id	sid	tid	token	lemma	pos
ejpam-3559	1	1	european	european	PROPN
ejpam-3559	1	2	journal	journal	PROPN
ejpam-3559	1	3	of	of	ADP
ejpam-3559	1	4	pure	pure	ADJ
ejpam-3559	1	5	and	and	CCONJ
ejpam-3559	1	6	applied	apply	VERB
ejpam-3559	1	7	mathematics	mathematic	NOUN
ejpam-3559	1	8	vol	vol	NOUN
ejpam-3559	1	9	.	.	PROPN
ejpam-3559	2	1	12	12	NUM
ejpam-3559	2	2	,	,	PUNCT
ejpam-3559	2	3	no	no	INTJ
ejpam-3559	2	4	.	.	NOUN
ejpam-3559	2	5	4	4	NUM
ejpam-3559	2	6	,	,	PUNCT
ejpam-3559	2	7	2019	2019	NUM
ejpam-3559	2	8	,	,	PUNCT
ejpam-3559	2	9	1524	1524	NUM
ejpam-3559	2	10	-	-	SYM
ejpam-3559	2	11	1532	1532	NUM
ejpam-3559	2	12	issn	issn	PROPN
ejpam-3559	2	13	1307	1307	NUM
ejpam-3559	2	14	-	-	SYM
ejpam-3559	2	15	5543	5543	NUM
ejpam-3559	2	16	–	–	PUNCT
ejpam-3559	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3559	2	18	published	publish	VERB
ejpam-3559	2	19	by	by	ADP
ejpam-3559	2	20	new	new	PROPN
ejpam-3559	2	21	york	york	PROPN
ejpam-3559	2	22	business	business	PROPN
ejpam-3559	2	23	global	global	ADJ
ejpam-3559	2	24	topologies	topology	NOUN
ejpam-3559	2	25	on	on	ADP
ejpam-3559	2	26	a	a	DET
ejpam-3559	2	27	hyper	hyper	ADJ
ejpam-3559	2	28	sum	sum	NOUN
ejpam-3559	2	29	and	and	CCONJ
ejpam-3559	2	30	hyper	hyper	ADJ
ejpam-3559	2	31	product	product	NOUN
ejpam-3559	2	32	of	of	ADP
ejpam-3559	2	33	two	two	NUM
ejpam-3559	2	34	hyper	hyper	ADJ
ejpam-3559	2	35	bck	bck	NOUN
ejpam-3559	2	36	-	-	PUNCT
ejpam-3559	2	37	algebras	algebras	PROPN
ejpam-3559	2	38	rachel	rachel	PROPN
ejpam-3559	2	39	m.	m.	PROPN
ejpam-3559	2	40	patangan1	patangan1	PROPN
ejpam-3559	2	41	,	,	PUNCT
ejpam-3559	2	42	sergio	sergio	PROPN
ejpam-3559	2	43	r.	r.	PROPN
ejpam-3559	2	44	canoy	canoy	PROPN
ejpam-3559	2	45	,	,	PUNCT
ejpam-3559	2	46	jr.2,∗	jr.2,∗	PROPN
ejpam-3559	2	47	1	1	NUM
ejpam-3559	2	48	department	department	NOUN
ejpam-3559	2	49	of	of	ADP
ejpam-3559	2	50	applied	apply	VERB
ejpam-3559	2	51	mathematics	mathematic	NOUN
ejpam-3559	2	52	,	,	PUNCT
ejpam-3559	2	53	college	college	NOUN
ejpam-3559	2	54	of	of	ADP
ejpam-3559	2	55	arts	art	NOUN
ejpam-3559	2	56	and	and	CCONJ
ejpam-3559	2	57	science	science	NOUN
ejpam-3559	2	58	,	,	PUNCT
ejpam-3559	2	59	agusan	agusan	ADJ
ejpam-3559	2	60	del	del	PROPN
ejpam-3559	2	61	sur	sur	PROPN
ejpam-3559	2	62	state	state	PROPN
ejpam-3559	2	63	college	college	PROPN
ejpam-3559	2	64	of	of	ADP
ejpam-3559	2	65	agriculture	agriculture	NOUN
ejpam-3559	2	66	and	and	CCONJ
ejpam-3559	2	67	technology	technology	NOUN
ejpam-3559	2	68	,	,	PUNCT
ejpam-3559	2	69	bunawan	bunawan	PROPN
ejpam-3559	2	70	,	,	PUNCT
ejpam-3559	2	71	agusan	agusan	PROPN
ejpam-3559	2	72	del	del	PROPN
ejpam-3559	2	73	sur	sur	PROPN
ejpam-3559	2	74	,	,	PUNCT
ejpam-3559	2	75	philippines	philippines	PROPN
ejpam-3559	2	76	2	2	NUM
ejpam-3559	2	77	department	department	NOUN
ejpam-3559	2	78	of	of	ADP
ejpam-3559	2	79	mathematics	mathematic	NOUN
ejpam-3559	2	80	and	and	CCONJ
ejpam-3559	2	81	statistics	statistic	NOUN
ejpam-3559	2	82	,	,	PUNCT
ejpam-3559	2	83	college	college	NOUN
ejpam-3559	2	84	of	of	ADP
ejpam-3559	2	85	science	science	NOUN
ejpam-3559	2	86	and	and	CCONJ
ejpam-3559	2	87	mathematics	mathematic	NOUN
ejpam-3559	2	88	,	,	PUNCT
ejpam-3559	2	89	center	center	NOUN
ejpam-3559	2	90	for	for	ADP
ejpam-3559	2	91	graph	graph	NOUN
ejpam-3559	2	92	theory	theory	NOUN
ejpam-3559	2	93	,	,	PUNCT
ejpam-3559	2	94	algebra	algebra	NOUN
ejpam-3559	2	95	,	,	PUNCT
ejpam-3559	2	96	and	and	CCONJ
ejpam-3559	2	97	analysis	analysis	NOUN
ejpam-3559	2	98	-	-	PUNCT
ejpam-3559	2	99	prism	prism	NOUN
ejpam-3559	2	100	,	,	PUNCT
ejpam-3559	2	101	mindanao	mindanao	PROPN
ejpam-3559	2	102	state	state	PROPN
ejpam-3559	2	103	university	university	PROPN
ejpam-3559	2	104	iligan	iligan	PROPN
ejpam-3559	2	105	institute	institute	PROPN
ejpam-3559	2	106	of	of	ADP
ejpam-3559	2	107	technology	technology	PROPN
ejpam-3559	2	108	,	,	PUNCT
ejpam-3559	2	109	9200	9200	NUM
ejpam-3559	2	110	,	,	PUNCT
ejpam-3559	2	111	iligan	iligan	ADJ
ejpam-3559	2	112	city	city	NOUN
ejpam-3559	2	113	,	,	PUNCT
ejpam-3559	2	114	philippines	philippine	NOUN
ejpam-3559	2	115	abstract	abstract	ADJ
ejpam-3559	2	116	.	.	PUNCT
ejpam-3559	3	1	given	give	VERB
ejpam-3559	3	2	a	a	DET
ejpam-3559	3	3	hyper	hyper	ADJ
ejpam-3559	3	4	bck	bck	NOUN
ejpam-3559	3	5	-	-	PUNCT
ejpam-3559	3	6	algebra	algebra	NOUN
ejpam-3559	3	7	(	(	PUNCT
ejpam-3559	3	8	h	h	NOUN
ejpam-3559	3	9	,	,	PUNCT
ejpam-3559	3	10	∗	∗	NOUN
ejpam-3559	3	11	,	,	PUNCT
ejpam-3559	3	12	0	0	NUM
ejpam-3559	3	13	)	)	PUNCT
ejpam-3559	3	14	,	,	PUNCT
ejpam-3559	3	15	each	each	PRON
ejpam-3559	3	16	of	of	ADP
ejpam-3559	3	17	the	the	DET
ejpam-3559	3	18	families	family	NOUN
ejpam-3559	3	19	bl(h	bl(h	PUNCT
ejpam-3559	3	20	)	)	PUNCT
ejpam-3559	3	21	=	=	PRON
ejpam-3559	3	22	{	{	PUNCT
ejpam-3559	3	23	lh(a	lh(a	NOUN
ejpam-3559	3	24	)	)	PUNCT
ejpam-3559	3	25	:	:	PUNCT
ejpam-3559	3	26	∅	∅	NOUN
ejpam-3559	3	27	6=	6=	ADP
ejpam-3559	3	28	a	a	DET
ejpam-3559	3	29	⊆	⊆	NUM
ejpam-3559	3	30	h	h	NOUN
ejpam-3559	3	31	}	}	PUNCT
ejpam-3559	3	32	and	and	CCONJ
ejpam-3559	3	33	br(h	br(h	NUM
ejpam-3559	3	34	)	)	PUNCT
ejpam-3559	3	35	=	=	PRON
ejpam-3559	3	36	{	{	PUNCT
ejpam-3559	3	37	rh(a	rh(a	NOUN
ejpam-3559	3	38	)	)	PUNCT
ejpam-3559	3	39	:	:	PUNCT
ejpam-3559	3	40	∅	∅	NOUN
ejpam-3559	3	41	6=	6=	ADP
ejpam-3559	3	42	a	a	DET
ejpam-3559	3	43	⊆	⊆	NUM
ejpam-3559	3	44	h	h	NOUN
ejpam-3559	3	45	}	}	PUNCT
ejpam-3559	3	46	forms	form	VERB
ejpam-3559	3	47	a	a	DET
ejpam-3559	3	48	base	base	NOUN
ejpam-3559	3	49	for	for	ADP
ejpam-3559	3	50	some	some	DET
ejpam-3559	3	51	topology	topology	NOUN
ejpam-3559	3	52	on	on	ADP
ejpam-3559	3	53	h	h	NOUN
ejpam-3559	3	54	,	,	PUNCT
ejpam-3559	3	55	where	where	SCONJ
ejpam-3559	3	56	lh(a	lh(a	NOUN
ejpam-3559	3	57	)	)	PUNCT
ejpam-3559	3	58	=	=	PRON
ejpam-3559	4	1	{	{	PUNCT
ejpam-3559	4	2	x	x	PUNCT
ejpam-3559	4	3	∈	∈	PROPN
ejpam-3559	4	4	h	h	NOUN
ejpam-3559	4	5	:	:	PUNCT
ejpam-3559	4	6	x	x	PUNCT
ejpam-3559	4	7	�	�	PROPN
ejpam-3559	4	8	a	a	PRON
ejpam-3559	4	9	,	,	PUNCT
ejpam-3559	4	10	∀a	∀a	NOUN
ejpam-3559	4	11	∈	∈	NOUN
ejpam-3559	4	12	a	a	PRON
ejpam-3559	4	13	}	}	PUNCT
ejpam-3559	4	14	and	and	CCONJ
ejpam-3559	4	15	rh(a	rh(a	NUM
ejpam-3559	4	16	)	)	PUNCT
ejpam-3559	4	17	=	=	PRON
ejpam-3559	4	18	{	{	PUNCT
ejpam-3559	4	19	x	x	PUNCT
ejpam-3559	4	20	∈	∈	PROPN
ejpam-3559	4	21	h	h	NOUN
ejpam-3559	4	22	:	:	PUNCT
ejpam-3559	4	23	a	a	DET
ejpam-3559	4	24	�	�	PROPN
ejpam-3559	4	25	x	x	X
ejpam-3559	4	26	,	,	PUNCT
ejpam-3559	4	27	∀a	∀a	NOUN
ejpam-3559	4	28	∈	∈	NOUN
ejpam-3559	4	29	a	a	X
ejpam-3559	4	30	}	}	PUNCT
ejpam-3559	4	31	for	for	ADP
ejpam-3559	4	32	any	any	DET
ejpam-3559	4	33	subset	subset	NOUN
ejpam-3559	4	34	a	a	PRON
ejpam-3559	4	35	of	of	ADP
ejpam-3559	4	36	h.	h.	NOUN
ejpam-3559	4	37	in	in	ADP
ejpam-3559	4	38	this	this	DET
ejpam-3559	4	39	paper	paper	NOUN
ejpam-3559	4	40	,	,	PUNCT
ejpam-3559	4	41	we	we	PRON
ejpam-3559	4	42	determine	determine	VERB
ejpam-3559	4	43	the	the	DET
ejpam-3559	4	44	bases	basis	NOUN
ejpam-3559	4	45	of	of	ADP
ejpam-3559	4	46	the	the	DET
ejpam-3559	4	47	topologies	topology	NOUN
ejpam-3559	4	48	induced	induce	VERB
ejpam-3559	4	49	by	by	ADP
ejpam-3559	4	50	the	the	DET
ejpam-3559	4	51	hyper	hyper	ADJ
ejpam-3559	4	52	sum	sum	NOUN
ejpam-3559	4	53	h1	h1	PROPN
ejpam-3559	4	54	⊕h2	⊕h2	NOUN
ejpam-3559	4	55	and	and	CCONJ
ejpam-3559	4	56	hyper	hyper	ADJ
ejpam-3559	4	57	product	product	NOUN
ejpam-3559	4	58	h1	h1	PROPN
ejpam-3559	4	59	×h2	×h2	PROPN
ejpam-3559	4	60	,	,	PUNCT
ejpam-3559	4	61	where	where	SCONJ
ejpam-3559	4	62	(	(	PUNCT
ejpam-3559	4	63	h1	h1	PROPN
ejpam-3559	4	64	,	,	PUNCT
ejpam-3559	4	65	∗1	∗1	PROPN
ejpam-3559	4	66	,	,	PUNCT
ejpam-3559	4	67	01	01	NUM
ejpam-3559	4	68	)	)	PUNCT
ejpam-3559	4	69	and	and	CCONJ
ejpam-3559	4	70	(	(	PUNCT
ejpam-3559	4	71	h2	h2	NOUN
ejpam-3559	4	72	,	,	PUNCT
ejpam-3559	4	73	∗2	∗2	PROPN
ejpam-3559	4	74	,	,	PUNCT
ejpam-3559	4	75	02	02	NUM
ejpam-3559	4	76	)	)	PUNCT
ejpam-3559	4	77	are	be	AUX
ejpam-3559	4	78	two	two	NUM
ejpam-3559	4	79	hyper	hyper	ADJ
ejpam-3559	4	80	bck	bck	NOUN
ejpam-3559	4	81	-	-	PUNCT
ejpam-3559	4	82	algebras	algebras	PROPN
ejpam-3559	4	83	.	.	PUNCT
ejpam-3559	5	1	2010	2010	NUM
ejpam-3559	5	2	mathematics	mathematic	NOUN
ejpam-3559	5	3	subject	subject	NOUN
ejpam-3559	5	4	classifications	classification	NOUN
ejpam-3559	5	5	:	:	PUNCT
ejpam-3559	5	6	06f35	06f35	NUM
ejpam-3559	5	7	,	,	PUNCT
ejpam-3559	5	8	03g25	03g25	NOUN
ejpam-3559	5	9	key	key	ADJ
ejpam-3559	5	10	words	word	NOUN
ejpam-3559	5	11	and	and	CCONJ
ejpam-3559	5	12	phrases	phrase	NOUN
ejpam-3559	5	13	:	:	PUNCT
ejpam-3559	5	14	hyper	hyper	ADJ
ejpam-3559	5	15	bck	bck	NOUN
ejpam-3559	5	16	-	-	PUNCT
ejpam-3559	5	17	algebra	algebra	NOUN
ejpam-3559	5	18	,	,	PUNCT
ejpam-3559	5	19	hyper	hyper	ADJ
ejpam-3559	5	20	sum	sum	NOUN
ejpam-3559	5	21	,	,	PUNCT
ejpam-3559	5	22	hyper	hyper	ADJ
ejpam-3559	5	23	product	product	NOUN
ejpam-3559	5	24	1	1	NUM
ejpam-3559	5	25	.	.	PUNCT
ejpam-3559	5	26	introduction	introduction	NOUN
ejpam-3559	5	27	although	although	SCONJ
ejpam-3559	5	28	algebra	algebra	NOUN
ejpam-3559	5	29	and	and	CCONJ
ejpam-3559	5	30	topology	topology	NOUN
ejpam-3559	5	31	seem	seem	VERB
ejpam-3559	5	32	to	to	PART
ejpam-3559	5	33	differ	differ	VERB
ejpam-3559	5	34	generally	generally	ADV
ejpam-3559	5	35	in	in	ADP
ejpam-3559	5	36	their	their	PRON
ejpam-3559	5	37	nature	nature	NOUN
ejpam-3559	5	38	,	,	PUNCT
ejpam-3559	5	39	they	they	PRON
ejpam-3559	5	40	appear	appear	VERB
ejpam-3559	5	41	together	together	ADV
ejpam-3559	5	42	in	in	ADP
ejpam-3559	5	43	some	some	DET
ejpam-3559	5	44	areas	area	NOUN
ejpam-3559	5	45	of	of	ADP
ejpam-3559	5	46	mathematics	mathematic	NOUN
ejpam-3559	5	47	such	such	ADJ
ejpam-3559	5	48	as	as	ADP
ejpam-3559	5	49	functional	functional	ADJ
ejpam-3559	5	50	analysis	analysis	NOUN
ejpam-3559	5	51	,	,	PUNCT
ejpam-3559	5	52	dynamical	dynamical	ADJ
ejpam-3559	5	53	systems	system	NOUN
ejpam-3559	5	54	,	,	PUNCT
ejpam-3559	5	55	and	and	CCONJ
ejpam-3559	5	56	representation	representation	NOUN
ejpam-3559	5	57	theory	theory	NOUN
ejpam-3559	5	58	.	.	PUNCT
ejpam-3559	6	1	previous	previous	ADJ
ejpam-3559	6	2	studies	study	NOUN
ejpam-3559	6	3	(	(	PUNCT
ejpam-3559	6	4	see	see	VERB
ejpam-3559	6	5	[	[	X
ejpam-3559	6	6	2	2	NUM
ejpam-3559	6	7	]	]	PUNCT
ejpam-3559	6	8	)	)	PUNCT
ejpam-3559	6	9	would	would	AUX
ejpam-3559	6	10	show	show	VERB
ejpam-3559	6	11	the	the	DET
ejpam-3559	6	12	blend	blend	NOUN
ejpam-3559	6	13	of	of	ADP
ejpam-3559	6	14	algebraic	algebraic	PROPN
ejpam-3559	6	15	and	and	CCONJ
ejpam-3559	6	16	of	of	ADP
ejpam-3559	6	17	topological	topological	ADJ
ejpam-3559	6	18	structures	structure	NOUN
ejpam-3559	6	19	.	.	PUNCT
ejpam-3559	7	1	indeed	indeed	ADV
ejpam-3559	7	2	,	,	PUNCT
ejpam-3559	7	3	there	there	PRON
ejpam-3559	7	4	are	be	VERB
ejpam-3559	7	5	various	various	ADJ
ejpam-3559	7	6	ways	way	NOUN
ejpam-3559	7	7	of	of	ADP
ejpam-3559	7	8	introducing	introduce	VERB
ejpam-3559	7	9	a	a	DET
ejpam-3559	7	10	a	a	DET
ejpam-3559	7	11	topological	topological	ADJ
ejpam-3559	7	12	structure	structure	NOUN
ejpam-3559	7	13	in	in	ADP
ejpam-3559	7	14	a	a	DET
ejpam-3559	7	15	given	give	VERB
ejpam-3559	7	16	algebraic	algebraic	ADJ
ejpam-3559	7	17	structure	structure	NOUN
ejpam-3559	7	18	.	.	PUNCT
ejpam-3559	8	1	for	for	ADP
ejpam-3559	8	2	example	example	NOUN
ejpam-3559	8	3	,	,	PUNCT
ejpam-3559	8	4	in	in	ADP
ejpam-3559	8	5	the	the	DET
ejpam-3559	8	6	definition	definition	NOUN
ejpam-3559	8	7	of	of	ADP
ejpam-3559	8	8	a	a	DET
ejpam-3559	8	9	topological	topological	ADJ
ejpam-3559	8	10	group	group	NOUN
ejpam-3559	8	11	,	,	PUNCT
ejpam-3559	8	12	the	the	DET
ejpam-3559	8	13	requirement	requirement	NOUN
ejpam-3559	8	14	imposed	impose	VERB
ejpam-3559	8	15	is	be	AUX
ejpam-3559	8	16	that	that	SCONJ
ejpam-3559	8	17	the	the	DET
ejpam-3559	8	18	topology	topology	NOUN
ejpam-3559	8	19	on	on	ADP
ejpam-3559	8	20	a	a	DET
ejpam-3559	8	21	given	give	VERB
ejpam-3559	8	22	group	group	NOUN
ejpam-3559	8	23	is	be	AUX
ejpam-3559	8	24	the	the	DET
ejpam-3559	8	25	one	one	NUM
ejpam-3559	8	26	that	that	PRON
ejpam-3559	8	27	makes	make	VERB
ejpam-3559	8	28	the	the	DET
ejpam-3559	8	29	multiplication	multiplication	NOUN
ejpam-3559	8	30	and	and	CCONJ
ejpam-3559	8	31	inversion	inversion	NOUN
ejpam-3559	8	32	maps	map	NOUN
ejpam-3559	8	33	continuous	continuous	ADJ
ejpam-3559	8	34	.	.	PUNCT
ejpam-3559	9	1	however	however	ADV
ejpam-3559	9	2	,	,	PUNCT
ejpam-3559	9	3	given	give	VERB
ejpam-3559	9	4	an	an	DET
ejpam-3559	9	5	algebraic	algebraic	ADJ
ejpam-3559	9	6	structure	structure	NOUN
ejpam-3559	9	7	(	(	PUNCT
ejpam-3559	9	8	or	or	CCONJ
ejpam-3559	9	9	hyperstructure	hyperstructure	NOUN
ejpam-3559	9	10	)	)	PUNCT
ejpam-3559	9	11	,	,	PUNCT
ejpam-3559	9	12	it	it	PRON
ejpam-3559	9	13	may	may	AUX
ejpam-3559	9	14	be	be	AUX
ejpam-3559	9	15	possible	possible	ADJ
ejpam-3559	9	16	to	to	PART
ejpam-3559	9	17	find	find	VERB
ejpam-3559	9	18	some	some	DET
ejpam-3559	9	19	family	family	NOUN
ejpam-3559	9	20	of	of	ADP
ejpam-3559	9	21	subsets	subset	NOUN
ejpam-3559	9	22	of	of	ADP
ejpam-3559	9	23	the	the	DET
ejpam-3559	9	24	underlying	underlie	VERB
ejpam-3559	9	25	set	set	NOUN
ejpam-3559	9	26	that	that	PRON
ejpam-3559	9	27	will	will	AUX
ejpam-3559	9	28	serve	serve	VERB
ejpam-3559	9	29	as	as	ADP
ejpam-3559	9	30	base	base	NOUN
ejpam-3559	9	31	for	for	ADP
ejpam-3559	9	32	some	some	DET
ejpam-3559	9	33	topology	topology	NOUN
ejpam-3559	9	34	on	on	ADP
ejpam-3559	9	35	the	the	DET
ejpam-3559	9	36	set	set	NOUN
ejpam-3559	9	37	.	.	PUNCT
ejpam-3559	10	1	this	this	DET
ejpam-3559	10	2	approach	approach	NOUN
ejpam-3559	10	3	can	can	AUX
ejpam-3559	10	4	then	then	ADV
ejpam-3559	10	5	give	give	VERB
ejpam-3559	10	6	rise	rise	NOUN
ejpam-3559	10	7	to	to	ADP
ejpam-3559	10	8	a	a	DET
ejpam-3559	10	9	structure	structure	NOUN
ejpam-3559	10	10	that	that	PRON
ejpam-3559	10	11	is	be	AUX
ejpam-3559	10	12	both	both	CCONJ
ejpam-3559	10	13	algebraic	algebraic	ADJ
ejpam-3559	10	14	and	and	CCONJ
ejpam-3559	10	15	topological	topological	ADJ
ejpam-3559	10	16	.	.	PUNCT
ejpam-3559	11	1	the	the	DET
ejpam-3559	11	2	present	present	ADJ
ejpam-3559	11	3	study	study	NOUN
ejpam-3559	11	4	considers	consider	VERB
ejpam-3559	11	5	an	an	DET
ejpam-3559	11	6	algebraic	algebraic	ADJ
ejpam-3559	11	7	structure	structure	NOUN
ejpam-3559	11	8	which	which	PRON
ejpam-3559	11	9	is	be	AUX
ejpam-3559	11	10	a	a	DET
ejpam-3559	11	11	decendant	decendant	NOUN
ejpam-3559	11	12	of	of	ADP
ejpam-3559	11	13	bckalgebra	bckalgebra	NOUN
ejpam-3559	11	14	,	,	PUNCT
ejpam-3559	11	15	an	an	DET
ejpam-3559	11	16	algebraic	algebraic	ADJ
ejpam-3559	11	17	structure	structure	NOUN
ejpam-3559	11	18	that	that	PRON
ejpam-3559	11	19	was	be	AUX
ejpam-3559	11	20	introduced	introduce	VERB
ejpam-3559	11	21	and	and	CCONJ
ejpam-3559	11	22	investigated	investigate	VERB
ejpam-3559	11	23	by	by	ADP
ejpam-3559	11	24	y.	y.	PROPN
ejpam-3559	11	25	imai	imai	PROPN
ejpam-3559	11	26	and	and	CCONJ
ejpam-3559	11	27	k.	k.	PROPN
ejpam-3559	11	28	iséki	iséki	PROPN
ejpam-3559	12	1	[	[	X
ejpam-3559	12	2	5	5	NUM
ejpam-3559	12	3	]	]	PUNCT
ejpam-3559	12	4	in	in	ADP
ejpam-3559	12	5	1966	1966	NUM
ejpam-3559	12	6	.	.	PUNCT
ejpam-3559	13	1	this	this	DET
ejpam-3559	13	2	variant	variant	NOUN
ejpam-3559	13	3	of	of	ADP
ejpam-3559	13	4	bck	bck	NOUN
ejpam-3559	13	5	-	-	PUNCT
ejpam-3559	13	6	algebra	algebra	NOUN
ejpam-3559	13	7	utilizes	utilize	VERB
ejpam-3559	13	8	the	the	DET
ejpam-3559	13	9	hyperstructure	hyperstructure	NOUN
ejpam-3559	13	10	theory	theory	NOUN
ejpam-3559	13	11	introduced	introduce	VERB
ejpam-3559	13	12	∗corresponding	∗corresponde	VERB
ejpam-3559	13	13	author	author	NOUN
ejpam-3559	13	14	.	.	PUNCT
ejpam-3559	14	1	doi	doi	NOUN
ejpam-3559	14	2	:	:	PUNCT
ejpam-3559	14	3	https://doi.org/10.29020/nybg.ejpam.v12i4.3559	https://doi.org/10.29020/nybg.ejpam.v12i4.3559	NUM
ejpam-3559	14	4	email	email	NOUN
ejpam-3559	14	5	addresses	address	NOUN
ejpam-3559	14	6	:	:	PUNCT
ejpam-3559	14	7	rhapsodistchelar@gmail.com	rhapsodistchelar@gmail.com	X
ejpam-3559	14	8	(	(	PUNCT
ejpam-3559	14	9	r.	r.	NOUN
ejpam-3559	14	10	patangan	patangan	PROPN
ejpam-3559	14	11	)	)	PUNCT
ejpam-3559	14	12	,	,	PUNCT
ejpam-3559	14	13	sergio.canoy@g.msuiit.edu.ph	sergio.canoy@g.msuiit.edu.ph	PROPN
ejpam-3559	14	14	(	(	PUNCT
ejpam-3559	14	15	s.	s.	PROPN
ejpam-3559	14	16	canoy	canoy	PROPN
ejpam-3559	14	17	,	,	PUNCT
ejpam-3559	14	18	jr	jr	PROPN
ejpam-3559	14	19	.	.	PUNCT
ejpam-3559	14	20	)	)	PUNCT
ejpam-3559	14	21	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3559	14	22	1524	1524	NUM
ejpam-3559	15	1	c	c	X
ejpam-3559	15	2	©	©	PROPN
ejpam-3559	15	3	2019	2019	NUM
ejpam-3559	15	4	ejpam	ejpam	NOUN
ejpam-3559	15	5	all	all	DET
ejpam-3559	15	6	rights	right	NOUN
ejpam-3559	15	7	reserved	reserve	VERB
ejpam-3559	15	8	.	.	PUNCT
ejpam-3559	16	1	r.	r.	PROPN
ejpam-3559	16	2	patangan	patangan	PROPN
ejpam-3559	16	3	,	,	PUNCT
ejpam-3559	16	4	s.	s.	PROPN
ejpam-3559	16	5	canoy	canoy	PROPN
ejpam-3559	16	6	,	,	PUNCT
ejpam-3559	16	7	jr	jr	PROPN
ejpam-3559	16	8	.	.	PROPN
ejpam-3559	16	9	/	/	SYM
ejpam-3559	16	10	eur	eur	PROPN
ejpam-3559	16	11	.	.	PUNCT
ejpam-3559	17	1	j.	j.	PROPN
ejpam-3559	17	2	pure	pure	PROPN
ejpam-3559	17	3	appl	appl	PROPN
ejpam-3559	17	4	.	.	PROPN
ejpam-3559	17	5	math	math	PROPN
ejpam-3559	17	6	,	,	PUNCT
ejpam-3559	17	7	12	12	NUM
ejpam-3559	17	8	(	(	PUNCT
ejpam-3559	17	9	4	4	NUM
ejpam-3559	17	10	)	)	PUNCT
ejpam-3559	17	11	(	(	PUNCT
ejpam-3559	17	12	2019	2019	NUM
