id	sid	tid	token	lemma	pos
ejpam-7061	1	1	european	european	PROPN
ejpam-7061	1	2	journal	journal	PROPN
ejpam-7061	1	3	of	of	ADP
ejpam-7061	1	4	pure	pure	ADJ
ejpam-7061	1	5	and	and	CCONJ
ejpam-7061	1	6	applied	applied	ADJ
ejpam-7061	1	7	mathematics	mathematic	NOUN
ejpam-7061	1	8	2025	2025	NUM
ejpam-7061	1	9	,	,	PUNCT
ejpam-7061	1	10	vol	vol	NOUN
ejpam-7061	1	11	.	.	PROPN
ejpam-7061	1	12	18	18	NUM
ejpam-7061	1	13	,	,	PUNCT
ejpam-7061	1	14	issue	issue	NOUN
ejpam-7061	1	15	4	4	NUM
ejpam-7061	1	16	,	,	PUNCT
ejpam-7061	1	17	article	article	NOUN
ejpam-7061	1	18	number	number	NOUN
ejpam-7061	1	19	7061	7061	NUM
ejpam-7061	1	20	issn	issn	PROPN
ejpam-7061	1	21	1307	1307	NUM
ejpam-7061	1	22	-	-	SYM
ejpam-7061	1	23	5543	5543	NUM
ejpam-7061	1	24	–	–	PUNCT
ejpam-7061	1	25	ejpam.com	ejpam.com	X
ejpam-7061	1	26	published	publish	VERB
ejpam-7061	1	27	by	by	ADP
ejpam-7061	1	28	new	new	PROPN
ejpam-7061	1	29	york	york	PROPN
ejpam-7061	1	30	business	business	PROPN
ejpam-7061	1	31	global	global	ADJ
ejpam-7061	1	32	left	leave	VERB
ejpam-7061	1	33	ideals	ideal	NOUN
ejpam-7061	1	34	and	and	CCONJ
ejpam-7061	1	35	l	l	NOUN
ejpam-7061	1	36	-	-	NOUN
ejpam-7061	1	37	classes	class	NOUN
ejpam-7061	1	38	in	in	ADP
ejpam-7061	1	39	the	the	DET
ejpam-7061	1	40	finite	finite	ADJ
ejpam-7061	1	41	direct	direct	ADJ
ejpam-7061	1	42	product	product	NOUN
ejpam-7061	1	43	of	of	ADP
ejpam-7061	1	44	semigroups	semigroup	NOUN
ejpam-7061	1	45	panuwat	panuwat	VERB
ejpam-7061	1	46	luangchaisri1	luangchaisri1	PROPN
ejpam-7061	1	47	,	,	PUNCT
ejpam-7061	1	48	ontima	ontima	PROPN
ejpam-7061	1	49	pankoon1	pankoon1	PROPN
ejpam-7061	1	50	,	,	PUNCT
ejpam-7061	1	51	thawhat	thawhat	PRON
ejpam-7061	1	52	changphas1,∗	changphas1,∗	NOUN
ejpam-7061	1	53	1	1	NUM
ejpam-7061	1	54	department	department	NOUN
ejpam-7061	1	55	of	of	ADP
ejpam-7061	1	56	mathematics	mathematic	NOUN
ejpam-7061	1	57	,	,	PUNCT
ejpam-7061	1	58	faculty	faculty	NOUN
ejpam-7061	1	59	of	of	ADP
ejpam-7061	1	60	science	science	PROPN
ejpam-7061	1	61	khon	khon	PROPN
ejpam-7061	1	62	kaen	kaen	PROPN
ejpam-7061	1	63	university	university	PROPN
ejpam-7061	1	64	,	,	PUNCT
ejpam-7061	1	65	khon	khon	PROPN
ejpam-7061	1	66	kaen	kaen	PROPN
ejpam-7061	1	67	40002	40002	NUM
ejpam-7061	1	68	,	,	PUNCT
ejpam-7061	1	69	thailand	thailand	PROPN
ejpam-7061	1	70	abstract	abstract	PROPN
ejpam-7061	1	71	.	.	PUNCT
ejpam-7061	2	1	let	let	VERB
ejpam-7061	2	2	si	si	X
ejpam-7061	2	3	be	be	AUX
ejpam-7061	2	4	a	a	DET
ejpam-7061	2	5	semigroup	semigroup	NOUN
ejpam-7061	2	6	for	for	ADP
ejpam-7061	2	7	all	all	PRON
ejpam-7061	2	8	i	i	PRON
ejpam-7061	2	9	∈	∈	PROPN
ejpam-7061	2	10	{	{	PUNCT
ejpam-7061	2	11	1	1	NUM
ejpam-7061	2	12	,	,	PUNCT
ejpam-7061	2	13	2	2	NUM
ejpam-7061	2	14	,	,	PUNCT
ejpam-7061	2	15	.	.	PUNCT
ejpam-7061	2	16	.	.	PUNCT
ejpam-7061	3	1	.	.	PUNCT
ejpam-7061	3	2	,	,	PUNCT
ejpam-7061	3	3	n	n	CCONJ
ejpam-7061	3	4	}	}	PUNCT
ejpam-7061	3	5	.	.	PUNCT
ejpam-7061	4	1	then	then	ADV
ejpam-7061	4	2	the	the	DET
ejpam-7061	4	3	cartesian	cartesian	ADJ
ejpam-7061	4	4	product	product	NOUN
ejpam-7061	4	5	of	of	ADP
ejpam-7061	4	6	s1	s1	NOUN
ejpam-7061	4	7	,	,	PUNCT
ejpam-7061	4	8	s2	s2	NOUN
ejpam-7061	4	9	,	,	PUNCT
ejpam-7061	4	10	.	.	PUNCT
ejpam-7061	4	11	.	.	PUNCT
ejpam-7061	4	12	.	.	PUNCT
ejpam-7061	5	1	,	,	PUNCT
ejpam-7061	5	2	sn	sn	PROPN
ejpam-7061	5	3	becomes	become	VERB
ejpam-7061	5	4	a	a	DET
ejpam-7061	5	5	semigroup	semigroup	NOUN
ejpam-7061	5	6	under	under	ADP
ejpam-7061	5	7	componentwise	componentwise	NOUN
ejpam-7061	5	8	multiplication	multiplication	NOUN
ejpam-7061	5	9	.	.	PUNCT
ejpam-7061	6	1	let	let	VERB
ejpam-7061	6	2	(	(	PUNCT
ejpam-7061	6	3	s1	s1	NOUN
ejpam-7061	6	4	,	,	PUNCT
ejpam-7061	6	5	s2	s2	NOUN
ejpam-7061	6	6	,	,	PUNCT
ejpam-7061	6	7	.	.	PUNCT
ejpam-7061	6	8	.	.	PUNCT
ejpam-7061	6	9	.	.	PUNCT
ejpam-7061	7	1	,	,	PUNCT
ejpam-7061	7	2	sn	sn	PROPN
ejpam-7061	7	3	)	)	PUNCT
ejpam-7061	7	4	∈	∈	PROPN
ejpam-7061	7	5	s1×s2	s1×s2	PROPN
ejpam-7061	7	6	·	·	PUNCT
ejpam-7061	7	7	·	·	PUNCT
ejpam-7061	7	8	·	·	PUNCT
ejpam-7061	7	9	×sn	×sn	PROPN
ejpam-7061	7	10	.	.	PUNCT
ejpam-7061	8	1	in	in	ADP
ejpam-7061	8	2	this	this	DET
ejpam-7061	8	3	paper	paper	NOUN
ejpam-7061	8	4	,	,	PUNCT
ejpam-7061	8	5	we	we	PRON
ejpam-7061	8	6	give	give	VERB
ejpam-7061	8	7	necessary	necessary	ADJ
ejpam-7061	8	8	and	and	CCONJ
ejpam-7061	8	9	sufficient	sufficient	ADJ
ejpam-7061	8	10	condition	condition	NOUN
ejpam-7061	8	11	when	when	SCONJ
ejpam-7061	8	12	the	the	DET
ejpam-7061	8	13	cartesian	cartesian	ADJ
ejpam-7061	8	14	product	product	NOUN
ejpam-7061	8	15	of	of	ADP
ejpam-7061	8	16	principal	principal	NOUN
ejpam-7061	8	17	left	leave	VERB
ejpam-7061	8	18	ideals	ideal	NOUN
ejpam-7061	8	19	l(s1)×	l(s1)×	PROPN
ejpam-7061	8	20	l(s2)×	l(s2)×	PROPN
ejpam-7061	8	21	·	·	PUNCT
ejpam-7061	8	22	·	·	PUNCT
ejpam-7061	8	23	·	·	PUNCT
ejpam-7061	9	1	×	×	NOUN
ejpam-7061	9	2	l(sn	l(sn	ADV
ejpam-7061	9	3	)	)	PUNCT
ejpam-7061	9	4	is	be	AUX
ejpam-7061	9	5	the	the	DET
ejpam-7061	9	6	principal	principal	NOUN
ejpam-7061	9	7	left	leave	VERB
ejpam-7061	9	8	ideal	ideal	ADJ
ejpam-7061	9	9	l((s1	l((s1	PROPN
ejpam-7061	9	10	,	,	PUNCT
ejpam-7061	9	11	s2	s2	PROPN
ejpam-7061	9	12	,	,	PUNCT
ejpam-7061	9	13	.	.	PUNCT
ejpam-7061	9	14	.	.	PUNCT
ejpam-7061	10	1	.	.	PUNCT
ejpam-7061	11	1	,	,	PUNCT
ejpam-7061	11	2	sn	sn	PROPN
ejpam-7061	11	3	)	)	PUNCT
ejpam-7061	11	4	)	)	PUNCT
ejpam-7061	12	1	and	and	CCONJ
ejpam-7061	12	2	the	the	DET
ejpam-7061	12	3	cartesian	cartesian	ADJ
ejpam-7061	12	4	product	product	NOUN
ejpam-7061	12	5	of	of	ADP
ejpam-7061	12	6	l	l	NOUN
ejpam-7061	12	7	-	-	PUNCT
ejpam-7061	12	8	classes	class	NOUN
ejpam-7061	12	9	ls1×ls2×	ls1×ls2×	X
ejpam-7061	12	10	·	·	PUNCT
ejpam-7061	12	11	·	·	PUNCT
ejpam-7061	13	1	·	·	PUNCT
ejpam-7061	13	2	×lsn	×lsn	NOUN
ejpam-7061	13	3	is	be	AUX
ejpam-7061	13	4	an	an	DET
ejpam-7061	13	5	l	l	NOUN
ejpam-7061	13	6	-	-	PUNCT
ejpam-7061	13	7	class	class	NOUN
ejpam-7061	13	8	l(s1,s2,	l(s1,s2,	NOUN
ejpam-7061	13	9	...	...	PUNCT
ejpam-7061	13	10	,sn	,sn	PUNCT
ejpam-7061	13	11	)	)	PUNCT
ejpam-7061	13	12	in	in	ADP
ejpam-7061	13	13	a	a	DET
ejpam-7061	13	14	semigroup	semigroup	ADJ
ejpam-7061	13	15	s1	s1	NOUN
ejpam-7061	13	16	×	×	NOUN
ejpam-7061	13	17	s2	s2	NOUN
ejpam-7061	13	18	×	×	NOUN
ejpam-7061	13	19	·	·	PUNCT
ejpam-7061	13	20	·	·	PUNCT
ejpam-7061	13	21	·	·	PUNCT
ejpam-7061	14	1	×	×	NOUN
ejpam-7061	14	2	sn	sn	PROPN
ejpam-7061	14	3	.	.	PROPN
ejpam-7061	14	4	2020	2020	NUM
ejpam-7061	14	5	mathematics	mathematics	PROPN
ejpam-7061	14	6	subject	subject	NOUN
ejpam-7061	14	7	classifications	classification	NOUN
ejpam-7061	14	8	:	:	PUNCT
ejpam-7061	14	9	20m12	20m12	NUM
ejpam-7061	14	10	key	key	ADJ
ejpam-7061	14	11	words	word	NOUN
ejpam-7061	14	12	and	and	CCONJ
ejpam-7061	14	13	phrases	phrase	NOUN
ejpam-7061	14	14	:	:	PUNCT
ejpam-7061	14	15	semigroup	semigroup	ADJ
ejpam-7061	14	16	,	,	PUNCT
ejpam-7061	14	17	direct	direct	ADJ
ejpam-7061	14	18	product	product	NOUN
ejpam-7061	14	19	,	,	PUNCT
ejpam-7061	14	20	principal	principal	NOUN
ejpam-7061	14	21	left	leave	VERB
ejpam-7061	14	22	ideal	ideal	ADJ
ejpam-7061	14	23	,	,	PUNCT
ejpam-7061	14	24	l	l	NOUN
ejpam-7061	14	25	-	-	NOUN
ejpam-7061	14	26	class	class	NOUN
ejpam-7061	14	27	1	1	NUM
ejpam-7061	14	28	.	.	PUNCT
ejpam-7061	15	1	introduction	introduction	NOUN
ejpam-7061	15	2	let	let	VERB
ejpam-7061	15	3	s	s	PRON
ejpam-7061	15	4	and	and	CCONJ
ejpam-7061	15	5	t	t	PROPN
ejpam-7061	15	6	be	be	AUX
ejpam-7061	15	7	semigroups	semigroup	NOUN
ejpam-7061	15	8	.	.	PUNCT
ejpam-7061	16	1	the	the	DET
ejpam-7061	16	2	cartesian	cartesian	ADJ
ejpam-7061	16	3	product	product	NOUN
ejpam-7061	16	4	s×t	s×t	PRON
ejpam-7061	16	5	becomes	become	VERB
ejpam-7061	16	6	a	a	DET
ejpam-7061	16	7	semigroup	semigroup	NOUN
ejpam-7061	16	8	under	under	ADP
ejpam-7061	16	9	a	a	DET
ejpam-7061	16	10	binary	binary	ADJ
ejpam-7061	16	11	operation	operation	NOUN
ejpam-7061	16	12	on	on	ADP
ejpam-7061	16	13	s	s	PRON
ejpam-7061	16	14	×	×	PROPN
ejpam-7061	16	15	t	t	NOUN
ejpam-7061	16	16	defined	define	VERB
ejpam-7061	16	17	by	by	ADP
ejpam-7061	16	18	(	(	PUNCT
ejpam-7061	16	19	s	s	PROPN
ejpam-7061	16	20	,	,	PUNCT
ejpam-7061	16	21	t)(s′	t)(s′	PROPN
ejpam-7061	16	22	,	,	PUNCT
ejpam-7061	16	23	t′	t′	NUM
ejpam-7061	16	24	)	)	PUNCT
ejpam-7061	17	1	=	=	PRON
ejpam-7061	17	2	(	(	PUNCT
ejpam-7061	17	3	ss′	ss′	PROPN
ejpam-7061	17	4	,	,	PUNCT
ejpam-7061	17	5	tt′	tt′	NOUN
ejpam-7061	17	6	)	)	PUNCT
ejpam-7061	17	7	for	for	ADP
ejpam-7061	17	8	all	all	DET
ejpam-7061	17	9	(	(	PUNCT
ejpam-7061	17	10	s	s	PROPN
ejpam-7061	17	11	,	,	PUNCT
ejpam-7061	17	12	t	t	PROPN
ejpam-7061	17	13	)	)	PUNCT
ejpam-7061	17	14	,	,	PUNCT
ejpam-7061	17	15	(	(	PUNCT
ejpam-7061	17	16	s′	s′	X
ejpam-7061	17	17	,	,	PUNCT
ejpam-7061	17	18	t′	t′	NUM
ejpam-7061	17	19	)	)	PUNCT
ejpam-7061	17	20	∈	∈	PROPN
ejpam-7061	17	21	s	s	PART
ejpam-7061	17	22	×	×	NOUN
ejpam-7061	17	23	t	t	NOUN
ejpam-7061	17	24	.	.	PUNCT
ejpam-7061	18	1	this	this	DET
ejpam-7061	18	2	semigroup	semigroup	PROPN
ejpam-7061	18	3	is	be	AUX
ejpam-7061	18	4	referred	refer	VERB
ejpam-7061	18	5	to	to	ADP
ejpam-7061	18	6	as	as	ADP
ejpam-7061	18	7	the	the	DET
ejpam-7061	18	8	direct	direct	ADJ
ejpam-7061	18	9	product	product	NOUN
ejpam-7061	18	10	of	of	ADP
ejpam-7061	18	11	s	s	NOUN
ejpam-7061	18	12	and	and	CCONJ
ejpam-7061	18	13	t	t	PROPN
ejpam-7061	18	14	.	.	PUNCT
ejpam-7061	19	1	a	a	DET
ejpam-7061	19	2	nonempty	nonempty	NOUN
ejpam-7061	19	3	subset	subset	VERB
ejpam-7061	19	4	a	a	PRON
ejpam-7061	19	5	of	of	ADP
ejpam-7061	19	6	s	s	NOUN
ejpam-7061	19	7	is	be	AUX
ejpam-7061	19	8	a	a	DET
ejpam-7061	19	9	left	left	ADJ
ejpam-7061	19	10	ideal	ideal	NOUN
ejpam-7061	19	11	of	of	ADP
ejpam-7061	19	12	s	s	PRON
ejpam-7061	19	13	if	if	SCONJ
ejpam-7061	19	14	sa	sa	PROPN
ejpam-7061	19	15	⊆	⊆	NUM
ejpam-7061	19	16	a.	a.	NOUN
ejpam-7061	19	17	for	for	ADP
ejpam-7061	19	18	any	any	DET
ejpam-7061	19	19	a	a	DET
ejpam-7061	19	20	∈	∈	ADJ
ejpam-7061	19	21	s	s	NOUN
ejpam-7061	19	22	,	,	PUNCT
ejpam-7061	19	23	the	the	DET
ejpam-7061	19	24	principal	principal	NOUN
ejpam-7061	19	25	left	leave	VERB
ejpam-7061	19	26	ideal	ideal	NOUN
ejpam-7061	19	27	of	of	ADP
ejpam-7061	19	28	s	s	AUX
ejpam-7061	19	29	generated	generate	VERB
ejpam-7061	19	30	by	by	ADP
ejpam-7061	19	31	a	a	PRON
ejpam-7061	19	32	,	,	PUNCT
ejpam-7061	19	33	denoted	denote	VERB
ejpam-7061	19	34	by	by	ADP
ejpam-7061	19	35	l(a	l(a	PROPN
ejpam-7061	19	36	)	)	PUNCT
ejpam-7061	19	37	,	,	PUNCT
ejpam-7061	19	38	is	be	AUX
ejpam-7061	19	39	the	the	DET
ejpam-7061	19	40	smallest	small	ADJ
ejpam-7061	19	41	left	leave	VERB
ejpam-7061	19	42	ideal	ideal	NOUN
ejpam-7061	19	43	of	of	ADP
ejpam-7061	19	44	s	s	AUX
ejpam-7061	19	45	containing	contain	VERB
ejpam-7061	19	46	a.	a.	NOUN
ejpam-7061	19	47	it	it	PRON
ejpam-7061	19	48	is	be	AUX
ejpam-7061	19	49	well	well	ADV
ejpam-7061	19	50	-	-	PUNCT
ejpam-7061	19	51	known	know	VERB
ejpam-7061	19	52	that	that	SCONJ
ejpam-7061	19	53	l(a	l(a	PROPN
ejpam-7061	19	54	)	)	PUNCT
ejpam-7061	19	55	=	=	PUNCT
ejpam-7061	20	1	a	a	DET
ejpam-7061	20	2	∪	∪	X
ejpam-7061	20	3	sa	sa	NOUN
ejpam-7061	20	4	.	.	PUNCT
ejpam-7061	21	1	a	a	DET
ejpam-7061	21	2	relation	relation	NOUN
ejpam-7061	21	3	l	l	NOUN
ejpam-7061	21	4	on	on	ADP
ejpam-7061	21	5	s	s	PROPN
ejpam-7061	21	6	is	be	AUX
ejpam-7061	21	7	then	then	ADV
ejpam-7061	21	8	defined	define	VERB
ejpam-7061	21	9	by	by	ADP
ejpam-7061	21	10	the	the	DET
ejpam-7061	21	11	rule	rule	NOUN
ejpam-7061	21	12	that	that	PRON
ejpam-7061	21	13	alb	alb	VERB
ejpam-7061	21	14	if	if	SCONJ
ejpam-7061	21	15	and	and	CCONJ
ejpam-7061	21	16	only	only	ADV
ejpam-7061	21	17	if	if	SCONJ
ejpam-7061	21	18	l(a	l(a	PROPN
ejpam-7061	21	19	)	)	PUNCT
ejpam-7061	21	20	=	=	SYM
ejpam-7061	21	21	l(b	l(b	PROPN
ejpam-7061	21	22	)	)	PUNCT
ejpam-7061	21	23	,	,	PUNCT
ejpam-7061	21	24	i.e.	i.e.	X
ejpam-7061	21	25	,	,	PUNCT
ejpam-7061	21	26	if	if	SCONJ
ejpam-7061	21	27	and	and	CCONJ
ejpam-7061	21	28	only	only	ADV
ejpam-7061	21	29	if	if	SCONJ
ejpam-7061	21	30	a	a	DET
ejpam-7061	21	31	∪	∪	NOUN
ejpam-7061	21	32	sa	sa	NOUN
ejpam-7061	21	33	=	=	SYM
ejpam-7061	21	34	b	b	PROPN
ejpam-7061	21	35	∪	∪	X
ejpam-7061	21	36	sb	sb	PROPN
ejpam-7061	21	37	.	.	PUNCT
ejpam-7061	22	1	it	it	PRON
ejpam-7061	22	2	is	be	AUX
ejpam-7061	22	3	one	one	NUM
ejpam-7061	22	4	of	of	ADP
ejpam-7061	22	5	green	green	PROPN
ejpam-7061	22	6	’s	’s	PART
ejpam-7061	22	7	equivalence	equivalence	NOUN
ejpam-7061	22	8	relations	relation	NOUN
ejpam-7061	22	9	.	.	PUNCT
ejpam-7061	23	1	then	then	ADV
ejpam-7061	23	2	the	the	DET
ejpam-7061	23	3	l	l	NOUN
ejpam-7061	23	4	-	-	NOUN
ejpam-7061	23	5	class	class	NOUN
ejpam-7061	23	6	of	of	ADP
ejpam-7061	23	7	s	s	NOUN
ejpam-7061	23	8	containing	contain	VERB
ejpam-7061	23	9	element	element	NOUN
ejpam-7061	23	10	a	a	PRON
ejpam-7061	23	11	will	will	AUX
ejpam-7061	23	12	be	be	AUX
ejpam-7061	23	13	written	write	VERB
ejpam-7061	23	14	by	by	ADP
ejpam-7061	23	15	la	la	PROPN
ejpam-7061	23	16	.	.	PUNCT
ejpam-7061	24	1	in	in	ADP
ejpam-7061	24	2	[	[	X
ejpam-7061	24	3	1	1	NUM
ejpam-7061	24	4	]	]	PUNCT
ejpam-7061	24	5	,	,	PUNCT
ejpam-7061	24	6	fabrici	fabrici	PROPN
ejpam-7061	24	7	considered	consider	VERB
ejpam-7061	24	8	a	a	DET
ejpam-7061	24	9	principal	principal	NOUN
ejpam-7061	24	10	left	leave	VERB
ejpam-7061	24	11	ideal	ideal	NOUN
ejpam-7061	24	12	and	and	CCONJ
ejpam-7061	24	13	a	a	DET
ejpam-7061	24	14	relation	relation	NOUN
ejpam-7061	24	15	l	l	NOUN
ejpam-7061	24	16	on	on	ADP
ejpam-7061	24	17	the	the	DET
ejpam-7061	24	18	direct	direct	ADJ
ejpam-7061	24	19	product	product	NOUN
ejpam-7061	24	20	of	of	ADP
ejpam-7061	24	21	two	two	NUM
ejpam-7061	24	22	semigroups	semigroup	NOUN
ejpam-7061	24	23	s	s	X
ejpam-7061	24	24	×	×	NOUN
ejpam-7061	24	25	t	t	NOUN
ejpam-7061	24	26	.	.	PUNCT
ejpam-7061	25	1	let	let	VERB
ejpam-7061	25	2	(	(	PUNCT
ejpam-7061	25	3	s	s	X
ejpam-7061	25	4	,	,	PUNCT
ejpam-7061	25	5	t	t	PROPN
ejpam-7061	25	6	)	)	PUNCT
ejpam-7061	25	7	∈	∈	PROPN
ejpam-7061	25	8	s	s	PART
ejpam-7061	25	9	×	×	NOUN
ejpam-7061	25	10	t	t	NOUN
ejpam-7061	25	11	.	.	PUNCT
ejpam-7061	26	1	∗corresponding	∗corresponde	VERB
ejpam-7061	26	2	author	author	NOUN
ejpam-7061	26	3	.	.	PUNCT
ejpam-7061	27	1	doi	doi	NOUN
ejpam-7061	27	2	:	:	PUNCT
ejpam-7061	27	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7061	https://doi.org/10.29020/nybg.ejpam.v18i4.7061	PROPN
ejpam-7061	27	4	email	email	NOUN
ejpam-7061	27	5	addresses	address	NOUN
ejpam-7061	27	6	:	:	PUNCT
ejpam-7061	27	7	panulu@kku.ac.th	panulu@kku.ac.th	PROPN
ejpam-7061	27	8	(	(	PUNCT
ejpam-7061	27	9	p.	p.	NOUN
ejpam-7061	27	10	luangchaisri	luangchaisri	PROPN
ejpam-7061	27	11	)	)	PUNCT
ejpam-7061	27	12	,	,	PUNCT
ejpam-7061	27	13	ontimapa@kkumail.com	ontimapa@kkumail.com	X
ejpam-7061	27	14	(	(	PUNCT
ejpam-7061	27	15	o.	o.	PROPN
ejpam-7061	27	16	pankoon	pankoon	PROPN
ejpam-7061	27	17	)	)	PUNCT
ejpam-7061	27	18	,	,	PUNCT
ejpam-7061	27	19	thacha@kku.ac.th	thacha@kku.ac.th	NOUN
ejpam-7061	27	20	(	(	PUNCT
ejpam-7061	27	21	t.	t.	NOUN
ejpam-7061	27	22	changphas	changphas	PROPN
ejpam-7061	27	23	)	)	PUNCT
ejpam-7061	27	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-7061	27	25	1	1	NUM
ejpam-7061	27	26	copyright	copyright	NOUN
ejpam-7061	27	27	:	:	PUNCT
ejpam-7061	28	1	©	©	PROPN
ejpam-7061	28	2	2025	2025	NUM
ejpam-7061	28	3	the	the	DET
ejpam-7061	28	4	author(s	author(s	NOUN
ejpam-7061	28	5	)	)	PUNCT
ejpam-7061	28	6	.	.	PUNCT
ejpam-7061	29	1	(	(	PUNCT
ejpam-7061	29	2	cc	cc	NOUN
ejpam-7061	29	3	by	by	ADP
ejpam-7061	29	4	-	-	PUNCT
ejpam-7061	29	5	nc	nc	PROPN
ejpam-7061	29	6	4.0	4.0	NUM
ejpam-7061	29	7	)	)	PUNCT
ejpam-7061	30	1	p.	p.	NOUN
ejpam-7061	30	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	30	3	,	,	PUNCT
ejpam-7061	30	4	o.	o.	PROPN
ejpam-7061	30	5	pankoon	pankoon	NOUN
ejpam-7061	30	6	,	,	PUNCT
ejpam-7061	30	7	t.	t.	PROPN
ejpam-7061	30	8	changphas	changphas	PROPN
ejpam-7061	30	9	/	/	SYM
ejpam-7061	30	10	eur	eur	PROPN
ejpam-7061	30	11	.	.	PUNCT
ejpam-7061	31	1	j.	j.	PROPN
ejpam-7061	31	2	pure	pure	PROPN
ejpam-7061	31	3	appl	appl	PROPN
ejpam-7061	31	4	.	.	PROPN
ejpam-7061	31	5	math	math	PROPN
ejpam-7061	31	6	,	,	PUNCT
ejpam-7061	31	7	18	18	NUM
ejpam-7061	31	8	(	(	PUNCT
ejpam-7061	31	9	4	4	NUM
ejpam-7061	31	10	)	)	PUNCT
ejpam-7061	31	11	(	(	PUNCT
ejpam-7061	31	12	2025	2025	NUM
ejpam-7061	31	13	)	)	PUNCT
ejpam-7061	31	14	,	,	PUNCT
ejpam-7061	31	15	7061	7061	NUM
ejpam-7061	31	16	2	2	NUM
ejpam-7061	31	17	of	of	ADP
ejpam-7061	31	18	10	10	NUM
ejpam-7061	31	19	necessary	necessary	ADJ
ejpam-7061	31	20	and	and	CCONJ
ejpam-7061	31	21	sufficient	sufficient	ADJ
ejpam-7061	31	22	condition	condition	NOUN
ejpam-7061	31	23	when	when	SCONJ
ejpam-7061	31	24	l(s)×	l(s)×	NOUN
ejpam-7061	31	25	l(t	l(t	PROPN
ejpam-7061	31	26	)	)	PUNCT
ejpam-7061	31	27	=	=	SYM
ejpam-7061	32	1	l((s	l((s	PROPN
ejpam-7061	32	2	,	,	PUNCT
ejpam-7061	32	3	t	t	PROPN
ejpam-7061	32	4	)	)	PUNCT
ejpam-7061	32	5	)	)	PUNCT
ejpam-7061	32	6	were	be	AUX
ejpam-7061	32	7	provided	provide	VERB
ejpam-7061	32	8	.	.	PUNCT
ejpam-7061	33	1	moreover	moreover	ADV
ejpam-7061	33	2	,	,	PUNCT
ejpam-7061	33	3	the	the	DET
ejpam-7061	33	4	author	author	NOUN
ejpam-7061	33	5	showed	show	VERB
ejpam-7061	33	6	necessary	necessary	ADJ
ejpam-7061	33	7	and	and	CCONJ
ejpam-7061	33	8	sufficient	sufficient	ADJ
ejpam-7061	33	9	condition	condition	NOUN
ejpam-7061	33	10	when	when	SCONJ
ejpam-7061	33	11	l(s	l(s	PROPN
ejpam-7061	33	12	,	,	PUNCT
ejpam-7061	33	13	t	t	PROPN
ejpam-7061	33	14	)	)	PUNCT
ejpam-7061	33	15	=	=	NOUN
ejpam-7061	34	1	ls	ls	ADJ
ejpam-7061	34	2	×lt	×lt	NOUN
ejpam-7061	34	3	in	in	ADP
ejpam-7061	34	4	s	s	NUM
ejpam-7061	34	5	×	×	PROPN
ejpam-7061	34	6	t	t	NOUN
ejpam-7061	34	7	.	.	PUNCT
ejpam-7061	35	1	the	the	DET
ejpam-7061	35	2	principal	principal	NOUN
ejpam-7061	35	3	(	(	PUNCT
ejpam-7061	35	4	two	two	NUM
ejpam-7061	35	5	-	-	PUNCT
ejpam-7061	35	6	sided	sided	ADJ
ejpam-7061	35	7	)	)	PUNCT
ejpam-7061	35	8	ideals	ideal	NOUN
ejpam-7061	35	9	on	on	ADP
ejpam-7061	35	10	the	the	DET
ejpam-7061	35	11	direct	direct	ADJ
ejpam-7061	35	12	product	product	NOUN
ejpam-7061	35	13	of	of	ADP
ejpam-7061	35	14	two	two	NUM
ejpam-7061	35	15	semigroups	semigroup	NOUN
ejpam-7061	35	16	were	be	AUX
ejpam-7061	35	17	considered	consider	VERB
ejpam-7061	35	18	in	in	ADP
ejpam-7061	35	19	the	the	DET
ejpam-7061	35	20	same	same	ADJ
ejpam-7061	35	21	way	way	NOUN
ejpam-7061	36	1	[	[	X
ejpam-7061	36	2	2	2	NUM
ejpam-7061	36	3	]	]	PUNCT
ejpam-7061	36	4	.	.	PUNCT
ejpam-7061	37	1	let	let	VERB
ejpam-7061	37	2	m	m	PRON
ejpam-7061	37	3	,	,	PUNCT
ejpam-7061	37	4	n	n	X
ejpam-7061	37	5	be	be	AUX
ejpam-7061	37	6	nonnegative	nonnegative	ADJ
ejpam-7061	37	7	integers	integer	NOUN
ejpam-7061	37	8	.	.	PUNCT
ejpam-7061	38	1	a	a	DET
ejpam-7061	38	2	subsemigroup	subsemigroup	NOUN
ejpam-7061	38	3	a	a	PRON
ejpam-7061	38	4	of	of	ADP
ejpam-7061	38	5	s	s	PRON
ejpam-7061	38	6	is	be	AUX
ejpam-7061	38	7	called	call	VERB
ejpam-7061	38	8	an	an	DET
ejpam-7061	38	9	(	(	PUNCT
ejpam-7061	38	10	m	m	PROPN
ejpam-7061	38	11	,	,	PUNCT
ejpam-7061	38	12	n)-ideal	n)-ideal	NOUN
ejpam-7061	38	13	of	of	ADP
ejpam-7061	38	14	s	s	PRON
ejpam-7061	38	15	if	if	SCONJ
ejpam-7061	38	16	amsan	amsan	ADJ
ejpam-7061	38	17	⊆	⊆	NUM
ejpam-7061	38	18	a	a	PRON
ejpam-7061	39	1	[	[	X
ejpam-7061	39	2	3	3	NUM
ejpam-7061	39	3	]	]	PUNCT
ejpam-7061	39	4	.	.	PUNCT
ejpam-7061	40	1	here	here	ADV
ejpam-7061	40	2	,	,	PUNCT
ejpam-7061	40	3	a0s	a0s	PROPN
ejpam-7061	40	4	=	=	PUNCT
ejpam-7061	40	5	sa0	sa0	NOUN
ejpam-7061	40	6	=	=	PUNCT
ejpam-7061	40	7	s.	s.	PROPN
ejpam-7061	40	8	this	this	DET
ejpam-7061	40	9	definition	definition	NOUN
ejpam-7061	40	10	is	be	AUX
ejpam-7061	40	11	a	a	DET
ejpam-7061	40	12	generalized	generalized	ADJ
ejpam-7061	40	13	form	form	NOUN
ejpam-7061	40	14	of	of	ADP
ejpam-7061	40	15	left	left	ADJ
ejpam-7061	40	16	ideals	ideal	NOUN
ejpam-7061	40	17	,	,	PUNCT
ejpam-7061	40	18	right	right	ADJ
ejpam-7061	40	19	ideals	ideal	NOUN
ejpam-7061	40	20	,	,	PUNCT
ejpam-7061	40	21	and	and	CCONJ
ejpam-7061	40	22	bi	bi	NOUN
ejpam-7061	40	23	-	-	NOUN
ejpam-7061	40	24	ideals	ideal	NOUN
ejpam-7061	40	25	.	.	PUNCT
ejpam-7061	41	1	for	for	ADP
ejpam-7061	41	2	any	any	DET
ejpam-7061	41	3	element	element	NOUN
ejpam-7061	41	4	a	a	PRON
ejpam-7061	41	5	in	in	ADP
ejpam-7061	41	6	s	s	PROPN
ejpam-7061	41	7	,	,	PUNCT
ejpam-7061	41	8	the	the	DET
ejpam-7061	41	9	smallest	small	ADJ
ejpam-7061	41	10	(	(	PUNCT
ejpam-7061	41	11	m	m	PROPN
ejpam-7061	41	12	,	,	PUNCT
ejpam-7061	41	13	n)-ideal	n)-ideal	NOUN
ejpam-7061	41	14	of	of	ADP
ejpam-7061	41	15	s	s	AUX
ejpam-7061	41	16	containing	contain	VERB
ejpam-7061	41	17	a	a	PRON
ejpam-7061	41	18	is	be	AUX
ejpam-7061	41	19	denoted	denote	VERB
ejpam-7061	41	20	by	by	ADP
ejpam-7061	41	21	[	[	X
ejpam-7061	41	22	a](m	a](m	NOUN
ejpam-7061	41	23	,	,	PUNCT
ejpam-7061	41	24	n	n	CCONJ
ejpam-7061	41	25	)	)	PUNCT
ejpam-7061	41	26	.	.	PUNCT
ejpam-7061	42	1	luangchaisri	luangchaisri	VERB
ejpam-7061	42	2	and	and	CCONJ
ejpam-7061	42	3	changphas	changpha	VERB
ejpam-7061	43	1	[	[	X
ejpam-7061	43	2	4	4	X
ejpam-7061	43	3	]	]	PUNCT
ejpam-7061	43	4	provided	provide	VERB
ejpam-7061	43	5	necessary	necessary	ADJ
ejpam-7061	43	6	and	and	CCONJ
ejpam-7061	43	7	sufficient	sufficient	ADJ
ejpam-7061	43	8	condition	condition	NOUN
ejpam-7061	43	9	for	for	ADP
ejpam-7061	43	10	[	[	X
ejpam-7061	43	11	s](m	s](m	NOUN
ejpam-7061	43	12	,	,	PUNCT
ejpam-7061	43	13	n	n	CCONJ
ejpam-7061	43	14	)	)	PUNCT
ejpam-7061	43	15	×	×	NOUN
ejpam-7061	44	1	[	[	X
ejpam-7061	44	2	t](m	t](m	NOUN
ejpam-7061	44	3	,	,	PUNCT
ejpam-7061	44	4	n	n	CCONJ
ejpam-7061	44	5	)	)	PUNCT
ejpam-7061	45	1	=	=	SYM
ejpam-7061	46	1	[	[	X
ejpam-7061	46	2	(	(	PUNCT
ejpam-7061	46	3	s	s	PROPN
ejpam-7061	46	4	,	,	PUNCT
ejpam-7061	46	5	t)](m	t)](m	NOUN
ejpam-7061	46	6	,	,	PUNCT
ejpam-7061	46	7	n	n	CCONJ
ejpam-7061	46	8	)	)	PUNCT
ejpam-7061	46	9	.	.	PUNCT
ejpam-7061	47	1	moreover	moreover	ADV
ejpam-7061	47	2	,	,	PUNCT
ejpam-7061	47	3	they	they	PRON
ejpam-7061	47	4	determined	determine	VERB
ejpam-7061	47	5	an	an	DET
ejpam-7061	47	6	equivalence	equivalence	NOUN
ejpam-7061	47	7	class	class	NOUN
ejpam-7061	47	8	on	on	ADP
ejpam-7061	47	9	a	a	DET
ejpam-7061	47	10	semigroup	semigroup	NOUN
ejpam-7061	47	11	s	s	NOUN
ejpam-7061	47	12	by	by	ADP
ejpam-7061	47	13	for	for	ADP
ejpam-7061	47	14	any	any	DET
ejpam-7061	47	15	x	x	SYM
ejpam-7061	47	16	∈	∈	PROPN
ejpam-7061	47	17	s	s	NOUN
ejpam-7061	47	18	,	,	PUNCT
ejpam-7061	47	19	j(m	j(m	PROPN
ejpam-7061	47	20	,	,	PUNCT
ejpam-7061	47	21	n),x	n),x	PROPN
ejpam-7061	47	22	=	=	PUNCT
ejpam-7061	47	23	{	{	PUNCT
ejpam-7061	47	24	y	y	PROPN
ejpam-7061	47	25	∈	∈	PROPN
ejpam-7061	47	26	s	s	VERB
ejpam-7061	47	27	|	|	ADV
ejpam-7061	47	28	[	[	X
ejpam-7061	47	29	x](m	x](m	NOUN
ejpam-7061	47	30	,	,	PUNCT
ejpam-7061	47	31	n	n	CCONJ
ejpam-7061	47	32	)	)	PUNCT
ejpam-7061	47	33	=	=	NOUN
ejpam-7061	48	1	[	[	X
ejpam-7061	48	2	y](m	y](m	NOUN
ejpam-7061	48	3	,	,	PUNCT
ejpam-7061	48	4	n	n	CCONJ
ejpam-7061	48	5	)	)	PUNCT
ejpam-7061	48	6	}	}	PUNCT
ejpam-7061	48	7	.	.	PUNCT
ejpam-7061	49	1	then	then	ADV
ejpam-7061	49	2	they	they	PRON
ejpam-7061	49	3	provided	provide	VERB
ejpam-7061	49	4	the	the	DET
ejpam-7061	49	5	conditions	condition	NOUN
ejpam-7061	49	6	for	for	ADP
ejpam-7061	49	7	j(m	j(m	PROPN
ejpam-7061	49	8	,	,	PUNCT
ejpam-7061	49	9	n),a	n),a	PROPN
ejpam-7061	49	10	×	×	PROPN
ejpam-7061	49	11	j(m	j(m	PROPN
ejpam-7061	49	12	,	,	PUNCT
ejpam-7061	49	13	n),b	n),b	PROPN
ejpam-7061	49	14	=	=	SYM
ejpam-7061	49	15	j(m	j(m	PROPN
ejpam-7061	49	16	,	,	PUNCT
ejpam-7061	49	17	n),(a	n),(a	ADJ
ejpam-7061	49	18	,	,	PUNCT
ejpam-7061	49	19	b	b	NOUN
ejpam-7061	49	20	)	)	PUNCT
ejpam-7061	49	21	.	.	PUNCT
ejpam-7061	50	1	a	a	DET
ejpam-7061	50	2	nonempty	nonempty	NOUN
ejpam-7061	50	3	subset	subset	VERB
ejpam-7061	50	4	q	q	NOUN
ejpam-7061	50	5	of	of	ADP
ejpam-7061	50	6	s	s	PROPN
ejpam-7061	50	7	is	be	AUX
ejpam-7061	50	8	called	call	VERB
ejpam-7061	50	9	a	a	DET
ejpam-7061	50	10	quasi	quasi	NOUN
ejpam-7061	50	11	-	-	NOUN
ejpam-7061	50	12	ideal	ideal	ADJ
ejpam-7061	50	13	of	of	ADP
ejpam-7061	50	14	s	s	PRON
ejpam-7061	50	15	if	if	SCONJ
ejpam-7061	50	16	qs	qs	ADP
ejpam-7061	50	17	∩	∩	NOUN
ejpam-7061	50	18	sq	sq	PROPN
ejpam-7061	50	19	⊆	⊆	NUM
ejpam-7061	50	20	q.	q.	NOUN
ejpam-7061	50	21	the	the	DET
ejpam-7061	50	22	concept	concept	NOUN
ejpam-7061	50	23	of	of	ADP
ejpam-7061	50	24	quasi	quasi	NOUN
ejpam-7061	50	25	-	-	NOUN
ejpam-7061	50	26	ideals	ideal	NOUN
ejpam-7061	50	27	was	be	AUX
ejpam-7061	50	28	introduced	introduce	VERB
ejpam-7061	50	29	by	by	ADP
ejpam-7061	50	30	steinfeld	steinfeld	PROPN
ejpam-7061	51	1	[	[	X
ejpam-7061	51	2	5	5	NUM
ejpam-7061	51	3	]	]	PUNCT
ejpam-7061	51	4	.	.	PUNCT
ejpam-7061	52	1	for	for	SCONJ
ejpam-7061	52	2	each	each	DET
ejpam-7061	52	3	a	a	DET
ejpam-7061	52	4	∈	∈	PROPN
ejpam-7061	52	5	s	s	NOUN
ejpam-7061	52	6	,	,	PUNCT
ejpam-7061	52	7	the	the	DET
ejpam-7061	52	8	principal	principal	ADJ
ejpam-7061	52	9	quasi	quasi	NOUN
ejpam-7061	52	10	-	-	NOUN
ejpam-7061	52	11	ideal	ideal	ADJ
ejpam-7061	52	12	of	of	ADP
ejpam-7061	52	13	s	s	AUX
ejpam-7061	52	14	generated	generate	VERB
ejpam-7061	52	15	by	by	ADP
ejpam-7061	52	16	a	a	PRON
ejpam-7061	52	17	is	be	AUX
ejpam-7061	52	18	denoted	denote	VERB
ejpam-7061	52	19	by	by	ADP
ejpam-7061	52	20	q(a	q(a	NOUN
ejpam-7061	52	21	)	)	PUNCT
ejpam-7061	52	22	.	.	PUNCT
ejpam-7061	53	1	luangchaisri	luangchaisri	VERB
ejpam-7061	53	2	et	et	PROPN
ejpam-7061	53	3	al	al	PROPN
ejpam-7061	53	4	.	.	PUNCT
ejpam-7061	54	1	[	[	X
ejpam-7061	54	2	6	6	NUM
ejpam-7061	54	3	]	]	PUNCT
ejpam-7061	54	4	considered	consider	VERB
ejpam-7061	54	5	necessary	necessary	ADJ
ejpam-7061	54	6	and	and	CCONJ
ejpam-7061	54	7	sufficient	sufficient	ADJ
ejpam-7061	54	8	condition	condition	NOUN
ejpam-7061	54	9	when	when	SCONJ
ejpam-7061	54	10	q(s)×q(t	q(s)×q(t	NOUN
ejpam-7061	54	11	)	)	PUNCT
ejpam-7061	54	12	=	=	SYM
ejpam-7061	54	13	q((s	q((s	PROPN
ejpam-7061	54	14	,	,	PUNCT
ejpam-7061	54	15	t	t	PROPN
ejpam-7061	54	16	)	)	PUNCT
ejpam-7061	54	17	)	)	PUNCT
ejpam-7061	54	18	.	.	PUNCT
ejpam-7061	55	1	moreover	moreover	ADV
ejpam-7061	55	2	,	,	PUNCT
ejpam-7061	55	3	they	they	PRON
ejpam-7061	55	4	characterized	characterize	VERB
ejpam-7061	55	5	when	when	SCONJ
ejpam-7061	55	6	the	the	DET
ejpam-7061	55	7	cartesian	cartesian	ADJ
ejpam-7061	55	8	product	product	NOUN
ejpam-7061	55	9	of	of	ADP
ejpam-7061	55	10	h	h	NOUN
ejpam-7061	55	11	-	-	PUNCT
