejpam-3548	400	3	will	will	AUX
ejpam-3548	400	4	lead	lead	VERB
ejpam-3548	400	5	to	to	ADP
ejpam-3548	400	6	some	some	DET
ejpam-3548	400	7	studies	study	NOUN
ejpam-3548	400	8	on	on	ADP
ejpam-3548	400	9	separation	separation	NOUN
ejpam-3548	400	10	axioms	axiom	NOUN
ejpam-3548	400	11	associated	associate	VERB
ejpam-3548	400	12	with	with	ADP
ejpam-3548	400	13	this	this	DET
ejpam-3548	400	14	kind	kind	NOUN
ejpam-3548	400	15	of	of	ADP
ejpam-3548	400	16	topological	topological	ADJ
ejpam-3548	400	17	space	space	NOUN
ejpam-3548	400	18	.	.	PUNCT
ejpam-3548	401	1	acknowledgements	acknowledgement	NOUN
ejpam-3548	401	2	this	this	DET
ejpam-3548	401	3	research	research	NOUN
ejpam-3548	401	4	is	be	AUX
ejpam-3548	401	5	funded	fund	VERB
ejpam-3548	401	6	by	by	ADP
ejpam-3548	401	7	the	the	DET
ejpam-3548	401	8	philippine	philippine	PROPN
ejpam-3548	401	9	department	department	PROPN
ejpam-3548	401	10	of	of	ADP
ejpam-3548	401	11	science	science	NOUN
ejpam-3548	401	12	and	and	CCONJ
ejpam-3548	401	13	technologyaccelerated	technologyaccelerated	ADJ
ejpam-3548	401	14	science	science	NOUN
ejpam-3548	401	15	and	and	CCONJ
ejpam-3548	401	16	technology	technology	NOUN
ejpam-3548	401	17	human	human	ADJ
ejpam-3548	401	18	resource	resource	NOUN
ejpam-3548	401	19	development	development	NOUN
ejpam-3548	401	20	program	program	NOUN
ejpam-3548	401	21	(	(	PUNCT
ejpam-3548	401	22	dostasthrdp	dostasthrdp	PROPN
ejpam-3548	401	23	)	)	PUNCT
ejpam-3548	401	24	.	.	PUNCT
ejpam-3548	402	1	references	reference	NOUN
ejpam-3548	402	2	1594	1594	NUM
ejpam-3548	402	3	references	reference	NOUN
ejpam-3548	402	4	[	[	X
ejpam-3548	402	5	1	1	NUM
ejpam-3548	402	6	]	]	X
ejpam-3548	402	7	s.s	s.s	PROPN
ejpam-3548	402	8	.	.	PROPN
ejpam-3548	402	9	ahn	ahn	PROPN
ejpam-3548	402	10	and	and	CCONJ
ejpam-3548	402	11	k.s	k.s	PROPN
ejpam-3548	402	12	.	.	PUNCT
ejpam-3548	403	1	so	so	ADV
ejpam-3548	403	2	.	.	PUNCT
ejpam-3548	404	1	on	on	ADP
ejpam-3548	404	2	ideals	ideal	NOUN
ejpam-3548	404	3	and	and	CCONJ
ejpam-3548	404	4	upper	upper	ADJ
ejpam-3548	404	5	sets	set	NOUN
ejpam-3548	404	6	in	in	ADP
ejpam-3548	404	7	be	be	NOUN
ejpam-3548	404	8	-	-	PUNCT
ejpam-3548	404	9	algebras	algebra	NOUN
ejpam-3548	404	10	.	.	PUNCT
ejpam-3548	405	1	scientiae	scientiae	PROPN
ejpam-3548	405	2	mathematicae	mathematicae	PROPN
ejpam-3548	405	3	japonicae	japonicae	PROPN
ejpam-3548	405	4	,	,	PUNCT
ejpam-3548	405	5	68(2):279–285	68(2):279–285	PROPN
