id	sid	tid	token	lemma	pos
ejpam-5976	1	1	european	european	PROPN
ejpam-5976	1	2	journal	journal	PROPN
ejpam-5976	1	3	of	of	ADP
ejpam-5976	1	4	pure	pure	ADJ
ejpam-5976	1	5	and	and	CCONJ
ejpam-5976	1	6	applied	applied	ADJ
ejpam-5976	1	7	mathematics	mathematic	NOUN
ejpam-5976	1	8	2025	2025	NUM
ejpam-5976	1	9	,	,	PUNCT
ejpam-5976	1	10	vol	vol	NOUN
ejpam-5976	1	11	.	.	PROPN
ejpam-5976	1	12	18	18	NUM
ejpam-5976	1	13	,	,	PUNCT
ejpam-5976	1	14	issue	issue	NOUN
ejpam-5976	1	15	2	2	NUM
ejpam-5976	1	16	,	,	PUNCT
ejpam-5976	1	17	article	article	NOUN
ejpam-5976	1	18	number	number	NOUN
ejpam-5976	1	19	5976	5976	NUM
ejpam-5976	1	20	issn	issn	VERB
ejpam-5976	1	21	1307	1307	NUM
ejpam-5976	1	22	-	-	SYM
ejpam-5976	1	23	5543	5543	NUM
ejpam-5976	1	24	–	–	PUNCT
ejpam-5976	1	25	ejpam.com	ejpam.com	X
ejpam-5976	1	26	published	publish	VERB
ejpam-5976	1	27	by	by	ADP
ejpam-5976	1	28	new	new	PROPN
ejpam-5976	1	29	york	york	PROPN
ejpam-5976	1	30	business	business	PROPN
ejpam-5976	1	31	global	global	PROPN
ejpam-5976	1	32	quasi	quasi	NOUN
ejpam-5976	1	33	-	-	NOUN
ejpam-5976	1	34	ideals	ideal	NOUN
ejpam-5976	1	35	and	and	CCONJ
ejpam-5976	1	36	h	h	NOUN
ejpam-5976	1	37	-	-	PUNCT
ejpam-5976	1	38	classes	class	NOUN
ejpam-5976	1	39	on	on	ADP
ejpam-5976	1	40	the	the	DET
ejpam-5976	1	41	direct	direct	ADJ
ejpam-5976	1	42	product	product	NOUN
ejpam-5976	1	43	of	of	ADP
ejpam-5976	1	44	two	two	NUM
ejpam-5976	1	45	semigroups	semigroup	NOUN
ejpam-5976	1	46	panuwat	panuwat	VERB
ejpam-5976	1	47	luangchaisri1	luangchaisri1	PROPN
ejpam-5976	1	48	,	,	PUNCT
ejpam-5976	1	49	ontima	ontima	PROPN
ejpam-5976	1	50	pankoon1	pankoon1	PROPN
ejpam-5976	1	51	,	,	PUNCT
ejpam-5976	1	52	thawhat	thawhat	PRON
ejpam-5976	1	53	changphas1,∗	changphas1,∗	NOUN
ejpam-5976	1	54	1	1	NUM
ejpam-5976	1	55	department	department	NOUN
ejpam-5976	1	56	of	of	ADP
ejpam-5976	1	57	mathematics	mathematic	NOUN
ejpam-5976	1	58	,	,	PUNCT
ejpam-5976	1	59	faculty	faculty	NOUN
ejpam-5976	1	60	of	of	ADP
ejpam-5976	1	61	science	science	PROPN
ejpam-5976	1	62	khon	khon	PROPN
ejpam-5976	1	63	kaen	kaen	PROPN
ejpam-5976	1	64	university	university	PROPN
ejpam-5976	1	65	,	,	PUNCT
ejpam-5976	1	66	khon	khon	PROPN
ejpam-5976	1	67	kaen	kaen	PROPN
ejpam-5976	1	68	40002	40002	NUM
ejpam-5976	1	69	,	,	PUNCT
ejpam-5976	1	70	thailand	thailand	PROPN
ejpam-5976	1	71	abstract	abstract	PROPN
ejpam-5976	1	72	.	.	PUNCT
ejpam-5976	2	1	let	let	VERB
ejpam-5976	2	2	s	s	PRON
ejpam-5976	2	3	be	be	AUX
ejpam-5976	2	4	a	a	DET
ejpam-5976	2	5	semigroup	semigroup	NOUN
ejpam-5976	2	6	and	and	CCONJ
ejpam-5976	2	7	x	x	SYM
ejpam-5976	2	8	∈	∈	PROPN
ejpam-5976	2	9	s.	s.	PROPN
ejpam-5976	2	10	the	the	DET
ejpam-5976	2	11	principal	principal	ADJ
ejpam-5976	2	12	quasi	quasi	NOUN
ejpam-5976	2	13	-	-	NOUN
ejpam-5976	2	14	ideal	ideal	ADJ
ejpam-5976	2	15	of	of	ADP
ejpam-5976	2	16	s	s	AUX
ejpam-5976	2	17	containing	contain	VERB
ejpam-5976	2	18	x	x	VERB
ejpam-5976	2	19	is	be	AUX
ejpam-5976	2	20	denoted	denote	VERB
ejpam-5976	2	21	by	by	ADP
ejpam-5976	2	22	q(x	q(x	NOUN
ejpam-5976	2	23	)	)	PUNCT
ejpam-5976	2	24	.	.	PUNCT
ejpam-5976	3	1	an	an	DET
ejpam-5976	3	2	h	h	NOUN
ejpam-5976	3	3	-	-	PUNCT
ejpam-5976	3	4	class	class	NOUN
ejpam-5976	3	5	of	of	ADP
ejpam-5976	3	6	s	s	NOUN
ejpam-5976	3	7	containing	contain	VERB
ejpam-5976	3	8	x	x	AUX
ejpam-5976	3	9	is	be	AUX
ejpam-5976	3	10	denoted	denote	VERB
ejpam-5976	3	11	by	by	ADP
ejpam-5976	3	12	hx	hx	PROPN
ejpam-5976	3	13	.	.	PUNCT
ejpam-5976	4	1	let	let	VERB
ejpam-5976	4	2	s1	s1	NOUN
ejpam-5976	4	3	,	,	PUNCT
ejpam-5976	4	4	s2	s2	X
ejpam-5976	4	5	be	be	VERB
ejpam-5976	4	6	semigroups	semigroup	NOUN
ejpam-5976	4	7	.	.	PUNCT
ejpam-5976	5	1	the	the	DET
ejpam-5976	5	2	direct	direct	ADJ
ejpam-5976	5	3	product	product	NOUN
ejpam-5976	5	4	s1×s2	s1×s2	PROPN
ejpam-5976	5	5	is	be	AUX
ejpam-5976	5	6	defined	define	VERB
ejpam-5976	5	7	as	as	ADP
ejpam-5976	5	8	the	the	DET
ejpam-5976	5	9	cartesian	cartesian	ADJ
ejpam-5976	5	10	product	product	NOUN
ejpam-5976	5	11	of	of	ADP
ejpam-5976	5	12	s1	s1	PROPN
ejpam-5976	5	13	and	and	CCONJ
ejpam-5976	5	14	s2	s2	PROPN
ejpam-5976	5	15	equipped	equip	VERB
ejpam-5976	5	16	with	with	ADP
ejpam-5976	5	17	the	the	DET
ejpam-5976	5	18	componentwise	componentwise	NOUN
ejpam-5976	5	19	binary	binary	NOUN
ejpam-5976	5	20	operation	operation	NOUN
ejpam-5976	5	21	.	.	PUNCT
ejpam-5976	6	1	let	let	VERB
ejpam-5976	6	2	(	(	PUNCT
ejpam-5976	6	3	a	a	PRON
ejpam-5976	6	4	,	,	PUNCT
ejpam-5976	6	5	b	b	NOUN
ejpam-5976	6	6	)	)	PUNCT
ejpam-5976	6	7	∈	∈	PROPN
ejpam-5976	6	8	s1×s2	s1×s2	PROPN
ejpam-5976	6	9	.	.	PUNCT
ejpam-5976	7	1	the	the	DET
ejpam-5976	7	2	direct	direct	ADJ
ejpam-5976	7	3	product	product	NOUN
ejpam-5976	7	4	of	of	ADP
ejpam-5976	7	5	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	7	6	)	)	PUNCT
ejpam-5976	7	7	need	need	VERB
ejpam-5976	7	8	not	not	PART
ejpam-5976	7	9	to	to	PART
ejpam-5976	7	10	be	be	AUX
ejpam-5976	7	11	q((a	q((a	PROPN
ejpam-5976	7	12	,	,	PUNCT
ejpam-5976	7	13	b	b	NOUN
ejpam-5976	7	14	)	)	PUNCT
ejpam-5976	7	15	)	)	PUNCT
ejpam-5976	7	16	.	.	PUNCT
ejpam-5976	8	1	in	in	ADP
ejpam-5976	8	2	this	this	DET
ejpam-5976	8	3	paper	paper	NOUN
ejpam-5976	8	4	,	,	PUNCT
ejpam-5976	8	5	we	we	PRON
ejpam-5976	8	6	provide	provide	VERB
ejpam-5976	8	7	necessary	necessary	ADJ
ejpam-5976	8	8	and	and	CCONJ
ejpam-5976	8	9	sufficient	sufficient	ADJ
ejpam-5976	8	10	conditions	condition	NOUN
ejpam-5976	8	11	when	when	SCONJ
ejpam-5976	8	12	q(a)×q(b	q(a)×q(b	ADV
ejpam-5976	8	13	)	)	PUNCT
ejpam-5976	8	14	=	=	SYM
ejpam-5976	8	15	q((a	q((a	PROPN
ejpam-5976	8	16	,	,	PUNCT
ejpam-5976	8	17	b	b	NOUN
ejpam-5976	8	18	)	)	PUNCT
ejpam-5976	8	19	)	)	PUNCT
ejpam-5976	8	20	and	and	CCONJ
ejpam-5976	8	21	the	the	DET
ejpam-5976	8	22	conditions	condition	NOUN
ejpam-5976	8	23	when	when	SCONJ
ejpam-5976	8	24	h(a	h(a	PROPN
ejpam-5976	8	25	,	,	PUNCT
ejpam-5976	8	26	b	b	NOUN
ejpam-5976	8	27	)	)	PUNCT
ejpam-5976	8	28	=	=	NOUN
ejpam-5976	8	29	ha	ha	INTJ
ejpam-5976	8	30	×hb	×hb	PROPN
ejpam-5976	8	31	.	.	PROPN
ejpam-5976	8	32	2020	2020	NUM
ejpam-5976	8	33	mathematics	mathematic	NOUN
ejpam-5976	8	34	subject	subject	NOUN
ejpam-5976	8	35	classifications	classification	NOUN
ejpam-5976	8	36	:	:	PUNCT
ejpam-5976	8	37	20m12	20m12	NUM
ejpam-5976	8	38	key	key	ADJ
ejpam-5976	8	39	words	word	NOUN
ejpam-5976	8	40	and	and	CCONJ
ejpam-5976	8	41	phrases	phrase	NOUN
ejpam-5976	8	42	:	:	PUNCT
ejpam-5976	8	43	semigroups	semigroup	NOUN
ejpam-5976	8	44	,	,	PUNCT
ejpam-5976	8	45	direct	direct	ADJ
ejpam-5976	8	46	product	product	NOUN
ejpam-5976	8	47	,	,	PUNCT
ejpam-5976	8	48	quasi	quasi	ADJ
ejpam-5976	8	49	-	-	ADJ
ejpam-5976	8	50	ideal	ideal	ADJ
ejpam-5976	8	51	,	,	PUNCT
ejpam-5976	8	52	h	h	NOUN
ejpam-5976	8	53	-	-	PUNCT
ejpam-5976	8	54	class	class	NOUN
ejpam-5976	8	55	,	,	PUNCT
ejpam-5976	8	56	maximal	maximal	ADJ
ejpam-5976	8	57	h	h	NOUN
ejpam-5976	8	58	-	-	PUNCT
ejpam-5976	8	59	class	class	NOUN
ejpam-5976	8	60	1	1	NUM
ejpam-5976	8	61	.	.	PUNCT
ejpam-5976	8	62	introduction	introduction	NOUN
ejpam-5976	8	63	let	let	VERB
ejpam-5976	8	64	s	s	PRON
ejpam-5976	8	65	be	be	AUX
ejpam-5976	8	66	a	a	DET
ejpam-5976	8	67	semigroup	semigroup	NOUN
ejpam-5976	8	68	.	.	PUNCT
ejpam-5976	9	1	a	a	DET
ejpam-5976	9	2	nonempty	nonempty	NOUN
ejpam-5976	9	3	subset	subset	VERB
ejpam-5976	9	4	q	q	NOUN
ejpam-5976	9	5	of	of	ADP
ejpam-5976	9	6	s	s	PROPN
ejpam-5976	9	7	is	be	AUX
ejpam-5976	9	8	called	call	VERB
ejpam-5976	9	9	a	a	DET
ejpam-5976	9	10	quasi	quasi	NOUN
ejpam-5976	9	11	-	-	NOUN
ejpam-5976	9	12	ideal	ideal	ADJ
ejpam-5976	9	13	of	of	ADP
ejpam-5976	9	14	s	s	PRON
ejpam-5976	9	15	if	if	SCONJ
ejpam-5976	9	16	qs	qs	ADP
ejpam-5976	9	17	∩	∩	NOUN
ejpam-5976	9	18	sq	sq	PROPN
ejpam-5976	9	19	⊆	⊆	NUM
ejpam-5976	9	20	q.	q.	NOUN
ejpam-5976	9	21	the	the	DET
ejpam-5976	9	22	concept	concept	NOUN
ejpam-5976	9	23	of	of	ADP
ejpam-5976	9	24	quasi	quasi	NOUN
ejpam-5976	9	25	-	-	NOUN
ejpam-5976	9	26	ideals	ideal	NOUN
ejpam-5976	9	27	was	be	AUX
ejpam-5976	9	28	introduced	introduce	VERB
ejpam-5976	9	29	by	by	ADP
ejpam-5976	9	30	steinfeld	steinfeld	PROPN
ejpam-5976	10	1	[	[	X
ejpam-5976	10	2	1	1	NUM
ejpam-5976	10	3	]	]	PUNCT
ejpam-5976	10	4	.	.	PUNCT
ejpam-5976	11	1	quasiideals	quasiideal	NOUN
ejpam-5976	11	2	have	have	AUX
ejpam-5976	11	3	been	be	AUX
ejpam-5976	11	4	studied	study	VERB
ejpam-5976	11	5	extensively	extensively	ADV
ejpam-5976	11	6	in	in	ADP
ejpam-5976	11	7	semigroup	semigroup	PROPN
ejpam-5976	11	8	theory	theory	NOUN
ejpam-5976	11	9	and	and	CCONJ
ejpam-5976	11	10	have	have	AUX
ejpam-5976	11	11	been	be	AUX
ejpam-5976	11	12	found	find	VERB
ejpam-5976	11	13	to	to	PART
ejpam-5976	11	14	have	have	VERB
ejpam-5976	11	15	many	many	ADJ
ejpam-5976	11	16	important	important	ADJ
ejpam-5976	11	17	applications	application	NOUN
ejpam-5976	11	18	and	and	CCONJ
ejpam-5976	11	19	connections	connection	NOUN
ejpam-5976	11	20	with	with	ADP
ejpam-5976	11	21	other	other	ADJ
ejpam-5976	11	22	algebraic	algebraic	ADJ
ejpam-5976	11	23	structures	structure	NOUN
ejpam-5976	11	24	.	.	PUNCT
ejpam-5976	12	1	one	one	NUM
ejpam-5976	12	2	such	such	ADJ
ejpam-5976	12	3	connection	connection	NOUN
ejpam-5976	12	4	is	be	AUX
ejpam-5976	12	5	with	with	ADP
ejpam-5976	12	6	congruences	congruence	NOUN
ejpam-5976	12	7	on	on	ADP
ejpam-5976	12	8	semigroups	semigroup	NOUN
ejpam-5976	12	9	.	.	PUNCT
ejpam-5976	13	1	it	it	PRON
ejpam-5976	13	2	was	be	AUX
ejpam-5976	13	3	proved	prove	VERB
ejpam-5976	13	4	by	by	ADP
ejpam-5976	13	5	steinfeld	steinfeld	PROPN
ejpam-5976	13	6	[	[	X
ejpam-5976	13	7	2	2	X
ejpam-5976	13	8	]	]	PUNCT
ejpam-5976	13	9	that	that	SCONJ
ejpam-5976	13	10	for	for	ADP
ejpam-5976	13	11	any	any	DET
ejpam-5976	13	12	a	a	NOUN
ejpam-5976	13	13	,	,	PUNCT
ejpam-5976	13	14	b	b	PROPN
ejpam-5976	13	15	∈	∈	PROPN
ejpam-5976	13	16	s	s	PROPN
ejpam-5976	13	17	,	,	PUNCT
ejpam-5976	13	18	ahb	ahb	NOUN
ejpam-5976	13	19	if	if	SCONJ
ejpam-5976	13	20	and	and	CCONJ
ejpam-5976	13	21	only	only	ADV
ejpam-5976	13	22	if	if	SCONJ
ejpam-5976	13	23	q(a	q(a	NOUN
ejpam-5976	13	24	)	)	PUNCT
ejpam-5976	13	25	=	=	SYM
ejpam-5976	13	26	q(b	q(b	ADJ
ejpam-5976	13	27	)	)	PUNCT
ejpam-5976	13	28	.	.	PUNCT
ejpam-5976	14	1	this	this	DET
ejpam-5976	14	2	result	result	NOUN
ejpam-5976	14	3	shows	show	VERB
ejpam-5976	14	4	that	that	SCONJ
ejpam-5976	14	5	a	a	DET
ejpam-5976	14	6	congruence	congruence	ADJ
ejpam-5976	14	7	h	h	NOUN
ejpam-5976	14	8	on	on	ADP
ejpam-5976	14	9	a	a	DET
ejpam-5976	14	10	semigroup	semigroup	NOUN
ejpam-5976	14	11	can	can	AUX
ejpam-5976	14	12	be	be	AUX
ejpam-5976	14	13	described	describe	VERB
ejpam-5976	14	14	in	in	ADP
ejpam-5976	14	15	terms	term	NOUN
ejpam-5976	14	16	of	of	ADP
ejpam-5976	14	17	the	the	DET
ejpam-5976	14	18	associated	associated	ADJ
ejpam-5976	14	19	quasi	quasi	NOUN
ejpam-5976	14	20	-	-	NOUN
ejpam-5976	14	21	ideal	ideal	ADJ
ejpam-5976	14	22	.	.	PUNCT
ejpam-5976	15	1	given	give	VERB
ejpam-5976	15	2	semigroups	semigroups	PROPN
ejpam-5976	15	3	s1	s1	PROPN
ejpam-5976	15	4	and	and	CCONJ
ejpam-5976	15	5	s2	s2	PROPN
ejpam-5976	15	6	,	,	PUNCT
ejpam-5976	15	7	the	the	DET
ejpam-5976	15	8	direct	direct	ADJ
ejpam-5976	15	9	product	product	NOUN
ejpam-5976	15	10	s1	s1	PROPN
ejpam-5976	15	11	×	×	NOUN
ejpam-5976	15	12	s2	s2	NOUN
ejpam-5976	15	13	is	be	AUX
ejpam-5976	15	14	defined	define	VERB
ejpam-5976	15	15	as	as	ADP
ejpam-5976	15	16	the	the	DET
ejpam-5976	15	17	cartesian	cartesian	ADJ
ejpam-5976	15	18	product	product	NOUN
ejpam-5976	15	19	of	of	ADP
ejpam-5976	15	20	s1	s1	PROPN
ejpam-5976	15	21	and	and	CCONJ
ejpam-5976	15	22	s2	s2	PROPN
ejpam-5976	15	23	,	,	PUNCT
ejpam-5976	15	24	equipped	equip	VERB
ejpam-5976	15	25	with	with	ADP
ejpam-5976	15	26	the	the	DET
ejpam-5976	15	27	componentwise	componentwise	NOUN
ejpam-5976	15	28	binary	binary	ADJ
ejpam-5976	15	29	operation	operation	NOUN
ejpam-5976	15	30	.	.	PUNCT
ejpam-5976	16	1	that	that	PRON
ejpam-5976	16	2	is	be	AUX
ejpam-5976	16	3	,	,	PUNCT
ejpam-5976	16	4	for	for	ADP
ejpam-5976	16	5	any	any	DET
ejpam-5976	16	6	(	(	PUNCT
ejpam-5976	16	7	s1	s1	NOUN
ejpam-5976	16	8	,	,	PUNCT
ejpam-5976	16	9	s2	s2	PROPN
ejpam-5976	16	10	)	)	PUNCT
ejpam-5976	16	11	,	,	PUNCT
ejpam-5976	16	12	(	(	PUNCT
ejpam-5976	16	13	t1	t1	NOUN
ejpam-5976	16	14	,	,	PUNCT
ejpam-5976	16	15	t2	t2	NOUN
ejpam-5976	16	16	)	)	PUNCT
ejpam-5976	16	17	∈	∈	PROPN
ejpam-5976	16	18	s1	s1	PROPN
ejpam-5976	16	19	×	×	PROPN
ejpam-5976	16	20	s2	s2	PROPN
ejpam-5976	16	21	,	,	PUNCT
ejpam-5976	16	22	we	we	PRON
ejpam-5976	16	23	define	define	VERB
ejpam-5976	16	24	(	(	PUNCT
ejpam-5976	16	25	s1	s1	NOUN
ejpam-5976	16	26	,	,	PUNCT
ejpam-5976	16	27	s2)(t1	s2)(t1	NOUN
ejpam-5976	16	28	,	,	PUNCT
ejpam-5976	16	29	t2	t2	NOUN
ejpam-5976	16	30	)	)	PUNCT
ejpam-5976	16	31	=	=	PUNCT
ejpam-5976	16	32	(	(	PUNCT
ejpam-5976	16	33	s1t1	s1t1	ADJ
ejpam-5976	16	34	,	,	PUNCT
ejpam-5976	16	35	s2t2	s2t2	PROPN
ejpam-5976	16	36	)	)	PUNCT
ejpam-5976	16	37	.	.	PUNCT
ejpam-5976	17	1	let	let	VERB
ejpam-5976	17	2	(	(	PUNCT
ejpam-5976	17	3	a	a	PRON
ejpam-5976	17	4	,	,	PUNCT
ejpam-5976	17	5	b	b	NOUN
ejpam-5976	17	6	)	)	PUNCT
ejpam-5976	17	7	∈	∈	PROPN
ejpam-5976	17	8	s1	s1	PROPN
ejpam-5976	17	9	×	×	PROPN
ejpam-5976	17	10	s2	s2	PROPN
ejpam-5976	17	11	.	.	PUNCT
ejpam-5976	18	1	fabrici	fabrici	PROPN
ejpam-5976	18	2	investigated	investigate	VERB
ejpam-5976	18	3	the	the	DET
ejpam-5976	18	4	structure	structure	NOUN
ejpam-5976	18	5	of	of	ADP
ejpam-5976	18	6	l	l	NOUN
ejpam-5976	18	7	-	-	NOUN
ejpam-5976	18	8	classes	class	NOUN
ejpam-5976	18	9	in	in	ADP
ejpam-5976	18	10	the	the	DET
ejpam-5976	18	11	direct	direct	ADJ
ejpam-5976	18	12	product	product	NOUN
ejpam-5976	18	13	of	of	ADP
ejpam-5976	18	14	s1	s1	NOUN
ejpam-5976	18	15	and	and	CCONJ
ejpam-5976	18	16	s2	s2	VERB
ejpam-5976	19	1	[	[	X
ejpam-5976	19	2	3	3	NUM
ejpam-5976	19	3	]	]	PUNCT
ejpam-5976	19	4	.	.	PUNCT
ejpam-5976	20	1	the	the	DET
ejpam-5976	20	2	author	author	NOUN
ejpam-5976	20	3	showed	show	VERB
ejpam-5976	20	4	that	that	SCONJ
ejpam-5976	20	5	l(a	l(a	PROPN
ejpam-5976	20	6	)	)	PUNCT
ejpam-5976	20	7	×	×	PROPN
ejpam-5976	20	8	l(b	l(b	PROPN
ejpam-5976	20	9	)	)	PUNCT
ejpam-5976	20	10	need	need	AUX
ejpam-5976	20	11	not	not	PART
ejpam-5976	20	12	be	be	AUX
ejpam-5976	20	13	equal	equal	ADJ
ejpam-5976	20	14	to	to	ADP
ejpam-5976	20	15	l((a	l((a	NOUN
ejpam-5976	20	16	,	,	PUNCT
ejpam-5976	20	17	b	b	NOUN
ejpam-5976	20	18	)	)	PUNCT
ejpam-5976	20	19	)	)	PUNCT
ejpam-5976	20	20	and	and	CCONJ
ejpam-5976	20	21	give	give	VERB
ejpam-5976	20	22	the	the	DET
ejpam-5976	20	23	conditions	condition	NOUN
ejpam-5976	20	24	when	when	SCONJ
ejpam-5976	20	25	l(a	l(a	PROPN
ejpam-5976	20	26	)	)	PUNCT
ejpam-5976	20	27	×	×	PROPN
ejpam-5976	20	28	l(b	l(b	PROPN
ejpam-5976	20	29	)	)	PUNCT
ejpam-5976	20	30	=	=	SYM
ejpam-5976	20	31	l((a	l((a	PROPN
ejpam-5976	20	32	,	,	PUNCT
ejpam-5976	20	33	b	b	NOUN
ejpam-5976	20	34	)	)	PUNCT
ejpam-5976	20	35	)	)	PUNCT
ejpam-5976	20	36	.	.	PUNCT
ejpam-5976	21	1	fabrici	fabrici	PROPN
ejpam-5976	21	2	also	also	ADV
ejpam-5976	21	3	investigated	investigate	VERB
ejpam-5976	21	4	the	the	DET
ejpam-5976	21	5	conditions	condition	NOUN
ejpam-5976	21	6	under	under	ADP
ejpam-5976	21	7	which	which	PRON
ejpam-5976	21	8	the	the	DET
ejpam-5976	21	9	direct	direct	ADJ
ejpam-5976	21	10	product	product	NOUN
ejpam-5976	21	11	of	of	ADP
ejpam-5976	21	12	l	l	NOUN
ejpam-5976	21	13	-	-	PUNCT
ejpam-5976	21	14	classes	class	NOUN
ejpam-5976	21	15	la	la	INTJ
ejpam-5976	21	16	×	×	NOUN
ejpam-5976	21	17	lb	lb	INTJ
ejpam-5976	21	18	is	be	AUX
ejpam-5976	21	19	an	an	DET
ejpam-5976	21	20	l	l	ADJ
ejpam-5976	21	21	-	-	PUNCT
ejpam-5976	21	22	class	class	NOUN
ejpam-5976	21	23	l(a	l(a	PROPN
ejpam-5976	21	24	,	,	PUNCT
ejpam-5976	21	25	b	b	NOUN
ejpam-5976	21	26	)	)	PUNCT
ejpam-5976	21	27	.	.	PUNCT
ejpam-5976	22	1	conditions	condition	NOUN
ejpam-5976	22	2	for	for	ADP
ejpam-5976	22	3	the	the	DET
ejpam-5976	22	4	direct	direct	ADJ
ejpam-5976	22	5	product	product	NOUN
ejpam-5976	22	6	of	of	ADP
ejpam-5976	22	7	j	j	PROPN
ejpam-5976	22	8	-classes	-classes	PROPN
ejpam-5976	22	9	ja	ja	PROPN
ejpam-5976	22	10	×	×	NOUN
ejpam-5976	22	11	jb	jb	PROPN
ejpam-5976	22	12	=	=	SYM
ejpam-5976	22	13	j(a	j(a	PROPN
ejpam-5976	22	14	,	,	PUNCT
ejpam-5976	22	15	b	b	NOUN
ejpam-5976	22	16	)	)	PUNCT
ejpam-5976	22	17	were	be	AUX
ejpam-5976	22	18	investigated	investigate	VERB
ejpam-5976	22	19	as	as	ADV
ejpam-5976	22	20	well	well	ADV
ejpam-5976	22	21	[	[	X
ejpam-5976	22	22	4	4	NUM
ejpam-5976	22	23	]	]	PUNCT
ejpam-5976	22	24	.	.	PUNCT
ejpam-5976	23	1	∗corresponding	∗corresponde	VERB
ejpam-5976	23	2	author	author	NOUN
ejpam-5976	23	3	.	.	PUNCT
ejpam-5976	24	1	doi	doi	NOUN
ejpam-5976	24	2	:	:	PUNCT
ejpam-5976	24	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5976	https://doi.org/10.29020/nybg.ejpam.v18i2.5976	NOUN
ejpam-5976	24	4	email	email	NOUN
ejpam-5976	24	5	addresses	address	NOUN
ejpam-5976	24	6	:	:	PUNCT
ejpam-5976	24	7	panulu@kku.ac.th	panulu@kku.ac.th	PROPN
ejpam-5976	24	8	(	(	PUNCT
ejpam-5976	24	9	p.	p.	NOUN
ejpam-5976	24	10	luangchaisri	luangchaisri	PROPN
ejpam-5976	24	11	)	)	PUNCT
ejpam-5976	24	12	,	,	PUNCT
ejpam-5976	24	13	ontimapa@kkumail.com	ontimapa@kkumail.com	X
ejpam-5976	24	14	(	(	PUNCT
ejpam-5976	24	15	o.	o.	PROPN
ejpam-5976	24	16	pankoon	pankoon	PROPN
ejpam-5976	24	17	)	)	PUNCT
ejpam-5976	24	18	,	,	PUNCT
ejpam-5976	24	19	thacha@kku.ac.th	thacha@kku.ac.th	NOUN
ejpam-5976	24	20	(	(	PUNCT
ejpam-5976	24	21	t.	t.	NOUN
ejpam-5976	24	22	changphas	changphas	PROPN
ejpam-5976	24	23	)	)	PUNCT
ejpam-5976	24	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5976	24	25	1	1	NUM
ejpam-5976	24	26	copyright	copyright	NOUN
ejpam-5976	24	27	:	:	PUNCT
ejpam-5976	25	1	©	©	PROPN
ejpam-5976	25	2	2025	2025	NUM
ejpam-5976	25	3	the	the	DET
ejpam-5976	25	4	author(s	author(s	NOUN
ejpam-5976	25	5	)	)	PUNCT
ejpam-5976	25	6	.	.	PUNCT
ejpam-5976	26	1	(	(	PUNCT
ejpam-5976	26	2	cc	cc	NOUN
ejpam-5976	26	3	by	by	ADP
ejpam-5976	26	4	-	-	PUNCT
ejpam-5976	26	5	nc	nc	PROPN
ejpam-5976	26	6	4.0	4.0	NUM
ejpam-5976	26	7	)	)	PUNCT
ejpam-5976	27	1	p.	p.	NOUN
ejpam-5976	27	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	27	3	,	,	PUNCT
ejpam-5976	27	4	o.	o.	PROPN
ejpam-5976	27	5	pankoon	pankoon	NOUN
ejpam-5976	27	6	,	,	PUNCT
ejpam-5976	27	7	t.	t.	PROPN
ejpam-5976	27	8	changphas	changphas	PROPN
ejpam-5976	27	9	/	/	SYM
ejpam-5976	27	10	eur	eur	PROPN
ejpam-5976	27	11	.	.	PUNCT
ejpam-5976	28	1	j.	j.	PROPN
ejpam-5976	28	2	pure	pure	PROPN
ejpam-5976	28	3	appl	appl	PROPN
ejpam-5976	28	4	.	.	PROPN
ejpam-5976	28	5	math	math	PROPN
ejpam-5976	28	6	,	,	PUNCT
ejpam-5976	28	7	18	18	NUM
ejpam-5976	28	8	(	(	PUNCT
ejpam-5976	28	9	2	2	NUM
ejpam-5976	28	10	)	)	PUNCT
ejpam-5976	28	11	(	(	PUNCT
ejpam-5976	28	12	2025	2025	NUM
ejpam-5976	28	13	)	)	PUNCT
ejpam-5976	28	14	,	,	PUNCT
ejpam-5976	28	15	5976	5976	NUM
ejpam-5976	28	16	2	2	NUM
ejpam-5976	28	17	of	of	ADP
ejpam-5976	28	18	14	14	NUM
ejpam-5976	28	19	let	let	VERB
ejpam-5976	28	20	m	m	PRON
ejpam-5976	28	21	,	,	PUNCT
ejpam-5976	28	22	n	n	X
ejpam-5976	28	23	be	be	AUX
ejpam-5976	28	24	nonnegative	nonnegative	ADJ
ejpam-5976	28	25	integers	integer	NOUN
ejpam-5976	28	26	.	.	PUNCT
ejpam-5976	29	1	a	a	DET
ejpam-5976	29	2	subsemigroup	subsemigroup	NOUN
ejpam-5976	29	3	a	a	PRON
ejpam-5976	29	4	of	of	ADP
ejpam-5976	29	5	s	s	PRON
ejpam-5976	29	6	is	be	AUX
ejpam-5976	29	7	called	call	VERB
ejpam-5976	29	8	an	an	DET
ejpam-5976	29	9	(	(	PUNCT
ejpam-5976	29	10	m	m	PROPN
ejpam-5976	29	11	,	,	PUNCT
ejpam-5976	29	12	n)-ideal	n)-ideal	NOUN
ejpam-5976	29	13	of	of	ADP
ejpam-5976	29	14	s	s	PRON
ejpam-5976	29	15	if	if	SCONJ
ejpam-5976	29	16	amsan	amsan	ADJ
ejpam-5976	29	17	⊆	⊆	NUM
ejpam-5976	29	18	a	a	PRON
ejpam-5976	30	1	[	[	X
ejpam-5976	30	2	5	5	NUM
ejpam-5976	30	3	]	]	PUNCT
ejpam-5976	30	4	.	.	PUNCT
ejpam-5976	31	1	here	here	ADV
ejpam-5976	31	2	,	,	PUNCT
ejpam-5976	31	3	a0s	a0s	PROPN
ejpam-5976	31	4	=	=	PUNCT
ejpam-5976	31	5	sa0	sa0	NOUN
ejpam-5976	31	6	=	=	PUNCT
ejpam-5976	31	7	s.	s.	PROPN
ejpam-5976	31	8	the	the	DET
ejpam-5976	31	9	smallest	small	ADJ
ejpam-5976	31	10	(	(	PUNCT
ejpam-5976	31	11	m	m	PROPN
ejpam-5976	31	12	,	,	PUNCT
ejpam-5976	31	13	n)-ideal	n)-ideal	NOUN
ejpam-5976	31	14	of	of	ADP
ejpam-5976	31	15	s	s	AUX
ejpam-5976	31	16	containing	contain	VERB
ejpam-5976	31	17	a	a	PRON
ejpam-5976	31	18	is	be	AUX
ejpam-5976	31	19	denoted	denote	VERB
ejpam-5976	31	20	by	by	ADP
ejpam-5976	31	21	[	[	X
ejpam-5976	31	22	a](m	a](m	NOUN
ejpam-5976	31	23	,	,	PUNCT
ejpam-5976	31	24	n	n	CCONJ
ejpam-5976	31	25	)	)	PUNCT
ejpam-5976	31	26	.	.	PUNCT
ejpam-5976	32	1	luangchaisri	luangchaisri	VERB
ejpam-5976	32	2	and	and	CCONJ
ejpam-5976	32	3	changphas	changpha	VERB
ejpam-5976	32	4	provided	provide	VERB
ejpam-5976	32	5	necessary	necessary	ADJ
ejpam-5976	32	6	and	and	CCONJ
ejpam-5976	32	7	sufficient	sufficient	ADJ
ejpam-5976	32	8	conditions	condition	NOUN
ejpam-5976	32	9	for	for	ADP
ejpam-5976	32	10	[	[	X
ejpam-5976	32	11	a](m	a](m	NOUN
ejpam-5976	32	12	,	,	PUNCT
ejpam-5976	32	13	n	n	CCONJ
ejpam-5976	32	14	)	)	PUNCT
ejpam-5976	32	15	×	×	NOUN
ejpam-5976	33	1	[	[	X
ejpam-5976	33	2	b](m	b](m	NOUN
ejpam-5976	33	3	,	,	PUNCT
ejpam-5976	33	4	n	n	CCONJ
ejpam-5976	33	5	)	)	PUNCT
ejpam-5976	33	6	=	=	SYM
ejpam-5976	34	1	[	[	X
ejpam-5976	34	2	(	(	PUNCT
ejpam-5976	34	3	a	a	PRON
ejpam-5976	34	4	,	,	PUNCT
ejpam-5976	34	5	b)](m	b)](m	NOUN
ejpam-5976	34	6	,	,	PUNCT
ejpam-5976	34	7	n	n	CCONJ
ejpam-5976	34	8	)	)	PUNCT
ejpam-5976	35	1	[	[	X
ejpam-5976	35	2	6	6	NUM
ejpam-5976	35	3	]	]	PUNCT
ejpam-5976	35	4	.	.	PUNCT
ejpam-5976	36	1	moreover	moreover	ADV
ejpam-5976	36	2	,	,	PUNCT
ejpam-5976	36	3	they	they	PRON
ejpam-5976	36	4	determined	determine	VERB
ejpam-5976	36	5	an	an	DET
ejpam-5976	36	6	equivalence	equivalence	NOUN
ejpam-5976	36	7	class	class	NOUN
ejpam-5976	36	8	on	on	ADP
ejpam-5976	36	9	a	a	DET
ejpam-5976	36	10	semigroup	semigroup	NOUN
ejpam-5976	36	11	s	s	NOUN
ejpam-5976	36	12	by	by	ADP
ejpam-5976	36	13	for	for	ADP
ejpam-5976	36	14	any	any	DET
ejpam-5976	36	15	x	x	SYM
ejpam-5976	36	16	∈	∈	PROPN
ejpam-5976	36	17	s	s	NOUN
ejpam-5976	36	18	,	,	PUNCT
ejpam-5976	36	19	j(m	j(m	PROPN
ejpam-5976	36	20	,	,	PUNCT
ejpam-5976	36	21	n),x	n),x	PROPN
ejpam-5976	36	22	=	=	PUNCT
ejpam-5976	36	23	{	{	PUNCT
ejpam-5976	36	24	y	y	PROPN
ejpam-5976	36	25	∈	∈	PROPN
ejpam-5976	36	26	s	s	VERB
ejpam-5976	36	27	|	|	ADV
ejpam-5976	36	28	[	[	X
ejpam-5976	36	29	x](m	x](m	NOUN
ejpam-5976	36	30	,	,	PUNCT
ejpam-5976	36	31	n	n	CCONJ
ejpam-5976	36	32	)	)	PUNCT
ejpam-5976	36	33	=	=	NOUN
ejpam-5976	37	1	[	[	X
ejpam-5976	37	2	y](m	y](m	NOUN
ejpam-5976	37	3	,	,	PUNCT
ejpam-5976	37	4	n	n	CCONJ
ejpam-5976	37	5	)	)	PUNCT
ejpam-5976	37	6	}	}	PUNCT
ejpam-5976	37	7	and	and	CCONJ
ejpam-5976	37	8	provided	provide	VERB
ejpam-5976	37	9	the	the	DET
ejpam-5976	37	10	condition	condition	NOUN
ejpam-5976	37	11	for	for	ADP
ejpam-5976	37	12	j(m	j(m	PROPN
ejpam-5976	37	13	,	,	PUNCT
ejpam-5976	37	14	n),a	n),a	PROPN
ejpam-5976	37	15	×	×	PROPN
ejpam-5976	37	16	j(m	j(m	PROPN
ejpam-5976	37	17	,	,	PUNCT
ejpam-5976	37	18	n),b	n),b	PROPN
ejpam-5976	37	19	=	=	SYM
ejpam-5976	37	20	j(m	j(m	PROPN
ejpam-5976	37	21	,	,	PUNCT
ejpam-5976	37	22	n),(a	n),(a	ADJ
ejpam-5976	37	23	,	,	PUNCT
ejpam-5976	37	24	b	b	NOUN
ejpam-5976	37	25	)	)	PUNCT
ejpam-5976	37	26	.	.	PUNCT
ejpam-5976	38	1	in	in	ADP
ejpam-5976	38	2	this	this	DET
ejpam-5976	38	3	paper	paper	NOUN
ejpam-5976	38	4	,	,	PUNCT
ejpam-5976	38	5	we	we	PRON
ejpam-5976	38	6	show	show	VERB
ejpam-5976	38	7	that	that	SCONJ
ejpam-5976	38	8	the	the	DET
ejpam-5976	38	9	direct	direct	ADJ
ejpam-5976	38	10	product	product	NOUN
ejpam-5976	38	11	q(a	q(a	NOUN
ejpam-5976	38	12	)	)	PUNCT
ejpam-5976	38	13	×q(b	×q(b	PROPN
ejpam-5976	38	14	)	)	PUNCT
ejpam-5976	38	15	need	need	VERB
ejpam-5976	38	16	not	not	PART
ejpam-5976	38	17	to	to	PART
ejpam-5976	38	18	be	be	AUX
ejpam-5976	38	19	q((a	q((a	PROPN
ejpam-5976	38	20	,	,	PUNCT
ejpam-5976	38	21	b	b	NOUN
ejpam-5976	38	22	)	)	PUNCT
ejpam-5976	38	23	)	)	PUNCT
ejpam-5976	38	24	.	.	PUNCT
ejpam-5976	39	1	in	in	ADP
ejpam-5976	39	2	addition	addition	NOUN
ejpam-5976	39	3	,	,	PUNCT
ejpam-5976	39	4	we	we	PRON
ejpam-5976	39	5	provide	provide	VERB
ejpam-5976	39	6	necessary	necessary	ADJ
ejpam-5976	39	7	and	and	CCONJ
ejpam-5976	39	8	sufficient	sufficient	ADJ
ejpam-5976	39	9	conditions	condition	NOUN
ejpam-5976	39	10	when	when	SCONJ
ejpam-5976	39	11	q(a)×q(b	q(a)×q(b	ADV
ejpam-5976	39	12	)	)	PUNCT
ejpam-5976	39	13	=	=	SYM
ejpam-5976	39	14	q((a	q((a	PROPN
ejpam-5976	39	15	,	,	PUNCT
ejpam-5976	39	16	b	b	NOUN
ejpam-5976	39	17	)	)	PUNCT
ejpam-5976	39	18	)	)	PUNCT
ejpam-5976	39	19	and	and	CCONJ
ejpam-5976	39	20	the	the	DET
ejpam-5976	39	21	condition	condition	NOUN
ejpam-5976	39	22	when	when	SCONJ
ejpam-5976	39	23	ha	ha	INTJ
ejpam-5976	39	24	×hb	×hb	PROPN
ejpam-5976	39	25	=	=	PUNCT
ejpam-5976	39	26	h(a	h(a	PROPN
ejpam-5976	39	27	,	,	PUNCT
ejpam-5976	39	28	b	b	NOUN
ejpam-5976	39	29	)	)	PUNCT
ejpam-5976	39	30	.	.	PUNCT
ejpam-5976	40	1	2	2	X
ejpam-5976	40	2	.	.	X
ejpam-5976	40	3	main	main	ADJ
ejpam-5976	40	4	results	result	NOUN
ejpam-5976	40	5	let	let	VERB
ejpam-5976	40	6	s	s	PRON
ejpam-5976	40	7	be	be	AUX
ejpam-5976	40	8	a	a	DET
ejpam-5976	40	9	semigroup	semigroup	NOUN
ejpam-5976	40	10	.	.	PUNCT
ejpam-5976	41	1	for	for	ADP
ejpam-5976	41	2	nonempty	nonempty	NOUN
ejpam-5976	41	3	subsets	subset	NOUN
ejpam-5976	41	4	a	a	DET
ejpam-5976	41	5	,	,	PUNCT
ejpam-5976	41	6	b	b	PROPN
ejpam-5976	41	7	of	of	ADP
ejpam-5976	41	8	s	s	PROPN
ejpam-5976	41	9	,	,	PUNCT
ejpam-5976	41	10	the	the	DET
ejpam-5976	41	11	set	set	ADJ
ejpam-5976	41	12	product	product	NOUN
ejpam-5976	41	13	of	of	ADP
ejpam-5976	41	14	a	a	PRON
ejpam-5976	41	15	and	and	CCONJ
ejpam-5976	41	16	b	b	NOUN
ejpam-5976	41	17	,	,	PUNCT
ejpam-5976	41	18	denoted	denote	VERB
ejpam-5976	41	19	by	by	ADP
ejpam-5976	41	20	ab	ab	PROPN
ejpam-5976	41	21	,	,	PUNCT
ejpam-5976	41	22	is	be	AUX
ejpam-5976	41	23	defined	define	VERB
ejpam-5976	41	24	by	by	ADP
ejpam-5976	41	25	ab	ab	PROPN
ejpam-5976	41	26	=	=	PUNCT
ejpam-5976	41	27	{	{	PUNCT
ejpam-5976	41	28	ab	ab	PROPN
ejpam-5976	41	29	|	|	ADV
ejpam-5976	41	30	a	a	DET
ejpam-5976	41	31	∈	∈	PROPN
ejpam-5976	41	32	a	a	PRON
ejpam-5976	41	33	and	and	CCONJ
ejpam-5976	41	34	b	b	NOUN
ejpam-5976	41	35	∈	∈	PROPN
ejpam-5976	41	36	b	b	NOUN
ejpam-5976	41	37	}	}	PUNCT
ejpam-5976	41	38	.	.	PUNCT
ejpam-5976	42	1	in	in	ADP
ejpam-5976	42	2	particular	particular	ADJ
ejpam-5976	42	3	,	,	PUNCT
ejpam-5976	42	4	we	we	PRON
ejpam-5976	42	5	write	write	VERB
ejpam-5976	42	6	ab	ab	PROPN
ejpam-5976	42	7	instead	instead	ADV
ejpam-5976	42	8	of	of	ADP
ejpam-5976	42	9	a{b	a{b	PROPN
ejpam-5976	42	10	}	}	PUNCT
ejpam-5976	42	11	and	and	CCONJ
ejpam-5976	42	12	write	write	VERB
ejpam-5976	42	13	ab	ab	PROPN
ejpam-5976	42	14	instead	instead	ADV
ejpam-5976	42	15	of	of	ADV
ejpam-5976	42	16	{	{	PUNCT
ejpam-5976	42	17	a}b	a}b	PROPN
ejpam-5976	42	18	.	.	PUNCT
ejpam-5976	43	1	definition	definition	NOUN
ejpam-5976	43	2	1	1	NUM
ejpam-5976	43	3	.	.	PUNCT
ejpam-5976	44	1	a	a	DET
ejpam-5976	44	2	nonempty	nonempty	NOUN
ejpam-5976	44	3	subset	subset	VERB
ejpam-5976	44	4	q	q	NOUN
ejpam-5976	44	5	of	of	ADP
ejpam-5976	44	6	a	a	DET
ejpam-5976	44	7	semigroup	semigroup	NOUN
ejpam-5976	44	8	s	s	PART
ejpam-5976	44	9	is	be	AUX
ejpam-5976	44	10	called	call	VERB
ejpam-5976	44	11	a	a	DET
ejpam-5976	44	12	quasi	quasi	NOUN
ejpam-5976	44	13	-	-	NOUN
ejpam-5976	44	14	ideal	ideal	ADJ
ejpam-5976	44	15	of	of	ADP
ejpam-5976	44	16	s	s	PRON
ejpam-5976	44	17	if	if	SCONJ
ejpam-5976	44	18	qs	qs	ADP
ejpam-5976	44	19	∩	∩	NOUN
ejpam-5976	44	20	sq	sq	PROPN
ejpam-5976	44	21	⊆	⊆	NUM
ejpam-5976	44	22	q.	q.	NOUN
ejpam-5976	44	23	definition	definition	NOUN
ejpam-5976	44	24	2	2	NUM
ejpam-5976	44	25	.	.	PUNCT
ejpam-5976	45	1	let	let	VERB
ejpam-5976	45	2	s	s	PRON
ejpam-5976	45	3	be	be	AUX
ejpam-5976	45	4	a	a	DET
ejpam-5976	45	5	semigroup	semigroup	NOUN
ejpam-5976	45	6	and	and	CCONJ
ejpam-5976	45	7	a	a	DET
ejpam-5976	45	8	∈	∈	PROPN
ejpam-5976	45	9	s.	s.	PROPN
ejpam-5976	45	10	the	the	DET
ejpam-5976	45	11	smallest	small	ADJ
ejpam-5976	45	12	quasi	quasi	NOUN
ejpam-5976	45	13	-	-	NOUN
ejpam-5976	45	14	ideal	ideal	ADJ
ejpam-5976	45	15	of	of	ADP
ejpam-5976	45	16	s	s	AUX
ejpam-5976	45	17	containing	contain	VERB
ejpam-5976	45	18	a	a	PRON
ejpam-5976	45	19	,	,	PUNCT
ejpam-5976	45	20	denoted	denote	VERB
ejpam-5976	45	21	by	by	ADP
ejpam-5976	45	22	q(a	q(a	NOUN
ejpam-5976	45	23	)	)	PUNCT
ejpam-5976	45	24	,	,	PUNCT
ejpam-5976	45	25	is	be	AUX
ejpam-5976	45	26	called	call	VERB
ejpam-5976	45	27	the	the	DET
ejpam-5976	45	28	principal	principal	ADJ
ejpam-5976	45	29	quasi	quasi	NOUN
ejpam-5976	45	30	-	-	NOUN
ejpam-5976	45	31	ideal	ideal	ADJ
ejpam-5976	45	32	of	of	ADP
ejpam-5976	45	33	s	s	AUX
ejpam-5976	45	34	generated	generate	VERB
ejpam-5976	45	35	by	by	ADP
ejpam-5976	45	36	a.	a.	NOUN
ejpam-5976	45	37	remark	remark	PROPN
ejpam-5976	45	38	1	1	NUM
ejpam-5976	45	39	.	.	PUNCT
ejpam-5976	46	1	let	let	VERB