ejpam-7061	55	12	classes	class	NOUN
ejpam-7061	55	13	hs	hs	NOUN
ejpam-7061	55	14	×ht	×ht	PROPN
ejpam-7061	55	15	is	be	AUX
ejpam-7061	55	16	an	an	DET
ejpam-7061	55	17	h	h	NOUN
ejpam-7061	55	18	-	-	PUNCT
ejpam-7061	55	19	class	class	NOUN
ejpam-7061	55	20	of	of	ADP
ejpam-7061	55	21	s	s	PRON
ejpam-7061	55	22	×	×	PROPN
ejpam-7061	55	23	t	t	NOUN
ejpam-7061	55	24	.	.	PUNCT
ejpam-7061	56	1	according	accord	VERB
ejpam-7061	56	2	to	to	ADP
ejpam-7061	56	3	the	the	DET
ejpam-7061	56	4	above	above	ADJ
ejpam-7061	56	5	examples	example	NOUN
ejpam-7061	56	6	,	,	PUNCT
ejpam-7061	56	7	we	we	PRON
ejpam-7061	56	8	can	can	AUX
ejpam-7061	56	9	observe	observe	VERB
ejpam-7061	56	10	the	the	DET
ejpam-7061	56	11	research	research	NOUN
ejpam-7061	56	12	line	line	NOUN
ejpam-7061	56	13	to	to	PART
ejpam-7061	56	14	study	study	VERB
ejpam-7061	56	15	various	various	ADJ
ejpam-7061	56	16	kinds	kind	NOUN
ejpam-7061	56	17	of	of	ADP
ejpam-7061	56	18	ideals	ideal	NOUN
ejpam-7061	56	19	and	and	CCONJ
ejpam-7061	56	20	equivalence	equivalence	NOUN
ejpam-7061	56	21	relations	relation	NOUN
ejpam-7061	56	22	on	on	ADP
ejpam-7061	56	23	the	the	DET
ejpam-7061	56	24	direct	direct	ADJ
ejpam-7061	56	25	product	product	NOUN
ejpam-7061	56	26	of	of	ADP
ejpam-7061	56	27	two	two	NUM
ejpam-7061	56	28	semigroups	semigroup	NOUN
ejpam-7061	56	29	.	.	PUNCT
ejpam-7061	57	1	in	in	ADP
ejpam-7061	57	2	this	this	DET
ejpam-7061	57	3	paper	paper	NOUN
ejpam-7061	57	4	,	,	PUNCT
ejpam-7061	57	5	we	we	PRON
ejpam-7061	57	6	consider	consider	VERB
ejpam-7061	57	7	these	these	DET
ejpam-7061	57	8	concepts	concept	NOUN
ejpam-7061	57	9	and	and	CCONJ
ejpam-7061	57	10	extend	extend	VERB
ejpam-7061	57	11	to	to	ADP
ejpam-7061	57	12	the	the	DET
ejpam-7061	57	13	finite	finite	ADJ
ejpam-7061	57	14	direct	direct	ADJ
ejpam-7061	57	15	product	product	NOUN
ejpam-7061	57	16	of	of	ADP
ejpam-7061	57	17	semigroups	semigroup	NOUN
ejpam-7061	57	18	.	.	PUNCT
ejpam-7061	58	1	the	the	DET
ejpam-7061	58	2	principal	principal	NOUN
ejpam-7061	58	3	left	leave	VERB
ejpam-7061	58	4	ideals	ideal	NOUN
ejpam-7061	58	5	and	and	CCONJ
ejpam-7061	58	6	l	l	NOUN
ejpam-7061	58	7	-	-	PUNCT
ejpam-7061	58	8	classes	class	NOUN
ejpam-7061	58	9	are	be	AUX
ejpam-7061	58	10	investigated	investigate	VERB
ejpam-7061	58	11	.	.	PUNCT
ejpam-7061	59	1	moreover	moreover	ADV
ejpam-7061	59	2	,	,	PUNCT
ejpam-7061	59	3	we	we	PRON
ejpam-7061	59	4	give	give	VERB
ejpam-7061	59	5	an	an	DET
ejpam-7061	59	6	example	example	NOUN
ejpam-7061	59	7	to	to	PART
ejpam-7061	59	8	show	show	VERB
ejpam-7061	59	9	that	that	SCONJ
ejpam-7061	59	10	the	the	DET
ejpam-7061	59	11	cartesian	cartesian	ADJ
ejpam-7061	59	12	product	product	NOUN
ejpam-7061	59	13	of	of	ADP
ejpam-7061	59	14	principal	principal	NOUN
ejpam-7061	59	15	left	leave	VERB
ejpam-7061	59	16	ideals	ideal	NOUN
ejpam-7061	59	17	need	need	AUX
ejpam-7061	59	18	not	not	PART
ejpam-7061	59	19	be	be	AUX
ejpam-7061	59	20	the	the	DET
ejpam-7061	59	21	principal	principal	NOUN
ejpam-7061	59	22	left	leave	VERB
ejpam-7061	59	23	ideal	ideal	ADJ
ejpam-7061	59	24	.	.	PUNCT
ejpam-7061	60	1	in	in	ADP
ejpam-7061	60	2	addition	addition	NOUN
ejpam-7061	60	3	,	,	PUNCT
ejpam-7061	60	4	an	an	DET
ejpam-7061	60	5	example	example	NOUN
ejpam-7061	60	6	for	for	ADP
ejpam-7061	60	7	l	l	NOUN
ejpam-7061	60	8	-	-	PUNCT
ejpam-7061	60	9	classes	class	NOUN
ejpam-7061	60	10	is	be	AUX
ejpam-7061	60	11	also	also	ADV
ejpam-7061	60	12	provided	provide	VERB
ejpam-7061	60	13	.	.	PUNCT
ejpam-7061	61	1	2	2	X
ejpam-7061	61	2	.	.	X
ejpam-7061	61	3	main	main	ADJ
ejpam-7061	61	4	results	result	NOUN
ejpam-7061	61	5	let	let	VERB
ejpam-7061	61	6	{	{	PUNCT
ejpam-7061	61	7	si	si	INTJ
ejpam-7061	62	1	|	|	ADV
ejpam-7061	62	2	i	i	PRON
ejpam-7061	62	3	∈	∈	PROPN
ejpam-7061	63	1	i	i	PRON
ejpam-7061	63	2	}	}	PUNCT
ejpam-7061	63	3	be	be	VERB
ejpam-7061	63	4	a	a	DET
ejpam-7061	63	5	family	family	NOUN
ejpam-7061	63	6	of	of	ADP
ejpam-7061	63	7	semigroups	semigroup	NOUN
ejpam-7061	63	8	indexed	index	VERB
ejpam-7061	63	9	by	by	ADP
ejpam-7061	63	10	the	the	DET
ejpam-7061	63	11	set	set	NOUN
ejpam-7061	63	12	i	i	PRON
ejpam-7061	63	13	=	=	PUNCT
ejpam-7061	63	14	{	{	PUNCT
ejpam-7061	63	15	1	1	NUM
ejpam-7061	63	16	,	,	PUNCT
ejpam-7061	63	17	2	2	NUM
ejpam-7061	63	18	,	,	PUNCT
ejpam-7061	63	19	.	.	PUNCT
ejpam-7061	63	20	.	.	PUNCT
ejpam-7061	64	1	.	.	PUNCT
ejpam-7061	64	2	,	,	PUNCT
ejpam-7061	64	3	n	n	CCONJ
ejpam-7061	64	4	}	}	PUNCT
ejpam-7061	64	5	.	.	PUNCT
ejpam-7061	65	1	then	then	ADV
ejpam-7061	65	2	s1	s1	PROPN
ejpam-7061	65	3	×	×	PROPN
ejpam-7061	65	4	s2	s2	NOUN
ejpam-7061	65	5	×	×	NOUN
ejpam-7061	65	6	·	·	PUNCT
ejpam-7061	65	7	·	·	PUNCT
ejpam-7061	65	8	·	·	PUNCT
ejpam-7061	66	1	×	×	NOUN
ejpam-7061	66	2	sn	sn	PROPN
ejpam-7061	66	3	becomes	become	VERB
ejpam-7061	66	4	a	a	DET
ejpam-7061	66	5	semigroup	semigroup	NOUN
ejpam-7061	66	6	under	under	ADP
ejpam-7061	66	7	a	a	DET
ejpam-7061	66	8	componentwise	componentwise	NOUN
ejpam-7061	66	9	multiplication	multiplication	NOUN
ejpam-7061	66	10	,	,	PUNCT
ejpam-7061	66	11	which	which	PRON
ejpam-7061	66	12	is	be	AUX
ejpam-7061	66	13	defined	define	VERB
ejpam-7061	66	14	by	by	ADP
ejpam-7061	66	15	(	(	PUNCT
ejpam-7061	66	16	s1	s1	NOUN
ejpam-7061	66	17	,	,	PUNCT
ejpam-7061	66	18	s2	s2	NOUN
ejpam-7061	66	19	,	,	PUNCT
ejpam-7061	66	20	.	.	PUNCT
ejpam-7061	66	21	.	.	PUNCT
ejpam-7061	67	1	.	.	PUNCT
ejpam-7061	68	1	,	,	PUNCT
ejpam-7061	68	2	sn)(s	sn)(s	NOUN
ejpam-7061	68	3	′	′	NUM
ejpam-7061	68	4	1	1	NUM
ejpam-7061	68	5	,	,	PUNCT
ejpam-7061	68	6	s	s	VERB
ejpam-7061	68	7	′	′	NOUN
ejpam-7061	68	8	2	2	NUM
ejpam-7061	68	9	,	,	PUNCT
ejpam-7061	68	10	.	.	PUNCT
ejpam-7061	68	11	.	.	PUNCT
ejpam-7061	69	1	.	.	PUNCT
ejpam-7061	70	1	,	,	PUNCT
ejpam-7061	70	2	s	s	VERB
ejpam-7061	70	3	′	′	NUM
ejpam-7061	70	4	n	n	CCONJ
ejpam-7061	70	5	)	)	PUNCT
ejpam-7061	70	6	=	=	SYM
ejpam-7061	71	1	(	(	PUNCT
ejpam-7061	71	2	s1s	s1s	PROPN
ejpam-7061	71	3	′	′	NUM
ejpam-7061	71	4	1	1	NUM
ejpam-7061	71	5	,	,	PUNCT
ejpam-7061	71	6	s2s	s2s	PROPN
ejpam-7061	71	7	′	′	NUM
ejpam-7061	71	8	2	2	NUM
ejpam-7061	71	9	,	,	PUNCT
ejpam-7061	71	10	.	.	PUNCT
ejpam-7061	71	11	.	.	PUNCT
ejpam-7061	71	12	.	.	PUNCT
ejpam-7061	72	1	,	,	PUNCT
ejpam-7061	72	2	sns	sns	PROPN
ejpam-7061	72	3	′	′	PROPN
ejpam-7061	72	4	n	n	CCONJ
ejpam-7061	72	5	)	)	PUNCT
ejpam-7061	72	6	for	for	ADP
ejpam-7061	72	7	all	all	PRON
ejpam-7061	72	8	(	(	PUNCT
ejpam-7061	72	9	s1	s1	NOUN
ejpam-7061	72	10	,	,	PUNCT
ejpam-7061	72	11	s2	s2	NOUN
ejpam-7061	72	12	,	,	PUNCT
ejpam-7061	72	13	.	.	PUNCT
ejpam-7061	72	14	.	.	PUNCT
ejpam-7061	73	1	.	.	PUNCT
ejpam-7061	74	1	,	,	PUNCT
ejpam-7061	74	2	sn	sn	PROPN
ejpam-7061	74	3	)	)	PUNCT
ejpam-7061	74	4	,	,	PUNCT
ejpam-7061	74	5	(	(	PUNCT
ejpam-7061	74	6	s	s	AUX
ejpam-7061	74	7	′	′	NUM
ejpam-7061	74	8	1	1	NUM
ejpam-7061	74	9	,	,	PUNCT
ejpam-7061	74	10	s	s	VERB
ejpam-7061	74	11	′	′	NOUN
ejpam-7061	74	12	2	2	NUM
ejpam-7061	74	13	,	,	PUNCT
ejpam-7061	74	14	.	.	PUNCT
ejpam-7061	74	15	.	.	PUNCT
ejpam-7061	75	1	.	.	PUNCT
ejpam-7061	76	1	,	,	PUNCT
ejpam-7061	76	2	s	s	VERB
ejpam-7061	76	3	′	′	NUM
ejpam-7061	76	4	n	n	CCONJ
ejpam-7061	76	5	)	)	PUNCT
ejpam-7061	76	6	∈	∈	PROPN
ejpam-7061	76	7	s1	s1	PROPN
ejpam-7061	76	8	×	×	PROPN
ejpam-7061	76	9	s2	s2	NOUN
ejpam-7061	76	10	×	×	NOUN
ejpam-7061	76	11	·	·	PUNCT
ejpam-7061	76	12	·	·	PUNCT
ejpam-7061	76	13	·	·	PUNCT
ejpam-7061	77	1	×	×	NOUN
ejpam-7061	77	2	sn	sn	PROPN
ejpam-7061	77	3	.	.	PUNCT
ejpam-7061	78	1	this	this	DET
ejpam-7061	78	2	semigroup	semigroup	NOUN
ejpam-7061	78	3	is	be	AUX
ejpam-7061	78	4	called	call	VERB
ejpam-7061	78	5	the	the	DET
ejpam-7061	78	6	direct	direct	ADJ
ejpam-7061	78	7	product	product	NOUN
ejpam-7061	78	8	of	of	ADP
ejpam-7061	78	9	{	{	PUNCT
ejpam-7061	78	10	si	si	INTJ
ejpam-7061	79	1	|	|	ADV
ejpam-7061	79	2	i	i	PRON
ejpam-7061	79	3	∈	∈	PROPN
ejpam-7061	79	4	i	i	PRON
ejpam-7061	79	5	}	}	PUNCT
ejpam-7061	79	6	.	.	PUNCT
ejpam-7061	80	1	note	note	VERB
ejpam-7061	80	2	that	that	SCONJ
ejpam-7061	80	3	the	the	DET
ejpam-7061	80	4	direct	direct	ADJ
ejpam-7061	80	5	product	product	NOUN
ejpam-7061	80	6	of	of	ADP
ejpam-7061	80	7	s1	s1	PROPN
ejpam-7061	80	8	is	be	AUX
ejpam-7061	80	9	trivially	trivially	ADV
ejpam-7061	80	10	the	the	DET
ejpam-7061	80	11	semigroup	semigroup	PROPN
ejpam-7061	80	12	s1	s1	PROPN
ejpam-7061	80	13	.	.	PUNCT
ejpam-7061	81	1	therefore	therefore	ADV
ejpam-7061	81	2	,	,	PUNCT
ejpam-7061	81	3	we	we	PRON
ejpam-7061	81	4	assume	assume	VERB
ejpam-7061	81	5	throughout	throughout	ADP
ejpam-7061	81	6	that	that	SCONJ
ejpam-7061	81	7	the	the	DET
ejpam-7061	81	8	indexed	index	VERB
ejpam-7061	81	9	set	set	NOUN
ejpam-7061	81	10	i	i	PRON
ejpam-7061	81	11	is	be	AUX
ejpam-7061	81	12	not	not	PART
ejpam-7061	81	13	a	a	DET
ejpam-7061	81	14	singleton	singleton	NOUN
ejpam-7061	81	15	.	.	PUNCT
ejpam-7061	82	1	if	if	SCONJ
ejpam-7061	82	2	li	li	PROPN
ejpam-7061	82	3	is	be	AUX
ejpam-7061	82	4	a	a	DET
ejpam-7061	82	5	left	left	ADJ
ejpam-7061	82	6	ideal	ideal	NOUN
ejpam-7061	82	7	of	of	ADP
ejpam-7061	82	8	si	si	PROPN
ejpam-7061	82	9	for	for	ADP
ejpam-7061	82	10	all	all	PRON
ejpam-7061	82	11	i	i	PRON
ejpam-7061	82	12	∈	∈	PROPN
ejpam-7061	83	1	i	i	PRON
ejpam-7061	83	2	,	,	PUNCT
ejpam-7061	83	3	then	then	ADV
ejpam-7061	83	4	the	the	DET
ejpam-7061	83	5	cartesian	cartesian	ADJ
ejpam-7061	83	6	product	product	NOUN
ejpam-7061	83	7	l1	l1	PROPN
ejpam-7061	83	8	×l2	×l2	PROPN
ejpam-7061	83	9	×	×	PROPN
ejpam-7061	83	10	·	·	PUNCT
ejpam-7061	83	11	·	·	PUNCT
ejpam-7061	83	12	·	·	PUNCT
ejpam-7061	84	1	×ln	×ln	ADV
ejpam-7061	84	2	is	be	AUX
ejpam-7061	84	3	a	a	DET
ejpam-7061	84	4	left	left	ADJ
ejpam-7061	84	5	ideal	ideal	NOUN
ejpam-7061	84	6	of	of	ADP
ejpam-7061	84	7	s1	s1	PROPN
ejpam-7061	84	8	×	×	PROPN
ejpam-7061	84	9	s2	s2	NOUN
ejpam-7061	84	10	×	×	NOUN
ejpam-7061	84	11	·	·	PUNCT
ejpam-7061	84	12	·	·	PUNCT
ejpam-7061	84	13	·	·	PUNCT
ejpam-7061	85	1	×	×	NOUN
ejpam-7061	85	2	sn	sn	INTJ
ejpam-7061	85	3	.	.	PUNCT
ejpam-7061	86	1	however	however	ADV
ejpam-7061	86	2	,	,	PUNCT
ejpam-7061	86	3	the	the	DET
ejpam-7061	86	4	cartesian	cartesian	ADJ
ejpam-7061	86	5	product	product	NOUN
ejpam-7061	86	6	of	of	ADP
ejpam-7061	86	7	principal	principal	NOUN
ejpam-7061	86	8	left	leave	VERB
ejpam-7061	86	9	ideals	ideal	NOUN
ejpam-7061	86	10	need	need	AUX
ejpam-7061	86	11	not	not	PART
ejpam-7061	86	12	be	be	AUX
ejpam-7061	86	13	the	the	DET
ejpam-7061	86	14	principal	principal	NOUN
ejpam-7061	86	15	left	leave	VERB
ejpam-7061	86	16	ideal	ideal	ADJ
ejpam-7061	86	17	.	.	PUNCT
ejpam-7061	87	1	this	this	PRON
ejpam-7061	87	2	is	be	AUX
ejpam-7061	87	3	clarified	clarify	VERB
ejpam-7061	87	4	by	by	ADP
ejpam-7061	87	5	the	the	DET
ejpam-7061	87	6	following	follow	VERB
ejpam-7061	87	7	example	example	NOUN
ejpam-7061	87	8	:	:	PUNCT
ejpam-7061	87	9	example	example	NOUN
ejpam-7061	88	1	1	1	X
ejpam-7061	88	2	.	.	PUNCT
ejpam-7061	89	1	let	let	VERB
ejpam-7061	89	2	s	s	VERB
ejpam-7061	89	3	=	=	NOUN
ejpam-7061	89	4	{	{	PUNCT
ejpam-7061	89	5	s1	s1	NOUN
ejpam-7061	89	6	,	,	PUNCT
ejpam-7061	89	7	s2	s2	PROPN
ejpam-7061	89	8	,	,	PUNCT
ejpam-7061	89	9	s3	s3	PROPN
ejpam-7061	89	10	,	,	PUNCT
ejpam-7061	89	11	s4	s4	PROPN
ejpam-7061	89	12	}	}	PUNCT
ejpam-7061	89	13	be	be	AUX
ejpam-7061	89	14	a	a	DET
ejpam-7061	89	15	semigroup	semigroup	NOUN
ejpam-7061	89	16	under	under	ADP
ejpam-7061	89	17	the	the	DET
ejpam-7061	89	18	following	follow	VERB
ejpam-7061	89	19	binary	binary	ADJ
ejpam-7061	89	20	operation	operation	NOUN
ejpam-7061	89	21	:	:	PUNCT
ejpam-7061	89	22	∗	∗	NOUN
ejpam-7061	89	23	s1	s1	PROPN
ejpam-7061	89	24	s2	s2	PROPN
ejpam-7061	89	25	s3	s3	PROPN
ejpam-7061	89	26	s4	s4	PROPN
ejpam-7061	89	27	s1	s1	PROPN
ejpam-7061	89	28	s1	s1	PROPN
ejpam-7061	89	29	s1	s1	PROPN
ejpam-7061	89	30	s1	s1	PROPN
ejpam-7061	89	31	s1	s1	PROPN
ejpam-7061	89	32	s2	s2	PROPN
ejpam-7061	89	33	s1	s1	PROPN
ejpam-7061	89	34	s2	s2	PROPN
ejpam-7061	89	35	s2	s2	PROPN
ejpam-7061	89	36	s4	s4	PROPN
ejpam-7061	89	37	s3	s3	PROPN
ejpam-7061	89	38	s1	s1	PROPN
ejpam-7061	89	39	s2	s2	PROPN
ejpam-7061	89	40	s2	s2	NOUN
ejpam-7061	89	41	s4	s4	PROPN
ejpam-7061	89	42	s4	s4	PROPN
ejpam-7061	89	43	s1	s1	PROPN
ejpam-7061	89	44	s4	s4	PROPN
ejpam-7061	89	45	s4	s4	PROPN
ejpam-7061	89	46	s2	s2	PROPN
ejpam-7061	89	47	p.	p.	PROPN
ejpam-7061	89	48	luangchaisri	luangchaisri	PROPN
ejpam-7061	89	49	,	,	PUNCT
ejpam-7061	90	1	o.	o.	PROPN
ejpam-7061	90	2	pankoon	pankoon	NOUN
ejpam-7061	90	3	,	,	PUNCT
ejpam-7061	90	4	t.	t.	PROPN
ejpam-7061	90	5	changphas	changphas	PROPN
ejpam-7061	90	6	/	/	SYM
ejpam-7061	90	7	eur	eur	PROPN
ejpam-7061	90	8	.	.	PUNCT
ejpam-7061	91	1	j.	j.	PROPN
ejpam-7061	91	2	pure	pure	PROPN
ejpam-7061	91	3	appl	appl	PROPN
ejpam-7061	91	4	.	.	PROPN
ejpam-7061	91	5	math	math	PROPN
ejpam-7061	91	6	,	,	PUNCT
ejpam-7061	91	7	18	18	NUM
ejpam-7061	91	8	(	(	PUNCT
ejpam-7061	91	9	4	4	NUM
ejpam-7061	91	10	)	)	PUNCT
ejpam-7061	91	11	(	(	PUNCT
ejpam-7061	91	12	2025	2025	NUM
ejpam-7061	91	13	)	)	PUNCT
ejpam-7061	91	14	,	,	PUNCT
ejpam-7061	91	15	7061	7061	NUM
ejpam-7061	91	16	3	3	NUM
ejpam-7061	91	17	of	of	ADP
ejpam-7061	91	18	10	10	NUM
ejpam-7061	91	19	this	this	DET
ejpam-7061	91	20	semigroup	semigroup	NOUN
ejpam-7061	91	21	is	be	AUX
ejpam-7061	91	22	applied	apply	VERB
ejpam-7061	91	23	from	from	ADP
ejpam-7061	91	24	[	[	X
ejpam-7061	91	25	7	7	NUM
ejpam-7061	91	26	]	]	PUNCT
ejpam-7061	91	27	.	.	PUNCT
ejpam-7061	92	1	since	since	SCONJ
ejpam-7061	92	2	(	(	PUNCT
ejpam-7061	92	3	s3	s3	PROPN
ejpam-7061	92	4	,	,	PUNCT
ejpam-7061	92	5	s1	s1	NOUN
ejpam-7061	92	6	)	)	PUNCT
ejpam-7061	92	7	∈	∈	PROPN
ejpam-7061	92	8	l(s3)×l(s3	l(s3)×l(s3	X
ejpam-7061	92	9	)	)	PUNCT
ejpam-7061	92	10	and	and	CCONJ
ejpam-7061	92	11	(	(	PUNCT
ejpam-7061	92	12	s3	s3	PROPN
ejpam-7061	92	13	,	,	PUNCT
ejpam-7061	92	14	s1	s1	NOUN
ejpam-7061	92	15	)	)	PUNCT
ejpam-7061	92	16	/∈	/∈	PUNCT
ejpam-7061	93	1	l((s3	l((s3	NOUN
ejpam-7061	93	2	,	,	PUNCT
ejpam-7061	93	3	s3	s3	PROPN
ejpam-7061	93	4	)	)	PUNCT
ejpam-7061	93	5	)	)	PUNCT
ejpam-7061	93	6	,	,	PUNCT
ejpam-7061	93	7	this	this	PRON
ejpam-7061	93	8	shows	show	VERB
ejpam-7061	93	9	that	that	SCONJ
ejpam-7061	93	10	l(s3)×l(s3	l(s3)×l(s3	X
ejpam-7061	93	11	)	)	PUNCT
ejpam-7061	93	12	̸=	̸=	PROPN
ejpam-7061	93	13	l((s3	l((s3	NOUN
ejpam-7061	93	14	,	,	PUNCT
ejpam-7061	93	15	s3	s3	PROPN
ejpam-7061	93	16	)	)	PUNCT
ejpam-7061	93	17	)	)	PUNCT
ejpam-7061	93	18	.	.	PUNCT
ejpam-7061	94	1	in	in	ADP
ejpam-7061	94	2	addition	addition	NOUN
ejpam-7061	94	3	,	,	PUNCT
ejpam-7061	94	4	since	since	SCONJ
ejpam-7061	94	5	(	(	PUNCT
ejpam-7061	94	6	s3	s3	PROPN
ejpam-7061	94	7	,	,	PUNCT
ejpam-7061	94	8	s3	s3	PROPN
ejpam-7061	94	9	)	)	PUNCT
ejpam-7061	94	10	∈	∈	PROPN
ejpam-7061	94	11	l(s3)×l(s3	l(s3)×l(s3	X
ejpam-7061	94	12	)	)	PUNCT
ejpam-7061	94	13	and	and	CCONJ
ejpam-7061	94	14	(	(	PUNCT
ejpam-7061	94	15	s3	s3	PROPN
ejpam-7061	94	16	,	,	PUNCT
ejpam-7061	94	17	s3	s3	PROPN
ejpam-7061	94	18	)	)	PUNCT
ejpam-7061	94	19	/∈	/∈	PUNCT
ejpam-7061	95	1	l((x	l((x	ADJ
ejpam-7061	95	2	,	,	PUNCT
ejpam-7061	95	3	y	y	PROPN
ejpam-7061	95	4	)	)	PUNCT
ejpam-7061	95	5	)	)	PUNCT
ejpam-7061	95	6	for	for	ADP
ejpam-7061	95	7	all	all	PRON
ejpam-7061	95	8	(	(	PUNCT
ejpam-7061	95	9	x	x	NOUN
ejpam-7061	95	10	,	,	PUNCT
ejpam-7061	95	11	y	y	NOUN
ejpam-7061	95	12	)	)	PUNCT
ejpam-7061	95	13	∈	∈	PROPN
ejpam-7061	95	14	s×s\{(s3	s×s\{(s3	PROPN
ejpam-7061	95	15	,	,	PUNCT
ejpam-7061	95	16	s3	s3	PROPN
ejpam-7061	95	17	)	)	PUNCT
ejpam-7061	95	18	}	}	PUNCT
ejpam-7061	95	19	,	,	PUNCT
ejpam-7061	95	20	it	it	PRON
ejpam-7061	95	21	follows	follow	VERB
ejpam-7061	95	22	that	that	SCONJ
ejpam-7061	95	23	l(s3)×l(s3	l(s3)×l(s3	X
ejpam-7061	95	24	)	)	PUNCT
ejpam-7061	95	25	̸=	̸=	PROPN
ejpam-7061	95	26	l((x	l((x	NOUN
ejpam-7061	95	27	,	,	PUNCT
ejpam-7061	95	28	y	y	PROPN
ejpam-7061	95	29	)	)	PUNCT
ejpam-7061	95	30	for	for	ADP
ejpam-7061	95	31	all	all	DET
ejpam-7061	95	32	(	(	PUNCT
ejpam-7061	95	33	x	x	NOUN
ejpam-7061	95	34	,	,	PUNCT
ejpam-7061	95	35	y	y	PROPN
ejpam-7061	95	36	)	)	PUNCT
ejpam-7061	95	37	∈	∈	PROPN
ejpam-7061	95	38	s	s	PART
ejpam-7061	95	39	×	×	NOUN
ejpam-7061	95	40	s.	s.	PROPN
ejpam-7061	95	41	hence	hence	ADV
ejpam-7061	95	42	,	,	PUNCT
ejpam-7061	95	43	l(s3	l(s3	PROPN
ejpam-7061	95	44	)	)	PUNCT
ejpam-7061	95	45	×	×	NOUN
ejpam-7061	95	46	l(s3	l(s3	NOUN
ejpam-7061	95	47	)	)	PUNCT
ejpam-7061	95	48	is	be	AUX
ejpam-7061	95	49	not	not	PART
ejpam-7061	95	50	a	a	DET
ejpam-7061	95	51	principal	principal	NOUN
ejpam-7061	95	52	left	leave	VERB
ejpam-7061	95	53	ideal	ideal	NOUN
ejpam-7061	95	54	of	of	ADP
ejpam-7061	95	55	a	a	DET
ejpam-7061	95	56	semigroup	semigroup	NOUN
ejpam-7061	95	57	s	s	PART
ejpam-7061	95	58	×	×	NOUN
ejpam-7061	95	59	s.	s.	PROPN
ejpam-7061	95	60	we	we	PRON
ejpam-7061	95	61	begin	begin	VERB
ejpam-7061	95	62	with	with	ADP
ejpam-7061	95	63	lemma	lemma	PROPN
ejpam-7061	95	64	1	1	NUM
ejpam-7061	95	65	to	to	PART
ejpam-7061	95	66	mention	mention	VERB
ejpam-7061	95	67	about	about	ADP
ejpam-7061	95	68	the	the	DET
ejpam-7061	95	69	inclusion	inclusion	NOUN
ejpam-7061	95	70	of	of	ADP
ejpam-7061	95	71	l((s1	l((s1	PROPN
ejpam-7061	95	72	,	,	PUNCT
ejpam-7061	95	73	s2	s2	PROPN
ejpam-7061	95	74	,	,	PUNCT
ejpam-7061	95	75	.	.	PUNCT
ejpam-7061	95	76	.	.	PUNCT
ejpam-7061	96	1	.	.	PUNCT
ejpam-7061	97	1	,	,	PUNCT
ejpam-7061	97	2	sn	sn	PROPN
ejpam-7061	97	3	)	)	PUNCT
ejpam-7061	97	4	)	)	PUNCT
ejpam-7061	98	1	and	and	CCONJ
ejpam-7061	98	2	l(s1	l(s1	ADJ
ejpam-7061	98	3	)	)	PUNCT
ejpam-7061	98	4	×	×	NOUN
ejpam-7061	98	5	l(s2	l(s2	NOUN
ejpam-7061	98	6	)	)	PUNCT
ejpam-7061	98	7	×	×	NOUN
ejpam-7061	98	8	·	·	PUNCT
ejpam-7061	98	9	·	·	PUNCT
ejpam-7061	98	10	·	·	PUNCT
ejpam-7061	99	1	×	×	NOUN
ejpam-7061	99	2	l(sn	l(sn	NUM
ejpam-7061	99	3	)	)	PUNCT
ejpam-7061	99	4	.	.	PUNCT
ejpam-7061	100	1	then	then	ADV
ejpam-7061	100	2	,	,	PUNCT
ejpam-7061	100	3	in	in	ADP
ejpam-7061	100	4	theorem	theorem	NOUN
ejpam-7061	100	5	1	1	NUM
ejpam-7061	100	6	,	,	PUNCT
ejpam-7061	100	7	we	we	PRON
ejpam-7061	100	8	give	give	VERB
ejpam-7061	100	9	a	a	DET
ejpam-7061	100	10	necessary	necessary	ADJ
ejpam-7061	100	11	and	and	CCONJ
ejpam-7061	100	12	sufficient	sufficient	ADJ
ejpam-7061	100	13	condition	condition	NOUN
ejpam-7061	100	14	when	when	SCONJ
ejpam-7061	100	15	l((s1	l((s1	PROPN
ejpam-7061	100	16	,	,	PUNCT
ejpam-7061	100	17	s2	s2	PROPN
ejpam-7061	100	18	,	,	PUNCT
ejpam-7061	100	19	.	.	PUNCT
ejpam-7061	100	20	.	.	PUNCT
ejpam-7061	100	21	.	.	PUNCT
ejpam-7061	101	1	,	,	PUNCT
ejpam-7061	101	2	sn	sn	PROPN
ejpam-7061	101	3	)	)	PUNCT
ejpam-7061	101	4	)	)	PUNCT
ejpam-7061	102	1	=	=	PUNCT
ejpam-7061	102	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	102	3	l(s2)×	l(s2)×	X
ejpam-7061	102	4	·	·	PUNCT
ejpam-7061	102	5	·	·	PUNCT
ejpam-7061	102	6	·	·	PUNCT
ejpam-7061	102	7	×	×	NOUN
ejpam-7061	102	8	l(sn	l(sn	NUM
ejpam-7061	102	9	)	)	PUNCT
ejpam-7061	102	10	.	.	PUNCT
ejpam-7061	103	1	lemma	lemma	PROPN
ejpam-7061	103	2	1	1	X
ejpam-7061	103	3	.	.	PUNCT
ejpam-7061	104	1	let	let	VERB
ejpam-7061	104	2	si	si	PART
ejpam-7061	104	3	be	be	AUX
ejpam-7061	104	4	a	a	DET
ejpam-7061	104	5	semigroup	semigroup	NOUN
ejpam-7061	104	6	and	and	CCONJ
ejpam-7061	104	7	let	let	VERB
ejpam-7061	104	8	si	si	PROPN
ejpam-7061	104	9	∈	∈	PROPN
ejpam-7061	104	10	si	si	X
ejpam-7061	104	11	where	where	SCONJ
ejpam-7061	104	12	i	i	PRON
ejpam-7061	104	13	∈	∈	PROPN
ejpam-7061	104	14	{	{	PUNCT
ejpam-7061	104	15	1	1	NUM
ejpam-7061	104	16	,	,	PUNCT
ejpam-7061	104	17	2	2	NUM
ejpam-7061	104	18	,	,	PUNCT
ejpam-7061	104	19	.	.	PUNCT
ejpam-7061	104	20	.	.	PUNCT
ejpam-7061	104	21	.	.	PUNCT
ejpam-7061	105	1	n	n	CCONJ
ejpam-7061	105	2	}	}	PUNCT
ejpam-7061	105	3	.	.	PUNCT
ejpam-7061	106	1	then	then	ADV
ejpam-7061	106	2	l((s1	l((s1	PROPN
ejpam-7061	106	3	,	,	PUNCT
ejpam-7061	106	4	s2	s2	PROPN
ejpam-7061	106	5	,	,	PUNCT
ejpam-7061	106	6	.	.	PUNCT
ejpam-7061	106	7	.	.	PUNCT
ejpam-7061	106	8	.	.	PUNCT
ejpam-7061	106	9	,	,	PUNCT
ejpam-7061	106	10	sn	sn	PROPN
ejpam-7061	106	11	)	)	PUNCT
ejpam-7061	106	12	)	)	PUNCT
ejpam-7061	107	1	⊆	⊆	NUM
ejpam-7061	107	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	107	3	l(s2)×	l(s2)×	X
ejpam-7061	107	4	·	·	PUNCT
ejpam-7061	107	5	·	·	PUNCT
ejpam-7061	107	6	·	·	PUNCT
ejpam-7061	107	7	×	×	ADJ
ejpam-7061	107	8	l(sn	l(sn	ADJ
ejpam-7061	107	9	)	)	PUNCT
ejpam-7061	107	10	proof	proof	NOUN
ejpam-7061	107	11	.	.	PUNCT
ejpam-7061	108	1	let	let	VERB
ejpam-7061	108	2	x	x	SYM
ejpam-7061	108	3	∈	∈	PROPN
ejpam-7061	108	4	l((s1	l((s1	PROPN
ejpam-7061	108	5	,	,	PUNCT
ejpam-7061	108	6	s2	s2	PROPN
ejpam-7061	108	7	,	,	PUNCT
ejpam-7061	108	8	.	.	PUNCT
ejpam-7061	108	9	.	.	PUNCT
ejpam-7061	109	1	.	.	PUNCT
ejpam-7061	110	1	,	,	PUNCT
ejpam-7061	110	2	sn	sn	PROPN
ejpam-7061	110	3	)	)	PUNCT
ejpam-7061	110	4	)	)	PUNCT
ejpam-7061	110	5	.	.	PUNCT
ejpam-7061	111	1	then	then	ADV
ejpam-7061	111	2	x	x	X
ejpam-7061	111	3	∈	∈	PROPN
ejpam-7061	111	4	(	(	PUNCT
ejpam-7061	111	5	s1	s1	NOUN
ejpam-7061	111	6	,	,	PUNCT
ejpam-7061	111	7	s2	s2	NOUN
ejpam-7061	111	8	,	,	PUNCT
ejpam-7061	111	9	.	.	PUNCT
ejpam-7061	111	10	.	.	PUNCT
ejpam-7061	111	11	.	.	PUNCT
ejpam-7061	111	12	,	,	PUNCT
ejpam-7061	111	13	sn	sn	PROPN
ejpam-7061	111	14	)	)	PUNCT
ejpam-7061	111	15	∪	∪	NOUN
ejpam-7061	111	16	(	(	PUNCT
ejpam-7061	111	17	s1s1	s1s1	NOUN
ejpam-7061	111	18	×	×	NOUN
ejpam-7061	111	19	s2s2	s2s2	NOUN
ejpam-7061	111	20	×	×	NOUN
ejpam-7061	111	21	·	·	PUNCT
ejpam-7061	111	22	·	·	PUNCT
ejpam-7061	111	23	·	·	PUNCT
ejpam-7061	111	24	×	×	NOUN
ejpam-7061	111	25	snsn	snsn	NOUN
ejpam-7061	111	26	)	)	PUNCT
ejpam-7061	111	27	⊆	⊆	NUM
ejpam-7061	111	28	(	(	PUNCT
ejpam-7061	111	29	s1	s1	PROPN
ejpam-7061	111	30	∪	∪	ADP
ejpam-7061	111	31	s1s1)×	s1s1)×	NOUN
ejpam-7061	111	32	(	(	PUNCT
ejpam-7061	111	33	s2	s2	PROPN
ejpam-7061	111	34	∪	∪	ADJ
ejpam-7061	111	35	s2s2)×	s2s2)×	NOUN
ejpam-7061	111	36	·	·	PUNCT
ejpam-7061	111	37	·	·	PUNCT
ejpam-7061	111	38	·	·	PUNCT
ejpam-7061	111	39	×	×	NOUN
ejpam-7061	111	40	(	(	PUNCT
ejpam-7061	111	41	sn	sn	PROPN
ejpam-7061	111	42	∪	∪	X
ejpam-7061	111	43	snsn	snsn	PROPN
ejpam-7061	111	44	)	)	PUNCT
ejpam-7061	111	45	=	=	PUNCT
ejpam-7061	111	46	l(s1)×	l(s1)×	NOUN
ejpam-7061	111	47	l(s2)×	l(s2)×	NOUN
ejpam-7061	111	48	·	·	PUNCT
ejpam-7061	111	49	·	·	PUNCT
ejpam-7061	111	50	·	·	PUNCT
ejpam-7061	111	51	×	×	NOUN
ejpam-7061	111	52	l(sn	l(sn	NUM
ejpam-7061	111	53	)	)	PUNCT
ejpam-7061	111	54	.	.	PUNCT
ejpam-7061	112	1	thus	thus	ADV
ejpam-7061	112	2	,	,	PUNCT
ejpam-7061	112	3	l((s1	l((s1	PROPN
ejpam-7061	112	4	,	,	PUNCT
ejpam-7061	112	5	s2	s2	PROPN
ejpam-7061	112	6	,	,	PUNCT
ejpam-7061	112	7	.	.	PUNCT
ejpam-7061	112	8	.	.	PUNCT
ejpam-7061	112	9	.	.	PUNCT
ejpam-7061	113	1	,	,	PUNCT
ejpam-7061	113	2	sn	sn	PROPN
ejpam-7061	113	3	)	)	PUNCT
ejpam-7061	113	4	)	)	PUNCT
ejpam-7061	114	1	⊆	⊆	NUM
ejpam-7061	114	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	114	3	l(s2)×	l(s2)×	X
ejpam-7061	114	4	·	·	PUNCT
ejpam-7061	114	5	·	·	PUNCT
ejpam-7061	114	6	·	·	PUNCT
ejpam-7061	114	7	×	×	NOUN
ejpam-7061	114	8	l(sn	l(sn	NUM
ejpam-7061	114	9	)	)	PUNCT
ejpam-7061	114	10	.	.	PUNCT
ejpam-7061	115	1	theorem	theorem	NOUN
ejpam-7061	115	2	1	1	NUM
ejpam-7061	115	3	.	.	PUNCT
ejpam-7061	116	1	let	let	VERB
ejpam-7061	116	2	si	si	PRON
ejpam-7061	116	3	be	be	AUX
ejpam-7061	116	4	a	a	DET
ejpam-7061	116	5	semigroup	semigroup	NOUN
ejpam-7061	116	6	and	and	CCONJ
ejpam-7061	116	7	let	let	VERB
ejpam-7061	116	8	si	si	PROPN
ejpam-7061	116	9	∈	∈	PROPN
ejpam-7061	116	10	si	si	X
ejpam-7061	116	11	,	,	PUNCT
ejpam-7061	116	12	where	where	SCONJ
ejpam-7061	116	13	i	i	PRON
ejpam-7061	116	14	∈	∈	PROPN
ejpam-7061	116	15	{	{	PUNCT
ejpam-7061	116	16	1	1	NUM
ejpam-7061	116	17	,	,	PUNCT
ejpam-7061	116	18	2	2	NUM
ejpam-7061	116	19	,	,	PUNCT
ejpam-7061	116	20	.	.	PUNCT
ejpam-7061	116	21	.	.	PUNCT
ejpam-7061	117	1	.	.	PUNCT
ejpam-7061	117	2	,	,	PUNCT
ejpam-7061	117	3	n	n	CCONJ
ejpam-7061	117	4	}	}	PUNCT
ejpam-7061	117	5	.	.	PUNCT
ejpam-7061	118	1	then	then	ADV
ejpam-7061	118	2	l((s1	l((s1	PROPN
ejpam-7061	118	3	,	,	PUNCT
ejpam-7061	118	4	s2	s2	PROPN
ejpam-7061	118	5	,	,	PUNCT
ejpam-7061	118	6	.	.	PUNCT
ejpam-7061	118	7	.	.	PUNCT
ejpam-7061	118	8	.	.	PUNCT
ejpam-7061	118	9	,	,	PUNCT
ejpam-7061	118	10	sn	sn	PROPN
ejpam-7061	118	11	)	)	PUNCT
ejpam-7061	118	12	)	)	PUNCT
ejpam-7061	119	1	=	=	PUNCT
ejpam-7061	119	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	119	3	l(s2)×	l(s2)×	X
ejpam-7061	119	4	·	·	PUNCT
ejpam-7061	119	5	·	·	PUNCT
ejpam-7061	119	6	·	·	PUNCT
ejpam-7061	119	7	×	×	NOUN
ejpam-7061	119	8	l(sn	l(sn	ADV
ejpam-7061	119	9	)	)	PUNCT
ejpam-7061	119	10	if	if	SCONJ
ejpam-7061	120	1	and	and	CCONJ
ejpam-7061	120	2	only	only	ADV
ejpam-7061	120	3	if	if	SCONJ
ejpam-7061	120	4	at	at	ADV
ejpam-7061	120	5	least	least	ADJ
ejpam-7061	120	6	one	one	NUM
ejpam-7061	120	7	of	of	ADP
ejpam-7061	120	8	the	the	DET
ejpam-7061	120	9	following	follow	VERB
ejpam-7061	120	10	conditions	condition	NOUN
ejpam-7061	120	11	is	be	AUX
ejpam-7061	120	12	satisfied	satisfied	ADJ
ejpam-7061	120	13	:	:	PUNCT
ejpam-7061	120	14	(	(	PUNCT
ejpam-7061	120	15	i	i	NOUN
ejpam-7061	120	16	)	)	PUNCT
ejpam-7061	120	17	si	si	PROPN
ejpam-7061	120	18	∈	∈	PROPN
ejpam-7061	120	19	sisi	sisi	PROPN
ejpam-7061	120	20	for	for	ADP
ejpam-7061	120	21	all	all	PRON
ejpam-7061	120	22	i	i	PRON
ejpam-7061	120	23	∈	∈	PROPN
ejpam-7061	120	24	{	{	PUNCT
ejpam-7061	120	25	1	1	NUM
ejpam-7061	120	26	,	,	PUNCT
ejpam-7061	120	27	2	2	NUM
ejpam-7061	120	28	,	,	PUNCT
ejpam-7061	120	29	.	.	PUNCT
ejpam-7061	120	30	.	.	PUNCT
ejpam-7061	121	1	.	.	PUNCT
ejpam-7061	122	1	,	,	PUNCT
ejpam-7061	122	2	n	n	CCONJ
ejpam-7061	122	3	}	}	PUNCT
ejpam-7061	122	4	;	;	PUNCT
ejpam-7061	122	5	(	(	PUNCT
ejpam-7061	122	6	ii	ii	NOUN
ejpam-7061	122	7	)	)	PUNCT
ejpam-7061	122	8	there	there	PRON
ejpam-7061	122	9	exists	exist	VERB
ejpam-7061	122	10	i	i	PRON
ejpam-7061	122	11	∈	∈	PROPN
ejpam-7061	122	12	{	{	PUNCT
ejpam-7061	122	13	1	1	NUM
ejpam-7061	122	14	,	,	PUNCT
ejpam-7061	122	15	2	2	NUM
ejpam-7061	122	16	,	,	PUNCT
ejpam-7061	122	17	.	.	PUNCT
ejpam-7061	122	18	.	.	PUNCT
ejpam-7061	123	1	.	.	PUNCT
ejpam-7061	124	1	,	,	PUNCT
ejpam-7061	125	1	n	n	CCONJ
ejpam-7061	125	2	}	}	PUNCT
ejpam-7061	125	3	such	such	ADJ
ejpam-7061	125	4	that	that	SCONJ
ejpam-7061	125	5	sjsj	sjsj	ADJ
ejpam-7061	125	6	=	=	SYM
ejpam-7061	125	7	{	{	PUNCT
ejpam-7061	125	8	sj	sj	NOUN
ejpam-7061	125	9	}	}	PUNCT
ejpam-7061	125	10	for	for	ADP
ejpam-7061	125	11	all	all	DET
