id	sid	tid	token	lemma	pos
ejpam-4301	1	1	european	european	PROPN
ejpam-4301	1	2	journal	journal	PROPN
ejpam-4301	1	3	of	of	ADP
ejpam-4301	1	4	pure	pure	ADJ
ejpam-4301	1	5	and	and	CCONJ
ejpam-4301	1	6	applied	apply	VERB
ejpam-4301	1	7	mathematics	mathematic	NOUN
ejpam-4301	1	8	vol	vol	NOUN
ejpam-4301	1	9	.	.	PROPN
ejpam-4301	2	1	15	15	NUM
ejpam-4301	2	2	,	,	PUNCT
ejpam-4301	2	3	no	no	INTJ
ejpam-4301	2	4	.	.	NOUN
ejpam-4301	2	5	2	2	NUM
ejpam-4301	2	6	,	,	PUNCT
ejpam-4301	2	7	2022	2022	NUM
ejpam-4301	2	8	,	,	PUNCT
ejpam-4301	2	9	478	478	NUM
ejpam-4301	2	10	-	-	SYM
ejpam-4301	2	11	485	485	NUM
ejpam-4301	2	12	issn	issn	PROPN
ejpam-4301	2	13	1307	1307	NUM
ejpam-4301	2	14	-	-	SYM
ejpam-4301	2	15	5543	5543	NUM
ejpam-4301	2	16	–	–	PUNCT
ejpam-4301	2	17	ejpam.com	ejpam.com	X
ejpam-4301	2	18	published	publish	VERB
ejpam-4301	2	19	by	by	ADP
ejpam-4301	2	20	new	new	PROPN
ejpam-4301	2	21	york	york	PROPN
ejpam-4301	2	22	business	business	PROPN
ejpam-4301	2	23	global	global	ADJ
ejpam-4301	2	24	e∗-essential	e∗-essential	ADJ
ejpam-4301	2	25	small	small	ADJ
ejpam-4301	2	26	submodules	submodule	NOUN
ejpam-4301	2	27	and	and	CCONJ
ejpam-4301	2	28	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	2	29	modules	module	NOUN
ejpam-4301	2	30	hiba	hiba	PROPN
ejpam-4301	2	31	r.	r.	PROPN
ejpam-4301	2	32	baanoon1,2,∗	baanoon1,2,∗	PROPN
ejpam-4301	2	33	,	,	PUNCT
ejpam-4301	2	34	wasan	wasan	PROPN
ejpam-4301	2	35	khalid2	khalid2	PROPN
ejpam-4301	2	36	1	1	NUM
ejpam-4301	2	37	mathematics	mathematics	PROPN
ejpam-4301	2	38	department	department	NOUN
ejpam-4301	2	39	,	,	PUNCT
ejpam-4301	2	40	college	college	NOUN
ejpam-4301	2	41	of	of	ADP
ejpam-4301	2	42	education	education	NOUN
ejpam-4301	2	43	,	,	PUNCT
ejpam-4301	2	44	university	university	NOUN
ejpam-4301	2	45	of	of	ADP
ejpam-4301	2	46	misan	misan	NOUN
ejpam-4301	2	47	,	,	PUNCT
ejpam-4301	2	48	iraq	iraq	PROPN
ejpam-4301	2	49	2	2	NUM
ejpam-4301	2	50	mathematics	mathematics	PROPN
ejpam-4301	2	51	department	department	NOUN
ejpam-4301	2	52	,	,	PUNCT
ejpam-4301	2	53	college	college	NOUN
ejpam-4301	2	54	of	of	ADP
ejpam-4301	2	55	science	science	NOUN
ejpam-4301	2	56	,	,	PUNCT
ejpam-4301	2	57	university	university	NOUN
ejpam-4301	2	58	of	of	ADP
ejpam-4301	2	59	baghdad	baghdad	PROPN
ejpam-4301	2	60	,	,	PUNCT
ejpam-4301	2	61	iraq	iraq	PROPN
ejpam-4301	2	62	abstract	abstract	NOUN
ejpam-4301	2	63	.	.	PUNCT
ejpam-4301	3	1	the	the	DET
ejpam-4301	3	2	purpose	purpose	NOUN
ejpam-4301	3	3	of	of	ADP
ejpam-4301	3	4	this	this	DET
ejpam-4301	3	5	paper	paper	NOUN
ejpam-4301	3	6	is	be	AUX
ejpam-4301	3	7	to	to	PART
ejpam-4301	3	8	introduce	introduce	VERB
ejpam-4301	3	9	the	the	DET
ejpam-4301	3	10	concepts	concept	NOUN
ejpam-4301	3	11	of	of	ADP
ejpam-4301	3	12	e∗-small	e∗-small	PROPN
ejpam-4301	3	13	essential	essential	ADJ
ejpam-4301	3	14	submodules	submodule	NOUN
ejpam-4301	3	15	,	,	PUNCT
ejpam-4301	3	16	e∗-radical	e∗-radical	ADJ
ejpam-4301	3	17	submodules	submodule	NOUN
ejpam-4301	3	18	,	,	PUNCT
ejpam-4301	3	19	and	and	CCONJ
ejpam-4301	3	20	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	3	21	modules	module	NOUN
ejpam-4301	3	22	as	as	ADP
ejpam-4301	3	23	a	a	DET
ejpam-4301	3	24	generalizations	generalization	NOUN
ejpam-4301	3	25	of	of	ADP
ejpam-4301	3	26	the	the	DET
ejpam-4301	3	27	concepts	concept	NOUN
ejpam-4301	3	28	of	of	ADP
ejpam-4301	3	29	small	small	ADJ
ejpam-4301	3	30	submodules	submodule	NOUN
ejpam-4301	3	31	,	,	PUNCT
ejpam-4301	3	32	radical	radical	ADJ
ejpam-4301	3	33	submodules	submodule	NOUN
ejpam-4301	3	34	,	,	PUNCT
ejpam-4301	3	35	and	and	CCONJ
ejpam-4301	3	36	hollow	hollow	ADJ
ejpam-4301	3	37	modules	module	NOUN
ejpam-4301	3	38	,	,	PUNCT
ejpam-4301	3	39	respectively	respectively	ADV
ejpam-4301	3	40	.	.	PUNCT
ejpam-4301	4	1	we	we	PRON
ejpam-4301	4	2	will	will	AUX
ejpam-4301	4	3	prove	prove	VERB
ejpam-4301	4	4	some	some	DET
ejpam-4301	4	5	properties	property	NOUN
ejpam-4301	4	6	of	of	ADP
ejpam-4301	4	7	these	these	DET
ejpam-4301	4	8	concepts	concept	NOUN
ejpam-4301	4	9	.	.	PUNCT
ejpam-4301	5	1	2020	2020	NUM
ejpam-4301	5	2	mathematics	mathematic	NOUN
ejpam-4301	5	3	subject	subject	NOUN
ejpam-4301	5	4	classifications	classification	NOUN
ejpam-4301	5	5	:	:	PUNCT
ejpam-4301	5	6	16d10	16d10	NUM
ejpam-4301	5	7	,	,	PUNCT
ejpam-4301	5	8	16d90	16d90	NUM
ejpam-4301	5	9	,	,	PUNCT
ejpam-4301	5	10	16d99	16d99	NUM
ejpam-4301	5	11	,	,	PUNCT
ejpam-4301	5	12	16s90	16s90	NUM
ejpam-4301	5	13	key	key	ADJ
ejpam-4301	5	14	words	word	NOUN
ejpam-4301	5	15	and	and	CCONJ
ejpam-4301	5	16	phrases	phrase	NOUN
ejpam-4301	5	17	:	:	PUNCT
ejpam-4301	5	18	e∗-small	e∗-small	NUM
ejpam-4301	5	19	essential	essential	ADJ
ejpam-4301	5	20	submodule	submodule	NOUN
ejpam-4301	5	21	,	,	PUNCT
ejpam-4301	5	22	small	small	ADJ
ejpam-4301	5	23	submodule	submodule	NOUN
ejpam-4301	5	24	,	,	PUNCT
ejpam-4301	5	25	e∗-radical	e∗-radical	ADJ
ejpam-4301	5	26	submodule	submodule	NOUN
ejpam-4301	5	27	,	,	PUNCT
ejpam-4301	5	28	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	5	29	module	module	NOUN
ejpam-4301	5	30	,	,	PUNCT
ejpam-4301	5	31	hollow	hollow	ADJ
ejpam-4301	5	32	module	module	NOUN
ejpam-4301	5	33	1	1	NUM
ejpam-4301	5	34	.	.	PUNCT
ejpam-4301	6	1	introduction	introduction	NOUN
ejpam-4301	6	2	let	let	VERB
ejpam-4301	6	3	r	r	PRON
ejpam-4301	6	4	be	be	AUX
ejpam-4301	6	5	a	a	DET
ejpam-4301	6	6	ring	ring	NOUN
ejpam-4301	6	7	with	with	ADP
ejpam-4301	6	8	identity	identity	NOUN
ejpam-4301	6	9	,	,	PUNCT
ejpam-4301	6	10	m	m	VERB
ejpam-4301	6	11	is	be	AUX
ejpam-4301	6	12	a	a	DET
ejpam-4301	6	13	right	right	ADJ
ejpam-4301	6	14	r	r	NOUN
ejpam-4301	6	15	-	-	PUNCT
ejpam-4301	6	16	module	module	NOUN
ejpam-4301	6	17	and	and	CCONJ
ejpam-4301	6	18	e(m	e(m	NOUN
ejpam-4301	6	19	)	)	PUNCT
ejpam-4301	6	20	the	the	DET
ejpam-4301	6	21	injective	injective	ADJ
ejpam-4301	6	22	hull	hull	NOUN
ejpam-4301	6	23	of	of	ADP
ejpam-4301	6	24	m	m	PROPN
ejpam-4301	6	25	.	.	PUNCT
ejpam-4301	7	1	a	a	DET
ejpam-4301	7	2	submodule	submodule	NOUN
ejpam-4301	7	3	n	n	PROPN
ejpam-4301	7	4	of	of	ADP
ejpam-4301	7	5	m	m	PROPN
ejpam-4301	7	6	is	be	AUX
ejpam-4301	7	7	called	call	VERB
ejpam-4301	7	8	a	a	DET
ejpam-4301	7	9	small	small	ADJ
ejpam-4301	7	10	submodule	submodule	NOUN
ejpam-4301	7	11	of	of	ADP
ejpam-4301	7	12	m	m	AUX
ejpam-4301	7	13	denoted	denote	VERB
ejpam-4301	7	14	(	(	PUNCT
ejpam-4301	7	15	n	n	X
ejpam-4301	7	16	�	�	PROPN
ejpam-4301	7	17	m	m	PROPN
ejpam-4301	7	18	)	)	PUNCT
ejpam-4301	7	19	if	if	SCONJ
ejpam-4301	7	20	for	for	ADP
ejpam-4301	7	21	any	any	DET
ejpam-4301	7	22	submodule	submodule	NOUN
ejpam-4301	7	23	a	a	PRON
ejpam-4301	7	24	of	of	ADP
ejpam-4301	7	25	m	m	PRON
ejpam-4301	7	26	such	such	ADJ
ejpam-4301	7	27	that	that	SCONJ
ejpam-4301	7	28	m	m	VERB
ejpam-4301	7	29	=	=	SYM
ejpam-4301	7	30	n	n	PROPN
ejpam-4301	8	1	+	+	CCONJ
ejpam-4301	8	2	a	a	X
ejpam-4301	8	3	,	,	PUNCT
ejpam-4301	8	4	we	we	PRON
ejpam-4301	8	5	have	have	VERB
ejpam-4301	8	6	a	a	DET
ejpam-4301	8	7	=	=	NOUN
ejpam-4301	8	8	m	m	NOUN
ejpam-4301	9	1	[	[	X
ejpam-4301	9	2	6	6	NUM
ejpam-4301	9	3	]	]	PUNCT
ejpam-4301	9	4	recall	recall	NOUN
ejpam-4301	9	5	that	that	SCONJ
ejpam-4301	9	6	a	a	DET
ejpam-4301	9	7	submodule	submodule	NOUN
ejpam-4301	9	8	a	a	PRON
ejpam-4301	9	9	of	of	ADP
ejpam-4301	9	10	r	r	NOUN
ejpam-4301	9	11	-	-	PUNCT
ejpam-4301	9	12	module	module	NOUN
ejpam-4301	9	13	b	b	NOUN
ejpam-4301	9	14	is	be	AUX
ejpam-4301	9	15	called	call	VERB
ejpam-4301	9	16	essential	essential	ADJ
ejpam-4301	9	17	in	in	ADP
ejpam-4301	9	18	b	b	NOUN
ejpam-4301	9	19	if	if	SCONJ
ejpam-4301	9	20	every	every	DET
ejpam-4301	9	21	nonzero	nonzero	PROPN
ejpam-4301	9	22	submodule	submodule	PROPN
ejpam-4301	9	23	of	of	ADP
ejpam-4301	9	24	b	b	PROPN
ejpam-4301	9	25	has	have	VERB
ejpam-4301	9	26	nonzero	nonzero	ADJ
ejpam-4301	9	27	intersection	intersection	NOUN
ejpam-4301	9	28	with	with	ADP
ejpam-4301	9	29	a	a	DET
ejpam-4301	9	30	[	[	X
ejpam-4301	9	31	6	6	NUM
ejpam-4301	9	32	]	]	PUNCT
ejpam-4301	9	33	,	,	PUNCT
ejpam-4301	9	34	[	[	X
ejpam-4301	9	35	4	4	X
ejpam-4301	9	36	]	]	PUNCT
ejpam-4301	9	37	and	and	CCONJ
ejpam-4301	9	38	[	[	X
ejpam-4301	9	39	5	5	NUM
ejpam-4301	9	40	]	]	PUNCT
ejpam-4301	9	41	.	.	PUNCT
ejpam-4301	10	1	oscan	oscan	PROPN
ejpam-4301	10	2	in	in	ADP
ejpam-4301	10	3	[	[	X
ejpam-4301	10	4	2	2	NUM
ejpam-4301	10	5	]	]	PUNCT
ejpam-4301	10	6	introduced	introduce	VERB
ejpam-4301	10	7	the	the	DET
ejpam-4301	10	8	concept	concept	NOUN
ejpam-4301	10	9	of	of	ADP
ejpam-4301	10	10	cosingular	cosingular	ADJ
ejpam-4301	10	11	submodule	submodule	NOUN
ejpam-4301	10	12	as	as	SCONJ
ejpam-4301	10	13	follows	follow	VERB
ejpam-4301	10	14	:	:	PUNCT
ejpam-4301	10	15	z∗(m	z∗(m	X
ejpam-4301	10	16	)	)	PUNCT
ejpam-4301	11	1	=	=	PRON
ejpam-4301	11	2	{	{	PUNCT
ejpam-4301	11	3	m	m	VERB
ejpam-4301	11	4	∈	∈	PROPN
ejpam-4301	11	5	m	m	VERB
ejpam-4301	11	6	|mr	|mr	PRON
ejpam-4301	11	7	�	�	PROPN
ejpam-4301	11	8	e(m	e(m	PROPN
ejpam-4301	11	9	)	)	PUNCT
ejpam-4301	11	10	}	}	PUNCT
ejpam-4301	11	11	.	.	PUNCT
ejpam-4301	12	1	an	an	DET
ejpam-4301	12	2	r	r	NOUN
ejpam-4301	12	3	-	-	PUNCT
ejpam-4301	12	4	module	module	NOUN
ejpam-4301	12	5	m	m	NOUN
ejpam-4301	12	6	is	be	AUX
ejpam-4301	12	7	called	call	VERB
ejpam-4301	12	8	cosingular	cosingular	ADJ
ejpam-4301	12	9	if	if	SCONJ
ejpam-4301	12	10	z∗(m	z∗(m	PROPN
ejpam-4301	12	11	)	)	PUNCT
ejpam-4301	13	1	=	=	PUNCT
ejpam-4301	14	1	m	m	PROPN
ejpam-4301	14	2	.	.	PUNCT
ejpam-4301	15	1	baanoon	baanoon	NOUN
ejpam-4301	15	2	and	and	CCONJ
ejpam-4301	15	3	khaild	khaild	NOUN
ejpam-4301	15	4	in	in	ADP
ejpam-4301	15	5	[	[	X
ejpam-4301	15	6	1	1	X
ejpam-4301	15	7	]	]	PUNCT
ejpam-4301	15	8	introduced	introduce	VERB
ejpam-4301	15	9	a	a	DET
ejpam-4301	15	10	type	type	NOUN
ejpam-4301	15	11	of	of	ADP
ejpam-4301	15	12	submodule	submodule	NOUN
ejpam-4301	15	13	which	which	PRON
ejpam-4301	15	14	called	call	VERB
ejpam-4301	15	15	e∗-essential	e∗-essential	PROPN
ejpam-4301	15	16	as	as	SCONJ
ejpam-4301	15	17	follows	follow	VERB
ejpam-4301	15	18	.	.	PUNCT
ejpam-4301	16	1	a	a	DET
ejpam-4301	16	2	submodule	submodule	NOUN
ejpam-4301	16	3	a	a	PRON
ejpam-4301	16	4	of	of	ADP
ejpam-4301	16	5	m	m	PROPN
ejpam-4301	16	6	is	be	AUX
ejpam-4301	16	7	said	say	VERB
ejpam-4301	16	8	to	to	PART
ejpam-4301	16	9	be	be	AUX
ejpam-4301	16	10	e∗-essential	e∗-essential	PROPN
ejpam-4301	16	11	in	in	ADP
ejpam-4301	16	12	m	m	PROPN
ejpam-4301	16	13	if	if	SCONJ
ejpam-4301	16	14	a∩b	a∩b	PROPN
ejpam-4301	16	15	6=	6=	ADP
ejpam-4301	16	16	0	0	NUM
ejpam-4301	16	17	for	for	ADP
ejpam-4301	16	18	each	each	DET
ejpam-4301	16	19	nonzero	nonzero	NOUN
ejpam-4301	16	20	cosingular	cosingular	PROPN
ejpam-4301	16	21	submodule	submodule	PROPN
ejpam-4301	16	22	b	b	PROPN
ejpam-4301	16	23	of	of	ADP
ejpam-4301	16	24	m	m	PROPN
ejpam-4301	16	25	.	.	PUNCT
ejpam-4301	17	1	denoted	denote	VERB
ejpam-4301	17	2	by	by	ADP
ejpam-4301	17	3	a	a	DET
ejpam-4301	17	4	≤e∗	≤e∗	PROPN
ejpam-4301	17	5	m	m	NOUN
ejpam-4301	17	6	.	.	PUNCT
ejpam-4301	18	1	as	as	ADP
ejpam-4301	18	2	in	in	ADP
ejpam-4301	18	3	[	[	X
ejpam-4301	18	4	7	7	NUM
ejpam-4301	18	5	]	]	PUNCT
ejpam-4301	18	6	,	,	PUNCT
ejpam-4301	18	7	we	we	PRON
ejpam-4301	18	8	will	will	AUX
ejpam-4301	18	9	used	use	VERB
ejpam-4301	18	10	e∗-essential	e∗-essential	PROPN
ejpam-4301	18	11	submodule	submodule	NOUN
ejpam-4301	18	12	that	that	PRON
ejpam-4301	18	13	appeared	appear	VERB
ejpam-4301	18	14	in	in	ADP
ejpam-4301	18	15	[	[	X
ejpam-4301	18	16	1	1	NUM
ejpam-4301	18	17	]	]	PUNCT
ejpam-4301	18	18	,	,	PUNCT
ejpam-4301	18	19	to	to	PART
ejpam-4301	18	20	present	present	VERB
ejpam-4301	18	21	a	a	DET
ejpam-4301	18	22	new	new	ADJ
ejpam-4301	18	23	generalization	generalization	NOUN
ejpam-4301	18	24	of	of	ADP
ejpam-4301	18	25	a	a	DET
ejpam-4301	18	26	small	small	ADJ
ejpam-4301	18	27	sumodule	sumodule	NOUN
ejpam-4301	18	28	namely	namely	ADV
ejpam-4301	18	29	e∗-essential	e∗-essential	ADJ
ejpam-4301	18	30	small	small	ADJ
ejpam-4301	18	31	submodule	submodule	NOUN
ejpam-4301	18	32	.	.	PUNCT
ejpam-4301	19	1	e∗-essential	e∗-essential	ADJ
ejpam-4301	19	2	small	small	ADJ
ejpam-4301	19	3	submodules	submodule	NOUN
ejpam-4301	19	4	leads	lead	VERB
ejpam-4301	19	5	us	we	PRON
ejpam-4301	19	6	to	to	PART
ejpam-4301	19	7	introduce	introduce	VERB
ejpam-4301	19	8	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	19	9	module	module	NOUN
ejpam-4301	19	10	as	as	ADP
ejpam-4301	19	11	a	a	DET
ejpam-4301	19	12	generalization	generalization	NOUN
ejpam-4301	19	13	of	of	ADP
ejpam-4301	19	14	hollow	hollow	ADJ
ejpam-4301	19	15	modules	module	NOUN
ejpam-4301	19	16	.	.	PUNCT
ejpam-4301	20	1	in	in	ADP
ejpam-4301	20	2	this	this	DET
ejpam-4301	20	3	paper	paper	NOUN
ejpam-4301	20	4	main	main	ADJ
ejpam-4301	20	5	properties	property	NOUN
ejpam-4301	20	6	of	of	ADP
ejpam-4301	20	7	these	these	DET
ejpam-4301	20	8	concepts	concept	NOUN
ejpam-4301	20	9	are	be	AUX
ejpam-4301	20	10	proved	prove	VERB
ejpam-4301	20	11	.	.	PUNCT
ejpam-4301	21	1	∗corresponding	∗corresponde	VERB
ejpam-4301	21	2	author	author	NOUN
ejpam-4301	21	3	.	.	PUNCT
ejpam-4301	22	1	doi	doi	NOUN
ejpam-4301	22	2	:	:	PUNCT
ejpam-4301	22	3	https://doi.org/10.29020/nybg.ejpam.v15i2.4301	https://doi.org/10.29020/nybg.ejpam.v15i2.4301	NUM
ejpam-4301	22	4	email	email	NOUN
ejpam-4301	22	5	addresses	address	NOUN
ejpam-4301	22	6	:	:	PUNCT
ejpam-4301	22	7	hibabaanoon@uomisan.edu.iq	hibabaanoon@uomisan.edu.iq	NOUN
ejpam-4301	22	8	(	(	PUNCT
ejpam-4301	22	9	h.r	h.r	PROPN
ejpam-4301	22	10	.	.	PROPN
ejpam-4301	22	11	baanoon	baanoon	PROPN
ejpam-4301	22	12	)	)	PUNCT
ejpam-4301	22	13	,	,	PUNCT
ejpam-4301	22	14	wasan.hasan@sc.uobaghdad.edu.iq	wasan.hasan@sc.uobaghdad.edu.iq	NOUN
ejpam-4301	22	15	(	(	PUNCT
ejpam-4301	22	16	w.	w.	PROPN
ejpam-4301	22	17	khalid	khalid	PROPN
ejpam-4301	22	18	)	)	PUNCT
ejpam-4301	22	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4301	23	1	478	478	NUM
ejpam-4301	24	1	©	©	PROPN
ejpam-4301	24	2	2022	2022	NUM
ejpam-4301	24	3	ejpam	ejpam	VERB
ejpam-4301	24	4	all	all	DET
ejpam-4301	24	5	rights	right	NOUN
ejpam-4301	24	6	reserved	reserve	VERB
ejpam-4301	24	7	.	.	PUNCT
ejpam-4301	25	1	h.r	h.r	PROPN
ejpam-4301	25	2	.	.	PROPN
ejpam-4301	25	3	baanoon	baanoon	PROPN
ejpam-4301	25	4	,	,	PUNCT
ejpam-4301	25	5	w.	w.	PROPN
ejpam-4301	25	6	khalid	khalid	PROPN
ejpam-4301	25	7	/	/	PUNCT
ejpam-4301	25	8	eur	eur	PROPN
ejpam-4301	25	9	.	.	PUNCT
ejpam-4301	26	1	j.	j.	PROPN
ejpam-4301	26	2	pure	pure	PROPN
ejpam-4301	26	3	appl	appl	PROPN
ejpam-4301	26	4	.	.	PROPN
ejpam-4301	26	5	math	math	PROPN
ejpam-4301	26	6	,	,	PUNCT
ejpam-4301	26	7	15	15	NUM
ejpam-4301	26	8	(	(	PUNCT
ejpam-4301	26	9	2	2	NUM
ejpam-4301	26	10	)	)	PUNCT
ejpam-4301	26	11	(	(	PUNCT
ejpam-4301	26	12	2022	2022	NUM
ejpam-4301	26	13	)	)	PUNCT
ejpam-4301	26	14	,	,	PUNCT
ejpam-4301	26	15	478	478	NUM
ejpam-4301	26	16	-	-	SYM
ejpam-4301	26	17	485	485	NUM
ejpam-4301	26	18	479	479	NUM
ejpam-4301	26	19	2	2	NUM
ejpam-4301	26	20	.	.	PUNCT
ejpam-4301	27	1	e∗-essential	e∗-essential	ADJ
ejpam-4301	27	2	small	small	ADJ
ejpam-4301	27	3	submodules	submodule	NOUN
ejpam-4301	27	4	in	in	ADP
ejpam-4301	27	5	this	this	DET
ejpam-4301	27	6	section	section	NOUN
ejpam-4301	27	7	,	,	PUNCT
ejpam-4301	27	8	so	so	ADV
ejpam-4301	27	9	one	one	NUM
ejpam-4301	27	10	generalization	generalization	NOUN
ejpam-4301	27	11	of	of	ADP
ejpam-4301	27	12	small	small	ADJ
ejpam-4301	27	13	submodules	submodule	NOUN
ejpam-4301	27	14	are	be	AUX
ejpam-4301	27	15	introduced	introduce	VERB
ejpam-4301	27	16	with	with	ADP
ejpam-4301	27	17	some	some	DET
ejpam-4301	27	18	properties	property	NOUN
ejpam-4301	27	19	.	.	PUNCT
ejpam-4301	28	1	recall	recall	VERB
ejpam-4301	28	2	that	that	SCONJ
ejpam-4301	28	3	a	a	DET
ejpam-4301	28	4	submodule	submodule	NOUN
ejpam-4301	28	5	a	a	PRON
ejpam-4301	28	6	of	of	ADP
ejpam-4301	28	7	m	m	PROPN
ejpam-4301	28	8	is	be	AUX
ejpam-4301	28	9	said	say	VERB
ejpam-4301	28	10	to	to	PART
ejpam-4301	28	11	be	be	AUX
ejpam-4301	28	12	e∗-essential	e∗-essential	PROPN
ejpam-4301	28	13	denoted	denote	VERB
ejpam-4301	28	14	by	by	ADP
ejpam-4301	28	15	a	a	DET
ejpam-4301	28	16	≤e∗	≤e∗	PROPN
ejpam-4301	28	17	m	m	NOUN
ejpam-4301	28	18	if	if	SCONJ
ejpam-4301	28	19	a	a	DET
ejpam-4301	28	20	∩b	∩b	NOUN
ejpam-4301	28	21	6=	6=	ADP
ejpam-4301	28	22	0	0	NUM
ejpam-4301	28	23	for	for	ADP
ejpam-4301	28	24	each	each	DET
ejpam-4301	28	25	nonzero	nonzero	NOUN
ejpam-4301	28	26	cosingular	cosingular	PROPN
ejpam-4301	28	27	submodule	submodule	PROPN
ejpam-4301	28	28	b	b	PROPN
ejpam-4301	28	29	of	of	ADP
ejpam-4301	28	30	m	m	PROPN
ejpam-4301	29	1	[	[	X
ejpam-4301	29	2	1	1	NUM
ejpam-4301	29	3	]	]	PUNCT
ejpam-4301	29	4	.	.	PUNCT
ejpam-4301	30	1	the	the	DET
ejpam-4301	30	2	following	follow	VERB
ejpam-4301	30	3	gives	give	VERB
ejpam-4301	30	4	some	some	DET
ejpam-4301	30	5	properties	property	NOUN
ejpam-4301	30	6	of	of	ADP
ejpam-4301	30	7	e∗-essential	e∗-essential	ADJ
ejpam-4301	30	8	submodules	submodule	NOUN
ejpam-4301	30	9	.	.	PUNCT
ejpam-4301	31	1	lemma	lemma	PROPN
ejpam-4301	31	2	1	1	NUM
ejpam-4301	31	3	.	.	PUNCT
ejpam-4301	32	1	[	[	X
ejpam-4301	32	2	1	1	X
ejpam-4301	32	3	]	]	PUNCT
ejpam-4301	32	4	let	let	VERB
ejpam-4301	32	5	m	m	PRON
ejpam-4301	32	6	be	be	AUX
ejpam-4301	32	7	an	an	DET
ejpam-4301	32	8	r	r	NOUN
ejpam-4301	32	9	-	-	PUNCT
ejpam-4301	32	10	module	module	NOUN
ejpam-4301	32	11	.	.	PUNCT
ejpam-4301	33	1	1	1	X
ejpam-4301	33	2	.	.	X
ejpam-4301	34	1	if	if	SCONJ
ejpam-4301	34	2	a	a	DET
ejpam-4301	34	3	≤	≤	NUM
ejpam-4301	34	4	b	b	NOUN
ejpam-4301	34	5	≤	≤	NUM
ejpam-4301	34	6	m	m	PROPN
ejpam-4301	34	7	,	,	PUNCT
ejpam-4301	34	8	then	then	ADV