ejpam-3559	17	13	)	)	PUNCT
ejpam-3559	17	14	,	,	PUNCT
ejpam-3559	17	15	1524	1524	NUM
ejpam-3559	17	16	-	-	SYM
ejpam-3559	17	17	1532	1532	NUM
ejpam-3559	17	18	1525	1525	NUM
ejpam-3559	17	19	by	by	ADP
ejpam-3559	17	20	f.	f.	PROPN
ejpam-3559	17	21	marty	marty	PROPN
ejpam-3559	18	1	[	[	X
ejpam-3559	18	2	7	7	X
ejpam-3559	18	3	]	]	PUNCT
ejpam-3559	18	4	at	at	ADP
ejpam-3559	18	5	the	the	DET
ejpam-3559	18	6	8th	8th	ADJ
ejpam-3559	18	7	congress	congress	PROPN
ejpam-3559	18	8	of	of	ADP
ejpam-3559	18	9	scandinavian	scandinavian	ADJ
ejpam-3559	18	10	mathematicians	mathematician	NOUN
ejpam-3559	18	11	in	in	ADP
ejpam-3559	18	12	1934	1934	NUM
ejpam-3559	18	13	.	.	PUNCT
ejpam-3559	19	1	specifically	specifically	ADV
ejpam-3559	19	2	,	,	PUNCT
ejpam-3559	19	3	y.b	y.b	PROPN
ejpam-3559	19	4	.	.	PROPN
ejpam-3559	19	5	jun	jun	PROPN
ejpam-3559	19	6	et	et	PROPN
ejpam-3559	19	7	al	al	PROPN
ejpam-3559	19	8	.	.	PUNCT
ejpam-3559	20	1	[	[	X
ejpam-3559	20	2	6	6	NUM
ejpam-3559	20	3	]	]	PUNCT
ejpam-3559	20	4	applied	apply	VERB
ejpam-3559	20	5	the	the	DET
ejpam-3559	20	6	hyperstructure	hyperstructure	NOUN
ejpam-3559	20	7	theory	theory	NOUN
ejpam-3559	20	8	to	to	PART
ejpam-3559	20	9	bck	bck	VERB
ejpam-3559	20	10	-	-	PUNCT
ejpam-3559	20	11	algebras	algebras	PROPN
ejpam-3559	20	12	and	and	CCONJ
ejpam-3559	20	13	introduced	introduce	VERB
ejpam-3559	20	14	the	the	DET
ejpam-3559	20	15	notion	notion	NOUN
ejpam-3559	20	16	of	of	ADP
ejpam-3559	20	17	a	a	DET
ejpam-3559	20	18	hyper	hyper	ADJ
ejpam-3559	20	19	bck	bck	NOUN
ejpam-3559	20	20	-	-	PUNCT
ejpam-3559	20	21	algebra	algebra	NOUN
ejpam-3559	20	22	.	.	PUNCT
ejpam-3559	21	1	recently	recently	ADV
ejpam-3559	21	2	,	,	PUNCT
ejpam-3559	21	3	patangan	patangan	NOUN
ejpam-3559	21	4	and	and	CCONJ
ejpam-3559	21	5	canoy	canoy	ADJ
ejpam-3559	21	6	[	[	X
ejpam-3559	21	7	8	8	NUM
ejpam-3559	21	8	,	,	PUNCT
ejpam-3559	21	9	9	9	NUM
ejpam-3559	21	10	]	]	PUNCT
ejpam-3559	21	11	showed	show	VERB
ejpam-3559	21	12	that	that	SCONJ
ejpam-3559	21	13	the	the	DET
ejpam-3559	21	14	families	family	NOUN
ejpam-3559	21	15	bl(h	bl(h	PUNCT
ejpam-3559	21	16	)	)	PUNCT
ejpam-3559	21	17	=	=	PRON
ejpam-3559	21	18	{	{	PUNCT
ejpam-3559	21	19	lh(a	lh(a	NOUN
ejpam-3559	21	20	)	)	PUNCT
ejpam-3559	21	21	:	:	PUNCT
ejpam-3559	21	22	∅	∅	NOUN
ejpam-3559	21	23	6=	6=	ADP
ejpam-3559	21	24	a	a	DET
ejpam-3559	21	25	⊆	⊆	NUM
ejpam-3559	21	26	h	h	NOUN
ejpam-3559	21	27	}	}	PUNCT
ejpam-3559	21	28	and	and	CCONJ
ejpam-3559	21	29	br(h	br(h	NUM
ejpam-3559	21	30	)	)	PUNCT
ejpam-3559	21	31	=	=	PRON
ejpam-3559	21	32	{	{	PUNCT
ejpam-3559	21	33	rh(a	rh(a	NOUN
ejpam-3559	21	34	)	)	PUNCT
ejpam-3559	21	35	:	:	PUNCT
ejpam-3559	21	36	∅	∅	NOUN
ejpam-3559	21	37	6=	6=	ADP
ejpam-3559	21	38	a	a	DET
ejpam-3559	21	39	⊆	⊆	NUM
ejpam-3559	21	40	h	h	NOUN
ejpam-3559	21	41	}	}	PUNCT
ejpam-3559	21	42	,	,	PUNCT
ejpam-3559	21	43	where	where	SCONJ
ejpam-3559	21	44	lh(a	lh(a	NOUN
ejpam-3559	21	45	)	)	PUNCT
ejpam-3559	21	46	=	=	PRON
ejpam-3559	21	47	{	{	PUNCT
ejpam-3559	21	48	x	x	PUNCT
ejpam-3559	21	49	∈	∈	PROPN
ejpam-3559	21	50	h	h	NOUN
ejpam-3559	21	51	:	:	PUNCT
ejpam-3559	21	52	x	x	PUNCT
ejpam-3559	21	53	�	�	PROPN
ejpam-3559	21	54	a	a	PRON
ejpam-3559	21	55	,	,	PUNCT
ejpam-3559	21	56	∀a	∀a	NOUN
ejpam-3559	21	57	∈	∈	NOUN
ejpam-3559	21	58	a	a	PRON
ejpam-3559	21	59	}	}	PUNCT
ejpam-3559	21	60	and	and	CCONJ
ejpam-3559	21	61	rh(a	rh(a	NUM
ejpam-3559	21	62	)	)	PUNCT
ejpam-3559	21	63	=	=	PRON
ejpam-3559	22	1	{	{	PUNCT
ejpam-3559	22	2	x	x	PUNCT
ejpam-3559	22	3	∈	∈	PROPN
ejpam-3559	22	4	h	h	NOUN
ejpam-3559	22	5	:	:	PUNCT
ejpam-3559	22	6	a	a	DET
ejpam-3559	22	7	�	�	PROPN
ejpam-3559	22	8	x	x	X
ejpam-3559	22	9	,	,	PUNCT
ejpam-3559	22	10	∀a	∀a	NOUN
ejpam-3559	22	11	∈	∈	NOUN
ejpam-3559	22	12	a	a	X
ejpam-3559	22	13	}	}	PUNCT
ejpam-3559	22	14	for	for	ADP
ejpam-3559	22	15	any	any	DET
ejpam-3559	22	16	subset	subset	NOUN
ejpam-3559	22	17	a	a	PRON
ejpam-3559	22	18	of	of	ADP
ejpam-3559	22	19	h	h	NOUN
ejpam-3559	22	20	,	,	PUNCT
ejpam-3559	22	21	are	be	AUX
ejpam-3559	22	22	bases	basis	NOUN
ejpam-3559	22	23	for	for	ADP
ejpam-3559	22	24	some	some	DET
ejpam-3559	22	25	topologies	topology	NOUN
ejpam-3559	22	26	on	on	ADP
ejpam-3559	22	27	a	a	DET
ejpam-3559	22	28	hyper	hyper	ADJ
ejpam-3559	22	29	bck	bck	NOUN
ejpam-3559	22	30	-	-	PUNCT
ejpam-3559	22	31	algebra	algebra	NOUN
ejpam-3559	22	32	(	(	PUNCT
ejpam-3559	22	33	h	h	NOUN
ejpam-3559	22	34	,	,	PUNCT
ejpam-3559	22	35	∗	∗	NOUN
ejpam-3559	22	36	,	,	PUNCT
ejpam-3559	22	37	0	0	NUM
ejpam-3559	22	38	)	)	PUNCT
ejpam-3559	22	39	.	.	PUNCT
ejpam-3559	23	1	thus	thus	ADV
ejpam-3559	23	2	,	,	PUNCT
ejpam-3559	23	3	given	give	VERB
ejpam-3559	23	4	a	a	DET
ejpam-3559	23	5	hyper	hyper	ADJ
ejpam-3559	23	6	bck	bck	NOUN
ejpam-3559	23	7	-	-	PUNCT
ejpam-3559	23	8	algebra	algebra	NOUN
ejpam-3559	23	9	,	,	PUNCT
ejpam-3559	23	10	two	two	NUM
ejpam-3559	23	11	different	different	ADJ
ejpam-3559	23	12	topological	topological	ADJ
ejpam-3559	23	13	structures	structure	NOUN
ejpam-3559	23	14	are	be	AUX
ejpam-3559	23	15	generated	generate	VERB
ejpam-3559	23	16	and	and	CCONJ
ejpam-3559	23	17	investigated	investigate	VERB
ejpam-3559	23	18	.	.	PUNCT
ejpam-3559	24	1	a	a	DET
ejpam-3559	24	2	hyper	hyper	ADJ
ejpam-3559	24	3	bck	bck	NOUN
ejpam-3559	24	4	-	-	PUNCT
ejpam-3559	24	5	algebra	algebra	NOUN
ejpam-3559	24	6	is	be	AUX
ejpam-3559	24	7	a	a	DET
ejpam-3559	24	8	nonempty	nonempty	ADV
ejpam-3559	24	9	set	set	VERB
ejpam-3559	24	10	h	h	NOUN
ejpam-3559	24	11	endowed	endow	VERB
ejpam-3559	24	12	with	with	ADP
ejpam-3559	24	13	a	a	DET
ejpam-3559	24	14	hyperoperation	hyperoperation	NOUN
ejpam-3559	24	15	“	"	PUNCT
ejpam-3559	24	16	∗	∗	NOUN
ejpam-3559	24	17	”	"	PUNCT
ejpam-3559	24	18	and	and	CCONJ
ejpam-3559	24	19	a	a	DET
ejpam-3559	24	20	constant	constant	ADJ
ejpam-3559	24	21	0	0	NUM
ejpam-3559	24	22	satisfying	satisfy	VERB
ejpam-3559	24	23	the	the	DET
ejpam-3559	24	24	following	follow	VERB
ejpam-3559	24	25	axioms	axiom	NOUN
ejpam-3559	24	26	:	:	PUNCT
ejpam-3559	24	27	for	for	ADP
ejpam-3559	24	28	all	all	DET
ejpam-3559	24	29	x	x	NOUN
ejpam-3559	24	30	,	,	PUNCT
ejpam-3559	24	31	y	y	PROPN
ejpam-3559	24	32	,	,	PUNCT
ejpam-3559	24	33	z	z	PROPN
ejpam-3559	24	34	∈	∈	PROPN
ejpam-3559	24	35	h	h	NOUN
ejpam-3559	24	36	,	,	PUNCT
ejpam-3559	24	37	(	(	PUNCT
ejpam-3559	24	38	h1	h1	PROPN
ejpam-3559	24	39	)	)	PUNCT
ejpam-3559	24	40	(	(	PUNCT
ejpam-3559	24	41	x	x	SYM
ejpam-3559	24	42	∗	∗	PROPN
ejpam-3559	24	43	z	z	NOUN
ejpam-3559	24	44	)	)	PUNCT
ejpam-3559	24	45	∗	∗	NOUN
ejpam-3559	24	46	(	(	PUNCT
ejpam-3559	24	47	y	y	PROPN
ejpam-3559	24	48	∗	∗	PROPN
ejpam-3559	24	49	z	z	PROPN
ejpam-3559	24	50	)	)	PUNCT
ejpam-3559	24	51	�	�	PROPN
ejpam-3559	24	52	x	x	PROPN
ejpam-3559	24	53	∗	∗	PROPN
ejpam-3559	24	54	y	y	PROPN
ejpam-3559	24	55	,	,	PUNCT
ejpam-3559	24	56	(	(	PUNCT
ejpam-3559	24	57	h2	h2	NOUN
ejpam-3559	24	58	)	)	PUNCT
ejpam-3559	24	59	(	(	PUNCT
ejpam-3559	24	60	x	x	SYM
ejpam-3559	24	61	∗	∗	PROPN
ejpam-3559	24	62	y	y	NOUN
ejpam-3559	24	63	)	)	PUNCT
ejpam-3559	24	64	∗	∗	NOUN
ejpam-3559	24	65	z	z	NOUN
ejpam-3559	24	66	=	=	SYM
ejpam-3559	24	67	(	(	PUNCT
ejpam-3559	24	68	x	x	X
ejpam-3559	24	69	∗	∗	PROPN
ejpam-3559	24	70	z	z	NOUN
ejpam-3559	24	71	)	)	PUNCT
ejpam-3559	24	72	∗	∗	PROPN
ejpam-3559	24	73	y	y	PROPN
ejpam-3559	24	74	,	,	PUNCT
ejpam-3559	24	75	(	(	PUNCT
ejpam-3559	24	76	h3	h3	NOUN
ejpam-3559	24	77	)	)	PUNCT
ejpam-3559	24	78	x	x	PUNCT
ejpam-3559	24	79	∗h	∗h	NOUN
ejpam-3559	24	80	�	�	PROPN
ejpam-3559	24	81	x	x	SYM
ejpam-3559	24	82	,	,	PUNCT
ejpam-3559	24	83	(	(	PUNCT
ejpam-3559	24	84	h4	h4	PROPN
ejpam-3559	24	85	)	)	PUNCT
ejpam-3559	24	86	x	x	NOUN
ejpam-3559	24	87	�	�	PROPN
ejpam-3559	24	88	y	y	PROPN
ejpam-3559	24	89	and	and	CCONJ
ejpam-3559	24	90	y	y	PROPN
ejpam-3559	24	91	�	�	PROPN
ejpam-3559	24	92	x	x	PUNCT
ejpam-3559	24	93	imply	imply	VERB
ejpam-3559	24	94	x	x	X
ejpam-3559	24	95	=	=	SYM
ejpam-3559	24	96	y	y	PROPN
ejpam-3559	24	97	,	,	PUNCT
ejpam-3559	24	98	where	where	SCONJ
ejpam-3559	24	99	for	for	ADP
ejpam-3559	24	100	every	every	DET
ejpam-3559	24	101	a	a	PROPN
ejpam-3559	24	102	,	,	PUNCT
ejpam-3559	24	103	b	b	PROPN
ejpam-3559	24	104	⊆	⊆	NUM
ejpam-3559	24	105	h	h	NOUN
ejpam-3559	24	106	,	,	PUNCT
ejpam-3559	24	107	a	a	DET
ejpam-3559	24	108	�	�	PROPN
ejpam-3559	24	109	b	b	PROPN
ejpam-3559	24	110	if	if	SCONJ
ejpam-3559	25	1	and	and	CCONJ
ejpam-3559	25	2	only	only	ADV
ejpam-3559	25	3	if	if	SCONJ
ejpam-3559	25	4	for	for	ADP
ejpam-3559	25	5	each	each	DET
ejpam-3559	25	6	a	a	DET
ejpam-3559	25	7	∈	∈	PROPN
ejpam-3559	25	8	a	a	PRON
ejpam-3559	25	9	,	,	PUNCT
ejpam-3559	25	10	there	there	PRON
ejpam-3559	25	11	exists	exist	VERB
ejpam-3559	25	12	b	b	PROPN
ejpam-3559	25	13	∈	∈	PROPN
ejpam-3559	25	14	b	b	NOUN
ejpam-3559	25	15	such	such	ADJ
ejpam-3559	25	16	that	that	DET
ejpam-3559	25	17	0	0	NUM
ejpam-3559	25	18	∈	∈	PROPN
ejpam-3559	25	19	a	a	DET
ejpam-3559	25	20	∗	∗	X
ejpam-3559	25	21	b.	b.	PROPN
ejpam-3559	25	22	in	in	ADP
ejpam-3559	25	23	particular	particular	ADJ
ejpam-3559	25	24	,	,	PUNCT
ejpam-3559	25	25	for	for	ADP
ejpam-3559	25	26	every	every	DET
ejpam-3559	25	27	x	x	NOUN
ejpam-3559	25	28	,	,	PUNCT
ejpam-3559	25	29	y	y	PROPN
ejpam-3559	25	30	∈	∈	PROPN
ejpam-3559	25	31	h	h	NOUN
ejpam-3559	25	32	,	,	PUNCT
ejpam-3559	25	33	x	x	PROPN
ejpam-3559	25	34	�	�	PROPN
ejpam-3559	25	35	y	y	PROPN
ejpam-3559	25	36	if	if	SCONJ
ejpam-3559	25	37	and	and	CCONJ
ejpam-3559	25	38	only	only	ADV
ejpam-3559	25	39	if	if	SCONJ
ejpam-3559	25	40	0	0	NUM
ejpam-3559	25	41	∈	∈	NOUN
ejpam-3559	25	42	x	x	X
ejpam-3559	25	43	∗	∗	NOUN
ejpam-3559	25	44	y.	y.	NOUN
ejpam-3559	25	45	in	in	ADP
ejpam-3559	25	46	such	such	ADJ
ejpam-3559	25	47	case	case	NOUN
ejpam-3559	25	48	,	,	PUNCT
ejpam-3559	25	49	we	we	PRON
ejpam-3559	25	50	call	call	VERB
ejpam-3559	25	51	“	"	PUNCT
ejpam-3559	25	52	�	�	PROPN
ejpam-3559	25	53	”	"	PUNCT
ejpam-3559	25	54	the	the	DET
ejpam-3559	25	55	hyper	hyper	ADJ
ejpam-3559	25	56	order	order	NOUN
ejpam-3559	25	57	in	in	ADP
ejpam-3559	25	58	h.	h.	PROPN
ejpam-3559	25	59	throughout	throughout	ADP
ejpam-3559	25	60	this	this	DET
ejpam-3559	25	61	study	study	NOUN
ejpam-3559	25	62	,	,	PUNCT
ejpam-3559	25	63	(	(	PUNCT
ejpam-3559	25	64	h1	h1	PROPN
ejpam-3559	25	65	,	,	PUNCT
ejpam-3559	25	66	∗1	∗1	PROPN
ejpam-3559	25	67	,	,	PUNCT
ejpam-3559	25	68	01	01	NUM
ejpam-3559	25	69	)	)	PUNCT
ejpam-3559	25	70	(	(	PUNCT
ejpam-3559	25	71	or	or	CCONJ
ejpam-3559	25	72	simply	simply	ADV
ejpam-3559	25	73	h1	h1	ADJ
ejpam-3559	25	74	)	)	PUNCT
ejpam-3559	25	75	and	and	CCONJ
ejpam-3559	25	76	(	(	PUNCT
ejpam-3559	25	77	h2	h2	NOUN
ejpam-3559	25	78	,	,	PUNCT
ejpam-3559	25	79	∗2	∗2	PROPN
ejpam-3559	25	80	,	,	PUNCT
ejpam-3559	25	81	02	02	NUM
ejpam-3559	25	82	)	)	PUNCT
ejpam-3559	25	83	(	(	PUNCT
ejpam-3559	25	84	or	or	CCONJ
ejpam-3559	25	85	simply	simply	ADV
ejpam-3559	25	86	h2	h2	NOUN
ejpam-3559	25	87	)	)	PUNCT
ejpam-3559	25	88	are	be	AUX
ejpam-3559	25	89	hyper	hyper	ADJ
ejpam-3559	25	90	bck	bck	NOUN
ejpam-3559	25	91	-	-	PUNCT
ejpam-3559	25	92	algebras	algebras	X
ejpam-3559	25	93	.	.	PUNCT
ejpam-3559	26	1	let	let	VERB
ejpam-3559	26	2	h	h	PRON
ejpam-3559	26	3	be	be	AUX
ejpam-3559	26	4	a	a	DET
ejpam-3559	26	5	hyper	hyper	ADJ
ejpam-3559	26	6	bck	bck	NOUN
ejpam-3559	26	7	-	-	PUNCT
ejpam-3559	26	8	algebra	algebra	NOUN
ejpam-3559	26	9	and	and	CCONJ
ejpam-3559	26	10	a	a	DET
ejpam-3559	26	11	⊆	⊆	NUM
ejpam-3559	26	12	h.	h.	NOUN
ejpam-3559	26	13	the	the	DET
ejpam-3559	26	14	sets	set	NOUN
ejpam-3559	26	15	lh(a	lh(a	NUM
ejpam-3559	26	16	)	)	PUNCT
ejpam-3559	26	17	and	and	CCONJ
ejpam-3559	27	1	rh(a	rh(a	NUM
ejpam-3559	27	2	)	)	PUNCT
ejpam-3559	27	3	are	be	AUX
ejpam-3559	27	4	given	give	VERB
ejpam-3559	27	5	as	as	SCONJ
ejpam-3559	27	6	follows	follow	VERB
ejpam-3559	27	7	:	:	PUNCT
ejpam-3559	27	8	lh(a	lh(a	NUM
ejpam-3559	27	9	)	)	PUNCT
ejpam-3559	27	10	:	:	PUNCT
ejpam-3559	28	1	=	=	SYM
ejpam-3559	28	2	{	{	PUNCT
ejpam-3559	28	3	x	x	SYM
ejpam-3559	28	4	∈	∈	NOUN
ejpam-3559	28	5	h	h	NOUN
ejpam-3559	29	1	|	|	ADV
ejpam-3559	29	2	x	x	X
ejpam-3559	29	3	�	�	PROPN
ejpam-3559	29	4	a	a	DET
ejpam-3559	29	5	∀a	∀a	X
ejpam-3559	29	6	∈	∈	NOUN
ejpam-3559	29	7	a	a	DET
ejpam-3559	29	8	}	}	PUNCT
ejpam-3559	29	9	=	=	SYM
ejpam-3559	29	10	{	{	PUNCT
ejpam-3559	29	11	x	x	PUNCT
ejpam-3559	29	12	∈	∈	NOUN
ejpam-3559	29	13	h	h	NOUN
ejpam-3559	30	1	|	|	ADV
ejpam-3559	30	2	0	0	NUM
ejpam-3559	30	3	∈	∈	NOUN
ejpam-3559	30	4	x	x	PUNCT
ejpam-3559	30	5	∗	∗	VERB
ejpam-3559	30	6	a	a	DET
ejpam-3559	30	7	∀a	∀a	NOUN
ejpam-3559	30	8	∈	∈	NOUN
ejpam-3559	30	9	a	a	NOUN
ejpam-3559	30	10	}	}	PUNCT
ejpam-3559	30	11	and	and	CCONJ
ejpam-3559	30	12	rh(a	rh(a	NUM
ejpam-3559	30	13	)	)	PUNCT
ejpam-3559	30	14	:	:	PUNCT
ejpam-3559	31	1	=	=	SYM
ejpam-3559	31	2	{	{	PUNCT
ejpam-3559	31	3	x	x	SYM
ejpam-3559	31	4	∈	∈	NOUN
ejpam-3559	31	5	h	h	NOUN
ejpam-3559	31	6	|	|	ADV
ejpam-3559	31	7	a	a	DET
ejpam-3559	31	8	�	�	PROPN
ejpam-3559	31	9	x	x	SYM
ejpam-3559	31	10	∀a	∀a	X
ejpam-3559	31	11	∈	∈	NOUN
ejpam-3559	31	12	a	a	DET
ejpam-3559	31	13	}	}	PUNCT
ejpam-3559	31	14	=	=	SYM
ejpam-3559	31	15	{	{	PUNCT
ejpam-3559	31	16	x	x	PUNCT
ejpam-3559	31	17	∈	∈	NOUN
ejpam-3559	31	18	h	h	NOUN
ejpam-3559	32	1	|	|	ADV
ejpam-3559	32	2	0	0	NUM
ejpam-3559	32	3	∈	∈	PROPN
ejpam-3559	32	4	a	a	DET
ejpam-3559	32	5	∗	∗	NOUN
ejpam-3559	32	6	x	x	SYM
ejpam-3559	32	7	∀a	∀a	X
ejpam-3559	32	8	∈	∈	NOUN
ejpam-3559	32	9	a	a	PRON
ejpam-3559	32	10	}	}	PUNCT
ejpam-3559	32	11	.	.	PUNCT
ejpam-3559	33	1	if	if	SCONJ
ejpam-3559	33	2	a	a	PRON
ejpam-3559	33	3	=	=	X
ejpam-3559	33	4	{	{	PUNCT
ejpam-3559	33	5	a	a	NOUN
ejpam-3559	33	6	}	}	PUNCT
ejpam-3559	33	7	,	,	PUNCT
ejpam-3559	33	8	we	we	PRON
ejpam-3559	33	9	write	write	VERB
ejpam-3559	33	10	lh({a	lh({a	PROPN
ejpam-3559	33	11	}	}	PUNCT
ejpam-3559	33	12	)	)	PUNCT
ejpam-3559	34	1	=	=	SYM
ejpam-3559	34	2	lh(a	lh(a	NOUN
ejpam-3559	34	3	)	)	PUNCT
ejpam-3559	34	4	and	and	CCONJ
ejpam-3559	34	5	rh({a	rh({a	NOUN
ejpam-3559	34	6	}	}	PUNCT
ejpam-3559	34	7	)	)	PUNCT
ejpam-3559	34	8	=	=	SYM
ejpam-3559	35	1	rh(a	rh(a	NUM
ejpam-3559	35	2	)	)	PUNCT
ejpam-3559	35	3	.	.	PUNCT
ejpam-3559	36	1	let	let	AUX
ejpam-3559	36	2	(	(	PUNCT
ejpam-3559	36	3	h1	h1	PROPN
ejpam-3559	36	4	,	,	PUNCT
ejpam-3559	36	5	∗1	∗1	PROPN
ejpam-3559	36	6	,	,	PUNCT
ejpam-3559	36	7	0	0	NUM
ejpam-3559	36	8	)	)	PUNCT
ejpam-3559	36	9	and	and	CCONJ
ejpam-3559	36	10	(	(	PUNCT
ejpam-3559	36	11	h2	h2	NOUN
ejpam-3559	36	12	,	,	PUNCT
ejpam-3559	36	13	∗2	∗2	PROPN
ejpam-3559	36	14	,	,	PUNCT
ejpam-3559	36	15	0	0	NUM
ejpam-3559	36	16	)	)	PUNCT
ejpam-3559	36	17	be	be	AUX
ejpam-3559	36	18	hyper	hyper	ADJ
ejpam-3559	36	19	bck	bck	NOUN
ejpam-3559	36	20	-	-	PUNCT
ejpam-3559	36	21	algebras	algebra	NOUN
ejpam-3559	36	22	such	such	ADJ
ejpam-3559	36	23	that	that	SCONJ
ejpam-3559	36	24	h1	h1	PROPN
ejpam-3559	36	25	∩	∩	ADJ
ejpam-3559	36	26	h2	h2	NOUN
ejpam-3559	36	27	=	=	PUNCT
ejpam-3559	36	28	{	{	PUNCT
ejpam-3559	36	29	0	0	NUM
ejpam-3559	36	30	}	}	PUNCT
ejpam-3559	36	31	and	and	CCONJ
ejpam-3559	36	32	h	h	NOUN
ejpam-3559	36	33	=	=	NOUN
ejpam-3559	36	34	h1	h1	PROPN
ejpam-3559	36	35	∪	∪	PROPN
ejpam-3559	36	36	h2	h2	PROPN
ejpam-3559	36	37	.	.	PUNCT
ejpam-3559	37	1	then	then	ADV
ejpam-3559	37	2	(	(	PUNCT
ejpam-3559	37	3	h	h	NOUN
ejpam-3559	37	4	,	,	PUNCT
ejpam-3559	37	5	∗	∗	NOUN
ejpam-3559	37	6	,	,	PUNCT
ejpam-3559	37	7	0	0	NUM
ejpam-3559	37	8	)	)	PUNCT
ejpam-3559	37	9	is	be	AUX
ejpam-3559	37	10	a	a	DET
ejpam-3559	37	11	hyper	hyper	ADJ
ejpam-3559	37	12	bck	bck	NOUN
ejpam-3559	37	13	-	-	PUNCT
ejpam-3559	37	14	algebra	algebra	NOUN
ejpam-3559	37	15	denoted	denote	VERB
ejpam-3559	37	16	by	by	ADP
ejpam-3559	37	17	h1	h1	PROPN
ejpam-3559	37	18	⊕	⊕	PROPN
ejpam-3559	37	19	h2	h2	PROPN
ejpam-3559	37	20	,	,	PUNCT
ejpam-3559	37	21	called	call	VERB
ejpam-3559	37	22	the	the	DET
ejpam-3559	37	23	hyper	hyper	ADJ
ejpam-3559	37	24	sum	sum	NOUN
ejpam-3559	37	25	,	,	PUNCT
ejpam-3559	37	26	where	where	SCONJ
ejpam-3559	37	27	the	the	DET
ejpam-3559	37	28	hyperoperation	hyperoperation	NOUN
ejpam-3559	37	29	“	"	PUNCT
ejpam-3559	37	30	∗	∗	NOUN
ejpam-3559	37	31	”	"	PUNCT
ejpam-3559	37	32	on	on	ADP
ejpam-3559	37	33	h	h	NOUN
ejpam-3559	37	34	is	be	AUX
ejpam-3559	37	35	defined	define	VERB
ejpam-3559	37	36	for	for	ADP
ejpam-3559	37	37	all	all	DET
ejpam-3559	37	38	x	x	NOUN
ejpam-3559	37	39	,	,	PUNCT
ejpam-3559	37	40	y	y	PROPN
ejpam-3559	37	41	∈	∈	PROPN
ejpam-3559	37	42	h	h	NOUN
ejpam-3559	37	43	by	by	ADV
ejpam-3559	37	44	,	,	PUNCT
ejpam-3559	37	45	x	x	PROPN
ejpam-3559	37	46	∗	∗	NOUN
ejpam-3559	38	1	y	y	NOUN
ejpam-3559	38	2	=	=	SYM
ejpam-3559	38	3			NOUN
ejpam-3559	38	4	x	x	X
ejpam-3559	38	5	∗1	∗1	PROPN
ejpam-3559	38	6	y	y	PROPN
ejpam-3559	38	7	if	if	SCONJ
ejpam-3559	38	8	x	x	PRON
ejpam-3559	38	9	,	,	PUNCT
ejpam-3559	38	10	y	y	PROPN
ejpam-3559	38	11	∈	∈	PROPN
ejpam-3559	38	12	h1	h1	PROPN
ejpam-3559	38	13	x	x	PUNCT
ejpam-3559	38	14	∗2	∗2	VERB
ejpam-3559	38	15	y	y	PROPN
ejpam-3559	38	16	if	if	SCONJ
ejpam-3559	38	17	x	x	PROPN
ejpam-3559	38	18	,	,	PUNCT
ejpam-3559	38	19	y	y	PROPN
ejpam-3559	38	20	∈	∈	PROPN
ejpam-3559	38	21	h2	h2	PROPN
ejpam-3559	38	22	{	{	PUNCT
ejpam-3559	38	23	x	x	NOUN
ejpam-3559	38	24	}	}	PUNCT
ejpam-3559	38	25	otherwise	otherwise	ADV
ejpam-3559	38	26	.	.	PUNCT
ejpam-3559	39	1	let	let	AUX
ejpam-3559	39	2	(	(	PUNCT
ejpam-3559	39	3	h1	h1	PROPN
ejpam-3559	39	4	,	,	PUNCT
ejpam-3559	39	5	∗1	∗1	PROPN
ejpam-3559	39	6	,	,	PUNCT
ejpam-3559	39	7	01	01	NUM
