id	sid	tid	token	lemma	pos
ejpam-4262	1	1	european	european	PROPN
ejpam-4262	1	2	journal	journal	PROPN
ejpam-4262	1	3	of	of	ADP
ejpam-4262	1	4	pure	pure	ADJ
ejpam-4262	1	5	and	and	CCONJ
ejpam-4262	1	6	applied	apply	VERB
ejpam-4262	1	7	mathematics	mathematic	NOUN
ejpam-4262	1	8	vol	vol	NOUN
ejpam-4262	1	9	.	.	PROPN
ejpam-4262	2	1	15	15	NUM
ejpam-4262	2	2	,	,	PUNCT
ejpam-4262	2	3	no	no	INTJ
ejpam-4262	2	4	.	.	NOUN
ejpam-4262	2	5	1	1	NUM
ejpam-4262	2	6	,	,	PUNCT
ejpam-4262	2	7	2022	2022	NUM
ejpam-4262	2	8	,	,	PUNCT
ejpam-4262	2	9	328	328	NUM
ejpam-4262	2	10	-	-	SYM
ejpam-4262	2	11	334	334	NUM
ejpam-4262	2	12	issn	issn	PROPN
ejpam-4262	2	13	1307	1307	NUM
ejpam-4262	2	14	-	-	SYM
ejpam-4262	2	15	5543	5543	NUM
ejpam-4262	2	16	–	–	PUNCT
ejpam-4262	2	17	ejpam.com	ejpam.com	X
ejpam-4262	2	18	published	publish	VERB
ejpam-4262	2	19	by	by	ADP
ejpam-4262	2	20	new	new	PROPN
ejpam-4262	2	21	york	york	PROPN
ejpam-4262	2	22	business	business	PROPN
ejpam-4262	2	23	global	global	PROPN
ejpam-4262	2	24	on	on	ADP
ejpam-4262	2	25	ϕ-β	ϕ-β	ADJ
ejpam-4262	2	26	-	-	ADJ
ejpam-4262	2	27	absorbing	absorbing	ADJ
ejpam-4262	2	28	submodules	submodule	NOUN
ejpam-4262	2	29	thawatchai	thawatchai	PROPN
ejpam-4262	2	30	khumprapussorn	khumprapussorn	PROPN
ejpam-4262	2	31	department	department	PROPN
ejpam-4262	2	32	of	of	ADP
ejpam-4262	2	33	mathematics	mathematic	NOUN
ejpam-4262	2	34	,	,	PUNCT
ejpam-4262	2	35	school	school	NOUN
ejpam-4262	2	36	of	of	ADP
ejpam-4262	2	37	science	science	NOUN
ejpam-4262	2	38	,	,	PUNCT
ejpam-4262	2	39	king	king	PROPN
ejpam-4262	2	40	mongkut	mongkut	PROPN
ejpam-4262	2	41	’s	’s	PROPN
ejpam-4262	2	42	institute	institute	PROPN
ejpam-4262	2	43	of	of	ADP
ejpam-4262	2	44	technology	technology	PROPN
ejpam-4262	2	45	ladkrabang	ladkrabang	PROPN
ejpam-4262	2	46	,	,	PUNCT
ejpam-4262	2	47	bangkok	bangkok	PROPN
ejpam-4262	2	48	10520	10520	NUM
ejpam-4262	2	49	,	,	PUNCT
ejpam-4262	2	50	thailand	thailand	PROPN
ejpam-4262	2	51	abstract	abstract	NOUN
ejpam-4262	2	52	.	.	PUNCT
ejpam-4262	3	1	in	in	ADP
ejpam-4262	3	2	this	this	DET
ejpam-4262	3	3	paper	paper	NOUN
ejpam-4262	3	4	,	,	PUNCT
ejpam-4262	3	5	we	we	PRON
ejpam-4262	3	6	extend	extend	VERB
ejpam-4262	3	7	the	the	DET
ejpam-4262	3	8	concept	concept	NOUN
ejpam-4262	3	9	of	of	ADP
ejpam-4262	3	10	β	β	ADJ
ejpam-4262	3	11	-	-	ADJ
ejpam-4262	3	12	absorbing	absorbing	ADJ
ejpam-4262	3	13	submodules	submodule	NOUN
ejpam-4262	3	14	to	to	ADP
ejpam-4262	3	15	ϕ-β	ϕ-β	ADJ
ejpam-4262	3	16	-	-	ADJ
ejpam-4262	3	17	absorbing	absorbing	ADJ
ejpam-4262	3	18	submodules	submodule	NOUN
ejpam-4262	3	19	over	over	ADP
ejpam-4262	3	20	a	a	DET
ejpam-4262	3	21	commutative	commutative	ADJ
ejpam-4262	3	22	ring	ring	NOUN
ejpam-4262	3	23	with	with	ADP
ejpam-4262	3	24	nonzero	nonzero	PROPN
ejpam-4262	3	25	identity	identity	NOUN
ejpam-4262	3	26	which	which	PRON
ejpam-4262	3	27	is	be	AUX
ejpam-4262	3	28	a	a	DET
ejpam-4262	3	29	generalization	generalization	NOUN
ejpam-4262	3	30	of	of	ADP
ejpam-4262	3	31	2	2	NUM
ejpam-4262	3	32	-	-	PUNCT
ejpam-4262	3	33	absorbing	absorbing	ADJ
ejpam-4262	3	34	submodules	submodule	NOUN
ejpam-4262	3	35	.	.	PUNCT
ejpam-4262	4	1	let	let	VERB
ejpam-4262	4	2	s(m	s(m	PROPN
ejpam-4262	4	3	)	)	PUNCT
ejpam-4262	4	4	be	be	VERB
ejpam-4262	4	5	the	the	DET
ejpam-4262	4	6	set	set	NOUN
ejpam-4262	4	7	of	of	ADP
ejpam-4262	4	8	all	all	DET
ejpam-4262	4	9	submodules	submodule	NOUN
ejpam-4262	4	10	ofm	ofm	PROPN
ejpam-4262	4	11	and	and	CCONJ
ejpam-4262	4	12	ϕ	ϕ	NOUN
ejpam-4262	4	13	:	:	PUNCT
ejpam-4262	4	14	s(m)→	s(m)→	NOUN
ejpam-4262	4	15	s(m)∪{∅	s(m)∪{∅	PROPN
ejpam-4262	4	16	}	}	PUNCT
ejpam-4262	4	17	be	be	AUX
ejpam-4262	4	18	a	a	DET
ejpam-4262	4	19	function	function	NOUN
ejpam-4262	4	20	.	.	PUNCT
ejpam-4262	5	1	a	a	DET
ejpam-4262	5	2	proper	proper	ADJ
ejpam-4262	5	3	submodule	submodule	NOUN
ejpam-4262	5	4	p	p	NOUN
ejpam-4262	5	5	of	of	ADP
ejpam-4262	5	6	m	m	PROPN
ejpam-4262	5	7	is	be	AUX
ejpam-4262	5	8	called	call	VERB
ejpam-4262	5	9	a	a	DET
ejpam-4262	5	10	ϕ-β	ϕ-β	ADJ
ejpam-4262	5	11	-	-	ADJ
ejpam-4262	5	12	absorbing	absorbing	ADJ
ejpam-4262	5	13	submodule	submodule	NOUN
ejpam-4262	5	14	,	,	PUNCT
ejpam-4262	5	15	if	if	SCONJ
ejpam-4262	5	16	for	for	ADP
ejpam-4262	5	17	each	each	DET
ejpam-4262	5	18	r	r	NOUN
ejpam-4262	5	19	,	,	PUNCT
ejpam-4262	5	20	s	s	NOUN
ejpam-4262	5	21	∈	∈	PROPN
ejpam-4262	5	22	r	r	NOUN
ejpam-4262	5	23	and	and	CCONJ
ejpam-4262	5	24	m	m	NOUN
ejpam-4262	5	25	∈m	∈m	NOUN
ejpam-4262	5	26	with	with	ADP
ejpam-4262	5	27	rsm	rsm	PROPN
ejpam-4262	5	28	∈	∈	PROPN
ejpam-4262	5	29	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	5	30	)	)	PUNCT
ejpam-4262	5	31	,	,	PUNCT
ejpam-4262	5	32	then	then	ADV
ejpam-4262	5	33	rs	rs	INTJ
ejpam-4262	5	34	+	+	CCONJ
ejpam-4262	5	35	rs	rs	PROPN
ejpam-4262	5	36	∈	∈	PROPN
ejpam-4262	5	37	(	(	PUNCT
ejpam-4262	5	38	p	p	X
ejpam-4262	5	39	:	:	PUNCT
ejpam-4262	5	40	m	m	NUM
ejpam-4262	5	41	)	)	PUNCT
ejpam-4262	5	42	or	or	CCONJ
ejpam-4262	5	43	r(m	r(m	PROPN
ejpam-4262	5	44	+	+	NUM
ejpam-4262	5	45	m	m	NOUN
ejpam-4262	5	46	)	)	PUNCT
ejpam-4262	5	47	∈	∈	PROPN
ejpam-4262	5	48	p	p	NOUN
ejpam-4262	5	49	or	or	CCONJ
ejpam-4262	5	50	s(m	s(m	NOUN
ejpam-4262	5	51	+	+	CCONJ
ejpam-4262	5	52	m	m	NOUN
ejpam-4262	5	53	)	)	PUNCT
ejpam-4262	5	54	∈	∈	PROPN
ejpam-4262	5	55	p	p	NOUN
ejpam-4262	5	56	.	.	PUNCT
ejpam-4262	6	1	some	some	PRON
ejpam-4262	6	2	of	of	ADP
ejpam-4262	6	3	the	the	DET
ejpam-4262	6	4	properties	property	NOUN
ejpam-4262	6	5	and	and	CCONJ
ejpam-4262	6	6	characterizations	characterization	NOUN
ejpam-4262	6	7	of	of	ADP
ejpam-4262	6	8	ϕ-β	ϕ-β	NOUN
ejpam-4262	6	9	-	-	ADJ
ejpam-4262	6	10	absorbing	absorbing	ADJ
ejpam-4262	6	11	submodules	submodule	NOUN
ejpam-4262	6	12	are	be	AUX
ejpam-4262	6	13	investigated	investigate	VERB
ejpam-4262	6	14	.	.	PUNCT
ejpam-4262	7	1	2020	2020	NUM
ejpam-4262	7	2	mathematics	mathematic	NOUN
ejpam-4262	7	3	subject	subject	NOUN
ejpam-4262	7	4	classifications	classification	NOUN
ejpam-4262	7	5	:	:	PUNCT
ejpam-4262	7	6	13c05	13c05	NUM
ejpam-4262	7	7	,	,	PUNCT
ejpam-4262	7	8	16d80	16d80	NUM
ejpam-4262	7	9	,	,	PUNCT
ejpam-4262	7	10	16d99	16d99	NUM
ejpam-4262	7	11	key	key	ADJ
ejpam-4262	7	12	words	word	NOUN
ejpam-4262	7	13	and	and	CCONJ
ejpam-4262	7	14	phrases	phrase	NOUN
ejpam-4262	7	15	:	:	PUNCT
ejpam-4262	7	16	2	2	NUM
ejpam-4262	7	17	-	-	PUNCT
ejpam-4262	7	18	absorbing	absorbing	ADJ
ejpam-4262	7	19	submodules	submodule	NOUN
ejpam-4262	7	20	,	,	PUNCT
ejpam-4262	7	21	ϕ-2	ϕ-2	ADV
ejpam-4262	7	22	-	-	PUNCT
ejpam-4262	7	23	absorbing	absorb	VERB
ejpam-4262	7	24	submodules	submodule	NOUN
ejpam-4262	7	25	,	,	PUNCT
ejpam-4262	7	26	β	β	NOUN
ejpam-4262	7	27	-	-	ADJ
ejpam-4262	7	28	absorbing	absorbing	ADJ
ejpam-4262	7	29	submodules	submodule	NOUN
ejpam-4262	7	30	,	,	PUNCT
ejpam-4262	7	31	ϕ-β	ϕ-β	NOUN
ejpam-4262	7	32	-	-	ADJ
ejpam-4262	7	33	absorbing	absorbing	ADJ
ejpam-4262	7	34	submodules	submodule	NOUN
ejpam-4262	7	35	1	1	NUM
ejpam-4262	7	36	.	.	PUNCT
ejpam-4262	7	37	introduction	introduction	NOUN
ejpam-4262	7	38	throughout	throughout	ADP
ejpam-4262	7	39	this	this	DET
ejpam-4262	7	40	paper	paper	NOUN
ejpam-4262	7	41	,	,	PUNCT
ejpam-4262	7	42	r	r	NOUN
ejpam-4262	7	43	will	will	AUX
ejpam-4262	7	44	denote	denote	VERB
ejpam-4262	7	45	a	a	DET
ejpam-4262	7	46	commutative	commutative	ADJ
ejpam-4262	7	47	ring	ring	NOUN
ejpam-4262	7	48	with	with	ADP
ejpam-4262	7	49	identity	identity	NOUN
ejpam-4262	7	50	and	and	CCONJ
ejpam-4262	7	51	all	all	DET
ejpam-4262	7	52	modules	module	NOUN
ejpam-4262	7	53	are	be	AUX
ejpam-4262	7	54	unital	unital	ADJ
ejpam-4262	7	55	left	left	ADJ
ejpam-4262	7	56	r	r	NOUN
ejpam-4262	7	57	-	-	PUNCT
ejpam-4262	7	58	modules	module	NOUN
ejpam-4262	7	59	.	.	PUNCT
ejpam-4262	8	1	we	we	PRON
ejpam-4262	8	2	recall	recall	VERB
ejpam-4262	8	3	that	that	SCONJ
ejpam-4262	8	4	a	a	DET
ejpam-4262	8	5	proper	proper	ADJ
ejpam-4262	8	6	submodule	submodule	NOUN
ejpam-4262	8	7	p	p	NOUN
ejpam-4262	8	8	of	of	ADP
ejpam-4262	8	9	a	a	DET
ejpam-4262	8	10	left	left	ADJ
ejpam-4262	8	11	r	r	NOUN
ejpam-4262	8	12	-	-	PUNCT
ejpam-4262	8	13	module	module	NOUN
ejpam-4262	8	14	m	m	NOUN
ejpam-4262	8	15	is	be	AUX
ejpam-4262	8	16	called	call	VERB
ejpam-4262	8	17	a	a	DET
ejpam-4262	8	18	prime	prime	ADJ
ejpam-4262	8	19	submodule	submodule	NOUN
ejpam-4262	8	20	of	of	ADP
ejpam-4262	8	21	m	m	PROPN
ejpam-4262	8	22	if	if	SCONJ
ejpam-4262	8	23	for	for	ADP
ejpam-4262	8	24	every	every	DET
ejpam-4262	8	25	r	r	NOUN
ejpam-4262	8	26	∈	∈	NOUN
ejpam-4262	8	27	r	r	NOUN
ejpam-4262	8	28	and	and	CCONJ
ejpam-4262	8	29	m	m	PROPN
ejpam-4262	8	30	∈	∈	PROPN
ejpam-4262	8	31	m	m	PROPN
ejpam-4262	8	32	,	,	PUNCT
ejpam-4262	8	33	rm	rm	PROPN
ejpam-4262	8	34	∈	∈	PROPN
ejpam-4262	8	35	p	p	PROPN
ejpam-4262	8	36	implies	imply	VERB
ejpam-4262	8	37	that	that	SCONJ
ejpam-4262	8	38	m	m	VERB
ejpam-4262	8	39	∈	∈	PROPN
ejpam-4262	8	40	p	p	NOUN
ejpam-4262	8	41	or	or	CCONJ
ejpam-4262	8	42	r	r	NOUN
ejpam-4262	8	43	∈	∈	PROPN
ejpam-4262	8	44	(	(	PUNCT
ejpam-4262	8	45	p	p	X
ejpam-4262	8	46	:	:	PUNCT
ejpam-4262	8	47	m	m	PROPN
ejpam-4262	8	48	)	)	PUNCT
ejpam-4262	8	49	.	.	PUNCT
ejpam-4262	9	1	various	various	ADJ
ejpam-4262	9	2	generalizations	generalization	NOUN
ejpam-4262	9	3	of	of	ADP
ejpam-4262	9	4	prime	prime	ADJ
ejpam-4262	9	5	submodules	submodule	NOUN
ejpam-4262	9	6	have	have	AUX
ejpam-4262	9	7	been	be	AUX
ejpam-4262	9	8	studied	study	VERB
ejpam-4262	9	9	.	.	PUNCT
ejpam-4262	10	1	for	for	ADP
ejpam-4262	10	2	example	example	NOUN
ejpam-4262	10	3	,	,	PUNCT
ejpam-4262	10	4	see	see	VERB
ejpam-4262	10	5	[	[	X
ejpam-4262	10	6	5	5	NUM
ejpam-4262	10	7	]	]	PUNCT
ejpam-4262	10	8	,	,	PUNCT
ejpam-4262	10	9	[	[	X
ejpam-4262	10	10	1	1	NUM
ejpam-4262	10	11	]	]	PUNCT
ejpam-4262	10	12	and	and	CCONJ
ejpam-4262	11	1	[	[	X
ejpam-4262	11	2	6	6	NUM
ejpam-4262	11	3	]	]	PUNCT
ejpam-4262	11	4	,	,	PUNCT
ejpam-4262	11	5	a	a	DET
ejpam-4262	11	6	proper	proper	ADJ
ejpam-4262	11	7	submodule	submodule	NOUN
ejpam-4262	11	8	p	p	NOUN
ejpam-4262	11	9	of	of	ADP
ejpam-4262	11	10	a	a	DET
ejpam-4262	11	11	left	left	ADJ
ejpam-4262	11	12	r	r	NOUN
ejpam-4262	11	13	-	-	PUNCT
ejpam-4262	11	14	module	module	NOUN
ejpam-4262	11	15	m	m	NOUN
ejpam-4262	11	16	is	be	AUX
ejpam-4262	11	17	called	call	VERB
ejpam-4262	11	18	a	a	DET
ejpam-4262	11	19	2	2	NUM
ejpam-4262	11	20	-	-	PUNCT
ejpam-4262	11	21	absorbing	absorbing	ADJ
ejpam-4262	11	22	(	(	PUNCT
ejpam-4262	11	23	resp	resp	NOUN
ejpam-4262	11	24	.	.	PUNCT
ejpam-4262	12	1	weakly	weakly	ADJ
ejpam-4262	12	2	2	2	NUM
ejpam-4262	12	3	-	-	PUNCT
ejpam-4262	12	4	absorbing	absorbing	ADJ
ejpam-4262	12	5	,	,	PUNCT
ejpam-4262	12	6	almost	almost	ADV
ejpam-4262	12	7	2	2	NUM
ejpam-4262	12	8	-	-	PUNCT
ejpam-4262	12	9	absorbing	absorbing	ADJ
ejpam-4262	12	10	)	)	PUNCT
ejpam-4262	12	11	submodule	submodule	NOUN
ejpam-4262	12	12	if	if	SCONJ
ejpam-4262	12	13	for	for	ADP
ejpam-4262	12	14	each	each	DET
ejpam-4262	12	15	r	r	NOUN
ejpam-4262	12	16	,	,	PUNCT
ejpam-4262	12	17	s	s	NOUN
ejpam-4262	12	18	∈	∈	PROPN
ejpam-4262	12	19	r	r	NOUN
ejpam-4262	12	20	and	and	CCONJ
ejpam-4262	12	21	every	every	DET
ejpam-4262	12	22	m	m	NOUN
ejpam-4262	12	23	∈m	∈m	NOUN
ejpam-4262	12	24	such	such	ADJ
ejpam-4262	12	25	that	that	SCONJ
ejpam-4262	12	26	rsm	rsm	PROPN
ejpam-4262	12	27	∈	∈	PROPN
ejpam-4262	12	28	p	p	X
ejpam-4262	12	29	(	(	PUNCT
ejpam-4262	12	30	resp	resp	NOUN
ejpam-4262	12	31	.	.	PUNCT
ejpam-4262	13	1	rsm	rsm	PROPN
ejpam-4262	13	2	∈	∈	PROPN
ejpam-4262	13	3	p\{0	p\{0	PROPN
ejpam-4262	13	4	}	}	PUNCT
ejpam-4262	13	5	,	,	PUNCT
ejpam-4262	13	6	rsm	rsm	PROPN
ejpam-4262	13	7	∈	∈	PROPN
ejpam-4262	13	8	p\(p	p\(p	X
ejpam-4262	13	9	:	:	PUNCT
ejpam-4262	13	10	m)p	m)p	X
ejpam-4262	13	11	)	)	PUNCT
ejpam-4262	13	12	,	,	PUNCT
ejpam-4262	13	13	we	we	PRON
ejpam-4262	13	14	have	have	VERB
ejpam-4262	13	15	rs	rs	PROPN
ejpam-4262	13	16	∈	∈	PROPN
ejpam-4262	13	17	(	(	PUNCT
ejpam-4262	13	18	p	p	X
ejpam-4262	13	19	:	:	PUNCT
ejpam-4262	13	20	m	m	X
ejpam-4262	13	21	)	)	PUNCT
ejpam-4262	13	22	or	or	CCONJ
ejpam-4262	13	23	rm	rm	PROPN
ejpam-4262	13	24	∈	∈	PROPN
ejpam-4262	13	25	p	p	PROPN
ejpam-4262	13	26	or	or	CCONJ
ejpam-4262	13	27	sm	sm	PROPN
ejpam-4262	13	28	∈	∈	PROPN
ejpam-4262	13	29	p	p	NOUN
ejpam-4262	13	30	.	.	PUNCT
ejpam-4262	14	1	according	accord	VERB
ejpam-4262	14	2	to	to	ADP
ejpam-4262	14	3	[	[	X
ejpam-4262	14	4	4	4	NUM
ejpam-4262	14	5	]	]	PUNCT
ejpam-4262	14	6	,	,	PUNCT
ejpam-4262	14	7	nz	nz	PROPN
ejpam-4262	14	8	is	be	AUX
ejpam-4262	14	9	a	a	DET
ejpam-4262	14	10	2	2	NUM
ejpam-4262	14	11	-	-	PUNCT
ejpam-4262	14	12	absorbing	absorb	VERB
ejpam-4262	14	13	submodule	submodule	NOUN
ejpam-4262	14	14	of	of	ADP
ejpam-4262	14	15	z	z	NOUN
ejpam-4262	14	16	if	if	SCONJ
ejpam-4262	15	1	and	and	CCONJ
ejpam-4262	15	2	only	only	ADV
ejpam-4262	15	3	if	if	SCONJ
ejpam-4262	15	4	n	n	PROPN
ejpam-4262	15	5	=	=	SYM
ejpam-4262	15	6	0	0	NUM
ejpam-4262	15	7	or	or	CCONJ
ejpam-4262	15	8	n	n	PROPN
ejpam-4262	15	9	is	be	AUX
ejpam-4262	15	10	a	a	DET
ejpam-4262	15	11	prime	prime	ADJ
ejpam-4262	15	12	number	number	NOUN
ejpam-4262	15	13	or	or	CCONJ
ejpam-4262	15	14	n	n	NOUN
ejpam-4262	15	15	=	=	SYM
ejpam-4262	15	16	pq	pq	NOUN
ejpam-4262	15	17	where	where	SCONJ
ejpam-4262	15	18	p	p	NOUN
ejpam-4262	15	19	and	and	CCONJ
ejpam-4262	15	20	q	q	NOUN
ejpam-4262	15	21	are	be	AUX
ejpam-4262	15	22	prime	prime	ADJ
ejpam-4262	15	23	numbers	number	NOUN
ejpam-4262	15	24	.	.	PUNCT
ejpam-4262	16	1	let	let	VERB
ejpam-4262	16	2	s(m	s(m	PROPN
ejpam-4262	16	3	)	)	PUNCT
ejpam-4262	16	4	be	be	VERB
ejpam-4262	16	5	the	the	DET
ejpam-4262	16	6	set	set	NOUN
ejpam-4262	16	7	of	of	ADP
ejpam-4262	16	8	all	all	DET
ejpam-4262	16	9	submodules	submodule	NOUN
ejpam-4262	16	10	of	of	ADP
ejpam-4262	16	11	m	m	PROPN
ejpam-4262	16	12	and	and	CCONJ
ejpam-4262	16	13	ϕ	ϕ	ADJ
ejpam-4262	16	14	:	:	PUNCT
ejpam-4262	16	15	s(m	s(m	NOUN
ejpam-4262	16	16	)	)	PUNCT
ejpam-4262	16	17	→	→	SYM
ejpam-4262	16	18	s(m	s(m	NOUN
ejpam-4262	16	19	)	)	PUNCT
ejpam-4262	16	20	∪	∪	NOUN
ejpam-4262	16	21	{	{	PUNCT
ejpam-4262	16	22	∅	∅	NOUN
ejpam-4262	16	23	}	}	PUNCT
ejpam-4262	16	24	be	be	AUX
ejpam-4262	16	25	a	a	DET
ejpam-4262	16	26	function	function	NOUN
ejpam-4262	16	27	.	.	PUNCT
ejpam-4262	17	1	in	in	ADP
ejpam-4262	17	2	this	this	DET
ejpam-4262	17	3	paper	paper	NOUN
ejpam-4262	17	4	,	,	PUNCT
ejpam-4262	17	5	we	we	PRON
ejpam-4262	17	6	assume	assume	VERB
ejpam-4262	17	7	that	that	SCONJ
ejpam-4262	17	8	ϕ(p	ϕ(p	PROPN
ejpam-4262	17	9	)	)	PUNCT
ejpam-4262	17	10	⊆	⊆	NUM
ejpam-4262	17	11	p	p	NOUN
ejpam-4262	17	12	.	.	PUNCT
ejpam-4262	18	1	in	in	ADP
ejpam-4262	18	2	[	[	X
ejpam-4262	18	3	3	3	NUM
ejpam-4262	18	4	]	]	PUNCT
ejpam-4262	18	5	,	,	PUNCT
ejpam-4262	18	6	the	the	DET
ejpam-4262	18	7	authors	author	NOUN
ejpam-4262	18	8	introduced	introduce	VERB
ejpam-4262	18	9	the	the	DET
ejpam-4262	18	10	concept	concept	NOUN
ejpam-4262	18	11	of	of	ADP
ejpam-4262	18	12	ϕ-2	ϕ-2	ADV
ejpam-4262	18	13	-	-	PUNCT
ejpam-4262	18	14	absorbing	absorb	VERB
ejpam-4262	18	15	submodule	submodule	NOUN
ejpam-4262	18	16	which	which	PRON
ejpam-4262	18	17	is	be	AUX
ejpam-4262	18	18	a	a	DET
ejpam-4262	18	19	generalization	generalization	NOUN
ejpam-4262	18	20	of	of	ADP
ejpam-4262	18	21	2	2	NUM
ejpam-4262	18	22	-	-	PUNCT
ejpam-4262	18	23	absorbing	absorbing	ADJ
ejpam-4262	18	24	submodules	submodule	NOUN
ejpam-4262	18	25	.	.	PUNCT
ejpam-4262	19	1	a	a	DET
ejpam-4262	19	2	proper	proper	ADJ
ejpam-4262	19	3	submodule	submodule	NOUN
ejpam-4262	19	4	p	p	NOUN
ejpam-4262	19	5	of	of	ADP
ejpam-4262	19	6	a	a	DET
ejpam-4262	19	7	left	left	ADJ
ejpam-4262	19	8	r	r	NOUN
ejpam-4262	19	9	-	-	PUNCT
ejpam-4262	19	10	module	module	NOUN
ejpam-4262	19	11	m	m	NOUN
ejpam-4262	19	12	is	be	AUX
ejpam-4262	19	13	a	a	DET
ejpam-4262	19	14	ϕ-2	ϕ-2	ADV
ejpam-4262	19	15	-	-	PUNCT
ejpam-4262	19	16	absorbing	absorb	VERB
ejpam-4262	19	17	submodule	submodule	NOUN
ejpam-4262	19	18	if	if	SCONJ
ejpam-4262	19	19	whenever	whenever	SCONJ
ejpam-4262	19	20	a	a	DET
ejpam-4262	19	21	,	,	PUNCT
ejpam-4262	19	22	b	b	X
ejpam-4262	19	23	∈	∈	PROPN
ejpam-4262	19	24	r	r	NOUN
ejpam-4262	19	25	,	,	PUNCT
ejpam-4262	19	26	m	m	NOUN
ejpam-4262	19	27	∈m	∈m	NOUN
ejpam-4262	19	28	with	with	ADP
ejpam-4262	19	29	abm	abm	PROPN
ejpam-4262	19	30	∈	∈	PROPN
ejpam-4262	19	31	p	p	PROPN
ejpam-4262	19	32	and	and	CCONJ
ejpam-4262	19	33	abm	abm	PROPN
ejpam-4262	19	34	/∈	/∈	PUNCT
ejpam-4262	19	35	ϕ(p	ϕ(p	PROPN
ejpam-4262	19	36	)	)	PUNCT
ejpam-4262	19	37	,	,	PUNCT
ejpam-4262	19	38	then	then	ADV
ejpam-4262	19	39	am	be	AUX
ejpam-4262	19	40	∈	∈	PROPN
ejpam-4262	19	41	p	p	NOUN
ejpam-4262	19	42	or	or	CCONJ
ejpam-4262	19	43	bm	bm	PROPN
ejpam-4262	19	44	∈	∈	PROPN
ejpam-4262	19	45	p	p	PROPN
ejpam-4262	19	46	or	or	CCONJ
ejpam-4262	19	47	ab	ab	PROPN
ejpam-4262	19	48	∈	∈	PROPN
ejpam-4262	19	49	(	(	PUNCT
ejpam-4262	19	50	p	p	X
ejpam-4262	19	51	:	:	PUNCT
ejpam-4262	19	52	m	m	PROPN
ejpam-4262	19	53	)	)	PUNCT
ejpam-4262	19	54	.	.	PUNCT
ejpam-4262	20	1	in	in	ADP
ejpam-4262	20	2	addition	addition	NOUN
ejpam-4262	20	3	,	,	PUNCT
ejpam-4262	20	4	the	the	DET
ejpam-4262	20	5	notion	notion	NOUN
ejpam-4262	20	6	of	of	ADP
ejpam-4262	20	7	ϕ-2	ϕ-2	ADV
ejpam-4262	20	8	-	-	PUNCT
ejpam-4262	20	9	absorbing	absorb	VERB
ejpam-4262	20	10	submodule	submodule	NOUN
ejpam-4262	20	11	is	be	AUX
ejpam-4262	20	12	also	also	ADV
ejpam-4262	20	13	a	a	DET
ejpam-4262	20	14	generalization	generalization	NOUN
ejpam-4262	20	15	of	of	ADP
ejpam-4262	20	16	both	both	CCONJ
ejpam-4262	20	17	weakly	weakly	ADJ
ejpam-4262	20	18	2	2	NUM
ejpam-4262	20	19	-	-	PUNCT
ejpam-4262	20	20	absorbing	absorb	VERB
ejpam-4262	20	21	submodule	submodule	NOUN
ejpam-4262	20	22	and	and	CCONJ
ejpam-4262	20	23	almost	almost	ADV
ejpam-4262	20	24	2	2	NUM
ejpam-4262	20	25	-	-	PUNCT
ejpam-4262	20	26	absorbing	absorb	VERB
ejpam-4262	20	27	submodule	submodule	NOUN
ejpam-4262	20	28	which	which	PRON
ejpam-4262	20	29	depends	depend	VERB
ejpam-4262	20	30	on	on	ADP
ejpam-4262	20	31	the	the	DET
ejpam-4262	20	32	definition	definition	NOUN
ejpam-4262	20	33	of	of	ADP
ejpam-4262	20	34	ϕ.	ϕ.	PROPN
ejpam-4262	20	35	doi	doi	PROPN
ejpam-4262	20	36	:	:	PUNCT
ejpam-4262	20	37	https://doi.org/10.29020/nybg.ejpam.v15i1.4262	https://doi.org/10.29020/nybg.ejpam.v15i1.4262	PROPN
ejpam-4262	20	38	email	email	NOUN
ejpam-4262	20	39	address	address	NOUN
ejpam-4262	20	40	:	:	PUNCT
ejpam-4262	20	41	thawatchai.kh@kmitl.ac.th	thawatchai.kh@kmitl.ac.th	PROPN
ejpam-4262	20	42	(	(	PUNCT
ejpam-4262	20	43	t.khumprapussorn	t.khumprapussorn	ADJ
ejpam-4262	20	44	)	)	PUNCT
ejpam-4262	20	45	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4262	21	1	328	328	NUM
ejpam-4262	21	2	©	©	PROPN
ejpam-4262	21	3	2022	2022	NUM
ejpam-4262	21	4	ejpam	ejpam	VERB