ejpam-4301	34	9	a	a	DET
ejpam-4301	34	10	≤e∗	≤e∗	PROPN
ejpam-4301	34	11	m	m	VERB
ejpam-4301	34	12	if	if	SCONJ
ejpam-4301	34	13	and	and	CCONJ
ejpam-4301	34	14	only	only	ADV
ejpam-4301	34	15	if	if	SCONJ
ejpam-4301	34	16	a	a	DET
ejpam-4301	34	17	≤e∗	≤e∗	PROPN
ejpam-4301	34	18	b	b	PROPN
ejpam-4301	34	19	≤e∗	≤e∗	PROPN
ejpam-4301	34	20	m	m	NOUN
ejpam-4301	34	21	2	2	NUM
ejpam-4301	34	22	.	.	PUNCT
ejpam-4301	35	1	let	let	VERB
ejpam-4301	35	2	f	f	NOUN
ejpam-4301	35	3	:	:	PUNCT
ejpam-4301	35	4	m	m	VERB
ejpam-4301	35	5	→	→	SYM
ejpam-4301	35	6	m	m	AUX
ejpam-4301	35	7	′	′	NUM
ejpam-4301	35	8	be	be	VERB
ejpam-4301	35	9	an	an	DET
ejpam-4301	35	10	r	r	NOUN
ejpam-4301	35	11	-	-	PUNCT
ejpam-4301	35	12	homomorphism	homomorphism	NOUN
ejpam-4301	35	13	.	.	PUNCT
ejpam-4301	36	1	if	if	SCONJ
ejpam-4301	36	2	a	a	DET
ejpam-4301	36	3	≤e∗	≤e∗	PROPN
ejpam-4301	36	4	m	m	NOUN
ejpam-4301	36	5	′	′	NOUN
ejpam-4301	36	6	,	,	PUNCT
ejpam-4301	36	7	then	then	ADV
ejpam-4301	36	8	f−1(a	f−1(a	PROPN
ejpam-4301	36	9	)	)	PUNCT
ejpam-4301	36	10	≤e∗	≤e∗	PROPN
ejpam-4301	36	11	m	m	NOUN
ejpam-4301	36	12	.	.	PUNCT
ejpam-4301	37	1	3	3	X
ejpam-4301	37	2	.	.	X
ejpam-4301	37	3	if	if	SCONJ
ejpam-4301	37	4	a	a	DET
ejpam-4301	37	5	≤e∗	≤e∗	PROPN
ejpam-4301	37	6	b	b	PROPN
ejpam-4301	37	7	≤	≤	NUM
ejpam-4301	37	8	m	m	PROPN
ejpam-4301	37	9	and	and	CCONJ
ejpam-4301	37	10	a	a	DET
ejpam-4301	37	11	′	′	NUM
ejpam-4301	37	12	≤e∗	≤e∗	PROPN
ejpam-4301	37	13	b	b	NOUN
ejpam-4301	37	14	′	′	NOUN
ejpam-4301	37	15	≤	≤	NUM
ejpam-4301	37	16	m	m	VERB
ejpam-4301	37	17	,	,	PUNCT
ejpam-4301	37	18	then	then	ADV
ejpam-4301	37	19	a	a	DET
ejpam-4301	37	20	∩a	∩a	PROPN
ejpam-4301	37	21	′	′	NUM
ejpam-4301	37	22	≤e∗	≤e∗	PROPN
ejpam-4301	38	1	b	b	NOUN
ejpam-4301	38	2	∩b	∩b	NOUN
ejpam-4301	38	3	′.	′.	NOUN
ejpam-4301	38	4	definition	definition	NOUN
ejpam-4301	38	5	1	1	X
ejpam-4301	38	6	.	.	PUNCT
ejpam-4301	39	1	let	let	VERB
ejpam-4301	39	2	m	m	PRON
ejpam-4301	39	3	be	be	AUX
ejpam-4301	39	4	an	an	DET
ejpam-4301	39	5	r	r	NOUN
ejpam-4301	39	6	-	-	PUNCT
ejpam-4301	39	7	module	module	NOUN
ejpam-4301	39	8	,	,	PUNCT
ejpam-4301	39	9	a	a	DET
ejpam-4301	39	10	submodule	submodule	NOUN
ejpam-4301	39	11	a	a	PRON
ejpam-4301	39	12	of	of	ADP
ejpam-4301	39	13	m	m	PROPN
ejpam-4301	39	14	is	be	AUX
ejpam-4301	39	15	said	say	VERB
ejpam-4301	39	16	to	to	PART
ejpam-4301	39	17	be	be	AUX
ejpam-4301	39	18	e∗-essential	e∗-essential	PROPN
ejpam-4301	39	19	small	small	ADJ
ejpam-4301	39	20	in	in	SCONJ
ejpam-4301	39	21	m	m	AUX
ejpam-4301	39	22	denoted	denote	VERB
ejpam-4301	39	23	by	by	ADP
ejpam-4301	39	24	a	a	DET
ejpam-4301	39	25	�	�	PROPN
ejpam-4301	39	26	e∗	e∗	NOUN
ejpam-4301	39	27	m	m	INTJ
ejpam-4301	39	28	,	,	PUNCT
ejpam-4301	39	29	if	if	SCONJ
ejpam-4301	39	30	whenever	whenever	SCONJ
ejpam-4301	39	31	m	m	VERB
ejpam-4301	39	32	=	=	PUNCT
ejpam-4301	39	33	a	a	DET
ejpam-4301	39	34	+	+	X
ejpam-4301	39	35	b	b	X
ejpam-4301	39	36	(	(	PUNCT
ejpam-4301	39	37	where	where	SCONJ
ejpam-4301	39	38	b	b	NOUN
ejpam-4301	39	39	is	be	AUX
ejpam-4301	39	40	an	an	DET
ejpam-4301	39	41	e∗-essential	e∗-essential	PROPN
ejpam-4301	39	42	submodule	submodule	NOUN
ejpam-4301	39	43	of	of	ADP
ejpam-4301	39	44	m	m	PROPN
ejpam-4301	39	45	)	)	PUNCT
ejpam-4301	39	46	implies	imply	VERB
ejpam-4301	39	47	that	that	SCONJ
ejpam-4301	39	48	m	m	PROPN
ejpam-4301	39	49	=	=	SYM
ejpam-4301	39	50	b.	b.	PROPN
ejpam-4301	39	51	examples	example	NOUN
ejpam-4301	39	52	and	and	CCONJ
ejpam-4301	39	53	remarks	remark	VERB
ejpam-4301	39	54	1	1	NUM
ejpam-4301	39	55	.	.	NOUN
ejpam-4301	39	56	1	1	NUM
ejpam-4301	39	57	.	.	X
ejpam-4301	40	1	every	every	DET
ejpam-4301	40	2	small	small	ADJ
ejpam-4301	40	3	submodule	submodule	NOUN
ejpam-4301	40	4	is	be	AUX
ejpam-4301	40	5	e∗-essential	e∗-essential	ADJ
ejpam-4301	40	6	small	small	ADJ
ejpam-4301	40	7	submodule	submodule	NOUN
ejpam-4301	40	8	,	,	PUNCT
ejpam-4301	40	9	but	but	CCONJ
ejpam-4301	40	10	the	the	DET
ejpam-4301	40	11	converse	converse	NOUN
ejpam-4301	40	12	need	need	VERB
ejpam-4301	40	13	not	not	PART
ejpam-4301	40	14	to	to	PART
ejpam-4301	40	15	be	be	AUX
ejpam-4301	40	16	true	true	ADJ
ejpam-4301	40	17	in	in	ADP
ejpam-4301	40	18	general	general	ADJ
ejpam-4301	40	19	.	.	PUNCT
ejpam-4301	41	1	for	for	ADP
ejpam-4301	41	2	example	example	NOUN
ejpam-4301	41	3	,	,	PUNCT
ejpam-4301	41	4	in	in	ADP
ejpam-4301	41	5	z6	z6	PROPN
ejpam-4301	41	6	as	as	ADP
ejpam-4301	41	7	a	a	DET
ejpam-4301	41	8	z	z	NOUN
ejpam-4301	41	9	-	-	PUNCT
ejpam-4301	41	10	module	module	NOUN
ejpam-4301	41	11	,	,	PUNCT
ejpam-4301	41	12	the	the	DET
ejpam-4301	41	13	only	only	ADJ
ejpam-4301	41	14	e∗-essential	e∗-essential	PROPN
ejpam-4301	41	15	submodule	submodule	NOUN
ejpam-4301	41	16	is	be	AUX
ejpam-4301	41	17	z6	z6	PROPN
ejpam-4301	42	1	[	[	X
ejpam-4301	42	2	1	1	NUM
ejpam-4301	42	3	]	]	PUNCT
ejpam-4301	42	4	.	.	PUNCT
ejpam-4301	43	1	so	so	ADV
ejpam-4301	43	2	,	,	PUNCT
ejpam-4301	43	3	every	every	DET
ejpam-4301	43	4	submodule	submodule	NOUN
ejpam-4301	43	5	of	of	ADP
ejpam-4301	43	6	z6	z6	PROPN
ejpam-4301	43	7	is	be	AUX
ejpam-4301	43	8	e∗-essential	e∗-essential	PROPN
ejpam-4301	43	9	small	small	ADJ
ejpam-4301	43	10	.	.	PUNCT
ejpam-4301	44	1	while	while	SCONJ
ejpam-4301	44	2	〈	〈	PROPN
ejpam-4301	44	3	2	2	NUM
ejpam-4301	44	4	〉	〉	PROPN
ejpam-4301	44	5	is	be	AUX
ejpam-4301	44	6	not	not	PART
ejpam-4301	44	7	a	a	DET
ejpam-4301	44	8	small	small	ADJ
ejpam-4301	44	9	submodule	submodule	NOUN
ejpam-4301	44	10	since	since	SCONJ
ejpam-4301	44	11	〈	〈	PROPN
ejpam-4301	44	12	2〉+	2〉+	NUM
ejpam-4301	44	13	〈	〈	PROPN
ejpam-4301	44	14	3	3	NUM
ejpam-4301	44	15	〉	〉	NOUN
ejpam-4301	44	16	=	=	SYM
ejpam-4301	44	17	z6	z6	PROPN
ejpam-4301	44	18	but	but	CCONJ
ejpam-4301	44	19	〈	〈	PROPN
ejpam-4301	44	20	3	3	NUM
ejpam-4301	44	21	〉	〉	PROPN
ejpam-4301	44	22	6=	6=	PROPN
ejpam-4301	44	23	z6	z6	PROPN
ejpam-4301	44	24	.	.	PUNCT
ejpam-4301	45	1	2	2	X
ejpam-4301	45	2	.	.	X
ejpam-4301	45	3	consider	consider	VERB
ejpam-4301	45	4	z4	z4	PROPN
ejpam-4301	45	5	as	as	ADP
ejpam-4301	45	6	a	a	DET
ejpam-4301	45	7	z	z	NOUN
ejpam-4301	45	8	-	-	PUNCT
ejpam-4301	45	9	module	module	NOUN
ejpam-4301	45	10	,	,	PUNCT
ejpam-4301	45	11	the	the	DET
ejpam-4301	45	12	submodles	submodle	NOUN
ejpam-4301	45	13	z4	z4	PROPN
ejpam-4301	45	14	and	and	CCONJ
ejpam-4301	45	15	〈	〈	PROPN
ejpam-4301	45	16	2	2	NUM
ejpam-4301	45	17	〉	〉	NOUN
ejpam-4301	45	18	are	be	AUX
ejpam-4301	45	19	cosingular[2	cosingular[2	ADP
ejpam-4301	45	20	]	]	X
ejpam-4301	45	21	and	and	CCONJ
ejpam-4301	45	22	e∗essential	e∗essential	ADJ
ejpam-4301	45	23	,	,	PUNCT
ejpam-4301	45	24	hence	hence	ADV
ejpam-4301	45	25	〈	〈	PROPN
ejpam-4301	45	26	2	2	NUM
ejpam-4301	45	27	〉	〉	PROPN
ejpam-4301	45	28	is	be	AUX
ejpam-4301	45	29	an	an	DET
ejpam-4301	45	30	e∗-essential	e∗-essential	ADJ
ejpam-4301	45	31	small	small	ADJ
ejpam-4301	45	32	submodule	submodule	NOUN
ejpam-4301	45	33	.	.	PUNCT
ejpam-4301	46	1	3	3	X
ejpam-4301	46	2	.	.	X
ejpam-4301	46	3	consider	consider	VERB
ejpam-4301	46	4	z6	z6	PROPN
ejpam-4301	46	5	as	as	ADP
ejpam-4301	46	6	a	a	DET
ejpam-4301	46	7	z6	z6	NOUN
ejpam-4301	46	8	-	-	PUNCT
ejpam-4301	46	9	module	module	NOUN
ejpam-4301	46	10	.	.	PUNCT
ejpam-4301	47	1	in	in	ADP
ejpam-4301	47	2	this	this	DET
ejpam-4301	47	3	module	module	NOUN
ejpam-4301	47	4	every	every	DET
ejpam-4301	47	5	submodule	submodule	NOUN
ejpam-4301	47	6	is	be	AUX
ejpam-4301	47	7	e∗-essential	e∗-essential	PROPN
ejpam-4301	47	8	[	[	PUNCT
ejpam-4301	47	9	1	1	NUM
ejpam-4301	47	10	]	]	PUNCT
ejpam-4301	47	11	,	,	PUNCT
ejpam-4301	47	12	so	so	ADV
ejpam-4301	47	13	〈	〈	PROPN
ejpam-4301	47	14	2	2	NUM
ejpam-4301	47	15	〉	〉	NOUN
ejpam-4301	47	16	+	+	CCONJ
ejpam-4301	47	17	〈	〈	PROPN
ejpam-4301	47	18	3	3	NUM
ejpam-4301	47	19	〉	〉	NOUN
ejpam-4301	47	20	=	=	SYM
ejpam-4301	47	21	z6	z6	PROPN
ejpam-4301	47	22	but	but	CCONJ
ejpam-4301	47	23	〈	〈	PROPN
ejpam-4301	47	24	3	3	NUM
ejpam-4301	47	25	〉	〉	PROPN
ejpam-4301	47	26	6=	6=	NUM
ejpam-4301	47	27	z6	z6	PROPN
ejpam-4301	47	28	.	.	PUNCT
ejpam-4301	48	1	therefore	therefore	ADV
ejpam-4301	48	2	,	,	PUNCT
ejpam-4301	48	3	〈	〈	PROPN
ejpam-4301	48	4	2	2	NUM
ejpam-4301	48	5	〉	〉	NOUN
ejpam-4301	48	6	is	be	AUX
ejpam-4301	48	7	not	not	PART
ejpam-4301	48	8	e∗-essential	e∗-essential	ADJ
ejpam-4301	48	9	small	small	ADJ
ejpam-4301	48	10	submodule	submodule	NOUN
ejpam-4301	48	11	.	.	PUNCT
ejpam-4301	49	1	thus	thus	ADV
ejpam-4301	49	2	,	,	PUNCT
ejpam-4301	49	3	e∗-essential	e∗-essential	PROPN
ejpam-4301	49	4	submodule	submodule	NOUN
ejpam-4301	49	5	need	need	AUX
ejpam-4301	49	6	not	not	PART
ejpam-4301	49	7	to	to	PART
ejpam-4301	49	8	be	be	AUX
ejpam-4301	49	9	e∗-essential	e∗-essential	PROPN
ejpam-4301	49	10	small	small	ADJ
ejpam-4301	49	11	.	.	PUNCT
ejpam-4301	50	1	4	4	X
ejpam-4301	50	2	.	.	X
ejpam-4301	50	3	let	let	VERB
ejpam-4301	50	4	m	m	PRON
ejpam-4301	50	5	be	be	AUX
ejpam-4301	50	6	an	an	DET
ejpam-4301	50	7	r	r	NOUN
ejpam-4301	50	8	-	-	PUNCT
ejpam-4301	50	9	module	module	NOUN
ejpam-4301	50	10	,	,	PUNCT
ejpam-4301	50	11	then	then	ADV
ejpam-4301	50	12	:	:	PUNCT
ejpam-4301	50	13	•	•	SCONJ
ejpam-4301	50	14	the	the	DET
ejpam-4301	50	15	trivial	trivial	ADJ
ejpam-4301	50	16	submodule	submodule	NOUN
ejpam-4301	50	17	is	be	AUX
ejpam-4301	50	18	always	always	ADV
ejpam-4301	50	19	e∗-essential	e∗-essential	ADJ
ejpam-4301	50	20	small	small	ADJ
ejpam-4301	50	21	in	in	ADP
ejpam-4301	50	22	m	m	PROPN
ejpam-4301	50	23	.	.	PUNCT
ejpam-4301	51	1	•	•	NUM
ejpam-4301	51	2	m	m	PROPN
ejpam-4301	51	3	�	�	PROPN
ejpam-4301	51	4	e∗	e∗	PROPN
ejpam-4301	51	5	m	m	NOUN
ejpam-4301	51	6	if	if	SCONJ
ejpam-4301	51	7	and	and	CCONJ
ejpam-4301	51	8	only	only	ADV
ejpam-4301	51	9	if	if	SCONJ
ejpam-4301	51	10	m	m	NOUN
ejpam-4301	51	11	is	be	AUX
ejpam-4301	51	12	a	a	DET
ejpam-4301	51	13	simple	simple	ADJ
ejpam-4301	51	14	module	module	NOUN
ejpam-4301	51	15	.	.	PUNCT
ejpam-4301	52	1	in	in	ADP
ejpam-4301	52	2	the	the	DET
ejpam-4301	52	3	following	following	NOUN
ejpam-4301	52	4	,	,	PUNCT
ejpam-4301	52	5	we	we	PRON
ejpam-4301	52	6	introduce	introduce	VERB
ejpam-4301	52	7	the	the	DET
ejpam-4301	52	8	basic	basic	ADJ
ejpam-4301	52	9	properties	property	NOUN
ejpam-4301	52	10	of	of	ADP
ejpam-4301	52	11	e∗-essential	e∗-essential	ADJ
ejpam-4301	52	12	small	small	ADJ
ejpam-4301	52	13	submodules	submodule	NOUN
ejpam-4301	52	14	.	.	PUNCT
ejpam-4301	53	1	proposition	proposition	NOUN
ejpam-4301	53	2	1	1	NUM
ejpam-4301	53	3	.	.	PUNCT
ejpam-4301	54	1	let	let	VERB
ejpam-4301	54	2	m	m	PRON
ejpam-4301	54	3	be	be	AUX
ejpam-4301	54	4	an	an	DET
ejpam-4301	54	5	r	r	NOUN
ejpam-4301	54	6	-	-	PUNCT
ejpam-4301	54	7	module	module	NOUN
ejpam-4301	54	8	,	,	PUNCT
ejpam-4301	54	9	n	n	CCONJ
ejpam-4301	54	10	a	a	DET
ejpam-4301	54	11	submodule	submodule	NOUN
ejpam-4301	54	12	of	of	ADP
ejpam-4301	54	13	m	m	PROPN
ejpam-4301	54	14	and	and	CCONJ
ejpam-4301	54	15	k	k	PROPN
ejpam-4301	54	16	a	a	DET
ejpam-4301	54	17	submodule	submodule	NOUN
ejpam-4301	54	18	of	of	ADP
ejpam-4301	54	19	n	n	PROPN
ejpam-4301	54	20	.	.	PUNCT
ejpam-4301	55	1	1	1	X
ejpam-4301	55	2	.	.	X
ejpam-4301	56	1	if	if	SCONJ
ejpam-4301	56	2	n	n	PROPN
ejpam-4301	56	3	�	�	PROPN
ejpam-4301	56	4	e∗	e∗	PROPN
ejpam-4301	56	5	m	m	PROPN
ejpam-4301	56	6	,	,	PUNCT
ejpam-4301	56	7	then	then	ADV
ejpam-4301	56	8	k	k	PROPN
ejpam-4301	56	9	�	�	PROPN
ejpam-4301	56	10	e∗	e∗	PROPN
ejpam-4301	56	11	m	m	PROPN
ejpam-4301	56	12	and	and	CCONJ
ejpam-4301	56	13	n	n	CCONJ
ejpam-4301	56	14	k	k	PROPN
ejpam-4301	56	15	�	�	PROPN
ejpam-4301	56	16	e∗	e∗	PROPN
ejpam-4301	56	17	m	m	PROPN
ejpam-4301	56	18	k	k	PROPN
ejpam-4301	56	19	.	.	PUNCT
ejpam-4301	57	1	2	2	X
ejpam-4301	57	2	.	.	X
ejpam-4301	57	3	if	if	SCONJ
ejpam-4301	57	4	k	k	PROPN
ejpam-4301	57	5	�	�	PROPN
ejpam-4301	57	6	e∗	e∗	PROPN
ejpam-4301	57	7	n	n	CCONJ
ejpam-4301	57	8	,	,	PUNCT
ejpam-4301	57	9	then	then	ADV
ejpam-4301	57	10	k	k	PROPN
ejpam-4301	57	11	�	�	PROPN
ejpam-4301	57	12	e∗	e∗	PROPN
ejpam-4301	57	13	m	m	PROPN
ejpam-4301	57	14	.	.	PUNCT
ejpam-4301	58	1	proof	proof	NOUN
ejpam-4301	58	2	.	.	PUNCT
ejpam-4301	59	1	h.r	h.r	PROPN
ejpam-4301	59	2	.	.	PROPN
ejpam-4301	59	3	baanoon	baanoon	PROPN
ejpam-4301	59	4	,	,	PUNCT
ejpam-4301	59	5	w.	w.	PROPN
ejpam-4301	59	6	khalid	khalid	PROPN
ejpam-4301	59	7	/	/	PUNCT
ejpam-4301	59	8	eur	eur	PROPN
ejpam-4301	59	9	.	.	PUNCT
ejpam-4301	60	1	j.	j.	PROPN
ejpam-4301	60	2	pure	pure	PROPN
ejpam-4301	60	3	appl	appl	PROPN
ejpam-4301	60	4	.	.	PROPN
ejpam-4301	60	5	math	math	PROPN
ejpam-4301	60	6	,	,	PUNCT
ejpam-4301	60	7	15	15	NUM
ejpam-4301	60	8	(	(	PUNCT
ejpam-4301	60	9	2	2	NUM
ejpam-4301	60	10	)	)	PUNCT
ejpam-4301	60	11	(	(	PUNCT
ejpam-4301	60	12	2022	2022	NUM
ejpam-4301	60	13	)	)	PUNCT
ejpam-4301	60	14	,	,	PUNCT
ejpam-4301	60	15	478	478	NUM
ejpam-4301	60	16	-	-	SYM
ejpam-4301	60	17	485	485	NUM
ejpam-4301	60	18	480	480	NUM
ejpam-4301	60	19	1	1	NUM
ejpam-4301	60	20	.	.	PUNCT
ejpam-4301	61	1	let	let	VERB
ejpam-4301	61	2	l	l	NOUN
ejpam-4301	61	3	be	be	AUX
ejpam-4301	61	4	an	an	DET
ejpam-4301	61	5	e∗-essential	e∗-essential	PROPN
ejpam-4301	61	6	submodule	submodule	NOUN
ejpam-4301	61	7	of	of	ADP
ejpam-4301	61	8	m	m	PRON
ejpam-4301	61	9	such	such	ADJ
ejpam-4301	61	10	that	that	SCONJ
ejpam-4301	61	11	k	k	PROPN
ejpam-4301	61	12	+	+	NOUN
ejpam-4301	61	13	l	l	NOUN
ejpam-4301	61	14	=	=	VERB
ejpam-4301	61	15	m	m	VERB
ejpam-4301	61	16	.	.	PUNCT
ejpam-4301	62	1	since	since	SCONJ
ejpam-4301	62	2	k	k	PROPN
ejpam-4301	62	3	≤	≤	PROPN
ejpam-4301	62	4	n	n	CCONJ
ejpam-4301	62	5	and	and	CCONJ
ejpam-4301	62	6	n	n	PRON
ejpam-4301	62	7	�	�	PROPN
ejpam-4301	62	8	e∗	e∗	PROPN
ejpam-4301	62	9	m	m	PROPN
ejpam-4301	62	10	,	,	PUNCT
ejpam-4301	62	11	l	l	PROPN
ejpam-4301	62	12	=	=	PUNCT
ejpam-4301	62	13	m	m	NOUN
ejpam-4301	62	14	.	.	PUNCT
ejpam-4301	63	1	thus	thus	ADV
ejpam-4301	63	2	k	k	PROPN
ejpam-4301	63	3	�	�	PROPN
ejpam-4301	63	4	e∗	e∗	PROPN
ejpam-4301	63	5	m	m	PROPN
ejpam-4301	63	6	.	.	PUNCT
ejpam-4301	64	1	now	now	ADV
ejpam-4301	64	2	,	,	PUNCT
ejpam-4301	64	3	to	to	PART
ejpam-4301	64	4	prove	prove	VERB
ejpam-4301	64	5	that	that	SCONJ
ejpam-4301	64	6	n	n	PROPN
ejpam-4301	64	7	k	k	PROPN
ejpam-4301	64	8	�	�	PROPN
ejpam-4301	64	9	e∗	e∗	PROPN
ejpam-4301	64	10	m	m	PROPN
ejpam-4301	64	11	k	k	NOUN
ejpam-4301	64	12	,	,	PUNCT
ejpam-4301	64	13	let	let	VERB
ejpam-4301	64	14	m	m	VERB
ejpam-4301	64	15	k	k	VERB
ejpam-4301	64	16	=	=	PUNCT
ejpam-4301	64	17	a	a	DET
ejpam-4301	64	18	k	k	PROPN
ejpam-4301	65	1	+	+	CCONJ
ejpam-4301	65	2	n	n	CCONJ
ejpam-4301	65	3	k	k	NOUN
ejpam-4301	65	4	where	where	SCONJ
ejpam-4301	65	5	a	a	DET
ejpam-4301	65	6	k	k	PROPN
ejpam-4301	65	7	is	be	AUX
ejpam-4301	65	8	an	an	DET
ejpam-4301	65	9	e∗-essential	e∗-essential	PROPN
ejpam-4301	65	10	submodule	submodule	NOUN
ejpam-4301	65	11	of	of	ADP
ejpam-4301	65	12	m	m	PROPN
ejpam-4301	65	13	k	k	NOUN
ejpam-4301	65	14	,	,	PUNCT
ejpam-4301	65	15	hence	hence	ADV
ejpam-4301	65	16	a	a	PRON
ejpam-4301	65	17	is	be	AUX
ejpam-4301	65	18	e∗-essential	e∗-essential	PROPN
ejpam-4301	65	19	submodule	submodule	NOUN
ejpam-4301	65	20	of	of	ADP
ejpam-4301	65	21	m	m	PRON
ejpam-4301	65	22	by	by	ADP
ejpam-4301	65	23	lemma	lemma	PROPN
ejpam-4301	65	24	1	1	NUM
ejpam-4301	65	25	,	,	PUNCT
ejpam-4301	65	26	and	and	CCONJ
ejpam-4301	65	27	m	m	PROPN
ejpam-4301	65	28	=	=	ADJ
ejpam-4301	65	29	a+n	a+n	NOUN
ejpam-4301	65	30	,	,	PUNCT
ejpam-4301	65	31	since	since	SCONJ
ejpam-4301	65	32	n	n	PROPN
ejpam-4301	65	33	�	�	PROPN
ejpam-4301	65	34	e∗	e∗	PROPN
ejpam-4301	65	35	m	m	PROPN
ejpam-4301	65	36	.	.	PUNCT
ejpam-4301	66	1	thus	thus	ADV
ejpam-4301	66	2	,	,	PUNCT
ejpam-4301	66	3	m	m	VERB
ejpam-4301	66	4	=	=	NOUN
ejpam-4301	66	5	a	a	PRON
ejpam-4301	66	6	implies	imply	VERB
ejpam-4301	66	7	that	that	SCONJ
ejpam-4301	66	8	a	a	DET
ejpam-4301	66	9	k	k	X
ejpam-4301	66	10	=	=	PUNCT
ejpam-4301	66	11	m	m	PROPN
ejpam-4301	66	12	k	k	X
ejpam-4301	66	13	.	.	PUNCT
ejpam-4301	67	1	therefore	therefore	ADV
ejpam-4301	67	2	,	,	PUNCT
ejpam-4301	67	3	n	n	PROPN
ejpam-4301	67	4	k	k	PROPN
ejpam-4301	67	5	�	�	PROPN
ejpam-4301	67	6	e∗	e∗	PROPN
ejpam-4301	67	7	m	m	PROPN
ejpam-4301	67	8	k	k	PROPN
ejpam-4301	67	9	.	.	PUNCT
ejpam-4301	68	1	2	2	X
ejpam-4301	68	2	.	.	X
ejpam-4301	68	3	let	let	VERB
ejpam-4301	68	4	l	l	NOUN
ejpam-4301	68	5	be	be	AUX
ejpam-4301	68	6	an	an	DET
ejpam-4301	68	7	e∗-essential	e∗-essential	PROPN
ejpam-4301	68	8	submodule	submodule	NOUN
ejpam-4301	68	9	of	of	ADP
ejpam-4301	68	10	m	m	PRON
ejpam-4301	68	11	such	such	ADJ
ejpam-4301	68	12	that	that	PRON
ejpam-4301	68	13	k+l	k+l	PROPN
ejpam-4301	68	14	=	=	X
ejpam-4301	68	15	m	m	NOUN
ejpam-4301	68	16	.	.	PUNCT
ejpam-4301	69	1	hence	hence	ADV
ejpam-4301	69	2	,	,	PUNCT
ejpam-4301	69	3	l∩n	l∩n	PROPN
ejpam-4301	69	4	≤e∗	≤e∗	PROPN
ejpam-4301	69	5	n	n	CCONJ
ejpam-4301	69	6	by	by	ADP
ejpam-4301	69	7	lemma	lemma	PROPN
ejpam-4301	69	8	1	1	NUM
ejpam-4301	69	9	,	,	PUNCT
ejpam-4301	69	10	and	and	CCONJ
ejpam-4301	69	11	k+(l∩n	k+(l∩n	PROPN
ejpam-4301	69	12	)	)	PUNCT
ejpam-4301	69	13	=	=	SYM
ejpam-4301	69	14	n∩(k+l	n∩(k+l	NOUN
ejpam-4301	69	15	)	)	PUNCT
ejpam-4301	69	16	=	=	SYM
ejpam-4301	69	17	n	n	CCONJ
ejpam-4301	69	18	,	,	PUNCT
ejpam-4301	69	19	since	since	SCONJ
ejpam-4301	69	20	k	k	PROPN
ejpam-4301	69	21	�	�	PROPN
ejpam-4301	69	22	e∗	e∗	PROPN
ejpam-4301	69	23	n	n	PROPN
ejpam-4301	69	24	.	.	PUNCT
ejpam-4301	70	1	thus	thus	ADV
ejpam-4301	70	2	,	,	PUNCT
ejpam-4301	70	3	l∩n	l∩n	PROPN
ejpam-4301	70	4	=	=	SYM
ejpam-4301	70	5	n	n	NOUN
ejpam-4301	70	6	,	,	PUNCT
ejpam-4301	70	7	n	n	CCONJ
ejpam-4301	70	8	≤	≤	NOUN
ejpam-4301	70	9	l.	l.	NOUN
ejpam-4301	71	1	so	so	ADV
ejpam-4301	71	2	k	k	PROPN
ejpam-4301	71	3	≤	≤	PROPN
ejpam-4301	71	4	l.	l.	PROPN