ejpam-3559	39	8	)	)	PUNCT
ejpam-3559	39	9	and	and	CCONJ
ejpam-3559	39	10	(	(	PUNCT
ejpam-3559	39	11	h2	h2	NOUN
ejpam-3559	39	12	,	,	PUNCT
ejpam-3559	39	13	∗2	∗2	PROPN
ejpam-3559	39	14	,	,	PUNCT
ejpam-3559	39	15	02	02	NUM
ejpam-3559	39	16	)	)	PUNCT
ejpam-3559	39	17	be	be	AUX
ejpam-3559	39	18	hyper	hyper	ADJ
ejpam-3559	39	19	bck	bck	NOUN
ejpam-3559	39	20	-	-	PUNCT
ejpam-3559	39	21	algebras	algebras	PROPN
ejpam-3559	39	22	and	and	CCONJ
ejpam-3559	39	23	h	h	NOUN
ejpam-3559	39	24	=	=	NOUN
ejpam-3559	39	25	h1	h1	PROPN
ejpam-3559	39	26	×h2	×h2	PROPN
ejpam-3559	39	27	.	.	PUNCT
ejpam-3559	40	1	define	define	VERB
ejpam-3559	40	2	a	a	DET
ejpam-3559	40	3	hyperoperation	hyperoperation	NOUN
ejpam-3559	40	4	“	"	PUNCT
ejpam-3559	40	5	∗	∗	NOUN
ejpam-3559	40	6	”	"	PUNCT
ejpam-3559	40	7	on	on	ADP
ejpam-3559	40	8	h	h	NOUN
ejpam-3559	40	9	as	as	SCONJ
ejpam-3559	40	10	follows	follow	VERB
ejpam-3559	40	11	:	:	PUNCT
ejpam-3559	40	12	for	for	ADP
ejpam-3559	40	13	all	all	DET
ejpam-3559	40	14	(	(	PUNCT
ejpam-3559	40	15	a1	a1	NOUN
ejpam-3559	40	16	,	,	PUNCT
ejpam-3559	40	17	b1	b1	NOUN
ejpam-3559	40	18	)	)	PUNCT
ejpam-3559	40	19	,	,	PUNCT
ejpam-3559	40	20	(	(	PUNCT
ejpam-3559	40	21	a2	a2	PROPN
ejpam-3559	40	22	,	,	PUNCT
ejpam-3559	40	23	b2	b2	NOUN
ejpam-3559	40	24	)	)	PUNCT
ejpam-3559	40	25	∈	∈	PROPN
ejpam-3559	40	26	h	h	NOUN
ejpam-3559	40	27	,	,	PUNCT
ejpam-3559	40	28	(	(	PUNCT
ejpam-3559	40	29	a1	a1	NOUN
ejpam-3559	40	30	,	,	PUNCT
ejpam-3559	40	31	b1)∗	b1)∗	NOUN
ejpam-3559	40	32	(	(	PUNCT
ejpam-3559	40	33	a2	a2	PROPN
ejpam-3559	40	34	,	,	PUNCT
ejpam-3559	40	35	b2	b2	NOUN
ejpam-3559	40	36	)	)	PUNCT
ejpam-3559	40	37	=	=	SYM
ejpam-3559	41	1	(	(	PUNCT
ejpam-3559	41	2	a1	a1	NOUN
ejpam-3559	41	3	∗1	∗1	PROPN
ejpam-3559	41	4	a2	a2	PROPN
ejpam-3559	41	5	,	,	PUNCT
ejpam-3559	41	6	b1	b1	NOUN
ejpam-3559	41	7	∗2	∗2	PROPN
ejpam-3559	41	8	b2	b2	NOUN
ejpam-3559	41	9	)	)	PUNCT
ejpam-3559	41	10	.	.	PUNCT
ejpam-3559	42	1	for	for	ADP
ejpam-3559	42	2	a	a	DET
ejpam-3559	42	3	⊆	⊆	NUM
ejpam-3559	42	4	h1	h1	NOUN
ejpam-3559	42	5	and	and	CCONJ
ejpam-3559	42	6	b	b	NOUN
ejpam-3559	42	7	⊆	⊆	NUM
ejpam-3559	42	8	h2	h2	NOUN
ejpam-3559	42	9	,	,	PUNCT
ejpam-3559	42	10	by	by	ADP
ejpam-3559	42	11	(	(	PUNCT
ejpam-3559	42	12	a	a	DET
ejpam-3559	42	13	,	,	PUNCT
ejpam-3559	42	14	b	b	NOUN
ejpam-3559	42	15	)	)	PUNCT
ejpam-3559	42	16	we	we	PRON
ejpam-3559	42	17	mean	mean	VERB
ejpam-3559	42	18	(	(	PUNCT
ejpam-3559	42	19	a	a	DET
ejpam-3559	42	20	,	,	PUNCT
ejpam-3559	42	21	b	b	NOUN
ejpam-3559	42	22	)	)	PUNCT
ejpam-3559	42	23	=	=	SYM
ejpam-3559	42	24	{	{	PUNCT
ejpam-3559	42	25	(	(	PUNCT
ejpam-3559	42	26	a	a	DET
ejpam-3559	42	27	,	,	PUNCT
ejpam-3559	42	28	b	b	NOUN
ejpam-3559	42	29	)	)	PUNCT
ejpam-3559	42	30	:	:	PUNCT
ejpam-3559	42	31	a	a	DET
ejpam-3559	42	32	∈	∈	PROPN
ejpam-3559	42	33	a	a	PRON
ejpam-3559	42	34	,	,	PUNCT
ejpam-3559	42	35	b	b	PROPN
ejpam-3559	42	36	∈	∈	PROPN
ejpam-3559	42	37	b	b	NOUN
ejpam-3559	42	38	}	}	PUNCT
ejpam-3559	42	39	,	,	PUNCT
ejpam-3559	42	40	0	0	X
ejpam-3559	42	41	=	=	SYM
ejpam-3559	42	42	(	(	PUNCT
ejpam-3559	42	43	01	01	NUM
ejpam-3559	42	44	,	,	PUNCT
ejpam-3559	42	45	02	02	NUM
ejpam-3559	42	46	)	)	PUNCT
ejpam-3559	42	47	and	and	CCONJ
ejpam-3559	42	48	(	(	PUNCT
ejpam-3559	42	49	a1	a1	NOUN
ejpam-3559	42	50	,	,	PUNCT
ejpam-3559	42	51	b1	b1	NOUN
ejpam-3559	42	52	)	)	PUNCT
ejpam-3559	42	53	�	�	PROPN
ejpam-3559	42	54	(	(	PUNCT
ejpam-3559	42	55	a2	a2	PROPN
ejpam-3559	42	56	,	,	PUNCT
ejpam-3559	42	57	b2	b2	NOUN
ejpam-3559	42	58	)	)	PUNCT
ejpam-3559	42	59	⇐	⇐	ADJ
ejpam-3559	42	60	⇒	⇒	NOUN
ejpam-3559	42	61	a1	a1	PROPN
ejpam-3559	42	62	�	�	PROPN
ejpam-3559	42	63	a2	a2	PROPN
ejpam-3559	42	64	and	and	CCONJ
ejpam-3559	42	65	b1	b1	PROPN
ejpam-3559	42	66	�	�	PROPN
ejpam-3559	42	67	b2	b2	PROPN
ejpam-3559	42	68	.	.	PUNCT
ejpam-3559	43	1	then	then	ADV
ejpam-3559	43	2	(	(	PUNCT
ejpam-3559	43	3	h	h	NOUN
ejpam-3559	43	4	,	,	PUNCT
ejpam-3559	43	5	∗	∗	NOUN
ejpam-3559	43	6	,	,	PUNCT
ejpam-3559	43	7	0	0	NUM
ejpam-3559	43	8	)	)	PUNCT
ejpam-3559	43	9	is	be	AUX
ejpam-3559	43	10	a	a	DET
ejpam-3559	43	11	hyper	hyper	ADJ
ejpam-3559	43	12	bck	bck	NOUN
ejpam-3559	43	13	-	-	PUNCT
ejpam-3559	43	14	algebra	algebra	NOUN
ejpam-3559	43	15	,	,	PUNCT
ejpam-3559	43	16	and	and	CCONJ
ejpam-3559	43	17	it	it	PRON
ejpam-3559	43	18	is	be	AUX
ejpam-3559	43	19	called	call	VERB
ejpam-3559	43	20	the	the	DET
ejpam-3559	43	21	hyper	hyper	ADJ
ejpam-3559	43	22	product	product	NOUN
ejpam-3559	43	23	of	of	ADP
ejpam-3559	43	24	h1	h1	NOUN
ejpam-3559	43	25	and	and	CCONJ
ejpam-3559	43	26	h2	h2	NOUN
ejpam-3559	43	27	.	.	PUNCT
ejpam-3559	44	1	2	2	X
ejpam-3559	44	2	.	.	X
ejpam-3559	44	3	known	know	VERB
ejpam-3559	44	4	results	result	NOUN
ejpam-3559	44	5	proposition	proposition	VERB
ejpam-3559	44	6	2.1	2.1	NUM
ejpam-3559	44	7	.	.	PUNCT
ejpam-3559	45	1	[	[	X
ejpam-3559	45	2	1	1	X
ejpam-3559	45	3	]	]	PUNCT
ejpam-3559	45	4	let	let	VERB
ejpam-3559	45	5	a	a	PRON
ejpam-3559	45	6	and	and	CCONJ
ejpam-3559	45	7	b	b	NOUN
ejpam-3559	45	8	be	be	AUX
ejpam-3559	45	9	subsets	subset	NOUN
ejpam-3559	45	10	of	of	ADP
ejpam-3559	45	11	a	a	DET
ejpam-3559	45	12	hyper	hyper	ADJ
ejpam-3559	45	13	bck	bck	NOUN
ejpam-3559	45	14	-	-	PUNCT
ejpam-3559	45	15	algebra	algebra	NOUN
ejpam-3559	45	16	h.	h.	NOUN
ejpam-3559	45	17	then	then	ADV
ejpam-3559	45	18	the	the	DET
ejpam-3559	45	19	following	follow	VERB
ejpam-3559	45	20	hold	hold	NOUN
ejpam-3559	45	21	:	:	PUNCT
ejpam-3559	45	22	r.	r.	PROPN
ejpam-3559	45	23	patangan	patangan	PROPN
ejpam-3559	45	24	,	,	PUNCT
ejpam-3559	45	25	s.	s.	PROPN
ejpam-3559	45	26	canoy	canoy	PROPN
ejpam-3559	45	27	,	,	PUNCT
ejpam-3559	45	28	jr	jr	PROPN
ejpam-3559	45	29	.	.	PROPN
ejpam-3559	45	30	/	/	SYM
ejpam-3559	45	31	eur	eur	PROPN
ejpam-3559	45	32	.	.	PUNCT
ejpam-3559	46	1	j.	j.	PROPN
ejpam-3559	46	2	pure	pure	PROPN
ejpam-3559	46	3	appl	appl	PROPN
ejpam-3559	46	4	.	.	PROPN
ejpam-3559	46	5	math	math	PROPN
ejpam-3559	46	6	,	,	PUNCT
ejpam-3559	46	7	12	12	NUM
ejpam-3559	46	8	(	(	PUNCT
ejpam-3559	46	9	4	4	NUM
ejpam-3559	46	10	)	)	PUNCT
ejpam-3559	46	11	(	(	PUNCT
ejpam-3559	46	12	2019	2019	NUM
ejpam-3559	46	13	)	)	PUNCT
ejpam-3559	46	14	,	,	PUNCT
ejpam-3559	46	15	1524	1524	NUM
ejpam-3559	46	16	-	-	SYM
ejpam-3559	46	17	1532	1532	NUM
ejpam-3559	46	18	1526	1526	NUM
ejpam-3559	46	19	(	(	PUNCT
ejpam-3559	46	20	i	i	NOUN
ejpam-3559	46	21	)	)	PUNCT
ejpam-3559	46	22	lh(∅	lh(∅	PROPN
ejpam-3559	46	23	)	)	PUNCT
ejpam-3559	47	1	=	=	SYM
ejpam-3559	47	2	h.	h.	PROPN
ejpam-3559	47	3	(	(	PUNCT
ejpam-3559	47	4	ii	ii	PROPN
ejpam-3559	47	5	)	)	PUNCT
ejpam-3559	47	6	lh(a	lh(a	NOUN
ejpam-3559	47	7	)	)	PUNCT
ejpam-3559	48	1	=	=	PUNCT
ejpam-3559	49	1	⋂	⋂	PROPN
ejpam-3559	49	2	a∈a	a∈a	ADJ
ejpam-3559	49	3	lh(a	lh(a	NOUN
ejpam-3559	49	4	)	)	PUNCT
ejpam-3559	49	5	.	.	PUNCT
ejpam-3559	50	1	(	(	PUNCT
ejpam-3559	50	2	iii	iii	X
ejpam-3559	50	3	)	)	PUNCT
ejpam-3559	50	4	for	for	ADP
ejpam-3559	50	5	any	any	DET
ejpam-3559	50	6	a	a	DET
ejpam-3559	50	7	⊆	⊆	NUM
ejpam-3559	50	8	h	h	NOUN
ejpam-3559	50	9	,	,	PUNCT
ejpam-3559	50	10	0	0	NUM
ejpam-3559	50	11	∈	∈	PROPN
ejpam-3559	50	12	lh(a	lh(a	NOUN
ejpam-3559	50	13	)	)	PUNCT
ejpam-3559	50	14	.	.	PUNCT
ejpam-3559	51	1	if	if	SCONJ
ejpam-3559	51	2	0	0	NUM
ejpam-3559	51	3	∈	∈	PROPN
ejpam-3559	51	4	a	a	PRON
ejpam-3559	51	5	,	,	PUNCT
ejpam-3559	51	6	then	then	ADV
ejpam-3559	51	7	lh(a	lh(a	NUM
ejpam-3559	51	8	)	)	PUNCT
ejpam-3559	51	9	=	=	PUNCT
ejpam-3559	51	10	{	{	PUNCT
ejpam-3559	51	11	0	0	NUM
ejpam-3559	51	12	}	}	PUNCT
ejpam-3559	51	13	.	.	PUNCT
ejpam-3559	52	1	proposition	proposition	NOUN
ejpam-3559	52	2	2.2	2.2	NUM
ejpam-3559	52	3	.	.	PUNCT
ejpam-3559	53	1	[	[	X
ejpam-3559	53	2	8	8	NUM
ejpam-3559	53	3	]	]	PUNCT
ejpam-3559	53	4	let	let	VERB
ejpam-3559	53	5	h	h	PRON
ejpam-3559	53	6	be	be	AUX
ejpam-3559	53	7	a	a	DET
ejpam-3559	53	8	hyper	hyper	ADJ
ejpam-3559	53	9	bck	bck	NOUN
ejpam-3559	53	10	-	-	PUNCT
ejpam-3559	53	11	algebra	algebra	NOUN
ejpam-3559	53	12	and	and	CCONJ
ejpam-3559	53	13	a	a	DET
ejpam-3559	53	14	⊆	⊆	NUM
ejpam-3559	53	15	h.	h.	NOUN
ejpam-3559	53	16	then	then	ADV
ejpam-3559	53	17	the	the	DET
ejpam-3559	53	18	following	follow	VERB
ejpam-3559	53	19	hold	hold	NOUN
ejpam-3559	53	20	:	:	PUNCT
ejpam-3559	53	21	(	(	PUNCT
ejpam-3559	53	22	i	i	NOUN
ejpam-3559	53	23	)	)	PUNCT
ejpam-3559	53	24	rh(a	rh(a	NUM
ejpam-3559	53	25	)	)	PUNCT
ejpam-3559	54	1	=	=	SYM
ejpam-3559	55	1	⋂	⋂	PROPN
ejpam-3559	55	2	a∈a	a∈a	ADJ
ejpam-3559	55	3	rh(a	rh(a	NUM
ejpam-3559	55	4	)	)	PUNCT
ejpam-3559	55	5	.	.	PUNCT
ejpam-3559	56	1	(	(	PUNCT
ejpam-3559	56	2	ii	ii	NOUN
ejpam-3559	56	3	)	)	PUNCT
ejpam-3559	56	4	for	for	ADP
ejpam-3559	56	5	any	any	DET
ejpam-3559	56	6	∅	∅	NOUN
ejpam-3559	56	7	6=	6=	ADP
ejpam-3559	56	8	a	a	DET
ejpam-3559	56	9	⊆	⊆	NUM
ejpam-3559	56	10	h	h	NOUN
ejpam-3559	56	11	such	such	ADJ
ejpam-3559	56	12	that	that	SCONJ
ejpam-3559	56	13	a	a	DET
ejpam-3559	56	14	6=	6=	NUM
ejpam-3559	56	15	{	{	PUNCT
ejpam-3559	56	16	0	0	NUM
ejpam-3559	56	17	}	}	PUNCT
ejpam-3559	56	18	,	,	PUNCT
ejpam-3559	56	19	0	0	NUM
ejpam-3559	56	20	/∈	/∈	PUNCT
ejpam-3559	57	1	rh(a	rh(a	NUM
ejpam-3559	57	2	)	)	PUNCT
ejpam-3559	57	3	.	.	PUNCT
ejpam-3559	58	1	(	(	PUNCT
ejpam-3559	58	2	iii	iii	NOUN
ejpam-3559	58	3	)	)	PUNCT
ejpam-3559	58	4	rh(x	rh(x	NUM
ejpam-3559	58	5	)	)	PUNCT
ejpam-3559	58	6	6=	6=	PRON
ejpam-3559	58	7	∅	∅	NOUN
ejpam-3559	58	8	∀x	∀x	NOUN
ejpam-3559	58	9	∈	∈	PROPN
ejpam-3559	58	10	h.	h.	NOUN
ejpam-3559	58	11	in	in	ADP
ejpam-3559	58	12	particular	particular	ADJ
ejpam-3559	58	13	,	,	PUNCT
ejpam-3559	58	14	x	x	SYM
ejpam-3559	58	15	∈	∈	NOUN
ejpam-3559	58	16	rh(x	rh(x	NUM
ejpam-3559	58	17	)	)	PUNCT
ejpam-3559	58	18	.	.	PUNCT
ejpam-3559	59	1	furthermore	furthermore	ADV
ejpam-3559	59	2	,	,	PUNCT
ejpam-3559	59	3	rh(x	rh(x	NUM
ejpam-3559	59	4	)	)	PUNCT
ejpam-3559	60	1	=	=	SYM
ejpam-3559	61	1	h	h	NOUN
ejpam-3559	61	2	if	if	SCONJ
ejpam-3559	61	3	and	and	CCONJ
ejpam-3559	61	4	only	only	ADV
ejpam-3559	61	5	if	if	SCONJ
ejpam-3559	61	6	x	x	SYM
ejpam-3559	61	7	=	=	SYM
ejpam-3559	61	8	0	0	NUM
ejpam-3559	61	9	∀x	∀x	X
ejpam-3559	61	10	∈	∈	PROPN
ejpam-3559	61	11	h.	h.	NOUN
ejpam-3559	61	12	theorem	theorem	VERB
ejpam-3559	61	13	2.3	2.3	NUM
ejpam-3559	61	14	.	.	PUNCT
ejpam-3559	62	1	[	[	X
ejpam-3559	62	2	9	9	NUM
ejpam-3559	62	3	]	]	PUNCT
ejpam-3559	62	4	the	the	DET
ejpam-3559	62	5	family	family	NOUN
ejpam-3559	62	6	bl(h	bl(h	PUNCT
ejpam-3559	62	7	)	)	PUNCT
ejpam-3559	62	8	=	=	PRON
ejpam-3559	62	9	{	{	PUNCT
ejpam-3559	62	10	lh(a	lh(a	NOUN
ejpam-3559	62	11	)	)	PUNCT
ejpam-3559	62	12	:	:	PUNCT
ejpam-3559	62	13	∅	∅	NOUN
ejpam-3559	62	14	6=	6=	ADP
ejpam-3559	62	15	a	a	DET
ejpam-3559	62	16	⊆	⊆	NUM
ejpam-3559	62	17	h	h	NOUN
ejpam-3559	62	18	}	}	PUNCT
ejpam-3559	62	19	where	where	SCONJ
ejpam-3559	62	20	h	h	NOUN
ejpam-3559	62	21	is	be	AUX
ejpam-3559	62	22	a	a	DET
ejpam-3559	62	23	hyper	hyper	ADJ
ejpam-3559	62	24	bckalgebra	bckalgebra	NOUN
ejpam-3559	62	25	,	,	PUNCT
ejpam-3559	62	26	is	be	AUX
ejpam-3559	62	27	a	a	DET
ejpam-3559	62	28	basis	basis	NOUN
ejpam-3559	62	29	for	for	ADP
ejpam-3559	62	30	some	some	DET
ejpam-3559	62	31	topology	topology	NOUN
ejpam-3559	62	32	on	on	ADP
ejpam-3559	62	33	h.	h.	PROPN
ejpam-3559	62	34	theorem	theorem	PROPN
ejpam-3559	62	35	2.4	2.4	NUM
ejpam-3559	62	36	.	.	PUNCT
ejpam-3559	63	1	[	[	X
ejpam-3559	63	2	8	8	NUM
ejpam-3559	63	3	]	]	PUNCT
ejpam-3559	63	4	the	the	DET
ejpam-3559	63	5	family	family	NOUN
ejpam-3559	63	6	br(h	br(h	NOUN
ejpam-3559	63	7	)	)	PUNCT
ejpam-3559	64	1	=	=	PRON
ejpam-3559	64	2	{	{	PUNCT
ejpam-3559	64	3	rh(a	rh(a	NOUN
ejpam-3559	64	4	)	)	PUNCT
ejpam-3559	64	5	:	:	PUNCT
ejpam-3559	64	6	∅	∅	NOUN
ejpam-3559	64	7	6=	6=	ADP
ejpam-3559	64	8	a	a	DET
ejpam-3559	64	9	⊆	⊆	NUM
ejpam-3559	64	10	h	h	NOUN
ejpam-3559	64	11	}	}	PUNCT
ejpam-3559	64	12	where	where	SCONJ
ejpam-3559	64	13	h	h	NOUN
ejpam-3559	64	14	is	be	AUX
ejpam-3559	64	15	a	a	DET
ejpam-3559	64	16	hyper	hyper	ADJ
ejpam-3559	64	17	bckalgebra	bckalgebra	NOUN
ejpam-3559	64	18	,	,	PUNCT
ejpam-3559	64	19	is	be	AUX
ejpam-3559	64	20	a	a	DET
ejpam-3559	64	21	basis	basis	NOUN
ejpam-3559	64	22	for	for	ADP
ejpam-3559	64	23	some	some	DET
ejpam-3559	64	24	topology	topology	NOUN
ejpam-3559	64	25	on	on	ADP
ejpam-3559	64	26	h.	h.	PROPN
ejpam-3559	64	27	3	3	X
ejpam-3559	64	28	.	.	PUNCT
ejpam-3559	64	29	bases	basis	NOUN
ejpam-3559	64	30	of	of	ADP
ejpam-3559	64	31	τl(h1	τl(h1	NUM
ejpam-3559	64	32	⊕h2	⊕h2	NUM
ejpam-3559	64	33	)	)	PUNCT
ejpam-3559	64	34	and	and	CCONJ
ejpam-3559	64	35	τr(h1	τr(h1	NOUN
ejpam-3559	64	36	⊕h2	⊕h2	NOUN
ejpam-3559	64	37	)	)	PUNCT
ejpam-3559	64	38	theorem	theorem	VERB
ejpam-3559	64	39	3.1	3.1	NUM
ejpam-3559	64	40	.	.	PUNCT
ejpam-3559	65	1	let	let	VERB
ejpam-3559	65	2	h	h	PRON
ejpam-3559	65	3	be	be	AUX
ejpam-3559	65	4	a	a	DET
ejpam-3559	65	5	hyper	hyper	ADJ
ejpam-3559	65	6	sum	sum	NOUN
ejpam-3559	65	7	of	of	ADP
ejpam-3559	65	8	hyper	hyper	ADJ
ejpam-3559	65	9	bck	bck	NOUN
ejpam-3559	65	10	-	-	PUNCT
ejpam-3559	65	11	algebras	algebras	ADJ
ejpam-3559	65	12	h1	h1	PROPN
ejpam-3559	65	13	and	and	CCONJ
ejpam-3559	65	14	h2	h2	PROPN
ejpam-3559	65	15	with	with	ADP
ejpam-3559	65	16	|h1|	|h1|	PROPN
ejpam-3559	65	17	≥	≥	NUM
ejpam-3559	65	18	2	2	NUM
ejpam-3559	65	19	and	and	CCONJ
ejpam-3559	65	20	|h2|	|h2|	PRON
ejpam-3559	65	21	≥	≥	NOUN
ejpam-3559	65	22	2	2	NUM
ejpam-3559	65	23	.	.	PUNCT
ejpam-3559	65	24	then	then	ADV
ejpam-3559	65	25	bl(h	bl(h	PUNCT
ejpam-3559	65	26	)	)	PUNCT
ejpam-3559	65	27	=	=	PUNCT
ejpam-3559	65	28	bl(h1	bl(h1	ADP
ejpam-3559	65	29	⊕h2	⊕h2	NOUN
ejpam-3559	65	30	)	)	PUNCT
ejpam-3559	65	31	=	=	SYM
ejpam-3559	66	1	bl(h1	bl(h1	NOUN
ejpam-3559	66	2	)	)	PUNCT
ejpam-3559	66	3	∪	∪	ADP
ejpam-3559	66	4	bl(h2	bl(h2	NOUN
ejpam-3559	66	5	)	)	PUNCT
ejpam-3559	66	6	.	.	PUNCT
ejpam-3559	67	1	proof	proof	NOUN
ejpam-3559	67	2	:	:	PUNCT
ejpam-3559	67	3	since	since	SCONJ
ejpam-3559	67	4	bl(h1	bl(h1	NOUN
ejpam-3559	67	5	)	)	PUNCT
ejpam-3559	67	6	⊆	⊆	NUM
ejpam-3559	67	7	bl(h	bl(h	NUM
ejpam-3559	67	8	)	)	PUNCT
ejpam-3559	67	9	and	and	CCONJ
ejpam-3559	67	10	bl(h2	bl(h2	NOUN
ejpam-3559	67	11	)	)	PUNCT
ejpam-3559	67	12	⊆	⊆	NUM
ejpam-3559	67	13	bl(h	bl(h	NUM
ejpam-3559	67	14	)	)	PUNCT
ejpam-3559	67	15	,	,	PUNCT
ejpam-3559	67	16	it	it	PRON
ejpam-3559	67	17	follows	follow	VERB
ejpam-3559	67	18	that	that	SCONJ
ejpam-3559	67	19	bl(h1	bl(h1	NOUN
ejpam-3559	67	20	)	)	PUNCT
ejpam-3559	67	21	∪	∪	ADP
ejpam-3559	67	22	bl(h2	bl(h2	NOUN
ejpam-3559	67	23	)	)	PUNCT
ejpam-3559	67	24	⊆	⊆	NUM
ejpam-3559	67	25	bl(h	bl(h	NUM
ejpam-3559	67	26	)	)	PUNCT
ejpam-3559	67	27	.	.	PUNCT
ejpam-3559	68	1	next	next	ADV
ejpam-3559	68	2	,	,	PUNCT
ejpam-3559	68	3	let	let	VERB
ejpam-3559	68	4	v	v	ADP
ejpam-3559	68	5	∈	∈	PROPN
ejpam-3559	68	6	bl(h	bl(h	PRON
ejpam-3559	68	7	)	)	PUNCT
ejpam-3559	68	8	.	.	PUNCT
ejpam-3559	69	1	then	then	ADV
ejpam-3559	69	2	there	there	PRON
ejpam-3559	69	3	exists	exist	VERB
ejpam-3559	69	4	a	a	DET
ejpam-3559	69	5	nonempty	nonempty	ADV
ejpam-3559	69	6	set	set	VERB
ejpam-3559	69	7	b	b	NOUN
ejpam-3559	69	8	⊆	⊆	NUM
ejpam-3559	69	9	h	h	NOUN
ejpam-3559	69	10	such	such	ADJ
ejpam-3559	69	11	that	that	DET
ejpam-3559	69	12	v	v	NOUN
ejpam-3559	69	13	=	=	PUNCT
ejpam-3559	69	14	lh(b	lh(b	NUM
ejpam-3559	69	15	)	)	PUNCT
ejpam-3559	69	16	.	.	PUNCT
ejpam-3559	70	1	let	let	VERB
ejpam-3559	70	2	b1	b1	NOUN
ejpam-3559	70	3	=	=	SYM
ejpam-3559	70	4	b	b	PROPN
ejpam-3559	70	5	∩h1	∩h1	PROPN
ejpam-3559	70	6	and	and	CCONJ
ejpam-3559	70	7	b2	b2	NOUN
ejpam-3559	70	8	=	=	SYM
ejpam-3559	70	9	b	b	PROPN
ejpam-3559	71	1	∩h2	∩h2	PROPN
ejpam-3559	71	2	.	.	PUNCT
ejpam-3559	72	1	if	if	SCONJ
ejpam-3559	72	2	v	v	NUM
ejpam-3559	72	3	=	=	SYM
ejpam-3559	72	4	{	{	PUNCT
ejpam-3559	72	5	0	0	NUM
ejpam-3559	72	6	}	}	PUNCT
ejpam-3559	72	7	,	,	PUNCT
ejpam-3559	72	8	then	then	ADV
ejpam-3559	72	9	by	by	ADP
ejpam-3559	72	10	proposition	proposition	NOUN
ejpam-3559	72	11	2.1(iii	2.1(iii	NUM
ejpam-3559	72	12	)	)	PUNCT
ejpam-3559	72	13	,	,	PUNCT
ejpam-3559	72	14	v	v	X
ejpam-3559	72	15	=	=	SYM
ejpam-3559	72	16	lh1(0	lh1(0	PROPN
ejpam-3559	72	17	)	)	PUNCT
ejpam-3559	72	18	∈	∈	PROPN
ejpam-3559	72	19	bl(h1)∪bl(h2	bl(h1)∪bl(h2	NOUN
ejpam-3559	72	20	)	)	PUNCT
ejpam-3559	72	21	.	.	PUNCT
ejpam-3559	73	1	so	so	ADV
ejpam-3559	73	2	,	,	PUNCT
ejpam-3559	73	3	suppose	suppose	VERB
ejpam-3559	73	4	that	that	SCONJ
ejpam-3559	73	5	v	v	ADP
ejpam-3559	73	6	6=	6=	PRON
ejpam-3559	73	7	{	{	PUNCT
ejpam-3559	73	8	0	0	NUM
ejpam-3559	73	9	}	}	PUNCT
ejpam-3559	73	10	.	.	PUNCT
ejpam-3559	74	1	suppose	suppose	VERB
ejpam-3559	74	2	further	far	ADV
ejpam-3559	74	3	that	that	PRON
ejpam-3559	74	4	b1	b1	VERB
ejpam-3559	74	5	6=	6=	SYM
ejpam-3559	74	6	∅	∅	NOUN
ejpam-3559	74	7	and	and	CCONJ
ejpam-3559	74	8	b2	b2	VERB
ejpam-3559	74	9	6=	6=	ADP
ejpam-3559	74	10	∅.	∅.	VERB
ejpam-3559	74	11	choose	choose	VERB
ejpam-3559	74	12	x	x	X
ejpam-3559	74	13	,	,	PUNCT
ejpam-3559	74	14	y	y	PROPN
ejpam-3559	74	15	∈	∈	PROPN
ejpam-3559	74	16	b	b	PROPN
ejpam-3559	74	17	such	such	ADJ
ejpam-3559	74	18	that	that	SCONJ
ejpam-3559	74	19	x	x	SYM
ejpam-3559	74	20	∈	∈	PROPN
ejpam-3559	74	21	b1	b1	NOUN
ejpam-3559	74	22	and	and	CCONJ
ejpam-3559	74	23	y	y	PROPN
ejpam-3559	74	24	∈	∈	PROPN
ejpam-3559	74	25	b2	b2	NOUN
ejpam-3559	74	26	.	.	PUNCT
ejpam-3559	75	1	pick	pick	VERB
ejpam-3559	75	2	u	u	PRON
ejpam-3559	75	3	∈	∈	PROPN
ejpam-3559	75	4	v	v	ADP
ejpam-3559	75	5	\	\	PUNCT
ejpam-3559	75	6	{	{	PUNCT
ejpam-3559	75	7	0	0	NUM
ejpam-3559	75	8	}	}	PUNCT