ejpam-7061	125	12	j	j	PROPN
ejpam-7061	125	13	∈	∈	PROPN
ejpam-7061	125	14	{	{	PUNCT
ejpam-7061	125	15	1	1	NUM
ejpam-7061	125	16	,	,	PUNCT
ejpam-7061	125	17	2	2	NUM
ejpam-7061	125	18	,	,	PUNCT
ejpam-7061	125	19	.	.	PUNCT
ejpam-7061	125	20	.	.	PUNCT
ejpam-7061	125	21	.	.	PUNCT
ejpam-7061	125	22	,	,	PUNCT
ejpam-7061	125	23	n	n	CCONJ
ejpam-7061	125	24	}	}	PUNCT
ejpam-7061	125	25	\	\	NOUN
ejpam-7061	125	26	{	{	PUNCT
ejpam-7061	125	27	i	i	NOUN
ejpam-7061	125	28	}	}	PUNCT
ejpam-7061	125	29	.	.	PUNCT
ejpam-7061	126	1	proof	proof	NOUN
ejpam-7061	126	2	.	.	PUNCT
ejpam-7061	127	1	suppose	suppose	VERB
ejpam-7061	127	2	(	(	PUNCT
ejpam-7061	127	3	i	i	NOUN
ejpam-7061	127	4	)	)	PUNCT
ejpam-7061	127	5	and	and	CCONJ
ejpam-7061	127	6	(	(	PUNCT
ejpam-7061	127	7	ii	ii	NOUN
ejpam-7061	127	8	)	)	PUNCT
ejpam-7061	127	9	do	do	AUX
ejpam-7061	127	10	not	not	PART
ejpam-7061	127	11	hold	hold	VERB
ejpam-7061	127	12	.	.	PUNCT
ejpam-7061	128	1	then	then	ADV
ejpam-7061	128	2	there	there	PRON
ejpam-7061	128	3	exists	exist	VERB
ejpam-7061	128	4	i	i	PRON
ejpam-7061	128	5	∈	∈	PROPN
ejpam-7061	128	6	{	{	PUNCT
ejpam-7061	128	7	1	1	NUM
ejpam-7061	128	8	,	,	PUNCT
ejpam-7061	128	9	2	2	NUM
ejpam-7061	128	10	,	,	PUNCT
ejpam-7061	128	11	.	.	PUNCT
ejpam-7061	128	12	.	.	PUNCT
ejpam-7061	129	1	.	.	PUNCT
ejpam-7061	130	1	,	,	PUNCT
ejpam-7061	131	1	n	n	CCONJ
ejpam-7061	131	2	}	}	PUNCT
ejpam-7061	132	1	such	such	ADJ
ejpam-7061	132	2	that	that	SCONJ
ejpam-7061	132	3	si	si	PROPN
ejpam-7061	132	4	/∈	/∈	PUNCT
ejpam-7061	132	5	sisi	sisi	PROPN
ejpam-7061	132	6	.	.	PUNCT
ejpam-7061	133	1	this	this	PRON
ejpam-7061	133	2	implies	imply	VERB
ejpam-7061	133	3	that	that	SCONJ
ejpam-7061	133	4	sisi	sisi	PROPN
ejpam-7061	133	5	̸=	̸=	PROPN
ejpam-7061	133	6	{	{	PUNCT
ejpam-7061	133	7	si	si	NOUN
ejpam-7061	133	8	}	}	PUNCT
ejpam-7061	133	9	.	.	PUNCT
ejpam-7061	134	1	since	since	SCONJ
ejpam-7061	134	2	(	(	PUNCT
ejpam-7061	134	3	ii	ii	NOUN
ejpam-7061	134	4	)	)	PUNCT
ejpam-7061	134	5	does	do	AUX
ejpam-7061	134	6	not	not	PART
ejpam-7061	134	7	hold	hold	VERB
ejpam-7061	134	8	,	,	PUNCT
ejpam-7061	134	9	there	there	PRON
ejpam-7061	134	10	exists	exist	VERB
ejpam-7061	134	11	j	j	PROPN
ejpam-7061	134	12	∈	∈	PROPN
ejpam-7061	134	13	{	{	PUNCT
ejpam-7061	134	14	1	1	NUM
ejpam-7061	134	15	,	,	PUNCT
ejpam-7061	134	16	2	2	NUM
ejpam-7061	134	17	,	,	PUNCT
ejpam-7061	134	18	.	.	PUNCT
ejpam-7061	134	19	.	.	PUNCT
ejpam-7061	135	1	.	.	PUNCT
ejpam-7061	136	1	,	,	PUNCT
ejpam-7061	136	2	n	n	CCONJ
ejpam-7061	136	3	}	}	PUNCT
ejpam-7061	136	4	such	such	ADJ
ejpam-7061	136	5	that	that	SCONJ
ejpam-7061	136	6	j	j	PROPN
ejpam-7061	136	7	̸=	̸=	PROPN
ejpam-7061	136	8	i	i	PRON
ejpam-7061	136	9	and	and	CCONJ
ejpam-7061	136	10	sjsj	sjsj	ADJ
ejpam-7061	136	11	̸=	̸=	PROPN
ejpam-7061	136	12	{	{	PUNCT
ejpam-7061	136	13	sj	sj	NOUN
ejpam-7061	136	14	}	}	PUNCT
ejpam-7061	136	15	.	.	PUNCT
ejpam-7061	137	1	without	without	ADP
ejpam-7061	137	2	loss	loss	NOUN
ejpam-7061	137	3	of	of	ADP
ejpam-7061	137	4	generality	generality	NOUN
ejpam-7061	137	5	,	,	PUNCT
ejpam-7061	137	6	we	we	PRON
ejpam-7061	137	7	let	let	VERB
ejpam-7061	137	8	s	s	PRON
ejpam-7061	137	9	=	=	PUNCT
ejpam-7061	137	10	(	(	PUNCT
ejpam-7061	137	11	s1	s1	PROPN
ejpam-7061	137	12	,	,	PUNCT
ejpam-7061	137	13	s2	s2	NOUN
ejpam-7061	137	14	,	,	PUNCT
ejpam-7061	137	15	.	.	PUNCT
ejpam-7061	137	16	.	.	PUNCT
ejpam-7061	138	1	.	.	PUNCT
ejpam-7061	139	1	,	,	PUNCT
ejpam-7061	139	2	si	si	INTJ
ejpam-7061	139	3	,	,	PUNCT
ejpam-7061	139	4	.	.	PUNCT
ejpam-7061	139	5	.	.	PUNCT
ejpam-7061	140	1	.	.	PUNCT
ejpam-7061	141	1	,	,	PUNCT
ejpam-7061	141	2	s	s	VERB
ejpam-7061	141	3	′	′	NUM
ejpam-7061	141	4	j	j	PROPN
ejpam-7061	141	5	,	,	PUNCT
ejpam-7061	141	6	.	.	PUNCT
ejpam-7061	141	7	.	.	PUNCT
ejpam-7061	142	1	.	.	PUNCT
ejpam-7061	143	1	,	,	PUNCT
ejpam-7061	143	2	sn	sn	PROPN
ejpam-7061	143	3	)	)	PUNCT
ejpam-7061	143	4	where	where	SCONJ
ejpam-7061	143	5	s′j	s′j	PROPN
ejpam-7061	143	6	∈	∈	PROPN
ejpam-7061	143	7	sjsj	sjsj	NOUN
ejpam-7061	143	8	\{sj	\{sj	NOUN
ejpam-7061	143	9	}	}	PUNCT
ejpam-7061	143	10	.	.	PUNCT
ejpam-7061	144	1	then	then	ADV
ejpam-7061	144	2	s	s	VERB
ejpam-7061	144	3	∈	∈	PROPN
ejpam-7061	144	4	l(s1)×l(s2)×	l(s1)×l(s2)×	X
ejpam-7061	144	5	·	·	PUNCT
ejpam-7061	144	6	·	·	PUNCT
ejpam-7061	144	7	·	·	PUNCT
ejpam-7061	144	8	×l(sn	×l(sn	NOUN
ejpam-7061	144	9	)	)	PUNCT
ejpam-7061	144	10	.	.	PUNCT
ejpam-7061	145	1	since	since	SCONJ
ejpam-7061	145	2	s	s	X
ejpam-7061	145	3	/∈	/∈	PUNCT
ejpam-7061	145	4	s1s1×s2s2×	s1s1×s2s2×	PROPN
ejpam-7061	145	5	·	·	PUNCT
ejpam-7061	145	6	·	·	PUNCT
ejpam-7061	145	7	·	·	PUNCT
ejpam-7061	145	8	×	×	NOUN
ejpam-7061	145	9	snsn	snsn	NOUN
ejpam-7061	145	10	and	and	CCONJ
ejpam-7061	145	11	s	s	PROPN
ejpam-7061	145	12	̸=	̸=	PROPN
ejpam-7061	145	13	(	(	PUNCT
ejpam-7061	145	14	s1	s1	PROPN
ejpam-7061	145	15	,	,	PUNCT
ejpam-7061	145	16	s2	s2	NOUN
ejpam-7061	145	17	,	,	PUNCT
ejpam-7061	145	18	.	.	PUNCT
ejpam-7061	145	19	.	.	PUNCT
ejpam-7061	145	20	.	.	PUNCT
ejpam-7061	146	1	,	,	PUNCT
ejpam-7061	146	2	sn	sn	PROPN
ejpam-7061	146	3	)	)	PUNCT
ejpam-7061	146	4	,	,	PUNCT
ejpam-7061	146	5	we	we	PRON
ejpam-7061	146	6	have	have	VERB
ejpam-7061	146	7	s	s	NOUN
ejpam-7061	146	8	/∈	/∈	X
ejpam-7061	146	9	l((s1	l((s1	PROPN
ejpam-7061	146	10	,	,	PUNCT
ejpam-7061	146	11	s2	s2	PROPN
ejpam-7061	146	12	,	,	PUNCT
ejpam-7061	146	13	.	.	PUNCT
ejpam-7061	146	14	.	.	PUNCT
ejpam-7061	147	1	.	.	PUNCT
ejpam-7061	148	1	,	,	PUNCT
ejpam-7061	148	2	sn	sn	PROPN
ejpam-7061	148	3	)	)	PUNCT
ejpam-7061	148	4	)	)	PUNCT
ejpam-7061	148	5	.	.	PUNCT
ejpam-7061	149	1	thus	thus	ADV
ejpam-7061	149	2	,	,	PUNCT
ejpam-7061	149	3	l((s1	l((s1	PROPN
ejpam-7061	149	4	,	,	PUNCT
ejpam-7061	149	5	s2	s2	PROPN
ejpam-7061	149	6	,	,	PUNCT
ejpam-7061	149	7	.	.	PUNCT
ejpam-7061	149	8	.	.	PUNCT
ejpam-7061	149	9	.	.	PUNCT
ejpam-7061	150	1	,	,	PUNCT
ejpam-7061	150	2	sn	sn	PROPN
ejpam-7061	150	3	)	)	PUNCT
ejpam-7061	150	4	)	)	PUNCT
ejpam-7061	151	1	̸=	̸=	PROPN
ejpam-7061	151	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	151	3	l(s2)×	l(s2)×	NOUN
ejpam-7061	151	4	·	·	PUNCT
ejpam-7061	151	5	·	·	PUNCT
ejpam-7061	151	6	·	·	PUNCT
ejpam-7061	151	7	×	×	NOUN
ejpam-7061	151	8	l(sn	l(sn	NUM
ejpam-7061	151	9	)	)	PUNCT
ejpam-7061	151	10	.	.	PUNCT
ejpam-7061	152	1	conversely	conversely	ADV
ejpam-7061	152	2	,	,	PUNCT
ejpam-7061	152	3	assume	assume	VERB
ejpam-7061	152	4	that	that	SCONJ
ejpam-7061	152	5	(	(	PUNCT
ejpam-7061	152	6	i	i	NOUN
ejpam-7061	152	7	)	)	PUNCT
ejpam-7061	152	8	or	or	CCONJ
ejpam-7061	152	9	(	(	PUNCT
ejpam-7061	152	10	ii	ii	NOUN
ejpam-7061	152	11	)	)	PUNCT
ejpam-7061	152	12	holds	hold	VERB
ejpam-7061	152	13	.	.	PUNCT
ejpam-7061	153	1	if	if	SCONJ
ejpam-7061	153	2	(	(	PUNCT
ejpam-7061	153	3	i	i	NOUN
ejpam-7061	153	4	)	)	PUNCT
ejpam-7061	153	5	holds	hold	VERB
ejpam-7061	153	6	,	,	PUNCT
ejpam-7061	153	7	then	then	ADV
ejpam-7061	153	8	we	we	PRON
ejpam-7061	153	9	have	have	VERB
ejpam-7061	153	10	l(s1)×	l(s1)×	NOUN
ejpam-7061	153	11	l(s2)×	l(s2)×	X
ejpam-7061	153	12	·	·	PUNCT
ejpam-7061	153	13	·	·	PUNCT
ejpam-7061	153	14	·	·	PUNCT
ejpam-7061	154	1	×	×	NOUN
ejpam-7061	154	2	l(sn	l(sn	ADJ
ejpam-7061	154	3	)	)	PUNCT
ejpam-7061	154	4	=	=	PRON
ejpam-7061	154	5	s1s1	s1s1	ADP
ejpam-7061	154	6	×	×	NOUN
ejpam-7061	154	7	s2s2	s2s2	NOUN
ejpam-7061	154	8	×	×	NOUN
ejpam-7061	154	9	·	·	PUNCT
ejpam-7061	154	10	·	·	PUNCT
ejpam-7061	154	11	·	·	PUNCT
ejpam-7061	155	1	×	×	NOUN
ejpam-7061	155	2	snsn	snsn	NOUN
ejpam-7061	155	3	⊆	⊆	NUM
ejpam-7061	155	4	l((s1	l((s1	NOUN
ejpam-7061	155	5	,	,	PUNCT
ejpam-7061	155	6	s2	s2	PROPN
ejpam-7061	155	7	,	,	PUNCT
ejpam-7061	155	8	.	.	PUNCT
ejpam-7061	155	9	.	.	PUNCT
ejpam-7061	155	10	.	.	PUNCT
ejpam-7061	156	1	,	,	PUNCT
ejpam-7061	156	2	sn	sn	PROPN
ejpam-7061	156	3	)	)	PUNCT
ejpam-7061	156	4	)	)	PUNCT
ejpam-7061	157	1	⊆	⊆	NUM
ejpam-7061	157	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	157	3	l(s2)×	l(s2)×	X
ejpam-7061	157	4	·	·	PUNCT
ejpam-7061	157	5	·	·	PUNCT
ejpam-7061	157	6	·	·	PUNCT
ejpam-7061	157	7	×	×	NOUN
ejpam-7061	157	8	l(sn	l(sn	NUM
ejpam-7061	157	9	)	)	PUNCT
ejpam-7061	157	10	.	.	PUNCT
ejpam-7061	158	1	p.	p.	PROPN
ejpam-7061	158	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	158	3	,	,	PUNCT
ejpam-7061	158	4	o.	o.	PROPN
ejpam-7061	158	5	pankoon	pankoon	NOUN
ejpam-7061	158	6	,	,	PUNCT
ejpam-7061	158	7	t.	t.	PROPN
ejpam-7061	158	8	changphas	changphas	PROPN
ejpam-7061	158	9	/	/	SYM
ejpam-7061	158	10	eur	eur	PROPN
ejpam-7061	158	11	.	.	PUNCT
ejpam-7061	159	1	j.	j.	PROPN
ejpam-7061	159	2	pure	pure	PROPN
ejpam-7061	159	3	appl	appl	PROPN
ejpam-7061	159	4	.	.	PROPN
ejpam-7061	159	5	math	math	PROPN
ejpam-7061	159	6	,	,	PUNCT
ejpam-7061	159	7	18	18	NUM
ejpam-7061	159	8	(	(	PUNCT
ejpam-7061	159	9	4	4	NUM
ejpam-7061	159	10	)	)	PUNCT
ejpam-7061	159	11	(	(	PUNCT
ejpam-7061	159	12	2025	2025	NUM
ejpam-7061	159	13	)	)	PUNCT
ejpam-7061	159	14	,	,	PUNCT
ejpam-7061	159	15	7061	7061	NUM
ejpam-7061	159	16	4	4	NUM
ejpam-7061	159	17	of	of	ADP
ejpam-7061	159	18	10	10	NUM
ejpam-7061	159	19	thus	thus	ADV
ejpam-7061	159	20	,	,	PUNCT
ejpam-7061	159	21	l(s1	l(s1	ADJ
ejpam-7061	159	22	)	)	PUNCT
ejpam-7061	159	23	×	×	NOUN
ejpam-7061	159	24	l(s2	l(s2	NOUN
ejpam-7061	159	25	)	)	PUNCT
ejpam-7061	159	26	×	×	NOUN
ejpam-7061	159	27	·	·	PUNCT
ejpam-7061	159	28	·	·	PUNCT
ejpam-7061	159	29	·	·	PUNCT
ejpam-7061	160	1	×	×	NOUN
ejpam-7061	160	2	l(sn	l(sn	ADJ
ejpam-7061	160	3	)	)	PUNCT
ejpam-7061	160	4	=	=	SYM
ejpam-7061	160	5	l((s1	l((s1	PROPN
ejpam-7061	160	6	,	,	PUNCT
ejpam-7061	160	7	s2	s2	PROPN
ejpam-7061	160	8	,	,	PUNCT
ejpam-7061	160	9	.	.	PUNCT
ejpam-7061	160	10	.	.	PUNCT
ejpam-7061	160	11	.	.	PUNCT
ejpam-7061	160	12	,	,	PUNCT
ejpam-7061	160	13	sn	sn	PROPN
ejpam-7061	160	14	)	)	PUNCT
ejpam-7061	160	15	)	)	PUNCT
ejpam-7061	160	16	.	.	PUNCT
ejpam-7061	161	1	meanwhile	meanwhile	ADV
ejpam-7061	161	2	,	,	PUNCT
ejpam-7061	161	3	if	if	SCONJ
ejpam-7061	161	4	(	(	PUNCT
ejpam-7061	161	5	ii	ii	NOUN
ejpam-7061	161	6	)	)	PUNCT
ejpam-7061	161	7	holds	hold	VERB
ejpam-7061	161	8	,	,	PUNCT
ejpam-7061	161	9	let	let	VERB
ejpam-7061	161	10	i	i	PRON
ejpam-7061	161	11	∈	∈	PROPN
ejpam-7061	161	12	{	{	PUNCT
ejpam-7061	161	13	1	1	NUM
ejpam-7061	161	14	,	,	PUNCT
ejpam-7061	161	15	2	2	NUM
ejpam-7061	161	16	,	,	PUNCT
ejpam-7061	161	17	.	.	PUNCT
ejpam-7061	161	18	.	.	PUNCT
ejpam-7061	162	1	.	.	PUNCT
ejpam-7061	163	1	,	,	PUNCT
ejpam-7061	163	2	n	n	CCONJ
ejpam-7061	163	3	}	}	PUNCT
ejpam-7061	163	4	be	be	AUX
ejpam-7061	163	5	the	the	DET
ejpam-7061	163	6	index	index	NOUN
ejpam-7061	163	7	such	such	ADJ
ejpam-7061	163	8	that	that	SCONJ
ejpam-7061	163	9	sjsj	sjsj	ADJ
ejpam-7061	163	10	=	=	SYM
ejpam-7061	163	11	{	{	PUNCT
ejpam-7061	163	12	sj	sj	NOUN
ejpam-7061	163	13	}	}	PUNCT
ejpam-7061	163	14	for	for	ADP
ejpam-7061	163	15	all	all	DET
ejpam-7061	163	16	j	j	PROPN
ejpam-7061	163	17	∈	∈	PROPN
ejpam-7061	163	18	{	{	PUNCT
ejpam-7061	163	19	1	1	NUM
ejpam-7061	163	20	,	,	PUNCT
ejpam-7061	163	21	2	2	NUM
ejpam-7061	163	22	,	,	PUNCT
ejpam-7061	163	23	.	.	PUNCT
ejpam-7061	163	24	.	.	PUNCT
ejpam-7061	164	1	.	.	PUNCT
ejpam-7061	165	1	,	,	PUNCT
ejpam-7061	165	2	n	n	CCONJ
ejpam-7061	165	3	}	}	PUNCT
ejpam-7061	165	4	\	\	NOUN
ejpam-7061	165	5	{	{	PUNCT
ejpam-7061	165	6	i	i	NOUN
ejpam-7061	165	7	}	}	PUNCT
ejpam-7061	165	8	.	.	PUNCT
ejpam-7061	166	1	then	then	ADV
ejpam-7061	166	2	l(s1)×	l(s1)×	PROPN
ejpam-7061	166	3	l(s2)×	l(s2)×	PROPN
ejpam-7061	166	4	·	·	PUNCT
ejpam-7061	166	5	·	·	PUNCT
ejpam-7061	166	6	·	·	PUNCT
ejpam-7061	167	1	×	×	NOUN
ejpam-7061	167	2	l(sn	l(sn	ADJ
ejpam-7061	167	3	)	)	PUNCT
ejpam-7061	167	4	=	=	SYM
ejpam-7061	167	5	{	{	PUNCT
ejpam-7061	167	6	s1	s1	PROPN
ejpam-7061	167	7	}	}	PUNCT
ejpam-7061	167	8	×	×	PROPN
ejpam-7061	167	9	{	{	PUNCT
ejpam-7061	167	10	s2	s2	PROPN
ejpam-7061	167	11	}	}	PUNCT
ejpam-7061	167	12	×	×	NOUN
ejpam-7061	167	13	·	·	PUNCT
ejpam-7061	167	14	·	·	PUNCT
ejpam-7061	167	15	·	·	PUNCT
ejpam-7061	167	16	×	×	NOUN
ejpam-7061	167	17	(	(	PUNCT
ejpam-7061	167	18	{	{	PUNCT
ejpam-7061	167	19	si	si	INTJ
ejpam-7061	167	20	}	}	PUNCT
ejpam-7061	167	21	∪	∪	ADJ
ejpam-7061	167	22	sisi)×	sisi)×	PROPN
ejpam-7061	167	23	·	·	PUNCT
ejpam-7061	167	24	·	·	PUNCT
ejpam-7061	167	25	·	·	PUNCT
ejpam-7061	167	26	×	×	NOUN
ejpam-7061	167	27	{	{	PUNCT
ejpam-7061	167	28	sn	sn	NOUN
ejpam-7061	167	29	}	}	PUNCT
ejpam-7061	167	30	=	=	SYM
ejpam-7061	167	31	(	(	PUNCT
ejpam-7061	167	32	s1	s1	NOUN
ejpam-7061	167	33	,	,	PUNCT
ejpam-7061	167	34	s2	s2	NOUN
ejpam-7061	167	35	,	,	PUNCT
ejpam-7061	167	36	.	.	PUNCT
ejpam-7061	167	37	.	.	PUNCT
ejpam-7061	167	38	.	.	PUNCT
ejpam-7061	168	1	,	,	PUNCT
ejpam-7061	168	2	sn	sn	PROPN
ejpam-7061	168	3	)	)	PUNCT
ejpam-7061	168	4	∪	∪	NOUN
ejpam-7061	168	5	(	(	PUNCT
ejpam-7061	168	6	s1s1	s1s1	NOUN
ejpam-7061	168	7	×	×	NOUN
ejpam-7061	168	8	s2s2	s2s2	NOUN
ejpam-7061	168	9	×	×	NOUN
ejpam-7061	168	10	·	·	PUNCT
ejpam-7061	168	11	·	·	PUNCT
ejpam-7061	168	12	·	·	PUNCT
ejpam-7061	169	1	×	×	PROPN
ejpam-7061	169	2	sisi	sisi	X
ejpam-7061	169	3	×	×	NOUN
ejpam-7061	169	4	·	·	PUNCT
ejpam-7061	169	5	·	·	PUNCT
ejpam-7061	169	6	·	·	PUNCT
ejpam-7061	169	7	×	×	NOUN
ejpam-7061	169	8	snsn	snsn	NOUN
ejpam-7061	169	9	)	)	PUNCT
ejpam-7061	169	10	=	=	SYM
ejpam-7061	169	11	l((s1	l((s1	PROPN
ejpam-7061	169	12	,	,	PUNCT
ejpam-7061	169	13	s2	s2	PROPN
ejpam-7061	169	14	,	,	PUNCT
ejpam-7061	169	15	.	.	PUNCT
ejpam-7061	169	16	.	.	PUNCT
ejpam-7061	169	17	.	.	PUNCT
ejpam-7061	169	18	,	,	PUNCT
ejpam-7061	169	19	sn	sn	PROPN
ejpam-7061	169	20	)	)	PUNCT
ejpam-7061	169	21	)	)	PUNCT
ejpam-7061	169	22	.	.	PUNCT
ejpam-7061	170	1	by	by	ADP
ejpam-7061	170	2	these	these	DET
ejpam-7061	170	3	two	two	NUM
ejpam-7061	170	4	cases	case	NOUN
ejpam-7061	170	5	,	,	PUNCT
ejpam-7061	170	6	we	we	PRON
ejpam-7061	170	7	conclude	conclude	VERB
ejpam-7061	170	8	that	that	SCONJ
ejpam-7061	170	9	l((s1	l((s1	NOUN
ejpam-7061	170	10	,	,	PUNCT
ejpam-7061	170	11	s2	s2	PROPN
ejpam-7061	170	12	,	,	PUNCT
ejpam-7061	170	13	.	.	PUNCT
ejpam-7061	170	14	.	.	PUNCT
ejpam-7061	170	15	.	.	PUNCT
ejpam-7061	171	1	,	,	PUNCT
ejpam-7061	171	2	sn	sn	PROPN
ejpam-7061	171	3	)	)	PUNCT
ejpam-7061	171	4	)	)	PUNCT
ejpam-7061	172	1	=	=	PUNCT
ejpam-7061	172	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	172	3	l(s2)×	l(s2)×	X
ejpam-7061	172	4	·	·	PUNCT
ejpam-7061	172	5	·	·	PUNCT
ejpam-7061	172	6	·	·	PUNCT
ejpam-7061	172	7	×	×	NOUN
ejpam-7061	172	8	l(sn	l(sn	NUM
ejpam-7061	172	9	)	)	PUNCT
ejpam-7061	172	10	.	.	PUNCT
ejpam-7061	173	1	remark	remark	PROPN
ejpam-7061	173	2	1	1	NUM
ejpam-7061	173	3	.	.	PUNCT
ejpam-7061	174	1	let	let	VERB
ejpam-7061	174	2	s	s	PRON
ejpam-7061	174	3	be	be	AUX
ejpam-7061	174	4	a	a	DET
ejpam-7061	174	5	semigroup	semigroup	NOUN
ejpam-7061	174	6	defined	define	VERB
ejpam-7061	174	7	as	as	ADP
ejpam-7061	174	8	in	in	ADP
ejpam-7061	174	9	example	example	NOUN
ejpam-7061	175	1	1	1	X
ejpam-7061	175	2	.	.	PUNCT
ejpam-7061	176	1	we	we	PRON
ejpam-7061	176	2	have	have	VERB
ejpam-7061	176	3	s3	s3	PROPN
ejpam-7061	176	4	/∈	/∈	PUNCT
ejpam-7061	176	5	{	{	PUNCT
ejpam-7061	176	6	s1	s1	NOUN
ejpam-7061	176	7	,	,	PUNCT
ejpam-7061	176	8	s2	s2	PROPN
ejpam-7061	176	9	,	,	PUNCT
ejpam-7061	176	10	s4	s4	PROPN
ejpam-7061	176	11	}	}	PUNCT
ejpam-7061	176	12	=	=	PUNCT
ejpam-7061	176	13	ss3	ss3	NOUN
ejpam-7061	176	14	.	.	PUNCT
ejpam-7061	177	1	thus	thus	ADV
ejpam-7061	177	2	,	,	PUNCT
ejpam-7061	177	3	we	we	PRON
ejpam-7061	177	4	immediately	immediately	ADV
ejpam-7061	177	5	obtain	obtain	VERB
ejpam-7061	177	6	from	from	ADP
ejpam-7061	177	7	theorem	theorem	ADJ
ejpam-7061	177	8	1	1	NUM
ejpam-7061	177	9	that	that	SCONJ
ejpam-7061	177	10	l(s3)×	l(s3)×	NOUN
ejpam-7061	177	11	l(s3	l(s3	NOUN
ejpam-7061	177	12	)	)	PUNCT
ejpam-7061	177	13	̸=	̸=	PROPN
ejpam-7061	177	14	l((s3	l((s3	NOUN
ejpam-7061	177	15	,	,	PUNCT
ejpam-7061	177	16	s3	s3	PROPN
ejpam-7061	177	17	)	)	PUNCT
ejpam-7061	177	18	)	)	PUNCT
ejpam-7061	177	19	.	.	PUNCT
ejpam-7061	178	1	in	in	ADP
ejpam-7061	178	2	theorem	theorem	NOUN
ejpam-7061	178	3	1	1	NUM
ejpam-7061	178	4	,	,	PUNCT
ejpam-7061	178	5	we	we	PRON
ejpam-7061	178	6	establish	establish	VERB
ejpam-7061	178	7	the	the	DET
ejpam-7061	178	8	sufficient	sufficient	ADJ
ejpam-7061	178	9	and	and	CCONJ
ejpam-7061	178	10	necessary	necessary	ADJ
ejpam-7061	178	11	condition	condition	NOUN
ejpam-7061	178	12	when	when	SCONJ
ejpam-7061	178	13	l((s1	l((s1	PROPN
ejpam-7061	178	14	,	,	PUNCT
ejpam-7061	178	15	s2	s2	PROPN
ejpam-7061	178	16	,	,	PUNCT
ejpam-7061	178	17	.	.	PUNCT
ejpam-7061	178	18	.	.	PUNCT
ejpam-7061	178	19	.	.	PUNCT
ejpam-7061	179	1	,	,	PUNCT
ejpam-7061	179	2	sn	sn	PROPN
ejpam-7061	179	3	)	)	PUNCT
ejpam-7061	179	4	)	)	PUNCT
ejpam-7061	180	1	=	=	PUNCT
ejpam-7061	180	2	l(s1)×l(s2)×	l(s1)×l(s2)×	X
ejpam-7061	180	3	·	·	PUNCT
ejpam-7061	180	4	·	·	PUNCT
ejpam-7061	180	5	·	·	PUNCT
ejpam-7061	180	6	×l(sn	×l(sn	NOUN
ejpam-7061	180	7	)	)	PUNCT
ejpam-7061	180	8	.	.	PUNCT
ejpam-7061	181	1	however	however	ADV
ejpam-7061	181	2	,	,	PUNCT
ejpam-7061	181	3	the	the	DET
ejpam-7061	181	4	negation	negation	NOUN
ejpam-7061	181	5	of	of	ADP
ejpam-7061	181	6	such	such	ADJ
ejpam-7061	181	7	condition	condition	NOUN
ejpam-7061	181	8	does	do	AUX
ejpam-7061	181	9	not	not	PART
ejpam-7061	181	10	ensure	ensure	VERB
ejpam-7061	181	11	that	that	SCONJ
ejpam-7061	181	12	l(s1)×l(s2)×	l(s1)×l(s2)×	NOUN
ejpam-7061	181	13	·	·	SYM
ejpam-7061	181	14	·	·	PUNCT
ejpam-7061	181	15	·	·	PUNCT
ejpam-7061	181	16	×l(sn	×l(sn	NOUN
ejpam-7061	181	17	)	)	PUNCT
ejpam-7061	181	18	is	be	AUX
ejpam-7061	181	19	not	not	PART
ejpam-7061	181	20	a	a	DET
ejpam-7061	181	21	principal	principal	NOUN
ejpam-7061	181	22	left	leave	VERB
ejpam-7061	181	23	ideal	ideal	ADJ
ejpam-7061	181	24	.	.	PUNCT
ejpam-7061	182	1	this	this	DET
ejpam-7061	182	2	assumption	assumption	NOUN
ejpam-7061	182	3	can	can	AUX
ejpam-7061	182	4	be	be	AUX
ejpam-7061	182	5	confirmed	confirm	VERB
ejpam-7061	182	6	by	by	ADP
ejpam-7061	182	7	the	the	DET
ejpam-7061	182	8	following	follow	VERB
ejpam-7061	182	9	theorem	theorem	PROPN
ejpam-7061	182	10	.	.	PUNCT
ejpam-7061	182	11	theorem	theorem	NOUN
ejpam-7061	182	12	2	2	NUM
ejpam-7061	182	13	.	.	PUNCT
ejpam-7061	183	1	let	let	VERB
ejpam-7061	183	2	si	si	PART
ejpam-7061	183	3	be	be	AUX
ejpam-7061	183	4	a	a	DET
ejpam-7061	183	5	semigroup	semigroup	NOUN
ejpam-7061	183	6	and	and	CCONJ
ejpam-7061	183	7	let	let	VERB
ejpam-7061	183	8	si	si	PROPN
ejpam-7061	183	9	∈	∈	PROPN
ejpam-7061	183	10	si	si	X
ejpam-7061	183	11	,	,	PUNCT
ejpam-7061	183	12	where	where	SCONJ
ejpam-7061	183	13	i	i	PRON
ejpam-7061	183	14	=	=	NOUN
ejpam-7061	183	15	1	1	NUM
ejpam-7061	183	16	,	,	PUNCT
ejpam-7061	183	17	2	2	NUM
ejpam-7061	183	18	,	,	PUNCT
ejpam-7061	183	19	.	.	PUNCT
ejpam-7061	183	20	.	.	PUNCT
ejpam-7061	184	1	.	.	PUNCT
ejpam-7061	185	1	,	,	PUNCT
ejpam-7061	185	2	n.	n.	NOUN
ejpam-7061	185	3	if	if	SCONJ
ejpam-7061	185	4	l((s1	l((s1	PROPN
ejpam-7061	185	5	,	,	PUNCT
ejpam-7061	185	6	s2	s2	PROPN
ejpam-7061	185	7	,	,	PUNCT
ejpam-7061	185	8	.	.	PUNCT
ejpam-7061	185	9	.	.	PUNCT
ejpam-7061	186	1	.	.	PUNCT
ejpam-7061	187	1	,	,	PUNCT
ejpam-7061	187	2	sn	sn	PROPN
ejpam-7061	187	3	)	)	PUNCT
ejpam-7061	187	4	)	)	PUNCT
ejpam-7061	188	1	̸=	̸=	PROPN
ejpam-7061	188	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	188	3	l(s2)×	l(s2)×	NOUN
ejpam-7061	188	4	·	·	PUNCT
ejpam-7061	188	5	·	·	PUNCT
ejpam-7061	188	6	·	·	PUNCT
ejpam-7061	188	7	×	×	NOUN
ejpam-7061	188	8	l(sn	l(sn	PROPN
ejpam-7061	188	9	)	)	PUNCT
ejpam-7061	188	10	,	,	PUNCT
ejpam-7061	188	11	then	then	ADV
ejpam-7061	188	12	l(s1)×	l(s1)×	PROPN
ejpam-7061	188	13	l(s2)×	l(s2)×	PROPN
ejpam-7061	188	14	·	·	PUNCT
ejpam-7061	188	15	·	·	PUNCT
ejpam-7061	188	16	·	·	PUNCT
ejpam-7061	188	17	×	×	NOUN
ejpam-7061	188	18	l(sn	l(sn	ADV
ejpam-7061	188	19	)	)	PUNCT
ejpam-7061	188	20	is	be	AUX
ejpam-7061	188	21	not	not	PART
ejpam-7061	188	22	a	a	DET
ejpam-7061	188	23	principal	principal	NOUN
ejpam-7061	188	24	left	leave	VERB
ejpam-7061	188	25	ideal	ideal	ADJ
ejpam-7061	188	26	.	.	PUNCT
ejpam-7061	189	1	proof	proof	NOUN
ejpam-7061	189	2	.	.	PUNCT
ejpam-7061	190	1	assume	assume	VERB
ejpam-7061	190	2	that	that	SCONJ
ejpam-7061	190	3	l((s1	l((s1	NOUN
ejpam-7061	190	4	,	,	PUNCT
ejpam-7061	190	5	s2	s2	PROPN
ejpam-7061	190	6	,	,	PUNCT
ejpam-7061	190	7	.	.	PUNCT
ejpam-7061	190	8	.	.	PUNCT
ejpam-7061	191	1	.	.	PUNCT
ejpam-7061	192	1	,	,	PUNCT
ejpam-7061	192	2	sn	sn	PROPN
ejpam-7061	192	3	)	)	PUNCT
ejpam-7061	192	4	)	)	PUNCT
ejpam-7061	193	1	̸=	̸=	PROPN
ejpam-7061	193	2	l(s1	l(s1	VERB
ejpam-7061	193	3	)	)	PUNCT
ejpam-7061	193	4	×	×	NOUN
ejpam-7061	193	5	l(s2	l(s2	NOUN
ejpam-7061	193	6	)	)	PUNCT
ejpam-7061	193	7	×	×	NOUN
ejpam-7061	193	8	·	·	PUNCT
ejpam-7061	193	9	·	·	PUNCT
ejpam-7061	193	10	·	·	PUNCT
ejpam-7061	193	11	×	×	NOUN
ejpam-7061	193	12	l(sn	l(sn	NUM
ejpam-7061	193	13	)	)	PUNCT
ejpam-7061	193	14	.	.	PUNCT
ejpam-7061	193	15	suppose	suppose	VERB
ejpam-7061	193	16	that	that	SCONJ
ejpam-7061	193	17	l(s1)×	l(s1)×	PROPN
ejpam-7061	193	18	l(s2)×	l(s2)×	PROPN
ejpam-7061	193	19	·	·	PUNCT
ejpam-7061	193	20	·	·	PUNCT
ejpam-7061	193	21	·	·	PUNCT
ejpam-7061	194	1	×	×	NOUN
ejpam-7061	194	2	l(sn	l(sn	ADV
ejpam-7061	194	3	)	)	PUNCT
ejpam-7061	194	4	is	be	AUX
ejpam-7061	194	5	a	a	DET
ejpam-7061	194	6	principal	principal	NOUN
ejpam-7061	194	7	left	leave	VERB
ejpam-7061	194	8	ideal	ideal	NOUN
ejpam-7061	194	9	of	of	ADP
ejpam-7061	194	10	s1	s1	PROPN
ejpam-7061	194	11	×	×	PROPN
ejpam-7061	194	12	s2	s2	NOUN
ejpam-7061	194	13	×	×	NOUN
ejpam-7061	194	14	·	·	PUNCT
ejpam-7061	194	15	·	·	PUNCT
ejpam-7061	194	16	·	·	PUNCT
ejpam-7061	195	1	×	×	NOUN
ejpam-7061	195	2	sn	sn	INTJ
ejpam-7061	195	3	.	.	PUNCT
ejpam-7061	196	1	then	then	ADV
ejpam-7061	196	2	l(s1)×	l(s1)×	PROPN
ejpam-7061	196	3	l(s2)×	l(s2)×	PROPN
ejpam-7061	196	4	·	·	PUNCT
ejpam-7061	196	5	·	·	PUNCT
ejpam-7061	196	6	·	·	PUNCT
ejpam-7061	197	1	×	×	NOUN
ejpam-7061	197	2	l(sn	l(sn	ADJ
ejpam-7061	197	3	)	)	PUNCT
ejpam-7061	197	4	=	=	SYM
ejpam-7061	197	5	l((t1	l((t1	NOUN
ejpam-7061	197	6	,	,	PUNCT
ejpam-7061	197	7	t2	t2	NOUN
ejpam-7061	197	8	,	,	PUNCT
ejpam-7061	197	9	.	.	PUNCT
ejpam-7061	197	10	.	.	PUNCT
ejpam-7061	197	11	.	.	PUNCT
ejpam-7061	197	12	,	,	PUNCT
ejpam-7061	197	13	tn	tn	PROPN
ejpam-7061	197	14	)	)	PUNCT
ejpam-7061	197	15	)	)	PUNCT
ejpam-7061	197	16	for	for	ADP
ejpam-7061	197	17	some	some	PRON
ejpam-7061	197	18	(	(	PUNCT
ejpam-7061	197	19	t1	t1	NOUN
ejpam-7061	197	20	,	,	PUNCT
ejpam-7061	197	21	t2	t2	NOUN
ejpam-7061	197	22	,	,	PUNCT
ejpam-7061	197	23	.	.	PUNCT
ejpam-7061	197	24	.	.	PUNCT
ejpam-7061	198	1	.	.	PUNCT
ejpam-7061	199	1	,	,	PUNCT
ejpam-7061	199	2	tn	tn	NOUN
ejpam-7061	199	3	)	)	PUNCT
ejpam-7061	199	4	∈	∈	PROPN
ejpam-7061	199	5	s1	s1	PROPN
ejpam-7061	199	6	×	×	PROPN
ejpam-7061	199	7	s2	s2	NOUN
ejpam-7061	199	8	×	×	NOUN
ejpam-7061	199	9	·	·	PUNCT
ejpam-7061	199	10	·	·	PUNCT
ejpam-7061	199	11	·	·	PUNCT
ejpam-7061	200	1	×	×	NOUN
ejpam-7061	200	2	sn	sn	INTJ
ejpam-7061	200	3	.	.	PUNCT
ejpam-7061	201	1	we	we	PRON
ejpam-7061	201	2	observe	observe	VERB
ejpam-7061	201	3	that	that	SCONJ
ejpam-7061	201	4	(	(	PUNCT
ejpam-7061	201	5	s1	s1	NOUN
ejpam-7061	201	6	,	,	PUNCT
ejpam-7061	201	7	s2	s2	NOUN
ejpam-7061	201	8	,	,	PUNCT
ejpam-7061	201	9	.	.	PUNCT
ejpam-7061	201	10	.	.	PUNCT
ejpam-7061	201	11	.	.	PUNCT
ejpam-7061	202	1	,	,	PUNCT
ejpam-7061	202	2	sn	sn	PROPN
ejpam-7061	202	3	)	)	PUNCT
ejpam-7061	202	4	∈	∈	PROPN
ejpam-7061	202	5	l(s1)×	l(s1)×	NOUN
ejpam-7061	202	6	l(s2)×	l(s2)×	X
ejpam-7061	202	7	·	·	PUNCT
ejpam-7061	202	8	·	·	PUNCT
ejpam-7061	202	9	·	·	PUNCT
ejpam-7061	203	1	×	×	NOUN
ejpam-7061	203	2	l(sn	l(sn	ADJ
ejpam-7061	203	3	)	)	PUNCT
ejpam-7061	203	4	=	=	SYM
ejpam-7061	203	5	l((t1	l((t1	NOUN
ejpam-7061	203	6	,	,	PUNCT
ejpam-7061	203	7	t2	t2	NOUN
ejpam-7061	203	8	,	,	PUNCT
ejpam-7061	203	9	.	.	PUNCT
ejpam-7061	203	10	.	.	PUNCT
ejpam-7061	203	11	.	.	PUNCT
ejpam-7061	203	12	,	,	PUNCT
ejpam-7061	203	13	tn	tn	PROPN
ejpam-7061	203	14	)	)	PUNCT
ejpam-7061	203	15	)	)	PUNCT
ejpam-7061	204	1	⊆	⊆	NUM
ejpam-7061	204	2	l(t1)×	l(t1)×	NOUN
ejpam-7061	204	3	l(t2)×	l(t2)×	NOUN
ejpam-7061	204	4	·	·	PUNCT
ejpam-7061	204	5	·	·	PUNCT
ejpam-7061	204	6	·	·	PUNCT
ejpam-7061	204	7	×	×	PROPN
ejpam-7061	204	8	l(tn	l(tn	NOUN
ejpam-7061	204	9	)	)	PUNCT
ejpam-7061	204	10	.	.	PUNCT
ejpam-7061	205	1	on	on	ADP
ejpam-7061	205	2	the	the	DET
ejpam-7061	205	3	same	same	ADJ
ejpam-7061	205	4	way	way	NOUN
ejpam-7061	205	5	,	,	PUNCT
ejpam-7061	205	6	we	we	PRON
ejpam-7061	205	7	also	also	ADV
ejpam-7061	205	8	obtain	obtain	VERB
ejpam-7061	205	9	(	(	PUNCT
ejpam-7061	205	10	t1	t1	NOUN
ejpam-7061	205	11	,	,	PUNCT
ejpam-7061	205	12	t2	t2	NOUN
ejpam-7061	205	13	,	,	PUNCT
ejpam-7061	205	14	.	.	PUNCT
ejpam-7061	205	15	.	.	PUNCT
ejpam-7061	206	1	.	.	PUNCT
ejpam-7061	207	1	,	,	PUNCT
ejpam-7061	207	2	tn	tn	PROPN
ejpam-7061	207	3	)	)	PUNCT
ejpam-7061	207	4	∈	∈	PROPN
ejpam-7061	207	5	l(s1)×l(s2)×	l(s1)×l(s2)×	NOUN
ejpam-7061	207	6	·	·	PUNCT
ejpam-7061	207	7	·	·	PUNCT
ejpam-7061	207	8	·	·	PUNCT
ejpam-7061	207	9	×l(sn	×l(sn	NOUN
ejpam-7061	207	10	)	)	PUNCT
ejpam-7061	207	11	.	.	PUNCT
ejpam-7061	208	1	these	these	PRON
ejpam-7061	208	2	imply	imply	VERB
ejpam-7061	208	3	that	that	SCONJ
ejpam-7061	208	4	s1s1	s1s1	ADP
ejpam-7061	208	5	×	×	NOUN
ejpam-7061	208	6	s2s2	s2s2	NOUN
ejpam-7061	208	7	×	×	NOUN
ejpam-7061	208	8	·	·	PUNCT
ejpam-7061	208	9	·	·	PUNCT
ejpam-7061	208	10	·	·	PUNCT
ejpam-7061	208	11	×	×	NOUN
ejpam-7061	208	12	snsn	snsn	NOUN
ejpam-7061	208	13	=	=	SYM
ejpam-7061	208	14	(	(	PUNCT
ejpam-7061	208	15	s1	s1	PROPN
ejpam-7061	208	16	×	×	PROPN
ejpam-7061	208	17	s2	s2	NOUN
ejpam-7061	208	18	×	×	NOUN
ejpam-7061	208	19	·	·	PUNCT
ejpam-7061	208	20	·	·	PUNCT
ejpam-7061	208	21	·	·	PUNCT
ejpam-7061	208	22	×	×	NOUN
ejpam-7061	208	23	sn)(s1	sn)(s1	NOUN
ejpam-7061	208	24	,	,	PUNCT
ejpam-7061	208	25	s2	s2	PROPN
ejpam-7061	208	26	,	,	PUNCT
ejpam-7061	208	27	.	.	PUNCT
ejpam-7061	208	28	.	.	PUNCT
ejpam-7061	208	29	.	.	PUNCT
ejpam-7061	209	1	,	,	PUNCT
ejpam-7061	209	2	sn	sn	PROPN
ejpam-7061	209	3	)	)	PUNCT
ejpam-7061	209	4	⊆	⊆	NUM
ejpam-7061	209	5	(	(	PUNCT
ejpam-7061	209	6	s1	s1	PROPN
ejpam-7061	209	7	×	×	PROPN
ejpam-7061	209	8	s2	s2	NOUN
ejpam-7061	209	9	×	×	NOUN
ejpam-7061	209	10	·	·	PUNCT
ejpam-7061	209	11	·	·	PUNCT
ejpam-7061	209	12	·	·	PUNCT
ejpam-7061	210	1	×	×	NOUN
ejpam-7061	210	2	sn)(l(t1)×	sn)(l(t1)×	PROPN
ejpam-7061	210	3	l(t2)×	l(t2)×	NOUN