ejpam-4262	21	5	all	all	DET
ejpam-4262	21	6	rights	right	NOUN
ejpam-4262	21	7	reserved	reserve	VERB
ejpam-4262	21	8	.	.	PUNCT
ejpam-4262	22	1	t.	t.	PROPN
ejpam-4262	22	2	khumprapussorn	khumprapussorn	PROPN
ejpam-4262	22	3	/	/	SYM
ejpam-4262	22	4	eur	eur	PROPN
ejpam-4262	22	5	.	.	PUNCT
ejpam-4262	23	1	j.	j.	PROPN
ejpam-4262	23	2	pure	pure	PROPN
ejpam-4262	23	3	appl	appl	PROPN
ejpam-4262	23	4	.	.	PROPN
ejpam-4262	23	5	math	math	PROPN
ejpam-4262	23	6	,	,	PUNCT
ejpam-4262	23	7	15	15	NUM
ejpam-4262	23	8	(	(	PUNCT
ejpam-4262	23	9	1	1	NUM
ejpam-4262	23	10	)	)	PUNCT
ejpam-4262	23	11	(	(	PUNCT
ejpam-4262	23	12	2022	2022	NUM
ejpam-4262	23	13	)	)	PUNCT
ejpam-4262	23	14	,	,	PUNCT
ejpam-4262	23	15	328	328	NUM
ejpam-4262	23	16	-	-	SYM
ejpam-4262	23	17	334	334	NUM
ejpam-4262	23	18	329	329	NUM
ejpam-4262	23	19	let	let	VERB
ejpam-4262	23	20	(	(	PUNCT
ejpam-4262	23	21	g,+	g,+	PROPN
ejpam-4262	23	22	)	)	PUNCT
ejpam-4262	23	23	be	be	AUX
ejpam-4262	23	24	a	a	DET
ejpam-4262	23	25	group	group	NOUN
ejpam-4262	23	26	and	and	CCONJ
ejpam-4262	23	27	h	h	NOUN
ejpam-4262	23	28	is	be	AUX
ejpam-4262	23	29	a	a	DET
ejpam-4262	23	30	subgroup	subgroup	NOUN
ejpam-4262	23	31	of	of	ADP
ejpam-4262	23	32	g.	g.	PROPN
ejpam-4262	23	33	we	we	PRON
ejpam-4262	23	34	denote	denote	VERB
ejpam-4262	23	35	the	the	DET
ejpam-4262	23	36	symbol	symbol	NOUN
ejpam-4262	23	37	β(h	β(h	NOUN
ejpam-4262	23	38	)	)	PUNCT
ejpam-4262	23	39	by	by	ADP
ejpam-4262	23	40	{	{	PUNCT
ejpam-4262	23	41	h	h	NOUN
ejpam-4262	24	1	+	+	CCONJ
ejpam-4262	24	2	h	h	NOUN
ejpam-4262	25	1	|	|	ADV
ejpam-4262	25	2	h	h	NOUN
ejpam-4262	25	3	∈	∈	PROPN
ejpam-4262	25	4	h	h	NOUN
ejpam-4262	25	5	}	}	PUNCT
ejpam-4262	25	6	and	and	CCONJ
ejpam-4262	25	7	α(h	α(h	NOUN
ejpam-4262	25	8	)	)	PUNCT
ejpam-4262	25	9	by	by	ADP
ejpam-4262	25	10	{	{	PUNCT
ejpam-4262	25	11	h	h	NOUN
ejpam-4262	25	12	|	|	ADV
ejpam-4262	26	1	h	h	NOUN
ejpam-4262	27	1	+	+	CCONJ
ejpam-4262	27	2	h	h	NOUN
ejpam-4262	27	3	∈	∈	PROPN
ejpam-4262	27	4	h	h	NOUN
ejpam-4262	27	5	}	}	PUNCT
ejpam-4262	27	6	.	.	PUNCT
ejpam-4262	28	1	we	we	PRON
ejpam-4262	28	2	see	see	VERB
ejpam-4262	28	3	that	that	SCONJ
ejpam-4262	28	4	β(h	β(h	NOUN
ejpam-4262	28	5	)	)	PUNCT
ejpam-4262	28	6	⊆	⊆	NUM
ejpam-4262	28	7	h	h	NOUN
ejpam-4262	28	8	⊆	⊆	NUM
ejpam-4262	28	9	α(h	α(h	NOUN
ejpam-4262	28	10	)	)	PUNCT
ejpam-4262	28	11	.	.	PUNCT
ejpam-4262	29	1	if	if	SCONJ
ejpam-4262	29	2	i	i	PRON
ejpam-4262	29	3	is	be	AUX
ejpam-4262	29	4	an	an	DET
ejpam-4262	29	5	ideal	ideal	NOUN
ejpam-4262	29	6	of	of	ADP
ejpam-4262	29	7	r	r	NOUN
ejpam-4262	29	8	,	,	PUNCT
ejpam-4262	29	9	then	then	ADV
ejpam-4262	29	10	both	both	PRON
ejpam-4262	29	11	of	of	ADP
ejpam-4262	29	12	α(i	α(i	PROPN
ejpam-4262	29	13	)	)	PUNCT
ejpam-4262	29	14	and	and	CCONJ
ejpam-4262	29	15	β(i	β(i	NOUN
ejpam-4262	29	16	)	)	PUNCT
ejpam-4262	29	17	are	be	AUX
ejpam-4262	29	18	ideals	ideal	NOUN
ejpam-4262	29	19	of	of	ADP
ejpam-4262	29	20	r.	r.	PROPN
ejpam-4262	29	21	moreover	moreover	ADV
ejpam-4262	29	22	,	,	PUNCT
ejpam-4262	29	23	if	if	SCONJ
ejpam-4262	29	24	n	n	PRON
ejpam-4262	29	25	is	be	AUX
ejpam-4262	29	26	a	a	DET
ejpam-4262	29	27	submodule	submodule	NOUN
ejpam-4262	29	28	of	of	ADP
ejpam-4262	29	29	m	m	PROPN
ejpam-4262	29	30	,	,	PUNCT
ejpam-4262	29	31	then	then	ADV
ejpam-4262	29	32	both	both	PRON
ejpam-4262	29	33	of	of	ADP
ejpam-4262	29	34	α(n	α(n	NOUN
ejpam-4262	29	35	)	)	PUNCT
ejpam-4262	29	36	and	and	CCONJ
ejpam-4262	29	37	β(n	β(n	NUM
ejpam-4262	29	38	)	)	PUNCT
ejpam-4262	29	39	are	be	AUX
ejpam-4262	29	40	submodules	submodule	NOUN
ejpam-4262	29	41	of	of	ADP
ejpam-4262	29	42	m	m	PRON
ejpam-4262	29	43	.	.	PUNCT
ejpam-4262	30	1	in	in	ADP
ejpam-4262	30	2	[	[	X
ejpam-4262	30	3	2	2	NUM
ejpam-4262	30	4	]	]	PUNCT
ejpam-4262	30	5	,	,	PUNCT
ejpam-4262	30	6	a	a	DET
ejpam-4262	30	7	proper	proper	ADJ
ejpam-4262	30	8	submodule	submodule	NOUN
ejpam-4262	30	9	p	p	NOUN
ejpam-4262	30	10	of	of	ADP
ejpam-4262	30	11	a	a	DET
ejpam-4262	30	12	left	left	ADJ
ejpam-4262	30	13	r	r	NOUN
ejpam-4262	30	14	-	-	PUNCT
ejpam-4262	30	15	module	module	NOUN
ejpam-4262	30	16	m	m	NOUN
ejpam-4262	30	17	is	be	AUX
ejpam-4262	30	18	called	call	VERB
ejpam-4262	30	19	β	β	VERB
ejpam-4262	30	20	-	-	ADJ
ejpam-4262	30	21	absorbing	absorbing	ADJ
ejpam-4262	30	22	if	if	SCONJ
ejpam-4262	30	23	for	for	SCONJ
ejpam-4262	30	24	any	any	DET
ejpam-4262	30	25	element	element	NOUN
ejpam-4262	30	26	r	r	NOUN
ejpam-4262	30	27	,	,	PUNCT
ejpam-4262	30	28	s	s	NOUN
ejpam-4262	30	29	∈	∈	PROPN
ejpam-4262	30	30	r	r	NOUN
ejpam-4262	30	31	and	and	CCONJ
ejpam-4262	30	32	m	m	PROPN
ejpam-4262	30	33	∈	∈	NOUN
ejpam-4262	30	34	m	m	VERB
ejpam-4262	30	35	such	such	ADJ
ejpam-4262	30	36	that	that	SCONJ
ejpam-4262	30	37	rsm	rsm	PROPN
ejpam-4262	30	38	∈	∈	PROPN
ejpam-4262	30	39	p	p	PROPN
ejpam-4262	30	40	,	,	PUNCT
ejpam-4262	30	41	we	we	PRON
ejpam-4262	30	42	have	have	AUX
ejpam-4262	30	43	rs	r	VERB
ejpam-4262	31	1	+	+	CCONJ
ejpam-4262	31	2	rs	rs	PROPN
ejpam-4262	31	3	∈	∈	PROPN
ejpam-4262	31	4	(	(	PUNCT
ejpam-4262	31	5	p	p	X
ejpam-4262	31	6	:	:	PUNCT
ejpam-4262	31	7	m	m	NUM
ejpam-4262	31	8	)	)	PUNCT
ejpam-4262	31	9	or	or	CCONJ
ejpam-4262	31	10	r(m	r(m	PROPN
ejpam-4262	31	11	+	+	NUM
ejpam-4262	31	12	m	m	NOUN
ejpam-4262	31	13	)	)	PUNCT
ejpam-4262	31	14	∈	∈	PROPN
ejpam-4262	31	15	p	p	NOUN
ejpam-4262	31	16	or	or	CCONJ
ejpam-4262	31	17	s(m	s(m	NOUN
ejpam-4262	31	18	+	+	CCONJ
ejpam-4262	31	19	m	m	NOUN
ejpam-4262	31	20	)	)	PUNCT
ejpam-4262	31	21	∈	∈	PROPN
ejpam-4262	31	22	p	p	NOUN
ejpam-4262	31	23	.	.	PUNCT
ejpam-4262	32	1	the	the	DET
ejpam-4262	32	2	characterization	characterization	NOUN
ejpam-4262	32	3	of	of	ADP
ejpam-4262	32	4	β	β	ADJ
ejpam-4262	32	5	-	-	ADJ
ejpam-4262	32	6	absorbing	absorbing	ADJ
ejpam-4262	32	7	submodule	submodule	NOUN
ejpam-4262	32	8	of	of	ADP
ejpam-4262	32	9	z	z	NOUN
ejpam-4262	32	10	-	-	PUNCT
ejpam-4262	32	11	module	module	NOUN
ejpam-4262	32	12	z	z	NOUN
ejpam-4262	32	13	was	be	AUX
ejpam-4262	32	14	also	also	ADV
ejpam-4262	32	15	given	give	VERB
ejpam-4262	32	16	.	.	PUNCT
ejpam-4262	33	1	on	on	ADP
ejpam-4262	33	2	the	the	DET
ejpam-4262	33	3	z	z	NOUN
ejpam-4262	33	4	-	-	PUNCT
ejpam-4262	33	5	module	module	NOUN
ejpam-4262	33	6	z	z	NOUN
ejpam-4262	33	7	,	,	PUNCT
ejpam-4262	33	8	nz	nz	PROPN
ejpam-4262	33	9	is	be	AUX
ejpam-4262	33	10	a	a	DET
ejpam-4262	33	11	β	β	NOUN
ejpam-4262	33	12	-	-	ADJ
ejpam-4262	33	13	absorbing	absorbing	ADJ
ejpam-4262	33	14	submodule	submodule	NOUN
ejpam-4262	33	15	of	of	ADP
ejpam-4262	33	16	z	z	NOUN
ejpam-4262	33	17	if	if	SCONJ
ejpam-4262	34	1	and	and	CCONJ
ejpam-4262	34	2	only	only	ADV
ejpam-4262	34	3	if	if	SCONJ
ejpam-4262	34	4	n	n	PROPN
ejpam-4262	34	5	=	=	SYM
ejpam-4262	34	6	0	0	NUM
ejpam-4262	34	7	or	or	CCONJ
ejpam-4262	34	8	n	n	CCONJ
ejpam-4262	34	9	=	=	SYM
ejpam-4262	34	10	32	32	NUM
ejpam-4262	34	11	or	or	CCONJ
ejpam-4262	34	12	n	n	PROPN
ejpam-4262	34	13	is	be	AUX
ejpam-4262	34	14	a	a	DET
ejpam-4262	34	15	prime	prime	ADJ
ejpam-4262	34	16	number	number	NOUN
ejpam-4262	34	17	or	or	CCONJ
ejpam-4262	34	18	n	n	NOUN
ejpam-4262	34	19	=	=	SYM
ejpam-4262	34	20	pq	pq	NOUN
ejpam-4262	34	21	where	where	SCONJ
ejpam-4262	34	22	p	p	NOUN
ejpam-4262	34	23	and	and	CCONJ
ejpam-4262	34	24	q	q	NOUN
ejpam-4262	34	25	are	be	AUX
ejpam-4262	34	26	prime	prime	ADJ
ejpam-4262	34	27	numbers	number	NOUN
ejpam-4262	34	28	or	or	CCONJ
ejpam-4262	34	29	n	n	NOUN
ejpam-4262	34	30	=	=	NOUN
ejpam-4262	34	31	23p	23p	NOUN
ejpam-4262	34	32	where	where	SCONJ
ejpam-4262	34	33	p	p	NOUN
ejpam-4262	34	34	is	be	AUX
ejpam-4262	34	35	prime	prime	ADJ
ejpam-4262	34	36	number	number	NOUN
ejpam-4262	34	37	or	or	CCONJ
ejpam-4262	34	38	n	n	NOUN
ejpam-4262	34	39	=	=	NOUN
ejpam-4262	34	40	2pq	2pq	NOUN
ejpam-4262	34	41	where	where	SCONJ
ejpam-4262	34	42	p	p	NOUN
ejpam-4262	34	43	and	and	CCONJ
ejpam-4262	34	44	q	q	NOUN
ejpam-4262	34	45	are	be	AUX
ejpam-4262	34	46	prime	prime	ADJ
ejpam-4262	34	47	numbers	number	NOUN
ejpam-4262	34	48	.	.	PUNCT
ejpam-4262	35	1	the	the	DET
ejpam-4262	35	2	characterization	characterization	NOUN
ejpam-4262	35	3	of	of	ADP
ejpam-4262	35	4	β	β	ADJ
ejpam-4262	35	5	-	-	ADJ
ejpam-4262	35	6	absorbing	absorbing	ADJ
ejpam-4262	35	7	submodule	submodule	NOUN
ejpam-4262	35	8	of	of	ADP
ejpam-4262	35	9	z	z	NOUN
ejpam-4262	35	10	-	-	PUNCT
ejpam-4262	35	11	module	module	NOUN
ejpam-4262	35	12	z	z	NOUN
ejpam-4262	35	13	explains	explain	VERB
ejpam-4262	35	14	that	that	SCONJ
ejpam-4262	35	15	β	β	X
ejpam-4262	35	16	-	-	ADJ
ejpam-4262	35	17	absorbing	absorbing	ADJ
ejpam-4262	35	18	submodules	submodule	NOUN
ejpam-4262	35	19	need	need	VERB
ejpam-4262	35	20	not	not	PART
ejpam-4262	35	21	to	to	PART
ejpam-4262	35	22	be	be	AUX
ejpam-4262	35	23	2	2	NUM
ejpam-4262	35	24	-	-	PUNCT
ejpam-4262	35	25	absorbing	absorbing	ADJ
ejpam-4262	35	26	submodules	submodule	NOUN
ejpam-4262	35	27	.	.	PUNCT
ejpam-4262	36	1	also	also	ADV
ejpam-4262	36	2	,	,	PUNCT
ejpam-4262	36	3	in	in	ADP
ejpam-4262	36	4	[	[	PUNCT
ejpam-4262	36	5	2	2	NUM
ejpam-4262	36	6	]	]	PUNCT
ejpam-4262	36	7	,	,	PUNCT
ejpam-4262	36	8	a	a	DET
ejpam-4262	36	9	proper	proper	ADJ
ejpam-4262	36	10	submodule	submodule	NOUN
ejpam-4262	36	11	p	p	NOUN
ejpam-4262	36	12	of	of	ADP
ejpam-4262	36	13	m	m	PROPN
ejpam-4262	36	14	is	be	AUX
ejpam-4262	36	15	a	a	DET
ejpam-4262	36	16	weakly	weakly	ADJ
ejpam-4262	36	17	β	β	NOUN
ejpam-4262	36	18	-	-	ADJ
ejpam-4262	36	19	absorbing	absorbing	ADJ
ejpam-4262	36	20	submodule	submodule	NOUN
ejpam-4262	36	21	of	of	ADP
ejpam-4262	36	22	m	m	PROPN
ejpam-4262	36	23	if	if	SCONJ
ejpam-4262	36	24	for	for	ADP
ejpam-4262	36	25	each	each	DET
ejpam-4262	36	26	r	r	NOUN
ejpam-4262	36	27	,	,	PUNCT
ejpam-4262	36	28	s	s	NOUN
ejpam-4262	36	29	∈	∈	PROPN
ejpam-4262	36	30	r	r	NOUN
ejpam-4262	36	31	and	and	CCONJ
ejpam-4262	36	32	every	every	DET
ejpam-4262	36	33	m	m	NOUN
ejpam-4262	36	34	∈m	∈m	NOUN
ejpam-4262	36	35	such	such	ADJ
ejpam-4262	36	36	that	that	SCONJ
ejpam-4262	36	37	rsm	rsm	PROPN
ejpam-4262	36	38	∈	∈	PROPN
ejpam-4262	36	39	p\{0	p\{0	PROPN
ejpam-4262	36	40	}	}	PUNCT
ejpam-4262	36	41	,	,	PUNCT
ejpam-4262	36	42	we	we	PRON
ejpam-4262	36	43	have	have	AUX
ejpam-4262	36	44	rs+	rs+	VERB
ejpam-4262	36	45	rs	rs	PROPN
ejpam-4262	36	46	∈	∈	PROPN
ejpam-4262	36	47	(	(	PUNCT
ejpam-4262	36	48	p	p	X
ejpam-4262	36	49	:	:	PUNCT
ejpam-4262	36	50	m	m	NOUN
ejpam-4262	36	51	)	)	PUNCT
ejpam-4262	36	52	or	or	CCONJ
ejpam-4262	36	53	r(m+m	r(m+m	NOUN
ejpam-4262	36	54	)	)	PUNCT
ejpam-4262	36	55	∈	∈	PROPN
ejpam-4262	36	56	p	p	NOUN
ejpam-4262	36	57	or	or	CCONJ
ejpam-4262	36	58	s(m+m	s(m+m	SYM
ejpam-4262	36	59	)	)	PUNCT
ejpam-4262	36	60	∈	∈	PROPN
ejpam-4262	36	61	p	p	NOUN
ejpam-4262	36	62	.	.	PUNCT
ejpam-4262	37	1	in	in	ADP
ejpam-4262	37	2	this	this	DET
ejpam-4262	37	3	research	research	NOUN
ejpam-4262	37	4	,	,	PUNCT
ejpam-4262	37	5	we	we	PRON
ejpam-4262	37	6	extend	extend	VERB
ejpam-4262	37	7	the	the	DET
ejpam-4262	37	8	notion	notion	NOUN
ejpam-4262	37	9	of	of	ADP
ejpam-4262	37	10	β	β	ADJ
ejpam-4262	37	11	-	-	ADJ
ejpam-4262	37	12	absorbing	absorbing	ADJ
ejpam-4262	37	13	submodules	submodule	NOUN
ejpam-4262	37	14	to	to	ADP
ejpam-4262	37	15	ϕ-β	ϕ-β	ADJ
ejpam-4262	37	16	-	-	ADJ
ejpam-4262	37	17	absorbing	absorbing	ADJ
ejpam-4262	37	18	submodules	submodule	NOUN
ejpam-4262	37	19	.	.	PUNCT
ejpam-4262	38	1	first	first	ADV
ejpam-4262	38	2	,	,	PUNCT
ejpam-4262	38	3	we	we	PRON
ejpam-4262	38	4	introduce	introduce	VERB
ejpam-4262	38	5	notions	notion	NOUN
ejpam-4262	38	6	of	of	ADP
ejpam-4262	38	7	ϕ-β	ϕ-β	NOUN
ejpam-4262	38	8	-	-	ADJ
ejpam-4262	38	9	absorbing	absorbing	ADJ
ejpam-4262	38	10	submodules	submodule	NOUN
ejpam-4262	38	11	.	.	PUNCT
ejpam-4262	39	1	a	a	DET
ejpam-4262	39	2	proper	proper	ADJ
ejpam-4262	39	3	submodule	submodule	NOUN
ejpam-4262	39	4	p	p	NOUN
ejpam-4262	39	5	of	of	ADP
ejpam-4262	39	6	a	a	DET
ejpam-4262	39	7	left	left	ADJ
ejpam-4262	39	8	r	r	NOUN
ejpam-4262	39	9	-	-	PUNCT
ejpam-4262	39	10	module	module	NOUN
ejpam-4262	39	11	m	m	NOUN
ejpam-4262	39	12	is	be	AUX
ejpam-4262	39	13	called	call	VERB
ejpam-4262	39	14	a	a	DET
ejpam-4262	39	15	ϕ-β	ϕ-β	ADJ
ejpam-4262	39	16	-	-	ADJ
ejpam-4262	39	17	absorbing	absorbing	ADJ
ejpam-4262	39	18	submodule	submodule	NOUN
ejpam-4262	39	19	of	of	ADP
ejpam-4262	39	20	m	m	PROPN
ejpam-4262	39	21	if	if	SCONJ
ejpam-4262	39	22	for	for	ADP
ejpam-4262	39	23	any	any	DET
ejpam-4262	39	24	element	element	NOUN
ejpam-4262	39	25	r	r	NOUN
ejpam-4262	39	26	,	,	PUNCT
ejpam-4262	39	27	s	s	NOUN
ejpam-4262	39	28	∈	∈	PROPN
ejpam-4262	39	29	r	r	NOUN
ejpam-4262	39	30	and	and	CCONJ
ejpam-4262	39	31	m	m	NOUN
ejpam-4262	39	32	∈m	∈m	NOUN
ejpam-4262	39	33	such	such	ADJ
ejpam-4262	39	34	that	that	SCONJ
ejpam-4262	39	35	rsm	rsm	PROPN
ejpam-4262	39	36	∈	∈	PROPN
ejpam-4262	39	37	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	39	38	)	)	PUNCT
ejpam-4262	39	39	,	,	PUNCT
ejpam-4262	39	40	we	we	PRON
ejpam-4262	39	41	have	have	VERB
ejpam-4262	39	42	rs+rs	rs+rs	ADJ
ejpam-4262	39	43	∈	∈	NOUN
ejpam-4262	39	44	(	(	PUNCT
ejpam-4262	39	45	p	p	X
ejpam-4262	39	46	:	:	PUNCT
ejpam-4262	39	47	m	m	NOUN
ejpam-4262	39	48	)	)	PUNCT
ejpam-4262	39	49	or	or	CCONJ
ejpam-4262	39	50	r(m+m	r(m+m	NOUN
ejpam-4262	39	51	)	)	PUNCT
ejpam-4262	39	52	∈	∈	PROPN
ejpam-4262	39	53	p	p	NOUN
ejpam-4262	39	54	or	or	CCONJ
ejpam-4262	39	55	s(m	s(m	NOUN
ejpam-4262	39	56	+	+	CCONJ
ejpam-4262	39	57	m	m	NOUN
ejpam-4262	39	58	)	)	PUNCT
ejpam-4262	39	59	∈	∈	PROPN
ejpam-4262	39	60	p	p	NOUN
ejpam-4262	39	61	.	.	PUNCT
ejpam-4262	40	1	in	in	ADP
ejpam-4262	40	2	case	case	NOUN
ejpam-4262	40	3	ϕ0(n	ϕ0(n	PROPN
ejpam-4262	40	4	)	)	PUNCT
ejpam-4262	40	5	=	=	SYM
ejpam-4262	40	6	{	{	PUNCT
ejpam-4262	40	7	0	0	NUM
ejpam-4262	40	8	}	}	PUNCT
ejpam-4262	40	9	for	for	ADP
ejpam-4262	40	10	all	all	DET
ejpam-4262	40	11	submodule	submodule	NOUN
ejpam-4262	40	12	n	n	PROPN
ejpam-4262	40	13	of	of	ADP
ejpam-4262	40	14	m	m	PROPN
ejpam-4262	40	15	,	,	PUNCT
ejpam-4262	40	16	we	we	PRON
ejpam-4262	40	17	have	have	VERB
ejpam-4262	40	18	weakly	weakly	ADJ
ejpam-4262	40	19	βabsorbing	βabsorbing	NOUN
ejpam-4262	40	20	submodules	submodule	NOUN
ejpam-4262	40	21	and	and	CCONJ
ejpam-4262	40	22	ϕ0	ϕ0	NOUN
ejpam-4262	40	23	-	-	PUNCT
ejpam-4262	40	24	β	β	NOUN
ejpam-4262	40	25	-	-	ADJ
ejpam-4262	40	26	absorbing	absorbing	ADJ
ejpam-4262	40	27	submodules	submodule	NOUN
ejpam-4262	40	28	are	be	AUX
ejpam-4262	40	29	equivalent	equivalent	ADJ
ejpam-4262	40	30	.	.	PUNCT
ejpam-4262	41	1	this	this	DET
ejpam-4262	41	2	case	case	NOUN
ejpam-4262	41	3	inspired	inspire	VERB
ejpam-4262	41	4	us	we	PRON
ejpam-4262	41	5	to	to	PART
ejpam-4262	41	6	investigate	investigate	VERB
ejpam-4262	41	7	some	some	DET
ejpam-4262	41	8	basic	basic	ADJ
ejpam-4262	41	9	properties	property	NOUN
ejpam-4262	41	10	of	of	ADP
ejpam-4262	41	11	ϕ-β	ϕ-β	NOUN
ejpam-4262	41	12	-	-	ADJ
ejpam-4262	41	13	absorbing	absorbing	ADJ
ejpam-4262	41	14	submodules	submodule	NOUN
ejpam-4262	41	15	in	in	ADP
ejpam-4262	41	16	section	section	NOUN
ejpam-4262	41	17	2	2	NUM
ejpam-4262	41	18	,	,	PUNCT
ejpam-4262	41	19	whereas	whereas	SCONJ
ejpam-4262	41	20	section	section	NOUN
ejpam-4262	41	21	3	3	NUM
ejpam-4262	41	22	contains	contain	VERB
ejpam-4262	41	23	the	the	DET
ejpam-4262	41	24	characterizations	characterization	NOUN
ejpam-4262	41	25	of	of	ADP
ejpam-4262	41	26	ϕ-β	ϕ-β	NOUN
ejpam-4262	41	27	-	-	ADJ
ejpam-4262	41	28	absorbing	absorbing	ADJ
ejpam-4262	41	29	submodules	submodule	NOUN
ejpam-4262	41	30	.	.	PUNCT
ejpam-4262	42	1	2	2	X
ejpam-4262	42	2	.	.	X
ejpam-4262	42	3	on	on	ADP
ejpam-4262	42	4	ϕ-β	ϕ-β	ADJ
ejpam-4262	42	5	-	-	ADJ
ejpam-4262	42	6	absorbing	absorbing	ADJ
ejpam-4262	42	7	submodules	submodule	NOUN
ejpam-4262	42	8	in	in	ADP
ejpam-4262	42	9	this	this	DET
ejpam-4262	42	10	section	section	NOUN
ejpam-4262	42	11	,	,	PUNCT
ejpam-4262	42	12	we	we	PRON
ejpam-4262	42	13	define	define	VERB
ejpam-4262	42	14	ϕ-β	ϕ-β	ADJ
ejpam-4262	42	15	-	-	ADJ
ejpam-4262	42	16	absorbing	absorbing	ADJ
ejpam-4262	42	17	submodules	submodule	NOUN
ejpam-4262	42	18	and	and	CCONJ
ejpam-4262	42	19	obtain	obtain	VERB
ejpam-4262	42	20	some	some	DET
ejpam-4262	42	21	related	relate	VERB
ejpam-4262	42	22	results	result	NOUN
ejpam-4262	42	23	.	.	PUNCT
ejpam-4262	43	1	definition	definition	NOUN
ejpam-4262	43	2	1	1	NUM
ejpam-4262	43	3	.	.	PUNCT
ejpam-4262	44	1	a	a	DET
ejpam-4262	44	2	proper	proper	ADJ
ejpam-4262	44	3	submodule	submodule	NOUN
ejpam-4262	44	4	p	p	NOUN
ejpam-4262	44	5	of	of	ADP
ejpam-4262	44	6	an	an	DET
ejpam-4262	44	7	r	r	NOUN
ejpam-4262	44	8	-	-	PUNCT
ejpam-4262	44	9	module	module	NOUN
ejpam-4262	44	10	m	m	NOUN
ejpam-4262	44	11	is	be	AUX
ejpam-4262	44	12	said	say	VERB
ejpam-4262	44	13	to	to	PART
ejpam-4262	44	14	be	be	AUX
ejpam-4262	44	15	a	a	DET
ejpam-4262	44	16	ϕ-β	ϕ-β	ADJ
ejpam-4262	44	17	-	-	ADJ
ejpam-4262	44	18	absorbing	absorbing	ADJ
ejpam-4262	44	19	submodule	submodule	NOUN
ejpam-4262	44	20	of	of	ADP
ejpam-4262	44	21	m	m	PRON
ejpam-4262	44	22	if	if	SCONJ
ejpam-4262	44	23	whenever	whenever	SCONJ
ejpam-4262	44	24	r	r	NOUN
ejpam-4262	44	25	,	,	PUNCT
ejpam-4262	44	26	s	s	NOUN
ejpam-4262	44	27	∈	∈	PROPN
ejpam-4262	44	28	r	r	NOUN
ejpam-4262	44	29	and	and	CCONJ
ejpam-4262	44	30	m	m	NOUN
ejpam-4262	44	31	∈m	∈m	NOUN
ejpam-4262	44	32	such	such	ADJ
ejpam-4262	44	33	that	that	SCONJ
ejpam-4262	44	34	rsm	rsm	PROPN
ejpam-4262	44	35	∈	∈	PROPN
ejpam-4262	44	36	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	44	37	)	)	PUNCT
ejpam-4262	44	38	,	,	PUNCT
ejpam-4262	44	39	then	then	ADV
ejpam-4262	44	40	rs+rs	rs+rs	VERB
ejpam-4262	44	41	∈	∈	NOUN
ejpam-4262	44	42	(	(	PUNCT
ejpam-4262	44	43	p	p	X
ejpam-4262	44	44	:	:	PUNCT
ejpam-4262	44	45	m	m	NOUN
ejpam-4262	44	46	)	)	PUNCT
ejpam-4262	44	47	or	or	CCONJ
ejpam-4262	44	48	r(m+m	r(m+m	NOUN
ejpam-4262	44	49	)	)	PUNCT
ejpam-4262	44	50	∈	∈	PROPN
ejpam-4262	44	51	p	p	NOUN
ejpam-4262	44	52	or	or	CCONJ
ejpam-4262	44	53	s(m+m	s(m+m	SYM
ejpam-4262	44	54	)	)	PUNCT
ejpam-4262	44	55	∈	∈	PROPN
ejpam-4262	44	56	p	p	NOUN
ejpam-4262	44	57	.	.	PUNCT
ejpam-4262	45	1	every	every	DET
ejpam-4262	45	2	β	β	X
ejpam-4262	45	3	-	-	ADJ
ejpam-4262	45	4	absorbing	absorbing	ADJ
ejpam-4262	45	5	is	be	AUX
ejpam-4262	45	6	a	a	DET
ejpam-4262	45	7	weakly	weakly	ADJ
ejpam-4262	45	8	β	β	NOUN
ejpam-4262	45	9	-	-	ADJ
ejpam-4262	45	10	absorbing	absorbing	ADJ
ejpam-4262	45	11	submodule	submodule	NOUN
ejpam-4262	45	12	but	but	CCONJ
ejpam-4262	45	13	the	the	DET
ejpam-4262	45	14	converse	converse	NOUN
ejpam-4262	45	15	does	do	AUX
ejpam-4262	45	16	not	not	PART
ejpam-4262	45	17	necessarity	necessarity	NOUN
ejpam-4262	45	18	hold	hold	VERB
ejpam-4262	45	19	.	.	PUNCT
ejpam-4262	46	1	as	as	SCONJ
ejpam-4262	46	2	mentioned	mention	VERB
ejpam-4262	46	3	above	above	ADV
ejpam-4262	46	4	,	,	PUNCT