ejpam-3559	75	9	.	.	PUNCT
ejpam-3559	76	1	then	then	ADV
ejpam-3559	76	2	u	u	PROPN
ejpam-3559	76	3	�	�	PROPN
ejpam-3559	76	4	x	x	SYM
ejpam-3559	76	5	and	and	CCONJ
ejpam-3559	76	6	u	u	PROPN
ejpam-3559	76	7	�	�	PROPN
ejpam-3559	76	8	y.	y.	NOUN
ejpam-3559	76	9	if	if	SCONJ
ejpam-3559	76	10	u	u	PROPN
ejpam-3559	76	11	∈	∈	PROPN
ejpam-3559	76	12	h1	h1	PROPN
ejpam-3559	76	13	,	,	PUNCT
ejpam-3559	76	14	then	then	ADV
ejpam-3559	76	15	u	u	NOUN
ejpam-3559	76	16	∗	∗	NOUN
ejpam-3559	76	17	y	y	NOUN
ejpam-3559	76	18	=	=	SYM
ejpam-3559	76	19	{	{	PUNCT
ejpam-3559	76	20	u	u	NOUN
ejpam-3559	76	21	}	}	PUNCT
ejpam-3559	76	22	.	.	PUNCT
ejpam-3559	77	1	if	if	SCONJ
ejpam-3559	77	2	u	u	PROPN
ejpam-3559	77	3	∈	∈	PROPN
ejpam-3559	77	4	h2	h2	NOUN
ejpam-3559	77	5	,	,	PUNCT
ejpam-3559	77	6	u	u	NOUN
ejpam-3559	77	7	∗	∗	NOUN
ejpam-3559	77	8	x	x	PUNCT
ejpam-3559	77	9	=	=	PRON
ejpam-3559	77	10	{	{	PUNCT
ejpam-3559	77	11	u	u	NOUN
ejpam-3559	77	12	}	}	PUNCT
ejpam-3559	77	13	.	.	PUNCT
ejpam-3559	78	1	in	in	ADP
ejpam-3559	78	2	both	both	DET
ejpam-3559	78	3	cases	case	NOUN
ejpam-3559	78	4	,	,	PUNCT
ejpam-3559	78	5	we	we	PRON
ejpam-3559	78	6	get	get	VERB
ejpam-3559	78	7	a	a	DET
ejpam-3559	78	8	contradiction	contradiction	NOUN
ejpam-3559	78	9	since	since	SCONJ
ejpam-3559	78	10	u	u	PROPN
ejpam-3559	78	11	6=	6=	PROPN
ejpam-3559	78	12	0	0	NUM
ejpam-3559	78	13	.	.	PUNCT
ejpam-3559	79	1	therefore	therefore	ADV
ejpam-3559	79	2	,	,	PUNCT
ejpam-3559	79	3	either	either	CCONJ
ejpam-3559	79	4	b1	b1	NOUN
ejpam-3559	79	5	=	=	SYM
ejpam-3559	79	6	∅	∅	NOUN
ejpam-3559	79	7	or	or	CCONJ
ejpam-3559	79	8	b2	b2	NOUN
ejpam-3559	79	9	=	=	NOUN
ejpam-3559	79	10	∅	∅	NOUN
ejpam-3559	79	11	,	,	PUNCT
ejpam-3559	79	12	say	say	VERB
ejpam-3559	79	13	b2	b2	NOUN
ejpam-3559	79	14	=	=	PUNCT
ejpam-3559	79	15	∅.	∅.	NOUN
ejpam-3559	79	16	then	then	ADV
ejpam-3559	79	17	b	b	NOUN
ejpam-3559	79	18	=	=	PUNCT
ejpam-3559	79	19	b1	b1	VERB
ejpam-3559	79	20	⊆	⊆	NUM
ejpam-3559	79	21	h1	h1	NOUN
ejpam-3559	79	22	.	.	PUNCT
ejpam-3559	80	1	hence	hence	ADV
ejpam-3559	80	2	,	,	PUNCT
ejpam-3559	80	3	v	v	NOUN
ejpam-3559	80	4	=	=	PUNCT
ejpam-3559	80	5	lh(b	lh(b	NOUN
ejpam-3559	80	6	)	)	PUNCT
ejpam-3559	80	7	=	=	SYM
ejpam-3559	80	8	lh1(b	lh1(b	PROPN
ejpam-3559	80	9	)	)	PUNCT
ejpam-3559	80	10	∈	∈	PROPN
ejpam-3559	80	11	bl(h1	bl(h1	NOUN
ejpam-3559	80	12	)	)	PUNCT
ejpam-3559	80	13	∪	∪	ADP
ejpam-3559	80	14	bl(h2	bl(h2	NOUN
ejpam-3559	80	15	)	)	PUNCT
ejpam-3559	80	16	.	.	PUNCT
ejpam-3559	81	1	therefore	therefore	ADV
ejpam-3559	81	2	,	,	PUNCT
ejpam-3559	81	3	bl(h	bl(h	NUM
ejpam-3559	81	4	)	)	PUNCT
ejpam-3559	81	5	=	=	SYM
ejpam-3559	81	6	bl(h1	bl(h1	NOUN
ejpam-3559	81	7	)	)	PUNCT
ejpam-3559	81	8	∪	∪	ADP
ejpam-3559	81	9	bl(h2	bl(h2	NOUN
ejpam-3559	81	10	)	)	PUNCT
ejpam-3559	81	11	.	.	PUNCT
ejpam-3559	82	1	theorem	theorem	ADJ
ejpam-3559	82	2	3.2	3.2	NUM
ejpam-3559	82	3	.	.	PUNCT
ejpam-3559	83	1	let	let	VERB
ejpam-3559	83	2	h	h	PRON
ejpam-3559	83	3	be	be	AUX
ejpam-3559	83	4	a	a	DET
ejpam-3559	83	5	hyper	hyper	ADJ
ejpam-3559	83	6	sum	sum	NOUN
ejpam-3559	83	7	of	of	ADP
ejpam-3559	83	8	hyper	hyper	ADJ
ejpam-3559	83	9	bck	bck	NOUN
ejpam-3559	83	10	-	-	PUNCT
ejpam-3559	83	11	algebras	algebras	ADJ
ejpam-3559	83	12	h1	h1	PROPN
ejpam-3559	83	13	and	and	CCONJ
ejpam-3559	83	14	h2	h2	PROPN
ejpam-3559	83	15	with	with	ADP
ejpam-3559	83	16	|h1|	|h1|	PROPN
ejpam-3559	83	17	≥	≥	NUM
ejpam-3559	83	18	2	2	NUM
ejpam-3559	83	19	and	and	CCONJ
ejpam-3559	83	20	|h2|	|h2|	PRON
ejpam-3559	83	21	≥	≥	NOUN
ejpam-3559	83	22	2	2	NUM
ejpam-3559	83	23	.	.	PUNCT
ejpam-3559	83	24	then	then	ADV
ejpam-3559	83	25	br(h	br(h	NUM
ejpam-3559	83	26	)	)	PUNCT
ejpam-3559	83	27	\	\	NOUN
ejpam-3559	84	1	{	{	PUNCT
ejpam-3559	84	2	∅	∅	NOUN
ejpam-3559	84	3	,	,	PUNCT
ejpam-3559	84	4	h	h	NOUN
ejpam-3559	84	5	}	}	PUNCT
ejpam-3559	84	6	=	=	SYM
ejpam-3559	84	7	(	(	PUNCT
ejpam-3559	84	8	br(h1	br(h1	NOUN
ejpam-3559	84	9	)	)	PUNCT
ejpam-3559	84	10	∪	∪	NOUN
ejpam-3559	84	11	br(h2	br(h2	NOUN
ejpam-3559	84	12	)	)	PUNCT
ejpam-3559	84	13	)	)	PUNCT
ejpam-3559	84	14	\	\	NOUN
ejpam-3559	84	15	{	{	PUNCT
ejpam-3559	84	16	∅	∅	NOUN
ejpam-3559	84	17	,	,	PUNCT
ejpam-3559	84	18	h1	h1	NOUN
ejpam-3559	84	19	,	,	PUNCT
ejpam-3559	84	20	h2	h2	PROPN
ejpam-3559	84	21	}	}	PUNCT
ejpam-3559	84	22	.	.	PUNCT
ejpam-3559	85	1	proof	proof	NOUN
ejpam-3559	85	2	:	:	PUNCT
ejpam-3559	85	3	let	let	VERB
ejpam-3559	85	4	p	p	PRON
ejpam-3559	85	5	∈	∈	PROPN
ejpam-3559	85	6	br(h1	br(h1	NOUN
ejpam-3559	85	7	)	)	PUNCT
ejpam-3559	85	8	\	\	NOUN
ejpam-3559	85	9	{	{	PUNCT
ejpam-3559	85	10	∅	∅	NOUN
ejpam-3559	85	11	,	,	PUNCT
ejpam-3559	85	12	h1	h1	PROPN
ejpam-3559	85	13	}	}	PUNCT
ejpam-3559	85	14	.	.	PUNCT
ejpam-3559	86	1	since	since	SCONJ
ejpam-3559	86	2	p	p	PROPN
ejpam-3559	86	3	6=	6=	ADP
ejpam-3559	86	4	h1	h1	PROPN
ejpam-3559	86	5	,	,	PUNCT
ejpam-3559	86	6	by	by	ADP
ejpam-3559	86	7	proposition	proposition	NOUN
ejpam-3559	86	8	2.2(iii	2.2(iii	NUM
ejpam-3559	86	9	)	)	PUNCT
ejpam-3559	86	10	,	,	PUNCT
ejpam-3559	86	11	there	there	PRON
ejpam-3559	86	12	exists	exist	VERB
ejpam-3559	86	13	a	a	DET
ejpam-3559	86	14	nonempty	nonempty	NOUN
ejpam-3559	86	15	set	set	VERB
ejpam-3559	86	16	a	a	DET
ejpam-3559	86	17	⊆	⊆	NUM
ejpam-3559	86	18	h1	h1	NOUN
ejpam-3559	86	19	\	\	NOUN
ejpam-3559	86	20	{	{	PUNCT
ejpam-3559	86	21	0	0	NUM
ejpam-3559	86	22	}	}	PUNCT
ejpam-3559	86	23	such	such	ADJ
ejpam-3559	86	24	that	that	SCONJ
ejpam-3559	86	25	p	p	X
ejpam-3559	86	26	=	=	PUNCT
ejpam-3559	86	27	rh1(a	rh1(a	PROPN
ejpam-3559	86	28	)	)	PUNCT
ejpam-3559	86	29	.	.	PUNCT
ejpam-3559	87	1	but	but	CCONJ
ejpam-3559	87	2	a	a	DET
ejpam-3559	87	3	⊆	⊆	NUM
ejpam-3559	87	4	h1	h1	NOUN
ejpam-3559	87	5	\	\	PUNCT
ejpam-3559	87	6	{	{	PUNCT
ejpam-3559	87	7	0	0	NUM
ejpam-3559	87	8	}	}	SYM
ejpam-3559	87	9	⊆	⊆	NUM
ejpam-3559	87	10	h	h	NOUN
ejpam-3559	87	11	,	,	PUNCT
ejpam-3559	87	12	thus	thus	ADV
ejpam-3559	87	13	,	,	PUNCT
ejpam-3559	87	14	p	p	X
ejpam-3559	87	15	=	=	PUNCT
ejpam-3559	87	16	rh1(a	rh1(a	PROPN
ejpam-3559	87	17	)	)	PUNCT
ejpam-3559	87	18	=	=	PUNCT
ejpam-3559	87	19	rh(a	rh(a	NOUN
ejpam-3559	87	20	)	)	PUNCT
ejpam-3559	87	21	.	.	PUNCT
ejpam-3559	88	1	since	since	SCONJ
ejpam-3559	88	2	a	a	DET
ejpam-3559	88	3	6=	6=	NUM
ejpam-3559	88	4	{	{	PUNCT
ejpam-3559	88	5	0	0	NUM
ejpam-3559	88	6	}	}	PUNCT
ejpam-3559	88	7	and	and	CCONJ
ejpam-3559	88	8	p	p	NOUN
ejpam-3559	88	9	6=	6=	NUM
ejpam-3559	88	10	∅	∅	NOUN
ejpam-3559	88	11	,	,	PUNCT
ejpam-3559	88	12	by	by	ADP
ejpam-3559	88	13	theorem	theorem	ADJ
ejpam-3559	88	14	2.2(iii	2.2(iii	NUM
ejpam-3559	88	15	)	)	PUNCT
ejpam-3559	88	16	and	and	CCONJ
ejpam-3559	88	17	definition	definition	NOUN
ejpam-3559	88	18	of	of	ADP
ejpam-3559	88	19	a	a	DET
ejpam-3559	88	20	hyper	hyper	ADJ
ejpam-3559	88	21	sum	sum	NOUN
ejpam-3559	88	22	,	,	PUNCT
ejpam-3559	88	23	p	p	NOUN
ejpam-3559	88	24	6=	6=	PROPN
ejpam-3559	88	25	h	h	NOUN
ejpam-3559	88	26	and	and	CCONJ
ejpam-3559	88	27	p	p	NOUN
ejpam-3559	88	28	6=	6=	NOUN
ejpam-3559	88	29	∅	∅	NOUN
ejpam-3559	88	30	in	in	ADP
ejpam-3559	88	31	h.	h.	PROPN
ejpam-3559	88	32	consequently	consequently	ADV
ejpam-3559	88	33	,	,	PUNCT
ejpam-3559	88	34	p	p	NOUN
ejpam-3559	88	35	=	=	NOUN
ejpam-3559	88	36	rh(a	rh(a	NUM
ejpam-3559	88	37	)	)	PUNCT
ejpam-3559	88	38	∈	∈	PROPN
ejpam-3559	88	39	br(h	br(h	NOUN
ejpam-3559	88	40	)	)	PUNCT
ejpam-3559	88	41	\	\	NOUN
ejpam-3559	88	42	{	{	PUNCT
ejpam-3559	88	43	∅	∅	NOUN
ejpam-3559	88	44	,	,	PUNCT
ejpam-3559	88	45	h	h	NOUN
ejpam-3559	88	46	}	}	PUNCT
ejpam-3559	88	47	.	.	PUNCT
ejpam-3559	89	1	similarly	similarly	ADV
ejpam-3559	89	2	,	,	PUNCT
ejpam-3559	89	3	if	if	SCONJ
ejpam-3559	89	4	q	q	X
ejpam-3559	89	5	∈	∈	PROPN
ejpam-3559	89	6	br(h2	br(h2	NOUN
ejpam-3559	89	7	)	)	PUNCT
ejpam-3559	89	8	\	\	NOUN
ejpam-3559	89	9	{	{	PUNCT
ejpam-3559	89	10	∅	∅	NOUN
ejpam-3559	89	11	,	,	PUNCT
ejpam-3559	89	12	h2	h2	NOUN
ejpam-3559	89	13	}	}	PUNCT
ejpam-3559	89	14	then	then	ADV
ejpam-3559	89	15	q	q	X
ejpam-3559	89	16	6=	6=	PROPN
ejpam-3559	89	17	h	h	NOUN
ejpam-3559	89	18	and	and	CCONJ
ejpam-3559	89	19	q	q	NOUN
ejpam-3559	89	20	6=	6=	NOUN
ejpam-3559	89	21	∅	∅	NOUN
ejpam-3559	89	22	in	in	ADP
ejpam-3559	89	23	h.	h.	PROPN
ejpam-3559	89	24	hence	hence	PROPN
ejpam-3559	89	25	,	,	PUNCT
ejpam-3559	89	26	q	q	NOUN
ejpam-3559	89	27	=	=	PUNCT
ejpam-3559	89	28	rh(b	rh(b	X
ejpam-3559	89	29	)	)	PUNCT
ejpam-3559	89	30	∈	∈	PROPN
ejpam-3559	89	31	br(h	br(h	NOUN
ejpam-3559	89	32	)	)	PUNCT
ejpam-3559	89	33	\	\	NOUN
ejpam-3559	89	34	{	{	PUNCT
ejpam-3559	89	35	∅	∅	NOUN
ejpam-3559	89	36	,	,	PUNCT
ejpam-3559	89	37	h	h	NOUN
ejpam-3559	89	38	}	}	PUNCT
ejpam-3559	89	39	.	.	PUNCT
ejpam-3559	90	1	accordingly	accordingly	ADV
ejpam-3559	90	2	,	,	PUNCT
ejpam-3559	90	3	(	(	PUNCT
ejpam-3559	90	4	br(h1	br(h1	NOUN
ejpam-3559	90	5	)	)	PUNCT
ejpam-3559	90	6	∪	∪	NOUN
ejpam-3559	90	7	br(h2	br(h2	NOUN
ejpam-3559	90	8	)	)	PUNCT
ejpam-3559	90	9	)	)	PUNCT
ejpam-3559	90	10	\	\	NOUN
ejpam-3559	91	1	{	{	PUNCT
ejpam-3559	91	2	∅	∅	NOUN
ejpam-3559	91	3	,	,	PUNCT
ejpam-3559	91	4	h1	h1	NOUN
ejpam-3559	91	5	,	,	PUNCT
ejpam-3559	91	6	h2	h2	PROPN
ejpam-3559	91	7	}	}	PUNCT
ejpam-3559	91	8	⊆	⊆	NUM
ejpam-3559	91	9	br(h	br(h	NUM
ejpam-3559	91	10	)	)	PUNCT
ejpam-3559	91	11	\	\	NOUN
ejpam-3559	91	12	{	{	PUNCT
ejpam-3559	91	13	∅	∅	NOUN
ejpam-3559	91	14	,	,	PUNCT
ejpam-3559	91	15	h	h	NOUN
ejpam-3559	91	16	}	}	PUNCT
ejpam-3559	91	17	.	.	PUNCT
ejpam-3559	92	1	r.	r.	PROPN
ejpam-3559	92	2	patangan	patangan	PROPN
ejpam-3559	92	3	,	,	PUNCT
ejpam-3559	92	4	s.	s.	PROPN
ejpam-3559	92	5	canoy	canoy	PROPN
ejpam-3559	92	6	,	,	PUNCT
ejpam-3559	92	7	jr	jr	PROPN
ejpam-3559	92	8	.	.	PROPN
ejpam-3559	92	9	/	/	SYM
ejpam-3559	92	10	eur	eur	PROPN
ejpam-3559	92	11	.	.	PUNCT
ejpam-3559	93	1	j.	j.	PROPN
ejpam-3559	93	2	pure	pure	PROPN
ejpam-3559	93	3	appl	appl	PROPN
ejpam-3559	93	4	.	.	PROPN
ejpam-3559	93	5	math	math	PROPN
ejpam-3559	93	6	,	,	PUNCT
ejpam-3559	93	7	12	12	NUM
ejpam-3559	93	8	(	(	PUNCT
ejpam-3559	93	9	4	4	NUM
ejpam-3559	93	10	)	)	PUNCT
ejpam-3559	93	11	(	(	PUNCT
ejpam-3559	93	12	2019	2019	NUM
ejpam-3559	93	13	)	)	PUNCT
ejpam-3559	93	14	,	,	PUNCT
ejpam-3559	93	15	1524	1524	NUM
ejpam-3559	93	16	-	-	SYM
ejpam-3559	93	17	1532	1532	NUM
ejpam-3559	93	18	1527	1527	NUM
ejpam-3559	93	19	next	next	ADV
ejpam-3559	93	20	,	,	PUNCT
ejpam-3559	93	21	let	let	VERB
ejpam-3559	93	22	u	u	PRON
ejpam-3559	93	23	∈	∈	PROPN
ejpam-3559	93	24	br(h	br(h	NOUN
ejpam-3559	93	25	)	)	PUNCT
ejpam-3559	93	26	\	\	NOUN
ejpam-3559	93	27	{	{	PUNCT
ejpam-3559	93	28	∅	∅	NOUN
ejpam-3559	93	29	,	,	PUNCT
ejpam-3559	93	30	h	h	NOUN
ejpam-3559	93	31	}	}	PUNCT
ejpam-3559	93	32	.	.	PUNCT
ejpam-3559	94	1	since	since	SCONJ
ejpam-3559	94	2	u	u	PROPN
ejpam-3559	94	3	6=	6=	PROPN
ejpam-3559	94	4	h	h	PROPN
ejpam-3559	94	5	,	,	PUNCT
ejpam-3559	94	6	by	by	ADP
ejpam-3559	94	7	proposition	proposition	NOUN
ejpam-3559	94	8	2.2(iii	2.2(iii	NUM
ejpam-3559	94	9	)	)	PUNCT
ejpam-3559	94	10	,	,	PUNCT
ejpam-3559	94	11	there	there	PRON
ejpam-3559	94	12	exists	exist	VERB
ejpam-3559	94	13	a	a	DET
ejpam-3559	94	14	nonempty	nonempty	NOUN
ejpam-3559	94	15	subset	subset	VERB
ejpam-3559	95	1	d	d	X
ejpam-3559	95	2	⊆	⊆	NUM
ejpam-3559	95	3	h	h	NOUN
ejpam-3559	95	4	\	\	PUNCT
ejpam-3559	95	5	{	{	PUNCT
ejpam-3559	95	6	0	0	NUM
ejpam-3559	95	7	}	}	PUNCT
ejpam-3559	95	8	such	such	ADJ
ejpam-3559	95	9	that	that	SCONJ
ejpam-3559	95	10	u	u	NOUN
ejpam-3559	95	11	=	=	X
ejpam-3559	95	12	rh(d	rh(d	X
ejpam-3559	95	13	)	)	PUNCT
ejpam-3559	95	14	.	.	PUNCT
ejpam-3559	96	1	let	let	VERB
ejpam-3559	96	2	d1	d1	PROPN
ejpam-3559	96	3	=	=	SYM
ejpam-3559	97	1	d	d	PROPN
ejpam-3559	97	2	∩	∩	X
ejpam-3559	97	3	(	(	PUNCT
ejpam-3559	97	4	h1	h1	PROPN
ejpam-3559	97	5	\	\	PROPN
ejpam-3559	97	6	{	{	PUNCT
ejpam-3559	97	7	0	0	NUM
ejpam-3559	97	8	}	}	PUNCT
ejpam-3559	97	9	)	)	PUNCT
ejpam-3559	97	10	and	and	CCONJ
ejpam-3559	97	11	d2	d2	PROPN
ejpam-3559	97	12	=	=	SYM
ejpam-3559	97	13	d	d	PROPN
ejpam-3559	97	14	∩	∩	NOUN
ejpam-3559	97	15	(	(	PUNCT
ejpam-3559	97	16	h2	h2	PROPN
ejpam-3559	97	17	\	\	PROPN
ejpam-3559	97	18	{	{	PUNCT
ejpam-3559	97	19	0	0	NUM
ejpam-3559	97	20	}	}	PUNCT
ejpam-3559	97	21	)	)	PUNCT
ejpam-3559	97	22	.	.	PUNCT
ejpam-3559	97	23	suppose	suppose	VERB
ejpam-3559	97	24	that	that	SCONJ
ejpam-3559	97	25	d1	d1	PROPN
ejpam-3559	97	26	6=	6=	ADP
ejpam-3559	97	27	∅	∅	NOUN
ejpam-3559	97	28	and	and	CCONJ
ejpam-3559	97	29	d2	d2	PROPN
ejpam-3559	97	30	6=	6=	ADP
ejpam-3559	97	31	∅.	∅.	ADV
ejpam-3559	97	32	choose	choose	VERB
ejpam-3559	97	33	any	any	DET
ejpam-3559	97	34	x	x	SYM
ejpam-3559	97	35	∈	∈	NOUN
ejpam-3559	97	36	d1	d1	NOUN
ejpam-3559	97	37	and	and	CCONJ
ejpam-3559	97	38	any	any	DET
ejpam-3559	97	39	y	y	PROPN
ejpam-3559	97	40	∈	∈	PROPN
ejpam-3559	97	41	d2	d2	PROPN
ejpam-3559	97	42	.	.	PUNCT
ejpam-3559	98	1	since	since	SCONJ
ejpam-3559	98	2	x	x	X
ejpam-3559	98	3	,	,	PUNCT
ejpam-3559	98	4	y	y	PROPN
ejpam-3559	98	5	∈	∈	PROPN
ejpam-3559	98	6	d	d	X
ejpam-3559	98	7	,	,	PUNCT
ejpam-3559	98	8	it	it	PRON
ejpam-3559	98	9	follows	follow	VERB
ejpam-3559	98	10	that	that	SCONJ
ejpam-3559	98	11	x	x	PUNCT
ejpam-3559	98	12	�	�	PROPN
ejpam-3559	98	13	u	u	PROPN
ejpam-3559	98	14	and	and	CCONJ
ejpam-3559	98	15	y	y	PROPN
ejpam-3559	98	16	�	�	PROPN
ejpam-3559	98	17	u	u	PROPN
ejpam-3559	98	18	for	for	ADP
ejpam-3559	98	19	all	all	PRON
ejpam-3559	98	20	u	u	PRON
ejpam-3559	98	21	∈	∈	PROPN
ejpam-3559	98	22	u	u	NOUN
ejpam-3559	98	23	.	.	PUNCT
ejpam-3559	99	1	pick	pick	VERB
ejpam-3559	99	2	w	w	PROPN
ejpam-3559	99	3	∈	∈	PROPN
ejpam-3559	99	4	u	u	NOUN
ejpam-3559	99	5	.	.	PUNCT
ejpam-3559	100	1	by	by	ADP
ejpam-3559	100	2	the	the	DET
ejpam-3559	100	3	definition	definition	NOUN
ejpam-3559	100	4	of	of	ADP
ejpam-3559	100	5	a	a	DET
ejpam-3559	100	6	hyper	hyper	ADJ
ejpam-3559	100	7	sum	sum	NOUN
ejpam-3559	100	8	,	,	PUNCT
ejpam-3559	100	9	if	if	SCONJ
ejpam-3559	100	10	w	w	PROPN
ejpam-3559	100	11	∈	∈	PROPN
ejpam-3559	100	12	h1	h1	PROPN
ejpam-3559	100	13	,	,	PUNCT
ejpam-3559	100	14	then	then	ADV
ejpam-3559	100	15	y	y	PROPN
ejpam-3559	100	16	∗	∗	VERB
ejpam-3559	100	17	w	w	PUNCT
ejpam-3559	100	18	=	=	PUNCT
ejpam-3559	100	19	{	{	PUNCT
ejpam-3559	100	20	y	y	NOUN
ejpam-3559	100	21	}	}	PUNCT
ejpam-3559	100	22	and	and	CCONJ
ejpam-3559	100	23	if	if	SCONJ
ejpam-3559	100	24	w	w	PROPN
ejpam-3559	100	25	∈	∈	PROPN
ejpam-3559	100	26	h2	h2	NOUN
ejpam-3559	100	27	,	,	PUNCT
ejpam-3559	100	28	then	then	ADV
ejpam-3559	100	29	x	x	X
ejpam-3559	100	30	∗w	∗w	PROPN
ejpam-3559	100	31	=	=	PUNCT
ejpam-3559	100	32	{	{	PUNCT
ejpam-3559	100	33	x	x	NOUN
ejpam-3559	100	34	}	}	PUNCT
ejpam-3559	100	35	.	.	PUNCT
ejpam-3559	101	1	since	since	SCONJ
ejpam-3559	101	2	x	x	PROPN
ejpam-3559	101	3	and	and	CCONJ
ejpam-3559	101	4	y	y	PROPN
ejpam-3559	101	5	are	be	AUX
ejpam-3559	101	6	nonzero	nonzero	ADJ
ejpam-3559	101	7	,	,	PUNCT
ejpam-3559	101	8	y	y	PROPN
ejpam-3559	101	9	6	6	NUM
ejpam-3559	101	10	�	�	PROPN
ejpam-3559	101	11	w	w	PROPN
ejpam-3559	101	12	and	and	CCONJ
ejpam-3559	101	13	x	x	SYM
ejpam-3559	101	14	6	6	NUM
ejpam-3559	101	15	�	�	PROPN
ejpam-3559	101	16	w	w	PROPN
ejpam-3559	101	17	,	,	PUNCT
ejpam-3559	101	18	a	a	DET
ejpam-3559	101	19	contradiction	contradiction	NOUN
ejpam-3559	101	20	.	.	PUNCT
ejpam-3559	102	1	thus	thus	ADV
ejpam-3559	102	2	,	,	PUNCT
ejpam-3559	102	3	either	either	CCONJ
ejpam-3559	102	4	d1	d1	NOUN
ejpam-3559	102	5	=	=	SYM
ejpam-3559	102	6	∅	∅	NOUN
ejpam-3559	102	7	or	or	CCONJ
ejpam-3559	102	8	d2	d2	NOUN
ejpam-3559	102	9	=	=	PUNCT
ejpam-3559	102	10	∅	∅	NOUN
ejpam-3559	102	11	,	,	PUNCT
ejpam-3559	102	12	that	that	ADV
ejpam-3559	102	13	is	is	ADV
ejpam-3559	102	14	,	,	PUNCT
ejpam-3559	102	15	either	either	CCONJ
ejpam-3559	102	16	d	d	PROPN
ejpam-3559	102	17	=	=	SYM
ejpam-3559	102	18	d1	d1	PROPN
ejpam-3559	102	19	or	or	CCONJ
ejpam-3559	102	20	d	d	NOUN
ejpam-3559	102	21	=	=	SYM
ejpam-3559	102	22	d2	d2	PROPN
ejpam-3559	102	23	.	.	PUNCT
ejpam-3559	103	1	if	if	SCONJ
ejpam-3559	103	2	d	d	NOUN
ejpam-3559	103	3	=	=	SYM
ejpam-3559	103	4	d1	d1	PROPN
ejpam-3559	103	5	then	then	ADV
ejpam-3559	103	6	u	u	X
ejpam-3559	103	7	=	=	NOUN
ejpam-3559	103	8	rh1(d1	rh1(d1	X
ejpam-3559	103	9	)	)	PUNCT
ejpam-3559	103	10	.	.	PUNCT
ejpam-3559	104	1	since	since	SCONJ
ejpam-3559	104	2	d1	d1	PROPN
ejpam-3559	104	3	6=	6=	X
ejpam-3559	104	4	{	{	PUNCT
ejpam-3559	104	5	0	0	NUM
ejpam-3559	104	6	}	}	PUNCT
ejpam-3559	104	7	and	and	CCONJ
ejpam-3559	104	8	u	u	PROPN
ejpam-3559	104	9	6=	6=	NOUN
ejpam-3559	104	10	∅	∅	NOUN
ejpam-3559	104	11	in	in	ADP
ejpam-3559	104	12	h	h	NOUN
ejpam-3559	104	13	,	,	PUNCT
ejpam-3559	104	14	by	by	ADP
ejpam-3559	104	15	theorem	theorem	ADJ
ejpam-3559	104	16	2.2(iii	2.2(iii	NUM
ejpam-3559	104	17	)	)	PUNCT
ejpam-3559	104	18	,	,	PUNCT
ejpam-3559	104	19	u	u	PROPN
ejpam-3559	104	20	6=	6=	ADP
ejpam-3559	104	21	h1	h1	NOUN
ejpam-3559	104	22	and	and	CCONJ
ejpam-3559	104	23	u	u	NOUN
ejpam-3559	104	24	6=	6=	NOUN
ejpam-3559	104	25	∅	∅	NOUN
ejpam-3559	104	26	in	in	ADP
ejpam-3559	104	27	h1	h1	PROPN
ejpam-3559	104	28	.	.	PUNCT
ejpam-3559	105	1	thus	thus	ADV
ejpam-3559	105	2	,	,	PUNCT
ejpam-3559	105	3	u	u	PROPN
ejpam-3559	105	4	=	=	NOUN
ejpam-3559	105	5	rh1(d1	rh1(d1	X
ejpam-3559	105	6	)	)	PUNCT
ejpam-3559	105	7	∈	∈	PROPN
ejpam-3559	105	8	br(h1	br(h1	NOUN
ejpam-3559	105	9	)	)	PUNCT
ejpam-3559	105	10	\	\	NOUN
ejpam-3559	105	11	{	{	PUNCT
ejpam-3559	105	12	∅	∅	NOUN
ejpam-3559	105	13	,	,	PUNCT
ejpam-3559	105	14	h1	h1	NOUN
ejpam-3559	105	15	}	}	PUNCT
ejpam-3559	105	16	⊆	⊆	NUM
ejpam-3559	105	17	[	[	SYM
