id	sid	tid	token	lemma	pos
ejpam-3126	1	1	european	european	PROPN
ejpam-3126	1	2	journal	journal	PROPN
ejpam-3126	1	3	of	of	ADP
ejpam-3126	1	4	pure	pure	ADJ
ejpam-3126	1	5	and	and	CCONJ
ejpam-3126	1	6	applied	apply	VERB
ejpam-3126	1	7	mathematics	mathematic	NOUN
ejpam-3126	1	8	vol	vol	NOUN
ejpam-3126	1	9	.	.	PUNCT
ejpam-3126	2	1	11	11	NUM
ejpam-3126	2	2	,	,	PUNCT
ejpam-3126	2	3	no	no	INTJ
ejpam-3126	2	4	.	.	NOUN
ejpam-3126	2	5	1	1	NUM
ejpam-3126	2	6	,	,	PUNCT
ejpam-3126	2	7	2018	2018	NUM
ejpam-3126	2	8	,	,	PUNCT
ejpam-3126	2	9	35	35	NUM
ejpam-3126	2	10	-	-	SYM
ejpam-3126	2	11	50	50	NUM
ejpam-3126	2	12	issn	issn	PROPN
ejpam-3126	2	13	1307	1307	NUM
ejpam-3126	2	14	-	-	SYM
ejpam-3126	2	15	5543	5543	NUM
ejpam-3126	2	16	–	–	PUNCT
ejpam-3126	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3126	2	18	published	publish	VERB
ejpam-3126	2	19	by	by	ADP
ejpam-3126	2	20	new	new	PROPN
ejpam-3126	2	21	york	york	PROPN
ejpam-3126	2	22	business	business	PROPN
ejpam-3126	2	23	global	global	ADJ
ejpam-3126	2	24	on	on	ADP
ejpam-3126	2	25	generalizations	generalization	NOUN
ejpam-3126	2	26	of	of	ADP
ejpam-3126	2	27	φ-2	φ-2	PROPN
ejpam-3126	2	28	-	-	PUNCT
ejpam-3126	2	29	absorbing	absorbing	ADJ
ejpam-3126	2	30	primary	primary	ADJ
ejpam-3126	2	31	submodules	submodule	NOUN
ejpam-3126	2	32	pairote	pairote	NOUN
ejpam-3126	2	33	yiarayong1,∗	yiarayong1,∗	NOUN
ejpam-3126	2	34	,	,	PUNCT
ejpam-3126	2	35	manoj	manoj	PROPN
ejpam-3126	2	36	siripitukdet1	siripitukdet1	NOUN
ejpam-3126	2	37	1	1	NUM
ejpam-3126	2	38	department	department	NOUN
ejpam-3126	2	39	of	of	ADP
ejpam-3126	2	40	mathematics	mathematic	NOUN
ejpam-3126	2	41	,	,	PUNCT
ejpam-3126	2	42	faculty	faculty	NOUN
ejpam-3126	2	43	of	of	ADP
ejpam-3126	2	44	science	science	NOUN
ejpam-3126	2	45	,	,	PUNCT
ejpam-3126	2	46	naresuan	naresuan	PROPN
ejpam-3126	2	47	university	university	NOUN
ejpam-3126	2	48	,	,	PUNCT
ejpam-3126	2	49	phitsanuloke	phitsanuloke	NOUN
ejpam-3126	2	50	65000	65000	NUM
ejpam-3126	2	51	,	,	PUNCT
ejpam-3126	2	52	thailand	thailand	PROPN
ejpam-3126	2	53	abstract	abstract	NOUN
ejpam-3126	2	54	.	.	PUNCT
ejpam-3126	3	1	let	let	VERB
ejpam-3126	3	2	φ	φ	NOUN
ejpam-3126	3	3	:	:	PUNCT
ejpam-3126	3	4	s(m)→	s(m)→	ADJ
ejpam-3126	3	5	s(m)∪	s(m)∪	NOUN
ejpam-3126	3	6	{	{	PUNCT
ejpam-3126	3	7	∅	∅	NOUN
ejpam-3126	3	8	}	}	PUNCT
ejpam-3126	3	9	be	be	AUX
ejpam-3126	3	10	a	a	DET
ejpam-3126	3	11	function	function	NOUN
ejpam-3126	3	12	where	where	SCONJ
ejpam-3126	3	13	s(m	s(m	NOUN
ejpam-3126	3	14	)	)	PUNCT
ejpam-3126	3	15	is	be	AUX
ejpam-3126	3	16	the	the	DET
ejpam-3126	3	17	set	set	NOUN
ejpam-3126	3	18	of	of	ADP
ejpam-3126	3	19	all	all	DET
ejpam-3126	3	20	submodules	submodule	NOUN
ejpam-3126	3	21	of	of	ADP
ejpam-3126	3	22	m	m	PRON
ejpam-3126	3	23	.	.	PUNCT
ejpam-3126	4	1	in	in	ADP
ejpam-3126	4	2	this	this	DET
ejpam-3126	4	3	paper	paper	NOUN
ejpam-3126	4	4	,	,	PUNCT
ejpam-3126	4	5	we	we	PRON
ejpam-3126	4	6	extend	extend	VERB
ejpam-3126	4	7	the	the	DET
ejpam-3126	4	8	concept	concept	NOUN
ejpam-3126	4	9	of	of	ADP
ejpam-3126	4	10	φ-2	φ-2	PROPN
ejpam-3126	4	11	-	-	PUNCT
ejpam-3126	4	12	absorbing	absorbing	ADJ
ejpam-3126	4	13	primary	primary	ADJ
ejpam-3126	4	14	submodules	submodule	NOUN
ejpam-3126	4	15	to	to	ADP
ejpam-3126	4	16	the	the	DET
ejpam-3126	4	17	context	context	NOUN
ejpam-3126	4	18	of	of	ADP
ejpam-3126	4	19	φ-2	φ-2	PROPN
ejpam-3126	4	20	-	-	PUNCT
ejpam-3126	4	21	absorbing	absorbing	ADJ
ejpam-3126	4	22	semi	semi	ADJ
ejpam-3126	4	23	-	-	ADJ
ejpam-3126	4	24	primary	primary	ADJ
ejpam-3126	4	25	submodules	submodule	NOUN
ejpam-3126	4	26	.	.	PUNCT
ejpam-3126	5	1	a	a	DET
ejpam-3126	5	2	proper	proper	ADJ
ejpam-3126	5	3	submodule	submodule	NOUN
ejpam-3126	5	4	n	n	PROPN
ejpam-3126	5	5	of	of	ADP
ejpam-3126	5	6	m	m	PROPN
ejpam-3126	5	7	is	be	AUX
ejpam-3126	5	8	called	call	VERB
ejpam-3126	5	9	a	a	DET
ejpam-3126	5	10	φ-2	φ-2	NOUN
ejpam-3126	5	11	-	-	PUNCT
ejpam-3126	5	12	absorbing	absorbing	ADJ
ejpam-3126	5	13	semi	semi	ADJ
ejpam-3126	5	14	-	-	ADJ
ejpam-3126	5	15	primary	primary	ADJ
ejpam-3126	5	16	submodule	submodule	NOUN
ejpam-3126	5	17	,	,	PUNCT
ejpam-3126	5	18	if	if	SCONJ
ejpam-3126	5	19	for	for	ADP
ejpam-3126	5	20	each	each	DET
ejpam-3126	5	21	m	m	NOUN
ejpam-3126	5	22	∈	∈	PROPN
ejpam-3126	5	23	m	m	NOUN
ejpam-3126	5	24	and	and	CCONJ
ejpam-3126	5	25	a1	a1	NOUN
ejpam-3126	5	26	,	,	PUNCT
ejpam-3126	5	27	a2	a2	PROPN
ejpam-3126	5	28	∈	∈	PROPN
ejpam-3126	5	29	r	r	NOUN
ejpam-3126	5	30	with	with	ADP
ejpam-3126	5	31	a1a2	a1a2	PROPN
ejpam-3126	5	32	m	m	NOUN
ejpam-3126	5	33	∈	∈	ADJ
ejpam-3126	5	34	n	n	CCONJ
ejpam-3126	5	35	−	−	PROPN
ejpam-3126	5	36	φ(n	φ(n	NOUN
ejpam-3126	5	37	)	)	PUNCT
ejpam-3126	5	38	,	,	PUNCT
ejpam-3126	5	39	then	then	ADV
ejpam-3126	5	40	a1a2	a1a2	ADP
ejpam-3126	5	41	∈	∈	NOUN
ejpam-3126	5	42	√	√	NUM
ejpam-3126	5	43	(	(	PUNCT
ejpam-3126	5	44	n	n	NUM
ejpam-3126	5	45	:	:	PUNCT
ejpam-3126	5	46	m	m	X
ejpam-3126	5	47	)	)	PUNCT
ejpam-3126	5	48	or	or	CCONJ
ejpam-3126	5	49	a1	a1	NOUN
ejpam-3126	5	50	m	m	PROPN
ejpam-3126	5	51	∈	∈	NOUN
ejpam-3126	5	52	n	n	NOUN
ejpam-3126	5	53	or	or	CCONJ
ejpam-3126	5	54	an2	an2	PROPN
ejpam-3126	5	55	m	m	NOUN
ejpam-3126	5	56	∈	∈	PROPN
ejpam-3126	5	57	n	n	NOUN
ejpam-3126	5	58	for	for	ADP
ejpam-3126	5	59	some	some	DET
ejpam-3126	5	60	positive	positive	ADJ
ejpam-3126	5	61	integer	integer	NOUN
ejpam-3126	5	62	n.	n.	NOUN
ejpam-3126	5	63	those	those	PRON
ejpam-3126	5	64	are	be	AUX
ejpam-3126	5	65	extended	extend	VERB
ejpam-3126	5	66	from	from	ADP
ejpam-3126	5	67	2	2	NUM
ejpam-3126	5	68	-	-	PUNCT
ejpam-3126	5	69	absorbing	absorb	VERB
ejpam-3126	5	70	primary	primary	NOUN
ejpam-3126	5	71	,	,	PUNCT
ejpam-3126	5	72	weakly	weakly	ADJ
ejpam-3126	5	73	2	2	NUM
ejpam-3126	5	74	-	-	PUNCT
ejpam-3126	5	75	absorbing	absorb	VERB
ejpam-3126	5	76	primary	primary	NOUN
ejpam-3126	5	77	,	,	PUNCT
ejpam-3126	5	78	almost	almost	ADV
ejpam-3126	5	79	2	2	NUM
ejpam-3126	5	80	-	-	PUNCT
ejpam-3126	5	81	absorbing	absorb	VERB
ejpam-3126	5	82	primary	primary	NOUN
ejpam-3126	5	83	,	,	PUNCT
ejpam-3126	5	84	φn-2	φn-2	NOUN
ejpam-3126	5	85	-	-	ADJ
ejpam-3126	5	86	absorbing	absorbing	ADJ
ejpam-3126	5	87	primary	primary	ADJ
ejpam-3126	5	88	,	,	PUNCT
ejpam-3126	5	89	ω-2	ω-2	ADV
ejpam-3126	5	90	-	-	PUNCT
ejpam-3126	5	91	absorbing	absorb	VERB
ejpam-3126	5	92	primary	primary	NOUN
ejpam-3126	5	93	and	and	CCONJ
ejpam-3126	5	94	φ-2	φ-2	ADJ
ejpam-3126	5	95	-	-	PUNCT
ejpam-3126	5	96	absorbing	absorbing	ADJ
ejpam-3126	5	97	primary	primary	ADJ
ejpam-3126	5	98	submodules	submodule	NOUN
ejpam-3126	5	99	,	,	PUNCT
ejpam-3126	5	100	respectively	respectively	ADV
ejpam-3126	5	101	.	.	PUNCT
ejpam-3126	6	1	some	some	DET
ejpam-3126	6	2	characterizations	characterization	NOUN
ejpam-3126	6	3	of	of	ADP
ejpam-3126	6	4	2	2	NUM
ejpam-3126	6	5	-	-	PUNCT
ejpam-3126	6	6	absorbing	absorbing	ADJ
ejpam-3126	6	7	semi	semi	ADJ
ejpam-3126	6	8	-	-	ADJ
ejpam-3126	6	9	primary	primary	ADJ
ejpam-3126	6	10	,	,	PUNCT
ejpam-3126	6	11	φn-2	φn-2	NOUN
ejpam-3126	6	12	-	-	PUNCT
ejpam-3126	6	13	absorbing	absorbing	ADJ
ejpam-3126	6	14	semi	semi	ADJ
ejpam-3126	6	15	-	-	ADJ
ejpam-3126	6	16	primary	primary	ADJ
ejpam-3126	6	17	and	and	CCONJ
ejpam-3126	6	18	φ-2	φ-2	NOUN
ejpam-3126	6	19	-	-	PUNCT
ejpam-3126	6	20	absorbing	absorbing	ADJ
ejpam-3126	6	21	semiprimary	semiprimary	ADJ
ejpam-3126	6	22	submodules	submodule	NOUN
ejpam-3126	6	23	are	be	AUX
ejpam-3126	6	24	obtained	obtain	VERB
ejpam-3126	6	25	.	.	PUNCT
ejpam-3126	7	1	moreover	moreover	ADV
ejpam-3126	7	2	,	,	PUNCT
ejpam-3126	7	3	we	we	PRON
ejpam-3126	7	4	investigate	investigate	VERB
ejpam-3126	7	5	relationships	relationship	NOUN
ejpam-3126	7	6	between	between	ADP
ejpam-3126	7	7	2	2	NUM
ejpam-3126	7	8	-	-	PUNCT
ejpam-3126	7	9	absorbing	absorbing	ADJ
ejpam-3126	7	10	semi	semi	ADJ
ejpam-3126	7	11	-	-	ADJ
ejpam-3126	7	12	primary	primary	ADJ
ejpam-3126	7	13	,	,	PUNCT
ejpam-3126	7	14	φn-2	φn-2	NOUN
ejpam-3126	7	15	-	-	PUNCT
ejpam-3126	7	16	absorbing	absorbing	ADJ
ejpam-3126	7	17	semi	semi	ADJ
ejpam-3126	7	18	-	-	ADJ
ejpam-3126	7	19	primary	primary	ADJ
ejpam-3126	7	20	and	and	CCONJ
ejpam-3126	7	21	φ	φ	VERB
ejpam-3126	7	22	-	-	ADJ
ejpam-3126	7	23	primary	primary	ADJ
ejpam-3126	7	24	submodules	submodule	NOUN
ejpam-3126	7	25	of	of	ADP
ejpam-3126	7	26	modules	module	NOUN
ejpam-3126	7	27	over	over	ADP
ejpam-3126	7	28	commutative	commutative	ADJ
ejpam-3126	7	29	rings	ring	NOUN
ejpam-3126	7	30	.	.	PUNCT
ejpam-3126	8	1	finally	finally	ADV
ejpam-3126	8	2	,	,	PUNCT
ejpam-3126	8	3	we	we	PRON
ejpam-3126	8	4	obtain	obtain	VERB
ejpam-3126	8	5	necessary	necessary	ADJ
ejpam-3126	8	6	and	and	CCONJ
ejpam-3126	8	7	sufficient	sufficient	ADJ
ejpam-3126	8	8	conditions	condition	NOUN
ejpam-3126	8	9	of	of	ADP
ejpam-3126	8	10	a	a	DET
ejpam-3126	8	11	φ-2	φ-2	NOUN
ejpam-3126	8	12	-	-	PUNCT
ejpam-3126	8	13	absorbing	absorbing	ADJ
ejpam-3126	8	14	semi	semi	ADJ
ejpam-3126	8	15	-	-	ADJ
ejpam-3126	8	16	primary	primary	ADJ
ejpam-3126	8	17	in	in	ADP
ejpam-3126	8	18	order	order	NOUN
ejpam-3126	8	19	to	to	PART
ejpam-3126	8	20	be	be	AUX
ejpam-3126	8	21	a	a	DET
ejpam-3126	8	22	2	2	NUM
ejpam-3126	8	23	-	-	PUNCT
ejpam-3126	8	24	absorbing	absorbing	ADJ
ejpam-3126	8	25	semi	semi	ADJ
ejpam-3126	8	26	-	-	ADJ
ejpam-3126	8	27	primary	primary	ADJ
ejpam-3126	8	28	submodule	submodule	NOUN
ejpam-3126	8	29	.	.	PUNCT
ejpam-3126	9	1	2010	2010	NUM
ejpam-3126	9	2	mathematics	mathematic	NOUN
ejpam-3126	9	3	subject	subject	NOUN
ejpam-3126	9	4	classifications	classification	NOUN
ejpam-3126	9	5	:	:	PUNCT
ejpam-3126	9	6	13c05	13c05	NUM
ejpam-3126	9	7	,	,	PUNCT
ejpam-3126	9	8	13c13	13c13	NUM
ejpam-3126	9	9	.	.	PUNCT
ejpam-3126	10	1	key	key	ADJ
ejpam-3126	10	2	words	word	NOUN
ejpam-3126	10	3	and	and	CCONJ
ejpam-3126	10	4	phrases	phrase	NOUN
ejpam-3126	10	5	:	:	PUNCT
ejpam-3126	10	6	2	2	NUM
ejpam-3126	10	7	-	-	PUNCT
ejpam-3126	10	8	absorbing	absorbing	ADJ
ejpam-3126	10	9	semi	semi	ADJ
ejpam-3126	10	10	-	-	ADJ
ejpam-3126	10	11	primary	primary	ADJ
ejpam-3126	10	12	submodule	submodule	NOUN
ejpam-3126	10	13	,	,	PUNCT
ejpam-3126	10	14	φα-2	φα-2	NOUN
ejpam-3126	10	15	-	-	PUNCT
ejpam-3126	10	16	absorbing	absorbing	ADJ
ejpam-3126	10	17	semi	semi	ADJ
ejpam-3126	10	18	-	-	ADJ
ejpam-3126	10	19	primary	primary	ADJ
ejpam-3126	10	20	submodule	submodule	NOUN
ejpam-3126	10	21	,	,	PUNCT
ejpam-3126	10	22	φ-2	φ-2	NOUN
ejpam-3126	10	23	-	-	PUNCT
ejpam-3126	10	24	absorbing	absorbing	ADJ
ejpam-3126	10	25	primary	primary	ADJ
ejpam-3126	10	26	submodule	submodule	NOUN
ejpam-3126	10	27	,	,	PUNCT
ejpam-3126	10	28	φ	φ	NOUN
ejpam-3126	10	29	-	-	ADJ
ejpam-3126	10	30	primary	primary	ADJ
ejpam-3126	10	31	ideal	ideal	NOUN
ejpam-3126	10	32	,	,	PUNCT
ejpam-3126	10	33	φ-2	φ-2	NOUN
ejpam-3126	10	34	-	-	PUNCT
ejpam-3126	10	35	absorbing	absorbing	ADJ
ejpam-3126	10	36	ideal	ideal	NOUN
ejpam-3126	10	37	.	.	PUNCT
ejpam-3126	11	1	1	1	X
ejpam-3126	11	2	.	.	X
ejpam-3126	11	3	introduction	introduction	NOUN
ejpam-3126	11	4	throughout	throughout	ADP
ejpam-3126	11	5	this	this	DET
ejpam-3126	11	6	paper	paper	NOUN
ejpam-3126	11	7	,	,	PUNCT
ejpam-3126	11	8	we	we	PRON
ejpam-3126	11	9	assume	assume	VERB
ejpam-3126	11	10	that	that	SCONJ
ejpam-3126	11	11	all	all	DET
ejpam-3126	11	12	rings	ring	NOUN
ejpam-3126	11	13	are	be	AUX
ejpam-3126	11	14	commutative	commutative	ADJ
ejpam-3126	11	15	with	with	ADP
ejpam-3126	11	16	a	a	DET
ejpam-3126	11	17	nonzero	nonzero	NOUN
ejpam-3126	11	18	identity	identity	NOUN
ejpam-3126	11	19	suppose	suppose	VERB
ejpam-3126	11	20	that	that	SCONJ
ejpam-3126	11	21	r	r	NOUN
ejpam-3126	11	22	is	be	AUX
ejpam-3126	11	23	a	a	DET
ejpam-3126	11	24	ring	ring	NOUN
ejpam-3126	11	25	and	and	CCONJ
ejpam-3126	11	26	m	m	NOUN
ejpam-3126	11	27	is	be	AUX
ejpam-3126	11	28	an	an	DET
ejpam-3126	11	29	r	r	NOUN
ejpam-3126	11	30	-	-	PUNCT
ejpam-3126	11	31	module	module	NOUN
ejpam-3126	11	32	the	the	DET
ejpam-3126	11	33	concept	concept	NOUN
ejpam-3126	11	34	of	of	ADP
ejpam-3126	11	35	ϕ-prime	ϕ-prime	NOUN
ejpam-3126	11	36	ideals	ideal	NOUN
ejpam-3126	11	37	,	,	PUNCT
ejpam-3126	11	38	a	a	DET
ejpam-3126	11	39	generalization	generalization	NOUN
ejpam-3126	11	40	of	of	ADP
ejpam-3126	11	41	prime	prime	ADJ
ejpam-3126	11	42	ideals	ideal	NOUN
ejpam-3126	11	43	was	be	AUX
ejpam-3126	11	44	introduced	introduce	VERB
ejpam-3126	11	45	and	and	CCONJ
ejpam-3126	11	46	investigated	investigate	VERB
ejpam-3126	11	47	in	in	ADP
ejpam-3126	11	48	[	[	X
ejpam-3126	11	49	2	2	NUM
ejpam-3126	11	50	]	]	PUNCT
ejpam-3126	11	51	.	.	PUNCT
ejpam-3126	12	1	let	let	VERB
ejpam-3126	12	2	ϕ	ϕ	NOUN
ejpam-3126	12	3	:	:	PUNCT
ejpam-3126	12	4	j	j	PROPN
ejpam-3126	12	5	(	(	PUNCT
ejpam-3126	12	6	r	r	NOUN
ejpam-3126	12	7	)	)	PUNCT
ejpam-3126	12	8	→	→	SYM
ejpam-3126	12	9	j	j	PROPN
ejpam-3126	12	10	(	(	PUNCT
ejpam-3126	12	11	r	r	NOUN
ejpam-3126	12	12	)	)	PUNCT
ejpam-3126	12	13	∪	∪	NOUN
ejpam-3126	12	14	{	{	PUNCT
ejpam-3126	12	15	∅	∅	NOUN
ejpam-3126	12	16	}	}	PUNCT
ejpam-3126	12	17	be	be	AUX
ejpam-3126	12	18	a	a	DET
ejpam-3126	12	19	function	function	NOUN
ejpam-3126	12	20	where	where	SCONJ
ejpam-3126	12	21	j	j	PROPN
ejpam-3126	12	22	(	(	PUNCT
ejpam-3126	12	23	r	r	NOUN
ejpam-3126	12	24	)	)	PUNCT
ejpam-3126	12	25	is	be	AUX
ejpam-3126	12	26	a	a	DET
ejpam-3126	12	27	set	set	NOUN
ejpam-3126	12	28	of	of	ADP
ejpam-3126	12	29	ideals	ideal	NOUN
ejpam-3126	12	30	of	of	ADP
ejpam-3126	12	31	r.	r.	PROPN
ejpam-3126	12	32	a	a	DET
ejpam-3126	12	33	proper	proper	ADJ
ejpam-3126	12	34	ideal	ideal	NOUN
ejpam-3126	12	35	i	i	PRON
ejpam-3126	12	36	of	of	ADP
ejpam-3126	12	37	r	r	NOUN
ejpam-3126	12	38	is	be	AUX
ejpam-3126	12	39	said	say	VERB
ejpam-3126	12	40	to	to	PART
ejpam-3126	12	41	be	be	AUX
ejpam-3126	12	42	ϕ-prime	ϕ-prime	ADJ
ejpam-3126	13	1	if	if	SCONJ
ejpam-3126	13	2	whenever	whenever	SCONJ
ejpam-3126	13	3	a1	a1	PROPN
ejpam-3126	13	4	,	,	PUNCT
ejpam-3126	13	5	a2	a2	PROPN
ejpam-3126	13	6	∈	∈	PROPN
ejpam-3126	13	7	r	r	NOUN
ejpam-3126	13	8	and	and	CCONJ
ejpam-3126	13	9	a1a2	a1a2	ADP
ejpam-3126	13	10	∈	∈	PROPN
ejpam-3126	13	11	i	i	PRON
ejpam-3126	13	12	−	−	PROPN
ejpam-3126	13	13	ϕ(i	ϕ(i	NUM
ejpam-3126	13	14	)	)	PUNCT
ejpam-3126	13	15	,	,	PUNCT
ejpam-3126	13	16	then	then	ADV
ejpam-3126	13	17	a1	a1	PROPN
ejpam-3126	13	18	∈	∈	PROPN
ejpam-3126	13	19	i	i	PROPN
ejpam-3126	13	20	or	or	CCONJ
ejpam-3126	13	21	a2	a2	PROPN
ejpam-3126	13	22	∈	∈	PROPN
ejpam-3126	13	23	i.	i.	NOUN
ejpam-3126	13	24	since	since	SCONJ
ejpam-3126	13	25	i	i	PRON
ejpam-3126	13	26	−	−	PROPN
ejpam-3126	13	27	ϕ(i	ϕ(i	PROPN
ejpam-3126	13	28	)	)	PUNCT
ejpam-3126	14	1	=	=	PUNCT
ejpam-3126	15	1	i	i	PRON
ejpam-3126	15	2	−	−	PROPN
ejpam-3126	16	1	(	(	PUNCT
ejpam-3126	16	2	i	i	PROPN
ejpam-3126	16	3	∩	∩	X
ejpam-3126	16	4	ϕ(i	ϕ(i	PROPN
ejpam-3126	16	5	)	)	PUNCT
ejpam-3126	16	6	)	)	PUNCT
ejpam-3126	16	7	,	,	PUNCT
ejpam-3126	16	8	so	so	CCONJ
ejpam-3126	16	9	without	without	ADP
ejpam-3126	16	10	loss	loss	NOUN
ejpam-3126	16	11	of	of	ADP
ejpam-3126	16	12	generality	generality	NOUN
ejpam-3126	16	13	,	,	PUNCT
ejpam-3126	16	14	throughout	throughout	ADP
ejpam-3126	16	15	this	this	DET
ejpam-3126	16	16	paper	paper	NOUN
ejpam-3126	16	17	we	we	PRON
ejpam-3126	16	18	will	will	AUX
ejpam-3126	16	19	consider	consider	VERB
ejpam-3126	16	20	ϕ(i	ϕ(i	X
ejpam-3126	16	21	)	)	PUNCT
ejpam-3126	17	1	⊆	⊆	NUM
ejpam-3126	17	2	i.	i.	NOUN
ejpam-3126	17	3	later	later	ADV
ejpam-3126	17	4	,	,	PUNCT
ejpam-3126	17	5	darani	darani	PROPN
ejpam-3126	17	6	[	[	X
ejpam-3126	17	7	7	7	X
ejpam-3126	17	8	]	]	PUNCT
ejpam-3126	17	9	gave	give	VERB
ejpam-3126	17	10	a	a	DET
ejpam-3126	17	11	generalization	generalization	NOUN
ejpam-3126	17	12	of	of	ADP
ejpam-3126	17	13	primary	primary	ADJ
ejpam-3126	17	14	ideals	ideal	NOUN
ejpam-3126	17	15	which	which	PRON
ejpam-3126	17	16	covers	cover	VERB
ejpam-3126	17	17	all	all	DET
ejpam-3126	17	18	the	the	DET
ejpam-3126	17	19	above	above	ADJ
ejpam-3126	17	20	mentioned	mention	VERB
ejpam-3126	17	21	definitions	definition	NOUN
ejpam-3126	17	22	.	.	PUNCT
ejpam-3126	18	1	he	he	PRON
ejpam-3126	18	2	defined	define	VERB
ejpam-3126	18	3	the	the	DET
ejpam-3126	18	4	a	a	DET
ejpam-3126	18	5	proper	proper	ADJ
ejpam-3126	18	6	ideal	ideal	NOUN
ejpam-3126	18	7	i	i	PRON
ejpam-3126	18	8	of	of	ADP
ejpam-3126	18	9	r	r	NOUN
ejpam-3126	18	10	is	be	AUX
ejpam-3126	18	11	said	say	VERB
ejpam-3126	18	12	to	to	PART
ejpam-3126	18	13	be	be	AUX
ejpam-3126	18	14	ϕ-primary	ϕ-primary	ADJ
ejpam-3126	18	15	if	if	SCONJ
ejpam-3126	18	16	for	for	ADP
ejpam-3126	18	17	a1	a1	NOUN
ejpam-3126	18	18	,	,	PUNCT
ejpam-3126	18	19	a2	a2	PROPN
ejpam-3126	18	20	∈	∈	PROPN
ejpam-3126	18	21	r	r	NOUN
ejpam-3126	18	22	with	with	ADP
ejpam-3126	18	23	a1a2	a1a2	INTJ
ejpam-3126	18	24	∈	∈	PROPN
ejpam-3126	19	1	i	i	PRON
ejpam-3126	19	2	−	−	PROPN
ejpam-3126	19	3	ϕ(i	ϕ(i	NUM
ejpam-3126	19	4	)	)	PUNCT
ejpam-3126	19	5	,	,	PUNCT
ejpam-3126	19	6	either	either	CCONJ
ejpam-3126	19	7	a1	a1	NOUN
ejpam-3126	19	8	∈	∈	PROPN
ejpam-3126	19	9	i	i	PRON
ejpam-3126	19	10	or	or	CCONJ
ejpam-3126	19	11	an2	an2	PROPN
ejpam-3126	19	12	∈	∈	PROPN
ejpam-3126	20	1	i	i	PRON
ejpam-3126	20	2	for	for	ADP
ejpam-3126	20	3	some	some	DET
ejpam-3126	20	4	∗corresponding	∗corresponde	VERB
ejpam-3126	20	5	author	author	NOUN
ejpam-3126	20	6	.	.	PUNCT
ejpam-3126	21	1	email	email	NOUN
ejpam-3126	21	2	addresses	address	NOUN
ejpam-3126	21	3	:	:	PUNCT
ejpam-3126	21	4	pairote0027@hotmail.com	pairote0027@hotmail.com	NOUN
ejpam-3126	21	5	(	(	PUNCT
ejpam-3126	21	6	pai	pai	PROPN
ejpam-3126	21	7	.	.	PROPN
ejpam-3126	21	8	yiarayong	yiarayong	PROPN
ejpam-3126	21	9	)	)	PUNCT
ejpam-3126	21	10	,	,	PUNCT
ejpam-3126	21	11	manojs@nu.ac.th	manojs@nu.ac.th	PROPN
ejpam-3126	21	12	(	(	PUNCT
ejpam-3126	21	13	m.	m.	NOUN
ejpam-3126	21	14	siripitukdet	siripitukdet	PROPN
ejpam-3126	21	15	)	)	PUNCT
ejpam-3126	21	16	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3126	22	1	35	35	NUM
ejpam-3126	22	2	c	c	NOUN
ejpam-3126	22	3	©	©	PROPN
ejpam-3126	22	4	2018	2018	NUM
ejpam-3126	22	5	ejpam	ejpam	VERB
ejpam-3126	22	6	all	all	DET
ejpam-3126	22	7	rights	right	NOUN
ejpam-3126	22	8	reserved	reserve	VERB
ejpam-3126	22	9	.	.	PUNCT
ejpam-3126	23	1	pai	pai	PROPN
ejpam-3126	23	2	.	.	PROPN
ejpam-3126	23	3	yiarayong	yiarayong	PROPN
ejpam-3126	23	4	,	,	PUNCT
ejpam-3126	23	5	m.	m.	NOUN
ejpam-3126	23	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	23	7	/	/	SYM
ejpam-3126	23	8	eur	eur	PROPN
ejpam-3126	23	9	.	.	PUNCT
ejpam-3126	24	1	j.	j.	PROPN
ejpam-3126	24	2	pure	pure	PROPN
ejpam-3126	24	3	appl	appl	PROPN
ejpam-3126	24	4	.	.	PROPN
ejpam-3126	24	5	math	math	PROPN
ejpam-3126	24	6	,	,	PUNCT
ejpam-3126	24	7	11	11	NUM
ejpam-3126	24	8	(	(	PUNCT
ejpam-3126	24	9	1	1	NUM
ejpam-3126	24	10	)	)	PUNCT
ejpam-3126	24	11	(	(	PUNCT
ejpam-3126	24	12	2018	2018	NUM
ejpam-3126	24	13	)	)	PUNCT
ejpam-3126	24	14	,	,	PUNCT
ejpam-3126	24	15	35	35	NUM
ejpam-3126	24	16	-	-	SYM
ejpam-3126	24	17	50	50	NUM
ejpam-3126	24	18	36	36	NUM
ejpam-3126	24	19	positive	positive	ADJ
ejpam-3126	24	20	integer	integer	NOUN
ejpam-3126	24	21	n.	n.	NOUN
ejpam-3126	24	22	thus	thus	ADV
ejpam-3126	24	23	a	a	DET
ejpam-3126	24	24	ϕ-prime	ϕ-prime	NOUN
ejpam-3126	24	25	ideal	ideal	NOUN
ejpam-3126	24	26	is	be	AUX
ejpam-3126	24	27	just	just	ADV
ejpam-3126	24	28	a	a	DET
ejpam-3126	24	29	ϕ-primary	ϕ-primary	ADJ
ejpam-3126	24	30	ideal	ideal	NOUN
ejpam-3126	24	31	.	.	PUNCT
ejpam-3126	25	1	in	in	ADP
ejpam-3126	25	2	[	[	X
ejpam-3126	25	3	9	9	NUM
ejpam-3126	25	4	]	]	PUNCT
ejpam-3126	25	5	,	,	PUNCT
ejpam-3126	25	6	ebrahimpour	ebrahimpour	NOUN
ejpam-3126	25	7	and	and	CCONJ
ejpam-3126	25	8	nekooei	nekooei	INTJ
ejpam-3126	25	9	called	call	VERB
ejpam-3126	25	10	a	a	DET
ejpam-3126	25	11	proper	proper	ADJ
ejpam-3126	25	12	ideal	ideal	NOUN
ejpam-3126	25	13	i	i	PRON
ejpam-3126	25	14	of	of	ADP
ejpam-3126	25	15	a	a	DET
ejpam-3126	25	16	commutative	commutative	ADJ
ejpam-3126	25	17	ring	ring	NOUN
ejpam-3126	25	18	r	r	NOUN
ejpam-3126	25	19	to	to	PART
ejpam-3126	25	20	be	be	AUX
ejpam-3126	25	21	ϕ-2	ϕ-2	ADV
ejpam-3126	25	22	-	-	PUNCT
ejpam-3126	25	23	absorbing	absorb	VERB
ejpam-3126	25	24	if	if	SCONJ
ejpam-3126	25	25	whenever	whenever	SCONJ
ejpam-3126	25	26	a1	a1	PROPN
ejpam-3126	25	27	,	,	PUNCT
ejpam-3126	25	28	a2	a2	PROPN
ejpam-3126	25	29	,	,	PUNCT
ejpam-3126	25	30	a3	a3	NOUN
ejpam-3126	25	31	∈	∈	PROPN
ejpam-3126	25	32	r	r	NOUN
ejpam-3126	25	33	and	and	CCONJ
ejpam-3126	25	34	a1a2a3	a1a2a3	VERB
ejpam-3126	25	35	∈	∈	PROPN
ejpam-3126	25	36	i	i	PRON
ejpam-3126	25	37	−	−	PROPN
ejpam-3126	25	38	ϕ(i	ϕ(i	NUM
ejpam-3126	25	39	)	)	PUNCT
ejpam-3126	25	40	,	,	PUNCT
ejpam-3126	26	1	either	either	CCONJ
ejpam-3126	26	2	a1a2	a1a2	ADP
ejpam-3126	26	3	∈	∈	X
ejpam-3126	26	4	i	i	PRON
ejpam-3126	26	5	or	or	CCONJ
ejpam-3126	26	6	a2a3	a2a3	PROPN
ejpam-3126	26	7	∈	∈	PROPN
ejpam-3126	26	8	i	i	PRON
ejpam-3126	26	9	or	or	CCONJ
ejpam-3126	26	10	a1a3	a1a3	ADP
ejpam-3126	26	11	∈	∈	PROPN
ejpam-3126	26	12	i.	i.	PROPN
ejpam-3126	26	13	badawi	badawi	PROPN
ejpam-3126	26	14	,	,	PUNCT
ejpam-3126	26	15	et	et	PROPN
ejpam-3126	26	16	al	al	PROPN
ejpam-3126	26	17	.	.	PUNCT
ejpam-3126	27	1	[	[	X
ejpam-3126	27	2	1	1	X
ejpam-3126	27	3	]	]	PUNCT
ejpam-3126	27	4	generalized	generalize	VERB
ejpam-3126	27	5	the	the	DET
ejpam-3126	27	6	concept	concept	NOUN
ejpam-3126	27	7	of	of	ADP
ejpam-3126	27	8	2	2	NUM
ejpam-3126	27	9	-	-	PUNCT
ejpam-3126	27	10	absorbing	absorbing	ADJ
ejpam-3126	27	11	primary	primary	ADJ
ejpam-3126	27	12	ideals	ideal	NOUN
ejpam-3126	27	13	to	to	ADP
ejpam-3126	27	14	ϕ-2	ϕ-2	ADV
ejpam-3126	27	15	-	-	PUNCT
ejpam-3126	27	16	absorbing	absorb	VERB
ejpam-3126	27	17	primary	primary	ADJ
ejpam-3126	27	18	ideals	ideal	NOUN
ejpam-3126	27	19	.	.	PUNCT
ejpam-3126	28	1	according	accord	VERB
ejpam-3126	28	2	to	to	ADP
ejpam-3126	28	3	their	their	PRON
ejpam-3126	28	4	definition	definition	NOUN
ejpam-3126	28	5	,	,	PUNCT
ejpam-3126	28	6	a	a	DET
ejpam-3126	28	7	proper	proper	ADJ
ejpam-3126	28	8	ideal	ideal	NOUN
ejpam-3126	28	9	i	i	PRON
ejpam-3126	28	10	of	of	ADP
ejpam-3126	28	11	r	r	NOUN
ejpam-3126	28	12	is	be	AUX
ejpam-3126	28	13	called	call	VERB
ejpam-3126	28	14	a	a	DET
ejpam-3126	28	15	ϕ-2	ϕ-2	ADV
ejpam-3126	28	16	-	-	PUNCT
ejpam-3126	28	17	absorbing	absorb	VERB
ejpam-3126	28	18	primary	primary	ADJ
ejpam-3126	28	19	ideal	ideal	NOUN
ejpam-3126	28	20	if	if	SCONJ
ejpam-3126	28	21	whenever	whenever	SCONJ
ejpam-3126	28	22	a1a2a3	a1a2a3	ADJ
ejpam-3126	28	23	∈	∈	NOUN
ejpam-3126	28	24	i	i	PRON
ejpam-3126	28	25	−	−	PROPN
ejpam-3126	28	26	ϕ(i	ϕ(i	NUM
ejpam-3126	28	27	)	)	PUNCT
ejpam-3126	28	28	for	for	ADP
ejpam-3126	28	29	a1	a1	PROPN
ejpam-3126	28	30	,	,	PUNCT
ejpam-3126	28	31	a2	a2	PROPN
ejpam-3126	28	32	,	,	PUNCT
ejpam-3126	28	33	a3	a3	NOUN
ejpam-3126	28	34	∈	∈	PROPN
ejpam-3126	28	35	r	r	NOUN
ejpam-3126	28	36	,	,	PUNCT
ejpam-3126	28	37	then	then	ADV
ejpam-3126	28	38	a1a2	a1a2	AUX
ejpam-3126	28	39	∈	∈	NOUN
ejpam-3126	28	40	i	i	PRON
ejpam-3126	28	41	or	or	CCONJ
ejpam-3126	28	42	a2a3	a2a3	PROPN
ejpam-3126	28	43	∈	∈	NOUN
ejpam-3126	28	44	√	√	VERB
ejpam-3126	28	45	i	i	PRON
ejpam-3126	28	46	or	or	CCONJ
ejpam-3126	28	47	a1a3	a1a3	ADP
ejpam-3126	28	48	∈	∈	NOUN
ejpam-3126	28	49	√	√	NOUN
ejpam-3126	28	50	i.	i.	NOUN
ejpam-3126	28	51	clearly	clearly	ADV
ejpam-3126	28	52	a	a	DET
ejpam-3126	28	53	ϕ-2	ϕ-2	ADV
ejpam-3126	28	54	-	-	PUNCT
ejpam-3126	28	55	absorbing	absorb	VERB
ejpam-3126	28	56	ideal	ideal	NOUN
ejpam-3126	28	57	of	of	ADP
ejpam-3126	28	58	r	r	NOUN
ejpam-3126	28	59	is	be	AUX
ejpam-3126	28	60	also	also	ADV
ejpam-3126	28	61	a	a	DET
ejpam-3126	28	62	ϕ-2	ϕ-2	ADV
ejpam-3126	28	63	-	-	PUNCT
ejpam-3126	28	64	absorbing	absorb	VERB
ejpam-3126	28	65	primary	primary	ADJ
ejpam-3126	28	66	ideal	ideal	NOUN
ejpam-3126	28	67	of	of	ADP
ejpam-3126	28	68	r.	r.	PROPN
ejpam-3126	28	69	other	other	ADJ
ejpam-3126	28	70	generalizations	generalization	NOUN
ejpam-3126	28	71	of	of	ADP
ejpam-3126	28	72	prime	prime	ADJ
ejpam-3126	28	73	ideals	ideal	NOUN
ejpam-3126	28	74	have	have	AUX
ejpam-3126	28	75	recently	recently	ADV
ejpam-3126	28	76	been	be	AUX
ejpam-3126	28	77	studied	study	VERB
ejpam-3126	28	78	in	in	ADP
ejpam-3126	28	79	[	[	X
ejpam-3126	28	80	4	4	NUM
ejpam-3126	28	81	,	,	PUNCT
ejpam-3126	28	82	3	3	NUM
ejpam-3126	28	83	,	,	PUNCT
ejpam-3126	28	84	5	5	NUM
ejpam-3126	28	85	,	,	PUNCT
ejpam-3126	28	86	6	6	NUM
ejpam-3126	28	87	]	]	PUNCT
ejpam-3126	28	88	.	.	PUNCT
ejpam-3126	29	1	the	the	DET
ejpam-3126	29	2	notion	notion	NOUN
ejpam-3126	29	3	of	of	ADP
ejpam-3126	29	4	φ	φ	PROPN
ejpam-3126	29	5	-	-	ADJ
ejpam-3126	29	6	prime	prime	ADJ
ejpam-3126	29	7	submodule	submodule	NOUN
ejpam-3126	29	8	,	,	PUNCT
ejpam-3126	29	9	which	which	PRON
ejpam-3126	29	10	is	be	AUX
ejpam-3126	29	11	a	a	DET
ejpam-3126	29	12	generalization	generalization	NOUN
ejpam-3126	29	13	of	of	ADP
ejpam-3126	29	14	prime	prime	ADJ
ejpam-3126	29	15	submodule	submodule	NOUN
ejpam-3126	29	16	,	,	PUNCT
ejpam-3126	29	17	was	be	AUX
ejpam-3126	29	18	introduced	introduce	VERB
ejpam-3126	29	19	by	by	ADP
ejpam-3126	29	20	zamani	zamani	PROPN
ejpam-3126	29	21	in	in	ADP
ejpam-3126	29	22	[	[	PUNCT
ejpam-3126	29	23	11	11	NUM
ejpam-3126	29	24	]	]	PUNCT
ejpam-3126	29	25	.	.	PUNCT
ejpam-3126	30	1	let	let	VERB
ejpam-3126	30	2	φ	φ	NOUN
ejpam-3126	30	3	:	:	PUNCT
ejpam-3126	30	4	s(m)→	s(m)→	NOUN
ejpam-3126	30	5	s(m	s(m	NOUN
ejpam-3126	30	6	)	)	PUNCT
ejpam-3126	30	7	∪	∪	ADP
ejpam-3126	30	8	{	{	PUNCT
ejpam-3126	30	9	∅	∅	NOUN
ejpam-3126	30	10	}	}	PUNCT
ejpam-3126	30	11	be	be	AUX
ejpam-3126	30	12	a	a	DET
ejpam-3126	30	13	function	function	NOUN
ejpam-3126	30	14	where	where	SCONJ
ejpam-3126	30	15	s(m	s(m	NOUN
ejpam-3126	30	16	)	)	PUNCT
ejpam-3126	30	17	is	be	AUX
ejpam-3126	30	18	a	a	DET
ejpam-3126	30	19	set	set	NOUN
ejpam-3126	30	20	of	of	ADP
ejpam-3126	30	21	all	all	DET
ejpam-3126	30	22	submodules	submodule	NOUN
ejpam-3126	30	23	of	of	ADP
ejpam-3126	30	24	m	m	PROPN
ejpam-3126	30	25	.	.	PUNCT
ejpam-3126	31	1	a	a	DET
ejpam-3126	31	2	proper	proper	ADJ
ejpam-3126	31	3	submodule	submodule	NOUN
ejpam-3126	31	4	n	n	PROPN
ejpam-3126	31	5	of	of	ADP
ejpam-3126	31	6	m	m	PROPN
ejpam-3126	31	7	is	be	AUX
ejpam-3126	31	8	called	call	VERB
ejpam-3126	31	9	φ	φ	NUM
ejpam-3126	31	10	-	-	ADJ
ejpam-3126	31	11	prime	prime	ADJ
ejpam-3126	31	12	submodule	submodule	NOUN
ejpam-3126	31	13	of	of	ADP
ejpam-3126	31	14	m	m	PROPN
ejpam-3126	31	15	if	if	SCONJ
ejpam-3126	31	16	whenever	whenever	SCONJ
ejpam-3126	31	17	a	a	DET
ejpam-3126	31	18	∈	∈	NOUN
ejpam-3126	31	19	r	r	NOUN
ejpam-3126	31	20	and	and	CCONJ
ejpam-3126	31	21	am	be	AUX
ejpam-3126	31	22	∈	∈	PROPN
ejpam-3126	31	23	n	n	CCONJ
ejpam-3126	31	24	−	−	PROPN
ejpam-3126	31	25	φ(n	φ(n	NOUN
ejpam-3126	31	26	)	)	PUNCT
ejpam-3126	31	27	,	,	PUNCT
ejpam-3126	31	28	then	then	ADV
ejpam-3126	31	29	m	m	VERB
ejpam-3126	31	30	∈	∈	PROPN
ejpam-3126	31	31	n	n	NOUN
ejpam-3126	31	32	or	or	CCONJ
ejpam-3126	31	33	a	a	DET
ejpam-3126	31	34	∈	∈	PROPN
ejpam-3126	31	35	(	(	PUNCT
ejpam-3126	31	36	n	n	NOUN
ejpam-3126	31	37	:	:	PUNCT
ejpam-3126	31	38	m	m	X
ejpam-3126	31	39	)	)	PUNCT
ejpam-3126	31	40	.	.	PUNCT
ejpam-3126	32	1	since	since	SCONJ
ejpam-3126	32	2	n	n	CCONJ
ejpam-3126	32	3	−	−	PROPN
ejpam-3126	32	4	φ(n	φ(n	ADJ
ejpam-3126	32	5	)	)	PUNCT
ejpam-3126	32	6	=	=	SYM
ejpam-3126	32	7	n	n	PRON
ejpam-3126	32	8	−	−	PROPN
ejpam-3126	32	9	(	(	PUNCT
ejpam-3126	32	10	n	n	X
ejpam-3126	32	11	∩	∩	ADJ
ejpam-3126	32	12	φ(i	φ(i	PROPN
ejpam-3126	32	13	)	)	PUNCT
ejpam-3126	32	14	)	)	PUNCT
ejpam-3126	32	15	,	,	PUNCT
ejpam-3126	32	16	without	without	ADP
ejpam-3126	32	17	loss	loss	NOUN
ejpam-3126	32	18	of	of	ADP
ejpam-3126	32	19	generality	generality	NOUN
ejpam-3126	32	20	we	we	PRON
ejpam-3126	32	21	may	may	AUX
ejpam-3126	32	22	assume	assume	VERB
ejpam-3126	32	23	that	that	SCONJ
ejpam-3126	32	24	φ(n	φ(n	NOUN
ejpam-3126	32	25	)	)	PUNCT
ejpam-3126	32	26	⊆	⊆	NUM
ejpam-3126	32	27	n	n	NOUN
ejpam-3126	32	28	.	.	PUNCT
ejpam-3126	33	1	recall	recall	VERB
ejpam-3126	33	2	that	that	SCONJ
ejpam-3126	33	3	a	a	DET
ejpam-3126	33	4	proper	proper	ADJ
ejpam-3126	33	5	submodule	submodule	NOUN
ejpam-3126	33	6	n	n	PROPN
ejpam-3126	33	7	of	of	ADP
ejpam-3126	33	8	m	m	PROPN
ejpam-3126	33	9	is	be	AUX
ejpam-3126	33	10	called	call	VERB
ejpam-3126	33	11	a	a	DET
ejpam-3126	33	12	φ	φ	ADJ
ejpam-3126	33	13	-	-	ADJ
ejpam-3126	33	14	primary	primary	ADJ
ejpam-3126	33	15	submodule	submodule	NOUN
ejpam-3126	33	16	submodule	submodule	NOUN
ejpam-3126	33	17	of	of	ADP
ejpam-3126	33	18	m	m	PRON
ejpam-3126	33	19	as	as	ADP
ejpam-3126	33	20	in	in	ADP
ejpam-3126	33	21	[	[	PUNCT
ejpam-3126	33	22	11	11	NUM
ejpam-3126	33	23	]	]	PUNCT
ejpam-3126	33	24	if	if	SCONJ
ejpam-3126	33	25	whenever	whenever	SCONJ
ejpam-3126	33	26	am	be	AUX
ejpam-3126	33	27	∈	∈	PROPN
ejpam-3126	33	28	n	n	CCONJ
ejpam-3126	33	29	−	−	PROPN
ejpam-3126	33	30	φ(n	φ(n	PROPN
ejpam-3126	33	31	)	)	PUNCT
ejpam-3126	33	32	for	for	ADP
ejpam-3126	33	33	some	some	DET
ejpam-3126	33	34	a	a	DET
ejpam-3126	33	35	∈	∈	PROPN
ejpam-3126	33	36	r	r	NOUN
ejpam-3126	33	37	,	,	PUNCT
ejpam-3126	33	38	m	m	VERB
ejpam-3126	33	39	∈	∈	ADJ
ejpam-3126	33	40	m	m	NOUN
ejpam-3126	33	41	,	,	PUNCT
ejpam-3126	33	42	then	then	ADV
ejpam-3126	33	43	m	m	VERB
ejpam-3126	33	44	∈	∈	PROPN
ejpam-3126	33	45	n	n	NOUN
ejpam-3126	33	46	or	or	CCONJ
ejpam-3126	33	47	an	an	DET
ejpam-3126	33	48	∈	∈	PROPN
ejpam-3126	33	49	(	(	PUNCT
ejpam-3126	33	50	n	n	NOUN
ejpam-3126	33	51	:	:	PUNCT
ejpam-3126	33	52	m	m	X
ejpam-3126	33	53	)	)	PUNCT
ejpam-3126	33	54	for	for	ADP
ejpam-3126	33	55	some	some	DET
ejpam-3126	33	56	positive	positive	ADJ
ejpam-3126	33	57	integer	integer	NOUN
ejpam-3126	33	58	n.	n.	NOUN
ejpam-3126	33	59	in	in	ADP
ejpam-3126	33	60	2017	2017	NUM
ejpam-3126	33	61	,	,	PUNCT
ejpam-3126	33	62	ebrahimpour	ebrahimpour	NOUN
ejpam-3126	33	63	and	and	CCONJ
ejpam-3126	33	64	mirzaee	mirzaee	PROPN
ejpam-3126	34	1	[	[	X
ejpam-3126	34	2	8	8	NUM
ejpam-3126	34	3	]	]	PUNCT
ejpam-3126	34	4	generalized	generalize	VERB
ejpam-3126	34	5	the	the	DET
ejpam-3126	34	6	concept	concept	NOUN
ejpam-3126	34	7	of	of	ADP
ejpam-3126	34	8	semiprime	semiprime	NOUN
ejpam-3126	34	9	submodules	submodule	NOUN
ejpam-3126	34	10	to	to	ADP
ejpam-3126	34	11	φ	φ	VERB
ejpam-3126	34	12	-	-	PUNCT
ejpam-3126	34	13	semiprime	semiprime	NOUN
ejpam-3126	34	14	submodules	submodule	NOUN
ejpam-3126	34	15	.	.	PUNCT
ejpam-3126	35	1	according	accord	VERB
ejpam-3126	35	2	to	to	ADP
ejpam-3126	35	3	their	their	PRON
ejpam-3126	35	4	definition	definition	NOUN
ejpam-3126	35	5	,	,	PUNCT
ejpam-3126	35	6	a	a	DET
ejpam-3126	35	7	proper	proper	ADJ
ejpam-3126	35	8	submodule	submodule	NOUN
ejpam-3126	35	9	n	n	PROPN
ejpam-3126	35	10	of	of	ADP
ejpam-3126	35	11	m	m	PROPN
ejpam-3126	35	12	is	be	AUX
ejpam-3126	35	13	called	call	VERB
ejpam-3126	35	14	a	a	DET
ejpam-3126	35	15	φ	φ	NUM
ejpam-3126	35	16	-	-	PUNCT
ejpam-3126	35	17	semiprime	semiprime	NOUN
ejpam-3126	35	18	submodule	submodule	NOUN
ejpam-3126	35	19	if	if	SCONJ
ejpam-3126	35	20	whenever	whenever	SCONJ
ejpam-3126	35	21	a2	a2	PROPN
ejpam-3126	35	22	m	m	PROPN
ejpam-3126	35	23	∈	∈	NOUN
ejpam-3126	35	24	n	n	CCONJ
ejpam-3126	35	25	−	−	PROPN
ejpam-3126	35	26	φ(n	φ(n	PROPN
ejpam-3126	35	27	)	)	PUNCT
ejpam-3126	35	28	for	for	ADP
ejpam-3126	35	29	a	a	DET
ejpam-3126	35	30	∈	∈	PROPN
ejpam-3126	35	31	r	r	NOUN
ejpam-3126	35	32	,	,	PUNCT
ejpam-3126	35	33	m	m	VERB
ejpam-3126	35	34	∈	∈	ADJ
ejpam-3126	35	35	m	m	NOUN
ejpam-3126	35	36	,	,	PUNCT
ejpam-3126	35	37	then	then	ADV
ejpam-3126	35	38	am	be	AUX
ejpam-3126	35	39	∈	∈	PROPN
ejpam-3126	35	40	n	n	NOUN
ejpam-3126	35	41	.	.	PUNCT
ejpam-3126	36	1	in	in	ADP
ejpam-3126	36	2	[	[	X
ejpam-3126	36	3	10	10	NUM
ejpam-3126	36	4	]	]	PUNCT
ejpam-3126	36	5	,	,	PUNCT
ejpam-3126	36	6	the	the	DET
ejpam-3126	36	7	concept	concept	NOUN
ejpam-3126	36	8	of	of	ADP
ejpam-3126	36	9	φ	φ	PROPN
ejpam-3126	36	10	-	-	NOUN
ejpam-3126	36	11	prime	prime	NOUN
ejpam-3126	36	12	and	and	CCONJ
ejpam-3126	36	13	φ	φ	VERB
ejpam-3126	36	14	-	-	ADJ
ejpam-3126	36	15	primary	primary	ADJ
ejpam-3126	36	16	submodules	submodule	NOUN
ejpam-3126	36	17	generalized	generalize	VERB
ejpam-3126	36	18	to	to	ADP
ejpam-3126	36	19	φ-2	φ-2	ADJ
ejpam-3126	36	20	-	-	PUNCT
ejpam-3126	36	21	absorbing	absorbing	ADJ
ejpam-3126	36	22	primary	primary	ADJ
ejpam-3126	36	23	submodule	submodule	NOUN
ejpam-3126	36	24	of	of	ADP
ejpam-3126	36	25	a	a	DET
ejpam-3126	36	26	module	module	NOUN
ejpam-3126	36	27	over	over	ADP
ejpam-3126	36	28	a	a	DET
ejpam-3126	36	29	commutative	commutative	ADJ
ejpam-3126	36	30	ring	ring	NOUN
ejpam-3126	36	31	.	.	PUNCT
ejpam-3126	37	1	let	let	VERB
ejpam-3126	37	2	n	n	PRON
ejpam-3126	37	3	be	be	AUX
ejpam-3126	37	4	a	a	DET
ejpam-3126	37	5	proper	proper	ADJ
ejpam-3126	37	6	submodule	submodule	NOUN
ejpam-3126	37	7	of	of	ADP
ejpam-3126	37	8	m	m	PROPN
ejpam-3126	37	9	.	.	PUNCT
ejpam-3126	38	1	n	n	PRON
ejpam-3126	38	2	is	be	AUX
ejpam-3126	38	3	said	say	VERB
ejpam-3126	38	4	to	to	PART
ejpam-3126	38	5	be	be	AUX
ejpam-3126	38	6	a	a	DET
ejpam-3126	38	7	φ-2	φ-2	ADJ
ejpam-3126	38	8	-	-	PUNCT
ejpam-3126	38	9	absorbing	absorbing	ADJ
ejpam-3126	38	10	primary	primary	ADJ
ejpam-3126	38	11	submodule	submodule	NOUN
ejpam-3126	38	12	of	of	ADP
ejpam-3126	38	13	m	m	PROPN
ejpam-3126	38	14	if	if	SCONJ
ejpam-3126	38	15	whenever	whenever	SCONJ
ejpam-3126	38	16	a1	a1	PROPN
ejpam-3126	38	17	,	,	PUNCT
ejpam-3126	38	18	a2	a2	PROPN
ejpam-3126	38	19	∈	∈	PROPN
ejpam-3126	38	20	r	r	NOUN
ejpam-3126	38	21	and	and	CCONJ
ejpam-3126	38	22	m	m	PROPN
ejpam-3126	38	23	∈	∈	NOUN
ejpam-3126	38	24	m	m	VERB
ejpam-3126	38	25	with	with	ADP
ejpam-3126	38	26	a1a2	a1a2	PROPN
ejpam-3126	38	27	m	m	NOUN
ejpam-3126	38	28	∈	∈	ADJ
ejpam-3126	38	29	n	n	CCONJ
ejpam-3126	38	30	−	−	PROPN
ejpam-3126	39	1	φ(n	φ(n	NOUN
ejpam-3126	40	1	)	)	PUNCT
ejpam-3126	41	1	,	,	PUNCT
ejpam-3126	41	2	then	then	ADV
ejpam-3126	41	3	a1a2	a1a2	ADP
ejpam-3126	41	4	∈	∈	NOUN
ejpam-3126	41	5	√	√	NUM
ejpam-3126	41	6	(	(	PUNCT
ejpam-3126	41	7	n	n	NUM
ejpam-3126	41	8	:	:	PUNCT
ejpam-3126	41	9	m	m	X
ejpam-3126	41	10	)	)	PUNCT
ejpam-3126	41	11	or	or	CCONJ
ejpam-3126	41	12	a1	a1	NOUN
ejpam-3126	41	13	m	m	PROPN
ejpam-3126	41	14	∈	∈	NOUN
ejpam-3126	41	15	n	n	NOUN
ejpam-3126	41	16	or	or	CCONJ
ejpam-3126	41	17	a2	a2	PROPN
ejpam-3126	41	18	m	m	NOUN
ejpam-3126	41	19	∈	∈	NOUN
ejpam-3126	41	20	n	n	NOUN
ejpam-3126	41	21	.	.	PUNCT
ejpam-3126	42	1	moreover	moreover	ADV
ejpam-3126	42	2	,	,	PUNCT
ejpam-3126	42	3	recall	recall	VERB
ejpam-3126	42	4	from	from	ADP
ejpam-3126	42	5	[	[	X
ejpam-3126	42	6	10	10	NUM
ejpam-3126	42	7	]	]	PUNCT
ejpam-3126	42	8	that	that	SCONJ
ejpam-3126	42	9	a	a	DET
ejpam-3126	42	10	proper	proper	ADJ
ejpam-3126	42	11	submodule	submodule	NOUN
ejpam-3126	42	12	n	n	PROPN
ejpam-3126	42	13	of	of	ADP
ejpam-3126	42	14	m	m	PROPN
ejpam-3126	42	15	is	be	AUX
ejpam-3126	42	16	said	say	VERB
ejpam-3126	42	17	to	to	PART
ejpam-3126	42	18	be	be	AUX
ejpam-3126	42	19	a	a	DET
ejpam-3126	42	20	φ-2	φ-2	ADJ
ejpam-3126	42	21	-	-	PUNCT
ejpam-3126	42	22	absorbing	absorb	VERB
ejpam-3126	42	23	submodule	submodule	NOUN
ejpam-3126	42	24	of	of	ADP
ejpam-3126	42	25	m	m	PROPN
ejpam-3126	42	26	if	if	SCONJ
ejpam-3126	42	27	whenever	whenever	SCONJ
ejpam-3126	42	28	a1	a1	PROPN
ejpam-3126	42	29	,	,	PUNCT
ejpam-3126	42	30	a2	a2	PROPN
ejpam-3126	42	31	∈	∈	PROPN
ejpam-3126	42	32	r	r	NOUN
ejpam-3126	42	33	and	and	CCONJ
ejpam-3126	42	34	m	m	PROPN
ejpam-3126	42	35	∈	∈	NOUN
ejpam-3126	42	36	m	m	VERB
ejpam-3126	42	37	with	with	ADP
ejpam-3126	42	38	a1a2	a1a2	PROPN
ejpam-3126	42	39	m	m	NOUN
ejpam-3126	42	40	∈	∈	ADJ
ejpam-3126	42	41	n	n	CCONJ
ejpam-3126	43	1	−	−	PROPN
ejpam-3126	43	2	φ(n	φ(n	ADJ
ejpam-3126	43	3	)	)	PUNCT
ejpam-3126	43	4	implies	imply	VERB
ejpam-3126	43	5	a1a2	a1a2	ADP
ejpam-3126	43	6	∈	∈	PROPN
ejpam-3126	43	7	(	(	PUNCT
ejpam-3126	43	8	n	n	NOUN
ejpam-3126	43	9	:	:	PUNCT
ejpam-3126	43	10	m	m	X
ejpam-3126	43	11	)	)	PUNCT
ejpam-3126	43	12	or	or	CCONJ
ejpam-3126	43	13	a1	a1	NOUN
ejpam-3126	43	14	m	m	PROPN
ejpam-3126	43	15	∈	∈	NOUN
ejpam-3126	43	16	n	n	NOUN
ejpam-3126	43	17	or	or	CCONJ
ejpam-3126	43	18	a2	a2	PROPN
ejpam-3126	43	19	m	m	NOUN
ejpam-3126	43	20	∈	∈	NOUN
ejpam-3126	43	21	n	n	NOUN
ejpam-3126	43	22	.	.	PUNCT
ejpam-3126	44	1	thus	thus	ADV
ejpam-3126	44	2	a	a	DET
ejpam-3126	44	3	φ-2	φ-2	ADJ
ejpam-3126	44	4	-	-	PUNCT
ejpam-3126	44	5	absorbing	absorb	VERB
ejpam-3126	44	6	submodule	submodule	NOUN
ejpam-3126	44	7	is	be	AUX
ejpam-3126	44	8	just	just	ADV
ejpam-3126	44	9	a	a	DET
ejpam-3126	44	10	φ-2	φ-2	ADJ
ejpam-3126	44	11	-	-	PUNCT
ejpam-3126	44	12	absorbing	absorbing	ADJ
ejpam-3126	44	13	primary	primary	ADJ
ejpam-3126	44	14	submodule	submodule	NOUN
ejpam-3126	44	15	.	.	PUNCT
ejpam-3126	45	1	in	in	ADP
ejpam-3126	45	2	this	this	DET
ejpam-3126	45	3	paper	paper	NOUN
ejpam-3126	45	4	,	,	PUNCT
ejpam-3126	45	5	we	we	PRON
ejpam-3126	45	6	extend	extend	VERB
ejpam-3126	45	7	the	the	DET
ejpam-3126	45	8	concept	concept	NOUN
ejpam-3126	45	9	of	of	ADP
ejpam-3126	45	10	φ-2	φ-2	PROPN
ejpam-3126	45	11	-	-	PUNCT
ejpam-3126	45	12	absorbing	absorbing	ADJ
ejpam-3126	45	13	primary	primary	ADJ
ejpam-3126	45	14	submodule	submodule	NOUN
ejpam-3126	45	15	to	to	ADP
ejpam-3126	45	16	the	the	DET
ejpam-3126	45	17	context	context	NOUN
ejpam-3126	45	18	of	of	ADP
ejpam-3126	45	19	φ-2	φ-2	PROPN
ejpam-3126	45	20	-	-	PUNCT
ejpam-3126	45	21	absorbing	absorbing	ADJ
ejpam-3126	45	22	semi	semi	ADJ
ejpam-3126	45	23	-	-	ADJ
ejpam-3126	45	24	primary	primary	ADJ
ejpam-3126	45	25	submodule	submodule	NOUN
ejpam-3126	45	26	.	.	PUNCT
ejpam-3126	46	1	let	let	VERB
ejpam-3126	46	2	φ	φ	PROPN
ejpam-3126	46	3	:	:	PUNCT
ejpam-3126	46	4	s(m	s(m	PROPN
ejpam-3126	46	5	)	)	PUNCT
ejpam-3126	46	6	→	→	SYM
ejpam-3126	46	7	s(m	s(m	NOUN
ejpam-3126	46	8	)	)	PUNCT
ejpam-3126	46	9	∪	∪	NOUN
ejpam-3126	46	10	{	{	PUNCT
ejpam-3126	46	11	∅	∅	NOUN
ejpam-3126	46	12	}	}	PUNCT
ejpam-3126	46	13	be	be	AUX
ejpam-3126	46	14	a	a	DET
ejpam-3126	46	15	function	function	NOUN
ejpam-3126	46	16	where	where	SCONJ
ejpam-3126	46	17	s(m	s(m	NOUN
ejpam-3126	46	18	)	)	PUNCT
ejpam-3126	46	19	is	be	AUX
ejpam-3126	46	20	the	the	DET
ejpam-3126	46	21	set	set	NOUN
ejpam-3126	46	22	of	of	ADP
ejpam-3126	46	23	all	all	DET
ejpam-3126	46	24	submodules	submodule	NOUN
ejpam-3126	46	25	of	of	ADP
ejpam-3126	46	26	m	m	PROPN
ejpam-3126	46	27	.	.	PUNCT
ejpam-3126	47	1	a	a	DET
ejpam-3126	47	2	proper	proper	ADJ
ejpam-3126	47	3	submodule	submodule	NOUN
ejpam-3126	47	4	n	n	PROPN
ejpam-3126	47	5	of	of	ADP
ejpam-3126	47	6	m	m	PROPN
ejpam-3126	47	7	is	be	AUX
ejpam-3126	47	8	called	call	VERB
ejpam-3126	47	9	a	a	DET
ejpam-3126	47	10	φ-2	φ-2	NOUN
ejpam-3126	47	11	-	-	PUNCT
ejpam-3126	47	12	absorbing	absorbing	ADJ
ejpam-3126	47	13	semi	semi	ADJ
ejpam-3126	47	14	-	-	ADJ
ejpam-3126	47	15	primary	primary	ADJ
ejpam-3126	47	16	submodule	submodule	NOUN
ejpam-3126	47	17	if	if	SCONJ
ejpam-3126	47	18	for	for	ADP
ejpam-3126	47	19	each	each	DET
ejpam-3126	47	20	m	m	NOUN
ejpam-3126	47	21	∈	∈	PROPN
ejpam-3126	47	22	m	m	NOUN
ejpam-3126	47	23	and	and	CCONJ
ejpam-3126	47	24	a1	a1	NOUN
ejpam-3126	47	25	,	,	PUNCT
ejpam-3126	47	26	a2	a2	PROPN
ejpam-3126	47	27	∈	∈	PROPN
ejpam-3126	47	28	r	r	NOUN
ejpam-3126	47	29	with	with	ADP
ejpam-3126	47	30	a1a2	a1a2	PROPN
ejpam-3126	47	31	m	m	NOUN
ejpam-3126	47	32	∈	∈	ADJ
ejpam-3126	47	33	n	n	CCONJ
ejpam-3126	47	34	−	−	PROPN
ejpam-3126	47	35	φ(n	φ(n	NOUN
ejpam-3126	47	36	)	)	PUNCT
ejpam-3126	47	37	,	,	PUNCT
ejpam-3126	47	38	then	then	ADV
ejpam-3126	47	39	a1a2	a1a2	ADP
ejpam-3126	47	40	∈	∈	NOUN
ejpam-3126	47	41	√	√	NUM
ejpam-3126	47	42	(	(	PUNCT
ejpam-3126	47	43	n	n	NUM
ejpam-3126	47	44	:	:	PUNCT
ejpam-3126	47	45	m	m	X
ejpam-3126	47	46	)	)	PUNCT
ejpam-3126	47	47	or	or	CCONJ
ejpam-3126	47	48	a1	a1	NOUN
ejpam-3126	47	49	m	m	PROPN
ejpam-3126	47	50	∈	∈	NOUN
ejpam-3126	47	51	n	n	NOUN
ejpam-3126	47	52	or	or	CCONJ
ejpam-3126	47	53	an2	an2	PROPN
ejpam-3126	47	54	m	m	NOUN
ejpam-3126	47	55	∈	∈	PROPN
ejpam-3126	47	56	n	n	NOUN
ejpam-3126	47	57	for	for	ADP
ejpam-3126	47	58	some	some	DET
ejpam-3126	47	59	positive	positive	ADJ
ejpam-3126	47	60	integer	integer	NOUN
ejpam-3126	47	61	n.	n.	NOUN
ejpam-3126	47	62	let	let	VERB
ejpam-3126	47	63	n	n	PRON
ejpam-3126	47	64	be	be	AUX
ejpam-3126	47	65	a	a	DET
ejpam-3126	47	66	φ-2	φ-2	NOUN
ejpam-3126	47	67	-	-	PUNCT
ejpam-3126	47	68	absorbing	absorbing	ADJ
ejpam-3126	47	69	semi	semi	ADJ
ejpam-3126	47	70	-	-	ADJ
ejpam-3126	47	71	primary	primary	ADJ
ejpam-3126	47	72	submodule	submodule	NOUN
ejpam-3126	47	73	of	of	ADP
ejpam-3126	47	74	m	m	PROPN
ejpam-3126	47	75	.	.	PUNCT
ejpam-3126	48	1	•	•	INTJ
ejpam-3126	48	2	if	if	SCONJ
ejpam-3126	48	3	φ(n	φ(n	ADJ
ejpam-3126	48	4	)	)	PUNCT
ejpam-3126	48	5	=	=	NOUN
ejpam-3126	48	6	∅	∅	NOUN
ejpam-3126	48	7	for	for	ADP
ejpam-3126	48	8	every	every	DET
ejpam-3126	48	9	n	n	PRON
ejpam-3126	48	10	∈	∈	PROPN
ejpam-3126	48	11	s(m	s(m	PROPN
ejpam-3126	48	12	)	)	PUNCT
ejpam-3126	49	1	,	,	PUNCT
ejpam-3126	49	2	then	then	ADV
ejpam-3126	49	3	we	we	PRON
ejpam-3126	49	4	say	say	VERB
ejpam-3126	49	5	that	that	SCONJ
ejpam-3126	49	6	φ	φ	PROPN
ejpam-3126	49	7	=	=	SYM
ejpam-3126	49	8	φ∅	φ∅	NOUN
ejpam-3126	49	9	and	and	CCONJ
ejpam-3126	49	10	n	n	ADV
ejpam-3126	49	11	is	be	AUX
ejpam-3126	49	12	called	call	VERB
ejpam-3126	49	13	a	a	DET
ejpam-3126	49	14	φ∅-2absorbing	φ∅-2absorbing	NOUN
ejpam-3126	49	15	semi	semi	ADJ
ejpam-3126	49	16	-	-	ADJ
ejpam-3126	49	17	primary	primary	ADJ
ejpam-3126	49	18	submodule	submodule	NOUN
ejpam-3126	49	19	of	of	ADP
ejpam-3126	49	20	m	m	PROPN
ejpam-3126	49	21	,	,	PUNCT
ejpam-3126	49	22	and	and	CCONJ
ejpam-3126	49	23	hence	hence	ADV
ejpam-3126	49	24	n	n	PRON
ejpam-3126	49	25	is	be	AUX
ejpam-3126	49	26	a	a	DET
ejpam-3126	49	27	2	2	NUM
ejpam-3126	49	28	-	-	PUNCT
ejpam-3126	49	29	absorbing	absorbing	ADJ
ejpam-3126	49	30	semi	semi	ADJ
ejpam-3126	49	31	-	-	ADJ
ejpam-3126	49	32	primary	primary	ADJ
ejpam-3126	49	33	submodule	submodule	NOUN
ejpam-3126	49	34	of	of	ADP
ejpam-3126	49	35	m	m	PROPN
ejpam-3126	49	36	.	.	PUNCT
ejpam-3126	50	1	•	•	INTJ
ejpam-3126	50	2	if	if	SCONJ
ejpam-3126	50	3	φ(n	φ(n	ADJ
ejpam-3126	50	4	)	)	PUNCT
ejpam-3126	50	5	=	=	PRON
ejpam-3126	50	6	{	{	PUNCT
ejpam-3126	50	7	0	0	NUM
ejpam-3126	50	8	}	}	PUNCT
ejpam-3126	50	9	for	for	ADP
ejpam-3126	50	10	every	every	DET
ejpam-3126	50	11	n	n	PRON
ejpam-3126	50	12	∈	∈	PROPN
ejpam-3126	50	13	s(m	s(m	PROPN
ejpam-3126	50	14	)	)	PUNCT
ejpam-3126	50	15	,	,	PUNCT
ejpam-3126	50	16	then	then	ADV
ejpam-3126	50	17	we	we	PRON
ejpam-3126	50	18	say	say	VERB
ejpam-3126	50	19	that	that	SCONJ
ejpam-3126	50	20	φ	φ	PROPN
ejpam-3126	50	21	=	=	SYM
ejpam-3126	50	22	φ0	φ0	PROPN
ejpam-3126	50	23	and	and	CCONJ
ejpam-3126	50	24	n	n	PROPN
ejpam-3126	50	25	is	be	AUX
ejpam-3126	50	26	called	call	VERB
ejpam-3126	50	27	a	a	DET
ejpam-3126	50	28	φ0	φ0	PROPN
ejpam-3126	50	29	-	-	PUNCT
ejpam-3126	50	30	2	2	NUM
ejpam-3126	50	31	-	-	PUNCT
ejpam-3126	50	32	absorbing	absorbing	ADJ
ejpam-3126	50	33	semi	semi	ADJ
ejpam-3126	50	34	-	-	ADJ
ejpam-3126	50	35	primary	primary	ADJ
ejpam-3126	50	36	submodule	submodule	NOUN
ejpam-3126	50	37	of	of	ADP
ejpam-3126	50	38	m	m	PROPN
ejpam-3126	50	39	,	,	PUNCT
ejpam-3126	50	40	and	and	CCONJ
ejpam-3126	50	41	hence	hence	ADV
ejpam-3126	50	42	n	n	PRON
ejpam-3126	50	43	is	be	AUX