ejpam-4301	71	5	hence	hence	ADV
ejpam-4301	71	6	,	,	PUNCT
ejpam-4301	71	7	l	l	PROPN
ejpam-4301	71	8	=	=	PUNCT
ejpam-4301	71	9	k	k	PROPN
ejpam-4301	72	1	+	+	PUNCT
ejpam-4301	72	2	l	l	NOUN
ejpam-4301	72	3	=	=	VERB
ejpam-4301	72	4	m	m	PROPN
ejpam-4301	72	5	.	.	PUNCT
ejpam-4301	73	1	therefore	therefore	ADV
ejpam-4301	73	2	,	,	PUNCT
ejpam-4301	73	3	k	k	PROPN
ejpam-4301	73	4	�	�	PROPN
ejpam-4301	73	5	e∗	e∗	PROPN
ejpam-4301	73	6	m	m	PROPN
ejpam-4301	73	7	.	.	PUNCT
ejpam-4301	74	1	proposition	proposition	NOUN
ejpam-4301	74	2	2	2	NUM
ejpam-4301	74	3	.	.	PUNCT
ejpam-4301	75	1	let	let	VERB
ejpam-4301	75	2	m	m	PRON
ejpam-4301	75	3	be	be	AUX
ejpam-4301	75	4	an	an	DET
ejpam-4301	75	5	r	r	NOUN
ejpam-4301	75	6	-	-	PUNCT
ejpam-4301	75	7	module	module	NOUN
ejpam-4301	75	8	,	,	PUNCT
ejpam-4301	75	9	k	k	PROPN
ejpam-4301	75	10	and	and	CCONJ
ejpam-4301	75	11	n	n	NUM
ejpam-4301	75	12	submodules	submodule	NOUN
ejpam-4301	75	13	of	of	ADP
ejpam-4301	75	14	m	m	NOUN
ejpam-4301	75	15	such	such	ADJ
ejpam-4301	75	16	that	that	SCONJ
ejpam-4301	75	17	k	k	PROPN
ejpam-4301	75	18	≤	≤	PROPN
ejpam-4301	75	19	n	n	ADV
ejpam-4301	75	20	.	.	PUNCT
ejpam-4301	76	1	if	if	SCONJ
ejpam-4301	76	2	k	k	PROPN
ejpam-4301	76	3	�	�	PROPN
ejpam-4301	76	4	e∗	e∗	PROPN
ejpam-4301	76	5	m	m	PROPN
ejpam-4301	76	6	and	and	CCONJ
ejpam-4301	76	7	n	n	PROPN
ejpam-4301	76	8	is	be	AUX
ejpam-4301	76	9	a	a	DET
ejpam-4301	76	10	direct	direct	ADJ
ejpam-4301	76	11	summand	summand	NOUN
ejpam-4301	76	12	e∗-essential	e∗-essential	PROPN
ejpam-4301	76	13	submodule	submodule	NOUN
ejpam-4301	76	14	of	of	ADP
ejpam-4301	76	15	m	m	PROPN
ejpam-4301	76	16	,	,	PUNCT
ejpam-4301	76	17	then	then	ADV
ejpam-4301	76	18	k	k	PROPN
ejpam-4301	76	19	�	�	PROPN
ejpam-4301	76	20	e∗	e∗	PROPN
ejpam-4301	76	21	n	n	PART
ejpam-4301	76	22	.	.	PUNCT
ejpam-4301	77	1	proof	proof	NOUN
ejpam-4301	77	2	.	.	PUNCT
ejpam-4301	78	1	let	let	VERB
ejpam-4301	78	2	l	l	NOUN
ejpam-4301	78	3	be	be	AUX
ejpam-4301	78	4	an	an	DET
ejpam-4301	78	5	e∗-essential	e∗-essential	PROPN
ejpam-4301	78	6	submodule	submodule	NOUN
ejpam-4301	78	7	of	of	ADP
ejpam-4301	78	8	n	n	PRON
ejpam-4301	78	9	such	such	ADJ
ejpam-4301	78	10	that	that	SCONJ
ejpam-4301	78	11	k	k	PROPN
ejpam-4301	78	12	+	+	NUM
ejpam-4301	78	13	l	l	NOUN
ejpam-4301	78	14	=	=	SYM
ejpam-4301	78	15	n	n	PROPN
ejpam-4301	78	16	.	.	PUNCT
ejpam-4301	79	1	since	since	SCONJ
ejpam-4301	79	2	n	n	NUM
ejpam-4301	79	3	is	be	AUX
ejpam-4301	79	4	a	a	DET
ejpam-4301	79	5	direct	direct	ADJ
ejpam-4301	79	6	summand	summand	NOUN
ejpam-4301	79	7	of	of	ADP
ejpam-4301	79	8	m	m	PROPN
ejpam-4301	79	9	,	,	PUNCT
ejpam-4301	79	10	there	there	PRON
ejpam-4301	79	11	exists	exist	VERB
ejpam-4301	79	12	a	a	DET
ejpam-4301	79	13	submodule	submodule	NOUN
ejpam-4301	79	14	n	n	ADV
ejpam-4301	79	15	′	′	NOUN
ejpam-4301	79	16	of	of	ADP
ejpam-4301	79	17	m	m	PRON
ejpam-4301	79	18	such	such	ADJ
ejpam-4301	79	19	that	that	SCONJ
ejpam-4301	79	20	m	m	VERB
ejpam-4301	79	21	=	=	SYM
ejpam-4301	79	22	n	n	PROPN
ejpam-4301	79	23	⊕	⊕	PROPN
ejpam-4301	79	24	n	n	CCONJ
ejpam-4301	79	25	′	′	NOUN
ejpam-4301	79	26	and	and	CCONJ
ejpam-4301	79	27	m	m	PROPN
ejpam-4301	79	28	=	=	SYM
ejpam-4301	79	29	(	(	PUNCT
ejpam-4301	79	30	k	k	X
ejpam-4301	79	31	+	+	PROPN
ejpam-4301	79	32	l	l	NOUN
ejpam-4301	79	33	)	)	PUNCT
ejpam-4301	79	34	⊕	⊕	PROPN
ejpam-4301	79	35	n	n	NOUN
ejpam-4301	80	1	′	′	NOUN
ejpam-4301	81	1	=	=	PUNCT
ejpam-4301	81	2	k	k	PROPN
ejpam-4301	82	1	+	+	PUNCT
ejpam-4301	82	2	(	(	PUNCT
ejpam-4301	82	3	l	l	NOUN
ejpam-4301	82	4	+	+	CCONJ
ejpam-4301	82	5	n	n	NUM
ejpam-4301	82	6	′	′	NUM
ejpam-4301	82	7	)	)	PUNCT
ejpam-4301	82	8	.	.	PUNCT
ejpam-4301	83	1	since	since	SCONJ
ejpam-4301	83	2	l	l	PROPN
ejpam-4301	83	3	≤e∗	≤e∗	PROPN
ejpam-4301	83	4	n	n	CCONJ
ejpam-4301	83	5	≤e∗	≤e∗	NOUN
ejpam-4301	83	6	m	m	PROPN
ejpam-4301	83	7	,	,	PUNCT
ejpam-4301	83	8	by	by	ADP
ejpam-4301	83	9	lemma	lemma	PROPN
ejpam-4301	83	10	1	1	NUM
ejpam-4301	83	11	,	,	PUNCT
ejpam-4301	83	12	this	this	PRON
ejpam-4301	83	13	implies	imply	VERB
ejpam-4301	83	14	that	that	SCONJ
ejpam-4301	83	15	l	l	NOUN
ejpam-4301	83	16	≤e∗	≤e∗	NOUN
ejpam-4301	83	17	m	m	NOUN
ejpam-4301	83	18	and	and	CCONJ
ejpam-4301	83	19	since	since	SCONJ
ejpam-4301	83	20	l	l	NOUN
ejpam-4301	83	21	≤	≤	X
ejpam-4301	83	22	l	l	NOUN
ejpam-4301	84	1	+	+	CCONJ
ejpam-4301	84	2	n	n	CCONJ
ejpam-4301	84	3	′	′	NOUN
ejpam-4301	84	4	≤	≤	NOUN
ejpam-4301	84	5	m	m	VERB
ejpam-4301	84	6	also	also	ADV
ejpam-4301	84	7	by	by	ADP
ejpam-4301	84	8	the	the	DET
ejpam-4301	84	9	same	same	ADJ
ejpam-4301	84	10	lemma	lemma	PROPN
ejpam-4301	84	11	,	,	PUNCT
ejpam-4301	84	12	this	this	PRON
ejpam-4301	84	13	implies	imply	VERB
ejpam-4301	84	14	that	that	SCONJ
ejpam-4301	84	15	l	l	PROPN
ejpam-4301	85	1	+	+	CCONJ
ejpam-4301	85	2	n	n	CCONJ
ejpam-4301	85	3	′	′	NUM
ejpam-4301	85	4	leqe∗m	leqe∗m	NOUN
ejpam-4301	85	5	.	.	PUNCT
ejpam-4301	86	1	k	k	PROPN
ejpam-4301	86	2	�	�	PROPN
ejpam-4301	86	3	e∗	e∗	PROPN
ejpam-4301	86	4	n	n	PRON
ejpam-4301	86	5	implies	imply	VERB
ejpam-4301	86	6	that	that	SCONJ
ejpam-4301	86	7	l	l	PROPN
ejpam-4301	87	1	+	+	NOUN
ejpam-4301	87	2	n	n	CCONJ
ejpam-4301	87	3	′	′	NOUN
ejpam-4301	87	4	=	=	NOUN
ejpam-4301	87	5	m	m	VERB
ejpam-4301	87	6	.	.	PUNCT
ejpam-4301	88	1	now	now	ADV
ejpam-4301	88	2	,	,	PUNCT
ejpam-4301	88	3	for	for	ADP
ejpam-4301	88	4	any	any	DET
ejpam-4301	88	5	n	n	PRON
ejpam-4301	88	6	∈	∈	PROPN
ejpam-4301	88	7	n	n	NOUN
ejpam-4301	88	8	,	,	PUNCT
ejpam-4301	88	9	there	there	PRON
ejpam-4301	88	10	exists	exist	VERB
ejpam-4301	88	11	l	l	NOUN
ejpam-4301	88	12	∈	∈	PROPN
ejpam-4301	88	13	l	l	NOUN
ejpam-4301	88	14	and	and	CCONJ
ejpam-4301	88	15	n	n	NUM
ejpam-4301	88	16	′	′	NUM
ejpam-4301	88	17	∈	∈	PROPN
ejpam-4301	89	1	n	n	ADV
ejpam-4301	89	2	′	′	NUM
ejpam-4301	90	1	such	such	ADJ
ejpam-4301	90	2	that	that	SCONJ
ejpam-4301	90	3	n	n	NOUN
ejpam-4301	90	4	=	=	SYM
ejpam-4301	90	5	l	l	NOUN
ejpam-4301	90	6	+	+	CCONJ
ejpam-4301	90	7	n	n	PRON
ejpam-4301	90	8	′	′	NUM
ejpam-4301	90	9	,	,	PUNCT
ejpam-4301	90	10	so	so	ADV
ejpam-4301	90	11	n	n	CCONJ
ejpam-4301	90	12	−	−	PROPN
ejpam-4301	90	13	l	l	NOUN
ejpam-4301	90	14	=	=	SYM
ejpam-4301	90	15	n	n	CCONJ
ejpam-4301	90	16	′	′	NUM
ejpam-4301	90	17	∈	∈	PROPN
ejpam-4301	90	18	n	n	PRON
ejpam-4301	90	19	∩	∩	X
ejpam-4301	90	20	n	n	CCONJ
ejpam-4301	90	21	′	′	NOUN
ejpam-4301	90	22	=	=	SYM
ejpam-4301	90	23	0	0	NUM
ejpam-4301	90	24	,	,	PUNCT
ejpam-4301	90	25	hence	hence	ADV
ejpam-4301	90	26	n	n	NOUN
ejpam-4301	90	27	=	=	SYM
ejpam-4301	90	28	l	l	NOUN
ejpam-4301	90	29	and	and	CCONJ
ejpam-4301	90	30	n	n	PRON
ejpam-4301	90	31	≤	≤	NOUN
ejpam-4301	90	32	l.	l.	NOUN
ejpam-4301	90	33	therefore	therefore	ADV
ejpam-4301	90	34	,	,	PUNCT
ejpam-4301	90	35	n	n	NOUN
ejpam-4301	90	36	=	=	SYM
ejpam-4301	90	37	l	l	NOUN
ejpam-4301	90	38	and	and	CCONJ
ejpam-4301	90	39	k	k	PROPN
ejpam-4301	90	40	�	�	PROPN
ejpam-4301	90	41	e∗	e∗	PROPN
ejpam-4301	90	42	n	n	CCONJ
ejpam-4301	90	43	the	the	DET
ejpam-4301	90	44	following	follow	VERB
ejpam-4301	90	45	proposition	proposition	NOUN
ejpam-4301	90	46	shows	show	VERB
ejpam-4301	90	47	that	that	SCONJ
ejpam-4301	90	48	,	,	PUNCT
ejpam-4301	90	49	the	the	DET
ejpam-4301	90	50	homorphic	homorphic	ADJ
ejpam-4301	90	51	image	image	NOUN
ejpam-4301	90	52	of	of	ADP
ejpam-4301	90	53	an	an	DET
ejpam-4301	90	54	e∗-essential	e∗-essential	ADJ
ejpam-4301	90	55	small	small	ADJ
ejpam-4301	90	56	submodule	submodule	NOUN
ejpam-4301	90	57	is	be	AUX
ejpam-4301	90	58	e∗-essential	e∗-essential	ADJ
ejpam-4301	90	59	small	small	ADJ
ejpam-4301	90	60	submodule	submodule	NOUN
ejpam-4301	90	61	.	.	PUNCT
ejpam-4301	91	1	proposition	proposition	NOUN
ejpam-4301	91	2	3	3	NUM
ejpam-4301	91	3	.	.	PUNCT
ejpam-4301	92	1	if	if	SCONJ
ejpam-4301	92	2	k	k	PROPN
ejpam-4301	92	3	�	�	PROPN
ejpam-4301	92	4	e∗	e∗	PROPN
ejpam-4301	92	5	m	m	PROPN
ejpam-4301	92	6	and	and	CCONJ
ejpam-4301	92	7	f	f	X
ejpam-4301	92	8	:	:	PUNCT
ejpam-4301	92	9	m	m	VERB
ejpam-4301	92	10	→	→	SYM
ejpam-4301	92	11	n	n	X
ejpam-4301	92	12	is	be	AUX
ejpam-4301	92	13	an	an	DET
ejpam-4301	92	14	r	r	NOUN
ejpam-4301	92	15	-	-	PUNCT
ejpam-4301	92	16	homomorphism	homomorphism	NOUN
ejpam-4301	92	17	,	,	PUNCT
ejpam-4301	92	18	then	then	ADV
ejpam-4301	92	19	f(k	f(k	VERB
ejpam-4301	92	20	)	)	PUNCT
ejpam-4301	92	21	�	�	PROPN
ejpam-4301	92	22	e∗	e∗	PROPN
ejpam-4301	92	23	n	n	NOUN
ejpam-4301	92	24	.	.	PUNCT
ejpam-4301	93	1	proof	proof	NOUN
ejpam-4301	93	2	.	.	PUNCT
ejpam-4301	94	1	let	let	VERB
ejpam-4301	94	2	l	l	NOUN
ejpam-4301	94	3	be	be	AUX
ejpam-4301	94	4	an	an	DET
ejpam-4301	94	5	e∗-essential	e∗-essential	PROPN
ejpam-4301	94	6	submodule	submodule	NOUN
ejpam-4301	94	7	of	of	ADP
ejpam-4301	94	8	n	n	PRON
ejpam-4301	94	9	such	such	ADJ
ejpam-4301	94	10	that	that	DET
ejpam-4301	94	11	f(k)+l	f(k)+l	NOUN
ejpam-4301	94	12	=	=	SYM
ejpam-4301	94	13	n	n	NOUN
ejpam-4301	94	14	.	.	PUNCT
ejpam-4301	95	1	hence	hence	ADV
ejpam-4301	95	2	f−1(l	f−1(l	NOUN
ejpam-4301	95	3	)	)	PUNCT
ejpam-4301	95	4	is	be	AUX
ejpam-4301	95	5	e∗-essential	e∗-essential	ADJ
ejpam-4301	95	6	in	in	ADP
ejpam-4301	95	7	m	m	PROPN
ejpam-4301	95	8	by	by	ADP
ejpam-4301	95	9	lemma	lemma	PROPN
ejpam-4301	95	10	1	1	NUM
ejpam-4301	95	11	.	.	PUNCT
ejpam-4301	96	1	let	let	VERB
ejpam-4301	96	2	m	m	PRON
ejpam-4301	96	3	∈	∈	VERB
ejpam-4301	96	4	m	m	NOUN
ejpam-4301	96	5	,	,	PUNCT
ejpam-4301	96	6	hence	hence	ADV
ejpam-4301	96	7	f(m	f(m	NOUN
ejpam-4301	96	8	)	)	PUNCT
ejpam-4301	96	9	∈	∈	PROPN
ejpam-4301	96	10	n	n	NOUN
ejpam-4301	96	11	=	=	PUNCT
ejpam-4301	96	12	f(k)+l	f(k)+l	NOUN
ejpam-4301	96	13	,	,	PUNCT
ejpam-4301	96	14	so	so	SCONJ
ejpam-4301	96	15	there	there	PRON
ejpam-4301	96	16	exist	exist	VERB
ejpam-4301	96	17	k	k	PROPN
ejpam-4301	96	18	∈	∈	PROPN
ejpam-4301	96	19	k	k	PROPN
ejpam-4301	96	20	and	and	CCONJ
ejpam-4301	96	21	l	l	NOUN
ejpam-4301	96	22	∈	∈	PROPN
ejpam-4301	96	23	l	l	NOUN
ejpam-4301	96	24	such	such	ADJ
ejpam-4301	96	25	that	that	SCONJ
ejpam-4301	96	26	f(m	f(m	PROPN
ejpam-4301	96	27	)	)	PUNCT
ejpam-4301	96	28	=	=	PUNCT
ejpam-4301	96	29	f(k	f(k	VERB
ejpam-4301	96	30	)	)	PUNCT
ejpam-4301	97	1	+	+	CCONJ
ejpam-4301	97	2	l.	l.	PROPN
ejpam-4301	97	3	thus	thus	ADV
ejpam-4301	97	4	,	,	PUNCT
ejpam-4301	97	5	l	l	PROPN
ejpam-4301	97	6	=	=	SYM
ejpam-4301	97	7	f(m	f(m	PROPN
ejpam-4301	97	8	−	−	PROPN
ejpam-4301	97	9	k	k	NOUN
ejpam-4301	97	10	)	)	PUNCT
ejpam-4301	98	1	so	so	ADV
ejpam-4301	98	2	,	,	PUNCT
ejpam-4301	98	3	m	m	VERB
ejpam-4301	98	4	−	−	PROPN
ejpam-4301	98	5	k	k	PROPN
ejpam-4301	98	6	∈	∈	PROPN
ejpam-4301	98	7	f−1(l	f−1(l	NOUN
ejpam-4301	98	8	)	)	PUNCT
ejpam-4301	98	9	and	and	CCONJ
ejpam-4301	98	10	m	m	PROPN
ejpam-4301	98	11	=	=	ADJ
ejpam-4301	98	12	m	m	VERB
ejpam-4301	98	13	−	−	NOUN
ejpam-4301	99	1	k	k	NOUN
ejpam-4301	100	1	+	+	CCONJ
ejpam-4301	100	2	k	k	PROPN
ejpam-4301	100	3	∈	∈	PROPN
ejpam-4301	100	4	k	k	PROPN
ejpam-4301	100	5	+	+	NUM
ejpam-4301	100	6	f−1(l	f−1(l	NOUN
ejpam-4301	100	7	)	)	PUNCT
ejpam-4301	100	8	.	.	PUNCT
ejpam-4301	101	1	hence	hence	ADV
ejpam-4301	101	2	,	,	PUNCT
ejpam-4301	101	3	k	k	PROPN
ejpam-4301	101	4	+	+	CCONJ
ejpam-4301	101	5	f−1(l	f−1(l	NOUN
ejpam-4301	101	6	)	)	PUNCT
ejpam-4301	101	7	=	=	PUNCT
ejpam-4301	101	8	m	m	VERB
ejpam-4301	101	9	since	since	SCONJ
ejpam-4301	101	10	k	k	PROPN
ejpam-4301	101	11	�	�	PROPN
ejpam-4301	101	12	e∗	e∗	PROPN
ejpam-4301	101	13	m	m	PROPN
ejpam-4301	101	14	.	.	PUNCT
ejpam-4301	102	1	thus	thus	ADV
ejpam-4301	102	2	f−1(l	f−1(l	NOUN
ejpam-4301	102	3	)	)	PUNCT
ejpam-4301	102	4	=	=	SYM
ejpam-4301	102	5	m	m	PROPN
ejpam-4301	102	6	and	and	CCONJ
ejpam-4301	102	7	f(m	f(m	PROPN
ejpam-4301	102	8	)	)	PUNCT
ejpam-4301	102	9	=	=	PUNCT
ejpam-4301	103	1	f(f−1(l	f(f−1(l	PROPN
ejpam-4301	103	2	)	)	PUNCT
ejpam-4301	103	3	)	)	PUNCT
ejpam-4301	104	1	=	=	SYM
ejpam-4301	104	2	f(l	f(l	VERB
ejpam-4301	104	3	)	)	PUNCT
ejpam-4301	104	4	∩	∩	ADJ
ejpam-4301	104	5	l	l	NOUN
ejpam-4301	104	6	,	,	PUNCT
ejpam-4301	104	7	hence	hence	ADV
ejpam-4301	104	8	f(m	f(m	PROPN
ejpam-4301	104	9	)	)	PUNCT
ejpam-4301	104	10	⊆	⊆	NUM
ejpam-4301	104	11	l	l	NOUN
ejpam-4301	104	12	i.e.	i.e.	X
ejpam-4301	104	13	f(k	f(k	VERB
ejpam-4301	104	14	)	)	PUNCT
ejpam-4301	104	15	⊆	⊆	NUM
ejpam-4301	104	16	l.	l.	PROPN
ejpam-4301	104	17	therefore	therefore	ADV
ejpam-4301	104	18	,	,	PUNCT
ejpam-4301	104	19	l	l	NOUN
ejpam-4301	104	20	=	=	PUNCT
ejpam-4301	104	21	f(k	f(k	VERB
ejpam-4301	104	22	)	)	PUNCT
ejpam-4301	105	1	+	+	NUM
ejpam-4301	105	2	l	l	NOUN
ejpam-4301	105	3	=	=	SYM
ejpam-4301	105	4	n	n	NOUN
ejpam-4301	105	5	and	and	CCONJ
ejpam-4301	105	6	f(k	f(k	VERB
ejpam-4301	105	7	)	)	PUNCT
ejpam-4301	105	8	�	�	PROPN
ejpam-4301	105	9	e∗	e∗	PROPN
ejpam-4301	105	10	n	n	PROPN
ejpam-4301	105	11	.	.	PUNCT
ejpam-4301	106	1	the	the	DET
ejpam-4301	106	2	sum	sum	NOUN
ejpam-4301	106	3	of	of	ADP
ejpam-4301	106	4	e∗-essential	e∗-essential	ADJ
ejpam-4301	106	5	small	small	ADJ
ejpam-4301	106	6	submodules	submodule	NOUN
ejpam-4301	106	7	is	be	AUX
ejpam-4301	106	8	e∗-essential	e∗-essential	ADJ
ejpam-4301	106	9	small	small	ADJ
ejpam-4301	106	10	submodule	submodule	NOUN
ejpam-4301	106	11	as	as	ADP
ejpam-4301	106	12	the	the	DET
ejpam-4301	106	13	following	follow	VERB
ejpam-4301	106	14	proposition	proposition	NOUN
ejpam-4301	106	15	shows	show	NOUN
ejpam-4301	106	16	.	.	PUNCT
ejpam-4301	107	1	proposition	proposition	NOUN
ejpam-4301	107	2	4	4	NUM
ejpam-4301	107	3	.	.	PUNCT
ejpam-4301	108	1	let	let	VERB
ejpam-4301	108	2	n	n	NOUN
ejpam-4301	108	3	and	and	CCONJ
ejpam-4301	108	4	l	l	PROPN
ejpam-4301	108	5	be	be	AUX
ejpam-4301	108	6	submodules	submodule	NOUN
ejpam-4301	108	7	of	of	ADP
ejpam-4301	108	8	an	an	DET
ejpam-4301	108	9	r	r	NOUN
ejpam-4301	108	10	-	-	PUNCT
ejpam-4301	108	11	module	module	NOUN
ejpam-4301	108	12	m	m	NOUN
ejpam-4301	108	13	.	.	PUNCT
ejpam-4301	109	1	then	then	ADV
ejpam-4301	109	2	n	n	PROPN
ejpam-4301	109	3	+	+	CCONJ
ejpam-4301	109	4	l	l	NOUN
ejpam-4301	109	5	�	�	PROPN
ejpam-4301	109	6	e∗	e∗	PROPN
ejpam-4301	109	7	m	m	NOUN
ejpam-4301	109	8	if	if	SCONJ
ejpam-4301	109	9	and	and	CCONJ
ejpam-4301	109	10	only	only	ADV
ejpam-4301	109	11	if	if	SCONJ
ejpam-4301	109	12	n	n	PROPN
ejpam-4301	109	13	�	�	PROPN
ejpam-4301	109	14	e∗	e∗	PROPN
ejpam-4301	109	15	m	m	PROPN
ejpam-4301	109	16	and	and	CCONJ
ejpam-4301	109	17	l	l	PROPN
ejpam-4301	109	18	�	�	PROPN
ejpam-4301	109	19	e∗	e∗	PROPN
ejpam-4301	109	20	m	m	PROPN
ejpam-4301	109	21	.	.	PUNCT
ejpam-4301	110	1	proof	proof	NOUN
ejpam-4301	110	2	.	.	PUNCT
ejpam-4301	111	1	⇒	⇒	NOUN
ejpam-4301	111	2	)	)	PUNCT
ejpam-4301	111	3	let	let	VERB
ejpam-4301	111	4	k	k	PRON
ejpam-4301	111	5	be	be	AUX
ejpam-4301	111	6	e∗-essential	e∗-essential	PROPN
ejpam-4301	111	7	in	in	ADP
ejpam-4301	111	8	m	m	PRON
ejpam-4301	111	9	such	such	ADJ
ejpam-4301	111	10	that	that	SCONJ
ejpam-4301	111	11	k	k	PROPN
ejpam-4301	112	1	+	+	NOUN
ejpam-4301	112	2	n	n	NOUN
ejpam-4301	112	3	=	=	NOUN
ejpam-4301	112	4	m	m	NOUN
ejpam-4301	112	5	.	.	PUNCT
ejpam-4301	113	1	so	so	ADV
ejpam-4301	113	2	,	,	PUNCT
ejpam-4301	113	3	k	k	PROPN
ejpam-4301	113	4	+	+	PROPN
ejpam-4301	113	5	n	n	NOUN
ejpam-4301	113	6	+	+	X
ejpam-4301	113	7	l	l	NOUN
ejpam-4301	114	1	=	=	VERB
ejpam-4301	114	2	m	m	VERB
ejpam-4301	114	3	.	.	PUNCT
ejpam-4301	115	1	by	by	ADP
ejpam-4301	115	2	assumption	assumption	NOUN
ejpam-4301	115	3	,	,	PUNCT
ejpam-4301	115	4	k	k	PROPN
ejpam-4301	115	5	=	=	VERB
ejpam-4301	115	6	m	m	PROPN
ejpam-4301	115	7	and	and	CCONJ
ejpam-4301	115	8	n	n	PRON
ejpam-4301	115	9	�	�	PROPN
ejpam-4301	115	10	e∗	e∗	PROPN
ejpam-4301	115	11	m	m	PROPN
ejpam-4301	115	12	.	.	PUNCT
ejpam-4301	116	1	similarly	similarly	ADV
ejpam-4301	116	2	for	for	ADP
ejpam-4301	116	3	l	l	PROPN
ejpam-4301	116	4	�	�	PROPN
ejpam-4301	116	5	e∗	e∗	PROPN
ejpam-4301	116	6	m	m	PROPN
ejpam-4301	116	7	.	.	PUNCT
ejpam-4301	117	1	⇐	⇐	ADJ
ejpam-4301	117	2	)	)	PUNCT
ejpam-4301	117	3	let	let	VERB
ejpam-4301	117	4	a	a	PRON
ejpam-4301	117	5	be	be	AUX
ejpam-4301	117	6	e∗-essential	e∗-essential	PROPN
ejpam-4301	117	7	in	in	ADP
ejpam-4301	117	8	m	m	PROPN
ejpam-4301	117	9	such	such	ADJ
ejpam-4301	117	10	that	that	SCONJ
ejpam-4301	117	11	n	n	PROPN
ejpam-4301	117	12	+	+	CCONJ
ejpam-4301	117	13	l+a	l+a	X
ejpam-4301	117	14	=	=	VERB
ejpam-4301	117	15	m	m	PROPN
ejpam-4301	117	16	,	,	PUNCT
ejpam-4301	117	17	m	m	VERB
ejpam-4301	117	18	=	=	SYM
ejpam-4301	117	19	n	n	PROPN
ejpam-4301	117	20	+	+	CCONJ
ejpam-4301	117	21	(	(	PUNCT
ejpam-4301	117	22	l+a	l+a	NUM
ejpam-4301	117	23	)	)	PUNCT
ejpam-4301	118	1	=	=	SYM
ejpam-4301	118	2	m	m	ADJ
ejpam-4301	118	3	,	,	PUNCT
ejpam-4301	118	4	since	since	SCONJ
ejpam-4301	118	5	a	a	DET
ejpam-4301	118	6	≤	≤	NUM
ejpam-4301	118	7	a+l	a+l	PROPN
ejpam-4301	118	8	≤	≤	NUM
ejpam-4301	118	9	m	m	PROPN
ejpam-4301	118	10	and	and	CCONJ
ejpam-4301	118	11	a	a	DET
ejpam-4301	118	12	≤e∗	≤e∗	NOUN
ejpam-4301	118	13	m	m	NOUN
ejpam-4301	118	14	by	by	ADP
ejpam-4301	118	15	lemma	lemma	PROPN
ejpam-4301	118	16	1	1	NUM
ejpam-4301	118	17	,	,	PUNCT
ejpam-4301	118	18	this	this	PRON
ejpam-4301	118	19	implies	imply	VERB
ejpam-4301	118	20	that	that	SCONJ
ejpam-4301	118	21	l+a	l+a	PROPN
ejpam-4301	118	22	=	=	PUNCT
ejpam-4301	118	23	m	m	NOUN
ejpam-4301	118	24	.	.	PUNCT
ejpam-4301	119	1	now	now	ADV
ejpam-4301	119	2	,	,	PUNCT
ejpam-4301	119	3	n	n	PROPN
ejpam-4301	119	4	�	�	PROPN
ejpam-4301	119	5	e∗	e∗	PROPN
ejpam-4301	119	6	m	m	PROPN
ejpam-4301	119	7	implies	imply	VERB
ejpam-4301	119	8	that	that	SCONJ
ejpam-4301	119	9	l+a+m	l+a+m	NOUN
ejpam-4301	119	10	and	and	CCONJ
ejpam-4301	119	11	l	l	NOUN
ejpam-4301	119	12	�	�	PROPN
ejpam-4301	119	13	e∗	e∗	PROPN
ejpam-4301	119	14	m	m	PROPN
ejpam-4301	119	15	implies	imply	VERB