ejpam-4262	46	5	β	β	ADJ
ejpam-4262	46	6	-	-	ADJ
ejpam-4262	46	7	absorbing	absorbing	ADJ
ejpam-4262	46	8	submodules	submodule	NOUN
ejpam-4262	46	9	and	and	CCONJ
ejpam-4262	46	10	weakly	weakly	ADJ
ejpam-4262	46	11	β	β	NOUN
ejpam-4262	46	12	-	-	ADJ
ejpam-4262	46	13	absorbing	absorbing	ADJ
ejpam-4262	46	14	submodules	submodule	NOUN
ejpam-4262	46	15	are	be	AUX
ejpam-4262	46	16	special	special	ADJ
ejpam-4262	46	17	cases	case	NOUN
ejpam-4262	46	18	of	of	ADP
ejpam-4262	46	19	ϕ-β	ϕ-β	NOUN
ejpam-4262	46	20	-	-	ADJ
ejpam-4262	46	21	absorbing	absorbing	ADJ
ejpam-4262	46	22	submodules	submodule	NOUN
ejpam-4262	46	23	.	.	PUNCT
ejpam-4262	47	1	theorem	theorem	NOUN
ejpam-4262	47	2	1	1	NUM
ejpam-4262	47	3	.	.	PUNCT
ejpam-4262	48	1	if	if	SCONJ
ejpam-4262	48	2	p	p	NOUN
ejpam-4262	48	3	is	be	AUX
ejpam-4262	48	4	a	a	DET
ejpam-4262	48	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	48	6	-	-	ADJ
ejpam-4262	48	7	absorbing	absorbing	ADJ
ejpam-4262	48	8	submodule	submodule	NOUN
ejpam-4262	48	9	of	of	ADP
ejpam-4262	48	10	m	m	PROPN
ejpam-4262	48	11	and	and	CCONJ
ejpam-4262	48	12	(	(	PUNCT
ejpam-4262	48	13	p	p	X
ejpam-4262	48	14	:	:	PUNCT
ejpam-4262	48	15	m)2β(p	m)2β(p	NOUN
ejpam-4262	48	16	)	)	PUNCT
ejpam-4262	48	17	⊈	⊈	PROPN
ejpam-4262	49	1	ϕ(p	ϕ(p	PROPN
ejpam-4262	49	2	)	)	PUNCT
ejpam-4262	49	3	,	,	PUNCT
ejpam-4262	49	4	then	then	ADV
ejpam-4262	49	5	p	p	NOUN
ejpam-4262	49	6	is	be	AUX
ejpam-4262	49	7	a	a	DET
ejpam-4262	49	8	β	β	NOUN
ejpam-4262	49	9	-	-	ADJ
ejpam-4262	49	10	absorbing	absorbing	ADJ
ejpam-4262	49	11	submodule	submodule	NOUN
ejpam-4262	49	12	of	of	ADP
ejpam-4262	49	13	m	m	PROPN
ejpam-4262	49	14	.	.	PUNCT
ejpam-4262	50	1	proof	proof	NOUN
ejpam-4262	50	2	.	.	PUNCT
ejpam-4262	51	1	assume	assume	VERB
ejpam-4262	51	2	that	that	SCONJ
ejpam-4262	51	3	p	p	NOUN
ejpam-4262	51	4	is	be	AUX
ejpam-4262	51	5	a	a	DET
ejpam-4262	51	6	ϕ-β	ϕ-β	ADJ
ejpam-4262	51	7	-	-	ADJ
ejpam-4262	51	8	absorbing	absorbing	ADJ
ejpam-4262	51	9	submodule	submodule	NOUN
ejpam-4262	51	10	of	of	ADP
ejpam-4262	51	11	m	m	PROPN
ejpam-4262	51	12	and	and	CCONJ
ejpam-4262	51	13	(	(	PUNCT
ejpam-4262	51	14	p	p	X
ejpam-4262	51	15	:	:	PUNCT
ejpam-4262	51	16	m)2β(p	m)2β(p	NOUN
ejpam-4262	51	17	)	)	PUNCT
ejpam-4262	51	18	⊈	⊈	PROPN
ejpam-4262	52	1	ϕ(p	ϕ(p	PROPN
ejpam-4262	52	2	)	)	PUNCT
ejpam-4262	52	3	.	.	PUNCT
ejpam-4262	53	1	let	let	VERB
ejpam-4262	53	2	r	r	NOUN
ejpam-4262	53	3	,	,	PUNCT
ejpam-4262	53	4	s	s	PART
ejpam-4262	53	5	∈	∈	PROPN
ejpam-4262	53	6	r	r	NOUN
ejpam-4262	53	7	and	and	CCONJ
ejpam-4262	53	8	m	m	PROPN
ejpam-4262	53	9	∈	∈	NOUN
ejpam-4262	53	10	m	m	AUX
ejpam-4262	53	11	be	be	VERB
ejpam-4262	53	12	such	such	ADJ
ejpam-4262	53	13	that	that	SCONJ
ejpam-4262	53	14	rsm	rsm	PROPN
ejpam-4262	53	15	∈	∈	PROPN
ejpam-4262	53	16	p	p	X
ejpam-4262	53	17	.	.	PUNCT
ejpam-4262	54	1	if	if	SCONJ
ejpam-4262	54	2	rsm	rsm	PROPN
ejpam-4262	54	3	/∈	/∈	PUNCT
ejpam-4262	54	4	ϕ(p	ϕ(p	PROPN
ejpam-4262	54	5	)	)	PUNCT
ejpam-4262	54	6	,	,	PUNCT
ejpam-4262	54	7	then	then	ADV
ejpam-4262	54	8	rs	rs	INTJ
ejpam-4262	54	9	+	+	CCONJ
ejpam-4262	54	10	rs	rs	PROPN
ejpam-4262	54	11	∈	∈	PROPN
ejpam-4262	54	12	(	(	PUNCT
ejpam-4262	54	13	p	p	X
ejpam-4262	54	14	:	:	PUNCT
ejpam-4262	54	15	m	m	NOUN
ejpam-4262	54	16	)	)	PUNCT
ejpam-4262	54	17	or	or	CCONJ
ejpam-4262	54	18	r(m+m	r(m+m	NOUN
ejpam-4262	54	19	)	)	PUNCT
ejpam-4262	54	20	∈	∈	PROPN
ejpam-4262	54	21	p	p	NOUN
ejpam-4262	54	22	or	or	CCONJ
ejpam-4262	54	23	s(m+m	s(m+m	SYM
ejpam-4262	54	24	)	)	PUNCT
ejpam-4262	54	25	∈	∈	PROPN
ejpam-4262	55	1	p	p	NOUN
ejpam-4262	55	2	.	.	PUNCT
ejpam-4262	56	1	next	next	ADV
ejpam-4262	56	2	,	,	PUNCT
ejpam-4262	56	3	assume	assume	VERB
ejpam-4262	56	4	that	that	SCONJ
ejpam-4262	56	5	rsm	rsm	PROPN
ejpam-4262	56	6	∈	∈	PROPN
ejpam-4262	56	7	ϕ(p	ϕ(p	PROPN
ejpam-4262	56	8	)	)	PUNCT
ejpam-4262	56	9	.	.	PUNCT
ejpam-4262	57	1	case	case	NOUN
ejpam-4262	57	2	1	1	X
ejpam-4262	57	3	.	.	PUNCT
ejpam-4262	57	4	rsp	rsp	PROPN
ejpam-4262	58	1	⊈	⊈	PROPN
ejpam-4262	58	2	ϕ(p	ϕ(p	PROPN
ejpam-4262	58	3	)	)	PUNCT
ejpam-4262	58	4	.	.	PUNCT
ejpam-4262	59	1	then	then	ADV
ejpam-4262	59	2	rsp0	rsp0	PROPN
ejpam-4262	59	3	/∈	/∈	PUNCT
ejpam-4262	60	1	ϕ(p	ϕ(p	PROPN
ejpam-4262	60	2	)	)	PUNCT
ejpam-4262	61	1	for	for	ADP
ejpam-4262	61	2	some	some	DET
ejpam-4262	61	3	p0	p0	NOUN
ejpam-4262	61	4	∈	∈	PROPN
ejpam-4262	61	5	p	p	NOUN
ejpam-4262	61	6	.	.	PUNCT
ejpam-4262	62	1	hence	hence	ADV
ejpam-4262	62	2	rs(m	rs(m	PROPN
ejpam-4262	62	3	+	+	CCONJ
ejpam-4262	62	4	p0	p0	NOUN
ejpam-4262	62	5	)	)	PUNCT
ejpam-4262	62	6	∈	∈	NOUN
ejpam-4262	62	7	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	62	8	)	)	PUNCT
ejpam-4262	62	9	.	.	PUNCT
ejpam-4262	63	1	since	since	SCONJ
ejpam-4262	63	2	p	p	NOUN
ejpam-4262	63	3	is	be	AUX
ejpam-4262	63	4	a	a	DET
ejpam-4262	63	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	63	6	-	-	ADJ
ejpam-4262	63	7	absorbing	absorbing	ADJ
ejpam-4262	63	8	submodule	submodule	NOUN
ejpam-4262	63	9	of	of	ADP
ejpam-4262	63	10	m	m	PRON
ejpam-4262	63	11	,	,	PUNCT
ejpam-4262	63	12	rs	rs	ADJ
ejpam-4262	63	13	+	+	CCONJ
ejpam-4262	63	14	rs	rs	PROPN
ejpam-4262	63	15	∈	∈	PROPN
ejpam-4262	63	16	(	(	PUNCT
ejpam-4262	63	17	p	p	X
ejpam-4262	63	18	:	:	PUNCT
ejpam-4262	63	19	m	m	NUM
ejpam-4262	63	20	)	)	PUNCT
ejpam-4262	63	21	or	or	CCONJ
ejpam-4262	63	22	r(m	r(m	ADJ
ejpam-4262	63	23	+	+	CCONJ
ejpam-4262	63	24	p0	p0	NOUN
ejpam-4262	63	25	+	+	CCONJ
ejpam-4262	63	26	m	m	VERB
ejpam-4262	63	27	+	+	X
ejpam-4262	63	28	p0	p0	NOUN
ejpam-4262	63	29	)	)	PUNCT
ejpam-4262	63	30	∈	∈	PROPN
ejpam-4262	63	31	p	p	NOUN
ejpam-4262	63	32	or	or	CCONJ
ejpam-4262	63	33	t.	t.	PROPN
ejpam-4262	63	34	khumprapussorn	khumprapussorn	PROPN
ejpam-4262	63	35	/	/	SYM
ejpam-4262	63	36	eur	eur	PROPN
ejpam-4262	63	37	.	.	PUNCT
ejpam-4262	64	1	j.	j.	PROPN
ejpam-4262	64	2	pure	pure	PROPN
ejpam-4262	64	3	appl	appl	PROPN
ejpam-4262	64	4	.	.	PROPN
ejpam-4262	64	5	math	math	PROPN
ejpam-4262	64	6	,	,	PUNCT
ejpam-4262	64	7	15	15	NUM
ejpam-4262	64	8	(	(	PUNCT
ejpam-4262	64	9	1	1	NUM
ejpam-4262	64	10	)	)	PUNCT
ejpam-4262	64	11	(	(	PUNCT
ejpam-4262	64	12	2022	2022	NUM
ejpam-4262	64	13	)	)	PUNCT
ejpam-4262	64	14	,	,	PUNCT
ejpam-4262	64	15	328	328	NUM
ejpam-4262	64	16	-	-	SYM
ejpam-4262	64	17	334	334	NUM
ejpam-4262	64	18	330	330	NUM
ejpam-4262	64	19	s(m	s(m	NOUN
ejpam-4262	64	20	+	+	CCONJ
ejpam-4262	64	21	p0	p0	NOUN
ejpam-4262	65	1	+	+	NOUN
ejpam-4262	65	2	m	m	VERB
ejpam-4262	65	3	+	+	ADJ
ejpam-4262	65	4	p0	p0	NOUN
ejpam-4262	65	5	)	)	PUNCT
ejpam-4262	65	6	∈	∈	PROPN
ejpam-4262	65	7	p	p	NOUN
ejpam-4262	65	8	.	.	PUNCT
ejpam-4262	66	1	since	since	SCONJ
ejpam-4262	66	2	p0	p0	PROPN
ejpam-4262	66	3	∈	∈	PROPN
ejpam-4262	66	4	p	p	NOUN
ejpam-4262	66	5	,	,	PUNCT
ejpam-4262	66	6	we	we	PRON
ejpam-4262	66	7	have	have	AUX
ejpam-4262	66	8	rs	r	VERB
ejpam-4262	66	9	+	+	CCONJ
ejpam-4262	66	10	rs	rs	PROPN
ejpam-4262	66	11	∈	∈	PROPN
ejpam-4262	66	12	(	(	PUNCT
ejpam-4262	66	13	p	p	X
ejpam-4262	66	14	:	:	PUNCT
ejpam-4262	66	15	m	m	NUM
ejpam-4262	66	16	)	)	PUNCT
ejpam-4262	66	17	or	or	CCONJ
ejpam-4262	66	18	r(m	r(m	ADJ
ejpam-4262	66	19	+	+	NOUN
ejpam-4262	66	20	m	m	NOUN
ejpam-4262	66	21	)	)	PUNCT
ejpam-4262	66	22	∈	∈	PROPN
ejpam-4262	66	23	p	p	NOUN
ejpam-4262	66	24	or	or	CCONJ
ejpam-4262	66	25	s(m+m	s(m+m	SYM
ejpam-4262	66	26	)	)	PUNCT
ejpam-4262	66	27	∈	∈	PROPN
ejpam-4262	66	28	p	p	NOUN
ejpam-4262	66	29	.	.	PUNCT
ejpam-4262	67	1	case	case	NOUN
ejpam-4262	67	2	2	2	NUM
ejpam-4262	67	3	.	.	PUNCT
ejpam-4262	67	4	rsp	rsp	PROPN
ejpam-4262	68	1	⊆	⊆	NUM
ejpam-4262	68	2	ϕ(p	ϕ(p	PROPN
ejpam-4262	68	3	)	)	PUNCT
ejpam-4262	68	4	.	.	PUNCT
ejpam-4262	69	1	subcase	subcase	VERB
ejpam-4262	69	2	2.1	2.1	NUM
ejpam-4262	69	3	s(p	s(p	PROPN
ejpam-4262	69	4	:	:	PUNCT
ejpam-4262	69	5	m)m	m)m	X
ejpam-4262	70	1	⊈	⊈	X
ejpam-4262	70	2	ϕ(p	ϕ(p	PROPN
ejpam-4262	70	3	)	)	PUNCT
ejpam-4262	70	4	.	.	PUNCT
ejpam-4262	71	1	there	there	PRON
ejpam-4262	71	2	exists	exist	VERB
ejpam-4262	71	3	an	an	DET
ejpam-4262	71	4	element	element	NOUN
ejpam-4262	71	5	a0	a0	NOUN
ejpam-4262	71	6	∈	∈	PROPN
ejpam-4262	71	7	(	(	PUNCT
ejpam-4262	71	8	p	p	X
ejpam-4262	71	9	:	:	PUNCT
ejpam-4262	71	10	m	m	X
ejpam-4262	71	11	)	)	PUNCT
ejpam-4262	71	12	such	such	ADJ
ejpam-4262	71	13	that	that	DET
ejpam-4262	71	14	sa0	sa0	NOUN
ejpam-4262	71	15	m	m	NOUN
ejpam-4262	71	16	/∈	/∈	PUNCT
ejpam-4262	71	17	ϕ(p	ϕ(p	PROPN
ejpam-4262	71	18	)	)	PUNCT
ejpam-4262	71	19	.	.	PUNCT
ejpam-4262	72	1	thus	thus	ADV
ejpam-4262	72	2	(	(	PUNCT
ejpam-4262	72	3	r	r	NOUN
ejpam-4262	72	4	+	+	CCONJ
ejpam-4262	72	5	a0)sm	a0)sm	ADV
ejpam-4262	72	6	=	=	SYM
ejpam-4262	72	7	rsm+a0sm	rsm+a0sm	NOUN
ejpam-4262	72	8	∈	∈	NOUN
ejpam-4262	72	9	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	72	10	)	)	PUNCT
ejpam-4262	72	11	.	.	PUNCT
ejpam-4262	73	1	since	since	SCONJ
ejpam-4262	73	2	p	p	NOUN
ejpam-4262	73	3	is	be	AUX
ejpam-4262	73	4	a	a	DET
ejpam-4262	73	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	73	6	-	-	ADJ
ejpam-4262	73	7	absorbing	absorbing	ADJ
ejpam-4262	73	8	submodule	submodule	NOUN
ejpam-4262	73	9	of	of	ADP
ejpam-4262	73	10	m	m	PRON
ejpam-4262	73	11	,	,	PUNCT
ejpam-4262	73	12	(	(	PUNCT
ejpam-4262	73	13	r+a0)s+(r+a0)s	r+a0)s+(r+a0)s	PROPN
ejpam-4262	73	14	∈	∈	PROPN
ejpam-4262	73	15	(	(	PUNCT
ejpam-4262	73	16	p	p	X
ejpam-4262	73	17	:	:	PUNCT
ejpam-4262	73	18	m	m	X
ejpam-4262	73	19	)	)	PUNCT
ejpam-4262	73	20	or	or	CCONJ
ejpam-4262	73	21	(	(	PUNCT
ejpam-4262	73	22	r	r	NOUN
ejpam-4262	73	23	+	+	CCONJ
ejpam-4262	73	24	a0)(m+m	a0)(m+m	NOUN
ejpam-4262	73	25	)	)	PUNCT
ejpam-4262	73	26	∈	∈	PROPN
ejpam-4262	73	27	p	p	NOUN
ejpam-4262	73	28	or	or	CCONJ
ejpam-4262	73	29	s(m+m	s(m+m	SYM
ejpam-4262	73	30	)	)	PUNCT
ejpam-4262	73	31	∈	∈	PROPN
ejpam-4262	73	32	p	p	NOUN
ejpam-4262	73	33	.	.	PUNCT
ejpam-4262	74	1	then	then	ADV
ejpam-4262	74	2	rs+	rs+	VERB
ejpam-4262	74	3	a0s+	a0s+	PROPN
ejpam-4262	74	4	rs+	rs+	PROPN
ejpam-4262	74	5	a0s	a0s	PROPN
ejpam-4262	74	6	∈	∈	PROPN
ejpam-4262	74	7	(	(	PUNCT
ejpam-4262	74	8	p	p	X
ejpam-4262	74	9	:	:	PUNCT
ejpam-4262	74	10	m	m	NUM
ejpam-4262	74	11	)	)	PUNCT
ejpam-4262	74	12	or	or	CCONJ
ejpam-4262	74	13	r(m	r(m	ADJ
ejpam-4262	74	14	+	+	NOUN
ejpam-4262	74	15	m	m	X
ejpam-4262	74	16	)	)	PUNCT
ejpam-4262	75	1	+	+	NUM
ejpam-4262	75	2	a0(m	a0(m	X
ejpam-4262	76	1	+	+	NOUN
ejpam-4262	76	2	m	m	NOUN
ejpam-4262	76	3	)	)	PUNCT
ejpam-4262	76	4	∈	∈	PROPN
ejpam-4262	76	5	p	p	NOUN
ejpam-4262	76	6	or	or	CCONJ
ejpam-4262	76	7	s(m	s(m	NOUN
ejpam-4262	76	8	+	+	PROPN
ejpam-4262	76	9	m	m	NOUN
ejpam-4262	76	10	)	)	PUNCT
ejpam-4262	76	11	∈	∈	PROPN
ejpam-4262	76	12	p	p	NOUN
ejpam-4262	76	13	.	.	PUNCT
ejpam-4262	77	1	since	since	SCONJ
ejpam-4262	77	2	a0	a0	PROPN
ejpam-4262	77	3	∈	∈	PROPN
ejpam-4262	77	4	(	(	PUNCT
ejpam-4262	77	5	p	p	X
ejpam-4262	77	6	:	:	PUNCT
ejpam-4262	77	7	m	m	PROPN
ejpam-4262	77	8	)	)	PUNCT
ejpam-4262	77	9	,	,	PUNCT
ejpam-4262	77	10	a0	a0	PROPN
ejpam-4262	77	11	m	m	PROPN
ejpam-4262	77	12	⊆	⊆	NUM
ejpam-4262	77	13	p	p	NOUN
ejpam-4262	77	14	.	.	PUNCT
ejpam-4262	78	1	this	this	PRON
ejpam-4262	78	2	implies	imply	VERB
ejpam-4262	78	3	that	that	SCONJ
ejpam-4262	78	4	a0s	a0s	PROPN
ejpam-4262	78	5	+	+	CCONJ
ejpam-4262	78	6	a0s	a0s	PROPN
ejpam-4262	78	7	∈	∈	PROPN
ejpam-4262	78	8	(	(	PUNCT
ejpam-4262	78	9	p	p	X
ejpam-4262	78	10	:	:	PUNCT
ejpam-4262	78	11	m	m	NUM
ejpam-4262	78	12	)	)	PUNCT
ejpam-4262	78	13	and	and	CCONJ
ejpam-4262	78	14	a0(m	a0(m	X
ejpam-4262	78	15	+	+	NOUN
ejpam-4262	78	16	m	m	VERB
ejpam-4262	78	17	)	)	PUNCT
ejpam-4262	78	18	∈	∈	PROPN
ejpam-4262	78	19	p	p	NOUN
ejpam-4262	78	20	.	.	PUNCT
ejpam-4262	79	1	therefore	therefore	ADV
ejpam-4262	79	2	rs	rs	INTJ
ejpam-4262	80	1	+	+	CCONJ
ejpam-4262	80	2	rs	rs	PROPN
ejpam-4262	80	3	∈	∈	PROPN
ejpam-4262	80	4	(	(	PUNCT
ejpam-4262	80	5	p	p	X
ejpam-4262	80	6	:	:	PUNCT
ejpam-4262	80	7	m	m	NOUN
ejpam-4262	80	8	)	)	PUNCT
ejpam-4262	80	9	or	or	CCONJ
ejpam-4262	80	10	r(m+m	r(m+m	NOUN
ejpam-4262	80	11	)	)	PUNCT
ejpam-4262	80	12	∈	∈	PROPN
ejpam-4262	80	13	p	p	NOUN
ejpam-4262	80	14	or	or	CCONJ
ejpam-4262	80	15	s(m+m	s(m+m	SYM
ejpam-4262	80	16	)	)	PUNCT
ejpam-4262	80	17	∈	∈	PROPN
ejpam-4262	80	18	p	p	NOUN
ejpam-4262	80	19	.	.	PUNCT
ejpam-4262	81	1	subcase	subcase	PROPN
ejpam-4262	81	2	2.2	2.2	NUM
ejpam-4262	81	3	s(p	s(p	NOUN
ejpam-4262	81	4	:	:	PUNCT
ejpam-4262	81	5	m)m	m)m	X
ejpam-4262	81	6	⊆	⊆	NUM
ejpam-4262	81	7	ϕ(p	ϕ(p	PROPN
ejpam-4262	81	8	)	)	PUNCT
ejpam-4262	81	9	.	.	PUNCT
ejpam-4262	82	1	since	since	SCONJ
ejpam-4262	82	2	(	(	PUNCT
ejpam-4262	82	3	p	p	X
ejpam-4262	82	4	:	:	PUNCT
ejpam-4262	82	5	m)2β(p	m)2β(p	NOUN
ejpam-4262	82	6	)	)	PUNCT
ejpam-4262	82	7	⊈	⊈	PROPN
ejpam-4262	82	8	ϕ(p	ϕ(p	PROPN
ejpam-4262	82	9	)	)	PUNCT
ejpam-4262	82	10	,	,	PUNCT
ejpam-4262	82	11	we	we	PRON
ejpam-4262	82	12	have	have	VERB
ejpam-4262	82	13	that	that	DET
ejpam-4262	82	14	kt(n+n	kt(n+n	NOUN
ejpam-4262	82	15	)	)	PUNCT
ejpam-4262	82	16	/∈	/∈	PUNCT
ejpam-4262	83	1	ϕ(p	ϕ(p	PROPN
ejpam-4262	83	2	)	)	PUNCT
ejpam-4262	84	1	for	for	ADP
ejpam-4262	84	2	some	some	DET
ejpam-4262	84	3	k	k	PROPN
ejpam-4262	84	4	,	,	PUNCT
ejpam-4262	84	5	t	t	PROPN
ejpam-4262	84	6	∈	∈	PROPN
ejpam-4262	84	7	(	(	PUNCT
ejpam-4262	84	8	p	p	X
ejpam-4262	84	9	:	:	PUNCT
ejpam-4262	84	10	m	m	NUM
ejpam-4262	84	11	)	)	PUNCT
ejpam-4262	84	12	and	and	CCONJ
ejpam-4262	84	13	n	n	DET
ejpam-4262	84	14	∈	∈	PROPN
ejpam-4262	84	15	p	p	NOUN
ejpam-4262	84	16	.	.	PUNCT
ejpam-4262	85	1	if	if	SCONJ
ejpam-4262	85	2	rkm	rkm	PROPN
ejpam-4262	85	3	/∈	/∈	PUNCT
ejpam-4262	85	4	ϕ(p	ϕ(p	PROPN
ejpam-4262	85	5	)	)	PUNCT
ejpam-4262	85	6	,	,	PUNCT
ejpam-4262	85	7	then	then	ADV
ejpam-4262	85	8	r(k	r(k	PROPN
ejpam-4262	85	9	+	+	CCONJ
ejpam-4262	85	10	s)m	s)m	ADJ
ejpam-4262	85	11	=	=	PUNCT
ejpam-4262	85	12	rkm+	rkm+	NOUN
ejpam-4262	85	13	rsm	rsm	PROPN
ejpam-4262	85	14	/∈	/∈	PUNCT
ejpam-4262	85	15	ϕ(p	ϕ(p	PROPN
ejpam-4262	85	16	)	)	PUNCT
ejpam-4262	85	17	.	.	PUNCT
ejpam-4262	86	1	since	since	SCONJ
ejpam-4262	86	2	p	p	NOUN
ejpam-4262	86	3	is	be	AUX
ejpam-4262	86	4	a	a	DET
ejpam-4262	86	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	86	6	-	-	ADJ
ejpam-4262	86	7	absorbing	absorbing	ADJ
ejpam-4262	86	8	submodule	submodule	NOUN
ejpam-4262	86	9	of	of	ADP
ejpam-4262	86	10	m	m	PRON
ejpam-4262	86	11	,	,	PUNCT
ejpam-4262	86	12	r(k	r(k	PROPN
ejpam-4262	86	13	+	+	PROPN
ejpam-4262	86	14	s	s	X
ejpam-4262	86	15	)	)	PUNCT
ejpam-4262	87	1	+	+	CCONJ
ejpam-4262	87	2	r(k	r(k	PROPN
ejpam-4262	87	3	+	+	CCONJ
ejpam-4262	87	4	s	s	X
ejpam-4262	87	5	)	)	PUNCT
ejpam-4262	87	6	∈	∈	PROPN
ejpam-4262	87	7	(	(	PUNCT
ejpam-4262	87	8	p	p	X
ejpam-4262	87	9	:	:	PUNCT
ejpam-4262	87	10	m	m	NOUN
ejpam-4262	87	11	)	)	PUNCT
ejpam-4262	87	12	or	or	CCONJ
ejpam-4262	87	13	r(m+m	r(m+m	NOUN
ejpam-4262	87	14	)	)	PUNCT
ejpam-4262	87	15	∈	∈	PROPN
ejpam-4262	87	16	p	p	NOUN
ejpam-4262	87	17	or	or	CCONJ
ejpam-4262	87	18	(	(	PUNCT
ejpam-4262	87	19	k	k	PROPN
ejpam-4262	87	20	+	+	CCONJ
ejpam-4262	87	21	s)(m+m	s)(m+m	NOUN
ejpam-4262	87	22	)	)	PUNCT
ejpam-4262	87	23	∈	∈	PROPN
ejpam-4262	87	24	p	p	NOUN
ejpam-4262	87	25	.	.	PUNCT
ejpam-4262	88	1	since	since	SCONJ
ejpam-4262	88	2	k	k	PROPN
ejpam-4262	88	3	∈	∈	PROPN
ejpam-4262	88	4	(	(	PUNCT
ejpam-4262	88	5	p	p	X
ejpam-4262	88	6	:	:	PUNCT
ejpam-4262	88	7	m	m	PROPN
ejpam-4262	88	8	)	)	PUNCT
ejpam-4262	88	9	,	,	PUNCT
ejpam-4262	88	10	rs	rs	X
ejpam-4262	88	11	+	+	CCONJ
ejpam-4262	88	12	rs	rs	PROPN
ejpam-4262	88	13	∈	∈	PROPN
ejpam-4262	88	14	(	(	PUNCT
ejpam-4262	88	15	p	p	X
ejpam-4262	88	16	:	:	PUNCT
ejpam-4262	88	17	m	m	NUM
ejpam-4262	88	18	)	)	PUNCT
ejpam-4262	88	19	or	or	CCONJ
ejpam-4262	88	20	r(m	r(m	ADJ
ejpam-4262	88	21	+	+	NOUN
ejpam-4262	88	22	m	m	NOUN
ejpam-4262	88	23	)	)	PUNCT
ejpam-4262	88	24	∈	∈	PROPN
ejpam-4262	88	25	p	p	NOUN
ejpam-4262	88	26	or	or	CCONJ
ejpam-4262	88	27	s(m	s(m	NOUN
ejpam-4262	88	28	+	+	PROPN
ejpam-4262	88	29	m	m	NOUN
ejpam-4262	88	30	)	)	PUNCT
ejpam-4262	88	31	∈	∈	PROPN
ejpam-4262	88	32	p	p	NOUN
ejpam-4262	88	33	.	.	PUNCT
ejpam-4262	89	1	similarly	similarly	ADV
ejpam-4262	89	2	,	,	PUNCT
ejpam-4262	89	3	if	if	SCONJ
ejpam-4262	89	4	rtm	rtm	NOUN
ejpam-4262	89	5	/∈	/∈	PUNCT
ejpam-4262	89	6	ϕ(p	ϕ(p	PROPN
ejpam-4262	89	7	)	)	PUNCT
ejpam-4262	89	8	,	,	PUNCT
ejpam-4262	89	9	then	then	ADV
ejpam-4262	89	10	r(t+	r(t+	ADJ
ejpam-4262	89	11	s)m	s)m	ADJ
ejpam-4262	89	12	=	=	PUNCT
ejpam-4262	89	13	rtm+	rtm+	PROPN
ejpam-4262	89	14	rsm	rsm	PROPN
ejpam-4262	89	15	/∈	/∈	PUNCT
ejpam-4262	89	16	ϕ(p	ϕ(p	PROPN
ejpam-4262	89	17	)	)	PUNCT
ejpam-4262	89	18	.	.	PUNCT
ejpam-4262	90	1	since	since	SCONJ
ejpam-4262	90	2	p	p	NOUN
ejpam-4262	90	3	is	be	AUX
ejpam-4262	90	4	a	a	DET
ejpam-4262	90	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	90	6	-	-	ADJ
ejpam-4262	90	7	absorbing	absorbing	ADJ
ejpam-4262	90	8	submodule	submodule	NOUN
ejpam-4262	90	9	of	of	ADP
ejpam-4262	90	10	m	m	PROPN
ejpam-4262	90	11	,	,	PUNCT
ejpam-4262	90	12	r(t	r(t	NOUN
ejpam-4262	90	13	+	+	CCONJ
ejpam-4262	90	14	s	s	X
ejpam-4262	90	15	)	)	PUNCT
ejpam-4262	91	1	+	+	NUM
ejpam-4262	91	2	r(t	r(t	NOUN
ejpam-4262	91	3	+	+	CCONJ
ejpam-4262	91	4	s	s	X
ejpam-4262	91	5	)	)	PUNCT
ejpam-4262	91	6	∈	∈	PROPN
ejpam-4262	91	7	(	(	PUNCT
ejpam-4262	91	8	p	p	X
ejpam-4262	91	9	:	:	PUNCT
ejpam-4262	91	10	m	m	NUM
ejpam-4262	91	11	)	)	PUNCT
ejpam-4262	91	12	or	or	CCONJ
ejpam-4262	91	13	r(m	r(m	PROPN
ejpam-4262	91	14	+	+	NUM
ejpam-4262	91	15	m	m	NOUN
ejpam-4262	91	16	)	)	PUNCT
ejpam-4262	91	17	∈	∈	PROPN
ejpam-4262	91	18	p	p	NOUN
ejpam-4262	91	19	or	or	CCONJ
ejpam-4262	91	20	(	(	PUNCT
ejpam-4262	91	21	t	t	PROPN
ejpam-4262	91	22	+	+	CCONJ
ejpam-4262	91	23	s)(m	s)(m	VERB
ejpam-4262	91	24	+	+	CCONJ
ejpam-4262	91	25	m	m	X
ejpam-4262	91	26	)	)	PUNCT
ejpam-4262	91	27	∈	∈	PROPN
ejpam-4262	91	28	p	p	NOUN
ejpam-4262	91	29	.	.	PUNCT
ejpam-4262	92	1	since	since	SCONJ
ejpam-4262	92	2	t	t	PROPN
ejpam-4262	92	3	∈	∈	PROPN
ejpam-4262	92	4	(	(	PUNCT
ejpam-4262	92	5	p	p	X
ejpam-4262	92	6	:	:	PUNCT
ejpam-4262	92	7	m	m	PROPN
ejpam-4262	92	8	)	)	PUNCT
ejpam-4262	92	9	,	,	PUNCT
ejpam-4262	92	10	rs	rs	X
ejpam-4262	92	11	+	+	CCONJ
ejpam-4262	92	12	rs	rs	PROPN
ejpam-4262	92	13	∈	∈	PROPN
ejpam-4262	92	14	(	(	PUNCT
ejpam-4262	92	15	p	p	X
ejpam-4262	92	16	:	:	PUNCT
ejpam-4262	92	17	m	m	NUM
ejpam-4262	92	18	)	)	PUNCT
ejpam-4262	92	19	or	or	CCONJ
ejpam-4262	92	20	r(m	r(m	PROPN
ejpam-4262	92	21	+	+	NUM
ejpam-4262	92	22	m	m	NOUN
ejpam-4262	92	23	)	)	PUNCT
ejpam-4262	92	24	∈	∈	PROPN
ejpam-4262	92	25	p	p	NOUN
ejpam-4262	92	26	or	or	CCONJ
ejpam-4262	92	27	s(m	s(m	NOUN
ejpam-4262	92	28	+	+	CCONJ
ejpam-4262	92	29	m	m	NOUN
ejpam-4262	92	30	)	)	PUNCT
ejpam-4262	92	31	∈	∈	PROPN
ejpam-4262	92	32	p	p	NOUN
ejpam-4262	92	33	.	.	PUNCT
ejpam-4262	93	1	from	from	ADP