ejpam-3126	50	44	a	a	DET
ejpam-3126	50	45	weakly	weakly	ADJ
ejpam-3126	50	46	2	2	NUM
ejpam-3126	50	47	-	-	PUNCT
ejpam-3126	50	48	absorbing	absorbing	ADJ
ejpam-3126	50	49	semi	semi	ADJ
ejpam-3126	50	50	-	-	ADJ
ejpam-3126	50	51	primary	primary	ADJ
ejpam-3126	50	52	submodule	submodule	NOUN
ejpam-3126	50	53	of	of	ADP
ejpam-3126	50	54	m	m	PROPN
ejpam-3126	50	55	.	.	PUNCT
ejpam-3126	51	1	•	•	INTJ
ejpam-3126	51	2	if	if	SCONJ
ejpam-3126	51	3	φ(n	φ(n	ADJ
ejpam-3126	51	4	)	)	PUNCT
ejpam-3126	51	5	=	=	SYM
ejpam-3126	52	1	n	n	CCONJ
ejpam-3126	52	2	for	for	ADP
ejpam-3126	52	3	every	every	DET
ejpam-3126	52	4	n	n	PRON
ejpam-3126	52	5	∈	∈	PROPN
ejpam-3126	52	6	s(m	s(m	PROPN
ejpam-3126	52	7	)	)	PUNCT
ejpam-3126	52	8	,	,	PUNCT
ejpam-3126	52	9	then	then	ADV
ejpam-3126	52	10	we	we	PRON
ejpam-3126	52	11	say	say	VERB
ejpam-3126	52	12	that	that	SCONJ
ejpam-3126	52	13	φ	φ	PROPN
ejpam-3126	52	14	=	=	SYM
ejpam-3126	52	15	φ1	φ1	PROPN
ejpam-3126	52	16	and	and	CCONJ
ejpam-3126	52	17	n	n	PROPN
ejpam-3126	52	18	is	be	AUX
ejpam-3126	52	19	called	call	VERB
ejpam-3126	52	20	a	a	DET
ejpam-3126	52	21	φ1	φ1	PROPN
ejpam-3126	52	22	-	-	PUNCT
ejpam-3126	52	23	2	2	NUM
ejpam-3126	52	24	-	-	PUNCT
ejpam-3126	52	25	absorbing	absorbing	ADJ
ejpam-3126	52	26	semi	semi	ADJ
ejpam-3126	52	27	-	-	ADJ
ejpam-3126	52	28	primary	primary	ADJ
ejpam-3126	52	29	submodule	submodule	NOUN
ejpam-3126	52	30	of	of	ADP
ejpam-3126	52	31	m	m	PROPN
ejpam-3126	52	32	.	.	PUNCT
ejpam-3126	53	1	it	it	PRON
ejpam-3126	53	2	is	be	AUX
ejpam-3126	53	3	easy	easy	ADJ
ejpam-3126	53	4	to	to	PART
ejpam-3126	53	5	see	see	VERB
ejpam-3126	53	6	that	that	SCONJ
ejpam-3126	53	7	every	every	DET
ejpam-3126	53	8	proper	proper	ADJ
ejpam-3126	53	9	pai	pai	PROPN
ejpam-3126	53	10	.	.	PROPN
ejpam-3126	53	11	yiarayong	yiarayong	PROPN
ejpam-3126	53	12	,	,	PUNCT
ejpam-3126	53	13	m.	m.	NOUN
ejpam-3126	53	14	siripitukdet	siripitukdet	NOUN
ejpam-3126	53	15	/	/	SYM
ejpam-3126	53	16	eur	eur	PROPN
ejpam-3126	53	17	.	.	PUNCT
ejpam-3126	54	1	j.	j.	PROPN
ejpam-3126	54	2	pure	pure	PROPN
ejpam-3126	54	3	appl	appl	PROPN
ejpam-3126	54	4	.	.	PROPN
ejpam-3126	54	5	math	math	PROPN
ejpam-3126	54	6	,	,	PUNCT
ejpam-3126	54	7	11	11	NUM
ejpam-3126	54	8	(	(	PUNCT
ejpam-3126	54	9	1	1	NUM
ejpam-3126	54	10	)	)	PUNCT
ejpam-3126	54	11	(	(	PUNCT
ejpam-3126	54	12	2018	2018	NUM
ejpam-3126	54	13	)	)	PUNCT
ejpam-3126	54	14	,	,	PUNCT
ejpam-3126	54	15	35	35	NUM
ejpam-3126	54	16	-	-	SYM
ejpam-3126	54	17	50	50	NUM
ejpam-3126	54	18	37	37	NUM
ejpam-3126	54	19	submodule	submodule	NOUN
ejpam-3126	54	20	is	be	AUX
ejpam-3126	54	21	φ1	φ1	NOUN
ejpam-3126	54	22	-	-	PUNCT
ejpam-3126	54	23	2	2	NUM
ejpam-3126	54	24	-	-	PUNCT
ejpam-3126	54	25	absorbing	absorbing	ADJ
ejpam-3126	54	26	semi	semi	ADJ
ejpam-3126	54	27	-	-	ADJ
ejpam-3126	54	28	primary	primary	ADJ
ejpam-3126	54	29	.	.	PUNCT
ejpam-3126	55	1	•	•	NOUN
ejpam-3126	55	2	if	if	SCONJ
ejpam-3126	55	3	φ(n	φ(n	NOUN
ejpam-3126	55	4	)	)	PUNCT
ejpam-3126	55	5	=	=	SYM
ejpam-3126	55	6	(	(	PUNCT
ejpam-3126	55	7	n	n	NUM
ejpam-3126	55	8	:	:	PUNCT
ejpam-3126	55	9	m)n	m)n	X
ejpam-3126	55	10	for	for	ADP
ejpam-3126	55	11	every	every	DET
ejpam-3126	55	12	n	n	PRON
ejpam-3126	55	13	∈	∈	PROPN
ejpam-3126	55	14	s(m	s(m	PROPN
ejpam-3126	55	15	)	)	PUNCT
ejpam-3126	56	1	,	,	PUNCT
ejpam-3126	56	2	then	then	ADV
ejpam-3126	56	3	we	we	PRON
ejpam-3126	56	4	say	say	VERB
ejpam-3126	56	5	that	that	SCONJ
ejpam-3126	56	6	φ	φ	PROPN
ejpam-3126	56	7	=	=	SYM
ejpam-3126	56	8	φ2	φ2	PROPN
ejpam-3126	56	9	and	and	CCONJ
ejpam-3126	56	10	n	n	PROPN
ejpam-3126	56	11	is	be	AUX
ejpam-3126	56	12	called	call	VERB
ejpam-3126	56	13	a	a	DET
ejpam-3126	56	14	φ2	φ2	PROPN
ejpam-3126	56	15	-	-	PUNCT
ejpam-3126	56	16	2	2	NUM
ejpam-3126	56	17	-	-	PUNCT
ejpam-3126	56	18	absorbing	absorbing	ADJ
ejpam-3126	56	19	semi	semi	ADJ
ejpam-3126	56	20	-	-	ADJ
ejpam-3126	56	21	primary	primary	ADJ
ejpam-3126	56	22	submodule	submodule	NOUN
ejpam-3126	56	23	of	of	ADP
ejpam-3126	56	24	m	m	PROPN
ejpam-3126	56	25	,	,	PUNCT
ejpam-3126	56	26	and	and	CCONJ
ejpam-3126	56	27	hence	hence	ADV
ejpam-3126	56	28	n	n	PRON
ejpam-3126	56	29	is	be	AUX
ejpam-3126	56	30	an	an	DET
ejpam-3126	56	31	almost	almost	ADV
ejpam-3126	56	32	2	2	NUM
ejpam-3126	56	33	-	-	PUNCT
ejpam-3126	56	34	absorbing	absorbing	ADJ
ejpam-3126	56	35	semi	semi	ADJ
ejpam-3126	56	36	-	-	ADJ
ejpam-3126	56	37	primary	primary	ADJ
ejpam-3126	56	38	submodule	submodule	NOUN
ejpam-3126	56	39	of	of	ADP
ejpam-3126	56	40	m	m	PROPN
ejpam-3126	56	41	.	.	PUNCT
ejpam-3126	57	1	•	•	INTJ
ejpam-3126	57	2	if	if	SCONJ
ejpam-3126	57	3	φ(n	φ(n	NOUN
ejpam-3126	57	4	)	)	PUNCT
ejpam-3126	57	5	=	=	PUNCT
ejpam-3126	57	6	(	(	PUNCT
ejpam-3126	58	1	n	n	CCONJ
ejpam-3126	58	2	:	:	PUNCT
ejpam-3126	58	3	m)n−1n	m)n−1n	ADJ
ejpam-3126	58	4	for	for	ADP
ejpam-3126	58	5	every	every	DET
ejpam-3126	58	6	n	n	PRON
ejpam-3126	58	7	∈	∈	PROPN
ejpam-3126	58	8	s(m	s(m	PROPN
ejpam-3126	58	9	)	)	PUNCT
ejpam-3126	58	10	,	,	PUNCT
ejpam-3126	58	11	then	then	ADV
ejpam-3126	58	12	we	we	PRON
ejpam-3126	58	13	say	say	VERB
ejpam-3126	58	14	that	that	SCONJ
ejpam-3126	58	15	φ	φ	PROPN
ejpam-3126	58	16	=	=	SYM
ejpam-3126	58	17	φn≥2	φn≥2	PROPN
ejpam-3126	58	18	and	and	CCONJ
ejpam-3126	58	19	n	n	PROPN
ejpam-3126	58	20	is	be	AUX
ejpam-3126	58	21	called	call	VERB
ejpam-3126	58	22	a	a	DET
ejpam-3126	58	23	φn-2	φn-2	NOUN
ejpam-3126	58	24	-	-	PUNCT
ejpam-3126	58	25	absorbing	absorbing	ADJ
ejpam-3126	58	26	semi	semi	ADJ
ejpam-3126	58	27	-	-	ADJ
ejpam-3126	58	28	primary	primary	ADJ
ejpam-3126	58	29	submodule	submodule	NOUN
ejpam-3126	58	30	of	of	ADP
ejpam-3126	58	31	m	m	PROPN
ejpam-3126	58	32	,	,	PUNCT
ejpam-3126	58	33	and	and	CCONJ
ejpam-3126	58	34	hence	hence	ADV
ejpam-3126	58	35	n	n	PRON
ejpam-3126	58	36	is	be	AUX
ejpam-3126	58	37	a	a	DET
ejpam-3126	58	38	n-2absorbing	n-2absorbing	NOUN
ejpam-3126	58	39	semi	semi	ADJ
ejpam-3126	58	40	-	-	ADJ
ejpam-3126	58	41	primary	primary	ADJ
ejpam-3126	58	42	submodule	submodule	NOUN
ejpam-3126	58	43	of	of	ADP
ejpam-3126	58	44	m	m	PROPN
ejpam-3126	58	45	.	.	PUNCT
ejpam-3126	59	1	•	•	INTJ
ejpam-3126	59	2	if	if	SCONJ
ejpam-3126	59	3	φ(n	φ(n	ADJ
ejpam-3126	59	4	)	)	PUNCT
ejpam-3126	59	5	=	=	SYM
ejpam-3126	60	1	∞⋂	∞⋂	PROPN
ejpam-3126	60	2	i=1	i=1	PROPN
ejpam-3126	60	3	(	(	PUNCT
ejpam-3126	60	4	n	n	X
ejpam-3126	60	5	:	:	PUNCT
ejpam-3126	60	6	m)in	m)in	PROPN
ejpam-3126	60	7	for	for	ADP
ejpam-3126	60	8	every	every	DET
ejpam-3126	60	9	n	n	PRON
ejpam-3126	60	10	∈	∈	PROPN
ejpam-3126	60	11	s(m	s(m	PROPN
ejpam-3126	60	12	)	)	PUNCT
ejpam-3126	60	13	,	,	PUNCT
ejpam-3126	60	14	then	then	ADV
ejpam-3126	60	15	we	we	PRON
ejpam-3126	60	16	say	say	VERB
ejpam-3126	60	17	that	that	SCONJ
ejpam-3126	60	18	φ	φ	PROPN
ejpam-3126	60	19	=	=	SYM
ejpam-3126	60	20	φω	φω	PROPN
ejpam-3126	60	21	and	and	CCONJ
ejpam-3126	60	22	n	n	PROPN
ejpam-3126	60	23	is	be	AUX
ejpam-3126	60	24	called	call	VERB
ejpam-3126	60	25	a	a	DET
ejpam-3126	60	26	φω-2	φω-2	ADV
ejpam-3126	60	27	-	-	PUNCT
ejpam-3126	60	28	absorbing	absorbing	ADJ
ejpam-3126	60	29	semi	semi	ADJ
ejpam-3126	60	30	-	-	ADJ
ejpam-3126	60	31	primary	primary	ADJ
ejpam-3126	60	32	submodule	submodule	NOUN
ejpam-3126	60	33	of	of	ADP
ejpam-3126	60	34	m	m	PROPN
ejpam-3126	60	35	,	,	PUNCT
ejpam-3126	60	36	and	and	CCONJ
ejpam-3126	60	37	hence	hence	ADV
ejpam-3126	60	38	n	n	PRON
ejpam-3126	60	39	is	be	AUX
ejpam-3126	60	40	a	a	DET
ejpam-3126	60	41	ω-2absorbing	ω-2absorbe	VERB
ejpam-3126	60	42	semi	semi	ADJ
ejpam-3126	60	43	-	-	ADJ
ejpam-3126	60	44	primary	primary	ADJ
ejpam-3126	60	45	submodule	submodule	NOUN
ejpam-3126	60	46	of	of	ADP
ejpam-3126	60	47	m	m	PROPN
ejpam-3126	60	48	.	.	PUNCT
ejpam-3126	61	1	in	in	ADP
ejpam-3126	61	2	section	section	NOUN
ejpam-3126	61	3	2	2	NUM
ejpam-3126	61	4	,	,	PUNCT
ejpam-3126	61	5	we	we	PRON
ejpam-3126	61	6	give	give	VERB
ejpam-3126	61	7	some	some	DET
ejpam-3126	61	8	basic	basic	ADJ
ejpam-3126	61	9	properties	property	NOUN
ejpam-3126	61	10	of	of	ADP
ejpam-3126	61	11	φ-2	φ-2	PROPN
ejpam-3126	61	12	-	-	PUNCT
ejpam-3126	61	13	absorbing	absorbing	ADJ
ejpam-3126	61	14	semi	semi	ADJ
ejpam-3126	61	15	-	-	ADJ
ejpam-3126	61	16	primary	primary	ADJ
ejpam-3126	61	17	submodules	submodule	NOUN
ejpam-3126	61	18	.	.	PUNCT
ejpam-3126	62	1	among	among	ADP
ejpam-3126	62	2	many	many	ADJ
ejpam-3126	62	3	results	result	NOUN
ejpam-3126	62	4	in	in	ADP
ejpam-3126	62	5	this	this	DET
ejpam-3126	62	6	paper	paper	NOUN
ejpam-3126	62	7	,	,	PUNCT
ejpam-3126	62	8	it	it	PRON
ejpam-3126	62	9	is	be	AUX
ejpam-3126	62	10	shown	show	VERB
ejpam-3126	62	11	that	that	SCONJ
ejpam-3126	62	12	n	n	PRON
ejpam-3126	62	13	is	be	AUX
ejpam-3126	62	14	a	a	DET
ejpam-3126	62	15	φ-2	φ-2	NOUN
ejpam-3126	62	16	-	-	PUNCT
ejpam-3126	62	17	absorbing	absorbing	ADJ
ejpam-3126	62	18	semi	semi	ADJ
ejpam-3126	62	19	-	-	ADJ
ejpam-3126	62	20	primary	primary	ADJ
ejpam-3126	62	21	submodule	submodule	NOUN
ejpam-3126	62	22	of	of	ADP
ejpam-3126	62	23	m	m	PROPN
ejpam-3126	62	24	if	if	SCONJ
ejpam-3126	62	25	and	and	CCONJ
ejpam-3126	62	26	only	only	ADV
ejpam-3126	62	27	if	if	SCONJ
ejpam-3126	62	28	for	for	ADP
ejpam-3126	62	29	every	every	DET
ejpam-3126	62	30	a1	a1	NOUN
ejpam-3126	62	31	,	,	PUNCT
ejpam-3126	62	32	a2	a2	PROPN
ejpam-3126	62	33	∈	∈	PROPN
ejpam-3126	62	34	r−	r−	PROPN
ejpam-3126	62	35	(	(	PUNCT
ejpam-3126	62	36	n	n	NUM
ejpam-3126	62	37	:	:	PUNCT
ejpam-3126	62	38	m	m	X
ejpam-3126	62	39	)	)	PUNCT
ejpam-3126	62	40	and	and	CCONJ
ejpam-3126	62	41	a1a2	a1a2	ADP
ejpam-3126	62	42	∈	∈	PROPN
ejpam-3126	62	43	r−	r−	PROPN
ejpam-3126	62	44	√	√	PROPN
ejpam-3126	62	45	(	(	PUNCT
ejpam-3126	62	46	n	n	NUM
ejpam-3126	62	47	:	:	PUNCT
ejpam-3126	62	48	m	m	X
ejpam-3126	62	49	)	)	PUNCT
ejpam-3126	62	50	,	,	PUNCT
ejpam-3126	62	51	(	(	PUNCT
ejpam-3126	62	52	n	n	X
ejpam-3126	62	53	:	:	PUNCT
ejpam-3126	62	54	a1a2	a1a2	ADJ
ejpam-3126	62	55	)	)	PUNCT
ejpam-3126	62	56	⊆	⊆	NUM
ejpam-3126	62	57	(	(	PUNCT
ejpam-3126	62	58	φ(n	φ(n	ADJ
ejpam-3126	62	59	)	)	PUNCT
ejpam-3126	62	60	:	:	PUNCT
ejpam-3126	63	1	a1a2	a1a2	ADJ
ejpam-3126	63	2	)	)	PUNCT
ejpam-3126	63	3	∪	∪	NOUN
ejpam-3126	63	4	(	(	PUNCT
ejpam-3126	63	5	n	n	NOUN
ejpam-3126	63	6	:	:	PUNCT
ejpam-3126	63	7	a1	a1	PROPN
ejpam-3126	63	8	)	)	PUNCT
ejpam-3126	63	9	∪	∪	NOUN
ejpam-3126	63	10	(	(	PUNCT
ejpam-3126	63	11	n	n	NOUN
ejpam-3126	63	12	:	:	PUNCT
ejpam-3126	63	13	an2	an2	PROPN
ejpam-3126	63	14	)	)	PUNCT
ejpam-3126	63	15	for	for	ADP
ejpam-3126	63	16	some	some	DET
ejpam-3126	63	17	positive	positive	ADJ
ejpam-3126	63	18	integer	integer	NOUN
ejpam-3126	63	19	n.	n.	NOUN
ejpam-3126	63	20	in	in	ADP
ejpam-3126	63	21	section	section	NOUN
ejpam-3126	63	22	3	3	NUM
ejpam-3126	63	23	,	,	PUNCT
ejpam-3126	63	24	we	we	PRON
ejpam-3126	63	25	study	study	VERB
ejpam-3126	63	26	the	the	DET
ejpam-3126	63	27	stability	stability	NOUN
ejpam-3126	63	28	of	of	ADP
ejpam-3126	63	29	φα-2	φα-2	NOUN
ejpam-3126	63	30	-	-	PUNCT
ejpam-3126	63	31	absorbing	absorbing	ADJ
ejpam-3126	63	32	semi	semi	ADJ
ejpam-3126	63	33	-	-	ADJ
ejpam-3126	63	34	primary	primary	ADJ
ejpam-3126	63	35	submodules	submodule	NOUN
ejpam-3126	63	36	.	.	PUNCT
ejpam-3126	64	1	moreover	moreover	ADV
ejpam-3126	64	2	,	,	PUNCT
ejpam-3126	64	3	we	we	PRON
ejpam-3126	64	4	investigate	investigate	VERB
ejpam-3126	64	5	relationships	relationship	NOUN
ejpam-3126	64	6	between	between	ADP
ejpam-3126	64	7	2	2	NUM
ejpam-3126	64	8	-	-	PUNCT
ejpam-3126	64	9	absorbing	absorbing	ADJ
ejpam-3126	64	10	semi	semi	ADJ
ejpam-3126	64	11	-	-	ADJ
ejpam-3126	64	12	primary	primary	ADJ
ejpam-3126	64	13	,	,	PUNCT
ejpam-3126	64	14	φ0	φ0	PROPN
ejpam-3126	64	15	-	-	PUNCT
ejpam-3126	64	16	2	2	NUM
ejpam-3126	64	17	-	-	PUNCT
ejpam-3126	64	18	absorbing	absorb	VERB
ejpam-3126	64	19	semiprimary	semiprimary	NOUN
ejpam-3126	64	20	,	,	PUNCT
ejpam-3126	64	21	φn-2	φn-2	NOUN
ejpam-3126	64	22	-	-	PUNCT
ejpam-3126	64	23	absorbing	absorbing	ADJ
ejpam-3126	64	24	semi	semi	ADJ
ejpam-3126	64	25	-	-	ADJ
ejpam-3126	64	26	primary	primary	ADJ
ejpam-3126	64	27	and	and	CCONJ
ejpam-3126	64	28	φ	φ	VERB
ejpam-3126	64	29	-	-	ADJ
ejpam-3126	64	30	primary	primary	ADJ
ejpam-3126	64	31	submodules	submodule	NOUN
ejpam-3126	64	32	of	of	ADP
ejpam-3126	64	33	modules	module	NOUN
ejpam-3126	64	34	over	over	ADP
ejpam-3126	64	35	commutative	commutative	ADJ
ejpam-3126	64	36	rings	ring	NOUN
ejpam-3126	64	37	.	.	PUNCT
ejpam-3126	65	1	finally	finally	ADV
ejpam-3126	65	2	,	,	PUNCT
ejpam-3126	65	3	we	we	PRON
ejpam-3126	65	4	obtain	obtain	VERB
ejpam-3126	65	5	necessary	necessary	ADJ
ejpam-3126	65	6	and	and	CCONJ
ejpam-3126	65	7	sufficient	sufficient	ADJ
ejpam-3126	65	8	conditions	condition	NOUN
ejpam-3126	65	9	of	of	ADP
ejpam-3126	65	10	a	a	DET
ejpam-3126	65	11	φ-2	φ-2	NOUN
ejpam-3126	65	12	-	-	PUNCT
ejpam-3126	65	13	absorbing	absorbing	ADJ
ejpam-3126	65	14	semi	semi	ADJ
ejpam-3126	65	15	-	-	ADJ
ejpam-3126	65	16	primary	primary	ADJ
ejpam-3126	65	17	in	in	ADP
ejpam-3126	65	18	order	order	NOUN
ejpam-3126	65	19	to	to	PART
ejpam-3126	65	20	be	be	AUX
ejpam-3126	65	21	a	a	DET
ejpam-3126	65	22	2	2	NUM
ejpam-3126	65	23	-	-	PUNCT
ejpam-3126	65	24	absorbing	absorbing	ADJ
ejpam-3126	65	25	semi	semi	ADJ
ejpam-3126	65	26	-	-	ADJ
ejpam-3126	65	27	primary	primary	ADJ
ejpam-3126	65	28	.	.	PUNCT
ejpam-3126	66	1	2	2	X
ejpam-3126	66	2	.	.	X
ejpam-3126	66	3	properties	property	NOUN
ejpam-3126	66	4	of	of	ADP
ejpam-3126	66	5	φ-2	φ-2	PROPN
ejpam-3126	66	6	-	-	PUNCT
ejpam-3126	66	7	absorbing	absorbing	ADJ
ejpam-3126	66	8	semi	semi	ADJ
ejpam-3126	66	9	-	-	ADJ
ejpam-3126	66	10	primary	primary	ADJ
ejpam-3126	66	11	submodules	submodule	NOUN
ejpam-3126	66	12	the	the	DET
ejpam-3126	66	13	results	result	NOUN
ejpam-3126	66	14	of	of	ADP
ejpam-3126	66	15	the	the	DET
ejpam-3126	66	16	following	follow	VERB
ejpam-3126	66	17	theorems	theorem	NOUN
ejpam-3126	66	18	seem	seem	VERB
ejpam-3126	66	19	to	to	PART
ejpam-3126	66	20	play	play	VERB
ejpam-3126	66	21	an	an	DET
ejpam-3126	66	22	important	important	ADJ
ejpam-3126	66	23	role	role	NOUN
ejpam-3126	66	24	to	to	PART
ejpam-3126	66	25	study	study	VERB
ejpam-3126	66	26	φclassical	φclassical	ADJ
ejpam-3126	66	27	semi	semi	ADJ
ejpam-3126	66	28	-	-	ADJ
ejpam-3126	66	29	primary	primary	ADJ
ejpam-3126	66	30	submodules	submodule	NOUN
ejpam-3126	66	31	of	of	ADP
ejpam-3126	66	32	modules	module	NOUN
ejpam-3126	66	33	over	over	ADP
ejpam-3126	66	34	commutative	commutative	ADJ
ejpam-3126	66	35	rings	ring	NOUN
ejpam-3126	66	36	;	;	PUNCT
ejpam-3126	66	37	these	these	DET
ejpam-3126	66	38	facts	fact	NOUN
ejpam-3126	66	39	will	will	AUX
ejpam-3126	66	40	be	be	AUX
ejpam-3126	66	41	used	use	VERB
ejpam-3126	66	42	frequently	frequently	ADV
ejpam-3126	66	43	and	and	CCONJ
ejpam-3126	66	44	normally	normally	ADV
ejpam-3126	66	45	we	we	PRON
ejpam-3126	66	46	shall	shall	AUX
ejpam-3126	66	47	make	make	VERB
ejpam-3126	66	48	no	no	DET
ejpam-3126	66	49	reference	reference	NOUN
ejpam-3126	66	50	to	to	ADP
ejpam-3126	66	51	this	this	DET
ejpam-3126	66	52	definition	definition	NOUN
ejpam-3126	66	53	.	.	PUNCT
ejpam-3126	67	1	definition	definition	NOUN
ejpam-3126	67	2	1	1	NUM
ejpam-3126	67	3	.	.	PUNCT
ejpam-3126	68	1	let	let	VERB
ejpam-3126	68	2	m	m	PRON
ejpam-3126	68	3	be	be	AUX
ejpam-3126	68	4	an	an	DET
ejpam-3126	68	5	r	r	NOUN
ejpam-3126	68	6	-	-	PUNCT
ejpam-3126	68	7	module	module	NOUN
ejpam-3126	68	8	and	and	CCONJ
ejpam-3126	68	9	let	let	VERB
ejpam-3126	68	10	φ	φ	PROPN
ejpam-3126	68	11	:	:	PUNCT
ejpam-3126	68	12	s(m	s(m	PROPN
ejpam-3126	68	13	)	)	PUNCT
ejpam-3126	68	14	→	→	SYM
ejpam-3126	68	15	s(m	s(m	NOUN
ejpam-3126	68	16	)	)	PUNCT
ejpam-3126	68	17	∪	∪	NOUN
ejpam-3126	68	18	{	{	PUNCT
ejpam-3126	68	19	∅	∅	NOUN
ejpam-3126	68	20	}	}	PUNCT
ejpam-3126	68	21	be	be	AUX
ejpam-3126	68	22	a	a	DET
ejpam-3126	68	23	function	function	NOUN
ejpam-3126	68	24	where	where	SCONJ
ejpam-3126	68	25	s(m	s(m	PROPN
ejpam-3126	68	26	)	)	PUNCT
ejpam-3126	68	27	be	be	VERB
ejpam-3126	68	28	a	a	DET
ejpam-3126	68	29	set	set	NOUN
ejpam-3126	68	30	of	of	ADP
ejpam-3126	68	31	all	all	DET
ejpam-3126	68	32	submodules	submodule	NOUN
ejpam-3126	68	33	of	of	ADP
ejpam-3126	68	34	m	m	PROPN
ejpam-3126	68	35	.	.	PUNCT
ejpam-3126	69	1	a	a	DET
ejpam-3126	69	2	proper	proper	ADJ
ejpam-3126	69	3	submodule	submodule	NOUN
ejpam-3126	69	4	n	n	PROPN
ejpam-3126	69	5	of	of	ADP
ejpam-3126	69	6	m	m	PROPN
ejpam-3126	69	7	is	be	AUX
ejpam-3126	69	8	called	call	VERB
ejpam-3126	69	9	a	a	DET
ejpam-3126	69	10	φ-2	φ-2	NOUN
ejpam-3126	69	11	-	-	PUNCT
ejpam-3126	69	12	absorbing	absorbing	ADJ
ejpam-3126	69	13	semi	semi	ADJ
ejpam-3126	69	14	-	-	ADJ
ejpam-3126	69	15	primary	primary	ADJ
ejpam-3126	69	16	submodule	submodule	NOUN
ejpam-3126	69	17	,	,	PUNCT
ejpam-3126	69	18	if	if	SCONJ
ejpam-3126	69	19	for	for	ADP
ejpam-3126	69	20	each	each	DET
ejpam-3126	69	21	m	m	NOUN
ejpam-3126	69	22	∈	∈	PROPN
ejpam-3126	69	23	m	m	NOUN
ejpam-3126	69	24	and	and	CCONJ
ejpam-3126	69	25	a1	a1	NOUN
ejpam-3126	69	26	,	,	PUNCT
ejpam-3126	69	27	a2	a2	PROPN
ejpam-3126	69	28	∈	∈	PROPN
ejpam-3126	69	29	r	r	NOUN
ejpam-3126	69	30	with	with	ADP
ejpam-3126	69	31	a1a2	a1a2	PROPN
ejpam-3126	69	32	m	m	NOUN
ejpam-3126	69	33	∈	∈	ADJ
ejpam-3126	69	34	n	n	CCONJ
ejpam-3126	69	35	−	−	PROPN
ejpam-3126	69	36	φ(n	φ(n	NOUN
ejpam-3126	69	37	)	)	PUNCT
ejpam-3126	69	38	,	,	PUNCT
ejpam-3126	69	39	then	then	ADV
ejpam-3126	69	40	a1a2	a1a2	ADP
ejpam-3126	69	41	∈	∈	NOUN
ejpam-3126	69	42	√	√	NUM
ejpam-3126	69	43	(	(	PUNCT
ejpam-3126	69	44	n	n	NUM
ejpam-3126	69	45	:	:	PUNCT
ejpam-3126	69	46	m	m	X
ejpam-3126	69	47	)	)	PUNCT
ejpam-3126	69	48	or	or	CCONJ
ejpam-3126	69	49	a1	a1	NOUN
ejpam-3126	69	50	m	m	PROPN
ejpam-3126	69	51	∈	∈	NOUN
ejpam-3126	69	52	n	n	NOUN
ejpam-3126	69	53	or	or	CCONJ
ejpam-3126	69	54	an2	an2	PROPN
ejpam-3126	69	55	m	m	NOUN
ejpam-3126	69	56	∈	∈	PROPN
ejpam-3126	69	57	n	n	NOUN
ejpam-3126	69	58	for	for	ADP
ejpam-3126	69	59	some	some	DET
ejpam-3126	69	60	positive	positive	ADJ
ejpam-3126	69	61	integer	integer	NOUN
ejpam-3126	69	62	n.	n.	NOUN
ejpam-3126	69	63	remark	remark	NOUN
ejpam-3126	69	64	1	1	NUM
ejpam-3126	69	65	.	.	PUNCT
ejpam-3126	70	1	it	it	PRON
ejpam-3126	70	2	is	be	AUX
ejpam-3126	70	3	easy	easy	ADJ
ejpam-3126	70	4	to	to	PART
ejpam-3126	70	5	see	see	VERB
ejpam-3126	70	6	that	that	SCONJ
ejpam-3126	70	7	every	every	DET
ejpam-3126	70	8	φ-2	φ-2	ADJ
ejpam-3126	70	9	-	-	PUNCT
ejpam-3126	70	10	absorbing	absorbing	ADJ
ejpam-3126	70	11	primary	primary	ADJ
ejpam-3126	70	12	submodule	submodule	NOUN
ejpam-3126	70	13	is	be	AUX
ejpam-3126	70	14	φ-2	φ-2	NOUN
ejpam-3126	70	15	-	-	PUNCT
ejpam-3126	70	16	absorbing	absorbing	ADJ
ejpam-3126	70	17	semi	semi	ADJ
ejpam-3126	70	18	-	-	ADJ
ejpam-3126	70	19	primary	primary	ADJ
ejpam-3126	70	20	.	.	PUNCT
ejpam-3126	71	1	the	the	DET
ejpam-3126	71	2	following	follow	VERB
ejpam-3126	71	3	example	example	NOUN
ejpam-3126	71	4	shows	show	VERB
ejpam-3126	71	5	that	that	SCONJ
ejpam-3126	71	6	the	the	DET
ejpam-3126	71	7	converse	converse	NOUN
ejpam-3126	71	8	of	of	ADP
ejpam-3126	71	9	remark	remark	NOUN
ejpam-3126	71	10	1	1	NUM
ejpam-3126	71	11	is	be	AUX
ejpam-3126	71	12	not	not	PART
ejpam-3126	71	13	true	true	ADJ
ejpam-3126	71	14	.	.	PUNCT
ejpam-3126	72	1	example	example	NOUN
ejpam-3126	73	1	1	1	NUM
ejpam-3126	73	2	.	.	PUNCT
ejpam-3126	73	3	let	let	VERB
ejpam-3126	73	4	r	r	NOUN
ejpam-3126	73	5	=	=	PUNCT
ejpam-3126	73	6	z	z	PROPN
ejpam-3126	73	7	and	and	CCONJ
ejpam-3126	73	8	m	m	PROPN
ejpam-3126	73	9	=	=	ADJ
ejpam-3126	73	10	z.	z.	PROPN
ejpam-3126	73	11	consider	consider	VERB
ejpam-3126	73	12	the	the	DET
ejpam-3126	73	13	submodule	submodule	NOUN
ejpam-3126	73	14	n	n	PROPN
ejpam-3126	73	15	=	=	PROPN
ejpam-3126	73	16	12z	12z	PROPN
ejpam-3126	73	17	of	of	ADP
ejpam-3126	73	18	m	m	PROPN
ejpam-3126	73	19	.	.	PUNCT
ejpam-3126	74	1	define	define	VERB
ejpam-3126	74	2	φ	φ	NOUN
ejpam-3126	74	3	:	:	PUNCT
ejpam-3126	74	4	s(m)→	s(m)→	NOUN
ejpam-3126	74	5	s(m)∪{∅	s(m)∪{∅	PROPN
ejpam-3126	74	6	}	}	PUNCT
ejpam-3126	74	7	by	by	ADP
ejpam-3126	74	8	φ(n	φ(n	NOUN
ejpam-3126	74	9	)	)	PUNCT
ejpam-3126	74	10	=	=	PRON
ejpam-3126	74	11	{	{	PUNCT
ejpam-3126	74	12	0	0	NUM
ejpam-3126	74	13	}	}	PUNCT
ejpam-3126	74	14	for	for	ADP
ejpam-3126	74	15	every	every	DET
ejpam-3126	74	16	n	n	PRON
ejpam-3126	74	17	∈	∈	PROPN
ejpam-3126	74	18	s(m	s(m	PROPN
ejpam-3126	74	19	)	)	PUNCT
ejpam-3126	74	20	.	.	PUNCT
ejpam-3126	75	1	it	it	PRON
ejpam-3126	75	2	is	be	AUX
ejpam-3126	75	3	easy	easy	ADJ
ejpam-3126	75	4	to	to	PART
ejpam-3126	75	5	see	see	VERB
ejpam-3126	75	6	that	that	SCONJ
ejpam-3126	75	7	n	n	PRON
ejpam-3126	75	8	is	be	AUX
ejpam-3126	75	9	a	a	DET
ejpam-3126	75	10	φ-2	φ-2	NOUN
ejpam-3126	75	11	-	-	PUNCT
ejpam-3126	75	12	absorbing	absorbing	ADJ
ejpam-3126	75	13	semi	semi	ADJ
ejpam-3126	75	14	-	-	ADJ
ejpam-3126	75	15	primary	primary	ADJ
ejpam-3126	75	16	submodule	submodule	NOUN
ejpam-3126	75	17	of	of	ADP
ejpam-3126	75	18	m	m	PROPN
ejpam-3126	75	19	.	.	PUNCT
ejpam-3126	76	1	notice	notice	VERB
ejpam-3126	76	2	that	that	SCONJ
ejpam-3126	76	3	2	2	NUM
ejpam-3126	76	4	·	·	SYM
ejpam-3126	76	5	2	2	NUM
ejpam-3126	76	6	·	·	SYM
ejpam-3126	76	7	3	3	NUM
ejpam-3126	76	8	∈	∈	NOUN
ejpam-3126	76	9	n−φ(n	n−φ(n	NOUN
ejpam-3126	76	10	)	)	PUNCT
ejpam-3126	76	11	,	,	PUNCT
ejpam-3126	76	12	but	but	CCONJ
ejpam-3126	76	13	2	2	NUM
ejpam-3126	76	14	·	·	SYM
ejpam-3126	76	15	3	3	NUM
ejpam-3126	76	16	6∈	6∈	NOUN
ejpam-3126	76	17	n	n	NOUN
ejpam-3126	76	18	and	and	CCONJ
ejpam-3126	76	19	(	(	PUNCT
ejpam-3126	76	20	2	2	NUM
ejpam-3126	76	21	·	·	SYM
ejpam-3126	76	22	2)n	2)n	NUM
ejpam-3126	76	23	6∈	6∈	NOUN
ejpam-3126	76	24	(	(	PUNCT
ejpam-3126	76	25	n	n	NUM
ejpam-3126	76	26	:	:	PUNCT
ejpam-3126	76	27	m	m	X
ejpam-3126	76	28	)	)	PUNCT
ejpam-3126	76	29	for	for	ADP
ejpam-3126	76	30	all	all	DET
ejpam-3126	76	31	positive	positive	ADJ
ejpam-3126	76	32	integer	integer	NOUN
ejpam-3126	76	33	n.	n.	NOUN
ejpam-3126	76	34	therefore	therefore	ADV
ejpam-3126	76	35	n	n	ADV
ejpam-3126	76	36	is	be	AUX
ejpam-3126	76	37	not	not	PART
ejpam-3126	76	38	a	a	DET
ejpam-3126	76	39	φ-2	φ-2	ADJ
ejpam-3126	76	40	-	-	PUNCT
ejpam-3126	76	41	absorbing	absorbing	ADJ
ejpam-3126	76	42	primary	primary	ADJ
ejpam-3126	76	43	submodule	submodule	NOUN
ejpam-3126	76	44	of	of	ADP
ejpam-3126	76	45	m	m	PROPN
ejpam-3126	76	46	.	.	PUNCT
ejpam-3126	77	1	pai	pai	PROPN
ejpam-3126	77	2	.	.	PROPN
ejpam-3126	77	3	yiarayong	yiarayong	PROPN
ejpam-3126	77	4	,	,	PUNCT
ejpam-3126	77	5	m.	m.	NOUN
ejpam-3126	77	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	77	7	/	/	SYM
ejpam-3126	77	8	eur	eur	PROPN
ejpam-3126	77	9	.	.	PUNCT
ejpam-3126	78	1	j.	j.	PROPN
ejpam-3126	78	2	pure	pure	PROPN
ejpam-3126	78	3	appl	appl	PROPN
ejpam-3126	78	4	.	.	PROPN
ejpam-3126	78	5	math	math	PROPN
ejpam-3126	78	6	,	,	PUNCT
ejpam-3126	78	7	11	11	NUM
ejpam-3126	78	8	(	(	PUNCT
ejpam-3126	78	9	1	1	NUM
ejpam-3126	78	10	)	)	PUNCT
ejpam-3126	78	11	(	(	PUNCT
ejpam-3126	78	12	2018	2018	NUM
ejpam-3126	78	13	)	)	PUNCT
ejpam-3126	78	14	,	,	PUNCT
ejpam-3126	78	15	35	35	NUM
ejpam-3126	78	16	-	-	SYM
ejpam-3126	78	17	50	50	NUM
ejpam-3126	78	18	38	38	NUM
ejpam-3126	78	19	theorem	theorem	NOUN
ejpam-3126	78	20	1	1	NUM
ejpam-3126	78	21	.	.	PUNCT
ejpam-3126	79	1	let	let	VERB
ejpam-3126	79	2	φ	φ	NOUN
ejpam-3126	79	3	:	:	PUNCT
ejpam-3126	79	4	s(m)→	s(m)→	NOUN
ejpam-3126	79	5	s(m	s(m	NOUN
ejpam-3126	79	6	)	)	PUNCT
ejpam-3126	79	7	∪	∪	ADP
ejpam-3126	79	8	{	{	PUNCT
ejpam-3126	79	9	∅	∅	NOUN
ejpam-3126	79	10	}	}	PUNCT
ejpam-3126	79	11	and	and	CCONJ
ejpam-3126	79	12	ϕ	ϕ	X
ejpam-3126	79	13	:	:	PUNCT
ejpam-3126	79	14	j	j	PROPN
ejpam-3126	79	15	(	(	PUNCT
ejpam-3126	79	16	r)→	r)→	PROPN
ejpam-3126	79	17	j	j	PROPN
ejpam-3126	79	18	(	(	PUNCT
ejpam-3126	79	19	r	r	NOUN
ejpam-3126	79	20	)	)	PUNCT
ejpam-3126	79	21	∪	∪	NOUN
ejpam-3126	79	22	{	{	PUNCT
ejpam-3126	79	23	∅	∅	NOUN
ejpam-3126	79	24	}	}	PUNCT
ejpam-3126	79	25	be	be	AUX
ejpam-3126	79	26	two	two	NUM
ejpam-3126	79	27	functions	function	NOUN
ejpam-3126	79	28	.	.	PUNCT
ejpam-3126	80	1	(	(	PUNCT
ejpam-3126	80	2	i	i	NOUN
ejpam-3126	80	3	)	)	PUNCT
ejpam-3126	80	4	if	if	SCONJ
ejpam-3126	80	5	n	n	PRON
ejpam-3126	80	6	is	be	AUX
ejpam-3126	80	7	a	a	DET
ejpam-3126	80	8	φ-2	φ-2	NOUN
ejpam-3126	80	9	-	-	PUNCT
ejpam-3126	80	10	absorbing	absorbing	ADJ
ejpam-3126	80	11	semi	semi	ADJ
ejpam-3126	80	12	-	-	ADJ
ejpam-3126	80	13	primary	primary	ADJ
ejpam-3126	80	14	submodule	submodule	NOUN
ejpam-3126	80	15	of	of	ADP
ejpam-3126	80	16	m	m	PROPN
ejpam-3126	80	17	,	,	PUNCT
ejpam-3126	80	18	then	then	ADV
ejpam-3126	80	19	(	(	PUNCT
ejpam-3126	80	20	n	n	X
ejpam-3126	80	21	:	:	PUNCT
ejpam-3126	80	22	m	m	X
ejpam-3126	80	23	)	)	PUNCT
ejpam-3126	80	24	is	be	AUX
ejpam-3126	80	25	a	a	DET
ejpam-3126	80	26	ϕ-2absorbing	ϕ-2absorbing	ADJ
ejpam-3126	80	27	primary	primary	ADJ
ejpam-3126	80	28	ideal	ideal	NOUN
ejpam-3126	80	29	of	of	ADP
ejpam-3126	80	30	r	r	NOUN
ejpam-3126	80	31	with	with	ADP
ejpam-3126	80	32	m	m	NOUN
ejpam-3126	80	33	∈m	∈m	NOUN
ejpam-3126	80	34	−n	−n	ADJ
ejpam-3126	80	35	and	and	CCONJ
ejpam-3126	80	36	(	(	PUNCT
ejpam-3126	80	37	φ(n	φ(n	PROPN
ejpam-3126	80	38	)	)	PUNCT
ejpam-3126	80	39	:	:	PUNCT
ejpam-3126	80	40	m	m	X
ejpam-3126	80	41	)	)	PUNCT
ejpam-3126	80	42	)	)	PUNCT
ejpam-3126	81	1	⊆	⊆	NUM
ejpam-3126	81	2	ϕ(n	ϕ(n	X
ejpam-3126	81	3	:	:	PUNCT
ejpam-3126	81	4	m	m	VERB
ejpam-3126	81	5	)	)	PUNCT
ejpam-3126	81	6	.	.	PUNCT
ejpam-3126	82	1	(	(	PUNCT
ejpam-3126	82	2	ii	ii	NOUN
ejpam-3126	82	3	)	)	PUNCT
ejpam-3126	82	4	for	for	ADP
ejpam-3126	82	5	every	every	DET
ejpam-3126	82	6	m	m	NOUN
ejpam-3126	82	7	∈m−n	∈m−n	NOUN
ejpam-3126	82	8	if	if	SCONJ
ejpam-3126	82	9	(	(	PUNCT
ejpam-3126	82	10	n	n	X
ejpam-3126	82	11	:	:	PUNCT
ejpam-3126	82	12	m	m	X
ejpam-3126	82	13	)	)	PUNCT
ejpam-3126	82	14	is	be	AUX
ejpam-3126	82	15	a	a	DET
ejpam-3126	82	16	ϕ-primary	ϕ-primary	ADJ
ejpam-3126	82	17	ideal	ideal	NOUN
ejpam-3126	82	18	of	of	ADP
ejpam-3126	82	19	r	r	NOUN
ejpam-3126	82	20	,	,	PUNCT
ejpam-3126	82	21	then	then	ADV
ejpam-3126	82	22	n	n	PRON
ejpam-3126	82	23	is	be	AUX
ejpam-3126	82	24	a	a	DET
ejpam-3126	82	25	φ-2	φ-2	NOUN
ejpam-3126	82	26	-	-	PUNCT
ejpam-3126	82	27	absorbing	absorbing	ADJ
ejpam-3126	82	28	semi	semi	ADJ
ejpam-3126	82	29	-	-	ADJ
ejpam-3126	82	30	primary	primary	ADJ
ejpam-3126	82	31	submodule	submodule	NOUN
ejpam-3126	82	32	of	of	ADP
ejpam-3126	82	33	m	m	PROPN
ejpam-3126	82	34	with	with	ADP
ejpam-3126	82	35	ϕ(n	ϕ(n	PROPN
ejpam-3126	82	36	:	:	PUNCT
ejpam-3126	82	37	m	m	X
ejpam-3126	82	38	)	)	PUNCT
ejpam-3126	82	39	⊆	⊆	X
ejpam-3126	82	40	(	(	PUNCT
ejpam-3126	82	41	φ(n	φ(n	PROPN
ejpam-3126	82	42	)	)	PUNCT
ejpam-3126	82	43	:	:	PUNCT
ejpam-3126	82	44	m	m	X
ejpam-3126	82	45	)	)	PUNCT
ejpam-3126	82	46	)	)	PUNCT
ejpam-3126	82	47	.	.	PUNCT
ejpam-3126	83	1	proof	proof	NOUN
ejpam-3126	83	2	.	.	PUNCT
ejpam-3126	84	1	1	1	X
ejpam-3126	84	2	.	.	X
ejpam-3126	84	3	let	let	VERB
ejpam-3126	84	4	a1	a1	NOUN
ejpam-3126	84	5	,	,	PUNCT
ejpam-3126	84	6	a2	a2	PROPN
ejpam-3126	84	7	,	,	PUNCT
ejpam-3126	84	8	a3	a3	NOUN
ejpam-3126	84	9	∈	∈	PROPN
ejpam-3126	84	10	r	r	NOUN
ejpam-3126	84	11	such	such	ADJ
ejpam-3126	84	12	that	that	DET
ejpam-3126	84	13	a1a2a3	a1a2a3	VERB
ejpam-3126	84	14	∈	∈	PROPN
ejpam-3126	84	15	(	(	PUNCT
ejpam-3126	84	16	n	n	NUM
ejpam-3126	84	17	:	:	PUNCT
ejpam-3126	84	18	m)−ϕ((n	m)−ϕ((n	NOUN
ejpam-3126	84	19	:	:	PUNCT
ejpam-3126	84	20	m	m	X
ejpam-3126	84	21	)	)	PUNCT
ejpam-3126	84	22	)	)	PUNCT
ejpam-3126	84	23	.	.	PUNCT
ejpam-3126	85	1	by	by	ADP
ejpam-3126	85	2	assumption	assumption	NOUN
ejpam-3126	85	3	,	,	PUNCT
ejpam-3126	85	4	a1a3(a2	a1a3(a2	X
ejpam-3126	85	5	m	m	NOUN
ejpam-3126	85	6	)	)	PUNCT
ejpam-3126	85	7	∈	∈	PROPN
ejpam-3126	85	8	n	n	PRON
ejpam-3126	85	9	−φ(n	−φ(n	NOUN
ejpam-3126	85	10	)	)	PUNCT
ejpam-3126	85	11	.	.	PUNCT
ejpam-3126	86	1	then	then	ADV
ejpam-3126	86	2	by	by	ADP
ejpam-3126	86	3	definition	definition	NOUN
ejpam-3126	86	4	1	1	NUM
ejpam-3126	86	5	,	,	PUNCT
ejpam-3126	86	6	a1a3	a1a3	ADP
ejpam-3126	86	7	∈	∈	NOUN
ejpam-3126	86	8	√	√	NUM
ejpam-3126	86	9	(	(	PUNCT
ejpam-3126	86	10	n	n	NUM
ejpam-3126	86	11	:	:	PUNCT
ejpam-3126	86	12	m	m	X
ejpam-3126	86	13	)	)	PUNCT
ejpam-3126	87	1	⊆	⊆	NUM
ejpam-3126	87	2	√	√	NUM
ejpam-3126	87	3	(	(	PUNCT
ejpam-3126	87	4	n	n	NUM
ejpam-3126	87	5	:	:	PUNCT
ejpam-3126	87	6	m	m	X
ejpam-3126	87	7	)	)	PUNCT
ejpam-3126	87	8	or	or	CCONJ
ejpam-3126	87	9	a1a2	a1a2	ADP
ejpam-3126	87	10	m	m	VERB
ejpam-3126	87	11	∈	∈	ADJ
ejpam-3126	87	12	n	n	NOUN
ejpam-3126	87	13	or	or	CCONJ
ejpam-3126	87	14	an3a2	an3a2	PROPN
ejpam-3126	87	15	m	m	NOUN
ejpam-3126	87	16	∈	∈	NOUN
ejpam-3126	87	17	n	n	NOUN
ejpam-3126	87	18	for	for	ADP
ejpam-3126	87	19	some	some	DET
ejpam-3126	87	20	positive	positive	ADJ
ejpam-3126	87	21	integer	integer	NOUN
ejpam-3126	87	22	n.	n.	NOUN
ejpam-3126	87	23	therefore	therefore	ADV
ejpam-3126	87	24	a1a2	a1a2	ADP
ejpam-3126	87	25	∈	∈	PROPN
ejpam-3126	87	26	(	(	PUNCT
ejpam-3126	87	27	n	n	NOUN
ejpam-3126	87	28	:	:	PUNCT
ejpam-3126	87	29	m	m	X
ejpam-3126	87	30	)	)	PUNCT
ejpam-3126	87	31	or	or	CCONJ
ejpam-3126	87	32	a2a3	a2a3	VERB
ejpam-3126	87	33	∈√	∈√	PROPN
ejpam-3126	87	34	(	(	PUNCT
ejpam-3126	87	35	n	n	NUM
ejpam-3126	87	36	:	:	PUNCT
ejpam-3126	87	37	m	m	X
ejpam-3126	87	38	)	)	PUNCT
ejpam-3126	87	39	or	or	CCONJ
ejpam-3126	87	40	a1a3	a1a3	PUNCT
ejpam-3126	87	41	∈	∈	NOUN
ejpam-3126	87	42	√	√	NUM
ejpam-3126	87	43	(	(	PUNCT
ejpam-3126	87	44	n	n	NUM
ejpam-3126	87	45	:	:	PUNCT
ejpam-3126	87	46	m	m	X
ejpam-3126	87	47	)	)	PUNCT
ejpam-3126	87	48	.	.	PUNCT
ejpam-3126	88	1	this	this	PRON
ejpam-3126	88	2	completes	complete	VERB
ejpam-3126	88	3	the	the	DET
ejpam-3126	88	4	proof	proof	NOUN
ejpam-3126	88	5	.	.	PUNCT
ejpam-3126	89	1	2	2	X
ejpam-3126	89	2	.	.	X
ejpam-3126	89	3	let	let	VERB
ejpam-3126	89	4	a1	a1	NOUN
ejpam-3126	89	5	,	,	PUNCT
ejpam-3126	89	6	a2	a2	PROPN
ejpam-3126	89	7	∈	∈	PROPN
ejpam-3126	89	8	r	r	NOUN
ejpam-3126	89	9	such	such	ADJ
ejpam-3126	89	10	that	that	DET
ejpam-3126	89	11	a1a2	a1a2	PROPN
ejpam-3126	89	12	m	m	VERB
ejpam-3126	89	13	∈	∈	ADJ
ejpam-3126	89	14	n	n	CCONJ
ejpam-3126	89	15	−	−	PROPN
ejpam-3126	89	16	φ(n	φ(n	NOUN
ejpam-3126	89	17	)	)	PUNCT
ejpam-3126	89	18	.	.	PUNCT
ejpam-3126	90	1	then	then	ADV
ejpam-3126	90	2	a1a2	a1a2	ADP
ejpam-3126	90	3	∈	∈	PROPN
ejpam-3126	90	4	(	(	PUNCT
ejpam-3126	90	5	n	n	NOUN
ejpam-3126	90	6	:	:	PUNCT
ejpam-3126	90	7	m	m	X
ejpam-3126	90	8	)	)	PUNCT
ejpam-3126	90	9	and	and	CCONJ
ejpam-3126	90	10	a1a2	a1a2	ADP
ejpam-3126	90	11	6∈	6∈	NOUN
ejpam-3126	90	12	(	(	PUNCT
ejpam-3126	90	13	φ(n	φ(n	PROPN
ejpam-3126	90	14	)	)	PUNCT
ejpam-3126	90	15	:	:	PUNCT
ejpam-3126	90	16	m	m	X
ejpam-3126	90	17	)	)	PUNCT
ejpam-3126	90	18	.	.	PUNCT
ejpam-3126	91	1	by	by	ADP
ejpam-3126	91	2	assumption	assumption	NOUN
ejpam-3126	91	3	,	,	PUNCT
ejpam-3126	91	4	a1a2	a1a2	PROPN
ejpam-3126	91	5	∈	∈	PROPN
ejpam-3126	91	6	(	(	PUNCT
ejpam-3126	91	7	n	n	NOUN
ejpam-3126	91	8	:	:	PUNCT
ejpam-3126	91	9	m	m	X
ejpam-3126	91	10	)	)	PUNCT
ejpam-3126	91	11	−	−	PROPN
ejpam-3126	91	12	ϕ((n	ϕ((n	NOUN
ejpam-3126	91	13	:	:	PUNCT
ejpam-3126	91	14	m	m	X
ejpam-3126	91	15	)	)	PUNCT
ejpam-3126	91	16	)	)	PUNCT
ejpam-3126	91	17	.	.	PUNCT
ejpam-3126	92	1	again	again	ADV
ejpam-3126	92	2	,	,	PUNCT
ejpam-3126	92	3	by	by	ADP
ejpam-3126	92	4	assumption	assumption	NOUN
ejpam-3126	92	5	,	,	PUNCT
ejpam-3126	92	6	a1	a1	NOUN
ejpam-3126	92	7	∈	∈	PROPN
ejpam-3126	92	8	(	(	PUNCT
ejpam-3126	92	9	n	n	NOUN
ejpam-3126	92	10	:	:	PUNCT
ejpam-3126	92	11	m	m	X
ejpam-3126	92	12	)	)	PUNCT
ejpam-3126	92	13	or	or	CCONJ
ejpam-3126	92	14	an2	an2	PROPN
ejpam-3126	92	15	∈	∈	PROPN
ejpam-3126	92	16	(	(	PUNCT
ejpam-3126	92	17	n	n	NOUN
ejpam-3126	92	18	:	:	PUNCT
ejpam-3126	92	19	m	m	X
ejpam-3126	92	20	)	)	PUNCT
ejpam-3126	92	21	for	for	ADP
ejpam-3126	92	22	some	some	DET
ejpam-3126	92	23	positive	positive	ADJ
ejpam-3126	92	24	integer	integer	NOUN
ejpam-3126	92	25	n.	n.	NOUN
ejpam-3126	92	26	this	this	PRON
ejpam-3126	92	27	completes	complete	VERB
ejpam-3126	92	28	the	the	DET
ejpam-3126	92	29	proof	proof	NOUN
ejpam-3126	92	30	.	.	PUNCT
ejpam-3126	93	1	the	the	DET
ejpam-3126	93	2	following	follow	VERB
ejpam-3126	93	3	example	example	NOUN
ejpam-3126	93	4	shows	show	VERB
ejpam-3126	93	5	that	that	SCONJ
ejpam-3126	93	6	the	the	DET
ejpam-3126	93	7	converse	converse	NOUN
ejpam-3126	93	8	of	of	ADP
ejpam-3126	93	9	theorem	theorem	NOUN
ejpam-3126	93	10	1	1	NUM
ejpam-3126	93	11	is	be	AUX
ejpam-3126	93	12	not	not	PART
ejpam-3126	93	13	true	true	ADJ
ejpam-3126	93	14	.	.	PUNCT
ejpam-3126	94	1	example	example	NOUN
ejpam-3126	94	2	2	2	NUM
ejpam-3126	94	3	.	.	NOUN
ejpam-3126	94	4	1	1	NUM
ejpam-3126	94	5	.	.	X
ejpam-3126	95	1	let	let	AUX
ejpam-3126	95	2	m	m	VERB
ejpam-3126	95	3	=	=	NOUN
ejpam-3126	95	4	z	z	NUM
ejpam-3126	95	5	×	×	PROPN
ejpam-3126	95	6	z	z	NOUN
ejpam-3126	95	7	×	×	NOUN
ejpam-3126	95	8	z	z	NOUN
ejpam-3126	95	9	be	be	AUX
ejpam-3126	95	10	an	an	DET
ejpam-3126	95	11	z	z	NOUN
ejpam-3126	95	12	-	-	PUNCT
ejpam-3126	95	13	module	module	NOUN
ejpam-3126	95	14	.	.	PUNCT
ejpam-3126	96	1	define	define	VERB
ejpam-3126	96	2	ϕ	ϕ	NOUN
ejpam-3126	96	3	:	:	PUNCT
ejpam-3126	96	4	j	j	PROPN
ejpam-3126	96	5	(	(	PUNCT
ejpam-3126	96	6	r	r	NOUN
ejpam-3126	96	7	)	)	PUNCT
ejpam-3126	96	8	→	→	SYM
ejpam-3126	96	9	j	j	PROPN
ejpam-3126	96	10	(	(	PUNCT
ejpam-3126	96	11	r	r	NOUN
ejpam-3126	96	12	)	)	PUNCT
ejpam-3126	96	13	∪	∪	NOUN
ejpam-3126	96	14	{	{	PUNCT
ejpam-3126	96	15	∅	∅	NOUN
ejpam-3126	96	16	}	}	PUNCT
ejpam-3126	96	17	by	by	ADP
ejpam-3126	96	18	ϕ(i	ϕ(i	PROPN
ejpam-3126	96	19	)	)	PUNCT
ejpam-3126	96	20	=	=	PRON
ejpam-3126	96	21	{	{	PUNCT
ejpam-3126	96	22	0	0	NUM
ejpam-3126	96	23	}	}	PUNCT
ejpam-3126	96	24	for	for	ADP
ejpam-3126	96	25	every	every	DET
ejpam-3126	96	26	i	i	PROPN
ejpam-3126	96	27	∈	∈	PROPN
ejpam-3126	96	28	j	j	PROPN
ejpam-3126	96	29	(	(	PUNCT
ejpam-3126	96	30	r	r	NOUN
ejpam-3126	96	31	)	)	PUNCT
ejpam-3126	96	32	.	.	PUNCT
ejpam-3126	97	1	consider	consider	VERB
ejpam-3126	97	2	the	the	DET
ejpam-3126	97	3	submodule	submodule	NOUN
ejpam-3126	97	4	n	n	NOUN
ejpam-3126	97	5	=	=	NUM
ejpam-3126	97	6	{	{	PUNCT
ejpam-3126	97	7	0	0	NUM
ejpam-3126	97	8	}	}	PUNCT
ejpam-3126	97	9	×	×	PROPN
ejpam-3126	97	10	12z	12z	X
ejpam-3126	97	11	×	×	PROPN
ejpam-3126	97	12	z	z	NOUN
ejpam-3126	97	13	of	of	ADP
ejpam-3126	97	14	m	m	PROPN
ejpam-3126	97	15	.	.	PUNCT
ejpam-3126	98	1	clearly	clearly	ADV
ejpam-3126	98	2	,	,	PUNCT
ejpam-3126	98	3	(	(	PUNCT
ejpam-3126	98	4	n	n	X
ejpam-3126	98	5	:	:	PUNCT
ejpam-3126	98	6	(	(	PUNCT
ejpam-3126	98	7	m1,m2,m3	m1,m2,m3	NOUN
ejpam-3126	98	8	)	)	PUNCT
ejpam-3126	98	9	)	)	PUNCT
ejpam-3126	99	1	=	=	PRON
ejpam-3126	99	2	{	{	PUNCT
ejpam-3126	99	3	0	0	NUM
ejpam-3126	99	4	}	}	PUNCT
ejpam-3126	99	5	is	be	AUX
ejpam-3126	99	6	a	a	DET
ejpam-3126	99	7	ϕ-2	ϕ-2	ADV
ejpam-3126	99	8	-	-	PUNCT
ejpam-3126	99	9	absorbing	absorb	VERB
ejpam-3126	99	10	primary	primary	ADJ
ejpam-3126	99	11	ideal	ideal	NOUN
ejpam-3126	99	12	of	of	ADP
ejpam-3126	99	13	j	j	PROPN
ejpam-3126	99	14	(	(	PUNCT
ejpam-3126	99	15	r	r	NOUN
ejpam-3126	99	16	)	)	PUNCT
ejpam-3126	99	17	,	,	PUNCT
ejpam-3126	99	18	where	where	SCONJ
ejpam-3126	99	19	(	(	PUNCT
ejpam-3126	99	20	m1,m2,m3	m1,m2,m3	ADJ
ejpam-3126	99	21	)	)	PUNCT
ejpam-3126	99	22	∈	∈	PROPN
ejpam-3126	99	23	m	m	VERB
ejpam-3126	99	24	−	−	NOUN
ejpam-3126	99	25	n	n	ADV
ejpam-3126	99	26	.	.	PUNCT
ejpam-3126	100	1	define	define	VERB
ejpam-3126	100	2	φ	φ	PROPN
ejpam-3126	100	3	:	:	PUNCT
ejpam-3126	100	4	s(m	s(m	PROPN
ejpam-3126	100	5	)	)	PUNCT
ejpam-3126	100	6	→	→	SYM
ejpam-3126	100	7	s(m	s(m	NOUN
ejpam-3126	100	8	)	)	PUNCT
ejpam-3126	100	9	∪	∪	ADP
ejpam-3126	100	10	{	{	PUNCT
ejpam-3126	100	11	∅	∅	NOUN
ejpam-3126	100	12	}	}	PUNCT
ejpam-3126	100	13	by	by	ADP
ejpam-3126	100	14	φ(n	φ(n	NOUN
ejpam-3126	100	15	)	)	PUNCT
ejpam-3126	100	16	=	=	PRON
ejpam-3126	100	17	{	{	PUNCT
ejpam-3126	100	18	(	(	PUNCT
ejpam-3126	100	19	0	0	NUM
ejpam-3126	100	20	,	,	PUNCT
ejpam-3126	100	21	0	0	NUM
ejpam-3126	100	22	,	,	PUNCT
ejpam-3126	100	23	0	0	NUM
ejpam-3126	100	24	)	)	PUNCT
ejpam-3126	100	25	}	}	PUNCT
ejpam-3126	100	26	for	for	ADP
ejpam-3126	100	27	every	every	DET
ejpam-3126	100	28	n	n	PRON
ejpam-3126	100	29	∈	∈	PROPN
ejpam-3126	100	30	s(m	s(m	PROPN
ejpam-3126	100	31	)	)	PUNCT
ejpam-3126	100	32	.	.	PUNCT
ejpam-3126	101	1	notice	notice	VERB
ejpam-3126	101	2	that	that	SCONJ
ejpam-3126	101	3	3	3	X
ejpam-3126	101	4	·	·	SYM
ejpam-3126	101	5	4(0	4(0	NUM
ejpam-3126	101	6	,	,	PUNCT
ejpam-3126	101	7	1	1	NUM
ejpam-3126	101	8	,	,	PUNCT
ejpam-3126	101	9	1	1	NUM
ejpam-3126	101	10	)	)	PUNCT
ejpam-3126	101	11	∈	∈	PROPN
ejpam-3126	101	12	n−φ(n	n−φ(n	NOUN
ejpam-3126	101	13	)	)	PUNCT
ejpam-3126	101	14	,	,	PUNCT
ejpam-3126	101	15	but	but	CCONJ
ejpam-3126	101	16	(	(	PUNCT
ejpam-3126	101	17	3	3	NUM
ejpam-3126	101	18	·	·	SYM
ejpam-3126	101	19	4	4	NUM
ejpam-3126	101	20	)	)	PUNCT
ejpam-3126	101	21	6∈	6∈	NOUN
ejpam-3126	101	22	√	√	NOUN
ejpam-3126	101	23	(	(	PUNCT
ejpam-3126	101	24	n	n	NUM
ejpam-3126	101	25	:	:	PUNCT
ejpam-3126	101	26	m	m	X
ejpam-3126	101	27	)	)	PUNCT
ejpam-3126	101	28	,	,	PUNCT
ejpam-3126	101	29	3(0	3(0	NUM
ejpam-3126	101	30	,	,	PUNCT
ejpam-3126	101	31	1	1	NUM
ejpam-3126	101	32	,	,	PUNCT
ejpam-3126	101	33	1	1	X
ejpam-3126	101	34	)	)	PUNCT
ejpam-3126	101	35	6∈	6∈	NOUN
ejpam-3126	101	36	n	n	NOUN
ejpam-3126	101	37	and	and	CCONJ
ejpam-3126	101	38	4n(0	4n(0	NUM
ejpam-3126	101	39	,	,	PUNCT
ejpam-3126	101	40	1	1	NUM
ejpam-3126	101	41	,	,	PUNCT
ejpam-3126	101	42	1	1	X
ejpam-3126	101	43	)	)	PUNCT
ejpam-3126	101	44	6∈	6∈	NOUN
ejpam-3126	101	45	n	n	NOUN
ejpam-3126	101	46	for	for	ADP
ejpam-3126	101	47	all	all	DET
ejpam-3126	101	48	positive	positive	ADJ
ejpam-3126	101	49	integer	integer	NOUN
ejpam-3126	101	50	n.	n.	NOUN
ejpam-3126	102	1	hence	hence	ADV
ejpam-3126	102	2	n	n	ADV
ejpam-3126	102	3	is	be	AUX
ejpam-3126	102	4	not	not	PART
ejpam-3126	102	5	a	a	DET
ejpam-3126	102	6	φ-2	φ-2	NOUN
ejpam-3126	102	7	-	-	PUNCT
ejpam-3126	102	8	absorbing	absorbing	ADJ
ejpam-3126	102	9	semi	semi	ADJ
ejpam-3126	102	10	-	-	ADJ
ejpam-3126	102	11	primary	primary	ADJ
ejpam-3126	102	12	submodule	submodule	NOUN
ejpam-3126	102	13	of	of	ADP
ejpam-3126	102	14	m	m	PROPN
ejpam-3126	102	15	.	.	PUNCT
ejpam-3126	103	1	2	2	X
ejpam-3126	103	2	.	.	X
ejpam-3126	103	3	let	let	VERB
ejpam-3126	103	4	m	m	VERB
ejpam-3126	103	5	=	=	NOUN
ejpam-3126	103	6	z12	z12	NUM
ejpam-3126	103	7	be	be	AUX
ejpam-3126	103	8	an	an	DET
ejpam-3126	103	9	z12	z12	NUM
ejpam-3126	103	10	-	-	PUNCT
ejpam-3126	103	11	module	module	NOUN
ejpam-3126	103	12	.	.	PUNCT
ejpam-3126	104	1	define	define	VERB
ejpam-3126	104	2	φ	φ	NOUN
ejpam-3126	104	3	:	:	PUNCT
ejpam-3126	104	4	s(m)→	s(m)→	NOUN
ejpam-3126	104	5	s(m)∪{∅	s(m)∪{∅	PROPN
ejpam-3126	104	6	}	}	PUNCT
ejpam-3126	104	7	by	by	ADP
ejpam-3126	104	8	φ(n	φ(n	NOUN
ejpam-3126	104	9	)	)	PUNCT
ejpam-3126	104	10	=	=	PRON
ejpam-3126	104	11	{	{	PUNCT
ejpam-3126	105	1	[	[	X
ejpam-3126	105	2	0	0	NUM
ejpam-3126	105	3	]	]	PUNCT
ejpam-3126	105	4	}	}	PUNCT
ejpam-3126	105	5	for	for	ADP
ejpam-3126	105	6	every	every	DET
ejpam-3126	105	7	n	n	PRON
ejpam-3126	105	8	∈	∈	PROPN
ejpam-3126	105	9	s(m	s(m	PROPN
ejpam-3126	105	10	)	)	PUNCT
ejpam-3126	105	11	.	.	PUNCT
ejpam-3126	106	1	consider	consider	VERB
ejpam-3126	106	2	the	the	DET
ejpam-3126	106	3	submodule	submodule	NOUN
ejpam-3126	106	4	n	n	NOUN
ejpam-3126	106	5	=	=	PRON
ejpam-3126	106	6	{	{	PUNCT
ejpam-3126	107	1	[	[	X
ejpam-3126	107	2	0	0	NUM
ejpam-3126	107	3	]	]	X
ejpam-3126	107	4	}	}	PUNCT
ejpam-3126	107	5	of	of	ADP
ejpam-3126	107	6	m	m	PROPN
ejpam-3126	107	7	.	.	PUNCT
ejpam-3126	108	1	clearly	clearly	ADV
ejpam-3126	108	2	,	,	PUNCT
ejpam-3126	108	3	n	n	PRON
ejpam-3126	108	4	is	be	AUX
ejpam-3126	108	5	a	a	DET
ejpam-3126	108	6	φ-2	φ-2	NOUN
ejpam-3126	108	7	-	-	PUNCT
ejpam-3126	108	8	absorbing	absorbing	ADJ
ejpam-3126	108	9	semi	semi	ADJ
ejpam-3126	108	10	-	-	ADJ
ejpam-3126	108	11	primary	primary	ADJ
ejpam-3126	108	12	submodule	submodule	NOUN
ejpam-3126	108	13	of	of	ADP
ejpam-3126	108	14	m	m	PROPN
ejpam-3126	108	15	.	.	PUNCT
ejpam-3126	109	1	define	define	VERB
ejpam-3126	109	2	ϕ	ϕ	NOUN
ejpam-3126	109	3	:	:	PUNCT
ejpam-3126	109	4	j	j	PROPN
ejpam-3126	109	5	(	(	PUNCT
ejpam-3126	109	6	r	r	NOUN
ejpam-3126	109	7	)	)	PUNCT
ejpam-3126	109	8	→	→	SYM
ejpam-3126	109	9	j	j	PROPN
ejpam-3126	109	10	(	(	PUNCT
ejpam-3126	109	11	r	r	NOUN
ejpam-3126	109	12	)	)	PUNCT
ejpam-3126	109	13	∪	∪	NOUN
ejpam-3126	109	14	{	{	PUNCT
ejpam-3126	109	15	∅	∅	NOUN
ejpam-3126	109	16	}	}	PUNCT
ejpam-3126	109	17	by	by	ADP
ejpam-3126	109	18	ϕ(i	ϕ(i	PROPN
ejpam-3126	109	19	)	)	PUNCT
ejpam-3126	110	1	=	=	NOUN
ejpam-3126	110	2	∅	∅	NOUN
ejpam-3126	110	3	for	for	ADP
ejpam-3126	110	4	every	every	DET
ejpam-3126	110	5	i	i	PROPN
ejpam-3126	110	6	∈	∈	PROPN
ejpam-3126	110	7	j	j	PROPN
ejpam-3126	110	8	(	(	PUNCT
ejpam-3126	110	9	r	r	NOUN
ejpam-3126	110	10	)	)	PUNCT
ejpam-3126	110	11	.	.	PUNCT
ejpam-3126	111	1	notice	notice	VERB
ejpam-3126	111	2	that	that	SCONJ
ejpam-3126	111	3	[	[	X
ejpam-3126	111	4	4][3	4][3	X
ejpam-3126	111	5	]	]	X
ejpam-3126	111	6	∈	∈	NOUN
ejpam-3126	111	7	{	{	PUNCT
ejpam-3126	111	8	[	[	X
ejpam-3126	111	9	0	0	NUM
ejpam-3126	111	10	]	]	X
ejpam-3126	111	11	}	}	PUNCT
ejpam-3126	111	12	=	=	SYM
ejpam-3126	111	13	(	(	PUNCT
ejpam-3126	111	14	n	n	NOUN
ejpam-3126	111	15	:	:	PUNCT
ejpam-3126	112	1	[	[	X
ejpam-3126	112	2	1	1	NUM
ejpam-3126	112	3	]	]	SYM
ejpam-3126	112	4	)	)	PUNCT
ejpam-3126	112	5	−	−	PROPN
ejpam-3126	112	6	ϕ((n	ϕ((n	NOUN
ejpam-3126	112	7	:	:	PUNCT
ejpam-3126	113	1	[	[	X
ejpam-3126	113	2	1	1	NUM
ejpam-3126	113	3	]	]	NUM
ejpam-3126	113	4	)	)	PUNCT
ejpam-3126	113	5	)	)	PUNCT
ejpam-3126	113	6	,	,	PUNCT
ejpam-3126	113	7	but	but	CCONJ
ejpam-3126	113	8	[	[	X
ejpam-3126	113	9	4	4	NUM
ejpam-3126	113	10	]	]	X
ejpam-3126	113	11	∈	∈	PROPN
ejpam-3126	113	12	(	(	PUNCT
ejpam-3126	113	13	n	n	NOUN
ejpam-3126	113	14	:	:	PUNCT
ejpam-3126	113	15	[	[	X
ejpam-3126	113	16	1	1	NUM
ejpam-3126	113	17	]	]	PUNCT
ejpam-3126	113	18	)	)	PUNCT
ejpam-3126	113	19	and	and	CCONJ
ejpam-3126	113	20	[	[	X
ejpam-3126	113	21	3]n	3]n	NUM
ejpam-3126	113	22	∈	∈	NOUN
ejpam-3126	113	23	(	(	PUNCT
ejpam-3126	113	24	n	n	NOUN
ejpam-3126	113	25	:	:	PUNCT
ejpam-3126	114	1	[	[	X
ejpam-3126	114	2	1	1	NUM
ejpam-3126	114	3	]	]	PUNCT
ejpam-3126	114	4	)	)	PUNCT
ejpam-3126	114	5	for	for	ADP
ejpam-3126	114	6	all	all	DET
ejpam-3126	114	7	positive	positive	ADJ
ejpam-3126	114	8	integer	integer	NOUN
ejpam-3126	114	9	n.	n.	NOUN
ejpam-3126	114	10	let	let	VERB
ejpam-3126	114	11	n	n	PRON
ejpam-3126	114	12	be	be	AUX
ejpam-3126	114	13	a	a	DET
ejpam-3126	114	14	submodule	submodule	NOUN
ejpam-3126	114	15	of	of	ADP
ejpam-3126	114	16	an	an	DET
ejpam-3126	114	17	r	r	NOUN
ejpam-3126	114	18	-	-	PUNCT
ejpam-3126	114	19	module	module	NOUN
ejpam-3126	114	20	m	m	NOUN
ejpam-3126	114	21	and	and	CCONJ
ejpam-3126	114	22	let	let	VERB
ejpam-3126	114	23	φ	φ	PROPN
ejpam-3126	114	24	:	:	PUNCT
ejpam-3126	114	25	s(m	s(m	PROPN
ejpam-3126	114	26	)	)	PUNCT
ejpam-3126	114	27	→	→	SYM
ejpam-3126	114	28	s(m	s(m	NOUN
ejpam-3126	114	29	)	)	PUNCT
ejpam-3126	114	30	∪	∪	NOUN
ejpam-3126	114	31	{	{	PUNCT
ejpam-3126	114	32	∅	∅	NOUN
ejpam-3126	114	33	}	}	PUNCT
ejpam-3126	114	34	be	be	AUX
ejpam-3126	114	35	a	a	DET