ejpam-3559	105	18	br(h1	br(h1	NOUN
ejpam-3559	105	19	)	)	PUNCT
ejpam-3559	105	20	∪	∪	ADP
ejpam-3559	105	21	br(h2	br(h2	NOUN
ejpam-3559	105	22	)	)	PUNCT
ejpam-3559	105	23	]	]	PUNCT
ejpam-3559	105	24	\	\	NOUN
ejpam-3559	105	25	{	{	PUNCT
ejpam-3559	105	26	∅	∅	NOUN
ejpam-3559	105	27	,	,	PUNCT
ejpam-3559	105	28	h1	h1	NOUN
ejpam-3559	105	29	,	,	PUNCT
ejpam-3559	105	30	h2	h2	PROPN
ejpam-3559	105	31	}	}	PUNCT
ejpam-3559	105	32	.	.	PUNCT
ejpam-3559	106	1	in	in	ADP
ejpam-3559	106	2	the	the	DET
ejpam-3559	106	3	same	same	ADJ
ejpam-3559	106	4	way	way	NOUN
ejpam-3559	106	5	,	,	PUNCT
ejpam-3559	106	6	if	if	SCONJ
ejpam-3559	106	7	d	d	PROPN
ejpam-3559	106	8	=	=	SYM
ejpam-3559	106	9	d2	d2	PROPN
ejpam-3559	106	10	then	then	ADV
ejpam-3559	106	11	u	u	NOUN
ejpam-3559	106	12	=	=	NOUN
ejpam-3559	106	13	rh2(d2	rh2(d2	X
ejpam-3559	106	14	)	)	PUNCT
ejpam-3559	106	15	∈	∈	NOUN
ejpam-3559	106	16	br(h2	br(h2	NOUN
ejpam-3559	106	17	)	)	PUNCT
ejpam-3559	106	18	\	\	NOUN
ejpam-3559	106	19	{	{	PUNCT
ejpam-3559	106	20	∅	∅	NOUN
ejpam-3559	106	21	,	,	PUNCT
ejpam-3559	106	22	h2	h2	NOUN
ejpam-3559	106	23	}	}	PUNCT
ejpam-3559	106	24	⊆	⊆	NUM
ejpam-3559	106	25	[	[	AUX
ejpam-3559	106	26	br(h1	br(h1	NOUN
ejpam-3559	106	27	)	)	PUNCT
ejpam-3559	106	28	∪	∪	ADP
ejpam-3559	106	29	br(h2	br(h2	NOUN
ejpam-3559	106	30	)	)	PUNCT
ejpam-3559	106	31	]	]	PUNCT
ejpam-3559	106	32	\	\	NOUN
ejpam-3559	106	33	{	{	PUNCT
ejpam-3559	106	34	∅	∅	NOUN
ejpam-3559	106	35	,	,	PUNCT
ejpam-3559	106	36	h1	h1	NOUN
ejpam-3559	106	37	,	,	PUNCT
ejpam-3559	106	38	h2	h2	PROPN
ejpam-3559	106	39	}	}	PUNCT
ejpam-3559	106	40	.	.	PUNCT
ejpam-3559	107	1	therefore	therefore	ADV
ejpam-3559	107	2	,	,	PUNCT
ejpam-3559	107	3	br(h	br(h	NUM
ejpam-3559	107	4	)	)	PUNCT
ejpam-3559	107	5	\	\	NOUN
ejpam-3559	107	6	{	{	PUNCT
ejpam-3559	107	7	∅	∅	NOUN
ejpam-3559	107	8	,	,	PUNCT
ejpam-3559	107	9	h	h	NOUN
ejpam-3559	107	10	}	}	PUNCT
ejpam-3559	107	11	=	=	SYM
ejpam-3559	107	12	(	(	PUNCT
ejpam-3559	107	13	br(h1	br(h1	NOUN
ejpam-3559	107	14	)	)	PUNCT
ejpam-3559	107	15	∪	∪	NOUN
ejpam-3559	107	16	br(h2	br(h2	NOUN
ejpam-3559	107	17	)	)	PUNCT
ejpam-3559	107	18	)	)	PUNCT
ejpam-3559	107	19	\	\	NOUN
ejpam-3559	107	20	{	{	PUNCT
ejpam-3559	107	21	∅	∅	NOUN
ejpam-3559	107	22	,	,	PUNCT
ejpam-3559	107	23	h1	h1	NOUN
ejpam-3559	107	24	,	,	PUNCT
ejpam-3559	107	25	h2	h2	PROPN
ejpam-3559	107	26	}	}	PUNCT
ejpam-3559	107	27	.	.	PUNCT
ejpam-3559	108	1	4	4	X
ejpam-3559	108	2	.	.	X
ejpam-3559	108	3	bases	basis	NOUN
ejpam-3559	108	4	of	of	ADP
ejpam-3559	108	5	τl(h1	τl(h1	PROPN
ejpam-3559	108	6	×h2	×h2	NOUN
ejpam-3559	108	7	)	)	PUNCT
ejpam-3559	108	8	and	and	CCONJ
ejpam-3559	108	9	τr(h1	τr(h1	NUM
ejpam-3559	108	10	×h2	×h2	PROPN
ejpam-3559	108	11	)	)	PUNCT
ejpam-3559	108	12	for	for	ADP
ejpam-3559	108	13	any	any	DET
ejpam-3559	108	14	∅	∅	NOUN
ejpam-3559	108	15	6=	6=	ADP
ejpam-3559	108	16	d	d	SYM
ejpam-3559	108	17	⊆	⊆	NUM
ejpam-3559	108	18	h1×h2	h1×h2	NOUN
ejpam-3559	108	19	,	,	PUNCT
ejpam-3559	108	20	the	the	DET
ejpam-3559	108	21	h1	h1	NOUN
ejpam-3559	108	22	-	-	PUNCT
ejpam-3559	108	23	projection	projection	NOUN
ejpam-3559	108	24	and	and	CCONJ
ejpam-3559	108	25	h2	h2	NOUN
ejpam-3559	108	26	-	-	PUNCT
ejpam-3559	108	27	projection	projection	NOUN
ejpam-3559	108	28	of	of	ADP
ejpam-3559	108	29	d	d	NOUN
ejpam-3559	108	30	are	be	AUX
ejpam-3559	108	31	,	,	PUNCT
ejpam-3559	108	32	respectively	respectively	ADV
ejpam-3559	108	33	,	,	PUNCT
ejpam-3559	108	34	the	the	DET
ejpam-3559	108	35	sets	set	NOUN
ejpam-3559	108	36	dh1	dh1	PROPN
ejpam-3559	108	37	=	=	SYM
ejpam-3559	108	38	{	{	PUNCT
ejpam-3559	108	39	x	x	PUNCT
ejpam-3559	108	40	∈	∈	PROPN
ejpam-3559	108	41	h1	h1	NOUN
ejpam-3559	108	42	:	:	PUNCT
ejpam-3559	109	1	(	(	PUNCT
ejpam-3559	109	2	x	x	X
ejpam-3559	109	3	,	,	PUNCT
ejpam-3559	109	4	y	y	NOUN
ejpam-3559	109	5	)	)	PUNCT
ejpam-3559	109	6	∈	∈	PROPN
ejpam-3559	109	7	d	d	NOUN
ejpam-3559	109	8	for	for	ADP
ejpam-3559	109	9	some	some	DET
ejpam-3559	109	10	y	y	PROPN
ejpam-3559	109	11	∈	∈	PROPN
ejpam-3559	109	12	h2	h2	PROPN
ejpam-3559	109	13	}	}	PUNCT
ejpam-3559	109	14	and	and	CCONJ
ejpam-3559	109	15	dh2	dh2	PROPN
ejpam-3559	109	16	=	=	PUNCT
ejpam-3559	109	17	{	{	PUNCT
ejpam-3559	109	18	y	y	PROPN
ejpam-3559	109	19	∈	∈	PROPN
ejpam-3559	109	20	h1	h1	PROPN
ejpam-3559	109	21	:	:	PUNCT
ejpam-3559	110	1	(	(	PUNCT
ejpam-3559	110	2	z	z	X
ejpam-3559	110	3	,	,	PUNCT
ejpam-3559	110	4	y	y	NOUN
ejpam-3559	110	5	)	)	PUNCT
ejpam-3559	110	6	∈	∈	PROPN
ejpam-3559	110	7	d	d	NOUN
ejpam-3559	110	8	for	for	ADP
ejpam-3559	110	9	some	some	DET
ejpam-3559	110	10	z	z	PROPN
ejpam-3559	110	11	∈	∈	PROPN
ejpam-3559	110	12	dh1	dh1	PROPN
ejpam-3559	110	13	}	}	PUNCT
ejpam-3559	110	14	.	.	PUNCT
ejpam-3559	111	1	now	now	ADV
ejpam-3559	111	2	,	,	PUNCT
ejpam-3559	111	3	for	for	SCONJ
ejpam-3559	111	4	each	each	DET
ejpam-3559	111	5	x	x	SYM
ejpam-3559	111	6	∈	∈	PROPN
ejpam-3559	111	7	s	s	PART
ejpam-3559	111	8	=	=	X
ejpam-3559	111	9	dh1	dh1	PROPN
ejpam-3559	111	10	,	,	PUNCT
ejpam-3559	111	11	let	let	VERB
ejpam-3559	111	12	tx	tx	VERB
ejpam-3559	111	13	=	=	PUNCT
ejpam-3559	111	14	{	{	PUNCT
ejpam-3559	111	15	y	y	PROPN
ejpam-3559	111	16	∈	∈	PROPN
ejpam-3559	111	17	dh2	dh2	NOUN
ejpam-3559	111	18	:	:	PUNCT
ejpam-3559	111	19	(	(	PUNCT
ejpam-3559	111	20	x	x	X
ejpam-3559	111	21	,	,	PUNCT
ejpam-3559	111	22	y	y	NOUN
ejpam-3559	111	23	)	)	PUNCT
ejpam-3559	111	24	∈	∈	PROPN
ejpam-3559	112	1	d	d	NOUN
ejpam-3559	112	2	}	}	PUNCT
ejpam-3559	112	3	.	.	PUNCT
ejpam-3559	113	1	then	then	ADV
ejpam-3559	113	2	d	d	X
ejpam-3559	113	3	=	=	PUNCT
ejpam-3559	113	4	⋃	⋃	PROPN
ejpam-3559	113	5	x∈s	x∈s	NOUN
ejpam-3559	114	1	[	[	X
ejpam-3559	114	2	{	{	PUNCT
ejpam-3559	114	3	x	x	NOUN
ejpam-3559	114	4	}	}	PUNCT
ejpam-3559	114	5	×	×	PROPN
ejpam-3559	114	6	tx	tx	PROPN
ejpam-3559	114	7	]	]	PUNCT
ejpam-3559	114	8	.	.	PUNCT
ejpam-3559	115	1	lemma	lemma	PROPN
ejpam-3559	115	2	4.1	4.1	NUM
ejpam-3559	115	3	.	.	PUNCT
ejpam-3559	116	1	let	let	VERB
ejpam-3559	116	2	{	{	PUNCT
ejpam-3559	116	3	aα	aα	NOUN
ejpam-3559	116	4	:	:	PUNCT
ejpam-3559	116	5	α	α	PROPN
ejpam-3559	116	6	∈	∈	PROPN
ejpam-3559	117	1	i	i	PRON
ejpam-3559	117	2	}	}	PUNCT
ejpam-3559	117	3	be	be	VERB
ejpam-3559	117	4	a	a	DET
ejpam-3559	117	5	collection	collection	NOUN
ejpam-3559	117	6	of	of	ADP
ejpam-3559	117	7	subsets	subset	NOUN
ejpam-3559	117	8	of	of	ADP
ejpam-3559	117	9	a	a	DET
ejpam-3559	117	10	hyper	hyper	ADJ
ejpam-3559	117	11	bck	bck	NOUN
ejpam-3559	117	12	-	-	PUNCT
ejpam-3559	117	13	algebra	algebra	NOUN
ejpam-3559	117	14	h.	h.	NOUN
ejpam-3559	117	15	then	then	ADV
ejpam-3559	117	16	⋂	⋂	PROPN
ejpam-3559	117	17	α∈i	α∈i	X
ejpam-3559	117	18	lh(aα	lh(aα	PROPN
ejpam-3559	117	19	)	)	PUNCT
ejpam-3559	117	20	=	=	SYM
ejpam-3559	117	21	lh	lh	NOUN
ejpam-3559	117	22	(	(	PUNCT
ejpam-3559	117	23	⋃	⋃	PROPN
ejpam-3559	117	24	α∈i	α∈i	NUM
ejpam-3559	117	25	aα	aα	NOUN
ejpam-3559	117	26	)	)	PUNCT
ejpam-3559	117	27	.	.	PUNCT
ejpam-3559	118	1	proof	proof	NOUN
ejpam-3559	118	2	:	:	PUNCT
ejpam-3559	118	3	let	let	VERB
ejpam-3559	118	4	{	{	PUNCT
ejpam-3559	118	5	aα	aα	NOUN
ejpam-3559	118	6	:	:	PUNCT
ejpam-3559	118	7	α	α	PROPN
ejpam-3559	118	8	∈	∈	PROPN
ejpam-3559	119	1	i	i	PRON
ejpam-3559	119	2	}	}	PUNCT
ejpam-3559	119	3	be	be	VERB
ejpam-3559	119	4	a	a	DET
ejpam-3559	119	5	collection	collection	NOUN
ejpam-3559	119	6	of	of	ADP
ejpam-3559	119	7	subsets	subset	NOUN
ejpam-3559	119	8	of	of	ADP
ejpam-3559	119	9	h.	h.	PROPN
ejpam-3559	119	10	then	then	ADV
ejpam-3559	120	1	x	x	SYM
ejpam-3559	120	2	∈	∈	PROPN
ejpam-3559	120	3	⋂	⋂	PROPN
ejpam-3559	120	4	α∈i	α∈i	ADJ
ejpam-3559	120	5	lh(aα)⇔	lh(aα)⇔	NOUN
ejpam-3559	120	6	x	x	PUNCT
ejpam-3559	120	7	∈	∈	PROPN
ejpam-3559	120	8	lh(aα	lh(aα	PROPN
ejpam-3559	120	9	)	)	PUNCT
ejpam-3559	120	10	for	for	ADP
ejpam-3559	120	11	all	all	DET
ejpam-3559	120	12	α	α	PRON
ejpam-3559	120	13	∈	∈	NOUN
ejpam-3559	120	14	i	i	PRON
ejpam-3559	120	15	⇔	⇔	PROPN
ejpam-3559	120	16	x	x	SYM
ejpam-3559	120	17	�	�	PROPN
ejpam-3559	120	18	a	a	PRON
ejpam-3559	120	19	for	for	ADP
ejpam-3559	120	20	all	all	DET
ejpam-3559	120	21	a	a	DET
ejpam-3559	120	22	∈	∈	ADJ
ejpam-3559	120	23	aα	aα	NOUN
ejpam-3559	120	24	and	and	CCONJ
ejpam-3559	120	25	for	for	ADP
ejpam-3559	120	26	all	all	DET
ejpam-3559	120	27	α	α	PRON
ejpam-3559	120	28	∈	∈	NOUN
ejpam-3559	120	29	i	i	PRON
ejpam-3559	120	30	⇔	⇔	PROPN
ejpam-3559	120	31	x	x	SYM
ejpam-3559	120	32	�	�	PROPN
ejpam-3559	120	33	a	a	PRON
ejpam-3559	120	34	for	for	ADP
ejpam-3559	120	35	all	all	DET
ejpam-3559	120	36	a	a	DET
ejpam-3559	120	37	∈	∈	NOUN
ejpam-3559	120	38	⋃	⋃	NOUN
ejpam-3559	120	39	α∈i	α∈i	NUM
ejpam-3559	120	40	aα	aα	NOUN
ejpam-3559	120	41	⇔	⇔	NOUN
ejpam-3559	120	42	x	x	SYM
ejpam-3559	120	43	∈	∈	PROPN
ejpam-3559	120	44	lh	lh	PROPN
ejpam-3559	120	45	(	(	PUNCT
ejpam-3559	120	46	⋃	⋃	PROPN
ejpam-3559	120	47	α∈i	α∈i	NUM
ejpam-3559	120	48	aα	aα	NOUN
ejpam-3559	120	49	)	)	PUNCT
ejpam-3559	120	50	.	.	PUNCT
ejpam-3559	121	1	therefore	therefore	ADV
ejpam-3559	121	2	,	,	PUNCT
ejpam-3559	121	3	the	the	DET
ejpam-3559	121	4	equality	equality	NOUN
ejpam-3559	121	5	is	be	AUX
ejpam-3559	121	6	true	true	ADJ
ejpam-3559	121	7	.	.	PUNCT
ejpam-3559	122	1	theorem	theorem	VERB
ejpam-3559	122	2	4.2	4.2	NUM
ejpam-3559	122	3	.	.	PUNCT
ejpam-3559	123	1	let	let	VERB
ejpam-3559	123	2	h	h	PRON
ejpam-3559	123	3	be	be	AUX
ejpam-3559	123	4	a	a	DET
ejpam-3559	123	5	hyper	hyper	ADJ
ejpam-3559	123	6	product	product	NOUN
ejpam-3559	123	7	of	of	ADP
ejpam-3559	123	8	hyper	hyper	ADJ
ejpam-3559	123	9	bck	bck	NOUN
ejpam-3559	123	10	-	-	PUNCT
ejpam-3559	123	11	algebras	algebras	ADJ
ejpam-3559	123	12	h1	h1	PROPN
ejpam-3559	123	13	and	and	CCONJ
ejpam-3559	123	14	h2	h2	PROPN
ejpam-3559	123	15	.	.	PUNCT
ejpam-3559	124	1	then	then	ADV
ejpam-3559	124	2	the	the	DET
ejpam-3559	124	3	following	follow	VERB
ejpam-3559	124	4	properties	property	NOUN
ejpam-3559	124	5	hold	hold	VERB
ejpam-3559	124	6	:	:	PUNCT
ejpam-3559	124	7	(	(	PUNCT
ejpam-3559	124	8	i	i	NOUN
ejpam-3559	124	9	)	)	PUNCT
ejpam-3559	124	10	lh(a×b	lh(a×b	PROPN
ejpam-3559	124	11	)	)	PUNCT
ejpam-3559	124	12	=	=	PUNCT
ejpam-3559	124	13	lh1(a)×	lh1(a)×	NOUN
ejpam-3559	124	14	lh2(b	lh2(b	ADJ
ejpam-3559	124	15	)	)	PUNCT
ejpam-3559	124	16	for	for	ADP
ejpam-3559	124	17	a	a	DET
ejpam-3559	124	18	⊆	⊆	NUM
ejpam-3559	124	19	h1	h1	NOUN
ejpam-3559	124	20	and	and	CCONJ
ejpam-3559	124	21	b	b	NOUN
ejpam-3559	124	22	⊆	⊆	NUM
ejpam-3559	124	23	h2	h2	NOUN
ejpam-3559	124	24	.	.	PUNCT
ejpam-3559	125	1	(	(	PUNCT
ejpam-3559	125	2	ii	ii	NOUN
ejpam-3559	125	3	)	)	PUNCT
ejpam-3559	125	4	if	if	SCONJ
ejpam-3559	125	5	{	{	PUNCT
ejpam-3559	125	6	aα	aα	NOUN
ejpam-3559	125	7	:	:	PUNCT
ejpam-3559	125	8	α	α	PROPN
ejpam-3559	125	9	∈	∈	PROPN
ejpam-3559	126	1	i	i	X
ejpam-3559	126	2	}	}	PUNCT
ejpam-3559	126	3	and	and	CCONJ
ejpam-3559	126	4	{	{	PUNCT
ejpam-3559	126	5	bα	bα	NOUN
ejpam-3559	126	6	:	:	PUNCT
ejpam-3559	126	7	α	α	PROPN
ejpam-3559	126	8	∈	∈	PROPN
ejpam-3559	127	1	i	i	PRON
ejpam-3559	127	2	}	}	PUNCT
ejpam-3559	127	3	are	be	AUX
ejpam-3559	127	4	collections	collection	NOUN
ejpam-3559	127	5	of	of	ADP
ejpam-3559	127	6	subsets	subset	NOUN
ejpam-3559	127	7	of	of	ADP
ejpam-3559	127	8	h1	h1	NOUN
ejpam-3559	127	9	and	and	CCONJ
ejpam-3559	127	10	h2	h2	NOUN
ejpam-3559	127	11	,	,	PUNCT
ejpam-3559	127	12	respectively	respectively	ADV
ejpam-3559	127	13	,	,	PUNCT
ejpam-3559	127	14	then⋂	then⋂	NUM
ejpam-3559	127	15	α∈i	α∈i	NUM
ejpam-3559	128	1	[	[	X
ejpam-3559	128	2	lh1(aα)×	lh1(aα)×	PROPN
ejpam-3559	128	3	lh2(bα	lh2(bα	NOUN
ejpam-3559	128	4	)	)	PUNCT
ejpam-3559	128	5	]	]	PUNCT
ejpam-3559	129	1	=	=	SYM
ejpam-3559	129	2	lh1	lh1	X
ejpam-3559	129	3	(	(	PUNCT
ejpam-3559	129	4	⋃	⋃	PROPN
ejpam-3559	129	5	α∈i	α∈i	NUM
ejpam-3559	129	6	aα	aα	NOUN
ejpam-3559	129	7	)	)	PUNCT
ejpam-3559	129	8	×	×	NOUN
ejpam-3559	129	9	lh2	lh2	NOUN
ejpam-3559	129	10	(	(	PUNCT
ejpam-3559	129	11	⋃	⋃	PROPN
ejpam-3559	129	12	α∈i	α∈i	NUM
ejpam-3559	129	13	bα	bα	NOUN
ejpam-3559	129	14	)	)	PUNCT
ejpam-3559	129	15	.	.	PUNCT
ejpam-3559	130	1	r.	r.	PROPN
ejpam-3559	130	2	patangan	patangan	PROPN
ejpam-3559	130	3	,	,	PUNCT
ejpam-3559	130	4	s.	s.	PROPN
ejpam-3559	130	5	canoy	canoy	PROPN
ejpam-3559	130	6	,	,	PUNCT
ejpam-3559	130	7	jr	jr	PROPN
ejpam-3559	130	8	.	.	PROPN
ejpam-3559	130	9	/	/	SYM
ejpam-3559	130	10	eur	eur	PROPN
ejpam-3559	130	11	.	.	PUNCT
ejpam-3559	131	1	j.	j.	PROPN
ejpam-3559	131	2	pure	pure	PROPN
ejpam-3559	131	3	appl	appl	PROPN
ejpam-3559	131	4	.	.	PROPN
ejpam-3559	131	5	math	math	PROPN
ejpam-3559	131	6	,	,	PUNCT
ejpam-3559	131	7	12	12	NUM
ejpam-3559	131	8	(	(	PUNCT
ejpam-3559	131	9	4	4	NUM
ejpam-3559	131	10	)	)	PUNCT
ejpam-3559	131	11	(	(	PUNCT
ejpam-3559	131	12	2019	2019	NUM
ejpam-3559	131	13	)	)	PUNCT
ejpam-3559	131	14	,	,	PUNCT
ejpam-3559	131	15	1524	1524	NUM
ejpam-3559	131	16	-	-	SYM
ejpam-3559	131	17	1532	1532	NUM
ejpam-3559	131	18	1528	1528	NUM
ejpam-3559	131	19	(	(	PUNCT
ejpam-3559	131	20	iii	iii	NOUN
ejpam-3559	131	21	)	)	PUNCT
ejpam-3559	131	22	if	if	SCONJ
ejpam-3559	131	23	d	d	PROPN
ejpam-3559	131	24	=	=	SYM
ejpam-3559	131	25	⋃	⋃	PROPN
ejpam-3559	131	26	x∈s	x∈s	NOUN
ejpam-3559	131	27	(	(	PUNCT
ejpam-3559	131	28	{	{	PUNCT
ejpam-3559	131	29	x	x	NOUN
ejpam-3559	131	30	}	}	PUNCT
ejpam-3559	131	31	×	×	PROPN
ejpam-3559	131	32	tx	tx	PROPN
ejpam-3559	131	33	)	)	PUNCT
ejpam-3559	131	34	,	,	PUNCT
ejpam-3559	131	35	where	where	SCONJ
ejpam-3559	131	36	s	s	VERB
ejpam-3559	131	37	⊆	⊆	NUM
ejpam-3559	131	38	h1	h1	NOUN
ejpam-3559	131	39	and	and	CCONJ
ejpam-3559	131	40	tx	tx	VERB
ejpam-3559	131	41	⊆	⊆	NUM
ejpam-3559	131	42	h2	h2	NOUN
ejpam-3559	131	43	for	for	ADP
ejpam-3559	131	44	each	each	DET
ejpam-3559	131	45	x	x	SYM
ejpam-3559	131	46	∈	∈	PROPN
ejpam-3559	131	47	s	s	NOUN
ejpam-3559	131	48	,	,	PUNCT
ejpam-3559	131	49	then	then	ADV
ejpam-3559	131	50	lh(d	lh(d	PRON
ejpam-3559	131	51	)	)	PUNCT
ejpam-3559	132	1	=	=	SYM
ejpam-3559	132	2	⋂	⋂	PROPN
ejpam-3559	132	3	x∈s	x∈s	X
ejpam-3559	132	4	(	(	PUNCT
ejpam-3559	132	5	lh1(x)×	lh1(x)×	NOUN
ejpam-3559	132	6	lh2(tx	lh2(tx	ADJ
ejpam-3559	132	7	)	)	PUNCT
ejpam-3559	132	8	)	)	PUNCT
ejpam-3559	133	1	=	=	SYM
ejpam-3559	133	2	lh1(s)×	lh1(s)×	NOUN
ejpam-3559	133	3	lh2	lh2	NOUN
ejpam-3559	133	4	(	(	PUNCT
ejpam-3559	133	5	⋃	⋃	PROPN
ejpam-3559	133	6	x∈s	x∈s	PROPN
ejpam-3559	133	7	tx	tx	PROPN
ejpam-3559	133	8	)	)	PUNCT
ejpam-3559	133	9	.	.	PUNCT
ejpam-3559	134	1	proof	proof	NOUN
ejpam-3559	134	2	:	:	PUNCT
ejpam-3559	134	3	(	(	PUNCT
ejpam-3559	134	4	i	i	NOUN
ejpam-3559	134	5	)	)	PUNCT
ejpam-3559	134	6	let	let	VERB
ejpam-3559	134	7	a	a	PRON
ejpam-3559	134	8	and	and	CCONJ
ejpam-3559	134	9	b	b	NOUN
ejpam-3559	134	10	be	be	AUX
ejpam-3559	134	11	subsets	subset	NOUN
ejpam-3559	134	12	of	of	ADP
ejpam-3559	134	13	h1	h1	NOUN
ejpam-3559	134	14	and	and	CCONJ
ejpam-3559	134	15	h2	h2	NOUN
ejpam-3559	134	16	,	,	PUNCT
ejpam-3559	134	17	respectively	respectively	ADV
ejpam-3559	134	18	.	.	PUNCT
ejpam-3559	135	1	then	then	ADV
ejpam-3559	135	2	lh(a×b	lh(a×b	NUM
ejpam-3559	135	3	)	)	PUNCT
ejpam-3559	135	4	=	=	PRON
ejpam-3559	135	5	{	{	PUNCT
ejpam-3559	135	6	(	(	PUNCT
ejpam-3559	135	7	x	x	NOUN
ejpam-3559	135	8	,	,	PUNCT
ejpam-3559	135	9	y	y	NOUN
ejpam-3559	135	10	)	)	PUNCT
ejpam-3559	135	11	∈	∈	PROPN
ejpam-3559	135	12	h1	h1	PROPN
ejpam-3559	135	13	×h2	×h2	PROPN
ejpam-3559	135	14	:	:	PUNCT
ejpam-3559	135	15	(	(	PUNCT
ejpam-3559	135	16	x	x	X
ejpam-3559	135	17	,	,	PUNCT
ejpam-3559	135	18	y	y	PROPN
ejpam-3559	135	19	)	)	PUNCT
ejpam-3559	135	20	�	�	PROPN
ejpam-3559	135	21	(	(	PUNCT
ejpam-3559	135	22	a	a	DET
ejpam-3559	135	23	,	,	PUNCT
ejpam-3559	135	24	b	b	NOUN
ejpam-3559	135	25	)	)	PUNCT
ejpam-3559	135	26	for	for	ADP
ejpam-3559	135	27	all	all	PRON
ejpam-3559	135	28	(	(	PUNCT
ejpam-3559	135	29	a	a	PRON
ejpam-3559	135	30	,	,	PUNCT
ejpam-3559	135	31	b	b	NOUN
ejpam-3559	135	32	)	)	PUNCT
ejpam-3559	135	33	∈	∈	NOUN
ejpam-3559	135	34	a×b	a×b	PROPN
ejpam-3559	135	35	}	}	PUNCT
ejpam-3559	135	36	=	=	SYM
ejpam-3559	135	37	{	{	PUNCT
ejpam-3559	135	38	(	(	PUNCT
ejpam-3559	135	39	x	x	NOUN
ejpam-3559	135	40	,	,	PUNCT
ejpam-3559	135	41	y	y	NOUN
ejpam-3559	135	42	)	)	PUNCT
ejpam-3559	135	43	∈	∈	PROPN
ejpam-3559	135	44	h1	h1	PROPN
ejpam-3559	135	45	×h2	×h2	PROPN
ejpam-3559	135	46	:	:	PUNCT
ejpam-3559	135	47	x	x	X
ejpam-3559	135	48	�	�	PROPN
ejpam-3559	135	49	a	a	PROPN
ejpam-3559	135	50	and	and	CCONJ
ejpam-3559	135	51	y	y	PROPN
ejpam-3559	135	52	�	�	PROPN
ejpam-3559	135	53	b	b	PROPN
ejpam-3559	135	54	∀a	∀a	X
ejpam-3559	135	55	∈	∈	PROPN
ejpam-3559	135	56	a	a	PRON
ejpam-3559	135	57	and	and	CCONJ
ejpam-3559	135	58	b	b	NOUN
ejpam-3559	135	59	∈	∈	ADJ
ejpam-3559	135	60	b	b	PROPN
ejpam-3559	135	61	}	}	PUNCT
ejpam-3559	135	62	=	=	SYM
ejpam-3559	135	63	{	{	PUNCT
ejpam-3559	135	64	x	x	PUNCT
ejpam-3559	135	65	∈	∈	PROPN
ejpam-3559	135	66	h1	h1	NOUN
ejpam-3559	135	67	:	:	PUNCT
ejpam-3559	135	68	x	x	X
ejpam-3559	135	69	�	�	PROPN
ejpam-3559	135	70	a	a	DET
ejpam-3559	135	71	∀a	∀a	X
ejpam-3559	135	72	∈	∈	NOUN
ejpam-3559	135	73	a	a	DET
ejpam-3559	135	74	}	}	PUNCT
ejpam-3559	135	75	×	×	NOUN
ejpam-3559	135	76	{	{	PUNCT
ejpam-3559	135	77	y	y	PROPN
ejpam-3559	135	78	∈	∈	PROPN
ejpam-3559	135	79	h2	h2	NOUN
ejpam-3559	135	80	:	:	PUNCT
ejpam-3559	135	81	y	y	PROPN
ejpam-3559	135	82	�	�	PROPN
ejpam-3559	135	83	b	b	PROPN
ejpam-3559	135	84	∀b	∀b	PROPN
ejpam-3559	135	85	∈	∈	PROPN
ejpam-3559	135	86	b	b	NOUN
ejpam-3559	135	87	}	}	PUNCT
ejpam-3559	135	88	=	=	PUNCT