ejpam-4262	93	2	now	now	ADV
ejpam-4262	93	3	on	on	ADV
ejpam-4262	93	4	,	,	PUNCT
ejpam-4262	93	5	we	we	PRON
ejpam-4262	93	6	assume	assume	VERB
ejpam-4262	93	7	that	that	SCONJ
ejpam-4262	93	8	rkm	rkm	PROPN
ejpam-4262	93	9	∈	∈	PROPN
ejpam-4262	93	10	ϕ(p	ϕ(p	PROPN
ejpam-4262	93	11	)	)	PUNCT
ejpam-4262	93	12	and	and	CCONJ
ejpam-4262	93	13	rtm	rtm	NOUN
ejpam-4262	93	14	∈	∈	PROPN
ejpam-4262	93	15	ϕ(p	ϕ(p	PROPN
ejpam-4262	93	16	)	)	PUNCT
ejpam-4262	93	17	.	.	PUNCT
ejpam-4262	94	1	(	(	PUNCT
ejpam-4262	94	2	1	1	X
ejpam-4262	94	3	)	)	PUNCT
ejpam-4262	94	4	if	if	SCONJ
ejpam-4262	94	5	ktm	ktm	PROPN
ejpam-4262	94	6	/∈	/∈	PUNCT
ejpam-4262	94	7	ϕ(p	ϕ(p	PROPN
ejpam-4262	94	8	)	)	PUNCT
ejpam-4262	94	9	,	,	PUNCT
ejpam-4262	94	10	then	then	ADV
ejpam-4262	94	11	(	(	PUNCT
ejpam-4262	94	12	k+	k+	X
ejpam-4262	94	13	r)(t+	r)(t+	NOUN
ejpam-4262	94	14	s)m	s)m	ADJ
ejpam-4262	94	15	/∈	/∈	PUNCT
ejpam-4262	94	16	ϕ(p	ϕ(p	PROPN
ejpam-4262	94	17	)	)	PUNCT
ejpam-4262	94	18	.	.	PUNCT
ejpam-4262	95	1	since	since	SCONJ
ejpam-4262	95	2	p	p	NOUN
ejpam-4262	95	3	is	be	AUX
ejpam-4262	95	4	a	a	DET
ejpam-4262	95	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	95	6	-	-	ADJ
ejpam-4262	95	7	absorbing	absorbing	ADJ
ejpam-4262	95	8	submodule	submodule	NOUN
ejpam-4262	95	9	of	of	ADP
ejpam-4262	95	10	m	m	PRON
ejpam-4262	95	11	,	,	PUNCT
ejpam-4262	95	12	(	(	PUNCT
ejpam-4262	95	13	k	k	X
ejpam-4262	95	14	+	+	X
ejpam-4262	95	15	r)(t	r)(t	X
ejpam-4262	95	16	+	+	X
ejpam-4262	95	17	s	s	X
ejpam-4262	95	18	)	)	PUNCT
ejpam-4262	96	1	+	+	CCONJ
ejpam-4262	96	2	(	(	PUNCT
ejpam-4262	96	3	k	k	X
ejpam-4262	96	4	+	+	X
ejpam-4262	96	5	r)(t	r)(t	ADJ
ejpam-4262	96	6	+	+	CCONJ
ejpam-4262	96	7	s	s	X
ejpam-4262	96	8	)	)	PUNCT
ejpam-4262	96	9	∈	∈	PROPN
ejpam-4262	96	10	(	(	PUNCT
ejpam-4262	96	11	p	p	X
ejpam-4262	96	12	:	:	PUNCT
ejpam-4262	96	13	m	m	X
ejpam-4262	96	14	)	)	PUNCT
ejpam-4262	96	15	or	or	CCONJ
ejpam-4262	96	16	(	(	PUNCT
ejpam-4262	96	17	k	k	PROPN
ejpam-4262	96	18	+	+	NOUN
ejpam-4262	96	19	r)(m	r)(m	VERB
ejpam-4262	96	20	+	+	PROPN
ejpam-4262	96	21	m	m	NOUN
ejpam-4262	96	22	)	)	PUNCT
ejpam-4262	96	23	∈	∈	PROPN
ejpam-4262	96	24	p	p	NOUN
ejpam-4262	96	25	or	or	CCONJ
ejpam-4262	96	26	(	(	PUNCT
ejpam-4262	96	27	t	t	PROPN
ejpam-4262	96	28	+	+	CCONJ
ejpam-4262	96	29	s)(m	s)(m	VERB
ejpam-4262	96	30	+	+	PROPN
ejpam-4262	96	31	m	m	NOUN
ejpam-4262	96	32	)	)	PUNCT
ejpam-4262	96	33	∈	∈	PROPN
ejpam-4262	96	34	p	p	NOUN
ejpam-4262	96	35	.	.	PUNCT
ejpam-4262	97	1	since	since	SCONJ
ejpam-4262	97	2	k	k	PROPN
ejpam-4262	97	3	,	,	PUNCT
ejpam-4262	97	4	t	t	PROPN
ejpam-4262	97	5	∈	∈	PROPN
ejpam-4262	97	6	(	(	PUNCT
ejpam-4262	97	7	p	p	X
ejpam-4262	97	8	:	:	PUNCT
ejpam-4262	97	9	m	m	PROPN
ejpam-4262	97	10	)	)	PUNCT
ejpam-4262	97	11	,	,	PUNCT
ejpam-4262	97	12	rs	rs	X
ejpam-4262	97	13	+	+	CCONJ
ejpam-4262	97	14	rs	rs	PROPN
ejpam-4262	97	15	∈	∈	PROPN
ejpam-4262	97	16	(	(	PUNCT
ejpam-4262	97	17	p	p	X
ejpam-4262	97	18	:	:	PUNCT
ejpam-4262	97	19	m	m	NUM
ejpam-4262	97	20	)	)	PUNCT
ejpam-4262	97	21	or	or	CCONJ
ejpam-4262	97	22	r(m	r(m	PROPN
ejpam-4262	97	23	+	+	NUM
ejpam-4262	97	24	m	m	NOUN
ejpam-4262	97	25	)	)	PUNCT
ejpam-4262	97	26	∈	∈	PROPN
ejpam-4262	97	27	p	p	NOUN
ejpam-4262	97	28	or	or	CCONJ
ejpam-4262	97	29	s(m	s(m	NOUN
ejpam-4262	97	30	+	+	CCONJ
ejpam-4262	97	31	m	m	NOUN
ejpam-4262	97	32	)	)	PUNCT
ejpam-4262	97	33	∈	∈	PROPN
ejpam-4262	97	34	p	p	NOUN
ejpam-4262	97	35	.	.	PUNCT
ejpam-4262	98	1	now	now	ADV
ejpam-4262	98	2	,	,	PUNCT
ejpam-4262	98	3	we	we	PRON
ejpam-4262	98	4	assume	assume	VERB
ejpam-4262	98	5	that	that	SCONJ
ejpam-4262	98	6	ktm	ktm	PROPN
ejpam-4262	98	7	∈	∈	PROPN
ejpam-4262	98	8	ϕ(p	ϕ(p	PROPN
ejpam-4262	98	9	)	)	PUNCT
ejpam-4262	98	10	.	.	PUNCT
ejpam-4262	99	1	(	(	PUNCT
ejpam-4262	99	2	2	2	X
ejpam-4262	99	3	)	)	PUNCT
ejpam-4262	99	4	subsubcase	subsubcase	VERB
ejpam-4262	99	5	2.2.1	2.2.1	NUM
ejpam-4262	99	6	kr(n+	kr(n+	NOUN
ejpam-4262	99	7	n	n	CCONJ
ejpam-4262	99	8	)	)	PUNCT
ejpam-4262	99	9	/∈	/∈	PUNCT
ejpam-4262	99	10	ϕ(p	ϕ(p	PROPN
ejpam-4262	99	11	)	)	PUNCT
ejpam-4262	99	12	or	or	CCONJ
ejpam-4262	99	13	st(n+	st(n+	PROPN
ejpam-4262	99	14	n	n	CCONJ
ejpam-4262	99	15	)	)	PUNCT
ejpam-4262	99	16	/∈	/∈	PUNCT
ejpam-4262	99	17	ϕ(p	ϕ(p	PROPN
ejpam-4262	99	18	)	)	PUNCT
ejpam-4262	99	19	.	.	PUNCT
ejpam-4262	100	1	suppose	suppose	VERB
ejpam-4262	100	2	that	that	SCONJ
ejpam-4262	100	3	kr(n+	kr(n+	PROPN
ejpam-4262	100	4	n	n	CCONJ
ejpam-4262	100	5	)	)	PUNCT
ejpam-4262	100	6	/∈	/∈	PUNCT
ejpam-4262	100	7	ϕ(p	ϕ(p	PROPN
ejpam-4262	100	8	)	)	PUNCT
ejpam-4262	100	9	.	.	PUNCT
ejpam-4262	101	1	then	then	ADV
ejpam-4262	101	2	r(s+	r(s+	VERB
ejpam-4262	101	3	k)(n+	k)(n+	PROPN
ejpam-4262	101	4	n+m	n+m	NUM
ejpam-4262	101	5	)	)	PUNCT
ejpam-4262	101	6	/∈	/∈	PUNCT
ejpam-4262	101	7	ϕ(p	ϕ(p	PROPN
ejpam-4262	101	8	)	)	PUNCT
ejpam-4262	101	9	.	.	PUNCT
ejpam-4262	102	1	since	since	SCONJ
ejpam-4262	102	2	p	p	NOUN
ejpam-4262	102	3	is	be	AUX
ejpam-4262	102	4	a	a	DET
ejpam-4262	102	5	ϕ-βabsorbing	ϕ-βabsorbe	VERB
ejpam-4262	102	6	submodule	submodule	NOUN
ejpam-4262	102	7	of	of	ADP
ejpam-4262	102	8	m	m	PROPN
ejpam-4262	102	9	,	,	PUNCT
ejpam-4262	102	10	r(s+k)+	r(s+k)+	X
ejpam-4262	102	11	r(s+k	r(s+k	PROPN
ejpam-4262	102	12	)	)	PUNCT
ejpam-4262	102	13	∈	∈	PROPN
ejpam-4262	102	14	(	(	PUNCT
ejpam-4262	102	15	p	p	X
ejpam-4262	102	16	:	:	PUNCT
ejpam-4262	102	17	m	m	NUM
ejpam-4262	102	18	)	)	PUNCT
ejpam-4262	102	19	or	or	CCONJ
ejpam-4262	102	20	r(n+n+m+n+n+m	r(n+n+m+n+n+m	X
ejpam-4262	102	21	)	)	PUNCT
ejpam-4262	102	22	∈	∈	PROPN
ejpam-4262	102	23	p	p	NOUN
ejpam-4262	102	24	or	or	CCONJ
ejpam-4262	102	25	(	(	PUNCT
ejpam-4262	102	26	s+k)(n+n+m+n+n+m	s+k)(n+n+m+n+n+m	ADJ
ejpam-4262	102	27	)	)	PUNCT
ejpam-4262	102	28	∈	∈	PROPN
ejpam-4262	102	29	p	p	NOUN
ejpam-4262	102	30	.	.	PUNCT
ejpam-4262	103	1	this	this	PRON
ejpam-4262	103	2	implies	imply	VERB
ejpam-4262	103	3	that	that	SCONJ
ejpam-4262	103	4	rs+rs	rs+rs	ADJ
ejpam-4262	103	5	∈	∈	NOUN
ejpam-4262	103	6	(	(	PUNCT
ejpam-4262	103	7	p	p	X
ejpam-4262	103	8	:	:	PUNCT
ejpam-4262	103	9	m	m	NOUN
ejpam-4262	103	10	)	)	PUNCT
ejpam-4262	103	11	or	or	CCONJ
ejpam-4262	103	12	r(m+m	r(m+m	NOUN
ejpam-4262	103	13	)	)	PUNCT
ejpam-4262	103	14	∈	∈	PROPN
ejpam-4262	103	15	p	p	NOUN
ejpam-4262	103	16	or	or	CCONJ
ejpam-4262	103	17	s(m+m	s(m+m	SYM
ejpam-4262	103	18	)	)	PUNCT
ejpam-4262	103	19	∈	∈	PROPN
ejpam-4262	103	20	p	p	NOUN
ejpam-4262	103	21	.	.	PUNCT
ejpam-4262	104	1	next	next	ADV
ejpam-4262	104	2	,	,	PUNCT
ejpam-4262	104	3	suppose	suppose	VERB
ejpam-4262	104	4	that	that	SCONJ
ejpam-4262	104	5	st(n+	st(n+	PROPN
ejpam-4262	104	6	n	n	CCONJ
ejpam-4262	104	7	)	)	PUNCT
ejpam-4262	104	8	/∈	/∈	PUNCT
ejpam-4262	104	9	ϕ(p	ϕ(p	PROPN
ejpam-4262	104	10	)	)	PUNCT
ejpam-4262	104	11	.	.	PUNCT
ejpam-4262	105	1	then	then	ADV
ejpam-4262	105	2	s(r	s(r	PROPN
ejpam-4262	105	3	+	+	CCONJ
ejpam-4262	105	4	t)(n+	t)(n+	NUM
ejpam-4262	105	5	n+m	n+m	NUM
ejpam-4262	105	6	)	)	PUNCT
ejpam-4262	105	7	/∈	/∈	PUNCT
ejpam-4262	105	8	ϕ(p	ϕ(p	PROPN
ejpam-4262	105	9	)	)	PUNCT
ejpam-4262	105	10	.	.	PUNCT
ejpam-4262	106	1	since	since	SCONJ
ejpam-4262	106	2	p	p	NOUN
ejpam-4262	106	3	is	be	AUX
ejpam-4262	106	4	a	a	DET
ejpam-4262	106	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	106	6	-	-	ADJ
ejpam-4262	106	7	absorbing	absorbing	ADJ
ejpam-4262	106	8	submodule	submodule	NOUN
ejpam-4262	106	9	of	of	ADP
ejpam-4262	106	10	m	m	PROPN
ejpam-4262	106	11	,	,	PUNCT
ejpam-4262	106	12	s(r+t)+s(r+t	s(r+t)+s(r+t	PROPN
ejpam-4262	106	13	)	)	PUNCT
ejpam-4262	106	14	∈	∈	PROPN
ejpam-4262	106	15	(	(	PUNCT
ejpam-4262	106	16	p	p	X
ejpam-4262	106	17	:	:	PUNCT
ejpam-4262	106	18	m	m	NOUN
ejpam-4262	106	19	)	)	PUNCT
ejpam-4262	106	20	or	or	CCONJ
ejpam-4262	106	21	s(n+n+m+n+n+m	s(n+n+m+n+n+m	PUNCT
ejpam-4262	106	22	)	)	PUNCT
ejpam-4262	106	23	∈	∈	PROPN
ejpam-4262	106	24	p	p	NOUN
ejpam-4262	106	25	or	or	CCONJ
ejpam-4262	106	26	(	(	PUNCT
ejpam-4262	106	27	r+t)(n+n+m+n+n+m	r+t)(n+n+m+n+n+m	NOUN
ejpam-4262	106	28	)	)	PUNCT
ejpam-4262	106	29	∈	∈	PROPN
ejpam-4262	106	30	p	p	NOUN
ejpam-4262	106	31	.	.	PUNCT
ejpam-4262	107	1	this	this	PRON
ejpam-4262	107	2	implies	imply	VERB
ejpam-4262	107	3	that	that	SCONJ
ejpam-4262	107	4	rs+rs	rs+rs	ADJ
ejpam-4262	107	5	∈	∈	NOUN
ejpam-4262	107	6	(	(	PUNCT
ejpam-4262	107	7	p	p	X
ejpam-4262	107	8	:	:	PUNCT
ejpam-4262	107	9	m	m	NOUN
ejpam-4262	107	10	)	)	PUNCT
ejpam-4262	107	11	or	or	CCONJ
ejpam-4262	107	12	r(m+m	r(m+m	NOUN
ejpam-4262	107	13	)	)	PUNCT
ejpam-4262	107	14	∈	∈	PROPN
ejpam-4262	107	15	p	p	NOUN
ejpam-4262	107	16	or	or	CCONJ
ejpam-4262	107	17	s(m+m	s(m+m	SYM
ejpam-4262	107	18	)	)	PUNCT
ejpam-4262	107	19	∈	∈	PROPN
ejpam-4262	107	20	p	p	NOUN
ejpam-4262	107	21	.	.	PUNCT
ejpam-4262	108	1	subsubcase	subsubcase	PROPN
ejpam-4262	108	2	2.2.2	2.2.2	NUM
ejpam-4262	108	3	kr(n+	kr(n+	PROPN
ejpam-4262	108	4	n	n	CCONJ
ejpam-4262	108	5	)	)	PUNCT
ejpam-4262	108	6	∈	∈	PROPN
ejpam-4262	108	7	ϕ(p	ϕ(p	PROPN
ejpam-4262	108	8	)	)	PUNCT
ejpam-4262	108	9	and	and	CCONJ
ejpam-4262	108	10	st(n+	st(n+	PROPN
ejpam-4262	108	11	n	n	CCONJ
ejpam-4262	108	12	)	)	PUNCT
ejpam-4262	108	13	∈	∈	PROPN
ejpam-4262	108	14	ϕ(p	ϕ(p	PROPN
ejpam-4262	108	15	)	)	PUNCT
ejpam-4262	108	16	.	.	PUNCT
ejpam-4262	109	1	then	then	ADV
ejpam-4262	109	2	(	(	PUNCT
ejpam-4262	109	3	s	s	AUX
ejpam-4262	109	4	+	+	NOUN
ejpam-4262	109	5	k)(r	k)(r	NOUN
ejpam-4262	109	6	+	+	X
ejpam-4262	109	7	t)(n	t)(n	X
ejpam-4262	109	8	+	+	CCONJ
ejpam-4262	109	9	n	n	PROPN
ejpam-4262	109	10	+	+	NUM
ejpam-4262	109	11	m	m	NOUN
ejpam-4262	109	12	)	)	PUNCT
ejpam-4262	109	13	/∈	/∈	PUNCT
ejpam-4262	109	14	ϕ(p	ϕ(p	PROPN
ejpam-4262	109	15	)	)	PUNCT
ejpam-4262	109	16	.	.	PUNCT
ejpam-4262	110	1	since	since	SCONJ
ejpam-4262	110	2	p	p	NOUN
ejpam-4262	110	3	is	be	AUX
ejpam-4262	110	4	a	a	DET
ejpam-4262	110	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	110	6	-	-	ADJ
ejpam-4262	110	7	absorbing	absorbing	ADJ
ejpam-4262	110	8	submodule	submodule	NOUN
ejpam-4262	110	9	of	of	ADP
ejpam-4262	110	10	m	m	PRON
ejpam-4262	110	11	,	,	PUNCT
ejpam-4262	110	12	(	(	PUNCT
ejpam-4262	110	13	s	s	VERB
ejpam-4262	110	14	+	+	NOUN
ejpam-4262	110	15	k)(r	k)(r	NOUN
ejpam-4262	110	16	+	+	ADJ
ejpam-4262	110	17	t	t	NOUN
ejpam-4262	110	18	)	)	PUNCT
ejpam-4262	111	1	+	+	CCONJ
ejpam-4262	111	2	(	(	PUNCT
ejpam-4262	111	3	s	s	VERB
ejpam-4262	111	4	+	+	NOUN
ejpam-4262	111	5	k)(r	k)(r	NOUN
ejpam-4262	111	6	+	+	NOUN
ejpam-4262	111	7	t	t	NOUN
ejpam-4262	111	8	)	)	PUNCT
ejpam-4262	111	9	∈	∈	PROPN
ejpam-4262	111	10	(	(	PUNCT
ejpam-4262	111	11	p	p	X
ejpam-4262	111	12	:	:	PUNCT
ejpam-4262	111	13	m	m	X
ejpam-4262	111	14	)	)	PUNCT
ejpam-4262	111	15	or	or	CCONJ
ejpam-4262	111	16	(	(	PUNCT
ejpam-4262	111	17	s	s	X
ejpam-4262	111	18	+	+	CCONJ
ejpam-4262	112	1	k)(n	k)(n	PROPN
ejpam-4262	113	1	+	+	CCONJ
ejpam-4262	114	1	n	n	PROPN
ejpam-4262	114	2	+	+	NOUN
ejpam-4262	114	3	m	m	VERB
ejpam-4262	114	4	+	+	ADJ
ejpam-4262	114	5	n	n	PROPN
ejpam-4262	114	6	+	+	CCONJ
ejpam-4262	114	7	n	n	PROPN
ejpam-4262	114	8	+	+	NOUN
ejpam-4262	114	9	m	m	NOUN
ejpam-4262	114	10	)	)	PUNCT
ejpam-4262	114	11	∈	∈	PROPN
ejpam-4262	114	12	p	p	NOUN
ejpam-4262	114	13	or	or	CCONJ
ejpam-4262	114	14	(	(	PUNCT
ejpam-4262	114	15	r	r	NOUN
ejpam-4262	114	16	+	+	NOUN
ejpam-4262	114	17	t)(n+	t)(n+	NOUN
ejpam-4262	114	18	n+m+	n+m+	NOUN
ejpam-4262	114	19	n+	n+	X
ejpam-4262	114	20	n+m	n+m	NUM
ejpam-4262	114	21	)	)	PUNCT
ejpam-4262	114	22	∈	∈	PROPN
ejpam-4262	115	1	p	p	NOUN
ejpam-4262	115	2	.	.	PUNCT
ejpam-4262	116	1	since	since	SCONJ
ejpam-4262	116	2	k	k	PROPN
ejpam-4262	116	3	,	,	PUNCT
ejpam-4262	116	4	t	t	PROPN
ejpam-4262	116	5	∈	∈	PROPN
ejpam-4262	116	6	(	(	PUNCT
ejpam-4262	116	7	p	p	X
ejpam-4262	116	8	:	:	PUNCT
ejpam-4262	116	9	m	m	PROPN
ejpam-4262	116	10	)	)	PUNCT
ejpam-4262	116	11	,	,	PUNCT
ejpam-4262	116	12	we	we	PRON
ejpam-4262	116	13	have	have	AUX
ejpam-4262	116	14	rs+	rs+	VERB
ejpam-4262	116	15	rs	rs	PROPN
ejpam-4262	116	16	∈	∈	PROPN
ejpam-4262	116	17	(	(	PUNCT
ejpam-4262	116	18	p	p	X
ejpam-4262	116	19	:	:	PUNCT
ejpam-4262	116	20	m	m	NOUN
ejpam-4262	116	21	)	)	PUNCT
ejpam-4262	116	22	or	or	CCONJ
ejpam-4262	116	23	r(m+m	r(m+m	NOUN
ejpam-4262	116	24	)	)	PUNCT
ejpam-4262	116	25	∈	∈	PROPN
ejpam-4262	116	26	p	p	NOUN
ejpam-4262	116	27	or	or	CCONJ
ejpam-4262	116	28	s(m+m	s(m+m	SYM
ejpam-4262	116	29	)	)	PUNCT
ejpam-4262	116	30	∈	∈	PROPN
ejpam-4262	116	31	p	p	NOUN
ejpam-4262	116	32	.	.	PUNCT
ejpam-4262	117	1	therefore	therefore	ADV
ejpam-4262	117	2	p	p	PROPN
ejpam-4262	117	3	is	be	AUX
ejpam-4262	117	4	a	a	DET
ejpam-4262	117	5	β	β	NOUN
ejpam-4262	117	6	-	-	ADJ
ejpam-4262	117	7	absorbing	absorbing	ADJ
ejpam-4262	117	8	submodule	submodule	NOUN
ejpam-4262	117	9	of	of	ADP
ejpam-4262	117	10	m	m	PROPN
ejpam-4262	117	11	.	.	PUNCT
ejpam-4262	118	1	t.	t.	PROPN
ejpam-4262	118	2	khumprapussorn	khumprapussorn	PROPN
ejpam-4262	118	3	/	/	SYM
ejpam-4262	118	4	eur	eur	PROPN
ejpam-4262	118	5	.	.	PUNCT
ejpam-4262	119	1	j.	j.	PROPN
ejpam-4262	119	2	pure	pure	PROPN
ejpam-4262	119	3	appl	appl	PROPN
ejpam-4262	119	4	.	.	PROPN
ejpam-4262	119	5	math	math	PROPN
ejpam-4262	119	6	,	,	PUNCT
ejpam-4262	119	7	15	15	NUM
ejpam-4262	119	8	(	(	PUNCT
ejpam-4262	119	9	1	1	NUM
ejpam-4262	119	10	)	)	PUNCT
ejpam-4262	119	11	(	(	PUNCT
ejpam-4262	119	12	2022	2022	NUM
ejpam-4262	119	13	)	)	PUNCT
ejpam-4262	119	14	,	,	PUNCT
ejpam-4262	119	15	328	328	NUM
ejpam-4262	119	16	-	-	SYM
ejpam-4262	119	17	334	334	NUM
ejpam-4262	119	18	331	331	NUM
ejpam-4262	119	19	corollary	corollary	ADJ
ejpam-4262	119	20	1	1	NUM
ejpam-4262	119	21	.	.	PUNCT
ejpam-4262	120	1	[	[	X
ejpam-4262	120	2	1	1	X
ejpam-4262	120	3	]	]	X
ejpam-4262	120	4	if	if	SCONJ
ejpam-4262	120	5	p	p	NOUN
ejpam-4262	120	6	is	be	AUX
ejpam-4262	120	7	a	a	DET
ejpam-4262	120	8	weakly	weakly	ADJ
ejpam-4262	120	9	β	β	NOUN
ejpam-4262	120	10	-	-	ADJ
ejpam-4262	120	11	absorbing	absorbing	ADJ
ejpam-4262	120	12	submodule	submodule	NOUN
ejpam-4262	120	13	of	of	ADP
ejpam-4262	120	14	m	m	PROPN
ejpam-4262	120	15	and	and	CCONJ
ejpam-4262	120	16	(	(	PUNCT
ejpam-4262	120	17	p	p	X
ejpam-4262	120	18	:	:	PUNCT
ejpam-4262	120	19	m)2β(p	m)2β(p	NOUN
ejpam-4262	120	20	)	)	PUNCT
ejpam-4262	120	21	̸=	̸=	PROPN
ejpam-4262	120	22	{	{	PUNCT
ejpam-4262	120	23	0	0	NUM
ejpam-4262	120	24	}	}	PUNCT
ejpam-4262	120	25	,	,	PUNCT
ejpam-4262	120	26	then	then	ADV
ejpam-4262	120	27	p	p	PROPN
ejpam-4262	120	28	is	be	AUX
ejpam-4262	120	29	a	a	DET
ejpam-4262	120	30	β	β	NOUN
ejpam-4262	120	31	-	-	ADJ
ejpam-4262	120	32	absorbing	absorbing	ADJ
ejpam-4262	120	33	submodule	submodule	NOUN
ejpam-4262	120	34	of	of	ADP
ejpam-4262	120	35	m	m	PROPN
ejpam-4262	120	36	.	.	PUNCT
ejpam-4262	121	1	next	next	ADV
ejpam-4262	121	2	,	,	PUNCT
ejpam-4262	121	3	we	we	PRON
ejpam-4262	121	4	use	use	VERB
ejpam-4262	121	5	the	the	DET
ejpam-4262	121	6	function	function	NOUN
ejpam-4262	121	7	ϕi	ϕi	ADP
ejpam-4262	121	8	:	:	PUNCT
ejpam-4262	121	9	s(m)→	s(m)→	ADJ
ejpam-4262	121	10	s(m	s(m	NOUN
ejpam-4262	121	11	)	)	PUNCT
ejpam-4262	121	12	∪	∪	ADP
ejpam-4262	121	13	{	{	PUNCT
ejpam-4262	121	14	∅	∅	NOUN
ejpam-4262	121	15	}	}	PUNCT
ejpam-4262	121	16	by	by	ADP
ejpam-4262	121	17	the	the	DET
ejpam-4262	121	18	following	follow	VERB
ejpam-4262	121	19	meaning	meaning	NOUN
ejpam-4262	121	20	,	,	PUNCT
ejpam-4262	121	21	for	for	ADP
ejpam-4262	121	22	any	any	DET
ejpam-4262	121	23	submodule	submodule	NOUN
ejpam-4262	121	24	n	n	PROPN
ejpam-4262	121	25	of	of	ADP
ejpam-4262	121	26	an	an	DET
ejpam-4262	121	27	r	r	NOUN
ejpam-4262	121	28	-	-	PUNCT
ejpam-4262	121	29	module	module	NOUN
ejpam-4262	121	30	m	m	NOUN
ejpam-4262	121	31	and	and	CCONJ
ejpam-4262	121	32	natural	natural	ADJ
ejpam-4262	121	33	number	number	NOUN
ejpam-4262	121	34	n	n	NOUN
ejpam-4262	121	35	with	with	ADP
ejpam-4262	121	36	n	n	PRON
ejpam-4262	121	37	≥	≥	NUM
ejpam-4262	121	38	2	2	NUM
ejpam-4262	121	39	,	,	PUNCT
ejpam-4262	121	40	ϕ∅(n	ϕ∅(n	PRON
ejpam-4262	121	41	)	)	PUNCT
ejpam-4262	121	42	=	=	PUNCT
ejpam-4262	121	43	∅	∅	NOUN
ejpam-4262	121	44	ϕ0(n	ϕ0(n	NOUN
ejpam-4262	121	45	)	)	PUNCT
ejpam-4262	121	46	=	=	SYM
ejpam-4262	121	47	{	{	PUNCT
ejpam-4262	121	48	0	0	NUM
ejpam-4262	121	49	}	}	SYM
ejpam-4262	121	50	ϕ1(n	ϕ1(n	PROPN
ejpam-4262	121	51	)	)	PUNCT
ejpam-4262	121	52	=	=	PUNCT
ejpam-4262	121	53	(	(	PUNCT
ejpam-4262	121	54	n	n	NOUN
ejpam-4262	121	55	:	:	PUNCT
ejpam-4262	121	56	m)β(n	m)β(n	X
ejpam-4262	121	57	)	)	PUNCT
ejpam-4262	121	58	ϕn(n	ϕn(n	PUNCT
ejpam-4262	121	59	)	)	PUNCT
ejpam-4262	121	60	=	=	PUNCT
ejpam-4262	121	61	(	(	PUNCT
ejpam-4262	121	62	n	n	NOUN
ejpam-4262	121	63	:	:	PUNCT
ejpam-4262	121	64	m)nβ(n	m)nβ(n	NOUN
ejpam-4262	121	65	)	)	PUNCT
ejpam-4262	121	66	ϕω(n	ϕω(n	NUM
ejpam-4262	121	67	)	)	PUNCT
ejpam-4262	122	1	=	=	SYM
ejpam-4262	122	2	∞⋂	∞⋂	PROPN
ejpam-4262	122	3	i=1	i=1	PROPN
ejpam-4262	123	1	(	(	PUNCT
ejpam-4262	123	2	n	n	PROPN
ejpam-4262	123	3	:	:	PUNCT
ejpam-4262	123	4	m)iβ(n	m)iβ(n	X
ejpam-4262	123	5	)	)	PUNCT
ejpam-4262	123	6	let	let	VERB
ejpam-4262	123	7	ϕ	ϕ	NOUN
ejpam-4262	123	8	:	:	PUNCT
ejpam-4262	123	9	s(m	s(m	PROPN
ejpam-4262	123	10	)	)	PUNCT
ejpam-4262	123	11	→	→	SYM
ejpam-4262	123	12	s(m	s(m	NOUN
ejpam-4262	123	13	)	)	PUNCT
ejpam-4262	123	14	∪	∪	ADP
ejpam-4262	123	15	{	{	PUNCT
ejpam-4262	123	16	∅	∅	NOUN
ejpam-4262	123	17	}	}	PUNCT
ejpam-4262	123	18	and	and	CCONJ
ejpam-4262	123	19	φ	φ	NUM
ejpam-4262	123	20	:	:	PUNCT
ejpam-4262	123	21	s(m	s(m	PROPN
ejpam-4262	123	22	)	)	PUNCT
ejpam-4262	123	23	→	→	SYM
ejpam-4262	123	24	s(m	s(m	NOUN
ejpam-4262	123	25	)	)	PUNCT
ejpam-4262	123	26	∪	∪	NOUN
ejpam-4262	123	27	{	{	PUNCT
ejpam-4262	123	28	∅	∅	NOUN
ejpam-4262	123	29	}	}	PUNCT
ejpam-4262	123	30	be	be	AUX
ejpam-4262	123	31	functions	function	NOUN
ejpam-4262	123	32	.	.	PUNCT
ejpam-4262	124	1	we	we	PRON
ejpam-4262	124	2	write	write	VERB
ejpam-4262	124	3	ϕ	ϕ	PROPN
ejpam-4262	124	4	≤	≤	PROPN
ejpam-4262	124	5	φ	φ	PROPN
ejpam-4262	124	6	if	if	SCONJ
ejpam-4262	124	7	ϕ(n	ϕ(n	X
ejpam-4262	124	8	)	)	PUNCT
ejpam-4262	124	9	⊆	⊆	NUM
ejpam-4262	124	10	φ(n	φ(n	NOUN
ejpam-4262	124	11	)	)	PUNCT
ejpam-4262	124	12	for	for	ADP
ejpam-4262	124	13	all	all	DET
ejpam-4262	124	14	n	n	PRON
ejpam-4262	124	15	∈	∈	PROPN
ejpam-4262	124	16	s(m	s(m	PROPN
ejpam-4262	124	17	)	)	PUNCT
ejpam-4262	124	18	.	.	PUNCT
ejpam-4262	125	1	then	then	ADV
ejpam-4262	125	2	ϕ∅	ϕ∅	PUNCT
ejpam-4262	125	3	≤	≤	ADJ
ejpam-4262	125	4	ϕ0	ϕ0	NOUN
ejpam-4262	125	5	≤	≤	NOUN
ejpam-4262	125	6	ϕω	ϕω	ADP
ejpam-4262	125	7	≤	≤	NUM
ejpam-4262	125	8	·	·	PUNCT
ejpam-4262	125	9	·	·	PUNCT
ejpam-4262	125	10	·	·	PUNCT
ejpam-4262	126	1	≤	≤	NUM
ejpam-4262	126	2	ϕn+1	ϕn+1	X
ejpam-4262	126	3	≤	≤	NOUN
ejpam-4262	126	4	ϕn	ϕn	ADP
ejpam-4262	126	5	≤	≤	NOUN
ejpam-4262	126	6	·	·	PUNCT
ejpam-4262	126	7	·	·	PUNCT