ejpam-3126	114	36	function	function	NOUN
ejpam-3126	114	37	.	.	PUNCT
ejpam-3126	115	1	define	define	VERB
ejpam-3126	115	2	φn	φn	ADP
ejpam-3126	115	3	:	:	PUNCT
ejpam-3126	115	4	s(m	s(m	PROPN
ejpam-3126	115	5	/	/	SYM
ejpam-3126	115	6	n)→	n)→	PROPN
ejpam-3126	115	7	s(m	s(m	PROPN
ejpam-3126	115	8	/	/	SYM
ejpam-3126	115	9	n	n	CCONJ
ejpam-3126	115	10	)	)	PUNCT
ejpam-3126	115	11	∪	∪	NOUN
ejpam-3126	115	12	{	{	PUNCT
ejpam-3126	115	13	∅	∅	NOUN
ejpam-3126	115	14	}	}	PUNCT
ejpam-3126	115	15	by	by	ADP
ejpam-3126	115	16	φn	φn	PROPN
ejpam-3126	115	17	(	(	PUNCT
ejpam-3126	115	18	k	k	NOUN
ejpam-3126	115	19	/	/	SYM
ejpam-3126	115	20	n	n	CCONJ
ejpam-3126	115	21	)	)	PUNCT
ejpam-3126	115	22	=	=	PRON
ejpam-3126	115	23	{	{	PUNCT
ejpam-3126	115	24	(	(	PUNCT
ejpam-3126	115	25	φ(k	φ(k	PROPN
ejpam-3126	115	26	)	)	PUNCT
ejpam-3126	116	1	+	+	ADV
ejpam-3126	116	2	n)/n	n)/n	PROPN
ejpam-3126	116	3	;	;	PUNCT
ejpam-3126	116	4	φ(k	φ(k	PROPN
ejpam-3126	116	5	)	)	PUNCT
ejpam-3126	116	6	6=	6=	ADP
ejpam-3126	116	7	∅	∅	NOUN
ejpam-3126	116	8	∅	∅	NOUN
ejpam-3126	116	9	;	;	PUNCT
ejpam-3126	116	10	φ(k	φ(k	PROPN
ejpam-3126	116	11	)	)	PUNCT
ejpam-3126	116	12	=	=	SYM
ejpam-3126	116	13	∅	∅	NOUN
ejpam-3126	116	14	,	,	PUNCT
ejpam-3126	116	15	for	for	ADP
ejpam-3126	116	16	every	every	DET
ejpam-3126	116	17	submodule	submodule	NOUN
ejpam-3126	116	18	k	k	PROPN
ejpam-3126	116	19	of	of	ADP
ejpam-3126	116	20	m	m	PROPN
ejpam-3126	116	21	with	with	ADP
ejpam-3126	116	22	n	n	PRON
ejpam-3126	116	23	⊆	⊆	NUM
ejpam-3126	116	24	k[11	k[11	PROPN
ejpam-3126	116	25	]	]	PUNCT
ejpam-3126	116	26	.	.	PUNCT
ejpam-3126	117	1	in	in	ADP
ejpam-3126	117	2	,	,	PUNCT
ejpam-3126	117	3	2010	2010	NUM
ejpam-3126	117	4	zamani	zamani	NOUN
ejpam-3126	117	5	in	in	ADP
ejpam-3126	117	6	[	[	X
ejpam-3126	117	7	11	11	NUM
ejpam-3126	117	8	]	]	PUNCT
ejpam-3126	117	9	gives	give	VERB
ejpam-3126	117	10	relations	relation	NOUN
ejpam-3126	117	11	between	between	ADP
ejpam-3126	117	12	φ	φ	PROPN
ejpam-3126	117	13	-	-	ADJ
ejpam-3126	117	14	prime	prime	ADJ
ejpam-3126	117	15	submodules	submodule	NOUN
ejpam-3126	117	16	of	of	ADP
ejpam-3126	117	17	m	m	PROPN
ejpam-3126	117	18	and	and	CCONJ
ejpam-3126	117	19	φn	φn	ADP
ejpam-3126	117	20	-prime	-prime	NOUN
ejpam-3126	117	21	submodules	submodule	NOUN
ejpam-3126	117	22	of	of	ADP
ejpam-3126	117	23	m	m	PROPN
ejpam-3126	117	24	/	/	SYM
ejpam-3126	117	25	n	n	PROPN
ejpam-3126	117	26	.	.	PUNCT
ejpam-3126	118	1	this	this	PRON
ejpam-3126	118	2	leads	lead	VERB
ejpam-3126	118	3	us	we	PRON
ejpam-3126	118	4	to	to	PART
ejpam-3126	118	5	give	give	VERB
ejpam-3126	118	6	relations	relation	NOUN
ejpam-3126	118	7	between	between	ADP
ejpam-3126	118	8	φ-2	φ-2	PROPN
ejpam-3126	118	9	-	-	PUNCT
ejpam-3126	118	10	absorbing	absorbing	ADJ
ejpam-3126	118	11	semi	semi	ADJ
ejpam-3126	118	12	-	-	ADJ
ejpam-3126	118	13	primary	primary	ADJ
ejpam-3126	118	14	submodules	submodule	NOUN
ejpam-3126	118	15	of	of	ADP
ejpam-3126	118	16	m	m	PROPN
ejpam-3126	118	17	and	and	CCONJ
ejpam-3126	118	18	φn	φn	ADP
ejpam-3126	118	19	-2	-2	ADV
ejpam-3126	118	20	-	-	PUNCT
ejpam-3126	118	21	absorbing	absorb	VERB
ejpam-3126	118	22	semi	semi	ADJ
ejpam-3126	118	23	-	-	ADJ
ejpam-3126	118	24	primary	primary	ADJ
ejpam-3126	118	25	submodules	submodule	NOUN
ejpam-3126	118	26	of	of	ADP
ejpam-3126	118	27	m	m	PROPN
ejpam-3126	118	28	/	/	SYM
ejpam-3126	118	29	n	n	PROPN
ejpam-3126	118	30	.	.	PUNCT
ejpam-3126	119	1	theorem	theorem	NOUN
ejpam-3126	119	2	2	2	NUM
ejpam-3126	119	3	.	.	PUNCT
ejpam-3126	120	1	let	let	VERB
ejpam-3126	120	2	φ	φ	NOUN
ejpam-3126	120	3	:	:	PUNCT
ejpam-3126	120	4	s(m)→	s(m)→	NOUN
ejpam-3126	120	5	s(m	s(m	NOUN
ejpam-3126	120	6	)	)	PUNCT
ejpam-3126	120	7	∪	∪	ADP
ejpam-3126	120	8	{	{	PUNCT
ejpam-3126	120	9	∅	∅	NOUN
ejpam-3126	120	10	}	}	PUNCT
ejpam-3126	120	11	be	be	AUX
ejpam-3126	120	12	a	a	DET
ejpam-3126	120	13	function	function	NOUN
ejpam-3126	120	14	and	and	CCONJ
ejpam-3126	120	15	let	let	VERB
ejpam-3126	120	16	n	n	PRON
ejpam-3126	120	17	,	,	PUNCT
ejpam-3126	120	18	k	k	X
ejpam-3126	120	19	be	be	VERB
ejpam-3126	120	20	two	two	NUM
ejpam-3126	120	21	submodules	submodule	NOUN
ejpam-3126	120	22	of	of	ADP
ejpam-3126	120	23	m	m	PROPN
ejpam-3126	120	24	with	with	ADP
ejpam-3126	120	25	n	n	PRON
ejpam-3126	120	26	⊆	⊆	NUM
ejpam-3126	120	27	k.	k.	NOUN
ejpam-3126	120	28	if	if	SCONJ
ejpam-3126	120	29	k	k	PROPN
ejpam-3126	120	30	is	be	AUX
ejpam-3126	120	31	a	a	DET
ejpam-3126	120	32	φ-2	φ-2	NOUN
ejpam-3126	120	33	-	-	PUNCT
ejpam-3126	120	34	absorbing	absorbing	ADJ
ejpam-3126	120	35	semi	semi	ADJ
ejpam-3126	120	36	-	-	ADJ
ejpam-3126	120	37	primary	primary	ADJ
ejpam-3126	120	38	submodule	submodule	NOUN
ejpam-3126	120	39	of	of	ADP
ejpam-3126	120	40	m	m	PROPN
ejpam-3126	120	41	,	,	PUNCT
ejpam-3126	120	42	then	then	ADV
ejpam-3126	120	43	k	k	X
ejpam-3126	120	44	/	/	SYM
ejpam-3126	120	45	n	n	PROPN
ejpam-3126	120	46	is	be	AUX
ejpam-3126	120	47	a	a	DET
ejpam-3126	120	48	φn	φn	VERB
ejpam-3126	120	49	-2	-2	ADV
ejpam-3126	120	50	-	-	PUNCT
ejpam-3126	120	51	absorbing	absorbing	ADJ
ejpam-3126	120	52	semi	semi	ADJ
ejpam-3126	120	53	-	-	ADJ
ejpam-3126	120	54	primary	primary	ADJ
ejpam-3126	120	55	submodule	submodule	NOUN
ejpam-3126	120	56	of	of	ADP
ejpam-3126	120	57	m	m	PROPN
ejpam-3126	120	58	/	/	SYM
ejpam-3126	120	59	n	n	PROPN
ejpam-3126	120	60	.	.	PUNCT
ejpam-3126	121	1	pai	pai	PROPN
ejpam-3126	121	2	.	.	PROPN
ejpam-3126	121	3	yiarayong	yiarayong	PROPN
ejpam-3126	121	4	,	,	PUNCT
ejpam-3126	121	5	m.	m.	NOUN
ejpam-3126	121	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	121	7	/	/	SYM
ejpam-3126	121	8	eur	eur	PROPN
ejpam-3126	121	9	.	.	PUNCT
ejpam-3126	122	1	j.	j.	PROPN
ejpam-3126	122	2	pure	pure	PROPN
ejpam-3126	122	3	appl	appl	PROPN
ejpam-3126	122	4	.	.	PROPN
ejpam-3126	122	5	math	math	PROPN
ejpam-3126	122	6	,	,	PUNCT
ejpam-3126	122	7	11	11	NUM
ejpam-3126	122	8	(	(	PUNCT
ejpam-3126	122	9	1	1	NUM
ejpam-3126	122	10	)	)	PUNCT
ejpam-3126	122	11	(	(	PUNCT
ejpam-3126	122	12	2018	2018	NUM
ejpam-3126	122	13	)	)	PUNCT
ejpam-3126	122	14	,	,	PUNCT
ejpam-3126	122	15	35	35	NUM
ejpam-3126	122	16	-	-	SYM
ejpam-3126	122	17	50	50	NUM
ejpam-3126	122	18	39	39	NUM
ejpam-3126	122	19	proof	proof	NOUN
ejpam-3126	122	20	.	.	PUNCT
ejpam-3126	123	1	let	let	VERB
ejpam-3126	123	2	a1	a1	NOUN
ejpam-3126	123	3	,	,	PUNCT
ejpam-3126	123	4	a2	a2	PROPN
ejpam-3126	123	5	∈	∈	PROPN
ejpam-3126	123	6	r	r	NOUN
ejpam-3126	123	7	and	and	CCONJ
ejpam-3126	123	8	m	m	PROPN
ejpam-3126	123	9	∈	∈	NOUN
ejpam-3126	123	10	m	m	VERB
ejpam-3126	124	1	such	such	ADJ
ejpam-3126	124	2	that	that	SCONJ
ejpam-3126	124	3	a1a2(m	a1a2(m	ADP
ejpam-3126	124	4	+	+	NUM
ejpam-3126	124	5	n	n	CCONJ
ejpam-3126	124	6	)	)	PUNCT
ejpam-3126	124	7	∈	∈	PROPN
ejpam-3126	124	8	(	(	PUNCT
ejpam-3126	124	9	k	k	NOUN
ejpam-3126	124	10	/	/	SYM
ejpam-3126	124	11	n	n	CCONJ
ejpam-3126	124	12	)	)	PUNCT
ejpam-3126	124	13	−	−	PROPN
ejpam-3126	125	1	φn	φn	INTJ
ejpam-3126	125	2	(	(	PUNCT
ejpam-3126	125	3	k	k	NOUN
ejpam-3126	125	4	/	/	SYM
ejpam-3126	125	5	n	n	CCONJ
ejpam-3126	125	6	)	)	PUNCT
ejpam-3126	125	7	.	.	PUNCT
ejpam-3126	126	1	then	then	ADV
ejpam-3126	126	2	a1a2	a1a2	ADP
ejpam-3126	126	3	m	m	VERB
ejpam-3126	126	4	∈	∈	ADJ
ejpam-3126	126	5	k	k	NOUN
ejpam-3126	126	6	−	−	PROPN
ejpam-3126	126	7	φ(k	φ(k	PROPN
ejpam-3126	126	8	)	)	PUNCT
ejpam-3126	126	9	.	.	PUNCT
ejpam-3126	127	1	by	by	ADP
ejpam-3126	127	2	definition	definition	NOUN
ejpam-3126	127	3	1	1	NUM
ejpam-3126	127	4	,	,	PUNCT
ejpam-3126	127	5	a1a2	a1a2	PROPN
ejpam-3126	127	6	∈	∈	NOUN
ejpam-3126	127	7	√	√	NUM
ejpam-3126	127	8	(	(	PUNCT
ejpam-3126	127	9	k	k	NOUN
ejpam-3126	127	10	:	:	PUNCT
ejpam-3126	127	11	m	m	X
ejpam-3126	127	12	)	)	PUNCT
ejpam-3126	127	13	or	or	CCONJ
ejpam-3126	127	14	a1	a1	NOUN
ejpam-3126	127	15	m	m	PROPN
ejpam-3126	127	16	∈	∈	NOUN
ejpam-3126	127	17	k	k	NOUN
ejpam-3126	127	18	or	or	CCONJ
ejpam-3126	127	19	an2	an2	PROPN
ejpam-3126	127	20	m	m	NOUN
ejpam-3126	127	21	∈	∈	PROPN
ejpam-3126	127	22	k	k	NOUN
ejpam-3126	127	23	for	for	ADP
ejpam-3126	127	24	some	some	DET
ejpam-3126	127	25	positive	positive	ADJ
ejpam-3126	127	26	integer	integer	NOUN
ejpam-3126	127	27	n.	n.	NOUN
ejpam-3126	127	28	clearly	clearly	ADV
ejpam-3126	127	29	,	,	PUNCT
ejpam-3126	127	30	a1a2	a1a2	PROPN
ejpam-3126	127	31	∈	∈	NOUN
ejpam-3126	127	32	√	√	ADP
ejpam-3126	127	33	(	(	PUNCT
ejpam-3126	127	34	k	k	NOUN
ejpam-3126	127	35	/	/	SYM
ejpam-3126	127	36	n	n	NUM
ejpam-3126	127	37	:	:	PUNCT
ejpam-3126	127	38	m	m	X
ejpam-3126	127	39	/	/	SYM
ejpam-3126	127	40	n	n	CCONJ
ejpam-3126	127	41	)	)	PUNCT
ejpam-3126	127	42	or	or	CCONJ
ejpam-3126	127	43	a1(m	a1(m	X
ejpam-3126	127	44	+	+	CCONJ
ejpam-3126	127	45	n	n	CCONJ
ejpam-3126	127	46	)	)	PUNCT
ejpam-3126	127	47	∈	∈	PROPN
ejpam-3126	127	48	k	k	PROPN
ejpam-3126	127	49	/	/	SYM
ejpam-3126	127	50	n	n	PROPN
ejpam-3126	127	51	or	or	CCONJ
ejpam-3126	127	52	an2	an2	PROPN
ejpam-3126	127	53	(	(	PUNCT
ejpam-3126	127	54	m+n	m+n	PROPN
ejpam-3126	127	55	)	)	PUNCT
ejpam-3126	127	56	∈	∈	PROPN
ejpam-3126	128	1	k	k	PROPN
ejpam-3126	128	2	/	/	SYM
ejpam-3126	128	3	n	n	PROPN
ejpam-3126	128	4	for	for	ADP
ejpam-3126	128	5	some	some	DET
ejpam-3126	128	6	positive	positive	ADJ
ejpam-3126	128	7	integer	integer	NOUN
ejpam-3126	128	8	n.	n.	NOUN
ejpam-3126	128	9	this	this	PRON
ejpam-3126	128	10	completes	complete	VERB
ejpam-3126	128	11	the	the	DET
ejpam-3126	128	12	proof	proof	NOUN
ejpam-3126	128	13	.	.	PUNCT
ejpam-3126	129	1	theorem	theorem	NOUN
ejpam-3126	129	2	3	3	X
ejpam-3126	129	3	.	.	PUNCT
ejpam-3126	130	1	let	let	VERB
ejpam-3126	130	2	φ	φ	NOUN
ejpam-3126	130	3	:	:	PUNCT
ejpam-3126	130	4	s(m)→	s(m)→	NOUN
ejpam-3126	130	5	s(m	s(m	NOUN
ejpam-3126	130	6	)	)	PUNCT
ejpam-3126	130	7	∪	∪	ADP
ejpam-3126	130	8	{	{	PUNCT
ejpam-3126	130	9	∅	∅	NOUN
ejpam-3126	130	10	}	}	PUNCT
ejpam-3126	130	11	be	be	AUX
ejpam-3126	130	12	a	a	DET
ejpam-3126	130	13	function	function	NOUN
ejpam-3126	130	14	and	and	CCONJ
ejpam-3126	130	15	let	let	VERB
ejpam-3126	130	16	n	n	PRON
ejpam-3126	130	17	,	,	PUNCT
ejpam-3126	130	18	k	k	X
ejpam-3126	130	19	be	be	VERB
ejpam-3126	130	20	two	two	NUM
ejpam-3126	130	21	submodules	submodule	NOUN
ejpam-3126	130	22	of	of	ADP
ejpam-3126	130	23	m	m	PROPN
ejpam-3126	130	24	.	.	PUNCT
ejpam-3126	131	1	if	if	SCONJ
ejpam-3126	131	2	n	n	PROPN
ejpam-3126	131	3	⊆	⊆	NUM
ejpam-3126	131	4	φ(k	φ(k	PROPN
ejpam-3126	131	5	)	)	PUNCT
ejpam-3126	131	6	and	and	CCONJ
ejpam-3126	131	7	k	k	PROPN
ejpam-3126	131	8	/	/	SYM
ejpam-3126	131	9	n	n	PROPN
ejpam-3126	131	10	is	be	AUX
ejpam-3126	131	11	a	a	DET
ejpam-3126	131	12	φn	φn	VERB
ejpam-3126	131	13	-2	-2	ADV
ejpam-3126	131	14	-	-	PUNCT
ejpam-3126	131	15	absorbing	absorbing	ADJ
ejpam-3126	131	16	semi	semi	ADJ
ejpam-3126	131	17	-	-	ADJ
ejpam-3126	131	18	primary	primary	ADJ
ejpam-3126	131	19	submodule	submodule	NOUN
ejpam-3126	131	20	of	of	ADP
ejpam-3126	131	21	m	m	PROPN
ejpam-3126	131	22	/	/	SYM
ejpam-3126	131	23	n	n	PROPN
ejpam-3126	131	24	,	,	PUNCT
ejpam-3126	131	25	then	then	ADV
ejpam-3126	131	26	k	k	PROPN
ejpam-3126	131	27	is	be	AUX
ejpam-3126	131	28	a	a	DET
ejpam-3126	131	29	φ-2	φ-2	NOUN
ejpam-3126	131	30	-	-	PUNCT
ejpam-3126	131	31	absorbing	absorbing	ADJ
ejpam-3126	131	32	semi	semi	ADJ
ejpam-3126	131	33	-	-	ADJ
ejpam-3126	131	34	primary	primary	ADJ
ejpam-3126	131	35	submodule	submodule	NOUN
ejpam-3126	131	36	of	of	ADP
ejpam-3126	131	37	m	m	PROPN
ejpam-3126	131	38	.	.	PUNCT
ejpam-3126	132	1	proof	proof	NOUN
ejpam-3126	132	2	.	.	PUNCT
ejpam-3126	133	1	let	let	VERB
ejpam-3126	133	2	a1	a1	NOUN
ejpam-3126	133	3	,	,	PUNCT
ejpam-3126	133	4	a2	a2	PROPN
ejpam-3126	133	5	∈	∈	PROPN
ejpam-3126	133	6	r	r	NOUN
ejpam-3126	133	7	and	and	CCONJ
ejpam-3126	133	8	m	m	NOUN
ejpam-3126	133	9	∈m	∈m	NOUN
ejpam-3126	133	10	such	such	ADJ
ejpam-3126	133	11	that	that	SCONJ
ejpam-3126	133	12	a1a2	a1a2	PROPN
ejpam-3126	133	13	m	m	VERB
ejpam-3126	133	14	∈	∈	ADJ
ejpam-3126	133	15	k	k	NOUN
ejpam-3126	133	16	−φ(k	−φ(k	PROPN
ejpam-3126	133	17	)	)	PUNCT
ejpam-3126	133	18	.	.	PUNCT
ejpam-3126	134	1	then	then	ADV
ejpam-3126	134	2	a1a2(m+n	a1a2(m+n	PROPN
ejpam-3126	134	3	)	)	PUNCT
ejpam-3126	134	4	∈	∈	PROPN
ejpam-3126	134	5	(	(	PUNCT
ejpam-3126	134	6	k	k	NOUN
ejpam-3126	134	7	−	−	PROPN
ejpam-3126	134	8	φ(k))/n	φ(k))/n	PROPN
ejpam-3126	134	9	.	.	PUNCT
ejpam-3126	135	1	by	by	ADP
ejpam-3126	135	2	definition	definition	NOUN
ejpam-3126	135	3	1	1	NUM
ejpam-3126	135	4	,	,	PUNCT
ejpam-3126	135	5	a1a2	a1a2	PROPN
ejpam-3126	135	6	∈	∈	NOUN
ejpam-3126	135	7	√	√	ADP
ejpam-3126	135	8	(	(	PUNCT
ejpam-3126	135	9	k	k	NOUN
ejpam-3126	135	10	/	/	SYM
ejpam-3126	135	11	n	n	NUM
ejpam-3126	135	12	:	:	PUNCT
ejpam-3126	135	13	m	m	X
ejpam-3126	135	14	/	/	SYM
ejpam-3126	135	15	n	n	CCONJ
ejpam-3126	135	16	)	)	PUNCT
ejpam-3126	135	17	or	or	CCONJ
ejpam-3126	135	18	a1(m	a1(m	X
ejpam-3126	135	19	+	+	CCONJ
ejpam-3126	135	20	n	n	CCONJ
ejpam-3126	135	21	)	)	PUNCT
ejpam-3126	135	22	∈	∈	PROPN
ejpam-3126	136	1	k	k	PROPN
ejpam-3126	136	2	/	/	SYM
ejpam-3126	136	3	n	n	PROPN
ejpam-3126	136	4	or	or	CCONJ
ejpam-3126	136	5	an2	an2	PROPN
ejpam-3126	136	6	(	(	PUNCT
ejpam-3126	136	7	m+n	m+n	PROPN
ejpam-3126	136	8	)	)	PUNCT
ejpam-3126	136	9	∈	∈	PROPN
ejpam-3126	136	10	k	k	PROPN
ejpam-3126	136	11	/	/	SYM
ejpam-3126	136	12	n	n	PROPN
ejpam-3126	136	13	for	for	ADP
ejpam-3126	136	14	some	some	DET
ejpam-3126	136	15	positive	positive	ADJ
ejpam-3126	136	16	integer	integer	NOUN
ejpam-3126	136	17	n.	n.	NOUN
ejpam-3126	136	18	clearly	clearly	ADV
ejpam-3126	136	19	,	,	PUNCT
ejpam-3126	136	20	a1a2	a1a2	PROPN
ejpam-3126	136	21	∈	∈	NOUN
ejpam-3126	136	22	√	√	NUM
ejpam-3126	136	23	(	(	PUNCT
ejpam-3126	136	24	k	k	NOUN
ejpam-3126	136	25	:	:	PUNCT
ejpam-3126	136	26	m	m	X
ejpam-3126	136	27	)	)	PUNCT
ejpam-3126	136	28	or	or	CCONJ
ejpam-3126	136	29	a1	a1	NOUN
ejpam-3126	136	30	m	m	PROPN
ejpam-3126	136	31	∈	∈	NOUN
ejpam-3126	136	32	k	k	NOUN
ejpam-3126	136	33	or	or	CCONJ
ejpam-3126	136	34	an2	an2	PROPN
ejpam-3126	136	35	m	m	NOUN
ejpam-3126	136	36	∈	∈	PROPN
ejpam-3126	136	37	k	k	NOUN
ejpam-3126	136	38	for	for	ADP
ejpam-3126	136	39	some	some	DET
ejpam-3126	136	40	positive	positive	ADJ
ejpam-3126	136	41	integer	integer	NOUN
ejpam-3126	136	42	n.	n.	NOUN
ejpam-3126	136	43	now	now	ADV
ejpam-3126	136	44	,	,	PUNCT
ejpam-3126	136	45	by	by	ADP
ejpam-3126	136	46	theorem	theorem	NOUN
ejpam-3126	136	47	2	2	NUM
ejpam-3126	136	48	and	and	CCONJ
ejpam-3126	136	49	theorem	theorem	VERB
ejpam-3126	136	50	3	3	NUM
ejpam-3126	136	51	,	,	PUNCT
ejpam-3126	136	52	we	we	PRON
ejpam-3126	136	53	have	have	VERB
ejpam-3126	136	54	the	the	DET
ejpam-3126	136	55	following	follow	VERB
ejpam-3126	136	56	corollary	corollary	NOUN
ejpam-3126	136	57	.	.	PUNCT
ejpam-3126	137	1	corollary	corollary	ADJ
ejpam-3126	137	2	1	1	NUM
ejpam-3126	137	3	.	.	PUNCT
ejpam-3126	138	1	let	let	VERB
ejpam-3126	138	2	φ	φ	NOUN
ejpam-3126	138	3	:	:	PUNCT
ejpam-3126	138	4	s(m)→	s(m)→	NOUN
ejpam-3126	138	5	s(m	s(m	NOUN
ejpam-3126	138	6	)	)	PUNCT
ejpam-3126	138	7	∪	∪	ADP
ejpam-3126	138	8	{	{	PUNCT
ejpam-3126	138	9	∅	∅	NOUN
ejpam-3126	138	10	}	}	PUNCT
ejpam-3126	138	11	be	be	AUX
ejpam-3126	138	12	a	a	DET
ejpam-3126	138	13	function	function	NOUN
ejpam-3126	138	14	and	and	CCONJ
ejpam-3126	138	15	let	let	VERB
ejpam-3126	138	16	n	n	PRON
ejpam-3126	138	17	,	,	PUNCT
ejpam-3126	138	18	k	k	X
ejpam-3126	138	19	be	be	VERB
ejpam-3126	138	20	two	two	NUM
ejpam-3126	138	21	submodules	submodule	NOUN
ejpam-3126	138	22	of	of	ADP
ejpam-3126	138	23	m	m	PROPN
ejpam-3126	138	24	with	with	ADP
ejpam-3126	138	25	n	n	PRON
ejpam-3126	138	26	⊆	⊆	NUM
ejpam-3126	138	27	φ(k	φ(k	PROPN
ejpam-3126	138	28	)	)	PUNCT
ejpam-3126	138	29	.	.	PUNCT
ejpam-3126	139	1	then	then	ADV
ejpam-3126	139	2	k	k	PROPN
ejpam-3126	139	3	is	be	AUX
ejpam-3126	139	4	a	a	DET
ejpam-3126	139	5	φ-2	φ-2	NOUN
ejpam-3126	139	6	-	-	PUNCT
ejpam-3126	139	7	absorbing	absorbing	ADJ
ejpam-3126	139	8	semi	semi	ADJ
ejpam-3126	139	9	-	-	ADJ
ejpam-3126	139	10	primary	primary	ADJ
ejpam-3126	139	11	submodule	submodule	NOUN
ejpam-3126	139	12	of	of	ADP
ejpam-3126	139	13	m	m	PROPN
ejpam-3126	139	14	if	if	SCONJ
ejpam-3126	140	1	and	and	CCONJ
ejpam-3126	140	2	only	only	ADV
ejpam-3126	140	3	if	if	SCONJ
ejpam-3126	140	4	k	k	PROPN
ejpam-3126	140	5	/	/	SYM
ejpam-3126	140	6	n	n	PROPN
ejpam-3126	140	7	is	be	AUX
ejpam-3126	140	8	a	a	DET
ejpam-3126	140	9	φn	φn	VERB
ejpam-3126	140	10	-2	-2	ADV
ejpam-3126	140	11	-	-	PUNCT
ejpam-3126	140	12	absorbing	absorbing	ADJ
ejpam-3126	140	13	semi	semi	ADJ
ejpam-3126	140	14	-	-	ADJ
ejpam-3126	140	15	primary	primary	ADJ
ejpam-3126	140	16	submodule	submodule	NOUN
ejpam-3126	140	17	of	of	ADP
ejpam-3126	140	18	m	m	PROPN
ejpam-3126	140	19	/	/	SYM
ejpam-3126	140	20	n	n	PROPN
ejpam-3126	140	21	.	.	PUNCT
ejpam-3126	141	1	proof	proof	NOUN
ejpam-3126	141	2	.	.	PUNCT
ejpam-3126	142	1	the	the	DET
ejpam-3126	142	2	proof	proof	NOUN
ejpam-3126	142	3	follows	follow	VERB
ejpam-3126	142	4	from	from	ADP
ejpam-3126	142	5	theorem	theorem	ADJ
ejpam-3126	142	6	2	2	NUM
ejpam-3126	142	7	,	,	PUNCT
ejpam-3126	142	8	3	3	NUM
ejpam-3126	142	9	.	.	X
ejpam-3126	142	10	zamani	zamani	NOUN
ejpam-3126	142	11	in	in	ADP
ejpam-3126	142	12	[	[	X
ejpam-3126	142	13	11	11	NUM
ejpam-3126	142	14	]	]	PUNCT
ejpam-3126	142	15	gives	give	VERB
ejpam-3126	142	16	relations	relation	NOUN
ejpam-3126	142	17	between	between	ADP
ejpam-3126	142	18	φ	φ	PROPN
ejpam-3126	142	19	-	-	ADJ
ejpam-3126	142	20	prime	prime	ADJ
ejpam-3126	142	21	submodules	submodule	NOUN
ejpam-3126	142	22	of	of	ADP
ejpam-3126	142	23	m	m	PROPN
ejpam-3126	142	24	and	and	CCONJ
ejpam-3126	142	25	φs	φs	ADP
ejpam-3126	142	26	-	-	PUNCT
ejpam-3126	142	27	prime	prime	ADJ
ejpam-3126	142	28	submodules	submodule	NOUN
ejpam-3126	142	29	of	of	ADP
ejpam-3126	142	30	s−1	s−1	PROPN
ejpam-3126	142	31	m	m	NOUN
ejpam-3126	142	32	.	.	PUNCT
ejpam-3126	143	1	this	this	PRON
ejpam-3126	143	2	leads	lead	VERB
ejpam-3126	143	3	us	we	PRON
ejpam-3126	143	4	to	to	PART
ejpam-3126	143	5	give	give	VERB
ejpam-3126	143	6	relations	relation	NOUN
ejpam-3126	143	7	between	between	ADP
ejpam-3126	143	8	φ-2	φ-2	PROPN
ejpam-3126	143	9	-	-	PUNCT
ejpam-3126	143	10	absorbing	absorbing	ADJ
ejpam-3126	143	11	semi	semi	ADJ
ejpam-3126	143	12	-	-	ADJ
ejpam-3126	143	13	primary	primary	ADJ
ejpam-3126	143	14	submodules	submodule	NOUN
ejpam-3126	143	15	of	of	ADP
ejpam-3126	143	16	m	m	PROPN
ejpam-3126	143	17	and	and	CCONJ
ejpam-3126	143	18	φs-2	φs-2	NOUN
ejpam-3126	143	19	-	-	ADJ
ejpam-3126	143	20	absorbing	absorbing	ADJ
ejpam-3126	143	21	semi	semi	ADJ
ejpam-3126	143	22	-	-	ADJ
ejpam-3126	143	23	primary	primary	ADJ
ejpam-3126	143	24	submodules	submodule	NOUN
ejpam-3126	143	25	of	of	ADP
ejpam-3126	143	26	s−1	s−1	PROPN
ejpam-3126	143	27	m	m	PROPN
ejpam-3126	143	28	.	.	PUNCT
ejpam-3126	144	1	theorem	theorem	ADJ
ejpam-3126	144	2	4	4	NUM
ejpam-3126	144	3	.	.	PUNCT
ejpam-3126	145	1	let	let	VERB
ejpam-3126	145	2	s	s	PRON
ejpam-3126	145	3	be	be	AUX
ejpam-3126	145	4	a	a	DET
ejpam-3126	145	5	multiplicative	multiplicative	ADJ
ejpam-3126	145	6	closed	close	VERB
ejpam-3126	145	7	subset	subset	NOUN
ejpam-3126	145	8	of	of	ADP
ejpam-3126	145	9	r	r	NOUN
ejpam-3126	145	10	and	and	CCONJ
ejpam-3126	145	11	let	let	VERB
ejpam-3126	145	12	φ	φ	PROPN
ejpam-3126	145	13	:	:	PUNCT
ejpam-3126	145	14	s(m)→	s(m)→	NOUN
ejpam-3126	145	15	s(m)∪{∅	s(m)∪{∅	PROPN
ejpam-3126	145	16	}	}	PUNCT
ejpam-3126	145	17	be	be	AUX
ejpam-3126	145	18	a	a	DET
ejpam-3126	145	19	function	function	NOUN
ejpam-3126	145	20	.	.	PUNCT
ejpam-3126	146	1	if	if	SCONJ
ejpam-3126	146	2	n	n	PRON
ejpam-3126	146	3	is	be	AUX
ejpam-3126	146	4	a	a	DET
ejpam-3126	146	5	φ-2	φ-2	NOUN
ejpam-3126	146	6	-	-	PUNCT
ejpam-3126	146	7	absorbing	absorbing	ADJ
ejpam-3126	146	8	semi	semi	ADJ
ejpam-3126	146	9	-	-	ADJ
ejpam-3126	146	10	primary	primary	ADJ
ejpam-3126	146	11	submodule	submodule	NOUN
ejpam-3126	146	12	of	of	ADP
ejpam-3126	146	13	m	m	PROPN
ejpam-3126	146	14	,	,	PUNCT
ejpam-3126	146	15	then	then	ADV
ejpam-3126	146	16	s−1n	s−1n	NOUN
ejpam-3126	146	17	is	be	AUX
ejpam-3126	146	18	a	a	DET
ejpam-3126	146	19	φs-2	φs-2	NOUN
ejpam-3126	146	20	-	-	ADJ
ejpam-3126	146	21	absorbing	absorbing	ADJ
ejpam-3126	146	22	semi	semi	ADJ
ejpam-3126	146	23	-	-	ADJ
ejpam-3126	146	24	primary	primary	ADJ
ejpam-3126	146	25	submodule	submodule	NOUN
ejpam-3126	146	26	of	of	ADP
ejpam-3126	146	27	s−1	s−1	PROPN
ejpam-3126	146	28	m	m	NOUN
ejpam-3126	146	29	.	.	PUNCT
ejpam-3126	147	1	proof	proof	NOUN
ejpam-3126	147	2	.	.	PUNCT
ejpam-3126	148	1	let	let	VERB
ejpam-3126	148	2	a1	a1	NOUN
ejpam-3126	148	3	,	,	PUNCT
ejpam-3126	148	4	a2	a2	PROPN
ejpam-3126	148	5	∈	∈	PROPN
ejpam-3126	148	6	r	r	NOUN
ejpam-3126	148	7	,	,	PUNCT
ejpam-3126	148	8	s1	s1	NOUN
ejpam-3126	148	9	,	,	PUNCT
ejpam-3126	148	10	s2	s2	PROPN
ejpam-3126	148	11	,	,	PUNCT
ejpam-3126	148	12	s3	s3	PROPN
ejpam-3126	148	13	∈	∈	PROPN
ejpam-3126	148	14	s	s	PART
ejpam-3126	148	15	and	and	CCONJ
ejpam-3126	148	16	m	m	VERB
ejpam-3126	148	17	∈m	∈m	NOUN
ejpam-3126	148	18	such	such	ADJ
ejpam-3126	148	19	that	that	DET
ejpam-3126	148	20	a1	a1	NOUN
ejpam-3126	148	21	s1	s1	PROPN
ejpam-3126	148	22	a2	a2	PROPN
ejpam-3126	148	23	s2	s2	PROPN
ejpam-3126	148	24	m	m	NOUN
ejpam-3126	148	25	s3	s3	NOUN
ejpam-3126	148	26	∈	∈	PROPN
ejpam-3126	148	27	s−1n−φs(s−1n	s−1n−φs(s−1n	NOUN
ejpam-3126	148	28	)	)	PUNCT
ejpam-3126	148	29	.	.	PUNCT
ejpam-3126	149	1	then	then	ADV
ejpam-3126	149	2	there	there	PRON
ejpam-3126	149	3	exists	exist	VERB
ejpam-3126	149	4	s	s	X
ejpam-3126	149	5	∈	∈	PROPN
ejpam-3126	149	6	s	s	VERB
ejpam-3126	149	7	such	such	ADJ
ejpam-3126	149	8	that	that	SCONJ
ejpam-3126	149	9	sa1a2	sa1a2	PROPN
ejpam-3126	149	10	m	m	VERB
ejpam-3126	149	11	∈	∈	NOUN
ejpam-3126	149	12	n	n	NOUN
ejpam-3126	149	13	.	.	PUNCT
ejpam-3126	150	1	if	if	SCONJ
ejpam-3126	150	2	sa1a2	sa1a2	NOUN
ejpam-3126	150	3	m	m	VERB
ejpam-3126	150	4	∈	∈	NOUN
ejpam-3126	150	5	φ(n	φ(n	NOUN
ejpam-3126	150	6	)	)	PUNCT
ejpam-3126	150	7	,	,	PUNCT
ejpam-3126	150	8	then	then	ADV
ejpam-3126	150	9	a1	a1	NOUN
ejpam-3126	150	10	s1	s1	PROPN
ejpam-3126	150	11	a2	a2	PROPN
ejpam-3126	150	12	s2	s2	PROPN
ejpam-3126	150	13	m	m	PROPN
ejpam-3126	150	14	s3	s3	NOUN
ejpam-3126	150	15	=	=	PUNCT
ejpam-3126	150	16	sa1a2	sa1a2	PROPN
ejpam-3126	150	17	m	m	PROPN
ejpam-3126	150	18	ss1s2s3	ss1s2s3	NOUN
ejpam-3126	150	19	∈	∈	NOUN
ejpam-3126	150	20	s−1φ(n	s−1φ(n	X
ejpam-3126	150	21	)	)	PUNCT
ejpam-3126	150	22	=	=	SYM
ejpam-3126	150	23	φs(s−1n	φs(s−1n	NOUN
ejpam-3126	150	24	)	)	PUNCT
ejpam-3126	150	25	,	,	PUNCT
ejpam-3126	150	26	a	a	DET
ejpam-3126	150	27	contradiction	contradiction	NOUN
ejpam-3126	150	28	.	.	PUNCT
ejpam-3126	151	1	now	now	ADV
ejpam-3126	151	2	if	if	SCONJ
ejpam-3126	151	3	sa1a2	sa1a2	PROPN
ejpam-3126	151	4	m	m	PROPN
ejpam-3126	151	5	6∈	6∈	NOUN
ejpam-3126	151	6	φ(n	φ(n	NOUN
ejpam-3126	151	7	)	)	PUNCT
ejpam-3126	151	8	,	,	PUNCT
ejpam-3126	151	9	then	then	ADV
ejpam-3126	151	10	a1a2(sm	a1a2(sm	PROPN
ejpam-3126	151	11	)	)	PUNCT
ejpam-3126	151	12	∈	∈	PROPN
ejpam-3126	151	13	n	n	CCONJ
ejpam-3126	151	14	−	−	PROPN
ejpam-3126	152	1	φ(n	φ(n	NOUN
ejpam-3126	152	2	)	)	PUNCT
ejpam-3126	152	3	.	.	PUNCT
ejpam-3126	153	1	by	by	ADP
ejpam-3126	153	2	definition	definition	NOUN
ejpam-3126	153	3	1	1	NUM
ejpam-3126	153	4	,	,	PUNCT
ejpam-3126	153	5	a1a2	a1a2	PROPN
ejpam-3126	153	6	∈	∈	NOUN
ejpam-3126	153	7	√	√	NUM
ejpam-3126	153	8	(	(	PUNCT
ejpam-3126	153	9	n	n	NUM
ejpam-3126	153	10	:	:	PUNCT
ejpam-3126	153	11	m	m	X
ejpam-3126	153	12	)	)	PUNCT
ejpam-3126	153	13	or	or	CCONJ
ejpam-3126	153	14	a1sm	a1sm	PUNCT
ejpam-3126	153	15	∈	∈	PROPN
ejpam-3126	153	16	n	n	ADP
ejpam-3126	153	17	or	or	CCONJ
ejpam-3126	153	18	an2sm	an2sm	PRON
ejpam-3126	153	19	∈	∈	NOUN
ejpam-3126	153	20	n	n	NOUN
ejpam-3126	153	21	for	for	ADP
ejpam-3126	153	22	some	some	DET
ejpam-3126	153	23	positive	positive	ADJ
ejpam-3126	153	24	integer	integer	NOUN
ejpam-3126	153	25	n.	n.	NOUN
ejpam-3126	153	26	if	if	SCONJ
ejpam-3126	153	27	a1sm	a1sm	SYM
ejpam-3126	153	28	∈	∈	PROPN
ejpam-3126	153	29	n	n	ADP
ejpam-3126	153	30	or	or	CCONJ
ejpam-3126	153	31	an2sm	an2sm	PRON
ejpam-3126	153	32	∈	∈	PROPN
ejpam-3126	153	33	n	n	NOUN
ejpam-3126	153	34	,	,	PUNCT
ejpam-3126	153	35	then	then	ADV
ejpam-3126	153	36	a1	a1	NOUN
ejpam-3126	153	37	s1	s1	PROPN
ejpam-3126	153	38	m	m	PROPN
ejpam-3126	153	39	s3	s3	NOUN
ejpam-3126	153	40	=	=	PUNCT
ejpam-3126	153	41	a1sm	a1sm	X
ejpam-3126	153	42	s1ss3	s1ss3	VERB
ejpam-3126	153	43	∈	∈	PROPN
ejpam-3126	153	44	s−1n	s−1n	NOUN
ejpam-3126	153	45	or	or	CCONJ
ejpam-3126	153	46	(	(	PUNCT
ejpam-3126	153	47	a2s2	a2s2	NOUN
ejpam-3126	153	48	)	)	PUNCT
ejpam-3126	154	1	n	n	PRON
ejpam-3126	154	2	ms3	ms3	NOUN
ejpam-3126	154	3	=	=	PROPN
ejpam-3126	154	4	an2	an2	PROPN
ejpam-3126	154	5	sm	sm	PROPN
ejpam-3126	154	6	sn2	sn2	PROPN
ejpam-3126	154	7	s3s	s3s	NOUN
ejpam-3126	154	8	∈	∈	PROPN
ejpam-3126	154	9	s−1n	s−1n	NOUN
ejpam-3126	154	10	.	.	PUNCT
ejpam-3126	155	1	now	now	ADV
ejpam-3126	155	2	if	if	SCONJ
ejpam-3126	155	3	a1a2	a1a2	ADP
ejpam-3126	155	4	∈	∈	NOUN
ejpam-3126	155	5	√	√	NUM
ejpam-3126	155	6	(	(	PUNCT
ejpam-3126	155	7	n	n	NUM
ejpam-3126	155	8	:	:	PUNCT
ejpam-3126	155	9	m	m	X
ejpam-3126	155	10	)	)	PUNCT
ejpam-3126	155	11	,	,	PUNCT
ejpam-3126	155	12	then	then	ADV
ejpam-3126	155	13	there	there	PRON
ejpam-3126	155	14	exists	exist	VERB
ejpam-3126	155	15	positive	positive	ADJ
ejpam-3126	155	16	integer	integer	NOUN
ejpam-3126	155	17	n1	n1	PROPN
ejpam-3126	155	18	such	such	ADJ
ejpam-3126	155	19	that	that	SCONJ
ejpam-3126	155	20	(	(	PUNCT
ejpam-3126	155	21	a1a2	a1a2	ADJ
ejpam-3126	155	22	)	)	PUNCT
ejpam-3126	155	23	n1	n1	NOUN
ejpam-3126	155	24	∈	∈	PROPN
ejpam-3126	155	25	(	(	PUNCT
ejpam-3126	155	26	n	n	NOUN
ejpam-3126	155	27	:	:	PUNCT
ejpam-3126	155	28	m	m	X
ejpam-3126	155	29	)	)	PUNCT
ejpam-3126	155	30	.	.	PUNCT
ejpam-3126	156	1	clearly	clearly	ADV
ejpam-3126	156	2	,	,	PUNCT
ejpam-3126	156	3	(	(	PUNCT
ejpam-3126	156	4	a1s1	a1s1	NOUN
ejpam-3126	156	5	a2	a2	PROPN
ejpam-3126	156	6	s2	s2	PROPN
ejpam-3126	156	7	)	)	PUNCT
ejpam-3126	156	8	n	n	CCONJ
ejpam-3126	156	9	∈	∈	NOUN
ejpam-3126	156	10	s−1(n	s−1(n	PRON
ejpam-3126	156	11	:	:	PUNCT
ejpam-3126	156	12	m	m	X
ejpam-3126	156	13	)	)	PUNCT
ejpam-3126	156	14	.	.	PUNCT
ejpam-3126	157	1	this	this	PRON
ejpam-3126	157	2	completes	complete	VERB
ejpam-3126	157	3	the	the	DET
ejpam-3126	157	4	proof	proof	NOUN
ejpam-3126	157	5	.	.	PUNCT
ejpam-3126	158	1	theorem	theorem	ADJ
ejpam-3126	158	2	5	5	NUM
ejpam-3126	158	3	.	.	PUNCT
ejpam-3126	159	1	let	let	VERB
ejpam-3126	159	2	s	s	PRON
ejpam-3126	159	3	be	be	AUX
ejpam-3126	159	4	a	a	DET
ejpam-3126	159	5	multiplicative	multiplicative	ADJ
ejpam-3126	159	6	closed	close	VERB
ejpam-3126	159	7	subset	subset	NOUN
ejpam-3126	159	8	of	of	ADP
ejpam-3126	159	9	r	r	NOUN
ejpam-3126	159	10	and	and	CCONJ
ejpam-3126	159	11	let	let	VERB
ejpam-3126	159	12	φ	φ	PROPN
ejpam-3126	159	13	:	:	PUNCT
ejpam-3126	159	14	s(m)→	s(m)→	NOUN
ejpam-3126	159	15	s(m)∪{∅	s(m)∪{∅	PROPN
ejpam-3126	159	16	}	}	PUNCT
ejpam-3126	159	17	be	be	AUX
ejpam-3126	159	18	a	a	DET
ejpam-3126	159	19	function	function	NOUN
ejpam-3126	159	20	.	.	PUNCT
ejpam-3126	160	1	if	if	SCONJ
ejpam-3126	160	2	s−1n	s−1n	NOUN
ejpam-3126	160	3	is	be	AUX
ejpam-3126	160	4	a	a	DET
ejpam-3126	160	5	φs-2	φs-2	NOUN
ejpam-3126	160	6	-	-	ADJ
ejpam-3126	160	7	absorbing	absorbing	ADJ
ejpam-3126	160	8	semi	semi	ADJ
ejpam-3126	160	9	-	-	ADJ
ejpam-3126	160	10	primary	primary	ADJ
ejpam-3126	160	11	submodule	submodule	NOUN
ejpam-3126	160	12	of	of	ADP
ejpam-3126	160	13	s−1	s−1	PROPN
ejpam-3126	160	14	m	m	NOUN
ejpam-3126	160	15	such	such	ADJ
ejpam-3126	160	16	that	that	SCONJ
ejpam-3126	160	17	s	s	NOUN
ejpam-3126	160	18	∩	∩	NOUN
ejpam-3126	160	19	zd(n	zd(n	NOUN
ejpam-3126	160	20	/	/	SYM
ejpam-3126	160	21	φ(n	φ(n	ADJ
ejpam-3126	160	22	)	)	PUNCT
ejpam-3126	160	23	)	)	PUNCT
ejpam-3126	161	1	=	=	NOUN
ejpam-3126	161	2	∅	∅	NOUN
ejpam-3126	161	3	and	and	CCONJ
ejpam-3126	161	4	s	s	X
ejpam-3126	161	5	∩	∩	ADJ
ejpam-3126	161	6	zd(m	zd(m	NUM
ejpam-3126	161	7	/	/	SYM
ejpam-3126	161	8	n	n	CCONJ
ejpam-3126	161	9	)	)	PUNCT
ejpam-3126	161	10	=	=	NOUN
ejpam-3126	161	11	∅	∅	NOUN
ejpam-3126	161	12	,	,	PUNCT
ejpam-3126	161	13	then	then	ADV
ejpam-3126	161	14	n	n	PRON
ejpam-3126	161	15	is	be	AUX
ejpam-3126	161	16	a	a	DET
ejpam-3126	161	17	φ-2	φ-2	NOUN
ejpam-3126	161	18	-	-	PUNCT
ejpam-3126	161	19	absorbing	absorbing	ADJ
ejpam-3126	161	20	semi	semi	ADJ
ejpam-3126	161	21	-	-	ADJ
ejpam-3126	161	22	primary	primary	ADJ
ejpam-3126	161	23	submodule	submodule	NOUN
ejpam-3126	161	24	of	of	ADP
ejpam-3126	161	25	m	m	PROPN
ejpam-3126	161	26	.	.	PUNCT
ejpam-3126	162	1	proof	proof	NOUN
ejpam-3126	162	2	.	.	PUNCT
ejpam-3126	163	1	let	let	VERB
ejpam-3126	163	2	a1	a1	NOUN
ejpam-3126	163	3	,	,	PUNCT
ejpam-3126	163	4	a2	a2	PROPN
ejpam-3126	163	5	∈	∈	PROPN
ejpam-3126	163	6	r	r	NOUN
ejpam-3126	163	7	and	and	CCONJ
ejpam-3126	163	8	m	m	NOUN
ejpam-3126	163	9	∈m	∈m	NOUN
ejpam-3126	163	10	such	such	ADJ
ejpam-3126	163	11	that	that	SCONJ
ejpam-3126	163	12	a1a2	a1a2	PROPN
ejpam-3126	163	13	m	m	NOUN
ejpam-3126	163	14	∈	∈	ADJ
ejpam-3126	163	15	n−φ(n	n−φ(n	NOUN
ejpam-3126	163	16	)	)	PUNCT
ejpam-3126	163	17	.	.	PUNCT
ejpam-3126	164	1	then	then	ADV
ejpam-3126	164	2	a1	a1	VERB
ejpam-3126	164	3	1	1	NUM
ejpam-3126	164	4	a2	a2	PROPN
ejpam-3126	164	5	1	1	NUM
ejpam-3126	164	6	m	m	NOUN
ejpam-3126	164	7	1	1	NUM
ejpam-3126	164	8	=	=	SYM
ejpam-3126	164	9	abm	abm	PROPN
ejpam-3126	164	10	1	1	NUM
ejpam-3126	164	11	∈	∈	PROPN
ejpam-3126	164	12	s−1n	s−1n	NOUN
ejpam-3126	164	13	.	.	PUNCT
ejpam-3126	165	1	if	if	SCONJ
ejpam-3126	165	2	a1	a1	PROPN
ejpam-3126	165	3	1	1	NUM
ejpam-3126	165	4	a2	a2	PROPN
ejpam-3126	165	5	1	1	NUM
ejpam-3126	165	6	m	m	NOUN
ejpam-3126	165	7	1	1	NUM
ejpam-3126	165	8	∈	∈	NOUN
ejpam-3126	165	9	φs(s−1n	φs(s−1n	NOUN
ejpam-3126	165	10	)	)	PUNCT
ejpam-3126	165	11	=	=	SYM
ejpam-3126	165	12	s−1φ(n	s−1φ(n	X
ejpam-3126	165	13	)	)	PUNCT
ejpam-3126	165	14	,	,	PUNCT
ejpam-3126	165	15	then	then	ADV
ejpam-3126	165	16	there	there	PRON
ejpam-3126	165	17	exists	exist	VERB
ejpam-3126	165	18	s	s	X
ejpam-3126	165	19	∈	∈	PROPN
ejpam-3126	165	20	s	s	VERB
ejpam-3126	165	21	such	such	ADJ
ejpam-3126	165	22	that	that	SCONJ
ejpam-3126	165	23	sa1a2	sa1a2	PROPN
ejpam-3126	165	24	m	m	PROPN
ejpam-3126	165	25	∈	∈	NOUN
ejpam-3126	165	26	φ(n	φ(n	NOUN
ejpam-3126	165	27	)	)	PUNCT
ejpam-3126	165	28	which	which	PRON
ejpam-3126	165	29	is	be	AUX
ejpam-3126	165	30	a	a	DET
ejpam-3126	165	31	contradiction	contradiction	NOUN
ejpam-3126	165	32	.	.	PUNCT
ejpam-3126	166	1	if	if	SCONJ
ejpam-3126	166	2	a1	a1	PROPN
ejpam-3126	166	3	1	1	NUM
ejpam-3126	166	4	a2	a2	PROPN
ejpam-3126	166	5	1	1	NUM
ejpam-3126	166	6	m	m	PROPN
ejpam-3126	166	7	1	1	NUM
ejpam-3126	166	8	6∈	6∈	NOUN
ejpam-3126	166	9	φs(s−1n	φs(s−1n	NOUN
ejpam-3126	166	10	)	)	PUNCT
ejpam-3126	166	11	,	,	PUNCT
ejpam-3126	166	12	then	then	ADV
ejpam-3126	166	13	a1	a1	VERB
ejpam-3126	166	14	1	1	NUM
ejpam-3126	166	15	a2	a2	PROPN
ejpam-3126	166	16	1	1	NUM
ejpam-3126	166	17	m	m	NOUN
ejpam-3126	166	18	1	1	NUM
ejpam-3126	166	19	∈	∈	NOUN
ejpam-3126	166	20	s	s	PART
ejpam-3126	166	21	−1n	−1n	NOUN
ejpam-3126	166	22	−	−	NOUN
ejpam-3126	166	23	φs(s−1n	φs(s−1n	NOUN
ejpam-3126	166	24	)	)	PUNCT
ejpam-3126	166	25	.	.	PUNCT
ejpam-3126	167	1	pai	pai	PROPN
ejpam-3126	167	2	.	.	PROPN
ejpam-3126	167	3	yiarayong	yiarayong	PROPN
ejpam-3126	167	4	,	,	PUNCT
ejpam-3126	167	5	m.	m.	NOUN
ejpam-3126	167	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	167	7	/	/	SYM
ejpam-3126	167	8	eur	eur	PROPN
ejpam-3126	167	9	.	.	PUNCT
ejpam-3126	168	1	j.	j.	PROPN
ejpam-3126	168	2	pure	pure	PROPN
ejpam-3126	168	3	appl	appl	PROPN
ejpam-3126	168	4	.	.	PROPN
ejpam-3126	168	5	math	math	PROPN
ejpam-3126	168	6	,	,	PUNCT
ejpam-3126	168	7	11	11	NUM
ejpam-3126	168	8	(	(	PUNCT
ejpam-3126	168	9	1	1	NUM
ejpam-3126	168	10	)	)	PUNCT
ejpam-3126	168	11	(	(	PUNCT
ejpam-3126	168	12	2018	2018	NUM
ejpam-3126	168	13	)	)	PUNCT
ejpam-3126	168	14	,	,	PUNCT
ejpam-3126	168	15	35	35	NUM
ejpam-3126	168	16	-	-	SYM
ejpam-3126	168	17	50	50	NUM
ejpam-3126	168	18	40	40	NUM
ejpam-3126	168	19	by	by	ADP
ejpam-3126	168	20	definition	definition	NOUN
ejpam-3126	168	21	1	1	NUM
ejpam-3126	168	22	,	,	PUNCT
ejpam-3126	168	23	a1	a1	NOUN
ejpam-3126	168	24	1	1	NUM
ejpam-3126	168	25	a2	a2	PROPN
ejpam-3126	168	26	1	1	NUM
ejpam-3126	168	27	∈	∈	NOUN
ejpam-3126	168	28	√	√	NUM
ejpam-3126	169	1	(	(	PUNCT
ejpam-3126	169	2	s−1n	s−1n	NOUN
ejpam-3126	169	3	:	:	PUNCT
ejpam-3126	169	4	s−1	s−1	PROPN
ejpam-3126	169	5	m	m	NOUN
ejpam-3126	169	6	)	)	PUNCT
ejpam-3126	169	7	or	or	CCONJ
ejpam-3126	169	8	a1	a1	VERB
ejpam-3126	169	9	1	1	NUM
ejpam-3126	169	10	m	m	NOUN
ejpam-3126	169	11	1	1	NUM
ejpam-3126	169	12	∈	∈	NOUN
ejpam-3126	169	13	s	s	PART
ejpam-3126	169	14	−1n	−1n	PROPN
ejpam-3126	169	15	or	or	CCONJ
ejpam-3126	169	16	(	(	PUNCT
ejpam-3126	169	17	a21	a21	NOUN
ejpam-3126	169	18	)	)	PUNCT
ejpam-3126	169	19	nm1	nm1	NOUN
ejpam-3126	170	1	∈	∈	NOUN
ejpam-3126	170	2	s	s	VERB
ejpam-3126	170	3	−1n	−1n	PROPN
ejpam-3126	170	4	for	for	ADP
ejpam-3126	170	5	some	some	DET
ejpam-3126	170	6	positive	positive	ADJ
ejpam-3126	170	7	integer	integer	NOUN
ejpam-3126	170	8	n.	n.	NOUN
ejpam-3126	170	9	if	if	SCONJ
ejpam-3126	170	10	a1	a1	PROPN
ejpam-3126	170	11	1	1	NUM
ejpam-3126	170	12	a2	a2	PROPN
ejpam-3126	170	13	1	1	NUM
ejpam-3126	170	14	∈	∈	NOUN
ejpam-3126	170	15	√	√	NUM
ejpam-3126	170	16	(	(	PUNCT
ejpam-3126	170	17	s−1n	s−1n	NOUN
ejpam-3126	170	18	:	:	PUNCT
ejpam-3126	170	19	s−1	s−1	PROPN
ejpam-3126	170	20	m	m	NOUN
ejpam-3126	170	21	)	)	PUNCT
ejpam-3126	170	22	,	,	PUNCT
ejpam-3126	170	23	then	then	ADV
ejpam-3126	170	24	(	(	PUNCT
ejpam-3126	170	25	a11	a11	PROPN
ejpam-3126	170	26	a2	a2	PROPN
ejpam-3126	170	27	1	1	NUM
ejpam-3126	170	28	)	)	PUNCT
ejpam-3126	170	29	n	n	X
ejpam-3126	170	30	∈	∈	NOUN
ejpam-3126	170	31	(	(	PUNCT
ejpam-3126	170	32	s−1n	s−1n	NOUN
ejpam-3126	170	33	:	:	PUNCT
ejpam-3126	170	34	s−1	s−1	PROPN
ejpam-3126	170	35	m	m	NOUN
ejpam-3126	170	36	)	)	PUNCT
ejpam-3126	170	37	for	for	ADP
ejpam-3126	170	38	some	some	DET
ejpam-3126	170	39	positive	positive	ADJ
ejpam-3126	170	40	integer	integer	NOUN
ejpam-3126	170	41	n.	n.	NOUN
ejpam-3126	170	42	thus	thus	ADV
ejpam-3126	170	43	there	there	PRON
ejpam-3126	170	44	exists	exist	VERB
ejpam-3126	170	45	s	s	X
ejpam-3126	170	46	∈	∈	NOUN
ejpam-3126	170	47	s	s	VERB
ejpam-3126	170	48	such	such	ADJ
ejpam-3126	170	49	that	that	DET
ejpam-3126	170	50	s(a1a2	s(a1a2	NOUN
ejpam-3126	170	51	)	)	PUNCT
ejpam-3126	170	52	nm	nm	NOUN
ejpam-3126	170	53	⊆	⊆	NUM
ejpam-3126	170	54	n	n	NOUN
ejpam-3126	170	55	for	for	ADP
ejpam-3126	170	56	some	some	DET
ejpam-3126	170	57	positive	positive	ADJ
ejpam-3126	170	58	integer	integer	NOUN
ejpam-3126	170	59	n.	n.	NOUN
ejpam-3126	170	60	by	by	ADP
ejpam-3126	170	61	assumption	assumption	NOUN
ejpam-3126	170	62	,	,	PUNCT
ejpam-3126	170	63	(	(	PUNCT
ejpam-3126	170	64	a1a2	a1a2	NOUN
ejpam-3126	170	65	)	)	PUNCT
ejpam-3126	170	66	nm	nm	ADV
ejpam-3126	171	1	⊆	⊆	NUM
ejpam-3126	171	2	n	n	NOUN
ejpam-3126	171	3	so	so	ADV
ejpam-3126	171	4	a1a2	a1a2	ADP
ejpam-3126	171	5	∈	∈	NOUN
ejpam-3126	171	6	√	√	NUM
ejpam-3126	171	7	(	(	PUNCT
ejpam-3126	171	8	n	n	NUM
ejpam-3126	171	9	:	:	PUNCT
ejpam-3126	171	10	m	m	X
ejpam-3126	171	11	)	)	PUNCT
ejpam-3126	171	12	.	.	PUNCT
ejpam-3126	172	1	in	in	ADP
ejpam-3126	172	2	view	view	NOUN
ejpam-3126	172	3	of	of	ADP
ejpam-3126	172	4	theorem	theorem	ADJ
ejpam-3126	172	5	4	4	NUM
ejpam-3126	172	6	and	and	CCONJ
ejpam-3126	172	7	theorem	theorem	VERB
ejpam-3126	172	8	5	5	NUM
ejpam-3126	172	9	,	,	PUNCT
ejpam-3126	172	10	we	we	PRON
ejpam-3126	172	11	have	have	VERB
ejpam-3126	172	12	the	the	DET
ejpam-3126	172	13	following	follow	VERB
ejpam-3126	172	14	result	result	NOUN
ejpam-3126	172	15	.	.	PUNCT
ejpam-3126	173	1	corollary	corollary	ADJ
ejpam-3126	173	2	2	2	NUM
ejpam-3126	173	3	.	.	PUNCT
ejpam-3126	174	1	let	let	VERB
ejpam-3126	174	2	s	s	PRON
ejpam-3126	174	3	be	be	AUX
ejpam-3126	174	4	a	a	DET
ejpam-3126	174	5	multiplicative	multiplicative	ADJ
ejpam-3126	174	6	closed	close	VERB
ejpam-3126	174	7	subset	subset	NOUN
ejpam-3126	174	8	of	of	ADP
ejpam-3126	174	9	r	r	NOUN
ejpam-3126	174	10	and	and	CCONJ
ejpam-3126	174	11	let	let	VERB
ejpam-3126	174	12	φ	φ	PROPN
ejpam-3126	174	13	:	:	PUNCT
ejpam-3126	174	14	s(m)→	s(m)→	NOUN
ejpam-3126	174	15	s(m)∪{∅	s(m)∪{∅	PROPN
ejpam-3126	174	16	}	}	PUNCT
ejpam-3126	174	17	be	be	AUX
ejpam-3126	174	18	a	a	DET
ejpam-3126	174	19	function	function	NOUN
ejpam-3126	174	20	with	with	ADP
ejpam-3126	174	21	s∩zd(n	s∩zd(n	NOUN
ejpam-3126	174	22	/	/	SYM
ejpam-3126	174	23	φ(n	φ(n	ADJ
ejpam-3126	174	24	)	)	PUNCT
ejpam-3126	174	25	)	)	PUNCT
ejpam-3126	175	1	=	=	NOUN
ejpam-3126	175	2	∅	∅	NOUN
ejpam-3126	175	3	and	and	CCONJ
ejpam-3126	175	4	s∩zd(m	s∩zd(m	NOUN
ejpam-3126	175	5	/	/	SYM
ejpam-3126	175	6	n	n	CCONJ
ejpam-3126	175	7	)	)	PUNCT
ejpam-3126	175	8	=	=	VERB
ejpam-3126	175	9	∅.	∅.	NOUN
ejpam-3126	175	10	then	then	ADV
ejpam-3126	175	11	n	n	PRON
ejpam-3126	175	12	is	be	AUX
ejpam-3126	175	13	a	a	DET
ejpam-3126	175	14	φ-2	φ-2	NOUN
ejpam-3126	175	15	-	-	PUNCT
ejpam-3126	175	16	absorbing	absorbing	ADJ
ejpam-3126	175	17	semi	semi	ADJ
ejpam-3126	175	18	-	-	ADJ
ejpam-3126	175	19	primary	primary	ADJ
ejpam-3126	175	20	submodule	submodule	NOUN
ejpam-3126	175	21	of	of	ADP
ejpam-3126	175	22	m	m	PROPN
ejpam-3126	175	23	if	if	SCONJ
ejpam-3126	175	24	and	and	CCONJ
ejpam-3126	175	25	only	only	ADV
ejpam-3126	175	26	if	if	SCONJ
ejpam-3126	175	27	s−1n	s−1n	NOUN
ejpam-3126	175	28	is	be	AUX
ejpam-3126	175	29	a	a	DET
ejpam-3126	175	30	φs-2	φs-2	NOUN
ejpam-3126	175	31	-	-	ADJ
ejpam-3126	175	32	absorbing	absorbing	ADJ
ejpam-3126	175	33	semi	semi	ADJ
ejpam-3126	175	34	-	-	ADJ
ejpam-3126	175	35	primary	primary	ADJ
ejpam-3126	175	36	submodule	submodule	NOUN
ejpam-3126	175	37	of	of	ADP
ejpam-3126	175	38	s−1	s−1	PROPN
ejpam-3126	175	39	m	m	NOUN
ejpam-3126	175	40	.	.	PUNCT
ejpam-3126	176	1	proof	proof	NOUN
ejpam-3126	176	2	.	.	PUNCT
ejpam-3126	177	1	the	the	DET
ejpam-3126	177	2	proof	proof	NOUN
ejpam-3126	177	3	follows	follow	VERB
ejpam-3126	177	4	from	from	ADP
ejpam-3126	177	5	theorem	theorem	ADJ
ejpam-3126	177	6	4	4	NUM
ejpam-3126	177	7	,	,	PUNCT
ejpam-3126	177	8	5	5	NUM
ejpam-3126	177	9	.	.	PUNCT
ejpam-3126	178	1	in	in	ADP
ejpam-3126	178	2	the	the	DET
ejpam-3126	178	3	following	follow	VERB
ejpam-3126	178	4	result	result	NOUN
ejpam-3126	178	5	,	,	PUNCT
ejpam-3126	178	6	we	we	PRON
ejpam-3126	178	7	give	give	VERB
ejpam-3126	178	8	an	an	DET
ejpam-3126	178	9	equivalent	equivalent	ADJ
ejpam-3126	178	10	definition	definition	NOUN
ejpam-3126	178	11	of	of	ADP
ejpam-3126	178	12	φ-2	φ-2	PROPN
ejpam-3126	178	13	-	-	PUNCT
ejpam-3126	178	14	absorbing	absorbing	ADJ
ejpam-3126	178	15	semi	semi	ADJ
ejpam-3126	178	16	-	-	ADJ
ejpam-3126	178	17	primary	primary	ADJ
ejpam-3126	178	18	submodules	submodule	NOUN
ejpam-3126	178	19	.	.	PUNCT
ejpam-3126	179	1	theorem	theorem	VERB
ejpam-3126	179	2	6	6	NUM
ejpam-3126	179	3	.	.	PUNCT
ejpam-3126	180	1	let	let	VERB
ejpam-3126	180	2	φ	φ	PROPN
ejpam-3126	180	3	:	:	PUNCT
ejpam-3126	180	4	s(m	s(m	PROPN
ejpam-3126	180	5	)	)	PUNCT
ejpam-3126	180	6	→	→	SYM
ejpam-3126	180	7	s(m	s(m	NOUN
ejpam-3126	180	8	)	)	PUNCT
ejpam-3126	180	9	∪	∪	NOUN
ejpam-3126	180	10	{	{	PUNCT
ejpam-3126	180	11	∅	∅	NOUN
ejpam-3126	180	12	}	}	PUNCT
ejpam-3126	180	13	be	be	AUX
ejpam-3126	180	14	a	a	DET
ejpam-3126	180	15	function	function	NOUN
ejpam-3126	180	16	.	.	PUNCT
ejpam-3126	181	1	the	the	DET
ejpam-3126	181	2	following	follow	VERB
ejpam-3126	181	3	conditions	condition	NOUN
ejpam-3126	181	4	are	be	AUX
ejpam-3126	181	5	equivalent	equivalent	ADJ
ejpam-3126	181	6	:	:	PUNCT
ejpam-3126	181	7	(	(	PUNCT
ejpam-3126	181	8	i	i	NOUN
ejpam-3126	181	9	)	)	PUNCT
ejpam-3126	181	10	n	n	PRON
ejpam-3126	181	11	is	be	AUX
ejpam-3126	181	12	a	a	DET
ejpam-3126	181	13	φ-2	φ-2	NOUN
ejpam-3126	181	14	-	-	PUNCT
ejpam-3126	181	15	absorbing	absorbing	ADJ
ejpam-3126	181	16	semi	semi	ADJ
ejpam-3126	181	17	-	-	ADJ
ejpam-3126	181	18	primary	primary	ADJ
ejpam-3126	181	19	submodule	submodule	NOUN
ejpam-3126	181	20	of	of	ADP
ejpam-3126	181	21	m	m	PROPN
ejpam-3126	181	22	.	.	PUNCT
ejpam-3126	182	1	(	(	PUNCT
ejpam-3126	182	2	ii	ii	NOUN
ejpam-3126	182	3	)	)	PUNCT
ejpam-3126	182	4	for	for	ADP
ejpam-3126	182	5	every	every	DET
ejpam-3126	182	6	a1	a1	NOUN
ejpam-3126	182	7	,	,	PUNCT
ejpam-3126	182	8	a2	a2	PROPN
ejpam-3126	182	9	∈	∈	PROPN
ejpam-3126	182	10	r−	r−	PROPN
ejpam-3126	182	11	(	(	PUNCT
ejpam-3126	182	12	n	n	NUM
ejpam-3126	182	13	:	:	PUNCT
ejpam-3126	182	14	m	m	X
ejpam-3126	182	15	)	)	PUNCT
ejpam-3126	182	16	if	if	SCONJ
ejpam-3126	182	17	a1a2	a1a2	PROPN
ejpam-3126	182	18	∈	∈	PROPN
ejpam-3126	182	19	r−	r−	PROPN
ejpam-3126	182	20	√	√	PROPN
ejpam-3126	182	21	(	(	PUNCT
ejpam-3126	182	22	n	n	NUM
ejpam-3126	182	23	:	:	PUNCT
ejpam-3126	182	24	m	m	X
ejpam-3126	182	25	)	)	PUNCT
ejpam-3126	182	26	,	,	PUNCT
ejpam-3126	182	27	then	then	ADV
ejpam-3126	182	28	(	(	PUNCT
ejpam-3126	182	29	n	n	X
ejpam-3126	182	30	:	:	PUNCT
ejpam-3126	182	31	a1a2	a1a2	ADJ
ejpam-3126	182	32	)	)	PUNCT
ejpam-3126	182	33	⊆	⊆	NUM
ejpam-3126	182	34	(	(	PUNCT
ejpam-3126	182	35	φ(n	φ(n	ADJ
ejpam-3126	182	36	)	)	PUNCT
ejpam-3126	182	37	:	:	PUNCT
ejpam-3126	182	38	a1a2	a1a2	ADJ
ejpam-3126	182	39	)	)	PUNCT
ejpam-3126	182	40	∪	∪	NOUN
ejpam-3126	182	41	(	(	PUNCT
ejpam-3126	182	42	n	n	NOUN
ejpam-3126	182	43	:	:	PUNCT
ejpam-3126	182	44	a1	a1	PROPN
ejpam-3126	182	45	)	)	PUNCT
ejpam-3126	182	46	∪	∪	NOUN
ejpam-3126	182	47	(	(	PUNCT
ejpam-3126	182	48	n	n	NOUN
ejpam-3126	182	49	:	:	PUNCT
ejpam-3126	182	50	an2	an2	PROPN
ejpam-3126	182	51	)	)	PUNCT
ejpam-3126	182	52	for	for	ADP
ejpam-3126	182	53	some	some	DET
ejpam-3126	182	54	positive	positive	ADJ
ejpam-3126	182	55	integer	integer	NOUN
ejpam-3126	182	56	n.	n.	NOUN
ejpam-3126	182	57	proof	proof	NOUN
ejpam-3126	182	58	.	.	PUNCT
ejpam-3126	183	1	(	(	PUNCT
ejpam-3126	183	2	i	i	PRON
ejpam-3126	183	3	⇒	⇒	AUX
ejpam-3126	183	4	ii	ii	PROPN
ejpam-3126	183	5	)	)	PUNCT
ejpam-3126	183	6	let	let	VERB
ejpam-3126	183	7	m	m	PRON
ejpam-3126	183	8	∈	∈	PROPN
ejpam-3126	183	9	(	(	PUNCT
ejpam-3126	183	10	n	n	NOUN
ejpam-3126	183	11	:	:	PUNCT
ejpam-3126	183	12	a1a2	a1a2	ADJ
ejpam-3126	183	13	)	)	PUNCT
ejpam-3126	183	14	.	.	PUNCT
ejpam-3126	184	1	then	then	ADV
ejpam-3126	184	2	a1a2	a1a2	ADP
ejpam-3126	184	3	m	m	VERB
ejpam-3126	184	4	∈	∈	ADJ
ejpam-3126	184	5	n	n	NOUN
ejpam-3126	184	6	.	.	PUNCT
ejpam-3126	185	1	if	if	SCONJ
ejpam-3126	185	2	a1a2	a1a2	ADP
ejpam-3126	185	3	m	m	VERB
ejpam-3126	185	4	∈	∈	NOUN
ejpam-3126	185	5	φ(n	φ(n	NOUN
ejpam-3126	185	6	)	)	PUNCT
ejpam-3126	185	7	,	,	PUNCT
ejpam-3126	185	8	then	then	ADV
ejpam-3126	185	9	m	m	VERB
ejpam-3126	185	10	∈	∈	NOUN
ejpam-3126	185	11	(	(	PUNCT
ejpam-3126	185	12	φ(n	φ(n	PROPN
ejpam-3126	185	13	)	)	PUNCT
ejpam-3126	185	14	:	:	PUNCT
ejpam-3126	185	15	a1a2	a1a2	ADJ
ejpam-3126	185	16	)	)	PUNCT
ejpam-3126	185	17	∪	∪	NOUN
ejpam-3126	185	18	(	(	PUNCT
ejpam-3126	185	19	n	n	NOUN
ejpam-3126	185	20	:	:	PUNCT
ejpam-3126	185	21	a1	a1	PROPN
ejpam-3126	185	22	)	)	PUNCT
ejpam-3126	185	23	∪	∪	NOUN
ejpam-3126	185	24	(	(	PUNCT