ejpam-3559	135	89	lh1(a)×	lh1(a)×	NOUN
ejpam-3559	135	90	lh2(b	lh2(b	ADJ
ejpam-3559	135	91	)	)	PUNCT
ejpam-3559	135	92	.	.	PUNCT
ejpam-3559	136	1	(	(	PUNCT
ejpam-3559	136	2	ii	ii	X
ejpam-3559	136	3	)	)	PUNCT
ejpam-3559	136	4	let	let	VERB
ejpam-3559	136	5	{	{	PUNCT
ejpam-3559	136	6	aα	aα	NOUN
ejpam-3559	136	7	:	:	PUNCT
ejpam-3559	136	8	α	α	PROPN
ejpam-3559	136	9	∈	∈	PROPN
ejpam-3559	137	1	i	i	X
ejpam-3559	137	2	}	}	PUNCT
ejpam-3559	137	3	and	and	CCONJ
ejpam-3559	137	4	{	{	PUNCT
ejpam-3559	137	5	bα	bα	NOUN
ejpam-3559	137	6	:	:	PUNCT
ejpam-3559	137	7	α	α	PROPN
ejpam-3559	137	8	∈	∈	PROPN
ejpam-3559	138	1	i	i	PRON
ejpam-3559	138	2	}	}	PUNCT
ejpam-3559	138	3	be	be	VERB
ejpam-3559	138	4	collections	collection	NOUN
ejpam-3559	138	5	of	of	ADP
ejpam-3559	138	6	subsets	subset	NOUN
ejpam-3559	138	7	of	of	ADP
ejpam-3559	138	8	h1	h1	NOUN
ejpam-3559	138	9	and	and	CCONJ
ejpam-3559	138	10	h2	h2	NOUN
ejpam-3559	138	11	,	,	PUNCT
ejpam-3559	138	12	respectively	respectively	ADV
ejpam-3559	138	13	,	,	PUNCT
ejpam-3559	138	14	and	and	CCONJ
ejpam-3559	138	15	let	let	VERB
ejpam-3559	138	16	k	k	PROPN
ejpam-3559	138	17	=	=	PUNCT
ejpam-3559	138	18	⋂	⋂	PROPN
ejpam-3559	138	19	α∈i	α∈i	NUM
ejpam-3559	139	1	[	[	X
ejpam-3559	139	2	lh1(aα)×	lh1(aα)×	PROPN
ejpam-3559	139	3	lh2(bα	lh2(bα	PROPN
ejpam-3559	139	4	)	)	PUNCT
ejpam-3559	139	5	]	]	PUNCT
ejpam-3559	139	6	.	.	PUNCT
ejpam-3559	140	1	then	then	ADV
ejpam-3559	140	2	(	(	PUNCT
ejpam-3559	140	3	x	x	X
ejpam-3559	140	4	,	,	PUNCT
ejpam-3559	140	5	y	y	NOUN
ejpam-3559	140	6	)	)	PUNCT
ejpam-3559	140	7	∈	∈	PROPN
ejpam-3559	140	8	k	k	PROPN
ejpam-3559	140	9	⇔	⇔	X
ejpam-3559	140	10	(	(	PUNCT
ejpam-3559	140	11	x	x	PROPN
ejpam-3559	140	12	,	,	PUNCT
ejpam-3559	140	13	y	y	NOUN
ejpam-3559	140	14	)	)	PUNCT
ejpam-3559	140	15	∈	∈	PROPN
ejpam-3559	140	16	lh1(aα)×	lh1(aα)×	PROPN
ejpam-3559	140	17	lh2(bα	lh2(bα	PROPN
ejpam-3559	140	18	)	)	PUNCT
ejpam-3559	140	19	∀α	∀α	VERB
ejpam-3559	140	20	∈	∈	PROPN
ejpam-3559	141	1	i	i	PRON
ejpam-3559	141	2	⇔	⇔	NOUN
ejpam-3559	141	3	x	x	X
ejpam-3559	141	4	∈	∈	PROPN
ejpam-3559	141	5	lh1(aα	lh1(aα	PROPN
ejpam-3559	141	6	)	)	PUNCT
ejpam-3559	141	7	and	and	CCONJ
ejpam-3559	141	8	y	y	PROPN
ejpam-3559	141	9	∈	∈	PROPN
ejpam-3559	141	10	lh2(bα	lh2(bα	PROPN
ejpam-3559	141	11	)	)	PUNCT
ejpam-3559	141	12	∀α	∀α	VERB
ejpam-3559	141	13	∈	∈	PROPN
ejpam-3559	142	1	i	i	PRON
ejpam-3559	142	2	⇔	⇔	PROPN
ejpam-3559	142	3	x	x	PROPN
ejpam-3559	142	4	�	�	PROPN
ejpam-3559	142	5	a	a	DET
ejpam-3559	142	6	∀a	∀a	NOUN
ejpam-3559	142	7	∈	∈	NOUN
ejpam-3559	142	8	aα	aα	NOUN
ejpam-3559	142	9	and	and	CCONJ
ejpam-3559	142	10	y	y	PROPN
ejpam-3559	142	11	�	�	PROPN
ejpam-3559	142	12	b	b	PROPN
ejpam-3559	142	13	∀b	∀b	NOUN
ejpam-3559	142	14	∈	∈	PROPN
ejpam-3559	142	15	bα	bα	NOUN
ejpam-3559	142	16	and	and	CCONJ
ejpam-3559	142	17	∀α	∀α	NOUN
ejpam-3559	142	18	∈	∈	PROPN
ejpam-3559	142	19	i	i	PRON
ejpam-3559	142	20	⇔	⇔	PROPN
ejpam-3559	142	21	x	x	PROPN
ejpam-3559	142	22	�	�	PROPN
ejpam-3559	142	23	a	a	DET
ejpam-3559	142	24	∀a	∀a	X
ejpam-3559	142	25	∈	∈	NOUN
ejpam-3559	142	26	⋃	⋃	NOUN
ejpam-3559	142	27	α∈i	α∈i	NUM
ejpam-3559	142	28	aα	aα	NOUN
ejpam-3559	142	29	and	and	CCONJ
ejpam-3559	142	30	y	y	PROPN
ejpam-3559	142	31	�	�	PROPN
ejpam-3559	142	32	b	b	PROPN
ejpam-3559	142	33	∀b	∀b	NOUN
ejpam-3559	142	34	∈	∈	PROPN
ejpam-3559	142	35	⋃	⋃	NOUN
ejpam-3559	142	36	α∈i	α∈i	NUM
ejpam-3559	142	37	bα	bα	PROPN
ejpam-3559	142	38	⇔	⇔	X
ejpam-3559	142	39	x	x	SYM
ejpam-3559	142	40	∈	∈	PROPN
ejpam-3559	142	41	lh1	lh1	NOUN
ejpam-3559	142	42	(	(	PUNCT
ejpam-3559	142	43	⋃	⋃	PROPN
ejpam-3559	142	44	α∈i	α∈i	NUM
ejpam-3559	142	45	aα	aα	NOUN
ejpam-3559	142	46	)	)	PUNCT
ejpam-3559	142	47	and	and	CCONJ
ejpam-3559	142	48	y	y	PROPN
ejpam-3559	142	49	∈	∈	PROPN
ejpam-3559	142	50	lh2	lh2	PROPN
ejpam-3559	142	51	(	(	PUNCT
ejpam-3559	142	52	⋃	⋃	PROPN
ejpam-3559	142	53	α∈i	α∈i	NUM
ejpam-3559	142	54	bα	bα	NOUN
ejpam-3559	142	55	)	)	PUNCT
ejpam-3559	142	56	⇔	⇔	X
ejpam-3559	142	57	(	(	PUNCT
ejpam-3559	142	58	x	x	NOUN
ejpam-3559	142	59	,	,	PUNCT
ejpam-3559	142	60	y	y	NOUN
ejpam-3559	142	61	)	)	PUNCT
ejpam-3559	142	62	∈	∈	NOUN
ejpam-3559	142	63	lh1	lh1	NOUN
ejpam-3559	142	64	(	(	PUNCT
ejpam-3559	142	65	⋃	⋃	PROPN
ejpam-3559	142	66	α∈i	α∈i	NUM
ejpam-3559	142	67	aα	aα	NOUN
ejpam-3559	142	68	)	)	PUNCT
ejpam-3559	142	69	×	×	NOUN
ejpam-3559	142	70	lh2	lh2	NOUN
ejpam-3559	142	71	(	(	PUNCT
ejpam-3559	142	72	⋃	⋃	PROPN
ejpam-3559	142	73	α∈i	α∈i	NUM
ejpam-3559	142	74	bα	bα	NOUN
ejpam-3559	142	75	)	)	PUNCT
ejpam-3559	142	76	.	.	PUNCT
ejpam-3559	143	1	therefore	therefore	ADV
ejpam-3559	143	2	,	,	PUNCT
ejpam-3559	143	3	the	the	DET
ejpam-3559	143	4	assertion	assertion	NOUN
ejpam-3559	143	5	is	be	AUX
ejpam-3559	143	6	true	true	ADJ
ejpam-3559	143	7	.	.	PUNCT
ejpam-3559	144	1	(	(	PUNCT
ejpam-3559	144	2	iii	iii	X
ejpam-3559	144	3	)	)	PUNCT
ejpam-3559	144	4	let	let	VERB
ejpam-3559	144	5	d	d	NOUN
ejpam-3559	144	6	=	=	PUNCT
ejpam-3559	145	1	⋃	⋃	PROPN
ejpam-3559	145	2	x∈s	x∈s	NOUN
ejpam-3559	145	3	(	(	PUNCT
ejpam-3559	145	4	{	{	PUNCT
ejpam-3559	145	5	x}×tx	x}×tx	NUM
ejpam-3559	145	6	)	)	PUNCT
ejpam-3559	145	7	,	,	PUNCT
ejpam-3559	145	8	where	where	SCONJ
ejpam-3559	145	9	s	s	VERB
ejpam-3559	145	10	⊆	⊆	NUM
ejpam-3559	145	11	h1	h1	NOUN
ejpam-3559	145	12	and	and	CCONJ
ejpam-3559	145	13	tx	tx	VERB
ejpam-3559	145	14	⊆	⊆	NUM
ejpam-3559	145	15	h2	h2	NOUN
ejpam-3559	145	16	for	for	ADP
ejpam-3559	145	17	each	each	DET
ejpam-3559	145	18	x	x	PROPN
ejpam-3559	145	19	∈	∈	PROPN
ejpam-3559	145	20	s.	s.	PROPN
ejpam-3559	145	21	then	then	ADV
ejpam-3559	145	22	by	by	ADP
ejpam-3559	145	23	lemma	lemma	PROPN
ejpam-3559	145	24	4.1	4.1	NUM
ejpam-3559	145	25	,	,	PUNCT
ejpam-3559	145	26	(	(	PUNCT
ejpam-3559	145	27	i	i	NOUN
ejpam-3559	145	28	)	)	PUNCT
ejpam-3559	145	29	,	,	PUNCT
ejpam-3559	145	30	and	and	CCONJ
ejpam-3559	145	31	(	(	PUNCT
ejpam-3559	145	32	ii	ii	NOUN
ejpam-3559	145	33	)	)	PUNCT
ejpam-3559	145	34	,	,	PUNCT
ejpam-3559	145	35	lh(d	lh(d	NOUN
ejpam-3559	145	36	)	)	PUNCT
ejpam-3559	145	37	=	=	SYM
ejpam-3559	146	1	lh	lh	PROPN
ejpam-3559	147	1	[	[	X
ejpam-3559	147	2	⋃	⋃	PROPN
ejpam-3559	147	3	x∈s	x∈s	NOUN
ejpam-3559	147	4	(	(	PUNCT
ejpam-3559	147	5	{	{	PUNCT
ejpam-3559	147	6	x	x	NOUN
ejpam-3559	147	7	}	}	PUNCT
ejpam-3559	147	8	×	×	PROPN
ejpam-3559	147	9	tx	tx	PROPN
ejpam-3559	147	10	)	)	PUNCT
ejpam-3559	147	11	]	]	PUNCT
ejpam-3559	148	1	=	=	SYM
ejpam-3559	148	2	⋂	⋂	PROPN
ejpam-3559	148	3	x∈s	x∈s	NOUN
ejpam-3559	149	1	[	[	X
ejpam-3559	149	2	lh({x	lh({x	X
ejpam-3559	149	3	}	}	PUNCT
ejpam-3559	149	4	×	×	PROPN
ejpam-3559	149	5	tx	tx	PROPN
ejpam-3559	149	6	)	)	PUNCT
ejpam-3559	149	7	]	]	PUNCT
ejpam-3559	150	1	=	=	SYM
ejpam-3559	150	2	⋂	⋂	PROPN
ejpam-3559	150	3	x∈s	x∈s	PROPN
ejpam-3559	151	1	[	[	X
ejpam-3559	151	2	lh1(x)×	lh1(x)×	NOUN
ejpam-3559	151	3	lh2(tx	lh2(tx	VERB
ejpam-3559	151	4	)	)	PUNCT
ejpam-3559	151	5	]	]	PUNCT
ejpam-3559	152	1	=	=	SYM
ejpam-3559	152	2	lh1(s)×	lh1(s)×	X
ejpam-3559	152	3	lh2	lh2	NOUN
ejpam-3559	152	4	(	(	PUNCT
ejpam-3559	152	5	⋃	⋃	PROPN
ejpam-3559	152	6	x∈s	x∈s	NOUN
ejpam-3559	152	7	tx	tx	PROPN
ejpam-3559	152	8	)	)	PUNCT
ejpam-3559	152	9	.	.	PUNCT
ejpam-3559	153	1	r.	r.	PROPN
ejpam-3559	153	2	patangan	patangan	PROPN
ejpam-3559	153	3	,	,	PUNCT
ejpam-3559	153	4	s.	s.	PROPN
ejpam-3559	153	5	canoy	canoy	PROPN
ejpam-3559	153	6	,	,	PUNCT
ejpam-3559	153	7	jr	jr	PROPN
ejpam-3559	153	8	.	.	PROPN
ejpam-3559	153	9	/	/	SYM
ejpam-3559	153	10	eur	eur	PROPN
ejpam-3559	153	11	.	.	PUNCT
ejpam-3559	154	1	j.	j.	PROPN
ejpam-3559	154	2	pure	pure	PROPN
ejpam-3559	154	3	appl	appl	PROPN
ejpam-3559	154	4	.	.	PROPN
ejpam-3559	154	5	math	math	PROPN
ejpam-3559	154	6	,	,	PUNCT
ejpam-3559	154	7	12	12	NUM
ejpam-3559	154	8	(	(	PUNCT
ejpam-3559	154	9	4	4	NUM
ejpam-3559	154	10	)	)	PUNCT
ejpam-3559	154	11	(	(	PUNCT
ejpam-3559	154	12	2019	2019	NUM
ejpam-3559	154	13	)	)	PUNCT
ejpam-3559	154	14	,	,	PUNCT
ejpam-3559	154	15	1524	1524	NUM
ejpam-3559	154	16	-	-	SYM
ejpam-3559	154	17	1532	1532	NUM
ejpam-3559	154	18	1529	1529	NUM
ejpam-3559	154	19	theorem	theorem	VERB
ejpam-3559	154	20	4.3	4.3	NUM
ejpam-3559	154	21	.	.	PUNCT
ejpam-3559	155	1	let	let	VERB
ejpam-3559	155	2	h	h	PRON
ejpam-3559	155	3	be	be	AUX
ejpam-3559	155	4	a	a	DET
ejpam-3559	155	5	hyper	hyper	ADJ
ejpam-3559	155	6	product	product	NOUN
ejpam-3559	155	7	of	of	ADP
ejpam-3559	155	8	hyper	hyper	ADJ
ejpam-3559	155	9	bck	bck	NOUN
ejpam-3559	155	10	-	-	PUNCT
ejpam-3559	155	11	algebras	algebras	ADJ
ejpam-3559	155	12	h1	h1	PROPN
ejpam-3559	155	13	and	and	CCONJ
ejpam-3559	155	14	h2	h2	PROPN
ejpam-3559	155	15	.	.	PUNCT
ejpam-3559	156	1	then	then	ADV
ejpam-3559	156	2	bl(h	bl(h	PUNCT
ejpam-3559	156	3	)	)	PUNCT
ejpam-3559	156	4	=	=	SYM
ejpam-3559	157	1	bl(h1)×	bl(h1)×	NUM
ejpam-3559	157	2	bl(h2	bl(h2	NOUN
ejpam-3559	157	3	)	)	PUNCT
ejpam-3559	157	4	.	.	PUNCT
ejpam-3559	158	1	proof	proof	NOUN
ejpam-3559	158	2	:	:	PUNCT
ejpam-3559	158	3	let	let	VERB
ejpam-3559	158	4	u	u	PRON
ejpam-3559	158	5	∈	∈	PROPN
ejpam-3559	158	6	bl(h	bl(h	PUNCT
ejpam-3559	158	7	)	)	PUNCT
ejpam-3559	158	8	.	.	PUNCT
ejpam-3559	159	1	then	then	ADV
ejpam-3559	159	2	there	there	PRON
ejpam-3559	159	3	exists	exist	VERB
ejpam-3559	159	4	a	a	DET
ejpam-3559	159	5	nonempty	nonempty	ADV
ejpam-3559	159	6	set	set	VERB
ejpam-3559	159	7	d	d	PROPN
ejpam-3559	159	8	⊆	⊆	NUM
ejpam-3559	159	9	h	h	NOUN
ejpam-3559	159	10	=	=	PRON
ejpam-3559	159	11	h1	h1	PROPN
ejpam-3559	159	12	×h2	×h2	PROPN
ejpam-3559	159	13	such	such	ADJ
ejpam-3559	159	14	that	that	SCONJ
ejpam-3559	159	15	u	u	NOUN
ejpam-3559	159	16	=	=	PRON
ejpam-3559	159	17	lh(d	lh(d	X
ejpam-3559	159	18	)	)	PUNCT
ejpam-3559	159	19	.	.	PUNCT
ejpam-3559	160	1	let	let	VERB
ejpam-3559	160	2	d	d	NOUN
ejpam-3559	160	3	=	=	PUNCT
ejpam-3559	160	4	⋃	⋃	PROPN
ejpam-3559	160	5	x∈s	x∈s	NOUN
ejpam-3559	160	6	(	(	PUNCT
ejpam-3559	160	7	{	{	PUNCT
ejpam-3559	160	8	x	x	NOUN
ejpam-3559	160	9	}	}	PUNCT
ejpam-3559	160	10	×	×	PROPN
ejpam-3559	160	11	tx	tx	PROPN
ejpam-3559	160	12	)	)	PUNCT
ejpam-3559	160	13	where	where	SCONJ
ejpam-3559	160	14	s	s	VERB
ejpam-3559	160	15	⊆	⊆	NUM
ejpam-3559	160	16	h1	h1	NOUN
ejpam-3559	160	17	and	and	CCONJ
ejpam-3559	160	18	tx	tx	VERB
ejpam-3559	160	19	⊆	⊆	NUM
ejpam-3559	160	20	h2	h2	NOUN
ejpam-3559	160	21	for	for	ADP
ejpam-3559	160	22	each	each	DET
ejpam-3559	160	23	x	x	PROPN
ejpam-3559	160	24	∈	∈	PROPN
ejpam-3559	160	25	s.	s.	PROPN
ejpam-3559	160	26	then	then	ADV
ejpam-3559	160	27	lh(d	lh(d	PRON
ejpam-3559	160	28	)	)	PUNCT
ejpam-3559	160	29	=	=	SYM
ejpam-3559	160	30	lh1(s)×lh2	lh1(s)×lh2	NOUN
ejpam-3559	160	31	(	(	PUNCT
ejpam-3559	160	32	⋃	⋃	PROPN
ejpam-3559	160	33	x∈s	x∈s	PROPN
ejpam-3559	160	34	tx	tx	PROPN
ejpam-3559	160	35	)	)	PUNCT
ejpam-3559	160	36	by	by	ADP
ejpam-3559	160	37	theorem	theorem	NOUN
ejpam-3559	160	38	4.2(iii	4.2(iii	NUM
ejpam-3559	160	39	)	)	PUNCT
ejpam-3559	160	40	.	.	PUNCT
ejpam-3559	161	1	hence	hence	ADV
ejpam-3559	161	2	,	,	PUNCT
ejpam-3559	161	3	u	u	PROPN
ejpam-3559	161	4	∈	∈	PROPN
ejpam-3559	161	5	bl(h1)×bl(h2	bl(h1)×bl(h2	PROPN
ejpam-3559	161	6	)	)	PUNCT
ejpam-3559	161	7	,	,	PUNCT
ejpam-3559	161	8	showing	show	VERB
ejpam-3559	161	9	that	that	PRON
ejpam-3559	161	10	bl(h	bl(h	PUNCT
ejpam-3559	161	11	)	)	PUNCT
ejpam-3559	161	12	⊆	⊆	NUM
ejpam-3559	161	13	bl(h1	bl(h1	NOUN
ejpam-3559	161	14	)	)	PUNCT
ejpam-3559	161	15	×	×	NOUN
ejpam-3559	161	16	bl(h2	bl(h2	NOUN
ejpam-3559	161	17	)	)	PUNCT
ejpam-3559	161	18	.	.	PUNCT
ejpam-3559	162	1	next	next	ADV
ejpam-3559	162	2	,	,	PUNCT
ejpam-3559	162	3	let	let	VERB
ejpam-3559	162	4	v	v	NUM
ejpam-3559	162	5	∈	∈	VERB
ejpam-3559	162	6	bl(h1	bl(h1	NOUN
ejpam-3559	162	7	)	)	PUNCT
ejpam-3559	162	8	×	×	NOUN
ejpam-3559	162	9	bl(h2	bl(h2	NOUN
ejpam-3559	162	10	)	)	PUNCT
ejpam-3559	162	11	.	.	PUNCT
ejpam-3559	163	1	then	then	ADV
ejpam-3559	163	2	there	there	PRON
ejpam-3559	163	3	exist	exist	VERB
ejpam-3559	163	4	nonempty	nonempty	NOUN
ejpam-3559	163	5	sets	set	VERB
ejpam-3559	163	6	a	a	DET
ejpam-3559	163	7	⊆	⊆	NUM
ejpam-3559	163	8	h1	h1	NOUN
ejpam-3559	163	9	and	and	CCONJ
ejpam-3559	163	10	b	b	NOUN
ejpam-3559	163	11	⊆	⊆	NUM
ejpam-3559	163	12	h2	h2	NOUN
ejpam-3559	163	13	such	such	ADJ
ejpam-3559	163	14	that	that	DET
ejpam-3559	163	15	v	v	NOUN
ejpam-3559	163	16	=	=	SYM
ejpam-3559	163	17	lh1(a)×lh2(b	lh1(a)×lh2(b	ADJ
ejpam-3559	163	18	)	)	PUNCT
ejpam-3559	163	19	=	=	SYM
ejpam-3559	163	20	lh(a×b	lh(a×b	PROPN
ejpam-3559	163	21	)	)	PUNCT
ejpam-3559	163	22	∈	∈	PROPN
ejpam-3559	163	23	bl(h	bl(h	PUNCT
ejpam-3559	163	24	)	)	PUNCT
ejpam-3559	163	25	by	by	ADP
ejpam-3559	163	26	theorem	theorem	NOUN
ejpam-3559	163	27	4.2(i	4.2(i	NUM
ejpam-3559	163	28	)	)	PUNCT
ejpam-3559	163	29	.	.	PUNCT
ejpam-3559	164	1	thus	thus	ADV
ejpam-3559	164	2	,	,	PUNCT
ejpam-3559	164	3	bl(h1	bl(h1	NOUN
ejpam-3559	164	4	)	)	PUNCT
ejpam-3559	164	5	×	×	NOUN
ejpam-3559	164	6	bl(h2	bl(h2	NOUN
ejpam-3559	164	7	)	)	PUNCT
ejpam-3559	164	8	⊆	⊆	NUM
ejpam-3559	164	9	bl(h	bl(h	NUM
ejpam-3559	164	10	)	)	PUNCT
ejpam-3559	164	11	.	.	PUNCT
ejpam-3559	165	1	therefore	therefore	ADV
ejpam-3559	165	2	,	,	PUNCT
ejpam-3559	165	3	bl(h	bl(h	PUNCT
ejpam-3559	165	4	)	)	PUNCT
ejpam-3559	165	5	=	=	SYM
ejpam-3559	165	6	bl(h1)×	bl(h1)×	NUM
ejpam-3559	165	7	bl(h2	bl(h2	NOUN
ejpam-3559	165	8	)	)	PUNCT
ejpam-3559	165	9	.	.	PUNCT
ejpam-3559	166	1	lemma	lemma	PROPN
ejpam-3559	166	2	4.4	4.4	NUM
ejpam-3559	166	3	.	.	PUNCT
ejpam-3559	167	1	let	let	VERB
ejpam-3559	167	2	{	{	PUNCT
ejpam-3559	167	3	aα	aα	NOUN
ejpam-3559	167	4	:	:	PUNCT
ejpam-3559	167	5	α	α	PROPN
ejpam-3559	167	6	∈	∈	PROPN
ejpam-3559	168	1	i	i	PRON
ejpam-3559	168	2	}	}	PUNCT
ejpam-3559	168	3	be	be	VERB
ejpam-3559	168	4	a	a	DET
ejpam-3559	168	5	collection	collection	NOUN
ejpam-3559	168	6	of	of	ADP
ejpam-3559	168	7	subsets	subset	NOUN
ejpam-3559	168	8	of	of	ADP
ejpam-3559	168	9	a	a	DET
ejpam-3559	168	10	hyper	hyper	ADJ
ejpam-3559	168	11	bck	bck	NOUN
ejpam-3559	168	12	-	-	PUNCT
ejpam-3559	168	13	algebra	algebra	NOUN
ejpam-3559	168	14	h.	h.	NOUN
ejpam-3559	168	15	then	then	ADV
ejpam-3559	168	16	⋂	⋂	PROPN
ejpam-3559	168	17	α∈i	α∈i	NUM
ejpam-3559	168	18	rh(aα	rh(aα	ADJ
ejpam-3559	168	19	)	)	PUNCT
ejpam-3559	168	20	=	=	SYM
ejpam-3559	168	21	rh	rh	PROPN
ejpam-3559	168	22	(	(	PUNCT
ejpam-3559	168	23	⋃	⋃	PROPN
ejpam-3559	168	24	α∈i	α∈i	NUM
ejpam-3559	168	25	aα	aα	NOUN
ejpam-3559	168	26	)	)	PUNCT
ejpam-3559	168	27	.	.	PUNCT
ejpam-3559	169	1	proof	proof	NOUN
ejpam-3559	169	2	:	:	PUNCT
ejpam-3559	169	3	let	let	VERB
ejpam-3559	169	4	{	{	PUNCT
ejpam-3559	169	5	aα	aα	NOUN
ejpam-3559	169	6	:	:	PUNCT
ejpam-3559	169	7	α	α	PROPN
ejpam-3559	169	8	∈	∈	PROPN
ejpam-3559	170	1	i	i	PRON
ejpam-3559	170	2	}	}	PUNCT
ejpam-3559	170	3	be	be	VERB
ejpam-3559	170	4	a	a	DET
ejpam-3559	170	5	collection	collection	NOUN
ejpam-3559	170	6	of	of	ADP
ejpam-3559	170	7	subsets	subset	NOUN
ejpam-3559	170	8	of	of	ADP
ejpam-3559	170	9	h.	h.	PROPN
ejpam-3559	170	10	then	then	ADV
ejpam-3559	170	11	x	x	SYM
ejpam-3559	170	12	∈	∈	PROPN
ejpam-3559	170	13	⋂	⋂	PROPN
ejpam-3559	170	14	α∈i	α∈i	ADJ
ejpam-3559	170	15	rh(aα)⇔	rh(aα)⇔	NOUN
ejpam-3559	170	16	x	x	SYM
ejpam-3559	170	17	∈	∈	PROPN
ejpam-3559	170	18	rh(aα	rh(aα	PROPN
ejpam-3559	170	19	)	)	PUNCT
ejpam-3559	170	20	for	for	ADP
ejpam-3559	170	21	all	all	DET
ejpam-3559	170	22	α	α	PRON
ejpam-3559	170	23	∈	∈	PROPN
ejpam-3559	170	24	i	i	PRON
ejpam-3559	170	25	⇔	⇔	PROPN
ejpam-3559	170	26	a	a	DET
ejpam-3559	170	27	�	�	PROPN
ejpam-3559	170	28	x	x	PUNCT
ejpam-3559	170	29	for	for	ADP
ejpam-3559	170	30	all	all	DET
ejpam-3559	170	31	a	a	DET
ejpam-3559	170	32	∈	∈	ADJ
ejpam-3559	170	33	aα	aα	NOUN
ejpam-3559	170	34	and	and	CCONJ
ejpam-3559	170	35	for	for	ADP
ejpam-3559	170	36	all	all	DET
ejpam-3559	170	37	α	α	PRON
ejpam-3559	170	38	∈	∈	NOUN
ejpam-3559	170	39	i	i	PRON
ejpam-3559	170	40	⇔	⇔	PROPN
ejpam-3559	170	41	a	a	DET
ejpam-3559	170	42	�	�	PROPN
ejpam-3559	170	43	x	x	PUNCT
ejpam-3559	170	44	for	for	ADP
ejpam-3559	170	45	all	all	DET
ejpam-3559	170	46	a	a	DET
ejpam-3559	170	47	∈	∈	NOUN
ejpam-3559	170	48	⋃	⋃	NOUN
ejpam-3559	170	49	α∈i	α∈i	NUM
ejpam-3559	170	50	aα	aα	NOUN
ejpam-3559	170	51	⇔	⇔	NOUN
ejpam-3559	170	52	x	x	SYM
ejpam-3559	170	53	∈	∈	PROPN
ejpam-3559	170	54	rh	rh	PROPN
ejpam-3559	170	55	(	(	PUNCT
ejpam-3559	170	56	⋃	⋃	PROPN
ejpam-3559	170	57	α∈i	α∈i	NUM
ejpam-3559	170	58	aα	aα	NOUN
ejpam-3559	170	59	)	)	PUNCT
ejpam-3559	170	60	.	.	PUNCT
ejpam-3559	171	1	therefore	therefore	ADV
ejpam-3559	171	2	,	,	PUNCT
ejpam-3559	171	3	the	the	DET
ejpam-3559	171	4	equality	equality	NOUN
ejpam-3559	171	5	holds	hold	VERB
ejpam-3559	171	6	.	.	PUNCT
ejpam-3559	172	1	theorem	theorem	VERB
ejpam-3559	172	2	4.5	4.5	NUM
ejpam-3559	172	3	.	.	PUNCT
ejpam-3559	173	1	let	let	VERB
ejpam-3559	173	2	h	h	PRON
ejpam-3559	173	3	be	be	AUX
ejpam-3559	173	4	a	a	DET
ejpam-3559	173	5	hyper	hyper	ADJ
ejpam-3559	173	6	product	product	NOUN
ejpam-3559	173	7	of	of	ADP
ejpam-3559	173	8	hyper	hyper	ADJ
ejpam-3559	173	9	bck	bck	NOUN
ejpam-3559	173	10	-	-	PUNCT
ejpam-3559	173	11	algebras	algebras	ADJ
ejpam-3559	173	12	h1	h1	PROPN
ejpam-3559	173	13	and	and	CCONJ
ejpam-3559	173	14	h2	h2	PROPN
ejpam-3559	173	15	.	.	PUNCT
ejpam-3559	174	1	then	then	ADV
ejpam-3559	174	2	the	the	DET
ejpam-3559	174	3	following	follow	VERB
ejpam-3559	174	4	properties	property	NOUN
ejpam-3559	174	5	hold	hold	VERB
ejpam-3559	174	6	:	:	PUNCT
ejpam-3559	174	7	(	(	PUNCT
ejpam-3559	174	8	i	i	NOUN
ejpam-3559	174	9	)	)	PUNCT