ejpam-5976	46	2	a	a	DET
ejpam-5976	46	3	∈	∈	PROPN
ejpam-5976	46	4	s.	s.	PROPN
ejpam-5976	46	5	then	then	ADV
ejpam-5976	46	6	q(a	q(a	PROPN
ejpam-5976	46	7	)	)	PUNCT
ejpam-5976	47	1	=	=	PRON
ejpam-5976	47	2	{	{	PUNCT
ejpam-5976	47	3	a	a	DET
ejpam-5976	47	4	}	}	PUNCT
ejpam-5976	47	5	∪	∪	NOUN
ejpam-5976	47	6	(	(	PUNCT
ejpam-5976	47	7	as	as	ADP
ejpam-5976	47	8	∩	∩	ADJ
ejpam-5976	47	9	sa	sa	PROPN
ejpam-5976	47	10	)	)	PUNCT
ejpam-5976	47	11	.	.	PUNCT
ejpam-5976	48	1	theorem	theorem	NOUN
ejpam-5976	48	2	1	1	NUM
ejpam-5976	48	3	.	.	PUNCT
ejpam-5976	49	1	if	if	SCONJ
ejpam-5976	49	2	q1	q1	PROPN
ejpam-5976	49	3	and	and	CCONJ
ejpam-5976	49	4	q2	q2	NOUN
ejpam-5976	49	5	are	be	AUX
ejpam-5976	49	6	quasi	quasi	NOUN
ejpam-5976	49	7	-	-	NOUN
ejpam-5976	49	8	ideals	ideal	NOUN
ejpam-5976	49	9	of	of	ADP
ejpam-5976	49	10	semigroups	semigroup	NOUN
ejpam-5976	49	11	s1	s1	PROPN
ejpam-5976	49	12	and	and	CCONJ
ejpam-5976	49	13	s2	s2	PROPN
ejpam-5976	49	14	,	,	PUNCT
ejpam-5976	49	15	respectively	respectively	ADV
ejpam-5976	49	16	,	,	PUNCT
ejpam-5976	49	17	then	then	ADV
ejpam-5976	49	18	q1	q1	PROPN
ejpam-5976	49	19	×q2	×q2	PROPN
ejpam-5976	49	20	is	be	AUX
ejpam-5976	49	21	a	a	DET
ejpam-5976	49	22	quasi	quasi	NOUN
ejpam-5976	49	23	-	-	NOUN
ejpam-5976	49	24	ideal	ideal	NOUN
ejpam-5976	49	25	of	of	ADP
ejpam-5976	49	26	s1	s1	PROPN
ejpam-5976	49	27	×	×	PROPN
ejpam-5976	49	28	s2	s2	PROPN
ejpam-5976	49	29	.	.	PUNCT
ejpam-5976	50	1	proof	proof	NOUN
ejpam-5976	50	2	.	.	PUNCT
ejpam-5976	51	1	assume	assume	VERB
ejpam-5976	51	2	that	that	SCONJ
ejpam-5976	51	3	q1	q1	PROPN
ejpam-5976	51	4	and	and	CCONJ
ejpam-5976	51	5	q2	q2	NOUN
ejpam-5976	51	6	be	be	VERB
ejpam-5976	51	7	quasi	quasi	NOUN
ejpam-5976	51	8	-	-	NOUN
ejpam-5976	51	9	ideals	ideal	NOUN
ejpam-5976	51	10	of	of	ADP
ejpam-5976	51	11	s1	s1	NOUN
ejpam-5976	51	12	and	and	CCONJ
ejpam-5976	51	13	s2	s2	PROPN
ejpam-5976	51	14	,	,	PUNCT
ejpam-5976	51	15	respectively	respectively	ADV
ejpam-5976	51	16	.	.	PUNCT
ejpam-5976	52	1	then	then	ADV
ejpam-5976	52	2	(	(	PUNCT
ejpam-5976	52	3	q1	q1	VERB
ejpam-5976	52	4	×q2)(s1	×q2)(s1	NOUN
ejpam-5976	52	5	×	×	PROPN
ejpam-5976	52	6	s2	s2	PROPN
ejpam-5976	52	7	)	)	PUNCT
ejpam-5976	52	8	∩	∩	NOUN
ejpam-5976	52	9	(	(	PUNCT
ejpam-5976	52	10	s1	s1	PROPN
ejpam-5976	52	11	×	×	PROPN
ejpam-5976	52	12	s2)(q1	s2)(q1	PROPN
ejpam-5976	52	13	×q2	×q2	PROPN
ejpam-5976	52	14	)	)	PUNCT
ejpam-5976	52	15	=	=	SYM
ejpam-5976	52	16	(	(	PUNCT
ejpam-5976	52	17	q1s1	q1s1	PROPN
ejpam-5976	52	18	×q2s2	×q2s2	NOUN
ejpam-5976	52	19	)	)	PUNCT
ejpam-5976	52	20	∩	∩	NOUN
ejpam-5976	52	21	(	(	PUNCT
ejpam-5976	52	22	s1q1	s1q1	PROPN
ejpam-5976	52	23	×	×	PROPN
ejpam-5976	52	24	s2q2	s2q2	NOUN
ejpam-5976	52	25	)	)	PUNCT
ejpam-5976	52	26	=	=	SYM
ejpam-5976	52	27	(	(	PUNCT
ejpam-5976	52	28	q1s1	q1s1	PROPN
ejpam-5976	52	29	∩	∩	PROPN
ejpam-5976	52	30	s1q1)×	s1q1)×	PROPN
ejpam-5976	52	31	(	(	PUNCT
ejpam-5976	52	32	q2s2	q2s2	PROPN
ejpam-5976	52	33	∩	∩	ADJ
ejpam-5976	52	34	s2q2	s2q2	NOUN
ejpam-5976	52	35	)	)	PUNCT
ejpam-5976	52	36	⊆	⊆	NUM
ejpam-5976	52	37	q1	q1	NOUN
ejpam-5976	52	38	×q2	×q2	PROPN
ejpam-5976	52	39	.	.	PUNCT
ejpam-5976	53	1	therefore	therefore	ADV
ejpam-5976	53	2	,	,	PUNCT
ejpam-5976	53	3	q1	q1	PROPN
ejpam-5976	53	4	×q2	×q2	PROPN
ejpam-5976	53	5	is	be	AUX
ejpam-5976	53	6	a	a	DET
ejpam-5976	53	7	quasi	quasi	NOUN
ejpam-5976	53	8	-	-	NOUN
ejpam-5976	53	9	ideal	ideal	NOUN
ejpam-5976	53	10	of	of	ADP
ejpam-5976	53	11	s1	s1	PROPN
ejpam-5976	53	12	×	×	PROPN
ejpam-5976	53	13	s2	s2	PROPN
ejpam-5976	53	14	.	.	PUNCT
ejpam-5976	54	1	the	the	DET
ejpam-5976	54	2	above	above	ADJ
ejpam-5976	54	3	theorem	theorem	ADJ
ejpam-5976	54	4	states	state	NOUN
ejpam-5976	54	5	that	that	SCONJ
ejpam-5976	54	6	the	the	DET
ejpam-5976	54	7	direct	direct	ADJ
ejpam-5976	54	8	product	product	NOUN
ejpam-5976	54	9	of	of	ADP
ejpam-5976	54	10	two	two	NUM
ejpam-5976	54	11	quasi	quasi	NOUN
ejpam-5976	54	12	-	-	NOUN
ejpam-5976	54	13	ideals	ideal	NOUN
ejpam-5976	54	14	is	be	AUX
ejpam-5976	54	15	a	a	DET
ejpam-5976	54	16	quasi	quasi	NOUN
ejpam-5976	54	17	-	-	NOUN
ejpam-5976	54	18	ideal	ideal	ADJ
ejpam-5976	54	19	.	.	PUNCT
ejpam-5976	55	1	however	however	ADV
ejpam-5976	55	2	,	,	PUNCT
ejpam-5976	55	3	it	it	PRON
ejpam-5976	55	4	is	be	AUX
ejpam-5976	55	5	important	important	ADJ
ejpam-5976	55	6	to	to	PART
ejpam-5976	55	7	note	note	VERB
ejpam-5976	55	8	that	that	SCONJ
ejpam-5976	55	9	the	the	DET
ejpam-5976	55	10	direct	direct	ADJ
ejpam-5976	55	11	product	product	NOUN
ejpam-5976	55	12	of	of	ADP
ejpam-5976	55	13	two	two	NUM
ejpam-5976	55	14	principal	principal	ADJ
ejpam-5976	55	15	quasi	quasi	NOUN
ejpam-5976	55	16	-	-	NOUN
ejpam-5976	55	17	ideals	ideal	NOUN
ejpam-5976	55	18	does	do	AUX
ejpam-5976	55	19	not	not	PART
ejpam-5976	55	20	necessarily	necessarily	ADV
ejpam-5976	55	21	result	result	VERB
ejpam-5976	55	22	in	in	ADP
ejpam-5976	55	23	a	a	DET
ejpam-5976	55	24	principal	principal	ADJ
ejpam-5976	55	25	quasi	quasi	NOUN
ejpam-5976	55	26	-	-	NOUN
ejpam-5976	55	27	ideal	ideal	ADJ
ejpam-5976	55	28	.	.	PUNCT
ejpam-5976	56	1	this	this	PRON
ejpam-5976	56	2	can	can	AUX
ejpam-5976	56	3	be	be	AUX
ejpam-5976	56	4	seen	see	VERB
ejpam-5976	56	5	in	in	ADP
ejpam-5976	56	6	the	the	DET
ejpam-5976	56	7	following	follow	VERB
ejpam-5976	56	8	example	example	NOUN
ejpam-5976	56	9	:	:	PUNCT
ejpam-5976	56	10	example	example	NOUN
ejpam-5976	57	1	1	1	NUM
ejpam-5976	57	2	.	.	PUNCT
ejpam-5976	58	1	(	(	PUNCT
ejpam-5976	58	2	[	[	X
ejpam-5976	58	3	7	7	NUM
ejpam-5976	58	4	]	]	PUNCT
ejpam-5976	58	5	)	)	PUNCT
ejpam-5976	58	6	consider	consider	VERB
ejpam-5976	58	7	two	two	NUM
ejpam-5976	58	8	semigroups	semigroup	NOUN
ejpam-5976	58	9	(	(	PUNCT
ejpam-5976	58	10	s1	s1	NOUN
ejpam-5976	58	11	,	,	PUNCT
ejpam-5976	58	12	∗	∗	NOUN
ejpam-5976	58	13	)	)	PUNCT
ejpam-5976	58	14	and	and	CCONJ
ejpam-5976	58	15	(	(	PUNCT
ejpam-5976	58	16	s2	s2	PROPN
ejpam-5976	58	17	,	,	PUNCT
ejpam-5976	58	18	·	·	PUNCT
ejpam-5976	58	19	)	)	PUNCT
ejpam-5976	58	20	,	,	PUNCT
ejpam-5976	58	21	where	where	SCONJ
ejpam-5976	58	22	s1	s1	NOUN
ejpam-5976	58	23	=	=	PUNCT
ejpam-5976	58	24	{	{	PUNCT
ejpam-5976	58	25	a1	a1	PROPN
ejpam-5976	58	26	,	,	PUNCT
ejpam-5976	58	27	a2	a2	PROPN
ejpam-5976	58	28	,	,	PUNCT
ejpam-5976	58	29	a3	a3	NOUN
ejpam-5976	58	30	,	,	PUNCT
ejpam-5976	58	31	a4	a4	NOUN
ejpam-5976	58	32	}	}	PUNCT
ejpam-5976	58	33	and	and	CCONJ
ejpam-5976	58	34	s2	s2	VERB
ejpam-5976	58	35	=	=	SYM
ejpam-5976	58	36	{	{	PUNCT
ejpam-5976	58	37	b1	b1	PROPN
ejpam-5976	58	38	,	,	PUNCT
ejpam-5976	58	39	b2	b2	NOUN
ejpam-5976	58	40	,	,	PUNCT
ejpam-5976	58	41	b3	b3	NOUN
ejpam-5976	58	42	,	,	PUNCT
ejpam-5976	58	43	b4	b4	NOUN
ejpam-5976	58	44	}	}	PUNCT
ejpam-5976	58	45	.	.	PUNCT
ejpam-5976	59	1	the	the	DET
ejpam-5976	59	2	binary	binary	PROPN
ejpam-5976	59	3	operation	operation	NOUN
ejpam-5976	59	4	∗	∗	NOUN
ejpam-5976	59	5	on	on	ADP
ejpam-5976	59	6	s1	s1	PROPN
ejpam-5976	59	7	and	and	CCONJ
ejpam-5976	59	8	the	the	DET
ejpam-5976	59	9	binary	binary	PROPN
ejpam-5976	59	10	operation	operation	NOUN
ejpam-5976	59	11	·	·	PUNCT
ejpam-5976	59	12	on	on	ADP
ejpam-5976	59	13	s2	s2	PROPN
ejpam-5976	59	14	are	be	AUX
ejpam-5976	59	15	defined	define	VERB
ejpam-5976	59	16	as	as	SCONJ
ejpam-5976	59	17	follows	follow	VERB
ejpam-5976	59	18	:	:	PUNCT
ejpam-5976	59	19	p.	p.	NOUN
ejpam-5976	59	20	luangchaisri	luangchaisri	VERB
ejpam-5976	59	21	,	,	PUNCT
ejpam-5976	59	22	o.	o.	PROPN
ejpam-5976	59	23	pankoon	pankoon	NOUN
ejpam-5976	59	24	,	,	PUNCT
ejpam-5976	59	25	t.	t.	PROPN
ejpam-5976	59	26	changphas	changphas	PROPN
ejpam-5976	59	27	/	/	SYM
ejpam-5976	59	28	eur	eur	PROPN
ejpam-5976	59	29	.	.	PUNCT
ejpam-5976	60	1	j.	j.	PROPN
ejpam-5976	60	2	pure	pure	PROPN
ejpam-5976	60	3	appl	appl	PROPN
ejpam-5976	60	4	.	.	PROPN
ejpam-5976	60	5	math	math	PROPN
ejpam-5976	60	6	,	,	PUNCT
ejpam-5976	60	7	18	18	NUM
ejpam-5976	60	8	(	(	PUNCT
ejpam-5976	60	9	2	2	NUM
ejpam-5976	60	10	)	)	PUNCT
ejpam-5976	60	11	(	(	PUNCT
ejpam-5976	60	12	2025	2025	NUM
ejpam-5976	60	13	)	)	PUNCT
ejpam-5976	60	14	,	,	PUNCT
ejpam-5976	60	15	5976	5976	NUM
ejpam-5976	60	16	3	3	NUM
ejpam-5976	60	17	of	of	ADP
ejpam-5976	60	18	14	14	NUM
ejpam-5976	60	19	∗	∗	NOUN
ejpam-5976	60	20	a1	a1	NOUN
ejpam-5976	60	21	a2	a2	PROPN
ejpam-5976	60	22	a3	a3	PROPN
ejpam-5976	60	23	a4	a4	CCONJ
ejpam-5976	60	24	a1	a1	NOUN
ejpam-5976	60	25	a1	a1	NOUN
ejpam-5976	60	26	a1	a1	NOUN
ejpam-5976	60	27	a1	a1	NOUN
ejpam-5976	60	28	a1	a1	NOUN
ejpam-5976	60	29	a2	a2	PROPN
ejpam-5976	60	30	a1	a1	PROPN
ejpam-5976	60	31	a2	a2	PROPN
ejpam-5976	60	32	a2	a2	PROPN
ejpam-5976	60	33	a4	a4	PROPN
ejpam-5976	60	34	a3	a3	NOUN
ejpam-5976	60	35	a1	a1	PROPN
ejpam-5976	60	36	a2	a2	PROPN
ejpam-5976	60	37	a2	a2	PROPN
ejpam-5976	60	38	a4	a4	NOUN
ejpam-5976	60	39	a4	a4	NOUN
ejpam-5976	60	40	a1	a1	NOUN
ejpam-5976	60	41	a4	a4	NOUN
ejpam-5976	60	42	a4	a4	NOUN
ejpam-5976	60	43	a2	a2	PROPN
ejpam-5976	60	44	·	·	PUNCT
ejpam-5976	60	45	b1	b1	PROPN
ejpam-5976	60	46	b2	b2	NOUN
ejpam-5976	60	47	b3	b3	PROPN
ejpam-5976	60	48	b4	b4	NOUN
ejpam-5976	60	49	b1	b1	NOUN
ejpam-5976	60	50	b1	b1	PROPN
ejpam-5976	60	51	b1	b1	PROPN
ejpam-5976	60	52	b1	b1	PROPN
ejpam-5976	60	53	b4	b4	PROPN
ejpam-5976	60	54	b2	b2	PROPN
ejpam-5976	60	55	b1	b1	NOUN
ejpam-5976	60	56	b2	b2	NOUN
ejpam-5976	60	57	b2	b2	NOUN
ejpam-5976	60	58	b4	b4	PROPN
ejpam-5976	60	59	b3	b3	PROPN
ejpam-5976	60	60	b1	b1	PROPN
ejpam-5976	60	61	b3	b3	PROPN
ejpam-5976	60	62	b3	b3	PROPN
ejpam-5976	60	63	b4	b4	PROPN
ejpam-5976	60	64	b4	b4	PROPN
ejpam-5976	60	65	b4	b4	PROPN
ejpam-5976	60	66	b4	b4	PROPN
ejpam-5976	60	67	b4	b4	PROPN
ejpam-5976	60	68	b1	b1	NOUN
ejpam-5976	60	69	we	we	PRON
ejpam-5976	60	70	observe	observe	VERB
ejpam-5976	60	71	that	that	SCONJ
ejpam-5976	60	72	(	(	PUNCT
ejpam-5976	60	73	a3	a3	NOUN
ejpam-5976	60	74	,	,	PUNCT
ejpam-5976	60	75	b4	b4	NOUN
ejpam-5976	60	76	)	)	PUNCT
ejpam-5976	60	77	∈	∈	PROPN
ejpam-5976	60	78	q(a3	q(a3	NOUN
ejpam-5976	60	79	)	)	PUNCT
ejpam-5976	60	80	×	×	PROPN
ejpam-5976	60	81	q(b3	q(b3	NOUN
ejpam-5976	60	82	)	)	PUNCT
ejpam-5976	60	83	,	,	PUNCT
ejpam-5976	60	84	but	but	CCONJ
ejpam-5976	60	85	(	(	PUNCT
ejpam-5976	60	86	a3	a3	NOUN
ejpam-5976	60	87	,	,	PUNCT
ejpam-5976	60	88	b4	b4	NOUN
ejpam-5976	60	89	)	)	PUNCT
ejpam-5976	60	90	/∈	/∈	PUNCT
ejpam-5976	61	1	q((a3	q((a3	NOUN
ejpam-5976	61	2	,	,	PUNCT
ejpam-5976	61	3	b3	b3	PROPN
ejpam-5976	61	4	)	)	PUNCT
ejpam-5976	61	5	)	)	PUNCT
ejpam-5976	61	6	.	.	PUNCT
ejpam-5976	62	1	thus	thus	ADV
ejpam-5976	62	2	,	,	PUNCT
ejpam-5976	62	3	we	we	PRON
ejpam-5976	62	4	have	have	VERB
ejpam-5976	62	5	q(a3	q(a3	NOUN
ejpam-5976	62	6	)	)	PUNCT
ejpam-5976	62	7	×	×	PROPN
ejpam-5976	62	8	q(b3	q(b3	NOUN
ejpam-5976	62	9	)	)	PUNCT
ejpam-5976	62	10	̸=	̸=	PROPN
ejpam-5976	62	11	q((a3	q((a3	NOUN
ejpam-5976	62	12	,	,	PUNCT
ejpam-5976	62	13	b3	b3	PROPN
ejpam-5976	62	14	)	)	PUNCT
ejpam-5976	62	15	)	)	PUNCT
ejpam-5976	62	16	.	.	PUNCT
ejpam-5976	63	1	furthermore	furthermore	ADV
ejpam-5976	63	2	,	,	PUNCT
ejpam-5976	63	3	for	for	ADP
ejpam-5976	63	4	any	any	DET
ejpam-5976	63	5	(	(	PUNCT
ejpam-5976	63	6	u	u	NOUN
ejpam-5976	63	7	,	,	PUNCT
ejpam-5976	63	8	v	v	NOUN
ejpam-5976	63	9	)	)	PUNCT
ejpam-5976	63	10	∈	∈	PROPN
ejpam-5976	63	11	s1	s1	NOUN
ejpam-5976	63	12	×	×	NOUN
ejpam-5976	63	13	s2	s2	NOUN
ejpam-5976	63	14	such	such	ADJ
ejpam-5976	63	15	that	that	SCONJ
ejpam-5976	63	16	(	(	PUNCT
ejpam-5976	63	17	u	u	NOUN
ejpam-5976	63	18	,	,	PUNCT
ejpam-5976	63	19	v	v	NOUN
ejpam-5976	63	20	)	)	PUNCT
ejpam-5976	63	21	̸=	̸=	PROPN
ejpam-5976	63	22	(	(	PUNCT
ejpam-5976	63	23	a3	a3	NOUN
ejpam-5976	63	24	,	,	PUNCT
ejpam-5976	63	25	b4	b4	NOUN
ejpam-5976	63	26	)	)	PUNCT
ejpam-5976	63	27	,	,	PUNCT
ejpam-5976	63	28	we	we	PRON
ejpam-5976	63	29	have	have	VERB
ejpam-5976	63	30	(	(	PUNCT
ejpam-5976	63	31	a3	a3	NOUN
ejpam-5976	63	32	,	,	PUNCT
ejpam-5976	63	33	b4	b4	NOUN
ejpam-5976	63	34	)	)	PUNCT
ejpam-5976	63	35	∈	∈	PROPN
ejpam-5976	63	36	q(a3)×q(b3	q(a3)×q(b3	NOUN
ejpam-5976	63	37	)	)	PUNCT
ejpam-5976	63	38	,	,	PUNCT
ejpam-5976	63	39	but	but	CCONJ
ejpam-5976	63	40	(	(	PUNCT
ejpam-5976	63	41	a3	a3	NOUN
ejpam-5976	63	42	,	,	PUNCT
ejpam-5976	63	43	b4	b4	NOUN
ejpam-5976	63	44	)	)	PUNCT
ejpam-5976	63	45	/∈	/∈	PUNCT
ejpam-5976	64	1	{	{	PUNCT
ejpam-5976	64	2	(	(	PUNCT
ejpam-5976	64	3	u	u	NOUN
ejpam-5976	64	4	,	,	PUNCT
ejpam-5976	64	5	v)}∪((us1∩s1u)×(vs2∩s2v	v)}∪((us1∩s1u)×(vs2∩s2v	PROPN
ejpam-5976	64	6	)	)	PUNCT
ejpam-5976	64	7	)	)	PUNCT
ejpam-5976	64	8	.	.	PUNCT
ejpam-5976	65	1	therefore	therefore	ADV
ejpam-5976	65	2	,	,	PUNCT
ejpam-5976	65	3	we	we	PRON
ejpam-5976	65	4	obtain	obtain	VERB
ejpam-5976	65	5	q(a3	q(a3	NOUN
ejpam-5976	65	6	)	)	PUNCT
ejpam-5976	65	7	×q(b3	×q(b3	PROPN
ejpam-5976	65	8	)	)	PUNCT
ejpam-5976	65	9	̸=	̸=	PROPN
ejpam-5976	65	10	q((u	q((u	NOUN
ejpam-5976	65	11	,	,	PUNCT
ejpam-5976	65	12	v	v	NOUN
ejpam-5976	65	13	)	)	PUNCT
ejpam-5976	65	14	)	)	PUNCT
ejpam-5976	65	15	,	,	PUNCT
ejpam-5976	65	16	which	which	PRON
ejpam-5976	65	17	implies	imply	VERB
ejpam-5976	65	18	that	that	PRON
ejpam-5976	65	19	q(a3	q(a3	NOUN
ejpam-5976	65	20	)	)	PUNCT
ejpam-5976	65	21	×q(b3	×q(b3	PROPN
ejpam-5976	65	22	)	)	PUNCT
ejpam-5976	65	23	is	be	AUX
ejpam-5976	65	24	not	not	PART
ejpam-5976	65	25	the	the	DET
ejpam-5976	65	26	principal	principal	ADJ
ejpam-5976	65	27	quasi	quasi	NOUN
ejpam-5976	65	28	-	-	NOUN
ejpam-5976	65	29	ideal	ideal	NOUN
ejpam-5976	65	30	of	of	ADP
ejpam-5976	65	31	s1	s1	PROPN
ejpam-5976	65	32	×	×	PROPN
ejpam-5976	65	33	s2	s2	PROPN
ejpam-5976	65	34	.	.	PUNCT
ejpam-5976	66	1	from	from	ADP
ejpam-5976	66	2	example	example	NOUN
ejpam-5976	66	3	1	1	NUM
ejpam-5976	66	4	,	,	PUNCT
ejpam-5976	66	5	we	we	PRON
ejpam-5976	66	6	see	see	VERB
ejpam-5976	66	7	that	that	SCONJ
ejpam-5976	66	8	q(a)×q(b	q(a)×q(b	ADV
ejpam-5976	66	9	)	)	PUNCT
ejpam-5976	66	10	need	need	VERB
ejpam-5976	66	11	not	not	PART
ejpam-5976	66	12	to	to	PART
ejpam-5976	66	13	be	be	AUX
ejpam-5976	66	14	q((a	q((a	PROPN
ejpam-5976	66	15	,	,	PUNCT
ejpam-5976	66	16	b	b	NOUN
ejpam-5976	66	17	)	)	PUNCT
ejpam-5976	66	18	)	)	PUNCT
ejpam-5976	66	19	.	.	PUNCT
ejpam-5976	67	1	next	next	ADV
ejpam-5976	67	2	,	,	PUNCT
ejpam-5976	67	3	we	we	PRON
ejpam-5976	67	4	present	present	VERB
ejpam-5976	67	5	the	the	DET
ejpam-5976	67	6	inclusion	inclusion	NOUN
ejpam-5976	67	7	of	of	ADP
ejpam-5976	67	8	q(a	q(a	NOUN
ejpam-5976	67	9	)	)	PUNCT
ejpam-5976	67	10	×	×	NOUN
ejpam-5976	67	11	q(b	q(b	ADJ
ejpam-5976	67	12	)	)	PUNCT
ejpam-5976	67	13	and	and	CCONJ
ejpam-5976	67	14	q((a	q((a	PROPN
ejpam-5976	67	15	,	,	PUNCT
ejpam-5976	67	16	b	b	NOUN
ejpam-5976	67	17	)	)	PUNCT
ejpam-5976	67	18	)	)	PUNCT
ejpam-5976	67	19	and	and	CCONJ
ejpam-5976	67	20	give	give	VERB
ejpam-5976	67	21	necessary	necessary	ADJ
ejpam-5976	67	22	and	and	CCONJ
ejpam-5976	67	23	sufficient	sufficient	ADJ
ejpam-5976	67	24	conditions	condition	NOUN
ejpam-5976	67	25	when	when	SCONJ
ejpam-5976	67	26	q(a)×q(b	q(a)×q(b	ADV
ejpam-5976	67	27	)	)	PUNCT
ejpam-5976	67	28	=	=	SYM
ejpam-5976	67	29	q((a	q((a	PROPN
ejpam-5976	67	30	,	,	PUNCT
ejpam-5976	67	31	b	b	NOUN
ejpam-5976	67	32	)	)	PUNCT
ejpam-5976	67	33	)	)	PUNCT
ejpam-5976	67	34	.	.	PUNCT
ejpam-5976	68	1	theorem	theorem	NOUN
ejpam-5976	68	2	2	2	NUM
ejpam-5976	68	3	.	.	PUNCT
ejpam-5976	69	1	let	let	VERB
ejpam-5976	69	2	(	(	PUNCT
ejpam-5976	69	3	a	a	PRON
ejpam-5976	69	4	,	,	PUNCT
ejpam-5976	69	5	b	b	NOUN
ejpam-5976	69	6	)	)	PUNCT
ejpam-5976	69	7	∈	∈	PROPN
ejpam-5976	69	8	s1	s1	PROPN
ejpam-5976	69	9	×	×	PROPN
ejpam-5976	69	10	s2	s2	PROPN
ejpam-5976	69	11	.	.	PUNCT
ejpam-5976	70	1	then	then	ADV
ejpam-5976	70	2	q((a	q((a	PROPN
ejpam-5976	70	3	,	,	PUNCT
ejpam-5976	70	4	b	b	NOUN
ejpam-5976	70	5	)	)	PUNCT
ejpam-5976	70	6	)	)	PUNCT
ejpam-5976	71	1	⊆	⊆	NUM
ejpam-5976	71	2	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	71	3	)	)	PUNCT
ejpam-5976	71	4	.	.	PUNCT
ejpam-5976	72	1	proof	proof	NOUN
ejpam-5976	72	2	.	.	PUNCT
ejpam-5976	73	1	assume	assume	VERB
ejpam-5976	73	2	that	that	SCONJ
ejpam-5976	73	3	(	(	PUNCT
ejpam-5976	73	4	a	a	PRON
ejpam-5976	73	5	,	,	PUNCT
ejpam-5976	73	6	b	b	NOUN
ejpam-5976	73	7	)	)	PUNCT
ejpam-5976	73	8	∈	∈	PROPN
ejpam-5976	73	9	s1	s1	PROPN
ejpam-5976	73	10	×	×	PROPN
ejpam-5976	73	11	s2	s2	PROPN
ejpam-5976	73	12	.	.	PUNCT
ejpam-5976	74	1	then	then	ADV
ejpam-5976	74	2	q((a	q((a	PROPN
ejpam-5976	74	3	,	,	PUNCT
ejpam-5976	74	4	b	b	NOUN
ejpam-5976	74	5	)	)	PUNCT
ejpam-5976	74	6	)	)	PUNCT
ejpam-5976	75	1	=	=	PRON
ejpam-5976	75	2	{	{	PUNCT
ejpam-5976	75	3	(	(	PUNCT
ejpam-5976	75	4	a	a	PRON
ejpam-5976	75	5	,	,	PUNCT
ejpam-5976	75	6	b	b	NOUN
ejpam-5976	75	7	)	)	PUNCT
ejpam-5976	75	8	}	}	PUNCT
ejpam-5976	75	9	∪	∪	X
ejpam-5976	75	10	(	(	PUNCT
ejpam-5976	75	11	(	(	PUNCT
ejpam-5976	75	12	a	a	PRON
ejpam-5976	75	13	,	,	PUNCT
ejpam-5976	75	14	b)(s1	b)(s1	NOUN
ejpam-5976	75	15	×	×	PROPN
ejpam-5976	75	16	s2	s2	NOUN
ejpam-5976	75	17	)	)	PUNCT
ejpam-5976	75	18	∩	∩	NOUN
ejpam-5976	75	19	(	(	PUNCT
ejpam-5976	75	20	s1	s1	PROPN
ejpam-5976	75	21	×	×	PROPN
ejpam-5976	75	22	s2)(a	s2)(a	NOUN
ejpam-5976	75	23	,	,	PUNCT
ejpam-5976	75	24	b	b	NOUN
ejpam-5976	75	25	)	)	PUNCT
ejpam-5976	75	26	)	)	PUNCT
ejpam-5976	76	1	=	=	PRON
ejpam-5976	76	2	{	{	PUNCT
ejpam-5976	76	3	(	(	PUNCT
ejpam-5976	76	4	a	a	PRON
ejpam-5976	76	5	,	,	PUNCT
ejpam-5976	76	6	b	b	NOUN
ejpam-5976	76	7	)	)	PUNCT
ejpam-5976	76	8	}	}	PUNCT
ejpam-5976	76	9	∪	∪	X
ejpam-5976	76	10	(	(	PUNCT
ejpam-5976	76	11	(	(	PUNCT
ejpam-5976	76	12	as1	as1	PROPN
ejpam-5976	76	13	×	×	PROPN
ejpam-5976	76	14	bs2	bs2	PROPN
ejpam-5976	76	15	)	)	PUNCT
ejpam-5976	76	16	∩	∩	NOUN
ejpam-5976	76	17	(	(	PUNCT
ejpam-5976	76	18	s1a×	s1a×	ADP
ejpam-5976	76	19	s2b	s2b	NOUN
ejpam-5976	76	20	)	)	PUNCT
ejpam-5976	76	21	)	)	PUNCT
ejpam-5976	77	1	=	=	PRON
ejpam-5976	77	2	{	{	PUNCT
ejpam-5976	77	3	(	(	PUNCT
ejpam-5976	77	4	a	a	PRON
ejpam-5976	77	5	,	,	PUNCT
ejpam-5976	77	6	b	b	NOUN
ejpam-5976	77	7	)	)	PUNCT
ejpam-5976	77	8	}	}	PUNCT
ejpam-5976	77	9	∪	∪	X
ejpam-5976	77	10	(	(	PUNCT
ejpam-5976	77	11	(	(	PUNCT
ejpam-5976	77	12	as1	as1	NOUN
ejpam-5976	77	13	∩	∩	X
ejpam-5976	77	14	s1a)×	s1a)×	X
ejpam-5976	77	15	(	(	PUNCT
ejpam-5976	77	16	bs2	bs2	PROPN
ejpam-5976	77	17	∩	∩	PROPN
ejpam-5976	77	18	s2b	s2b	PROPN
ejpam-5976	77	19	)	)	PUNCT
ejpam-5976	77	20	)	)	PUNCT
ejpam-5976	78	1	⊆	⊆	X
ejpam-5976	78	2	(	(	PUNCT
ejpam-5976	78	3	{	{	PUNCT
ejpam-5976	78	4	a	a	PRON
ejpam-5976	78	5	}	}	PUNCT
ejpam-5976	78	6	∪	∪	X
ejpam-5976	78	7	(	(	PUNCT
ejpam-5976	78	8	as1	as1	NOUN
ejpam-5976	78	9	∩	∩	NOUN
ejpam-5976	78	10	s1a))×	s1a))×	X
ejpam-5976	78	11	(	(	PUNCT
ejpam-5976	78	12	{	{	PUNCT
ejpam-5976	78	13	b	b	NOUN
ejpam-5976	78	14	}	}	PUNCT
ejpam-5976	78	15	∪	∪	NOUN
ejpam-5976	78	16	(	(	PUNCT
ejpam-5976	78	17	bs2	bs2	PROPN
ejpam-5976	78	18	∩	∩	PROPN
ejpam-5976	78	19	s2b	s2b	PROPN
ejpam-5976	78	20	)	)	PUNCT
ejpam-5976	78	21	)	)	PUNCT
ejpam-5976	79	1	=	=	PUNCT
ejpam-5976	79	2	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	79	3	)	)	PUNCT
ejpam-5976	79	4	.	.	PUNCT
ejpam-5976	80	1	therefore	therefore	ADV
ejpam-5976	80	2	,	,	PUNCT
ejpam-5976	80	3	q((a	q((a	PROPN
ejpam-5976	80	4	,	,	PUNCT
ejpam-5976	80	5	b	b	NOUN
ejpam-5976	80	6	)	)	PUNCT
ejpam-5976	80	7	)	)	PUNCT
ejpam-5976	81	1	⊆	⊆	NUM
ejpam-5976	81	2	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	81	3	)	)	PUNCT
ejpam-5976	81	4	.	.	PUNCT
ejpam-5976	82	1	theorem	theorem	NOUN
ejpam-5976	82	2	3	3	X
ejpam-5976	82	3	.	.	PUNCT
ejpam-5976	83	1	let	let	VERB
ejpam-5976	83	2	(	(	PUNCT
ejpam-5976	83	3	a	a	PRON
ejpam-5976	83	4	,	,	PUNCT
ejpam-5976	83	5	b	b	NOUN
ejpam-5976	83	6	)	)	PUNCT
ejpam-5976	83	7	∈	∈	PROPN
ejpam-5976	83	8	s1	s1	PROPN
ejpam-5976	83	9	×	×	PROPN
ejpam-5976	83	10	s2	s2	PROPN
ejpam-5976	83	11	.	.	PUNCT
ejpam-5976	84	1	then	then	ADV
ejpam-5976	84	2	q((a	q((a	PROPN
ejpam-5976	84	3	,	,	PUNCT
ejpam-5976	84	4	b	b	NOUN
ejpam-5976	84	5	)	)	PUNCT
ejpam-5976	84	6	)	)	PUNCT
ejpam-5976	85	1	=	=	SYM
ejpam-5976	85	2	q(a	q(a	PROPN
ejpam-5976	85	3	)	)	PUNCT
ejpam-5976	85	4	×	×	NOUN
ejpam-5976	85	5	q(b	q(b	NOUN
ejpam-5976	85	6	)	)	PUNCT
ejpam-5976	85	7	if	if	SCONJ
ejpam-5976	85	8	and	and	CCONJ
ejpam-5976	85	9	only	only	ADV
ejpam-5976	85	10	if	if	SCONJ
ejpam-5976	85	11	at	at	ADV
ejpam-5976	85	12	least	least	ADJ
ejpam-5976	85	13	one	one	NUM
ejpam-5976	85	14	of	of	ADP
ejpam-5976	85	15	the	the	DET
ejpam-5976	85	16	following	follow	VERB
ejpam-5976	85	17	conditions	condition	NOUN
ejpam-5976	85	18	holds	hold	VERB
ejpam-5976	85	19	:	:	PUNCT
ejpam-5976	85	20	(	(	PUNCT
ejpam-5976	85	21	1	1	X
ejpam-5976	85	22	)	)	PUNCT
ejpam-5976	85	23	as1	as1	NOUN
ejpam-5976	85	24	∩	∩	NOUN
ejpam-5976	85	25	s1a	s1a	NOUN
ejpam-5976	85	26	=	=	PUNCT
ejpam-5976	85	27	{	{	PUNCT
ejpam-5976	85	28	a	a	NOUN
ejpam-5976	85	29	}	}	PUNCT
ejpam-5976	85	30	;	;	PUNCT
ejpam-5976	85	31	(	(	PUNCT
ejpam-5976	85	32	2	2	X
ejpam-5976	85	33	)	)	PUNCT
ejpam-5976	85	34	bs2	bs2	NOUN
ejpam-5976	85	35	∩	∩	PROPN
ejpam-5976	85	36	s2b	s2b	PROPN
ejpam-5976	85	37	=	=	PUNCT
ejpam-5976	85	38	{	{	PUNCT
ejpam-5976	85	39	b	b	NOUN
ejpam-5976	85	40	}	}	PUNCT
ejpam-5976	85	41	;	;	PUNCT
ejpam-5976	85	42	(	(	PUNCT
ejpam-5976	85	43	3	3	X
ejpam-5976	85	44	)	)	PUNCT
ejpam-5976	85	45	a	a	DET
ejpam-5976	85	46	∈	∈	PROPN
ejpam-5976	85	47	as1	as1	NOUN
ejpam-5976	85	48	∩	∩	PROPN
ejpam-5976	85	49	s1a	s1a	PROPN
ejpam-5976	85	50	and	and	CCONJ
ejpam-5976	85	51	b	b	X
ejpam-5976	85	52	∈	∈	PROPN
ejpam-5976	85	53	bs2	bs2	PROPN
ejpam-5976	85	54	∩	∩	PROPN
ejpam-5976	85	55	s2b	s2b	PROPN
ejpam-5976	85	56	.	.	PUNCT
ejpam-5976	86	1	proof	proof	NOUN
ejpam-5976	86	2	.	.	PUNCT
ejpam-5976	87	1	assume	assume	VERB
ejpam-5976	87	2	that	that	SCONJ
ejpam-5976	87	3	(	(	PUNCT
ejpam-5976	87	4	1	1	NUM
ejpam-5976	87	5	)	)	PUNCT
ejpam-5976	87	6	,	,	PUNCT
ejpam-5976	87	7	(	(	PUNCT
ejpam-5976	87	8	2	2	NUM
ejpam-5976	87	9	)	)	PUNCT
ejpam-5976	87	10	,	,	PUNCT
ejpam-5976	87	11	and	and	CCONJ
ejpam-5976	87	12	(	(	PUNCT
ejpam-5976	87	13	3	3	X
ejpam-5976	87	14	)	)	PUNCT
ejpam-5976	87	15	do	do	AUX
ejpam-5976	87	16	not	not	PART
ejpam-5976	87	17	hold	hold	VERB
ejpam-5976	87	18	.	.	PUNCT
ejpam-5976	88	1	then	then	ADV
ejpam-5976	88	2	we	we	PRON
ejpam-5976	88	3	have	have	VERB
ejpam-5976	88	4	two	two	NUM
ejpam-5976	88	5	cases	case	NOUN
ejpam-5976	88	6	to	to	PART
ejpam-5976	88	7	consider	consider	VERB
ejpam-5976	88	8	:	:	PUNCT
ejpam-5976	88	9	(	(	PUNCT
ejpam-5976	88	10	i	i	NOUN
ejpam-5976	88	11	)	)	PUNCT
ejpam-5976	88	12	a	a	DET
ejpam-5976	88	13	/∈	/∈	PROPN
ejpam-5976	88	14	as1	as1	NOUN
ejpam-5976	88	15	∩	∩	PROPN
ejpam-5976	88	16	s1a	s1a	PROPN
ejpam-5976	88	17	and	and	CCONJ
ejpam-5976	88	18	bs2	bs2	PROPN
ejpam-5976	88	19	∩	∩	PROPN
ejpam-5976	88	20	s2b	s2b	PROPN
ejpam-5976	88	21	̸=	̸=	PROPN
ejpam-5976	88	22	{	{	PUNCT
ejpam-5976	88	23	b	b	NOUN
ejpam-5976	88	24	}	}	PUNCT
ejpam-5976	88	25	;	;	PUNCT
ejpam-5976	88	26	(	(	PUNCT
ejpam-5976	88	27	ii	ii	NOUN
ejpam-5976	88	28	)	)	PUNCT
ejpam-5976	88	29	b	b	PROPN
ejpam-5976	88	30	/∈	/∈	PUNCT
ejpam-5976	88	31	bs2	bs2	PROPN
ejpam-5976	88	32	∩	∩	PROPN
ejpam-5976	88	33	s2b	s2b	PROPN
ejpam-5976	88	34	and	and	CCONJ
ejpam-5976	88	35	as1	as1	PROPN
ejpam-5976	88	36	∩	∩	PROPN
ejpam-5976	88	37	s1a	s1a	PROPN
ejpam-5976	88	38	̸=	̸=	PROPN
ejpam-5976	88	39	{	{	PUNCT
ejpam-5976	88	40	a	a	X
ejpam-5976	88	41	}	}	PUNCT
ejpam-5976	88	42	.	.	PUNCT
ejpam-5976	89	1	if	if	SCONJ
ejpam-5976	89	2	(	(	PUNCT
ejpam-5976	89	3	i	i	NOUN
ejpam-5976	89	4	)	)	PUNCT
ejpam-5976	89	5	holds	hold	VERB
ejpam-5976	89	6	,	,	PUNCT
ejpam-5976	89	7	then	then	ADV
ejpam-5976	89	8	there	there	PRON
ejpam-5976	89	9	exists	exist	VERB
ejpam-5976	89	10	v	v	ADP
ejpam-5976	89	11	∈	∈	PROPN
ejpam-5976	89	12	bs2	bs2	NOUN
ejpam-5976	89	13	∩	∩	NOUN
ejpam-5976	89	14	s2b	s2b	VERB
ejpam-5976	89	15	such	such	ADJ
ejpam-5976	89	16	that	that	PRON
ejpam-5976	89	17	v	v	ADP
ejpam-5976	89	18	̸=	̸=	PROPN
ejpam-5976	89	19	b.	b.	NOUN
ejpam-5976	90	1	then	then	ADV
ejpam-5976	90	2	(	(	PUNCT
ejpam-5976	90	3	a	a	DET
ejpam-5976	90	4	,	,	PUNCT
ejpam-5976	90	5	v	v	NOUN
ejpam-5976	90	6	)	)	PUNCT
ejpam-5976	90	7	̸=	̸=	PROPN
ejpam-5976	90	8	(	(	PUNCT
ejpam-5976	90	9	a	a	DET
ejpam-5976	90	10	,	,	PUNCT
ejpam-5976	90	11	b	b	NOUN
ejpam-5976	90	12	)	)	PUNCT
ejpam-5976	90	13	and	and	CCONJ
ejpam-5976	90	14	(	(	PUNCT
ejpam-5976	90	15	a	a	DET
ejpam-5976	90	16	,	,	PUNCT
ejpam-5976	90	17	v	v	NOUN
ejpam-5976	90	18	)	)	PUNCT
ejpam-5976	90	19	/∈	/∈	PUNCT
ejpam-5976	90	20	(	(	PUNCT
ejpam-5976	90	21	as1	as1	NOUN
ejpam-5976	90	22	∩	∩	X
ejpam-5976	90	23	s1a)×	s1a)×	X
ejpam-5976	90	24	(	(	PUNCT
ejpam-5976	90	25	bs2	bs2	PROPN
ejpam-5976	90	26	∩	∩	PROPN
ejpam-5976	90	27	s2b	s2b	PROPN
ejpam-5976	90	28	)	)	PUNCT
ejpam-5976	90	29	.	.	PUNCT
ejpam-5976	91	1	thus	thus	ADV
ejpam-5976	91	2	,	,	PUNCT
ejpam-5976	91	3	(	(	PUNCT
ejpam-5976	91	4	a	a	DET
ejpam-5976	91	5	,	,	PUNCT
ejpam-5976	91	6	v	v	NOUN
ejpam-5976	91	7	)	)	PUNCT
ejpam-5976	91	8	/∈	/∈	PUNCT
ejpam-5976	92	1	q((a	q((a	NOUN
ejpam-5976	92	2	,	,	PUNCT
ejpam-5976	92	3	b	b	NOUN
ejpam-5976	92	4	)	)	PUNCT
ejpam-5976	92	5	)	)	PUNCT
ejpam-5976	92	6	.	.	PUNCT
ejpam-5976	93	1	since	since	SCONJ
ejpam-5976	93	2	(	(	PUNCT
ejpam-5976	93	3	a	a	PRON
ejpam-5976	93	4	,	,	PUNCT
ejpam-5976	93	5	v	v	NOUN
ejpam-5976	93	6	)	)	PUNCT
ejpam-5976	93	7	∈	∈	PROPN
ejpam-5976	93	8	q(a)×q(b	q(a)×q(b	PROPN
ejpam-5976	93	9	)	)	PUNCT
ejpam-5976	93	10	,	,	PUNCT
ejpam-5976	93	11	it	it	PRON
ejpam-5976	93	12	follows	follow	VERB
ejpam-5976	93	13	that	that	SCONJ
ejpam-5976	93	14	q((a	q((a	NOUN
ejpam-5976	93	15	,	,	PUNCT
ejpam-5976	93	16	b	b	NOUN
ejpam-5976	93	17	)	)	PUNCT
ejpam-5976	93	18	)	)	PUNCT
ejpam-5976	93	19	̸=	̸=	PROPN
ejpam-5976	93	20	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	93	21	)	)	PUNCT
ejpam-5976	93	22	.	.	PUNCT
ejpam-5976	94	1	we	we	PRON
ejpam-5976	94	2	can	can	AUX
ejpam-5976	94	3	prove	prove	VERB
ejpam-5976	94	4	similarly	similarly	ADV
ejpam-5976	94	5	that	that	SCONJ
ejpam-5976	94	6	q((a	q((a	NOUN
ejpam-5976	94	7	,	,	PUNCT
ejpam-5976	94	8	b	b	NOUN
ejpam-5976	94	9	)	)	PUNCT
ejpam-5976	94	10	)	)	PUNCT
ejpam-5976	95	1	̸=	̸=	PROPN
ejpam-5976	95	2	q(a)×q(b	q(a)×q(b	ADV
ejpam-5976	95	3	)	)	PUNCT
ejpam-5976	95	4	when	when	SCONJ
ejpam-5976	95	5	(	(	PUNCT
ejpam-5976	95	6	ii	ii	NOUN
ejpam-5976	95	7	)	)	PUNCT
ejpam-5976	95	8	holds	hold	VERB
ejpam-5976	95	9	.	.	PUNCT
ejpam-5976	96	1	p.	p.	NOUN
ejpam-5976	96	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	96	3	,	,	PUNCT
ejpam-5976	96	4	o.	o.	PROPN
ejpam-5976	96	5	pankoon	pankoon	NOUN
ejpam-5976	96	6	,	,	PUNCT
ejpam-5976	96	7	t.	t.	PROPN
ejpam-5976	96	8	changphas	changphas	PROPN
ejpam-5976	96	9	/	/	SYM
ejpam-5976	96	10	eur	eur	PROPN
ejpam-5976	96	11	.	.	PUNCT
ejpam-5976	97	1	j.	j.	PROPN
ejpam-5976	97	2	pure	pure	PROPN
ejpam-5976	97	3	appl	appl	PROPN
ejpam-5976	97	4	.	.	PROPN
ejpam-5976	97	5	math	math	PROPN
ejpam-5976	97	6	,	,	PUNCT
ejpam-5976	97	7	18	18	NUM
ejpam-5976	97	8	(	(	PUNCT
ejpam-5976	97	9	2	2	NUM
ejpam-5976	97	10	)	)	PUNCT
ejpam-5976	97	11	(	(	PUNCT
ejpam-5976	97	12	2025	2025	NUM
ejpam-5976	97	13	)	)	PUNCT
ejpam-5976	97	14	,	,	PUNCT
ejpam-5976	97	15	5976	5976	NUM
ejpam-5976	97	16	4	4	NUM
ejpam-5976	97	17	of	of	ADP
ejpam-5976	97	18	14	14	NUM
ejpam-5976	97	19	conversely	conversely	ADV
ejpam-5976	97	20	,	,	PUNCT
ejpam-5976	97	21	assume	assume	VERB
ejpam-5976	97	22	that	that	SCONJ
ejpam-5976	97	23	(	(	PUNCT
ejpam-5976	97	24	1	1	X
ejpam-5976	97	25	)	)	PUNCT
ejpam-5976	97	26	holds	hold	VERB
ejpam-5976	97	27	.	.	PUNCT
ejpam-5976	98	1	then	then	ADV
ejpam-5976	98	2	we	we	PRON
ejpam-5976	98	3	get	get	VERB
ejpam-5976	98	4	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	98	5	)	)	PUNCT
ejpam-5976	98	6	=	=	SYM
ejpam-5976	98	7	(	(	PUNCT
ejpam-5976	98	8	{	{	PUNCT
ejpam-5976	98	9	a	a	PRON
ejpam-5976	98	10	}	}	PUNCT
ejpam-5976	98	11	∪	∪	X
ejpam-5976	98	12	(	(	PUNCT
ejpam-5976	98	13	as1	as1	NOUN
ejpam-5976	98	14	∩	∩	NOUN
ejpam-5976	98	15	s1a))×	s1a))×	X
ejpam-5976	98	16	(	(	PUNCT
ejpam-5976	98	17	{	{	PUNCT
ejpam-5976	98	18	b	b	NOUN
ejpam-5976	98	19	}	}	PUNCT
ejpam-5976	98	20	∪	∪	NOUN
ejpam-5976	98	21	(	(	PUNCT
ejpam-5976	98	22	bs2	bs2	PROPN
ejpam-5976	98	23	∩	∩	PROPN
ejpam-5976	98	24	s2b	s2b	PROPN
ejpam-5976	98	25	)	)	PUNCT
ejpam-5976	98	26	)	)	PUNCT
ejpam-5976	99	1	=	=	PRON
ejpam-5976	99	2	{	{	PUNCT
ejpam-5976	99	3	a	a	DET
ejpam-5976	99	4	}	}	PUNCT
ejpam-5976	99	5	×	×	NOUN
ejpam-5976	99	6	(	(	PUNCT
ejpam-5976	99	7	{	{	PUNCT
ejpam-5976	99	8	b	b	NOUN
ejpam-5976	99	9	}	}	PUNCT
ejpam-5976	99	10	∪	∪	NOUN
ejpam-5976	99	11	(	(	PUNCT
ejpam-5976	99	12	bs2	bs2	PROPN
ejpam-5976	99	13	∩	∩	PROPN
ejpam-5976	99	14	s2b	s2b	PROPN
ejpam-5976	99	15	)	)	PUNCT
ejpam-5976	99	16	)	)	PUNCT
ejpam-5976	100	1	=	=	PRON
ejpam-5976	100	2	{	{	PUNCT
ejpam-5976	100	3	(	(	PUNCT
ejpam-5976	100	4	a	a	PRON
ejpam-5976	100	5	,	,	PUNCT
ejpam-5976	100	6	b	b	NOUN
ejpam-5976	100	7	)	)	PUNCT
ejpam-5976	100	8	}	}	PUNCT
ejpam-5976	100	9	∪	∪	X
ejpam-5976	100	10	(	(	PUNCT
ejpam-5976	100	11	{	{	PUNCT
ejpam-5976	100	12	a	a	DET
ejpam-5976	100	13	}	}	PUNCT