ejpam-3126	185	25	n	n	NOUN
ejpam-3126	185	26	:	:	PUNCT
ejpam-3126	185	27	an2	an2	PROPN
ejpam-3126	185	28	)	)	PUNCT
ejpam-3126	185	29	for	for	ADP
ejpam-3126	185	30	some	some	DET
ejpam-3126	185	31	positive	positive	ADJ
ejpam-3126	185	32	integer	integer	NOUN
ejpam-3126	185	33	n.	n.	NOUN
ejpam-3126	185	34	if	if	SCONJ
ejpam-3126	185	35	a1a2	a1a2	PROPN
ejpam-3126	185	36	m	m	VERB
ejpam-3126	185	37	6∈	6∈	NOUN
ejpam-3126	185	38	φ(n	φ(n	ADJ
ejpam-3126	185	39	)	)	PUNCT
ejpam-3126	185	40	,	,	PUNCT
ejpam-3126	185	41	then	then	ADV
ejpam-3126	185	42	a1a2	a1a2	ADP
ejpam-3126	185	43	m	m	VERB
ejpam-3126	185	44	∈	∈	ADJ
ejpam-3126	185	45	n	n	CCONJ
ejpam-3126	185	46	−	−	PROPN
ejpam-3126	185	47	φ(n	φ(n	NOUN
ejpam-3126	185	48	)	)	PUNCT
ejpam-3126	185	49	.	.	PUNCT
ejpam-3126	186	1	by	by	ADP
ejpam-3126	186	2	definition	definition	NOUN
ejpam-3126	186	3	1	1	NUM
ejpam-3126	186	4	,	,	PUNCT
ejpam-3126	186	5	a1a2	a1a2	PROPN
ejpam-3126	186	6	∈	∈	NOUN
ejpam-3126	186	7	√	√	NUM
ejpam-3126	186	8	(	(	PUNCT
ejpam-3126	186	9	n	n	NUM
ejpam-3126	186	10	:	:	PUNCT
ejpam-3126	186	11	m	m	X
ejpam-3126	186	12	)	)	PUNCT
ejpam-3126	186	13	or	or	CCONJ
ejpam-3126	186	14	a1	a1	NOUN
ejpam-3126	186	15	m	m	PROPN
ejpam-3126	186	16	∈	∈	NOUN
ejpam-3126	186	17	n	n	NOUN
ejpam-3126	186	18	or	or	CCONJ
ejpam-3126	186	19	an2	an2	PROPN
ejpam-3126	186	20	m	m	NOUN
ejpam-3126	186	21	∈	∈	PROPN
ejpam-3126	186	22	n	n	NOUN
ejpam-3126	186	23	for	for	ADP
ejpam-3126	186	24	some	some	DET
ejpam-3126	186	25	positive	positive	ADJ
ejpam-3126	186	26	integer	integer	NOUN
ejpam-3126	186	27	n.	n.	NOUN
ejpam-3126	186	28	by	by	ADP
ejpam-3126	186	29	assumption	assumption	NOUN
ejpam-3126	186	30	,	,	PUNCT
ejpam-3126	186	31	m	m	VERB
ejpam-3126	186	32	∈	∈	ADJ
ejpam-3126	186	33	(	(	PUNCT
ejpam-3126	186	34	n	n	NOUN
ejpam-3126	186	35	:	:	PUNCT
ejpam-3126	186	36	a1	a1	PROPN
ejpam-3126	186	37	)	)	PUNCT
ejpam-3126	186	38	or	or	CCONJ
ejpam-3126	186	39	m	m	PROPN
ejpam-3126	186	40	∈	∈	NOUN
ejpam-3126	186	41	(	(	PUNCT
ejpam-3126	186	42	n	n	NOUN
ejpam-3126	186	43	:	:	PUNCT
ejpam-3126	186	44	an2	an2	PROPN
ejpam-3126	186	45	)	)	PUNCT
ejpam-3126	186	46	for	for	ADP
ejpam-3126	186	47	some	some	DET
ejpam-3126	186	48	positive	positive	ADJ
ejpam-3126	186	49	integer	integer	NOUN
ejpam-3126	186	50	n.	n.	NOUN
ejpam-3126	186	51	therefore	therefore	ADV
ejpam-3126	186	52	(	(	PUNCT
ejpam-3126	186	53	n	n	NUM
ejpam-3126	186	54	:	:	PUNCT
ejpam-3126	186	55	a1a2	a1a2	ADJ
ejpam-3126	186	56	)	)	PUNCT
ejpam-3126	186	57	=	=	SYM
ejpam-3126	186	58	(	(	PUNCT
ejpam-3126	186	59	φ(n	φ(n	ADJ
ejpam-3126	186	60	)	)	PUNCT
ejpam-3126	186	61	:	:	PUNCT
ejpam-3126	186	62	a1a2	a1a2	ADJ
ejpam-3126	186	63	)	)	PUNCT
ejpam-3126	186	64	∪	∪	NOUN
ejpam-3126	186	65	(	(	PUNCT
ejpam-3126	186	66	n	n	NOUN
ejpam-3126	186	67	:	:	PUNCT
ejpam-3126	186	68	a1	a1	PROPN
ejpam-3126	186	69	)	)	PUNCT
ejpam-3126	186	70	∪	∪	NOUN
ejpam-3126	186	71	(	(	PUNCT
ejpam-3126	186	72	n	n	NOUN
ejpam-3126	186	73	:	:	PUNCT
ejpam-3126	186	74	an2	an2	PROPN
ejpam-3126	186	75	)	)	PUNCT
ejpam-3126	186	76	for	for	ADP
ejpam-3126	186	77	some	some	DET
ejpam-3126	186	78	positive	positive	ADJ
ejpam-3126	186	79	integer	integer	NOUN
ejpam-3126	186	80	n.	n.	NOUN
ejpam-3126	186	81	(	(	PUNCT
ejpam-3126	186	82	ii⇒	ii⇒	INTJ
ejpam-3126	186	83	i	i	NOUN
ejpam-3126	186	84	)	)	PUNCT
ejpam-3126	186	85	it	it	PRON
ejpam-3126	186	86	is	be	AUX
ejpam-3126	186	87	obvious	obvious	ADJ
ejpam-3126	186	88	.	.	PUNCT
ejpam-3126	187	1	corollary	corollary	ADJ
ejpam-3126	187	2	3	3	X
ejpam-3126	187	3	.	.	PUNCT
ejpam-3126	188	1	let	let	VERB
ejpam-3126	188	2	φ	φ	PROPN
ejpam-3126	188	3	:	:	PUNCT
ejpam-3126	188	4	s(m	s(m	PROPN
ejpam-3126	188	5	)	)	PUNCT
ejpam-3126	188	6	→	→	SYM
ejpam-3126	188	7	s(m	s(m	NOUN
ejpam-3126	188	8	)	)	PUNCT
ejpam-3126	188	9	∪	∪	NOUN
ejpam-3126	188	10	{	{	PUNCT
ejpam-3126	188	11	∅	∅	NOUN
ejpam-3126	188	12	}	}	PUNCT
ejpam-3126	188	13	be	be	AUX
ejpam-3126	188	14	a	a	DET
ejpam-3126	188	15	function	function	NOUN
ejpam-3126	188	16	.	.	PUNCT
ejpam-3126	189	1	the	the	DET
ejpam-3126	189	2	following	follow	VERB
ejpam-3126	189	3	conditions	condition	NOUN
ejpam-3126	189	4	are	be	AUX
ejpam-3126	189	5	equivalent	equivalent	ADJ
ejpam-3126	189	6	:	:	PUNCT
ejpam-3126	189	7	(	(	PUNCT
ejpam-3126	189	8	i	i	NOUN
ejpam-3126	189	9	)	)	PUNCT
ejpam-3126	189	10	n	n	PRON
ejpam-3126	189	11	is	be	AUX
ejpam-3126	189	12	a	a	DET
ejpam-3126	189	13	φ-2	φ-2	NOUN
ejpam-3126	189	14	-	-	PUNCT
ejpam-3126	189	15	absorbing	absorbing	ADJ
ejpam-3126	189	16	semi	semi	ADJ
ejpam-3126	189	17	-	-	ADJ
ejpam-3126	189	18	primary	primary	ADJ
ejpam-3126	189	19	submodule	submodule	NOUN
ejpam-3126	189	20	of	of	ADP
ejpam-3126	189	21	m	m	PROPN
ejpam-3126	189	22	.	.	PUNCT
ejpam-3126	190	1	(	(	PUNCT
ejpam-3126	190	2	ii	ii	NOUN
ejpam-3126	190	3	)	)	PUNCT
ejpam-3126	190	4	for	for	ADP
ejpam-3126	190	5	every	every	DET
ejpam-3126	190	6	a	a	DET
ejpam-3126	190	7	∈	∈	PROPN
ejpam-3126	190	8	r	r	NOUN
ejpam-3126	190	9	−	−	PROPN
ejpam-3126	190	10	(	(	PUNCT
ejpam-3126	190	11	n	n	NUM
ejpam-3126	190	12	:	:	PUNCT
ejpam-3126	190	13	m	m	X
ejpam-3126	190	14	)	)	PUNCT
ejpam-3126	190	15	and	and	CCONJ
ejpam-3126	190	16	every	every	DET
ejpam-3126	190	17	ideal	ideal	NOUN
ejpam-3126	190	18	i	i	PRON
ejpam-3126	190	19	of	of	ADP
ejpam-3126	190	20	r	r	NOUN
ejpam-3126	190	21	such	such	ADJ
ejpam-3126	190	22	that	that	SCONJ
ejpam-3126	190	23	i	i	PRON
ejpam-3126	190	24	6⊆	6⊆	NUM
ejpam-3126	190	25	(	(	PUNCT
ejpam-3126	190	26	n	n	NUM
ejpam-3126	190	27	:	:	PUNCT
ejpam-3126	190	28	m	m	X
ejpam-3126	190	29	)	)	PUNCT
ejpam-3126	190	30	if	if	SCONJ
ejpam-3126	190	31	ai	ai	VERB
ejpam-3126	190	32	6⊆	6⊆	NUM
ejpam-3126	190	33	√	√	NUM
ejpam-3126	190	34	(	(	PUNCT
ejpam-3126	190	35	n	n	NUM
ejpam-3126	190	36	:	:	PUNCT
ejpam-3126	190	37	m	m	X
ejpam-3126	190	38	)	)	PUNCT
ejpam-3126	190	39	,	,	PUNCT
ejpam-3126	190	40	then	then	ADV
ejpam-3126	190	41	(	(	PUNCT
ejpam-3126	190	42	n	n	X
ejpam-3126	190	43	:	:	PUNCT
ejpam-3126	190	44	ai	ai	VERB
ejpam-3126	190	45	)	)	PUNCT
ejpam-3126	190	46	⊆	⊆	NUM
ejpam-3126	190	47	(	(	PUNCT
ejpam-3126	190	48	φ(n	φ(n	PROPN
ejpam-3126	190	49	)	)	PUNCT
ejpam-3126	190	50	:	:	PUNCT
ejpam-3126	190	51	ai	ai	VERB
ejpam-3126	190	52	)	)	PUNCT
ejpam-3126	190	53	∪	∪	NOUN
ejpam-3126	190	54	(	(	PUNCT
ejpam-3126	190	55	n	n	NOUN
ejpam-3126	190	56	:	:	PUNCT
ejpam-3126	190	57	a	a	X
ejpam-3126	190	58	)	)	PUNCT
ejpam-3126	190	59	∪	∪	NOUN
ejpam-3126	190	60	(	(	PUNCT
ejpam-3126	190	61	n	n	NUM
ejpam-3126	190	62	:	:	PUNCT
ejpam-3126	190	63	in	in	ADP
ejpam-3126	190	64	)	)	PUNCT
ejpam-3126	190	65	for	for	ADP
ejpam-3126	190	66	some	some	DET
ejpam-3126	190	67	positive	positive	ADJ
ejpam-3126	190	68	integer	integer	NOUN
ejpam-3126	190	69	n.	n.	NOUN
ejpam-3126	190	70	(	(	PUNCT
ejpam-3126	190	71	iii	iii	NOUN
ejpam-3126	190	72	)	)	PUNCT
ejpam-3126	190	73	for	for	ADP
ejpam-3126	190	74	every	every	DET
ejpam-3126	190	75	ideals	ideal	NOUN
ejpam-3126	190	76	i	i	PRON
ejpam-3126	190	77	,	,	PUNCT
ejpam-3126	190	78	j	j	PROPN
ejpam-3126	190	79	of	of	ADP
ejpam-3126	190	80	r	r	NOUN
ejpam-3126	190	81	such	such	ADJ
ejpam-3126	190	82	that	that	SCONJ
ejpam-3126	190	83	i	i	PRON
ejpam-3126	190	84	,	,	PUNCT
ejpam-3126	190	85	j	j	PROPN
ejpam-3126	190	86	6⊆	6⊆	NUM
ejpam-3126	190	87	(	(	PUNCT
ejpam-3126	190	88	n	n	NUM
ejpam-3126	190	89	:	:	PUNCT
ejpam-3126	190	90	m	m	X
ejpam-3126	190	91	)	)	PUNCT
ejpam-3126	190	92	if	if	SCONJ
ejpam-3126	190	93	ij	ij	NUM
ejpam-3126	190	94	6⊆	6⊆	NUM
ejpam-3126	190	95	√	√	PROPN
ejpam-3126	190	96	(	(	PUNCT
ejpam-3126	190	97	n	n	NUM
ejpam-3126	190	98	:	:	PUNCT
ejpam-3126	190	99	m	m	X
ejpam-3126	190	100	)	)	PUNCT
ejpam-3126	190	101	,	,	PUNCT
ejpam-3126	190	102	then	then	ADV
ejpam-3126	190	103	(	(	PUNCT
ejpam-3126	190	104	n	n	X
ejpam-3126	190	105	:	:	PUNCT
ejpam-3126	190	106	ij	ij	X
ejpam-3126	190	107	)	)	PUNCT
ejpam-3126	190	108	⊆	⊆	NUM
ejpam-3126	190	109	(	(	PUNCT
ejpam-3126	190	110	φ(n	φ(n	ADJ
ejpam-3126	190	111	)	)	PUNCT
ejpam-3126	190	112	:	:	PUNCT
ejpam-3126	190	113	ij	ij	X
ejpam-3126	190	114	)	)	PUNCT
ejpam-3126	190	115	∪	∪	NOUN
ejpam-3126	190	116	(	(	PUNCT
ejpam-3126	190	117	n	n	NOUN
ejpam-3126	190	118	:	:	PUNCT
ejpam-3126	190	119	i	i	NOUN
ejpam-3126	190	120	)	)	PUNCT
ejpam-3126	190	121	∪	∪	VERB
ejpam-3126	190	122	(	(	PUNCT
ejpam-3126	190	123	n	n	NUM
ejpam-3126	190	124	:	:	PUNCT
ejpam-3126	190	125	jn	jn	PROPN
ejpam-3126	190	126	)	)	PUNCT
ejpam-3126	190	127	for	for	ADP
ejpam-3126	190	128	some	some	DET
ejpam-3126	190	129	positive	positive	ADJ
ejpam-3126	190	130	integer	integer	NOUN
ejpam-3126	190	131	n.	n.	NOUN
ejpam-3126	190	132	proof	proof	NOUN
ejpam-3126	190	133	.	.	PUNCT
ejpam-3126	191	1	the	the	DET
ejpam-3126	191	2	proof	proof	NOUN
ejpam-3126	191	3	is	be	AUX
ejpam-3126	191	4	similar	similar	ADJ
ejpam-3126	191	5	to	to	AUX
ejpam-3126	191	6	theorem	theorem	VERB
ejpam-3126	191	7	6	6	NUM
ejpam-3126	191	8	.	.	PUNCT
ejpam-3126	192	1	the	the	DET
ejpam-3126	192	2	following	follow	VERB
ejpam-3126	192	3	theorem	theorem	NOUN
ejpam-3126	192	4	offers	offer	VERB
ejpam-3126	192	5	a	a	DET
ejpam-3126	192	6	characterization	characterization	NOUN
ejpam-3126	192	7	of	of	ADP
ejpam-3126	192	8	φ-2	φ-2	PROPN
ejpam-3126	192	9	-	-	PUNCT
ejpam-3126	192	10	absorbing	absorbing	ADJ
ejpam-3126	192	11	semi	semi	ADJ
ejpam-3126	192	12	-	-	ADJ
ejpam-3126	192	13	primary	primary	ADJ
ejpam-3126	192	14	submodules	submodule	NOUN
ejpam-3126	192	15	.	.	PUNCT
ejpam-3126	193	1	pai	pai	PROPN
ejpam-3126	193	2	.	.	PROPN
ejpam-3126	193	3	yiarayong	yiarayong	PROPN
ejpam-3126	193	4	,	,	PUNCT
ejpam-3126	193	5	m.	m.	NOUN
ejpam-3126	193	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	193	7	/	/	SYM
ejpam-3126	193	8	eur	eur	PROPN
ejpam-3126	193	9	.	.	PUNCT
ejpam-3126	194	1	j.	j.	PROPN
ejpam-3126	194	2	pure	pure	PROPN
ejpam-3126	194	3	appl	appl	PROPN
ejpam-3126	194	4	.	.	PROPN
ejpam-3126	194	5	math	math	PROPN
ejpam-3126	194	6	,	,	PUNCT
ejpam-3126	194	7	11	11	NUM
ejpam-3126	194	8	(	(	PUNCT
ejpam-3126	194	9	1	1	NUM
ejpam-3126	194	10	)	)	PUNCT
ejpam-3126	194	11	(	(	PUNCT
ejpam-3126	194	12	2018	2018	NUM
ejpam-3126	194	13	)	)	PUNCT
ejpam-3126	194	14	,	,	PUNCT
ejpam-3126	194	15	35	35	NUM
ejpam-3126	194	16	-	-	SYM
ejpam-3126	194	17	50	50	NUM
ejpam-3126	194	18	41	41	NUM
ejpam-3126	194	19	theorem	theorem	NOUN
ejpam-3126	194	20	7	7	NUM
ejpam-3126	194	21	.	.	PUNCT
ejpam-3126	195	1	let	let	VERB
ejpam-3126	195	2	φ	φ	PROPN
ejpam-3126	195	3	:	:	PUNCT
ejpam-3126	195	4	s(m	s(m	PROPN
ejpam-3126	195	5	)	)	PUNCT
ejpam-3126	195	6	→	→	SYM
ejpam-3126	195	7	s(m	s(m	NOUN
ejpam-3126	195	8	)	)	PUNCT
ejpam-3126	195	9	∪	∪	NOUN
ejpam-3126	195	10	{	{	PUNCT
ejpam-3126	195	11	∅	∅	NOUN
ejpam-3126	195	12	}	}	PUNCT
ejpam-3126	195	13	be	be	AUX
ejpam-3126	195	14	a	a	DET
ejpam-3126	195	15	function	function	NOUN
ejpam-3126	195	16	.	.	PUNCT
ejpam-3126	196	1	the	the	DET
ejpam-3126	196	2	following	follow	VERB
ejpam-3126	196	3	conditions	condition	NOUN
ejpam-3126	196	4	are	be	AUX
ejpam-3126	196	5	equivalent	equivalent	ADJ
ejpam-3126	196	6	:	:	PUNCT
ejpam-3126	196	7	(	(	PUNCT
ejpam-3126	196	8	i	i	NOUN
ejpam-3126	196	9	)	)	PUNCT
ejpam-3126	196	10	n	n	PRON
ejpam-3126	196	11	is	be	AUX
ejpam-3126	196	12	a	a	DET
ejpam-3126	196	13	φ-2	φ-2	NOUN
ejpam-3126	196	14	-	-	PUNCT
ejpam-3126	196	15	absorbing	absorbing	ADJ
ejpam-3126	196	16	semi	semi	ADJ
ejpam-3126	196	17	-	-	ADJ
ejpam-3126	196	18	primary	primary	ADJ
ejpam-3126	196	19	submodule	submodule	NOUN
ejpam-3126	196	20	of	of	ADP
ejpam-3126	196	21	m	m	PROPN
ejpam-3126	196	22	.	.	PUNCT
ejpam-3126	197	1	(	(	PUNCT
ejpam-3126	197	2	ii	ii	NOUN
ejpam-3126	197	3	)	)	PUNCT
ejpam-3126	197	4	for	for	ADP
ejpam-3126	197	5	every	every	DET
ejpam-3126	197	6	a	a	DET
ejpam-3126	197	7	∈	∈	PROPN
ejpam-3126	197	8	r	r	NOUN
ejpam-3126	197	9	−	−	PROPN
ejpam-3126	197	10	(	(	PUNCT
ejpam-3126	197	11	n	n	NUM
ejpam-3126	197	12	:	:	PUNCT
ejpam-3126	197	13	m	m	X
ejpam-3126	197	14	)	)	PUNCT
ejpam-3126	197	15	and	and	CCONJ
ejpam-3126	198	1	m	m	PROPN
ejpam-3126	198	2	∈	∈	ADJ
ejpam-3126	198	3	m	m	VERB
ejpam-3126	198	4	if	if	SCONJ
ejpam-3126	198	5	am	be	AUX
ejpam-3126	198	6	6∈	6∈	NUM
ejpam-3126	198	7	n	n	NOUN
ejpam-3126	198	8	,	,	PUNCT
ejpam-3126	198	9	then	then	ADV
ejpam-3126	198	10	(	(	PUNCT
ejpam-3126	198	11	n	n	X
ejpam-3126	198	12	:	:	PUNCT
ejpam-3126	198	13	am	be	AUX
ejpam-3126	198	14	)	)	PUNCT
ejpam-3126	198	15	⊆	⊆	X
ejpam-3126	198	16	(	(	PUNCT
ejpam-3126	198	17	φ(n	φ(n	PROPN
ejpam-3126	198	18	)	)	PUNCT
ejpam-3126	198	19	:	:	PUNCT
ejpam-3126	198	20	am	be	AUX
ejpam-3126	198	21	)	)	PUNCT
ejpam-3126	198	22	∪	∪	ADV
ejpam-3126	198	23	(	(	PUNCT
ejpam-3126	198	24	√	√	NUM
ejpam-3126	198	25	(	(	PUNCT
ejpam-3126	198	26	(	(	PUNCT
ejpam-3126	198	27	n	n	NUM
ejpam-3126	198	28	:	:	PUNCT
ejpam-3126	198	29	m	m	X
ejpam-3126	198	30	)	)	PUNCT
ejpam-3126	198	31	:	:	PUNCT
ejpam-3126	198	32	a	a	X
ejpam-3126	198	33	)	)	PUNCT
ejpam-3126	198	34	∪	∪	ADP
ejpam-3126	198	35	√	√	NUM
ejpam-3126	198	36	(	(	PUNCT
ejpam-3126	198	37	n	n	NUM
ejpam-3126	198	38	:	:	PUNCT
ejpam-3126	198	39	m	m	X
ejpam-3126	198	40	)	)	PUNCT
ejpam-3126	198	41	.	.	PUNCT
ejpam-3126	199	1	proof	proof	NOUN
ejpam-3126	199	2	.	.	PUNCT
ejpam-3126	200	1	(	(	PUNCT
ejpam-3126	200	2	i	i	PRON
ejpam-3126	200	3	⇒	⇒	AUX
ejpam-3126	200	4	ii	ii	PROPN
ejpam-3126	200	5	)	)	PUNCT
ejpam-3126	200	6	let	let	VERB
ejpam-3126	200	7	a	a	DET
ejpam-3126	200	8	∈	∈	NOUN
ejpam-3126	200	9	r	r	NOUN
ejpam-3126	200	10	−	−	PROPN
ejpam-3126	200	11	(	(	PUNCT
ejpam-3126	200	12	n	n	NUM
ejpam-3126	200	13	:	:	PUNCT
ejpam-3126	200	14	m	m	X
ejpam-3126	200	15	)	)	PUNCT
ejpam-3126	200	16	and	and	CCONJ
ejpam-3126	200	17	m	m	PROPN
ejpam-3126	200	18	∈	∈	NOUN
ejpam-3126	200	19	m	m	VERB
ejpam-3126	200	20	such	such	ADJ
ejpam-3126	200	21	that	that	PRON
ejpam-3126	200	22	am	be	AUX
ejpam-3126	200	23	6∈	6∈	NOUN
ejpam-3126	200	24	n	n	PROPN
ejpam-3126	200	25	.	.	PUNCT
ejpam-3126	201	1	assume	assume	VERB
ejpam-3126	201	2	that	that	SCONJ
ejpam-3126	201	3	r	r	NOUN
ejpam-3126	201	4	∈	∈	PROPN
ejpam-3126	201	5	(	(	PUNCT
ejpam-3126	201	6	n	n	NUM
ejpam-3126	201	7	:	:	PUNCT
ejpam-3126	201	8	am	be	AUX
ejpam-3126	201	9	)	)	PUNCT
ejpam-3126	201	10	.	.	PUNCT
ejpam-3126	202	1	if	if	SCONJ
ejpam-3126	202	2	ram	ram	VERB
ejpam-3126	202	3	6∈	6∈	NOUN
ejpam-3126	202	4	φ(n	φ(n	NOUN
ejpam-3126	202	5	)	)	PUNCT
ejpam-3126	202	6	,	,	PUNCT
ejpam-3126	202	7	then	then	ADV
ejpam-3126	202	8	ram	ram	VERB
ejpam-3126	202	9	∈	∈	PROPN
ejpam-3126	202	10	n	n	PRON
ejpam-3126	202	11	−	−	PROPN
ejpam-3126	202	12	φ(n	φ(n	NOUN
ejpam-3126	202	13	)	)	PUNCT
ejpam-3126	202	14	.	.	PUNCT
ejpam-3126	203	1	by	by	ADP
ejpam-3126	203	2	definition	definition	NOUN
ejpam-3126	203	3	1	1	NUM
ejpam-3126	203	4	,	,	PUNCT
ejpam-3126	203	5	ar	ar	NOUN
ejpam-3126	203	6	∈	∈	PROPN
ejpam-3126	203	7	√	√	NUM
ejpam-3126	203	8	(	(	PUNCT
ejpam-3126	203	9	n	n	NUM
ejpam-3126	203	10	:	:	PUNCT
ejpam-3126	203	11	m	m	X
ejpam-3126	203	12	)	)	PUNCT
ejpam-3126	203	13	or	or	CCONJ
ejpam-3126	203	14	am	be	AUX
ejpam-3126	203	15	∈	∈	PROPN
ejpam-3126	203	16	n	n	PRON
ejpam-3126	203	17	or	or	CCONJ
ejpam-3126	203	18	rnm	rnm	NOUN
ejpam-3126	203	19	∈	∈	PROPN
ejpam-3126	203	20	n	n	NOUN
ejpam-3126	203	21	for	for	ADP
ejpam-3126	203	22	some	some	DET
ejpam-3126	203	23	positive	positive	ADJ
ejpam-3126	203	24	integer	integer	NOUN
ejpam-3126	203	25	n.	n.	NOUN
ejpam-3126	203	26	by	by	ADP
ejpam-3126	203	27	assumption	assumption	NOUN
ejpam-3126	203	28	,	,	PUNCT
ejpam-3126	203	29	r	r	NOUN
ejpam-3126	203	30	∈	∈	PROPN
ejpam-3126	203	31	(	(	PUNCT
ejpam-3126	203	32	√	√	NUM
ejpam-3126	203	33	(	(	PUNCT
ejpam-3126	203	34	n	n	NUM
ejpam-3126	203	35	:	:	PUNCT
ejpam-3126	203	36	m	m	X
ejpam-3126	203	37	)	)	PUNCT
ejpam-3126	203	38	:	:	PUNCT
ejpam-3126	203	39	a	a	X
ejpam-3126	203	40	)	)	PUNCT
ejpam-3126	203	41	∪	∪	ADP
ejpam-3126	203	42	√	√	NUM
ejpam-3126	203	43	(	(	PUNCT
ejpam-3126	203	44	n	n	NUM
ejpam-3126	203	45	:	:	PUNCT
ejpam-3126	203	46	m	m	X
ejpam-3126	203	47	)	)	PUNCT
ejpam-3126	203	48	⊆	⊆	X
ejpam-3126	203	49	(	(	PUNCT
ejpam-3126	203	50	φ(n	φ(n	PROPN
ejpam-3126	203	51	)	)	PUNCT
ejpam-3126	203	52	:	:	PUNCT
ejpam-3126	203	53	am	be	AUX
ejpam-3126	203	54	)	)	PUNCT
ejpam-3126	203	55	∪	∪	ADV
ejpam-3126	203	56	(	(	PUNCT
ejpam-3126	203	57	√	√	NUM
ejpam-3126	203	58	(	(	PUNCT
ejpam-3126	203	59	(	(	PUNCT
ejpam-3126	203	60	n	n	NUM
ejpam-3126	203	61	:	:	PUNCT
ejpam-3126	203	62	m	m	X
ejpam-3126	203	63	)	)	PUNCT
ejpam-3126	203	64	:	:	PUNCT
ejpam-3126	203	65	a	a	X
ejpam-3126	203	66	)	)	PUNCT
ejpam-3126	203	67	∪	∪	ADP
ejpam-3126	203	68	√	√	NUM
ejpam-3126	203	69	(	(	PUNCT
ejpam-3126	203	70	n	n	NUM
ejpam-3126	203	71	:	:	PUNCT
ejpam-3126	203	72	m	m	X
ejpam-3126	203	73	)	)	PUNCT
ejpam-3126	203	74	.	.	PUNCT
ejpam-3126	204	1	now	now	ADV
ejpam-3126	204	2	if	if	SCONJ
ejpam-3126	204	3	ram	ram	VERB
ejpam-3126	204	4	∈	∈	NOUN
ejpam-3126	204	5	φ(n	φ(n	NOUN
ejpam-3126	204	6	)	)	PUNCT
ejpam-3126	204	7	,	,	PUNCT
ejpam-3126	204	8	then	then	ADV
ejpam-3126	204	9	r	r	NOUN
ejpam-3126	204	10	∈	∈	PROPN
ejpam-3126	204	11	(	(	PUNCT
ejpam-3126	204	12	φ(n	φ(n	PROPN
ejpam-3126	204	13	)	)	PUNCT
ejpam-3126	204	14	:	:	PUNCT
ejpam-3126	204	15	am	be	AUX
ejpam-3126	204	16	)	)	PUNCT
ejpam-3126	204	17	⊆	⊆	X
ejpam-3126	204	18	(	(	PUNCT
ejpam-3126	204	19	φ(n	φ(n	PROPN
ejpam-3126	204	20	)	)	PUNCT
ejpam-3126	204	21	:	:	PUNCT
ejpam-3126	204	22	am	be	AUX
ejpam-3126	204	23	)	)	PUNCT
ejpam-3126	204	24	∪	∪	ADV
ejpam-3126	204	25	(	(	PUNCT
ejpam-3126	204	26	√	√	NUM
ejpam-3126	204	27	(	(	PUNCT
ejpam-3126	204	28	(	(	PUNCT
ejpam-3126	204	29	n	n	NUM
ejpam-3126	204	30	:	:	PUNCT
ejpam-3126	204	31	m	m	X
ejpam-3126	204	32	)	)	PUNCT
ejpam-3126	204	33	:	:	PUNCT
ejpam-3126	204	34	a	a	X
ejpam-3126	204	35	)	)	PUNCT
ejpam-3126	204	36	∪	∪	ADP
ejpam-3126	204	37	√	√	NUM
ejpam-3126	204	38	(	(	PUNCT
ejpam-3126	204	39	n	n	NUM
ejpam-3126	204	40	:	:	PUNCT
ejpam-3126	204	41	m	m	X
ejpam-3126	204	42	)	)	PUNCT
ejpam-3126	204	43	.	.	PUNCT
ejpam-3126	205	1	(	(	PUNCT
ejpam-3126	205	2	ii⇒	ii⇒	INTJ
ejpam-3126	205	3	i	i	NOUN
ejpam-3126	205	4	)	)	PUNCT
ejpam-3126	206	1	it	it	PRON
ejpam-3126	206	2	is	be	AUX
ejpam-3126	206	3	obvious	obvious	ADJ
ejpam-3126	206	4	.	.	PUNCT
ejpam-3126	207	1	corollary	corollary	ADJ
ejpam-3126	207	2	4	4	NUM
ejpam-3126	207	3	.	.	PUNCT
ejpam-3126	208	1	let	let	VERB
ejpam-3126	208	2	φ	φ	PROPN
ejpam-3126	208	3	:	:	PUNCT
ejpam-3126	208	4	s(m	s(m	PROPN
ejpam-3126	208	5	)	)	PUNCT
ejpam-3126	208	6	→	→	SYM
ejpam-3126	208	7	s(m	s(m	NOUN
ejpam-3126	208	8	)	)	PUNCT
ejpam-3126	208	9	∪	∪	NOUN
ejpam-3126	208	10	{	{	PUNCT
ejpam-3126	208	11	∅	∅	NOUN
ejpam-3126	208	12	}	}	PUNCT
ejpam-3126	208	13	be	be	AUX
ejpam-3126	208	14	a	a	DET
ejpam-3126	208	15	function	function	NOUN
ejpam-3126	208	16	.	.	PUNCT
ejpam-3126	209	1	the	the	DET
ejpam-3126	209	2	following	follow	VERB
ejpam-3126	209	3	conditions	condition	NOUN
ejpam-3126	209	4	are	be	AUX
ejpam-3126	209	5	equivalent	equivalent	ADJ
ejpam-3126	209	6	:	:	PUNCT
ejpam-3126	209	7	(	(	PUNCT
ejpam-3126	209	8	i	i	NOUN
ejpam-3126	209	9	)	)	PUNCT
ejpam-3126	209	10	n	n	PRON
ejpam-3126	209	11	is	be	AUX
ejpam-3126	209	12	a	a	DET
ejpam-3126	209	13	φ-2	φ-2	NOUN
ejpam-3126	209	14	-	-	PUNCT
ejpam-3126	209	15	absorbing	absorbing	ADJ
ejpam-3126	209	16	semi	semi	ADJ
ejpam-3126	209	17	-	-	ADJ
ejpam-3126	209	18	primary	primary	ADJ
ejpam-3126	209	19	submodule	submodule	NOUN
ejpam-3126	209	20	of	of	ADP
ejpam-3126	209	21	m	m	PROPN
ejpam-3126	209	22	.	.	PUNCT
ejpam-3126	210	1	(	(	PUNCT
ejpam-3126	210	2	ii	ii	NOUN
ejpam-3126	210	3	)	)	PUNCT
ejpam-3126	210	4	for	for	ADP
ejpam-3126	210	5	every	every	DET
ejpam-3126	210	6	ideal	ideal	NOUN
ejpam-3126	210	7	i	i	PRON
ejpam-3126	210	8	of	of	ADP
ejpam-3126	210	9	r	r	NOUN
ejpam-3126	211	1	such	such	ADJ
ejpam-3126	211	2	that	that	SCONJ
ejpam-3126	211	3	i	i	PRON
ejpam-3126	211	4	⊆	⊆	NUM
ejpam-3126	211	5	r	r	NOUN
ejpam-3126	211	6	−	−	PROPN
ejpam-3126	211	7	(	(	PUNCT
ejpam-3126	211	8	n	n	NUM
ejpam-3126	211	9	:	:	PUNCT
ejpam-3126	211	10	m	m	X
ejpam-3126	211	11	)	)	PUNCT
ejpam-3126	211	12	and	and	CCONJ
ejpam-3126	211	13	m	m	PROPN
ejpam-3126	211	14	∈	∈	ADJ
ejpam-3126	211	15	m	m	VERB
ejpam-3126	211	16	if	if	SCONJ
ejpam-3126	211	17	i	i	PRON
ejpam-3126	211	18	m	m	VERB
ejpam-3126	211	19	6⊆	6⊆	NUM
ejpam-3126	211	20	n	n	PROPN
ejpam-3126	211	21	,	,	PUNCT
ejpam-3126	211	22	then	then	ADV
ejpam-3126	211	23	(	(	PUNCT
ejpam-3126	211	24	n	n	X
ejpam-3126	211	25	:	:	PUNCT
ejpam-3126	211	26	i	i	PRON
ejpam-3126	211	27	m	m	VERB
ejpam-3126	211	28	)	)	PUNCT
ejpam-3126	212	1	⊆	⊆	NUM
ejpam-3126	212	2	(	(	PUNCT
ejpam-3126	212	3	φ(n	φ(n	PROPN
ejpam-3126	212	4	)	)	PUNCT
ejpam-3126	212	5	:	:	PUNCT
ejpam-3126	213	1	i	i	PRON
ejpam-3126	213	2	m	m	VERB
ejpam-3126	213	3	)	)	PUNCT
ejpam-3126	213	4	∪	∪	ADV
ejpam-3126	213	5	(	(	PUNCT
ejpam-3126	213	6	√	√	NUM
ejpam-3126	213	7	(	(	PUNCT
ejpam-3126	213	8	n	n	NUM
ejpam-3126	213	9	:	:	PUNCT
ejpam-3126	213	10	m	m	X
ejpam-3126	213	11	)	)	PUNCT
ejpam-3126	213	12	:	:	PUNCT
ejpam-3126	213	13	i	i	NOUN
ejpam-3126	213	14	)	)	PUNCT
ejpam-3126	213	15	∪	∪	ADP
ejpam-3126	213	16	√	√	PROPN
ejpam-3126	213	17	(	(	PUNCT
ejpam-3126	213	18	n	n	NUM
ejpam-3126	213	19	:	:	PUNCT
ejpam-3126	213	20	m	m	X
ejpam-3126	213	21	)	)	PUNCT
ejpam-3126	213	22	.	.	PUNCT
ejpam-3126	214	1	proof	proof	NOUN
ejpam-3126	214	2	.	.	PUNCT
ejpam-3126	215	1	the	the	DET
ejpam-3126	215	2	proof	proof	NOUN
ejpam-3126	215	3	is	be	AUX
ejpam-3126	215	4	similar	similar	ADJ
ejpam-3126	215	5	to	to	AUX
ejpam-3126	215	6	theorem	theorem	VERB
ejpam-3126	215	7	7	7	NUM
ejpam-3126	215	8	.	.	NOUN
ejpam-3126	215	9	3	3	NUM
ejpam-3126	215	10	.	.	PUNCT
ejpam-3126	215	11	properties	property	NOUN
ejpam-3126	215	12	of	of	ADP
ejpam-3126	215	13	φα-2	φα-2	NOUN
ejpam-3126	215	14	-	-	PUNCT
ejpam-3126	215	15	absorbing	absorbing	ADJ
ejpam-3126	215	16	semi	semi	ADJ
ejpam-3126	215	17	-	-	ADJ
ejpam-3126	215	18	primary	primary	ADJ
ejpam-3126	215	19	submodules	submodule	NOUN
ejpam-3126	215	20	we	we	PRON
ejpam-3126	215	21	start	start	VERB
ejpam-3126	215	22	with	with	ADP
ejpam-3126	215	23	the	the	DET
ejpam-3126	215	24	following	follow	VERB
ejpam-3126	215	25	theorem	theorem	NOUN
ejpam-3126	215	26	that	that	PRON
ejpam-3126	215	27	gives	give	VERB
ejpam-3126	215	28	a	a	DET
ejpam-3126	215	29	relation	relation	NOUN
ejpam-3126	215	30	between	between	ADP
ejpam-3126	215	31	φα-2	φα-2	NOUN
ejpam-3126	215	32	-	-	PUNCT
ejpam-3126	215	33	absorbing	absorb	VERB
ejpam-3126	215	34	semiprimary	semiprimary	NOUN
ejpam-3126	215	35	and	and	CCONJ
ejpam-3126	215	36	φ-2	φ-2	NOUN
ejpam-3126	215	37	-	-	PUNCT
ejpam-3126	215	38	absorbing	absorbing	ADJ
ejpam-3126	215	39	semi	semi	ADJ
ejpam-3126	215	40	-	-	ADJ
ejpam-3126	215	41	primary	primary	ADJ
ejpam-3126	215	42	submodule	submodule	NOUN
ejpam-3126	215	43	.	.	PUNCT
ejpam-3126	216	1	our	our	PRON
ejpam-3126	216	2	starting	start	VERB
ejpam-3126	216	3	points	point	NOUN
ejpam-3126	216	4	are	be	AUX
ejpam-3126	216	5	the	the	DET
ejpam-3126	216	6	following	follow	VERB
ejpam-3126	216	7	definitions	definition	NOUN
ejpam-3126	216	8	:	:	PUNCT
ejpam-3126	216	9	definition	definition	NOUN
ejpam-3126	216	10	2	2	NUM
ejpam-3126	216	11	.	.	PUNCT
ejpam-3126	217	1	let	let	VERB
ejpam-3126	217	2	m	m	PRON
ejpam-3126	217	3	be	be	AUX
ejpam-3126	217	4	an	an	DET
ejpam-3126	217	5	r	r	NOUN
ejpam-3126	217	6	-	-	PUNCT
ejpam-3126	217	7	module	module	NOUN
ejpam-3126	217	8	and	and	CCONJ
ejpam-3126	217	9	let	let	VERB
ejpam-3126	217	10	s(m	s(m	PROPN
ejpam-3126	217	11	)	)	PUNCT
ejpam-3126	217	12	be	be	VERB
ejpam-3126	217	13	the	the	DET
ejpam-3126	217	14	set	set	NOUN
ejpam-3126	217	15	of	of	ADP
ejpam-3126	217	16	all	all	DET
ejpam-3126	217	17	submodules	submodule	NOUN
ejpam-3126	217	18	of	of	ADP
ejpam-3126	217	19	m	m	PROPN
ejpam-3126	217	20	.	.	PUNCT
ejpam-3126	218	1	define	define	VERB
ejpam-3126	218	2	the	the	DET
ejpam-3126	218	3	following	follow	VERB
ejpam-3126	218	4	functions	function	NOUN
ejpam-3126	218	5	φα	φα	X
ejpam-3126	218	6	:	:	PUNCT
ejpam-3126	218	7	s(m	s(m	PROPN
ejpam-3126	218	8	)	)	PUNCT
ejpam-3126	218	9	→	→	SYM
ejpam-3126	218	10	s(m	s(m	NOUN
ejpam-3126	218	11	)	)	PUNCT
ejpam-3126	218	12	∪	∪	ADP
ejpam-3126	218	13	{	{	PUNCT
ejpam-3126	218	14	∅	∅	NOUN
ejpam-3126	218	15	}	}	PUNCT
ejpam-3126	218	16	and	and	CCONJ
ejpam-3126	218	17	the	the	DET
ejpam-3126	218	18	corresponding	corresponding	ADJ
ejpam-3126	218	19	φα-2absorbing	φα-2absorbing	ADJ
ejpam-3126	218	20	semi	semi	ADJ
ejpam-3126	218	21	-	-	ADJ
ejpam-3126	218	22	primary	primary	ADJ
ejpam-3126	218	23	submodules	submodule	NOUN
ejpam-3126	218	24	:	:	PUNCT
ejpam-3126	218	25	•	•	ADP
ejpam-3126	218	26	if	if	SCONJ
ejpam-3126	218	27	φ(n	φ(n	ADJ
ejpam-3126	218	28	)	)	PUNCT
ejpam-3126	218	29	=	=	NOUN
ejpam-3126	218	30	∅	∅	NOUN
ejpam-3126	218	31	for	for	ADP
ejpam-3126	218	32	every	every	DET
ejpam-3126	218	33	n	n	PRON
ejpam-3126	218	34	∈	∈	PROPN
ejpam-3126	218	35	s(m	s(m	PROPN
ejpam-3126	218	36	)	)	PUNCT
ejpam-3126	218	37	,	,	PUNCT
ejpam-3126	218	38	then	then	ADV
ejpam-3126	218	39	we	we	PRON
ejpam-3126	218	40	say	say	VERB
ejpam-3126	218	41	that	that	SCONJ
ejpam-3126	218	42	φ	φ	PROPN
ejpam-3126	218	43	=	=	SYM
ejpam-3126	218	44	φ∅	φ∅	NOUN
ejpam-3126	218	45	and	and	CCONJ
ejpam-3126	218	46	n	n	ADV
ejpam-3126	218	47	is	be	AUX
ejpam-3126	218	48	called	call	VERB
ejpam-3126	218	49	a	a	DET
ejpam-3126	218	50	φ∅-2absorbing	φ∅-2absorbing	NOUN
ejpam-3126	218	51	semi	semi	ADJ
ejpam-3126	218	52	-	-	ADJ
ejpam-3126	218	53	primary	primary	ADJ
ejpam-3126	218	54	submodule	submodule	NOUN
ejpam-3126	218	55	of	of	ADP
ejpam-3126	218	56	m	m	PROPN
ejpam-3126	218	57	,	,	PUNCT
ejpam-3126	218	58	and	and	CCONJ
ejpam-3126	218	59	hence	hence	ADV
ejpam-3126	218	60	n	n	PRON
ejpam-3126	218	61	is	be	AUX
ejpam-3126	218	62	a	a	DET
ejpam-3126	218	63	2	2	NUM
ejpam-3126	218	64	-	-	PUNCT
ejpam-3126	218	65	absorbing	absorbing	ADJ
ejpam-3126	218	66	semi	semi	ADJ
ejpam-3126	218	67	-	-	ADJ
ejpam-3126	218	68	primary	primary	ADJ
ejpam-3126	218	69	submodule	submodule	NOUN
ejpam-3126	218	70	of	of	ADP
ejpam-3126	218	71	m	m	PROPN
ejpam-3126	218	72	.	.	PUNCT
ejpam-3126	219	1	•	•	INTJ
ejpam-3126	219	2	if	if	SCONJ
ejpam-3126	219	3	φ(n	φ(n	ADJ
ejpam-3126	219	4	)	)	PUNCT
ejpam-3126	219	5	=	=	PRON
ejpam-3126	219	6	{	{	PUNCT
ejpam-3126	219	7	0	0	NUM
ejpam-3126	219	8	}	}	PUNCT
ejpam-3126	219	9	for	for	ADP
ejpam-3126	219	10	every	every	DET
ejpam-3126	219	11	n	n	PRON
ejpam-3126	219	12	∈	∈	PROPN
ejpam-3126	219	13	s(m	s(m	PROPN
ejpam-3126	219	14	)	)	PUNCT
ejpam-3126	219	15	,	,	PUNCT
ejpam-3126	219	16	then	then	ADV
ejpam-3126	219	17	we	we	PRON
ejpam-3126	219	18	say	say	VERB
ejpam-3126	219	19	that	that	SCONJ
ejpam-3126	219	20	φ	φ	PROPN
ejpam-3126	219	21	=	=	SYM
ejpam-3126	219	22	φ0	φ0	PROPN
ejpam-3126	219	23	and	and	CCONJ
ejpam-3126	219	24	n	n	PROPN
ejpam-3126	219	25	is	be	AUX
ejpam-3126	219	26	called	call	VERB
ejpam-3126	219	27	a	a	DET
ejpam-3126	219	28	φ0	φ0	PROPN
ejpam-3126	219	29	-	-	PUNCT
ejpam-3126	219	30	2	2	NUM
ejpam-3126	219	31	-	-	PUNCT
ejpam-3126	219	32	absorbing	absorbing	ADJ
ejpam-3126	219	33	semi	semi	ADJ
ejpam-3126	219	34	-	-	ADJ
ejpam-3126	219	35	primary	primary	ADJ
ejpam-3126	219	36	submodule	submodule	NOUN
ejpam-3126	219	37	of	of	ADP
ejpam-3126	219	38	m	m	PROPN
ejpam-3126	219	39	,	,	PUNCT
ejpam-3126	219	40	and	and	CCONJ
ejpam-3126	219	41	hence	hence	ADV
ejpam-3126	219	42	n	n	PRON
ejpam-3126	219	43	is	be	AUX
ejpam-3126	219	44	a	a	DET
ejpam-3126	219	45	weakly	weakly	ADJ
ejpam-3126	219	46	2	2	NUM
ejpam-3126	219	47	-	-	PUNCT
ejpam-3126	219	48	absorbing	absorbing	ADJ
ejpam-3126	219	49	semi	semi	ADJ
ejpam-3126	219	50	-	-	ADJ
ejpam-3126	219	51	primary	primary	ADJ
ejpam-3126	219	52	submodule	submodule	NOUN
ejpam-3126	219	53	of	of	ADP
ejpam-3126	219	54	m	m	PROPN
ejpam-3126	219	55	.	.	PUNCT
ejpam-3126	220	1	•	•	INTJ
ejpam-3126	220	2	if	if	SCONJ
ejpam-3126	220	3	φ(n	φ(n	ADJ
ejpam-3126	220	4	)	)	PUNCT
ejpam-3126	220	5	=	=	SYM
ejpam-3126	221	1	n	n	CCONJ
ejpam-3126	221	2	for	for	ADP
ejpam-3126	221	3	every	every	DET
ejpam-3126	221	4	n	n	PRON
ejpam-3126	221	5	∈	∈	PROPN
ejpam-3126	221	6	s(m	s(m	PROPN
ejpam-3126	221	7	)	)	PUNCT
ejpam-3126	221	8	,	,	PUNCT
ejpam-3126	221	9	then	then	ADV
ejpam-3126	221	10	we	we	PRON
ejpam-3126	221	11	say	say	VERB
ejpam-3126	221	12	that	that	SCONJ
ejpam-3126	221	13	φ	φ	PROPN
ejpam-3126	221	14	=	=	SYM
ejpam-3126	221	15	φ1	φ1	PROPN
ejpam-3126	221	16	and	and	CCONJ
ejpam-3126	221	17	n	n	PROPN
ejpam-3126	221	18	is	be	AUX
ejpam-3126	221	19	called	call	VERB
ejpam-3126	221	20	a	a	DET
ejpam-3126	221	21	φ1	φ1	PROPN
ejpam-3126	221	22	-	-	PUNCT
ejpam-3126	221	23	2	2	NUM
ejpam-3126	221	24	-	-	PUNCT
ejpam-3126	221	25	absorbing	absorbing	ADJ
ejpam-3126	221	26	semi	semi	ADJ
ejpam-3126	221	27	-	-	ADJ
ejpam-3126	221	28	primary	primary	ADJ
ejpam-3126	221	29	submodule	submodule	NOUN
ejpam-3126	221	30	of	of	ADP
ejpam-3126	221	31	m	m	PROPN
ejpam-3126	221	32	.	.	PUNCT
ejpam-3126	222	1	pai	pai	PROPN
ejpam-3126	222	2	.	.	PROPN
ejpam-3126	222	3	yiarayong	yiarayong	PROPN
ejpam-3126	222	4	,	,	PUNCT
ejpam-3126	222	5	m.	m.	NOUN
ejpam-3126	222	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	222	7	/	/	SYM
ejpam-3126	222	8	eur	eur	PROPN
ejpam-3126	222	9	.	.	PUNCT
ejpam-3126	223	1	j.	j.	PROPN
ejpam-3126	223	2	pure	pure	PROPN
ejpam-3126	223	3	appl	appl	PROPN
ejpam-3126	223	4	.	.	PROPN
ejpam-3126	223	5	math	math	PROPN
ejpam-3126	223	6	,	,	PUNCT
ejpam-3126	223	7	11	11	NUM
ejpam-3126	223	8	(	(	PUNCT
ejpam-3126	223	9	1	1	NUM
ejpam-3126	223	10	)	)	PUNCT
ejpam-3126	223	11	(	(	PUNCT
ejpam-3126	223	12	2018	2018	NUM
ejpam-3126	223	13	)	)	PUNCT
ejpam-3126	223	14	,	,	PUNCT
ejpam-3126	223	15	35	35	NUM
ejpam-3126	223	16	-	-	SYM
ejpam-3126	223	17	50	50	NUM
ejpam-3126	223	18	42	42	NUM
ejpam-3126	223	19	•	•	NOUN
ejpam-3126	223	20	if	if	SCONJ
ejpam-3126	223	21	φ(n	φ(n	ADJ
ejpam-3126	223	22	)	)	PUNCT
ejpam-3126	223	23	=	=	SYM
ejpam-3126	223	24	(	(	PUNCT
ejpam-3126	223	25	n	n	NUM
ejpam-3126	223	26	:	:	PUNCT
ejpam-3126	223	27	m)n	m)n	X
ejpam-3126	223	28	for	for	ADP
ejpam-3126	223	29	every	every	DET
ejpam-3126	223	30	n	n	PRON
ejpam-3126	223	31	∈	∈	PROPN
ejpam-3126	223	32	s(m	s(m	PROPN
ejpam-3126	223	33	)	)	PUNCT
ejpam-3126	224	1	,	,	PUNCT
ejpam-3126	224	2	then	then	ADV
ejpam-3126	224	3	we	we	PRON
ejpam-3126	224	4	say	say	VERB
ejpam-3126	224	5	that	that	SCONJ
ejpam-3126	224	6	φ	φ	PROPN
ejpam-3126	224	7	=	=	SYM
ejpam-3126	224	8	φ2	φ2	PROPN
ejpam-3126	224	9	and	and	CCONJ
ejpam-3126	224	10	n	n	PROPN
ejpam-3126	224	11	is	be	AUX
ejpam-3126	224	12	called	call	VERB
ejpam-3126	224	13	a	a	DET
ejpam-3126	224	14	φ2	φ2	PROPN
ejpam-3126	224	15	-	-	PUNCT
ejpam-3126	224	16	2	2	NUM
ejpam-3126	224	17	-	-	PUNCT
ejpam-3126	224	18	absorbing	absorbing	ADJ
ejpam-3126	224	19	semi	semi	ADJ
ejpam-3126	224	20	-	-	ADJ
ejpam-3126	224	21	primary	primary	ADJ
ejpam-3126	224	22	submodule	submodule	NOUN
ejpam-3126	224	23	of	of	ADP
ejpam-3126	224	24	m	m	PROPN
ejpam-3126	224	25	,	,	PUNCT
ejpam-3126	224	26	and	and	CCONJ
ejpam-3126	224	27	hence	hence	ADV
ejpam-3126	224	28	n	n	PRON
ejpam-3126	224	29	is	be	AUX
ejpam-3126	224	30	an	an	DET
ejpam-3126	224	31	almost	almost	ADV
ejpam-3126	224	32	2	2	NUM
ejpam-3126	224	33	-	-	PUNCT
ejpam-3126	224	34	absorbing	absorbing	ADJ
ejpam-3126	224	35	semi	semi	ADJ
ejpam-3126	224	36	-	-	ADJ
ejpam-3126	224	37	primary	primary	ADJ
ejpam-3126	224	38	submodule	submodule	NOUN
ejpam-3126	224	39	of	of	ADP
ejpam-3126	224	40	m	m	PROPN
ejpam-3126	224	41	.	.	PUNCT
ejpam-3126	225	1	•	•	INTJ
ejpam-3126	225	2	if	if	SCONJ
ejpam-3126	225	3	φ(n	φ(n	NOUN
ejpam-3126	225	4	)	)	PUNCT
ejpam-3126	225	5	=	=	PUNCT
ejpam-3126	225	6	(	(	PUNCT
ejpam-3126	226	1	n	n	CCONJ
ejpam-3126	226	2	:	:	PUNCT
ejpam-3126	226	3	m)n−1n	m)n−1n	ADJ
ejpam-3126	226	4	for	for	ADP
ejpam-3126	226	5	every	every	DET
ejpam-3126	226	6	n	n	PRON
ejpam-3126	226	7	∈	∈	PROPN
ejpam-3126	226	8	s(m	s(m	PROPN
ejpam-3126	226	9	)	)	PUNCT
ejpam-3126	226	10	,	,	PUNCT
ejpam-3126	226	11	then	then	ADV
ejpam-3126	226	12	we	we	PRON
ejpam-3126	226	13	say	say	VERB
ejpam-3126	226	14	that	that	SCONJ
ejpam-3126	226	15	φ	φ	PROPN
ejpam-3126	226	16	=	=	SYM
ejpam-3126	226	17	φn≥2	φn≥2	PROPN
ejpam-3126	226	18	and	and	CCONJ
ejpam-3126	226	19	n	n	PROPN
ejpam-3126	226	20	is	be	AUX
ejpam-3126	226	21	called	call	VERB
ejpam-3126	226	22	a	a	DET
ejpam-3126	226	23	φn-2	φn-2	NOUN
ejpam-3126	226	24	-	-	PUNCT
ejpam-3126	226	25	absorbing	absorbing	ADJ
ejpam-3126	226	26	semi	semi	ADJ
ejpam-3126	226	27	-	-	ADJ
ejpam-3126	226	28	primary	primary	ADJ
ejpam-3126	226	29	submodule	submodule	NOUN
ejpam-3126	226	30	of	of	ADP
ejpam-3126	226	31	m	m	PROPN
ejpam-3126	226	32	,	,	PUNCT
ejpam-3126	226	33	and	and	CCONJ
ejpam-3126	226	34	hence	hence	ADV
ejpam-3126	226	35	n	n	PRON
ejpam-3126	226	36	is	be	AUX
ejpam-3126	226	37	a	a	DET
ejpam-3126	226	38	n-2absorbing	n-2absorbing	NOUN
ejpam-3126	226	39	semi	semi	ADJ
ejpam-3126	226	40	-	-	ADJ
ejpam-3126	226	41	primary	primary	ADJ
ejpam-3126	226	42	submodule	submodule	NOUN
ejpam-3126	226	43	of	of	ADP
ejpam-3126	226	44	m	m	PROPN
ejpam-3126	226	45	.	.	PUNCT
ejpam-3126	227	1	•	•	INTJ
ejpam-3126	227	2	if	if	SCONJ
ejpam-3126	227	3	φ(n	φ(n	ADJ
ejpam-3126	227	4	)	)	PUNCT
ejpam-3126	227	5	=	=	SYM
ejpam-3126	228	1	∞⋂	∞⋂	PROPN
ejpam-3126	228	2	i=1	i=1	PROPN
ejpam-3126	228	3	(	(	PUNCT
ejpam-3126	228	4	n	n	X
ejpam-3126	228	5	:	:	PUNCT
ejpam-3126	228	6	m)in	m)in	PROPN
ejpam-3126	228	7	for	for	ADP
ejpam-3126	228	8	every	every	DET
ejpam-3126	228	9	n	n	PRON
ejpam-3126	228	10	∈	∈	PROPN
ejpam-3126	228	11	s(m	s(m	PROPN
ejpam-3126	228	12	)	)	PUNCT
ejpam-3126	228	13	,	,	PUNCT
ejpam-3126	228	14	then	then	ADV
ejpam-3126	228	15	we	we	PRON
ejpam-3126	228	16	say	say	VERB
ejpam-3126	228	17	that	that	SCONJ
ejpam-3126	228	18	φ	φ	PROPN
ejpam-3126	228	19	=	=	SYM
ejpam-3126	228	20	φω	φω	PROPN
ejpam-3126	228	21	and	and	CCONJ
ejpam-3126	228	22	n	n	PROPN
ejpam-3126	228	23	is	be	AUX
ejpam-3126	228	24	called	call	VERB
ejpam-3126	228	25	a	a	DET
ejpam-3126	228	26	φω-2	φω-2	ADV
ejpam-3126	228	27	-	-	PUNCT
ejpam-3126	228	28	absorbing	absorbing	ADJ
ejpam-3126	228	29	semi	semi	ADJ
ejpam-3126	228	30	-	-	ADJ
ejpam-3126	228	31	primary	primary	ADJ
ejpam-3126	228	32	submodule	submodule	NOUN
ejpam-3126	228	33	of	of	ADP
ejpam-3126	228	34	m	m	PROPN
ejpam-3126	228	35	,	,	PUNCT
ejpam-3126	228	36	and	and	CCONJ
ejpam-3126	228	37	hence	hence	ADV
ejpam-3126	228	38	n	n	PRON
ejpam-3126	228	39	is	be	AUX
ejpam-3126	228	40	a	a	DET
ejpam-3126	228	41	ω-2absorbing	ω-2absorbe	VERB
ejpam-3126	228	42	semi	semi	ADJ
ejpam-3126	228	43	-	-	ADJ
ejpam-3126	228	44	primary	primary	ADJ
ejpam-3126	228	45	submodule	submodule	NOUN
ejpam-3126	228	46	of	of	ADP
ejpam-3126	228	47	m	m	PROPN
ejpam-3126	228	48	.	.	PUNCT
ejpam-3126	229	1	remark	remark	PROPN
ejpam-3126	229	2	2	2	NUM
ejpam-3126	229	3	.	.	PUNCT
ejpam-3126	230	1	let	let	VERB
ejpam-3126	230	2	m	m	PRON
ejpam-3126	230	3	be	be	AUX
ejpam-3126	230	4	an	an	DET
ejpam-3126	230	5	r	r	NOUN
ejpam-3126	230	6	-	-	PUNCT
ejpam-3126	230	7	module	module	NOUN
ejpam-3126	230	8	and	and	CCONJ
ejpam-3126	230	9	let	let	VERB
ejpam-3126	230	10	s(m	s(m	PROPN
ejpam-3126	230	11	)	)	PUNCT
ejpam-3126	230	12	be	be	AUX
ejpam-3126	230	13	a	a	DET
ejpam-3126	230	14	set	set	NOUN
ejpam-3126	230	15	of	of	ADP
ejpam-3126	230	16	all	all	DET
ejpam-3126	230	17	submodules	submodule	NOUN
ejpam-3126	230	18	of	of	ADP
ejpam-3126	230	19	m	m	PROPN
ejpam-3126	230	20	.	.	PUNCT
ejpam-3126	231	1	for	for	ADP
ejpam-3126	231	2	two	two	NUM
ejpam-3126	231	3	functions	function	NOUN
ejpam-3126	231	4	φα	φα	ADP
ejpam-3126	231	5	,	,	PUNCT
ejpam-3126	231	6	φβ	φβ	PRON
ejpam-3126	231	7	:	:	PUNCT
ejpam-3126	231	8	s(m	s(m	NOUN
ejpam-3126	231	9	)	)	PUNCT
ejpam-3126	231	10	→	→	SYM
ejpam-3126	231	11	s(m	s(m	NOUN
ejpam-3126	231	12	)	)	PUNCT
ejpam-3126	231	13	∪	∪	ADP
ejpam-3126	231	14	{	{	PUNCT
ejpam-3126	231	15	∅	∅	NOUN
ejpam-3126	231	16	}	}	PUNCT
ejpam-3126	231	17	.	.	PUNCT
ejpam-3126	232	1	we	we	PRON
ejpam-3126	232	2	define	define	VERB
ejpam-3126	232	3	φα	φα	ADP
ejpam-3126	232	4	≤	≤	NOUN
ejpam-3126	232	5	φβ	φβ	ADP
ejpam-3126	232	6	,	,	PUNCT
ejpam-3126	232	7	if	if	SCONJ
ejpam-3126	232	8	φα(n	φα(n	NOUN
ejpam-3126	232	9	)	)	PUNCT
ejpam-3126	232	10	⊆	⊆	NUM
ejpam-3126	232	11	φβ(n	φβ(n	NUM
ejpam-3126	232	12	)	)	PUNCT
ejpam-3126	232	13	for	for	ADP
ejpam-3126	232	14	all	all	PRON
ejpam-3126	232	15	n	n	DET
ejpam-3126	232	16	∈	∈	NOUN
ejpam-3126	232	17	s(m)[11	s(m)[11	NOUN
ejpam-3126	232	18	]	]	PUNCT
ejpam-3126	232	19	.	.	PUNCT
ejpam-3126	233	1	observe	observe	VERB
ejpam-3126	233	2	that	that	PRON
ejpam-3126	233	3	φ∅	φ∅	ADV
ejpam-3126	233	4	≤	≤	X
ejpam-3126	233	5	φ0	φ0	PROPN
ejpam-3126	233	6	≤	≤	PROPN
ejpam-3126	233	7	φω	φω	ADP
ejpam-3126	233	8	≤	≤	NOUN
ejpam-3126	233	9	.	.	PUNCT
ejpam-3126	233	10	.	.	PUNCT
ejpam-3126	233	11	.	.	PUNCT
ejpam-3126	234	1	≤	≤	NUM
ejpam-3126	234	2	φn+1	φn+1	NOUN
ejpam-3126	234	3	≤	≤	NOUN
ejpam-3126	234	4	φn	φn	ADP
ejpam-3126	234	5	≤	≤	NOUN
ejpam-3126	234	6	.	.	PUNCT
ejpam-3126	234	7	.	.	PUNCT
ejpam-3126	234	8	.	.	PUNCT
ejpam-3126	235	1	≤	≤	PROPN
ejpam-3126	235	2	φ2	φ2	PROPN
ejpam-3126	235	3	≤	≤	PROPN
ejpam-3126	235	4	φ1	φ1	PROPN
ejpam-3126	235	5	.	.	PUNCT
ejpam-3126	236	1	notice	notice	VERB
ejpam-3126	236	2	that	that	SCONJ
ejpam-3126	236	3	for	for	ADP
ejpam-3126	236	4	an	an	DET
ejpam-3126	236	5	r	r	NOUN
ejpam-3126	236	6	-	-	PUNCT
ejpam-3126	236	7	module	module	NOUN
ejpam-3126	236	8	m	m	NOUN
ejpam-3126	236	9	,	,	PUNCT
ejpam-3126	236	10	the	the	DET
ejpam-3126	236	11	zero	zero	NUM
ejpam-3126	236	12	submodule	submodule	NOUN
ejpam-3126	236	13	{	{	PUNCT
ejpam-3126	236	14	0	0	NUM
ejpam-3126	236	15	}	}	PUNCT
ejpam-3126	236	16	is	be	AUX
ejpam-3126	236	17	always	always	ADV
ejpam-3126	236	18	a	a	DET
ejpam-3126	236	19	φ0	φ0	PROPN
ejpam-3126	236	20	-	-	PUNCT
ejpam-3126	236	21	2	2	NUM
ejpam-3126	236	22	-	-	PUNCT
ejpam-3126	236	23	absorbing	absorbing	ADJ
ejpam-3126	236	24	semi	semi	ADJ
ejpam-3126	236	25	-	-	ADJ
ejpam-3126	236	26	primary	primary	ADJ
ejpam-3126	236	27	submodule	submodule	NOUN
ejpam-3126	236	28	.	.	PUNCT
ejpam-3126	237	1	in	in	ADP
ejpam-3126	237	2	the	the	DET
ejpam-3126	237	3	following	follow	VERB
ejpam-3126	237	4	example	example	NOUN
ejpam-3126	237	5	,	,	PUNCT
ejpam-3126	237	6	we	we	PRON
ejpam-3126	237	7	give	give	VERB
ejpam-3126	237	8	a	a	DET
ejpam-3126	237	9	module	module	NOUN
ejpam-3126	237	10	in	in	ADP
ejpam-3126	237	11	which	which	PRON
ejpam-3126	237	12	a	a	DET
ejpam-3126	237	13	φ0	φ0	PROPN
ejpam-3126	237	14	-	-	PUNCT
ejpam-3126	237	15	2absorbing	2absorbing	NOUN
ejpam-3126	237	16	semi	semi	ADJ
ejpam-3126	237	17	-	-	ADJ
ejpam-3126	237	18	primary	primary	ADJ
ejpam-3126	237	19	submodule	submodule	NOUN
ejpam-3126	237	20	is	be	AUX
ejpam-3126	237	21	not	not	PART
ejpam-3126	237	22	φ-2	φ-2	ADV
ejpam-3126	237	23	-	-	PUNCT
ejpam-3126	237	24	absorbing	absorbing	ADJ
ejpam-3126	237	25	semi	semi	ADJ
ejpam-3126	237	26	-	-	ADJ
ejpam-3126	237	27	primary	primary	ADJ
ejpam-3126	237	28	.	.	PUNCT
ejpam-3126	238	1	example	example	NOUN
ejpam-3126	239	1	3	3	X
ejpam-3126	239	2	.	.	PUNCT
ejpam-3126	239	3	let	let	VERB
ejpam-3126	239	4	r	r	NOUN
ejpam-3126	239	5	=	=	PUNCT
ejpam-3126	239	6	z	z	PROPN
ejpam-3126	239	7	and	and	CCONJ
ejpam-3126	239	8	m	m	PROPN
ejpam-3126	239	9	=	=	ADJ
ejpam-3126	239	10	z30	z30	PROPN
ejpam-3126	239	11	×	×	PROPN
ejpam-3126	239	12	z30	z30	NOUN
ejpam-3126	239	13	.	.	PUNCT
ejpam-3126	240	1	consider	consider	VERB
ejpam-3126	240	2	the	the	DET
ejpam-3126	240	3	submodule	submodule	NOUN
ejpam-3126	240	4	n	n	NOUN
ejpam-3126	240	5	=	=	PRON
ejpam-3126	240	6	{	{	PUNCT
ejpam-3126	241	1	[	[	X
ejpam-3126	241	2	0	0	NUM
ejpam-3126	241	3	]	]	PUNCT
ejpam-3126	241	4	}	}	PUNCT
ejpam-3126	241	5	×	×	PROPN
ejpam-3126	241	6	z30	z30	NOUN
ejpam-3126	241	7	of	of	ADP
ejpam-3126	241	8	m	m	PROPN
ejpam-3126	241	9	.	.	PUNCT
ejpam-3126	242	1	define	define	VERB
ejpam-3126	242	2	φ	φ	NOUN
ejpam-3126	242	3	:	:	PUNCT
ejpam-3126	242	4	s(m	s(m	PROPN
ejpam-3126	242	5	)	)	PUNCT
ejpam-3126	242	6	→	→	SYM
ejpam-3126	242	7	s(m	s(m	NOUN
ejpam-3126	242	8	)	)	PUNCT
ejpam-3126	242	9	∪	∪	ADP
ejpam-3126	242	10	{	{	PUNCT
ejpam-3126	242	11	∅	∅	NOUN
ejpam-3126	242	12	}	}	PUNCT
ejpam-3126	242	13	by	by	ADP
ejpam-3126	242	14	φ(n	φ(n	NOUN
ejpam-3126	242	15	)	)	PUNCT
ejpam-3126	242	16	=	=	PRON
ejpam-3126	242	17	{	{	PUNCT
ejpam-3126	243	1	[	[	X
ejpam-3126	243	2	0	0	NUM
ejpam-3126	243	3	]	]	PUNCT
ejpam-3126	243	4	}	}	PUNCT
ejpam-3126	243	5	×	×	NOUN
ejpam-3126	243	6	{	{	PUNCT
ejpam-3126	243	7	[	[	X
ejpam-3126	243	8	0	0	NUM
ejpam-3126	243	9	]	]	PUNCT
ejpam-3126	243	10	}	}	PUNCT
ejpam-3126	243	11	for	for	ADP
ejpam-3126	243	12	every	every	DET
ejpam-3126	243	13	n	n	PRON
ejpam-3126	243	14	∈	∈	PROPN
ejpam-3126	243	15	s(m	s(m	PROPN
ejpam-3126	243	16	)	)	PUNCT
ejpam-3126	243	17	.	.	PUNCT
ejpam-3126	244	1	it	it	PRON
ejpam-3126	244	2	is	be	AUX
ejpam-3126	244	3	easy	easy	ADJ
ejpam-3126	244	4	to	to	PART
ejpam-3126	244	5	see	see	VERB
ejpam-3126	244	6	that	that	SCONJ
ejpam-3126	244	7	n	n	PRON
ejpam-3126	244	8	is	be	AUX
ejpam-3126	244	9	a	a	DET
ejpam-3126	244	10	φ0	φ0	PROPN
ejpam-3126	244	11	-	-	PUNCT
ejpam-3126	244	12	2	2	NUM
ejpam-3126	244	13	-	-	PUNCT
ejpam-3126	244	14	absorbing	absorbing	ADJ
ejpam-3126	244	15	semi	semi	ADJ
ejpam-3126	244	16	-	-	ADJ
ejpam-3126	244	17	primary	primary	ADJ
ejpam-3126	244	18	submodule	submodule	NOUN
ejpam-3126	244	19	of	of	ADP
ejpam-3126	244	20	m	m	PROPN
ejpam-3126	244	21	.	.	PUNCT
ejpam-3126	245	1	notice	notice	VERB
ejpam-3126	245	2	that	that	SCONJ
ejpam-3126	245	3	(	(	PUNCT
ejpam-3126	245	4	2	2	NUM
ejpam-3126	245	5	·	·	PUNCT
ejpam-3126	245	6	3)([5	3)([5	NUM
ejpam-3126	245	7	]	]	PUNCT
ejpam-3126	245	8	,	,	PUNCT
ejpam-3126	246	1	[	[	X
ejpam-3126	246	2	1	1	NUM
ejpam-3126	246	3	]	]	PUNCT
ejpam-3126	246	4	)	)	PUNCT
ejpam-3126	246	5	∈	∈	PROPN
ejpam-3126	246	6	n	n	CCONJ
ejpam-3126	246	7	−	−	PROPN
ejpam-3126	246	8	φ(n	φ(n	NOUN
ejpam-3126	246	9	)	)	PUNCT
ejpam-3126	246	10	,	,	PUNCT
ejpam-3126	246	11	but	but	CCONJ
ejpam-3126	246	12	2	2	X
ejpam-3126	246	13	·	·	SYM
ejpam-3126	246	14	3	3	NUM
ejpam-3126	246	15	6∈	6∈	NOUN
ejpam-3126	246	16	√	√	NOUN
ejpam-3126	246	17	(	(	PUNCT
ejpam-3126	246	18	n	n	NUM
ejpam-3126	246	19	:	:	PUNCT
ejpam-3126	246	20	m	m	X
ejpam-3126	246	21	)	)	PUNCT
ejpam-3126	246	22	,	,	PUNCT
ejpam-3126	246	23	2([5	2([5	NUM
ejpam-3126	246	24	]	]	PUNCT
ejpam-3126	246	25	,	,	PUNCT
ejpam-3126	247	1	[	[	X
ejpam-3126	247	2	1	1	NUM
ejpam-3126	247	3	]	]	PUNCT
ejpam-3126	247	4	)	)	PUNCT
ejpam-3126	247	5	6∈	6∈	NOUN
ejpam-3126	247	6	{	{	PUNCT
ejpam-3126	248	1	[	[	X
ejpam-3126	248	2	0]}×z30	0]}×z30	NOUN
ejpam-3126	248	3	and	and	CCONJ
ejpam-3126	248	4	3n([5	3n([5	NUM
ejpam-3126	248	5	]	]	X
ejpam-3126	248	6	,	,	PUNCT
ejpam-3126	248	7	[	[	X
ejpam-3126	248	8	1	1	NUM
ejpam-3126	248	9	]	]	PUNCT
ejpam-3126	248	10	)	)	PUNCT
ejpam-3126	248	11	6∈	6∈	NOUN
ejpam-3126	248	12	{	{	PUNCT
ejpam-3126	249	1	[	[	X
ejpam-3126	249	2	0	0	NUM
ejpam-3126	249	3	]	]	PUNCT
ejpam-3126	249	4	}	}	PUNCT
ejpam-3126	249	5	×	×	PROPN
ejpam-3126	249	6	z30	z30	NOUN
ejpam-3126	249	7	for	for	ADP
ejpam-3126	249	8	all	all	DET
ejpam-3126	249	9	positive	positive	ADJ
ejpam-3126	249	10	integer	integer	NOUN
ejpam-3126	249	11	n.	n.	NOUN
ejpam-3126	249	12	therefore	therefore	ADV