ejpam-4262	126	8	·	·	PUNCT
ejpam-4262	127	1	≤	≤	NUM
ejpam-4262	127	2	ϕ2	ϕ2	ADV
ejpam-4262	127	3	≤	≤	NUM
ejpam-4262	127	4	ϕ1	ϕ1	NOUN
ejpam-4262	127	5	.	.	PUNCT
ejpam-4262	128	1	proposition	proposition	NOUN
ejpam-4262	128	2	1	1	NUM
ejpam-4262	128	3	.	.	PUNCT
ejpam-4262	129	1	let	let	VERB
ejpam-4262	129	2	ϕ	ϕ	NOUN
ejpam-4262	129	3	:	:	PUNCT
ejpam-4262	129	4	s(m	s(m	PROPN
ejpam-4262	129	5	)	)	PUNCT
ejpam-4262	129	6	→	→	SYM
ejpam-4262	129	7	s(m	s(m	NOUN
ejpam-4262	129	8	)	)	PUNCT
ejpam-4262	129	9	∪	∪	ADP
ejpam-4262	129	10	{	{	PUNCT
ejpam-4262	129	11	∅	∅	NOUN
ejpam-4262	129	12	}	}	PUNCT
ejpam-4262	129	13	and	and	CCONJ
ejpam-4262	129	14	φ	φ	NUM
ejpam-4262	129	15	:	:	PUNCT
ejpam-4262	129	16	s(m	s(m	PROPN
ejpam-4262	129	17	)	)	PUNCT
ejpam-4262	129	18	→	→	SYM
ejpam-4262	129	19	s(m	s(m	NOUN
ejpam-4262	129	20	)	)	PUNCT
ejpam-4262	129	21	∪	∪	NOUN
ejpam-4262	129	22	{	{	PUNCT
ejpam-4262	129	23	∅	∅	NOUN
ejpam-4262	129	24	}	}	PUNCT
ejpam-4262	129	25	be	be	AUX
ejpam-4262	129	26	functions	function	NOUN
ejpam-4262	129	27	such	such	ADJ
ejpam-4262	129	28	that	that	PRON
ejpam-4262	129	29	ϕ	ϕ	PROPN
ejpam-4262	129	30	≤	≤	NUM
ejpam-4262	129	31	φ	φ	NOUN
ejpam-4262	129	32	.	.	PUNCT
ejpam-4262	130	1	if	if	SCONJ
ejpam-4262	130	2	p	p	NOUN
ejpam-4262	130	3	is	be	AUX
ejpam-4262	130	4	a	a	DET
ejpam-4262	130	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	130	6	-	-	ADJ
ejpam-4262	130	7	absorbing	absorbing	ADJ
ejpam-4262	130	8	submodule	submodule	NOUN
ejpam-4262	130	9	of	of	ADP
ejpam-4262	130	10	m	m	PROPN
ejpam-4262	130	11	,	,	PUNCT
ejpam-4262	130	12	then	then	ADV
ejpam-4262	130	13	p	p	NOUN
ejpam-4262	130	14	is	be	AUX
ejpam-4262	130	15	a	a	DET
ejpam-4262	130	16	φ	φ	PROPN
ejpam-4262	130	17	-	-	PUNCT
ejpam-4262	130	18	β	β	ADJ
ejpam-4262	130	19	-	-	ADJ
ejpam-4262	130	20	absorbing	absorbing	ADJ
ejpam-4262	130	21	submodule	submodule	NOUN
ejpam-4262	130	22	of	of	ADP
ejpam-4262	130	23	m	m	PROPN
ejpam-4262	130	24	.	.	PUNCT
ejpam-4262	131	1	proof	proof	NOUN
ejpam-4262	131	2	.	.	PUNCT
ejpam-4262	132	1	this	this	DET
ejpam-4262	132	2	proof	proof	NOUN
ejpam-4262	132	3	is	be	AUX
ejpam-4262	132	4	straightforward	straightforward	ADJ
ejpam-4262	132	5	.	.	PUNCT
ejpam-4262	133	1	let	let	VERB
ejpam-4262	133	2	m1	m1	PROPN
ejpam-4262	133	3	be	be	AUX
ejpam-4262	133	4	a	a	DET
ejpam-4262	133	5	r1	r1	NOUN
ejpam-4262	133	6	-	-	PUNCT
ejpam-4262	133	7	module	module	NOUN
ejpam-4262	133	8	and	and	CCONJ
ejpam-4262	133	9	m2	m2	PROPN
ejpam-4262	133	10	be	be	VERB
ejpam-4262	133	11	a	a	DET
ejpam-4262	133	12	r2	r2	NOUN
ejpam-4262	133	13	-	-	PUNCT
ejpam-4262	133	14	module	module	NOUN
ejpam-4262	133	15	.	.	PUNCT
ejpam-4262	134	1	then	then	ADV
ejpam-4262	134	2	m	m	VERB
ejpam-4262	134	3	=	=	ADJ
ejpam-4262	134	4	m1×m2	m1×m2	PROPN
ejpam-4262	134	5	is	be	AUX
ejpam-4262	134	6	an	an	DET
ejpam-4262	134	7	r1×r2module	r1×r2module	NOUN
ejpam-4262	134	8	by	by	ADP
ejpam-4262	134	9	(	(	PUNCT
ejpam-4262	134	10	r1	r1	PROPN
ejpam-4262	134	11	,	,	PUNCT
ejpam-4262	134	12	r2)(m1,m2	r2)(m1,m2	NOUN
ejpam-4262	134	13	)	)	PUNCT
ejpam-4262	134	14	=	=	PUNCT
ejpam-4262	134	15	(	(	PUNCT
ejpam-4262	134	16	r1m1	r1m1	NOUN
ejpam-4262	134	17	,	,	PUNCT
ejpam-4262	134	18	r2m2	r2m2	NOUN
ejpam-4262	134	19	)	)	PUNCT
ejpam-4262	134	20	.	.	PUNCT
ejpam-4262	135	1	next	next	ADV
ejpam-4262	135	2	,	,	PUNCT
ejpam-4262	135	3	let	let	VERB
ejpam-4262	135	4	ϕ	ϕ	NOUN
ejpam-4262	135	5	:	:	PUNCT
ejpam-4262	135	6	s(m	s(m	PROPN
ejpam-4262	135	7	)	)	PUNCT
ejpam-4262	135	8	→	→	SYM
ejpam-4262	135	9	s(m	s(m	NOUN
ejpam-4262	135	10	)	)	PUNCT
ejpam-4262	135	11	∪	∪	NOUN
ejpam-4262	135	12	{	{	PUNCT
ejpam-4262	135	13	∅	∅	NOUN
ejpam-4262	135	14	}	}	PUNCT
ejpam-4262	135	15	be	be	AUX
ejpam-4262	135	16	a	a	DET
ejpam-4262	135	17	function	function	NOUN
ejpam-4262	135	18	and	and	CCONJ
ejpam-4262	135	19	p	p	NOUN
ejpam-4262	135	20	be	be	AUX
ejpam-4262	135	21	a	a	DET
ejpam-4262	135	22	submodule	submodule	NOUN
ejpam-4262	135	23	of	of	ADP
ejpam-4262	135	24	m1	m1	PROPN
ejpam-4262	135	25	.	.	PUNCT
ejpam-4262	136	1	then	then	ADV
ejpam-4262	136	2	(	(	PUNCT
ejpam-4262	136	3	p	p	NOUN
ejpam-4262	136	4	×m2)\ϕ(p	×m2)\ϕ(p	NOUN
ejpam-4262	136	5	×m2	×m2	NOUN
ejpam-4262	136	6	)	)	PUNCT
ejpam-4262	136	7	⊆	⊆	NUM
ejpam-4262	136	8	(	(	PUNCT
ejpam-4262	136	9	p	p	NOUN
ejpam-4262	136	10	×m2)\({0	×m2)\({0	PROPN
ejpam-4262	136	11	}	}	PUNCT
ejpam-4262	136	12	×m2	×m2	PROPN
ejpam-4262	136	13	)	)	PUNCT
ejpam-4262	136	14	=	=	SYM
ejpam-4262	136	15	(	(	PUNCT
ejpam-4262	136	16	p\{0})×m2	p\{0})×m2	NOUN
ejpam-4262	136	17	.	.	PUNCT
ejpam-4262	137	1	we	we	PRON
ejpam-4262	137	2	have	have	VERB
ejpam-4262	137	3	the	the	DET
ejpam-4262	137	4	following	follow	VERB
ejpam-4262	137	5	results	result	NOUN
ejpam-4262	137	6	.	.	PUNCT
ejpam-4262	138	1	proposition	proposition	NOUN
ejpam-4262	138	2	2	2	NUM
ejpam-4262	138	3	.	.	PUNCT
ejpam-4262	139	1	let	let	VERB
ejpam-4262	139	2	r	r	NOUN
ejpam-4262	139	3	=	=	SYM
ejpam-4262	139	4	r1×r2	r1×r2	PROPN
ejpam-4262	139	5	and	and	CCONJ
ejpam-4262	139	6	m	m	PROPN
ejpam-4262	139	7	=	=	ADJ
ejpam-4262	139	8	m1×m2	m1×m2	PROPN
ejpam-4262	139	9	and	and	CCONJ
ejpam-4262	139	10	let	let	VERB
ejpam-4262	139	11	ϕ	ϕ	NOUN
ejpam-4262	139	12	:	:	PUNCT
ejpam-4262	139	13	s(m)→	s(m)→	ADJ
ejpam-4262	139	14	s(m)∪	s(m)∪	NOUN
ejpam-4262	139	15	{	{	PUNCT
ejpam-4262	139	16	∅	∅	NOUN
ejpam-4262	139	17	}	}	PUNCT
ejpam-4262	139	18	be	be	AUX
ejpam-4262	139	19	a	a	DET
ejpam-4262	139	20	function	function	NOUN
ejpam-4262	139	21	.	.	PUNCT
ejpam-4262	140	1	if	if	SCONJ
ejpam-4262	140	2	p	p	NOUN
ejpam-4262	140	3	is	be	AUX
ejpam-4262	140	4	a	a	DET
ejpam-4262	140	5	ϕ0	ϕ0	NOUN
ejpam-4262	140	6	-	-	PUNCT
ejpam-4262	140	7	β	β	NOUN
ejpam-4262	140	8	-	-	ADJ
ejpam-4262	140	9	absorbing	absorbing	ADJ
ejpam-4262	140	10	submodule	submodule	NOUN
ejpam-4262	140	11	of	of	ADP
ejpam-4262	140	12	m1	m1	PROPN
ejpam-4262	140	13	with	with	ADP
ejpam-4262	140	14	{	{	PUNCT
ejpam-4262	140	15	0	0	NUM
ejpam-4262	140	16	}	}	PUNCT
ejpam-4262	140	17	×m2	×m2	NOUN
ejpam-4262	140	18	⊆	⊆	NUM
ejpam-4262	140	19	ϕ(p	ϕ(p	PROPN
ejpam-4262	140	20	×m2	×m2	PROPN
ejpam-4262	140	21	)	)	PUNCT
ejpam-4262	140	22	,	,	PUNCT
ejpam-4262	140	23	then	then	ADV
ejpam-4262	140	24	p	p	NOUN
ejpam-4262	140	25	×m2	×m2	PROPN
ejpam-4262	140	26	is	be	AUX
ejpam-4262	140	27	a	a	DET
ejpam-4262	140	28	ϕ-β	ϕ-β	ADJ
ejpam-4262	140	29	-	-	ADJ
ejpam-4262	140	30	absorbing	absorbing	ADJ
ejpam-4262	140	31	submodule	submodule	NOUN
ejpam-4262	140	32	of	of	ADP
ejpam-4262	140	33	m	m	PROPN
ejpam-4262	140	34	.	.	PUNCT
ejpam-4262	141	1	proof	proof	NOUN
ejpam-4262	141	2	.	.	PUNCT
ejpam-4262	142	1	assume	assume	VERB
ejpam-4262	142	2	that	that	SCONJ
ejpam-4262	142	3	p	p	NOUN
ejpam-4262	142	4	is	be	AUX
ejpam-4262	142	5	a	a	DET
ejpam-4262	142	6	ϕ0	ϕ0	NOUN
ejpam-4262	142	7	-	-	PUNCT
ejpam-4262	142	8	β	β	NOUN
ejpam-4262	142	9	-	-	ADJ
ejpam-4262	142	10	absorbing	absorbing	ADJ
ejpam-4262	142	11	submodule	submodule	NOUN
ejpam-4262	142	12	of	of	ADP
ejpam-4262	142	13	m1	m1	PROPN
ejpam-4262	142	14	with	with	ADP
ejpam-4262	142	15	{	{	PUNCT
ejpam-4262	142	16	0	0	NUM
ejpam-4262	142	17	}	}	PUNCT
ejpam-4262	142	18	×	×	NOUN
ejpam-4262	142	19	m2	m2	PROPN
ejpam-4262	142	20	⊆	⊆	NUM
ejpam-4262	142	21	ϕ(p	ϕ(p	PROPN
ejpam-4262	142	22	×	×	PROPN
ejpam-4262	142	23	m2	m2	PROPN
ejpam-4262	142	24	)	)	PUNCT
ejpam-4262	142	25	.	.	PUNCT
ejpam-4262	143	1	let	let	VERB
ejpam-4262	143	2	(	(	PUNCT
ejpam-4262	143	3	r1	r1	NOUN
ejpam-4262	143	4	,	,	PUNCT
ejpam-4262	143	5	r2	r2	PROPN
ejpam-4262	143	6	)	)	PUNCT
ejpam-4262	143	7	,	,	PUNCT
ejpam-4262	143	8	(	(	PUNCT
ejpam-4262	143	9	s1	s1	NOUN
ejpam-4262	143	10	,	,	PUNCT
ejpam-4262	143	11	s2	s2	PROPN
ejpam-4262	143	12	)	)	PUNCT
ejpam-4262	143	13	∈	∈	PROPN
ejpam-4262	143	14	r1	r1	NOUN
ejpam-4262	143	15	×	×	NOUN
ejpam-4262	143	16	r2	r2	PROPN
ejpam-4262	143	17	and	and	CCONJ
ejpam-4262	143	18	(	(	PUNCT
ejpam-4262	143	19	m1,m2	m1,m2	PROPN
ejpam-4262	143	20	)	)	PUNCT
ejpam-4262	143	21	∈	∈	PROPN
ejpam-4262	143	22	m1	m1	PROPN
ejpam-4262	143	23	×	×	PROPN
ejpam-4262	143	24	m2	m2	PROPN
ejpam-4262	143	25	be	be	VERB
ejpam-4262	143	26	such	such	ADJ
ejpam-4262	143	27	that	that	SCONJ
ejpam-4262	143	28	(	(	PUNCT
ejpam-4262	143	29	r1	r1	NOUN
ejpam-4262	143	30	,	,	PUNCT
ejpam-4262	143	31	r2)(s1	r2)(s1	NOUN
ejpam-4262	143	32	,	,	PUNCT
ejpam-4262	143	33	s2)(m1,m2	s2)(m1,m2	NOUN
ejpam-4262	143	34	)	)	PUNCT
ejpam-4262	143	35	∈	∈	PROPN
ejpam-4262	143	36	(	(	PUNCT
ejpam-4262	143	37	p	p	NOUN
ejpam-4262	143	38	×	×	NOUN
ejpam-4262	143	39	m2)\ϕ(p	m2)\ϕ(p	NUM
ejpam-4262	143	40	×	×	NOUN
ejpam-4262	143	41	m2	m2	PROPN
ejpam-4262	143	42	)	)	PUNCT
ejpam-4262	143	43	.	.	PUNCT
ejpam-4262	144	1	then	then	ADV
ejpam-4262	144	2	r1s1m1	r1s1m1	PROPN
ejpam-4262	144	3	∈	∈	PROPN
ejpam-4262	144	4	p\{0	p\{0	PROPN
ejpam-4262	144	5	}	}	PUNCT
ejpam-4262	144	6	.	.	PUNCT
ejpam-4262	145	1	since	since	SCONJ
ejpam-4262	145	2	p	p	NOUN
ejpam-4262	145	3	is	be	AUX
ejpam-4262	145	4	a	a	DET
ejpam-4262	145	5	ϕ0	ϕ0	NOUN
ejpam-4262	145	6	-	-	PUNCT
ejpam-4262	145	7	β	β	NOUN
ejpam-4262	145	8	-	-	ADJ
ejpam-4262	145	9	absorbing	absorbing	ADJ
ejpam-4262	145	10	submodule	submodule	NOUN
ejpam-4262	145	11	of	of	ADP
ejpam-4262	145	12	m1	m1	NOUN
ejpam-4262	145	13	,	,	PUNCT
ejpam-4262	145	14	r1s1	r1s1	NOUN
ejpam-4262	145	15	+	+	SYM
ejpam-4262	145	16	r1s1	r1s1	NOUN
ejpam-4262	145	17	∈	∈	NOUN
ejpam-4262	145	18	(	(	PUNCT
ejpam-4262	145	19	p	p	X
ejpam-4262	145	20	:	:	PUNCT
ejpam-4262	145	21	m	m	X
ejpam-4262	145	22	)	)	PUNCT
ejpam-4262	145	23	or	or	CCONJ
ejpam-4262	145	24	r1(m1	r1(m1	NOUN
ejpam-4262	145	25	+	+	CCONJ
ejpam-4262	145	26	m1	m1	NOUN
ejpam-4262	145	27	)	)	PUNCT
ejpam-4262	145	28	∈	∈	PROPN
ejpam-4262	145	29	p	p	NOUN
ejpam-4262	145	30	or	or	CCONJ
ejpam-4262	145	31	s1(m1+m1	s1(m1+m1	NUM
ejpam-4262	145	32	)	)	PUNCT
ejpam-4262	145	33	∈	∈	PROPN
ejpam-4262	145	34	p	p	NOUN
ejpam-4262	145	35	.	.	PUNCT
ejpam-4262	146	1	this	this	PRON
ejpam-4262	146	2	implies	imply	VERB
ejpam-4262	146	3	that	that	SCONJ
ejpam-4262	146	4	(	(	PUNCT
ejpam-4262	146	5	r1	r1	NOUN
ejpam-4262	146	6	,	,	PUNCT
ejpam-4262	146	7	r2)(s1	r2)(s1	NOUN
ejpam-4262	146	8	,	,	PUNCT
ejpam-4262	146	9	s2)+	s2)+	PROPN
ejpam-4262	146	10	(	(	PUNCT
ejpam-4262	146	11	r1	r1	PROPN
ejpam-4262	146	12	,	,	PUNCT
ejpam-4262	146	13	r2)(s1	r2)(s1	NOUN
ejpam-4262	146	14	,	,	PUNCT
ejpam-4262	146	15	s2	s2	PROPN
ejpam-4262	146	16	)	)	PUNCT
ejpam-4262	146	17	∈	∈	PROPN
ejpam-4262	146	18	(	(	PUNCT
ejpam-4262	146	19	p	p	NOUN
ejpam-4262	146	20	×m2	×m2	NOUN
ejpam-4262	146	21	:	:	PUNCT
ejpam-4262	146	22	m1×m2	m1×m2	PROPN
ejpam-4262	146	23	)	)	PUNCT
ejpam-4262	146	24	or	or	CCONJ
ejpam-4262	146	25	(	(	PUNCT
ejpam-4262	146	26	r1	r1	PROPN
ejpam-4262	146	27	,	,	PUNCT
ejpam-4262	146	28	r2)[(m1,m2	r2)[(m1,m2	PROPN
ejpam-4262	146	29	)	)	PUNCT
ejpam-4262	146	30	+	+	CCONJ
ejpam-4262	146	31	(	(	PUNCT
ejpam-4262	146	32	m1,m2	m1,m2	PROPN
ejpam-4262	146	33	)	)	PUNCT
ejpam-4262	146	34	]	]	PUNCT
ejpam-4262	147	1	∈	∈	PROPN
ejpam-4262	147	2	p	p	X
ejpam-4262	147	3	×	×	PROPN
ejpam-4262	147	4	m2	m2	PROPN
ejpam-4262	147	5	or	or	CCONJ
ejpam-4262	147	6	(	(	PUNCT
ejpam-4262	147	7	s1	s1	NOUN
ejpam-4262	147	8	,	,	PUNCT
ejpam-4262	147	9	s2)[(m1,m2	s2)[(m1,m2	NOUN
ejpam-4262	147	10	)	)	PUNCT
ejpam-4262	147	11	+	+	CCONJ
ejpam-4262	147	12	(	(	PUNCT
ejpam-4262	147	13	m1,m2	m1,m2	PROPN
ejpam-4262	147	14	)	)	PUNCT
ejpam-4262	147	15	]	]	PUNCT
ejpam-4262	148	1	∈	∈	PROPN
ejpam-4262	148	2	p	p	X
ejpam-4262	148	3	×	×	PROPN
ejpam-4262	148	4	m2	m2	PROPN
ejpam-4262	148	5	.	.	PUNCT
ejpam-4262	149	1	therefore	therefore	ADV
ejpam-4262	149	2	p	p	NOUN
ejpam-4262	149	3	×m2	×m2	PROPN
ejpam-4262	149	4	is	be	AUX
ejpam-4262	149	5	a	a	DET
ejpam-4262	149	6	ϕ-β	ϕ-β	ADJ
ejpam-4262	149	7	-	-	ADJ
ejpam-4262	149	8	absorbing	absorbing	ADJ
ejpam-4262	149	9	submodule	submodule	NOUN
ejpam-4262	149	10	of	of	ADP
ejpam-4262	149	11	m	m	PROPN
ejpam-4262	149	12	.	.	PUNCT
ejpam-4262	150	1	t.	t.	PROPN
ejpam-4262	150	2	khumprapussorn	khumprapussorn	PROPN
ejpam-4262	150	3	/	/	SYM
ejpam-4262	150	4	eur	eur	PROPN
ejpam-4262	150	5	.	.	PUNCT
ejpam-4262	151	1	j.	j.	PROPN
ejpam-4262	151	2	pure	pure	PROPN
ejpam-4262	151	3	appl	appl	PROPN
ejpam-4262	151	4	.	.	PROPN
ejpam-4262	151	5	math	math	PROPN
ejpam-4262	151	6	,	,	PUNCT
ejpam-4262	151	7	15	15	NUM
ejpam-4262	151	8	(	(	PUNCT
ejpam-4262	151	9	1	1	NUM
ejpam-4262	151	10	)	)	PUNCT
ejpam-4262	151	11	(	(	PUNCT
ejpam-4262	151	12	2022	2022	NUM
ejpam-4262	151	13	)	)	PUNCT
ejpam-4262	151	14	,	,	PUNCT
ejpam-4262	151	15	328	328	NUM
ejpam-4262	151	16	-	-	SYM
ejpam-4262	151	17	334	334	NUM
ejpam-4262	151	18	332	332	NUM
ejpam-4262	151	19	proposition	proposition	NOUN
ejpam-4262	151	20	3	3	X
ejpam-4262	151	21	.	.	PUNCT
ejpam-4262	152	1	let	let	VERB
ejpam-4262	152	2	mi	mi	PROPN
ejpam-4262	152	3	be	be	AUX
ejpam-4262	152	4	an	an	DET
ejpam-4262	152	5	ri	ri	NOUN
ejpam-4262	152	6	-	-	PUNCT
ejpam-4262	152	7	module	module	NOUN
ejpam-4262	152	8	and	and	CCONJ
ejpam-4262	152	9	ϕmi	ϕmi	NOUN
ejpam-4262	152	10	:	:	PUNCT
ejpam-4262	153	1	s(mi	s(mi	VERB
ejpam-4262	153	2	)	)	PUNCT
ejpam-4262	153	3	→	→	SYM
ejpam-4262	153	4	s(mi	s(mi	NOUN
ejpam-4262	153	5	)	)	PUNCT
ejpam-4262	153	6	∪	∪	ADP
ejpam-4262	153	7	{	{	PUNCT
ejpam-4262	153	8	∅	∅	NOUN
ejpam-4262	153	9	}	}	PUNCT
ejpam-4262	153	10	be	be	AUX
ejpam-4262	153	11	a	a	DET
ejpam-4262	153	12	function	function	NOUN
ejpam-4262	153	13	where	where	SCONJ
ejpam-4262	153	14	i	i	PRON
ejpam-4262	153	15	=	=	NOUN
ejpam-4262	153	16	1	1	NUM
ejpam-4262	153	17	,	,	PUNCT
ejpam-4262	153	18	2	2	NUM
ejpam-4262	153	19	.	.	X
ejpam-4262	153	20	for	for	ADP
ejpam-4262	153	21	this	this	DET
ejpam-4262	153	22	result	result	NOUN
ejpam-4262	153	23	,	,	PUNCT
ejpam-4262	153	24	we	we	PRON
ejpam-4262	153	25	define	define	VERB
ejpam-4262	153	26	ϕ	ϕ	NOUN
ejpam-4262	153	27	:	:	PUNCT
ejpam-4262	153	28	s(m1	s(m1	NOUN
ejpam-4262	153	29	×	×	PROPN
ejpam-4262	153	30	m2	m2	PROPN
ejpam-4262	153	31	)	)	PUNCT
ejpam-4262	153	32	→	→	SYM
ejpam-4262	153	33	s(m1	s(m1	NOUN
ejpam-4262	153	34	×	×	PROPN
ejpam-4262	153	35	m2	m2	PROPN
ejpam-4262	153	36	)	)	PUNCT
ejpam-4262	153	37	∪	∪	ADP
ejpam-4262	153	38	{	{	PUNCT
ejpam-4262	153	39	∅	∅	NOUN
ejpam-4262	153	40	}	}	PUNCT
ejpam-4262	153	41	by	by	ADP
ejpam-4262	153	42	ϕ	ϕ	NOUN
ejpam-4262	153	43	=	=	PUNCT
ejpam-4262	154	1	ϕm1	ϕm1	PROPN
ejpam-4262	154	2	×	×	PROPN
ejpam-4262	154	3	ϕm2	ϕm2	PROPN
ejpam-4262	154	4	.	.	PUNCT
ejpam-4262	155	1	if	if	SCONJ
ejpam-4262	155	2	p1	p1	PROPN
ejpam-4262	155	3	×	×	NOUN
ejpam-4262	155	4	p2	p2	PROPN
ejpam-4262	155	5	is	be	AUX
ejpam-4262	155	6	a	a	DET
ejpam-4262	155	7	ϕ-β	ϕ-β	ADJ
ejpam-4262	155	8	-	-	ADJ
ejpam-4262	155	9	absorbing	absorbing	ADJ
ejpam-4262	155	10	submodule	submodule	NOUN
ejpam-4262	155	11	of	of	ADP
ejpam-4262	155	12	m1	m1	PROPN
ejpam-4262	155	13	×	×	PROPN
ejpam-4262	155	14	m2	m2	PROPN
ejpam-4262	155	15	,	,	PUNCT
ejpam-4262	155	16	then	then	ADV
ejpam-4262	155	17	pi	pi	NOUN
ejpam-4262	155	18	is	be	AUX
ejpam-4262	155	19	a	a	DET
ejpam-4262	155	20	ϕmi	ϕmi	NOUN
ejpam-4262	155	21	-	-	PUNCT
ejpam-4262	155	22	β	β	NOUN
ejpam-4262	155	23	-	-	ADJ
ejpam-4262	155	24	absorbing	absorbing	ADJ
ejpam-4262	155	25	submodule	submodule	NOUN
ejpam-4262	155	26	of	of	ADP
ejpam-4262	155	27	mi	mi	PROPN
ejpam-4262	155	28	.	.	PROPN
ejpam-4262	155	29	proof	proof	PROPN
ejpam-4262	155	30	.	.	PUNCT
ejpam-4262	156	1	assume	assume	VERB
ejpam-4262	156	2	that	that	SCONJ
ejpam-4262	156	3	p1	p1	NOUN
ejpam-4262	156	4	×	×	NOUN
ejpam-4262	156	5	p2	p2	PROPN
ejpam-4262	156	6	is	be	AUX
ejpam-4262	156	7	a	a	DET
ejpam-4262	156	8	ϕ-β	ϕ-β	ADJ
ejpam-4262	156	9	-	-	ADJ
ejpam-4262	156	10	absorbing	absorbing	ADJ
ejpam-4262	156	11	submodule	submodule	NOUN
ejpam-4262	156	12	of	of	ADP
ejpam-4262	156	13	m1	m1	PROPN
ejpam-4262	156	14	×	×	PROPN
ejpam-4262	156	15	m2	m2	PROPN
ejpam-4262	156	16	.	.	PUNCT
ejpam-4262	156	17	to	to	PART
ejpam-4262	156	18	show	show	VERB
ejpam-4262	156	19	that	that	SCONJ
ejpam-4262	156	20	p1	p1	NOUN
ejpam-4262	156	21	is	be	AUX
ejpam-4262	156	22	a	a	DET
ejpam-4262	156	23	ϕm1	ϕm1	PROPN
ejpam-4262	156	24	-	-	PUNCT
ejpam-4262	156	25	β	β	NOUN
ejpam-4262	156	26	-	-	ADJ
ejpam-4262	156	27	absorbing	absorbing	ADJ
ejpam-4262	156	28	submodule	submodule	NOUN
ejpam-4262	156	29	of	of	ADP
ejpam-4262	156	30	m1	m1	NOUN
ejpam-4262	156	31	,	,	PUNCT
ejpam-4262	156	32	let	let	VERB
ejpam-4262	156	33	r	r	NOUN
ejpam-4262	156	34	,	,	PUNCT
ejpam-4262	156	35	s	s	PART
ejpam-4262	156	36	∈	∈	PROPN
ejpam-4262	156	37	r1	r1	NOUN
ejpam-4262	156	38	and	and	CCONJ
ejpam-4262	156	39	m	m	PROPN
ejpam-4262	156	40	∈	∈	PROPN
ejpam-4262	156	41	m1	m1	NOUN
ejpam-4262	156	42	be	be	AUX
ejpam-4262	156	43	such	such	ADJ
ejpam-4262	156	44	that	that	SCONJ
ejpam-4262	156	45	rsm	rsm	PROPN
ejpam-4262	156	46	∈	∈	PROPN
ejpam-4262	156	47	p1\ϕm1(p1	p1\ϕm1(p1	NOUN
ejpam-4262	156	48	)	)	PUNCT
ejpam-4262	156	49	.	.	PUNCT
ejpam-4262	157	1	since	since	SCONJ
ejpam-4262	157	2	ϕ(p1	ϕ(p1	PROPN
ejpam-4262	157	3	×	×	NOUN
ejpam-4262	157	4	p2	p2	NOUN
ejpam-4262	157	5	)	)	PUNCT
ejpam-4262	157	6	=	=	SYM
ejpam-4262	157	7	ϕm1(p1	ϕm1(p1	NOUN
ejpam-4262	157	8	)	)	PUNCT
ejpam-4262	157	9	×	×	NOUN
ejpam-4262	157	10	ϕm2(p2	ϕm2(p2	NUM
ejpam-4262	157	11	)	)	PUNCT
ejpam-4262	157	12	,	,	PUNCT
ejpam-4262	157	13	we	we	PRON
ejpam-4262	157	14	have	have	VERB
ejpam-4262	157	15	(	(	PUNCT
ejpam-4262	157	16	r	r	NOUN
ejpam-4262	157	17	,	,	PUNCT
ejpam-4262	157	18	1)(s	1)(s	NUM
ejpam-4262	157	19	,	,	PUNCT
ejpam-4262	157	20	1)(m	1)(m	NUM
ejpam-4262	157	21	,	,	PUNCT
ejpam-4262	157	22	0	0	NUM
ejpam-4262	157	23	)	)	PUNCT
ejpam-4262	157	24	=	=	SYM
ejpam-4262	157	25	(	(	PUNCT
ejpam-4262	157	26	rsm	rsm	PROPN
ejpam-4262	157	27	,	,	PUNCT
ejpam-4262	157	28	0	0	NUM
ejpam-4262	157	29	)	)	PUNCT
ejpam-4262	157	30	∈	∈	PROPN
ejpam-4262	157	31	(	(	PUNCT
ejpam-4262	157	32	p1×p2)\ϕ(p1×p2	p1×p2)\ϕ(p1×p2	PROPN
ejpam-4262	157	33	)	)	PUNCT
ejpam-4262	157	34	.	.	PUNCT
ejpam-4262	158	1	since	since	SCONJ
ejpam-4262	158	2	p1×p2	p1×p2	NOUN
ejpam-4262	158	3	is	be	AUX
ejpam-4262	158	4	a	a	DET
ejpam-4262	158	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	158	6	-	-	ADJ
ejpam-4262	158	7	absorbing	absorbing	ADJ
ejpam-4262	158	8	submodule	submodule	NOUN
ejpam-4262	158	9	of	of	ADP
ejpam-4262	158	10	m1×m2	m1×m2	PROPN
ejpam-4262	158	11	,	,	PUNCT
ejpam-4262	158	12	(	(	PUNCT
ejpam-4262	158	13	r	r	NOUN
ejpam-4262	158	14	,	,	PUNCT
ejpam-4262	158	15	1)(s	1)(s	NUM
ejpam-4262	158	16	,	,	PUNCT
ejpam-4262	158	17	1	1	NUM
ejpam-4262	158	18	)	)	PUNCT
ejpam-4262	158	19	+	+	CCONJ
ejpam-4262	158	20	(	(	PUNCT
ejpam-4262	158	21	r	r	NOUN
ejpam-4262	158	22	,	,	PUNCT
ejpam-4262	158	23	1)(s	1)(s	NUM
ejpam-4262	158	24	,	,	PUNCT
ejpam-4262	158	25	1	1	NUM
ejpam-4262	158	26	)	)	PUNCT
ejpam-4262	158	27	∈	∈	PROPN