ejpam-7061	210	4	·	·	PUNCT
ejpam-7061	210	5	·	·	PUNCT
ejpam-7061	210	6	·	·	PUNCT
ejpam-7061	210	7	×	×	PROPN
ejpam-7061	210	8	l(tn	l(tn	NOUN
ejpam-7061	210	9	)	)	PUNCT
ejpam-7061	210	10	)	)	PUNCT
ejpam-7061	211	1	=	=	PUNCT
ejpam-7061	212	1	s1t1	s1t1	NUM
ejpam-7061	212	2	×	×	NOUN
ejpam-7061	212	3	s2t2	s2t2	NUM
ejpam-7061	212	4	×	×	NOUN
ejpam-7061	212	5	·	·	PUNCT
ejpam-7061	212	6	·	·	PUNCT
ejpam-7061	212	7	·	·	PUNCT
ejpam-7061	213	1	×	×	NOUN
ejpam-7061	213	2	sntn	sntn	ADJ
ejpam-7061	213	3	=	=	SYM
ejpam-7061	213	4	(	(	PUNCT
ejpam-7061	213	5	s1	s1	PROPN
ejpam-7061	213	6	×	×	PROPN
ejpam-7061	213	7	s2	s2	NOUN
ejpam-7061	213	8	×	×	NOUN
ejpam-7061	213	9	·	·	PUNCT
ejpam-7061	213	10	·	·	PUNCT
ejpam-7061	213	11	·	·	PUNCT
ejpam-7061	214	1	×	×	PROPN
ejpam-7061	214	2	sn)(t1	sn)(t1	NOUN
ejpam-7061	214	3	,	,	PUNCT
ejpam-7061	214	4	t2	t2	NOUN
ejpam-7061	214	5	,	,	PUNCT
ejpam-7061	214	6	.	.	PUNCT
ejpam-7061	214	7	.	.	PUNCT
ejpam-7061	214	8	.	.	PUNCT
ejpam-7061	214	9	,	,	PUNCT
ejpam-7061	214	10	tn	tn	PROPN
ejpam-7061	214	11	)	)	PUNCT
ejpam-7061	214	12	⊆	⊆	NUM
ejpam-7061	214	13	(	(	PUNCT
ejpam-7061	214	14	s1	s1	PROPN
ejpam-7061	214	15	×	×	PROPN
ejpam-7061	214	16	s2	s2	NOUN
ejpam-7061	214	17	×	×	NOUN
ejpam-7061	214	18	·	·	PUNCT
ejpam-7061	214	19	·	·	PUNCT
ejpam-7061	214	20	·	·	PUNCT
ejpam-7061	215	1	×	×	NOUN
ejpam-7061	215	2	sn)(l(s1)×	sn)(l(s1)×	PROPN
ejpam-7061	215	3	l(s2)×	l(s2)×	PROPN
ejpam-7061	215	4	·	·	PUNCT
ejpam-7061	215	5	·	·	PUNCT
ejpam-7061	215	6	·	·	PUNCT
ejpam-7061	215	7	×	×	NOUN
ejpam-7061	215	8	l(sn	l(sn	ADJ
ejpam-7061	215	9	)	)	PUNCT
ejpam-7061	215	10	)	)	PUNCT
ejpam-7061	216	1	=	=	PRON
ejpam-7061	216	2	s1s1	s1s1	ADP
ejpam-7061	216	3	×	×	NOUN
ejpam-7061	216	4	s2s2	s2s2	NOUN
ejpam-7061	216	5	×	×	NOUN
ejpam-7061	216	6	·	·	PUNCT
ejpam-7061	216	7	·	·	PUNCT
ejpam-7061	216	8	·	·	PUNCT
ejpam-7061	216	9	×	×	NOUN
ejpam-7061	216	10	snsn	snsn	NOUN
ejpam-7061	216	11	.	.	PUNCT
ejpam-7061	217	1	thus	thus	ADV
ejpam-7061	217	2	,	,	PUNCT
ejpam-7061	217	3	s1s1	s1s1	VERB
ejpam-7061	217	4	×	×	NOUN
ejpam-7061	217	5	s2s2	s2s2	NOUN
ejpam-7061	217	6	×	×	NOUN
ejpam-7061	217	7	·	·	PUNCT
ejpam-7061	217	8	·	·	PUNCT
ejpam-7061	217	9	·	·	PUNCT
ejpam-7061	218	1	×	×	NOUN
ejpam-7061	218	2	snsn	snsn	NOUN
ejpam-7061	218	3	=	=	PUNCT
ejpam-7061	219	1	s1t1	s1t1	NUM
ejpam-7061	219	2	×	×	NOUN
ejpam-7061	219	3	s2t2	s2t2	NUM
ejpam-7061	219	4	×	×	NOUN
ejpam-7061	219	5	·	·	PUNCT
ejpam-7061	219	6	·	·	PUNCT
ejpam-7061	219	7	·	·	PUNCT
ejpam-7061	220	1	×	×	NOUN
ejpam-7061	220	2	sntn	sntn	NOUN
ejpam-7061	220	3	.	.	PUNCT
ejpam-7061	221	1	since	since	SCONJ
ejpam-7061	221	2	(	(	PUNCT
ejpam-7061	221	3	s1	s1	NOUN
ejpam-7061	221	4	,	,	PUNCT
ejpam-7061	221	5	s2	s2	NOUN
ejpam-7061	221	6	,	,	PUNCT
ejpam-7061	221	7	.	.	PUNCT
ejpam-7061	221	8	.	.	PUNCT
ejpam-7061	221	9	.	.	PUNCT
ejpam-7061	222	1	,	,	PUNCT
ejpam-7061	222	2	sn	sn	X
ejpam-7061	222	3	)	)	PUNCT
ejpam-7061	222	4	̸=	̸=	PROPN
ejpam-7061	222	5	(	(	PUNCT
ejpam-7061	222	6	t1	t1	PROPN
ejpam-7061	222	7	,	,	PUNCT
ejpam-7061	222	8	t2	t2	NOUN
ejpam-7061	222	9	,	,	PUNCT
ejpam-7061	222	10	.	.	PUNCT
ejpam-7061	222	11	.	.	PUNCT
ejpam-7061	223	1	.	.	PUNCT
ejpam-7061	224	1	,	,	PUNCT
ejpam-7061	224	2	tn	tn	PROPN
ejpam-7061	224	3	)	)	PUNCT
ejpam-7061	224	4	and	and	CCONJ
ejpam-7061	224	5	(	(	PUNCT
ejpam-7061	224	6	s1	s1	NOUN
ejpam-7061	224	7	,	,	PUNCT
ejpam-7061	224	8	s2	s2	NOUN
ejpam-7061	224	9	,	,	PUNCT
ejpam-7061	224	10	.	.	PUNCT
ejpam-7061	224	11	.	.	PUNCT
ejpam-7061	225	1	.	.	PUNCT
ejpam-7061	226	1	,	,	PUNCT
ejpam-7061	226	2	sn	sn	PROPN
ejpam-7061	226	3	)	)	PUNCT
ejpam-7061	226	4	∈	∈	PROPN
ejpam-7061	226	5	l((t1	l((t1	NOUN
ejpam-7061	226	6	,	,	PUNCT
ejpam-7061	226	7	t2	t2	NOUN
ejpam-7061	226	8	,	,	PUNCT
ejpam-7061	226	9	.	.	PUNCT
ejpam-7061	226	10	.	.	PUNCT
ejpam-7061	226	11	.	.	PUNCT
ejpam-7061	227	1	,	,	PUNCT
ejpam-7061	227	2	tn	tn	PROPN
ejpam-7061	227	3	)	)	PUNCT
ejpam-7061	227	4	)	)	PUNCT
ejpam-7061	227	5	,	,	PUNCT
ejpam-7061	227	6	we	we	PRON
ejpam-7061	227	7	have	have	VERB
ejpam-7061	227	8	that	that	DET
ejpam-7061	227	9	(	(	PUNCT
ejpam-7061	227	10	s1	s1	NOUN
ejpam-7061	227	11	,	,	PUNCT
ejpam-7061	227	12	s2	s2	NOUN
ejpam-7061	227	13	,	,	PUNCT
ejpam-7061	227	14	.	.	PUNCT
ejpam-7061	227	15	.	.	PUNCT
ejpam-7061	228	1	.	.	PUNCT
ejpam-7061	229	1	,	,	PUNCT
ejpam-7061	229	2	sn	sn	PROPN
ejpam-7061	229	3	)	)	PUNCT
ejpam-7061	229	4	∈	∈	PROPN
ejpam-7061	230	1	s1t1	s1t1	CCONJ
ejpam-7061	230	2	×	×	NOUN
ejpam-7061	230	3	s2t2	s2t2	NUM
ejpam-7061	230	4	×	×	NOUN
ejpam-7061	230	5	·	·	PUNCT
ejpam-7061	230	6	·	·	PUNCT
ejpam-7061	230	7	·	·	PUNCT
ejpam-7061	231	1	×	×	NOUN
ejpam-7061	231	2	sntn	sntn	NOUN
ejpam-7061	231	3	=	=	PUNCT
ejpam-7061	231	4	s1s1	s1s1	ADP
ejpam-7061	231	5	×	×	NOUN
ejpam-7061	231	6	s2s2	s2s2	NOUN
ejpam-7061	231	7	×	×	NOUN
ejpam-7061	231	8	·	·	PUNCT
ejpam-7061	231	9	·	·	PUNCT
ejpam-7061	231	10	·	·	PUNCT
ejpam-7061	231	11	×	×	NOUN
ejpam-7061	231	12	snsn	snsn	NOUN
ejpam-7061	231	13	.	.	PUNCT
ejpam-7061	232	1	p.	p.	NOUN
ejpam-7061	232	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	232	3	,	,	PUNCT
ejpam-7061	233	1	o.	o.	PROPN
ejpam-7061	233	2	pankoon	pankoon	NOUN
ejpam-7061	233	3	,	,	PUNCT
ejpam-7061	233	4	t.	t.	PROPN
ejpam-7061	233	5	changphas	changphas	PROPN
ejpam-7061	233	6	/	/	SYM
ejpam-7061	233	7	eur	eur	PROPN
ejpam-7061	233	8	.	.	PUNCT
ejpam-7061	234	1	j.	j.	PROPN
ejpam-7061	234	2	pure	pure	PROPN
ejpam-7061	234	3	appl	appl	PROPN
ejpam-7061	234	4	.	.	PROPN
ejpam-7061	234	5	math	math	PROPN
ejpam-7061	234	6	,	,	PUNCT
ejpam-7061	234	7	18	18	NUM
ejpam-7061	234	8	(	(	PUNCT
ejpam-7061	234	9	4	4	NUM
ejpam-7061	234	10	)	)	PUNCT
ejpam-7061	234	11	(	(	PUNCT
ejpam-7061	234	12	2025	2025	NUM
ejpam-7061	234	13	)	)	PUNCT
ejpam-7061	234	14	,	,	PUNCT
ejpam-7061	234	15	7061	7061	NUM
ejpam-7061	234	16	5	5	NUM
ejpam-7061	234	17	of	of	ADP
ejpam-7061	234	18	10	10	NUM
ejpam-7061	234	19	by	by	ADP
ejpam-7061	234	20	theorem	theorem	NOUN
ejpam-7061	234	21	1(i	1(i	NUM
ejpam-7061	234	22	)	)	PUNCT
ejpam-7061	234	23	,	,	PUNCT
ejpam-7061	234	24	l((s1	l((s1	PROPN
ejpam-7061	234	25	,	,	PUNCT
ejpam-7061	234	26	s2	s2	PROPN
ejpam-7061	234	27	,	,	PUNCT
ejpam-7061	234	28	.	.	PUNCT
ejpam-7061	234	29	.	.	PUNCT
ejpam-7061	235	1	.	.	PUNCT
ejpam-7061	236	1	,	,	PUNCT
ejpam-7061	236	2	sn	sn	PROPN
ejpam-7061	236	3	)	)	PUNCT
ejpam-7061	236	4	)	)	PUNCT
ejpam-7061	237	1	=	=	SYM
ejpam-7061	237	2	l(s1	l(s1	ADJ
ejpam-7061	237	3	)	)	PUNCT
ejpam-7061	237	4	×	×	NOUN
ejpam-7061	237	5	l(s2	l(s2	NOUN
ejpam-7061	237	6	)	)	PUNCT
ejpam-7061	237	7	×	×	NOUN
ejpam-7061	237	8	·	·	PUNCT
ejpam-7061	237	9	·	·	PUNCT
ejpam-7061	237	10	·	·	PUNCT
ejpam-7061	238	1	×	×	NOUN
ejpam-7061	238	2	l(sn	l(sn	NUM
ejpam-7061	238	3	)	)	PUNCT
ejpam-7061	238	4	.	.	PUNCT
ejpam-7061	239	1	this	this	PRON
ejpam-7061	239	2	contradicts	contradict	VERB
ejpam-7061	239	3	to	to	ADP
ejpam-7061	239	4	assumption	assumption	NOUN
ejpam-7061	239	5	.	.	PUNCT
ejpam-7061	240	1	therefore	therefore	ADV
ejpam-7061	240	2	,	,	PUNCT
ejpam-7061	240	3	l(s1)×	l(s1)×	NOUN
ejpam-7061	240	4	l(s2)×	l(s2)×	NOUN
ejpam-7061	240	5	·	·	PUNCT
ejpam-7061	240	6	·	·	PUNCT
ejpam-7061	240	7	·	·	PUNCT
ejpam-7061	240	8	×	×	NOUN
ejpam-7061	240	9	l(sn	l(sn	ADV
ejpam-7061	240	10	)	)	PUNCT
ejpam-7061	240	11	is	be	AUX
ejpam-7061	240	12	not	not	PART
ejpam-7061	240	13	a	a	DET
ejpam-7061	240	14	principal	principal	NOUN
ejpam-7061	240	15	left	leave	VERB
ejpam-7061	240	16	ideal	ideal	ADJ
ejpam-7061	240	17	.	.	PUNCT
ejpam-7061	241	1	the	the	DET
ejpam-7061	241	2	following	follow	VERB
ejpam-7061	241	3	example	example	NOUN
ejpam-7061	241	4	shows	show	VERB
ejpam-7061	241	5	that	that	SCONJ
ejpam-7061	241	6	the	the	DET
ejpam-7061	241	7	cartesian	cartesian	ADJ
ejpam-7061	241	8	product	product	NOUN
ejpam-7061	241	9	of	of	ADP
ejpam-7061	241	10	l	l	NOUN
ejpam-7061	241	11	-	-	PUNCT
ejpam-7061	241	12	classes	class	NOUN
ejpam-7061	241	13	need	need	AUX
ejpam-7061	241	14	not	not	PART
ejpam-7061	241	15	be	be	AUX
ejpam-7061	241	16	an	an	DET
ejpam-7061	241	17	l	l	NOUN
ejpam-7061	241	18	-	-	NOUN
ejpam-7061	241	19	class	class	NOUN
ejpam-7061	241	20	.	.	PUNCT
ejpam-7061	241	21	example	example	NOUN
ejpam-7061	242	1	2	2	NUM
ejpam-7061	242	2	.	.	X
ejpam-7061	242	3	from	from	ADP
ejpam-7061	242	4	the	the	DET
ejpam-7061	242	5	definition	definition	NOUN
ejpam-7061	242	6	of	of	ADP
ejpam-7061	242	7	the	the	DET
ejpam-7061	242	8	semigroup	semigroup	NOUN
ejpam-7061	242	9	s	s	X
ejpam-7061	242	10	in	in	ADP
ejpam-7061	242	11	example	example	NOUN
ejpam-7061	242	12	1	1	NUM
ejpam-7061	242	13	,	,	PUNCT
ejpam-7061	242	14	we	we	PRON
ejpam-7061	242	15	have	have	VERB
ejpam-7061	242	16	ls2	ls2	PROPN
ejpam-7061	242	17	×	×	PROPN
ejpam-7061	242	18	ls3	ls3	PROPN
ejpam-7061	242	19	̸=	̸=	PROPN
ejpam-7061	242	20	l(s2,s3	l(s2,s3	ADV
ejpam-7061	242	21	)	)	PUNCT
ejpam-7061	242	22	.	.	PUNCT
ejpam-7061	243	1	indeed	indeed	ADV
ejpam-7061	243	2	:	:	PUNCT
ejpam-7061	243	3	we	we	PRON
ejpam-7061	243	4	have	have	VERB
ejpam-7061	243	5	that	that	PRON
ejpam-7061	243	6	l(s2,s3	l(s2,s3	ADJ
ejpam-7061	243	7	)	)	PUNCT
ejpam-7061	243	8	=	=	SYM
ejpam-7061	243	9	{	{	PUNCT
ejpam-7061	243	10	(	(	PUNCT
ejpam-7061	243	11	s2	s2	PROPN
ejpam-7061	243	12	,	,	PUNCT
ejpam-7061	243	13	s3	s3	PROPN
ejpam-7061	243	14	)	)	PUNCT
ejpam-7061	243	15	}	}	PUNCT
ejpam-7061	243	16	and	and	CCONJ
ejpam-7061	243	17	ls2	ls2	PROPN
ejpam-7061	243	18	×	×	PROPN
ejpam-7061	243	19	ls3	ls3	PROPN
ejpam-7061	243	20	=	=	PRON
ejpam-7061	243	21	{	{	PUNCT
ejpam-7061	243	22	s2	s2	PROPN
ejpam-7061	243	23	,	,	PUNCT
ejpam-7061	243	24	s4	s4	PROPN
ejpam-7061	243	25	}	}	PUNCT
ejpam-7061	243	26	×	×	PROPN
ejpam-7061	243	27	{	{	PUNCT
ejpam-7061	243	28	s3	s3	PROPN
ejpam-7061	243	29	}	}	PUNCT
ejpam-7061	243	30	=	=	SYM
ejpam-7061	243	31	{	{	PUNCT
ejpam-7061	243	32	(	(	PUNCT
ejpam-7061	243	33	s2	s2	PROPN
ejpam-7061	243	34	,	,	PUNCT
ejpam-7061	243	35	s3	s3	PROPN
ejpam-7061	243	36	)	)	PUNCT
ejpam-7061	243	37	,	,	PUNCT
ejpam-7061	243	38	(	(	PUNCT
ejpam-7061	243	39	s4	s4	PROPN
ejpam-7061	243	40	,	,	PUNCT
ejpam-7061	243	41	s3	s3	PROPN
ejpam-7061	243	42	)	)	PUNCT
ejpam-7061	243	43	}	}	PUNCT
ejpam-7061	243	44	.	.	PUNCT
ejpam-7061	244	1	since	since	SCONJ
ejpam-7061	244	2	(	(	PUNCT
ejpam-7061	244	3	s4	s4	PROPN
ejpam-7061	244	4	,	,	PUNCT
ejpam-7061	244	5	s3	s3	PROPN
ejpam-7061	244	6	)	)	PUNCT
ejpam-7061	244	7	∈	∈	PROPN
ejpam-7061	244	8	ls2	ls2	PROPN
ejpam-7061	244	9	×	×	PROPN
ejpam-7061	244	10	ls3	ls3	PROPN
ejpam-7061	244	11	and	and	CCONJ
ejpam-7061	244	12	(	(	PUNCT
ejpam-7061	244	13	s4	s4	PROPN
ejpam-7061	244	14	,	,	PUNCT
ejpam-7061	244	15	s3	s3	PROPN
ejpam-7061	244	16	)	)	PUNCT
ejpam-7061	244	17	/∈	/∈	PUNCT
ejpam-7061	245	1	l(s2,s3	l(s2,s3	ADJ
ejpam-7061	245	2	)	)	PUNCT
ejpam-7061	245	3	,	,	PUNCT
ejpam-7061	245	4	we	we	PRON
ejpam-7061	245	5	get	get	VERB
ejpam-7061	245	6	ls2	ls2	PROPN
ejpam-7061	245	7	×	×	NOUN
ejpam-7061	245	8	ls3	ls3	PROPN
ejpam-7061	245	9	̸=	̸=	PROPN
ejpam-7061	245	10	l(s2,s3	l(s2,s3	ADV
ejpam-7061	245	11	)	)	PUNCT
ejpam-7061	245	12	.	.	PUNCT
ejpam-7061	246	1	this	this	PRON
ejpam-7061	246	2	shows	show	VERB
ejpam-7061	246	3	that	that	SCONJ
ejpam-7061	246	4	ls2	ls2	PROPN
ejpam-7061	246	5	×	×	PROPN
ejpam-7061	246	6	ls3	ls3	PROPN
ejpam-7061	246	7	is	be	AUX
ejpam-7061	246	8	not	not	PART
ejpam-7061	246	9	an	an	DET
ejpam-7061	246	10	l	l	NOUN
ejpam-7061	246	11	-	-	NOUN
ejpam-7061	246	12	class	class	NOUN
ejpam-7061	246	13	.	.	PUNCT
ejpam-7061	247	1	next	next	ADV
ejpam-7061	247	2	,	,	PUNCT
ejpam-7061	247	3	we	we	PRON
ejpam-7061	247	4	present	present	VERB
ejpam-7061	247	5	theorem	theorem	VERB
ejpam-7061	247	6	3	3	NUM
ejpam-7061	247	7	to	to	PART
ejpam-7061	247	8	mention	mention	VERB
ejpam-7061	247	9	the	the	DET
ejpam-7061	247	10	conclusion	conclusion	NOUN
ejpam-7061	247	11	of	of	ADP
ejpam-7061	247	12	ls1	ls1	PROPN
ejpam-7061	247	13	×	×	PROPN
ejpam-7061	247	14	ls2	ls2	PROPN
ejpam-7061	247	15	×	×	PROPN
ejpam-7061	247	16	·	·	PUNCT
ejpam-7061	247	17	·	·	PUNCT
ejpam-7061	247	18	·	·	PUNCT
ejpam-7061	248	1	×	×	NOUN
ejpam-7061	248	2	lsn	lsn	NOUN
ejpam-7061	248	3	and	and	CCONJ
ejpam-7061	248	4	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	248	5	...	...	PUNCT
ejpam-7061	248	6	,sn	,sn	PUNCT
ejpam-7061	248	7	)	)	PUNCT
ejpam-7061	248	8	.	.	PUNCT
ejpam-7061	249	1	then	then	ADV
ejpam-7061	249	2	we	we	PRON
ejpam-7061	249	3	give	give	VERB
ejpam-7061	249	4	a	a	DET
ejpam-7061	249	5	necessary	necessary	ADJ
ejpam-7061	249	6	and	and	CCONJ
ejpam-7061	249	7	sufficient	sufficient	ADJ
ejpam-7061	249	8	condition	condition	NOUN
ejpam-7061	249	9	when	when	SCONJ
ejpam-7061	249	10	ls1×ls2×	ls1×ls2×	X
ejpam-7061	249	11	·	·	PUNCT
ejpam-7061	249	12	·	·	PUNCT
ejpam-7061	249	13	·	·	PUNCT
ejpam-7061	249	14	×lsn	×lsn	NOUN
ejpam-7061	249	15	=	=	SYM
ejpam-7061	249	16	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	249	17	...	...	PUNCT
ejpam-7061	249	18	,sn	,sn	PUNCT
ejpam-7061	249	19	)	)	PUNCT
ejpam-7061	249	20	in	in	ADP
ejpam-7061	249	21	theorem	theorem	NOUN
ejpam-7061	249	22	4	4	NUM
ejpam-7061	249	23	.	.	PUNCT
ejpam-7061	250	1	furthermore	furthermore	ADV
ejpam-7061	250	2	,	,	PUNCT
ejpam-7061	250	3	we	we	PRON
ejpam-7061	250	4	provide	provide	VERB
ejpam-7061	250	5	the	the	DET
ejpam-7061	250	6	relation	relation	NOUN
ejpam-7061	250	7	between	between	ADP
ejpam-7061	250	8	the	the	DET
ejpam-7061	250	9	cartesian	cartesian	ADJ
ejpam-7061	250	10	product	product	NOUN
ejpam-7061	250	11	of	of	ADP
ejpam-7061	250	12	principal	principal	NOUN
ejpam-7061	250	13	left	leave	VERB
ejpam-7061	250	14	ideals	ideal	NOUN
ejpam-7061	250	15	and	and	CCONJ
ejpam-7061	250	16	the	the	DET
ejpam-7061	250	17	cartesian	cartesian	ADJ
ejpam-7061	250	18	product	product	NOUN
ejpam-7061	250	19	of	of	ADP
ejpam-7061	250	20	l	l	NOUN
ejpam-7061	250	21	-	-	PUNCT
ejpam-7061	250	22	classes	class	NOUN
ejpam-7061	250	23	in	in	ADP
ejpam-7061	250	24	theorem	theorem	NOUN
ejpam-7061	250	25	5	5	NUM
ejpam-7061	250	26	.	.	PUNCT
ejpam-7061	250	27	theorem	theorem	NOUN
ejpam-7061	250	28	3	3	X
ejpam-7061	250	29	.	.	PUNCT
ejpam-7061	251	1	let	let	VERB
ejpam-7061	251	2	si	si	PRON
ejpam-7061	251	3	be	be	AUX
ejpam-7061	251	4	a	a	DET
ejpam-7061	251	5	semigroup	semigroup	NOUN
ejpam-7061	251	6	and	and	CCONJ
ejpam-7061	251	7	let	let	VERB
ejpam-7061	251	8	si	si	PROPN
ejpam-7061	251	9	∈	∈	PROPN
ejpam-7061	251	10	si	si	X
ejpam-7061	251	11	where	where	SCONJ
ejpam-7061	251	12	i	i	PRON
ejpam-7061	251	13	=	=	NOUN
ejpam-7061	251	14	1	1	NUM
ejpam-7061	251	15	,	,	PUNCT
ejpam-7061	251	16	2	2	NUM
ejpam-7061	251	17	,	,	PUNCT
ejpam-7061	251	18	.	.	PUNCT
ejpam-7061	251	19	.	.	PUNCT
ejpam-7061	252	1	.	.	PUNCT
ejpam-7061	253	1	,	,	PUNCT
ejpam-7061	253	2	n.	n.	PROPN
ejpam-7061	253	3	then	then	ADV
ejpam-7061	253	4	the	the	DET
ejpam-7061	253	5	following	follow	VERB
ejpam-7061	253	6	statements	statement	NOUN
ejpam-7061	253	7	hold	hold	VERB
ejpam-7061	253	8	:	:	PUNCT
ejpam-7061	253	9	(	(	PUNCT
ejpam-7061	253	10	i	i	NOUN
ejpam-7061	253	11	)	)	PUNCT
ejpam-7061	253	12	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	253	13	...	...	PUNCT
ejpam-7061	253	14	,sn	,sn	PUNCT
ejpam-7061	253	15	)	)	PUNCT
ejpam-7061	254	1	⊆	⊆	NUM
ejpam-7061	254	2	ls1	ls1	NOUN
ejpam-7061	254	3	×	×	PROPN
ejpam-7061	254	4	ls2	ls2	PROPN
ejpam-7061	254	5	×	×	PROPN
ejpam-7061	254	6	·	·	PUNCT
ejpam-7061	254	7	·	·	PUNCT
ejpam-7061	254	8	·	·	PUNCT
ejpam-7061	254	9	×	×	NOUN
ejpam-7061	254	10	lsn	lsn	NOUN
ejpam-7061	254	11	;	;	PUNCT
ejpam-7061	254	12	(	(	PUNCT
ejpam-7061	254	13	ii	ii	NOUN
ejpam-7061	254	14	)	)	PUNCT
ejpam-7061	254	15	if	if	SCONJ
ejpam-7061	254	16	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	254	17	...	...	PUNCT
ejpam-7061	254	18	,sn	,sn	PUNCT
ejpam-7061	254	19	)	)	PUNCT
ejpam-7061	254	20	̸=	̸=	PROPN
ejpam-7061	254	21	ls1	ls1	VERB
ejpam-7061	254	22	×	×	PROPN
ejpam-7061	254	23	ls2	ls2	PROPN
ejpam-7061	254	24	×	×	PROPN
ejpam-7061	254	25	·	·	PUNCT
ejpam-7061	254	26	·	·	PUNCT
ejpam-7061	254	27	·	·	PUNCT
ejpam-7061	254	28	×	×	NOUN
ejpam-7061	254	29	lsn	lsn	NOUN
ejpam-7061	254	30	,	,	PUNCT
ejpam-7061	254	31	then	then	ADV
ejpam-7061	254	32	ls1	ls1	VERB
ejpam-7061	254	33	×	×	PROPN
ejpam-7061	254	34	ls2	ls2	PROPN
ejpam-7061	254	35	×	×	PROPN
ejpam-7061	254	36	·	·	PUNCT
ejpam-7061	254	37	·	·	PUNCT
ejpam-7061	254	38	·	·	PUNCT
ejpam-7061	255	1	×	×	NOUN
ejpam-7061	255	2	lsn	lsn	NOUN
ejpam-7061	255	3	contains	contain	VERB
ejpam-7061	255	4	at	at	ADV
ejpam-7061	255	5	least	least	ADV
ejpam-7061	255	6	two	two	NUM
ejpam-7061	255	7	l	l	NOUN
ejpam-7061	255	8	-	-	PUNCT
ejpam-7061	255	9	classes	class	NOUN
ejpam-7061	255	10	in	in	ADP
ejpam-7061	255	11	s1	s1	PROPN
ejpam-7061	255	12	×	×	PROPN
ejpam-7061	255	13	s2	s2	NOUN
ejpam-7061	255	14	×	×	NOUN
ejpam-7061	255	15	·	·	PUNCT
ejpam-7061	255	16	·	·	PUNCT
ejpam-7061	255	17	·	·	PUNCT
ejpam-7061	256	1	×	×	NOUN
ejpam-7061	256	2	sn	sn	INTJ
ejpam-7061	256	3	.	.	PUNCT
ejpam-7061	257	1	proof	proof	NOUN
ejpam-7061	257	2	.	.	PUNCT
ejpam-7061	258	1	(	(	PUNCT
ejpam-7061	258	2	i	i	NOUN
ejpam-7061	258	3	)	)	PUNCT
ejpam-7061	258	4	let	let	VERB
ejpam-7061	258	5	(	(	PUNCT
ejpam-7061	258	6	t1	t1	NOUN
ejpam-7061	258	7	,	,	PUNCT
ejpam-7061	258	8	t2	t2	NOUN
ejpam-7061	258	9	,	,	PUNCT
ejpam-7061	258	10	.	.	PUNCT
ejpam-7061	258	11	.	.	PUNCT
ejpam-7061	259	1	.	.	PUNCT
ejpam-7061	260	1	,	,	PUNCT
ejpam-7061	260	2	tn	tn	PROPN
ejpam-7061	260	3	)	)	PUNCT
ejpam-7061	260	4	∈	∈	PROPN
ejpam-7061	260	5	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	260	6	...	...	PUNCT
ejpam-7061	260	7	,sn	,sn	PUNCT
ejpam-7061	260	8	)	)	PUNCT
ejpam-7061	260	9	.	.	PUNCT
ejpam-7061	261	1	then	then	ADV
ejpam-7061	261	2	l((t1	l((t1	PROPN
ejpam-7061	261	3	,	,	PUNCT
ejpam-7061	261	4	t2	t2	NOUN
ejpam-7061	261	5	,	,	PUNCT
ejpam-7061	261	6	.	.	PUNCT
ejpam-7061	261	7	.	.	PUNCT
ejpam-7061	261	8	.	.	PUNCT
ejpam-7061	262	1	,	,	PUNCT
ejpam-7061	262	2	tn	tn	PROPN
ejpam-7061	262	3	)	)	PUNCT
ejpam-7061	262	4	)	)	PUNCT
ejpam-7061	263	1	=	=	PUNCT
ejpam-7061	263	2	l((s1	l((s1	PROPN
ejpam-7061	263	3	,	,	PUNCT
ejpam-7061	263	4	s2	s2	PROPN
ejpam-7061	263	5	,	,	PUNCT
ejpam-7061	263	6	.	.	PUNCT
ejpam-7061	263	7	.	.	PUNCT
ejpam-7061	263	8	.	.	PUNCT
ejpam-7061	263	9	,	,	PUNCT
ejpam-7061	263	10	sn	sn	PROPN
ejpam-7061	263	11	)	)	PUNCT
ejpam-7061	263	12	)	)	PUNCT
ejpam-7061	263	13	.	.	PUNCT
ejpam-7061	264	1	this	this	PRON
ejpam-7061	264	2	implies	imply	VERB
ejpam-7061	264	3	that	that	SCONJ
ejpam-7061	264	4	(	(	PUNCT
ejpam-7061	264	5	s1	s1	NOUN
ejpam-7061	264	6	,	,	PUNCT
ejpam-7061	264	7	s2	s2	NOUN
ejpam-7061	264	8	,	,	PUNCT
ejpam-7061	264	9	.	.	PUNCT
ejpam-7061	264	10	.	.	PUNCT
ejpam-7061	265	1	.	.	PUNCT
ejpam-7061	266	1	,	,	PUNCT
ejpam-7061	266	2	sn	sn	PROPN
ejpam-7061	266	3	)	)	PUNCT
ejpam-7061	266	4	∈	∈	PROPN
ejpam-7061	266	5	l((t1	l((t1	NOUN
ejpam-7061	266	6	,	,	PUNCT
ejpam-7061	266	7	t2	t2	NOUN
ejpam-7061	266	8	,	,	PUNCT
ejpam-7061	266	9	.	.	PUNCT
ejpam-7061	266	10	.	.	PUNCT
ejpam-7061	266	11	.	.	PUNCT
ejpam-7061	267	1	,	,	PUNCT
ejpam-7061	267	2	tn	tn	PROPN
ejpam-7061	267	3	)	)	PUNCT
ejpam-7061	267	4	)	)	PUNCT
ejpam-7061	268	1	⊆	⊆	NUM
ejpam-7061	268	2	l(t1)×	l(t1)×	NOUN
ejpam-7061	268	3	l(t2)×	l(t2)×	NOUN
ejpam-7061	268	4	·	·	PUNCT
ejpam-7061	268	5	·	·	PUNCT
ejpam-7061	268	6	·	·	PUNCT
ejpam-7061	268	7	×	×	PROPN
ejpam-7061	268	8	l(tn	l(tn	NOUN
ejpam-7061	268	9	)	)	PUNCT
ejpam-7061	268	10	and	and	CCONJ
ejpam-7061	268	11	(	(	PUNCT
ejpam-7061	268	12	t1	t1	NOUN
ejpam-7061	268	13	,	,	PUNCT
ejpam-7061	268	14	t2	t2	NOUN
ejpam-7061	268	15	,	,	PUNCT
ejpam-7061	268	16	.	.	PUNCT
ejpam-7061	268	17	.	.	PUNCT
ejpam-7061	268	18	.	.	PUNCT
ejpam-7061	269	1	,	,	PUNCT
ejpam-7061	269	2	tn	tn	PROPN
ejpam-7061	269	3	)	)	PUNCT
ejpam-7061	269	4	∈	∈	PROPN
ejpam-7061	269	5	l((s1	l((s1	PROPN
ejpam-7061	269	6	,	,	PUNCT
ejpam-7061	269	7	s2	s2	PROPN
ejpam-7061	269	8	,	,	PUNCT
ejpam-7061	269	9	.	.	PUNCT
ejpam-7061	269	10	.	.	PUNCT
ejpam-7061	269	11	.	.	PUNCT
ejpam-7061	270	1	,	,	PUNCT
ejpam-7061	270	2	sn	sn	PROPN
ejpam-7061	270	3	)	)	PUNCT
ejpam-7061	270	4	)	)	PUNCT
ejpam-7061	271	1	⊆	⊆	NUM
ejpam-7061	271	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	271	3	l(s2)×	l(s2)×	X
ejpam-7061	271	4	·	·	PUNCT
ejpam-7061	271	5	·	·	PUNCT
ejpam-7061	271	6	·	·	PUNCT
ejpam-7061	271	7	×	×	NOUN
ejpam-7061	271	8	l(sn	l(sn	NUM
ejpam-7061	271	9	)	)	PUNCT
ejpam-7061	271	10	.	.	PUNCT
ejpam-7061	272	1	thus	thus	ADV
ejpam-7061	272	2	,	,	PUNCT
ejpam-7061	272	3	si	si	PROPN
ejpam-7061	272	4	∈	∈	PROPN
ejpam-7061	272	5	l(ti	l(ti	PROPN
ejpam-7061	272	6	)	)	PUNCT
ejpam-7061	272	7	and	and	CCONJ
ejpam-7061	272	8	ti	ti	PROPN
ejpam-7061	272	9	∈	∈	PROPN
ejpam-7061	272	10	l(si	l(si	PROPN
ejpam-7061	272	11	)	)	PUNCT
ejpam-7061	272	12	for	for	ADP
ejpam-7061	272	13	all	all	DET
ejpam-7061	272	14	i	i	PRON
ejpam-7061	272	15	=	=	NOUN
ejpam-7061	272	16	1	1	NUM
ejpam-7061	272	17	,	,	PUNCT
ejpam-7061	272	18	2	2	NUM
ejpam-7061	272	19	,	,	PUNCT
ejpam-7061	272	20	.	.	PUNCT
ejpam-7061	272	21	.	.	PUNCT
ejpam-7061	273	1	.	.	PUNCT
ejpam-7061	274	1	,	,	PUNCT
ejpam-7061	274	2	n.	n.	PROPN
ejpam-7061	274	3	it	it	PRON
ejpam-7061	274	4	follows	follow	VERB
ejpam-7061	274	5	that	that	DET
ejpam-7061	274	6	l(si	l(si	PROPN
ejpam-7061	274	7	)	)	PUNCT
ejpam-7061	275	1	=	=	PUNCT
ejpam-7061	275	2	si	si	X
ejpam-7061	275	3	∪	∪	X
ejpam-7061	275	4	sisi	sisi	X
ejpam-7061	275	5	⊆	⊆	NUM
ejpam-7061	275	6	l(ti	l(ti	NOUN
ejpam-7061	275	7	)	)	PUNCT
ejpam-7061	275	8	∪	∪	ADP
ejpam-7061	275	9	sil(ti	sil(ti	NOUN
ejpam-7061	275	10	)	)	PUNCT
ejpam-7061	275	11	=	=	SYM
ejpam-7061	275	12	l(ti	l(ti	PROPN
ejpam-7061	275	13	)	)	PUNCT
ejpam-7061	275	14	.	.	PUNCT
ejpam-7061	276	1	similarly	similarly	ADV
ejpam-7061	276	2	,	,	PUNCT
ejpam-7061	276	3	we	we	PRON
ejpam-7061	276	4	obtain	obtain	VERB
ejpam-7061	276	5	l(ti	l(ti	NOUN
ejpam-7061	276	6	)	)	PUNCT
ejpam-7061	276	7	⊆	⊆	NUM
ejpam-7061	276	8	l(si	l(si	PROPN
ejpam-7061	276	9	)	)	PUNCT
ejpam-7061	276	10	.	.	PUNCT
ejpam-7061	277	1	thus	thus	ADV
ejpam-7061	277	2	,	,	PUNCT
ejpam-7061	277	3	l(si	l(si	PROPN
ejpam-7061	277	4	)	)	PUNCT
ejpam-7061	277	5	=	=	SYM
ejpam-7061	277	6	l(ti	l(ti	PROPN
ejpam-7061	277	7	)	)	PUNCT
ejpam-7061	277	8	for	for	ADP
ejpam-7061	277	9	all	all	DET
ejpam-7061	277	10	i	i	PRON
ejpam-7061	277	11	=	=	NOUN
ejpam-7061	277	12	1	1	NUM
ejpam-7061	277	13	,	,	PUNCT
ejpam-7061	277	14	2	2	NUM
ejpam-7061	277	15	,	,	PUNCT
ejpam-7061	277	16	.	.	PUNCT
ejpam-7061	277	17	.	.	PUNCT
ejpam-7061	278	1	.	.	PUNCT
ejpam-7061	279	1	,	,	PUNCT
ejpam-7061	279	2	n.	n.	PROPN
ejpam-7061	279	3	therefore	therefore	ADV
ejpam-7061	279	4	,	,	PUNCT
ejpam-7061	279	5	(	(	PUNCT
ejpam-7061	279	6	t1	t1	NOUN
ejpam-7061	279	7	,	,	PUNCT
ejpam-7061	279	8	t2	t2	NOUN
ejpam-7061	279	9	,	,	PUNCT
ejpam-7061	279	10	.	.	PUNCT
ejpam-7061	279	11	.	.	PUNCT
ejpam-7061	280	1	.	.	PUNCT
ejpam-7061	281	1	,	,	PUNCT
ejpam-7061	281	2	tn	tn	NOUN
ejpam-7061	281	3	)	)	PUNCT
ejpam-7061	281	4	∈	∈	PROPN
ejpam-7061	281	5	ls1	ls1	NOUN
ejpam-7061	281	6	×	×	PROPN
ejpam-7061	281	7	ls2	ls2	PROPN
ejpam-7061	281	8	×	×	PROPN
ejpam-7061	281	9	·	·	PUNCT
ejpam-7061	281	10	·	·	PUNCT
ejpam-7061	281	11	·	·	PUNCT
ejpam-7061	282	1	×	×	NOUN
ejpam-7061	282	2	lsn	lsn	NOUN
ejpam-7061	282	3	.	.	PUNCT
ejpam-7061	283	1	(	(	PUNCT
ejpam-7061	283	2	ii	ii	NOUN
ejpam-7061	283	3	)	)	PUNCT
ejpam-7061	283	4	assume	assume	VERB
ejpam-7061	283	5	that	that	SCONJ
ejpam-7061	283	6	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	283	7	...	...	PUNCT
ejpam-7061	283	8	,sn	,sn	PUNCT
ejpam-7061	283	9	)	)	PUNCT
ejpam-7061	283	10	̸=	̸=	PROPN
ejpam-7061	283	11	ls1×ls2×	ls1×ls2×	X
ejpam-7061	283	12	·	·	PUNCT
ejpam-7061	283	13	·	·	PUNCT
ejpam-7061	283	14	·	·	PUNCT
ejpam-7061	283	15	×lsn	×lsn	NOUN
ejpam-7061	283	16	.	.	PUNCT
ejpam-7061	284	1	by	by	ADP
ejpam-7061	284	2	(	(	PUNCT
ejpam-7061	284	3	i	i	NOUN
ejpam-7061	284	4	)	)	PUNCT
ejpam-7061	284	5	,	,	PUNCT
ejpam-7061	284	6	there	there	PRON
ejpam-7061	284	7	exists	exist	VERB
ejpam-7061	284	8	(	(	PUNCT
ejpam-7061	284	9	t1	t1	NOUN
ejpam-7061	284	10	,	,	PUNCT
ejpam-7061	284	11	t2	t2	NOUN
ejpam-7061	284	12	,	,	PUNCT
ejpam-7061	284	13	.	.	PUNCT
ejpam-7061	284	14	.	.	PUNCT
ejpam-7061	285	1	.	.	PUNCT
ejpam-7061	286	1	,	,	PUNCT
ejpam-7061	286	2	tn	tn	NOUN
ejpam-7061	286	3	)	)	PUNCT
ejpam-7061	286	4	∈	∈	PROPN
ejpam-7061	286	5	ls1	ls1	NOUN
ejpam-7061	286	6	×	×	PROPN
ejpam-7061	286	7	ls2	ls2	PROPN
ejpam-7061	286	8	×	×	PROPN
ejpam-7061	286	9	·	·	PUNCT
ejpam-7061	286	10	·	·	PUNCT
ejpam-7061	286	11	·	·	PUNCT
ejpam-7061	287	1	×	×	NOUN
ejpam-7061	287	2	lsn	lsn	ADP
ejpam-7061	287	3	such	such	ADJ
ejpam-7061	287	4	that	that	PRON
ejpam-7061	287	5	(	(	PUNCT
ejpam-7061	287	6	t1	t1	NOUN
ejpam-7061	287	7	,	,	PUNCT
ejpam-7061	287	8	t2	t2	NOUN
ejpam-7061	287	9	,	,	PUNCT
ejpam-7061	287	10	.	.	PUNCT
ejpam-7061	287	11	.	.	PUNCT
ejpam-7061	288	1	.	.	PUNCT
ejpam-7061	289	1	,	,	PUNCT
ejpam-7061	289	2	tn	tn	PROPN
ejpam-7061	289	3	)	)	PUNCT
ejpam-7061	289	4	/∈	/∈	PUNCT
ejpam-7061	290	1	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	290	2	...	...	PUNCT
ejpam-7061	290	3	,sn	,sn	PUNCT
ejpam-7061	290	4	)	)	PUNCT
ejpam-7061	290	5	.	.	PUNCT
ejpam-7061	291	1	then	then	ADV
ejpam-7061	291	2	lti	lti	PROPN
ejpam-7061	291	3	=	=	PROPN
ejpam-7061	291	4	lsi	lsi	PROPN
ejpam-7061	291	5	for	for	ADP
ejpam-7061	291	6	all	all	PRON
ejpam-7061	291	7	i	i	PRON
ejpam-7061	291	8	∈	∈	PROPN
ejpam-7061	291	9	{	{	PUNCT
ejpam-7061	291	10	1	1	NUM
ejpam-7061	291	11	,	,	PUNCT
ejpam-7061	291	12	2	2	NUM
ejpam-7061	291	13	,	,	PUNCT
ejpam-7061	291	14	.	.	PUNCT
ejpam-7061	291	15	.	.	PUNCT
ejpam-7061	291	16	.	.	PUNCT
ejpam-7061	291	17	,	,	PUNCT
ejpam-7061	291	18	n	n	CCONJ
ejpam-7061	291	19	}	}	PUNCT
ejpam-7061	291	20	.	.	PUNCT
ejpam-7061	292	1	thus	thus	ADV
ejpam-7061	292	2	,	,	PUNCT
ejpam-7061	292	3	l(t1,t2,	l(t1,t2,	NUM
ejpam-7061	292	4	...	...	PUNCT
ejpam-7061	292	5	,tn	,tn	PUNCT
ejpam-7061	292	6	)	)	PUNCT
ejpam-7061	292	7	⊆	⊆	NUM
ejpam-7061	292	8	lt1	lt1	NOUN
ejpam-7061	292	9	×	×	PROPN
ejpam-7061	292	10	lt2	lt2	PROPN
ejpam-7061	292	11	×	×	PROPN
ejpam-7061	292	12	·	·	PUNCT
ejpam-7061	292	13	·	·	PUNCT
ejpam-7061	292	14	·	·	PUNCT
ejpam-7061	292	15	×	×	PROPN
ejpam-7061	292	16	ltn	ltn	PROPN
ejpam-7061	292	17	=	=	PUNCT
ejpam-7061	292	18	ls1	ls1	VERB
ejpam-7061	292	19	×	×	PROPN
ejpam-7061	292	20	ls2	ls2	PROPN
ejpam-7061	292	21	×	×	PROPN
ejpam-7061	292	22	·	·	PUNCT
ejpam-7061	292	23	·	·	PUNCT
ejpam-7061	292	24	·	·	PUNCT
ejpam-7061	292	25	×	×	NOUN
ejpam-7061	292	26	lsn	lsn	NOUN