ejpam-3126	249	13	n	n	ADV
ejpam-3126	249	14	is	be	AUX
ejpam-3126	249	15	not	not	PART
ejpam-3126	249	16	a	a	DET
ejpam-3126	249	17	φ-2	φ-2	NOUN
ejpam-3126	249	18	-	-	PUNCT
ejpam-3126	249	19	absorbing	absorbing	ADJ
ejpam-3126	249	20	semi	semi	ADJ
ejpam-3126	249	21	-	-	ADJ
ejpam-3126	249	22	primary	primary	ADJ
ejpam-3126	249	23	submodule	submodule	NOUN
ejpam-3126	249	24	of	of	ADP
ejpam-3126	249	25	m	m	PROPN
ejpam-3126	249	26	.	.	PUNCT
ejpam-3126	250	1	we	we	PRON
ejpam-3126	250	2	are	be	AUX
ejpam-3126	250	3	finding	find	VERB
ejpam-3126	250	4	additional	additional	ADJ
ejpam-3126	250	5	condition	condition	NOUN
ejpam-3126	250	6	to	to	PART
ejpam-3126	250	7	show	show	VERB
ejpam-3126	250	8	that	that	SCONJ
ejpam-3126	250	9	a	a	DET
ejpam-3126	250	10	2	2	NUM
ejpam-3126	250	11	-	-	PUNCT
ejpam-3126	250	12	absorbing	absorbing	ADJ
ejpam-3126	250	13	semi	semi	ADJ
ejpam-3126	250	14	-	-	ADJ
ejpam-3126	250	15	primary	primary	ADJ
ejpam-3126	250	16	submodule	submodule	NOUN
ejpam-3126	250	17	is	be	AUX
ejpam-3126	250	18	a	a	DET
ejpam-3126	250	19	φ-2	φ-2	NOUN
ejpam-3126	250	20	-	-	PUNCT
ejpam-3126	250	21	absorbing	absorbing	ADJ
ejpam-3126	250	22	semi	semi	ADJ
ejpam-3126	250	23	-	-	ADJ
ejpam-3126	250	24	primary	primary	ADJ
ejpam-3126	250	25	submodule	submodule	NOUN
ejpam-3126	250	26	of	of	ADP
ejpam-3126	250	27	an	an	DET
ejpam-3126	250	28	r	r	NOUN
ejpam-3126	250	29	-	-	PUNCT
ejpam-3126	250	30	module	module	NOUN
ejpam-3126	250	31	.	.	PUNCT
ejpam-3126	251	1	theorem	theorem	NOUN
ejpam-3126	251	2	8	8	NUM
ejpam-3126	251	3	.	.	PUNCT
ejpam-3126	252	1	let	let	VERB
ejpam-3126	252	2	φ	φ	PROPN
ejpam-3126	252	3	:	:	PUNCT
ejpam-3126	252	4	s(m	s(m	PROPN
ejpam-3126	252	5	)	)	PUNCT
ejpam-3126	252	6	→	→	SYM
ejpam-3126	252	7	s(m	s(m	NOUN
ejpam-3126	252	8	)	)	PUNCT
ejpam-3126	252	9	∪	∪	NOUN
ejpam-3126	252	10	{	{	PUNCT
ejpam-3126	252	11	∅	∅	NOUN
ejpam-3126	252	12	}	}	PUNCT
ejpam-3126	252	13	be	be	AUX
ejpam-3126	252	14	a	a	DET
ejpam-3126	252	15	function	function	NOUN
ejpam-3126	252	16	and	and	CCONJ
ejpam-3126	252	17	let	let	VERB
ejpam-3126	252	18	φ(n	φ(n	NOUN
ejpam-3126	252	19	)	)	PUNCT
ejpam-3126	252	20	is	be	AUX
ejpam-3126	252	21	a	a	DET
ejpam-3126	252	22	2	2	NUM
ejpam-3126	252	23	-	-	PUNCT
ejpam-3126	252	24	absorbing	absorbing	ADJ
ejpam-3126	252	25	semi	semi	ADJ
ejpam-3126	252	26	-	-	ADJ
ejpam-3126	252	27	primary	primary	ADJ
ejpam-3126	252	28	submodule	submodule	NOUN
ejpam-3126	252	29	of	of	ADP
ejpam-3126	252	30	m	m	PROPN
ejpam-3126	252	31	.	.	PUNCT
ejpam-3126	253	1	then	then	ADV
ejpam-3126	253	2	n	n	PRON
ejpam-3126	253	3	is	be	AUX
ejpam-3126	253	4	a	a	DET
ejpam-3126	253	5	φ-2	φ-2	NOUN
ejpam-3126	253	6	-	-	PUNCT
ejpam-3126	253	7	absorbing	absorbing	ADJ
ejpam-3126	253	8	semi	semi	ADJ
ejpam-3126	253	9	-	-	ADJ
ejpam-3126	253	10	primary	primary	ADJ
ejpam-3126	253	11	submodule	submodule	NOUN
ejpam-3126	253	12	of	of	ADP
ejpam-3126	253	13	m	m	PROPN
ejpam-3126	253	14	if	if	SCONJ
ejpam-3126	254	1	and	and	CCONJ
ejpam-3126	254	2	only	only	ADV
ejpam-3126	254	3	if	if	SCONJ
ejpam-3126	254	4	n	n	PRON
ejpam-3126	254	5	is	be	AUX
ejpam-3126	254	6	a	a	DET
ejpam-3126	254	7	2	2	NUM
ejpam-3126	254	8	-	-	PUNCT
ejpam-3126	254	9	absorbing	absorbing	ADJ
ejpam-3126	254	10	semi	semi	ADJ
ejpam-3126	254	11	-	-	ADJ
ejpam-3126	254	12	primary	primary	ADJ
ejpam-3126	254	13	submodule	submodule	NOUN
ejpam-3126	254	14	of	of	ADP
ejpam-3126	254	15	m	m	PROPN
ejpam-3126	254	16	.	.	PUNCT
ejpam-3126	255	1	proof	proof	NOUN
ejpam-3126	255	2	.	.	PUNCT
ejpam-3126	256	1	suppose	suppose	VERB
ejpam-3126	256	2	that	that	SCONJ
ejpam-3126	256	3	n	n	PRON
ejpam-3126	256	4	is	be	AUX
ejpam-3126	256	5	a	a	DET
ejpam-3126	256	6	2	2	NUM
ejpam-3126	256	7	-	-	PUNCT
ejpam-3126	256	8	absorbing	absorbing	ADJ
ejpam-3126	256	9	semi	semi	ADJ
ejpam-3126	256	10	-	-	ADJ
ejpam-3126	256	11	primary	primary	ADJ
ejpam-3126	256	12	submodule	submodule	NOUN
ejpam-3126	256	13	of	of	ADP
ejpam-3126	256	14	m	m	PROPN
ejpam-3126	256	15	.	.	PUNCT
ejpam-3126	257	1	clearly	clearly	ADV
ejpam-3126	257	2	,	,	PUNCT
ejpam-3126	257	3	n	n	PRON
ejpam-3126	257	4	is	be	AUX
ejpam-3126	257	5	a	a	DET
ejpam-3126	257	6	φ-2	φ-2	NOUN
ejpam-3126	257	7	-	-	PUNCT
ejpam-3126	257	8	absorbing	absorbing	ADJ
ejpam-3126	257	9	semi	semi	ADJ
ejpam-3126	257	10	-	-	ADJ
ejpam-3126	257	11	primary	primary	ADJ
ejpam-3126	257	12	submodule	submodule	NOUN
ejpam-3126	257	13	of	of	ADP
ejpam-3126	257	14	m	m	PROPN
ejpam-3126	257	15	.	.	PUNCT
ejpam-3126	258	1	conversely	conversely	ADV
ejpam-3126	258	2	,	,	PUNCT
ejpam-3126	258	3	assume	assume	VERB
ejpam-3126	258	4	that	that	SCONJ
ejpam-3126	258	5	n	n	PRON
ejpam-3126	258	6	is	be	AUX
ejpam-3126	258	7	a	a	DET
ejpam-3126	258	8	φ-2	φ-2	NOUN
ejpam-3126	258	9	-	-	PUNCT
ejpam-3126	258	10	absorbing	absorbing	ADJ
ejpam-3126	258	11	semi	semi	ADJ
ejpam-3126	258	12	-	-	ADJ
ejpam-3126	258	13	primary	primary	ADJ
ejpam-3126	258	14	submodule	submodule	NOUN
ejpam-3126	258	15	of	of	ADP
ejpam-3126	258	16	m	m	PROPN
ejpam-3126	258	17	.	.	PUNCT
ejpam-3126	259	1	let	let	VERB
ejpam-3126	259	2	a1	a1	NOUN
ejpam-3126	259	3	,	,	PUNCT
ejpam-3126	259	4	a2	a2	PROPN
ejpam-3126	259	5	∈	∈	PROPN
ejpam-3126	259	6	r	r	NOUN
ejpam-3126	259	7	and	and	CCONJ
ejpam-3126	259	8	m	m	PROPN
ejpam-3126	259	9	∈	∈	NOUN
ejpam-3126	259	10	m	m	VERB
ejpam-3126	260	1	such	such	ADJ
ejpam-3126	260	2	that	that	SCONJ
ejpam-3126	260	3	a1a2	a1a2	PROPN
ejpam-3126	260	4	m	m	VERB
ejpam-3126	260	5	∈	∈	NOUN
ejpam-3126	260	6	n	n	NOUN
ejpam-3126	260	7	.	.	PUNCT
ejpam-3126	261	1	if	if	SCONJ
ejpam-3126	261	2	a1a2	a1a2	PROPN
ejpam-3126	261	3	m	m	VERB
ejpam-3126	261	4	6∈	6∈	NOUN
ejpam-3126	261	5	φ(n	φ(n	ADJ
ejpam-3126	261	6	)	)	PUNCT
ejpam-3126	261	7	,	,	PUNCT
ejpam-3126	261	8	then	then	ADV
ejpam-3126	261	9	a1a2	a1a2	ADP
ejpam-3126	261	10	m	m	NOUN
ejpam-3126	261	11	∈	∈	ADJ
ejpam-3126	261	12	n−φ(n	n−φ(n	NOUN
ejpam-3126	261	13	)	)	PUNCT
ejpam-3126	261	14	.	.	PUNCT
ejpam-3126	262	1	by	by	ADP
ejpam-3126	262	2	definition	definition	NOUN
ejpam-3126	262	3	1	1	NUM
ejpam-3126	262	4	,	,	PUNCT
ejpam-3126	262	5	a1a2	a1a2	PROPN
ejpam-3126	262	6	∈	∈	NOUN
ejpam-3126	262	7	√	√	NUM
ejpam-3126	262	8	(	(	PUNCT
ejpam-3126	262	9	n	n	NUM
ejpam-3126	262	10	:	:	PUNCT
ejpam-3126	262	11	m	m	X
ejpam-3126	262	12	)	)	PUNCT
ejpam-3126	262	13	or	or	CCONJ
ejpam-3126	262	14	a1	a1	NOUN
ejpam-3126	262	15	m	m	PROPN
ejpam-3126	262	16	∈	∈	NOUN
ejpam-3126	262	17	n	n	NOUN
ejpam-3126	262	18	or	or	CCONJ
ejpam-3126	262	19	an2	an2	PROPN
ejpam-3126	262	20	m	m	NOUN
ejpam-3126	262	21	∈	∈	PROPN
ejpam-3126	262	22	n	n	NOUN
ejpam-3126	262	23	for	for	ADP
ejpam-3126	262	24	some	some	DET
ejpam-3126	262	25	positive	positive	ADJ
ejpam-3126	262	26	integer	integer	NOUN
ejpam-3126	262	27	n.	n.	NOUN
ejpam-3126	262	28	now	now	ADV
ejpam-3126	262	29	if	if	SCONJ
ejpam-3126	262	30	a1a2	a1a2	ADP
ejpam-3126	262	31	m	m	VERB
ejpam-3126	262	32	∈	∈	NOUN
ejpam-3126	262	33	φ(n	φ(n	NOUN
ejpam-3126	262	34	)	)	PUNCT
ejpam-3126	263	1	,	,	PUNCT
ejpam-3126	263	2	then	then	ADV
ejpam-3126	263	3	a1a2	a1a2	ADP
ejpam-3126	263	4	∈	∈	NOUN
ejpam-3126	263	5	√	√	NUM
ejpam-3126	263	6	(	(	PUNCT
ejpam-3126	263	7	n	n	NUM
ejpam-3126	263	8	:	:	PUNCT
ejpam-3126	263	9	m	m	X
ejpam-3126	263	10	)	)	PUNCT
ejpam-3126	263	11	or	or	CCONJ
ejpam-3126	263	12	a1	a1	NOUN
ejpam-3126	263	13	m	m	PROPN
ejpam-3126	263	14	∈	∈	NOUN
ejpam-3126	263	15	n	n	NOUN
ejpam-3126	263	16	or	or	CCONJ
ejpam-3126	263	17	an2	an2	PROPN
ejpam-3126	263	18	m	m	NOUN
ejpam-3126	263	19	∈	∈	PROPN
ejpam-3126	263	20	n	n	NOUN
ejpam-3126	263	21	for	for	ADP
ejpam-3126	263	22	some	some	DET
ejpam-3126	263	23	positive	positive	ADJ
ejpam-3126	263	24	integer	integer	NOUN
ejpam-3126	263	25	n.	n.	NOUN
ejpam-3126	263	26	further	far	ADV
ejpam-3126	263	27	,	,	PUNCT
ejpam-3126	263	28	we	we	PRON
ejpam-3126	263	29	give	give	VERB
ejpam-3126	263	30	another	another	DET
ejpam-3126	263	31	characterization	characterization	NOUN
ejpam-3126	263	32	of	of	ADP
ejpam-3126	263	33	φα-2	φα-2	NOUN
ejpam-3126	263	34	-	-	PUNCT
ejpam-3126	263	35	absorbing	absorbing	ADJ
ejpam-3126	263	36	semi	semi	ADJ
ejpam-3126	263	37	-	-	ADJ
ejpam-3126	263	38	primary	primary	ADJ
ejpam-3126	263	39	submodule	submodule	NOUN
ejpam-3126	263	40	of	of	ADP
ejpam-3126	263	41	m	m	PROPN
ejpam-3126	263	42	.	.	PUNCT
ejpam-3126	264	1	theorem	theorem	ADJ
ejpam-3126	264	2	9	9	NUM
ejpam-3126	264	3	.	.	PUNCT
ejpam-3126	265	1	let	let	VERB
ejpam-3126	265	2	φ2	φ2	PROPN
ejpam-3126	265	3	:	:	PUNCT
ejpam-3126	265	4	s(m	s(m	PROPN
ejpam-3126	265	5	)	)	PUNCT
ejpam-3126	265	6	→	→	SYM
ejpam-3126	265	7	s(m	s(m	NOUN
ejpam-3126	265	8	)	)	PUNCT
ejpam-3126	265	9	∪	∪	NOUN
ejpam-3126	265	10	{	{	PUNCT
ejpam-3126	265	11	∅	∅	NOUN
ejpam-3126	265	12	}	}	PUNCT
ejpam-3126	265	13	be	be	AUX
ejpam-3126	265	14	a	a	DET
ejpam-3126	265	15	function	function	NOUN
ejpam-3126	265	16	and	and	CCONJ
ejpam-3126	265	17	let	let	VERB
ejpam-3126	265	18	(	(	PUNCT
ejpam-3126	265	19	0	0	NUM
ejpam-3126	265	20	:	:	PUNCT
ejpam-3126	265	21	rk	rk	NOUN
ejpam-3126	265	22	)	)	PUNCT
ejpam-3126	265	23	⊆	⊆	NUM
ejpam-3126	265	24	rkm	rkm	PROPN
ejpam-3126	265	25	6=	6=	PROPN
ejpam-3126	265	26	m	m	PROPN
ejpam-3126	265	27	,	,	PUNCT
ejpam-3126	265	28	where	where	SCONJ
ejpam-3126	265	29	r	r	PROPN
ejpam-3126	265	30	∈	∈	PROPN
ejpam-3126	265	31	r.	r.	PROPN
ejpam-3126	265	32	then	then	ADV
ejpam-3126	265	33	rkm	rkm	PROPN
ejpam-3126	265	34	is	be	AUX
ejpam-3126	265	35	a	a	DET
ejpam-3126	265	36	φ2	φ2	PROPN
ejpam-3126	265	37	-	-	PUNCT
ejpam-3126	265	38	2	2	NUM
ejpam-3126	265	39	-	-	PUNCT
ejpam-3126	265	40	absorbing	absorbing	ADJ
ejpam-3126	265	41	semi	semi	ADJ
ejpam-3126	265	42	-	-	ADJ
ejpam-3126	265	43	primary	primary	ADJ
ejpam-3126	265	44	submodule	submodule	NOUN
ejpam-3126	265	45	of	of	ADP
ejpam-3126	265	46	m	m	PROPN
ejpam-3126	265	47	if	if	SCONJ
ejpam-3126	266	1	and	and	CCONJ
ejpam-3126	266	2	only	only	ADV
ejpam-3126	266	3	if	if	SCONJ
ejpam-3126	266	4	it	it	PRON
ejpam-3126	266	5	is	be	AUX
ejpam-3126	266	6	a	a	DET
ejpam-3126	266	7	2	2	NUM
ejpam-3126	266	8	-	-	PUNCT
ejpam-3126	266	9	absorbing	absorbing	ADJ
ejpam-3126	266	10	semi	semi	ADJ
ejpam-3126	266	11	-	-	ADJ
ejpam-3126	266	12	primary	primary	ADJ
ejpam-3126	266	13	submodule	submodule	NOUN
ejpam-3126	266	14	of	of	ADP
ejpam-3126	266	15	m	m	PROPN
ejpam-3126	266	16	.	.	PUNCT
ejpam-3126	267	1	pai	pai	PROPN
ejpam-3126	267	2	.	.	PROPN
ejpam-3126	267	3	yiarayong	yiarayong	PROPN
ejpam-3126	267	4	,	,	PUNCT
ejpam-3126	267	5	m.	m.	NOUN
ejpam-3126	267	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	267	7	/	/	SYM
ejpam-3126	267	8	eur	eur	PROPN
ejpam-3126	267	9	.	.	PUNCT
ejpam-3126	268	1	j.	j.	PROPN
ejpam-3126	268	2	pure	pure	PROPN
ejpam-3126	268	3	appl	appl	PROPN
ejpam-3126	268	4	.	.	PROPN
ejpam-3126	268	5	math	math	PROPN
ejpam-3126	268	6	,	,	PUNCT
ejpam-3126	268	7	11	11	NUM
ejpam-3126	268	8	(	(	PUNCT
ejpam-3126	268	9	1	1	NUM
ejpam-3126	268	10	)	)	PUNCT
ejpam-3126	268	11	(	(	PUNCT
ejpam-3126	268	12	2018	2018	NUM
ejpam-3126	268	13	)	)	PUNCT
ejpam-3126	268	14	,	,	PUNCT
ejpam-3126	268	15	35	35	NUM
ejpam-3126	268	16	-	-	SYM
ejpam-3126	268	17	50	50	NUM
ejpam-3126	268	18	43	43	NUM
ejpam-3126	268	19	proof	proof	NOUN
ejpam-3126	268	20	.	.	PUNCT
ejpam-3126	268	21	suppose	suppose	VERB
ejpam-3126	268	22	that	that	SCONJ
ejpam-3126	268	23	rkm	rkm	PROPN
ejpam-3126	268	24	is	be	AUX
ejpam-3126	268	25	a	a	DET
ejpam-3126	268	26	2	2	NUM
ejpam-3126	268	27	-	-	PUNCT
ejpam-3126	268	28	absorbing	absorbing	ADJ
ejpam-3126	268	29	semi	semi	ADJ
ejpam-3126	268	30	-	-	ADJ
ejpam-3126	268	31	primary	primary	ADJ
ejpam-3126	268	32	submodule	submodule	NOUN
ejpam-3126	268	33	of	of	ADP
ejpam-3126	268	34	m	m	PROPN
ejpam-3126	268	35	.	.	PUNCT
ejpam-3126	269	1	clearly	clearly	ADV
ejpam-3126	269	2	,	,	PUNCT
ejpam-3126	269	3	rkm	rkm	PROPN
ejpam-3126	269	4	is	be	AUX
ejpam-3126	269	5	a	a	DET
ejpam-3126	269	6	φ2	φ2	PROPN
ejpam-3126	269	7	-	-	PUNCT
ejpam-3126	269	8	2	2	NUM
ejpam-3126	269	9	-	-	PUNCT
ejpam-3126	269	10	absorbing	absorbing	ADJ
ejpam-3126	269	11	semi	semi	ADJ
ejpam-3126	269	12	-	-	ADJ
ejpam-3126	269	13	primary	primary	ADJ
ejpam-3126	269	14	submodule	submodule	NOUN
ejpam-3126	269	15	of	of	ADP
ejpam-3126	269	16	m	m	PROPN
ejpam-3126	269	17	.	.	PUNCT
ejpam-3126	270	1	conversely	conversely	ADV
ejpam-3126	270	2	,	,	PUNCT
ejpam-3126	270	3	assume	assume	VERB
ejpam-3126	270	4	that	that	SCONJ
ejpam-3126	270	5	rkm	rkm	PROPN
ejpam-3126	270	6	is	be	AUX
ejpam-3126	270	7	a	a	DET
ejpam-3126	270	8	φ2	φ2	PROPN
ejpam-3126	270	9	-	-	PUNCT
ejpam-3126	270	10	2	2	NUM
ejpam-3126	270	11	-	-	PUNCT
ejpam-3126	270	12	absorbing	absorbing	ADJ
ejpam-3126	270	13	semi	semi	ADJ
ejpam-3126	270	14	-	-	ADJ
ejpam-3126	270	15	primary	primary	ADJ
ejpam-3126	270	16	submodule	submodule	NOUN
ejpam-3126	270	17	of	of	ADP
ejpam-3126	270	18	m	m	PROPN
ejpam-3126	270	19	.	.	PUNCT
ejpam-3126	271	1	let	let	VERB
ejpam-3126	271	2	a1	a1	NOUN
ejpam-3126	271	3	,	,	PUNCT
ejpam-3126	271	4	a2	a2	PROPN
ejpam-3126	271	5	∈	∈	PROPN
ejpam-3126	271	6	r	r	NOUN
ejpam-3126	271	7	and	and	CCONJ
ejpam-3126	271	8	m	m	PROPN
ejpam-3126	271	9	∈	∈	NOUN
ejpam-3126	271	10	m	m	VERB
ejpam-3126	271	11	such	such	ADJ
ejpam-3126	271	12	that	that	SCONJ
ejpam-3126	271	13	a1a2	a1a2	PROPN
ejpam-3126	271	14	m	m	PROPN
ejpam-3126	271	15	∈	∈	PROPN
ejpam-3126	271	16	rkm	rkm	PROPN
ejpam-3126	271	17	.	.	PUNCT
ejpam-3126	272	1	if	if	SCONJ
ejpam-3126	272	2	a1a2	a1a2	PROPN
ejpam-3126	272	3	m	m	VERB
ejpam-3126	272	4	6∈	6∈	NUM
ejpam-3126	272	5	φ2(rkm	φ2(rkm	NUM
ejpam-3126	272	6	)	)	PUNCT
ejpam-3126	272	7	,	,	PUNCT
ejpam-3126	272	8	then	then	ADV
ejpam-3126	272	9	a1a2	a1a2	ADP
ejpam-3126	272	10	m	m	PROPN
ejpam-3126	272	11	∈	∈	PROPN
ejpam-3126	272	12	rkm	rkm	PROPN
ejpam-3126	272	13	−	−	PROPN
ejpam-3126	272	14	φ2(rkm	φ2(rkm	PROPN
ejpam-3126	272	15	)	)	PUNCT
ejpam-3126	272	16	.	.	PUNCT
ejpam-3126	273	1	by	by	ADP
ejpam-3126	273	2	definition	definition	NOUN
ejpam-3126	273	3	1	1	NUM
ejpam-3126	273	4	,	,	PUNCT
ejpam-3126	273	5	a1a2	a1a2	PROPN
ejpam-3126	273	6	∈	∈	NOUN
ejpam-3126	273	7	√	√	NUM
ejpam-3126	273	8	(	(	PUNCT
ejpam-3126	273	9	n	n	NUM
ejpam-3126	273	10	:	:	PUNCT
ejpam-3126	273	11	m	m	X
ejpam-3126	273	12	)	)	PUNCT
ejpam-3126	273	13	or	or	CCONJ
ejpam-3126	273	14	a1	a1	NOUN
ejpam-3126	273	15	m	m	PROPN
ejpam-3126	273	16	∈	∈	NOUN
ejpam-3126	273	17	n	n	NOUN
ejpam-3126	273	18	or	or	CCONJ
ejpam-3126	273	19	an2	an2	PROPN
ejpam-3126	273	20	m	m	NOUN
ejpam-3126	273	21	∈	∈	PROPN
ejpam-3126	273	22	n	n	NOUN
ejpam-3126	273	23	for	for	ADP
ejpam-3126	273	24	some	some	DET
ejpam-3126	273	25	positive	positive	ADJ
ejpam-3126	273	26	integer	integer	NOUN
ejpam-3126	273	27	n.	n.	NOUN
ejpam-3126	273	28	assume	assume	VERB
ejpam-3126	273	29	that	that	SCONJ
ejpam-3126	273	30	a1a2	a1a2	ADP
ejpam-3126	273	31	m	m	VERB
ejpam-3126	273	32	∈	∈	NOUN
ejpam-3126	273	33	φ2(rkm	φ2(rkm	PRON
ejpam-3126	273	34	)	)	PUNCT
ejpam-3126	273	35	.	.	PUNCT
ejpam-3126	274	1	since	since	SCONJ
ejpam-3126	274	2	a1a2	a1a2	PROPN
ejpam-3126	274	3	m	m	PROPN
ejpam-3126	274	4	,	,	PUNCT
ejpam-3126	274	5	r	r	PROPN
ejpam-3126	274	6	ka2	ka2	PROPN
ejpam-3126	274	7	m	m	PROPN
ejpam-3126	274	8	∈	∈	PROPN
ejpam-3126	274	9	rkm	rkm	PROPN
ejpam-3126	274	10	,	,	PUNCT
ejpam-3126	274	11	we	we	PRON
ejpam-3126	274	12	have	have	VERB
ejpam-3126	274	13	(	(	PUNCT
ejpam-3126	274	14	a1	a1	NOUN
ejpam-3126	274	15	+	+	CCONJ
ejpam-3126	274	16	rk)a2	rk)a2	PROPN
ejpam-3126	274	17	m	m	PROPN
ejpam-3126	274	18	∈	∈	PROPN
ejpam-3126	274	19	rkm	rkm	PROPN
ejpam-3126	274	20	.	.	PUNCT
ejpam-3126	275	1	if	if	SCONJ
ejpam-3126	275	2	(	(	PUNCT
ejpam-3126	275	3	a1	a1	NOUN
ejpam-3126	275	4	+	+	CCONJ
ejpam-3126	275	5	rk)a2	rk)a2	PROPN
ejpam-3126	275	6	m	m	VERB
ejpam-3126	275	7	6∈	6∈	PROPN
ejpam-3126	275	8	φ2(rkm	φ2(rkm	NUM
ejpam-3126	275	9	)	)	PUNCT
ejpam-3126	275	10	,	,	PUNCT
ejpam-3126	275	11	then	then	ADV
ejpam-3126	275	12	(	(	PUNCT
ejpam-3126	275	13	a1	a1	NOUN
ejpam-3126	275	14	+	+	CCONJ
ejpam-3126	275	15	rk)a2	rk)a2	PROPN
ejpam-3126	275	16	m	m	PROPN
ejpam-3126	275	17	∈	∈	PROPN
ejpam-3126	275	18	rkm	rkm	PROPN
ejpam-3126	275	19	−	−	PROPN
ejpam-3126	275	20	φ2(rkm	φ2(rkm	PROPN
ejpam-3126	275	21	)	)	PUNCT
ejpam-3126	275	22	.	.	PUNCT
ejpam-3126	276	1	then	then	ADV
ejpam-3126	276	2	by	by	ADP
ejpam-3126	276	3	definition	definition	NOUN
ejpam-3126	276	4	1	1	NUM
ejpam-3126	276	5	,	,	PUNCT
ejpam-3126	276	6	a1a2	a1a2	PROPN
ejpam-3126	276	7	∈	∈	NOUN
ejpam-3126	276	8	√	√	NUM
ejpam-3126	276	9	(	(	PUNCT
ejpam-3126	276	10	rkm	rkm	PROPN
ejpam-3126	276	11	:	:	PUNCT
ejpam-3126	276	12	m	m	PROPN
ejpam-3126	276	13	)	)	PUNCT
ejpam-3126	276	14	or	or	CCONJ
ejpam-3126	276	15	a1	a1	PROPN
ejpam-3126	276	16	m	m	PROPN
ejpam-3126	276	17	∈	∈	NOUN
ejpam-3126	276	18	rkm	rkm	PROPN
ejpam-3126	276	19	or	or	CCONJ
ejpam-3126	276	20	an2	an2	PROPN
ejpam-3126	276	21	m	m	PROPN
ejpam-3126	276	22	∈	∈	PROPN
ejpam-3126	276	23	rkm	rkm	NOUN
ejpam-3126	276	24	for	for	ADP
ejpam-3126	276	25	some	some	DET
ejpam-3126	276	26	positive	positive	ADJ
ejpam-3126	276	27	integer	integer	NOUN
ejpam-3126	276	28	n.	n.	NOUN
ejpam-3126	276	29	now	now	ADV
ejpam-3126	276	30	if	if	SCONJ
ejpam-3126	276	31	(	(	PUNCT
ejpam-3126	276	32	a1	a1	NOUN
ejpam-3126	276	33	+	+	CCONJ
ejpam-3126	276	34	rk)a2	rk)a2	PROPN
ejpam-3126	276	35	m	m	VERB
ejpam-3126	276	36	∈	∈	NOUN
ejpam-3126	276	37	φ2(rkm	φ2(rkm	NUM
ejpam-3126	276	38	)	)	PUNCT
ejpam-3126	276	39	,	,	PUNCT
ejpam-3126	276	40	then	then	ADV
ejpam-3126	276	41	rka2	rka2	PROPN
ejpam-3126	276	42	m	m	PROPN
ejpam-3126	276	43	∈	∈	PROPN
ejpam-3126	276	44	φ2(rkm	φ2(rkm	PRON
ejpam-3126	276	45	)	)	PUNCT
ejpam-3126	276	46	.	.	PUNCT
ejpam-3126	277	1	then	then	ADV
ejpam-3126	277	2	there	there	PRON
ejpam-3126	277	3	exists	exist	VERB
ejpam-3126	277	4	m0	m0	PROPN
ejpam-3126	277	5	∈	∈	PROPN
ejpam-3126	277	6	(	(	PUNCT
ejpam-3126	277	7	rkm	rkm	PROPN
ejpam-3126	277	8	:	:	PUNCT
ejpam-3126	277	9	m)m	m)m	NOUN
ejpam-3126	277	10	such	such	ADJ
ejpam-3126	277	11	that	that	SCONJ
ejpam-3126	277	12	rka2	rka2	PROPN
ejpam-3126	277	13	m	m	PROPN
ejpam-3126	277	14	=	=	ADJ
ejpam-3126	277	15	rkm0	rkm0	NOUN
ejpam-3126	277	16	.	.	PUNCT
ejpam-3126	278	1	by	by	ADP
ejpam-3126	278	2	assumption	assumption	NOUN
ejpam-3126	278	3	,	,	PUNCT
ejpam-3126	278	4	a2m−m0	a2m−m0	PUNCT
ejpam-3126	278	5	∈	∈	PROPN
ejpam-3126	278	6	rkm	rkm	PROPN
ejpam-3126	278	7	.	.	PUNCT
ejpam-3126	279	1	hence	hence	ADV
ejpam-3126	279	2	a2	a2	PROPN
ejpam-3126	279	3	m	m	PROPN
ejpam-3126	279	4	∈	∈	PROPN
ejpam-3126	279	5	rkm	rkm	PROPN
ejpam-3126	279	6	.	.	PUNCT
ejpam-3126	280	1	theorem	theorem	VERB
ejpam-3126	280	2	10	10	NUM
ejpam-3126	280	3	.	.	PUNCT
ejpam-3126	281	1	let	let	VERB
ejpam-3126	281	2	φ	φ	NOUN
ejpam-3126	281	3	:	:	PUNCT
ejpam-3126	281	4	s(m)→	s(m)→	NOUN
ejpam-3126	281	5	s(m)∪{∅	s(m)∪{∅	PROPN
ejpam-3126	281	6	}	}	PUNCT
ejpam-3126	281	7	be	be	AUX
ejpam-3126	281	8	a	a	DET
ejpam-3126	281	9	function	function	NOUN
ejpam-3126	281	10	and	and	CCONJ
ejpam-3126	281	11	let	let	VERB
ejpam-3126	281	12	n	n	PRON
ejpam-3126	281	13	,	,	PUNCT
ejpam-3126	281	14	k	k	X
ejpam-3126	281	15	be	be	VERB
ejpam-3126	281	16	two	two	NUM
ejpam-3126	281	17	submodules	submodule	NOUN
ejpam-3126	281	18	of	of	ADP
ejpam-3126	281	19	m	m	PROPN
ejpam-3126	281	20	with	with	ADP
ejpam-3126	281	21	n	n	PRON
ejpam-3126	281	22	⊆	⊆	NUM
ejpam-3126	281	23	k.	k.	NOUN
ejpam-3126	282	1	if	if	SCONJ
ejpam-3126	282	2	φ(k	φ(k	PROPN
ejpam-3126	282	3	)	)	PUNCT
ejpam-3126	283	1	⊆	⊆	NUM
ejpam-3126	283	2	n	n	PROPN
ejpam-3126	283	3	and	and	CCONJ
ejpam-3126	283	4	k	k	PROPN
ejpam-3126	283	5	is	be	AUX
ejpam-3126	283	6	a	a	DET
ejpam-3126	283	7	φ-2	φ-2	NOUN
ejpam-3126	283	8	-	-	PUNCT
ejpam-3126	283	9	absorbing	absorbing	ADJ
ejpam-3126	283	10	semi	semi	ADJ
ejpam-3126	283	11	-	-	ADJ
ejpam-3126	283	12	primary	primary	ADJ
ejpam-3126	283	13	submodule	submodule	NOUN
ejpam-3126	283	14	of	of	ADP
ejpam-3126	283	15	m	m	PROPN
ejpam-3126	283	16	,	,	PUNCT
ejpam-3126	283	17	then	then	ADV
ejpam-3126	283	18	k	k	X
ejpam-3126	283	19	/	/	SYM
ejpam-3126	283	20	n	n	PROPN
ejpam-3126	283	21	is	be	AUX
ejpam-3126	283	22	a	a	DET
ejpam-3126	283	23	φ0	φ0	PROPN
ejpam-3126	283	24	-	-	PUNCT
ejpam-3126	283	25	2	2	NUM
ejpam-3126	283	26	-	-	PUNCT
ejpam-3126	283	27	absorbing	absorbing	ADJ
ejpam-3126	283	28	semi	semi	ADJ
ejpam-3126	283	29	-	-	ADJ
ejpam-3126	283	30	primary	primary	ADJ
ejpam-3126	283	31	submodule	submodule	NOUN
ejpam-3126	283	32	of	of	ADP
ejpam-3126	283	33	m	m	PROPN
ejpam-3126	283	34	/	/	SYM
ejpam-3126	283	35	n	n	PROPN
ejpam-3126	283	36	.	.	PUNCT
ejpam-3126	284	1	proof	proof	NOUN
ejpam-3126	284	2	.	.	PUNCT
ejpam-3126	285	1	let	let	VERB
ejpam-3126	285	2	a1	a1	NOUN
ejpam-3126	285	3	,	,	PUNCT
ejpam-3126	285	4	a2	a2	PROPN
ejpam-3126	285	5	∈	∈	PROPN
ejpam-3126	285	6	r	r	NOUN
ejpam-3126	285	7	and	and	CCONJ
ejpam-3126	285	8	m	m	PROPN
ejpam-3126	285	9	∈	∈	NOUN
ejpam-3126	285	10	m	m	VERB
ejpam-3126	285	11	such	such	ADJ
ejpam-3126	285	12	that	that	SCONJ
ejpam-3126	285	13	a1a2	a1a2	PROPN
ejpam-3126	285	14	m	m	VERB
ejpam-3126	285	15	+	+	NUM
ejpam-3126	285	16	n	n	CCONJ
ejpam-3126	285	17	∈	∈	PROPN
ejpam-3126	285	18	k	k	PROPN
ejpam-3126	285	19	/	/	SYM
ejpam-3126	285	20	n	n	PRON
ejpam-3126	285	21	−	−	PROPN
ejpam-3126	285	22	(	(	PUNCT
ejpam-3126	285	23	φn	φn	NOUN
ejpam-3126	285	24	)	)	PUNCT
ejpam-3126	285	25	0(k	0(k	NUM
ejpam-3126	285	26	/	/	SYM
ejpam-3126	285	27	n	n	CCONJ
ejpam-3126	285	28	)	)	PUNCT
ejpam-3126	285	29	.	.	PUNCT
ejpam-3126	286	1	since	since	SCONJ
ejpam-3126	286	2	φ(k	φ(k	PROPN
ejpam-3126	286	3	)	)	PUNCT
ejpam-3126	286	4	⊆	⊆	NUM
ejpam-3126	286	5	n	n	NOUN
ejpam-3126	286	6	,	,	PUNCT
ejpam-3126	286	7	we	we	PRON
ejpam-3126	286	8	have	have	VERB
ejpam-3126	286	9	a1a2	a1a2	PROPN
ejpam-3126	286	10	m	m	VERB
ejpam-3126	286	11	6∈	6∈	PROPN
ejpam-3126	286	12	φ(k	φ(k	PROPN
ejpam-3126	286	13	)	)	PUNCT
ejpam-3126	286	14	.	.	PUNCT
ejpam-3126	287	1	clearly	clearly	ADV
ejpam-3126	287	2	,	,	PUNCT
ejpam-3126	287	3	a1a2	a1a2	ADP
ejpam-3126	287	4	m	m	VERB
ejpam-3126	287	5	∈	∈	ADJ
ejpam-3126	287	6	k	k	NOUN
ejpam-3126	288	1	−	−	PROPN
ejpam-3126	288	2	φ(k	φ(k	PROPN
ejpam-3126	288	3	)	)	PUNCT
ejpam-3126	288	4	.	.	PUNCT
ejpam-3126	289	1	by	by	ADP
ejpam-3126	289	2	definition	definition	NOUN
ejpam-3126	289	3	1	1	NUM
ejpam-3126	289	4	,	,	PUNCT
ejpam-3126	289	5	a1a2	a1a2	PROPN
ejpam-3126	289	6	∈	∈	NOUN
ejpam-3126	289	7	√	√	NUM
ejpam-3126	289	8	(	(	PUNCT
ejpam-3126	289	9	k	k	NOUN
ejpam-3126	289	10	:	:	PUNCT
ejpam-3126	289	11	m	m	X
ejpam-3126	289	12	)	)	PUNCT
ejpam-3126	289	13	or	or	CCONJ
ejpam-3126	289	14	a1	a1	NOUN
ejpam-3126	289	15	m	m	PROPN
ejpam-3126	289	16	∈	∈	NOUN
ejpam-3126	289	17	k	k	NOUN
ejpam-3126	289	18	or	or	CCONJ
ejpam-3126	289	19	an2	an2	PROPN
ejpam-3126	289	20	m	m	NOUN
ejpam-3126	289	21	∈	∈	PROPN
ejpam-3126	289	22	k	k	NOUN
ejpam-3126	289	23	for	for	ADP
ejpam-3126	289	24	some	some	DET
ejpam-3126	289	25	positive	positive	ADJ
ejpam-3126	289	26	integer	integer	NOUN
ejpam-3126	289	27	n.	n.	NOUN
ejpam-3126	289	28	thus	thus	ADV
ejpam-3126	289	29	a1a2	a1a2	ADP
ejpam-3126	289	30	∈√	∈√	PROPN
ejpam-3126	289	31	(	(	PUNCT
ejpam-3126	289	32	k	k	NOUN
ejpam-3126	289	33	/	/	SYM
ejpam-3126	289	34	n	n	NUM
ejpam-3126	289	35	:	:	PUNCT
ejpam-3126	289	36	m	m	X
ejpam-3126	289	37	/	/	SYM
ejpam-3126	289	38	n	n	CCONJ
ejpam-3126	289	39	)	)	PUNCT
ejpam-3126	289	40	or	or	CCONJ
ejpam-3126	289	41	a1(m+n	a1(m+n	NOUN
ejpam-3126	289	42	)	)	PUNCT
ejpam-3126	289	43	∈	∈	PROPN
ejpam-3126	290	1	k	k	PROPN
ejpam-3126	290	2	/	/	SYM
ejpam-3126	290	3	n	n	PROPN
ejpam-3126	290	4	or	or	CCONJ
ejpam-3126	290	5	an2	an2	PROPN
ejpam-3126	290	6	(	(	PUNCT
ejpam-3126	290	7	m+n	m+n	PROPN
ejpam-3126	290	8	)	)	PUNCT
ejpam-3126	290	9	∈	∈	PROPN
ejpam-3126	290	10	k	k	PROPN
ejpam-3126	290	11	/	/	SYM
ejpam-3126	290	12	n	n	PROPN
ejpam-3126	290	13	for	for	ADP
ejpam-3126	290	14	some	some	DET
ejpam-3126	290	15	positive	positive	ADJ
ejpam-3126	290	16	integer	integer	NOUN
ejpam-3126	290	17	n.	n.	NOUN
ejpam-3126	290	18	theorem	theorem	VERB
ejpam-3126	290	19	11	11	NUM
ejpam-3126	290	20	.	.	PUNCT
ejpam-3126	291	1	let	let	VERB
ejpam-3126	291	2	φ	φ	PROPN
ejpam-3126	291	3	:	:	PUNCT
ejpam-3126	291	4	s(m	s(m	PROPN
ejpam-3126	291	5	)	)	PUNCT
ejpam-3126	291	6	→	→	SYM
ejpam-3126	291	7	s(m	s(m	NOUN
ejpam-3126	291	8	)	)	PUNCT
ejpam-3126	291	9	∪	∪	NOUN
ejpam-3126	291	10	{	{	PUNCT
ejpam-3126	291	11	∅	∅	NOUN
ejpam-3126	291	12	}	}	PUNCT
ejpam-3126	291	13	be	be	AUX
ejpam-3126	291	14	a	a	DET
ejpam-3126	291	15	function	function	NOUN
ejpam-3126	291	16	,	,	PUNCT
ejpam-3126	291	17	n	n	CCONJ
ejpam-3126	291	18	be	be	AUX
ejpam-3126	291	19	a	a	DET
ejpam-3126	291	20	φ-2	φ-2	ADJ
ejpam-3126	291	21	-	-	PUNCT
ejpam-3126	291	22	absorbing	absorbing	ADJ
ejpam-3126	291	23	semiprimary	semiprimary	ADJ
ejpam-3126	291	24	submodule	submodule	NOUN
ejpam-3126	291	25	of	of	ADP
ejpam-3126	291	26	m	m	PRON
ejpam-3126	291	27	and	and	CCONJ
ejpam-3126	291	28	let	let	VERB
ejpam-3126	291	29	k	k	PRON
ejpam-3126	291	30	be	be	AUX
ejpam-3126	291	31	a	a	DET
ejpam-3126	291	32	submodule	submodule	NOUN
ejpam-3126	291	33	of	of	ADP
ejpam-3126	291	34	m	m	PROPN
ejpam-3126	291	35	with	with	ADP
ejpam-3126	291	36	n	n	PRON
ejpam-3126	291	37	⊆	⊆	NUM
ejpam-3126	291	38	k.	k.	NOUN
ejpam-3126	291	39	if	if	SCONJ
ejpam-3126	291	40	φ(n	φ(n	NOUN
ejpam-3126	291	41	)	)	PUNCT
ejpam-3126	291	42	⊆	⊆	NUM
ejpam-3126	291	43	φ(k	φ(k	PROPN
ejpam-3126	291	44	)	)	PUNCT
ejpam-3126	291	45	and	and	CCONJ
ejpam-3126	291	46	k	k	PROPN
ejpam-3126	291	47	/	/	SYM
ejpam-3126	291	48	n	n	PROPN
ejpam-3126	291	49	is	be	AUX
ejpam-3126	291	50	a	a	DET
ejpam-3126	291	51	φ0	φ0	PROPN
ejpam-3126	291	52	-	-	PUNCT
ejpam-3126	291	53	2	2	NUM
ejpam-3126	291	54	-	-	PUNCT
ejpam-3126	291	55	absorbing	absorbing	ADJ
ejpam-3126	291	56	semi	semi	ADJ
ejpam-3126	291	57	-	-	ADJ
ejpam-3126	291	58	primary	primary	ADJ
ejpam-3126	291	59	submodule	submodule	NOUN
ejpam-3126	291	60	of	of	ADP
ejpam-3126	291	61	m	m	PROPN
ejpam-3126	291	62	/	/	SYM
ejpam-3126	291	63	n	n	PROPN
ejpam-3126	291	64	,	,	PUNCT
ejpam-3126	291	65	then	then	ADV
ejpam-3126	291	66	k	k	PROPN
ejpam-3126	291	67	is	be	AUX
ejpam-3126	291	68	a	a	DET
ejpam-3126	291	69	φ-2	φ-2	NOUN
ejpam-3126	291	70	-	-	PUNCT
ejpam-3126	291	71	absorbing	absorbing	ADJ
ejpam-3126	291	72	semi	semi	ADJ
ejpam-3126	291	73	-	-	ADJ
ejpam-3126	291	74	primary	primary	ADJ
ejpam-3126	291	75	submodule	submodule	NOUN
ejpam-3126	291	76	of	of	ADP
ejpam-3126	291	77	m	m	PROPN
ejpam-3126	291	78	.	.	PUNCT
ejpam-3126	292	1	proof	proof	NOUN
ejpam-3126	292	2	.	.	PUNCT
ejpam-3126	293	1	let	let	VERB
ejpam-3126	293	2	a1	a1	NOUN
ejpam-3126	293	3	,	,	PUNCT
ejpam-3126	293	4	a2	a2	PROPN
ejpam-3126	293	5	∈	∈	PROPN
ejpam-3126	293	6	r	r	NOUN
ejpam-3126	293	7	and	and	CCONJ
ejpam-3126	293	8	m	m	PROPN
ejpam-3126	293	9	∈	∈	NOUN
ejpam-3126	293	10	m	m	VERB
ejpam-3126	293	11	such	such	ADJ
ejpam-3126	293	12	that	that	SCONJ
ejpam-3126	293	13	a1a2	a1a2	PROPN
ejpam-3126	293	14	m	m	VERB
ejpam-3126	293	15	∈	∈	ADJ
ejpam-3126	293	16	k	k	NOUN
ejpam-3126	293	17	−	−	PROPN
ejpam-3126	293	18	φ(k	φ(k	PROPN
ejpam-3126	293	19	)	)	PUNCT
ejpam-3126	293	20	.	.	PUNCT
ejpam-3126	294	1	by	by	ADP
ejpam-3126	294	2	assumption	assumption	NOUN
ejpam-3126	294	3	,	,	PUNCT
ejpam-3126	294	4	a1a2	a1a2	PROPN
ejpam-3126	294	5	m	m	PROPN
ejpam-3126	294	6	6∈	6∈	NOUN
ejpam-3126	294	7	φ(n	φ(n	ADJ
ejpam-3126	294	8	)	)	PUNCT
ejpam-3126	294	9	.	.	PUNCT
ejpam-3126	295	1	if	if	SCONJ
ejpam-3126	295	2	a1a2	a1a2	PROPN
ejpam-3126	295	3	m	m	VERB
ejpam-3126	295	4	∈	∈	ADJ
ejpam-3126	295	5	n	n	NOUN
ejpam-3126	295	6	,	,	PUNCT
ejpam-3126	295	7	then	then	ADV
ejpam-3126	295	8	a1a2	a1a2	ADP
ejpam-3126	295	9	m	m	VERB
ejpam-3126	295	10	∈	∈	ADJ
ejpam-3126	295	11	n	n	CCONJ
ejpam-3126	295	12	−	−	PROPN
ejpam-3126	295	13	φ(n	φ(n	NOUN
ejpam-3126	295	14	)	)	PUNCT
ejpam-3126	295	15	.	.	PUNCT
ejpam-3126	296	1	by	by	ADP
ejpam-3126	296	2	definition	definition	NOUN
ejpam-3126	296	3	1	1	NUM
ejpam-3126	296	4	,	,	PUNCT
ejpam-3126	296	5	a1a2	a1a2	ADP
ejpam-3126	296	6	∈√	∈√	PROPN
ejpam-3126	296	7	(	(	PUNCT
ejpam-3126	296	8	n	n	NUM
ejpam-3126	296	9	:	:	PUNCT
ejpam-3126	296	10	m	m	X
ejpam-3126	296	11	)	)	PUNCT
ejpam-3126	297	1	⊆	⊆	NUM
ejpam-3126	297	2	√	√	NUM
ejpam-3126	297	3	(	(	PUNCT
ejpam-3126	297	4	k	k	NOUN
ejpam-3126	297	5	:	:	PUNCT
ejpam-3126	297	6	m	m	X
ejpam-3126	297	7	)	)	PUNCT
ejpam-3126	297	8	or	or	CCONJ
ejpam-3126	297	9	a1	a1	NOUN
ejpam-3126	297	10	m	m	PROPN
ejpam-3126	297	11	∈	∈	NOUN
ejpam-3126	297	12	n	n	ADP
ejpam-3126	297	13	⊆	⊆	NUM
ejpam-3126	297	14	k	k	PROPN
ejpam-3126	297	15	or	or	CCONJ
ejpam-3126	297	16	an2	an2	PROPN
ejpam-3126	297	17	m	m	NOUN
ejpam-3126	297	18	∈	∈	PROPN
ejpam-3126	297	19	n	n	ADP
ejpam-3126	297	20	⊆	⊆	NUM
ejpam-3126	297	21	k	k	NOUN
ejpam-3126	297	22	for	for	ADP
ejpam-3126	297	23	some	some	DET
ejpam-3126	297	24	positive	positive	ADJ
ejpam-3126	297	25	integer	integer	NOUN
ejpam-3126	297	26	n.	n.	NOUN
ejpam-3126	297	27	if	if	SCONJ
ejpam-3126	297	28	a1a2	a1a2	PROPN
ejpam-3126	297	29	m	m	VERB
ejpam-3126	297	30	6∈	6∈	NUM
ejpam-3126	297	31	n	n	NOUN
ejpam-3126	297	32	,	,	PUNCT
ejpam-3126	297	33	then	then	ADV
ejpam-3126	297	34	a1a2(m+n	a1a2(m+n	PROPN
ejpam-3126	297	35	)	)	PUNCT
ejpam-3126	297	36	6∈	6∈	PROPN
ejpam-3126	297	37	φ0(k	φ0(k	NOUN
ejpam-3126	297	38	/	/	SYM
ejpam-3126	297	39	n	n	CCONJ
ejpam-3126	297	40	)	)	PUNCT
ejpam-3126	297	41	.	.	PUNCT
ejpam-3126	298	1	therefore	therefore	ADV
ejpam-3126	298	2	a1a2(m+n	a1a2(m+n	PROPN
ejpam-3126	298	3	)	)	PUNCT
ejpam-3126	298	4	∈	∈	PROPN
ejpam-3126	299	1	k	k	PROPN
ejpam-3126	299	2	/	/	SYM
ejpam-3126	299	3	n	n	CCONJ
ejpam-3126	299	4	−φ0(k	−φ0(k	NOUN
ejpam-3126	299	5	/	/	SYM
ejpam-3126	299	6	n	n	CCONJ
ejpam-3126	299	7	)	)	PUNCT
ejpam-3126	299	8	.	.	PUNCT
ejpam-3126	300	1	by	by	ADP
ejpam-3126	300	2	definition	definition	NOUN
ejpam-3126	300	3	1	1	NUM
ejpam-3126	300	4	,	,	PUNCT
ejpam-3126	300	5	a1a2	a1a2	PROPN
ejpam-3126	300	6	∈	∈	NOUN
ejpam-3126	300	7	√	√	ADP
ejpam-3126	300	8	(	(	PUNCT
ejpam-3126	300	9	k	k	NOUN
ejpam-3126	300	10	/	/	SYM
ejpam-3126	300	11	n	n	NUM
ejpam-3126	300	12	:	:	PUNCT
ejpam-3126	300	13	m	m	X
ejpam-3126	300	14	/	/	SYM
ejpam-3126	300	15	n	n	CCONJ
ejpam-3126	300	16	)	)	PUNCT
ejpam-3126	300	17	or	or	CCONJ
ejpam-3126	300	18	a1(m+n	a1(m+n	NOUN
ejpam-3126	300	19	)	)	PUNCT
ejpam-3126	300	20	∈	∈	PROPN
ejpam-3126	300	21	k	k	PROPN
ejpam-3126	300	22	/	/	SYM
ejpam-3126	300	23	n	n	PROPN
ejpam-3126	300	24	or	or	CCONJ
ejpam-3126	300	25	an2	an2	PROPN
ejpam-3126	300	26	(	(	PUNCT
ejpam-3126	300	27	m+n	m+n	PROPN
ejpam-3126	300	28	)	)	PUNCT
ejpam-3126	300	29	∈	∈	PROPN
ejpam-3126	301	1	k	k	PROPN
ejpam-3126	301	2	/	/	SYM
ejpam-3126	301	3	n	n	PROPN
ejpam-3126	301	4	for	for	ADP
ejpam-3126	301	5	some	some	DET
ejpam-3126	301	6	positive	positive	ADJ
ejpam-3126	301	7	integer	integer	NOUN
ejpam-3126	301	8	n.	n.	NOUN
ejpam-3126	301	9	this	this	PRON
ejpam-3126	301	10	completes	complete	VERB
ejpam-3126	301	11	the	the	DET
ejpam-3126	301	12	proof	proof	NOUN
ejpam-3126	301	13	.	.	PUNCT
ejpam-3126	302	1	as	as	ADP
ejpam-3126	302	2	an	an	DET
ejpam-3126	302	3	immediate	immediate	ADJ
ejpam-3126	302	4	consequence	consequence	NOUN
ejpam-3126	302	5	of	of	ADP
ejpam-3126	302	6	theorem	theorem	ADJ
ejpam-3126	302	7	10	10	NUM
ejpam-3126	302	8	and	and	CCONJ
ejpam-3126	302	9	theorem	theorem	VERB
ejpam-3126	302	10	11	11	NUM
ejpam-3126	302	11	we	we	PRON
ejpam-3126	302	12	have	have	VERB
ejpam-3126	302	13	the	the	DET
ejpam-3126	302	14	next	next	ADJ
ejpam-3126	302	15	corollary	corollary	NOUN
ejpam-3126	302	16	.	.	PUNCT
ejpam-3126	303	1	corollary	corollary	ADJ
ejpam-3126	303	2	5	5	NUM
ejpam-3126	303	3	.	.	PUNCT
ejpam-3126	304	1	let	let	VERB
ejpam-3126	304	2	φ	φ	NOUN
ejpam-3126	304	3	:	:	PUNCT
ejpam-3126	304	4	s(m)→	s(m)→	NOUN
ejpam-3126	304	5	s(m	s(m	NOUN
ejpam-3126	304	6	)	)	PUNCT
ejpam-3126	304	7	∪	∪	ADP
ejpam-3126	304	8	{	{	PUNCT
ejpam-3126	304	9	∅	∅	NOUN
ejpam-3126	304	10	}	}	PUNCT
ejpam-3126	304	11	be	be	AUX
ejpam-3126	304	12	a	a	DET
ejpam-3126	304	13	function	function	NOUN
ejpam-3126	304	14	and	and	CCONJ
ejpam-3126	304	15	let	let	VERB
ejpam-3126	304	16	n	n	PRON
ejpam-3126	304	17	,	,	PUNCT
ejpam-3126	304	18	k	k	X
ejpam-3126	304	19	be	be	VERB
ejpam-3126	304	20	two	two	NUM
ejpam-3126	304	21	submodules	submodule	NOUN
ejpam-3126	304	22	of	of	ADP
ejpam-3126	304	23	m	m	PROPN
ejpam-3126	304	24	with	with	ADP
ejpam-3126	304	25	n	n	PRON
ejpam-3126	304	26	⊆	⊆	NUM
ejpam-3126	304	27	k.	k.	NOUN
ejpam-3126	304	28	then	then	ADV
ejpam-3126	304	29	n	n	PROPN
ejpam-3126	304	30	is	be	AUX
ejpam-3126	304	31	a	a	DET
ejpam-3126	304	32	φ-2	φ-2	NOUN
ejpam-3126	304	33	-	-	PUNCT
ejpam-3126	304	34	absorbing	absorbing	ADJ
ejpam-3126	304	35	semi	semi	ADJ
ejpam-3126	304	36	-	-	ADJ
ejpam-3126	304	37	primary	primary	ADJ
ejpam-3126	304	38	submodule	submodule	NOUN
ejpam-3126	304	39	of	of	ADP
ejpam-3126	304	40	m	m	PROPN
ejpam-3126	304	41	if	if	SCONJ
ejpam-3126	305	1	and	and	CCONJ
ejpam-3126	305	2	only	only	ADV
ejpam-3126	305	3	if	if	SCONJ
ejpam-3126	305	4	n	n	CCONJ
ejpam-3126	305	5	/	/	SYM
ejpam-3126	305	6	φ(n	φ(n	ADJ
ejpam-3126	305	7	)	)	PUNCT
ejpam-3126	305	8	is	be	AUX
ejpam-3126	305	9	a	a	DET
ejpam-3126	305	10	φ0	φ0	PROPN
ejpam-3126	305	11	-	-	PUNCT
ejpam-3126	305	12	2	2	NUM
ejpam-3126	305	13	-	-	PUNCT
ejpam-3126	305	14	absorbing	absorbing	ADJ
ejpam-3126	305	15	semi	semi	ADJ
ejpam-3126	305	16	-	-	ADJ
ejpam-3126	305	17	primary	primary	ADJ
ejpam-3126	305	18	submodule	submodule	NOUN
ejpam-3126	305	19	of	of	ADP
ejpam-3126	305	20	m	m	PROPN
ejpam-3126	305	21	/	/	SYM
ejpam-3126	305	22	φ(n	φ(n	ADJ
ejpam-3126	305	23	)	)	PUNCT
ejpam-3126	305	24	.	.	PUNCT
ejpam-3126	306	1	proof	proof	NOUN
ejpam-3126	306	2	.	.	PUNCT
ejpam-3126	307	1	it	it	PRON
ejpam-3126	307	2	is	be	AUX
ejpam-3126	307	3	straightforward	straightforward	ADJ
ejpam-3126	307	4	by	by	ADP
ejpam-3126	307	5	theorem	theorem	ADJ
ejpam-3126	307	6	10	10	NUM
ejpam-3126	307	7	and	and	CCONJ
ejpam-3126	307	8	theorem	theorem	VERB
ejpam-3126	307	9	11	11	NUM
ejpam-3126	307	10	.	.	PUNCT
ejpam-3126	308	1	theorem	theorem	NOUN
ejpam-3126	308	2	12	12	NUM
ejpam-3126	308	3	.	.	PUNCT
ejpam-3126	309	1	let	let	VERB
ejpam-3126	309	2	φα	φα	X
ejpam-3126	309	3	:	:	PUNCT
ejpam-3126	309	4	s(m)→	s(m)→	NOUN
ejpam-3126	309	5	s(m	s(m	NOUN
ejpam-3126	309	6	)	)	PUNCT
ejpam-3126	309	7	∪	∪	ADP
ejpam-3126	309	8	{	{	PUNCT
ejpam-3126	309	9	∅	∅	NOUN
ejpam-3126	309	10	}	}	PUNCT
ejpam-3126	309	11	be	be	AUX
ejpam-3126	309	12	a	a	DET
ejpam-3126	309	13	function	function	NOUN
ejpam-3126	309	14	.	.	PUNCT
ejpam-3126	310	1	then	then	ADV
ejpam-3126	310	2	the	the	DET
ejpam-3126	310	3	following	follow	VERB
ejpam-3126	310	4	hold	hold	NOUN
ejpam-3126	310	5	.	.	PUNCT
ejpam-3126	311	1	(	(	PUNCT
ejpam-3126	311	2	i	i	NOUN
ejpam-3126	311	3	)	)	PUNCT
ejpam-3126	311	4	if	if	SCONJ
ejpam-3126	311	5	n	n	PRON
ejpam-3126	311	6	is	be	AUX
ejpam-3126	311	7	a	a	DET
ejpam-3126	311	8	φβ-2	φβ-2	VERB
ejpam-3126	311	9	-	-	PUNCT
ejpam-3126	311	10	absorbing	absorbing	ADJ
ejpam-3126	311	11	semi	semi	ADJ
ejpam-3126	311	12	-	-	ADJ
ejpam-3126	311	13	primary	primary	ADJ
ejpam-3126	311	14	submodule	submodule	NOUN
ejpam-3126	311	15	of	of	ADP
ejpam-3126	311	16	m	m	PROPN
ejpam-3126	311	17	such	such	ADJ
ejpam-3126	311	18	that	that	SCONJ
ejpam-3126	311	19	φβ	φβ	ADJ
ejpam-3126	311	20	≤	≤	NOUN
ejpam-3126	311	21	φγ	φγ	ADP
ejpam-3126	311	22	,	,	PUNCT
ejpam-3126	311	23	then	then	ADV
ejpam-3126	311	24	n	n	PRON
ejpam-3126	311	25	is	be	AUX
ejpam-3126	311	26	a	a	DET
ejpam-3126	311	27	φγ-2	φγ-2	ADV
ejpam-3126	311	28	-	-	PUNCT
ejpam-3126	311	29	absorbing	absorbing	ADJ
ejpam-3126	311	30	semi	semi	ADJ
ejpam-3126	311	31	-	-	ADJ
ejpam-3126	311	32	primary	primary	ADJ
ejpam-3126	311	33	submodule	submodule	NOUN
ejpam-3126	311	34	of	of	ADP
ejpam-3126	311	35	m	m	PROPN
ejpam-3126	311	36	.	.	PUNCT
ejpam-3126	312	1	pai	pai	PROPN
ejpam-3126	312	2	.	.	PROPN
ejpam-3126	312	3	yiarayong	yiarayong	PROPN
ejpam-3126	312	4	,	,	PUNCT
ejpam-3126	312	5	m.	m.	NOUN
ejpam-3126	312	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	312	7	/	/	SYM
ejpam-3126	312	8	eur	eur	PROPN
ejpam-3126	312	9	.	.	PUNCT
ejpam-3126	313	1	j.	j.	PROPN
ejpam-3126	313	2	pure	pure	PROPN
ejpam-3126	313	3	appl	appl	PROPN
ejpam-3126	313	4	.	.	PROPN
ejpam-3126	313	5	math	math	PROPN
ejpam-3126	313	6	,	,	PUNCT
ejpam-3126	313	7	11	11	NUM
ejpam-3126	313	8	(	(	PUNCT
ejpam-3126	313	9	1	1	NUM
ejpam-3126	313	10	)	)	PUNCT
ejpam-3126	313	11	(	(	PUNCT
ejpam-3126	313	12	2018	2018	NUM
ejpam-3126	313	13	)	)	PUNCT
ejpam-3126	313	14	,	,	PUNCT
ejpam-3126	313	15	35	35	NUM
ejpam-3126	313	16	-	-	SYM
ejpam-3126	313	17	50	50	NUM
ejpam-3126	313	18	44	44	NUM
ejpam-3126	313	19	(	(	PUNCT
ejpam-3126	313	20	ii	ii	NOUN
ejpam-3126	313	21	)	)	PUNCT
ejpam-3126	313	22	if	if	SCONJ
ejpam-3126	313	23	n	n	PRON
ejpam-3126	313	24	is	be	AUX
ejpam-3126	313	25	a	a	DET
ejpam-3126	313	26	φ∅-2	φ∅-2	NOUN
ejpam-3126	313	27	-	-	PUNCT
ejpam-3126	313	28	absorbing	absorbing	ADJ
ejpam-3126	313	29	semi	semi	ADJ
ejpam-3126	313	30	-	-	ADJ
ejpam-3126	313	31	primary	primary	ADJ
ejpam-3126	313	32	submodule	submodule	NOUN
ejpam-3126	313	33	of	of	ADP
ejpam-3126	313	34	m	m	PROPN
ejpam-3126	313	35	,	,	PUNCT
ejpam-3126	313	36	then	then	ADV
ejpam-3126	313	37	n	n	PRON
ejpam-3126	313	38	is	be	AUX
ejpam-3126	313	39	a	a	DET
ejpam-3126	313	40	φ0	φ0	PROPN
ejpam-3126	313	41	-	-	PUNCT
ejpam-3126	313	42	2	2	NUM
ejpam-3126	313	43	-	-	PUNCT
ejpam-3126	313	44	absorbing	absorbing	ADJ
ejpam-3126	313	45	semi	semi	ADJ
ejpam-3126	313	46	-	-	ADJ
ejpam-3126	313	47	primary	primary	ADJ
ejpam-3126	313	48	submodule	submodule	NOUN
ejpam-3126	313	49	of	of	ADP
ejpam-3126	313	50	m	m	PROPN
ejpam-3126	313	51	.	.	PUNCT
ejpam-3126	314	1	(	(	PUNCT
ejpam-3126	314	2	iii	iii	X
ejpam-3126	314	3	)	)	PUNCT
ejpam-3126	314	4	if	if	SCONJ
ejpam-3126	314	5	n	n	PRON
ejpam-3126	314	6	is	be	AUX
ejpam-3126	314	7	a	a	DET
ejpam-3126	314	8	φ0	φ0	PROPN
ejpam-3126	314	9	-	-	PUNCT
ejpam-3126	314	10	2	2	NUM
ejpam-3126	314	11	-	-	PUNCT
ejpam-3126	314	12	absorbing	absorbing	ADJ
ejpam-3126	314	13	semi	semi	ADJ
ejpam-3126	314	14	-	-	ADJ
ejpam-3126	314	15	primary	primary	ADJ
ejpam-3126	314	16	submodule	submodule	NOUN
ejpam-3126	314	17	of	of	ADP
ejpam-3126	314	18	m	m	PROPN
ejpam-3126	314	19	,	,	PUNCT
ejpam-3126	314	20	then	then	ADV
ejpam-3126	314	21	n	n	PRON
ejpam-3126	314	22	is	be	AUX
ejpam-3126	314	23	an	an	DET
ejpam-3126	314	24	ω-2	ω-2	NOUN
ejpam-3126	314	25	-	-	PUNCT
ejpam-3126	314	26	absorbing	absorb	VERB
ejpam-3126	314	27	semi	semi	ADJ
ejpam-3126	314	28	-	-	ADJ
ejpam-3126	314	29	primary	primary	ADJ
ejpam-3126	314	30	submodule	submodule	NOUN
ejpam-3126	314	31	of	of	ADP
ejpam-3126	314	32	m	m	PROPN
ejpam-3126	314	33	.	.	PUNCT
ejpam-3126	315	1	(	(	PUNCT
ejpam-3126	315	2	iv	iv	X
ejpam-3126	315	3	)	)	PUNCT
ejpam-3126	315	4	if	if	SCONJ
ejpam-3126	315	5	n	n	PRON
ejpam-3126	315	6	is	be	AUX
ejpam-3126	315	7	a	a	DET
ejpam-3126	315	8	φω-2	φω-2	ADV
ejpam-3126	315	9	-	-	PUNCT
ejpam-3126	315	10	absorbing	absorb	VERB
ejpam-3126	315	11	semi	semi	ADJ
ejpam-3126	315	12	-	-	ADJ
ejpam-3126	315	13	primary	primary	ADJ
ejpam-3126	315	14	submodule	submodule	NOUN
ejpam-3126	315	15	of	of	ADP
ejpam-3126	315	16	m	m	PROPN
ejpam-3126	315	17	,	,	PUNCT
ejpam-3126	315	18	then	then	ADV
ejpam-3126	315	19	n	n	PRON
ejpam-3126	315	20	is	be	AUX
ejpam-3126	315	21	a	a	DET
ejpam-3126	315	22	φn-2	φn-2	ADV
ejpam-3126	315	23	-	-	PUNCT
ejpam-3126	315	24	absorbing	absorbing	ADJ
ejpam-3126	315	25	semi	semi	ADJ
ejpam-3126	315	26	-	-	ADJ
ejpam-3126	315	27	primary	primary	ADJ
ejpam-3126	315	28	submodule	submodule	NOUN
ejpam-3126	315	29	of	of	ADP
ejpam-3126	315	30	m	m	PROPN
ejpam-3126	315	31	.	.	PUNCT
ejpam-3126	316	1	proof	proof	NOUN
ejpam-3126	316	2	.	.	PUNCT
ejpam-3126	317	1	i.	i.	PROPN
ejpam-3126	317	2	let	let	VERB
ejpam-3126	317	3	a1	a1	PROPN
ejpam-3126	317	4	,	,	PUNCT
ejpam-3126	317	5	a2	a2	PROPN
ejpam-3126	317	6	∈	∈	PROPN
ejpam-3126	317	7	r	r	NOUN
ejpam-3126	317	8	and	and	CCONJ
ejpam-3126	317	9	m	m	PROPN
ejpam-3126	317	10	∈	∈	NOUN
ejpam-3126	317	11	m	m	VERB
ejpam-3126	317	12	such	such	ADJ
ejpam-3126	317	13	that	that	SCONJ
ejpam-3126	317	14	a1a2	a1a2	PROPN
ejpam-3126	317	15	m	m	VERB
ejpam-3126	317	16	∈	∈	ADJ
ejpam-3126	317	17	n	n	PRON
ejpam-3126	317	18	−	−	NOUN
ejpam-3126	317	19	φγ(n	φγ(n	NUM
ejpam-3126	317	20	)	)	PUNCT
ejpam-3126	317	21	.	.	PUNCT
ejpam-3126	318	1	by	by	ADP
ejpam-3126	318	2	assumption	assumption	NOUN
ejpam-3126	318	3	,	,	PUNCT
ejpam-3126	318	4	φβ(n	φβ(n	NOUN
ejpam-3126	318	5	)	)	PUNCT
ejpam-3126	318	6	⊆	⊆	NUM
ejpam-3126	318	7	φγ(n	φγ(n	NUM
ejpam-3126	318	8	)	)	PUNCT
ejpam-3126	318	9	.	.	PUNCT
ejpam-3126	319	1	then	then	ADV
ejpam-3126	319	2	a1a2	a1a2	ADP
ejpam-3126	319	3	m	m	VERB
ejpam-3126	319	4	∈	∈	ADJ
ejpam-3126	319	5	n	n	PRON
ejpam-3126	319	6	−	−	NOUN
ejpam-3126	319	7	φγ(n	φγ(n	NUM
ejpam-3126	319	8	)	)	PUNCT
ejpam-3126	319	9	⊆	⊆	NUM
ejpam-3126	319	10	n	n	NUM
ejpam-3126	319	11	−	−	PROPN
ejpam-3126	319	12	φβ(n	φβ(n	NUM
ejpam-3126	319	13	)	)	PUNCT
ejpam-3126	319	14	.	.	PUNCT
ejpam-3126	320	1	then	then	ADV
ejpam-3126	320	2	by	by	ADP
ejpam-3126	320	3	definition	definition	NOUN
ejpam-3126	320	4	1	1	NUM
ejpam-3126	320	5	,	,	PUNCT
ejpam-3126	320	6	a1a2	a1a2	PROPN
ejpam-3126	320	7	∈	∈	NOUN
ejpam-3126	320	8	√	√	NUM
ejpam-3126	320	9	(	(	PUNCT
ejpam-3126	320	10	n	n	NUM
ejpam-3126	320	11	:	:	PUNCT
ejpam-3126	320	12	m	m	X
ejpam-3126	320	13	)	)	PUNCT
ejpam-3126	320	14	or	or	CCONJ
ejpam-3126	320	15	a1	a1	NOUN
ejpam-3126	320	16	m	m	PROPN
ejpam-3126	320	17	∈	∈	NOUN
ejpam-3126	320	18	n	n	NOUN
ejpam-3126	320	19	or	or	CCONJ
ejpam-3126	320	20	an2	an2	PROPN
ejpam-3126	320	21	m	m	NOUN
ejpam-3126	320	22	∈	∈	PROPN
ejpam-3126	320	23	n	n	NOUN
ejpam-3126	320	24	for	for	ADP
ejpam-3126	320	25	some	some	DET
ejpam-3126	320	26	positive	positive	ADJ
ejpam-3126	320	27	integer	integer	NOUN
ejpam-3126	320	28	n.	n.	PROPN
ejpam-3126	320	29	ii	ii	PROPN
ejpam-3126	320	30	,	,	PUNCT
ejpam-3126	320	31	iii	iii	PROPN
ejpam-3126	320	32	,	,	PUNCT
ejpam-3126	320	33	iv	iv	X
ejpam-3126	320	34	.	.	PUNCT
ejpam-3126	321	1	it	it	PRON
ejpam-3126	321	2	is	be	AUX
ejpam-3126	321	3	obvious	obvious	ADJ
ejpam-3126	321	4	.	.	PUNCT
ejpam-3126	322	1	from	from	ADP
ejpam-3126	322	2	the	the	DET
ejpam-3126	322	3	above	above	ADJ
ejpam-3126	322	4	definitions	definition	NOUN
ejpam-3126	322	5	we	we	PRON
ejpam-3126	322	6	obtain	obtain	VERB
ejpam-3126	322	7	immediately	immediately	ADV
ejpam-3126	322	8	the	the	DET
ejpam-3126	322	9	following	follow	VERB
ejpam-3126	322	10	implication	implication	NOUN
ejpam-3126	322	11	chart	chart	NOUN
ejpam-3126	322	12	for	for	ADP
ejpam-3126	322	13	the	the	DET
ejpam-3126	322	14	considered	consider	VERB
ejpam-3126	322	15	types	type	NOUN
ejpam-3126	322	16	of	of	ADP
ejpam-3126	322	17	submodules	submodule	NOUN
ejpam-3126	322	18	:	:	PUNCT
ejpam-3126	322	19	2	2	NUM
ejpam-3126	322	20	-	-	PUNCT
ejpam-3126	322	21	absorbing	absorbing	ADJ
ejpam-3126	322	22	semi	semi	ADJ
ejpam-3126	322	23	-	-	ADJ
ejpam-3126	322	24	primary	primary	ADJ
ejpam-3126	322	25	⇒	⇒	NOUN
ejpam-3126	322	26	weakly	weakly	ADJ
ejpam-3126	322	27	2	2	NUM
ejpam-3126	322	28	-	-	PUNCT
ejpam-3126	322	29	absorbing	absorbing	ADJ
ejpam-3126	322	30	semi	semi	ADJ
ejpam-3126	322	31	-	-	ADJ