ejpam-4262	158	28	(	(	PUNCT
ejpam-4262	158	29	p1	p1	NOUN
ejpam-4262	158	30	×m2	×m2	NOUN
ejpam-4262	158	31	:	:	PUNCT
ejpam-4262	158	32	m1	m1	PROPN
ejpam-4262	158	33	×m2	×m2	PROPN
ejpam-4262	158	34	)	)	PUNCT
ejpam-4262	158	35	or	or	CCONJ
ejpam-4262	158	36	(	(	PUNCT
ejpam-4262	158	37	r	r	NOUN
ejpam-4262	158	38	,	,	PUNCT
ejpam-4262	158	39	1)[(m	1)[(m	NUM
ejpam-4262	158	40	,	,	PUNCT
ejpam-4262	158	41	0	0	NUM
ejpam-4262	158	42	)	)	PUNCT
ejpam-4262	159	1	+	+	CCONJ
ejpam-4262	159	2	(	(	PUNCT
ejpam-4262	159	3	m	m	PROPN
ejpam-4262	159	4	,	,	PUNCT
ejpam-4262	159	5	0	0	NUM
ejpam-4262	159	6	)	)	PUNCT
ejpam-4262	159	7	]	]	PUNCT
ejpam-4262	160	1	∈	∈	PROPN
ejpam-4262	160	2	p1	p1	NOUN
ejpam-4262	160	3	×m2	×m2	NOUN
ejpam-4262	160	4	or	or	CCONJ
ejpam-4262	160	5	(	(	PUNCT
ejpam-4262	160	6	s	s	X
ejpam-4262	160	7	,	,	PUNCT
ejpam-4262	160	8	1)[(m	1)[(m	NUM
ejpam-4262	160	9	,	,	PUNCT
ejpam-4262	160	10	0	0	NUM
ejpam-4262	160	11	)	)	PUNCT
ejpam-4262	160	12	+	+	CCONJ
ejpam-4262	160	13	(	(	PUNCT
ejpam-4262	160	14	m	m	PROPN
ejpam-4262	160	15	,	,	PUNCT
ejpam-4262	160	16	0	0	NUM
ejpam-4262	160	17	)	)	PUNCT
ejpam-4262	160	18	]	]	PUNCT
ejpam-4262	160	19	∈	∈	PROPN
ejpam-4262	160	20	p1	p1	NOUN
ejpam-4262	160	21	×m2	×m2	PROPN
ejpam-4262	160	22	.	.	PUNCT
ejpam-4262	161	1	this	this	PRON
ejpam-4262	161	2	implies	imply	VERB
ejpam-4262	161	3	that	that	SCONJ
ejpam-4262	161	4	rs+	rs+	VERB
ejpam-4262	161	5	rs	rs	PROPN
ejpam-4262	161	6	∈	∈	PROPN
ejpam-4262	161	7	(	(	PUNCT
ejpam-4262	161	8	p1	p1	NOUN
ejpam-4262	161	9	:	:	PUNCT
ejpam-4262	161	10	m1	m1	PROPN
ejpam-4262	161	11	)	)	PUNCT
ejpam-4262	161	12	or	or	CCONJ
ejpam-4262	161	13	r(m+m	r(m+m	NOUN
ejpam-4262	161	14	)	)	PUNCT
ejpam-4262	161	15	∈	∈	PROPN
ejpam-4262	161	16	p1	p1	NOUN
ejpam-4262	161	17	or	or	CCONJ
ejpam-4262	161	18	s(m	s(m	PROPN
ejpam-4262	161	19	+	+	CCONJ
ejpam-4262	161	20	m	m	NOUN
ejpam-4262	161	21	)	)	PUNCT
ejpam-4262	161	22	∈	∈	PROPN
ejpam-4262	161	23	p1	p1	NOUN
ejpam-4262	161	24	.	.	PUNCT
ejpam-4262	162	1	hence	hence	ADV
ejpam-4262	162	2	p1	p1	PROPN
ejpam-4262	162	3	is	be	AUX
ejpam-4262	162	4	a	a	DET
ejpam-4262	162	5	ϕm1	ϕm1	PROPN
ejpam-4262	162	6	-	-	PUNCT
ejpam-4262	162	7	β	β	NOUN
ejpam-4262	162	8	-	-	ADJ
ejpam-4262	162	9	absorbing	absorbing	ADJ
ejpam-4262	162	10	submodule	submodule	NOUN
ejpam-4262	162	11	of	of	ADP
ejpam-4262	162	12	m1	m1	PROPN
ejpam-4262	162	13	.	.	PUNCT
ejpam-4262	163	1	similarly	similarly	ADV
ejpam-4262	163	2	,	,	PUNCT
ejpam-4262	163	3	we	we	PRON
ejpam-4262	163	4	can	can	AUX
ejpam-4262	163	5	show	show	VERB
ejpam-4262	163	6	that	that	SCONJ
ejpam-4262	163	7	p2	p2	PROPN
ejpam-4262	163	8	is	be	AUX
ejpam-4262	163	9	a	a	DET
ejpam-4262	163	10	ϕm2	ϕm2	VERB
ejpam-4262	163	11	-	-	PUNCT
ejpam-4262	163	12	β	β	NOUN
ejpam-4262	163	13	-	-	ADJ
ejpam-4262	163	14	absorbing	absorbing	ADJ
ejpam-4262	163	15	submodule	submodule	NOUN
ejpam-4262	163	16	of	of	ADP
ejpam-4262	163	17	m2	m2	PROPN
ejpam-4262	163	18	.	.	PROPN
ejpam-4262	164	1	3	3	X
ejpam-4262	164	2	.	.	PUNCT
ejpam-4262	164	3	characterizations	characterization	NOUN
ejpam-4262	164	4	of	of	ADP
ejpam-4262	164	5	ϕ-β	ϕ-β	NOUN
ejpam-4262	164	6	-	-	ADJ
ejpam-4262	164	7	absorbing	absorbing	ADJ
ejpam-4262	164	8	submodules	submodule	NOUN
ejpam-4262	164	9	the	the	DET
ejpam-4262	164	10	purpose	purpose	NOUN
ejpam-4262	164	11	of	of	ADP
ejpam-4262	164	12	this	this	DET
ejpam-4262	164	13	section	section	NOUN
ejpam-4262	164	14	is	be	AUX
ejpam-4262	164	15	to	to	PART
ejpam-4262	164	16	investigate	investigate	VERB
ejpam-4262	164	17	characterizations	characterization	NOUN
ejpam-4262	164	18	of	of	ADP
ejpam-4262	164	19	ϕ-β	ϕ-β	NOUN
ejpam-4262	164	20	-	-	ADJ
ejpam-4262	164	21	absorbing	absorbing	ADJ
ejpam-4262	164	22	submodules	submodule	NOUN
ejpam-4262	164	23	.	.	PUNCT
ejpam-4262	165	1	for	for	ADP
ejpam-4262	165	2	a	a	DET
ejpam-4262	165	3	submodule	submodule	NOUN
ejpam-4262	165	4	n	n	PROPN
ejpam-4262	165	5	of	of	ADP
ejpam-4262	165	6	a	a	DET
ejpam-4262	165	7	left	left	ADJ
ejpam-4262	165	8	r	r	NOUN
ejpam-4262	165	9	-	-	PUNCT
ejpam-4262	165	10	module	module	NOUN
ejpam-4262	165	11	m	m	NOUN
ejpam-4262	165	12	and	and	CCONJ
ejpam-4262	165	13	r	r	NOUN
ejpam-4262	165	14	∈	∈	PROPN
ejpam-4262	165	15	r	r	NOUN
ejpam-4262	165	16	,	,	PUNCT
ejpam-4262	165	17	we	we	PRON
ejpam-4262	165	18	define	define	VERB
ejpam-4262	165	19	the	the	DET
ejpam-4262	165	20	symbol	symbol	NOUN
ejpam-4262	165	21	nr	nr	PRON
ejpam-4262	165	22	by	by	ADP
ejpam-4262	165	23	{	{	PUNCT
ejpam-4262	165	24	m	m	NOUN
ejpam-4262	165	25	∈m	∈m	NOUN
ejpam-4262	165	26	|	|	ADV
ejpam-4262	165	27	rm	rm	NOUN
ejpam-4262	165	28	∈	∈	PROPN
ejpam-4262	165	29	n	n	CCONJ
ejpam-4262	165	30	}	}	PUNCT
ejpam-4262	165	31	.	.	PUNCT
ejpam-4262	166	1	theorem	theorem	NOUN
ejpam-4262	166	2	2	2	NUM
ejpam-4262	166	3	.	.	PUNCT
ejpam-4262	167	1	let	let	VERB
ejpam-4262	167	2	p	p	PRON
ejpam-4262	167	3	be	be	AUX
ejpam-4262	167	4	a	a	DET
ejpam-4262	167	5	submodule	submodule	NOUN
ejpam-4262	167	6	of	of	ADP
ejpam-4262	167	7	m	m	PROPN
ejpam-4262	167	8	.	.	PUNCT
ejpam-4262	168	1	then	then	ADV
ejpam-4262	168	2	the	the	DET
ejpam-4262	168	3	following	follow	VERB
ejpam-4262	168	4	statements	statement	NOUN
ejpam-4262	168	5	are	be	AUX
ejpam-4262	168	6	equivalent	equivalent	ADJ
ejpam-4262	168	7	:	:	PUNCT
ejpam-4262	168	8	(	(	PUNCT
ejpam-4262	168	9	i	i	NOUN
ejpam-4262	168	10	)	)	PUNCT
ejpam-4262	168	11	p	p	NOUN
ejpam-4262	168	12	is	be	AUX
ejpam-4262	168	13	a	a	DET
ejpam-4262	168	14	ϕ-β	ϕ-β	ADJ
ejpam-4262	168	15	-	-	ADJ
ejpam-4262	168	16	absorbing	absorbing	ADJ
ejpam-4262	168	17	submodule	submodule	NOUN
ejpam-4262	168	18	of	of	ADP
ejpam-4262	168	19	m	m	PROPN
ejpam-4262	168	20	.	.	PUNCT
ejpam-4262	169	1	(	(	PUNCT
ejpam-4262	169	2	ii	ii	NOUN
ejpam-4262	169	3	)	)	PUNCT
ejpam-4262	169	4	for	for	ADP
ejpam-4262	169	5	all	all	DET
ejpam-4262	169	6	r	r	NOUN
ejpam-4262	169	7	,	,	PUNCT
ejpam-4262	169	8	s	s	NOUN
ejpam-4262	169	9	∈	∈	PROPN
ejpam-4262	169	10	r	r	NOUN
ejpam-4262	169	11	,	,	PUNCT
ejpam-4262	169	12	if	if	SCONJ
ejpam-4262	169	13	rs+	rs+	NOUN
ejpam-4262	169	14	rs	rs	NOUN
ejpam-4262	169	15	/∈	/∈	PUNCT
ejpam-4262	170	1	(	(	PUNCT
ejpam-4262	170	2	p	p	X
ejpam-4262	170	3	:	:	PUNCT
ejpam-4262	170	4	m	m	PROPN
ejpam-4262	170	5	)	)	PUNCT
ejpam-4262	170	6	,	,	PUNCT
ejpam-4262	170	7	then	then	ADV
ejpam-4262	170	8	prs	prs	VERB
ejpam-4262	170	9	⊆	⊆	NUM
ejpam-4262	170	10	α(pr	α(pr	NOUN
ejpam-4262	170	11	)	)	PUNCT
ejpam-4262	170	12	∪	∪	ADP
ejpam-4262	170	13	α(ps	α(ps	NOUN
ejpam-4262	170	14	)	)	PUNCT
ejpam-4262	170	15	∪	∪	ADP
ejpam-4262	170	16	ϕ(p	ϕ(p	PROPN
ejpam-4262	170	17	)	)	PUNCT
ejpam-4262	170	18	rs	rs	NOUN
ejpam-4262	170	19	.	.	PUNCT
ejpam-4262	171	1	proof	proof	NOUN
ejpam-4262	171	2	.	.	PUNCT
ejpam-4262	172	1	(	(	PUNCT
ejpam-4262	172	2	i	i	NOUN
ejpam-4262	172	3	)	)	PUNCT
ejpam-4262	172	4	→	→	SYM
ejpam-4262	172	5	(	(	PUNCT
ejpam-4262	172	6	ii	ii	NOUN
ejpam-4262	172	7	)	)	PUNCT
ejpam-4262	172	8	assume	assume	VERB
ejpam-4262	172	9	that	that	SCONJ
ejpam-4262	172	10	p	p	NOUN
ejpam-4262	172	11	is	be	AUX
ejpam-4262	172	12	a	a	DET
ejpam-4262	172	13	ϕ-β	ϕ-β	ADJ
ejpam-4262	172	14	-	-	ADJ
ejpam-4262	172	15	absorbing	absorbing	ADJ
ejpam-4262	172	16	submodule	submodule	NOUN
ejpam-4262	172	17	of	of	ADP
ejpam-4262	172	18	m	m	PROPN
ejpam-4262	172	19	.	.	PUNCT
ejpam-4262	173	1	let	let	VERB
ejpam-4262	173	2	r	r	NOUN
ejpam-4262	173	3	,	,	PUNCT
ejpam-4262	173	4	s	s	PART
ejpam-4262	173	5	∈	∈	NOUN
ejpam-4262	173	6	r	r	NOUN
ejpam-4262	173	7	be	be	VERB
ejpam-4262	174	1	such	such	ADJ
ejpam-4262	174	2	that	that	DET
ejpam-4262	174	3	rs	rs	NOUN
ejpam-4262	175	1	+	+	CCONJ
ejpam-4262	175	2	rs	rs	ADJ
ejpam-4262	175	3	/∈	/∈	PUNCT
ejpam-4262	176	1	(	(	PUNCT
ejpam-4262	176	2	p	p	X
ejpam-4262	176	3	:	:	PUNCT
ejpam-4262	176	4	m	m	NUM
ejpam-4262	176	5	)	)	PUNCT
ejpam-4262	176	6	and	and	CCONJ
ejpam-4262	176	7	m	m	PROPN
ejpam-4262	176	8	∈	∈	PROPN
ejpam-4262	176	9	prs	prs	NOUN
ejpam-4262	176	10	.	.	PUNCT
ejpam-4262	177	1	then	then	ADV
ejpam-4262	177	2	rsm	rsm	PROPN
ejpam-4262	177	3	∈	∈	PROPN
ejpam-4262	177	4	p	p	X
ejpam-4262	177	5	.	.	PUNCT
ejpam-4262	178	1	if	if	SCONJ
ejpam-4262	178	2	rsm	rsm	PROPN
ejpam-4262	178	3	∈	∈	PROPN
ejpam-4262	178	4	ϕ(p	ϕ(p	PROPN
ejpam-4262	178	5	)	)	PUNCT
ejpam-4262	178	6	,	,	PUNCT
ejpam-4262	178	7	then	then	ADV
ejpam-4262	178	8	m	m	VERB
ejpam-4262	178	9	∈	∈	ADJ
ejpam-4262	178	10	ϕ(p	ϕ(p	PROPN
ejpam-4262	178	11	)	)	PUNCT
ejpam-4262	178	12	rs	rs	NOUN
ejpam-4262	178	13	.	.	PUNCT
ejpam-4262	178	14	assume	assume	VERB
ejpam-4262	178	15	that	that	SCONJ
ejpam-4262	178	16	rsm	rsm	PROPN
ejpam-4262	178	17	∈	∈	PROPN
ejpam-4262	178	18	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	178	19	)	)	PUNCT
ejpam-4262	178	20	.	.	PUNCT
ejpam-4262	179	1	since	since	SCONJ
ejpam-4262	179	2	p	p	NOUN
ejpam-4262	179	3	is	be	AUX
ejpam-4262	179	4	a	a	DET
ejpam-4262	179	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	179	6	-	-	ADJ
ejpam-4262	179	7	absorbing	absorbing	ADJ
ejpam-4262	179	8	submodule	submodule	NOUN
ejpam-4262	179	9	of	of	ADP
ejpam-4262	179	10	m	m	PROPN
ejpam-4262	179	11	and	and	CCONJ
ejpam-4262	179	12	rs+	rs+	VERB
ejpam-4262	179	13	rs	rs	PROPN
ejpam-4262	179	14	/∈	/∈	PUNCT
ejpam-4262	180	1	(	(	PUNCT
ejpam-4262	180	2	p	p	X
ejpam-4262	180	3	:	:	PUNCT
ejpam-4262	180	4	m	m	NOUN
ejpam-4262	180	5	)	)	PUNCT
ejpam-4262	180	6	,	,	PUNCT
ejpam-4262	180	7	r(m+m	r(m+m	NOUN
ejpam-4262	180	8	)	)	PUNCT
ejpam-4262	180	9	∈	∈	PROPN
ejpam-4262	180	10	p	p	NOUN
ejpam-4262	180	11	or	or	CCONJ
ejpam-4262	180	12	s(m+m	s(m+m	SYM
ejpam-4262	180	13	)	)	PUNCT
ejpam-4262	180	14	∈	∈	PROPN
ejpam-4262	180	15	p	p	NOUN
ejpam-4262	180	16	.	.	PUNCT
ejpam-4262	181	1	thus	thus	ADV
ejpam-4262	181	2	m	m	PROPN
ejpam-4262	181	3	∈	∈	NOUN
ejpam-4262	181	4	α(pr	α(pr	NOUN
ejpam-4262	181	5	)	)	PUNCT
ejpam-4262	181	6	or	or	CCONJ
ejpam-4262	181	7	m	m	PROPN
ejpam-4262	181	8	∈	∈	NOUN
ejpam-4262	181	9	α(ps	α(p	NOUN
ejpam-4262	181	10	)	)	PUNCT
ejpam-4262	181	11	.	.	PUNCT
ejpam-4262	182	1	this	this	PRON
ejpam-4262	182	2	shows	show	VERB
ejpam-4262	182	3	that	that	SCONJ
ejpam-4262	182	4	prs	prs	VERB
ejpam-4262	182	5	⊆	⊆	NUM
ejpam-4262	182	6	α(pr	α(pr	NOUN
ejpam-4262	182	7	)	)	PUNCT
ejpam-4262	182	8	∪	∪	ADP
ejpam-4262	182	9	α(ps	α(ps	NOUN
ejpam-4262	182	10	)	)	PUNCT
ejpam-4262	182	11	∪	∪	ADP
ejpam-4262	182	12	ϕ(p	ϕ(p	PROPN
ejpam-4262	182	13	)	)	PUNCT
ejpam-4262	182	14	rs	rs	NOUN
ejpam-4262	182	15	.	.	PUNCT
ejpam-4262	183	1	(	(	PUNCT
ejpam-4262	183	2	ii)→	ii)→	NOUN
ejpam-4262	183	3	(	(	PUNCT
ejpam-4262	183	4	i	i	NOUN
ejpam-4262	183	5	)	)	PUNCT
ejpam-4262	183	6	assume	assume	VERB
ejpam-4262	183	7	that	that	SCONJ
ejpam-4262	183	8	(	(	PUNCT
ejpam-4262	183	9	ii	ii	NOUN
ejpam-4262	183	10	)	)	PUNCT
ejpam-4262	183	11	holds	hold	VERB
ejpam-4262	183	12	.	.	PUNCT
ejpam-4262	184	1	let	let	VERB
ejpam-4262	184	2	r	r	NOUN
ejpam-4262	184	3	,	,	PUNCT
ejpam-4262	184	4	s	s	PART
ejpam-4262	184	5	∈	∈	PROPN
ejpam-4262	184	6	r	r	NOUN
ejpam-4262	184	7	and	and	CCONJ
ejpam-4262	184	8	m	m	PROPN
ejpam-4262	184	9	∈m	∈m	NOUN
ejpam-4262	184	10	be	be	AUX
ejpam-4262	184	11	such	such	ADJ
ejpam-4262	184	12	that	that	SCONJ
ejpam-4262	184	13	rsm	rsm	PROPN
ejpam-4262	184	14	∈	∈	PROPN
ejpam-4262	184	15	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	184	16	)	)	PUNCT
ejpam-4262	184	17	and	and	CCONJ
ejpam-4262	184	18	rs	rs	INTJ
ejpam-4262	185	1	+	+	CCONJ
ejpam-4262	185	2	rs	rs	ADJ
ejpam-4262	185	3	/∈	/∈	PUNCT
ejpam-4262	186	1	(	(	PUNCT
ejpam-4262	186	2	p	p	X
ejpam-4262	186	3	:	:	PUNCT
ejpam-4262	186	4	m	m	PROPN
ejpam-4262	186	5	)	)	PUNCT
ejpam-4262	186	6	.	.	PUNCT
ejpam-4262	187	1	then	then	ADV
ejpam-4262	187	2	m	m	PROPN
ejpam-4262	187	3	∈	∈	PROPN
ejpam-4262	187	4	prs	prs	NOUN
ejpam-4262	187	5	.	.	PUNCT
ejpam-4262	188	1	by	by	ADP
ejpam-4262	188	2	our	our	PRON
ejpam-4262	188	3	assumptions	assumption	NOUN
ejpam-4262	188	4	,	,	PUNCT
ejpam-4262	188	5	m	m	PROPN
ejpam-4262	188	6	∈	∈	NOUN
ejpam-4262	188	7	α(pr	α(pr	NOUN
ejpam-4262	188	8	)	)	PUNCT
ejpam-4262	188	9	∪	∪	ADP
ejpam-4262	188	10	α(ps	α(p	NOUN
ejpam-4262	188	11	)	)	PUNCT
ejpam-4262	188	12	.	.	PUNCT
ejpam-4262	189	1	hence	hence	ADV
ejpam-4262	189	2	r(m+m	r(m+m	NOUN
ejpam-4262	189	3	)	)	PUNCT
ejpam-4262	189	4	∈	∈	PROPN
ejpam-4262	189	5	p	p	NOUN
ejpam-4262	189	6	or	or	CCONJ
ejpam-4262	189	7	s(m+m	s(m+m	SYM
ejpam-4262	189	8	)	)	PUNCT
ejpam-4262	189	9	∈	∈	PROPN
ejpam-4262	190	1	p	p	NOUN
ejpam-4262	190	2	.	.	PUNCT
ejpam-4262	191	1	therefore	therefore	ADV
ejpam-4262	191	2	p	p	PROPN
ejpam-4262	191	3	is	be	AUX
ejpam-4262	191	4	a	a	DET
ejpam-4262	191	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	191	6	-	-	ADJ
ejpam-4262	191	7	absorbing	absorbing	ADJ
ejpam-4262	191	8	submodule	submodule	NOUN
ejpam-4262	191	9	of	of	ADP
ejpam-4262	191	10	m	m	PROPN
ejpam-4262	191	11	.	.	PUNCT
ejpam-4262	192	1	for	for	ADP
ejpam-4262	192	2	a	a	DET
ejpam-4262	192	3	submodule	submodule	NOUN
ejpam-4262	192	4	n	n	PROPN
ejpam-4262	192	5	of	of	ADP
ejpam-4262	192	6	a	a	DET
ejpam-4262	192	7	left	left	ADJ
ejpam-4262	192	8	r	r	NOUN
ejpam-4262	192	9	-	-	PUNCT
ejpam-4262	192	10	module	module	NOUN
ejpam-4262	192	11	m	m	NOUN
ejpam-4262	192	12	and	and	CCONJ
ejpam-4262	192	13	m	m	NOUN
ejpam-4262	192	14	∈m	∈m	NOUN
ejpam-4262	192	15	,	,	PUNCT
ejpam-4262	192	16	we	we	PRON
ejpam-4262	192	17	define	define	VERB
ejpam-4262	192	18	the	the	DET
ejpam-4262	192	19	symbol	symbol	NOUN
ejpam-4262	192	20	(	(	PUNCT
ejpam-4262	192	21	n	n	NOUN
ejpam-4262	192	22	:	:	PUNCT
ejpam-4262	192	23	m	m	X
ejpam-4262	192	24	)	)	PUNCT
ejpam-4262	192	25	by	by	ADP
ejpam-4262	192	26	{	{	PUNCT
ejpam-4262	192	27	r	r	NOUN
ejpam-4262	192	28	∈	∈	PROPN
ejpam-4262	192	29	r	r	NOUN
ejpam-4262	192	30	|	|	NOUN
ejpam-4262	192	31	rm	rm	PROPN
ejpam-4262	192	32	∈	∈	PROPN
ejpam-4262	192	33	n	n	CCONJ
ejpam-4262	192	34	}	}	PUNCT
ejpam-4262	192	35	.	.	PUNCT
ejpam-4262	193	1	theorem	theorem	NOUN
ejpam-4262	193	2	3	3	X
ejpam-4262	193	3	.	.	PUNCT
ejpam-4262	194	1	let	let	VERB
ejpam-4262	194	2	p	p	PRON
ejpam-4262	194	3	be	be	AUX
ejpam-4262	194	4	a	a	DET
ejpam-4262	194	5	submodule	submodule	NOUN
ejpam-4262	194	6	of	of	ADP
ejpam-4262	194	7	m	m	PROPN
ejpam-4262	194	8	.	.	PUNCT
ejpam-4262	195	1	then	then	ADV
ejpam-4262	195	2	the	the	DET
ejpam-4262	195	3	following	follow	VERB
ejpam-4262	195	4	statements	statement	NOUN
ejpam-4262	195	5	are	be	AUX
ejpam-4262	195	6	equivalent	equivalent	ADJ
ejpam-4262	195	7	:	:	PUNCT
ejpam-4262	195	8	(	(	PUNCT
ejpam-4262	195	9	i	i	NOUN
ejpam-4262	195	10	)	)	PUNCT
ejpam-4262	195	11	p	p	NOUN
ejpam-4262	195	12	is	be	AUX
ejpam-4262	195	13	a	a	DET
ejpam-4262	195	14	ϕ-β	ϕ-β	ADJ
ejpam-4262	195	15	-	-	ADJ
ejpam-4262	195	16	absorbing	absorbing	ADJ
ejpam-4262	195	17	submodule	submodule	NOUN
ejpam-4262	195	18	of	of	ADP
ejpam-4262	195	19	m	m	PROPN
ejpam-4262	195	20	.	.	PUNCT
ejpam-4262	196	1	(	(	PUNCT
ejpam-4262	196	2	ii	ii	NOUN
ejpam-4262	196	3	)	)	PUNCT
ejpam-4262	196	4	for	for	ADP
ejpam-4262	196	5	all	all	DET
ejpam-4262	196	6	s	s	PART
ejpam-4262	196	7	∈	∈	NOUN
ejpam-4262	196	8	r	r	NOUN
ejpam-4262	196	9	and	and	CCONJ
ejpam-4262	196	10	m	m	NOUN
ejpam-4262	196	11	∈m	∈m	NOUN
ejpam-4262	196	12	,	,	PUNCT
ejpam-4262	196	13	if	if	SCONJ
ejpam-4262	196	14	sm	sm	PROPN
ejpam-4262	196	15	/∈	/∈	PUNCT
ejpam-4262	197	1	α(p	α(p	PROPN
ejpam-4262	197	2	)	)	PUNCT
ejpam-4262	198	1	,	,	PUNCT
ejpam-4262	198	2	then	then	ADV
ejpam-4262	198	3	(	(	PUNCT
ejpam-4262	198	4	p	p	X
ejpam-4262	198	5	:	:	PUNCT
ejpam-4262	198	6	sm	sm	NOUN
ejpam-4262	198	7	)	)	PUNCT
ejpam-4262	198	8	⊆	⊆	NUM
ejpam-4262	198	9	α((p	α((p	NOUN
ejpam-4262	198	10	:	:	PUNCT
ejpam-4262	198	11	sm	sm	PROPN
ejpam-4262	198	12	)	)	PUNCT
ejpam-4262	198	13	)	)	PUNCT
ejpam-4262	198	14	∪	∪	ADP
ejpam-4262	198	15	α((p	α((p	NOUN
ejpam-4262	198	16	:	:	PUNCT
ejpam-4262	198	17	m	m	X
ejpam-4262	198	18	)	)	PUNCT
ejpam-4262	198	19	)	)	PUNCT
ejpam-4262	198	20	∪	∪	ADV
ejpam-4262	198	21	(	(	PUNCT
ejpam-4262	198	22	ϕ(p	ϕ(p	PROPN
ejpam-4262	198	23	)	)	PUNCT
ejpam-4262	198	24	:	:	PUNCT
ejpam-4262	198	25	sm	sm	NOUN
ejpam-4262	198	26	)	)	PUNCT
ejpam-4262	198	27	.	.	PUNCT
ejpam-4262	199	1	t.	t.	PROPN
ejpam-4262	199	2	khumprapussorn	khumprapussorn	PROPN
ejpam-4262	199	3	/	/	SYM
ejpam-4262	199	4	eur	eur	PROPN
ejpam-4262	199	5	.	.	PUNCT
ejpam-4262	200	1	j.	j.	PROPN
ejpam-4262	200	2	pure	pure	PROPN
ejpam-4262	200	3	appl	appl	PROPN
ejpam-4262	200	4	.	.	PROPN
ejpam-4262	200	5	math	math	PROPN
ejpam-4262	200	6	,	,	PUNCT
ejpam-4262	200	7	15	15	NUM
ejpam-4262	200	8	(	(	PUNCT
ejpam-4262	200	9	1	1	NUM
ejpam-4262	200	10	)	)	PUNCT
ejpam-4262	200	11	(	(	PUNCT
ejpam-4262	200	12	2022	2022	NUM
ejpam-4262	200	13	)	)	PUNCT
ejpam-4262	200	14	,	,	PUNCT
ejpam-4262	200	15	328	328	NUM
ejpam-4262	200	16	-	-	SYM
ejpam-4262	200	17	334	334	NUM
ejpam-4262	200	18	333	333	NUM
ejpam-4262	200	19	proof	proof	NOUN
ejpam-4262	200	20	.	.	PUNCT
ejpam-4262	201	1	(	(	PUNCT
ejpam-4262	201	2	i	i	NOUN
ejpam-4262	201	3	)	)	PUNCT
ejpam-4262	201	4	→	→	SYM
ejpam-4262	201	5	(	(	PUNCT
ejpam-4262	201	6	ii	ii	NOUN
ejpam-4262	201	7	)	)	PUNCT
ejpam-4262	201	8	assume	assume	VERB
ejpam-4262	201	9	that	that	SCONJ
ejpam-4262	201	10	p	p	NOUN
ejpam-4262	201	11	is	be	AUX
ejpam-4262	201	12	a	a	DET
ejpam-4262	201	13	ϕ-β	ϕ-β	ADJ
ejpam-4262	201	14	-	-	ADJ
ejpam-4262	201	15	absorbing	absorbing	ADJ
ejpam-4262	201	16	submodule	submodule	NOUN
ejpam-4262	201	17	of	of	ADP
ejpam-4262	201	18	m	m	PROPN
ejpam-4262	201	19	.	.	PUNCT
ejpam-4262	202	1	let	let	VERB
ejpam-4262	202	2	s	s	PRON
ejpam-4262	202	3	∈	∈	NOUN
ejpam-4262	202	4	r	r	NOUN
ejpam-4262	202	5	and	and	CCONJ
ejpam-4262	202	6	m	m	PROPN
ejpam-4262	202	7	∈	∈	NOUN
ejpam-4262	202	8	m	m	AUX
ejpam-4262	202	9	be	be	VERB
ejpam-4262	202	10	such	such	ADJ
ejpam-4262	202	11	that	that	SCONJ
ejpam-4262	202	12	sm	sm	PROPN
ejpam-4262	202	13	/∈	/∈	PUNCT
ejpam-4262	203	1	α(p	α(p	PROPN
ejpam-4262	203	2	)	)	PUNCT
ejpam-4262	203	3	.	.	PUNCT
ejpam-4262	204	1	let	let	VERB
ejpam-4262	204	2	r	r	NOUN
ejpam-4262	204	3	∈	∈	PROPN
ejpam-4262	204	4	(	(	PUNCT
ejpam-4262	204	5	p	p	X
ejpam-4262	204	6	:	:	PUNCT
ejpam-4262	204	7	sm	sm	NOUN
ejpam-4262	204	8	)	)	PUNCT
ejpam-4262	204	9	.	.	PUNCT
ejpam-4262	205	1	then	then	ADV
ejpam-4262	205	2	rsm	rsm	PROPN
ejpam-4262	205	3	∈	∈	PROPN
ejpam-4262	205	4	p	p	X
ejpam-4262	205	5	.	.	PUNCT
ejpam-4262	206	1	if	if	SCONJ
ejpam-4262	206	2	rsm	rsm	PROPN
ejpam-4262	206	3	∈	∈	PROPN
ejpam-4262	206	4	ϕ(p	ϕ(p	PROPN
ejpam-4262	206	5	)	)	PUNCT
ejpam-4262	206	6	,	,	PUNCT
ejpam-4262	206	7	then	then	ADV
ejpam-4262	206	8	r	r	NOUN
ejpam-4262	206	9	∈	∈	PROPN
ejpam-4262	206	10	(	(	PUNCT
ejpam-4262	206	11	ϕ(p	ϕ(p	PROPN
ejpam-4262	206	12	)	)	PUNCT
ejpam-4262	206	13	:	:	PUNCT
ejpam-4262	206	14	sm	sm	NOUN
ejpam-4262	206	15	)	)	PUNCT