ejpam-3559	174	10	rh(a×b	rh(a×b	PROPN
ejpam-3559	174	11	)	)	PUNCT
ejpam-3559	174	12	=	=	SYM
ejpam-3559	174	13	rh1(a)×rh2(b	rh1(a)×rh2(b	NOUN
ejpam-3559	174	14	)	)	PUNCT
ejpam-3559	174	15	for	for	ADP
ejpam-3559	174	16	a	a	DET
ejpam-3559	174	17	⊆	⊆	NUM
ejpam-3559	174	18	h1	h1	NOUN
ejpam-3559	174	19	and	and	CCONJ
ejpam-3559	174	20	b	b	NOUN
ejpam-3559	174	21	⊆	⊆	NUM
ejpam-3559	174	22	h2	h2	NOUN
ejpam-3559	174	23	.	.	PUNCT
ejpam-3559	175	1	(	(	PUNCT
ejpam-3559	175	2	ii	ii	NOUN
ejpam-3559	175	3	)	)	PUNCT
ejpam-3559	175	4	if	if	SCONJ
ejpam-3559	175	5	{	{	PUNCT
ejpam-3559	175	6	aα	aα	NOUN
ejpam-3559	175	7	:	:	PUNCT
ejpam-3559	175	8	α	α	PROPN
ejpam-3559	175	9	∈	∈	PROPN
ejpam-3559	176	1	i	i	X
ejpam-3559	176	2	}	}	PUNCT
ejpam-3559	176	3	and	and	CCONJ
ejpam-3559	176	4	{	{	PUNCT
ejpam-3559	176	5	bα	bα	NOUN
ejpam-3559	176	6	:	:	PUNCT
ejpam-3559	176	7	α	α	PROPN
ejpam-3559	176	8	∈	∈	PROPN
ejpam-3559	177	1	i	i	PRON
ejpam-3559	177	2	}	}	PUNCT
ejpam-3559	177	3	are	be	AUX
ejpam-3559	177	4	collections	collection	NOUN
ejpam-3559	177	5	of	of	ADP
ejpam-3559	177	6	subsets	subset	NOUN
ejpam-3559	177	7	of	of	ADP
ejpam-3559	177	8	h1	h1	NOUN
ejpam-3559	177	9	and	and	CCONJ
ejpam-3559	177	10	h2	h2	NOUN
ejpam-3559	177	11	,	,	PUNCT
ejpam-3559	177	12	respectively	respectively	ADV
ejpam-3559	177	13	,	,	PUNCT
ejpam-3559	177	14	then⋂	then⋂	NUM
ejpam-3559	177	15	α∈i	α∈i	NOUN
ejpam-3559	177	16	[	[	X
ejpam-3559	177	17	rh1(aα)×rh2(bα	rh1(aα)×rh2(bα	NOUN
ejpam-3559	177	18	)	)	PUNCT
ejpam-3559	177	19	]	]	PUNCT
ejpam-3559	178	1	=	=	X
ejpam-3559	178	2	rh1	rh1	X
ejpam-3559	178	3	(	(	PUNCT
ejpam-3559	178	4	⋃	⋃	PROPN
ejpam-3559	178	5	α∈i	α∈i	NUM
ejpam-3559	178	6	aα	aα	NOUN
ejpam-3559	178	7	)	)	PUNCT
ejpam-3559	178	8	×rh2	×rh2	PUNCT
ejpam-3559	178	9	(	(	PUNCT
ejpam-3559	178	10	⋃	⋃	VERB
ejpam-3559	178	11	α∈i	α∈i	NUM
ejpam-3559	178	12	bα	bα	NOUN
ejpam-3559	178	13	)	)	PUNCT
ejpam-3559	178	14	.	.	PUNCT
ejpam-3559	179	1	(	(	PUNCT
ejpam-3559	179	2	iii	iii	X
ejpam-3559	179	3	)	)	PUNCT
ejpam-3559	179	4	if	if	SCONJ
ejpam-3559	179	5	e	e	NOUN
ejpam-3559	179	6	=	=	PUNCT
ejpam-3559	179	7	⋃	⋃	PROPN
ejpam-3559	179	8	x∈p	x∈p	NOUN
ejpam-3559	179	9	(	(	PUNCT
ejpam-3559	179	10	{	{	PUNCT
ejpam-3559	179	11	x	x	NOUN
ejpam-3559	179	12	}	}	PUNCT
ejpam-3559	179	13	×	×	PROPN
ejpam-3559	179	14	tx	tx	PROPN
ejpam-3559	179	15	)	)	PUNCT
ejpam-3559	179	16	,	,	PUNCT
ejpam-3559	179	17	where	where	SCONJ
ejpam-3559	179	18	p	p	PROPN
ejpam-3559	179	19	⊆	⊆	NUM
ejpam-3559	179	20	h1	h1	NOUN
ejpam-3559	179	21	and	and	CCONJ
ejpam-3559	179	22	tx	tx	VERB
ejpam-3559	179	23	⊆	⊆	NUM
ejpam-3559	179	24	h2	h2	NOUN
ejpam-3559	179	25	for	for	ADP
ejpam-3559	179	26	each	each	DET
ejpam-3559	179	27	x	x	SYM
ejpam-3559	179	28	∈	∈	PROPN
ejpam-3559	179	29	p	p	X
ejpam-3559	179	30	,	,	PUNCT
ejpam-3559	179	31	then	then	ADV
ejpam-3559	179	32	rh(e	rh(e	VERB
ejpam-3559	179	33	)	)	PUNCT
ejpam-3559	179	34	=	=	SYM
ejpam-3559	180	1	⋂	⋂	PROPN
ejpam-3559	180	2	x∈p	x∈p	NUM
ejpam-3559	180	3	(	(	PUNCT
ejpam-3559	180	4	rh1(x)×rh2(tx	rh1(x)×rh2(tx	NUM
ejpam-3559	180	5	)	)	PUNCT
ejpam-3559	180	6	)	)	PUNCT
ejpam-3559	181	1	=	=	SYM
ejpam-3559	181	2	rh1(p	rh1(p	PROPN
ejpam-3559	181	3	)	)	PUNCT
ejpam-3559	181	4	×rh2	×rh2	CCONJ
ejpam-3559	181	5	(	(	PUNCT
ejpam-3559	181	6	⋃	⋃	PROPN
ejpam-3559	181	7	x∈p	x∈p	NOUN
ejpam-3559	181	8	tx	tx	PROPN
ejpam-3559	181	9	)	)	PUNCT
ejpam-3559	181	10	.	.	PUNCT
ejpam-3559	182	1	proof	proof	NOUN
ejpam-3559	182	2	:	:	PUNCT
ejpam-3559	182	3	r.	r.	PROPN
ejpam-3559	182	4	patangan	patangan	PROPN
ejpam-3559	182	5	,	,	PUNCT
ejpam-3559	182	6	s.	s.	PROPN
ejpam-3559	182	7	canoy	canoy	PROPN
ejpam-3559	182	8	,	,	PUNCT
ejpam-3559	182	9	jr	jr	PROPN
ejpam-3559	182	10	.	.	PROPN
ejpam-3559	182	11	/	/	SYM
ejpam-3559	182	12	eur	eur	PROPN
ejpam-3559	182	13	.	.	PUNCT
ejpam-3559	183	1	j.	j.	PROPN
ejpam-3559	183	2	pure	pure	PROPN
ejpam-3559	183	3	appl	appl	PROPN
ejpam-3559	183	4	.	.	PROPN
ejpam-3559	183	5	math	math	PROPN
ejpam-3559	183	6	,	,	PUNCT
ejpam-3559	183	7	12	12	NUM
ejpam-3559	183	8	(	(	PUNCT
ejpam-3559	183	9	4	4	NUM
ejpam-3559	183	10	)	)	PUNCT
ejpam-3559	183	11	(	(	PUNCT
ejpam-3559	183	12	2019	2019	NUM
ejpam-3559	183	13	)	)	PUNCT
ejpam-3559	183	14	,	,	PUNCT
ejpam-3559	183	15	1524	1524	NUM
ejpam-3559	183	16	-	-	SYM
ejpam-3559	183	17	1532	1532	NUM
ejpam-3559	183	18	1530	1530	NUM
ejpam-3559	183	19	(	(	PUNCT
ejpam-3559	183	20	i	i	NOUN
ejpam-3559	183	21	)	)	PUNCT
ejpam-3559	183	22	let	let	VERB
ejpam-3559	183	23	a	a	PRON
ejpam-3559	183	24	and	and	CCONJ
ejpam-3559	183	25	b	b	NOUN
ejpam-3559	183	26	be	be	AUX
ejpam-3559	183	27	subsets	subset	NOUN
ejpam-3559	183	28	of	of	ADP
ejpam-3559	183	29	h1	h1	NOUN
ejpam-3559	183	30	and	and	CCONJ
ejpam-3559	183	31	h2	h2	NOUN
ejpam-3559	183	32	,	,	PUNCT
ejpam-3559	183	33	respectively	respectively	ADV
ejpam-3559	183	34	.	.	PUNCT
ejpam-3559	184	1	then	then	ADV
ejpam-3559	184	2	rh(a×b	rh(a×b	NUM
ejpam-3559	184	3	)	)	PUNCT
ejpam-3559	184	4	=	=	PRON
ejpam-3559	184	5	{	{	PUNCT
ejpam-3559	184	6	(	(	PUNCT
ejpam-3559	184	7	x	x	NOUN
ejpam-3559	184	8	,	,	PUNCT
ejpam-3559	184	9	y	y	NOUN
ejpam-3559	184	10	)	)	PUNCT
ejpam-3559	184	11	∈	∈	PROPN
ejpam-3559	184	12	h1	h1	PROPN
ejpam-3559	184	13	×h2	×h2	PROPN
ejpam-3559	184	14	:	:	PUNCT
ejpam-3559	184	15	(	(	PUNCT
ejpam-3559	184	16	a	a	PRON
ejpam-3559	184	17	,	,	PUNCT
ejpam-3559	184	18	b	b	NOUN
ejpam-3559	184	19	)	)	PUNCT
ejpam-3559	184	20	�	�	PROPN
ejpam-3559	184	21	(	(	PUNCT
ejpam-3559	184	22	x	x	NOUN
ejpam-3559	184	23	,	,	PUNCT
ejpam-3559	184	24	y	y	NOUN
ejpam-3559	184	25	)	)	PUNCT
ejpam-3559	184	26	for	for	ADP
ejpam-3559	184	27	all	all	PRON
ejpam-3559	184	28	(	(	PUNCT
ejpam-3559	184	29	a	a	PRON
ejpam-3559	184	30	,	,	PUNCT
ejpam-3559	184	31	b	b	NOUN
ejpam-3559	184	32	)	)	PUNCT
ejpam-3559	184	33	∈	∈	NOUN
ejpam-3559	184	34	a×b	a×b	PROPN
ejpam-3559	184	35	}	}	PUNCT
ejpam-3559	184	36	=	=	SYM
ejpam-3559	184	37	{	{	PUNCT
ejpam-3559	184	38	(	(	PUNCT
ejpam-3559	184	39	x	x	NOUN
ejpam-3559	184	40	,	,	PUNCT
ejpam-3559	184	41	y	y	NOUN
ejpam-3559	184	42	)	)	PUNCT
ejpam-3559	184	43	∈	∈	PROPN
ejpam-3559	184	44	h1	h1	PROPN
ejpam-3559	184	45	×h2	×h2	PROPN
ejpam-3559	184	46	:	:	PUNCT
ejpam-3559	184	47	a	a	DET
ejpam-3559	184	48	�	�	PROPN
ejpam-3559	184	49	x	x	SYM
ejpam-3559	184	50	and	and	CCONJ
ejpam-3559	184	51	b	b	PROPN
ejpam-3559	184	52	�	�	PROPN
ejpam-3559	184	53	y	y	PROPN
ejpam-3559	184	54	∀a	∀a	NOUN
ejpam-3559	184	55	∈	∈	PROPN
ejpam-3559	184	56	a	a	PRON
ejpam-3559	184	57	and	and	CCONJ
ejpam-3559	184	58	b	b	NOUN
ejpam-3559	184	59	∈	∈	ADJ
ejpam-3559	184	60	b	b	PROPN
ejpam-3559	184	61	}	}	PUNCT
ejpam-3559	184	62	=	=	SYM
ejpam-3559	184	63	{	{	PUNCT
ejpam-3559	184	64	x	x	PUNCT
ejpam-3559	184	65	∈	∈	PROPN
ejpam-3559	184	66	h1	h1	NOUN
ejpam-3559	184	67	:	:	PUNCT
ejpam-3559	184	68	a	a	DET
ejpam-3559	184	69	�	�	PROPN
ejpam-3559	184	70	x	x	SYM
ejpam-3559	184	71	∀a	∀a	X
ejpam-3559	184	72	∈	∈	NOUN
ejpam-3559	184	73	a	a	DET
ejpam-3559	184	74	}	}	PUNCT
ejpam-3559	184	75	×	×	NOUN
ejpam-3559	184	76	{	{	PUNCT
ejpam-3559	184	77	y	y	PROPN
ejpam-3559	184	78	∈	∈	PROPN
ejpam-3559	184	79	h2	h2	NOUN
ejpam-3559	184	80	:	:	PUNCT
ejpam-3559	184	81	b	b	X
ejpam-3559	184	82	�	�	PROPN
ejpam-3559	184	83	y	y	PROPN
ejpam-3559	184	84	∀b	∀b	PROPN
ejpam-3559	184	85	∈	∈	PROPN
ejpam-3559	184	86	b	b	NOUN
ejpam-3559	184	87	}	}	PUNCT
ejpam-3559	184	88	=	=	SYM
ejpam-3559	184	89	rh1(a)×rh2(b	rh1(a)×rh2(b	NOUN
ejpam-3559	184	90	)	)	PUNCT
ejpam-3559	184	91	.	.	PUNCT
ejpam-3559	185	1	(	(	PUNCT
ejpam-3559	185	2	ii	ii	X
ejpam-3559	185	3	)	)	PUNCT
ejpam-3559	185	4	let	let	VERB
ejpam-3559	185	5	{	{	PUNCT
ejpam-3559	185	6	aα	aα	NOUN
ejpam-3559	185	7	:	:	PUNCT
ejpam-3559	185	8	α	α	PROPN
ejpam-3559	185	9	∈	∈	PROPN
ejpam-3559	186	1	i	i	X
ejpam-3559	186	2	}	}	PUNCT
ejpam-3559	186	3	and	and	CCONJ
ejpam-3559	186	4	{	{	PUNCT
ejpam-3559	186	5	bα	bα	NOUN
ejpam-3559	186	6	:	:	PUNCT
ejpam-3559	186	7	α	α	PROPN
ejpam-3559	186	8	∈	∈	PROPN
ejpam-3559	187	1	i	i	PRON
ejpam-3559	187	2	}	}	PUNCT
ejpam-3559	187	3	be	be	VERB
ejpam-3559	187	4	collections	collection	NOUN
ejpam-3559	187	5	of	of	ADP
ejpam-3559	187	6	subsets	subset	NOUN
ejpam-3559	187	7	of	of	ADP
ejpam-3559	187	8	h1	h1	NOUN
ejpam-3559	187	9	and	and	CCONJ
ejpam-3559	187	10	h2	h2	NOUN
ejpam-3559	187	11	,	,	PUNCT
ejpam-3559	187	12	respectively	respectively	ADV
ejpam-3559	187	13	,	,	PUNCT
ejpam-3559	187	14	and	and	CCONJ
ejpam-3559	187	15	let	let	VERB
ejpam-3559	187	16	q	q	NOUN
ejpam-3559	187	17	=	=	SYM
ejpam-3559	187	18	⋂	⋂	NUM
ejpam-3559	187	19	α∈i	α∈i	NOUN
ejpam-3559	187	20	[	[	X
ejpam-3559	187	21	rh1(aα)×rh2(bα	rh1(aα)×rh2(bα	NOUN
ejpam-3559	187	22	)	)	PUNCT
ejpam-3559	187	23	]	]	PUNCT
ejpam-3559	187	24	.	.	PUNCT
ejpam-3559	188	1	then	then	ADV
ejpam-3559	188	2	(	(	PUNCT
ejpam-3559	188	3	x	x	X
ejpam-3559	188	4	,	,	PUNCT
ejpam-3559	188	5	y	y	NOUN
ejpam-3559	188	6	)	)	PUNCT
ejpam-3559	188	7	∈	∈	PROPN
ejpam-3559	188	8	q⇔	q⇔	NOUN
ejpam-3559	188	9	(	(	PUNCT
ejpam-3559	188	10	x	x	X
ejpam-3559	188	11	,	,	PUNCT
ejpam-3559	188	12	y	y	NOUN
ejpam-3559	188	13	)	)	PUNCT
ejpam-3559	188	14	∈	∈	PROPN
ejpam-3559	188	15	rh1(aα)×rh2(bα	rh1(aα)×rh2(bα	NOUN
ejpam-3559	188	16	)	)	PUNCT
ejpam-3559	188	17	∀α	∀α	VERB
ejpam-3559	188	18	∈	∈	PROPN
ejpam-3559	189	1	i	i	PRON
ejpam-3559	189	2	⇔	⇔	NOUN
ejpam-3559	189	3	x	x	X
ejpam-3559	189	4	∈	∈	PROPN
ejpam-3559	189	5	rh1(aα	rh1(aα	PROPN
ejpam-3559	189	6	)	)	PUNCT
ejpam-3559	189	7	and	and	CCONJ
ejpam-3559	189	8	y	y	PROPN
ejpam-3559	189	9	∈	∈	PROPN
ejpam-3559	189	10	rh2(bα	rh2(bα	PROPN
ejpam-3559	189	11	)	)	PUNCT
ejpam-3559	189	12	∀α	∀α	VERB
ejpam-3559	189	13	∈	∈	PROPN
ejpam-3559	190	1	i	i	PRON
ejpam-3559	190	2	⇔	⇔	PROPN
ejpam-3559	190	3	a	a	DET
ejpam-3559	190	4	�	�	PROPN
ejpam-3559	190	5	x	x	SYM
ejpam-3559	190	6	∀a	∀a	NOUN
ejpam-3559	190	7	∈	∈	NOUN
ejpam-3559	190	8	aα	aα	NOUN
ejpam-3559	190	9	and	and	CCONJ
ejpam-3559	190	10	b	b	PROPN
ejpam-3559	190	11	�	�	PROPN
ejpam-3559	190	12	y	y	PROPN
ejpam-3559	190	13	∀b	∀b	NOUN
ejpam-3559	190	14	∈	∈	PROPN
ejpam-3559	190	15	bα	bα	NOUN
ejpam-3559	190	16	and	and	CCONJ
ejpam-3559	190	17	∀α	∀α	NOUN
ejpam-3559	190	18	∈	∈	PROPN
ejpam-3559	191	1	i	i	PRON
ejpam-3559	191	2	⇔	⇔	PROPN
ejpam-3559	191	3	a	a	DET
ejpam-3559	191	4	�	�	PROPN
ejpam-3559	191	5	x	x	SYM
ejpam-3559	191	6	∀a	∀a	NOUN
ejpam-3559	191	7	∈	∈	NOUN
ejpam-3559	191	8	⋃	⋃	NOUN
ejpam-3559	191	9	α∈i	α∈i	NUM
ejpam-3559	191	10	aα	aα	NOUN
ejpam-3559	191	11	and	and	CCONJ
ejpam-3559	191	12	b	b	PROPN
ejpam-3559	191	13	�	�	PROPN
ejpam-3559	191	14	y	y	PROPN
ejpam-3559	191	15	∀b	∀b	PROPN
ejpam-3559	191	16	∈	∈	PROPN
ejpam-3559	191	17	⋃	⋃	NOUN
ejpam-3559	191	18	α∈i	α∈i	NUM
ejpam-3559	191	19	bα	bα	PROPN
ejpam-3559	191	20	⇔	⇔	X
ejpam-3559	191	21	x	x	SYM
ejpam-3559	191	22	∈	∈	PROPN
ejpam-3559	191	23	rh1	rh1	NOUN
ejpam-3559	191	24	(	(	PUNCT
ejpam-3559	191	25	⋃	⋃	PROPN
ejpam-3559	191	26	α∈i	α∈i	NUM
ejpam-3559	191	27	aα	aα	NOUN
ejpam-3559	191	28	)	)	PUNCT
ejpam-3559	191	29	and	and	CCONJ
ejpam-3559	191	30	y	y	PROPN
ejpam-3559	191	31	∈	∈	PROPN
ejpam-3559	191	32	rh2	rh2	X
ejpam-3559	191	33	(	(	PUNCT
ejpam-3559	191	34	⋃	⋃	VERB
ejpam-3559	191	35	α∈i	α∈i	ADJ
ejpam-3559	191	36	bα	bα	NOUN
ejpam-3559	191	37	)	)	PUNCT
ejpam-3559	191	38	⇔	⇔	X
ejpam-3559	191	39	(	(	PUNCT
ejpam-3559	191	40	x	x	NOUN
ejpam-3559	191	41	,	,	PUNCT
ejpam-3559	191	42	y	y	NOUN
ejpam-3559	191	43	)	)	PUNCT
ejpam-3559	191	44	∈	∈	PROPN
ejpam-3559	191	45	rh1	rh1	NOUN
ejpam-3559	191	46	(	(	PUNCT
ejpam-3559	191	47	⋃	⋃	PROPN
ejpam-3559	191	48	α∈i	α∈i	NUM
ejpam-3559	191	49	aα	aα	NOUN
ejpam-3559	191	50	)	)	PUNCT
ejpam-3559	191	51	×rh2	×rh2	PUNCT
ejpam-3559	191	52	(	(	PUNCT
ejpam-3559	191	53	⋃	⋃	VERB
ejpam-3559	191	54	α∈i	α∈i	NUM
ejpam-3559	191	55	bα	bα	NOUN
ejpam-3559	191	56	)	)	PUNCT
ejpam-3559	191	57	.	.	PUNCT
ejpam-3559	192	1	therefore	therefore	ADV
ejpam-3559	192	2	,	,	PUNCT
ejpam-3559	192	3	the	the	DET
ejpam-3559	192	4	equality	equality	NOUN
ejpam-3559	192	5	holds	hold	VERB
ejpam-3559	192	6	.	.	PUNCT
ejpam-3559	193	1	(	(	PUNCT
ejpam-3559	193	2	iii	iii	X
ejpam-3559	193	3	)	)	PUNCT
ejpam-3559	193	4	let	let	VERB
ejpam-3559	193	5	e	e	NOUN
ejpam-3559	193	6	=	=	PUNCT
ejpam-3559	193	7	⋃	⋃	PROPN
ejpam-3559	193	8	x∈p	x∈p	NOUN
ejpam-3559	193	9	(	(	PUNCT
ejpam-3559	193	10	{	{	PUNCT
ejpam-3559	193	11	x	x	NOUN
ejpam-3559	193	12	}	}	PUNCT
ejpam-3559	193	13	×	×	PROPN
ejpam-3559	193	14	tx	tx	PROPN
ejpam-3559	193	15	)	)	PUNCT
ejpam-3559	193	16	,	,	PUNCT
ejpam-3559	193	17	where	where	SCONJ
ejpam-3559	193	18	p	p	PROPN
ejpam-3559	193	19	⊆	⊆	NUM
ejpam-3559	193	20	h1	h1	NOUN
ejpam-3559	193	21	and	and	CCONJ
ejpam-3559	193	22	tx	tx	VERB
ejpam-3559	193	23	⊆	⊆	NUM
ejpam-3559	193	24	h2	h2	NOUN
ejpam-3559	193	25	for	for	ADP
ejpam-3559	193	26	each	each	DET
ejpam-3559	193	27	x	x	SYM
ejpam-3559	193	28	∈	∈	PROPN
ejpam-3559	193	29	p	p	NOUN
ejpam-3559	193	30	.	.	PUNCT
ejpam-3559	194	1	then	then	ADV
ejpam-3559	194	2	by	by	ADP
ejpam-3559	194	3	lemma	lemma	PROPN
ejpam-3559	194	4	4.4	4.4	NUM
ejpam-3559	194	5	,	,	PUNCT
ejpam-3559	194	6	(	(	PUNCT
ejpam-3559	194	7	i	i	NOUN
ejpam-3559	194	8	)	)	PUNCT
ejpam-3559	194	9	,	,	PUNCT
ejpam-3559	194	10	and	and	CCONJ
ejpam-3559	194	11	(	(	PUNCT
ejpam-3559	194	12	ii	ii	NOUN
ejpam-3559	194	13	)	)	PUNCT
ejpam-3559	194	14	,	,	PUNCT
ejpam-3559	194	15	rh(e	rh(e	NUM
ejpam-3559	194	16	)	)	PUNCT
ejpam-3559	194	17	=	=	SYM
ejpam-3559	194	18	rh	rh	PROPN
ejpam-3559	195	1	[	[	X
ejpam-3559	195	2	⋃	⋃	PROPN
ejpam-3559	195	3	x∈p	x∈p	NOUN
ejpam-3559	195	4	(	(	PUNCT
ejpam-3559	195	5	{	{	PUNCT
ejpam-3559	195	6	x	x	NOUN
ejpam-3559	195	7	}	}	PUNCT
ejpam-3559	195	8	×	×	PROPN
ejpam-3559	195	9	tx	tx	PROPN
ejpam-3559	195	10	)	)	PUNCT
ejpam-3559	195	11	]	]	PUNCT
ejpam-3559	196	1	=	=	PUNCT
ejpam-3559	196	2	⋂	⋂	PROPN
ejpam-3559	196	3	x∈p	x∈p	NOUN
ejpam-3559	196	4	[	[	X
ejpam-3559	196	5	rh({x	rh({x	NOUN
ejpam-3559	196	6	}	}	PUNCT
ejpam-3559	196	7	×	×	PROPN
ejpam-3559	196	8	tx	tx	PROPN
ejpam-3559	196	9	)	)	PUNCT
ejpam-3559	196	10	]	]	PUNCT
ejpam-3559	197	1	=	=	SYM
ejpam-3559	197	2	⋂	⋂	PROPN
ejpam-3559	197	3	x∈p	x∈p	NUM
ejpam-3559	197	4	[	[	X
ejpam-3559	197	5	rh1(x)×rh2(tx	rh1(x)×rh2(tx	X
ejpam-3559	197	6	)	)	PUNCT
ejpam-3559	197	7	]	]	PUNCT
ejpam-3559	198	1	=	=	SYM
ejpam-3559	198	2	rh1(p	rh1(p	PROPN
ejpam-3559	198	3	)	)	PUNCT
ejpam-3559	198	4	×rh2	×rh2	PUNCT
ejpam-3559	198	5	(	(	PUNCT
ejpam-3559	198	6	⋃	⋃	VERB
ejpam-3559	198	7	x∈p	x∈p	NOUN
ejpam-3559	198	8	tx	tx	PROPN
ejpam-3559	198	9	)	)	PUNCT
ejpam-3559	198	10	.	.	PUNCT
ejpam-3559	199	1	theorem	theorem	VERB
ejpam-3559	199	2	4.6	4.6	NUM
ejpam-3559	199	3	.	.	PUNCT
ejpam-3559	200	1	let	let	VERB
ejpam-3559	200	2	h	h	PRON
ejpam-3559	200	3	be	be	AUX
ejpam-3559	200	4	a	a	DET
ejpam-3559	200	5	hyper	hyper	ADJ
ejpam-3559	200	6	product	product	NOUN
ejpam-3559	200	7	of	of	ADP
ejpam-3559	200	8	hyper	hyper	ADJ
ejpam-3559	200	9	bck	bck	NOUN
ejpam-3559	200	10	-	-	PUNCT
ejpam-3559	200	11	algebras	algebras	ADJ
ejpam-3559	200	12	h1	h1	PROPN
ejpam-3559	200	13	and	and	CCONJ
ejpam-3559	200	14	h2	h2	PROPN
ejpam-3559	200	15	.	.	PUNCT
ejpam-3559	201	1	then	then	ADV
ejpam-3559	201	2	br(h	br(h	NUM
ejpam-3559	201	3	)	)	PUNCT
ejpam-3559	201	4	=	=	PUNCT
ejpam-3559	201	5	br(h1)×	br(h1)×	ADP
ejpam-3559	201	6	br(h2	br(h2	NOUN
ejpam-3559	201	7	)	)	PUNCT
ejpam-3559	201	8	.	.	PUNCT
ejpam-3559	202	1	proof	proof	NOUN
ejpam-3559	202	2	:	:	PUNCT
ejpam-3559	202	3	let	let	VERB
ejpam-3559	202	4	d	d	X
ejpam-3559	202	5	∈	∈	PROPN
ejpam-3559	202	6	br(h	br(h	NOUN
ejpam-3559	202	7	)	)	PUNCT
ejpam-3559	202	8	.	.	PUNCT
ejpam-3559	203	1	then	then	ADV
ejpam-3559	203	2	there	there	PRON
ejpam-3559	203	3	exists	exist	VERB
ejpam-3559	203	4	a	a	DET
ejpam-3559	203	5	nonempty	nonempty	ADV
ejpam-3559	203	6	set	set	VERB
ejpam-3559	203	7	e	e	NOUN
ejpam-3559	203	8	⊆	⊆	NUM
ejpam-3559	203	9	h	h	NOUN
ejpam-3559	203	10	=	=	PRON
ejpam-3559	203	11	h1	h1	PROPN
ejpam-3559	203	12	×h2	×h2	PROPN
ejpam-3559	203	13	such	such	ADJ
ejpam-3559	203	14	that	that	SCONJ
ejpam-3559	203	15	d	d	NOUN
ejpam-3559	203	16	=	=	PUNCT
ejpam-3559	203	17	rh(e	rh(e	PROPN
ejpam-3559	203	18	)	)	PUNCT
ejpam-3559	203	19	.	.	PUNCT
ejpam-3559	204	1	let	let	VERB
ejpam-3559	204	2	e	e	NOUN
ejpam-3559	204	3	=	=	PUNCT
ejpam-3559	204	4	⋃	⋃	VERB
ejpam-3559	204	5	x∈p	x∈p	NOUN
ejpam-3559	204	6	(	(	PUNCT
ejpam-3559	204	7	{	{	PUNCT
ejpam-3559	204	8	x	x	NOUN
ejpam-3559	204	9	}	}	PUNCT
ejpam-3559	204	10	×	×	PROPN
ejpam-3559	204	11	tx	tx	PROPN
ejpam-3559	204	12	)	)	PUNCT
ejpam-3559	204	13	where	where	SCONJ
ejpam-3559	204	14	p	p	NOUN
ejpam-3559	204	15	⊆	⊆	NUM
ejpam-3559	204	16	h1	h1	NOUN
ejpam-3559	204	17	and	and	CCONJ
ejpam-3559	204	18	tx	tx	VERB
ejpam-3559	204	19	⊆	⊆	NUM
ejpam-3559	204	20	h2	h2	NOUN
ejpam-3559	204	21	for	for	ADP
ejpam-3559	204	22	each	each	DET
ejpam-3559	204	23	x	x	SYM
ejpam-3559	204	24	∈	∈	PROPN
ejpam-3559	204	25	p	p	NOUN
ejpam-3559	204	26	.	.	PUNCT
ejpam-3559	205	1	references	reference	NOUN
ejpam-3559	205	2	1531	1531	NUM
ejpam-3559	205	3	then	then	ADV
ejpam-3559	205	4	lh(e	lh(e	PUNCT
ejpam-3559	205	5	)	)	PUNCT
ejpam-3559	206	1	=	=	SYM
ejpam-3559	206	2	rh1(p	rh1(p	PROPN
ejpam-3559	206	3	)	)	PUNCT
ejpam-3559	206	4	×	×	PROPN
ejpam-3559	206	5	rh2	rh2	NOUN
ejpam-3559	206	6	(	(	PUNCT
ejpam-3559	206	7	⋃	⋃	PROPN
ejpam-3559	206	8	x∈p	x∈p	ADJ
ejpam-3559	206	9	tx	tx	NOUN
ejpam-3559	206	10	)	)	PUNCT
ejpam-3559	206	11	∈	∈	PROPN
ejpam-3559	206	12	br(h1	br(h1	NOUN
ejpam-3559	206	13	)	)	PUNCT
ejpam-3559	206	14	×	×	NOUN
ejpam-3559	206	15	br(h2	br(h2	NOUN