ejpam-7061	292	27	.	.	PUNCT
ejpam-7061	293	1	since	since	SCONJ
ejpam-7061	293	2	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	293	3	...	...	PUNCT
ejpam-7061	293	4	,sn	,sn	PUNCT
ejpam-7061	293	5	)	)	PUNCT
ejpam-7061	293	6	and	and	CCONJ
ejpam-7061	293	7	l(t1,t2,	l(t1,t2,	NUM
ejpam-7061	293	8	...	...	PUNCT
ejpam-7061	293	9	,tn	,tn	PUNCT
ejpam-7061	293	10	)	)	PUNCT
ejpam-7061	293	11	are	be	AUX
ejpam-7061	293	12	difference	difference	NOUN
ejpam-7061	293	13	,	,	PUNCT
ejpam-7061	293	14	we	we	PRON
ejpam-7061	293	15	obtain	obtain	VERB
ejpam-7061	293	16	that	that	PRON
ejpam-7061	293	17	ls1	ls1	VERB
ejpam-7061	293	18	×	×	PROPN
ejpam-7061	293	19	ls2	ls2	PROPN
ejpam-7061	293	20	×	×	PROPN
ejpam-7061	293	21	·	·	PUNCT
ejpam-7061	293	22	·	·	PUNCT
ejpam-7061	293	23	·	·	PUNCT
ejpam-7061	294	1	×	×	NOUN
ejpam-7061	294	2	lsn	lsn	NOUN
ejpam-7061	294	3	contains	contain	VERB
ejpam-7061	294	4	at	at	ADV
ejpam-7061	294	5	least	least	ADV
ejpam-7061	294	6	two	two	NUM
ejpam-7061	294	7	l	l	NOUN
ejpam-7061	294	8	-	-	PUNCT
ejpam-7061	294	9	classes	class	NOUN
ejpam-7061	294	10	of	of	ADP
ejpam-7061	294	11	s1	s1	PROPN
ejpam-7061	294	12	×	×	PROPN
ejpam-7061	294	13	s2	s2	NOUN
ejpam-7061	294	14	×	×	NOUN
ejpam-7061	294	15	.	.	PUNCT
ejpam-7061	294	16	.	.	PUNCT
ejpam-7061	295	1	.×	.×	PROPN
ejpam-7061	295	2	sn	sn	PROPN
ejpam-7061	295	3	.	.	PUNCT
ejpam-7061	296	1	p.	p.	PROPN
ejpam-7061	296	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	296	3	,	,	PUNCT
ejpam-7061	297	1	o.	o.	PROPN
ejpam-7061	297	2	pankoon	pankoon	NOUN
ejpam-7061	297	3	,	,	PUNCT
ejpam-7061	297	4	t.	t.	PROPN
ejpam-7061	297	5	changphas	changphas	PROPN
ejpam-7061	297	6	/	/	SYM
ejpam-7061	297	7	eur	eur	PROPN
ejpam-7061	297	8	.	.	PUNCT
ejpam-7061	298	1	j.	j.	PROPN
ejpam-7061	298	2	pure	pure	PROPN
ejpam-7061	298	3	appl	appl	PROPN
ejpam-7061	298	4	.	.	PROPN
ejpam-7061	298	5	math	math	PROPN
ejpam-7061	298	6	,	,	PUNCT
ejpam-7061	298	7	18	18	NUM
ejpam-7061	298	8	(	(	PUNCT
ejpam-7061	298	9	4	4	NUM
ejpam-7061	298	10	)	)	PUNCT
ejpam-7061	298	11	(	(	PUNCT
ejpam-7061	298	12	2025	2025	NUM
ejpam-7061	298	13	)	)	PUNCT
ejpam-7061	298	14	,	,	PUNCT
ejpam-7061	298	15	7061	7061	NUM
ejpam-7061	298	16	6	6	NUM
ejpam-7061	298	17	of	of	ADP
ejpam-7061	298	18	10	10	NUM
ejpam-7061	298	19	theorem	theorem	NOUN
ejpam-7061	298	20	4	4	NUM
ejpam-7061	298	21	.	.	PUNCT
ejpam-7061	299	1	let	let	VERB
ejpam-7061	299	2	si	si	PRON
ejpam-7061	299	3	be	be	AUX
ejpam-7061	299	4	a	a	DET
ejpam-7061	299	5	semigroup	semigroup	NOUN
ejpam-7061	299	6	and	and	CCONJ
ejpam-7061	299	7	let	let	VERB
ejpam-7061	299	8	si	si	PROPN
ejpam-7061	299	9	∈	∈	PROPN
ejpam-7061	299	10	si	si	X
ejpam-7061	300	1	where	where	SCONJ
ejpam-7061	300	2	i	i	PRON
ejpam-7061	300	3	=	=	NOUN
ejpam-7061	300	4	1	1	NUM
ejpam-7061	300	5	,	,	PUNCT
ejpam-7061	300	6	2	2	NUM
ejpam-7061	300	7	,	,	PUNCT
ejpam-7061	300	8	.	.	PUNCT
ejpam-7061	300	9	.	.	PUNCT
ejpam-7061	301	1	.	.	PUNCT
ejpam-7061	302	1	,	,	PUNCT
ejpam-7061	302	2	n.	n.	PROPN
ejpam-7061	302	3	then	then	ADV
ejpam-7061	302	4	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	302	5	...	...	PUNCT
ejpam-7061	302	6	,sn	,sn	PUNCT
ejpam-7061	302	7	)	)	PUNCT
ejpam-7061	303	1	=	=	PUNCT
ejpam-7061	303	2	ls1	ls1	VERB
ejpam-7061	303	3	×	×	PROPN
ejpam-7061	303	4	ls2	ls2	PROPN
ejpam-7061	303	5	×	×	PROPN
ejpam-7061	303	6	·	·	PUNCT
ejpam-7061	303	7	·	·	PUNCT
ejpam-7061	303	8	·	·	PUNCT
ejpam-7061	304	1	×	×	NOUN
ejpam-7061	304	2	lsn	lsn	NOUN
ejpam-7061	304	3	if	if	SCONJ
ejpam-7061	304	4	and	and	CCONJ
ejpam-7061	304	5	only	only	ADV
ejpam-7061	304	6	if	if	SCONJ
ejpam-7061	304	7	at	at	ADV
ejpam-7061	304	8	least	least	ADJ
ejpam-7061	304	9	one	one	NUM
ejpam-7061	304	10	of	of	ADP
ejpam-7061	304	11	the	the	DET
ejpam-7061	304	12	following	follow	VERB
ejpam-7061	304	13	conditions	condition	NOUN
ejpam-7061	304	14	is	be	AUX
ejpam-7061	304	15	satisfied	satisfied	ADJ
ejpam-7061	304	16	:	:	PUNCT
ejpam-7061	304	17	(	(	PUNCT
ejpam-7061	304	18	i	i	NOUN
ejpam-7061	304	19	)	)	PUNCT
ejpam-7061	304	20	lsi	lsi	PROPN
ejpam-7061	304	21	=	=	SYM
ejpam-7061	304	22	{	{	PUNCT
ejpam-7061	304	23	si	si	X
ejpam-7061	304	24	}	}	PUNCT
ejpam-7061	304	25	for	for	ADP
ejpam-7061	304	26	all	all	PRON
ejpam-7061	304	27	i	i	PRON
ejpam-7061	304	28	∈	∈	PROPN
ejpam-7061	304	29	{	{	PUNCT
ejpam-7061	304	30	1	1	NUM
ejpam-7061	304	31	,	,	PUNCT
ejpam-7061	304	32	2	2	NUM
ejpam-7061	304	33	,	,	PUNCT
ejpam-7061	304	34	.	.	PUNCT
ejpam-7061	304	35	.	.	PUNCT
ejpam-7061	305	1	.	.	PUNCT
ejpam-7061	306	1	,	,	PUNCT
ejpam-7061	306	2	n	n	CCONJ
ejpam-7061	306	3	}	}	PUNCT
ejpam-7061	306	4	;	;	PUNCT
ejpam-7061	306	5	(	(	PUNCT
ejpam-7061	306	6	ii	ii	NOUN
ejpam-7061	306	7	)	)	PUNCT
ejpam-7061	306	8	si	si	PROPN
ejpam-7061	306	9	∈	∈	PROPN
ejpam-7061	306	10	sisi	sisi	PROPN
ejpam-7061	306	11	for	for	ADP
ejpam-7061	306	12	all	all	PRON
ejpam-7061	306	13	i	i	PRON
ejpam-7061	306	14	∈	∈	PROPN
ejpam-7061	306	15	{	{	PUNCT
ejpam-7061	306	16	1	1	NUM
ejpam-7061	306	17	,	,	PUNCT
ejpam-7061	306	18	2	2	NUM
ejpam-7061	306	19	,	,	PUNCT
ejpam-7061	306	20	.	.	PUNCT
ejpam-7061	306	21	.	.	PUNCT
ejpam-7061	307	1	.	.	PUNCT
ejpam-7061	307	2	,	,	PUNCT
ejpam-7061	308	1	n	n	CCONJ
ejpam-7061	308	2	}	}	PUNCT
ejpam-7061	308	3	.	.	PUNCT
ejpam-7061	309	1	proof	proof	NOUN
ejpam-7061	309	2	.	.	PUNCT
ejpam-7061	310	1	assume	assume	VERB
ejpam-7061	310	2	that	that	SCONJ
ejpam-7061	310	3	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	310	4	...	...	PUNCT
ejpam-7061	310	5	,sn	,sn	PUNCT
ejpam-7061	310	6	)	)	PUNCT
ejpam-7061	310	7	=	=	PUNCT
ejpam-7061	311	1	ls1×ls2×	ls1×ls2×	X
ejpam-7061	311	2	·	·	PUNCT
ejpam-7061	311	3	·	·	PUNCT
ejpam-7061	311	4	·	·	PUNCT
ejpam-7061	311	5	×lsn	×lsn	NOUN
ejpam-7061	311	6	.	.	PUNCT
ejpam-7061	312	1	if	if	SCONJ
ejpam-7061	312	2	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	312	3	...	...	PUNCT
ejpam-7061	312	4	,sn	,sn	PUNCT
ejpam-7061	312	5	)	)	PUNCT
ejpam-7061	313	1	=	=	PRON
ejpam-7061	313	2	{	{	PUNCT
ejpam-7061	313	3	(	(	PUNCT
ejpam-7061	313	4	s1	s1	NOUN
ejpam-7061	313	5	,	,	PUNCT
ejpam-7061	313	6	s2	s2	NOUN
ejpam-7061	313	7	,	,	PUNCT
ejpam-7061	313	8	.	.	PUNCT
ejpam-7061	313	9	.	.	PUNCT
ejpam-7061	313	10	.	.	PUNCT
ejpam-7061	314	1	,	,	PUNCT
ejpam-7061	314	2	sn	sn	PROPN
ejpam-7061	314	3	)	)	PUNCT
ejpam-7061	314	4	}	}	PUNCT
ejpam-7061	314	5	,	,	PUNCT
ejpam-7061	314	6	then	then	ADV
ejpam-7061	314	7	by	by	ADP
ejpam-7061	314	8	assumption	assumption	NOUN
ejpam-7061	314	9	,	,	PUNCT
ejpam-7061	314	10	we	we	PRON
ejpam-7061	314	11	have	have	AUX
ejpam-7061	314	12	ls1	ls1	VERB
ejpam-7061	314	13	×	×	PROPN
ejpam-7061	314	14	ls2	ls2	PROPN
ejpam-7061	314	15	×	×	PROPN
ejpam-7061	314	16	·	·	PUNCT
ejpam-7061	314	17	·	·	PUNCT
ejpam-7061	314	18	·	·	PUNCT
ejpam-7061	315	1	×	×	NOUN
ejpam-7061	315	2	lsn	lsn	NOUN
ejpam-7061	315	3	=	=	SYM
ejpam-7061	315	4	{	{	PUNCT
ejpam-7061	315	5	(	(	PUNCT
ejpam-7061	315	6	s1	s1	NOUN
ejpam-7061	315	7	,	,	PUNCT
ejpam-7061	315	8	s2	s2	NOUN
ejpam-7061	315	9	,	,	PUNCT
ejpam-7061	315	10	.	.	PUNCT
ejpam-7061	315	11	.	.	PUNCT
ejpam-7061	315	12	.	.	PUNCT
ejpam-7061	316	1	,	,	PUNCT
ejpam-7061	316	2	sn	sn	PROPN
ejpam-7061	316	3	)	)	PUNCT
ejpam-7061	316	4	}	}	PUNCT
ejpam-7061	316	5	.	.	PUNCT
ejpam-7061	317	1	thus	thus	ADV
ejpam-7061	317	2	,	,	PUNCT
ejpam-7061	317	3	lsi	lsi	PROPN
ejpam-7061	317	4	=	=	SYM
ejpam-7061	317	5	{	{	PUNCT
ejpam-7061	317	6	si	si	X
ejpam-7061	317	7	}	}	PUNCT
ejpam-7061	317	8	for	for	ADP
ejpam-7061	317	9	all	all	PRON
ejpam-7061	317	10	i	i	PRON
ejpam-7061	317	11	∈	∈	PROPN
ejpam-7061	317	12	{	{	PUNCT
ejpam-7061	317	13	1	1	NUM
ejpam-7061	317	14	,	,	PUNCT
ejpam-7061	317	15	2	2	NUM
ejpam-7061	317	16	,	,	PUNCT
ejpam-7061	317	17	.	.	PUNCT
ejpam-7061	317	18	.	.	PUNCT
ejpam-7061	317	19	.	.	PUNCT
ejpam-7061	317	20	,	,	PUNCT
ejpam-7061	317	21	n	n	CCONJ
ejpam-7061	317	22	}	}	PUNCT
ejpam-7061	317	23	.	.	PUNCT
ejpam-7061	318	1	suppose	suppose	VERB
ejpam-7061	318	2	that	that	SCONJ
ejpam-7061	318	3	there	there	PRON
ejpam-7061	318	4	exists	exist	VERB
ejpam-7061	318	5	(	(	PUNCT
ejpam-7061	318	6	t1	t1	NOUN
ejpam-7061	318	7	,	,	PUNCT
ejpam-7061	318	8	t2	t2	NOUN
ejpam-7061	318	9	,	,	PUNCT
ejpam-7061	318	10	.	.	PUNCT
ejpam-7061	318	11	.	.	PUNCT
ejpam-7061	319	1	.	.	PUNCT
ejpam-7061	320	1	,	,	PUNCT
ejpam-7061	320	2	tn	tn	PROPN
ejpam-7061	320	3	)	)	PUNCT
ejpam-7061	320	4	∈	∈	PROPN
ejpam-7061	320	5	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	320	6	...	...	PUNCT
ejpam-7061	320	7	,sn	,sn	PUNCT
ejpam-7061	320	8	)	)	PUNCT
ejpam-7061	321	1	such	such	ADJ
ejpam-7061	321	2	that	that	SCONJ
ejpam-7061	321	3	(	(	PUNCT
ejpam-7061	321	4	t1	t1	NOUN
ejpam-7061	321	5	,	,	PUNCT
ejpam-7061	321	6	t2	t2	NOUN
ejpam-7061	321	7	,	,	PUNCT
ejpam-7061	321	8	.	.	PUNCT
ejpam-7061	321	9	.	.	PUNCT
ejpam-7061	321	10	.	.	PUNCT
ejpam-7061	321	11	,	,	PUNCT
ejpam-7061	321	12	tn	tn	PROPN
ejpam-7061	321	13	)	)	PUNCT
ejpam-7061	321	14	̸=	̸=	PROPN
ejpam-7061	321	15	(	(	PUNCT
ejpam-7061	321	16	s1	s1	PROPN
ejpam-7061	321	17	,	,	PUNCT
ejpam-7061	321	18	s2	s2	NOUN
ejpam-7061	321	19	,	,	PUNCT
ejpam-7061	321	20	.	.	PUNCT
ejpam-7061	321	21	.	.	PUNCT
ejpam-7061	321	22	.	.	PUNCT
ejpam-7061	321	23	,	,	PUNCT
ejpam-7061	321	24	sn	sn	PROPN
ejpam-7061	321	25	)	)	PUNCT
ejpam-7061	321	26	.	.	PUNCT
ejpam-7061	322	1	then	then	ADV
ejpam-7061	322	2	l((t1	l((t1	PROPN
ejpam-7061	322	3	,	,	PUNCT
ejpam-7061	322	4	t2	t2	NOUN
ejpam-7061	322	5	,	,	PUNCT
ejpam-7061	322	6	.	.	PUNCT
ejpam-7061	322	7	.	.	PUNCT
ejpam-7061	322	8	.	.	PUNCT
ejpam-7061	323	1	,	,	PUNCT
ejpam-7061	323	2	tn	tn	PROPN
ejpam-7061	323	3	)	)	PUNCT
ejpam-7061	323	4	)	)	PUNCT
ejpam-7061	324	1	=	=	PUNCT
ejpam-7061	324	2	l((s1	l((s1	PROPN
ejpam-7061	324	3	,	,	PUNCT
ejpam-7061	324	4	s2	s2	PROPN
ejpam-7061	324	5	,	,	PUNCT
ejpam-7061	324	6	.	.	PUNCT
ejpam-7061	324	7	.	.	PUNCT
ejpam-7061	324	8	.	.	PUNCT
ejpam-7061	324	9	,	,	PUNCT
ejpam-7061	324	10	sn	sn	PROPN
ejpam-7061	324	11	)	)	PUNCT
ejpam-7061	324	12	)	)	PUNCT
ejpam-7061	324	13	.	.	PUNCT
ejpam-7061	325	1	since	since	SCONJ
ejpam-7061	325	2	(	(	PUNCT
ejpam-7061	325	3	t1	t1	NOUN
ejpam-7061	325	4	,	,	PUNCT
ejpam-7061	325	5	t2	t2	NOUN
ejpam-7061	325	6	,	,	PUNCT
ejpam-7061	325	7	.	.	PUNCT
ejpam-7061	325	8	.	.	PUNCT
ejpam-7061	325	9	.	.	PUNCT
ejpam-7061	325	10	,	,	PUNCT
ejpam-7061	325	11	tn	tn	PROPN
ejpam-7061	325	12	)	)	PUNCT
ejpam-7061	325	13	∈	∈	PROPN
ejpam-7061	325	14	l((s1	l((s1	PROPN
ejpam-7061	325	15	,	,	PUNCT
ejpam-7061	325	16	s2	s2	PROPN
ejpam-7061	325	17	,	,	PUNCT
ejpam-7061	325	18	.	.	PUNCT
ejpam-7061	325	19	.	.	PUNCT
ejpam-7061	325	20	.	.	PUNCT
ejpam-7061	326	1	,	,	PUNCT
ejpam-7061	326	2	sn	sn	PROPN
ejpam-7061	326	3	)	)	PUNCT
ejpam-7061	326	4	)	)	PUNCT
ejpam-7061	327	1	and	and	CCONJ
ejpam-7061	327	2	(	(	PUNCT
ejpam-7061	327	3	t1	t1	NOUN
ejpam-7061	327	4	,	,	PUNCT
ejpam-7061	327	5	t2	t2	NOUN
ejpam-7061	327	6	,	,	PUNCT
ejpam-7061	327	7	.	.	PUNCT
ejpam-7061	327	8	.	.	PUNCT
ejpam-7061	328	1	.	.	PUNCT
ejpam-7061	329	1	,	,	PUNCT
ejpam-7061	329	2	tn	tn	PROPN
ejpam-7061	329	3	)	)	PUNCT
ejpam-7061	329	4	̸=	̸=	PROPN
ejpam-7061	329	5	(	(	PUNCT
ejpam-7061	329	6	s1	s1	PROPN
ejpam-7061	329	7	,	,	PUNCT
ejpam-7061	329	8	s2	s2	NOUN
ejpam-7061	329	9	,	,	PUNCT
ejpam-7061	329	10	.	.	PUNCT
ejpam-7061	329	11	.	.	PUNCT
ejpam-7061	329	12	.	.	PUNCT
ejpam-7061	330	1	,	,	PUNCT
ejpam-7061	330	2	sn	sn	PROPN
ejpam-7061	330	3	)	)	PUNCT
ejpam-7061	330	4	,	,	PUNCT
ejpam-7061	330	5	we	we	PRON
ejpam-7061	330	6	have	have	VERB
ejpam-7061	330	7	that	that	PRON
ejpam-7061	330	8	(	(	PUNCT
ejpam-7061	330	9	t1	t1	NOUN
ejpam-7061	330	10	,	,	PUNCT
ejpam-7061	330	11	t2	t2	NOUN
ejpam-7061	330	12	,	,	PUNCT
ejpam-7061	330	13	.	.	PUNCT
ejpam-7061	330	14	.	.	PUNCT
ejpam-7061	331	1	.	.	PUNCT
ejpam-7061	332	1	,	,	PUNCT
ejpam-7061	332	2	tn	tn	PROPN
ejpam-7061	332	3	)	)	PUNCT
ejpam-7061	332	4	∈	∈	PROPN
ejpam-7061	332	5	s1s1	s1s1	VERB
ejpam-7061	332	6	×	×	NOUN
ejpam-7061	332	7	s2s2	s2s2	NOUN
ejpam-7061	332	8	×	×	NOUN
ejpam-7061	332	9	·	·	PUNCT
ejpam-7061	332	10	·	·	PUNCT
ejpam-7061	333	1	·	·	PUNCT
ejpam-7061	333	2	×	×	NOUN
ejpam-7061	333	3	snsn	snsn	NOUN
ejpam-7061	333	4	.	.	PUNCT
ejpam-7061	334	1	similarly	similarly	ADV
ejpam-7061	334	2	,	,	PUNCT
ejpam-7061	334	3	we	we	PRON
ejpam-7061	334	4	get	get	VERB
ejpam-7061	334	5	(	(	PUNCT
ejpam-7061	334	6	s1	s1	NOUN
ejpam-7061	334	7	,	,	PUNCT
ejpam-7061	334	8	s2	s2	NOUN
ejpam-7061	334	9	,	,	PUNCT
ejpam-7061	334	10	.	.	PUNCT
ejpam-7061	334	11	.	.	PUNCT
ejpam-7061	334	12	.	.	PUNCT
ejpam-7061	335	1	,	,	PUNCT
ejpam-7061	335	2	sn	sn	PROPN
ejpam-7061	335	3	)	)	PUNCT
ejpam-7061	335	4	∈	∈	PROPN
ejpam-7061	336	1	s1t1	s1t1	CCONJ
ejpam-7061	336	2	×	×	NOUN
ejpam-7061	336	3	s2t2	s2t2	NUM
ejpam-7061	336	4	×	×	NOUN
ejpam-7061	336	5	·	·	PUNCT
ejpam-7061	336	6	·	·	PUNCT
ejpam-7061	336	7	·	·	PUNCT
ejpam-7061	337	1	×	×	NOUN
ejpam-7061	337	2	sntn	sntn	NOUN
ejpam-7061	337	3	.	.	PUNCT
ejpam-7061	338	1	thus	thus	ADV
ejpam-7061	338	2	,	,	PUNCT
ejpam-7061	338	3	(	(	PUNCT
ejpam-7061	338	4	s1	s1	NOUN
ejpam-7061	338	5	,	,	PUNCT
ejpam-7061	338	6	s2	s2	NOUN
ejpam-7061	338	7	,	,	PUNCT
ejpam-7061	338	8	.	.	PUNCT
ejpam-7061	338	9	.	.	PUNCT
ejpam-7061	338	10	.	.	PUNCT
ejpam-7061	339	1	,	,	PUNCT
ejpam-7061	339	2	sn	sn	PROPN
ejpam-7061	339	3	)	)	PUNCT
ejpam-7061	339	4	∈	∈	PROPN
ejpam-7061	340	1	s1t1	s1t1	CCONJ
ejpam-7061	340	2	×	×	NOUN
ejpam-7061	340	3	s2t2	s2t2	NUM
ejpam-7061	340	4	×	×	NOUN
ejpam-7061	340	5	·	·	PUNCT
ejpam-7061	340	6	·	·	PUNCT
ejpam-7061	340	7	·	·	PUNCT
ejpam-7061	341	1	×	×	NOUN
ejpam-7061	341	2	sntn	sntn	ADJ
ejpam-7061	341	3	=	=	SYM
ejpam-7061	341	4	(	(	PUNCT
ejpam-7061	341	5	s1	s1	PROPN
ejpam-7061	341	6	×	×	PROPN
ejpam-7061	341	7	s2	s2	NOUN
ejpam-7061	341	8	×	×	NOUN
ejpam-7061	341	9	·	·	PUNCT
ejpam-7061	341	10	·	·	PUNCT
ejpam-7061	341	11	·	·	PUNCT
ejpam-7061	342	1	×	×	PROPN
ejpam-7061	342	2	sn)(t1	sn)(t1	NOUN
ejpam-7061	342	3	,	,	PUNCT
ejpam-7061	342	4	t2	t2	NOUN
ejpam-7061	342	5	,	,	PUNCT
ejpam-7061	342	6	.	.	PUNCT
ejpam-7061	342	7	.	.	PUNCT
ejpam-7061	342	8	.	.	PUNCT
ejpam-7061	342	9	,	,	PUNCT
ejpam-7061	342	10	tn	tn	PROPN
ejpam-7061	342	11	)	)	PUNCT
ejpam-7061	342	12	⊆	⊆	NUM
ejpam-7061	342	13	(	(	PUNCT
ejpam-7061	342	14	s1	s1	PROPN
ejpam-7061	342	15	×	×	PROPN
ejpam-7061	342	16	s2	s2	NOUN
ejpam-7061	342	17	×	×	NOUN
ejpam-7061	342	18	·	·	PUNCT
ejpam-7061	342	19	·	·	PUNCT
ejpam-7061	342	20	·	·	PUNCT
ejpam-7061	343	1	×	×	NOUN
ejpam-7061	343	2	sn)(s1s1	sn)(s1s1	NOUN
ejpam-7061	343	3	×	×	NOUN
ejpam-7061	343	4	s2s2	s2s2	NOUN
ejpam-7061	343	5	×	×	NOUN
ejpam-7061	343	6	·	·	PUNCT
ejpam-7061	343	7	·	·	PUNCT
ejpam-7061	343	8	·	·	PUNCT
ejpam-7061	343	9	×	×	NOUN
ejpam-7061	343	10	snsn	snsn	NOUN
ejpam-7061	343	11	)	)	PUNCT
ejpam-7061	343	12	=	=	VERB
ejpam-7061	344	1	s1s1s1	s1s1s1	PROPN
ejpam-7061	344	2	×	×	NOUN
ejpam-7061	344	3	s2s2s2	s2s2s2	PROPN
ejpam-7061	344	4	×	×	PROPN
ejpam-7061	344	5	·	·	PUNCT
ejpam-7061	344	6	·	·	PUNCT
ejpam-7061	344	7	·	·	PUNCT
ejpam-7061	345	1	×	×	NOUN
ejpam-7061	345	2	snsnsn	snsnsn	NOUN
ejpam-7061	345	3	⊆	⊆	NUM
ejpam-7061	345	4	s1s1	s1s1	NOUN
ejpam-7061	345	5	×	×	NOUN
ejpam-7061	345	6	s2s2	s2s2	NOUN
ejpam-7061	345	7	×	×	NOUN
ejpam-7061	345	8	·	·	PUNCT
ejpam-7061	345	9	·	·	PUNCT
ejpam-7061	345	10	·	·	PUNCT
ejpam-7061	345	11	×	×	NOUN
ejpam-7061	345	12	snsn	snsn	NOUN
ejpam-7061	345	13	.	.	PUNCT
ejpam-7061	346	1	therefore	therefore	ADV
ejpam-7061	346	2	,	,	PUNCT
ejpam-7061	346	3	si	si	PROPN
ejpam-7061	346	4	∈	∈	PROPN
ejpam-7061	346	5	sisi	sisi	PROPN
ejpam-7061	346	6	for	for	ADP
ejpam-7061	346	7	all	all	PRON
ejpam-7061	346	8	i	i	PRON
ejpam-7061	346	9	∈	∈	PROPN
ejpam-7061	346	10	{	{	PUNCT
ejpam-7061	346	11	1	1	NUM
ejpam-7061	346	12	,	,	PUNCT
ejpam-7061	346	13	2	2	NUM
ejpam-7061	346	14	,	,	PUNCT
ejpam-7061	346	15	.	.	PUNCT
ejpam-7061	346	16	.	.	PUNCT
ejpam-7061	347	1	.	.	PUNCT
ejpam-7061	347	2	,	,	PUNCT
ejpam-7061	347	3	n	n	CCONJ
ejpam-7061	347	4	}	}	PUNCT
ejpam-7061	347	5	.	.	PUNCT
ejpam-7061	348	1	conversely	conversely	ADV
ejpam-7061	348	2	,	,	PUNCT
ejpam-7061	348	3	assume	assume	VERB
ejpam-7061	348	4	that	that	SCONJ
ejpam-7061	348	5	(	(	PUNCT
ejpam-7061	348	6	i	i	NOUN
ejpam-7061	348	7	)	)	PUNCT
ejpam-7061	348	8	or	or	CCONJ
ejpam-7061	348	9	(	(	PUNCT
ejpam-7061	348	10	ii	ii	NOUN
ejpam-7061	348	11	)	)	PUNCT
ejpam-7061	348	12	holds	hold	VERB
ejpam-7061	348	13	.	.	PUNCT
ejpam-7061	349	1	if	if	SCONJ
ejpam-7061	349	2	(	(	PUNCT
ejpam-7061	349	3	i	i	NOUN
ejpam-7061	349	4	)	)	PUNCT
ejpam-7061	349	5	holds	hold	VERB
ejpam-7061	349	6	,	,	PUNCT
ejpam-7061	349	7	then	then	ADV
ejpam-7061	349	8	we	we	PRON
ejpam-7061	349	9	get	get	VERB
ejpam-7061	349	10	ls1	ls1	ADJ
ejpam-7061	349	11	×	×	PROPN
ejpam-7061	349	12	ls2	ls2	PROPN
ejpam-7061	349	13	×	×	PROPN
ejpam-7061	349	14	·	·	PUNCT
ejpam-7061	349	15	·	·	PUNCT
ejpam-7061	349	16	·	·	PUNCT
ejpam-7061	350	1	×	×	NOUN
ejpam-7061	350	2	lsn	lsn	NOUN
ejpam-7061	350	3	=	=	SYM
ejpam-7061	350	4	{	{	PUNCT
ejpam-7061	350	5	(	(	PUNCT
ejpam-7061	350	6	s1	s1	NOUN
ejpam-7061	350	7	,	,	PUNCT
ejpam-7061	350	8	s2	s2	NOUN
ejpam-7061	350	9	,	,	PUNCT
ejpam-7061	350	10	.	.	PUNCT
ejpam-7061	350	11	.	.	PUNCT
ejpam-7061	350	12	.	.	PUNCT
ejpam-7061	351	1	,	,	PUNCT
ejpam-7061	351	2	sn	sn	PROPN
ejpam-7061	351	3	)	)	PUNCT
ejpam-7061	351	4	}	}	PUNCT
ejpam-7061	352	1	⊆	⊆	NUM
ejpam-7061	352	2	l(s1,s2,	l(s1,s2,	NOUN
ejpam-7061	352	3	...	...	PUNCT
ejpam-7061	352	4	,sn	,sn	PUNCT
ejpam-7061	352	5	)	)	PUNCT
ejpam-7061	353	1	⊆	⊆	NUM
ejpam-7061	353	2	ls1	ls1	NOUN
ejpam-7061	353	3	×	×	PROPN
ejpam-7061	353	4	ls2	ls2	PROPN
ejpam-7061	353	5	×	×	PROPN
ejpam-7061	353	6	·	·	PUNCT
ejpam-7061	353	7	·	·	PUNCT
ejpam-7061	353	8	·	·	PUNCT
ejpam-7061	353	9	×	×	NOUN
ejpam-7061	353	10	lsn	lsn	NOUN
ejpam-7061	353	11	.	.	PUNCT
ejpam-7061	354	1	thus	thus	ADV
ejpam-7061	354	2	,	,	PUNCT
ejpam-7061	354	3	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	354	4	...	...	PUNCT
ejpam-7061	354	5	,sn	,sn	PUNCT
ejpam-7061	354	6	)	)	PUNCT
ejpam-7061	355	1	=	=	PUNCT
ejpam-7061	355	2	ls1	ls1	VERB
ejpam-7061	355	3	×	×	PROPN
ejpam-7061	355	4	ls2	ls2	PROPN
ejpam-7061	355	5	×	×	PROPN
ejpam-7061	355	6	·	·	PUNCT
ejpam-7061	355	7	·	·	PUNCT
ejpam-7061	355	8	·	·	PUNCT
ejpam-7061	356	1	×	×	NOUN
ejpam-7061	356	2	lsn	lsn	NOUN
ejpam-7061	356	3	.	.	PUNCT
ejpam-7061	357	1	assume	assume	VERB
ejpam-7061	357	2	that	that	SCONJ
ejpam-7061	357	3	(	(	PUNCT
ejpam-7061	357	4	ii	ii	NOUN
ejpam-7061	357	5	)	)	PUNCT
ejpam-7061	357	6	holds	hold	VERB
ejpam-7061	357	7	.	.	PUNCT
ejpam-7061	358	1	then	then	ADV
ejpam-7061	358	2	we	we	PRON
ejpam-7061	358	3	have	have	VERB
ejpam-7061	358	4	l((s1	l((s1	NOUN
ejpam-7061	358	5	,	,	PUNCT
ejpam-7061	358	6	s2	s2	PROPN
ejpam-7061	358	7	,	,	PUNCT
ejpam-7061	358	8	.	.	PUNCT
ejpam-7061	358	9	.	.	PUNCT
ejpam-7061	358	10	.	.	PUNCT
ejpam-7061	359	1	,	,	PUNCT
ejpam-7061	359	2	sn	sn	PROPN
ejpam-7061	359	3	)	)	PUNCT
ejpam-7061	359	4	)	)	PUNCT
ejpam-7061	360	1	=	=	PRON
ejpam-7061	360	2	s1s1	s1s1	ADP
ejpam-7061	360	3	×	×	NOUN
ejpam-7061	360	4	s2s2	s2s2	NOUN
ejpam-7061	360	5	×	×	NOUN
ejpam-7061	360	6	·	·	PUNCT
ejpam-7061	360	7	·	·	PUNCT
ejpam-7061	360	8	·	·	PUNCT
ejpam-7061	360	9	×	×	NOUN
ejpam-7061	360	10	snsn	snsn	NOUN
ejpam-7061	360	11	.	.	PUNCT
ejpam-7061	361	1	let	let	VERB
ejpam-7061	361	2	(	(	PUNCT
ejpam-7061	361	3	t1	t1	NOUN
ejpam-7061	361	4	,	,	PUNCT
ejpam-7061	361	5	t2	t2	NOUN
ejpam-7061	361	6	,	,	PUNCT
ejpam-7061	361	7	.	.	PUNCT
ejpam-7061	361	8	.	.	PUNCT
ejpam-7061	362	1	.	.	PUNCT
ejpam-7061	363	1	,	,	PUNCT
ejpam-7061	363	2	tn	tn	NOUN
ejpam-7061	363	3	)	)	PUNCT
ejpam-7061	363	4	∈	∈	PROPN
ejpam-7061	363	5	ls1	ls1	NOUN
ejpam-7061	363	6	×	×	PROPN
ejpam-7061	363	7	ls2	ls2	PROPN
ejpam-7061	363	8	×	×	PROPN
ejpam-7061	363	9	·	·	PUNCT
ejpam-7061	363	10	·	·	PUNCT
ejpam-7061	363	11	·	·	PUNCT
ejpam-7061	364	1	×	×	NOUN
ejpam-7061	364	2	lsn	lsn	NOUN
ejpam-7061	364	3	.	.	PUNCT
ejpam-7061	365	1	for	for	ADP
ejpam-7061	365	2	each	each	DET
ejpam-7061	365	3	i	i	PRON
ejpam-7061	365	4	∈	∈	PROPN
ejpam-7061	365	5	{	{	PUNCT
ejpam-7061	365	6	1	1	NUM
ejpam-7061	365	7	,	,	PUNCT
ejpam-7061	365	8	2	2	NUM
ejpam-7061	365	9	,	,	PUNCT
ejpam-7061	365	10	.	.	PUNCT
ejpam-7061	365	11	.	.	PUNCT
ejpam-7061	365	12	.	.	PUNCT
ejpam-7061	365	13	,	,	PUNCT
ejpam-7061	365	14	n	n	CCONJ
ejpam-7061	365	15	}	}	PUNCT
ejpam-7061	365	16	,	,	PUNCT
ejpam-7061	365	17	we	we	PRON
ejpam-7061	365	18	have	have	AUX
ejpam-7061	365	19	siti	siti	NOUN
ejpam-7061	365	20	⊆	⊆	NUM
ejpam-7061	365	21	l(ti	l(ti	NOUN
ejpam-7061	365	22	)	)	PUNCT
ejpam-7061	365	23	=	=	SYM
ejpam-7061	365	24	l(si	l(si	PROPN
ejpam-7061	365	25	)	)	PUNCT
ejpam-7061	365	26	=	=	PUNCT
ejpam-7061	365	27	sisi	sisi	PROPN
ejpam-7061	365	28	⊆	⊆	X
ejpam-7061	365	29	si(l(ti	si(l(ti	NOUN
ejpam-7061	365	30	)	)	PUNCT
ejpam-7061	365	31	)	)	PUNCT
ejpam-7061	366	1	=	=	PUNCT
ejpam-7061	366	2	siti	siti	NOUN
ejpam-7061	366	3	.	.	PUNCT
ejpam-7061	367	1	thus	thus	ADV
ejpam-7061	367	2	,	,	PUNCT
ejpam-7061	367	3	sisi	sisi	PROPN
ejpam-7061	367	4	=	=	SYM
ejpam-7061	367	5	siti	siti	PROPN
ejpam-7061	367	6	.	.	PUNCT
ejpam-7061	368	1	that	that	PRON
ejpam-7061	368	2	is	be	AUX
ejpam-7061	368	3	ti	ti	PROPN
ejpam-7061	368	4	∈	∈	PROPN
ejpam-7061	368	5	l(ti	l(ti	PROPN
ejpam-7061	368	6	)	)	PUNCT
ejpam-7061	368	7	=	=	SYM
ejpam-7061	368	8	l(si	l(si	PROPN
ejpam-7061	368	9	)	)	PUNCT
ejpam-7061	368	10	=	=	PUNCT
ejpam-7061	368	11	sisi	sisi	X
ejpam-7061	368	12	=	=	SYM
ejpam-7061	368	13	siti	siti	NOUN
ejpam-7061	368	14	for	for	ADP
ejpam-7061	368	15	all	all	PRON
ejpam-7061	368	16	i	i	PRON
ejpam-7061	368	17	∈	∈	PROPN
ejpam-7061	368	18	{	{	PUNCT
ejpam-7061	368	19	1	1	NUM
ejpam-7061	368	20	,	,	PUNCT
ejpam-7061	368	21	2	2	NUM
ejpam-7061	368	22	,	,	PUNCT
ejpam-7061	368	23	.	.	PUNCT
ejpam-7061	368	24	.	.	PUNCT
ejpam-7061	369	1	.	.	PUNCT
ejpam-7061	369	2	,	,	PUNCT
ejpam-7061	369	3	n	n	CCONJ
ejpam-7061	369	4	}	}	PUNCT
ejpam-7061	369	5	.	.	PUNCT
ejpam-7061	370	1	this	this	PRON
ejpam-7061	370	2	implies	imply	VERB
ejpam-7061	370	3	l((t1	l((t1	PROPN
ejpam-7061	370	4	,	,	PUNCT
ejpam-7061	370	5	t2	t2	NOUN
ejpam-7061	370	6	,	,	PUNCT
ejpam-7061	370	7	.	.	PUNCT
ejpam-7061	370	8	.	.	PUNCT
ejpam-7061	371	1	.	.	PUNCT
ejpam-7061	372	1	,	,	PUNCT
ejpam-7061	372	2	tn	tn	PROPN
ejpam-7061	372	3	)	)	PUNCT
ejpam-7061	372	4	)	)	PUNCT
ejpam-7061	373	1	=	=	PRON
ejpam-7061	373	2	(	(	PUNCT
ejpam-7061	373	3	t1	t1	NOUN
ejpam-7061	373	4	,	,	PUNCT
ejpam-7061	373	5	t2	t2	NOUN
ejpam-7061	373	6	,	,	PUNCT
ejpam-7061	373	7	.	.	PUNCT
ejpam-7061	373	8	.	.	PUNCT
ejpam-7061	374	1	.	.	PUNCT
ejpam-7061	375	1	,	,	PUNCT
ejpam-7061	375	2	tn	tn	NOUN
ejpam-7061	375	3	)	)	PUNCT
ejpam-7061	375	4	∪	∪	NOUN
ejpam-7061	375	5	(	(	PUNCT
ejpam-7061	375	6	s1t1	s1t1	INTJ
ejpam-7061	375	7	×	×	NOUN
ejpam-7061	375	8	s2t2	s2t2	NUM
ejpam-7061	375	9	×	×	NOUN
ejpam-7061	375	10	·	·	PUNCT
ejpam-7061	375	11	·	·	PUNCT
ejpam-7061	375	12	·	·	PUNCT
ejpam-7061	376	1	×	×	NOUN
ejpam-7061	376	2	sntn	sntn	ADJ
ejpam-7061	376	3	)	)	PUNCT
ejpam-7061	377	1	p.	p.	NOUN
ejpam-7061	377	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	377	3	,	,	PUNCT
ejpam-7061	377	4	o.	o.	PROPN
ejpam-7061	377	5	pankoon	pankoon	NOUN
ejpam-7061	377	6	,	,	PUNCT
ejpam-7061	377	7	t.	t.	PROPN
ejpam-7061	377	8	changphas	changphas	PROPN
ejpam-7061	377	9	/	/	SYM
ejpam-7061	377	10	eur	eur	PROPN
ejpam-7061	377	11	.	.	PUNCT
ejpam-7061	378	1	j.	j.	PROPN
ejpam-7061	378	2	pure	pure	PROPN
ejpam-7061	378	3	appl	appl	PROPN
ejpam-7061	378	4	.	.	PROPN
ejpam-7061	378	5	math	math	PROPN
ejpam-7061	378	6	,	,	PUNCT
ejpam-7061	378	7	18	18	NUM
ejpam-7061	378	8	(	(	PUNCT
ejpam-7061	378	9	4	4	NUM
ejpam-7061	378	10	)	)	PUNCT
ejpam-7061	378	11	(	(	PUNCT
ejpam-7061	378	12	2025	2025	NUM
ejpam-7061	378	13	)	)	PUNCT
ejpam-7061	378	14	,	,	PUNCT
ejpam-7061	378	15	7061	7061	NUM
ejpam-7061	378	16	7	7	NUM
ejpam-7061	378	17	of	of	ADP
ejpam-7061	378	18	10	10	NUM
ejpam-7061	378	19	=	=	SYM
ejpam-7061	378	20	s1t1	s1t1	NUM
ejpam-7061	378	21	×	×	NOUN
ejpam-7061	378	22	s2t2	s2t2	NUM
ejpam-7061	378	23	×	×	NOUN
ejpam-7061	378	24	·	·	PUNCT
ejpam-7061	378	25	·	·	PUNCT
ejpam-7061	378	26	·	·	PUNCT
ejpam-7061	379	1	×	×	NOUN
ejpam-7061	379	2	sntn	sntn	NOUN
ejpam-7061	379	3	=	=	PUNCT
ejpam-7061	379	4	s1s1	s1s1	ADP
ejpam-7061	379	5	×	×	PROPN
ejpam-7061	379	6	s2t2	s2t2	NUM
ejpam-7061	379	7	×	×	NOUN
ejpam-7061	379	8	·	·	PUNCT
ejpam-7061	379	9	·	·	PUNCT
ejpam-7061	379	10	·	·	PUNCT
ejpam-7061	379	11	×	×	NOUN
ejpam-7061	379	12	snsn	snsn	NOUN
ejpam-7061	379	13	=	=	SYM
ejpam-7061	379	14	l((s1	l((s1	PROPN
ejpam-7061	379	15	,	,	PUNCT
ejpam-7061	379	16	s2	s2	PROPN
ejpam-7061	379	17	,	,	PUNCT
ejpam-7061	379	18	.	.	PUNCT
ejpam-7061	379	19	.	.	PUNCT
ejpam-7061	379	20	.	.	PUNCT
ejpam-7061	379	21	,	,	PUNCT
ejpam-7061	379	22	sn	sn	PROPN
ejpam-7061	379	23	)	)	PUNCT
ejpam-7061	379	24	)	)	PUNCT
ejpam-7061	379	25	.	.	PUNCT
ejpam-7061	380	1	thus	thus	ADV
ejpam-7061	380	2	,	,	PUNCT
ejpam-7061	380	3	(	(	PUNCT
ejpam-7061	380	4	t1	t1	NOUN
ejpam-7061	380	5	,	,	PUNCT
ejpam-7061	380	6	t2	t2	NOUN
ejpam-7061	380	7	,	,	PUNCT
ejpam-7061	380	8	.	.	PUNCT
ejpam-7061	380	9	.	.	PUNCT
ejpam-7061	381	1	.	.	PUNCT
ejpam-7061	382	1	,	,	PUNCT
ejpam-7061	382	2	tn	tn	PROPN
ejpam-7061	382	3	)	)	PUNCT
ejpam-7061	382	4	∈	∈	PROPN
ejpam-7061	382	5	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	382	6	...	...	PUNCT
ejpam-7061	382	7	,sn	,sn	PUNCT
ejpam-7061	382	8	)	)	PUNCT
ejpam-7061	382	9	.	.	PUNCT
ejpam-7061	383	1	hence	hence	ADV
ejpam-7061	383	2	ls1×ls2×	ls1×ls2×	X
ejpam-7061	383	3	·	·	PUNCT
ejpam-7061	383	4	·	·	PUNCT
ejpam-7061	383	5	·	·	PUNCT
ejpam-7061	383	6	×lsn	×lsn	NOUN
ejpam-7061	383	7	⊆	⊆	NUM
ejpam-7061	383	8	l(s1,s2,	l(s1,s2,	SYM
ejpam-7061	383	9	...	...	PUNCT
ejpam-7061	383	10	,sn	,sn	PUNCT
ejpam-7061	383	11	)	)	PUNCT
ejpam-7061	383	12	.	.	PUNCT
ejpam-7061	384	1	the	the	DET
ejpam-7061	384	2	opposite	opposite	ADJ
ejpam-7061	384	3	inclusion	inclusion	NOUN
ejpam-7061	384	4	is	be	AUX
ejpam-7061	384	5	obtained	obtain	VERB
ejpam-7061	384	6	by	by	ADP
ejpam-7061	384	7	theorem	theorem	NOUN
ejpam-7061	384	8	3(i	3(i	NUM
ejpam-7061	384	9	)	)	PUNCT
ejpam-7061	384	10	.	.	PUNCT
ejpam-7061	385	1	therefore	therefore	ADV
ejpam-7061	385	2	,	,	PUNCT
ejpam-7061	385	3	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	385	4	...	...	PUNCT
ejpam-7061	385	5	,sn	,sn	PUNCT
ejpam-7061	385	6	)	)	PUNCT
ejpam-7061	385	7	=	=	PUNCT
ejpam-7061	385	8	ls1	ls1	VERB
ejpam-7061	385	9	×	×	PROPN
ejpam-7061	385	10	ls2	ls2	PROPN
ejpam-7061	385	11	×	×	PROPN
ejpam-7061	385	12	·	·	PUNCT