ejpam-4262	206	16	.	.	PUNCT
ejpam-4262	207	1	assume	assume	VERB
ejpam-4262	207	2	that	that	SCONJ
ejpam-4262	207	3	rsm	rsm	PROPN
ejpam-4262	207	4	∈	∈	PROPN
ejpam-4262	207	5	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	207	6	)	)	PUNCT
ejpam-4262	207	7	.	.	PUNCT
ejpam-4262	208	1	since	since	SCONJ
ejpam-4262	208	2	p	p	NOUN
ejpam-4262	208	3	is	be	AUX
ejpam-4262	208	4	a	a	DET
ejpam-4262	208	5	ϕ-β	ϕ-β	ADJ
ejpam-4262	208	6	-	-	ADJ
ejpam-4262	208	7	absorbing	absorbing	ADJ
ejpam-4262	208	8	submodule	submodule	NOUN
ejpam-4262	208	9	of	of	ADP
ejpam-4262	208	10	m	m	PROPN
ejpam-4262	208	11	and	and	CCONJ
ejpam-4262	208	12	sm	sm	PROPN
ejpam-4262	208	13	/∈	/∈	PUNCT
ejpam-4262	209	1	α(p	α(p	PROPN
ejpam-4262	209	2	)	)	PUNCT
ejpam-4262	209	3	,	,	PUNCT
ejpam-4262	210	1	rs	rs	INTJ
ejpam-4262	210	2	+	+	CCONJ
ejpam-4262	210	3	rs	rs	PROPN
ejpam-4262	210	4	∈	∈	PROPN
ejpam-4262	210	5	(	(	PUNCT
ejpam-4262	210	6	p	p	X
ejpam-4262	210	7	:	:	PUNCT
ejpam-4262	210	8	m	m	NUM
ejpam-4262	210	9	)	)	PUNCT
ejpam-4262	210	10	or	or	CCONJ
ejpam-4262	210	11	r(m	r(m	PROPN
ejpam-4262	210	12	+	+	NUM
ejpam-4262	210	13	m	m	NOUN
ejpam-4262	210	14	)	)	PUNCT
ejpam-4262	210	15	∈	∈	PROPN
ejpam-4262	210	16	p	p	NOUN
ejpam-4262	210	17	.	.	PUNCT
ejpam-4262	211	1	thus	thus	ADV
ejpam-4262	211	2	r	r	NOUN
ejpam-4262	211	3	∈	∈	PROPN
ejpam-4262	211	4	α((p	α((p	NOUN
ejpam-4262	212	1	:	:	PUNCT
ejpam-4262	213	1	sm	sm	X
ejpam-4262	213	2	)	)	PUNCT
ejpam-4262	213	3	)	)	PUNCT
ejpam-4262	214	1	or	or	CCONJ
ejpam-4262	214	2	r	r	NOUN
ejpam-4262	214	3	∈	∈	PROPN
ejpam-4262	214	4	α((p	α((p	NOUN
ejpam-4262	215	1	:	:	PUNCT
ejpam-4262	216	1	m	m	X
ejpam-4262	216	2	)	)	PUNCT
ejpam-4262	216	3	)	)	PUNCT
ejpam-4262	216	4	.	.	PUNCT
ejpam-4262	217	1	this	this	PRON
ejpam-4262	217	2	proves	prove	VERB
ejpam-4262	217	3	that	that	SCONJ
ejpam-4262	217	4	(	(	PUNCT
ejpam-4262	217	5	p	p	X
ejpam-4262	217	6	:	:	PUNCT
ejpam-4262	217	7	sm	sm	NOUN
ejpam-4262	217	8	)	)	PUNCT
ejpam-4262	217	9	⊆	⊆	NUM
ejpam-4262	217	10	α((p	α((p	NOUN
ejpam-4262	217	11	:	:	PUNCT
ejpam-4262	217	12	sm	sm	PROPN
ejpam-4262	217	13	)	)	PUNCT
ejpam-4262	217	14	)	)	PUNCT
ejpam-4262	217	15	∪	∪	ADP
ejpam-4262	217	16	α((p	α((p	NOUN
ejpam-4262	217	17	:	:	PUNCT
ejpam-4262	217	18	m	m	X
ejpam-4262	217	19	)	)	PUNCT
ejpam-4262	217	20	)	)	PUNCT
ejpam-4262	217	21	∪	∪	ADV
ejpam-4262	217	22	(	(	PUNCT
ejpam-4262	217	23	ϕ(p	ϕ(p	PROPN
ejpam-4262	217	24	)	)	PUNCT
ejpam-4262	217	25	:	:	PUNCT
ejpam-4262	217	26	sm	sm	NOUN
ejpam-4262	217	27	)	)	PUNCT
ejpam-4262	217	28	.	.	PUNCT
ejpam-4262	218	1	(	(	PUNCT
ejpam-4262	218	2	ii	ii	NOUN
ejpam-4262	218	3	)	)	PUNCT
ejpam-4262	218	4	→	→	SYM
ejpam-4262	218	5	(	(	PUNCT
ejpam-4262	218	6	i	i	NOUN
ejpam-4262	218	7	)	)	PUNCT
ejpam-4262	218	8	assume	assume	VERB
ejpam-4262	218	9	that	that	SCONJ
ejpam-4262	218	10	(	(	PUNCT
ejpam-4262	218	11	ii	ii	NOUN
ejpam-4262	218	12	)	)	PUNCT
ejpam-4262	218	13	holds	hold	VERB
ejpam-4262	218	14	.	.	PUNCT
ejpam-4262	219	1	let	let	VERB
ejpam-4262	219	2	r	r	NOUN
ejpam-4262	219	3	,	,	PUNCT
ejpam-4262	219	4	s	s	PART
ejpam-4262	219	5	∈	∈	PROPN
ejpam-4262	219	6	r	r	NOUN
ejpam-4262	219	7	and	and	CCONJ
ejpam-4262	219	8	m	m	PROPN
ejpam-4262	219	9	∈	∈	NOUN
ejpam-4262	219	10	m	m	AUX
ejpam-4262	219	11	be	be	VERB
ejpam-4262	219	12	such	such	ADJ
ejpam-4262	219	13	that	that	SCONJ
ejpam-4262	219	14	rsm	rsm	PROPN
ejpam-4262	219	15	∈	∈	PROPN
ejpam-4262	219	16	p\ϕ(p	p\ϕ(p	NOUN
ejpam-4262	219	17	)	)	PUNCT
ejpam-4262	220	1	and	and	CCONJ
ejpam-4262	220	2	sm	sm	INTJ
ejpam-4262	220	3	/∈	/∈	PUNCT
ejpam-4262	220	4	α(p	α(p	PROPN
ejpam-4262	220	5	)	)	PUNCT
ejpam-4262	220	6	.	.	PUNCT
ejpam-4262	221	1	then	then	ADV
ejpam-4262	221	2	r	r	NOUN
ejpam-4262	221	3	∈	∈	PROPN
ejpam-4262	221	4	(	(	PUNCT
ejpam-4262	221	5	p	p	X
ejpam-4262	221	6	:	:	PUNCT
ejpam-4262	221	7	sm	sm	NOUN
ejpam-4262	221	8	)	)	PUNCT
ejpam-4262	221	9	and	and	CCONJ
ejpam-4262	221	10	r	r	NOUN
ejpam-4262	221	11	/∈	/∈	PUNCT
ejpam-4262	221	12	(	(	PUNCT
ejpam-4262	221	13	ϕ(p	ϕ(p	PROPN
ejpam-4262	221	14	)	)	PUNCT
ejpam-4262	221	15	:	:	PUNCT
ejpam-4262	221	16	sm	sm	X
ejpam-4262	221	17	)	)	PUNCT
ejpam-4262	221	18	.	.	PUNCT
ejpam-4262	222	1	this	this	PRON
ejpam-4262	222	2	implies	imply	VERB
ejpam-4262	222	3	that	that	SCONJ
ejpam-4262	222	4	r	r	NOUN
ejpam-4262	222	5	∈	∈	PROPN
ejpam-4262	222	6	α((p	α((p	NOUN
ejpam-4262	222	7	:	:	PUNCT
ejpam-4262	222	8	sm	sm	PROPN
ejpam-4262	222	9	)	)	PUNCT
ejpam-4262	222	10	)	)	PUNCT
ejpam-4262	222	11	∪	∪	ADP
ejpam-4262	222	12	α((p	α((p	NOUN
ejpam-4262	222	13	:	:	PUNCT
ejpam-4262	222	14	m	m	X
ejpam-4262	222	15	)	)	PUNCT
ejpam-4262	222	16	)	)	PUNCT
ejpam-4262	222	17	.	.	PUNCT
ejpam-4262	223	1	hence	hence	ADV
ejpam-4262	223	2	rs	rs	INTJ
ejpam-4262	224	1	+	+	CCONJ
ejpam-4262	224	2	rs	rs	PROPN
ejpam-4262	224	3	∈	∈	PROPN
ejpam-4262	224	4	(	(	PUNCT
ejpam-4262	224	5	p	p	X
ejpam-4262	224	6	:	:	PUNCT
ejpam-4262	224	7	m	m	NUM
ejpam-4262	224	8	)	)	PUNCT
ejpam-4262	224	9	or	or	CCONJ
ejpam-4262	224	10	r(m	r(m	ADJ
ejpam-4262	224	11	+	+	NOUN
ejpam-4262	224	12	m	m	NOUN
ejpam-4262	224	13	)	)	PUNCT
ejpam-4262	224	14	∈	∈	PROPN
ejpam-4262	224	15	p	p	NOUN
ejpam-4262	224	16	.	.	PUNCT
ejpam-4262	225	1	this	this	PRON
ejpam-4262	225	2	shows	show	VERB
ejpam-4262	225	3	that	that	SCONJ
ejpam-4262	225	4	p	p	NOUN
ejpam-4262	225	5	is	be	AUX
ejpam-4262	225	6	a	a	DET
ejpam-4262	225	7	ϕ-β	ϕ-β	ADJ
ejpam-4262	225	8	-	-	ADJ
ejpam-4262	225	9	absorbing	absorbing	ADJ
ejpam-4262	225	10	submodule	submodule	NOUN
ejpam-4262	225	11	of	of	ADP
ejpam-4262	225	12	m	m	PROPN
ejpam-4262	225	13	.	.	PUNCT
ejpam-4262	226	1	by	by	ADP
ejpam-4262	226	2	the	the	DET
ejpam-4262	226	3	definition	definition	NOUN
ejpam-4262	226	4	of	of	ADP
ejpam-4262	226	5	ϕ1	ϕ1	NOUN
ejpam-4262	226	6	,	,	PUNCT
ejpam-4262	226	7	a	a	DET
ejpam-4262	226	8	proper	proper	ADJ
ejpam-4262	226	9	submodule	submodule	NOUN
ejpam-4262	226	10	p	p	NOUN
ejpam-4262	226	11	of	of	ADP
ejpam-4262	226	12	m	m	PROPN
ejpam-4262	226	13	is	be	AUX
ejpam-4262	226	14	said	say	VERB
ejpam-4262	226	15	to	to	PART
ejpam-4262	226	16	be	be	AUX
ejpam-4262	226	17	ϕ1	ϕ1	NOUN
ejpam-4262	226	18	-	-	PUNCT
ejpam-4262	226	19	β	β	NOUN
ejpam-4262	226	20	-	-	ADJ
ejpam-4262	226	21	absorbing	absorbing	ADJ
ejpam-4262	226	22	provided	provide	VERB
ejpam-4262	226	23	for	for	ADP
ejpam-4262	226	24	each	each	DET
ejpam-4262	226	25	r	r	NOUN
ejpam-4262	226	26	,	,	PUNCT
ejpam-4262	226	27	s	s	NOUN
ejpam-4262	226	28	∈	∈	PROPN
ejpam-4262	226	29	r	r	NOUN
ejpam-4262	226	30	and	and	CCONJ
ejpam-4262	226	31	m	m	NOUN
ejpam-4262	226	32	∈m	∈m	NOUN
ejpam-4262	226	33	,	,	PUNCT
ejpam-4262	226	34	if	if	SCONJ
ejpam-4262	226	35	rsm	rsm	PROPN
ejpam-4262	226	36	∈	∈	PROPN
ejpam-4262	226	37	p\(p	p\(p	X
ejpam-4262	226	38	:	:	PUNCT
ejpam-4262	226	39	m)β(p	m)β(p	ADJ
ejpam-4262	226	40	)	)	PUNCT
ejpam-4262	226	41	,	,	PUNCT
ejpam-4262	226	42	then	then	ADV
ejpam-4262	226	43	rs+	rs+	VERB
ejpam-4262	226	44	rs	rs	PROPN
ejpam-4262	226	45	∈	∈	PROPN
ejpam-4262	226	46	(	(	PUNCT
ejpam-4262	226	47	p	p	X
ejpam-4262	226	48	:	:	PUNCT
ejpam-4262	226	49	m	m	NUM
ejpam-4262	226	50	)	)	PUNCT
ejpam-4262	226	51	or	or	CCONJ
ejpam-4262	226	52	r(m	r(m	ADJ
ejpam-4262	226	53	+	+	NOUN
ejpam-4262	226	54	m	m	NOUN
ejpam-4262	226	55	)	)	PUNCT
ejpam-4262	226	56	∈	∈	PROPN
ejpam-4262	226	57	p	p	NOUN
ejpam-4262	226	58	or	or	CCONJ
ejpam-4262	226	59	s(m	s(m	NOUN
ejpam-4262	226	60	+	+	PROPN
ejpam-4262	226	61	m	m	NOUN
ejpam-4262	226	62	)	)	PUNCT
ejpam-4262	226	63	∈	∈	PROPN
ejpam-4262	226	64	p	p	NOUN
ejpam-4262	226	65	.	.	PUNCT
ejpam-4262	227	1	for	for	ADP
ejpam-4262	227	2	an	an	DET
ejpam-4262	227	3	element	element	NOUN
ejpam-4262	227	4	m	m	NOUN
ejpam-4262	227	5	of	of	ADP
ejpam-4262	227	6	an	an	DET
ejpam-4262	227	7	r	r	NOUN
ejpam-4262	227	8	-	-	PUNCT
ejpam-4262	227	9	module	module	NOUN
ejpam-4262	227	10	m	m	NOUN
ejpam-4262	227	11	,	,	PUNCT
ejpam-4262	227	12	we	we	PRON
ejpam-4262	227	13	recall	recall	VERB
ejpam-4262	227	14	the	the	DET
ejpam-4262	227	15	symbol	symbol	NOUN
ejpam-4262	227	16	that	that	PRON
ejpam-4262	227	17	(	(	PUNCT
ejpam-4262	227	18	{	{	PUNCT
ejpam-4262	227	19	0	0	NUM
ejpam-4262	227	20	}	}	PUNCT
ejpam-4262	227	21	:	:	PUNCT
ejpam-4262	227	22	m	m	X
ejpam-4262	227	23	)	)	PUNCT
ejpam-4262	228	1	=	=	PRON
ejpam-4262	228	2	{	{	PUNCT
ejpam-4262	228	3	r	r	NOUN
ejpam-4262	228	4	∈	∈	PROPN
ejpam-4262	228	5	r	r	NOUN
ejpam-4262	228	6	|	|	NOUN
ejpam-4262	228	7	rm	rm	NOUN
ejpam-4262	228	8	=	=	PUNCT
ejpam-4262	228	9	{	{	PUNCT
ejpam-4262	228	10	0	0	NUM
ejpam-4262	228	11	}	}	PUNCT
ejpam-4262	228	12	}	}	PUNCT
ejpam-4262	228	13	and	and	CCONJ
ejpam-4262	228	14	denote	denote	VERB
ejpam-4262	228	15	(	(	PUNCT
ejpam-4262	228	16	{	{	PUNCT
ejpam-4262	228	17	0	0	NUM
ejpam-4262	228	18	}	}	PUNCT
ejpam-4262	228	19	:	:	PUNCT
ejpam-4262	228	20	m	m	X
ejpam-4262	228	21	)	)	PUNCT
ejpam-4262	228	22	by	by	ADP
ejpam-4262	228	23	(	(	PUNCT
ejpam-4262	228	24	0	0	NUM
ejpam-4262	228	25	:	:	PUNCT
ejpam-4262	228	26	m	m	X
ejpam-4262	228	27	)	)	PUNCT
ejpam-4262	228	28	for	for	ADP
ejpam-4262	228	29	short	short	ADJ
ejpam-4262	228	30	.	.	PUNCT
ejpam-4262	229	1	finally	finally	ADV
ejpam-4262	229	2	,	,	PUNCT
ejpam-4262	229	3	we	we	PRON
ejpam-4262	229	4	show	show	VERB
ejpam-4262	229	5	some	some	DET
ejpam-4262	229	6	assumptions	assumption	NOUN
ejpam-4262	229	7	which	which	PRON
ejpam-4262	229	8	β	β	X
ejpam-4262	229	9	-	-	ADJ
ejpam-4262	229	10	absorbing	absorbing	ADJ
ejpam-4262	229	11	submodules	submodule	NOUN
ejpam-4262	229	12	and	and	CCONJ
ejpam-4262	229	13	ϕ1	ϕ1	NOUN
ejpam-4262	229	14	-	-	PUNCT
ejpam-4262	229	15	β	β	NOUN
ejpam-4262	229	16	-	-	ADJ
ejpam-4262	229	17	absorbing	absorbing	ADJ
ejpam-4262	229	18	submodules	submodule	NOUN
ejpam-4262	229	19	are	be	AUX
ejpam-4262	229	20	equivalent	equivalent	ADJ
ejpam-4262	229	21	.	.	PUNCT
ejpam-4262	230	1	theorem	theorem	ADJ
ejpam-4262	230	2	4	4	NUM
ejpam-4262	230	3	.	.	PUNCT
ejpam-4262	231	1	let	let	VERB
ejpam-4262	231	2	m	m	PRON
ejpam-4262	231	3	be	be	AUX
ejpam-4262	231	4	a	a	DET
ejpam-4262	231	5	nonzero	nonzero	ADJ
ejpam-4262	231	6	element	element	NOUN
ejpam-4262	231	7	of	of	ADP
ejpam-4262	231	8	an	an	DET
ejpam-4262	231	9	r	r	NOUN
ejpam-4262	231	10	-	-	PUNCT
ejpam-4262	231	11	module	module	NOUN
ejpam-4262	231	12	m	m	NOUN
ejpam-4262	231	13	such	such	ADJ
ejpam-4262	231	14	that	that	SCONJ
ejpam-4262	231	15	(	(	PUNCT
ejpam-4262	231	16	0	0	NUM
ejpam-4262	231	17	:	:	PUNCT
ejpam-4262	231	18	m	m	X
ejpam-4262	231	19	)	)	PUNCT
ejpam-4262	231	20	=	=	PRON
ejpam-4262	231	21	{	{	PUNCT
ejpam-4262	231	22	0	0	NUM
ejpam-4262	231	23	}	}	PUNCT
ejpam-4262	231	24	and	and	CCONJ
ejpam-4262	231	25	rm	rm	PROPN
ejpam-4262	231	26	̸=	̸=	PROPN
ejpam-4262	231	27	m	m	PROPN
ejpam-4262	231	28	.	.	PUNCT
ejpam-4262	232	1	then	then	ADV
ejpam-4262	232	2	rm	rm	PROPN
ejpam-4262	232	3	is	be	AUX
ejpam-4262	232	4	a	a	DET
ejpam-4262	232	5	β	β	NOUN
ejpam-4262	232	6	-	-	ADJ
ejpam-4262	232	7	absorbing	absorbing	ADJ
ejpam-4262	232	8	submodule	submodule	NOUN
ejpam-4262	232	9	of	of	ADP
ejpam-4262	232	10	m	m	PROPN
ejpam-4262	232	11	if	if	SCONJ
ejpam-4262	233	1	and	and	CCONJ
ejpam-4262	233	2	only	only	ADV
ejpam-4262	233	3	if	if	SCONJ
ejpam-4262	233	4	rm	rm	PROPN
ejpam-4262	233	5	is	be	AUX
ejpam-4262	233	6	a	a	DET
ejpam-4262	233	7	ϕ1	ϕ1	NOUN
ejpam-4262	233	8	-	-	PUNCT
ejpam-4262	233	9	βabsorbing	βabsorbe	VERB
ejpam-4262	233	10	submodule	submodule	NOUN
ejpam-4262	233	11	of	of	ADP
ejpam-4262	233	12	m	m	PROPN
ejpam-4262	233	13	.	.	PUNCT
ejpam-4262	234	1	proof	proof	NOUN
ejpam-4262	234	2	.	.	PUNCT
ejpam-4262	235	1	(	(	PUNCT
ejpam-4262	235	2	→	→	NOUN
ejpam-4262	235	3	)	)	PUNCT
ejpam-4262	235	4	this	this	PRON
ejpam-4262	235	5	is	be	AUX
ejpam-4262	235	6	obvious	obvious	ADJ
ejpam-4262	235	7	.	.	PUNCT
ejpam-4262	236	1	(	(	PUNCT
ejpam-4262	236	2	←	←	PROPN
ejpam-4262	236	3	)	)	PUNCT
ejpam-4262	236	4	assume	assume	VERB
ejpam-4262	236	5	that	that	SCONJ
ejpam-4262	236	6	rm	rm	PROPN
ejpam-4262	236	7	is	be	AUX
ejpam-4262	236	8	not	not	PART
ejpam-4262	236	9	a	a	DET
ejpam-4262	236	10	β	β	NOUN
ejpam-4262	236	11	-	-	ADJ
ejpam-4262	236	12	absorbing	absorbing	ADJ
ejpam-4262	236	13	submodule	submodule	NOUN
ejpam-4262	236	14	of	of	ADP
ejpam-4262	236	15	m	m	PROPN
ejpam-4262	236	16	.	.	PUNCT
ejpam-4262	237	1	then	then	ADV
ejpam-4262	237	2	there	there	PRON
ejpam-4262	237	3	are	be	VERB
ejpam-4262	237	4	r	r	NOUN
ejpam-4262	237	5	,	,	PUNCT
ejpam-4262	237	6	s	s	NOUN
ejpam-4262	237	7	∈	∈	PROPN
ejpam-4262	237	8	r	r	NOUN
ejpam-4262	237	9	and	and	CCONJ
ejpam-4262	237	10	x	x	AUX
ejpam-4262	237	11	∈m	∈m	NOUN
ejpam-4262	237	12	such	such	ADJ
ejpam-4262	237	13	that	that	DET
ejpam-4262	237	14	rsx	rsx	PROPN
ejpam-4262	237	15	∈	∈	PROPN
ejpam-4262	237	16	rm	rm	NOUN
ejpam-4262	237	17	and	and	CCONJ
ejpam-4262	237	18	rs+rs	rs+rs	NOUN
ejpam-4262	237	19	/∈	/∈	PUNCT
ejpam-4262	238	1	(	(	PUNCT
ejpam-4262	238	2	rm	rm	NOUN
ejpam-4262	238	3	:	:	PUNCT
ejpam-4262	238	4	m	m	PROPN
ejpam-4262	238	5	)	)	PUNCT
ejpam-4262	238	6	and	and	CCONJ
ejpam-4262	238	7	r(x+x	r(x+x	PROPN
ejpam-4262	238	8	)	)	PUNCT
ejpam-4262	238	9	/∈	/∈	PUNCT
ejpam-4262	239	1	rm	rm	NOUN
ejpam-4262	239	2	and	and	CCONJ
ejpam-4262	239	3	s(x+x	s(x+x	PROPN
ejpam-4262	239	4	)	)	PUNCT
ejpam-4262	239	5	/∈	/∈	PUNCT
ejpam-4262	240	1	rm	rm	NOUN
ejpam-4262	240	2	.	.	PUNCT
ejpam-4262	241	1	if	if	SCONJ
ejpam-4262	241	2	rsx	rsx	PROPN
ejpam-4262	241	3	/∈	/∈	PUNCT
ejpam-4262	242	1	(	(	PUNCT
ejpam-4262	242	2	rm	rm	NOUN
ejpam-4262	242	3	:	:	PUNCT
ejpam-4262	242	4	m)β(rm	m)β(rm	NUM
ejpam-4262	242	5	)	)	PUNCT
ejpam-4262	242	6	,	,	PUNCT
ejpam-4262	242	7	then	then	ADV
ejpam-4262	242	8	we	we	PRON
ejpam-4262	242	9	are	be	AUX
ejpam-4262	242	10	done	do	VERB
ejpam-4262	242	11	.	.	PUNCT
ejpam-4262	243	1	assume	assume	VERB
ejpam-4262	243	2	that	that	SCONJ
ejpam-4262	243	3	rsx	rsx	PROPN
ejpam-4262	243	4	∈	∈	PROPN
ejpam-4262	243	5	(	(	PUNCT
ejpam-4262	243	6	rm	rm	NOUN
ejpam-4262	243	7	:	:	PUNCT
ejpam-4262	243	8	m)β(rm	m)β(rm	NUM
ejpam-4262	243	9	)	)	PUNCT
ejpam-4262	243	10	.	.	PUNCT
ejpam-4262	244	1	since	since	SCONJ
ejpam-4262	244	2	r(x	r(x	PROPN
ejpam-4262	244	3	+	+	CCONJ
ejpam-4262	244	4	x	x	NOUN
ejpam-4262	244	5	)	)	PUNCT
ejpam-4262	244	6	/∈	/∈	PUNCT
ejpam-4262	245	1	rm	rm	PROPN
ejpam-4262	245	2	,	,	PUNCT
ejpam-4262	245	3	r(x	r(x	PROPN
ejpam-4262	245	4	+	+	CCONJ
ejpam-4262	245	5	x	x	PUNCT
ejpam-4262	246	1	+	+	NUM
ejpam-4262	246	2	m	m	VERB
ejpam-4262	246	3	+	+	NUM
ejpam-4262	246	4	m	m	VERB
ejpam-4262	246	5	)	)	PUNCT
ejpam-4262	246	6	/∈	/∈	PUNCT
ejpam-4262	247	1	rm	rm	PROPN
ejpam-4262	247	2	.	.	PUNCT
ejpam-4262	248	1	since	since	SCONJ
ejpam-4262	248	2	rsx	rsx	PROPN
ejpam-4262	248	3	∈	∈	PROPN
ejpam-4262	248	4	rm	rm	PROPN
ejpam-4262	248	5	and	and	CCONJ
ejpam-4262	248	6	rsm	rsm	PROPN
ejpam-4262	248	7	∈	∈	PROPN
ejpam-4262	248	8	rm	rm	PROPN
ejpam-4262	248	9	,	,	PUNCT
ejpam-4262	248	10	rs(x	rs(x	PUNCT
ejpam-4262	248	11	+	+	CCONJ
ejpam-4262	248	12	m	m	X
ejpam-4262	248	13	)	)	PUNCT
ejpam-4262	248	14	∈	∈	PROPN
ejpam-4262	248	15	rm	rm	NOUN
ejpam-4262	248	16	.	.	PUNCT
ejpam-4262	249	1	if	if	SCONJ
ejpam-4262	249	2	rs(x	rs(x	PUNCT
ejpam-4262	249	3	+	+	CCONJ
ejpam-4262	249	4	m	m	NOUN
ejpam-4262	249	5	)	)	PUNCT
ejpam-4262	249	6	/∈	/∈	PUNCT
ejpam-4262	250	1	(	(	PUNCT
ejpam-4262	250	2	rm	rm	NOUN
ejpam-4262	250	3	:	:	PUNCT
ejpam-4262	250	4	m)β(rm	m)β(rm	NUM
ejpam-4262	250	5	)	)	PUNCT
ejpam-4262	250	6	,	,	PUNCT
ejpam-4262	250	7	then	then	ADV
ejpam-4262	250	8	we	we	PRON
ejpam-4262	250	9	are	be	AUX
ejpam-4262	250	10	done	do	VERB
ejpam-4262	250	11	.	.	PUNCT
ejpam-4262	250	12	suppose	suppose	VERB
ejpam-4262	250	13	that	that	SCONJ
ejpam-4262	250	14	rs(x	rs(x	PUNCT
ejpam-4262	250	15	+	+	NUM
ejpam-4262	250	16	m	m	X
ejpam-4262	250	17	)	)	PUNCT
ejpam-4262	250	18	∈	∈	PROPN
ejpam-4262	250	19	(	(	PUNCT
ejpam-4262	250	20	rm	rm	NOUN
ejpam-4262	250	21	:	:	PUNCT
ejpam-4262	250	22	m)β(rm	m)β(rm	NUM
ejpam-4262	250	23	)	)	PUNCT
ejpam-4262	250	24	.	.	PUNCT
ejpam-4262	251	1	since	since	SCONJ
ejpam-4262	251	2	rsx	rsx	PROPN
ejpam-4262	251	3	∈	∈	PROPN
ejpam-4262	251	4	(	(	PUNCT
ejpam-4262	251	5	rm	rm	NOUN
ejpam-4262	251	6	:	:	PUNCT
ejpam-4262	251	7	m)β(rm	m)β(rm	NUM
ejpam-4262	251	8	)	)	PUNCT
ejpam-4262	251	9	,	,	PUNCT
ejpam-4262	251	10	rsm	rsm	PROPN
ejpam-4262	251	11	∈	∈	PROPN
ejpam-4262	251	12	(	(	PUNCT
ejpam-4262	251	13	rm	rm	NOUN
ejpam-4262	251	14	:	:	PUNCT
ejpam-4262	251	15	m)β(rm	m)β(rm	NUM
ejpam-4262	251	16	)	)	PUNCT
ejpam-4262	251	17	.	.	PUNCT
ejpam-4262	251	18	note	note	VERB
ejpam-4262	251	19	that	that	SCONJ
ejpam-4262	251	20	β(rm	β(rm	NOUN
ejpam-4262	251	21	)	)	PUNCT
ejpam-4262	251	22	=	=	SYM
ejpam-4262	251	23	β(r)m	β(r)m	NOUN
ejpam-4262	251	24	.	.	PUNCT
ejpam-4262	252	1	hence	hence	ADV
ejpam-4262	252	2	rsm	rsm	PROPN
ejpam-4262	252	3	∈	∈	PROPN
ejpam-4262	252	4	(	(	PUNCT
ejpam-4262	252	5	rm	rm	NOUN
ejpam-4262	252	6	:	:	PUNCT
ejpam-4262	252	7	m)β(r)m	m)β(r)m	NOUN
ejpam-4262	252	8	.	.	PUNCT
ejpam-4262	253	1	thus	thus	ADV
ejpam-4262	253	2	rsm	rsm	X
ejpam-4262	253	3	=	=	SYM
ejpam-4262	253	4	tm	tm	NOUN
ejpam-4262	253	5	for	for	ADP
ejpam-4262	253	6	some	some	DET
ejpam-4262	253	7	t	t	NOUN
ejpam-4262	253	8	∈	∈	PROPN
ejpam-4262	253	9	(	(	PUNCT
ejpam-4262	253	10	rm	rm	NOUN
ejpam-4262	253	11	:	:	PUNCT
ejpam-4262	253	12	m)β(r	m)β(r	PROPN
ejpam-4262	253	13	)	)	PUNCT
ejpam-4262	253	14	.	.	PUNCT
ejpam-4262	254	1	so	so	ADV
ejpam-4262	254	2	rs	rs	ADV
ejpam-4262	254	3	=	=	SYM
ejpam-4262	254	4	t	t	NOUN
ejpam-4262	254	5	∈	∈	PROPN
ejpam-4262	254	6	(	(	PUNCT
ejpam-4262	254	7	rm	rm	NOUN
ejpam-4262	254	8	:	:	PUNCT
ejpam-4262	254	9	m)β(r	m)β(r	PROPN
ejpam-4262	254	10	)	)	PUNCT
ejpam-4262	254	11	⊆	⊆	NUM
ejpam-4262	254	12	(	(	PUNCT
ejpam-4262	254	13	rm	rm	NOUN
ejpam-4262	254	14	:	:	PUNCT
ejpam-4262	254	15	m	m	PROPN
ejpam-4262	254	16	)	)	PUNCT
ejpam-4262	254	17	.	.	PUNCT
ejpam-4262	255	1	consequently	consequently	ADV
ejpam-4262	255	2	,	,	PUNCT
ejpam-4262	255	3	rs+	rs+	VERB
ejpam-4262	255	4	rs	rs	PROPN
ejpam-4262	255	5	∈	∈	PROPN
ejpam-4262	255	6	(	(	PUNCT
ejpam-4262	255	7	rm	rm	NOUN
ejpam-4262	255	8	:	:	PUNCT
ejpam-4262	255	9	m	m	PROPN
ejpam-4262	255	10	)	)	PUNCT
ejpam-4262	255	11	which	which	PRON
ejpam-4262	255	12	is	be	AUX
ejpam-4262	255	13	a	a	DET
ejpam-4262	255	14	contradiction	contradiction	NOUN
ejpam-4262	255	15	.	.	PUNCT
ejpam-4262	256	1	for	for	ADP
ejpam-4262	256	2	each	each	DET
ejpam-4262	256	3	r	r	NOUN
ejpam-4262	256	4	∈	∈	NOUN
ejpam-4262	256	5	r	r	NOUN
ejpam-4262	256	6	,	,	PUNCT
ejpam-4262	256	7	we	we	PRON
ejpam-4262	256	8	would	would	AUX
ejpam-4262	256	9	like	like	VERB
ejpam-4262	256	10	to	to	PART
ejpam-4262	256	11	remind	remind	VERB
ejpam-4262	256	12	that	that	SCONJ
ejpam-4262	256	13	{	{	PUNCT
ejpam-4262	256	14	0}r	0}r	NUM
ejpam-4262	256	15	=	=	SYM
ejpam-4262	256	16	{	{	PUNCT