ejpam-4301	119	16	that	that	SCONJ
ejpam-4301	119	17	a	a	DET
ejpam-4301	119	18	=	=	NOUN
ejpam-4301	119	19	m	m	NOUN
ejpam-4301	119	20	.	.	PUNCT
ejpam-4301	120	1	therefore	therefore	ADV
ejpam-4301	120	2	,	,	PUNCT
ejpam-4301	120	3	n	n	PROPN
ejpam-4301	120	4	+	+	CCONJ
ejpam-4301	120	5	l	l	NOUN
ejpam-4301	120	6	�	�	PROPN
ejpam-4301	120	7	e∗	e∗	PROPN
ejpam-4301	120	8	m	m	PROPN
ejpam-4301	120	9	.	.	PUNCT
ejpam-4301	121	1	the	the	DET
ejpam-4301	121	2	following	follow	VERB
ejpam-4301	121	3	corollary	corollary	NOUN
ejpam-4301	121	4	follows	follow	VERB
ejpam-4301	121	5	from	from	ADP
ejpam-4301	121	6	proposition	proposition	NOUN
ejpam-4301	121	7	3	3	NUM
ejpam-4301	121	8	and	and	CCONJ
ejpam-4301	121	9	proposition	proposition	NOUN
ejpam-4301	121	10	4	4	NUM
ejpam-4301	121	11	.	.	PUNCT
ejpam-4301	121	12	corollary	corollary	ADJ
ejpam-4301	121	13	1	1	NUM
ejpam-4301	121	14	.	.	PUNCT
ejpam-4301	122	1	let	let	VERB
ejpam-4301	122	2	m	m	NOUN
ejpam-4301	122	3	=	=	VERB
ejpam-4301	122	4	m1	m1	PROPN
ejpam-4301	122	5	⊕m2	⊕m2	PROPN
ejpam-4301	122	6	and	and	CCONJ
ejpam-4301	122	7	ki	ki	PROPN
ejpam-4301	122	8	a	a	DET
ejpam-4301	122	9	submodule	submodule	NOUN
ejpam-4301	122	10	of	of	ADP
ejpam-4301	122	11	mi	mi	PROPN
ejpam-4301	122	12	,	,	PUNCT
ejpam-4301	122	13	i	i	NOUN
ejpam-4301	122	14	=	=	NOUN
ejpam-4301	122	15	1	1	NUM
ejpam-4301	122	16	,	,	PUNCT
ejpam-4301	122	17	2	2	NUM
ejpam-4301	122	18	.	.	PUNCT
ejpam-4301	123	1	then	then	ADV
ejpam-4301	123	2	ki	ki	PROPN
ejpam-4301	123	3	�	�	PROPN
ejpam-4301	123	4	e∗	e∗	PROPN
ejpam-4301	123	5	mi	mi	PROPN
ejpam-4301	123	6	,	,	PUNCT
ejpam-4301	123	7	i	i	NOUN
ejpam-4301	123	8	=	=	NOUN
ejpam-4301	123	9	1	1	NUM
ejpam-4301	123	10	,	,	PUNCT
ejpam-4301	123	11	2	2	NUM
ejpam-4301	123	12	if	if	SCONJ
ejpam-4301	123	13	and	and	CCONJ
ejpam-4301	123	14	only	only	ADV
ejpam-4301	123	15	if	if	SCONJ
ejpam-4301	123	16	k1	k1	PROPN
ejpam-4301	123	17	⊕k2	⊕k2	NUM
ejpam-4301	123	18	�	�	PROPN
ejpam-4301	123	19	e∗	e∗	PROPN
ejpam-4301	123	20	m1	m1	PROPN
ejpam-4301	123	21	⊕m2	⊕m2	PROPN
ejpam-4301	123	22	.	.	PROPN
ejpam-4301	124	1	h.r	h.r	PROPN
ejpam-4301	124	2	.	.	PROPN
ejpam-4301	124	3	baanoon	baanoon	PROPN
ejpam-4301	124	4	,	,	PUNCT
ejpam-4301	124	5	w.	w.	PROPN
ejpam-4301	124	6	khalid	khalid	PROPN
ejpam-4301	124	7	/	/	PUNCT
ejpam-4301	124	8	eur	eur	PROPN
ejpam-4301	124	9	.	.	PUNCT
ejpam-4301	125	1	j.	j.	PROPN
ejpam-4301	125	2	pure	pure	PROPN
ejpam-4301	125	3	appl	appl	PROPN
ejpam-4301	125	4	.	.	PROPN
ejpam-4301	125	5	math	math	PROPN
ejpam-4301	125	6	,	,	PUNCT
ejpam-4301	125	7	15	15	NUM
ejpam-4301	125	8	(	(	PUNCT
ejpam-4301	125	9	2	2	NUM
ejpam-4301	125	10	)	)	PUNCT
ejpam-4301	125	11	(	(	PUNCT
ejpam-4301	125	12	2022	2022	NUM
ejpam-4301	125	13	)	)	PUNCT
ejpam-4301	125	14	,	,	PUNCT
ejpam-4301	125	15	478	478	NUM
ejpam-4301	125	16	-	-	SYM
ejpam-4301	125	17	485	485	NUM
ejpam-4301	125	18	481	481	NUM
ejpam-4301	125	19	3	3	NUM
ejpam-4301	125	20	.	.	PUNCT
ejpam-4301	126	1	e∗radical	e∗radical	PROPN
ejpam-4301	126	2	submodule	submodule	PROPN
ejpam-4301	126	3	recall	recall	VERB
ejpam-4301	126	4	that	that	PRON
ejpam-4301	126	5	for	for	ADP
ejpam-4301	126	6	an	an	DET
ejpam-4301	126	7	r	r	NOUN
ejpam-4301	126	8	-	-	PUNCT
ejpam-4301	126	9	module	module	NOUN
ejpam-4301	126	10	m	m	NOUN
ejpam-4301	126	11	,	,	PUNCT
ejpam-4301	126	12	if	if	SCONJ
ejpam-4301	126	13	m	m	PROPN
ejpam-4301	126	14	has	have	VERB
ejpam-4301	126	15	maximal	maximal	ADJ
ejpam-4301	126	16	submodule	submodule	NOUN
ejpam-4301	126	17	,	,	PUNCT
ejpam-4301	126	18	then	then	ADV
ejpam-4301	126	19	the	the	DET
ejpam-4301	126	20	radical	radical	NOUN
ejpam-4301	126	21	of	of	ADP
ejpam-4301	126	22	m	m	PROPN
ejpam-4301	126	23	is	be	AUX
ejpam-4301	126	24	the	the	DET
ejpam-4301	126	25	intersection	intersection	NOUN
ejpam-4301	126	26	of	of	ADP
ejpam-4301	126	27	all	all	DET
ejpam-4301	126	28	maximal	maximal	ADJ
ejpam-4301	126	29	submodules	submodule	NOUN
ejpam-4301	126	30	of	of	ADP
ejpam-4301	126	31	m	m	VERB
ejpam-4301	126	32	dented	dent	VERB
ejpam-4301	126	33	by	by	ADP
ejpam-4301	126	34	rad(m	rad(m	PROPN
ejpam-4301	126	35	)	)	PUNCT
ejpam-4301	127	1	[	[	X
ejpam-4301	127	2	6	6	NUM
ejpam-4301	127	3	]	]	PUNCT
ejpam-4301	127	4	.	.	PUNCT
ejpam-4301	128	1	we	we	PRON
ejpam-4301	128	2	generalize	generalize	VERB
ejpam-4301	128	3	this	this	DET
ejpam-4301	128	4	concept	concept	NOUN
ejpam-4301	128	5	as	as	ADP
ejpam-4301	128	6	the	the	DET
ejpam-4301	128	7	following	following	NOUN
ejpam-4301	128	8	:	:	PUNCT
ejpam-4301	128	9	definition	definition	NOUN
ejpam-4301	128	10	2	2	NUM
ejpam-4301	128	11	.	.	PUNCT
ejpam-4301	129	1	let	let	VERB
ejpam-4301	129	2	m	m	PRON
ejpam-4301	129	3	be	be	AUX
ejpam-4301	129	4	r	r	NOUN
ejpam-4301	129	5	-	-	PUNCT
ejpam-4301	129	6	module	module	NOUN
ejpam-4301	129	7	.	.	PUNCT
ejpam-4301	130	1	then	then	ADV
ejpam-4301	130	2	the	the	DET
ejpam-4301	130	3	intersection	intersection	NOUN
ejpam-4301	130	4	of	of	ADP
ejpam-4301	130	5	all	all	DET
ejpam-4301	130	6	e∗-essential	e∗-essential	PROPN
ejpam-4301	130	7	maximal	maximal	ADJ
ejpam-4301	130	8	submodule	submodule	NOUN
ejpam-4301	130	9	of	of	ADP
ejpam-4301	130	10	m	m	PROPN
ejpam-4301	130	11	is	be	AUX
ejpam-4301	130	12	called	call	VERB
ejpam-4301	130	13	e∗radical	e∗radical	ADJ
ejpam-4301	130	14	submodule	submodule	NOUN
ejpam-4301	130	15	denoted	denote	VERB
ejpam-4301	130	16	by	by	ADP
ejpam-4301	130	17	rad(m	rad(m	PROPN
ejpam-4301	130	18	)	)	PUNCT
ejpam-4301	130	19	e∗	e∗	NOUN
ejpam-4301	130	20	.	.	PUNCT
ejpam-4301	131	1	if	if	SCONJ
ejpam-4301	131	2	m	m	PROPN
ejpam-4301	131	3	has	have	VERB
ejpam-4301	131	4	no	no	DET
ejpam-4301	131	5	e∗-essential	e∗-essential	PROPN
ejpam-4301	131	6	maximal	maximal	ADJ
ejpam-4301	131	7	submodule	submodule	NOUN
ejpam-4301	131	8	,	,	PUNCT
ejpam-4301	131	9	then	then	ADV
ejpam-4301	131	10	rad(m	rad(m	PROPN
ejpam-4301	131	11	)	)	PUNCT
ejpam-4301	131	12	e∗	e∗	NOUN
ejpam-4301	132	1	=	=	NOUN
ejpam-4301	132	2	m	m	PROPN
ejpam-4301	132	3	.	.	PUNCT
ejpam-4301	133	1	the	the	DET
ejpam-4301	133	2	following	follow	VERB
ejpam-4301	133	3	proposition	proposition	NOUN
ejpam-4301	133	4	gives	give	VERB
ejpam-4301	133	5	the	the	DET
ejpam-4301	133	6	relationship	relationship	NOUN
ejpam-4301	133	7	between	between	ADP
ejpam-4301	133	8	e∗-essential	e∗-essential	ADJ
ejpam-4301	133	9	small	small	ADJ
ejpam-4301	133	10	submodules	submodule	NOUN
ejpam-4301	133	11	and	and	CCONJ
ejpam-4301	133	12	e∗-essential	e∗-essential	ADJ
ejpam-4301	133	13	maximal	maximal	ADJ
ejpam-4301	133	14	submodules	submodule	NOUN
ejpam-4301	133	15	.	.	PUNCT
ejpam-4301	134	1	proposition	proposition	NOUN
ejpam-4301	134	2	5	5	NUM
ejpam-4301	134	3	.	.	PUNCT
ejpam-4301	135	1	let	let	VERB
ejpam-4301	135	2	m	m	PRON
ejpam-4301	135	3	be	be	AUX
ejpam-4301	135	4	an	an	DET
ejpam-4301	135	5	r	r	NOUN
ejpam-4301	135	6	-	-	PUNCT
ejpam-4301	135	7	module	module	NOUN
ejpam-4301	135	8	and	and	CCONJ
ejpam-4301	135	9	m	m	NOUN
ejpam-4301	135	10	∈	∈	PROPN
ejpam-4301	135	11	m	m	NOUN
ejpam-4301	135	12	,	,	PUNCT
ejpam-4301	135	13	then	then	ADV
ejpam-4301	135	14	〈	〈	PROPN
ejpam-4301	135	15	m	m	PROPN
ejpam-4301	135	16	〉	〉	PROPN
ejpam-4301	135	17	is	be	AUX
ejpam-4301	135	18	not	not	PART
ejpam-4301	135	19	e∗-essential	e∗-essential	ADJ
ejpam-4301	135	20	small	small	ADJ
ejpam-4301	135	21	if	if	SCONJ
ejpam-4301	136	1	and	and	CCONJ
ejpam-4301	136	2	only	only	ADV
ejpam-4301	136	3	if	if	SCONJ
ejpam-4301	136	4	there	there	PRON
ejpam-4301	136	5	exists	exist	VERB
ejpam-4301	136	6	an	an	DET
ejpam-4301	136	7	e∗-essential	e∗-essential	PROPN
ejpam-4301	136	8	maximal	maximal	ADJ
ejpam-4301	136	9	submodule	submodule	NOUN
ejpam-4301	136	10	n	n	PROPN
ejpam-4301	136	11	of	of	ADP
ejpam-4301	136	12	m	m	PROPN
ejpam-4301	136	13	with	with	ADP
ejpam-4301	136	14	m	m	PROPN
ejpam-4301	136	15	/∈	/∈	PUNCT
ejpam-4301	137	1	n	n	NOUN
ejpam-4301	137	2	.	.	PUNCT
ejpam-4301	138	1	proof	proof	NOUN
ejpam-4301	138	2	.	.	PUNCT
ejpam-4301	139	1	⇒	⇒	NOUN
ejpam-4301	139	2	)	)	PUNCT
ejpam-4301	139	3	consider	consider	VERB
ejpam-4301	139	4	the	the	DET
ejpam-4301	139	5	set	set	NOUN
ejpam-4301	139	6	γ	γ	X
ejpam-4301	139	7	=	=	PRON
ejpam-4301	139	8	{	{	PUNCT
ejpam-4301	139	9	b|b	b|b	NOUN
ejpam-4301	139	10	is	be	AUX
ejpam-4301	139	11	a	a	DET
ejpam-4301	139	12	proper	proper	ADJ
ejpam-4301	139	13	e∗-essential	e∗-essential	PROPN
ejpam-4301	139	14	submodule	submodule	NOUN
ejpam-4301	139	15	of	of	ADP
ejpam-4301	139	16	m	m	PROPN
ejpam-4301	139	17	and	and	CCONJ
ejpam-4301	139	18	〈	〈	PROPN
ejpam-4301	139	19	m	m	PROPN
ejpam-4301	140	1	〉	〉	NOUN
ejpam-4301	140	2	+	+	PROPN
ejpam-4301	140	3	b	b	NOUN
ejpam-4301	140	4	=	=	SYM
ejpam-4301	140	5	m	m	NOUN
ejpam-4301	140	6	}	}	PUNCT
ejpam-4301	140	7	.	.	PUNCT
ejpam-4301	141	1	since	since	SCONJ
ejpam-4301	141	2	〈	〈	PROPN
ejpam-4301	141	3	m	m	PROPN
ejpam-4301	141	4	〉	〉	PROPN
ejpam-4301	141	5	is	be	AUX
ejpam-4301	141	6	not	not	PART
ejpam-4301	141	7	e∗-essential	e∗-essential	PROPN
ejpam-4301	141	8	small	small	ADJ
ejpam-4301	141	9	,	,	PUNCT
ejpam-4301	141	10	there	there	PRON
ejpam-4301	141	11	exists	exist	VERB
ejpam-4301	141	12	b	b	NOUN
ejpam-4301	141	13	′	′	NUM
ejpam-4301	141	14	≤e∗	≤e∗	PROPN
ejpam-4301	141	15	m	m	VERB
ejpam-4301	141	16	such	such	ADJ
ejpam-4301	142	1	that	that	SCONJ
ejpam-4301	142	2	〈	〈	PROPN
ejpam-4301	142	3	m	m	PROPN
ejpam-4301	142	4	〉	〉	NOUN
ejpam-4301	142	5	+	+	NOUN
ejpam-4301	142	6	b	b	NOUN
ejpam-4301	142	7	′	′	NUM
ejpam-4301	143	1	=	=	VERB
ejpam-4301	143	2	m	m	PROPN
ejpam-4301	143	3	and	and	CCONJ
ejpam-4301	143	4	b	b	X
ejpam-4301	143	5	′	′	NUM
ejpam-4301	143	6	6=	6=	NUM
ejpam-4301	143	7	m	m	PRON
ejpam-4301	143	8	,	,	PUNCT
ejpam-4301	143	9	hence	hence	ADV
ejpam-4301	143	10	γ	γ	PROPN
ejpam-4301	143	11	6=	6=	PROPN
ejpam-4301	143	12	φ	φ	PROPN
ejpam-4301	143	13	.	.	PUNCT
ejpam-4301	144	1	let	let	VERB
ejpam-4301	144	2	{	{	PUNCT
ejpam-4301	144	3	cα}α∈λ	cα}α∈λ	VERB
ejpam-4301	144	4	be	be	AUX
ejpam-4301	144	5	a	a	DET
ejpam-4301	144	6	chain	chain	NOUN
ejpam-4301	144	7	in	in	ADP
ejpam-4301	144	8	γ	γ	NOUN
ejpam-4301	144	9	,	,	PUNCT
ejpam-4301	144	10	hence	hence	ADV
ejpam-4301	144	11	∪α∈λcα	∪α∈λcα	PROPN
ejpam-4301	144	12	is	be	AUX
ejpam-4301	144	13	a	a	DET
ejpam-4301	144	14	proper	proper	ADJ
ejpam-4301	144	15	submodule	submodule	NOUN
ejpam-4301	144	16	and	and	CCONJ
ejpam-4301	144	17	since	since	SCONJ
ejpam-4301	144	18	cα	cα	ADP
ejpam-4301	144	19	≤	≤	NUM
ejpam-4301	144	20	∪α∈λcα	∪α∈λcα	NOUN
ejpam-4301	144	21	≤	≤	NUM
ejpam-4301	144	22	m	m	VERB
ejpam-4301	144	23	for	for	ADP
ejpam-4301	144	24	each	each	DET
ejpam-4301	144	25	α	α	NOUN
ejpam-4301	144	26	∈	∈	PROPN
ejpam-4301	144	27	λ	λ	X
ejpam-4301	144	28	with	with	ADP
ejpam-4301	144	29	cα	cα	ADP
ejpam-4301	144	30	≤e∗	≤e∗	PROPN
ejpam-4301	144	31	m	m	NOUN
ejpam-4301	144	32	,	,	PUNCT
ejpam-4301	144	33	then	then	ADV
ejpam-4301	144	34	∪α∈λcα	∪α∈λcα	PROPN
ejpam-4301	144	35	≤e∗	≤e∗	PROPN
ejpam-4301	144	36	m	m	VERB
ejpam-4301	144	37	with	with	ADP
ejpam-4301	144	38	〈	〈	PROPN
ejpam-4301	144	39	m	m	PROPN
ejpam-4301	144	40	〉	〉	NOUN
ejpam-4301	144	41	+	+	CCONJ
ejpam-4301	144	42	∪α∈λcα	∪α∈λcα	NOUN
ejpam-4301	144	43	=	=	NOUN
ejpam-4301	144	44	m	m	PROPN
ejpam-4301	144	45	.	.	PUNCT
ejpam-4301	145	1	so	so	ADV
ejpam-4301	145	2	,	,	PUNCT
ejpam-4301	145	3	by	by	ADP
ejpam-4301	145	4	zorn	zorn	PROPN
ejpam-4301	145	5	’s	’s	PART
ejpam-4301	145	6	lemma	lemma	PROPN
ejpam-4301	145	7	,	,	PUNCT
ejpam-4301	145	8	γ	γ	PROPN
ejpam-4301	145	9	has	have	VERB
ejpam-4301	145	10	a	a	DET
ejpam-4301	145	11	maximal	maximal	ADJ
ejpam-4301	145	12	element	element	NOUN
ejpam-4301	145	13	say	say	VERB
ejpam-4301	145	14	b0	b0	NOUN
ejpam-4301	145	15	.	.	PUNCT
ejpam-4301	146	1	we	we	PRON
ejpam-4301	146	2	claim	claim	VERB
ejpam-4301	146	3	that	that	SCONJ
ejpam-4301	146	4	b0	b0	NOUN
ejpam-4301	146	5	is	be	AUX
ejpam-4301	146	6	maximal	maximal	ADJ
ejpam-4301	146	7	in	in	ADP
ejpam-4301	146	8	m	m	PROPN
ejpam-4301	146	9	.	.	PUNCT
ejpam-4301	147	1	otherwise	otherwise	ADV
ejpam-4301	147	2	if	if	SCONJ
ejpam-4301	147	3	b0	b0	VERB
ejpam-4301	147	4	�	�	PROPN
ejpam-4301	147	5	c	c	PROPN
ejpam-4301	147	6	6	6	NUM
ejpam-4301	147	7	m	m	NOUN
ejpam-4301	147	8	,	,	PUNCT
ejpam-4301	147	9	then	then	ADV
ejpam-4301	147	10	m	m	VERB
ejpam-4301	147	11	=	=	PUNCT
ejpam-4301	147	12	b0+〈m	b0+〈m	NUM
ejpam-4301	147	13	〉	〉	PROPN
ejpam-4301	147	14	≤	≤	NUM
ejpam-4301	147	15	c+〈m	c+〈m	VERB
ejpam-4301	147	16	〉	〉	PROPN
ejpam-4301	147	17	≤	≤	NUM
ejpam-4301	147	18	m	m	VERB
ejpam-4301	147	19	.	.	PUNCT
ejpam-4301	148	1	thus	thus	ADV
ejpam-4301	148	2	,	,	PUNCT
ejpam-4301	148	3	〈	〈	PROPN
ejpam-4301	148	4	m〉+c	m〉+c	NOUN
ejpam-4301	148	5	=	=	SYM
ejpam-4301	148	6	m	m	NOUN
ejpam-4301	148	7	and	and	CCONJ
ejpam-4301	148	8	since	since	SCONJ
ejpam-4301	148	9	b0	b0	VERB
ejpam-4301	148	10	6e∗	6e∗	PROPN
ejpam-4301	148	11	m	m	NOUN
ejpam-4301	148	12	,	,	PUNCT
ejpam-4301	148	13	hence	hence	ADV
ejpam-4301	148	14	c	c	PROPN
ejpam-4301	148	15	6e∗	6e∗	NUM
ejpam-4301	148	16	m	m	NOUN
ejpam-4301	148	17	.	.	PUNCT
ejpam-4301	149	1	now	now	ADV
ejpam-4301	149	2	,	,	PUNCT
ejpam-4301	149	3	if	if	SCONJ
ejpam-4301	149	4	c	c	PROPN
ejpam-4301	149	5	6=	6=	ADP
ejpam-4301	149	6	m	m	VERB
ejpam-4301	149	7	,	,	PUNCT
ejpam-4301	149	8	hence	hence	ADV
ejpam-4301	149	9	c	c	NOUN
ejpam-4301	149	10	∈	∈	PROPN
ejpam-4301	149	11	γ	γ	NOUN
ejpam-4301	149	12	which	which	PRON
ejpam-4301	149	13	is	be	AUX
ejpam-4301	149	14	a	a	DET
ejpam-4301	149	15	contradiction	contradiction	NOUN
ejpam-4301	149	16	.	.	PUNCT
ejpam-4301	150	1	thus	thus	ADV
ejpam-4301	150	2	,	,	PUNCT
ejpam-4301	150	3	c	c	PROPN
ejpam-4301	150	4	=	=	NOUN
ejpam-4301	150	5	m	m	PROPN
ejpam-4301	150	6	.	.	PUNCT
ejpam-4301	151	1	so	so	ADV
ejpam-4301	151	2	b0	b0	VERB
ejpam-4301	151	3	6e∗	6e∗	PROPN
ejpam-4301	151	4	m	m	NOUN
ejpam-4301	151	5	which	which	PRON
ejpam-4301	151	6	is	be	AUX
ejpam-4301	151	7	maximal	maximal	ADJ
ejpam-4301	151	8	in	in	ADP
ejpam-4301	151	9	m	m	PROPN
ejpam-4301	151	10	.	.	PUNCT
ejpam-4301	152	1	now	now	ADV
ejpam-4301	152	2	,	,	PUNCT
ejpam-4301	152	3	if	if	SCONJ
ejpam-4301	152	4	m	m	PROPN
ejpam-4301	152	5	∈	∈	PROPN
ejpam-4301	152	6	b0	b0	NOUN
ejpam-4301	152	7	,	,	PUNCT
ejpam-4301	152	8	then	then	ADV
ejpam-4301	152	9	〈	〈	PROPN
ejpam-4301	152	10	m	m	PROPN
ejpam-4301	152	11	〉	〉	PROPN
ejpam-4301	152	12	⊆	⊆	NUM
ejpam-4301	152	13	b0	b0	NOUN
ejpam-4301	152	14	and	and	CCONJ
ejpam-4301	152	15	since	since	SCONJ
ejpam-4301	152	16	〈	〈	PROPN
ejpam-4301	152	17	m	m	PROPN
ejpam-4301	152	18	〉	〉	NOUN
ejpam-4301	152	19	+	+	SYM
ejpam-4301	152	20	b0	b0	NOUN
ejpam-4301	152	21	=	=	NOUN
ejpam-4301	152	22	m	m	PROPN
ejpam-4301	152	23	,	,	PUNCT
ejpam-4301	152	24	we	we	PRON
ejpam-4301	152	25	have	have	AUX
ejpam-4301	152	26	b0	b0	NOUN
ejpam-4301	152	27	=	=	VERB
ejpam-4301	152	28	m	m	VERB
ejpam-4301	152	29	which	which	PRON
ejpam-4301	152	30	is	be	AUX
ejpam-4301	152	31	a	a	DET
ejpam-4301	152	32	contradiction	contradiction	NOUN
ejpam-4301	152	33	.	.	PUNCT
ejpam-4301	153	1	so	so	ADV
ejpam-4301	153	2	,	,	PUNCT
ejpam-4301	153	3	m	m	NOUN
ejpam-4301	153	4	/∈	/∈	PUNCT
ejpam-4301	153	5	b0	b0	NOUN
ejpam-4301	153	6	i.e.	i.e.	X
ejpam-4301	153	7	there	there	PRON
ejpam-4301	153	8	exists	exist	VERB
ejpam-4301	153	9	an	an	DET
ejpam-4301	153	10	e∗-essential	e∗-essential	PROPN
ejpam-4301	153	11	maximal	maximal	ADJ
ejpam-4301	153	12	submodule	submodule	NOUN
ejpam-4301	153	13	of	of	ADP
ejpam-4301	153	14	m	m	PRON
ejpam-4301	153	15	that	that	PRON
ejpam-4301	153	16	does	do	AUX
ejpam-4301	153	17	not	not	PART
ejpam-4301	153	18	contain	contain	VERB
ejpam-4301	153	19	m.	m.	NOUN
ejpam-4301	153	20	⇐	⇐	NOUN
ejpam-4301	153	21	)	)	PUNCT
ejpam-4301	153	22	to	to	PART
ejpam-4301	153	23	show	show	VERB
ejpam-4301	153	24	that	that	SCONJ
ejpam-4301	153	25	〈	〈	PROPN
ejpam-4301	153	26	x	x	SYM
ejpam-4301	153	27	〉	〉	PROPN
ejpam-4301	153	28	is	be	AUX
ejpam-4301	153	29	not	not	PART
ejpam-4301	153	30	e∗-essential	e∗-essential	ADJ
ejpam-4301	153	31	small	small	ADJ
ejpam-4301	153	32	in	in	ADP
ejpam-4301	153	33	m	m	PROPN
ejpam-4301	153	34	.	.	PUNCT
ejpam-4301	154	1	if	if	SCONJ
ejpam-4301	154	2	not	not	PART
ejpam-4301	154	3	,	,	PUNCT
ejpam-4301	154	4	then	then	ADV
ejpam-4301	154	5	as	as	SCONJ
ejpam-4301	154	6	x	x	X
ejpam-4301	154	7	/∈	/∈	PROPN
ejpam-4301	154	8	n	n	PROPN
ejpam-4301	154	9	and	and	CCONJ
ejpam-4301	154	10	n	n	ADV
ejpam-4301	154	11	is	be	AUX
ejpam-4301	154	12	as	as	ADV
ejpam-4301	154	13	maximal	maximal	ADJ
ejpam-4301	154	14	submodule	submodule	NOUN
ejpam-4301	154	15	we	we	PRON
ejpam-4301	154	16	have	have	VERB
ejpam-4301	154	17	〈	〈	PROPN
ejpam-4301	154	18	x〉+n	x〉+n	X
ejpam-4301	154	19	=	=	NOUN
ejpam-4301	154	20	m	m	PROPN
ejpam-4301	154	21	.	.	PUNCT
ejpam-4301	155	1	now	now	ADV
ejpam-4301	155	2	,	,	PUNCT
ejpam-4301	155	3	〈	〈	PROPN
ejpam-4301	155	4	x	x	PROPN
ejpam-4301	155	5	〉	〉	PROPN
ejpam-4301	155	6	�	�	PROPN
ejpam-4301	155	7	e∗	e∗	PROPN
ejpam-4301	155	8	m	m	PROPN
ejpam-4301	155	9	and	and	CCONJ
ejpam-4301	155	10	n	n	PRON
ejpam-4301	155	11	≤e∗	≤e∗	NOUN
ejpam-4301	155	12	m	m	VERB
ejpam-4301	155	13	implies	imply	VERB
ejpam-4301	155	14	that	that	SCONJ
ejpam-4301	155	15	n	n	NOUN
ejpam-4301	155	16	=	=	VERB
ejpam-4301	155	17	m	m	VERB
ejpam-4301	155	18	which	which	PRON
ejpam-4301	155	19	is	be	AUX
ejpam-4301	155	20	a	a	DET
ejpam-4301	155	21	contraindication	contraindication	NOUN
ejpam-4301	155	22	.	.	PUNCT
ejpam-4301	156	1	therefore	therefore	ADV
ejpam-4301	156	2	,	,	PUNCT
ejpam-4301	156	3	〈	〈	PROPN
ejpam-4301	156	4	x	x	PART
ejpam-4301	156	5	〉	〉	PROPN
ejpam-4301	156	6	is	be	AUX
ejpam-4301	156	7	not	not	PART
ejpam-4301	156	8	e∗-essential	e∗-essential	ADJ
ejpam-4301	156	9	small	small	ADJ