ejpam-3126	322	32	primary	primary	ADJ
ejpam-3126	322	33	⇒	⇒	NOUN
ejpam-3126	322	34	ω-2	ω-2	NOUN
ejpam-3126	322	35	-	-	PUNCT
ejpam-3126	322	36	absorbing	absorb	VERB
ejpam-3126	322	37	semi	semi	ADJ
ejpam-3126	322	38	-	-	ADJ
ejpam-3126	322	39	primary	primary	ADJ
ejpam-3126	322	40	⇒	⇒	NOUN
ejpam-3126	322	41	φn≥2	φn≥2	PROPN
ejpam-3126	322	42	-	-	PUNCT
ejpam-3126	322	43	2	2	NUM
ejpam-3126	322	44	-	-	PUNCT
ejpam-3126	322	45	absorbing	absorbing	ADJ
ejpam-3126	322	46	semi	semi	ADJ
ejpam-3126	322	47	-	-	ADJ
ejpam-3126	322	48	primary	primary	ADJ
ejpam-3126	322	49	⇒	⇒	NOUN
ejpam-3126	322	50	almost	almost	ADV
ejpam-3126	322	51	2	2	NUM
ejpam-3126	322	52	-	-	PUNCT
ejpam-3126	322	53	absorbing	absorbing	ADJ
ejpam-3126	322	54	semi	semi	ADJ
ejpam-3126	322	55	-	-	ADJ
ejpam-3126	322	56	primary	primary	ADJ
ejpam-3126	322	57	theorem	theorem	NOUN
ejpam-3126	322	58	13	13	NUM
ejpam-3126	322	59	.	.	PUNCT
ejpam-3126	323	1	let	let	VERB
ejpam-3126	323	2	φ	φ	NUM
ejpam-3126	323	3	,	,	PUNCT
ejpam-3126	323	4	φ3	φ3	NOUN
ejpam-3126	323	5	:	:	PUNCT
ejpam-3126	323	6	s(m	s(m	NUM
ejpam-3126	323	7	)	)	PUNCT
ejpam-3126	323	8	→	→	SYM
ejpam-3126	323	9	s(m	s(m	NOUN
ejpam-3126	323	10	)	)	PUNCT
ejpam-3126	323	11	∪	∪	NOUN
ejpam-3126	323	12	{	{	PUNCT
ejpam-3126	323	13	∅	∅	NOUN
ejpam-3126	323	14	}	}	PUNCT
ejpam-3126	323	15	be	be	AUX
ejpam-3126	323	16	two	two	NUM
ejpam-3126	323	17	functions	function	NOUN
ejpam-3126	323	18	and	and	CCONJ
ejpam-3126	323	19	let	let	VERB
ejpam-3126	323	20	n	n	PRON
ejpam-3126	323	21	be	be	AUX
ejpam-3126	323	22	a	a	DET
ejpam-3126	323	23	φ-2absorbing	φ-2absorbing	NOUN
ejpam-3126	323	24	semi	semi	ADJ
ejpam-3126	323	25	-	-	ADJ
ejpam-3126	323	26	primary	primary	ADJ
ejpam-3126	323	27	submodule	submodule	NOUN
ejpam-3126	323	28	.	.	PUNCT
ejpam-3126	324	1	if	if	SCONJ
ejpam-3126	324	2	φ3	φ3	PROPN
ejpam-3126	324	3	6≤	6≤	NUM
ejpam-3126	324	4	φ	φ	NUM
ejpam-3126	324	5	,	,	PUNCT
ejpam-3126	324	6	then	then	ADV
ejpam-3126	324	7	n	n	PRON
ejpam-3126	324	8	is	be	AUX
ejpam-3126	324	9	a	a	DET
ejpam-3126	324	10	2	2	NUM
ejpam-3126	324	11	-	-	PUNCT
ejpam-3126	324	12	absorbing	absorbing	ADJ
ejpam-3126	324	13	semi	semi	ADJ
ejpam-3126	324	14	-	-	ADJ
ejpam-3126	324	15	primary	primary	ADJ
ejpam-3126	324	16	submodule	submodule	NOUN
ejpam-3126	324	17	of	of	ADP
ejpam-3126	324	18	m	m	PROPN
ejpam-3126	324	19	.	.	PUNCT
ejpam-3126	325	1	proof	proof	NOUN
ejpam-3126	325	2	.	.	PUNCT
ejpam-3126	326	1	let	let	VERB
ejpam-3126	326	2	a1	a1	NOUN
ejpam-3126	326	3	,	,	PUNCT
ejpam-3126	326	4	a2	a2	PROPN
ejpam-3126	326	5	∈	∈	PROPN
ejpam-3126	326	6	r	r	NOUN
ejpam-3126	326	7	and	and	CCONJ
ejpam-3126	326	8	m	m	PROPN
ejpam-3126	326	9	∈	∈	NOUN
ejpam-3126	326	10	m	m	VERB
ejpam-3126	326	11	such	such	ADJ
ejpam-3126	326	12	that	that	SCONJ
ejpam-3126	326	13	a1a2	a1a2	PROPN
ejpam-3126	326	14	m	m	VERB
ejpam-3126	326	15	∈	∈	NOUN
ejpam-3126	326	16	n	n	NOUN
ejpam-3126	326	17	.	.	PUNCT
ejpam-3126	327	1	if	if	SCONJ
ejpam-3126	327	2	a1a2	a1a2	PROPN
ejpam-3126	327	3	m	m	VERB
ejpam-3126	327	4	6∈	6∈	NOUN
ejpam-3126	327	5	φ(n	φ(n	ADJ
ejpam-3126	327	6	)	)	PUNCT
ejpam-3126	327	7	,	,	PUNCT
ejpam-3126	327	8	then	then	ADV
ejpam-3126	327	9	a1a2	a1a2	ADP
ejpam-3126	327	10	m	m	VERB
ejpam-3126	327	11	∈	∈	ADJ
ejpam-3126	327	12	n	n	CCONJ
ejpam-3126	327	13	−	−	PROPN
ejpam-3126	327	14	φ(n	φ(n	NOUN
ejpam-3126	327	15	)	)	PUNCT
ejpam-3126	327	16	.	.	PUNCT
ejpam-3126	328	1	by	by	ADP
ejpam-3126	328	2	definition	definition	NOUN
ejpam-3126	328	3	1	1	NUM
ejpam-3126	328	4	,	,	PUNCT
ejpam-3126	328	5	a1a2	a1a2	PROPN
ejpam-3126	328	6	∈	∈	NOUN
ejpam-3126	328	7	√	√	NUM
ejpam-3126	328	8	(	(	PUNCT
ejpam-3126	328	9	n	n	NUM
ejpam-3126	328	10	:	:	PUNCT
ejpam-3126	328	11	m	m	X
ejpam-3126	328	12	)	)	PUNCT
ejpam-3126	328	13	or	or	CCONJ
ejpam-3126	328	14	a1	a1	NOUN
ejpam-3126	328	15	m	m	PROPN
ejpam-3126	328	16	∈	∈	NOUN
ejpam-3126	328	17	n	n	NOUN
ejpam-3126	328	18	or	or	CCONJ
ejpam-3126	328	19	an2	an2	PROPN
ejpam-3126	328	20	m	m	NOUN
ejpam-3126	328	21	∈	∈	PROPN
ejpam-3126	328	22	n	n	NOUN
ejpam-3126	328	23	for	for	ADP
ejpam-3126	328	24	some	some	DET
ejpam-3126	328	25	positive	positive	ADJ
ejpam-3126	328	26	integer	integer	NOUN
ejpam-3126	328	27	n.	n.	NOUN
ejpam-3126	328	28	next	next	ADV
ejpam-3126	328	29	,	,	PUNCT
ejpam-3126	328	30	let	let	VERB
ejpam-3126	328	31	a1a2	a1a2	ADP
ejpam-3126	328	32	m	m	NOUN
ejpam-3126	328	33	∈	∈	NOUN
ejpam-3126	328	34	φ(n	φ(n	NOUN
ejpam-3126	328	35	)	)	PUNCT
ejpam-3126	328	36	.	.	PUNCT
ejpam-3126	329	1	in	in	ADP
ejpam-3126	329	2	this	this	DET
ejpam-3126	329	3	case	case	NOUN
ejpam-3126	329	4	,	,	PUNCT
ejpam-3126	329	5	we	we	PRON
ejpam-3126	329	6	may	may	AUX
ejpam-3126	329	7	assume	assume	VERB
ejpam-3126	329	8	that	that	SCONJ
ejpam-3126	329	9	a1a2n	a1a2n	SCONJ
ejpam-3126	329	10	⊆	⊆	NUM
ejpam-3126	329	11	φ(n	φ(n	NOUN
ejpam-3126	329	12	)	)	PUNCT
ejpam-3126	329	13	,	,	PUNCT
ejpam-3126	329	14	because	because	SCONJ
ejpam-3126	329	15	if	if	SCONJ
ejpam-3126	329	16	a1a2n	a1a2n	ADP
ejpam-3126	329	17	6⊆	6⊆	NOUN
ejpam-3126	329	18	φ(n	φ(n	NOUN
ejpam-3126	329	19	)	)	PUNCT
ejpam-3126	329	20	then	then	ADV
ejpam-3126	329	21	there	there	PRON
ejpam-3126	329	22	exists	exist	VERB
ejpam-3126	329	23	m0	m0	PROPN
ejpam-3126	329	24	∈	∈	PROPN
ejpam-3126	329	25	n	n	PRON
ejpam-3126	329	26	such	such	ADJ
ejpam-3126	329	27	that	that	SCONJ
ejpam-3126	329	28	a1a2m0	a1a2m0	PROPN
ejpam-3126	329	29	6∈	6∈	PROPN
ejpam-3126	329	30	φ(n	φ(n	PROPN
ejpam-3126	329	31	)	)	PUNCT
ejpam-3126	329	32	.	.	PUNCT
ejpam-3126	330	1	clearly	clearly	ADV
ejpam-3126	330	2	,	,	PUNCT
ejpam-3126	330	3	a1	a1	PROPN
ejpam-3126	330	4	m	m	NOUN
ejpam-3126	330	5	∈	∈	NOUN
ejpam-3126	330	6	n	n	NOUN
ejpam-3126	330	7	or	or	CCONJ
ejpam-3126	330	8	a1a2	a1a2	INTJ
ejpam-3126	330	9	∈	∈	NOUN
ejpam-3126	330	10	√	√	NUM
ejpam-3126	330	11	(	(	PUNCT
ejpam-3126	330	12	n	n	NUM
ejpam-3126	330	13	:	:	PUNCT
ejpam-3126	330	14	m	m	X
ejpam-3126	330	15	)	)	PUNCT
ejpam-3126	330	16	or	or	CCONJ
ejpam-3126	330	17	an2	an2	NUM
ejpam-3126	330	18	m	m	NOUN
ejpam-3126	330	19	∈	∈	PROPN
ejpam-3126	330	20	n	n	NOUN
ejpam-3126	330	21	for	for	ADP
ejpam-3126	330	22	some	some	DET
ejpam-3126	330	23	positive	positive	ADJ
ejpam-3126	330	24	integer	integer	NOUN
ejpam-3126	330	25	n.	n.	NOUN
ejpam-3126	330	26	second	second	NOUN
ejpam-3126	331	1	we	we	PRON
ejpam-3126	331	2	may	may	AUX
ejpam-3126	331	3	assume	assume	VERB
ejpam-3126	331	4	that	that	SCONJ
ejpam-3126	331	5	(	(	PUNCT
ejpam-3126	331	6	n	n	X
ejpam-3126	331	7	:	:	PUNCT
ejpam-3126	331	8	m)2	m)2	PROPN
ejpam-3126	331	9	m	m	VERB
ejpam-3126	331	10	⊆	⊆	NUM
ejpam-3126	331	11	φ(n	φ(n	NOUN
ejpam-3126	331	12	)	)	PUNCT
ejpam-3126	331	13	.	.	PUNCT
ejpam-3126	332	1	if	if	SCONJ
ejpam-3126	332	2	this	this	PRON
ejpam-3126	332	3	is	be	AUX
ejpam-3126	332	4	not	not	PART
ejpam-3126	332	5	the	the	DET
ejpam-3126	332	6	case	case	NOUN
ejpam-3126	332	7	,	,	PUNCT
ejpam-3126	332	8	there	there	PRON
ejpam-3126	332	9	exist	exist	VERB
ejpam-3126	332	10	r1	r1	NOUN
ejpam-3126	332	11	,	,	PUNCT
ejpam-3126	332	12	r2	r2	PROPN
ejpam-3126	332	13	∈	∈	PROPN
ejpam-3126	332	14	(	(	PUNCT
ejpam-3126	332	15	n	n	NOUN
ejpam-3126	332	16	:	:	PUNCT
ejpam-3126	332	17	m	m	X
ejpam-3126	332	18	)	)	PUNCT
ejpam-3126	332	19	such	such	ADJ
ejpam-3126	332	20	that	that	SCONJ
ejpam-3126	332	21	(	(	PUNCT
ejpam-3126	332	22	a1	a1	NOUN
ejpam-3126	332	23	+	+	X
ejpam-3126	332	24	r1)(a2	r1)(a2	PRON
ejpam-3126	332	25	+	+	CCONJ
ejpam-3126	332	26	r2)m	r2)m	VERB
ejpam-3126	332	27	6∈	6∈	NOUN
ejpam-3126	332	28	φ(n	φ(n	ADJ
ejpam-3126	332	29	)	)	PUNCT
ejpam-3126	332	30	.	.	PUNCT
ejpam-3126	333	1	by	by	ADP
ejpam-3126	333	2	assumption	assumption	NOUN
ejpam-3126	333	3	,	,	PUNCT
ejpam-3126	333	4	a1	a1	PROPN
ejpam-3126	333	5	m	m	VERB
ejpam-3126	333	6	∈	∈	NOUN
ejpam-3126	333	7	n	n	NOUN
ejpam-3126	333	8	or	or	CCONJ
ejpam-3126	333	9	a1a2	a1a2	INTJ
ejpam-3126	333	10	∈	∈	NOUN
ejpam-3126	333	11	√	√	NUM
ejpam-3126	333	12	(	(	PUNCT
ejpam-3126	333	13	n	n	NUM
ejpam-3126	333	14	:	:	PUNCT
ejpam-3126	333	15	m	m	X
ejpam-3126	333	16	)	)	PUNCT
ejpam-3126	333	17	or	or	CCONJ
ejpam-3126	333	18	an2	an2	PROPN
ejpam-3126	333	19	∈	∈	PROPN
ejpam-3126	333	20	√	√	NUM
ejpam-3126	333	21	(	(	PUNCT
ejpam-3126	333	22	n	n	NUM
ejpam-3126	333	23	:	:	PUNCT
ejpam-3126	333	24	m	m	X
ejpam-3126	333	25	)	)	PUNCT
ejpam-3126	333	26	for	for	ADP
ejpam-3126	333	27	some	some	DET
ejpam-3126	333	28	positive	positive	ADJ
ejpam-3126	333	29	integer	integer	NOUN
ejpam-3126	333	30	n.	n.	NOUN
ejpam-3126	333	31	again	again	ADV
ejpam-3126	333	32	,	,	PUNCT
ejpam-3126	333	33	by	by	ADP
ejpam-3126	333	34	assumption	assumption	NOUN
ejpam-3126	333	35	,	,	PUNCT
ejpam-3126	333	36	(	(	PUNCT
ejpam-3126	333	37	n	n	X
ejpam-3126	333	38	:	:	PUNCT
ejpam-3126	333	39	m)2n	m)2n	NOUN
ejpam-3126	333	40	6⊆	6⊆	PROPN
ejpam-3126	333	41	φ(n	φ(n	PROPN
ejpam-3126	333	42	)	)	PUNCT
ejpam-3126	333	43	.	.	PUNCT
ejpam-3126	334	1	there	there	PRON
ejpam-3126	334	2	exist	exist	VERB
ejpam-3126	334	3	r1	r1	NOUN
ejpam-3126	334	4	,	,	PUNCT
ejpam-3126	334	5	r2	r2	PROPN
ejpam-3126	334	6	∈	∈	PROPN
ejpam-3126	334	7	(	(	PUNCT
ejpam-3126	334	8	n	n	NOUN
ejpam-3126	334	9	:	:	PUNCT
ejpam-3126	334	10	m	m	X
ejpam-3126	334	11	)	)	PUNCT
ejpam-3126	334	12	and	and	CCONJ
ejpam-3126	334	13	m0	m0	PROPN
ejpam-3126	334	14	∈	∈	PROPN
ejpam-3126	334	15	n	n	PRON
ejpam-3126	334	16	such	such	ADJ
ejpam-3126	334	17	that	that	SCONJ
ejpam-3126	334	18	r1r2m0	r1r2m0	PROPN
ejpam-3126	334	19	6∈	6∈	PROPN
ejpam-3126	334	20	φ(n	φ(n	PROPN
ejpam-3126	334	21	)	)	PUNCT
ejpam-3126	334	22	.	.	PUNCT
ejpam-3126	335	1	thus	thus	ADV
ejpam-3126	335	2	by	by	ADP
ejpam-3126	335	3	definition	definition	NOUN
ejpam-3126	335	4	1	1	NUM
ejpam-3126	335	5	,	,	PUNCT
ejpam-3126	335	6	a1	a1	PROPN
ejpam-3126	335	7	m	m	NOUN
ejpam-3126	335	8	∈	∈	NOUN
ejpam-3126	335	9	n	n	NOUN
ejpam-3126	335	10	or	or	CCONJ
ejpam-3126	335	11	a1a2	a1a2	INTJ
ejpam-3126	335	12	∈	∈	NOUN
ejpam-3126	335	13	√	√	NUM
ejpam-3126	335	14	(	(	PUNCT
ejpam-3126	335	15	n	n	NUM
ejpam-3126	335	16	:	:	PUNCT
ejpam-3126	335	17	m	m	X
ejpam-3126	335	18	)	)	PUNCT
ejpam-3126	335	19	or	or	CCONJ
ejpam-3126	335	20	an2	an2	NUM
ejpam-3126	335	21	m	m	NOUN
ejpam-3126	335	22	∈	∈	PROPN
ejpam-3126	335	23	n	n	NOUN
ejpam-3126	335	24	for	for	ADP
ejpam-3126	335	25	some	some	DET
ejpam-3126	335	26	positive	positive	ADJ
ejpam-3126	335	27	integer	integer	NOUN
ejpam-3126	335	28	n.	n.	NOUN
ejpam-3126	335	29	corollary	corollary	NOUN
ejpam-3126	335	30	6	6	NUM
ejpam-3126	335	31	.	.	PUNCT
ejpam-3126	336	1	let	let	VERB
ejpam-3126	336	2	φn	φn	VERB
ejpam-3126	336	3	:	:	PUNCT
ejpam-3126	336	4	s(m)→	s(m)→	NOUN
ejpam-3126	336	5	s(m	s(m	NOUN
ejpam-3126	336	6	)	)	PUNCT
ejpam-3126	336	7	∪	∪	ADP
ejpam-3126	336	8	{	{	PUNCT
ejpam-3126	336	9	∅	∅	NOUN
ejpam-3126	336	10	}	}	PUNCT
ejpam-3126	336	11	be	be	AUX
ejpam-3126	336	12	a	a	DET
ejpam-3126	336	13	function	function	NOUN
ejpam-3126	336	14	and	and	CCONJ
ejpam-3126	336	15	let	let	VERB
ejpam-3126	336	16	n	n	PRON
ejpam-3126	336	17	be	be	AUX
ejpam-3126	336	18	a	a	DET
ejpam-3126	336	19	φ0	φ0	PROPN
ejpam-3126	336	20	-	-	PUNCT
ejpam-3126	336	21	2	2	NUM
ejpam-3126	336	22	-	-	PUNCT
ejpam-3126	336	23	absorbing	absorbing	ADJ
ejpam-3126	336	24	semi	semi	ADJ
ejpam-3126	336	25	-	-	ADJ
ejpam-3126	336	26	primary	primary	ADJ
ejpam-3126	336	27	submodule	submodule	NOUN
ejpam-3126	336	28	of	of	ADP
ejpam-3126	336	29	m	m	PROPN
ejpam-3126	336	30	.	.	PUNCT
ejpam-3126	337	1	if	if	SCONJ
ejpam-3126	337	2	φ3	φ3	NOUN
ejpam-3126	337	3	6=	6=	NUM
ejpam-3126	337	4	φ0	φ0	PROPN
ejpam-3126	337	5	,	,	PUNCT
ejpam-3126	337	6	then	then	ADV
ejpam-3126	337	7	n	n	PRON
ejpam-3126	337	8	is	be	AUX
ejpam-3126	337	9	a	a	DET
ejpam-3126	337	10	2	2	NUM
ejpam-3126	337	11	-	-	PUNCT
ejpam-3126	337	12	absorbing	absorbing	ADJ
ejpam-3126	337	13	semi	semi	ADJ
ejpam-3126	337	14	-	-	ADJ
ejpam-3126	337	15	primary	primary	ADJ
ejpam-3126	337	16	submodule	submodule	NOUN
ejpam-3126	337	17	of	of	ADP
ejpam-3126	337	18	m	m	PROPN
ejpam-3126	337	19	.	.	PUNCT
ejpam-3126	338	1	proof	proof	NOUN
ejpam-3126	338	2	.	.	PUNCT
ejpam-3126	339	1	similar	similar	ADJ
ejpam-3126	339	2	to	to	ADP
ejpam-3126	339	3	the	the	DET
ejpam-3126	339	4	proof	proof	NOUN
ejpam-3126	339	5	of	of	ADP
ejpam-3126	339	6	theorem	theorem	ADJ
ejpam-3126	339	7	13	13	NUM
ejpam-3126	339	8	.	.	PUNCT
ejpam-3126	339	9	theorem	theorem	NOUN
ejpam-3126	339	10	14	14	NUM
ejpam-3126	339	11	.	.	PUNCT
ejpam-3126	340	1	let	let	VERB
ejpam-3126	340	2	φ	φ	NOUN
ejpam-3126	340	3	,	,	PUNCT
ejpam-3126	340	4	φ4	φ4	NOUN
ejpam-3126	340	5	:	:	PUNCT
ejpam-3126	340	6	s(m)→	s(m)→	NOUN
ejpam-3126	340	7	s(m	s(m	NOUN
ejpam-3126	340	8	)	)	PUNCT
ejpam-3126	340	9	∪	∪	ADP
ejpam-3126	340	10	{	{	PUNCT
ejpam-3126	340	11	∅	∅	NOUN
ejpam-3126	340	12	}	}	PUNCT
ejpam-3126	340	13	be	be	AUX
ejpam-3126	340	14	two	two	NUM
ejpam-3126	340	15	functions	function	NOUN
ejpam-3126	340	16	.	.	PUNCT
ejpam-3126	341	1	if	if	SCONJ
ejpam-3126	341	2	n	n	PRON
ejpam-3126	341	3	is	be	AUX
ejpam-3126	341	4	a	a	DET
ejpam-3126	341	5	φ-2	φ-2	NOUN
ejpam-3126	341	6	-	-	PUNCT
ejpam-3126	341	7	absorbing	absorbing	ADJ
ejpam-3126	341	8	semi	semi	ADJ
ejpam-3126	341	9	-	-	ADJ
ejpam-3126	341	10	primary	primary	ADJ
ejpam-3126	341	11	submodule	submodule	NOUN
ejpam-3126	341	12	such	such	ADJ
ejpam-3126	341	13	that	that	SCONJ
ejpam-3126	341	14	φ	φ	PROPN
ejpam-3126	341	15	≤	≤	PUNCT
ejpam-3126	341	16	φ4	φ4	NOUN
ejpam-3126	341	17	,	,	PUNCT
ejpam-3126	341	18	then	then	ADV
ejpam-3126	341	19	n	n	PRON
ejpam-3126	341	20	is	be	AUX
ejpam-3126	341	21	a	a	DET
ejpam-3126	341	22	ω-2	ω-2	NOUN
ejpam-3126	341	23	-	-	PUNCT
ejpam-3126	341	24	absorbing	absorb	VERB
ejpam-3126	341	25	semi	semi	ADJ
ejpam-3126	341	26	-	-	ADJ
ejpam-3126	341	27	primary	primary	ADJ
ejpam-3126	341	28	submodule	submodule	NOUN
ejpam-3126	341	29	of	of	ADP
ejpam-3126	341	30	m	m	PROPN
ejpam-3126	341	31	.	.	PUNCT
ejpam-3126	342	1	pai	pai	PROPN
ejpam-3126	342	2	.	.	PROPN
ejpam-3126	342	3	yiarayong	yiarayong	PROPN
ejpam-3126	342	4	,	,	PUNCT
ejpam-3126	342	5	m.	m.	NOUN
ejpam-3126	342	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	342	7	/	/	SYM
ejpam-3126	342	8	eur	eur	PROPN
ejpam-3126	342	9	.	.	PUNCT
ejpam-3126	343	1	j.	j.	PROPN
ejpam-3126	343	2	pure	pure	PROPN
ejpam-3126	343	3	appl	appl	PROPN
ejpam-3126	343	4	.	.	PROPN
ejpam-3126	343	5	math	math	PROPN
ejpam-3126	343	6	,	,	PUNCT
ejpam-3126	343	7	11	11	NUM
ejpam-3126	343	8	(	(	PUNCT
ejpam-3126	343	9	1	1	NUM
ejpam-3126	343	10	)	)	PUNCT
ejpam-3126	343	11	(	(	PUNCT
ejpam-3126	343	12	2018	2018	NUM
ejpam-3126	343	13	)	)	PUNCT
ejpam-3126	343	14	,	,	PUNCT
ejpam-3126	343	15	35	35	NUM
ejpam-3126	343	16	-	-	SYM
ejpam-3126	343	17	50	50	NUM
ejpam-3126	343	18	45	45	NUM
ejpam-3126	343	19	proof	proof	NOUN
ejpam-3126	343	20	.	.	PUNCT
ejpam-3126	344	1	if	if	SCONJ
ejpam-3126	344	2	n	n	PRON
ejpam-3126	344	3	is	be	AUX
ejpam-3126	344	4	a	a	DET
ejpam-3126	344	5	2	2	NUM
ejpam-3126	344	6	-	-	PUNCT
ejpam-3126	344	7	absorbing	absorbing	ADJ
ejpam-3126	344	8	semi	semi	ADJ
ejpam-3126	344	9	-	-	ADJ
ejpam-3126	344	10	primary	primary	ADJ
ejpam-3126	344	11	submodule	submodule	NOUN
ejpam-3126	344	12	of	of	ADP
ejpam-3126	344	13	m	m	PROPN
ejpam-3126	344	14	,	,	PUNCT
ejpam-3126	344	15	then	then	ADV
ejpam-3126	344	16	there	there	PRON
ejpam-3126	344	17	is	be	VERB
ejpam-3126	344	18	nothing	nothing	PRON
ejpam-3126	344	19	to	to	PART
ejpam-3126	344	20	prove	prove	VERB
ejpam-3126	344	21	.	.	PUNCT
ejpam-3126	345	1	assume	assume	VERB
ejpam-3126	345	2	that	that	SCONJ
ejpam-3126	345	3	n	n	PRON
ejpam-3126	345	4	is	be	AUX
ejpam-3126	345	5	not	not	PART
ejpam-3126	345	6	a	a	DET
ejpam-3126	345	7	2	2	NUM
ejpam-3126	345	8	-	-	PUNCT
ejpam-3126	345	9	absorbing	absorbing	ADJ
ejpam-3126	345	10	semi	semi	ADJ
ejpam-3126	345	11	-	-	ADJ
ejpam-3126	345	12	primary	primary	ADJ
ejpam-3126	345	13	submodule	submodule	NOUN
ejpam-3126	345	14	of	of	ADP
ejpam-3126	345	15	m	m	PROPN
ejpam-3126	345	16	.	.	PUNCT
ejpam-3126	346	1	then	then	ADV
ejpam-3126	346	2	by	by	ADP
ejpam-3126	346	3	theorem	theorem	NOUN
ejpam-3126	346	4	13	13	NUM
ejpam-3126	346	5	,	,	PUNCT
ejpam-3126	346	6	(	(	PUNCT
ejpam-3126	346	7	n	n	X
ejpam-3126	346	8	:	:	PUNCT
ejpam-3126	346	9	m)2n	m)2n	PROPN
ejpam-3126	346	10	=	=	PUNCT
ejpam-3126	346	11	φ3(n	φ3(n	PROPN
ejpam-3126	346	12	)	)	PUNCT
ejpam-3126	346	13	⊆	⊆	NUM
ejpam-3126	346	14	φ(n	φ(n	NOUN
ejpam-3126	346	15	)	)	PUNCT
ejpam-3126	346	16	⊆	⊆	NUM
ejpam-3126	346	17	(	(	PUNCT
ejpam-3126	346	18	n	n	NUM
ejpam-3126	346	19	:	:	PUNCT
ejpam-3126	346	20	m)3n	m)3n	NOUN
ejpam-3126	346	21	.	.	PUNCT
ejpam-3126	347	1	this	this	PRON
ejpam-3126	347	2	implies	imply	VERB
ejpam-3126	347	3	that	that	SCONJ
ejpam-3126	347	4	φ(n	φ(n	VERB
ejpam-3126	347	5	)	)	PUNCT
ejpam-3126	347	6	=	=	PUNCT
ejpam-3126	347	7	(	(	PUNCT
ejpam-3126	347	8	n	n	NOUN
ejpam-3126	347	9	:	:	PUNCT
ejpam-3126	347	10	m)2n	m)2n	PROPN
ejpam-3126	348	1	=	=	SYM
ejpam-3126	349	1	(	(	PUNCT
ejpam-3126	349	2	n	n	X
ejpam-3126	349	3	:	:	PUNCT
ejpam-3126	349	4	m)3n	m)3n	NOUN
ejpam-3126	349	5	.	.	PUNCT
ejpam-3126	350	1	thus	thus	ADV
ejpam-3126	350	2	φ(n	φ(n	NOUN
ejpam-3126	350	3	)	)	PUNCT
ejpam-3126	350	4	=	=	PUNCT
ejpam-3126	350	5	(	(	PUNCT
ejpam-3126	350	6	n	n	X
ejpam-3126	350	7	:	:	PUNCT
ejpam-3126	350	8	m)in	m)in	PROPN
ejpam-3126	350	9	for	for	ADP
ejpam-3126	350	10	all	all	PRON
ejpam-3126	350	11	i	i	PRON
ejpam-3126	350	12	≥	≥	VERB
ejpam-3126	350	13	3	3	NUM
ejpam-3126	350	14	.	.	PUNCT
ejpam-3126	350	15	theorem	theorem	NOUN
ejpam-3126	350	16	15	15	NUM
ejpam-3126	350	17	.	.	PUNCT
ejpam-3126	351	1	let	let	VERB
ejpam-3126	351	2	m	m	PRON
ejpam-3126	351	3	be	be	AUX
ejpam-3126	351	4	a	a	DET
ejpam-3126	351	5	multiplication	multiplication	NOUN
ejpam-3126	351	6	r	r	NOUN
ejpam-3126	351	7	-	-	PUNCT
ejpam-3126	351	8	module	module	NOUN
ejpam-3126	351	9	and	and	CCONJ
ejpam-3126	351	10	φ	φ	NOUN
ejpam-3126	351	11	:	:	PUNCT
ejpam-3126	351	12	s(m	s(m	PROPN
ejpam-3126	351	13	)	)	PUNCT
ejpam-3126	351	14	→	→	SYM
ejpam-3126	351	15	s(m	s(m	NOUN
ejpam-3126	351	16	)	)	PUNCT
ejpam-3126	351	17	∪	∪	NOUN
ejpam-3126	351	18	{	{	PUNCT
ejpam-3126	351	19	∅	∅	NOUN
ejpam-3126	351	20	}	}	PUNCT
ejpam-3126	351	21	be	be	AUX
ejpam-3126	351	22	a	a	DET
ejpam-3126	351	23	function	function	NOUN
ejpam-3126	351	24	.	.	PUNCT
ejpam-3126	352	1	then	then	ADV
ejpam-3126	352	2	the	the	DET
ejpam-3126	352	3	following	follow	VERB
ejpam-3126	352	4	properties	property	NOUN
ejpam-3126	352	5	hold	hold	VERB
ejpam-3126	352	6	.	.	PUNCT
ejpam-3126	353	1	(	(	PUNCT
ejpam-3126	353	2	i	i	NOUN
ejpam-3126	353	3	)	)	PUNCT
ejpam-3126	353	4	if	if	SCONJ
ejpam-3126	353	5	n	n	PRON
ejpam-3126	353	6	is	be	AUX
ejpam-3126	353	7	a	a	DET
ejpam-3126	353	8	φ-2	φ-2	NOUN
ejpam-3126	353	9	-	-	PUNCT
ejpam-3126	353	10	absorbing	absorbing	ADJ
ejpam-3126	353	11	semi	semi	ADJ
ejpam-3126	353	12	-	-	ADJ
ejpam-3126	353	13	primary	primary	ADJ
ejpam-3126	353	14	submodule	submodule	NOUN
ejpam-3126	353	15	of	of	ADP
ejpam-3126	353	16	m	m	PROPN
ejpam-3126	353	17	with	with	ADP
ejpam-3126	353	18	n3	n3	PROPN
ejpam-3126	353	19	6⊆	6⊆	PROPN
ejpam-3126	353	20	φ(n	φ(n	PROPN
ejpam-3126	353	21	)	)	PUNCT
ejpam-3126	353	22	,	,	PUNCT
ejpam-3126	353	23	then	then	ADV
ejpam-3126	353	24	n	n	PRON
ejpam-3126	353	25	is	be	AUX
ejpam-3126	353	26	a	a	DET
ejpam-3126	353	27	2	2	NUM
ejpam-3126	353	28	-	-	PUNCT
ejpam-3126	353	29	absorbing	absorbing	ADJ
ejpam-3126	353	30	semi	semi	ADJ
ejpam-3126	353	31	-	-	ADJ
ejpam-3126	353	32	primary	primary	ADJ
ejpam-3126	353	33	submodule	submodule	NOUN
ejpam-3126	353	34	of	of	ADP
ejpam-3126	353	35	m	m	PROPN
ejpam-3126	353	36	.	.	PUNCT
ejpam-3126	354	1	(	(	PUNCT
ejpam-3126	354	2	ii	ii	NOUN
ejpam-3126	354	3	)	)	PUNCT
ejpam-3126	354	4	if	if	SCONJ
ejpam-3126	354	5	n	n	PRON
ejpam-3126	354	6	is	be	AUX
ejpam-3126	354	7	a	a	DET
ejpam-3126	354	8	φn-2	φn-2	ADV
ejpam-3126	354	9	-	-	PUNCT
ejpam-3126	354	10	absorbing	absorbing	ADJ
ejpam-3126	354	11	semi	semi	ADJ
ejpam-3126	354	12	-	-	ADJ
ejpam-3126	354	13	primary	primary	ADJ
ejpam-3126	354	14	submodule	submodule	NOUN
ejpam-3126	354	15	of	of	ADP
ejpam-3126	354	16	m	m	PROPN
ejpam-3126	354	17	with	with	ADP
ejpam-3126	354	18	n3	n3	PROPN
ejpam-3126	354	19	6=	6=	PROPN
ejpam-3126	354	20	nn	nn	PROPN
ejpam-3126	354	21	for	for	ADP
ejpam-3126	354	22	all	all	DET
ejpam-3126	354	23	n	n	PRON
ejpam-3126	354	24	≥	≥	NOUN
ejpam-3126	354	25	3	3	NUM
ejpam-3126	354	26	,	,	PUNCT
ejpam-3126	354	27	then	then	ADV
ejpam-3126	354	28	n	n	PRON
ejpam-3126	354	29	is	be	AUX
ejpam-3126	354	30	a	a	DET
ejpam-3126	354	31	2	2	NUM
ejpam-3126	354	32	-	-	PUNCT
ejpam-3126	354	33	absorbing	absorbing	ADJ
ejpam-3126	354	34	semi	semi	ADJ
ejpam-3126	354	35	-	-	ADJ
ejpam-3126	354	36	primary	primary	ADJ
ejpam-3126	354	37	submodule	submodule	NOUN
ejpam-3126	354	38	of	of	ADP
ejpam-3126	354	39	m	m	PROPN
ejpam-3126	354	40	.	.	PUNCT
ejpam-3126	355	1	proof	proof	NOUN
ejpam-3126	355	2	.	.	PUNCT
ejpam-3126	356	1	1	1	X
ejpam-3126	356	2	.	.	X
ejpam-3126	356	3	suppose	suppose	VERB
ejpam-3126	356	4	that	that	SCONJ
ejpam-3126	356	5	n	n	PRON
ejpam-3126	356	6	is	be	AUX
ejpam-3126	356	7	a	a	DET
ejpam-3126	356	8	φ-2	φ-2	NOUN
ejpam-3126	356	9	-	-	PUNCT
ejpam-3126	356	10	absorbing	absorbing	ADJ
ejpam-3126	356	11	semi	semi	ADJ
ejpam-3126	356	12	-	-	ADJ
ejpam-3126	356	13	primary	primary	ADJ
ejpam-3126	356	14	submodule	submodule	NOUN
ejpam-3126	356	15	of	of	ADP
ejpam-3126	356	16	m	m	PRON
ejpam-3126	356	17	that	that	PRON
ejpam-3126	356	18	is	be	AUX
ejpam-3126	356	19	not	not	PART
ejpam-3126	356	20	2	2	NUM
ejpam-3126	356	21	-	-	PUNCT
ejpam-3126	356	22	absorbing	absorbing	ADJ
ejpam-3126	356	23	semi	semi	ADJ
ejpam-3126	356	24	-	-	ADJ
ejpam-3126	356	25	primary	primary	ADJ
ejpam-3126	356	26	.	.	PUNCT
ejpam-3126	357	1	clearly	clearly	ADV
ejpam-3126	357	2	,	,	PUNCT
ejpam-3126	357	3	n	n	PROPN
ejpam-3126	357	4	=	=	SYM
ejpam-3126	357	5	(	(	PUNCT
ejpam-3126	357	6	n	n	NUM
ejpam-3126	357	7	:	:	PUNCT
ejpam-3126	357	8	m)m	m)m	X
ejpam-3126	357	9	.	.	PUNCT
ejpam-3126	358	1	then	then	ADV
ejpam-3126	358	2	by	by	ADP
ejpam-3126	358	3	theorem	theorem	NOUN
ejpam-3126	358	4	13	13	NUM
ejpam-3126	358	5	,	,	PUNCT
ejpam-3126	358	6	n3	n3	NOUN
ejpam-3126	358	7	=	=	SYM
ejpam-3126	358	8	(	(	PUNCT
ejpam-3126	358	9	n	n	NOUN
ejpam-3126	358	10	:	:	PUNCT
ejpam-3126	358	11	m)3	m)3	NOUN
ejpam-3126	358	12	m	m	NOUN
ejpam-3126	358	13	=	=	X
ejpam-3126	358	14	(	(	PUNCT
ejpam-3126	358	15	n	n	NOUN
ejpam-3126	358	16	:	:	PUNCT
ejpam-3126	358	17	m)2((n	m)2((n	NOUN
ejpam-3126	358	18	:	:	PUNCT
ejpam-3126	358	19	m)m	m)m	X
ejpam-3126	358	20	)	)	PUNCT
ejpam-3126	358	21	=	=	SYM
ejpam-3126	358	22	(	(	PUNCT
ejpam-3126	358	23	n	n	NOUN
ejpam-3126	358	24	:	:	PUNCT
ejpam-3126	358	25	m)2n	m)2n	PROPN
ejpam-3126	358	26	=	=	PUNCT
ejpam-3126	358	27	φ3(n	φ3(n	PROPN
ejpam-3126	358	28	)	)	PUNCT
ejpam-3126	358	29	⊆	⊆	NUM
ejpam-3126	358	30	φ(n	φ(n	NOUN
ejpam-3126	358	31	)	)	PUNCT
ejpam-3126	358	32	.	.	PUNCT
ejpam-3126	359	1	2	2	X
ejpam-3126	359	2	.	.	X
ejpam-3126	359	3	suppose	suppose	VERB
ejpam-3126	359	4	that	that	SCONJ
ejpam-3126	359	5	n	n	PRON
ejpam-3126	359	6	is	be	AUX
ejpam-3126	359	7	a	a	DET
ejpam-3126	359	8	φn-2	φn-2	ADV
ejpam-3126	359	9	-	-	PUNCT
ejpam-3126	359	10	absorbing	absorbing	ADJ
ejpam-3126	359	11	semi	semi	ADJ
ejpam-3126	359	12	-	-	ADJ
ejpam-3126	359	13	primary	primary	ADJ
ejpam-3126	359	14	submodule	submodule	NOUN
ejpam-3126	359	15	of	of	ADP
ejpam-3126	359	16	m	m	PRON
ejpam-3126	359	17	that	that	PRON
ejpam-3126	359	18	is	be	AUX
ejpam-3126	359	19	not	not	PART
ejpam-3126	359	20	2absorbing	2absorbe	VERB
ejpam-3126	359	21	semi	semi	ADJ
ejpam-3126	359	22	-	-	ADJ
ejpam-3126	359	23	primary	primary	ADJ
ejpam-3126	359	24	.	.	PUNCT
ejpam-3126	360	1	clearly	clearly	ADV
ejpam-3126	360	2	,	,	PUNCT
ejpam-3126	360	3	nn	nn	PROPN
ejpam-3126	360	4	⊆	⊆	NUM
ejpam-3126	360	5	n3	n3	NOUN
ejpam-3126	360	6	,	,	PUNCT
ejpam-3126	360	7	for	for	ADP
ejpam-3126	360	8	all	all	DET
ejpam-3126	360	9	n	n	PRON
ejpam-3126	360	10	≥	≥	NOUN
ejpam-3126	360	11	3	3	NUM
ejpam-3126	360	12	.	.	PUNCT
ejpam-3126	360	13	then	then	ADV
ejpam-3126	360	14	by	by	ADP
ejpam-3126	360	15	parts	part	NOUN
ejpam-3126	360	16	1	1	NUM
ejpam-3126	360	17	,	,	PUNCT
ejpam-3126	360	18	n3	n3	NOUN
ejpam-3126	360	19	⊆	⊆	NUM
ejpam-3126	360	20	φn(n	φn(n	NUM
ejpam-3126	360	21	)	)	PUNCT
ejpam-3126	360	22	=	=	PUNCT
ejpam-3126	360	23	(	(	PUNCT
ejpam-3126	360	24	n	n	NOUN
ejpam-3126	360	25	:	:	PUNCT
ejpam-3126	360	26	m)n−1n	m)n−1n	ADJ
ejpam-3126	360	27	=	=	SYM
ejpam-3126	360	28	(	(	PUNCT
ejpam-3126	360	29	n	n	NOUN
ejpam-3126	360	30	:	:	PUNCT
ejpam-3126	360	31	m)nm	m)nm	PROPN
ejpam-3126	360	32	=	=	SYM
ejpam-3126	360	33	nn	nn	PROPN
ejpam-3126	360	34	.	.	PROPN
ejpam-3126	360	35	hence	hence	ADV
ejpam-3126	360	36	n3	n3	PROPN
ejpam-3126	360	37	=	=	SYM
ejpam-3126	360	38	nn	nn	PROPN
ejpam-3126	360	39	.	.	PROPN
ejpam-3126	361	1	this	this	PRON
ejpam-3126	361	2	completes	complete	VERB
ejpam-3126	361	3	the	the	DET
ejpam-3126	361	4	proof	proof	NOUN
ejpam-3126	361	5	.	.	PUNCT
ejpam-3126	362	1	theorem	theorem	VERB
ejpam-3126	362	2	16	16	NUM
ejpam-3126	362	3	.	.	PUNCT
ejpam-3126	363	1	let	let	VERB
ejpam-3126	363	2	ψi	ψi	VERB
ejpam-3126	363	3	:	:	PUNCT
ejpam-3126	363	4	s(mi	s(mi	VERB
ejpam-3126	363	5	)	)	PUNCT
ejpam-3126	363	6	→	→	SYM
ejpam-3126	363	7	s(mi	s(mi	NOUN
ejpam-3126	363	8	)	)	PUNCT
ejpam-3126	363	9	∪	∪	ADP
ejpam-3126	363	10	{	{	PUNCT
ejpam-3126	363	11	∅	∅	NOUN
ejpam-3126	363	12	}	}	PUNCT
ejpam-3126	363	13	be	be	AUX
ejpam-3126	363	14	a	a	DET
ejpam-3126	363	15	function	function	NOUN
ejpam-3126	363	16	with	with	ADP
ejpam-3126	363	17	φ	φ	PROPN
ejpam-3126	363	18	=	=	SYM
ejpam-3126	363	19	ψ1	ψ1	ADJ
ejpam-3126	363	20	×	×	NOUN
ejpam-3126	363	21	ψ2	ψ2	NOUN
ejpam-3126	363	22	.	.	PUNCT
ejpam-3126	364	1	then	then	ADV
ejpam-3126	364	2	the	the	DET
ejpam-3126	364	3	following	follow	VERB
ejpam-3126	364	4	statements	statement	NOUN
ejpam-3126	364	5	are	be	AUX
ejpam-3126	364	6	equivalent	equivalent	ADJ
ejpam-3126	364	7	:	:	PUNCT
ejpam-3126	364	8	(	(	PUNCT
ejpam-3126	364	9	i	i	NOUN
ejpam-3126	364	10	)	)	PUNCT
ejpam-3126	364	11	n1	n1	PROPN
ejpam-3126	364	12	×m2	×m2	NOUN
ejpam-3126	364	13	is	be	AUX
ejpam-3126	364	14	a	a	DET
ejpam-3126	364	15	φ-2	φ-2	NOUN
ejpam-3126	364	16	-	-	PUNCT
ejpam-3126	364	17	absorbing	absorbing	ADJ
ejpam-3126	364	18	semi	semi	ADJ
ejpam-3126	364	19	-	-	ADJ
ejpam-3126	364	20	primary	primary	ADJ
ejpam-3126	364	21	submodule	submodule	NOUN
ejpam-3126	364	22	of	of	ADP
ejpam-3126	364	23	m1	m1	PROPN
ejpam-3126	364	24	×m2	×m2	PROPN
ejpam-3126	364	25	.	.	PUNCT
ejpam-3126	365	1	(	(	PUNCT
ejpam-3126	365	2	ii	ii	NOUN
ejpam-3126	365	3	)	)	PUNCT
ejpam-3126	365	4	(	(	PUNCT
ejpam-3126	365	5	a	a	X
ejpam-3126	365	6	)	)	PUNCT
ejpam-3126	365	7	n1	n1	NOUN
ejpam-3126	365	8	is	be	AUX
ejpam-3126	365	9	a	a	DET
ejpam-3126	365	10	ψ1	ψ1	NOUN
ejpam-3126	365	11	-	-	PUNCT
ejpam-3126	365	12	2	2	NUM
ejpam-3126	365	13	-	-	PUNCT
ejpam-3126	365	14	absorbing	absorbing	ADJ
ejpam-3126	365	15	semi	semi	ADJ
ejpam-3126	365	16	-	-	ADJ
ejpam-3126	365	17	primary	primary	ADJ
ejpam-3126	365	18	submodule	submodule	NOUN
ejpam-3126	365	19	of	of	ADP
ejpam-3126	365	20	m1	m1	PROPN
ejpam-3126	365	21	.	.	PUNCT
ejpam-3126	366	1	(	(	PUNCT
ejpam-3126	366	2	b	b	X
ejpam-3126	366	3	)	)	PUNCT
ejpam-3126	366	4	for	for	ADP
ejpam-3126	366	5	each	each	DET
ejpam-3126	366	6	a1	a1	NOUN
ejpam-3126	366	7	,	,	PUNCT
ejpam-3126	366	8	a2	a2	PROPN
ejpam-3126	366	9	∈	∈	PROPN
ejpam-3126	366	10	r	r	NOUN
ejpam-3126	366	11	and	and	CCONJ
ejpam-3126	366	12	m	m	PROPN
ejpam-3126	366	13	∈	∈	PROPN
ejpam-3126	366	14	m1	m1	NOUN
ejpam-3126	366	15	such	such	ADJ
ejpam-3126	366	16	that	that	SCONJ
ejpam-3126	366	17	a1a2	a1a2	PROPN
ejpam-3126	366	18	m	m	NOUN
ejpam-3126	366	19	∈	∈	ADJ
ejpam-3126	366	20	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	366	21	)	)	PUNCT
ejpam-3126	366	22	if	if	SCONJ
ejpam-3126	366	23	a1a2	a1a2	ADP
ejpam-3126	366	24	6∈√	6∈√	NUM
ejpam-3126	366	25	(	(	PUNCT
ejpam-3126	366	26	n1	n1	NOUN
ejpam-3126	366	27	:	:	PUNCT
ejpam-3126	366	28	m1	m1	NOUN
ejpam-3126	366	29	)	)	PUNCT
ejpam-3126	366	30	and	and	CCONJ
ejpam-3126	366	31	a1	a1	PROPN
ejpam-3126	366	32	m	m	PROPN
ejpam-3126	366	33	6∈	6∈	NOUN
ejpam-3126	366	34	n1	n1	NOUN
ejpam-3126	366	35	,	,	PUNCT
ejpam-3126	366	36	a	a	DET
ejpam-3126	366	37	n	n	NUM
ejpam-3126	366	38	2	2	NUM
ejpam-3126	366	39	m	m	NOUN
ejpam-3126	366	40	6∈	6∈	NOUN
ejpam-3126	366	41	n1	n1	NOUN
ejpam-3126	366	42	for	for	ADP
ejpam-3126	366	43	all	all	DET
ejpam-3126	366	44	positive	positive	ADJ
ejpam-3126	366	45	integer	integer	NOUN
ejpam-3126	366	46	n	n	CCONJ
ejpam-3126	366	47	,	,	PUNCT
ejpam-3126	366	48	then	then	ADV
ejpam-3126	366	49	a1a2	a1a2	ADP
ejpam-3126	366	50	∈	∈	PROPN
ejpam-3126	366	51	(	(	PUNCT
ejpam-3126	366	52	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	366	53	)	)	PUNCT
ejpam-3126	366	54	:	:	PUNCT
ejpam-3126	366	55	m2	m2	PROPN
ejpam-3126	366	56	)	)	PUNCT
ejpam-3126	366	57	.	.	PUNCT
ejpam-3126	367	1	proof	proof	NOUN
ejpam-3126	367	2	.	.	PUNCT
ejpam-3126	368	1	(	(	PUNCT
ejpam-3126	368	2	i⇒	i⇒	NOUN
ejpam-3126	368	3	ii	ii	NUM
ejpam-3126	368	4	)	)	PUNCT
ejpam-3126	368	5	.	.	PUNCT
ejpam-3126	369	1	(	(	PUNCT
ejpam-3126	369	2	a	a	X
ejpam-3126	369	3	)	)	PUNCT
ejpam-3126	369	4	.	.	PUNCT
ejpam-3126	370	1	it	it	PRON
ejpam-3126	370	2	is	be	AUX
ejpam-3126	370	3	obvious	obvious	ADJ
ejpam-3126	370	4	.	.	PUNCT
ejpam-3126	371	1	(	(	PUNCT
ejpam-3126	371	2	b	b	NOUN
ejpam-3126	371	3	)	)	PUNCT
ejpam-3126	371	4	.	.	PUNCT
ejpam-3126	372	1	let	let	VERB
ejpam-3126	373	1	a1a2	a1a2	VERB
ejpam-3126	373	2	m	m	NOUN
ejpam-3126	373	3	∈	∈	ADJ
ejpam-3126	373	4	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	373	5	)	)	PUNCT
ejpam-3126	373	6	,	,	PUNCT
ejpam-3126	373	7	a1	a1	PROPN
ejpam-3126	373	8	m	m	PROPN
ejpam-3126	373	9	6∈	6∈	NOUN
ejpam-3126	373	10	n1	n1	PROPN
ejpam-3126	373	11	and	and	CCONJ
ejpam-3126	373	12	an2	an2	PROPN
ejpam-3126	373	13	m	m	PROPN
ejpam-3126	373	14	6∈	6∈	NOUN
ejpam-3126	373	15	n1	n1	NOUN
ejpam-3126	373	16	,	,	PUNCT
ejpam-3126	373	17	where	where	SCONJ
ejpam-3126	373	18	a1	a1	NOUN
ejpam-3126	373	19	,	,	PUNCT
ejpam-3126	373	20	a2	a2	PROPN
ejpam-3126	373	21	∈	∈	PROPN
ejpam-3126	373	22	r	r	NOUN
ejpam-3126	373	23	and	and	CCONJ
ejpam-3126	373	24	m	m	PROPN
ejpam-3126	373	25	∈	∈	PROPN
ejpam-3126	373	26	m1	m1	NOUN
ejpam-3126	373	27	.	.	PUNCT
ejpam-3126	373	28	suppose	suppose	VERB
ejpam-3126	373	29	that	that	SCONJ
ejpam-3126	373	30	a1a2	a1a2	PRON
ejpam-3126	373	31	6∈	6∈	NOUN
ejpam-3126	373	32	(	(	PUNCT
ejpam-3126	373	33	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	373	34	)	)	PUNCT
ejpam-3126	373	35	:	:	PUNCT
ejpam-3126	373	36	m2	m2	PROPN
ejpam-3126	373	37	)	)	PUNCT
ejpam-3126	373	38	.	.	PUNCT
ejpam-3126	374	1	there	there	PRON
ejpam-3126	374	2	exists	exist	VERB
ejpam-3126	374	3	m2	m2	PROPN
ejpam-3126	374	4	∈	∈	PROPN
ejpam-3126	374	5	m2	m2	PROPN
ejpam-3126	374	6	such	such	ADJ
ejpam-3126	374	7	that	that	SCONJ
ejpam-3126	374	8	a1a2m2	a1a2m2	PROPN
ejpam-3126	374	9	/∈	/∈	PUNCT
ejpam-3126	374	10	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	374	11	)	)	PUNCT
ejpam-3126	374	12	.	.	PUNCT
ejpam-3126	375	1	thus	thus	ADV
ejpam-3126	375	2	a1a2(m	a1a2(m	ADP
ejpam-3126	375	3	,	,	PUNCT
ejpam-3126	375	4	m2	m2	PROPN
ejpam-3126	375	5	)	)	PUNCT
ejpam-3126	375	6	∈	∈	PROPN
ejpam-3126	375	7	n1	n1	NOUN
ejpam-3126	375	8	×m2	×m2	NOUN
ejpam-3126	375	9	−	−	NOUN
ejpam-3126	375	10	φ(n1	φ(n1	NOUN
ejpam-3126	375	11	×m2	×m2	NOUN
ejpam-3126	375	12	)	)	PUNCT
ejpam-3126	375	13	.	.	PUNCT
ejpam-3126	376	1	by	by	ADP
ejpam-3126	376	2	part	part	NOUN
ejpam-3126	376	3	(	(	PUNCT
ejpam-3126	376	4	1	1	NUM
ejpam-3126	376	5	)	)	PUNCT
ejpam-3126	376	6	,	,	PUNCT
ejpam-3126	376	7	i.e.	i.e.	X
ejpam-3126	376	8	,	,	PUNCT
ejpam-3126	376	9	a1a2	a1a2	PROPN
ejpam-3126	376	10	∈	∈	NOUN
ejpam-3126	376	11	√	√	NUM
ejpam-3126	376	12	(	(	PUNCT
ejpam-3126	376	13	n1	n1	PROPN
ejpam-3126	376	14	:	:	PUNCT
ejpam-3126	376	15	m1	m1	NOUN
ejpam-3126	376	16	)	)	PUNCT
ejpam-3126	376	17	or	or	CCONJ
ejpam-3126	376	18	a1	a1	NOUN
ejpam-3126	376	19	m	m	PROPN
ejpam-3126	376	20	∈	∈	NOUN
ejpam-3126	376	21	n1	n1	NOUN
ejpam-3126	376	22	or	or	CCONJ
ejpam-3126	376	23	an2	an2	PROPN
ejpam-3126	376	24	m	m	NOUN
ejpam-3126	376	25	∈	∈	NOUN
ejpam-3126	376	26	n1	n1	NOUN
ejpam-3126	376	27	which	which	PRON
ejpam-3126	376	28	is	be	AUX
ejpam-3126	376	29	a	a	DET
ejpam-3126	376	30	contradiction	contradiction	NOUN
ejpam-3126	376	31	.	.	PUNCT
ejpam-3126	377	1	(	(	PUNCT
ejpam-3126	377	2	ii	ii	NOUN
ejpam-3126	377	3	⇒	⇒	NOUN
ejpam-3126	377	4	i	i	PRON
ejpam-3126	377	5	)	)	PUNCT
ejpam-3126	377	6	.	.	PUNCT
ejpam-3126	378	1	let	let	VERB
ejpam-3126	378	2	a1	a1	NOUN
ejpam-3126	378	3	,	,	PUNCT
ejpam-3126	378	4	a2	a2	PROPN
ejpam-3126	378	5	∈	∈	PROPN
ejpam-3126	378	6	r	r	NOUN
ejpam-3126	378	7	and	and	CCONJ
ejpam-3126	378	8	(	(	PUNCT
ejpam-3126	378	9	m1,m2	m1,m2	PROPN
ejpam-3126	378	10	)	)	PUNCT
ejpam-3126	378	11	∈	∈	PROPN
ejpam-3126	378	12	m1	m1	NOUN
ejpam-3126	378	13	×m2	×m2	NOUN
ejpam-3126	378	14	such	such	ADJ
ejpam-3126	378	15	that	that	DET
ejpam-3126	378	16	a1a2(m1,m2	a1a2(m1,m2	NOUN
ejpam-3126	378	17	)	)	PUNCT
ejpam-3126	378	18	∈	∈	PROPN
ejpam-3126	378	19	n1	n1	PROPN
ejpam-3126	378	20	×	×	PROPN
ejpam-3126	378	21	m2	m2	PROPN
ejpam-3126	378	22	−	−	PROPN
ejpam-3126	378	23	φ(n1	φ(n1	NOUN
ejpam-3126	378	24	×	×	PROPN
ejpam-3126	378	25	m2	m2	PROPN
ejpam-3126	378	26	)	)	PUNCT
ejpam-3126	378	27	.	.	PUNCT
ejpam-3126	379	1	if	if	SCONJ
ejpam-3126	379	2	a1a2m1	a1a2m1	PROPN
ejpam-3126	379	3	6∈	6∈	NOUN
ejpam-3126	379	4	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	379	5	)	)	PUNCT
ejpam-3126	379	6	,	,	PUNCT
ejpam-3126	379	7	then	then	ADV
ejpam-3126	379	8	a1a2m1	a1a2m1	PROPN
ejpam-3126	379	9	∈	∈	PROPN
ejpam-3126	379	10	n1	n1	PROPN
ejpam-3126	379	11	−	−	PROPN
ejpam-3126	379	12	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	379	13	)	)	PUNCT
ejpam-3126	379	14	.	.	PUNCT
ejpam-3126	380	1	by	by	ADP
ejpam-3126	380	2	part	part	NOUN
ejpam-3126	380	3	(	(	PUNCT
ejpam-3126	380	4	a	a	NOUN
ejpam-3126	380	5	)	)	PUNCT
ejpam-3126	380	6	,	,	PUNCT
ejpam-3126	380	7	a1a2	a1a2	PROPN
ejpam-3126	380	8	∈	∈	NOUN
ejpam-3126	380	9	√	√	NUM
ejpam-3126	380	10	(	(	PUNCT
ejpam-3126	380	11	n1	n1	NOUN
ejpam-3126	380	12	×m2	×m2	NOUN
ejpam-3126	380	13	:	:	PUNCT
ejpam-3126	380	14	m1	m1	PROPN
ejpam-3126	380	15	×m2	×m2	PROPN
ejpam-3126	380	16	)	)	PUNCT
ejpam-3126	380	17	or	or	CCONJ
ejpam-3126	380	18	a1(m1,m2	a1(m1,m2	NOUN
ejpam-3126	380	19	)	)	PUNCT
ejpam-3126	380	20	=	=	PUNCT
ejpam-3126	380	21	(	(	PUNCT
ejpam-3126	380	22	a1m1	a1m1	NOUN
ejpam-3126	380	23	,	,	PUNCT
ejpam-3126	380	24	a1m2	a1m2	NOUN
ejpam-3126	380	25	)	)	PUNCT
ejpam-3126	380	26	∈	∈	PROPN
ejpam-3126	380	27	n1×m2	n1×m2	NOUN
ejpam-3126	380	28	or	or	CCONJ
ejpam-3126	380	29	an2	an2	PROPN
ejpam-3126	380	30	(	(	PUNCT
ejpam-3126	380	31	m1,m2	m1,m2	PROPN
ejpam-3126	380	32	)	)	PUNCT
ejpam-3126	380	33	=	=	SYM
ejpam-3126	380	34	(	(	PUNCT
ejpam-3126	380	35	an2m1	an2m1	PROPN
ejpam-3126	380	36	,	,	PUNCT
ejpam-3126	380	37	a	a	DET
ejpam-3126	380	38	n	n	PRON
ejpam-3126	380	39	2m2	2m2	NUM
ejpam-3126	380	40	)	)	PUNCT
ejpam-3126	380	41	∈	∈	PROPN
ejpam-3126	380	42	n1	n1	NOUN
ejpam-3126	380	43	×m2	×m2	PROPN
ejpam-3126	380	44	,	,	PUNCT
ejpam-3126	380	45	and	and	CCONJ
ejpam-3126	380	46	thus	thus	ADV
ejpam-3126	380	47	we	we	PRON
ejpam-3126	380	48	are	be	AUX
ejpam-3126	380	49	done	do	VERB
ejpam-3126	380	50	.	.	PUNCT
ejpam-3126	381	1	if	if	SCONJ
ejpam-3126	381	2	a1a2m1	a1a2m1	PROPN
ejpam-3126	381	3	∈	∈	PROPN
ejpam-3126	381	4	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	381	5	)	)	PUNCT
ejpam-3126	381	6	,	,	PUNCT
ejpam-3126	381	7	then	then	ADV
ejpam-3126	381	8	a1a2m2	a1a2m2	PROPN
ejpam-3126	381	9	6∈	6∈	PROPN
ejpam-3126	381	10	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	381	11	)	)	PUNCT
ejpam-3126	381	12	.	.	PUNCT
ejpam-3126	382	1	therefore	therefore	ADV
ejpam-3126	382	2	a1a2	a1a2	PRON
ejpam-3126	382	3	6∈	6∈	PROPN
ejpam-3126	382	4	(	(	PUNCT
ejpam-3126	382	5	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	382	6	)	)	PUNCT
ejpam-3126	382	7	:	:	PUNCT
ejpam-3126	382	8	m2	m2	PROPN
ejpam-3126	382	9	)	)	PUNCT
ejpam-3126	382	10	.	.	PUNCT
ejpam-3126	383	1	by	by	ADP
ejpam-3126	383	2	part	part	NOUN
ejpam-3126	383	3	(	(	PUNCT
ejpam-3126	383	4	b	b	NOUN
ejpam-3126	383	5	)	)	PUNCT
ejpam-3126	383	6	,	,	PUNCT
ejpam-3126	383	7	a1a2	a1a2	PROPN
ejpam-3126	383	8	∈	∈	NOUN
ejpam-3126	383	9	√	√	NUM
ejpam-3126	383	10	(	(	PUNCT
ejpam-3126	383	11	n1	n1	NOUN
ejpam-3126	383	12	×m2	×m2	NOUN
ejpam-3126	383	13	:	:	PUNCT
ejpam-3126	383	14	m1	m1	PROPN
ejpam-3126	383	15	×m2	×m2	PROPN
ejpam-3126	383	16	)	)	PUNCT
ejpam-3126	383	17	or	or	CCONJ
ejpam-3126	383	18	a1(m1,m2	a1(m1,m2	NOUN
ejpam-3126	383	19	)	)	PUNCT
ejpam-3126	383	20	∈	∈	NOUN
ejpam-3126	383	21	n1	n1	NOUN
ejpam-3126	383	22	×m2	×m2	NOUN
ejpam-3126	383	23	or	or	CCONJ
ejpam-3126	383	24	an2	an2	PROPN
ejpam-3126	383	25	(	(	PUNCT
ejpam-3126	383	26	m1,m2	m1,m2	PROPN
ejpam-3126	383	27	)	)	PUNCT
ejpam-3126	383	28	∈	∈	PROPN
ejpam-3126	383	29	n1	n1	NOUN
ejpam-3126	383	30	×m2	×m2	PROPN
ejpam-3126	383	31	.	.	PUNCT
ejpam-3126	384	1	corollary	corollary	ADJ
ejpam-3126	384	2	7	7	NUM
ejpam-3126	384	3	.	.	PUNCT
ejpam-3126	385	1	let	let	VERB
ejpam-3126	385	2	ψi	ψi	NOUN
ejpam-3126	385	3	:	:	PUNCT
ejpam-3126	385	4	s(mi	s(mi	VERB
ejpam-3126	385	5	)	)	PUNCT
ejpam-3126	385	6	→	→	SYM
ejpam-3126	385	7	s(mi	s(mi	NOUN
ejpam-3126	385	8	)	)	PUNCT
ejpam-3126	385	9	∪	∪	ADP
ejpam-3126	385	10	{	{	PUNCT
ejpam-3126	385	11	∅	∅	NOUN
ejpam-3126	385	12	}	}	PUNCT
ejpam-3126	385	13	be	be	AUX
ejpam-3126	385	14	a	a	DET
ejpam-3126	385	15	function	function	NOUN
ejpam-3126	385	16	with	with	ADP
ejpam-3126	385	17	φ	φ	PROPN
ejpam-3126	385	18	=	=	SYM
ejpam-3126	385	19	ψ1	ψ1	ADJ
ejpam-3126	385	20	×	×	NOUN
ejpam-3126	385	21	ψ2	ψ2	NOUN
ejpam-3126	385	22	.	.	PUNCT
ejpam-3126	386	1	then	then	ADV
ejpam-3126	386	2	the	the	DET
ejpam-3126	386	3	following	follow	VERB
ejpam-3126	386	4	conditions	condition	NOUN
ejpam-3126	386	5	are	be	AUX
ejpam-3126	386	6	equivalent	equivalent	ADJ
ejpam-3126	386	7	:	:	PUNCT
ejpam-3126	386	8	pai	pai	PROPN
ejpam-3126	386	9	.	.	PROPN
ejpam-3126	386	10	yiarayong	yiarayong	PROPN
ejpam-3126	386	11	,	,	PUNCT
ejpam-3126	386	12	m.	m.	NOUN
ejpam-3126	386	13	siripitukdet	siripitukdet	NOUN
ejpam-3126	386	14	/	/	SYM
ejpam-3126	386	15	eur	eur	PROPN
ejpam-3126	386	16	.	.	PUNCT
ejpam-3126	387	1	j.	j.	PROPN
ejpam-3126	387	2	pure	pure	PROPN
ejpam-3126	387	3	appl	appl	PROPN
ejpam-3126	387	4	.	.	PROPN
ejpam-3126	387	5	math	math	PROPN
ejpam-3126	387	6	,	,	PUNCT
ejpam-3126	387	7	11	11	NUM
ejpam-3126	387	8	(	(	PUNCT
ejpam-3126	387	9	1	1	NUM
ejpam-3126	387	10	)	)	PUNCT
ejpam-3126	387	11	(	(	PUNCT
ejpam-3126	387	12	2018	2018	NUM
ejpam-3126	387	13	)	)	PUNCT
ejpam-3126	387	14	,	,	PUNCT
ejpam-3126	387	15	35	35	NUM
ejpam-3126	387	16	-	-	SYM
ejpam-3126	387	17	50	50	NUM
ejpam-3126	387	18	46	46	NUM
ejpam-3126	387	19	(	(	PUNCT
ejpam-3126	387	20	i	i	NOUN
ejpam-3126	387	21	)	)	PUNCT
ejpam-3126	387	22	m1	m1	PROPN
ejpam-3126	387	23	×n2	×n2	PROPN
ejpam-3126	387	24	is	be	AUX
ejpam-3126	387	25	a	a	DET
ejpam-3126	387	26	φ-2	φ-2	NOUN
ejpam-3126	387	27	-	-	PUNCT
ejpam-3126	387	28	absorbing	absorbing	ADJ
ejpam-3126	387	29	semi	semi	ADJ
ejpam-3126	387	30	-	-	ADJ
ejpam-3126	387	31	primary	primary	ADJ
ejpam-3126	387	32	submodule	submodule	NOUN
ejpam-3126	387	33	of	of	ADP
ejpam-3126	387	34	m1	m1	PROPN
ejpam-3126	387	35	×m2	×m2	PROPN
ejpam-3126	387	36	.	.	PUNCT
ejpam-3126	388	1	(	(	PUNCT
ejpam-3126	388	2	ii	ii	NOUN
ejpam-3126	388	3	)	)	PUNCT
ejpam-3126	388	4	(	(	PUNCT
ejpam-3126	388	5	a	a	X
ejpam-3126	388	6	)	)	PUNCT
ejpam-3126	388	7	n2	n2	NOUN
ejpam-3126	388	8	is	be	AUX
ejpam-3126	388	9	a	a	DET
ejpam-3126	388	10	ψ2	ψ2	NOUN
ejpam-3126	388	11	-	-	PUNCT
ejpam-3126	388	12	2	2	NUM
ejpam-3126	388	13	-	-	PUNCT
ejpam-3126	388	14	absorbing	absorbing	ADJ
ejpam-3126	388	15	semi	semi	ADJ
ejpam-3126	388	16	-	-	ADJ
ejpam-3126	388	17	primary	primary	ADJ
ejpam-3126	388	18	submodule	submodule	NOUN
ejpam-3126	388	19	of	of	ADP
ejpam-3126	388	20	m2	m2	PROPN
ejpam-3126	388	21	.	.	PUNCT
ejpam-3126	389	1	(	(	PUNCT
ejpam-3126	389	2	b	b	X
ejpam-3126	389	3	)	)	PUNCT
ejpam-3126	389	4	for	for	ADP
ejpam-3126	389	5	each	each	DET
ejpam-3126	389	6	a1	a1	NOUN
ejpam-3126	389	7	,	,	PUNCT
ejpam-3126	389	8	a2	a2	PROPN
ejpam-3126	389	9	∈	∈	PROPN
ejpam-3126	389	10	r	r	NOUN
ejpam-3126	389	11	and	and	CCONJ
ejpam-3126	389	12	m	m	PROPN
ejpam-3126	389	13	∈	∈	PROPN
ejpam-3126	389	14	m2	m2	PROPN
ejpam-3126	389	15	such	such	ADJ
ejpam-3126	389	16	that	that	SCONJ
ejpam-3126	389	17	a1a2	a1a2	PROPN
ejpam-3126	389	18	m	m	NOUN
ejpam-3126	389	19	∈	∈	NOUN
ejpam-3126	389	20	ψ2(n2	ψ2(n2	NOUN
ejpam-3126	389	21	)	)	PUNCT
ejpam-3126	389	22	,	,	PUNCT
ejpam-3126	389	23	if	if	SCONJ
ejpam-3126	389	24	a1a2	a1a2	ADP
ejpam-3126	389	25	6∈√	6∈√	NUM
ejpam-3126	389	26	(	(	PUNCT
ejpam-3126	389	27	n2	n2	NOUN
ejpam-3126	389	28	:	:	PUNCT
ejpam-3126	389	29	m2	m2	PROPN
ejpam-3126	389	30	)	)	PUNCT
ejpam-3126	389	31	,	,	PUNCT
ejpam-3126	389	32	a1	a1	PROPN
ejpam-3126	389	33	m	m	PROPN
ejpam-3126	389	34	6∈	6∈	NOUN
ejpam-3126	389	35	n2	n2	NOUN
ejpam-3126	389	36	and	and	CCONJ
ejpam-3126	389	37	an2	an2	PROPN
ejpam-3126	389	38	m	m	PROPN
ejpam-3126	389	39	6∈	6∈	NOUN
ejpam-3126	389	40	n2	n2	NOUN
ejpam-3126	389	41	for	for	ADP
ejpam-3126	389	42	all	all	DET
ejpam-3126	389	43	positive	positive	ADJ
ejpam-3126	389	44	integer	integer	NOUN
ejpam-3126	389	45	n	n	CCONJ
ejpam-3126	389	46	,	,	PUNCT
ejpam-3126	389	47	then	then	ADV
ejpam-3126	389	48	a1a2	a1a2	ADP
ejpam-3126	389	49	∈	∈	PROPN
ejpam-3126	389	50	(	(	PUNCT
ejpam-3126	389	51	ψ1(m1	ψ1(m1	NOUN
ejpam-3126	389	52	)	)	PUNCT
ejpam-3126	389	53	:	:	PUNCT
ejpam-3126	389	54	m1	m1	NOUN
ejpam-3126	389	55	)	)	PUNCT
ejpam-3126	389	56	.	.	PUNCT
ejpam-3126	390	1	proof	proof	NOUN
ejpam-3126	390	2	.	.	PUNCT
ejpam-3126	391	1	similar	similar	ADJ
ejpam-3126	391	2	to	to	ADP
ejpam-3126	391	3	the	the	DET
ejpam-3126	391	4	proof	proof	NOUN
ejpam-3126	391	5	of	of	ADP
ejpam-3126	391	6	theorem	theorem	ADJ
ejpam-3126	391	7	16	16	NUM
ejpam-3126	391	8	.	.	PUNCT
ejpam-3126	391	9	theorem	theorem	NOUN
ejpam-3126	391	10	17	17	NUM
ejpam-3126	391	11	.	.	PUNCT
ejpam-3126	392	1	let	let	VERB
ejpam-3126	392	2	ψi	ψi	VERB
ejpam-3126	392	3	:	:	PUNCT
ejpam-3126	392	4	s(mi)→	s(mi)→	PROPN
ejpam-3126	392	5	s(mi	s(mi	PROPN
ejpam-3126	392	6	)	)	PUNCT
ejpam-3126	392	7	∪	∪	ADP
ejpam-3126	392	8	{	{	PUNCT
ejpam-3126	392	9	∅	∅	NOUN
ejpam-3126	392	10	}	}	PUNCT
ejpam-3126	392	11	is	be	AUX
ejpam-3126	392	12	a	a	DET
ejpam-3126	392	13	function	function	NOUN
ejpam-3126	392	14	with	with	ADP
ejpam-3126	392	15	φ	φ	PROPN
ejpam-3126	392	16	=	=	SYM
ejpam-3126	392	17	ψ1	ψ1	ADJ
ejpam-3126	392	18	×	×	NOUN
ejpam-3126	392	19	.	.	PUNCT
ejpam-3126	392	20	.	.	PUNCT
ejpam-3126	393	1	.×	.×	PROPN
ejpam-3126	393	2	ψk	ψk	VERB
ejpam-3126	393	3	.	.	PUNCT
ejpam-3126	394	1	then	then	ADV
ejpam-3126	394	2	the	the	DET
ejpam-3126	394	3	following	follow	VERB
ejpam-3126	394	4	conditions	condition	NOUN
ejpam-3126	394	5	are	be	AUX
ejpam-3126	394	6	equivalent	equivalent	ADJ
ejpam-3126	394	7	:	:	PUNCT
ejpam-3126	394	8	(	(	PUNCT
ejpam-3126	394	9	i	i	NOUN