ejpam-5976	100	14	×	×	NOUN
ejpam-5976	100	15	(	(	PUNCT
ejpam-5976	100	16	bs2	bs2	PROPN
ejpam-5976	100	17	∩	∩	PROPN
ejpam-5976	100	18	s2b	s2b	PROPN
ejpam-5976	100	19	)	)	PUNCT
ejpam-5976	100	20	)	)	PUNCT
ejpam-5976	101	1	=	=	PRON
ejpam-5976	101	2	{	{	PUNCT
ejpam-5976	101	3	(	(	PUNCT
ejpam-5976	101	4	a	a	PRON
ejpam-5976	101	5	,	,	PUNCT
ejpam-5976	101	6	b	b	NOUN
ejpam-5976	101	7	)	)	PUNCT
ejpam-5976	101	8	}	}	PUNCT
ejpam-5976	101	9	∪	∪	X
ejpam-5976	101	10	(	(	PUNCT
ejpam-5976	101	11	(	(	PUNCT
ejpam-5976	101	12	as1	as1	NOUN
ejpam-5976	101	13	∩	∩	X
ejpam-5976	101	14	s1a)×	s1a)×	X
ejpam-5976	101	15	(	(	PUNCT
ejpam-5976	101	16	bs2	bs2	PROPN
ejpam-5976	101	17	∩	∩	PROPN
ejpam-5976	101	18	s2b	s2b	PROPN
ejpam-5976	101	19	)	)	PUNCT
ejpam-5976	101	20	)	)	PUNCT
ejpam-5976	102	1	=	=	SYM
ejpam-5976	102	2	q((a	q((a	PROPN
ejpam-5976	102	3	,	,	PUNCT
ejpam-5976	102	4	b	b	NOUN
ejpam-5976	102	5	)	)	PUNCT
ejpam-5976	102	6	)	)	PUNCT
ejpam-5976	102	7	.	.	PUNCT
ejpam-5976	103	1	on	on	ADP
ejpam-5976	103	2	the	the	DET
ejpam-5976	103	3	same	same	ADJ
ejpam-5976	103	4	way	way	NOUN
ejpam-5976	103	5	,	,	PUNCT
ejpam-5976	103	6	we	we	PRON
ejpam-5976	103	7	have	have	VERB
ejpam-5976	103	8	q(a	q(a	NOUN
ejpam-5976	103	9	)	)	PUNCT
ejpam-5976	103	10	×	×	NOUN
ejpam-5976	103	11	q(b	q(b	ADJ
ejpam-5976	103	12	)	)	PUNCT
ejpam-5976	103	13	=	=	SYM
ejpam-5976	103	14	q((a	q((a	PROPN
ejpam-5976	103	15	,	,	PUNCT
ejpam-5976	103	16	b	b	NOUN
ejpam-5976	103	17	)	)	PUNCT
ejpam-5976	103	18	)	)	PUNCT
ejpam-5976	103	19	when	when	SCONJ
ejpam-5976	103	20	the	the	DET
ejpam-5976	103	21	condition	condition	NOUN
ejpam-5976	103	22	(	(	PUNCT
ejpam-5976	103	23	2	2	X
ejpam-5976	103	24	)	)	PUNCT
ejpam-5976	103	25	is	be	AUX
ejpam-5976	103	26	satisfied	satisfied	ADJ
ejpam-5976	103	27	.	.	PUNCT
ejpam-5976	104	1	assume	assume	VERB
ejpam-5976	104	2	that	that	SCONJ
ejpam-5976	104	3	the	the	DET
ejpam-5976	104	4	condition	condition	NOUN
ejpam-5976	104	5	(	(	PUNCT
ejpam-5976	104	6	3	3	X
ejpam-5976	104	7	)	)	PUNCT
ejpam-5976	104	8	occurs	occur	VERB
ejpam-5976	104	9	.	.	PUNCT
ejpam-5976	105	1	then	then	ADV
ejpam-5976	105	2	we	we	PRON
ejpam-5976	105	3	have	have	VERB
ejpam-5976	105	4	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	105	5	)	)	PUNCT
ejpam-5976	105	6	=	=	SYM
ejpam-5976	105	7	(	(	PUNCT
ejpam-5976	105	8	{	{	PUNCT
ejpam-5976	105	9	a	a	PRON
ejpam-5976	105	10	}	}	PUNCT
ejpam-5976	105	11	∪	∪	X
ejpam-5976	105	12	(	(	PUNCT
ejpam-5976	105	13	as1	as1	NOUN
ejpam-5976	105	14	∩	∩	NOUN
ejpam-5976	105	15	s1a))×	s1a))×	X
ejpam-5976	105	16	(	(	PUNCT
ejpam-5976	105	17	{	{	PUNCT
ejpam-5976	105	18	b	b	NOUN
ejpam-5976	105	19	}	}	PUNCT
ejpam-5976	105	20	∪	∪	NOUN
ejpam-5976	105	21	(	(	PUNCT
ejpam-5976	105	22	bs2	bs2	PROPN
ejpam-5976	105	23	∩	∩	PROPN
ejpam-5976	105	24	s2b	s2b	PROPN
ejpam-5976	105	25	)	)	PUNCT
ejpam-5976	105	26	)	)	PUNCT
ejpam-5976	106	1	=	=	SYM
ejpam-5976	106	2	(	(	PUNCT
ejpam-5976	106	3	as1	as1	NOUN
ejpam-5976	106	4	∩	∩	X
ejpam-5976	106	5	s1a)×	s1a)×	X
ejpam-5976	106	6	(	(	PUNCT
ejpam-5976	106	7	bs2	bs2	PROPN
ejpam-5976	106	8	∩	∩	PROPN
ejpam-5976	106	9	s2b	s2b	PROPN
ejpam-5976	106	10	)	)	PUNCT
ejpam-5976	106	11	=	=	SYM
ejpam-5976	106	12	q((a	q((a	PROPN
ejpam-5976	106	13	,	,	PUNCT
ejpam-5976	106	14	b	b	NOUN
ejpam-5976	106	15	)	)	PUNCT
ejpam-5976	106	16	)	)	PUNCT
ejpam-5976	106	17	.	.	PUNCT
ejpam-5976	107	1	therefore	therefore	ADV
ejpam-5976	107	2	,	,	PUNCT
ejpam-5976	107	3	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	107	4	)	)	PUNCT
ejpam-5976	107	5	=	=	SYM
ejpam-5976	107	6	q((a	q((a	PROPN
ejpam-5976	107	7	,	,	PUNCT
ejpam-5976	107	8	b	b	NOUN
ejpam-5976	107	9	)	)	PUNCT
ejpam-5976	107	10	)	)	PUNCT
ejpam-5976	107	11	.	.	PUNCT
ejpam-5976	108	1	the	the	DET
ejpam-5976	108	2	following	follow	VERB
ejpam-5976	108	3	theorem	theorem	NOUN
ejpam-5976	108	4	shows	show	VERB
ejpam-5976	108	5	that	that	SCONJ
ejpam-5976	108	6	if	if	SCONJ
ejpam-5976	108	7	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	108	8	)	)	PUNCT
ejpam-5976	108	9	̸=	̸=	PROPN
ejpam-5976	108	10	q((a	q((a	PROPN
ejpam-5976	108	11	,	,	PUNCT
ejpam-5976	108	12	b	b	NOUN
ejpam-5976	108	13	)	)	PUNCT
ejpam-5976	108	14	)	)	PUNCT
ejpam-5976	108	15	,	,	PUNCT
ejpam-5976	108	16	then	then	ADV
ejpam-5976	108	17	q(a)×q(b	q(a)×q(b	X
ejpam-5976	108	18	)	)	PUNCT
ejpam-5976	108	19	is	be	AUX
ejpam-5976	108	20	not	not	PART
ejpam-5976	108	21	the	the	DET
ejpam-5976	108	22	principal	principal	ADJ
ejpam-5976	108	23	quasi	quasi	NOUN
ejpam-5976	108	24	ideal	ideal	PROPN
ejpam-5976	108	25	.	.	PUNCT
ejpam-5976	109	1	theorem	theorem	VERB
ejpam-5976	109	2	4	4	NUM
ejpam-5976	109	3	.	.	PUNCT
ejpam-5976	110	1	if	if	SCONJ
ejpam-5976	110	2	q(a	q(a	NOUN
ejpam-5976	110	3	)	)	PUNCT
ejpam-5976	110	4	×	×	NOUN
ejpam-5976	110	5	q(b	q(b	ADJ
ejpam-5976	110	6	)	)	PUNCT
ejpam-5976	111	1	̸=	̸=	PROPN
ejpam-5976	111	2	q((a	q((a	PROPN
ejpam-5976	111	3	,	,	PUNCT
ejpam-5976	111	4	b	b	NOUN
ejpam-5976	111	5	)	)	PUNCT
ejpam-5976	111	6	)	)	PUNCT
ejpam-5976	111	7	,	,	PUNCT
ejpam-5976	111	8	then	then	ADV
ejpam-5976	111	9	q(a	q(a	NOUN
ejpam-5976	111	10	)	)	PUNCT
ejpam-5976	111	11	×	×	NOUN
ejpam-5976	111	12	q(b	q(b	ADJ
ejpam-5976	111	13	)	)	PUNCT
ejpam-5976	111	14	̸=	̸=	PROPN
ejpam-5976	111	15	q((u	q((u	NOUN
ejpam-5976	111	16	,	,	PUNCT
ejpam-5976	111	17	v	v	NOUN
ejpam-5976	111	18	)	)	PUNCT
ejpam-5976	111	19	)	)	PUNCT
ejpam-5976	111	20	for	for	ADP
ejpam-5976	111	21	all	all	DET
ejpam-5976	111	22	(	(	PUNCT
ejpam-5976	111	23	u	u	NOUN
ejpam-5976	111	24	,	,	PUNCT
ejpam-5976	111	25	v	v	NOUN
ejpam-5976	111	26	)	)	PUNCT
ejpam-5976	111	27	∈	∈	PROPN
ejpam-5976	111	28	s1	s1	PROPN
ejpam-5976	111	29	×	×	PROPN
ejpam-5976	111	30	s2	s2	PROPN
ejpam-5976	111	31	.	.	PUNCT
ejpam-5976	112	1	proof	proof	NOUN
ejpam-5976	112	2	.	.	PUNCT
ejpam-5976	113	1	assume	assume	VERB
ejpam-5976	113	2	that	that	SCONJ
ejpam-5976	113	3	q(a	q(a	NOUN
ejpam-5976	113	4	)	)	PUNCT
ejpam-5976	113	5	×	×	NOUN
ejpam-5976	113	6	q(b	q(b	ADJ
ejpam-5976	113	7	)	)	PUNCT
ejpam-5976	113	8	̸=	̸=	PROPN
ejpam-5976	113	9	q((a	q((a	PROPN
ejpam-5976	113	10	,	,	PUNCT
ejpam-5976	113	11	b	b	NOUN
ejpam-5976	113	12	)	)	PUNCT
ejpam-5976	113	13	)	)	PUNCT
ejpam-5976	113	14	and	and	CCONJ
ejpam-5976	113	15	q(a	q(a	PROPN
ejpam-5976	113	16	)	)	PUNCT
ejpam-5976	113	17	×	×	NOUN
ejpam-5976	113	18	q(b	q(b	ADJ
ejpam-5976	113	19	)	)	PUNCT
ejpam-5976	113	20	=	=	SYM
ejpam-5976	114	1	q((u	q((u	NOUN
ejpam-5976	114	2	,	,	PUNCT
ejpam-5976	114	3	v	v	NOUN
ejpam-5976	114	4	)	)	PUNCT
ejpam-5976	114	5	)	)	PUNCT
ejpam-5976	114	6	for	for	ADP
ejpam-5976	114	7	some	some	PRON
ejpam-5976	114	8	(	(	PUNCT
ejpam-5976	114	9	u	u	NOUN
ejpam-5976	114	10	,	,	PUNCT
ejpam-5976	114	11	v	v	NOUN
ejpam-5976	114	12	)	)	PUNCT
ejpam-5976	114	13	∈	∈	PROPN
ejpam-5976	114	14	s1	s1	PROPN
ejpam-5976	114	15	×	×	PROPN
ejpam-5976	114	16	s2	s2	PROPN
ejpam-5976	114	17	.	.	PUNCT
ejpam-5976	115	1	then	then	ADV
ejpam-5976	115	2	we	we	PRON
ejpam-5976	115	3	obtain	obtain	VERB
ejpam-5976	115	4	the	the	DET
ejpam-5976	115	5	following	following	NOUN
ejpam-5976	115	6	:	:	PUNCT
ejpam-5976	115	7	(	(	PUNCT
ejpam-5976	115	8	a	a	PRON
ejpam-5976	115	9	,	,	PUNCT
ejpam-5976	115	10	b	b	NOUN
ejpam-5976	115	11	)	)	PUNCT
ejpam-5976	115	12	∈	∈	PROPN
ejpam-5976	115	13	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	115	14	)	)	PUNCT
ejpam-5976	115	15	=	=	PUNCT
ejpam-5976	115	16	q((u	q((u	NOUN
ejpam-5976	115	17	,	,	PUNCT
ejpam-5976	115	18	v	v	NOUN
ejpam-5976	115	19	)	)	PUNCT
ejpam-5976	115	20	)	)	PUNCT
ejpam-5976	116	1	⊆	⊆	NUM
ejpam-5976	116	2	q(u)×q(v	q(u)×q(v	NOUN
ejpam-5976	116	3	)	)	PUNCT
ejpam-5976	116	4	(	(	PUNCT
ejpam-5976	116	5	u	u	NOUN
ejpam-5976	116	6	,	,	PUNCT
ejpam-5976	116	7	v	v	NOUN
ejpam-5976	116	8	)	)	PUNCT
ejpam-5976	116	9	∈	∈	PROPN
ejpam-5976	116	10	q((u	q((u	NOUN
ejpam-5976	116	11	,	,	PUNCT
ejpam-5976	116	12	v	v	NOUN
ejpam-5976	116	13	)	)	PUNCT
ejpam-5976	116	14	)	)	PUNCT
ejpam-5976	117	1	=	=	PUNCT
ejpam-5976	117	2	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	117	3	)	)	PUNCT
ejpam-5976	117	4	.	.	PUNCT
ejpam-5976	118	1	this	this	PRON
ejpam-5976	118	2	implies	imply	VERB
ejpam-5976	118	3	that	that	SCONJ
ejpam-5976	118	4	(	(	PUNCT
ejpam-5976	118	5	as1	as1	NOUN
ejpam-5976	118	6	∩	∩	X
ejpam-5976	118	7	s1a)×	s1a)×	X
ejpam-5976	118	8	(	(	PUNCT
ejpam-5976	118	9	bs2	bs2	PROPN
ejpam-5976	118	10	∩	∩	PROPN
ejpam-5976	118	11	s2b	s2b	PROPN
ejpam-5976	118	12	)	)	PUNCT
ejpam-5976	118	13	⊆	⊆	NUM
ejpam-5976	118	14	(	(	PUNCT
ejpam-5976	118	15	q(u)s1	q(u)s1	PROPN
ejpam-5976	118	16	∩	∩	ADJ
ejpam-5976	118	17	s1q(u))×	s1q(u))×	PROPN
ejpam-5976	118	18	(	(	PUNCT
ejpam-5976	118	19	q(v)s2	q(v)s2	PROPN
ejpam-5976	118	20	∩	∩	PROPN
ejpam-5976	118	21	s2q(v	s2q(v	PROPN
ejpam-5976	118	22	)	)	PUNCT
ejpam-5976	118	23	)	)	PUNCT
ejpam-5976	119	1	=	=	PRON
ejpam-5976	119	2	(	(	PUNCT
ejpam-5976	119	3	us1	us1	PROPN
ejpam-5976	119	4	∩	∩	PROPN
ejpam-5976	119	5	s1u)×	s1u)×	PROPN
ejpam-5976	119	6	(	(	PUNCT
ejpam-5976	119	7	vs2	vs2	NOUN
ejpam-5976	119	8	∩	∩	NOUN
ejpam-5976	119	9	s2v	s2v	NOUN
ejpam-5976	119	10	)	)	PUNCT
ejpam-5976	119	11	⊆	⊆	NUM
ejpam-5976	119	12	(	(	PUNCT
ejpam-5976	119	13	q(a)s1	q(a)s1	PROPN
ejpam-5976	119	14	∩	∩	PROPN
ejpam-5976	119	15	s1q(a))×	s1q(a))×	PROPN
ejpam-5976	119	16	(	(	PUNCT
ejpam-5976	119	17	q(b)s2	q(b)s2	PROPN
ejpam-5976	119	18	∩	∩	PROPN
ejpam-5976	119	19	s2q(b	s2q(b	NOUN
ejpam-5976	119	20	)	)	PUNCT
ejpam-5976	119	21	)	)	PUNCT
ejpam-5976	120	1	=	=	SYM
ejpam-5976	120	2	(	(	PUNCT
ejpam-5976	120	3	as1	as1	NOUN
ejpam-5976	120	4	∩	∩	X
ejpam-5976	120	5	s1a)×	s1a)×	X
ejpam-5976	120	6	(	(	PUNCT
ejpam-5976	120	7	bs2	bs2	PROPN
ejpam-5976	120	8	∩	∩	PROPN
ejpam-5976	120	9	s2b	s2b	PROPN
ejpam-5976	120	10	)	)	PUNCT
ejpam-5976	120	11	.	.	PUNCT
ejpam-5976	121	1	since	since	SCONJ
ejpam-5976	121	2	(	(	PUNCT
ejpam-5976	121	3	a	a	DET
ejpam-5976	121	4	,	,	PUNCT
ejpam-5976	121	5	b	b	NOUN
ejpam-5976	121	6	)	)	PUNCT
ejpam-5976	121	7	̸=	̸=	PROPN
ejpam-5976	121	8	(	(	PUNCT
ejpam-5976	121	9	u	u	NOUN
ejpam-5976	121	10	,	,	PUNCT
ejpam-5976	121	11	v	v	NOUN
ejpam-5976	121	12	)	)	PUNCT
ejpam-5976	121	13	and	and	CCONJ
ejpam-5976	121	14	(	(	PUNCT
ejpam-5976	121	15	a	a	PRON
ejpam-5976	121	16	,	,	PUNCT
ejpam-5976	121	17	b	b	NOUN
ejpam-5976	121	18	)	)	PUNCT
ejpam-5976	121	19	∈	∈	PROPN
ejpam-5976	121	20	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	121	21	)	)	PUNCT
ejpam-5976	121	22	=	=	PUNCT
ejpam-5976	121	23	q((u	q((u	NOUN
ejpam-5976	121	24	,	,	PUNCT
ejpam-5976	121	25	v	v	NOUN
ejpam-5976	121	26	)	)	PUNCT
ejpam-5976	121	27	)	)	PUNCT
ejpam-5976	121	28	,	,	PUNCT
ejpam-5976	121	29	it	it	PRON
ejpam-5976	121	30	follows	follow	VERB
ejpam-5976	121	31	that	that	SCONJ
ejpam-5976	121	32	(	(	PUNCT
ejpam-5976	121	33	a	a	PRON
ejpam-5976	121	34	,	,	PUNCT
ejpam-5976	121	35	b	b	NOUN
ejpam-5976	121	36	)	)	PUNCT
ejpam-5976	121	37	∈	∈	PROPN
ejpam-5976	121	38	(	(	PUNCT
ejpam-5976	121	39	us1	us1	PROPN
ejpam-5976	121	40	∩	∩	PROPN
ejpam-5976	121	41	s1u)×	s1u)×	PROPN
ejpam-5976	121	42	(	(	PUNCT
ejpam-5976	121	43	vs2	vs2	NOUN
ejpam-5976	121	44	∩	∩	NOUN
ejpam-5976	121	45	s2v	s2v	NOUN
ejpam-5976	121	46	)	)	PUNCT
ejpam-5976	121	47	=	=	SYM
ejpam-5976	121	48	(	(	PUNCT
ejpam-5976	121	49	as1	as1	NOUN
ejpam-5976	121	50	∩	∩	X
ejpam-5976	121	51	s1a)×	s1a)×	X
ejpam-5976	121	52	(	(	PUNCT
ejpam-5976	121	53	bs2	bs2	PROPN
ejpam-5976	121	54	∩	∩	PROPN
ejpam-5976	121	55	s2b	s2b	PROPN
ejpam-5976	121	56	)	)	PUNCT
ejpam-5976	121	57	.	.	PUNCT
ejpam-5976	122	1	by	by	ADP
ejpam-5976	122	2	theorem	theorem	NOUN
ejpam-5976	122	3	3	3	NUM
ejpam-5976	122	4	,	,	PUNCT
ejpam-5976	122	5	we	we	PRON
ejpam-5976	122	6	have	have	VERB
ejpam-5976	122	7	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	122	8	)	)	PUNCT
ejpam-5976	122	9	=	=	SYM
ejpam-5976	122	10	q((a	q((a	PROPN
ejpam-5976	122	11	,	,	PUNCT
ejpam-5976	122	12	b	b	NOUN
ejpam-5976	122	13	)	)	PUNCT
ejpam-5976	122	14	)	)	PUNCT
ejpam-5976	122	15	,	,	PUNCT
ejpam-5976	122	16	which	which	PRON
ejpam-5976	122	17	contradicts	contradict	VERB
ejpam-5976	122	18	the	the	DET
ejpam-5976	122	19	hypothesis	hypothesis	NOUN
ejpam-5976	122	20	.	.	PUNCT
ejpam-5976	123	1	p.	p.	PROPN
ejpam-5976	123	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	123	3	,	,	PUNCT
ejpam-5976	123	4	o.	o.	PROPN
ejpam-5976	123	5	pankoon	pankoon	NOUN
ejpam-5976	123	6	,	,	PUNCT
ejpam-5976	123	7	t.	t.	PROPN
ejpam-5976	123	8	changphas	changphas	PROPN
ejpam-5976	123	9	/	/	SYM
ejpam-5976	123	10	eur	eur	PROPN
ejpam-5976	123	11	.	.	PUNCT
ejpam-5976	124	1	j.	j.	PROPN
ejpam-5976	124	2	pure	pure	PROPN
ejpam-5976	124	3	appl	appl	PROPN
ejpam-5976	124	4	.	.	PROPN
ejpam-5976	124	5	math	math	PROPN
ejpam-5976	124	6	,	,	PUNCT
ejpam-5976	124	7	18	18	NUM
ejpam-5976	124	8	(	(	PUNCT
ejpam-5976	124	9	2	2	NUM
ejpam-5976	124	10	)	)	PUNCT
ejpam-5976	124	11	(	(	PUNCT
ejpam-5976	124	12	2025	2025	NUM
ejpam-5976	124	13	)	)	PUNCT
ejpam-5976	124	14	,	,	PUNCT
ejpam-5976	124	15	5976	5976	NUM
ejpam-5976	124	16	5	5	NUM
ejpam-5976	124	17	of	of	ADP
ejpam-5976	124	18	14	14	NUM
ejpam-5976	124	19	the	the	DET
ejpam-5976	124	20	concept	concept	NOUN
ejpam-5976	124	21	of	of	ADP
ejpam-5976	124	22	green	green	PROPN
ejpam-5976	124	23	’s	’s	PART
ejpam-5976	124	24	relations	relation	NOUN
ejpam-5976	124	25	were	be	AUX
ejpam-5976	124	26	introduced	introduce	VERB
ejpam-5976	124	27	by	by	ADP
ejpam-5976	124	28	green	green	ADJ
ejpam-5976	124	29	[	[	X
ejpam-5976	124	30	8	8	NUM
ejpam-5976	124	31	]	]	PUNCT
ejpam-5976	124	32	.	.	PUNCT
ejpam-5976	125	1	it	it	PRON
ejpam-5976	125	2	can	can	AUX
ejpam-5976	125	3	be	be	AUX
ejpam-5976	125	4	found	find	VERB
ejpam-5976	125	5	in	in	ADP
ejpam-5976	125	6	many	many	ADJ
ejpam-5976	125	7	textbooks	textbook	NOUN
ejpam-5976	125	8	on	on	ADP
ejpam-5976	125	9	semigroup	semigroup	PROPN
ejpam-5976	125	10	theory	theory	NOUN
ejpam-5976	125	11	(	(	PUNCT
ejpam-5976	125	12	e.g.	e.g.	ADV
ejpam-5976	125	13	[	[	X
ejpam-5976	125	14	9	9	NUM
ejpam-5976	125	15	]	]	PUNCT
ejpam-5976	125	16	,	,	PUNCT
ejpam-5976	125	17	[	[	X
ejpam-5976	125	18	10	10	NUM
ejpam-5976	125	19	]	]	NUM
ejpam-5976	125	20	)	)	PUNCT
ejpam-5976	125	21	.	.	PUNCT
ejpam-5976	126	1	equivalence	equivalence	NOUN
ejpam-5976	126	2	relations	relation	NOUN
ejpam-5976	126	3	l	l	PROPN
ejpam-5976	126	4	and	and	CCONJ
ejpam-5976	126	5	r	r	NOUN
ejpam-5976	126	6	on	on	ADP
ejpam-5976	126	7	a	a	DET
ejpam-5976	126	8	semigroup	semigroup	NOUN
ejpam-5976	126	9	s	s	PRON
ejpam-5976	126	10	are	be	AUX
ejpam-5976	126	11	defined	define	VERB
ejpam-5976	126	12	by	by	ADP
ejpam-5976	126	13	alb	alb	PROPN
ejpam-5976	126	14	⇔	⇔	PROPN
ejpam-5976	126	15	l(a	l(a	PROPN
ejpam-5976	126	16	)	)	PUNCT
ejpam-5976	126	17	=	=	SYM
ejpam-5976	126	18	l(b	l(b	PROPN
ejpam-5976	126	19	)	)	PUNCT
ejpam-5976	126	20	arb	arb	PROPN
ejpam-5976	126	21	⇔	⇔	PROPN
ejpam-5976	126	22	r(a	r(a	PROPN
ejpam-5976	126	23	)	)	PUNCT
ejpam-5976	126	24	=	=	SYM
ejpam-5976	126	25	r(b	r(b	PROPN
ejpam-5976	126	26	)	)	PUNCT
ejpam-5976	126	27	,	,	PUNCT
ejpam-5976	126	28	where	where	SCONJ
ejpam-5976	126	29	l(u	l(u	PROPN
ejpam-5976	126	30	)	)	PUNCT
ejpam-5976	126	31	(	(	PUNCT
ejpam-5976	126	32	respectively	respectively	ADV
ejpam-5976	126	33	,	,	PUNCT
ejpam-5976	126	34	r(u	r(u	PROPN
ejpam-5976	126	35	)	)	PUNCT
ejpam-5976	126	36	)	)	PUNCT
ejpam-5976	126	37	is	be	AUX
ejpam-5976	126	38	the	the	DET
ejpam-5976	126	39	principal	principal	NOUN
ejpam-5976	126	40	left	leave	VERB
ejpam-5976	126	41	(	(	PUNCT
ejpam-5976	126	42	respectively	respectively	ADV
ejpam-5976	126	43	,	,	PUNCT
ejpam-5976	126	44	right	right	ADJ
ejpam-5976	126	45	)	)	PUNCT
ejpam-5976	126	46	ideal	ideal	NOUN
ejpam-5976	126	47	of	of	ADP
ejpam-5976	126	48	s	s	PRON
ejpam-5976	126	49	containing	contain	VERB
ejpam-5976	126	50	u.	u.	NOUN
ejpam-5976	126	51	an	an	DET
ejpam-5976	126	52	equivalence	equivalence	NOUN
ejpam-5976	126	53	relation	relation	NOUN
ejpam-5976	126	54	h	h	NOUN
ejpam-5976	126	55	on	on	ADP
ejpam-5976	126	56	s	s	PROPN
ejpam-5976	126	57	is	be	AUX
ejpam-5976	126	58	define	define	ADJ
ejpam-5976	126	59	by	by	ADP
ejpam-5976	126	60	h	h	NOUN
ejpam-5976	126	61	=	=	SYM
ejpam-5976	126	62	l	l	NOUN
ejpam-5976	126	63	∩r	∩r	NOUN
ejpam-5976	126	64	.	.	PUNCT
ejpam-5976	127	1	for	for	ADP
ejpam-5976	127	2	any	any	DET
ejpam-5976	127	3	a	a	DET
ejpam-5976	127	4	∈	∈	ADJ
ejpam-5976	127	5	s	s	NOUN
ejpam-5976	127	6	,	,	PUNCT
ejpam-5976	127	7	let	let	VERB
ejpam-5976	127	8	ha	ha	INTJ
ejpam-5976	127	9	denote	denote	VERB
ejpam-5976	127	10	an	an	DET
ejpam-5976	127	11	h	h	NOUN
ejpam-5976	127	12	-	-	PUNCT
ejpam-5976	127	13	class	class	NOUN
ejpam-5976	127	14	of	of	ADP
ejpam-5976	127	15	s	s	PRON
ejpam-5976	127	16	containing	contain	VERB
ejpam-5976	127	17	a	a	PRON
ejpam-5976	127	18	,	,	PUNCT
ejpam-5976	127	19	that	that	ADV
ejpam-5976	127	20	is	is	ADV
ejpam-5976	127	21	,	,	PUNCT
ejpam-5976	127	22	ha	ha	INTJ
ejpam-5976	127	23	=	=	X
ejpam-5976	127	24	{	{	PUNCT
ejpam-5976	127	25	b	b	X
ejpam-5976	127	26	∈	∈	PROPN
ejpam-5976	127	27	s	s	VERB
ejpam-5976	127	28	|	|	NOUN
ejpam-5976	127	29	ahb	ahb	NOUN
ejpam-5976	127	30	}	}	PUNCT
ejpam-5976	127	31	.	.	PUNCT
ejpam-5976	128	1	lemma	lemma	PROPN
ejpam-5976	128	2	1	1	NUM
ejpam-5976	128	3	.	.	PUNCT
ejpam-5976	129	1	(	(	PUNCT
ejpam-5976	129	2	[	[	X
ejpam-5976	129	3	2	2	NUM
ejpam-5976	129	4	]	]	PUNCT
ejpam-5976	129	5	)	)	PUNCT
ejpam-5976	129	6	let	let	VERB
ejpam-5976	129	7	a	a	DET
ejpam-5976	129	8	,	,	PUNCT
ejpam-5976	129	9	b	b	X
ejpam-5976	129	10	∈	∈	PROPN
ejpam-5976	129	11	s.	s.	PROPN
ejpam-5976	129	12	then	then	ADV
ejpam-5976	129	13	ahb	ahb	PRON
ejpam-5976	130	1	if	if	SCONJ
ejpam-5976	130	2	and	and	CCONJ
ejpam-5976	130	3	only	only	ADV
ejpam-5976	130	4	if	if	SCONJ
ejpam-5976	130	5	q(a	q(a	NOUN
ejpam-5976	130	6	)	)	PUNCT
ejpam-5976	130	7	=	=	SYM
ejpam-5976	130	8	q(b	q(b	ADJ
ejpam-5976	130	9	)	)	PUNCT
ejpam-5976	130	10	.	.	PUNCT
ejpam-5976	131	1	this	this	PRON
ejpam-5976	131	2	implies	imply	VERB
ejpam-5976	131	3	that	that	SCONJ
ejpam-5976	131	4	ha	ha	INTJ
ejpam-5976	131	5	=	=	SYM
ejpam-5976	131	6	hb	hb	X
ejpam-5976	131	7	if	if	SCONJ
ejpam-5976	131	8	and	and	CCONJ
ejpam-5976	131	9	only	only	ADV
ejpam-5976	131	10	if	if	SCONJ
ejpam-5976	131	11	q(a	q(a	NOUN
ejpam-5976	131	12	)	)	PUNCT
ejpam-5976	131	13	=	=	SYM
ejpam-5976	131	14	q(b	q(b	ADJ
ejpam-5976	131	15	)	)	PUNCT
ejpam-5976	131	16	.	.	PUNCT
ejpam-5976	132	1	we	we	PRON
ejpam-5976	132	2	give	give	VERB
ejpam-5976	132	3	an	an	DET
ejpam-5976	132	4	example	example	NOUN
ejpam-5976	132	5	to	to	PART
ejpam-5976	132	6	show	show	VERB
ejpam-5976	132	7	that	that	SCONJ
ejpam-5976	132	8	the	the	DET
ejpam-5976	132	9	direct	direct	ADJ
ejpam-5976	132	10	product	product	NOUN
ejpam-5976	132	11	of	of	ADP
ejpam-5976	132	12	two	two	NUM
ejpam-5976	132	13	h	h	NOUN
ejpam-5976	132	14	-	-	PUNCT
ejpam-5976	132	15	classes	class	NOUN
ejpam-5976	132	16	need	need	VERB
ejpam-5976	132	17	not	not	PART
ejpam-5976	132	18	to	to	PART
ejpam-5976	132	19	be	be	AUX
ejpam-5976	132	20	an	an	DET
ejpam-5976	132	21	h	h	NOUN
ejpam-5976	132	22	-	-	PUNCT
ejpam-5976	132	23	class	class	NOUN
ejpam-5976	132	24	as	as	ADP
ejpam-5976	132	25	the	the	DET
ejpam-5976	132	26	following	follow	VERB
ejpam-5976	132	27	illustrative	illustrative	ADJ
ejpam-5976	132	28	example	example	NOUN
ejpam-5976	132	29	.	.	PUNCT
ejpam-5976	133	1	example	example	NOUN
ejpam-5976	134	1	2	2	NUM
ejpam-5976	134	2	.	.	PUNCT
ejpam-5976	135	1	(	(	PUNCT
ejpam-5976	135	2	[	[	X
ejpam-5976	135	3	7	7	NUM
ejpam-5976	135	4	]	]	PUNCT
ejpam-5976	135	5	)	)	PUNCT
ejpam-5976	135	6	let	let	VERB
ejpam-5976	135	7	(	(	PUNCT
ejpam-5976	135	8	s1	s1	NOUN
ejpam-5976	135	9	,	,	PUNCT
ejpam-5976	135	10	∗	∗	NOUN
ejpam-5976	135	11	)	)	PUNCT
ejpam-5976	135	12	and	and	CCONJ
ejpam-5976	135	13	(	(	PUNCT
ejpam-5976	135	14	s2	s2	PROPN
ejpam-5976	135	15	,	,	PUNCT
ejpam-5976	135	16	·	·	PUNCT
ejpam-5976	135	17	)	)	PUNCT
ejpam-5976	135	18	be	be	AUX
ejpam-5976	135	19	semigroups	semigroup	NOUN
ejpam-5976	135	20	where	where	SCONJ
ejpam-5976	135	21	s1	s1	NOUN
ejpam-5976	135	22	=	=	PUNCT
ejpam-5976	135	23	{	{	PUNCT
ejpam-5976	135	24	a1	a1	PROPN
ejpam-5976	135	25	,	,	PUNCT
ejpam-5976	135	26	a2	a2	PROPN
ejpam-5976	135	27	,	,	PUNCT
ejpam-5976	135	28	a3	a3	NOUN
ejpam-5976	135	29	,	,	PUNCT
ejpam-5976	135	30	a4	a4	NOUN
ejpam-5976	135	31	}	}	PUNCT
ejpam-5976	135	32	and	and	CCONJ
ejpam-5976	135	33	s2	s2	VERB
ejpam-5976	135	34	=	=	SYM
ejpam-5976	135	35	{	{	PUNCT
ejpam-5976	135	36	b1	b1	PROPN
ejpam-5976	135	37	,	,	PUNCT
ejpam-5976	135	38	b2	b2	NOUN
ejpam-5976	135	39	,	,	PUNCT
ejpam-5976	135	40	b3	b3	NOUN
ejpam-5976	135	41	,	,	PUNCT
ejpam-5976	135	42	b4	b4	NOUN
ejpam-5976	135	43	}	}	PUNCT
ejpam-5976	135	44	together	together	ADV
ejpam-5976	135	45	with	with	ADP
ejpam-5976	135	46	∗	∗	NOUN
ejpam-5976	135	47	:	:	PUNCT
ejpam-5976	135	48	s1	s1	PROPN
ejpam-5976	135	49	×	×	PROPN
ejpam-5976	135	50	s1	s1	PROPN
ejpam-5976	135	51	→	→	SYM
ejpam-5976	135	52	s1	s1	PROPN
ejpam-5976	135	53	and	and	CCONJ
ejpam-5976	135	54	·	·	PUNCT
ejpam-5976	135	55	:	:	PUNCT
ejpam-5976	135	56	s2	s2	VERB
ejpam-5976	135	57	×	×	PROPN
ejpam-5976	135	58	s2	s2	PROPN
ejpam-5976	135	59	→	→	SYM
ejpam-5976	135	60	s2	s2	NOUN
ejpam-5976	135	61	defined	define	VERB
ejpam-5976	135	62	by	by	ADP
ejpam-5976	135	63	∗	∗	NOUN
ejpam-5976	135	64	a1	a1	NOUN
ejpam-5976	135	65	a2	a2	PROPN
ejpam-5976	135	66	a3	a3	PROPN
ejpam-5976	135	67	a4	a4	CCONJ
ejpam-5976	135	68	a1	a1	NOUN
ejpam-5976	135	69	a1	a1	NOUN
ejpam-5976	135	70	a1	a1	NOUN
ejpam-5976	135	71	a1	a1	NOUN
ejpam-5976	135	72	a1	a1	NOUN
ejpam-5976	135	73	a2	a2	PROPN
ejpam-5976	135	74	a1	a1	PROPN
ejpam-5976	135	75	a2	a2	PROPN
ejpam-5976	135	76	a3	a3	PROPN
ejpam-5976	135	77	a4	a4	PROPN
ejpam-5976	135	78	a3	a3	NOUN
ejpam-5976	135	79	a1	a1	NOUN
ejpam-5976	135	80	a3	a3	NOUN
ejpam-5976	135	81	a2	a2	PROPN
ejpam-5976	135	82	a4	a4	NOUN
ejpam-5976	135	83	a4	a4	NOUN
ejpam-5976	135	84	a4	a4	NOUN
ejpam-5976	135	85	a4	a4	NOUN
ejpam-5976	135	86	a4	a4	NOUN
ejpam-5976	135	87	a4	a4	NOUN
ejpam-5976	135	88	·	·	PUNCT
ejpam-5976	135	89	b1	b1	NOUN
ejpam-5976	135	90	b2	b2	NOUN
ejpam-5976	135	91	b3	b3	PROPN
ejpam-5976	135	92	b4	b4	NOUN
ejpam-5976	135	93	b1	b1	VERB
ejpam-5976	135	94	b1	b1	PROPN
ejpam-5976	135	95	b1	b1	PROPN
ejpam-5976	135	96	b3	b3	PROPN
ejpam-5976	135	97	b4	b4	PROPN
ejpam-5976	135	98	b2	b2	NOUN
ejpam-5976	135	99	b1	b1	NOUN
ejpam-5976	135	100	b1	b1	PROPN
ejpam-5976	135	101	b3	b3	PROPN
ejpam-5976	135	102	b4	b4	PROPN
ejpam-5976	135	103	b3	b3	PROPN
ejpam-5976	135	104	b3	b3	PROPN
ejpam-5976	135	105	b3	b3	PROPN
ejpam-5976	135	106	b4	b4	PROPN
ejpam-5976	135	107	b1	b1	NOUN
ejpam-5976	135	108	b4	b4	PROPN
ejpam-5976	135	109	b4	b4	PROPN
ejpam-5976	135	110	b4	b4	PROPN
ejpam-5976	135	111	b1	b1	PROPN
ejpam-5976	135	112	b3	b3	PROPN
ejpam-5976	135	113	since	since	SCONJ
ejpam-5976	135	114	(	(	PUNCT
ejpam-5976	135	115	a3	a3	NOUN
ejpam-5976	135	116	,	,	PUNCT
ejpam-5976	135	117	b2	b2	NOUN
ejpam-5976	135	118	)	)	PUNCT
ejpam-5976	135	119	∈	∈	PROPN
ejpam-5976	135	120	ha2×hb2	ha2×hb2	NOUN
ejpam-5976	135	121	and	and	CCONJ
ejpam-5976	135	122	(	(	PUNCT
ejpam-5976	135	123	a3	a3	NOUN
ejpam-5976	135	124	,	,	PUNCT
ejpam-5976	135	125	b2	b2	NOUN
ejpam-5976	135	126	)	)	PUNCT
ejpam-5976	135	127	/∈	/∈	PUNCT
ejpam-5976	136	1	h(a2,b2	h(a2,b2	PROPN
ejpam-5976	136	2	)	)	PUNCT
ejpam-5976	136	3	,	,	PUNCT
ejpam-5976	136	4	we	we	PRON
ejpam-5976	136	5	have	have	VERB
ejpam-5976	136	6	that	that	DET
ejpam-5976	136	7	ha2×hb2	ha2×hb2	NOUN
ejpam-5976	137	1	⊈	⊈	PROPN
ejpam-5976	137	2	h(a2,b2	h(a2,b2	NOUN
ejpam-5976	137	3	)	)	PUNCT
ejpam-5976	137	4	.	.	PUNCT
ejpam-5976	138	1	thus	thus	ADV
ejpam-5976	138	2	,	,	PUNCT
ejpam-5976	138	3	ha2	ha2	PROPN
ejpam-5976	138	4	×hb2	×hb2	PUNCT
ejpam-5976	138	5	is	be	AUX
ejpam-5976	138	6	not	not	PART
ejpam-5976	138	7	an	an	DET
ejpam-5976	138	8	h	h	NOUN
ejpam-5976	138	9	-	-	PUNCT
ejpam-5976	138	10	class	class	NOUN
ejpam-5976	138	11	of	of	ADP
ejpam-5976	138	12	s1	s1	PROPN
ejpam-5976	138	13	×	×	PROPN
ejpam-5976	138	14	s2	s2	PROPN
ejpam-5976	138	15	.	.	PUNCT
ejpam-5976	139	1	theorem	theorem	VERB
ejpam-5976	139	2	5	5	NUM
ejpam-5976	139	3	.	.	PUNCT
ejpam-5976	140	1	let	let	VERB
ejpam-5976	140	2	(	(	PUNCT
ejpam-5976	140	3	a	a	PRON
ejpam-5976	140	4	,	,	PUNCT
ejpam-5976	140	5	b	b	NOUN
ejpam-5976	140	6	)	)	PUNCT
ejpam-5976	140	7	∈	∈	PROPN
ejpam-5976	140	8	s1	s1	PROPN
ejpam-5976	140	9	×	×	PROPN
ejpam-5976	140	10	s2	s2	PROPN
ejpam-5976	140	11	.	.	PUNCT
ejpam-5976	141	1	then	then	ADV
ejpam-5976	141	2	(	(	PUNCT
ejpam-5976	141	3	1	1	X
ejpam-5976	141	4	)	)	PUNCT
ejpam-5976	141	5	h(a	h(a	PROPN
ejpam-5976	141	6	,	,	PUNCT
ejpam-5976	141	7	b	b	NOUN
ejpam-5976	141	8	)	)	PUNCT
ejpam-5976	141	9	⊆	⊆	NUM
ejpam-5976	141	10	ha	ha	INTJ
ejpam-5976	141	11	×hb	×hb	PROPN
ejpam-5976	141	12	;	;	PUNCT
ejpam-5976	141	13	(	(	PUNCT
ejpam-5976	141	14	2	2	X
ejpam-5976	141	15	)	)	PUNCT
ejpam-5976	141	16	if	if	SCONJ
ejpam-5976	141	17	h(a	h(a	PROPN
ejpam-5976	141	18	,	,	PUNCT
ejpam-5976	141	19	b	b	NOUN
ejpam-5976	141	20	)	)	PUNCT
ejpam-5976	141	21	⊊	⊊	VERB
ejpam-5976	141	22	ha	ha	INTJ
ejpam-5976	141	23	×hb	×hb	PROPN
ejpam-5976	141	24	,	,	PUNCT
ejpam-5976	141	25	then	then	ADV
ejpam-5976	141	26	ha	ha	INTJ
ejpam-5976	141	27	×hb	×hb	PROPN
ejpam-5976	141	28	is	be	AUX
ejpam-5976	141	29	a	a	DET
ejpam-5976	141	30	union	union	NOUN
ejpam-5976	141	31	of	of	ADP
ejpam-5976	141	32	at	at	ADV
ejpam-5976	141	33	least	least	ADV
ejpam-5976	141	34	two	two	NUM
ejpam-5976	141	35	h	h	NOUN
ejpam-5976	141	36	-	-	PUNCT
ejpam-5976	141	37	classes	class	NOUN
ejpam-5976	141	38	.	.	PUNCT
ejpam-5976	142	1	proof	proof	NOUN
ejpam-5976	142	2	.	.	PUNCT
ejpam-5976	143	1	(	(	PUNCT
ejpam-5976	143	2	1	1	NUM
ejpam-5976	143	3	)	)	PUNCT
ejpam-5976	143	4	.	.	PUNCT
ejpam-5976	144	1	let	let	VERB
ejpam-5976	144	2	(	(	PUNCT
ejpam-5976	144	3	u	u	NOUN
ejpam-5976	144	4	,	,	PUNCT
ejpam-5976	144	5	v	v	NOUN
ejpam-5976	144	6	)	)	PUNCT
ejpam-5976	144	7	∈	∈	PROPN
ejpam-5976	144	8	h(a	h(a	PROPN
ejpam-5976	144	9	,	,	PUNCT
ejpam-5976	144	10	b	b	NOUN
ejpam-5976	144	11	)	)	PUNCT
ejpam-5976	144	12	.	.	PUNCT
ejpam-5976	145	1	then	then	ADV
ejpam-5976	145	2	,	,	PUNCT
ejpam-5976	145	3	q((u	q((u	ADV
ejpam-5976	145	4	,	,	PUNCT
ejpam-5976	145	5	v	v	NOUN
ejpam-5976	145	6	)	)	PUNCT
ejpam-5976	145	7	)	)	PUNCT
ejpam-5976	146	1	=	=	SYM
ejpam-5976	146	2	q((a	q((a	PROPN
ejpam-5976	146	3	,	,	PUNCT
ejpam-5976	146	4	b	b	NOUN
ejpam-5976	146	5	)	)	PUNCT
ejpam-5976	146	6	)	)	PUNCT
ejpam-5976	146	7	.	.	PUNCT
ejpam-5976	147	1	we	we	PRON
ejpam-5976	147	2	obtain	obtain	VERB
ejpam-5976	147	3	that	that	PRON
ejpam-5976	147	4	q(u	q(u	ADP
ejpam-5976	147	5	)	)	PUNCT
ejpam-5976	147	6	=	=	SYM
ejpam-5976	147	7	q(a	q(a	PROPN
ejpam-5976	147	8	)	)	PUNCT
ejpam-5976	147	9	and	and	CCONJ
ejpam-5976	147	10	q(v	q(v	NOUN
ejpam-5976	147	11	)	)	PUNCT
ejpam-5976	147	12	=	=	PUNCT
ejpam-5976	147	13	q(b	q(b	ADJ
ejpam-5976	147	14	)	)	PUNCT
ejpam-5976	147	15	.	.	PUNCT
ejpam-5976	148	1	indeed	indeed	ADV
ejpam-5976	148	2	,	,	PUNCT
ejpam-5976	148	3	(	(	PUNCT
ejpam-5976	148	4	u	u	NOUN
ejpam-5976	148	5	,	,	PUNCT
ejpam-5976	148	6	v	v	NOUN
ejpam-5976	148	7	)	)	PUNCT
ejpam-5976	148	8	∈	∈	PROPN
ejpam-5976	148	9	q((u	q((u	NOUN
ejpam-5976	148	10	,	,	PUNCT
ejpam-5976	148	11	v	v	NOUN
ejpam-5976	148	12	)	)	PUNCT
ejpam-5976	148	13	)	)	PUNCT
ejpam-5976	149	1	=	=	SYM
ejpam-5976	149	2	q((a	q((a	PROPN
ejpam-5976	149	3	,	,	PUNCT
ejpam-5976	149	4	b	b	NOUN
ejpam-5976	149	5	)	)	PUNCT
ejpam-5976	149	6	)	)	PUNCT
ejpam-5976	150	1	⊆	⊆	NUM
ejpam-5976	150	2	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	150	3	)	)	PUNCT
ejpam-5976	150	4	p.	p.	NOUN
ejpam-5976	150	5	luangchaisri	luangchaisri	PROPN
ejpam-5976	150	6	,	,	PUNCT
ejpam-5976	150	7	o.	o.	PROPN
ejpam-5976	150	8	pankoon	pankoon	NOUN
ejpam-5976	150	9	,	,	PUNCT
ejpam-5976	150	10	t.	t.	PROPN
ejpam-5976	150	11	changphas	changphas	PROPN
ejpam-5976	150	12	/	/	SYM
ejpam-5976	150	13	eur	eur	PROPN
ejpam-5976	150	14	.	.	PUNCT
ejpam-5976	151	1	j.	j.	PROPN
ejpam-5976	151	2	pure	pure	PROPN
ejpam-5976	151	3	appl	appl	PROPN
ejpam-5976	151	4	.	.	PROPN
ejpam-5976	151	5	math	math	PROPN
ejpam-5976	151	6	,	,	PUNCT
ejpam-5976	151	7	18	18	NUM
ejpam-5976	151	8	(	(	PUNCT
ejpam-5976	151	9	2	2	NUM
ejpam-5976	151	10	)	)	PUNCT
ejpam-5976	151	11	(	(	PUNCT
ejpam-5976	151	12	2025	2025	NUM
ejpam-5976	151	13	)	)	PUNCT
ejpam-5976	151	14	,	,	PUNCT
ejpam-5976	151	15	5976	5976	NUM
ejpam-5976	151	16	6	6	NUM
ejpam-5976	151	17	of	of	ADP
ejpam-5976	151	18	14	14	NUM
ejpam-5976	151	19	and	and	CCONJ
ejpam-5976	151	20	(	(	PUNCT
ejpam-5976	151	21	a	a	PRON
ejpam-5976	151	22	,	,	PUNCT
ejpam-5976	151	23	b	b	NOUN
ejpam-5976	151	24	)	)	PUNCT
ejpam-5976	151	25	∈	∈	PROPN
ejpam-5976	151	26	q((a	q((a	PROPN
ejpam-5976	151	27	,	,	PUNCT
ejpam-5976	151	28	b	b	NOUN
ejpam-5976	151	29	)	)	PUNCT
ejpam-5976	151	30	)	)	PUNCT
ejpam-5976	152	1	=	=	SYM
ejpam-5976	152	2	q((u	q((u	NOUN
ejpam-5976	152	3	,	,	PUNCT
ejpam-5976	152	4	v	v	NOUN
ejpam-5976	152	5	)	)	PUNCT
ejpam-5976	152	6	)	)	PUNCT
ejpam-5976	153	1	⊆	⊆	NUM
ejpam-5976	153	2	q(u)×q(v	q(u)×q(v	NOUN