ejpam-4301	156	10	submodule	submodule	NOUN
ejpam-4301	156	11	of	of	ADP
ejpam-4301	156	12	m	m	PROPN
ejpam-4301	156	13	.	.	PUNCT
ejpam-4301	157	1	examples	example	NOUN
ejpam-4301	157	2	and	and	CCONJ
ejpam-4301	157	3	remarks	remark	VERB
ejpam-4301	157	4	2	2	NUM
ejpam-4301	157	5	.	.	NOUN
ejpam-4301	157	6	1	1	NUM
ejpam-4301	157	7	.	.	X
ejpam-4301	158	1	let	let	VERB
ejpam-4301	158	2	m	m	PRON
ejpam-4301	158	3	be	be	AUX
ejpam-4301	158	4	an	an	DET
ejpam-4301	158	5	r	r	NOUN
ejpam-4301	158	6	-	-	PUNCT
ejpam-4301	158	7	module	module	NOUN
ejpam-4301	158	8	,	,	PUNCT
ejpam-4301	158	9	then	then	ADV
ejpam-4301	158	10	rad(m	rad(m	NUM
ejpam-4301	158	11	)	)	PUNCT
ejpam-4301	158	12	≤	≤	NOUN
ejpam-4301	158	13	rad(m	rad(m	PROPN
ejpam-4301	158	14	)	)	PUNCT
ejpam-4301	158	15	e∗	e∗	NOUN
ejpam-4301	158	16	.	.	PUNCT
ejpam-4301	159	1	but	but	CCONJ
ejpam-4301	159	2	the	the	DET
ejpam-4301	159	3	converse	converse	NOUN
ejpam-4301	159	4	need	need	VERB
ejpam-4301	159	5	not	not	PART
ejpam-4301	159	6	to	to	PART
ejpam-4301	159	7	be	be	AUX
ejpam-4301	159	8	true	true	ADJ
ejpam-4301	159	9	in	in	ADP
ejpam-4301	159	10	general	general	ADJ
ejpam-4301	159	11	.	.	PUNCT
ejpam-4301	160	1	for	for	ADP
ejpam-4301	160	2	example	example	NOUN
ejpam-4301	160	3	:	:	PUNCT
ejpam-4301	160	4	consider	consider	VERB
ejpam-4301	160	5	z6	z6	NOUN
ejpam-4301	160	6	as	as	ADP
ejpam-4301	160	7	a	a	DET
ejpam-4301	160	8	z	z	NOUN
ejpam-4301	160	9	-	-	PUNCT
ejpam-4301	160	10	module	module	NOUN
ejpam-4301	160	11	,	,	PUNCT
ejpam-4301	160	12	rad(z6	rad(z6	PROPN
ejpam-4301	160	13	)	)	PUNCT
ejpam-4301	160	14	=	=	PUNCT
ejpam-4301	160	15	{	{	PUNCT
ejpam-4301	160	16	0	0	NUM
ejpam-4301	160	17	}	}	PUNCT
ejpam-4301	160	18	.	.	PUNCT
ejpam-4301	161	1	when	when	SCONJ
ejpam-4301	161	2	rad(z6	rad(z6	PROPN
ejpam-4301	161	3	)	)	PUNCT
ejpam-4301	161	4	e∗	e∗	PROPN
ejpam-4301	161	5	=	=	SYM
ejpam-4301	161	6	z6	z6	PROPN
ejpam-4301	161	7	,	,	PUNCT
ejpam-4301	161	8	since	since	SCONJ
ejpam-4301	161	9	the	the	DET
ejpam-4301	161	10	maximal	maximal	ADJ
ejpam-4301	161	11	submodules	submodule	NOUN
ejpam-4301	161	12	of	of	ADP
ejpam-4301	161	13	z6	z6	PROPN
ejpam-4301	161	14	are	be	AUX
ejpam-4301	161	15	〈	〈	PROPN
ejpam-4301	161	16	2	2	NUM
ejpam-4301	161	17	〉	〉	NUM
ejpam-4301	161	18	and	and	CCONJ
ejpam-4301	161	19	〈	〈	PROPN
ejpam-4301	161	20	3	3	NUM
ejpam-4301	161	21	〉	〉	NOUN
ejpam-4301	161	22	while	while	SCONJ
ejpam-4301	161	23	the	the	DET
ejpam-4301	161	24	only	only	ADJ
ejpam-4301	161	25	e∗-essntial	e∗-essntial	ADJ
ejpam-4301	161	26	submodule	submodule	NOUN
ejpam-4301	161	27	is	be	AUX
ejpam-4301	161	28	z6	z6	PROPN
ejpam-4301	162	1	[	[	X
ejpam-4301	162	2	1	1	NUM
ejpam-4301	162	3	]	]	PUNCT
ejpam-4301	162	4	.	.	PUNCT
ejpam-4301	163	1	2	2	X
ejpam-4301	163	2	.	.	X
ejpam-4301	163	3	in	in	ADP
ejpam-4301	163	4	z4	z4	PROPN
ejpam-4301	163	5	as	as	ADP
ejpam-4301	163	6	a	a	DET
ejpam-4301	163	7	z	z	NOUN
ejpam-4301	163	8	-	-	PUNCT
ejpam-4301	163	9	module	module	NOUN
ejpam-4301	163	10	rad(z4	rad(z4	NOUN
ejpam-4301	163	11	)	)	PUNCT
ejpam-4301	163	12	e∗	e∗	NOUN
ejpam-4301	163	13	=	=	SYM
ejpam-4301	163	14	{	{	PUNCT
ejpam-4301	163	15	0	0	NUM
ejpam-4301	163	16	,	,	PUNCT
ejpam-4301	163	17	2	2	NUM
ejpam-4301	163	18	}	}	PUNCT
ejpam-4301	163	19	.	.	PUNCT
ejpam-4301	164	1	since	since	SCONJ
ejpam-4301	164	2	all	all	DET
ejpam-4301	164	3	submodules	submodule	NOUN
ejpam-4301	164	4	of	of	ADP
ejpam-4301	164	5	z4	z4	PROPN
ejpam-4301	164	6	are	be	AUX
ejpam-4301	164	7	:	:	PUNCT
ejpam-4301	164	8	{	{	PUNCT
ejpam-4301	164	9	0	0	NUM
ejpam-4301	164	10	}	}	PUNCT
ejpam-4301	164	11	,	,	PUNCT
ejpam-4301	164	12	{	{	PUNCT
ejpam-4301	164	13	0	0	NUM
ejpam-4301	164	14	,	,	PUNCT
ejpam-4301	164	15	2	2	NUM
ejpam-4301	164	16	}	}	PUNCT
ejpam-4301	164	17	and	and	CCONJ
ejpam-4301	164	18	z4	z4	NOUN
ejpam-4301	164	19	.	.	PUNCT
ejpam-4301	165	1	hence	hence	ADV
ejpam-4301	165	2	,	,	PUNCT
ejpam-4301	165	3	the	the	DET
ejpam-4301	165	4	e∗-essential	e∗-essential	PROPN
ejpam-4301	165	5	submodule	submodule	NOUN
ejpam-4301	165	6	of	of	ADP
ejpam-4301	165	7	z4	z4	PROPN
ejpam-4301	165	8	are	be	AUX
ejpam-4301	165	9	:	:	PUNCT
ejpam-4301	165	10	{	{	PUNCT
ejpam-4301	165	11	0	0	NUM
ejpam-4301	165	12	,	,	PUNCT
ejpam-4301	165	13	2	2	NUM
ejpam-4301	165	14	}	}	PUNCT
ejpam-4301	165	15	and	and	CCONJ
ejpam-4301	165	16	z4	z4	NOUN
ejpam-4301	165	17	.	.	PUNCT
ejpam-4301	166	1	thus	thus	ADV
ejpam-4301	166	2	,	,	PUNCT
ejpam-4301	166	3	the	the	DET
ejpam-4301	166	4	only	only	ADJ
ejpam-4301	166	5	e∗-essential	e∗-essential	ADJ
ejpam-4301	166	6	maximal	maximal	ADJ
ejpam-4301	166	7	submodule	submodule	NOUN
ejpam-4301	166	8	is	be	AUX
ejpam-4301	166	9	{	{	PUNCT
ejpam-4301	166	10	0	0	NUM
ejpam-4301	166	11	,	,	PUNCT
ejpam-4301	166	12	2	2	NUM
ejpam-4301	166	13	}	}	PUNCT
ejpam-4301	166	14	.	.	PUNCT
ejpam-4301	167	1	h.r	h.r	PROPN
ejpam-4301	167	2	.	.	PROPN
ejpam-4301	167	3	baanoon	baanoon	PROPN
ejpam-4301	167	4	,	,	PUNCT
ejpam-4301	167	5	w.	w.	PROPN
ejpam-4301	167	6	khalid	khalid	PROPN
ejpam-4301	167	7	/	/	PUNCT
ejpam-4301	167	8	eur	eur	PROPN
ejpam-4301	167	9	.	.	PUNCT
ejpam-4301	168	1	j.	j.	PROPN
ejpam-4301	168	2	pure	pure	PROPN
ejpam-4301	168	3	appl	appl	PROPN
ejpam-4301	168	4	.	.	PROPN
ejpam-4301	168	5	math	math	PROPN
ejpam-4301	168	6	,	,	PUNCT
ejpam-4301	168	7	15	15	NUM
ejpam-4301	168	8	(	(	PUNCT
ejpam-4301	168	9	2	2	NUM
ejpam-4301	168	10	)	)	PUNCT
ejpam-4301	168	11	(	(	PUNCT
ejpam-4301	168	12	2022	2022	NUM
ejpam-4301	168	13	)	)	PUNCT
ejpam-4301	168	14	,	,	PUNCT
ejpam-4301	168	15	478	478	NUM
ejpam-4301	168	16	-	-	SYM
ejpam-4301	168	17	485	485	NUM
ejpam-4301	168	18	482	482	NUM
ejpam-4301	168	19	theorem	theorem	NOUN
ejpam-4301	168	20	1	1	NUM
ejpam-4301	168	21	.	.	PUNCT
ejpam-4301	169	1	let	let	VERB
ejpam-4301	169	2	m	m	PRON
ejpam-4301	169	3	be	be	AUX
ejpam-4301	169	4	an	an	DET
ejpam-4301	169	5	r	r	NOUN
ejpam-4301	169	6	-	-	PUNCT
ejpam-4301	169	7	module	module	NOUN
ejpam-4301	169	8	,	,	PUNCT
ejpam-4301	169	9	then	then	ADV
ejpam-4301	169	10	rad(m	rad(m	PROPN
ejpam-4301	169	11	)	)	PUNCT
ejpam-4301	169	12	e∗	e∗	NOUN
ejpam-4301	169	13	=	=	SYM
ejpam-4301	169	14	∑	∑	PROPN
ejpam-4301	169	15	n	n	PROPN
ejpam-4301	169	16	n	n	PRON
ejpam-4301	169	17	�	�	PROPN
ejpam-4301	169	18	e∗m	e∗m	PUNCT
ejpam-4301	169	19	.	.	PUNCT
ejpam-4301	170	1	proof	proof	NOUN
ejpam-4301	170	2	.	.	PUNCT
ejpam-4301	171	1	let	let	VERB
ejpam-4301	171	2	m	m	PRON
ejpam-4301	171	3	/∈	/∈	VERB
ejpam-4301	172	1	rad(m	rad(m	ADJ
ejpam-4301	172	2	)	)	PUNCT
ejpam-4301	172	3	e∗	e∗	NOUN
ejpam-4301	172	4	then	then	ADV
ejpam-4301	172	5	there	there	PRON
ejpam-4301	172	6	exists	exist	VERB
ejpam-4301	172	7	an	an	DET
ejpam-4301	172	8	e∗-essential	e∗-essential	PROPN
ejpam-4301	172	9	maximal	maximal	ADJ
ejpam-4301	172	10	n	n	NOUN
ejpam-4301	172	11	of	of	ADP
ejpam-4301	172	12	m	m	PRON
ejpam-4301	172	13	such	such	ADJ
ejpam-4301	172	14	that	that	SCONJ
ejpam-4301	172	15	m	m	VERB
ejpam-4301	172	16	/∈	/∈	PUNCT
ejpam-4301	173	1	n	n	INTJ
ejpam-4301	173	2	.	.	PUNCT
ejpam-4301	174	1	hence	hence	ADV
ejpam-4301	174	2	by	by	ADP
ejpam-4301	174	3	proposition	proposition	NOUN
ejpam-4301	174	4	5	5	NUM
ejpam-4301	174	5	,	,	PUNCT
ejpam-4301	174	6	we	we	PRON
ejpam-4301	174	7	have	have	VERB
ejpam-4301	174	8	that	that	PRON
ejpam-4301	174	9	〈	〈	PROPN
ejpam-4301	174	10	m	m	PROPN
ejpam-4301	174	11	〉	〉	PROPN
ejpam-4301	174	12	is	be	AUX
ejpam-4301	174	13	not	not	PART
ejpam-4301	174	14	e∗-essential	e∗-essential	PROPN
ejpam-4301	174	15	small	small	ADJ
ejpam-4301	174	16	.	.	PUNCT
ejpam-4301	175	1	thus	thus	ADV
ejpam-4301	175	2	,	,	PUNCT
ejpam-4301	175	3	m	m	VERB
ejpam-4301	175	4	/∈	/∈	ADJ
ejpam-4301	175	5	∑	∑	PUNCT
ejpam-4301	175	6	{	{	PUNCT
ejpam-4301	175	7	n	n	CCONJ
ejpam-4301	175	8	|n	|n	X
ejpam-4301	175	9	�	�	PROPN
ejpam-4301	175	10	e∗	e∗	PROPN
ejpam-4301	175	11	m	m	PROPN
ejpam-4301	175	12	}	}	PUNCT
ejpam-4301	175	13	.	.	PUNCT
ejpam-4301	176	1	therefore	therefore	ADV
ejpam-4301	176	2	,	,	PUNCT
ejpam-4301	176	3	∑	∑	ADP
ejpam-4301	176	4	{	{	PUNCT
ejpam-4301	176	5	n	n	CCONJ
ejpam-4301	176	6	|n	|n	X
ejpam-4301	176	7	�	�	PROPN
ejpam-4301	176	8	e∗	e∗	PROPN
ejpam-4301	176	9	m	m	PROPN
ejpam-4301	176	10	}	}	PUNCT
ejpam-4301	176	11	⊆	⊆	NUM
ejpam-4301	176	12	rad(m	rad(m	NUM
ejpam-4301	176	13	)	)	PUNCT
ejpam-4301	176	14	e∗	e∗	NOUN
ejpam-4301	176	15	.	.	PUNCT
ejpam-4301	177	1	now	now	ADV
ejpam-4301	177	2	,	,	PUNCT
ejpam-4301	177	3	let	let	VERB
ejpam-4301	177	4	x	x	X
ejpam-4301	177	5	∈	∈	PROPN
ejpam-4301	177	6	rad(m	rad(m	PROPN
ejpam-4301	177	7	)	)	PUNCT
ejpam-4301	177	8	e∗	e∗	NOUN
ejpam-4301	177	9	and	and	CCONJ
ejpam-4301	177	10	x	x	NOUN
ejpam-4301	177	11	/∈	/∈	PUNCT
ejpam-4301	177	12	∑	∑	PUNCT
ejpam-4301	177	13	{	{	PUNCT
ejpam-4301	177	14	n	n	CCONJ
ejpam-4301	177	15	|n	|n	X
ejpam-4301	177	16	�	�	PROPN
ejpam-4301	177	17	e∗	e∗	PROPN
ejpam-4301	177	18	m	m	PROPN
ejpam-4301	177	19	}	}	PUNCT
ejpam-4301	177	20	.	.	PUNCT
ejpam-4301	178	1	hence	hence	ADV
ejpam-4301	178	2	,	,	PUNCT
ejpam-4301	178	3	〈	〈	PROPN
ejpam-4301	178	4	x	x	PART
ejpam-4301	178	5	〉	〉	PROPN
ejpam-4301	178	6	is	be	AUX
ejpam-4301	178	7	not	not	PART
ejpam-4301	178	8	e∗-essential	e∗-essential	ADJ
ejpam-4301	178	9	small	small	ADJ
ejpam-4301	178	10	and	and	CCONJ
ejpam-4301	178	11	by	by	ADP
ejpam-4301	178	12	proposition	proposition	NOUN
ejpam-4301	178	13	5	5	NUM
ejpam-4301	178	14	,	,	PUNCT
ejpam-4301	178	15	there	there	PRON
ejpam-4301	178	16	exists	exist	VERB
ejpam-4301	178	17	an	an	DET
ejpam-4301	178	18	e∗-essential	e∗-essential	PROPN
ejpam-4301	178	19	maximal	maximal	ADJ
ejpam-4301	178	20	submodule	submodule	NOUN
ejpam-4301	178	21	k	k	PROPN
ejpam-4301	178	22	of	of	ADP
ejpam-4301	178	23	m	m	PROPN
ejpam-4301	178	24	such	such	ADJ
ejpam-4301	178	25	that	that	SCONJ
ejpam-4301	178	26	x	x	PROPN
ejpam-4301	178	27	/∈	/∈	PUNCT
ejpam-4301	179	1	k	k	PROPN
ejpam-4301	179	2	but	but	CCONJ
ejpam-4301	179	3	rad(m	rad(m	PROPN
ejpam-4301	179	4	)	)	PUNCT
ejpam-4301	179	5	e∗	e∗	NOUN
ejpam-4301	179	6	≤	≤	PROPN
ejpam-4301	179	7	k	k	X
ejpam-4301	179	8	which	which	PRON
ejpam-4301	179	9	is	be	AUX
ejpam-4301	179	10	a	a	DET
ejpam-4301	179	11	contradiction	contradiction	NOUN
ejpam-4301	179	12	.	.	PUNCT
ejpam-4301	180	1	thus	thus	ADV
ejpam-4301	180	2	,	,	PUNCT
ejpam-4301	180	3	x	x	SYM
ejpam-4301	180	4	∈	∈	PROPN
ejpam-4301	180	5	∑	∑	PUNCT
ejpam-4301	180	6	{	{	PUNCT
ejpam-4301	180	7	n	n	CCONJ
ejpam-4301	180	8	|n	|n	X
ejpam-4301	180	9	�	�	PROPN
ejpam-4301	180	10	e∗	e∗	PROPN
ejpam-4301	180	11	m	m	PROPN
ejpam-4301	180	12	}	}	PUNCT
ejpam-4301	180	13	and	and	CCONJ
ejpam-4301	180	14	rad(m	rad(m	ADJ
ejpam-4301	180	15	)	)	PUNCT
ejpam-4301	180	16	e∗	e∗	NOUN
ejpam-4301	180	17	≤	≤	PROPN
ejpam-4301	180	18	∑	∑	PUNCT
ejpam-4301	180	19	{	{	PUNCT
ejpam-4301	180	20	n	n	CCONJ
ejpam-4301	180	21	|n	|n	X
ejpam-4301	180	22	�	�	PROPN
ejpam-4301	180	23	e∗	e∗	PROPN
ejpam-4301	180	24	m	m	PROPN
ejpam-4301	180	25	}	}	PUNCT
ejpam-4301	180	26	.	.	PUNCT
ejpam-4301	180	27	therefore	therefore	ADV
ejpam-4301	180	28	,	,	PUNCT
ejpam-4301	180	29	rad(m	rad(m	PROPN
ejpam-4301	180	30	)	)	PUNCT
ejpam-4301	180	31	e∗	e∗	NOUN
ejpam-4301	180	32	=	=	SYM
ejpam-4301	180	33	∑	∑	PUNCT
ejpam-4301	180	34	{	{	PUNCT
ejpam-4301	180	35	n	n	CCONJ
ejpam-4301	180	36	|n	|n	X
ejpam-4301	180	37	�	�	PROPN
ejpam-4301	180	38	e∗	e∗	PROPN
ejpam-4301	180	39	m	m	PRON
ejpam-4301	180	40	}	}	PUNCT
ejpam-4301	180	41	.	.	PUNCT
ejpam-4301	181	1	proposition	proposition	NOUN
ejpam-4301	181	2	6	6	NUM
ejpam-4301	181	3	.	.	PUNCT
ejpam-4301	182	1	if	if	SCONJ
ejpam-4301	182	2	f	f	PROPN
ejpam-4301	182	3	:	:	PUNCT
ejpam-4301	182	4	m	m	VERB
ejpam-4301	182	5	→	→	SYM
ejpam-4301	182	6	m	m	AUX
ejpam-4301	182	7	′	′	NOUN
ejpam-4301	182	8	is	be	AUX
ejpam-4301	182	9	an	an	DET
ejpam-4301	182	10	r	r	NOUN
ejpam-4301	182	11	-	-	PUNCT
ejpam-4301	182	12	homomorphism	homomorphism	NOUN
ejpam-4301	182	13	,	,	PUNCT
ejpam-4301	182	14	then	then	ADV
ejpam-4301	182	15	f(rad(m	f(rad(m	ADJ
ejpam-4301	182	16	)	)	PUNCT
ejpam-4301	182	17	e∗	e∗	NOUN
ejpam-4301	182	18	)	)	PUNCT
ejpam-4301	182	19	≤	≤	PUNCT
ejpam-4301	183	1	rad(m	rad(m	NUM
ejpam-4301	183	2	′	′	NOUN
ejpam-4301	183	3	)	)	PUNCT
ejpam-4301	183	4	e∗	e∗	PROPN
ejpam-4301	183	5	.	.	PUNCT
ejpam-4301	184	1	in	in	ADP
ejpam-4301	184	2	particular	particular	ADJ
ejpam-4301	184	3	,	,	PUNCT
ejpam-4301	184	4	rad(m	rad(m	PROPN
ejpam-4301	184	5	)	)	PUNCT
ejpam-4301	184	6	e∗	e∗	NOUN
ejpam-4301	184	7	is	be	AUX
ejpam-4301	184	8	a	a	DET
ejpam-4301	184	9	fully	fully	ADV
ejpam-4301	184	10	invariant	invariant	ADJ
ejpam-4301	184	11	submodule	submodule	NOUN
ejpam-4301	184	12	of	of	ADP
ejpam-4301	184	13	m	m	PROPN
ejpam-4301	184	14	.	.	PUNCT
ejpam-4301	185	1	proof	proof	NOUN
ejpam-4301	185	2	.	.	PUNCT
ejpam-4301	186	1	by	by	ADP
ejpam-4301	186	2	therorm	therorm	NOUN
ejpam-4301	186	3	1	1	NUM
ejpam-4301	186	4	,	,	PUNCT
ejpam-4301	186	5	rad(m	rad(m	NUM
ejpam-4301	186	6	)	)	PUNCT
ejpam-4301	186	7	e∗	e∗	NOUN
ejpam-4301	186	8	=	=	SYM
ejpam-4301	186	9	∑	∑	PUNCT
ejpam-4301	186	10	k	k	PROPN
ejpam-4301	186	11	k	k	X
ejpam-4301	186	12	�	�	PROPN
ejpam-4301	186	13	e∗m	e∗m	PUNCT
ejpam-4301	186	14	.	.	PUNCT
ejpam-4301	187	1	hence	hence	ADV
ejpam-4301	187	2	,	,	PUNCT
ejpam-4301	187	3	f(rad(m	f(rad(m	ADJ
ejpam-4301	187	4	)	)	PUNCT
ejpam-4301	187	5	e∗	e∗	NOUN
ejpam-4301	187	6	)	)	PUNCT
ejpam-4301	188	1	=	=	PUNCT
ejpam-4301	188	2	∑	∑	PUNCT
ejpam-4301	188	3	f(k	f(k	ADJ
ejpam-4301	188	4	)	)	PUNCT
ejpam-4301	188	5	k	k	X
ejpam-4301	188	6	�	�	PROPN
ejpam-4301	188	7	e∗m	e∗m	PUNCT
ejpam-4301	188	8	.	.	PUNCT
ejpam-4301	189	1	by	by	ADP
ejpam-4301	189	2	proposition	proposition	NOUN
ejpam-4301	189	3	3	3	NUM
ejpam-4301	189	4	,	,	PUNCT
ejpam-4301	189	5	since	since	SCONJ
ejpam-4301	189	6	k	k	PROPN
ejpam-4301	189	7	�	�	PROPN
ejpam-4301	189	8	e∗	e∗	PROPN
ejpam-4301	189	9	m	m	PROPN
ejpam-4301	189	10	then	then	ADV
ejpam-4301	189	11	f(k	f(k	VERB
ejpam-4301	189	12	)	)	PUNCT
ejpam-4301	189	13	�	�	PROPN
ejpam-4301	189	14	e∗	e∗	PROPN
ejpam-4301	189	15	m	m	PROPN
ejpam-4301	189	16	′.	′.	PROPN
ejpam-4301	189	17	thus	thus	ADV
ejpam-4301	189	18	,	,	PUNCT
ejpam-4301	189	19	∑	∑	ADV
ejpam-4301	189	20	f(k	f(k	VERB
ejpam-4301	189	21	)	)	PUNCT
ejpam-4301	189	22	k	k	X
ejpam-4301	189	23	�	�	PROPN
ejpam-4301	189	24	e∗m	e∗m	PUNCT
ejpam-4301	189	25	≤	≤	NUM
ejpam-4301	189	26	rad(m	rad(m	PROPN
ejpam-4301	189	27	′	′	NOUN
ejpam-4301	189	28	)	)	PUNCT
ejpam-4301	189	29	e∗	e∗	NOUN
ejpam-4301	189	30	and	and	CCONJ
ejpam-4301	189	31	f(rad(m	f(rad(m	ADJ
ejpam-4301	189	32	)	)	PUNCT
ejpam-4301	189	33	)	)	PUNCT
ejpam-4301	189	34	e∗	e∗	PROPN
ejpam-4301	189	35	≤	≤	PROPN
ejpam-4301	190	1	rad(m	rad(m	PROPN
ejpam-4301	190	2	′	′	NOUN
ejpam-4301	190	3	)	)	PUNCT
ejpam-4301	190	4	e∗	e∗	PROPN
ejpam-4301	190	5	.	.	PUNCT
ejpam-4301	191	1	corollary	corollary	ADJ
ejpam-4301	191	2	2	2	NUM
ejpam-4301	191	3	.	.	PUNCT
ejpam-4301	192	1	let	let	VERB
ejpam-4301	192	2	m	m	PRON
ejpam-4301	192	3	be	be	AUX
ejpam-4301	192	4	an	an	DET
ejpam-4301	192	5	r	r	NOUN
ejpam-4301	192	6	-	-	PUNCT
ejpam-4301	192	7	module	module	NOUN
ejpam-4301	192	8	and	and	CCONJ
ejpam-4301	192	9	n	n	CCONJ
ejpam-4301	192	10	be	be	VERB
ejpam-4301	192	11	a	a	DET
ejpam-4301	192	12	submodule	submodule	NOUN
ejpam-4301	192	13	of	of	ADP
ejpam-4301	192	14	m	m	PROPN
ejpam-4301	192	15	,	,	PUNCT
ejpam-4301	192	16	then	then	ADV
ejpam-4301	192	17	:	:	PUNCT
ejpam-4301	192	18	1	1	X
ejpam-4301	192	19	.	.	X
ejpam-4301	192	20	rad(n	rad(n	NOUN
ejpam-4301	192	21	)	)	PUNCT
ejpam-4301	192	22	e∗	e∗	PROPN
ejpam-4301	192	23	≤	≤	PROPN
ejpam-4301	192	24	rad(m	rad(m	PROPN
ejpam-4301	192	25	)	)	PUNCT
ejpam-4301	192	26	e∗	e∗	NOUN
ejpam-4301	192	27	.	.	PUNCT
ejpam-4301	193	1	2	2	X
ejpam-4301	193	2	.	.	X
ejpam-4301	193	3	rad(m	rad(m	ADJ
ejpam-4301	193	4	)	)	PUNCT
ejpam-4301	193	5	e∗	e∗	NOUN
ejpam-4301	193	6	n	n	CCONJ
ejpam-4301	193	7	≤	≤	PROPN
ejpam-4301	193	8	rad(mn	rad(mn	NOUN
ejpam-4301	193	9	)	)	PUNCT
ejpam-4301	193	10	e∗	e∗	PROPN
ejpam-4301	193	11	.	.	PUNCT
ejpam-4301	194	1	4	4	X
ejpam-4301	194	2	.	.	X
ejpam-4301	194	3	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	194	4	modules	module	NOUN
ejpam-4301	194	5	recall	recall	VERB
ejpam-4301	194	6	that	that	SCONJ
ejpam-4301	194	7	a	a	DET
ejpam-4301	194	8	non	non	ADJ
ejpam-4301	194	9	-	-	ADJ
ejpam-4301	194	10	zeror	zeror	NOUN
ejpam-4301	194	11	-	-	PUNCT
ejpam-4301	194	12	modulem	modulem	NOUN
ejpam-4301	194	13	is	be	AUX
ejpam-4301	194	14	called	call	VERB
ejpam-4301	194	15	a	a	DET
ejpam-4301	194	16	hollow	hollow	ADJ
ejpam-4301	194	17	module	module	NOUN
ejpam-4301	194	18	if	if	SCONJ
ejpam-4301	194	19	every	every	DET
ejpam-4301	194	20	proper	proper	ADJ
ejpam-4301	194	21	submodule	submodule	NOUN
ejpam-4301	194	22	of	of	ADP
ejpam-4301	194	23	m	m	PROPN
ejpam-4301	194	24	is	be	AUX
ejpam-4301	194	25	small	small	ADJ
ejpam-4301	194	26	in	in	ADP
ejpam-4301	194	27	m	m	PROPN
ejpam-4301	194	28	[	[	X
ejpam-4301	194	29	3	3	NUM
ejpam-4301	194	30	]	]	PUNCT
ejpam-4301	194	31	.	.	PUNCT
ejpam-4301	195	1	in	in	ADP
ejpam-4301	195	2	this	this	DET
ejpam-4301	195	3	section	section	NOUN
ejpam-4301	195	4	we	we	PRON
ejpam-4301	195	5	introduce	introduce	VERB
ejpam-4301	195	6	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	195	7	modules	module	NOUN
ejpam-4301	195	8	as	as	ADP
ejpam-4301	195	9	a	a	DET
ejpam-4301	195	10	generalization	generalization	NOUN
ejpam-4301	195	11	of	of	ADP
ejpam-4301	195	12	hollow	hollow	ADJ
ejpam-4301	195	13	modules	module	NOUN
ejpam-4301	195	14	and	and	CCONJ
ejpam-4301	195	15	investigate	investigate	VERB
ejpam-4301	195	16	some	some	PRON
ejpam-4301	195	17	of	of	ADP
ejpam-4301	195	18	their	their	PRON