ejpam-3548	405	6	,	,	PUNCT
ejpam-3548	405	7	2008	2008	NUM
ejpam-3548	405	8	.	.	PUNCT
ejpam-3548	406	1	[	[	X
ejpam-3548	406	2	2	2	X
ejpam-3548	406	3	]	]	PUNCT
ejpam-3548	406	4	j.	j.	PROPN
ejpam-3548	406	5	dugundji	dugundji	PROPN
ejpam-3548	406	6	.	.	PUNCT
ejpam-3548	406	7	topology	topology	PROPN
ejpam-3548	406	8	.	.	PUNCT
ejpam-3548	407	1	prentice	prentice	PROPN
ejpam-3548	407	2	hall	hall	PROPN
ejpam-3548	407	3	of	of	ADP
ejpam-3548	407	4	india	india	PROPN
ejpam-3548	407	5	private	private	PROPN
ejpam-3548	407	6	ltd	ltd	PROPN
ejpam-3548	407	7	.	.	PROPN
ejpam-3548	407	8	,	,	PUNCT
ejpam-3548	407	9	new	new	PROPN
ejpam-3548	407	10	delhi	delhi	PROPN
ejpam-3548	407	11	,	,	PUNCT
ejpam-3548	407	12	1975	1975	NUM
ejpam-3548	407	13	.	.	PUNCT
ejpam-3548	408	1	[	[	X
ejpam-3548	408	2	3	3	NUM
ejpam-3548	408	3	]	]	X
ejpam-3548	408	4	y.	y.	PROPN
ejpam-3548	408	5	imai	imai	PROPN
ejpam-3548	408	6	and	and	CCONJ
ejpam-3548	408	7	k.	k.	PROPN
ejpam-3548	408	8	iséki	iséki	PROPN
ejpam-3548	408	9	.	.	PROPN
ejpam-3548	409	1	on	on	ADP
ejpam-3548	409	2	axiom	axiom	NOUN
ejpam-3548	409	3	systems	system	NOUN
ejpam-3548	409	4	of	of	ADP
ejpam-3548	409	5	propositional	propositional	ADJ
ejpam-3548	409	6	calculi	calculi	PROPN
ejpam-3548	409	7	xiv	xiv	PROPN
ejpam-3548	409	8	.	.	PUNCT
ejpam-3548	409	9	proc	proc	PROPN
ejpam-3548	409	10	.	.	PUNCT
ejpam-3548	410	1	japan	japan	PROPN
ejpam-3548	410	2	academy	academy	PROPN
ejpam-3548	410	3	,	,	PUNCT
ejpam-3548	410	4	42:19–22	42:19–22	NUM
ejpam-3548	410	5	,	,	PUNCT
ejpam-3548	410	6	1996	1996	NUM
ejpam-3548	410	7	.	.	PUNCT
ejpam-3548	411	1	[	[	X
ejpam-3548	411	2	4	4	NUM
ejpam-3548	411	3	]	]	X
ejpam-3548	411	4	h.s	h.s	PROPN
ejpam-3548	411	5	.	.	PROPN
ejpam-3548	411	6	kim	kim	PROPN
ejpam-3548	411	7	and	and	CCONJ
ejpam-3548	411	8	y.h	y.h	PROPN
ejpam-3548	411	9	.	.	PROPN
ejpam-3548	411	10	kim	kim	PROPN
ejpam-3548	411	11	.	.	PUNCT
ejpam-3548	412	1	on	on	ADP
ejpam-3548	412	2	be	be	AUX
ejpam-3548	412	3	-	-	PUNCT
ejpam-3548	412	4	algebras	algebra	NOUN
ejpam-3548	412	5	.	.	PUNCT
ejpam-3548	413	1	scientiae	scientiae	PROPN
ejpam-3548	413	2	mathematicae	mathematicae	VERB
ejpam-3548	413	3	japonicae	japonicae	PROPN
ejpam-3548	413	4	online	online	ADV
ejpam-3548	413	5	,	,	PUNCT
ejpam-3548	413	6	pages	page	NOUN
ejpam-3548	413	7	1299–1302	1299–1302	NUM
ejpam-3548	413	8	,	,	PUNCT
ejpam-3548	413	9	2004	2004	NUM
ejpam-3548	413	10	.	.	PUNCT
ejpam-3548	414	1	[	[	X
ejpam-3548	414	2	5	5	X