ejpam-5976	153	3	)	)	PUNCT
ejpam-5976	153	4	.	.	PUNCT
ejpam-5976	154	1	by	by	ADP
ejpam-5976	154	2	lemma	lemma	PROPN
ejpam-5976	154	3	1	1	NUM
ejpam-5976	154	4	,	,	PUNCT
ejpam-5976	154	5	we	we	PRON
ejpam-5976	154	6	have	have	VERB
ejpam-5976	154	7	(	(	PUNCT
ejpam-5976	154	8	u	u	NOUN
ejpam-5976	154	9	,	,	PUNCT
ejpam-5976	154	10	v	v	NOUN
ejpam-5976	154	11	)	)	PUNCT
ejpam-5976	154	12	∈	∈	PROPN
ejpam-5976	154	13	ha	ha	INTJ
ejpam-5976	154	14	×hb	×hb	PROPN
ejpam-5976	154	15	.	.	PUNCT
ejpam-5976	155	1	(	(	PUNCT
ejpam-5976	155	2	2	2	NUM
ejpam-5976	155	3	)	)	PUNCT
ejpam-5976	155	4	.	.	PUNCT
ejpam-5976	156	1	assume	assume	VERB
ejpam-5976	156	2	that	that	SCONJ
ejpam-5976	156	3	h(a	h(a	PROPN
ejpam-5976	156	4	,	,	PUNCT
ejpam-5976	156	5	b	b	NOUN
ejpam-5976	156	6	)	)	PUNCT
ejpam-5976	156	7	⊊	⊊	VERB
ejpam-5976	156	8	ha	ha	INTJ
ejpam-5976	156	9	×hb	×hb	PROPN
ejpam-5976	156	10	.	.	PUNCT
ejpam-5976	157	1	there	there	PRON
ejpam-5976	157	2	is	be	VERB
ejpam-5976	157	3	(	(	PUNCT
ejpam-5976	157	4	u	u	NOUN
ejpam-5976	157	5	,	,	PUNCT
ejpam-5976	157	6	v	v	NOUN
ejpam-5976	157	7	)	)	PUNCT
ejpam-5976	157	8	∈	∈	PROPN
ejpam-5976	157	9	s1	s1	NOUN
ejpam-5976	157	10	×	×	NOUN
ejpam-5976	157	11	s2	s2	NOUN
ejpam-5976	157	12	such	such	ADJ
ejpam-5976	157	13	that	that	SCONJ
ejpam-5976	157	14	(	(	PUNCT
ejpam-5976	157	15	u	u	NOUN
ejpam-5976	157	16	,	,	PUNCT
ejpam-5976	157	17	v	v	NOUN
ejpam-5976	157	18	)	)	PUNCT
ejpam-5976	157	19	∈	∈	PROPN
ejpam-5976	157	20	ha	ha	INTJ
ejpam-5976	157	21	×hb	×hb	PROPN
ejpam-5976	157	22	and	and	CCONJ
ejpam-5976	157	23	(	(	PUNCT
ejpam-5976	157	24	u	u	NOUN
ejpam-5976	157	25	,	,	PUNCT
ejpam-5976	157	26	v	v	NOUN
ejpam-5976	157	27	)	)	PUNCT
ejpam-5976	157	28	/∈	/∈	PUNCT
ejpam-5976	158	1	h(a	h(a	PROPN
ejpam-5976	158	2	,	,	PUNCT
ejpam-5976	158	3	b	b	NOUN
ejpam-5976	158	4	)	)	PUNCT
ejpam-5976	158	5	.	.	PUNCT
ejpam-5976	159	1	by	by	ADP
ejpam-5976	159	2	(	(	PUNCT
ejpam-5976	159	3	1	1	NUM
ejpam-5976	159	4	)	)	PUNCT
ejpam-5976	159	5	,	,	PUNCT
ejpam-5976	159	6	we	we	PRON
ejpam-5976	159	7	have	have	VERB
ejpam-5976	159	8	that	that	SCONJ
ejpam-5976	159	9	h(u	h(u	PROPN
ejpam-5976	159	10	,	,	PUNCT
ejpam-5976	159	11	v	v	NOUN
ejpam-5976	159	12	)	)	PUNCT
ejpam-5976	159	13	⊆	⊆	NUM
ejpam-5976	159	14	hu	hu	PROPN
ejpam-5976	159	15	×hv	×hv	PROPN
ejpam-5976	159	16	=	=	PROPN
ejpam-5976	159	17	ha	ha	INTJ
ejpam-5976	159	18	×hb	×hb	PROPN
ejpam-5976	159	19	.	.	PROPN
ejpam-5976	160	1	therefore	therefore	ADV
ejpam-5976	160	2	,	,	PUNCT
ejpam-5976	160	3	h(a	h(a	PROPN
ejpam-5976	160	4	,	,	PUNCT
ejpam-5976	160	5	b	b	NOUN
ejpam-5976	160	6	)	)	PUNCT
ejpam-5976	160	7	and	and	CCONJ
ejpam-5976	160	8	h(u	h(u	PROPN
ejpam-5976	160	9	,	,	PUNCT
ejpam-5976	160	10	v	v	NOUN
ejpam-5976	160	11	)	)	PUNCT
ejpam-5976	160	12	are	be	AUX
ejpam-5976	160	13	difference	difference	NOUN
ejpam-5976	160	14	classes	class	NOUN
ejpam-5976	160	15	contained	contain	VERB
ejpam-5976	160	16	in	in	ADP
ejpam-5976	160	17	ha	ha	INTJ
ejpam-5976	160	18	×hb	×hb	PROPN
ejpam-5976	160	19	.	.	PROPN
ejpam-5976	160	20	theorem	theorem	NOUN
ejpam-5976	160	21	6	6	NUM
ejpam-5976	160	22	.	.	PUNCT
ejpam-5976	161	1	let	let	VERB
ejpam-5976	161	2	(	(	PUNCT
ejpam-5976	161	3	a	a	PRON
ejpam-5976	161	4	,	,	PUNCT
ejpam-5976	161	5	b	b	NOUN
ejpam-5976	161	6	)	)	PUNCT
ejpam-5976	161	7	∈	∈	PROPN
ejpam-5976	161	8	s1	s1	PROPN
ejpam-5976	161	9	×	×	PROPN
ejpam-5976	161	10	s2	s2	PROPN
ejpam-5976	161	11	.	.	PUNCT
ejpam-5976	162	1	then	then	ADV
ejpam-5976	162	2	h(a	h(a	PROPN
ejpam-5976	162	3	,	,	PUNCT
ejpam-5976	162	4	b	b	NOUN
ejpam-5976	162	5	)	)	PUNCT
ejpam-5976	162	6	=	=	NOUN
ejpam-5976	163	1	ha	ha	INTJ
ejpam-5976	163	2	×	×	INTJ
ejpam-5976	163	3	hb	hb	X
ejpam-5976	163	4	if	if	SCONJ
ejpam-5976	163	5	and	and	CCONJ
ejpam-5976	163	6	only	only	ADV
ejpam-5976	163	7	if	if	SCONJ
ejpam-5976	163	8	at	at	ADV
ejpam-5976	163	9	least	least	ADJ
ejpam-5976	163	10	one	one	NUM
ejpam-5976	163	11	of	of	ADP
ejpam-5976	163	12	the	the	DET
ejpam-5976	163	13	following	follow	VERB
ejpam-5976	163	14	conditions	condition	NOUN
ejpam-5976	163	15	holds	hold	VERB
ejpam-5976	163	16	:	:	PUNCT
ejpam-5976	163	17	(	(	PUNCT
ejpam-5976	163	18	1	1	X
ejpam-5976	163	19	)	)	PUNCT
ejpam-5976	163	20	ha	ha	NOUN
ejpam-5976	164	1	=	=	X
ejpam-5976	164	2	{	{	PUNCT
ejpam-5976	164	3	a	a	NOUN
ejpam-5976	164	4	}	}	PUNCT
ejpam-5976	164	5	and	and	CCONJ
ejpam-5976	164	6	hb	hb	X
ejpam-5976	164	7	=	=	PUNCT
ejpam-5976	164	8	{	{	PUNCT
ejpam-5976	164	9	b	b	NOUN
ejpam-5976	164	10	}	}	PUNCT
ejpam-5976	164	11	;	;	PUNCT
ejpam-5976	164	12	(	(	PUNCT
ejpam-5976	164	13	2	2	X
ejpam-5976	164	14	)	)	PUNCT
ejpam-5976	164	15	a	a	DET
ejpam-5976	164	16	∈	∈	PROPN
ejpam-5976	164	17	as1	as1	NOUN
ejpam-5976	164	18	∩	∩	PROPN
ejpam-5976	164	19	s1a	s1a	PROPN
ejpam-5976	164	20	and	and	CCONJ
ejpam-5976	164	21	b	b	X
ejpam-5976	164	22	∈	∈	PROPN
ejpam-5976	164	23	bs2	bs2	PROPN
ejpam-5976	164	24	∩	∩	PROPN
ejpam-5976	164	25	s2b	s2b	PROPN
ejpam-5976	164	26	.	.	PUNCT
ejpam-5976	165	1	proof	proof	NOUN
ejpam-5976	165	2	.	.	PUNCT
ejpam-5976	166	1	assume	assume	VERB
ejpam-5976	166	2	that	that	SCONJ
ejpam-5976	166	3	h(a	h(a	PROPN
ejpam-5976	166	4	,	,	PUNCT
ejpam-5976	166	5	b	b	NOUN
ejpam-5976	166	6	)	)	PUNCT
ejpam-5976	166	7	=	=	NOUN
ejpam-5976	166	8	ha	ha	INTJ
ejpam-5976	166	9	×hb	×hb	PROPN
ejpam-5976	166	10	.	.	PROPN
ejpam-5976	167	1	if	if	SCONJ
ejpam-5976	167	2	h(a	h(a	PROPN
ejpam-5976	167	3	,	,	PUNCT
ejpam-5976	167	4	b	b	NOUN
ejpam-5976	167	5	)	)	PUNCT
ejpam-5976	167	6	=	=	SYM
ejpam-5976	167	7	{	{	PUNCT
ejpam-5976	167	8	(	(	PUNCT
ejpam-5976	167	9	a	a	DET
ejpam-5976	167	10	,	,	PUNCT
ejpam-5976	167	11	b	b	NOUN
ejpam-5976	167	12	)	)	PUNCT
ejpam-5976	167	13	}	}	PUNCT
ejpam-5976	167	14	,	,	PUNCT
ejpam-5976	167	15	then	then	ADV
ejpam-5976	167	16	we	we	PRON
ejpam-5976	167	17	have	have	VERB
ejpam-5976	167	18	ha	ha	INTJ
ejpam-5976	167	19	×hb	×hb	PROPN
ejpam-5976	167	20	=	=	SYM
ejpam-5976	167	21	h(a	h(a	PROPN
ejpam-5976	167	22	,	,	PUNCT
ejpam-5976	167	23	b	b	NOUN
ejpam-5976	167	24	)	)	PUNCT
ejpam-5976	167	25	=	=	SYM
ejpam-5976	167	26	{	{	PUNCT
ejpam-5976	167	27	(	(	PUNCT
ejpam-5976	167	28	a	a	PRON
ejpam-5976	167	29	,	,	PUNCT
ejpam-5976	167	30	b	b	NOUN
ejpam-5976	167	31	)	)	PUNCT
ejpam-5976	167	32	}	}	PUNCT
ejpam-5976	167	33	=	=	SYM
ejpam-5976	167	34	{	{	PUNCT
ejpam-5976	167	35	a	a	PRON
ejpam-5976	167	36	}	}	PUNCT
ejpam-5976	167	37	×	×	NOUN
ejpam-5976	167	38	{	{	PUNCT
ejpam-5976	167	39	b	b	NOUN
ejpam-5976	167	40	}	}	PUNCT
ejpam-5976	167	41	.	.	PUNCT
ejpam-5976	168	1	hence	hence	ADV
ejpam-5976	168	2	,	,	PUNCT
ejpam-5976	168	3	ha	ha	INTJ
ejpam-5976	168	4	=	=	X
ejpam-5976	168	5	{	{	PUNCT
ejpam-5976	168	6	a	a	NOUN
ejpam-5976	168	7	}	}	PUNCT
ejpam-5976	168	8	and	and	CCONJ
ejpam-5976	168	9	hb	hb	X
ejpam-5976	168	10	=	=	PUNCT
ejpam-5976	168	11	{	{	PUNCT
ejpam-5976	168	12	b	b	NOUN
ejpam-5976	168	13	}	}	PUNCT
ejpam-5976	168	14	.	.	PUNCT
ejpam-5976	169	1	if	if	SCONJ
ejpam-5976	169	2	h(a	h(a	PROPN
ejpam-5976	169	3	,	,	PUNCT
ejpam-5976	169	4	b	b	NOUN
ejpam-5976	169	5	)	)	PUNCT
ejpam-5976	169	6	̸=	̸=	PROPN
ejpam-5976	169	7	{	{	PUNCT
ejpam-5976	169	8	(	(	PUNCT
ejpam-5976	169	9	a	a	DET
ejpam-5976	169	10	,	,	PUNCT
ejpam-5976	169	11	b	b	NOUN
ejpam-5976	169	12	)	)	PUNCT
ejpam-5976	169	13	}	}	PUNCT
ejpam-5976	169	14	,	,	PUNCT
ejpam-5976	169	15	then	then	ADV
ejpam-5976	169	16	there	there	PRON
ejpam-5976	169	17	exists	exist	VERB
ejpam-5976	169	18	(	(	PUNCT
ejpam-5976	169	19	u	u	NOUN
ejpam-5976	169	20	,	,	PUNCT
ejpam-5976	169	21	v	v	NOUN
ejpam-5976	169	22	)	)	PUNCT
ejpam-5976	169	23	∈	∈	PROPN
ejpam-5976	169	24	s1	s1	NOUN
ejpam-5976	169	25	×	×	NOUN
ejpam-5976	169	26	s2	s2	NOUN
ejpam-5976	169	27	such	such	ADJ
ejpam-5976	169	28	that	that	SCONJ
ejpam-5976	169	29	(	(	PUNCT
ejpam-5976	169	30	u	u	NOUN
ejpam-5976	169	31	,	,	PUNCT
ejpam-5976	169	32	v	v	NOUN
ejpam-5976	169	33	)	)	PUNCT
ejpam-5976	169	34	̸=	̸=	PROPN
ejpam-5976	169	35	(	(	PUNCT
ejpam-5976	169	36	a	a	DET
ejpam-5976	169	37	,	,	PUNCT
ejpam-5976	169	38	b	b	NOUN
ejpam-5976	169	39	)	)	PUNCT
ejpam-5976	169	40	and	and	CCONJ
ejpam-5976	169	41	q((u	q((u	NOUN
ejpam-5976	169	42	,	,	PUNCT
ejpam-5976	169	43	v	v	NOUN
ejpam-5976	169	44	)	)	PUNCT
ejpam-5976	169	45	)	)	PUNCT
ejpam-5976	170	1	=	=	SYM
ejpam-5976	170	2	q((a	q((a	PROPN
ejpam-5976	170	3	,	,	PUNCT
ejpam-5976	170	4	b	b	NOUN
ejpam-5976	170	5	)	)	PUNCT
ejpam-5976	170	6	)	)	PUNCT
ejpam-5976	170	7	.	.	PUNCT
ejpam-5976	171	1	since	since	SCONJ
ejpam-5976	171	2	(	(	PUNCT
ejpam-5976	171	3	u	u	NOUN
ejpam-5976	171	4	,	,	PUNCT
ejpam-5976	171	5	v	v	NOUN
ejpam-5976	171	6	)	)	PUNCT
ejpam-5976	171	7	∈	∈	PROPN
ejpam-5976	171	8	h(a	h(a	PROPN
ejpam-5976	171	9	,	,	PUNCT
ejpam-5976	171	10	b	b	NOUN
ejpam-5976	171	11	)	)	PUNCT
ejpam-5976	171	12	=	=	NOUN
ejpam-5976	171	13	ha	ha	INTJ
ejpam-5976	171	14	×hb	×hb	PROPN
ejpam-5976	171	15	,	,	PUNCT
ejpam-5976	171	16	we	we	PRON
ejpam-5976	171	17	have	have	VERB
ejpam-5976	171	18	q(u	q(u	NOUN
ejpam-5976	171	19	)	)	PUNCT
ejpam-5976	171	20	=	=	SYM
ejpam-5976	171	21	q(a	q(a	PROPN
ejpam-5976	171	22	)	)	PUNCT
ejpam-5976	171	23	and	and	CCONJ
ejpam-5976	171	24	q(v	q(v	NOUN
ejpam-5976	171	25	)	)	PUNCT
ejpam-5976	171	26	=	=	PUNCT
ejpam-5976	171	27	q(b	q(b	ADJ
ejpam-5976	171	28	)	)	PUNCT
ejpam-5976	171	29	.	.	PUNCT
ejpam-5976	172	1	then	then	ADV
ejpam-5976	172	2	(	(	PUNCT
ejpam-5976	172	3	us1	us1	PROPN
ejpam-5976	172	4	∩	∩	PROPN
ejpam-5976	172	5	s1u)×	s1u)×	PROPN
ejpam-5976	172	6	(	(	PUNCT
ejpam-5976	172	7	vs2	vs2	NOUN
ejpam-5976	172	8	∩	∩	NOUN
ejpam-5976	172	9	s2v	s2v	NOUN
ejpam-5976	172	10	)	)	PUNCT
ejpam-5976	172	11	=	=	SYM
ejpam-5976	172	12	(	(	PUNCT
ejpam-5976	172	13	as1	as1	NOUN
ejpam-5976	172	14	∩	∩	X
ejpam-5976	172	15	s1a)×	s1a)×	X
ejpam-5976	172	16	(	(	PUNCT
ejpam-5976	172	17	bs2	bs2	PROPN
ejpam-5976	172	18	∩	∩	PROPN
ejpam-5976	172	19	s2b	s2b	PROPN
ejpam-5976	172	20	)	)	PUNCT
ejpam-5976	172	21	.	.	PUNCT
ejpam-5976	173	1	the	the	DET
ejpam-5976	173	2	assumptions	assumption	NOUN
ejpam-5976	173	3	(	(	PUNCT
ejpam-5976	173	4	a	a	PRON
ejpam-5976	173	5	,	,	PUNCT
ejpam-5976	173	6	b	b	NOUN
ejpam-5976	173	7	)	)	PUNCT
ejpam-5976	173	8	∈	∈	PROPN
ejpam-5976	173	9	q((a	q((a	PROPN
ejpam-5976	173	10	,	,	PUNCT
ejpam-5976	173	11	b	b	NOUN
ejpam-5976	173	12	)	)	PUNCT
ejpam-5976	173	13	)	)	PUNCT
ejpam-5976	174	1	=	=	SYM
ejpam-5976	174	2	q((u	q((u	NOUN
ejpam-5976	174	3	,	,	PUNCT
ejpam-5976	174	4	v	v	NOUN
ejpam-5976	174	5	)	)	PUNCT
ejpam-5976	174	6	)	)	PUNCT
ejpam-5976	174	7	and	and	CCONJ
ejpam-5976	174	8	(	(	PUNCT
ejpam-5976	174	9	a	a	DET
ejpam-5976	174	10	,	,	PUNCT
ejpam-5976	174	11	b	b	NOUN
ejpam-5976	174	12	)	)	PUNCT
ejpam-5976	174	13	̸=	̸=	PROPN
ejpam-5976	174	14	(	(	PUNCT
ejpam-5976	174	15	u	u	NOUN
ejpam-5976	174	16	,	,	PUNCT
ejpam-5976	174	17	v	v	NOUN
ejpam-5976	174	18	)	)	PUNCT
ejpam-5976	174	19	lead	lead	NOUN
ejpam-5976	174	20	to	to	ADP
ejpam-5976	174	21	(	(	PUNCT
ejpam-5976	174	22	a	a	PRON
ejpam-5976	174	23	,	,	PUNCT
ejpam-5976	174	24	b	b	NOUN
ejpam-5976	174	25	)	)	PUNCT
ejpam-5976	174	26	∈	∈	PROPN
ejpam-5976	174	27	(	(	PUNCT
ejpam-5976	174	28	us1	us1	PROPN
ejpam-5976	174	29	∩	∩	PROPN
ejpam-5976	174	30	s1u)×	s1u)×	PROPN
ejpam-5976	174	31	(	(	PUNCT
ejpam-5976	174	32	vs2	vs2	NOUN
ejpam-5976	174	33	∩	∩	NOUN
ejpam-5976	174	34	s2v	s2v	NOUN
ejpam-5976	174	35	)	)	PUNCT
ejpam-5976	174	36	=	=	SYM
ejpam-5976	174	37	(	(	PUNCT
ejpam-5976	174	38	as1	as1	NOUN
ejpam-5976	174	39	∩	∩	X
ejpam-5976	174	40	s1a)×	s1a)×	X
ejpam-5976	174	41	(	(	PUNCT
ejpam-5976	174	42	bs2	bs2	PROPN
ejpam-5976	174	43	∩	∩	PROPN
ejpam-5976	174	44	s2b	s2b	PROPN
ejpam-5976	174	45	)	)	PUNCT
ejpam-5976	174	46	.	.	PUNCT
ejpam-5976	175	1	thus	thus	ADV
ejpam-5976	175	2	,	,	PUNCT
ejpam-5976	175	3	a	a	DET
ejpam-5976	175	4	∈	∈	PROPN
ejpam-5976	175	5	as1	as1	NOUN
ejpam-5976	175	6	∩	∩	PROPN
ejpam-5976	175	7	s1a	s1a	PROPN
ejpam-5976	175	8	and	and	CCONJ
ejpam-5976	175	9	b	b	X
ejpam-5976	175	10	∈	∈	PROPN
ejpam-5976	175	11	bs2	bs2	PROPN
ejpam-5976	175	12	∩	∩	PROPN
ejpam-5976	175	13	s2b	s2b	PROPN
ejpam-5976	175	14	.	.	PUNCT
ejpam-5976	176	1	conversely	conversely	ADV
ejpam-5976	176	2	,	,	PUNCT
ejpam-5976	176	3	let	let	VERB
ejpam-5976	176	4	(	(	PUNCT
ejpam-5976	176	5	u	u	NOUN
ejpam-5976	176	6	,	,	PUNCT
ejpam-5976	176	7	v	v	NOUN
ejpam-5976	176	8	)	)	PUNCT
ejpam-5976	176	9	∈	∈	PROPN
ejpam-5976	176	10	ha	ha	INTJ
ejpam-5976	176	11	×hb	×hb	PROPN
ejpam-5976	176	12	.	.	PUNCT
ejpam-5976	177	1	if	if	SCONJ
ejpam-5976	177	2	(	(	PUNCT
ejpam-5976	177	3	1	1	X
ejpam-5976	177	4	)	)	PUNCT
ejpam-5976	177	5	holds	hold	VERB
ejpam-5976	177	6	,	,	PUNCT
ejpam-5976	177	7	then	then	ADV
ejpam-5976	177	8	(	(	PUNCT
ejpam-5976	177	9	u	u	NOUN
ejpam-5976	177	10	,	,	PUNCT
ejpam-5976	177	11	v	v	NOUN
ejpam-5976	177	12	)	)	PUNCT
ejpam-5976	177	13	=	=	SYM
ejpam-5976	177	14	(	(	PUNCT
ejpam-5976	177	15	a	a	PRON
ejpam-5976	177	16	,	,	PUNCT
ejpam-5976	177	17	b	b	NOUN
ejpam-5976	177	18	)	)	PUNCT
ejpam-5976	177	19	∈	∈	PROPN
ejpam-5976	177	20	h(a	h(a	PROPN
ejpam-5976	177	21	,	,	PUNCT
ejpam-5976	177	22	b	b	NOUN
ejpam-5976	177	23	)	)	PUNCT
ejpam-5976	177	24	.	.	PUNCT
ejpam-5976	177	25	suppose	suppose	VERB
ejpam-5976	177	26	that	that	SCONJ
ejpam-5976	177	27	(	(	PUNCT
ejpam-5976	177	28	2	2	X
ejpam-5976	177	29	)	)	PUNCT
ejpam-5976	177	30	holds	hold	NOUN
ejpam-5976	177	31	.	.	PUNCT
ejpam-5976	178	1	by	by	ADP
ejpam-5976	178	2	theorem	theorem	NOUN
ejpam-5976	178	3	3	3	NUM
ejpam-5976	178	4	,	,	PUNCT
ejpam-5976	178	5	we	we	PRON
ejpam-5976	178	6	have	have	VERB
ejpam-5976	178	7	q((a	q((a	NOUN
ejpam-5976	178	8	,	,	PUNCT
ejpam-5976	178	9	b	b	NOUN
ejpam-5976	178	10	)	)	PUNCT
ejpam-5976	178	11	)	)	PUNCT
ejpam-5976	179	1	=	=	SYM
ejpam-5976	179	2	q(a	q(a	PROPN
ejpam-5976	179	3	)	)	PUNCT
ejpam-5976	179	4	×	×	NOUN
ejpam-5976	179	5	q(b	q(b	ADJ
ejpam-5976	179	6	)	)	PUNCT
ejpam-5976	179	7	.	.	PUNCT
ejpam-5976	180	1	since	since	SCONJ
ejpam-5976	180	2	(	(	PUNCT
ejpam-5976	180	3	u	u	NOUN
ejpam-5976	180	4	,	,	PUNCT
ejpam-5976	180	5	v	v	NOUN
ejpam-5976	180	6	)	)	PUNCT
ejpam-5976	180	7	∈	∈	PROPN
ejpam-5976	180	8	ha	ha	INTJ
ejpam-5976	180	9	×hb	×hb	PROPN
ejpam-5976	180	10	,	,	PUNCT
ejpam-5976	180	11	it	it	PRON
ejpam-5976	180	12	follows	follow	VERB
ejpam-5976	180	13	that	that	SCONJ
ejpam-5976	180	14	q(a	q(a	NOUN
ejpam-5976	180	15	)	)	PUNCT
ejpam-5976	180	16	=	=	SYM
ejpam-5976	181	1	q(u	q(u	NOUN
ejpam-5976	181	2	)	)	PUNCT
ejpam-5976	181	3	and	and	CCONJ
ejpam-5976	181	4	q(b	q(b	ADJ
ejpam-5976	181	5	)	)	PUNCT
ejpam-5976	181	6	=	=	SYM
ejpam-5976	181	7	q(v	q(v	NOUN
ejpam-5976	181	8	)	)	PUNCT
ejpam-5976	181	9	.	.	PUNCT
ejpam-5976	182	1	p.	p.	NOUN
ejpam-5976	182	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	182	3	,	,	PUNCT
ejpam-5976	182	4	o.	o.	PROPN
ejpam-5976	182	5	pankoon	pankoon	NOUN
ejpam-5976	182	6	,	,	PUNCT
ejpam-5976	182	7	t.	t.	PROPN
ejpam-5976	182	8	changphas	changphas	PROPN
ejpam-5976	182	9	/	/	SYM
ejpam-5976	182	10	eur	eur	PROPN
ejpam-5976	182	11	.	.	PUNCT
ejpam-5976	183	1	j.	j.	PROPN
ejpam-5976	183	2	pure	pure	PROPN
ejpam-5976	183	3	appl	appl	PROPN
ejpam-5976	183	4	.	.	PROPN
ejpam-5976	183	5	math	math	PROPN
ejpam-5976	183	6	,	,	PUNCT
ejpam-5976	183	7	18	18	NUM
ejpam-5976	183	8	(	(	PUNCT
ejpam-5976	183	9	2	2	NUM
ejpam-5976	183	10	)	)	PUNCT
ejpam-5976	183	11	(	(	PUNCT
ejpam-5976	183	12	2025	2025	NUM
ejpam-5976	183	13	)	)	PUNCT
ejpam-5976	183	14	,	,	PUNCT
ejpam-5976	183	15	5976	5976	NUM
ejpam-5976	183	16	7	7	NUM
ejpam-5976	183	17	of	of	ADP
ejpam-5976	183	18	14	14	NUM
ejpam-5976	183	19	thus	thus	ADV
ejpam-5976	183	20	,	,	PUNCT
ejpam-5976	183	21	q((u	q((u	NOUN
ejpam-5976	183	22	,	,	PUNCT
ejpam-5976	183	23	v	v	NOUN
ejpam-5976	183	24	)	)	PUNCT
ejpam-5976	183	25	)	)	PUNCT
ejpam-5976	184	1	⊆	⊆	NUM
ejpam-5976	184	2	q(u)×q(v	q(u)×q(v	NOUN
ejpam-5976	184	3	)	)	PUNCT
ejpam-5976	184	4	=	=	SYM
ejpam-5976	184	5	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	184	6	)	)	PUNCT
ejpam-5976	184	7	=	=	SYM
ejpam-5976	184	8	q((a	q((a	PROPN
ejpam-5976	184	9	,	,	PUNCT
ejpam-5976	184	10	b	b	NOUN
ejpam-5976	184	11	)	)	PUNCT
ejpam-5976	184	12	)	)	PUNCT
ejpam-5976	184	13	.	.	PUNCT
ejpam-5976	185	1	on	on	ADP
ejpam-5976	185	2	the	the	DET
ejpam-5976	185	3	other	other	ADJ
ejpam-5976	185	4	hand	hand	NOUN
ejpam-5976	185	5	,	,	PUNCT
ejpam-5976	185	6	q(a	q(a	PROPN
ejpam-5976	185	7	,	,	PUNCT
ejpam-5976	185	8	b	b	NOUN
ejpam-5976	185	9	)	)	PUNCT
ejpam-5976	185	10	=	=	SYM
ejpam-5976	185	11	q(a)×q(b	q(a)×q(b	X
ejpam-5976	185	12	)	)	PUNCT
ejpam-5976	185	13	=	=	SYM
ejpam-5976	185	14	(	(	PUNCT
ejpam-5976	185	15	as1	as1	NOUN
ejpam-5976	185	16	∩	∩	X
ejpam-5976	185	17	s1a)×	s1a)×	X
ejpam-5976	185	18	(	(	PUNCT
ejpam-5976	185	19	bs2	bs2	PROPN
ejpam-5976	185	20	∩	∩	PROPN
ejpam-5976	185	21	s2b	s2b	PROPN
ejpam-5976	185	22	)	)	PUNCT
ejpam-5976	185	23	=	=	PUNCT
ejpam-5976	186	1	(	(	PUNCT
ejpam-5976	186	2	q(a)s1	q(a)s1	PROPN
ejpam-5976	186	3	∩	∩	PROPN
ejpam-5976	186	4	s1q(a))×	s1q(a))×	PROPN
ejpam-5976	186	5	(	(	PUNCT
ejpam-5976	186	6	q(b)s2	q(b)s2	PROPN
ejpam-5976	186	7	∩	∩	PROPN
ejpam-5976	186	8	s2q(b	s2q(b	NOUN
ejpam-5976	186	9	)	)	PUNCT
ejpam-5976	186	10	)	)	PUNCT
ejpam-5976	187	1	=	=	PUNCT
ejpam-5976	187	2	(	(	PUNCT
ejpam-5976	187	3	q(u)s1	q(u)s1	PROPN
ejpam-5976	187	4	∩	∩	ADJ
ejpam-5976	187	5	s1q(u))×	s1q(u))×	PROPN
ejpam-5976	187	6	(	(	PUNCT
ejpam-5976	187	7	q(v)s2	q(v)s2	PROPN
ejpam-5976	187	8	∩	∩	PROPN
ejpam-5976	187	9	s2q(v	s2q(v	PROPN
ejpam-5976	187	10	)	)	PUNCT
ejpam-5976	187	11	)	)	PUNCT
ejpam-5976	188	1	=	=	PRON
ejpam-5976	189	1	(	(	PUNCT
ejpam-5976	189	2	us1	us1	PROPN
ejpam-5976	189	3	∩	∩	PROPN
ejpam-5976	189	4	s1u)×	s1u)×	PROPN
ejpam-5976	189	5	(	(	PUNCT
ejpam-5976	189	6	vs2	vs2	NOUN
ejpam-5976	189	7	∩	∩	NOUN
ejpam-5976	189	8	s2v	s2v	NOUN
ejpam-5976	189	9	)	)	PUNCT
ejpam-5976	189	10	=	=	PUNCT
ejpam-5976	189	11	(	(	PUNCT
ejpam-5976	189	12	u	u	NOUN
ejpam-5976	189	13	,	,	PUNCT
ejpam-5976	189	14	v)(s1	v)(s1	PROPN
ejpam-5976	189	15	×	×	PROPN
ejpam-5976	189	16	s2	s2	PROPN
ejpam-5976	189	17	)	)	PUNCT
ejpam-5976	189	18	∩	∩	NOUN
ejpam-5976	189	19	(	(	PUNCT
ejpam-5976	189	20	s1	s1	PROPN
ejpam-5976	189	21	×	×	PROPN
ejpam-5976	189	22	s2)(u	s2)(u	PROPN
ejpam-5976	189	23	,	,	PUNCT
ejpam-5976	189	24	v	v	NOUN
ejpam-5976	189	25	)	)	PUNCT
ejpam-5976	189	26	⊆	⊆	NUM
ejpam-5976	189	27	q((u	q((u	NOUN
ejpam-5976	189	28	,	,	PUNCT
ejpam-5976	189	29	v	v	NOUN
ejpam-5976	189	30	)	)	PUNCT
ejpam-5976	189	31	)	)	PUNCT
ejpam-5976	189	32	.	.	PUNCT
ejpam-5976	190	1	thus	thus	ADV
ejpam-5976	190	2	,	,	PUNCT
ejpam-5976	190	3	q((a	q((a	PROPN
ejpam-5976	190	4	,	,	PUNCT
ejpam-5976	190	5	b	b	NOUN
ejpam-5976	190	6	)	)	PUNCT
ejpam-5976	190	7	)	)	PUNCT
ejpam-5976	191	1	=	=	SYM
ejpam-5976	191	2	q((u	q((u	NOUN
ejpam-5976	191	3	,	,	PUNCT
ejpam-5976	191	4	v	v	NOUN
ejpam-5976	191	5	)	)	PUNCT
ejpam-5976	191	6	)	)	PUNCT
ejpam-5976	191	7	.	.	PUNCT
ejpam-5976	192	1	this	this	PRON
ejpam-5976	192	2	means	mean	VERB
ejpam-5976	192	3	(	(	PUNCT
ejpam-5976	192	4	u	u	NOUN
ejpam-5976	192	5	,	,	PUNCT
ejpam-5976	192	6	v	v	NOUN
ejpam-5976	192	7	)	)	PUNCT
ejpam-5976	192	8	∈	∈	PROPN
ejpam-5976	192	9	h(a	h(a	PROPN
ejpam-5976	192	10	,	,	PUNCT
ejpam-5976	192	11	b	b	NOUN
ejpam-5976	192	12	)	)	PUNCT
ejpam-5976	192	13	.	.	PUNCT
ejpam-5976	193	1	by	by	ADP
ejpam-5976	193	2	these	these	DET
ejpam-5976	193	3	two	two	NUM
ejpam-5976	193	4	cases	case	NOUN
ejpam-5976	193	5	,	,	PUNCT
ejpam-5976	193	6	we	we	PRON
ejpam-5976	193	7	conclude	conclude	VERB
ejpam-5976	193	8	that	that	SCONJ
ejpam-5976	193	9	ha	ha	INTJ
ejpam-5976	193	10	×hb	×hb	PROPN
ejpam-5976	193	11	⊆	⊆	NUM
ejpam-5976	193	12	h(a	h(a	PROPN
ejpam-5976	193	13	,	,	PUNCT
ejpam-5976	193	14	b	b	NOUN
ejpam-5976	193	15	)	)	PUNCT
ejpam-5976	193	16	.	.	PUNCT
ejpam-5976	194	1	by	by	ADP
ejpam-5976	194	2	theorem	theorem	NOUN
ejpam-5976	194	3	5	5	NUM
ejpam-5976	194	4	,	,	PUNCT
ejpam-5976	194	5	we	we	PRON
ejpam-5976	194	6	have	have	VERB
ejpam-5976	194	7	h(a	h(a	PROPN
ejpam-5976	194	8	,	,	PUNCT
ejpam-5976	194	9	b	b	NOUN
ejpam-5976	194	10	)	)	PUNCT
ejpam-5976	194	11	⊆	⊆	NUM
ejpam-5976	194	12	ha	ha	INTJ
ejpam-5976	194	13	×hb	×hb	PROPN
ejpam-5976	194	14	.	.	PUNCT
ejpam-5976	195	1	therefore	therefore	ADV
ejpam-5976	195	2	,	,	PUNCT
ejpam-5976	195	3	ha	ha	INTJ
ejpam-5976	195	4	×hb	×hb	PROPN
ejpam-5976	195	5	=	=	PUNCT
ejpam-5976	195	6	h(a	h(a	PROPN
ejpam-5976	195	7	,	,	PUNCT
ejpam-5976	195	8	b	b	NOUN
ejpam-5976	195	9	)	)	PUNCT
ejpam-5976	195	10	.	.	PUNCT
ejpam-5976	196	1	theorem	theorem	ADJ
ejpam-5976	196	2	7	7	NUM
ejpam-5976	196	3	.	.	PUNCT
ejpam-5976	197	1	let	let	VERB
ejpam-5976	197	2	(	(	PUNCT
ejpam-5976	197	3	a	a	PRON
ejpam-5976	197	4	,	,	PUNCT
ejpam-5976	197	5	b	b	NOUN
ejpam-5976	197	6	)	)	PUNCT
ejpam-5976	197	7	∈	∈	PROPN
ejpam-5976	197	8	s1	s1	PROPN
ejpam-5976	197	9	×	×	PROPN
ejpam-5976	197	10	s2	s2	PROPN
ejpam-5976	197	11	.	.	PUNCT
ejpam-5976	198	1	if	if	SCONJ
ejpam-5976	198	2	q((a	q((a	PROPN
ejpam-5976	198	3	,	,	PUNCT
ejpam-5976	198	4	b	b	NOUN
ejpam-5976	198	5	)	)	PUNCT
ejpam-5976	198	6	)	)	PUNCT
ejpam-5976	199	1	=	=	PUNCT
ejpam-5976	199	2	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	199	3	)	)	PUNCT
ejpam-5976	199	4	,	,	PUNCT
ejpam-5976	199	5	then	then	ADV
ejpam-5976	199	6	h(a	h(a	PROPN
ejpam-5976	199	7	,	,	PUNCT
ejpam-5976	199	8	b	b	NOUN
ejpam-5976	199	9	)	)	PUNCT
ejpam-5976	199	10	=	=	NOUN
ejpam-5976	199	11	ha	ha	INTJ
ejpam-5976	199	12	×hb	×hb	PROPN
ejpam-5976	199	13	.	.	PROPN
ejpam-5976	199	14	proof	proof	NOUN
ejpam-5976	199	15	.	.	PUNCT
ejpam-5976	200	1	assume	assume	VERB
ejpam-5976	200	2	that	that	SCONJ
ejpam-5976	200	3	q((a	q((a	NOUN
ejpam-5976	200	4	,	,	PUNCT
ejpam-5976	200	5	b	b	NOUN
ejpam-5976	200	6	)	)	PUNCT
ejpam-5976	200	7	)	)	PUNCT
ejpam-5976	201	1	=	=	SYM
ejpam-5976	201	2	q(a	q(a	PROPN
ejpam-5976	201	3	)	)	PUNCT
ejpam-5976	201	4	×	×	NOUN
ejpam-5976	201	5	q(b	q(b	NOUN
ejpam-5976	201	6	)	)	PUNCT
ejpam-5976	201	7	.	.	PUNCT
ejpam-5976	202	1	by	by	ADP
ejpam-5976	202	2	theorem	theorem	NOUN
ejpam-5976	202	3	3	3	NUM
ejpam-5976	202	4	,	,	PUNCT
ejpam-5976	202	5	we	we	PRON
ejpam-5976	202	6	have	have	VERB
ejpam-5976	202	7	one	one	NUM
ejpam-5976	202	8	of	of	ADP
ejpam-5976	202	9	the	the	DET
ejpam-5976	202	10	following	follow	VERB
ejpam-5976	202	11	conditions	condition	NOUN
ejpam-5976	202	12	holds	hold	VERB
ejpam-5976	202	13	:	:	PUNCT
ejpam-5976	202	14	(	(	PUNCT
ejpam-5976	202	15	i	i	NOUN
ejpam-5976	202	16	)	)	PUNCT
ejpam-5976	202	17	as1	as1	NOUN
ejpam-5976	202	18	∩	∩	NOUN
ejpam-5976	202	19	s1a	s1a	NOUN
ejpam-5976	202	20	=	=	PUNCT
ejpam-5976	202	21	{	{	PUNCT
ejpam-5976	202	22	a	a	NOUN
ejpam-5976	202	23	}	}	PUNCT
ejpam-5976	202	24	;	;	PUNCT
ejpam-5976	202	25	(	(	PUNCT
ejpam-5976	202	26	ii	ii	NOUN
ejpam-5976	202	27	)	)	PUNCT
ejpam-5976	202	28	bs2	bs2	PROPN
ejpam-5976	202	29	∩	∩	PROPN
ejpam-5976	203	1	s2b	s2b	PROPN
ejpam-5976	203	2	=	=	PUNCT
ejpam-5976	203	3	{	{	PUNCT
ejpam-5976	203	4	b	b	NOUN
ejpam-5976	203	5	}	}	PUNCT
ejpam-5976	203	6	;	;	PUNCT
ejpam-5976	203	7	(	(	PUNCT
ejpam-5976	203	8	iii	iii	X
ejpam-5976	203	9	)	)	PUNCT
ejpam-5976	203	10	a	a	DET
ejpam-5976	203	11	∈	∈	PROPN
ejpam-5976	203	12	as1	as1	NOUN
ejpam-5976	203	13	∩	∩	PROPN
ejpam-5976	203	14	s1a	s1a	PROPN
ejpam-5976	203	15	and	and	CCONJ
ejpam-5976	203	16	b	b	X
ejpam-5976	203	17	∈	∈	PROPN
ejpam-5976	203	18	bs2	bs2	PROPN
ejpam-5976	203	19	∩	∩	PROPN
ejpam-5976	203	20	s2b	s2b	PROPN
ejpam-5976	203	21	.	.	PUNCT
ejpam-5976	203	22	assume	assume	VERB
ejpam-5976	203	23	that	that	SCONJ
ejpam-5976	203	24	as1	as1	PROPN
ejpam-5976	203	25	∩	∩	NOUN
ejpam-5976	203	26	s1a	s1a	NOUN
ejpam-5976	203	27	=	=	PUNCT
ejpam-5976	203	28	{	{	PUNCT
ejpam-5976	203	29	a	a	NOUN
ejpam-5976	203	30	}	}	PUNCT
ejpam-5976	203	31	.	.	PUNCT
ejpam-5976	204	1	then	then	ADV
ejpam-5976	204	2	ha	ha	INTJ
ejpam-5976	204	3	=	=	X
ejpam-5976	204	4	{	{	PUNCT
ejpam-5976	204	5	a	a	NOUN
ejpam-5976	204	6	}	}	PUNCT
ejpam-5976	204	7	.	.	PUNCT
ejpam-5976	205	1	if	if	SCONJ
ejpam-5976	205	2	b	b	PROPN
ejpam-5976	205	3	∈	∈	PROPN
ejpam-5976	205	4	bs2	bs2	PROPN
ejpam-5976	205	5	∩	∩	PROPN
ejpam-5976	205	6	s2b	s2b	PROPN
ejpam-5976	205	7	,	,	PUNCT
ejpam-5976	205	8	we	we	PRON
ejpam-5976	205	9	have	have	VERB
ejpam-5976	205	10	that	that	DET
ejpam-5976	205	11	h(a	h(a	PROPN
ejpam-5976	205	12	,	,	PUNCT
ejpam-5976	205	13	b	b	NOUN
ejpam-5976	205	14	)	)	PUNCT
ejpam-5976	205	15	=	=	NOUN
ejpam-5976	206	1	ha	ha	INTJ
ejpam-5976	206	2	×	×	NOUN
ejpam-5976	206	3	hb	hb	X
ejpam-5976	206	4	by	by	ADP
ejpam-5976	206	5	theorem	theorem	NOUN
ejpam-5976	206	6	6	6	NUM
ejpam-5976	206	7	(	(	PUNCT
ejpam-5976	206	8	2	2	NUM
ejpam-5976	206	9	)	)	PUNCT
ejpam-5976	206	10	.	.	PUNCT
ejpam-5976	207	1	if	if	SCONJ
ejpam-5976	207	2	b	b	X
ejpam-5976	207	3	/∈	/∈	PROPN
ejpam-5976	207	4	bs2	bs2	PROPN
ejpam-5976	207	5	∩	∩	PROPN
ejpam-5976	207	6	s2b	s2b	PROPN
ejpam-5976	207	7	,	,	PUNCT
ejpam-5976	207	8	then	then	ADV
ejpam-5976	207	9	we	we	PRON
ejpam-5976	207	10	obtain	obtain	VERB
ejpam-5976	207	11	that	that	PRON
ejpam-5976	207	12	hb	hb	X
ejpam-5976	208	1	=	=	PUNCT
ejpam-5976	208	2	{	{	PUNCT
ejpam-5976	208	3	b	b	NOUN
ejpam-5976	208	4	}	}	PUNCT
ejpam-5976	208	5	.	.	PUNCT
ejpam-5976	209	1	hence	hence	ADV
ejpam-5976	209	2	,	,	PUNCT
ejpam-5976	209	3	h(a	h(a	PROPN
ejpam-5976	209	4	,	,	PUNCT
ejpam-5976	209	5	b	b	NOUN
ejpam-5976	209	6	)	)	PUNCT
ejpam-5976	209	7	=	=	NOUN
ejpam-5976	209	8	ha	ha	INTJ
ejpam-5976	209	9	×	×	NOUN
ejpam-5976	209	10	hb	hb	X
ejpam-5976	209	11	by	by	ADP
ejpam-5976	209	12	theorem	theorem	NOUN
ejpam-5976	209	13	6	6	NUM
ejpam-5976	209	14	(	(	PUNCT
ejpam-5976	209	15	1	1	NUM
ejpam-5976	209	16	)	)	PUNCT
ejpam-5976	209	17	.	.	PUNCT
ejpam-5976	210	1	the	the	DET
ejpam-5976	210	2	case	case	NOUN
ejpam-5976	210	3	bs2	bs2	PROPN
ejpam-5976	210	4	∩	∩	PROPN
ejpam-5976	210	5	s2b	s2b	NOUN
ejpam-5976	210	6	=	=	SYM
ejpam-5976	210	7	{	{	PUNCT
ejpam-5976	210	8	b	b	NOUN
ejpam-5976	210	9	}	}	PUNCT
ejpam-5976	210	10	can	can	AUX
ejpam-5976	210	11	be	be	AUX
ejpam-5976	210	12	proved	prove	VERB
ejpam-5976	210	13	similarly	similarly	ADV
ejpam-5976	210	14	.	.	PUNCT
ejpam-5976	211	1	the	the	DET
ejpam-5976	211	2	case	case	NOUN
ejpam-5976	211	3	a	a	DET
ejpam-5976	211	4	∈	∈	PROPN
ejpam-5976	211	5	as1	as1	NOUN
ejpam-5976	211	6	∩	∩	PROPN
ejpam-5976	211	7	s1a	s1a	PROPN
ejpam-5976	211	8	and	and	CCONJ
ejpam-5976	211	9	b	b	X
ejpam-5976	211	10	∈	∈	PROPN
ejpam-5976	211	11	bs2	bs2	PROPN
ejpam-5976	211	12	∩	∩	NOUN
ejpam-5976	211	13	s2b	s2b	NOUN
ejpam-5976	211	14	is	be	AUX
ejpam-5976	211	15	obtained	obtain	VERB
ejpam-5976	211	16	directly	directly	ADV
ejpam-5976	211	17	from	from	ADP
ejpam-5976	211	18	theorem	theorem	ADJ
ejpam-5976	211	19	6	6	NUM
ejpam-5976	211	20	(	(	PUNCT
ejpam-5976	211	21	2	2	NUM
ejpam-5976	211	22	)	)	PUNCT
ejpam-5976	211	23	.	.	PUNCT
ejpam-5976	212	1	therefore	therefore	ADV
ejpam-5976	212	2	,	,	PUNCT
ejpam-5976	212	3	h(a	h(a	PROPN
ejpam-5976	212	4	,	,	PUNCT
ejpam-5976	212	5	b	b	NOUN
ejpam-5976	212	6	)	)	PUNCT
ejpam-5976	212	7	=	=	NOUN
ejpam-5976	212	8	ha	ha	INTJ
ejpam-5976	212	9	×hb	×hb	PROPN
ejpam-5976	212	10	.	.	PUNCT
ejpam-5976	213	1	however	however	ADV
ejpam-5976	213	2	,	,	PUNCT
ejpam-5976	213	3	the	the	DET
ejpam-5976	213	4	converse	converse	NOUN
ejpam-5976	213	5	of	of	ADP
ejpam-5976	213	6	theorem	theorem	NOUN
ejpam-5976	213	7	7	7	NUM
ejpam-5976	213	8	is	be	AUX
ejpam-5976	213	9	not	not	PART
ejpam-5976	213	10	true	true	ADJ
ejpam-5976	213	11	in	in	ADP
ejpam-5976	213	12	general	general	ADJ
ejpam-5976	213	13	.	.	PUNCT
ejpam-5976	214	1	that	that	PRON
ejpam-5976	214	2	is	be	AUX
ejpam-5976	214	3	,	,	PUNCT
ejpam-5976	214	4	h(a	h(a	PROPN
ejpam-5976	214	5	,	,	PUNCT
ejpam-5976	214	6	b	b	NOUN
ejpam-5976	214	7	)	)	PUNCT
ejpam-5976	214	8	=	=	VERB
ejpam-5976	215	1	ha	ha	INTJ
ejpam-5976	215	2	×hb	×hb	PROPN
ejpam-5976	215	3	does	do	AUX
ejpam-5976	215	4	not	not	PART
ejpam-5976	215	5	implies	imply	VERB
ejpam-5976	215	6	that	that	SCONJ
ejpam-5976	215	7	q((a	q((a	NOUN
ejpam-5976	215	8	,	,	PUNCT
ejpam-5976	215	9	b	b	NOUN
ejpam-5976	215	10	)	)	PUNCT
ejpam-5976	215	11	)	)	PUNCT
ejpam-5976	215	12	=	=	PUNCT
ejpam-5976	216	1	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	216	2	)	)	PUNCT
ejpam-5976	216	3	.	.	PUNCT
ejpam-5976	217	1	example	example	NOUN
ejpam-5976	218	1	3	3	NUM
ejpam-5976	218	2	.	.	PUNCT
ejpam-5976	219	1	(	(	PUNCT
ejpam-5976	219	2	[	[	X
ejpam-5976	219	3	7	7	NUM
ejpam-5976	219	4	]	]	PUNCT
ejpam-5976	219	5	)	)	PUNCT
ejpam-5976	219	6	let	let	VERB
ejpam-5976	219	7	(	(	PUNCT
ejpam-5976	219	8	s1	s1	NOUN
ejpam-5976	219	9	,	,	PUNCT
ejpam-5976	219	10	∗	∗	NOUN
ejpam-5976	219	11	)	)	PUNCT
ejpam-5976	219	12	and	and	CCONJ
ejpam-5976	219	13	(	(	PUNCT
ejpam-5976	219	14	s2	s2	PROPN
ejpam-5976	219	15	,	,	PUNCT
ejpam-5976	219	16	·	·	PUNCT
ejpam-5976	219	17	)	)	PUNCT
ejpam-5976	219	18	be	be	AUX
ejpam-5976	219	19	semigroups	semigroup	NOUN
ejpam-5976	219	20	where	where	SCONJ
ejpam-5976	219	21	s1	s1	NOUN
ejpam-5976	219	22	=	=	PUNCT
ejpam-5976	219	23	{	{	PUNCT
ejpam-5976	219	24	a1	a1	PROPN
ejpam-5976	219	25	,	,	PUNCT
ejpam-5976	219	26	a2	a2	PROPN
ejpam-5976	219	27	,	,	PUNCT