ejpam-4301	195	19	properties	property	NOUN
ejpam-4301	195	20	.	.	PUNCT
ejpam-4301	196	1	definition	definition	NOUN
ejpam-4301	196	2	3	3	NUM
ejpam-4301	196	3	.	.	PUNCT
ejpam-4301	196	4	a	a	DET
ejpam-4301	196	5	non	non	ADJ
ejpam-4301	196	6	zero	zero	NUM
ejpam-4301	196	7	r	r	NOUN
ejpam-4301	196	8	-	-	PUNCT
ejpam-4301	196	9	module	module	NOUN
ejpam-4301	196	10	m	m	NOUN
ejpam-4301	196	11	is	be	AUX
ejpam-4301	196	12	called	call	VERB
ejpam-4301	196	13	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	196	14	module	module	NOUN
ejpam-4301	196	15	if	if	SCONJ
ejpam-4301	196	16	every	every	DET
ejpam-4301	196	17	proper	proper	ADJ
ejpam-4301	196	18	submodule	submodule	NOUN
ejpam-4301	196	19	of	of	ADP
ejpam-4301	196	20	m	m	PROPN
ejpam-4301	196	21	is	be	AUX
ejpam-4301	196	22	e∗-essential	e∗-essential	PROPN
ejpam-4301	196	23	small	small	ADJ
ejpam-4301	196	24	in	in	ADP
ejpam-4301	196	25	m	m	PROPN
ejpam-4301	196	26	.	.	PUNCT
ejpam-4301	197	1	examples	example	NOUN
ejpam-4301	197	2	and	and	CCONJ
ejpam-4301	197	3	remarks	remark	VERB
ejpam-4301	197	4	3	3	NUM
ejpam-4301	197	5	.	.	NOUN
ejpam-4301	197	6	1	1	NUM
ejpam-4301	197	7	.	.	X
ejpam-4301	198	1	every	every	DET
ejpam-4301	198	2	hollow	hollow	ADJ
ejpam-4301	198	3	module	module	NOUN
ejpam-4301	198	4	is	be	AUX
ejpam-4301	198	5	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	198	6	module	module	NOUN
ejpam-4301	198	7	.	.	PUNCT
ejpam-4301	199	1	but	but	CCONJ
ejpam-4301	199	2	the	the	DET
ejpam-4301	199	3	converse	converse	NOUN
ejpam-4301	199	4	need	need	VERB
ejpam-4301	199	5	not	not	PART
ejpam-4301	199	6	to	to	PART
ejpam-4301	199	7	be	be	AUX
ejpam-4301	199	8	true	true	ADJ
ejpam-4301	199	9	in	in	ADP
ejpam-4301	199	10	general	general	ADJ
ejpam-4301	199	11	.	.	PUNCT
ejpam-4301	200	1	for	for	ADP
ejpam-4301	200	2	example	example	NOUN
ejpam-4301	200	3	:	:	PUNCT
ejpam-4301	200	4	in	in	ADP
ejpam-4301	200	5	z6	z6	PROPN
ejpam-4301	200	6	as	as	ADP
ejpam-4301	200	7	z	z	NOUN
ejpam-4301	200	8	-	-	PUNCT
ejpam-4301	200	9	module	module	NOUN
ejpam-4301	200	10	every	every	DET
ejpam-4301	200	11	proper	proper	ADJ
ejpam-4301	200	12	submodule	submodule	NOUN
ejpam-4301	200	13	is	be	AUX
ejpam-4301	200	14	e∗-essential	e∗-essential	PROPN
ejpam-4301	200	15	small	small	ADJ
ejpam-4301	200	16	,	,	PUNCT
ejpam-4301	200	17	hence	hence	ADV
ejpam-4301	200	18	z6	z6	PROPN
ejpam-4301	200	19	is	be	AUX
ejpam-4301	200	20	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	200	21	module	module	NOUN
ejpam-4301	200	22	,	,	PUNCT
ejpam-4301	200	23	but	but	CCONJ
ejpam-4301	200	24	it	it	PRON
ejpam-4301	200	25	is	be	AUX
ejpam-4301	200	26	not	not	PART
ejpam-4301	200	27	hollow	hollow	ADJ
ejpam-4301	200	28	,	,	PUNCT
ejpam-4301	200	29	since	since	SCONJ
ejpam-4301	200	30	〈	〈	PROPN
ejpam-4301	200	31	2	2	NUM
ejpam-4301	200	32	〉	〉	NOUN
ejpam-4301	200	33	is	be	AUX
ejpam-4301	200	34	not	not	PART
ejpam-4301	200	35	small	small	ADJ
ejpam-4301	200	36	submodule	submodule	NOUN
ejpam-4301	200	37	.	.	PUNCT
ejpam-4301	201	1	h.r	h.r	PROPN
ejpam-4301	201	2	.	.	PROPN
ejpam-4301	201	3	baanoon	baanoon	PROPN
ejpam-4301	201	4	,	,	PUNCT
ejpam-4301	201	5	w.	w.	PROPN
ejpam-4301	201	6	khalid	khalid	PROPN
ejpam-4301	201	7	/	/	PUNCT
ejpam-4301	201	8	eur	eur	PROPN
ejpam-4301	201	9	.	.	PUNCT
ejpam-4301	202	1	j.	j.	PROPN
ejpam-4301	202	2	pure	pure	PROPN
ejpam-4301	202	3	appl	appl	PROPN
ejpam-4301	202	4	.	.	PROPN
ejpam-4301	202	5	math	math	PROPN
ejpam-4301	202	6	,	,	PUNCT
ejpam-4301	202	7	15	15	NUM
ejpam-4301	202	8	(	(	PUNCT
ejpam-4301	202	9	2	2	NUM
ejpam-4301	202	10	)	)	PUNCT
ejpam-4301	202	11	(	(	PUNCT
ejpam-4301	202	12	2022	2022	NUM
ejpam-4301	202	13	)	)	PUNCT
ejpam-4301	202	14	,	,	PUNCT
ejpam-4301	202	15	478	478	NUM
ejpam-4301	202	16	-	-	SYM
ejpam-4301	202	17	485	485	NUM
ejpam-4301	202	18	483	483	NUM
ejpam-4301	202	19	2	2	NUM
ejpam-4301	202	20	.	.	PUNCT
ejpam-4301	202	21	consider	consider	VERB
ejpam-4301	202	22	z6	z6	PROPN
ejpam-4301	202	23	as	as	ADP
ejpam-4301	202	24	a	a	DET
ejpam-4301	202	25	z6	z6	NOUN
ejpam-4301	202	26	-	-	PUNCT
ejpam-4301	202	27	module	module	NOUN
ejpam-4301	202	28	.	.	PUNCT
ejpam-4301	203	1	since	since	SCONJ
ejpam-4301	203	2	〈	〈	PROPN
ejpam-4301	203	3	2	2	NUM
ejpam-4301	203	4	〉	〉	NOUN
ejpam-4301	203	5	is	be	AUX
ejpam-4301	203	6	not	not	PART
ejpam-4301	203	7	an	an	DET
ejpam-4301	203	8	e∗-essential	e∗-essential	ADJ
ejpam-4301	203	9	small	small	ADJ
ejpam-4301	203	10	submodule	submodule	NOUN
ejpam-4301	203	11	.	.	PUNCT
ejpam-4301	204	1	thus	thus	ADV
ejpam-4301	204	2	,	,	PUNCT
ejpam-4301	204	3	z6	z6	PROPN
ejpam-4301	204	4	is	be	AUX
ejpam-4301	204	5	not	not	PART
ejpam-4301	204	6	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	204	7	module	module	NOUN
ejpam-4301	204	8	.	.	PUNCT
ejpam-4301	205	1	3	3	X
ejpam-4301	205	2	.	.	X
ejpam-4301	205	3	the	the	DET
ejpam-4301	205	4	direct	direct	ADJ
ejpam-4301	205	5	sum	sum	NOUN
ejpam-4301	205	6	of	of	ADP
ejpam-4301	205	7	two	two	NUM
ejpam-4301	205	8	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	205	9	modules	module	NOUN
ejpam-4301	205	10	need	need	VERB
ejpam-4301	205	11	not	not	PART
ejpam-4301	205	12	to	to	PART
ejpam-4301	205	13	be	be	AUX
ejpam-4301	205	14	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	205	15	.	.	PUNCT
ejpam-4301	206	1	for	for	ADP
ejpam-4301	206	2	example	example	NOUN
ejpam-4301	206	3	:	:	PUNCT
ejpam-4301	206	4	z4	z4	PROPN
ejpam-4301	206	5	as	as	ADP
ejpam-4301	206	6	a	a	DET
ejpam-4301	206	7	z	z	NOUN
ejpam-4301	206	8	-	-	PUNCT
ejpam-4301	206	9	module	module	NOUN
ejpam-4301	206	10	is	be	AUX
ejpam-4301	206	11	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	206	12	since	since	SCONJ
ejpam-4301	206	13	〈	〈	PROPN
ejpam-4301	206	14	2	2	NUM
ejpam-4301	206	15	〉	〉	NUM
ejpam-4301	206	16	and	and	CCONJ
ejpam-4301	206	17	z4	z4	PROPN
ejpam-4301	206	18	are	be	AUX
ejpam-4301	206	19	the	the	DET
ejpam-4301	206	20	only	only	ADJ
ejpam-4301	206	21	e∗-essntial	e∗-essntial	ADJ
ejpam-4301	206	22	submodules	submodule	NOUN
ejpam-4301	206	23	.	.	PUNCT
ejpam-4301	207	1	so	so	ADV
ejpam-4301	207	2	all	all	DET
ejpam-4301	207	3	the	the	DET
ejpam-4301	207	4	proper	proper	ADJ
ejpam-4301	207	5	submodules	submodule	NOUN
ejpam-4301	207	6	are	be	AUX
ejpam-4301	207	7	e∗-essntial	e∗-essntial	ADJ
ejpam-4301	207	8	small	small	ADJ
ejpam-4301	207	9	.	.	PUNCT
ejpam-4301	208	1	also	also	ADV
ejpam-4301	208	2	,	,	PUNCT
ejpam-4301	208	3	z3	z3	PROPN
ejpam-4301	208	4	as	as	ADP
ejpam-4301	208	5	a	a	DET
ejpam-4301	208	6	z	z	NOUN
ejpam-4301	208	7	-	-	PUNCT
ejpam-4301	208	8	module	module	NOUN
ejpam-4301	208	9	is	be	AUX
ejpam-4301	208	10	e∗hollow	e∗hollow	ADJ
ejpam-4301	208	11	since	since	SCONJ
ejpam-4301	208	12	the	the	DET
ejpam-4301	208	13	only	only	ADJ
ejpam-4301	208	14	e∗-essntial	e∗-essntial	ADJ
ejpam-4301	208	15	submodule	submodule	NOUN
ejpam-4301	208	16	is	be	AUX
ejpam-4301	208	17	z3	z3	PROPN
ejpam-4301	208	18	it	it	PRON
ejpam-4301	208	19	self	self	NOUN
ejpam-4301	208	20	.	.	PUNCT
ejpam-4301	209	1	but	but	CCONJ
ejpam-4301	209	2	z4	z4	PROPN
ejpam-4301	209	3	⊕z3	⊕z3	PART
ejpam-4301	209	4	'	'	PART
ejpam-4301	209	5	z12	z12	PROPN
ejpam-4301	209	6	and	and	CCONJ
ejpam-4301	209	7	z12	z12	PROPN
ejpam-4301	209	8	is	be	AUX
ejpam-4301	209	9	not	not	PART
ejpam-4301	209	10	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	209	11	.	.	PUNCT
ejpam-4301	210	1	since	since	SCONJ
ejpam-4301	210	2	the	the	DET
ejpam-4301	210	3	only	only	ADJ
ejpam-4301	210	4	e∗-essntial	e∗-essntial	ADJ
ejpam-4301	210	5	submodule	submodule	NOUN
ejpam-4301	210	6	of	of	ADP
ejpam-4301	210	7	z12	z12	PROPN
ejpam-4301	210	8	are	be	AUX
ejpam-4301	210	9	〈	〈	PROPN
ejpam-4301	210	10	2	2	NUM
ejpam-4301	210	11	〉	〉	NUM
ejpam-4301	210	12	and	and	CCONJ
ejpam-4301	210	13	z12	z12	NUM
ejpam-4301	210	14	with	with	ADP
ejpam-4301	210	15	〈	〈	PROPN
ejpam-4301	210	16	3〉+	3〉+	PROPN
ejpam-4301	210	17	〈	〈	PROPN
ejpam-4301	210	18	2	2	NUM
ejpam-4301	210	19	〉	〉	NOUN
ejpam-4301	210	20	=	=	SYM
ejpam-4301	210	21	z12	z12	PROPN
ejpam-4301	210	22	but	but	CCONJ
ejpam-4301	210	23	〈	〈	PROPN
ejpam-4301	210	24	2	2	NUM
ejpam-4301	210	25	〉	〉	PROPN
ejpam-4301	210	26	6=	6=	NUM
ejpam-4301	210	27	z12	z12	PROPN
ejpam-4301	210	28	.	.	PROPN
ejpam-4301	211	1	4	4	NUM
ejpam-4301	211	2	.	.	X
ejpam-4301	212	1	any	any	DET
ejpam-4301	212	2	r	r	NOUN
ejpam-4301	212	3	-	-	PUNCT
ejpam-4301	212	4	module	module	NOUN
ejpam-4301	212	5	which	which	PRON
ejpam-4301	212	6	has	have	VERB
ejpam-4301	212	7	no	no	DET
ejpam-4301	212	8	proper	proper	ADJ
ejpam-4301	212	9	e∗-essential	e∗-essential	PROPN
ejpam-4301	212	10	submodule	submodule	NOUN
ejpam-4301	212	11	is	be	AUX
ejpam-4301	212	12	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	212	13	.	.	PUNCT
ejpam-4301	213	1	proposition	proposition	NOUN
ejpam-4301	213	2	7	7	NUM
ejpam-4301	213	3	.	.	PUNCT
ejpam-4301	214	1	the	the	DET
ejpam-4301	214	2	epimorphic	epimorphic	ADJ
ejpam-4301	214	3	image	image	NOUN
ejpam-4301	214	4	of	of	ADP
ejpam-4301	214	5	an	an	DET
ejpam-4301	214	6	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	214	7	module	module	NOUN
ejpam-4301	214	8	is	be	AUX
ejpam-4301	214	9	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	214	10	.	.	PUNCT
ejpam-4301	215	1	proof	proof	NOUN
ejpam-4301	215	2	.	.	PUNCT
ejpam-4301	216	1	let	let	VERB
ejpam-4301	216	2	f	f	NOUN
ejpam-4301	216	3	:	:	PUNCT
ejpam-4301	216	4	m	m	VERB
ejpam-4301	216	5	→	→	SYM
ejpam-4301	216	6	m	m	AUX
ejpam-4301	216	7	′	′	NUM
ejpam-4301	216	8	be	be	VERB
ejpam-4301	216	9	an	an	DET
ejpam-4301	216	10	r	r	NOUN
ejpam-4301	216	11	-	-	PUNCT
ejpam-4301	216	12	epimorphism	epimorphism	NOUN
ejpam-4301	216	13	,	,	PUNCT
ejpam-4301	216	14	with	with	ADP
ejpam-4301	216	15	m	m	DET
ejpam-4301	216	16	an	an	DET
ejpam-4301	216	17	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	216	18	module	module	NOUN
ejpam-4301	216	19	.	.	PUNCT
ejpam-4301	217	1	let	let	VERB
ejpam-4301	217	2	b	b	X
ejpam-4301	217	3	be	be	AUX
ejpam-4301	217	4	a	a	DET
ejpam-4301	217	5	proper	proper	ADJ
ejpam-4301	217	6	submodule	submodule	NOUN
ejpam-4301	217	7	of	of	ADP
ejpam-4301	217	8	m	m	PROPN
ejpam-4301	217	9	′.	′.	NOUN
ejpam-4301	217	10	hence	hence	ADV
ejpam-4301	217	11	f−1(b	f−1(b	PROPN
ejpam-4301	217	12	)	)	PUNCT
ejpam-4301	217	13	is	be	AUX
ejpam-4301	217	14	a	a	DET
ejpam-4301	217	15	proper	proper	ADJ
ejpam-4301	217	16	submodule	submodule	NOUN
ejpam-4301	217	17	of	of	ADP
ejpam-4301	217	18	m	m	PROPN
ejpam-4301	217	19	.	.	PUNCT
ejpam-4301	218	1	since	since	SCONJ
ejpam-4301	218	2	if	if	SCONJ
ejpam-4301	218	3	not	not	PART
ejpam-4301	218	4	,	,	PUNCT
ejpam-4301	218	5	f−1(b	f−1(b	PROPN
ejpam-4301	218	6	)	)	PUNCT
ejpam-4301	218	7	=	=	VERB
ejpam-4301	218	8	m	m	NOUN
ejpam-4301	218	9	implies	imply	VERB
ejpam-4301	218	10	that	that	SCONJ
ejpam-4301	218	11	ff−1(b	ff−1(b	ADJ
ejpam-4301	218	12	)	)	PUNCT
ejpam-4301	218	13	=	=	SYM
ejpam-4301	218	14	b	b	X
ejpam-4301	218	15	=	=	NOUN
ejpam-4301	218	16	m	m	NOUN
ejpam-4301	218	17	′	′	NOUN
ejpam-4301	218	18	which	which	PRON
ejpam-4301	218	19	is	be	AUX
ejpam-4301	218	20	a	a	DET
ejpam-4301	218	21	contradiction	contradiction	NOUN
ejpam-4301	218	22	.	.	PUNCT
ejpam-4301	219	1	since	since	SCONJ
ejpam-4301	219	2	m	m	PROPN
ejpam-4301	219	3	is	be	AUX
ejpam-4301	219	4	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	219	5	then	then	ADV
ejpam-4301	219	6	f−1(b	f−1(b	PROPN
ejpam-4301	219	7	)	)	PUNCT
ejpam-4301	219	8	is	be	AUX
ejpam-4301	219	9	e∗-essential	e∗-essential	PROPN
ejpam-4301	219	10	small	small	ADJ
ejpam-4301	219	11	.	.	PUNCT
ejpam-4301	220	1	by	by	ADP
ejpam-4301	220	2	proposition	proposition	NOUN
ejpam-4301	220	3	3	3	NUM
ejpam-4301	220	4	,	,	PUNCT
ejpam-4301	220	5	ff−1(b	ff−1(b	ADJ
ejpam-4301	220	6	)	)	PUNCT
ejpam-4301	220	7	=	=	SYM
ejpam-4301	220	8	b	b	NOUN
ejpam-4301	220	9	is	be	AUX
ejpam-4301	220	10	an	an	DET
ejpam-4301	220	11	e∗-essential	e∗-essential	ADJ
ejpam-4301	220	12	small	small	ADJ
ejpam-4301	220	13	submodule	submodule	NOUN
ejpam-4301	220	14	.	.	PUNCT
ejpam-4301	221	1	therefore	therefore	ADV
ejpam-4301	221	2	,	,	PUNCT
ejpam-4301	221	3	m	m	VERB
ejpam-4301	221	4	′is	′is	ADP
ejpam-4301	221	5	e∗-hollow	e∗-hollow	PROPN
ejpam-4301	221	6	.	.	PUNCT
ejpam-4301	221	7	corollary	corollary	ADJ
ejpam-4301	221	8	3	3	X
ejpam-4301	221	9	.	.	PUNCT
ejpam-4301	222	1	if	if	SCONJ
ejpam-4301	222	2	m	m	NOUN
ejpam-4301	222	3	is	be	AUX
ejpam-4301	222	4	an	an	DET
ejpam-4301	222	5	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	222	6	module	module	NOUN
ejpam-4301	222	7	,	,	PUNCT
ejpam-4301	222	8	then	then	ADV
ejpam-4301	222	9	m	m	VERB
ejpam-4301	222	10	n	n	ADJ
ejpam-4301	222	11	is	be	AUX
ejpam-4301	222	12	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	222	13	for	for	ADP
ejpam-4301	222	14	any	any	DET
ejpam-4301	222	15	proper	proper	ADJ
ejpam-4301	222	16	submodule	submodule	NOUN
ejpam-4301	222	17	n	n	PROPN
ejpam-4301	222	18	of	of	ADP
ejpam-4301	222	19	m	m	PROPN
ejpam-4301	222	20	.	.	PUNCT
ejpam-4301	223	1	remark	remark	PROPN
ejpam-4301	223	2	1	1	NUM
ejpam-4301	223	3	.	.	PUNCT
ejpam-4301	224	1	the	the	DET
ejpam-4301	224	2	converse	converse	NOUN
ejpam-4301	224	3	of	of	ADP
ejpam-4301	224	4	the	the	DET
ejpam-4301	224	5	above	above	ADJ
ejpam-4301	224	6	corollary	corollary	ADJ
ejpam-4301	224	7	need	need	NOUN
ejpam-4301	224	8	not	not	PART
ejpam-4301	224	9	to	to	PART
ejpam-4301	224	10	be	be	AUX
ejpam-4301	224	11	true	true	ADJ
ejpam-4301	224	12	in	in	ADP
ejpam-4301	224	13	general	general	ADJ
ejpam-4301	224	14	.	.	PUNCT
ejpam-4301	225	1	for	for	ADP
ejpam-4301	225	2	example	example	NOUN
ejpam-4301	225	3	:	:	PUNCT
ejpam-4301	225	4	consider	consider	VERB
ejpam-4301	225	5	z24	z24	PROPN
ejpam-4301	225	6	as	as	ADP
ejpam-4301	225	7	a	a	DET
ejpam-4301	225	8	z	z	NOUN
ejpam-4301	225	9	-	-	PUNCT
ejpam-4301	225	10	module	module	NOUN
ejpam-4301	225	11	which	which	PRON
ejpam-4301	225	12	is	be	AUX
ejpam-4301	225	13	not	not	PART
ejpam-4301	225	14	e∗-hollow	e∗-hollow	VERB
ejpam-4301	225	15	.	.	PUNCT
ejpam-4301	226	1	since	since	SCONJ
ejpam-4301	226	2	every	every	DET
ejpam-4301	226	3	submodule	submodule	NOUN
ejpam-4301	226	4	of	of	ADP
ejpam-4301	226	5	z24	z24	PROPN
ejpam-4301	226	6	is	be	AUX
ejpam-4301	226	7	cosingular	cosingular	ADJ
ejpam-4301	226	8	then	then	ADV
ejpam-4301	226	9	the	the	DET
ejpam-4301	226	10	only	only	ADJ
ejpam-4301	226	11	e∗-essntial	e∗-essntial	ADJ
ejpam-4301	226	12	submodule	submodule	NOUN
ejpam-4301	226	13	of	of	ADP
ejpam-4301	226	14	z24	z24	PROPN
ejpam-4301	226	15	are	be	AUX
ejpam-4301	226	16	〈	〈	PROPN
ejpam-4301	226	17	0	0	NUM
ejpam-4301	226	18	〉	〉	NUM
ejpam-4301	226	19	,	,	PUNCT
ejpam-4301	226	20	〈	〈	PROPN
ejpam-4301	226	21	2	2	NUM
ejpam-4301	226	22	〉	〉	NUM
ejpam-4301	226	23	,	,	PUNCT
ejpam-4301	226	24	〈	〈	PROPN
ejpam-4301	226	25	4	4	NUM
ejpam-4301	226	26	〉	〉	NUM
ejpam-4301	226	27	,	,	PUNCT
ejpam-4301	226	28	and	and	CCONJ
ejpam-4301	226	29	z24	z24	PROPN
ejpam-4301	226	30	.	.	PUNCT
ejpam-4301	227	1	since	since	SCONJ
ejpam-4301	227	2	〈	〈	PROPN
ejpam-4301	227	3	3〉+	3〉+	PROPN
ejpam-4301	227	4	〈	〈	PROPN
ejpam-4301	227	5	2	2	NUM
ejpam-4301	227	6	〉	〉	NOUN
ejpam-4301	227	7	=	=	NOUN
ejpam-4301	227	8	z24	z24	PROPN
ejpam-4301	227	9	and	and	CCONJ
ejpam-4301	227	10	〈	〈	PROPN
ejpam-4301	227	11	2	2	NUM
ejpam-4301	227	12	〉	〉	PROPN
ejpam-4301	227	13	6=	6=	NUM
ejpam-4301	227	14	z24	z24	PROPN
ejpam-4301	227	15	we	we	PRON
ejpam-4301	227	16	have	have	VERB
ejpam-4301	227	17	that	that	SCONJ
ejpam-4301	227	18	〈	〈	PROPN
ejpam-4301	227	19	3	3	NUM
ejpam-4301	227	20	〉	〉	PROPN
ejpam-4301	227	21	is	be	AUX
ejpam-4301	227	22	not	not	PART
ejpam-4301	227	23	e∗-essential	e∗-essential	PROPN
ejpam-4301	227	24	small	small	ADJ
ejpam-4301	227	25	.	.	PUNCT
ejpam-4301	228	1	but	but	CCONJ
ejpam-4301	228	2	z24	z24	PROPN
ejpam-4301	228	3	〈	〈	PROPN
ejpam-4301	228	4	4	4	NUM
ejpam-4301	228	5	〉	〉	NOUN
ejpam-4301	228	6	is	be	AUX
ejpam-4301	228	7	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	228	8	module	module	NOUN
ejpam-4301	228	9	since	since	SCONJ
ejpam-4301	228	10	z24	z24	PROPN
ejpam-4301	228	11	〈	〈	PROPN
ejpam-4301	228	12	4	4	NUM
ejpam-4301	228	13	〉	〉	PROPN
ejpam-4301	228	14	'	'	PUNCT
ejpam-4301	228	15	z4	z4	PROPN
ejpam-4301	228	16	.	.	PUNCT
ejpam-4301	229	1	the	the	DET
ejpam-4301	229	2	following	follow	VERB
ejpam-4301	229	3	proposition	proposition	NOUN
ejpam-4301	229	4	shows	show	VERB
ejpam-4301	229	5	that	that	SCONJ
ejpam-4301	229	6	under	under	ADP
ejpam-4301	229	7	certain	certain	ADJ
ejpam-4301	229	8	conditions	condition	NOUN
ejpam-4301	229	9	the	the	DET
ejpam-4301	229	10	converse	converse	NOUN
ejpam-4301	229	11	of	of	ADP
ejpam-4301	229	12	corollary	corollary	ADJ
ejpam-4301	229	13	3	3	NUM
ejpam-4301	229	14	is	be	AUX
ejpam-4301	229	15	true	true	ADJ
ejpam-4301	229	16	.	.	PUNCT
ejpam-4301	230	1	recall	recall	VERB
ejpam-4301	230	2	that	that	SCONJ
ejpam-4301	230	3	a	a	DET
ejpam-4301	230	4	submodule	submodule	NOUN
ejpam-4301	230	5	a	a	PRON
ejpam-4301	230	6	of	of	ADP
ejpam-4301	230	7	a	a	DET
ejpam-4301	230	8	module	module	NOUN
ejpam-4301	230	9	m	m	VERB
ejpam-4301	230	10	is	be	AUX
ejpam-4301	230	11	called	call	VERB
ejpam-4301	230	12	e∗-closed	e∗-close	VERB
ejpam-4301	230	13	if	if	SCONJ
ejpam-4301	230	14	a	a	PRON
ejpam-4301	230	15	has	have	VERB
ejpam-4301	230	16	no	no	DET
ejpam-4301	230	17	proper	proper	ADJ
ejpam-4301	230	18	e∗-essential	e∗-essential	ADJ
ejpam-4301	230	19	extension	extension	NOUN
ejpam-4301	230	20	inside	inside	ADP
ejpam-4301	230	21	m	m	PROPN
ejpam-4301	231	1	[	[	X
ejpam-4301	231	2	1	1	NUM
ejpam-4301	231	3	]	]	PUNCT
ejpam-4301	231	4	.	.	PUNCT
ejpam-4301	232	1	lemma	lemma	PROPN
ejpam-4301	232	2	2	2	NUM