ejpam-3126	394	10	)	)	PUNCT
ejpam-3126	394	11	m1×	m1×	NOUN
ejpam-3126	394	12	.	.	PUNCT
ejpam-3126	394	13	.	.	PUNCT
ejpam-3126	395	1	.×mi−1×ni×mi+1×	.×mi−1×ni×mi+1×	PROPN
ejpam-3126	395	2	.	.	PUNCT
ejpam-3126	395	3	.	.	PUNCT
ejpam-3126	396	1	.×mk	.×mk	NOUN
ejpam-3126	396	2	is	be	AUX
ejpam-3126	396	3	a	a	DET
ejpam-3126	396	4	φ-2	φ-2	NOUN
ejpam-3126	396	5	-	-	PUNCT
ejpam-3126	396	6	absorbing	absorbing	ADJ
ejpam-3126	396	7	semi	semi	ADJ
ejpam-3126	396	8	-	-	ADJ
ejpam-3126	396	9	primary	primary	ADJ
ejpam-3126	396	10	submodule	submodule	NOUN
ejpam-3126	396	11	of	of	ADP
ejpam-3126	396	12	m1	m1	PROPN
ejpam-3126	396	13	×	×	PROPN
ejpam-3126	396	14	.	.	PUNCT
ejpam-3126	396	15	.	.	PUNCT
ejpam-3126	397	1	.×mk	.×mk	NOUN
ejpam-3126	397	2	.	.	PUNCT
ejpam-3126	398	1	(	(	PUNCT
ejpam-3126	398	2	ii	ii	NOUN
ejpam-3126	398	3	)	)	PUNCT
ejpam-3126	398	4	(	(	PUNCT
ejpam-3126	398	5	a	a	X
ejpam-3126	398	6	)	)	PUNCT
ejpam-3126	398	7	ni	ni	PROPN
ejpam-3126	398	8	is	be	AUX
ejpam-3126	398	9	a	a	DET
ejpam-3126	398	10	ψi-2	ψi-2	ADV
ejpam-3126	398	11	-	-	PUNCT
ejpam-3126	398	12	absorbing	absorbing	ADJ
ejpam-3126	398	13	semi	semi	ADJ
ejpam-3126	398	14	-	-	ADJ
ejpam-3126	398	15	primary	primary	ADJ
ejpam-3126	398	16	submodule	submodule	NOUN
ejpam-3126	398	17	of	of	ADP
ejpam-3126	398	18	mi	mi	PROPN
ejpam-3126	398	19	.	.	PROPN
ejpam-3126	398	20	(	(	PUNCT
ejpam-3126	398	21	b	b	NOUN
ejpam-3126	398	22	)	)	PUNCT
ejpam-3126	398	23	for	for	ADP
ejpam-3126	398	24	each	each	DET
ejpam-3126	398	25	a1	a1	NOUN
ejpam-3126	398	26	,	,	PUNCT
ejpam-3126	398	27	a2	a2	PROPN
ejpam-3126	398	28	∈	∈	PROPN
ejpam-3126	398	29	r	r	NOUN
ejpam-3126	398	30	and	and	CCONJ
ejpam-3126	398	31	m	m	PROPN
ejpam-3126	398	32	∈	∈	PROPN
ejpam-3126	398	33	mi	mi	PROPN
ejpam-3126	398	34	such	such	ADJ
ejpam-3126	398	35	that	that	SCONJ
ejpam-3126	398	36	a1a2	a1a2	PROPN
ejpam-3126	398	37	m	m	PROPN
ejpam-3126	398	38	∈	∈	ADJ
ejpam-3126	398	39	ψi(ni	ψi(ni	PROPN
ejpam-3126	398	40	)	)	PUNCT
ejpam-3126	398	41	,	,	PUNCT
ejpam-3126	398	42	if	if	SCONJ
ejpam-3126	398	43	a1a2	a1a2	ADP
ejpam-3126	398	44	6∈√	6∈√	NUM
ejpam-3126	398	45	(	(	PUNCT
ejpam-3126	398	46	ni	ni	PROPN
ejpam-3126	398	47	:	:	PUNCT
ejpam-3126	398	48	mi	mi	PROPN
ejpam-3126	398	49	)	)	PUNCT
ejpam-3126	398	50	,	,	PUNCT
ejpam-3126	398	51	a1	a1	PROPN
ejpam-3126	398	52	m	m	PROPN
ejpam-3126	398	53	6∈	6∈	PROPN
ejpam-3126	398	54	ni	ni	PROPN
ejpam-3126	398	55	and	and	CCONJ
ejpam-3126	398	56	an2	an2	PROPN
ejpam-3126	398	57	m	m	PROPN
ejpam-3126	398	58	6∈	6∈	PROPN
ejpam-3126	398	59	ni	ni	PROPN
ejpam-3126	398	60	for	for	ADP
ejpam-3126	398	61	all	all	DET
ejpam-3126	398	62	positive	positive	ADJ
ejpam-3126	398	63	integer	integer	NOUN
ejpam-3126	398	64	n	n	CCONJ
ejpam-3126	398	65	,	,	PUNCT
ejpam-3126	398	66	then	then	ADV
ejpam-3126	398	67	there	there	PRON
ejpam-3126	398	68	exists	exist	VERB
ejpam-3126	398	69	j	j	PROPN
ejpam-3126	398	70	∈	∈	PROPN
ejpam-3126	398	71	{	{	PUNCT
ejpam-3126	398	72	1	1	NUM
ejpam-3126	398	73	,	,	PUNCT
ejpam-3126	398	74	2	2	NUM
ejpam-3126	398	75	,	,	PUNCT
ejpam-3126	398	76	.	.	PUNCT
ejpam-3126	398	77	.	.	PUNCT
ejpam-3126	399	1	.	.	PUNCT
ejpam-3126	400	1	,	,	PUNCT
ejpam-3126	400	2	k	k	X
ejpam-3126	400	3	}	}	PUNCT
ejpam-3126	400	4	such	such	ADJ
ejpam-3126	400	5	that	that	SCONJ
ejpam-3126	400	6	a1a2	a1a2	PROPN
ejpam-3126	400	7	∈	∈	PROPN
ejpam-3126	400	8	(	(	PUNCT
ejpam-3126	400	9	ψj(mj	ψj(mj	PROPN
ejpam-3126	400	10	)	)	PUNCT
ejpam-3126	400	11	:	:	PUNCT
ejpam-3126	400	12	mj	mj	PROPN
ejpam-3126	400	13	)	)	PUNCT
ejpam-3126	400	14	.	.	PUNCT
ejpam-3126	401	1	proof	proof	NOUN
ejpam-3126	401	2	.	.	PUNCT
ejpam-3126	402	1	similar	similar	ADJ
ejpam-3126	402	2	to	to	ADP
ejpam-3126	402	3	the	the	DET
ejpam-3126	402	4	proof	proof	NOUN
ejpam-3126	402	5	of	of	ADP
ejpam-3126	402	6	theorem	theorem	NOUN
ejpam-3126	402	7	16	16	NUM
ejpam-3126	402	8	.	.	PUNCT
ejpam-3126	403	1	next	next	ADV
ejpam-3126	403	2	,	,	PUNCT
ejpam-3126	403	3	let	let	VERB
ejpam-3126	403	4	ri	ri	PRON
ejpam-3126	403	5	be	be	AUX
ejpam-3126	403	6	a	a	DET
ejpam-3126	403	7	commutative	commutative	ADJ
ejpam-3126	403	8	ring	ring	NOUN
ejpam-3126	403	9	with	with	ADP
ejpam-3126	403	10	identity	identity	NOUN
ejpam-3126	403	11	and	and	CCONJ
ejpam-3126	403	12	let	let	VERB
ejpam-3126	403	13	mi	mi	PROPN
ejpam-3126	403	14	be	be	AUX
ejpam-3126	403	15	an	an	DET
ejpam-3126	403	16	ri	ri	NOUN
ejpam-3126	403	17	-	-	PUNCT
ejpam-3126	403	18	module	module	NOUN
ejpam-3126	403	19	.	.	PUNCT
ejpam-3126	404	1	then	then	ADV
ejpam-3126	404	2	m1×m2	m1×m2	PROPN
ejpam-3126	404	3	is	be	AUX
ejpam-3126	404	4	an	an	DET
ejpam-3126	404	5	r1×r2	r1×r2	NOUN
ejpam-3126	404	6	-	-	PUNCT
ejpam-3126	404	7	module	module	NOUN
ejpam-3126	404	8	and	and	CCONJ
ejpam-3126	404	9	each	each	DET
ejpam-3126	404	10	submodule	submodule	NOUN
ejpam-3126	404	11	of	of	ADP
ejpam-3126	404	12	m1×m2	m1×m2	PROPN
ejpam-3126	404	13	is	be	AUX
ejpam-3126	404	14	of	of	ADP
ejpam-3126	404	15	the	the	DET
ejpam-3126	404	16	form	form	NOUN
ejpam-3126	404	17	n1×n2	n1×n2	NOUN
ejpam-3126	404	18	for	for	ADP
ejpam-3126	404	19	some	some	DET
ejpam-3126	404	20	submodules	submodule	NOUN
ejpam-3126	404	21	n1	n1	NOUN
ejpam-3126	404	22	of	of	ADP
ejpam-3126	404	23	m1	m1	PROPN
ejpam-3126	404	24	and	and	CCONJ
ejpam-3126	404	25	n2	n2	NOUN
ejpam-3126	404	26	of	of	ADP
ejpam-3126	404	27	m2	m2	PROPN
ejpam-3126	404	28	.	.	PUNCT
ejpam-3126	405	1	next	next	ADV
ejpam-3126	405	2	we	we	PRON
ejpam-3126	405	3	show	show	VERB
ejpam-3126	405	4	that	that	SCONJ
ejpam-3126	405	5	,	,	PUNCT
ejpam-3126	405	6	if	if	SCONJ
ejpam-3126	405	7	n1	n1	PROPN
ejpam-3126	405	8	is	be	AUX
ejpam-3126	405	9	a	a	DET
ejpam-3126	405	10	(	(	PUNCT
ejpam-3126	405	11	ψ1)0	ψ1)0	NOUN
ejpam-3126	405	12	-	-	PUNCT
ejpam-3126	405	13	2	2	NUM
ejpam-3126	405	14	-	-	PUNCT
ejpam-3126	405	15	absorbing	absorbing	ADJ
ejpam-3126	405	16	semi	semi	ADJ
ejpam-3126	405	17	-	-	ADJ
ejpam-3126	405	18	primary	primary	ADJ
ejpam-3126	405	19	submodule	submodule	NOUN
ejpam-3126	405	20	of	of	ADP
ejpam-3126	405	21	m1	m1	PROPN
ejpam-3126	405	22	,	,	PUNCT
ejpam-3126	405	23	then	then	ADV
ejpam-3126	405	24	n1×m2	n1×m2	PROPN
ejpam-3126	405	25	is	be	AUX
ejpam-3126	405	26	a	a	DET
ejpam-3126	405	27	φ-2	φ-2	NOUN
ejpam-3126	405	28	-	-	PUNCT
ejpam-3126	405	29	absorbing	absorbing	ADJ
ejpam-3126	405	30	semi	semi	ADJ
ejpam-3126	405	31	-	-	ADJ
ejpam-3126	405	32	primary	primary	ADJ
ejpam-3126	405	33	submodule	submodule	NOUN
ejpam-3126	405	34	if	if	SCONJ
ejpam-3126	405	35	{	{	PUNCT
ejpam-3126	405	36	0}×m2	0}×m2	NOUN
ejpam-3126	405	37	⊆	⊆	NUM
ejpam-3126	405	38	ψ1×ψ2(n1×m2	ψ1×ψ2(n1×m2	NOUN
ejpam-3126	405	39	)	)	PUNCT
ejpam-3126	405	40	.	.	PUNCT
ejpam-3126	406	1	first	first	ADV
ejpam-3126	406	2	,	,	PUNCT
ejpam-3126	406	3	we	we	PRON
ejpam-3126	406	4	would	would	AUX
ejpam-3126	406	5	like	like	VERB
ejpam-3126	406	6	to	to	PART
ejpam-3126	406	7	show	show	VERB
ejpam-3126	406	8	that	that	SCONJ
ejpam-3126	406	9	,	,	PUNCT
ejpam-3126	406	10	n1	n1	PROPN
ejpam-3126	406	11	is	be	AUX
ejpam-3126	406	12	a	a	DET
ejpam-3126	406	13	ψ1	ψ1	NOUN
ejpam-3126	406	14	-	-	PUNCT
ejpam-3126	406	15	2	2	NUM
ejpam-3126	406	16	-	-	PUNCT
ejpam-3126	406	17	absorbing	absorbing	ADJ
ejpam-3126	406	18	semi	semi	ADJ
ejpam-3126	406	19	-	-	ADJ
ejpam-3126	406	20	primary	primary	ADJ
ejpam-3126	406	21	submodule	submodule	NOUN
ejpam-3126	406	22	of	of	ADP
ejpam-3126	406	23	m1	m1	PROPN
ejpam-3126	406	24	if	if	SCONJ
ejpam-3126	406	25	n1	n1	NOUN
ejpam-3126	406	26	×m2	×m2	NOUN
ejpam-3126	406	27	is	be	AUX
ejpam-3126	406	28	a	a	DET
ejpam-3126	406	29	φ-2	φ-2	NOUN
ejpam-3126	406	30	-	-	PUNCT
ejpam-3126	406	31	absorbing	absorbing	ADJ
ejpam-3126	406	32	semi	semi	ADJ
ejpam-3126	406	33	-	-	ADJ
ejpam-3126	406	34	primary	primary	ADJ
ejpam-3126	406	35	submodule	submodule	NOUN
ejpam-3126	406	36	of	of	ADP
ejpam-3126	406	37	m1	m1	PROPN
ejpam-3126	406	38	×m2	×m2	PROPN
ejpam-3126	406	39	.	.	PUNCT
ejpam-3126	407	1	theorem	theorem	NOUN
ejpam-3126	407	2	18	18	NUM
ejpam-3126	407	3	.	.	PUNCT
ejpam-3126	408	1	let	let	VERB
ejpam-3126	408	2	ψi	ψi	VERB
ejpam-3126	408	3	:	:	PUNCT
ejpam-3126	408	4	s(mi)→	s(mi)→	PROPN
ejpam-3126	408	5	s(mi)∪{∅	s(mi)∪{∅	PROPN
ejpam-3126	408	6	}	}	PUNCT
ejpam-3126	408	7	be	be	AUX
ejpam-3126	408	8	a	a	DET
ejpam-3126	408	9	function	function	NOUN
ejpam-3126	408	10	with	with	ADP
ejpam-3126	408	11	φ	φ	PROPN
ejpam-3126	408	12	=	=	SYM
ejpam-3126	408	13	ψ1×ψ2	ψ1×ψ2	PROPN
ejpam-3126	408	14	.	.	PUNCT
ejpam-3126	409	1	if	if	SCONJ
ejpam-3126	409	2	n1×m2	n1×m2	PROPN
ejpam-3126	409	3	is	be	AUX
ejpam-3126	409	4	a	a	DET
ejpam-3126	409	5	φ-2	φ-2	NOUN
ejpam-3126	409	6	-	-	PUNCT
ejpam-3126	409	7	absorbing	absorbing	ADJ
ejpam-3126	409	8	semi	semi	ADJ
ejpam-3126	409	9	-	-	ADJ
ejpam-3126	409	10	primary	primary	ADJ
ejpam-3126	409	11	submodule	submodule	NOUN
ejpam-3126	409	12	of	of	ADP
ejpam-3126	409	13	m1	m1	PROPN
ejpam-3126	409	14	×	×	PROPN
ejpam-3126	409	15	m2	m2	PROPN
ejpam-3126	409	16	,	,	PUNCT
ejpam-3126	409	17	then	then	ADV
ejpam-3126	409	18	n1	n1	PROPN
ejpam-3126	409	19	is	be	AUX
ejpam-3126	409	20	a	a	DET
ejpam-3126	409	21	ψ1	ψ1	NOUN
ejpam-3126	409	22	-	-	PUNCT
ejpam-3126	409	23	2	2	NUM
ejpam-3126	409	24	-	-	PUNCT
ejpam-3126	409	25	absorbing	absorbing	ADJ
ejpam-3126	409	26	semi	semi	ADJ
ejpam-3126	409	27	-	-	ADJ
ejpam-3126	409	28	primary	primary	ADJ
ejpam-3126	409	29	submodule	submodule	NOUN
ejpam-3126	409	30	of	of	ADP
ejpam-3126	409	31	m1	m1	PROPN
ejpam-3126	409	32	.	.	PUNCT
ejpam-3126	410	1	proof	proof	NOUN
ejpam-3126	410	2	.	.	PUNCT
ejpam-3126	411	1	let	let	VERB
ejpam-3126	411	2	a1	a1	NOUN
ejpam-3126	411	3	,	,	PUNCT
ejpam-3126	411	4	a2	a2	PROPN
ejpam-3126	411	5	∈	∈	PROPN
ejpam-3126	411	6	r1	r1	PROPN
ejpam-3126	411	7	and	and	CCONJ
ejpam-3126	411	8	m	m	PROPN
ejpam-3126	411	9	∈	∈	PROPN
ejpam-3126	411	10	m1	m1	NOUN
ejpam-3126	411	11	such	such	ADJ
ejpam-3126	411	12	that	that	SCONJ
ejpam-3126	411	13	a1a2	a1a2	PROPN
ejpam-3126	411	14	m	m	NOUN
ejpam-3126	411	15	∈	∈	ADJ
ejpam-3126	411	16	n1	n1	NOUN
ejpam-3126	411	17	−	−	PROPN
ejpam-3126	411	18	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	411	19	)	)	PUNCT
ejpam-3126	411	20	.	.	PUNCT
ejpam-3126	412	1	then	then	ADV
ejpam-3126	412	2	(	(	PUNCT
ejpam-3126	412	3	a1	a1	PROPN
ejpam-3126	412	4	,	,	PUNCT
ejpam-3126	412	5	0)(a2	0)(a2	NOUN
ejpam-3126	412	6	,	,	PUNCT
ejpam-3126	412	7	0)(m	0)(m	NUM
ejpam-3126	412	8	,	,	PUNCT
ejpam-3126	412	9	0	0	NUM
ejpam-3126	412	10	)	)	PUNCT
ejpam-3126	412	11	∈	∈	NOUN
ejpam-3126	412	12	n1	n1	NOUN
ejpam-3126	412	13	×m2	×m2	NOUN
ejpam-3126	412	14	−	−	NOUN
ejpam-3126	412	15	ψ1(n1	ψ1(n1	SYM
ejpam-3126	412	16	)	)	PUNCT
ejpam-3126	412	17	×	×	NOUN
ejpam-3126	412	18	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	412	19	)	)	PUNCT
ejpam-3126	412	20	.	.	PUNCT
ejpam-3126	413	1	by	by	ADP
ejpam-3126	413	2	definition	definition	NOUN
ejpam-3126	413	3	1	1	NUM
ejpam-3126	413	4	,	,	PUNCT
ejpam-3126	413	5	a1a2	a1a2	PROPN
ejpam-3126	413	6	∈	∈	NOUN
ejpam-3126	413	7	√	√	NUM
ejpam-3126	413	8	(	(	PUNCT
ejpam-3126	413	9	n1	n1	PROPN
ejpam-3126	413	10	:	:	PUNCT
ejpam-3126	413	11	m1	m1	NOUN
ejpam-3126	413	12	)	)	PUNCT
ejpam-3126	413	13	or	or	CCONJ
ejpam-3126	413	14	a1	a1	NOUN
ejpam-3126	413	15	m	m	PROPN
ejpam-3126	413	16	∈	∈	NOUN
ejpam-3126	413	17	n1	n1	NOUN
ejpam-3126	413	18	or	or	CCONJ
ejpam-3126	413	19	an2	an2	PROPN
ejpam-3126	413	20	m	m	NOUN
ejpam-3126	413	21	∈	∈	PROPN
ejpam-3126	413	22	n1	n1	NOUN
ejpam-3126	413	23	.	.	PUNCT
ejpam-3126	414	1	lemma	lemma	PROPN
ejpam-3126	414	2	1	1	X
ejpam-3126	414	3	.	.	PUNCT
ejpam-3126	415	1	let	let	VERB
ejpam-3126	415	2	ψi	ψi	VERB
ejpam-3126	415	3	:	:	PUNCT
ejpam-3126	415	4	s(mi	s(mi	VERB
ejpam-3126	415	5	)	)	PUNCT
ejpam-3126	415	6	→	→	SYM
ejpam-3126	415	7	s(mi	s(mi	NOUN
ejpam-3126	415	8	)	)	PUNCT
ejpam-3126	415	9	∪	∪	ADP
ejpam-3126	415	10	{	{	PUNCT
ejpam-3126	415	11	∅	∅	NOUN
ejpam-3126	415	12	}	}	PUNCT
ejpam-3126	415	13	be	be	AUX
ejpam-3126	415	14	a	a	DET
ejpam-3126	415	15	function	function	NOUN
ejpam-3126	415	16	.	.	PUNCT
ejpam-3126	416	1	if	if	SCONJ
ejpam-3126	416	2	n1	n1	PROPN
ejpam-3126	416	3	is	be	AUX
ejpam-3126	416	4	a	a	DET
ejpam-3126	416	5	(	(	PUNCT
ejpam-3126	416	6	ψ1)0	ψ1)0	NOUN
ejpam-3126	416	7	-	-	PUNCT
ejpam-3126	416	8	2	2	NUM
ejpam-3126	416	9	-	-	PUNCT
ejpam-3126	416	10	absorbing	absorbing	ADJ
ejpam-3126	416	11	semi	semi	ADJ
ejpam-3126	416	12	-	-	ADJ
ejpam-3126	416	13	primary	primary	ADJ
ejpam-3126	416	14	submodule	submodule	NOUN
ejpam-3126	416	15	of	of	ADP
ejpam-3126	416	16	m1	m1	PROPN
ejpam-3126	416	17	such	such	ADJ
ejpam-3126	416	18	that	that	SCONJ
ejpam-3126	416	19	{	{	PUNCT
ejpam-3126	416	20	0}×m2	0}×m2	NOUN
ejpam-3126	416	21	⊆	⊆	NUM
ejpam-3126	416	22	ψ1×ψ2(n1×m2	ψ1×ψ2(n1×m2	NOUN
ejpam-3126	416	23	)	)	PUNCT
ejpam-3126	416	24	,	,	PUNCT
ejpam-3126	416	25	then	then	ADV
ejpam-3126	416	26	n1×m2	n1×m2	PROPN
ejpam-3126	416	27	is	be	AUX
ejpam-3126	416	28	a	a	DET
ejpam-3126	416	29	ψ1	ψ1	ADJ
ejpam-3126	416	30	×	×	NOUN
ejpam-3126	416	31	ψ2	ψ2	NOUN
ejpam-3126	416	32	-	-	PUNCT
ejpam-3126	416	33	2	2	NUM
ejpam-3126	416	34	-	-	PUNCT
ejpam-3126	416	35	absorbing	absorbing	ADJ
ejpam-3126	416	36	semi	semi	ADJ
ejpam-3126	416	37	-	-	ADJ
ejpam-3126	416	38	primary	primary	ADJ
ejpam-3126	416	39	submodule	submodule	NOUN
ejpam-3126	416	40	of	of	ADP
ejpam-3126	416	41	m1	m1	PROPN
ejpam-3126	416	42	×m2	×m2	PROPN
ejpam-3126	416	43	.	.	PUNCT
ejpam-3126	417	1	proof	proof	NOUN
ejpam-3126	417	2	.	.	PUNCT
ejpam-3126	418	1	let	let	VERB
ejpam-3126	418	2	(	(	PUNCT
ejpam-3126	418	3	a1	a1	NOUN
ejpam-3126	418	4	,	,	PUNCT
ejpam-3126	418	5	b1	b1	NOUN
ejpam-3126	418	6	)	)	PUNCT
ejpam-3126	418	7	,	,	PUNCT
ejpam-3126	418	8	(	(	PUNCT
ejpam-3126	418	9	a2	a2	PROPN
ejpam-3126	418	10	,	,	PUNCT
ejpam-3126	418	11	b2	b2	NOUN
ejpam-3126	418	12	)	)	PUNCT
ejpam-3126	418	13	∈	∈	PROPN
ejpam-3126	418	14	r1	r1	PROPN
ejpam-3126	418	15	×r2	×r2	PROPN
ejpam-3126	418	16	and	and	CCONJ
ejpam-3126	418	17	(	(	PUNCT
ejpam-3126	418	18	m1,m2	m1,m2	PROPN
ejpam-3126	418	19	)	)	PUNCT
ejpam-3126	418	20	∈m1	∈m1	ADJ
ejpam-3126	418	21	×m2	×m2	NOUN
ejpam-3126	418	22	such	such	ADJ
ejpam-3126	418	23	that	that	SCONJ
ejpam-3126	418	24	(	(	PUNCT
ejpam-3126	418	25	a1	a1	PROPN
ejpam-3126	418	26	,	,	PUNCT
ejpam-3126	418	27	b1)(a2	b1)(a2	PROPN
ejpam-3126	418	28	,	,	PUNCT
ejpam-3126	418	29	b2)(m1,m2	b2)(m1,m2	PROPN
ejpam-3126	418	30	)	)	PUNCT
ejpam-3126	418	31	∈	∈	PROPN
ejpam-3126	418	32	n1	n1	NOUN
ejpam-3126	418	33	×m2	×m2	NOUN
ejpam-3126	418	34	−	−	NOUN
ejpam-3126	418	35	φ(n1	φ(n1	NOUN
ejpam-3126	418	36	×m2	×m2	NOUN
ejpam-3126	418	37	)	)	PUNCT
ejpam-3126	418	38	.	.	PUNCT
ejpam-3126	419	1	pai	pai	PROPN
ejpam-3126	419	2	.	.	PROPN
ejpam-3126	419	3	yiarayong	yiarayong	PROPN
ejpam-3126	419	4	,	,	PUNCT
ejpam-3126	419	5	m.	m.	NOUN
ejpam-3126	419	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	419	7	/	/	SYM
ejpam-3126	419	8	eur	eur	PROPN
ejpam-3126	419	9	.	.	PUNCT
ejpam-3126	420	1	j.	j.	PROPN
ejpam-3126	420	2	pure	pure	PROPN
ejpam-3126	420	3	appl	appl	PROPN
ejpam-3126	420	4	.	.	PROPN
ejpam-3126	420	5	math	math	PROPN
ejpam-3126	420	6	,	,	PUNCT
ejpam-3126	420	7	11	11	NUM
ejpam-3126	420	8	(	(	PUNCT
ejpam-3126	420	9	1	1	NUM
ejpam-3126	420	10	)	)	PUNCT
ejpam-3126	420	11	(	(	PUNCT
ejpam-3126	420	12	2018	2018	NUM
ejpam-3126	420	13	)	)	PUNCT
ejpam-3126	420	14	,	,	PUNCT
ejpam-3126	420	15	35	35	NUM
ejpam-3126	420	16	-	-	SYM
ejpam-3126	420	17	50	50	NUM
ejpam-3126	420	18	47	47	NUM
ejpam-3126	420	19	by	by	ADP
ejpam-3126	420	20	assumption	assumption	NOUN
ejpam-3126	420	21	,	,	PUNCT
ejpam-3126	420	22	(	(	PUNCT
ejpam-3126	420	23	a1a2m1	a1a2m1	PROPN
ejpam-3126	420	24	,	,	PUNCT
ejpam-3126	420	25	b1b2m2	b1b2m2	NOUN
ejpam-3126	420	26	)	)	PUNCT
ejpam-3126	420	27	6∈	6∈	NOUN
ejpam-3126	420	28	{	{	PUNCT
ejpam-3126	420	29	0	0	NUM
ejpam-3126	420	30	}	}	PUNCT
ejpam-3126	420	31	×m2	×m2	NOUN
ejpam-3126	420	32	.	.	PUNCT
ejpam-3126	421	1	clearly	clearly	ADV
ejpam-3126	421	2	,	,	PUNCT
ejpam-3126	421	3	a1a2m1	a1a2m1	PROPN
ejpam-3126	421	4	∈	∈	PROPN
ejpam-3126	421	5	n1	n1	PROPN
ejpam-3126	421	6	−	−	PROPN
ejpam-3126	421	7	(	(	PUNCT
ejpam-3126	421	8	ψ1)0(n1	ψ1)0(n1	PROPN
ejpam-3126	421	9	)	)	PUNCT
ejpam-3126	421	10	.	.	PUNCT
ejpam-3126	422	1	by	by	ADP
ejpam-3126	422	2	definition	definition	NOUN
ejpam-3126	422	3	1	1	NUM
ejpam-3126	422	4	,	,	PUNCT
ejpam-3126	422	5	a1a2	a1a2	PROPN
ejpam-3126	422	6	∈	∈	NOUN
ejpam-3126	422	7	√	√	NUM
ejpam-3126	422	8	(	(	PUNCT
ejpam-3126	422	9	n1	n1	PROPN
ejpam-3126	422	10	:	:	PUNCT
ejpam-3126	422	11	m1	m1	NOUN
ejpam-3126	422	12	)	)	PUNCT
ejpam-3126	422	13	or	or	CCONJ
ejpam-3126	422	14	a1m1	a1m1	VERB
ejpam-3126	422	15	∈	∈	NOUN
ejpam-3126	422	16	n1	n1	NOUN
ejpam-3126	422	17	or	or	CCONJ
ejpam-3126	422	18	an2m1	an2m1	PROPN
ejpam-3126	422	19	∈	∈	PROPN
ejpam-3126	422	20	n1	n1	NOUN
ejpam-3126	422	21	for	for	ADP
ejpam-3126	422	22	some	some	DET
ejpam-3126	422	23	positive	positive	ADJ
ejpam-3126	422	24	integer	integer	NOUN
ejpam-3126	422	25	n.	n.	NOUN
ejpam-3126	422	26	therefore	therefore	ADV
ejpam-3126	422	27	n1	n1	PROPN
ejpam-3126	422	28	×m2	×m2	PROPN
ejpam-3126	422	29	is	be	AUX
ejpam-3126	422	30	a	a	DET
ejpam-3126	422	31	φ-2	φ-2	NOUN
ejpam-3126	422	32	-	-	PUNCT
ejpam-3126	422	33	absorbing	absorbing	ADJ
ejpam-3126	422	34	semi	semi	ADJ
ejpam-3126	422	35	-	-	ADJ
ejpam-3126	422	36	primary	primary	ADJ
ejpam-3126	422	37	submodule	submodule	NOUN
ejpam-3126	422	38	of	of	ADP
ejpam-3126	422	39	m1	m1	PROPN
ejpam-3126	422	40	×m2	×m2	PROPN
ejpam-3126	422	41	.	.	PUNCT
ejpam-3126	423	1	corollary	corollary	ADJ
ejpam-3126	423	2	8	8	NUM
ejpam-3126	423	3	.	.	PUNCT
ejpam-3126	424	1	let	let	VERB
ejpam-3126	424	2	ψi	ψi	VERB
ejpam-3126	424	3	:	:	PUNCT
ejpam-3126	424	4	s(mi	s(mi	VERB
ejpam-3126	424	5	)	)	PUNCT
ejpam-3126	424	6	→	→	SYM
ejpam-3126	424	7	s(mi	s(mi	NOUN
ejpam-3126	424	8	)	)	PUNCT
ejpam-3126	424	9	∪	∪	ADP
ejpam-3126	424	10	{	{	PUNCT
ejpam-3126	424	11	∅	∅	NOUN
ejpam-3126	424	12	}	}	PUNCT
ejpam-3126	424	13	be	be	AUX
ejpam-3126	424	14	a	a	DET
ejpam-3126	424	15	function	function	NOUN
ejpam-3126	424	16	with	with	ADP
ejpam-3126	424	17	φ	φ	PROPN
ejpam-3126	424	18	=	=	SYM
ejpam-3126	424	19	ψ1	ψ1	ADJ
ejpam-3126	424	20	×	×	NOUN
ejpam-3126	424	21	ψ2	ψ2	NOUN
ejpam-3126	424	22	.	.	PUNCT
ejpam-3126	425	1	if	if	SCONJ
ejpam-3126	425	2	n2	n2	ADJ
ejpam-3126	425	3	is	be	AUX
ejpam-3126	425	4	a	a	DET
ejpam-3126	425	5	(	(	PUNCT
ejpam-3126	425	6	ψ2)0	ψ2)0	NUM
ejpam-3126	425	7	-	-	PUNCT
ejpam-3126	425	8	2	2	NUM
ejpam-3126	425	9	-	-	PUNCT
ejpam-3126	425	10	absorbing	absorbing	ADJ
ejpam-3126	425	11	semi	semi	ADJ
ejpam-3126	425	12	-	-	ADJ
ejpam-3126	425	13	primary	primary	ADJ
ejpam-3126	425	14	submodule	submodule	NOUN
ejpam-3126	425	15	of	of	ADP
ejpam-3126	425	16	m2	m2	PROPN
ejpam-3126	425	17	such	such	ADJ
ejpam-3126	425	18	that	that	SCONJ
ejpam-3126	425	19	m1×{0	m1×{0	PRON
ejpam-3126	425	20	}	}	PUNCT
ejpam-3126	425	21	⊆	⊆	NUM
ejpam-3126	425	22	ψ1×ψ2(m1×n2	ψ1×ψ2(m1×n2	NOUN
ejpam-3126	425	23	)	)	PUNCT
ejpam-3126	425	24	,	,	PUNCT
ejpam-3126	425	25	then	then	ADV
ejpam-3126	425	26	m1	m1	PROPN
ejpam-3126	425	27	×n2	×n2	PROPN
ejpam-3126	425	28	is	be	AUX
ejpam-3126	425	29	a	a	DET
ejpam-3126	425	30	φ-2	φ-2	NOUN
ejpam-3126	425	31	-	-	PUNCT
ejpam-3126	425	32	absorbing	absorbing	ADJ
ejpam-3126	425	33	semi	semi	ADJ
ejpam-3126	425	34	-	-	ADJ
ejpam-3126	425	35	primary	primary	ADJ
ejpam-3126	425	36	submodule	submodule	NOUN
ejpam-3126	425	37	of	of	ADP
ejpam-3126	425	38	m1	m1	PROPN
ejpam-3126	425	39	×m2	×m2	PROPN
ejpam-3126	425	40	.	.	PUNCT
ejpam-3126	426	1	proof	proof	NOUN
ejpam-3126	426	2	.	.	PUNCT
ejpam-3126	427	1	similar	similar	ADJ
ejpam-3126	427	2	to	to	ADP
ejpam-3126	427	3	the	the	DET
ejpam-3126	427	4	proof	proof	NOUN
ejpam-3126	427	5	of	of	ADP
ejpam-3126	427	6	lemma	lemma	PROPN
ejpam-3126	427	7	1	1	NUM
ejpam-3126	427	8	.	.	PUNCT
ejpam-3126	427	9	theorem	theorem	NOUN
ejpam-3126	427	10	19	19	NUM
ejpam-3126	427	11	.	.	PUNCT
ejpam-3126	428	1	let	let	VERB
ejpam-3126	428	2	ψi	ψi	NOUN
ejpam-3126	428	3	:	:	PUNCT
ejpam-3126	428	4	s(mi	s(mi	VERB
ejpam-3126	428	5	)	)	PUNCT
ejpam-3126	428	6	→	→	SYM
ejpam-3126	428	7	s(mi	s(mi	NOUN
ejpam-3126	428	8	)	)	PUNCT
ejpam-3126	428	9	∪	∪	ADP
ejpam-3126	428	10	{	{	PUNCT
ejpam-3126	428	11	∅	∅	NOUN
ejpam-3126	428	12	}	}	PUNCT
ejpam-3126	428	13	be	be	AUX
ejpam-3126	428	14	a	a	DET
ejpam-3126	428	15	function	function	NOUN
ejpam-3126	428	16	with	with	ADP
ejpam-3126	428	17	φ	φ	PROPN
ejpam-3126	428	18	=	=	SYM
ejpam-3126	428	19	ψ1	ψ1	ADJ
ejpam-3126	428	20	×	×	NOUN
ejpam-3126	428	21	.	.	PUNCT
ejpam-3126	428	22	.	.	PUNCT
ejpam-3126	428	23	.	.	PUNCT
ejpam-3126	429	1	×	×	NOUN
ejpam-3126	429	2	ψk	ψk	NOUN
ejpam-3126	429	3	and	and	CCONJ
ejpam-3126	429	4	m1	m1	PROPN
ejpam-3126	429	5	×	×	NOUN
ejpam-3126	429	6	.	.	PUNCT
ejpam-3126	429	7	.	.	PUNCT
ejpam-3126	429	8	.	.	PUNCT
ejpam-3126	430	1	×	×	NOUN
ejpam-3126	430	2	mi−1	mi−1	PROPN
ejpam-3126	430	3	×	×	NOUN
ejpam-3126	430	4	{	{	PUNCT
ejpam-3126	430	5	0	0	NUM
ejpam-3126	430	6	}	}	PUNCT
ejpam-3126	430	7	×	×	NOUN
ejpam-3126	430	8	mi+1	mi+1	INTJ
ejpam-3126	430	9	×	×	NOUN
ejpam-3126	430	10	.	.	PUNCT
ejpam-3126	430	11	.	.	PUNCT
ejpam-3126	431	1	.	.	PUNCT
ejpam-3126	432	1	×	×	PROPN
ejpam-3126	432	2	mk	mk	NOUN
ejpam-3126	432	3	⊆	⊆	NUM
ejpam-3126	432	4	φ(m1	φ(m1	NOUN
ejpam-3126	432	5	×	×	VERB
ejpam-3126	432	6	.	.	PUNCT
ejpam-3126	432	7	.	.	PUNCT
ejpam-3126	432	8	.	.	PUNCT
ejpam-3126	433	1	×	×	NOUN
ejpam-3126	434	1	mi−1	mi−1	PROPN
ejpam-3126	434	2	×	×	NOUN
ejpam-3126	434	3	ni	ni	NOUN
ejpam-3126	434	4	×	×	NOUN
ejpam-3126	434	5	mi+1	mi+1	INTJ
ejpam-3126	434	6	×	×	NOUN
ejpam-3126	434	7	.	.	PUNCT
ejpam-3126	434	8	.	.	PUNCT
ejpam-3126	434	9	.	.	PUNCT
ejpam-3126	435	1	×mk	×mk	NOUN
ejpam-3126	435	2	)	)	PUNCT
ejpam-3126	435	3	.	.	PUNCT
ejpam-3126	436	1	then	then	ADV
ejpam-3126	436	2	nj	nj	PROPN
ejpam-3126	436	3	is	be	AUX
ejpam-3126	436	4	a	a	DET
ejpam-3126	436	5	(	(	PUNCT
ejpam-3126	436	6	ψj)0	ψj)0	NOUN
ejpam-3126	436	7	-	-	PUNCT
ejpam-3126	436	8	2	2	NUM
ejpam-3126	436	9	-	-	PUNCT
ejpam-3126	436	10	absorbing	absorbing	ADJ
ejpam-3126	436	11	semi	semi	ADJ
ejpam-3126	436	12	-	-	ADJ
ejpam-3126	436	13	primary	primary	ADJ
ejpam-3126	436	14	submodule	submodule	NOUN
ejpam-3126	436	15	of	of	ADP
ejpam-3126	436	16	mj	mj	PROPN
ejpam-3126	436	17	if	if	SCONJ
ejpam-3126	437	1	and	and	CCONJ
ejpam-3126	437	2	only	only	ADV
ejpam-3126	437	3	if	if	SCONJ
ejpam-3126	437	4	m1×m2×	m1×m2×	PROPN
ejpam-3126	437	5	.	.	PUNCT
ejpam-3126	437	6	.	.	PUNCT
ejpam-3126	438	1	.×mi−1×ni×mi+1×	.×mi−1×ni×mi+1×	PROPN
ejpam-3126	438	2	.	.	PUNCT
ejpam-3126	438	3	.	.	PUNCT
ejpam-3126	439	1	.×mk	.×mk	NOUN
ejpam-3126	439	2	is	be	AUX
ejpam-3126	439	3	a	a	DET
ejpam-3126	439	4	φ-2	φ-2	NOUN
ejpam-3126	439	5	-	-	PUNCT
ejpam-3126	439	6	absorbing	absorbing	ADJ
ejpam-3126	439	7	semi	semi	ADJ
ejpam-3126	439	8	-	-	ADJ
ejpam-3126	439	9	primary	primary	ADJ
ejpam-3126	439	10	submodule	submodule	NOUN
ejpam-3126	439	11	of	of	ADP
ejpam-3126	439	12	m1	m1	PROPN
ejpam-3126	439	13	×m2	×m2	PROPN
ejpam-3126	439	14	×	×	VERB
ejpam-3126	439	15	.	.	PUNCT
ejpam-3126	439	16	.	.	PUNCT
ejpam-3126	440	1	.×mk	.×mk	NOUN
ejpam-3126	440	2	.	.	PUNCT
ejpam-3126	441	1	proof	proof	NOUN
ejpam-3126	441	2	.	.	PUNCT
ejpam-3126	442	1	similar	similar	ADJ
ejpam-3126	442	2	to	to	ADP
ejpam-3126	442	3	the	the	DET
ejpam-3126	442	4	proof	proof	NOUN
ejpam-3126	442	5	of	of	ADP
ejpam-3126	442	6	theorem	theorem	ADJ
ejpam-3126	442	7	18	18	NUM
ejpam-3126	442	8	and	and	CCONJ
ejpam-3126	442	9	lemma	lemma	PROPN
ejpam-3126	442	10	1	1	NUM
ejpam-3126	442	11	.	.	PUNCT
ejpam-3126	442	12	theorem	theorem	NOUN
ejpam-3126	442	13	20	20	NUM
ejpam-3126	442	14	.	.	PUNCT
ejpam-3126	443	1	let	let	VERB
ejpam-3126	443	2	(	(	PUNCT
ejpam-3126	443	3	ψi)n	ψi)n	NOUN
ejpam-3126	443	4	:	:	PUNCT
ejpam-3126	443	5	s(mi	s(mi	PROPN
ejpam-3126	443	6	)	)	PUNCT
ejpam-3126	443	7	→	→	SYM
ejpam-3126	443	8	s(mi	s(mi	NOUN
ejpam-3126	443	9	)	)	PUNCT
ejpam-3126	443	10	∪	∪	ADP
ejpam-3126	443	11	{	{	PUNCT
ejpam-3126	443	12	∅	∅	NOUN
ejpam-3126	443	13	}	}	PUNCT
ejpam-3126	443	14	be	be	AUX
ejpam-3126	443	15	a	a	DET
ejpam-3126	443	16	function	function	NOUN
ejpam-3126	443	17	with	with	ADP
ejpam-3126	443	18	φn	φn	NOUN
ejpam-3126	443	19	=	=	PUNCT
ejpam-3126	443	20	(	(	PUNCT
ejpam-3126	443	21	ψ1)n	ψ1)n	PROPN
ejpam-3126	443	22	×	×	PROPN
ejpam-3126	443	23	(	(	PUNCT
ejpam-3126	443	24	ψ2)n	ψ2)n	NOUN
ejpam-3126	443	25	.	.	PUNCT
ejpam-3126	444	1	if	if	SCONJ
ejpam-3126	444	2	n1	n1	PROPN
ejpam-3126	444	3	is	be	AUX
ejpam-3126	444	4	a	a	DET
ejpam-3126	444	5	(	(	PUNCT
ejpam-3126	444	6	ψ1)0	ψ1)0	NOUN
ejpam-3126	444	7	-	-	PUNCT
ejpam-3126	444	8	2	2	NUM
ejpam-3126	444	9	-	-	PUNCT
ejpam-3126	444	10	absorbing	absorbing	ADJ
ejpam-3126	444	11	semi	semi	ADJ
ejpam-3126	444	12	-	-	ADJ
ejpam-3126	444	13	primary	primary	ADJ
ejpam-3126	444	14	submodule	submodule	NOUN
ejpam-3126	444	15	of	of	ADP
ejpam-3126	444	16	m1	m1	PROPN
ejpam-3126	444	17	such	such	ADJ
ejpam-3126	444	18	that	that	PRON
ejpam-3126	444	19	(	(	PUNCT
ejpam-3126	444	20	ψ2)3(m2	ψ2)3(m2	NOUN
ejpam-3126	444	21	)	)	PUNCT
ejpam-3126	444	22	=	=	SYM
ejpam-3126	444	23	m2	m2	PROPN
ejpam-3126	444	24	,	,	PUNCT
ejpam-3126	444	25	then	then	ADV
ejpam-3126	444	26	n1	n1	NOUN
ejpam-3126	444	27	×m2	×m2	NOUN
ejpam-3126	444	28	is	be	AUX
ejpam-3126	444	29	a	a	DET
ejpam-3126	444	30	φ3	φ3	NOUN
ejpam-3126	444	31	-	-	PUNCT
ejpam-3126	444	32	2	2	NUM
ejpam-3126	444	33	-	-	PUNCT
ejpam-3126	444	34	absorbing	absorbing	ADJ
ejpam-3126	444	35	semi	semi	ADJ
ejpam-3126	444	36	-	-	ADJ
ejpam-3126	444	37	primary	primary	ADJ
ejpam-3126	444	38	submodule	submodule	NOUN
ejpam-3126	444	39	of	of	ADP
ejpam-3126	444	40	m1	m1	PROPN
ejpam-3126	444	41	×m2	×m2	PROPN
ejpam-3126	444	42	.	.	PUNCT
ejpam-3126	445	1	proof	proof	NOUN
ejpam-3126	445	2	.	.	PUNCT
ejpam-3126	446	1	if	if	SCONJ
ejpam-3126	446	2	n1	n1	PROPN
ejpam-3126	446	3	is	be	AUX
ejpam-3126	446	4	a	a	DET
ejpam-3126	446	5	2	2	NUM
ejpam-3126	446	6	-	-	PUNCT
ejpam-3126	446	7	absorbing	absorbing	ADJ
ejpam-3126	446	8	semi	semi	ADJ
ejpam-3126	446	9	-	-	ADJ
ejpam-3126	446	10	primary	primary	ADJ
ejpam-3126	446	11	submodule	submodule	NOUN
ejpam-3126	446	12	of	of	ADP
ejpam-3126	446	13	m1	m1	PROPN
ejpam-3126	446	14	,	,	PUNCT
ejpam-3126	446	15	then	then	ADV
ejpam-3126	446	16	n1	n1	NOUN
ejpam-3126	446	17	×m2	×m2	NOUN
ejpam-3126	446	18	is	be	AUX
ejpam-3126	446	19	a	a	DET
ejpam-3126	446	20	2absorbing	2absorbing	NUM
ejpam-3126	446	21	semi	semi	ADJ
ejpam-3126	446	22	-	-	ADJ
ejpam-3126	446	23	primary	primary	ADJ
ejpam-3126	446	24	submodule	submodule	NOUN
ejpam-3126	446	25	of	of	ADP
ejpam-3126	446	26	m1	m1	PROPN
ejpam-3126	446	27	×m2	×m2	PROPN
ejpam-3126	446	28	.	.	PUNCT
ejpam-3126	447	1	clearly	clearly	ADV
ejpam-3126	447	2	,	,	PUNCT
ejpam-3126	447	3	n1	n1	PROPN
ejpam-3126	447	4	×m2	×m2	NOUN
ejpam-3126	447	5	is	be	AUX
ejpam-3126	447	6	a	a	DET
ejpam-3126	447	7	φ3	φ3	NOUN
ejpam-3126	447	8	-	-	PUNCT
ejpam-3126	447	9	2	2	NUM
ejpam-3126	447	10	-	-	PUNCT
ejpam-3126	447	11	absorbing	absorbing	ADJ
ejpam-3126	447	12	semi	semi	ADJ
ejpam-3126	447	13	-	-	ADJ
ejpam-3126	447	14	primary	primary	ADJ
ejpam-3126	447	15	submodule	submodule	NOUN
ejpam-3126	447	16	of	of	ADP
ejpam-3126	447	17	m1	m1	PROPN
ejpam-3126	447	18	×m2	×m2	PROPN
ejpam-3126	447	19	.	.	PUNCT
ejpam-3126	448	1	assume	assume	VERB
ejpam-3126	448	2	that	that	SCONJ
ejpam-3126	448	3	n1	n1	NOUN
ejpam-3126	448	4	is	be	AUX
ejpam-3126	448	5	not	not	PART
ejpam-3126	448	6	2	2	NUM
ejpam-3126	448	7	-	-	PUNCT
ejpam-3126	448	8	absorbing	absorbing	ADJ
ejpam-3126	448	9	semi	semi	ADJ
ejpam-3126	448	10	-	-	ADJ
ejpam-3126	448	11	primary	primary	ADJ
ejpam-3126	448	12	.	.	PUNCT
ejpam-3126	449	1	by	by	ADP
ejpam-3126	449	2	corollary	corollary	ADJ
ejpam-3126	449	3	6	6	NUM
ejpam-3126	449	4	,	,	PUNCT
ejpam-3126	449	5	(	(	PUNCT
ejpam-3126	449	6	ψ1)3	ψ1)3	NOUN
ejpam-3126	449	7	≤	≤	NUM
ejpam-3126	449	8	(	(	PUNCT
ejpam-3126	449	9	ψ1)0	ψ1)0	VERB
ejpam-3126	449	10	so	so	ADV
ejpam-3126	449	11	(	(	PUNCT
ejpam-3126	449	12	n1	n1	NOUN
ejpam-3126	449	13	:	:	PUNCT
ejpam-3126	449	14	m1	m1	NOUN
ejpam-3126	449	15	)	)	PUNCT
ejpam-3126	449	16	2n1	2n1	NUM
ejpam-3126	449	17	=	=	SYM
ejpam-3126	449	18	{	{	PUNCT
ejpam-3126	449	19	0	0	NUM
ejpam-3126	449	20	}	}	PUNCT
ejpam-3126	449	21	.	.	PUNCT
ejpam-3126	450	1	therefore	therefore	ADV
ejpam-3126	450	2	(	(	PUNCT
ejpam-3126	450	3	ψ1)3×(ψ2)3(n1×m2	ψ1)3×(ψ2)3(n1×m2	NOUN
ejpam-3126	450	4	)	)	PUNCT
ejpam-3126	450	5	=	=	SYM
ejpam-3126	450	6	(	(	PUNCT
ejpam-3126	450	7	ψ1)3(n1	ψ1)3(n1	PROPN
ejpam-3126	450	8	)	)	PUNCT
ejpam-3126	450	9	×	×	NOUN
ejpam-3126	450	10	(	(	PUNCT
ejpam-3126	450	11	ψ2)3(m2	ψ2)3(m2	NOUN
ejpam-3126	450	12	)	)	PUNCT
ejpam-3126	450	13	=	=	PRON
ejpam-3126	450	14	{	{	PUNCT
ejpam-3126	450	15	0	0	NUM
ejpam-3126	450	16	}	}	PUNCT
ejpam-3126	450	17	×m2	×m2	NOUN
ejpam-3126	450	18	.	.	PUNCT
ejpam-3126	451	1	now	now	ADV
ejpam-3126	451	2	,	,	PUNCT
ejpam-3126	451	3	by	by	ADP
ejpam-3126	451	4	lemma	lemma	PROPN
ejpam-3126	451	5	1	1	NUM
ejpam-3126	451	6	,	,	PUNCT
ejpam-3126	451	7	n1	n1	NOUN
ejpam-3126	451	8	×m2	×m2	NOUN
ejpam-3126	451	9	is	be	AUX
ejpam-3126	451	10	a	a	DET
ejpam-3126	451	11	(	(	PUNCT
ejpam-3126	451	12	ψ1)3	ψ1)3	NUM
ejpam-3126	451	13	×	×	NOUN
ejpam-3126	451	14	(	(	PUNCT
ejpam-3126	451	15	ψ2)3	ψ2)3	NOUN
ejpam-3126	451	16	-	-	ADJ
ejpam-3126	451	17	2absorbing	2absorbing	ADJ
ejpam-3126	451	18	semi	semi	ADJ
ejpam-3126	451	19	-	-	ADJ
ejpam-3126	451	20	primary	primary	ADJ
ejpam-3126	451	21	submodule	submodule	NOUN
ejpam-3126	451	22	of	of	ADP
ejpam-3126	451	23	m1	m1	PROPN
ejpam-3126	451	24	×m2	×m2	PROPN
ejpam-3126	451	25	.	.	PUNCT
ejpam-3126	452	1	corollary	corollary	ADJ
ejpam-3126	452	2	9	9	NUM
ejpam-3126	452	3	.	.	PUNCT
ejpam-3126	453	1	let	let	VERB
ejpam-3126	453	2	(	(	PUNCT
ejpam-3126	453	3	ψi)n	ψi)n	NOUN
ejpam-3126	453	4	:	:	PUNCT
ejpam-3126	453	5	s(mi	s(mi	PROPN
ejpam-3126	453	6	)	)	PUNCT
ejpam-3126	453	7	→	→	SYM
ejpam-3126	453	8	s(mi	s(mi	NOUN
ejpam-3126	453	9	)	)	PUNCT
ejpam-3126	453	10	∪	∪	ADP
ejpam-3126	453	11	{	{	PUNCT
ejpam-3126	453	12	∅	∅	NOUN
ejpam-3126	453	13	}	}	PUNCT
ejpam-3126	453	14	be	be	AUX
ejpam-3126	453	15	a	a	DET
ejpam-3126	453	16	function	function	NOUN
ejpam-3126	453	17	with	with	ADP
ejpam-3126	453	18	φn	φn	NOUN
ejpam-3126	453	19	=	=	PUNCT
ejpam-3126	453	20	(	(	PUNCT
ejpam-3126	453	21	ψ1)n	ψ1)n	PROPN
ejpam-3126	453	22	×	×	PROPN
ejpam-3126	453	23	(	(	PUNCT
ejpam-3126	453	24	ψ2)n	ψ2)n	NOUN
ejpam-3126	453	25	.	.	PUNCT
ejpam-3126	454	1	if	if	SCONJ
ejpam-3126	454	2	n2	n2	PROPN
ejpam-3126	454	3	is	be	AUX
ejpam-3126	454	4	a	a	DET
ejpam-3126	454	5	(	(	PUNCT
ejpam-3126	454	6	ψ2)0	ψ2)0	NUM
ejpam-3126	454	7	-	-	PUNCT
ejpam-3126	454	8	2	2	NUM
ejpam-3126	454	9	-	-	PUNCT
ejpam-3126	454	10	absorbing	absorbing	ADJ
ejpam-3126	454	11	semi	semi	ADJ
ejpam-3126	454	12	-	-	ADJ
ejpam-3126	454	13	primary	primary	ADJ
ejpam-3126	454	14	submodule	submodule	NOUN
ejpam-3126	454	15	of	of	ADP
ejpam-3126	454	16	m2	m2	PROPN
ejpam-3126	454	17	such	such	ADJ
ejpam-3126	454	18	that	that	SCONJ
ejpam-3126	454	19	(	(	PUNCT
ejpam-3126	454	20	ψ1)3(m1	ψ1)3(m1	PROPN
ejpam-3126	454	21	)	)	PUNCT
ejpam-3126	454	22	=	=	SYM
ejpam-3126	454	23	m1	m1	NOUN
ejpam-3126	454	24	,	,	PUNCT
ejpam-3126	454	25	then	then	ADV
ejpam-3126	454	26	m1	m1	PROPN
ejpam-3126	454	27	×n2	×n2	PROPN
ejpam-3126	454	28	is	be	AUX
ejpam-3126	454	29	a	a	DET
ejpam-3126	454	30	φ3	φ3	NOUN
ejpam-3126	454	31	-	-	PUNCT
ejpam-3126	454	32	2	2	NUM
ejpam-3126	454	33	-	-	PUNCT
ejpam-3126	454	34	absorbing	absorbing	ADJ
ejpam-3126	454	35	semi	semi	ADJ
ejpam-3126	454	36	-	-	ADJ
ejpam-3126	454	37	primary	primary	ADJ
ejpam-3126	454	38	submodule	submodule	NOUN
ejpam-3126	454	39	of	of	ADP
ejpam-3126	454	40	m1	m1	PROPN
ejpam-3126	454	41	×m2	×m2	PROPN
ejpam-3126	454	42	.	.	PUNCT
ejpam-3126	455	1	proof	proof	NOUN
ejpam-3126	455	2	.	.	PUNCT
ejpam-3126	456	1	similar	similar	ADJ
ejpam-3126	456	2	to	to	ADP
ejpam-3126	456	3	the	the	DET
ejpam-3126	456	4	proof	proof	NOUN
ejpam-3126	456	5	of	of	ADP
ejpam-3126	456	6	theorem	theorem	ADJ
ejpam-3126	456	7	20	20	NUM
ejpam-3126	456	8	.	.	PUNCT
ejpam-3126	456	9	theorem	theorem	NOUN
ejpam-3126	456	10	21	21	NUM
ejpam-3126	456	11	.	.	PUNCT
ejpam-3126	457	1	let	let	VERB
ejpam-3126	457	2	(	(	PUNCT
ejpam-3126	457	3	ψi)n	ψi)n	PROPN
ejpam-3126	457	4	:	:	PUNCT
ejpam-3126	457	5	s(mi)→	s(mi)→	PROPN
ejpam-3126	457	6	s(mi)∪{∅	s(mi)∪{∅	PROPN
ejpam-3126	457	7	}	}	PUNCT
ejpam-3126	457	8	be	be	AUX
ejpam-3126	457	9	a	a	DET
ejpam-3126	457	10	function	function	NOUN
ejpam-3126	457	11	with	with	ADP
ejpam-3126	457	12	φn	φn	NOUN
ejpam-3126	457	13	=	=	PUNCT
ejpam-3126	457	14	(	(	PUNCT
ejpam-3126	457	15	ψ1)n×.	ψ1)n×.	X
ejpam-3126	457	16	.	.	PUNCT
ejpam-3126	458	1	.×(ψk)n	.×(ψk)n	PROPN
ejpam-3126	458	2	.	.	PUNCT
ejpam-3126	459	1	if	if	SCONJ
ejpam-3126	459	2	nj	nj	PROPN
ejpam-3126	459	3	is	be	AUX
ejpam-3126	459	4	a	a	DET
ejpam-3126	459	5	(	(	PUNCT
ejpam-3126	459	6	ψj)0	ψj)0	NOUN
ejpam-3126	459	7	-	-	PUNCT
ejpam-3126	459	8	2	2	NUM
ejpam-3126	459	9	-	-	PUNCT
ejpam-3126	459	10	absorbing	absorbing	ADJ
ejpam-3126	459	11	semi	semi	ADJ
ejpam-3126	459	12	-	-	ADJ
ejpam-3126	459	13	primary	primary	ADJ
ejpam-3126	459	14	submodule	submodule	NOUN
ejpam-3126	459	15	of	of	ADP
ejpam-3126	459	16	mj	mj	PROPN
ejpam-3126	460	1	such	such	ADJ
ejpam-3126	460	2	that	that	PRON
ejpam-3126	460	3	(	(	PUNCT
ejpam-3126	460	4	ψj)3(mj	ψj)3(mj	NOUN
ejpam-3126	460	5	)	)	PUNCT
ejpam-3126	460	6	=	=	SYM
ejpam-3126	460	7	mj	mj	PROPN
ejpam-3126	460	8	,	,	PUNCT
ejpam-3126	460	9	then	then	ADV
ejpam-3126	460	10	m1	m1	PROPN
ejpam-3126	460	11	×m2	×m2	PROPN
ejpam-3126	460	12	×	×	VERB
ejpam-3126	460	13	.	.	PUNCT
ejpam-3126	460	14	.	.	PUNCT
ejpam-3126	460	15	.	.	PUNCT
ejpam-3126	461	1	×mi−1	×mi−1	PROPN
ejpam-3126	461	2	×	×	PROPN
ejpam-3126	461	3	ni	ni	PROPN
ejpam-3126	461	4	×mi+1	×mi+1	PROPN
ejpam-3126	461	5	×	×	NOUN
ejpam-3126	461	6	.	.	PUNCT
ejpam-3126	461	7	.	.	PUNCT
ejpam-3126	461	8	.	.	PUNCT
ejpam-3126	462	1	×mk	×mk	NOUN
ejpam-3126	462	2	is	be	AUX
ejpam-3126	462	3	a	a	DET
ejpam-3126	462	4	φ3	φ3	NOUN
ejpam-3126	462	5	-	-	PUNCT
ejpam-3126	462	6	2	2	NUM
ejpam-3126	462	7	-	-	PUNCT
ejpam-3126	462	8	absorbing	absorbing	ADJ
ejpam-3126	462	9	semi	semi	ADJ
ejpam-3126	462	10	-	-	ADJ
ejpam-3126	462	11	primary	primary	ADJ
ejpam-3126	462	12	submodule	submodule	NOUN
ejpam-3126	462	13	of	of	ADP
ejpam-3126	462	14	m1	m1	PROPN
ejpam-3126	462	15	×	×	PROPN
ejpam-3126	462	16	.	.	PUNCT
ejpam-3126	462	17	.	.	PUNCT
ejpam-3126	463	1	.×mk	.×mk	NOUN
ejpam-3126	463	2	.	.	PUNCT
ejpam-3126	464	1	proof	proof	NOUN
ejpam-3126	464	2	.	.	PUNCT
ejpam-3126	465	1	similar	similar	ADJ
ejpam-3126	465	2	to	to	ADP
ejpam-3126	465	3	the	the	DET
ejpam-3126	465	4	proof	proof	NOUN
ejpam-3126	465	5	of	of	ADP
ejpam-3126	465	6	theorem	theorem	ADJ
ejpam-3126	465	7	20	20	NUM
ejpam-3126	465	8	and	and	CCONJ
ejpam-3126	465	9	corollary	corollary	ADJ
ejpam-3126	465	10	9	9	NUM
ejpam-3126	465	11	.	.	PUNCT
ejpam-3126	465	12	theorem	theorem	NOUN
ejpam-3126	465	13	22	22	NUM
ejpam-3126	465	14	.	.	PUNCT
ejpam-3126	466	1	let	let	VERB
ejpam-3126	466	2	ψi	ψi	NOUN
ejpam-3126	466	3	:	:	PUNCT
ejpam-3126	466	4	s(mi	s(mi	VERB
ejpam-3126	466	5	)	)	PUNCT
ejpam-3126	466	6	→	→	SYM
ejpam-3126	466	7	s(mi	s(mi	NOUN
ejpam-3126	466	8	)	)	PUNCT
ejpam-3126	466	9	∪	∪	ADP
ejpam-3126	466	10	{	{	PUNCT
ejpam-3126	466	11	∅	∅	NOUN
ejpam-3126	466	12	}	}	PUNCT
ejpam-3126	466	13	be	be	AUX
ejpam-3126	466	14	a	a	DET
ejpam-3126	466	15	function	function	NOUN
ejpam-3126	466	16	with	with	ADP
ejpam-3126	466	17	φ	φ	PROPN
ejpam-3126	466	18	=	=	SYM
ejpam-3126	466	19	ψ1	ψ1	ADJ
ejpam-3126	466	20	×	×	NOUN
ejpam-3126	466	21	ψ2	ψ2	NOUN
ejpam-3126	466	22	.	.	PUNCT
ejpam-3126	467	1	then	then	ADV
ejpam-3126	467	2	the	the	DET
ejpam-3126	467	3	following	follow	VERB
ejpam-3126	467	4	conditions	condition	NOUN
ejpam-3126	467	5	are	be	AUX
ejpam-3126	467	6	equivalent	equivalent	ADJ
ejpam-3126	467	7	:	:	PUNCT
ejpam-3126	467	8	pai	pai	PROPN
ejpam-3126	467	9	.	.	PROPN
ejpam-3126	467	10	yiarayong	yiarayong	PROPN
ejpam-3126	467	11	,	,	PUNCT
ejpam-3126	467	12	m.	m.	NOUN
ejpam-3126	467	13	siripitukdet	siripitukdet	NOUN
ejpam-3126	467	14	/	/	SYM
ejpam-3126	467	15	eur	eur	PROPN
ejpam-3126	467	16	.	.	PUNCT
ejpam-3126	468	1	j.	j.	PROPN
ejpam-3126	468	2	pure	pure	PROPN
ejpam-3126	468	3	appl	appl	PROPN
ejpam-3126	468	4	.	.	PROPN
ejpam-3126	468	5	math	math	PROPN
ejpam-3126	468	6	,	,	PUNCT
ejpam-3126	468	7	11	11	NUM
ejpam-3126	468	8	(	(	PUNCT
ejpam-3126	468	9	1	1	NUM
ejpam-3126	468	10	)	)	PUNCT
ejpam-3126	468	11	(	(	PUNCT
ejpam-3126	468	12	2018	2018	NUM
ejpam-3126	468	13	)	)	PUNCT
ejpam-3126	468	14	,	,	PUNCT
ejpam-3126	468	15	35	35	NUM
ejpam-3126	468	16	-	-	SYM
ejpam-3126	468	17	50	50	NUM
ejpam-3126	468	18	48	48	NUM
ejpam-3126	468	19	(	(	PUNCT
ejpam-3126	468	20	i	i	NOUN
ejpam-3126	468	21	)	)	PUNCT
ejpam-3126	468	22	n1	n1	PROPN
ejpam-3126	468	23	is	be	AUX
ejpam-3126	468	24	a	a	DET
ejpam-3126	468	25	2	2	NUM
ejpam-3126	468	26	-	-	PUNCT
ejpam-3126	468	27	absorbing	absorbing	ADJ
ejpam-3126	468	28	semi	semi	ADJ
ejpam-3126	468	29	-	-	ADJ
ejpam-3126	468	30	primary	primary	ADJ
ejpam-3126	468	31	submodule	submodule	NOUN
ejpam-3126	468	32	of	of	ADP
ejpam-3126	468	33	m1	m1	PROPN
ejpam-3126	468	34	.	.	PUNCT
ejpam-3126	469	1	(	(	PUNCT
ejpam-3126	469	2	ii	ii	NOUN
ejpam-3126	469	3	)	)	PUNCT
ejpam-3126	469	4	n1	n1	NOUN
ejpam-3126	469	5	×m2	×m2	NOUN
ejpam-3126	469	6	is	be	AUX
ejpam-3126	469	7	a	a	DET
ejpam-3126	469	8	2	2	NUM
ejpam-3126	469	9	-	-	PUNCT
ejpam-3126	469	10	absorbing	absorbing	ADJ
ejpam-3126	469	11	semi	semi	ADJ
ejpam-3126	469	12	-	-	ADJ
ejpam-3126	469	13	primary	primary	ADJ
ejpam-3126	469	14	submodule	submodule	NOUN
ejpam-3126	469	15	of	of	ADP
ejpam-3126	469	16	m1	m1	PROPN
ejpam-3126	469	17	×m2	×m2	PROPN
ejpam-3126	469	18	.	.	PUNCT
ejpam-3126	470	1	(	(	PUNCT
ejpam-3126	470	2	iii	iii	X
ejpam-3126	470	3	)	)	PUNCT
ejpam-3126	470	4	n1	n1	NOUN
ejpam-3126	470	5	×m2	×m2	NOUN
ejpam-3126	470	6	is	be	AUX
ejpam-3126	470	7	a	a	DET
ejpam-3126	470	8	φ-2	φ-2	NOUN
ejpam-3126	470	9	-	-	PUNCT
ejpam-3126	470	10	absorbing	absorbing	ADJ
ejpam-3126	470	11	semi	semi	ADJ
ejpam-3126	470	12	-	-	ADJ
ejpam-3126	470	13	primary	primary	ADJ
ejpam-3126	470	14	submodule	submodule	NOUN
ejpam-3126	470	15	of	of	ADP
ejpam-3126	470	16	m1	m1	PROPN
ejpam-3126	470	17	×m2	×m2	PROPN
ejpam-3126	470	18	,	,	PUNCT
ejpam-3126	470	19	where	where	SCONJ
ejpam-3126	470	20	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	470	21	)	)	PUNCT
ejpam-3126	470	22	6=	6=	NUM
ejpam-3126	470	23	m2	m2	PROPN
ejpam-3126	470	24	.	.	PUNCT
ejpam-3126	470	25	proof	proof	NOUN
ejpam-3126	470	26	.	.	PUNCT
ejpam-3126	471	1	(	(	PUNCT
ejpam-3126	471	2	1	1	NUM
ejpam-3126	471	3	⇒	⇒	NOUN
ejpam-3126	471	4	2	2	NUM
ejpam-3126	471	5	)	)	PUNCT
ejpam-3126	471	6	.	.	PUNCT
ejpam-3126	472	1	let	let	VERB
ejpam-3126	472	2	(	(	PUNCT
ejpam-3126	472	3	a1	a1	NOUN
ejpam-3126	472	4	,	,	PUNCT
ejpam-3126	472	5	b1	b1	NOUN
ejpam-3126	472	6	)	)	PUNCT
ejpam-3126	472	7	,	,	PUNCT
ejpam-3126	472	8	(	(	PUNCT
ejpam-3126	472	9	a2	a2	PROPN
ejpam-3126	472	10	,	,	PUNCT
ejpam-3126	472	11	b2	b2	NOUN
ejpam-3126	472	12	)	)	PUNCT
ejpam-3126	472	13	∈	∈	PROPN
ejpam-3126	472	14	r1	r1	NOUN
ejpam-3126	472	15	×	×	NOUN
ejpam-3126	472	16	r2	r2	PROPN
ejpam-3126	472	17	and	and	CCONJ
ejpam-3126	472	18	(	(	PUNCT
ejpam-3126	472	19	m1,m2	m1,m2	PROPN
ejpam-3126	472	20	)	)	PUNCT
ejpam-3126	472	21	∈	∈	PROPN
ejpam-3126	472	22	m1	m1	NOUN
ejpam-3126	472	23	×m2	×m2	NOUN
ejpam-3126	472	24	such	such	ADJ
ejpam-3126	472	25	that	that	SCONJ
ejpam-3126	472	26	(	(	PUNCT
ejpam-3126	472	27	a1	a1	PROPN
ejpam-3126	472	28	,	,	PUNCT
ejpam-3126	472	29	b1)(a2	b1)(a2	PROPN
ejpam-3126	472	30	,	,	PUNCT
ejpam-3126	472	31	b2)(m1,m2	b2)(m1,m2	PROPN
ejpam-3126	472	32	)	)	PUNCT
ejpam-3126	472	33	∈	∈	PROPN
ejpam-3126	472	34	n1	n1	PROPN
ejpam-3126	472	35	×	×	PROPN
ejpam-3126	472	36	m2	m2	PROPN
ejpam-3126	472	37	.	.	PUNCT
ejpam-3126	473	1	clearly	clearly	ADV
ejpam-3126	473	2	,	,	PUNCT
ejpam-3126	473	3	a1a2m1	a1a2m1	PROPN
ejpam-3126	473	4	∈	∈	PROPN
ejpam-3126	473	5	n1	n1	PROPN
ejpam-3126	473	6	.	.	PUNCT
ejpam-3126	474	1	by	by	ADP
ejpam-3126	474	2	definition	definition	NOUN
ejpam-3126	474	3	2	2	NUM
ejpam-3126	474	4	,	,	PUNCT
ejpam-3126	474	5	a1a2	a1a2	ADP
ejpam-3126	474	6	∈√	∈√	PROPN
ejpam-3126	474	7	(	(	PUNCT
ejpam-3126	474	8	n1	n1	PROPN
ejpam-3126	474	9	:	:	PUNCT
ejpam-3126	474	10	m1	m1	NOUN
ejpam-3126	474	11	)	)	PUNCT
ejpam-3126	474	12	or	or	CCONJ
ejpam-3126	474	13	a1m1	a1m1	VERB
ejpam-3126	474	14	∈	∈	NOUN
ejpam-3126	474	15	n1	n1	NOUN
ejpam-3126	474	16	or	or	CCONJ
ejpam-3126	474	17	an2m1	an2m1	PROPN
ejpam-3126	474	18	∈	∈	PROPN
ejpam-3126	474	19	n1	n1	NOUN
ejpam-3126	474	20	for	for	ADP
ejpam-3126	474	21	some	some	DET
ejpam-3126	474	22	positive	positive	ADJ
ejpam-3126	474	23	integer	integer	NOUN
ejpam-3126	474	24	n.	n.	NOUN
ejpam-3126	474	25	this	this	PRON
ejpam-3126	474	26	completes	complete	VERB
ejpam-3126	474	27	the	the	DET
ejpam-3126	474	28	proof	proof	NOUN
ejpam-3126	474	29	.	.	PUNCT
ejpam-3126	475	1	(	(	PUNCT
ejpam-3126	475	2	2⇒	2⇒	NOUN
ejpam-3126	475	3	3	3	NUM
ejpam-3126	475	4	)	)	PUNCT
ejpam-3126	475	5	.	.	PUNCT
ejpam-3126	476	1	it	it	PRON
ejpam-3126	476	2	is	be	AUX
ejpam-3126	476	3	easy	easy	ADJ
ejpam-3126	476	4	to	to	PART
ejpam-3126	476	5	see	see	VERB
ejpam-3126	476	6	that	that	SCONJ
ejpam-3126	476	7	every	every	DET
ejpam-3126	476	8	2	2	NUM
ejpam-3126	476	9	-	-	PUNCT
ejpam-3126	476	10	absorbing	absorbing	ADJ
ejpam-3126	476	11	primary	primary	ADJ
ejpam-3126	476	12	submodule	submodule	NOUN
ejpam-3126	476	13	is	be	AUX
ejpam-3126	476	14	φ-2	φ-2	NOUN
ejpam-3126	476	15	-	-	PUNCT
ejpam-3126	476	16	absorbing	absorbing	ADJ
ejpam-3126	476	17	semi	semi	ADJ
ejpam-3126	476	18	-	-	ADJ
ejpam-3126	476	19	primary	primary	ADJ
ejpam-3126	476	20	.	.	PUNCT
ejpam-3126	477	1	(	(	PUNCT
ejpam-3126	477	2	3	3	NUM
ejpam-3126	477	3	⇒	⇒	NOUN
ejpam-3126	477	4	1	1	NUM
ejpam-3126	477	5	)	)	PUNCT
ejpam-3126	477	6	.	.	PUNCT
ejpam-3126	478	1	let	let	VERB
ejpam-3126	478	2	a1	a1	NOUN
ejpam-3126	478	3	,	,	PUNCT
ejpam-3126	478	4	a2	a2	PROPN
ejpam-3126	478	5	∈	∈	PROPN
ejpam-3126	478	6	r1	r1	PROPN
ejpam-3126	478	7	and	and	CCONJ
ejpam-3126	478	8	m	m	PROPN
ejpam-3126	478	9	∈	∈	PROPN
ejpam-3126	478	10	m1	m1	NOUN
ejpam-3126	478	11	such	such	ADJ
ejpam-3126	478	12	that	that	SCONJ
ejpam-3126	478	13	a1a2	a1a2	PROPN
ejpam-3126	478	14	m	m	NOUN
ejpam-3126	478	15	∈	∈	NOUN
ejpam-3126	478	16	n1	n1	NOUN
ejpam-3126	478	17	.	.	PUNCT
ejpam-3126	479	1	by	by	ADP
ejpam-3126	479	2	assumption	assumption	NOUN
ejpam-3126	479	3	,	,	PUNCT
ejpam-3126	479	4	there	there	PRON
ejpam-3126	479	5	exists	exist	VERB
ejpam-3126	479	6	m2	m2	PROPN
ejpam-3126	479	7	∈	∈	PROPN
ejpam-3126	479	8	m2	m2	PROPN
ejpam-3126	479	9	such	such	ADJ
ejpam-3126	479	10	that	that	SCONJ
ejpam-3126	479	11	m2	m2	PROPN
ejpam-3126	479	12	6∈	6∈	PROPN
ejpam-3126	479	13	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	479	14	)	)	PUNCT
ejpam-3126	479	15	.	.	PUNCT
ejpam-3126	480	1	since	since	SCONJ
ejpam-3126	480	2	φ(n1	φ(n1	NOUN
ejpam-3126	480	3	×m2	×m2	PROPN
ejpam-3126	480	4	)	)	PUNCT
ejpam-3126	480	5	⊆	⊆	NUM
ejpam-3126	480	6	m1	m1	PROPN