ejpam-3548	414	3	]	]	X
ejpam-3548	414	4	k.h	k.h	PROPN
ejpam-3548	414	5	.	.	PROPN
ejpam-3548	414	6	kim	kim	PROPN
ejpam-3548	414	7	and	and	CCONJ
ejpam-3548	414	8	y.h	y.h	PROPN
ejpam-3548	414	9	.	.	PROPN
ejpam-3548	414	10	yon	yon	PROPN
ejpam-3548	414	11	.	.	PUNCT
ejpam-3548	415	1	dual	dual	ADJ
ejpam-3548	415	2	bck	bck	NOUN
ejpam-3548	415	3	-	-	PUNCT
ejpam-3548	415	4	algebra	algebra	PROPN
ejpam-3548	415	5	and	and	CCONJ
ejpam-3548	415	6	mv	mv	NOUN
ejpam-3548	415	7	-	-	NOUN
ejpam-3548	415	8	algebra	algebra	NOUN
ejpam-3548	415	9	.	.	PUNCT
ejpam-3548	416	1	scientiae	scientiae	PROPN
ejpam-3548	416	2	mathematicae	mathematicae	VERB
ejpam-3548	416	3	japonicae	japonicae	PROPN
ejpam-3548	416	4	online	online	ADV
ejpam-3548	416	5	,	,	PUNCT
ejpam-3548	416	6	pages	page	NOUN
ejpam-3548	416	7	393–399	393–399	NUM
ejpam-3548	416	8	,	,	PUNCT
ejpam-3548	416	9	2007	2007	NUM
ejpam-3548	416	10	.	.	PUNCT
ejpam-3548	417	1	[	[	X
ejpam-3548	417	2	6	6	NUM
ejpam-3548	417	3	]	]	PUNCT
ejpam-3548	417	4	s.	s.	PROPN
ejpam-3548	417	5	lipschutz	lipschutz	PROPN
ejpam-3548	417	6	.	.	PUNCT
ejpam-3548	418	1	schaum	schaum	PROPN
ejpam-3548	418	2	’s	’s	PART
ejpam-3548	418	3	outines	outine	NOUN
ejpam-3548	418	4	:	:	PUNCT
ejpam-3548	418	5	general	general	ADJ
ejpam-3548	418	6	topology	topology	NOUN
ejpam-3548	418	7	.	.	PUNCT
ejpam-3548	419	1	schaum	schaum	PROPN
ejpam-3548	419	2	pub	pub	PROPN
ejpam-3548	419	3	.	.	PUNCT
ejpam-3548	420	1	co.	co.	PROPN
ejpam-3548	420	2	,	,	PUNCT
ejpam-3548	420	3	new	new	PROPN
ejpam-3548	420	4	york	york	PROPN
ejpam-3548	420	5	,	,	PUNCT
ejpam-3548	420	6	1965	1965	NUM
ejpam-3548	420	7	.	.	PUNCT
ejpam-3548	421	1	[	[	X
ejpam-3548	421	2	7	7	X
ejpam-3548	421	3	]	]	X
ejpam-3548	421	4	s.	s.	PROPN
ejpam-3548	421	5	mehrshad	mehrshad	VERB
ejpam-3548	421	6	and	and	CCONJ
ejpam-3548	421	7	j.	j.	PROPN
ejpam-3548	421	8	golzarpoor	golzarpoor	PROPN
ejpam-3548	421	9	.	.	PUNCT
ejpam-3548	422	1	on	on	ADP
ejpam-3548	422	2	topological	topological	ADJ
ejpam-3548	422	3	be	be	NOUN
ejpam-3548	422	4	-	-	PUNCT
ejpam-3548	422	5	algebras	algebra	NOUN
ejpam-3548	422	6	.	.	PUNCT
ejpam-3548	423	1	mathematica	mathematica	PROPN
ejpam-3548	423	2	moravica	moravica	PROPN
ejpam-3548	423	3	,	,	PUNCT
ejpam-3548	423	4	21(2):1–13	21(2):1–13	NUM
ejpam-3548	423	5	,	,	PUNCT
ejpam-3548	423	6	2017	2017	NUM
ejpam-3548	423	7	.	.	PUNCT
ejpam-3548	424	1	[	[	X
ejpam-3548	424	2	8	8	NUM
ejpam-3548	424	3	]	]	X
ejpam-3548	424	4	s.r	s.r	PROPN