ejpam-7061	385	13	·	·	PUNCT
ejpam-7061	385	14	·	·	PUNCT
ejpam-7061	386	1	×	×	NOUN
ejpam-7061	386	2	lsn	lsn	NOUN
ejpam-7061	386	3	.	.	PUNCT
ejpam-7061	387	1	theorem	theorem	ADJ
ejpam-7061	387	2	5	5	NUM
ejpam-7061	387	3	.	.	PUNCT
ejpam-7061	388	1	let	let	VERB
ejpam-7061	388	2	si	si	PRON
ejpam-7061	388	3	be	be	AUX
ejpam-7061	388	4	a	a	DET
ejpam-7061	388	5	semigroup	semigroup	NOUN
ejpam-7061	388	6	and	and	CCONJ
ejpam-7061	388	7	let	let	VERB
ejpam-7061	388	8	si	si	PROPN
ejpam-7061	388	9	∈	∈	PROPN
ejpam-7061	388	10	si	si	X
ejpam-7061	388	11	where	where	SCONJ
ejpam-7061	388	12	i	i	PRON
ejpam-7061	388	13	∈	∈	PROPN
ejpam-7061	388	14	{	{	PUNCT
ejpam-7061	388	15	1	1	NUM
ejpam-7061	388	16	,	,	PUNCT
ejpam-7061	388	17	2	2	NUM
ejpam-7061	388	18	,	,	PUNCT
ejpam-7061	388	19	.	.	PUNCT
ejpam-7061	388	20	.	.	PUNCT
ejpam-7061	389	1	.	.	PUNCT
ejpam-7061	389	2	,	,	PUNCT
ejpam-7061	389	3	n	n	CCONJ
ejpam-7061	389	4	}	}	PUNCT
ejpam-7061	389	5	.	.	PUNCT
ejpam-7061	390	1	if	if	SCONJ
ejpam-7061	390	2	l((s1	l((s1	PROPN
ejpam-7061	390	3	,	,	PUNCT
ejpam-7061	390	4	s2	s2	PROPN
ejpam-7061	390	5	,	,	PUNCT
ejpam-7061	390	6	.	.	PUNCT
ejpam-7061	390	7	.	.	PUNCT
ejpam-7061	390	8	.	.	PUNCT
ejpam-7061	390	9	,	,	PUNCT
ejpam-7061	390	10	sn	sn	PROPN
ejpam-7061	390	11	)	)	PUNCT
ejpam-7061	390	12	)	)	PUNCT
ejpam-7061	391	1	=	=	PUNCT
ejpam-7061	391	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	391	3	l(s2)×	l(s2)×	X
ejpam-7061	391	4	·	·	PUNCT
ejpam-7061	391	5	·	·	PUNCT
ejpam-7061	391	6	·	·	PUNCT
ejpam-7061	391	7	×	×	NOUN
ejpam-7061	391	8	l(sn	l(sn	PROPN
ejpam-7061	391	9	)	)	PUNCT
ejpam-7061	391	10	,	,	PUNCT
ejpam-7061	391	11	then	then	ADV
ejpam-7061	391	12	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	391	13	...	...	PUNCT
ejpam-7061	391	14	,sn	,sn	PUNCT
ejpam-7061	391	15	)	)	PUNCT
ejpam-7061	392	1	=	=	PUNCT
ejpam-7061	392	2	ls1	ls1	VERB
ejpam-7061	392	3	×	×	PROPN
ejpam-7061	392	4	ls2	ls2	PROPN
ejpam-7061	392	5	×	×	PROPN
ejpam-7061	392	6	·	·	PUNCT
ejpam-7061	392	7	·	·	PUNCT
ejpam-7061	392	8	·	·	PUNCT
ejpam-7061	393	1	×	×	NOUN
ejpam-7061	393	2	lsn	lsn	NOUN
ejpam-7061	393	3	.	.	PUNCT
ejpam-7061	394	1	proof	proof	NOUN
ejpam-7061	394	2	.	.	PUNCT
ejpam-7061	395	1	assume	assume	VERB
ejpam-7061	395	2	that	that	SCONJ
ejpam-7061	395	3	l((s1	l((s1	NOUN
ejpam-7061	395	4	,	,	PUNCT
ejpam-7061	395	5	s2	s2	PROPN
ejpam-7061	395	6	,	,	PUNCT
ejpam-7061	395	7	.	.	PUNCT
ejpam-7061	395	8	.	.	PUNCT
ejpam-7061	395	9	.	.	PUNCT
ejpam-7061	396	1	,	,	PUNCT
ejpam-7061	396	2	sn	sn	PROPN
ejpam-7061	396	3	)	)	PUNCT
ejpam-7061	396	4	)	)	PUNCT
ejpam-7061	397	1	=	=	PUNCT
ejpam-7061	397	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	397	3	l(s2)×	l(s2)×	X
ejpam-7061	397	4	·	·	PUNCT
ejpam-7061	397	5	·	·	PUNCT
ejpam-7061	397	6	·	·	PUNCT
ejpam-7061	397	7	×	×	NOUN
ejpam-7061	397	8	l(sn	l(sn	NUM
ejpam-7061	397	9	)	)	PUNCT
ejpam-7061	397	10	.	.	PUNCT
ejpam-7061	398	1	by	by	ADP
ejpam-7061	398	2	theorem	theorem	NOUN
ejpam-7061	398	3	1	1	NUM
ejpam-7061	398	4	,	,	PUNCT
ejpam-7061	398	5	there	there	PRON
ejpam-7061	398	6	are	be	VERB
ejpam-7061	398	7	two	two	NUM
ejpam-7061	398	8	possible	possible	ADJ
ejpam-7061	398	9	cases	case	NOUN
ejpam-7061	398	10	,	,	PUNCT
ejpam-7061	398	11	as	as	SCONJ
ejpam-7061	398	12	follows	follow	VERB
ejpam-7061	398	13	.	.	PUNCT
ejpam-7061	399	1	case	case	NOUN
ejpam-7061	399	2	1	1	NUM
ejpam-7061	399	3	:	:	PUNCT
ejpam-7061	399	4	si	si	PROPN
ejpam-7061	399	5	∈	∈	PROPN
ejpam-7061	399	6	sisi	sisi	PROPN
ejpam-7061	399	7	for	for	ADP
ejpam-7061	399	8	all	all	PRON
ejpam-7061	399	9	i	i	PRON
ejpam-7061	399	10	∈	∈	PROPN
ejpam-7061	399	11	{	{	PUNCT
ejpam-7061	399	12	1	1	NUM
ejpam-7061	399	13	,	,	PUNCT
ejpam-7061	399	14	2	2	NUM
ejpam-7061	399	15	,	,	PUNCT
ejpam-7061	399	16	.	.	PUNCT
ejpam-7061	399	17	.	.	PUNCT
ejpam-7061	400	1	.	.	PUNCT
ejpam-7061	400	2	,	,	PUNCT
ejpam-7061	400	3	n	n	CCONJ
ejpam-7061	400	4	}	}	PUNCT
ejpam-7061	400	5	.	.	PUNCT
ejpam-7061	401	1	we	we	PRON
ejpam-7061	401	2	obtain	obtain	VERB
ejpam-7061	401	3	from	from	ADP
ejpam-7061	401	4	theorem	theorem	ADJ
ejpam-7061	401	5	4(ii	4(ii	NUM
ejpam-7061	401	6	)	)	PUNCT
ejpam-7061	401	7	that	that	PRON
ejpam-7061	401	8	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	401	9	...	...	PUNCT
ejpam-7061	401	10	,sn	,sn	PUNCT
ejpam-7061	401	11	)	)	PUNCT
ejpam-7061	402	1	=	=	PUNCT
ejpam-7061	402	2	ls1	ls1	VERB
ejpam-7061	402	3	×	×	PROPN
ejpam-7061	402	4	ls2	ls2	PROPN
ejpam-7061	402	5	×	×	PROPN
ejpam-7061	402	6	·	·	PUNCT
ejpam-7061	402	7	·	·	PUNCT
ejpam-7061	402	8	·	·	PUNCT
ejpam-7061	403	1	×	×	NOUN
ejpam-7061	403	2	lsn	lsn	NOUN
ejpam-7061	403	3	.	.	PUNCT
ejpam-7061	404	1	case	case	NOUN
ejpam-7061	404	2	2	2	NUM
ejpam-7061	404	3	:	:	PUNCT
ejpam-7061	404	4	there	there	PRON
ejpam-7061	404	5	exists	exist	VERB
ejpam-7061	404	6	i	i	PRON
ejpam-7061	404	7	∈	∈	PROPN
ejpam-7061	404	8	{	{	PUNCT
ejpam-7061	404	9	1	1	NUM
ejpam-7061	404	10	,	,	PUNCT
ejpam-7061	404	11	2	2	NUM
ejpam-7061	404	12	,	,	PUNCT
ejpam-7061	404	13	.	.	PUNCT
ejpam-7061	404	14	.	.	PUNCT
ejpam-7061	405	1	.	.	PUNCT
ejpam-7061	406	1	,	,	PUNCT
ejpam-7061	407	1	n	n	CCONJ
ejpam-7061	407	2	}	}	PUNCT
ejpam-7061	407	3	such	such	ADJ
ejpam-7061	407	4	that	that	PRON
ejpam-7061	407	5	for	for	ADP
ejpam-7061	407	6	each	each	DET
ejpam-7061	407	7	j	j	PROPN
ejpam-7061	407	8	∈	∈	PROPN
ejpam-7061	407	9	{	{	PUNCT
ejpam-7061	407	10	1	1	NUM
ejpam-7061	407	11	,	,	PUNCT
ejpam-7061	407	12	2	2	NUM
ejpam-7061	407	13	,	,	PUNCT
ejpam-7061	407	14	.	.	PUNCT
ejpam-7061	407	15	.	.	PUNCT
ejpam-7061	407	16	.	.	PUNCT
ejpam-7061	408	1	,	,	PUNCT
ejpam-7061	408	2	n}\{i	n}\{i	NOUN
ejpam-7061	408	3	}	}	PUNCT
ejpam-7061	408	4	,	,	PUNCT
ejpam-7061	408	5	sjsj	sjsj	ADJ
ejpam-7061	408	6	=	=	SYM
ejpam-7061	408	7	{	{	PUNCT
ejpam-7061	408	8	sj	sj	NOUN
ejpam-7061	408	9	}	}	PUNCT
ejpam-7061	408	10	,	,	PUNCT
ejpam-7061	408	11	which	which	PRON
ejpam-7061	408	12	yields	yield	VERB
ejpam-7061	408	13	l(sj	l(sj	PROPN
ejpam-7061	408	14	)	)	PUNCT
ejpam-7061	408	15	=	=	PRON
ejpam-7061	408	16	{	{	PUNCT
ejpam-7061	408	17	sj	sj	NOUN
ejpam-7061	408	18	}	}	PUNCT
ejpam-7061	408	19	and	and	CCONJ
ejpam-7061	408	20	lsj	lsj	NOUN
ejpam-7061	408	21	=	=	SYM
ejpam-7061	408	22	{	{	PUNCT
ejpam-7061	408	23	sj	sj	NOUN
ejpam-7061	408	24	}	}	PUNCT
ejpam-7061	408	25	.	.	PUNCT
ejpam-7061	409	1	by	by	ADP
ejpam-7061	409	2	focusing	focus	VERB
ejpam-7061	409	3	on	on	ADP
ejpam-7061	409	4	the	the	DET
ejpam-7061	409	5	index	index	NOUN
ejpam-7061	409	6	i	i	PRON
ejpam-7061	409	7	,	,	PUNCT
ejpam-7061	409	8	if	if	SCONJ
ejpam-7061	409	9	si	si	PROPN
ejpam-7061	409	10	∈	∈	PROPN
ejpam-7061	409	11	sisi	sisi	NOUN
ejpam-7061	409	12	,	,	PUNCT
ejpam-7061	409	13	we	we	PRON
ejpam-7061	409	14	obtain	obtain	VERB
ejpam-7061	409	15	from	from	ADP
ejpam-7061	409	16	theorem	theorem	ADJ
ejpam-7061	409	17	4(ii	4(ii	NUM
ejpam-7061	409	18	)	)	PUNCT
ejpam-7061	409	19	that	that	PRON
ejpam-7061	409	20	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	409	21	...	...	PUNCT
ejpam-7061	409	22	,sn	,sn	PUNCT
ejpam-7061	409	23	)	)	PUNCT
ejpam-7061	410	1	=	=	PUNCT
ejpam-7061	410	2	ls1	ls1	VERB
ejpam-7061	410	3	×	×	PROPN
ejpam-7061	410	4	ls2	ls2	PROPN
ejpam-7061	410	5	×	×	PROPN
ejpam-7061	410	6	·	·	PUNCT
ejpam-7061	410	7	·	·	PUNCT
ejpam-7061	410	8	·	·	PUNCT
ejpam-7061	411	1	×	×	NOUN
ejpam-7061	411	2	lsn	lsn	NOUN
ejpam-7061	411	3	.	.	PUNCT
ejpam-7061	412	1	on	on	ADP
ejpam-7061	412	2	the	the	DET
ejpam-7061	412	3	other	other	ADJ
ejpam-7061	412	4	hand	hand	NOUN
ejpam-7061	412	5	,	,	PUNCT
ejpam-7061	412	6	we	we	PRON
ejpam-7061	412	7	suppose	suppose	VERB
ejpam-7061	412	8	that	that	SCONJ
ejpam-7061	412	9	si	si	PROPN
ejpam-7061	412	10	/∈	/∈	PUNCT
ejpam-7061	412	11	sisi	sisi	PROPN
ejpam-7061	412	12	.	.	PUNCT
ejpam-7061	413	1	we	we	PRON
ejpam-7061	413	2	will	will	AUX
ejpam-7061	413	3	prove	prove	VERB
ejpam-7061	413	4	that	that	SCONJ
ejpam-7061	413	5	lsi	lsi	PROPN
ejpam-7061	413	6	=	=	SYM
ejpam-7061	413	7	{	{	PUNCT
ejpam-7061	413	8	si	si	NOUN
ejpam-7061	413	9	}	}	PUNCT
ejpam-7061	413	10	by	by	ADP
ejpam-7061	413	11	a	a	DET
ejpam-7061	413	12	contradiction	contradiction	NOUN
ejpam-7061	413	13	.	.	PUNCT
ejpam-7061	414	1	suppose	suppose	VERB
ejpam-7061	414	2	that	that	SCONJ
ejpam-7061	414	3	there	there	PRON
ejpam-7061	414	4	exists	exist	VERB
ejpam-7061	414	5	ti	ti	X
ejpam-7061	414	6	∈	∈	PROPN
ejpam-7061	414	7	si	si	X
ejpam-7061	414	8	such	such	ADJ
ejpam-7061	414	9	that	that	DET
ejpam-7061	414	10	ti	ti	NOUN
ejpam-7061	414	11	̸=	̸=	PROPN
ejpam-7061	414	12	si	si	X
ejpam-7061	414	13	and	and	CCONJ
ejpam-7061	414	14	l(ti	l(ti	PROPN
ejpam-7061	414	15	)	)	PUNCT
ejpam-7061	414	16	=	=	SYM
ejpam-7061	414	17	l(si	l(si	PROPN
ejpam-7061	414	18	)	)	PUNCT
ejpam-7061	414	19	,	,	PUNCT
ejpam-7061	414	20	which	which	PRON
ejpam-7061	414	21	implies	imply	VERB
ejpam-7061	414	22	siti	siti	NOUN
ejpam-7061	414	23	=	=	SYM
ejpam-7061	414	24	sisi	sisi	PROPN
ejpam-7061	414	25	.	.	PUNCT
ejpam-7061	415	1	since	since	SCONJ
ejpam-7061	415	2	si	si	PROPN
ejpam-7061	415	3	∈	∈	PROPN
ejpam-7061	415	4	l(si	l(si	PROPN
ejpam-7061	415	5	)	)	PUNCT
ejpam-7061	415	6	=	=	SYM
ejpam-7061	415	7	l(ti	l(ti	PROPN
ejpam-7061	415	8	)	)	PUNCT
ejpam-7061	415	9	and	and	CCONJ
ejpam-7061	415	10	si	si	PROPN
ejpam-7061	415	11	̸=	̸=	PROPN
ejpam-7061	415	12	ti	ti	NOUN
ejpam-7061	415	13	,	,	PUNCT
ejpam-7061	415	14	we	we	PRON
ejpam-7061	415	15	get	get	VERB
ejpam-7061	415	16	si	si	X
ejpam-7061	415	17	∈	∈	PROPN
ejpam-7061	415	18	siti	siti	NOUN
ejpam-7061	415	19	=	=	SYM
ejpam-7061	415	20	sisi	sisi	PROPN
ejpam-7061	415	21	,	,	PUNCT
ejpam-7061	415	22	which	which	PRON
ejpam-7061	415	23	is	be	AUX
ejpam-7061	415	24	a	a	DET
ejpam-7061	415	25	contradiction	contradiction	NOUN
ejpam-7061	415	26	.	.	PUNCT
ejpam-7061	416	1	therefore	therefore	ADV
ejpam-7061	416	2	,	,	PUNCT
ejpam-7061	416	3	lsi	lsi	PROPN
ejpam-7061	416	4	=	=	SYM
ejpam-7061	416	5	{	{	PUNCT
ejpam-7061	416	6	si	si	NOUN
ejpam-7061	416	7	}	}	PUNCT
ejpam-7061	416	8	.	.	PUNCT
ejpam-7061	417	1	we	we	PRON
ejpam-7061	417	2	thus	thus	ADV
ejpam-7061	417	3	conclude	conclude	VERB
ejpam-7061	417	4	by	by	ADP
ejpam-7061	417	5	theorem	theorem	NOUN
ejpam-7061	417	6	4(i	4(i	NUM
ejpam-7061	417	7	)	)	PUNCT
ejpam-7061	417	8	that	that	PRON
ejpam-7061	417	9	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	417	10	...	...	PUNCT
ejpam-7061	417	11	,sn	,sn	PUNCT
ejpam-7061	417	12	)	)	PUNCT
ejpam-7061	418	1	=	=	PUNCT
ejpam-7061	418	2	ls1	ls1	VERB
ejpam-7061	418	3	×	×	PROPN
ejpam-7061	418	4	ls2	ls2	PROPN
ejpam-7061	418	5	×	×	PROPN
ejpam-7061	418	6	·	·	PUNCT
ejpam-7061	418	7	·	·	PUNCT
ejpam-7061	418	8	·	·	PUNCT
ejpam-7061	419	1	×	×	NOUN
ejpam-7061	419	2	lsn	lsn	NOUN
ejpam-7061	419	3	as	as	SCONJ
ejpam-7061	419	4	required	require	VERB
ejpam-7061	419	5	.	.	PUNCT
ejpam-7061	420	1	let	let	VERB
ejpam-7061	420	2	s	s	PRON
ejpam-7061	420	3	be	be	AUX
ejpam-7061	420	4	a	a	DET
ejpam-7061	420	5	semigroup	semigroup	NOUN
ejpam-7061	420	6	.	.	PUNCT
ejpam-7061	421	1	since	since	SCONJ
ejpam-7061	421	2	the	the	DET
ejpam-7061	421	3	relation	relation	NOUN
ejpam-7061	421	4	l	l	NOUN
ejpam-7061	421	5	on	on	ADP
ejpam-7061	421	6	s	s	PROPN
ejpam-7061	421	7	is	be	AUX
ejpam-7061	421	8	defined	define	VERB
ejpam-7061	421	9	in	in	ADP
ejpam-7061	421	10	terms	term	NOUN
ejpam-7061	421	11	of	of	ADP
ejpam-7061	421	12	left	left	ADJ
ejpam-7061	421	13	ideals	ideal	NOUN
ejpam-7061	421	14	of	of	ADP
ejpam-7061	421	15	s	s	PROPN
ejpam-7061	421	16	,	,	PUNCT
ejpam-7061	421	17	the	the	DET
ejpam-7061	421	18	order	order	NOUN
ejpam-7061	421	19	among	among	ADP
ejpam-7061	421	20	these	these	DET
ejpam-7061	421	21	left	leave	VERB
ejpam-7061	421	22	ideals	ideal	NOUN
ejpam-7061	421	23	induces	induce	VERB
ejpam-7061	421	24	a	a	DET
ejpam-7061	421	25	partial	partial	ADJ
ejpam-7061	421	26	order	order	NOUN
ejpam-7061	421	27	among	among	ADP
ejpam-7061	421	28	their	their	PRON
ejpam-7061	421	29	equivalence	equivalence	NOUN
ejpam-7061	421	30	classes	class	NOUN
ejpam-7061	421	31	,	,	PUNCT
ejpam-7061	421	32	that	that	ADV
ejpam-7061	421	33	is	is	ADV
ejpam-7061	421	34	,	,	PUNCT
ejpam-7061	421	35	la	la	ADV
ejpam-7061	421	36	≤	≤	ADJ
ejpam-7061	421	37	lb	lb	ADP
ejpam-7061	421	38	if	if	SCONJ
ejpam-7061	421	39	l(a	l(a	PROPN
ejpam-7061	421	40	)	)	PUNCT
ejpam-7061	421	41	⊆	⊆	NUM
ejpam-7061	421	42	l(b	l(b	PROPN
ejpam-7061	421	43	)	)	PUNCT
ejpam-7061	421	44	for	for	ADP
ejpam-7061	421	45	all	all	DET
ejpam-7061	421	46	a	a	DET
ejpam-7061	421	47	,	,	PUNCT
ejpam-7061	421	48	b	b	X
ejpam-7061	421	49	∈	∈	PROPN
ejpam-7061	421	50	s.	s.	PROPN
ejpam-7061	421	51	then	then	ADV
ejpam-7061	421	52	an	an	DET
ejpam-7061	421	53	l	l	NOUN
ejpam-7061	421	54	-	-	PUNCT
ejpam-7061	421	55	class	class	NOUN
ejpam-7061	421	56	ls	ls	NOUN
ejpam-7061	421	57	of	of	ADP
ejpam-7061	421	58	s	s	PRON
ejpam-7061	421	59	is	be	AUX
ejpam-7061	421	60	maximal	maximal	ADJ
ejpam-7061	421	61	if	if	SCONJ
ejpam-7061	421	62	there	there	PRON
ejpam-7061	421	63	is	be	VERB
ejpam-7061	421	64	no	no	DET
ejpam-7061	421	65	u	u	NOUN
ejpam-7061	421	66	∈	∈	NOUN
ejpam-7061	421	67	s	s	VERB
ejpam-7061	421	68	such	such	ADJ
ejpam-7061	421	69	that	that	SCONJ
ejpam-7061	421	70	l(s	l(s	PROPN
ejpam-7061	421	71	)	)	PUNCT
ejpam-7061	421	72	⊊	⊊	VERB
ejpam-7061	421	73	l(u	l(u	PROPN
ejpam-7061	421	74	)	)	PUNCT
ejpam-7061	421	75	.	.	PUNCT
ejpam-7061	422	1	lemma	lemma	PROPN
ejpam-7061	422	2	2	2	X
ejpam-7061	422	3	.	.	PUNCT
ejpam-7061	423	1	let	let	VERB
ejpam-7061	423	2	s	s	PRON
ejpam-7061	423	3	be	be	AUX
ejpam-7061	423	4	a	a	DET
ejpam-7061	423	5	semigroup	semigroup	NOUN
ejpam-7061	423	6	and	and	CCONJ
ejpam-7061	423	7	let	let	VERB
ejpam-7061	423	8	s	s	NOUN
ejpam-7061	423	9	,	,	PUNCT
ejpam-7061	423	10	u	u	PROPN
ejpam-7061	423	11	∈	∈	PROPN
ejpam-7061	423	12	s.	s.	PROPN
ejpam-7061	423	13	if	if	SCONJ
ejpam-7061	423	14	l(s	l(s	PROPN
ejpam-7061	423	15	)	)	PUNCT
ejpam-7061	423	16	⊊	⊊	VERB
ejpam-7061	423	17	l(u	l(u	PROPN
ejpam-7061	423	18	)	)	PUNCT
ejpam-7061	423	19	,	,	PUNCT
ejpam-7061	423	20	then	then	ADV
ejpam-7061	423	21	the	the	DET
ejpam-7061	423	22	following	following	ADJ
ejpam-7061	423	23	statements	statement	NOUN
ejpam-7061	423	24	hold	hold	VERB
ejpam-7061	423	25	:	:	PUNCT
ejpam-7061	423	26	(	(	PUNCT
ejpam-7061	423	27	i	i	NOUN
ejpam-7061	423	28	)	)	PUNCT
ejpam-7061	423	29	u	u	NOUN
ejpam-7061	423	30	/∈	/∈	PUNCT
ejpam-7061	423	31	l(s	l(s	PROPN
ejpam-7061	423	32	)	)	PUNCT
ejpam-7061	423	33	;	;	PUNCT
ejpam-7061	423	34	(	(	PUNCT
ejpam-7061	423	35	ii	ii	NOUN
ejpam-7061	423	36	)	)	PUNCT
ejpam-7061	423	37	l(s	l(s	PROPN
ejpam-7061	423	38	)	)	PUNCT
ejpam-7061	423	39	⊆	⊆	NUM
ejpam-7061	423	40	su	su	NOUN
ejpam-7061	423	41	.	.	PUNCT
ejpam-7061	423	42	proof	proof	NOUN
ejpam-7061	423	43	.	.	PUNCT
ejpam-7061	424	1	assume	assume	VERB
ejpam-7061	424	2	that	that	SCONJ
ejpam-7061	424	3	l(s	l(s	PROPN
ejpam-7061	424	4	)	)	PUNCT
ejpam-7061	424	5	⊊	⊊	VERB
ejpam-7061	424	6	l(u	l(u	PROPN
ejpam-7061	424	7	)	)	PUNCT
ejpam-7061	424	8	.	.	PUNCT
ejpam-7061	425	1	to	to	PART
ejpam-7061	425	2	prove	prove	VERB
ejpam-7061	425	3	(	(	PUNCT
ejpam-7061	425	4	i	i	NOUN
ejpam-7061	425	5	)	)	PUNCT
ejpam-7061	425	6	,	,	PUNCT
ejpam-7061	425	7	suppose	suppose	VERB
ejpam-7061	425	8	u	u	PRON
ejpam-7061	425	9	∈	∈	PROPN
ejpam-7061	425	10	l(s	l(s	PROPN
ejpam-7061	425	11	)	)	PUNCT
ejpam-7061	425	12	.	.	PUNCT
ejpam-7061	426	1	then	then	ADV
ejpam-7061	426	2	l(u	l(u	PROPN
ejpam-7061	426	3	)	)	PUNCT
ejpam-7061	426	4	⊆	⊆	NUM
ejpam-7061	426	5	l(s	l(s	PROPN
ejpam-7061	426	6	)	)	PUNCT
ejpam-7061	426	7	⊊	⊊	VERB
ejpam-7061	426	8	l(u	l(u	PROPN
ejpam-7061	426	9	)	)	PUNCT
ejpam-7061	426	10	.	.	PUNCT
ejpam-7061	427	1	this	this	DET
ejpam-7061	427	2	contradiction	contradiction	NOUN
ejpam-7061	427	3	implies	imply	VERB
ejpam-7061	427	4	u	u	NOUN
ejpam-7061	427	5	/∈	/∈	PUNCT
ejpam-7061	427	6	l(s	l(s	PROPN
ejpam-7061	427	7	)	)	PUNCT
ejpam-7061	427	8	.	.	PUNCT
ejpam-7061	428	1	to	to	PART
ejpam-7061	428	2	prove	prove	VERB
ejpam-7061	428	3	(	(	PUNCT
ejpam-7061	428	4	ii	ii	NOUN
ejpam-7061	428	5	)	)	PUNCT
ejpam-7061	428	6	,	,	PUNCT
ejpam-7061	428	7	let	let	VERB
ejpam-7061	428	8	t	t	PROPN
ejpam-7061	428	9	∈	∈	PROPN
ejpam-7061	428	10	l(s	l(s	PROPN
ejpam-7061	428	11	)	)	PUNCT
ejpam-7061	428	12	.	.	PUNCT
ejpam-7061	429	1	it	it	PRON
ejpam-7061	429	2	follows	follow	VERB
ejpam-7061	429	3	that	that	SCONJ
ejpam-7061	429	4	t	t	PROPN
ejpam-7061	429	5	∈	∈	PROPN
ejpam-7061	429	6	l(u	l(u	PROPN
ejpam-7061	429	7	)	)	PUNCT
ejpam-7061	430	1	=	=	SYM
ejpam-7061	430	2	u	u	PROPN
ejpam-7061	430	3	∪	∪	VERB
ejpam-7061	430	4	su	su	PROPN
ejpam-7061	430	5	.	.	PUNCT
ejpam-7061	431	1	if	if	SCONJ
ejpam-7061	431	2	t	t	PROPN
ejpam-7061	431	3	=	=	SYM
ejpam-7061	431	4	u	u	PROPN
ejpam-7061	431	5	,	,	PUNCT
ejpam-7061	431	6	then	then	ADV
ejpam-7061	431	7	l(s	l(s	PROPN
ejpam-7061	431	8	)	)	PUNCT
ejpam-7061	431	9	⊊	⊊	VERB
ejpam-7061	431	10	l(u	l(u	PROPN
ejpam-7061	431	11	)	)	PUNCT
ejpam-7061	431	12	=	=	SYM
ejpam-7061	431	13	l(t	l(t	PROPN
ejpam-7061	431	14	)	)	PUNCT
ejpam-7061	431	15	.	.	PUNCT
ejpam-7061	432	1	by	by	ADP
ejpam-7061	432	2	(	(	PUNCT
ejpam-7061	432	3	i	i	NOUN
ejpam-7061	432	4	)	)	PUNCT
ejpam-7061	432	5	,	,	PUNCT
ejpam-7061	432	6	we	we	PRON
ejpam-7061	432	7	get	get	VERB
ejpam-7061	432	8	that	that	DET
ejpam-7061	432	9	t	t	NOUN
ejpam-7061	432	10	/∈	/∈	PUNCT
ejpam-7061	433	1	l(s	l(s	PROPN
ejpam-7061	433	2	)	)	PUNCT
ejpam-7061	433	3	,	,	PUNCT
ejpam-7061	433	4	which	which	PRON
ejpam-7061	433	5	is	be	AUX
ejpam-7061	433	6	a	a	DET
ejpam-7061	433	7	contradiction	contradiction	NOUN
ejpam-7061	433	8	.	.	PUNCT
ejpam-7061	434	1	thus	thus	ADV
ejpam-7061	434	2	,	,	PUNCT
ejpam-7061	434	3	t	t	PROPN
ejpam-7061	434	4	∈	∈	PROPN
ejpam-7061	434	5	su	su	PROPN
ejpam-7061	434	6	.	.	PUNCT
ejpam-7061	435	1	p.	p.	PROPN
ejpam-7061	435	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	435	3	,	,	PUNCT
ejpam-7061	436	1	o.	o.	PROPN
ejpam-7061	436	2	pankoon	pankoon	NOUN
ejpam-7061	436	3	,	,	PUNCT
ejpam-7061	436	4	t.	t.	PROPN
ejpam-7061	436	5	changphas	changphas	PROPN
ejpam-7061	436	6	/	/	SYM
ejpam-7061	436	7	eur	eur	PROPN
ejpam-7061	436	8	.	.	PUNCT
ejpam-7061	437	1	j.	j.	PROPN
ejpam-7061	437	2	pure	pure	PROPN
ejpam-7061	437	3	appl	appl	PROPN
ejpam-7061	437	4	.	.	PROPN
ejpam-7061	437	5	math	math	PROPN
ejpam-7061	437	6	,	,	PUNCT
ejpam-7061	437	7	18	18	NUM
ejpam-7061	437	8	(	(	PUNCT
ejpam-7061	437	9	4	4	NUM
ejpam-7061	437	10	)	)	PUNCT
ejpam-7061	437	11	(	(	PUNCT
ejpam-7061	437	12	2025	2025	NUM
ejpam-7061	437	13	)	)	PUNCT
ejpam-7061	437	14	,	,	PUNCT
ejpam-7061	437	15	7061	7061	NUM
ejpam-7061	437	16	8	8	NUM
ejpam-7061	437	17	of	of	ADP
ejpam-7061	437	18	10	10	NUM
ejpam-7061	437	19	theorem	theorem	NOUN
ejpam-7061	437	20	6	6	NUM
ejpam-7061	437	21	.	.	PUNCT
ejpam-7061	438	1	let	let	VERB
ejpam-7061	438	2	si	si	PRON
ejpam-7061	438	3	be	be	AUX
ejpam-7061	438	4	a	a	DET
ejpam-7061	438	5	semigroup	semigroup	NOUN
ejpam-7061	438	6	and	and	CCONJ
ejpam-7061	438	7	let	let	VERB
ejpam-7061	438	8	si	si	PROPN
ejpam-7061	438	9	∈	∈	PROPN
ejpam-7061	438	10	si	si	X
ejpam-7061	438	11	where	where	SCONJ
ejpam-7061	438	12	i	i	PRON
ejpam-7061	438	13	∈	∈	PROPN
ejpam-7061	438	14	{	{	PUNCT
ejpam-7061	438	15	1	1	NUM
ejpam-7061	438	16	,	,	PUNCT
ejpam-7061	438	17	2	2	NUM
ejpam-7061	438	18	,	,	PUNCT
ejpam-7061	438	19	.	.	PUNCT
ejpam-7061	438	20	.	.	PUNCT
ejpam-7061	439	1	.	.	PUNCT
ejpam-7061	439	2	,	,	PUNCT
ejpam-7061	439	3	n	n	CCONJ
ejpam-7061	439	4	}	}	PUNCT
ejpam-7061	439	5	.	.	PUNCT
ejpam-7061	440	1	if	if	SCONJ
ejpam-7061	440	2	(	(	PUNCT
ejpam-7061	440	3	s1	s1	NOUN
ejpam-7061	440	4	,	,	PUNCT
ejpam-7061	440	5	s2	s2	NOUN
ejpam-7061	440	6	,	,	PUNCT
ejpam-7061	440	7	.	.	PUNCT
ejpam-7061	440	8	.	.	PUNCT
ejpam-7061	440	9	.	.	PUNCT
ejpam-7061	440	10	,	,	PUNCT
ejpam-7061	440	11	sn	sn	PROPN
ejpam-7061	440	12	)	)	PUNCT
ejpam-7061	440	13	∈	∈	PROPN
ejpam-7061	440	14	s1s1×s2s2×	s1s1×s2s2×	X
ejpam-7061	440	15	·	·	PUNCT
ejpam-7061	440	16	·	·	PUNCT
ejpam-7061	440	17	·	·	PUNCT
ejpam-7061	440	18	×snsn	×snsn	PROPN
ejpam-7061	440	19	,	,	PUNCT
ejpam-7061	440	20	then	then	ADV
ejpam-7061	440	21	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	440	22	...	...	PUNCT
ejpam-7061	440	23	,sn	,sn	PUNCT
ejpam-7061	440	24	)	)	PUNCT
ejpam-7061	440	25	is	be	AUX
ejpam-7061	440	26	a	a	DET
ejpam-7061	440	27	maximal	maximal	ADJ
ejpam-7061	440	28	l	l	NOUN
ejpam-7061	440	29	-	-	NOUN
ejpam-7061	440	30	class	class	NOUN
ejpam-7061	440	31	if	if	SCONJ
ejpam-7061	441	1	and	and	CCONJ
ejpam-7061	441	2	only	only	ADV
ejpam-7061	441	3	if	if	SCONJ
ejpam-7061	441	4	lsi	lsi	PROPN
ejpam-7061	441	5	is	be	AUX
ejpam-7061	441	6	a	a	DET
ejpam-7061	441	7	maximal	maximal	ADJ
ejpam-7061	441	8	l	l	NOUN
ejpam-7061	441	9	-	-	NOUN
ejpam-7061	441	10	class	class	NOUN
ejpam-7061	441	11	for	for	ADP
ejpam-7061	441	12	all	all	PRON
ejpam-7061	441	13	i	i	PRON
ejpam-7061	441	14	∈	∈	PROPN
ejpam-7061	441	15	{	{	PUNCT
ejpam-7061	441	16	1	1	NUM
ejpam-7061	441	17	,	,	PUNCT
ejpam-7061	441	18	2	2	NUM
ejpam-7061	441	19	,	,	PUNCT
ejpam-7061	441	20	.	.	PUNCT
ejpam-7061	441	21	.	.	PUNCT
ejpam-7061	442	1	.	.	PUNCT
ejpam-7061	442	2	,	,	PUNCT
ejpam-7061	443	1	n	n	CCONJ
ejpam-7061	443	2	}	}	PUNCT
ejpam-7061	443	3	.	.	PUNCT
ejpam-7061	444	1	proof	proof	NOUN
ejpam-7061	444	2	.	.	PUNCT
ejpam-7061	445	1	assume	assume	VERB
ejpam-7061	445	2	that	that	SCONJ
ejpam-7061	445	3	(	(	PUNCT
ejpam-7061	445	4	s1	s1	NOUN
ejpam-7061	445	5	,	,	PUNCT
ejpam-7061	445	6	s2	s2	NOUN
ejpam-7061	445	7	,	,	PUNCT
ejpam-7061	445	8	.	.	PUNCT
ejpam-7061	445	9	.	.	PUNCT
ejpam-7061	446	1	.	.	PUNCT
ejpam-7061	447	1	,	,	PUNCT
ejpam-7061	447	2	sn	sn	PROPN
ejpam-7061	447	3	)	)	PUNCT
ejpam-7061	447	4	∈	∈	PROPN
ejpam-7061	447	5	s1s1	s1s1	VERB
ejpam-7061	447	6	×	×	NOUN
ejpam-7061	447	7	s2s2	s2s2	NOUN
ejpam-7061	447	8	×	×	NOUN
ejpam-7061	447	9	·	·	PUNCT
ejpam-7061	447	10	·	·	PUNCT
ejpam-7061	447	11	·	·	PUNCT
ejpam-7061	447	12	×	×	NOUN
ejpam-7061	447	13	snsn	snsn	NOUN
ejpam-7061	447	14	.	.	PUNCT
ejpam-7061	448	1	by	by	ADP
ejpam-7061	448	2	theorem	theorem	NOUN
ejpam-7061	448	3	1(i	1(i	NUM
ejpam-7061	448	4	)	)	PUNCT
ejpam-7061	448	5	,	,	PUNCT
ejpam-7061	448	6	we	we	PRON
ejpam-7061	448	7	have	have	VERB
ejpam-7061	448	8	l((s1	l((s1	NOUN
ejpam-7061	448	9	,	,	PUNCT
ejpam-7061	448	10	s2	s2	PROPN
ejpam-7061	448	11	,	,	PUNCT
ejpam-7061	448	12	.	.	PUNCT
ejpam-7061	448	13	.	.	PUNCT
ejpam-7061	449	1	.	.	PUNCT
ejpam-7061	450	1	,	,	PUNCT
ejpam-7061	450	2	sn	sn	PROPN
ejpam-7061	450	3	)	)	PUNCT
ejpam-7061	450	4	)	)	PUNCT
ejpam-7061	451	1	=	=	PUNCT
ejpam-7061	451	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	451	3	l(s2)×	l(s2)×	X
ejpam-7061	451	4	·	·	PUNCT
ejpam-7061	451	5	·	·	PUNCT
ejpam-7061	451	6	·	·	PUNCT
ejpam-7061	451	7	×	×	NOUN
ejpam-7061	451	8	l(sn	l(sn	NUM
ejpam-7061	451	9	)	)	PUNCT
ejpam-7061	451	10	.	.	PUNCT
ejpam-7061	452	1	suppose	suppose	VERB
ejpam-7061	452	2	that	that	SCONJ
ejpam-7061	452	3	lsi	lsi	PROPN
ejpam-7061	452	4	is	be	AUX
ejpam-7061	452	5	not	not	PART
ejpam-7061	452	6	a	a	DET
ejpam-7061	452	7	maximal	maximal	ADJ
ejpam-7061	452	8	l	l	NOUN
ejpam-7061	452	9	-	-	NOUN
ejpam-7061	452	10	class	class	NOUN
ejpam-7061	452	11	for	for	ADP
ejpam-7061	452	12	some	some	DET
ejpam-7061	452	13	i	i	PRON
ejpam-7061	452	14	∈	∈	PROPN
ejpam-7061	452	15	{	{	PUNCT
ejpam-7061	452	16	1	1	NUM
ejpam-7061	452	17	,	,	PUNCT
ejpam-7061	452	18	2	2	NUM
ejpam-7061	452	19	,	,	PUNCT
ejpam-7061	452	20	.	.	PUNCT
ejpam-7061	452	21	.	.	PUNCT
ejpam-7061	453	1	.	.	PUNCT
ejpam-7061	453	2	,	,	PUNCT
ejpam-7061	453	3	n	n	CCONJ
ejpam-7061	453	4	}	}	PUNCT
ejpam-7061	453	5	.	.	PUNCT
ejpam-7061	454	1	then	then	ADV
ejpam-7061	454	2	there	there	PRON
ejpam-7061	454	3	exists	exist	VERB
ejpam-7061	454	4	ui	ui	PROPN
ejpam-7061	454	5	∈	∈	PROPN
ejpam-7061	454	6	si	si	INTJ
ejpam-7061	454	7	such	such	ADJ
ejpam-7061	454	8	that	that	DET
ejpam-7061	454	9	l(si	l(si	PROPN
ejpam-7061	454	10	)	)	PUNCT
ejpam-7061	454	11	⊊	⊊	VERB
ejpam-7061	454	12	l(ui	l(ui	PROPN
ejpam-7061	454	13	)	)	PUNCT
ejpam-7061	454	14	.	.	PUNCT
ejpam-7061	455	1	by	by	ADP
ejpam-7061	455	2	lemma	lemma	PROPN
ejpam-7061	455	3	2(i	2(i	NUM
ejpam-7061	455	4	)	)	PUNCT
ejpam-7061	455	5	,	,	PUNCT
ejpam-7061	455	6	ui	ui	PROPN
ejpam-7061	455	7	/∈	/∈	PUNCT
ejpam-7061	455	8	sisi	sisi	PROPN
ejpam-7061	455	9	and	and	CCONJ
ejpam-7061	455	10	sisi	sisi	ADJ
ejpam-7061	455	11	⊆	⊆	NUM
ejpam-7061	455	12	siui	siui	NOUN
ejpam-7061	455	13	.	.	PUNCT
ejpam-7061	456	1	thus	thus	ADV
ejpam-7061	456	2	,	,	PUNCT
ejpam-7061	456	3	l((s1	l((s1	PROPN
ejpam-7061	456	4	,	,	PUNCT
ejpam-7061	456	5	s2	s2	PROPN
ejpam-7061	456	6	,	,	PUNCT
ejpam-7061	456	7	.	.	PUNCT
ejpam-7061	456	8	.	.	PUNCT
ejpam-7061	456	9	.	.	PUNCT
ejpam-7061	457	1	,	,	PUNCT
ejpam-7061	457	2	si	si	INTJ
ejpam-7061	457	3	,	,	PUNCT
ejpam-7061	457	4	.	.	PUNCT
ejpam-7061	457	5	.	.	PUNCT
ejpam-7061	458	1	.	.	PUNCT
ejpam-7061	459	1	,	,	PUNCT
ejpam-7061	459	2	sn	sn	PROPN
ejpam-7061	459	3	)	)	PUNCT
ejpam-7061	459	4	)	)	PUNCT
ejpam-7061	460	1	=	=	PUNCT
ejpam-7061	460	2	l(s1)×	l(s1)×	NOUN
ejpam-7061	460	3	l(s2)×	l(s2)×	X
ejpam-7061	460	4	·	·	PUNCT
ejpam-7061	460	5	·	·	PUNCT
ejpam-7061	460	6	·	·	PUNCT
ejpam-7061	460	7	×	×	NOUN
ejpam-7061	460	8	l(si)×	l(si)×	X
ejpam-7061	460	9	·	·	PUNCT
ejpam-7061	460	10	·	·	PUNCT
ejpam-7061	460	11	·	·	PUNCT
ejpam-7061	460	12	×	×	NOUN
ejpam-7061	460	13	l(sn	l(sn	ADJ
ejpam-7061	460	14	)	)	PUNCT
ejpam-7061	460	15	=	=	PRON
ejpam-7061	460	16	s1s1	s1s1	ADP
ejpam-7061	460	17	×	×	NOUN
ejpam-7061	460	18	s2s2	s2s2	NOUN
ejpam-7061	460	19	×	×	NOUN
ejpam-7061	460	20	·	·	PUNCT
ejpam-7061	460	21	·	·	PUNCT
ejpam-7061	460	22	·	·	PUNCT
ejpam-7061	461	1	×	×	PROPN
ejpam-7061	461	2	sisi	sisi	X
ejpam-7061	461	3	×	×	NOUN
ejpam-7061	461	4	·	·	PUNCT
ejpam-7061	461	5	·	·	PUNCT
ejpam-7061	461	6	·	·	PUNCT
ejpam-7061	461	7	×	×	NOUN
ejpam-7061	461	8	snsn	snsn	NOUN
ejpam-7061	461	9	⊊	⊊	VERB
ejpam-7061	461	10	(	(	PUNCT
ejpam-7061	461	11	s1	s1	NOUN
ejpam-7061	461	12	,	,	PUNCT
ejpam-7061	461	13	s2	s2	NOUN
ejpam-7061	461	14	,	,	PUNCT
ejpam-7061	461	15	.	.	PUNCT
ejpam-7061	461	16	.	.	PUNCT
ejpam-7061	461	17	.	.	PUNCT
ejpam-7061	462	1	,	,	PUNCT
ejpam-7061	462	2	ui	ui	PROPN
ejpam-7061	462	3	,	,	PUNCT
ejpam-7061	462	4	.	.	PUNCT
ejpam-7061	462	5	.	.	PUNCT
ejpam-7061	463	1	.	.	PUNCT
ejpam-7061	464	1	,	,	PUNCT
ejpam-7061	464	2	sn	sn	PROPN
ejpam-7061	464	3	)	)	PUNCT
ejpam-7061	464	4	∪	∪	NOUN
ejpam-7061	464	5	(	(	PUNCT
ejpam-7061	464	6	s1s1	s1s1	NOUN
ejpam-7061	464	7	×	×	NOUN
ejpam-7061	464	8	s2s2	s2s2	NOUN