ejpam-3126	480	7	×	×	PROPN
ejpam-3126	480	8	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	480	9	)	)	PUNCT
ejpam-3126	480	10	,	,	PUNCT
ejpam-3126	480	11	we	we	PRON
ejpam-3126	480	12	have	have	VERB
ejpam-3126	480	13	(	(	PUNCT
ejpam-3126	480	14	a1	a1	NOUN
ejpam-3126	480	15	,	,	PUNCT
ejpam-3126	480	16	1)(a2	1)(a2	NUM
ejpam-3126	480	17	,	,	PUNCT
ejpam-3126	480	18	1)(m	1)(m	NUM
ejpam-3126	480	19	,	,	PUNCT
ejpam-3126	480	20	m2	m2	PROPN
ejpam-3126	480	21	)	)	PUNCT
ejpam-3126	480	22	∈	∈	PROPN
ejpam-3126	480	23	n1×m2−ψ1×ψ2(n1×m2	n1×m2−ψ1×ψ2(n1×m2	PROPN
ejpam-3126	480	24	)	)	PUNCT
ejpam-3126	480	25	.	.	PUNCT
ejpam-3126	481	1	by	by	ADP
ejpam-3126	481	2	definition	definition	NOUN
ejpam-3126	481	3	1	1	NUM
ejpam-3126	481	4	,	,	PUNCT
ejpam-3126	481	5	a1a2	a1a2	PROPN
ejpam-3126	481	6	∈	∈	NOUN
ejpam-3126	481	7	√	√	NUM
ejpam-3126	481	8	(	(	PUNCT
ejpam-3126	481	9	n1	n1	PROPN
ejpam-3126	481	10	:	:	PUNCT
ejpam-3126	481	11	m1	m1	NOUN
ejpam-3126	481	12	)	)	PUNCT
ejpam-3126	481	13	or	or	CCONJ
ejpam-3126	481	14	a1	a1	NOUN
ejpam-3126	481	15	m	m	PROPN
ejpam-3126	481	16	∈	∈	NOUN
ejpam-3126	481	17	n1	n1	NOUN
ejpam-3126	481	18	or	or	CCONJ
ejpam-3126	481	19	an2	an2	PROPN
ejpam-3126	481	20	m	m	NOUN
ejpam-3126	481	21	∈	∈	PROPN
ejpam-3126	481	22	n1	n1	NOUN
ejpam-3126	481	23	.	.	PUNCT
ejpam-3126	482	1	corollary	corollary	ADJ
ejpam-3126	482	2	10	10	NUM
ejpam-3126	482	3	.	.	PUNCT
ejpam-3126	483	1	let	let	VERB
ejpam-3126	483	2	ψi	ψi	NOUN
ejpam-3126	483	3	:	:	PUNCT
ejpam-3126	483	4	s(mi	s(mi	VERB
ejpam-3126	483	5	)	)	PUNCT
ejpam-3126	483	6	→	→	SYM
ejpam-3126	483	7	s(mi	s(mi	NOUN
ejpam-3126	483	8	)	)	PUNCT
ejpam-3126	483	9	∪	∪	ADP
ejpam-3126	483	10	{	{	PUNCT
ejpam-3126	483	11	∅	∅	NOUN
ejpam-3126	483	12	}	}	PUNCT
ejpam-3126	483	13	be	be	AUX
ejpam-3126	483	14	a	a	DET
ejpam-3126	483	15	function	function	NOUN
ejpam-3126	483	16	φ	φ	NOUN
ejpam-3126	483	17	=	=	SYM
ejpam-3126	483	18	ψ1	ψ1	ADJ
ejpam-3126	483	19	×	×	NOUN
ejpam-3126	483	20	ψ2	ψ2	NOUN
ejpam-3126	483	21	.	.	PUNCT
ejpam-3126	484	1	then	then	ADV
ejpam-3126	484	2	the	the	DET
ejpam-3126	484	3	following	follow	VERB
ejpam-3126	484	4	conditions	condition	NOUN
ejpam-3126	484	5	are	be	AUX
ejpam-3126	484	6	equivalent	equivalent	ADJ
ejpam-3126	484	7	:	:	PUNCT
ejpam-3126	484	8	(	(	PUNCT
ejpam-3126	484	9	i	i	NOUN
ejpam-3126	484	10	)	)	PUNCT
ejpam-3126	484	11	n2	n2	PROPN
ejpam-3126	484	12	is	be	AUX
ejpam-3126	484	13	a	a	DET
ejpam-3126	484	14	2	2	NUM
ejpam-3126	484	15	-	-	PUNCT
ejpam-3126	484	16	absorbing	absorbing	ADJ
ejpam-3126	484	17	semi	semi	ADJ
ejpam-3126	484	18	-	-	ADJ
ejpam-3126	484	19	primary	primary	ADJ
ejpam-3126	484	20	submodule	submodule	NOUN
ejpam-3126	484	21	of	of	ADP
ejpam-3126	484	22	m2	m2	PROPN
ejpam-3126	484	23	.	.	PUNCT
ejpam-3126	485	1	(	(	PUNCT
ejpam-3126	485	2	ii	ii	NOUN
ejpam-3126	485	3	)	)	PUNCT
ejpam-3126	485	4	m1	m1	PROPN
ejpam-3126	485	5	×n2	×n2	PROPN
ejpam-3126	485	6	is	be	AUX
ejpam-3126	485	7	a	a	DET
ejpam-3126	485	8	2	2	NUM
ejpam-3126	485	9	-	-	PUNCT
ejpam-3126	485	10	absorbing	absorbing	ADJ
ejpam-3126	485	11	semi	semi	ADJ
ejpam-3126	485	12	-	-	ADJ
ejpam-3126	485	13	primary	primary	ADJ
ejpam-3126	485	14	submodule	submodule	NOUN
ejpam-3126	485	15	of	of	ADP
ejpam-3126	485	16	m1	m1	PROPN
ejpam-3126	485	17	×m2	×m2	PROPN
ejpam-3126	485	18	.	.	PUNCT
ejpam-3126	486	1	(	(	PUNCT
ejpam-3126	486	2	iii	iii	X
ejpam-3126	486	3	)	)	PUNCT
ejpam-3126	486	4	m1	m1	PROPN
ejpam-3126	486	5	×n2	×n2	PROPN
ejpam-3126	486	6	is	be	AUX
ejpam-3126	486	7	a	a	DET
ejpam-3126	486	8	φ-2	φ-2	NOUN
ejpam-3126	486	9	-	-	PUNCT
ejpam-3126	486	10	absorbing	absorbing	ADJ
ejpam-3126	486	11	semi	semi	ADJ
ejpam-3126	486	12	-	-	ADJ
ejpam-3126	486	13	primary	primary	ADJ
ejpam-3126	486	14	submodule	submodule	NOUN
ejpam-3126	486	15	of	of	ADP
ejpam-3126	486	16	m1	m1	PROPN
ejpam-3126	486	17	×m2	×m2	PROPN
ejpam-3126	486	18	,	,	PUNCT
ejpam-3126	486	19	where	where	SCONJ
ejpam-3126	486	20	ψ1(m1	ψ1(m1	NOUN
ejpam-3126	486	21	)	)	PUNCT
ejpam-3126	486	22	6=	6=	NOUN
ejpam-3126	486	23	m1	m1	NOUN
ejpam-3126	486	24	.	.	PUNCT
ejpam-3126	487	1	proof	proof	NOUN
ejpam-3126	487	2	.	.	PUNCT
ejpam-3126	488	1	similar	similar	ADJ
ejpam-3126	488	2	to	to	ADP
ejpam-3126	488	3	the	the	DET
ejpam-3126	488	4	proof	proof	NOUN
ejpam-3126	488	5	of	of	ADP
ejpam-3126	488	6	theorem	theorem	ADJ
ejpam-3126	488	7	22	22	NUM
ejpam-3126	488	8	.	.	PUNCT
ejpam-3126	488	9	theorem	theorem	NOUN
ejpam-3126	488	10	23	23	NUM
ejpam-3126	488	11	.	.	PUNCT
ejpam-3126	489	1	let	let	VERB
ejpam-3126	489	2	ψi	ψi	VERB
ejpam-3126	489	3	:	:	PUNCT
ejpam-3126	489	4	s(mi)→	s(mi)→	PROPN
ejpam-3126	489	5	s(mi)∪	s(mi)∪	PROPN
ejpam-3126	489	6	{	{	PUNCT
ejpam-3126	489	7	∅	∅	NOUN
ejpam-3126	489	8	}	}	PUNCT
ejpam-3126	489	9	be	be	AUX
ejpam-3126	489	10	a	a	DET
ejpam-3126	489	11	function	function	NOUN
ejpam-3126	489	12	with	with	ADP
ejpam-3126	489	13	φ	φ	PROPN
ejpam-3126	489	14	=	=	SYM
ejpam-3126	489	15	ψ1	ψ1	ADJ
ejpam-3126	489	16	×	×	NOUN
ejpam-3126	489	17	.	.	PUNCT
ejpam-3126	489	18	.	.	PUNCT
ejpam-3126	490	1	.×ψk	.×ψk	PROPN
ejpam-3126	490	2	.	.	PUNCT
ejpam-3126	491	1	then	then	ADV
ejpam-3126	491	2	the	the	DET
ejpam-3126	491	3	following	follow	VERB
ejpam-3126	491	4	conditions	condition	NOUN
ejpam-3126	491	5	are	be	AUX
ejpam-3126	491	6	equivalent	equivalent	ADJ
ejpam-3126	491	7	:	:	PUNCT
ejpam-3126	491	8	(	(	PUNCT
ejpam-3126	491	9	i	i	NOUN
ejpam-3126	491	10	)	)	PUNCT
ejpam-3126	491	11	ni	ni	PROPN
ejpam-3126	491	12	is	be	AUX
ejpam-3126	491	13	a	a	DET
ejpam-3126	491	14	2	2	NUM
ejpam-3126	491	15	-	-	PUNCT
ejpam-3126	491	16	absorbing	absorbing	ADJ
ejpam-3126	491	17	semi	semi	ADJ
ejpam-3126	491	18	-	-	ADJ
ejpam-3126	491	19	primary	primary	ADJ
ejpam-3126	491	20	submodule	submodule	NOUN
ejpam-3126	491	21	of	of	ADP
ejpam-3126	491	22	mi	mi	PROPN
ejpam-3126	491	23	.	.	PROPN
ejpam-3126	491	24	(	(	PUNCT
ejpam-3126	491	25	ii	ii	NOUN
ejpam-3126	491	26	)	)	PUNCT
ejpam-3126	491	27	m1×m2×.	m1×m2×.	PROPN
ejpam-3126	491	28	.	.	PUNCT
ejpam-3126	492	1	.×mi−1×ni×mi+1×.	.×mi−1×ni×mi+1×.	PROPN
ejpam-3126	492	2	.	.	PUNCT
ejpam-3126	493	1	.×mk	.×mk	NOUN
ejpam-3126	493	2	is	be	AUX
ejpam-3126	493	3	a	a	DET
ejpam-3126	493	4	2	2	NUM
ejpam-3126	493	5	-	-	PUNCT
ejpam-3126	493	6	absorbing	absorbing	ADJ
ejpam-3126	493	7	semi	semi	ADJ
ejpam-3126	493	8	-	-	ADJ
ejpam-3126	493	9	primary	primary	ADJ
ejpam-3126	493	10	submodule	submodule	NOUN
ejpam-3126	493	11	of	of	ADP
ejpam-3126	493	12	m1	m1	PROPN
ejpam-3126	493	13	×	×	PROPN
ejpam-3126	493	14	.	.	PUNCT
ejpam-3126	493	15	.	.	PUNCT
ejpam-3126	494	1	.×m2	.×m2	PROPN
ejpam-3126	494	2	.	.	PUNCT
ejpam-3126	495	1	(	(	PUNCT
ejpam-3126	495	2	iii	iii	X
ejpam-3126	495	3	)	)	PUNCT
ejpam-3126	495	4	m1	m1	NOUN
ejpam-3126	495	5	×m2	×m2	NOUN
ejpam-3126	495	6	×	×	VERB
ejpam-3126	495	7	.	.	PUNCT
ejpam-3126	495	8	.	.	PUNCT
ejpam-3126	495	9	.	.	PUNCT
ejpam-3126	496	1	×mi−1	×mi−1	PROPN
ejpam-3126	496	2	×	×	PROPN
ejpam-3126	496	3	ni	ni	PROPN
ejpam-3126	496	4	×mi+1	×mi+1	PROPN
ejpam-3126	496	5	×	×	NOUN
ejpam-3126	496	6	.	.	PUNCT
ejpam-3126	496	7	.	.	PUNCT
ejpam-3126	496	8	.	.	PUNCT
ejpam-3126	497	1	×mk	×mk	NOUN
ejpam-3126	497	2	is	be	AUX
ejpam-3126	497	3	a	a	DET
ejpam-3126	497	4	φ-2	φ-2	NOUN
ejpam-3126	497	5	-	-	PUNCT
ejpam-3126	497	6	absorbing	absorbing	ADJ
ejpam-3126	497	7	semi	semi	ADJ
ejpam-3126	497	8	-	-	ADJ
ejpam-3126	497	9	primary	primary	ADJ
ejpam-3126	497	10	submodule	submodule	NOUN
ejpam-3126	497	11	of	of	ADP
ejpam-3126	497	12	m1	m1	PROPN
ejpam-3126	497	13	×	×	PROPN
ejpam-3126	497	14	.	.	PUNCT
ejpam-3126	497	15	.	.	PUNCT
ejpam-3126	498	1	.×m2	.×m2	PUNCT
ejpam-3126	499	1	with	with	ADP
ejpam-3126	499	2	ψj(mj	ψj(mj	PROPN
ejpam-3126	499	3	)	)	PUNCT
ejpam-3126	499	4	6=	6=	PROPN
ejpam-3126	499	5	mj	mj	PROPN
ejpam-3126	499	6	.	.	PUNCT
ejpam-3126	499	7	proof	proof	NOUN
ejpam-3126	499	8	.	.	PUNCT
ejpam-3126	500	1	similar	similar	ADJ
ejpam-3126	500	2	to	to	ADP
ejpam-3126	500	3	the	the	DET
ejpam-3126	500	4	proof	proof	NOUN
ejpam-3126	500	5	of	of	ADP
ejpam-3126	500	6	theorem	theorem	ADJ
ejpam-3126	500	7	22	22	NUM
ejpam-3126	500	8	and	and	CCONJ
ejpam-3126	500	9	corollary	corollary	ADJ
ejpam-3126	500	10	10	10	NUM
ejpam-3126	500	11	.	.	PUNCT
ejpam-3126	500	12	theorem	theorem	NOUN
ejpam-3126	500	13	24	24	NUM
ejpam-3126	500	14	.	.	PUNCT
ejpam-3126	501	1	let	let	VERB
ejpam-3126	501	2	ψi	ψi	VERB
ejpam-3126	501	3	:	:	PUNCT
ejpam-3126	501	4	s(mi	s(mi	VERB
ejpam-3126	501	5	)	)	PUNCT
ejpam-3126	501	6	→	→	SYM
ejpam-3126	501	7	s(mi	s(mi	NOUN
ejpam-3126	501	8	)	)	PUNCT
ejpam-3126	501	9	∪	∪	ADP
ejpam-3126	501	10	{	{	PUNCT
ejpam-3126	501	11	∅	∅	NOUN
ejpam-3126	501	12	}	}	PUNCT
ejpam-3126	501	13	be	be	AUX
ejpam-3126	501	14	a	a	DET
ejpam-3126	501	15	function	function	NOUN
ejpam-3126	501	16	with	with	ADP
ejpam-3126	501	17	ψ2(m2	ψ2(m2	NOUN
ejpam-3126	501	18	)	)	PUNCT
ejpam-3126	501	19	=	=	SYM
ejpam-3126	501	20	m2	m2	PROPN
ejpam-3126	501	21	and	and	CCONJ
ejpam-3126	501	22	φ	φ	NOUN
ejpam-3126	501	23	=	=	SYM
ejpam-3126	501	24	ψ1	ψ1	ADJ
ejpam-3126	501	25	×	×	NOUN
ejpam-3126	501	26	ψ2	ψ2	NOUN
ejpam-3126	501	27	.	.	PUNCT
ejpam-3126	502	1	then	then	ADV
ejpam-3126	502	2	n1	n1	NOUN
ejpam-3126	502	3	×m2	×m2	NOUN
ejpam-3126	502	4	is	be	AUX
ejpam-3126	502	5	a	a	DET
ejpam-3126	502	6	φ-2	φ-2	NOUN
ejpam-3126	502	7	-	-	PUNCT
ejpam-3126	502	8	absorbing	absorbing	ADJ
ejpam-3126	502	9	semi	semi	ADJ
ejpam-3126	502	10	-	-	ADJ
ejpam-3126	502	11	primary	primary	ADJ
ejpam-3126	502	12	submodule	submodule	NOUN
ejpam-3126	502	13	of	of	ADP
ejpam-3126	502	14	m1	m1	PROPN
ejpam-3126	502	15	×m2	×m2	PROPN
ejpam-3126	502	16	if	if	SCONJ
ejpam-3126	502	17	and	and	CCONJ
ejpam-3126	502	18	only	only	ADV
ejpam-3126	502	19	if	if	SCONJ
ejpam-3126	502	20	n1	n1	PROPN
ejpam-3126	502	21	is	be	AUX
ejpam-3126	502	22	a	a	DET
ejpam-3126	502	23	ψ1	ψ1	NOUN
ejpam-3126	502	24	-	-	PUNCT
ejpam-3126	502	25	2	2	NUM
ejpam-3126	502	26	-	-	PUNCT
ejpam-3126	502	27	absorbing	absorbing	ADJ
ejpam-3126	502	28	semi	semi	ADJ
ejpam-3126	502	29	-	-	ADJ
ejpam-3126	502	30	primary	primary	ADJ
ejpam-3126	502	31	submodule	submodule	NOUN
ejpam-3126	502	32	of	of	ADP
ejpam-3126	502	33	m1	m1	PROPN
ejpam-3126	502	34	.	.	PUNCT
ejpam-3126	503	1	pai	pai	PROPN
ejpam-3126	503	2	.	.	PROPN
ejpam-3126	503	3	yiarayong	yiarayong	PROPN
ejpam-3126	503	4	,	,	PUNCT
ejpam-3126	503	5	m.	m.	NOUN
ejpam-3126	503	6	siripitukdet	siripitukdet	NOUN
ejpam-3126	503	7	/	/	SYM
ejpam-3126	503	8	eur	eur	PROPN
ejpam-3126	503	9	.	.	PUNCT
ejpam-3126	504	1	j.	j.	PROPN
ejpam-3126	504	2	pure	pure	PROPN
ejpam-3126	504	3	appl	appl	PROPN
ejpam-3126	504	4	.	.	PROPN
ejpam-3126	504	5	math	math	PROPN
ejpam-3126	504	6	,	,	PUNCT
ejpam-3126	504	7	11	11	NUM
ejpam-3126	504	8	(	(	PUNCT
ejpam-3126	504	9	1	1	NUM
ejpam-3126	504	10	)	)	PUNCT
ejpam-3126	504	11	(	(	PUNCT
ejpam-3126	504	12	2018	2018	NUM
ejpam-3126	504	13	)	)	PUNCT
ejpam-3126	504	14	,	,	PUNCT
ejpam-3126	504	15	35	35	NUM
ejpam-3126	504	16	-	-	SYM
ejpam-3126	504	17	50	50	NUM
ejpam-3126	504	18	49	49	NUM
ejpam-3126	504	19	proof	proof	NOUN
ejpam-3126	504	20	.	.	PUNCT
ejpam-3126	505	1	the	the	DET
ejpam-3126	505	2	proof	proof	NOUN
ejpam-3126	505	3	is	be	AUX
ejpam-3126	505	4	clear	clear	ADJ
ejpam-3126	505	5	.	.	PUNCT
ejpam-3126	506	1	corollary	corollary	ADJ
ejpam-3126	506	2	11	11	NUM
ejpam-3126	506	3	.	.	PUNCT
ejpam-3126	507	1	let	let	VERB
ejpam-3126	507	2	ψi	ψi	VERB
ejpam-3126	507	3	:	:	PUNCT
ejpam-3126	507	4	s(mi	s(mi	VERB
ejpam-3126	507	5	)	)	PUNCT
ejpam-3126	507	6	→	→	SYM
ejpam-3126	507	7	s(mi	s(mi	NOUN
ejpam-3126	507	8	)	)	PUNCT
ejpam-3126	507	9	∪	∪	ADP
ejpam-3126	507	10	{	{	PUNCT
ejpam-3126	507	11	∅	∅	NOUN
ejpam-3126	507	12	}	}	PUNCT
ejpam-3126	507	13	be	be	AUX
ejpam-3126	507	14	a	a	DET
ejpam-3126	507	15	function	function	NOUN
ejpam-3126	507	16	with	with	ADP
ejpam-3126	507	17	ψ1(m1	ψ1(m1	NOUN
ejpam-3126	507	18	)	)	PUNCT
ejpam-3126	507	19	=	=	SYM
ejpam-3126	507	20	m1	m1	PROPN
ejpam-3126	507	21	and	and	CCONJ
ejpam-3126	507	22	φ	φ	NOUN
ejpam-3126	507	23	=	=	SYM
ejpam-3126	507	24	ψ1	ψ1	ADJ
ejpam-3126	507	25	×	×	NOUN
ejpam-3126	507	26	ψ2	ψ2	NOUN
ejpam-3126	507	27	.	.	PUNCT
ejpam-3126	508	1	then	then	ADV
ejpam-3126	508	2	m1	m1	PROPN
ejpam-3126	508	3	×	×	PROPN
ejpam-3126	508	4	n2	n2	NOUN
ejpam-3126	508	5	is	be	AUX
ejpam-3126	508	6	a	a	DET
ejpam-3126	508	7	φ-2	φ-2	NOUN
ejpam-3126	508	8	-	-	PUNCT
ejpam-3126	508	9	absorbing	absorbing	ADJ
ejpam-3126	508	10	semi	semi	ADJ
ejpam-3126	508	11	-	-	ADJ
ejpam-3126	508	12	primary	primary	ADJ
ejpam-3126	508	13	submodule	submodule	NOUN
ejpam-3126	508	14	of	of	ADP
ejpam-3126	508	15	m1	m1	PROPN
ejpam-3126	508	16	×m2	×m2	PROPN
ejpam-3126	508	17	if	if	SCONJ
ejpam-3126	508	18	and	and	CCONJ
ejpam-3126	508	19	only	only	ADV
ejpam-3126	508	20	if	if	SCONJ
ejpam-3126	508	21	n2	n2	ADJ
ejpam-3126	508	22	is	be	AUX
ejpam-3126	508	23	a	a	DET
ejpam-3126	508	24	ψ2	ψ2	NOUN
ejpam-3126	508	25	-	-	PUNCT
ejpam-3126	508	26	2	2	NUM
ejpam-3126	508	27	-	-	PUNCT
ejpam-3126	508	28	absorbing	absorbing	ADJ
ejpam-3126	508	29	semi	semi	ADJ
ejpam-3126	508	30	-	-	ADJ
ejpam-3126	508	31	primary	primary	ADJ
ejpam-3126	508	32	submodule	submodule	NOUN
ejpam-3126	508	33	of	of	ADP
ejpam-3126	508	34	m2	m2	PROPN
ejpam-3126	508	35	.	.	PUNCT
ejpam-3126	509	1	proof	proof	NOUN
ejpam-3126	509	2	.	.	PUNCT
ejpam-3126	510	1	similar	similar	ADJ
ejpam-3126	510	2	to	to	ADP
ejpam-3126	510	3	the	the	DET
ejpam-3126	510	4	proof	proof	NOUN
ejpam-3126	510	5	of	of	ADP
ejpam-3126	510	6	theorem	theorem	ADJ
ejpam-3126	510	7	24	24	NUM
ejpam-3126	510	8	.	.	PUNCT
ejpam-3126	510	9	theorem	theorem	VERB
ejpam-3126	510	10	25	25	NUM
ejpam-3126	510	11	.	.	PUNCT
ejpam-3126	511	1	let	let	VERB
ejpam-3126	511	2	ψi	ψi	NOUN
ejpam-3126	511	3	:	:	PUNCT
ejpam-3126	511	4	s(mi	s(mi	VERB
ejpam-3126	511	5	)	)	PUNCT
ejpam-3126	511	6	→	→	SYM
ejpam-3126	511	7	s(mi	s(mi	NOUN
ejpam-3126	511	8	)	)	PUNCT
ejpam-3126	511	9	∪	∪	ADP
ejpam-3126	511	10	{	{	PUNCT
ejpam-3126	511	11	∅	∅	NOUN
ejpam-3126	511	12	}	}	PUNCT
ejpam-3126	511	13	be	be	AUX
ejpam-3126	511	14	a	a	DET
ejpam-3126	511	15	function	function	NOUN
ejpam-3126	511	16	with	with	ADP
ejpam-3126	511	17	ψj(mj	ψj(mj	PROPN
ejpam-3126	511	18	)	)	PUNCT
ejpam-3126	512	1	=	=	SYM
ejpam-3126	512	2	mj	mj	PROPN
ejpam-3126	512	3	and	and	CCONJ
ejpam-3126	512	4	φ	φ	NOUN
ejpam-3126	512	5	=	=	SYM
ejpam-3126	512	6	ψ1	ψ1	ADJ
ejpam-3126	512	7	×	×	NOUN
ejpam-3126	512	8	.	.	PUNCT
ejpam-3126	512	9	.	.	PUNCT
ejpam-3126	512	10	.	.	PUNCT
ejpam-3126	513	1	×	×	PROPN
ejpam-3126	513	2	ψk	ψk	NOUN
ejpam-3126	513	3	.	.	PUNCT
ejpam-3126	514	1	then	then	ADV
ejpam-3126	514	2	m1	m1	PROPN
ejpam-3126	514	3	×	×	PROPN
ejpam-3126	514	4	m2	m2	PROPN
ejpam-3126	514	5	×	×	NOUN
ejpam-3126	514	6	.	.	PUNCT
ejpam-3126	514	7	.	.	PUNCT
ejpam-3126	514	8	.	.	PUNCT
ejpam-3126	515	1	×	×	NOUN
ejpam-3126	516	1	mi−1	mi−1	PROPN
ejpam-3126	516	2	×	×	NOUN
ejpam-3126	516	3	ni	ni	NOUN
ejpam-3126	516	4	×	×	NOUN
ejpam-3126	516	5	mi+1	mi+1	INTJ
ejpam-3126	516	6	×	×	NOUN
ejpam-3126	516	7	.	.	PUNCT
ejpam-3126	516	8	.	.	PUNCT
ejpam-3126	516	9	.	.	PUNCT
ejpam-3126	517	1	×	×	PROPN
ejpam-3126	517	2	mk	mk	PROPN
ejpam-3126	517	3	is	be	AUX
ejpam-3126	517	4	a	a	DET
ejpam-3126	517	5	φ-2absorbing	φ-2absorbe	VERB
ejpam-3126	517	6	semi	semi	ADJ
ejpam-3126	517	7	-	-	ADJ
ejpam-3126	517	8	primary	primary	ADJ
ejpam-3126	517	9	submodule	submodule	NOUN
ejpam-3126	517	10	of	of	ADP
ejpam-3126	517	11	m1	m1	PROPN
ejpam-3126	517	12	×	×	PROPN
ejpam-3126	517	13	.	.	PUNCT
ejpam-3126	517	14	.	.	PUNCT
ejpam-3126	518	1	.×mk	.×mk	PUNCT
ejpam-3126	519	1	if	if	SCONJ
ejpam-3126	519	2	and	and	CCONJ
ejpam-3126	519	3	only	only	ADV
ejpam-3126	519	4	if	if	SCONJ
ejpam-3126	519	5	ni	ni	PROPN
ejpam-3126	519	6	is	be	AUX
ejpam-3126	519	7	a	a	DET
ejpam-3126	519	8	ψi-2	ψi-2	ADV
ejpam-3126	519	9	-	-	PUNCT
ejpam-3126	519	10	absorbing	absorbing	ADJ
ejpam-3126	519	11	semi	semi	ADJ
ejpam-3126	519	12	-	-	ADJ
ejpam-3126	519	13	primary	primary	ADJ
ejpam-3126	519	14	submodule	submodule	NOUN
ejpam-3126	519	15	of	of	ADP
ejpam-3126	519	16	mi	mi	PROPN
ejpam-3126	519	17	.	.	PROPN
ejpam-3126	519	18	proof	proof	NOUN
ejpam-3126	519	19	.	.	PUNCT
ejpam-3126	520	1	similar	similar	ADJ
ejpam-3126	520	2	to	to	ADP
ejpam-3126	520	3	the	the	DET
ejpam-3126	520	4	proof	proof	NOUN
ejpam-3126	520	5	of	of	ADP
ejpam-3126	520	6	theorem	theorem	ADJ
ejpam-3126	520	7	24	24	NUM
ejpam-3126	520	8	and	and	CCONJ
ejpam-3126	520	9	corollary	corollary	ADJ
ejpam-3126	520	10	11	11	NUM
ejpam-3126	520	11	.	.	PUNCT
ejpam-3126	521	1	theorem	theorem	NOUN
ejpam-3126	521	2	26	26	NUM
ejpam-3126	521	3	.	.	PUNCT
ejpam-3126	522	1	let	let	VERB
ejpam-3126	522	2	ni	ni	PROPN
ejpam-3126	522	3	be	be	AUX
ejpam-3126	522	4	a	a	DET
ejpam-3126	522	5	proper	proper	ADJ
ejpam-3126	522	6	submodule	submodule	NOUN
ejpam-3126	522	7	of	of	ADP
ejpam-3126	522	8	mi	mi	PROPN
ejpam-3126	522	9	and	and	CCONJ
ejpam-3126	522	10	let	let	VERB
ejpam-3126	522	11	ψi	ψi	ADP
ejpam-3126	522	12	:	:	PUNCT
ejpam-3126	522	13	s(mi)→	s(mi)→	PROPN
ejpam-3126	522	14	s(mi	s(mi	PROPN
ejpam-3126	522	15	)	)	PUNCT
ejpam-3126	522	16	∪	∪	ADP
ejpam-3126	522	17	{	{	PUNCT
ejpam-3126	522	18	∅	∅	NOUN
ejpam-3126	522	19	}	}	PUNCT
ejpam-3126	522	20	be	be	AUX
ejpam-3126	522	21	a	a	DET
ejpam-3126	522	22	function	function	NOUN
ejpam-3126	522	23	with	with	ADP
ejpam-3126	522	24	φ	φ	PROPN
ejpam-3126	522	25	=	=	SYM
ejpam-3126	522	26	ψ1	ψ1	ADJ
ejpam-3126	522	27	×	×	NOUN
ejpam-3126	522	28	ψ2	ψ2	NOUN
ejpam-3126	522	29	.	.	PUNCT
ejpam-3126	523	1	if	if	SCONJ
ejpam-3126	523	2	n1	n1	PROPN
ejpam-3126	523	3	×	×	PROPN
ejpam-3126	523	4	n2	n2	NOUN
ejpam-3126	523	5	is	be	AUX
ejpam-3126	523	6	a	a	DET
ejpam-3126	523	7	φ-2	φ-2	NOUN
ejpam-3126	523	8	-	-	PUNCT
ejpam-3126	523	9	absorbing	absorbing	ADJ
ejpam-3126	523	10	semi	semi	ADJ
ejpam-3126	523	11	-	-	ADJ
ejpam-3126	523	12	primary	primary	ADJ
ejpam-3126	523	13	submodule	submodule	NOUN
ejpam-3126	523	14	of	of	ADP
ejpam-3126	523	15	m1	m1	PROPN
ejpam-3126	523	16	×m2	×m2	PROPN
ejpam-3126	523	17	,	,	PUNCT
ejpam-3126	523	18	then	then	ADV
ejpam-3126	523	19	(	(	PUNCT
ejpam-3126	523	20	i	i	NOUN
ejpam-3126	523	21	)	)	PUNCT
ejpam-3126	523	22	n1	n1	PROPN
ejpam-3126	523	23	is	be	AUX
ejpam-3126	523	24	a	a	DET
ejpam-3126	523	25	ψ1	ψ1	NOUN
ejpam-3126	523	26	-	-	PUNCT
ejpam-3126	523	27	2	2	NUM
ejpam-3126	523	28	-	-	PUNCT
ejpam-3126	523	29	absorbing	absorbing	ADJ
ejpam-3126	523	30	semi	semi	ADJ
ejpam-3126	523	31	-	-	ADJ
ejpam-3126	523	32	primary	primary	ADJ
ejpam-3126	523	33	submodule	submodule	NOUN
ejpam-3126	523	34	of	of	ADP
ejpam-3126	523	35	m1	m1	PROPN
ejpam-3126	523	36	,	,	PUNCT
ejpam-3126	523	37	(	(	PUNCT
ejpam-3126	523	38	ii	ii	NOUN
ejpam-3126	523	39	)	)	PUNCT
ejpam-3126	523	40	n2	n2	NOUN
ejpam-3126	523	41	is	be	AUX
ejpam-3126	523	42	a	a	DET
ejpam-3126	523	43	ψ2	ψ2	NOUN
ejpam-3126	523	44	-	-	PUNCT
ejpam-3126	523	45	2	2	NUM
ejpam-3126	523	46	-	-	PUNCT
ejpam-3126	523	47	absorbing	absorbing	ADJ
ejpam-3126	523	48	semi	semi	ADJ
ejpam-3126	523	49	-	-	ADJ
ejpam-3126	523	50	primary	primary	ADJ
ejpam-3126	523	51	submodule	submodule	NOUN
ejpam-3126	523	52	of	of	ADP
ejpam-3126	523	53	m2	m2	PROPN
ejpam-3126	523	54	.	.	PUNCT
ejpam-3126	524	1	proof	proof	NOUN
ejpam-3126	524	2	.	.	PUNCT
ejpam-3126	525	1	the	the	DET
ejpam-3126	525	2	proof	proof	NOUN
ejpam-3126	525	3	is	be	AUX
ejpam-3126	525	4	clear	clear	ADJ
ejpam-3126	525	5	.	.	PUNCT
ejpam-3126	526	1	the	the	DET
ejpam-3126	526	2	next	next	ADJ
ejpam-3126	526	3	theorem	theorem	NOUN
ejpam-3126	526	4	gives	give	VERB
ejpam-3126	526	5	conditions	condition	NOUN
ejpam-3126	526	6	for	for	ADP
ejpam-3126	526	7	a	a	DET
ejpam-3126	526	8	φ-2	φ-2	NOUN
ejpam-3126	526	9	-	-	PUNCT
ejpam-3126	526	10	absorbing	absorbing	ADJ
ejpam-3126	526	11	semi	semi	ADJ
ejpam-3126	526	12	-	-	ADJ
ejpam-3126	526	13	primary	primary	ADJ
ejpam-3126	526	14	to	to	PART
ejpam-3126	526	15	be	be	AUX
ejpam-3126	526	16	2	2	NUM
ejpam-3126	526	17	-	-	PUNCT
ejpam-3126	526	18	absorbing	absorbing	ADJ
ejpam-3126	526	19	semi	semi	ADJ
ejpam-3126	526	20	-	-	ADJ
ejpam-3126	526	21	primary	primary	ADJ
ejpam-3126	526	22	.	.	PUNCT
ejpam-3126	527	1	theorem	theorem	NOUN
ejpam-3126	527	2	27	27	NUM
ejpam-3126	527	3	.	.	PUNCT
ejpam-3126	528	1	let	let	VERB
ejpam-3126	528	2	ψi	ψi	VERB
ejpam-3126	528	3	:	:	PUNCT
ejpam-3126	528	4	s(mi	s(mi	VERB
ejpam-3126	528	5	)	)	PUNCT
ejpam-3126	528	6	→	→	SYM
ejpam-3126	528	7	s(mi	s(mi	NOUN
ejpam-3126	528	8	)	)	PUNCT
ejpam-3126	528	9	∪	∪	ADP
ejpam-3126	528	10	{	{	PUNCT
ejpam-3126	528	11	∅	∅	NOUN
ejpam-3126	528	12	}	}	PUNCT
ejpam-3126	528	13	be	be	AUX
ejpam-3126	528	14	a	a	DET
ejpam-3126	528	15	function	function	NOUN
ejpam-3126	528	16	with	with	ADP
ejpam-3126	528	17	ψi(mi	ψi(mi	PROPN
ejpam-3126	528	18	)	)	PUNCT
ejpam-3126	528	19	6=	6=	PROPN
ejpam-3126	528	20	mi	mi	PROPN
ejpam-3126	528	21	,	,	PUNCT
ejpam-3126	528	22	φ	φ	NOUN
ejpam-3126	528	23	=	=	SYM
ejpam-3126	528	24	ψ1	ψ1	ADJ
ejpam-3126	528	25	×	×	NOUN
ejpam-3126	528	26	ψ2	ψ2	NOUN
ejpam-3126	528	27	×	×	NOUN
ejpam-3126	528	28	ψ3	ψ3	NOUN
ejpam-3126	528	29	.	.	PUNCT
ejpam-3126	529	1	if	if	SCONJ
ejpam-3126	529	2	n	n	PRON
ejpam-3126	529	3	is	be	AUX
ejpam-3126	529	4	a	a	DET
ejpam-3126	529	5	φ-2	φ-2	NOUN
ejpam-3126	529	6	-	-	PUNCT
ejpam-3126	529	7	absorbing	absorbing	ADJ
ejpam-3126	529	8	semi	semi	ADJ
ejpam-3126	529	9	-	-	ADJ
ejpam-3126	529	10	primary	primary	ADJ
ejpam-3126	529	11	submodule	submodule	NOUN
ejpam-3126	529	12	of	of	ADP
ejpam-3126	529	13	m1	m1	PROPN
ejpam-3126	529	14	×m2	×m2	PROPN
ejpam-3126	529	15	×m3	×m3	PROPN
ejpam-3126	529	16	,	,	PUNCT
ejpam-3126	529	17	then	then	ADV
ejpam-3126	529	18	n	n	NOUN
ejpam-3126	529	19	=	=	PUNCT
ejpam-3126	529	20	φ(n	φ(n	ADJ
ejpam-3126	529	21	)	)	PUNCT
ejpam-3126	529	22	or	or	CCONJ
ejpam-3126	529	23	n	n	PRON
ejpam-3126	529	24	is	be	AUX
ejpam-3126	529	25	a	a	DET
ejpam-3126	529	26	2	2	NUM
ejpam-3126	529	27	-	-	PUNCT
ejpam-3126	529	28	absorbing	absorbing	ADJ
ejpam-3126	529	29	semi	semi	ADJ
ejpam-3126	529	30	-	-	ADJ
ejpam-3126	529	31	primary	primary	ADJ
ejpam-3126	529	32	submodule	submodule	NOUN
ejpam-3126	529	33	of	of	ADP
ejpam-3126	529	34	m1	m1	PROPN
ejpam-3126	529	35	×m2	×m2	PROPN
ejpam-3126	529	36	×m3	×m3	PROPN
ejpam-3126	529	37	.	.	PUNCT
ejpam-3126	530	1	proof	proof	NOUN
ejpam-3126	530	2	.	.	PUNCT
ejpam-3126	531	1	suppose	suppose	VERB
ejpam-3126	531	2	that	that	SCONJ
ejpam-3126	531	3	n	n	PRON
ejpam-3126	531	4	is	be	AUX
ejpam-3126	531	5	a	a	DET
ejpam-3126	531	6	φ-2	φ-2	NOUN
ejpam-3126	531	7	-	-	PUNCT
ejpam-3126	531	8	absorbing	absorbing	ADJ
ejpam-3126	531	9	semi	semi	ADJ
ejpam-3126	531	10	-	-	ADJ
ejpam-3126	531	11	primary	primary	ADJ
ejpam-3126	531	12	submodule	submodule	NOUN
ejpam-3126	531	13	of	of	ADP
ejpam-3126	531	14	m1	m1	PROPN
ejpam-3126	531	15	×	×	PROPN
ejpam-3126	531	16	m2	m2	PROPN
ejpam-3126	531	17	×	×	PROPN
ejpam-3126	531	18	m3	m3	PROPN
ejpam-3126	531	19	that	that	PRON
ejpam-3126	531	20	is	be	AUX
ejpam-3126	531	21	not	not	PART
ejpam-3126	531	22	2	2	NUM
ejpam-3126	531	23	-	-	PUNCT
ejpam-3126	531	24	absorbing	absorbing	ADJ
ejpam-3126	531	25	semi	semi	ADJ
ejpam-3126	531	26	-	-	ADJ
ejpam-3126	531	27	primary	primary	ADJ
ejpam-3126	531	28	.	.	PUNCT
ejpam-3126	532	1	now	now	ADV
ejpam-3126	532	2	suppose	suppose	VERB
ejpam-3126	532	3	that	that	SCONJ
ejpam-3126	532	4	n1	n1	PROPN
ejpam-3126	532	5	×	×	NOUN
ejpam-3126	532	6	n2	n2	ADJ
ejpam-3126	532	7	×	×	NOUN
ejpam-3126	532	8	n3	n3	NOUN
ejpam-3126	532	9	=	=	SYM
ejpam-3126	532	10	n	n	CCONJ
ejpam-3126	532	11	6=	6=	ADP
ejpam-3126	532	12	ψ1	ψ1	ADJ
ejpam-3126	532	13	×	×	NOUN
ejpam-3126	532	14	ψ2	ψ2	NOUN
ejpam-3126	532	15	×	×	NOUN
ejpam-3126	532	16	ψ3(n	ψ3(n	NUM
ejpam-3126	532	17	)	)	PUNCT
ejpam-3126	532	18	.	.	PUNCT
ejpam-3126	533	1	thus	thus	ADV
ejpam-3126	533	2	ni	ni	PROPN
ejpam-3126	533	3	6=	6=	PROPN
ejpam-3126	533	4	ψi(ni	ψi(ni	PROPN
ejpam-3126	533	5	)	)	PUNCT
ejpam-3126	533	6	for	for	ADP
ejpam-3126	533	7	some	some	DET
ejpam-3126	533	8	i	i	NOUN
ejpam-3126	533	9	=	=	NOUN
ejpam-3126	533	10	1	1	NUM
ejpam-3126	533	11	,	,	PUNCT
ejpam-3126	533	12	2	2	NUM
ejpam-3126	533	13	,	,	PUNCT
ejpam-3126	533	14	3	3	NUM
ejpam-3126	533	15	.	.	X
ejpam-3126	534	1	we	we	PRON
ejpam-3126	534	2	may	may	AUX
ejpam-3126	534	3	assume	assume	VERB
ejpam-3126	534	4	that	that	SCONJ
ejpam-3126	534	5	n1	n1	PROPN
ejpam-3126	534	6	6=	6=	CCONJ
ejpam-3126	534	7	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	534	8	)	)	PUNCT
ejpam-3126	534	9	.	.	PUNCT
ejpam-3126	535	1	there	there	PRON
ejpam-3126	535	2	exists	exist	VERB
ejpam-3126	535	3	m1	m1	PROPN
ejpam-3126	535	4	∈	∈	PROPN
ejpam-3126	535	5	n1	n1	PROPN
ejpam-3126	535	6	such	such	ADJ
ejpam-3126	535	7	that	that	DET
ejpam-3126	535	8	m1	m1	PROPN
ejpam-3126	535	9	/∈	/∈	PUNCT
ejpam-3126	536	1	ψ1(n1	ψ1(n1	NOUN
ejpam-3126	536	2	)	)	PUNCT
ejpam-3126	536	3	.	.	PUNCT
ejpam-3126	537	1	assume	assume	VERB
ejpam-3126	537	2	that	that	SCONJ
ejpam-3126	537	3	n2	n2	PROPN
ejpam-3126	537	4	6=	6=	NUM
ejpam-3126	537	5	m2	m2	PROPN
ejpam-3126	537	6	and	and	CCONJ
ejpam-3126	537	7	n3	n3	PROPN
ejpam-3126	537	8	6=	6=	PROPN
ejpam-3126	537	9	m3	m3	PROPN
ejpam-3126	537	10	.	.	PUNCT
ejpam-3126	538	1	thus	thus	ADV
ejpam-3126	538	2	there	there	PRON
ejpam-3126	538	3	exist	exist	VERB
ejpam-3126	538	4	m2	m2	PROPN
ejpam-3126	538	5	∈	∈	PROPN
ejpam-3126	538	6	m2	m2	PROPN
ejpam-3126	538	7	and	and	CCONJ
ejpam-3126	538	8	m3	m3	PROPN
ejpam-3126	538	9	∈	∈	PROPN
ejpam-3126	538	10	m3	m3	PROPN
ejpam-3126	538	11	such	such	ADJ
ejpam-3126	538	12	that	that	SCONJ
ejpam-3126	538	13	m2	m2	PROPN
ejpam-3126	538	14	6∈	6∈	PROPN
ejpam-3126	538	15	n2	n2	PROPN
ejpam-3126	538	16	and	and	CCONJ
ejpam-3126	538	17	m3	m3	PROPN
ejpam-3126	538	18	6∈	6∈	PROPN
ejpam-3126	538	19	n3	n3	PROPN
ejpam-3126	538	20	.	.	PUNCT
ejpam-3126	539	1	since	since	SCONJ
ejpam-3126	539	2	(	(	PUNCT
ejpam-3126	539	3	1	1	NUM
ejpam-3126	539	4	,	,	PUNCT
ejpam-3126	539	5	0	0	NUM
ejpam-3126	539	6	,	,	PUNCT
ejpam-3126	539	7	1)(1	1)(1	NUM
ejpam-3126	539	8	,	,	PUNCT
ejpam-3126	539	9	1	1	NUM
ejpam-3126	539	10	,	,	PUNCT
ejpam-3126	539	11	0)(m1,m2,m3	0)(m1,m2,m3	NOUN
ejpam-3126	539	12	)	)	PUNCT
ejpam-3126	539	13	6∈	6∈	NOUN
ejpam-3126	539	14	ψ1	ψ1	ADJ
ejpam-3126	539	15	×	×	NOUN
ejpam-3126	539	16	ψ2	ψ2	NOUN
ejpam-3126	539	17	×	×	NOUN
ejpam-3126	539	18	ψ3(n1	ψ3(n1	NUM
ejpam-3126	539	19	×	×	NOUN
ejpam-3126	539	20	n2	n2	ADJ
ejpam-3126	539	21	×	×	PROPN
ejpam-3126	539	22	n3	n3	NOUN
ejpam-3126	539	23	)	)	PUNCT
ejpam-3126	539	24	,	,	PUNCT
ejpam-3126	539	25	we	we	PRON
ejpam-3126	539	26	have	have	VERB
ejpam-3126	539	27	(	(	PUNCT
ejpam-3126	539	28	1	1	NUM
ejpam-3126	539	29	,	,	PUNCT
ejpam-3126	539	30	0	0	NUM
ejpam-3126	539	31	,	,	PUNCT
ejpam-3126	539	32	1)(1	1)(1	NUM
ejpam-3126	539	33	,	,	PUNCT
ejpam-3126	539	34	1	1	NUM
ejpam-3126	539	35	,	,	PUNCT
ejpam-3126	539	36	0)(m1,m2,m3	0)(m1,m2,m3	PROPN
ejpam-3126	539	37	)	)	PUNCT
ejpam-3126	539	38	∈	∈	PROPN
ejpam-3126	539	39	n	n	PRON
ejpam-3126	539	40	−φ(n	−φ(n	NOUN
ejpam-3126	539	41	)	)	PUNCT
ejpam-3126	539	42	.	.	PUNCT
ejpam-3126	540	1	by	by	ADP
ejpam-3126	540	2	definition	definition	NOUN
ejpam-3126	540	3	1	1	NUM
ejpam-3126	540	4	,	,	PUNCT
ejpam-3126	540	5	m2	m2	PROPN
ejpam-3126	540	6	∈	∈	PROPN
ejpam-3126	540	7	n2	n2	NOUN
ejpam-3126	540	8	or	or	CCONJ
ejpam-3126	540	9	m3	m3	PROPN
ejpam-3126	540	10	∈	∈	PROPN
ejpam-3126	540	11	n3	n3	NOUN
ejpam-3126	540	12	,	,	PUNCT
ejpam-3126	540	13	a	a	DET
ejpam-3126	540	14	contradiction	contradiction	NOUN
ejpam-3126	540	15	.	.	PUNCT
ejpam-3126	541	1	therefore	therefore	ADV
ejpam-3126	541	2	n	n	ADV
ejpam-3126	541	3	=	=	PUNCT
ejpam-3126	541	4	n1×m2×n3	n1×m2×n3	ADJ
ejpam-3126	541	5	or	or	CCONJ
ejpam-3126	541	6	n	n	NOUN
ejpam-3126	541	7	=	=	PRON
ejpam-3126	541	8	n1×n2×m3	n1×n2×m3	NOUN
ejpam-3126	541	9	.	.	PUNCT
ejpam-3126	542	1	if	if	SCONJ
ejpam-3126	542	2	n	n	NOUN
ejpam-3126	542	3	=	=	SYM
ejpam-3126	542	4	n1×m2×n3	n1×m2×n3	PROPN
ejpam-3126	542	5	,	,	PUNCT
ejpam-3126	542	6	then	then	ADV
ejpam-3126	542	7	(	(	PUNCT
ejpam-3126	542	8	0	0	NUM
ejpam-3126	542	9	,	,	PUNCT
ejpam-3126	542	10	1	1	NUM
ejpam-3126	542	11	,	,	PUNCT
ejpam-3126	542	12	0	0	NUM
ejpam-3126	542	13	)	)	PUNCT
ejpam-3126	542	14	∈	∈	NOUN
ejpam-3126	542	15	(	(	PUNCT
ejpam-3126	542	16	n	n	NOUN
ejpam-3126	542	17	:	:	PUNCT
ejpam-3126	542	18	m1×m2×m3	m1×m2×m3	NOUN
ejpam-3126	542	19	)	)	PUNCT
ejpam-3126	542	20	.	.	PUNCT
ejpam-3126	543	1	by	by	ADP
ejpam-3126	543	2	theorem	theorem	NOUN
ejpam-3126	543	3	13	13	NUM
ejpam-3126	543	4	,	,	PUNCT
ejpam-3126	543	5	{	{	PUNCT
ejpam-3126	543	6	0}×m2×{0	0}×m2×{0	NUM
ejpam-3126	543	7	}	}	PUNCT
ejpam-3126	543	8	=	=	SYM
ejpam-3126	543	9	(	(	PUNCT
ejpam-3126	543	10	0	0	NUM
ejpam-3126	543	11	,	,	PUNCT
ejpam-3126	543	12	1	1	NUM
ejpam-3126	543	13	,	,	PUNCT
ejpam-3126	543	14	0)2n	0)2n	NOUN
ejpam-3126	543	15	⊆	⊆	NUM
ejpam-3126	543	16	(	(	PUNCT
ejpam-3126	543	17	n	n	NUM
ejpam-3126	543	18	:	:	PUNCT
ejpam-3126	543	19	n1×	n1×	NOUN
ejpam-3126	543	20	m2×n3	m2×n3	PROPN
ejpam-3126	543	21	)	)	PUNCT
ejpam-3126	543	22	2n	2n	NUM
ejpam-3126	543	23	=	=	SYM
ejpam-3126	543	24	(	(	PUNCT
ejpam-3126	543	25	ψ1)3×	ψ1)3×	PUNCT
ejpam-3126	543	26	(	(	PUNCT
ejpam-3126	543	27	ψ2)3×	ψ2)3×	X
ejpam-3126	543	28	(	(	PUNCT
ejpam-3126	543	29	ψ3)3(n	ψ3)3(n	NOUN
ejpam-3126	543	30	)	)	PUNCT
ejpam-3126	543	31	⊆	⊆	NUM
ejpam-3126	543	32	ψ1×ψ2×ψ3(n	ψ1×ψ2×ψ3(n	NOUN
ejpam-3126	543	33	)	)	PUNCT
ejpam-3126	543	34	=	=	SYM
ejpam-3126	543	35	ψ1(n1)×ψ2(m2)×ψ3(n3	ψ1(n1)×ψ2(m2)×ψ3(n3	NOUN
ejpam-3126	543	36	)	)	PUNCT
ejpam-3126	543	37	,	,	PUNCT
ejpam-3126	543	38	which	which	PRON
ejpam-3126	543	39	is	be	AUX
ejpam-3126	543	40	a	a	DET
ejpam-3126	543	41	contradiction	contradiction	NOUN
ejpam-3126	543	42	.	.	PUNCT
ejpam-3126	544	1	this	this	PRON
ejpam-3126	544	2	completes	complete	VERB
ejpam-3126	544	3	the	the	DET
ejpam-3126	544	4	proof	proof	NOUN
ejpam-3126	544	5	.	.	PUNCT
ejpam-3126	545	1	the	the	DET
ejpam-3126	545	2	above	above	ADJ
ejpam-3126	545	3	theorem	theorem	NOUN
ejpam-3126	545	4	shows	show	VERB
ejpam-3126	545	5	the	the	DET
ejpam-3126	545	6	relationship	relationship	NOUN
ejpam-3126	545	7	between	between	ADP
ejpam-3126	545	8	2	2	NUM
ejpam-3126	545	9	-	-	PUNCT
ejpam-3126	545	10	absorbing	absorbing	ADJ
ejpam-3126	545	11	semi	semi	ADJ
ejpam-3126	545	12	-	-	ADJ
ejpam-3126	545	13	primary	primary	ADJ
ejpam-3126	545	14	and	and	CCONJ
ejpam-3126	545	15	φ-2absorbing	φ-2absorbe	VERB
ejpam-3126	545	16	semi	semi	ADJ
ejpam-3126	545	17	-	-	ADJ
ejpam-3126	545	18	primary	primary	ADJ
ejpam-3126	545	19	submodules	submodule	NOUN
ejpam-3126	545	20	in	in	ADP
ejpam-3126	545	21	r1	r1	PROPN
ejpam-3126	545	22	×r2	×r2	PROPN
ejpam-3126	545	23	×r3	×r3	PROPN
ejpam-3126	545	24	-	-	PUNCT
ejpam-3126	545	25	modules	module	NOUN
ejpam-3126	545	26	.	.	PUNCT
ejpam-3126	546	1	from	from	ADP
ejpam-3126	546	2	the	the	DET
ejpam-3126	546	3	above	above	ADJ
ejpam-3126	546	4	theorem	theorem	NOUN
ejpam-3126	546	5	,	,	PUNCT
ejpam-3126	546	6	we	we	PRON
ejpam-3126	546	7	have	have	VERB
ejpam-3126	546	8	the	the	DET
ejpam-3126	546	9	following	follow	VERB
ejpam-3126	546	10	corollary	corollary	NOUN
ejpam-3126	546	11	.	.	PUNCT
ejpam-3126	547	1	references	reference	VERB
ejpam-3126	547	2	50	50	NUM
ejpam-3126	547	3	corollary	corollary	ADJ
ejpam-3126	547	4	12	12	NUM
ejpam-3126	547	5	.	.	PUNCT
ejpam-3126	548	1	let	let	VERB
ejpam-3126	548	2	ψi	ψi	VERB
ejpam-3126	548	3	:	:	PUNCT
ejpam-3126	548	4	s(mi	s(mi	VERB
ejpam-3126	548	5	)	)	PUNCT
ejpam-3126	548	6	→	→	SYM
ejpam-3126	548	7	s(mi	s(mi	NOUN
ejpam-3126	548	8	)	)	PUNCT
ejpam-3126	548	9	∪	∪	ADP
ejpam-3126	548	10	{	{	PUNCT
ejpam-3126	548	11	∅	∅	NOUN
ejpam-3126	548	12	}	}	PUNCT
ejpam-3126	548	13	be	be	AUX
ejpam-3126	548	14	a	a	DET
ejpam-3126	548	15	function	function	NOUN
ejpam-3126	548	16	with	with	ADP
ejpam-3126	548	17	ψi(mi	ψi(mi	PROPN
ejpam-3126	548	18	)	)	PUNCT
ejpam-3126	548	19	6=	6=	PROPN
ejpam-3126	548	20	mi	mi	PROPN
ejpam-3126	548	21	,	,	PUNCT
ejpam-3126	548	22	φ	φ	NOUN
ejpam-3126	548	23	=	=	SYM
ejpam-3126	548	24	ψ1	ψ1	ADJ
ejpam-3126	548	25	×	×	NOUN
ejpam-3126	548	26	ψ2	ψ2	NOUN
ejpam-3126	548	27	×	×	NOUN
ejpam-3126	548	28	ψ3	ψ3	NOUN
ejpam-3126	548	29	and	and	CCONJ
ejpam-3126	548	30	n	n	CCONJ
ejpam-3126	548	31	6=	6=	ADP
ejpam-3126	548	32	φ(n	φ(n	NOUN
ejpam-3126	548	33	)	)	PUNCT
ejpam-3126	548	34	.	.	PUNCT
ejpam-3126	549	1	then	then	ADV
ejpam-3126	549	2	n	n	PRON
ejpam-3126	549	3	is	be	AUX
ejpam-3126	549	4	a	a	DET
ejpam-3126	549	5	φ-2	φ-2	NOUN
ejpam-3126	549	6	-	-	PUNCT
ejpam-3126	549	7	absorbing	absorbing	ADJ
ejpam-3126	549	8	semi	semi	ADJ
ejpam-3126	549	9	-	-	ADJ
ejpam-3126	549	10	primary	primary	ADJ
ejpam-3126	549	11	submodule	submodule	NOUN
ejpam-3126	549	12	of	of	ADP
ejpam-3126	549	13	m1×m2×m3	m1×m2×m3	NOUN
ejpam-3126	549	14	if	if	SCONJ
ejpam-3126	549	15	and	and	CCONJ
ejpam-3126	549	16	only	only	ADV
ejpam-3126	549	17	if	if	SCONJ
ejpam-3126	549	18	n	n	PRON
ejpam-3126	549	19	is	be	AUX
ejpam-3126	549	20	a	a	DET
ejpam-3126	549	21	2	2	NUM
ejpam-3126	549	22	-	-	PUNCT
ejpam-3126	549	23	absorbing	absorbing	ADJ
ejpam-3126	549	24	semi	semi	ADJ
ejpam-3126	549	25	-	-	ADJ
ejpam-3126	549	26	primary	primary	ADJ
ejpam-3126	549	27	submodule	submodule	NOUN
ejpam-3126	549	28	of	of	ADP
ejpam-3126	549	29	m1×m2×m3	m1×m2×m3	NOUN
ejpam-3126	549	30	.	.	PUNCT
ejpam-3126	550	1	proof	proof	NOUN
ejpam-3126	550	2	.	.	PUNCT
ejpam-3126	551	1	this	this	PRON
ejpam-3126	551	2	follows	follow	VERB
ejpam-3126	551	3	from	from	ADP
ejpam-3126	551	4	theorem	theorem	ADJ
ejpam-3126	551	5	27	27	NUM
ejpam-3126	551	6	.	.	PUNCT
ejpam-3126	552	1	references	reference	NOUN
ejpam-3126	552	2	[	[	X
ejpam-3126	552	3	1	1	NUM
ejpam-3126	552	4	]	]	X
ejpam-3126	552	5	e	e	X
ejpam-3126	552	6	a	a	DET
ejpam-3126	552	7	ugurlu	ugurlu	ADJ
ejpam-3126	552	8	g	g	PROPN
ejpam-3126	552	9	ulucak	ulucak	NOUN
ejpam-3126	552	10	a	a	DET
ejpam-3126	552	11	badawi	badawi	NOUN
ejpam-3126	552	12	,	,	PUNCT
ejpam-3126	552	13	u	u	NOUN
ejpam-3126	552	14	tekir	tekir	NOUN
ejpam-3126	552	15	and	and	CCONJ
ejpam-3126	552	16	e	e	NOUN
ejpam-3126	552	17	y	y	PROPN
ejpam-3126	552	18	celikel	celikel	PROPN
ejpam-3126	552	19	.	.	PUNCT
ejpam-3126	553	1	generalizations	generalization	NOUN
ejpam-3126	553	2	of	of	ADP
ejpam-3126	553	3	2absorbing	2absorbing	NUM
ejpam-3126	553	4	primary	primary	ADJ
ejpam-3126	553	5	ideals	ideal	NOUN
ejpam-3126	553	6	of	of	ADP
ejpam-3126	553	7	commutative	commutative	ADJ
ejpam-3126	553	8	rings	ring	NOUN
ejpam-3126	553	9	.	.	PUNCT
ejpam-3126	554	1	turkish	turkish	ADJ
ejpam-3126	554	2	journal	journal	PROPN
ejpam-3126	554	3	of	of	ADP
ejpam-3126	554	4	mathematics	mathematic	NOUN
ejpam-3126	554	5	,	,	PUNCT
ejpam-3126	554	6	40:703	40:703	NUM
ejpam-3126	554	7	–	–	PUNCT
ejpam-3126	554	8	717	717	NUM
ejpam-3126	554	9	,	,	PUNCT
ejpam-3126	554	10	2016	2016	NUM
ejpam-3126	554	11	.	.	PUNCT
ejpam-3126	555	1	[	[	X
ejpam-3126	555	2	2	2	NUM
ejpam-3126	555	3	]	]	PUNCT
ejpam-3126	555	4	d	d	NOUN
ejpam-3126	555	5	d	d	X
ejpam-3126	555	6	anderson	anderson	PROPN
ejpam-3126	555	7	and	and	CCONJ
ejpam-3126	555	8	m	m	PROPN
ejpam-3126	555	9	batanieh	batanieh	ADJ
ejpam-3126	555	10	.	.	PUNCT
ejpam-3126	556	1	generalizations	generalization	NOUN
ejpam-3126	556	2	of	of	ADP
ejpam-3126	556	3	prime	prime	ADJ
ejpam-3126	556	4	ideals	ideal	NOUN
ejpam-3126	556	5	.	.	PUNCT
ejpam-3126	557	1	communications	communication	NOUN
ejpam-3126	557	2	in	in	ADP
ejpam-3126	557	3	algebra	algebra	NOUN
ejpam-3126	557	4	,	,	PUNCT
ejpam-3126	557	5	36:686	36:686	NUM
ejpam-3126	557	6	–	–	PUNCT
ejpam-3126	557	7	696	696	NUM
ejpam-3126	557	8	,	,	PUNCT
ejpam-3126	557	9	2008	2008	NUM
ejpam-3126	557	10	.	.	PUNCT
ejpam-3126	558	1	[	[	X
ejpam-3126	558	2	3	3	X
ejpam-3126	558	3	]	]	X
ejpam-3126	558	4	d	d	NOUN
ejpam-3126	558	5	d	d	X
ejpam-3126	558	6	anderson	anderson	PROPN
ejpam-3126	558	7	and	and	CCONJ
ejpam-3126	558	8	e.	e.	PROPN
ejpam-3126	558	9	smith	smith	PROPN
ejpam-3126	558	10	.	.	PUNCT
ejpam-3126	559	1	weakly	weakly	ADJ
ejpam-3126	559	2	prime	prime	ADJ
ejpam-3126	559	3	ideals	ideal	NOUN
ejpam-3126	559	4	.	.	PUNCT
ejpam-3126	560	1	houston	houston	PROPN
ejpam-3126	560	2	journal	journal	PROPN
ejpam-3126	560	3	of	of	ADP
ejpam-3126	560	4	mathematics	mathematic	NOUN
ejpam-3126	560	5	,	,	PUNCT
ejpam-3126	560	6	29:831	29:831	NUM
ejpam-3126	560	7	–	–	PUNCT
ejpam-3126	560	8	840	840	NUM
ejpam-3126	560	9	,	,	PUNCT
ejpam-3126	560	10	2003	2003	NUM
ejpam-3126	560	11	.	.	PUNCT
ejpam-3126	561	1	[	[	X
ejpam-3126	561	2	4	4	NUM
ejpam-3126	561	3	]	]	X
ejpam-3126	561	4	d	d	X
ejpam-3126	561	5	f	f	PROPN
ejpam-3126	561	6	anderson	anderson	PROPN
ejpam-3126	561	7	and	and	CCONJ
ejpam-3126	561	8	a	a	DET
ejpam-3126	561	9	badawi	badawi	NOUN
ejpam-3126	561	10	.	.	PUNCT
ejpam-3126	562	1	on	on	ADP
ejpam-3126	562	2	n	n	CCONJ
ejpam-3126	562	3	-	-	PUNCT
ejpam-3126	562	4	absorbing	absorbing	ADJ
ejpam-3126	562	5	ideals	ideal	NOUN
ejpam-3126	562	6	of	of	ADP
ejpam-3126	562	7	commutative	commutative	ADJ
ejpam-3126	562	8	rings	ring	NOUN
ejpam-3126	562	9	.	.	PUNCT
ejpam-3126	563	1	communications	communication	NOUN
ejpam-3126	563	2	in	in	ADP
ejpam-3126	563	3	algebra	algebra	NOUN
ejpam-3126	563	4	,	,	PUNCT
ejpam-3126	563	5	39(5):1646	39(5):1646	NUM
ejpam-3126	563	6	–	–	PUNCT
ejpam-3126	563	7	1672	1672	NUM
ejpam-3126	563	8	,	,	PUNCT
ejpam-3126	563	9	2011	2011	NUM
ejpam-3126	563	10	.	.	PUNCT
ejpam-3126	564	1	[	[	X
ejpam-3126	564	2	5	5	NUM
ejpam-3126	564	3	]	]	PUNCT
ejpam-3126	564	4	a	a	DET
ejpam-3126	564	5	badawi	badawi	NOUN
ejpam-3126	564	6	.	.	PUNCT
ejpam-3126	565	1	on	on	ADP
ejpam-3126	565	2	2	2	NUM
ejpam-3126	565	3	-	-	PUNCT
ejpam-3126	565	4	absorbing	absorbing	ADJ
ejpam-3126	565	5	ideals	ideal	NOUN
ejpam-3126	565	6	of	of	ADP
ejpam-3126	565	7	commutative	commutative	ADJ
ejpam-3126	565	8	rings	ring	NOUN
ejpam-3126	565	9	.	.	PUNCT
ejpam-3126	566	1	bulletin	bulletin	NOUN
ejpam-3126	566	2	of	of	ADP
ejpam-3126	566	3	the	the	DET
ejpam-3126	566	4	australian	australian	ADJ
ejpam-3126	566	5	mathematical	mathematical	ADJ
ejpam-3126	566	6	society	society	NOUN
ejpam-3126	566	7	,	,	PUNCT
ejpam-3126	566	8	75:417	75:417	NUM
ejpam-3126	566	9	–	–	PUNCT
ejpam-3126	566	10	429	429	NUM
ejpam-3126	566	11	,	,	PUNCT
ejpam-3126	566	12	2007	2007	NUM
ejpam-3126	566	13	.	.	PUNCT
ejpam-3126	567	1	[	[	X
ejpam-3126	567	2	6	6	NUM
ejpam-3126	567	3	]	]	PUNCT
ejpam-3126	567	4	a	a	DET
ejpam-3126	567	5	badawi	badawi	NOUN
ejpam-3126	567	6	and	and	CCONJ
ejpam-3126	567	7	a	a	DET
ejpam-3126	567	8	y	y	PROPN
ejpam-3126	567	9	darani	darani	PROPN
ejpam-3126	567	10	.	.	PUNCT
ejpam-3126	568	1	on	on	ADP
ejpam-3126	568	2	weakly	weakly	ADJ
ejpam-3126	568	3	2	2	NUM
ejpam-3126	568	4	-	-	PUNCT
ejpam-3126	568	5	absorbing	absorbing	ADJ
ejpam-3126	568	6	ideals	ideal	NOUN
ejpam-3126	568	7	of	of	ADP
ejpam-3126	568	8	commutative	commutative	ADJ
ejpam-3126	568	9	rings	ring	NOUN
ejpam-3126	568	10	.	.	PUNCT
ejpam-3126	569	1	houston	houston	PROPN
ejpam-3126	569	2	journal	journal	PROPN
ejpam-3126	569	3	of	of	ADP
ejpam-3126	569	4	mathematics	mathematic	NOUN
ejpam-3126	569	5	,	,	PUNCT
ejpam-3126	569	6	39:441	39:441	NUM
ejpam-3126	569	7	–	–	PUNCT
ejpam-3126	569	8	452	452	NUM
ejpam-3126	569	9	,	,	PUNCT
ejpam-3126	569	10	2013	2013	NUM
ejpam-3126	569	11	.	.	PUNCT
ejpam-3126	570	1	[	[	X
ejpam-3126	570	2	7	7	X
ejpam-3126	570	3	]	]	X
ejpam-3126	570	4	a	a	DET
ejpam-3126	570	5	y	y	PROPN
ejpam-3126	570	6	darani	darani	PROPN
ejpam-3126	570	7	.	.	PUNCT
ejpam-3126	571	1	generalizations	generalization	NOUN
ejpam-3126	571	2	of	of	ADP
ejpam-3126	571	3	primary	primary	ADJ
ejpam-3126	571	4	ideals	ideal	NOUN
ejpam-3126	571	5	in	in	ADP
ejpam-3126	571	6	commutative	commutative	ADJ
ejpam-3126	571	7	rings	ring	NOUN
ejpam-3126	571	8	.	.	PUNCT
ejpam-3126	572	1	novi	novi	PROPN
ejpam-3126	572	2	sad	sad	PROPN
ejpam-3126	572	3	journal	journal	PROPN
ejpam-3126	572	4	of	of	ADP
ejpam-3126	572	5	mathematics	mathematic	NOUN
ejpam-3126	572	6	,	,	PUNCT
ejpam-3126	572	7	42(1):27–35	42(1):27–35	NUM
ejpam-3126	572	8	,	,	PUNCT
ejpam-3126	572	9	2012	2012	NUM
ejpam-3126	572	10	.	.	PUNCT
ejpam-3126	573	1	[	[	X
ejpam-3126	573	2	8	8	NUM
ejpam-3126	573	3	]	]	X
ejpam-3126	573	4	m	m	VERB
ejpam-3126	573	5	ebrahimpour	ebrahimpour	NOUN
ejpam-3126	573	6	and	and	CCONJ
ejpam-3126	573	7	f	f	PROPN
ejpam-3126	573	8	mirzaee	mirzaee	PROPN
ejpam-3126	573	9	.	.	PUNCT
ejpam-3126	574	1	on	on	ADP
ejpam-3126	574	2	φ	φ	VERB
ejpam-3126	574	3	-	-	PUNCT
ejpam-3126	574	4	semiprime	semiprime	NOUN
ejpam-3126	574	5	submodules	submodule	NOUN
ejpam-3126	574	6	.	.	PUNCT
ejpam-3126	575	1	journal	journal	NOUN
ejpam-3126	575	2	of	of	ADP
ejpam-3126	575	3	the	the	DET
ejpam-3126	575	4	korean	korean	PROPN
ejpam-3126	575	5	mathematical	mathematical	ADJ
ejpam-3126	575	6	society	society	NOUN
ejpam-3126	575	7	,	,	PUNCT
ejpam-3126	575	8	54(4):1099	54(4):1099	NUM
ejpam-3126	575	9	–	–	PUNCT
ejpam-3126	575	10	1108	1108	NUM
ejpam-3126	575	11	,	,	PUNCT
ejpam-3126	575	12	2017	2017	NUM
ejpam-3126	575	13	.	.	PUNCT
ejpam-3126	576	1	[	[	X
ejpam-3126	576	2	9	9	NUM
ejpam-3126	576	3	]	]	SYM
ejpam-3126	576	4	m	m	VERB
ejpam-3126	576	5	ebrahimpour	ebrahimpour	ADJ
ejpam-3126	576	6	and	and	CCONJ
ejpam-3126	576	7	r	r	NOUN
ejpam-3126	576	8	nekooei	nekooei	NOUN
ejpam-3126	576	9	.	.	PUNCT
ejpam-3126	577	1	on	on	ADP
ejpam-3126	577	2	generalizations	generalization	NOUN
ejpam-3126	577	3	of	of	ADP
ejpam-3126	577	4	prime	prime	ADJ
ejpam-3126	577	5	ideals	ideal	NOUN
ejpam-3126	577	6	.	.	PUNCT
ejpam-3126	578	1	communications	communication	NOUN
ejpam-3126	578	2	in	in	ADP
ejpam-3126	578	3	algebra	algebra	NOUN
ejpam-3126	578	4	,	,	PUNCT
ejpam-3126	578	5	40:1268	40:1268	NUM
ejpam-3126	578	6	–	–	PUNCT
ejpam-3126	578	7	1279	1279	NUM
ejpam-3126	578	8	,	,	PUNCT
ejpam-3126	578	9	2012	2012	NUM
ejpam-3126	578	10	.	.	PUNCT
ejpam-3126	579	1	[	[	X
ejpam-3126	579	2	10	10	NUM
ejpam-3126	579	3	]	]	X
ejpam-3126	579	4	r	r	NOUN
ejpam-3126	579	5	moradi	moradi	NOUN
ejpam-3126	579	6	and	and	CCONJ
ejpam-3126	579	7	m	m	VERB
ejpam-3126	579	8	ebrahimpour	ebrahimpour	ADJ
ejpam-3126	579	9	.	.	PUNCT
ejpam-3126	580	1	on	on	ADP
ejpam-3126	580	2	φ-2	φ-2	PROPN
ejpam-3126	580	3	-	-	PUNCT
ejpam-3126	580	4	absorbing	absorbing	ADJ
ejpam-3126	580	5	primary	primary	ADJ
ejpam-3126	580	6	submodules	submodule	NOUN
ejpam-3126	580	7	.	.	PUNCT
ejpam-3126	581	1	acta	acta	PROPN
ejpam-3126	581	2	mathematica	mathematica	PROPN
ejpam-3126	581	3	vietnamica	vietnamica	PROPN
ejpam-3126	581	4	,	,	PUNCT
ejpam-3126	581	5	42:27	42:27	NUM
ejpam-3126	581	6	–	–	PUNCT
ejpam-3126	581	7	35	35	NUM
ejpam-3126	581	8	,	,	PUNCT
ejpam-3126	581	9	2017	2017	NUM
ejpam-3126	581	10	.	.	PUNCT
ejpam-3126	582	1	[	[	X
ejpam-3126	582	2	11	11	NUM
ejpam-3126	582	3	]	]	SYM
ejpam-3126	582	4	n	n	X
ejpam-3126	582	5	zamani	zamani	PROPN
ejpam-3126	582	6	.	.	PUNCT
ejpam-3126	583	1	φ	φ	VERB
ejpam-3126	583	2	-	-	ADJ
ejpam-3126	583	3	prime	prime	ADJ
ejpam-3126	583	4	submodules	submodule	NOUN
ejpam-3126	583	5	.	.	PUNCT
ejpam-3126	584	1	glasgow	glasgow	PROPN
ejpam-3126	584	2	mathematical	mathematical	ADJ
ejpam-3126	584	3	journal	journal	NOUN
ejpam-3126	584	4	,	,	PUNCT
ejpam-3126	584	5	52(2):253	52(2):253	NUM
ejpam-3126	584	6	–	–	PUNCT
ejpam-3126	584	7	259	259	NUM
ejpam-3126	584	8	,	,	PUNCT
ejpam-3126	584	9	2010	2010	NUM
ejpam-3126	584	10	.	.	PUNCT