ejpam-4301	232	3	.	.	PUNCT
ejpam-4301	233	1	[	[	X
ejpam-4301	233	2	1	1	X
ejpam-4301	233	3	]	]	X
ejpam-4301	233	4	if	if	SCONJ
ejpam-4301	233	5	b	b	PROPN
ejpam-4301	233	6	≤	≤	X
ejpam-4301	233	7	k	k	NOUN
ejpam-4301	233	8	are	be	AUX
ejpam-4301	233	9	submodules	submodule	NOUN
ejpam-4301	233	10	of	of	ADP
ejpam-4301	233	11	an	an	DET
ejpam-4301	233	12	r	r	NOUN
ejpam-4301	233	13	-	-	PUNCT
ejpam-4301	233	14	module	module	NOUN
ejpam-4301	233	15	m	m	NOUN
ejpam-4301	233	16	such	such	ADJ
ejpam-4301	233	17	that	that	SCONJ
ejpam-4301	233	18	b	b	NOUN
ejpam-4301	233	19	is	be	AUX
ejpam-4301	233	20	e∗-closed	e∗-close	VERB
ejpam-4301	233	21	in	in	ADP
ejpam-4301	233	22	m	m	PROPN
ejpam-4301	233	23	and	and	CCONJ
ejpam-4301	233	24	k	k	PROPN
ejpam-4301	233	25	is	be	AUX
ejpam-4301	233	26	e∗-essential	e∗-essential	PROPN
ejpam-4301	233	27	in	in	ADP
ejpam-4301	233	28	m	m	PROPN
ejpam-4301	233	29	,	,	PUNCT
ejpam-4301	233	30	then	then	ADV
ejpam-4301	233	31	k	k	PROPN
ejpam-4301	233	32	b	b	PROPN
ejpam-4301	233	33	≤e∗	≤e∗	PROPN
ejpam-4301	233	34	m	m	PROPN
ejpam-4301	233	35	b	b	PROPN
ejpam-4301	233	36	.	.	PUNCT
ejpam-4301	234	1	proposition	proposition	NOUN
ejpam-4301	234	2	8	8	NUM
ejpam-4301	234	3	.	.	PUNCT
ejpam-4301	235	1	let	let	VERB
ejpam-4301	235	2	m	m	PRON
ejpam-4301	235	3	be	be	AUX
ejpam-4301	235	4	an	an	DET
ejpam-4301	235	5	r	r	NOUN
ejpam-4301	235	6	-	-	PUNCT
ejpam-4301	235	7	module	module	NOUN
ejpam-4301	235	8	.	.	PUNCT
ejpam-4301	236	1	if	if	SCONJ
ejpam-4301	236	2	m	m	VERB
ejpam-4301	236	3	n	n	VERB
ejpam-4301	236	4	is	be	AUX
ejpam-4301	236	5	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	236	6	with	with	ADP
ejpam-4301	236	7	n	n	PRON
ejpam-4301	236	8	is	be	AUX
ejpam-4301	236	9	a	a	DET
ejpam-4301	236	10	proper	proper	ADJ
ejpam-4301	236	11	small	small	ADJ
ejpam-4301	236	12	e∗-closed	e∗-close	VERB
ejpam-4301	236	13	submodule	submodule	NOUN
ejpam-4301	236	14	,	,	PUNCT
ejpam-4301	236	15	then	then	ADV
ejpam-4301	236	16	m	m	VERB
ejpam-4301	236	17	is	be	AUX
ejpam-4301	236	18	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	236	19	.	.	PUNCT
ejpam-4301	237	1	proof	proof	NOUN
ejpam-4301	237	2	.	.	PUNCT
ejpam-4301	238	1	let	let	VERB
ejpam-4301	238	2	l	l	NOUN
ejpam-4301	238	3	be	be	AUX
ejpam-4301	238	4	a	a	DET
ejpam-4301	238	5	proper	proper	ADJ
ejpam-4301	238	6	submodule	submodule	NOUN
ejpam-4301	238	7	of	of	ADP
ejpam-4301	238	8	m	m	PROPN
ejpam-4301	238	9	and	and	CCONJ
ejpam-4301	238	10	k	k	X
ejpam-4301	238	11	an	an	DET
ejpam-4301	238	12	e∗-essential	e∗-essential	PROPN
ejpam-4301	238	13	submodule	submodule	NOUN
ejpam-4301	238	14	of	of	ADP
ejpam-4301	238	15	m	m	PRON
ejpam-4301	238	16	such	such	ADJ
ejpam-4301	238	17	that	that	SCONJ
ejpam-4301	238	18	l+k	l+k	PROPN
ejpam-4301	238	19	=	=	PUNCT
ejpam-4301	239	1	m	m	VERB
ejpam-4301	239	2	.	.	PUNCT
ejpam-4301	240	1	then	then	ADV
ejpam-4301	240	2	m	m	VERB
ejpam-4301	240	3	n	n	NOUN
ejpam-4301	240	4	=	=	PUNCT
ejpam-4301	240	5	l+n	l+n	PROPN
ejpam-4301	240	6	n	n	PROPN
ejpam-4301	240	7	+	+	SYM
ejpam-4301	240	8	k+n	k+n	PROPN
ejpam-4301	240	9	n	n	PRON
ejpam-4301	240	10	implies	imply	VERB
ejpam-4301	240	11	that	that	SCONJ
ejpam-4301	240	12	m	m	VERB
ejpam-4301	240	13	6=	6=	NUM
ejpam-4301	240	14	l+n	l+n	PROPN
ejpam-4301	240	15	.	.	PUNCT
ejpam-4301	241	1	for	for	ADP
ejpam-4301	241	2	if	if	SCONJ
ejpam-4301	241	3	m	m	VERB
ejpam-4301	241	4	=	=	NOUN
ejpam-4301	241	5	l+n	l+n	PROPN
ejpam-4301	241	6	with	with	ADP
ejpam-4301	241	7	n	n	CCONJ
ejpam-4301	241	8	a	a	DET
ejpam-4301	241	9	small	small	ADJ
ejpam-4301	241	10	submodule	submodule	NOUN
ejpam-4301	241	11	of	of	ADP
ejpam-4301	241	12	m	m	PROPN
ejpam-4301	241	13	i.e.	i.e.	X
ejpam-4301	241	14	m	m	ADJ
ejpam-4301	241	15	=	=	VERB
ejpam-4301	241	16	l	l	NOUN
ejpam-4301	241	17	which	which	PRON
ejpam-4301	241	18	is	be	AUX
ejpam-4301	241	19	a	a	DET
ejpam-4301	241	20	contradiction	contradiction	NOUN
ejpam-4301	241	21	.	.	PUNCT
ejpam-4301	242	1	thus	thus	ADV
ejpam-4301	242	2	,	,	PUNCT
ejpam-4301	242	3	m	m	VERB
ejpam-4301	242	4	n	n	PRON
ejpam-4301	242	5	6=	6=	NUM
ejpam-4301	242	6	l+n	l+n	PROPN
ejpam-4301	242	7	n	n	PROPN
ejpam-4301	242	8	.	.	PUNCT
ejpam-4301	243	1	since	since	ADV
ejpam-4301	243	2	,	,	PUNCT
ejpam-4301	243	3	n	n	PRON
ejpam-4301	243	4	≤e∗	≤e∗	NOUN
ejpam-4301	243	5	m	m	VERB
ejpam-4301	243	6	then	then	ADV
ejpam-4301	243	7	by	by	ADP
ejpam-4301	243	8	lemma	lemma	PROPN
ejpam-4301	243	9	1	1	NUM
ejpam-4301	243	10	,	,	PUNCT
ejpam-4301	243	11	n	n	PRON
ejpam-4301	243	12	≤ce∗	≤ce∗	PROPN
ejpam-4301	243	13	k	k	X
ejpam-4301	243	14	+	+	PROPN
ejpam-4301	243	15	n	n	PROPN
ejpam-4301	243	16	≤e∗	≤e∗	NOUN
ejpam-4301	243	17	m	m	NOUN
ejpam-4301	243	18	,	,	PUNCT
ejpam-4301	243	19	and	and	CCONJ
ejpam-4301	243	20	by	by	ADP
ejpam-4301	243	21	lemma	lemma	PROPN
ejpam-4301	243	22	2	2	NUM
ejpam-4301	243	23	,	,	PUNCT
ejpam-4301	243	24	k+n	k+n	PROPN
ejpam-4301	243	25	n	n	NUM
ejpam-4301	243	26	≤e∗	≤e∗	NOUN
ejpam-4301	243	27	m	m	NOUN
ejpam-4301	243	28	n	n	NOUN
ejpam-4301	243	29	.	.	PUNCT
ejpam-4301	244	1	since	since	SCONJ
ejpam-4301	244	2	m	m	NOUN
ejpam-4301	244	3	n	n	VERB
ejpam-4301	244	4	is	be	AUX
ejpam-4301	244	5	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	244	6	,	,	PUNCT
ejpam-4301	244	7	then	then	ADV
ejpam-4301	244	8	k+n	k+n	PROPN
ejpam-4301	244	9	n	n	NOUN
ejpam-4301	244	10	=	=	SYM
ejpam-4301	244	11	m	m	NOUN
ejpam-4301	244	12	n	n	NOUN
ejpam-4301	244	13	,	,	PUNCT
ejpam-4301	244	14	and	and	CCONJ
ejpam-4301	244	15	m	m	PROPN
ejpam-4301	245	1	=	=	SYM
ejpam-4301	245	2	k	k	PROPN
ejpam-4301	246	1	+	+	CCONJ
ejpam-4301	246	2	n	n	X
ejpam-4301	246	3	because	because	SCONJ
ejpam-4301	246	4	n	n	PROPN
ejpam-4301	246	5	�	�	PROPN
ejpam-4301	246	6	m	m	PROPN
ejpam-4301	246	7	.	.	PUNCT
ejpam-4301	247	1	therefore	therefore	ADV
ejpam-4301	247	2	,	,	PUNCT
ejpam-4301	247	3	k	k	PROPN
ejpam-4301	247	4	=	=	VERB
ejpam-4301	247	5	m	m	VERB
ejpam-4301	247	6	and	and	CCONJ
ejpam-4301	247	7	m	m	VERB
ejpam-4301	247	8	is	be	AUX
ejpam-4301	247	9	e∗-hollow	e∗-hollow	PROPN
ejpam-4301	247	10	.	.	PUNCT
ejpam-4301	248	1	h.r	h.r	PROPN
ejpam-4301	248	2	.	.	PROPN
ejpam-4301	248	3	baanoon	baanoon	PROPN
ejpam-4301	248	4	,	,	PUNCT
ejpam-4301	248	5	w.	w.	PROPN
ejpam-4301	248	6	khalid	khalid	PROPN
ejpam-4301	248	7	/	/	PUNCT
ejpam-4301	248	8	eur	eur	PROPN
ejpam-4301	248	9	.	.	PUNCT
ejpam-4301	249	1	j.	j.	PROPN
ejpam-4301	249	2	pure	pure	PROPN
ejpam-4301	249	3	appl	appl	PROPN
ejpam-4301	249	4	.	.	PROPN
ejpam-4301	249	5	math	math	PROPN
ejpam-4301	249	6	,	,	PUNCT
ejpam-4301	249	7	15	15	NUM
ejpam-4301	249	8	(	(	PUNCT
ejpam-4301	249	9	2	2	NUM
ejpam-4301	249	10	)	)	PUNCT
ejpam-4301	249	11	(	(	PUNCT
ejpam-4301	249	12	2022	2022	NUM
ejpam-4301	249	13	)	)	PUNCT
ejpam-4301	249	14	,	,	PUNCT
ejpam-4301	249	15	478	478	NUM
ejpam-4301	249	16	-	-	SYM
ejpam-4301	249	17	485	485	NUM
ejpam-4301	249	18	484	484	NUM
ejpam-4301	249	19	proposition	proposition	NOUN
ejpam-4301	249	20	9	9	NUM
ejpam-4301	249	21	.	.	PUNCT
ejpam-4301	250	1	let	let	VERB
ejpam-4301	250	2	m	m	PRON
ejpam-4301	250	3	be	be	AUX
ejpam-4301	250	4	an	an	DET
ejpam-4301	250	5	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	250	6	module	module	NOUN
ejpam-4301	250	7	,	,	PUNCT
ejpam-4301	250	8	if	if	SCONJ
ejpam-4301	250	9	m	m	PROPN
ejpam-4301	250	10	has	have	AUX
ejpam-4301	250	11	proper	proper	ADJ
ejpam-4301	250	12	a	a	DET
ejpam-4301	250	13	e∗-essential	e∗-essential	PROPN
ejpam-4301	250	14	submodule	submodule	NOUN
ejpam-4301	250	15	n	n	PROPN
ejpam-4301	250	16	and	and	CCONJ
ejpam-4301	250	17	m	m	PRON
ejpam-4301	250	18	n	n	PRON
ejpam-4301	250	19	is	be	AUX
ejpam-4301	250	20	finitely	finitely	ADV
ejpam-4301	250	21	generated	generate	VERB
ejpam-4301	250	22	then	then	ADV
ejpam-4301	250	23	m	m	VERB
ejpam-4301	250	24	is	be	AUX
ejpam-4301	250	25	finitely	finitely	ADV
ejpam-4301	250	26	generated	generate	VERB
ejpam-4301	250	27	.	.	PUNCT
ejpam-4301	251	1	proof	proof	NOUN
ejpam-4301	251	2	.	.	PUNCT
ejpam-4301	252	1	since	since	SCONJ
ejpam-4301	252	2	m	m	NOUN
ejpam-4301	252	3	n	n	VERB
ejpam-4301	252	4	is	be	AUX
ejpam-4301	252	5	finitely	finitely	ADV
ejpam-4301	252	6	generated	generate	VERB
ejpam-4301	252	7	there	there	PRON
ejpam-4301	252	8	are	be	VERB
ejpam-4301	252	9	x1	x1	PROPN
ejpam-4301	252	10	,	,	PUNCT
ejpam-4301	252	11	x2	x2	PROPN
ejpam-4301	252	12	,	,	PUNCT
ejpam-4301	252	13	...	...	PUNCT
ejpam-4301	252	14	,	,	PUNCT
ejpam-4301	252	15	xn	xn	PROPN
ejpam-4301	252	16	∈	∈	PROPN
ejpam-4301	252	17	m	m	VERB
ejpam-4301	252	18	such	such	ADJ
ejpam-4301	252	19	that	that	SCONJ
ejpam-4301	252	20	m	m	VERB
ejpam-4301	252	21	n	n	NOUN
ejpam-4301	252	22	=	=	SYM
ejpam-4301	252	23	〈	〈	PROPN
ejpam-4301	252	24	x1	x1	PROPN
ejpam-4301	252	25	+	+	SYM
ejpam-4301	252	26	n	n	CCONJ
ejpam-4301	252	27	,	,	PUNCT
ejpam-4301	252	28	x2	x2	PROPN
ejpam-4301	252	29	+	+	CCONJ
ejpam-4301	252	30	n	n	CCONJ
ejpam-4301	252	31	,	,	PUNCT
ejpam-4301	252	32	...	...	PUNCT
ejpam-4301	252	33	,	,	PUNCT
ejpam-4301	252	34	xn	xn	PROPN
ejpam-4301	253	1	+	+	CCONJ
ejpam-4301	253	2	n	n	PRON
ejpam-4301	253	3	〉	〉	NOUN
ejpam-4301	253	4	.	.	PUNCT
ejpam-4301	254	1	we	we	PRON
ejpam-4301	254	2	claim	claim	VERB
ejpam-4301	254	3	that	that	SCONJ
ejpam-4301	254	4	m	m	VERB
ejpam-4301	254	5	=	=	SYM
ejpam-4301	254	6	〈	〈	PROPN
ejpam-4301	254	7	x1	x1	PROPN
ejpam-4301	254	8	,	,	PUNCT
ejpam-4301	254	9	x2	x2	PROPN
ejpam-4301	254	10	,	,	PUNCT
ejpam-4301	254	11	...	...	PUNCT
ejpam-4301	254	12	,	,	PUNCT
ejpam-4301	254	13	xn	xn	PROPN
ejpam-4301	254	14	〉	〉	NUM
ejpam-4301	254	15	.	.	PUNCT
ejpam-4301	255	1	let	let	VERB
ejpam-4301	255	2	m	m	PRON
ejpam-4301	255	3	∈	∈	VERB
ejpam-4301	255	4	m	m	NOUN
ejpam-4301	255	5	,	,	PUNCT
ejpam-4301	255	6	hence	hence	ADV
ejpam-4301	255	7	m	m	VERB
ejpam-4301	255	8	+	+	SYM
ejpam-4301	255	9	n	n	CCONJ
ejpam-4301	255	10	∈	∈	NOUN
ejpam-4301	255	11	m	m	VERB
ejpam-4301	255	12	n	n	NOUN
ejpam-4301	255	13	and	and	CCONJ
ejpam-4301	255	14	m	m	PROPN
ejpam-4301	255	15	+	+	CCONJ
ejpam-4301	255	16	n	n	CCONJ
ejpam-4301	255	17	=	=	SYM
ejpam-4301	255	18	(	(	PUNCT
ejpam-4301	255	19	x1r1	x1r1	PROPN
ejpam-4301	255	20	+	+	CCONJ
ejpam-4301	255	21	x2r2	x2r2	X
ejpam-4301	256	1	+	+	CCONJ
ejpam-4301	256	2	...	...	PUNCT
ejpam-4301	257	1	+	+	NUM
ejpam-4301	257	2	xnrn	xnrn	NOUN
ejpam-4301	257	3	)	)	PUNCT
ejpam-4301	258	1	+	+	CCONJ
ejpam-4301	258	2	n	n	CCONJ
ejpam-4301	258	3	for	for	ADP
ejpam-4301	258	4	some	some	DET
ejpam-4301	258	5	r1	r1	NOUN
ejpam-4301	258	6	,	,	PUNCT
ejpam-4301	258	7	r2	r2	PROPN
ejpam-4301	258	8	,	,	PUNCT
ejpam-4301	258	9	...	...	PUNCT
ejpam-4301	258	10	,	,	PUNCT
ejpam-4301	258	11	rn	rn	PROPN
ejpam-4301	258	12	∈	∈	PROPN
ejpam-4301	258	13	r.	r.	PROPN
ejpam-4301	259	1	so	so	ADV
ejpam-4301	259	2	,	,	PUNCT
ejpam-4301	259	3	m	m	VERB
ejpam-4301	259	4	−	−	PROPN
ejpam-4301	259	5	(	(	PUNCT
ejpam-4301	259	6	x1r1	x1r1	PROPN
ejpam-4301	260	1	+	+	CCONJ
ejpam-4301	260	2	x2r2	x2r2	X
ejpam-4301	261	1	+	+	CCONJ
ejpam-4301	261	2	...	...	PUNCT
ejpam-4301	262	1	+	+	NUM
ejpam-4301	262	2	xnrn	xnrn	NOUN
ejpam-4301	262	3	)	)	PUNCT
ejpam-4301	262	4	∈	∈	PROPN
ejpam-4301	262	5	n	n	ADV
ejpam-4301	262	6	.	.	PUNCT
ejpam-4301	263	1	let	let	VERB
ejpam-4301	263	2	n	n	NOUN
ejpam-4301	263	3	=	=	VERB
ejpam-4301	263	4	m	m	VERB
ejpam-4301	263	5	−	−	PROPN
ejpam-4301	263	6	(	(	PUNCT
ejpam-4301	263	7	x1r1	x1r1	PROPN
ejpam-4301	263	8	+	+	CCONJ
ejpam-4301	263	9	x2r2	x2r2	X
ejpam-4301	264	1	+	+	CCONJ
ejpam-4301	264	2	...	...	PUNCT
ejpam-4301	265	1	+	+	NUM
ejpam-4301	265	2	xnrn	xnrn	NOUN
ejpam-4301	265	3	)	)	PUNCT
ejpam-4301	265	4	where	where	SCONJ
ejpam-4301	265	5	n	n	X
ejpam-4301	265	6	∈	∈	PROPN
ejpam-4301	265	7	n	n	CCONJ
ejpam-4301	265	8	,	,	PUNCT
ejpam-4301	265	9	hence	hence	ADV
ejpam-4301	265	10	m	m	VERB
ejpam-4301	265	11	=	=	PUNCT
ejpam-4301	265	12	(	(	PUNCT
ejpam-4301	265	13	x1r1	x1r1	PROPN
ejpam-4301	265	14	+	+	CCONJ
ejpam-4301	265	15	x2r2	x2r2	X
ejpam-4301	266	1	+	+	CCONJ
ejpam-4301	266	2	...	...	PUNCT
ejpam-4301	267	1	+	+	NUM
ejpam-4301	267	2	xnrn	xnrn	NOUN
ejpam-4301	267	3	)	)	PUNCT
ejpam-4301	268	1	+	+	CCONJ
ejpam-4301	268	2	n.	n.	PROPN
ejpam-4301	268	3	thus	thus	ADV
ejpam-4301	268	4	,	,	PUNCT
ejpam-4301	268	5	m	m	VERB
ejpam-4301	268	6	=	=	PUNCT
ejpam-4301	268	7	〈	〈	PROPN
ejpam-4301	268	8	x1	x1	PROPN
ejpam-4301	268	9	,	,	PUNCT
ejpam-4301	268	10	x2	x2	PROPN
ejpam-4301	268	11	,	,	PUNCT
ejpam-4301	268	12	...	...	PUNCT
ejpam-4301	268	13	,	,	PUNCT
ejpam-4301	268	14	xn	xn	PROPN
ejpam-4301	268	15	〉	〉	PROPN
ejpam-4301	268	16	+	+	CCONJ
ejpam-4301	268	17	n	n	NOUN
ejpam-4301	268	18	.	.	PUNCT
ejpam-4301	269	1	if	if	SCONJ
ejpam-4301	269	2	〈	〈	PROPN
ejpam-4301	269	3	x1	x1	PROPN
ejpam-4301	269	4	,	,	PUNCT
ejpam-4301	269	5	x2	x2	PROPN
ejpam-4301	269	6	,	,	PUNCT
ejpam-4301	269	7	...	...	PUNCT
ejpam-4301	269	8	,	,	PUNCT
ejpam-4301	269	9	xn	xn	PROPN
ejpam-4301	269	10	〉	〉	PROPN
ejpam-4301	269	11	6=	6=	ADP
ejpam-4301	269	12	m	m	PROPN
ejpam-4301	269	13	,	,	PUNCT
ejpam-4301	269	14	then	then	ADV
ejpam-4301	269	15	〈	〈	PROPN
ejpam-4301	269	16	x1	x1	PROPN
ejpam-4301	269	17	,	,	PUNCT
ejpam-4301	269	18	x2	x2	PROPN
ejpam-4301	269	19	,	,	PUNCT
ejpam-4301	269	20	...	...	PUNCT
ejpam-4301	269	21	,	,	PUNCT
ejpam-4301	269	22	xn	xn	PROPN
ejpam-4301	269	23	〉	〉	PROPN
ejpam-4301	269	24	�	�	PROPN
ejpam-4301	269	25	e∗	e∗	PROPN
ejpam-4301	269	26	m	m	PROPN
ejpam-4301	269	27	,	,	PUNCT
ejpam-4301	269	28	since	since	SCONJ
ejpam-4301	269	29	n	n	PROPN
ejpam-4301	269	30	�	�	PROPN
ejpam-4301	269	31	e∗	e∗	PROPN
ejpam-4301	269	32	m	m	PROPN
ejpam-4301	269	33	.	.	PUNCT
ejpam-4301	270	1	hence	hence	ADV
ejpam-4301	270	2	,	,	PUNCT
ejpam-4301	270	3	m	m	VERB
ejpam-4301	270	4	=	=	SYM
ejpam-4301	270	5	n	n	X
ejpam-4301	270	6	which	which	PRON
ejpam-4301	270	7	is	be	AUX
ejpam-4301	270	8	a	a	DET
ejpam-4301	270	9	contradiction	contradiction	NOUN
ejpam-4301	270	10	.	.	PUNCT
ejpam-4301	271	1	therefore	therefore	ADV
ejpam-4301	271	2	,	,	PUNCT
ejpam-4301	271	3	m	m	VERB
ejpam-4301	271	4	=	=	PUNCT
ejpam-4301	271	5	〈	〈	PROPN
ejpam-4301	271	6	x1	x1	PROPN
ejpam-4301	271	7	,	,	PUNCT
ejpam-4301	271	8	x2	x2	PROPN
ejpam-4301	271	9	,	,	PUNCT
ejpam-4301	271	10	...	...	PUNCT
ejpam-4301	271	11	,	,	PUNCT
ejpam-4301	271	12	xn	xn	PROPN
ejpam-4301	271	13	〉	〉	NUM
ejpam-4301	271	14	.	.	PUNCT
ejpam-4301	272	1	the	the	DET
ejpam-4301	272	2	following	follow	VERB
ejpam-4301	272	3	proposition	proposition	NOUN
ejpam-4301	272	4	is	be	AUX
ejpam-4301	272	5	a	a	DET
ejpam-4301	272	6	characterizes	characterize	VERB
ejpam-4301	272	7	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	272	8	modules	module	NOUN
ejpam-4301	272	9	.	.	PUNCT
ejpam-4301	273	1	proposition	proposition	NOUN
ejpam-4301	273	2	10	10	NUM
ejpam-4301	273	3	.	.	PUNCT
ejpam-4301	274	1	an	an	DET
ejpam-4301	274	2	r	r	NOUN
ejpam-4301	274	3	-	-	PUNCT
ejpam-4301	274	4	module	module	NOUN
ejpam-4301	274	5	m	m	NOUN
ejpam-4301	274	6	is	be	AUX
ejpam-4301	274	7	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	274	8	module	module	NOUN
ejpam-4301	274	9	if	if	SCONJ
ejpam-4301	274	10	and	and	CCONJ
ejpam-4301	274	11	only	only	ADV
ejpam-4301	274	12	if	if	SCONJ
ejpam-4301	274	13	every	every	DET
ejpam-4301	274	14	proper	proper	ADJ
ejpam-4301	274	15	e∗essential	e∗essential	ADJ
ejpam-4301	274	16	submodule	submodule	NOUN
ejpam-4301	274	17	of	of	ADP
ejpam-4301	274	18	m	m	PROPN
ejpam-4301	274	19	is	be	AUX
ejpam-4301	274	20	small	small	ADJ
ejpam-4301	274	21	in	in	ADP
ejpam-4301	274	22	m	m	PROPN
ejpam-4301	274	23	.	.	PUNCT
ejpam-4301	275	1	proof	proof	NOUN
ejpam-4301	275	2	.	.	PUNCT
ejpam-4301	276	1	⇒	⇒	NOUN
ejpam-4301	276	2	)	)	PUNCT
ejpam-4301	276	3	clear	clear	ADJ
ejpam-4301	276	4	⇐	⇐	NOUN
ejpam-4301	276	5	)	)	PUNCT
ejpam-4301	276	6	let	let	VERB
ejpam-4301	276	7	a	a	PRON
ejpam-4301	276	8	be	be	AUX
ejpam-4301	276	9	a	a	DET
ejpam-4301	276	10	proper	proper	ADJ
ejpam-4301	276	11	submodule	submodule	NOUN
ejpam-4301	276	12	of	of	ADP
ejpam-4301	276	13	m	m	PROPN
ejpam-4301	276	14	and	and	CCONJ
ejpam-4301	276	15	b	b	DET
ejpam-4301	276	16	an	an	DET
ejpam-4301	276	17	e∗-essential	e∗-essential	PROPN
ejpam-4301	276	18	submodule	submodule	NOUN
ejpam-4301	276	19	of	of	ADP
ejpam-4301	276	20	m	m	PRON
ejpam-4301	276	21	such	such	ADJ
ejpam-4301	276	22	that	that	SCONJ
ejpam-4301	276	23	a+b	a+b	PROPN
ejpam-4301	276	24	=	=	SYM
ejpam-4301	276	25	m	m	NOUN
ejpam-4301	276	26	.	.	PUNCT
ejpam-4301	277	1	if	if	SCONJ
ejpam-4301	277	2	b	b	PROPN
ejpam-4301	277	3	6=	6=	PROPN
ejpam-4301	277	4	m	m	VERB
ejpam-4301	277	5	then	then	ADV
ejpam-4301	277	6	b	b	NOUN
ejpam-4301	277	7	is	be	AUX
ejpam-4301	277	8	a	a	DET
ejpam-4301	277	9	proper	proper	ADJ
ejpam-4301	277	10	e∗-essential	e∗-essential	PROPN
ejpam-4301	277	11	submodule	submodule	NOUN
ejpam-4301	277	12	of	of	ADP
ejpam-4301	277	13	m	m	PROPN
ejpam-4301	277	14	and	and	CCONJ
ejpam-4301	277	15	by	by	ADP
ejpam-4301	277	16	assumption	assumption	NOUN
ejpam-4301	277	17	b	b	PROPN
ejpam-4301	277	18	is	be	AUX
ejpam-4301	277	19	small	small	ADJ
ejpam-4301	277	20	.	.	PUNCT
ejpam-4301	278	1	hence	hence	ADV
ejpam-4301	278	2	a	a	DET
ejpam-4301	278	3	=	=	NOUN
ejpam-4301	278	4	m	m	VERB
ejpam-4301	278	5	which	which	PRON
ejpam-4301	278	6	is	be	AUX
ejpam-4301	278	7	a	a	DET
ejpam-4301	278	8	contradiction	contradiction	NOUN
ejpam-4301	278	9	.	.	PUNCT
ejpam-4301	279	1	thus	thus	ADV
ejpam-4301	279	2	,	,	PUNCT
ejpam-4301	279	3	b	b	X
ejpam-4301	279	4	=	=	SYM
ejpam-4301	279	5	m	m	PROPN
ejpam-4301	279	6	and	and	CCONJ
ejpam-4301	279	7	a	a	PRON
ejpam-4301	279	8	is	be	AUX
ejpam-4301	279	9	e∗-essential	e∗-essential	PROPN
ejpam-4301	279	10	small	small	ADJ
ejpam-4301	279	11	in	in	ADP
ejpam-4301	279	12	m	m	PROPN
ejpam-4301	279	13	.	.	PUNCT