ejpam-7061	464	9	×	×	NOUN
ejpam-7061	464	10	·	·	PUNCT
ejpam-7061	464	11	·	·	PUNCT
ejpam-7061	464	12	·	·	PUNCT
ejpam-7061	465	1	×	×	NOUN
ejpam-7061	465	2	siui	siui	NOUN
ejpam-7061	465	3	×	×	NOUN
ejpam-7061	465	4	·	·	PUNCT
ejpam-7061	465	5	·	·	PUNCT
ejpam-7061	465	6	·	·	PUNCT
ejpam-7061	465	7	×	×	NOUN
ejpam-7061	465	8	snsn	snsn	NOUN
ejpam-7061	465	9	)	)	PUNCT
ejpam-7061	465	10	=	=	SYM
ejpam-7061	465	11	l((s1	l((s1	PROPN
ejpam-7061	465	12	,	,	PUNCT
ejpam-7061	465	13	s2	s2	PROPN
ejpam-7061	465	14	,	,	PUNCT
ejpam-7061	465	15	.	.	PUNCT
ejpam-7061	465	16	.	.	PUNCT
ejpam-7061	465	17	.	.	PUNCT
ejpam-7061	466	1	,	,	PUNCT
ejpam-7061	466	2	ui	ui	PROPN
ejpam-7061	466	3	,	,	PUNCT
ejpam-7061	466	4	.	.	PUNCT
ejpam-7061	466	5	.	.	PUNCT
ejpam-7061	466	6	.	.	PUNCT
ejpam-7061	467	1	,	,	PUNCT
ejpam-7061	467	2	sn	sn	PROPN
ejpam-7061	467	3	)	)	PUNCT
ejpam-7061	467	4	)	)	PUNCT
ejpam-7061	467	5	.	.	PUNCT
ejpam-7061	468	1	therefore	therefore	ADV
ejpam-7061	468	2	,	,	PUNCT
ejpam-7061	468	3	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	468	4	...	...	PUNCT
ejpam-7061	468	5	,sn	,sn	PUNCT
ejpam-7061	468	6	)	)	PUNCT
ejpam-7061	468	7	is	be	AUX
ejpam-7061	468	8	not	not	PART
ejpam-7061	468	9	a	a	DET
ejpam-7061	468	10	maximal	maximal	ADJ
ejpam-7061	468	11	l	l	NOUN
ejpam-7061	468	12	-	-	NOUN
ejpam-7061	468	13	class	class	NOUN
ejpam-7061	468	14	.	.	PUNCT
ejpam-7061	469	1	conversely	conversely	ADV
ejpam-7061	469	2	,	,	PUNCT
ejpam-7061	469	3	assume	assume	VERB
ejpam-7061	469	4	that	that	SCONJ
ejpam-7061	469	5	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	469	6	...	...	PUNCT
ejpam-7061	469	7	,sn	,sn	PUNCT
ejpam-7061	469	8	)	)	PUNCT
ejpam-7061	469	9	is	be	AUX
ejpam-7061	469	10	not	not	PART
ejpam-7061	469	11	a	a	DET
ejpam-7061	469	12	maximal	maximal	ADJ
ejpam-7061	469	13	l	l	NOUN
ejpam-7061	469	14	-	-	NOUN
ejpam-7061	469	15	class	class	NOUN
ejpam-7061	469	16	.	.	PUNCT
ejpam-7061	470	1	then	then	ADV
ejpam-7061	470	2	there	there	PRON
ejpam-7061	470	3	exists	exist	VERB
ejpam-7061	470	4	(	(	PUNCT
ejpam-7061	470	5	u1	u1	NOUN
ejpam-7061	470	6	,	,	PUNCT
ejpam-7061	470	7	u2	u2	NOUN
ejpam-7061	470	8	,	,	PUNCT
ejpam-7061	470	9	.	.	PUNCT
ejpam-7061	470	10	.	.	PUNCT
ejpam-7061	470	11	.	.	PUNCT
ejpam-7061	471	1	,	,	PUNCT
ejpam-7061	471	2	un	un	PROPN
ejpam-7061	471	3	)	)	PUNCT
ejpam-7061	471	4	∈	∈	PROPN
ejpam-7061	471	5	s1	s1	NOUN
ejpam-7061	471	6	×	×	PROPN
ejpam-7061	471	7	s2	s2	NOUN
ejpam-7061	471	8	×	×	NOUN
ejpam-7061	471	9	·	·	PUNCT
ejpam-7061	471	10	·	·	PUNCT
ejpam-7061	471	11	·	·	PUNCT
ejpam-7061	472	1	×	×	NOUN
ejpam-7061	472	2	sn	sn	INTJ
ejpam-7061	472	3	such	such	ADJ
ejpam-7061	472	4	that	that	SCONJ
ejpam-7061	472	5	l((s1	l((s1	NOUN
ejpam-7061	472	6	,	,	PUNCT
ejpam-7061	472	7	s2	s2	PROPN
ejpam-7061	472	8	,	,	PUNCT
ejpam-7061	472	9	.	.	PUNCT
ejpam-7061	472	10	.	.	PUNCT
ejpam-7061	472	11	.	.	PUNCT
ejpam-7061	473	1	,	,	PUNCT
ejpam-7061	473	2	sn	sn	PROPN
ejpam-7061	473	3	)	)	PUNCT
ejpam-7061	473	4	)	)	PUNCT
ejpam-7061	474	1	⊊	⊊	VERB
ejpam-7061	474	2	l((u1	l((u1	NOUN
ejpam-7061	474	3	,	,	PUNCT
ejpam-7061	474	4	u2	u2	PROPN
ejpam-7061	474	5	,	,	PUNCT
ejpam-7061	474	6	.	.	PUNCT
ejpam-7061	474	7	.	.	PUNCT
ejpam-7061	475	1	.	.	PUNCT
ejpam-7061	476	1	,	,	PUNCT
ejpam-7061	476	2	un	un	PROPN
ejpam-7061	476	3	)	)	PUNCT
ejpam-7061	476	4	)	)	PUNCT
ejpam-7061	476	5	.	.	PUNCT
ejpam-7061	477	1	by	by	ADP
ejpam-7061	477	2	lemma	lemma	PROPN
ejpam-7061	477	3	2(i	2(i	NUM
ejpam-7061	477	4	)	)	PUNCT
ejpam-7061	477	5	,	,	PUNCT
ejpam-7061	477	6	we	we	PRON
ejpam-7061	477	7	obtain	obtain	VERB
ejpam-7061	477	8	(	(	PUNCT
ejpam-7061	477	9	u1	u1	NOUN
ejpam-7061	477	10	,	,	PUNCT
ejpam-7061	477	11	u2	u2	NOUN
ejpam-7061	477	12	,	,	PUNCT
ejpam-7061	477	13	.	.	PUNCT
ejpam-7061	477	14	.	.	PUNCT
ejpam-7061	477	15	.	.	PUNCT
ejpam-7061	478	1	,	,	PUNCT
ejpam-7061	478	2	un	un	PROPN
ejpam-7061	478	3	)	)	PUNCT
ejpam-7061	478	4	/∈	/∈	PUNCT
ejpam-7061	479	1	l((s1	l((s1	PROPN
ejpam-7061	479	2	,	,	PUNCT
ejpam-7061	479	3	s2	s2	PROPN
ejpam-7061	479	4	,	,	PUNCT
ejpam-7061	479	5	.	.	PUNCT
ejpam-7061	479	6	.	.	PUNCT
ejpam-7061	479	7	.	.	PUNCT
ejpam-7061	480	1	,	,	PUNCT
ejpam-7061	480	2	sn	sn	PROPN
ejpam-7061	480	3	)	)	PUNCT
ejpam-7061	480	4	)	)	PUNCT
ejpam-7061	481	1	=	=	PRON
ejpam-7061	481	2	(	(	PUNCT
ejpam-7061	481	3	s1	s1	NOUN
ejpam-7061	481	4	,	,	PUNCT
ejpam-7061	481	5	s2	s2	NOUN
ejpam-7061	481	6	,	,	PUNCT
ejpam-7061	481	7	.	.	PUNCT
ejpam-7061	481	8	.	.	PUNCT
ejpam-7061	482	1	.	.	PUNCT
ejpam-7061	483	1	,	,	PUNCT
ejpam-7061	483	2	sn	sn	PROPN
ejpam-7061	483	3	)	)	PUNCT
ejpam-7061	483	4	∪	∪	NOUN
ejpam-7061	483	5	(	(	PUNCT
ejpam-7061	483	6	s1s1	s1s1	NOUN
ejpam-7061	483	7	×	×	NOUN
ejpam-7061	483	8	s2s2	s2s2	NOUN
ejpam-7061	483	9	×	×	NOUN
ejpam-7061	483	10	·	·	PUNCT
ejpam-7061	483	11	·	·	PUNCT
ejpam-7061	483	12	·	·	PUNCT
ejpam-7061	483	13	×	×	NOUN
ejpam-7061	483	14	snsn	snsn	NOUN
ejpam-7061	483	15	)	)	PUNCT
ejpam-7061	484	1	=	=	PRON
ejpam-7061	484	2	s1s1	s1s1	ADP
ejpam-7061	484	3	×	×	NOUN
ejpam-7061	484	4	s2s2	s2s2	NOUN
ejpam-7061	484	5	×	×	NOUN
ejpam-7061	484	6	·	·	PUNCT
ejpam-7061	484	7	·	·	PUNCT
ejpam-7061	484	8	·	·	PUNCT
ejpam-7061	484	9	×	×	NOUN
ejpam-7061	484	10	snsn	snsn	NOUN
ejpam-7061	484	11	.	.	PUNCT
ejpam-7061	485	1	this	this	PRON
ejpam-7061	485	2	implies	imply	VERB
ejpam-7061	485	3	that	that	PRON
ejpam-7061	485	4	ui	ui	PROPN
ejpam-7061	485	5	/∈	/∈	PUNCT
ejpam-7061	485	6	sisi	sisi	PROPN
ejpam-7061	485	7	for	for	ADP
ejpam-7061	485	8	some	some	DET
ejpam-7061	485	9	i	i	PRON
ejpam-7061	485	10	∈	∈	PROPN
ejpam-7061	485	11	{	{	PUNCT
ejpam-7061	485	12	1	1	NUM
ejpam-7061	485	13	,	,	PUNCT
ejpam-7061	485	14	2	2	NUM
ejpam-7061	485	15	,	,	PUNCT
ejpam-7061	485	16	.	.	PUNCT
ejpam-7061	485	17	.	.	PUNCT
ejpam-7061	486	1	.	.	PUNCT
ejpam-7061	486	2	,	,	PUNCT
ejpam-7061	487	1	n	n	CCONJ
ejpam-7061	487	2	}	}	PUNCT
ejpam-7061	487	3	.	.	PUNCT
ejpam-7061	488	1	it	it	PRON
ejpam-7061	488	2	follows	follow	VERB
ejpam-7061	488	3	by	by	ADP
ejpam-7061	488	4	assumption	assumption	NOUN
ejpam-7061	488	5	that	that	SCONJ
ejpam-7061	488	6	ui	ui	PROPN
ejpam-7061	488	7	̸=	̸=	PROPN
ejpam-7061	488	8	si	si	PROPN
ejpam-7061	488	9	.	.	PROPN
ejpam-7061	488	10	therefore	therefore	ADV
ejpam-7061	488	11	,	,	PUNCT
ejpam-7061	488	12	l(si	l(si	PROPN
ejpam-7061	488	13	)	)	PUNCT
ejpam-7061	488	14	⊆	⊆	NUM
ejpam-7061	488	15	l(ui	l(ui	PROPN
ejpam-7061	488	16	)	)	PUNCT
ejpam-7061	488	17	and	and	CCONJ
ejpam-7061	488	18	ui	ui	PROPN
ejpam-7061	488	19	/∈	/∈	PUNCT
ejpam-7061	489	1	si	si	PROPN
ejpam-7061	489	2	∪	∪	X
ejpam-7061	489	3	sisi	sisi	X
ejpam-7061	489	4	=	=	SYM
ejpam-7061	489	5	l(si	l(si	PROPN
ejpam-7061	489	6	)	)	PUNCT
ejpam-7061	489	7	.	.	PUNCT
ejpam-7061	490	1	these	these	DET
ejpam-7061	490	2	imply	imply	VERB
ejpam-7061	490	3	l(si	l(si	PROPN
ejpam-7061	490	4	)	)	PUNCT
ejpam-7061	490	5	⊊	⊊	VERB
ejpam-7061	490	6	l(ui	l(ui	PROPN
ejpam-7061	490	7	)	)	PUNCT
ejpam-7061	490	8	.	.	PUNCT
ejpam-7061	491	1	thus	thus	ADV
ejpam-7061	491	2	,	,	PUNCT
ejpam-7061	491	3	lsi	lsi	PROPN
ejpam-7061	491	4	is	be	AUX
ejpam-7061	491	5	not	not	PART
ejpam-7061	491	6	a	a	DET
ejpam-7061	491	7	maximal	maximal	ADJ
ejpam-7061	491	8	l	l	NOUN
ejpam-7061	491	9	-	-	NOUN
ejpam-7061	491	10	class	class	NOUN
ejpam-7061	491	11	.	.	PUNCT
ejpam-7061	492	1	corollary	corollary	ADJ
ejpam-7061	492	2	1	1	NUM
ejpam-7061	492	3	.	.	PUNCT
ejpam-7061	493	1	let	let	VERB
ejpam-7061	493	2	si	si	PRON
ejpam-7061	493	3	be	be	AUX
ejpam-7061	493	4	a	a	DET
ejpam-7061	493	5	semigroup	semigroup	NOUN
ejpam-7061	493	6	and	and	CCONJ
ejpam-7061	493	7	let	let	VERB
ejpam-7061	493	8	si	si	PROPN
ejpam-7061	493	9	∈	∈	PROPN
ejpam-7061	493	10	si	si	X
ejpam-7061	493	11	where	where	SCONJ
ejpam-7061	493	12	i	i	PRON
ejpam-7061	493	13	∈	∈	PROPN
ejpam-7061	493	14	{	{	PUNCT
ejpam-7061	493	15	1	1	NUM
ejpam-7061	493	16	,	,	PUNCT
ejpam-7061	493	17	2	2	NUM
ejpam-7061	493	18	,	,	PUNCT
ejpam-7061	493	19	.	.	PUNCT
ejpam-7061	493	20	.	.	PUNCT
ejpam-7061	494	1	.	.	PUNCT
ejpam-7061	494	2	,	,	PUNCT
ejpam-7061	494	3	n	n	CCONJ
ejpam-7061	494	4	}	}	PUNCT
ejpam-7061	494	5	.	.	PUNCT
ejpam-7061	495	1	if	if	SCONJ
ejpam-7061	495	2	(	(	PUNCT
ejpam-7061	495	3	s1	s1	NOUN
ejpam-7061	495	4	,	,	PUNCT
ejpam-7061	495	5	s2	s2	NOUN
ejpam-7061	495	6	,	,	PUNCT
ejpam-7061	495	7	.	.	PUNCT
ejpam-7061	495	8	.	.	PUNCT
ejpam-7061	495	9	.	.	PUNCT
ejpam-7061	495	10	,	,	PUNCT
ejpam-7061	495	11	sn	sn	PROPN
ejpam-7061	495	12	)	)	PUNCT
ejpam-7061	495	13	∈	∈	PROPN
ejpam-7061	495	14	s1s1×s2s2×	s1s1×s2s2×	PROPN
ejpam-7061	495	15	·	·	PUNCT
ejpam-7061	495	16	·	·	PUNCT
ejpam-7061	495	17	·	·	PUNCT
ejpam-7061	495	18	×snsn	×snsn	PROPN
ejpam-7061	495	19	,	,	PUNCT
ejpam-7061	495	20	then	then	ADV
ejpam-7061	495	21	ls1×ls2×	ls1×ls2×	X
ejpam-7061	495	22	·	·	PUNCT
ejpam-7061	495	23	·	·	PUNCT
ejpam-7061	495	24	·	·	PUNCT
ejpam-7061	495	25	×lsi	×lsi	NOUN
ejpam-7061	495	26	is	be	AUX
ejpam-7061	495	27	a	a	DET
ejpam-7061	495	28	maximal	maximal	ADJ
ejpam-7061	495	29	l	l	NOUN
ejpam-7061	495	30	-	-	NOUN
ejpam-7061	495	31	class	class	NOUN
ejpam-7061	495	32	if	if	SCONJ
ejpam-7061	495	33	and	and	CCONJ
ejpam-7061	495	34	only	only	ADV
ejpam-7061	495	35	if	if	SCONJ
ejpam-7061	495	36	lsi	lsi	PROPN
ejpam-7061	495	37	is	be	AUX
ejpam-7061	495	38	a	a	DET
ejpam-7061	495	39	maximal	maximal	ADJ
ejpam-7061	495	40	l	l	NOUN
ejpam-7061	495	41	-	-	NOUN
ejpam-7061	495	42	class	class	NOUN
ejpam-7061	495	43	for	for	ADP
ejpam-7061	495	44	all	all	PRON
ejpam-7061	495	45	i	i	PRON
ejpam-7061	495	46	∈	∈	PROPN
ejpam-7061	495	47	{	{	PUNCT
ejpam-7061	495	48	1	1	NUM
ejpam-7061	495	49	,	,	PUNCT
ejpam-7061	495	50	2	2	NUM
ejpam-7061	495	51	,	,	PUNCT
ejpam-7061	495	52	.	.	PUNCT
ejpam-7061	495	53	.	.	PUNCT
ejpam-7061	496	1	.	.	PUNCT
ejpam-7061	496	2	,	,	PUNCT
ejpam-7061	497	1	n	n	CCONJ
ejpam-7061	497	2	}	}	PUNCT
ejpam-7061	497	3	.	.	PUNCT
ejpam-7061	498	1	proof	proof	NOUN
ejpam-7061	498	2	.	.	PUNCT
ejpam-7061	499	1	the	the	DET
ejpam-7061	499	2	proof	proof	NOUN
ejpam-7061	499	3	is	be	AUX
ejpam-7061	499	4	obtained	obtain	VERB
ejpam-7061	499	5	directly	directly	ADV
ejpam-7061	499	6	from	from	ADP
ejpam-7061	499	7	theorem	theorem	ADJ
ejpam-7061	499	8	4	4	NUM
ejpam-7061	499	9	and	and	CCONJ
ejpam-7061	499	10	theorem	theorem	VERB
ejpam-7061	499	11	6	6	NUM
ejpam-7061	499	12	definition	definition	NOUN
ejpam-7061	499	13	1	1	NUM
ejpam-7061	499	14	.	.	PUNCT
ejpam-7061	500	1	let	let	VERB
ejpam-7061	500	2	s	s	PRON
ejpam-7061	500	3	be	be	AUX
ejpam-7061	500	4	a	a	DET
ejpam-7061	500	5	semigroup	semigroup	NOUN
ejpam-7061	500	6	.	.	PUNCT
ejpam-7061	501	1	an	an	DET
ejpam-7061	501	2	element	element	NOUN
ejpam-7061	501	3	s	s	PART
ejpam-7061	501	4	∈	∈	NOUN
ejpam-7061	501	5	s	s	VERB
ejpam-7061	501	6	is	be	AUX
ejpam-7061	501	7	decomposable	decomposable	ADJ
ejpam-7061	501	8	if	if	SCONJ
ejpam-7061	501	9	s	s	PROPN
ejpam-7061	501	10	∈	∈	PROPN
ejpam-7061	501	11	s2	s2	PROPN
ejpam-7061	501	12	.	.	PUNCT
ejpam-7061	502	1	if	if	SCONJ
ejpam-7061	502	2	an	an	DET
ejpam-7061	502	3	element	element	NOUN
ejpam-7061	502	4	s	s	PART
ejpam-7061	502	5	∈	∈	NOUN
ejpam-7061	502	6	s	s	PART
ejpam-7061	502	7	is	be	AUX
ejpam-7061	502	8	not	not	PART
ejpam-7061	502	9	decomposable	decomposable	ADJ
ejpam-7061	502	10	,	,	PUNCT
ejpam-7061	502	11	we	we	PRON
ejpam-7061	502	12	say	say	VERB
ejpam-7061	502	13	that	that	PRON
ejpam-7061	502	14	s	s	VERB
ejpam-7061	502	15	is	be	AUX
ejpam-7061	502	16	indecomposable	indecomposable	ADJ
ejpam-7061	502	17	.	.	PUNCT
ejpam-7061	503	1	p.	p.	NOUN
ejpam-7061	503	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	503	3	,	,	PUNCT
ejpam-7061	503	4	o.	o.	PROPN
ejpam-7061	503	5	pankoon	pankoon	NOUN
ejpam-7061	503	6	,	,	PUNCT
ejpam-7061	503	7	t.	t.	PROPN
ejpam-7061	503	8	changphas	changphas	PROPN
ejpam-7061	503	9	/	/	SYM
ejpam-7061	503	10	eur	eur	PROPN
ejpam-7061	503	11	.	.	PUNCT
ejpam-7061	504	1	j.	j.	PROPN
ejpam-7061	504	2	pure	pure	PROPN
ejpam-7061	504	3	appl	appl	PROPN
ejpam-7061	504	4	.	.	PROPN
ejpam-7061	504	5	math	math	PROPN
ejpam-7061	504	6	,	,	PUNCT
ejpam-7061	504	7	18	18	NUM
ejpam-7061	504	8	(	(	PUNCT
ejpam-7061	504	9	4	4	NUM
ejpam-7061	504	10	)	)	PUNCT
ejpam-7061	504	11	(	(	PUNCT
ejpam-7061	504	12	2025	2025	NUM
ejpam-7061	504	13	)	)	PUNCT
ejpam-7061	504	14	,	,	PUNCT
ejpam-7061	504	15	7061	7061	NUM
ejpam-7061	504	16	9	9	NUM
ejpam-7061	504	17	of	of	ADP
ejpam-7061	504	18	10	10	NUM
ejpam-7061	504	19	in	in	ADP
ejpam-7061	504	20	[	[	X
ejpam-7061	504	21	1	1	NUM
ejpam-7061	504	22	]	]	PUNCT
ejpam-7061	505	1	,	,	PUNCT
ejpam-7061	505	2	the	the	DET
ejpam-7061	505	3	author	author	NOUN
ejpam-7061	505	4	remarked	remark	VERB
ejpam-7061	505	5	that	that	SCONJ
ejpam-7061	505	6	if	if	SCONJ
ejpam-7061	505	7	an	an	DET
ejpam-7061	505	8	element	element	NOUN
ejpam-7061	505	9	s	s	NOUN
ejpam-7061	505	10	of	of	ADP
ejpam-7061	505	11	a	a	DET
ejpam-7061	505	12	semigroup	semigroup	NOUN
ejpam-7061	505	13	s	s	VERB
ejpam-7061	505	14	is	be	AUX
ejpam-7061	505	15	indecomposable	indecomposable	ADJ
ejpam-7061	505	16	,	,	PUNCT
ejpam-7061	505	17	then	then	ADV
ejpam-7061	505	18	ls	ls	ADJ
ejpam-7061	505	19	=	=	PUNCT
ejpam-7061	505	20	{	{	PUNCT
ejpam-7061	505	21	s	s	NOUN
ejpam-7061	505	22	}	}	PUNCT
ejpam-7061	505	23	.	.	PUNCT
ejpam-7061	506	1	in	in	ADP
ejpam-7061	506	2	the	the	DET
ejpam-7061	506	3	direct	direct	ADJ
ejpam-7061	506	4	product	product	NOUN
ejpam-7061	506	5	of	of	ADP
ejpam-7061	506	6	{	{	PUNCT
ejpam-7061	506	7	si	si	INTJ
ejpam-7061	507	1	|	|	ADV
ejpam-7061	507	2	i	i	PRON
ejpam-7061	507	3	∈	∈	PROPN
ejpam-7061	507	4	{	{	PUNCT
ejpam-7061	507	5	1	1	NUM
ejpam-7061	507	6	,	,	PUNCT
ejpam-7061	507	7	2	2	NUM
ejpam-7061	507	8	,	,	PUNCT
ejpam-7061	507	9	.	.	PUNCT
ejpam-7061	507	10	.	.	PUNCT
ejpam-7061	507	11	.	.	PUNCT
ejpam-7061	507	12	,	,	PUNCT
ejpam-7061	507	13	n	n	CCONJ
ejpam-7061	507	14	}	}	PUNCT
ejpam-7061	507	15	}	}	PUNCT
ejpam-7061	507	16	we	we	PRON
ejpam-7061	507	17	have	have	VERB
ejpam-7061	507	18	that	that	SCONJ
ejpam-7061	507	19	if	if	SCONJ
ejpam-7061	507	20	(	(	PUNCT
ejpam-7061	507	21	s1	s1	NOUN
ejpam-7061	507	22	,	,	PUNCT
ejpam-7061	507	23	s2	s2	NOUN
ejpam-7061	507	24	,	,	PUNCT
ejpam-7061	507	25	.	.	PUNCT
ejpam-7061	507	26	.	.	PUNCT
ejpam-7061	507	27	.	.	PUNCT
ejpam-7061	508	1	,	,	PUNCT
ejpam-7061	508	2	sn	sn	PROPN
ejpam-7061	508	3	)	)	PUNCT
ejpam-7061	508	4	∈	∈	PROPN
ejpam-7061	508	5	s1×s2×	s1×s2×	PROPN
ejpam-7061	508	6	·	·	PUNCT
ejpam-7061	508	7	·	·	PUNCT
ejpam-7061	508	8	·	·	PUNCT
ejpam-7061	508	9	×sn	×sn	PROPN
ejpam-7061	508	10	is	be	AUX
ejpam-7061	508	11	indecomposable	indecomposable	ADJ
ejpam-7061	508	12	,	,	PUNCT
ejpam-7061	508	13	then	then	ADV
ejpam-7061	508	14	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	508	15	...	...	PUNCT
ejpam-7061	508	16	,sn	,sn	PUNCT
ejpam-7061	508	17	)	)	PUNCT
ejpam-7061	509	1	=	=	PRON
ejpam-7061	509	2	{	{	PUNCT
ejpam-7061	509	3	(	(	PUNCT
ejpam-7061	509	4	s1	s1	NOUN
ejpam-7061	509	5	,	,	PUNCT
ejpam-7061	509	6	s2	s2	NOUN
ejpam-7061	509	7	,	,	PUNCT
ejpam-7061	509	8	.	.	PUNCT
ejpam-7061	509	9	.	.	PUNCT
ejpam-7061	509	10	.	.	PUNCT
ejpam-7061	510	1	,	,	PUNCT
ejpam-7061	510	2	sn	sn	PROPN
ejpam-7061	510	3	)	)	PUNCT
ejpam-7061	510	4	}	}	PUNCT
ejpam-7061	510	5	and	and	CCONJ
ejpam-7061	510	6	si	si	X
ejpam-7061	510	7	is	be	AUX
ejpam-7061	510	8	indecomposable	indecomposable	ADJ
ejpam-7061	510	9	for	for	ADP
ejpam-7061	510	10	some	some	DET
ejpam-7061	510	11	i	i	PRON
ejpam-7061	510	12	∈	∈	PROPN
ejpam-7061	510	13	{	{	PUNCT
ejpam-7061	510	14	1	1	NUM
ejpam-7061	510	15	,	,	PUNCT
ejpam-7061	510	16	2	2	NUM
ejpam-7061	510	17	,	,	PUNCT
ejpam-7061	510	18	.	.	PUNCT
ejpam-7061	510	19	.	.	PUNCT
ejpam-7061	511	1	.	.	PUNCT
ejpam-7061	511	2	,	,	PUNCT
ejpam-7061	511	3	n	n	CCONJ
ejpam-7061	511	4	}	}	PUNCT
ejpam-7061	511	5	.	.	PUNCT
ejpam-7061	512	1	on	on	ADP
ejpam-7061	512	2	the	the	DET
ejpam-7061	512	3	other	other	ADJ
ejpam-7061	512	4	hand	hand	NOUN
ejpam-7061	512	5	,	,	PUNCT
ejpam-7061	512	6	if	if	SCONJ
ejpam-7061	512	7	there	there	PRON
ejpam-7061	512	8	exists	exist	VERB
ejpam-7061	512	9	i	i	PRON
ejpam-7061	512	10	∈	∈	PROPN
ejpam-7061	512	11	{	{	PUNCT
ejpam-7061	512	12	1	1	NUM
ejpam-7061	512	13	,	,	PUNCT
ejpam-7061	512	14	2	2	NUM
ejpam-7061	512	15	,	,	PUNCT
ejpam-7061	512	16	.	.	PUNCT
ejpam-7061	512	17	.	.	PUNCT
ejpam-7061	513	1	.	.	PUNCT
ejpam-7061	514	1	,	,	PUNCT
ejpam-7061	515	1	n	n	CCONJ
ejpam-7061	515	2	}	}	PUNCT
ejpam-7061	515	3	such	such	ADJ
ejpam-7061	515	4	that	that	SCONJ
ejpam-7061	515	5	si	si	PROPN
ejpam-7061	515	6	is	be	AUX
ejpam-7061	515	7	indecomposable	indecomposable	ADJ
ejpam-7061	515	8	,	,	PUNCT
ejpam-7061	515	9	then	then	ADV
ejpam-7061	515	10	we	we	PRON
ejpam-7061	515	11	get	get	VERB
ejpam-7061	515	12	that	that	DET
ejpam-7061	515	13	(	(	PUNCT
ejpam-7061	515	14	s1	s1	NOUN
ejpam-7061	515	15	,	,	PUNCT
ejpam-7061	515	16	s2	s2	NOUN
ejpam-7061	515	17	,	,	PUNCT
ejpam-7061	515	18	.	.	PUNCT
ejpam-7061	515	19	.	.	PUNCT
ejpam-7061	516	1	.	.	PUNCT
ejpam-7061	517	1	,	,	PUNCT
ejpam-7061	517	2	sn	sn	PROPN
ejpam-7061	517	3	)	)	PUNCT
ejpam-7061	517	4	is	be	AUX
ejpam-7061	517	5	indecomposable	indecomposable	ADJ
ejpam-7061	517	6	.	.	PUNCT
ejpam-7061	518	1	next	next	ADV
ejpam-7061	518	2	,	,	PUNCT
ejpam-7061	518	3	we	we	PRON
ejpam-7061	518	4	consider	consider	VERB
ejpam-7061	518	5	relationships	relationship	NOUN
ejpam-7061	518	6	between	between	ADP
ejpam-7061	518	7	indecomposable	indecomposable	ADJ
ejpam-7061	518	8	elements	element	NOUN
ejpam-7061	518	9	and	and	CCONJ
ejpam-7061	518	10	maximal	maximal	ADJ
ejpam-7061	518	11	l	l	NOUN
ejpam-7061	518	12	-	-	PUNCT
ejpam-7061	518	13	classes	class	NOUN
ejpam-7061	518	14	.	.	PUNCT
ejpam-7061	519	1	theorem	theorem	NOUN
ejpam-7061	519	2	7	7	NUM
ejpam-7061	519	3	.	.	PUNCT
ejpam-7061	520	1	let	let	VERB
ejpam-7061	520	2	si	si	PRON
ejpam-7061	520	3	be	be	AUX
ejpam-7061	520	4	a	a	DET
ejpam-7061	520	5	semigroup	semigroup	NOUN
ejpam-7061	520	6	and	and	CCONJ
ejpam-7061	520	7	let	let	VERB
ejpam-7061	520	8	si	si	PROPN
ejpam-7061	520	9	∈	∈	PROPN
ejpam-7061	520	10	si	si	X
ejpam-7061	520	11	where	where	SCONJ
ejpam-7061	520	12	i	i	PRON
ejpam-7061	520	13	∈	∈	PROPN
ejpam-7061	520	14	{	{	PUNCT
ejpam-7061	520	15	1	1	NUM
ejpam-7061	520	16	,	,	PUNCT
ejpam-7061	520	17	2	2	NUM
ejpam-7061	520	18	,	,	PUNCT
ejpam-7061	520	19	.	.	PUNCT
ejpam-7061	520	20	.	.	PUNCT
ejpam-7061	521	1	.	.	PUNCT
ejpam-7061	521	2	,	,	PUNCT
ejpam-7061	521	3	n	n	CCONJ
ejpam-7061	521	4	}	}	PUNCT
ejpam-7061	521	5	.	.	PUNCT
ejpam-7061	522	1	(	(	PUNCT
ejpam-7061	522	2	i	i	NOUN
ejpam-7061	522	3	)	)	PUNCT
ejpam-7061	522	4	if	if	SCONJ
ejpam-7061	522	5	(	(	PUNCT
ejpam-7061	522	6	s1	s1	NOUN
ejpam-7061	522	7	,	,	PUNCT
ejpam-7061	522	8	s2	s2	NOUN
ejpam-7061	522	9	,	,	PUNCT
ejpam-7061	522	10	.	.	PUNCT
ejpam-7061	522	11	.	.	PUNCT
ejpam-7061	522	12	.	.	PUNCT
ejpam-7061	523	1	,	,	PUNCT
ejpam-7061	523	2	sn	sn	PROPN
ejpam-7061	523	3	)	)	PUNCT
ejpam-7061	523	4	is	be	AUX
ejpam-7061	523	5	indecomposable	indecomposable	ADJ
ejpam-7061	523	6	,	,	PUNCT
ejpam-7061	523	7	then	then	ADV
ejpam-7061	523	8	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	523	9	...	...	PUNCT
ejpam-7061	523	10	,sn	,sn	PUNCT
ejpam-7061	523	11	)	)	PUNCT
ejpam-7061	524	1	is	be	AUX
ejpam-7061	524	2	a	a	DET
ejpam-7061	524	3	maximal	maximal	ADJ
ejpam-7061	524	4	l	l	NOUN
ejpam-7061	524	5	-	-	NOUN
ejpam-7061	524	6	class	class	NOUN
ejpam-7061	524	7	.	.	PUNCT
ejpam-7061	525	1	(	(	PUNCT
ejpam-7061	525	2	ii	ii	NOUN
ejpam-7061	525	3	)	)	PUNCT
ejpam-7061	525	4	if	if	SCONJ
ejpam-7061	525	5	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	525	6	...	...	PUNCT
ejpam-7061	525	7	,sn	,sn	PUNCT
ejpam-7061	525	8	)	)	PUNCT
ejpam-7061	525	9	is	be	AUX
ejpam-7061	525	10	a	a	DET
ejpam-7061	525	11	maximal	maximal	ADJ
ejpam-7061	525	12	l	l	NOUN
ejpam-7061	525	13	-	-	NOUN
ejpam-7061	525	14	class	class	NOUN
ejpam-7061	525	15	of	of	ADP
ejpam-7061	525	16	s1	s1	PROPN
ejpam-7061	525	17	×	×	PROPN
ejpam-7061	525	18	s2	s2	NOUN
ejpam-7061	525	19	×	×	NOUN
ejpam-7061	525	20	·	·	PUNCT
ejpam-7061	525	21	·	·	PUNCT
ejpam-7061	525	22	·	·	PUNCT
ejpam-7061	526	1	×	×	NOUN
ejpam-7061	526	2	sn	sn	PROPN
ejpam-7061	526	3	and	and	CCONJ
ejpam-7061	526	4	(	(	PUNCT
ejpam-7061	526	5	s1	s1	NOUN
ejpam-7061	526	6	,	,	PUNCT
ejpam-7061	526	7	s2	s2	NOUN
ejpam-7061	526	8	,	,	PUNCT
ejpam-7061	526	9	.	.	PUNCT
ejpam-7061	526	10	.	.	PUNCT
ejpam-7061	526	11	.	.	PUNCT
ejpam-7061	527	1	,	,	PUNCT
ejpam-7061	527	2	sn	sn	PROPN
ejpam-7061	527	3	)	)	PUNCT
ejpam-7061	527	4	/∈	/∈	PUNCT
ejpam-7061	528	1	s1s1	s1s1	PROPN
ejpam-7061	528	2	×	×	NOUN
ejpam-7061	528	3	s2s2	s2s2	NOUN
ejpam-7061	528	4	×	×	NOUN
ejpam-7061	528	5	·	·	PUNCT
ejpam-7061	528	6	·	·	PUNCT
ejpam-7061	529	1	·	·	PUNCT
ejpam-7061	529	2	×	×	NOUN
ejpam-7061	529	3	snsn	snsn	NOUN
ejpam-7061	529	4	,	,	PUNCT
ejpam-7061	529	5	then	then	ADV
ejpam-7061	529	6	(	(	PUNCT
ejpam-7061	529	7	s1	s1	NOUN
ejpam-7061	529	8	,	,	PUNCT
ejpam-7061	529	9	s2	s2	NOUN
ejpam-7061	529	10	,	,	PUNCT
ejpam-7061	529	11	.	.	PUNCT
ejpam-7061	529	12	.	.	PUNCT
ejpam-7061	529	13	.	.	PUNCT
ejpam-7061	530	1	,	,	PUNCT
ejpam-7061	530	2	sn	sn	PROPN
ejpam-7061	530	3	)	)	PUNCT
ejpam-7061	530	4	is	be	AUX
ejpam-7061	530	5	indecomposable	indecomposable	ADJ
ejpam-7061	530	6	.	.	PUNCT
ejpam-7061	531	1	proof	proof	NOUN
ejpam-7061	531	2	.	.	PUNCT
ejpam-7061	532	1	(	(	PUNCT
ejpam-7061	532	2	i	i	NOUN
ejpam-7061	532	3	)	)	PUNCT
ejpam-7061	532	4	assume	assume	VERB
ejpam-7061	532	5	that	that	SCONJ
ejpam-7061	532	6	(	(	PUNCT
ejpam-7061	532	7	s1	s1	NOUN
ejpam-7061	532	8	,	,	PUNCT
ejpam-7061	532	9	s2	s2	NOUN
ejpam-7061	532	10	,	,	PUNCT
ejpam-7061	532	11	.	.	PUNCT
ejpam-7061	532	12	.	.	PUNCT
ejpam-7061	533	1	.	.	PUNCT
ejpam-7061	534	1	,	,	PUNCT
ejpam-7061	534	2	sn	sn	PROPN
ejpam-7061	534	3	)	)	PUNCT
ejpam-7061	534	4	is	be	AUX
ejpam-7061	534	5	indecomposable	indecomposable	ADJ
ejpam-7061	534	6	.	.	PUNCT
ejpam-7061	535	1	suppose	suppose	VERB
ejpam-7061	535	2	that	that	SCONJ
ejpam-7061	535	3	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	535	4	...	...	PUNCT
ejpam-7061	535	5	,sn	,sn	PUNCT
ejpam-7061	535	6	)	)	PUNCT
ejpam-7061	535	7	is	be	AUX
ejpam-7061	535	8	not	not	PART
ejpam-7061	535	9	a	a	DET
ejpam-7061	535	10	maximal	maximal	ADJ
ejpam-7061	535	11	l	l	NOUN
ejpam-7061	535	12	-	-	NOUN
ejpam-7061	535	13	class	class	NOUN
ejpam-7061	535	14	.	.	PUNCT
ejpam-7061	536	1	then	then	ADV
ejpam-7061	536	2	there	there	PRON
ejpam-7061	536	3	exists	exist	VERB
ejpam-7061	536	4	(	(	PUNCT
ejpam-7061	536	5	u1	u1	NOUN
ejpam-7061	536	6	,	,	PUNCT
ejpam-7061	536	7	u2	u2	NOUN
ejpam-7061	536	8	,	,	PUNCT
ejpam-7061	536	9	.	.	PUNCT
ejpam-7061	536	10	.	.	PUNCT
ejpam-7061	536	11	.	.	PUNCT
ejpam-7061	537	1	,	,	PUNCT
ejpam-7061	537	2	un	un	PROPN
ejpam-7061	537	3	)	)	PUNCT
ejpam-7061	537	4	∈	∈	PROPN
ejpam-7061	537	5	s1	s1	NOUN
ejpam-7061	537	6	×	×	PROPN
ejpam-7061	537	7	s2	s2	NOUN
ejpam-7061	537	8	×	×	NOUN
ejpam-7061	537	9	·	·	PUNCT
ejpam-7061	537	10	·	·	PUNCT
ejpam-7061	537	11	·	·	PUNCT
ejpam-7061	538	1	×	×	NOUN
ejpam-7061	538	2	sn	sn	INTJ
ejpam-7061	538	3	such	such	ADJ
ejpam-7061	538	4	that	that	SCONJ
ejpam-7061	538	5	l((s1	l((s1	NOUN
ejpam-7061	538	6	,	,	PUNCT
ejpam-7061	538	7	s2	s2	PROPN
ejpam-7061	538	8	,	,	PUNCT
ejpam-7061	538	9	.	.	PUNCT
ejpam-7061	538	10	.	.	PUNCT
ejpam-7061	538	11	.	.	PUNCT
ejpam-7061	539	1	,	,	PUNCT
ejpam-7061	539	2	sn	sn	PROPN
ejpam-7061	539	3	)	)	PUNCT
ejpam-7061	539	4	)	)	PUNCT
ejpam-7061	540	1	⊊	⊊	VERB
ejpam-7061	540	2	l((u1	l((u1	NOUN
ejpam-7061	540	3	,	,	PUNCT
ejpam-7061	540	4	u2	u2	PROPN
ejpam-7061	540	5	,	,	PUNCT
ejpam-7061	540	6	.	.	PUNCT
ejpam-7061	540	7	.	.	PUNCT
ejpam-7061	541	1	.	.	PUNCT
ejpam-7061	542	1	,	,	PUNCT
ejpam-7061	542	2	un	un	PROPN
ejpam-7061	542	3	)	)	PUNCT
ejpam-7061	542	4	)	)	PUNCT
ejpam-7061	542	5	.	.	PUNCT
ejpam-7061	543	1	by	by	ADP
ejpam-7061	543	2	lemma	lemma	PROPN
ejpam-7061	543	3	2(ii	2(ii	NUM
ejpam-7061	543	4	)	)	PUNCT
ejpam-7061	543	5	,	,	PUNCT
ejpam-7061	543	6	we	we	PRON
ejpam-7061	543	7	obtain	obtain	VERB
ejpam-7061	543	8	that	that	PRON
ejpam-7061	543	9	(	(	PUNCT
ejpam-7061	543	10	s1	s1	NOUN
ejpam-7061	543	11	,	,	PUNCT
ejpam-7061	543	12	s2	s2	NOUN
ejpam-7061	543	13	,	,	PUNCT
ejpam-7061	543	14	.	.	PUNCT
ejpam-7061	543	15	.	.	PUNCT
ejpam-7061	543	16	.	.	PUNCT
ejpam-7061	544	1	,	,	PUNCT
ejpam-7061	544	2	sn	sn	PROPN
ejpam-7061	544	3	)	)	PUNCT
ejpam-7061	544	4	∈	∈	PROPN
ejpam-7061	544	5	l((s1	l((s1	PROPN
ejpam-7061	544	6	,	,	PUNCT
ejpam-7061	544	7	s2	s2	PROPN
ejpam-7061	544	8	,	,	PUNCT
ejpam-7061	544	9	.	.	PUNCT
ejpam-7061	544	10	.	.	PUNCT
ejpam-7061	544	11	.	.	PUNCT
ejpam-7061	545	1	,	,	PUNCT
ejpam-7061	545	2	sn	sn	PROPN
ejpam-7061	545	3	)	)	PUNCT
ejpam-7061	545	4	)	)	PUNCT
ejpam-7061	546	1	⊆	⊆	X
ejpam-7061	546	2	s1u1	s1u1	PRON
ejpam-7061	546	3	×	×	NOUN
ejpam-7061	546	4	s2u2	s2u2	CCONJ
ejpam-7061	546	5	×	×	NOUN
ejpam-7061	546	6	·	·	PUNCT
ejpam-7061	546	7	·	·	PUNCT
ejpam-7061	546	8	·	·	PUNCT
ejpam-7061	547	1	×	×	NOUN
ejpam-7061	547	2	snun	snun	NOUN
ejpam-7061	547	3	⊆	⊆	NUM
ejpam-7061	547	4	s2	s2	NOUN
ejpam-7061	547	5	1	1	NUM
ejpam-7061	547	6	×	×	NOUN
ejpam-7061	547	7	s2	s2	NOUN
ejpam-7061	547	8	2	2	NUM
ejpam-7061	547	9	×	×	NOUN
ejpam-7061	547	10	·	·	PUNCT
ejpam-7061	547	11	·	·	PUNCT
ejpam-7061	547	12	·	·	PUNCT
ejpam-7061	548	1	×	×	NOUN
ejpam-7061	548	2	s2	s2	PROPN
ejpam-7061	548	3	n.	n.	NOUN
ejpam-7061	548	4	this	this	PRON
ejpam-7061	548	5	contradicts	contradict	VERB
ejpam-7061	548	6	to	to	ADP
ejpam-7061	548	7	an	an	DET
ejpam-7061	548	8	assumption	assumption	NOUN
ejpam-7061	548	9	.	.	PUNCT
ejpam-7061	549	1	therefore	therefore	ADV
ejpam-7061	549	2	,	,	PUNCT
ejpam-7061	549	3	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	549	4	...	...	PUNCT
ejpam-7061	549	5	,sn	,sn	PUNCT
ejpam-7061	549	6	)	)	PUNCT
ejpam-7061	549	7	is	be	AUX
ejpam-7061	549	8	a	a	DET
ejpam-7061	549	9	maximal	maximal	ADJ
ejpam-7061	549	10	l	l	NOUN
ejpam-7061	549	11	-	-	NOUN
ejpam-7061	549	12	class	class	NOUN
ejpam-7061	549	13	.	.	PUNCT
ejpam-7061	550	1	(	(	PUNCT
ejpam-7061	550	2	ii	ii	NOUN
ejpam-7061	550	3	)	)	PUNCT
ejpam-7061	550	4	assume	assume	VERB
ejpam-7061	550	5	that	that	SCONJ
ejpam-7061	550	6	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	550	7	...	...	PUNCT
ejpam-7061	550	8	,sn	,sn	PUNCT
ejpam-7061	550	9	)	)	PUNCT
ejpam-7061	550	10	is	be	AUX
ejpam-7061	550	11	a	a	DET
ejpam-7061	550	12	maximal	maximal	ADJ
ejpam-7061	550	13	l	l	NOUN
ejpam-7061	550	14	-	-	NOUN
ejpam-7061	550	15	class	class	NOUN
ejpam-7061	550	16	and	and	CCONJ