ejpam-5976	219	28	a3	a3	NOUN
ejpam-5976	219	29	,	,	PUNCT
ejpam-5976	219	30	a4	a4	NOUN
ejpam-5976	219	31	}	}	PUNCT
ejpam-5976	219	32	and	and	CCONJ
ejpam-5976	219	33	s2	s2	VERB
ejpam-5976	219	34	=	=	SYM
ejpam-5976	219	35	{	{	PUNCT
ejpam-5976	219	36	b1	b1	PROPN
ejpam-5976	219	37	,	,	PUNCT
ejpam-5976	219	38	b2	b2	NOUN
ejpam-5976	219	39	,	,	PUNCT
ejpam-5976	219	40	b3	b3	NOUN
ejpam-5976	219	41	,	,	PUNCT
ejpam-5976	219	42	b4	b4	NOUN
ejpam-5976	219	43	}	}	PUNCT
ejpam-5976	219	44	.	.	PUNCT
ejpam-5976	220	1	the	the	DET
ejpam-5976	220	2	binary	binary	PROPN
ejpam-5976	220	3	operations	operation	NOUN
ejpam-5976	220	4	∗	∗	NOUN
ejpam-5976	220	5	:	:	PUNCT
ejpam-5976	220	6	s1	s1	NOUN
ejpam-5976	220	7	×	×	PROPN
ejpam-5976	220	8	s1	s1	NOUN
ejpam-5976	220	9	−→	−→	NOUN
ejpam-5976	220	10	s1	s1	NOUN
ejpam-5976	220	11	and	and	CCONJ
ejpam-5976	220	12	·	·	PUNCT
ejpam-5976	220	13	:	:	PUNCT
ejpam-5976	220	14	s2	s2	VERB
ejpam-5976	220	15	×	×	PROPN
ejpam-5976	220	16	s2	s2	NOUN
ejpam-5976	220	17	−→	−→	NOUN
ejpam-5976	220	18	s2	s2	NOUN
ejpam-5976	220	19	are	be	AUX
ejpam-5976	220	20	defined	define	VERB
ejpam-5976	220	21	as	as	ADP
ejpam-5976	220	22	:	:	PUNCT
ejpam-5976	220	23	∗	∗	NOUN
ejpam-5976	220	24	a1	a1	NOUN
ejpam-5976	220	25	a2	a2	PROPN
ejpam-5976	220	26	a3	a3	PROPN
ejpam-5976	220	27	a4	a4	CCONJ
ejpam-5976	220	28	a1	a1	NOUN
ejpam-5976	220	29	a1	a1	NOUN
ejpam-5976	220	30	a1	a1	NOUN
ejpam-5976	220	31	a1	a1	NOUN
ejpam-5976	220	32	a1	a1	NOUN
ejpam-5976	220	33	a2	a2	PROPN
ejpam-5976	220	34	a1	a1	NOUN
ejpam-5976	220	35	a1	a1	NOUN
ejpam-5976	220	36	a1	a1	NOUN
ejpam-5976	220	37	a1	a1	NOUN
ejpam-5976	220	38	a3	a3	NOUN
ejpam-5976	220	39	a1	a1	NOUN
ejpam-5976	220	40	a1	a1	NOUN
ejpam-5976	220	41	a1	a1	NOUN
ejpam-5976	220	42	a2	a2	PROPN
ejpam-5976	220	43	a4	a4	NOUN
ejpam-5976	220	44	a1	a1	NOUN
ejpam-5976	220	45	a1	a1	NOUN
ejpam-5976	220	46	a2	a2	PROPN
ejpam-5976	220	47	a2	a2	PROPN
ejpam-5976	220	48	·	·	PUNCT
ejpam-5976	220	49	b1	b1	PROPN
ejpam-5976	220	50	b2	b2	NOUN
ejpam-5976	220	51	b3	b3	PROPN
ejpam-5976	220	52	b4	b4	NOUN
ejpam-5976	220	53	b1	b1	NOUN
ejpam-5976	220	54	b1	b1	PROPN
ejpam-5976	220	55	b1	b1	PROPN
ejpam-5976	220	56	b1	b1	PROPN
ejpam-5976	220	57	b1	b1	PROPN
ejpam-5976	220	58	b2	b2	PROPN
ejpam-5976	220	59	b1	b1	NOUN
ejpam-5976	220	60	b1	b1	NOUN
ejpam-5976	220	61	b1	b1	PROPN
ejpam-5976	220	62	b1	b1	PROPN
ejpam-5976	220	63	b3	b3	PROPN
ejpam-5976	220	64	b1	b1	PROPN
ejpam-5976	220	65	b1	b1	PROPN
ejpam-5976	220	66	b3	b3	PROPN
ejpam-5976	220	67	b3	b3	PROPN
ejpam-5976	220	68	b4	b4	PROPN
ejpam-5976	220	69	b1	b1	PROPN
ejpam-5976	220	70	b1	b1	PROPN
ejpam-5976	220	71	b3	b3	PROPN
ejpam-5976	220	72	b3	b3	PROPN
ejpam-5976	220	73	we	we	PRON
ejpam-5976	220	74	have	have	VERB
ejpam-5976	220	75	ha3	ha3	NOUN
ejpam-5976	220	76	=	=	SYM
ejpam-5976	220	77	{	{	PUNCT
ejpam-5976	220	78	a3	a3	NOUN
ejpam-5976	220	79	}	}	PUNCT
ejpam-5976	220	80	and	and	CCONJ
ejpam-5976	220	81	hb4	hb4	PROPN
ejpam-5976	220	82	=	=	SYM
ejpam-5976	220	83	{	{	PUNCT
ejpam-5976	220	84	b4	b4	NOUN
ejpam-5976	220	85	}	}	PUNCT
ejpam-5976	220	86	.	.	PUNCT
ejpam-5976	221	1	hence	hence	ADV
ejpam-5976	221	2	h(a3,b4	h(a3,b4	NOUN
ejpam-5976	221	3	)	)	PUNCT
ejpam-5976	221	4	=	=	PUNCT
ejpam-5976	222	1	ha3	ha3	ADJ
ejpam-5976	222	2	×	×	NOUN
ejpam-5976	222	3	hb4	hb4	NOUN
ejpam-5976	222	4	by	by	ADP
ejpam-5976	222	5	theorem	theorem	NOUN
ejpam-5976	222	6	6	6	NUM
ejpam-5976	222	7	.	.	PUNCT
ejpam-5976	223	1	since	since	SCONJ
ejpam-5976	223	2	(	(	PUNCT
ejpam-5976	223	3	a3	a3	NOUN
ejpam-5976	223	4	,	,	PUNCT
ejpam-5976	223	5	b1	b1	NOUN
ejpam-5976	223	6	)	)	PUNCT
ejpam-5976	223	7	∈	∈	PROPN
ejpam-5976	223	8	q(a3	q(a3	NOUN
ejpam-5976	223	9	)	)	PUNCT
ejpam-5976	223	10	×	×	PROPN
ejpam-5976	223	11	q(b4	q(b4	NOUN
ejpam-5976	223	12	)	)	PUNCT
ejpam-5976	223	13	but	but	CCONJ
ejpam-5976	223	14	(	(	PUNCT
ejpam-5976	223	15	a3	a3	NOUN
ejpam-5976	223	16	,	,	PUNCT
ejpam-5976	223	17	b1	b1	NOUN
ejpam-5976	223	18	)	)	PUNCT
ejpam-5976	223	19	/∈	/∈	PUNCT
ejpam-5976	224	1	q((a3	q((a3	NOUN
ejpam-5976	224	2	,	,	PUNCT
ejpam-5976	224	3	b4	b4	NOUN
ejpam-5976	224	4	)	)	PUNCT
ejpam-5976	224	5	)	)	PUNCT
ejpam-5976	224	6	,	,	PUNCT
ejpam-5976	224	7	it	it	PRON
ejpam-5976	224	8	follows	follow	VERB
ejpam-5976	224	9	that	that	DET
ejpam-5976	224	10	q((a3	q((a3	NOUN
ejpam-5976	224	11	,	,	PUNCT
ejpam-5976	224	12	b4	b4	NOUN
ejpam-5976	224	13	)	)	PUNCT
ejpam-5976	224	14	)	)	PUNCT
ejpam-5976	225	1	̸=	̸=	PROPN
ejpam-5976	225	2	q(a3)×q(b4	q(a3)×q(b4	NOUN
ejpam-5976	225	3	)	)	PUNCT
ejpam-5976	225	4	.	.	PUNCT
ejpam-5976	226	1	p.	p.	NOUN
ejpam-5976	226	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	226	3	,	,	PUNCT
ejpam-5976	226	4	o.	o.	PROPN
ejpam-5976	226	5	pankoon	pankoon	NOUN
ejpam-5976	226	6	,	,	PUNCT
ejpam-5976	226	7	t.	t.	PROPN
ejpam-5976	226	8	changphas	changphas	PROPN
ejpam-5976	226	9	/	/	SYM
ejpam-5976	226	10	eur	eur	PROPN
ejpam-5976	226	11	.	.	PUNCT
ejpam-5976	227	1	j.	j.	PROPN
ejpam-5976	227	2	pure	pure	PROPN
ejpam-5976	227	3	appl	appl	PROPN
ejpam-5976	227	4	.	.	PROPN
ejpam-5976	227	5	math	math	PROPN
ejpam-5976	227	6	,	,	PUNCT
ejpam-5976	227	7	18	18	NUM
ejpam-5976	227	8	(	(	PUNCT
ejpam-5976	227	9	2	2	NUM
ejpam-5976	227	10	)	)	PUNCT
ejpam-5976	227	11	(	(	PUNCT
ejpam-5976	227	12	2025	2025	NUM
ejpam-5976	227	13	)	)	PUNCT
ejpam-5976	227	14	,	,	PUNCT
ejpam-5976	227	15	5976	5976	NUM
ejpam-5976	227	16	8	8	NUM
ejpam-5976	227	17	of	of	ADP
ejpam-5976	227	18	14	14	NUM
ejpam-5976	227	19	theorem	theorem	NOUN
ejpam-5976	227	20	8	8	NUM
ejpam-5976	227	21	.	.	PUNCT
ejpam-5976	228	1	let	let	VERB
ejpam-5976	228	2	(	(	PUNCT
ejpam-5976	228	3	a	a	PRON
ejpam-5976	228	4	,	,	PUNCT
ejpam-5976	228	5	b	b	NOUN
ejpam-5976	228	6	)	)	PUNCT
ejpam-5976	228	7	∈	∈	PROPN
ejpam-5976	228	8	s1	s1	PROPN
ejpam-5976	228	9	×	×	PROPN
ejpam-5976	228	10	s2	s2	PROPN
ejpam-5976	228	11	.	.	PUNCT
ejpam-5976	229	1	if	if	SCONJ
ejpam-5976	229	2	|	|	ADV
ejpam-5976	229	3	ha	ha	INTJ
ejpam-5976	229	4	|	|	ADV
ejpam-5976	229	5	>	>	X
ejpam-5976	229	6	1	1	NUM
ejpam-5976	230	1	and	and	CCONJ
ejpam-5976	230	2	|	|	ADV
ejpam-5976	230	3	hb	hb	X
ejpam-5976	230	4	|	|	ADV
ejpam-5976	230	5	>	>	X
ejpam-5976	230	6	1	1	NUM
ejpam-5976	230	7	,	,	PUNCT
ejpam-5976	230	8	then	then	ADV
ejpam-5976	230	9	(	(	PUNCT
ejpam-5976	230	10	1	1	X
ejpam-5976	230	11	)	)	PUNCT
ejpam-5976	230	12	a	a	DET
ejpam-5976	230	13	∈	∈	PROPN
ejpam-5976	230	14	as1	as1	NOUN
ejpam-5976	230	15	∩	∩	PROPN
ejpam-5976	230	16	s1a	s1a	PROPN
ejpam-5976	230	17	and	and	CCONJ
ejpam-5976	230	18	b	b	X
ejpam-5976	230	19	∈	∈	PROPN
ejpam-5976	230	20	bs2	bs2	PROPN
ejpam-5976	230	21	∩	∩	PROPN
ejpam-5976	230	22	s2b	s2b	PROPN
ejpam-5976	230	23	;	;	PUNCT
ejpam-5976	230	24	(	(	PUNCT
ejpam-5976	230	25	2	2	X
ejpam-5976	230	26	)	)	PUNCT
ejpam-5976	230	27	h(a	h(a	PROPN
ejpam-5976	230	28	,	,	PUNCT
ejpam-5976	230	29	b	b	NOUN
ejpam-5976	230	30	)	)	PUNCT
ejpam-5976	230	31	=	=	NOUN
ejpam-5976	230	32	ha	ha	INTJ
ejpam-5976	230	33	×hb	×hb	PROPN
ejpam-5976	230	34	.	.	PROPN
ejpam-5976	230	35	proof	proof	NOUN
ejpam-5976	230	36	.	.	PUNCT
ejpam-5976	231	1	(	(	PUNCT
ejpam-5976	231	2	1	1	X
ejpam-5976	231	3	)	)	PUNCT
ejpam-5976	231	4	assume	assume	VERB
ejpam-5976	231	5	that	that	SCONJ
ejpam-5976	232	1	|	|	ADV
ejpam-5976	232	2	ha	ha	INTJ
ejpam-5976	233	1	|	|	ADV
ejpam-5976	233	2	>	>	X
ejpam-5976	233	3	1	1	NUM
ejpam-5976	234	1	and	and	CCONJ
ejpam-5976	234	2	|	|	ADV
ejpam-5976	234	3	hb	hb	X
ejpam-5976	234	4	|	|	ADV
ejpam-5976	234	5	>	>	X
ejpam-5976	234	6	1	1	NUM
ejpam-5976	234	7	.	.	PUNCT
ejpam-5976	235	1	then	then	ADV
ejpam-5976	235	2	there	there	PRON
ejpam-5976	235	3	exist	exist	VERB
ejpam-5976	235	4	u	u	NOUN
ejpam-5976	235	5	∈	∈	PROPN
ejpam-5976	235	6	ha	ha	INTJ
ejpam-5976	235	7	and	and	CCONJ
ejpam-5976	235	8	v	v	NOUN
ejpam-5976	235	9	∈	∈	NOUN
ejpam-5976	235	10	hb	hb	ADP
ejpam-5976	235	11	such	such	ADJ
ejpam-5976	235	12	that	that	SCONJ
ejpam-5976	235	13	u	u	PROPN
ejpam-5976	235	14	̸=	̸=	PROPN
ejpam-5976	235	15	a	a	PRON
ejpam-5976	235	16	and	and	CCONJ
ejpam-5976	235	17	v	v	ADP
ejpam-5976	235	18	̸=	̸=	PROPN
ejpam-5976	235	19	b.	b.	NOUN
ejpam-5976	236	1	it	it	PRON
ejpam-5976	236	2	can	can	AUX
ejpam-5976	236	3	be	be	AUX
ejpam-5976	236	4	observed	observe	VERB
ejpam-5976	236	5	that	that	SCONJ
ejpam-5976	236	6	(	(	PUNCT
ejpam-5976	236	7	a	a	PRON
ejpam-5976	236	8	,	,	PUNCT
ejpam-5976	236	9	b	b	NOUN
ejpam-5976	236	10	)	)	PUNCT
ejpam-5976	236	11	∈	∈	PROPN
ejpam-5976	236	12	(	(	PUNCT
ejpam-5976	236	13	us1	us1	PROPN
ejpam-5976	236	14	∩	∩	PROPN
ejpam-5976	236	15	s1u)×	s1u)×	PROPN
ejpam-5976	236	16	(	(	PUNCT
ejpam-5976	236	17	vs2	vs2	NOUN
ejpam-5976	236	18	∩	∩	NOUN
ejpam-5976	236	19	s2v	s2v	NOUN
ejpam-5976	236	20	)	)	PUNCT
ejpam-5976	236	21	and	and	CCONJ
ejpam-5976	236	22	(	(	PUNCT
ejpam-5976	236	23	u	u	NOUN
ejpam-5976	236	24	,	,	PUNCT
ejpam-5976	236	25	v	v	NOUN
ejpam-5976	236	26	)	)	PUNCT
ejpam-5976	236	27	∈	∈	PROPN
ejpam-5976	236	28	(	(	PUNCT
ejpam-5976	236	29	as1	as1	NOUN
ejpam-5976	236	30	∩	∩	NOUN
ejpam-5976	236	31	s1a)×	s1a)×	X
ejpam-5976	236	32	(	(	PUNCT
ejpam-5976	236	33	bs2	bs2	PROPN
ejpam-5976	236	34	∩	∩	PROPN
ejpam-5976	236	35	s2b	s2b	PROPN
ejpam-5976	236	36	)	)	PUNCT
ejpam-5976	236	37	.	.	PUNCT
ejpam-5976	237	1	thus	thus	ADV
ejpam-5976	237	2	,	,	PUNCT
ejpam-5976	237	3	(	(	PUNCT
ejpam-5976	237	4	a	a	PRON
ejpam-5976	237	5	,	,	PUNCT
ejpam-5976	237	6	b	b	NOUN
ejpam-5976	237	7	)	)	PUNCT
ejpam-5976	237	8	∈	∈	PROPN
ejpam-5976	237	9	(	(	PUNCT
ejpam-5976	237	10	us1	us1	PROPN
ejpam-5976	237	11	∩	∩	PROPN
ejpam-5976	237	12	s1u)×	s1u)×	PROPN
ejpam-5976	237	13	(	(	PUNCT
ejpam-5976	237	14	vs2	vs2	NOUN
ejpam-5976	237	15	∩	∩	NOUN
ejpam-5976	237	16	s2v	s2v	NOUN
ejpam-5976	237	17	)	)	PUNCT
ejpam-5976	237	18	=	=	PUNCT
ejpam-5976	237	19	(	(	PUNCT
ejpam-5976	237	20	q(u)s1	q(u)s1	PROPN
ejpam-5976	237	21	∩	∩	ADJ
ejpam-5976	237	22	s1q(u))×	s1q(u))×	PROPN
ejpam-5976	237	23	(	(	PUNCT
ejpam-5976	237	24	q(v)s2	q(v)s2	PROPN
ejpam-5976	237	25	∩	∩	PROPN
ejpam-5976	237	26	s2q(v	s2q(v	PROPN
ejpam-5976	237	27	)	)	PUNCT
ejpam-5976	237	28	)	)	PUNCT
ejpam-5976	238	1	=	=	PRON
ejpam-5976	238	2	(	(	PUNCT
ejpam-5976	238	3	q(a)s1	q(a)s1	PROPN
ejpam-5976	238	4	∩	∩	PROPN
ejpam-5976	238	5	s1q(a))×	s1q(a))×	PROPN
ejpam-5976	238	6	(	(	PUNCT
ejpam-5976	238	7	q(b)s2	q(b)s2	PROPN
ejpam-5976	238	8	∩	∩	PROPN
ejpam-5976	238	9	s2q(b	s2q(b	NOUN
ejpam-5976	238	10	)	)	PUNCT
ejpam-5976	238	11	)	)	PUNCT
ejpam-5976	239	1	=	=	SYM
ejpam-5976	240	1	(	(	PUNCT
ejpam-5976	240	2	as1	as1	NOUN
ejpam-5976	240	3	∩	∩	X
ejpam-5976	240	4	s1a)×	s1a)×	X
ejpam-5976	240	5	(	(	PUNCT
ejpam-5976	240	6	bs2	bs2	PROPN
ejpam-5976	240	7	∩	∩	PROPN
ejpam-5976	240	8	s2b	s2b	PROPN
ejpam-5976	240	9	)	)	PUNCT
ejpam-5976	240	10	.	.	PUNCT
ejpam-5976	241	1	therefore	therefore	ADV
ejpam-5976	241	2	,	,	PUNCT
ejpam-5976	241	3	a	a	DET
ejpam-5976	241	4	∈	∈	PROPN
ejpam-5976	241	5	as1	as1	NOUN
ejpam-5976	241	6	∩	∩	PROPN
ejpam-5976	241	7	s1a	s1a	PROPN
ejpam-5976	241	8	and	and	CCONJ
ejpam-5976	241	9	b	b	X
ejpam-5976	241	10	∈	∈	PROPN
ejpam-5976	241	11	bs2	bs2	PROPN
ejpam-5976	241	12	∩	∩	PROPN
ejpam-5976	241	13	s2b	s2b	PROPN
ejpam-5976	241	14	.	.	PUNCT
ejpam-5976	242	1	(	(	PUNCT
ejpam-5976	242	2	2	2	X
ejpam-5976	242	3	)	)	PUNCT
ejpam-5976	242	4	it	it	PRON
ejpam-5976	242	5	is	be	AUX
ejpam-5976	242	6	obtained	obtain	VERB
ejpam-5976	242	7	immediately	immediately	ADV
ejpam-5976	242	8	by	by	ADP
ejpam-5976	242	9	theorem	theorem	NOUN
ejpam-5976	242	10	6	6	NUM
ejpam-5976	242	11	.	.	PUNCT
ejpam-5976	243	1	we	we	PRON
ejpam-5976	243	2	obtain	obtain	VERB
ejpam-5976	243	3	that	that	SCONJ
ejpam-5976	243	4	the	the	DET
ejpam-5976	243	5	reverse	reverse	NOUN
ejpam-5976	243	6	of	of	ADP
ejpam-5976	243	7	theorem	theorem	ADJ
ejpam-5976	243	8	8	8	NUM
ejpam-5976	243	9	is	be	AUX
ejpam-5976	243	10	not	not	PART
ejpam-5976	243	11	true	true	ADJ
ejpam-5976	243	12	as	as	ADP
ejpam-5976	243	13	the	the	DET
ejpam-5976	243	14	following	follow	VERB
ejpam-5976	243	15	example	example	NOUN
ejpam-5976	243	16	.	.	PUNCT
ejpam-5976	244	1	example	example	NOUN
ejpam-5976	245	1	4	4	NUM
ejpam-5976	245	2	.	.	PUNCT
ejpam-5976	246	1	[	[	X
ejpam-5976	246	2	7	7	X
ejpam-5976	246	3	]	]	X
ejpam-5976	246	4	let	let	NOUN
ejpam-5976	246	5	(	(	PUNCT
ejpam-5976	246	6	s1	s1	NOUN
ejpam-5976	246	7	,	,	PUNCT
ejpam-5976	246	8	∗	∗	NOUN
ejpam-5976	246	9	)	)	PUNCT
ejpam-5976	246	10	and	and	CCONJ
ejpam-5976	246	11	(	(	PUNCT
ejpam-5976	246	12	s2	s2	PROPN
ejpam-5976	246	13	,	,	PUNCT
ejpam-5976	246	14	·	·	PUNCT
ejpam-5976	246	15	)	)	PUNCT
ejpam-5976	246	16	be	be	AUX
ejpam-5976	246	17	semigroups	semigroup	NOUN
ejpam-5976	246	18	where	where	SCONJ
ejpam-5976	246	19	s1	s1	NOUN
ejpam-5976	246	20	=	=	PUNCT
ejpam-5976	246	21	{	{	PUNCT
ejpam-5976	246	22	a1	a1	PROPN
ejpam-5976	246	23	,	,	PUNCT
ejpam-5976	246	24	a2	a2	PROPN
ejpam-5976	246	25	,	,	PUNCT
ejpam-5976	246	26	a3	a3	NOUN
ejpam-5976	246	27	,	,	PUNCT
ejpam-5976	246	28	a4	a4	NOUN
ejpam-5976	246	29	}	}	PUNCT
ejpam-5976	246	30	and	and	CCONJ
ejpam-5976	246	31	s2	s2	VERB
ejpam-5976	246	32	=	=	SYM
ejpam-5976	246	33	{	{	PUNCT
ejpam-5976	246	34	b1	b1	PROPN
ejpam-5976	246	35	,	,	PUNCT
ejpam-5976	246	36	b2	b2	NOUN
ejpam-5976	246	37	,	,	PUNCT
ejpam-5976	246	38	b3	b3	NOUN
ejpam-5976	246	39	,	,	PUNCT
ejpam-5976	246	40	b4	b4	NOUN
ejpam-5976	246	41	}	}	PUNCT
ejpam-5976	246	42	.	.	PUNCT
ejpam-5976	247	1	the	the	DET
ejpam-5976	247	2	binary	binary	PROPN
ejpam-5976	247	3	operations	operation	NOUN
ejpam-5976	247	4	∗	∗	NOUN
ejpam-5976	247	5	:	:	PUNCT
ejpam-5976	247	6	s1	s1	NOUN
ejpam-5976	247	7	×	×	PROPN
ejpam-5976	247	8	s1	s1	NOUN
ejpam-5976	247	9	−→	−→	NOUN
ejpam-5976	247	10	s1	s1	NOUN
ejpam-5976	247	11	and	and	CCONJ
ejpam-5976	247	12	·	·	PUNCT
ejpam-5976	247	13	:	:	PUNCT
ejpam-5976	247	14	s2	s2	VERB
ejpam-5976	247	15	×	×	PROPN
ejpam-5976	247	16	s2	s2	NOUN
ejpam-5976	247	17	−→	−→	NOUN
ejpam-5976	247	18	s2	s2	NOUN
ejpam-5976	247	19	are	be	AUX
ejpam-5976	247	20	defined	define	VERB
ejpam-5976	247	21	as	as	ADP
ejpam-5976	247	22	:	:	PUNCT
ejpam-5976	247	23	∗	∗	NOUN
ejpam-5976	247	24	a1	a1	NOUN
ejpam-5976	247	25	a2	a2	PROPN
ejpam-5976	247	26	a3	a3	PROPN
ejpam-5976	247	27	a4	a4	CCONJ
ejpam-5976	247	28	a1	a1	NOUN
ejpam-5976	247	29	a1	a1	NOUN
ejpam-5976	247	30	a1	a1	NOUN
ejpam-5976	247	31	a1	a1	NOUN
ejpam-5976	247	32	a1	a1	NOUN
ejpam-5976	247	33	a2	a2	PROPN
ejpam-5976	247	34	a1	a1	PROPN
ejpam-5976	247	35	a2	a2	PROPN
ejpam-5976	247	36	a2	a2	PROPN
ejpam-5976	247	37	a2	a2	PROPN
ejpam-5976	247	38	a3	a3	PROPN
ejpam-5976	247	39	a1	a1	PROPN
ejpam-5976	247	40	a2	a2	PROPN
ejpam-5976	247	41	a2	a2	PROPN
ejpam-5976	247	42	a2	a2	PROPN
ejpam-5976	247	43	a4	a4	NOUN
ejpam-5976	247	44	a1	a1	NOUN
ejpam-5976	247	45	a4	a4	NOUN
ejpam-5976	247	46	a4	a4	NOUN
ejpam-5976	247	47	a4	a4	NOUN
ejpam-5976	247	48	·	·	PUNCT
ejpam-5976	247	49	b1	b1	NOUN
ejpam-5976	247	50	b2	b2	NOUN
ejpam-5976	247	51	b3	b3	PROPN
ejpam-5976	247	52	b4	b4	NOUN
ejpam-5976	247	53	b1	b1	NOUN
ejpam-5976	247	54	b1	b1	PROPN
ejpam-5976	247	55	b1	b1	PROPN
ejpam-5976	247	56	b1	b1	PROPN
ejpam-5976	247	57	b1	b1	PROPN
ejpam-5976	247	58	b2	b2	PROPN
ejpam-5976	247	59	b1	b1	NOUN
ejpam-5976	247	60	b2	b2	NOUN
ejpam-5976	247	61	b2	b2	NOUN
ejpam-5976	247	62	b2	b2	NOUN
ejpam-5976	247	63	b3	b3	PROPN
ejpam-5976	247	64	b1	b1	NOUN
ejpam-5976	247	65	b2	b2	NOUN
ejpam-5976	247	66	b2	b2	NOUN
ejpam-5976	247	67	b2	b2	NOUN
ejpam-5976	247	68	b4	b4	NOUN
ejpam-5976	247	69	b1	b1	NOUN
ejpam-5976	247	70	b2	b2	NOUN
ejpam-5976	247	71	b2	b2	NOUN
ejpam-5976	247	72	b2	b2	NOUN
ejpam-5976	247	73	we	we	PRON
ejpam-5976	247	74	have	have	VERB
ejpam-5976	247	75	that	that	DET
ejpam-5976	247	76	h(a1,b2	h(a1,b2	NOUN
ejpam-5976	247	77	)	)	PUNCT
ejpam-5976	247	78	=	=	PRON
ejpam-5976	247	79	{	{	PUNCT
ejpam-5976	247	80	(	(	PUNCT
ejpam-5976	247	81	a1	a1	NOUN
ejpam-5976	247	82	,	,	PUNCT
ejpam-5976	247	83	b2	b2	NOUN
ejpam-5976	247	84	)	)	PUNCT
ejpam-5976	247	85	}	}	PUNCT
ejpam-5976	247	86	=	=	SYM
ejpam-5976	247	87	{	{	PUNCT
ejpam-5976	247	88	a1	a1	PROPN
ejpam-5976	247	89	}	}	PUNCT
ejpam-5976	247	90	×	×	NOUN
ejpam-5976	247	91	{	{	PUNCT
ejpam-5976	247	92	b2	b2	NOUN
ejpam-5976	247	93	}	}	PUNCT
ejpam-5976	247	94	=	=	SYM
ejpam-5976	247	95	ha1	ha1	NOUN
ejpam-5976	247	96	×hb2	×hb2	PUNCT
ejpam-5976	248	1	but	but	CCONJ
ejpam-5976	248	2	|	|	ADV
ejpam-5976	248	3	ha1	ha1	PROPN
ejpam-5976	248	4	|=|	|=|	PROPN
ejpam-5976	248	5	hb2	hb2	NOUN
ejpam-5976	248	6	|=	|=	NOUN
ejpam-5976	248	7	1	1	NUM
ejpam-5976	248	8	.	.	PUNCT
ejpam-5976	248	9	corollary	corollary	ADJ
ejpam-5976	248	10	1	1	NUM
ejpam-5976	248	11	.	.	PUNCT
ejpam-5976	249	1	let	let	VERB
ejpam-5976	249	2	(	(	PUNCT
ejpam-5976	249	3	a	a	PRON
ejpam-5976	249	4	,	,	PUNCT
ejpam-5976	249	5	b	b	NOUN
ejpam-5976	249	6	)	)	PUNCT
ejpam-5976	249	7	∈	∈	PROPN
ejpam-5976	249	8	s1	s1	PROPN
ejpam-5976	249	9	×	×	PROPN
ejpam-5976	249	10	s2	s2	PROPN
ejpam-5976	249	11	.	.	PUNCT
ejpam-5976	250	1	if	if	SCONJ
ejpam-5976	250	2	ha	ha	INTJ
ejpam-5976	250	3	×hb	×hb	PROPN
ejpam-5976	250	4	is	be	AUX
ejpam-5976	250	5	a	a	DET
ejpam-5976	250	6	union	union	NOUN
ejpam-5976	250	7	of	of	ADP
ejpam-5976	250	8	at	at	ADV
ejpam-5976	250	9	least	least	ADV
ejpam-5976	250	10	two	two	NUM
ejpam-5976	250	11	h	h	NOUN
ejpam-5976	250	12	-	-	PUNCT
ejpam-5976	250	13	classes	class	NOUN
ejpam-5976	250	14	,	,	PUNCT
ejpam-5976	250	15	then	then	ADV
ejpam-5976	250	16	(	(	PUNCT
ejpam-5976	250	17	1	1	X
ejpam-5976	250	18	)	)	PUNCT
ejpam-5976	251	1	|	|	ADV
ejpam-5976	251	2	ha	ha	INTJ
ejpam-5976	252	1	|	|	ADV
ejpam-5976	252	2	>	>	X
ejpam-5976	252	3	1	1	NUM
ejpam-5976	252	4	and	and	CCONJ
ejpam-5976	252	5	hb	hb	X
ejpam-5976	253	1	=	=	PUNCT
ejpam-5976	253	2	{	{	PUNCT
ejpam-5976	253	3	b	b	NOUN
ejpam-5976	253	4	}	}	PUNCT
ejpam-5976	253	5	or	or	CCONJ
ejpam-5976	253	6	(	(	PUNCT
ejpam-5976	253	7	2	2	X
ejpam-5976	253	8	)	)	PUNCT
ejpam-5976	253	9	ha	ha	NOUN
ejpam-5976	254	1	=	=	X
ejpam-5976	254	2	{	{	PUNCT
ejpam-5976	254	3	a	a	NOUN
ejpam-5976	254	4	}	}	PUNCT
ejpam-5976	254	5	and	and	CCONJ
ejpam-5976	254	6	|	|	ADV
ejpam-5976	254	7	hb	hb	X
ejpam-5976	255	1	|	|	ADV
ejpam-5976	255	2	>	>	X
ejpam-5976	255	3	1	1	NUM
ejpam-5976	255	4	.	.	PUNCT
ejpam-5976	255	5	theorem	theorem	NOUN
ejpam-5976	255	6	9	9	NUM
ejpam-5976	255	7	.	.	PUNCT
ejpam-5976	256	1	let	let	VERB
ejpam-5976	256	2	(	(	PUNCT
ejpam-5976	256	3	a	a	PRON
ejpam-5976	256	4	,	,	PUNCT
ejpam-5976	256	5	b	b	NOUN
ejpam-5976	256	6	)	)	PUNCT
ejpam-5976	256	7	∈	∈	PROPN
ejpam-5976	256	8	s1	s1	PROPN
ejpam-5976	256	9	×	×	PROPN
ejpam-5976	256	10	s2	s2	PROPN
ejpam-5976	256	11	.	.	PUNCT
ejpam-5976	257	1	then	then	ADV
ejpam-5976	257	2	ha	ha	INTJ
ejpam-5976	257	3	×hb	×hb	PROPN
ejpam-5976	257	4	is	be	AUX
ejpam-5976	257	5	a	a	DET
ejpam-5976	257	6	union	union	NOUN
ejpam-5976	257	7	of	of	ADP
ejpam-5976	257	8	at	at	ADV
ejpam-5976	257	9	least	least	ADV
ejpam-5976	257	10	two	two	NUM
ejpam-5976	257	11	h	h	NOUN
ejpam-5976	257	12	-	-	PUNCT
ejpam-5976	257	13	classes	class	NOUN
ejpam-5976	257	14	if	if	SCONJ
ejpam-5976	257	15	and	and	CCONJ
ejpam-5976	257	16	only	only	ADV
ejpam-5976	257	17	if	if	SCONJ
ejpam-5976	257	18	(	(	PUNCT
ejpam-5976	257	19	1	1	X
ejpam-5976	257	20	)	)	PUNCT
ejpam-5976	258	1	|	|	ADV
ejpam-5976	258	2	ha	ha	INTJ
ejpam-5976	259	1	|	|	ADV
ejpam-5976	259	2	>	>	X
ejpam-5976	259	3	1	1	NUM
ejpam-5976	259	4	,	,	PUNCT
ejpam-5976	259	5	hb	hb	X
ejpam-5976	259	6	=	=	PUNCT
ejpam-5976	259	7	{	{	PUNCT
ejpam-5976	259	8	b	b	NOUN
ejpam-5976	259	9	}	}	PUNCT
ejpam-5976	259	10	and	and	CCONJ
ejpam-5976	259	11	b	b	PROPN
ejpam-5976	259	12	/∈	/∈	PUNCT
ejpam-5976	260	1	bs2	bs2	PROPN
ejpam-5976	260	2	∩	∩	PROPN
ejpam-5976	260	3	s2b	s2b	PROPN
ejpam-5976	260	4	or	or	CCONJ
ejpam-5976	260	5	(	(	PUNCT
ejpam-5976	260	6	2	2	NUM
ejpam-5976	260	7	)	)	PUNCT
ejpam-5976	260	8	a	a	DET
ejpam-5976	260	9	/∈	/∈	NOUN
ejpam-5976	260	10	as1	as1	NOUN
ejpam-5976	260	11	∩	∩	NOUN
ejpam-5976	260	12	s1a	s1a	PROPN
ejpam-5976	260	13	,	,	PUNCT
ejpam-5976	260	14	ha	ha	INTJ
ejpam-5976	260	15	=	=	X
ejpam-5976	260	16	{	{	PUNCT
ejpam-5976	260	17	a	a	NOUN
ejpam-5976	260	18	}	}	PUNCT
ejpam-5976	260	19	and	and	CCONJ
ejpam-5976	260	20	|	|	ADV
ejpam-5976	260	21	hb	hb	X
ejpam-5976	261	1	|	|	ADV
ejpam-5976	261	2	>	>	X
ejpam-5976	261	3	1	1	NUM
ejpam-5976	261	4	.	.	PUNCT
ejpam-5976	262	1	p.	p.	NOUN
ejpam-5976	262	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	262	3	,	,	PUNCT
ejpam-5976	262	4	o.	o.	PROPN
ejpam-5976	262	5	pankoon	pankoon	NOUN
ejpam-5976	262	6	,	,	PUNCT
ejpam-5976	262	7	t.	t.	PROPN
ejpam-5976	262	8	changphas	changphas	PROPN
ejpam-5976	262	9	/	/	SYM
ejpam-5976	262	10	eur	eur	PROPN
ejpam-5976	262	11	.	.	PUNCT
ejpam-5976	263	1	j.	j.	PROPN
ejpam-5976	263	2	pure	pure	PROPN
ejpam-5976	263	3	appl	appl	PROPN
ejpam-5976	263	4	.	.	PROPN
ejpam-5976	263	5	math	math	PROPN
ejpam-5976	263	6	,	,	PUNCT
ejpam-5976	263	7	18	18	NUM
ejpam-5976	263	8	(	(	PUNCT
ejpam-5976	263	9	2	2	NUM
ejpam-5976	263	10	)	)	PUNCT
ejpam-5976	263	11	(	(	PUNCT
ejpam-5976	263	12	2025	2025	NUM
ejpam-5976	263	13	)	)	PUNCT
ejpam-5976	263	14	,	,	PUNCT
ejpam-5976	263	15	5976	5976	NUM
ejpam-5976	263	16	9	9	NUM
ejpam-5976	263	17	of	of	ADP
ejpam-5976	263	18	14	14	NUM
ejpam-5976	263	19	proof	proof	NOUN
ejpam-5976	263	20	.	.	PUNCT
ejpam-5976	264	1	assume	assume	VERB
ejpam-5976	264	2	that	that	SCONJ
ejpam-5976	264	3	ha	ha	INTJ
ejpam-5976	264	4	×hb	×hb	PROPN
ejpam-5976	264	5	is	be	AUX
ejpam-5976	264	6	a	a	DET
ejpam-5976	264	7	union	union	NOUN
ejpam-5976	264	8	of	of	ADP
ejpam-5976	264	9	at	at	ADV
ejpam-5976	264	10	least	least	ADV
ejpam-5976	264	11	two	two	NUM
ejpam-5976	264	12	h	h	NOUN
ejpam-5976	264	13	-	-	PUNCT
ejpam-5976	264	14	classes	class	NOUN
ejpam-5976	264	15	.	.	PUNCT
ejpam-5976	265	1	by	by	ADP
ejpam-5976	265	2	corollary	corollary	ADJ
ejpam-5976	265	3	1	1	NUM
ejpam-5976	265	4	,	,	PUNCT
ejpam-5976	265	5	we	we	PRON
ejpam-5976	265	6	have	have	VERB
ejpam-5976	265	7	either	either	CCONJ
ejpam-5976	265	8	|	|	ADV
ejpam-5976	266	1	ha	ha	INTJ
ejpam-5976	266	2	|	|	ADV
ejpam-5976	266	3	>	>	X
ejpam-5976	266	4	1	1	NUM
ejpam-5976	266	5	and	and	CCONJ
ejpam-5976	266	6	hb	hb	X
ejpam-5976	267	1	=	=	PUNCT
ejpam-5976	267	2	{	{	PUNCT
ejpam-5976	267	3	b	b	NOUN
ejpam-5976	267	4	}	}	PUNCT
ejpam-5976	267	5	or	or	CCONJ
ejpam-5976	267	6	ha	ha	INTJ
ejpam-5976	267	7	=	=	X
ejpam-5976	267	8	{	{	PUNCT
ejpam-5976	267	9	a	a	NOUN
ejpam-5976	267	10	}	}	PUNCT
ejpam-5976	267	11	and	and	CCONJ
ejpam-5976	267	12	|	|	ADV
ejpam-5976	267	13	hb	hb	X
ejpam-5976	268	1	|	|	ADV
ejpam-5976	268	2	>	>	X
ejpam-5976	268	3	1	1	NUM
ejpam-5976	268	4	.	.	PUNCT
ejpam-5976	268	5	case	case	NOUN
ejpam-5976	268	6	1	1	NUM
ejpam-5976	268	7	:	:	PUNCT
ejpam-5976	268	8	|	|	ADV
ejpam-5976	268	9	ha	ha	INTJ
ejpam-5976	269	1	|	|	ADV
ejpam-5976	269	2	>	>	X
ejpam-5976	269	3	1	1	NUM
ejpam-5976	269	4	and	and	CCONJ
ejpam-5976	269	5	hb	hb	NOUN
ejpam-5976	270	1	=	=	PUNCT
ejpam-5976	270	2	{	{	PUNCT
ejpam-5976	270	3	b	b	NOUN
ejpam-5976	270	4	}	}	PUNCT
ejpam-5976	270	5	.	.	PUNCT
ejpam-5976	271	1	we	we	PRON
ejpam-5976	271	2	have	have	VERB
ejpam-5976	271	3	that	that	PRON
ejpam-5976	271	4	a	a	DET
ejpam-5976	271	5	∈	∈	PROPN
ejpam-5976	271	6	as1	as1	NOUN
ejpam-5976	271	7	∩	∩	PROPN
ejpam-5976	271	8	s1a	s1a	PROPN
ejpam-5976	271	9	.	.	PUNCT
ejpam-5976	271	10	suppose	suppose	VERB
ejpam-5976	271	11	that	that	SCONJ
ejpam-5976	271	12	b	b	PROPN
ejpam-5976	271	13	∈	∈	PROPN
ejpam-5976	271	14	bs2∩s2b	bs2∩s2b	NOUN
ejpam-5976	271	15	.	.	PUNCT
ejpam-5976	272	1	form	form	NOUN
ejpam-5976	272	2	theorem	theorem	VERB
ejpam-5976	272	3	3	3	NUM
ejpam-5976	272	4	,	,	PUNCT
ejpam-5976	272	5	we	we	PRON
ejpam-5976	272	6	haveha×hb	haveha×hb	ADP
ejpam-5976	272	7	=	=	SYM
ejpam-5976	272	8	h(a	h(a	PROPN
ejpam-5976	272	9	,	,	PUNCT
ejpam-5976	272	10	b	b	NOUN
ejpam-5976	272	11	)	)	PUNCT
ejpam-5976	272	12	which	which	PRON
ejpam-5976	272	13	contradicts	contradict	VERB
ejpam-5976	272	14	to	to	ADP
ejpam-5976	272	15	the	the	DET
ejpam-5976	272	16	assumption	assumption	NOUN
ejpam-5976	272	17	.	.	PUNCT
ejpam-5976	273	1	therefore	therefore	ADV
ejpam-5976	273	2	,	,	PUNCT
ejpam-5976	273	3	b	b	X
ejpam-5976	273	4	/∈	/∈	PROPN
ejpam-5976	273	5	bs2	bs2	PROPN
ejpam-5976	273	6	∩	∩	PROPN
ejpam-5976	273	7	s2b	s2b	PROPN
ejpam-5976	273	8	.	.	PUNCT
ejpam-5976	273	9	case	case	NOUN
ejpam-5976	273	10	2	2	NUM
ejpam-5976	273	11	:	:	PUNCT
ejpam-5976	273	12	ha	ha	INTJ
ejpam-5976	273	13	=	=	X
ejpam-5976	273	14	{	{	PUNCT
ejpam-5976	273	15	a	a	NOUN
ejpam-5976	273	16	}	}	PUNCT
ejpam-5976	273	17	and	and	CCONJ
ejpam-5976	273	18	|	|	ADV
ejpam-5976	273	19	hb	hb	X
ejpam-5976	274	1	|	|	ADV
ejpam-5976	274	2	>	>	X
ejpam-5976	274	3	1	1	NUM
ejpam-5976	274	4	.	.	PUNCT
ejpam-5976	274	5	we	we	PRON
ejpam-5976	274	6	can	can	AUX
ejpam-5976	274	7	prove	prove	VERB
ejpam-5976	274	8	similarly	similarly	ADV
ejpam-5976	274	9	case	case	NOUN
ejpam-5976	274	10	1	1	NUM
ejpam-5976	274	11	.	.	PUNCT
ejpam-5976	275	1	conversely	conversely	ADV
ejpam-5976	275	2	,	,	PUNCT
ejpam-5976	275	3	we	we	PRON
ejpam-5976	275	4	assume	assume	VERB
ejpam-5976	275	5	that	that	SCONJ
ejpam-5976	275	6	(	(	PUNCT
ejpam-5976	275	7	1	1	X
ejpam-5976	275	8	)	)	PUNCT
ejpam-5976	275	9	or	or	CCONJ
ejpam-5976	275	10	(	(	PUNCT
ejpam-5976	275	11	2	2	X
ejpam-5976	275	12	)	)	PUNCT
ejpam-5976	275	13	holds	hold	VERB
ejpam-5976	275	14	.	.	PUNCT
ejpam-5976	276	1	if	if	SCONJ
ejpam-5976	276	2	(	(	PUNCT
ejpam-5976	276	3	1	1	X
ejpam-5976	276	4	)	)	PUNCT
ejpam-5976	276	5	holds	hold	VERB
ejpam-5976	276	6	,	,	PUNCT
ejpam-5976	276	7	then	then	ADV
ejpam-5976	276	8	h(a	h(a	PROPN
ejpam-5976	276	9	,	,	PUNCT
ejpam-5976	276	10	b	b	NOUN
ejpam-5976	276	11	)	)	PUNCT
ejpam-5976	276	12	̸=	̸=	PROPN
ejpam-5976	276	13	ha	ha	INTJ
ejpam-5976	276	14	×	×	NOUN
ejpam-5976	276	15	hb	hb	X
ejpam-5976	276	16	by	by	ADP
ejpam-5976	276	17	theorem	theorem	NOUN
ejpam-5976	276	18	6	6	NUM
ejpam-5976	276	19	.	.	PUNCT
ejpam-5976	277	1	hence	hence	ADV
ejpam-5976	277	2	,	,	PUNCT
ejpam-5976	277	3	h(a	h(a	PROPN
ejpam-5976	277	4	,	,	PUNCT
ejpam-5976	277	5	b	b	NOUN
ejpam-5976	277	6	)	)	PUNCT
ejpam-5976	277	7	⊊	⊊	AUX
ejpam-5976	277	8	ha	ha	INTJ
ejpam-5976	277	9	×hb	×hb	PROPN
ejpam-5976	277	10	.	.	PROPN
ejpam-5976	277	11	by	by	ADP
ejpam-5976	277	12	theorem	theorem	NOUN
ejpam-5976	277	13	5	5	NUM
ejpam-5976	277	14	,	,	PUNCT
ejpam-5976	277	15	we	we	PRON
ejpam-5976	277	16	obtain	obtain	VERB
ejpam-5976	277	17	that	that	SCONJ
ejpam-5976	277	18	ha	ha	INTJ
ejpam-5976	277	19	×hb	×hb	PROPN
ejpam-5976	277	20	is	be	AUX
ejpam-5976	277	21	a	a	DET
ejpam-5976	277	22	union	union	NOUN
ejpam-5976	277	23	of	of	ADP
ejpam-5976	277	24	at	at	ADV
ejpam-5976	277	25	least	least	ADV
ejpam-5976	277	26	two	two	NUM
ejpam-5976	277	27	h	h	NOUN
ejpam-5976	277	28	-	-	PUNCT
ejpam-5976	277	29	classes	class	NOUN
ejpam-5976	277	30	.	.	PUNCT
ejpam-5976	278	1	we	we	PRON
ejpam-5976	278	2	can	can	AUX
ejpam-5976	278	3	proceed	proceed	VERB
ejpam-5976	278	4	analogously	analogously	ADV
ejpam-5976	278	5	in	in	ADP
ejpam-5976	278	6	another	another	DET
ejpam-5976	278	7	case	case	NOUN
ejpam-5976	278	8	.	.	PUNCT
ejpam-5976	279	1	let	let	VERB
ejpam-5976	279	2	u	u	PRON
ejpam-5976	279	3	∈	∈	PROPN
ejpam-5976	279	4	s.	s.	PROPN
ejpam-5976	279	5	an	an	DET
ejpam-5976	279	6	l	l	NOUN
ejpam-5976	279	7	-	-	NOUN
ejpam-5976	279	8	class	class	NOUN
ejpam-5976	279	9	(	(	PUNCT
ejpam-5976	279	10	respectively	respectively	ADV
ejpam-5976	279	11	,	,	PUNCT
ejpam-5976	279	12	r	r	NOUN
ejpam-5976	279	13	-	-	PUNCT
ejpam-5976	279	14	class	class	NOUN
ejpam-5976	279	15	)	)	PUNCT
ejpam-5976	279	16	of	of	ADP
ejpam-5976	279	17	s	s	AUX
ejpam-5976	279	18	containing	contain	VERB
ejpam-5976	279	19	u	u	NOUN
ejpam-5976	279	20	is	be	AUX
ejpam-5976	279	21	denoted	denote	VERB
ejpam-5976	279	22	by	by	ADP
ejpam-5976	279	23	lu	lu	PROPN
ejpam-5976	279	24	(	(	PUNCT
ejpam-5976	279	25	respectively	respectively	ADV
ejpam-5976	279	26	,	,	PUNCT
ejpam-5976	279	27	ru	ru	PROPN
ejpam-5976	279	28	)	)	PUNCT
ejpam-5976	279	29	.	.	PUNCT
ejpam-5976	280	1	since	since	SCONJ
ejpam-5976	280	2	relations	relation	NOUN
ejpam-5976	280	3	l	l	NOUN
ejpam-5976	280	4	and	and	CCONJ
ejpam-5976	280	5	r	r	NOUN
ejpam-5976	280	6	on	on	ADP
ejpam-5976	280	7	s	s	NOUN
ejpam-5976	280	8	are	be	AUX
ejpam-5976	280	9	defined	define	VERB
ejpam-5976	280	10	in	in	ADP
ejpam-5976	280	11	terms	term	NOUN
ejpam-5976	280	12	of	of	ADP
ejpam-5976	280	13	ideals	ideal	NOUN
ejpam-5976	280	14	,	,	PUNCT
ejpam-5976	280	15	the	the	DET
ejpam-5976	280	16	inclusion	inclusion	NOUN
ejpam-5976	280	17	order	order	NOUN
ejpam-5976	280	18	among	among	ADP
ejpam-5976	280	19	these	these	DET
ejpam-5976	280	20	ideals	ideal	NOUN
ejpam-5976	280	21	induces	induce	VERB
ejpam-5976	280	22	a	a	DET
ejpam-5976	280	23	partial	partial	ADJ
ejpam-5976	280	24	order	order	NOUN
ejpam-5976	280	25	among	among	ADP
ejpam-5976	280	26	equivalence	equivalence	NOUN
ejpam-5976	280	27	classes	class	NOUN
ejpam-5976	280	28	.	.	PUNCT
ejpam-5976	281	1	for	for	ADP
ejpam-5976	281	2	any	any	DET
ejpam-5976	281	3	a	a	PRON
ejpam-5976	281	4	,	,	PUNCT
ejpam-5976	281	5	b	b	PROPN
ejpam-5976	281	6	∈	∈	PROPN