ejpam-4262	256	17	m	m	NOUN
ejpam-4262	256	18	∈m	∈m	NOUN
ejpam-4262	256	19	|	|	ADV
ejpam-4262	256	20	rm	rm	NOUN
ejpam-4262	256	21	=	=	NOUN
ejpam-4262	256	22	0	0	NUM
ejpam-4262	256	23	}	}	PUNCT
ejpam-4262	256	24	.	.	PUNCT
ejpam-4262	257	1	theorem	theorem	NOUN
ejpam-4262	257	2	5	5	NUM
ejpam-4262	257	3	.	.	PUNCT
ejpam-4262	258	1	let	let	VERB
ejpam-4262	258	2	m	m	PRON
ejpam-4262	258	3	be	be	AUX
ejpam-4262	258	4	an	an	DET
ejpam-4262	258	5	r	r	NOUN
ejpam-4262	258	6	-	-	PUNCT
ejpam-4262	258	7	module	module	NOUN
ejpam-4262	258	8	and	and	CCONJ
ejpam-4262	258	9	r	r	NOUN
ejpam-4262	258	10	∈	∈	NOUN
ejpam-4262	258	11	r	r	NOUN
ejpam-4262	258	12	be	be	VERB
ejpam-4262	258	13	such	such	ADJ
ejpam-4262	258	14	that	that	SCONJ
ejpam-4262	258	15	rm	rm	PROPN
ejpam-4262	258	16	̸=	̸=	PROPN
ejpam-4262	258	17	m	m	PROPN
ejpam-4262	258	18	and	and	CCONJ
ejpam-4262	258	19	{	{	PUNCT
ejpam-4262	258	20	0}r	0}r	NOUN
ejpam-4262	258	21	⊆	⊆	NUM
ejpam-4262	258	22	β(rm	β(rm	NOUN
ejpam-4262	258	23	)	)	PUNCT
ejpam-4262	258	24	.	.	PUNCT
ejpam-4262	259	1	then	then	ADV
ejpam-4262	259	2	rm	rm	PROPN
ejpam-4262	259	3	is	be	AUX
ejpam-4262	259	4	a	a	DET
ejpam-4262	259	5	β	β	NOUN
ejpam-4262	259	6	-	-	ADJ
ejpam-4262	259	7	absorbing	absorbing	ADJ
ejpam-4262	259	8	submodule	submodule	NOUN
ejpam-4262	259	9	of	of	ADP
ejpam-4262	259	10	m	m	PROPN
ejpam-4262	259	11	if	if	SCONJ
ejpam-4262	260	1	and	and	CCONJ
ejpam-4262	260	2	only	only	ADV
ejpam-4262	260	3	if	if	SCONJ
ejpam-4262	260	4	rm	rm	PROPN
ejpam-4262	260	5	is	be	AUX
ejpam-4262	260	6	a	a	DET
ejpam-4262	260	7	ϕ1	ϕ1	NOUN
ejpam-4262	260	8	-	-	PUNCT
ejpam-4262	260	9	β	β	NOUN
ejpam-4262	260	10	-	-	ADJ
ejpam-4262	260	11	absorbing	absorbing	ADJ
ejpam-4262	260	12	submodule	submodule	NOUN
ejpam-4262	260	13	of	of	ADP
ejpam-4262	260	14	m	m	PROPN
ejpam-4262	260	15	.	.	PUNCT
ejpam-4262	261	1	proof	proof	NOUN
ejpam-4262	261	2	.	.	PUNCT
ejpam-4262	262	1	(	(	PUNCT
ejpam-4262	262	2	→	→	NOUN
ejpam-4262	262	3	)	)	PUNCT
ejpam-4262	262	4	this	this	PRON
ejpam-4262	262	5	is	be	AUX
ejpam-4262	262	6	obvious	obvious	ADJ
ejpam-4262	262	7	.	.	PUNCT
ejpam-4262	263	1	(	(	PUNCT
ejpam-4262	263	2	←	←	PROPN
ejpam-4262	263	3	)	)	PUNCT
ejpam-4262	263	4	assume	assume	VERB
ejpam-4262	263	5	that	that	SCONJ
ejpam-4262	263	6	rm	rm	PROPN
ejpam-4262	263	7	is	be	AUX
ejpam-4262	263	8	a	a	DET
ejpam-4262	263	9	ϕ1	ϕ1	NOUN
ejpam-4262	263	10	-	-	PUNCT
ejpam-4262	263	11	β	β	NOUN
ejpam-4262	263	12	-	-	ADJ
ejpam-4262	263	13	absorbing	absorbing	ADJ
ejpam-4262	263	14	submodule	submodule	NOUN
ejpam-4262	263	15	of	of	ADP
ejpam-4262	263	16	m	m	PROPN
ejpam-4262	263	17	.	.	PUNCT
ejpam-4262	264	1	let	let	VERB
ejpam-4262	264	2	a	a	DET
ejpam-4262	264	3	,	,	PUNCT
ejpam-4262	264	4	b	b	X
ejpam-4262	264	5	∈	∈	PROPN
ejpam-4262	264	6	r	r	NOUN
ejpam-4262	264	7	and	and	CCONJ
ejpam-4262	264	8	m	m	PROPN
ejpam-4262	264	9	∈	∈	NOUN
ejpam-4262	264	10	m	m	VERB
ejpam-4262	264	11	such	such	ADJ
ejpam-4262	264	12	that	that	SCONJ
ejpam-4262	264	13	abm	abm	PROPN
ejpam-4262	264	14	∈	∈	PROPN
ejpam-4262	264	15	rm	rm	PROPN
ejpam-4262	264	16	.	.	PUNCT
ejpam-4262	265	1	there	there	PRON
ejpam-4262	265	2	are	be	VERB
ejpam-4262	265	3	2	2	NUM
ejpam-4262	265	4	cases	case	NOUN
ejpam-4262	265	5	to	to	PART
ejpam-4262	265	6	be	be	AUX
ejpam-4262	265	7	considered	consider	VERB
ejpam-4262	265	8	:	:	PUNCT
ejpam-4262	265	9	(	(	PUNCT
ejpam-4262	265	10	i	i	NOUN
ejpam-4262	265	11	)	)	PUNCT
ejpam-4262	265	12	abm	abm	PROPN
ejpam-4262	265	13	/∈	/∈	PUNCT
ejpam-4262	266	1	(	(	PUNCT
ejpam-4262	266	2	rm	rm	NOUN
ejpam-4262	266	3	:	:	PUNCT
ejpam-4262	266	4	m)β(rm	m)β(rm	NUM
ejpam-4262	266	5	)	)	PUNCT
ejpam-4262	266	6	,	,	PUNCT
ejpam-4262	266	7	(	(	PUNCT
ejpam-4262	266	8	ii	ii	NOUN
ejpam-4262	266	9	)	)	PUNCT
ejpam-4262	266	10	abm	abm	PROPN
ejpam-4262	266	11	∈	∈	PROPN
ejpam-4262	266	12	(	(	PUNCT
ejpam-4262	266	13	rm	rm	NOUN
ejpam-4262	266	14	:	:	PUNCT
ejpam-4262	266	15	m)β(rm	m)β(rm	NUM
ejpam-4262	266	16	)	)	PUNCT
ejpam-4262	266	17	.	.	PUNCT
ejpam-4262	267	1	references	reference	NOUN
ejpam-4262	267	2	334	334	NUM
ejpam-4262	267	3	first	first	ADJ
ejpam-4262	267	4	,	,	PUNCT
ejpam-4262	267	5	we	we	PRON
ejpam-4262	267	6	consider	consider	VERB
ejpam-4262	267	7	case	case	NOUN
ejpam-4262	267	8	(	(	PUNCT
ejpam-4262	267	9	i	i	NOUN
ejpam-4262	267	10	)	)	PUNCT
ejpam-4262	267	11	.	.	PUNCT
ejpam-4262	268	1	since	since	SCONJ
ejpam-4262	268	2	rm	rm	PROPN
ejpam-4262	268	3	is	be	AUX
ejpam-4262	268	4	a	a	DET
ejpam-4262	268	5	ϕ1	ϕ1	NOUN
ejpam-4262	268	6	-	-	PUNCT
ejpam-4262	268	7	β	β	NOUN
ejpam-4262	268	8	-	-	ADJ
ejpam-4262	268	9	absorbing	absorbing	ADJ
ejpam-4262	268	10	submodule	submodule	NOUN
ejpam-4262	268	11	of	of	ADP
ejpam-4262	268	12	m	m	PROPN
ejpam-4262	268	13	,	,	PUNCT
ejpam-4262	268	14	ab	ab	PROPN
ejpam-4262	269	1	+	+	CCONJ
ejpam-4262	269	2	ab	ab	PROPN
ejpam-4262	269	3	∈	∈	PROPN
ejpam-4262	269	4	(	(	PUNCT
ejpam-4262	269	5	rm	rm	NOUN
ejpam-4262	269	6	:	:	PUNCT
ejpam-4262	269	7	m	m	PROPN
ejpam-4262	269	8	)	)	PUNCT
ejpam-4262	269	9	or	or	CCONJ
ejpam-4262	269	10	a(m	a(m	PROPN
ejpam-4262	269	11	+	+	NUM
ejpam-4262	269	12	m	m	NOUN
ejpam-4262	269	13	)	)	PUNCT
ejpam-4262	269	14	∈	∈	PROPN
ejpam-4262	269	15	rm	rm	NOUN
ejpam-4262	269	16	or	or	CCONJ
ejpam-4262	269	17	b(m	b(m	PROPN
ejpam-4262	269	18	+	+	CCONJ
ejpam-4262	269	19	m	m	NOUN
ejpam-4262	269	20	)	)	PUNCT
ejpam-4262	269	21	∈	∈	PROPN
ejpam-4262	269	22	rm	rm	NOUN
ejpam-4262	269	23	.	.	PUNCT
ejpam-4262	270	1	next	next	ADJ
ejpam-4262	270	2	,	,	PUNCT
ejpam-4262	270	3	case	case	NOUN
ejpam-4262	270	4	(	(	PUNCT
ejpam-4262	270	5	ii	ii	NOUN
ejpam-4262	270	6	)	)	PUNCT
ejpam-4262	270	7	is	be	AUX
ejpam-4262	270	8	considered	consider	VERB
ejpam-4262	270	9	.	.	PUNCT
ejpam-4262	271	1	since	since	SCONJ
ejpam-4262	271	2	abm	abm	PROPN
ejpam-4262	271	3	∈	∈	PROPN
ejpam-4262	271	4	rm	rm	PROPN
ejpam-4262	271	5	and	and	CCONJ
ejpam-4262	271	6	rbm	rbm	PROPN
ejpam-4262	271	7	∈	∈	PROPN
ejpam-4262	271	8	rm	rm	PROPN
ejpam-4262	271	9	,	,	PUNCT
ejpam-4262	271	10	(	(	PUNCT
ejpam-4262	271	11	ab	ab	PROPN
ejpam-4262	271	12	+	+	PROPN
ejpam-4262	271	13	rb)m	rb)m	PROPN
ejpam-4262	271	14	∈	∈	PROPN
ejpam-4262	271	15	rm	rm	NOUN
ejpam-4262	271	16	.	.	PUNCT
ejpam-4262	272	1	if	if	SCONJ
ejpam-4262	272	2	(	(	PUNCT
ejpam-4262	272	3	ab	ab	PROPN
ejpam-4262	272	4	+	+	PROPN
ejpam-4262	272	5	rb)m	rb)m	PROPN
ejpam-4262	272	6	/∈	/∈	PUNCT
ejpam-4262	272	7	(	(	PUNCT
ejpam-4262	272	8	rm	rm	NOUN
ejpam-4262	272	9	:	:	PUNCT
ejpam-4262	272	10	m)β(rm	m)β(rm	NUM
ejpam-4262	272	11	)	)	PUNCT
ejpam-4262	272	12	,	,	PUNCT
ejpam-4262	272	13	then	then	ADV
ejpam-4262	272	14	(	(	PUNCT
ejpam-4262	272	15	a	a	DET
ejpam-4262	272	16	+	+	X
ejpam-4262	272	17	r)b	r)b	X
ejpam-4262	272	18	+	+	CCONJ
ejpam-4262	272	19	(	(	PUNCT
ejpam-4262	272	20	a	a	DET
ejpam-4262	272	21	+	+	X
ejpam-4262	272	22	r)b	r)b	X
ejpam-4262	272	23	∈	∈	NOUN
ejpam-4262	272	24	(	(	PUNCT
ejpam-4262	272	25	rm	rm	NOUN
ejpam-4262	272	26	:	:	PUNCT
ejpam-4262	272	27	m	m	PROPN
ejpam-4262	272	28	)	)	PUNCT
ejpam-4262	272	29	or	or	CCONJ
ejpam-4262	272	30	(	(	PUNCT
ejpam-4262	272	31	a	a	DET
ejpam-4262	272	32	+	+	NOUN
ejpam-4262	272	33	r)(m	r)(m	NOUN
ejpam-4262	272	34	+	+	NUM
ejpam-4262	272	35	m	m	X
ejpam-4262	272	36	)	)	PUNCT
ejpam-4262	272	37	∈	∈	PROPN
ejpam-4262	272	38	rm	rm	NOUN
ejpam-4262	272	39	or	or	CCONJ
ejpam-4262	272	40	b(m	b(m	PROPN
ejpam-4262	272	41	+	+	CCONJ
ejpam-4262	272	42	m	m	NOUN
ejpam-4262	272	43	)	)	PUNCT
ejpam-4262	272	44	∈	∈	PROPN
ejpam-4262	272	45	rm	rm	NOUN
ejpam-4262	272	46	.	.	PUNCT
ejpam-4262	273	1	since	since	SCONJ
ejpam-4262	273	2	rb	rb	NOUN
ejpam-4262	273	3	+	+	CCONJ
ejpam-4262	273	4	rb	rb	X
ejpam-4262	273	5	∈	∈	PROPN
ejpam-4262	273	6	(	(	PUNCT
ejpam-4262	273	7	rm	rm	NOUN
ejpam-4262	273	8	:	:	PUNCT
ejpam-4262	273	9	m	m	PROPN
ejpam-4262	273	10	)	)	PUNCT
ejpam-4262	273	11	and	and	CCONJ
ejpam-4262	273	12	r(m	r(m	PROPN
ejpam-4262	273	13	+	+	NUM
ejpam-4262	273	14	m	m	NOUN
ejpam-4262	273	15	)	)	PUNCT
ejpam-4262	273	16	∈	∈	PROPN
ejpam-4262	273	17	rm	rm	PROPN
ejpam-4262	273	18	,	,	PUNCT
ejpam-4262	273	19	ab	ab	PROPN
ejpam-4262	273	20	+	+	CCONJ
ejpam-4262	273	21	ab	ab	PROPN
ejpam-4262	273	22	∈	∈	PROPN
ejpam-4262	273	23	(	(	PUNCT
ejpam-4262	273	24	rm	rm	NOUN
ejpam-4262	273	25	:	:	PUNCT
ejpam-4262	273	26	m	m	PROPN
ejpam-4262	273	27	)	)	PUNCT
ejpam-4262	273	28	or	or	CCONJ
ejpam-4262	273	29	a(m	a(m	PROPN
ejpam-4262	273	30	+	+	NUM
ejpam-4262	273	31	m	m	NOUN
ejpam-4262	273	32	)	)	PUNCT
ejpam-4262	273	33	∈	∈	PROPN
ejpam-4262	273	34	rm	rm	NOUN
ejpam-4262	273	35	or	or	CCONJ
ejpam-4262	273	36	b(m	b(m	PROPN
ejpam-4262	273	37	+	+	CCONJ
ejpam-4262	273	38	m	m	NOUN
ejpam-4262	273	39	)	)	PUNCT
ejpam-4262	273	40	∈	∈	PROPN
ejpam-4262	273	41	rm	rm	PROPN
ejpam-4262	273	42	.	.	PUNCT
ejpam-4262	274	1	assume	assume	VERB
ejpam-4262	274	2	that	that	SCONJ
ejpam-4262	274	3	(	(	PUNCT
ejpam-4262	274	4	ab	ab	PROPN
ejpam-4262	274	5	+	+	PROPN
ejpam-4262	274	6	rb)m	rb)m	PROPN
ejpam-4262	274	7	∈	∈	PROPN
ejpam-4262	274	8	(	(	PUNCT
ejpam-4262	274	9	rm	rm	NOUN
ejpam-4262	274	10	:	:	PUNCT
ejpam-4262	274	11	m)β(rm	m)β(rm	NUM
ejpam-4262	274	12	)	)	PUNCT
ejpam-4262	274	13	.	.	PUNCT
ejpam-4262	275	1	since	since	SCONJ
ejpam-4262	275	2	β(rm	β(rm	NOUN
ejpam-4262	275	3	)	)	PUNCT
ejpam-4262	275	4	=	=	SYM
ejpam-4262	275	5	rβ(m	rβ(m	NOUN
ejpam-4262	275	6	)	)	PUNCT
ejpam-4262	275	7	,	,	PUNCT
ejpam-4262	275	8	(	(	PUNCT
ejpam-4262	275	9	ab	ab	PROPN
ejpam-4262	275	10	+	+	CCONJ
ejpam-4262	275	11	rb)m	rb)m	PROPN
ejpam-4262	275	12	and	and	CCONJ
ejpam-4262	275	13	abm	abm	PROPN
ejpam-4262	275	14	are	be	AUX
ejpam-4262	275	15	elements	element	NOUN
ejpam-4262	275	16	of	of	ADP
ejpam-4262	275	17	r(rm	r(rm	NOUN
ejpam-4262	275	18	:	:	PUNCT
ejpam-4262	275	19	m)β(m	m)β(m	PROPN
ejpam-4262	275	20	)	)	PUNCT
ejpam-4262	275	21	.	.	PUNCT
ejpam-4262	276	1	this	this	PRON
ejpam-4262	276	2	implies	imply	VERB
ejpam-4262	276	3	that	that	SCONJ
ejpam-4262	276	4	rbm	rbm	PROPN
ejpam-4262	276	5	∈	∈	PROPN
ejpam-4262	276	6	r(rm	r(rm	NOUN
ejpam-4262	276	7	:	:	PUNCT
ejpam-4262	276	8	m)β(m	m)β(m	PROPN
ejpam-4262	276	9	)	)	PUNCT
ejpam-4262	276	10	.	.	PUNCT
ejpam-4262	277	1	thus	thus	ADV
ejpam-4262	277	2	rbm	rbm	PROPN
ejpam-4262	277	3	=	=	SYM
ejpam-4262	277	4	ry	ry	PROPN
ejpam-4262	277	5	for	for	ADP
ejpam-4262	277	6	some	some	DET
ejpam-4262	277	7	y	y	PROPN
ejpam-4262	277	8	∈	∈	PROPN
ejpam-4262	277	9	(	(	PUNCT
ejpam-4262	277	10	rm	rm	NOUN
ejpam-4262	277	11	:	:	PUNCT
ejpam-4262	277	12	m)β(m	m)β(m	PROPN
ejpam-4262	277	13	)	)	PUNCT
ejpam-4262	277	14	.	.	PUNCT
ejpam-4262	278	1	so	so	ADV
ejpam-4262	278	2	bm	bm	PROPN
ejpam-4262	278	3	−	−	PROPN
ejpam-4262	278	4	y	y	PROPN
ejpam-4262	278	5	∈	∈	PROPN
ejpam-4262	278	6	{	{	PUNCT
ejpam-4262	278	7	0}r	0}r	NOUN
ejpam-4262	278	8	⊆	⊆	NUM
ejpam-4262	278	9	β(rm	β(rm	NOUN
ejpam-4262	278	10	)	)	PUNCT
ejpam-4262	278	11	.	.	PUNCT
ejpam-4262	279	1	therefore	therefore	ADV
ejpam-4262	279	2	bm	bm	PROPN
ejpam-4262	279	3	=	=	PRON
ejpam-4262	279	4	(	(	PUNCT
ejpam-4262	279	5	bm−y)+y	bm−y)+y	PROPN
ejpam-4262	279	6	∈	∈	PROPN
ejpam-4262	279	7	(	(	PUNCT
ejpam-4262	279	8	rm	rm	NOUN
ejpam-4262	279	9	:	:	PUNCT
ejpam-4262	279	10	m)β(rm)+{0}r	m)β(rm)+{0}r	VERB
ejpam-4262	279	11	⊆	⊆	NUM
ejpam-4262	279	12	β(rm	β(rm	NOUN
ejpam-4262	279	13	)	)	PUNCT
ejpam-4262	279	14	.	.	PUNCT
ejpam-4262	280	1	we	we	PRON
ejpam-4262	280	2	have	have	VERB
ejpam-4262	280	3	b(m+m	b(m+m	PROPN
ejpam-4262	280	4	)	)	PUNCT
ejpam-4262	280	5	∈	∈	PROPN
ejpam-4262	280	6	rm	rm	NOUN
ejpam-4262	280	7	.	.	PUNCT
ejpam-4262	281	1	therefore	therefore	ADV
ejpam-4262	281	2	rm	rm	PROPN
ejpam-4262	281	3	is	be	AUX
ejpam-4262	281	4	a	a	DET
ejpam-4262	281	5	β	β	NOUN
ejpam-4262	281	6	-	-	ADJ
ejpam-4262	281	7	absorbing	absorbing	ADJ
ejpam-4262	281	8	submodule	submodule	NOUN
ejpam-4262	281	9	of	of	ADP
ejpam-4262	281	10	m	m	PROPN
ejpam-4262	281	11	.	.	PUNCT
ejpam-4262	282	1	acknowledgements	acknowledgement	NOUN
ejpam-4262	282	2	this	this	DET
ejpam-4262	282	3	work	work	NOUN
ejpam-4262	282	4	was	be	AUX
ejpam-4262	282	5	supported	support	VERB
ejpam-4262	282	6	by	by	ADP
ejpam-4262	282	7	school	school	NOUN
ejpam-4262	282	8	of	of	ADP
ejpam-4262	282	9	science	science	NOUN
ejpam-4262	282	10	,	,	PUNCT
ejpam-4262	282	11	king	king	PROPN
ejpam-4262	282	12	mongkut	mongkut	PROPN
ejpam-4262	282	13	’s	’s	PROPN
ejpam-4262	282	14	institute	institute	PROPN
ejpam-4262	282	15	of	of	ADP
ejpam-4262	282	16	technology	technology	PROPN
ejpam-4262	282	17	ladkrabang	ladkrabang	VERB
ejpam-4262	282	18	under	under	ADP
ejpam-4262	282	19	grant	grant	VERB
ejpam-4262	282	20	no.2565	no.2565	ADV
ejpam-4262	282	21	-	-	PUNCT
ejpam-4262	282	22	02	02	NUM
ejpam-4262	282	23	-	-	PUNCT
ejpam-4262	282	24	05	05	NUM
ejpam-4262	282	25	-	-	PUNCT
ejpam-4262	282	26	001	001	NOUN
ejpam-4262	282	27	.	.	PUNCT
ejpam-4262	283	1	references	reference	NOUN
ejpam-4262	283	2	[	[	X
ejpam-4262	283	3	1	1	X
ejpam-4262	283	4	]	]	PUNCT
ejpam-4262	283	5	a	a	DET
ejpam-4262	283	6	darani	darani	PROPN
ejpam-4262	283	7	and	and	CCONJ
ejpam-4262	283	8	f	f	PROPN
ejpam-4262	283	9	soheilnia	soheilnia	NOUN
ejpam-4262	283	10	.	.	PUNCT
ejpam-4262	284	1	2	2	NUM
ejpam-4262	284	2	-	-	PUNCT
ejpam-4262	284	3	absorbing	absorbing	ADJ
ejpam-4262	284	4	and	and	CCONJ
ejpam-4262	284	5	weakly	weakly	ADJ
ejpam-4262	284	6	2	2	NUM
ejpam-4262	284	7	-	-	PUNCT
ejpam-4262	284	8	absorbing	absorbing	ADJ
ejpam-4262	284	9	submodules	submodule	NOUN
ejpam-4262	284	10	.	.	PUNCT
ejpam-4262	285	1	thai	thai	PROPN
ejpam-4262	285	2	journal	journal	PROPN
ejpam-4262	285	3	of	of	ADP
ejpam-4262	285	4	mathematics	mathematic	NOUN
ejpam-4262	285	5	,	,	PUNCT
ejpam-4262	285	6	9:577–584	9:577–584	NUM
ejpam-4262	285	7	,	,	PUNCT
ejpam-4262	285	8	2011	2011	NUM
ejpam-4262	285	9	.	.	PUNCT
ejpam-4262	286	1	[	[	X
ejpam-4262	286	2	2	2	NUM
ejpam-4262	286	3	]	]	PUNCT
ejpam-4262	286	4	t	t	PROPN
ejpam-4262	286	5	khumprapussorn	khumprapussorn	PROPN
ejpam-4262	286	6	.	.	PUNCT
ejpam-4262	287	1	on	on	ADP
ejpam-4262	287	2	β	β	ADJ
ejpam-4262	287	3	-	-	ADJ
ejpam-4262	287	4	absorbing	absorbing	ADJ
ejpam-4262	287	5	submodules	submodule	NOUN
ejpam-4262	287	6	.	.	PUNCT
ejpam-4262	288	1	international	international	ADJ
ejpam-4262	288	2	journal	journal	PROPN
ejpam-4262	288	3	of	of	ADP
ejpam-4262	288	4	mathematics	mathematic	NOUN
ejpam-4262	288	5	and	and	CCONJ
ejpam-4262	288	6	computer	computer	NOUN
ejpam-4262	288	7	science	science	NOUN
ejpam-4262	288	8	,	,	PUNCT
ejpam-4262	288	9	15:809–720	15:809–720	NUM
ejpam-4262	288	10	,	,	PUNCT
ejpam-4262	288	11	2020	2020	NUM
ejpam-4262	288	12	.	.	PUNCT
ejpam-4262	289	1	[	[	X
ejpam-4262	289	2	3	3	NUM
ejpam-4262	289	3	]	]	X
ejpam-4262	289	4	h	h	NOUN
ejpam-4262	289	5	mostafanasab	mostafanasab	VERB
ejpam-4262	289	6	,	,	PUNCT
ejpam-4262	289	7	u	u	NOUN
ejpam-4262	289	8	tekir	tekir	NOUN
ejpam-4262	289	9	,	,	PUNCT
ejpam-4262	289	10	e	e	NOUN
ejpam-4262	289	11	celikel	celikel	NOUN
ejpam-4262	289	12	,	,	PUNCT
ejpam-4262	289	13	e	e	X
ejpam-4262	289	14	ugurlu	ugurlu	NOUN
ejpam-4262	289	15	,	,	PUNCT
ejpam-4262	289	16	g	g	PROPN
ejpam-4262	289	17	ulucak	ulucak	NOUN
ejpam-4262	289	18	,	,	PUNCT
ejpam-4262	289	19	and	and	CCONJ
ejpam-4262	289	20	a	a	DET
ejpam-4262	289	21	darani	darani	PROPN
ejpam-4262	289	22	.	.	PUNCT
ejpam-4262	290	1	generalizations	generalization	NOUN
ejpam-4262	290	2	of	of	ADP
ejpam-4262	290	3	2	2	NUM
ejpam-4262	290	4	-	-	PUNCT
ejpam-4262	290	5	absorbing	absorbing	ADJ
ejpam-4262	290	6	and	and	CCONJ
ejpam-4262	290	7	2	2	NUM
ejpam-4262	290	8	-	-	PUNCT
ejpam-4262	290	9	absorbing	absorbing	ADJ
ejpam-4262	290	10	primary	primary	ADJ
ejpam-4262	290	11	submodules	submodule	NOUN
ejpam-4262	290	12	.	.	PUNCT
ejpam-4262	291	1	hacet	hacet	PROPN
ejpam-4262	291	2	.	.	PUNCT
ejpam-4262	292	1	j.	j.	PROPN
ejpam-4262	292	2	math	math	PROPN
ejpam-4262	292	3	.	.	PUNCT
ejpam-4262	293	1	stat	stat	PROPN
ejpam-4262	293	2	.	.	PUNCT
ejpam-4262	293	3	,	,	PUNCT
ejpam-4262	293	4	48:1001–1016	48:1001–1016	NUM
ejpam-4262	293	5	,	,	PUNCT
ejpam-4262	293	6	2019	2019	NUM
ejpam-4262	293	7	.	.	PUNCT
ejpam-4262	294	1	[	[	X
ejpam-4262	294	2	4	4	X
ejpam-4262	294	3	]	]	X
ejpam-4262	294	4	sh	sh	PROPN
ejpam-4262	294	5	payrovi	payrovi	PROPN
ejpam-4262	294	6	and	and	CCONJ
ejpam-4262	294	7	s	s	VERB
ejpam-4262	294	8	babaei	babaei	NOUN
ejpam-4262	294	9	.	.	PUNCT
ejpam-4262	295	1	on	on	ADP
ejpam-4262	295	2	2	2	NUM
ejpam-4262	295	3	-	-	PUNCT
ejpam-4262	295	4	absorbing	absorbing	ADJ
ejpam-4262	295	5	submodules	submodule	NOUN
ejpam-4262	295	6	.	.	PUNCT
ejpam-4262	296	1	algebra	algebra	NOUN
ejpam-4262	296	2	colloquium	colloquium	NOUN
ejpam-4262	296	3	(	(	PUNCT
ejpam-4262	296	4	spec	spec	PROPN
ejpam-4262	296	5	1	1	NUM
ejpam-4262	296	6	)	)	PUNCT
ejpam-4262	296	7	,	,	PUNCT
ejpam-4262	296	8	19:913–920	19:913–920	NUM
ejpam-4262	296	9	,	,	PUNCT
ejpam-4262	296	10	2012	2012	NUM
ejpam-4262	296	11	.	.	PUNCT
ejpam-4262	297	1	[	[	X
ejpam-4262	297	2	5	5	X
ejpam-4262	297	3	]	]	X
ejpam-4262	297	4	sh	sh	PROPN
ejpam-4262	297	5	payrovi	payrovi	PROPN
ejpam-4262	297	6	and	and	CCONJ
ejpam-4262	297	7	s	s	VERB
ejpam-4262	297	8	babaei	babaei	NOUN
ejpam-4262	297	9	.	.	PUNCT
ejpam-4262	298	1	on	on	ADP
ejpam-4262	298	2	the	the	DET
ejpam-4262	298	3	2	2	NUM
ejpam-4262	298	4	-	-	PUNCT
ejpam-4262	298	5	absorbing	absorbing	ADJ
ejpam-4262	298	6	submodules	submodule	NOUN
ejpam-4262	298	7	.	.	PUNCT
ejpam-4262	299	1	iranian	iranian	ADJ
ejpam-4262	299	2	journal	journal	PROPN
ejpam-4262	299	3	of	of	ADP
ejpam-4262	299	4	mathematical	mathematical	ADJ
ejpam-4262	299	5	sciences	sciences	PROPN
ejpam-4262	299	6	and	and	CCONJ
ejpam-4262	299	7	informatics	informatic	NOUN
ejpam-4262	299	8	,	,	PUNCT
ejpam-4262	299	9	10:131–137	10:131–137	NUM
ejpam-4262	299	10	,	,	PUNCT
ejpam-4262	299	11	2015	2015	NUM
ejpam-4262	299	12	.	.	PUNCT
ejpam-4262	300	1	[	[	X
ejpam-4262	300	2	6	6	NUM
ejpam-4262	300	3	]	]	PUNCT
ejpam-4262	300	4	m	m	VERB
ejpam-4262	300	5	yasein	yasein	NOUN
ejpam-4262	301	1	and	and	CCONJ
ejpam-4262	301	2	r	r	PROPN
ejpam-4262	301	3	abu	abu	PROPN
ejpam-4262	301	4	-	-	PUNCT
ejpam-4262	301	5	dawwas	dawwas	PROPN
ejpam-4262	301	6	.	.	PUNCT
ejpam-4262	302	1	on	on	ADP
ejpam-4262	302	2	almost	almost	ADV
ejpam-4262	302	3	2	2	NUM
ejpam-4262	302	4	-	-	PUNCT
ejpam-4262	302	5	absorbing	absorbing	ADJ
ejpam-4262	302	6	submodules	submodule	NOUN
ejpam-4262	302	7	.	.	PUNCT
ejpam-4262	303	1	italian	italian	ADJ
ejpam-4262	303	2	journal	journal	NOUN
ejpam-4262	303	3	of	of	ADP
ejpam-4262	303	4	pure	pure	ADJ
ejpam-4262	303	5	and	and	CCONJ
ejpam-4262	303	6	applied	applied	ADJ
ejpam-4262	303	7	mathematics	mathematic	NOUN
ejpam-4262	303	8	,	,	PUNCT
ejpam-4262	303	9	36:923–928	36:923–928	NUM
ejpam-4262	303	10	,	,	PUNCT
ejpam-4262	303	11	2016	2016	NUM
ejpam-4262	303	12	.	.	PUNCT