ejpam-7061	550	17	(	(	PUNCT
ejpam-7061	550	18	s1	s1	NOUN
ejpam-7061	550	19	,	,	PUNCT
ejpam-7061	550	20	s2	s2	NOUN
ejpam-7061	550	21	,	,	PUNCT
ejpam-7061	550	22	.	.	PUNCT
ejpam-7061	550	23	.	.	PUNCT
ejpam-7061	551	1	.	.	PUNCT
ejpam-7061	552	1	,	,	PUNCT
ejpam-7061	552	2	sn	sn	PROPN
ejpam-7061	552	3	)	)	PUNCT
ejpam-7061	552	4	/∈	/∈	PUNCT
ejpam-7061	553	1	s1s1	s1s1	PROPN
ejpam-7061	553	2	×	×	NOUN
ejpam-7061	553	3	s2s2	s2s2	NOUN
ejpam-7061	553	4	×	×	NOUN
ejpam-7061	553	5	·	·	PUNCT
ejpam-7061	553	6	·	·	PUNCT
ejpam-7061	553	7	·	·	PUNCT
ejpam-7061	553	8	×snsn	×snsn	PROPN
ejpam-7061	553	9	.	.	PUNCT
ejpam-7061	554	1	suppose	suppose	VERB
ejpam-7061	554	2	that	that	SCONJ
ejpam-7061	554	3	(	(	PUNCT
ejpam-7061	554	4	s1	s1	NOUN
ejpam-7061	554	5	,	,	PUNCT
ejpam-7061	554	6	s2	s2	NOUN
ejpam-7061	554	7	,	,	PUNCT
ejpam-7061	554	8	.	.	PUNCT
ejpam-7061	554	9	.	.	PUNCT
ejpam-7061	555	1	.	.	PUNCT
ejpam-7061	556	1	,	,	PUNCT
ejpam-7061	556	2	sn	sn	PROPN
ejpam-7061	556	3	)	)	PUNCT
ejpam-7061	556	4	is	be	AUX
ejpam-7061	556	5	decomposable	decomposable	ADJ
ejpam-7061	556	6	.	.	PUNCT
ejpam-7061	557	1	then	then	ADV
ejpam-7061	557	2	there	there	PRON
ejpam-7061	557	3	exists	exist	VERB
ejpam-7061	557	4	(	(	PUNCT
ejpam-7061	557	5	t1	t1	NOUN
ejpam-7061	557	6	,	,	PUNCT
ejpam-7061	557	7	t2	t2	NOUN
ejpam-7061	557	8	,	,	PUNCT
ejpam-7061	557	9	.	.	PUNCT
ejpam-7061	557	10	.	.	PUNCT
ejpam-7061	558	1	.	.	PUNCT
ejpam-7061	559	1	,	,	PUNCT
ejpam-7061	559	2	tn	tn	PROPN
ejpam-7061	559	3	)	)	PUNCT
ejpam-7061	560	1	such	such	ADJ
ejpam-7061	560	2	that	that	SCONJ
ejpam-7061	560	3	(	(	PUNCT
ejpam-7061	560	4	s1	s1	NOUN
ejpam-7061	560	5	,	,	PUNCT
ejpam-7061	560	6	s2	s2	NOUN
ejpam-7061	560	7	,	,	PUNCT
ejpam-7061	560	8	.	.	PUNCT
ejpam-7061	560	9	.	.	PUNCT
ejpam-7061	560	10	.	.	PUNCT
ejpam-7061	560	11	,	,	PUNCT
ejpam-7061	560	12	sn	sn	PROPN
ejpam-7061	560	13	)	)	PUNCT
ejpam-7061	560	14	∈	∈	PROPN
ejpam-7061	560	15	(	(	PUNCT
ejpam-7061	560	16	s1	s1	PROPN
ejpam-7061	560	17	×	×	PROPN
ejpam-7061	560	18	s2	s2	NOUN
ejpam-7061	560	19	×	×	NOUN
ejpam-7061	560	20	·	·	PUNCT
ejpam-7061	560	21	·	·	PUNCT
ejpam-7061	560	22	·	·	PUNCT
ejpam-7061	561	1	×	×	PROPN
ejpam-7061	561	2	sn)(t1	sn)(t1	NOUN
ejpam-7061	561	3	,	,	PUNCT
ejpam-7061	561	4	t2	t2	NOUN
ejpam-7061	561	5	,	,	PUNCT
ejpam-7061	561	6	.	.	PUNCT
ejpam-7061	561	7	.	.	PUNCT
ejpam-7061	561	8	.	.	PUNCT
ejpam-7061	561	9	,	,	PUNCT
ejpam-7061	561	10	tn	tn	PROPN
ejpam-7061	561	11	)	)	PUNCT
ejpam-7061	561	12	.	.	PUNCT
ejpam-7061	562	1	this	this	PRON
ejpam-7061	562	2	implies	imply	VERB
ejpam-7061	562	3	that	that	SCONJ
ejpam-7061	562	4	l((s1	l((s1	NOUN
ejpam-7061	562	5	,	,	PUNCT
ejpam-7061	562	6	s2	s2	PROPN
ejpam-7061	562	7	,	,	PUNCT
ejpam-7061	562	8	.	.	PUNCT
ejpam-7061	562	9	.	.	PUNCT
ejpam-7061	562	10	.	.	PUNCT
ejpam-7061	563	1	,	,	PUNCT
ejpam-7061	563	2	sn	sn	PROPN
ejpam-7061	563	3	)	)	PUNCT
ejpam-7061	563	4	)	)	PUNCT
ejpam-7061	564	1	⊆	⊆	NUM
ejpam-7061	564	2	l((t1	l((t1	NOUN
ejpam-7061	564	3	,	,	PUNCT
ejpam-7061	564	4	t2	t2	NOUN
ejpam-7061	564	5	,	,	PUNCT
ejpam-7061	564	6	.	.	PUNCT
ejpam-7061	564	7	.	.	PUNCT
ejpam-7061	564	8	.	.	PUNCT
ejpam-7061	564	9	,	,	PUNCT
ejpam-7061	564	10	tn	tn	PROPN
ejpam-7061	564	11	)	)	PUNCT
ejpam-7061	564	12	)	)	PUNCT
ejpam-7061	564	13	.	.	PUNCT
ejpam-7061	565	1	we	we	PRON
ejpam-7061	565	2	observe	observe	VERB
ejpam-7061	565	3	that	that	SCONJ
ejpam-7061	565	4	(	(	PUNCT
ejpam-7061	565	5	t1	t1	NOUN
ejpam-7061	565	6	,	,	PUNCT
ejpam-7061	565	7	t2	t2	NOUN
ejpam-7061	565	8	,	,	PUNCT
ejpam-7061	565	9	.	.	PUNCT
ejpam-7061	565	10	.	.	PUNCT
ejpam-7061	566	1	.	.	PUNCT
ejpam-7061	567	1	,	,	PUNCT
ejpam-7061	567	2	tn	tn	PROPN
ejpam-7061	567	3	)	)	PUNCT
ejpam-7061	567	4	̸=	̸=	PROPN
ejpam-7061	567	5	(	(	PUNCT
ejpam-7061	567	6	s1	s1	PROPN
ejpam-7061	567	7	,	,	PUNCT
ejpam-7061	567	8	s2	s2	NOUN
ejpam-7061	567	9	,	,	PUNCT
ejpam-7061	567	10	.	.	PUNCT
ejpam-7061	567	11	.	.	PUNCT
ejpam-7061	567	12	.	.	PUNCT
ejpam-7061	568	1	,	,	PUNCT
ejpam-7061	568	2	sn	sn	PROPN
ejpam-7061	568	3	)	)	PUNCT
ejpam-7061	568	4	.	.	PUNCT
ejpam-7061	569	1	suppose	suppose	VERB
ejpam-7061	569	2	that	that	SCONJ
ejpam-7061	569	3	(	(	PUNCT
ejpam-7061	569	4	t1	t1	NOUN
ejpam-7061	569	5	,	,	PUNCT
ejpam-7061	569	6	t2	t2	NOUN
ejpam-7061	569	7	,	,	PUNCT
ejpam-7061	569	8	.	.	PUNCT
ejpam-7061	569	9	.	.	PUNCT
ejpam-7061	570	1	.	.	PUNCT
ejpam-7061	571	1	,	,	PUNCT
ejpam-7061	571	2	tn	tn	PROPN
ejpam-7061	571	3	)	)	PUNCT
ejpam-7061	571	4	∈	∈	PROPN
ejpam-7061	571	5	(	(	PUNCT
ejpam-7061	571	6	s1×s2×	s1×s2×	X
ejpam-7061	571	7	·	·	PUNCT
ejpam-7061	571	8	·	·	PUNCT
ejpam-7061	571	9	·	·	PUNCT
ejpam-7061	572	1	×	×	NOUN
ejpam-7061	572	2	sn)(s1	sn)(s1	NOUN
ejpam-7061	572	3	,	,	PUNCT
ejpam-7061	572	4	s2	s2	PROPN
ejpam-7061	572	5	,	,	PUNCT
ejpam-7061	572	6	.	.	PUNCT
ejpam-7061	572	7	.	.	PUNCT
ejpam-7061	572	8	.	.	PUNCT
ejpam-7061	572	9	,	,	PUNCT
ejpam-7061	572	10	sn	sn	PROPN
ejpam-7061	572	11	)	)	PUNCT
ejpam-7061	572	12	,	,	PUNCT
ejpam-7061	572	13	it	it	PRON
ejpam-7061	572	14	follows	follow	VERB
ejpam-7061	572	15	that	that	SCONJ
ejpam-7061	572	16	(	(	PUNCT
ejpam-7061	572	17	s1	s1	NOUN
ejpam-7061	572	18	,	,	PUNCT
ejpam-7061	572	19	s2	s2	NOUN
ejpam-7061	572	20	,	,	PUNCT
ejpam-7061	572	21	.	.	PUNCT
ejpam-7061	572	22	.	.	PUNCT
ejpam-7061	573	1	.	.	PUNCT
ejpam-7061	574	1	,	,	PUNCT
ejpam-7061	574	2	sn	sn	PROPN
ejpam-7061	574	3	)	)	PUNCT
ejpam-7061	574	4	∈	∈	PROPN
ejpam-7061	575	1	s1t1	s1t1	CCONJ
ejpam-7061	575	2	×	×	NOUN
ejpam-7061	575	3	s2t2	s2t2	NUM
ejpam-7061	575	4	×	×	NOUN
ejpam-7061	575	5	·	·	PUNCT
ejpam-7061	575	6	·	·	PUNCT
ejpam-7061	575	7	·	·	PUNCT
ejpam-7061	576	1	×	×	NOUN
ejpam-7061	576	2	sntn	sntn	NOUN
ejpam-7061	576	3	=	=	PUNCT
ejpam-7061	576	4	s1s1	s1s1	ADP
ejpam-7061	576	5	×	×	NOUN
ejpam-7061	576	6	s2s2	s2s2	NOUN
ejpam-7061	576	7	×	×	NOUN
ejpam-7061	576	8	·	·	PUNCT
ejpam-7061	576	9	·	·	PUNCT
ejpam-7061	576	10	·	·	PUNCT
ejpam-7061	576	11	×	×	NOUN
ejpam-7061	576	12	snsn	snsn	NOUN
ejpam-7061	576	13	.	.	PUNCT
ejpam-7061	577	1	this	this	PRON
ejpam-7061	577	2	contradicts	contradict	VERB
ejpam-7061	577	3	to	to	ADP
ejpam-7061	577	4	our	our	PRON
ejpam-7061	577	5	assumption	assumption	NOUN
ejpam-7061	577	6	.	.	PUNCT
ejpam-7061	578	1	thus	thus	ADV
ejpam-7061	578	2	,	,	PUNCT
ejpam-7061	578	3	(	(	PUNCT
ejpam-7061	578	4	t1	t1	NOUN
ejpam-7061	578	5	,	,	PUNCT
ejpam-7061	578	6	t2	t2	NOUN
ejpam-7061	578	7	,	,	PUNCT
ejpam-7061	578	8	.	.	PUNCT
ejpam-7061	578	9	.	.	PUNCT
ejpam-7061	579	1	.	.	PUNCT
ejpam-7061	580	1	,	,	PUNCT
ejpam-7061	580	2	tn	tn	PROPN
ejpam-7061	580	3	)	)	PUNCT
ejpam-7061	580	4	/∈	/∈	PUNCT
ejpam-7061	581	1	l((s1	l((s1	PROPN
ejpam-7061	581	2	,	,	PUNCT
ejpam-7061	581	3	s2	s2	PROPN
ejpam-7061	581	4	,	,	PUNCT
ejpam-7061	581	5	.	.	PUNCT
ejpam-7061	581	6	.	.	PUNCT
ejpam-7061	581	7	.	.	PUNCT
ejpam-7061	582	1	,	,	PUNCT
ejpam-7061	582	2	sn	sn	PROPN
ejpam-7061	582	3	)	)	PUNCT
ejpam-7061	582	4	)	)	PUNCT
ejpam-7061	582	5	.	.	PUNCT
ejpam-7061	583	1	therefore	therefore	ADV
ejpam-7061	583	2	,	,	PUNCT
ejpam-7061	583	3	l((s1	l((s1	PROPN
ejpam-7061	583	4	,	,	PUNCT
ejpam-7061	583	5	s2	s2	PROPN
ejpam-7061	583	6	,	,	PUNCT
ejpam-7061	583	7	.	.	PUNCT
ejpam-7061	583	8	.	.	PUNCT
ejpam-7061	583	9	.	.	PUNCT
ejpam-7061	584	1	,	,	PUNCT
ejpam-7061	584	2	sn	sn	PROPN
ejpam-7061	584	3	)	)	PUNCT
ejpam-7061	584	4	)	)	PUNCT
ejpam-7061	585	1	⊊	⊊	VERB
ejpam-7061	585	2	l((t1	l((t1	NOUN
ejpam-7061	585	3	,	,	PUNCT
ejpam-7061	585	4	t2	t2	NOUN
ejpam-7061	585	5	,	,	PUNCT
ejpam-7061	585	6	.	.	PUNCT
ejpam-7061	585	7	.	.	PUNCT
ejpam-7061	586	1	.	.	PUNCT
ejpam-7061	587	1	,	,	PUNCT
ejpam-7061	587	2	tn	tn	PROPN
ejpam-7061	587	3	)	)	PUNCT
ejpam-7061	587	4	)	)	PUNCT
ejpam-7061	587	5	.	.	PUNCT
ejpam-7061	588	1	this	this	PRON
ejpam-7061	588	2	contradicts	contradict	VERB
ejpam-7061	588	3	to	to	ADP
ejpam-7061	588	4	maximality	maximality	PROPN
ejpam-7061	588	5	of	of	ADP
ejpam-7061	588	6	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	588	7	...	...	PUNCT
ejpam-7061	588	8	,sn	,sn	PROPN
ejpam-7061	588	9	)	)	PUNCT
ejpam-7061	588	10	.	.	PUNCT
ejpam-7061	589	1	hence	hence	ADV
ejpam-7061	589	2	,	,	PUNCT
ejpam-7061	589	3	(	(	PUNCT
ejpam-7061	589	4	s1	s1	NOUN
ejpam-7061	589	5	,	,	PUNCT
ejpam-7061	589	6	s2	s2	NOUN
ejpam-7061	589	7	,	,	PUNCT
ejpam-7061	589	8	.	.	PUNCT
ejpam-7061	589	9	.	.	PUNCT
ejpam-7061	589	10	.	.	PUNCT
ejpam-7061	590	1	,	,	PUNCT
ejpam-7061	590	2	sn	sn	PROPN
ejpam-7061	590	3	)	)	PUNCT
ejpam-7061	590	4	is	be	AUX
ejpam-7061	590	5	indecomposable	indecomposable	ADJ
ejpam-7061	590	6	.	.	PUNCT
ejpam-7061	591	1	according	accord	VERB
ejpam-7061	591	2	to	to	ADP
ejpam-7061	591	3	the	the	DET
ejpam-7061	591	4	observation	observation	NOUN
ejpam-7061	591	5	of	of	ADP
ejpam-7061	591	6	indecomposable	indecomposable	ADJ
ejpam-7061	591	7	elements	element	NOUN
ejpam-7061	591	8	on	on	ADP
ejpam-7061	591	9	a	a	DET
ejpam-7061	591	10	direct	direct	ADJ
ejpam-7061	591	11	product	product	NOUN
ejpam-7061	591	12	of	of	ADP
ejpam-7061	591	13	semigroups	semigroup	NOUN
ejpam-7061	591	14	together	together	ADV
ejpam-7061	591	15	with	with	ADP
ejpam-7061	591	16	theorem	theorem	NOUN
ejpam-7061	591	17	7	7	NUM
ejpam-7061	591	18	,	,	PUNCT
ejpam-7061	591	19	we	we	PRON
ejpam-7061	591	20	have	have	VERB
ejpam-7061	591	21	the	the	DET
ejpam-7061	591	22	following	follow	VERB
ejpam-7061	591	23	corollary	corollary	NOUN
ejpam-7061	591	24	.	.	PUNCT
ejpam-7061	592	1	corollary	corollary	ADJ
ejpam-7061	592	2	2	2	NUM
ejpam-7061	592	3	.	.	PUNCT
ejpam-7061	593	1	let	let	VERB
ejpam-7061	593	2	si	si	PART
ejpam-7061	593	3	be	be	AUX
ejpam-7061	593	4	a	a	DET
ejpam-7061	593	5	semigroup	semigroup	NOUN
ejpam-7061	593	6	and	and	CCONJ
ejpam-7061	593	7	let	let	VERB
ejpam-7061	593	8	si	si	PROPN
ejpam-7061	593	9	∈	∈	PROPN
ejpam-7061	593	10	si	si	X
ejpam-7061	593	11	where	where	SCONJ
ejpam-7061	593	12	i	i	PRON
ejpam-7061	593	13	∈	∈	PROPN
ejpam-7061	593	14	{	{	PUNCT
ejpam-7061	593	15	1	1	NUM
ejpam-7061	593	16	,	,	PUNCT
ejpam-7061	593	17	2	2	NUM
ejpam-7061	593	18	,	,	PUNCT
ejpam-7061	593	19	.	.	PUNCT
ejpam-7061	593	20	.	.	PUNCT
ejpam-7061	594	1	.	.	PUNCT
ejpam-7061	594	2	,	,	PUNCT
ejpam-7061	594	3	n	n	CCONJ
ejpam-7061	594	4	}	}	PUNCT
ejpam-7061	594	5	.	.	PUNCT
ejpam-7061	595	1	if	if	SCONJ
ejpam-7061	595	2	(	(	PUNCT
ejpam-7061	595	3	s1	s1	NOUN
ejpam-7061	595	4	,	,	PUNCT
ejpam-7061	595	5	s2	s2	NOUN
ejpam-7061	595	6	,	,	PUNCT
ejpam-7061	595	7	.	.	PUNCT
ejpam-7061	595	8	.	.	PUNCT
ejpam-7061	595	9	.	.	PUNCT
ejpam-7061	595	10	,	,	PUNCT
ejpam-7061	595	11	sn	sn	PROPN
ejpam-7061	595	12	)	)	PUNCT
ejpam-7061	595	13	∈	∈	PROPN
ejpam-7061	595	14	s1s1	s1s1	VERB
ejpam-7061	595	15	×	×	NOUN
ejpam-7061	595	16	s2s2	s2s2	NOUN
ejpam-7061	595	17	×	×	NOUN
ejpam-7061	595	18	·	·	PUNCT
ejpam-7061	595	19	·	·	PUNCT
ejpam-7061	595	20	·	·	PUNCT
ejpam-7061	595	21	×	×	NOUN
ejpam-7061	595	22	snsn	snsn	NOUN
ejpam-7061	595	23	,	,	PUNCT
ejpam-7061	595	24	then	then	ADV
ejpam-7061	595	25	l(s1,s2,	l(s1,s2,	PROPN
ejpam-7061	595	26	...	...	PUNCT
ejpam-7061	595	27	,sn	,sn	PUNCT
ejpam-7061	595	28	)	)	PUNCT
ejpam-7061	595	29	is	be	AUX
ejpam-7061	595	30	a	a	DET
ejpam-7061	595	31	maximal	maximal	ADJ
ejpam-7061	595	32	l	l	NOUN
ejpam-7061	595	33	-	-	NOUN
ejpam-7061	595	34	class	class	NOUN
ejpam-7061	595	35	in	in	ADP
ejpam-7061	595	36	s1	s1	PROPN
ejpam-7061	595	37	×	×	PROPN
ejpam-7061	595	38	s2	s2	NOUN
ejpam-7061	595	39	×	×	NOUN
ejpam-7061	595	40	·	·	PUNCT
ejpam-7061	595	41	·	·	PUNCT
ejpam-7061	595	42	·	·	PUNCT
ejpam-7061	596	1	×	×	NOUN
ejpam-7061	596	2	sn	sn	INTJ
ejpam-7061	596	3	if	if	SCONJ
ejpam-7061	597	1	and	and	CCONJ
ejpam-7061	597	2	only	only	ADV
ejpam-7061	597	3	if	if	SCONJ
ejpam-7061	597	4	si	si	PROPN
ejpam-7061	597	5	∈	∈	PROPN
ejpam-7061	597	6	si	si	X
ejpam-7061	597	7	is	be	AUX
ejpam-7061	597	8	indecomposable	indecomposable	ADJ
ejpam-7061	597	9	for	for	ADP
ejpam-7061	597	10	some	some	DET
ejpam-7061	597	11	i	i	PRON
ejpam-7061	597	12	∈	∈	PROPN
ejpam-7061	597	13	{	{	PUNCT
ejpam-7061	597	14	1	1	NUM
ejpam-7061	597	15	,	,	PUNCT
ejpam-7061	597	16	2	2	NUM
ejpam-7061	597	17	,	,	PUNCT
ejpam-7061	597	18	.	.	PUNCT
ejpam-7061	597	19	.	.	PUNCT
ejpam-7061	598	1	.	.	PUNCT
ejpam-7061	598	2	,	,	PUNCT
ejpam-7061	599	1	n	n	CCONJ
ejpam-7061	599	2	}	}	PUNCT
ejpam-7061	599	3	.	.	PUNCT
ejpam-7061	600	1	p.	p.	NOUN
ejpam-7061	600	2	luangchaisri	luangchaisri	PROPN
ejpam-7061	600	3	,	,	PUNCT
ejpam-7061	600	4	o.	o.	PROPN
ejpam-7061	600	5	pankoon	pankoon	NOUN
ejpam-7061	600	6	,	,	PUNCT
ejpam-7061	600	7	t.	t.	PROPN
ejpam-7061	600	8	changphas	changphas	PROPN
ejpam-7061	600	9	/	/	SYM
ejpam-7061	600	10	eur	eur	PROPN
ejpam-7061	600	11	.	.	PUNCT
ejpam-7061	601	1	j.	j.	PROPN
ejpam-7061	601	2	pure	pure	PROPN
ejpam-7061	601	3	appl	appl	PROPN
ejpam-7061	601	4	.	.	PROPN
ejpam-7061	601	5	math	math	PROPN
ejpam-7061	601	6	,	,	PUNCT
ejpam-7061	601	7	18	18	NUM
ejpam-7061	601	8	(	(	PUNCT
ejpam-7061	601	9	4	4	NUM
ejpam-7061	601	10	)	)	PUNCT
ejpam-7061	601	11	(	(	PUNCT
ejpam-7061	601	12	2025	2025	NUM
ejpam-7061	601	13	)	)	PUNCT
ejpam-7061	601	14	,	,	PUNCT
ejpam-7061	601	15	7061	7061	NUM
ejpam-7061	601	16	10	10	NUM
ejpam-7061	601	17	of	of	ADP
ejpam-7061	601	18	10	10	NUM
ejpam-7061	601	19	3	3	NUM
ejpam-7061	601	20	.	.	PUNCT
ejpam-7061	602	1	conclusions	conclusion	NOUN
ejpam-7061	602	2	in	in	ADP
ejpam-7061	602	3	this	this	DET
ejpam-7061	602	4	paper	paper	NOUN
ejpam-7061	603	1	,	,	PUNCT
ejpam-7061	603	2	we	we	PRON
ejpam-7061	603	3	studied	study	VERB
ejpam-7061	603	4	the	the	DET
ejpam-7061	603	5	cartesian	cartesian	ADJ
ejpam-7061	603	6	product	product	NOUN
ejpam-7061	603	7	of	of	ADP
ejpam-7061	603	8	principal	principal	NOUN
ejpam-7061	603	9	left	leave	VERB
ejpam-7061	603	10	ideals	ideal	NOUN
ejpam-7061	603	11	and	and	CCONJ
ejpam-7061	603	12	the	the	DET
ejpam-7061	603	13	cartesian	cartesian	ADJ
ejpam-7061	603	14	product	product	NOUN
ejpam-7061	603	15	of	of	ADP
ejpam-7061	603	16	l	l	NOUN
ejpam-7061	603	17	-	-	NOUN
ejpam-7061	603	18	classes	class	NOUN
ejpam-7061	603	19	in	in	ADP
ejpam-7061	603	20	the	the	DET
ejpam-7061	603	21	direct	direct	ADJ
ejpam-7061	603	22	product	product	NOUN
ejpam-7061	603	23	of	of	ADP
ejpam-7061	603	24	n	n	CCONJ
ejpam-7061	603	25	semigroups	semigroup	NOUN
ejpam-7061	603	26	(	(	PUNCT
ejpam-7061	603	27	n	n	CCONJ
ejpam-7061	603	28	≥	≥	NOUN
ejpam-7061	603	29	2	2	NUM
ejpam-7061	603	30	)	)	PUNCT
ejpam-7061	603	31	.	.	PUNCT
ejpam-7061	604	1	we	we	PRON
ejpam-7061	604	2	gave	give	VERB
ejpam-7061	604	3	an	an	DET
ejpam-7061	604	4	explicit	explicit	ADJ
ejpam-7061	604	5	counterexample	counterexample	NOUN
ejpam-7061	604	6	showing	show	VERB
ejpam-7061	604	7	that	that	SCONJ
ejpam-7061	604	8	the	the	DET
ejpam-7061	604	9	cartesian	cartesian	ADJ
ejpam-7061	604	10	product	product	NOUN
ejpam-7061	604	11	of	of	ADP
ejpam-7061	604	12	principal	principal	ADJ
ejpam-7061	604	13	left	leave	VERB
ejpam-7061	604	14	ideals	ideal	NOUN
ejpam-7061	604	15	is	be	AUX
ejpam-7061	604	16	not	not	PART
ejpam-7061	604	17	necessarily	necessarily	ADV
ejpam-7061	604	18	a	a	DET
ejpam-7061	604	19	principal	principal	NOUN
ejpam-7061	604	20	left	leave	VERB
ejpam-7061	604	21	ideal	ideal	ADJ
ejpam-7061	604	22	.	.	PUNCT
ejpam-7061	605	1	similarly	similarly	ADV
ejpam-7061	605	2	,	,	PUNCT
ejpam-7061	605	3	an	an	DET
ejpam-7061	605	4	explicit	explicit	ADJ
ejpam-7061	605	5	counterexample	counterexample	NOUN
ejpam-7061	605	6	for	for	ADP
ejpam-7061	605	7	the	the	DET
ejpam-7061	605	8	cartesian	cartesian	ADJ
ejpam-7061	605	9	product	product	NOUN
ejpam-7061	605	10	of	of	ADP
ejpam-7061	605	11	l	l	NOUN
ejpam-7061	605	12	-	-	PUNCT
ejpam-7061	605	13	classes	class	NOUN
ejpam-7061	605	14	was	be	AUX
ejpam-7061	605	15	provided	provide	VERB
ejpam-7061	605	16	.	.	PUNCT
ejpam-7061	606	1	we	we	PRON
ejpam-7061	606	2	established	establish	VERB
ejpam-7061	606	3	the	the	DET
ejpam-7061	606	4	two	two	NUM
ejpam-7061	606	5	main	main	ADJ
ejpam-7061	606	6	results	result	NOUN
ejpam-7061	606	7	,	,	PUNCT
ejpam-7061	606	8	consisting	consist	VERB
ejpam-7061	606	9	of	of	ADP
ejpam-7061	606	10	a	a	DET
ejpam-7061	606	11	necessary	necessary	ADJ
ejpam-7061	606	12	and	and	CCONJ
ejpam-7061	606	13	sufficient	sufficient	ADJ
ejpam-7061	606	14	condition	condition	NOUN
ejpam-7061	606	15	for	for	ADP
ejpam-7061	606	16	the	the	DET
ejpam-7061	606	17	cartesian	cartesian	ADJ
ejpam-7061	606	18	product	product	NOUN
ejpam-7061	606	19	of	of	ADP
ejpam-7061	606	20	principal	principal	NOUN
ejpam-7061	606	21	left	leave	VERB
ejpam-7061	606	22	ideals	ideal	NOUN
ejpam-7061	606	23	to	to	PART
ejpam-7061	606	24	be	be	AUX
ejpam-7061	606	25	a	a	DET
ejpam-7061	606	26	principal	principal	NOUN
ejpam-7061	606	27	left	leave	VERB
ejpam-7061	606	28	ideal	ideal	ADJ
ejpam-7061	606	29	,	,	PUNCT
ejpam-7061	606	30	and	and	CCONJ
ejpam-7061	606	31	a	a	DET
ejpam-7061	606	32	necessary	necessary	ADJ
ejpam-7061	606	33	and	and	CCONJ
ejpam-7061	606	34	sufficient	sufficient	ADJ
ejpam-7061	606	35	condition	condition	NOUN
ejpam-7061	606	36	for	for	ADP
ejpam-7061	606	37	the	the	DET
ejpam-7061	606	38	cartesian	cartesian	ADJ
ejpam-7061	606	39	product	product	NOUN
ejpam-7061	606	40	of	of	ADP
ejpam-7061	606	41	l	l	NOUN
ejpam-7061	606	42	-	-	PUNCT
ejpam-7061	606	43	classes	class	NOUN
ejpam-7061	606	44	to	to	PART
ejpam-7061	606	45	be	be	AUX
ejpam-7061	606	46	an	an	DET
ejpam-7061	606	47	l	l	NOUN
ejpam-7061	606	48	-	-	NOUN
ejpam-7061	606	49	class	class	NOUN
ejpam-7061	606	50	in	in	ADP
ejpam-7061	606	51	the	the	DET
ejpam-7061	606	52	direct	direct	ADJ
ejpam-7061	606	53	product	product	NOUN
ejpam-7061	606	54	of	of	ADP
ejpam-7061	606	55	n	n	PRON
ejpam-7061	606	56	semigroups	semigroup	NOUN
ejpam-7061	606	57	.	.	PUNCT
ejpam-7061	607	1	in	in	ADP
ejpam-7061	607	2	addition	addition	NOUN
ejpam-7061	607	3	,	,	PUNCT
ejpam-7061	607	4	the	the	DET
ejpam-7061	607	5	condition	condition	NOUN
ejpam-7061	607	6	when	when	SCONJ
ejpam-7061	607	7	the	the	DET
ejpam-7061	607	8	cartesian	cartesian	ADJ
ejpam-7061	607	9	product	product	NOUN
ejpam-7061	607	10	of	of	ADP
ejpam-7061	607	11	maximal	maximal	ADJ
ejpam-7061	607	12	l	l	NOUN
ejpam-7061	607	13	-	-	PUNCT
ejpam-7061	607	14	classes	class	NOUN
ejpam-7061	607	15	is	be	AUX
ejpam-7061	607	16	maximal	maximal	ADJ
ejpam-7061	607	17	was	be	AUX
ejpam-7061	607	18	also	also	ADV
ejpam-7061	607	19	investigated	investigate	VERB
ejpam-7061	607	20	.	.	PUNCT
ejpam-7061	608	1	acknowledgements	acknowledgement	NOUN
ejpam-7061	608	2	this	this	DET
ejpam-7061	608	3	work	work	NOUN
ejpam-7061	608	4	(	(	PUNCT
ejpam-7061	608	5	grant	grant	VERB
ejpam-7061	608	6	no	no	INTJ
ejpam-7061	608	7	.	.	PUNCT
ejpam-7061	609	1	rgns	rgn	VERB
ejpam-7061	609	2	65	65	NUM
ejpam-7061	609	3	-	-	PUNCT
ejpam-7061	609	4	054	054	NUM
ejpam-7061	609	5	)	)	PUNCT
ejpam-7061	609	6	was	be	AUX
ejpam-7061	609	7	supported	support	VERB
ejpam-7061	609	8	by	by	ADP
ejpam-7061	609	9	office	office	NOUN
ejpam-7061	609	10	of	of	ADP
ejpam-7061	609	11	the	the	DET
ejpam-7061	609	12	permanent	permanent	ADJ
ejpam-7061	609	13	secretary	secretary	NOUN
ejpam-7061	609	14	,	,	PUNCT
ejpam-7061	609	15	ministry	ministry	PROPN
ejpam-7061	609	16	of	of	ADP
ejpam-7061	609	17	higher	high	ADJ
ejpam-7061	609	18	education	education	NOUN
ejpam-7061	609	19	,	,	PUNCT
ejpam-7061	609	20	science	science	NOUN
ejpam-7061	609	21	,	,	PUNCT
ejpam-7061	609	22	research	research	NOUN
ejpam-7061	609	23	and	and	CCONJ
ejpam-7061	609	24	innovation	innovation	NOUN
ejpam-7061	609	25	(	(	PUNCT
ejpam-7061	609	26	ops	op	NOUN
ejpam-7061	609	27	mhesi	mhesi	PROPN
ejpam-7061	609	28	)	)	PUNCT
ejpam-7061	609	29	,	,	PUNCT
ejpam-7061	609	30	thailand	thailand	PROPN
ejpam-7061	609	31	science	science	PROPN
ejpam-7061	609	32	research	research	PROPN
ejpam-7061	609	33	and	and	CCONJ
ejpam-7061	609	34	innovation	innovation	NOUN
ejpam-7061	609	35	(	(	PUNCT
ejpam-7061	609	36	tsri	tsri	ADJ
ejpam-7061	609	37	)	)	PUNCT
ejpam-7061	609	38	and	and	CCONJ
ejpam-7061	609	39	khon	khon	PROPN
ejpam-7061	609	40	kaen	kaen	PROPN
ejpam-7061	609	41	university	university	PROPN
ejpam-7061	609	42	.	.	PUNCT
ejpam-7061	610	1	references	reference	NOUN
ejpam-7061	610	2	[	[	X
ejpam-7061	610	3	1	1	NUM
ejpam-7061	610	4	]	]	PUNCT
ejpam-7061	610	5	i.	i.	NOUN
ejpam-7061	610	6	fabrici	fabrici	PROPN
ejpam-7061	610	7	.	.	PUNCT
ejpam-7061	611	1	one	one	NUM
ejpam-7061	611	2	-	-	PUNCT
ejpam-7061	611	3	sided	sided	ADJ
ejpam-7061	611	4	principal	principal	ADJ
ejpam-7061	611	5	ideals	ideal	NOUN
ejpam-7061	611	6	in	in	ADP
ejpam-7061	611	7	the	the	DET
ejpam-7061	611	8	direct	direct	ADJ
ejpam-7061	611	9	product	product	NOUN
ejpam-7061	611	10	of	of	ADP
ejpam-7061	611	11	two	two	NUM
ejpam-7061	611	12	semigroups	semigroup	NOUN
ejpam-7061	611	13	.	.	PUNCT
ejpam-7061	612	1	mathematica	mathematica	PROPN
ejpam-7061	612	2	bohemica	bohemica	PROPN
ejpam-7061	612	3	,	,	PUNCT
ejpam-7061	612	4	118(4):337–342	118(4):337–342	NUM
ejpam-7061	612	5	,	,	PUNCT
ejpam-7061	612	6	1993	1993	NUM
ejpam-7061	612	7	.	.	PUNCT
ejpam-7061	613	1	[	[	X
ejpam-7061	613	2	2	2	NUM
ejpam-7061	613	3	]	]	PUNCT
ejpam-7061	613	4	i.	i.	NOUN
ejpam-7061	613	5	fabrici	fabrici	PROPN
ejpam-7061	613	6	.	.	PUNCT
ejpam-7061	614	1	principal	principal	ADJ
ejpam-7061	614	2	two	two	NUM
ejpam-7061	614	3	-	-	PUNCT
ejpam-7061	614	4	sided	sided	ADJ
ejpam-7061	614	5	ideals	ideal	NOUN
ejpam-7061	614	6	in	in	ADP
ejpam-7061	614	7	the	the	DET
ejpam-7061	614	8	direct	direct	ADJ
ejpam-7061	614	9	product	product	NOUN
ejpam-7061	614	10	of	of	ADP
ejpam-7061	614	11	two	two	NUM
ejpam-7061	614	12	semigroups	semigroup	NOUN
ejpam-7061	614	13	.	.	PUNCT
ejpam-7061	615	1	czechoslovak	czechoslovak	ADJ
ejpam-7061	615	2	mathematical	mathematical	PROPN
ejpam-7061	615	3	journal	journal	PROPN
ejpam-7061	615	4	,	,	PUNCT
ejpam-7061	615	5	41(3):411–421	41(3):411–421	PROPN
ejpam-7061	615	6	,	,	PUNCT
ejpam-7061	615	7	1991	1991	NUM
ejpam-7061	615	8	.	.	PUNCT
ejpam-7061	616	1	[	[	X
ejpam-7061	616	2	3	3	X
ejpam-7061	616	3	]	]	X
ejpam-7061	616	4	s.	s.	PROPN
ejpam-7061	616	5	lajos	lajos	PROPN
ejpam-7061	616	6	.	.	PUNCT
ejpam-7061	617	1	generalized	generalized	ADJ
ejpam-7061	617	2	ideals	ideal	NOUN
ejpam-7061	617	3	in	in	ADP
ejpam-7061	617	4	semigroups	semigroup	NOUN
ejpam-7061	617	5	.	.	PUNCT
ejpam-7061	618	1	acta	acta	PROPN
ejpam-7061	618	2	scientiarum	scientiarum	PROPN
ejpam-7061	618	3	mathematicarum	mathematicarum	PROPN
ejpam-7061	618	4	,	,	PUNCT
ejpam-7061	618	5	22:217–222	22:217–222	PROPN
ejpam-7061	618	6	,	,	PUNCT
ejpam-7061	618	7	1961	1961	NUM
ejpam-7061	618	8	.	.	PUNCT
ejpam-7061	619	1	[	[	X
ejpam-7061	619	2	4	4	X
ejpam-7061	619	3	]	]	PUNCT
ejpam-7061	619	4	p.	p.	NOUN
ejpam-7061	619	5	luangchaisri	luangchaisri	VERB
ejpam-7061	619	6	and	and	CCONJ
ejpam-7061	619	7	t.	t.	PROPN
ejpam-7061	619	8	changphas	changphas	PROPN
ejpam-7061	619	9	.	.	PUNCT
ejpam-7061	620	1	on	on	ADP
ejpam-7061	620	2	the	the	DET
ejpam-7061	620	3	principal	principal	NOUN
ejpam-7061	620	4	(	(	PUNCT
ejpam-7061	620	5	m	m	PROPN
ejpam-7061	620	6	,	,	PUNCT
ejpam-7061	620	7	n)-ideals	n)-ideal	NOUN
ejpam-7061	620	8	in	in	ADP
ejpam-7061	620	9	the	the	DET
ejpam-7061	620	10	direct	direct	ADJ
ejpam-7061	620	11	product	product	NOUN
ejpam-7061	620	12	of	of	ADP
ejpam-7061	620	13	two	two	NUM
ejpam-7061	620	14	semigroups	semigroup	NOUN
ejpam-7061	620	15	.	.	PUNCT
ejpam-7061	621	1	quasigroups	quasigroup	NOUN
ejpam-7061	621	2	and	and	CCONJ
ejpam-7061	621	3	related	related	ADJ
ejpam-7061	621	4	systems	system	NOUN
ejpam-7061	621	5	,	,	PUNCT
ejpam-7061	621	6	24:75–80	24:75–80	NUM
ejpam-7061	621	7	,	,	PUNCT
ejpam-7061	621	8	2016	2016	NUM
ejpam-7061	621	9	.	.	PUNCT
ejpam-7061	622	1	[	[	X
ejpam-7061	622	2	5	5	NUM
ejpam-7061	622	3	]	]	PUNCT
ejpam-7061	622	4	o.	o.	PROPN
ejpam-7061	622	5	steinfeld	steinfeld	PROPN
ejpam-7061	622	6	.	.	PUNCT
ejpam-7061	623	1	über	über	PROPN
ejpam-7061	623	2	die	die	VERB
ejpam-7061	623	3	quasiideale	quasiideale	PROPN
ejpam-7061	623	4	von	von	PROPN
ejpam-7061	623	5	halbgruppen	halbgruppen	PROPN
ejpam-7061	623	6	.	.	PUNCT
ejpam-7061	624	1	publicationes	publicatione	NOUN
ejpam-7061	624	2	mathematicae	mathematicae	PROPN
ejpam-7061	624	3	debrecen	debrecen	PROPN
ejpam-7061	624	4	,	,	PUNCT
ejpam-7061	624	5	4:262–275	4:262–275	PROPN
ejpam-7061	624	6	,	,	PUNCT
ejpam-7061	624	7	1956	1956	NUM
ejpam-7061	624	8	.	.	PUNCT
ejpam-7061	625	1	[	[	X
ejpam-7061	625	2	6	6	NUM
ejpam-7061	625	3	]	]	PUNCT
ejpam-7061	625	4	p.	p.	NOUN
ejpam-7061	625	5	luangchaisri	luangchaisri	PROPN
ejpam-7061	625	6	,	,	PUNCT
ejpam-7061	625	7	o.	o.	PROPN
ejpam-7061	625	8	pankoon	pankoon	PROPN
ejpam-7061	625	9	and	and	CCONJ
ejpam-7061	625	10	t.	t.	PROPN
ejpam-7061	625	11	changphas	changphas	PROPN
ejpam-7061	625	12	.	.	PUNCT
ejpam-7061	626	1	quasi	quasi	ADJ
ejpam-7061	626	2	-	-	NOUN
ejpam-7061	626	3	ideals	ideal	NOUN
ejpam-7061	626	4	andh	andh	NOUN
ejpam-7061	626	5	-	-	PUNCT
ejpam-7061	626	6	classes	class	NOUN
ejpam-7061	626	7	on	on	ADP
ejpam-7061	626	8	the	the	DET
ejpam-7061	626	9	direct	direct	ADJ
ejpam-7061	626	10	product	product	NOUN
ejpam-7061	626	11	of	of	ADP
ejpam-7061	626	12	two	two	NUM
ejpam-7061	626	13	semigroups	semigroup	NOUN
ejpam-7061	626	14	.	.	PUNCT
ejpam-7061	627	1	european	european	ADJ
ejpam-7061	627	2	journal	journal	PROPN
ejpam-7061	627	3	of	of	ADP
ejpam-7061	627	4	pure	pure	ADJ
ejpam-7061	627	5	and	and	CCONJ
ejpam-7061	627	6	applied	applied	ADJ
ejpam-7061	627	7	mathematics	mathematic	NOUN
ejpam-7061	627	8	,	,	PUNCT
ejpam-7061	627	9	18(2):58–59	18(2):58–59	NUM
ejpam-7061	627	10	,	,	PUNCT
ejpam-7061	627	11	2025	2025	NUM
ejpam-7061	627	12	.	.	PUNCT
ejpam-7061	628	1	[	[	X
ejpam-7061	628	2	7	7	X
ejpam-7061	628	3	]	]	X
ejpam-7061	628	4	g.	g.	PROPN
ejpam-7061	628	5	e.	e.	PROPN
ejpam-7061	628	6	forsythe	forsythe	PROPN
ejpam-7061	628	7	.	.	PUNCT
ejpam-7061	629	1	swac	swac	PROPN
ejpam-7061	629	2	computes	compute	VERB
ejpam-7061	629	3	126	126	NUM
ejpam-7061	629	4	distinct	distinct	ADJ
ejpam-7061	629	5	semigroups	semigroup	NOUN
ejpam-7061	629	6	of	of	ADP
ejpam-7061	629	7	order	order	NOUN
ejpam-7061	629	8	4	4	NUM
ejpam-7061	629	9	.	.	PUNCT
ejpam-7061	629	10	proceedings	proceeding	NOUN
ejpam-7061	629	11	of	of	ADP
ejpam-7061	629	12	the	the	DET
ejpam-7061	629	13	american	american	PROPN
ejpam-7061	629	14	mathematical	mathematical	PROPN
ejpam-7061	629	15	society	society	NOUN
ejpam-7061	629	16	,	,	PUNCT
ejpam-7061	629	17	6(3):443–447	6(3):443–447	NOUN
ejpam-7061	629	18	,	,	PUNCT
ejpam-7061	629	19	1955	1955	NUM
ejpam-7061	629	20	.	.	PUNCT