ejpam-5976	281	7	s	s	PROPN
ejpam-5976	281	8	,	,	PUNCT
ejpam-5976	281	9	la	la	ADJ
ejpam-5976	281	10	≤	≤	NOUN
ejpam-5976	281	11	lb	lb	ADP
ejpam-5976	281	12	⇔	⇔	PROPN
ejpam-5976	281	13	l(a	l(a	PROPN
ejpam-5976	281	14	)	)	PUNCT
ejpam-5976	281	15	⊆	⊆	NUM
ejpam-5976	281	16	l(b	l(b	PROPN
ejpam-5976	281	17	)	)	PUNCT
ejpam-5976	281	18	ra	ra	PROPN
ejpam-5976	281	19	≤	≤	PROPN
ejpam-5976	281	20	rb	rb	PROPN
ejpam-5976	281	21	⇔	⇔	X
ejpam-5976	281	22	r(a	r(a	PROPN
ejpam-5976	281	23	)	)	PUNCT
ejpam-5976	281	24	⊆	⊆	NUM
ejpam-5976	281	25	r(b	r(b	NOUN
ejpam-5976	281	26	)	)	PUNCT
ejpam-5976	281	27	.	.	PUNCT
ejpam-5976	282	1	these	these	DET
ejpam-5976	282	2	partial	partial	ADJ
ejpam-5976	282	3	orders	order	NOUN
ejpam-5976	282	4	induce	induce	VERB
ejpam-5976	282	5	a	a	DET
ejpam-5976	282	6	partial	partial	ADJ
ejpam-5976	282	7	order	order	NOUN
ejpam-5976	282	8	among	among	ADP
ejpam-5976	282	9	h	h	NOUN
ejpam-5976	282	10	-	-	PUNCT
ejpam-5976	282	11	classes	class	NOUN
ejpam-5976	282	12	as	as	SCONJ
ejpam-5976	282	13	follows	follow	VERB
ejpam-5976	282	14	:	:	PUNCT
ejpam-5976	282	15	ha	ha	INTJ
ejpam-5976	282	16	≤	≤	NUM
ejpam-5976	282	17	hb	hb	X
ejpam-5976	282	18	⇔	⇔	PROPN
ejpam-5976	282	19	la	la	PROPN
ejpam-5976	282	20	≤	≤	PROPN
ejpam-5976	283	1	lb	lb	ADP
ejpam-5976	283	2	and	and	CCONJ
ejpam-5976	283	3	ra	ra	PROPN
ejpam-5976	283	4	≤	≤	NUM
ejpam-5976	283	5	rb	rb	NOUN
ejpam-5976	283	6	.	.	PUNCT
ejpam-5976	284	1	we	we	PRON
ejpam-5976	284	2	say	say	VERB
ejpam-5976	284	3	that	that	SCONJ
ejpam-5976	284	4	ha	ha	INTJ
ejpam-5976	284	5	is	be	AUX
ejpam-5976	284	6	maximal	maximal	ADJ
ejpam-5976	284	7	if	if	SCONJ
ejpam-5976	284	8	there	there	PRON
ejpam-5976	284	9	is	be	VERB
ejpam-5976	284	10	no	no	DET
ejpam-5976	284	11	u	u	NOUN
ejpam-5976	284	12	∈	∈	NOUN
ejpam-5976	284	13	s	s	VERB
ejpam-5976	284	14	such	such	ADJ
ejpam-5976	284	15	that	that	SCONJ
ejpam-5976	284	16	ha	ha	INTJ
ejpam-5976	284	17	≤	≤	NUM
ejpam-5976	284	18	hu	hu	PROPN
ejpam-5976	285	1	and	and	CCONJ
ejpam-5976	285	2	ha	ha	INTJ
ejpam-5976	285	3	̸=	̸=	PROPN
ejpam-5976	285	4	hu	hu	PROPN
ejpam-5976	285	5	.	.	PUNCT
ejpam-5976	286	1	equivalently	equivalently	ADV
ejpam-5976	286	2	,	,	PUNCT
ejpam-5976	286	3	ha	ha	INTJ
ejpam-5976	286	4	is	be	AUX
ejpam-5976	286	5	maximal	maximal	ADJ
ejpam-5976	286	6	if	if	SCONJ
ejpam-5976	286	7	and	and	CCONJ
ejpam-5976	286	8	only	only	ADV
ejpam-5976	286	9	if	if	SCONJ
ejpam-5976	286	10	there	there	PRON
ejpam-5976	286	11	is	be	VERB
ejpam-5976	286	12	no	no	DET
ejpam-5976	286	13	u	u	NOUN
ejpam-5976	286	14	∈	∈	NOUN
ejpam-5976	286	15	s	s	VERB
ejpam-5976	286	16	such	such	ADJ
ejpam-5976	286	17	that	that	SCONJ
ejpam-5976	286	18	ha	ha	INTJ
ejpam-5976	286	19	≤	≤	NUM
ejpam-5976	286	20	hu	hu	PROPN
ejpam-5976	286	21	and	and	CCONJ
ejpam-5976	286	22	q(a	q(a	PROPN
ejpam-5976	286	23	)	)	PUNCT
ejpam-5976	286	24	̸=	̸=	PROPN
ejpam-5976	286	25	q(u	q(u	NOUN
ejpam-5976	286	26	)	)	PUNCT
ejpam-5976	286	27	.	.	PUNCT
ejpam-5976	287	1	a	a	DET
ejpam-5976	287	2	characterization	characterization	NOUN
ejpam-5976	287	3	of	of	ADP
ejpam-5976	287	4	a	a	DET
ejpam-5976	287	5	maximal	maximal	ADJ
ejpam-5976	287	6	h	h	NOUN
ejpam-5976	287	7	-	-	PUNCT
ejpam-5976	287	8	class	class	NOUN
ejpam-5976	287	9	is	be	AUX
ejpam-5976	287	10	indicated	indicate	VERB
ejpam-5976	287	11	as	as	ADP
ejpam-5976	287	12	the	the	DET
ejpam-5976	287	13	following	follow	VERB
ejpam-5976	287	14	theorem	theorem	NOUN
ejpam-5976	287	15	.	.	PUNCT
ejpam-5976	287	16	theorem	theorem	PROPN
ejpam-5976	287	17	10	10	NUM
ejpam-5976	287	18	.	.	PUNCT
ejpam-5976	288	1	let	let	VERB
ejpam-5976	288	2	a	a	DET
ejpam-5976	288	3	∈	∈	NOUN
ejpam-5976	288	4	s.	s.	PROPN
ejpam-5976	288	5	then	then	ADV
ejpam-5976	288	6	ha	ha	INTJ
ejpam-5976	288	7	is	be	AUX
ejpam-5976	288	8	a	a	DET
ejpam-5976	288	9	maximal	maximal	ADJ
ejpam-5976	288	10	if	if	SCONJ
ejpam-5976	289	1	and	and	CCONJ
ejpam-5976	289	2	only	only	ADV
ejpam-5976	289	3	if	if	SCONJ
ejpam-5976	289	4	there	there	PRON
ejpam-5976	289	5	is	be	VERB
ejpam-5976	289	6	no	no	DET
ejpam-5976	289	7	b	b	PROPN
ejpam-5976	289	8	∈	∈	NOUN
ejpam-5976	289	9	s	s	VERB
ejpam-5976	289	10	such	such	ADJ
ejpam-5976	289	11	that	that	SCONJ
ejpam-5976	289	12	q(a	q(a	PROPN
ejpam-5976	289	13	)	)	PUNCT
ejpam-5976	289	14	⊊	⊊	VERB
ejpam-5976	289	15	q(b	q(b	ADJ
ejpam-5976	289	16	)	)	PUNCT
ejpam-5976	289	17	.	.	PUNCT
ejpam-5976	290	1	proof	proof	NOUN
ejpam-5976	290	2	.	.	PUNCT
ejpam-5976	291	1	suppose	suppose	VERB
ejpam-5976	291	2	that	that	SCONJ
ejpam-5976	291	3	there	there	PRON
ejpam-5976	291	4	exists	exist	VERB
ejpam-5976	291	5	b	b	PROPN
ejpam-5976	291	6	∈	∈	PROPN
ejpam-5976	291	7	s	s	VERB
ejpam-5976	291	8	such	such	ADJ
ejpam-5976	291	9	that	that	SCONJ
ejpam-5976	291	10	q(a	q(a	PROPN
ejpam-5976	291	11	)	)	PUNCT
ejpam-5976	291	12	⊊	⊊	VERB
ejpam-5976	291	13	q(b),which	q(b),which	PROPN
ejpam-5976	291	14	implies	imply	VERB
ejpam-5976	291	15	that	that	SCONJ
ejpam-5976	291	16	ha	ha	INTJ
ejpam-5976	291	17	̸=	̸=	PROPN
ejpam-5976	291	18	hb	hb	PROPN
ejpam-5976	291	19	.	.	PUNCT
ejpam-5976	292	1	since	since	SCONJ
ejpam-5976	292	2	a	a	DET
ejpam-5976	292	3	∈	∈	PROPN
ejpam-5976	292	4	q(a	q(a	PROPN
ejpam-5976	292	5	)	)	PUNCT
ejpam-5976	292	6	⊊	⊊	VERB
ejpam-5976	292	7	q(b	q(b	ADV
ejpam-5976	292	8	)	)	PUNCT
ejpam-5976	292	9	=	=	SYM
ejpam-5976	292	10	l(b	l(b	PROPN
ejpam-5976	292	11	)	)	PUNCT
ejpam-5976	292	12	∩r(b	∩r(b	NOUN
ejpam-5976	292	13	)	)	PUNCT
ejpam-5976	292	14	,	,	PUNCT
ejpam-5976	292	15	we	we	PRON
ejpam-5976	292	16	have	have	VERB
ejpam-5976	292	17	a	a	DET
ejpam-5976	292	18	∈	∈	PROPN
ejpam-5976	292	19	l(b	l(b	PROPN
ejpam-5976	292	20	)	)	PUNCT
ejpam-5976	292	21	and	and	CCONJ
ejpam-5976	292	22	a	a	DET
ejpam-5976	292	23	∈	∈	PROPN
ejpam-5976	292	24	r(b	r(b	NOUN
ejpam-5976	292	25	)	)	PUNCT
ejpam-5976	292	26	.	.	PUNCT
ejpam-5976	293	1	hence	hence	ADV
ejpam-5976	293	2	,	,	PUNCT
ejpam-5976	293	3	l(a	l(a	PROPN
ejpam-5976	293	4	)	)	PUNCT
ejpam-5976	293	5	⊆	⊆	NUM
ejpam-5976	293	6	l(b	l(b	PROPN
ejpam-5976	293	7	)	)	PUNCT
ejpam-5976	293	8	and	and	CCONJ
ejpam-5976	293	9	r(a	r(a	PROPN
ejpam-5976	293	10	)	)	PUNCT
ejpam-5976	293	11	⊆	⊆	NUM
ejpam-5976	293	12	r(b	r(b	NOUN
ejpam-5976	293	13	)	)	PUNCT
ejpam-5976	293	14	.	.	PUNCT
ejpam-5976	294	1	it	it	PRON
ejpam-5976	294	2	follows	follow	VERB
ejpam-5976	294	3	that	that	SCONJ
ejpam-5976	294	4	la	la	ADV
ejpam-5976	294	5	⩽	⩽	ADJ
ejpam-5976	294	6	lb	lb	ADP
ejpam-5976	294	7	and	and	CCONJ
ejpam-5976	294	8	ra	ra	PROPN
ejpam-5976	294	9	⩽	⩽	PROPN
ejpam-5976	294	10	rb	rb	PROPN
ejpam-5976	294	11	.	.	PUNCT
ejpam-5976	295	1	thus	thus	ADV
ejpam-5976	295	2	,	,	PUNCT
ejpam-5976	295	3	ha	ha	INTJ
ejpam-5976	295	4	is	be	AUX
ejpam-5976	295	5	not	not	PART
ejpam-5976	295	6	maximal	maximal	ADJ
ejpam-5976	295	7	.	.	PUNCT
ejpam-5976	296	1	conversely	conversely	ADV
ejpam-5976	296	2	,	,	PUNCT
ejpam-5976	296	3	assume	assume	VERB
ejpam-5976	296	4	thatha	thatha	PROPN
ejpam-5976	296	5	is	be	AUX
ejpam-5976	296	6	not	not	PART
ejpam-5976	296	7	maximal	maximal	ADJ
ejpam-5976	296	8	.	.	PUNCT
ejpam-5976	297	1	then	then	ADV
ejpam-5976	297	2	there	there	PRON
ejpam-5976	297	3	exists	exist	VERB
ejpam-5976	297	4	b	b	PROPN
ejpam-5976	297	5	∈	∈	PROPN
ejpam-5976	297	6	s	s	VERB
ejpam-5976	297	7	such	such	ADJ
ejpam-5976	297	8	thatha	thatha	NOUN
ejpam-5976	297	9	⩽	⩽	PROPN
ejpam-5976	297	10	hb	hb	PROPN
ejpam-5976	298	1	and	and	CCONJ
ejpam-5976	298	2	ha	ha	INTJ
ejpam-5976	298	3	̸=	̸=	PROPN
ejpam-5976	298	4	hb	hb	PROPN
ejpam-5976	298	5	.	.	PUNCT
ejpam-5976	299	1	thus	thus	ADV
ejpam-5976	299	2	,	,	PUNCT
ejpam-5976	299	3	la	la	PROPN
ejpam-5976	299	4	⩽	⩽	ADJ
ejpam-5976	299	5	lb	lb	ADP
ejpam-5976	299	6	and	and	CCONJ
ejpam-5976	299	7	ra	ra	PROPN
ejpam-5976	299	8	⩽	⩽	NOUN
ejpam-5976	299	9	rb	rb	PROPN
ejpam-5976	299	10	.	.	PUNCT
ejpam-5976	300	1	these	these	PRON
ejpam-5976	300	2	imply	imply	VERB
ejpam-5976	300	3	l(a	l(a	PROPN
ejpam-5976	300	4	)	)	PUNCT
ejpam-5976	300	5	⊆	⊆	NUM
ejpam-5976	300	6	l(b	l(b	PROPN
ejpam-5976	300	7	)	)	PUNCT
ejpam-5976	300	8	r(a	r(a	PROPN
ejpam-5976	300	9	)	)	PUNCT
ejpam-5976	300	10	⊆	⊆	NUM
ejpam-5976	300	11	r(b	r(b	NOUN
ejpam-5976	300	12	)	)	PUNCT
ejpam-5976	300	13	q(a	q(a	PROPN
ejpam-5976	300	14	)	)	PUNCT
ejpam-5976	300	15	̸=	̸=	PROPN
ejpam-5976	300	16	q(b	q(b	ADJ
ejpam-5976	300	17	)	)	PUNCT
ejpam-5976	300	18	.	.	PUNCT
ejpam-5976	301	1	p.	p.	NOUN
ejpam-5976	301	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	301	3	,	,	PUNCT
ejpam-5976	301	4	o.	o.	PROPN
ejpam-5976	301	5	pankoon	pankoon	NOUN
ejpam-5976	301	6	,	,	PUNCT
ejpam-5976	301	7	t.	t.	PROPN
ejpam-5976	301	8	changphas	changphas	PROPN
ejpam-5976	301	9	/	/	SYM
ejpam-5976	301	10	eur	eur	PROPN
ejpam-5976	301	11	.	.	PUNCT
ejpam-5976	302	1	j.	j.	PROPN
ejpam-5976	302	2	pure	pure	PROPN
ejpam-5976	302	3	appl	appl	PROPN
ejpam-5976	302	4	.	.	PROPN
ejpam-5976	302	5	math	math	PROPN
ejpam-5976	302	6	,	,	PUNCT
ejpam-5976	302	7	18	18	NUM
ejpam-5976	302	8	(	(	PUNCT
ejpam-5976	302	9	2	2	NUM
ejpam-5976	302	10	)	)	PUNCT
ejpam-5976	302	11	(	(	PUNCT
ejpam-5976	302	12	2025	2025	NUM
ejpam-5976	302	13	)	)	PUNCT
ejpam-5976	302	14	,	,	PUNCT
ejpam-5976	302	15	5976	5976	NUM
ejpam-5976	302	16	10	10	NUM
ejpam-5976	302	17	of	of	ADP
ejpam-5976	302	18	14	14	NUM
ejpam-5976	302	19	hence	hence	ADV
ejpam-5976	302	20	,	,	PUNCT
ejpam-5976	302	21	q(a	q(a	PROPN
ejpam-5976	302	22	)	)	PUNCT
ejpam-5976	302	23	=	=	SYM
ejpam-5976	302	24	l(a	l(a	X
ejpam-5976	302	25	)	)	PUNCT
ejpam-5976	302	26	∩r(a	∩r(a	PROPN
ejpam-5976	302	27	)	)	PUNCT
ejpam-5976	302	28	⊆	⊆	NUM
ejpam-5976	302	29	l(b	l(b	PROPN
ejpam-5976	302	30	)	)	PUNCT
ejpam-5976	302	31	∩r(b	∩r(b	NOUN
ejpam-5976	302	32	)	)	PUNCT
ejpam-5976	302	33	=	=	PUNCT
ejpam-5976	303	1	q(b	q(b	ADJ
ejpam-5976	303	2	)	)	PUNCT
ejpam-5976	303	3	.	.	PUNCT
ejpam-5976	304	1	therefore	therefore	ADV
ejpam-5976	304	2	,	,	PUNCT
ejpam-5976	304	3	q(a	q(a	PROPN
ejpam-5976	304	4	)	)	PUNCT
ejpam-5976	304	5	⊊	⊊	VERB
ejpam-5976	304	6	q(b	q(b	ADJ
ejpam-5976	304	7	)	)	PUNCT
ejpam-5976	304	8	.	.	PUNCT
ejpam-5976	305	1	theorem	theorem	VERB
ejpam-5976	305	2	11	11	NUM
ejpam-5976	305	3	.	.	PUNCT
ejpam-5976	306	1	let	let	VERB
ejpam-5976	306	2	(	(	PUNCT
ejpam-5976	306	3	a	a	PRON
ejpam-5976	306	4	,	,	PUNCT
ejpam-5976	306	5	b	b	NOUN
ejpam-5976	306	6	)	)	PUNCT
ejpam-5976	306	7	∈	∈	PROPN
ejpam-5976	306	8	s1	s1	NOUN
ejpam-5976	306	9	×	×	NOUN
ejpam-5976	306	10	s2	s2	NOUN
ejpam-5976	306	11	such	such	ADJ
ejpam-5976	306	12	that	that	SCONJ
ejpam-5976	306	13	(	(	PUNCT
ejpam-5976	306	14	a	a	PRON
ejpam-5976	306	15	,	,	PUNCT
ejpam-5976	306	16	b	b	NOUN
ejpam-5976	306	17	)	)	PUNCT
ejpam-5976	306	18	∈	∈	PROPN
ejpam-5976	306	19	(	(	PUNCT
ejpam-5976	306	20	as1	as1	NOUN
ejpam-5976	306	21	∩	∩	NOUN
ejpam-5976	306	22	s1a	s1a	PROPN
ejpam-5976	306	23	)	)	PUNCT
ejpam-5976	306	24	×	×	NOUN
ejpam-5976	306	25	(	(	PUNCT
ejpam-5976	306	26	bs2	bs2	PROPN
ejpam-5976	306	27	∩	∩	PROPN
ejpam-5976	306	28	s2b	s2b	PROPN
ejpam-5976	306	29	)	)	PUNCT
ejpam-5976	306	30	.	.	PUNCT
ejpam-5976	307	1	then	then	ADV
ejpam-5976	307	2	h(a	h(a	PROPN
ejpam-5976	307	3	,	,	PUNCT
ejpam-5976	307	4	b	b	X
ejpam-5976	307	5	)	)	PUNCT
ejpam-5976	307	6	is	be	AUX
ejpam-5976	307	7	maximal	maximal	ADJ
ejpam-5976	307	8	if	if	SCONJ
ejpam-5976	307	9	and	and	CCONJ
ejpam-5976	307	10	only	only	ADV
ejpam-5976	307	11	if	if	SCONJ
ejpam-5976	307	12	ha	ha	INTJ
ejpam-5976	307	13	and	and	CCONJ
ejpam-5976	307	14	hb	hb	NOUN
ejpam-5976	307	15	are	be	AUX
ejpam-5976	307	16	maximal	maximal	ADJ
ejpam-5976	307	17	.	.	PUNCT
ejpam-5976	308	1	proof	proof	NOUN
ejpam-5976	308	2	.	.	PUNCT
ejpam-5976	309	1	assume	assume	VERB
ejpam-5976	309	2	that	that	SCONJ
ejpam-5976	309	3	h(a	h(a	PROPN
ejpam-5976	309	4	,	,	PUNCT
ejpam-5976	309	5	b	b	X
ejpam-5976	309	6	)	)	PUNCT
ejpam-5976	309	7	is	be	AUX
ejpam-5976	309	8	not	not	PART
ejpam-5976	309	9	maximal	maximal	ADJ
ejpam-5976	309	10	.	.	PUNCT
ejpam-5976	310	1	then	then	ADV
ejpam-5976	310	2	there	there	PRON
ejpam-5976	310	3	exists	exist	VERB
ejpam-5976	310	4	(	(	PUNCT
ejpam-5976	310	5	u	u	NOUN
ejpam-5976	310	6	,	,	PUNCT
ejpam-5976	310	7	v	v	NOUN
ejpam-5976	310	8	)	)	PUNCT
ejpam-5976	310	9	∈	∈	PROPN
ejpam-5976	310	10	s1×s2	s1×s2	PROPN
ejpam-5976	310	11	such	such	ADJ
ejpam-5976	310	12	that	that	SCONJ
ejpam-5976	310	13	q((a	q((a	NOUN
ejpam-5976	310	14	,	,	PUNCT
ejpam-5976	310	15	b	b	NOUN
ejpam-5976	310	16	)	)	PUNCT
ejpam-5976	310	17	)	)	PUNCT
ejpam-5976	310	18	⊊	⊊	VERB
ejpam-5976	310	19	q((u	q((u	NOUN
ejpam-5976	310	20	,	,	PUNCT
ejpam-5976	310	21	v	v	NOUN
ejpam-5976	310	22	)	)	PUNCT
ejpam-5976	310	23	)	)	PUNCT
ejpam-5976	310	24	.	.	PUNCT
ejpam-5976	311	1	by	by	ADP
ejpam-5976	311	2	theorem	theorem	NOUN
ejpam-5976	311	3	3	3	NUM
ejpam-5976	311	4	,	,	PUNCT
ejpam-5976	311	5	we	we	PRON
ejpam-5976	311	6	have	have	VERB
ejpam-5976	311	7	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	311	8	)	)	PUNCT
ejpam-5976	311	9	=	=	SYM
ejpam-5976	311	10	q((a	q((a	PROPN
ejpam-5976	311	11	,	,	PUNCT
ejpam-5976	311	12	b	b	NOUN
ejpam-5976	311	13	)	)	PUNCT
ejpam-5976	311	14	)	)	PUNCT
ejpam-5976	311	15	⊊	⊊	VERB
ejpam-5976	311	16	q((u	q((u	NOUN
ejpam-5976	311	17	,	,	PUNCT
ejpam-5976	311	18	v	v	NOUN
ejpam-5976	311	19	)	)	PUNCT
ejpam-5976	311	20	)	)	PUNCT
ejpam-5976	312	1	⊆	⊆	NUM
ejpam-5976	312	2	q(u)×q(v	q(u)×q(v	NOUN
ejpam-5976	312	3	)	)	PUNCT
ejpam-5976	312	4	.	.	PUNCT
ejpam-5976	313	1	this	this	PRON
ejpam-5976	313	2	means	mean	VERB
ejpam-5976	313	3	that	that	SCONJ
ejpam-5976	313	4	q(a	q(a	NOUN
ejpam-5976	313	5	)	)	PUNCT
ejpam-5976	313	6	⊊	⊊	VERB
ejpam-5976	313	7	q(u	q(u	NOUN
ejpam-5976	313	8	)	)	PUNCT
ejpam-5976	313	9	or	or	CCONJ
ejpam-5976	313	10	q(b	q(b	ADJ
ejpam-5976	313	11	)	)	PUNCT
ejpam-5976	313	12	⊊	⊊	VERB
ejpam-5976	313	13	q(v	q(v	NOUN
ejpam-5976	313	14	)	)	PUNCT
ejpam-5976	313	15	.	.	PUNCT
ejpam-5976	314	1	thus	thus	ADV
ejpam-5976	314	2	,	,	PUNCT
ejpam-5976	314	3	ha	ha	INTJ
ejpam-5976	314	4	or	or	CCONJ
ejpam-5976	314	5	hb	hb	PROPN
ejpam-5976	314	6	is	be	AUX
ejpam-5976	314	7	not	not	PART
ejpam-5976	314	8	maximal	maximal	ADJ
ejpam-5976	314	9	.	.	PUNCT
ejpam-5976	315	1	conversely	conversely	ADV
ejpam-5976	315	2	,	,	PUNCT
ejpam-5976	315	3	suppose	suppose	VERB
ejpam-5976	315	4	that	that	SCONJ
ejpam-5976	315	5	ha	ha	INTJ
ejpam-5976	315	6	is	be	AUX
ejpam-5976	315	7	not	not	PART
ejpam-5976	315	8	maximal	maximal	ADJ
ejpam-5976	315	9	.	.	PUNCT
ejpam-5976	316	1	then	then	ADV
ejpam-5976	316	2	there	there	PRON
ejpam-5976	316	3	exists	exist	VERB
ejpam-5976	316	4	u	u	NOUN
ejpam-5976	316	5	∈	∈	PROPN
ejpam-5976	316	6	s1	s1	NOUN
ejpam-5976	316	7	such	such	ADJ
ejpam-5976	316	8	that	that	SCONJ
ejpam-5976	316	9	q(a	q(a	PROPN
ejpam-5976	316	10	)	)	PUNCT
ejpam-5976	316	11	⊊	⊊	VERB
ejpam-5976	316	12	q(u	q(u	NOUN
ejpam-5976	316	13	)	)	PUNCT
ejpam-5976	316	14	.	.	PUNCT
ejpam-5976	317	1	this	this	PRON
ejpam-5976	317	2	implies	imply	VERB
ejpam-5976	317	3	that	that	SCONJ
ejpam-5976	317	4	q(a	q(a	NOUN
ejpam-5976	317	5	)	)	PUNCT
ejpam-5976	317	6	⊆	⊆	NUM
ejpam-5976	317	7	us1	us1	PROPN
ejpam-5976	317	8	∩	∩	ADJ
ejpam-5976	317	9	s1u	s1u	NOUN
ejpam-5976	317	10	and	and	CCONJ
ejpam-5976	317	11	u	u	NOUN
ejpam-5976	317	12	/∈	/∈	PUNCT
ejpam-5976	317	13	q(a	q(a	PROPN
ejpam-5976	317	14	)	)	PUNCT
ejpam-5976	317	15	.	.	PUNCT
ejpam-5976	318	1	case	case	NOUN
ejpam-5976	318	2	1	1	NUM
ejpam-5976	318	3	:	:	PUNCT
ejpam-5976	318	4	u	u	PROPN
ejpam-5976	318	5	∈	∈	PROPN
ejpam-5976	318	6	us1	us1	PROPN
ejpam-5976	318	7	∩	∩	PROPN
ejpam-5976	318	8	s1u	s1u	PROPN
ejpam-5976	318	9	.	.	PUNCT
ejpam-5976	319	1	then	then	ADV
ejpam-5976	319	2	q((a	q((a	PROPN
ejpam-5976	319	3	,	,	PUNCT
ejpam-5976	319	4	b	b	NOUN
ejpam-5976	319	5	)	)	PUNCT
ejpam-5976	319	6	)	)	PUNCT
ejpam-5976	320	1	=	=	SYM
ejpam-5976	320	2	q(a)×q(b	q(a)×q(b	X
ejpam-5976	320	3	)	)	PUNCT
ejpam-5976	320	4	⊊	⊊	VERB
ejpam-5976	320	5	q(u)×q(b	q(u)×q(b	ADV
ejpam-5976	320	6	)	)	PUNCT
ejpam-5976	320	7	=	=	SYM
ejpam-5976	320	8	q((u	q((u	NOUN
ejpam-5976	320	9	,	,	PUNCT
ejpam-5976	320	10	b	b	NOUN
ejpam-5976	320	11	)	)	PUNCT
ejpam-5976	320	12	)	)	PUNCT
ejpam-5976	320	13	.	.	PUNCT
ejpam-5976	321	1	case	case	NOUN
ejpam-5976	321	2	2	2	NUM
ejpam-5976	321	3	:	:	PUNCT
ejpam-5976	321	4	u	u	PROPN
ejpam-5976	321	5	/∈	/∈	PROPN
ejpam-5976	322	1	us1	us1	PROPN
ejpam-5976	322	2	∩	∩	PROPN
ejpam-5976	322	3	s1u	s1u	PROPN
ejpam-5976	322	4	.	.	PUNCT
ejpam-5976	323	1	then	then	ADV
ejpam-5976	323	2	q((a	q((a	PROPN
ejpam-5976	323	3	,	,	PUNCT
ejpam-5976	323	4	b	b	NOUN
ejpam-5976	323	5	)	)	PUNCT
ejpam-5976	323	6	)	)	PUNCT
ejpam-5976	324	1	=	=	SYM
ejpam-5976	324	2	q(a)×q(b	q(a)×q(b	X
ejpam-5976	324	3	)	)	PUNCT
ejpam-5976	324	4	⊊	⊊	VERB
ejpam-5976	324	5	{	{	PUNCT
ejpam-5976	324	6	(	(	PUNCT
ejpam-5976	324	7	u	u	NOUN
ejpam-5976	324	8	,	,	PUNCT
ejpam-5976	324	9	b	b	NOUN
ejpam-5976	324	10	)	)	PUNCT
ejpam-5976	324	11	}	}	PUNCT
ejpam-5976	324	12	∪	∪	ADV
ejpam-5976	324	13	(	(	PUNCT
ejpam-5976	324	14	q(a)×q(b	q(a)×q(b	ADJ
ejpam-5976	324	15	)	)	PUNCT
ejpam-5976	324	16	)	)	PUNCT
ejpam-5976	325	1	=	=	PRON
ejpam-5976	325	2	{	{	PUNCT
ejpam-5976	325	3	(	(	PUNCT
ejpam-5976	325	4	u	u	NOUN
ejpam-5976	325	5	,	,	PUNCT
ejpam-5976	325	6	b	b	NOUN
ejpam-5976	325	7	)	)	PUNCT
ejpam-5976	325	8	}	}	PUNCT
ejpam-5976	325	9	∪	∪	X
ejpam-5976	325	10	(	(	PUNCT
ejpam-5976	325	11	(	(	PUNCT
ejpam-5976	325	12	as1	as1	NOUN
ejpam-5976	325	13	∩	∩	X
ejpam-5976	325	14	s1a)×	s1a)×	X
ejpam-5976	325	15	(	(	PUNCT
ejpam-5976	325	16	bs2	bs2	PROPN
ejpam-5976	325	17	∩	∩	PROPN
ejpam-5976	325	18	s2b	s2b	PROPN
ejpam-5976	325	19	)	)	PUNCT
ejpam-5976	325	20	)	)	PUNCT
ejpam-5976	326	1	⊆	⊆	X
ejpam-5976	326	2	{	{	PUNCT
ejpam-5976	326	3	(	(	PUNCT
ejpam-5976	326	4	u	u	NOUN
ejpam-5976	326	5	,	,	PUNCT
ejpam-5976	326	6	b	b	NOUN
ejpam-5976	326	7	)	)	PUNCT
ejpam-5976	326	8	}	}	PUNCT
ejpam-5976	326	9	∪	∪	X
ejpam-5976	326	10	(	(	PUNCT
ejpam-5976	326	11	(	(	PUNCT
ejpam-5976	326	12	us1	us1	PROPN
ejpam-5976	326	13	∩	∩	PROPN
ejpam-5976	326	14	s1u)×	s1u)×	PROPN
ejpam-5976	326	15	(	(	PUNCT
ejpam-5976	326	16	bs2	bs2	PROPN
ejpam-5976	326	17	∩	∩	PROPN
ejpam-5976	326	18	s2b	s2b	PROPN
ejpam-5976	326	19	)	)	PUNCT
ejpam-5976	326	20	)	)	PUNCT
ejpam-5976	327	1	=	=	PUNCT
ejpam-5976	327	2	q((u	q((u	NOUN
ejpam-5976	327	3	,	,	PUNCT
ejpam-5976	327	4	b	b	NOUN
ejpam-5976	327	5	)	)	PUNCT
ejpam-5976	327	6	)	)	PUNCT
ejpam-5976	327	7	.	.	PUNCT
ejpam-5976	328	1	therefore	therefore	ADV
ejpam-5976	328	2	,	,	PUNCT
ejpam-5976	328	3	h(a	h(a	PROPN
ejpam-5976	328	4	,	,	PUNCT
ejpam-5976	328	5	b	b	NOUN
ejpam-5976	328	6	)	)	PUNCT
ejpam-5976	328	7	is	be	AUX
ejpam-5976	328	8	not	not	PART
ejpam-5976	328	9	maximal	maximal	ADJ
ejpam-5976	328	10	.	.	PUNCT
ejpam-5976	329	1	we	we	PRON
ejpam-5976	329	2	obtain	obtain	VERB
ejpam-5976	329	3	the	the	DET
ejpam-5976	329	4	same	same	ADJ
ejpam-5976	329	5	conclusion	conclusion	NOUN
ejpam-5976	329	6	when	when	SCONJ
ejpam-5976	329	7	hb	hb	PROPN
ejpam-5976	329	8	is	be	AUX
ejpam-5976	329	9	not	not	PART
ejpam-5976	329	10	maximal	maximal	ADJ
ejpam-5976	329	11	.	.	PUNCT
ejpam-5976	330	1	lemma	lemma	PROPN
ejpam-5976	330	2	2	2	X
ejpam-5976	330	3	.	.	PUNCT
ejpam-5976	331	1	let	let	VERB
ejpam-5976	331	2	(	(	PUNCT
ejpam-5976	331	3	a	a	PRON
ejpam-5976	331	4	,	,	PUNCT
ejpam-5976	331	5	b	b	NOUN
ejpam-5976	331	6	)	)	PUNCT
ejpam-5976	331	7	∈	∈	PROPN
ejpam-5976	331	8	s1	s1	NOUN
ejpam-5976	331	9	×	×	NOUN
ejpam-5976	331	10	s2	s2	NOUN
ejpam-5976	331	11	such	such	ADJ
ejpam-5976	331	12	that	that	SCONJ
ejpam-5976	331	13	(	(	PUNCT
ejpam-5976	331	14	a	a	DET
ejpam-5976	331	15	,	,	PUNCT
ejpam-5976	331	16	b	b	NOUN
ejpam-5976	331	17	)	)	PUNCT
ejpam-5976	331	18	/∈	/∈	PUNCT
ejpam-5976	332	1	(	(	PUNCT
ejpam-5976	332	2	as1	as1	NOUN
ejpam-5976	332	3	∩	∩	NOUN
ejpam-5976	332	4	s1a	s1a	PROPN
ejpam-5976	332	5	)	)	PUNCT
ejpam-5976	332	6	×	×	NOUN
ejpam-5976	332	7	(	(	PUNCT
ejpam-5976	332	8	bs2	bs2	PROPN
ejpam-5976	332	9	∩	∩	PROPN
ejpam-5976	332	10	s2b	s2b	PROPN
ejpam-5976	332	11	)	)	PUNCT
ejpam-5976	332	12	.	.	PUNCT
ejpam-5976	333	1	if	if	SCONJ
ejpam-5976	333	2	ha	ha	INTJ
ejpam-5976	333	3	and	and	CCONJ
ejpam-5976	333	4	hb	hb	NOUN
ejpam-5976	333	5	are	be	AUX
ejpam-5976	333	6	maximal	maximal	ADJ
ejpam-5976	333	7	,	,	PUNCT
ejpam-5976	333	8	then	then	ADV
ejpam-5976	333	9	h(a	h(a	PROPN
ejpam-5976	333	10	,	,	PUNCT
ejpam-5976	333	11	b	b	X
ejpam-5976	333	12	)	)	PUNCT
ejpam-5976	333	13	is	be	AUX
ejpam-5976	333	14	maximal	maximal	ADJ
ejpam-5976	333	15	.	.	PUNCT
ejpam-5976	334	1	p.	p.	NOUN
ejpam-5976	334	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	334	3	,	,	PUNCT
ejpam-5976	334	4	o.	o.	PROPN
ejpam-5976	334	5	pankoon	pankoon	NOUN
ejpam-5976	334	6	,	,	PUNCT
ejpam-5976	334	7	t.	t.	PROPN
ejpam-5976	334	8	changphas	changphas	PROPN
ejpam-5976	334	9	/	/	SYM
ejpam-5976	334	10	eur	eur	PROPN
ejpam-5976	334	11	.	.	PUNCT
ejpam-5976	335	1	j.	j.	PROPN
ejpam-5976	335	2	pure	pure	PROPN
ejpam-5976	335	3	appl	appl	PROPN
ejpam-5976	335	4	.	.	PROPN
ejpam-5976	335	5	math	math	PROPN
ejpam-5976	335	6	,	,	PUNCT
ejpam-5976	335	7	18	18	NUM
ejpam-5976	335	8	(	(	PUNCT
ejpam-5976	335	9	2	2	NUM
ejpam-5976	335	10	)	)	PUNCT
ejpam-5976	335	11	(	(	PUNCT
ejpam-5976	335	12	2025	2025	NUM
ejpam-5976	335	13	)	)	PUNCT
ejpam-5976	335	14	,	,	PUNCT
ejpam-5976	335	15	5976	5976	NUM
ejpam-5976	335	16	11	11	NUM
ejpam-5976	335	17	of	of	ADP
ejpam-5976	335	18	14	14	NUM
ejpam-5976	335	19	proof	proof	NOUN
ejpam-5976	335	20	.	.	PUNCT
ejpam-5976	336	1	assume	assume	VERB
ejpam-5976	336	2	that	that	SCONJ
ejpam-5976	336	3	h(a	h(a	PROPN
ejpam-5976	336	4	,	,	PUNCT
ejpam-5976	336	5	b	b	X
ejpam-5976	336	6	)	)	PUNCT
ejpam-5976	336	7	is	be	AUX
ejpam-5976	336	8	not	not	PART
ejpam-5976	336	9	maximal	maximal	ADJ
ejpam-5976	336	10	.	.	PUNCT
ejpam-5976	337	1	then	then	ADV
ejpam-5976	337	2	there	there	PRON
ejpam-5976	337	3	exists	exist	VERB
ejpam-5976	337	4	(	(	PUNCT
ejpam-5976	337	5	u	u	NOUN
ejpam-5976	337	6	,	,	PUNCT
ejpam-5976	337	7	v	v	NOUN
ejpam-5976	337	8	)	)	PUNCT
ejpam-5976	337	9	∈	∈	PROPN
ejpam-5976	337	10	s1×s2	s1×s2	PROPN
ejpam-5976	337	11	such	such	ADJ
ejpam-5976	337	12	that	that	SCONJ
ejpam-5976	337	13	q((a	q((a	NOUN
ejpam-5976	337	14	,	,	PUNCT
ejpam-5976	337	15	b	b	NOUN
ejpam-5976	337	16	)	)	PUNCT
ejpam-5976	337	17	)	)	PUNCT
ejpam-5976	337	18	⊊	⊊	VERB
ejpam-5976	337	19	q((u	q((u	NOUN
ejpam-5976	337	20	,	,	PUNCT
ejpam-5976	337	21	v	v	NOUN
ejpam-5976	337	22	)	)	PUNCT
ejpam-5976	337	23	)	)	PUNCT
ejpam-5976	337	24	.	.	PUNCT
ejpam-5976	338	1	it	it	PRON
ejpam-5976	338	2	follows	follow	VERB
ejpam-5976	338	3	that	that	SCONJ
ejpam-5976	338	4	(	(	PUNCT
ejpam-5976	338	5	a	a	DET
ejpam-5976	338	6	,	,	PUNCT
ejpam-5976	338	7	b	b	NOUN
ejpam-5976	338	8	)	)	PUNCT
ejpam-5976	338	9	̸=	̸=	PROPN
ejpam-5976	338	10	(	(	PUNCT
ejpam-5976	338	11	u	u	NOUN
ejpam-5976	338	12	,	,	PUNCT
ejpam-5976	338	13	v	v	NOUN
ejpam-5976	338	14	)	)	PUNCT
ejpam-5976	338	15	and	and	CCONJ
ejpam-5976	338	16	(	(	PUNCT
ejpam-5976	338	17	a	a	PRON
ejpam-5976	338	18	,	,	PUNCT
ejpam-5976	338	19	b	b	NOUN
ejpam-5976	338	20	)	)	PUNCT
ejpam-5976	338	21	∈	∈	PROPN
ejpam-5976	338	22	(	(	PUNCT
ejpam-5976	338	23	us1∩s1u)×(vs2∩s2v	us1∩s1u)×(vs2∩s2v	NOUN
ejpam-5976	338	24	)	)	PUNCT
ejpam-5976	338	25	.	.	PUNCT
ejpam-5976	339	1	we	we	PRON
ejpam-5976	339	2	obtain	obtain	VERB
ejpam-5976	339	3	q(a	q(a	NOUN
ejpam-5976	339	4	)	)	PUNCT
ejpam-5976	339	5	⊆	⊆	NUM
ejpam-5976	339	6	q(u	q(u	NOUN
ejpam-5976	339	7	)	)	PUNCT
ejpam-5976	339	8	and	and	CCONJ
ejpam-5976	339	9	q(b	q(b	ADJ
ejpam-5976	339	10	)	)	PUNCT
ejpam-5976	339	11	⊆	⊆	NUM
ejpam-5976	339	12	q(v	q(v	NOUN
ejpam-5976	339	13	)	)	PUNCT
ejpam-5976	339	14	.	.	PUNCT
ejpam-5976	340	1	if	if	SCONJ
ejpam-5976	340	2	q(a	q(a	NOUN
ejpam-5976	340	3	)	)	PUNCT
ejpam-5976	340	4	=	=	SYM
ejpam-5976	341	1	q(u	q(u	NOUN
ejpam-5976	341	2	)	)	PUNCT
ejpam-5976	341	3	and	and	CCONJ
ejpam-5976	341	4	q(b	q(b	ADJ
ejpam-5976	341	5	)	)	PUNCT
ejpam-5976	341	6	=	=	SYM
ejpam-5976	341	7	q(v	q(v	NOUN
ejpam-5976	341	8	)	)	PUNCT
ejpam-5976	341	9	,	,	PUNCT
ejpam-5976	341	10	then	then	ADV
ejpam-5976	341	11	we	we	PRON
ejpam-5976	341	12	have	have	VERB
ejpam-5976	341	13	(	(	PUNCT
ejpam-5976	341	14	a	a	PRON
ejpam-5976	341	15	,	,	PUNCT
ejpam-5976	341	16	b	b	NOUN
ejpam-5976	341	17	)	)	PUNCT
ejpam-5976	341	18	∈	∈	PROPN
ejpam-5976	341	19	(	(	PUNCT
ejpam-5976	341	20	us1	us1	PROPN
ejpam-5976	341	21	∩	∩	PROPN
ejpam-5976	341	22	s1u)×	s1u)×	PROPN
ejpam-5976	341	23	(	(	PUNCT
ejpam-5976	341	24	vs2	vs2	NOUN
ejpam-5976	341	25	∩	∩	NOUN
ejpam-5976	341	26	s2v	s2v	NOUN
ejpam-5976	341	27	)	)	PUNCT
ejpam-5976	341	28	=	=	PUNCT
ejpam-5976	342	1	(	(	PUNCT
ejpam-5976	342	2	q(u)s1	q(u)s1	PROPN
ejpam-5976	342	3	∩	∩	ADJ
ejpam-5976	342	4	s1q(u))×	s1q(u))×	PROPN
ejpam-5976	342	5	(	(	PUNCT
ejpam-5976	342	6	q(v)s2	q(v)s2	PROPN
ejpam-5976	342	7	∩	∩	PROPN
ejpam-5976	342	8	s2q(v	s2q(v	PROPN
ejpam-5976	342	9	)	)	PUNCT
ejpam-5976	342	10	)	)	PUNCT
ejpam-5976	343	1	=	=	PRON
ejpam-5976	343	2	(	(	PUNCT
ejpam-5976	343	3	q(a)s1	q(a)s1	PROPN
ejpam-5976	343	4	∩	∩	PROPN
ejpam-5976	343	5	s1q(a))×	s1q(a))×	PROPN
ejpam-5976	343	6	(	(	PUNCT
ejpam-5976	343	7	q(b)s2	q(b)s2	PROPN
ejpam-5976	343	8	∩	∩	PROPN
ejpam-5976	343	9	s2q(b	s2q(b	NOUN
ejpam-5976	343	10	)	)	PUNCT
ejpam-5976	343	11	)	)	PUNCT
ejpam-5976	344	1	=	=	SYM
ejpam-5976	345	1	(	(	PUNCT
ejpam-5976	345	2	as1	as1	NOUN
ejpam-5976	345	3	∩	∩	X
ejpam-5976	345	4	s1a)×	s1a)×	X
ejpam-5976	345	5	(	(	PUNCT
ejpam-5976	345	6	bs2	bs2	PROPN
ejpam-5976	345	7	∩	∩	PROPN
ejpam-5976	345	8	s2b	s2b	PROPN
ejpam-5976	345	9	)	)	PUNCT
ejpam-5976	345	10	.	.	PUNCT
ejpam-5976	346	1	this	this	PRON
ejpam-5976	346	2	contradicts	contradict	VERB
ejpam-5976	346	3	to	to	ADP
ejpam-5976	346	4	(	(	PUNCT
ejpam-5976	346	5	a	a	DET
ejpam-5976	346	6	,	,	PUNCT
ejpam-5976	346	7	b	b	NOUN
ejpam-5976	346	8	)	)	PUNCT
ejpam-5976	346	9	/∈	/∈	PUNCT
ejpam-5976	347	1	(	(	PUNCT
ejpam-5976	347	2	as1	as1	NOUN
ejpam-5976	347	3	∩	∩	X
ejpam-5976	347	4	s1a)×	s1a)×	X
ejpam-5976	347	5	(	(	PUNCT
ejpam-5976	347	6	bs2	bs2	PROPN
ejpam-5976	347	7	∩	∩	PROPN
ejpam-5976	347	8	s2b	s2b	PROPN
ejpam-5976	347	9	)	)	PUNCT
ejpam-5976	347	10	.	.	PUNCT
ejpam-5976	348	1	we	we	PRON
ejpam-5976	348	2	therefore	therefore	ADV
ejpam-5976	348	3	conclude	conclude	VERB
ejpam-5976	348	4	that	that	SCONJ
ejpam-5976	348	5	q(a	q(a	NOUN
ejpam-5976	348	6	)	)	PUNCT
ejpam-5976	348	7	⊊	⊊	VERB
ejpam-5976	348	8	q(u	q(u	NOUN
ejpam-5976	348	9	)	)	PUNCT
ejpam-5976	348	10	or	or	CCONJ
ejpam-5976	348	11	q(b	q(b	ADJ
ejpam-5976	348	12	)	)	PUNCT
ejpam-5976	348	13	⊊	⊊	VERB
ejpam-5976	348	14	q(v	q(v	NOUN
ejpam-5976	348	15	)	)	PUNCT
ejpam-5976	348	16	.	.	PUNCT
ejpam-5976	349	1	thus	thus	ADV
ejpam-5976	349	2	,	,	PUNCT
ejpam-5976	349	3	ha	ha	INTJ
ejpam-5976	349	4	or	or	CCONJ
ejpam-5976	349	5	hb	hb	PROPN
ejpam-5976	349	6	is	be	AUX
ejpam-5976	349	7	not	not	PART
ejpam-5976	349	8	maximal	maximal	ADJ
ejpam-5976	349	9	.	.	PUNCT
ejpam-5976	350	1	lemma	lemma	PROPN
ejpam-5976	350	2	3	3	X
ejpam-5976	350	3	.	.	PUNCT
ejpam-5976	351	1	let	let	VERB
ejpam-5976	351	2	h(a	h(a	PROPN
ejpam-5976	351	3	,	,	PUNCT
ejpam-5976	351	4	b	b	NOUN
ejpam-5976	351	5	)	)	PUNCT
ejpam-5976	351	6	and	and	CCONJ
ejpam-5976	351	7	h(u	h(u	PROPN
ejpam-5976	351	8	,	,	PUNCT
ejpam-5976	351	9	v	v	NOUN
ejpam-5976	351	10	)	)	PUNCT
ejpam-5976	351	11	be	be	AUX
ejpam-5976	351	12	difference	difference	NOUN
ejpam-5976	351	13	h	h	NOUN
ejpam-5976	351	14	-	-	PUNCT
ejpam-5976	351	15	classes	class	NOUN
ejpam-5976	351	16	contained	contain	VERB
ejpam-5976	351	17	in	in	ADP
ejpam-5976	351	18	ha	ha	INTJ
ejpam-5976	351	19	×hb	×hb	PROPN
ejpam-5976	351	20	.	.	PUNCT
ejpam-5976	352	1	if	if	SCONJ
ejpam-5976	352	2	(	(	PUNCT
ejpam-5976	352	3	a	a	DET
ejpam-5976	352	4	,	,	PUNCT
ejpam-5976	352	5	b	b	NOUN
ejpam-5976	352	6	)	)	PUNCT
ejpam-5976	352	7	/∈	/∈	PUNCT
ejpam-5976	353	1	(	(	PUNCT
ejpam-5976	353	2	as1	as1	NOUN
ejpam-5976	353	3	∩	∩	X
ejpam-5976	353	4	s1a)×	s1a)×	X
ejpam-5976	353	5	(	(	PUNCT
ejpam-5976	353	6	bs2	bs2	PROPN
ejpam-5976	353	7	∩	∩	PROPN
ejpam-5976	353	8	s2b	s2b	PROPN
ejpam-5976	353	9	)	)	PUNCT
ejpam-5976	353	10	,	,	PUNCT
ejpam-5976	353	11	then	then	ADV
ejpam-5976	353	12	(	(	PUNCT
ejpam-5976	353	13	u	u	NOUN
ejpam-5976	353	14	,	,	PUNCT
ejpam-5976	353	15	v	v	NOUN
ejpam-5976	353	16	)	)	PUNCT
ejpam-5976	353	17	/∈	/∈	PUNCT
ejpam-5976	354	1	(	(	PUNCT
ejpam-5976	354	2	us1	us1	PROPN
ejpam-5976	354	3	∩	∩	PROPN
ejpam-5976	354	4	s1u)×	s1u)×	PROPN
ejpam-5976	354	5	(	(	PUNCT
ejpam-5976	354	6	vs2	vs2	NOUN
ejpam-5976	354	7	∩	∩	NOUN
ejpam-5976	354	8	s2v	s2v	NOUN
ejpam-5976	354	9	)	)	PUNCT
ejpam-5976	354	10	.	.	PUNCT
ejpam-5976	355	1	proof	proof	NOUN
ejpam-5976	355	2	.	.	PUNCT
ejpam-5976	356	1	assume	assume	VERB
ejpam-5976	356	2	that	that	SCONJ
ejpam-5976	356	3	(	(	PUNCT
ejpam-5976	356	4	u	u	NOUN
ejpam-5976	356	5	,	,	PUNCT
ejpam-5976	356	6	v	v	NOUN
ejpam-5976	356	7	)	)	PUNCT
ejpam-5976	356	8	∈	∈	PROPN
ejpam-5976	356	9	(	(	PUNCT
ejpam-5976	356	10	us1	us1	PROPN
ejpam-5976	356	11	∩	∩	PROPN
ejpam-5976	356	12	s1u)×	s1u)×	PROPN
ejpam-5976	356	13	(	(	PUNCT
ejpam-5976	356	14	vs2	vs2	NOUN
ejpam-5976	356	15	∩	∩	NOUN