ejpam-3548	424	5	.	.	PROPN
ejpam-3548	424	6	mukkamala	mukkamala	PROPN
ejpam-3548	424	7	.	.	PUNCT
ejpam-3548	425	1	a	a	DET
ejpam-3548	425	2	course	course	NOUN
ejpam-3548	425	3	in	in	ADP
ejpam-3548	425	4	be	be	NOUN
ejpam-3548	425	5	-	-	PUNCT
ejpam-3548	425	6	algebra	algebra	NOUN
ejpam-3548	425	7	.	.	PUNCT
ejpam-3548	426	1	springer	springer	NOUN
ejpam-3548	426	2	nature	nature	PROPN
ejpam-3548	426	3	singapore	singapore	PROPN
ejpam-3548	426	4	pte	pte	PROPN
ejpam-3548	426	5	ltd	ltd	PROPN
ejpam-3548	426	6	.	.	PROPN
ejpam-3548	426	7	,	,	PUNCT
ejpam-3548	426	8	singapore	singapore	PROPN
ejpam-3548	426	9	,	,	PUNCT
ejpam-3548	426	10	2018	2018	NUM
ejpam-3548	426	11	.	.	PUNCT
ejpam-3548	427	1	[	[	X
ejpam-3548	427	2	9	9	NUM
ejpam-3548	427	3	]	]	X
ejpam-3548	427	4	s.z	s.z	PROPN
ejpam-3548	427	5	.	.	PROPN
ejpam-3548	427	6	song	song	NOUN
ejpam-3548	427	7	and	and	CCONJ
ejpam-3548	427	8	et.al	et.al	NOUN
ejpam-3548	427	9	.	.	PUNCT
ejpam-3548	428	1	fuzzy	fuzzy	ADJ
ejpam-3548	428	2	ideals	ideal	NOUN
ejpam-3548	428	3	in	in	ADP
ejpam-3548	428	4	be	be	NOUN
ejpam-3548	428	5	-	-	PUNCT
ejpam-3548	428	6	algebras	algebra	NOUN
ejpam-3548	428	7	.	.	PUNCT
ejpam-3548	429	1	bull	bull	NOUN
ejpam-3548	429	2	.	.	PUNCT
ejpam-3548	430	1	malays	malays	PROPN
ejpam-3548	430	2	.	.	PUNCT
ejpam-3548	431	1	math	math	NOUN
ejpam-3548	431	2	.	.	PUNCT
ejpam-3548	432	1	sci	sci	PROPN
ejpam-3548	432	2	.	.	PROPN
ejpam-3548	432	3	soc	soc	PROPN
ejpam-3548	432	4	.	.	PUNCT
ejpam-3548	432	5	,	,	PUNCT
ejpam-3548	432	6	33:147–153	33:147–153	NUM
ejpam-3548	432	7	,	,	PUNCT
ejpam-3548	432	8	2010	2010	NUM
ejpam-3548	432	9	.	.	PUNCT
ejpam-3548	433	1	[	[	X
ejpam-3548	433	2	10	10	NUM
ejpam-3548	433	3	]	]	X
ejpam-3548	433	4	l.a	l.a	PROPN
ejpam-3548	433	5	.	.	PROPN
ejpam-3548	433	6	steen	steen	PROPN
ejpam-3548	433	7	and	and	CCONJ
ejpam-3548	433	8	j.a	j.a	PROPN
ejpam-3548	433	9	.	.	PROPN
ejpam-3548	434	1	seebach	seebach	PROPN
ejpam-3548	434	2	jr	jr	PROPN
ejpam-3548	434	3	.	.	PROPN
ejpam-3548	434	4	counterexamples	counterexample	NOUN
ejpam-3548	434	5	in	in	ADP
ejpam-3548	434	6	topology	topology	NOUN
ejpam-3548	434	7	.	.	PUNCT
ejpam-3548	435	1	new	new	PROPN
ejpam-3548	435	2	york	york	PROPN
ejpam-3548	435	3	:	:	PUNCT
ejpam-3548	435	4	springerverlag	springerverlag	PROPN
ejpam-3548	435	5	,	,	PUNCT
ejpam-3548	435	6	berlin	berlin	PROPN
ejpam-3548	435	7	,	,	PUNCT
ejpam-3548	435	8	1978	1978	NUM
ejpam-3548	435	9	.	.	PUNCT