ejpam-3559	206	16	)	)	PUNCT
ejpam-3559	206	17	by	by	ADP
ejpam-3559	206	18	theorem	theorem	ADJ
ejpam-3559	206	19	4.5(iii	4.5(iii	NUM
ejpam-3559	206	20	)	)	PUNCT
ejpam-3559	206	21	.	.	PUNCT
ejpam-3559	207	1	hence	hence	ADV
ejpam-3559	207	2	,	,	PUNCT
ejpam-3559	207	3	br(h	br(h	NUM
ejpam-3559	207	4	)	)	PUNCT
ejpam-3559	207	5	⊆	⊆	NUM
ejpam-3559	207	6	br(h1)×br(h2	br(h1)×br(h2	NOUN
ejpam-3559	207	7	)	)	PUNCT
ejpam-3559	207	8	.	.	PUNCT
ejpam-3559	208	1	next	next	ADV
ejpam-3559	208	2	,	,	PUNCT
ejpam-3559	208	3	suppose	suppose	VERB
ejpam-3559	208	4	that	that	SCONJ
ejpam-3559	208	5	f	f	PROPN
ejpam-3559	208	6	∈	∈	PROPN
ejpam-3559	208	7	br(h1)×br(h2	br(h1)×br(h2	PROPN
ejpam-3559	208	8	)	)	PUNCT
ejpam-3559	208	9	.	.	PUNCT
ejpam-3559	209	1	then	then	ADV
ejpam-3559	209	2	there	there	PRON
ejpam-3559	209	3	exist	exist	VERB
ejpam-3559	209	4	nonempty	nonempty	NOUN
ejpam-3559	209	5	sets	set	NOUN
ejpam-3559	209	6	o	o	NOUN
ejpam-3559	209	7	⊆	⊆	NUM
ejpam-3559	209	8	h1	h1	NOUN
ejpam-3559	209	9	and	and	CCONJ
ejpam-3559	209	10	u	u	NOUN
ejpam-3559	209	11	⊆	⊆	NUM
ejpam-3559	209	12	h2	h2	NOUN
ejpam-3559	209	13	such	such	ADJ
ejpam-3559	209	14	that	that	SCONJ
ejpam-3559	209	15	f	f	PROPN
ejpam-3559	209	16	=	=	SYM
ejpam-3559	209	17	rh1(o	rh1(o	PROPN
ejpam-3559	209	18	)	)	PUNCT
ejpam-3559	209	19	×	×	NOUN
ejpam-3559	209	20	rh2(u	rh2(u	PROPN
ejpam-3559	209	21	)	)	PUNCT
ejpam-3559	209	22	=	=	SYM
ejpam-3559	209	23	rh(o	rh(o	NUM
ejpam-3559	209	24	×	×	PROPN
ejpam-3559	209	25	u	u	NOUN
ejpam-3559	209	26	)	)	PUNCT
ejpam-3559	209	27	by	by	ADP
ejpam-3559	209	28	theorem	theorem	NOUN
ejpam-3559	209	29	4.5(i	4.5(i	NUM
ejpam-3559	209	30	)	)	PUNCT
ejpam-3559	209	31	.	.	PUNCT
ejpam-3559	210	1	thus	thus	ADV
ejpam-3559	210	2	,	,	PUNCT
ejpam-3559	210	3	f	f	PROPN
ejpam-3559	210	4	∈	∈	PROPN
ejpam-3559	210	5	br(h	br(h	NOUN
ejpam-3559	210	6	)	)	PUNCT
ejpam-3559	210	7	,	,	PUNCT
ejpam-3559	210	8	showing	show	VERB
ejpam-3559	210	9	that	that	DET
ejpam-3559	210	10	br(h1)×br(h2	br(h1)×br(h2	NOUN
ejpam-3559	210	11	)	)	PUNCT
ejpam-3559	210	12	⊆	⊆	NUM
ejpam-3559	210	13	br(h	br(h	NUM
ejpam-3559	210	14	)	)	PUNCT
ejpam-3559	210	15	.	.	PUNCT
ejpam-3559	211	1	therefore	therefore	ADV
ejpam-3559	211	2	,	,	PUNCT
ejpam-3559	211	3	br(h	br(h	NOUN
ejpam-3559	211	4	)	)	PUNCT
ejpam-3559	211	5	=	=	PUNCT
ejpam-3559	212	1	br(h1)×	br(h1)×	ADP
ejpam-3559	212	2	br(h2	br(h2	NOUN
ejpam-3559	212	3	)	)	PUNCT
ejpam-3559	212	4	.	.	PUNCT
ejpam-3559	213	1	conclusion	conclusion	NOUN
ejpam-3559	213	2	:	:	PUNCT
ejpam-3559	213	3	this	this	DET
ejpam-3559	213	4	study	study	NOUN
ejpam-3559	213	5	shows	show	VERB
ejpam-3559	213	6	that	that	SCONJ
ejpam-3559	213	7	,	,	PUNCT
ejpam-3559	213	8	indeed	indeed	ADV
ejpam-3559	213	9	,	,	PUNCT
ejpam-3559	213	10	a	a	DET
ejpam-3559	213	11	topological	topological	ADJ
ejpam-3559	213	12	structure	structure	NOUN
ejpam-3559	213	13	may	may	AUX
ejpam-3559	213	14	be	be	AUX
ejpam-3559	213	15	generated	generate	VERB
ejpam-3559	213	16	from	from	ADP
ejpam-3559	213	17	a	a	DET
ejpam-3559	213	18	given	give	VERB
ejpam-3559	213	19	(	(	PUNCT
ejpam-3559	213	20	hyper	hyper	ADJ
ejpam-3559	213	21	)	)	PUNCT
ejpam-3559	213	22	algebraic	algebraic	ADJ
ejpam-3559	213	23	structure	structure	NOUN
ejpam-3559	213	24	by	by	ADP
ejpam-3559	213	25	considering	consider	VERB
ejpam-3559	213	26	some	some	DET
ejpam-3559	213	27	family	family	NOUN
ejpam-3559	213	28	of	of	ADP
ejpam-3559	213	29	subsets	subset	NOUN
ejpam-3559	213	30	of	of	ADP
ejpam-3559	213	31	the	the	DET
ejpam-3559	213	32	underlying	underlying	ADJ
ejpam-3559	213	33	set	set	NOUN
ejpam-3559	213	34	of	of	ADP
ejpam-3559	213	35	the	the	DET
ejpam-3559	213	36	structure	structure	NOUN
ejpam-3559	213	37	that	that	PRON
ejpam-3559	213	38	would	would	AUX
ejpam-3559	213	39	qualify	qualify	VERB
ejpam-3559	213	40	as	as	ADP
ejpam-3559	213	41	a	a	DET
ejpam-3559	213	42	base	base	NOUN
ejpam-3559	213	43	for	for	ADP
ejpam-3559	213	44	some	some	DET
ejpam-3559	213	45	topology	topology	NOUN
ejpam-3559	213	46	on	on	ADP
ejpam-3559	213	47	the	the	DET
ejpam-3559	213	48	set	set	NOUN
ejpam-3559	213	49	.	.	PUNCT
ejpam-3559	214	1	the	the	DET
ejpam-3559	214	2	topology	topology	NOUN
ejpam-3559	214	3	generated	generate	VERB
ejpam-3559	214	4	in	in	ADP
ejpam-3559	214	5	this	this	DET
ejpam-3559	214	6	way	way	NOUN
ejpam-3559	214	7	need	need	AUX
ejpam-3559	214	8	not	not	PART
ejpam-3559	214	9	coincide	coincide	VERB
ejpam-3559	214	10	with	with	ADP
ejpam-3559	214	11	the	the	DET
ejpam-3559	214	12	topology	topology	NOUN
ejpam-3559	214	13	for	for	SCONJ
ejpam-3559	214	14	which	which	DET
ejpam-3559	214	15	continuity	continuity	NOUN
ejpam-3559	214	16	is	be	AUX
ejpam-3559	214	17	imposed	impose	VERB
ejpam-3559	214	18	on	on	ADP
ejpam-3559	214	19	some	some	DET
ejpam-3559	214	20	hyperoperations	hyperoperation	NOUN
ejpam-3559	214	21	associated	associate	VERB
ejpam-3559	214	22	with	with	ADP
ejpam-3559	214	23	the	the	DET
ejpam-3559	214	24	algebraic	algebraic	ADJ
ejpam-3559	214	25	structure	structure	NOUN
ejpam-3559	214	26	.	.	PUNCT
ejpam-3559	215	1	in	in	ADP
ejpam-3559	215	2	this	this	DET
ejpam-3559	215	3	study	study	NOUN
ejpam-3559	215	4	,	,	PUNCT
ejpam-3559	215	5	the	the	DET
ejpam-3559	215	6	authors	author	NOUN
ejpam-3559	215	7	,	,	PUNCT
ejpam-3559	215	8	using	use	VERB
ejpam-3559	215	9	the	the	DET
ejpam-3559	215	10	construction	construction	NOUN
ejpam-3559	215	11	of	of	ADP
ejpam-3559	215	12	a	a	DET
ejpam-3559	215	13	topological	topological	ADJ
ejpam-3559	215	14	structure	structure	NOUN
ejpam-3559	215	15	they	they	PRON
ejpam-3559	215	16	introduced	introduce	VERB
ejpam-3559	215	17	,	,	PUNCT
ejpam-3559	215	18	are	be	AUX
ejpam-3559	215	19	able	able	ADJ
ejpam-3559	215	20	to	to	PART
ejpam-3559	215	21	determine	determine	VERB
ejpam-3559	215	22	the	the	DET
ejpam-3559	215	23	bases	basis	NOUN
ejpam-3559	215	24	of	of	ADP
ejpam-3559	215	25	the	the	DET
ejpam-3559	215	26	topologies	topology	NOUN
ejpam-3559	215	27	generated	generate	VERB
ejpam-3559	215	28	by	by	ADP
ejpam-3559	215	29	the	the	DET
ejpam-3559	215	30	hyper	hyper	ADJ
ejpam-3559	215	31	sum	sum	NOUN
ejpam-3559	215	32	and	and	CCONJ
ejpam-3559	215	33	hyper	hyper	ADJ
ejpam-3559	215	34	product	product	NOUN
ejpam-3559	215	35	of	of	ADP
ejpam-3559	215	36	two	two	NUM
ejpam-3559	215	37	hyper	hyper	ADJ
ejpam-3559	215	38	bck	bck	NOUN
ejpam-3559	215	39	-	-	PUNCT
ejpam-3559	215	40	algebras	algebra	NOUN
ejpam-3559	215	41	.	.	PUNCT
ejpam-3559	216	1	acknowledgements	acknowledgement	NOUN
ejpam-3559	216	2	this	this	DET
ejpam-3559	216	3	research	research	NOUN
ejpam-3559	216	4	is	be	AUX
ejpam-3559	216	5	funded	fund	VERB
ejpam-3559	216	6	by	by	ADP
ejpam-3559	216	7	the	the	DET
ejpam-3559	216	8	philippine	philippine	PROPN
ejpam-3559	216	9	department	department	PROPN
ejpam-3559	216	10	of	of	ADP
ejpam-3559	216	11	science	science	NOUN
ejpam-3559	216	12	and	and	CCONJ
ejpam-3559	216	13	technology	technology	NOUN
ejpam-3559	216	14	accelerated	accelerate	VERB
ejpam-3559	216	15	science	science	NOUN
ejpam-3559	216	16	and	and	CCONJ
ejpam-3559	216	17	technology	technology	NOUN
ejpam-3559	216	18	human	human	ADJ
ejpam-3559	216	19	resource	resource	NOUN
ejpam-3559	216	20	development	development	NOUN
ejpam-3559	216	21	program	program	NOUN
ejpam-3559	216	22	(	(	PUNCT
ejpam-3559	216	23	dostasthrdp	dostasthrdp	PROPN
ejpam-3559	216	24	)	)	PUNCT
ejpam-3559	216	25	and	and	CCONJ
ejpam-3559	216	26	msu	msu	PROPN
ejpam-3559	216	27	-	-	PUNCT
ejpam-3559	216	28	iligan	iligan	PROPN
ejpam-3559	216	29	institute	institute	PROPN
ejpam-3559	216	30	of	of	ADP
ejpam-3559	216	31	technology	technology	PROPN
ejpam-3559	216	32	.	.	PUNCT
ejpam-3559	217	1	references	reference	NOUN
ejpam-3559	217	2	[	[	X
ejpam-3559	217	3	1	1	NUM
ejpam-3559	217	4	]	]	PUNCT
ejpam-3559	217	5	j	j	PROPN
ejpam-3559	217	6	albaracin	albaracin	PROPN
ejpam-3559	217	7	and	and	CCONJ
ejpam-3559	217	8	j	j	PROPN
ejpam-3559	217	9	vilela	vilela	NOUN
ejpam-3559	217	10	.	.	PUNCT
ejpam-3559	218	1	zero	zero	NUM
ejpam-3559	218	2	divisor	divisor	NOUN
ejpam-3559	218	3	graph	graph	NOUN
ejpam-3559	218	4	of	of	ADP
ejpam-3559	218	5	finite	finite	ADJ
ejpam-3559	218	6	hyper	hyper	ADJ
ejpam-3559	218	7	bck	bck	NOUN
ejpam-3559	218	8	-	-	PUNCT
ejpam-3559	218	9	algebra	algebra	NOUN
ejpam-3559	218	10	involving	involve	VERB
ejpam-3559	218	11	hyperatoms	hyperatom	NOUN
ejpam-3559	218	12	.	.	PUNCT
ejpam-3559	219	1	far	far	PROPN
ejpam-3559	219	2	east	east	PROPN
ejpam-3559	219	3	journal	journal	PROPN
ejpam-3559	219	4	of	of	ADP
ejpam-3559	219	5	mathematical	mathematical	ADJ
ejpam-3559	219	6	sciences	science	NOUN
ejpam-3559	219	7	,	,	PUNCT
ejpam-3559	219	8	103(4	103(4	NUM
ejpam-3559	219	9	):	):	PUNCT
ejpam-3559	219	10	743	743	NUM
ejpam-3559	219	11	-	-	SYM
ejpam-3559	219	12	755	755	NUM
ejpam-3559	219	13	,	,	PUNCT
ejpam-3559	219	14	2018	2018	NUM
ejpam-3559	219	15	.	.	PUNCT
ejpam-3559	220	1	[	[	X
ejpam-3559	220	2	2	2	X
ejpam-3559	220	3	]	]	PUNCT
ejpam-3559	220	4	a	a	DET
ejpam-3559	220	5	arhangel’skii	arhangel’skii	ADJ
ejpam-3559	220	6	and	and	CCONJ
ejpam-3559	220	7	m	m	NOUN
ejpam-3559	220	8	tkachenko	tkachenko	NOUN
ejpam-3559	220	9	.	.	PUNCT
ejpam-3559	221	1	topological	topological	ADJ
ejpam-3559	221	2	groups	group	NOUN
ejpam-3559	221	3	and	and	CCONJ
ejpam-3559	221	4	related	related	ADJ
ejpam-3559	221	5	structures	structure	NOUN
ejpam-3559	221	6	.	.	PUNCT
ejpam-3559	222	1	world	world	NOUN
ejpam-3559	222	2	scientific	scientific	PROPN
ejpam-3559	222	3	,	,	PUNCT
ejpam-3559	222	4	2008	2008	NUM
ejpam-3559	222	5	.	.	PUNCT
ejpam-3559	223	1	[	[	X
ejpam-3559	223	2	3	3	NUM
ejpam-3559	223	3	]	]	X
ejpam-3559	223	4	r	r	NOUN
ejpam-3559	223	5	borzooie	borzooie	NOUN
ejpam-3559	223	6	,	,	PUNCT
ejpam-3559	223	7	a	a	DET
ejpam-3559	223	8	hasankhani	hasankhani	PROPN
ejpam-3559	223	9	,	,	PUNCT
ejpam-3559	223	10	m	m	PROPN
ejpam-3559	223	11	zahedi	zahedi	PROPN
ejpam-3559	223	12	,	,	PUNCT
ejpam-3559	223	13	and	and	CCONJ
ejpam-3559	223	14	y	y	PROPN
ejpam-3559	223	15	jun	jun	PROPN
ejpam-3559	223	16	.	.	PROPN
ejpam-3559	224	1	on	on	ADP
ejpam-3559	224	2	hyper	hyper	ADJ
ejpam-3559	224	3	k	k	NOUN
ejpam-3559	224	4	-	-	PUNCT
ejpam-3559	224	5	algebras	algebras	PROPN
ejpam-3559	224	6	.	.	PUNCT
ejpam-3559	225	1	mathematicae	mathematicae	PROPN
ejpam-3559	225	2	japonicae	japonicae	PROPN
ejpam-3559	225	3	,	,	PUNCT
ejpam-3559	225	4	52(1	52(1	NOUN
ejpam-3559	225	5	):	):	PUNCT
ejpam-3559	225	6	113	113	NUM
ejpam-3559	225	7	-	-	SYM
ejpam-3559	225	8	121	121	NUM
ejpam-3559	225	9	,	,	PUNCT
ejpam-3559	225	10	2000	2000	NUM
ejpam-3559	225	11	.	.	PUNCT
ejpam-3559	226	1	[	[	X
ejpam-3559	226	2	4	4	NUM
ejpam-3559	226	3	]	]	PUNCT
ejpam-3559	226	4	h	h	NOUN
ejpam-3559	226	5	harizavi	harizavi	NOUN
ejpam-3559	226	6	.	.	PUNCT
ejpam-3559	227	1	on	on	ADP
ejpam-3559	227	2	direct	direct	ADJ
ejpam-3559	227	3	sum	sum	NOUN
ejpam-3559	227	4	of	of	ADP
ejpam-3559	227	5	branches	branch	NOUN
ejpam-3559	227	6	in	in	ADP
ejpam-3559	227	7	hyper	hyper	ADJ
ejpam-3559	227	8	bck	bck	NOUN
ejpam-3559	227	9	-	-	PUNCT
ejpam-3559	227	10	algebras	algebras	PROPN
ejpam-3559	227	11	.	.	PUNCT
ejpam-3559	227	12	iranian	iranian	PROPN
ejpam-3559	227	13	journal	journal	PROPN
ejpam-3559	227	14	of	of	ADP
ejpam-3559	227	15	mathematical	mathematical	ADJ
ejpam-3559	227	16	sciences	sciences	PROPN
ejpam-3559	227	17	and	and	CCONJ
ejpam-3559	227	18	informatics	informatic	NOUN
ejpam-3559	227	19	,	,	PUNCT
ejpam-3559	227	20	11(2	11(2	NUM
ejpam-3559	227	21	):	):	PUNCT
ejpam-3559	227	22	43	43	NUM
ejpam-3559	227	23	-	-	SYM
ejpam-3559	227	24	55	55	NUM
ejpam-3559	227	25	,	,	PUNCT
ejpam-3559	227	26	2016	2016	NUM
ejpam-3559	227	27	.	.	PUNCT
ejpam-3559	228	1	[	[	X
ejpam-3559	228	2	5	5	NUM
ejpam-3559	228	3	]	]	X
ejpam-3559	228	4	y	y	PROPN
ejpam-3559	228	5	imai	imai	PROPN
ejpam-3559	228	6	and	and	CCONJ
ejpam-3559	228	7	k	k	PROPN
ejpam-3559	228	8	iséki	iséki	PROPN
ejpam-3559	228	9	.	.	PROPN
ejpam-3559	228	10	on	on	ADP
ejpam-3559	228	11	axiom	axiom	NOUN
ejpam-3559	228	12	systems	system	NOUN
ejpam-3559	228	13	of	of	ADP
ejpam-3559	228	14	propositional	propositional	ADJ
ejpam-3559	228	15	calculi	calculi	PROPN
ejpam-3559	228	16	xiv	xiv	PROPN
ejpam-3559	228	17	.	.	PUNCT
ejpam-3559	229	1	proc	proc	PROPN
ejpam-3559	229	2	.	.	PUNCT
ejpam-3559	230	1	japan	japan	PROPN
ejpam-3559	230	2	academy	academy	PROPN
ejpam-3559	230	3	,	,	PUNCT
ejpam-3559	230	4	42	42	NUM
ejpam-3559	230	5	:	:	SYM
ejpam-3559	230	6	19	19	NUM
ejpam-3559	230	7	-	-	SYM
ejpam-3559	230	8	22	22	NUM
ejpam-3559	230	9	,	,	PUNCT
ejpam-3559	230	10	1966	1966	NUM
ejpam-3559	230	11	.	.	PUNCT
ejpam-3559	231	1	[	[	X
ejpam-3559	231	2	6	6	NUM
ejpam-3559	231	3	]	]	X
ejpam-3559	231	4	y	y	PROPN
ejpam-3559	231	5	jun	jun	PROPN
ejpam-3559	231	6	,	,	PUNCT
ejpam-3559	231	7	m	m	PROPN
ejpam-3559	231	8	zahedi	zahedi	PROPN
ejpam-3559	231	9	,	,	PUNCT
ejpam-3559	231	10	x	x	X
ejpam-3559	231	11	xin	xin	PROPN
ejpam-3559	231	12	,	,	PUNCT
ejpam-3559	231	13	and	and	CCONJ
ejpam-3559	231	14	r	r	NOUN
ejpam-3559	231	15	borzooei	borzooei	ADJ
ejpam-3559	231	16	.	.	PUNCT
ejpam-3559	232	1	on	on	ADP
ejpam-3559	232	2	hyper	hyper	ADJ
ejpam-3559	232	3	bck	bck	NOUN
ejpam-3559	232	4	-	-	PUNCT
ejpam-3559	232	5	algebras	algebras	PROPN
ejpam-3559	232	6	.	.	PUNCT
ejpam-3559	233	1	italian	italian	ADJ
ejpam-3559	233	2	journal	journal	NOUN
ejpam-3559	233	3	of	of	ADP
ejpam-3559	233	4	pure	pure	ADJ
ejpam-3559	233	5	and	and	CCONJ
ejpam-3559	233	6	applied	applied	ADJ
ejpam-3559	233	7	mathematics	mathematic	NOUN
ejpam-3559	233	8	,	,	PUNCT
ejpam-3559	233	9	8	8	NUM
ejpam-3559	233	10	:	:	SYM
ejpam-3559	233	11	127	127	NUM
ejpam-3559	233	12	-	-	SYM
ejpam-3559	233	13	136	136	NUM
ejpam-3559	233	14	,	,	PUNCT
ejpam-3559	233	15	2000	2000	NUM
ejpam-3559	233	16	.	.	PUNCT
ejpam-3559	234	1	[	[	X
ejpam-3559	234	2	7	7	X
ejpam-3559	234	3	]	]	X
ejpam-3559	234	4	f.	f.	PROPN
ejpam-3559	234	5	marty	marty	PROPN
ejpam-3559	234	6	.	.	PUNCT
ejpam-3559	235	1	sun	sun	PROPN
ejpam-3559	235	2	une	une	PROPN
ejpam-3559	235	3	generalization	generalization	NOUN
ejpam-3559	235	4	da	da	PROPN
ejpam-3559	235	5	la	la	PROPN
ejpam-3559	235	6	notion	notion	NOUN
ejpam-3559	235	7	de	de	PROPN
ejpam-3559	235	8	group	group	NOUN
ejpam-3559	235	9	.	.	PUNCT
ejpam-3559	236	1	stockholm	stockholm	PROPN
ejpam-3559	236	2	:	:	PUNCT
ejpam-3559	236	3	8th	8th	ADJ
ejpam-3559	236	4	congress	congress	PROPN
ejpam-3559	236	5	math	math	NOUN
ejpam-3559	236	6	.	.	PUNCT
ejpam-3559	237	1	scandinaves	scandinave	NOUN
ejpam-3559	237	2	,	,	PUNCT
ejpam-3559	237	3	pages	page	NOUN
ejpam-3559	237	4	45	45	NUM
ejpam-3559	237	5	-	-	SYM
ejpam-3559	237	6	49	49	NUM
ejpam-3559	237	7	,	,	PUNCT
ejpam-3559	237	8	1934	1934	NUM
ejpam-3559	237	9	.	.	PUNCT
ejpam-3559	238	1	[	[	X
ejpam-3559	238	2	8	8	NUM
ejpam-3559	238	3	]	]	X
ejpam-3559	238	4	r	r	NOUN
ejpam-3559	238	5	patangan	patangan	NOUN
ejpam-3559	238	6	and	and	CCONJ
ejpam-3559	238	7	canoy	canoy	ADJ
ejpam-3559	238	8	,	,	PUNCT
ejpam-3559	238	9	s	s	PART
ejpam-3559	238	10	canoy	canoy	NOUN
ejpam-3559	238	11	,	,	PUNCT
ejpam-3559	238	12	jr	jr	PROPN
ejpam-3559	238	13	.	.	PUNCT
ejpam-3559	239	1	a	a	DET
ejpam-3559	239	2	topology	topology	NOUN
ejpam-3559	239	3	on	on	ADP
ejpam-3559	239	4	a	a	DET
ejpam-3559	239	5	hyper	hyper	ADJ
ejpam-3559	239	6	bck	bck	NOUN
ejpam-3559	239	7	-	-	PUNCT
ejpam-3559	239	8	algebra	algebra	NOUN
ejpam-3559	239	9	.	.	PUNCT
ejpam-3559	240	1	jp	jp	PROPN
ejpam-3559	240	2	journal	journal	PROPN
ejpam-3559	240	3	of	of	ADP
ejpam-3559	240	4	algebra	algebra	PROPN
ejpam-3559	240	5	,	,	PUNCT
ejpam-3559	240	6	number	number	NOUN
ejpam-3559	240	7	theory	theory	NOUN
ejpam-3559	240	8	and	and	CCONJ
ejpam-3559	240	9	applications	application	NOUN
ejpam-3559	240	10	,	,	PUNCT
ejpam-3559	240	11	40	40	NUM
ejpam-3559	240	12	:	:	PUNCT
ejpam-3559	240	13	787	787	NUM
ejpam-3559	240	14	-	-	SYM
ejpam-3559	240	15	797	797	NUM
ejpam-3559	240	16	,	,	PUNCT
ejpam-3559	240	17	2018	2018	NUM
ejpam-3559	240	18	.	.	PUNCT
ejpam-3559	241	1	references	reference	NOUN
ejpam-3559	241	2	1532	1532	NUM
ejpam-3559	241	3	[	[	X
ejpam-3559	241	4	9	9	NUM
ejpam-3559	241	5	]	]	SYM
ejpam-3559	241	6	r	r	NOUN
ejpam-3559	241	7	patangan	patangan	NOUN
ejpam-3559	241	8	and	and	CCONJ
ejpam-3559	241	9	s	s	VERB
ejpam-3559	241	10	canoy	canoy	NOUN
ejpam-3559	241	11	,	,	PUNCT
ejpam-3559	241	12	jr	jr	PROPN
ejpam-3559	241	13	.	.	PUNCT
ejpam-3559	242	1	a	a	DET
ejpam-3559	242	2	topology	topology	NOUN
ejpam-3559	242	3	on	on	ADP
ejpam-3559	242	4	a	a	DET
ejpam-3559	242	5	hyper	hyper	ADJ
ejpam-3559	242	6	bck	bck	NOUN
ejpam-3559	242	7	-	-	PUNCT
ejpam-3559	242	8	algebra	algebra	NOUN
ejpam-3559	242	9	via	via	ADP
ejpam-3559	242	10	left	left	ADJ
ejpam-3559	242	11	application	application	NOUN
ejpam-3559	242	12	of	of	ADP
ejpam-3559	242	13	a	a	DET
ejpam-3559	242	14	hyper	hyper	ADJ
ejpam-3559	242	15	order	order	NOUN
ejpam-3559	242	16	.	.	PUNCT
ejpam-3559	243	1	jp	jp	PROPN
ejpam-3559	243	2	journal	journal	PROPN
ejpam-3559	243	3	of	of	ADP
ejpam-3559	243	4	algebra	algebra	PROPN
ejpam-3559	243	5	,	,	PUNCT
ejpam-3559	243	6	number	number	NOUN
ejpam-3559	243	7	theory	theory	NOUN
ejpam-3559	243	8	and	and	CCONJ
ejpam-3559	243	9	applications	application	NOUN
ejpam-3559	243	10	,	,	PUNCT
ejpam-3559	243	11	40	40	NUM
ejpam-3559	243	12	:	:	SYM
ejpam-3559	243	13	321	321	NUM
ejpam-3559	243	14	-	-	SYM
ejpam-3559	243	15	332	332	NUM
ejpam-3559	243	16	,	,	PUNCT
ejpam-3559	243	17	2018	2018	NUM
ejpam-3559	243	18	.	.	PUNCT