ejpam-5976	356	16	s2v	s2v	NOUN
ejpam-5976	356	17	)	)	PUNCT
ejpam-5976	356	18	.	.	PUNCT
ejpam-5976	357	1	since	since	SCONJ
ejpam-5976	357	2	h(u	h(u	PROPN
ejpam-5976	357	3	,	,	PUNCT
ejpam-5976	357	4	v	v	NOUN
ejpam-5976	357	5	)	)	PUNCT
ejpam-5976	357	6	⊆	⊆	NUM
ejpam-5976	357	7	ha	ha	INTJ
ejpam-5976	357	8	×	×	PROPN
ejpam-5976	357	9	hb	hb	PROPN
ejpam-5976	357	10	,	,	PUNCT
ejpam-5976	357	11	we	we	PRON
ejpam-5976	357	12	have	have	VERB
ejpam-5976	357	13	hu	hu	PROPN
ejpam-5976	357	14	×	×	PROPN
ejpam-5976	357	15	hv	hv	PROPN
ejpam-5976	358	1	=	=	NOUN
ejpam-5976	358	2	ha	ha	INTJ
ejpam-5976	358	3	×	×	PROPN
ejpam-5976	358	4	hb	hb	PROPN
ejpam-5976	358	5	.	.	PUNCT
ejpam-5976	359	1	this	this	PRON
ejpam-5976	359	2	implies	imply	VERB
ejpam-5976	359	3	q(u	q(u	NOUN
ejpam-5976	359	4	)	)	PUNCT
ejpam-5976	359	5	=	=	SYM
ejpam-5976	359	6	q(a	q(a	PROPN
ejpam-5976	359	7	)	)	PUNCT
ejpam-5976	359	8	and	and	CCONJ
ejpam-5976	359	9	q(v	q(v	NOUN
ejpam-5976	359	10	)	)	PUNCT
ejpam-5976	359	11	=	=	PUNCT
ejpam-5976	359	12	q(b	q(b	ADJ
ejpam-5976	359	13	)	)	PUNCT
ejpam-5976	359	14	.	.	PUNCT
ejpam-5976	360	1	thus	thus	ADV
ejpam-5976	360	2	,	,	PUNCT
ejpam-5976	360	3	(	(	PUNCT
ejpam-5976	360	4	a	a	PRON
ejpam-5976	360	5	,	,	PUNCT
ejpam-5976	360	6	b	b	NOUN
ejpam-5976	360	7	)	)	PUNCT
ejpam-5976	360	8	∈	∈	PROPN
ejpam-5976	360	9	q(a)×q(b	q(a)×q(b	PROPN
ejpam-5976	360	10	)	)	PUNCT
ejpam-5976	360	11	=	=	SYM
ejpam-5976	360	12	q(u)×q(v	q(u)×q(v	NOUN
ejpam-5976	360	13	)	)	PUNCT
ejpam-5976	360	14	=	=	PUNCT
ejpam-5976	360	15	(	(	PUNCT
ejpam-5976	360	16	us1	us1	PROPN
ejpam-5976	360	17	∩	∩	PROPN
ejpam-5976	360	18	s1u)×	s1u)×	PROPN
ejpam-5976	360	19	(	(	PUNCT
ejpam-5976	360	20	vs2	vs2	NOUN
ejpam-5976	360	21	∩	∩	NOUN
ejpam-5976	360	22	s2v	s2v	NOUN
ejpam-5976	360	23	)	)	PUNCT
ejpam-5976	360	24	=	=	PUNCT
ejpam-5976	360	25	(	(	PUNCT
ejpam-5976	360	26	q(u)s1	q(u)s1	PROPN
ejpam-5976	360	27	∩	∩	ADJ
ejpam-5976	360	28	s1q(u))×	s1q(u))×	PROPN
ejpam-5976	360	29	(	(	PUNCT
ejpam-5976	360	30	q(v)s2	q(v)s2	PROPN
ejpam-5976	360	31	∩	∩	PROPN
ejpam-5976	360	32	s2q(v	s2q(v	PROPN
ejpam-5976	360	33	)	)	PUNCT
ejpam-5976	360	34	)	)	PUNCT
ejpam-5976	361	1	=	=	PRON
ejpam-5976	361	2	(	(	PUNCT
ejpam-5976	361	3	q(a)s1	q(a)s1	PROPN
ejpam-5976	361	4	∩	∩	PROPN
ejpam-5976	361	5	s1q(a))×	s1q(a))×	PROPN
ejpam-5976	361	6	(	(	PUNCT
ejpam-5976	361	7	q(b)s2	q(b)s2	PROPN
ejpam-5976	361	8	∩	∩	PROPN
ejpam-5976	361	9	s2q(b	s2q(b	NOUN
ejpam-5976	361	10	)	)	PUNCT
ejpam-5976	361	11	)	)	PUNCT
ejpam-5976	362	1	=	=	SYM
ejpam-5976	363	1	(	(	PUNCT
ejpam-5976	363	2	as1	as1	NOUN
ejpam-5976	363	3	∩	∩	X
ejpam-5976	363	4	s1a)×	s1a)×	X
ejpam-5976	363	5	(	(	PUNCT
ejpam-5976	363	6	bs2	bs2	PROPN
ejpam-5976	363	7	∩	∩	PROPN
ejpam-5976	363	8	s2b	s2b	PROPN
ejpam-5976	363	9	)	)	PUNCT
ejpam-5976	363	10	.	.	PUNCT
ejpam-5976	364	1	lemma	lemma	PROPN
ejpam-5976	364	2	4	4	X
ejpam-5976	364	3	.	.	PUNCT
ejpam-5976	365	1	let	let	VERB
ejpam-5976	365	2	ha	ha	INTJ
ejpam-5976	365	3	and	and	CCONJ
ejpam-5976	365	4	hb	hb	PROPN
ejpam-5976	365	5	be	be	AUX
ejpam-5976	365	6	maximal	maximal	ADJ
ejpam-5976	365	7	h	h	NOUN
ejpam-5976	365	8	-	-	PUNCT
ejpam-5976	365	9	classes	class	NOUN
ejpam-5976	365	10	of	of	ADP
ejpam-5976	365	11	s1	s1	PROPN
ejpam-5976	365	12	and	and	CCONJ
ejpam-5976	365	13	s2	s2	PROPN
ejpam-5976	365	14	,	,	PUNCT
ejpam-5976	365	15	respectively	respectively	ADV
ejpam-5976	365	16	.	.	PUNCT
ejpam-5976	366	1	(	(	PUNCT
ejpam-5976	366	2	1	1	X
ejpam-5976	366	3	)	)	PUNCT
ejpam-5976	366	4	if	if	SCONJ
ejpam-5976	366	5	ha	ha	X
ejpam-5976	366	6	=	=	X
ejpam-5976	366	7	{	{	PUNCT
ejpam-5976	366	8	a	a	NOUN
ejpam-5976	366	9	}	}	PUNCT
ejpam-5976	366	10	and	and	CCONJ
ejpam-5976	366	11	hb	hb	X
ejpam-5976	366	12	=	=	PUNCT
ejpam-5976	366	13	{	{	PUNCT
ejpam-5976	366	14	b	b	NOUN
ejpam-5976	366	15	}	}	PUNCT
ejpam-5976	366	16	,	,	PUNCT
ejpam-5976	366	17	then	then	ADV
ejpam-5976	366	18	ha	ha	INTJ
ejpam-5976	366	19	×hb	×hb	PROPN
ejpam-5976	366	20	=	=	SYM
ejpam-5976	366	21	h(a	h(a	PROPN
ejpam-5976	366	22	,	,	PUNCT
ejpam-5976	366	23	b	b	NOUN
ejpam-5976	366	24	)	)	PUNCT
ejpam-5976	366	25	is	be	AUX
ejpam-5976	366	26	maximal	maximal	ADJ
ejpam-5976	366	27	.	.	PUNCT
ejpam-5976	367	1	(	(	PUNCT
ejpam-5976	367	2	2	2	X
ejpam-5976	367	3	)	)	PUNCT
ejpam-5976	367	4	if	if	SCONJ
ejpam-5976	367	5	|	|	ADV
ejpam-5976	367	6	ha	ha	INTJ
ejpam-5976	367	7	|	|	ADV
ejpam-5976	367	8	>	>	X
ejpam-5976	367	9	1	1	NUM
ejpam-5976	368	1	and	and	CCONJ
ejpam-5976	368	2	|	|	ADV
ejpam-5976	368	3	hb	hb	X
ejpam-5976	368	4	|	|	ADV
ejpam-5976	368	5	>	>	X
ejpam-5976	368	6	1	1	NUM
ejpam-5976	368	7	,	,	PUNCT
ejpam-5976	368	8	then	then	ADV
ejpam-5976	368	9	ha	ha	INTJ
ejpam-5976	368	10	×hb	×hb	PROPN
ejpam-5976	368	11	=	=	SYM
ejpam-5976	368	12	h(a	h(a	PROPN
ejpam-5976	368	13	,	,	PUNCT
ejpam-5976	368	14	b	b	NOUN
ejpam-5976	368	15	)	)	PUNCT
ejpam-5976	368	16	is	be	AUX
ejpam-5976	368	17	maximal	maximal	ADJ
ejpam-5976	368	18	.	.	PUNCT
ejpam-5976	369	1	proof	proof	NOUN
ejpam-5976	369	2	.	.	PUNCT
ejpam-5976	370	1	(	(	PUNCT
ejpam-5976	370	2	1	1	X
ejpam-5976	370	3	)	)	PUNCT
ejpam-5976	370	4	assume	assume	VERB
ejpam-5976	370	5	that	that	SCONJ
ejpam-5976	371	1	ha	ha	INTJ
ejpam-5976	371	2	=	=	X
ejpam-5976	371	3	{	{	PUNCT
ejpam-5976	371	4	a	a	NOUN
ejpam-5976	371	5	}	}	PUNCT
ejpam-5976	371	6	and	and	CCONJ
ejpam-5976	371	7	hb	hb	X
ejpam-5976	371	8	=	=	PUNCT
ejpam-5976	371	9	{	{	PUNCT
ejpam-5976	371	10	b	b	NOUN
ejpam-5976	371	11	}	}	PUNCT
ejpam-5976	371	12	.	.	PUNCT
ejpam-5976	372	1	let	let	VERB
ejpam-5976	372	2	(	(	PUNCT
ejpam-5976	372	3	u	u	NOUN
ejpam-5976	372	4	,	,	PUNCT
ejpam-5976	372	5	v	v	NOUN
ejpam-5976	372	6	)	)	PUNCT
ejpam-5976	372	7	∈	∈	PROPN
ejpam-5976	372	8	s1	s1	NOUN
ejpam-5976	372	9	×	×	NOUN
ejpam-5976	372	10	s2	s2	NOUN
ejpam-5976	372	11	such	such	ADJ
ejpam-5976	372	12	that	that	DET
ejpam-5976	372	13	q((a	q((a	NOUN
ejpam-5976	372	14	,	,	PUNCT
ejpam-5976	372	15	b	b	NOUN
ejpam-5976	372	16	)	)	PUNCT
ejpam-5976	372	17	)	)	PUNCT
ejpam-5976	373	1	⊆	⊆	NUM
ejpam-5976	373	2	q((u	q((u	NOUN
ejpam-5976	373	3	,	,	PUNCT
ejpam-5976	373	4	v	v	NOUN
ejpam-5976	373	5	)	)	PUNCT
ejpam-5976	373	6	)	)	PUNCT
ejpam-5976	373	7	.	.	PUNCT
ejpam-5976	374	1	if	if	SCONJ
ejpam-5976	374	2	(	(	PUNCT
ejpam-5976	374	3	u	u	NOUN
ejpam-5976	374	4	,	,	PUNCT
ejpam-5976	374	5	v	v	NOUN
ejpam-5976	374	6	)	)	PUNCT
ejpam-5976	374	7	/∈	/∈	PUNCT
ejpam-5976	375	1	q((a	q((a	NOUN
ejpam-5976	375	2	,	,	PUNCT
ejpam-5976	375	3	b	b	NOUN
ejpam-5976	375	4	)	)	PUNCT
ejpam-5976	375	5	)	)	PUNCT
ejpam-5976	375	6	,	,	PUNCT
ejpam-5976	375	7	then	then	ADV
ejpam-5976	375	8	we	we	PRON
ejpam-5976	375	9	have	have	VERB
ejpam-5976	375	10	(	(	PUNCT
ejpam-5976	375	11	u	u	NOUN
ejpam-5976	375	12	,	,	PUNCT
ejpam-5976	375	13	v	v	NOUN
ejpam-5976	375	14	)	)	PUNCT
ejpam-5976	375	15	̸=	̸=	PROPN
ejpam-5976	375	16	(	(	PUNCT
ejpam-5976	375	17	a	a	DET
ejpam-5976	375	18	,	,	PUNCT
ejpam-5976	375	19	b	b	NOUN
ejpam-5976	375	20	)	)	PUNCT
ejpam-5976	375	21	,	,	PUNCT
ejpam-5976	375	22	that	that	ADV
ejpam-5976	375	23	is	is	ADV
ejpam-5976	375	24	,	,	PUNCT
ejpam-5976	375	25	u	u	PROPN
ejpam-5976	375	26	̸=	̸=	PROPN
ejpam-5976	375	27	a	a	PRON
ejpam-5976	375	28	or	or	CCONJ
ejpam-5976	375	29	v	v	ADP
ejpam-5976	375	30	̸=	̸=	PROPN
ejpam-5976	375	31	b.	b.	PROPN
ejpam-5976	375	32	assume	assume	VERB
ejpam-5976	375	33	that	that	SCONJ
ejpam-5976	375	34	u	u	PRON
ejpam-5976	375	35	̸=	̸=	PROPN
ejpam-5976	375	36	a.	a.	NOUN
ejpam-5976	375	37	then	then	ADV
ejpam-5976	375	38	we	we	PRON
ejpam-5976	375	39	have	have	VERB
ejpam-5976	375	40	two	two	NUM
ejpam-5976	375	41	cases	case	NOUN
ejpam-5976	375	42	to	to	PART
ejpam-5976	375	43	consider	consider	VERB
ejpam-5976	375	44	:	:	PUNCT
ejpam-5976	375	45	p.	p.	PROPN
ejpam-5976	375	46	luangchaisri	luangchaisri	PROPN
ejpam-5976	375	47	,	,	PUNCT
ejpam-5976	375	48	o.	o.	PROPN
ejpam-5976	375	49	pankoon	pankoon	NOUN
ejpam-5976	375	50	,	,	PUNCT
ejpam-5976	375	51	t.	t.	PROPN
ejpam-5976	375	52	changphas	changphas	PROPN
ejpam-5976	375	53	/	/	SYM
ejpam-5976	375	54	eur	eur	PROPN
ejpam-5976	375	55	.	.	PUNCT
ejpam-5976	376	1	j.	j.	PROPN
ejpam-5976	376	2	pure	pure	PROPN
ejpam-5976	376	3	appl	appl	PROPN
ejpam-5976	376	4	.	.	PROPN
ejpam-5976	376	5	math	math	PROPN
ejpam-5976	376	6	,	,	PUNCT
ejpam-5976	376	7	18	18	NUM
ejpam-5976	376	8	(	(	PUNCT
ejpam-5976	376	9	2	2	NUM
ejpam-5976	376	10	)	)	PUNCT
ejpam-5976	376	11	(	(	PUNCT
ejpam-5976	376	12	2025	2025	NUM
ejpam-5976	376	13	)	)	PUNCT
ejpam-5976	376	14	,	,	PUNCT
ejpam-5976	376	15	5976	5976	NUM
ejpam-5976	376	16	12	12	NUM
ejpam-5976	376	17	of	of	ADP
ejpam-5976	376	18	14	14	NUM
ejpam-5976	376	19	(	(	PUNCT
ejpam-5976	376	20	i	i	NOUN
ejpam-5976	376	21	)	)	PUNCT
ejpam-5976	376	22	if	if	SCONJ
ejpam-5976	376	23	u	u	PROPN
ejpam-5976	376	24	/∈	/∈	PROPN
ejpam-5976	376	25	as1	as1	PROPN
ejpam-5976	376	26	∩	∩	PROPN
ejpam-5976	376	27	s1a	s1a	PROPN
ejpam-5976	376	28	,	,	PUNCT
ejpam-5976	376	29	then	then	ADV
ejpam-5976	376	30	u	u	X
ejpam-5976	376	31	/∈	/∈	PUNCT
ejpam-5976	376	32	q(a	q(a	PROPN
ejpam-5976	376	33	)	)	PUNCT
ejpam-5976	376	34	.	.	PUNCT
ejpam-5976	377	1	this	this	PRON
ejpam-5976	377	2	implies	imply	VERB
ejpam-5976	377	3	that	that	SCONJ
ejpam-5976	377	4	q(a	q(a	NOUN
ejpam-5976	377	5	)	)	PUNCT
ejpam-5976	377	6	⊊	⊊	VERB
ejpam-5976	377	7	q(u	q(u	NOUN
ejpam-5976	377	8	)	)	PUNCT
ejpam-5976	377	9	which	which	PRON
ejpam-5976	377	10	contradicts	contradict	VERB
ejpam-5976	377	11	to	to	AUX
ejpam-5976	377	12	ha	ha	INTJ
ejpam-5976	377	13	is	be	AUX
ejpam-5976	377	14	maximal	maximal	ADJ
ejpam-5976	377	15	.	.	PUNCT
ejpam-5976	378	1	(	(	PUNCT
ejpam-5976	378	2	ii	ii	NOUN
ejpam-5976	378	3	)	)	PUNCT
ejpam-5976	378	4	if	if	SCONJ
ejpam-5976	378	5	u	u	PROPN
ejpam-5976	378	6	∈	∈	PROPN
ejpam-5976	378	7	as1	as1	NOUN
ejpam-5976	378	8	∩	∩	PROPN
ejpam-5976	378	9	s1a	s1a	PROPN
ejpam-5976	378	10	,	,	PUNCT
ejpam-5976	378	11	then	then	ADV
ejpam-5976	378	12	q(a	q(a	NOUN
ejpam-5976	378	13	)	)	PUNCT
ejpam-5976	378	14	=	=	SYM
ejpam-5976	378	15	q(u	q(u	NOUN
ejpam-5976	378	16	)	)	PUNCT
ejpam-5976	378	17	.	.	PUNCT
ejpam-5976	379	1	thus	thus	ADV
ejpam-5976	379	2	,	,	PUNCT
ejpam-5976	379	3	u	u	PROPN
ejpam-5976	379	4	∈	∈	PROPN
ejpam-5976	379	5	ha	ha	INTJ
ejpam-5976	379	6	which	which	PRON
ejpam-5976	379	7	contradicts	contradict	VERB
ejpam-5976	379	8	to	to	ADP
ejpam-5976	379	9	ha	ha	INTJ
ejpam-5976	379	10	=	=	X
ejpam-5976	379	11	{	{	PUNCT
ejpam-5976	379	12	a	a	NOUN
ejpam-5976	379	13	}	}	PUNCT
ejpam-5976	379	14	.	.	PUNCT
ejpam-5976	380	1	the	the	DET
ejpam-5976	380	2	case	case	NOUN
ejpam-5976	380	3	v	v	ADP
ejpam-5976	380	4	̸=	̸=	PROPN
ejpam-5976	380	5	b	b	NOUN
ejpam-5976	380	6	can	can	AUX
ejpam-5976	380	7	be	be	AUX
ejpam-5976	380	8	proved	prove	VERB
ejpam-5976	380	9	similarly	similarly	ADV
ejpam-5976	380	10	.	.	PUNCT
ejpam-5976	381	1	therefore	therefore	ADV
ejpam-5976	381	2	,	,	PUNCT
ejpam-5976	381	3	(	(	PUNCT
ejpam-5976	381	4	u	u	NOUN
ejpam-5976	381	5	,	,	PUNCT
ejpam-5976	381	6	v	v	NOUN
ejpam-5976	381	7	)	)	PUNCT
ejpam-5976	381	8	∈	∈	PROPN
ejpam-5976	381	9	q((a	q((a	PROPN
ejpam-5976	381	10	,	,	PUNCT
ejpam-5976	381	11	b	b	NOUN
ejpam-5976	381	12	)	)	PUNCT
ejpam-5976	381	13	)	)	PUNCT
ejpam-5976	381	14	.	.	PUNCT
ejpam-5976	382	1	thus	thus	ADV
ejpam-5976	382	2	,	,	PUNCT
ejpam-5976	382	3	q((u	q((u	NOUN
ejpam-5976	382	4	,	,	PUNCT
ejpam-5976	382	5	v	v	NOUN
ejpam-5976	382	6	)	)	PUNCT
ejpam-5976	382	7	)	)	PUNCT
ejpam-5976	383	1	=	=	SYM
ejpam-5976	383	2	q((a	q((a	PROPN
ejpam-5976	383	3	,	,	PUNCT
ejpam-5976	383	4	b	b	NOUN
ejpam-5976	383	5	)	)	PUNCT
ejpam-5976	383	6	)	)	PUNCT
ejpam-5976	383	7	.	.	PUNCT
ejpam-5976	384	1	this	this	PRON
ejpam-5976	384	2	means	mean	VERB
ejpam-5976	384	3	that	that	SCONJ
ejpam-5976	384	4	h(a	h(a	PROPN
ejpam-5976	384	5	,	,	PUNCT
ejpam-5976	384	6	b	b	X
ejpam-5976	384	7	)	)	PUNCT
ejpam-5976	384	8	is	be	AUX
ejpam-5976	384	9	maximal	maximal	ADJ
ejpam-5976	384	10	.	.	PUNCT
ejpam-5976	385	1	(	(	PUNCT
ejpam-5976	385	2	2	2	X
ejpam-5976	385	3	)	)	PUNCT
ejpam-5976	385	4	assume	assume	VERB
ejpam-5976	385	5	that	that	SCONJ
ejpam-5976	386	1	|	|	ADV
ejpam-5976	386	2	ha	ha	INTJ
ejpam-5976	387	1	|	|	ADV
ejpam-5976	387	2	>	>	X
ejpam-5976	387	3	1	1	NUM
ejpam-5976	388	1	and	and	CCONJ
ejpam-5976	388	2	|	|	ADV
ejpam-5976	388	3	hb	hb	X
ejpam-5976	388	4	|	|	ADV
ejpam-5976	388	5	>	>	X
ejpam-5976	388	6	1	1	NUM
ejpam-5976	388	7	.	.	PUNCT
ejpam-5976	389	1	then	then	ADV
ejpam-5976	389	2	a	a	DET
ejpam-5976	389	3	∈	∈	PROPN
ejpam-5976	389	4	as1	as1	NOUN
ejpam-5976	389	5	∩	∩	PROPN
ejpam-5976	389	6	s1a	s1a	PROPN
ejpam-5976	389	7	and	and	CCONJ
ejpam-5976	389	8	b	b	X
ejpam-5976	389	9	∈	∈	PROPN
ejpam-5976	389	10	bs2	bs2	PROPN
ejpam-5976	389	11	∩	∩	PROPN
ejpam-5976	389	12	s2b	s2b	PROPN
ejpam-5976	389	13	.	.	PUNCT
ejpam-5976	390	1	these	these	PRON
ejpam-5976	390	2	imply	imply	VERB
ejpam-5976	390	3	that	that	SCONJ
ejpam-5976	390	4	ha	ha	INTJ
ejpam-5976	390	5	×hb	×hb	PROPN
ejpam-5976	390	6	=	=	SYM
ejpam-5976	390	7	h(a	h(a	PROPN
ejpam-5976	390	8	,	,	PUNCT
ejpam-5976	390	9	b	b	NOUN
ejpam-5976	390	10	)	)	PUNCT
ejpam-5976	390	11	.	.	PUNCT
ejpam-5976	391	1	by	by	ADP
ejpam-5976	391	2	theorem	theorem	NOUN
ejpam-5976	391	3	11	11	NUM
ejpam-5976	391	4	,	,	PUNCT
ejpam-5976	391	5	the	the	DET
ejpam-5976	391	6	direct	direct	ADJ
ejpam-5976	391	7	product	product	NOUN
ejpam-5976	391	8	ha	ha	INTJ
ejpam-5976	391	9	×hb	×hb	PROPN
ejpam-5976	391	10	is	be	AUX
ejpam-5976	391	11	maximal	maximal	ADJ
ejpam-5976	391	12	.	.	PUNCT
ejpam-5976	392	1	theorem	theorem	NOUN
ejpam-5976	392	2	12	12	NUM
ejpam-5976	392	3	.	.	PUNCT
ejpam-5976	393	1	let	let	VERB
ejpam-5976	393	2	(	(	PUNCT
ejpam-5976	393	3	a	a	PRON
ejpam-5976	393	4	,	,	PUNCT
ejpam-5976	393	5	b	b	NOUN
ejpam-5976	393	6	)	)	PUNCT
ejpam-5976	393	7	∈	∈	PROPN
ejpam-5976	393	8	s1	s1	PROPN
ejpam-5976	393	9	×	×	PROPN
ejpam-5976	393	10	s2	s2	PROPN
ejpam-5976	393	11	.	.	PUNCT
ejpam-5976	394	1	if	if	SCONJ
ejpam-5976	394	2	ha	ha	INTJ
ejpam-5976	394	3	and	and	CCONJ
ejpam-5976	394	4	hb	hb	NOUN
ejpam-5976	394	5	are	be	AUX
ejpam-5976	394	6	maximal	maximal	ADJ
ejpam-5976	394	7	,	,	PUNCT
ejpam-5976	394	8	then	then	ADV
ejpam-5976	394	9	one	one	NUM
ejpam-5976	394	10	of	of	ADP
ejpam-5976	394	11	the	the	DET
ejpam-5976	394	12	following	follow	VERB
ejpam-5976	394	13	conditions	condition	NOUN
ejpam-5976	394	14	holds	hold	VERB
ejpam-5976	394	15	:	:	PUNCT
ejpam-5976	394	16	(	(	PUNCT
ejpam-5976	394	17	1	1	X
ejpam-5976	394	18	)	)	PUNCT
ejpam-5976	395	1	ha	ha	INTJ
ejpam-5976	396	1	×hb	×hb	PROPN
ejpam-5976	396	2	is	be	AUX
ejpam-5976	396	3	maximal	maximal	ADJ
ejpam-5976	396	4	;	;	PUNCT
ejpam-5976	396	5	(	(	PUNCT
ejpam-5976	396	6	2	2	X
ejpam-5976	396	7	)	)	PUNCT
ejpam-5976	397	1	ha	ha	INTJ
ejpam-5976	397	2	×hb	×hb	PROPN
ejpam-5976	397	3	is	be	AUX
ejpam-5976	397	4	the	the	DET
ejpam-5976	397	5	union	union	NOUN
ejpam-5976	397	6	of	of	ADP
ejpam-5976	397	7	at	at	ADV
ejpam-5976	397	8	least	least	ADV
ejpam-5976	397	9	two	two	NUM
ejpam-5976	397	10	maximal	maximal	ADJ
ejpam-5976	397	11	h	h	NOUN
ejpam-5976	397	12	-	-	PUNCT
ejpam-5976	397	13	classes	class	NOUN
ejpam-5976	397	14	in	in	ADP
ejpam-5976	397	15	s1	s1	PROPN
ejpam-5976	397	16	×	×	PROPN
ejpam-5976	397	17	s2	s2	PROPN
ejpam-5976	397	18	.	.	PUNCT
ejpam-5976	398	1	proof	proof	NOUN
ejpam-5976	398	2	.	.	PUNCT
ejpam-5976	399	1	assume	assume	VERB
ejpam-5976	399	2	that	that	SCONJ
ejpam-5976	399	3	ha	ha	INTJ
ejpam-5976	399	4	and	and	CCONJ
ejpam-5976	399	5	hb	hb	PROPN
ejpam-5976	399	6	are	be	AUX
ejpam-5976	399	7	maximal	maximal	ADJ
ejpam-5976	399	8	.	.	PUNCT
ejpam-5976	400	1	by	by	ADP
ejpam-5976	400	2	lemma	lemma	PROPN
ejpam-5976	400	3	4	4	NUM
ejpam-5976	400	4	,	,	PUNCT
ejpam-5976	400	5	ha	ha	INTJ
ejpam-5976	400	6	×	×	NOUN
ejpam-5976	400	7	hb	hb	X
ejpam-5976	400	8	=	=	SYM
ejpam-5976	400	9	h(a	h(a	PROPN
ejpam-5976	400	10	,	,	PUNCT
ejpam-5976	400	11	b	b	NOUN
ejpam-5976	400	12	)	)	PUNCT
ejpam-5976	400	13	is	be	AUX
ejpam-5976	400	14	maximal	maximal	ADJ
ejpam-5976	400	15	when	when	SCONJ
ejpam-5976	400	16	one	one	NUM
ejpam-5976	400	17	of	of	ADP
ejpam-5976	400	18	the	the	DET
ejpam-5976	400	19	following	follow	VERB
ejpam-5976	400	20	conditions	condition	NOUN
ejpam-5976	400	21	holds	hold	VERB
ejpam-5976	400	22	:	:	PUNCT
ejpam-5976	400	23	(	(	PUNCT
ejpam-5976	400	24	i	i	NOUN
ejpam-5976	400	25	)	)	PUNCT
ejpam-5976	401	1	|	|	ADV
ejpam-5976	401	2	ha	ha	INTJ
ejpam-5976	401	3	|=	|=	NUM
ejpam-5976	401	4	1	1	NUM
ejpam-5976	401	5	and	and	CCONJ
ejpam-5976	401	6	|	|	ADV
ejpam-5976	401	7	hb	hb	X
ejpam-5976	401	8	|=	|=	PUNCT
ejpam-5976	401	9	1	1	NUM
ejpam-5976	401	10	or	or	CCONJ
ejpam-5976	401	11	(	(	PUNCT
ejpam-5976	401	12	ii	ii	NOUN
ejpam-5976	401	13	)	)	PUNCT
ejpam-5976	402	1	|	|	ADV
ejpam-5976	403	1	ha	ha	INTJ
ejpam-5976	403	2	|	|	ADV
ejpam-5976	403	3	>	>	X
ejpam-5976	403	4	1	1	NUM
ejpam-5976	404	1	and	and	CCONJ
ejpam-5976	404	2	|	|	ADV
ejpam-5976	404	3	hb	hb	X
ejpam-5976	404	4	|	|	ADV
ejpam-5976	404	5	>	>	X
ejpam-5976	404	6	1	1	X
ejpam-5976	404	7	.	.	PUNCT
ejpam-5976	404	8	assume	assume	VERB
ejpam-5976	404	9	that	that	SCONJ
ejpam-5976	405	1	|	|	ADV
ejpam-5976	405	2	ha	ha	INTJ
ejpam-5976	406	1	|	|	ADV
ejpam-5976	406	2	>	>	X
ejpam-5976	406	3	1	1	NUM
ejpam-5976	407	1	and	and	CCONJ
ejpam-5976	407	2	|	|	ADV
ejpam-5976	407	3	hb	hb	X
ejpam-5976	407	4	|=	|=	PUNCT
ejpam-5976	407	5	1	1	X
ejpam-5976	407	6	.	.	PUNCT
ejpam-5976	408	1	if	if	SCONJ
ejpam-5976	408	2	(	(	PUNCT
ejpam-5976	408	3	a	a	PRON
ejpam-5976	408	4	,	,	PUNCT
ejpam-5976	408	5	b	b	NOUN
ejpam-5976	408	6	)	)	PUNCT
ejpam-5976	408	7	∈	∈	PROPN
ejpam-5976	408	8	(	(	PUNCT
ejpam-5976	408	9	as1	as1	NOUN
ejpam-5976	408	10	∩	∩	NOUN
ejpam-5976	408	11	s1a)×	s1a)×	X
ejpam-5976	408	12	(	(	PUNCT
ejpam-5976	408	13	bs2	bs2	PROPN
ejpam-5976	408	14	∩	∩	PROPN
ejpam-5976	408	15	s2b	s2b	PROPN
ejpam-5976	408	16	)	)	PUNCT
ejpam-5976	408	17	,	,	PUNCT
ejpam-5976	408	18	then	then	ADV
ejpam-5976	408	19	we	we	PRON
ejpam-5976	408	20	obtain	obtain	VERB
ejpam-5976	408	21	that	that	PRON
ejpam-5976	408	22	ha	ha	INTJ
ejpam-5976	408	23	×hb	×hb	PROPN
ejpam-5976	408	24	=	=	SYM
ejpam-5976	408	25	h(a	h(a	PROPN
ejpam-5976	408	26	,	,	PUNCT
ejpam-5976	408	27	b	b	NOUN
ejpam-5976	408	28	)	)	PUNCT
ejpam-5976	408	29	by	by	ADP
ejpam-5976	408	30	theorem	theorem	NOUN
ejpam-5976	408	31	6	6	NUM
ejpam-5976	408	32	.	.	PUNCT
ejpam-5976	408	33	by	by	ADP
ejpam-5976	408	34	theorem	theorem	NOUN
ejpam-5976	408	35	11	11	NUM
ejpam-5976	408	36	,	,	PUNCT
ejpam-5976	408	37	ha	ha	INTJ
ejpam-5976	408	38	×hb	×hb	PROPN
ejpam-5976	408	39	is	be	AUX
ejpam-5976	408	40	maximal	maximal	ADJ
ejpam-5976	408	41	.	.	PUNCT
ejpam-5976	409	1	on	on	ADP
ejpam-5976	409	2	the	the	DET
ejpam-5976	409	3	other	other	ADJ
ejpam-5976	409	4	hand	hand	NOUN
ejpam-5976	409	5	,	,	PUNCT
ejpam-5976	409	6	assume	assume	VERB
ejpam-5976	409	7	that	that	SCONJ
ejpam-5976	409	8	(	(	PUNCT
ejpam-5976	409	9	a	a	DET
ejpam-5976	409	10	,	,	PUNCT
ejpam-5976	409	11	b	b	NOUN
ejpam-5976	409	12	)	)	PUNCT
ejpam-5976	409	13	/∈	/∈	PUNCT
ejpam-5976	410	1	(	(	PUNCT
ejpam-5976	410	2	as1	as1	NOUN
ejpam-5976	410	3	∩	∩	X
ejpam-5976	410	4	s1a)×	s1a)×	X
ejpam-5976	410	5	(	(	PUNCT
ejpam-5976	410	6	bs2	bs2	PROPN
ejpam-5976	410	7	∩	∩	PROPN
ejpam-5976	410	8	s2b	s2b	PROPN
ejpam-5976	410	9	)	)	PUNCT
ejpam-5976	410	10	.	.	PUNCT
ejpam-5976	411	1	then	then	ADV
ejpam-5976	411	2	h(a	h(a	PROPN
ejpam-5976	411	3	,	,	PUNCT
ejpam-5976	411	4	b	b	NOUN
ejpam-5976	411	5	)	)	PUNCT
ejpam-5976	411	6	⊊	⊊	VERB
ejpam-5976	411	7	ha	ha	INTJ
ejpam-5976	411	8	×hb	×hb	PROPN
ejpam-5976	411	9	.	.	PROPN
ejpam-5976	411	10	by	by	ADP
ejpam-5976	411	11	theorem	theorem	NOUN
ejpam-5976	411	12	5	5	NUM
ejpam-5976	411	13	,	,	PUNCT
ejpam-5976	411	14	the	the	DET
ejpam-5976	411	15	direct	direct	ADJ
ejpam-5976	411	16	product	product	NOUN
ejpam-5976	411	17	ha	ha	INTJ
ejpam-5976	411	18	×hb	×hb	PROPN
ejpam-5976	411	19	contains	contain	VERB
ejpam-5976	411	20	at	at	ADV
ejpam-5976	411	21	least	least	ADV
ejpam-5976	411	22	two	two	NUM
ejpam-5976	411	23	h	h	NOUN
ejpam-5976	411	24	-	-	PUNCT
ejpam-5976	411	25	classes	class	NOUN
ejpam-5976	411	26	in	in	ADP
ejpam-5976	411	27	s1	s1	PROPN
ejpam-5976	411	28	×	×	PROPN
ejpam-5976	411	29	s2	s2	PROPN
ejpam-5976	411	30	.	.	PUNCT
ejpam-5976	412	1	let	let	VERB
ejpam-5976	412	2	h(u	h(u	PROPN
ejpam-5976	412	3	,	,	PUNCT
ejpam-5976	412	4	v	v	NOUN
ejpam-5976	412	5	)	)	PUNCT
ejpam-5976	412	6	be	be	AUX
ejpam-5976	412	7	arbitrary	arbitrary	ADJ
ejpam-5976	412	8	h	h	NOUN
ejpam-5976	412	9	-	-	PUNCT
ejpam-5976	412	10	class	class	NOUN
ejpam-5976	412	11	of	of	ADP
ejpam-5976	412	12	s1	s1	PROPN
ejpam-5976	412	13	×	×	PROPN
ejpam-5976	412	14	s2	s2	NOUN
ejpam-5976	412	15	contained	contain	VERB
ejpam-5976	412	16	in	in	ADP
ejpam-5976	412	17	ha	ha	INTJ
ejpam-5976	412	18	×	×	PROPN
ejpam-5976	412	19	hb	hb	PROPN
ejpam-5976	412	20	.	.	PUNCT
ejpam-5976	413	1	by	by	ADP
ejpam-5976	413	2	lemma	lemma	PROPN
ejpam-5976	413	3	3	3	NUM
ejpam-5976	413	4	,	,	PUNCT
ejpam-5976	413	5	we	we	PRON
ejpam-5976	413	6	have	have	VERB
ejpam-5976	413	7	(	(	PUNCT
ejpam-5976	413	8	u	u	NOUN
ejpam-5976	413	9	,	,	PUNCT
ejpam-5976	413	10	v	v	NOUN
ejpam-5976	413	11	)	)	PUNCT
ejpam-5976	413	12	/∈	/∈	PUNCT
ejpam-5976	414	1	(	(	PUNCT
ejpam-5976	414	2	us1	us1	PROPN
ejpam-5976	414	3	∩	∩	PROPN
ejpam-5976	414	4	s1u)×	s1u)×	PROPN
ejpam-5976	414	5	(	(	PUNCT
ejpam-5976	414	6	vs2	vs2	NOUN
ejpam-5976	414	7	∩	∩	NOUN
ejpam-5976	414	8	s2v	s2v	NOUN
ejpam-5976	414	9	)	)	PUNCT
ejpam-5976	414	10	.	.	PUNCT
ejpam-5976	415	1	suppose	suppose	VERB
ejpam-5976	415	2	that	that	SCONJ
ejpam-5976	415	3	h(u	h(u	PROPN
ejpam-5976	415	4	,	,	PUNCT
ejpam-5976	415	5	v	v	NOUN
ejpam-5976	415	6	)	)	PUNCT
ejpam-5976	415	7	is	be	AUX
ejpam-5976	415	8	not	not	PART
ejpam-5976	415	9	maximal	maximal	ADJ
ejpam-5976	415	10	.	.	PUNCT
ejpam-5976	416	1	by	by	ADP
ejpam-5976	416	2	lemma	lemma	PROPN
ejpam-5976	416	3	2	2	NUM
ejpam-5976	416	4	,	,	PUNCT
ejpam-5976	416	5	ha	ha	INTJ
ejpam-5976	416	6	=	=	PUNCT
ejpam-5976	416	7	hu	hu	PROPN
ejpam-5976	416	8	is	be	AUX
ejpam-5976	416	9	not	not	PART
ejpam-5976	416	10	maximal	maximal	ADJ
ejpam-5976	416	11	or	or	CCONJ
ejpam-5976	416	12	hb	hb	X
ejpam-5976	416	13	=	=	NOUN
ejpam-5976	416	14	hv	hv	PROPN
ejpam-5976	416	15	is	be	AUX
ejpam-5976	416	16	not	not	PART
ejpam-5976	416	17	maximal	maximal	ADJ
ejpam-5976	416	18	.	.	PUNCT
ejpam-5976	417	1	this	this	PRON
ejpam-5976	417	2	contradicts	contradict	VERB
ejpam-5976	417	3	to	to	ADP
ejpam-5976	417	4	assumption	assumption	NOUN
ejpam-5976	417	5	.	.	PUNCT
ejpam-5976	418	1	therefore	therefore	ADV
ejpam-5976	418	2	,	,	PUNCT
ejpam-5976	418	3	h(u	h(u	PROPN
ejpam-5976	418	4	,	,	PUNCT
ejpam-5976	418	5	v	v	NOUN
ejpam-5976	418	6	)	)	PUNCT
ejpam-5976	418	7	is	be	AUX
ejpam-5976	418	8	maximal	maximal	ADJ
ejpam-5976	418	9	.	.	PUNCT
ejpam-5976	419	1	the	the	DET
ejpam-5976	419	2	case	case	NOUN
ejpam-5976	419	3	|	|	ADV
ejpam-5976	419	4	ha	ha	INTJ
ejpam-5976	419	5	|=	|=	PUNCT
ejpam-5976	419	6	1	1	NUM
ejpam-5976	419	7	and	and	CCONJ
ejpam-5976	419	8	|	|	ADV
ejpam-5976	419	9	hb	hb	X
ejpam-5976	420	1	|	|	ADV
ejpam-5976	420	2	>	>	X
ejpam-5976	420	3	1	1	NUM
ejpam-5976	420	4	can	can	AUX
ejpam-5976	420	5	be	be	AUX
ejpam-5976	420	6	proved	prove	VERB
ejpam-5976	420	7	similarly	similarly	ADV
ejpam-5976	420	8	.	.	PUNCT
ejpam-5976	421	1	definition	definition	NOUN
ejpam-5976	421	2	3	3	NUM
ejpam-5976	421	3	.	.	PUNCT
ejpam-5976	422	1	let	let	VERB
ejpam-5976	422	2	s	s	PRON
ejpam-5976	422	3	be	be	AUX
ejpam-5976	422	4	a	a	DET
ejpam-5976	422	5	semigroup	semigroup	NOUN
ejpam-5976	422	6	.	.	PUNCT
ejpam-5976	423	1	an	an	DET
ejpam-5976	423	2	element	element	NOUN
ejpam-5976	423	3	a	a	DET
ejpam-5976	423	4	∈	∈	NOUN
ejpam-5976	423	5	s	s	VERB
ejpam-5976	423	6	is	be	AUX
ejpam-5976	423	7	decomposable	decomposable	ADJ
ejpam-5976	423	8	if	if	SCONJ
ejpam-5976	423	9	there	there	PRON
ejpam-5976	423	10	are	be	VERB
ejpam-5976	423	11	u	u	NOUN
ejpam-5976	423	12	,	,	PUNCT
ejpam-5976	423	13	v	v	PROPN
ejpam-5976	423	14	∈	∈	NOUN
ejpam-5976	423	15	s	s	VERB
ejpam-5976	424	1	such	such	ADJ
ejpam-5976	424	2	that	that	SCONJ
ejpam-5976	424	3	a	a	DET
ejpam-5976	424	4	=	=	NOUN
ejpam-5976	424	5	uv	uv	NOUN
ejpam-5976	424	6	.	.	PUNCT
ejpam-5976	425	1	we	we	PRON
ejpam-5976	425	2	say	say	VERB
ejpam-5976	425	3	that	that	SCONJ
ejpam-5976	425	4	b	b	X
ejpam-5976	425	5	∈	∈	NOUN
ejpam-5976	425	6	s	s	VERB
ejpam-5976	425	7	is	be	AUX
ejpam-5976	425	8	indecomposable	indecomposable	ADJ
ejpam-5976	425	9	if	if	SCONJ
ejpam-5976	425	10	b	b	NOUN
ejpam-5976	425	11	is	be	AUX
ejpam-5976	425	12	not	not	PART
ejpam-5976	425	13	decomposable	decomposable	ADJ
ejpam-5976	425	14	,	,	PUNCT
ejpam-5976	425	15	equivalently	equivalently	ADV
ejpam-5976	425	16	,	,	PUNCT
ejpam-5976	425	17	b	b	PROPN
ejpam-5976	425	18	∈	∈	PROPN
ejpam-5976	425	19	s	s	PART
ejpam-5976	425	20	\	\	PROPN
ejpam-5976	425	21	s2	s2	PROPN
ejpam-5976	425	22	.	.	PUNCT
ejpam-5976	425	23	theorem	theorem	VERB
ejpam-5976	425	24	13	13	NUM
ejpam-5976	425	25	.	.	PUNCT
ejpam-5976	426	1	let	let	VERB
ejpam-5976	426	2	(	(	PUNCT
ejpam-5976	426	3	a	a	PRON
ejpam-5976	426	4	,	,	PUNCT
ejpam-5976	426	5	b	b	NOUN
ejpam-5976	426	6	)	)	PUNCT
ejpam-5976	426	7	∈	∈	PROPN
ejpam-5976	426	8	s1	s1	PROPN
ejpam-5976	426	9	×	×	PROPN
ejpam-5976	426	10	s2	s2	PROPN
ejpam-5976	426	11	.	.	PUNCT
ejpam-5976	427	1	if	if	SCONJ
ejpam-5976	427	2	(	(	PUNCT
ejpam-5976	427	3	a	a	DET
ejpam-5976	427	4	,	,	PUNCT
ejpam-5976	427	5	b	b	NOUN
ejpam-5976	427	6	)	)	PUNCT
ejpam-5976	427	7	is	be	AUX
ejpam-5976	427	8	indecomposable	indecomposable	ADJ
ejpam-5976	427	9	,	,	PUNCT
ejpam-5976	427	10	then	then	ADV
ejpam-5976	427	11	h(a	h(a	PROPN
ejpam-5976	427	12	,	,	PUNCT
ejpam-5976	427	13	b	b	X
ejpam-5976	427	14	)	)	PUNCT
ejpam-5976	427	15	is	be	AUX
ejpam-5976	427	16	a	a	DET
ejpam-5976	427	17	maximal	maximal	ADJ
ejpam-5976	427	18	h	h	NOUN
ejpam-5976	427	19	-	-	PUNCT
ejpam-5976	427	20	class	class	NOUN
ejpam-5976	427	21	of	of	ADP
ejpam-5976	427	22	s1	s1	PROPN
ejpam-5976	427	23	×	×	PROPN
ejpam-5976	427	24	s2	s2	PROPN
ejpam-5976	427	25	.	.	PUNCT
ejpam-5976	428	1	proof	proof	NOUN
ejpam-5976	428	2	.	.	PUNCT
ejpam-5976	429	1	assume	assume	VERB
ejpam-5976	429	2	that	that	SCONJ
ejpam-5976	429	3	h(a	h(a	PROPN
ejpam-5976	429	4	,	,	PUNCT
ejpam-5976	429	5	b	b	X
ejpam-5976	429	6	)	)	PUNCT
ejpam-5976	429	7	is	be	AUX
ejpam-5976	429	8	not	not	PART
ejpam-5976	429	9	a	a	DET
ejpam-5976	429	10	maximal	maximal	ADJ
ejpam-5976	429	11	h	h	NOUN
ejpam-5976	429	12	-	-	PUNCT
ejpam-5976	429	13	class	class	NOUN
ejpam-5976	429	14	.	.	PUNCT
ejpam-5976	430	1	this	this	PRON
ejpam-5976	430	2	means	mean	VERB
ejpam-5976	430	3	that	that	SCONJ
ejpam-5976	430	4	there	there	PRON
ejpam-5976	430	5	exists	exist	VERB
ejpam-5976	430	6	(	(	PUNCT
ejpam-5976	430	7	u	u	NOUN
ejpam-5976	430	8	,	,	PUNCT
ejpam-5976	430	9	v	v	NOUN
ejpam-5976	430	10	)	)	PUNCT
ejpam-5976	430	11	∈	∈	PROPN
ejpam-5976	430	12	s1	s1	NOUN
ejpam-5976	430	13	×	×	NOUN
ejpam-5976	430	14	s2	s2	NOUN
ejpam-5976	430	15	such	such	ADJ
ejpam-5976	430	16	that	that	PRON
ejpam-5976	430	17	q((a	q((a	NOUN
ejpam-5976	430	18	,	,	PUNCT
ejpam-5976	430	19	b	b	NOUN
ejpam-5976	430	20	)	)	PUNCT
ejpam-5976	430	21	)	)	PUNCT
ejpam-5976	430	22	⊊	⊊	VERB
ejpam-5976	430	23	q((u	q((u	NOUN
ejpam-5976	430	24	,	,	PUNCT
ejpam-5976	430	25	v	v	NOUN
ejpam-5976	430	26	)	)	PUNCT
ejpam-5976	430	27	)	)	PUNCT
ejpam-5976	430	28	.	.	PUNCT
ejpam-5976	431	1	p.	p.	NOUN
ejpam-5976	431	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	431	3	,	,	PUNCT
ejpam-5976	431	4	o.	o.	PROPN
ejpam-5976	431	5	pankoon	pankoon	NOUN
ejpam-5976	431	6	,	,	PUNCT
ejpam-5976	431	7	t.	t.	PROPN
ejpam-5976	431	8	changphas	changphas	PROPN
ejpam-5976	431	9	/	/	SYM
ejpam-5976	431	10	eur	eur	PROPN
ejpam-5976	431	11	.	.	PUNCT
ejpam-5976	432	1	j.	j.	PROPN
ejpam-5976	432	2	pure	pure	PROPN
ejpam-5976	432	3	appl	appl	PROPN
ejpam-5976	432	4	.	.	PROPN
ejpam-5976	432	5	math	math	PROPN
ejpam-5976	432	6	,	,	PUNCT
ejpam-5976	432	7	18	18	NUM
ejpam-5976	432	8	(	(	PUNCT
ejpam-5976	432	9	2	2	NUM
ejpam-5976	432	10	)	)	PUNCT
ejpam-5976	432	11	(	(	PUNCT
ejpam-5976	432	12	2025	2025	NUM
ejpam-5976	432	13	)	)	PUNCT
ejpam-5976	432	14	,	,	PUNCT
ejpam-5976	432	15	5976	5976	NUM
ejpam-5976	432	16	13	13	NUM
ejpam-5976	432	17	of	of	ADP
ejpam-5976	432	18	14	14	NUM
ejpam-5976	432	19	since	since	SCONJ
ejpam-5976	432	20	(	(	PUNCT
ejpam-5976	432	21	a	a	DET
ejpam-5976	432	22	,	,	PUNCT
ejpam-5976	432	23	b	b	NOUN
ejpam-5976	432	24	)	)	PUNCT
ejpam-5976	432	25	̸=	̸=	PROPN
ejpam-5976	432	26	(	(	PUNCT
ejpam-5976	432	27	u	u	NOUN
ejpam-5976	432	28	,	,	PUNCT
ejpam-5976	432	29	v	v	NOUN
ejpam-5976	432	30	)	)	PUNCT
ejpam-5976	432	31	and	and	CCONJ
ejpam-5976	432	32	(	(	PUNCT
ejpam-5976	432	33	a	a	PRON
ejpam-5976	432	34	,	,	PUNCT
ejpam-5976	432	35	b	b	NOUN
ejpam-5976	432	36	)	)	PUNCT
ejpam-5976	432	37	∈	∈	PROPN
ejpam-5976	432	38	q((u	q((u	NOUN
ejpam-5976	432	39	,	,	PUNCT
ejpam-5976	432	40	v	v	NOUN
ejpam-5976	432	41	)	)	PUNCT
ejpam-5976	432	42	)	)	PUNCT
ejpam-5976	432	43	,	,	PUNCT
ejpam-5976	432	44	we	we	PRON