ejpam-4301	280	1	therefore	therefore	ADV
ejpam-4301	280	2	,	,	PUNCT
ejpam-4301	280	3	m	m	VERB
ejpam-4301	280	4	is	be	AUX
ejpam-4301	280	5	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	280	6	.	.	PUNCT
ejpam-4301	281	1	definition	definition	NOUN
ejpam-4301	281	2	4	4	NUM
ejpam-4301	281	3	.	.	PUNCT
ejpam-4301	282	1	let	let	VERB
ejpam-4301	282	2	m	m	PRON
ejpam-4301	282	3	be	be	AUX
ejpam-4301	282	4	an	an	DET
ejpam-4301	282	5	r	r	NOUN
ejpam-4301	282	6	-	-	PUNCT
ejpam-4301	282	7	module	module	NOUN
ejpam-4301	282	8	.	.	PUNCT
ejpam-4301	283	1	a	a	DET
ejpam-4301	283	2	submodule	submodule	NOUN
ejpam-4301	283	3	a	a	PRON
ejpam-4301	283	4	of	of	ADP
ejpam-4301	283	5	m	m	PROPN
ejpam-4301	283	6	is	be	AUX
ejpam-4301	283	7	called	call	VERB
ejpam-4301	283	8	e∗-coclosed	e∗-coclosed	ADJ
ejpam-4301	283	9	if	if	SCONJ
ejpam-4301	283	10	whenever	whenever	SCONJ
ejpam-4301	283	11	b	b	PROPN
ejpam-4301	283	12	≤	≤	PROPN
ejpam-4301	283	13	a	a	PRON
ejpam-4301	283	14	,	,	PUNCT
ejpam-4301	283	15	a	a	DET
ejpam-4301	283	16	b	b	PROPN
ejpam-4301	283	17	�	�	PROPN
ejpam-4301	283	18	e∗	e∗	PROPN
ejpam-4301	283	19	m	m	PROPN
ejpam-4301	283	20	b	b	PROPN
ejpam-4301	283	21	,	,	PUNCT
ejpam-4301	283	22	implies	imply	VERB
ejpam-4301	283	23	that	that	SCONJ
ejpam-4301	283	24	a	a	DET
ejpam-4301	283	25	=	=	X
ejpam-4301	283	26	b.	b.	PROPN
ejpam-4301	283	27	one	one	NOUN
ejpam-4301	283	28	may	may	AUX
ejpam-4301	283	29	ask	ask	VERB
ejpam-4301	283	30	a	a	DET
ejpam-4301	283	31	question	question	NOUN
ejpam-4301	283	32	.	.	PUNCT
ejpam-4301	283	33	is	be	AUX
ejpam-4301	283	34	any	any	DET
ejpam-4301	283	35	submodule	submodule	NOUN
ejpam-4301	283	36	of	of	ADP
ejpam-4301	283	37	an	an	DET
ejpam-4301	283	38	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	283	39	module	module	NOUN
ejpam-4301	283	40	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	283	41	?	?	PUNCT
ejpam-4301	284	1	the	the	DET
ejpam-4301	284	2	following	follow	VERB
ejpam-4301	284	3	proportion	proportion	NOUN
ejpam-4301	284	4	gives	give	VERB
ejpam-4301	284	5	a	a	DET
ejpam-4301	284	6	partial	partial	ADJ
ejpam-4301	284	7	answer	answer	NOUN
ejpam-4301	284	8	.	.	PUNCT
ejpam-4301	285	1	proposition	proposition	NOUN
ejpam-4301	285	2	11	11	NUM
ejpam-4301	285	3	.	.	PUNCT
ejpam-4301	286	1	let	let	VERB
ejpam-4301	286	2	m	m	PRON
ejpam-4301	286	3	be	be	AUX
ejpam-4301	286	4	an	an	DET
ejpam-4301	286	5	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	286	6	r	r	NOUN
ejpam-4301	286	7	-	-	PUNCT
ejpam-4301	286	8	module	module	NOUN
ejpam-4301	286	9	.	.	PUNCT
ejpam-4301	287	1	1	1	X
ejpam-4301	287	2	.	.	X
ejpam-4301	287	3	an	an	DET
ejpam-4301	287	4	e∗-essential	e∗-essential	PROPN
ejpam-4301	287	5	direct	direct	ADJ
ejpam-4301	287	6	summand	summand	NOUN
ejpam-4301	287	7	of	of	ADP
ejpam-4301	287	8	an	an	DET
ejpam-4301	287	9	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	287	10	module	module	NOUN
ejpam-4301	287	11	is	be	AUX
ejpam-4301	287	12	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	287	13	.	.	PUNCT
ejpam-4301	288	1	2	2	NUM
ejpam-4301	288	2	.	.	X
ejpam-4301	288	3	an	an	DET
ejpam-4301	288	4	e∗-coclosed	e∗-coclosed	ADJ
ejpam-4301	288	5	submodule	submodule	NOUN
ejpam-4301	288	6	of	of	ADP
ejpam-4301	288	7	an	an	DET
ejpam-4301	288	8	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	288	9	is	be	AUX
ejpam-4301	288	10	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	288	11	.	.	PUNCT
ejpam-4301	289	1	proof	proof	NOUN
ejpam-4301	289	2	.	.	PUNCT
ejpam-4301	290	1	1	1	X
ejpam-4301	290	2	.	.	X
ejpam-4301	290	3	let	let	VERB
ejpam-4301	290	4	a	a	DET
ejpam-4301	290	5	be	be	AUX
ejpam-4301	290	6	an	an	DET
ejpam-4301	290	7	e∗-essential	e∗-essential	PROPN
ejpam-4301	290	8	direct	direct	ADJ
ejpam-4301	290	9	summand	summand	NOUN
ejpam-4301	290	10	of	of	ADP
ejpam-4301	290	11	m	m	PROPN
ejpam-4301	290	12	and	and	CCONJ
ejpam-4301	290	13	b	b	DET
ejpam-4301	290	14	a	a	DET
ejpam-4301	290	15	proper	proper	ADJ
ejpam-4301	290	16	submodule	submodule	NOUN
ejpam-4301	290	17	of	of	ADP
ejpam-4301	290	18	a	a	PRON
ejpam-4301	290	19	with	with	ADP
ejpam-4301	290	20	l	l	PROPN
ejpam-4301	290	21	≤e∗	≤e∗	PROPN
ejpam-4301	290	22	a	a	DET
ejpam-4301	290	23	such	such	ADJ
ejpam-4301	290	24	that	that	DET
ejpam-4301	290	25	b	b	NOUN
ejpam-4301	290	26	+	+	NUM
ejpam-4301	290	27	l	l	NOUN
ejpam-4301	290	28	=	=	X
ejpam-4301	290	29	a.	a.	NOUN
ejpam-4301	290	30	since	since	SCONJ
ejpam-4301	290	31	l	l	PROPN
ejpam-4301	290	32	≤e∗	≤e∗	PROPN
ejpam-4301	290	33	a	a	DET
ejpam-4301	290	34	≤e∗	≤e∗	PROPN
ejpam-4301	290	35	m	m	NOUN
ejpam-4301	290	36	,	,	PUNCT
ejpam-4301	290	37	then	then	ADV
ejpam-4301	290	38	by	by	ADP
ejpam-4301	290	39	lemma	lemma	PROPN
ejpam-4301	290	40	1	1	NUM
ejpam-4301	290	41	,	,	PUNCT
ejpam-4301	290	42	l	l	PROPN
ejpam-4301	290	43	≤e∗	≤e∗	PROPN
ejpam-4301	290	44	m	m	VERB
ejpam-4301	290	45	.	.	PUNCT
ejpam-4301	291	1	also	also	ADV
ejpam-4301	291	2	,	,	PUNCT
ejpam-4301	291	3	since	since	SCONJ
ejpam-4301	291	4	a	a	PRON
ejpam-4301	291	5	is	be	AUX
ejpam-4301	291	6	a	a	DET
ejpam-4301	291	7	direct	direct	ADJ
ejpam-4301	291	8	summand	summand	NOUN
ejpam-4301	291	9	of	of	ADP
ejpam-4301	291	10	m	m	PROPN
ejpam-4301	291	11	,	,	PUNCT
ejpam-4301	291	12	there	there	PRON
ejpam-4301	291	13	is	be	VERB
ejpam-4301	291	14	a	a	DET
ejpam-4301	291	15	submodule	submodule	NOUN
ejpam-4301	291	16	a	a	DET
ejpam-4301	291	17	′	′	NOUN
ejpam-4301	291	18	of	of	ADP
ejpam-4301	291	19	m	m	NOUN
ejpam-4301	291	20	such	such	ADJ
ejpam-4301	291	21	that	that	SCONJ
ejpam-4301	291	22	a⊕	a⊕	PROPN
ejpam-4301	291	23	a	a	DET
ejpam-4301	291	24	′	′	NUM
ejpam-4301	292	1	=	=	NOUN
ejpam-4301	292	2	m	m	NOUN
ejpam-4301	292	3	.	.	PUNCT
ejpam-4301	293	1	thus	thus	ADV
ejpam-4301	293	2	,	,	PUNCT
ejpam-4301	293	3	m	m	VERB
ejpam-4301	293	4	=	=	SYM
ejpam-4301	293	5	b	b	PROPN
ejpam-4301	293	6	+	+	CCONJ
ejpam-4301	293	7	l+	l+	X
ejpam-4301	293	8	a	a	DET
ejpam-4301	293	9	′	′	NOUN
ejpam-4301	293	10	with	with	ADP
ejpam-4301	293	11	l+	l+	NOUN
ejpam-4301	293	12	a	a	DET
ejpam-4301	293	13	′	′	NUM
ejpam-4301	293	14	≤e∗	≤e∗	NOUN
ejpam-4301	293	15	m	m	NOUN
ejpam-4301	293	16	and	and	CCONJ
ejpam-4301	293	17	hence	hence	ADV
ejpam-4301	293	18	b	b	PROPN
ejpam-4301	293	19	is	be	AUX
ejpam-4301	293	20	a	a	DET
ejpam-4301	293	21	proper	proper	ADJ
ejpam-4301	293	22	submodule	submodule	NOUN
ejpam-4301	293	23	of	of	ADP
ejpam-4301	293	24	m	m	PROPN
ejpam-4301	293	25	.	.	PUNCT
ejpam-4301	294	1	this	this	PRON
ejpam-4301	294	2	implies	imply	VERB
ejpam-4301	294	3	that	that	SCONJ
ejpam-4301	294	4	b	b	NOUN
ejpam-4301	294	5	is	be	AUX
ejpam-4301	294	6	e∗-essential	e∗-essential	PROPN
ejpam-4301	294	7	small	small	ADJ
ejpam-4301	294	8	in	in	ADP
ejpam-4301	294	9	m	m	PROPN
ejpam-4301	294	10	.	.	PUNCT
ejpam-4301	295	1	hence	hence	ADV
ejpam-4301	295	2	,	,	PUNCT
ejpam-4301	295	3	m	m	VERB
ejpam-4301	295	4	=	=	SYM
ejpam-4301	295	5	l+a	l+a	NUM
ejpam-4301	295	6	′	′	NUM
ejpam-4301	295	7	and	and	CCONJ
ejpam-4301	295	8	a	a	DET
ejpam-4301	295	9	=	=	X
ejpam-4301	295	10	a	a	DET
ejpam-4301	295	11	∩m	∩m	NOUN
ejpam-4301	295	12	=	=	PUNCT
ejpam-4301	296	1	a	a	DET
ejpam-4301	296	2	∩	∩	NOUN
ejpam-4301	296	3	(	(	PUNCT
ejpam-4301	296	4	l	l	NOUN
ejpam-4301	296	5	+	+	NOUN
ejpam-4301	296	6	a	a	DET
ejpam-4301	296	7	′	′	NOUN
ejpam-4301	296	8	)	)	PUNCT
ejpam-4301	296	9	=	=	PUNCT
ejpam-4301	297	1	l	l	NOUN
ejpam-4301	298	1	+	+	CCONJ
ejpam-4301	298	2	(	(	PUNCT
ejpam-4301	298	3	a	a	DET
ejpam-4301	298	4	∩	∩	NOUN
ejpam-4301	298	5	a	a	DET
ejpam-4301	298	6	′	′	NOUN
ejpam-4301	298	7	)	)	PUNCT
ejpam-4301	299	1	=	=	SYM
ejpam-4301	299	2	l.	l.	PROPN
ejpam-4301	299	3	therefore	therefore	ADV
ejpam-4301	299	4	,	,	PUNCT
ejpam-4301	299	5	b	b	PROPN
ejpam-4301	299	6	is	be	AUX
ejpam-4301	299	7	e∗-essential	e∗-essential	PROPN
ejpam-4301	299	8	small	small	ADJ
ejpam-4301	299	9	in	in	ADP
ejpam-4301	299	10	a	a	PRON
ejpam-4301	299	11	and	and	CCONJ
ejpam-4301	299	12	a	a	PRON
ejpam-4301	299	13	is	be	AUX
ejpam-4301	299	14	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	299	15	.	.	PUNCT
ejpam-4301	300	1	2	2	X
ejpam-4301	300	2	.	.	X
ejpam-4301	300	3	let	let	VERB
ejpam-4301	300	4	a	a	PRON
ejpam-4301	300	5	be	be	AUX
ejpam-4301	300	6	a	a	DET
ejpam-4301	300	7	e∗-coclosed	e∗-coclose	VERB
ejpam-4301	300	8	submodule	submodule	NOUN
ejpam-4301	300	9	of	of	ADP
ejpam-4301	300	10	m	m	PROPN
ejpam-4301	300	11	and	and	CCONJ
ejpam-4301	300	12	b	b	DET
ejpam-4301	300	13	a	a	DET
ejpam-4301	300	14	proper	proper	ADJ
ejpam-4301	300	15	submodule	submodule	NOUN
ejpam-4301	300	16	of	of	ADP
ejpam-4301	300	17	a	a	PRON
ejpam-4301	300	18	with	with	ADP
ejpam-4301	300	19	c	c	PROPN
ejpam-4301	300	20	an	an	DET
ejpam-4301	300	21	e∗-essential	e∗-essential	PROPN
ejpam-4301	300	22	submodule	submodule	NOUN
ejpam-4301	300	23	of	of	ADP
ejpam-4301	300	24	a	a	DET
ejpam-4301	300	25	such	such	ADJ
ejpam-4301	300	26	that	that	DET
ejpam-4301	300	27	b	b	NOUN
ejpam-4301	301	1	+	+	CCONJ
ejpam-4301	301	2	c	c	NOUN
ejpam-4301	301	3	=	=	PUNCT
ejpam-4301	301	4	a.	a.	NOUN
ejpam-4301	301	5	since	since	SCONJ
ejpam-4301	301	6	m	m	PROPN
ejpam-4301	301	7	is	be	AUX
ejpam-4301	301	8	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	301	9	then	then	ADV
ejpam-4301	301	10	by	by	ADP
ejpam-4301	301	11	corollary	corollary	ADJ
ejpam-4301	301	12	3	3	NUM
ejpam-4301	301	13	,	,	PUNCT
ejpam-4301	301	14	m	m	VERB
ejpam-4301	301	15	c	c	NOUN
ejpam-4301	301	16	is	be	AUX
ejpam-4301	301	17	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	301	18	.	.	PUNCT
ejpam-4301	302	1	now	now	ADV
ejpam-4301	302	2	,	,	PUNCT
ejpam-4301	302	3	a	a	DET
ejpam-4301	302	4	c	c	NOUN
ejpam-4301	302	5	is	be	AUX
ejpam-4301	302	6	a	a	DET
ejpam-4301	302	7	proper	proper	ADJ
ejpam-4301	302	8	submodule	submodule	NOUN
ejpam-4301	302	9	of	of	ADP
ejpam-4301	302	10	m	m	PROPN
ejpam-4301	302	11	c	c	NOUN
ejpam-4301	302	12	implies	imply	VERB
ejpam-4301	302	13	that	that	SCONJ
ejpam-4301	302	14	a	a	DET
ejpam-4301	302	15	c	c	NOUN
ejpam-4301	302	16	is	be	AUX
ejpam-4301	302	17	e∗-essential	e∗-essential	PROPN
ejpam-4301	302	18	small	small	ADJ
ejpam-4301	302	19	of	of	ADP
ejpam-4301	302	20	m	m	PROPN
ejpam-4301	302	21	c	c	NOUN
ejpam-4301	302	22	since	since	SCONJ
ejpam-4301	302	23	a	a	PRON
ejpam-4301	302	24	is	be	AUX
ejpam-4301	302	25	e∗-coclosed	e∗-coclosed	ADJ
ejpam-4301	302	26	.	.	PUNCT
ejpam-4301	303	1	thus	thus	ADV
ejpam-4301	303	2	a	a	DET
ejpam-4301	303	3	=	=	SYM
ejpam-4301	303	4	c	c	PROPN
ejpam-4301	303	5	and	and	CCONJ
ejpam-4301	303	6	b	b	PROPN
ejpam-4301	303	7	is	be	AUX
ejpam-4301	303	8	e∗-essential	e∗-essential	PROPN
ejpam-4301	303	9	small	small	ADJ
ejpam-4301	303	10	of	of	ADP
ejpam-4301	303	11	a.the	a.the	DET
ejpam-4301	303	12	case	case	NOUN
ejpam-4301	303	13	a	a	DET
ejpam-4301	303	14	c	c	NOUN
ejpam-4301	304	1	=	=	VERB
ejpam-4301	304	2	m	m	VERB
ejpam-4301	304	3	c	c	NOUN
ejpam-4301	304	4	,	,	PUNCT
ejpam-4301	304	5	implies	imply	VERB
ejpam-4301	304	6	that	that	SCONJ
ejpam-4301	304	7	a	a	DET
ejpam-4301	304	8	=	=	NOUN
ejpam-4301	304	9	m	m	NOUN
ejpam-4301	304	10	.	.	PUNCT
ejpam-4301	305	1	thus	thus	ADV
ejpam-4301	305	2	a	a	PRON
ejpam-4301	305	3	is	be	AUX
ejpam-4301	305	4	e∗-hollow	e∗-hollow	NOUN
ejpam-4301	305	5	.	.	PUNCT
ejpam-4301	306	1	references	reference	NOUN
ejpam-4301	306	2	485	485	NUM
ejpam-4301	306	3	acknowledgements	acknowledgement	NOUN
ejpam-4301	306	4	the	the	DET
ejpam-4301	306	5	authors	author	NOUN
ejpam-4301	306	6	would	would	AUX
ejpam-4301	306	7	like	like	VERB
ejpam-4301	306	8	to	to	PART
ejpam-4301	306	9	thank	thank	VERB
ejpam-4301	306	10	the	the	DET
ejpam-4301	306	11	reviewers	reviewer	NOUN
ejpam-4301	306	12	for	for	ADP
ejpam-4301	306	13	their	their	PRON
ejpam-4301	306	14	invaluable	invaluable	ADJ
ejpam-4301	306	15	comments	comment	NOUN
ejpam-4301	306	16	and	and	CCONJ
ejpam-4301	306	17	suggestions	suggestion	NOUN
ejpam-4301	306	18	that	that	PRON
ejpam-4301	306	19	led	lead	VERB
ejpam-4301	306	20	to	to	ADP
ejpam-4301	306	21	this	this	DET
ejpam-4301	306	22	improved	improve	VERB
ejpam-4301	306	23	version	version	NOUN
ejpam-4301	306	24	of	of	ADP
ejpam-4301	306	25	the	the	DET
ejpam-4301	306	26	paper	paper	NOUN
ejpam-4301	306	27	.	.	PUNCT
ejpam-4301	307	1	references	reference	NOUN
ejpam-4301	307	2	[	[	X
ejpam-4301	307	3	1	1	NUM
ejpam-4301	307	4	]	]	PUNCT
ejpam-4301	307	5	h.	h.	PROPN
ejpam-4301	307	6	baanoon	baanoon	PROPN
ejpam-4301	307	7	and	and	CCONJ
ejpam-4301	307	8	w.	w.	PROPN
ejpam-4301	307	9	khalid	khalid	PROPN
ejpam-4301	307	10	.	.	PUNCT
ejpam-4301	308	1	e*-essential	e*-essential	ADJ
ejpam-4301	308	2	submodule	submodule	NOUN
ejpam-4301	308	3	.	.	PUNCT
ejpam-4301	309	1	european	european	PROPN
ejpam-4301	309	2	journal	journal	PROPN
ejpam-4301	309	3	of	of	ADP
ejpam-4301	309	4	pure	pure	ADJ
ejpam-4301	309	5	and	and	CCONJ
ejpam-4301	309	6	applied	applied	ADJ
ejpam-4301	309	7	mathematics	mathematic	NOUN
ejpam-4301	309	8	,	,	PUNCT
ejpam-4301	309	9	15(1):224–228	15(1):224–228	PROPN
ejpam-4301	309	10	,	,	PUNCT
ejpam-4301	309	11	2022	2022	NUM
ejpam-4301	309	12	.	.	PUNCT
ejpam-4301	310	1	[	[	X
ejpam-4301	310	2	2	2	X
ejpam-4301	310	3	]	]	PUNCT
ejpam-4301	310	4	a	a	DET
ejpam-4301	310	5	çiğdem	çiğdem	NOUN
ejpam-4301	310	6	özcan	özcan	PROPN
ejpam-4301	310	7	.	.	PUNCT
ejpam-4301	311	1	modules	module	NOUN
ejpam-4301	311	2	with	with	ADP
ejpam-4301	311	3	small	small	ADJ
ejpam-4301	311	4	cyclic	cyclic	ADJ
ejpam-4301	311	5	submodules	submodule	NOUN
ejpam-4301	311	6	in	in	ADP
ejpam-4301	311	7	their	their	PRON
ejpam-4301	311	8	injective	injective	ADJ
ejpam-4301	311	9	hulls	hull	NOUN
ejpam-4301	311	10	.	.	PUNCT
ejpam-4301	312	1	2002	2002	NUM
ejpam-4301	312	2	.	.	PUNCT
ejpam-4301	313	1	[	[	X
ejpam-4301	313	2	3	3	X
ejpam-4301	313	3	]	]	X
ejpam-4301	313	4	patrick	patrick	PROPN
ejpam-4301	313	5	fleury	fleury	PROPN
ejpam-4301	313	6	.	.	PUNCT
ejpam-4301	314	1	hollow	hollow	ADJ
ejpam-4301	314	2	modules	module	NOUN
ejpam-4301	314	3	and	and	CCONJ
ejpam-4301	314	4	local	local	ADJ
ejpam-4301	314	5	endomorphism	endomorphism	NOUN
ejpam-4301	314	6	rings	ring	NOUN
ejpam-4301	314	7	.	.	PUNCT
ejpam-4301	315	1	pacific	pacific	PROPN
ejpam-4301	315	2	journal	journal	PROPN
ejpam-4301	315	3	of	of	ADP
ejpam-4301	315	4	mathematics	mathematic	NOUN
ejpam-4301	315	5	,	,	PUNCT
ejpam-4301	315	6	53(2):379–385	53(2):379–385	PROPN
ejpam-4301	315	7	,	,	PUNCT
ejpam-4301	315	8	1974	1974	NUM
ejpam-4301	315	9	.	.	PUNCT
ejpam-4301	316	1	[	[	X
ejpam-4301	316	2	4	4	NUM
ejpam-4301	316	3	]	]	X
ejpam-4301	316	4	kenneth	kenneth	PROPN
ejpam-4301	316	5	goodearl	goodearl	PROPN
ejpam-4301	316	6	.	.	PROPN
ejpam-4301	316	7	ring	ring	PROPN
ejpam-4301	316	8	theory	theory	PROPN
ejpam-4301	316	9	:	:	PUNCT
ejpam-4301	316	10	nonsingular	nonsingular	ADJ
ejpam-4301	316	11	rings	ring	NOUN
ejpam-4301	316	12	and	and	CCONJ
ejpam-4301	316	13	modules	module	NOUN
ejpam-4301	316	14	,	,	PUNCT
ejpam-4301	316	15	volume	volume	NOUN
ejpam-4301	316	16	33	33	NUM
ejpam-4301	316	17	.	.	PUNCT
ejpam-4301	317	1	crc	crc	PROPN
ejpam-4301	317	2	press	press	PROPN
ejpam-4301	317	3	,	,	PUNCT
ejpam-4301	317	4	1976	1976	NUM
ejpam-4301	317	5	.	.	PUNCT
ejpam-4301	318	1	[	[	X
ejpam-4301	318	2	5	5	X
ejpam-4301	318	3	]	]	X
ejpam-4301	318	4	michiel	michiel	PROPN
ejpam-4301	318	5	hazewinkel	hazewinkel	PROPN
ejpam-4301	318	6	,	,	PUNCT
ejpam-4301	318	7	nadiya	nadiya	PROPN
ejpam-4301	318	8	gubareni	gubareni	PROPN
ejpam-4301	318	9	,	,	PUNCT
ejpam-4301	318	10	and	and	CCONJ
ejpam-4301	318	11	vladimir	vladimir	PROPN
ejpam-4301	318	12	v	v	ADP
ejpam-4301	318	13	kirichenko	kirichenko	PROPN
ejpam-4301	318	14	.	.	PUNCT
ejpam-4301	319	1	algebras	algebras	PROPN
ejpam-4301	319	2	,	,	PUNCT
ejpam-4301	319	3	rings	ring	NOUN
ejpam-4301	319	4	and	and	CCONJ
ejpam-4301	319	5	modules	module	NOUN
ejpam-4301	319	6	,	,	PUNCT
ejpam-4301	319	7	volume	volume	NOUN
ejpam-4301	319	8	1	1	NUM
ejpam-4301	319	9	.	.	PUNCT
ejpam-4301	319	10	springer	springer	PROPN
ejpam-4301	319	11	science	science	PROPN
ejpam-4301	319	12	&	&	CCONJ
ejpam-4301	319	13	business	business	NOUN
ejpam-4301	319	14	media	medium	NOUN
ejpam-4301	319	15	,	,	PUNCT
ejpam-4301	319	16	2004	2004	NUM
ejpam-4301	319	17	.	.	PUNCT
ejpam-4301	320	1	[	[	X
ejpam-4301	320	2	6	6	NUM
ejpam-4301	320	3	]	]	X
ejpam-4301	320	4	friedrich	friedrich	PROPN
ejpam-4301	320	5	kasch	kasch	PROPN
ejpam-4301	320	6	.	.	PUNCT
ejpam-4301	320	7	modules	module	NOUN
ejpam-4301	320	8	and	and	CCONJ
ejpam-4301	320	9	rings	ring	NOUN
ejpam-4301	320	10	,	,	PUNCT
ejpam-4301	320	11	volume	volume	NOUN
ejpam-4301	320	12	17	17	NUM
ejpam-4301	320	13	.	.	PUNCT
ejpam-4301	321	1	academic	academic	ADJ
ejpam-4301	321	2	press	press	NOUN
ejpam-4301	321	3	,	,	PUNCT
ejpam-4301	321	4	1982	1982	NUM
ejpam-4301	321	5	.	.	PUNCT
ejpam-4301	322	1	[	[	X
ejpam-4301	322	2	7	7	X
ejpam-4301	322	3	]	]	X
ejpam-4301	322	4	dx	dx	PROPN
ejpam-4301	322	5	zhou	zhou	PROPN
ejpam-4301	322	6	and	and	CCONJ
ejpam-4301	322	7	xr	xr	PROPN
ejpam-4301	322	8	zhang	zhang	PROPN
ejpam-4301	322	9	.	.	PUNCT
ejpam-4301	323	1	small	small	ADJ
ejpam-4301	323	2	-	-	PUNCT
ejpam-4301	323	3	essential	essential	ADJ
ejpam-4301	323	4	submodules	submodule	NOUN
ejpam-4301	323	5	and	and	CCONJ
ejpam-4301	323	6	morita	morita	PROPN
ejpam-4301	323	7	duality	duality	PROPN
ejpam-4301	323	8	.	.	PUNCT
ejpam-4301	324	1	southeast	southeast	ADJ
ejpam-4301	324	2	asian	asian	ADJ
ejpam-4301	324	3	bulletin	bulletin	NOUN
ejpam-4301	324	4	of	of	ADP
ejpam-4301	324	5	mathematics	mathematic	NOUN
ejpam-4301	324	6	,	,	PUNCT
ejpam-4301	324	7	35(6	35(6	NUM
ejpam-4301	324	8	)	)	PUNCT
ejpam-4301	324	9	,	,	PUNCT
ejpam-4301	324	10	2011	2011	NUM
ejpam-4301	324	11	.	.	PUNCT