ejpam-5976	432	45	have	have	VERB
ejpam-5976	432	46	(	(	PUNCT
ejpam-5976	432	47	a	a	PRON
ejpam-5976	432	48	,	,	PUNCT
ejpam-5976	432	49	b	b	NOUN
ejpam-5976	432	50	)	)	PUNCT
ejpam-5976	432	51	∈	∈	PROPN
ejpam-5976	432	52	(	(	PUNCT
ejpam-5976	432	53	us1	us1	PROPN
ejpam-5976	432	54	∩	∩	PROPN
ejpam-5976	432	55	s1u)×	s1u)×	PROPN
ejpam-5976	432	56	(	(	PUNCT
ejpam-5976	432	57	vs2	vs2	NOUN
ejpam-5976	432	58	∩	∩	NOUN
ejpam-5976	432	59	s2v	s2v	NOUN
ejpam-5976	432	60	)	)	PUNCT
ejpam-5976	432	61	.	.	PUNCT
ejpam-5976	433	1	therefore	therefore	ADV
ejpam-5976	433	2	,	,	PUNCT
ejpam-5976	433	3	(	(	PUNCT
ejpam-5976	433	4	a	a	DET
ejpam-5976	433	5	,	,	PUNCT
ejpam-5976	433	6	b	b	NOUN
ejpam-5976	433	7	)	)	PUNCT
ejpam-5976	433	8	is	be	AUX
ejpam-5976	433	9	decomposable	decomposable	ADJ
ejpam-5976	433	10	.	.	PUNCT
ejpam-5976	434	1	the	the	DET
ejpam-5976	434	2	following	follow	VERB
ejpam-5976	434	3	example	example	NOUN
ejpam-5976	434	4	shows	show	VERB
ejpam-5976	434	5	that	that	SCONJ
ejpam-5976	434	6	the	the	DET
ejpam-5976	434	7	reverse	reverse	NOUN
ejpam-5976	434	8	of	of	ADP
ejpam-5976	434	9	the	the	DET
ejpam-5976	434	10	above	above	ADJ
ejpam-5976	434	11	theorem	theorem	NOUN
ejpam-5976	434	12	is	be	AUX
ejpam-5976	434	13	not	not	PART
ejpam-5976	434	14	generally	generally	ADV
ejpam-5976	434	15	true	true	ADJ
ejpam-5976	434	16	.	.	PUNCT
ejpam-5976	435	1	example	example	NOUN
ejpam-5976	435	2	5	5	NUM
ejpam-5976	435	3	.	.	PUNCT
ejpam-5976	436	1	[	[	X
ejpam-5976	436	2	7	7	X
ejpam-5976	436	3	]	]	X
ejpam-5976	436	4	let	let	VERB
ejpam-5976	436	5	s	s	PRON
ejpam-5976	436	6	=	=	PUNCT
ejpam-5976	436	7	{	{	PUNCT
ejpam-5976	436	8	a	a	DET
ejpam-5976	436	9	,	,	PUNCT
ejpam-5976	436	10	b	b	NOUN
ejpam-5976	436	11	,	,	PUNCT
ejpam-5976	436	12	c	c	NOUN
ejpam-5976	436	13	,	,	PUNCT
ejpam-5976	436	14	d	d	AUX
ejpam-5976	436	15	}	}	PUNCT
ejpam-5976	436	16	be	be	AUX
ejpam-5976	436	17	a	a	DET
ejpam-5976	436	18	semigroup	semigroup	NOUN
ejpam-5976	436	19	with	with	ADP
ejpam-5976	436	20	the	the	DET
ejpam-5976	436	21	binary	binary	PROPN
ejpam-5976	436	22	operation	operation	NOUN
ejpam-5976	436	23	∗	∗	NOUN
ejpam-5976	436	24	on	on	ADP
ejpam-5976	436	25	s	s	PRON
ejpam-5976	436	26	defined	define	VERB
ejpam-5976	436	27	by	by	ADP
ejpam-5976	436	28	∗	∗	NOUN
ejpam-5976	437	1	a	a	DET
ejpam-5976	437	2	b	b	NOUN
ejpam-5976	437	3	c	c	NOUN
ejpam-5976	437	4	d	d	NOUN
ejpam-5976	437	5	a	a	PRON
ejpam-5976	437	6	a	a	DET
ejpam-5976	437	7	a	a	DET
ejpam-5976	437	8	a	a	DET
ejpam-5976	437	9	a	a	DET
ejpam-5976	437	10	b	b	NOUN
ejpam-5976	437	11	a	a	DET
ejpam-5976	437	12	a	a	DET
ejpam-5976	437	13	a	a	NOUN
ejpam-5976	437	14	a	a	DET
ejpam-5976	437	15	c	c	NOUN
ejpam-5976	437	16	a	a	DET
ejpam-5976	437	17	b	b	NOUN
ejpam-5976	437	18	c	c	NOUN
ejpam-5976	437	19	c	c	NOUN
ejpam-5976	438	1	d	d	NOUN
ejpam-5976	438	2	a	a	PRON
ejpam-5976	438	3	b	b	NOUN
ejpam-5976	438	4	c	c	NOUN
ejpam-5976	438	5	c	c	NOUN
ejpam-5976	438	6	then	then	ADV
ejpam-5976	438	7	h(b	h(b	PROPN
ejpam-5976	438	8	,	,	PUNCT
ejpam-5976	438	9	b	b	NOUN
ejpam-5976	438	10	)	)	PUNCT
ejpam-5976	438	11	is	be	AUX
ejpam-5976	438	12	maximal	maximal	ADJ
ejpam-5976	438	13	,	,	PUNCT
ejpam-5976	438	14	but	but	CCONJ
ejpam-5976	438	15	(	(	PUNCT
ejpam-5976	438	16	b	b	X
ejpam-5976	438	17	,	,	PUNCT
ejpam-5976	438	18	b	b	NOUN
ejpam-5976	438	19	)	)	PUNCT
ejpam-5976	438	20	is	be	AUX
ejpam-5976	438	21	decomposable	decomposable	ADJ
ejpam-5976	438	22	.	.	PUNCT
ejpam-5976	439	1	3	3	X
ejpam-5976	439	2	.	.	X
ejpam-5976	439	3	conclusions	conclusion	NOUN
ejpam-5976	439	4	in	in	ADP
ejpam-5976	439	5	this	this	DET
ejpam-5976	439	6	paper	paper	NOUN
ejpam-5976	439	7	,	,	PUNCT
ejpam-5976	439	8	we	we	PRON
ejpam-5976	439	9	studied	study	VERB
ejpam-5976	439	10	quasi	quasi	NOUN
ejpam-5976	439	11	-	-	NOUN
ejpam-5976	439	12	ideals	ideal	NOUN
ejpam-5976	439	13	and	and	CCONJ
ejpam-5976	439	14	h	h	NOUN
ejpam-5976	439	15	-	-	PUNCT
ejpam-5976	439	16	classes	class	NOUN
ejpam-5976	439	17	in	in	ADP
ejpam-5976	439	18	the	the	DET
ejpam-5976	439	19	direct	direct	ADJ
ejpam-5976	439	20	product	product	NOUN
ejpam-5976	439	21	of	of	ADP
ejpam-5976	439	22	two	two	NUM
ejpam-5976	439	23	semigroups	semigroup	NOUN
ejpam-5976	439	24	s1	s1	PROPN
ejpam-5976	439	25	×	×	PROPN
ejpam-5976	439	26	s2	s2	NOUN
ejpam-5976	439	27	by	by	ADP
ejpam-5976	439	28	focusing	focus	VERB
ejpam-5976	439	29	on	on	ADP
ejpam-5976	439	30	being	be	AUX
ejpam-5976	439	31	the	the	DET
ejpam-5976	439	32	principal	principal	ADJ
ejpam-5976	439	33	quasi	quasi	NOUN
ejpam-5976	439	34	-	-	NOUN
ejpam-5976	439	35	ideal	ideal	NOUN
ejpam-5976	439	36	of	of	ADP
ejpam-5976	439	37	q(a	q(a	NOUN
ejpam-5976	439	38	)	)	PUNCT
ejpam-5976	439	39	×	×	NOUN
ejpam-5976	439	40	q(b	q(b	ADJ
ejpam-5976	439	41	)	)	PUNCT
ejpam-5976	439	42	and	and	CCONJ
ejpam-5976	439	43	being	be	AUX
ejpam-5976	439	44	the	the	DET
ejpam-5976	439	45	h	h	NOUN
ejpam-5976	439	46	-	-	PUNCT
ejpam-5976	439	47	class	class	NOUN
ejpam-5976	439	48	of	of	ADP
ejpam-5976	439	49	ha	ha	INTJ
ejpam-5976	439	50	×hb	×hb	PROPN
ejpam-5976	439	51	in	in	ADP
ejpam-5976	439	52	s1	s1	PROPN
ejpam-5976	439	53	×	×	PROPN
ejpam-5976	439	54	s2	s2	PROPN
ejpam-5976	439	55	,	,	PUNCT
ejpam-5976	439	56	where	where	SCONJ
ejpam-5976	439	57	q(a	q(a	NOUN
ejpam-5976	439	58	)	)	PUNCT
ejpam-5976	439	59	and	and	CCONJ
ejpam-5976	439	60	q(b	q(b	ADV
ejpam-5976	439	61	)	)	PUNCT
ejpam-5976	439	62	are	be	AUX
ejpam-5976	439	63	the	the	DET
ejpam-5976	439	64	principal	principal	ADJ
ejpam-5976	439	65	quasi	quasi	NOUN
ejpam-5976	439	66	-	-	NOUN
ejpam-5976	439	67	ideal	ideal	NOUN
ejpam-5976	439	68	of	of	ADP
ejpam-5976	439	69	s1	s1	NOUN
ejpam-5976	439	70	generated	generate	VERB
ejpam-5976	439	71	by	by	ADP
ejpam-5976	439	72	a	a	DET
ejpam-5976	439	73	∈	∈	PROPN
ejpam-5976	439	74	s1	s1	NOUN
ejpam-5976	439	75	and	and	CCONJ
ejpam-5976	439	76	the	the	DET
ejpam-5976	439	77	principal	principal	ADJ
ejpam-5976	439	78	quasi	quasi	NOUN
ejpam-5976	439	79	-	-	NOUN
ejpam-5976	439	80	ideal	ideal	NOUN
ejpam-5976	439	81	of	of	ADP
ejpam-5976	439	82	s2	s2	NOUN
ejpam-5976	439	83	generated	generate	VERB
ejpam-5976	439	84	by	by	ADP
ejpam-5976	439	85	b	b	PROPN
ejpam-5976	439	86	∈	∈	PROPN
ejpam-5976	439	87	s2	s2	PROPN
ejpam-5976	439	88	,	,	PUNCT
ejpam-5976	439	89	respectively	respectively	ADV
ejpam-5976	439	90	.	.	PUNCT
ejpam-5976	440	1	similarly	similarly	ADV
ejpam-5976	440	2	,	,	PUNCT
ejpam-5976	440	3	ha	ha	INTJ
ejpam-5976	440	4	and	and	CCONJ
ejpam-5976	440	5	hb	hb	PROPN
ejpam-5976	440	6	are	be	AUX
ejpam-5976	440	7	an	an	DET
ejpam-5976	440	8	h	h	NOUN
ejpam-5976	440	9	-	-	PUNCT
ejpam-5976	440	10	class	class	NOUN
ejpam-5976	440	11	of	of	ADP
ejpam-5976	440	12	s1	s1	NOUN
ejpam-5976	440	13	containing	contain	VERB
ejpam-5976	440	14	a	a	DET
ejpam-5976	440	15	∈	∈	NOUN
ejpam-5976	440	16	s1	s1	NOUN
ejpam-5976	440	17	and	and	CCONJ
ejpam-5976	440	18	an	an	DET
ejpam-5976	440	19	h	h	NOUN
ejpam-5976	440	20	-	-	PUNCT
ejpam-5976	440	21	class	class	NOUN
ejpam-5976	440	22	of	of	ADP
ejpam-5976	440	23	s2	s2	NOUN
ejpam-5976	440	24	containing	contain	VERB
ejpam-5976	440	25	b	b	PROPN
ejpam-5976	440	26	∈	∈	PROPN
ejpam-5976	440	27	s2	s2	PROPN
ejpam-5976	440	28	,	,	PUNCT
ejpam-5976	440	29	respectively	respectively	ADV
ejpam-5976	440	30	.	.	PUNCT
ejpam-5976	441	1	first	first	ADV
ejpam-5976	441	2	,	,	PUNCT
ejpam-5976	441	3	we	we	PRON
ejpam-5976	441	4	proved	prove	VERB
ejpam-5976	441	5	that	that	SCONJ
ejpam-5976	441	6	q(a	q(a	NOUN
ejpam-5976	441	7	)	)	PUNCT
ejpam-5976	441	8	×	×	NOUN
ejpam-5976	441	9	q(b	q(b	ADV
ejpam-5976	441	10	)	)	PUNCT
ejpam-5976	441	11	is	be	AUX
ejpam-5976	441	12	a	a	DET
ejpam-5976	441	13	quasi	quasi	NOUN
ejpam-5976	441	14	-	-	NOUN
ejpam-5976	441	15	ideal	ideal	ADJ
ejpam-5976	441	16	in	in	ADP
ejpam-5976	441	17	s1	s1	PROPN
ejpam-5976	441	18	×	×	PROPN
ejpam-5976	441	19	s2	s2	PROPN
ejpam-5976	441	20	,	,	PUNCT
ejpam-5976	441	21	and	and	CCONJ
ejpam-5976	441	22	we	we	PRON
ejpam-5976	441	23	provided	provide	VERB
ejpam-5976	441	24	an	an	DET
ejpam-5976	441	25	example	example	NOUN
ejpam-5976	441	26	to	to	PART
ejpam-5976	441	27	indicate	indicate	VERB
ejpam-5976	441	28	that	that	SCONJ
ejpam-5976	441	29	it	it	PRON
ejpam-5976	441	30	is	be	AUX
ejpam-5976	441	31	not	not	PART
ejpam-5976	441	32	guaranteed	guarantee	VERB
ejpam-5976	441	33	to	to	PART
ejpam-5976	441	34	be	be	AUX
ejpam-5976	441	35	the	the	DET
ejpam-5976	441	36	principal	principal	ADJ
ejpam-5976	441	37	quasi	quasi	ADJ
ejpam-5976	441	38	-	-	ADJ
ejpam-5976	441	39	ideal	ideal	ADJ
ejpam-5976	441	40	generated	generate	VERB
ejpam-5976	441	41	by	by	ADP
ejpam-5976	441	42	(	(	PUNCT
ejpam-5976	441	43	a	a	DET
ejpam-5976	441	44	,	,	PUNCT
ejpam-5976	441	45	b	b	NOUN
ejpam-5976	441	46	)	)	PUNCT
ejpam-5976	441	47	.	.	PUNCT
ejpam-5976	442	1	furthermore	furthermore	ADV
ejpam-5976	442	2	,	,	PUNCT
ejpam-5976	442	3	we	we	PRON
ejpam-5976	442	4	ensured	ensure	VERB
ejpam-5976	442	5	that	that	SCONJ
ejpam-5976	442	6	if	if	SCONJ
ejpam-5976	442	7	q(a	q(a	NOUN
ejpam-5976	442	8	)	)	PUNCT
ejpam-5976	442	9	×	×	NOUN
ejpam-5976	442	10	q(b	q(b	ADJ
ejpam-5976	442	11	)	)	PUNCT
ejpam-5976	442	12	is	be	AUX
ejpam-5976	442	13	not	not	PART
ejpam-5976	442	14	the	the	DET
ejpam-5976	442	15	principal	principal	ADJ
ejpam-5976	442	16	quasi	quasi	ADJ
ejpam-5976	442	17	-	-	ADJ
ejpam-5976	442	18	ideal	ideal	ADJ
ejpam-5976	442	19	generated	generate	VERB
ejpam-5976	442	20	by	by	ADP
ejpam-5976	442	21	(	(	PUNCT
ejpam-5976	442	22	a	a	DET
ejpam-5976	442	23	,	,	PUNCT
ejpam-5976	442	24	b	b	NOUN
ejpam-5976	442	25	)	)	PUNCT
ejpam-5976	442	26	,	,	PUNCT
ejpam-5976	442	27	then	then	ADV
ejpam-5976	442	28	it	it	PRON
ejpam-5976	442	29	is	be	AUX
ejpam-5976	442	30	not	not	PART
ejpam-5976	442	31	the	the	DET
ejpam-5976	442	32	principal	principal	ADJ
ejpam-5976	442	33	quasi	quasi	ADJ
ejpam-5976	442	34	-	-	ADJ
ejpam-5976	442	35	ideal	ideal	ADJ
ejpam-5976	442	36	generated	generate	VERB
ejpam-5976	442	37	by	by	ADP
ejpam-5976	442	38	any	any	DET
ejpam-5976	442	39	other	other	ADJ
ejpam-5976	442	40	element	element	NOUN
ejpam-5976	442	41	in	in	ADP
ejpam-5976	442	42	s1×s2	s1×s2	PROPN
ejpam-5976	442	43	.	.	PUNCT
ejpam-5976	443	1	we	we	PRON
ejpam-5976	443	2	then	then	ADV
ejpam-5976	443	3	characterized	characterize	VERB
ejpam-5976	443	4	when	when	SCONJ
ejpam-5976	443	5	the	the	DET
ejpam-5976	443	6	direct	direct	ADJ
ejpam-5976	443	7	product	product	NOUN
ejpam-5976	443	8	of	of	ADP
ejpam-5976	443	9	two	two	NUM
ejpam-5976	443	10	principal	principal	ADJ
ejpam-5976	443	11	quasi	quasi	NOUN
ejpam-5976	443	12	-	-	NOUN
ejpam-5976	443	13	ideals	ideal	NOUN
ejpam-5976	443	14	is	be	AUX
ejpam-5976	443	15	the	the	DET
ejpam-5976	443	16	principal	principal	ADJ
ejpam-5976	443	17	quasi	quasi	NOUN
ejpam-5976	443	18	-	-	NOUN
ejpam-5976	443	19	ideal	ideal	ADJ
ejpam-5976	443	20	by	by	ADP
ejpam-5976	443	21	presenting	present	VERB
ejpam-5976	443	22	the	the	DET
ejpam-5976	443	23	sufficient	sufficient	ADJ
ejpam-5976	443	24	and	and	CCONJ
ejpam-5976	443	25	necessary	necessary	ADJ
ejpam-5976	443	26	conditions	condition	NOUN
ejpam-5976	443	27	for	for	ADP
ejpam-5976	443	28	q(a	q(a	NOUN
ejpam-5976	443	29	)	)	PUNCT
ejpam-5976	443	30	×	×	NOUN
ejpam-5976	443	31	q(b	q(b	ADJ
ejpam-5976	443	32	)	)	PUNCT
ejpam-5976	443	33	=	=	SYM
ejpam-5976	443	34	q((a	q((a	PROPN
ejpam-5976	443	35	,	,	PUNCT
ejpam-5976	443	36	b	b	NOUN
ejpam-5976	443	37	)	)	PUNCT
ejpam-5976	443	38	)	)	PUNCT
ejpam-5976	443	39	.	.	PUNCT
ejpam-5976	444	1	in	in	ADP
ejpam-5976	444	2	the	the	DET
ejpam-5976	444	3	context	context	NOUN
ejpam-5976	444	4	of	of	ADP
ejpam-5976	444	5	the	the	DET
ejpam-5976	444	6	h	h	NOUN
ejpam-5976	444	7	-	-	PUNCT
ejpam-5976	444	8	class	class	NOUN
ejpam-5976	444	9	,	,	PUNCT
ejpam-5976	444	10	we	we	PRON
ejpam-5976	444	11	gave	give	VERB
ejpam-5976	444	12	an	an	DET
ejpam-5976	444	13	example	example	NOUN
ejpam-5976	444	14	to	to	PART
ejpam-5976	444	15	demonstrate	demonstrate	VERB
ejpam-5976	444	16	that	that	SCONJ
ejpam-5976	444	17	ha	ha	INTJ
ejpam-5976	444	18	×	×	NOUN
ejpam-5976	444	19	hb	hb	PROPN
ejpam-5976	444	20	is	be	AUX
ejpam-5976	444	21	not	not	PART
ejpam-5976	444	22	guaranteed	guarantee	VERB
ejpam-5976	444	23	to	to	PART
ejpam-5976	444	24	be	be	AUX
ejpam-5976	444	25	an	an	DET
ejpam-5976	444	26	h	h	NOUN
ejpam-5976	444	27	-	-	PUNCT
ejpam-5976	444	28	class	class	NOUN
ejpam-5976	444	29	in	in	ADP
ejpam-5976	444	30	s1	s1	PROPN
ejpam-5976	444	31	×	×	PROPN
ejpam-5976	444	32	s2	s2	PROPN
ejpam-5976	444	33	.	.	PUNCT
ejpam-5976	445	1	the	the	DET
ejpam-5976	445	2	relation	relation	NOUN
ejpam-5976	445	3	between	between	ADP
ejpam-5976	445	4	the	the	DET
ejpam-5976	445	5	h	h	NOUN
ejpam-5976	445	6	-	-	PUNCT
ejpam-5976	445	7	class	class	NOUN
ejpam-5976	445	8	h(a	h(a	PROPN
ejpam-5976	445	9	,	,	PUNCT
ejpam-5976	445	10	b	b	NOUN
ejpam-5976	445	11	)	)	PUNCT
ejpam-5976	445	12	and	and	CCONJ
ejpam-5976	445	13	the	the	DET
ejpam-5976	445	14	direct	direct	ADJ
ejpam-5976	445	15	product	product	NOUN
ejpam-5976	445	16	ha	ha	INTJ
ejpam-5976	445	17	×hb	×hb	PROPN
ejpam-5976	445	18	is	be	AUX
ejpam-5976	445	19	explained	explain	VERB
ejpam-5976	445	20	.	.	PUNCT
ejpam-5976	446	1	after	after	ADP
ejpam-5976	446	2	that	that	PRON
ejpam-5976	446	3	,	,	PUNCT
ejpam-5976	446	4	we	we	PRON
ejpam-5976	446	5	provided	provide	VERB
ejpam-5976	446	6	the	the	DET
ejpam-5976	446	7	necessary	necessary	ADJ
ejpam-5976	446	8	and	and	CCONJ
ejpam-5976	446	9	sufficient	sufficient	ADJ
ejpam-5976	446	10	conditions	condition	NOUN
ejpam-5976	446	11	for	for	ADP
ejpam-5976	446	12	ha	ha	INTJ
ejpam-5976	446	13	×	×	NOUN
ejpam-5976	446	14	hb	hb	PROPN
ejpam-5976	446	15	=	=	SYM
ejpam-5976	446	16	h(a	h(a	PROPN
ejpam-5976	446	17	,	,	PUNCT
ejpam-5976	446	18	b	b	NOUN
ejpam-5976	446	19	)	)	PUNCT
ejpam-5976	446	20	.	.	PUNCT
ejpam-5976	447	1	the	the	DET
ejpam-5976	447	2	connection	connection	NOUN
ejpam-5976	447	3	between	between	ADP
ejpam-5976	447	4	the	the	DET
ejpam-5976	447	5	two	two	NUM
ejpam-5976	447	6	main	main	ADJ
ejpam-5976	447	7	points	point	NOUN
ejpam-5976	447	8	is	be	AUX
ejpam-5976	447	9	as	as	SCONJ
ejpam-5976	447	10	follows	follow	VERB
ejpam-5976	447	11	:	:	PUNCT
ejpam-5976	447	12	if	if	SCONJ
ejpam-5976	447	13	q(a	q(a	NOUN
ejpam-5976	447	14	)	)	PUNCT
ejpam-5976	447	15	×	×	NOUN
ejpam-5976	447	16	q(b	q(b	ADV
ejpam-5976	447	17	)	)	PUNCT
ejpam-5976	447	18	is	be	AUX
ejpam-5976	447	19	the	the	DET
ejpam-5976	447	20	principal	principal	ADJ
ejpam-5976	447	21	quasi	quasi	NOUN
ejpam-5976	447	22	-	-	NOUN
ejpam-5976	447	23	ideal	ideal	ADJ
ejpam-5976	447	24	,	,	PUNCT
ejpam-5976	447	25	we	we	PRON
ejpam-5976	447	26	can	can	AUX
ejpam-5976	447	27	ensure	ensure	VERB
ejpam-5976	447	28	that	that	SCONJ
ejpam-5976	447	29	ha	ha	INTJ
ejpam-5976	447	30	×hb	×hb	PROPN
ejpam-5976	447	31	is	be	AUX
ejpam-5976	447	32	an	an	DET
ejpam-5976	447	33	h	h	NOUN
ejpam-5976	447	34	-	-	PUNCT
ejpam-5976	447	35	class	class	NOUN
ejpam-5976	447	36	in	in	ADP
ejpam-5976	447	37	s1	s1	PROPN
ejpam-5976	447	38	×	×	PROPN
ejpam-5976	447	39	s2	s2	PROPN
ejpam-5976	447	40	.	.	PUNCT
ejpam-5976	448	1	finally	finally	ADV
ejpam-5976	448	2	,	,	PUNCT
ejpam-5976	448	3	we	we	PRON
ejpam-5976	448	4	also	also	ADV
ejpam-5976	448	5	studied	study	VERB
ejpam-5976	448	6	the	the	DET
ejpam-5976	448	7	maximal	maximal	ADJ
ejpam-5976	448	8	h	h	NOUN
ejpam-5976	448	9	-	-	PUNCT
ejpam-5976	448	10	class	class	NOUN
ejpam-5976	448	11	.	.	PUNCT
ejpam-5976	449	1	we	we	PRON
ejpam-5976	449	2	proved	prove	VERB
ejpam-5976	449	3	under	under	ADP
ejpam-5976	449	4	some	some	DET
ejpam-5976	449	5	conditions	condition	NOUN
ejpam-5976	449	6	that	that	SCONJ
ejpam-5976	449	7	the	the	DET
ejpam-5976	449	8	maximality	maximality	NOUN
ejpam-5976	449	9	of	of	ADP
ejpam-5976	449	10	ha	ha	INTJ
ejpam-5976	449	11	and	and	CCONJ
ejpam-5976	449	12	hb	hb	PROPN
ejpam-5976	449	13	implies	imply	VERB
ejpam-5976	449	14	the	the	DET
ejpam-5976	449	15	maximality	maximality	NOUN
ejpam-5976	449	16	of	of	ADP
ejpam-5976	449	17	h(a	h(a	PROPN
ejpam-5976	449	18	,	,	PUNCT
ejpam-5976	449	19	b	b	NOUN
ejpam-5976	449	20	)	)	PUNCT
ejpam-5976	449	21	.	.	PUNCT
ejpam-5976	450	1	furthermore	furthermore	ADV
ejpam-5976	450	2	,	,	PUNCT
ejpam-5976	450	3	the	the	DET
ejpam-5976	450	4	converse	converse	NOUN
ejpam-5976	450	5	of	of	ADP
ejpam-5976	450	6	this	this	DET
ejpam-5976	450	7	statement	statement	NOUN
ejpam-5976	450	8	also	also	ADV
ejpam-5976	450	9	holds	hold	VERB
ejpam-5976	450	10	.	.	PUNCT
ejpam-5976	451	1	in	in	ADP
ejpam-5976	451	2	addition	addition	NOUN
ejpam-5976	451	3	,	,	PUNCT
ejpam-5976	451	4	we	we	PRON
ejpam-5976	451	5	investigated	investigate	VERB
ejpam-5976	451	6	the	the	DET
ejpam-5976	451	7	properties	property	NOUN
ejpam-5976	451	8	of	of	ADP
ejpam-5976	451	9	the	the	DET
ejpam-5976	451	10	direct	direct	ADJ
ejpam-5976	451	11	product	product	NOUN
ejpam-5976	451	12	of	of	ADP
ejpam-5976	451	13	two	two	NUM
ejpam-5976	451	14	maximal	maximal	ADJ
ejpam-5976	451	15	h	h	NOUN
ejpam-5976	451	16	-	-	PUNCT
ejpam-5976	451	17	classes	class	NOUN
ejpam-5976	451	18	ha	ha	INTJ
ejpam-5976	451	19	×	×	NOUN
ejpam-5976	451	20	hb	hb	PROPN
ejpam-5976	451	21	and	and	CCONJ
ejpam-5976	451	22	studied	study	VERB
ejpam-5976	451	23	the	the	DET
ejpam-5976	451	24	maximal	maximal	ADJ
ejpam-5976	451	25	h	h	NOUN
ejpam-5976	451	26	-	-	PUNCT
ejpam-5976	451	27	class	class	NOUN
ejpam-5976	451	28	of	of	ADP
ejpam-5976	451	29	h(a	h(a	PROPN
ejpam-5976	451	30	,	,	PUNCT
ejpam-5976	451	31	b	b	NOUN
ejpam-5976	451	32	)	)	PUNCT
ejpam-5976	451	33	through	through	ADP
ejpam-5976	451	34	the	the	DET
ejpam-5976	451	35	maximal	maximal	ADJ
ejpam-5976	451	36	h	h	NOUN
ejpam-5976	451	37	-	-	PUNCT
ejpam-5976	451	38	class	class	NOUN
ejpam-5976	451	39	of	of	ADP
ejpam-5976	451	40	ha	ha	INTJ
ejpam-5976	451	41	and	and	CCONJ
ejpam-5976	451	42	hb	hb	PROPN
ejpam-5976	451	43	,	,	PUNCT
ejpam-5976	451	44	including	include	VERB
ejpam-5976	451	45	the	the	DET
ejpam-5976	451	46	element	element	NOUN
ejpam-5976	451	47	,	,	PUNCT
ejpam-5976	451	48	namely	namely	ADV
ejpam-5976	451	49	,	,	PUNCT
ejpam-5976	451	50	the	the	DET
ejpam-5976	451	51	indecomposable	indecomposable	ADJ
ejpam-5976	451	52	element	element	NOUN
ejpam-5976	451	53	.	.	PUNCT
ejpam-5976	452	1	p.	p.	NOUN
ejpam-5976	452	2	luangchaisri	luangchaisri	PROPN
ejpam-5976	452	3	,	,	PUNCT
ejpam-5976	452	4	o.	o.	PROPN
ejpam-5976	452	5	pankoon	pankoon	NOUN
ejpam-5976	452	6	,	,	PUNCT
ejpam-5976	452	7	t.	t.	PROPN
ejpam-5976	452	8	changphas	changphas	PROPN
ejpam-5976	452	9	/	/	SYM
ejpam-5976	452	10	eur	eur	PROPN
ejpam-5976	452	11	.	.	PUNCT
ejpam-5976	453	1	j.	j.	PROPN
ejpam-5976	453	2	pure	pure	PROPN
ejpam-5976	453	3	appl	appl	PROPN
ejpam-5976	453	4	.	.	PROPN
ejpam-5976	453	5	math	math	PROPN
ejpam-5976	453	6	,	,	PUNCT
ejpam-5976	453	7	18	18	NUM
ejpam-5976	453	8	(	(	PUNCT
ejpam-5976	453	9	2	2	NUM
ejpam-5976	453	10	)	)	PUNCT
ejpam-5976	453	11	(	(	PUNCT
ejpam-5976	453	12	2025	2025	NUM
ejpam-5976	453	13	)	)	PUNCT
ejpam-5976	453	14	,	,	PUNCT
ejpam-5976	453	15	5976	5976	NUM
ejpam-5976	453	16	14	14	NUM
ejpam-5976	453	17	of	of	ADP
ejpam-5976	453	18	14	14	NUM
ejpam-5976	453	19	acknowledgements	acknowledgement	NOUN
ejpam-5976	453	20	this	this	DET
ejpam-5976	453	21	work	work	NOUN
ejpam-5976	453	22	(	(	PUNCT
ejpam-5976	453	23	grant	grant	VERB
ejpam-5976	453	24	no	no	INTJ
ejpam-5976	453	25	.	.	PUNCT
ejpam-5976	453	26	rgns	rgn	VERB
ejpam-5976	453	27	65	65	NUM
ejpam-5976	453	28	-	-	PUNCT
ejpam-5976	453	29	054	054	NUM
ejpam-5976	453	30	)	)	PUNCT
ejpam-5976	453	31	was	be	AUX
ejpam-5976	453	32	supported	support	VERB
ejpam-5976	453	33	by	by	ADP
ejpam-5976	453	34	office	office	NOUN
ejpam-5976	453	35	of	of	ADP
ejpam-5976	453	36	the	the	DET
ejpam-5976	453	37	permanent	permanent	ADJ
ejpam-5976	453	38	secretary	secretary	NOUN
ejpam-5976	453	39	,	,	PUNCT
ejpam-5976	453	40	ministry	ministry	PROPN
ejpam-5976	453	41	of	of	ADP
ejpam-5976	453	42	higher	high	ADJ
ejpam-5976	453	43	education	education	NOUN
ejpam-5976	453	44	,	,	PUNCT
ejpam-5976	453	45	science	science	NOUN
ejpam-5976	453	46	,	,	PUNCT
ejpam-5976	453	47	research	research	NOUN
ejpam-5976	453	48	and	and	CCONJ
ejpam-5976	453	49	innovation	innovation	NOUN
ejpam-5976	453	50	(	(	PUNCT
ejpam-5976	453	51	ops	op	NOUN
ejpam-5976	453	52	mhesi	mhesi	PROPN
ejpam-5976	453	53	)	)	PUNCT
ejpam-5976	453	54	,	,	PUNCT
ejpam-5976	453	55	thailand	thailand	PROPN
ejpam-5976	453	56	science	science	PROPN
ejpam-5976	453	57	research	research	PROPN
ejpam-5976	453	58	and	and	CCONJ
ejpam-5976	453	59	innovation	innovation	NOUN
ejpam-5976	453	60	(	(	PUNCT
ejpam-5976	453	61	tsri	tsri	ADJ
ejpam-5976	453	62	)	)	PUNCT
ejpam-5976	453	63	and	and	CCONJ
ejpam-5976	453	64	khon	khon	PROPN
ejpam-5976	453	65	kaen	kaen	PROPN
ejpam-5976	453	66	university	university	PROPN
ejpam-5976	453	67	.	.	PUNCT
ejpam-5976	454	1	references	reference	NOUN
ejpam-5976	454	2	[	[	X
ejpam-5976	454	3	1	1	NUM
ejpam-5976	454	4	]	]	PUNCT
ejpam-5976	454	5	o	o	X
ejpam-5976	454	6	steinfeld	steinfeld	PROPN
ejpam-5976	454	7	.	.	PUNCT
ejpam-5976	455	1	über	über	PROPN
ejpam-5976	455	2	die	die	VERB
ejpam-5976	455	3	quasiideale	quasiideale	PROPN
ejpam-5976	455	4	von	von	PROPN
ejpam-5976	455	5	halbgruppend	halbgruppend	PROPN
ejpam-5976	455	6	.	.	PUNCT
ejpam-5976	456	1	publ	publ	PROPN
ejpam-5976	456	2	.	.	PUNCT
ejpam-5976	457	1	math	math	NOUN
ejpam-5976	457	2	.	.	PUNCT
ejpam-5976	458	1	debrecen	debrecen	PROPN
ejpam-5976	458	2	,	,	PUNCT
ejpam-5976	458	3	4:262	4:262	NUM
ejpam-5976	458	4	–	–	PUNCT
ejpam-5976	458	5	275	275	NUM
ejpam-5976	458	6	,	,	PUNCT
ejpam-5976	458	7	1956	1956	NUM
ejpam-5976	458	8	.	.	PUNCT
ejpam-5976	459	1	[	[	X
ejpam-5976	459	2	2	2	NUM
ejpam-5976	459	3	]	]	PUNCT
ejpam-5976	459	4	o	o	X
ejpam-5976	459	5	steinfeld	steinfeld	PROPN
ejpam-5976	459	6	.	.	PUNCT
ejpam-5976	460	1	quasi	quasi	ADJ
ejpam-5976	460	2	-	-	NOUN
ejpam-5976	460	3	ideal	ideal	ADJ
ejpam-5976	460	4	in	in	ADP
ejpam-5976	460	5	rings	ring	NOUN
ejpam-5976	460	6	and	and	CCONJ
ejpam-5976	460	7	semigroups	semigroup	NOUN
ejpam-5976	460	8	.	.	PUNCT
ejpam-5976	461	1	semigroup	semigroup	PROPN
ejpam-5976	461	2	forum	forum	PROPN
ejpam-5976	461	3	,	,	PUNCT
ejpam-5976	461	4	19:371	19:371	NUM
ejpam-5976	461	5	–	–	PUNCT
ejpam-5976	461	6	372	372	NUM
ejpam-5976	461	7	,	,	PUNCT
ejpam-5976	461	8	1980	1980	NUM
ejpam-5976	461	9	.	.	PUNCT
ejpam-5976	462	1	[	[	X
ejpam-5976	462	2	3	3	X
ejpam-5976	462	3	]	]	X
ejpam-5976	462	4	i	i	PRON
ejpam-5976	462	5	fabrici	fabrici	NOUN
ejpam-5976	462	6	.	.	PUNCT
ejpam-5976	463	1	one	one	NUM
ejpam-5976	463	2	-	-	PUNCT
ejpam-5976	463	3	sided	sided	ADJ
ejpam-5976	463	4	principal	principal	ADJ
ejpam-5976	463	5	ideals	ideal	NOUN
ejpam-5976	463	6	in	in	ADP
ejpam-5976	463	7	the	the	DET
ejpam-5976	463	8	direct	direct	ADJ
ejpam-5976	463	9	product	product	NOUN
ejpam-5976	463	10	of	of	ADP
ejpam-5976	463	11	two	two	NUM
ejpam-5976	463	12	semigroups	semigroup	NOUN
ejpam-5976	463	13	.	.	PUNCT
ejpam-5976	464	1	mathematica	mathematica	PROPN
ejpam-5976	464	2	bohemica	bohemica	PROPN
ejpam-5976	464	3	,	,	PUNCT
ejpam-5976	464	4	118(4):337	118(4):337	NUM
ejpam-5976	464	5	–	–	PUNCT
ejpam-5976	464	6	342	342	NUM
ejpam-5976	464	7	,	,	PUNCT
ejpam-5976	464	8	1993	1993	NUM
ejpam-5976	464	9	.	.	PUNCT
ejpam-5976	465	1	[	[	X
ejpam-5976	465	2	4	4	X
ejpam-5976	465	3	]	]	X
ejpam-5976	465	4	i	i	PRON
ejpam-5976	465	5	fabrici	fabrici	PROPN
ejpam-5976	465	6	.	.	PUNCT
ejpam-5976	466	1	principal	principal	ADJ
ejpam-5976	466	2	two	two	NUM
ejpam-5976	466	3	-	-	PUNCT
ejpam-5976	466	4	sided	sided	ADJ
ejpam-5976	466	5	ideals	ideal	NOUN
ejpam-5976	466	6	in	in	ADP
ejpam-5976	466	7	the	the	DET
ejpam-5976	466	8	direct	direct	ADJ
ejpam-5976	466	9	product	product	NOUN
ejpam-5976	466	10	of	of	ADP
ejpam-5976	466	11	two	two	NUM
ejpam-5976	466	12	semigroups	semigroup	NOUN
ejpam-5976	466	13	.	.	PUNCT
ejpam-5976	467	1	czechoslovak	czechoslovak	ADJ
ejpam-5976	467	2	mathematical	mathematical	PROPN
ejpam-5976	467	3	journal	journal	NOUN
ejpam-5976	467	4	,	,	PUNCT
ejpam-5976	467	5	41(3):411	41(3):411	PROPN
ejpam-5976	467	6	–	–	PUNCT
ejpam-5976	467	7	421	421	NUM
ejpam-5976	467	8	,	,	PUNCT
ejpam-5976	467	9	1991	1991	NUM
ejpam-5976	467	10	.	.	PUNCT
ejpam-5976	468	1	[	[	X
ejpam-5976	468	2	5	5	NUM
ejpam-5976	468	3	]	]	PUNCT
ejpam-5976	468	4	s	s	PART
ejpam-5976	468	5	lajos	lajos	NOUN
ejpam-5976	468	6	.	.	PUNCT
ejpam-5976	469	1	generalized	generalized	ADJ
ejpam-5976	469	2	ideals	ideal	NOUN
ejpam-5976	469	3	in	in	ADP
ejpam-5976	469	4	semigroups	semigroup	NOUN
ejpam-5976	469	5	.	.	PUNCT
ejpam-5976	470	1	acta	acta	PROPN
ejpam-5976	470	2	sci	sci	PROPN
ejpam-5976	470	3	.	.	PROPN
ejpam-5976	470	4	math	math	PROPN
ejpam-5976	470	5	.	.	PUNCT
ejpam-5976	471	1	szeged	szeged	PROPN
ejpam-5976	471	2	,	,	PUNCT
ejpam-5976	471	3	22:217	22:217	NUM
ejpam-5976	471	4	–	–	PUNCT
ejpam-5976	471	5	222	222	NUM
ejpam-5976	471	6	,	,	PUNCT
ejpam-5976	471	7	1961	1961	NUM
ejpam-5976	471	8	.	.	PUNCT
ejpam-5976	472	1	[	[	X
ejpam-5976	472	2	6	6	NUM
ejpam-5976	472	3	]	]	PUNCT
ejpam-5976	472	4	p	p	X
ejpam-5976	472	5	luangchaisri	luangchaisri	VERB
ejpam-5976	472	6	and	and	CCONJ
ejpam-5976	472	7	t	t	PROPN
ejpam-5976	472	8	changphas	changpha	NOUN
ejpam-5976	472	9	.	.	PUNCT
ejpam-5976	473	1	on	on	ADP
ejpam-5976	473	2	the	the	DET
ejpam-5976	473	3	principal	principal	NOUN
ejpam-5976	473	4	(	(	PUNCT
ejpam-5976	473	5	m	m	PROPN
ejpam-5976	473	6	,	,	PUNCT
ejpam-5976	473	7	n)-ideals	n)-ideal	NOUN
ejpam-5976	473	8	in	in	ADP
ejpam-5976	473	9	the	the	DET
ejpam-5976	473	10	direct	direct	ADJ
ejpam-5976	473	11	product	product	NOUN
ejpam-5976	473	12	of	of	ADP
ejpam-5976	473	13	two	two	NUM
ejpam-5976	473	14	semigroups	semigroup	NOUN
ejpam-5976	473	15	.	.	PUNCT
ejpam-5976	474	1	quasigroups	quasigroup	NOUN
ejpam-5976	474	2	and	and	CCONJ
ejpam-5976	474	3	related	related	ADJ
ejpam-5976	474	4	systems	system	NOUN
ejpam-5976	474	5	,	,	PUNCT
ejpam-5976	474	6	24:75	24:75	NUM
ejpam-5976	474	7	–	–	PUNCT
ejpam-5976	474	8	80	80	NUM
ejpam-5976	474	9	,	,	PUNCT
ejpam-5976	474	10	2016	2016	NUM
ejpam-5976	474	11	.	.	PUNCT
ejpam-5976	475	1	[	[	X
ejpam-5976	475	2	7	7	X
ejpam-5976	475	3	]	]	X
ejpam-5976	475	4	g	g	PROPN
ejpam-5976	475	5	forsythe	forsythe	PROPN
ejpam-5976	475	6	.	.	PUNCT
ejpam-5976	476	1	swac	swac	PROPN
ejpam-5976	476	2	computes	compute	VERB
ejpam-5976	476	3	126	126	NUM
ejpam-5976	476	4	distinct	distinct	ADJ
ejpam-5976	476	5	semigroups	semigroup	NOUN
ejpam-5976	476	6	of	of	ADP
ejpam-5976	476	7	order	order	NOUN
ejpam-5976	476	8	4	4	NUM
ejpam-5976	476	9	.	.	PUNCT
ejpam-5976	476	10	proceedings	proceeding	NOUN
ejpam-5976	476	11	of	of	ADP
ejpam-5976	476	12	the	the	DET
ejpam-5976	476	13	american	american	PROPN
ejpam-5976	476	14	mathematical	mathematical	PROPN
ejpam-5976	476	15	society	society	NOUN
ejpam-5976	476	16	,	,	PUNCT
ejpam-5976	476	17	6(3):443	6(3):443	NUM
ejpam-5976	476	18	,	,	PUNCT
ejpam-5976	476	19	1995	1995	NUM
ejpam-5976	476	20	.	.	PUNCT
ejpam-5976	477	1	[	[	X
ejpam-5976	477	2	8	8	NUM
ejpam-5976	477	3	]	]	X
ejpam-5976	477	4	j	j	PROPN
ejpam-5976	477	5	a	a	DET
ejpam-5976	477	6	green	green	NOUN
ejpam-5976	477	7	.	.	PUNCT
ejpam-5976	478	1	on	on	ADP
ejpam-5976	478	2	the	the	DET
ejpam-5976	478	3	structure	structure	NOUN
ejpam-5976	478	4	of	of	ADP
ejpam-5976	478	5	semigroups	semigroup	NOUN
ejpam-5976	478	6	.	.	PUNCT
ejpam-5976	479	1	ann	ann	PROPN
ejpam-5976	479	2	.	.	PROPN
ejpam-5976	479	3	of	of	ADP
ejpam-5976	479	4	math	math	NOUN
ejpam-5976	479	5	,	,	PUNCT
ejpam-5976	479	6	54(1):163	54(1):163	NUM
ejpam-5976	479	7	–	–	PUNCT
ejpam-5976	479	8	172	172	NUM
ejpam-5976	479	9	,	,	PUNCT
ejpam-5976	479	10	1951	1951	NUM
ejpam-5976	479	11	.	.	PUNCT
ejpam-5976	480	1	[	[	X
ejpam-5976	480	2	9	9	NUM
ejpam-5976	480	3	]	]	X
ejpam-5976	480	4	a	a	DET
ejpam-5976	480	5	clifford	clifford	PROPN
ejpam-5976	480	6	and	and	CCONJ
ejpam-5976	480	7	g	g	PROPN
ejpam-5976	480	8	preston	preston	PROPN
ejpam-5976	480	9	.	.	PUNCT
ejpam-5976	481	1	the	the	DET
ejpam-5976	481	2	algebraic	algebraic	PROPN
ejpam-5976	481	3	theory	theory	NOUN
ejpam-5976	481	4	of	of	ADP
ejpam-5976	481	5	semigroups	semigroup	NOUN
ejpam-5976	481	6	.	.	PUNCT
ejpam-5976	482	1	providence	providence	NOUN
ejpam-5976	482	2	:	:	PUNCT
ejpam-5976	482	3	american	american	PROPN
ejpam-5976	482	4	mathematical	mathematical	PROPN
ejpam-5976	482	5	society	society	NOUN
ejpam-5976	482	6	,	,	PUNCT
ejpam-5976	482	7	1961	1961	NUM
ejpam-5976	482	8	.	.	PUNCT
ejpam-5976	483	1	[	[	X
ejpam-5976	483	2	10	10	NUM
ejpam-5976	483	3	]	]	X
ejpam-5976	483	4	j	j	PROPN
ejpam-5976	483	5	m	m	PROPN
ejpam-5976	483	6	howie	howie	NOUN
ejpam-5976	483	7	.	.	PUNCT
ejpam-5976	484	1	fundamentals	fundamental	NOUN
ejpam-5976	484	2	of	of	ADP
ejpam-5976	484	3	semigroup	semigroup	PROPN
ejpam-5976	484	4	theory	theory	NOUN
ejpam-5976	484	5	.	.	PUNCT
ejpam-5976	485	1	clarendon	clarendon	PROPN
ejpam-5976	485	2	press	press	PROPN
ejpam-5976	485	3	,	,	PUNCT
ejpam-5976	485	4	oxford	oxford	NOUN
ejpam-5976	485	5	,	,	PUNCT
ejpam-5976	485	6	1995	1995	NUM
ejpam-5976	485	7	.	.	PUNCT